\begin{align*} \frac { \delta F } { \delta m } ( m ' , x ) - \frac { \delta F } { \delta m } ( m , x ) = \int _ { 0 } ^ { 1 } \int _ { \mathcal { M } } \frac { \delta ^ 2 F } { \delta m ^ 2 } ( m + \lambda ( m ' - m ) , x , x ' ) \left ( m ' - m \right ) ( \mathrm { d } x ' ) \ , \mathrm { d } \lambda . \end{align*}


\begin{gather*} \Phi ^ - ( \nu , m ) \coloneqq \{ \beta \in P ( \R ^ m ) : \beta = \phi ( \nu ) \phi \in \mathcal { M } \} , \\ \Phi ^ + ( \mu , n ) \coloneqq \{ \alpha \in P ( \R ^ n ) : \mu = \phi ( \alpha ) \phi \in \mathcal { M } \} , \end{gather*}


\begin{align*} \alpha _ i = \nu \left ( \frac { d } { d x } ( x - a _ i ) \right ) - \nu ( x - a _ i ) = \nu ( 1 ) - \nu ( x - a _ i ) = - \nu ( x - a _ i ) . \end{align*}


\begin{align*} \| A _ s ( \mathrm { i , j } ) + x B _ s ( \mathrm { i , j } ) \| _ { ( p , k ) } ^ p = 2 ^ { s - 1 } + a _ s x ^ p \hbox { f o r } 0 < x < 1 , \end{align*}


\begin{align*} \left . \begin{array} { l l } \xi = ( \xi _ { \infty } ) _ { \beta } , \eta = ( \eta _ { \infty } ) _ { \beta } , \end{array} \right . \end{align*}


\begin{align*} \xi = \prod _ { v \in V ( X ) } c ( v ) \sum _ { T \subset X } \prod _ { e \in E ( T ) } c ( e ) ^ { - 1 } , \end{align*}


\begin{align*} L _ a = \frac { 1 } { n } \sum _ { j , k = 1 } ^ n a _ { j \bar k } \frac { \partial ^ 2 } { \partial z _ j \partial \bar z _ k } . \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 } } = \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( \frac { y - ( 1 + n ) y ^ { n + 1 } + n y ^ { n + 2 } } { ( 1 - y ) ^ 2 } \right ) \frac { z ^ n } { n ^ 2 } \right \} \end{align*}


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


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


\begin{align*} \rho \ , u _ t - \Delta u + ( - \Delta ) ^ s u = \ , 0 \textrm { i n } \ ; \ ; S _ T \ , , \end{align*}


\begin{align*} e ^ { \pm H \widetilde { d } } = e ^ { \pm H { \rm R e } ( q ) } e ^ { \pm H \widetilde { e } } \end{align*}


\begin{align*} L _ { 0 } w = 0 B _ { R } \implies \left \Vert D w \right \Vert _ { L ^ { \infty } ( B _ { R / 4 } ) } \leq C _ { 3 } ( \lambda , n , p _ 0 , R ) \left \Vert w \right \Vert _ { L ^ { p _ 0 } ( B _ { R } ) } . \end{align*}


\begin{align*} \mathcal { F } ( f ) ( \xi ) : = \int _ { \mathbb { R } ^ n } e ^ { - i x \cdot \xi } f ( x ) \ : x . \end{align*}


\begin{align*} \sigma \cdot ( x , w , z ) = ( \sigma \cdot x , \sigma \cdot w , z ) \ . \end{align*}


\begin{align*} \int _ M \frac { S } { m - 1 } w ^ 2 = \int _ M | \nabla w | ^ 2 \ , . \end{align*}


\begin{align*} \lambda _ t = \lambda _ { j _ \ell } = \lambda _ { j _ \ell ^ * } = \pm 1 . \end{align*}


\begin{align*} ( \Lambda _ { 2 , 0 } ) ^ { \sharp } = S ^ { n - 1 } _ 0 \times C ' \end{align*}


\begin{align*} I _ { 1 , j } = \sum _ { k = 1 } ^ d ( \nabla \varphi _ { j } ^ { ( k ) } \cdot \nabla \theta _ { \epsilon } , \varphi _ { j } ^ { ( k ) } ) , \ \ \ I _ { 2 , j } = ( i ( \mathbf { c } \cdot \nabla ) \varphi _ j , \theta _ { \epsilon } \varphi _ j ) , \ \ ( \varphi _ j = ( \varphi _ j ^ { ( 1 ) } , \cdots , \varphi _ j ^ { ( d ) } ) ) \end{align*}


\begin{align*} \rho _ w ( 1 + , w ) = \overline { \rho _ w ( 1 - , \overline { w } ) } = \overline { \rho _ w ( 1 - , w ) } \end{align*}


\begin{align*} - \frac { 1 } { 8 e ^ 7 } \frac { 1 } { p } < - | S | \log r = - | S | \end{align*}


\begin{align*} \widetilde { B } ( \xi ) = e ^ { - i \frac { \pi + 2 \theta } { 4 } } \end{align*}


\begin{align*} x ^ 4 + x y + 2 y ^ 3 = 0 . \end{align*}


\begin{align*} D _ { r , z } u ( t , x ) = & \sum _ { j \geq 1 } \sum _ { n \geq j } I _ { n - 1 } ( f _ j ^ { ( n ) } \big ( \cdot , r , z , t , x ) \big ) \\ = & \sum _ { j \geq 1 } I _ { j - 1 } \big ( f _ { j - 1 } ( \cdot , r , z ) \big ) \sum _ { n \geq j } I _ { n - j } \big ( g _ { n - j } ( \cdot , r , z , t , x ) \big ) . \end{align*}


\begin{align*} \varphi _ { \lambda , \ell } ( x _ 1 , \dotsc , x _ L ) = \min _ { x ' , d ( x ' ) = \ell } \sum _ { \ell = 1 } ^ { L } \lambda _ \ell \| x _ \ell - x ' \| _ { 2 } . \end{align*}


\begin{align*} [ b _ \ell ( s ) , D _ k ^ * ( t ) ] & = \int _ { \Lambda ^ * } \chi ^ \perp ( p ) \chi ^ \perp ( p - k ) \chi ( p - \ell ) e ^ { i ( t - s ) E _ p } a _ { p - \ell } ( s ) a _ { p - k } ( t ) \d p \\ & + \int _ { \Lambda ^ * } \chi ( h ) \chi ( h + k ) \chi ^ \perp ( h + \ell ) e ^ { i ( t - s ) E _ h } a _ { h + \ell } ( s ) a _ { h + k } ( t ) \d h \ . \end{align*}


\begin{align*} a _ 0 \ = \ \begin{cases} 3 , & , \\ 8 , & . \end{cases} \end{align*}


\begin{align*} w _ 0 ( r e ^ { i \theta } ) : = i c - I + \frac { 1 } { 2 \pi } \langle h _ b , P _ r ( \theta - \cdot ) \rangle . \end{align*}


\begin{align*} s = \left ( \begin{smallmatrix} a _ { 1 , 1 } & 0 \\ 0 & a _ { 2 , 2 } \end{smallmatrix} \right ) = \left ( \begin{smallmatrix} a _ { 1 , 1 } & 0 \\ 0 & a _ { 1 , 1 } ^ { - 1 } \end{smallmatrix} \right ) \ , \ , s = \left ( \begin{smallmatrix} 0 & a _ { 1 , 2 } \\ - a _ { 2 , 1 } & 0 \end{smallmatrix} \right ) = \left ( \begin{smallmatrix} 0 & a _ { 1 , 2 } \\ - a _ { 1 , 2 } ^ { - 1 } & 0 \end{smallmatrix} \right ) . \end{align*}


\begin{align*} B = \begin{bmatrix} A & I _ n \\ I _ n & A \end{bmatrix} , \end{align*}


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


\begin{align*} \lim _ { r \to \infty } d \left ( \bigcup _ { s \leq r } \operatorname { P e r } ( x , s ) \right ) = 1 . \end{align*}


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


\begin{align*} & \ , \cos ^ 3 \alpha \ , \bigg ( \ ! ( 1 + \Delta / R ) ^ 3 \bigg ( 1 + \frac { 3 } { 2 } \tan ^ 2 \alpha \ , ( \Delta / R ) \big ( 2 + O ( \Delta / R ) \big ) + \tan ^ 4 \alpha \ , O ( \Delta ^ 2 / R ^ 2 ) \ ! \bigg ) - 1 \bigg ) \\ & = \cos ^ 3 \alpha \ , \Big ( \ ! ( 1 + \Delta / R ) ^ 3 - 1 + ( 1 + \Delta / R ) ^ 3 ( \Delta / R ) \tan ^ 2 \alpha \ , \big ( 3 + O ( \Delta / R ) + \tan ^ 2 \alpha \ , O ( \Delta / R ) \big ) \ ! \Big ) . \end{align*}


\begin{align*} B _ { V ^ 1 L ^ { p , q } ( \Sigma , \mu ) } \subseteq \bigcup _ { k = 1 } ^ m \big ( g _ k + r B _ { L ^ { p ^ * , q } ( \Sigma , \mu ) } \big ) . \end{align*}


\begin{align*} \lim _ { m \rightarrow \infty } \sum _ { 0 \leq n < 2 ^ { m } } ( 1 - \frac { 2 n } { 2 ^ { m + 1 } } ) ^ { \frac { \gamma ( i + 1 ) - 1 - \delta - \frac { \epsilon } { 2 } } { 1 - \gamma + \delta + \epsilon } } \frac { 1 } { 2 ^ { m + 1 } } = \frac { 1 } { 2 } \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { \frac { \gamma ( i + 1 ) - \delta - 1 - \frac { \epsilon } { 2 } } { 1 - \gamma + \delta + \epsilon } } \mathrm { d } x , \end{align*}


\begin{align*} \mu ( \bigcap _ { s \in U } B _ s ) & = \int \prod _ s \chi _ { B _ s } ( \{ i _ j \} _ { j \in s } ) \ , d \mu ( i _ 1 , \ldots , i _ n ) \\ & = \int \chi _ { B _ t } ( \{ i _ j \} _ { j \in t } ) \int \prod _ { s \neq t } \chi _ { B _ s } ( \{ i _ j \} _ { j \in s } ) \ , d \mu \ , d \mu ( \{ i _ j \} _ { j \in [ n ] \setminus t } ) . \end{align*}


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


\begin{align*} z _ t & : = M _ t ( \hat { x } , \bar { y } ) ( \hat { x } - \bar { y } ) , \\ M _ t ( \hat { x } , \bar { y } ) & : = A _ \gamma ( \hat { x } , \hat { x } ) + t \big ( A _ \gamma ( \hat { x } , \bar { y } ) - A _ \gamma ( \hat { x } , \hat { x } ) \big ) \end{align*}


\begin{align*} \sigma _ n = \frac { 1 } { \abs { W _ n } } \sum _ { a \in W _ n } \phi _ a . \end{align*}


\begin{align*} | \langle h ^ { = d } , g \rangle | \leq e \ , \mu _ { p } ( h ) \mu _ { p } ( g ) \left ( 4 0 0 \ , r \ , \sqrt { q } \cdot \sqrt { A } \right ) ^ { d } . \end{align*}


\begin{align*} \tilde t = \alpha ( s _ 2 ) + \frac { d ( s _ 2 ) } { d ( s _ 1 ) } ( \tilde t - \alpha ( s _ 1 ) ) . \end{align*}


\begin{align*} \sigma ( A ) = C \mbox { a n d } \sigma ( B ) = D . \end{align*}


\begin{align*} L i _ 2 ( - \phi ) = - \frac { \pi ^ 2 } { 1 0 } - ( \log \phi ) ^ 2 , \end{align*}


\begin{align*} P \left ( \tau _ N < s _ N ( \rho ) \right ) = P \left ( \sup _ { t \leq s _ N ( \rho ) } Z _ 0 ( t ) \geq N \right ) \leq \frac { 1 } { N ^ 2 } E \left [ Z _ 0 \left ( s _ N ( \rho ) \right ) ^ 2 \right ] = O \left ( N ^ { 2 ( \rho - 1 ) } \right ) . \end{align*}


\begin{align*} \varphi ( q _ { n * } ( \xi , \ldots , \xi , a , b ) ) = q _ { n * } ( \varphi ( \xi ) , \ldots , \varphi ( \xi ) , \varphi ( a ) , \varphi ( b ) ) , \end{align*}


\begin{align*} \cos ( \theta ) = \frac { \tanh ( \Delta _ { k - 1 } ) } { \tanh ( r _ { k - 1 } ) } , \sin ( \theta ) = \frac { \sinh ( r _ k ) } { \sinh ( r _ { k - 1 } ) } , \cosh ( r _ { k - 1 } ) = \cosh ( r _ k ) \cosh ( \Delta _ { k - 1 } ) . \end{align*}


\begin{align*} \Psi ( M _ n ) = 1 - \frac { 2 ( \gamma - 1 ) } { \gamma ( 2 - \gamma ) } \frac { 1 } { M _ n ^ 2 } , \end{align*}


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


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta ) = \inf _ { \lambda \in \mathbb { R } ^ 2 _ + } \left \{ \langle \lambda , \delta \rangle + \sup _ { \pi \in \bar { \Gamma } } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda \ , d \pi \right \} . \end{align*}


\begin{align*} \delta ^ { \prime } = x \alpha _ { m + 1 } \cdots \alpha _ k \delta \pi _ j \pi _ { j + 1 } \cdots \pi _ { l - 1 } \end{align*}


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


\begin{align*} \underset { x ^ { \prime } \in \mathbb { R } ^ d } { \arg \max } \ \varphi _ { \ell } ( x ^ { \prime } , \lambda _ { \ell } ) = \frac { 1 } { 2 } a _ \ell { \lambda _ { \mathrm { D } , \ell } ^ { \star } } ^ { - 1 } V _ { 2 , X Y } , \ell = 1 , 2 . \end{align*}


\begin{align*} \Psi ( \tilde h d X _ i + \mathcal J ) = \frac { a _ i h } { Q ' _ i } \frac { Q ' _ i } { a _ i } + I _ \beta = h + I _ \beta . \end{align*}


\begin{align*} v ( T x ) - v ( x ) = \varphi _ f ( x ) I , \end{align*}


\begin{align*} L _ C ( U ) = L _ C ( 2 ( I ^ - T ^ { - 1 } ) ^ T \Sigma ( \sqrt { | \Lambda ^ - | } W ^ - - R \sqrt { \Lambda ^ + } W ^ + - S G ) ) , \end{align*}


\begin{align*} \mathcal { H } = R ( B + i ) = R ( A + i ) + R ( B \vert _ V + i ) . \end{align*}


\begin{align*} t ( [ z _ 0 , g ] _ { \mathbf { K } } ) = [ z _ 0 , t g ] _ { \mathbf { K } } . \end{align*}


\begin{align*} r ' ( \varphi ( b ) , x ) = i ( x ) s ' ( \varphi ( b ) ) = i ( x ) s ( b ) = r ( b , x ) , \end{align*}


\begin{align*} & \forall \ , x : n \forall \ , y : n \forall \ , z : n ( ( x + _ n y ) + _ n z = x + _ n ( y + _ n z ) ) \\ & \forall \ , x : n ( x + _ n 0 _ n = x ) \\ & \forall \ , x : n \forall \ , y : n ( x + _ n y = y + _ n x ) \\ & \forall \ , x : n ( x + _ n x = 0 _ n ) \\ & \forall \ , x : n ( 1 \cdot _ n x = x ) \\ & \forall \ , x : n \forall \ , y : n \forall \ , a : 0 ( a \cdot _ n ( x + _ n y ) = a \cdot _ n x + _ n a \cdot _ n y ) \\ & \forall \ , x : n \forall \ , a : 0 \forall \ , b : 0 ( ( a + _ 0 b ) \cdot _ n x = a \cdot _ n x + _ n b \cdot _ n x ) \end{align*}


\begin{align*} \{ \nu _ \eta \} = \mathcal { P } ( X _ \eta ) = \mathcal { P } ( X _ { \eta ^ * } ) = \{ \nu _ { \eta ^ * } \} . \end{align*}


\begin{align*} \sup _ { \pi \in \Pi ( \mu _ 1 , \dotsc , \mu _ L ) } \int _ { \mathcal { X } } f d \pi = \inf _ { ( \phi _ \ell ) _ { \ell \in [ L ] } \in \Phi _ f } \left \{ \sum _ { \ell = 1 } ^ { L } \int _ { \mathcal { X } _ \ell } \phi _ \ell d \mu _ \ell \right \} . \end{align*}


\begin{align*} \tilde { \mathbf r } _ m = & \mathbf F \mathbf r _ m = \sum _ { n \in \mathcal N } \alpha _ n g _ n ^ { \frac { 1 } { 2 } } \mathbf F \mathbf H _ { n , m } \mathbf F ^ H \tilde { \mathbf s } _ n + \tilde { \mathbf n } _ m \\ = & \sum _ { n \in \mathcal N } \alpha _ n g _ n ^ { \frac { 1 } { 2 } } { \rm d i a g } ( \tilde { \mathbf { s } } _ n ) \mathbf { F } ( \mathbf H _ { n , m } ) _ { : , 1 } + \tilde { \mathbf n } _ m , m \in \mathcal { M } , \end{align*}


\begin{align*} \Delta [ G _ { \alpha } ( u ) ] = G _ { \alpha } ' ( u ) \Delta u + G _ { \alpha } '' ( u ) | \nabla u | ^ 2 \ , \ , \R ^ N . \end{align*}


\begin{align*} \widetilde { u } _ n : = \sum _ { k = n } ^ { N ( n ) } \alpha _ { n , k } u _ { \lambda _ k } . \end{align*}


\begin{align*} f _ { N , j , n } ( b _ 1 , \ldots , b _ N ) : = \sum _ { M \in \Z } \left ( ( - 1 ) ^ M \sum _ { u \in \mathbb { Z } } a ^ { j - u } q ^ { ( M - u ) ( n - u ) + ( j - u ) ( n - N ) } \begin{bmatrix} M \\ u \end{bmatrix} \begin{bmatrix} N - M \\ j - u \end{bmatrix} \right ) e _ M ( b _ 1 , \dots , b _ N ) , \end{align*}


\begin{align*} L = A \otimes _ P \overline { L } . \end{align*}


\begin{align*} \psi ( x ) : = \begin{cases} - \varphi ( x ) & x \in \mathbb { R } ^ n _ + \\ \infty & x \notin \mathbb { R } ^ n _ + . \end{cases} \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty e ^ { \mu b - \frac { \mu ^ 2 } { 2 } } \ , \frac { 1 } { \sqrt { 2 \pi } } \ , e ^ { - \frac { b ^ 2 } { 2 } } \ , \mathrm { d } b = 1 . \end{align*}


\begin{align*} A ^ { \ast } ( x ) = A \left ( x , \ldots , x \right ) \left ( x \in G \right ) . \end{align*}


\begin{align*} \lambda _ \chi ^ { \rm s t i f f } : H ^ { 1 / 2 } ( \Gamma ; \C ^ 3 ) \times H ^ { 1 / 2 } ( \Gamma ; \C ^ 3 ) \to \R , \lambda _ \chi ^ { \rm s t i f f } ( \vect g , \vect h ) : = a _ { 0 , \chi } ^ { \rm s t i f f } ( \Pi _ \chi ^ { \rm s t i f f } \vect g , \Pi _ \chi ^ { \rm s t i f f } \vect h ) . \end{align*}


\begin{align*} a \overline { b } = \frac { 1 } { 3 } \sum _ { t = 0 } ^ 2 e ^ { \frac { 2 \pi i t } { 3 } } \vert a + e ^ { \frac { - 2 \pi i t } { 3 } } b \vert ^ 2 , ~ ~ a , b \in \mathbb { C } \end{align*}


\begin{align*} P _ 0 ^ { ( \alpha , \beta ) } ( z ) = 1 , \end{align*}


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


\begin{align*} d = \texttt { w p i n i t i a l } + 1 + \texttt { m a g } \Bigl \{ 3 \Bigl ( \frac { t } { 2 \pi } \Bigr ) ^ { 3 / 2 } \log \frac { t } { 2 \pi } \Bigr \} \end{align*}


\begin{align*} \mu _ C \big ( Q _ 1 ( x , y , z , w ) \big ) = Q _ 2 ( x , \alpha x - y , z , \alpha z - w ) \end{align*}


\begin{align*} & \forall \ , x : n ( h _ n ( x ) = \top _ 1 \ast _ { 1 n } x ) \end{align*}


\begin{align*} ( \underline { X } , \underline { A } ) ^ { \sigma } = \{ ( x _ 1 , \ldots , x _ m ) \in \prod ^ m _ { i = 1 } X _ i \mid x _ i \in A _ i i \notin \sigma \} . \end{align*}


\begin{align*} \mathcal { T } = \bigsqcup _ { i = 1 } ^ { r ' } \mathcal { T } _ i ' . \end{align*}


\begin{align*} \P ( \omega _ 1 ^ { \bar w } = \omega _ 1 ) \le \ell / ( n - \ell ) . \end{align*}


\begin{align*} M ( z ) \Gamma _ 0 \vect u = \Gamma _ 1 \vect u . \end{align*}


\begin{align*} \frac { 1 } { \Delta } \ , a _ { ( p , \lambda _ 2 , \dots , \lambda _ n ) } = s _ { ( k - ( n - 1 ) , \lambda _ 2 , \dots , \lambda _ n ) } \ . \end{align*}


\begin{align*} \int _ J | \varphi ( x ) | d x \leq \left \{ \begin{array} { c l } \frac { \norm { \varphi } _ { L ^ 1 ( I ) } | J | } { | I | } + \frac { 2 ^ { a + 3 } p _ a ( \varphi ) | J | ^ { 1 - a } } { a ( 1 - a ) } & 0 < a < 1 , \\ \frac { \norm { \varphi } _ { L ^ 1 ( I ) } | J | } { | I | } + 4 p _ a ( \varphi ) | J | ( 1 + \log \frac { | I | } { | J | } ) & a = 0 . \end{array} \right . \end{align*}


\begin{align*} H ^ 1 ( A , q ) : = H ^ 1 ( A _ { \overline { \mathbb { F } } _ \ell } ^ \vee , \mathbb { Z } _ q ( 1 ) ) , \ , H ^ 1 ( A , \ell ) : = H ^ 1 _ \mathrm { c r y s } ( A ^ \vee / W ( { \kappa } ) ) \end{align*}


\begin{align*} \varphi _ { \lambda } \left ( x _ { 1 } , t \right ) = \exp \left [ 2 \lambda \left ( x _ { 1 } ^ { 2 } - \alpha \left ( t - T / 2 \right ) ^ { 2 } \right ) \right ] . \end{align*}


\begin{align*} \widetilde { Z } ( \mathcal { C } ) : = \{ \alpha \in \mathcal { C } | \alpha v = v \alpha ' \ \forall v \in M \} \quad \mathrm { a n d } \quad \Gamma ( M , q ) : = \{ \alpha \in \mathcal { C } ^ { \times } | \alpha V \alpha '^ { - 1 } = V \} \end{align*}


\begin{align*} Q _ { [ i ] } = \begin{pmatrix} \tilde { S } _ { \alpha _ i \alpha _ i } & \tilde { S } _ { \alpha _ i \beta _ i } \\ - \tilde { S } _ { \alpha _ i \beta _ i } & \tilde { S } _ { \alpha _ i \alpha _ i } \end{pmatrix} + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} Q _ { [ i ] } \coloneqq \begin{pmatrix} U & V \\ - V & U \end{pmatrix} \end{align*}


\begin{align*} \partial _ x \left ( A _ t ^ { - 1 } ( x ) \right ) = \frac { 1 } { 1 - \gamma _ x \left ( A _ t ^ { - 1 } ( x ) , t \right ) } , \partial _ x ^ 2 \left ( A _ t ^ { - 1 } ( x ) \right ) = \frac { \gamma _ { x x } \left ( A _ t ^ { - 1 } ( x ) , t \right ) } { \left ( 1 - \gamma _ x \left ( A _ t ^ { - 1 } ( x ) , t \right ) \right ) ^ 3 } , \end{align*}


\begin{align*} g _ { - 1 , - 1 } \textrm { i s E g e r v \ ' a r y e q u i v a l e n t t o } \begin{cases} g _ { - 1 , 1 } & \textrm { i f a n d o n l y i f } n + m \textrm { i s a n e v e n n u m b e r } , \\ g _ { 1 , 1 } & \textrm { i f a n d o n l y i f } n \textrm { i s a n e v e n n u m b e r } , \\ g _ { 1 , - 1 } & \textrm { i f a n d o n l y i f } m \textrm { i s a n e v e n n u m b e r } , \\ \end{cases} \end{align*}


\begin{align*} \Lambda ^ { \rm h o m } \vect \xi : \vect \zeta , \Lambda ^ { \rm h o m } \vect \xi : \vect \xi \geq \eta | \vect \xi | ^ 2 , \forall \vect \xi , \vect \zeta \in \R ^ { 3 \times 3 } , \vect \xi ^ \top = \vect \xi , \vect \zeta ^ \top = \vect \zeta , \end{align*}


\begin{align*} \frac { \textup { M M D } _ { n , m } ^ { 2 } } { \surd { \textup { v a r } ( \Delta _ { 1 } ) } } = \frac { \Delta _ 1 + \sum ^ { l - 1 } _ { s = 2 } \{ \Delta _ s - E ( \Delta _ s ) \} + \{ \widetilde { \Delta } _ l - E ( \widetilde { \Delta } _ l ) \} + ^ 2 ( P _ X , P _ Y ) } { \surd { \textup { v a r } ( \Delta _ { 1 } ) } } . \end{align*}


\begin{align*} \lim _ { r \nearrow 1 } \int _ 0 ^ { 2 \pi } | T _ D ( f ) ( e ^ { i \theta } ) - T _ D ( f ) ( r e ^ { i \theta } ) | ^ \gamma \ , d \theta = 0 , \end{align*}


\begin{align*} \Pi _ { \psi } ( g ) f ( [ 1 , y ] ) = \pi _ { \psi } ( g ) f _ 1 ( y ) = \int _ { X ^ { \ast } } \psi ( \langle y , y ^ { \ast } \rangle ) f _ 1 ( y ^ { \ast } \omega ^ { - 1 } ) d y ^ { \ast } . \end{align*}


\begin{align*} \begin{aligned} u _ n & \rightharpoonup u & & X , \\ T u _ n & \rightarrow v \enspace & & X ^ { * } , \\ \end{aligned} \enspace \Longrightarrow \enspace u _ n \rightarrow u X . \end{align*}


\begin{align*} C _ { \alpha , n } ^ { a , + } ( \varphi ) & = ( - 1 ) ^ { n } C _ { \alpha } ^ + ( D ^ n \varphi ) : = ( - 1 ) ^ { n + 1 } \lim _ { x \searrow l _ \alpha } D ^ { n + 1 } \varphi ( x ) ( x - l _ \alpha ) ^ { 1 + a } , \\ C _ { \alpha , n } ^ { a , - } ( \varphi ) & = C _ { \alpha } ^ - ( D ^ n \varphi ) : = \lim _ { x \nearrow r _ \alpha } D ^ { n + 1 } \varphi ( x ) ( r _ \alpha - x ) ^ { 1 + a } \end{align*}


\begin{align*} { \rm I m } ( e ^ { i \pi s } \kappa ( t ) ) = - { \rm I m } ( e ^ { i \pi ( 1 + s ) } \kappa ( t ) ) = 0 . \end{align*}


\begin{align*} | | \sum _ { j } c _ j a _ j | | _ { a t } : = \left ( \inf \sum _ { j } | c _ j | ^ p \right ) ^ { 1 / p } , \end{align*}


\begin{align*} \nu ( A ) = \int _ { H \times \{ 0 , 1 \} ^ \Z } \mathbf { 1 } _ A ( \underline { \varphi } ( h ) + x \cdot ( \varphi ( h ) - \underline { \varphi } ( h ) ) ) \ , d \rho ( h , x ) . \end{align*}


\begin{align*} e _ n ^ { ( k ) } = \frac 1 { k ! } \sum _ { I \vDash n } \varphi ^ I \end{align*}


\begin{align*} \sum _ { \substack { m , v \in \mathcal { M } \\ m p = v } } f ( m ) f ( v ) \le \sum _ { \substack { m ' , v ' \in \mathcal { M } ' \\ | p m ' / v ' - 1 | \le 3 / T } } r ( m ' ) r ( v ' ) . \end{align*}


\begin{align*} f = \sum _ { i = 0 } ^ r \left ( \sum _ { j = 1 } ^ { s _ i } a _ { i , j } \textbf { Q } ^ { \lambda _ { i , j } } \right ) q ^ i = \sum _ { i = 0 } ^ r \sum _ { j = 1 } ^ { s _ i } a _ { i , j } \textbf { Q } ^ { \lambda ' _ { i , j } } , \end{align*}


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


\begin{align*} c = c ( \theta ) : = \inf _ { j , n } \left \{ \lvert { \theta ^ n } _ j ( \mathcal A ) \rvert : 0 \le j < \ell ^ n \right \} . \end{align*}


\begin{align*} H _ b ( X , Y ) = \langle \langle P X , Y \rangle \rangle = \int _ a ^ b \langle P X , Y \rangle d s . \end{align*}


\begin{align*} \beta ^ { \prime } = \underset { \delta ^ { \prime } } { \underbrace { d _ p d _ { p + 1 } \cdots d _ { q - 1 } } } d _ q d _ { q + 1 } \cdots d _ l x \pi _ j \cdots \pi _ n . \end{align*}


\begin{align*} \langle D _ u \theta ( u ^ * ) , v \rangle _ \mathcal { U } + \langle \bar { \lambda } ^ * , S v \rangle = 0 \forall \ v \in \mathcal { U } . \end{align*}


\begin{align*} S _ { j , + } ^ k ( t ) \approx \nu \int _ 0 ^ t Z _ 0 ( s ) p _ { j , + } ^ k ( t - s ) d s = : \bar { S } _ { j , + } ^ k ( t ) . \end{align*}


\begin{align*} \mathcal { L } \left ( \frac { 1 } { 2 \theta _ i ^ 2 T _ i ^ * ( u ) } \Bigg | \mathcal { Z } _ u , Z _ i ^ * ( u ) = z _ i / \theta _ i \right ) = I G \left ( \frac { 1 } { 2 \theta _ i z _ i } , \frac { 1 } { 2 z _ i ^ 2 } \right ) . \end{align*}


\begin{align*} \nu ^ * ( x ) = \frac { 1 } { Z ( \nu ^ * , \mu ^ * ) } \exp { \left ( - \frac { 2 } { \sigma ^ 2 } \frac { \delta F } { \delta \nu } ( \nu ^ * , \mu ^ * , x ) - U ^ { \pi } ( x ) \right ) } , \end{align*}


\begin{align*} \beta = q _ { i + } ( \beta _ 1 ) q _ { i - } ( \beta _ 2 ) . \end{align*}


\begin{align*} \delta _ r ^ A = \exp ( \ln ( r ) A ) , r > 0 , \end{align*}


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


\begin{align*} \Sigma \times \{ 0 \} = W _ 1 \cup W _ 2 \cup _ Y W _ 3 \cup W _ 4 \end{align*}


\begin{align*} R _ m ( u , 0 ) = u \cdot R _ m ( u , 1 ) ( m \geq 1 ) . \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 , \\ \tilde { u } = 0 & \Omega ^ c \end{cases} \end{align*}


\begin{align*} f ( z _ 1 , \ldots , z _ m ) = \sum _ { ( k _ 1 , \ldots , k _ m ) \in \Z ^ m _ { \geq 0 } } c _ { k _ 1 , \ldots , k _ m } z _ 1 ^ { k _ 1 } \cdots z _ m ^ { k _ m } , \end{align*}


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


\begin{align*} m = m _ n \xrightarrow [ ] { } + \infty , \frac { m } { \sqrt n } \xrightarrow [ ] { } 0 \end{align*}


\begin{align*} e ( \theta ) = C _ v \theta + a ( \theta ) q ^ 2 , \qquad p ( \rho , \theta ) = R \rho \theta , \end{align*}


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


\begin{align*} \lim _ { n \to \infty } z _ n ( t , x ) = v ( t , x ) L ^ r ( \Omega ; H ) \end{align*}


\begin{align*} \begin{dcases} L \psi \le - \frac { \alpha _ r } { r ^ { 2 s } } & B _ r ( x _ r ) \setminus \overline { B _ { r / 2 } ( x _ r ) } , \\ \psi \ge \frac { \alpha _ r } { r ^ { 2 s } } ( r ^ 2 - | x - x _ r | ^ 2 ) _ + ^ s & \\ \psi \le \alpha _ r C & B _ { r / 2 } ( x _ r ) , \\ \psi = 0 & \R ^ n \setminus B _ r ( x _ r ) . \end{dcases} \end{align*}


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


\begin{align*} f ( p ) : = \frac { 1 } { \sqrt { | \log { ( 2 \sigma - 1 ) } | } } , \ ; p \in P . \end{align*}


\begin{align*} h _ \psi ^ { \alpha / z } = h _ \varphi ^ { ( \alpha - 1 ) / 2 z } x h _ \varphi ^ { ( \alpha - 1 ) / 2 z } . \end{align*}


\begin{align*} ( \mathcal { T } \varphi , v ) _ { L ^ 2 ( \mathrm { M } _ 0 ) } = ( \varphi | \overline { \mathfrak { I } \overline { w } } ) _ { H ^ { 1 / 2 } _ \Gamma ( \mathrm { M } ) } . \end{align*}


\begin{align*} c ( x ) - c ( \hat { y } ) & = \nabla c ( x ) \cdot ( x - \hat { y } ) + R _ 2 ( x , \hat { y } ) \\ & = \nabla _ { x ^ \prime } c ( x ) \cdot ( x ^ \prime - y ^ \prime ) + \partial _ { x _ n } c ( x ) ( x _ n + y _ n ) + R _ 2 ( x , \hat { y } ) , \end{align*}


\begin{align*} R _ { s , j } & = \left ( 1 + 2 ^ { \mathfrak { c } | l | } \lambda ^ { \gamma + \epsilon _ 3 - \epsilon _ 4 } \right ) \sum _ { m _ j \in \mathcal { M } _ j } \sum _ { k _ j \in \mathbb { Z } ^ 2 } \left | a _ { j , m _ j , k _ j , s } \right | ^ 2 \\ & \lesssim 2 ^ { \mathfrak { c } | l | } \lambda ^ { \gamma + \epsilon _ 3 - \epsilon _ 4 } \sum _ { m _ j \in \mathcal { M } _ j } F _ { j , m _ j } ( s ) . \end{align*}


\begin{align*} \partial \big ( ( n _ v ) _ { v \in V ( G ) } \big ) = \big ( n _ { v _ E } - n _ { v ' _ E } \big ) _ { E \in E ( G ) } . \end{align*}


\begin{align*} \tilde { y } _ { s } = \zeta + \int _ { s } ^ { T } \tilde { f } \left ( s , \tilde { y } _ { s } , \tilde { z } _ { s } , \mathbb { P } _ { \left ( \tilde { y } _ { s } , \tilde { z } _ { s } \right ) } \right ) \mathrm { d } s - \int _ { s } ^ { T } \tilde { z } _ { s } \mathrm { d } W _ { s } ^ { t _ { 0 } } . \end{align*}


\begin{align*} \mathcal J _ { \ell } = \{ n _ 0 + 1 , n _ 0 + 2 , \dots , \ell - 1 , \ell \} . \end{align*}


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


\begin{align*} \vec { \delta } _ { 1 0 2 } \cdot \vec { \delta } _ { 2 0 3 } = 1 + \cos \alpha _ { 1 0 2 } + \cos \alpha _ { 1 0 3 } + \cos \alpha _ { 2 0 3 } . \end{align*}


\begin{align*} \big | L ( f , k ) \big | ^ 2 = \frac { 2 } { \left ( ( k - 1 ) ! \right ) ^ 2 } \sum _ { l , \ , m \geq 1 } G _ k \left ( \frac { l m } { p } \right ) \frac { \lambda _ f ( l ) \lambda _ f ( m ) } { \sqrt { l m } } , \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*} Z _ q ( B _ n ) = \left \{ \begin{aligned} 2 & \mbox { i f } q = 0 \\ n & \mbox { f o r } q \geq 1 . \end{aligned} \right . \end{align*}


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


\begin{align*} F _ { j - 1 } = 4 \alpha ^ 2 h _ { j } ( \tanh ( \alpha x _ j ) - \tanh ( \alpha x _ { j - 1 } ) - \alpha h _ j \tanh ( \alpha x _ { j - 1 } ) \tanh ( \alpha x _ j ) ) , \end{align*}


\begin{align*} \lambda ^ { ( i ) } = ( \lambda _ 1 , \lambda _ 2 , \ldots , \lambda _ j , i , \lambda _ { j + 1 } + 1 , \ldots , \lambda _ { n - d - 1 } + 1 ) \in B _ { n - d , k } \ , \backslash \ , B _ { n - d - 1 , k } . \end{align*}


\begin{align*} \frac { \omega ^ 2 } { r ^ 2 } \ , \frac { \partial z } { \partial \phi } ( \phi , \omega , r ; t ) = \frac { \partial } { \partial s _ 1 } F ( 0 , p ( \phi + \omega t ) ) \cdot p ' ( \phi ) + \frac { \partial } { \partial s _ 2 } F ( 0 , p ( \phi + \omega t ) ) \cdot p ' ( \phi + \omega t ) . \end{align*}


\begin{align*} \partial _ { x _ n } u ( \hat { x } ) _ { \big | \{ x _ n = 0 \} } = \partial _ { x _ n } \big ( c \circ F _ \gamma \big ) ( \hat { x } ) _ { \big | \{ x _ n = 0 \} } = \nabla c ( x ^ \prime , \gamma ( x ^ \prime ) ) \cdot ( - \mathbf { n } ( x ^ \prime , \gamma ( x ^ \prime ) ) ) = 0 , \end{align*}


\begin{align*} \tilde { y } _ { t _ { k , m } + 2 ^ { 2 - 2 k } } ^ { K } \circ \big ( \tilde { y } _ { t _ { k , m } } ^ { K } \big ) ^ { - 1 } \begin{aligned} [ t ] & = \begin{cases} y _ { 2 \tau _ k } ^ { ( i _ k ; L _ k ) } & k = K \ ; ( \mathrm { m o d } \ ; 2 ) , \\ \mathrm { I d } & . \end{cases} \\ & = : Y _ { k , m } ^ K , \end{aligned} \end{align*}


\begin{align*} \sup _ { k \leq n } | S _ k - T _ k | = o ( n ^ { 1 / p } ( \log n ) ^ { 1 / 2 - 1 / p } ) \end{align*}


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


\begin{align*} x _ 1 - x _ 3 = ( n + m ) \delta + 2 \pi z _ 1 \quad \textrm { a n d } x _ 2 - x _ 3 = - m \delta + 2 \pi z _ 2 \textrm { f o r } \delta \in \mathbb { R } , \end{align*}


\begin{align*} \Lambda _ A ( s ) = \prod _ { n = 1 } ^ N ( 1 - s e ^ { - \i \theta _ n } ) , \end{align*}


\begin{align*} F ( z ) = I + i ( P _ 0 - z - i M h W ) ^ { - 1 } M h W , \end{align*}


\begin{align*} \langle f _ b , \varphi \rangle : = \lim _ { r \nearrow 1 } \int _ 0 ^ { 2 \pi } f ( r e ^ { i \theta } ) \ , \varphi ( \theta ) \ , d \theta \end{align*}


\begin{align*} Q _ { 1 , \vec { v } _ { s , t , f , 0 } ^ { ( 1 ) } } = \left ( \left ( \begin{array} { r r r } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right ) , \left ( \begin{array} { r r r } 0 & t & 1 \\ 1 & t ^ { 2 } + s & t \\ 0 & s t + 1 & s \end{array} \right ) , \left ( \begin{array} { r r r } 0 & 1 & 0 \\ 0 & t & 1 \\ 1 & s & 0 \end{array} \right ) \right ) , \end{align*}


\begin{gather*} \partial _ \xi ^ \alpha f ( \xi ) \sim \sum _ { j \geq 0 } \partial _ \xi ^ \alpha f _ { m - j } ( \xi ) \\ g ^ { B , \alpha } ( \xi ) : = \partial _ x ^ \alpha \big | _ { x = 0 } \Big ( \alpha _ { - x } \big ( g ( \xi + B x ) \big ) \Big ) \sim \sum _ { j \geq 0 } g _ { m ' - j } ^ { B , \alpha } ( \xi ) , \end{gather*}


\begin{align*} \sum _ { 0 \leq j < N ^ + } \varphi ( T ^ i y ) = \sum _ { l = 0 } ^ k \sum _ { 0 \leq j < q ( l ) } S ( l ) \varphi ( ( T ^ { ( l ) } ) ^ j ( y ( l ) ) ) q ( l ) \leq \| Z ( l + 1 ) \| . \end{align*}


\begin{align*} \langle b \rangle = \int _ { 0 } ^ { \infty } x ^ { b - 1 } \ , d x \end{align*}


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


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


\begin{align*} f ( z _ 1 , \ldots , z _ { m } ) = a _ 0 + \left ( \sum _ { S \in \mathcal { Q } _ m } \left ( \prod _ { i \in S } z _ i \right ) f _ S ( z _ S ) \right ) + f ^ * ( z _ 1 , \ldots , z _ { m } ) , \end{align*}


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


\begin{align*} ( \nabla _ { X } { f } ) \ , \xi _ i = - { f } \ , \nabla _ { X } \ , \xi _ i . \end{align*}


\begin{align*} M _ b ( \theta ) = \mathbb { E } _ { \Phi | \Phi ( \mathcal { A } ) > 0 } \{ P _ s ^ b ( \theta ) \} \Pr ( \Phi ( \mathcal { A } ) > 0 ) . \end{align*}


\begin{align*} a _ \varepsilon ( x _ n ) : = \int _ { B _ { \delta } ( x ^ \prime ) \times ( 0 , \delta ) } \rho _ \varepsilon ( | x - \hat { y } | ) y . \end{align*}


\begin{align*} \left | \sum _ { i = 1 } ^ k A _ { i } \right | \le \frac { k } { 4 } I + \sum _ { i = 1 } ^ k | A _ { i } | . \end{align*}


\begin{align*} \left . \begin{array} { l l } \phi _ { \mu } = \mu ^ { \frac { N } { 2 } } \phi ( \mu x ) , \psi _ { \mu } = \mu ^ { \frac { N } { 2 } } \psi ( \mu x ) , \end{array} \right . \end{align*}


\begin{align*} \langle x _ 1 , x _ 2 , \dots , x _ 7 : x _ i x _ j x _ i ^ { - 1 } = x _ { 5 i + 3 j \bmod 7 } \rangle . \end{align*}


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


\begin{align*} \begin{pmatrix} W _ { \mathcal { I } \mathcal { J } } & X _ { \mathcal { I } \mathcal { J } } \\ Y _ { \mathcal { I } \mathcal { J } } & Z _ { \mathcal { I } \mathcal { J } } \end{pmatrix} . \end{align*}


\begin{align*} \beta _ { 1 , X } ( k ) = \norm { \sup _ { \norm { f } _ \infty \leq 1 } \abs { P _ { X _ k \lvert \mathcal F _ 0 } ( f ) - P _ { X _ k } ( f ) } } _ 1 . \end{align*}


\begin{align*} \mathrm { A n n } ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) = \lbrace T \in \mathbb { T } \mid T ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) = 0 \rbrace \end{align*}


\begin{align*} \widetilde { \varepsilon _ 5 } ( m ) : = \min \Bigl \{ 4 F _ m , \frac { ( \pi ^ 2 B _ 1 a ) ^ { m / 3 } } { 3 1 6 \ , A ( \sigma ) } \Bigl ( \frac { m ! } { \Gamma ( m / 3 + 2 ) } \Bigr ) ^ { 1 / 2 } \ , \varepsilon _ 4 \Bigr \} . \end{align*}


\begin{align*} \frac { ( r _ t ) ^ l \Delta t } { [ ( r _ x ) ^ { l } \Delta x ] ^ s } = \bigg [ \frac { r _ t } { ( r _ x ) ^ s } \bigg ] ^ l \frac { \Delta t } { \Delta x ^ s } = \frac { \Delta t } { \Delta x ^ s } . \end{align*}


\begin{align*} V ( x , \omega ) = \sum _ i W ( x - x _ i ) . \end{align*}


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


\begin{align*} p _ r = x _ 1 ^ r + x _ 2 ^ r + \cdots , \end{align*}


\begin{align*} \mathbf { h } = \zeta \mathbf { \hat h } + \sqrt { 1 - \zeta ^ 2 } \Delta \mathbf { h } , \end{align*}


\begin{align*} \pi _ { \psi } [ \begin{pmatrix} a & 0 \\ 0 & a ^ { \ast - 1 } \end{pmatrix} , t ] f ( y ) = t | \det ( a ) | ^ { 1 / 2 } f ( y a ) , \end{align*}


\begin{align*} \lim _ { t \to 0 } \frac { d _ { M \cap \partial W _ 0 } ( x , z ) } { d _ X ( x , z ) } = 1 \end{align*}


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


\begin{align*} w _ b = H _ b + ( \widetilde { T } ( f ) ) _ b \end{align*}


\begin{align*} \Phi ( \pi ) = \tilde { \delta } \mu _ k \tilde { \alpha } _ k \mu _ { k - 1 } \tilde { \alpha } _ { k - 1 } \cdots \mu _ 1 \tilde { \alpha } _ 1 \mu _ 0 \end{align*}


\begin{align*} B _ \gamma ^ { ( \alpha , \beta ) } w _ { \gamma } ^ { ( \alpha , \beta ) } ( z ) + A _ \gamma ^ { ( \alpha , \beta ) } ( z ) w _ { \gamma + 1 } ^ { ( \alpha , \beta ) } ( z ) + w _ { \gamma + 2 } ^ { ( \alpha , \beta ) } ( z ) = 0 , \end{align*}


\begin{align*} \{ T > t \} T = \inf \{ s \geq 0 : Z _ t ( K ) \geq 1 \} . \end{align*}


\begin{align*} \bar { d } _ { 0 , i } = D _ 0 h _ { 0 , i } , \forall i \in I . \end{align*}


\begin{align*} T \left ( \frac { 2 } { f ^ { 2 } - 1 } \right ) = T \left ( \frac { 1 } { f - 1 } - \frac { 1 } { f + 1 } \right ) = T \left ( \frac { 1 } { f - 1 } \right ) - T \left ( \frac { 1 } { f + 1 } \right ) \end{align*}


\begin{align*} \omega _ \Pi = k | \Sigma | . \end{align*}


\begin{align*} L _ { \varphi } ( a , r , s , A ) : & = \frac { 1 } { | P _ { r , s , A } | K _ { P _ { r , s , A } , \varphi } ( a ) } \\ & = \inf \left \{ \frac { \int _ { a + P _ { r , s , A } } | f | ^ 2 e ^ { - \varphi } } { | P _ { r , s , A } | e ^ { - \varphi ( a ) } } ~ \middle | ~ f \in A ^ 2 ( a + P _ { r , s , A } , \varphi ) \And f ( a ) = 1 \right \} , \end{align*}


\begin{align*} \eta \times \widetilde { \mathbb { P } } _ { \mu } \left ( \left \{ L _ { x } \omega _ { n } \in S ( x , \omega ) \mbox { f o r i n f i n i t e l y m a n y } n \in \mathbb { N } \right \} \right ) = 0 . \end{align*}


\begin{align*} \nu ( g ' ) = v ( p ) + ( p - 1 ) \nu ( x ) = v ( p ) . \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } t ^ { \frac { d + 2 \beta } { \alpha } \frac { q - 1 } { q } } \| P _ t ^ { \Gamma } f \| _ { q , M _ \Gamma } = 0 . \end{align*}


\begin{align*} \begin{aligned} - \Delta _ p u + ( - \Delta _ p ) ^ s u = f \ ; \ ; \Omega , \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \mathbb { E } \left [ \rho _ 2 ( Y _ 2 , y _ 2 ) \right ] - 1 & \leq \mathbb { E } \left [ \boldsymbol { d } _ { \mathcal { S } _ 2 } ( S _ 2 , s _ 2 ) ^ { p _ 2 } \right ] = \boldsymbol { W } _ { p _ 2 } \left ( \gamma _ { 2 , 3 } ^ { \eta _ 1 , \delta } , \delta _ { s _ 2 } \right ) ^ { p _ 2 } \leq \left [ \delta ^ { 1 / p _ 2 } + \boldsymbol { W } _ { p _ 2 } \left ( \mu _ 2 , \delta _ { s _ 2 } \right ) \right ] ^ { p _ 2 } , \\ \end{aligned} \end{align*}


\begin{align*} f = f _ 0 + f _ 1 q + \ldots + f _ n q ^ n \end{align*}


\begin{align*} u : = \overline { u } + \overline { m } \end{align*}


\begin{align*} \aligned \int _ { \mathbb { R } ^ N } ( I _ \mu * | { u _ \nu } | ^ { p } ) | { v _ \nu } | ^ { q } = & ( 1 + o _ \nu ( 1 ) ) [ ( \gamma _ p + \gamma _ q ) \nu D ^ { \frac { 2 } { \gamma _ p + \gamma _ q } } _ 0 ] ^ { \frac { \gamma _ p + \gamma _ q } { 2 - \gamma _ p - \gamma _ q } } \\ = & ( 1 + o _ \nu ( 1 ) ) \nu ^ { \frac { \gamma _ p + \gamma _ q } { 2 - \gamma _ p - \gamma _ q } } \int _ { \mathbb { R } ^ N } ( I _ \mu * | { u _ 0 } | ^ { p } ) | { v _ 0 } | ^ { q } . \endaligned \end{align*}


\begin{align*} r _ b ( n ) \ : = \ ( a _ 0 a _ 1 \cdots a _ { L - 1 } ) _ b . \end{align*}


\begin{gather*} \begin{aligned} \left [ T _ { i j } , T _ { j k } \right ] & = T _ { i k } & & i \neq k , \\ \left [ T _ { i j } , T _ { k \ell } \right ] & = 1 & & i \neq \ell , j \neq k , \\ & & & i , j , k \in \{ 1 , \dots , n \} \\ ( T _ { 1 2 } T _ { 2 1 } ^ { - 1 } T _ { 1 2 } ) ^ 4 & = 1 . \end{aligned} \end{gather*}


\begin{align*} \aligned \min \{ p , q \} \left \{ \begin{array} { l l l } > \frac { 6 - \mu } { 2 } 2 < \mu < 3 , \ & N = 3 , \\ < 2 , \ & N = 4 , \end{array} \right . \endaligned \end{align*}


\begin{align*} \int _ M ( 1 + w ) | T ^ \varphi | ^ 2 = 0 , \int _ M w ( 1 + w ) | T ^ \varphi | ^ 2 = \int _ M ( 1 + w ) ^ 2 | T ^ \varphi | ^ 2 . \end{align*}


\begin{align*} \exp ( t x ) H = \exp ( t y ) H \mbox { f o r a l l } t \in \R \end{align*}


\begin{align*} \Theta ( x _ 1 , \dots , x _ n ) = \frac { c | x _ 1 | } { \log | x _ 1 | } \end{align*}


\begin{align*} p ( \phi + \vartheta ) = \frac { 1 } { y _ 0 } \vartheta + o ( \vartheta ) p ' ( \phi + \vartheta ) = \frac { 1 } { y _ 0 } + o ( 1 ) , \qquad \vartheta \to 0 . \end{align*}


\begin{align*} \begin{aligned} \displaystyle \int _ M \sigma _ k ( \kappa ) \mathrm { d } \mu _ g = & \int _ { \mathbb { S } ^ n } \sum \limits _ { m = 0 } ^ k \left [ \left ( A _ 0 ^ m + A _ 1 ^ m u + A _ 2 ^ m u ^ 2 + A ^ m | \nabla u | ^ 2 \right ) \sigma _ m ( D ^ 2 u ) + B ^ m u ^ i u _ j [ T _ m ] _ i ^ j ( D ^ 2 u ) \right ] \mathrm { d } A \\ & + O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 + O ( \varepsilon ) \| \nabla u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 , \end{aligned} \end{align*}


\begin{align*} \int _ { \mathcal { T } } \left [ \sup _ { x \in \mathcal { X } } \varphi ( t , x ) \right ] d \mu ( t ) = \sup _ { \pi \in \Gamma ( \mu , \varphi ) } \int _ { \mathcal { T } \times \mathcal { X } } \varphi \left ( t , x \right ) d \pi ( t , x ) . \end{align*}


\begin{align*} x = \gamma _ 2 y ^ 2 + \gamma _ 1 y + \gamma _ 0 . \end{align*}


\begin{align*} f ( z ) = \sum _ { n \ge 1 } f _ { n } n ^ { \frac { \lambda - 1 } { 2 } } e ( n z ) . \end{align*}


\begin{align*} ( z ( t ) , - \dot z ( t ) ) \in \mathcal { M } ^ { - 1 } \mathcal { L } = \{ ( z , w ) ~ \vert ~ ( w , v ) \in \mathcal { M } , ~ ( z , v ) \in \mathcal { L } \} , \end{align*}


\begin{align*} \left . \begin{array} { l l } \mathcal { M } : = \left \{ ( u , v ) \in H ^ 1 ( \mathbb { R } ^ N ) \times H ^ 1 ( \mathbb { R } ^ N ) \backslash \{ ( 0 , 0 ) \} , Q ( u , v ) = 0 \right \} , \end{array} \right . \end{align*}


\begin{align*} \delta _ { i _ 1 i _ 2 \cdots i _ m i _ { m + 1 } \cdots i _ k } ^ { j _ 1 j _ 2 \cdots j _ m j _ { m + 1 } \cdots j _ k } \delta ^ { i _ { m + 1 } } _ { j _ { m + 1 } } \cdots \delta ^ { i _ k } _ { j _ k } = \delta _ { i _ 1 i _ 2 \cdots i _ m } ^ { j _ 1 j _ 2 \cdots j _ m } { n - m \choose k - m } ( k - m ) ! , \end{align*}


\begin{align*} \begin{cases} \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) \tilde { u } = 0 & \Omega , \\ \mathcal { L } \tilde { u } = 0 & \Omega ^ c , \\ \tilde { u } = 0 & Q , \end{cases} \end{align*}


\begin{gather*} \mathsf { d } _ { c c } ( [ X _ { n 1 } , \dots , X _ { n j } ] _ c ) \leq \sum _ { i = 1 } ^ { j - 1 } 2 ^ i \mathsf { d } _ { c c } ( X _ { n i } ) + 2 ^ { j - 1 } \mathsf { d } _ { c c } ( X _ { n j } ) . \end{gather*}


\begin{align*} u = \sum _ { s , t = 1 } ^ { d _ 1 } \alpha _ { \mathrm { s t } } X _ s \otimes X _ t = \sum _ { s , t = 1 } ^ { d _ 1 } \big ( \alpha _ { \mathrm { s t } } ^ 1 X _ s \big ) \otimes \big ( \alpha _ { \mathrm { s t } } ^ 2 X _ t \big ) = : \sum _ { n = 1 } ^ { d _ 1 ^ 2 } X _ { n 1 } \otimes X _ { n 2 } , \end{align*}


\begin{align*} 1 + \frac { z f ^ { \prime \prime } ( z ) } { f ^ { \prime } ( z ) } = w ( z ) + \sqrt { 1 + w ^ 2 ( z ) } , \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } M ( t ) = \begin{cases} \dfrac { \nu Y } { \lambda } , & p = 0 , \\ - \dfrac { \nu q \log ( q ) Y } { \lambda p } , & 0 < p < 1 . \end{cases} \end{align*}


\begin{align*} \nu _ i \left ( l a _ l Q _ i ^ { l - 1 } \right ) = \nu _ i ( f ) + \alpha _ i . \end{align*}


\begin{align*} m _ \nu ( a , b ) = \inf _ { \mathcal { O } _ \nu ( a , b ) } \max _ { t \in \mathbb { R } } J _ \nu ( t \star ( u , v ) ) . \end{align*}


\begin{align*} \Delta w + \frac { S ^ \varphi } { m - 1 } w = 0 \ , . \end{align*}


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


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


\begin{align*} \chi e ^ { - i t h ^ { - 1 } \tilde { P } _ h ( 0 ) } q _ 1 e ^ { - ( t _ 0 - t ) X } \tilde { R } _ h ( z ) \chi = \chi e ^ { - t X } q _ 1 e ^ { - ( t _ 0 - t ) X } \tilde { R } _ h ( z ) \chi = 0 . \end{align*}


\begin{align*} \lim _ { R \to \infty } \lambda _ ( B _ R ) = \frac { ( n - 1 ) ^ 2 \kappa ^ 2 } { 4 } , \end{align*}


\begin{align*} M = \mathcal { Q } _ { \ell , \vec { v } } \left ( M _ { \bullet , \ell } \right ) \end{align*}


\begin{align*} \frac { R } { ( L + 1 ) ( 1 - R ) } \le \frac { 1 } { L + 1 } \frac { 1 } { 1 - R } \le \frac { \log q } { 2 \log d ( \chi ) } \frac { 4 \log d ( \chi ) } { \log q \log ( d ( \chi ) \log d ( \chi ) / n ) } = \frac { 2 } { \log ( d ( \chi ) \log d ( \chi ) / n ) } \le 1 \end{align*}


\begin{align*} \frac { C _ k ( p ) } { a ^ k } = \frac { 1 } { ( \pi ^ 2 a ) ^ { k } } \sum _ { j = 0 } ^ { \lfloor 3 k / 2 \rfloor } \Bigl ( \frac { \pi } { 2 i } \Bigr ) ^ j d ^ { ( k ) } _ j F ^ { ( 3 k - 2 j ) } ( p ) \end{align*}


\begin{align*} \alpha _ { i , j } = \sum _ { v = 0 } ^ { N + q } C _ v \alpha _ { i , j - v - 1 } , C _ v \in \mathbb { C } , j \ge N + q + 1 , \end{align*}


\begin{align*} \dot { \widetilde { x } } = \widetilde { f } ( \widetilde { x } ) , \dot { \widetilde { y } } = \widetilde { f } ( \widetilde { y } ) , \dot { \widetilde { z } } = \frac { \widetilde { f } ( \widetilde { x } ) - \widetilde { f } ( \widetilde { y } ) } { \epsilon } , \end{align*}


\begin{align*} \eta : = C _ v \ln \theta - R \ln \rho - \left ( \frac { Z ( \theta ) } { 2 \theta } \right ) ^ \prime q ^ 2 . \end{align*}


\begin{align*} \int _ { - R } ^ R \int _ { - R } ^ R e ^ { - i \xi ( x - y ) } d x d y = \left | \int _ { - R } ^ { R } e ^ { - i \xi x } d x \right | ^ 2 = \frac { 4 \sin ^ 2 ( R | \xi | ) } { | \xi | ^ 2 } = 4 \pi R \ell _ R ( \xi ) \end{align*}


\begin{align*} \mathfrak { p } _ { I } = \mathfrak { m } _ { I } \oplus \mathfrak { a } _ { I } \oplus \mathfrak { n } ^ { I } . \end{align*}


\begin{align*} \| f ^ { = d } \| _ 2 ^ 2 \le \left ( \frac { 1 0 0 r } { d } \right ) ^ d \gamma _ 1 \cdot \gamma \log ^ { d } \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \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*} ( M \vec x ) _ { i ^ * } - ( M \vec x ) _ 1 = y _ { i ^ * } - y _ 1 . \end{align*}


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


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


\begin{align*} q ( n ) \ = \ \frac { 1 0 ^ { n - 1 } } { n + 1 } , \quad . \end{align*}


\begin{align*} L i _ 2 ( z ) = - \int _ { 0 } ^ { z } \frac { \log ( 1 - t ) } { t } d t . \end{align*}


\begin{align*} \big | \Im L ' / \Im \widetilde { L } \big | = \left | \frac { \langle u _ 1 ' \cdots u _ n ' \rangle } { \langle u _ 1 ' \cdots u _ { n - 1 } ' \rangle } \right | = \frac { c _ v } { c ( v _ n ) } . \end{align*}


\begin{align*} [ D \widetilde { \varphi } : D \tau _ M ] _ t = \lambda ^ { \varphi } ( t ) , t \in \mathbb { R } , \end{align*}


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


\begin{align*} A ^ * \phi ( x , \sigma ) = \left ( \tfrac { \kappa } { 2 } \partial _ { x x } - \sigma \lambda \partial _ x \right ) \phi ( x , \sigma ) + \sum _ { \sigma ' \in S } c ( \sigma , \sigma ' ) \big ( \phi ( x , \sigma ' ) - \phi ( x , \sigma ) \big ) . \end{align*}


\begin{align*} I _ x : = \left \{ \lambda _ x / 2 ^ k : k \in \N , \ , \ , \mathcal { H } ^ 1 \left ( J \cap B _ { \lambda _ x / 2 ^ k } ( x ) \right ) \geq \eta \lambda _ x / 2 ^ k \right \} \end{align*}


\begin{align*} \lambda _ { i j } = \sum _ { k = 1 } ^ m \left ( \beta _ { i , k } - \beta _ { j , k } \right ) + \gamma _ i + \alpha _ { i j } \leq 4 m p _ { \max } , \forall j \in \{ 1 , \ldots , n \} . \end{align*}


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


\begin{align*} 5 \cdot 1 0 ^ m - 1 \ = \ 4 \underbrace { 9 \cdots 9 } _ { m } , \end{align*}


\begin{align*} 1 = \sum _ { i = 1 } ^ n ( \gamma _ i - \alpha _ i ) z _ i = \sum _ { i = 1 } ^ n ( \gamma _ i - \alpha _ i ) \sum _ { k = 1 } ^ { N _ 1 } u _ k e _ { k i } = \sum _ { k = 1 } ^ { N _ 1 } u _ k \sum _ { i = 1 } ^ n ( \gamma _ i - \alpha _ i ) e _ { k i } = \sum _ { k = 1 } ^ { N _ 1 } u _ k e _ k , \end{align*}


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


\begin{align*} T _ x ^ \ast \mathcal { M } = E ^ * _ 0 ( x ) \oplus E ^ * _ s ( x ) \oplus E ^ * _ u ( x ) , \end{align*}


\begin{align*} \bar { \lambda } = \int _ { { \rm T r e e } _ { F } ^ { H _ { 0 } } } \left ( \zeta _ { H } \right ) _ { \ast } \nu d \rho ( H ) . \end{align*}


\begin{align*} V ( \theta , \dot \theta ) = \frac { 1 } { 2 } \dot \theta ^ 2 + 1 - \cos ( \theta ) + \alpha ( 1 - \cos ^ 3 ( \theta ) ) , \end{align*}


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


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


\begin{align*} \psi _ { \mu } & : \mathcal { P } _ { \mu } ( X ) \to \mathcal { P } _ { H } ( X ) \\ \eta & \mapsto \lambda = \boldsymbol { \beta } _ { \eta } ( H ) , \end{align*}


\begin{align*} \tilde { \varphi } ( v ) = \begin{cases} \varphi ( v ) , & v \in T , \\ w _ { i } , & v = v _ { i } . \end{cases} \end{align*}


\begin{align*} \bigcup p ^ \in = p ^ \wedge . \end{align*}


\begin{align*} A B E P \leq \frac { 1 } { \log _ 2 N _ t } \sum _ { \hat n _ t = 1 } ^ { N _ t } \sum _ { n _ t \neq \hat n _ t } ^ { N _ t } \bar P _ i N ( \hat n _ t \to n _ t ) , \ \ \ i \in \{ a , b \} \end{align*}


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


\begin{align*} \pi = \cdots y z \cdots \beta _ 1 ( \pi ) \cdots x \cdots \qquad \phi ( x , \pi ) = \cdots y x z \cdots \beta _ 1 ( \pi ) \cdots . \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { \infty } \omega ( 2 ^ { - j } ) & \approx \int _ { 0 } ^ { 1 } \omega ( t ) \frac { \mathrm { d } t } { t } < \infty . \end{align*}


\begin{align*} | S ( k ) D ^ j \varphi ( x ) | & = | D ^ j v ( T ^ { Q _ \alpha ( k ) } ( x ) ) - D ^ j v ( x ) | \leq \left \{ \begin{array} { l l } \| D ^ { j } v \| _ { C ^ 1 } | I ^ { ( k ) } | & 0 \leq j < m \\ \| D ^ { m } v \| _ { C ^ b } | I ^ { ( k ) } | ^ b & j = m . \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} & \underset { S _ { n } } { } ~ R _ { M M S E } \left ( S _ { n } \right ) \\ & \ \left \| \textbf { P } _ c \left ( S _ { n } \right ) \right \| ^ { 2 } \leq P . \end{aligned} \end{align*}


\begin{align*} \phi = \Tilde { \phi } _ { e } + \sum _ { m = 1 } ^ { k } C _ { p _ m } h ^ { p _ m } , \end{align*}


\begin{align*} \tau = \min \left \{ \frac { s } { 2 } , \frac { 1 - s } { 2 } \right \} . \end{align*}


\begin{align*} \phi _ t ( p , q ) : = e _ t ( \gamma _ { p , q } ) = \exp _ q ( ( t - 1 ) \lambda _ p ) , \lambda _ p \in T ^ * _ q M p = \exp _ q ( - \lambda _ p ) . \end{align*}


\begin{align*} J ( g p _ z , i ) & = \sqrt { ( c y i + c x + d y ^ { - 1 } \det p _ z ) \det ( g p _ z ) } \\ & = \sqrt { y ^ { - 1 } \det g ( c z + d ) } \\ & = J ( g , z ) J ( p _ z , i ) . \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*} H _ { i , t _ 0 + \epsilon } = H _ { i , t _ 0 - \epsilon } \cup N ( L _ i ) \end{align*}


\begin{align*} \widehat { \varphi } ^ { ( i ) } ( x _ i , t ) : = \frac { 1 } { \pi h ^ 2 _ i } \int _ { \partial \Upsilon _ i } \varphi ^ { ( i ) } ( x _ i , \bar { \xi } _ i , t \big ) \ , d \sigma _ { \bar { \xi } _ i } . \end{align*}


\begin{align*} W ^ { = 1 } [ f ] = p ( 1 - p ) \approx p \end{align*}


\begin{align*} \norm { \widetilde { \psi } _ m } _ { X _ { \mu _ 0 , \xi _ 0 } } \leq \frac { C } { \mu _ 0 } \norm { g } _ { Y _ { \mu _ 0 , \xi _ 0 } } , m = 0 , 1 , \ldots , N . \end{align*}


\begin{align*} \sup _ { \pi \in \Gamma ( \mathcal { G } , \varphi ) } \int _ { \mathcal { T } \times \mathcal { X } } \varphi ( t , x ) d \pi ( t , x ) = \sup _ { \mu \in \mathcal { G } } \left \{ \int _ { \mathcal { T } } \left [ \sup _ { x \in \mathcal { X } } \varphi ( t , x ) \right ] d \mu ( t ) \right \} . \end{align*}


\begin{align*} = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \prod _ { j = 1 } ^ { M } { L i _ { s _ j } } ^ { ( d _ j ) } \left ( e ^ { i \theta } \right ) \prod _ { k = 1 } ^ { N } { L i _ { t _ k } } ^ { ( e _ k ) } \left ( e ^ { - i \theta } \right ) d \theta , \end{align*}


\begin{align*} a _ 0 \ \equiv \ b _ 0 + 4 \ \equiv \ \begin{cases} 7 , & , \\ 1 2 , & . \end{cases} \end{align*}


\begin{align*} \sum _ { 1 \le k \le L } \frac { R ^ { k } } { k } \min \{ q ^ { k / 2 } , d ( \chi ) \} = \sum _ { 1 \le k \le L } \frac { ( R \sqrt { q } ) ^ k } { k } \le \sum _ { 1 \le k \le L } ( R \sqrt { q } ) ^ k \le \frac { ( R \sqrt { q } ) ^ L } { 1 - ( R \sqrt { q } ) ^ { - 1 } } \le 6 d ( \chi ) R ^ L \end{align*}


\begin{align*} f ( \theta ) = ( - 1 ) ^ k e ^ { - \i k \theta } \prod _ { j = 1 } ^ { 2 k } ( e ^ { \i \theta } - e ^ { \i \alpha _ j } ) . \end{align*}


\begin{align*} f ^ * _ { n , l } ( z _ 1 , \ldots , z _ m ) : = f ^ * _ { n , l - 1 } ( z _ 1 , \ldots , z _ m ) + \delta _ { n , l } z _ 1 ^ { j _ 1 } \cdots z _ m ^ { j _ m } A _ n ( z _ 1 , \ldots , z _ m ) \end{align*}


\begin{align*} 0 = & - \lambda h ( x _ 0 ) = M h ( x _ 0 ) = \sum _ { y \sim x _ 0 } M _ { x _ 0 y } h ( y ) \\ = & \sum _ { \substack { y \sim x _ 0 \\ y \in D _ j \setminus D _ i } } M _ { x _ 0 y } h ( y ) = \left ( 1 - \frac { a _ j } { a _ i } \right ) \sum _ { \substack { y \sim x _ 0 \\ y \in D _ j \setminus D _ i } } M _ { x _ 0 y } f ( y ) . \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle \dfrac { 6 } { N } \left ( \dfrac { a _ 2 } { 2 m _ 1 } \int _ { \mathbb { R } ^ N } | \nabla \xi | ^ 2 d x + \dfrac { a _ 1 } { 4 m _ 2 } \int _ { \mathbb { R } ^ N } | \nabla \eta | ^ 2 d x \right ) - \dfrac { 3 } { 2 } a _ 1 a _ 2 \int _ { \mathbb { R } ^ N } \eta \xi ^ 2 d x = 0 . \end{array} \right . \end{align*}


\begin{align*} \sigma _ { L ^ 2 _ N } \left ( \mathcal { A } [ \phi ] \right ) = \bigcup _ { \xi \in \Omega _ N } \sigma _ { L ^ 2 _ { \rm p e r } ( 0 , T ) } \left ( \mathcal { A } _ \xi [ \phi ] \right ) , \end{align*}


\begin{align*} \begin{array} { r c l } \mathrm { d } x ^ { 1 } \\ \mathrm { d } x ^ { 2 } \\ \mathrm { d } x ^ { 3 } \\ \mathrm { d } \varphi ^ { 1 } & = & \mathrm { d } x ^ { 1 } \\ \mathrm { d } \varphi ^ { 2 } & = & \mathrm { d } x ^ { 2 } \\ \mathrm { d } \delta ( \varphi ^ { 1 } ) & = & \mathrm { d } x ^ { 1 } + \mathrm { d } u ^ { 1 } \\ \mathrm { d } \delta ( \varphi ^ { 2 } ) & = & \frac { 1 } { u ^ { 1 } + 1 } \mathrm { d } x ^ { 3 } - \frac { x ^ { 3 } } { ( u ^ { 1 } + 1 ) ^ { 2 } } \mathrm { d } u ^ { 1 } \end{array} \end{align*}


\begin{align*} \phi ( ( \overline { \mathbb { Q } } _ { \ell } ) _ { \widetilde { M } } ) = \phi ( \mathcal { I C } _ { \widetilde { M } } ) = 0 . \end{align*}


\begin{align*} \mathcal { G } ^ M _ { r } ( K , \gamma ) \cap B _ { \rho ( r ) } ( p ) = C N _ { r } ( \zeta ) \cap B _ { \rho ( r ) } ( p ) , \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle \int _ { \mathbb { R } ^ N } \left ( a _ 2 | \phi ( t , x ) | ^ 2 + a _ 1 | \psi ( t , x ) | ^ 2 \right ) d x = \int _ { \mathbb { R } ^ N } \left ( a _ 2 | \phi _ 0 ( x ) | ^ 2 + a _ 1 | \psi _ 0 ( x ) | ^ 2 \right ) d x ; \end{array} \right . \end{align*}


\begin{align*} f ( V A V ^ * ) = V f ( A ) V ^ * \end{align*}


\begin{align*} f ( \ell ) = \left \lceil e ^ { a \ell ^ { 3 / 2 } } \right \rceil \leq \left \lceil e ^ { a ( n + 1 ) ^ { 3 / 2 } } \right \rceil = f ( n + 1 ) . \end{align*}


\begin{align*} \sigma ^ 2 = \sum _ { s = - \infty } ^ { \infty } \Theta _ { \infty } ( s ) = \frac { 2 } { \zeta _ N ( m + n ) } \sum _ { s = - \infty } ^ { \infty } \left ( \sum _ { \substack { p \geq 1 \\ g c d ( p , N ) = 1 } } \sum _ { \substack { q \geq 1 \\ q = p ( m o d \ { N } ) } } \int _ { \R ^ { m + n } } \chi ( p a ^ s x ) \chi ( q x ) \ , d x \right ) . \end{align*}


\begin{align*} M _ { \Gamma } ( x ) = | x | ^ { \beta } M _ { \Gamma } ( x / | x | ) , x \in \Gamma . \end{align*}


\begin{align*} \alpha ( x ) = \alpha _ { d } ( \pi ( x ) ) , \end{align*}


\begin{align*} ( W , \lambda _ 1 , \phi _ 1 ) = | W _ 1 ^ { \flat } : = W _ 0 ^ { \flat } \cup h _ { n - 1 } ; \ , \mathcal { L } | \ , \cup \ , h _ n \end{align*}


\begin{align*} \Pr ( \Phi ( \mathcal { A } ) > 0 ) = 1 - \exp \left ( - 2 \pi \lambda R _ { m i n } R _ S \right ) . \end{align*}


\begin{align*} R _ 2 ( x , \hat { y } ) : = \sum _ { | \beta | = 2 } \frac { 2 } { \beta ! } \left ( \int _ { 0 } ^ 1 ( 1 - t ) ^ \beta c ( \hat { y } + t ( x - \hat { y } ) ) t \right ) ( x - \hat { y } ) ^ \beta . \end{align*}


\begin{align*} \psi _ { i } ^ { N } ( \mathbf { x } _ { N } , \mathbf { m } _ { N } ) = \frac { 1 } { N } \stackrel [ j = 1 ] { N } { \sum } m _ { i } m _ { j } S ( x _ { j } - x _ { i } ) \end{align*}


\begin{align*} \begin{aligned} I _ { 2 , 2 } \leq \frac { c \lambda ^ { \tilde { p } - 1 } } { \mathcal { M } ^ { \frac { 2 s } { n } } } \sum _ { \mathcal { K } \times I \in \mathcal { A } } \int _ { \lambda _ { 0 } } ^ { N \mathcal { M } ^ { - 1 } } \mu _ { \tau , t } \left ( \mathcal { K } \times I \right ) \ , d \lambda \end{aligned} \end{align*}


\begin{align*} a _ { t } ^ { \prime } x ^ { \mu } - b _ { t } ^ { \mu } = \left ( 1 - \mu \right ) \left ( a _ { t } ^ { \prime } \overline { x } - \overline { b } _ { t } \right ) + \mu \left ( a _ { t } ^ { \prime } x - b _ { t } \right ) \leq 0 , \end{align*}


\begin{align*} n _ g ^ * : = \sum _ { \lambda = 1 } ^ \kappa n _ \lambda ^ * , n _ g : = \sum _ { \ell = 1 } ^ k n _ \ell - 2 k . \end{align*}


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


\begin{align*} \epsilon \int \limits _ { \Omega } U ^ T ( D _ i ) _ { x _ i } d \Omega = \epsilon \oint \limits _ { \partial \Omega } U ^ T ( n _ i D _ i ) d s - \epsilon \int \limits _ { \Omega } U _ { x _ i } ^ T D _ i d \Omega . \end{align*}


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


\begin{align*} E ( \phi ) : = \frac { 1 } { n + 1 } \int _ \Omega ( - \phi ) ( d d ^ c \phi ) ^ n . \end{align*}


\begin{align*} \mathsf { s o r t } ( \sigma ( A ) ) : = \mu ( A , k ) \ldots \mu ( A , 2 ) \mu ( A , 1 ) . \end{align*}


\begin{align*} E _ y = ( - w + a ) ( - w + b ) - ( - w + c ) ( - y ) w . \end{align*}


\begin{align*} w ( b ) = \sup _ { \psi \in \S ( B ) } | \psi ( b ) | \end{align*}


\begin{align*} C ( \gamma _ 1 , \gamma _ 2 ) + C ( \gamma _ 2 , \gamma _ 3 ) = C ( \gamma _ 1 , \gamma _ 3 ) . \end{align*}


\begin{align*} & { _ { 2 } F _ { 1 } } \left [ \begin{array} { c c c c c c c c } a , 1 - a \\ b \end{array} ; \frac { 1 } { 2 } \right ] = \frac { \Gamma ( \frac { b } { 2 } ) \Gamma ( \frac { 1 + b } { 2 } ) } { \Gamma ( \frac { a + b } { 2 } ) \Gamma ( \frac { 1 - a + b } { 2 } ) } . \end{align*}


\begin{align*} W _ i ^ { \natural } = W ^ { \flat } \cup h _ { i , 1 } \cup \cdots \cup h _ { i , k _ i } , \end{align*}


\begin{align*} \iota _ \pm = \left ( \frac { \sqrt { \lambda ^ { 2 } - \cos ^ { 2 } ( \Delta \xi / 2 ) } \pm \sin ( \Delta \xi / 2 ) } { \sqrt { \lambda ^ { 2 } - 1 } } \right ) ^ { 2 } \end{align*}


\begin{align*} U _ k = 1 e _ k = 0 \prod _ { k = 1 } ^ { N } U _ k ^ { e _ k } \leq q ^ { 1 + e _ M } , \end{align*}


\begin{align*} g = \begin{bmatrix} g _ { 1 1 } & g _ { 1 2 } \\ \ 0 & g _ { 2 2 } \end{bmatrix} \end{align*}


\begin{align*} \big \{ l ( s ) = \{ y = s + k x \} \ , : \ , s \in \R \big \} \end{align*}


\begin{align*} \| \phi ( A _ 1 \otimes \cdots \otimes A _ m ) \| _ { ( p , k ) } = \| A _ 1 \otimes \cdots \otimes A _ m \| _ { ( p , k ) } \end{align*}


\begin{align*} x _ i v _ j = - v _ { 5 i + 3 j \bmod 7 } , i , j \in \{ 1 , 2 , \dots , 7 \} . \end{align*}


\begin{align*} Q = ( \Vert \eta \Vert _ { { 4 . 5 } } + \Vert v \Vert _ { { 4 } } + 1 ) ^ 2 \end{align*}


\begin{align*} g ( y ) : = \sum _ { n \geq 1 } \varphi _ { n } \frac { y ^ n } { n ! } = - \varphi _ 0 . \end{align*}


\begin{align*} \begin{cases} C ^ \varphi _ { i j k } = - C ^ \varphi _ { i k j } \ , C ^ \varphi _ { i k k } = 0 \\ C ^ \varphi _ { k k i } = \alpha \varphi ^ a _ { k k } \varphi ^ a _ i \\ C ^ \varphi _ { i j k } + C ^ \varphi _ { j k i } + C ^ \varphi _ { k i j } = 0 \ , . \end{cases} \end{align*}


\begin{align*} h = \begin{bmatrix} 1 & 0 \\ 0 & - 1 \end{bmatrix} , e ^ + = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} , e ^ - = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} , J ^ + = \tfrac { 1 } { 2 } \begin{bmatrix} - i & 1 \\ 1 & i \end{bmatrix} , J ^ - = \tfrac { 1 } { 2 } \begin{bmatrix} i & 1 \\ 1 & - i \end{bmatrix} , J ^ 0 = \begin{bmatrix} 0 & 1 \\ - 1 & 0 \end{bmatrix} . \end{align*}


\begin{align*} \tilde A = { \rm I } _ { N _ { S } } + ( a _ { j } ) _ { i , j } . \end{align*}


\begin{align*} \Theta ( \delta ) = \left [ \min _ { \gamma \in \Sigma ( \delta ) } \int _ \mathcal { S } f ( y _ 1 , y _ 2 ) \ , d \gamma ( s ) , \max _ { \gamma \in \Sigma ( \delta ) } \int _ \mathcal { S } f ( y _ 1 , y _ 2 ) \ , d \gamma ( s ) \right ] , \end{align*}


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


\begin{align*} A ^ \pm _ { n , k , i } = \begin{bmatrix} A _ { n , k , i } & A _ { n , k , i } \\ A _ { n , k , i } & A _ { n , k , i } \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \otimes A _ { n , k , i } . \end{align*}


\begin{align*} \widehat { \mu } ( t ) = \exp \left [ - \frac { 1 } { 2 } a t ^ 2 + i b t + \int _ { - \infty } ^ { \infty } \left ( e ^ { i t \lambda } - 1 - \frac { i t \lambda } { 1 + \lambda ^ 2 } \right ) \nu ( d \lambda ) \right ] , \end{align*}


\begin{align*} \det \left [ \binom { 2 m } { j - i + m } - \binom { 2 m } { m - i - j - 1 } \right ] _ { i , j = 0 } ^ { n - 1 } = \prod _ { 1 \leq i \leq j \leq m - 1 } \frac { 2 n + i + j } { i + j } . \end{align*}


\begin{align*} { { \boldsymbol { d } } _ k } = { { \boldsymbol { W } } _ k } { { \boldsymbol { x } } _ k } + { { \boldsymbol { e } } _ k } \end{align*}


\begin{align*} y ( q ^ 2 x ) - ( q x \sin ( 2 q x ) + \cos ( x ) ) y ( q x ) + x \cos ( x ) \sin ( 2 x ) y ( x ) = 0 . \end{align*}


\begin{align*} \left \langle X , \nabla _ X X \right \rangle = \frac { 1 } { 2 } \nabla _ X \left ( | X | ^ 2 \right ) . \end{align*}


\begin{align*} \lim _ { | x | \rightarrow \infty } \nabla \cdot \varphi _ 3 ( x ) = 0 . \end{align*}


\begin{align*} \langle A ( v - s d ) , v - s d \rangle = \langle A v , v \rangle - 2 s \langle A v , d \rangle + s ^ 2 \langle A d , d \rangle = \langle A v , v \rangle - 2 s \| d \| ^ 2 + s ^ 2 \langle A d , d \rangle . \end{align*}


\begin{align*} p _ { k } : Q \to G _ { 2 } : = \left \{ M \in G L ( 2 , \mathbb { R } ) , \left | \det ( M ) \right | = 1 \right \} \end{align*}


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


\begin{align*} \limsup _ { n \to \infty } \frac { y ^ { ( i ) } _ n ( I _ 1 , J _ 1 ) } { y ^ { ( i + 1 ) } _ n ( I _ 1 , J _ 1 ) } = \infty , \end{align*}


\begin{align*} \ < \Psi , [ \N ^ \ell , V _ { F B } ( t ) ] \Psi \ > = 2 \mathrm { I m } \int _ { \Lambda ^ * } \hat V ( k ) \langle \Psi , [ \N ^ \ell , D ^ * _ k ( t ) b _ k ( t ) ] \Psi \rangle \ . \end{align*}


\begin{align*} \det \ , D ^ 2 v _ 0 = c _ 0 \chi _ { _ { \Omega ^ * _ 0 } } \quad \R ^ n \end{align*}


\begin{align*} X _ \varphi : = \overline { \varphi ( H ) } \end{align*}


\begin{align*} \sum _ I \ < Y _ F , S ^ I \ > M _ I ( X ) = \Gamma _ F ( X ) \end{align*}


\begin{align*} \widetilde { T } _ l ( f _ 1 , f _ 2 ) ( x , y ) = \int _ \mathbb { R } \ ! f _ 1 \left ( x + \widetilde { P } _ 1 ( t ) , y \right ) f _ 2 \left ( x , y + \widetilde { P } _ 2 ( t ) \right ) \zeta ( x , y , t ) \ , \mathrm { d } t . \end{align*}


\begin{align*} \cos \alpha _ { 1 0 3 } = \cos \alpha _ { 2 0 4 } \end{align*}


\begin{align*} i \frac { \partial \psi _ 1 } { \partial t } & = \Big ( - \frac { \partial ^ 2 } { \partial x ^ 2 } - a _ 1 ( | \psi _ 1 | ^ 2 + | \psi _ 2 | ^ 2 ) - U _ 1 ( x ) \Big ) \psi _ 1 , \ \\ i \frac { \partial \psi _ 2 } { \partial t } & = \Big ( - \frac { \partial ^ 2 } { \partial x ^ 2 } - a _ 2 ( | \psi _ 1 | ^ 2 + | \psi _ 2 | ^ 2 ) - U _ 2 ( x ) \Big ) \psi _ 2 , \ \end{align*}


\begin{align*} \| u \| _ { L ^ { \phi } } = \inf \left \{ \lambda > 0 : \int _ { \R ^ d } \phi \left ( \frac { | u ( x ) | } { \lambda } \right ) d x \leq 1 \right \} . \end{align*}


\begin{align*} \hat \phi ( 0 ) - \phi ( 0 ) + \int _ { - \infty } ^ { \infty } \hat \phi ( x ) | x | d x = \int _ { - \infty } ^ { \infty } \phi ( x ) W _ { \rm U } ^ 1 ( x ) d x = \int _ { - \infty } ^ { \infty } \phi ( x ) \left ( 1 - \frac { \sin ^ 2 ( \pi x ) } { ( \pi x ) ^ 2 } \right ) d x . \end{align*}


\begin{align*} | \delta _ t ( \lambda y ) - 2 t / \pi | = t | \delta _ 1 ( t \lambda y ) - \delta _ 1 ( 0 ) | \leq C t ( t \lambda | y | ) ^ 2 \ . \end{align*}


\begin{align*} \phi ( \theta ) = \begin{pmatrix} i \lambda & g ( \theta ) \\ - g ( \theta ) ^ { - 1 } & i \lambda \end{pmatrix} , \end{align*}


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


\begin{align*} \mathcal { E C S } _ { { \rm { I } } } = \mathcal { E C S } _ { 0 0 1 } \cup \mathcal { E C S } _ { 0 1 0 } \cup \mathcal { E C S } _ { 1 0 0 } , \end{align*}


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


\begin{align*} \Lambda _ 2 : = ( \Lambda _ { 2 , + } ) ^ { \sharp } \cup ( \Lambda _ { 2 , 0 } ) ^ { \sharp } \cup ( \Lambda _ { 2 , - } ) ^ { \sharp } \subset \Gamma , \end{align*}


\begin{align*} { \mathcal { C } } _ { \alpha } = B _ 1 ^ { ( \alpha ) } \otimes L + B _ 2 ^ { ( \alpha ) } \otimes ( - 2 I _ { m } ) . \end{align*}


\begin{align*} K _ 1 ( t ) = q ( t ) \tilde { K } _ 1 ( t ) q ( t ) \ , , K _ 2 ( t ) = q ( t ) \otimes q ( t ) \tilde { K } _ 2 ( t ) \end{align*}


\begin{align*} \lim _ { h \to + \infty } \gamma _ h \mathcal { H } ^ 1 \left ( \left \{ \widehat { f _ h } \neq \widehat { g _ h } \right \} \cap \partial B _ \rho \right ) = 0 \end{align*}


\begin{align*} \lim _ { \lambda \rightarrow + 0 } K _ { \omega , \mathbf { c } } ( V ^ { \lambda } ) = 2 \omega M ( V ) > 0 . \end{align*}


\begin{align*} \mathcal { H } ^ 1 \left ( J _ { w _ 1 } \cap \cup _ { i \in \N } B ^ i _ 1 \right ) \leq \liminf _ { h \to + \infty } \mathcal { H } ^ 1 \left ( J _ { w ^ h _ 1 } \cap \cup _ { i \in \N } B ^ i _ 1 \right ) = 0 \end{align*}


\begin{align*} i _ { \boldsymbol { a } } = q . \end{align*}


\begin{align*} \Delta _ g w = 0 \ ; \mathrm { i n } \ ; \mathrm { M } , w _ { | \partial \mathrm { M } } = 1 . \end{align*}


\begin{align*} H ^ s ( \mathbb { R } ^ n ) = \{ u \in \mathcal { S } ' ( \mathbb { R } ^ n ) : \{ \xi \mapsto ( 1 + | \xi | ^ 2 ) ^ { s / 2 } \hat { u } ( \xi ) \} \in L ^ 2 ( \mathbb { R } ^ n ) \} , \end{align*}


\begin{align*} \omega _ { n \pm } : = \omega _ { \pm } ( 1 / n ) = \left ( \sqrt { \omega } \pm \frac { 1 } { n } \right ) ^ 2 , \ \ \mathbf { c } _ { n \pm } : = \mathbf { c } _ { \pm } ( 1 / n ) = \frac { \mathbf { c } } { \sqrt { \omega } } \left ( \sqrt { \omega } \pm \frac { 1 } { n } \right ) . \end{align*}


\begin{align*} ( A _ 2 - \overline { A _ 2 } ) z ^ { n + m } + ( B _ 2 - \overline { B _ 2 } ) \overline { z } ^ m + ( C _ 2 - \overline { C _ 2 } ) = 0 \textrm { f o r a l l } z \in \mathbb { C } . \end{align*}


\begin{align*} \mathfrak { S } ( f _ k ) = k + r - 1 - | \mathcal { F } | + c _ 1 , \ , \ , \ , \ , \ , \overline { \mathfrak { S } } ( f _ k ) = n - k + 1 - | \mathcal { F } | + c _ 2 , \end{align*}


\begin{align*} v ( b _ { \ell i } ) = - \min \limits _ j v ( b _ { \ell i j } ) . \end{align*}


\begin{align*} \phi = \Tilde { \phi } _ { e } + C _ { \Tilde { q } } h ^ { \Tilde { q } } . \end{align*}


\begin{align*} ( 1 + \eta _ r ) ( 1 + \eta _ s ) = ( 1 + \eta _ { r + s } ) . \end{align*}


\begin{align*} \overline { W } _ t = W _ t + \int _ 0 ^ t C _ { s } \ , d s . \end{align*}


\begin{align*} \Big ( \sum _ { e \in E ( \mathbb Z ^ d ) } p _ e ^ 2 \Big ) ^ d \le \Big ( \sum _ { e \in E ( \mathbb Z ^ d ) } q _ e p _ e \Big ) ^ d \le C n ( \log n ) ^ { \frac { d ( d + 1 ) } { 2 } } \Big ( \sum _ { e \in E ( \mathbb Z ^ d ) } p _ e ^ 2 \Big ) ^ { \frac { d - 1 } { 2 } } . \end{align*}


\begin{align*} \hat Z _ { s , \ell } = \Phi ^ { s } _ { 2 } ( y ) - \Phi ^ { \ell } _ { 2 } ( x ) . \end{align*}


\begin{align*} L i _ 3 ( 1 ) = \zeta ( 3 ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta _ { p } u & = v ^ { k _ 1 } \cdot | \nabla u | ^ { \alpha } & \quad \mbox { i n } \Omega , \\ \Delta _ { p } v & = v ^ { k _ 2 } \cdot | \nabla u | ^ { k _ 3 } & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \zeta ( s ) : = \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ s } \Re ( s ) > 1 , \end{align*}


\begin{align*} J _ { \overrightarrow 0 } ( t ) = J _ { \overrightarrow 0 } ( t ^ { ( 1 ) } ) ( \mathrm { s e c t } ( \delta / 2 ) ) \cdot J _ { \mathrm { s e c t } ( \delta / 2 ) } ( t ^ { ( 2 ) } ) \end{align*}


\begin{align*} \phi ^ { \alpha , n } ( g ) ^ \ast & \left ( S , f _ \beta \otimes \rho _ { n } ( E ^ \beta ) \right ) \\ = & - \left ( S , \overline { g ^ { S ^ \flat } } \cdot f _ \beta \otimes \rho _ n ( [ E ^ \alpha , E ^ \beta ] ) \right ) + \left \langle ( S , f _ \beta \otimes \rho _ { n } ( E ^ \beta ) ) , \phi ^ { \alpha , n } ( g ) 1 \right \rangle 1 . \end{align*}


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


\begin{align*} \mathcal { I } ( \delta ) = \sup _ { \gamma \in \Sigma ( \delta ) } \int _ { \mathcal { S } } f \ , d \gamma , \end{align*}


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


\begin{align*} l ' ( \varphi ( b ) , x ) = s ' ( \varphi ( b ) ) i ' ( x ) = s ( b ) i ( x ) = l ( b , x ) \end{align*}


\begin{align*} \sigma _ { m n } : = \frac { 1 } { P _ { m n } } \sum _ { i = 0 } ^ m \sum _ { j = 0 } ^ { n } p _ { i j } u _ { i j } \end{align*}


\begin{align*} H _ { \alpha } ^ { \mathrm { X Y } } = - \frac \alpha 2 \sum _ { j = 0 } ^ { M - 1 } \left ( ( 1 + \gamma ) \sigma ^ x _ j \sigma ^ x _ { j + 1 } + ( 1 - \gamma ) \sigma ^ y _ j \sigma ^ y _ { j + 1 } \right ) - \sum _ { j = 0 } ^ { M - 1 } \sigma ^ z _ j , \end{align*}


\begin{align*} D ^ { k - 1 } ( f ) = c F + D ^ { k - 1 } ( h ) + d 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*} J ( \overline { u } ) = { 1 \over 2 } \overline { Y } _ 0 x + { 1 \over 2 } \int _ 0 ^ T L _ { t } \ , d \overline { \mu } _ { t } . \end{align*}


\begin{align*} \beta = \begin{cases} \frac { 1 } { 2 } - \delta & { \rm i f ~ } r \ge 4 \\ \frac { 1 } { 3 } - \delta & { \rm i f ~ } r = 3 \ ; . \end{cases} \end{align*}


\begin{align*} g ^ { \flat } ( p , t ) = ( \bar g ^ { \flat } ( p ) , \chi _ g ( p ) \cdot t ) \end{align*}


\begin{align*} \begin{aligned} \rho ( u ) = \log ( \theta ) + B ^ * ( u ) + \frac { 1 } { 2 } u + \log \left ( \frac { T ^ { \infty } - T ( u ) } { T ^ { \infty } } \right ) . \end{aligned} \end{align*}


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


\begin{align*} \begin{cases} L _ { \lambda } [ X ] = - X '' + ( R - \lambda I ) X = 0 , \\ X ( a ) = X ( b ) = 0 \end{cases} \end{align*}


\begin{align*} \mathrm { D o m } \left ( \mathcal { H } _ { \hbar , V } \right ) : = \left \{ u \in \ell ^ { 2 } ( \hbar \mathbb { Z } ^ { n } ) : ( I + \mathcal { H } _ { \hbar , V } ) u \in \ell ^ { 2 } ( \hbar \mathbb { Z } ^ { n } ) \right \} . \end{align*}


\begin{align*} \liminf _ { N \to \infty } \frac { 2 } { N ^ 2 } \log \P _ { N , r _ N } ( U _ \varepsilon ^ x ) \geq - \lim _ { N \to \infty } \frac { 2 } { N ^ 2 } \int _ { U _ \varepsilon ^ x } \left ( \log \frac { \d \P _ { N , h _ N } } { \d \P _ { N , r _ N } } \right ) \d \P _ { N , h _ N } = - I _ r ( h ) . \end{align*}


\begin{align*} T _ w ( M , \bar { u } ) = \{ v \in \mathcal { U } : \ \exists \{ u _ n \} \subset M , \ \{ t _ n \} \subset \mathbb { R } ^ + , \ u _ n \rightarrow \bar { u } , \ t _ n \rightarrow 0 ^ + , \frac { 1 } { t _ n } ( u _ n - \bar { u } ) \rightharpoonup v \} \end{align*}


\begin{align*} \lambda _ { - k } ( f ( A _ { \cal S } ) ) = & \min _ { \{ h \in { \cal F } \ , : \ , \Vert h \Vert = 1 \} } \langle h , f ( A _ { \cal S } ) h \rangle \\ = & \min _ { \{ h \in { \cal F } \ , : \ , \Vert h \Vert = 1 \} } f ( \langle h , A _ { \cal S } h \rangle ) \\ \le & \inf _ { \{ h \in { \cal F } \ , : \ , \Vert h \Vert = 1 \} } \langle h , f ( A ) h \rangle \le \lambda _ { - k } ( f ( A ) _ { \cal S } ) \end{align*}


\begin{align*} 1 / m _ { \boldsymbol { \nu } , q } = \int _ 0 ^ { 1 / ( q \mathrm { k } ^ * ) } \Pi ( q t ) \dd t , \end{align*}


\begin{align*} \mathbf { T } _ { K ^ \circ } = \mathrm { R e s } _ { K / \mathbb { Q } } ( \mathbb { G } _ m ) \cap \mathbf { G } _ { \mathbb { Q } } , \mathbf { T } _ { K ^ \circ } ^ 1 = \mathbf { T } _ { K ^ \circ } \cap \mathbf { G } ^ 1 _ { \mathbb { Q } } . \end{align*}


\begin{align*} ( \Delta + \lambda - q _ 1 ) w = \chi ( q _ 2 - q _ 1 ) v _ 2 + [ \Delta , \psi ] v . \end{align*}


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


\begin{align*} h ( Z , \lambda ) = \int _ { G ^ { \mathbb { N } } } \int _ { X } \log \frac { d \omega _ { 1 } . \eta } { d \eta } \left ( x \right ) d \eta _ { \omega } ( x ) d \mathbb { P } _ { \mu } ( \omega ) + & \int _ { G ^ { \mathbb { N } } } \int _ { X } \log \frac { d \left ( \omega _ { 1 } \lambda \right ) _ { x } } { d \lambda _ { x } } \left ( \psi ( x , \omega ) \right ) d \eta _ { \omega } ( x ) d \mathbb { P } _ { \mu } ( \omega ) = { \rm I } + { \rm I I } . \end{align*}


\begin{align*} \| e _ t \varphi ( a ) e _ t - \varphi ( a ) \| & = \| v _ t v _ t ^ * \varphi ( a ) v _ t v _ t ^ * - \varphi ( a ) \| \\ & \leq \| v _ t v _ t ^ * \varphi ( a ) v _ t v _ t ^ * - v _ t \varphi ( \alpha _ { t ^ { - 1 } } ( a ) ) v _ t ^ * \| + \| v _ t \varphi ( \alpha _ { t ^ { - 1 } } ( a ) ) v _ t ^ * - \varphi ( a ) \| \\ & \leq \| v _ t ^ * \varphi ( a ) v _ t - \varphi ( \alpha _ { t ^ { - 1 } } ( a ) ) \| \end{align*}


\begin{align*} \Lambda = \frac { m - 2 } { 2 } \lambda \end{align*}


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


\begin{align*} R _ { \theta _ 0 } ( z _ 1 , z _ 2 , u ) = ( z _ 1 \cos \theta _ 0 - z _ 2 \sin \theta _ 0 , z _ 1 \sin \theta _ 0 + z _ 2 \cos \theta _ 0 , u ) . \end{align*}


\begin{align*} & \left | \left \lbrace ( x , y , t ) \in \ ! K : \left | \widetilde { P } _ 1 ' ( t ) \alpha ( x + \widetilde { P } _ 1 ( t ) , y ) - \widetilde { P } _ 2 ' ( t ) \beta ( x , y + \widetilde { P } _ 2 ( t ) ) \right | \leq \ ! \varepsilon \ ! \right \rbrace \right | \\ \lesssim & 2 ^ { c | l | } \varepsilon ^ { 1 / 3 5 } \end{align*}


\begin{align*} \bigcup _ { \ell \in \mathbb { N } } \left \{ h _ { \mu } \left ( Z , \lambda _ { \ell , p } \right ) : p \in [ 0 , 1 ] \right \} = \left ( h _ { \mu } \left ( G / P _ { I } \right ) , h _ { \mu } \left ( G / P _ { I ' } \right ) \right ] . \end{align*}


\begin{align*} \begin{cases} \overset \cdot a _ 0 = a _ 0 a _ 1 a _ 2 , \overset \cdot a _ 1 = a _ 1 a _ 2 a _ 3 , \\ \overset \cdot a _ 2 = a _ 2 a _ 3 a _ 4 \ , - \ , a _ 2 a _ 1 a _ 0 , \\ \overset \cdot a _ 3 = - a _ 3 a _ 2 a _ 1 , \ ; \overset \cdot a _ 4 = - a _ 4 a _ 3 a _ 2 ; \end{cases} \end{align*}


\begin{align*} \Pi _ { \psi } ( [ w ' , t ] ) f ( \epsilon , w ) = \psi ( t + \epsilon \tfrac { \langle w , w ' \rangle } { 2 } ) f ( \epsilon , w + w ' ) , \end{align*}


\begin{align*} \beta ( a , b , c ) & = \sigma _ 2 \sigma _ 1 ^ { a + 1 } \sigma _ 2 \sigma _ 1 ^ { b + 1 } \sigma _ 2 \sigma _ 1 ^ { c + 1 } = \sigma _ 2 \sigma _ 1 ( \sigma _ 2 \sigma _ 1 ) \sigma _ 2 ^ a \sigma _ 1 ^ { b - 1 } \sigma _ 2 ^ c ( \sigma _ 1 \sigma _ 2 ) \sigma _ 1 = \Delta _ 3 \sigma _ 1 \sigma _ 2 ^ a \sigma _ 1 ^ { b - 1 } \sigma _ 2 ^ c \Delta _ 3 . \end{align*}


\begin{align*} h _ { d } ( X _ 0 , X _ 1 , . . . , X _ k ) \overset { } { = } \sum \limits _ { 0 \le i _ 1 \leqslant i _ 2 \leqslant \cdots \leqslant i _ d \le k } X _ { i _ 1 } \cdot X _ { i _ 2 } \cdots X _ { i _ d } \ , , \end{align*}


\begin{align*} \Psi ( h , x ) = ( h , \widehat { x } _ { \varphi ( h ) - \underline { \varphi } ( h ) } ) \end{align*}


\begin{align*} \sigma _ k \left [ f \left ( \sum _ { i = 1 } ^ m Z ^ * _ i A _ i Z _ i \right ) \right ] \le \sigma _ k \left [ \sum _ { i = 1 } ^ m Z ^ * _ i f ( A ) _ i Z _ i \right ] , k = 1 , \ldots , n , \end{align*}


\begin{align*} 0 = c ( \beta _ i - \beta _ j ) + \alpha _ { i j } - \lambda _ { i j } + \gamma _ { i } , \quad \forall i \neq j . \end{align*}


\begin{align*} \frac { d } { d \lambda } S _ { \omega , \mathbf { c } } ( \Phi _ 0 ^ { \lambda } ) = - ( \sqrt { \lambda } - 1 ) ( 2 L ( \Phi _ 0 ) + \mathbf { c } \cdot \mathbf { P } ( \Phi _ 0 ) ) \left ( \lambda + \frac { \mathbf { c } \cdot \mathbf { P } ( \Phi _ 0 ) } { 2 L ( \Phi _ 0 ) + \mathbf { c } \cdot \mathbf { P } ( \Phi _ 0 ) } \sqrt { \lambda } + \frac { \mathbf { c } \cdot \mathbf { P } ( \Phi _ 0 ) } { 2 L ( \Phi _ 0 ) + \mathbf { c } \cdot \mathbf { P } ( \Phi _ 0 ) } \right ) \end{align*}


\begin{align*} \nu ( g ' ) - \nu _ j ( g ' ) = \beta _ j - \tilde { \beta } _ j = ( \beta _ j - \alpha _ { i _ 0 } ) - ( \tilde { \beta } _ j - \alpha _ { i _ 0 } ) \in \Delta . \end{align*}


\begin{align*} S ( z ) = \left ( I - z \mathcal { A } _ 0 ^ { - 1 } \right ) ^ { - 1 } \Pi , M ( z ) = \Gamma _ 1 \left ( I - z \mathcal { A } _ 0 ^ { - 1 } \right ) ^ { - 1 } \Pi . \end{align*}


\begin{align*} x ^ r & = \int ^ { \infty } _ 0 ( s + x ) ^ { - 1 } d \mu ( s ) , \end{align*}


\begin{align*} \sigma _ k ( \lambda ) = \sum \limits _ { 1 \leqslant i _ 1 < i _ 2 < \cdots < i _ k \leqslant n } \lambda _ { i _ 1 } \lambda _ { i _ 2 } \cdots \lambda _ { i _ k } \end{align*}


\begin{align*} \Lambda _ { q _ \mathfrak { c } , 0 } ^ 0 = \Lambda _ { 0 , 0 } ^ 0 . \end{align*}


\begin{align*} R _ p = \left \lbrace \left ( \begin{array} { c c c } 1 & r \\ 0 & p \end{array} \right ) \mid r = 0 , 1 , \ldots , p - 1 \right \rbrace \cup \left \lbrace \left ( \begin{array} { c c c } p & 0 \\ 0 & 1 \end{array} \right ) \right \rbrace . \end{align*}


\begin{align*} C _ D = \mu ( B _ 1 \cap \Sigma ) , \end{align*}


\begin{align*} 2 ^ l I & \gtrsim _ \tau \int _ { [ 0 , 1 ] ^ 3 } \ ! \ ! f ( x , y ) f \left ( x + P _ 1 ( t ) , y \right ) f \left ( x , y + P _ 2 ( t ) \right ) \tau _ l ( t ) \ , \mathrm { d } t \mathrm { d } x \mathrm { d } y \\ & = I _ 1 + I _ 2 + I _ 3 , \end{align*}


\begin{align*} ( x , y ) & \mapsto [ x , y ] = x \ast y - y \ast x \\ ( x , y ) & \mapsto x \circ y = x \ast y + y \ast x \end{align*}


\begin{align*} N \coloneqq ( d + 1 ) \cdot \binom { n + h } { n } \ , . \end{align*}


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


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


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


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


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


\begin{align*} D ^ { n } \mathfrak { r } _ { a , n } ( \varphi ) = D ^ { n } \varphi - \sum _ { 1 \leq i < i _ a } d ^ + _ { i , 0 } ( D ^ n \varphi ) h _ { i } . \end{align*}


\begin{align*} 0 \geq \frac { b } { 2 } \sum _ { p = 1 } ^ n u ^ { p \bar p } - \frac { A _ 1 ( 1 + \Vert \nabla v \Vert ^ 2 + \Delta v ) } { g \Delta u } . \end{align*}


\begin{align*} & { \textstyle \min _ { x \leq y \leq x + B } \{ K + G _ n ( y ) - G _ n ( x ) - V _ n ( x ) + V _ n ( y ) \} } \\ [ - 5 p t ] & { \textstyle = K + \min _ { x \leq y \leq x + B } G _ n ( y ) - G _ n ( x ) > 0 . } \end{align*}


\begin{align*} \Gamma \left ( s \right ) = \sqrt { \pi } - \sqrt { \pi } \left ( 2 \log ( 2 ) + \gamma \right ) \left ( s - \frac { 1 } { 2 } \right ) + O \left ( s - \frac { 1 } { 2 } \right ) ^ { 2 } . \end{align*}


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


\begin{align*} \left \{ \begin{array} { l } \frac { d } { d s } \chi _ s ( x _ 0 , \xi _ 0 ) = H _ { p _ 2 } \big ( \chi _ s ( x _ 0 , \xi _ 0 ) \big ) , \\ \chi _ 0 ( x _ 0 , \xi _ 0 ) = ( x _ 0 , \xi _ 0 ) , \end{array} \right . \end{align*}


\begin{align*} \begin{cases} \frac { d } { d t } U ( t ) = B _ { u ( t ) } \ , U ( t ) \\ U ( 0 ) = \operatorname { I d } \end{cases} \ , . \end{align*}


\begin{align*} 2 ^ { 4 - 2 s } c ^ { \frac { 1 } { 2 } - s } \sin ( \pi s ) \sum _ { n = 1 } ^ { \infty } \frac { p ^ { \prime 2 } + \lambda _ { n } ^ { \prime 2 } } { p ^ { \prime } \left ( p ^ { \prime } + \frac { 1 } { \pi } \right ) + \lambda _ { n } ^ { \prime 2 } } \ , \lambda _ { n } ^ { \prime 1 - 2 s } \ , \intop _ { 0 } ^ { \infty } \ , \frac { y ^ { - s } ( y + 1 ) ^ { - s } } { \sigma \left ( \lambda _ { n } ^ { \prime } \ , ( 2 y + 1 ) \ , \sqrt { c } \right ) e ^ { 2 \pi \lambda _ { n } ^ { \prime } ( 2 y + 1 ) \sqrt { c } } - 1 } d y , \end{align*}


\begin{align*} p ( \gamma _ { ( m , n ) } ) = \Gamma ( \gamma _ { ( m , n ) } ; a , b ) \end{align*}


\begin{align*} \min \frac { 1 } { 2 } x ^ 2 \mbox { s . t . } \ \left \{ \begin{aligned} x & = 1 ; \\ 2 x & = 2 . \end{aligned} \right . \end{align*}


\begin{align*} = \exp \left \{ \left ( \sum _ { g = 1 } ^ { \infty } \frac { v ^ g } { g ^ q } \right ) \left ( \sum _ { h = 1 } ^ { \infty } \frac { w ^ h } { h ^ r } \right ) \left ( \sum _ { i = 1 } ^ { \infty } \frac { x ^ i } { i ^ s } \right ) \left ( \sum _ { j = 1 } ^ { \infty } \frac { y ^ j } { j ^ t } \right ) \left ( \sum _ { k = 1 } ^ { \infty } \frac { z ^ k } { k ^ u } \right ) \right \} . \end{align*}


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


\begin{align*} \mathbf { c } _ 0 \cdot \mathbf { P } ( \Phi ) = - 2 ( L ( \Phi ) + N ( \Phi ) ) = - 2 E ( \Phi ) \end{align*}


\begin{align*} q _ 1 = D v ( p ) \cdot e _ 1 \geq \frac { v ( p ) - v ( z ) } { | p - z | } = \frac { h } { p _ 1 } \geq C \frac { h } { s ^ 2 } . \end{align*}


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


\begin{align*} \Delta \left ( \frac { m } { a } \right ) = \begin{cases} 1 , & m = 0 \\ 0 , & m \neq 0 . \end{cases} \end{align*}


\begin{align*} \mu ( A \cdot \{ b _ 1 , b _ 2 , b _ 3 \} ) = \mu ( X ) + \mu ( ( A \cdot b _ 3 ) \setminus X ) \geq \mu ( A ) + \mu ( B ) , \end{align*}


\begin{align*} A _ { k , 1 } ( \varphi _ 1 ( x ) , \ldots , \varphi _ k ( x ) ) = \sum _ { j \geq 0 } a _ { k , j } \frac { x ^ j } { j ! } . \end{align*}


\begin{align*} f ( \Phi ( t ) ) - f ( \Phi ( 0 ) ) = f ( \rho ( v ) , T ( v ) ) - f ( \rho ( u ) , T ( u ) ) , \end{align*}


\begin{align*} \liminf _ { n \rightarrow \infty } \pi ^ { n } _ 1 ( U _ 1 ) = \liminf _ { n \rightarrow \infty } \pi ^ { n } ( U _ 1 \times \mathcal { S } _ 2 ) \geq \pi ^ \infty ( U _ 1 \times \mathcal { S } _ 2 ) = \pi ^ \infty _ 1 ( U _ 1 ) . \end{align*}


\begin{align*} \Phi ( X ) = \sum _ { i = 1 } ^ m \sum _ { j = 1 } ^ n Z _ { i , j } ^ * X Z _ { i , j } \end{align*}


\begin{align*} x _ 0 = [ z _ 0 , g _ 0 ] _ { \mathbf { K } } \in X _ { \mathbf { K } } ( \mathbb { C } ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta _ p u & = c _ 1 | x | ^ { m _ 1 } \cdot v ^ { k _ 1 } \cdot | \nabla u | ^ { \alpha } & \quad \mbox { i n } \Omega , \\ \Delta _ p v & = c _ 2 | x | ^ { m _ 2 } \cdot v ^ { k _ 2 } \cdot | \nabla u | ^ { k _ 3 } & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


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


\begin{align*} y _ { s } = \zeta + \int _ { s } ^ { T } \tilde { f } \left ( s , y _ { s } , z _ { s } , \mathbb { P } _ { \left ( y _ { s } , z _ { s } \right ) } \right ) \mathrm { d } s - \int _ { s } ^ { T } z _ { s } \mathrm { d } W _ { s } . \end{align*}


\begin{align*} \tfrac { d } { d t } e ^ { t X } m = X ( m ) & & m \in M \end{align*}


\begin{align*} V ( \gamma , \sigma , t ) [ x ] & = \psi \left ( x - \gamma ( x ) - \frac { 1 } { N } \sigma , t \right ) - \phi ( x ) , \end{align*}


\begin{align*} \frac { \partial \mathrm { R } _ { \mathrm { D } } ( \lambda , \delta ) } { \partial \lambda _ \ell } = \delta _ { \ell } - \frac { V _ \ell / V _ { \ell , X X } } { 4 \lambda _ \ell ^ 2 } - \frac { 1 } { 4 } V _ o ^ { \top } \Lambda _ \lambda ^ { - 1 } V _ { \ell , X X } ^ { - 1 } \Lambda _ \lambda ^ { - 1 } V _ o . \end{align*}


\begin{align*} \mathfrak { a } _ q ( u , v ) = \int _ { \mathrm M } ( \nabla u | \overline { \nabla v } ) _ g d \mu + \int _ { \mathrm { M } } q u \overline { v } d \mu , u , v \in H _ 0 ^ 1 ( \mathrm { M } ) , \end{align*}


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


\begin{align*} S _ 1 = \sqrt { \pi } \Gamma ( \nu + 1 / 2 ) \sum _ { k = 0 } ^ \infty \frac { ( \nu + 1 / 2 ) _ k } { k ! } \bigg ( \frac { \beta } { \alpha } \bigg ) ^ { 2 k } = \frac { \sqrt { \pi } \Gamma ( \nu + 1 / 2 ) } { ( 1 - \beta ^ 2 / \alpha ^ 2 ) ^ { \nu + 1 / 2 } } , \end{align*}


\begin{align*} I ( z ) & \leq | \partial ^ 2 u _ { k } ( z ) - \partial ^ 2 u _ { k } ( 0 ) | + | \partial ^ 2 u _ { k } ( 0 ) - \partial ^ 2 u ( 0 ) | + \\ & \qquad \qquad \quad + | \partial ^ 2 u ( z ) - \partial ^ 2 u _ { k } ( z ) | = : I _ { 1 } ( z ) + I _ { 2 } ( z ) + I _ { 3 } ( z ) . \end{align*}


\begin{align*} v ( n ) \ = \ \sum _ { \substack { 1 \leq i \leq k , \\ \alpha _ i = 1 } } p _ i + \sum _ { \substack { 1 \leq i \leq k , \\ \alpha _ i \geq 2 } } \left ( p _ i + \alpha _ i \right ) . \end{align*}


\begin{align*} \Xi _ { 1 } ^ { 2 } = \sum _ { i \le N } ( \max _ { j \le n } g ^ { \prime 2 } _ { i j } ) \Vert B _ { i } \Vert ^ { 2 } _ { 2 } , \end{align*}


\begin{align*} \mu _ j \tilde { S } _ { \gamma _ i \gamma _ j } & = \mu _ i J _ { 2 | \alpha _ i | } \tilde { S } _ { \gamma _ i \gamma _ j } J ^ T _ { 2 | \alpha _ j | } + \mathcal { O } ( \| H \| ) . \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 ^ 2 } { n ^ 3 } } \end{align*}


\begin{align*} \sigma : = \max \left \{ \frac { 1 } { 2 | \alpha | } , \frac { 1 } { | \beta | } , \frac { 1 } { | \gamma | } \right \} = \left ( \min \{ 2 | \alpha | , | \beta | , | \gamma | \} \right ) ^ { - 1 } > 0 \end{align*}


\begin{align*} = \exp \left \{ L i _ q ( v ) L i _ r ( w ) L i _ { s } ( x ) L i _ { t } ( y ) L i _ u ( z ) \right \} . \end{align*}


\begin{align*} Z _ \infty : = \{ z \in \{ 0 , 1 \} ^ \Z : | z \cap ( - \infty , 0 ] | = | z \cap [ 0 , \infty ) | = \infty \} . \end{align*}


\begin{align*} 0 < v \left ( a ^ { - 1 } \right ) \leq \mu ( q ) = \gamma _ i , \mbox { f o r s o m e } i , 1 \leq i \leq \epsilon , \end{align*}


\begin{align*} A & \ll | d _ 3 | | r _ 3 | ^ { n / 2 + 3 / 2 + \varepsilon } \widehat { C } _ 1 \widehat { C } _ 2 \left ( | r _ 3 | ^ { n / 2 - 1 } + \widehat { V } ^ { n - 2 } \right ) \end{align*}


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


\begin{align*} \langle A ( ( - \Delta ) ^ \alpha ) u , v \rangle = \left \langle \alpha _ 1 , ( ( - \Delta ) ^ { a _ 1 } u ) v \right \rangle + \left \langle \alpha _ 2 , ( ( - \Delta ) ^ { a _ 2 } u ) v \right \rangle + \cdots + \left \langle \alpha _ p , ( ( - \Delta ) ^ { a _ p } u ) v \right \rangle . \end{align*}


\begin{align*} v _ \varpi = \nu _ { \varpi } ( \Delta _ { F _ 2 } ) . \end{align*}


\begin{align*} \widetilde { \gamma } _ { \xi } = \frac { 1 - \overline { a } \xi } { \xi - a } \ , \frac { 1 - a \xi } { \xi - \overline { a } } \ , \gamma _ { \xi } . \end{align*}


\begin{align*} \Big ( \sum _ { e \in E ( \mathbb Z ^ d ) } q _ e p _ e \Big ) ^ d = \mu ^ d \le C ( 8 / 3 ^ d ) ^ k \| q \| ^ { d - 1 } \sum _ { e \in E ( \mathbb Z ^ d ) } q _ e \le C \| q \| ^ { d - 1 } \sum _ { e \in E ( \mathbb Z ^ d ) } q _ e , \end{align*}


\begin{align*} - \varepsilon \ , \partial _ { \boldsymbol { \nu } _ \varepsilon } \mathfrak { U } _ { M } ^ { ( \varepsilon ) } = \varepsilon ^ { \alpha } \varphi ^ { ( 0 ) } _ \varepsilon \ \Big ( \partial \Omega ^ { ( 0 ) } _ { \varepsilon , \gamma } \setminus \Big \{ \bigcup _ { i = 1 } ^ 3 \overline { \Upsilon _ \varepsilon ^ { ( i ) } ( 2 \ell _ 0 \varepsilon ^ \gamma ) } \Big \} \Big ) \times ( 0 , T ) . \end{align*}


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


\begin{align*} A ( t ) = ( \l ( t ) , \l ( t ) , \l ( t ) ) , \end{align*}


\begin{align*} \mathfrak { W } ( f _ k ) = m \leq k . \end{align*}


\begin{align*} & J ( a , z ) ( t ) = J _ { \mathrm { s e c t } ( t ) } ( a ) ( z ) = J _ { \mathrm { s e c t } ( t ) } ( a ^ { ( 1 ) } ) ( \mathrm { s e c t } ( \delta / 2 ) ) J _ { \mathrm { s e c t } ( \delta / 2 ) } ( a ^ { ( 2 ) } ) ( z ) \\ & = J ( a ^ { ( 1 ) } , \mathrm { s e c t } ( \delta / 2 ) ) ( t ) \cdot J _ { \mathrm { s e c t } ( \delta / 2 ) } ( a ^ { ( 2 ) } ) ( z ) \end{align*}


\begin{align*} & \left | \left \lbrace ( x , y , t ) \in K : \left | \widetilde { P } _ 1 ' ( t ) \alpha ( x + \widetilde { P } _ 1 ( t ) , y ) - \widetilde { P } _ 2 ' ( t ) \beta ( x , y + \widetilde { P } _ 2 ( t ) ) \right | \leq \varepsilon \right \rbrace \right | \\ \lesssim & 2 ^ { c | l | } \varepsilon ^ { \frac { 1 } { 7 ( 8 ( d _ 1 + d _ 2 ) - 1 7 ) } } \end{align*}


\begin{gather*} \Z ^ { \# S - 1 } = \bigsqcup _ { M , N \in \Z ^ 2 } \mathcal { Z } _ { M , N } \\ : = \bigsqcup _ { M , N \in \Z ^ 2 } \left \{ ( v _ \nu ) _ { \nu \in S _ \sigma } \in \Z ^ { \# S - 1 } \ , : \ , \sum _ { \nu \in S _ \sigma } \min ( 0 , v _ \nu ) = - M , \sum _ { \nu \in S _ \sigma } \max ( 0 , v _ \nu ) = N \right \} \end{gather*}


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


\begin{align*} C _ { m , n } = \frac { 1 } { { n \choose m } } \sum _ { 1 \leq i _ { 1 } < i _ { 2 } < \ldots < i _ { m } \leq n } \left ( \frac { X _ { i _ { 1 } } + \ldots + X _ { i _ { m } } } { m } - \theta ( m ) \right ) \otimes \left ( \frac { X _ { i _ { 1 } } + \ldots + X _ { i _ { m } } } { m } - \theta ( m ) \right ) , \end{align*}


\begin{align*} \partial _ i g _ { \gamma , i } ( x ) = - \frac { X _ { \gamma , i } ( x ) } { f _ { \gamma , i } ' ( g _ { \gamma , i } ( x ) ) } , i \in \{ 1 , 2 \} . \end{align*}


\begin{align*} M ( f ) _ f = 0 & \ ; \Leftrightarrow \ ; \exists k : \N . \ ; \\ & \ ; \Leftrightarrow \ ; \exists k : \N . \ ; \forall n : \N . \ ; f ^ k \in ( f ^ n ) \subseteq R \\ & \ ; \Leftrightarrow \ ; \exists k : \N . \ ; f ^ k \in ( f ^ { k + 1 } ) \subseteq R \rlap { . } \end{align*}


\begin{align*} g ( z ) : = u | A | z ^ { n + m } + v | B | \overline { z } ^ m + | C | \textrm { f o r a l l } z \in \mathbb { C } . \end{align*}


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


\begin{align*} P \bigl ( S - E ( S ) \ge v \bigr ) \le \exp \Biggl ( - \frac { 2 v ^ { 2 } } { \sum _ { j = 1 } ^ { J } ( b _ { j } - a _ { j } ) ^ { 2 } } \Biggr ) . \end{align*}


\begin{align*} ( f \sharp g ) _ { m + m ' - j } ( \xi ) = \sum _ { k + l + | \alpha | = j } \frac { ( - i ) ^ { | \alpha | } } { \alpha ! } ( \partial _ \xi ^ \alpha f _ { m - k } ) ( \xi ) g _ { m ' - l } ^ { B , \alpha } ( \xi ) , j \geq 0 . \end{align*}


\begin{align*} \overline { \nabla } _ { X } \xi = - A _ { \xi } X + \nabla _ { X } ^ { \bot } \xi \end{align*}


\begin{align*} g ( s _ + ) = \frac { 1 } { 4 } ( \sqrt { C - 4 } + \sqrt { C } ) \end{align*}


\begin{align*} \sum _ I \ < Y _ F , \Lambda ^ I \ > M _ I ( X ) = ( - 1 ) ^ { | F | } \Gamma _ F ( - X ) = : \chi _ F ( X ) . \end{align*}


\begin{align*} f = - \log w , \mu = 1 , \lambda ( x ) = - \frac { S ^ \varphi } { m - 1 } \frac { e ^ f } { m } . \end{align*}


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


\begin{align*} \begin{cases} \widetilde { W } ^ { ( u ) } = W ( K ^ { ( u ) } ) ^ { - 1 } = W + W ( H ^ { ( u ) } ) ^ { - 1 } W , \\ \widetilde \eta ^ { ( u ) } = \widetilde { W } ^ { ( u ) } T ( u ) \eta + \eta , \\ \widetilde \theta _ i ^ { ( u ) } = e ^ { \rho _ i ( u ) } i \in V \end{cases} \end{align*}


\begin{align*} \omega _ i ( r ) = \prod \limits _ { g \mid \rho _ i ( r ) } \frac { - \kappa _ i ( ( g , r ) ) } { g } . \end{align*}


\begin{align*} ( h _ p ^ { - 1 } ) ^ \vee \overline { \lambda } _ 0 h _ p ^ { - 1 } = ( Y _ \mathbb { Q } ^ { - 1 } ) ^ \vee Y _ p ^ \vee \overline { \lambda } _ 0 Y _ p Y _ \mathbb { Q } ^ { - 1 } , ( h _ q ^ { - 1 } ) ^ \vee \overline { \lambda } _ 0 h _ q ^ { - 1 } = ( Y _ \mathbb { Q } ^ { - 1 } ) ^ \vee Y _ q ^ \vee \overline { \lambda } _ 0 Y _ q Y _ \mathbb { Q } ^ { - 1 } , \ , \forall q \ne p . \end{align*}


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


\begin{align*} \lim _ { n \to \infty } D _ \Theta X _ \cdot ^ { ( n ) } = D _ \Theta X _ \cdot \textrm { ~ ~ w e a k l y ~ i n ~ } L ^ p ( \Omega \times [ 0 , T ] ) . \end{align*}


\begin{align*} \max _ { A \in \mathcal { P } _ { x , n } } \left | 1 - \frac { \beta _ { x } \left ( A \right ) } { q _ { x , n } ^ { t } ( A ) } \right | \le \varepsilon _ { x , n } ( t ) \quad \textrm { w i t h } \quad \lim _ { t \to \infty } \varepsilon _ { x , n } ( t ) = 0 . \end{align*}


\begin{align*} h ( \overline { f } _ m ( x _ m ) , \overline { f } _ m ( x _ 0 ) ) \geqslant \varepsilon _ 0 , m = 1 , 2 , \ldots . \end{align*}


\begin{align*} q ( 1 ) < q ( 2 ) < \cdots < q ( u ) = z _ { t ( m ) } ( r ) = n t ( m ) + s ( r ) < n t ( m ) + n \leq n \beta + n . \end{align*}


\begin{align*} \rho ( \gamma ) g ( \rho , y ) = g ( \rho , \gamma y ) \rho _ 0 ( y ) \end{align*}


\begin{align*} \alpha = \alpha ( \{ \beta _ j \mid j \in I ' \} ) = \alpha ( \{ \tilde { \beta } _ j \mid j \in I ' \} ) , \end{align*}


\begin{align*} U _ { \lambda } = \left \{ ( x , y , t ) \in \mathcal { Q } _ { r _ { 1 } } \ , : \ , | D ^ { \tau } d _ { s } u ( x , y , t ) | \geq \lambda \right \} \subset \left ( \bigcup \limits _ { i \in \mathbb { N } } \mathcal { Q } _ { 5 \rho _ { { i } } } \left ( z _ { i } \right ) \right ) \bigcup \left ( \bigcup \limits _ { \mathcal { K } \times I \in \mathcal { A } } \mathcal { K } \times I \right ) \end{align*}


\begin{align*} \sum \limits _ { \substack { m = 1 \\ ( m . n ^ s ) _ s = 1 } } ^ { n ^ s } ( m - 1 , n ^ s ) _ s = \Phi _ s ( n ^ s ) \tau _ s ( n ^ s ) . \end{align*}


\begin{align*} \mathfrak { s } _ j ^ 2 = \frac { 1 } { 2 } + { \rm O } ( e ^ { - { c \sqrt { j } } } ) . \end{align*}