\begin{align*} c _ { \ell } ( s _ \ell , s _ \ell ' ) = ( s _ \ell - s _ \ell ' ) ^ { \top } V _ \ell ^ { - 1 } ( s _ \ell - s _ \ell ' ) , \end{align*}


\begin{align*} \begin{aligned} \mathcal { I } \left ( \lambda \delta + ( 1 - \lambda ) \delta ^ \prime \right ) & \geq \lambda \sup _ { \gamma \in \Sigma ( \delta ) } \int _ { \mathcal { S } } f ( s ) d \gamma ( s ) + ( 1 - \lambda ) \sup _ { \gamma ^ \prime \in \Sigma ( \delta ^ \prime ) } \int _ { \mathcal { S } } f ( s ) d \gamma ^ { \prime } ( s ) \\ & \geq \lambda \mathcal { I } ( \delta ) + ( 1 - \lambda ) \mathcal { I } ( \delta ^ \prime ) . \end{aligned} \end{align*}


\begin{align*} \frac { A _ y ( f + T ) } { A ( f + T ) } = \frac { 1 } { T } + \frac { Q ' ( T ) } { Q ( T ) } \ , \cdot \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( 1 - z ^ n \right ) ^ { \frac { m ^ 2 } { n ^ 3 } } = \sqrt { 1 - z } \ ; \exp \left \{ \frac { - 1 } { 6 } L i _ 2 ( z ) - \frac { 1 } { 3 } \frac { z } { 1 - z } \right \} . \end{align*}


\begin{align*} F _ m ( p ) : = \begin{cases} \min _ { v } & c \displaystyle \sum _ { i = 1 } ^ n \sum _ { k = 1 } ^ m p _ i ( \phi _ { k , i } ( x _ i ) - d _ { i , k } ) \cdot \P ( \omega _ k ) + \sum _ { i \neq j } \alpha _ { i j } v _ { i j } \\ s . t \quad & \begin{cases} - v \leq 0 \\ \sum _ { j \neq i } v _ { i j } - x _ { 0 i } \leq 0 , \forall i \in I . \end{cases} \end{cases} \end{align*}


\begin{align*} \int _ { k _ \epsilon } ^ k | s | ^ { p - 2 } ( s - k ) _ - \ , d s = \tfrac { 1 } { p - 1 } \ , \mathfrak g _ - ( k _ \epsilon , k ) \geq \tfrac { 1 } { \boldsymbol \gamma ( p ) } \big ( | k _ \epsilon | + | k | \big ) ^ { p - 2 } ( k - k _ \epsilon ) ^ 2 \geq \tfrac { 1 } { \boldsymbol \gamma ( p ) } M ^ p . \end{align*}


\begin{align*} \| D _ { S , x } f \| _ p \le \sum _ { T \subseteq S } \| [ E _ { T } [ f ] ] _ { S \to x } \| _ p \leq \sum _ { T \subseteq S } r ^ { | S | } \gamma = ( 2 r ) ^ { | S | } \gamma , \end{align*}


\begin{align*} X _ S ( k , n , \omega ) : = { \rm G r } _ { ( k \omega , \dots , k \omega ) } ( U _ { \omega n , S } ) . \end{align*}


\begin{align*} \bar \Psi _ n = ( ( X _ \bullet \triangleright X _ \bullet ) \cdots ) \triangleright X _ \bullet . \end{align*}


\begin{align*} \begin{pmatrix} f & a e + b f \\ c e + d f & p e + q f \end{pmatrix} \end{align*}


\begin{align*} z f ^ { \prime } ( z ) = \frac { 1 } { 2 } ( p ( z ) + 1 ) f ( z ) , z \in \mathbb { D } \end{align*}


\begin{align*} { \rm I n t } ( \gamma ) = \left \{ w \in \C \setminus \{ 0 \} ; | w | < R \left ( \frac { w } { | w | } \right ) \right \} \cup \{ 0 \} . \end{align*}


\begin{align*} f ( z ) - P _ { \infty } ( z ) = ( e ^ { - \varepsilon y } ) y \rightarrow \infty \epsilon > 0 . \end{align*}


\begin{align*} P ^ \omega ( C _ t ^ c \mid S ) & = \frac { 1 } { P ^ \omega ( S ) } \left [ P ^ \omega \left ( C _ t ^ c \cap S \cap A _ { m ( t ) } \right ) + P ^ \omega \left ( C _ t ^ c \cap S \cap A _ { m ( t ) } ^ c \right ) \right ] \\ & \leq c ( \omega ) \left [ P ^ \omega \left ( C _ t ^ c \mid A _ { m ( t ) } \right ) + P ^ \omega \left ( A _ { m ( t ) } ^ c \mid S \right ) \right ] . \end{align*}


\begin{align*} \rho ( \xi , w ) = \rho ( \overline { \xi } , \overline { w } ) \end{align*}


\begin{align*} \sqrt { \alpha } \left ( \frac { \gamma } { 4 } - \frac { \log ( 4 \beta ) } { 4 } + \sum _ { n = 1 } ^ { \infty } d ( n ) \ , K _ { 0 } ( 2 n \alpha ) \right ) = \sqrt { \beta } \left ( \frac { \gamma } { 4 } - \frac { \log ( 4 \alpha ) } { 4 } + \sum _ { n = 1 } ^ { \infty } d ( n ) \ , K _ { 0 } ( 2 n \beta ) \right ) . \end{align*}


\begin{align*} ( R \otimes 1 _ L ) ( \alpha \otimes \varphi _ L ) \Delta ( l ) & = ( \alpha \otimes \varphi _ L ) ( R \otimes 1 _ L ) \Delta ( l ) , \\ \tau \circ ( 1 _ L \otimes R ) ( \alpha \otimes \varphi _ L ) \Delta ( l ) & = ( \varphi _ L \otimes \alpha ) ( \tau \circ ( 1 _ L \otimes R ) ) \Delta ( l ) . \end{align*}


\begin{align*} \begin{cases} \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) u = 0 & \mathbb { R } ^ n , \\ u = 0 & W \end{cases} \end{align*}


\begin{align*} ( U ^ T P U / 2 ) _ t + ( U ^ T A _ i U ) _ { x _ i } = ( U _ { x _ i } ^ T A _ i U - U ^ T B _ i U _ { x _ i } ) - U ^ T C U . \end{align*}


\begin{align*} \mathcal { S } ( I L , J M , c ) = \sum _ { ( a , c ) = 1 } e \left ( \frac { a I L + \bar { a } J M } { c } \right ) , \end{align*}


\begin{align*} \sum _ { k \in [ n ] , k \neq j } t _ k v _ k & = \lambda v _ j , \\ t _ { k } ^ * v _ j & = \lambda v _ { k } , \forall k \in [ n ] , k \neq j . \end{align*}


\begin{align*} f ( x ) = \begin{cases} \frac { 1 } { 1 - a ^ d / b ^ d } f _ 0 ( x ) & | x | \leq b \\ 1 & | x | \geq b \end{cases} \end{align*}


\begin{align*} - D _ u f ( u ^ * ) \in \overline { L ( \mathcal { K } , u ^ * ) } ^ \circ = L ^ \circ ( \mathcal { K } , u ^ * ) . \end{align*}


\begin{align*} y _ { [ a ^ { j } ] } ^ { j } = v ^ { j } \ , , j = 1 , \ldots , m \end{align*}


\begin{align*} \begin{aligned} \frac { M ^ { p - 2 } } { 2 ^ { p ( n + 3 ) } } \int _ { \widetilde { K } _ n } ( u - \tilde { k } _ n ) _ - ^ 2 \ , d x + \underset { \widetilde { Q } _ n } { \iint } \frac { | ( u - \tilde { k } _ n ) _ { - } ( x , t ) - ( u - \tilde { k } _ n ) _ { - } | ^ p } { | x - y | ^ { N + s p } } \ , d x \ , d y \ , d t \leq \boldsymbol \gamma \frac { 2 ^ { n ( N + 2 p ) } } { \theta \varrho ^ { s p } } M ^ { p } | A _ n | . \end{aligned} \end{align*}


\begin{align*} s _ - = \sqrt { \frac { 1 } { 2 } - \frac { 1 } { 2 } \sqrt { \frac { C - 4 } { C } } } \end{align*}


\begin{align*} & \lim _ { x \rightarrow 0 ^ - } \bigg | \frac { r ( x ) } { x ^ { l + 1 } } \bigg | = \bigg | \frac { h ^ { ( l + 1 ) } ( 0 ) } { ( l + 1 ) ! } \bigg | . \end{align*}


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


\begin{align*} \widehat { \pi } _ { 1 , 2 } = ( S _ 1 , S _ 2 ) = \gamma ^ { \eta _ 0 , \delta } , \widehat { \pi } _ { 1 , 3 } = ( S _ 1 , \widetilde { S } _ 1 ) = \nu \in \Pi ( \gamma _ 1 ^ { \eta , \delta } , \gamma _ 1 ^ { \star } ) , \end{align*}


\begin{align*} \begin{cases} P ( ( - \Delta ) ^ s ) u _ j + q _ j u _ j = 0 & \Omega , \\ u _ j = f _ j & \Omega ^ c . \end{cases} \end{align*}


\begin{align*} \textbf { y } = \sqrt { \rho _ { f } } \textbf { G } ^ T \textbf { P } \textbf { x } + \textbf { w } , \end{align*}


\begin{align*} \sup _ { \omega \in P } \omega ( a ^ * a ) = \| a \| ^ 2 , a \in A . \end{align*}


\begin{align*} S A = - a S \nabla E a - A S \nabla \tilde \eta a - \tilde a S \nabla \tilde \eta A - R a - \tilde r A - A , \end{align*}


\begin{align*} v _ { 0 , \lambda } ( t , x ) : = ( \tau , z ) \mapsto K _ { t - \tau } ( x , z ) \ , \sigma ( u _ \lambda ( \tau , z ) ) \ , 1 _ { [ 0 , t ] } ( \tau ) . \end{align*}


\begin{align*} & \int _ { \mathcal { A } ^ l _ m } \bigg | f _ 0 ^ { ' } \Big ( U _ { \mu ^ i _ m , \xi ^ i _ m } ( x ) \Big ) \psi _ { \mu ^ i _ m , \xi ^ i _ m } ^ h ( x ) u _ m ( x ) \bigg | d x \\ \leq & C | \psi _ { \mu ^ i _ m , \xi ^ i _ m } ^ h | _ { \frac { 2 n } { n - 2 } } | u _ m | _ { \frac { 2 n } { n - 2 } } \Big ( \int _ { \mathcal { A } ^ l _ m } U _ { \mu ^ i _ m , \xi ^ i _ m } ^ { \frac { 2 n } { n - 2 } } d x \Big ) ^ { \frac { 2 } { n } } = O \bigg ( \Big ( \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big ) ^ { \frac { n } { n - 2 } } \bigg ) . \end{align*}


\begin{align*} B = \sum _ { i = 1 } ^ { n - r } b _ i Y _ i ^ 2 , \end{align*}


\begin{align*} \begin{aligned} \psi ( \beta , \mu ) & = \sup _ { \tilde \rho } \left [ \tilde \rho \mu - \phi ( \beta , \tilde \rho ) \right ] \geq \rho _ 0 \mu - \phi ( \beta , \rho _ 0 ) \\ \psi _ 0 ( \beta , \mu ) & = \sup _ { \tilde \rho } \left [ \tilde \rho \mu - \phi _ 0 ( \beta , \tilde \rho ) \right ] = \rho _ 0 \mu - \phi _ 0 ( \beta , \rho _ 0 ) \end{aligned} \end{align*}


\begin{align*} A = A ( \alpha ) = \langle x _ 1 , x _ 2 , x _ 3 \rangle / ( \alpha x _ 3 x _ 1 + x _ 1 x _ 3 , \alpha x _ 3 x _ 2 - x _ 2 x _ 3 , x _ 1 ^ 2 - x _ 2 ^ 2 ) . \end{align*}


\begin{align*} L i _ 2 ( 1 ) = - \frac { \pi ^ 2 } { 6 } , \end{align*}


\begin{align*} T _ 0 : = L _ 0 + C \hbox { a n d } \widetilde { T } _ 0 : = \widetilde { L } _ 0 + C ^ * \ ; . \end{align*}


\begin{align*} \| v _ t \| _ { L ^ 2 ( \mathrm { M } _ 0 ) } ^ 2 & = ( u | \partial _ \nu w _ t ) _ { L ^ 2 ( \partial \mathrm { M } _ 0 ) } - ( \partial _ \nu u | w _ t ) _ { L ^ 2 ( \partial \mathrm { M } _ 0 ) } \\ & \le \| u \| _ { L ^ 2 ( \partial \mathrm { M } _ 0 ) } \| \partial _ \nu w _ t \| _ { L ^ 2 ( \partial \mathrm { M } _ 0 ) } + \| \partial _ \nu u \| _ { L ^ 2 ( \partial \mathrm { M } _ 0 ) } \| w _ t \| _ { L ^ 2 ( \partial \mathrm { M } _ 0 ) } . \end{align*}


\begin{align*} e ^ { r X ^ M } m = ( e ^ { r X } \tau ( m ) ) \cdot m & & r \in \R , m \in M . \end{align*}


\begin{align*} n ^ { - 1 } g u ^ { p \bar q } u _ { p \bar q j } = g _ j + g _ v v _ j , \end{align*}


\begin{align*} \begin{aligned} \bar P _ b \approx & \frac { 1 } { 1 2 } \int _ 0 ^ \infty \exp \left ( - { \frac { \rho \zeta ^ 2 x } { 4 ( 1 + \rho ( 1 - \zeta ^ 2 ) \sigma _ e ^ 2 L ) } } \right ) f ( x ) d x \\ & + \frac { 1 } { 4 } \int _ 0 ^ \infty \exp \left ( - { \frac { \rho \zeta ^ 2 x } { 3 ( 1 + \rho ( 1 - \zeta ^ 2 ) \sigma _ e ^ 2 L ) } } \right ) f ( x ) d x . \end{aligned} \end{align*}


\begin{align*} \bar { \alpha } ( \Omega ) = \inf \left \lbrace \dfrac { \mathrm { V o l } \left ( \Omega \Delta B _ \rho ( x ) \right ) } { \mathrm { V o l } ( B _ { \rho } ) } : x \in \mathbb { R } ^ { n + 1 } , \mathrm { V o l } ( \Omega ) = \mathrm { V o l } ( B _ { \rho } ) \right \rbrace \end{align*}


\begin{align*} & U ( p ) \left ( \sum _ { n \in \mathbb { Z } } C ( n ) q ^ { n } \right ) : = \sum _ { n \in \mathbb { Z } } C ( p n ) q ^ { n } \\ & V ( p ) \left ( \sum _ { n \in \mathbb { Z } } C ( n ) q ^ { n } \right ) : = \sum _ { n \in \mathbb { Z } } C ( n / p ) q ^ { n } \\ & D ^ { k - 1 } \left ( \sum _ { n \in \mathbb { Z } } C ( n ) q ^ { n } \right ) : = \sum _ { n \in \mathbb { Z } } n ^ { k - 1 } C ( n ) q ^ { n } . \end{align*}


\begin{align*} - \mathrm { d i v } \Big ( a \big ( H ( \nabla u ) \big ) \ , \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla u ) \Big ) = f \quad \Omega \ , . \end{align*}


\begin{align*} \beta = \alpha \left ( \left \{ \left . \tilde { \beta } _ j \ \right | \ j \in I ' \right \} \right ) . \end{align*}


\begin{align*} F _ X ( x ) = 1 & - \frac { ( 1 - \beta ^ 2 / \alpha ^ 2 ) ^ { \nu + 1 / 2 } } { 2 \sqrt { \pi } \Gamma ( \nu + 1 / 2 ) } \sum _ { k = 0 } ^ \infty \frac { 1 } { k ! } \bigg ( \frac { 2 \beta } { \alpha } \bigg ) ^ k \Gamma \bigg ( \frac { k + 1 } { 2 } \bigg ) \Gamma \bigg ( \nu + \frac { k + 1 } { 2 } \bigg ) \tilde { G } _ { \nu + k , \nu } ( \alpha ( x - \mu ) ) , \end{align*}


\begin{align*} Z _ q ( K _ { n , m } ) & = \left \{ \begin{aligned} & \min ( n , m ) & \mbox { i f } q = 0 \\ & n + m - 2 & \mbox { f o r } q \geq 1 . \end{aligned} \right . \end{align*}


\begin{align*} L _ { I } = \left \{ \left [ \begin{array} { c c c c } M _ { 1 } \\ & M _ { 2 } \\ & & \ddots \\ & & & M _ { \ell } \end{array} \right ] , \ M _ { j } \in G L ( d _ { j } , \mathbb { R } ) , \ \det ( M _ { 1 } ) \ldots \det ( M _ { \ell } ) = 1 \right \} . \end{align*}


\begin{align*} \Lambda _ f ( s , \alpha ) = \gamma ( s ) \sum _ { n \geq 1 } \frac { f _ n e ( n \alpha ) } { n ^ s } . \end{align*}


\begin{align*} ( \underbar { x } ' , \underbar { y } ) = ( x _ 1 , \ldots , x _ m + c ' , \ldots , x _ N , y _ 1 , \ldots , y _ N ) . \end{align*}


\begin{align*} A _ { S } [ \partial _ { t } \gamma _ { S } ] ( e ) = G _ { V , 1 } ( e ) \ , . \end{align*}


\begin{align*} \dfrac { d } { d t } \| U \| ^ 2 _ { I _ 2 \otimes P _ x } + \vec U ^ T ( I _ 2 \otimes B ) { \bf A } \vec U = ( { \bf D _ x } \vec U ) ^ T ( I _ 2 \otimes P _ x ) { \bf A } \vec U - ( { \bf A } \vec U ) ^ T ( I _ 2 \otimes P _ x ) { \bf D _ x } \vec U , \end{align*}


\begin{align*} S ( n , \alpha ) = \sum _ { i = 0 } ^ { n } S ( i , \alpha \cdot \chi _ { 0 } ) \alpha ( T ) ^ { n - i } = \sum _ { i = 0 } ^ { n } S ( i , \iota _ \alpha ) \alpha ( T ) ^ { n - i } . \end{align*}


\begin{align*} X _ N ( L , \alpha ) : = R _ N ^ 2 ( L , \alpha , \Delta ) - \langle R _ N ^ 2 ( L , \alpha , \Delta ) \rangle = \sum _ { k \ne 0 } b _ { k , N } ( L ) e ( k \alpha ) . \end{align*}


\begin{align*} E _ p = F _ 1 ( E _ { p } ) \supsetneq F _ 2 ( E _ { p } ) \supsetneq \cdots \supsetneq F _ { l _ p } ( E _ { p } ) \supsetneq F _ { l _ { p + 1 } } ( E _ { p } ) = 0 \end{align*}


\begin{align*} W ^ T \Lambda W + 2 U ^ T ( J ^ - T ^ { - 1 } ) ^ T \Sigma ( \sqrt { | \Lambda ^ - | } W ^ - - R \sqrt { \Lambda ^ + } W ^ + ) ) = W ^ T \Lambda W + 2 ( W ^ - ) ^ T \Sigma ( \sqrt { | \Lambda ^ - | } W ^ - - R \sqrt { \Lambda ^ + } W ^ + ) . \end{align*}


\begin{align*} \left \| \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} \right \| _ { ( k ) } & \le \left \| \frac { A + B } { 2 } + ( r + \omega ) I \right \| _ { ( k ) } + \left \| \frac { A + B } { 2 } - r I \right \| _ { ( k ) } \\ & = \left \| A + B + \omega I \right \| _ { ( k ) } . \end{align*}


\begin{align*} \left < \phi , Y _ t ^ N \right > = \left < \phi ( \cdot , 1 ) , Y _ { t , 1 } ^ N \right > + \left < \phi ( \cdot , - 1 ) , Y _ { t , - 1 } ^ N \right > . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta ) : = \sup _ { \gamma \in \Sigma _ { \mathrm { D } } ( \delta ) } \int _ { \mathcal { V } } g \ , d \gamma , \delta \in \mathbb { R } _ { + } ^ 2 , \end{align*}


\begin{align*} \| F \| _ X \leq K _ X \| F - F _ N \| _ X + K _ X \sum _ { n = 1 } ^ \infty K _ X ^ n \| f _ n \| _ X \end{align*}


\begin{align*} \partial _ t \phi _ { A } + \frac { 1 } { 2 } \vert \nabla \phi _ { A } \vert ^ 2 = - \frac { p _ { A } } { \rho _ { A } } - g x _ { 2 } \ , . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c w ) ^ n = f ^ n \omega ^ n & \textnormal { i n } & \Omega , \\ w = 0 & \textnormal { o n } & \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} \Lambda _ k : = \bigcup _ { \lambda \vdash k } 2 \lambda + 1 . \end{align*}


\begin{align*} \| f ^ { = d } \| _ q \leq \gamma ' \rho ^ { - d } = \gamma \left ( 9 9 \cdot 2 \ , r \sqrt { q } \sqrt { \frac { \log ( 1 / \gamma ) } { d } } \right ) ^ d \leq \gamma \left ( 2 0 0 r \sqrt { q } \cdot \max \left \{ \sqrt { q } , \sqrt { \log ( 1 / \gamma ) } \right \} \right ) ^ d , \end{align*}


\begin{align*} A D _ { \lambda } ^ { d } = \left \{ \mathcal { K } \times I \in \mathcal { A } \ , : E _ { p , \tau } \left ( u \ , ; \ , { K } ^ { d } \times I \right ) \leq \frac { \lambda } { 1 6 } \right \} \end{align*}


\begin{align*} V _ k ^ { j , \tau } ( T , \bar { s } ) & = O ( e ^ { ( \max \{ \lambda _ { i } , \lambda _ 1 a \} - \lambda _ 1 + 8 \tau ( 1 + \lambda _ 1 ) ) r ( 0 , k ) } ) \\ W _ k ^ { j , \tau } ( T , \bar { s } ) & = O ( e ^ { ( \max \{ \lambda _ { i } - \lambda _ 1 , \lambda _ 1 a - \lambda _ 1 , - \lambda _ j \} + 9 \tau ( 1 + \lambda _ 1 ) ) r ( 0 , k ) } ) . \end{align*}


\begin{align*} V _ I = \{ e _ i \ | \ i \in I \} \subset \C ^ n \ . \end{align*}


\begin{align*} g = ( d \psi + \hat \sigma ) ^ 2 + \hat { g } \end{align*}


\begin{align*} | \mathcal { P } _ k | & = ( 2 k - 1 ) ! ! . \end{align*}


\begin{align*} \lim _ { k \rightarrow \omega } \| u ^ k _ { g h } - u ^ k _ g u ^ k _ h \| & = 0 , g , h \in G , \intertext { a n d } \lim _ { k \rightarrow \omega } \mathrm { t r } _ G ( u ^ k _ g ) & = 0 , g \in G \setminus \{ 1 \} . \end{align*}


\begin{align*} & \vec { \mathcal { A } } = \mathcal { A } , \\ & \overleftrightarrow { \mathcal { A } } = \overleftarrow { \mathcal { A } } = \coprod _ { n \geq 0 } \{ f : [ n ] \to [ n ] \mid f \in \mathcal { A } \} . \end{align*}


\begin{align*} P \left ( \tau _ N < s _ N ( \rho ) \right ) = O \big ( N ^ { 2 ( \rho - 1 ) } \big ) . \end{align*}


\begin{align*} e ^ { - \phi } = 1 - \phi + O ( \phi ^ 2 ) . \end{align*}


\begin{align*} \begin{bmatrix} x _ { 1 , k + 1 } \\ y _ { 1 , k } ^ 1 \end{bmatrix} = \begin{bmatrix} A _ 1 & B _ 1 \\ C _ 1 & D _ 1 \end{bmatrix} \begin{bmatrix} x _ { 1 , k } \\ u _ { 1 , k } ^ 1 \end{bmatrix} , \begin{bmatrix} x _ { 2 , k + 1 } \\ y _ { 2 , k } ^ 1 \end{bmatrix} = \begin{bmatrix} A _ 2 & B _ 2 \\ C _ 2 & D _ 2 \end{bmatrix} \begin{bmatrix} x _ { 2 , k } \\ u _ { 2 , k } ^ 1 \end{bmatrix} , \end{align*}


\begin{align*} \Lambda _ { , 2 } ( k , p ) = \left ( \frac { n - p } { p } \right ) ^ p \prod _ { s = 1 } ^ k \left ( \frac { n } { p } - 2 s - 1 \right ) ^ p \left ( \frac { n ( p - 1 ) } { p } + 2 s - 1 \right ) ^ p . \end{align*}


\begin{align*} p _ i ( x _ i ) : = \frac { 1 } { 1 + e ^ { \alpha _ i h _ i + \beta _ i x _ i + \gamma _ i } } , \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \partial ^ { 2 } _ { t } v ( t , x ) + a ( t ) \mathcal { H } _ { V } v ( t , x ) + q ( t ) v ( t , x ) = f ( t , x ) , ( t , x ) \in ( 0 , T ] \times \mathbb { R } ^ { n } , \\ v ( 0 , x ) = u _ { 0 } ( x ) , x \in \mathbb { R } ^ { n } , \\ \partial _ { t } v ( 0 , x ) = u _ 1 ( x ) , x \in \mathbb { R } ^ { n } , \end{array} \right . \end{align*}


\begin{align*} \mathcal { H } ^ 1 \left ( J \cap B _ { 2 s \rho } \right ) \geq \mathcal { H } ^ 1 \left ( J \cap B _ { 2 r _ x } ( x ) \right ) = \mathcal { H } ^ 1 \left ( J \cap B _ { \lambda _ x } ( x ) \right ) \geq \eta \lambda _ x > \eta ( 1 - s ) \rho , \end{align*}


\begin{align*} G = \bigsqcup _ { i = 1 } ^ d g _ { i } H . \end{align*}


\begin{align*} \mathbf { v } \cdot \mathbf { n } = 0 \ \ \ \ \ , \partial D , \end{align*}


\begin{align*} \tilde { \chi } _ j \Pi _ \lambda ( 1 - \chi _ j ) u = 0 . \end{align*}


\begin{align*} f \left ( \sigma ( w ) \right ) : = \delta _ { \sigma K } - \frac { | \mathcal { S } | } { | \Omega | } \sum _ { x K \in G / K } \delta _ { \sigma x K } . \end{align*}


\begin{align*} \mathcal { R } \big ( a \mid b \big ) = a \log \tfrac { a } { b } + ( 1 - a ) \log \tfrac { 1 - a } { 1 - b } \end{align*}


\begin{align*} \widetilde { c } _ { X ^ { \ast } } ( g _ 1 ^ { \alpha } , g _ 2 ^ { \alpha } ) = \widetilde { c } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) \nu ( \alpha , g _ 1 ) ^ { - 1 } \nu ( \alpha , g _ 2 ) ^ { - 1 } \nu ( \alpha , g _ 1 g _ 2 ) . \end{align*}


\begin{align*} \left ( 1 + \log ( n - 1 ) \right ) \frac { 1 } { n } \leq 1 , \quad \sum _ { k = 1 } ^ { n - 2 } \frac { 1 } { k } \leq \log n , \quad \sum _ { k = 1 } ^ { n - 2 } \log k \leq n \log n . \end{align*}


\begin{align*} X = \left \{ f \in L ^ { \infty } ( \Omega \times ( 0 , T ) ) \ , \bigr | \ , f \leq K + 1 \ , \mbox { a . e . i n } \ , \Omega \times ( 0 , T ) \right \} . \end{align*}


\begin{align*} U ^ T _ n \tilde A U _ n - \epsilon ( U ^ T _ n \tilde F + \tilde F ^ T U _ n ) \tilde F = ( \tilde F _ n , \tilde F _ { \tau } , 0 ) ^ T \end{align*}


\begin{align*} \lim \limits _ { k \rightarrow + \infty } \langle D _ u \theta ( u ^ { k + 1 } ) , u ^ * - u ^ { k + 1 } \rangle _ \mathcal { U } = 0 . \end{align*}


\begin{align*} \int _ \Omega f \ , d x = 0 . \end{align*}


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


\begin{align*} u _ n ( x \Omega ) = \sigma _ n ( x ) \Omega , ~ ~ x \in M . \end{align*}


\begin{align*} \tau ( ( \sigma ( 1 ) , \sigma ( 2 ) , \ldots , \sigma ( k ) ) ) = ( 1 , 2 , \ldots , k ) = \hat { A } . \end{align*}


\begin{align*} \nabla w = ( \partial _ \nu w ) \ , \nu \quad \partial \Omega , \end{align*}


\begin{align*} Q = \left \{ \begin{array} { l r } \min \{ S _ k - x , B \} & s _ { k - 1 } < x \leq s _ k , \\ 0 & x > s _ { m } ; \end{array} \right . \end{align*}


\begin{align*} \psi ( x , y ) = \psi ( x , z ) \frac { Q ( z ) } { Q ( y ) } + \sum _ { y < p \le z } \psi ( x / p , p ^ - ) \cdot \frac { 1 } { p } \prod _ { y < q < p } \left ( 1 - \frac { 1 } { q } \right ) , \end{align*}


\begin{align*} \frac { \partial V _ { n + k + 1 } } { \partial \bar { c } _ { n - i } } = ( - 1 ) ^ { n - i + 1 } c _ { k + i + 1 } \ . \end{align*}


\begin{align*} \tilde Q _ \ell = \sum \limits _ { j = 1 } ^ { s _ \ell } b _ { \ell i j } \tilde { \textbf { Q } } ^ { \lambda _ j } \end{align*}


\begin{align*} \xi _ { 2 - k } ( H ) = \xi _ { 2 - k } ( f ) = \dfrac { g } { \left \| g \right \| ^ { 2 } } \not = 0 . \end{align*}


\begin{align*} f _ { \mathcal J } ( a ) = \begin{cases} a , & n \notin J _ a , \\ n + a + 1 - x , & n \in J _ a , J _ { a + 1 } = \tau _ 1 ( J _ a \setminus \{ n \} ) \cup \{ x \} . \end{cases} \end{align*}


\begin{align*} X \qquad \Leftrightarrow \qquad \mathcal { C } = \mathsf { C } \mathbb { P } \qquad \Leftrightarrow \qquad \mathcal { C } \mathsf { C } \mathbb { P } . \end{align*}


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


\begin{align*} D = 1 - \Delta . \end{align*}


\begin{align*} M _ n ( f ) : = \frac { 1 } { n ^ d } \sum _ { ( j _ 1 , \ldots , j _ d ) \in \Z ^ d _ + \cap Q _ n } S _ 1 ^ { j _ 1 } \cdots S _ d ^ { j _ d } ( M _ 1 ( f ) ) \end{align*}


\begin{align*} K \subset \bigcup _ { i = 1 } ^ r \{ \psi \in \S ( B ) : \psi ( f _ { \lambda _ i } ) < 1 - 1 / { m _ i } \} \subset \{ \psi \in \S ( B ) : \psi ( f ) < \gamma \} . \end{align*}


\begin{align*} \langle h _ 3 \rangle = W _ 1 , \langle h _ 2 , h _ 3 \rangle = W _ 2 , \langle h _ 1 , h _ 2 , h _ 3 \rangle = W _ 3 . \end{align*}


\begin{align*} \begin{cases} P ( ( - \Delta ) ^ s ) _ j u _ j + q u _ j : = \sum _ { i = 1 } ^ M \alpha _ i ^ j ( - \Delta ) ^ { s _ i } u _ j + q u _ j = 0 & \Omega , \\ u _ j = f _ j & \Omega ^ c \end{cases} \end{align*}


\begin{align*} \xi _ i = \xi + \mu _ i \sigma _ i , \quad \mbox { f o r \ s o m e \ p o i n t s } \ \sigma _ i \in \R ^ n , \end{align*}


\begin{align*} P \left ( M _ t > \gamma t \right ) = P \left ( \exists \ , u \ : \in \mathcal { N } _ t : \sup _ { 0 \le s \le t } | Y _ u ( s ) | > \gamma t \right ) \le E [ N _ t ] \ : \mathbf { P } _ 0 \left ( \sup _ { 0 \le s \le t } | X _ s | > \gamma t \right ) . \end{align*}


\begin{align*} \beta _ i \in \begin{cases} \{ 0 \} & \mbox { i f } d _ i - x _ i < y _ i \\ \{ p _ i \} & \mbox { i f } d _ i - x _ i > y _ i \\ [ 0 , p _ i ] & \mbox { i f } d _ i - x _ i = y _ i \end{cases} . \end{align*}


\begin{align*} - \frac { \binom { n - k } { k } \binom { n - 1 } { k - 1 } } { \binom { n - 1 } { k } } . \end{align*}


\begin{align*} m _ 2 = 2 m _ 1 . \end{align*}


\begin{align*} \begin{cases} u = - \widetilde { P } _ 1 ( t _ 1 ) + \widetilde { P } _ 1 ( t _ 2 ) - \widetilde { P } _ 1 ( t _ 3 ) , \\ v = \widetilde { P } _ 2 ( t _ 1 ) - \widetilde { P } _ 2 ( t _ 2 ) + \widetilde { P } _ 2 ( t _ 3 ) , \\ w = t _ 1 . \end{cases} \end{align*}


\begin{align*} \Tilde { z } _ { k , \mathrm { a v g } } & = \sum _ { k ' = k - K } ^ { k + K } \left ( \sum _ { l = 0 } ^ { L - 1 } z ' _ { k ' , l } \right ) ^ { \mu } \\ \varphi _ { k , \mathrm { e s t } , \mathrm { m o d } } & = \frac { 1 } { \mu } \mathrm { u n w r a p } \left ( \arg \left ( \Tilde { z } _ { k , \mathrm { a v g } } \right ) \right ) . \end{align*}


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


\begin{align*} \delta _ r ( z ) = E _ I ( 0 , \delta _ r ( L o g ( z ) ) ) . \end{align*}


\begin{align*} \mathcal S = \{ i \in I \mid \alpha \neq \alpha ( \{ \beta _ j \mid j \geq i \} ) \} . \end{align*}


\begin{align*} ( f \sharp g ) _ { m + m ' } ( \xi ) = f _ m ( \xi ) g _ { m ' } ( \xi ) . \end{align*}


\begin{align*} w ( z ) = \Phi ( z ) + T ( f ) ( z ) , \end{align*}


\begin{align*} W _ { h , q } ( i ) = \begin{cases} h + 1 & i = q ; \\ 1 & k - j + h \leq i \leq k - 2 i \neq q ; \\ l - j + 1 & i = k - 1 ; \\ 0 & . \end{cases} \end{align*}


\begin{align*} v \cdot u = & \sinh ( \phi - \theta ) = \left ( \begin{array} { c } \cosh ( \phi - \theta ) \\ \sinh ( \phi - \theta ) \\ \end{array} \right ) \cdot \left ( \begin{array} { c } 0 \\ 1 \\ \end{array} \right ) , \end{align*}


\begin{align*} S _ { \omega , \mathbf { c } } ( \Psi _ { \omega } ) = \omega ^ { 2 - \frac { d } { 2 } } S _ { 1 , \frac { \mathbf { c } } { \sqrt { \omega } } } ( \Psi ) . \end{align*}


\begin{align*} r ^ \delta u ' ( r ) ^ { p - 1 - \alpha } = \frac { \delta } { n - 1 } \int _ 0 ^ r s ^ \delta f _ 1 ( s ) g _ 1 ( v ( s ) ) \ , d s \leq \frac { \delta } { n - 1 } \ , f _ 1 ( r ) g _ 1 ( v ( r ) ) \int _ 0 ^ r s ^ \delta \ , d s \end{align*}


\begin{align*} g ( \nabla _ { X } \ , \xi _ i , \ \xi _ k ) = 0 , X \in T M , \ 1 \le i , k \le p . \end{align*}


\begin{align*} u _ i - u _ j + u _ j - u _ k = u _ i - u _ k \rlap { . } \end{align*}


\begin{align*} \left [ P \right ] _ { i , j } ^ { } = \begin{dcases} q _ i ^ { } & \mbox { i f } \ , j = i ; \\ p _ i ^ { } & \mbox { i f } \ , j = i + 1 ; \\ 0 & \mbox { o t h e r w i s e } ; \end{dcases} \end{align*}


\begin{align*} f = f _ 0 + f _ 1 q + \ldots + f _ r q ^ r \mbox { w i t h } \deg ( f _ i ) < \deg ( q ) \mbox { f o r e v e r y } i , 0 \leq i \leq r , \end{align*}


\begin{align*} \lim _ { i \to \infty } \frac { 1 } { \ell _ i } \sum _ { n \leq \ell _ i } \mathbf { 1 } _ { \underline { C } _ K } ( R ^ n ( \Delta ( 0 ) ) ) \mathbf { 1 } _ { C _ K } ( R ^ n ( \Delta ( 0 ) ) ) = m _ H ( \underline { C } _ K \cap C _ K ) . \end{align*}


\begin{align*} \Pr ( \widetilde { \mathbf { X } } _ { t } = \widetilde { \mathbf { x } } ^ * _ { t } ) = p _ 0 \cdot p _ 1 \cdots p _ { k - 1 } \cdot q _ 0 ^ { c _ 0 ^ { * } } \cdot q _ 1 ^ { c _ 1 ^ { * } } \cdots q _ k ^ { c _ k ^ { * } } . \end{align*}


\begin{align*} \frac { t e ^ { x t } } { e ^ t - 1 } = \sum _ { n = 0 } ^ \infty B _ n ( x ) \cdot \frac { t ^ n } { n ! } . \end{align*}


\begin{align*} \Omega _ s = \{ \omega \in \Omega : \exists \ : \ell _ 1 = \ell _ 1 ( \omega ) , \ : \forall \ : \ell \geq \ell _ 1 , \ : [ - \ell , \ell ] ^ d \ : \ : R _ \ell \} . \end{align*}


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


\begin{align*} \mathcal { M } _ { q _ 1 , \alpha } f = \mathcal { M } _ { q _ 2 , \alpha } f \end{align*}


\begin{align*} \hat { \nabla } = D _ { \theta } + \sum ^ r _ { i = 1 } g _ i ( z , \theta ) \check { \alpha } _ i + \sum ^ r _ { i = 1 } { a _ i ( z , \theta ) } e _ i , \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta _ 1 , 0 ) = \widehat { \mathcal { I } } _ { \mathrm { D } } ( \delta _ 1 , \delta _ 2 ) = \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \left [ \langle \lambda , \delta \rangle + \sup _ { \varpi \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \int _ { \mathcal { V } } g _ \lambda \left ( s _ 1 , s _ 2 \right ) d \varpi ( s _ 1 , s _ 2 ) \right ] , \end{align*}


\begin{align*} \lim _ { N \to \infty } N ^ { - 1 } S _ j ( \tau _ N ) = \nu \int _ 0 ^ \infty e ^ { - \lambda s } p _ j ( s ) d s , \end{align*}


\begin{align*} \liminf _ { m \rightarrow \infty } \int _ \Omega f _ \epsilon ^ { ' } ( V _ m ) u _ m ^ 2 d x = C > 0 , \end{align*}


\begin{align*} \zeta _ { Q } \left ( s \right ) = \frac { 2 \pi } { \sqrt { | d | } } \ , \frac { 1 } { s - 1 } + \frac { 2 \pi } { \sqrt { | d | } } \ , \left ( 2 \gamma - \log \left ( \frac { | d | } { a } \right ) - 4 \log \ , ( | \eta ( \tau ) | ) \right ) + O ( s - 1 ) , \end{align*}


\begin{align*} s _ { 1 } ( A ) = \sup _ { x \in \mathbb { S } ^ { n - 1 } } \Vert A x \Vert _ { 2 } , s _ { n } ( A ) = \inf _ { x \in \mathbb { S } ^ { n - 1 } } \Vert A x \Vert _ { 2 } . \end{align*}


\begin{align*} D ( t , \mu ) : = \Sigma _ t \setminus R ( \Sigma _ t ' , \mu ) \end{align*}


\begin{align*} T _ 5 \geq & - C s ^ 2 \lambda ^ 4 \iint _ Q \xi \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 | u | ^ 2 d x d t - C \lambda ^ 2 \iint _ { Q } \xi \left | A \nabla u \cdot \nabla \eta \right | ^ 2 d x d t \\ & - C s ^ 2 \lambda ^ 3 \int _ 0 ^ T \int _ { \omega } \xi | u | ^ 2 d x d t - C \lambda \iint _ Q \xi A \nabla u \cdot \nabla u d x d t . \end{align*}


\begin{align*} \widetilde { M } _ i : = M _ i \bar { \rtimes } _ { \sigma ^ { \varphi _ i } } \mathbb { R } , \widetilde { M _ 1 \bar { \otimes } M _ 2 } : = ( M _ 1 \bar { \otimes } M _ 2 ) \bar { \rtimes } _ { \sigma ^ { \varphi _ 1 \bar { \otimes } \varphi _ 2 } } \mathbb { R } \end{align*}


\begin{align*} \partial _ t u _ \varepsilon - \varepsilon \ , \Delta _ x u _ \varepsilon + \mathrm { d i v } _ x \big ( \overrightarrow { V _ \varepsilon } \ , u _ \varepsilon \big ) = 0 , \end{align*}


\begin{align*} i \partial _ t u + \partial _ x ^ 2 u + i \partial _ x ( | u | ^ 2 u ) = 0 , \ \ \ ( t , x ) \in \R \times \R , \end{align*}


\begin{align*} \beta _ n = \sum _ { j \leq n } \frac { \alpha _ j } { ( q ) _ { n - j } ( a q ) _ { n + j } } \ ; \ ; \ ; \ ; \forall \ , n \in \mathbb { Z } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ k a _ i \leq \sum _ { i = 1 } ^ k b _ i \end{align*}


\begin{align*} \phi ( E _ { i i } \otimes E _ { j j } ) = E _ { i i } \otimes E _ { j j } \end{align*}


\begin{align*} \mathcal { S } = \left \{ \left ( A ^ \pm , \underline { A } ^ \pm \right ) : \ A \in \binom { [ n ] } { k } \mbox { a n d } n \in A \right \} . \end{align*}


\begin{align*} y ( q x ) = h ( x ) y ( x ) \quad h ( x ) \in \mathcal O ( \C ) \backslash \{ 0 \} \end{align*}


\begin{align*} \frac { 1 } { e _ i ^ * ( u ) } = \frac { \theta _ i } { e ^ { \rho _ i ( u ) } } - \frac { \theta _ i Z _ i ( u ) } { T _ i ^ { \infty } } . \end{align*}


\begin{align*} m ( r _ n , \frac { 1 } { A } ) & = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \log \left | \frac { 1 } { A ( r e ^ { \iota \theta } ) } \right | d \theta \\ & = 0 . \end{align*}


\begin{align*} E ^ * : = \{ y : y \cdot x \leq h \ \ x \in E \} . \end{align*}


\begin{align*} & p _ a ( S ( l ) D \varphi ) \leq C p _ a ( D \varphi ) 0 < a < 1 , \\ & p _ a ( S ( l ) D \varphi ) \leq C ( 1 + \log \| Q ( l ) \| ) p _ a ( D \varphi ) a = 0 . \end{align*}


\begin{align*} ( d d ^ c u ) ^ n = \psi ( \cdot , u ) \omega ^ n . \end{align*}


\begin{align*} \Omega ( \varepsilon ) : = \{ \mathbf { t } \in I ^ 3 : | F ( \mathbf { t } ) | \leq \varepsilon \} \textrm { f o r a n y $ \varepsilon > 0 $ } . \end{align*}


\begin{align*} \nu ( g ' ) = \nu _ i ( g ' ) = \nu ( g ' ( a _ i ) ) \mbox { f o r e v e r y } i \in I ^ * \mbox { w i t h } i \geq i _ 0 . \end{align*}


\begin{align*} F = \frac { f ' } { f } , \end{align*}


\begin{align*} f ( g ( x y ) ) + ( f ( x y ) ) g + \alpha _ 1 \big ( f ( g ( x y ) ) + f ( ( x y ) g ) \big ) + \alpha _ 2 \big ( g ( f ( x y ) ) + g ( ( x y ) f ) \big ) \\ = ( f ( g x ) ) y + ( ( f x ) g ) y + \alpha _ 1 \big ( ( f ( g x ) ) y + ( f ( x g ) ) y \big ) + \alpha _ 2 \big ( ( g ( f x ) ) y + ( g ( x f ) ) y \big ) \\ - ( f ( g y ) ) x + ( ( f y ) g ) x + \alpha _ 1 \big ( ( f ( g y ) ) x + ( f ( y g ) ) x \big ) + \alpha _ 2 \big ( ( g ( f y ) ) x + ( g ( y f ) ) x \big ) . \end{align*}


\begin{align*} { { \hat V } _ { \alpha , \beta } } \left ( { { \boldsymbol { X } } , { \boldsymbol { Y } } } \right ) = { { \hat V } _ { \alpha , \beta } } \left ( { \boldsymbol { e } } \right ) = \frac { 1 } { { { N ^ 2 } } } \sum \limits _ { i = 1 } ^ N { \sum \limits _ { j = 1 } ^ N { { G _ { \alpha , \beta } } \left ( { { e _ i } - { e _ j } } \right ) } } , \end{align*}


\begin{align*} \textsf { E } \Vert B A \Vert ^ { p } = \textsf { E } \Vert B A \Vert ^ { p } \mathbb { I } _ { ( M \le C _ { 1 } ( \varepsilon ) ) } + \sum _ { k = 1 } ^ { \infty } \textsf { E } \Vert B A \Vert ^ { p } \mathbb { I } _ { ( 2 ^ { k - 1 } C _ { 1 } ( \varepsilon ) < M \le 2 ^ { k } C _ { 1 } ( \varepsilon ) ) } . \end{align*}


\begin{align*} \textsf { E } \Vert B A \Vert ^ { p } & = \textsf { E } \Vert B ( A - \textsf { E } \tilde { A } ) \Vert ^ { p } = \textsf { E } \Vert \textsf { E } ( B ( A - \tilde { A } ) | A ) \Vert ^ { p } \le \textsf { E } \Vert B ( A - \tilde { A } ) \Vert ^ { p } \\ & = \textsf { E } \Vert B ( \varepsilon _ { i j } ( a _ { i j } - \tilde { a } _ { i j } ) ) _ { N \times n } \Vert ^ { p } \le 2 ^ { p } \textsf { E } \Vert B ( \varepsilon _ { i j } a _ { i j } ) _ { N \times n } \Vert ^ { p } . \end{align*}


\begin{align*} \mu _ { \omega , \mathbf { c } } \le \frac { 1 } { 6 } L _ { \omega , \mathbf { c } } ( \lambda U ) = \frac { \lambda ^ 2 } { 6 } L _ { \omega , \mathbf { c } } ( U ) < \frac { 1 } { 6 } L _ { \omega , \mathbf { c } } ( U ) . \end{align*}


\begin{align*} \begin{bmatrix} A ^ * A & A ^ * B \\ B ^ * A & B ^ * B \end{bmatrix} \end{align*}


\begin{align*} \Psi _ { \omega } ( x ) : = \omega ^ { \frac { 1 } { 2 } } \Psi ( \omega ^ { \frac { 1 } { 2 } } x ) . \end{align*}


\begin{align*} F _ { \lambda , k , r } ( c _ 1 , c _ 2 ; b _ 1 , \dots , b _ \lambda ; a ; q ) = G _ { \lambda , k , r } ( c _ 1 , c _ 2 ; b _ 1 , \dots , b _ \lambda ; a ; q ) , \end{align*}


\begin{align*} h _ 1 ( z ) : = A _ 1 z ^ { n + m } + B _ 1 \overline { z } ^ m + C _ 1 \textrm { f o r a l l } z \in \mathbb { C } , \end{align*}


\begin{align*} g ( z ) : = | A | z ^ { n + m } - | B | \overline { z } ^ m - | C | , z \in \mathbb { C } , \end{align*}


\begin{align*} \left | \frac { \partial z } { \partial \phi } \det ( M _ 2 ) \right | ( \phi , \omega , r ; t ) , \ , \left | \frac { \partial z } { \partial r } \det ( M _ 3 ) \right | ( \phi , \omega , r ; t ) = o ( t ^ 5 ) , t \to 0 . \end{align*}


\begin{align*} b _ { i , n + 1 } = b _ { \eta ( i ) , \eta ( n + 1 ) } = b _ { i , n } . \end{align*}


\begin{align*} F _ X ( x ) & = \frac { ( 1 - \beta ^ 2 / \alpha ^ 2 ) ^ { \nu + 1 / 2 } } { 2 \sqrt { \pi } \Gamma ( \nu + 1 / 2 ) } \sum _ { k = 0 } ^ \infty \frac { ( - 1 ) ^ k } { k ! } \bigg ( \frac { 2 \beta } { \alpha } \bigg ) ^ k \Gamma \bigg ( \frac { k + 1 } { 2 } \bigg ) \Gamma \bigg ( \nu + \frac { k + 1 } { 2 } \bigg ) \tilde { G } _ { \nu + k , \nu } ( - \alpha ( x - \mu ) ) . \end{align*}


\begin{align*} f _ A ( x ) = \mu _ H ( ( [ \psi ^ { - 1 } ( x ) ] ^ { - 1 } \cdot A ) \cap H ) \ \ \ \ x \in \psi ( A ) , \end{align*}


\begin{align*} { \rm I I } _ { x } = - \sum _ { s \in G } \mu ( s ) \frac { d s \nu } { d \nu } ( \pi ( x ) ) H \left ( \xi _ { n - 1 } ^ { s ^ { - 1 } . x } \right ) . \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n \left ( e ^ { 2 \alpha n } + 1 \right ) } + 2 \ , \sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 2 n - 1 ) } \ , \frac { 1 } { \left ( e ^ { \beta ( 2 n - 1 ) } - 1 \right ) } = \frac { \alpha } { 4 } - \log ( 2 ) - \frac { 1 } { 4 } \log \left ( \frac { \alpha } { \beta } \right ) . \end{align*}


\begin{align*} \begin{cases} L _ { \lambda } [ X ] = - X '' + ( R - \lambda I ) X = 0 , \\ X ( a ) = X ( t ) = 0 \end{cases} \end{align*}


\begin{align*} W ^ T \Lambda W = ( W ^ + ) ^ T \Lambda ^ + W ^ + - ( W ^ - ) ^ T | \Lambda ^ - | W ^ - = ( \sqrt { \Lambda ^ + } W ^ + ) ^ T ( I - R ^ T R ) ( \sqrt { \Lambda ^ + } W ^ + ) \geq 0 \end{align*}


\begin{align*} a ( j ) = \begin{cases} b , & \mbox { i f } j = 1 \\ f ( r + t ( j - 1 ) ) , & \mbox { i f } j \in \{ 2 , \cdots , m + 1 \} . \end{cases} \end{align*}


\begin{align*} \bar { \rm P } _ { m , i } = { m \choose i } \mathbb { P } ( E _ i ) , \end{align*}


\begin{align*} \zeta _ { p , p ^ { \prime } } ( s , \ , c ) = c ^ { - s } \ , \zeta _ { p ^ { \prime } , p } \left ( s , \ , \frac { 1 } { c } \right ) . \end{align*}


\begin{align*} S _ x = \{ \psi \in \S ( B ) : \psi ( x ^ * x ) = 1 \} . \end{align*}


\begin{align*} R _ { \mathbf { \varepsilon } \bullet } = ( - 1 ) ^ { l ( I ) - 1 } R _ I ( R _ \emptyset = 1 , \ R _ { \bullet } = R _ 1 ) \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*} P U _ t + ( A _ i ( V ) U ) _ { x _ i } + B _ i ( V ) U _ { x _ i } + C ( V ) U = \epsilon ( D _ i ( U ) ) _ { x _ i } , t \geq 0 , \vec x = ( x _ 1 , x _ 2 , . . , x _ k ) \in \Omega \end{align*}


\begin{align*} \| 1 + d Z \| _ { q } ^ { q } \ge 1 + \binom { \lfloor q \rfloor } { 2 } d ^ 2 \ge 1 + \frac { q ^ 2 } { 6 } d ^ 2 . \end{align*}


\begin{align*} \mathcal { A } = \Re \mathcal { A } + i \Im \mathcal { A } , \Re \mathcal { A } = \frac { \mathcal { A } + \mathcal { A } ^ * } { 2 } , \Im \mathcal { A } = \frac { \mathcal { A } - \mathcal { A } ^ * } { 2 i } , \end{align*}


\begin{align*} x _ j + \log \left ( \sum _ { k = 1 } ^ n a _ { j k } x _ k \right ) , j = 1 , \ldots , n . \end{align*}


\begin{align*} 0 = ( ( \lambda _ 3 v _ i ) _ { \vec { v } } ) _ \ell \end{align*}


\begin{align*} \begin{cases} 2 \lambda _ 2 + \frac { 3 } { 2 } \lambda _ 2 \lambda _ { 2 , 3 } = \lambda _ 1 \\ 2 \lambda _ 2 + \lambda _ 2 \lambda _ { 2 , 4 } = \lambda _ 1 \\ \lambda _ 1 + \lambda _ 2 = \lambda _ { 2 , 3 } + \lambda _ { 2 , 4 } = 1 \end{cases} , \end{align*}


\begin{align*} f _ { Z } ( z ) = \frac { z ^ { \kappa - 1 } ( 1 - z ) ^ { \beta - 1 } } { B ( \kappa , \beta ) } , \end{align*}


\begin{align*} \mathbf { T } _ { K ^ \circ } ( \widehat { \mathbb { Z } } ) = \mathbf { T } _ { K ^ \circ } ( \mathbb { A } _ f ) \cap \mathbf { G } ( \widehat { \mathbb { Z } } ) , \mathbf { T } _ { K ^ \circ } ^ 1 ( \widehat { \mathbb { Z } } ) = \mathbf { T } _ { K ^ \circ } ( \widehat { \mathbb { Z } } ) \cap \mathbf { G } ^ 1 ( \widehat { \mathbb { Z } } ) . \end{align*}


\begin{align*} b _ { i , n } b _ { \tau ( i ) , n } = b _ { i , n } b _ { \tau ( i ) , \tau ( n ) } = b _ { i , n } b _ { i , n } \ge 0 . \end{align*}


\begin{align*} \Gamma = \bigcup _ { i = 1 } ^ { \epsilon } \left ( \gamma _ i + \Delta \right ) \mbox { a n d } 0 = \gamma _ 1 < \ldots < \gamma _ \epsilon < \Delta _ { > 0 } . \end{align*}


\begin{align*} E ( t ) = P _ s ( \hat { L } _ p + \hat { L } _ t ) + \alpha P _ s ( \hat { L } _ p + \hat { L } _ { \mathrm { f } } ) , \end{align*}


\begin{align*} \min f ( u ^ * ) + \langle p _ { u ^ * } , u - u ^ * \rangle _ { \mathcal { U } } + \frac { c } { 2 } \| u - u ^ * \| _ { \mathcal { U } } ^ 2 \mbox { s . t . } \ \ S _ { u ^ * } u \in K = \mathcal { K } - G ( u ^ * ) + S _ { u ^ * } u ^ * \end{align*}


\begin{align*} \nu ^ { ( n ) } _ t ( x ) = e ^ { - \alpha t } \nu _ 0 ( x ) + \int _ 0 ^ t \alpha e ^ { - \alpha ( t - s ) } \Psi ( \nu ^ { ( n - 1 ) } _ s , \mu ^ { ( n - 1 ) } _ s ) ( x ) \mathrm { d } s , \end{align*}


\begin{align*} \liminf _ { n \rightarrow + \infty } \left | v _ n \right | _ p ^ p \leq C \left ( \liminf _ { n \rightarrow + \infty } \left | v _ n \right | _ 2 \right ) ^ { \left ( 1 - \beta _ p \right ) p } K ^ { \beta _ p p } \beta _ p = N \left ( \frac { 1 } { 2 } - \frac { 1 } { p } \right ) \end{align*}


\begin{align*} \nu _ i ( Q _ j ) - \nu _ i ( Q ' _ j ) = \alpha _ i . \end{align*}


\begin{align*} P _ { { \cal F } ^ \perp } { \cal S } = P _ { { \cal F } ^ \perp } { \cal S } P _ { { \cal F } ^ \perp } \end{align*}


\begin{align*} Q ( r _ 1 , \theta ) \approx & \frac { \pi ^ 2 \lambda R _ S ( R _ { m a x } - r _ 1 ) } { N R _ E } \sum _ { n = 1 } ^ N \sqrt { 1 - \phi _ n ^ 2 } c _ n \\ & \times \left ( 1 - \prod _ { m = 1 } ^ { M } { \left ( 1 + \frac { m \eta \theta r _ 1 ^ { \alpha } } { M c _ n ^ { \alpha } } \right ) ^ { - M b _ m } } \right ) \end{align*}


\begin{align*} \vect u ( x ) = ( 2 \pi \varepsilon ) ^ { - 3 / 2 } \int _ { Y ' } { \rm e } ^ { { \rm i } \chi \cdot x / \varepsilon } ( \mathcal { G } _ \varepsilon \vect u ) ( x / \varepsilon , \chi ) d \chi . \end{align*}


\begin{align*} \zeta _ { \infty , 0 } ( s , \ , c ) = \frac { \pi } { \sqrt { c } } \ , \frac { 1 } { s - 1 } + \frac { \pi } { \sqrt { c } } \left ( 2 \gamma - \log \left ( \frac { c } { 4 } \right ) + 8 \ , \sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 n - 1 } \cdot \frac { 1 } { e ^ { \pi \sqrt { c } ( 2 n - 1 ) } - 1 } \right ) + O \left ( s - 1 \right ) . \end{align*}


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


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


\begin{align*} \limsup _ { \ell \to \infty } h _ { \mu } ( Z _ { 0 } , \lambda _ { \ell , 0 } ) = 0 \end{align*}


\begin{align*} b ( \xi ) = | \xi - 1 | ^ { \delta _ 1 } | \xi + 1 | ^ { \delta _ { - 1 } } \widetilde { b } ( \xi ) \end{align*}


\begin{align*} \rho e _ \theta \theta _ t + \left ( \rho u e _ \theta - \frac { 2 a ( \theta ) } { Z ( \theta ) } q \right ) \theta _ x + R \rho \theta u _ x + q _ x = \frac { 2 a ( \theta ) } { \tau _ 1 ( \theta ) } q ^ 2 + \frac { 1 } { \mu } S ^ 2 . \end{align*}


\begin{align*} u _ { t _ \alpha t _ \beta } ( 0 ) = - u _ { x _ n } ( 0 ) \rho _ { t _ \alpha t _ \beta } ( 0 ) , ~ ~ \alpha , \beta < 2 n . \end{align*}


\begin{align*} \Sigma _ N ^ 2 ( L , \alpha ) = L + o ( L ) \end{align*}


\begin{align*} f _ { Z } ( z ) = \frac { z ^ { \kappa - 1 } ( 1 - z ) ^ { \beta - 1 } } { B ( \kappa , \beta ) } , \end{align*}


\begin{align*} \frac { i } { z ^ 2 } \left ( \frac { 1 } { z - \gamma } + \frac { 1 } { \gamma } \right ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - i t \gamma } - 1 } { \gamma ^ 2 } \ , e ^ { i z t } \ , d t , \Im ( z ) > \Im ( \gamma ) \end{align*}


\begin{align*} J ( f ; x ) = J ( f _ 1 , \ldots , f _ n ; x _ 1 , \ldots , x _ n ) : = \left | \begin{array} { c c c } \frac { \partial f _ 1 } { \partial x _ 1 } & \ldots & \frac { \partial f _ n } { \partial x _ 1 } \\ \ldots & \ldots & \ldots \\ \frac { \partial f _ 1 } { \partial x _ n } & \ldots & \frac { \partial f _ n } { \partial x _ n } \\ \end{array} \right | \in \C ^ { * } = \C \setminus \{ 0 \} . \end{align*}


\begin{align*} ( M _ i ) _ { k , l } : = \delta _ { k , l + s _ { i + 1 } - s _ i } = \begin{cases} 1 & { \rm i f } \ k = l + s _ { i + 1 } - s _ i , \\ 0 & { \rm o t h e r w i s e } \end{cases} \end{align*}


\begin{align*} \Big ( \frac { d } { d t } \Big ) ^ { s } \det ( p _ { i , j } ( t ) ) _ { i , j = 1 , \ldots , k } = \sum _ { \substack { l _ 1 + \cdots + l _ k = s \\ l _ 1 \geq 0 , \ldots , l _ k \geq 0 } } \binom { s } { l _ 1 , \ldots , l _ k } \det \Big ( p _ { i , j } ^ { ( l _ { i } ) } ( t ) \Big ) _ { i , j = 1 , \ldots , k } , \end{align*}


\begin{align*} \mathcal { T } : = \cap _ { n = 1 } ^ { \infty } \sigma \left ( \xi _ { n } , \xi _ { n + 1 } , \xi _ { n + 2 } , \ldots \right ) . \end{align*}


\begin{align*} \theta _ { 3 / 2 } ( z , \epsilon ) & = \sum _ { n \in \Z } n e ^ { i \epsilon \pi n ^ 2 z } , \end{align*}


\begin{align*} \begin{aligned} - d y _ { ( n , t ) } & = \bigg [ G \big ( y _ { ( n , t ) } , z _ { ( n , t ) } , \tilde { z } _ { ( n , t ) } , \gamma _ { ( n , t , e ) } \big ) + \varphi _ n ( t ) \bigg ] d t - z _ { ( n , t ) } d W _ t - \tilde { z } _ { ( n , t ) } d \xi _ t \\ & \quad - \int _ { \mathcal { E } } \gamma _ { ( n , t , e ) } \tilde { N } ( d e , d t ) , \ t \in [ 0 , \infty ) . \end{aligned} \end{align*}


\begin{align*} \P _ x ( \tau _ { \Gamma } > t ) & = \P _ { t ^ { - 1 / \alpha } x } ( \tau _ { \Gamma } > 1 ) \\ & \leq c \left ( 1 + \left ( t ^ { - 1 / \alpha } | x | \right ) ^ { \alpha - \beta } \right ) M _ { \Gamma } ( t ^ { - 1 / \alpha } x ) \\ & = c \left ( t ^ { - \beta / \alpha } + t ^ { - 1 } | x | ^ { \alpha - \beta } \right ) M _ { \Gamma } ( x ) . \end{align*}


\begin{align*} { \bf c t } _ \kappa ( t ) = \begin{cases} \frac { 1 } { t } , & \kappa = 0 , \\ \kappa \coth ( \kappa t ) , & \kappa > 0 . \end{cases} \end{align*}


\begin{align*} T _ n ( \phi ) = T _ n ( \phi _ 0 ) T _ n ( \phi _ 1 ) \cdots T _ n ( \phi _ R ) + P _ n K _ 1 P _ n + W _ n K _ 2 W _ n + C _ n \end{align*}


\begin{align*} \sum \limits _ { k = 1 } ^ { + \infty } \left ( \mathcal { D } _ { \eta ^ { k } } ( \eta ^ { k + 1 } , \eta ^ { k } ; \lambda ^ { k } ) + \beta \| r ^ { k + 1 } \| ^ 2 \right ) \leq \mathcal { D } _ { \eta ^ { 1 } } ( \hat { \eta } , \eta ^ { 1 } ; \lambda ^ { 1 } ) < + \infty , \end{align*}


\begin{align*} \overline { i } : P ^ { \pm } _ { > 0 } ( \R ) \longrightarrow \mathcal { H } ^ { \pm } ; g = \begin{pmatrix} a & b \\ 0 & ( \det g ) a ^ { - 1 } \end{pmatrix} \longmapsto a b \det g + i a ^ 2 \det g . \end{align*}


\begin{align*} \overline S : = \{ s _ i + \Delta \in \Gamma / \Delta \mid i \in I _ 1 \} \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \partial _ t u + ( - \Delta + \varrho | x | ^ 2 ) ^ { \beta } u & = & f ( u ) , ( x , t ) \in \R ^ d \times ( 0 , \infty ) , \\ u ( x , 0 ) & = & u _ 0 ( x ) , \end{array} \right . \end{align*}


\begin{align*} \int _ { \R ^ N } | F ( B \sum _ { j = 1 } ^ \ell w _ j ) - F ( \sum _ { j = 1 } ^ \ell w _ j ) | \leq & | B - 1 | \int _ { \R ^ N } \sum _ { j = 1 } ^ \ell w _ j | f ( \theta \sum _ { j = 1 } ^ \ell w _ j ) | \leq C e ^ { - \frac { \sigma \xi ( \boldsymbol p ) ^ 2 } 8 } , \end{align*}


\begin{align*} \xi _ { k , \psi } ( f ) & : = \psi ( p _ k ( f ) ) , \end{align*}


\begin{align*} { \rm I m } ( f ( t ) ) = 0 { \rm \ f o r \ } t \le 0 \end{align*}


\begin{align*} \lim _ { k \to \infty } p _ k ( M , \omega ) = 1 \end{align*}


\begin{align*} \phi ( x ) = \int _ { \Omega } G ( x - y ) ( \rho _ { + } - \rho _ { - } ) ( y ) \ , d y , \end{align*}


\begin{align*} w _ 1 : = \begin{cases} \phi ( u ) & \mbox { o n } \cup _ { i \in \N } B ^ i _ 1 , \\ u & \mbox { o t h e r w i s e } , \end{cases} \end{align*}


\begin{align*} a y ^ m = b x ^ n + c . \end{align*}


\begin{align*} T _ { h } Y : = ( t _ { 1 } + h , \ldots , t _ { s } + h , h , \ldots , h ) \in \mathbb { N } ^ k . \end{align*}


\begin{align*} { \rm I m } ( f ( \xi ) ) = 0 { \rm \ f o r \ } \xi \in L \end{align*}


\begin{align*} \lim _ { m \rightarrow \infty } \dfrac { ( D ^ { k - 1 } \widetilde { \mathcal { F } } _ { \gamma } ) | U ( p ^ { 2 m + 1 } ) } { d _ { \gamma } ( p ^ { 2 m + 1 } ) } = g . \end{align*}


\begin{align*} \zeta _ { \infty , p ^ { \prime } } ( s , c ) : = \sum _ { m , n \neq 0 } \frac { p ^ { \prime 2 } + \lambda _ { n } ^ { \prime 2 } } { p ^ { \prime } ( p ^ { \prime } + \frac { 1 } { \pi } ) + \lambda _ { n } ^ { \prime 2 } } \ , \frac { 1 } { \left ( m ^ { 2 } + c \lambda _ { n } ^ { \prime 2 } \right ) ^ { s } } , \ , \ , \ , \ , ( s ) > 1 . \end{align*}


\begin{align*} \frac { c ^ { \sigma } _ i } { c ^ { \mu } _ i } = \frac { c ^ { \sigma } _ i } { c ^ { \tau } _ i } \cdot \frac { c ^ { \tau } _ i } { c ^ { \mu } _ i } = \frac { c ^ { \tau } _ i } { c ^ { \mu } _ i } \cdot \frac { c ^ { \sigma } _ i } { c ^ { \tau } _ i } \end{align*}


\begin{align*} \lim _ { j \to \infty } \| h _ j \| _ { p ^ * , q , \mu } = 0 . \end{align*}


\begin{align*} - \Delta \xi _ i = \gamma _ i \xi _ i \ ; \mathcal { D } _ a , \xi _ i = 0 \ ; \Gamma _ a , \end{align*}


\begin{align*} \mathbf r _ m = & \mathbf S \mathbf B \mathbf { G } ^ { \frac { 1 } { 2 } } \mathbf h _ m + \mathbf n _ m , m \in \mathcal { M } , \end{align*}


\begin{align*} \partial _ t u & = \Delta w , \\ w & = \mathcal { L } _ \varepsilon ^ \Omega c _ \varepsilon + \Delta c + f ^ \prime ( c _ \varepsilon ) - f ^ \prime ( c ) . \end{align*}


\begin{align*} \varphi _ 1 , \ \varphi _ 2 , \ \varphi _ 3 \in \left ( \bigcap _ { m = 1 } ^ { \infty } H ^ m ( \R ^ d ) \right ) ^ d . \end{align*}


\begin{align*} \tilde { S } ( a ) = \left \{ u \in S ( a ) : I _ \epsilon ( u ) \leq \Upsilon _ { 0 , a } + h ( \epsilon ) \right \} . \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \infty } \frac { x } { n ( n - x ) } : = - \Psi ( 1 - x ) - \gamma , \textmd { w h e r e } \gamma : = \lim _ { N \rightarrow \infty } \left ( \sum _ { n = 1 } ^ { N } \frac { 1 } { n } - \log N \right ) \end{align*}


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


\begin{align*} \sigma _ J = \sigma _ { j _ 1 + 1 } \sigma _ { j _ 2 + 2 } \dots \sigma _ { j _ k + k } ( a ) . \end{align*}


\begin{align*} H ( \mathcal { S } ) _ i = \# \{ k : \nu _ k \le i < \nu _ k + p _ k \} . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { M } } ( \mu _ 1 , \mu _ 2 ) : = \sup _ { \gamma \in \Pi ( \mu _ 1 , \mu _ 2 ) } \int _ { \mathcal { V } } g \ , d \gamma . \end{align*}


\begin{align*} ( - \Delta ) ^ s u ( x ) & = C _ { N , s } \cdot \mathrm { P . V . } \int _ { \R ^ N } \frac { u ( y ) - u ( x ) } { | x - y | ^ { N + 2 s } } \ , d y \\ & = C _ { N , s } \cdot \lim _ { \varepsilon \to 0 ^ + } \int _ { \{ | x - y | \geq \varepsilon \} } \frac { u ( y ) - u ( x ) } { | x - y | ^ { N + 2 s } } \ , d y , \end{align*}


\begin{align*} V ^ { \perp } : = \{ v \in V | ( u , v ) = 0 \ \forall u \in V \} , \qquad \mathrm { a s \ w e l l \ a s \ s e t } \qquad \overline { V } : = V / V ^ { \perp } . \end{align*}


\begin{align*} M ^ { \sharp } f ( x ) & = \sup _ { Q \ni x } \inf _ { c } \dfrac { 1 } { | Q | } \int _ { Q } | f ( y ) - c | \mathrm { d } y \approx \sup _ { Q \ni x } \dfrac { 1 } { | Q | } \int _ { Q } | f ( y ) - f _ { Q } | \mathrm { d } y , \end{align*}


\begin{align*} \aligned m _ { r , \nu } ( a , b ) & \leq m _ { r , \nu } ( | \phi | _ 2 , | \phi | _ 2 ) \leq \max _ { t \in \mathbb { R } } J _ \nu ( t \star ( \phi , \phi ) ) \\ & = \max _ { t \in \mathbb { R } } \Bigl [ 2 E ( t \star \phi ) - \nu e ^ { ( \gamma _ p + \gamma _ q ) t } \int _ { \mathbb { R } ^ N } ( I _ \mu \ast | \phi | ^ { p } ) | \phi | ^ { q } \Bigr ] : = \max _ { t \in \mathbb { R } } g ( t ) , \endaligned \end{align*}


\begin{align*} p _ t ^ { T \Gamma } ( T x , T y ) = p _ t ^ \Gamma ( x , y ) , x , y \in \Gamma , \ ; t > 0 . \end{align*}


\begin{align*} \psi ( v _ n ) = \frac { \gamma _ 1 \pi ^ n + \gamma _ 2 \pi ^ l } { 2 } , \end{align*}


\begin{align*} \sup _ { x \in G \setminus \{ e \} } \sup _ { r > 0 } \frac { \| \delta _ { r } ^ { A } x \| } { \| \delta _ { r } ^ { B } x \| } = \sup _ { x \in G \setminus \{ e \} } \sup _ { r > 0 } \frac { \| \delta _ { 1 / r } ^ { A } x \| } { \| \delta _ { 1 / r } ^ { B } x \| } \leq R , \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*} \tilde { x } _ { t } = \tilde { x } _ { 0 } + \int _ { 0 } ^ { t } \tilde { b } _ { r } \mathrm { d } r + \int _ { 0 } ^ { t } \tilde { \sigma } _ { r } \mathrm { d } W _ { r } , 0 \leq t \leq T . \end{align*}


\begin{align*} \ell E _ { \ell ^ { - 1 } \alpha } = \inf _ { \gamma \in \Gamma } \sup _ { \boldsymbol s \in S } \mathbb { J } ( \gamma ( \boldsymbol { s } ) ) . \end{align*}


\begin{align*} B _ v : = \{ [ v + L _ n ] \} \cup \{ [ v + w + L _ { n m } ] \mid w \in ( L _ n / L _ { n m } ) ' \} . \end{align*}


\begin{align*} p ( z ) = 1 + \sum _ { n = 1 } ^ { \infty } c _ n z ^ n \end{align*}


\begin{align*} c _ 0 + c _ 1 + c _ 2 = | \mathcal { F } | , \end{align*}


\begin{align*} \widetilde { W } _ { h , q } ( i ) = \begin{cases} h _ { 0 } + h + 1 & i = q ; \\ 1 & h - 1 \leq i \leq k - 2 i \neq q \\ l - h _ { 0 } - k & i = k - 1 ; \\ 0 & . \end{cases} \end{align*}


\begin{align*} A = { \rm d i a g } \left ( A _ 1 , \cdots , A _ k \right ) , A _ k = \begin{pmatrix} B _ k & 1 & 0 & \cdots & 0 \\ 0 & B _ k & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & B _ k \end{pmatrix} , \end{align*}


\begin{align*} \Delta ^ k \left [ f ( x ) \right ] _ { x = \alpha } = \sum _ { j = 0 } ^ k ( - 1 ) ^ { k - j } \binom { k } { j } f ( \alpha + j ) . \end{align*}


\begin{align*} B _ { L , k , 1 } & = \sum _ { L \le d < 2 L } \sum _ { \substack { p \mid k , \ , \phi _ e ( q ) \\ e \mid d \\ e > d / T } } \log \gcd ( p ^ { \nu _ p ( k ) } , \phi _ e ( q ) ) , \\ B _ { L , k , 2 } & = \sum _ { L \le d < 2 L } \sum _ { \substack { p \mid k , \ , \phi _ e ( q ) \\ e \mid d \\ e \le d / T } } \log \gcd ( p ^ { \nu _ p ( k ) } , \phi _ e ( q ) ) . \end{align*}


\begin{align*} \widetilde { \rho } _ w ( \theta , \kappa ( \theta ) ) = \xi ^ { - N } e ^ { u + i v } ( \theta ) , \end{align*}


\begin{align*} \sqrt { \frac { 1 - p } { p } } \hat f ( \{ i \} ) = \mu _ p ( f _ { i \to 1 } ) - \mu _ p ( f ) \geq 2 \mu _ p ( f ) . \end{align*}


\begin{align*} \Omega ^ * _ 0 = H ^ * \times \omega ^ * \end{align*}


\begin{align*} \sigma _ n ( x ) : = \frac { 1 } { \abs { W _ n } } \sum _ { a \in W _ n } \phi ( a ) ( x ) , ~ n \in \N , \end{align*}


\begin{align*} \gamma _ 1 ( a ) = \gamma _ 1 ( L _ a , \Omega ) = \sup \{ \gamma \geq 0 : ~ ~ \exists \phi < 0 , L _ a \phi + \gamma \phi f \geq 0 \} , \end{align*}


\begin{align*} \sup _ { y _ 2 ' \in \mathcal { Y } _ 2 } \{ - y _ 2 ' d ( x ' ) - \lambda _ 2 | y _ 2 - y _ 2 ' | \} = \begin{cases} \infty & 0 \le \lambda _ 2 < 1 \\ - y _ 2 d ( x ' ) & \lambda _ 2 \ge 1 \end{cases} , \end{align*}


\begin{align*} [ D \psi : D \varphi ] _ t = s ( \psi ) [ D \chi : D \varphi ] _ t , t \in \mathbb { R } . \end{align*}


\begin{align*} \ast ( d x ^ 0 \wedge d x ^ 1 ) = d x ^ 2 \wedge d x ^ 3 , \ast ( d x ^ 0 \wedge d x ^ 2 ) = d x ^ 3 \wedge d x ^ 1 , \ast ( d x ^ 0 \wedge d x ^ 3 ) = d x ^ 1 \wedge d x ^ 2 . \end{align*}


\begin{align*} Q _ { \mu } ( s ) = \intop _ { \mu } ^ { \infty } t ^ { s - 1 } e ^ { - t } d t \end{align*}


\begin{align*} a = \varliminf _ { m \to \infty } \big ( d ( m S , \N ^ n \setminus m S ) \big ) ^ { 1 / { m } } > 0 . \end{align*}


\begin{align*} ( \vec U , \vec V ) _ { \Omega } = \vec U ^ T P _ { \Omega } \vec V \approx \int \limits _ { \Omega } U ^ T V d \Omega , ( \vec U , \vec U ) _ { \Omega } = \| \vec U \| ^ 2 _ { \Omega } = \vec U ^ T P _ { \Omega } \vec U \approx \int \limits _ { \Omega } U ^ T U d \Omega = \| U \| ^ 2 _ { \Omega } . \end{align*}


\begin{align*} y _ i ' = y _ { \sigma ( i ) } \quad b _ { i , j } ' = b _ { \sigma ( i ) , \sigma ( j ) } \quad 1 \le i , j \le n . \end{align*}


\begin{align*} \| f ^ { = d } \| _ 2 ^ 2 \le ( e / \rho ) ^ { d } \gamma _ 1 \cdot \gamma . \end{align*}


\begin{align*} \varphi _ \ell ( x , \lambda _ \ell ) = \frac { Q _ { \ell , Y Y } ^ { - 1 } } { 4 \lambda _ { \ell } } + a _ \ell x ^ { \top } V _ { \ell , X X } ^ { - 1 } V _ { \ell , X Y } - \lambda _ { \ell } x ^ { \top } V _ { \ell , X X } ^ { - 1 } x . \end{align*}


\begin{align*} F _ { k + 1 } ( O [ g _ 1 ] ) = \partial ( F _ k ( O [ g _ 1 ] ) ) + \mathrm { S p a n } _ { \mathbb C } ( 1 , g _ 1 , \ldots , g _ { k + 1 } ) . \end{align*}


\begin{align*} 0 = \langle A h , g \rangle _ { \mu _ p } = \sum _ { d = 0 } ^ n \left ( - \frac { p } { 1 - p } \right ) ^ d \langle h ^ { = d } , g \rangle \geq \mu _ { p } ( h ) \mu _ { p } ( g ) - \sum _ { d = 1 } ^ n \left ( \frac { p } { 1 - p } \right ) ^ d | \langle h ^ { = d } , g \rangle | . \end{align*}


\begin{align*} \mbox { $ \mathcal { S } _ + = \left \{ A ^ + \in \mathcal { S } : \ A \in \binom { [ n ] } { k } \mbox { a n d } a \in A \right \} $ a n d $ \mathcal { S } _ - = \left \{ A ^ - \in \mathcal { S } : \ A \in \binom { [ n ] } { k } \mbox { a n d } b \in A \right \} $ . } \end{align*}


\begin{align*} \limsup _ { \rho \to 0 } \omega ( \rho ) \log \left ( \frac { 1 } { \rho } \right ) = 0 . \end{align*}


\begin{align*} \texttt { w p t c o e f [ k ] } : = \max \Bigl \{ \texttt { m a g } \Bigl ( 6 8 \frac { ( L + 2 ) A ( \sigma ) } { \varepsilon _ 4 } \frac { \Gamma ( k + 1 / 2 ) ^ { 1 / 2 } } { ( B _ 1 a \sqrt { \pi } ) ^ k } \Bigr ) , \texttt { m a g } ( 4 0 ( L + 2 ) ) \Bigr \} . \end{align*}


\begin{align*} \mathcal { V } \left ( t , \mu \right ) = V \left ( t , \xi \right ) , \mathbb { P } \end{align*}


\begin{align*} \sum _ { \substack { j , l \in \mathcal { J } \\ j \not = l } } r ( m _ j ) r ( n _ l ) \Phi \left ( | j - l | - 1 \right ) \le \sum _ { \substack { j , l \in \mathcal { J } \\ j \not = l } } r ( m _ j ) ^ { 2 } \Phi \left ( | j - l | - 1 \right ) . \end{align*}


\begin{align*} \abs { \lambda ^ { ( s ) } _ h - \lambda ^ { ( 1 / 2 ) } _ h } \stackrel { \eqref { e q : d e f - l a m b d a - h - s } } { = } t _ h \abs { \widetilde { \lambda } ^ { ( s ) } _ h - \widetilde { \lambda } ^ { ( 1 / 2 ) } _ h } \leq t _ h \left ( \abs { \widetilde { \lambda } ^ { ( s ) } _ h - 1 } + \abs { 1 - \widetilde { \lambda } ^ { ( 1 / 2 ) } _ h } \right ) \stackrel { \eqref { e q : C a L e - t r i c k - c o n d i t i o n s } } { \lesssim } 1 . \end{align*}


\begin{align*} f = \sum _ { n } c _ n a _ n \end{align*}


\begin{align*} ( f _ \ell ) _ { \lambda _ \ell } ( y _ \ell ) = \sup _ { y _ \ell ^ \prime \in \mathcal { Y } _ \ell } \left \{ f _ \ell ( y _ \ell ^ { \prime } ) - \lambda _ \ell c _ { Y _ \ell } ( y _ { \ell } , y _ { \ell } ' ) \right \} . \end{align*}


\begin{align*} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \to \begin{pmatrix} - b & a \\ - d & c \end{pmatrix} . \end{align*}


\begin{align*} ( \prod _ { i = 1 } ^ m b ( i ) \ast f ( t ( i ) ) ) \ast b ( m + 1 ) \in A \cap \sigma ( W ) . \end{align*}


\begin{align*} \int _ { I ^ { ( l ) } } S ( k , l ) \varphi ( x ) \ , d x = \int _ { I ^ { ( k ) } } \varphi ( x ) \ , d x . \end{align*}


\begin{align*} M _ n = \sum _ { k = 0 } ^ n ( n - k + 4 ) A _ k B _ { n - k } , N _ n = \sum _ { k = 0 } ^ n A _ k B _ { n - k } \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c \bar u ) ^ n \leq \psi ( z , \bar u ) \omega ^ n & \textnormal { o n } & \Omega , \\ \bar u = 0 & \textnormal { o n } & \partial \Omega . \end{array} \right . \end{align*}


\begin{align*} \langle \eta , \partial _ u ( H ^ { ( u ) } ) ^ { - 1 } \eta \rangle & = \langle T ( u ) ^ { - 1 } ( H ^ { ( u ) } ) ^ { - 1 } \eta , e ^ { 2 \rho ( u ) } T ( u ) ^ { - 1 } ( H ^ { ( u ) } ) ^ { - 1 } \eta \rangle \\ & = \langle \widetilde \eta ^ { ( u ) } , e ^ { 2 \rho ( u ) } \widetilde \eta ^ { ( u ) } \rangle = \sum _ { i \in V } ( \widetilde \eta _ i ^ { ( u ) } ) ^ 2 e ^ { 2 \rho _ i ( u ) } . \end{align*}


\begin{align*} \begin{aligned} A ( t , \rho ) & \leq S ( t , \rho ) + \int _ 0 ^ t \Big [ S ( \tau , \rho ) + \int _ 0 ^ \tau A ( \tau ' , \rho ) \ , d \tau ' \Big ] \ , d \tau \\ & \leq S ( t , \rho ) + \int _ 0 ^ t S ( \tau , \rho ) \ , d \tau + \int _ 0 ^ t ( t - \tau ) A ( \tau , \rho ) \ , d \tau . \end{aligned} \end{align*}


\begin{align*} \mathcal N ( p ) : = \inf \{ s > 0 : C _ s ( p ) = + \infty \} = \sup \{ s > 0 : C _ s ( p ) = 0 \} , \end{align*}


\begin{align*} W ^ { \rm K M S } : = \{ m \in M \ : \alpha _ { i t } ( m ) \in \Xi \mbox { f o r } 0 < t < \pi \} . \end{align*}


\begin{align*} L ( t ) = \frac { n - 1 } { t } , W ( t ) = \frac { C } { t ^ { ( 2 - \gamma ) p } } , w = t ^ { \gamma p } , G ( t ) = \frac { a } { t ^ { 2 p - 2 } } \quad \mbox { a n d } H ( t ) = \frac { b } { t } , \qquad \forall t \in ( 0 , \sup _ \Omega \rho ) , \end{align*}


\begin{align*} v = 1 - u \ \ \in C ^ \infty ( \overline \Omega ) \end{align*}


\begin{align*} \rho _ t ( a _ p ^ * a _ q ) = \delta ( p - q ) \Big ( \chi ( p ) + \big [ \chi ( p ) - \chi ^ \perp ( p ) \big ] f _ t ( p ) \Big ) \end{align*}


\begin{align*} \int _ \Omega f _ 0 ( V ) \psi ^ h _ { \mu _ j , \xi _ j } d x & = \int _ { \Omega \setminus B ( \xi , \rho ) } f _ 0 ( V ) \psi ^ h _ { \mu _ j , \xi _ j } d x + \sum _ { l = 1 } ^ k \int _ { \mathcal { A } _ l } f _ 0 ( V ) \psi ^ h _ { \mu _ j , \xi _ j } d x . \end{align*}


\begin{align*} \int _ { s } ^ { t } S _ { t - \tau } Y _ { \tau } \circ \mathrm { d } \mathbf { X } _ { \tau } : = \lim _ { \substack { | \pi | \rightarrow 0 , \\ \pi = \lbrace \tau _ 0 = s , \tau _ { 1 } , . . . \tau _ { m } = t \rbrace } } \sum _ { 0 \leq j < m } \big { [ } S _ { t - \tau _ j } Y _ { \tau _ j } \circ ( \delta X ) _ { \tau _ j , \tau _ { j + 1 } } + S _ { t - \tau _ j } Y ^ { \prime } _ { \tau _ j } \circ \mathbb { X } _ { \tau _ j , \tau _ { j + 1 } } \big { ] } . \end{align*}


\begin{align*} \lambda _ 1 ^ n = \frac { \int _ \Omega ( - \varphi _ 1 ) ( d d ^ c \varphi _ 1 ) ^ n } { \int _ \Omega ( - \varphi _ 1 ) ^ { n + 1 } d \nu _ { f } } = \inf \left \{ \frac { \int _ \Omega ( - \phi ) ( d d ^ c \phi ) ^ n } { \int _ \Omega ( - \phi ) ^ { n + 1 } d \nu _ { f } } ; \phi \in \mathcal E ^ 1 ( \Omega ) , \phi \neq 0 \right \} \cdot \end{align*}


\begin{align*} | f ^ { ' } _ \epsilon ( V ) - f ^ { ' } _ 0 ( V ) | _ { \frac { n } { 2 } } = O \bigg ( \epsilon \ln \Big | \ln \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big | \bigg ) . \end{align*}


\begin{align*} T _ { n , m } ( x ) : & = \left [ \binom { x + m } { j - i + m } - \binom { x + m } { m - i - j - 1 } \right ] _ { i , j = 0 } ^ { n - 1 } . \end{align*}


\begin{align*} a _ 0 \ \equiv \ b _ 0 + 2 \ \equiv \ \begin{cases} 4 , & , \\ 9 , & . \end{cases} \end{align*}


\begin{align*} x = L + K L ^ \theta + O ( 1 ) = L + K \big ( x + O ( x ^ \theta ) \big ) ^ \theta + O ( 1 ) = L + K x ^ \theta + O ( 1 + x ^ { 2 \theta - 1 } ) . \end{align*}


\begin{align*} = \exp \left \{ \sum _ { n = 1 } ^ { \infty } A ( n , x ) B ( n , y ) \frac { z ^ n } { n ^ 4 } \right \} \end{align*}


\begin{align*} \begin{cases} a _ { i , k } + b _ { i , k } + c _ { i , k } = 1 \\ d _ { i , k } - x _ i = r _ { i , k } + s _ { i , k } + t _ { i , k } \\ - C a _ { i , k } \leq r _ { i , k } \leq 0 \\ 0 \leq s _ { i , k } \leq \bar { y } b _ { i , k } \\ \bar { y } c _ { i , k } \leq t _ { i , k } \leq C c _ { i , k } \\ \beta _ { i , k } = - c \P ( \omega _ k ) \left ( \frac { s _ { i , k } } { \bar { y } } + c _ { i , k } \right ) \end{cases} \end{align*}


\begin{align*} v ( \psi _ 0 ( a ) ) & = v \left ( \sum _ { j = 1 } ^ { n } \psi _ j ( a ) \varphi ( a ) ^ j \right ) \\ & = v ( \varphi ( a ) ) + v \left ( \sum _ { j = 1 } ^ { n } \psi _ j ( a ) \varphi ( a ) ^ { j - 1 } \right ) \\ & \geq v ( \varphi ( a ) ) > 0 \end{align*}


\begin{align*} [ P , g ] = g ( P \otimes \lbrace 0 , i \infty \rbrace ) . \end{align*}


\begin{align*} \mathfrak { S } ( f _ k ) = \mathfrak { S } ( f _ k | _ { V \setminus \mathcal { F } } ) = \tilde { k } + \tilde { r } - 1 , \ , \ , \ , \ , \ , \overline { \mathfrak { S } } ( f _ k ) = \overline { \mathfrak { S } } ( f _ k | _ { V \setminus \mathcal { F } } ) = n - | \mathcal { F } | - \tilde { k } + 1 . \end{align*}


\begin{align*} \bigvee ^ k _ { s = 1 } c ^ { \sigma } _ { i _ s } \imath _ { i _ s } \colon \bigvee ^ k _ { s = 1 } \Sigma X _ { i _ s } \longrightarrow ( \underline { \Sigma X } , \underline { \ast } ) ^ { \partial \sigma , c } . \end{align*}


\begin{align*} g = \left ( { \begin{array} { c c } a & b \\ c & d \\ \end{array} } \right ) = ( a b | c d ) = ( a , b | c , d ) \ , , a , b , c , d \in \Z _ q \ , , \det ( g ) = a d - b c = 1 \ , , \end{align*}


\begin{align*} \Pr ( X _ t = k \ , \vert \ , X _ 0 = r ) = \frac { ( n - r ) _ { k - r + 1 } } { n ^ t } \frac { 1 } { ( k - r ) ! } \Delta ^ { k - r } \left [ \frac { x ^ t } { n - x } \right ] _ { x = r } ^ { } . \end{align*}


\begin{align*} \nu _ { g } = \lim _ { i \to \infty } \sum _ { s \in F } \delta _ { \left \{ H _ { 0 } g s \xi \right \} } \mu ^ { ( n _ { i } ) } ( s ) = \lim _ { i \to \infty } \sum _ { s \in F } \delta _ { \left \{ g H _ { 0 } s \xi \right \} } \mu ^ { ( n _ { i } ) } ( s ) = g . \nu _ { e } . \end{align*}


\begin{align*} ( g . \lambda ) _ { x } = \theta ( x , \cdot ) _ { \ast } \left ( \overline { \mathbb { P } } _ { \mu , x , g } \right ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } u _ { t t } + \alpha f ( u _ { t } ) + \beta u = \lambda g ( u ) & \mbox { f o r $ t > 0 $ } , \\ u ( 0 ) = u _ { 0 } , \\ u _ { t } ( 0 ) = v _ { 0 } , \end{array} \right . \end{align*}


\begin{align*} \mathfrak { d } ( \tau ) _ { \sigma } = \begin{cases} p & \tau = \sigma \\ 1 & \mbox { e l s e } . \end{cases} \end{align*}


\begin{align*} H _ { q } ( n ) ( f ) : = \begin{vmatrix} a _ { n } & a _ { n + 1 } & \cdots & a _ { n + q - 1 } \\ a _ { n + 1 } & a _ { n + 2 } & \cdots & a _ { n + q } \\ \vdots & \vdots & \vdots & \vdots \\ a _ { n + q - 1 } & a _ { n + q } & \cdots & a _ { n + 2 q - 2 } \end{vmatrix} . \end{align*}


\begin{align*} \mathcal { T } ( V , q ) : = \{ \alpha \in \mathcal { C } | \alpha V \alpha ^ { * } \subseteq V , \ N ( \alpha ) \in \mathbb { F } \} . \end{align*}


\begin{align*} & \beta \circ \psi _ V = \psi _ V \circ \beta , \\ & T \circ \psi _ V = \psi _ L \circ T ' , \\ & \psi _ V ( \rho ( x , y ) u ) = \rho ( \psi _ L ( x ) , \psi _ L ( y ) ) \psi _ V ( u ) . \end{align*}


\begin{align*} f = \sum _ { i = 1 } ^ r a _ i \tilde { \textbf { Q } } ^ { \lambda _ i } . \end{align*}


\begin{align*} C _ { m } = E \left \{ \left ( \frac { X _ { 1 } + \ldots + X _ { m } } { m } - \theta ( m ) \right ) \otimes \left ( \frac { X _ { 1 } + \ldots + X _ { m } } { m } - \theta ( m ) \right ) \right \} . \end{align*}


\begin{align*} C _ 1 ( \lambda _ 1 \rho _ 0 ) = C _ 2 ( \lambda _ 2 \rho _ 0 ) = C _ 3 ( \lambda _ 3 \rho _ 0 ) \end{align*}


\begin{align*} \psi ( x ) = - \sum \limits _ { \substack { ( y , y ' ) \in \partial \Lambda \\ y \in \Lambda , y ' \notin \Lambda } } G _ { \omega , \Lambda , E } ( x , y ) \psi ( y ' ) . \end{align*}


\begin{align*} v _ h ( y ) : = \frac { 1 } { h } v ( T _ h ^ { - 1 } y ) . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ { 3 } \sum \limits _ { k = 0 } ^ { + \infty } \varepsilon ^ { \alpha _ i + k - 1 } \ , \left ( w _ { \alpha _ i + k - 1 } ^ { ( i ) } ( x _ i ) + u _ { \alpha _ i + k - 1 } ^ { ( i ) } \big ( x _ i , \tfrac { \overline { x } _ i } { \varepsilon } , t \big ) \right ) + \sum \limits _ { k = 0 } ^ { + \infty } \varepsilon ^ { k } \ , \left ( w _ { k } ^ { ( i ) } ( x _ i ) + u _ { k } ^ { ( i ) } \big ( x _ i , \tfrac { \overline { x } _ i } { \varepsilon } , t \big ) \right ) . \end{align*}


\begin{align*} \begin{array} { l } C _ n ( x ) = \min \left \{ \begin{array} { l } L _ n ( x ) + \int _ 0 ^ \infty C _ { n - 1 } ( x - \xi ) f _ n ( \xi ) \mathrm { d } \xi , \\ \min _ { x < y \leq x + B } \{ K + v ( y - x ) + L _ n ( y ) + \int _ 0 ^ \infty C _ { n - 1 } ( y - \xi ) f _ n ( \xi ) \mathrm { d } \xi \} \end{array} \right \} , \\ G _ n ( x ) = v x + L _ n ( x ) + \int _ 0 ^ \infty C _ { n - 1 } ( x - \xi ) f _ n ( \xi ) \mathrm { d } \xi , \\ C _ n ( x ) = - v x + \min \{ G _ n ( x ) , K + \min _ { x \leq y \leq x + B } G _ n ( y ) \} . \end{array} \end{align*}


\begin{align*} B ^ { k + 1 } = \{ p \} \cup B ^ k \setminus \{ q \} \end{align*}


\begin{align*} \gathered a _ { i k } \in \mathbb C , a _ { i k } = 0 , k > i + r , i > k + q , \\ a _ { i \ , i + r } = 1 , a _ { i + q \ , i } \ne 0 , i \ge 0 ; \endgathered \end{align*}


\begin{align*} T _ { \rho } ^ { [ m - 1 ] } = T _ { \rho } ^ { ( m - 1 ) } \circ \cdots \circ T _ { \rho } ^ { ( 1 ) } . \end{align*}


\begin{align*} L _ { C F } = 3 M K \end{align*}


\begin{align*} \mathbb { M } _ k : = \mathbb { Z } [ X , Y ] _ { k - 2 } \otimes _ { \mathbb { Z } } \mathbb { M } _ 2 , \end{align*}


\begin{align*} \log ( A _ 1 ) & = \log ( A + h ) \\ & = \log A + o ( 1 ) \\ & \geq r ^ \gamma + o ( 1 ) . \end{align*}


\begin{align*} S _ { \mathrm { o f f } } : = \sum _ { i = 1 } ^ D \sum _ { j = 1 } ^ D \alpha _ i \bar { \alpha } _ j \sum _ { p | c } \frac { \mathcal { S } ( i , j , c ) } { c } \ , J _ { 2 k - 1 } \left ( \frac { 4 \pi \sqrt { i j } } { c } \right ) \ , \big | L ( f , n + 1 ) \big | ^ 2 . \end{align*}


\begin{align*} \lim _ { r \to \infty } \frac { u ( r ) } { u _ 0 ( r ) } = \lim _ { r \to \infty } \frac { v ( r ) } { v _ 0 ( r ) } = \lim _ { r \to \infty } \frac { u ' ( r ) } { u _ 0 ' ( r ) } = \lim _ { r \to \infty } \frac { v ' ( r ) } { v _ 0 ' ( r ) } = 1 . \end{align*}


\begin{align*} \xi ^ { ( 0 ) } = \sup \bigg \{ a \ge 0 \colon \lim _ { n \to \infty } \sup _ { v \in \mathbb { Z } ^ d , \ , \| v \| \ge n } \mathbb { P } \big ( \gamma ( 0 , v ) \subset \mathrm { c y l } ( 0 , v , \| v \| ^ a ) \big ) < 1 \bigg \} , \end{align*}


\begin{align*} a _ { - } = \lim _ { y \rightarrow - \infty } a ( y ) a _ { + } = \lim _ { y \rightarrow + \infty } a ( y ) \end{align*}


\begin{align*} \prod _ { i \in V } \phi _ i ( u ) ^ { 3 / 2 } \sqrt { | H _ \beta | } = \prod _ { i \in V } \phi _ i ( u ) ^ 2 \sqrt { | \widetilde { H } ^ { ( u ) } | } \sqrt { | K ^ { ( u ) } | } , \end{align*}


\begin{align*} S _ { y ' } : = \{ ( y , r ) \} \times { \rm T r e e } _ { F , H _ { 0 } } & \to \mathbb { R } _ { \ge 0 } \\ x & \mapsto H _ { \alpha _ { x , g } \parallel \alpha _ { x , e } } \left ( \mathcal { P } _ { x , n } \right ) \end{align*}


\begin{align*} \begin{aligned} & z ^ { \vec x ^ 1 } _ { r _ 1 } - z ^ { \vec x ^ 2 } _ { r _ 1 } \geq 0 , \\ & z ^ { \vec x ^ 1 } _ { r _ 2 } - z ^ { \vec x ^ 2 } _ { r _ 2 } \leq 0 , \end{aligned} \end{align*}


\begin{align*} y \big ( \bigl \{ X _ { n i } ^ { ( j + 1 ) } \bigr \} \big ) = Z _ { j + 1 } + \sum _ { l = j + 2 } ^ k A _ l , \end{align*}


\begin{align*} \ < \Psi , [ \N ^ \ell , V _ B ( t ) ] \Psi \ > = \mathrm { I m } \int \hat V ( k ) \ < \Psi , [ \N ^ \ell , b _ k ( t ) b _ { - k } ( t ) ] \Psi \ > \d k \forall t \in \R \ . \end{align*}


\begin{align*} \Omega ( b ^ \ast ) = ( J _ n \Omega ( b ) J _ n ^ { - 1 } ) ^ { \mathrm { t } } , \forall b \in \mathcal { R } _ p . \end{align*}


\begin{align*} f ( w , z ) = \exp \left ( \prod _ { k = 1 } ^ n P _ k ( w , z ) \right ) \end{align*}


\begin{align*} v ^ 3 z + 1 = u ^ 3 . \end{align*}


\begin{align*} { \rm I m } \ W _ { \vec { k } , \vec { l } } ( \vec { x } , \vec { y } ) + { \rm I m } \ W _ { \vec { k } , \vec { l } } ( \vec { y } , \vec { x } ) = 0 , \end{align*}


\begin{align*} \prod _ { \ell \in \mathcal { S } } \prod _ { i = 1 } ^ { r ( \ell ) } \mathbf { G } ^ 1 [ \ell , \mathcal { T } ( \ell ) _ i ] \backslash \left ( \mathbf { G } ^ 1 [ \ell , \mathcal { T } ( \ell ) _ i ] \Delta ^ { \mathcal { T } ( \ell ) _ i } ( \mathbf { G } ^ 1 ( \mathbb { Q } _ p ) ) \mathbf { K } ^ 1 [ \ell ] / \mathbf { K } ^ 1 [ \ell ] \right ) , \end{align*}


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


\begin{align*} \frac { d } { d t } E ^ { \mbox { t o t a l } } = - \triangle , \end{align*}


\begin{align*} T _ \lambda ^ { \vec { v } } T _ { \widehat { \lambda } } ^ { \vec { v } } = T _ { \widehat { \lambda } } ^ { \vec { v } } T _ \lambda ^ { \vec { v } } \ ; \ ; T _ { \lambda ^ m } ^ { \vec { v } } = \left ( T _ \lambda ^ { \vec { v } } \right ) ^ m , \end{align*}


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


\begin{align*} \aligned m _ \nu ( a , b ) = & \inf _ { \mathcal { P } _ \nu ( a , b ) } \Bigl \{ \frac { \gamma _ p + \gamma _ q - 2 } { 2 ( \gamma _ p + \gamma _ q ) } ( | \nabla u | ^ 2 _ 2 + | \nabla v | ^ 2 _ 2 ) \\ & + \frac { 2 2 ^ * _ \mu - \gamma _ p - \gamma _ q } { 2 2 ^ * _ \mu ( \gamma _ p + \gamma _ q ) } \int _ { \mathbb { R } ^ N } [ ( I _ \mu \ast | u | ^ { 2 ^ * _ \mu } ) | u | ^ { 2 ^ * _ \mu } + ( I _ \mu \ast | v | ^ { 2 ^ * _ \mu } ) | v | ^ { 2 ^ * _ \mu } ] \Bigr \} > 0 . \endaligned \end{align*}


\begin{align*} _ { n , m } ^ { 2 } = \Delta _ { 0 } + \Delta _ { 1 } + \widetilde { \Delta } _ { 2 } , \end{align*}


\begin{align*} B _ 1 : = e ^ { - C _ 0 ^ 2 - 2 } \sup _ { \bar \Omega } ( 1 + | \nabla v | ^ 2 ) . \end{align*}


\begin{align*} \frac { d ^ n q _ 1 } { d t ^ n } \Big | _ { t = 0 } = 0 \frac { \partial ^ n \varphi ^ { ( 0 ) } } { \partial t ^ n } \Big | _ { t = 0 } = 0 . \end{align*}


\begin{align*} \begin{aligned} \int _ { \R ^ N } \rho ( x ) G _ \alpha [ u ( x , \tau ) ] v ( x , \tau ) \ , d x & \leq \int _ \epsilon ^ \tau \int _ { \R ^ N } G _ \alpha ( u ) \big [ \Delta v - ( - \Delta ) ^ s v + \rho v _ t \big ] \ , d x d t \\ & + \int _ { \R ^ N } \rho ( x ) v ( x , \epsilon ) G _ \alpha [ u ( x , \epsilon ) ] \ , d x \ , . \end{aligned} \end{align*}


\begin{align*} f ( x ) = e ^ { - z | x | ^ 2 + \xi \cdot x } \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*} g ( x _ 1 , \dots , x _ { k + 1 } ) = \sum _ { \emptyset \neq I \subsetneq [ k + 1 ] } a _ I ( x _ 1 , \dots , x _ { k + 1 } ) \left ( \prod _ { i \in I } x _ i \right ) ' \end{align*}


\begin{align*} T ^ m _ j & = \left \{ ( A _ 0 \subsetneq \dots \subsetneq A _ m ) \in ( K ^ n _ k ) _ m \mid \begin{array} { l } A _ m = [ n ] \\ k \not \in A _ { j - 1 } \\ A _ { j } \supsetneq A _ { j - 1 } \cup \{ k \} \end{array} \right \} . \end{align*}


\begin{align*} \Theta _ \chi \left ( { \vect u } ^ { \rm s o f t } \oplus { \vect u } ^ { \rm s t i f f } \right ) = { \vect u } ^ { \rm s o f t } \oplus P _ { \widehat { \mathcal { H } } _ \chi ^ { \rm s t i f f } } \vect u ^ { \rm s t i f f } . \end{align*}


\begin{align*} h _ { \mu } ( M , \bar { \lambda } _ { \ell , p } ) = h _ { \mu } ( M , { \bold \alpha } _ { p } ) . \end{align*}


\begin{align*} V _ { \mu } \otimes V _ { \nu } = \bigoplus _ { \substack { 0 \leq k _ 1 \leq \cdots \leq k _ n \\ [ 1 p t ] \mu + \nu - \sum _ { i \in I } k _ i \alpha _ i \in P _ { + } } } V _ { \mu + \nu - \sum _ { i = 1 } ^ n k _ i \alpha _ i } . \end{align*}


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


\begin{align*} C _ r = \| d _ r \| _ \infty . \end{align*}


\begin{align*} \alpha = q \quad \mbox { a n d } \beta = \frac { q } { q - 1 } , \end{align*}


\begin{align*} \lim _ { t \to 0 ^ + } b ( t ) = 0 . \end{align*}


\begin{align*} \chi _ \delta ^ { ( i ) } ( x _ i ) = \left \{ \begin{array} { l l } 1 , & \ \ x _ i \ge \ell _ i - \delta , \\ 0 , & \ \ x _ i \le \ell _ i - 2 \delta , \end{array} \right . i \in \{ 2 , 3 \} , \end{align*}


\begin{align*} \sum _ F \chi _ F ( \alpha ) X _ F = \lambda _ 1 ( A ) ^ \alpha = \exp \left \{ \alpha \sum _ { n \ge 1 } ( - 1 ) ^ { n - 1 } \varphi _ n \right \} \end{align*}


\begin{align*} d ( u , v ) = d ( u ^ { ( 0 ) } , v ) \leq n _ 0 + \frac { t + 1 } { t - 2 } \cdot \frac { n } { \log n _ 0 } = \big ( 1 + o ( 1 ) \big ) \frac { n } { \log n } , \end{align*}


\begin{align*} \sum _ { n = 1 } ^ N 1 \cdot { \rm P r o b } ( T _ n ) . \end{align*}


\begin{align*} d \eta ^ i = \Phi , 1 \le i \le p . \end{align*}


\begin{align*} \lim \limits _ { k \rightarrow + \infty } u ^ k = \bar { u } \mbox { a n d } \lim \limits _ { k \rightarrow + \infty } \zeta ^ k = \bar { \zeta } . \end{align*}


\begin{align*} \mathcal { O } _ k ( \mathsf { G L } _ { m | n } ) = k [ t _ { i j } : 1 \le i , j \le m + n ] _ { \det _ 0 } . \end{align*}


\begin{align*} h = \sum _ { n } c _ n a _ n . \end{align*}


\begin{align*} \pi _ { \psi } [ \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} , t ] f ( y ) = t \psi ( \tfrac { 1 } { 2 } \langle y , y b \rangle ) f ( y ) , \end{align*}


\begin{align*} \| h _ \beta \| _ { L ^ q _ \omega } & \lesssim \| g ^ { 1 / r } \| _ { L ^ q _ \omega } = \| g \| ^ { 1 / r } _ { L ^ { \tilde q } _ \omega } \\ & \lesssim \| ( h _ \alpha ) ^ r \| _ { L ^ { \tilde p } _ \omega } \\ & = \| h _ \alpha \| _ { L ^ p _ \omega } , \end{align*}


\begin{align*} r ( m _ j ) : = \left ( \sum _ { \substack { ( 1 - T ^ { - 1 } ) ^ { j - 1 } \le n \le ( 1 + T ^ { - 1 } ) ^ { j + 2 } \\ n \in \mathcal { M } } } f ( n ) ^ 2 \right ) ^ { 1 / 2 } , \textrm { f o r e v e r y } j \in \mathcal { J } \end{align*}


\begin{align*} \pi _ 1 ( \tilde { \mathcal { X } } _ { j _ 0 + 1 } ) = \pi _ 1 \left ( X ^ { \mathcal { X } _ { j _ 0 } , \mathcal { N } _ { j _ 0 } } _ { \tau _ { H , w _ { j _ 0 } } } \right ) \leqslant \pi _ 1 ( w _ { j _ 0 } ) + ( v _ + - 2 \delta ) H . \end{align*}


\begin{align*} S _ { \omega , \mathbf { c } } ( U ) & : = E ( U ) + \omega Q ( U ) + \mathbf { c } \cdot \mathbf { P } ( U ) , \\ K _ { \omega , \mathbf { c } } ( U ) & : = \partial _ { \lambda } S _ { \omega , \mathbf { c } } ( \lambda U ) | _ { \lambda = 1 } = 2 L ( U ) + 3 N ( U ) + 2 \omega Q ( U ) + 2 \mathbf { c } \cdot \mathbf { P } ( U ) , \\ L _ { \omega , \mathbf { c } } ( U ) & : = K _ { \omega , \mathbf { c } } ( U ) - 3 N ( U ) = 2 L ( U ) + 2 \omega Q ( U ) + 2 \mathbf { c } \cdot \mathbf { P } ( U ) . \end{align*}


\begin{align*} ( \delta Z ) _ { s , t } = S _ { t - s } Z _ { s } + \int _ { s } ^ { t } S _ { t - \tau } F ( Z _ { \tau } ) \mathrm { d } \tau + \int _ { s } ^ { t } S _ { t - \tau } G ( Z _ { \tau } ) \circ \mathrm { d } \mathbf { X } _ { \tau } , \ \ Z _ { 0 } = z _ 0 , \ s , t \in \mathbb { R } , \end{align*}


\begin{align*} \tilde { b } . e _ i \otimes t & = \sum _ { j } e _ j b _ { j , i } e _ i . e _ i \otimes t \\ & = ( e _ 1 + \dots + e _ m ) \otimes ( \sum _ { j } e _ j b _ { j , i } e _ i . t ) \\ & = ( e _ 1 + \dots + e _ m ) \otimes e _ \ell t ' \\ & = e _ \ell \otimes t ' . \end{align*}


\begin{align*} \rho _ { w } ( j , w ) = \overline { w } - 2 R _ j \left ( - \frac { 1 } { 2 } \frac { \overline { w } ^ 2 } { | w | ^ 3 } ( R _ j ) _ { \overline { z } } + \frac { 1 } { 2 } \frac { 1 } { | w | } ( R _ j ) _ { z } \right ) \end{align*}


\begin{align*} c \not \precsim E _ d \setminus \{ d \} \ \Rightarrow \ d \leq c \qquad c \precsim d \ \Rightarrow \ c = d c ^ \precsim \in \mathsf { S } \mathbb { P } , \end{align*}


\begin{align*} \overline { C } = \max _ { p _ n } & \sum _ { n } c _ { n } \ , p _ { n } \\ & \sum _ n p _ n = \sum _ n d _ n \end{align*}


\begin{align*} & \rho ( \alpha ( x ) , \alpha ( y ) ) \circ \beta = \beta \circ \rho ( x , y ) , \\ & \rho ( \alpha ( x ) , \alpha ( y ) ) \rho ( a , b ) - \rho ( \alpha ( a ) , \alpha ( b ) ) \rho ( x , y ) = ( \rho ( [ x , y , a ] , \alpha ( b ) ) - \rho ( [ x , y , b ] , \alpha ( a ) ) ) \circ \beta , \\ & \rho ( [ x , y , a ] , \alpha ( b ) ) \circ \beta - \rho ( \alpha ( y ) , \alpha ( a ) ) \rho ( x , b ) = \rho ( \alpha ( a ) , \alpha ( x ) ) \rho ( y , b ) + \rho ( \alpha ( x ) , \alpha ( y ) ) \rho ( a , b ) , \end{align*}


\begin{align*} \sum _ { \beta = 1 } ^ { N ( \gamma ) } d _ \beta ^ \gamma \phi ^ { \alpha _ 1 ^ { \gamma , \beta } , n } ( F _ 1 ) \cdots \phi ^ { \alpha _ m ^ { \gamma , \beta } , n } ( F _ m ) 1 & = \left ( S , \prod _ { i = 1 } ^ m f _ i \otimes \rho _ n ( E ^ \gamma ) \right ) . \end{align*}


\begin{align*} ( \psi _ 1 \circ \theta _ 1 ) \cdot \chi _ { f _ 1 } = ( \psi _ 2 \circ \theta _ 2 ) \cdot \chi _ { f _ 2 } . \end{align*}


\begin{align*} \hat { d } ( x , t ) = \begin{cases} d ( 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*} \| D _ { S , x } f \| _ p \le \sum _ { T \subseteq S } \| [ E _ { T } [ f ] ] _ { S \to x } \| _ p \leq \sum _ { T \subseteq S } r ^ { | S | } \gamma = ( 2 r ) ^ { | S | } \gamma , \end{align*}


\begin{align*} A _ q u = - \Delta u + q , u \in D ( A ) = H _ \Delta ( \mathrm { M } ) \cap H _ 0 ^ 1 ( \mathrm { M } ) . \end{align*}


\begin{align*} \Omega _ s = \{ \omega \in \Omega : \exists \ : t _ 0 \ : \ : \forall \ : t \geq t _ 0 , \ : \ : \forall \ : x \in C ( 0 , k t ) \ : \ : \exists \ : y \in B ( x , \rho ( t ) ) \ : \ : \ : B \left ( y , R _ { \rho ( t ) } + b \right ) \subseteq K ^ c \} , \end{align*}


\begin{align*} \lambda ' = \varphi _ { A ' } ^ \vee Y _ q ^ \vee \overline { \lambda } _ 0 Y _ q \varphi _ { A ' } . \end{align*}


\begin{align*} ( \prod _ { i = 1 } ^ u d ( i ) \ast h _ l ( q ( i ) ) ) \ast d ( u + 1 ) = ( \prod _ { i = 1 } ^ u d ( i ) \ast f _ l ( \delta ( q ( i ) ) ) \ast d ( u + 1 ) \notin A _ 1 \cap \sigma ( L _ 1 ) . \end{align*}


\begin{align*} v ( x ) = ( 1 - | x | ^ 2 ) _ + ^ s \end{align*}


\begin{align*} F ( k , t + 1 , n ) = \begin{dcases} \frac { n - k } { n } F ( k - 1 , t , n ) + \frac { k } { n } F ( k , t , n ) & \textnormal { i f } \ ; k < t ; \\ 1 & \textnormal { i f } \ ; k \ge t ; \end{dcases} \end{align*}


\begin{align*} g ( \kappa ) = \left \langle \beta , \kappa \right \rangle , \beta , \kappa \in \mathbb { R } ^ { d } . \end{align*}


\begin{align*} \lim \limits _ { k \rightarrow + \infty } \langle \lambda ^ k , S v \rangle = \langle \lambda ^ * , S v \rangle \forall \ v \in \mathcal { U } . \end{align*}


\begin{align*} \begin{aligned} b _ { k , N } ( L ) & = \int _ 0 ^ 1 R _ N ^ 2 ( L , \alpha , \Delta ) e ( - k \alpha ) \ , d \alpha \\ & = \frac { L } { N ^ 2 } \sum _ { n \in \mathbb { Z } } \widehat { \Delta } \left ( \frac { L n } { N } \right ) \sum _ { 1 \leq i \not = j \leq N } \int _ 0 ^ 1 e ( ( n a _ i - n a _ j - k ) \alpha ) \ , d \alpha \\ & = \frac { L } { N ^ 2 } \sum _ { n \not = 0 } \sum _ { \substack { 1 \leq i \not = j \leq N \\ n ( a _ i - a _ j ) = k } } \widehat { \Delta } \left ( \frac { L n } { N } \right ) , \end{aligned} \end{align*}


\begin{align*} D _ { \varrho } = \left \{ s \in \cap _ { \alpha \in \varrho } \ker ( \alpha ) : \alpha ( s ) < 0 \mbox { f o r a l l } \alpha \in \Delta - \varrho \right \} . \end{align*}


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


\begin{align*} \frac { \partial z } { \partial \omega } ( \phi , \omega , r ; t ) = ( 1 + O ( \varepsilon ) ) k t ^ 3 , \forall \ , t \in [ - \rho , \rho ] . \end{align*}


\begin{align*} h = \sum _ n c _ n a _ n . \end{align*}


\begin{align*} S _ { v e } = \left \{ \begin{array} { c c } 1 , & s ( e ) = v , \\ 0 , & s ( e ) \neq v , \end{array} \right . , T _ { v e } = \left \{ \begin{array} { c c } 1 , & t ( e ) = v , \\ 0 , & t ( e ) \neq v . \end{array} \right . \end{align*}


\begin{align*} ( a , b , t ) \longmapsto \left \{ \begin{bmatrix} ( t - b ) / 2 & & a \\ a & & ( t + b ) / 2 \end{bmatrix} \right \} _ { \Gamma _ { p } } , \end{align*}


\begin{align*} \Gamma = \bigcup _ { i = 1 } ^ \epsilon \left ( \gamma _ i + v K \right ) . \end{align*}


\begin{align*} \Psi ( u , \beta , A ) : = \mathcal { G } ( u , \beta , A ) - \Phi ( u , \beta , A ) , \end{align*}


\begin{align*} A _ I = \bigcap _ { i \in I } A _ i \hat { A } _ I = q ^ { - 1 } ( A _ I ) \end{align*}


\begin{align*} \left . \begin{array} { c } \left ( x , x ^ { \ast } \right ) , \left ( y , y ^ { \ast } \right ) \in \mathrm { g p h } T \\ \left \langle x - y , x ^ { \ast } - y ^ { \ast } \right \rangle = 0 \end{array} \right \} \Rightarrow h ( x , y ^ { \ast } ) + h ( y , x ^ { \ast } ) = h ( x , x ^ { \ast } ) + h ( y , y ^ { \ast } ) . \end{align*}


\begin{align*} c _ 1 = \sqrt { \frac { 3 } { 2 } } - 1 . \end{align*}


\begin{align*} \widetilde { \psi } : = \widehat { \psi } \circ T _ M \end{align*}


\begin{align*} \mathrm { d } Z _ t = A Z _ t \ , \mathrm { d } t + F ( Z _ t ) \ , \mathrm { d } t + G ( Z _ t ) \circ \mathrm { d } \mathbf { X } _ t , \ \ \ \ Z _ { 0 } = z _ 0 \in \mathcal { B } _ { \alpha } \end{align*}


\begin{align*} \| S ( k ) \varphi \| _ { \sup } = O ( e ^ { ( - \rho - \lambda _ 1 + 2 \tau ( \lambda _ 1 + \rho + 1 ) ) r ( 0 , k ) } ) c ( D \varphi ) . \end{align*}


\begin{align*} \int _ { B _ { \rho } } \left \vert w - ( w ) _ { \rho } \right \vert ^ { p _ 0 } & \leq \int _ { B _ { \rho } } \left \vert w - w ( 0 ) \right \vert ^ { p _ 0 } \\ & \leq \int _ { 0 } ^ { \rho } \int _ { \mathbb { S } ^ { n - 1 } } \tau ^ { p _ 0 } \left \vert D w \right \vert ^ { p _ 0 } \tau ^ { n - 1 } d \tau d \phi \\ & = C _ { 8 } ( n , p _ 0 , \lambda ) \rho ^ { n + p _ 0 } \left \| D w \right \| _ { L ^ { \infty } ( B _ { 1 / 4 } ) } ^ { p _ 0 } . \end{align*}


\begin{align*} \widetilde { \mathcal { P } } ( \gamma , \sigma , t ) = \mathcal { P } \left ( V \left ( \gamma , \sigma , t \right ) , \gamma \right ) , \widetilde { \mathcal { Q } } ( \gamma , \sigma , t ) = \mathcal { Q } \left ( V \left ( \gamma , \sigma , t \right ) , \gamma \right ) , \end{align*}


\begin{align*} T = W P _ { { \cal F } ^ \perp } { \cal S } W ^ { - 1 } , \varphi = W P _ { { \cal F } ^ \perp } 1 _ { \mathbb D } . \end{align*}


\begin{align*} v _ j = \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 ) v _ i . \end{align*}


\begin{align*} Z = \sum _ { j = 1 } ^ k A _ j \otimes B _ j \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } ( s , \lambda k ) _ { k } = \frac { 1 } { \left ( 1 + \frac { 2 } { \lambda } \right ) ^ { s } } , \end{align*}


\begin{align*} \lim _ { K \to \infty } \overline { d } _ { ( \ell _ i ) } ( \{ n \in \N : \eta _ { K } ( n ) \neq \eta ( n ) \} ) = 0 \end{align*}


\begin{align*} \nu _ p \left ( \sum _ { n = 0 } ^ { p ^ { \ell - t + k } - 1 } [ c ] ^ s ( n ) \right ) = \begin{cases} \ell - 1 & a _ k = a _ { k - 1 } = \cdots = a _ 0 = p - 1 \\ \geq \ell & \end{cases} \end{align*}


\begin{align*} \int _ M d d ^ c \omega _ 0 ^ { n - k - 1 } \wedge \omega ^ k \stackrel { \eqref { t h m : v a i s m a n - p p - e q 1 } } { = } - ( n - k - 1 ) \int _ M ( - d J \theta ) ^ { n - k } \wedge \omega ^ k < 0 . \end{align*}


\begin{align*} \left ( f , g \right ) : = \sum _ { k \in \hbar \mathbb { Z } ^ { n } } f ( k ) \overline { g ( k ) } , \end{align*}


\begin{align*} \left . \begin{array} { l l } \xi ^ * = r ^ \frac { N - 1 } { 2 } \xi , \eta ^ * = r ^ \frac { N - 1 } { 2 } \eta . \end{array} \right . \end{align*}


\begin{align*} t _ \mu = \pi s _ { j _ 1 } \dots s _ { j _ l } , \pi \in \Pi , \ j _ 1 , \dotsc , j _ l \in I \cup \{ 0 \} , \ l = \ell ( t _ \mu ) \end{align*}


\begin{align*} K = \bigcap \limits _ { t \in \mathbb { R } } \varphi ^ t ( V ) . \end{align*}


\begin{align*} \tilde { \mathbf { n } } _ \lambda = \max \left ( \mathbf { n } _ \lambda ^ 2 , \sqrt { \lambda } \mathbf { b } _ \lambda \right ) . \end{align*}


\begin{align*} U ( \vec { a } , \Lambda ) & \phi ^ { \alpha , n } ( f ) U ( \vec { a } , \Lambda ) ^ { - 1 } \left ( S , g _ \beta \otimes \rho _ n ( E ^ \beta ) \right ) \\ & = \left ( S , f ( \Lambda ^ { - 1 } ( \cdot - \vec { a } ) ) ^ { S ^ \flat } ( \cdot ) ( \bar { \Lambda } [ \hat { H } ( \rho ) \hat { f } _ 0 \pm \hat { P } ( \rho ) \hat { f } _ 1 ] ) \cdot g _ \beta \otimes \rho _ n ( E ^ \beta ) \right ) , \end{align*}


\begin{align*} Z _ { h , q } ( i ) = \begin{cases} h - 1 & i = q ; \\ 1 & k - j + h \leq i \leq k - 2 i \neq q ; \\ l - j + 1 & i = k - 1 ; \\ 0 & , \end{cases} \end{align*}


\begin{align*} \left ( T _ \lambda ^ { \vec { v } } \right ) _ { i , j } = x _ { j - i + 1 } + x _ { j - i + 2 } \alpha _ { n - 1 } + \dots + x _ j \alpha _ { n - i + 1 } \end{align*}


\begin{align*} p \mapsto H _ { n , K } ( p ) : = \int _ { Y } \int _ { \pi ^ { - 1 } ( y ) } H ( \theta _ { L _ { x } \ast } \mathbb { P } _ { \mu , n | K } ^ { y } ) d \eta _ { p } ^ { y } ( x ) d \nu ( y ) . \end{align*}


\begin{align*} \Pr ( X _ t = j ) = \Pr ( \widetilde { X } _ t = j ) \end{align*}


\begin{align*} G ( a ) = \left \{ \begin{array} { l l } a + \bar { k } , & \hbox { $ 1 \leq a \leq \bar { l } $ ; } \\ a + \bar { k } + \underline { k } , & \hbox { $ \bar { l } + 1 \leq a \leq \bar { k } $ ; } \\ a + \underline { k } + \bar { k } - \bar { l } \equiv a + \underline { l } , & \hbox { $ r + 1 \leq a \leq r + \underline { k } $ . } \end{array} \right . \end{align*}


\begin{align*} R _ N ^ 2 ( L , f ) : = \frac { 1 } { N } \sum _ { 1 \leq i \neq j \leq N } \sum _ { m \in \mathbb { Z } } f \left ( \frac { x _ i - x _ j + m } { \ell } \right ) . \end{align*}


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


\begin{align*} Q ( \Psi _ { \omega } ) = \omega ^ { 1 - \frac { d } { 2 } } Q ( \Psi ) , \ \ E ( \Psi _ { \omega } ) = \omega ^ { 2 - \frac { d } { 2 } } E ( \Psi ) , \ \ \mathbf { P } ( \Psi _ { \omega } ) = \omega ^ { \frac { 3 - d } { 2 } } \mathbf { P } ( \Psi ) . \end{align*}


\begin{align*} \overline { \psi } _ { y _ 1 , y _ 2 } = \norm { \cdot } _ 1 - \lim _ { n \to \infty } S _ n ( \psi _ { y _ 1 , y _ 2 } ) . \end{align*}


\begin{align*} S _ { \mathrm { m a i n } } \geq \sum _ { L = 1 } ^ { \infty } \phi ( L ) | y _ L | ^ 2 \left ( \log \left ( \frac { p ^ { \frac { ( ( k - 1 ) ! ) ^ 2 } { 2 } } } { D } \right ) + c _ 0 - c _ 1 \right ) , \end{align*}


\begin{align*} n \psi ^ { - 1 \slash n } \Delta F = L ( H ) + \frac { 1 } { n } \sum _ { \ell = 1 } ^ n u ^ { j \bar k } u ^ { p \bar q } u _ { \ell j \bar k } u _ { \bar \ell p \bar q } - \sum _ { \ell = 1 } ^ n u ^ { j \bar q } u ^ { p \bar k } u _ { \ell j \bar k } u _ { \bar \ell p \bar q } . \end{align*}


\begin{align*} \partial _ i V & = \frac { \lambda _ i } { 1 + \lambda _ i ^ 2 } V \\ \partial _ { i i } V & = \frac { 1 } { ( 1 + \lambda _ i ^ 2 ) ^ 2 } V \\ \partial _ { i j } V & = \frac { \lambda _ j \lambda _ i } { ( 1 + \lambda _ j ^ 2 ) ( 1 + \lambda _ i ^ 2 ) } V . \end{align*}


\begin{align*} & ( \mathcal { N } _ 1 , \mathcal { N } _ 1 ' ) , & \mathcal C _ 1 = \left ( \begin{matrix} \mathfrak { s } ^ 2 & \rho \\ \rho & \mathfrak { s } ^ 2 \end{matrix} \right ) , \\ & ( \mathcal { N } _ 2 , \mathcal { N } _ 2 ' ) , & \mathcal C _ 2 = \left ( \begin{matrix} \mathfrak { s } ^ 2 + | \rho | & 0 \\ 0 & \mathfrak { s } ^ 2 + | \rho | \end{matrix} \right ) . \end{align*}


\begin{align*} ( \ast ) & = w _ j C ^ \varphi _ { k j i } R ^ \varphi _ { i k } + \frac { 1 } { 2 } \langle \nabla w , \nabla | T ^ \varphi | ^ 2 \rangle + \frac { 1 } { 2 } ( 1 + w ) | C ^ \varphi | ^ 2 \\ & w _ { t i } R ^ \varphi _ { i k , t k } - \alpha w _ { t i } R ^ \varphi _ { i j } \varphi ^ a _ j \varphi ^ a _ t - \alpha w _ t R ^ \varphi _ { j k } \varphi ^ a _ { j t } \varphi ^ a _ k \ , . \end{align*}


\begin{align*} \int _ M | T ^ \varphi | ^ 2 | \nabla w | ^ 2 = 0 \ , . \end{align*}


\begin{align*} \underset { t \to \infty } { \lim } P ^ \omega ( D _ t ^ c \mid C _ t ) = 0 . \end{align*}


\begin{align*} H _ { \Sigma _ t } = t | \lambda | \phi + O ( t ^ 2 ) , \end{align*}


\begin{align*} w ( z ) = \Phi ( z ) + T ( f ) ( z ) , \end{align*}


\begin{align*} \langle ( \varphi _ \alpha , \varphi _ \beta ) , ( \varphi _ { \alpha ' } , \varphi _ { \beta ' } ) \rangle = ( \varphi _ \alpha \circ \varphi _ { \alpha ' } , \varphi _ { \beta ' } \circ \varphi _ \beta ) = ( \varphi _ { \alpha \alpha ' } , \varphi _ { \beta \beta ' } ) . \end{align*}


\begin{align*} \Theta _ L \left ( \tau , g ; \begin{pmatrix} \alpha \\ \beta \end{pmatrix} ; p \right ) = \sum _ { \gamma \in D _ L } \theta _ { L + \gamma } \left ( \tau , g ; \begin{pmatrix} \alpha \\ \beta \end{pmatrix} ; p \right ) \mathfrak { e } _ { \gamma } . \end{align*}


\begin{align*} f ( x _ { 1 } , \cdots , x _ { n } ) = n ! \cdot \exp \big ( - \sum _ { i \le n } x _ { i } \big ) \mathbf { 1 } _ { \{ x _ { 1 } \le \cdots \le x _ { n } \} } . \end{align*}


\begin{align*} F _ { | g _ i | ^ 2 } ( x ) = \gamma ( M , M x ) / \Gamma ( M ) = 1 - \sum _ { i = 0 } ^ M \frac { ( M x ) ^ i } { i ! } e ^ { - M x } , \end{align*}


\begin{align*} \tau ( i ) & = 2 n - i \end{align*}


\begin{align*} ( \theta , p _ \theta , z , s ) \mapsto ( x _ 1 , x _ 2 , y _ 1 , y _ 2 ) = ( e ^ s \cos \theta , e ^ s \sin \theta , z \cos \theta - p _ \theta \sin \theta , z \sin \theta + p _ \theta \cos \theta ) . \end{align*}


\begin{align*} C ( s , N , \mu , r ) = C ( N , \mu ) = \pi ^ { \frac { \mu } { 2 } } \frac { \Gamma ( \frac { N - \mu } { 2 } ) } { \Gamma ( N - \frac { \mu } { 2 } ) } \Bigl ( \frac { \Gamma ( \frac N 2 ) } { \Gamma ( N ) } \Bigr ) ^ { - 1 + \frac { \mu } { N } } . \end{align*}


\begin{align*} M = ( c ( m ) \frac { q _ n } { \vert \vert p - a \vert \vert ^ { m + 2 } } ) ( I - ( m + 2 ) v v ^ T ) , \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c v _ \lambda ) ^ n \leq C ^ n f ^ n \omega ^ n \leq ( d d ^ c ( C u _ 0 ) ) ^ n & \textnormal { i n } & \Omega \\ v _ \lambda = 0 & \textnormal { o n } & \partial \Omega . \end{array} \right . \end{align*}


\begin{align*} J _ { 0 } \le \| \theta _ { \epsilon } \| _ { L ^ { \infty } ( | x | < R ) } \| \nabla \cdot \varphi _ 3 \| _ { L ^ 2 } \| \varphi _ 1 \| _ { L ^ 4 } \| \varphi _ 2 \| _ { L ^ 4 } \le C \theta _ { \epsilon } ( R ) \prod _ { j = 1 } ^ 3 \| \varphi _ j \| _ { H ^ 1 } . \end{align*}


\begin{align*} \lim _ { p , p ^ { \prime } \rightarrow \infty } \tilde { \zeta } _ { p , p ^ { \prime } } ( s , c ) = \sum _ { m , n \neq 0 } \frac { 1 } { \left ( m ^ { 2 } + c n ^ { 2 } \right ) ^ { s } } \end{align*}


\begin{align*} | F _ k ( x , y ) | = | f ( x _ i ( y + y _ k ) ) | \leq \frac { C } { | y + y _ k | ^ N } , \end{align*}


\begin{align*} \phi _ l = \Tilde { \phi } _ { e } + \sum _ { m = 1 } ^ { k } D _ { p _ m } ( r _ x ^ { l - 1 } \Delta x ) ^ { p _ m } \end{align*}


\begin{align*} \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} \quad { \mathrm { a n d } } \begin{bmatrix} B & X ^ * \\ X & A \end{bmatrix} \end{align*}


\begin{align*} \frac { i } { n } \binom { n - i } { n - k } + \frac { ( n - i ) } { n } \binom { n - i - 1 } { n - k } = \frac { k } { n } \binom { n - i } { n - k } \end{align*}


\begin{align*} \theta _ \tau : \mathcal { H } \ni h \mapsto ( \langle h , \tau _ j \rangle ) _ { j = 1 } ^ n \in \mathbb { C } ^ n , \theta _ \omega : \mathcal { H } \ni h \mapsto ( \langle h , \omega _ j \rangle ) _ { j = 1 } ^ n \in \mathbb { C } ^ n . \end{align*}


\begin{align*} h _ { \mu } \left ( M , \boldsymbol { \alpha } _ { p } \right ) & : = h _ { \mu } ( Y , \nu ) + \int _ { G } \int _ { X } D \left ( \alpha _ { x , g } \parallel \alpha _ { x , e } \right ) \varphi _ { g } ( \pi ( x ) ) d \eta _ { p } ( x ) d \mu ( g ) \\ & = h _ { \mu } ( Y , \nu ) + \int _ { G } \int _ { Y } \int _ { S _ { y } } D \left ( \alpha _ { x , g } \parallel \alpha _ { x , e } \right ) d \eta _ { p } ^ { y } d g \nu ( y ) d \mu ( g ) \end{align*}


\begin{align*} \chi = \begin{bmatrix} \chi _ 1 , \dots , \chi _ r \end{bmatrix} ^ T , \end{align*}


\begin{align*} \forall x \geq R : F ^ { \mathrm { i n } } ( \pm x ) = \pm \frac { 1 } { 2 } . \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*} ( \partial _ j \partial _ i V ^ i ) \ , V ^ j = ( \partial _ j \partial _ i V ^ j ) \ , V ^ i . \end{align*}


\begin{gather*} 0 = \int _ a ^ t \| X ' \| ^ 2 + \langle ( R - \lambda I ) X , X \rangle d s . \end{gather*}


\begin{align*} a _ i = b _ { \sigma ( i ) } , \mbox { f o r } i \in \{ 1 , 2 , \ldots , k \} . \end{align*}


\begin{align*} & 0 _ 0 \neq \top _ 0 \\ & \forall \ , x : n ( x = 0 _ 0 \vee x = \top _ 0 ) \end{align*}


\begin{align*} ( \psi _ 1 ' \circ \theta _ 1 ' ) \cdot \chi _ { f _ 1 ' } = ( \psi _ 2 ' \circ \theta _ 2 ' ) \cdot \chi _ { f _ 2 ' } \end{align*}


\begin{align*} \Psi ( v _ 1 , \ldots , v _ { 2 f } ) = \Psi ( w _ 1 , \ldots , w _ { 2 f } ) . \end{align*}


\begin{align*} g \varrho g ^ { - 1 } = \ \left [ \begin{array} { c c c c } \varrho _ 1 & a _ { 1 2 } & \cdots & a _ { 1 n } \\ 0 & \varrho _ 2 & \cdots & a _ { 2 n } \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \varrho _ n \end{array} \right ] \end{align*}


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


\begin{align*} \phi ( t ) & = \phi _ 0 ( t ) u _ { B _ 1 , \tau _ 1 } ( t ) \cdots u _ { B _ R , \tau _ R } ( t ) \end{align*}


\begin{align*} \left ( \prod _ { x : X } M _ x \right ) _ f = \left ( \prod _ { x : X } M _ x \right ) ^ { D ( f ) } \end{align*}


\begin{align*} \Gamma ( f ^ \sharp ) \circ \phi _ F & = [ e _ { R } \circ ( f ^ \sharp _ 1 ) \circ e ^ { - 1 } _ { k ( F ) } ] \circ [ e _ { k ( F ) } \circ \rho _ F ] = e _ { R } \circ ( f ^ \sharp _ 1 ) \circ \rho _ F \\ & = e _ { R } \circ [ l _ R \circ f \circ ( \phi _ F ) ^ { - 1 } \circ e _ { k ( F ) } ] \circ \rho _ F \\ & = f \circ ( \phi _ F ) ^ { - 1 } \circ [ e _ { k ( F ) } \circ \rho _ F ] \\ & = f \circ ( \phi _ F ) ^ { - 1 } \circ \phi _ F = f . \end{align*}


\begin{align*} \Delta _ { n , k } = \varphi _ { n - k } ( \tilde S _ { k - 1 } + Z _ k ) - \varphi _ { n - k } ( \tilde S _ { k - 1 } + Y _ k ) \ , . \end{align*}


\begin{align*} D = \sum D _ N ( x - x _ \ast ) ^ N , \ \ W = \sum W _ N ( x - x _ \ast ) ^ N \end{align*}


\begin{align*} m ( x _ 1 - x _ 3 ) + ( n + m ) ( x _ 2 - x _ 3 ) = 2 \pi k \textrm { f o r s o m e } k \in \mathbb { Z } . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } 0 & = & \psi ^ { \prime } \left ( t \right ) - \frac { 1 } { 4 } \Pi _ { 5 } ^ { \top } \left ( t \right ) \Pi _ { 2 } ^ { - 1 } \left ( t \right ) \Pi _ { 5 } \left ( t \right ) \\ \varphi \left ( T \right ) & = & 0 , \end{array} \right . \end{align*}


\begin{align*} \lim _ \lambda \| e ^ * _ \lambda ( b - \omega ( b ) I ) e _ \lambda \| = 0 , b \in B . \end{align*}


\begin{align*} C : = \varphi ^ { - 1 } ( A ) = \bigcap _ { s \in S } R ^ { - s } W ^ c . \end{align*}


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


\begin{align*} \mathrm { E n d } _ { \mathbb { C } } ( A _ g ) = \{ x \in K \mid x g \Lambda _ 0 \subset g \Lambda _ 0 \} . \end{align*}


\begin{align*} C _ { h } ( n ) = C _ { f } ^ { + } ( n ) \end{align*}


\begin{align*} \varGamma _ J = \frac { Z } { Z _ J } F \varGamma F , F = \bigoplus _ { n = 0 } ^ \infty F _ n , F _ n = \prod _ { 1 \leq i < j \leq n } f ( x _ i - x _ j ) , \end{align*}


\begin{align*} c ( d , \nu ) : = \lambda _ d \left ( \frac { d } { \nu \omega _ d } \right ) ^ { - 2 / d } . \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle \xi _ { r r } ( 0 ) = - \dfrac { 2 m _ 1 Q _ 1 ( 0 ) } { N } , \quad \eta _ { r r } ( 0 ) = - \dfrac { 2 m _ 2 Q _ 2 ( 0 ) } { N } . \end{array} \right . \end{align*}


\begin{align*} \ J _ \nu ( u ^ * , v ^ * ) \leq J _ \nu ( u , v ) , \ P _ \nu ( u ^ * , v ^ * ) \leq P _ \nu ( u , v ) = 0 . \end{align*}


\begin{align*} S ^ { - 1 } \left ( \begin{bmatrix} \frac { 1 } { \sqrt { 2 | U _ n | \sqrt { U _ 1 } } } & 0 \\ 0 & \frac { 1 } { \sqrt { 2 | U _ n | \sqrt { U _ 1 } } } \end{bmatrix} \begin{bmatrix} U _ 1 ^ 2 + U _ n ^ 2 \\ U _ n U _ { \tau } \end{bmatrix} - \begin{bmatrix} R _ 1 \\ R _ 2 \end{bmatrix} \frac { 1 } { \sqrt { 2 | U _ n | \sqrt { U _ 1 } } } U _ 1 ^ 2 \right ) = G . \end{align*}


\begin{align*} 2 L ( \Phi ) + \left ( \frac { d } { 2 } + 1 \right ) N ( \Phi ) + \mathbf { c } \cdot \mathbf { P } ( \Phi ) = 0 \end{align*}


\begin{align*} & \sum \limits _ { i = 1 } ^ k \sum \limits _ { l = 0 } ^ n c ^ { i l } _ m \langle P \psi ^ l _ { \mu _ m ^ i , \xi _ m ^ i } , P \psi ^ h _ { \mu _ m ^ j , \xi _ m ^ j } \rangle \\ = & c ^ { j h } _ m \Big ( c _ h ( 1 + o ( 1 ) ) \Big ) + O ( 1 ) \sum \limits _ { l = 0 , l \neq h } ^ n c ^ { j l } _ m + o \bigg ( \Big ( \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big ) ^ { \frac { n } { n - 2 } } \bigg ) \sum \limits _ { i = 1 , i \neq j } ^ k \sum \limits _ { l = 0 } ^ n c ^ { i l } _ m . \end{align*}