\begin{align*} W _ { \Omega } : = \left \{ \left ( x , \left ( L _ { x } \omega _ { 1 } , L _ { x } \omega _ { 2 } , \dots \right ) \right ) : x \in X , ( \omega _ { 1 } , \omega _ { 2 } \dots ) \in G ^ { \mathbb { N } } \right \} = \bigsqcup _ { x \in X } \{ x \} \times ( L _ { x } \backslash G ) ^ { \mathbb { N } } \end{align*}


\begin{align*} \sum _ { x \in \Z ^ { d } } \sum _ { w \in \mathcal { W } _ n ^ { S A W } ( 0 , x ) } \prod _ { j = 0 } ^ { n - 1 } D ( w _ { j } , w _ { j + 1 } ) \leq \left ( \sum _ { z \neq 0 } D ( 0 , z ) \right ) ^ { n } , \end{align*}


\begin{align*} Q \simeq P \oplus \begin{pmatrix} I & 0 \\ R & 0 \end{pmatrix} . \end{align*}


\begin{align*} \Phi _ t ^ \omega = \{ x \in \mathbb { R } ^ d : B ( x , r ( t ) ) \subseteq K ^ c ( \omega ) \} , \quad \ : \ : \widehat { \Phi } _ t ^ \omega = \Phi _ t ^ \omega \cap [ - k t , k t ] ^ d . \end{align*}


\begin{align*} \widetilde { \mathcal { M } } _ { \omega , c _ * } : = \bigcup _ { | c | \le c _ * } \mathcal { M } _ { \omega , c } , \end{align*}


\begin{align*} \lim _ { \Delta \to 0 } E \big | \bar { S } ^ k _ { j , + , \Delta } ( t ) - \bar { S } ^ k _ { j , + } ( t ) \big | = 0 , t \geq 0 . \end{align*}


\begin{align*} \begin{cases} P ( ( - \Delta ) ^ s ) _ 1 \tilde { u } + q \tilde { u } = \left ( \alpha _ m ^ 2 - \alpha _ m ^ 1 \right ) ( - \Delta ) ^ { s _ m } u _ 2 & \Omega , \\ \mathcal { L } \tilde { u } = 0 & \Omega ^ c , \\ \tilde { u } = 0 & \Omega ^ c . \end{cases} \end{align*}


\begin{align*} \lim _ { h \to + \infty } \left ( \frac { 1 } { \gamma _ h \sigma _ h } \right ) ^ { \frac { p _ h ( y ) } { p _ h ^ 0 } - 1 } = 1 . \end{align*}


\begin{align*} \lambda ^ { - 2 \gamma } \ ! & \int _ { \mathbb { R } ^ 4 } \ ! f _ { 1 , m _ 1 } \left ( x + \widetilde { P } _ 1 ( t + s ) , y \right ) \overline { f _ { 1 , m _ 1 } \left ( x + \widetilde { P } _ 1 ( t ) , y \right ) } \\ & f _ { 2 , m _ 2 } \left ( x , y + \widetilde { P } _ 2 ( t + s ) \right ) \overline { f _ { 2 , m _ 2 } \left ( x , y + \widetilde { P } _ 2 ( t ) \right ) } \widetilde { \zeta } _ \mathbf { m } ( x , y , t , s ) \ , \mathrm { d } x \mathrm { d } y \mathrm { d } t \mathrm { d } s , \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c u ) ^ n = \psi _ \varepsilon ( \cdot , u ) \ , \omega ^ n & \textnormal { i n } & \Omega \\ u = \phi & \textnormal { o n } & \partial \Omega . \end{array} \right . \end{align*}


\begin{align*} ( n - 1 ) T ( 1 ) f + n B ( A ( f ) , A ( 1 ) ) = 0 \end{align*}


\begin{align*} \begin{aligned} f ( x y ) & = ( f x ) y - ( f y ) x \\ ( x y ) f & = ( x f ) y + x ( y f ) \\ x ( f y ) & = x ( y f ) \end{aligned} \end{align*}


\begin{align*} R ( t ) + 1 = R _ { \ell ( t ) } + 1 \asymp R _ 0 [ \log ( \ell ( t ) ) ] ^ { 1 / d } \asymp R _ 0 [ \log t ] ^ { 1 / d } , \end{align*}


\begin{align*} u ( t ) = e ^ { - t H ^ { \beta } } u _ 0 + \int _ 0 ^ t e ^ { - ( t - s ) H ^ { \beta } } \ , f ( u ( s ) ) \ , d s \end{align*}


\begin{align*} a _ { j } & = \dfrac { \cosh ( \alpha x _ { j - 1 } ) } { \cosh ( \alpha x _ j ) } E _ { j - 1 } = \dfrac { h _ j } { 6 } + O \left ( ( \alpha \overline { h } ) ^ 2 \right ) ; \alpha \to 0 , \\ c _ { j } & = \dfrac { \cosh ( \alpha x _ { j + 1 } ) } { \cosh ( \alpha x _ j ) } E _ { j } = \dfrac { h _ { j + 1 } } { 6 } + O \left ( ( \alpha \overline { h } ) ^ 2 \right ) ; \alpha \to 0 , \\ d _ { j } & = \dfrac { y _ { j + 1 } - y _ j } { h _ { j + 1 } } - \dfrac { y _ { j } - y _ { j - 1 } } { h _ j } . \end{align*}


\begin{align*} \aligned \xi _ F ( s ) & = s ^ { m _ F } ( s - 1 ) ^ { m _ F } Q ^ s \prod _ { j = 1 } ^ { r } \Gamma ( \lambda _ { j } s + \mu _ { j } ) \ , F ( s ) , \endaligned \end{align*}


\begin{align*} \varphi : \mathbb { T } \to \bigoplus _ { n = 0 } ^ { 2 k - 2 } \prod _ { [ f ] \in \mathcal { E } _ n } \mathbb { T } \slash \mathrm { A n n } _ { [ f ] _ n } . \end{align*}


\begin{align*} L _ X = r _ X \circ ( I d - \iota _ X ) \circ \tau _ X , L _ X ( v ) = \sum _ { h \in T _ v X } ( v - r _ X ( \iota _ X ( h ) ) ) . \end{align*}


\begin{align*} N ( U ) = \frac { 1 } { 3 } ( K _ { \omega , \mathbf { c } } ( U ) - L _ { \omega , \mathbf { c } } ( U ) ) < - 2 \mu _ { \omega , \mathbf { c } } . \end{align*}


\begin{align*} \left \Vert \partial _ { t } ^ { s } \widetilde { u } \right \Vert _ { H ^ { 2 , 1 } \left ( Q _ { \varepsilon , T } \right ) } , \left \Vert \partial _ { t } ^ { s } \widetilde { m } \right \Vert _ { H ^ { 2 , 1 } \left ( Q _ { \varepsilon , T } \right ) } , \left \Vert \widetilde { k } \right \Vert _ { L _ { 2 } \left ( \Omega \right ) } \leq C \delta ^ { 1 - \rho } , \forall \delta \in \left ( 0 , \delta _ { 0 } \right ) , s = 0 , 1 , 2 . \end{align*}


\begin{align*} M _ b ( \theta ) = \mathbb { E } _ { \Phi } \{ P _ s ^ b ( \theta ) \} \end{align*}


\begin{align*} \Psi = \begin{bmatrix} \psi ( x _ 1 ) & \psi ( x _ 2 ) & \cdots & \psi ( x _ n ) \\ \psi ( x _ 1 ) u _ { 1 , 1 } & \psi ( x _ 2 ) u _ { 1 , 2 } & \cdots & \psi ( x _ n ) u _ { 1 , n } \\ \vdots & \vdots & \ddots & \vdots \\ \psi ( x _ 1 ) u _ { m , 1 } & \psi ( x _ 2 ) u _ { m , 2 } & \cdots & \psi ( x _ n ) u _ { m , n } \end{bmatrix} , \end{align*}


\begin{align*} \mathrm { d i v } ( \nabla V \ , V ) = \partial _ j \Big ( V ^ i \ , \partial _ i V ^ j \Big ) = \partial _ j V ^ i \ , \partial _ i V ^ j + V ^ i \ , \partial _ j \partial _ i V ^ j = \mathrm { t r } ( ( \nabla V ) ^ 2 ) + V ^ i \ , \partial _ j \partial _ i V ^ j \ , , \end{align*}


\begin{align*} { \bf { b } } ( p _ s ) : = b _ { e } - b _ { f } . \end{align*}


\begin{align*} \begin{aligned} B ^ m = & \left . ( - 1 ) ^ m \rho ^ { m + 2 } { n - m \choose k - m } \dfrac { k + n - 2 m } { n - m } \left [ \frac { \phi '^ { k - m } } { D ^ { k + 1 } } \phi ^ { n - m - 1 } \right ] \right | _ { u = | \nabla u | ^ 2 = 0 } \\ = & ( - 1 ) ^ m \rho ^ { m + 2 } { n - m \choose k - m } \dfrac { k + n - 2 m } { n - m } \phi '^ { k - m } ( \rho ) \phi ^ { n - m - 1 } ( \rho ) . \end{aligned} \end{align*}


\begin{align*} \gamma ' + \varphi \gamma = 0 \sigma ' + \varphi \sigma + q = 0 . \end{align*}


\begin{align*} ( A f ) ( x ) = - \frac { \sqrt { 2 } } { 2 } \Delta _ { h } f ( x ) = \frac { \sqrt { 2 } } { 2 } \left ( f ( x ) - f ( x + h ) \right ) \end{align*}


\begin{align*} \sum _ { s = 1 } ^ { m } \ , \mu _ s \ , ^ { c } _ { 0 } { D } ^ { \alpha _ s } _ { t } u ( x , t ) - \Delta u ( x , t ) & = f ( x , t ) & \mbox { i n } & \Omega \times ( 0 , T ] , \\ u ( x , t ) & = 0 & \mbox { o n } & \partial \Omega \times ( 0 , T ] , \\ u ( x , 0 ) & = u _ 0 ( x ) & \mbox { i n } & \Omega , \end{align*}


\begin{align*} \overline { \xi } ( x ) = \sum _ { \lambda = 0 } ^ { \infty } \alpha _ { \lambda + 1 } ^ { ( 1 ) } x ^ \lambda . \end{align*}


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


\begin{align*} \operatorname { W e i g h t } ( \pi ) : & = \operatorname { s i g n } ( \pi ) \ , a _ { 1 , \pi _ 1 } \cdots a _ { n , \pi _ n } \ ; , \end{align*}


\begin{align*} z _ h ^ { ( s ) } : = t _ h \widetilde { z } _ h ^ { ( s ) } . \end{align*}


\begin{align*} H _ { \alpha \parallel \beta } \left ( \mathcal { P } \right ) : = \sum _ { A \in \mathcal { P } } \alpha ( A ) \log \frac { \alpha ( A ) } { \beta ( A ) } . \end{align*}


\begin{align*} \mathrm { f p } ( T ) & : = \{ v \in V ( T ) : \mathrm { d e s c d } ( v ) = \mathrm { d e p t h } ( v ) \} \\ \mathrm { a d t } ( T ) & : = \{ ( v , w ) \in V ( T ) : \mathrm { d e s c d } ( w ) = \mathrm { d e p t h } ( v ) , w v \} . \end{align*}


\begin{align*} & \mathrm { d i v } _ T \Big ( h \ , \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \Big ) \ , h \ , \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \cdot \nu \\ & - h ^ 2 \ , \nabla _ T \Big [ h \ , \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \Big ] \ , \Big [ \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \Big ] _ T \cdot \nu = h \ , H ( \nu ) \ , H ^ 2 ( \nabla w ) \ , \mathrm { t r } \ , \mathcal { B } ^ H \end{align*}


\begin{align*} \lim _ { \mu \to 0 } \frac { n + \mu } { \mu } C _ n ^ \mu ( x ) = \epsilon _ n T _ n ( x ) , \end{align*}


\begin{align*} d \overline { \Pi } _ { \psi } ( e ^ + ) f ( [ \epsilon , x ] ) & = \frac { d } { d t } \Big \{ \Pi _ { \psi } ( [ e ^ { t e ^ + } , 1 ] ) f ( [ \epsilon , x ] ) \Big \} \Big | _ { t = 0 } \\ & = \frac { d } { d t } \Big \{ e ^ { \pi i \epsilon x ^ 2 t } f ( [ \epsilon , x ] ) \Big \} \Big | _ { t = 0 } \\ & = \pi i \epsilon x ^ 2 f ( [ \epsilon , x ] ) . \end{align*}


\begin{align*} \triangle _ 1 = \varepsilon ^ { \frac { 1 } { 8 ( d _ 1 + d _ 2 ) - 1 7 } } \textrm { a n d } \triangle _ 2 = \varepsilon ^ { \frac { 4 ( d _ 1 + d _ 2 ) - 1 0 } { 8 ( d _ 1 + d _ 2 ) - 1 7 } } . \end{align*}


\begin{align*} & 0 \geq - \int _ { \Omega } \langle \nabla u , \nabla u ^ - \rangle \ , d x = \int _ { \Omega } | \nabla u ^ - | ^ 2 \ , d x \geq 0 . \end{align*}


\begin{align*} 0 \leq | \nabla v _ n | \leq | \nabla u _ \mu ^ \bigstar | & \lim _ { n \to \infty } | \nabla v _ n - \nabla u _ \mu ^ \bigstar | ( x ) = 0 . \end{align*}


\begin{align*} | \mathcal { A } ( r _ 1 , \dots , r _ n ) | \le \frac { q ^ { 2 n } } { \prod _ { j = 1 } ^ t p ^ { r _ j n } _ j } \ , . \end{align*}


\begin{align*} \begin{cases} | \alpha Y - \beta X | & = d ( v _ i , v _ { i - 1 } ) - 1 , \\ \gamma Y - \delta X & = - 1 . \end{cases} \end{align*}


\begin{align*} h ( v ) = \begin{cases} 0 & v \in [ - 1 , 0 ) , \\ 2 ( b + c v ^ { 2 } ) & v \in [ 0 , 1 ] . \end{cases} \end{align*}


\begin{align*} \Omega _ { n , k } : = \left \{ ( S , \underline { S } ) ^ \pm : \ S \in \binom { [ n ] } { k } \right \} . \end{align*}


\begin{align*} ( U ( A + B ) V ) ^ { * } ( U C V ) = 0 \hbox { a n d } ( U ( A + B ) V ) ( U C V ) ^ * = 0 , \end{align*}


\begin{align*} \begin{cases} - \Delta v + q v = 0 & \Omega , \\ v = f & \partial \Omega \end{cases} \end{align*}


\begin{align*} \mathrm { d e g } ( \nabla V ) + \mathrm { d e g } ( F ) = \mathrm { d e g } ( V ) - 1 + \mathrm { d e g } ( F ) \geq \mathrm { d e g } ( V ) , \end{align*}


\begin{align*} P \Phi _ t + ( A \Phi ) _ x + A ^ T \Phi _ x + ( B \Phi ) _ y + B ^ T \Phi _ y = 0 , \end{align*}


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


\begin{align*} U _ { n } ^ { - 1 } \Sigma = \left [ \begin{array} { c | c } U _ { n - 1 } ^ { - 1 } & \mathbf { 0 } \\ \hline \mathbf { 0 ' } & 1 \end{array} \right ] , \Sigma ^ { - 1 } U _ { n } = \left [ \begin{array} { c | c } U _ { n - 1 } & \mathbf { 0 } \\ \hline \mathbf { 0 ' } & 1 \end{array} \right ] . \end{align*}


\begin{align*} \begin{dcases} L v = k & \\ v = 0 & , \end{dcases} \end{align*}


\begin{align*} \begin{array} { c } \sum _ { k = 1 } ^ m \left ( \beta _ { i , k } - \beta _ { j , k } \right ) + \alpha _ { i j } + \gamma _ i = 0 \\ \implies \\ \gamma _ i = \sum _ { k = 1 } ^ m \left ( \beta _ { j , k } - \beta _ { i , k } \right ) - \alpha _ { i j } \leq 2 m p _ { \max } , \end{array} \end{align*}


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


\begin{align*} t _ 4 = t v _ 1 \quad c ^ 2 t ^ { k - 1 } t _ 1 t _ 2 t _ 3 t _ 4 = p q . \end{align*}


\begin{align*} \theta ( a , b ) ( \tau ) = \Theta ( a \tau + b , \tau ) \end{align*}


\begin{align*} \| V _ n - V \| _ { \mathcal { H } ^ 1 } ^ 2 & \lesssim L _ { \omega , \mathbf { c } } ( V _ n - V ) \\ & = L _ { \omega , \mathbf { c } } ( V _ n ) - L _ { \omega , \mathbf { c } } ( V ) - ( L _ { \omega , \mathbf { c } } ( V _ n ) - L _ { \omega , \mathbf { c } } ( V _ n - V ) - L _ { \omega , \mathbf { c } } ( V ) ) \\ & \rightarrow 0 . \end{align*}


\begin{align*} & \left | \frac { \delta F } { \delta \nu } ( \nu _ t , \mu _ t , x ) + \frac { \sigma ^ 2 } { 2 } \log \left ( \frac { \nu _ t ( x ) } { \pi ( x ) } \right ) - \frac { \sigma ^ 2 } { 2 } \operatorname { D _ { K L } } ( \nu _ t | \pi ) \right | \\ & \leq 3 C _ { \nu } + \frac { \sigma ^ 2 } { 2 } \left ( \max \{ | \log r _ { 1 , \nu } | , \log R _ { 1 , \nu } \} + 2 \log R _ { \nu } \right ) = : C _ { V , \nu } , \end{align*}


\begin{align*} \frac { \int _ 0 ^ \infty e ^ { - \lambda s } p _ j ( s ) d s } { \int _ 0 ^ \infty e ^ { - \lambda s } ( 1 - p _ 0 ( s ) ) d s } & = \begin{cases} \dfrac 1 { j ( j + 1 ) } , & p = 0 , \\ \displaystyle - \frac { p } { \log ( q ) } \int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } ( 1 - y ) y ^ { j - 1 } d y , & 0 < p < 1 , \end{cases} \end{align*}


\begin{align*} \dot \gamma ( t ) = \sum _ { i = 1 } ^ k u _ i ( t ) X _ i ( \gamma ( t ) ) , \qquad t \in [ 0 , 1 ] . \end{align*}


\begin{align*} \det ( I - \mathcal { P } _ \gamma ^ n | _ { E _ s } ) > 0 , ( - 1 ) ^ s \det ( I - \mathcal { P } _ \gamma ^ n | _ { E _ s } ) = \det ( I - \mathcal { P } _ \gamma ^ { - n } | _ { E _ s } ) \det ( \mathcal { P } _ \gamma ^ n | _ { E _ s } ) > 0 . \end{align*}


\begin{align*} \theta ( b ) = P _ H \pi ( b ) | _ H , b \in B \end{align*}


\begin{align*} | \triangle _ { \rm D } ^ { - \alpha / 2 } f ( x ) | & = \bigg | \int _ { \mathbb R ^ N } K _ \alpha ( x , y ) f ( y ) d \omega ( y ) \bigg | \\ & \leq \int _ { d ( x , y ) \leq s } \frac { d ( x , y ) ^ { 2 + \alpha } } { \| x - y \| ^ 2 } \frac 1 { \omega ( B ( x , d ( x , y ) ) ) } | f ( y ) | d \omega ( y ) \\ & \quad + \int _ { d ( x , y ) > s } \frac { d ( x , y ) ^ { 2 + \alpha } } { \| x - y \| ^ 2 } \frac 1 { \omega ( B ( x , d ( x , y ) ) ) } | f ( y ) | d \omega ( y ) \\ & = : J _ 1 + J _ 2 . \end{align*}


\begin{align*} u \sim \begin{cases} u - e _ i + e _ { m + j } & \\ u + e _ i - e _ { m + j } & \end{cases} \end{align*}


\begin{align*} c & = \sum \limits _ { ( \rho _ { 1 } , \dots , \rho _ { m } ) \in \rho } T _ { \alpha } ( f _ { 1 } ^ { \rho _ { 1 } } , \dots , f _ { m } ^ { \rho _ { m } } ) ( x ) , \end{align*}


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


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


\begin{align*} \begin{array} { r l } g _ { w ( l ) } ( y ) = & \prod \limits _ { i = 1 } ^ n h _ { d _ i } ( n y + i ) \\ = & ( \prod \limits _ { p = 1 } ^ r ( \prod \limits _ { i \in L ( p ) } h _ { b _ i } ( n y + i ) ) \ast h _ l ( n y + s ( p ) ) ) \ast \prod \limits _ { i \in { L ( r + 1 ) } } h _ { b _ i } ( n y + i ) \\ = & ( \prod \limits _ { p = 1 } ^ r c _ y ( p ) \ast h _ l ( z _ y ( p ) ) ) \ast c _ y ( r + 1 ) \end{array} \end{align*}


\begin{align*} 4 D _ a = 4 D _ b = 4 D _ c = D _ a + D _ b + D _ c = 0 . \end{align*}


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


\begin{align*} x ^ n + x ^ k y ^ l + y ^ m = 0 , \end{align*}


\begin{align*} \begin{aligned} J \subset \Lambda _ { 2 ^ { \frac { 2 } { s } } \times 1 0 \sqrt { n } \mathfrak { l } \left ( \mathcal { H } \right ) } ( t _ { 0 } ) . \end{aligned} \end{align*}


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


\begin{align*} f _ \epsilon ^ { ' } \Big ( V _ m ( \mu _ m ^ i y + \xi _ m ) \Big ) = & f _ \epsilon ^ { ' } \Big ( P U _ { \mu _ { i n } , \xi _ { i n } } ( \mu _ m ^ i y + \xi _ m ) + \sum \limits _ { i = 1 , j \neq i } ^ k P U _ { \mu _ { j m } , \xi _ { j m } } ( \mu _ m ^ i y + \xi _ m ) \Big ) \\ = & f _ \epsilon ^ { ' } \Big ( ( \mu _ m ^ { i } ) ^ { - \frac { n - 2 } { 2 } } U ( y - \sigma _ { i n } ) + U _ { \mu _ { j m } , \xi _ { j m } } ( \mu _ m ^ i y + \xi _ m ) + o ( 1 ) \Big ) , \end{align*}


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


\begin{align*} p ( z ) = \frac { 1 + ( \overline { \tau _ 2 } \tau _ 3 + \overline { \tau _ 1 } \tau _ 2 + \tau _ 1 ) z + ( \overline { \tau _ 1 } \tau _ 3 + \tau _ 1 \overline { \tau _ 2 } \tau _ 3 + \tau _ 2 ) z ^ 2 + \tau _ 3 z ^ 3 } { 1 + ( \overline { \tau _ 2 } \tau _ 3 + \overline { \tau _ 1 } \tau _ 2 - \tau _ 1 ) z + ( \overline { \tau _ 1 } \tau _ 3 - \tau _ 1 \overline { \tau _ 2 } \tau _ 3 - \tau _ 2 ) z ^ 2 - \tau _ 3 z ^ 3 } , \ ; \ ; z \in \mathbb { D } . \end{align*}


\begin{align*} L i _ n ( z ) = \sum _ { r = 1 } ^ { \infty } \frac { z ^ r } { r ^ n } , \ ; \ ; | z | \leq 1 . \end{align*}


\begin{align*} g _ x g _ y ^ { - 1 } & = g _ { x + z } g _ { y + z } ^ { - 1 } , \\ g _ 0 g _ k ^ { - 1 } & = \gamma ^ k , \\ g _ x \gamma & = \gamma ^ \alpha g _ x \end{align*}


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


\begin{align*} f ( x ) & = - a x + \sum _ { k = 1 } ^ \infty \frac { x ^ k \log ^ k ( 1 + a ) } { k ! } , \end{align*}


\begin{align*} u ^ 3 v ^ 2 = w ^ 3 + 1 \end{align*}


\begin{align*} \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda \ , d \widetilde { \pi } = \int _ { \mathcal { V } } g \ , d \gamma - \int _ { \mathcal { S } _ { 1 } \times \mathcal { S } _ { 1 } } \lambda _ 1 c _ { 1 } \ , d \nu _ { 1 } - \int _ { \mathcal { S } _ { 2 } \times \mathcal { S } _ { 2 } } \lambda _ 2 c _ { 2 } \ , d \nu _ { 2 } < \infty , \end{align*}


\begin{align*} \pi ( f ) \varphi = \alpha _ ! ( f \otimes \varphi ) & & f \in C _ c ^ \infty ( \R ) , \varphi \in C _ c ^ \infty ( M ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { 3 } h _ i ^ 2 ( 0 ) \ , \mathrm { v } _ i = 0 \end{align*}


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


\begin{align*} \varphi _ K \circ R = \sigma \circ \varphi _ K \underline { \varphi } _ K \circ R = \sigma \circ \underline { \varphi } _ K . \end{align*}


\begin{align*} 0 _ { n } = \sum \limits _ { i = 1 } ^ { m } \left \Vert a _ { i } \right \Vert ^ { 2 } \nabla d _ { S _ { i } } ^ { 2 } \left ( x ^ { 0 } \right ) = 2 \sum \limits _ { i = 1 } ^ { m } \left \Vert a _ { i } \right \Vert d _ { S _ { i } } \left ( x ^ { 0 } \right ) a _ { i } = 2 \sum \limits _ { i = 1 } ^ { m } \left [ a _ { i } ^ { \prime } x ^ { 0 } - \overline { b } _ { i } \right ] _ { + } a _ { i } ; \end{align*}


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


\begin{align*} \sum _ { j = j _ 0 } ^ { j _ 1 } \mathfrak { E } _ p ( \partial u , B _ { j + 1 } ) \leq \frac { 1 } { 3 2 } \left ( \frac { \sigma } { 2 } \right ) ^ { \frac { d } { p } } \lambda . \end{align*}


\begin{align*} \tan ( \pi y ) = - \frac { y } { p } , \ , \ , \ , \ , \tan ( \pi y ) = - \frac { y } { p ^ { \prime } } . \end{align*}


\begin{align*} \Omega _ { s , t } = - \Omega _ { t , s } . \end{align*}


\begin{align*} \big ( \mathrm { d i v } \ , V \big ) ^ 2 = \mathrm { t r } \big ( ( \nabla V ) ^ 2 \big ) + \mathrm { d i v } \Big ( V \ , \mathrm { d i v } V - \nabla V \ , V \Big ) \ , . \end{align*}


\begin{align*} H _ i ( ( 2 , 2 n ) , \mathbb { Z } ) = \begin{cases} H _ i ( G ( 2 , 2 n ) , \mathbb { Z } ) , & i \leq 4 n - 5 , \\ H _ { i + 2 } ( G ( 2 , 2 n ) , \mathbb { Z } ) , & i \geq 4 n - 4 . \end{cases} \end{align*}


\begin{align*} \exp \left [ 2 \lambda \left ( 3 b ^ { 2 } / 2 + \alpha \left ( T / 2 - \varepsilon \right ) ^ { 2 } \right ) \right ] = \exp \left ( 2 \lambda d \right ) , \end{align*}


\begin{align*} w _ b = \Phi _ { 0 , b } + \widetilde { \Psi } _ b . \end{align*}


\begin{align*} f _ j Q _ i ' = a _ j + b _ j Q _ i \end{align*}


\begin{gather*} \sum _ { \{ S \colon a \cup b \to \{ U , V , U V \} \ , | \ , S ( b ) = \{ U \} \} } \Gamma _ S - \sum _ { \{ S ' \colon a \cup b \to \{ U , V , U V \} \ , | \ , S ' ( a ) = \{ U \} \} } \Gamma _ { S ' } . \end{gather*}


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


\begin{align*} \texttt { r s b o u n d } : = c \sqrt { \pi } / b a . \end{align*}


\begin{align*} \sum _ { p = 1 } ^ n u ^ { p \bar p } \leq A _ 1 g ^ { - 1 } . \end{align*}


\begin{align*} \sigma _ { j + 1 } = \inf _ { r > 0 } \Big \{ \tilde Z _ { \sigma _ { j } + 2 r } \in \{ \sqrt { m } - 1 , \sqrt { m } , \eta ( m ) - 1 , \eta ( m ) \} \Big \} \end{align*}


\begin{align*} \left . \begin{array} { l l } S ( u , v ) = & \displaystyle \dfrac { a _ 2 } { 2 m _ 1 } \int _ { \mathbb { R } ^ N } | \nabla u | ^ 2 d x + \dfrac { a _ 1 } { 4 m _ 2 } \int _ { \mathbb { R } ^ N } | \nabla v | ^ 2 d x \\ \\ & \displaystyle + a _ 2 \omega _ 1 \int _ { \mathbb { R } ^ N } | u | ^ 2 d x + \frac { a _ 1 } { 2 } \omega _ 2 \int _ { \mathbb { R } ^ N } | v | ^ 2 d x + a _ 1 a _ 2 \int _ { \mathbb { R } ^ N } v u ^ 2 d x . \end{array} \right . \end{align*}


\begin{align*} \zeta _ \delta = \frac { v } { w _ \delta } \ , \Omega \ , . \end{align*}


\begin{align*} \begin{aligned} & \leq C \mathbb { E } \int _ 0 ^ \infty \big ( | y _ n - y | ^ 2 + | z _ n - z | ^ 2 + | \tilde { z } _ n - \tilde { z } | ^ 2 + | | \gamma _ n - \gamma | | ^ 2 \big ) e ^ { - \beta s } d s \rightarrow 0 , \ \ n \rightarrow \infty , \end{aligned} \end{align*}


\begin{align*} \lim _ { r \to 0 ^ + } \frac { A ( r ) ^ 2 } { A ' ( r ) } \cdot \left [ \mathcal W ( r ) ^ { p - 1 - \alpha } \right ] ' & = \lim _ { r \to 0 ^ + } A ( r ) \cdot \left [ Q _ 1 ( \mathcal V ( r ) , v _ 0 ( r ) ) - \mathcal W ( r ) ^ { p - 1 - \alpha } \right ] \\ & = \lim _ { r \to 0 ^ + } \left [ r ^ \delta u _ 0 ' ( r ) ^ { p - 1 - \alpha } Q _ 1 ( \mathcal V ( r ) , v _ 0 ( r ) ) - r ^ \delta u ' ( r ) ^ { p - 1 - \alpha } \right ] \end{align*}


\begin{align*} u ^ i ( z ) = \zeta _ i + \ln ' ( p ^ i ( z ) ) . \end{align*}


\begin{align*} L _ { \omega , \mathbf { c } } ( V _ n ) = L _ { \omega , \mathbf { c } } ( U _ n ) \rightarrow 6 \mu _ { \omega , \mathbf { c } } \end{align*}


\begin{align*} \displaystyle \bigoplus _ { n = 0 } ^ { 2 k - 2 } \bigcap _ { [ f ] \in \mathcal { E } _ n } \mathrm { A n n } _ { [ f ] _ n } = \displaystyle \bigoplus _ { n = 0 } ^ { 2 k - 2 } \mathrm { A n n } ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ N } f ( x ) d \omega ( x ) = 0 . \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*} \alpha _ i & \coloneqq \{ j : d _ j ( A ) = \mu _ i , j = 1 , \ldots , n \} , \\ \beta _ i & \coloneqq \{ j + n : j \in \alpha _ i \} , \\ \gamma _ i & \coloneqq \alpha _ i \cup \beta _ i . \end{align*}


\begin{align*} \delta ^ L _ { n m } = 1 0 0 \frac { F _ { n m } + F _ { m n } } { 2 F ^ 0 _ { n m } } \end{align*}


\begin{align*} \Omega ( X _ 1 , r ) = ( n + 1 ) - 1 = n , \end{align*}


\begin{align*} S ( \beta ) & = \Delta _ { N + 1 } ( \beta _ 0 \sigma _ N ) \Delta _ { N + 1 } = ( \sigma _ 1 \dots \sigma _ N ) ( \Delta _ N \beta _ 0 \Delta _ N ) ( \sigma _ N \dots \sigma _ 1 ) \sigma _ 1 \mathrel { \dot { = } } \beta ( \sigma _ N \dots \sigma _ 2 \sigma _ 1 ^ 3 \sigma _ 2 \dots \sigma _ N ) , \end{align*}


\begin{align*} \Theta _ { \infty } ( s ) & : = \frac { 1 } { \zeta _ N ( m + n ) } \left ( \sum _ { \substack { k _ 1 \geq 1 \\ g c d ( k _ 1 , q ) = 1 } } \sum _ { \substack { k _ 2 \in \Z \setminus \{ 0 \} \\ k _ 2 = k _ 1 ( m o d \ { q } ) } } \int _ { \R ^ { m + n } } \left ( ( \chi \circ a ^ s ) ( k _ 1 x ) \chi ( k _ 2 x ) \right ) \ , d x \right ) . \end{align*}


\begin{align*} \Sigma ( \delta ) = \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \boldsymbol { K } _ \ell ( \mu _ { \ell , L + 1 } , \gamma _ { \ell , L + 1 } ) \le \delta _ \ell \ell \in [ L ] \} , \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta ) = \mathbb { E } \left [ Y _ 2 \right ] - \mathbb { E } \left [ Y _ 1 \right ] + \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \mathrm { R } _ { \mathrm { D } } ( \lambda , \delta ) , \end{align*}


\begin{gather*} \| u _ n \| _ { \varepsilon _ n } , \ \int _ { \R ^ N } f _ 1 ( u _ n ) u _ n , \ \int _ { \R ^ N } F _ 1 ( u _ n ) , \ \int _ { \R ^ N } \bar f _ 2 ( u _ n ) u _ n \ \int _ { \R ^ N } \bar F _ 2 ( u _ n ) , \ | \lambda _ n | \leq C _ 0 , \\ \int _ { \R ^ N \setminus \cup _ { j = 1 } ^ \ell B ( \Upsilon _ j ( u _ n ) , \frac { 1 } { 1 0 } \xi _ 1 ( \Upsilon ( u _ n ) ) ) } \left ( | \nabla u _ n | ^ 2 + u _ n ^ 2 \right ) \mathrm d x \leq C e ^ { - c \xi _ 1 ( \Upsilon ( u _ n ) ) } + o _ n ( 1 ) . \end{gather*}


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


\begin{align*} \alpha ( \mathcal { U } , \mathcal { V } ) = \sup \{ | { \ \mathbb { P } } ( U \cap V ) - { \mathbb { P } } ( U ) { \mathbb { P } } ( V ) | : U \in \mathcal { U } , V \in \mathcal { V } \} \ , . \end{align*}


\begin{align*} \mathcal { F } ( \mathbf { h } _ t ) = \mathcal { G } ( \tilde { \mathcal { F } } ( \mathbf { h } _ t ) , \mathbf { h } _ t ) , \end{align*}


\begin{align*} K = L \cap ( K _ 1 \oplus \langle z , z ' \rangle ) , \end{align*}


\begin{align*} B ( t ) = t ^ p , \end{align*}


\begin{align*} \eta _ { \omega } = \eta ^ { \boldsymbol { \beta } _ { Y } ( \omega ) } \mbox { f o r } \mathbb { P } _ { \mu } \mbox { - a . e . } \omega . \end{align*}


\begin{align*} f ( F ) = \sum _ { G \ge F } G . \end{align*}


\begin{align*} \partial _ t w = - i h ^ { - 1 } ( - i h X - i W ) w , w | _ { t = 0 } = f . \end{align*}


\begin{align*} T _ { ( 1 - p ) \rightarrow p } A \chi ^ { p } _ S = \left ( - \frac { p } { 1 - p } \right ) ^ { | S | } \chi ^ { p } _ S . \end{align*}


\begin{align*} e _ { | x | \bar { \otimes } | y | } ( \Lambda ) : = \int _ { [ 0 , \infty ) ^ 2 } \mathbf { 1 } _ \Lambda ( \lambda _ 1 \lambda _ 2 ) \ , e _ { | x | , | y | } ( d \lambda _ 1 , d \lambda _ 2 ) , \Lambda \subset [ 0 , \infty ) . \end{align*}


\begin{align*} \begin{cases} | - b + 0 \cdot a | & = d ( v _ 1 , v _ 2 ) - 1 , \\ 0 \cdot b - a & = - 1 , \end{cases} \end{align*}


\begin{align*} \left \{ \begin{aligned} x ^ { k + 1 } - 1 & = \frac { 1 } { 1 + 5 \beta } ( x ^ { k } - 1 ) ; \\ \lambda ^ { k + 1 } + \frac { 1 } { 5 } \begin{pmatrix} 1 \\ 2 \end{pmatrix} & = J \left [ \lambda ^ k + \frac { 1 } { 5 } \begin{pmatrix} 1 \\ 2 \end{pmatrix} \right ] ; \\ \lambda ^ { k + 1 } _ 1 + 2 \lambda ^ { k + 1 } _ 2 + 1 & = \frac { 1 } { 1 + 5 \beta } ( \lambda ^ { k } + 2 \lambda ^ { k } + 1 ) , \end{aligned} \right . \end{align*}


\begin{align*} e _ 1 \cdot [ X : Y : Z ] & = [ \zeta _ 5 ^ 2 X : \zeta _ 5 ^ 4 Y : Z ] \\ e _ 2 \cdot [ X : Y : Z ] & = [ \zeta _ 5 X : \zeta _ 5 ^ 3 Y : Z ] . \end{align*}


\begin{align*} a x ^ n + b x ^ k y ^ l + c y ^ m = 0 \end{align*}


\begin{align*} \begin{aligned} \ 1 ( x \in [ 0 , \Delta ^ { - 1 } ] ) & \leq ( 1 + C e ^ { - \Delta ^ { A - 1 } } ) | \mathcal D ^ + _ { \Delta , A } ( x ) | ^ 2 \\ | \mathcal D ^ + _ { \Delta , A } ( x ) | ^ 2 & \leq \mathbf { 1 } ( x \in [ - \Delta ^ { - A / 2 } , \Delta ^ { - 1 } + \Delta ^ { - A / 2 } ] ) + C e ^ { - \Delta ^ { A - 1 } } , \end{aligned} \end{align*}


\begin{align*} v _ t = \sum _ { j \in N _ t ^ 0 } a _ j \varphi _ j . \end{align*}


\begin{align*} u _ i ( z ) = \zeta _ i + \ln ' \big [ p _ i ( z ) \big ] , 1 , \dots , r . \end{align*}


\begin{align*} \omega _ r = \mathrm { C o n v } \{ F ^ 1 _ r , \ldots , F ^ p _ r , \sigma _ r \} + G \end{align*}


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


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


\begin{align*} \begin{array} { l l } ( f \circ _ i g ) = y ( f ; \overbrace { i d , \cdots , i d , \underbrace { g } _ { i - } i d \cdots , i d } ^ { m - } ) , { f o r } f \in \Theta ( m ) \end{array} \end{align*}


\begin{align*} \operatorname { v o l } ( B _ { c c } ( R ) ) = \operatorname { v o l } ( B _ { c c } ( 1 ) ) R ^ Q , \end{align*}


\begin{align*} e = c _ 1 e _ 1 + c _ 2 e _ 2 + \cdots + c _ { n - 1 } e _ { n - 1 } . \end{align*}


\begin{align*} \partial ^ 2 _ v \psi _ { \delta _ { x _ 1 } } ( x _ 1 ) = g '' ( 0 ) \ , . \end{align*}


\begin{align*} \langle S _ n ' ( y ' ) x \Omega _ { \rho } , \Omega _ { \rho } \rangle _ { \rho } = \langle y ' S _ n ( x ) \Omega _ { \rho } , \Omega _ { \rho } \rangle _ { \rho } , x \in M , y ' \in M ' . \end{align*}


\begin{align*} \Sigma _ { \mathrm { D } } ( \delta ) = \{ \gamma \in \mathcal { P } ( \mathcal { V } ) : \boldsymbol { K } _ \ell ( \mu _ \ell , \gamma _ \ell ) \le \delta _ \ell , \ \forall \ell \in [ L ] \} \end{align*}


\begin{align*} \left \lvert \sum _ { l \equiv v ( \bmod a ) } \binom n l \alpha ^ l ( 1 - \alpha ) ^ { n - l } - \frac 1 a \right \rvert = O \left ( n ^ { - \frac 1 2 } \right ) . \end{align*}


\begin{align*} p _ s = \exp \left [ - \frac { c ( d , \nu ) t } { ( \log t ) ^ { 2 / d } } ( 1 + o ( 1 ) ) \right ] . \end{align*}


\begin{align*} y ^ r \cdot y ^ q + r z ^ { r - 1 } \cdot ( z ^ q x - z x ^ q ) + ( z ^ r + k x ^ r ) \cdot ( - x ^ q ) = 0 . \end{align*}


\begin{align*} \ker \sigma \cap \Im [ \pi _ 1 ( F ) \to \pi _ 1 ( X ) ] = H \cap \Im [ \pi _ 1 ( F ) \to \pi _ 1 ( X ) ] . \end{align*}


\begin{align*} \begin{aligned} \sum _ { n \ne 0 } \widehat { \Delta } ( a n ) ^ { 2 } & \leq \frac { 1 } { a } \sum _ { | m | \leq a } \left ( 1 - \frac { | m | } { a } \right ) - 1 = \frac { 1 } { a } \left ( 2 \lfloor a \rfloor + 1 - \frac { \lfloor a \rfloor ( \lfloor a \rfloor + 1 ) } { a } \right ) - 1 \\ & = \frac { 1 } { a } \left ( \left ( 1 + \lfloor a \rfloor - a \right ) + \frac { \lfloor a \rfloor } { a } \left ( a - \lfloor a \rfloor - 1 \right ) \right ) < \frac { 1 } { a } , \end{aligned} \end{align*}


\begin{align*} A : = \left ( \begin{array} { c c c c c } - \alpha _ { k - 1 } / \alpha _ k & - \alpha _ { k - 2 } / \alpha _ k & \ldots & - \alpha _ { 1 } / \alpha _ k & - \alpha _ { 0 } / \alpha _ k \\ 1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & \ldots & 0 & 0 \\ 0 & 0 & \ddots & \vdots & \vdots \\ 0 & 0 & \ldots & 1 & 0 \\ \end{array} \right ) \end{align*}


\begin{align*} ( \theta _ i Z _ i ^ * ( u ) ) _ { u \geq 0 } = ( Z _ i ( u ) ) _ { u \geq 0 } . \end{align*}


\begin{align*} \phi ( a _ \lambda ^ * a _ \lambda ) \geq \omega ( a _ \lambda ^ * a _ \lambda ) = 1 \end{align*}


\begin{align*} \Sigma _ c f ( n ) : = \begin{cases} c n = 0 \\ f ( n - 1 ) + \Sigma _ c f ( n - 1 ) n \ge 1 . \end{cases} \end{align*}


\begin{align*} \overline { \overline { \Pi } } _ { \psi } [ h _ { - 1 } , t ] f ( [ \epsilon , x ] ) = t f ( [ - \epsilon , x ] ) . \end{align*}


\begin{align*} \gamma _ { k , \ell } = \frac { 1 } { 4 } \cdot \frac { \delta _ k } { 8 \cdot 2 ^ { - \ell } \pi ^ { \ell - 2 k } F _ m } = \frac { \sqrt { 2 \pi } } { 1 2 8 } \Bigl ( \frac { \pi a ^ 2 } { 8 } \Bigr ) ^ { k / 2 } \frac { 2 ^ { 2 \ell } \varepsilon _ 4 } { \Gamma \bigl ( \frac { 3 k - 2 \ell + 1 } { 2 } \bigr ) } \end{align*}


\begin{align*} | f ( x ) - f ( y ) | : = \varepsilon _ 0 \ , . \end{align*}


\begin{align*} \min \to \left \{ \sum \limits ^ { m } _ { i = 1 } f _ { i } ( v ) \ \vrule \ \bigcap \limits ^ { m } _ { i = 1 } X _ { i } \right \} , \end{align*}


\begin{align*} \alpha = \alpha \left ( \left \{ \left . \alpha _ i \ \right | \ i \in I \right \} \right ) \mbox { a n d } \beta = \alpha \left ( \left \{ \left . \beta _ i \ \right | \ i \in I \right \} \right ) . \end{align*}


\begin{align*} F _ { i + \frac { 1 } { 2 } } ^ { m } = \max ( \vartheta _ { } ^ { i + 1 } , 0 ) \Big [ \widetilde { m } _ { i } + \frac { \Delta x } { 2 } ( m _ { x } ) _ { i } \Big ] + \min ( \vartheta _ { m } ^ { i + 1 } , 0 ) \Big [ \widetilde { m } _ { i } - \frac { \Delta x } { 2 } ( m _ { x } ) ^ { i } \Big ] , \end{align*}


\begin{align*} b _ k = u _ k | _ F , 1 \leq k \leq n . \end{align*}


\begin{align*} X = \frac { x } { \rho } Y = \frac { Y } { \rho } T = \frac { t } { \delta \rho ^ { s p } } w = \frac { ( u - M ) _ { - } } { M } \end{align*}


\begin{align*} \int _ { Y } { \rm I } \left ( \mathbb { P } _ { \mu , 1 } ^ { y } , \mathbb { P } _ { \mu , n } ^ { y } \right ) d \nu ( y ) = { \rm I } \left ( \mathbb { P } _ { \mu , 1 } , \mathbb { P } _ { \mu , n } \right ) - h _ { \mu } ( Y , \nu ) . \end{align*}


\begin{align*} \mathbb { E } \{ Z \} = \frac { \kappa } { \kappa + \beta } , \mathbb { E } \{ Z ^ 2 \} = \frac { \kappa ( \kappa + 1 ) } { \kappa + \beta ( \kappa + \beta + 1 ) } . \end{align*}


\begin{align*} \langle D _ u \theta ( u ^ * ) , v \rangle _ \mathcal { U } + \lim \limits _ { k \rightarrow + \infty } \langle \lambda ^ { k } , S v \rangle = 0 \forall \ v \in \mathcal { U } , \end{align*}


\begin{align*} \mathcal { H } ^ 1 \left ( J _ { w ^ h _ 1 } \cap \cup _ { i \in \N } B ^ i _ 1 \right ) = \mathcal { H } ^ 1 \left ( J _ u \cap \cup _ { i \geq h + 1 } B _ 1 ^ i \right ) , \end{align*}


\begin{align*} \left | 1 + \rho d X \right | ^ { q } \le 1 + q \rho d X + \binom { q } { 2 } \rho ^ 2 d ^ 2 X ^ 2 + \binom { q } { 3 } \rho ^ 3 | d X | ^ 3 ( 1 + | \rho d X | ) ^ { q - 3 } . \end{align*}


\begin{align*} [ P ] \ : = \ \begin{cases} 1 , & , \\ 0 , & . \end{cases} \end{align*}


\begin{align*} \int _ M ( d J \theta ) ^ { n - p } \wedge \omega = \int _ M ( J \theta \wedge ( d J \theta ) ^ { n - p - 1 } ) \wedge d \omega = 0 , \end{align*}


\begin{align*} \delta _ { \square } ( h _ 1 , h _ 2 ) = \inf _ { \phi \in \mathcal { M } } d _ { \square } ( h _ 1 , h _ 2 ^ { \phi } ) , \end{align*}


\begin{align*} ( A \otimes B ) ( C \otimes D ) = A C \otimes B D , \end{align*}


\begin{align*} \mathbf { x } \sim \mathbf { y } \Leftrightarrow \mathbf { x } = a \odot \mathbf { y } ; \end{align*}


\begin{align*} u ( r ) = \varepsilon A \left [ \left ( \frac { \sigma } { r } \right ) ^ { n } - \left ( \frac { \sigma } { r } \right ) ^ { m } \right ] , \end{align*}


\begin{align*} S _ { \omega _ { \pm } , \mathbf { c } _ { \pm } } ( U _ 0 ) = h ( \mp \tau _ 0 ) - \tau _ 0 ^ 2 \left ( \frac { h '' ( \theta ) } { 2 } - Q ( \Phi _ { \omega , \mathbf { c } } ) \right ) + O ( \delta ) \le h ( \mp \tau _ 0 ) - \tau _ 0 ^ 2 F ( \theta ) + O ( \delta ) . \end{align*}


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


\begin{align*} P ( x , y ) = \frac { y } { x ^ 2 + y ^ 2 } \end{align*}


\begin{align*} & G _ 0 ( \bar { d } , \bar { \sigma } , \xi ) = \alpha _ n a _ 1 d _ 1 ^ { n - 2 } \varphi ( \xi ) + a _ 3 \sum \limits _ { i = 1 } ^ { k - 1 } \Big ( \frac { d _ { i + 1 } } { d _ i } \Big ) ^ { \frac { n - 2 } { 2 } } g ( \sigma _ i ) - a _ 4 \sum \limits _ { i = 1 } ^ k \frac { 2 } { 2 i - 1 } | \ln d _ i | , \\ & G _ h ( d _ 1 , \xi ) = \frac { \alpha _ n } { 2 } a _ 2 \partial _ { \xi _ h } \varphi ( \xi ) d _ 1 ^ { n - 2 } , \quad \mbox { f o r } \ \ h = 1 , \cdots , n . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { n } \binom { n } { k } = 2 ^ n \end{align*}


\begin{align*} d _ { p } \left ( \overline { b } , \Theta _ { c } \right ) ^ { p } \leq \inf _ { x \in C } \sum \limits _ { i = 1 } ^ { m } \left ( u _ { i } \right ) ^ { p } d _ { S _ { i } } ^ { p } \left ( x \right ) = \inf _ { x \in \mathbb { R } ^ { n } } \sum \limits _ { i = 1 } ^ { m } \left ( u _ { i } \right ) ^ { p } d _ { S _ { i } } ^ { p } \left ( x \right ) + I _ { C } \left ( x \right ) , \end{align*}


\begin{align*} \left \{ \begin{array} { l l l } \mathrm { d } \zeta ( s ) & = & - g ( \kappa ( s ) ) \mathrm { d } s + \kappa ( s ) \mathrm { d } W ( s ) , \\ \zeta ( T ) & = & \chi , 0 \leq s \leq T , \end{array} \right . \end{align*}


\begin{align*} \left \{ \begin{array} { c } - \dbinom { 0 _ { n } } { 2 h ^ { 0 } } = \dbinom { A ^ { \prime } } { - I _ { m } } \lambda + \dbinom { 0 _ { n \times m } } { - I _ { m } } \mu , \\ \left ( A x ^ { 0 } - \overline { b } - h ^ { 0 } \right ) ^ { \prime } \lambda = 0 , - \left ( h ^ { 0 } \right ) ^ { \prime } \mu = 0 , \\ A x ^ { 0 } - \overline { b } - h ^ { 0 } \leq 0 _ { m } , ~ h ^ { 0 } \geq 0 _ { m } . \end{array} \right . \end{align*}


\begin{align*} | b ( x ) - b ( y ) | & = | b ( x ) - m _ b ( \tilde B ) + m _ b ( \widetilde B ) - b ( y ) | \\ & = | b ( x ) - m _ b ( \widetilde B ) | + | m _ b ( \widetilde B ) - b ( y ) | \ge | b ( x ) - m _ b ( \widetilde B ) | . \end{align*}


\begin{align*} r _ n ^ { [ 2 ] } ( h ) : = \frac { 2 ^ { 2 p + 1 } \cdot y _ { 4 n } \left ( \frac { h } { 4 } \right ) - 3 \cdot 2 ^ p \cdot y _ { 2 n } \left ( \frac { h } { 2 } \right ) + y _ n ( h ) } { \left ( 2 ^ p - 1 \right ) \left ( 2 ^ { p + 1 } - 1 \right ) } \end{align*}


\begin{align*} \sum _ { k = 1 } ^ m ( p _ i \beta _ { i , k } - p _ j \beta _ { j , k } ) + \alpha _ { i j } - \lambda _ { i j } + \gamma _ { i } = 0 , \forall i , j \in \{ 1 , \ldots , n \} , \ i \neq j \end{align*}


\begin{align*} \Sigma _ N ^ 2 ( L , \alpha ) = L - L ^ 2 + L R _ N ^ 2 ( L , \alpha , \Delta ) . \end{align*}


\begin{align*} ( \exists f \in F _ i ) , ( \prod \limits _ { i = 1 } ^ m a ( i ) \ast f ( t ( i ) ) \ast a ( m + 1 ) \notin A _ i \cap \sigma ( L _ i ) . \end{align*}


\begin{align*} \tau ( \beta _ g ( x ) Y ) = \tau ( x \hat { \beta } _ g ( Y ) ) x \in M Y \in L ^ 1 ( M , \tau ) . \end{align*}


\begin{align*} C o n ( X \cdot Y ) = C o n ( X ) C o n ( Y ) , \end{align*}


\begin{align*} \sum _ { k = 1 } ^ m ( p _ i \beta _ { i , k } - p _ j \beta _ { j , k } ) + \alpha _ { i j } - \lambda _ { i j } + \gamma _ { i } = 0 , \forall j \neq i . \end{align*}


\begin{align*} \psi S A = - a S \nabla ( \psi E ) a + a ( S \nabla ( \psi E ) - \psi S \nabla E ) a - \psi A S \nabla \tilde \eta a - \psi \tilde a S \nabla \tilde \eta A - \psi R a - \psi \tilde r A - \psi A . \end{align*}


\begin{align*} \left \vert \pi _ 1 ( X _ n ) \left ( \frac { 1 } { \pi _ 2 ( X _ n ) } - \frac { 1 } { H _ n } \right ) \right \vert = \left \vert \frac { \pi _ 1 ( X _ n ) } { \pi _ 2 ( X _ n ) } \right \vert \frac { H _ n - \pi _ 2 ( X _ n ) } { H _ n } \leqslant \frac { n } { H _ n / 2 } \frac { H _ n ^ { 1 / 2 } } { H _ n } = 2 n H _ n ^ { - 3 / 2 } . \end{align*}


\begin{align*} \frac { X ( t ) } { \tau - t } = \frac { \theta } { \tau } e ^ { \widehat { B } ( U ( t ) ) + \frac { 1 } { 2 } U ( t ) } . \end{align*}


\begin{align*} & E \left [ Z _ 0 ( \Delta \ell _ 1 ) \left ( \nu \Delta Z _ 0 ( \Delta \ell _ 2 ) p ^ k _ { j , + } ( t - \Delta \ell _ 2 ) - W ^ k _ { \ell _ 2 \Delta , t } ( j ) \right ) \right ] \\ & = E \left [ E \left [ Z _ 0 ( \Delta \ell _ 1 ) \big ( \nu \Delta Z _ 0 ( \Delta \ell _ 2 ) p ^ k _ { j , + } ( t - \Delta \ell _ 2 ) - E \left [ W ^ k _ { \ell _ 2 \Delta , t } ( j ) | { \cal F } _ { ( \ell _ 2 + 1 ) \Delta } \right ] \big ) | { \cal F } _ { ( \ell _ 1 + 1 ) \Delta } \right ] \right ] . \end{align*}


\begin{align*} \forall ( y , \xi ) \in B _ 1 \times \R ^ k , f _ h ( y , \xi ) : = ( \gamma _ h \sigma _ h ) ^ { 1 - \frac { p _ h ( y ) } { p _ h ^ 0 } } \abs { \xi } ^ { p _ h ( y ) } , \end{align*}


\begin{align*} \lambda _ k [ f ( A _ { \mathcal { S ' } } ) ] & = \min _ { h \in { \mathcal { F } } ; \ \Vert h \Vert = 1 } \langle h , f ( A _ { \mathcal { F } } ) h \rangle \\ & = \min \{ f ( \lambda _ 1 ( A _ { \mathcal { F } } ) ) \ , ; \ , f ( \lambda _ k ( A _ { \mathcal { F } } ) ) \} \\ & = \min _ { h \in { \mathcal { F } } ; \ \Vert h \Vert = 1 } f ( \langle h , A _ { \mathcal { F } } h \rangle ) \\ & = \min _ { h \in { \mathcal { F } } ; \ \Vert h \Vert = 1 } f ( \langle h , A h \rangle ) \end{align*}


\begin{align*} \underset { \delta \rightarrow 0 } { \lim } \left | I I I _ { \delta } \right | \leq 2 \underset { \overline { t } \in [ 0 , T ] } { \sup } \left | A _ { - } ( \overline { t } , \overline { t } , \cdot ) \right | _ { \mathrm { L i p } } ( \tau - \sigma ) = \underset { \overline { t } \in [ 0 , T ] } { \sup } \left | A ( \overline { t } , \cdot ) - \widetilde { A } ( \overline { t } , \cdot ) \right | _ { \mathrm { L i p } } ( \tau - \sigma ) . \end{align*}


\begin{align*} x & = \varepsilon ^ { - 1 } ( R _ { n } R _ { n + 1 } ) \log ^ { 2 N - 3 } \left ( \left ( \frac { R _ { n } R _ { n + 1 } } { \varepsilon } \right ) ^ { \frac { R _ { n } R _ { n + 1 } } { 2 } } \frac { N - 1 } { \eta } \right ) , \\ y & = \log ^ 4 \left ( \left ( \frac { R _ { n } R _ { n + 1 } } { \varepsilon } \right ) ^ { \frac { 1 } { 2 } } \log ^ { N - 1 } \left ( \left ( \frac { R _ { n } R _ { n + 1 } } { \varepsilon } \right ) ^ { \frac { R _ { n } R _ { n + 1 } } { 2 } } \frac { N - 1 } { \eta } \right ) \right ) \log \prod _ { j \ne n } I _ j , \end{align*}


\begin{align*} \lambda ^ 2 = \sum _ { k \in [ n ] , k \neq j } ^ n ( t _ k t _ k ^ * ) = \sum _ { k \in [ n ] , k \neq j } ^ n \norm { t _ k } ^ 2 . \end{align*}


\begin{align*} e _ 1 \cdot [ X : Y : Z ] & = [ \zeta _ 5 X : Y : Z ] \\ e _ 2 \cdot [ X : Y : Z ] & = [ X : \zeta _ 5 Y : Z ] . \end{align*}


\begin{align*} \lambda \left ( \delta \right ) \geq \lambda _ { 1 } = \lambda _ { 1 } \left ( N , \varepsilon , \Omega , T , c \right ) \geq 1 , \end{align*}


\begin{align*} \psi _ { C } ( x ) = x - \log x - 1 + \sum _ { k = 1 } ^ { \infty } R _ { k } ( x ) , \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } g _ F ( t ) \ , e ^ { i z t } \ , d t = \frac { 1 } { z ^ 2 } \frac { \xi _ F ' } { \xi _ F } \left ( \frac { 1 } { 2 } - i z \right ) , \Im ( z ) > 1 / 2 \end{align*}


\begin{align*} u _ g ( x \Omega _ { \rho } ) = T _ g ( x ) \Omega _ { \rho } , x \in M _ s , g \in G . \end{align*}


\begin{align*} \mathsf { M } _ { x _ 0 } ^ { \sigma ^ s } ( \Gamma _ X \times \Gamma _ A ) = \int _ { \Gamma _ X } \sigma ^ s ( \Gamma _ A | x ) \mathsf { M } _ { x _ 0 } ^ { \sigma ^ s } ( d x \times \textbf { A } ) , ~ \Gamma _ X \in { \cal B } ( \textbf { X } ) , ~ \Gamma _ A \in { \cal B } ( \textbf { A } ) \end{align*}


\begin{align*} x y - z t = 0 \end{align*}


\begin{align*} V + \phi - i ^ * [ f _ \epsilon ( V + \phi ) ] = \sum _ { i = 1 } ^ k \sum _ { l = 0 } ^ n c _ { i l } P \psi ^ l _ { \mu _ i , \xi _ i } , \end{align*}


\begin{align*} K = \begin{bmatrix} A & B _ 1 & \cdots & B _ m \end{bmatrix} , \end{align*}


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


\begin{align*} W _ { j } ^ { \prime } = \sum _ { i \le N } g _ { i j } \tilde { g } _ { i j } B _ { i } , \end{align*}


\begin{align*} R \rho u _ x = - R \rho ( ( \ln \rho ) _ t + u ( \ln \rho ) _ x ) \end{align*}


\begin{align*} a = g _ 1 g _ 2 \cdots g _ s \end{align*}


\begin{align*} \wp ( z + w ) = \frac { 1 } { 4 } \left ( \frac { \wp ' ( z ) - \wp ' ( w ) } { \wp ( z ) - \wp ( w ) } \right ) ^ 2 - \wp ( z ) - \wp ( w ) . \end{align*}


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


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


\begin{align*} t _ n = \frac { \det A _ n } { \det A _ { n - 1 } } , \end{align*}


\begin{align*} { { \boldsymbol { \gamma } } _ { i ; k } } = { \boldsymbol { h } } \left ( { { { \boldsymbol { \xi } } _ { i ; k | k - 1 } } } \right ) , \left ( { i = 1 , 2 , \cdots , 2 n } \right ) . \end{align*}


\begin{align*} I _ { m , p } ^ { \star ( 1 ) } ( s , x ) = \frac { ( - 1 ) ^ { m } } { 2 \pi i } \intop _ { \sigma - i \infty } ^ { \sigma + i \infty } \Gamma ( z ) \ , \Gamma ( s - z ) \ , \left ( \frac { x } { m } \right ) ^ { 2 z } d z = \frac { ( - 1 ) ^ { m } \Gamma ( s ) x ^ { 2 s } } { \left ( x ^ { 2 } + m ^ { 2 } \right ) ^ { s } } . \end{align*}


\begin{align*} L : = \lim _ { n \to \infty } T ^ n = \begin{bmatrix} I & 0 \\ 0 & A \end{bmatrix} , \end{align*}


\begin{align*} \mathfrak { N } _ { f , a } \geq & \theta ( d ) ^ 2 \Theta ( g ) ^ 2 \Bigg [ \lambda \left ( \frac { 2 \sum _ { i = 1 } ^ { r } \theta ( p _ i ) + 2 \sum _ { i = 1 } ^ { s } \Theta ( g _ i ) + 1 } { \lambda } \right ) \left ( - ( 2 n + 1 ) W ( d ) ^ 2 W ( g ) ^ 2 q ^ { m / 2 } \right ) \\ + & \left \{ q ^ { m - 1 } - ( n + 1 ) - ( 2 n + 1 ) q ^ { m / 2 } \right \} \Bigg ] \end{align*}


\begin{align*} T = \psi _ { S , Z } ( T ) = S ( I \otimes T ) Z ^ * . \end{align*}


\begin{align*} H \equiv \frac { \hbar ^ 2 } { 2 } \sum _ { i = 1 } ^ N ( - \Delta _ { x _ i } ) + \lambda \sum _ { i < j } ^ N V ( x _ i - x _ j ) \ . \end{align*}


\begin{align*} \begin{cases} - \Delta u = f ( u ) , \ \ 0 < u < 1 , & \ \ D , \\ u = 1 , \ \ | \nabla u | = c , & \ \ , \partial \Omega _ 1 , \\ u = 0 , \ \ | \nabla u | = 0 , & \ \ , \partial \Omega _ 2 . \end{cases} \end{align*}


\begin{align*} \partial _ { t } \mu _ { S } = \cal A _ { D , N } ^ { - 1 } { \rm R H S } = 2 ( { \rm I } _ { N _ { B } } - \cal R ) ^ { - 1 } \tilde { \cal A } _ { - 1 } { \rm R H S } = 2 \sum _ { k = 0 } ^ { + \infty } \cal R ^ k \tilde { \cal A } _ { - 1 } { \rm R H S } . \end{align*}


\begin{align*} P = \sum _ { I \in \mathbb { N } _ 0 ^ n } c _ I \eta _ A ^ I , \eta _ A ^ I : = \eta _ { 1 , A } ^ { i _ 1 } \cdots \eta _ { n , A } ^ { i _ n } , \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d t } \| U \| ^ 2 _ P + \epsilon ( \| U _ x \| ^ 2 _ P + \| U _ y \| ^ 2 _ P ) + \frac { 1 } { 2 } \oint \limits _ { \partial \Omega } U ^ T ( n _ 1 A + n _ 2 B ) U - 2 \epsilon U ^ T ( n _ 1 P U _ x + n _ 2 P U _ y ) \\ \ d s = 0 . \end{align*}


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


\begin{align*} \nu _ t \left ( x \right ) \log \frac { \nu _ t \left ( x \right ) } { e ^ { - U ^ { \pi } \left ( x \right ) } } = \frac { \nu _ t \left ( x \right ) } { e ^ { - U ^ { \pi } \left ( x \right ) } } e ^ { - U ^ { \pi } \left ( x \right ) } \log \frac { \nu _ t \left ( x \right ) } { e ^ { - U ^ { \pi } \left ( x \right ) } } \geq - \frac 1 e e ^ { - U ^ { \pi } \left ( x \right ) } = : - f _ 2 \left ( x \right ) , \end{align*}


\begin{align*} M & = \left ( ( - \infty , 0 ] \times \{ - 1 , 1 \} \right ) \cup \left ( ( 0 , \infty ) \times \{ 0 \} \right ) \\ U _ + & = \left ( ( - \infty , 0 ] \times \{ 1 \} \right ) \cup \left ( ( 0 , \infty ) \times \{ 0 \} \right ) \\ U _ - & = \left ( ( - \infty , 0 ] \times \{ - 1 \} \right ) \cup \left ( ( 0 , \infty ) \times \{ 0 \} \right ) \end{align*}


\begin{align*} H _ { 0 , \Sigma } ^ 1 ( \mathrm { M } ) = \{ u \in H ^ 1 ( \mathrm { M } ) ; \ ; u _ { | \Sigma } = 0 \} . \end{align*}


\begin{align*} D ^ { 1 / 3 } \psi = X \cdot X ( \psi ) + Y \cdot Y ( \psi ) + Z \cdot T ( \psi ) + T \cdot Z ( \psi ) + { \textstyle \frac 1 2 } X Y Z \cdot \psi . \end{align*}


\begin{align*} x _ { k } = \sum _ { j = 0 } ^ { k } Q ( j , k ) \Delta x _ j . \end{align*}


\begin{align*} W ^ { 1 , p ( \cdot ) } \left ( \Omega , \mathcal { M } \right ) : = \left \{ W ^ { 1 , p ( \cdot ) } \left ( \Omega , \R ^ k \right ) : u ( x ) \in \mathcal { M } \ , \ , \mbox { f o r a . e . } x \in \Omega \right \} . \end{align*}


\begin{align*} h _ i ( m ) . v = ( \sum _ { j = 1 } ^ { k _ i } p _ { i , j } b _ { i , j } ^ m ) v \end{align*}


\begin{align*} \sup \{ | v ( x ) - v ( y ) | : x , y \in I \} \leq 2 \sum _ { l = 0 } ^ \infty \norm { Z ( l + 1 ) } \norm { S ( l ) \varphi } _ { \sup } . \end{align*}


\begin{align*} q ^ { \binom { k } { 2 } } [ k ] _ q ! S [ n , k ] = \sum _ { \ell = 1 } ^ k q ^ { k ( k - \ell ) } A _ { n , \ell - 1 } ( q ) { n - \ell \brack k - \ell } _ { q } \end{align*}


\begin{align*} a _ i \geq C _ \epsilon h _ 0 ^ { \frac { 1 } { 2 } + \epsilon } , \ \ i = 1 , \cdots , n - 1 . \end{align*}


\begin{align*} M = \langle S _ 1 \rangle \oplus \cdots \oplus \langle S _ s \rangle . \end{align*}


\begin{align*} \sum _ { l = 1 } ^ s b _ l \frac { d } { d x } ( \tilde Q _ l ) \in I _ \alpha . \end{align*}


\begin{align*} \mbox { i f } \ h \in L ^ 1 _ { r a d } ( \R ^ n ) , \ , i \neq l , \int _ { \frac { \mathcal { A } _ m ^ l - \xi _ m } { \mu _ m ^ i } } h ( | x | ) d x = O \bigg ( \Big ( \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big ) ^ { \frac { n } { n - 2 } } \bigg ) . \end{align*}


\begin{align*} \partial _ t u ( t , x ) \ , = \ , \frac { 1 } { 2 } \Delta _ x u ( t , x ) \ , + \ , \beta \ , \dot { W } ( t , x ) \ , u ( t , x ) \ , , \end{align*}


\begin{align*} \tilde { q } = \frac { 1 } { 2 } \left ( q + \frac { 2 s } { s - \tau } \right ) \end{align*}


\begin{align*} x = x _ 0 + \frac { m } { ( n , m ) } u , y = y _ 0 + \frac { n } { ( n , m ) } u , u \geq 0 , \end{align*}


\begin{align*} \overline { \Pi } _ { \psi } [ h _ { - 1 } , t ] f ( [ \epsilon , x ] ) = t f ( [ - \epsilon , x ] ) . \end{align*}


\begin{align*} 3 L + 4 < \frac { 8 } { 2 5 } a ^ 2 = \frac { 4 t } { 2 5 \pi } \quad 3 L + 2 + \sigma \ge 0 . \end{align*}


\begin{align*} w ( y ' , y _ n ) = \begin{cases} v \circ \psi ^ { - 1 } ( y ' , y _ n ) & \\ v \circ \psi ^ { - 1 } ( y ' , - y _ n ) & \end{cases} \end{align*}


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


\begin{align*} w ( z ) = \Phi _ 0 ( z ) + \Psi ( z ) , \end{align*}


\begin{align*} f ( A ) = { \mathrm { s o t } } \lim _ { n \to \infty } f ( A _ n ) \end{align*}


\begin{align*} u _ \mathrm { l i n } ( t , r ) = \begin{cases*} \frac { 1 + r - t } { r } & i f $ t - 1 \leq r < t $ , \\ \frac { 1 - r + t } { r } & i f $ t \leq r < t + 1 $ , \\ 0 & o t h e r w i s e . \end{cases*} \end{align*}


\begin{align*} y _ { C 1 } = \sqrt { 1 - \alpha } h _ { A C } x + h _ { C C } + n _ { C 1 } , \end{align*}


\begin{align*} f ( \tau ) = - \Delta u ( \sigma _ y ( g ^ { - 1 } ( \tau ) ) ) \ \ \ \ \ \tau \in ( 0 , u ( 0 ) ) . \end{align*}


\begin{align*} \delta = \begin{cases} \frac { 3 - 2 \beta } { 4 } & \beta \ge \frac 1 2 \\ \frac 1 2 & \beta \le \frac 1 2 \end{cases} \end{align*}


\begin{align*} \nu _ i \left ( f ' \right ) = \nu _ i ( f ) + \alpha _ i . \end{align*}


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


\begin{align*} \left . \begin{array} { l l } \displaystyle \dfrac { d ^ 2 } { d t ^ 2 } G ( t ) = \dfrac { 2 } { m _ 1 } R e Q ( \bar { \phi } , \psi ) , \end{array} \right . \end{align*}


\begin{align*} \underline { f } ( x ) : = \begin{cases} 1 - \eta ^ { - 1 } \psi ( x ) , & x \in [ 0 , \varepsilon _ 2 ] , \\ 0 , & x \in ( \varepsilon _ 2 , \infty ) \end{cases} \end{align*}


\begin{align*} \lim _ { l \to \infty } \norm { B _ l ( \nu _ k ) - \overline { \nu } } = 0 . \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 } m \cdots ( m - i + 1 ) \mathbb { P } _ K ^ { m } \gamma _ i \gamma _ { m , i } \\ & + { M - j \choose i } \mathbb { P } _ { \rm T X } ^ i \left ( 1 - \mathbb { P } _ { \rm T X } \right ) ^ { M - j - i } \frac { M - j - i } { M - j } i ! \mathbb { P } _ K ^ { i } \gamma _ i , \end{align*}


\begin{align*} \sigma _ { | X _ 1 } = \sigma _ { | X _ 0 \setminus \{ k \} } \mbox { i s a n e v e n p e r m u t a t i o n } . \end{align*}


\begin{align*} \mathcal { O } ( k ) : = \int _ { \Lambda ^ { * } } \ 1 _ \S ( p ) g ( p ) a _ { p + k } ^ * a _ p \d p \end{align*}


\begin{align*} D ( t _ 1 , \ldots , t _ r ) : = \min \{ t _ i , | t _ i - t _ j | : 1 \leq i \neq j \leq r \} . \end{align*}


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


\begin{align*} F _ X ( x ) & = F _ X ( 0 ) + M \mathrm { s g n } ( x ) \int _ 0 ^ x \mathrm { e } ^ { \beta t } | t | ^ \nu K _ { \nu } ( \alpha | t | ) \ , \mathrm { d } t \\ & = F _ X ( 0 ) + M \mathrm { s g n } ( x ) \sum _ { k = 0 } ^ \infty \frac { \beta ^ k } { k ! } \int _ 0 ^ x ( - 1 ) ^ k | t | ^ { \nu + k } K _ \nu ( \alpha | t | ) \ , \mathrm { d } t \\ & = F _ X ( 0 ) + M \sum _ { k = 0 } ^ \infty \frac { \beta ^ k } { k ! } ( \mathrm { s g n } ( x ) ) ^ { k + 1 } \int _ 0 ^ { | x | } t ^ { \nu + k } K _ \nu ( \alpha t ) \ , \mathrm { d } t . \end{align*}


\begin{align*} H = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \end{align*}


\begin{align*} \| f w \| _ { L ^ p ( \Omega ) } = \Big ( \int _ \Omega \ ! | f | ^ p w ^ p \ , \mathrm { d } \mu \Big ) ^ { \frac { 1 } { p } } , \end{align*}


\begin{align*} ( a ) _ n = \frac { \Gamma ( a + n ) } { \Gamma ( a ) } , \end{align*}


\begin{align*} \phi _ \sqsupset ( S ) = S ^ { \sqsubset \vartriangleleft } . \end{align*}


\begin{align*} a x _ i ^ n u ^ { n l ' - k l ' - l n ' } v ^ { n m ' - k m ' - l k ' } + b x _ i ^ k y _ i ^ l + c y _ i ^ m u ^ { m n ' - k l ' - l n ' } v ^ { m k ' - k m ' - l k ' } = 0 . \end{align*}


\begin{align*} \Psi \Phi v = A ^ { - 1 } B D ^ { - 1 } C v = A ^ { - 1 } A v = v . \end{align*}


\begin{align*} S _ n ( \mu ) = \frac { 1 } { n + 1 } \sum _ { k = 0 } ^ { n } \sigma _ k ( \mu ) . \end{align*}


\begin{align*} ( H _ { \omega } \phi ) ( n ) : = \sum _ { | m - n | = 1 } ( \phi ( m ) - \phi ( n ) ) + V _ { \omega } ( n ) \phi ( n ) . \end{align*}


\begin{align*} M = \left ( \begin{array} { c c c c } c _ 1 v _ 1 & c _ 1 v _ 2 & \ldots & c _ 1 v _ n \\ c _ 2 v _ 1 & c _ 2 v _ 2 & \ldots & c _ 2 v _ n \\ \ldots & \ldots & \ldots & \ldots \\ c _ n v _ 1 & c _ n v _ 2 & \ldots & c _ n v _ n \\ \end{array} \right ) = c \otimes v , \end{align*}


\begin{align*} H ( \alpha _ 1 , \ldots , \alpha _ m ) : = H ( [ \alpha _ 1 : \cdots : \alpha _ m : 1 ] ) . \end{align*}


\begin{align*} ( A z , H _ { J ^ { c } } ) = \Vert \tilde { W } z \Vert _ { 2 } . \end{align*}


\begin{align*} \int _ { C \times X } f d \lambda = \int _ { C } \int _ { X } f ( c , x ) d \beta _ { c } ( x ) d \nu _ { C } ( c ) . \end{align*}


\begin{align*} \tilde { D } = D + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} ( 1 - \Delta ) w _ j & = 0 \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ : \mathbb { R } ^ n _ + , \\ \mathbf { n } \cdot \nabla w _ j & = \mathbf { n } \cdot \nabla \tilde { c } _ j \ ; \ ; \ ; \partial \mathbb { R } ^ n _ + \end{align*}


\begin{align*} & R _ h ^ 0 ( z ) - \tilde { R } _ h ( z ) = - i \tilde { R } _ h ( z ) W R _ h ^ 0 ( z ) \\ & = i h ^ 2 \left ( \int _ 0 ^ \infty e ^ { - i s h ^ { - 1 } ( - i h X - i ( Q _ \infty + q _ 1 + W ) - z ) } d s \right ) W \left ( \int _ 0 ^ \infty e ^ { - i t h ^ { - 1 } ( - i h X - i ( Q _ \infty + q _ 1 ) - z ) } d t \right ) . \end{align*}


\begin{align*} \phi ( u ) ( x - y ) = \phi ( u ) ( x ) - \phi ( u ) ( y ) = \left ( \nabla \phi ( u ) \right ) ( x - y ) = u ( x ) - u ( y ) . \end{align*}


\begin{align*} P _ { \mu , x } ^ { n } \left ( L _ { x } g ^ { - 1 } , A \right ) = \int _ { G } { \bf 1 } _ { g . A } \left ( g . x , L _ { g . x } s \right ) \frac { d s \nu } { d \nu } ( \pi ( g . x ) ) d \mu ^ { ( n ) } ( s ) = P _ { \mu , g . x } ^ { n } \left ( L _ { g . x } , g . A \right ) . \end{align*}


\begin{align*} \max _ { a , b } f ( a , b ) = f \left ( \left ( \frac { n } { p } - 2 + \gamma \right ) ^ { p - 1 } \left ( \frac { n ( p - 1 ) } { p } - \gamma \right ) ^ { p - 1 } , \frac { n } { p } - 2 + \gamma \right ) = \left ( \frac { n } { p } - 2 + \gamma \right ) ^ p \left ( \frac { n ( p - 1 ) } { p } - \gamma \right ) ^ p , \end{align*}


\begin{align*} \boldsymbol { K } _ 1 ( \gamma _ 1 ^ { \eta _ 0 , \delta } , \gamma _ 1 ^ \star ) = \int _ { \mathcal { S } _ 1 \times \mathcal { S } _ 1 } c _ 1 \ , d \nu . \end{align*}


\begin{align*} \Psi \left ( - \frac 1 2 , x ' , 0 \right ) = \lim _ { m } U _ m \left ( y _ m ^ x , 0 \right ) = 0 | x ' | < \frac 1 4 , \end{align*}


\begin{align*} \mathcal { F } : & L ^ 2 ( \R ) \longrightarrow L ^ 2 ( \R ) ; \\ & f \longmapsto \mathcal { F } ( f ) ( x ) = \int _ { \R } \psi ( - x t ) f ( t ) d t \end{align*}


\begin{align*} \Sigma = \sqrt { | \Lambda ^ - | } . \end{align*}


\begin{align*} \bar { F } _ { P _ s } ( \theta , x ) \overset { \Delta } { = } \Pr \left ( P _ s ( \theta ) > x \right ) , x \in [ 0 , 1 ] , \end{align*}


\begin{align*} [ x _ 1 , \dots , x _ n ] _ c : = \psi _ n ( x _ 1 , \dots , x _ n ) . \end{align*}


\begin{align*} S & = \left \{ A \in \binom { [ n ] } { k } : \ n \in A \right \} . \end{align*}


\begin{align*} F ( \tau ) : = \inf _ { \Phi \in \mathcal { M } _ { \omega , \mathbf { c } } ^ * ( \eta ) } \left ( \frac { h '' ( \tau ) } { 2 } - Q ( \Phi ) \right ) . \end{align*}


\begin{align*} ( g ( x ) ) K [ x ] = \ker \ , ( _ \eta ) . \end{align*}


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


\begin{align*} \pi = \alpha _ 1 \alpha _ 2 \cdots \alpha _ k \beta . \end{align*}


\begin{align*} \lim _ { p \rightarrow 0 ^ { + } } \mathcal { K } _ { \nu , p } ( x ) = \frac { \Gamma \left ( \frac { 1 } { 2 } - \nu \right ) ( 2 x ) ^ { \nu } } { \sqrt { \pi } } \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \ , n ^ { \nu } \ , K _ { \nu } ( x n ) . \end{align*}


\begin{align*} \int _ V u h d \mu = 0 , \quad \forall u \in L ^ q ( V ) . \end{align*}


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


\begin{align*} P _ s ( \theta ) & \overset { \Delta } { = } \left ( > \theta , \Phi ( \mathcal { A } ) > 0 | \Phi \right ) \\ & = \left ( > \theta | \Phi , \Phi ( \mathcal { A } ) > 0 \right ) \mathbf { 1 } ( \Phi ( \mathcal { A } ) > 0 | \Phi ) , \end{align*}


\begin{align*} \nabla \big ( T _ { t } ( u ) \big ) = \chi _ { \{ | u | < t \} } Z _ u \hbox { a . e . i n $ \Omega $ } \end{align*}


\begin{align*} \Theta ^ 2 ( \Omega _ { H / S } | _ Y ) ^ { - 1 } \otimes \Theta ^ 2 ( N _ { Y / H } ) = { 1 \over 2 } \sum _ j E _ j b _ j \end{align*}


\begin{align*} [ \xi _ i , \xi _ j ] ^ \bot = 0 , 1 \le i , j \le p . \end{align*}


\begin{align*} p \cdot { \rm d i s t } ( \eta , K ) = { \rm d i s t } ( \eta , K ) . \end{align*}


\begin{align*} \lim _ { \Delta \rightarrow 0 } \boldsymbol { W } _ { p } \left ( \pi ^ \prime , \frac { \nu ( t _ 0 ) - \Delta \pi ^ \prime } { 1 - \Delta } \right ) = \boldsymbol { W } _ { p } \left ( \pi ^ \prime , \nu ( t _ 0 ) \right ) < \infty . \end{align*}


\begin{align*} \theta _ i ^ 2 T _ i ^ * ( u ) = \frac { T _ i ^ { \infty } T _ i ( u ) } { T _ i ^ { \infty } - T _ i ( u ) } . \end{align*}


\begin{align*} \dot { y } = f ( y ) , \dot { \xi } = f ^ \prime ( y ) \xi , \dot { \eta } = f ^ \prime ( y ) \eta . \end{align*}


\begin{align*} p _ 0 : = p ( x _ 0 ) . \end{align*}


\begin{align*} c ( \tau , j ) _ i ^ { \sigma } = \begin{cases} p & j = i \mbox { a n d } \tau \subseteq \sigma \\ 1 & \mbox { e l s e } \end{cases} \end{align*}


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


\begin{align*} \alpha = \frac { \left ( 1 + \beta \right ) b ^ { 2 } } { \varepsilon \left ( T - \varepsilon \right ) } . \end{align*}


\begin{align*} W ^ T \Lambda W = ( \sqrt { \Lambda ^ + } W ^ + ) ^ T ( \sqrt { \Lambda ^ + } W ^ + ) - ( R ( \sqrt { \Lambda ^ + } W ^ + ) + S G ) ^ T ( R ( \sqrt { \Lambda ^ + } W ^ + ) + S G ) . \end{align*}


\begin{align*} N _ { p \alpha + k } ( \xi , t ) = \sum \limits _ { i = 1 } ^ 3 \big ( w ^ { ( i ) } _ { p \alpha + k } ( 0 , t ) + \Psi ^ { ( i ) } _ { p \alpha + k } ( \xi _ i , t ) \big ) \ , \chi _ { \ell _ 0 } ( \xi _ i ) + \widetilde { N } _ { p \alpha + k } ( \xi , t ) , \end{align*}


\begin{align*} \sigma ( A ^ + ) = A ^ + \mbox { a n d } \sigma ( \underline { A } ^ + ) = \underline { A } ^ + . \end{align*}


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


\begin{align*} \pi _ t ( b ) = \pi ( b \delta _ t ) , b \in B _ t \end{align*}


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


\begin{align*} \mathrm { i n d } ( H _ b ) = \sum _ { \lambda < 0 } \mathrm { n u l } ( H _ b - \lambda I ) = \sum _ { \lambda \in ( \lambda _ 0 , 0 ) } \mathrm { n u l } ( H _ b - \lambda I ) = \sum _ { t \in ( a , b ) } \mathrm { n u l } ( H _ t ) , \end{align*}


\begin{align*} z ^ { c \odot \vec x } _ j & = \begin{cases} ( M ( c \odot \vec x ) _ { i } - \max _ k ( M ( c \odot \vec x ) _ k ) = M ( \vec x ) _ { i } - \max _ k ( M \vec x ) _ k & j \in J _ i \\ 0 \qquad & j \notin \cup _ i J _ i , \end{cases} \end{align*}


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


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


\begin{align*} Z _ k = \begin{cases} X _ { - ( \epsilon _ i - \epsilon _ { i + 1 } ) } ^ { p - 1 } X _ { \epsilon _ { i + 1 } + \epsilon _ j } & \beta _ k = ( \epsilon _ i + \epsilon _ j ) \in \Phi _ { 1 } \cr X _ { - ( \epsilon _ i - \epsilon _ { i + 1 } ) } X ^ { p - 1 } _ { \epsilon _ { i + 1 } - \epsilon _ j } & \beta _ k = \epsilon _ i - \epsilon _ j \in \Phi _ 0 ^ + \backslash \Pi _ 0 \end{cases} i < j . \end{align*}


\begin{align*} \left ( \det ( u _ { j \bar k } ) \right ) ^ { 1 / n } = \psi ^ { 1 \slash n } ( z , u ( z ) ) , \end{align*}


\begin{align*} \widetilde { S } _ { \alpha , \beta } ^ { \ , \log } f ( x ) & = x ^ { - \beta } S _ { \alpha , \beta } ^ { \log } \big ( ( \cdot ) ^ { \beta } f \big ) ( x ) \\ & \simeq x ^ { - \beta } \sup _ { t > 2 x } \int _ { t / 2 } ^ { t } ( t + x - z ) ^ { - \alpha - 1 / 2 } \log \Big ( 2 + \frac { x } { t - z } \Big ) \abs { f ( z ) } \ , d z , x > 0 . \end{align*}


\begin{align*} \begin{aligned} ( f _ { \mathcal { S } } ) _ \lambda ( s _ 1 , s _ 2 ) & \leq \sup _ { ( y _ 1 ^ \prime , y _ 2 ^ \prime , x ^ \prime ) \in \mathcal { S } } \left \{ f _ 1 ( y _ 1 ^ \prime ) + f _ 2 ( y _ 2 ^ \prime ) - \sum _ { 1 \leq \ell \leq 2 } \lambda _ \ell c _ { Y _ \ell } \left ( y _ \ell , y _ \ell ^ \prime \right ) \right \} \\ & = ( f _ 1 ) _ { \lambda _ 1 } ( y _ 1 ) + ( f _ 2 ) _ { \lambda _ 2 } ( y _ 2 ) . \end{aligned} \end{align*}


\begin{align*} ( N ^ { \ , ( 1 ) } ( X , Y ) ) ^ \bot = 2 \ , g ( X , f \widetilde { Q } Y ) \ , \bar \xi \ , ; \end{align*}


\begin{align*} a _ { i } : = \frac { b _ { i - p + 1 } } { a _ { i - p + 1 } \cdots a _ { i - 1 } } , i \ge I + p , \end{align*}


\begin{align*} \tilde { \mu } _ \delta \{ \gamma r \to \gamma s \} ( \gamma U ) = \tilde { \mu } _ \delta \{ r \to s \} ( U ) . \end{align*}


\begin{align*} \left ( \frac { 3 3 r } { \sqrt { d } } \right ) ^ d \gamma _ 1 \log ^ { d / 2 } \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) = \gamma ' _ d < \| f ^ { = d } \| _ 2 \leq \left ( \frac { 3 3 \sqrt { d } } { d \log ^ { 1 / 2 } \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) } \right ) ^ { d } \gamma _ 1 \log ^ { d } \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) , \end{align*}


\begin{align*} d ( x , ( 1 - t ) x + t y ) = t ^ 2 \max _ { 0 \le j \le n - 1 } ( x - y ) ^ 2 \cdot ( ( 1 - t ) x + t y ) ^ j \cdot x ^ { n - 1 - j } \lesssim t ^ 2 d ( x , y ) \end{align*}


\begin{align*} \sigma _ Y ^ * P & \simeq \{ ( v , y ) \in P \times Y \ , \mid \ , h ( v ) = \sigma _ Y ( y ) \} \\ & = \{ ( v , y ) \ , \mid \ , y = ( \sigma _ Y \circ h ) ( v ) \} \\ & = \{ ( v , \left ( \sigma _ Y \circ h ) ( v ) \right ) \ , \mid \ , v \in P \} \end{align*}


\begin{align*} \det ( I - \mathcal { P } _ \gamma ^ n | _ { E _ s } ) > 0 , ( - 1 ) ^ { s + n } \det ( I - \mathcal { P } _ \gamma ^ n | _ { E _ s } ) = \det ( I - \mathcal { P } _ \gamma ^ { - n } | _ { E _ s } ) ( - 1 ) ^ n \det ( \mathcal { P } _ \gamma ^ n | _ { E _ s } ) > 0 . \end{align*}


\begin{align*} F \left ( \xi , r , p , P \right ) = - \mathcal { H } \left ( \xi , r , p , P \right ) . \end{align*}


\begin{align*} 0 \geq \sum _ { p = 1 } ^ n u ^ { p \bar { p } } G _ { p \bar { p } } = \sum _ { p = 1 } ^ n u ^ { p \bar { p } } \left ( \frac { \beta _ { p \bar { p } } } { \beta } - \frac { | \beta _ p | ^ 2 } { \beta ^ 2 } + u u _ { p \bar p } + \vert u _ p \vert ^ 2 + B \right ) . \end{align*}


\begin{align*} \log \mu _ t ^ { ( n ) } ( y ) & = e ^ { - \frac { \sigma ^ 2 } { 2 } t } \log \mu _ 0 ( y ) \\ & + \int _ 0 ^ t \frac { \sigma ^ 2 } { 2 } e ^ { - \frac { \sigma ^ 2 } { 2 } ( t - s ) } \left ( \frac { 2 } { \sigma ^ 2 } \frac { \delta F } { \delta \mu } ( \nu _ s ^ { ( n - 1 ) } , \mu _ s ^ { ( n - 1 ) } , y ) + \log \rho ( y ) + \operatorname { D _ { K L } } ( \mu _ s ^ { ( n - 1 ) } | \rho ) \right ) \mathrm { d } s . \end{align*}


\begin{align*} \tau ( \alpha ^ * _ g ( x ) Y ) = \tau ( x \alpha _ g ( Y ) ) x \in M Y \in L ^ 1 ( M , \tau ) . \end{align*}


\begin{align*} 2 \ , g ( ( \nabla _ { \xi _ i } { f } ) Y , \xi _ j ) \overset { \eqref { 3 . 1 A A } } = N ^ { \ , ( 5 ) } ( \xi _ i , Y , \xi _ j ) \overset { \eqref { K K } } = 0 . \end{align*}


\begin{align*} a _ i = \frac { \Delta _ { i + 1 } \Delta _ { i - 1 } } { \Delta _ i ^ 2 } , b _ i = \frac { \Delta _ { i + p } \Delta _ { i - 1 } } { \Delta _ { i + p - 1 } \Delta _ i } , i \ge 0 , \end{align*}


\begin{align*} T ( f ^ { 2 } ) = 2 f T ( f ) + 2 B ( A ( f ) , A ( f ) ) \end{align*}


\begin{align*} \left [ U \right ] _ { { i , k } } = \begin{cases} \binom { n - i } { n - k } & \textnormal { i f } \ , i \le k ; \\ 0 & \textnormal { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} \overline { v } _ \lambda ( t ) - \overline { v } _ \mu ( t ) & = \int _ 0 ^ t S ( t - s ) \bigl ( F _ \lambda ( s ) \overline { v } _ \lambda ( s ) - F _ \mu ( s ) \overline { v } _ \mu ( s ) \bigr ) \ , d s \\ & + \int _ 0 ^ t S ( t - s ) \sigma ' ( u ( s ) ) \bigl ( \overline { v } _ \lambda ( s ) - \overline { v } _ \mu ( s ) \bigr ) B \ , d W ( s ) , \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( 1 - y ^ m z ^ n \right ) ^ { \frac { m ^ 3 } { n ^ 4 } } = \left ( \frac { 1 } { 1 - y z } \right ) ^ { \frac { y } { 1 - y } } \end{align*}


\begin{align*} g ( t ) = C ( 1 + O ( \varepsilon ) ) f ( t ) , \forall \ , t \in [ - \rho , \rho ] . \end{align*}


\begin{align*} \bar { d } = ( d _ 1 , \cdots , d _ k ) \in \R ^ k _ + , \ \bar { \sigma } = ( \sigma _ 1 , \cdots , \sigma _ k ) \in ( \R _ + ^ { n } ) ^ k . \end{align*}


\begin{align*} { \rm U } ( L ) = \bigoplus _ { \nu } { \rm U } _ \mathbf { \nu } , \end{align*}


\begin{align*} \begin{cases} ( \bar { \mathbf { z } } , t _ 1 ) \in \mathcal { E } , \\ ( \bar { \mathbf { z } } + \mathbf { p } ( t _ 1 ) - \mathbf { p } ( t _ 2 ) , t _ 2 ) \in \mathcal { E } , \\ ( \bar { \mathbf { z } } + \mathbf { p } ( t _ 1 ) - \mathbf { p } ( t _ 2 ) , t _ 3 ) \in \mathcal { E } , \end{cases} \end{align*}


\begin{align*} f _ \Theta ( a ) = \underbrace { \sum _ { i \in S ( \eta ) } [ \Theta ^ { - 1 } y ] _ i U ( a ) _ i } _ { \eqqcolon f _ 1 ( \eta , a ) } + \underbrace { \sum _ { i \notin S ( \eta ) } [ \Theta ^ { - 1 } y ] _ i U ( a ) _ i } _ { \eqqcolon f _ 2 ( \eta , a ) } . \end{align*}


\begin{align*} \{ p t \} = X ^ 0 \subset X ^ 1 \subset X ^ 2 \subset \dots \subset X ^ m . \end{align*}


\begin{align*} { \mathcal { B } = \{ \ell ^ i z _ k \mid 1 \leq k \leq s , \ , 0 \leq i \leq p _ k - 1 \} } , \end{align*}


\begin{align*} d _ 3 ' = e _ 3 ' f _ 3 ' , \quad f ' _ 3 = \prod _ { \nu _ \varpi ( d _ 3 ' ) = \nu _ { \varpi } ( e _ 3 ) + 1 } \varpi , \end{align*}


\begin{align*} \tilde { S } ^ T ( A + H ) \tilde { S } = \tilde { D } \oplus \tilde { D } , \end{align*}


\begin{align*} \left \{ \begin{array} { l } W _ { \varepsilon } ^ { \prime } ( t , \xi ) = i \langle \xi \rangle A _ { \varepsilon } ( t ) W _ { \varepsilon } ( t , \xi ) + i \langle \xi \rangle ^ { - 1 } Q _ { \varepsilon } ( t ) W _ { \varepsilon } ( t , \xi ) + G _ { \varepsilon } ( t , \xi ) , \quad \xi \in \mathcal { I } _ { \hbar } , \\ W _ { \varepsilon } ( 0 , \xi ) = 0 , \xi \in \mathcal { I } _ { \hbar } , \end{array} \right . \end{align*}


\begin{align*} \mathbf { A } _ { 3 k + i + j - 1 } : = \xi _ k \left ( \mathbf { e } _ i \otimes \mathbf { e } _ j - \mathbf { e } _ j \otimes \mathbf { e } _ i \right ) . \end{align*}


\begin{align*} n ^ { - 1 } \left | \sum _ { p = 1 } ^ n u ^ { p \bar p } u _ { p \bar p \bar j } \right | ^ 2 - \sum _ { p , q = 1 } ^ n u ^ { p \bar p } u ^ { q \bar q } | u _ { p \bar q j } | ^ 2 \leq n ^ { - 1 } \left | \sum _ { p = 1 } ^ n u ^ { p \bar p } u _ { p \bar p \bar j } \right | ^ 2 - \sum _ { p = 1 } ^ n | u ^ { p \bar p } u _ { p \bar p j } | ^ 2 \leq 0 . \end{align*}


\begin{align*} \sigma _ { \beta _ N } ^ 2 = \frac { 1 } { R _ N } \bigg ( 1 - \frac { \theta _ N } { \log N } \bigg ) 1 \ll \theta _ N \ll \log N \ , . \end{align*}


\begin{align*} \prod _ { k = 1 } ^ { N _ 1 } U _ k ^ { e _ k } = \frac { m } { C } \end{align*}


\begin{align*} g ( z ) = z \exp \left ( \int _ { 0 } ^ { z } \frac { x ^ 2 + \sqrt { 1 + x ^ 4 } - 1 } { x } d x \right ) = z + \frac { z ^ 3 } { 2 } + \frac { z ^ 5 } { 4 } + \cdots . \end{align*}


\begin{gather*} \boldsymbol { \Phi } ( \mathbf { a } , t ) = \mathbf { a } , \ , \mathbf { U } ( \mathbf { a } , t ) = \mathbf { 0 } , \ , \mathbf { R } ( \mathbf { a } , t ) \mathbf { = I } , \ , \mathbf { W } ( \mathbf { a } , t ) \mathbf { = I , \ } , \ , \mathbf { Z } ( \mathbf { a } , t ) = \mathbf { 0 . } \end{gather*}


\begin{align*} \inf _ { n \in \N } \alpha _ n = \inf _ { n \in \N } \beta _ n . \end{align*}


\begin{align*} \psi ( G ) = \pi ( G ) . \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*} w _ b = \sum _ { n } c _ n a _ n + T ( f ) _ b . \end{align*}


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


\begin{align*} S _ { \alpha } ^ k ( X ) : = \{ s \in C ^ { 2 k - 2 } [ a , b ] : ( D ^ 2 - \alpha ^ 2 ) ^ k s = 0 [ a , b ] \setminus X \} . \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*} \lambda ( X ' ) = ( \varphi _ { A ( X ) } ^ { - 1 } \varphi _ { A ( X ' ) } ) ^ \vee \lambda ( X ) ( \varphi _ { A ( X ) } ^ { - 1 } \varphi _ { A ( X ' ) } ) , \psi ^ { \square } ( X ' ) = \psi ( X ) ^ { \square } ( \varphi _ { A ( X ) } ^ { - 1 } \varphi _ { A ( X ' ) } ) . \end{align*}


\begin{align*} \lim _ { s \to 1 } \sup \left \{ d ( x ) : s \le u ( x ) < 1 \right \} = 0 . \end{align*}


\begin{align*} S \ast g : = \int _ 0 ^ \cdot S ( \cdot - s ) g ( s ) \ , d s S \diamond G : = \int _ 0 ^ \cdot S ( \cdot - s ) G ( s ) \ , d W ( s ) . \end{align*}


\begin{align*} R : = \mathrm { E n d } _ { \kappa } ( E ) = \mathrm { E n d } _ { \overline { \mathbb { F } } _ \ell } ( E ) . \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 } { e ^ { \lambda _ { n } ^ { 2 } x } + 1 } = \frac { 1 } { 4 \left ( 1 + \frac { 1 } { \pi p } \right ) } + \frac { 1 } { 2 } \sqrt { \frac { \pi } { x } } \left ( \sqrt { 2 } - 1 \right ) \zeta \left ( \frac { 1 } { 2 } \right ) + \sqrt { \frac { \pi } { 2 x } } \ , \sum _ { n = 1 } ^ { \infty } \frac { G _ { p } \left ( \frac { \pi ( 2 n - 1 ) } { x } \right ) } { \sqrt { 2 n - 1 } } , \end{align*}


\begin{align*} \partial _ { t } \mu ( t , x ) - \mathrm { d i v } ( \mu \nabla V \star \mu ) ( t , x ) = 0 , \mu ( 0 , \cdot ) = \mu ^ { i n } \end{align*}


\begin{align*} \theta - \ln \theta - 1 = \frac { 1 } { 2 \xi ^ 2 } ( \theta - 1 ) ^ 2 , \\ \rho \ln \rho - \rho + 1 = \frac { 1 } { 2 \eta } ( \rho - 1 ) ^ 2 , \end{align*}


\begin{align*} \int _ 0 ^ x t ^ \mu K _ \nu ( a t ) \ , \mathrm { d } t & = \frac { 2 ^ { \mu - 1 } } { a ^ \mu } \Gamma \bigg ( \frac { \mu - \nu + 1 } { 2 } \bigg ) \Gamma \bigg ( \frac { \mu + \nu + 1 } { 2 } \bigg ) G _ { \mu , \nu } ( a x ) , \\ \int _ x ^ \infty t ^ \mu K _ \nu ( a t ) \ , \mathrm { d } t & = \frac { 2 ^ { \mu - 1 } } { a ^ \mu } \Gamma \bigg ( \frac { \mu - \nu + 1 } { 2 } \bigg ) \Gamma \bigg ( \frac { \mu + \nu + 1 } { 2 } \bigg ) \tilde { G } _ { \mu , \nu } ( a x ) , \end{align*}


\begin{align*} L _ F & = \{ x \in X : x ^ \in \subseteq F \} = X \setminus \bigcup ( C \setminus F ) . \\ X _ F & = \{ x \in X : x ^ \in = F \} = L _ F \setminus \bigcup _ { G \subsetneqq F } L _ G . \end{align*}


\begin{align*} \sum _ { u \in V , u \not = v } \left ( d ( v , u ) - 1 \right ) + \sum _ { 1 \le i \le r ' } \left ( d ( u _ i , v ) - 1 \right ) \ge ( r - 1 ) ^ 2 \end{align*}


\begin{align*} y _ { D 2 } ^ { } = \left \{ \begin{array} { c c c c c c c c c c } h _ { C D } z + n _ { D 2 } , & \mbox { i f } \hat { x } = 1 ; \\ \sqrt { 2 - \alpha } h _ { C D } e ^ { \iota \frac { \pi } { M } } z + n _ { D 2 } , & \mbox { i f } \hat { x } = 0 ; \\ \end{array} \right . \end{align*}


\begin{align*} L ( \Phi ) = \frac { d + 2 } { 4 - d } \omega Q ( \Phi ) + \frac { d - 1 } { 4 - d } \mathbf { c } \cdot \mathbf { P } ( \Phi ) , \ \ \ \ \ \ N ( \Phi ) = - \frac { 2 } { 4 - d } \left ( 2 \omega Q ( \Phi ) + \mathbf { c } \cdot \mathbf { P } ( \Phi ) \right ) . \end{align*}


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


\begin{align*} ( - \Delta ) ^ s V ( x ) = \vartheta | x | ^ { - \beta - 2 s } \quad , \end{align*}


\begin{align*} - \Delta u = f ( u ) \ \ \ \ \ , D . \end{align*}


\begin{align*} \Sigma _ { \mathrm { D } } ( \delta ) = \Pi \left ( \mathcal { B } _ 1 , \mathcal { B } _ 2 \right ) , \mathcal { B } _ 1 = B _ { p _ 1 } ( \mu _ 1 , \delta _ 1 ^ { 1 / p _ 1 } ) \mathcal { B } _ 2 = B _ { p _ 2 } ( \mu _ 2 , \delta _ 2 ^ { 1 / p _ 2 } ) . \end{align*}


\begin{align*} \mathcal { X } = S ( \mathcal { U } ) - \mathcal { K } _ { K , \zeta ^ * } , \end{align*}


\begin{align*} \lim _ { \lambda , \kappa \to 1 ^ + } \limsup _ { m , n \to \infty } \max _ { \substack { m \leq i \leq m ^ \lambda \\ n \leq j \leq n ^ \kappa } } | u _ { i j } - u _ { i n } | = 0 , \end{align*}


\begin{align*} \| \theta ^ { n } \| & \le C \ , \big ( h ^ 2 + N ^ { - \min \{ 2 , \ , r \alpha \} } \big ) , \\ \end{align*}


\begin{align*} L _ { 0 , u v } = \begin{cases} | \{ \mbox { n o n - l o o p e d g e s a t } u \} | + 4 | \{ \mbox { o d d l o o p s a t } u \} | + 2 | \{ \mbox { o d d l e g s a t } u \} | + | \mbox { n u l l l e g s a t } u \} | , & u = v , \\ | \{ \mbox { o d d e d g e s b e t w e e n } u \mbox { a n d } v \} | - | \{ \mbox { e v e n e d g e s b e t w e e n } u \mbox { a n d } v \} | , & u \neq v . \end{cases} \end{align*}


\begin{align*} P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} , A = \begin{bmatrix} u & 0 & 1 \\ 0 & u & 0 \\ 1 & 0 & 0 \end{bmatrix} , B = \begin{bmatrix} v & 0 & 0 \\ 0 & v & 1 \\ 0 & 1 & 0 \end{bmatrix} . \end{align*}


\begin{align*} A = \left [ \begin{matrix} M _ 1 & 0 & N _ 1 ^ T \\ 0 & M _ 2 & N _ 2 ^ T \\ N _ 1 & N _ 2 & M _ 3 \end{matrix} \right ] \end{align*}


\begin{align*} & \int _ { \Omega \setminus B ( \xi , \rho ) } \Big | V ^ { p - 1 } - \sum \limits _ { i = 1 } ^ k ( - 1 ) ^ i ( P U _ { \mu _ i , \xi _ i } ) ^ { p - 1 } \Big | ^ { \frac { n } { 2 } } d x \\ \leq & \sum \limits _ { i = 1 } ^ k \int _ { \Omega \setminus B ( \xi , \rho ) } U _ { \mu _ i , \xi _ i } ^ { ( p - 1 ) \frac { n } { 2 } } d x \leq C \sum \limits _ { i = 1 } ^ k \mu _ i ^ n = O \bigg ( \Big ( \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big ) ^ { \frac { n } { n - 2 } } \bigg ) . \end{align*}


\begin{align*} q ( x , \xi _ x ) : = \sum _ { i , j = 1 } ^ d g ^ { i j } ( x ) \xi _ i \xi _ j \geq C | \xi _ x | ^ 2 , \textnormal { f o r a l l } ( x , \xi _ x ) \in \Omega _ x \times \R ^ d . \end{align*}


\begin{align*} ( \phi \cdot \psi ) ( y ) = \phi ( y ) \cdot \psi ( y ) . \end{align*}


\begin{align*} A = \sum _ { j = 1 } ^ k \lambda _ j ^ { \uparrow } ( A ) P _ j \end{align*}


\begin{align*} S T O ( z _ 1 ) : = \left \{ \begin{array} { l } \displaystyle \min _ { x , y } \ , \ , \mathbb { E } \left [ \theta \left ( x , y , z _ 1 , \zeta ( \omega ) \right ) \right ] \\ \left \{ \begin{array} { l l } x \in X & \\ y \in D ( x ) . & \end{array} \right . \end{array} \right . \end{align*}


\begin{align*} \inf _ { | z | = r \in J _ 1 } \log | A ( z ) | \geq r ^ \gamma , \end{align*}


\begin{align*} \Psi _ N ( t , x _ 1 , . . . , x _ N ) = \Psi _ N ( t , x _ { \sigma ( 1 ) } , . . . , x _ { \sigma ( N ) } ) \forall \sigma \in S _ N , \ ; \forall x _ i \in \R ^ 2 \ , , \end{align*}


\begin{align*} \textup { M M D } ^ 2 ( P _ X , P _ Y ) = O ( p ^ { - 2 r + 2 } ) . \end{align*}


\begin{align*} h = k M Y ^ { k - 1 } + R _ { k - 2 } ( Y , M ) . \end{align*}


\begin{align*} \gamma \big ( ( n _ E ) _ { E \in \mathbf { E } } , ( s _ i ) _ { 1 \le i \le r } , ( \ell _ { E , v } ) _ { v \in E \in \mathbf { E } } \big ) = { } & \big ( ( s _ i ) _ { 1 \le i \le r } , ( \ell _ { E , v _ E } + \ell _ { E , v ' _ E } ) _ { E \in \mathbf { E } } \big ) , \\ \delta \big ( ( n _ { E , v } ) _ { v \in E \in \mathbf { E } } \big ) = { } & \big ( n _ { E , v _ E } - n _ { E , v ' _ E } \big ) _ { E \in \mathbf { E } } . \end{align*}


\begin{align*} f _ i = \sum _ { j = 1 } ^ { s _ i } a _ { i , j } \textbf { Q } ^ { \lambda _ { i , j } } \end{align*}


\begin{align*} \widetilde { \mathcal { F } } _ { \gamma } : = f ^ { + } - \gamma E _ { g | V ( p ) } . \end{align*}


\begin{align*} y _ { B 2 } = \sqrt { 2 - \alpha } h _ { C B } z e ^ { \frac { \iota \pi } { M } } + n _ { B 2 } . \end{align*}


\begin{align*} \lim _ { \lambda \to 1 ^ + } \limsup _ { m , n \to \infty } \max _ { m \leq i \leq m ^ \lambda } | u _ { i n } - u _ { m n } | = 0 , \end{align*}


\begin{align*} g ( \kappa ) = 0 \Longleftrightarrow \kappa = 0 . \end{align*}


\begin{align*} & \frac { 2 } { \pi } \int _ { - ( \log { t } ) ^ { 2 } } ^ { ( \log { t } ) ^ { 2 } } \log \zeta ( \sigma + i ( t + u ) ) \left ( \frac { \sin { ( u / 2 ) } } { u } \right ) ^ 2 ( 1 + \cos ( \theta + u \log { x } ) ) d u \\ & = \frac { 1 } { 2 } e ^ { i \theta } \sum _ { e ^ { - 1 / 2 } x \le n \le e ^ { 1 / 2 } x } \frac { \Lambda ( n ) } { n ^ { \sigma + i t } \log { n } } \left ( \frac { 1 } { 2 } - \left | \log \frac { n } { x } \right | \right ) + O \left ( \frac { x } { ( \log { t } ) ^ 2 } \right ) . \end{align*}


\begin{align*} \log \det ( u _ { i \bar { j } } ) = \log \psi ( z , u ) = : f ( z , u ) \end{align*}


\begin{align*} { \bf { b } } \widetilde { \mathbb { S } } _ { \lambda ^ i } { \bf { b } } \widetilde { \mathbb { S } } _ { \lambda ^ j } & = \sum _ { \lambda ^ q \succeq \lambda ^ i , \lambda ^ j } \sum _ { P : \delta P = \Delta _ { \sigma _ r \lambda ^ i ~ \sigma _ r \lambda ^ j } ^ { \sigma _ r \lambda ^ q } } \sum _ { s = 0 } ^ { | P | } a ^ { ( P ) } _ s ( { \bf { b } } \widetilde { \mathbb { S } } _ { \lambda ^ 1 } ) ^ s { \bf { b } } \widetilde { \mathbb { S } } _ { \lambda ^ q } \\ \end{align*}


\begin{align*} ( \prod \limits _ { i = 1 } ^ m a ( i ) \ast g _ w ( t ( i ) ) \ast a ( m + 1 ) \in A \cap \sigma ( L ) \mbox { f o r a l l } g _ w \in G . \end{align*}


\begin{align*} 1 = \| T v \| ^ 2 = \langle \psi _ { Y , Z } ( T ) v , T v \rangle = \langle ( I \otimes T ) Z ^ * v , Y ^ * T v \rangle . \end{align*}


\begin{align*} \lim _ { l \to \infty } \norm { B _ l ' ( y _ 1 ' - B _ k ' ( y _ 1 ' ) ) } \leq \frac { 1 } { m ( I _ k ) } \int _ { I _ k } \lim _ { l \to \infty } \frac { m ( I _ l \Delta h I _ l ) } { m ( I _ l ) } d m ( h ) = 0 . \end{align*}


\begin{align*} \alpha _ { 1 } \left \Vert x - P _ { 1 } ( x ) \right \Vert ^ { p - 2 } \left ( x - P _ { 1 } ( x ) \right ) + \alpha _ { 2 } \left \Vert x - P _ { 2 } ( x ) \right \Vert ^ { p - 2 } \left ( x - P _ { 2 } ( x ) \right ) = 0 , \end{align*}


\begin{align*} \mathcal J _ 2 = \left ( \left . h _ i d \left ( \sum \limits _ { j = 1 } ^ { s _ \ell } b _ { i _ { } i j } { \textbf { X } } ^ { \lambda _ j } \right ) \ \right | \ i \in I \setminus \{ i _ { } \} \right ) . \end{align*}


\begin{align*} \int _ \Omega \nabla u ^ * \cdot \nabla v \ d x - \lambda ^ * \int _ { \Omega } u ^ * v \ d x \forall \ v \in H _ 0 ^ 1 ( \Omega ) \mbox { a n d } \| u ^ * \| _ { L ^ 2 ( \Omega ) } = 1 . \end{align*}


\begin{align*} \tilde a _ 0 ( x ) = \begin{cases} f ( x ) & \mbox { i f } x \in \mathcal { B } ( e , \theta ) , \\ 1 & \mbox { i f } x \in \mathcal { B } ( x _ { 1 } , \epsilon ) , \\ 0 & \mbox { i f } x \notin \mathcal { B } ( e , \theta ) \cup \mathcal { B } ( x _ { 1 } , \epsilon ) . \end{cases} \end{align*}


\begin{align*} r ' = \frac { \sqrt { d } } { \log ^ { 1 / 2 } ( 1 / \sqrt { \gamma } ) } \mbox { a n d } \gamma ' = \left ( \frac { 3 3 \cdot ( 2 r ) } { \sqrt { d } } \right ) ^ d \gamma \log ^ { d / 2 } \left ( \frac { 1 } { \sqrt { \gamma } } \right ) . \end{align*}


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


\begin{align*} ( q ; q ) _ \infty ^ { \mathrm { r k } ( \mathfrak { s } ) } \prod _ { \alpha \in \Delta _ + } ( 1 - x ^ \alpha ) \prod _ { \alpha \in \Delta } ( q x ^ \alpha ; q ) _ \infty & = \sum _ { k = 0 } ^ \infty a _ k ( x ) q ^ k . \end{align*}


\begin{align*} X - a Y & = \ell \\ - a X + Y & = m \end{align*}


\begin{align*} \lambda = { \rm m a x } _ { j ' < j _ { \varepsilon , 0 } } b _ { j ' } \rho ^ { j ' } . \end{align*}


\begin{gather*} \left \| f ( \cdot , t ) \right \| _ { L ^ p ( \mathbb { T } ^ d ) } \le \left \| f _ 0 \right \| _ { L ^ p } , \\ \int _ { \mathbb { T } ^ d } f ( \cdot , t ) \ ; d x = \int _ { \mathbb { T } ^ d } f _ 0 \ ; d x . \end{gather*}


\begin{align*} \begin{aligned} - d \hat { y } _ t & = \big [ g ^ 1 ( x , y _ 1 , z _ 1 , \tilde { z } _ 1 , \gamma _ { ( 1 , t , e ) } ) - g ^ 2 ( x , y _ 2 , z _ 2 , \tilde { z } _ 2 , \gamma _ { ( 2 , t , e ) } ) \big ] d t \\ & \quad - \hat { z } _ t d W _ t - \hat { \tilde { z } } _ t d \xi _ t - \int _ { \mathcal { E } } \hat { \gamma } _ { t , e } \tilde { N } ( d e , d t ) , \ t \in [ 0 , \infty ] . \end{aligned} \end{align*}


\begin{align*} q - I _ { \boldsymbol { a } } ( z ) = h ( z ) ( 1 - z a _ { j _ 1 } ) ^ { \beta } , z \in \mathbb { D } _ 1 \cap B ( 1 / a _ { j _ 1 } , r _ 1 ) . \end{align*}


\begin{align*} X _ 1 = \partial _ x - \frac { y } 2 \partial _ z , X _ 2 = \partial _ y + \frac { x } 2 \partial _ z , \end{align*}


\begin{align*} \dot \lambda _ t = \vec H ( \lambda _ t ) . \end{align*}


\begin{align*} \rho _ { \theta } ( \theta , f ( \theta ) ) + 2 { \rm R e } ( \rho _ w ( \theta , f ( \theta ) ) \frac { \partial f } { \partial \theta } ( \theta ) ) = 0 . \end{align*}


\begin{align*} P _ n ( \cos \theta _ 1 \cos \theta _ 2 \pm \sin \theta _ 1 \sin \theta _ 2 \cos \phi ) = \sum _ { k = - n } ^ n ( \pm 1 ) ^ k \frac { ( n - k ) ! } { ( n + k ) ! } { \sf P } _ n ^ k ( \cos \theta _ 1 ) { \sf P } _ n ^ k ( \cos \theta _ 2 ) \cos ( k \phi ) , \end{align*}


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


\begin{align*} \begin{aligned} \mathcal { L } V ( x ) & = C \big [ \beta ( \beta + 2 - N ) | x | ^ { - \beta - 2 } - \vartheta | x | ^ { - \beta - 2 s } \big ] \\ & \leq - C \vartheta | x | ^ { - \beta - 2 s } \leq - C \vartheta | x | ^ { - \alpha } \quad , \end{aligned} \end{align*}


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


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


\begin{align*} f = f _ 1 + f _ 2 + \sum _ { j = 1 } ^ { | G | - 1 } f _ { \sigma _ j } , \quad \mbox { w h e r e } f _ 1 = f \chi _ { 5 B } , \ f _ 2 = f \chi _ { ( { \mathcal O } ( 5 B ) ) ^ c } , \ f _ { \sigma _ j } = f \chi _ { U _ j } . \end{align*}


\begin{align*} \{ \underline { a } \colon \underline { a } / \varpi ^ k \in L ( \varpi ^ k \underline { c } ) \} = & \{ \underline { d } \colon ( \underline { d } , \varpi ) = 1 , | \underline { d } | < | \varpi | ^ k \} \setminus \\ & \{ a \underline { c } ^ \bot + \varpi \underline { d } \colon ( a , \varpi ) = 1 , | a | < | \varpi | , | \underline { d } | < | \varpi | ^ { k - 1 } \} . \end{align*}


\begin{align*} g _ k = \sum _ { i = 1 } ^ n ( \alpha _ i - \beta _ i ) g _ { k i } , f _ k = \sum _ { i = 1 } ^ n ( \beta _ i - \gamma _ i ) f _ { k i } , e _ k = \sum _ { i = 1 } ^ n ( \gamma _ i - \alpha _ i ) e _ { k i } , \end{align*}


\begin{align*} \left \{ \begin{array} { l } \mathcal { X } _ j ^ y = X ^ y _ { \tau _ { j H _ k } } ; \\ \mathcal { N } _ j ^ y = N ^ y _ { \tau _ { j H _ k } } . \end{array} \right . \end{align*}


\begin{align*} \mathcal Y _ k ( h ) = \mathcal S _ { k } ( h ) - \mathcal S _ { k - 1 } ( h ) , k \leq n _ { \mathcal L } . \end{align*}


\begin{align*} \int _ { B _ { r } } ( b ^ { i j , k l } ) _ { r } w _ { i j } \eta _ { k l } d x & = 0 , \forall \eta \in C _ { 0 } ^ { \infty } ( B _ { r } ) \\ w & = f \partial B _ { r } \\ D w & = D f \partial B _ { r } . \end{align*}


\begin{align*} T _ k = \sup \big \{ t \geq 0 : E _ t ( u _ k ) \subset B ( k ) \big \} . \end{align*}


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


\begin{align*} I _ { \omega , \mathbf { c } } ( \Phi ) = 0 , \ \ \mathbf { c } \cdot \mathbf { P } ( \Phi ) \ge 0 . \end{align*}


\begin{align*} \Theta _ r & = a X _ 1 + b X _ 2 + c Y _ 1 + d Y _ 2 \end{align*}


\begin{align*} ^ { c } _ { 0 } { D } ^ { \alpha } _ { t } u ( x , t ) - a ( l ( u ) ) \ : \Delta u ( x , t ) & = f ( x , t ) & \mbox { i n } & \Omega \times ( 0 , T ] , \\ u ( x , t ) & = 0 & \mbox { o n } & \partial \Omega \times ( 0 , T ] , \\ u ( x , 0 ) & = u _ 0 ( x ) & \mbox { i n } & \Omega , \end{align*}


\begin{align*} R _ k ' ( t ) & = k \sum _ j \nu ( j ) M _ j ( \boldsymbol { a } , t ) ^ k \frac { M ' _ j ( \boldsymbol { a } , t ) } { M _ j ( \boldsymbol { a } , t ) } \\ & = k \left ( q R _ { k + 1 } ( t ) + \varphi ( t ) R _ k ( t ) \right ) . \end{align*}


\begin{align*} T _ { b ^ { \mu } } \left ( x ^ { \mu } \right ) = T _ { \overline { b } } \left ( \overline { x } \right ) \cap T _ { b } \left ( x \right ) , \mu \in ] 0 , 1 [ . \end{align*}


\begin{align*} x _ { j } ( v ) = x _ { 0 , j } + \sum _ { i \neq j } v _ { i j } - \sum _ { k \neq j } v _ { j k } , \quad \forall j \in I . \end{align*}


\begin{align*} \lambda ' = \varphi _ { A ' } ^ \vee Y _ { \mathbb { Q } } ^ \vee \overline { \lambda } _ 0 Y _ { \mathbb { Q } } \varphi _ { A ' } ; \end{align*}


\begin{align*} y z t = x ^ 2 + 1 . \end{align*}


\begin{align*} \Vert \xi \Vert _ { \psi _ { 2 } } = \inf \{ K > 0 : \textsf { E } \exp ( \frac { \xi ^ { 2 } } { K ^ { 2 } } ) \le 2 \} . \end{align*}


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


\begin{align*} \psi ( a b ^ * ) = \langle \pi ( a b ^ * ) \xi , \xi \rangle , a , b \in A . \end{align*}


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


\begin{align*} P _ 0 = \frac { h } { i } X - i ( Q _ \infty + q _ 1 ) \in \Psi ^ 1 _ h , \Tilde { P } _ h ( z ) = P _ 0 - i M h W - z \in \Psi ^ 1 _ h . \end{align*}


\begin{align*} D ^ t = i \sqrt { \pi c } \left ( \begin{array} { c c } 0 & \mp 2 \\ 3 t - 1 & 0 \end{array} \right ) . \end{align*}


\begin{align*} \psi ( z _ 1 , \ldots , z _ m ) : = \frac { f ( z _ 1 , \ldots , z _ m ) + \overline { f ( \overline { z _ 1 } , \ldots , \overline { z _ m } ) } } { 2 } . \end{align*}


\begin{align*} J _ { 2 ( k + \ell ) } & = \left ( J _ { 2 k } ^ T \diamond O _ { 2 k , 2 \ell } \right ) ^ T \diamond \left ( O _ { 2 \ell , 2 k } \diamond J _ { 2 \ell } ^ T \right ) ^ T , \end{align*}


\begin{align*} \left ( \frac { \pi } { \sqrt { c } } \right ) ^ { - s } \Gamma ( s ) \ , \eta _ { p } ( 2 s ) = c ^ { 1 / 4 } \left [ \frac { 1 } { 1 + \frac { 1 } { \pi p } } \cdot \frac { 1 } { 2 s - 1 } + C _ { p } ^ { ( 2 ) } - \frac { 2 e ^ { 2 \pi p } } { 1 + \frac { 1 } { \pi p } } Q _ { 2 \pi p } ( 0 ) - \frac { 1 } { 2 } \left ( \gamma + \log \left ( \frac { 4 \pi } { \sqrt { c } } \right ) \right ) + O \left ( s - \frac { 1 } { 2 } \right ) \right ] . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } \Delta w = \frac { \mu _ 2 } { 3 } f \theta _ B & \textnormal { i n } & \Omega \\ w = 0 & \textnormal { o n } & \partial \Omega , \end{array} \right . \end{align*}


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


\begin{align*} \mu \left ( \left ( \Omega _ \delta \cap \left ( \overline { J _ u } \setminus J _ u \right ) \setminus \Sigma _ \delta \right ) \right ) = \mathcal { H } ^ 1 ( \underbrace { J _ u \cap ( \Omega _ \delta \cap \left ( \overline { J _ u } \setminus J _ u \right ) \setminus \Sigma _ \delta } _ { = \varnothing } ) ) = 0 , \end{align*}


\begin{align*} \d ( x , y ) : = F ( x ) - F ( y ) - \langle F ' ( y ) , x - y \rangle . \end{align*}


\begin{align*} \mathfrak { I } _ 2 u - \mathfrak { I } _ 1 u = & T \beta ^ { - 1 } ( \rho ( a , b ) u ) - \alpha ^ { - 1 } ( [ a , b , T u ] ) \\ = & T \rho ( a , b ) \beta ^ { - 1 } ( u ) - [ a , b , T \beta ^ { - 1 } ( u ) ] \\ = & \wp ( a , b ) u \in \mathcal { B } ^ 1 _ T ( V , L ) , \end{align*}


\begin{align*} E _ { i + 1 } \mathcal { T } _ { q _ i } ( v _ l ^ { ( i ) } ) = \mathcal { T } _ { q _ { i + 1 } } E _ i ( v _ l ^ { ( i ) } ) \mathrm { f o r } \ \mathrm { a l l } \ i \in \mathbb { Z } _ r , \ , l \in [ \omega n ] . \end{align*}


\begin{align*} D ( k ) = \frac { 1 } { \abs { k } ^ { 2 \alpha } } + \abs { k } ^ { 2 \gamma } , \end{align*}


\begin{align*} \widetilde { \Gamma } ( M , q ) : = \{ \alpha \in \mathcal { C } ^ { \times } | \alpha ( \mathbb { F } \oplus V ) \alpha '^ { - 1 } = \mathbb { F } \oplus V \} . \end{align*}


\begin{align*} c _ { \ell } ( s _ \ell , s _ \ell ' ) = c _ { Y _ \ell } ( y _ \ell , y _ \ell ' ) + b ( x , x ' ) , \end{align*}


\begin{align*} \mathcal { B } _ s ( u , v ) = \int _ { \Omega } \langle \nabla u , \nabla v \rangle \ , d x + \frac { C _ { N , s } } { 2 } \iint _ { \R ^ { 2 N } } \frac { ( \hat { u } ( x ) - \hat { u } ( y ) ) ( \hat { u } ( x ) - \hat { u } ( y ) } { | x - y | ^ { N + 2 s } } \ , d x \ , d y + \int _ \Omega V ( x ) u v \ , d x , \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ n } J _ \varepsilon ( | x | ) x _ i \ : x = 0 \end{align*}


\begin{align*} \underline { d } ( x , y ) : = \liminf _ { n \to \infty } \frac { 1 } { n } | \{ 1 \leq i \leq n : x ( i ) \neq y ( i ) \} | \end{align*}


\begin{align*} \| \{ \gamma _ j \} _ { j = 1 } ^ \infty \| _ { \ell _ { q ' } } = \Big ( \sum _ { j = 1 } ^ \infty | \gamma _ j | ^ { q ' } \Big ) ^ \frac 1 { q ' } \leq \varepsilon _ 2 . \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } \frac { \log \norm { S ( k ) \mathfrak { r } _ { a , n } ( \varphi ) } _ { \sup } } { k } = \lim _ { k \rightarrow \infty } \frac { \log \frac { \norm { S ( k ) \mathfrak { r } _ { a , n } ( \varphi ) } _ { L ^ 1 ( I ^ { ( k ) } ) } } { | I ^ { ( k ) } | } } { k } = ( a - n ) \lambda _ 1 . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 2 d + 2 } } \left ( f _ { \mathcal { S } } \right ) _ \lambda d \pi = \mathbb { E } [ Y _ 2 ] - \mathbb { E } [ Y _ 1 ] + \mathrm { R } ( \lambda , \delta ) , \end{align*}


\begin{align*} \lim _ { h \to + \infty } \mathcal { G } ( u _ h , \beta _ h , B _ s ) = : \Lambda ( s ) \end{align*}


\begin{align*} F _ { I } = \left \{ x \in \mathbb { R } ^ { m n } \ | \ x _ { s } - x _ { t } = \mathbf { 0 } , \ i = ( s , t ) \in I \right \} , \end{align*}


\begin{align*} { T _ { \varepsilon } ^ { \vec { v } } } = { T _ { \widehat { \varepsilon } } ^ { \widehat { \vec { v } } } } . \end{align*}


\begin{align*} \tilde { S } _ { \gamma _ i \gamma _ i } = M _ { [ 1 ] } \diamond \cdots \diamond M _ { [ | \alpha _ i | ] } = N _ { [ i ] } + \mathcal { O } ( \| H \| ^ 2 ) . \end{align*}


\begin{align*} \mathrm { S h } ( V ) = \oplus _ { k \geq 0 } [ \underbrace { \left ( \begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \right ) | \ldots | \left ( \begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \right ) } _ { k } | \mathrm { S h } ^ * ( V ) ] . \end{align*}


\begin{align*} \lim _ { h \to + \infty } t _ h \abs { 1 - \frac { \lambda _ h ^ { ( 1 / 2 ) } } { t _ h } } = d , \end{align*}


\begin{align*} 1 + q \rho d X + \binom { q } { 2 } \rho ^ 2 d ^ 2 X ^ 2 + e \frac { q ( q - 2 ) } { 6 } \rho ^ 3 | d X | ^ 2 , \end{align*}


\begin{align*} X ^ \dag = X ' ( w ) . \end{align*}


\begin{align*} p ( z ) = \frac { 1 + \tau _ 1 z } { 1 - \tau _ 1 z } , \ ; \ ; z \in \mathbb { D } . \end{align*}


\begin{align*} \gamma ' = \left ( \frac { 1 0 0 r } { \sqrt { d } } \right ) ^ d \gamma _ 1 \log ^ { d / 2 } \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) . \end{align*}


\begin{align*} \| u \| _ { L ^ 2 ( \mathrm { M } ) } ^ 2 = \sum _ { k \ge 1 } \frac { | ( f | \phi _ q ^ k ) | ^ 2 } { | \lambda _ q ^ k - \lambda | ^ 2 } . \end{align*}


\begin{align*} | \varpi | ^ { 2 m + k } N _ 3 ( \varpi ^ k ) \ll | \varpi | ^ { m + n k } + | \varpi | ^ { m + n k } \sum _ { l = 0 } ^ { k - 1 } | \varpi | ^ { ( n / 6 - 2 ) ( l - k ) } \ll | \varpi | ^ { m + n k } + | \varpi | ^ { m + n k } \end{align*}


\begin{align*} \mathcal { A } \vect u = z \vect u , \Gamma _ 1 \vect u = \mathcal { B } ( z ) \Gamma _ 0 \vect u , \end{align*}


\begin{align*} \left ( { D } ^ { \alpha } _ { N } U ^ { n - \sigma } , v _ h \right ) \ , - \ , a \big ( l ( U ^ { n , \sigma } ) \big ) \ , ( \Delta _ h U ^ { n , \sigma } , v _ h ) \ , = & \ , ( f ^ { n - \sigma } , v _ h ) , \ ; \ , \forall v _ h \in V _ h . \\ \end{align*}


\begin{align*} \mathcal { R } ( v , \gamma , \gamma _ t , \sigma _ t ) & = - \gamma _ t v - \frac { 1 } { N } \sigma _ t v + \beta \mathcal { J } \left [ \frac { \gamma _ { x x } } { \left ( 1 - \gamma _ x \right ) ^ 2 } v - \frac { \gamma _ x ^ 2 } { 1 - \gamma _ x } \phi ' \right ] , \\ \mathcal { P } ( v , \gamma ) & = - \beta \mathcal { J } \left [ \gamma _ x + \frac { \gamma _ x } { 1 - \gamma _ x } \right ] v , \end{align*}


\begin{gather*} a _ i ( n ) \ : = \ a _ i , \\ L ( n ) \ : = \ L , \end{gather*}


\begin{align*} \beta _ { \epsilon } ( t ) = \frac { 1 } { 1 + 2 i t \epsilon ^ 2 } , \end{align*}


\begin{align*} \dim W ^ { \mathrm { u } } ( p ^ - ) = 2 n - k ^ - , \dim W ^ { \mathrm { s } } ( p ^ + ) = k ^ + . \end{align*}


\begin{align*} x = \frac { m } { ( n , m ) } u , y = \frac { n } { ( n , m ) } u , \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle G ' ( t ) = - 4 I m \int _ { \mathbb { R } ^ N } \left ( \dfrac { a _ 2 } { 2 m _ 1 } x \phi \nabla \overline { \phi } + \dfrac { a _ 1 } { 2 m _ 2 } x \psi \nabla \overline { \psi } \right ) d x , \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} \phi _ \lambda ( v , s ^ { \prime } ) & = f ( s ^ { \prime } ) - \lambda _ 1 \boldsymbol { d } _ { \mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ { \prime } ) ^ { p _ 1 } - \lambda _ 2 \boldsymbol { d } _ { \mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ { \prime } ) ^ { p _ 2 } \\ & \geq B + ( B - \lambda _ 1 ) \boldsymbol { d } _ { \mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \prime ) ^ { p _ 1 } + ( B - \lambda _ 2 ) \boldsymbol { d } _ { \mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \prime ) ^ { p _ 2 } \geq B . \end{aligned} \end{align*}