\begin{align*} \sigma = \alpha _ 1 \alpha _ 2 \cdots \alpha _ m \underset { \beta ^ { \prime } } { \underbrace { x \alpha _ { m + 1 } \cdots \alpha _ k \delta \pi _ j \cdots \pi _ { l - 1 } \pi _ { l + 1 } \cdots \pi _ n } . } \end{align*}


\begin{align*} u \in C ^ 1 \left ( \Omega \setminus \overline { J _ u } \right ) . \end{align*}


\begin{align*} & E ( \hat \beta _ l ) = \sqrt { \frac { \pi } { 4 \kappa + 4 } } e ^ { - \frac { \kappa } { 2 } } \left [ ( 1 + \kappa ) I _ 0 \left ( \frac { \kappa } { 2 } \right ) + \kappa I _ 1 \left ( \frac { \kappa } { 2 } \right ) \right ] , \\ & V a r ( \hat \beta _ l ) = 1 - { E } ^ 2 ( \hat \beta _ { l } ) , \end{align*}


\begin{align*} \mu _ 0 ( f ) = \min \limits _ { j \in \{ 1 , \dots , s \} } \left \{ \nu \left ( b _ j \textbf { Q } ^ { \lambda _ j } \right ) \right \} . \end{align*}


\begin{align*} \begin{cases} \mathcal { L } u = 0 & \Omega ^ c , \\ u = 0 & W \subset \Omega ^ c , \end{cases} \end{align*}


\begin{align*} S _ { n _ { b } } = \underset { S _ { n } \in S _ { n \left ( j \right ) } } { \textup { a r g m a x } } \left \{ R _ { c } \left ( S _ { n } \right ) \right \} \end{align*}


\begin{align*} ( p - 1 - \alpha ) \lambda - k _ 1 \mu & = p - \alpha + m _ 1 , \\ ( p - 1 - k _ 2 ) \mu - k _ 3 \lambda & = p - k _ 3 + m _ 2 . \end{align*}


\begin{align*} | E _ { s } | & = 2 \cdot b _ { k - 2 } \cdot 2 ^ { \mathfrak z ( \lfloor b _ { k - 3 } \cdots b _ 0 \rfloor _ 2 ) } \\ & = b _ { k - 2 } \cdot 2 ^ { \mathfrak z ( s ) } = \begin{cases} 2 ^ { \mathfrak z ( s ) } & b _ { k - 2 } = 1 , \\ 0 & . \end{cases} \end{align*}


\begin{align*} \mu _ { K S } ^ \delta ( f ) ( \tau ) = \prod _ { D \mid N } u ( f ) ( D \tau ) ^ { c ^ 2 - 1 } . \end{align*}


\begin{align*} G ( t ) = G ( 0 ) > 0 . \end{align*}


\begin{align*} \Delta \Phi = 0 \ ; \mathrm { i n } \ ; \mathrm { M } , \Phi _ { | \partial M } = u _ { | \partial M } . \end{align*}


\begin{align*} \mathcal { E } ^ { N L } _ \varepsilon ( c ) : = \frac { 1 } { 4 } \int _ { \Omega } \int _ { \Omega } J _ \varepsilon ( x - y ) \big | c ( x ) - c ( y ) \big | ^ 2 \ : y x + \int _ { \Omega } f ( c ( x ) ) \ : x . \end{align*}


\begin{align*} ( - \Delta ) ^ s w \ , = \ , 0 \textrm { i n } \ ; \ ; \R ^ N \ , . \end{align*}


\begin{align*} [ z ^ { n - k } ] \prod \limits _ { i = 1 } ^ { n - \ell } \left ( 1 + \frac { z q } { q ^ { 2 i } } \right ) = q ^ { ( n - k ) ( 2 \ell - n - k ) } { n - \ell \brack n - k } _ { q ^ 2 } , \end{align*}


\begin{align*} L ( t ) = \frac { n - 1 } { t } , G ( t ) = \frac { n ( n - 4 ) } { 4 t ^ 2 } , H ( t ) = \frac { n - 4 } { 2 t } \quad \mbox { a n d } W ( t ) = \frac { n ( n - 8 ) } { 4 } , \forall t > 0 , \end{align*}


\begin{align*} G ( U _ n ( t _ n ) ) & = ( 4 - 2 d ) \omega Q ( U _ n ( t _ n ) ) + ( 3 - d ) \mathbf { c } \cdot \mathbf { P } ( U _ n ( t _ n ) ) \\ & \ = ( 4 - 2 d ) \omega Q ( U _ n ( 0 ) ) + ( 3 - d ) \mathbf { c } \cdot \mathbf { P } ( U _ n ( 0 ) ) \\ & \ = ( 4 - 2 d ) \omega Q ( \Phi _ { \omega , \mathbf { c } , n } ) + ( 3 - d ) \mathbf { c } \cdot \mathbf { P } ( \Phi _ { \omega , \mathbf { c } , n } ) + O ( \delta _ n ) . \end{align*}


\begin{align*} h _ { i j } = \dfrac { 1 } { D } \left [ 2 \phi ' \rho ^ 2 u _ i u _ j + \phi ^ 2 \phi ' s _ { i j } - \phi \rho u _ { i j } \right ] , \end{align*}


\begin{align*} = \exp \left \{ \frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } ( x y L i _ 1 ( x y z ) - 2 x y ^ 2 L i _ 1 ( x y z ) + x y ^ 3 L i _ 1 ( x y z ) ) \right \} \end{align*}


\begin{align*} \frac { \mu _ { \tau , t } \left ( \mathcal { Q } _ { R } ( x _ { 0 } , y _ { 0 } ) ) \right ) } { \mu _ { \tau , t } \left ( \mathcal { Q } _ { \rho } ( x _ { 0 } , y _ { 0 } ) \right ) } = \left ( \frac { R } { \rho } \right ) ^ { n + 2 s + 2 \tau } . \end{align*}


\begin{align*} \oplus _ { \mu \colon \rho ( \mu ) = \lambda } \Delta _ { \mu } ^ { m _ \mu } \twoheadrightarrow \imath _ { \leq \lambda } \imath ^ { * } _ { \leq \lambda } M / \imath _ { < \lambda } \imath ^ { * } _ { < \lambda } M = ( M ) _ { \not < \lambda } / ( M ) _ { \not \leq \lambda } , \end{align*}


\begin{align*} ( E , \sigma ^ { ( \mathcal { I } , r e d ) } ( H _ { \omega , \Lambda _ { L } ( x _ 0 ) } ) ) \leq e ^ { - \frac { \hat { m } } { K } L } = e ^ { - \frac { \hat m } { 9 } \frac { L } { 1 0 0 } } \end{align*}


\begin{align*} G ^ { \mathcal { T } ' } = \left \{ ( g _ { \sigma } ) \in \prod _ { \sigma \in \mathcal { T } } \mid g _ { \sigma } = 1 , \ , \forall \sigma \notin \mathcal { T } ' \right \} \end{align*}


\begin{align*} n ! = \sum _ { \pi \in S _ n } 1 \ ; . \end{align*}


\begin{align*} d f _ { i j } + \sum _ { k } f _ { k j } \circ f _ { i k } = 0 \end{align*}


\begin{align*} \mathcal { M } \subset \mathcal { B } ^ k _ M : = \left \{ y \in \R ^ k : \abs { y } < M \right \} . \end{align*}


\begin{align*} A u = \mathcal { O } ( h ^ \infty ) _ { C ^ \infty } . \end{align*}


\begin{align*} \nu ( h ) = \nu _ i ( h ) \mbox { a n d } \nu _ i ( f _ { \tilde { \textbf { Q } } } ) \geq 0 . \end{align*}


\begin{align*} \textsf { P } \{ g _ { i } ^ { 2 } > t \} = \textsf { P } \{ \vert g _ { i } \vert > \sqrt { t } \} \sim e ^ { - t / 2 } , \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \infty } \frac { p ^ { 2 } + \lambda _ { n } ^ { 2 } } { p \left ( p + \frac { 1 } { \pi } \right ) + \lambda _ { n } ^ { 2 } } \cdot \frac { 1 } { \sigma ^ { \prime } \left ( \frac { \lambda _ { n } ^ { 2 } x } { 2 \pi } \right ) e ^ { \lambda _ { n } ^ { 2 } x } - 1 } = \frac { 1 } { 2 \pi i } \ , \intop _ { \mu - i \infty } ^ { \mu + i \infty } \frac { \zeta _ { p ^ { \prime } } ( 1 - s ) \ , \zeta _ { p } ( 2 s ) } { 2 \cos \left ( \frac { \pi s } { 2 } \right ) } \ , \left ( \frac { x } { 2 \pi } \right ) ^ { - s } \ , d s . \end{align*}


\begin{align*} D _ \Theta = \begin{pmatrix} 2 & 1 \\ - 1 & 0 \end{pmatrix} D _ Y = \begin{pmatrix} 1 & 0 \\ - 1 & 1 \end{pmatrix} \end{align*}


\begin{align*} \mathbf { \hat h } = \sqrt { \frac { \kappa } { \kappa + 1 } } \mathbf { \hat h } ^ { L o S } + \sqrt { \frac { 1 } { \kappa + 1 } } \mathbf { \hat h } ^ { N L o S } , \end{align*}


\begin{align*} Q _ { \varepsilon , T } = \Omega \times \left ( \varepsilon , T - \varepsilon \right ) . \end{align*}


\begin{align*} L ^ { \rm s t i f f ( s o f t ) } _ \varepsilon : = \bigl \{ \vect f \in L ^ 2 ( \R ^ 3 ; \R ^ 3 ) : \ \vect f \equiv 0 \mbox { o n } \Omega _ { \rm s o f t ( s t i f f ) } ^ { \varepsilon } \bigr \} . \end{align*}


\begin{align*} \theta _ { 1 / 2 } ( z , \epsilon ) & = \sum _ { n \in \Z } e ^ { i \epsilon \pi n ^ 2 z } , \end{align*}


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


\begin{align*} { \mbox { $ u \ge w = \psi $ i n $ B _ r ( x _ r ) \setminus \overline { B _ { r / 2 } ( x _ r ) } $ . } } \end{align*}


\begin{align*} \frac { d } { d t } E ( \phi _ t ) = \int _ \Omega ( - \dot { \phi } _ t ) ( d d ^ c \phi _ t ) ^ n , \ , t \in I . \end{align*}


\begin{align*} S _ 1 = O ( | \varpi | ^ { n + 3 } ) . \end{align*}


\begin{align*} D _ { T } ( n ) : = \max _ { f \in B _ T ( n ) } \{ \deg ( f ) \} . \end{align*}


\begin{align*} & - \sum _ { e \in \delta _ { } ( i ) } \omega ( e ) v _ e + \sum _ { e \in \delta _ { } ( i ) } \omega ( e ) v _ e = 0 & ( i \in \hat { V } ) , \\ & \sum _ { e \in \hat { E } ^ \circ } \omega ( e ) z ( e ) v _ e = 0 , \end{align*}


\begin{align*} T \circ \alpha = \alpha \circ T . \end{align*}


\begin{align*} c z ^ { 2 } - ( a - d ) z - b = 0 . \end{align*}


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


\begin{align*} \Big \| \sum _ { j = 1 } ^ \infty \alpha _ j u _ j \chi _ { B _ { R _ { j + 1 } } } \Big \| _ { p ^ * , q , \mu } \leq \varepsilon _ 2 \| \{ \alpha _ j \} _ { j = 1 } ^ \infty \| _ { \ell _ q } . \end{align*}


\begin{align*} \sum \limits _ { \substack { b = 1 } } ^ { n ^ s } ( b , n ^ s ) _ s e ^ { \frac { 2 \pi i b j } { n ^ s } } & = \sum \limits _ { \substack { d ^ s \mid n ^ s \\ \frac { n ^ s } { d ^ s } \mid j } } \Phi _ s ( { d ^ s } ) { \frac { n ^ s } { d ^ s } } \\ & = \sum \limits _ { \substack { d ^ s \mid n ^ s \\ { d ^ s } \mid j } } \Phi _ s ( { \frac { n ^ s } { d ^ s } } ) { { d ^ s } } \\ & = \sum \limits _ { \substack { { d ^ s } \mid ( j , n ^ s ) _ s } } \Phi _ s ( { \frac { n ^ s } { d ^ s } } ) { { d ^ s } } . \end{align*}


\begin{align*} x _ i - x _ j = 0 , \ , \ , \{ i , j \} \in E . \end{align*}


\begin{align*} F = a _ { Q 0 } + a _ { Q 1 } Q + \ldots + a _ { Q D } Q ^ D \end{align*}


\begin{align*} ( \sigma _ U ) _ * ( ( f _ \sigma ^ * \widehat { \sigma _ L } ) \circ ( \sigma _ U ^ * \phi ) ) = ( \sigma _ U ) _ * ( f _ \sigma ^ * \widehat { \sigma _ L } ) \circ ( \sigma _ U ) _ * ( \sigma _ U ^ * \phi ) . \end{align*}


\begin{align*} \eta _ 1 ^ n : = \inf \left \{ \frac { E ( \phi ) } { I _ g ( \phi ) } ; \phi \in \mathcal E ^ 1 ( \Omega ) , w \neq 0 \right \} , \end{align*}


\begin{align*} x \otimes a & = x a ( 1 + \eta _ 1 ) , & a \otimes x & = a x ( 1 + \eta _ 1 ) . \end{align*}


\begin{align*} \partial _ { t } F + \partial _ { x } ( A ( t , F ) ) = \mathbf { S } [ F ] ( t , x ) . \end{align*}


\begin{align*} \widetilde \lambda _ h ^ { ( s ) } : = \abs { \avg { \widetilde { z } _ h ^ { ( s ) } } _ { B _ s } } \end{align*}


\begin{align*} \mathbf { p } ( f ) : = ( P _ 1 ( f ) , \cdots P _ d ( f ) ) , \ \ P _ k ( f ) : = - \frac { 1 } { 2 } { \rm R e } ( i f , \partial _ k f ) _ { L ^ 2 ( \R ^ d ) } \ \ ( k = 1 , \cdots d ) \end{align*}


\begin{align*} g ( x ) = L - 2 I _ { m } e ^ { \mathbf { i } x } + L e ^ { 2 \mathbf { i } x } . \end{align*}


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


\begin{align*} 0 = P ( ( - \Delta ) ^ s ) \tilde { u } + ( q _ 1 - q _ 2 ) u _ 1 + q _ 2 \tilde { u } = ( q _ 1 - q _ 2 ) u _ 1 \Omega . \end{align*}


\begin{align*} \phi _ m - i ^ * [ f _ \epsilon ^ { ' } ( V _ m ) \phi _ m ] = h _ m + \omega _ m , \end{align*}


\begin{align*} \mathrm { r a n k } ( \partial _ { ( x , u ) } f ) = n \ , , \end{align*}


\begin{align*} \mathcal { E C S } _ { { \rm { I I I } } } = \{ S ( p , q , a , b , c , d ) \in \mathcal { E C } : p , a , c \ne 0 \} . \end{align*}


\begin{align*} \begin{cases} \Delta _ { g } ^ 2 u = \lambda _ { } u , & \Omega , \\ u = \frac { \partial u } { \partial \bf n } = 0 , & \partial \Omega , \end{cases} \end{align*}


\begin{align*} \alpha _ { g } \not = 0 . \end{align*}


\begin{align*} \sup _ { \pi \in \bar { \Gamma } } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ { \lambda } \ , d \pi = \sup _ { \gamma \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \int _ { \mathcal { V } } g _ \lambda ( v ) \ , d \gamma ( v ) , \end{align*}


\begin{align*} \Pi ^ { \bot } _ { \mu , \xi } \Big ( V + \phi - i ^ * [ f _ \epsilon ( V + \phi ] \Big ) = 0 , \end{align*}


\begin{align*} \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } \leq x \right ) - \Phi ( x ) & = - \left [ \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } > x \right ) - ( 1 - \Phi ( x ) ) \right ] \\ & \leq - \left [ ( 1 - \Phi ( x ) ) \exp \left \{ - C \frac { 1 + x ^ { 3 } } { \sqrt { n } } \right \} - ( 1 - \Phi ( x ) ) \right ] \\ & = - ( 1 - \Phi ( x ) ) \left ( \exp \left \{ - C \frac { 1 + x ^ { 3 } } { \sqrt { n } } \right \} - 1 \right ) . \end{align*}


\begin{align*} g _ m ( u , x ) = \frac { 1 } { ( 1 - u ) ^ { m + 1 } } p _ m \left ( \frac { u } { 1 - u } , x \right ) . \end{align*}


\begin{align*} h ^ i _ j = \dfrac { \phi ' \delta ^ i _ j } { D } - \dfrac { \rho u ^ i _ j } { D \phi } + \dfrac { \phi ' \rho ^ 2 u ^ i u _ j } { D ^ 3 } + \dfrac { \rho ^ 3 u ^ i u _ k u ^ k _ j } { D ^ 3 \phi } , \end{align*}


\begin{align*} h _ { \mu } \left ( X , \eta \right ) = - \int _ { G } \int _ { X } \log \frac { d g ^ { - 1 } \eta ' } { d \eta ' } ( x ) d \eta ( x ) d \mu ( g ) . \end{align*}


\begin{align*} \mathcal { M } _ { q _ j , \alpha ^ j } : H ^ { s _ M } ( \Omega ^ c ) \to H ^ { - s _ M } ( \Omega ) , f \mapsto \left . \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) u _ f \right | _ { \Omega } , j = 1 , 2 , \end{align*}


\begin{align*} 0 \leq J ( u ^ { n } ) \le J _ n ( u ^ { n } ) \le J _ n ( u ^ { a } ) = J ( u ^ { a } ) \le M , \end{align*}


\begin{align*} Y \stackrel { \mathcal { D } } { = } p \delta _ 0 + q \xi , \end{align*}


\begin{align*} F _ 2 ( s ) - F _ 1 ( s ) = \frac { 1 } { 2 } s ^ { 2 } \log s ^ { 2 } , s \in \mathbb { R } , \end{align*}


\begin{align*} \| f ^ { = d } \| _ 2 ^ 2 \leq \| f ^ { = d } \| _ { q } \| f \| _ { q ' } \leq \rho ^ { - d } \gamma \cdot \gamma _ 1 \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) ^ \theta . \end{align*}


\begin{align*} Y = \frac { Q ' } { Q } + V ' . \end{align*}


\begin{align*} a _ 0 \ = \ \begin{cases} 7 , & , \\ 2 , & . \end{cases} \end{align*}


\begin{align*} \mu ^ \delta ( f ) = \sum n _ D \cdot ( \mu ( f \mid L R ( D ) ) \mid U L ( D ) ) . \end{align*}


\begin{align*} X _ { N , M } ^ { ( i ) } & = \frac { \sqrt { \theta _ N } } { N } \sum _ { z \in \Z ^ 2 } \big \{ Z _ { ( \frac { i - 1 } { M } N , \frac { i } { M } N ] , \beta _ N } ^ { \omega } ( z ) - 1 \big \} \ , \varphi _ N \big ( \tfrac { z } { \sqrt { N } } \big ) \ , . \end{align*}


\begin{align*} d _ { \square } ( h _ 1 , h _ 2 ) = \sup _ { S , T \subset [ 0 , 1 ] } \bigg | \int _ { S \times T } \d x \ , \d y \ , \big [ h _ 1 ( x , y ) - h _ 2 ( x , y ) \big ] \bigg | , \end{align*}


\begin{align*} \frac { p a } { p + \gamma - 1 } = \frac { p } { p + \gamma - 1 } \frac { ( p + \gamma - 1 ) ( 1 - \frac { 1 } { p } ) } { p + \gamma - 1 + \frac { 2 } { n } - 1 } < 1 , \end{align*}


\begin{align*} c ( ( y _ 1 , \dotsc , y _ L ) , ( y _ 1 ' , \dotsc , y _ L ' ) ) = \sum _ { \ell = 1 } ^ L | y _ \ell - y _ \ell ' | . \end{align*}


\begin{align*} A _ 1 = \frac { 1 } { 4 } \left ( \alpha + \frac { | \mathbf { c } | ^ 2 } { 8 \omega } \right ) , \ \ A _ 2 = \frac { 1 } { 4 } \left ( \beta + \frac { | \mathbf { c } | ^ 2 } { 4 \omega } \right ) , \ \ A _ 3 = \frac { 1 } { 4 } \left ( \gamma + \frac { | \mathbf { c } | ^ 2 } { 4 \omega } \right ) , \end{align*}


\begin{align*} \dfrac { \| \phi ^ n ( \rho ( 1 + u ) ) \| _ { L ^ { \infty } ( \mathbb { S } ^ n ) } } { \phi ^ n ( \rho ) } = 1 + O ( \varepsilon ) . \end{align*}


\begin{align*} \mathcal { C } _ { \alpha , \beta } ( \mathcal { M } ) : = \{ ( v , w ) ~ \vert ~ ( \alpha ( v + w ) , - \beta ( v - w ) ) \in \mathcal { M } \} . \end{align*}


\begin{align*} \mathbf { A } : \mathbf { B } = \sum _ { i , j = 1 } ^ 3 \mathbf { A } _ { i j } \mathbf { B } _ { i j } , \end{align*}


\begin{align*} \chi _ K \varphi ^ { - t * } \chi _ K = \chi _ K \varphi _ K ^ { - t * } \chi _ K : C ^ \infty ( M ) \to C ^ \infty ( M ) , \end{align*}


\begin{align*} f ( \lambda ) : = \lambda ^ 2 L ( U ) + \lambda ^ { \frac { d } { 2 } + 1 } N ( U ) + \omega M ( U ) + \lambda \mathbf { c } \cdot \mathbf { P } ( U ) . \end{align*}


\begin{align*} \nu _ { n + 1 } \geq \ , \min \{ \langle L _ { u } g \mid g \rangle \ ; \| g \| _ { L ^ 2 } = 1 \ , , \ ; g \in E ^ { \bot } \cap H ^ \frac 1 2 _ + \} \ , . \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta y & = u \ \mbox { i n } \ \Omega , \\ y & = 0 \ \mbox { o n } \ \partial \Omega , \\ \end{aligned} \right . \end{align*}


\begin{align*} \sum _ { j \in \mathcal J } \int _ { \{ x \in \Sigma : u ^ * _ \mu ( b _ j ) < | u ( x ) | < u ^ * _ \mu ( a _ j ) \} } | \nabla u ( x ) | ^ q \dd \mu ( x ) & = \int _ { \bigcup _ { j \in \mathcal J } \{ x \in \Sigma : u ^ * _ \mu ( b _ j ) < | u ( x ) | < u ^ * _ \mu ( a _ j ) \} } | \nabla u ( x ) | ^ q \dd \mu ( x ) \\ & \leq \int _ 0 ^ { \sum _ { j \in \mathcal J } ( b _ j - a _ j ) } ( \nabla u ) _ \mu ^ * ( s ) ^ q \dd s \\ & = \int _ 0 ^ t ( \nabla u ) _ \mu ^ * ( s ) ^ q \dd s . \end{align*}


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


\begin{align*} \tilde { Q } _ 1 \circ \partial & = \partial \circ Q _ 1 \circ f , \\ \tilde { Q } _ 2 \circ \partial & = \partial \circ \frac 1 { f } \circ Q _ 2 , \\ \tilde { Q } _ 1 \circ \tilde { Q } _ 2 \circ \partial & = \partial \circ Q _ 1 \circ Q _ 2 . \end{align*}


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


\begin{align*} e ( L / K , v ) = ( v L : v K ) . \end{align*}


\begin{align*} C _ K : = \bigcap _ { s \in S } R ^ { - s } W _ K ^ c . \end{align*}


\begin{align*} \sum _ { i } \mathrm { v } _ i \ , h _ i ^ 2 \ , w _ { \alpha } ^ { ( i ) } ( 0 , t ) = \boldsymbol { d _ { \alpha } } ( t ) \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } \frac { \log \norm { Q ( k ) h _ i } } { k } = \lambda _ i 1 \leq | i | \leq g , \ \lim _ { k \rightarrow \infty } \frac { \log \norm { Q ( k ) c _ s } } { k } = 0 1 \leq s < \gamma . \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle ( 1 + 2 \Lambda ) \dfrac { a _ 2 } { 2 m _ 1 } \int _ { \mathbb { R } ^ N } | \nabla \xi | ^ 2 d x + a _ 2 \omega _ 1 \int _ { \mathbb { R } ^ N } \xi ^ 2 d x - \left ( 1 + \dfrac { N } { 2 } \Lambda \right ) a _ 1 a _ 2 \int _ { \mathbb { R } ^ N } \xi ^ 2 \eta d x = 0 , \end{array} \right . \end{align*}


\begin{align*} V _ { e } = B \cup R _ 0 \cup _ i F _ i , \end{align*}


\begin{align*} \textsf { E } \xi = \int _ { 0 } ^ { \infty } \textsf { P } \{ \xi > t \} \ , d t . \end{align*}


\begin{align*} d y _ \lambda ^ h + A y _ \lambda ^ h = \bigl ( \sigma ( u _ \lambda ) Q h + \sigma ' ( u _ \lambda ) y _ \lambda ^ h B \ , d W , y _ \lambda ^ h ( 0 ) = 0 . \end{align*}


\begin{align*} F _ 0 ( p ) = \sum _ { q \in U } \delta ( p - q ) \ . \end{align*}


\begin{align*} S _ i ( f ' ) = \{ l - 1 \mid l \in S _ i ( f ) \setminus \{ 0 \} \mbox { a n d } p \nmid l \} . \end{align*}


\begin{align*} \lvert A ( D _ i ) \rvert - \lvert A ( D _ { i + 1 } ) \rvert = \deg _ { D _ i } ( v ) \le 2 \left \lfloor \frac { n - i } 2 \right \rfloor - ( r - 2 ) \end{align*}


\begin{align*} H ( \sigma ) = - \beta \sum _ { \{ u , v \} \in E } \ 1 ( \sigma _ u = \sigma _ v ) . \end{align*}


\begin{align*} \sigma _ J ( a ) = \begin{cases} a - k , & a - k \notin J , \\ a + n - k , & a - k \in J . \end{cases} . \end{align*}


\begin{align*} \partial _ { \mathbf { n } } c & = 0 \ ; \ ; \ ; \ ; \partial \Omega \times ( 0 , T ) , \\ c | _ { t = 0 } & = c _ 0 \ ; \ ; \ ; \Omega . \end{align*}


\begin{align*} A _ { \omega } : = \sup _ { \Phi \in \widetilde { \mathcal { M } } _ { \omega , c _ 1 ( \omega ) } } \| \Phi \| _ { \mathcal { H } ^ 1 } \in ( 0 , \infty ) . \end{align*}


\begin{align*} U = \overline { \xi } ( U _ 1 ) \circ \overline { \xi } ( U _ 2 ) , \end{align*}


\begin{align*} \Delta ( f ) = \varphi \left ( f \odot \frac { 1 } { 1 - t _ 1 \cdots t _ n } \right ) \ , , \end{align*}


\begin{align*} \mu _ { \tau , t } \left ( \mathcal { Q } _ { R } ( x _ { 0 } ) \right ) = c _ { n } \frac { R ^ { n + 2 s + 2 \tau } } { \tau } . \end{align*}


\begin{align*} | J _ j | & \le \int _ { \R ^ d } | \nabla \cdot \varphi _ 3 | \theta _ { \epsilon } | \varphi _ 1 | | \varphi _ 2 | d x = \int _ { | x | \ge R } | \nabla \cdot \varphi _ 3 | \theta _ { \epsilon } | \varphi _ 1 | | \varphi _ 2 | d x + \int _ { | x | < R } | \nabla \cdot \varphi _ 3 | \theta _ { \epsilon } | \varphi _ 1 | | \varphi _ 2 | d x = : J _ { \infty } + J _ { 0 } . \end{align*}


\begin{align*} p _ x & = \partial _ x z ( x , y ) , & p _ y & = \partial _ y z ( x , y ) . \end{align*}


\begin{align*} \Phi ( x , y ) = \Phi ( x , z ) + \sum _ { y < p \le z } \sum _ { v \ge 1 } \Phi ( x / p ^ { v } , p ) \end{align*}


\begin{align*} W = \bigoplus _ { j = 1 } ^ { \beta } \left \{ Q _ j \otimes I _ n \right \} \end{align*}


\begin{align*} \theta _ \alpha ^ j + \theta _ \beta ^ j = \rho _ j . \end{align*}


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


\begin{align*} \int _ { \{ h _ j \geq k ^ { n + 1 } \} } h _ j d V _ g = \int _ { \{ u _ j \leq - k \} } ( - u _ j ) ^ { n + 1 } g d V \end{align*}


\begin{align*} \int _ { I _ \alpha ^ { ( k ) } } \Big | \sum _ { i = 0 } ^ { N - 1 } D \varphi ( T ^ i x ) \Big | \ , d x \leq \sum _ { l = 0 } ^ { k } \sum _ { 0 \leq i < q ( l ) } \int _ { ( T ^ { ( l ) } ) ^ i J _ l } | S ( l ) D \varphi ( x ) | d x . \end{align*}


\begin{align*} \prod _ { \substack { l , m , n \geq 1 \\ l , m \leq n ; \ , \gcd ( l , m , n ) = 1 } } \left ( \frac { 1 } { 1 - x ^ l y ^ m z ^ n } \right ) ^ { \frac { 1 } { l ^ a m ^ b n ^ c } } = \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( \sum _ { l = 1 } ^ { n } \frac { x ^ l } { l ^ a } \right ) \left ( \sum _ { m = 1 } ^ { n } \frac { y ^ m } { m ^ b } \right ) \frac { z ^ n } { n ^ c } \right \} . \end{align*}


\begin{align*} J _ { 2 | \alpha _ i | } & = \tilde { S } ^ T _ { \gamma _ i } J _ { 2 n } \tilde { S } _ { \gamma _ i } \\ & = \sum _ { k = 1 } ^ r \tilde { S } _ { \gamma _ k \gamma _ i } ^ T J _ { 2 | \alpha _ k | } \tilde { S } _ { \gamma _ k \gamma _ i } \\ & = \tilde { S } _ { \gamma _ i \gamma _ i } ^ T J _ { 2 | \alpha _ i | } \tilde { S } _ { \gamma _ i \gamma _ i } + \sum _ { k \neq i , k = 1 } ^ r \tilde { S } _ { \gamma _ k \gamma _ i } ^ T J _ { 2 | \alpha _ k | } \tilde { S } _ { \gamma _ k \gamma _ i } . \end{align*}


\begin{align*} \mathcal { L } _ { \hbar } u ( k ) : = \sum \limits _ { j = 1 } ^ { n } \left ( u \left ( k + \hbar v _ { j } \right ) + u \left ( k - \hbar v _ { j } \right ) \right ) - 2 n u ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \end{align*}


\begin{align*} g ( { f } X , Y ) = - g ( X , { f } Y ) , g ( Q X , Y ) = g ( X , Q Y ) . \end{align*}


\begin{align*} m \ = \ L ( f ) \ \le \ m + 1 - \beta , \end{align*}


\begin{align*} \mathcal S ( t ) \triangleq \bigg \{ ( \mathbf { x } , \mathbf { y } ) | & \mathbf { x } , \mathbf { y } \in \{ 0 , 1 \} ^ { P - 1 } , x _ p + y _ p \leq 1 , p \in \mathcal { P } \backslash \{ P \} , \\ & \sum _ { p \in \mathcal { P } \backslash \{ P \} } ( x _ p + 2 y _ p ) = t \bigg \} , \\ h ( \mathbf { z } , t ) \triangleq & \sum _ { ( \mathbf { x } , \mathbf { y } ) \in \mathcal S ( t ) } \prod _ { p = 1 } ^ { P - 1 } 2 ^ { x _ p } z _ p ^ { x _ p + 2 y _ p } , \ \mathbf { z } \in \mathbb { R } ^ { P - 1 } _ { + + } , \end{align*}


\begin{align*} e ^ \xi S = \frac { n - 1 } { \sqrt { 1 + | D _ \tau \xi | ^ 2 } } - { \rm d i v } _ \tau \bigg ( \frac { D _ \tau \xi } { \sqrt { 1 + | D _ \tau \xi | ^ 2 } } \bigg ) , \end{align*}


\begin{align*} \chi _ \ell ( x ) = \sum _ { n \in \mathbb { Z } } \chi \left ( \frac { x - x _ 0 + n } { \ell } \right ) . \end{align*}


\begin{align*} \mathcal { A } ( f , g ) = T ( f \cdot g ) - f T ( g ) - T ( f ) g - 2 B ( A ( f ) , A ( g ) ) \left ( f , g \in P \right ) . \end{align*}


\begin{align*} { \rm P } _ { j , j + i } = & \sum ^ { M - j } _ { m = i + 1 } { M - j \choose m } \mathbb { P } _ { \rm T X } ^ m \left ( 1 - \mathbb { P } _ { \rm T X } \right ) ^ { M - j - m } \\ & \times \frac { M - j - i } { M - j } \bar { \rm P } _ { m , i } . \end{align*}


\begin{align*} \int _ { S _ { y } } f _ { n } ( y ) d \eta _ { p } ^ { y } = \lim _ { m \to \infty } \int _ { S _ { y } } f _ { n } ( y ) d \eta _ { p _ { m } } ^ { y } \le \liminf _ { m \to \infty } \int _ { S _ { y } } D \left ( \alpha _ { x , g } \parallel \alpha _ { x , e } \right ) d \eta _ { p _ { m } } ^ { y } . \end{align*}


\begin{align*} \frac { x ^ \lambda } { ( q ; q ) _ \infty ^ { \mathrm { r k } ( \mathfrak { g } ) } \prod _ { \alpha \in \Delta _ + } ( 1 - x ^ \alpha ) \prod _ { \alpha \in \Delta } ( q x ^ \alpha ; q ) _ \infty } = \sum _ { \mu \in P } m _ { \lambda \mu } ( q ) \frac { E _ \mu ( x , q , 0 ) } { ( q ) _ \mu } . \end{align*}


\begin{align*} f _ { 2 , m , \sharp } ( x , y ) = \sum _ { n = 1 } ^ { \mathcal { N } _ 2 } h _ { 2 , n , m } ( x , y ) e ^ { i \beta _ { n , m } ( x ) y } \end{align*}


\begin{align*} ( - ) ^ \ast \colon B \rightarrow B , b \mapsto b ^ \ast : = \mathrm { T r d } ( b ) - b . \end{align*}


\begin{align*} \Lambda ( f , s ) = \left ( \frac { \sqrt { N } } { 2 \pi } \right ) ^ s \Gamma ( s ) \ , L ( f , s ) \end{align*}


\begin{align*} \begin{pmatrix} f _ { 0 0 } & f _ { 0 1 } & f _ { 0 2 } & \ldots \\ f _ { 1 0 } & f _ { 1 1 } & f _ { 1 2 } & \ldots \\ \vdots & \vdots & & \end{pmatrix} \end{align*}


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


\begin{align*} c _ r ( n ) & = \sum \limits _ { \substack { { m = 1 } \\ ( m , r ) = 1 } } ^ { r } e ^ { \frac { 2 \pi i m n } { r } } \end{align*}


\begin{align*} \mathcal I = \mathcal I _ 1 + \mathcal I _ 2 . \end{align*}


\begin{align*} \exp \left ( - \frac { 0 . 0 0 0 1 } { p } \right ) \leq \mu _ p ( f ) = \frac { \mu _ p ( f ) } { r ^ { 0 } } \leq \frac { \mu _ p ( f _ { S \rightarrow x } ) } { r ^ { | S | } } \leq e ^ { - | S | } , \end{align*}


\begin{align*} H _ 2 ( z ) : = A _ 2 z ^ { n ' + m ' } + B _ 2 \overline { z } ^ { m ' } + C _ 2 \textrm { f o r a l l } z \in \mathbb { C } \end{align*}


\begin{align*} ( f ( g h ) ) x & = ( ( f g ) h ) x - ( ( f h ) g ) x \\ x ( f ( g h ) ) & = x ( ( f g ) h ) - x ( ( f h ) g ) . \end{align*}


\begin{align*} \sum _ { 1 \leq j \leq n } u _ { j p } u _ { \bar j } + u _ p u _ { p \bar p } = - \beta ( u u _ p + B \bar z _ p ) , \end{align*}


\begin{align*} \chi ( t , x , s , y ) & = b _ { \varepsilon } \left ( \frac { x - y } { 2 } \right ) b _ { \varepsilon } \left ( \frac { t - s } { 2 } \right ) g \left ( \frac { x + y } { 2 } \right ) h _ { \delta } \left ( \frac { t + s } { 2 } \right ) \\ & = b _ { \varepsilon } \left ( \overline { y } \right ) b _ { \varepsilon } \left ( \overline { s } \right ) g \left ( \overline { x } \right ) h _ { \delta } \left ( \overline { t } \right ) . \end{align*}


\begin{align*} \nu _ i ( f ) = \min _ { 1 \leq j \leq r } \left \{ v \left ( \tilde { b } _ j \right ) \right \} \mbox { a n d } \alpha _ \ell \geq \gamma \mbox { f o r e v e r y } \ell \in I _ 0 ( f , i ) , \end{align*}


\begin{align*} G ( 2 , d ) : = \left ( \begin{array} { c c } s t ^ d & - s ^ { d + 1 } \\ t ^ { d + 1 } & - s ^ d t \\ \end{array} \right ) \equiv \left ( \begin{array} { c c } 1 & - 1 \\ 1 & - 1 \\ \end{array} \right ) \odot \left ( \begin{array} { c c } s t ^ d & s ^ { d + 1 } \\ t ^ { d + 1 } & s ^ d t \\ \end{array} \right ) . \end{align*}


\begin{align*} \mathcal { E } _ { s } & : C ( X ) \to \tilde { L } ^ { \infty } ( X ) \\ \mathcal { E } _ { s } f \left ( u , v , x _ { 0 } \right ) & = \lim _ { N \to \infty } \frac { 1 } { N + 1 } \sum _ { n = 0 } ^ { N } s ^ { n } . \left ( f \circ \xi \right ) \left ( u , v , x _ { 0 } \right ) = \mathbb { E } _ { \lambda } \left ( \tilde { f } \left ( u , \cdot \right ) | \mathcal { F } ^ { s } \right ) ( x _ { 0 } ) . \end{align*}


\begin{align*} A = \lambda _ 1 ^ { \uparrow } ( A ) E _ 1 + \lambda _ 2 ^ { \uparrow } ( A ) ( E _ 2 - E _ 1 ) + \lambda _ 3 ^ { \uparrow } ( A ) ( E _ 3 - E _ 2 ) + . . . . + \lambda _ n ^ { \uparrow } ( A ) ( E _ n - E _ { n - 1 } ) \end{align*}


\begin{align*} ( t _ 1 , t _ 2 , \dots , t _ n , t ) \sum _ { i = 0 } ^ m a _ { i } e _ { \lambda ^ i } = \sum _ { i = 0 } ^ m t { \cdot } t _ { \lambda ^ i } a _ { i } e _ { \lambda ^ i } , \end{align*}


\begin{align*} \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} = U \begin{bmatrix} \frac { A + B } { 2 } + { \mathrm { R e } \ , } X & 0 \\ 0 & 0 \end{bmatrix} U ^ * + V \begin{bmatrix} 0 & 0 \\ 0 & \frac { A + B } { 2 } - { \mathrm { R e } \ , } X \end{bmatrix} V ^ * \end{align*}


\begin{align*} s = \lambda _ 0 a _ 0 + \cdots + \lambda _ n a _ n , \lambda _ j \geq 0 , \ j = 0 , \dots , n , \sum _ { k = 0 } ^ n \lambda _ k = n + 1 . \end{align*}


\begin{align*} & E \left [ \left ( Z _ 0 ( a ) e ^ { - \lambda a } - Y \right ) ^ 2 \right ] = O \left ( e ^ { - \lambda a } \right ) , \\ & E \left [ \left ( Z _ 0 ( a ) e ^ { - \lambda a } - Z _ 0 ( b ) e ^ { - \lambda b } \right ) ^ 2 \right ] = O \left ( e ^ { - \lambda a } \right ) , \end{align*}


\begin{align*} \tilde { \mathcal { L } } ( \varphi ) = \tau ^ { - 1 } \langle c , A \psi - \phi \rangle + \tau ^ { - 1 } \sum _ { i = 1 } ^ m \langle c , B _ i \psi \rangle u _ i , \end{align*}


\begin{align*} \begin{cases} 2 \lambda ^ { ( i ) } + 1 & : = [ 2 \lambda _ 1 , 2 \lambda _ 2 , \ldots , 2 \lambda _ i + 1 , \ldots , 2 \lambda _ t ] , \ \mbox { f o r } i \in \mathcal { I } ( \lambda ) \\ 2 \lambda ^ { ( t + 1 ) } + 1 & : = [ 2 \lambda _ 1 , 2 \lambda _ 2 , \ldots , 2 \lambda _ t , 1 ] . \end{cases} \end{align*}


\begin{align*} n \ = \ p _ { 1 } ^ { \alpha _ 1 } \cdots p _ { k } ^ { \alpha _ k } , \end{align*}


\begin{align*} h = \frac { 1 } { 2 } \begin{pmatrix} 1 & 0 \\ 0 & - 1 \end{pmatrix} , \theta ( x ) = - x ^ \top , \mbox { w e h a v e } \tau = \theta \tau _ h , \begin{pmatrix} a & b \\ c & - a \end{pmatrix} \mapsto \begin{pmatrix} - a & c \\ b & a \end{pmatrix} . \end{align*}


\begin{align*} k _ { e x \left ( 1 \right ) } = \underset { k \in S _ { n \left ( 1 \right ) } } { { \arg \min } } ~ \textbf { g } _ { k } ^ { H } \textbf { g } _ { k } \end{align*}


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


\begin{align*} \begin{cases} \overset \cdot b _ 0 = b _ 0 ( b _ 1 + b _ 2 ) , \\ \overset \cdot b _ 1 = b _ 1 ( b _ 2 + b _ 3 - b _ 1 ) , \\ \overset \cdot b _ 2 = b _ 2 ( b _ 3 - b _ 1 - b _ 0 ) , \\ \overset \cdot b _ 3 = b _ 3 ( - b _ 2 - b _ 1 ) . \end{cases} \end{align*}


\begin{align*} M = \pm \left ( { T _ { \varepsilon _ 1 } ^ { \vec { v } } } ^ T \right ) ^ { m _ 1 } \end{align*}


\begin{align*} M ( g a , g b ) = M ( g a _ 1 , \dots , g a _ n , g b _ 1 , \dots , g b _ m ) = M ( a , b ) \end{align*}


\begin{align*} h \left ( Z , \lambda \right ) - h \left ( M , \bar { \lambda } \right ) = \lim _ { n \to \infty } - \frac { 1 } { n } \log P _ { \mu , x } ^ { \zeta , n } \left ( L _ { x } , L _ { x } \omega _ { n } \right ) . \end{align*}


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


\begin{align*} \left \| P _ 1 u \right \| ^ 2 + \left \| P _ 2 u \right \| ^ 2 + 2 \left ( P _ 1 u , P _ 2 u \right ) = \left \| e ^ { - s \sigma } f \right \| ^ 2 . \end{align*}


\begin{align*} \Sigma : \ t ^ p \mapsto \Sigma _ 0 ^ t s ^ p \delta s = \frac { B _ { p + 1 } ( t ) - B _ { p + 1 } ( 0 ) } { p + 1 } \end{align*}


\begin{align*} A ( q ) = \frac 1 2 y _ 0 q + O ( q ^ 2 ) , \qquad q \to 0 . \end{align*}


\begin{align*} d \eta ^ j ( \xi _ i , \ , \cdot ) = 0 ; \end{align*}


\begin{align*} C ( \gamma _ 1 , \gamma _ 2 ) ( f ) = \tilde { \mu } _ \delta \{ \gamma _ 1 i \infty \to \gamma _ 2 i \infty \} ( \phi ^ \ast f ) . \end{align*}


\begin{align*} \delta _ { 1 / r } ^ { A } \delta _ { r } ^ { B } \{ x \in G \colon \| x \| = 1 \} \subseteq \mathcal { B } ^ { A } ( e , D ) . \end{align*}


\begin{align*} \Delta _ g \Theta = 0 , \end{align*}


\begin{align*} f _ \nu ( z ) : = \frac { 1 } { | z | ^ { \nu } } \ , f \left ( \frac { 1 } { z } \right ) . \end{align*}


\begin{align*} N _ { d r } ( m + 1 ) - N _ { d r } ( m ) & = a ^ 2 + a d + \left ( \sum _ { i = 1 } ^ k x _ i ^ { \prime } - \sum _ { i = 1 } ^ k x _ i \right ) a \\ & \geq \left ( a + d + \sum _ { i = 1 } ^ { k - 1 } x _ i ^ { \prime } - \sum _ { i = 1 } ^ { k - 1 } x _ i \right ) a \\ & \geq ( a + d - k ) a \geq 0 . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { n } \Big ( q _ i ( p - x _ { i } ) \ss _ i ^ { m + 2 } \Big ) = 0 . \end{align*}


\begin{align*} s _ 1 ( T _ v ) & = \sum _ { j , k = 1 } ^ n T _ v ( e _ j , \bar e _ j , e _ k , \bar e _ k ) = \sum _ { j , k = 1 } ^ n R ( e _ j , \bar e _ j , v , \bar v ) R ( e _ k , \bar e _ k , v , \bar v ) = r ( v , \bar v ) ^ 2 , \\ s _ 2 ( T _ v ) & = \sum _ { j , k = 1 } ^ n T _ v ( e _ j , \bar e _ k , e _ k , \bar e _ j ) = \sum _ { j , k = 1 } ^ n | R ( e _ j , \bar e _ k , v , \bar v ) | ^ 2 . \end{align*}


\begin{align*} \begin{aligned} - \Delta u & = f , \\ u & = 0 , \end{aligned} \end{align*}


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


\begin{align*} \pi _ n ( \cdot ) = \frac { \pi \left ( \cdot \cap \left ( A _ { 1 n } \right ) \right ) } { \pi \left ( A _ { 1 n } \right ) } . \end{align*}


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


\begin{align*} \langle E _ { \lambda } ( x , q , 0 ) , E _ { \mu } ( x ^ { - 1 } , q ^ { - 1 } , \infty ) \rangle _ 0 = ( q ) _ \lambda \delta _ { \lambda , \mu } , \end{align*}


\begin{align*} \texttt { w p r s s u m } = \max \Bigl \{ \texttt { m a g } ( 1 0 ( L + 1 ) ) , \texttt { m a g } \Bigl ( \frac { 4 . 4 c \sqrt { \pi } } { \varepsilon _ 2 b a } ( L + 3 ) ^ 2 \Bigr ) \Bigr \} + 1 . \end{align*}


\begin{align*} & \kappa _ 1 \circ \kappa _ n = w \kappa _ { n + 1 } + ( 1 - w ) \kappa _ { n - 1 } ~ a n d \\ & \kappa _ 0 ( x ) = x ~ x \in E . \end{align*}


\begin{align*} \langle \varphi , \psi \rangle : = & \int _ { \mathbb { R } ^ n } \varphi \psi \ , d x , \varphi , \psi \in L ^ 2 ( \mathbb { R } ^ n ) , \\ ( \varphi , \psi ) _ { \Omega } : = & \int _ { \Omega } \varphi \psi \ , d x , \varphi , \psi \in L ^ 2 ( \Omega ) . \end{align*}


\begin{align*} L ( \gamma ) + \frac { 1 } { 2 } L ( \gamma ^ 2 ) = \frac { 4 } { 7 } . \end{align*}


\begin{align*} z _ 0 = ( \Phi ( b _ { n + 1 } ) \cdots \Phi ( b _ { 2 n } ) ) ^ { - 1 } ( \Phi ( b _ 1 ) \cdots \Phi ( b _ n ) ) \in \mathbf { X } . \end{align*}


\begin{align*} \mu _ n = \mu _ n G \left ( u _ n \right ) = \mu _ n \left \langle G ^ { \prime } ( u _ n ) , u _ n \right \rangle = \left \langle F ^ { \prime } ( u _ n ) , u _ n \right \rangle = F \left ( u _ n \right ) = \beta _ n . \end{align*}


\begin{align*} \lim _ { N \to \infty } N ^ { - 1 } S _ j ( \tau _ N ) = \frac { \nu } { \lambda } \frac 1 { j ( j + 1 ) } . \end{align*}


\begin{align*} A r ^ { n + m } + B r ^ m + C = 0 \textrm { a n d } ( n + m ) A r ^ { n + m - 1 } + m B r ^ { m - 1 } = 0 , \end{align*}


\begin{align*} ( N / M ) _ f = ( N / M ) ^ { D ( f ) } . \end{align*}


\begin{align*} \sup _ { \substack { \pi , \\ \pi = \lbrace \kappa _ 0 = s , \kappa _ { 1 } , . . . \kappa _ { m } = t \rbrace } } \big { \lbrace } \sum _ { j } \big { [ } | h _ { \kappa _ { j } , \kappa _ { j + 1 } } | ^ { \frac { 1 } { { \gamma } ^ { \prime } } } \big { ] } \big { \rbrace } < \infty . \end{align*}


\begin{align*} \partial _ { l } \div \eta & = ( \delta _ { k j } - a _ { k j } ) \partial _ { l k } \eta _ j + \int _ 0 ^ t ( \partial _ t a _ { k j } \partial _ { l k } \eta _ j + \partial _ l a _ { k j } \partial _ k v _ j ) \ , \d s , \end{align*}


\begin{align*} I [ u ] = \int _ \Omega B \big ( | \nabla u | \big ) \ , d x - \int _ \Omega f u \ , d x \end{align*}


\begin{align*} \mu _ D \Big ( \mu _ C \big ( Q _ 1 ( x , y , z , w ) \big ) \Big ) & = Q _ 1 \big ( \alpha ( \alpha x - y ) - x , \alpha x - y , \alpha ( \alpha z - w ) - z , \alpha z - w \big ) \\ & : = Q _ 1 ( x ' , y ' , z ' , w ' ) \end{align*}


\begin{align*} L ( f , s ) = \sum _ { n \geq 1 } \frac { a _ f ( n ) } { n ^ s } \mathrm { R e } ( s ) \gg 0 . \end{align*}


\begin{align*} \pi = \alpha _ 1 \alpha _ 2 \cdots \alpha _ k \underset { \beta } { \underbrace { \delta \pi _ j \cdots \pi _ { l - 1 } x \pi _ { l + 1 } \cdots \pi _ n } } . \end{align*}


\begin{align*} \mathrm { d o m } \left ( T _ { 1 } \cap \left ( - T _ { 2 } \right ) \right ) = \left \{ x \in X \mid 0 \in \left ( T _ { 1 } + T _ { 2 } \right ) \left ( x \right ) \right \} , \end{align*}


\begin{align*} \alpha ( v + w ) = M _ 1 ^ H z , - \beta ( v - w ) = M _ 2 ^ H z \end{align*}


\begin{align*} \Phi ( t ) = \left ( s ( 2 \coth ( \kappa t ) - 2 ) \phi ' ( t ) + s ^ 2 ( 1 - 2 \coth ( \kappa t ) ) \phi ( t ) \right ) ^ 2 \left ( e ^ { - \kappa t } \sinh ( \kappa t ) \right ) ^ { n - 1 } , \end{align*}


\begin{align*} R _ N ^ 2 ( L , \alpha , \Delta ) = \frac { L } { N ^ 2 } \sum _ { n \in \mathbb { Z } } \widehat { \Delta } \left ( \frac { L n } { N } \right ) \sum _ { 1 \leq i \neq j \leq N } e ( n \alpha ( a _ i - a _ j ) ) ; \end{align*}


\begin{align*} U \xi _ { \nu } = U \pi _ { \nu } ( f ) \xi _ { \nu } = \pi _ { \mu } U \xi _ { \nu } . \end{align*}


\begin{align*} \lambda ^ { \varphi } ( t ) a = \sigma ^ { \varphi } _ t ( a ) \lambda ^ { \varphi } ( t ) , t \in \mathbb { R } , a \in M . \end{align*}


\begin{align*} \mu _ t ( y ) = e ^ { - \alpha t } \mu _ 0 ( y ) + \int _ 0 ^ t \alpha e ^ { - \alpha ( t - s ) } \Phi ( \nu _ s , \mu _ s ) ( y ) \mathrm { d } s . \end{align*}


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


\begin{align*} \lim _ \lambda \| b e _ \lambda \| = 0 , b \in L _ \omega . \end{align*}


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


\begin{align*} \alpha _ t ( g H ) = \exp ( t h ) g H . \end{align*}


\begin{align*} X _ { t g } = t X _ g . \end{align*}


\begin{align*} V = \{ ( x H , y H ) : \ x H \in \Omega \setminus \mathcal { S } \mbox { a n d } y H \in \mathcal { S } \} . \end{align*}


\begin{align*} \overline { \operatorname { S p G } _ N } = \overline { G _ M \cap \mathcal H } , \end{align*}


\begin{align*} d X ( t ) = d B ( t ) + \frac { 1 } { X ( t ) } d t - \frac { X ( t ) } { \tau - t } d t , \end{align*}


\begin{align*} \sup _ { j , k \leq n } \| S _ k - S _ j \| \leq 2 \sup _ j \| S _ j \| = 2 U _ n . \end{align*}


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


\begin{align*} & Z ^ { \prime } _ { s } = G ( Z _ s ) Z ^ { \# } _ { s , t } = Z _ { s , t } - G ( Z _ s ) \circ ( \delta X ) _ { s , t } . \end{align*}


\begin{align*} i _ b = \inf _ { t > 0 } \frac { t \ , b ' ( t ) } { b ( t ) } \quad s _ b = \sup _ { t > 0 } \frac { t \ , b ' ( t ) } { b ( t ) } , \end{align*}


\begin{align*} \| \iota ( g ) \| _ { X ^ \ast } = \sup _ { \| f \| _ X = 1 } \left | \int _ \Omega \ ! f g \ , \mathrm { d } \mu \right | \leq \sup _ { \| f \| _ X = 1 } \int _ \Omega \ ! | f g | \ , \mathrm { d } \mu = \| g \| _ { X ' } , \end{align*}


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


\begin{align*} N _ k = \sum _ { i = 1 } ^ { k - 2 } N _ { i } + 2 \end{align*}


\begin{align*} ( \Delta + \mu - q ) u = f \ ; \mathrm { i n } \ ; M , ( \partial _ \nu u \mp i a ) u = \varphi \ ; \mathrm { o n } \ ; \partial M , \end{align*}


\begin{align*} \frac { 2 \sqrt { \pi } \ , c ^ { \frac { 1 } { 2 } - s } } { \Gamma ( s ) } \cdot \frac { \Gamma \left ( s - \frac { 1 } { 2 } \right ) } { 1 + \frac { 1 } { \pi p } } \ , \zeta _ { p ^ { \prime } } ( 2 s - 1 ) = \frac { \pi } { \sqrt { c } \left ( 1 + \frac { 1 } { \pi p } \right ) } \ , \frac { 1 } { s - 1 } + \frac { \pi } { \sqrt { c } \left ( 1 + \frac { 1 } { \pi p } \right ) } \left ( 2 C _ { p ^ { \prime } } ^ { ( 1 ) } - \log \left ( 4 c \right ) \right ) + O \left ( s - 1 \right ) . \end{align*}


\begin{align*} Q ( G , Z , \lambda , \mathcal { G } _ w ) : = \sum _ { ( w _ { \mathbf { U } } , \mathbf { U } ) \in \mathcal { G } _ w } e ^ { - \lambda \mathfrak { l } ( \mathbf { U } ) } < p < 1 . \end{align*}


\begin{align*} H _ { n , d } ( s ) = ( s ^ { \mathcal { I } - e _ j + d e _ k } ) , j , k = 1 , \ldots , n , \end{align*}


\begin{align*} \begin{cases} \partial _ { t } z - ( y ^ { \alpha _ y } \partial _ { x x } z + x ^ { \alpha _ x } \partial _ { y y } z ) = \chi _ { \omega } g , & \mbox { i n } \ Q , \\ z ( x , y , t ) = 0 \ \mbox { o r } \ A \nabla z \cdot \nu = 0 , & \mbox { o n } \ \Sigma , \\ z ( x , y , 0 ) = z _ { 0 } ( x , y ) , & \mbox { i n } \ \Omega , \end{cases} \end{align*}


\begin{align*} \Theta ( i ) = W _ F ( i , 0 ) + W _ F ( i , 1 ) + \hdots + W _ F ( i , i - 1 ) . \end{align*}


\begin{align*} P ^ \omega \left ( A _ t ^ c \mid E _ t \right ) \leq P \left ( \big | Z _ s ^ { B _ s } \big | < e ^ { \delta t } \right ) \leq P \left ( \big | Z _ s ^ { B _ s } \big | < e ^ { - \sqrt { \beta / 2 } \ , \hat { r } ( s ) } p _ s e ^ { \beta s } \right ) = e ^ { - \sqrt { \beta / 2 } \ , \hat { r } ( s ) ( 1 + o ( 1 ) ) } , \end{align*}


\begin{align*} \frac { \textup { M M D } _ { n , m } ^ { 2 } } { \surd { \textup { v a r } ( \Delta _ { 1 } ) } } = \frac { \Delta _ 0 + T _ 1 + \Delta _ 1 + \{ \widetilde { \Delta } _ { 2 } - E ( \widetilde { \Delta } _ { 2 } ) \} + \{ E ( \widetilde { \Delta } _ { 2 } ) - T _ 1 \} } { \surd { ( \Delta _ 1 ) } } . \end{align*}


\begin{align*} \textsf { P } \{ \sum _ { i = 1 } ^ { n } a _ { i } \eta _ { i } \ge t \} \le \exp \big ( - c \min ( \frac { t ^ { 2 } } { K ^ { 2 } \Vert a \Vert _ { 2 } ^ { 2 } } , \frac { t } { K \Vert a \Vert _ { \infty } } ) \big ) , \end{align*}


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


\begin{align*} \frac { b _ n } { b _ i } \geq C _ \epsilon h _ 0 ^ { \epsilon - \eta } \rightarrow \infty \ \ h _ 0 \rightarrow 0 \ \ i = 1 , \cdots , n - 1 . \end{align*}


\begin{align*} \Phi _ { \mathfrak { p } } ^ { ( i ) } ( x _ i , \bar { \xi } _ i , t ) : = \big ( w ^ { ( i ) } _ { \mathfrak { p } - 1 } + u ^ { ( i ) } _ { \mathfrak { p } - 1 } \big ) \ , \overline { V } ^ { ( i ) } \boldsymbol { \boldsymbol { \cdot } } \bar { \nu } _ { \bar { \xi } _ i } , \end{align*}


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


\begin{align*} p _ { v } ( B ) : = \mathbb { P } _ { \mu } \left ( B \cap \left [ v \omega _ { t } ; \omega _ { t } ^ { - 1 } \omega _ { t + 1 } \right ] = \emptyset \mbox { f o r a l l } t \in \mathbb { N } \right ) > 0 . \end{align*}


\begin{align*} \left | \sum _ { f \in \mathcal { M } _ { k , q } } \chi ( f ) \Lambda ( f ) \right | \le \sum _ { f \in \mathcal { M } _ { k , q } } \Lambda ( f ) = q ^ k \end{align*}


\begin{align*} U A _ { s } ( \mathrm { i , j } ) V = I _ { 2 ^ { s - 1 } } \oplus 0 _ { N - 2 ^ { s - 1 } } \hbox { a n d } A _ { s } ( \mathrm { i , j } ) \perp B _ s ( \mathrm { i , j } ) . \end{align*}


\begin{align*} \overline { F } ( k , t + 1 ; p , n ) = \begin{dcases} \frac { ( n - k ) p } { n } \overline { F } ( k - 1 , t ; p , n ) + \frac { n - ( n - k ) p } { n } \overline { F } ( k , t ; p , n ) & \textnormal { i f } \ ; k \le t ; \\ 0 & \textnormal { i f } \ ; k > t . \end{dcases} \end{align*}


\begin{align*} t ^ n = \sum _ { k = 0 } ^ { n } S ( n , k ) ( t ) _ k , \end{align*}


\begin{align*} \Pi _ { x e _ \mathbf { n } } = ( \cdot - x ) ^ \mathbf { n } . \end{align*}


\begin{align*} \tilde { v } ( t ) : = \psi \left ( t \right ) - \phi , \end{align*}


\begin{align*} f _ n ( x ) : = f ( 0 ) + \int _ 0 ^ x f ' ( y ) \chi _ n ( y ) \ , d y , \end{align*}


\begin{align*} \overline { F } ( k , t + 1 , \mathbf { p } _ { n + 1 } ) = \begin{dcases} p _ { k } ^ { } \overline { F } ( k - 1 , t , \mathbf { p } _ { n + 1 } ) + ( 1 - p _ { k } ^ { } ) \overline { F } ( k , t , \mathbf { p } _ { n + 1 } ) & \textnormal { i f } \ ; k \le t ; \\ 0 & \textnormal { i f } \ ; k > t ; \end{dcases} \end{align*}


\begin{align*} f = \sum _ { j = 1 } ^ s b _ j \textbf { X } ^ { \lambda _ j } . \end{align*}


\begin{align*} \exp ( Q _ { j } ) = \exp ( A ) ^ j \exp ( B ) ^ { - \lfloor \varepsilon j \rfloor - d _ { j } } , j \geq 1 , \end{align*}


\begin{align*} \varphi _ x ( \pi ) \coloneqq \begin{cases} w _ 1 w _ 4 x w _ 2 w _ 5 , & x \pi , \\ \pi , & x \pi . \end{cases} \end{align*}


\begin{align*} \aligned | \nabla \tilde { u } _ n | ^ 2 _ 2 + | \nabla \tilde { v } _ n | ^ 2 _ 2 = \int _ { \mathbb { R } ^ N } ( I _ \mu \ast | \tilde { u } _ n | ^ { 2 ^ * _ \mu } ) | \tilde { u } _ n | ^ { 2 ^ * _ \mu } + \int _ { \mathbb { R } ^ N } ( I _ \mu \ast | \tilde { v } _ n | ^ { 2 ^ * _ \mu } ) | \tilde { v } _ n | ^ { 2 ^ * _ \mu } + o _ n ( 1 ) . \endaligned \end{align*}


\begin{align*} v ( z ) | _ { W ^ i } = \begin{pmatrix} p ^ i ( z ) ^ { - 1 } & 0 \\ 0 & p ^ i ( z ) \prod _ { j \neq i , j \in I _ w \cup I _ g } p ^ j ( z ) ^ { a _ { j i } } \prod _ { j \neq i , j \in I _ b } p ^ j ( z ) ^ { a _ { j i } / 2 } \end{pmatrix} \begin{pmatrix} 1 & - \frac { q ^ i _ { - } ( z ) } { q ^ i _ { + } ( z ) } \\ 0 & 1 \end{pmatrix} \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c , d , e ) = 1 \\ a , b , c , d , e \geq 1 } } \left ( \frac { 1 } { 1 - v ^ a w ^ b x ^ c y ^ d z ^ e } \right ) ^ { \frac { 1 } { a ^ q b ^ r c ^ s d ^ t e ^ u } } \end{align*}


\begin{align*} \overline { \Pi } _ { \psi } ( g ) = \Pi _ { \psi } ( g ) m _ { X ^ { \ast } } ( g ) . \end{align*}


\begin{align*} \int _ { \frac { - 1 } { 2 \log T _ k } } ^ { \frac { 1 } { 2 \log T _ k } } p ^ { - i t } \ , d t & = \frac { 2 \sin \bigl ( \frac { \log p } { 2 \log T _ k } \bigr ) } { \log p } , \ , \ , \int _ { \frac { - 1 } { 2 \log T _ k } } ^ { \frac { 1 } { 2 \log T _ k } } p ^ { - 2 i t } \ , d t = \frac { 1 } { \log T _ k } + O \biggl ( \frac { ( \log p ) ^ 2 } { ( \log T _ k ) ^ 3 } \biggr ) . \end{align*}


\begin{align*} \int \limits _ E u \circ f ( x ) | J ( x , f ) | ~ d m ( x ) = \int \limits _ { \mathbb R ^ n \setminus \varphi ( S ) } u ( y ) N _ f ( E , y ) ~ d m ( y ) \end{align*}


\begin{align*} I _ { \omega , \mathbf { c } } ( \Phi ) = \frac { d } { d \lambda } S _ { \omega , \mathbf { c } } ( \Phi ^ { \lambda } ) | _ { \lambda = 1 } \end{align*}


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


\begin{align*} P _ { \phi } = \prod _ { i : v _ i \in V _ E } P _ i \prod _ { i : v _ i \in V _ O } P _ i \prod _ { i : v _ i \in V _ O } P _ i \prod _ { i : v _ i \in V _ E } P _ i . \end{align*}


\begin{align*} N _ 2 \coloneqq ( d ' + 1 ) \cdot \min \left \{ \prod _ { i = 1 } ^ n ( h ' _ i + 1 ) , \binom { n + h ' } { n } \right \} \ , . \end{align*}


\begin{align*} \hat { P } \sum _ { n = 0 } ^ \infty v _ n : = \sum _ { n = 1 } ^ \infty \hat { P } ( \rho _ n ) v _ n , \hat { H } \sum _ { n = 0 } ^ \infty v _ n : = \sum _ { n = 1 } ^ \infty \hat { H } ( \rho _ n ) v _ n , \end{align*}


\begin{align*} ( \prod _ { j = 1 } ^ m a ( j ) \ast g _ { w ( l ) } t ( j ) ) \ast a ( m + 1 ) = ( \prod _ { i = 1 } ^ u d ( i ) \ast h _ l ( q ( i ) ) ) \ast d ( u + 1 ) . \end{align*}


\begin{align*} - \Delta u + V ( x ) u = \lambda u + g ( u ) , \mathbb { R } ^ N , \end{align*}


\begin{align*} & \int _ 0 ^ { \infty } e ^ { - \lambda s } p _ j ( s ) d s = \frac 1 { \lambda } \int _ 0 ^ { \infty } \frac { q ^ 2 e ^ { - \lambda s } } { ( 1 - p e ^ { - \lambda s } ) ^ 2 } \cdot \left ( \frac { 1 - e ^ { - \lambda s } } { 1 - p e ^ { - \lambda { s } } } \right ) ^ { j - 1 } \cdot \lambda e ^ { - \lambda s } d s . \end{align*}


\begin{align*} \mathcal { I } _ { k + 1 } ^ s = \{ T \in \mathcal { I } _ k : s \langle g _ k , \log _ { x _ k } ( y ) \rangle \geq 0 , \forall y \in B ( c _ T , \epsilon r ) \} . \end{align*}


\begin{align*} \mathbf { G } ^ 1 [ \ell ] : = \Omega ( \mathcal { G } _ { \overline { \psi } _ 0 } ^ { 1 , \mathbf { K } } ) , \mathbf { K } ^ 1 [ \ell ] : = \Omega ( \mathbf { K } _ { \overline { \psi } _ 0 , p } ^ 1 ) . \end{align*}


\begin{align*} \nu \left ( \frac { d } { d x } ( f _ { \tilde { \textbf { Q } } } ) \right ) \geq \nu ( g ' ) - b = \nu ( g ' ) - \nu _ i ( g ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} d x _ t & = \tilde { b } ( x _ t ) d t + \sigma ( x _ t ) d W _ t + \tilde { \sigma } ( x _ t ) d \xi _ t + \int _ \mathcal { E } l ( x _ { t - } , e ) \tilde { N } ( d e , d t ) , \ t \in [ 0 , \infty ) , \\ x _ 0 & = x _ 0 , \end{aligned} \right . \end{align*}


\begin{align*} g ( s ) = f ( \alpha _ * , s ) , 0 \leq s \leq 1 . \end{align*}


\begin{align*} \overline g = E _ f ^ { - 1 } s . \end{align*}


\begin{align*} B + v = \big \{ y \in \mathbb R ^ n \colon y = t \omega ( 1 + v ( \omega ) ) \omega \in \mathbb S ^ { n - 1 } , \ , 0 \le t < 1 \big \} . \end{align*}


\begin{align*} T Z _ j ^ * w = T Z _ j ^ * p ( Z ^ * ) v = Y _ j ^ * p ( Y ^ * ) T v = Y _ j ^ * T p ( Z ^ * ) v = Y _ j ^ * T w . \end{align*}


\begin{align*} a ( s , s ' ) & = z ( s ) \big ( 1 + \mathcal { O } ( s \sigma ^ 2 ) \big ) + \widetilde z ( s ' ) \big ( 1 + \mathcal { O } ( s ' h ^ 2 ) \big ) \\ & = \frac { 1 } { 2 } \sigma ^ 2 + \frac { 5 } { 2 } h ^ 2 + \mathcal { O } ( s \sigma ^ 2 + s ' h ^ 2 ) \\ b ( s , s ' ) & = \zeta ( s ) \big ( 1 + \mathcal { O } ( s \sigma ^ 2 ) \big ) + \widetilde \zeta ( s ' ) \big ( 1 + \mathcal { O } ( s \sigma ^ 2 ) \big ) \\ & = - i ( K - K _ 1 + K _ 2 - K _ 3 + K _ 4 - K _ 5 ) + \mathcal { O } ( s \sigma ^ 2 ) , \\ c ( s , s ' ) & = \mathcal { O } ( s ) . \end{align*}


\begin{align*} \vartheta _ L ( f ; p ) ( g ) = ( f ( \tau ) , \Theta _ L ( \tau , g ; p ) ) _ \tau \end{align*}


\begin{align*} \langle g , B g \rangle = \sum _ { \{ i , j \} \in E } ( - M _ { i j } ) f _ k ( i ) f _ k ( j ) ( g ( i ) - g ( j ) ) ^ 2 = \sum _ { \{ i , j \} \in E } a _ { i j } ( g ( i ) - g ( j ) ) ^ 2 \end{align*}


\begin{align*} F ( [ \zeta ] ) = \langle f , \zeta \rangle \forall \ [ \zeta ] \in [ \mathcal { X } ] . \end{align*}


\begin{align*} \mathcal { N } ( [ \zeta ^ * ] , [ K ] ) = \{ \tilde { F } \in [ \mathcal { X } ] ' : \ - \tilde { F } ( [ \zeta ] - [ \zeta ^ * ] ) \geq 0 \forall \ [ \zeta ] \in [ K ] \} . \end{align*}


\begin{align*} \left \Vert \frac { d \tau ( y ) ^ { - 1 } ( g \nu _ { P } ) ^ { y } } { d \bar { m } _ { K \cap Q } } \right \Vert _ { \infty } = \left \Vert \frac { d ( g \nu _ { P } ) ^ { y } } { d \left ( \tau ( y ) . \bar { m } _ { K } \right ) ^ { y } } \right \Vert _ { \infty } \le \left \Vert \frac { d ( g \nu _ { P } ) } { d \left ( \tau ( y ) . \bar { m } _ { K } \right ) } \right \Vert _ { \infty } \left \Vert \frac { d \left ( \tau ( y ) . \bar { m } _ { K } \right ) } { d \left ( g \nu _ { P } \right ) } \right \Vert _ { \infty } . \end{align*}


\begin{align*} \norm { \phi } _ X \leq C \norm { g - \sum _ { j = 0 } ^ { N } \dfrac { \displaystyle \int _ { \R ^ N } g Z _ j } { \displaystyle \int _ { \R ^ N } H _ j Z _ j } H _ j } _ Y . \end{align*}


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


\begin{align*} \left ( Z _ i ( u ) , \frac { 1 } { T _ i ( v ) } \right ) _ { u \geq 0 , v \geq 0 , i \in V } = \left ( \theta _ i Z ^ * _ i ( u ) , \frac { 1 } { T _ i ^ { \infty } } + \frac { 1 } { \theta _ i ^ 2 T _ i ^ * ( v ) } \right ) _ { u \geq 0 , v \geq 0 , i \in V } . \end{align*}


\begin{align*} \log ( 1 - x ) = - \sum _ { n = 1 } ^ \infty \frac { x ^ n } { n } . \end{align*}


\begin{align*} \frac { \partial \phi } { \partial t } + a \frac { \partial \phi } { \partial x } = 0 , \end{align*}


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


\begin{align*} \textrm { $ ( $ { \sc w o t } $ ) $ } \lim _ { n \to \infty } T ^ n = \begin{bmatrix} P \oplus 0 & 0 \\ 0 & A \end{bmatrix} . \end{align*}


\begin{align*} - a & : = ( - 1 ) \cdot a , \\ a \cdot 0 & = 0 \cdot a : = 0 , \\ a + 0 & = 0 + a = \{ a \} , \\ a + ( - a ) & = \Gamma ( R ) , \\ \mbox { f o r } & a , b \ne 0 , a \ne - b \mbox { d e f i n e } \\ a + b & : = \{ c \in \Gamma ( R ) : \mbox { t h e r e e x i s t } d \in G ( R ) \mbox { s u c h t h a t } \\ & a \cdot b = c \cdot d \in G ( R ) \mbox { a n d } l _ R ( a ) l _ R ( b ) = l _ R ( c ) l _ R ( d ) \in R _ 2 \} . \end{align*}


\begin{align*} \psi ( 0 ) = \phi + v _ 0 , \end{align*}


\begin{align*} \begin{array} { l l } \displaystyle \sum _ { i = 1 } ^ 3 \left ( C _ i ( \lambda ' _ i \rho _ 0 ) - C _ i ( \lambda _ i '' \rho _ 0 ) \right ) ( \lambda _ i ' - \lambda _ i '' ) > 0 \\ \displaystyle ( \lambda ' _ 1 , \lambda ' _ 2 , \lambda ' _ 3 ) \neq ( \lambda '' _ 1 , \lambda '' _ 2 , \lambda '' _ 3 ) \ \mbox { c o n v e x t r i p l e s a n d f o r a n y } \rho _ 0 > 0 . \end{array} \end{align*}


\begin{align*} \left \{ Q _ { 2 ^ { j _ { 0 } } \rho _ { z _ { i } } } ( z _ { i } ) \right \} _ { i \in \mathbb { N } } D _ { \kappa \lambda } \subset \bigcup _ { z _ { 0 } \in D _ { \kappa \lambda } } Q _ { \rho _ { z _ { 0 } } } ( z _ { 0 } ) \subset \bigcup \limits _ { i = 1 } ^ { \infty } Q _ { 5 \times 2 ^ { j _ { 0 } } \rho _ { z _ { i } } } ( z _ { i } ) . \end{align*}


\begin{align*} v _ { \gamma , 2 } ( x ) = ( \log 2 + \log ( x - \tfrac 1 2 ) ) g _ { \beta , 2 } ( x ) \forall x \in ( \tfrac 1 2 , 1 ] . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b ) = 1 \\ a , b \geq 1 } } \left ( 1 - \frac { z ^ b } { \phi ^ { 2 a } } \right ) ^ { \frac { b ^ { 2 } } { a ^ 3 } } = \exp \left \{ \left ( \frac { 4 } { 5 } \zeta ( 3 ) - \frac { 2 } { 1 5 } \pi ^ 2 \log \phi + \frac { 2 } { 3 } ( \log \phi ) ^ 3 \right ) \left ( \frac { z ( 1 + z ) } { ( 1 - z ) ^ 3 } \right ) \right \} , \end{align*}


\begin{align*} I = P _ { V } x ^ * x | _ V & = P _ V x ^ * P _ V x | _ V + P _ V x ^ * P _ { V ^ \perp } x | V \\ & = | \phi ( x ) | ^ 2 I + z ^ * z \end{align*}


\begin{align*} w ( z ) = \frac { h ( z ) - 1 } { h ( z ) + 1 } . \end{align*}