\begin{align*} \Vert g _ { > \iota } \Vert _ p & \lesssim R ^ { Q / q } \Vert g _ { > \iota } \Vert _ 2 \\ & \le R ^ { Q / q } \bigg \Vert \sum _ { \ell = \iota + 1 } ^ \infty F _ \ell ^ { ( \iota ) } ( L , U ) f \bigg \Vert _ 2 \\ & \lesssim 2 ^ { \iota ( Q - d _ 2 ) / q } \norm { F ^ { ( \iota ) } \psi } _ 2 \norm { f } _ p \\ & \lesssim 2 ^ { - \varepsilon \iota } \norm { F ^ { ( \iota ) } } _ { L ^ 2 _ s } \norm { f } _ p \end{align*}


\begin{align*} f _ t ( y ) : = \left \{ \begin{aligned} & y - t & & \ ; y \geq t \\ & 0 & & \ ; y \leq t \end{aligned} \right . \end{align*}


\begin{align*} \begin{array} { c } \rho v _ { t t } - \alpha v _ { x x } + \gamma \beta p _ { x x } = 0 , \\ \mu p _ { t t } - \beta p _ { x x } + \gamma \beta v _ { x x } = 0 , \end{array} \end{align*}


\begin{align*} & \int _ { D _ j } | u _ { m , \epsilon , j } | ^ { 2 } e ^ { v _ { \epsilon } ( \Psi _ m ) - \varphi _ { m } - \Psi _ m } \leq \int _ { D _ j } v '' _ { \epsilon } ( \Psi _ m ) | F | ^ 2 e ^ { - \phi - \varphi _ { m } - \Psi _ m } . \end{align*}


\begin{align*} \begin{array} [ c ] { c } i _ { \frac { \partial } { \partial r } } \overline { \omega } ^ { n + 1 } = ( \Pi ^ { \ast } \omega _ { h } ) ^ { n } \wedge \eta = \omega ^ { n } \wedge \eta . \end{array} \end{align*}


\begin{align*} & d \left ( x _ 1 ^ { a _ 1 } \cdots x _ n ^ { a _ n } ( ( a _ i + 1 ) x _ 1 d x _ 2 \wedge \cdots \wedge d x _ n - ( - 1 ) ^ { i - 1 } ( a _ 1 + 1 ) x _ i d x _ 1 \wedge \cdots \wedge \widehat { d x _ i } \wedge \cdots \wedge d x _ n \right ) \\ = { } & x _ 1 ^ { a _ 1 } \cdots x _ n ^ { a _ n } ( ( a _ 1 + 1 ) ( a _ i + 1 ) - ( a _ i + 1 ) ( a _ 1 + 1 ) ) d x _ 1 \wedge \cdots \wedge d x _ n \\ = { } & 0 . \end{align*}


\begin{align*} D _ { q ^ { - 1 } } v ( x ) - q A ( x ) v ( x ) = 1 . \end{align*}


\begin{align*} \cal T ( p , t ) = \log \frac { | \hat { \sf g } _ t v | } { | v | } . \end{align*}


\begin{align*} \tilde p = \tilde p _ \delta = \sum _ { n = 0 } ^ { \infty } p _ n . \end{align*}


\begin{align*} \rho ( t , 0 ) = \rho ( t , V _ F ) , t > 0 . \end{align*}


\begin{align*} \phi _ \zeta ( M _ f ) : = M _ f S _ \zeta , f \in \Z ^ { m | n } . \end{align*}


\begin{align*} \| u \| _ { k , \alpha } : = \displaystyle \sum _ { l _ { 1 } + 2 l _ { 2 } \leq k } \left \| | \widetilde { \nabla } ^ { \ell _ 1 } \partial ^ { \ell _ { 2 } } _ { s } u | _ { \widetilde { g } } \right \| _ { \alpha } . \end{align*}


\begin{align*} \bar \psi _ j ( x ) = \bar \psi _ j ( x _ 1 , \ldots , x _ n ) : = \sum _ { i = 1 } ^ n x _ i \varphi ^ j _ i . \end{align*}


\begin{align*} R ( N _ { 1 } , N _ { 2 } ) = \sum \log p _ { 1 } \cdots \log p _ { 1 0 } \end{align*}


\begin{align*} b _ j = \frac { ( b _ i - 1 ) a _ j } { a _ i - 1 } . \end{align*}


\begin{align*} \texttt { l } _ { g f } = \sum _ { x \in W ^ g } \ell _ { g \cdot x , f } q ^ { - \ell ( x ) } , f , g \in \Z ^ { m | n } _ { \zeta - } . \end{align*}


\begin{align*} \tilde { p } ( t , U _ F , g ) - \tilde { p } ( t , 0 , g ) = 0 , g \in \mathbb { R } , t > 0 , N ( t ) = \int _ { 0 } ^ { \infty } g \tilde { p } ( t , U _ F , g ) d g . \end{align*}


\begin{align*} c _ { n } ^ { \ast } = \frac { \Gamma ( n / 2 ) } { ( 2 \pi ) ^ { n / 2 } } \left [ \int _ { 0 } ^ { \infty } s ^ { n / 2 - 1 } \overline { G } _ { n } ( s ) \mathrm { d } s \right ] ^ { - 1 } \end{align*}


\begin{align*} \partial _ i L ( h _ \theta ) - a _ { i 0 } = \sum _ { k , j > 0 } a _ { i k j } Q _ \theta ^ j . \end{align*}


\begin{align*} j ( a * b ) ( \gamma ) = \sum _ { \eta \in G ^ { u } } j ( a ) ( \eta ) \ , j ( b ) ( \eta ^ { - 1 } \gamma ) . \end{align*}


\begin{align*} \limsup _ { n \to \infty } \sup _ { x \in \R ^ N } \int _ { x + [ 0 , 1 ) ^ N } u _ n = \lim _ { k \to \infty } \int _ { x _ k + [ 0 , 1 ) ^ N } u _ { n _ k } . \end{align*}


\begin{align*} \frac { 1 } { 2 } \{ g ( \nabla _ X Z , Y ) + g ( \nabla _ Y Z , X ) \} + \frac { 2 \mu } { \lambda ^ 2 } h ( \tilde { X } , \tilde { Y } ) = 0 . \end{align*}


\begin{align*} z \in \mathrm { F i x } ( S _ { N } ) = \mathrm { F i x } \left ( \mathcal { T } _ { \left \{ C _ { m , r } \left ( N \right ) \left ( j \right ) \right \} _ { j = 1 } ^ { r } } \right ) = \cap _ { j = 1 } ^ { r } C _ { f \left ( \left ( r - 1 \right ) N + j - 1 \right ) } \subset C _ { f \left ( \left ( r - 1 \right ) N + k \right ) } = C _ { p } . \end{align*}


\begin{align*} \mathrm { D T K } _ { ( p , q ) } ( X ) = \frac { \mathrm { E } \left [ ( X - \mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { 4 } | x _ { p } < X < x _ { q } \right ] } { \mathrm { D T V } _ { ( p , q ) } ^ { 2 } ( X ) } - 3 , \end{align*}


\begin{align*} \mu _ { n } = D _ { b _ { n } } \mu ^ { \boxplus n } \boxplus \delta _ { a _ { n } } \end{align*}


\begin{align*} \det ( \sigma ^ 2 I _ n - A ^ H A ) = 0 . \end{align*}


\begin{align*} \int _ { \mathbb R ^ n } f ( x ) d x = \int _ { S ^ { n - 1 } } \int _ 0 ^ \infty f ( \mathcal O + r \cdot \xi ) r ^ { n - 1 } d r d \mathcal H ^ { n - 1 } . \end{align*}


\begin{align*} \lambda R _ \lambda f + \tilde R _ \lambda \lambda q R _ \lambda f - R _ \lambda \psi - R _ \lambda q R _ \lambda \psi - \tilde R _ \lambda q f = f + \tilde R _ \lambda q f - \tilde R _ \lambda q f = f . \end{align*}


\begin{align*} \phi _ { n + 1 , \lambda } ( x ) = x \sum _ { k = 0 } ^ { n } \binom { n } { k } \phi _ { k , \lambda } ( x ) ( 1 - \lambda ) _ { n - k , \lambda } , ( n \ge 0 ) . \end{align*}


\begin{align*} \alpha _ { n , i } ( f ^ { n } , g ^ { n } _ { i } ) \coloneqq \Pr \left ( \hat { H } _ { i } = 1 | H = 0 \right ) i \in \{ 1 , 2 \} , \end{align*}


\begin{align*} S _ { k } ( c ; \xi ) = & { \sum _ { a \ , \hbox { \tiny m o d } \ , k } } ^ { \ ! \ ! \ ! * } \sum _ { { \tiny \begin{tabular} { c } $ z \ , \hbox { \tiny m o d } \ , k \ell $ \\ $ z = \xi ( \ell ) $ \end{tabular} } } e _ { k \ell } \big ( a \ell F _ h ( z ) + c \cdot z \big ) \\ = & { \sum _ { a \ , \hbox { \tiny m o d } \ , k } } ^ { \ ! \ ! \ ! * } \ ; \sum _ { z ' \ , \hbox { \tiny m o d } \ , k } e _ { k \ell } \big ( a \ell F _ h ( \xi + \ell z ' ) + c \cdot ( \xi + \ell z ' ) \big ) . \end{align*}


\begin{align*} I _ \alpha ( n ) : = \int _ 1 ^ \infty ( \log z ) ^ n e ^ { - \alpha z } \ , d z \qquad ( \alpha > 0 ) , \end{align*}


\begin{align*} \partial _ t v _ k = \frac { 1 } { m } \nabla \cdot ( m \nabla v _ k ^ k ) + \nabla v _ k \cdot \vec { b } + F v _ k Q _ T . \end{align*}


\begin{align*} \left ( \bigcup _ { i = 1 } ^ d S _ { d , i } \right ) \cup \left ( \bigcup _ { j = 1 } ^ e T _ { e , j } \right ) \cup \left ( \bigcup _ { \ell = 1 } ^ e U _ { e , \ell } \right ) = [ 1 , ( 4 k + 2 ) N ] , \end{align*}


\begin{align*} d V _ { \min } = ( 1 - \| \beta \| _ { \alpha } ) ^ { n + 1 } d V _ { \alpha } . \end{align*}


\begin{align*} & V ( Q ( I ) ) = \left \{ \left . \frac { e _ { i _ 1 } + \cdots + e _ { i _ k } } { k - s + d } \right | s - d + 1 \leq k \leq s , \ 1 \leq i _ 1 < \cdots < i _ k \leq s \right \} , \\ & V ( Q ( I ^ \vee ) ) = \left \{ \left . \frac { e _ { j _ 1 } + \cdots + e _ { j _ { \ell } } } { \ell - d + 1 } \right | d \leq \ell \leq s , \ 1 \leq j _ 1 < \cdots < j _ \ell \leq s \right \} . \end{align*}


\begin{align*} \Delta _ n ' : = \sup _ { t \in [ 0 , T _ n ] } \sup _ { i , j = 1 , . . . , d } | \varphi _ { n , i } ( t ) - \varphi _ { n , j } ( t ) | . \end{align*}


\begin{align*} B _ { \rm S T } \ = \ \{ t ^ { n } , \ n \in \mathbb { N } \} . \end{align*}


\begin{align*} E ( X _ 1 ^ 2 ) = \varepsilon ^ 2 \big [ P ( \boldsymbol { X } = \mathbf { s } ( 1 ) ) + P ( \boldsymbol { X } = \mathbf { s } ( 4 ) ) \big ] . \end{align*}


\begin{align*} \lim _ { z \rightarrow _ { \sphericalangle } \alpha } \left | \frac { f ( z ) - f ^ * ( \alpha ) } { z - \alpha } \right | = + \infty , \end{align*}


\begin{align*} A Y + Y A ^ { * } \leq 0 \ \ B = - A Y C ^ { * } . \end{align*}


\begin{align*} \left | \underset { \Gamma _ p } { \int \sum } \overline { \widehat { \eta } } _ k \prod _ { \substack { j = 1 \\ o d d } } ^ { p } \widehat { v } _ { k _ j } \prod _ { \substack { j = 2 \\ e v e n } } ^ p \overline { \widehat { v } } _ { k _ j } \ , d \Gamma \right | . \end{align*}


\begin{align*} \frac { d Y } { d t } = A Y , Y ( t ) = \begin{pmatrix} y _ { 1 1 } ( t ) & y _ { 1 2 } ( t ) \\ y _ { 2 1 } ( t ) & y _ { 2 2 } ( t ) \end{pmatrix} \in \mathrm { S L } ( 2 , \mathbb { R } ) , Y ( 0 ) = I _ 2 , \end{align*}


\begin{align*} ( L y _ i ) ( v ) & = - \sum \limits _ { e \in E _ v } \frac { \delta _ E ( e ) } { \delta _ V ( v ) } \frac { 1 } { | e | } y _ i ( v ) . \end{align*}


\begin{align*} \big ( \frac { v + n w - n t \tau / 2 } { w - t \tau / 2 } \big ) > 0 , \big ( \frac { v + n w - n t \tau / 2 } { w + t \tau / 2 } \big ) > 0 , \big ( \frac { v + n w - n t \tau / 2 } { - t } \big ) > 0 , \\ \big ( \frac { - t \tau / 2 } { w - t \tau / 2 } \big ) > 0 , \big ( \frac { - t \tau / 2 } { w + t \tau / 2 } \big ) > 0 , \big ( \frac { - t \tau / 2 } { - t } \big ) = ( \tau / 2 ) > 0 . \end{align*}


\begin{align*} \frac 1 2 \sum _ { i \in \N } m ( ( \mu ^ i _ n ) _ { n \in \sigma _ i ( \N ) } ) \le \sum _ { i \in \N } \norm { \mu ^ i } = \lim _ { \ell \to \infty } \norm { \mu _ { \sigma ( \ell ) } ^ b } \leq \liminf _ { \ell \to \infty } \norm { \mu _ { \sigma ( \ell ) } } < \infty . \end{align*}


\begin{align*} \int _ { \Xi _ { \mu } } e ^ { W _ { \mathbf { y } } / \hbar } f d x _ 1 \wedge \cdots \wedge d x _ a = \delta _ { \mu 0 } + t _ { \mu } ( \mathbf { y } ) \hbar ^ { - 1 } + O ( \hbar ^ { - 2 } ) \end{align*}


\begin{align*} i \delta \lambda ^ n u ^ n _ x = - i \lambda ^ n w ^ n + c \left ( ( 1 - m ) w ^ n + m \int _ 0 ^ { \infty } \sigma ( s ) \eta ^ n ( s ) d s \right ) _ { x x } + f ^ { 5 , n } + \delta f ^ { 1 , n } _ x \end{align*}


\begin{align*} x _ { e } = - \sum _ { u \in S ( v _ e ) } x _ { \phi ( u ) } + \sum _ { u \in S ( r _ e ) } x _ { \phi ( u ) } , \end{align*}


\begin{align*} \theta _ { i } : = \phi \min \nolimits _ { 0 < \gamma \leq \gamma ^ { \max } _ { i } } \{ \gamma ^ { - 1 } + K \varphi _ { i } ( \gamma ) \} , \forall i \in \mathcal { N } , \end{align*}


\begin{align*} \theta _ { t + 1 } = \theta _ { t + 1 / 2 } - \beta _ t w _ { \rm a } ( t ) . \end{align*}


\begin{align*} \Tilde { \mathcal { N } } ( f ) ( v _ n ) = \operatorname { N } _ { n + 1 , n } ( f ( v _ { n + 1 } ) ) = \operatorname { N } _ { n + 1 , n } ( \beta ' ) = \beta . \end{align*}


\begin{align*} f ( \gamma \cup \delta ) = f ( \gamma ) \star f ( \delta ) , \end{align*}


\begin{align*} ( \mbox { H o r } P _ U V ) _ { ( x , t ) } = 0 . \end{align*}


\begin{align*} n / 2 - 5 \gamma n \leq | X ' _ { R } | , | Y ' _ { R } | \leq n / 2 + 5 \gamma n \ , . \end{align*}


\begin{align*} \begin{array} { l l } \frac { 1 } { 2 } \{ g ( \nabla _ U \xi , V ) + g ( \nabla _ V \xi , U ) \} + R i c ^ \nu ( U , V ) - ( m - n ) \| H \| ^ 2 g ( U , V ) \\ + d i v ( H ) g ( U , V ) + \mu g ( U , V ) = 0 . \end{array} \end{align*}


\begin{align*} T _ 0 ( d ) = \frac { \Gamma ( \frac s 2 ) ^ 2 d ^ s L ( s , \chi _ d ) ^ 2 } { \Gamma ( s ) \zeta ( 2 s ) } . \end{align*}


\begin{align*} ( \mathbf { U } ^ 0 , \mathbf { B } ^ 0 , P ^ 0 ) = ( 1 , 0 , \sigma , 0 , 0 ) , \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty M _ k ( n ) q ^ n = \sum _ { n = k } ^ \infty \frac { q ^ { { k \choose 2 } + ( k + 1 ) n } } { ( q ; q ) _ n } \begin{bmatrix} n - 1 \\ k - 1 \end{bmatrix} . \end{align*}


\begin{align*} f ( B ( v ) _ q ) = \sum _ { b \in B ( v ) _ q } \frac { 1 } { b \log b } \ & \ge \ \frac { 1 } { 1 + v } \sum _ { b \in B ( v ) _ q } \frac { 1 } { b \log P ( b ) } . \end{align*}


\begin{align*} ( D _ q f ) ( x ) = \frac { f ( x ) - f ( q x ) } { ( 1 - q ) x } , x \neq 0 , \end{align*}


\begin{align*} \frac { 1 } { \sigma _ { k - 1 } } | P ( x _ k ) | = \frac { 1 } { 2 \sigma _ { k - 1 } } \left ( \norm { g ( x _ k ) } ^ 2 + \norm { \left [ - X ( x _ k ) \right ] _ { + } } _ { \mathrm { F } } ^ 2 \right ) \to 0 . \end{align*}


\begin{align*} \int _ R ^ \infty Q ( r \omega ) | u ( r ) | ^ 2 \ , d r & = \int _ 0 ^ \infty W ( r ) | \tilde u ( r ) | ^ 2 \ , d r \leq 4 \left ( \sup _ { s > 0 } s \int _ s ^ \infty W ( t ) \ , d t \right ) \int _ 0 ^ \infty | \tilde u ' ( r ) | ^ 2 \ , d r \\ & = 4 \left ( \sup _ { s > R } s \int _ s ^ \infty Q ( t \omega ) \ , d t \right ) \left ( \int _ R ^ \infty | \tilde u ' ( r ) | ^ 2 \ , d r + c _ R | u ( R ) | ^ 2 \right ) . \end{align*}


\begin{align*} I _ { ( X _ - , D _ - ) } ( y , z ) = z e ^ { t _ - / z } \sum _ { d \in \mathbb K _ { - } } \tilde y ^ { d } \left ( \prod _ { i \in M _ 0 } \frac { \prod _ { a \leq 0 , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } { \prod _ { a \leq D _ i \cdot d , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } \right ) \textbf { 1 } _ { [ - d ] } I _ { D _ - , d } , \end{align*}


\begin{align*} e ^ { i t } \cdot ( r , \theta ) : = ( r , \theta + t ) , \end{align*}


\begin{align*} \mathcal Q ( x ) ( v ) & = \sum \limits _ { e \in E _ v } \frac { ( a v g ( x ) ) ( e ) } { | e | } \frac { \delta _ E ( e ) } { \delta _ V ( v ) } \\ & = \sum \limits _ { e \in E _ v } \sum \limits _ { u \in e } x ( u ) \\ & = \sum \limits _ { e \in E } B _ { v e } \sum \limits _ { u \in V } B _ { u e } x ( u ) = ( ( B B ^ T ) x ) ( v ) , \end{align*}


\begin{align*} Q ^ { n , 6 } _ \tau = Q ^ { n , 7 } _ \tau + Q ^ { n , 8 } _ \tau , \end{align*}


\begin{align*} \tau _ { - x ^ i _ n } \mu ^ { i - 1 } _ n = \tau _ { - x ^ i _ n } \mu _ n - \sum _ { 0 \le j < i } \tau _ { - x ^ i _ n + x ^ j _ n } \mu ^ j , \end{align*}


\begin{align*} P _ { k , n } ( z ) = \sum _ { j = k + 1 } ^ n ( - z ) ^ { j - k - 1 } \binom { j - 1 } { k } { n \brack j } . \end{align*}


\begin{align*} \label [ I e q ] { e q : m c k e a n s d e c a n o n i c a l } \dd \mathsf { X } ^ i _ t = b ( \mathsf { X } ^ i _ t , f _ t ) \dd t + \dd \overline { B } ^ i _ t , \end{align*}


\begin{align*} A ( t ) = ( I - \kappa e ^ { - a ( t , s ) } ) ^ { - 1 } \partial _ t a ( t , s ) \end{align*}


\begin{align*} b _ { 2 k } ( M ) = h _ k ( P ) . \end{align*}


\begin{align*} F ( y ) = \int _ y ^ 1 \frac { \dd t } { \rho ( t ) } \in [ 0 , + \infty ] . \end{align*}


\begin{align*} \partial _ t u - \Delta u = f , u ( 0 , x ) = u _ 0 ( x ) . \end{align*}


\begin{align*} \Tilde { \sigma _ i } \Tilde { \Phi } _ L ( \beta ) = h _ i \Tilde { \sigma _ j } . \end{align*}


\begin{align*} f _ { X } ( x ) : = \frac { c _ { 1 } } { \sigma } g _ { 1 } \left \{ \frac { 1 } { 2 } \left ( \frac { x - \mu } { \sigma } \right ) \right \} , ~ x \in \mathbb { R } , \end{align*}


\begin{align*} q _ A \geq \frac { 2 } { 2 - \beta } + ( \ell _ { \delta } ( A ) - 1 ) + o ( 1 ) = \ell _ { \delta } ( A ) + \beta ' + o ( 1 ) . \end{align*}


\begin{align*} \mathbb { Z } _ { r } & \rightarrow \mathbb { C } ^ { 2 } \\ \gamma & \mapsto \begin{bmatrix} e ^ { 2 \pi i \frac { 1 } { r } } & 0 \\ 0 & e ^ { 2 \pi i \frac { a } { r } } , \end{bmatrix} \end{align*}


\begin{align*} ( a , x ) \ast ( b , y ) = ( a b f ( x , y ) , x y ) . \end{align*}


\begin{align*} F = I n v \circ ( 0 , 1 , \gamma ) , \ c , \gamma \not \in \{ 0 , 1 \} , \ n \ge 4 \end{align*}


\begin{align*} J : = \omega _ C \otimes S ^ \vee \end{align*}


\begin{align*} \lim \limits _ { n \rightarrow \infty } \sup \limits _ { t \geq 0 } \| P _ { \delta } ( W _ { n ; \textrm { f a r } } ) e ^ { - i t H } \varphi \| = 0 \end{align*}


\begin{align*} \{ I _ 1 , \ldots , I _ h \} = { \bf a } ( \{ Q _ 1 , \ldots , Q _ h \} ) . \end{align*}


\begin{align*} \frac { \partial ^ 2 C ^ 2 } { \partial z ^ i _ { \epsilon } \partial z ^ j _ { \eta } } = \frac { \partial } { \partial z ^ j _ { \eta } } \left ( 2 C \frac { \partial C } { \partial z ^ i _ { \epsilon } } \right ) = 2 \frac { \partial C } { \partial z ^ i _ { \epsilon } } \frac { \partial C } { \partial z ^ j _ { \eta } } + 2 C \frac { \partial ^ 2 C } { \partial z ^ i _ { \epsilon } \partial z ^ j _ { \eta } } . \end{align*}


\begin{align*} L _ t [ u ] = \exp \left ( \mp i \frac { p + 1 } { 4 \pi } \int _ 0 ^ t \int _ \mathbb { T } | u | ^ { p - 1 } ( x , s ) \ , d x d s \right ) , \end{align*}


\begin{align*} ( A _ h u , v ) _ { L ^ 2 ( \Omega ) } = ( \nabla u , \nabla v ) _ { L ^ 2 ( \Omega ) } u , v \in V _ h . \end{align*}


\begin{align*} \dfrac { \partial \det A } { \partial a _ { k k } } = \det \left ( A _ { k | k } \right ) , \ \ \dfrac { \partial \det A } { \partial a _ { k \l } } = ( - 1 ) ^ { k + \l } 2 \det \left ( A _ { k | \l } \right ) . \end{align*}


\begin{align*} \int _ { \Xi _ { \mu } } x ^ { \mu ' } e ^ { ( x _ 1 ^ { r _ 1 } + \cdots + x _ n ^ { r _ n } ) / \hbar } \Omega = \delta _ { \mu \mu ' } . \end{align*}


\begin{align*} \begin{aligned} y _ { p , \lambda } ( x ) & = Y _ { \lambda } ( x , p ) - e ^ { x } \phi _ { p , \lambda } ( x ) \\ & = \sum _ { n = 2 } ^ { \infty } \Big ( ( 1 ) _ { p , \lambda } + ( 2 ) _ { p , \lambda } + \cdots + ( n - 1 ) _ { p , \lambda } \Big ) \frac { x ^ { n } } { n ! } , ( p \in \mathbb { N } ) . \end{aligned} \end{align*}


\begin{align*} w \lim \limits _ { t \to \infty } e ^ { - i H _ a t } ( I _ d - J _ a ) P _ c ( H _ a ) U ( t , 0 ) \psi ( 0 ) = 0 , \end{align*}


\begin{align*} \rho _ { p , \ell } ( \beta _ h ) \phi _ w ( g ^ { - 1 } , u ) = & \int _ { X _ { p , \ell } } \phi _ w \ , d \mu _ { p , \ell } \\ & \ ; + O _ { p , w , \ell , \delta , \theta , \epsilon } \Big ( \| g \| _ E ^ \theta h ^ { 1 - ( \theta - 4 ) \delta } + \| g \| _ E h ^ { - 1 / 2 + 3 \delta + \epsilon } \Big ) . \end{align*}


\begin{align*} \mathcal { G } ( v , x ) & = x - \sum _ { w \in c ( v ) } \dfrac { 1 } { \mathcal { G } ( w , x ) } ( \forall v \in V ( T ) ) , \end{align*}


\begin{align*} R _ { p } ( h , \mathcal O ) : = \int _ { X \setminus B ( \mathcal O , 1 ) } h ^ { \frac { p } { 1 - p } } d \mu R _ { 1 } ( h , \mathcal O ) : = \| h ^ { - 1 } \| _ { L ^ \infty ( X \setminus B ( \mathcal O , 1 ) ) } . \end{align*}


\begin{align*} \delta ( e ^ { - z } ) = z + \O ( z ^ { p + 1 } ) \end{align*}


\begin{align*} | t | d _ n ^ { \gamma | t | } ( t \mathcal { K } ) _ X = d _ n ^ { \gamma } ( \mathcal { K } ) _ X , \mbox { w h e r e } { t \mathcal K } : = \{ t f : \ , f \in \mathcal K \} . \end{align*}


\begin{align*} b _ { i | v } = 2 / v . \end{align*}


\begin{align*} & \mu _ i ( x ) - \mu _ i ( y ) = r _ i x _ i \left ( 1 - \sum _ { j = 1 } ^ { d } a _ { i j } x _ j ^ + \right ) - r _ i y _ i \left ( 1 - \sum _ { j = 1 } ^ { d } a _ { i j } y _ j ^ + \right ) \\ & = r _ i ( x _ i - y _ i ) - r _ i \left [ \left ( x _ i \sum _ { j = 1 } ^ { d } a _ { i j } x _ j ^ + \right ) - \left ( y _ i \sum _ { j = 1 } ^ { d } a _ { i j } y _ j ^ + \right ) \right ] . \end{align*}


\begin{align*} { \bar \Omega } _ { A B } = d \theta _ { A B } - \sum _ { C = 1 } ^ { N + 1 } \theta _ { A C } \wedge \theta _ { C B } - \theta _ { A \ , N + 2 } \wedge \theta _ { N + 2 \ , B } \end{align*}


\begin{align*} \sum _ { k = N } ^ M a _ k \bigg ( \sum _ { i = k } ^ M a _ i \bigg ) ^ s b _ k & = \sum _ { k = N + 1 } ^ M ( b _ k - b _ { k - 1 } ) \sum _ { i = k } ^ M a _ i \big ( \sum _ { j = i } ^ M a _ j \big ) ^ s + \bigg ( \sum _ { k = N } ^ M a _ k \big ( \sum _ { i = k } ^ M a _ i \big ) ^ s \bigg ) b _ N \\ & \approx \sum _ { k = N + 1 } ^ M ( b _ k - b _ { k - 1 } ) \bigg ( \sum _ { i = k } ^ M a _ i \bigg ) ^ { s + 1 } + \bigg ( \sum _ { k = N } ^ M a _ k \bigg ) ^ { s + 1 } b _ N , \end{align*}


\begin{align*} c _ k = \int _ { x _ k } ^ { x _ { k + 1 } } u ( t ) \bigg ( \int _ t ^ { \infty } u \bigg ) ^ { \frac { q } { p - q } } d t , \end{align*}


\begin{align*} \sup _ { 0 < h < 1 } \bigg \{ R ^ { \frac { N } { p } - 2 } \sum _ { 1 < | z _ \gamma | \leq R } | z _ \gamma | ^ 2 \omega _ { \gamma , h } + R ^ { \frac { N } { p } } \sum _ { | z _ \gamma | > R } \omega _ { \gamma , h } \bigg \} = o _ R ( 1 ) , \end{align*}


\begin{align*} \log G ( z \ , | \ , \omega _ 1 , \widetilde { \omega } _ 1 , \omega _ 2 ) = \int _ C \frac { - e ^ { ( z + \overline \omega _ 1 ) s } } { ( e ^ { \omega _ 1 s } - 1 ) ( e ^ { \widetilde \omega _ 1 s } - 1 ) ( e ^ { \omega _ 2 s } - 1 ) } \cdot \frac { d s } { s } , \end{align*}


\begin{align*} \sum _ { j ( \neq i ) } \frac { 1 } { \lambda _ i - \lambda _ j } = \frac { \sum _ { j ( \neq i ) } \prod _ { k ( \neq i , j ) } ( \lambda _ i - \lambda _ k ) } { \prod _ { j ( \neq i ) } ( \lambda _ i - \lambda _ j ) } , \end{align*}


\begin{align*} - \frac { 1 } { 2 } g ( A _ X Y + A _ Y X , U ) + R i c ( X , Y ) + \mu \frac { 1 } { \lambda ^ 2 } h ( \tilde { X } , \tilde { Y } ) = 0 , \end{align*}


\begin{align*} \lim _ { N \to \infty } Q _ { N } \Big [ \varepsilon _ 0 \frac { \gamma } { \beta + \gamma } N ^ { 4 / 3 } ( \log N ) & ^ { - 2 / 3 } \leq R _ { N , d , D } \\ & \leq K _ { 0 } \Big ( \frac { \beta + \gamma } { \beta } \Big ) ^ { 1 / 2 } N ^ { 4 / 3 } ( \log N ) ^ { 4 / 3 } \Big ] = 1 . \end{align*}


\begin{align*} \begin{aligned} c _ n & : = - \frac { \int _ { { \C } _ 1 } \left ( \sum _ { i = 2 } ^ 4 f ( | y - h _ n t _ { i } | ) ( y - h _ n t _ { i } ) \cdot t _ { i } \psi _ n ( y ) \right ) d y } { \int _ { { \C } _ 1 } \left ( \sum _ { i = 1 } ^ 4 f ( | y - h _ n t _ { i } | ) ( y - h _ n t _ { i } ) \cdot t _ { i } \right ) ^ 2 d y } . \end{aligned} \end{align*}


\begin{align*} \mathcal E = \mathcal E ( \mathbb C P ^ 2 , \mathbb C P ^ 1 , \pi ) \ : = \ \mathbb C P ^ 1 \hookrightarrow \mathcal E \xrightarrow { \pi } \mathbb C P ^ 2 \ . \end{align*}


\begin{align*} \prod _ { k \in F } f ^ { - 1 } \circ ( \chi ^ { - 1 } ) ^ k \circ f \circ \chi ^ k & = \prod _ { k \in F } ( ( \chi ^ { - 1 } ) ^ k ) ^ f \circ \chi ^ k \\ & = \prod _ { k \in F } ( ( \chi ^ { - 1 } ) ^ k ) ^ f \circ \prod _ { k \in F } \chi ^ k \\ & = [ f , \chi ^ F ] , \end{align*}


\begin{align*} [ G , G ] = [ N , G ] . \end{align*}


\begin{align*} l _ \pm = \dim H ^ 2 ( X _ \pm ; \mathbb R ) = r - \# ( S _ \pm ) l = r - 1 - \# ( S _ 0 ) . \end{align*}


\begin{align*} G _ { N } ( z ) : = \prod \limits _ { k = 1 } ^ { N - 1 } \Bigg ( \frac { \mu _ { k } ^ { 2 } - z } { \lambda _ { k + 1 } ^ { 2 } - z } \Bigg ) \end{align*}


\begin{align*} \sup _ { k \leq M - 1 } 2 ^ { - \frac { k } { p } } \bigg ( \int _ { x _ k } ^ { \infty } u \bigg ) ^ { \frac { 1 } { q } } V _ r ( 0 , x _ k ) \approx \sup _ { k \leq M - 1 } 2 ^ { - \frac { k } { p } } \bigg ( \int _ { x _ k } ^ { \infty } u \bigg ) ^ { \frac { 1 } { q } } V _ r ( x _ { k - 1 } , x _ k ) = A _ 1 . \end{align*}


\begin{align*} \hat { \Gamma } _ { ( Y _ + , D _ { Y , + } ) } : = \bigoplus _ { \vec d \in ( \mathbb Z ) ^ { | I _ + | } , f \in \mathbb K _ { + } / \mathbb L } \frac { \prod _ { i \in \bar { I } _ + } \Gamma ( 1 + \bar D _ i - \langle D _ i \cdot f \rangle ) } { \prod _ { i = 1 } ^ k \Gamma ( 1 + v _ i ) } \prod _ { i \in I _ + , d _ i < 0 } \frac { 1 } { \bar D _ i - d _ i } \textbf { 1 } _ { f } [ \textbf 1 ] _ { ( d _ i ) _ { i \in I _ + } } . \end{align*}


\begin{align*} u _ n '' = ( \lambda _ n - u _ n ^ { p - 2 } ) u _ n \implies - u _ n '' + \frac { \lambda _ n } { 2 } u _ n \le 0 \end{align*}


\begin{align*} \big \{ h \in \dot { H } ^ { - 1 , p } ( \rho ) : \| h - \ell \| _ { \dot { H } ^ { - 1 , p } ( \rho ) } \le r \big \} = \bigcap _ { \substack { \varphi \in C _ 0 ^ { \infty } : \\ \| \varphi \| _ { \dot { H } ^ { 1 , q } ( \rho ) } \le 1 } } \big \{ h \in \dot { H } ^ { - 1 , p } ( \rho ) : | ( h - \ell ) ( \varphi ) | \le r \big \} . \end{align*}


\begin{align*} \mathbf { T ^ 2 _ { k } } = \int _ { B _ { 1 / 2 } } \nabla ( \eta _ r ( U _ k - U _ 0 ) ) \cdot \nabla ( U _ 0 + V ) \ , d x = \int _ { B _ { 1 / 2 } } & \eta _ r \nabla ( U _ 0 + V ) \cdot \nabla ( U _ k - U _ 0 ) \ , d x \\ & + \int _ { B _ { 1 / 2 } } ( U _ k - U _ 0 ) \nabla \eta _ r \cdot \nabla ( U _ 0 + V ) \ , d x . \end{align*}


\begin{align*} F ( 4 \lambda - 1 , 2 , 2 ) = 4 \lambda \end{align*}


\begin{align*} \tilde { \mu } _ { V X ^ { ( n ) } } ( A ) : = \mathbb { P } \Big ( \Big ( \frac { 1 } { 1 + \alpha } \Big ) ^ { 1 / p } \sum _ { j = 1 } ^ n Z _ j V _ { \bullet , j } \in A \Big ) , \end{align*}


\begin{align*} B ( | v - v _ * | , \cos \theta ) = | v - v _ * | ^ \gamma b \left ( \cos \theta \right ) , \gamma = \frac { s - ( 2 d - 1 ) } { s - 1 } , \end{align*}


\begin{align*} \Sigma _ { \mu \boxtimes \nu } ( z ) = \Sigma _ { \mu } ( z ) \Sigma _ { \nu } ( z ) , \quad | z | < \min \{ \rho _ { \mu } , \rho _ { \nu } \} . \end{align*}


\begin{align*} \mu \int _ { \Omega } \varepsilon y _ t : \varepsilon v + \nu \int _ { \Omega } \left ( 0 , \abs { \varepsilon y _ t } - g \right ) \ , \dfrac { \varepsilon y _ t } { \abs { \varepsilon y _ t } } : \varepsilon v = \int _ { \Omega } ( u + t h ) \cdot v , \forall v \in Y . \end{align*}


\begin{align*} H e s s f ( X , Y ) = g ( h _ f ( X ) , Y ) , ~ \forall X , Y \in \Gamma ( T M ) . \end{align*}


\begin{align*} \int _ 0 ^ { c _ 1 } h ' \left ( \abs { v } ^ 2 + \abs { u _ x + b v _ x } ^ 2 \right ) d x = o ( 1 ) . \end{align*}


\begin{align*} { \rm d } ( { \rm L } _ b ) = \frac { 1 } { b } \prod _ { p < P ( b ) } \Big ( 1 - \frac { 1 } { p } \Big ) = \frac { e ^ { - \gamma } + o _ y ( 1 ) } { b \log P ( b ) } . \end{align*}


\begin{align*} \kappa _ \gamma ( y , v ) : = \sum _ { \ell = - 1 } ^ \iota \int _ { A _ \ell ( y ) } | K _ \ell ^ { ( \iota ) } ( ( x , u ) ^ { - 1 } ( y , v ) ) | \ , d ( x , u ) . \end{align*}


\begin{align*} & ( | \eta _ \delta ^ \perp | ^ 2 * \chi _ { B _ { n } } ) ( x ) = \int \limits _ { B _ { n } } | \eta _ \delta ^ \perp ( y - x ) | ^ 2 \ , d y \leq C \int \limits _ { B _ { n } } \| x - y \| ^ { - \ell } \ , d y \leq C \int \limits _ { B _ { n } } ( 2 n - \| y \| ) ^ { - \ell } \ , d y \\ & \leq C \int \limits _ { 0 } ^ n ( 2 n - r ) ^ { - \ell } r ^ { d - k - 1 } \ , d r \leq C n ^ { - \ell + d - k } \end{align*}


\begin{align*} & \int \frac { 1 } { ( - x ^ 2 ; q ^ 2 ) _ \infty } \left ( q x [ n - 1 ] _ q + c [ n ] _ q \right ) \widetilde { h } _ n ( x ; q ) d _ q x = \\ & \frac { 1 - q } { ( - x ^ 2 ; q ^ 2 ) _ \infty } \left ( \widetilde { h } _ n ( x ; q ) - q ^ { 1 - n } [ n ] _ q ( c + x ) \widetilde { h } _ { n - 1 } ( x ; q ) \right ) , \end{align*}


\begin{align*} G _ t ( x ) : = \frac { 1 } { \sqrt { 2 \pi C ( t ) } } \exp \left ( - \frac { ( x - B ( t ) ) ^ 2 } { 2 C ( t ) } \right ) \end{align*}


\begin{align*} [ v ] & = [ \lambda _ { \delta } u _ 1 + \mu _ { \delta } u _ 2 ] \in S \cap [ \rho _ 2 ] , \\ [ w ] & = [ \lambda _ { \theta } u _ 1 + \mu _ { \theta } u _ 2 ] \in S \cap [ \rho _ 2 ] . \end{align*}


\begin{align*} \max \left \{ \frac { p q } { n p + q } , 1 \right \} = 1 . \end{align*}


\begin{align*} \mathcal { F } ( v , x ) & = x - \beta ( v ) - \sum _ { w \in c ( v ) } \dfrac { 1 } { \mathcal { F } ( w , x ) } ( \forall v \in V ( T ) ) , \end{align*}


\begin{align*} U _ N : \mathcal { H } _ N \to \ell ^ 2 ( \mathcal { F } _ \perp ) , U _ N \psi _ N = \bigoplus _ { s = 0 } ^ N \ ; \bigoplus _ { d = - N + s , - N + s + 2 , \dots } ^ { N - s - 2 , N - s } \Phi _ { s , d } . \end{align*}


\begin{align*} = \sum _ { p ^ { \nu } \le x ^ { 1 / k } } ( \log p ) \left ( \frac { x } { p ^ { \nu k } } P _ { k - 1 } \Big ( \log \frac { x } { p ^ { \nu k } } \Big ) + O \Big ( \Big ( \frac { x } { p ^ { \nu k } } \Big ) ^ { \theta _ k + \varepsilon } \Big ) \right ) . \end{align*}


\begin{align*} \gamma _ i = \textrm { m i n } \{ \gamma _ { i , 1 } , \gamma _ { i , 2 } , . . . , \gamma _ { i , l } , . . . , \gamma _ { i , L } \} , \end{align*}


\begin{gather*} ( \gamma , ( \lambda , \mu ) ) ( \gamma ' , ( \lambda ' , \mu ' ) ) = \left ( \gamma \gamma ' , ( \lambda , \mu ) \gamma ' + ( \lambda ' , \mu ' ) \right ) . \end{gather*}


\begin{align*} A _ { C , \Delta } ( p ) = \frac { 1 } { n } - \left ( \Omega \cdot C \right ) _ p . \end{align*}


\begin{align*} \mathcal A _ \omega : = \{ I \subset \{ 1 , \ldots , m \} | \omega \in \angle _ I \} . \end{align*}


\begin{align*} \sigma _ { B H } ( x ) = \frac { v o l ( B ^ n ( 1 ) ) } { v o l \left \lbrace ( y ^ i ) \in T _ x M : F ( x , y ) < 1 \right \rbrace } , \end{align*}


\begin{align*} T _ 0 ( d ) = \frac { \Gamma ( \frac s 2 ) ^ 2 } { 4 \Gamma ( s ) } d ^ s L ( s , \chi _ d ) \sum _ { c = 1 } ^ \infty \frac { K ^ + ( d , 0 ; 4 c ) } { c ^ { s + 1 / 2 } } . \end{align*}


\begin{align*} \cosh R = \cosh r \cosh r ' - \sinh r \sinh r ' \cos \theta _ R ( r , r ' ) . \end{align*}


\begin{align*} 1 2 \sqrt \pi | b ( 1 ) | ^ 2 = \langle \varphi , \varphi \rangle ^ { - 1 } \int _ 0 ^ \infty \varphi ( i y ) y ^ { - 1 } \ , d y . \end{align*}


\begin{align*} Q : = S _ { j _ { f } } \dots S _ { 0 } \end{align*}


\begin{align*} d _ n ^ { \gamma _ n } ( \mathcal K ) _ X \geq C ( \log _ 2 n ) ^ { - \alpha } , n = 1 , 2 , \dots . \end{align*}


\begin{align*} & - \int _ { \partial \Sigma \cap L } e ^ { - i \hat { \theta } } d ( f _ L F ) - \int _ { \partial \Sigma \cap L _ 0 } e ^ { - i \hat { \theta } } d ( f _ { L _ 0 } F ) \\ & = \sum _ { } e ^ { - i \hat { \theta } } F f | ^ + _ - , \end{align*}


\begin{align*} R ( G ) = \bigoplus _ { \rho \in \operatorname { I r r } ( G ) } \Q \ , \rho \ ; , \end{align*}


\begin{align*} \Omega _ { \mu } = \{ z \in \mathbb { H } : z + \varphi _ { \mu } ( z ) \in \mathbb { H } \} . \end{align*}


\begin{align*} \Sigma _ { \mu } ( z ) = \gamma \exp \left [ \int _ { [ 0 , + \infty ] } \frac { 1 + t z } { z - t } \ , d \sigma ( t ) \right ] , z \in \mathbb { C } \backslash \mathbb { R } _ { + } , \end{align*}


\begin{align*} \limsup _ { x \to \infty } \frac { \sup _ { y \in \Z ^ d , \lvert y \rvert \leq x } Y ( t , y ) } { f ( x ) } = \infty \limsup _ { x \to \infty } \frac { \sup _ { y \in \Z ^ d , \lvert y \rvert \leq x } Y ( t , y ) } { f ( x ) } = 0 , \end{align*}


\begin{align*} Z ( L ) = \int _ L \Omega . \end{align*}


\begin{align*} \sup _ { A \textnormal { p r i m i t i v e } } \overline { { \rm d } } ( A ) \ ; = \ ; \frac { 1 } { 2 } . \end{align*}


\begin{align*} f = \sum _ { j = 0 } ^ \infty f _ j f _ j : = f \chi _ { B _ j } . \end{align*}


\begin{align*} L ^ { n , 2 } _ \tau + L ^ { n , 3 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) ( s - \eta _ n ( s ) ) ^ \alpha \Lambda ^ n _ s d s , \end{align*}


\begin{align*} I _ { k , \ell } ( c ) = ( h \ell ) ^ 4 \int _ { \mathbb { R } ^ 4 } w ( \ell x ) H \big ( h ^ { - 1 } k , \ell ^ 2 N ( x ) - 1 \big ) e _ { k } ( - h c \cdot x ) \ , d x . \end{align*}


\begin{align*} f ( q ^ k x ) = q ^ { \frac { k ( k - 1 ) } { 2 } } x ^ k f ( x ) ( k \in \mathbb { N } _ 0 ) . \end{align*}


\begin{align*} K _ { - } = \{ s + i t \in K : t \le f _ { \nu } ( s _ { 0 } ) - \varepsilon \} \end{align*}


\begin{align*} s ^ 2 U - \Delta U = 0 \Omega ^ + , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , T ] } E \left [ | Q ^ { n , 3 } _ \tau | ^ 2 \right ] = 0 . \end{align*}


\begin{align*} { } _ { } ^ z T _ { k , j ; n , l } ^ { \mu ; m , 2 } ( B ) ( t , x , \zeta ) = \int _ { 0 } ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \big ( ( t - s ) \mathbf { P } _ 3 \big ( \mathfrak { K } ^ { \mu , B } _ { k ; n } ( y , \omega , v , \zeta ) \big ) + \mathbf { P } _ 3 \big ( \mathfrak { K } ^ { e r r ; \mu , B } _ { k ; n } ( y , v , \zeta ) \big ) \big ) f ( s , x - y + ( t - s ) \omega , v ) \end{align*}


\begin{align*} \overline { G } _ { ( 1 ) } ( u ) = \frac { m } { m - 1 } \left ( 1 + \frac { 2 u } { m } \right ) ^ { - ( m - 1 ) / 2 } \end{align*}


\begin{align*} h ( \varphi ) = \int _ { \R ^ d } \varphi d h = \int _ { \R ^ d } \varphi d \left ( \frac { \mu _ t - \nu _ t } { t } \right ) = \frac { 1 } { t } \int _ { \R ^ d } \varphi d ( \mu _ t - \nu _ t ) . \end{align*}


\begin{align*} \Gamma S = S ^ * \Gamma ; \end{align*}


\begin{align*} S _ 1 \leq \sum \limits _ { i = 0 } ^ { \infty } \mu \left ( \left [ a _ i , \infty \right ) \right ) < \infty . \end{align*}


\begin{align*} \frac { \sum _ { k = 1 } ^ N f ( S _ k p / q ) - E ( f ) N } { \sqrt { N } } \overset { d } { \to } \mathcal { N } ( 0 , \sigma ^ 2 ) \end{align*}


\begin{align*} k \coth ( k a / 2 ) = k ' \coth ( k ' a / 2 ) \end{align*}


\begin{align*} \lambda _ { \pm } = \frac { \sqrt { 1 + \left ( \frac { 4 \beta \hslash ^ { 2 } } { 3 a ^ { 2 } } \right ) ^ { 2 } } \pm 1 } { 4 \beta \hslash ^ { 2 } / 3 a ^ { 2 } } , \end{align*}


\begin{align*} T _ 0 ^ { x , \alpha } = \infty , \ \ a . s . , \end{align*}


\begin{align*} H _ r = ( - 1 ) ^ { r - 1 } \Delta ^ r ( \widehat { \mu } ( n ) - 2 \log \widehat { \mu } ( n ) ) - \sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { r - 1 } \Delta ^ r \Bigl ( \dfrac { 1 } { k \widehat { \mu } ( n ) ^ k } \Bigr ) \end{align*}


\begin{align*} \| f _ t \circ \psi \| = \int _ { - x _ 0 } ^ { x _ 0 } ( \psi ( x ) - t ) d x \leq \| f _ { t ' } \circ \psi \| + ( t ' - t ) \int _ { - x _ 0 } ^ { x _ 0 } d x < \infty . \end{align*}


\begin{align*} D _ { r } p ( x ) = \frac { 1 } { r ! } \Big ( \frac { d } { d x } \Big ) ^ { r } \bigg [ x ^ { r } p ( x ) \bigg ] . \end{align*}


\begin{align*} ( f ^ \dagger f , f ^ \dagger f ) = | f ^ \dagger f | ^ 2 = \sum _ { \mathcal { I } } \sum _ { \mathcal { J } } \left ( f ^ \mathcal { I } f ^ \mathcal { J } \right ) ^ 2 \end{align*}


\begin{align*} \sum _ { n \geq 0 } S _ n ^ { ( p ) } S _ n ^ { ( p ) } = \sum _ { \gamma \in \hat { \Gamma } _ 0 ^ n } \left | \gamma ' ( 0 ) \right | ^ p \left | \hat { \rho } ( \gamma ) ' ( 0 ) \right | ^ { 2 - p } . \end{align*}


\begin{align*} H ^ k ( V , T _ V \otimes \mathcal { O } ( - Y _ { \vert { V } } ) ) & = H ^ k ( V , \Omega _ V ^ { n - 1 } \otimes \mathcal { O } ( - Y _ { \vert { V } } - K _ { \mathbb { P } \vert { V } } ) ) \\ & \subset H ^ k ( V , \Omega _ V ^ { n - 1 } \otimes \mathcal { O } ( - m Y _ { \vert { V } } - m K _ { \mathbb { P } \vert { V } } ) ) \end{align*}


\begin{align*} \sum _ { k = N } ^ { M } a _ k \bigg ( \sum _ { j = k } ^ { M } a _ j \bigg ) ^ { \beta - 1 } \approx \bigg ( \sum _ { k = N } ^ { M } a _ k \bigg ) ^ { \beta } \end{align*}


\begin{align*} f _ { W | S } ^ * ( w | s ) = \delta ( w - g ^ { - 1 } ( s | \rho , \lambda ) ) , \end{align*}


\begin{align*} \eta \circ \lambda \circ ( \alpha _ i ) _ { \ast } = ( \beta ^ { ( 2 ) } _ i ) _ { \ast } \circ ( \iota ^ { A _ i } _ 2 ) _ { \ast } = \mu \circ \rho \circ ( \alpha _ i ) _ { \ast } . \end{align*}


\begin{align*} N ( t ) & = \int _ { 0 } ^ { \infty } g p ( t , V _ F , g ) d g = \frac { 1 } { V _ F } \int _ { 0 } ^ { \infty } g \left ( \sum _ { k = - \infty } ^ { + \infty } p _ k ( t , g ) \right ) d g . \\ & = \sum _ { k = - \infty } ^ { + \infty } \frac { 1 } { V _ F } \int _ { 0 } ^ { \infty } g p _ k ( t , g ) d g = \sum _ { k = - \infty } ^ { + \infty } N _ k ( t ) , \end{align*}


\begin{align*} \mathrm { D T V } _ { ( p , q ) } ( X ) = \mathrm { E } [ ( X - \mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { 2 } | x _ { p } < X < x _ { q } ] , \end{align*}


\begin{align*} T _ { k , l } ( s ) = \min \{ \max \{ s - k , 0 \} , l - k \} \hbox { i n } [ 0 , + \infty ) \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } t ^ { \frac { d - 2 \delta } { \alpha } ( 1 - \frac { 1 } { q } ) } \| \tilde P _ t f \| _ { q , h } = 0 . \end{align*}


\begin{align*} h _ { \mu , \nu } : = ( g _ { j , \mu , 1 } \circ \pi , \ , \ldots , \ , g _ { j , \mu , \nu - 1 } \circ \pi , \ , d g _ { j , \mu , \nu } \circ \iota , \ , g _ { j , \mu , \nu + 1 } \circ \pi , \ , \ldots , \ , g _ { j , \mu , k } \circ \pi ) . \end{align*}


\begin{align*} \begin{aligned} & v _ h \to v \quad \textrm { s t r o n g l y i n } H ^ { 1 } _ { \# } \quad \textrm { a s } h \to 0 , \\ & q _ h \to q \quad \textrm { s t r o n g l y i n } L ^ 2 _ { \# } \quad \textrm { a s } h \to 0 ; \end{aligned} \end{align*}


\begin{align*} \begin{aligned} h _ 1 - \dfrac { \mathrm { t r } _ g h _ 1 } { n } \cdot g & = - A _ 1 ( g ) + \dfrac { \mathrm { t r } _ g A _ 1 ( g ) } { n } \cdot g , \\ h _ 2 - \dfrac { \mathrm { t r } _ g h _ 2 } { n } \cdot g & = - A _ 2 ( g ) + \dfrac { \mathrm { t r } _ g A _ 2 ( g ) } { n } \cdot g - A _ { 1 , 1 } ( h _ 1 ) + \dfrac { \mathrm { t r } _ g A _ { 1 , 1 } ( h _ 1 ) } { n } \cdot g , \\ \cdots \cdots & = \cdots \cdots . \end{aligned} \end{align*}


\begin{align*} \gamma _ \rho : = \frac { \int \limits _ { \mathbb R ^ n } W _ \rho ( x ) \sum _ { i = 2 } ^ d W ^ 2 _ \rho ( \hat \Theta _ i ( x ) ) \partial _ \rho W _ \rho ( x ) d x } { \int \limits _ { \mathbb R ^ n } \ ( \partial _ \rho W _ \rho ( x ) \ ) ^ 2 d x } . \end{align*}


\begin{align*} \operatorname { c o d i m } \Phi \left ( { A } \right ) = \frac { | \Delta _ { 1 } | - | \Delta _ { 1 } \backslash { A } ^ { \perp } | } { 2 } - | { A } | = \frac { | \Delta _ { 1 } \cap { A } ^ { \perp } | } { 2 } - | { A } | . \end{align*}


\begin{align*} C _ R ( J / I ) = \left ( \bigcap _ { e > 0 } J ^ { [ p ^ e ] } : _ S ( I ^ { [ p ^ e ] } : _ S I ) \right ) / I . \end{align*}


\begin{gather*} \theta _ { m , r } ^ 0 ( \tau ) : = \theta _ { m , r } ( \tau , 0 ) \end{gather*}


\begin{align*} \tau ^ { \alpha } ( T _ 1 , \dots , T _ a ) = \left ( \tau ^ { \alpha _ 1 } ( T _ 1 ) , \dots , \tau ^ { \alpha _ a } ( T _ a ) \right ) , \end{align*}


\begin{align*} x | _ S = ( x _ { s _ 1 } , x _ { s _ 2 } , \dots ) \end{align*}


\begin{align*} F ^ { \{ \alpha \} } = \frac { F } { 1 + \alpha F } , \end{align*}


\begin{align*} ( ( \mu \boxplus \nu ) _ { } ) = ( p _ { \mu \boxplus \nu } ) = \overline { h ( V ) } . \end{align*}


\begin{align*} T [ f \alpha + g ] = T [ f ] \alpha + T [ g ] \end{align*}


\begin{align*} \Lambda = ( \lambda _ 1 , \lambda _ 2 , \ldots , \lambda _ m ) . \end{align*}


\begin{align*} \begin{aligned} B ( n , q ) = B ( n , q ; \chi _ { 0 } , \ldots , \chi _ { 0 } ) , A ( n , q ) = \frac { B ( n , q ) } { \varphi ^ { 5 } ( q ) } , \mathfrak { S } ( n ) = \sum \limits _ { q = 1 } ^ { \infty } A ( n , q ) . \end{aligned} \end{align*}


\begin{align*} p ^ * ( v , g ) : = \frac { 1 } { V _ F } \frac { 1 } { \sqrt { 2 \pi c ^ * } } \exp \left ( - \frac { ( g - b ^ * ) ^ 2 } { 2 c ^ * } \right ) , \end{align*}


\begin{align*} & \inf _ { \mathbb { R } ^ n } f > 0 \ \ f \in C ^ 3 ( \mathbb { R } ^ n \setminus D ) , \\ & \exists \ \beta > 2 \ \ \limsup _ { | x | \rightarrow \infty } | x | ^ { \beta + m } | D ^ m ( f ( x ) - 1 ) | < \infty , \ m = 0 , 1 , 2 , 3 , \end{align*}


\begin{align*} m = n _ { k _ 1 } + n _ { k _ 2 } + \ldots + n _ { k _ l } , \end{align*}


\begin{align*} C = P _ n \Lambda \otimes _ { P _ { n - 1 } \Lambda } \Q \end{align*}


\begin{align*} g _ e ^ 1 = \sigma v _ e ^ 1 . \end{align*}


\begin{align*} \chi ^ { \epsilon , L } _ t ( \lambda , \mu , m ^ 2 ) : = \max _ { x \in \Lambda _ { \epsilon , L } } \epsilon ^ d \sum _ { y \in \Lambda _ { \epsilon , L } } \big < \varphi _ x \varphi _ y \big > ^ { \epsilon , L } _ { \lambda , \mu + 1 / t , m ^ 2 } = \epsilon ^ d \sum _ { x \in \Lambda _ { \epsilon , L } } \big < \varphi _ 0 \varphi _ x \big > ^ { \epsilon , L } _ { \lambda , \mu + 1 / t , m ^ 2 } , \end{align*}


\begin{align*} \lim _ { y _ { \mathrm { r } + 1 } \rightarrow 0 } \bar I _ { \mathcal O _ { \mathbb P } ( - 1 ) \oplus \mathcal O _ { \mathbb P } ( 1 - \sum _ { i = 1 } ^ k w _ i ) } ( y ) = 0 , \end{align*}


\begin{align*} \zeta = \omega ( J _ X \frac { \partial u } { \partial s } + \sqrt { - 1 } \frac { \partial u } { \partial s } , Y ) d s = \omega ( J _ X \frac { \partial u } { \partial s } , Y ) d s . \end{align*}


\begin{align*} a ' b ' = a ' g ' ~ ~ ~ \mbox { a n d } ~ ~ ~ a '' b '' = a '' g '' . \end{align*}


\begin{align*} \begin{array} [ c ] { c } g _ { i } ^ { T } = \frac { 1 } { 2 } d \eta _ { i } = \pi ^ { \ast } ( h _ { i } ) = \pi ^ { \ast } ( \omega _ { h _ { i } } ) \end{array} \end{align*}


\begin{align*} y \leq \frac { t f ( y ) } { f ( t ) } = \frac { t \max ( - f ( y ) , 0 ) } { - f ( t ) } \leq \frac { t \max ( - f ( y ) , 0 ) } { - f ( t ) } + \frac { t ' f _ t ( y ) } { t ' - t } \end{align*}


\begin{gather*} \bigwedge _ { i = 1 } ^ k \bigwedge _ { j = 1 } ^ { n - k - 1 } \neg a _ { i , j } \lor a _ { i , j + 1 } \qquad \bigwedge _ { i = 0 } ^ k \bigwedge _ { j = 1 } ^ { n - k } \neg a _ { i , j } \lor a _ { i + 1 , j } \lor \neg b _ { i + j } \\ \bigwedge _ { i = 1 } ^ { k - 1 } \bigwedge _ { j = 1 } ^ { n - k } a _ { i , j } \lor \neg a _ { i + 1 , j } \qquad \bigwedge _ { i = 1 } ^ k \bigwedge _ { j = 0 } ^ { n - k } a _ { i , j } \lor \neg a _ { i , j + 1 } \lor b _ { i + j } \end{gather*}


\begin{align*} N = \sum _ { j = 1 } ^ n ( a _ j \partial _ { b _ j } - b _ j \partial _ { a _ j } ) . \end{align*}


\begin{align*} \begin{aligned} \min _ { 0 { } \leq { } i { } \leq { } N } & \| \nabla f ( x _ i ) \| ^ 2 { } \leq { } \ , 2 L \big ( f ( x _ 0 ) - f _ * \big ) \max \left \{ ( 1 - h ) ^ { 2 N } \ , \ \frac { 1 } { q + N \frac { h ( 2 - h ) ( 2 - \kappa h ) } { 2 - h \left ( 1 + \kappa \right ) } } \right \} \end{aligned} \end{align*}


\begin{align*} Y _ { k + 1 } = e ^ { \Omega _ k } \ , Y _ k . \end{align*}


\begin{align*} \frac 2 5 \sum _ x \pi ( x ) ( \mathbb { P } _ x [ T _ A \le t ] ) ^ { 2 } & \le \sum _ x \pi ( x ) ( \mathbb { P } _ x [ T _ A \le \tau ] ) ^ { 2 } \le \sum _ x \pi ( x ) \left ( \sum _ { a \in A } \mathbb { P } _ x [ T _ a \le \tau ] \right ) ^ { 2 } \\ & \le k ^ { 2 } \sum _ x \pi ( x ) \mathbb { ( P } _ x [ T _ o \le \tau ] ) ^ { 2 } , \end{align*}


\begin{align*} \partial _ s u + J ( t , u ) ( \partial _ t u - X _ H ) = 0 , \end{align*}


\begin{align*} | \psi _ { n , 1 } ( u , s _ 2 ) | = | ( s _ 2 - \eta _ n ( u ) ) ^ { \alpha } - ( s _ 2 - u ) ^ \alpha | \le \frac { \alpha } n \delta ^ { \alpha - 1 } . \end{align*}


\begin{align*} e _ { \Phi ^ { \mathfrak { b } , \gamma } } \left ( \rho \right ) = \int _ { \mathbb { S } \times \mathbb { D } _ { \varepsilon } } e _ { \mathcal { K } _ { \gamma } \left ( \Psi , f \right ) } \left ( \rho \right ) \mathfrak { b } \left ( \mathrm { d } \left ( \Psi , f \right ) \right ) \ , \rho \in E _ { 1 } \ . \end{align*}


\begin{align*} \bigg ( \sum _ { k = - \infty } ^ { M - 1 } \int _ { x _ k } ^ { x _ { k + 1 } } \bigg ( \int _ { x _ k } ^ t f ^ r v \bigg ) ^ { \frac { q } { r } } u ( t ) d t \bigg ) ^ { \frac { 1 } { q } } \leq \mathfrak { C '' } \bigg ( \sum _ { k = - \infty } ^ { M - 1 } 2 ^ k \bigg ( \int _ { x _ k } ^ { x _ { k + 1 } } f \bigg ) ^ p \bigg ) ^ { \frac { 1 } { p } } \end{align*}


\begin{align*} Q ( \varphi ; U , \mathcal { F } ) : = \int _ { \mathcal { F } } \left ( | \varphi ' | ^ 2 + ( \lambda - ( p - 1 ) \rho | U | ^ { p - 2 } ) \varphi ^ 2 \right ) \ , d x , \forall \varphi \in H ^ 1 ( \mathcal { F } ) \cap C _ c ( \mathcal { F } ) . \end{align*}


\begin{align*} a _ { \mathrm { p w } } ( u _ { \mathrm { n c } } , v _ { \mathrm { n c } } ) = \lambda _ h \Big ( \frac { u _ { \mathrm { n c } } } { 1 - \lambda _ h \kappa _ { m } ^ 2 h _ { \mathcal { T } } ^ { 2 { m } } } , v _ { \mathrm { n c } } \Big ) _ { L ^ 2 ( \Omega ) } \qquad v _ { \mathrm { n c } } \in V ( \mathcal { T } ) \end{align*}


\begin{align*} \Delta ( f \circ \delta ) = f ' ( \delta ) \Delta \delta + f '' ( \delta ) \norm { \nabla \delta } ^ 2 . \end{align*}


\begin{align*} & \| \varphi _ { n ; \textrm { i n } } \| = \| P _ { \delta } ( W _ { n ; \textrm { i n } } ) e ^ { - i t _ n H } \varphi \| \\ & \leq \| P _ { \delta } ( W _ { n ; \textrm { i n } } ) ( e ^ { - i t _ n H } - e ^ { - i t _ n H _ 0 } ) \varphi \| + \| P _ { \delta } ( W _ { n ; \textrm { i n } } ) e ^ { - i t _ n H _ 0 } \varphi \| \end{align*}


\begin{align*} V _ L = \bigoplus _ i V _ { \lambda _ i , 0 } \qquad V _ L = \bigoplus _ i V _ { \lambda _ i , 1 } . \end{align*}


\begin{align*} E ( \Psi _ t ) ( 0 , - \frac { 1 } { 2 } h ) + Q ( v , v ) = v \end{align*}


\begin{align*} \hat G ( 0 ) = A U _ 0 - f ( 0 ) , \hat G ( t _ j ) = 0 , \ ; j = 1 , \dots , N . \end{align*}


\begin{align*} \begin{pmatrix} 1 & p & q & \cdots & s \\ p & q & r & \cdots & t \end{pmatrix} . \end{align*}


\begin{align*} E V _ { \mathcal { S } } ( B , \nu ) = ( A _ { \mathcal { R } _ { B , \nu } } , \mu _ { \mathcal { R } _ { B , \nu } } ) , \end{align*}


\begin{align*} \mathrm { D T S } _ { ( p , q ) } ( X ) = \frac { \mathrm { E } \left [ ( X - \mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { 3 } | x _ { p } < X < x _ { q } \right ] } { \mathrm { D T V } _ { ( p , q ) } ^ { 3 / 2 } ( X ) } , \end{align*}


\begin{align*} \tilde { Y } _ O ( P , G ) & : = \sup \{ \| \phi _ 1 * \phi _ 2 * \cdots * \phi _ N \| _ { q ( P ) } \mid ( \phi _ 1 , \phi _ 2 , \cdots , \phi _ N ) \in \tilde { \mathcal { B } } \} , \\ \tilde { Y } _ R ( P , G ) & : = \inf \{ \| \phi _ 1 * \phi _ 2 * \cdots * \phi _ N \| _ { q ( P ) } \mid ( \phi _ 1 , \phi _ 2 , \cdots , \phi _ N ) \in \tilde { \mathcal { B } } \} . \end{align*}


\begin{align*} & \Vert \kappa _ { m } ^ { - 1 } h _ { \mathcal { T } } ^ { - { m } } ( \psi - \phi ) \Vert _ { L ^ 2 ( \Omega ) } ^ 2 = \Vert \kappa _ { m } ^ { - 1 } h _ { \mathcal { T } } ^ { - { m } } \delta \phi \Vert ^ 2 _ { L ^ 2 ( \Omega ) } = \lambda _ h ^ 2 \kappa _ { m } ^ 2 \Vert h _ { \mathcal { T } } ^ { m } ( 1 + \delta ) \phi \Vert ^ 2 _ { L ^ 2 ( \Omega ) } . \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \infty } | a _ n | \phi _ { n } ( r ) \leq \sum _ { n = 1 } ^ { \infty } M ( n ) \phi _ { n } ( r ) . \end{align*}


\begin{align*} \hat { A } ( z ) = \lambda ^ A \frac { z - a ^ A } { 1 - \overline { a ^ A } z } , \end{align*}


\begin{align*} \tilde U = \tilde \Xi \tilde \Lambda \tilde K \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 } \dfrac { \mathsf { m } ( E + t H ) - \mathsf { m } ( E ) } { t } & = \lim _ { t \rightarrow 0 } \dfrac { E + t H } { \abs { E + t H } } \left ( \dfrac { \abs { E + t H } - \abs { E } } { t } \right ) = \dfrac { E } { \abs { E } ^ 2 } E : H . \end{align*}


\begin{align*} C ^ 1 _ { T , \beta } = 2 \max ( 2 ^ { 1 - \beta } \Gamma ( 1 - \beta ) T ^ { \beta } , 1 ) \end{align*}


\begin{align*} \omega ^ { \prime } ( h ( \alpha ) ) = \frac { 1 } { H ^ { \prime } ( \alpha ) } = \frac { 1 } { 1 - g ( \alpha ) } = \begin{cases} 1 / ( 1 - I _ { 2 } I _ { 3 } ) & \alpha \in A , \\ 1 / ( 1 - I _ { 1 } I _ { 4 } ) & \alpha \in B \cup C . \end{cases} \end{align*}


\begin{align*} \limsup _ n \sum _ { i < k _ n } \mu _ n ( \tilde B _ n ^ i ) & \leq \limsup _ n \sum _ { i < k _ n } \mu _ n ( B _ n ^ i ) \\ & = \sum _ { i < k } \norm { \mu ^ i } \leq \sum _ { i < k } \liminf _ n \mu _ n ( \tilde B _ n ^ i ) \leq \liminf _ n \sum _ { i < k _ n } \mu _ n ( \tilde B _ n ^ i ) \end{align*}


\begin{align*} & D _ H B \equiv i [ H , B ] + \frac { \partial B } { \partial t } . \\ & B = B ^ * , \end{align*}


\begin{align*} b \star ( a _ 1 a _ 2 ) = l ( b a _ 1 a _ 2 ) = & l ( a ' b a _ 2 ) \stackrel { ( L 2 ) } = a ' \cdot l ( b a _ 2 ) \\ = & l ( b a _ 1 ) \cdot l ( b a _ 2 ) = ( b \star a _ 1 ) ( b \star a _ 2 ) . \end{align*}


\begin{align*} \psi = \psi _ k \circ \psi _ { k - 1 } \circ \cdots \circ \psi _ 1 , \end{align*}


\begin{align*} \lVert e _ Q a _ 1 ^ * u _ n a _ 2 e _ Q \rVert _ { \infty , 1 } & = \lVert E _ Q ( a _ 1 ^ * u _ n c _ a ) \rVert _ 1 \to 0 . \end{align*}


\begin{align*} X _ { Q , \bullet } = - X _ { Q ^ { + r } , \bullet } = - X _ { Q ^ { + s } , \bullet } . \end{align*}


\begin{align*} \rho _ { u + \delta } ( x ) = x - ( u + \delta , x ) ( u + \delta ) . \end{align*}


\begin{align*} \bar { x } _ i : = x _ i - \frac { - \kappa } { 1 - \kappa } \frac { h _ i } { L } U \in \big [ x _ { i + 1 } , x _ { i } \big ] , i \in \big \{ 0 , \dots , N - 1 \big \} . \end{align*}


\begin{align*} \mathbb { E } ( \tau ) L ^ k ( f , \mu ) & \geq \int _ { \tilde \Omega ^ k } d ( f ( e ) , f ( \gamma _ { \tau ( \tilde \omega ) } ^ k ( \tilde \omega ) ) ) d \mathbb { P } ( \tilde \omega ) \\ & = \sum _ { i = 1 } ^ { \infty } b _ i \left ( \int _ { \Gamma } d ( f ( e ) , f ( \gamma ) ) d \mu ^ { i k } ( \gamma ) \right ) \end{align*}


\begin{align*} \Delta _ 1 ( a , b ) = \frac { \pi } { 2 b } \left ( \ln \left ( \frac { a } { b } \right ) - \gamma \right ) + \frac { 2 } { b } \beta ^ { \prime } ( 1 ) . \end{align*}


\begin{align*} \dim ( \ker \bar { \partial } \subset W ^ { 1 , 2 ; \mu } ) = \dim \Gamma ( \mathbb { P } ^ 1 ( \R ) , \mathcal { E } \otimes \mathcal { O } ( \frac { \mu } { \pi } ) ) = K ( \frac { \mu } { \pi } + 1 ) + \sum _ 1 ^ K n _ i . \end{align*}


\begin{align*} & \big \langle \sum _ { i \in A _ 1 \cup A _ 2 } e _ i , \ , e _ { i _ 1 } + \cdots + e _ { i _ k } \big \rangle = | ( A _ 1 \cup A _ 2 ) \cap \{ i _ 1 , \ldots , i _ k \} | = d - s + k , \mbox { a n d } \\ & e _ i \in H \mbox { f o r } i \in \{ 1 , \ldots , s \} \setminus \{ i _ 1 , \ldots , i _ k \} . \end{align*}


\begin{align*} \frac { d ^ { 2 } \omega } { d t ^ { 2 } } + \omega - \frac { \mu a ^ { 2 } } { \lambda } = 0 , \end{align*}


\begin{align*} ( z f ' ( z ) ) ' = G ' ( z ) \psi ( \omega ( z ) ) . \end{align*}


\begin{align*} \mathrm { T C S } _ { q } ( X ) = \frac { \mathrm { E } \left [ ( X - \mathrm { T C E } _ { q } ( X ) ) ^ { 3 } | X > x _ { q } \right ] } { \mathrm { T V } _ { q } ^ { 3 / 2 } ( X ) } \end{align*}


\begin{align*} \left \{ \begin{aligned} \ \d u ^ { n } ( t ) = & \left [ ( 1 + i \alpha ) \Delta u ^ { n } ( t ) - ( \gamma ( t ) + i \beta ) P _ { n } | u ^ { n } ( t ) | ^ { 2 } u ^ { n } ( t ) + P _ n f ( t , u ^ { n } ( t ) ) \right ] \d t \\ & + P _ n g ( t , u ^ { n } ( t ) ) \d W ^ { n } ( t ) \\ \ u ^ { n } ( s ) = & P _ { n } \zeta _ { s } . \end{aligned} \right . \end{align*}


\begin{align*} \mbox { H o r } P _ { \frac { \partial } { \partial m _ i } } \frac { \partial } { \partial m _ i } = \frac { 1 } { 2 n \sigma } \frac { \partial } { \partial \sigma } \mbox { o r } - \frac { 1 } { 2 n \sigma } \frac { \partial } { \partial \sigma } , \end{align*}


\begin{align*} ( A , X ) \ast ( B , Y ) = ( A B + h _ s ( X , Y ) E _ { 1 3 } , X Y ) \end{align*}


\begin{align*} g ( X , Y ) = \sum _ { i = 1 } ^ { m } g ( X , e _ i ) g ( Y , e _ i ) . \end{align*}


\begin{align*} \Delta ^ 1 _ h f ( x ) = f ( x + h ) - f ( x ) , \Delta ^ 2 _ h f ( x ) = f ( x + 2 h ) - 2 f ( x + h ) + f ( x ) , \end{align*}


\begin{align*} 0 = & \nabla _ 1 k _ { 2 3 } - \nabla _ 2 k _ { 1 3 } , \\ 0 = & \nabla _ \alpha k _ { \beta \beta } - \nabla _ \beta k _ { \alpha \beta } , \\ 0 = & \nabla _ \alpha k _ { 3 3 } - \nabla _ 3 k _ { \alpha 3 } . \end{align*}


\begin{align*} G _ X ( \alpha ) = e ^ { \lambda _ Z ( e ^ { \alpha } - 1 - \alpha ) } \ , . \ , \end{align*}


\begin{align*} \sum _ { \gamma \in \mathcal { V } _ { [ n , \ell ] } } w ( \gamma ) = \sum _ { k \in \mathbb { Z } } a _ { 0 } ^ { ( \ell ) } A ^ { ( 1 ) } _ { [ k - 1 , \ell - 1 ] } W _ { [ n - k - 1 , 0 ] } , 1 \leq \ell \leq p . \end{align*}


\begin{align*} & \left \{ ( x , y ) \in I _ { n , \beta } ( w ) \times J _ { n , \beta } ( k ) : | T _ { \beta } ^ n x - f ( x , y ) | < \varphi ( n ) \right \} \\ & \subset \left \{ x \in I _ { n , \beta } ( w ) : | T _ { \beta } ^ n x - f ( x , k \varphi ( n ) / \beta ^ n ) | < 2 \varphi ( n ) \right \} \times J _ { n , \beta } ( k ) \\ & : = \tilde { I } _ { n , \beta } ( w , k ) \times J _ { n , \beta } ( k ) , \end{align*}


\begin{align*} I _ { K _ X } ( q ) = ( d H ) q ^ { H / z } \sum _ { \substack { n \in \mathbb Q _ { \geq 0 } \\ \exists j : n w _ j \in \mathbb Z } } q ^ n \frac { \prod _ { k = 1 } ^ { n d } ( d H + k z ) ( - 1 ) ^ { n d ^ \prime } \prod _ { k = 1 } ^ { n d ^ \prime } ( d ^ \prime H + k z ) } { \prod _ { i = 1 } ^ N \prod _ { 0 < k \leq n w _ i , \langle k \rangle = \langle n w _ i \rangle } ( w _ i H + k z ) } \textbf { 1 } _ { \langle - n \rangle } . \end{align*}


\begin{align*} M _ f S _ \zeta = M _ { f \cdot \tau } H _ { \tau ^ { - 1 } } S _ \zeta = q ^ { - \ell ( \tau ) } M _ { f \cdot \tau } S _ \zeta = q ^ { - \ell ( \tau ) } \widetilde { N } _ { f \cdot \tau } . \end{align*}


\begin{align*} \varphi ( \alpha , \beta ) = \left ( \left | \operatorname { d e t } \left ( H ^ { * } ( \overline { \alpha } ) \right ) \right | - 1 \right ) \left ( \left | \operatorname { d e t } \left ( H ^ { * } ( \overline { \beta } ) \right ) \right | - 1 \right ) . \end{align*}


\begin{align*} \sum _ { \gamma \in \mathcal { L } _ { [ n , j ] } } w ( \gamma ) = a _ { 0 } ^ { ( j ) } A _ { [ n - 1 , j ] } , 0 \leq j \leq p . \end{align*}


\begin{align*} g ( t ) = e ^ { - k t } , k > 0 , \end{align*}


\begin{align*} \C _ { B _ n } ( \sigma _ 1 ) = \langle \sigma _ 1 , \sigma _ 3 , \sigma _ 4 , \ldots , \sigma _ { n - 1 } , \sigma _ 2 \sigma _ 1 \sigma _ 1 \sigma _ 2 \rangle . \end{align*}


\begin{align*} ( A + B ) ^ \dagger = A ^ \dagger + B ^ \dagger , ( \lambda A ) ^ \dagger = \lambda ^ \dagger A ^ \dagger = \lambda A ^ \dagger , A ^ { \dagger \dagger } = A , ( A B ) ^ \dagger = B ^ \dagger A ^ \dagger , \end{align*}


\begin{align*} 0 = \nabla ^ * \nabla \omega - 2 W _ + ( \omega ) + \frac { s } { 3 } \omega \end{align*}


\begin{align*} Q ( f ) = \sum _ { \substack { a , b \geq 0 \\ a + b \geq s } } c _ { a , b } \phi ^ a ( f - r ) ^ b \ , \end{align*}


\begin{align*} \begin{array} { l c c } & \mbox { m a t r i x n o r m } & \mbox { L i p s c h i t z c o n s t a n t } \\ \mbox { B r e g m a n C o n d a t - - V \ ~ u } & \| A \| _ { 1 , 2 } = \sqrt 2 & L _ 1 = \max _ { i , j } | ( C ^ T C ) _ { i j } | \\ \mbox { B r e g m a n P D 3 O } & \| A \| _ 2 \leq 2 & L _ 2 = \| C \| _ 2 ^ 2 . \end{array} \end{align*}


\begin{align*} \lambda _ n = \frac { 1 } { n } \sum _ { k = 1 } ^ { n - 1 } \delta _ { n - k } + \frac { 1 } { n } \lambda _ 1 , n \geq 2 . \end{align*}


\begin{align*} u _ \infty ( x ) = \frac { 1 } { \rho _ f ( x ) } \int _ \R v f _ \infty \d v = 0 . \end{align*}


\begin{align*} A ^ n _ t = n ^ { \alpha + \frac 1 2 } \int ^ t _ 0 [ ( t - \eta _ n ( s ) ) ^ { \alpha } - ( t - s ) ^ { \alpha } ] \sigma ( X ^ n _ { \eta _ { n } ( s ) } ) \ , d W _ s , \end{align*}


\begin{align*} \varepsilon : = \varepsilon _ 7 : = \min \big \{ \varepsilon _ 6 , ( 2 \Lambda _ 1 ^ 2 M _ 3 ) ^ { - 1 / ( 2 \sigma ) } , ( ( c _ 3 - \beta ) / ( 1 + 1 / \beta ) ) ^ { 1 / ( 2 \sigma ) } \big \} . \end{align*}


\begin{align*} \sum _ { G \in I ^ { C _ \ell } } \mu ( U _ { C _ \ell , G } ) = \lim _ { g \to \infty } \lim _ { L \to \infty } \sum _ { G \in I ^ { C _ \ell } } \mu _ { g , L } ( U _ { C _ \ell , G } ) = 1 . \end{align*}


\begin{align*} C _ 1 \sum _ { i = 1 } ^ p \int _ { f _ i ^ { - 1 } ( E _ i ) } f _ i ^ \ast \rho ^ { - \delta - \frac { n } { p } + j } | \partial _ j f _ i ^ \ast u | d \mathbf { x } \leq \int _ M \rho ^ { - \delta - \frac { n } { p } + j } | \hat \nabla ^ j u | d V _ { \hat g } \leq C _ 2 \sum _ { i = 1 } ^ p \int _ { f _ i ^ { - 1 } ( E _ i ) } f _ i ^ \ast \rho ^ { - \delta - \frac { n } { p } + j } | \partial _ j f _ i ^ \ast u | d \mathbf { x } \end{align*}


\begin{align*} \frac { 1 } { 2 } ( L _ Z g ) ( X , Y ) + \frac { 1 } { \lambda ^ 2 } \{ R i c ^ N ( \tilde { X } , \tilde { Y } ) + f _ 3 h ( \tilde { X } , \tilde { Y } ) \} = 0 , \end{align*}


\begin{align*} k ' = \frac { 1 } { 2 \hslash \sqrt { \beta / 3 } } \sqrt { \frac { \sqrt { \frac { 1 6 } { 3 } m \beta | E | } + 1 } { 2 } } . \end{align*}


\begin{align*} K _ 0 ( x ) & = - \left [ \ln \left ( \frac { | x | } 2 \right ) + \gamma \right ] I _ 0 ( x ) + \sum \limits _ { k = 1 } ^ \infty b _ k x ^ { 2 k } , \end{align*}


\begin{align*} \mathcal { K } ^ { n } _ { 0 , j } = \left \{ \pi _ { \bar { X } , \bar { Y } _ { [ 0 : 2 ] } , \bar { Z } _ j } \in \mathcal { P } _ { n } \left ( \mathcal { X } \times \mathcal { Y } _ { [ 0 : 2 ] } \times \mathcal { Z } _ j \right ) \ : \ \pi _ { \bar { X } , \bar { Y } _ { [ 0 : 2 ] } } \stackrel { \delta ^ { \prime } _ { n } } { \approx } p _ { X , Y _ { [ 0 : 2 ] } } \wedge \pi _ { \bar { Y } _ { 0 } , \bar { Y } _ { j } , \bar { Z } _ j } \stackrel { \delta ^ { \prime } _ { n } } { \approx } p _ { Y _ { 0 } , Y _ { j } , Z _ j } \right \} , \end{align*}


\begin{align*} M _ 1 = f _ k ( i ) f _ k ( i + 1 ) + O _ \alpha \Big ( \frac { \tau _ 3 ( i ) \tau _ 3 ( i + 1 ) } { \sqrt { i } } \Big ) . \end{align*}


\begin{align*} \mathcal { L } ( V ) = \mathbb { C } [ t ^ { \frac { 1 } { T } } , t ^ { - \frac { 1 } { T } } ] \otimes V . \end{align*}


\begin{align*} \P ( Y ( t , x ) > R ) & \leq e ^ { - t \lvert D \rvert } R ^ { - 1 - \frac 2 d } \sum _ { N = 1 } ^ \infty \frac { ( C \log R ) ^ N } { N ! N ^ N } \sum _ { L = N } ^ \infty \frac { C ^ { L - N } } { ( L - N ) ! } \leq C R ^ { - 1 - \frac 2 d } e ^ { C ( \log R ) ^ { 1 / 2 } } , \end{align*}


\begin{align*} \sum _ { k = \ell } ^ { \ell + m } \eta _ k ^ 2 & \le \Lambda _ c \eta _ \ell ^ 2 \quad \quad \eta _ { \ell + m } ^ 2 \le \frac { q _ c ^ m } { 1 - q _ c } \eta _ { \ell } ^ 2 \quad \ell , m \in \mathbb { N } _ 0 . \end{align*}


\begin{align*} E = b ^ 2 \sum \limits _ { k = 1 } ^ { 3 } ( - 1 ) ^ { \gamma + \tau } z ^ { k } _ { \tilde { \gamma } } z ^ { k } _ { \tilde { \tau } } \left [ \cos ^ 2 \theta z ^ { 1 } _ { \gamma } z ^ { 1 } _ { \tau } + \sin ^ 2 \theta z ^ { 2 } _ { \gamma } z ^ { 2 } _ { \tau } + 2 \sin \theta \cos \theta z ^ { 1 } _ { \gamma } z ^ { 2 } _ { \tau } \right ] . \end{align*}


\begin{align*} \phi ( \pi ) : = \phi ( \sigma _ \pi , \lambda _ \pi ) \vert _ { W _ \pi } \in ( W _ \pi ) . \end{align*}


\begin{align*} B _ { n } ( x _ 1 , x _ 2 , \ldots , x _ n ) = \sum _ { \pi ( n ) } \frac { n ! } { \prod _ { r = 1 } ^ n k _ r ! } \prod _ { r = 1 } ^ { n } \left ( \frac { x _ r } { r ! } \right ) ^ { k _ r } , \end{align*}


\begin{align*} f ' ( \gamma ( 0 ) ) = f ' ( 0 ) \frac { \rho ( \gamma ) ' ( 0 ) } { \gamma ' ( 0 ) } . \end{align*}


\begin{align*} \langle \otimes _ { l = 1 } ^ { m } \mathfrak { c } \rangle _ { t } = \exp \biggl ( - \frac { t } { 2 } \sum \limits _ { j = 1 } ^ N \sum \limits _ { i , p = 1 } ^ m \otimes _ { l = 1 } ^ m ( E K _ { j } ) ^ { \delta _ { i l } + \delta _ { p l } } \biggr ) \langle \otimes _ { l = 1 } ^ { m } \mathfrak { c } \rangle _ { 0 } , \end{align*}


\begin{align*} \lim _ { t \to \infty } \frac { s _ 0 L _ t } { X _ t } = 1 , \end{align*}


\begin{align*} I _ { \bar { b } } ^ t + b _ k { a } _ \delta , \qquad { \rm w i t h ~ c a r d i n a l i t y } ~ ~ | I _ { \bar { b } } ^ t + b _ k { a } _ \delta | = | I _ { \bar { b } } ^ t | = q ^ { d _ { \bar { b } } t } ; \end{align*}


\begin{align*} \frac { \log ( 3 M ^ 2 / \delta ) } { 2 \varepsilon ^ 2 } = \frac { \log ( M ) } { \varepsilon ^ 2 } + \frac { \log \left ( 3 / \delta \right ) } { 2 \varepsilon ^ 2 } . \end{align*}


\begin{align*} p ^ * ( g ) : = \frac { 1 } { \sqrt { 2 \pi c ^ * } } \exp \left ( - \frac { ( g - b ^ * ) ^ 2 } { 2 c ^ * } \right ) , \end{align*}


\begin{align*} \int \frac { d x } { a + b \sin ^ 2 ( x ) } = \frac { \mathrm { s i g n } ( a ) } { \sqrt { a ( a + b ) } } \arctan \Big ( \sqrt { \frac { a + b } { a } } \tan ( x ) \Big ) . \end{align*}


\begin{align*} & \limsup _ { R \rightarrow \infty } \int _ { B _ { R } ^ { c } ( 0 ) } \frac { | u ( x ) - u ( y ) | ^ { p - 1 } } { | x - y | ^ { n + \sigma p } } d y \\ \leq & \limsup _ { R \rightarrow \infty } \left ( \frac { R } { R - | x | } \right ) ^ { n + \sigma p } \int _ { B _ { R } ^ { c } ( 0 ) } \frac { C ( p ) ( u ^ { p - 1 } ( x ) + u ^ { p - 1 } ( y ) ) } { | y | ^ { n + \sigma p } } d y = 0 , \end{align*}


\begin{align*} \begin{array} [ c ] { c } \frac { \partial } { \partial t } \omega ( t ) = - \mathrm { R i c } _ { \omega ( t ) } ^ { T } - \omega ( t ) , \omega ( 0 ) = \omega _ { 0 } . \end{array} \end{align*}


\begin{align*} \xi = \int ( \delta _ x - \delta _ y ) \ a ( d x , d y ) \ . \end{align*}


\begin{align*} a _ k \psi ^ { ( n + 1 ) } ( k _ 1 , \ldots , k _ n ) = \sqrt { n + 1 } \psi ^ { ( n + 1 ) } ( k , k _ 1 , \ldots , k _ n ) \end{align*}


\begin{align*} F ( x , y ) = \left ( \varphi ^ * \tilde { F } \right ) ( x , y ) = \tilde { F } \left ( \varphi ( x ) , \varphi _ * ( y ) \right ) , \forall ( x , y ) \in T M . \end{align*}


\begin{align*} u _ { r , \alpha } : = \frac { u _ { 2 \alpha + 1 } + u _ { 2 \alpha + 2 } } { \sqrt { 2 } } , u _ { \ell , \alpha } : = \frac { u _ { 2 \alpha + 1 } - u _ { 2 \alpha + 2 } } { \sqrt { 2 } } \end{align*}


\begin{align*} k _ { \pm } ^ { 2 } = \frac { 1 + \sqrt { 1 \pm 1 6 i m \beta \lambda / 3 } } { 4 \beta \hslash ^ { 2 } / 3 } \end{align*}


\begin{align*} n e ^ { - \alpha R } \kappa \sqrt { \ss } \int _ { \widehat { r } } ^ R e ^ { - \beta r _ 0 } \sinh ( \alpha r _ 0 ) d r _ 0 = \begin{cases} O \Big ( n \Big ( \frac { \kappa \sqrt { \ss } } { e ^ { \beta R } } \Big ) ^ { 1 \wedge \frac { \alpha } { \beta } } \Big ) , & \\ [ 8 p t ] \Theta \Big ( n \frac { \kappa \sqrt { \ss } } { e ^ { \beta R } } \log \big ( \frac { e ^ { \beta R } } { \kappa \sqrt { \ss } } \big ) \Big ) , & \end{cases} \end{align*}


\begin{align*} \eta ( x , z ) = f ( x ) + h ( x ) + \gamma \| A x - b \| + b ^ T z + ( f + h ) ^ * ( - A ^ T z ) \end{align*}


\begin{align*} \Phi ( t _ 2 , t _ 1 ) = \Phi ( t _ 2 , \tau ) \Phi ( \tau , t _ 1 ) \quad t _ 2 \geq \tau \geq t _ 1 \geq 0 . \end{align*}


\begin{align*} \Delta ( z ) = ( - 1 ) ^ { r m } \times \ z ^ { r m ( n + 1 ) } \ \ P _ { r m } ( z ) \times \ 1 / z ^ { r m ( n + 1 ) } \ \ \Delta _ { 1 , r m + 1 } ( z ) \enspace . \end{align*}


\begin{align*} \Lambda = \left ( \lambda _ 1 , \lambda _ 2 , \ldots , \lambda _ m \right ) \end{align*}


\begin{align*} \frac { \partial E } { \partial z ^ i _ { \epsilon } } v ^ i = 2 b ^ 2 f ( x ^ 1 ) \cos \left ( x ^ 2 - \theta \right ) \Big [ - \delta _ { \epsilon 1 } f ( x ^ 1 ) f ' ( x ^ 1 ) \cos \left ( x ^ 2 - \theta \right ) + \delta _ { \epsilon 2 } \left \lbrace 1 + f '^ 2 \left ( x ^ 1 \right ) \right \rbrace \sin \left ( x ^ 2 - \theta \right ) \Big ] . \end{align*}


\begin{align*} & A ( X \otimes X ) = 0 , & & ( X ^ * \otimes X ^ * ) A = 0 , & & ( \Psi \otimes \Psi ) ( 1 - A ) = 0 , & & ( 1 - A ) ( \Psi ^ * \otimes \Psi ^ * ) = 0 , \end{align*}


\begin{align*} \int _ { \R } g ( x ) d x = 0 . \end{align*}


\begin{align*} \begin{aligned} h _ * = \arg \max _ { 0 < h { } \leq { } \bar { h } ( \kappa ) } p ( h , \kappa ) . \end{aligned} \end{align*}


\begin{align*} \beta ( \xi _ 1 , \xi _ 2 ) : = \lim _ { n \to \infty } \langle \ , [ U ( \gamma _ n ) ] ( \xi _ 1 ) , \xi _ 2 \ , \rangle \end{align*}


\begin{align*} \mathcal { J } _ { 2 , 1 } = - 2 \mathcal { I } _ { 2 , 1 } + 2 \iint _ { Q _ T } [ \nabla m \cdot \nabla p ] | \nabla p | ^ 2 - p \Delta p [ \nabla \cdot ( m \nabla p ) ] . \end{align*}


\begin{align*} A \phi ( x ) = \nabla \cdot ( Q \nabla \phi ) ( x ) + a \cdot \nabla \phi ( x ) + \int _ { \R ^ d } \left ( \phi ( x + y ) - \phi ( x ) - \nabla \phi ( x ) y 1 _ { \{ | y | < 1 \} } \right ) \nu ( d y ) , \end{align*}


\begin{align*} \left ( X = \C ^ n , W = \sum _ { i = 1 } ^ n x _ i ^ { r _ i } \right ) . \end{align*}


\begin{align*} P ( T , x ) & = \prod _ { v \in V ( T ) } \mathcal { G } ( v , x ) \\ & = \mathcal { G } ( r ( T ) , x ) \prod _ { j = 1 } ^ { d } \left ( \prod _ { v \in V ( T _ j ) } \mathcal { G } ( v , x ) \right ) \\ & = \mathcal { G } ( r ( T ) , x ) \prod _ { j = 1 } ^ { d } P ( T _ j , x ) . \end{align*}


\begin{align*} \gg ^ { \mu } = \left \{ g \in \gg \mid \left [ h , g \right ] = \mu ( h ) g \right \} . \end{align*}


\begin{align*} \tilde { \lambda } _ i ^ k : = \lambda _ i \big ( \sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) \big ) . \end{align*}


\begin{align*} \dfrac { ( f ^ { \tilde { \phi } ^ { - n } } \circ \rho _ { \eta } ( X ) ) ' } { f ^ { \tilde { \phi } ^ { - n } } \circ \rho _ { \eta } ( X ) } = \eta \dfrac { ( f ^ { \tilde { \phi } ^ { - n } } ) ' } { f ^ { \tilde { \phi } ^ { - n } } } \circ \rho _ { \eta } ( X ) . \end{align*}


\begin{align*} ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \ldots , y _ i ) = 0 ( 1 \le k \le i ) \end{align*}


\begin{align*} | I | \leqslant \sum _ { i = 1 } ^ m r _ i ( Y _ i ) \end{align*}


\begin{align*} G _ { k , j ; n , l , r } ^ { \star ; m , i ; p , q } ( t , x , \zeta ) : = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ 0 ^ \pi \int _ { 0 } ^ { 2 \pi } \big [ i \mu \big ( \omega ^ { m ; E } _ { j , l , r } ( t - s , v , \omega ) + \hat { \zeta } \times \omega ^ { m ; B } _ { j , l , r } ( t - s , v , \omega ) \big ) K _ { k ; n } ^ { } ( \mathfrak { m } ) ( y , \zeta ) + \big ( c ^ { a ; m , E } _ { j , l , r } ( t - s , v , \omega ) \end{align*}


\begin{align*} \int _ { D } e ^ { - i \varphi } q ( x ) u _ { 1 } u _ 2 d x = \int _ { D } q ( x ) \ ; e ^ { i x \cdot \xi } \ , d x + \mathcal { R } ' _ { 1 } ( \xi , s ) + \mathcal { R } ' _ { 2 } ( \xi , s ) , \end{align*}


\begin{align*} S _ { 1 } = \left \{ \sum _ { i = 1 } ^ { n } a _ { i } \alpha _ { i } \mid 0 \leq a _ { i } < p q , \ a _ { i } \in \mathbb { Z } , \ 1 \leq i \leq n \right \} . \end{align*}


\begin{align*} & N : = M + 1 , \ \textup { f o r t h e e x p a n s i o n o f } I ^ { \frac { n } { 2 } } _ { y } \circ E ^ { \frac { n } { 2 } } _ { x , M } f ^ { - \alpha } , \\ & N ' : = 2 ( M + 1 ) , \ \textup { f o r t h e e x p a n s i o n o f } E ^ { \frac { n } { 2 } } _ { y , M } \circ E ^ { \frac { n } { 2 } } _ { x , M } f ^ { - \alpha } . \end{align*}


\begin{align*} \left | \mathcal K _ { \kappa , 2 } \right | = \frac 1 2 \left ( \sum _ { i = 0 } ^ { \lfloor \kappa / 2 \rfloor } ( \kappa - 2 i + 1 ) + \binom { \kappa + 3 } { 3 } \right ) . \end{align*}


\begin{align*} { B } w ^ \prime ( t ) - A ^ \ast w ( t ) - Q ^ * B w ( t ) = g ( t ) , \ t \in \ , [ 0 , T ] , w ( T ) = 0 . \end{align*}


\begin{align*} \Omega _ { \rho , \theta } = \left \{ r e ^ { i t } : \rho < r < \frac { 1 } { \rho } , t \in ( \theta , 2 \pi - \theta ) \right \} , \end{align*}


\begin{align*} \partial _ { i _ 1 } ^ { k _ 1 } \ldots \partial _ { i _ s } ^ { k _ s } S ( h ) = n ! \cdot \partial _ { i _ 1 } ^ { k _ 1 } \ldots \partial _ { i _ s } ^ { k _ s } V o l ( h ) = 0 . \end{align*}


\begin{align*} L _ { B } ( x ^ { n } , b _ { \mathcal { T } } ) = \frac { 1 } { p ^ { U } _ { \mathcal { T } } } \sum _ { \pi _ { \bar { Y } _ { \mathcal { T } } | \bar { X } } \in \mathcal { P } _ { n } ( \mathcal { Y } _ { \mathcal { T } } | \pi _ { \bar { X } } ) } N _ { \pi _ { \bar { Y } _ { \mathcal { T } } | \bar { X } } } ( x ^ n , b _ { \mathcal { T } } ) l _ { \pi _ { \bar { Y } _ { \mathcal { T } } | \bar { X } } } ( x ^ n ) , \end{align*}


\begin{align*} A _ { [ n , j ] } = \sum _ { k = i } ^ { n - j + i } \sum _ { \gamma \in \widetilde { \mathcal { D } } _ { [ n , j , k ] } } w ( \gamma ) = \sum _ { k = i } ^ { n - j + i } A _ { [ k , i ] } A _ { [ n - k - 1 , j - i - 1 ] } ^ { ( i + 1 ) } , n \geq j , \end{align*}


\begin{align*} q _ { i } ^ \star = \left \{ \begin{array} { l l } 1 & i \leq \lfloor t I \rfloor \\ t I - \lfloor t I \rfloor & i = \lfloor t I \rfloor + 1 \\ 0 & i > \lfloor t I \rfloor + 1 . \end{array} \right . \end{align*}


\begin{align*} \partial _ { i _ 1 } ^ { k _ 1 } \ldots \partial _ { i _ s } ^ { k _ s } S ( h ) = - \sum _ { l \ne i _ j } \langle \chi , \Lambda ( \rho _ { l } ) \rangle \partial _ l \partial _ { i _ 1 } ^ { k _ 1 - 1 } \ldots \partial _ { i _ s } ^ { k _ s } S ( h ) . \end{align*}


\begin{align*} w _ t = \nabla \lambda _ t \cdot \nabla p + ( k - 1 ) \left [ w ( w + F ) + ( \nabla w + \nabla F ) \cdot ( p \nabla \lambda + 2 \nabla p ) + p ( \Delta w + \Delta F ) \right ] + \end{align*}


\begin{align*} L _ { \Gamma } : \begin{cases} \ell ^ { 2 } ( V ) \rightarrow \ell ^ 2 ( V ) \\ L _ \Gamma \varphi ( v ) = \varphi ( v ) - \frac { 1 } { | \mathcal { E } ( \{ v \} ) | } \sum _ { \{ v , w \} \in E } \varphi ( w ) \end{cases} \end{align*}


\begin{align*} f ( x ) = x ^ 3 + a x ^ 2 . \end{align*}


\begin{align*} t ( v \Uparrow u ) = \cup \{ t ( u ' ) \mid u ' \in { \downarrow _ 1 } ( v ) : q ( u ) \prec q ( u ' ) \} \cup \{ i \} . \end{align*}


\begin{align*} { \mathfrak X } ( G / H \times { \mathbb R } ^ { k } ) ^ { G } = { \mathfrak X } ( G / H \times { \mathbb R } ^ { k } ) ^ { \mathfrak g } , \end{align*}


\begin{align*} | \langle A \cdot u , v \rangle | & = | \langle u , A ^ T \cdot v \rangle | \leq \| u \| _ { - r } \| A ^ T \cdot v \| _ r \leq c \| u \| _ { - r } \| A ^ T \| _ s \| v \| _ r = c \| u \| _ { - r } \| A \| _ s , \end{align*}


\begin{align*} \label [ I e q ] { e q : m c k e a n m a r t n i g a l e p r o b l e m } M ^ \varphi _ t : = \varphi ( \mathsf { X } _ t ) - \varphi ( \mathsf { X } _ 0 ) - \int _ 0 ^ t L _ { f _ s } \varphi ( \mathsf { X } _ s ) \dd s , \end{align*}


\begin{align*} \lim _ { t \to \infty } \frac { B ( X _ t ) } { t } = \lim _ { t \to \infty } \frac { B ( \| Z _ t \| _ { d + 1 } ) } { t } = \frac { s _ 0 \sigma ^ 2 } { 2 c _ 0 } , , \end{align*}


\begin{align*} F \cdot F ^ { ( n ) } = [ n + 1 ] F ^ { ( n + 1 ) } , \end{align*}


\begin{align*} u _ t = \Delta _ x u - | u | ^ { q - 1 } u . \end{align*}


\begin{align*} \lim _ { \mu \to + \infty } \Phi _ \rho ( \mu , 1 ) = \| u \| _ { \infty , B _ 1 ( 0 ) } = \sup _ { B _ 1 ( 0 ) } u . \end{align*}


\begin{align*} { \lim _ { | y | \to \infty } \left ( e ^ { | y | } | y | ^ { \left ( \frac { N - 1 } { 2 } \right ) } { U } _ 0 ( | y | ) \right ) = \alpha , } \end{align*}


\begin{align*} H _ N : = \sum _ { j = 1 } ^ N \big ( - \Delta _ { x _ j } + V _ \mathrm { D W } ( x _ j ) \big ) + \frac { \lambda } { { N - 1 } } \sum _ { 1 \le i < j \le N } w ( x _ i - x _ j ) \end{align*}


\begin{align*} \theta _ k = \frac { \tilde \theta _ k } { 2 } \geq \frac { - L + \sqrt { L ^ 2 + 4 \delta ^ 2 \beta \| A \| ^ 2 } } { 4 \tau _ { k - 1 } \beta \| A \| ^ 2 } . \end{align*}


\begin{align*} C _ 1 ( [ x , y ] _ \mathfrak { g } , z ) + \nabla _ { \varrho ( x ) } ^ 2 ( C _ 1 ( y , z ) ) + \circlearrowleft ( x , y , z ) = d t _ 2 ( C _ 2 ( x , y , z ) ) \end{align*}


\begin{align*} P ( u ) \cdot v _ 0 = \begin{bmatrix} \nabla u \\ \nabla \nabla u \end{bmatrix} v _ 0 = \begin{bmatrix} f \\ h \end{bmatrix} , v _ 0 ( x ) \perp \mathrm { K e r } P ( u ) ( x ) \end{align*}


\begin{align*} \begin{array} { l l l } \mbox { B r e g m a n p r i m a l C o n d a t - - V \ ~ { u } a l g o r i t h m } & \tilde x = 2 x ^ { ( k + 1 ) } - x ^ { ( k ) } & \tilde z = z ^ { ( k ) } \\ \mbox { B r e g m a n d u a l C o n d a t - - V \ ~ { u } a l g o r i t h m } & \tilde x = x ^ { ( k ) } & \tilde z = 2 z ^ { ( k + 1 ) } - z ^ { ( k ) } . \end{array} \end{align*}


\begin{align*} \begin{array} [ t ] { l l } \mbox { m i n i m i z e } & f ( x ) + h ( x ) \\ \mbox { s u b j e c t t o } & A x = b \end{array} \begin{array} [ t ] { l l } \mbox { m a x i m i z e } & - b ^ T z - ( f + h ) ^ * ( - A ^ T z ) . \end{array} \end{align*}


\begin{align*} d _ n ( { t \mathcal K } , N ) _ X = | t | d _ n ( { \mathcal K } , N ) _ X \ \mbox { a n d } \ d _ n ( t \mathcal { K } ) _ X = | t | d _ n ( \mathcal { K } ) _ X , \end{align*}


\begin{align*} x \in \ , & \mathfrak { g } [ 1 ] \longmapsto \Phi _ 0 ( x ) = { \mathcal { L } } _ { \varrho ( x ) } \in \mathfrak { X } _ 0 ( E ) [ 1 ] \\ x \wedge y \in \wedge ^ 2 & \mathfrak g [ 1 ] \longmapsto \Phi _ 1 ( x , y ) = \iota _ { \chi ( x , y ) } \in \mathfrak { X } _ { - 1 } ( E ) [ 1 ] \end{align*}


\begin{align*} f ( y ) = f _ + ' ( 0 ) y + \int _ { 0 } ^ { \infty } f _ t ( y ) d \nu ( t ) \end{align*}


\begin{align*} \lim _ { n \to \infty } \min _ { 1 \le i \le k _ { n } } \mu _ { n , i } ( \{ \zeta \in \mathbb { T } : | \zeta - 1 | < \varepsilon \} ) = 1 \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \overline { \chi } _ t - A _ 2 \overline { \chi } _ { i i } - ( B _ 2 + g ' C _ 2 + D _ 2 ) \overline { \chi } _ i = a _ { 0 2 } \varphi ^ { q } - \lambda _ { 0 2 } \sqrt { A _ 2 } ^ \alpha | \chi _ i | ^ { \alpha } , & t > 0 , \ 0 < i < g _ 0 , \\ \overline { \chi } _ i ( t , 0 ) = \overline { \chi } ( t , g _ 0 ) = 0 , & t > 0 , \\ \overline { \chi } ( 0 , i ) = \chi _ 0 ( i ) , & 0 \leq i \leq g _ 0 , \\ \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} x & \longmapsto x \exp ( ( a + 1 ) g x ^ a y ^ b ) \\ y & \longmapsto y \exp ( - ( a + 1 ) g x ^ a y ^ b ) . \end{aligned} \end{align*}


\begin{align*} I _ { 1 } = \int _ { \mathbb { R } } \frac { d \nu ( s ) } { s ^ { 2 } } = \lim _ { y \rightarrow 0 ^ { + } } \frac { - \Im G _ { \nu } ( i y ) } { y } \in ( 0 , + \infty ] \end{align*}


\begin{align*} \varphi _ { \mu _ { 1 } \boxplus \mu _ { 2 } } = \varphi _ { \mu _ { 1 } } + \varphi _ { \mu _ { 2 } } \end{align*}


\begin{align*} \delta _ { } : = D ^ { 1 / 2 } . \end{align*}


\begin{align*} \| \nabla \varphi \| _ { L ^ q ( \nu _ t ; \R ^ d ) } ^ q = \int _ { \R ^ d } | \nabla \varphi | ^ q d \mu + t \int _ { \R ^ d } | \nabla \varphi | ^ q d h _ 2 = \| \nabla \varphi \| _ { L ^ q ( \mu ; \R ^ d ) } ^ q + O ( t ) , \ t \downarrow 0 . \end{align*}


\begin{align*} H _ v : = \left [ \partial _ 1 F ( v ) , \dots , \partial _ n F ( v ) \right ] . \end{align*}


\begin{align*} ( ( - \Delta ) ^ s + q _ { \gamma } ) v & = 0 \quad \quad \Omega \\ v & = f \quad \quad \Omega _ e \end{align*}


\begin{align*} \partial _ t u _ R + \Delta ^ 2 u _ R = \Delta f _ R ( u _ R ) + \sigma ( u _ R ) \dot { W } , R \ge 1 , \end{align*}


\begin{align*} v ' = v _ 0 - \frac { \langle v _ 0 , w \rangle _ { \mathbb { R } ^ N } } { \langle w , w \rangle _ { \mathbb { R } ^ N } } w . \end{align*}


\begin{align*} \sum _ { \gamma \in \mathcal { S } _ { 1 } ( n + 1 , 0 ) } w ( \gamma ) = \sum _ { k = 0 } ^ { p } a _ { q } ^ { ( k ) } \Big ( \sum _ { \gamma \in \mathcal { S } _ { 1 } ( n , k ) } w ( \gamma ) \Big ) = \sum _ { k = 0 } ^ { p } a _ { q } ^ { ( k ) } \langle \mathcal { H } _ { q } ^ { n } e _ { k } , e _ { 0 } \rangle = \langle \mathcal { H } _ { q } ^ { n + 1 } e _ { 0 } , e _ { 0 } \rangle \end{align*}


\begin{align*} \hat { u _ p ^ 0 } = u _ p ^ 0 + \delta , \hat { h _ p ^ 0 } = h _ p ^ 0 + \sigma \end{align*}


\begin{align*} \hat { A } : = J ^ { - 1 } A J , \end{align*}


\begin{align*} ( \phi ^ t _ { X _ H } ) ^ * \omega _ Q = \omega _ Q \ ; . \end{align*}


\begin{align*} \Phi ( \eta _ { \mu } ( z ) ) = z \end{align*}


\begin{align*} \mathrm { M D T E } _ { ( \boldsymbol { a } , \boldsymbol { b } ) } ( \mathbf { P } ) = \left ( \begin{array} { c c c c c c c c c c c } 3 . 4 4 5 9 \\ 4 . 6 0 4 7 \end{array} \right ) , \mathrm { M D T C o v } _ { ( \boldsymbol { a } , \boldsymbol { b } ) } ( \mathbf { P } ) = \left ( \begin{array} { c c c c c c c c c c c } 0 . 6 0 0 8 & 0 . 2 4 3 6 \\ 0 . 2 4 3 6 & 0 . 2 7 6 0 \end{array} \right ) . \end{align*}


\begin{align*} w _ 0 = \sum _ { j = 0 } ^ J \sum _ { i = 0 } ^ J \alpha _ j { \sf A } _ { j i } e _ i ( z ) + \sum _ { j = 0 } ^ J \alpha _ j e _ { j , J } ^ \bot . \end{align*}


\begin{align*} U = \sqrt { \frac { 2 L \ \Delta } { 1 + \sum \limits _ { i = 0 } ^ { N - 1 } \Big ( 2 h _ i - h _ i ^ 2 \frac { - \kappa } { 1 - \kappa } \Big ) } } \end{align*}


\begin{align*} \frac { c _ { n } } { c _ { n - 2 , \xi _ { \boldsymbol { s } k } , \xi _ { \boldsymbol { t } l } } ^ { \ast \ast } } = & \frac { 2 ^ { ( n - 6 ) / 2 } } { \Gamma ( n - 1 ) \pi } \bigg \{ \int _ { 0 } ^ { \infty } t ^ { \frac { n - 4 } { 2 } } \left [ ( 2 + 3 \sqrt { 2 } ) \left ( t + \frac { \xi _ { \boldsymbol { s } , k } ^ { 2 } } { 2 } + \frac { \xi _ { \boldsymbol { t } , l } ^ { 2 } } { 2 } \right ) + 3 \right ] \exp \left ( - \sqrt { 2 t + \xi _ { \boldsymbol { s } , k } ^ { 2 } + \xi _ { \boldsymbol { t } , l } ^ { 2 } } \right ) \mathrm { d } t \bigg \} . \end{align*}


\begin{align*} \norm { h } ^ 2 = u ( a ) . \end{align*}


\begin{align*} J ^ { e s s ; 0 } _ { k , m ' } ( t , x ) : = \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \mathcal { K } _ { k , j } ^ { e s s } ( y , \omega , v ) f ( s , x - y + ( t - s ) \omega , v ) \varphi _ { m ' ; - k } ( t - s ) \varphi _ { m ; - 1 0 M _ t } ( t - s ) d \omega d y d v \big | _ { s = 0 } ^ t \end{align*}


\begin{align*} F _ { s } ( \alpha _ i / z ) = \sum _ { k = 0 } ^ { \infty } \dfrac { 1 } { ( k + x + 1 ) ^ { r - s } } \cdot \dfrac { \alpha ^ { k + 1 } _ i } { z ^ { k + 1 } } = \Phi _ { r - s } ( x , \alpha _ i / z ) \ \ ( 1 \le i \le m , 0 \le s \le r - 1 ) \enspace , \end{align*}


\begin{gather*} y _ { n } : = Q \left ( y _ { n - 1 } \right ) \end{gather*}


\begin{align*} \{ \sigma : \Gamma \to \{ \pm 1 \} \ | \ \sigma ( \gamma _ 1 + \gamma _ 2 ) = ( - 1 ) ^ { \langle \gamma _ 1 , \gamma _ 2 \rangle } \ \sigma ( \gamma _ 1 ) \sigma ( \gamma _ 2 ) \} \ . \end{align*}


\begin{align*} v o l ( D ^ 2 _ x ( 1 ) ) : = \frac { v o l ( B ^ 2 ( 1 ) ) b ^ 2 A ^ { \epsilon \eta } z ^ 3 _ { \epsilon } z ^ 3 _ { \eta } } { 2 \sqrt { \det A } } . \end{align*}


\begin{align*} & \sum _ { x \in \xi + \ell \mathbb { Z } ^ 4 } w ( h ^ { - 1 } x ) e _ k ( a F _ h ( x ) ) H \big ( h ^ { - 1 } k , h ^ { - 2 } F _ h ( x ) \big ) \\ = & \sum _ { { \tiny b \ , \hbox { \tiny m o d } \ , k } } \sum _ { { \tiny \begin{tabular} { c } $ z \ , \hbox { \tiny m o d } \ , k \ell $ \\ $ z = \xi ( \ell ) , z = b ( k ) $ \end{tabular} } } e _ k ( a F _ h ( b ) ) \left ( \sum _ { y \in \mathbb { Z } ^ 4 } f ( y ) \right ) , \end{align*}


\begin{align*} c = ( c _ 0 , c _ 1 , \cdots , c _ { n - 1 } ) \in C ~ \implies ~ \rho ( c ) = ( c _ { n - 1 } , c _ 0 , \cdots , c _ { n - 2 } ) \in C . \end{align*}


\begin{align*} \rho ^ { \rm ( s t ) } = \exp \biggl ( \frac 1 2 \mathfrak { a } ^ { T } M \mathfrak { a } + s \biggr ) \end{align*}


\begin{align*} b \nabla v = 0 \ \ \ \ \Omega \nabla v = \nabla u = 0 \omega _ b . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb P \big ( R _ 2 ^ { ( n ) } \ge 3 ^ { - 2 p } \big ) = 0 . \end{align*}


\begin{align*} \widehat { T _ \ell } [ v _ 1 , \cdots , v _ p ] _ k = \sum _ { \substack { k = k _ 1 - k _ 2 + \cdots - k _ { p - 1 } + k _ p \\ | k _ \ell | \gg k _ 2 ^ * \\ C a s e \ , B } } \frac { \prod _ { \substack { j = 1 \\ o d d } } ^ { p } \widehat { v } _ { k _ i } \prod _ { \substack { j = 2 \\ e v e n } } ^ p \overline { \widehat { v } _ { k _ i } } } { \Phi ( k , k _ 1 \cdots , k _ p ) } , \end{align*}


\begin{align*} D _ A w ( t ) = v ( t ) \end{align*}


\begin{align*} I = \{ l \mid 1 \leq l \leq d \mbox { s u c h t h a t } l = p ^ i \mbox { f o r s o m e } i \in \N \} \end{align*}


\begin{align*} \mbox { c a r d } \ , \{ \mbox { u p s t e p s i n } \ , \gamma \} = p \times \mbox { c a r d } \ , \{ \mbox { d o w n s t e p s i n } \ , \gamma \} + j . \end{align*}


\begin{align*} \tilde { \mu } _ { V X ^ { ( n ) } } ( A ) : = \mathbb { P } \Big ( \Big ( \frac { 1 } { \beta } \Big ) ^ { 1 / p } \sum _ { j = 1 } ^ n Z _ j V _ { \bullet , j } \in A \Big ) , \end{align*}


\begin{align*} \mathbb K : = \left \{ f \in \mathbb L \otimes \mathbb Q | \left \{ i \in \{ 1 , \ldots , m \} | D _ i \cdot f \in \mathbb Z \right \} \in \mathcal A \right \} . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 3 c _ i = 0 . \end{align*}


\begin{align*} f ( A ) = \sum _ { a \in A } \frac { 1 } { a \log { a } } < \sum _ { a \in A } \frac { 1 } { a \log P ( a ) } \approx e ^ \gamma \sum _ { a \in A } \frac { 1 } { a } \prod _ { p < P ( a ) } \Big ( 1 - \frac { 1 } { p } \Big ) . \end{align*}


\begin{align*} \{ 1 , \ldots , k + 1 \} = R _ 1 \cup \cdots \cup R _ p \end{align*}


\begin{align*} x _ { e } = - \sum _ { \{ i , c \} \in E ( \mathsf { P } _ { n + 2 } ) \atop b < i < c } x _ { i c } + \sum _ { \{ j , d \} \in E ( \mathsf { P } _ { n + 2 } ) \atop c \leq j < d } x _ { j d } . \end{align*}


\begin{align*} \label [ I e q ] { e q : P P l a w e m p i r i c a l p r o c e s s } F ^ N _ t = ( \boldsymbol { \mu } _ N ) _ \# S ^ N _ t ( f ^ N _ 0 ) , \overline { F } { } ^ N _ t : = ( \overline { S } _ t \circ \boldsymbol { \mu } _ N ) _ \# f ^ N _ 0 , \end{align*}


\begin{align*} k = \frac { \sqrt { \sqrt { 1 + \frac { 1 6 } { 3 } \beta m E } - 1 } } { 2 \hslash \sqrt { \beta / 3 } } , k ' = \frac { \sqrt { \sqrt { 1 + \frac { 1 6 } { 3 } \beta m E } + 1 } } { 2 \hslash \sqrt { \beta / 3 } } . \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } \bigg ( \int _ t ^ { \infty } u \bigg ) ^ { \frac { 1 } { 1 - q } } V _ r ( 0 , t ) ^ { \frac { q } { 1 - q } } = 0 . \end{align*}


\begin{align*} \| \varphi ( \cdot + a ) \| _ { \dot { H } ^ { 1 , q } ( \mu ^ { - a } * \gamma _ { \sigma } ) } ^ q & = \int _ { \R ^ d } | \nabla \varphi ( \cdot + a ) | ^ q d ( \mu ^ { - a } * \gamma _ { \sigma } ) \\ & = \int _ { \R ^ d } | \nabla \varphi | ^ q d ( \mu * \gamma _ { \sigma } ) = \| \varphi \| _ { \dot { H } ^ { 1 , q } ( \mu * \gamma _ { \sigma } ) } ^ q . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ { m } j | c _ j | = O \left ( m v \left ( \frac { 1 } { m } \right ) \right ) , \end{align*}


\begin{align*} \max \left \{ \sum _ { j = 1 } ^ m F ( x ^ j ) : \pi \in \Pi \land x ^ j \in P ( \chi _ { H ^ j ( \pi ) } ) \right \} \end{align*}


\begin{align*} \kappa _ 2 ^ 2 \int _ 0 ^ \tau \left | \sum _ { i = 1 } ^ Q \rho _ i \mathbf { 1 } _ { [ 0 , t _ i ] } ( \tau ) ( t _ i - s ) ^ { \alpha } ( \sigma ' \sigma ) ( X _ s ) \right | ^ 2 d s . \end{align*}


\begin{align*} \Delta _ { h } : = \frac { 1 } { 2 } \frac { \partial ^ 2 } { \partial r ^ 2 } + \frac { \alpha } { 2 } \frac { 1 } { \tanh ( \alpha r ) } \frac { \partial } { \partial r } + \frac { 1 } { 2 \sinh ^ 2 ( \beta r ) } \frac { \partial ^ 2 } { \partial \theta ^ 2 } \end{align*}


\begin{align*} \int _ { G } f ( g ) d g = \int _ S \int _ 0 ^ \infty f ( \gamma _ \xi ( t ) ) t ^ { Q - 1 } d t d \sigma ( \xi ) \end{align*}


\begin{align*} \alpha _ { i } \cdot \varpi _ J = 0 , \end{align*}


\begin{align*} \Delta ( z ) = \sum _ { n = 1 } ^ \infty \tau ( n ) q ^ n , \end{align*}


\begin{align*} E _ { k , t } ^ { } = \sum _ { i = 1 } ^ n \tau \xi ^ { } _ k \left ( f ^ { } _ { k , t , i } \right ) ^ 3 , \end{align*}


\begin{align*} \left [ u , \left [ x , z \right ] \right ] = \left ( - 1 \right ) ^ { p \left ( u \right ) } \left [ x , \left [ u , z \right ] \right ] \in \left [ x , \gg \right ] . \end{align*}


\begin{align*} \iota _ { \Lambda } ^ * o _ { 0 , k _ 1 , k _ 2 , 1 , \{ ( a _ i , b _ i ) \} _ { i \in I } } = o _ N \otimes \left ( o _ { v _ 1 } \boxtimes o _ { v _ 2 } \right ) , \end{align*}


\begin{align*} \vec { b } \vec { A } _ k = c _ k \equiv 0 \pmod { u _ k } \ , \end{align*}


\begin{align*} R ( U , Y ) Z = \nabla _ U \nabla _ Y Z - \nabla _ Y \nabla _ U Z - \nabla _ { [ U , Y ] } Z , \end{align*}


\begin{align*} \mathbb { E } ( X _ i ) = \mathbb { P } ( P _ i \ { \rm { i s } } \ k - { \rm { f r e e } } ) , \end{align*}


\begin{align*} \int _ { \{ v \cdot n _ x < 0 \} } R ( u \to v ; x ) | v \cdot n _ x | \d v = 1 . \end{align*}


\begin{align*} \Delta ( E ) = E \otimes 1 + K \otimes E , \Delta ( F ) = 1 \otimes F + F \otimes K ^ { - 1 } , \Delta ( K ) = K \otimes K . \end{align*}


\begin{align*} \limsup _ { t \to T } ( T - t ) ^ \frac { 1 } { p - 1 } \| u ( t ) \| _ \infty & < \infty ( ) , \\ \limsup _ { t \to T } ( T - t ) ^ \frac { 1 } { p - 1 } \| u ( t ) \| _ \infty & = \infty ( ) . \end{align*}


\begin{align*} \begin{array} { l l } \frac { 1 } { 2 } \{ g ( \nabla _ U \xi , V ) + g ( \nabla _ V \xi , U ) \} + R i c ^ \nu ( U , V ) - ( m - n ) g ( U , V ) \| H \| ^ 2 \\ + \sum \limits _ { j = 1 } ^ { n } g ( \nabla _ { X _ j } H , X _ j ) g ( U , V ) + \mu g ( U , V ) = 0 . \end{array} \end{align*}


\begin{align*} \chi ^ { \epsilon , L } ( \lambda , \mu ) : = \epsilon ^ d \sum _ { x \in \Lambda _ { \epsilon , L } } \langle \varphi _ 0 \varphi _ x \rangle ^ { \epsilon , L } _ { \lambda , \mu } \leq \bar \chi . \end{align*}


\begin{align*} B _ { \gamma } \colon H ^ s ( \R ^ n ) \times H ^ s ( \R ^ n ) \to \R , B _ { \gamma } ( u , v ) \vcentcolon = \int _ { \R ^ { 2 n } } \Theta _ { \gamma } \nabla ^ s u \cdot \nabla ^ s v \ , d x d y \end{align*}


\begin{align*} \langle q , v ^ 2 \rangle = \langle q _ s , v ^ 2 \rangle + \langle q _ 0 , v ^ 2 \rangle \geq \langle q _ s , v ^ 2 \rangle \end{align*}


\begin{align*} ( \cos ( t \sqrt L ) f , g ) = 0 \quad | t | < d _ { \mathrm { C C } } ( U , V ) . \end{align*}


\begin{align*} \| e _ B ( T - T _ n ) e _ B \| _ { \infty , 1 } = \sup _ { a , b \in ( B ) _ 1 } \langle e _ B ( T - T _ n ) e _ B \hat a , b \rangle \leq \| T - T _ n \| _ { \infty , 1 } , \end{align*}


\begin{align*} \phi ( N \times \{ 0 \} ) = S ^ { 1 } \cdot x . \end{align*}


\begin{align*} \begin{aligned} x _ i { } & = { } \frac { U _ * } { L } + \frac { U _ * } { L } \sum \limits _ { j = \textbf { \textit { i } } } ^ { N - 1 } h _ j \\ g _ i { } & = { } U _ * \\ f _ i { } & = { } \Delta - \frac { U _ * ^ 2 } { 2 L } \sum \limits _ { j = 0 } ^ { i - 1 } h _ j \ , \Big ( 2 - h _ j \frac { - \kappa } { 1 - \kappa } \Big ) \end{aligned} \end{align*}


\begin{align*} M _ { r } ^ { N } ( G ) \leq \sum _ { n = N } ^ { \infty } r ^ { 2 n - 1 } = \frac { 1 } { r } \left ( \frac { r ^ { 2 N } } { 1 - r ^ 2 } \right ) . \end{align*}


\begin{align*} \partial _ s V _ s & = - \partial _ s X _ s - [ V _ s , \Phi _ { s * } X ] \\ & = - \partial _ s X _ s + [ X _ s , \widehat \psi _ { s * } X ] \end{align*}


\begin{align*} f ( x ) = x ^ 4 + a x ^ 3 . \end{align*}


\begin{align*} \beta ^ { \prime } ( 0 ) = \frac { \gamma } { 2 } + \frac { 1 } { 2 } \ln \left ( \frac { \pi } { 2 } \right ) - \frac { 2 } { \pi } \beta ^ { \prime } ( 1 ) . \end{align*}


\begin{align*} \eta = \eta ^ \parallel \otimes \eta ^ \perp \end{align*}


\begin{align*} \dot { x } = f ( t , x ) + g ( t , x ) u \end{align*}


\begin{align*} K _ 1 ^ 2 \partial _ i \bar { \partial } _ j \log K _ 1 & = K _ 2 ^ 2 \partial _ i \bar { \partial } _ j \log K _ 2 + K _ 3 ^ 2 \partial _ i \bar { \partial } _ j \log K _ 3 \\ & \quad \quad + K _ 2 \partial _ i \bar { \partial } _ j K _ 3 + K _ 3 \partial _ i \bar { \partial } _ j K _ 2 - \partial _ i K _ 2 \bar { \partial } _ j K _ 3 - \partial _ i K _ 3 \bar { \partial } _ j K _ 2 . \end{align*}


\begin{align*} T _ k ( l , i , j , \alpha ) = & f _ k ( j ) + O _ { \alpha } \Big ( \frac { \tau _ 3 ( j ) } { \sqrt { j - i } } \Big ) , \end{align*}


\begin{align*} \begin{bmatrix} \omega _ i ^ 0 \\ a _ { i } ^ 0 \end{bmatrix} = S ^ { - 1 } \begin{bmatrix} a _ { x , i } + \dot { y } _ i \dot { \psi } _ i \\ a _ { y , i } - \dot { x } _ i \dot { \psi } _ i \end{bmatrix} , \end{align*}


\begin{align*} \widetilde { U } ( t ) = U ^ { j - 1 } + \frac { t - t _ { j - 1 } } { t _ j - t _ { j - 1 } } \Big ( U ^ { j } - U ^ { j - 1 } \Big ) = \frac { t - t _ { j - 1 } } { \Delta t _ j } U ^ j + \frac { t _ j - t } { \Delta t _ j } U ^ { j - 1 } , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , T ] } E [ | R ^ { n , 2 } _ \tau | ^ 2 ] = 0 . \end{align*}


\begin{align*} J ( x ) = A + \lambda L = \int _ { t _ 0 } ^ { t _ 1 } h ( x ^ 1 , x ^ 2 , \dot { x } ^ 1 , \dot { x } ^ 2 , \lambda ) d t , \end{align*}


\begin{align*} L _ { \sigma , \lambda } \circ \phi ( - \lambda , - \sigma ) = \phi ( \sigma , \lambda ) \circ L _ { \sigma , \lambda } . \end{align*}


\begin{align*} C _ s ( \tau ) & = e ( \tau ) - \sum _ { i < s + 1 } ( - 1 ) ^ i A _ s ( \tau ^ i ) , \\ S _ s ( \tau ) & = d \left ( C _ s ( \tau ) * F ( \tau ) \right ) , \textnormal { a n d } \\ A _ { s + 1 } ( \tau ) & = ( - 1 ) ^ { s + 1 } C _ s ( \tau ) * F ( \tau ) . \end{align*}


\begin{align*} { \cal J } \left [ X \right ] = { \rm D e t } \left ( { \frac { { \delta { A _ { i j } } \left ( t \right ) } } { { \delta { X _ { k p } } \left ( { t ' } \right ) } } } \right ) \end{align*}


\begin{align*} B : = \bigcup _ { a \in A } \phi _ a ^ { - 1 } ( A \setminus \{ a \} ) . \end{align*}


\begin{align*} A = \left [ \begin{matrix} B & { \bf 0 } \\ { \bf 0 } & 1 / 2 \end{matrix} \right ] , B = \left [ \begin{matrix} 1 & 1 \\ \epsilon & 1 \end{matrix} \right ] . \end{align*}


\begin{align*} g _ 1 < g _ 2 \Leftrightarrow \begin{cases} g _ 1 ^ { - 1 } g _ 2 \notin H \ & p ( g _ 1 ) \prec p ( g _ 2 ) , \\ g _ 1 ^ { - 1 } g _ 2 \in H & s ^ { - 1 } g _ 1 < _ H s ^ { - 1 } g _ 2 \textit { w h e r e $ s \in S $ i s s u c h t h a t $ s H = g _ 1 H = g _ 2 H $ } . \end{cases} \end{align*}


\begin{align*} Z _ n = Z _ n ^ * \leq \alpha Z ^ * _ { n - 1 } \end{align*}


\begin{align*} m = M ^ { - 1 } \cdot a . \end{align*}


\begin{align*} K _ { \tilde X } = f _ - ^ * K _ { X _ - } + \left ( 1 + \sum _ { i \in M _ - } D _ i \cdot e \right ) E = f _ + ^ * K _ { X _ + } + \left ( 1 - \sum _ { i \in M _ + } D _ i \cdot e \right ) E . \end{align*}


\begin{align*} \{ \Xi _ { \mu } : = \Xi ^ { r _ 1 } _ { \mu _ 1 } \times \cdots \times \Xi ^ { r _ n } _ { \mu _ n } \ , | \ , \mu \in D \} . \end{align*}


\begin{align*} \overline F _ p = \sum _ { i \in I \cup \{ 0 \} } a _ i Q _ \theta ^ i \end{align*}


\begin{align*} \mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) = & \sum _ { \substack { 0 \leq l \leq i \\ \gcd ( l , i ) \ { \rm { i s } } \ k - { \rm { f r e e } } } } C _ { \alpha } ( i , l ) \sum _ { \substack { 0 \leq m _ 1 \leq 1 \\ \gcd ( l + m _ 1 , i + 1 ) \ { \rm { i s } } \ k - { \rm { f r e e } } } } C _ { \alpha } ( 1 , m _ 1 ) \\ & \Big ( f _ k ( j ) f _ k ( j + 1 ) + O _ { \alpha } \Big ( \frac { \tau _ 3 ( j ) \tau _ 3 ( j + 1 ) } { \sqrt { j - i - 1 } } \Big ) \Big ) . \end{align*}


\begin{align*} \left \lbrace \begin{aligned} \frac { d x } { d t } & = 5 \sin 1 0 t - x + y , \\ \frac { d y } { d t } & = 5 \cos 1 0 t + x + y , \end{aligned} \right . \end{align*}


\begin{align*} u \times v \mapsto \langle u | v \rangle _ t = e ^ { t ( ( \tilde { f } ^ { | u \rangle } , \tilde { f } ^ { | v \rangle } ) _ { { } _ { \mathfrak { J } } } + 4 \pi ) } = e ^ { - t 4 \pi ( \lambda \textrm { c o t h } \lambda - 1 ) } \end{align*}


\begin{align*} \cap _ { i = 1 } ^ { m } C _ { i } \subset \cap _ { n = 0 } ^ { \infty } \mathrm { F i x } ( S _ { n } ) . \end{align*}


\begin{align*} s _ 0 = 1 , s _ 1 = n H , s _ 2 = \left [ n ^ 2 - \binom { n } { 2 } \right ] H ^ 2 , s _ 3 = \left [ n ^ 3 + \binom { n } { 3 } - 2 n \binom { n } { 2 } \right ] H ^ 3 . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \widehat { \omega } _ { t } = \frac { 1 } { T _ { 0 } } ( ( T _ { 0 } - t ) \omega _ { 0 } - t \psi ^ { \ast } \omega _ { N } ) \in \lbrack \omega ( t ) ] _ { B } = [ \omega _ { 0 } ] _ { B } - t c _ { 1 } ^ { B } ( M ) . \end{array} \end{align*}


\begin{align*} G [ u ] : = G ( \cdot , u , D u ) = 0 , { \rm o n } \ \partial \Omega , \end{align*}


\begin{align*} p _ { \varepsilon } ( \rho ) = \begin{cases} p ( \rho ) & \mbox { f o r } \rho \in [ 0 , \overline { \rho } - \varepsilon ] \\ p ( \overline { \rho } - \varepsilon ) + | ( \rho - \overline { \rho } + \varepsilon ) ^ { + } | ^ { \gamma } & \mbox { f o r } \rho \in ( \overline { \rho } - \varepsilon , \infty ) , \end{cases} \end{align*}


\begin{align*} M _ { \pm } : = \{ i \in \{ 1 , \ldots , m \} : \pm D _ i \cdot e > 0 \} ; \end{align*}


\begin{align*} \int _ 0 ^ \infty f ^ p ( x ) w ( x ) d x & \leq \frac { \gamma A } { ( \beta + 1 ) ^ { p / p _ 0 } } \int _ 0 ^ \infty g ^ p ( x ) w ( x ) d x \\ & = K \int _ 0 ^ \infty g ^ p ( x ) w ( x ) d x , \end{align*}


\begin{align*} N ( b , c ) : = \frac { 1 } { V _ F } \int _ { 0 } ^ { + \infty } g \frac { 1 } { \sqrt { 2 \pi c } } \exp \left ( - \frac { ( g - b ) ^ 2 } { 2 c } \right ) d g \geq 0 , b \in \mathbb { R } , \ , c > 0 . \end{align*}


\begin{align*} - \Delta _ p u _ i - \lambda ^ i | u _ i | ^ { p - 2 } u _ i = f \Omega , \end{align*}


\begin{align*} \nabla f \cdot \nabla f ( \lambda ) = 2 \sum _ { k = 1 } ^ N \det \left ( ( \lambda I _ N - H ) _ { k | k } \right ) ^ 2 + 4 \sum _ { k = 1 } ^ { N - 1 } \det \left ( ( \lambda I _ N - H ) _ { k | k + 1 } \right ) ^ 2 . \end{align*}


\begin{align*} ( u , v , h , g ) = ( - \phi _ y , \phi _ x , - \psi _ y , \psi _ x ) . \end{align*}


\begin{align*} { \bar \Omega } _ { n + 1 \ , n + 2 } = c ( p ) \ , \omega _ { n + 1 } \wedge \omega _ { n + 2 } \end{align*}


\begin{align*} \frac { \sigma _ { r - 1 } ( n ) } { n ^ { r - 1 } \zeta ( r ) } = \sum _ { m = 1 } ^ { \infty } \frac { c _ m ( n ) } { m ^ r } \end{align*}


\begin{align*} H _ { l , r } ^ { + } = \begin{cases} H _ l H _ { l + 1 } \cdots H _ { r } & \hbox { i f } l \leq r , \\ H _ l H _ { l - 1 } \cdots H _ { r } & \hbox { i f } l > r , \end{cases} \end{align*}


\begin{align*} \dim \left [ \gg ^ { \mu } , x \right ] = \frac { \dim \gg ^ { \mu } } { 2 } . \end{align*}


\begin{align*} & \gcd ( F _ n , F _ m ) = F _ { \gcd ( n , m ) } \mbox { f o r a l l $ m , n \ge 0 $ , $ ( m , n ) \ne ( 0 , 0 ) $ } ; \\ & F _ { n + m } = F _ { n - 1 } F _ m + F _ { n } F _ { m + 1 } \mbox { f o r a l l $ m \ge 0 $ a n d $ n \ge 1 $ } ; \\ & F _ n ^ 2 - F _ { n - r } F _ { n + r } = ( - 1 ) ^ { n - r } F _ r ^ 2 \mbox { f o r a l l $ n \ge r \ge 0 $ } \mbox { ( C a t a l a n i d e n t i t y ) } ; \\ & F _ n ^ 2 - F _ { n - 1 } F _ { n + 1 } = ( - 1 ) ^ { n - 1 } \mbox { f o r a l l $ n \ge 1 $ } \mbox { ( C a s s i n i i d e n t i t y ) } . \end{align*}


\begin{align*} Q ( T , x ) & = \prod _ { j = 1 } ^ { l ( T ) } \mathcal { H } _ j ^ { n ( T , j ) } . \end{align*}


\begin{align*} \limsup _ { n \to \infty } \sup _ { V \in \mathbb { V } _ { n , k } } \mathbb { E } \Big \lVert \sum _ { j = 1 } ^ n n ^ { 1 / p } \frac { Z _ j } { ( \lVert Z ^ { ( n ) } \rVert _ p ^ p + W _ n ) ^ { 1 / p } } V _ { \bullet , j } \Big \rVert _ 2 < \infty , \end{align*}


\begin{align*} \frac { d \mu } { d m } ( \xi ) = \lim _ { r \uparrow 1 } \Re \frac { 1 + \eta _ { \mu } ( r \ , \overline { \xi } ) } { 1 - \eta _ { \mu } ( r \ , \overline { \xi } ) } , \quad \xi \in \mathbb { T } . \end{align*}


\begin{align*} \sigma _ { H T } ( x ) = \frac { 1 } { v o l ( B ^ n ( 1 ) ) } \int _ { F ( x , y ) < 1 } \det ( g _ { i j } ( x , y ) ) d y . \end{align*}


\begin{align*} [ u ] _ { \alpha } : = \sup _ { M ^ 2 _ T } \left \{ \dfrac { | u ( p , t ) - u ( p ' , t ' ) | } { d ( p , p ' ) ^ { \alpha } + | t - t ' | ^ { \alpha / 2 } } \right \} < \infty , \end{align*}


\begin{align*} A ^ * \varphi ( x ) = \int _ { \R ^ d } \varphi ( x - y ) \nu ( d y ) \end{align*}


\begin{align*} \left ( \varrho , u \right ) | _ { t = 0 } = \left ( \varrho _ { 0 } ( x ) , u _ { 0 } ( x ) \right ) , \ x \in \R ^ { d } . \end{align*}


\begin{align*} \mathcal { H } _ { l ( T ) } & = x - 1 , \\ \mathcal { H } _ j & = x - ( c _ j + 1 ) - \dfrac { c _ j } { \mathcal { H } _ { j + 1 } } ( \forall j \in \overline { 2 , l ( T ) - 1 } ) , \\ \mathcal { H } _ 1 & = x - c _ 1 - \dfrac { c _ 1 } { \mathcal { H } _ 2 } , \end{align*}


\begin{gather*} \check h ( \tau ) : = h _ 0 ( 4 \tau ) + h _ 1 ( 4 \tau ) = \sum _ D C _ \phi ( D ) q ^ { - D } , \end{gather*}


\begin{align*} \Xi _ n ( \sigma ) : = [ \theta _ { i j } ] _ { 1 \leq i , j \leq n } \end{align*}


\begin{align*} \psi _ { B _ { k } } = B _ { k } \left [ \frac { \sin ( k x ) } { \sin ( k a / 2 ) } - \frac { \sinh ( k ' x ) } { \sinh ( k ' a / 2 ) } \right ] , \end{align*}


\begin{align*} W ( t ) = \sum _ { n = \ell + 1 } ^ \infty \widetilde { B _ 0 } S _ n ( t ) \in \mathcal { C } _ 1 , \end{align*}


\begin{align*} \begin{array} [ c ] { c } \omega _ { M } : = \psi ^ { \ast } \omega _ { N } + \frac { \sqrt { - 1 } } { 2 \pi } \frac { \varepsilon _ { 0 } } { r _ { S ^ { 1 } } ^ { 2 } } \sum _ { i , j = 1 } ^ { 2 } ( \delta _ { i j } - \frac { \overline { z } _ { i } z _ { j } } { r _ { S ^ { 1 } } ^ { 2 } } ) d z _ { i } \wedge d \overline { z } _ { j } \end{array} \end{align*}


\begin{align*} d V _ { \min } = \sigma _ { \min } ( x ) d x , \end{align*}


\begin{align*} \sum _ { n = 1 } ^ \infty \tau ( n ) q ^ n = q \prod _ { j = 1 } ^ \infty ( 1 - q ^ j ) ^ { 2 4 } \end{align*}


\begin{gather*} \overline { \varrho _ m ( \gamma , \upsilon ) } \theta _ m | _ { \frac 1 2 , m } ( \gamma , \upsilon ) = \theta _ m \end{gather*}


\begin{align*} E _ 0 = E , E _ { j + 1 } = E _ j \cap ( 2 ^ { - k _ j } \oplus E _ j ) , j = 0 , 1 , \ldots , l - 1 . \end{align*}


\begin{align*} \sup _ { N \leq k \leq M } a _ k \bigg ( \sum _ { i = k } ^ M b _ i \bigg ) ^ \alpha \approx \sup _ { N \leq k \leq M } a _ k b _ k ^ \alpha . \end{align*}


\begin{align*} \| \hat { u } _ n \| _ { p - 1 } = 1 , \ \sup _ { n \in \mathbb N } \| a _ n \| _ \infty < \infty , \ \| \hat { f } _ n \| _ { L ^ 1 ( \Omega ) } + \| \hat { g } _ n \| _ \infty \to 0 \hbox { a s } n \to + \infty \end{align*}


\begin{align*} \Psi \left ( F _ * ^ e \left ( c ( I ^ { [ p ^ e ] } : I ) \right ) \right ) = ( F _ * ^ e c \cdot \Psi ) ( I ^ { [ p ^ e ] } : I ) \not \subseteq J . \end{align*}


\begin{align*} R ( W , U , Y , Z ) = g ( R ( W , U ) Y , Z ) . \end{align*}


\begin{align*} & ( a ; q ) _ n = \begin{cases} 1 , & \\ ( 1 - a ) ( 1 - a q ) \cdots ( 1 - a q ^ { n - 1 } ) , & \end{cases} \\ & ( a ; q ) _ \infty = \lim _ { n \to \infty } ( a ; q ) _ n . \end{align*}


\begin{align*} e v _ { \mathcal { S } } ( B ) = A _ { \mathcal { R } _ { B } } \in \mathcal { B } ^ * ( X ) , \end{align*}


\begin{align*} [ u + D \mathcal { L } ( V , g ) , v + D \mathcal { L } ( V , g ) ] = u _ 0 v + D \mathcal { L } ( V , g ) \end{align*}


\begin{align*} S ^ m ( f _ 1 , \dots , f _ m ) ( x ) & \ge C _ m | x | ^ { 1 - m } \int _ { B ^ { m - 1 } ( 0 , \frac 1 { 3 0 0 m } ) } | \tilde { u } | ^ { - \frac 1 p } \left ( - \log ( | \tilde { u } | ) \right ) ^ { - \frac { 2 } { p } } d \tilde { u } \\ & \gtrsim \begin{cases} | x | ^ { 1 - m } & \frac 1 p = m - 1 , \\ \infty & \frac 1 p > m - 1 . \end{cases} \end{align*}


\begin{align*} \mathcal { H } ^ s _ { \infty } ( F ) & \ll \liminf _ { N \to \infty } \sum _ { n = N } ^ { \infty } \beta _ 1 ^ n \beta _ 2 ^ n \beta _ 2 ^ { - n ( 1 - ( 1 + \theta _ 1 ) \log _ { \beta _ 2 } \beta _ 1 ) } \beta _ 1 ^ { - n ( 1 + \theta _ { 1 } ) s } \\ & = \liminf _ { N \to \infty } \sum _ { n = N } ^ { \infty } \beta _ 2 ^ { n \left ( 2 + \theta _ 1 - ( 1 + \theta _ { 1 } ) s \right ) \log _ { \beta _ 2 } \beta _ 1 } \\ & = 0 . \end{align*}


\begin{align*} x _ { * } \in \cap _ { j = 1 } ^ { k } \mathrm { F i x } ( T _ { j } ) \subset \mathrm { F i x } ( Q ) = \cap _ { n = 1 } ^ { j _ { f } } \mathrm { F i x } ( S _ { n } ) = \cap _ { i = 1 } ^ { m } C _ { i } . \end{align*}


\begin{align*} g ^ c _ { k - 2 } ( z , \omega _ 1 , \widetilde \omega _ 1 ) = \int _ { c \cdot C } \frac { - e ^ { ( z + \overline \omega _ 1 ) s } \ , s ^ { k - 2 } } { ( e ^ { \omega _ 1 s } - 1 ) ( e ^ { \widetilde \omega _ 1 s } - 1 ) } \ d s , \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } \| \chi _ { S _ { t ^ \beta } ^ c } J U _ 0 ( t ) f \| = 0 \end{align*}


\begin{align*} g _ { r , x } ( z ) & = g ( x ) \mathrm { d i s t } ( x , z ) \leq \frac { r } { 2 } , \\ g _ { r , x } & = g X \setminus B ( x , r ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : O m e g a i n f t y } \Omega _ \infty \left ( f ^ N , f \right ) : = \mathcal { W } _ p { \left ( \mathrm { L a w } ( \mu _ { \mathcal { X } ^ N } ) , \delta _ f \right ) } \underset { N \to + \infty } { \longrightarrow } 0 , \end{align*}


\begin{align*} S _ { d , i } = \{ i , d + i , 2 d + i , \ldots , ( 4 k - 1 ) d + i \} . \end{align*}


\begin{align*} r _ l = \sum _ { \tau \in Y _ l } 2 m _ { \tau } . \end{align*}


\begin{align*} ( R f ^ { i _ 1 \dots i _ k } ) _ { p _ 1 q _ 1 \dots p _ { m - k } q _ { m - k } } = 0 \mbox { i n } U . \end{align*}


\begin{align*} J _ W : = \left ( \exp \left ( 2 \pi i q _ 1 \right ) , \ldots , \exp \left ( 2 \pi i q _ N \right ) \right ) \in ( \mathbb C ^ \times ) ^ N . \end{align*}


\begin{align*} M _ { \geq \kappa } ( H ^ s \to H ^ { - s } ) \vcentcolon = \{ \ , a \in M ( H ^ s \to H ^ { - s } ) \ , ; \ , a ( f ^ 2 ) \geq \kappa \norm { f } _ { L ^ 2 ( \R ^ n ) } ^ 2 \forall f \in C _ c ^ \infty ( \R ^ n ) \ , \} . \end{align*}


\begin{align*} I - \Sigma \Sigma ^ * = q q ^ * , \end{align*}


\begin{align*} \tanh ( k ' a / 2 ) = k ' a / 2 \end{align*}