\begin{align*} S _ r ( a , b ) & = \int _ 0 ^ \infty y ^ { - r / 2 } \exp ( - a y ) \sum _ { n = 0 } ^ \infty ( - 1 ) ^ n \frac { \Gamma ( b ) } { \Gamma ( b - n ) n ! } \exp ( - n y ) \mathrm { d } y \\ & = \sum _ { n = 0 } ^ \infty ( - 1 ) ^ n \frac { \Gamma ( b ) } { \Gamma ( b - n ) n ! } \int _ 0 ^ \infty y ^ { - r / 2 } \exp \left \{ - ( a + n ) y \right \} \mathrm { d } y . \end{align*}


\begin{align*} ( \sigma _ F ) _ { i , j } = \ < D F _ i , D F _ j \ > _ { \mathcal { H } } , i , j = 1 , \ldots , n . \end{align*}


\begin{align*} \varphi ( f \circ g ) = ( ( f \circ g ) _ { 1 1 } , \ldots , ( f \circ g ) _ { n n } ) = ( f _ { 1 1 } \circ g _ { 1 1 } , \ldots , f _ { n n } \circ g _ { n n } ) \\ = ( f _ { 1 1 } , \ldots , f _ { n n } ) \circ ( g _ { 1 1 } , \ldots , g _ { n n } ) = \varphi ( f ) \circ \varphi ( g ) . \end{align*}


\begin{align*} \operatorname { E } ( X ^ r ) = & \frac { 2 \theta } { \operatorname { B } ( a , b ) } \int _ 0 ^ \infty x ^ { r - 3 } \exp \left ( - \frac { a \theta } { x ^ 2 } \right ) \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } \mathrm { d } x . \end{align*}


\begin{align*} ( p _ n f , v _ n ) = ( f , v _ n ) v _ n \in X _ n . \end{align*}


\begin{align*} a _ { n - i } ( \gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \end{align*}


\begin{align*} & \hphantom { { } = { } } D _ { ( o , o ) } ( \{ f , g \} . \{ f ' , g ' \} ) \\ * & = \begin{pmatrix} D f ( o ) D f ' ( o ) \\ D g ( o ) D f ' ( o ) + D f ( o ) D g ' ( o ) & D f ( o ) D f ' ( o ) \end{pmatrix} \\ * & = \begin{pmatrix} D f ( o ) \\ D g ( o ) & D f ( o ) \end{pmatrix} \begin{pmatrix} D f ' ( o ) \\ D g ' ( o ) & D f ' ( o ) \end{pmatrix} . \end{align*}


\begin{align*} \frac { \partial p } { \partial x } & = h ^ x ( x , 0 ) + \int _ 0 ^ y \frac { \partial h ^ y } { \partial x } ( x , s ) \ , d s \\ & = h ^ x ( x , 0 ) + \int _ 0 ^ y g ^ { x y } ( x , s ) \ , d s \end{align*}


\begin{align*} A _ { p , N } & \rightarrow \sum _ { A = 0 } ^ { 2 n - 1 } \int _ S \left [ v _ p ( d _ 1 v _ { p } \wedge \alpha ) _ A \tau ( \mathbf { { n } } ) _ { A 0 } + v _ { p } ( d _ 0 v _ p \wedge \alpha ) _ A \tau ( \mathbf { { n } } ) _ { A 1 } \right ] d S \\ & = \int _ B ( d _ 0 v _ p \wedge d _ 1 v _ { p } + v _ p d _ 0 d _ 1 v _ { p } ) \wedge \alpha + \int _ B ( d _ 1 v _ { p } \wedge d _ 0 v _ p + v _ p d _ 1 d _ 0 v _ { p } ) \wedge \alpha = 0 , \end{align*}


\begin{align*} \Big \| \sum \limits _ { k = 2 } ^ { n } \mu ( k ) ( I - S ) h _ k - ( 1 - z ) \Big \| _ { X } \longrightarrow 0 n \to \infty . \end{align*}


\begin{align*} \beta _ { \bar { \mathbf { g } } } \mathbf { g } ( \partial _ t ) & = 0 \\ \beta _ { \bar { \mathbf { g } } } \mathbf { g } ( \partial _ a ) & = \beta _ { \bar g } g ( \partial _ a ) + \bar u ^ { - 2 } u \partial _ a u - \bar u ^ { - 1 } g ( \nabla _ { \bar g } \bar u , \partial _ a ) \mbox { f o r $ a = 1 , \dots , n $ } . \end{align*}


\begin{align*} & w ( x ; z ) = \sum _ { n = 0 } ^ \infty w _ n ( x ; z ) = \sum _ { n = 0 } ^ \infty F ^ n e _ 1 . \end{align*}


\begin{align*} X _ { i ; j } V _ j & = \omega _ { i j } V _ j \\ \omega _ { i j ; \ell } V _ \ell & = - R _ { k \ell i j } X _ k V _ \ell + \tfrac { 1 } { 2 } \left ( h _ { i j , \ell } + h _ { i \ell , j } - h _ { j \ell , i } \right ) V _ \ell \end{align*}


\begin{align*} \mu _ w ( \bigoplus _ { j = l } ^ { n - 1 } K _ j ' ) = \frac { ( \bigoplus _ { j = l } ^ { n - 1 } K _ j ' ) } { ( \sum \limits _ { j = l } ^ { n - 1 } w _ j ) r } \geq \frac { \chi ( E ) } { r } \end{align*}


\begin{align*} & d _ { \mathrm { s t } } = \int d z \sum _ { \alpha \in \Delta _ + } e _ { \alpha } ( z ) \otimes \psi _ { \alpha _ i } ^ * ( z ) - \frac { 1 } { 2 } \sum _ { \alpha , \beta , \gamma \in \Delta _ + } c _ { \alpha , \beta } ^ \gamma : \psi _ \gamma ( z ) \psi _ \alpha ^ * ( z ) \psi _ \beta ^ * ( z ) : , \\ & d _ { \chi } = \int d z \sum _ { \alpha \in \Pi } \psi ^ * _ \alpha ( z ) = \sum _ { \alpha \in \Pi } \psi ^ * _ { \alpha , 1 } . \end{align*}


\begin{align*} \left \{ \begin{array} { l } - \Delta \Psi + \Psi = \lambda \left ( \Psi - W x _ { 2 } \right ) _ { + } , \ , \Pi , \\ \Psi \rightarrow 0 r \rightarrow \infty , \Psi = 0 , \partial \Pi , \\ \Psi \left ( x _ { 1 } , x _ { 2 } \right ) = - \Psi \left ( x _ { 1 } , - x _ { 2 } \right ) , \forall \ , x \in \R ^ { 2 } . \end{array} \right . \end{align*}


\begin{align*} Q _ \delta u : = u - \sum _ { j = 1 } ^ m \sum _ { h = 1 } ^ 2 C _ { j , h } \frac { \partial } { \partial z _ { j , h } } ( - \delta ^ 2 ( K ( z _ j ) \nabla V _ { \delta , Z , j } ) ) , \end{align*}


\begin{align*} K _ r ( B ) = K _ r ( B _ { [ n ] , [ r ] } ) \times \R ^ { m - r } \mbox { a n d } \sigma _ m ( K _ r ( B ) ) = \sigma _ r ( K _ r ( B _ { [ n ] , [ r ] } ) ) , r \in [ m - 1 ] , \end{align*}


\begin{align*} R ^ i _ { n _ 0 ; l _ 1 , \ldots , l _ { n _ 0 } } = \underset { n _ 0 \geq j \geq i } { \bigsqcup } Q ^ j _ { n _ 0 ; l _ 1 , \ldots , l _ { n _ 0 } } . \end{align*}


\begin{align*} r ( k ) = \begin{cases} - 2 i k r _ 1 ( z ) & | k | \leq 1 \\ ( 2 i k ) ^ { - 1 } r _ 2 ( z ) & | k | \geq 1 . \end{cases} \end{align*}


\begin{align*} c _ 1 ( q ) = \begin{cases} c _ { 1 , 2 } ( q ) , & 0 < q \leq q _ 1 ^ * , \\ c _ { 1 , \infty } ( q ) , & q _ 1 ^ * \leq q < 2 , \end{cases} \end{align*}


\begin{align*} 0 = \int _ { B _ 1 ( 0 ) } \phi ^ { p - 1 } _ + \frac { \partial \phi } { \partial x _ h } \hat { u } _ i . \end{align*}


\begin{align*} \dfrac { p } { p + 1 } \displaystyle \sum _ { n = 1 } ^ \infty \dfrac { 2 ^ { \omega ( \gcd ( n , p ) ) } \cdot 2 r _ { Q ^ \ast } ( n ) ^ 2 } { n } \displaystyle \sum _ { d = 1 } ^ \infty \psi \left ( d \sqrt { \dfrac { n } { p } } \right ) \leq \frac { 1 } { \min Q ^ { * } } + 3 2 1 6 . 6 5 2 4 \frac { M ( 2 5 . 0 9 p ^ { 3 5 / 6 } ) } { p ^ { 1 / 4 } } . \end{align*}


\begin{align*} \varphi ^ g ( n ) = \varphi ( n ^ g ) \forall g \in G , \ , n \in N . \end{align*}


\begin{align*} & \Delta : = ( \langle u _ * , ( \lambda x _ v + ( 1 - \lambda ) x _ w ) \rangle ) ^ 2 + \lambda ( 1 - \lambda ) \| u _ * \| ^ 2 \| x _ v - x _ w \| ^ 2 \ge 0 , \\ \end{align*}


\begin{align*} A = \{ n \in \N : T ^ n a \in E \} . \end{align*}


\begin{align*} H ^ { ( k ) } \coloneqq \begin{cases} H & \\ H ^ * & \end{cases} \end{align*}


\begin{align*} \begin{cases} \partial _ t w + \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \nabla ^ \perp \varphi \cdot \nu | _ { \partial \Omega } = v _ n | _ { \partial \Omega } . \end{cases} \end{align*}


\begin{align*} \begin{aligned} | F ^ { i j } ( \nabla _ \beta u ) _ { i j } | \leq \ , & | \nabla _ \beta f ^ { 1 / k } | + C \left ( f ^ { 1 / k } + b _ \alpha \sum _ { i = 1 } ^ n F ^ { i i } \right ) \\ \leq \ , & C m ^ { 1 / k } m ^ { - 1 / 2 ( k - 1 ) } + C b _ \alpha \sum _ { i = 1 } ^ n F ^ { i i } \\ \leq \ , & C b _ \alpha ^ { - 1 / 2 } m ^ { 1 / k } + C b _ \alpha \sum _ { i = 1 } ^ n F ^ { i i } \end{aligned} \end{align*}


\begin{align*} \sum \limits _ { j = l } ^ { n - 1 } \chi _ j \leqslant ( \sum \limits _ { j = l } ^ { n } ) \end{align*}


\begin{align*} F = \{ x \in [ 0 , 1 ) : E ( x ) \} . \end{align*}


\begin{align*} | \mathfrak { I } _ { q _ 1 , \ldots , q _ 4 , \kappa } ( \ell ) | \leq \frac { C } { \ell ^ { 3 } } , \end{align*}


\begin{align*} ( f _ { I _ 1 } ( \mathbf { x } ) , f _ { I _ 2 } ( \mathbf { x } ) , \cdots , f _ { I _ s } ( \mathbf { x } ) ) = ( f _ { J _ 1 } ( \mathbf { y } ) , f _ { J _ 2 } ( \mathbf { y } ) , \cdots , f _ { J _ s } ( \mathbf { y } ) ) . \end{align*}


\begin{align*} l ( y ' ) : = u ( x _ 0 ) + \sum _ { \beta = 1 } ^ { n - 1 } \nabla _ \beta u ( x _ 0 ) y _ \beta . \end{align*}


\begin{align*} ( \sigma , \sigma ( 1 ) , \ldots , \sigma ( r - 1 ) ) = [ \sigma , \sigma ( r - 1 ) ] = \Delta \end{align*}


\begin{align*} \Sigma ^ { \pm } ( s - w , \phi , F _ { \pm } ) = & \lim _ { z \downarrow 0 } \int _ 0 ^ \infty w _ z ( \lambda ) \lambda ^ { 4 ( 1 - s + w ) } \int _ { - \infty } ^ \infty \int _ { 0 } ^ \infty \phi ( n _ { u } ^ t a _ { \frac { 1 } { t } } ) \\ & \times \sum _ { m = 1 } ^ \infty \sum _ { n = 0 } ^ { 3 m - 1 } \hat { F } ( \lambda a _ { \frac { 1 } { t } } n _ u \cdot ( 0 , 0 , 3 m , n ) ) \frac { d \lambda } { \lambda } \frac { d t } { t ^ 3 } d u . \end{align*}


\begin{align*} \d Y _ t ^ i = \bigg ( b ( Y _ t ^ i ) + \frac 1 { p - 1 } \sum _ { j \neq i , j \in \C } K ( Y _ t ^ i - Y _ t ^ j ) \bigg ) \d t + \sigma \d W _ t ^ i , ~ ~ i \in \C , ~ ~ t \in [ t _ n , t _ { n + 1 } ) . \end{align*}


\begin{align*} & \psi ^ \pm ( x , t ; k ) = e ^ { - i c _ - \sigma _ { 3 } } \mu ^ \pm ( x , t ; k ) e ^ { i c \sigma _ 3 } , \\ & \mu ^ \pm ( x , t ; k ) = e ^ { - i c _ - \sigma _ { 3 } } \varphi ^ \pm ( x , t ; k ) e ^ { - i \left ( k ^ 2 x + \eta ^ 2 t \right ) \sigma _ 3 } C ^ \pm ( k ) e ^ { i \left ( k ^ 2 x + \eta ^ 2 t \right ) \sigma _ 3 } e ^ { i c \sigma _ 3 } . \end{align*}


\begin{align*} - L \eta = G \textrm { i n } \ \mathbb { R } ^ { 2 } _ { + } , \eta = 0 \textrm { o n } \ \partial \mathbb { R } ^ { 2 } _ { + } , \end{align*}


\begin{align*} X _ \ell = \sum _ { q \ge 0 } X _ \ell [ q ] , X _ \ell [ q ] : = \frac { b _ q } { q ! } \int _ { \mathbb S ^ d } H _ q ( T _ \ell ( x ) ) \ , d x , \end{align*}


\begin{align*} A ' _ n ( j ) = A _ n ( j _ V ) \end{align*}


\begin{align*} & \lim _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \int _ { \{ z \in M : - \Psi ( z ) \in I _ { i , n } \backslash U _ k \} } | \tilde { F } | ^ 2 _ h \\ \le & \limsup _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \frac { 1 } { \inf _ { I _ { i , n } \backslash U _ k } c ( t ) } \int _ { \{ z \in M : - \Psi ( z ) \in I _ { i , n } \backslash U _ k \} } | \tilde F | ^ 2 _ h c ( - \Psi ) . \end{align*}


\begin{align*} & \Delta x _ j - \bar u ^ { - 1 } \nabla \bar u \cdot \nabla x _ j = 0 \\ & x _ j ( p ) = 0 \ ; \mbox { a n d } \ ; \frac { \partial x _ j } { \partial y _ i } ( p ) = \delta _ { i j } . \end{align*}


\begin{align*} S _ { i , j } = & X \sum _ { \substack { \alpha \leq Z \\ ( \alpha , 2 ) = 1 } } \frac { \mu ( \alpha ) } { 2 \alpha ^ 2 } \sum _ { m \geq 1 } \sum _ { \substack { ( p , 2 \alpha ) = 1 } } \frac { a _ { \pi } ( p ) \log p } { p } \hat { \phi } \left ( \frac { \log p } { M \log X } \right ) \left ( \frac { 2 j m } { p } \right ) \widetilde { W } _ { i } \left ( \frac { m X } { \alpha ^ 2 p } \right ) . \end{align*}


\begin{align*} \vec { b } = & \sum _ { i = j } ^ { n - 1 } b _ i \vec { e } _ i \end{align*}


\begin{align*} u ( t ) = e ^ { \kappa t \Delta } u _ 0 + \int _ 0 ^ t e ^ { \kappa ( t - s ) \Delta } \Delta _ p u ( s ) \ , \ , d s + \sum _ { k \in \Z ^ d } \sum _ { i = 1 } ^ { d - 1 } \int _ 0 ^ t e ^ { \kappa ( t - s ) \Delta } \xi _ { k , i } \cdot \nabla u ( s ) \ , d B _ s ^ { k , i } . \end{align*}


\begin{align*} \Tilde { x } \in \mathcal { S } ^ n : = \{ x \in \mathbb { R } ^ n \ | \ \| x \| ^ 2 = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 \} . \end{align*}


\begin{align*} \kappa _ { p , d } = 2 ^ { 1 - p } \frac { \Gamma \left ( \frac { d - p } { 2 } \right ) } { \Gamma \left ( \frac { d } { 2 } \right ) \Gamma \left ( \frac { p } { 2 } \right ) } . \end{align*}


\begin{align*} M ^ d \times ( - \infty , 0 ) \ni ( x , z ) \mapsto \mathcal { S } ( x , z ) = ( x , \varrho ( x , z ) ) \in \Omega , \end{align*}


\begin{align*} ( I - F ) \left ( \Psi _ - - e ^ { - i c _ - ( x ) \sigma _ 3 } \right ) = \begin{pmatrix} 0 & n \\ m & 0 \end{pmatrix} , \end{align*}


\begin{align*} \tilde { u } ^ { [ n ] } ( x ) = \sum _ i w ^ { [ n ] } _ i \sigma _ i ( x ) , \end{align*}


\begin{align*} \dot \phi _ i = \omega _ i + \frac 1 M \sum _ { j = 1 } ^ M a _ { i j } \sin ( \phi _ j - \phi _ i ) , \end{align*}


\begin{align*} p _ k = \log _ 2 \binom { 2 k } { k } . \end{align*}


\begin{align*} \tilde { h } _ { i j } = h _ { i j } - B _ { i j } . \end{align*}


\begin{align*} V \oplus W = \bigoplus _ { i \in \mathcal { I } } ( V _ i \oplus W _ i ) , V \otimes W = \bigoplus _ { k \in \mathcal { I } } \left ( \bigoplus _ { i + j = k } V _ i \otimes W _ j \right ) . \end{align*}


\begin{align*} \boldsymbol { \alpha } _ t ^ * : = ( \alpha _ t ^ { * , i } ) _ { 1 \leq i \leq N } = \left ( - R ^ { - 1 } B y _ t ^ { * , i } - h ( \mu ^ { N , i } _ { \boldsymbol { \alpha } ^ * _ t } ) \right ) _ { 1 \leq i \leq N } \end{align*}


\begin{align*} \boxed { \scriptstyle \sf H r a n k ~ 1 } \ , : = \ , \boxed { \aligned 0 & \ , \neq \ , F _ { x x } \\ 0 & \ , \equiv \ , F _ { x x } \ , F _ { y y } - F _ { x y } ^ 2 \endaligned } , \end{align*}


\begin{align*} h _ { k , S , ( n , g ) } ( \tau , z ; \tau _ 0 , z _ 0 ) : = \det ( \tau - \bar { \tau } _ 0 ) ^ { - k } e \left ( - ( \tau - \bar { \tau } _ 0 ) ^ { - 1 } S [ z - \overline { z } _ 0 ] \right ) . \end{align*}


\begin{align*} \div X = 0 . \end{align*}


\begin{align*} \lim _ { N \to \infty } \dfrac { | ( \Phi _ N - t ) \cap \Phi _ N | } { | \Phi _ N | } = 1 \end{align*}


\begin{align*} \dot { s } \varepsilon ( z ) \sum _ { x \in \bar { \mathbb { F } } _ q } a _ x \varepsilon ( x ) \xi _ 1 = \sum _ { x \in \bar { \mathbb { F } } _ q } a _ x ( \varepsilon ( x ( z ^ 2 - z ) ^ { - 1 } - z ^ { - 1 } ) + \varepsilon ( x z ^ { - 1 } ) - \varepsilon ( x ( z - 1 ) ^ { - 1 } ) ) \xi _ 1 \end{align*}


\begin{align*} [ \phi _ { \lambda } \psi ] _ { \mu } a = \phi _ { \lambda } ( \psi _ { \mu - \lambda } a ) - \psi _ { \mu - \lambda } ( \phi _ { \lambda } a ) , \forall \ a \in \mathcal { A } , \end{align*}


\begin{align*} ( d - a ) x = b . \end{align*}


\begin{align*} \begin{cases} \Delta _ \Psi ( - v ) \geq f ' ( u ) ( - v ) & A ( r _ { 1 } , r _ { 2 } ) \\ ( - v ) = 0 & \partial A ( r _ { 1 } , r _ { 2 } ) . \end{cases} \end{align*}


\begin{align*} \hat \Phi _ 2 ( ( a , 0 ) ^ \top , s ) & = \langle z _ 2 ^ * , \Phi ( a u _ \ell , p ( s ) ) \rangle \\ & = \langle z _ 2 ^ * , Q F ^ q ( a u _ \ell + \psi ( a u _ \ell , p ( s ) ) , p ( s ) ) \rangle \\ & = 0 , \end{align*}


\begin{align*} Q ( u ) = F ^ { - 1 } ( u ) = \sqrt { - \frac { \theta } { \log ( \operatorname { I } ^ { - 1 } _ { u } ( a , b ) ) } } , 0 < u < 1 , \end{align*}


\begin{align*} F ( X ) = \sum _ { i = 1 } ^ n f _ i ( x _ i ) , ~ G ( Y ) = \sum _ { i = 1 } ^ n g _ i ( y _ i ) . \end{align*}


\begin{align*} \nabla _ X \psi = \frac { 1 } { 2 } X _ 1 \cdot V \cdot \psi - \frac { 1 } { 2 } S ( X ) \cdot N \cdot \psi . \end{align*}


\begin{align*} H _ { j , n } = \frac { \sqrt { n ! } } { \Gamma ( \beta ) \sqrt { \left ( \beta \right ) ^ { \left ( n \right ) } } } \int _ { 0 } ^ { \infty } x ^ { j } L _ { n } ^ { ( \beta ) } ( x ) x ^ { \beta - 1 } \exp ( - x ) d x . \end{align*}


\begin{align*} \begin{cases} \Delta _ \Psi u = f ( u ) & A ( r _ { 1 } , r _ { 2 } ) \\ u \equiv c _ { 1 } & \{ r _ { 1 } \} \times N \\ u \equiv c _ { 2 } & \{ r _ { 2 } \} \times N . \end{cases} \end{align*}


\begin{align*} \mathcal D _ \pi = \sum _ { \widetilde { K } \ \ \widetilde { L } } ( e _ { \widetilde { K } } - 1 ) [ \widetilde { K } ] \in H _ 1 ( \widetilde M , \mathbb Z / 2 \mathbb Z ) \end{align*}


\begin{align*} \begin{aligned} [ D _ * ^ q , M _ { z ^ q } ] & = D ^ q _ * M _ { z ^ q } ( f ) ( z ) - M _ { z ^ q } D _ * ^ q ( f ) ( z ) \\ & = D ^ q _ * ( z ^ q f ) ( z ) - z ^ q D _ * ^ q ( f ) ( z ) . \end{aligned} \end{align*}


\begin{align*} L ^ { * } : = \left \lbrace y \in E \mid h \left ( L , y \right ) \subset { \mathcal O } _ K \right \rbrace . \end{align*}


\begin{align*} \begin{aligned} \mathcal { g } _ { D } ( v , w ) & : = \mathcal { W } \bigl ( \hat v \hat w D ^ { - n } \bigr ) , \\ \mathcal { G } _ { D } ( v , w ) & : = \mathcal { W } \bigl ( \hat v ( D \hat w + \hat w D ) D ^ { - n + 1 } \bigr ) \\ & \ , \ , = \mathcal { W } \bigl ( ( D \hat v + \hat v D ) \hat w D ^ { - n + 1 } \bigr ) , \end{aligned} \end{align*}


\begin{align*} \frac { \dd } { \dd t } \int _ { \mathbb { R } ^ { 3 } } \Phi _ + \frac { G } { r ^ { 2 } } \dd x = - \mu \int _ { \mathbb { R } ^ { 3 } } \nabla \times B \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x , \end{align*}


\begin{align*} & \Delta : = ( \langle u _ * , ( \lambda x _ v + ( 1 - \lambda ) x _ w ) \rangle ) ^ 2 + \lambda ( 1 - \lambda ) \| u _ * \| ^ 2 \| x _ v - x _ w \| ^ 2 \ge 0 , \\ \end{align*}


\begin{align*} | W _ i \cap C ^ { ( i ) } | = \prod _ { l = 1 } ^ { i } \frac { \beta _ { j + l - 1 } } { c _ { l } } . \end{align*}


\begin{align*} K ( c _ 1 , \dots , c _ n , 0 ) = & p _ i \circ H ( c _ 1 , \dots , c _ n , m ) \\ = & p _ i \circ \iota _ { U _ k } ( c 1 , \dots , c _ n ) = p _ i ( c _ 1 , \dots , c _ n ) , \end{align*}


\begin{align*} J a = \frac { { { c _ p } \left ( { { T _ w } - { T _ { s a t } } } \right ) } } { { { h _ { l v } } } } , \end{align*}


\begin{align*} & \int _ { \mathbb { R } ^ { N } } \omega \phi d x \leq C ( p ) \biggr ( C T ^ { - \frac { N } { 2 } } R ^ N + k C T ^ { - \frac { N } { 2 } } ( \ln R ) ^ \frac { 2 - N } { 2 } + C ( \ln R ) ^ \frac { 2 - N } { 2 } \biggr ) . \end{align*}


\begin{align*} \begin{aligned} | u _ { \tilde { \nu } ( y ' ) } ( y ) | \ , & \leq | u _ { \tilde { \nu } ( 0 ) } ( 0 ) | \\ & + \sum _ { \beta = \alpha + 1 } ^ { n - 1 } \sup _ { \xi \in \partial \omega \cap \partial \Omega } \left ( | \nabla _ { \tilde { \nu } \tau _ \beta } u ( \xi ) | + \sum _ { \gamma = 1 } ^ { n - 1 } | \rho _ { \gamma \beta } ( \xi ) u _ \gamma ( \xi ) | \right ) \cdot | y _ \beta | \leq C b _ \alpha \end{aligned} \end{align*}


\begin{align*} { \rho ^ { \circ } ( K ) } = { \sup _ { x \in K } \rho ( \{ x \} ) } < { \infty } , \end{align*}


\begin{align*} \langle \langle \varphi , \varphi ' \rangle \rangle = \tau \langle \langle \varphi ' , \varphi \rangle \rangle \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \inf _ { k \in { \mathbb Z } } \left ( \frac { \mu ( f ^ { k - n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \tag * { $ \mathcal { U E } 2 $ } \end{align*}


\begin{align*} X : p \mapsto \left . \frac { d } { d t } \right \vert _ { t = 0 } \Big ( \exp ( t \frak { X } ) ( p ) \Big ) . \end{align*}


\begin{align*} \begin{aligned} \mathbb { E } ^ { \boldsymbol { \xi } } [ | r ^ { N , i } ( t , \boldsymbol { x } ^ * _ t ) | ^ 2 ] \leq & \frac { C } { N ^ 2 } \Big ( 1 + \frac { 1 } { N } \underset { j = 1 } { \overset { N } \sum } \mathbb { E } ^ { \boldsymbol { \xi } } [ | x ^ { * , i } _ t - x ^ { * , j } _ t | ^ 2 ] \Big ) \\ \leq & \frac { C } { N ^ 2 } \Big ( 1 + \frac { 1 } { N } \underset { j = 1 } { \overset { N } \sum } | \xi ^ { i } - \xi ^ { j } | ^ 2 ] \Big ) . \end{aligned} \end{align*}


\begin{align*} \lim _ { m \to \infty } T ^ { c _ 2 ( m ) } x _ { 0 0 } = x _ { 0 1 } \quad \textup { a n d } \lim _ { n \to \infty } T ^ { c _ 1 ( n ) } x _ { 0 1 } = x _ { 1 1 } , \end{align*}


\begin{align*} \begin{aligned} & \int _ { - \infty } ^ { \infty } \bar { u } _ y e ^ { - 2 i z y } \mathrm { d } y = \bar { u } ( y ) e ^ { - 2 i z y } | _ { - \infty } ^ { + \infty } + 2 i z \int _ { - \infty } ^ { \infty } \bar { u } e ^ { - 2 i z y } \mathrm { d } y = i z \widehat { \bar { u } } ( z ) . \end{aligned} \end{align*}


\begin{align*} | f ( \lambda ) | \leq \underbrace { \frac { 1 } { | a _ k | } } _ { = : C _ p } \sup _ { | z | = 1 } | p ( \lambda + z \cdot v ) f ( \lambda + z \cdot v ) | \leq C _ p \sup _ { \mu \in \C ^ 2 , | \mu | \leq 1 } | p ( \lambda + \mu ) f ( \lambda + \mu ) | , \end{align*}


\begin{align*} \delta \in [ 0 , 1 ] \frac { 1 } { p } = \big ( \frac { 1 } { \alpha } - \frac { 1 } { d } \big ) \delta + \frac { 1 - \delta } { 2 } . \end{align*}


\begin{align*} h ( T ) & = y q \log ( y q ) + \sum _ { k = 1 } ^ { r } \left ( \Delta ( i _ k ) + \Delta ( j _ k ) \right ) \\ & \le y q \log ( y q ) + \sum _ { k = 1 } ^ { r } \left ( \Delta ( q + 1 ) + \Delta ( k ) \right ) \\ & = h ( T ' ) , \end{align*}


\begin{align*} { \bar F _ i } = - { { w } _ i } { \bar F } , \end{align*}


\begin{align*} \Upsilon ( \cdot ) = \Phi ( \cdot ) + \Psi ( \cdot ) \end{align*}


\begin{align*} \begin{aligned} b _ { 1 , \delta } ( x ^ \star ) = & \ , 2 \int _ { 0 } ^ { 1 } h ( x ^ \star + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \\ = & \ , 2 \int _ { 0 } ^ { 1 / 2 } h ( x ^ \star + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \\ & - 2 \int _ { 0 } ^ { 1 / 2 } h ( x ^ \star - \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \end{aligned} \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \sum _ { 1 \le \beta \le N } ( P ^ L _ { \alpha \beta } ) _ t ( x ' - y ' ) C _ { \beta } \ , d y ' = C _ { \alpha } , \quad \forall ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} \det ( D ^ 2 ( a _ k \hat C _ { q _ k , N } ) ) = c _ k \delta _ { q _ k } . \end{align*}


\begin{align*} \mathcal { M } _ { ( i , j ) } [ m , n ] : = \mathcal { M } _ { ( i + m , j + n ) } \ , , \end{align*}


\begin{align*} \varphi ( t ) : = \mathrm { K L } \left ( \rho _ { t } \mid \pi \right ) \end{align*}


\begin{align*} \bar { x } _ n - x _ n ^ 0 = \frac { - \varepsilon _ 1 \bar { \xi } _ 1 + | \hat { x } | \bar { \xi } _ 2 } { \sqrt { \varepsilon _ 1 ^ 2 + | \hat { x } | ^ 2 } } \leq | D \rho ( 0 , \tilde { x } ) | | \bar { \xi } _ 1 | + | \bar { \xi } _ 2 | \leq C \epsilon _ 0 \delta ^ 2 b _ \alpha \end{align*}


\begin{align*} \sigma ( \alpha + \beta ) = \sigma ( \alpha ) + \sigma ( \beta ) \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x & \leq C \int _ { B _ { 2 \rho } } r ^ { 2 - n } u _ r ^ 2 \d x + C \varepsilon ^ 2 \rho \int _ { B _ { 2 \rho } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \\ & \rho \leq 1 / 2 \varepsilon \leq \varepsilon _ 0 . \end{align*}


\begin{align*} \Gamma _ { k } = \{ \lambda \in \mathbb { R } ^ { n } : \sigma _ { j } ( \lambda ) > 0 , j = 1 , \ldots , k \} . \end{align*}


\begin{align*} v = 0 \partial \O . \end{align*}


\begin{align*} \dot { { x } } _ a & = - \gamma \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \\ \dot { \bar { y } } _ a & = - \gamma \ , \bar { y } _ a + \gamma \ , \dfrac { a _ { 0 , \delta } ( { x } _ a ) } { 2 } \end{align*}


\begin{align*} X ' \coloneqq \{ B ' \mid A / E \subset B ' \subset C / E , \ \dim B ' = b - e , \ B ' \cap D / E = 0 \} . \end{align*}


\begin{align*} \left | \frac { e ^ { \kappa s } - 1 } { s } \right | & = \left | \sum _ { n = 0 } ^ \infty \frac { \kappa ^ { n + 1 } s ^ n } { ( n + 1 ) ! } \right | \leq e ^ { \frac { 3 } { 4 } \kappa } \kappa , \end{align*}


\begin{align*} \langle \overline { \alpha } ' _ 1 , \overline { \alpha } ' _ 1 \rangle \langle \overline { \alpha } ' _ 2 , \overline { \alpha } ' _ 2 \rangle < \langle \overline { \alpha } ' _ 1 , \overline { \alpha } ' _ 2 \rangle ^ 2 = \langle \alpha ' _ 1 , \alpha ' _ 2 \rangle ^ 2 = \cosh ( \theta ) ^ 2 = 1 - s . \end{align*}


\begin{align*} \mathcal { J } _ 2 = \biggl ( \int _ 0 ^ T \eta ^ { - \frac { 1 } { p - 1 } } \left | \eta _ t \right | ^ { \frac { p } { p - 1 } } d t \biggr ) \biggl ( \int _ { \mathbb { R } ^ { N } } \phi ^ { - \frac { 1 } { p - 1 } } \left | \Delta \phi \right | ^ { \frac { p } { p - 1 } } d x \biggr ) . \end{align*}


\begin{align*} \tau \lim _ { n \to \infty } T ( t ) ( x _ n - x ) = 0 \end{align*}


\begin{align*} | \mathfrak { I } _ { q _ 1 , \ldots , q _ 4 , \kappa } ( \ell ) | \leq \frac { C } { \ell ^ { 3 } } , \end{align*}


\begin{align*} a _ I : = \sum _ { w \in W _ I } w \textrm { a n d } b _ I : = \sum _ { w \in W _ I } \epsilon ( w ) \ , w \ , , \end{align*}


\begin{align*} c _ n ( a , b ) = \frac { ( - 1 ) ^ n \Gamma ( a + b ) } { ( a + n ) \Gamma ( a ) \Gamma ( b - n ) n ! } , \end{align*}


\begin{align*} \log \det ( u _ { , i j } ) = - v _ j x ^ j + u _ { , i } \xi ^ i + c . \end{align*}


\begin{align*} x \Longleftrightarrow y = \frac { x - c } { \rho } , \end{align*}


\begin{align*} \abs { \mathcal { T } } \le 2 \sum _ { i = 1 } ^ { C _ { B C } } \abs { \mathcal { B } ^ { ( i ) } } . \end{align*}


\begin{align*} h ( q , r _ 0 ) = - \hat W _ { r _ 0 } ( q ) = - \frac { 2 } { \pi q } \sin ( 2 \pi q r _ 0 ) . \end{align*}


\begin{align*} \mathcal { S } ^ { k - 1 } : = \{ \underline { x } \in [ 0 , 1 ] ^ { k } : | | \underline { x } | | = 1 \} , ~ \mathcal { D } ^ { k } ( A ) : = \{ \rho ( \underline { a } ) : \underline { a } \in A ^ { k } \} ~ \mbox { a n d } ~ \mathcal { D } ^ { \underline { k } } ( A ) : = \{ \rho ( \underline { a } ) : \underline { a } \in A ^ { \underline { k } } \} , \end{align*}


\begin{align*} E [ t ] ^ { \vee } = E ^ { \vee } [ t ^ { - 1 } ] t > 0 . \end{align*}


\begin{align*} p _ l ( s \cdot z ) = s ^ l p _ l ( z ) , \ ; \ ; \forall l = 0 , \dots , k . \end{align*}


\begin{align*} \langle f \rangle ( a ) = \lim _ { { T ' } \to \infty } \frac { 1 } { { T ' } } \int _ 0 ^ { T ' } f ( \Phi _ { - t \Lambda } a ) d t , a \in \C ^ n . \end{align*}


\begin{align*} \| \mu _ 1 - \mu _ 2 \| _ { L , M } ^ * : = \sup _ { f \in C _ b ( M ) , \ , | f | _ L \leqslant 1 } \Big | \langle f , \mu _ 1 \rangle - \langle f , \mu _ 2 \rangle \Big | \le 2 , \end{align*}


\begin{align*} x ^ { j } \allowbreak = \allowbreak j ! \sum _ { k = 0 } ^ { j } ( - 1 ) ^ { k } \frac { ( \beta ) ^ { ( j ) } } { ( j - k ) ! ( \beta ) ^ { ( k ) } } L _ { k } ^ { ( \beta ) } ( x ) , \end{align*}


\begin{align*} k ^ 2 f _ j ( \mathbf 0 ) \int _ { \mathcal C ^ h } e ^ { \rho \cdot \mathbf x } = I _ 1 + I _ 2 + I _ 3 + J _ 1 + J _ 2 , \end{align*}


\begin{align*} \mathcal { L } _ \Phi : = { \big \{ F ( z , \omega ) = f ( \Phi ^ { - 1 } ( z , \omega ) , \omega ) ; f \in L ^ 2 _ { \rm l o c } ( \mathbb { R } ^ n ; L ^ 2 ( \Omega ) ) \ ; \ ; \big \} } \end{align*}


\begin{align*} \begin{aligned} \int _ { 0 } ^ { 1 / 2 } \left [ h ( x + \delta \sin ( 2 \pi t ) ) - h ( x ) \right ] \sin ( 2 \pi t ) \ , d t \\ \int _ { 0 } ^ { 1 / 2 } \left [ h ( x ) - h ( x - \delta \sin ( 2 \pi t ) ) \right ] \sin ( 2 \pi t ) \ , d t . \end{aligned} \end{align*}


\begin{align*} F _ n ( \eta ) : = T _ n - \frac { 1 } { \eta } I _ n , \eta \in \Omega . \end{align*}


\begin{gather*} P ( A ) = P _ 0 ( A ) = 0 \quad A \in \mathcal { R } P ^ * ( A ) = 0 . \end{gather*}


\begin{align*} A _ t \cdot B \dot { g } ( \Phi ) = \partial _ t \left ( g ( \Phi ) \frac { G } { r ^ { 2 } } \right ) + \nabla \cdot ( g ( \Phi ) \nabla \theta \times A _ t ) . \end{align*}


\begin{align*} B ( p ) = \frac { 1 } { p \mu } \int _ { 0 } ^ { Q ( p ) } x f ( x ) \mathrm { d } x = \frac { 1 } { p \mu } J ( Q ( p ) ) L ( p ) = \frac { 1 } { \mu } \int _ { 0 } ^ { Q ( p ) } x f ( x ) \mathrm { d } x = \frac { 1 } { \mu } J ( Q ( p ) ) , \end{align*}


\begin{align*} \sum _ { \gamma } f _ { \frac { 1 } { 2 } } ( t - \gamma ) \leq \frac { \widehat { h } ( 0 ) } { 2 \pi } \log t + \frac { 1 } { \pi } \sum _ { n = 2 } ^ \infty \frac { \Lambda ( n ) } { \sqrt { n } } \left | \widehat { h } \left ( \frac { \log n } { 2 \pi } \right ) \right | . \end{align*}


\begin{align*} \widetilde { x } ( [ 0 , \ldots , k + 1 ] \otimes [ 1 , \ldots , \ell ] ) = x ( [ 0 , \ldots , k ] \otimes [ 0 , 1 , \ldots , \ell ] ) = 0 . \end{align*}


\begin{align*} \max _ { v \in \Lambda } e ( \alpha ^ { ( v ) } , G ^ { ( v ) } ) = e ( 0 , G ) + 2 H - 3 / 2 \end{align*}


\begin{align*} N = m _ 1 s _ 1 + \cdots + m _ L s _ L \end{align*}


\begin{align*} c _ { b } = \mathop { \inf } \limits _ { ( u , v ) \in E \setminus \{ ( 0 , 0 ) \} } \mathop { \max } \limits _ { t > 0 } I ( \gamma _ { u , v } ( t ) ) . \end{align*}


\begin{align*} d ( h _ z ) = \frac { 2 } { \mu } \mu _ z h _ z d z + \frac { \sqrt { \tau _ 0 } } { 2 } \left ( \frac { 1 } { 2 } d \overline { z } + ( \alpha - I \beta ) d z \right ) \end{align*}


\begin{align*} I = k ^ 2 f _ { j } ( \mathbf 0 ) \int _ { \mathcal K ^ { h } } e ^ { \eta \cdot \mathbf x } \mathrm d \mathbf x , \end{align*}


\begin{align*} u _ n = T _ n f _ n . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { \pi ( a ) } q _ i \neq \prod _ { i = 1 } ^ { \pi ( a ) } p _ i ^ { \alpha _ i } . \end{align*}


\begin{align*} G ( z ) = \frac { 1 } { \varkappa } A ( z ) \bigg ( 1 - \sum _ n \frac { a _ n \bar b _ n } { z - t _ n } \bigg ) \end{align*}


\begin{align*} ( n - 2 ) r < \sum \limits _ { i = 2 } ^ n \chi _ i , \end{align*}


\begin{align*} \Delta _ g \phi & = g ^ { p q } \phi _ { , p q } - \frac { 1 } { 2 } \xi ^ p \phi _ { , p } + \frac { 1 } { 2 } g ^ { p q } v _ p \phi _ { , q } \end{align*}


\begin{align*} \log \int _ { \R ^ n } e ^ { f ( x ) } \ , d x - \int _ { \R ^ n } f \ , d Q + H ( Q ) = H ( Q \ , | \ , P ) \end{align*}


\begin{align*} \hat W _ r ( m _ 2 ) = \frac { m _ 1 } { m _ 2 } \hat W _ { \frac { r m _ 2 } { m _ 1 } } ( m _ 1 ) . \end{align*}


\begin{align*} \rho _ N = \sum _ { j = 1 } ^ { N } w _ { N , j } \delta _ { \theta _ { N , j } } \end{align*}


\begin{align*} \dfrac { \partial b _ { 1 , \delta } ( x ) } { \partial x } = \int _ { 0 } ^ { 1 } \left . \dfrac { \partial h ( s ) } { \partial s } \right | _ { x + \delta \sin ( 2 \pi t ) } \sin ( 2 \pi t ) d t \end{align*}


\begin{align*} \bigg | \frac { \partial h _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) } { \partial z _ { i , \hbar } } \bigg | \leq C , \ \ 1 \leq i , j \leq m , \hbar = 1 , 2 , \end{align*}


\begin{align*} ( \tilde h , \tilde v ) : = ( \hat h , \hat v ) + \big ( L _ Y \bar g , Y ( \bar u ) \big ) \end{align*}


\begin{align*} \nabla _ X \varphi = - \frac { 1 } { 2 } \sqrt { | c _ 1 | } \ X _ 1 \cdot \nu _ 1 \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi \end{align*}


\begin{align*} \begin{aligned} \gamma _ 2 ^ \star : = & \ , \dfrac { d } { 6 M _ \mathcal { D } T e ^ { L _ \mathcal { D } \bar { t } } } \\ \gamma _ 3 ^ \star : = & \ , \dfrac { d } { 9 M _ \mathcal { D } T \left ( e ^ { L _ \mathcal { D } \bar { t } } - 1 \right ) } \\ \gamma _ 4 ^ \star : = & \ , \dfrac { d } { 6 M _ \mathcal { D } T } , \end{aligned} \end{align*}


\begin{align*} [ \partial a _ \lambda b ] & = - \lambda [ a _ \lambda b ] , \ \ [ a _ \lambda \partial b ] = ( \partial + \lambda ) [ a _ \lambda b ] \ \ \mbox { ( c o n f o r m a l \ s e s q u i l i n e a r i t y ) } , \\ { [ a _ \lambda b ] } & = - [ b _ { - \lambda - \partial } a ] \ \ \mbox { ( s k e w - s y m m e t r y ) } , \\ { [ a _ \lambda [ b _ \mu c ] ] } & = [ [ a _ \lambda b ] _ { \lambda + \mu } c ] + [ b _ \mu [ a _ \lambda c ] ] \ \ \mbox { ( J a c o b i \ i d e n t i t y ) } . \end{align*}


\begin{align*} \gamma ( x _ n ^ D ; w ) \ge \tau ( x _ n ^ D ) : = 8 \frac { | \Gamma _ D | \mathrm { N } _ I \lambda _ d ^ 2 } { | \Omega | \mathrm { N } _ D \lambda _ 1 } \frac { | \nabla w ( x _ n ^ D ) | ^ 2 } { | \nabla w ( y _ n ) | ^ 2 } , \end{align*}


\begin{align*} f _ n ( t , z , x ) : = \sup _ { \| q \| \le n } \Big ( q \cdot z - g ( t , q , x ) \Big ) . \end{align*}


\begin{align*} \mathcal { W } \ , ( \Delta ^ { - m + 1 } ) = \frac { n - 2 } { 1 2 } v _ { n - 1 } \int _ { M } R ( g ) ~ v o l _ { g } , \end{align*}


\begin{align*} M ^ 2 ( t ) = 2 a t . \end{align*}


\begin{align*} \Psi ' & = ( \sigma _ { S } \times \sigma _ { X } ) \circ \tau _ { * } \Phi \circ ( \sigma _ { S } \times \sigma _ { X } ) ^ { - 1 } \circ \Psi \circ \Phi ^ { - 1 } \end{align*}


\begin{align*} z = ( 2 - \alpha - \beta ) \frac { \phi ( z ) } { [ 1 - \Phi ( z ) ] } + ( \alpha - 1 ) \frac { \phi ( z ) } { \Phi ( z ) [ 1 - \Phi ( z ) ] } . \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) = ( f _ n , v _ n ) \quad \ , v \in X _ n , \end{align*}


\begin{align*} ( U ^ \top U ) _ { i , i } - \frac { m } { n } = \frac { \sqrt { m - m ^ 2 / n } } { \sqrt { 1 - 1 / n } } \bigg ( t _ i ^ 2 - \frac { 1 } { n } \bigg ) \mbox { f o r a l l } i \in \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} L _ X g ( \nu , \cdot ) = h ( \nu , \cdot ) \mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \Sigma . \end{align*}


\begin{align*} u \ \mbox { i s b o u n d e d i n } \ B _ { R } \ \mbox { a n d } \ \lim \limits _ { | x | \nearrow R } v ( x ) = \infty \end{align*}


\begin{align*} f ( z , t ) = \int _ \Omega \int _ { \C ^ n } \widehat { f } ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \ , d \nu ( a ) . \end{align*}


\begin{align*} a \wedge b : = ( a \cdot b ) a = ( b \cdot a ) b . \end{align*}


\begin{align*} - \Delta p + \zeta p & = \overline { y } - g , \\ \langle p + \alpha \overline { u } , u - \overline { u } \rangle & \ge 0 u \in \mathcal { C } _ { a d } . \end{align*}


\begin{align*} \sigma ( x _ i ) = x _ { \sigma ( i ) } , \ \sigma ( \epsilon ) = \epsilon , \sigma \in S _ n , \ 1 \leq i \leq n . \end{align*}


\begin{align*} h _ = ( \sum _ { j = 1 } ^ m \lambda _ j \nabla g _ j , \lambda _ 1 g _ 1 , \dots , \lambda _ m g _ m , \sum _ { j = 1 } ^ m \lambda _ j ^ 2 - 1 ) \ , , \end{align*}


\begin{align*} a = - \frac { 1 } { \mu } \left ( \partial _ y \mu \ d x - \partial _ x \mu \ d y \right ) = ( \frac { 1 } { \mu } d \mu - \frac { 2 } { \mu } \mu _ z d z ) I \end{align*}


\begin{align*} \frac { x ^ { q + 1 } + x } { y ^ { q + 1 } + y } = d , \end{align*}


\begin{align*} D _ { n + 1 } ( x ) = m _ \tau ( x ) D _ n ( R ^ { t } x ) . \end{align*}


\begin{align*} \mathcal { M } = \frac { T _ { \rm s } \psi _ { 1 } } { \sqrt { 2 \pi } } \mathcal { W } _ 2 + \frac { T _ { \rm s } \psi _ { 2 } } { \sqrt { 2 \pi } } e ^ { - \frac { \psi _ { 2 } \tau _ { \rm d } } { N _ { \rm a } } } \underbrace { \int _ { \kappa _ { \rm b } } ^ { \kappa _ { \rm t } } e ^ { - \frac { 1 } { 2 } x ^ 2 - \frac { \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } x } \ , \mathrm { d } x } _ { \mathcal { M } _ 1 } . \end{align*}


\begin{gather*} P ( c ) = \inf _ { Q \in K } Q ( c ) \quad \quad P \in K \end{gather*}


\begin{align*} W _ { U _ { \tau _ k , R } } ( x ) = W \circ R _ { \tau _ k } ( x ) \end{align*}


\begin{align*} ( \mu ) _ \infty = \prod _ { n = 1 } ^ \infty ( 1 - z ^ { n - 1 } \mu ) . \end{align*}


\begin{align*} \begin{aligned} & \int _ { \Omega } \nabla n \cdot ( S ( x , n , c ) \cdot \nabla c ) \ , \varphi ^ { 3 } \\ & \le \| S _ { 0 } \| _ { \mathcal { C } ( [ 0 , \gamma ] ) } C _ { 2 } ^ { \frac { 1 } { 4 } } C _ { 3 } ^ { \frac { 1 } { 4 } } \| \nabla c \| _ { L ^ { 2 } ( \Omega \cap B _ { \delta } ) } ^ { \frac { 1 } { 4 } } \int _ { \Omega } \frac { | \nabla n | ^ { 2 } } { n } \varphi ^ { 3 } + \frac { 1 } { 5 } \int _ { \Omega } \frac { | \nabla n | ^ { 2 } } { n } \varphi ^ { 3 } + M . \end{aligned} \end{align*}


\begin{align*} h = 2 \mu \int _ { \mathbb { R } ^ { 3 } } ( \phi - \phi _ \infty ) _ { + } ^ { 2 } \frac { 1 } { r ^ { 2 } } \dd x . \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x & \leq C \int _ { B _ { 8 \rho } \setminus B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x \\ & \rho \leq 1 / 8 \varepsilon \leq \varepsilon _ 0 . \end{align*}


\begin{align*} \begin{aligned} \left \| M _ { + , 1 2 } ( x ; z ) \right \| _ { L _ { z } ^ { 2 } } \leq c \left \| \mathcal { P } ^ + \left ( \bar { r } _ 1 ( z ) \mathrm { e } ^ { - 2 i z x } \right ) \right \| _ { L _ { z } ^ 2 } , \end{aligned} \end{align*}


\begin{align*} \eta ( t ) = \nu \left ( \frac { t } { T } \right ) , \ , \ , \ , t > 0 , \end{align*}


\begin{align*} K _ M : = \bigg \{ f \in L ^ p ( [ 0 , T ] , H ) : \int _ 0 ^ T \Vert f ( t ) \Vert _ V ^ p d t \leq M \bigg \} . \end{align*}


\begin{align*} H _ { m i n } ( E \otimes F _ { \ell - 1 } ) = H _ { m i n } ( E \otimes F _ 1 ) . \end{align*}


\begin{align*} X _ { M ( n ) } ( x _ 1 , x _ 2 ) = x _ n . \end{align*}


\begin{align*} \begin{aligned} < u \times n , \xi > _ { \partial \Omega } & = \int _ { \Omega } \nabla \times u \cdot \xi \dd x - \int _ { \Omega } u \cdot \nabla \times \xi \dd x , \\ < u \cdot n , \varphi > _ { \partial \Omega } & = \int _ { \Omega } u \cdot \nabla \varphi \dd x + \int _ { \Omega } \nabla \cdot u \varphi \dd x , \end{aligned} \end{align*}


\begin{align*} \mathcal { M } = \{ Z = ( z _ 1 , z _ 2 , \cdots , z _ m ) \in \mathbb { R } ^ { ( 2 m ) } \mid z _ i \in B _ { \bar { \rho } } ( x _ { 0 , i } ) , \ i = 1 , \cdots , m \} . \end{align*}


\begin{align*} \left \langle X , \Gamma _ { ( g , u ) } ( X ) \right \rangle = - \tfrac { 1 } { 2 } \Delta | X | ^ 2 - \tfrac { 1 } { 2 } u ^ { - 1 } { \nabla u } \cdot \nabla | X | _ g ^ 2 + | \nabla X | ^ 2 + u ^ { - 2 } ( X ( u ) ) ^ 2 . \end{align*}


\begin{align*} \epsilon _ 1 b _ { n - 1 } u _ n ( x ) \leq u ( y ) - u ( x ) + \sum _ { \alpha = 1 } ^ { n - 1 } ( y _ \alpha - x _ \alpha ) u _ \alpha ( x ) \leq C b _ { n - 1 } ^ 2 \end{align*}


\begin{align*} x ( [ \mathbf { a ' } , \mathbf { a } ] \otimes [ 0 , \mathbf { b } ] ) = 0 \end{align*}


\begin{align*} x _ i ( g ) = x ( f _ i ( g ) ) \forall g \in G . \end{align*}


\begin{align*} \mathcal D _ { \psi , z ' } ( g ) = \left ( \frac { d } { d \bar z } g \circ T ^ { - 1 } \right ) \circ T . \end{align*}


\begin{align*} \begin{aligned} \partial _ t u - \nabla \cdot ( A \nabla u ) = & \ F , & ( 0 , T ] \times \Omega , \\ u ( 0 , \cdot ) = & \ u _ 0 , \ , & \Omega , \\ u = & \ g _ D ^ { } , & ( 0 , T ] \times \Gamma _ D ^ { } , \\ \mathbf { n } \cdot A \nabla u = & \ g _ N ^ { } , & ( 0 , T ] \times \Gamma _ N ^ { } , \end{aligned} \end{align*}


\begin{align*} e _ c \left [ i \right ] = a _ c { \left [ i \right ] } - a _ c ^ { \left ( o \right ) } . \end{align*}


\begin{align*} b ( \sigma \gamma ) = \pi ( \sigma ) b ( \gamma ) + b ( \sigma ) \end{align*}


\begin{align*} & \psi ^ { \pm } ( x , t ; k ) = \mathrm { e } ^ { - i c _ \pm ( x ) \sigma _ 3 } + \mathcal { O } ( k ^ { - 1 } ) , k \rightarrow \infty , \end{align*}


\begin{align*} \begin{aligned} \abs { c _ 3 ( S ^ 3 \setminus S _ i ) } & \le \abs { c _ 2 ( S ^ 2 \setminus S _ i ) } + \abs { c _ 3 ( P ^ 3 \setminus S _ i ) } \\ & \le ( 2 k - t ' - \abs { S \cap S _ i } - q _ i - s _ 1 ) + ( s _ 1 - t _ i ) = 2 k - t ' - \abs { S \cap S _ i } - p _ i . \end{aligned} \end{align*}


\begin{align*} \Big ( \frac p q \Big ) \Big ( \frac q p \Big ) = ( - 1 ) ^ { \mu + \nu } . \end{align*}


\begin{align*} \{ A ( \xi ) \} \cap \mathrm { K e r } ( B ) = \{ 0 \} , \forall \xi \in \R ^ * . \end{align*}


\begin{align*} \tilde { \nabla } \xi = d \xi + \omega \xi . \end{align*}


\begin{align*} \begin{array} { r l } \min \limits _ { x \in X } \{ f ( x ) : g _ i ( x ) \leq 0 , \ i = 1 , \ldots , m _ 1 ; h _ j ( x ) = 0 , \ j = 1 , \ldots , m _ 2 \} . \end{array} \end{align*}


\begin{align*} \Pi _ { \mathrm { k e r } A } B ( v _ 1 , v _ 2 ) + \Pi _ { \mathrm { k e r } A } B ( v _ 2 , v _ 1 ) = 0 \end{align*}


\begin{align*} \mathbf { w } = \frac { w } { k } \overrightarrow { \zeta } , \end{align*}


\begin{align*} ( \partial - \nabla ) _ X Y = - \sum _ { i = 1 , 2 } \sqrt { c _ i } \ \langle X _ i , Y \rangle \nu _ i + B ( X , Y _ T ) - B ^ * ( X , Y _ N ) \end{align*}


\begin{align*} \| \langle x \rangle I _ 1 ^ { \prime } ( x ) \| _ { L ^ 2 ( \mathbb { R } ^ + ) } = 2 \pi ^ { - 1 } \| \langle x \rangle \widehat { \bar { r } _ 1 ( z ) } ( 2 x ) \| _ { L ^ 2 ( \mathbb { R } ) } \leq c \| r _ 1 ( z ) \| _ { H ^ 1 ( \mathbb { R } ) } , \end{align*}


\begin{align*} Q _ { i j } ( h ) = \partial ^ 2 _ { i j } h - \Gamma _ { j i } ^ i \partial _ i h - \Gamma _ { i j } ^ j \partial _ j h + g _ { i j } h = 0 , \forall i < j , \end{align*}


\begin{align*} \frac { \partial ^ 2 \mathbb { E } \left [ \varepsilon | \hat { \mathbf { H } } \right ] } { \partial a _ c \partial a _ c } > 0 ~ { \rm a n d } ~ \frac { \partial ^ 2 \mathbb { E } \left [ \varepsilon | \hat { \mathbf { H } } \right ] } { \partial a _ i \partial a _ i } > 0 , ~ i = 1 , 2 , \ldots , K \end{align*}


\begin{align*} N ( \eta ) : = \frac { \operatorname { v o l } ( B ^ { n - 1 } ) \operatorname { v o l } ( \partial \Omega ) } { ( 2 \pi ) ^ { n - 1 } } \eta ^ { n - 1 } + o ( \eta ^ { n - 1 } ) \ \eta \to + \infty , \end{align*}


\begin{align*} [ \Delta ^ M _ \Psi , \Delta ^ N _ \Gamma ] = 0 . \end{align*}


\begin{gather*} \lim _ k \sup _ { P \in M } P \bigl \{ \abs { f } \ , 1 ( \abs { f } > k ) \bigr \} = 0 \quad \quad f \in F . \end{gather*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta v _ { l } + u _ { l } v _ { l } = f , & x \in \Omega , \\ \nabla v _ { l } \cdot \nu + v _ { l } = \eta , & x \in \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} H _ { \Psi } = H - g ( \nabla \Psi , \vec \nu ) \end{align*}


\begin{align*} v _ \epsilon ( t ) : = & \int _ { - \infty } ^ { t } ( \int _ { - \infty } ^ { t _ 1 } ( \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ) ( s ) d s ) d t _ 1 \\ & - \int _ { - \infty } ^ { - t _ 0 } ( \int _ { - \infty } ^ { t _ 1 } ( \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ) ( s ) d s ) d t _ 1 - t _ 0 , \end{align*}


\begin{align*} ( \tau ^ { * } \alpha _ { \underline { S } \times \underline { X } } \circ \tau ^ { * } \Psi \circ \alpha _ { \underline { S } \times \underline { X } } ) \circ \Psi = \mathrm { i d } _ { ( \underline { S } \times X ) _ { \mathbb { C } } } . \end{align*}


\begin{align*} T \in Z ( V ) \iff \mu _ T ( x , \mu _ T ( y , z ) ) = 0 \forall x , y , z \in V . \end{align*}


\begin{align*} F _ { + } ( x ) = G ( x ) + \frac { 1 } { 2 } \left [ - L , G ( x ) \right ] + \frac { 1 } { 1 2 } \left [ - L , \left [ - L , G ( x ) \right ] \right ] + \cdots . \end{align*}


\begin{align*} \partial ^ 2 _ { x _ i x _ j } \phi = \partial ^ 2 _ { x _ i x _ j } \psi + \partial ^ 2 _ { x _ i y } \psi \Gamma _ j + \partial ^ 2 _ { x _ j y } \psi \Gamma _ i + \partial ^ 2 _ y \psi \Gamma _ j \Gamma _ i + \partial _ y \psi \partial ^ 2 _ { x _ j x _ i } T - \partial _ y T \partial ^ 3 _ { x _ j x _ i y } T + \partial _ { y } \Gamma _ i \partial _ { y } \Gamma _ j \end{align*}


\begin{align*} \{ v _ 1 , \ldots , v _ k \} \in G \bigcap \limits _ { i = 1 } ^ k S _ { v _ i } = \emptyset . \end{align*}


\begin{align*} \overline { P } _ { \rm T x } = \delta \left [ f ( \kappa _ b ) - f ( \kappa _ t ) + \kappa _ t Q ( \kappa _ t ) + \kappa _ b Q ( - \kappa _ b ) \right ] + P _ { \rm b i a s } , \end{align*}


\begin{align*} - \Delta u = F \quad \Omega , u = g \quad \partial \Omega \end{align*}


\begin{align*} { p _ { E O S } } = \frac { { \rho R T } } { { 1 - b \rho } } - \frac { { a \xi \left ( T \right ) { \rho ^ 2 } } } { { 1 + 2 b \rho - { b ^ 2 } { \rho ^ 2 } } } , \end{align*}


\begin{align*} S ^ { p q , r s } _ k = \frac { \partial ^ 2 S _ k [ D ^ 2 u ] } { \partial r _ { p q } \partial r _ { r s } } . \end{align*}


\begin{align*} \theta ^ j _ { \alpha \beta } ( x ' , t ; y ' ) : = t \ , \partial _ j K ^ L _ { \alpha \beta } ( x ' - y ' , t ) , x ' , y ' \in \R ^ { n - 1 } , \ , t > 0 . \end{align*}


\begin{align*} \alpha _ i + \bar { \alpha } _ j = 1 . \end{align*}


\begin{align*} S t _ { D N D C } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( f , z ) - f ( z ) | | _ { \infty } = 0 \end{align*}


\begin{align*} { \bf P } ( \lim _ { n \to \infty } \kappa _ n = \kappa ) = 1 . \end{align*}


\begin{align*} & \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ { n } \cdot B _ { n } \dd x = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ { 0 , n } \cdot B _ { 0 , n } \dd x = h _ n , \\ & \mathcal { I } _ { h _ n } \leq \frac { 1 } { 2 } \int _ { \Omega } \left ( | u _ { n } | ^ { 2 } + | B _ { n } | ^ { 2 } \right ) \dd x \leq \frac { 1 } { 2 } \int _ { \Omega } \left ( | u _ { 0 , n } | ^ { 2 } + | B _ { 0 , n } | ^ { 2 } \right ) \dd x , \end{align*}


\begin{align*} \begin{aligned} f ( t , s ) = F _ { \boldsymbol { \omega } \boldsymbol { \eta } } \big ( L ^ { - 1 } _ { \boldsymbol { \omega } } ( t ) , K ^ { - 1 } _ { \boldsymbol { \eta } } ( s ) , f \big ( L ^ { - 1 } _ { \boldsymbol { \omega } } ( t ) , K ^ { - 1 } _ { \boldsymbol { \eta } } ( s ) \big ) \big ) , ~ ~ ~ ~ ~ \forall ~ ( t , s ) \in L _ { \boldsymbol { \omega } } ( S G ' ) \times K _ { \boldsymbol { \eta } } ( S G '' ) , \end{aligned} \end{align*}


\begin{align*} \mathcal { H } _ \Phi : = { \big \{ F ( z , \omega ) = f ( \Phi ^ { - 1 } ( z , \omega ) , \omega ) ; \ ; f \in H ^ 1 _ { \rm l o c } ( \mathbb { R } ^ n ; L ^ 2 ( \Omega ) ) \ ; \ ; \big \} } \end{align*}


\begin{align*} Z g \left ( X , Y \right ) = g \left ( \nabla _ { Z } X , Y \right ) + g \left ( X , \nabla _ { Z } ^ { \ast } Y \right ) \end{align*}


\begin{align*} h _ { j _ p + 1 } = \rho _ p h _ { j _ p } + \frac { \rho _ p ( \rho _ p - 1 ) } { 2 } \end{align*}


\begin{align*} W ( \xi _ 0 , k , x , \mathsf { L } ^ k ) = W ( 1 - \xi _ 0 , k , x , \mathsf { R } ^ k ) , ~ \forall \xi _ 0 \in [ 0 , 1 ] , \forall x \in \mathbb { R } , \end{align*}


\begin{align*} \Omega _ \delta : = \{ x \in \Omega : | x - x _ 0 | < \delta \} \end{align*}


\begin{align*} \left ( 2 n - \beta \left ( 3 \beta ^ { 2 } + 4 \right ) + 1 \right ) \left ( n ^ { 2 } + \left ( 2 \beta + 1 \right ) n + \beta ^ { 2 } + \beta + 1 \right ) = 0 \end{align*}


\begin{gather*} \Gamma _ 0 = \bigl \{ P \in \Gamma : ( \pi _ 1 , \ldots , \pi _ n ) P \bigr \} . \end{gather*}


\begin{align*} ( \Lambda - 1 + \delta ) \sum _ { j = 1 } ^ k ( \Lambda + \varepsilon ) ^ { - j } > 1 , \end{align*}


\begin{align*} \pi ( x , \lambda ) = x \ , , \ , \forall x \in \R ^ n \ , , \ , \forall \lambda \in \R ^ { m } \ , . \end{align*}


\begin{align*} w = 1 + 2 \gamma _ 1 a _ 1 + 2 \gamma _ 2 a _ 2 > 0 \end{align*}


\begin{align*} ( D ^ t ) ^ N ( \psi ) = \sum _ { \substack { \alpha = ( \alpha _ 0 , . . . , \alpha _ k ) \\ | \alpha | = N } } C _ { N , \alpha } \frac { \partial ^ { \alpha _ 0 } \psi \partial ^ { \alpha _ 1 + 1 } \phi \partial ^ { \alpha _ 2 + 1 } \phi . . . \partial ^ { \alpha _ k + 1 } \phi } { \lambda ^ N ( \phi ' ) ^ { N + k } } . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \Delta _ \Psi \omega = f ( u _ 2 ) - f ( u _ 1 ) & \textnormal { i n i n t } M \\ \omega = 0 & \textnormal { o n } \ \partial M , \end{array} \right . \end{align*}


\begin{align*} \int f ( x ) \ , \overline { g ( x ) } \ , d x = h \sum _ n f ( n h ) \ , \overline { g ( n h ) } . \end{align*}


\begin{align*} \mathcal { L } ( u ^ h ( t , x ) ) = \mathcal { L } \left ( \sum _ i w _ i ( t ) \sigma _ i ( x ) \right ) = \sum _ i w _ i ( t ) \mathcal { L } ( \sigma _ i ( x ) ) . \end{align*}


\begin{align*} [ \Delta _ { \Psi } , X ] = 0 , \O \end{align*}


\begin{align*} T _ { W , \widetilde { K } } ( \mu ^ t ) & = \frac 1 2 \int _ { [ 0 , 1 ] \times \R } \int _ { [ 0 , 1 ] \times \R } W ( u , v ) K ( x - y ) \mu ^ t ( d u , d x ) \mu ^ t ( d v , d y ) \\ & = \frac 1 2 \int _ { [ 0 , 1 ] \times \R } \int _ { [ 0 , 1 ] \times \R } W ( u , v ) K \big ( ( 1 - t ) ( x - y ) + t ( T _ u ( x ) - T _ u ( y ) ) \big ) \mu ^ 0 ( d u , d x ) \mu ^ 0 ( d v , d y ) . \end{align*}


\begin{align*} \beta _ n : = ( n + 1 ) h _ A + n ( \widetilde b + a ) + a \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) \| _ { \mathcal { A } } & \le \beta ( \| x ( \bar { t } / \gamma ) \| _ { \mathcal { A } } , 0 ) + \tilde { d } \\ & \le \beta ( \underline { \rho } , 0 ) + \tilde { d } \\ & \le d & & \forall \ , t \in I _ 1 \end{aligned} \end{align*}


\begin{align*} \sup _ { n \in \mathbb { N } } \sup _ { j \in [ n ] ^ V } \sup _ { s _ 1 , . . . , s _ i \in [ 0 , T ] } \norm { D ^ i _ { s _ 1 , . . . , s _ i } A _ n ( j ) } _ { L ^ p ( P ) } < \infty \end{align*}


\begin{align*} \tau ^ * = \bar { \tau } \Leftrightarrow \dfrac { | \lambda _ i | } { | \lambda _ { i + 1 } | } = \dfrac { | \lambda _ { i + 1 } | } { | \lambda _ { i + 2 } | } \quad \forall i \in [ 1 , p - 2 ] \quad \forall i \in [ p + 1 , n - 2 ] ) , \end{align*}


\begin{align*} - \Delta z _ { n } + \frac { F ( y _ { n } ) - F ( y ) } { t _ { n } } = h _ { n } . \end{align*}


\begin{align*} | F ^ { i j } ( \nabla _ \alpha \varphi ) _ { i j } | \leq C \sqrt { b _ { n - 1 } } \sum _ { i = 1 } ^ n F ^ { i i } \mbox { i n } \omega _ 1 . \end{align*}


\begin{align*} \widehat { f } ( \xi ) : = \int _ G f ( x ) \ , \overline { \xi ( x ) } \ , d \mu ( x ) \end{align*}


\begin{align*} \mathbf { y ' } ^ H \mathbf { y } ' = & y _ c ^ * y _ c + y _ 1 ^ * y _ 1 + \cdots + y _ M ^ * y _ M = \left ( \sum _ { l = 1 } ^ M y _ l ^ * \right ) \left ( \sum _ { j = 1 } ^ M y _ j \right ) + \sum _ { i = 1 } ^ M y _ i ^ * y _ i . \end{align*}


\begin{align*} \alpha _ { i j k } : = \frac { 1 } { 2 } ( c _ { i j k } + c _ { k i j } + c _ { k j i } ) , [ e _ { i } , e _ { j } ] = c _ { i j k } e _ { k } . \end{align*}


\begin{align*} \lambda = \sum _ { s = 1 } ^ q \langle \sum _ { r = 1 } ^ p \alpha _ { r s } e _ r , \sum _ { \mu = 1 } ^ p \alpha _ { \mu s } X e _ { \mu } \rangle , \end{align*}


\begin{align*} Q ( X ) = \sum _ { 0 \le i \ne j \le n - 1 } B _ { i j } X ^ { 2 ^ i + 2 ^ j } + \sum _ { 0 \le i \le n - 1 } C _ i X ^ { 2 ^ i } + D . \end{align*}


\begin{align*} \mu _ r ( \psi ( \cdot ) ) = \mu _ r ( \cdot ) : = \sum _ { i = 1 } ^ { \infty } i ^ r \psi _ i ( \cdot ) . \end{align*}


\begin{align*} \left \| \frac { 1 } { s ^ { \alpha } } \right \| _ { L ^ { q } ( \mathbb { R } ) } = C \frac { ( r r ' ) ^ { \alpha } } { | r - r ' | ^ { 2 \alpha - 1 / q } } , \alpha > \frac { 1 } { 2 q } , \end{align*}


\begin{align*} \sigma ( \lambda \alpha ) = \sigma ( \lambda ) \sigma ( \alpha ) \end{align*}


\begin{align*} R = \frac { n - 1 } { n } \mathcal { H } ^ 2 . \end{align*}


\begin{align*} ( I - Q ^ \dagger ) F ^ { q , \dagger } _ v ( v + \psi ^ \dagger ( v , p ) , p ) [ v _ 1 + \psi ^ \dagger _ v ( v , p ) v _ 1 ] = 0 \end{align*}


\begin{align*} \bar { y } _ n : = y _ n - \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( x ' _ 0 ) y _ \beta . \end{align*}


\begin{align*} a _ { i , d } ( \underline { b } ) + a ' _ { i , d } ( \underline { b } ) = x _ { i , d } i , d , \end{align*}


\begin{align*} \int _ W | \Delta \rho | \leq \int _ W \left ( | C - \Delta \rho | + C \right ) = \int _ W ( 2 C - \Delta \rho ) = 2 C \ , _ n ( W ) + _ { n - 1 } ( \partial W ) . \end{align*}


\begin{align*} ( X _ 4 \cup \{ c _ 4 \} ) \cap { \cal N } = ( X _ 1 \cup \{ c _ 1 \} ) \cap { \cal N } = \{ c _ 2 , c _ 3 \} \cap { \cal N } = \emptyset . \end{align*}


\begin{align*} \partial _ \tau \gamma = \frac { c \bar { K } } { 4 \pi } \mathbf { b } _ { \gamma ( \tau ) } . \end{align*}


\begin{align*} c _ \varepsilon = I _ \varepsilon ( u _ \varepsilon ) \geq \inf \limits _ { w \in \mathcal { N } _ \varepsilon } I _ \varepsilon ( w ) . \end{align*}


\begin{align*} L ( p ) = \log F ( p , 8 / 3 ) , R ( p ) = \log \left ( e ^ { - p / 6 } \tilde G ( p , 2 ) \right ) \end{align*}


\begin{align*} 2 ^ { \star } : = \frac { n - 1 } { n - 2 } 2 \end{align*}


\begin{align*} \frac { \sum \limits _ { i = 1 } ^ { j + 1 } \chi _ i - r ( j + 1 ) } { \chi } < \sum \limits _ { i = 1 } ^ { j + 1 } w _ i < \frac { \sum \limits _ { i = 1 } ^ { j + 1 } \chi _ i - r j } { \chi } . \end{align*}


\begin{align*} r _ { b , k } ( 1 + \alpha _ { b , k } ) R _ b ( j _ b ) = 0 . \end{align*}


\begin{align*} \widehat { f } ( a , z ) = R _ k ^ { n - 1 } ( - \lambda , f ) e _ { k , \lambda } ^ { n - 1 } ( z , 0 ) \end{align*}


\begin{align*} \begin{aligned} I _ 6 & \lesssim \int _ { \Gamma _ h ^ \pm } \vert e ^ { \rho \cdot \mathbf x } \psi ( \mathbf x ) \vert \mathrm d \mathbf x \lesssim \tau ^ { - \frac { 1 7 } { 1 2 } } , \mbox { a s $ \tau \rightarrow \infty $ . } \end{aligned} \end{align*}


\begin{align*} \langle - \Delta z + \zeta z , p \rangle = \langle \nabla z , \nabla p \rangle + \langle \zeta z , p \rangle = \langle - \Delta p , z \rangle + \langle \zeta p , z \rangle = \langle \overline { y } - g , z \rangle , \ ; \ ; z \in Y . \end{align*}


\begin{align*} - \Delta u = \lambda u & D , \\ [ 1 m m ] u = 0 & \partial D . \end{align*}


\begin{align*} u = \arg \min _ { w \in L _ 2 ( 0 , T ; \mathcal { V } ) } \bigg ( | | \partial _ t w + \mathcal { L } w - F | | ^ 2 _ { L _ 2 ( 0 , T ; \mathcal { H } ) } + | | \alpha w + \beta \mathbf { n } \cdot \nabla w - g | | ^ 2 _ { L _ 2 ( 0 , T ; \partial \mathcal { H } ) } + | | w ( 0 , \cdot ) - u _ 0 | | ^ 2 _ { \mathcal { H } } \bigg ) , \end{align*}


\begin{align*} \nabla _ X \varphi = \frac { 1 } { 2 } \sqrt { c _ 1 } \ X _ 1 \cdot \nu _ 1 \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi \end{align*}


\begin{align*} v = ( 1 , x ) ^ T \mbox { o r } v = ( x , 1 ) ^ T . \end{align*}


\begin{align*} \left | \Pi _ { \mathrm { k e r } A ^ \perp } \frac { d ^ { \ell - 1 } } { d t ^ { \ell - 1 } } X _ t | _ { t = t _ 0 } \right | \ge \frac { \gamma ^ 3 } { C _ J } K ^ \ell . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { m } \gamma _ { k } f _ { k } ( x ) \ge 0 \ \ \forall \ x \in \mathbb { R } ^ { n } . \end{align*}


\begin{align*} \mathcal F = \big \{ L ( s , \pi \otimes \chi _ { 8 d } ) : d \big \} . \end{align*}


\begin{align*} \begin{aligned} \partial _ t u + \mathcal { L } u = & \ F , ( 0 , T ] \times \Omega \\ u ( 0 , \cdot ) = & \ u _ 0 , \ , \Omega \\ u = & \ g _ D ^ { } , ( 0 , T ] \times \Gamma _ D ^ { } , \\ \mathbf { n } \cdot \nabla u = & \ g _ N ^ { } , ( 0 , T ] \times \Gamma _ N ^ { } , \end{aligned} \end{align*}


\begin{align*} L F = - \sum _ { q \geq 1 } q J _ q ( F ) . \end{align*}


\begin{align*} ( \Lambda - 1 + \delta ) \sum _ { j = 1 } ^ k ( \Lambda + \varepsilon ) ^ { - j } > 1 , \end{align*}


\begin{align*} \sum _ { K \ \ L } \ \sum _ { \widetilde { K } \ \ \pi ^ { - 1 } ( K ) } ( e _ { \widetilde { K } } - 1 ) \langle [ \widetilde { K } ] , \alpha \rangle = 0 . \end{align*}


\begin{align*} g ( z ) = ( 2 \pi ) ^ { - n } \ , | \lambda | ^ n \ , \sum _ { k = 0 } ^ \infty g \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ( z ) . \end{align*}


\begin{align*} I _ f ( p , q ) & = \sum _ { i \in A _ { 1 + \kappa , \infty } } q _ i f \left ( \frac { p _ i } { q _ i } \right ) + \sum _ { i \in A _ { 1 , 1 + \kappa } } q _ i f \left ( \frac { p _ i } { q _ i } \right ) \\ & + \sum _ { i \in A _ { 1 / ( 1 + \kappa ) , 1 } } q _ i f \left ( \frac { p _ i } { q _ i } \right ) + \sum _ { i \in A _ { 0 , 1 / ( 1 + \kappa ) } } q _ i f \left ( \frac { p _ i } { q _ i } \right ) , \end{align*}


\begin{align*} I ( \mu _ m ) & = H ( \mu ^ m \ , | \ , \overline { \mu } ) = \int _ { [ 0 , 1 ] \times \R } \log \frac { d \mu ^ m } { d \overline { \mu } } \ , d \mu ^ m \\ & = \int _ { [ 0 , 1 ] \times \R } \log \frac { d \mu ^ m } { d \mu } \ , d \mu ^ m + \int _ { [ 0 , 1 ] \times \R } \log \frac { d \mu } { d \overline { \mu } } \ , d \mu ^ m . \end{align*}


\begin{align*} \begin{aligned} & \sum _ { \begin{subarray} { c } r _ 1 + \cdots + r _ { n + 1 } = r \\ r _ i \in \mathbb { Z } _ { \ge 0 } \end{subarray} } Z _ { I } ( 1 + r _ 1 , \{ 2 + r _ i \} _ { i = 2 } ^ { n + 1 } ; ( \alpha , \beta ) ) \\ = & \sum _ { \begin{subarray} { c } r _ 1 + \cdots + r _ { n } = r \\ r _ i \in \mathbb { Z } _ { \ge 0 } \end{subarray} } Z _ { I I } ( \{ 2 + r _ i \} _ { i = 1 } ^ { n - 1 } , 3 + r _ n ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \begin{array} { r l } & a _ { n + 1 } \in f ( V ( h ) ) \\ \Leftrightarrow & \exists a \in V ( h ) \ , : \ , a _ { n + 1 } = f ( a ) \\ \Leftrightarrow & \exists a \in \R ^ { n } \ , : \ , ( a , a _ { n + 1 } ) \in V ( h , x _ { n + 1 } - f ) \\ \Leftrightarrow & a _ { n + 1 } \in \hat \pi ( V ( h , x _ { n + 1 } - f ) ) \ , . \\ \end{array} \end{align*}


\begin{align*} & \mathcal { B } ^ { \rm n r e } _ { h , r } ( \sigma ) = \Big \{ F \in \mathcal { B } _ { h , r } : F ( x ) = \mathcal { T } _ { N } F ( x ) \Big \} , \\ & \mathcal { B } ^ { \rm r e } _ { h , r } ( \sigma ) = \Big \{ F \in \mathcal { B } _ { h , r } : F ( x ) = \mathcal { R } _ { N } F ( x ) \Big \} . \end{align*}


\begin{align*} \psi ^ - ( x , t ; k ) = \psi ^ + ( x , t ; k ) e ^ { - i k ^ 2 x \widehat { \sigma } _ 3 } S ( k ) , \end{align*}


\begin{align*} \beta ^ t _ { a , b } ( z ) \gamma ^ t _ { a , b } ( w ) = ( z - w ) ^ { - 1 } \end{align*}


\begin{align*} \begin{aligned} f _ n = K i n e t i c ( n ) + F l u i d ( n ) \ , , \end{aligned} \end{align*}


\begin{align*} w _ 1 ( \eta ) = \epsilon + \begin{cases} \frac { 3 } { 2 } - \frac { 3 \eta } { 2 } & ( 0 \leq \eta \leq \frac { 1 } { 2 } ) \\ 1 - \frac { \eta } { 2 } & ( \frac { 1 } { 2 } \leq \eta \leq 1 ) \\ \frac { 3 } { 2 } - \eta & ( 1 \leq \eta \leq \frac { 3 } { 2 } ) \\ 0 & ( \frac { 3 } { 2 } \leq \eta \leq 2 ) \end{cases} . \end{align*}


\begin{align*} \Delta t b _ k \mathcal { L } ( \tilde { u } ^ { [ n + k ] } ( x ) ) + a _ k \tilde { u } ^ { [ n + k ] } ( x ) = - \sum _ { j = 0 } ^ { k - 1 } a _ j \tilde { u } ^ { [ n + j ] } ( x ) + \Delta t b _ k f ( t _ { n + k } , x ) , \end{align*}


\begin{align*} v = 0 \partial A ( r _ { 1 } , r _ { 2 } ) . \end{align*}


\begin{align*} d _ 1 K _ { m , \epsilon } = \frac { \kappa _ m d _ 1 | q | ^ 2 } { ( | q | ^ 2 + \epsilon ) ^ { \kappa _ m + 1 } } , \end{align*}


\begin{align*} \frac { d } { d t } ( \mathcal { Q } ^ { r } ( t ) ) = \theta _ { 1 } \varrho _ { 0 } ^ { 1 - \sigma } ( r - 1 ) ( \mathcal { M } ^ { r } ( t ) ) ^ { \sigma + 1 } \ge \theta _ { 1 } \varrho _ { 0 } ^ { 1 - \sigma } ( r - 1 ) ( \mathcal { Q } ^ { r } ( t ) ) ^ { \sigma + 1 } , t \in [ \delta , t _ { 0 } ) . \end{align*}


\begin{align*} \lim _ { t \to 0 } j ( t ) = \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x > 0 . \end{align*}


\begin{align*} J _ 3 \leq \begin{cases} C t _ m ^ { \alpha - 1 - \frac { \alpha \zeta } { 2 \beta } } h ^ { \min \{ \frac { 1 } { 2 } + \frac { \alpha } { 2 \beta } ( r + \lambda - \kappa ) , 1 \} } , & \mbox { i f } \alpha + \gamma = 1 , \\ C t _ m ^ { \alpha - 1 - \frac { \alpha \zeta } { 2 \beta } } h ^ { \min \{ \frac { \alpha } { 2 \beta } \min \{ \kappa , r + \lambda \} + ( \gamma - \frac { 1 } { 2 } ) ^ { + } , 1 - \varepsilon \} } , & \mbox { i f } \alpha + \gamma \neq 1 , \end{cases} \end{align*}


\begin{align*} 0 & = \int _ \Sigma \Big \langle Q ( h , v ) , \big ( k ^ \intercal , H ' ( k ) \big ) \Big \rangle d \sigma \\ & = \int _ \Sigma \Big \langle \big ( v A + \bar u A ' ( h ) - \nu ( v ) g ^ \intercal , 2 v \big ) , \big ( - 2 \zeta A ' ( h ) , \zeta A \cdot A ' ( h ) \big ) \Big \rangle \ , d \sigma \\ & = - \int _ \Sigma 2 \bar u \zeta | A ' ( h ) | ^ 2 \ , d \sigma \end{align*}


\begin{align*} h _ { n + 1 } = r _ n h _ n + \frac { r _ n ( r _ n - 1 ) } { 2 } . \end{align*}


\begin{align*} h ( x , y ) \leq \frac { 1 } { 2 \pi } \ln \frac { 1 } { \max \{ | x - y | , d i s t ( x , \partial \Omega ) , d i s t ( y , \partial \Omega ) \} } , \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ H u = - ( K _ H ( x ) \nabla u ) = \frac { 1 } { \varepsilon ^ 2 } \left ( u - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ & x \in \Omega , \\ u = 0 , \ \ & x \in \partial \Omega , \end{cases} \end{align*}


\begin{align*} H ( q , r ) = \frac { \hat W _ r ( 0 ) + \hat W _ r ( 2 q ) - 2 \hat W _ r ( q ) } { \hat W _ r ( q ) } = u ( q r ) , \end{align*}


\begin{align*} & \mathbb { E } \widetilde { g } _ { \delta , L , \mu } ( x ) = g _ { \delta , L , \mu } ( x ) , \\ & \mathbb { E } \| \widetilde { g } _ { \delta , L , \mu } ( x ) - g _ { \delta , L , \mu } ( x ) \| ^ 2 \leq \frac { \sum _ { i = 1 } ^ n \sigma _ { f , i } ^ 2 } { n ^ 2 r } = \frac { \sigma _ f ^ 2 } { n r } . \end{align*}


\begin{align*} R _ { k } = \log _ 2 \left ( 1 + \gamma _ { k } \right ) \end{align*}


\begin{align*} & \log \left ( \prod _ { \mathrm { N } ( v ) \leq q ^ { m } } \det ( 1 - \rho ( \mathrm { F r o b } _ { v } ) \mathrm { N } ( v ) ^ { - s } ) ^ { - 1 } \right ) \\ & = \sum _ { 1 \leq k \deg ( v ) \leq m } \frac { \mathrm { t r } ( \rho ( \mathrm { F r o b } _ { v } ) ^ { k } ) } { k } q ^ { - k \deg ( v ) s } + \sum _ { k = 2 } ^ { \infty } \frac { 1 } { k } \sum _ { \frac { m } { k } < \deg ( v ) \leq m } \mathrm { t r } ( \rho ( \mathrm { F r o b } _ { v } ) ^ { k } ) q ^ { - k \deg ( v ) s } . \end{align*}


\begin{align*} I ( \varphi , \overline { \mathbf { R } } ) = \dd \int _ { \Omega } \left [ W _ { \rm { m p } } ( \overline { \mathbf { U } } ) + W _ { \rm { c u r v } } ( \boldsymbol { \alpha } ) \right ] d V { \to } \textrm { \ \ m i n . } { \rm w . r . t . } ( \varphi , \overline { \mathbf { R } } ) \ , , \end{align*}


\begin{align*} \int _ { W } g _ { u _ j } ^ p \ , d \mu & \le \int _ { U } g _ { u _ j } ^ p \eta ^ p \ , d \mu = \lim _ { k \to \infty } \int _ { V _ k } g _ { u _ j } ^ p \eta _ k ^ p \ , d \mu \\ & \le \lim _ { k \to \infty } \int _ { V _ k } ( 2 ^ p \eta ^ p _ k g _ f ^ p + ( 4 p M ) ^ p g _ { \eta _ k } ^ p ) \ , d \mu \le \int _ { U } ( 2 ^ p g _ f ^ p + ( 4 p M ) ^ p g _ { \eta } ^ p ) \ , d \mu < \infty , \end{align*}


\begin{align*} \chi ( S ' _ b ) + \chi ( F ' _ b ) + \chi ( h ( F ' _ b ) ) & \ge \chi ( S ' _ b ) + \chi ( F ' _ b ) + \chi ( h ( F ' _ b ) ) + \chi ( \Sigma ) \\ & = \chi ( S ) + \chi ( F ) + \chi ( h ( F ) ) . \end{align*}


\begin{align*} ( r + \nu ) V ( s , i ) = H \left ( s , i , p , q \right ) . \end{align*}


\begin{align*} M ( \lambda _ 1 / \lambda _ 2 ) = M ( \lambda _ 2 / \lambda _ 1 ) = | \lambda _ 1 \lambda _ 2 | \max ( 1 , | \lambda _ 1 / \lambda _ 2 | ) \max ( 1 , | \lambda _ 2 / \lambda _ 1 | ) = | \lambda _ 1 | ^ 2 . \end{align*}


\begin{align*} \langle f , g \rangle = f _ 1 g _ 4 - \frac { 1 } { 3 } f _ 2 g _ 3 + \frac { 1 } { 3 } f _ 3 g _ 2 - f _ 4 g _ 1 . \end{align*}


\begin{align*} K _ { p , d } = 2 ^ { - p } \pi ^ { - d / 2 } \frac { \Gamma \left ( \frac { d - p } { 2 } \right ) } { \Gamma \left ( \frac { p } { 2 } \right ) } . \end{align*}


\begin{align*} \begin{aligned} \| f _ { j } \| _ { C ^ { \alpha } ( \mathcal C ^ h ) } & \leq \| q \| _ { C ^ { \alpha } ( \mathcal C ^ h ) } \sup _ { \mathcal C ^ h } \vert v _ { j } \vert + \| v _ { j } \| _ { C ^ { \alpha } ( \mathcal C ^ h ) } \sup _ { \mathcal C ^ h } \vert q - 1 \vert . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} x _ n - y _ n = \ , & \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( y ' ) ( x _ \beta - y _ \beta ) + O ( | x ' - y ' | ^ 2 ) \\ \geq \ , & \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( y ' ) ( x _ \beta - y _ \beta ) - \kappa | x ' - y ' | ^ 2 \end{aligned} \end{align*}


\begin{align*} [ V ] & = \frac { [ \mathbb C _ x ] } { D } \\ D & = \prod _ { n = 1 } ^ \infty ( 1 - e ^ { - n \delta } ) ^ { r } \prod _ { n = 1 } ^ \infty \prod _ { \alpha \in \Delta _ + ^ 0 } ( 1 - e ^ { - \alpha } e ^ { ( 1 - n ) \delta } ) ( 1 - e ^ { \alpha } e ^ { - n \delta } ) . \end{align*}


\begin{align*} \begin{cases} \Delta _ \Psi u = f ( u ) & A ( r _ { 0 } , + \infty ) \\ u \equiv c _ { 0 } & \{ r _ { 0 } \} \times N . \end{cases} \end{align*}


\begin{align*} \dot y = a y '' + M y , y ( 0 , x ) = y _ 0 ( x ) , \end{align*}


\begin{align*} - ( K _ H ( x ) \nabla u _ { \bar { x } } ) = \frac { 1 } { \varepsilon ^ 2 } \left ( u _ { \bar { x } } - q _ { \bar { x } } \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ \ \ \ \Omega _ { \bar { x } } . \end{align*}


\begin{align*} s _ p s _ q f = s _ p f \qquad t _ p t _ q f = t _ p f \end{align*}


\begin{align*} \int _ 0 ^ T \left | \frac { \zeta ' } { \zeta } ( \sigma + i t ) \right | ^ 2 d t = O \left ( \frac { T } { ( \sigma - 1 / 2 ) ^ 2 } \right ) . \end{align*}


\begin{align*} t _ \theta : = \begin{bmatrix} \cos \theta \dfrac { 1 } { \sqrt { k } } \mathbb { 1 } _ k \\ \hline \sin \theta \dfrac { 1 } { \sqrt { l } } \mathbb { 1 } _ l \end{bmatrix} \mbox { a n d } U _ \theta : = \begin{bmatrix} \ ; \cos \theta \sqrt { \dfrac { m } { k } } V & \vline & \sin \theta \sqrt { \dfrac { m } { l } } W \ ; \end{bmatrix} , \end{align*}


\begin{align*} \Delta ^ 2 u = \lambda u & D , \\ [ 1 m m ] u = \frac { \partial u } { \partial \nu } = 0 & \partial D . \end{align*}


\begin{align*} G = i \ \overline { v } \ \widehat { v } . \end{align*}


\begin{align*} \widetilde { \varphi } ^ { H } _ { n , p } & = \varphi ^ { H } _ { n , p } - \psi _ { n , p } \ , , \\ \widetilde { \varphi } ^ { V } _ { n , p } & = \varphi ^ { V } _ { n , p } - \rho _ { n , p } \ , . \end{align*}


\begin{align*} ( d _ i y ) ( [ \mathbf { a } ] \otimes [ \mathbf { b } ] ) = y ( d ^ i [ \mathbf { a } ] \otimes [ \mathbf { b } ] ) = 0 , \end{align*}


\begin{align*} & \mathbb { E } \Vert Y _ n ^ M ( t + \delta ) - Y _ n ^ M ( t ) \Vert _ { H } ^ 2 \\ = & \mathbb { E } \int _ { t \wedge \tau _ n ^ M } ^ { ( t + \delta ) \wedge \tau _ n ^ M } 2 \langle A ( r , Y _ n ( r ) ) , Y _ n ( r ) - Y _ n ( t \wedge \tau _ n ^ M ) \rangle d r \\ & + \mathbb { E } \int _ { t \wedge \tau _ n ^ M } ^ { ( t + \delta ) \wedge \tau _ n ^ M } \Vert P _ n B ( r , Y _ n ( r ) ) Q _ n \Vert _ { L _ 2 } ^ 2 d r . \end{align*}


\begin{align*} C _ 1 = \sum _ { X \in S ( 0 ) } \Theta _ { \theta } \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = - q ^ { 8 } ( 1 - q ) ^ 3 ( 1 - q ^ 2 ) . \end{align*}


\begin{align*} f _ { y , \mathbf { X } } ( \beta ) : = \sum _ { i = 1 } ^ p V ( \beta _ i ) - \frac { 1 } { 2 \sigma ^ 2 } \left | y - \mathbf { X } \beta \right | ^ 2 . \end{align*}


\begin{align*} \omega ( \vec { x } ) = \frac { c } { \sqrt { 1 + \norm { \vec { a } } ^ 2 } + \vec { a } \cdot \vec { x } } \end{align*}


\begin{align*} \min \{ c _ { d , 2 } ( q ) , c _ { d , \infty } ( q ) \} = \begin{cases} c _ { d , 2 } ( q ) , & - ( d - 1 ) < q \leq q _ d ^ * , \\ c _ { d , \infty } ( q ) , & q _ d ^ * \leq q \leq 2 , \end{cases} \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n \binom n k \frac { L _ { j k } } { L _ j ^ k } B _ { n - k } = 0 , \qquad \mbox { $ n $ o d d } . \end{align*}


\begin{align*} \langle C , C \rangle = \sum _ { i = 1 } ^ s | c _ { i } | ^ { 2 } \langle g _ { i } , g _ { i } \rangle . \end{align*}


\begin{align*} \langle - \Delta z + \zeta z , p \rangle = \langle \nabla z , \nabla p \rangle + \langle \zeta z , p \rangle = \langle - \Delta p , z \rangle + \langle \zeta p , z \rangle = \langle \overline { y } - g , z \rangle , \ ; \ ; z \in Y . \end{align*}


\begin{align*} _ { \rm r e f } [ V _ \lambda ] = q ^ { - \frac { c } { 2 4 } } q ^ { \frac { ( \lambda | \lambda + 2 \rho ) } { 2 ( k + h ^ \vee ) } - \lambda ( x + y ) } \ \prod _ { i = 1 } ^ L X _ i \ \prod _ { 1 \leq i < j \leq L } X _ { i , j } \end{align*}


\begin{align*} [ V _ \lambda ] = q ^ { - \frac { c } { 2 4 } } q ^ { \frac { ( \lambda | \lambda + 2 \rho ) } { 2 ( k + h ^ \vee ) } - \lambda ( x ) } \ \prod _ { n = 1 } ^ \infty \prod _ { \alpha \in \Delta _ { } ^ { 0 } } ( 1 - q ^ { n - \epsilon ( \alpha ) } ) ^ { - 1 } \prod _ { \alpha \in \Delta _ { } ^ { + , \sharp } } ( 1 - e ^ { \alpha ( h ) } q ^ { n - \epsilon ( \alpha ) } ) ^ { - 1 } ( 1 - e ^ { - \alpha ( h ) } q ^ { n + \epsilon ( \alpha ) - 1 } ) ^ { - 1 } , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \partial _ t \chi + \gamma \nabla \cdot ( \chi \nabla \Phi ) , & \Omega \times ( 0 , T ] , \\ & \partial _ t \Phi + \frac { \gamma } { 2 } \lvert \nabla \Phi \rvert ^ 2 + \gamma \chi = 0 , & \Omega \times ( 0 , T ] , \end{aligned} \right . \end{align*}


\begin{align*} \chi ( A ) = \left ( \begin{array} { c c } A _ 1 & A _ 2 \\ \overline { A _ 2 } & \overline { A _ 1 } \\ \end{array} \right ) . \end{align*}


\begin{align*} \gamma _ { t - 1 } \alpha _ t ^ 2 = ( 1 - \alpha _ t ) \alpha _ { t - 1 } ^ 2 \gamma _ t + \mu \alpha _ t \gamma _ t \gamma _ { t - 1 } , \end{align*}


\begin{align*} F ( x ) = \frac { 2 } { \pi } \arcsin \left \{ \exp \left ( - \frac { \theta } { 2 x ^ 2 } \right ) \right \} . \end{align*}


\begin{align*} \begin{cases} - ( K _ H ( x ) \nabla v ) = \frac { 1 } { \varepsilon ^ 2 } \left ( v - q _ { \hat { x } } \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ \ \ & \ \Omega _ { \hat { x } } . \\ v = 0 , \ \ \ \ & \ \partial \Omega _ { \hat { x } } . \end{cases} \end{align*}


\begin{align*} c _ { \N } : = \mathop { \inf } \limits _ { ( u , v ) \in \N } I ( u , v ) . \end{align*}


\begin{align*} ( d / d t ) v _ k + i \lambda _ k v _ k = 0 , \ ; \ ; 1 \leqslant k \leqslant n . \end{align*}


\begin{align*} L _ X g ( \nu , \cdot ) = h ( \nu , \cdot ) \mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \Sigma . \end{align*}


\begin{align*} y _ { b e n } ( R ) = y _ { n b } ( R ) , a y ' _ { b e n } ( R ) = b y ' _ { n b } ( R ) . \end{align*}


\begin{align*} \beta _ i + \bar { \beta _ j } = 1 \ , \ , { \rm a n d } \ , \ , \beta _ u + \bar { \beta } _ 4 = 1 . \end{align*}


\begin{align*} P _ { n , k } ( z ) = \sum _ { i = 0 } ^ { k + d _ n } \alpha _ { n , k , i } \binom { z } { i } \end{align*}


\begin{align*} A = \mathcal { N } ( A ) - \mathcal { R } ( A ) . \end{align*}


\begin{align*} \Phi _ { t } + u \cdot \nabla \Phi & = 0 \textrm { i n } \ \mathbb { R } ^ { 3 } \times ( 0 , T ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ k a _ i \neq 1 \ , \ , { \rm a n d } \ , \ , \sum _ { i \in \Omega _ t ^ o } a _ i - \sum _ { i \in \Omega _ t ^ e } a _ i \neq 1 . \end{align*}


\begin{align*} Z ( v , s ) & = \Psi ^ q + v + \psi _ v ( 0 , p _ 0 ) v + s \psi _ p ( 0 , p _ 0 ) p ' ( 0 ) + \mathcal O ( \norm { ( v , s ) } ^ 2 ) \\ & = \Psi ^ q + v + \mathcal O ( \norm { ( v , s ) } ^ 2 ) , \end{align*}


\begin{align*} T ( \lambda u ) = u . \end{align*}


\begin{align*} \int _ { \C ^ n } | g ( z ) | ^ 2 d z = ( 2 \pi ) ^ { - 2 n } \ , | \lambda | ^ { 2 n } \ , \sum _ { k = 0 } ^ \infty \int _ { \C ^ n } | g \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ( z ) | ^ 2 d z . \end{align*}


\begin{align*} \partial _ t \gamma = \frac { c } { 4 \pi } | \ln \varepsilon | ( \partial _ s \gamma \times \partial _ { s s } \gamma ) = \frac { c \bar { K } } { 4 \pi } | \ln \varepsilon | \mathbf { b } _ { \gamma ( t ) } , \end{align*}


\begin{align*} \delta _ { m , n } & = \frac { m } { n } \bigg ( 1 + \sqrt { n - 1 } \sqrt { \frac { n - m } { m } } \bigg ) < \frac { m } { n } \sqrt { 1 + n - 1 } \sqrt { 1 + \frac { n - m } { m } } = \sqrt { m } . \end{align*}


\begin{align*} \left | \dfrac { \partial a _ { 0 , \delta } ( x ) } { \partial x } \right | _ { x = x _ a } \le 2 \int _ 0 ^ 1 \left | \left . \dfrac { \partial h ( x ) } { \partial x } \right | _ { x = x _ a + \delta u ( t ) } \right | \ , d t \le 2 L _ r . \end{align*}


\begin{align*} \{ w ' \in \Omega : O ( \chi _ { n , m } ' ( w ' , M , \cdot ) ) = O ( \chi _ { n , m } ' ( w , M , \cdot ) ) \} \end{align*}


\begin{align*} - \int _ 0 ^ \infty u \phi = \int _ 0 ^ \infty f \phi ' , \qquad \phi \in C _ 0 ^ \infty ( 0 , \infty ) , \end{align*}


\begin{align*} u _ { g , n } = \sum _ { i = 0 } ^ n ( - 2 ) ^ i w _ { g , n - i } , \end{align*}


\begin{align*} x = q _ i ' - p _ i \end{align*}


\begin{align*} \begin{cases} \partial _ t w + \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \varphi | _ { \partial \Omega } = 0 . \end{cases} \end{align*}


\begin{align*} | | \mathcal { B } _ q ( \psi _ m ) | | _ { Q M L _ q ( \mathbb { H } ) } = 1 = | | \psi _ m | | _ { L ^ 2 ( \mathbb { R , , \mathbb { H } } ) } . \end{align*}


\begin{align*} \tilde G ( p , s ) = s ^ { - p / 2 } 2 ^ { 3 p / 2 - 1 } \Gamma ( p / 2 ) D ( p ) \end{align*}


\begin{align*} \Delta t b _ k \mathcal { L } ( \tilde { u } ^ { [ n + k ] } ( x ) ) + a _ k \tilde { u } ^ { [ n + k ] } ( x ) = - \sum _ { j = 0 } ^ { k - 1 } a _ j \tilde { u } ^ { [ n + j ] } ( x ) + \Delta t b _ k f ( t _ { n + k } , x ) , \end{align*}


\begin{align*} f ( r ) \frac { d W _ + ( r ) } { d r } = W _ + ( r ) W _ - ( r ) + E _ 1 - E _ 0 . \end{align*}


\begin{align*} - n ^ p - \sum _ { K = 1 } ^ { p } \binom { p } { K } \ , B _ K \ , n ^ { p - K } . \end{align*}


\begin{align*} \Gamma _ { i k } ^ k = \left ( \log \det g ^ { 1 / 2 } \right ) _ { , i } = \frac { 1 } { 2 } ( - v _ i + u _ { , i q } \xi ^ q ) . \end{align*}


\begin{align*} s \cdot ( x , y , z , \zeta ) = ( a x + b y , c x + d y , ( \det s ) z , ( \det s ) \zeta ) . \end{align*}


\begin{align*} & \phi _ i ( v w ) = v w + ( v w | e _ i ) y = v w + ( v | w e _ i ) y \\ & = v w + ( v | w ) y = ( v + ( v | e _ i ) y ) * ( w + ( w | e _ i ) y ) = \phi _ i ( v ) * \phi _ i ( w ) . \end{align*}


\begin{align*} X + \xi \in E = E ( \mathfrak { g } ) \coloneqq \mathfrak { g } \oplus \mathfrak { g } ^ * \end{align*}


\begin{align*} d \geq 2 : 0 < s < s _ s \leq - \frac d 2 , d = 1 : s < \frac 1 6 s \ne - \frac 1 2 , \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \theta ) = E _ P \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] , \end{align*}


\begin{align*} ( d _ { i } y ) ( [ \mathbf { a } ] \otimes [ \mathbf { b } ] ) = 0 , \end{align*}


\begin{align*} W ( r ) = \frac { \xi } { r } f + \eta + \zeta \frac { r } { f } + \frac { \sigma } { f ^ 2 } , \xi \le 0 , \sigma > 0 , \end{align*}


\begin{align*} [ l , r ] \otimes [ l ' , r ' ] = \bigoplus \limits _ { \nu = \vert l - l ' \vert \mbox { s t e p } 2 } ^ { \mathrm { m i n \left ( l + l ' , 2 d - 4 - l - l ' \right ) } } [ \nu , r + r ' ] . \end{align*}


\begin{align*} t ( v ' ) t ( w '' ) = t ( w ' ) t ( v '' ) \end{align*}


\begin{align*} R _ 2 = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 2 } } \Theta \begin{bmatrix} m _ 1 ' & X \\ 0 & m _ 2 ' \end{bmatrix} \overline { \psi _ A ( { m _ 1 ' } ^ { - 1 } X } ) = \theta ( a ) q ^ 8 ( 1 - q ) . \end{align*}


\begin{align*} d = 2 n + 1 = 4 l + 3 \equiv 3 \mod 4 \end{align*}


\begin{align*} \lim _ { n \to \infty } \| p _ n x \| _ { X _ n } = \| x \| _ X , \ , x \in X \lim _ { n \to \infty } \| q _ n y \| _ { Y _ n } = \| y \| _ Y , \ , y \in Y . \end{align*}


\begin{align*} c _ { k + 1 } = \sum _ { i = 1 } ^ { t - 1 } c _ i a _ i + c _ { k } \sum _ { i \in \Omega _ t ^ e } a _ i + c _ { k + 1 } \sum _ { i \in \Omega _ t ^ o } a _ i , \end{align*}


\begin{align*} \nabla _ i A _ { j k } = \nabla _ j A _ { i k } . \end{align*}


\begin{align*} \alpha ( n + 4 ) = s _ 1 s _ 2 s _ 1 \cdots ( \alpha _ { \langle n + 4 \rangle } ) , \alpha ( - ( n + 1 ) ) = s _ 2 s _ 1 s _ 2 \cdots ( \alpha _ { \langle - ( n + 1 ) \rangle } ) , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { m } \gamma _ { k } ( f _ { k } ( x ) + C _ { k } ) \ge 0 \ \ \forall \ x \in \mathbb { R } ^ { n } , \ \ C _ k \in \mathbb { R } _ + . \end{align*}


\begin{align*} A ^ * ( y ) = \left [ \hat { A } \circ \hat { \sigma } ^ { - 1 } + W \circ \hat { \sigma } ^ { - 1 } - W \right ] ( y | x ) , \end{align*}


\begin{align*} \mathcal { N } _ \varepsilon = & \{ u \in \mathcal { H } ( \Omega ) \setminus \{ 0 \} \mid \langle I ' _ \varepsilon ( u ) , u \rangle = 0 \} \\ = & \left \{ u \in \mathcal { H } ( \Omega ) \setminus \{ 0 \} \mid \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x = \frac { 1 } { \varepsilon ^ 2 } \int _ { \Omega } \left ( u - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + u d x \right \} . \end{align*}


\begin{align*} M _ n ^ { \psi } ( Q ) / n = T _ { W _ { n J } , \psi } ( \mu _ n ( Q ) ) - I ( \mu _ n ( Q ) ) . \end{align*}


\begin{align*} [ X + \xi , Y + \eta ] _ H = [ X , Y ] - \mathrm { a d } _ X ^ * \eta - \iota _ Y d \xi + H ( X , Y , \cdot ) \in \mathfrak { g } \oplus \mathfrak { g } ^ * , \end{align*}


\begin{align*} \frac { \mathcal { H } ^ 2 _ { m a x } } { \mathcal { H } ^ 2 _ { m i n } } = \frac { \operatorname { R } _ { m a x } } { \operatorname { R } _ { m i n } } & \to 1 , \\ \operatorname { d i a m } ( \Sigma _ t ) & \to 0 . \end{align*}


\begin{align*} x ^ { t + 1 } - y ^ t = \ & ( 1 - \alpha _ t ) x ^ t + \alpha _ t z ^ { t + 1 } - y ^ t = ( 1 - \alpha _ t ) ( x ^ t - y ^ t ) + \alpha _ t z ^ { t + 1 } - \alpha _ t y ^ t \\ \overset { \eqref { l 2 - e 2 } } { = } \ & \alpha _ t ( 1 - \beta _ t ) ( y ^ t - z ^ t ) + \alpha _ t z ^ { t + 1 } - \alpha _ t y ^ t = \alpha _ t \left ( z ^ { t + 1 } - \beta _ t y ^ t - ( 1 - \beta _ t ) z ^ t \right ) . \end{align*}


\begin{align*} \left | \dfrac { \gamma \alpha \omega - \gamma \omega } { - \gamma \alpha \omega } \right | = \dfrac { | \alpha \omega - \omega | } { | - \gamma \alpha \omega | | c \omega + d | | c \alpha \omega + d | } = \dfrac { 2 ^ { - 1 } } { | a \alpha \omega + b | | c \omega + d | } . \end{align*}


\begin{align*} \prod ^ e _ { \substack { i = 1 \\ ( i , e ) = 1 } } \Phi _ { n , e } ^ { ( i ) } ( x , y ) = \Phi _ { n e } ( x , y ) . \end{align*}


\begin{align*} W _ r ( x ) : = \begin{cases} 1 & \min ( x , 1 - x ) \le r \\ 0 & \end{cases} . \end{align*}


\begin{align*} \mu _ { ( \boldsymbol { x } , \boldsymbol { \phi } ( t , \boldsymbol { x } ) ) } ^ { N , i } + \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } h \Big ( \mu _ { ( \boldsymbol { x } , \boldsymbol { \phi } ( t , \boldsymbol { x } ) ) } ^ { N , j } \Big ) = - R ^ { - 1 } B \Big [ P _ t \nu ^ { N , i } _ { \boldsymbol { x } } + \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } \Phi ( t , \nu ^ { N , j } _ { \boldsymbol { x } } ) \Big ] \end{align*}


\begin{align*} \Delta _ g & = \frac { \partial ^ 2 } { \partial x _ n ^ 2 } + \Gamma ^ \alpha _ { n \alpha } \frac { \partial } { \partial x _ n } + g ^ { \alpha \beta } \frac { \partial ^ 2 } { \partial x _ \alpha \partial x _ \beta } + \biggl ( g ^ { \alpha \beta } \Gamma ^ \gamma _ { \gamma \alpha } + \frac { \partial g ^ { \alpha \beta } } { \partial x _ \alpha } \biggr ) \frac { \partial } { \partial x _ \beta } . \end{align*}


\begin{align*} \int _ T \gamma = \sum _ { \widetilde { K } \ \ \pi ^ { - 1 } ( K ) } ( e _ { \widetilde { K } } - 1 ) \langle [ \widetilde { K } ] , \alpha \rangle . \end{align*}


\begin{align*} \begin{aligned} N _ 1 [ K [ \hat { f } ] ] < k N _ 2 [ \hat { f } ] \ , , \ N _ 3 ^ { ( r + 1 / 2 ) } [ K [ \hat { f } ] ] < k N _ 3 ^ { ( r ) } [ \hat { f } ] \end{aligned} \end{align*}


\begin{align*} a = - \frac { 1 } { \mu } \left ( \partial _ y \mu \ d x - \partial _ x \mu \ d y \right ) = ( \frac { 1 } { \mu } d \mu - \frac { 2 } { \mu } \mu _ z d z ) I \end{align*}


\begin{align*} \begin{aligned} \dot { x } ( t ) & - \dot { z } ( t ) = \\ & - \gamma \left ( h ( x ( t ) + \delta u ( t ) ) u ( t ) - \dfrac { b _ { 1 , \delta } ( x _ a ( t ) ) } { 2 } \right ) \\ & - \left . \gamma ^ 2 \dfrac { \partial \epsilon ( x , t ) } { \partial x } \right | _ { x = x _ a ( t ) } \dfrac { b _ { 1 , \delta } ( x _ a ( t ) ) } { 2 } \\ & + \gamma \left . \dfrac { \partial \epsilon ( x , t ) } { \partial t } \right | _ { x = x _ a ( t ) } . \end{aligned} \end{align*}


\begin{align*} \varphi _ 1 ( x ) = u ( x ) + t v _ 1 ( x ) , \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) - x _ a ( t ) \| \le & \ , \| x ( 0 ) - z ( 0 ) \| e ^ { \gamma L _ \mathcal { D } t } \\ & \ , + \gamma 3 M _ \mathcal { D } T \left ( e ^ { \gamma L _ \mathcal { D } t } - 1 \right ) \\ & \ , + \gamma 2 M _ \mathcal { D } T . \end{aligned} \end{align*}


\begin{align*} \mathcal { U } _ { \lambda } ( \tau ) = : \Theta _ { \lambda } ( t ) + \frac { 1 } { 2 } \varrho _ { 0 } \lambda ^ { \beta - 1 } \int _ { \tau } ^ { t } \Theta _ { \lambda } ( s ) \ d s \leq \Theta _ { \lambda } ( \tau ) , 0 \leq t < \tau \leq t _ { 0 } < T _ { \mathrm { g e l } } . \end{align*}


\begin{align*} r ^ \vee ( \kappa + h ^ \vee ) ( \check { \kappa } + \check { h } ^ \vee ) = 1 , r ^ \vee ( \check { \kappa } + n + \check { h } ^ \vee ) ( \varkappa + h ^ \vee ) = 1 . \end{align*}


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \eta = \varepsilon ( - z ^ { - 1 } ) \dot { s } h ( - z ) \varepsilon ( - z ^ { - 1 } ) \varepsilon ( 1 ) \eta = \varepsilon ( - z ^ { - 1 } ) \dot { s } \varepsilon ( z ^ 2 - z ) \eta . \end{align*}


\begin{align*} \Lambda _ C : = \{ ( x , y ) \in \Lambda : x \in C \} \end{align*}


\begin{align*} \mu _ { \rm a } ( x [ n ] ) = \alpha x [ n ] + w _ { \rm d } [ n ] , \end{align*}


\begin{align*} ( u , v ) = \langle u , v \rangle , \ \ \ \forall \ , u \in H , \ \ \forall \ , v \in V . \end{align*}


\begin{align*} \varphi ^ { \pm } ( x , t ; k ) = I + k \int _ { \pm \infty } ^ { x } e ^ { - 2 i k ^ 2 ( x - y ) \widehat { \sigma } _ 3 } P _ y ( y ) \varphi ^ { \pm } ( y , t ; k ) d y . \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 + } \sup _ { n \in \mathbb { N } } \mathbb { E } \int _ 0 ^ { T - \delta } \Vert Y _ n ^ M ( t + \delta ) - Y _ n ^ M ( t ) \Vert _ { H } ^ { \alpha } d t = 0 , \end{align*}


\begin{align*} u = \left ( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right ) , \end{align*}


\begin{align*} \textup { S t o C } : = \min \{ \alpha + \gamma - \frac { 1 } { 2 } - \frac { \alpha } { 2 \beta } ( \kappa - r ) , 1 - \varepsilon \} \leq \min \Big \{ \alpha + \frac { 1 } { 2 } , 1 - \varepsilon \Big \} . \end{align*}


\begin{align*} S = \left ( \begin{array} { c c } 1 / 2 + \alpha & \beta \\ \beta & 1 / 2 - \alpha \end{array} \right ) \end{align*}


\begin{align*} \prod _ { 1 \leq j \leq N , n _ j \mid n _ 1 } \vert \eta _ { j , ( 1 ) } \vert _ F = \frac { 1 } { p ^ { D } } \end{align*}


\begin{align*} \begin{aligned} ( \Delta u ) ^ { k } \wedge \beta _ n ^ { n - k } \geq 0 , k = 1 , 2 , \dots , m . \end{aligned} \end{align*}


\begin{align*} \Delta = - \frac { 1 } { \sqrt { ( g ) } } \partial _ { a } \bigl ( \sqrt { ( g ) } g ^ { a b } \partial _ { b } \bigr ) , \end{align*}


\begin{align*} R = \max _ { q \in D } { \rm d i s t } ( q , p ) \end{align*}


\begin{align*} [ v _ a , v _ b ] = \varepsilon _ c L v _ c , \quad \forall \quad \mbox { c y c l i c } ( a , b , c ) \in \mathfrak { S } _ 3 . \end{align*}


\begin{align*} \mathcal { P } _ s = \{ V _ 1 \subset V _ 2 \subset \ldots V _ { l - 1 } \subset V _ { l + 1 } \subset \ldots \subset V _ { n } = \mathbb { C } ^ n ; \dim _ { \mathbb { C } } ( V _ i ) = i \} . \end{align*}


\begin{align*} 0 \leq \beta _ t = \mu \gamma _ t \alpha _ t ^ { - 1 } \leq \sqrt { \mu \gamma _ t } \leq 1 . \end{align*}


\begin{align*} \Delta ( \gamma ) : = \{ \pmb { \theta } : \ , \max _ { i , j } | \theta _ i - \theta _ j | \le \gamma \} . \end{align*}


\begin{align*} \begin{array} { l l l } f _ L ^ M ( \mathcal { M } _ { \mathbb { C } , \mathsf { S } } ( a ) ) \circ u & = & G ( L _ a , R _ a ) ( f _ L ( s ) \circ t ) \\ & = & f _ L ( L _ a \circ s ) \circ t \\ & = & f _ L ( a \bullet _ { \mu _ S } s ) \circ t \\ & = & f _ L ( a ) \circ f _ L ( s ) \circ t \\ & = & f _ L ( a ) \circ u . \end{array} \end{align*}


\begin{align*} v _ \epsilon { '' } ( t ) = \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ( t ) , \end{align*}


\begin{align*} b ( n , d , k ) = \max \{ \vartheta ( G ) : \ : \} \end{align*}


\begin{align*} \eta ( x ) = \sum _ { k = 1 } ^ \infty a _ k \sin ( 2 \pi k x ) + b _ k ( 1 - \cos ( 2 \pi k x ) ) , \end{align*}


\begin{align*} c _ { \psi } = \sin ^ 2 ( \alpha ) \int _ 0 ^ { 2 \pi } \frac { \psi _ 0 \cos \theta + \psi _ 1 \sin \theta } { \left ( \dfrac { \cos \theta } { r _ 0 ^ 2 } - \dfrac { \sin \theta \cos \alpha } { r _ 0 r _ 1 } \right ) \psi _ 0 + \left ( \dfrac { \sin \theta } { r ^ 2 _ 1 } - \dfrac { \cos \theta \cos \alpha } { r _ 0 r _ 1 } \right ) \psi _ 1 } d \theta \end{align*}


\begin{align*} t = r _ 0 + r _ 1 q + \cdots + r _ { n - 1 } q ^ { n - 1 } , \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty \left | \frac { \theta _ 0 ( e ^ { \kappa + \tau } ) - \theta _ 0 ( e ^ \tau ) - \delta e ^ \tau } { e ^ { \sigma \tau } } \right | ^ 2 d \tau = \frac { 1 } { 2 \pi } \int _ { - \infty } ^ \infty \left | \frac { e ^ { \kappa s } - 1 } { s } \right | ^ 2 \left | \frac { \zeta ' } { \zeta } ( s ) + g ( s ) \right | ^ 2 d t . \end{align*}


\begin{align*} \mathbf d = \frac { \mathbf z - s \omega _ 1 } { \vert \mathbf z - s \omega _ 1 \vert } = \frac { \mathbf z - s \omega _ 1 } { \sqrt { 1 + s ^ 2 } } , \end{align*}


\begin{align*} x \cdot x = x \cdot 1 = 1 , 1 \cdot x = x \end{align*}


\begin{align*} D = \{ c \} \cup \{ d \in C _ i \mid \varphi _ { \sigma , s } ^ L ( d ) = a \} \end{align*}


\begin{align*} ( \phi x ) _ \epsilon = x _ { \phi ( \epsilon ) } \end{align*}


\begin{align*} & m = \int _ { - \infty } ^ x e ^ { 2 i z ( x - y ) } w ( y ) d y , w ( x ) = - \partial _ x ( u _ x e ^ { i c _ - ( x ) } ) , \\ & n = \int _ { - \infty } ^ x e ^ { - 2 i z ( x - y ) } u _ y ( y ) e ^ { i c _ - ( x ) } d y . \end{align*}


\begin{align*} L _ X g ( V , \cdot ) = h ( V , \cdot ) \mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \Sigma . \end{align*}


\begin{align*} c _ s ^ 2 \partial _ { z 1 } T = - \frac { 1 } { \Delta t } { \varsigma } _ 3 { m } _ 3 ^ { ( 1 ) } , \end{align*}


\begin{align*} x _ n - y _ n \geq ( 2 - \epsilon ) b _ { n - 1 } \mbox { o n } \partial \omega \cap \{ x _ n = 2 b _ { n - 1 } \} . \end{align*}


\begin{align*} \int _ { \Omega } A \cdot B \dd x = \sum _ { j } \frac { 1 } { \lambda _ j } c _ j ^ { 2 } = \sum _ { j } \frac { 1 } { f _ j ^ { + } } c _ j ^ { 2 } + \sum _ { j } \frac { 1 } { f _ j ^ { - } } c _ j ^ { 2 } . \end{align*}


\begin{align*} J ( z ) = & \frac { \sqrt \theta } { B ( a , b ) } \sum _ { n = 0 } ^ \infty \frac { ( - 1 ) ^ n } { \Gamma ( b - n ) n ! } \int _ { \theta / z ^ 2 } ^ { \infty } y ^ { - 1 / 2 } \exp \{ - ( a + n ) y \} \mathrm { d } y \\ = & \sqrt { \pi \theta } \frac { \Gamma ( b ) } { B ( a , b ) } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ n } { \Gamma ( b - n ) n ! } \frac { 1 } { \sqrt { a + n } } \operatorname { e r f c } ( \sqrt { \theta / z ^ 2 } \sqrt { a + n } ) , \end{align*}


\begin{align*} \mathcal { H } _ m ( u ) ( q _ 0 ) = \lambda _ { n } ( q _ 0 ) \sum \limits _ { 1 \leq j _ 2 < \dots < j _ m \leq n - 1 } \lambda _ { j _ 1 } ( q _ 0 ) \cdots \lambda _ { j _ { m - 1 } } ( q _ 0 ) + \sum \limits _ { 1 \leq j _ 1 < \dots < j _ m \leq n - 1 } \lambda _ { j _ 1 } ( q _ 0 ) \cdots \lambda _ { j _ m } ( q _ 0 ) . \end{align*}


\begin{align*} F ^ q ( B _ \phi \eta , p ) = B _ \phi F ^ q ( \eta , p ) \end{align*}


\begin{align*} - K _ F = ( n - 1 ) A _ X \vert _ F , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \dot { P } _ t + 2 A P _ t - B ^ 2 R ^ { - 1 } P ^ 2 _ t + Q = 0 , \\ & P _ T = G , \end{aligned} \right . \end{align*}


\begin{align*} \frac { \partial ^ 2 v } { \partial n _ e ^ 2 } = n _ e \cdot ( \triangle v ) n _ e \end{align*}


\begin{align*} ( M N T _ { 2 } ) _ { i j } = \left \{ \begin{array} { c c } { i + 2 j \choose 3 j + 1 } & i \geq j \geq 0 ; \\ 0 & i < j , \end{array} \right . \end{align*}


\begin{align*} \left ( P \circ Q \right ) = - \left ( Q \circ P \right ) \ , , \end{align*}


\begin{align*} \beta = 3 \left ( 2 i - 1 \right ) \end{align*}


\begin{align*} K \oplus _ 2 L \supset K \widetilde { + } _ 2 L : = \cup _ { u \in S ^ { n - 1 } } \left [ 0 , \sqrt { \rho _ K ^ 2 ( u ) + \rho _ L ^ 2 ( u ) } u \right ] . \end{align*}


\begin{align*} p ' _ { j } \circ p _ { 1 \ldots ( n - 1 ) } \circ f \circ \iota _ { 1 \ldots ( n - 1 ) } \circ u ' _ { j } = p _ { j } \circ f \circ u _ { j } \circ \iota _ { 1 \ldots ( n - 1 ) } = p _ { j } \circ f \circ \iota _ { 1 \ldots ( n - 1 ) } \\ = p ' _ { j } \circ p _ { 1 \ldots ( n - 1 ) } \circ f \circ \iota _ { 1 \ldots ( n - 1 ) } ~ ~ ~ ~ \forall j = 1 , \ldots , n - 1 . \end{align*}


\begin{align*} \frac { \partial h } { \partial t } \leq \frac { 1 } { f _ { \min } } h [ h ^ { - \varepsilon } \beta _ { 0 } - f _ { \min } ] . \end{align*}


\begin{align*} \mathcal { J } ' ( \overline { u } ; h ) = \langle \overline { y } - g , S ' ( \overline { u } ; h ) \rangle + \alpha \langle \overline { u } , h \rangle \ge 0 , h \in T _ { \mathcal { C } _ { a d } } ( \overline { u } ) . \end{align*}


\begin{align*} H _ { m i n } ( E \otimes F _ { \ell - 1 } ) = H _ { m i n } ( E \otimes F _ 1 ) . \end{align*}


\begin{align*} \begin{aligned} | ( \langle \boldsymbol { \nu } , G \rangle - \bar G ) \cdot ( \langle \boldsymbol { \nu } , R \rangle - \bar R ) | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; . \end{aligned} \end{align*}


\begin{align*} 1 = - \sum _ { \emptyset \neq I \subset \{ 1 , \dots , m \} } ( - 1 ) ^ { | I | } , \end{align*}


\begin{align*} y _ n = \left [ g , x ^ { n } \right ] = \left [ g , x ^ { n - 1 } \right ] \left [ g , x \right ] ^ { x ^ { n - 1 } } = y _ { n - 1 } y \left [ y , x ^ { n - 1 } \right ] \end{align*}


\begin{align*} \begin{cases} \Delta _ { \Psi } u = f ( u ) & \O \\ u = c _ { j } & ( \partial \O ) _ { j } \end{cases} \end{align*}


\begin{align*} ( \nabla u ( x _ { 0 } ) ) ^ { \top } = 0 . \end{align*}


\begin{align*} | | u _ N | | _ { L ^ \infty ( B _ { L s _ { \delta _ N , i } } ( z _ { N , i } ) ) } = o ( 1 ) . \end{align*}


\begin{align*} M _ p = \max \{ d ( x _ 0 , p ) , d ( u , p ) \} . \end{align*}


\begin{align*} \hat \Phi ^ \dagger ( a , s ) = h ( a , s ) . \end{align*}


\begin{align*} \left | \sum _ { n = 1 } ^ { \infty } S _ N ( n ) \int _ { \frac { 1 } { n + 1 } } ^ { \frac { 1 } { n } } x ^ { s - 1 } d x \right | \leq \frac { 1 } { \Re s } \sum _ { n = 1 } ^ { \infty } \left | S _ N ( n ) \right | \left | \frac { 1 } { n ^ { \Re ( s ) } } - \frac { 1 } { ( n + 1 ) ^ { \Re ( s ) } } \right | . \end{align*}


\begin{align*} \rho { c _ v } \frac { { \partial T } } { { \partial t } } + \rho { c _ v } { \bf { u } } \cdot \nabla T = \nabla \cdot \left ( { \lambda \nabla T } \right ) - T { \left ( { \frac { { \partial { p _ { E O S } } } } { { \partial T } } } \right ) _ \rho } \nabla \cdot { \bf { u } } . \end{align*}


\begin{align*} \begin{aligned} \int _ { \mathbb { R } } s _ 1 ( z ) s _ 2 ( z ) d z = \int _ { \mathbb { R } } \left [ \mathcal { P } ^ { - } ( s J ) + s J \right ] \left [ \mathcal { P } ^ { - } ( s J ) \right ] ^ { H } \mathrm { d } z = 0 . \end{aligned} \end{align*}


\begin{align*} { U _ \theta } ^ \top U _ \theta = \begin{bmatrix} \cos ^ 2 \theta \dfrac { m } { k } V ^ \top V & \vline & \cos \theta \sin \theta \dfrac { m } { \sqrt { k l } } V ^ \top W \\ \hline \cos \theta \sin \theta \dfrac { m } { \sqrt { k l } } W ^ \top V & \vline & \sin ^ 2 \theta \dfrac { m } { l } W ^ \top W \end{bmatrix} . \end{align*}


\begin{align*} \begin{aligned} \frac { 1 } { 2 } \sum _ { \beta = 1 } ^ { \alpha } b _ \beta \bar { x } _ \beta ^ 2 \leq C b _ \alpha \bar { x } _ n + b _ \alpha O ( | \bar { x } ' | ^ 3 ) \leq C b _ \alpha \bar { x } _ n + C \delta ^ 3 b _ \alpha ^ 2 . \end{aligned} \end{align*}


\begin{align*} \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) = \int \eta ( \lambda | \bar { U } ; x , t ) d \boldsymbol { \nu } ( \lambda ) \end{align*}


\begin{align*} \Delta \rho _ 1 ( x ) \leq \psi _ { K , - H } ( \rho _ 1 ( x ) ) = - \psi _ { K , H } ( \rho ( x ) ) . \end{align*}


\begin{align*} p ' _ { 1 } \circ ( p _ { 1 2 } \circ f \circ \iota _ { 1 2 } ) = p _ { 1 } \circ f \circ \iota _ { 1 2 } = p _ { 1 } \circ f \circ l _ { 1 } \circ \iota _ { 1 2 } = p _ { 1 } \circ f \circ \iota _ { 1 2 } \circ l ' _ { 1 } \\ = p ' _ { 1 } \circ ( p _ { 1 2 } \circ f \circ \iota _ { 1 2 } ) \circ l ' _ { 1 } , \end{align*}


\begin{align*} \sum _ { f \in \mathcal F _ { N , k } } a _ f ( n ) = \begin{cases} \frac { | \mathcal F _ { N , k } | } { n ^ { 1 / 2 } } + \O \left ( n \sigma _ 0 ( n ) 4 ^ { \nu ( N ) } \right ) , & n \\ \O \left ( n \sigma _ 0 ( n ) 4 ^ { \nu ( N ) } \right ) , & . \end{cases} \end{align*}


\begin{align*} r ^ { 2 } < \phi ( z , r ) \leq \phi ( z ' , r ) - \phi ( z ' , 0 ) = \int _ { 0 } ^ { r } \partial _ { r ' } \phi ( z ' , r ' ) \dd r . \end{align*}


\begin{align*} H _ { j , n } = E X ^ { j } h _ { n } ( X ) = \frac { 1 } { \sqrt { n ! } } E X ^ { j } H _ { n } ( X ) = \left \{ \begin{array} { c c c } 0 & & n > j j - n \\ \frac { j ! } { 2 ^ { ( j - n ) / 2 } ( ( j - n ) / 2 ) ! \sqrt { n ! } } & & j - n \end{array} \right . \end{align*}


\begin{align*} F ^ { i j } v _ { i j } \geq \delta _ 1 \left ( \frac { 1 } { b _ { n - 1 } ^ { 1 / k } } + \sum _ { i = 1 } ^ n F ^ { i i } \right ) \mbox { i n } \omega . \end{align*}


\begin{align*} \begin{aligned} \varphi & = S ( t ) \varphi _ 0 + \int _ { 0 } ^ { t } S ( t - s ) \mathbb { P } \nabla \cdot F ( s ) \dd s . \end{aligned} \end{align*}


\begin{align*} h ( x , y ) \leq \frac { 1 } { 2 \pi } \ln \frac { 1 } { \max \{ | x - y | , d i s t ( x , \partial \Omega ) , d i s t ( y , \partial \Omega ) \} } , \end{align*}


\begin{align*} i _ { k + 1 } ( B _ { I , [ m ] } ) & = { \rm a r g m a x } _ { i \in I \backslash I _ k ( B _ { I , [ m ] } ) } | \langle v _ { k + 1 } ( B _ { I , [ m ] } ) , B _ { i , [ m ] } ^ \top \rangle | \\ & = { \rm a r g m a x } _ { i \in [ n ] \backslash I _ k ( B ) } | \langle v _ { k + 1 } ( B ) , B _ { i , [ m ] } ^ \top \rangle | = i _ { k + 1 } ( B ) . \end{align*}


\begin{align*} d z _ t = ( \Pi _ { \mathrm { k e r } A } ( B ( y _ t , z _ t ) + B ( z _ t , y _ t ) + B ( y _ t , y _ t ) - A y _ t ) d t + \Pi _ { \mathrm { k e r } A } \sigma d W _ t . \end{align*}


\begin{align*} \int _ { M \setminus \Omega } \bigg \langle S ' ( h , v ) , \ , \kappa _ 0 ( \bar { g } , \bar { u } , X ) \bigg \rangle _ { \bar g } \ , d \mathrm { v o l } _ { \bar g } & = 0 \end{align*}


\begin{align*} \int _ { 0 } ^ { T } \int _ { \mathbb { R } ^ { 3 } } A _ t \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \chi _ R \rho \dd x \dd t = - \int _ { 0 } ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi _ + \frac { G } { r ^ { 2 } } \chi _ R \dot { \rho } \dd x \dd t - \int _ { 0 } ^ { T } \int _ { \mathbb { R } ^ { 3 } } ( \Phi _ + \nabla \theta \times A _ t ) \cdot \nabla \chi _ R \rho \dd x \dd t . \end{align*}


\begin{align*} t = \sum _ { j = 1 } ^ \infty \frac { b _ j } { 2 ^ j } \end{align*}


\begin{align*} - L \phi = \mu ^ { 2 } ( \phi - \phi _ \infty ) _ { + } \textrm { i n } \ \mathbb { R } ^ { 2 } _ { + } , \phi = 0 \textrm { o n } \ \partial \mathbb { R } ^ { 2 } _ { + } , \end{align*}


\begin{align*} \frac { 1 } { q } = \sum _ { n = 0 } ^ { s - 1 } \frac { a _ n } { \beta ^ { n + 1 } } + \left ( \sum _ { n = s } ^ { s + t - 1 } \frac { a _ n } { \beta ^ { n + 1 } } \right ) \frac { \beta ^ t } { \beta ^ t - 1 } \end{align*}


\begin{align*} \Delta _ + ^ 0 = \{ \alpha _ i - \alpha _ j | 1 \leq 1 < j \leq r \} \end{align*}


\begin{align*} & ( r + \nu ) v ( s , i ) - H \left ( s , i , \partial _ s v ( s , i ) , \partial _ i v ( s , i ) \right ) = 0 , \\ - & ( r + \nu ) v ( s , i ) + H \left ( s , i , \partial _ s v ( s , i ) , \partial _ i v ( s , i ) \right ) = 0 , \end{align*}


\begin{align*} \int _ { \mathfrak X ( n ) } | \omega ( n ) | = ( \mathfrak h ( n ) _ 0 ) _ ! \int _ { \mathfrak Y ( n ) } | \mathfrak h ( n ) _ { \eta } ^ * \omega ( n ) | . \end{align*}


\begin{align*} \varepsilon \circ i ( B ) = A \varepsilon ( B ) A ^ { - 1 } . \end{align*}


\begin{align*} d ^ * ( \lfloor x \rfloor ) & = q ( c ( \lfloor x \rfloor ) ) \\ & = q ( c _ b c _ b c _ a c _ a ) \\ & = b ^ 2 a ^ 2 \end{align*}


\begin{align*} \left [ A ( y _ 1 x ) - A ( y _ 1 x ' ) \right ] + \sum _ { n = 2 } ^ \infty \left [ A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) \right ] \end{align*}


\begin{align*} \nabla _ V \big ( \nabla _ Y \hat X - \omega ( Y ) \big ) & = \big ( \nabla _ Y \omega - R ( Y , \hat X ) \big ) V - T _ h ( V , Y ) \\ \nabla _ V \left ( \nabla _ Y \omega - R ( Y , \hat X ) \right ) & = - R \big ( \nabla _ Y \hat X - \omega ( Y ) , V \big ) \\ & - R ( \omega ( Y ) , V ) + \nabla _ Y ( T _ h ( V ) ) + R ( V , Y ) \omega \\ & + ( \nabla _ { \hat X } R ) ( V , Y ) - R ( Y , \omega ( V ) ) . \end{align*}


\begin{align*} \varrho - \mathcal { P } ^ { - } ( \varrho J ) = I , \end{align*}


\begin{align*} ( 2 \pi ) ^ { - n } | \lambda | ^ n \ , \int _ { \C ^ n } \widehat { f } ( a , w ) \varphi _ { k , \lambda } ^ { n - 1 } ( z - w ) e ^ { \frac { i } { 2 } \lambda \Im ( z \cdot \bar { w } ) } d w = f ^ { - \lambda } \ast _ { - \lambda } \varphi _ { k , \lambda } ^ { n - 1 } ( z ) = \widehat { f } ( a , z ) . \end{align*}


\begin{align*} \Gamma _ { ( g , u ) } ( X ) = - \Delta _ g X - u ^ { - 1 } \nabla X ( \nabla _ g u , \cdot ) + u ^ { - 2 } X ( u ) d u . \end{align*}


\begin{align*} \nabla _ X \psi _ 2 = - \frac { 1 } { 2 } X _ 1 \cdot \psi _ 2 - \frac { 1 } { 2 } B ( X ) \cdot \psi _ 2 . \end{align*}


\begin{align*} \mu _ n = \frac { 1 } { \left [ G : K \right ] } \sum _ { g \in G / K } g _ * \nu _ n \end{align*}


\begin{align*} f ( x ) = \sum _ { | I | = 0 , 1 , \dots , } \frac { 1 } { | I | ! } \partial ^ I f ( p ) ( x - p ) ^ I \mbox { f o r a l l } x \in U \end{align*}


\begin{align*} \epsilon ( \varphi ^ * ( f ) ) = \epsilon ( f ) \circ \tilde { \varphi } . \end{align*}


\begin{align*} \mathbb P \left [ \frac 1 n \sum _ { i = 1 } ^ n ( X _ i - \mathbb E [ X _ i ] ) < - \epsilon \right ] \le \exp \left ( - \frac { n \epsilon ^ 2 } { 2 \sigma ^ 2 } \right ) \end{align*}


\begin{align*} W '' = e ^ { - \lambda ^ * T } \big ( W ' + W ^ D \big ) . \end{align*}


\begin{align*} \begin{aligned} \Big | \langle \boldsymbol { \nu } , \eta _ t \rangle - \bar \eta _ t + \langle \boldsymbol { \nu } , q _ { \alpha , x _ \alpha } \rangle - \bar q _ { \alpha , x _ \alpha } \Big | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; , \end{aligned} \end{align*}


\begin{align*} ( H _ { \lambda \exp , \alpha } \psi ) _ { n } = \psi _ { n + 1 } + \psi _ { n - 1 } + \lambda { \rm e } ^ { { \rm i } ( x + n \alpha ) } \psi _ { n } , \end{align*}


\begin{align*} \mu _ { \rm s } ( x [ n ] ) = \frac { \lambda _ \mathrm { a } [ n ] T _ { \rm s } } { N _ { \rm a } } \exp \left ( - \frac { \lambda _ \mathrm { a } [ n ] \tau _ { \rm d } } { N _ { \rm a } } \right ) , \end{align*}


\begin{align*} h _ { 1 1 i i } = h _ { i i 1 1 } + \kappa _ 1 ^ 2 \kappa _ i - \kappa _ 1 \kappa _ i ^ 2 - \kappa _ 1 + \kappa _ i . \end{align*}


\begin{align*} C ^ { \alpha } _ { i } G _ { \alpha \beta } ^ T H ^ { \beta } _ 1 \overline { G _ { \alpha \beta } } = G _ { \alpha \beta } ^ T H ^ { \beta } _ 1 \overline { G _ { \alpha \beta } } ( \overline { C ^ { \alpha } _ { i } } ^ { T } ) \end{align*}


\begin{align*} \int _ { \mathfrak Y ( n ) } | \mathfrak h ( n ) _ { \eta } ^ * \omega ( n ) | = ( \phi _ 0 ) _ ! \int _ { S m ( \widetilde { \mathfrak Y ( n ) } ) } | \phi _ { \eta } ^ * \mathfrak h ( n ) _ { \eta } ^ * \omega ( n ) | . \end{align*}


\begin{align*} s ( a , b ) = \{ d \mid b : d \le a \} . \end{align*}


\begin{align*} h ^ \intercal = 0 , A ' ( h ) = 0 , ( \nabla ^ k _ \nu A ) ' ( h ) = 0 \mbox { o n } \hat \Sigma \end{align*}


\begin{align*} \varphi ( - t ) ^ 2 G ( t ) H ( - t ) + \varphi ( t ) ^ 2 G ( t ) H ( t ) = 2 \varphi ( t ^ 4 ) ^ 2 G ( t ^ 4 ) G ( t ) . \end{align*}


\begin{align*} \int _ { \Omega } \bold { A } _ j \cdot \bold { B } _ k \dd x = \frac { 1 } { \lambda _ k } \int _ { \Omega } \bold { A } _ j \cdot \nabla \times \bold { B } _ k \dd x = \frac { 1 } { \lambda _ k } \int _ { \Omega } \bold { B } _ j \cdot \bold { B } _ k \dd x = \frac { 1 } { \lambda _ k } \delta _ { j , k } . \end{align*}


\begin{align*} I _ { \theta h } \leq E [ t ( \theta ) b ] = t ( \theta ) ^ { 2 } E [ b ] = t ( \theta ) ^ { 2 } I _ { h } < \theta I _ h , \end{align*}


\begin{align*} \begin{aligned} | K _ 2 ^ \chi f ( v ) | \leq W _ 1 ( v ) + W _ 2 ( v ) \ , , \end{aligned} \end{align*}


\begin{align*} L _ \delta \omega _ \delta = l _ { 1 , \delta } + l _ { 2 , \delta } + R _ \delta ( \omega _ \delta ) , \end{align*}


\begin{align*} ( p _ n f , v _ n ) = ( f , v _ n ) v _ n \in X _ n . \end{align*}


\begin{align*} V ( J ) = \{ x \in \R ^ n \ , : \ , p ( x ) = 0 \ , , \ , \forall p \in J \} \ , . \end{align*}


\begin{align*} \sigma _ { k - 1 } ( b ) u _ { n n } ( 0 ) - \sum _ { \alpha \leq n - 1 } u _ { n \alpha } ^ 2 ( 0 ) \sigma _ { k - 2 ; \alpha } ( b ) + \sigma _ k ( b ) = f ( 0 ) . \end{align*}