\begin{align*} X ^ { m } [ T ] = \lim _ { \ell \to \infty } X ^ { B _ { \ell } } _ { B _ m } [ T ] = X ^ { B _ { M _ m } } _ { B _ m } [ T ] . \end{align*}


\begin{align*} v ( x ) & : = \left ( 1 - \frac \theta 2 \right ) ^ { - n - 2 } u ( a ) \zeta ( x ) - u ( x ) x \in \R ^ n . \end{align*}


\begin{align*} \varphi _ \delta ( t ) = \exp \left ( - t ^ { \frac { \delta } { 2 } } \right ) , t \in ( 0 , + \infty ) , \end{align*}


\begin{align*} \sum _ { l } ( \partial _ { t _ i } \varphi ^ l ) ( \partial _ { t _ j } \varphi ^ l ) = \delta _ { i j } , \end{align*}


\begin{align*} \frac { 2 a \ , e ^ { 2 \gamma a } \left ( e ^ { 2 n \gamma a } - n \ , e ^ { n \gamma a } - 1 \right ) } { \left ( e ^ { a \gamma n } - 1 \right ) ^ 3 } = 0 \Longleftrightarrow e ^ { 2 n \gamma a } - n \ , e ^ { n \gamma a } - 1 = 0 . \end{align*}


\begin{align*} \forall ~ \ell \in \{ 1 , \dots , k - 1 \} , i \in \{ 1 , \ldots , 4 \} ~ \exists ~ ( \ell ' , j ) \neq ( \ell , i ) ~ p _ { \ell , i } = p _ { \ell ' , j } . \end{align*}


\begin{align*} \mathcal { F } ( T ^ \alpha _ t ( g ) ) ( \xi ) = \mathcal { F } ( g ) ( e ^ { - t } \xi ) \frac { \varphi _ \alpha ( \xi ) } { \varphi _ \alpha ( e ^ { - t } \xi ) } . \end{align*}


\begin{align*} \check H ^ q _ { c t } ( \Z ^ n ; A ) = \begin{cases} A \oplus A & n = 1 , q = 0 \\ A & n \not = 1 , q = 0 \vee q = n - 1 \\ 0 & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} W _ k , \zeta _ k = \varphi _ R ( A , b , \lbrace x _ j : j \leq k \rbrace , \lbrace W _ j : j < k \rbrace , \lbrace \zeta _ j : j < k \rbrace ) \in \mathbb { R } ^ { n \times n _ k } \times \mathfrak { Z } . \end{align*}


\begin{align*} \P ( w \ge r ) = \P ( \log ( 1 + \gamma h ) \ge r ) = \P ( \gamma h \ge e ^ r - 1 ) = \Lambda ( e ^ r - 1 ) . \end{align*}


\begin{align*} \begin{aligned} & \lim _ { n \to \infty } P _ { n , k } ( \mathcal { S } ( n , k ; M _ n ) ) = k \sum _ { l = 1 } ^ k \binom k l ( \frac c { 1 - c } ) ^ l \sum _ { s = 1 } ^ \infty ( 1 - c ) ^ { k ( s + 1 ) } \frac 1 { k ( s + 1 ) - l } + \\ & ( 1 - c ) ^ k \sum _ { l = 1 } ^ { k - 1 } \binom k l ( \frac c { 1 - c } ) ^ l , \ \ M _ n \sim c n , \ c \in ( 0 , 1 ) . \end{aligned} \end{align*}


\begin{align*} \tau _ t ( a _ n \cdots a _ 1 v b _ 1 \cdots b _ m ) = t \cdot \tau ' ( a _ n \cdots a _ 1 v b _ 1 \cdots b _ m ) + o ( t ) . \end{align*}


\begin{align*} \forall x , y \in \mathrm { d o m } A ( x = y \rightarrow H ( A x , A y ) = 0 ) . \end{align*}


\begin{align*} C _ { T } ^ { 2 n - 1 } ( V , L ) = \begin{cases} C ^ { 2 n - 1 } ( V , L ) , n \geq 1 , \\ L \wedge L , \quad \ ; \ ; \ ; n = 0 . \end{cases} \end{align*}


\begin{align*} \vartheta _ { \mathbf { d } } : \mathbb { C } [ x _ { \alpha } : \ , \alpha \in [ n ] ] \longrightarrow \mathbb { C } ( \mathbf { t } ) , \quad \vartheta _ { \mathbf { d } } ( x _ { \alpha } ) : = \mathbf { d } ( \mathbf { t } ) _ { \alpha } , \ , \alpha \in [ n ] \end{align*}


\begin{align*} P ( x , \omega ) & = \sum _ { \gamma _ 1 \leq \alpha } \sum _ { \gamma _ 2 \leq \beta } \begin{pmatrix} \alpha \\ \gamma _ 1 \end{pmatrix} \begin{pmatrix} \beta \\ \gamma _ 2 \end{pmatrix} | x ^ { \gamma _ 2 } | \ , | ( 2 \pi i \omega ) ^ { \gamma _ 1 } | \\ & = \prod _ { k = 1 } ^ d ( 1 + | x _ k | ) ^ { \beta _ k } ( 1 + 2 \pi | \omega _ k | ) ^ { \alpha _ k } . \end{align*}


\begin{align*} K ( p ) > 0 \quad \forall p \in \overline { \Omega } \setminus \{ p _ 1 , \ , \cdots \ , , p _ n \} \ \ \mathrm { a n d } \qquad \\ \mathrm { o r d e r } _ { p _ j } ( K ) = 2 \ , \mathrm { o r d e r } _ { p _ j } ( e ) = 2 \ , \mathrm { o r d e r } _ { p _ j } ( g ) \ \ \forall j \in \{ 1 , \cdots , n \} \ , , \end{align*}


\begin{align*} \partial _ z ( \kappa | \partial _ t \eta ^ L | ^ 2 ) = 0 \quad x _ 0 \in \ell , \end{align*}


\begin{align*} { m \left ( \bigcup _ { r > 0 } A ( r ) \right ) = m ( A ) . } \end{align*}


\begin{align*} D _ { 2 , q } : = | \nabla \varphi ( x _ q , t _ q ) | \left ( m ( \{ u ( \cdot , t _ q ) < u ( z _ q , t _ q ) \} ) - m ( \{ u _ { \star , \lambda } ( \cdot , t _ 0 ) < u _ { \star , \lambda } ( x _ 0 , t _ 0 ) \} ) \right ) . \end{align*}


\begin{align*} \sum _ { n \leq x } d _ k ( n + a ( n ) ) = x Q _ k ( \log x ) + O ( x ^ { 3 / 4 + \varepsilon } ) , \end{align*}


\begin{align*} \Psi ' = ( \gamma - 1 - ( k - j ) ) \Phi _ { \gamma - 1 - ( k - j ) - 1 , j } - \Phi _ { \gamma - 1 - ( k - j ) , j + 1 } . \end{align*}


\begin{align*} \int _ { 0 } ^ { + \infty } \dfrac { e ^ { - t } } { \sqrt { 1 - e ^ { - 2 t } } } d t = \frac { \pi } { 2 } . \end{align*}


\begin{align*} \mathcal { I } _ 1 ^ * ( h , k ) = J _ { 1 1 } + J _ { 2 1 } + J _ { 2 3 } + J _ { 3 1 } + J _ { 3 3 } + O \big ( X ^ { - \frac { 1 } { 2 } + \varepsilon } Q ^ { \frac { 5 } { 2 } } ( h k ) ^ { \varepsilon } \big ) + O \big ( X ^ { \varepsilon } Q ^ { \frac { 3 } { 2 } + \varepsilon } ( h k ) ^ { \varepsilon } \big ) , \end{align*}


\begin{align*} I \bigg ( X , \frac { P } { Q } \bigg ) \ = \ I \bigg ( X _ + , \frac { P } { Q } \bigg ) - I \bigg ( X _ - , - \frac { P } { Q } \bigg ) . \end{align*}


\begin{align*} R e ( z ^ * A w ^ 0 ) = c _ 2 \ ; \ ; \forall z \in S ^ { m } _ \alpha \Rightarrow R e ( ( z ^ 0 ) ^ * A w ^ 0 ) = c _ 2 \end{align*}


\begin{align*} - \frac { \log _ { \lambda } ( 1 - t ) } { 1 - t } = \sum _ { n = 1 } ^ { \infty } H _ { n , \lambda } t ^ { n } , ( \mathrm { s e e } \ [ 9 ] ) . \end{align*}


\begin{align*} L ' _ { b } \wedge L ' _ { p c } = L ' _ { i u b } . \end{align*}


\begin{align*} \int _ { \R ^ n _ + } \frac { \dd z } { \vert e _ 1 + z \vert ^ { n + 2 s } } & = \int _ 0 ^ \infty \frac 1 { ( z _ 1 + 1 ) ^ { 1 + 2 s } } \left ( \int _ { \R ^ { n - 1 } } \frac { \dd \tilde z ' } { \big ( 1 + \vert \tilde z ' \vert ^ 2 \big ) ^ { \frac { n + 2 s } 2 } } \right ) \dd z _ 1 . \end{align*}


\begin{align*} \mathcal { T } ( T ) u _ 0 = u ( T , \cdot ; 0 , u _ 0 ) \forall \ , u _ 0 \in \mathcal { E } . \end{align*}


\begin{align*} \iota ( \eta , \xi ) ( x , y ) = \Lambda \left ( \begin{pmatrix} ( \eta ^ { r _ 0 } \circ \xi ) ^ { r _ 1 } \circ \eta ( x ) \\ ( \eta ^ { r _ 0 } \circ \xi ) ^ { r _ 1 - 1 } \circ \eta ( x ) \end{pmatrix} , \begin{pmatrix} \eta ^ { r _ 0 } \circ \xi ( x ) \\ x \end{pmatrix} \right ) . \end{align*}


\begin{align*} \mathbf { w } _ 2 = \sqrt { \frac { 1 } { N _ s } } \textbf { a } _ 2 ^ * ( \theta _ d ) . \end{align*}


\begin{align*} & \partial _ y \Bigg [ \bigg ( \bigg | \sum _ { k = 1 } ^ n \sigma _ k \R _ k \bigg | ^ { p - 1 } - \sum _ { k = 1 } ^ n \R _ k ^ { p - 1 } \bigg ) \eta _ i \frac { \tau _ i ( A _ { i j , 0 } ) } { x _ { i j } ^ 2 } \Bigg ] \\ & = \partial _ y \Bigg [ \bigg ( \sum ^ n _ { \substack { k = 1 , \\ k \not = i } } \frac { ( p - 1 ) \kappa _ 0 \sigma _ k \tau _ i ( Q ^ { p - 2 } A _ { i j , 0 } ) } { x ^ 2 _ { i k } x ^ 2 _ { i j } } \bigg ) \eta _ i \Bigg ] + O ( \Gamma ) = \frac { \partial _ y ( \tau _ i f _ { i j , 4 } ) } { x ^ 2 _ { i j } } + O ( \Gamma ) , \end{align*}


\begin{align*} \begin{array} { c } X _ { p _ 1 } ( \Omega ) = \left \{ f : \Omega \rightarrow \R : \ ; \left \| \frac { \nabla ^ r f } { g } \right \| _ { L _ { p _ 1 } ( \Omega ) } < \infty \right \} , \\ \| f \| _ { X _ { p _ 1 } ( \Omega ) } = \left \| \frac { \nabla ^ r f } { g } \right \| _ { L _ { p _ 1 } ( \Omega ) } . \end{array} \end{align*}


\begin{align*} \pi ( a ) ( b \otimes \chi ) = a b \otimes \chi , \pi ( \omega ) ( b \otimes \chi ) = b _ { ( 2 ) } \otimes \omega ( b _ { ( 3 ) } - S ^ { - 1 } ( b _ { ( 1 ) } ) ) \chi , \end{align*}


\begin{align*} A ( \alpha ) = \sum m _ { i } A ( \alpha _ { i } ) = \sum m _ { i } \int _ { \alpha _ { i } } \lambda . \end{align*}


\begin{align*} M [ u ] = M [ Q ] , E [ u ] = E [ Q ] , \| \nabla u _ 0 \| _ 2 > \| \nabla Q \| _ 2 , \end{align*}


\begin{align*} J X ' ( \theta ) = \lambda H ( \theta ) X ( \theta ) , \end{align*}


\begin{align*} 2 e ( x y ) + 4 ( e x ) ( e y ) = x y . \end{align*}


\begin{align*} \varphi _ \mu ( r \theta ) = \exp \left ( - \frac { r ^ 2 } { 2 } \langle \theta ; \Sigma ( \theta ) \rangle \right ) . \end{align*}


\begin{align*} \alpha = 1 + \alpha _ 1 t + \alpha _ { 2 } t ^ { 2 } + \cdots + \alpha _ { n - 1 } t ^ { n - 1 } \end{align*}


\begin{align*} \lambda _ { n - k + 1 } ( M _ i ^ 2 ) \leq \max _ { f \in V ^ i } \left \{ \frac { \langle f , M _ i ^ 2 f \rangle } { \langle f , f \rangle } \right \} = \max _ { f \in V ^ i } \left \{ \frac { \norm { ( M - \lambda _ i I ) f } _ 2 ^ 2 } { \langle f , f \rangle } \right \} \leq c _ i ^ 2 \end{align*}


\begin{align*} \widetilde { \left ( \overline { L } Z \right ) } \ , \frac { \partial \widetilde { h } ^ { j } } { \partial \overline { \zeta } } = \frac { \widetilde { A } } { \widetilde { C } } \ , \widetilde { h } ^ { j } - \frac { \widetilde { B } } { \widetilde { C } } \ , \overline { \widetilde { h } ^ { j } } + \widetilde { M } , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \Phi ^ { \pm \infty } _ x + i k \sigma _ 3 \Phi ^ { \pm \infty } = Q ^ { \pm \infty } \Phi ^ { \pm \infty } , \\ & \Phi ^ { \pm \infty } _ t + 4 i k ^ 3 \sigma _ 3 \Phi ^ { \pm \infty } = V ^ { \pm \infty } \Phi ^ { \pm \infty } , \end{aligned} \right . \end{align*}


\begin{align*} \lambda _ h = \lambda _ 0 + { \lambda _ 0 } ^ 2 \large \left \langle ( T _ 0 - T _ h ( \lambda _ 0 ) ) u _ 0 , u _ 0 \large \right \rangle + \mathcal { O } ( h ^ 2 ) \end{align*}


\begin{align*} f ( z ) & : = \mathrm { e } _ { \pi _ { \lambda , \beta } } ( z ; 0 ) = \frac { 1 } { l _ { \pi _ { \lambda , \beta } } ( \log ( 1 + z ) ) } = \frac { 1 } { E _ { \beta } ( \lambda z ) } , \\ g ( z ) & : = \mathrm { \mathrm { e } } ^ { x \log ( 1 + z ) } . \end{align*}


\begin{align*} \norm { f } _ m = \sup \{ | \partial ^ \alpha f ( x ) | \mid x \in \Omega , | \alpha | \leq m \} . \end{align*}


\begin{align*} | | F | | _ y = \underset { ( x , y ) \in W } { \operatorname { s u p } } | | \partial _ y F ( x , y ) | | . \end{align*}


\begin{align*} \limsup _ { M \rightarrow \infty } \limsup _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { | x | < M n } \sup _ { s \in [ 0 , T ] } | [ - ( - \Delta ) ^ { \gamma / 2 } G _ s ] ( \tfrac { x - 1 } { n } ) - [ - ( - \Delta ) ^ { \gamma / 2 } G _ s ] ( \tfrac { x } { n } ) | \leq \limsup _ { M \rightarrow \infty } \limsup _ { n \rightarrow \infty } \frac { 2 M K } { n } = 0 . \end{align*}


\begin{align*} \gamma ^ \prime = 3 2 4 - 1 2 \Theta - 4 \Delta \ , ; t = 4 8 0 - 1 2 \Theta \ , ; t ^ \prime = 1 2 0 \ , . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ \infty ( - 1 ) ^ k \frac { \rho _ k } { k } Y ^ k & = - \sum _ { i = 1 } ^ n \sum _ { k = 1 } ^ \infty ( - 1 ) ^ { k - 1 } \frac { ( X _ i Y ) ^ k } { k } = - \sum _ { i = 1 } ^ n \log ( 1 + X _ i Y ) \overset { \eqref { f u n c l o g } } { = } - \log \Bigl ( \prod _ { i = 1 } ^ n ( 1 + X _ i Y ) \Bigr ) \\ & \overset { \eqref { v i e t a 1 } } { = } - \log \Bigl ( 1 + \sum _ { i = 1 } ^ n \sigma _ i Y ^ i \Bigr ) = \sum _ { l = 1 } ^ \infty \frac { ( - 1 ) ^ l } { l } \Bigl ( \sum _ { i = 1 } ^ n \sigma _ i Y ^ i \Bigr ) ^ l . \end{align*}


\begin{align*} x _ 1 ^ \prime = \pm x _ 2 \ , , x _ 2 ^ \prime = x _ 1 \ , , y _ 1 ^ \prime = \pm y _ 2 \ , , y _ 2 ^ \prime = y _ 1 \ , . \end{align*}


\begin{align*} \sum _ { p \leq x , f ( x ) = q } \mu _ { P } ( p , x ) = \sum _ { y \leq q , \ , g ( y ) = p } \mu _ { Q } ( y , q ) \end{align*}


\begin{align*} \beta _ \varepsilon ( t , x ) = \varepsilon \log u _ \varepsilon ( t , x ) , \quad u _ \varepsilon ( t , x ) = \exp \left ( \frac { \beta _ \varepsilon ( t , x ) } { \varepsilon } \right ) , \end{align*}


\begin{align*} [ u , v , w ] _ T = D _ { \rho } ( T u , T v ) w + \theta _ { \rho } ( T v , T w ) u - \theta _ { \rho } ( T u , T w ) v . \end{align*}


\begin{align*} \lim _ { l \to \infty } \limsup _ { n \to \infty } \| r _ n ^ l \| _ { L ^ s ( \mathbb { R } ^ N ) } = 0 , \forall s \in \left ( 2 , 2 ^ * \right ) , 2 ^ * = \begin{cases} \frac { 2 N } { N - 2 } , & N \ge 3 , \\ + \infty , & N = 1 , 2 . \end{cases} \end{align*}


\begin{align*} J = J _ 0 + \frac \partial { \partial x } \otimes d ^ c \left ( \frac 1 2 \log ( 1 + \left | w \right | ^ 2 ) \right ) + \frac \partial { \partial y } \otimes d ^ c \left ( \frac 1 2 \log ( 1 + \left | w \right | ^ 2 ) \right ) \circ J _ 0 . \end{align*}


\begin{align*} K _ { \ell } \otimes \Z _ v & = \begin{cases} \displaystyle { \bigoplus _ { j = 0 } ^ 1 } K _ { \ell , p _ i , j } ( \pi ^ j ) & ( v = p _ i \ \mathrm { f o r } \ i = k ' + 1 , \cdots , k ) , \\ K _ { \ell , v , 0 } & ( \mathrm { o t h e r w i s e } ) , \end{cases} \end{align*}


\begin{align*} C = C ' \big ( 1 + \rho ^ { 2 s } \| c \| _ { L ^ \infty ( B _ \rho ^ + ) } \big ) \end{align*}


\begin{align*} | \alpha ( t ) | = \frac { 1 } { \| Q \| _ 2 ^ 2 } \Big | \int Q h _ 1 d x \Big | + O ( \widetilde { \delta } ^ 2 ) . \end{align*}


\begin{align*} \mathrm { E C C } ( Y , \lambda , \Gamma ) : = \bigoplus _ { \alpha : \mathrm { a d m i s s i b e \ , \ , o r b i t \ , \ , s e t \ , \ , w i t h \ , \ , } { [ \alpha ] = \Gamma } } \mathbb { Z } _ { 2 } \langle \alpha \rangle . \end{align*}


\begin{align*} \eta _ n ( B ) = \frac { 1 } { \mu ( A ' _ n ) } \mu ( B \cap A ' _ n ) \le \frac { 1 } { \mu ( A ' _ n ) } \mu \Big ( \bigcup _ { m \in \mathcal { I } _ n \atop B \cap B _ m \ne \varnothing } B _ m \Big ) \le \frac { \mu ( ( 1 + \theta ) B ) } { \mu ( A ' _ n ) } < \frac { C ^ 2 ( 1 + \theta ) ^ s \mu ( B ) } { C ^ { - 2 } 5 ^ { - s } ( 1 - 1 / 2 n ) } , \end{align*}


\begin{align*} \textbf { w } _ m = \left [ e ^ { j \psi _ { 1 m } } , e ^ { j \psi _ { 2 m } } , \ldots , e ^ { j \psi _ { N _ s m } } \right ] ^ T . \end{align*}


\begin{align*} \Big | \frac { 1 } { T } \int _ { t _ 0 ( \varepsilon ) } ^ T ( 8 s _ c ( p - 1 ) \delta ( t ) + A _ R ( u ( t ) ) d t \Big | = \Big | \frac { 1 } { T } \int _ { t _ 0 ( \varepsilon ) } ^ T \ddot { y } _ R ( t ) d t \Big | \le \frac { C R } { T } , \end{align*}


\begin{align*} h _ t : I _ 0 & \to I _ 0 \\ ( x , y ) & \mapsto \begin{cases} ( x + t , y - t ) & t < y \\ ( x + y , 0 ) & t \ge y . \end{cases} \end{align*}


\begin{align*} \alpha ' = a _ 0 + a _ 1 \alpha + \cdots + a _ { d - 1 } \alpha ^ { d - 1 } \end{align*}


\begin{align*} \mathcal { F } ( \mathbf { h } _ * , \mathbf { g } _ * ) \leq \liminf _ { N \to \infty } \mathcal { F } _ N ( \mathbf { h } _ N , \mathbf { g } _ N ) \leq \limsup _ { N \to \infty } \mathcal { F } _ N ( \mathbf { h } _ N , \mathbf { g } _ N ) \leq \lim _ { N \to \infty } \mathcal { F } _ N ( \mathbf { h } , \mathbf { g } ) = \mathcal { F } ( \mathbf { h } , \mathbf { g } ) . \end{align*}


\begin{align*} \int _ { X _ E } r _ l ( V ) \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} r _ l ( \mathcal { S } ) ( g ) \ , d g = \int _ { X _ F } r _ l ( V ) \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} r _ l ( \mathcal { S } ) ( g ) \ , d g \end{align*}


\begin{align*} \left [ \begin{array} { c } \rho ( \alpha z ) \\ u _ { 0 2 } \end{array} \right ] = R \left [ \begin{array} { c } \alpha z \\ u _ { 0 1 } \end{array} \right ] \mu ( \alpha z ) ^ { - 1 } = R \alpha \left [ \begin{array} { c } z \\ u _ { 0 1 } \end{array} \right ] \lambda ( \alpha , z ) ^ { - 1 } \mu ( \alpha z ) ^ { - 1 } = \end{align*}


\begin{align*} \frac { y _ j } { t ^ { d + 1 } } \partial _ i \eta \left ( \frac { y } { t } \right ) = \partial _ i \left ( \frac { y _ j } { t ^ d } \eta \left ( \frac { y } { t } \right ) \right ) , \end{align*}


\begin{align*} \begin{aligned} \widetilde { U } ( \widetilde { \mathcal { O } } f ) & = \widetilde { U } \Big [ \Big ( \frac { d ^ 2 } { d x ^ 2 } + 2 x \frac { d } { d x } \Big ) f \Big ] \\ & = \Big ( \frac { d } { d x } + x \Big ) \Big ( \frac { d } { d x } - x \Big ) \widetilde { U } f \\ & = - \mathcal { H } ( \widetilde { U } f ) - \widetilde { U } f . \end{aligned} \end{align*}


\begin{align*} d _ { \beta } d ^ c _ { \beta } F & = - \frac { 1 } { 2 } \Big ( \Delta _ { g _ 0 } u + \big \langle 2 t _ 0 \alpha _ 0 - \theta _ { g _ 0 } , d u \big \rangle _ { g _ 0 } + t _ 0 ( 1 - t _ 0 ) | \alpha _ 0 | _ { g _ 0 } ^ 2 u \Big ) e ^ { t _ 0 f } \ , F _ 0 \wedge F _ 0 . \end{align*}


\begin{align*} \boxed { ( 1 + \alpha ) ^ c = \sum _ { k = 0 } ^ \infty \binom { c } { k } \alpha ^ k . } \end{align*}


\begin{align*} \mathcal { U } _ L = \left \{ \exists \ , n \in \N , n \geq L \colon | X _ n - n v | > \tfrac 1 2 \varepsilon n \right \} \end{align*}


\begin{align*} D _ { s } X _ { t } ^ { x ; l ^ \epsilon } = \nabla _ { x } X _ { t } ^ { x ; l ^ \epsilon } ( \nabla _ { x } X _ { \gamma ^ { \epsilon } _ { s } } ^ { x ; l ^ \epsilon } ) ^ { - 1 } . \end{align*}


\begin{align*} \Delta _ { \mathbf { L } ( \mathbf { t } ) } ( \mathcal { I } ) \cdot \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } ) = \mathbf { t } ^ { \mathbf { m } _ { 0 } } \cdot \Delta _ { \mathbf { L } ( \mathbf { 1 } ) } ( \mathcal { I } ) \cdot \Delta _ { \mathbf { R } ( \mathbf { 1 } ) } ( \mathcal { I } ) \cdot \prod _ { \alpha \in \mathcal { I } } \mathbf { t } ^ { \psi ( \alpha ) } , \quad \mathcal { I } \in \wp _ { k } [ n ] . \end{align*}


\begin{align*} J _ { 0 } ( \hat { \alpha } \cup { ( \gamma , N ) } , \hat { \alpha } \cup { ( \delta , 1 ) \cup { ( \gamma , N - p _ { i } ) } } ) = 1 , \end{align*}


\begin{align*} A \otimes ^ I ( A \otimes ^ I A ) = ( A \otimes ^ I A ) \otimes ^ I A = A \otimes ^ I A \otimes ^ I A , \end{align*}


\begin{align*} \inf _ { ( \rho , m ) } \sup _ { S \in H ^ 1 } \mathcal L ( \rho , m , S ) & = \inf _ { ( \rho , m ) } \{ \mathcal A ( \rho , m ) + \mathbb I _ { C _ F ( \rho ^ a , \rho ^ b ) } ( \rho , m ) \} \\ & = \inf _ { ( \rho , m ) \in C _ F ( \rho ^ 0 , \rho ^ 1 ) } \{ \mathcal A ( \rho , m ) \} . \end{align*}


\begin{gather*} D _ { U } \big ( \langle F _ { t } ( W ) , G _ { t } ( W ) \rangle \big ) = \langle D _ { U } F _ { t } ( W ) , G _ { t } ( W ) \rangle + \langle F _ { t } ( W ) , D _ { U } G _ { t } ( W ) \rangle , \intertext { a n d } D _ { U } \nabla \phi ( F _ { t } ( W ) ) = \nabla _ { D _ { U } F _ { t } ( W ) } \nabla \phi ( F _ { t } ( W ) ) . \end{gather*}


\begin{align*} \theta ( U ) = 1 , \theta ^ c ( V ) = 1 , \theta ( V ) = 0 , \theta ^ c ( U ) = 0 . \end{align*}


\begin{align*} \eta _ { { \rm r h s } , 1 } ( E ) = h _ E \Vert f - \Pi _ { E , q - 1 } f \Vert _ { 0 , E } \ , . \end{align*}


\begin{align*} \dim \mathbb { S } ^ \lambda = \frac { n ! } { \prod _ { i , j } h _ { i , j } ( \lambda ) } , \end{align*}


\begin{align*} C _ p q \left | \dot { \xi } ( s ) \right | ^ { q - 1 } = \left | D u ( \xi ( s ) ) \right | = ( f ( \xi ( s ) ) - u ( \xi ( s ) ) ) ^ \frac { 1 } { p } . \end{align*}


\begin{align*} \partial _ t c _ A = \partial _ t ( \zeta \circ d _ \Gamma ) ( c ^ { i n } - \chi _ + + \chi _ - ) + ( \zeta \circ d _ \Gamma ) \theta _ { 0 } ^ { \prime } ( \rho ) ( - \frac { V } { \varepsilon } - \partial _ { t } ^ { \Gamma } h _ { \varepsilon } ) + ( \zeta \circ d _ \Gamma ) ( \varepsilon ^ 2 \partial _ t c ^ { i n } _ 2 + \varepsilon ^ 3 \partial _ t c ^ { i n } _ 3 ) \end{align*}


\begin{align*} \sqrt { y ( t ) } - \sqrt { y ( 0 ) } = \int _ 0 ^ t \frac { \dot { y } ( s ) } { \sqrt { y ( s ) } } d s \le - C \left ( \dot { y } ( t ) - \dot { y } ( 0 ) \right ) \le C \dot { y } ( 0 ) \end{align*}


\begin{align*} \widehat { g } ( \omega ) = C \ , \prod _ { k = 1 } ^ M \frac { e ^ { 2 \pi i \delta _ k \omega } } { 1 + 2 \pi i \delta _ k \omega } , M \in \N . \end{align*}


\begin{align*} - \phi _ { t , u v } = ( \phi _ u \cdot \phi _ v ) \phi _ t + ( \phi _ u \cdot \phi _ { t , v } + \phi _ v \cdot \phi _ { t , u } ) \phi + \frac { 1 } { 4 } a \phi _ { t t } = : G ( t , x ) , \end{align*}


\begin{align*} \int _ { \mathbf { H } } F ( \mathbf { h } ) \ , d \mathbf { h } = \int _ { \mathbf { H } } F ( \mathbf { h } _ 0 \bullet \mathbf { h } ) \ , d \mathbf { h } \end{align*}


\begin{align*} f ( t , x ) : = e ^ { - i t H } u ( t , x ) , H : = - \partial _ { x x } + V , \end{align*}


\begin{align*} X _ v [ T ] : = \lim _ { m \to \infty } X ^ { m } _ v [ T ] = X ^ { m _ v } _ v [ T ] , v \in V _ G , \end{align*}


\begin{align*} \tau _ { \varphi } ( x , y ) = \frac { \varphi ( x ) - \varphi ( y ) } { \| \varphi ( x ) - \varphi ( y ) \| } . \end{align*}


\begin{align*} B _ t + J B _ x + \mu \Lambda ^ \alpha B = 0 , \ \ \ B _ x = \mathcal H J . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ j i { r \choose { i - 1 } } { { r - i } \choose { j - i } } ( - 1 ) ^ { j - i } = r + 1 . \end{align*}


\begin{align*} \sum _ { i \in \mathcal { I } _ { n } ^ { \sigma } } ( \mu _ n ^ { \sigma } ( i ) ) ^ q \prod _ { \ell = 1 } ^ n \big ( \lambda _ i ^ { ( \sigma _ { \ell } ) } \big ) ^ { T _ { \ell } ^ { \boldsymbol { \mu } , \sigma } ( q ) - T _ { \ell - 1 } ^ { \boldsymbol { \mu } , \sigma } ( q ) } = 1 . \end{align*}


\begin{align*} S ' ( H , f ) = \sum _ { 1 \neq h \in H } s ( h , f ) \hbox { a n d } N ( f , H ) : = - 3 + \frac { 2 } { f } + 1 2 S ' ( H , f ) . \end{align*}


\begin{align*} \varphi _ S ( a ) = \varphi ( a \delta _ { \varphi } ) , \varphi _ { S ^ { - 1 } } ( a ) = \varphi ( \delta _ { \varphi } a ) , \forall a \in A . \end{align*}


\begin{align*} \dot { \gamma } ( t ) = - 2 u ' \left ( \gamma ( t ) \right ) = 2 \left ( f \left ( \gamma ( t ) \right ) - u \left ( \gamma ( t ) \right ) \right ) ^ \frac { 1 } { 2 } \end{align*}


\begin{align*} \Phi ( x , y , t ) : = u ( x , t ) - v ( y , t ) - \frac { | x - y | ^ 4 } { \varepsilon ^ 4 } - \alpha ( | x | ^ 2 + | y | ^ 2 ) - \frac { \lambda } { T - t } \end{align*}


\begin{align*} v \xi ^ { - 1 } = s _ 1 \cdots s _ k s _ { k + 1 } \cdots s _ m u \xi ^ { - 1 } \ \ z \eta ^ { - 1 } = s _ { k + 1 } \cdots s _ m u \xi ^ { - 1 } , \end{align*}


\begin{align*} V _ g ( M _ \eta T _ \xi f ) ( x , \omega ) & = \langle M _ \eta T _ \xi f , M _ \omega T _ x g \rangle \\ & = \langle f , T _ { - \xi } M _ { - \eta } M _ \omega T _ x g \rangle \\ & = \langle f , e ^ { 2 \pi i \xi \cdot ( - \eta + \omega ) } M _ { \omega - \eta } T _ { x - \xi } g \rangle \\ & = e ^ { - 2 \pi i \xi \cdot ( \omega - \eta ) } V _ g f ( x - \xi , \omega - \eta ) \end{align*}


\begin{align*} \mathbf { f } ( g ) = j ( k _ { \infty } , z _ 0 ) ^ { - k } \sum _ { \tau \in S } \det ( q _ { \infty } ) ^ { - k } c ( \tau , q ; \mathbf { f } ) e _ { \infty } ( \lambda ( q ^ { \ast } \tau q ) z _ 0 ) e _ { \mathbb { A } } ( \lambda ( \tau \sigma ) ) . \end{align*}


\begin{align*} \alpha = 1 + D t ^ m + \alpha _ { m + 1 } t ^ { m + 1 } + \cdots + \alpha _ { m n } t ^ { m n } \end{align*}


\begin{align*} \underline { k } = j + 1 . \end{align*}


\begin{align*} \mathfrak { c } _ { \mathcal { J } } : = ( \mathfrak { c } _ { r } ) _ { \gamma } ^ { l } \times ( \mathfrak { c } _ { c } ) _ { l } ^ { \gamma } . \end{align*}


\begin{align*} \nabla _ { j } u & = h _ { j l } \bar { g } ( \lambda \partial _ { r } , e _ { l } ) , \\ \nabla _ { i } \nabla _ { j } u & = \nabla _ { i } h _ { j l } \bar { g } ( \lambda \partial _ { r } , e _ { l } ) + \lambda ' h _ { i j } - h _ { i l } h _ { j l } \bar { g } ( \lambda \partial _ { r } , \nu ) \\ & = \nabla _ { l } h _ { i j } \bar { g } ( \lambda \partial _ { r } , e _ { l } ) + \bar { R } _ { \nu j l i } \bar { g } ( \lambda \partial _ { r } , e _ { l } ) + \lambda ' h _ { i j } - u h _ { i l } h _ { j l } . \end{align*}


\begin{align*} m ^ { ( 1 ) } ( x , t , k ) = m ( x , t , k ) e ^ { i t [ g ( k , \xi ) - \theta ( k , \xi ) ] \sigma _ 3 } . \end{align*}


\begin{align*} Q ( x ) = \sum _ { k = 0 } ^ m { n + m + 1 \choose k } x ^ k ( 1 - x ) ^ { m - k } = \sum _ { k = 0 } ^ m { n + k \choose k } x ^ k . \end{align*}


\begin{align*} \widetilde { e } _ m ^ { M , N } = k \sum _ { j = 1 } ^ m E _ { k , N } ^ { m - j + 1 } P _ N A P ( F ( X ( t _ j ) ) - F ( X _ j ^ { M , N } ) ) . \end{align*}


\begin{align*} | \mathcal { V } _ { k } ( \ell ) | = \binom { d } { 2 k - \ell } \cdot \binom { 2 k - \ell - 1 } { k - 1 } \cdot \binom { k - 1 } { \ell } . \end{align*}


\begin{align*} \alpha ' = 1 + D t ^ m + \alpha _ { 2 m } ' t ^ { 2 m } + \cdots + \alpha _ { m n } ' t ^ { m n } \end{align*}


\begin{align*} \psi _ \xi ( u ) : = \varphi ( u ) \wedge \varphi ( u + \xi ) , \end{align*}


\begin{align*} ( s - t ) ^ 2 - \frac { s - t } { h ( s ) } + r ^ 2 = 0 . \end{align*}


\begin{align*} \left \langle \mathrm { P V } \left ( \tfrac { 1 } { x } \right ) , f \right \rangle = \lim _ { \epsilon \searrow 0 } \int _ { \mathbb { R } \setminus ( - \epsilon , \epsilon ) } \frac { f ( x ) } { x } \ , \mathrm { d } x . \end{align*}


\begin{align*} k _ { P _ H } = \frac { \left ( \mathbb { E } [ Y ] - \frac { 1 } { 1 + e ^ { a b } } \right ) ^ 2 } { \mathbb { E } [ Y ^ 2 ] - \mathbb { E } [ Y ] ^ 2 } , \end{align*}


\begin{align*} G _ m ( t ) : = \begin{cases} \log t & m = k \\ - t ^ { 2 - \frac { 2 k } { m } } & m < k . \end{cases} \end{align*}


\begin{align*} p = \exp \left ( - ( e ^ r - 1 ) / c \right ) \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } \frac { 1 } { | W ' _ R | } \int _ { W ' _ R } \chi _ \epsilon ( x , \omega ) \eta ( x ) d x = 0 . \end{align*}


\begin{align*} a : = - 6 \frac { \sigma _ 3 } { \sigma _ 1 } . \end{align*}


\begin{align*} \left ( b _ 2 \mathbb { N } + b _ 1 \Z \right ) \cap \pi \Z = \emptyset \ , . \end{align*}


\begin{align*} \partial _ t u + u \cdot \nabla u - \mu _ 1 \Delta u + \nabla \pi & = \frac { \beta } { k } \mathrm { d i v } \tau , \\ \mathrm { d i v } u & = 0 , \end{align*}


\begin{align*} \ \ & \mbox { $ c _ i = q ' _ i $ f o r $ i = 1 , 2 , \ldots , m u $ , w h e r e $ q ' _ i $ i s d e f i n e d i n ( \ref { q } ) . } \\ \ \ & \mbox { T h e c o n d i t i o n E M U f o r $ m \Delta _ { \overline { t } , \overline { u } , \overline { s } } $ i s n o t s a t i s f i e d i f a n d o n l y i f t h e r e e x i s t s $ i $ } \\ & \mbox { s a t i s f y i n g $ 0 < i < m u $ a n d $ e _ i > i $ . } \end{align*}


\begin{align*} \bar { v } = v _ { \min } + \frac { v _ { \max } - v _ { \min } } { \sqrt { \ell _ k } } \leq v _ k + \frac { \varepsilon _ \bullet } { \sqrt { \ell _ k } } \leq v _ { k + 1 } \end{align*}


\begin{align*} A ^ { 0 } u _ { t } ^ { \epsilon } + A ^ { i } \partial _ { i } u ^ { \epsilon } + D u ^ { \epsilon } - B ^ { i j } \partial _ { i } \partial _ { j } u ^ { \epsilon } = f ^ { \epsilon } + F ^ { \epsilon } , \end{align*}


\begin{align*} \C \cap L ^ 0 _ + = \{ 0 \} , \end{align*}


\begin{align*} \psi ^ { ( - 1 ) } = \Phi \psi . \end{align*}


\begin{align*} \hat { \mu } _ \phi ( k ) = \frac { 1 } { Z ( \phi ) } \frac { \phi ^ k } { g ( k ) ! } , k \in \mathbb { N } _ { 0 } , \end{align*}


\begin{align*} \mathrm { e } _ { \pi _ { \lambda , \beta } } ( z ; \cdot ) : \mathbb { R } \longrightarrow \mathbb { C } , \ ; x \mapsto \mathrm { e } _ { \pi _ { \lambda , \beta } } ( z ; x ) = { \displaystyle \frac { \mathrm { e } ^ { x z } } { l _ { \pi _ { \lambda , \beta } } ( z ) } } = { \displaystyle \frac { \mathrm { e } ^ { x z } } { E _ { \beta } ( \lambda ( \mathrm { e } ^ { z } - 1 ) ) } } . \end{align*}


\begin{align*} { \| \mathcal { F } f ( t , \cdot ) \| } _ { L ^ \infty } = { \| w ( t , \cdot ) \| } _ { L ^ \infty } \leq ( C _ 0 / 2 ) \varepsilon + C \varepsilon ^ 3 \leq C _ 0 \varepsilon . \end{align*}


\begin{align*} B ^ { - 1 } \norm { f } _ \mathcal { H } ^ 2 \leq \langle S ^ { - 1 } f , f \rangle = \sum _ { \gamma \in \Gamma } | \langle f , S ^ { - 1 } e _ \gamma \rangle | ^ 2 \leq A ^ { - 1 } \norm { f } _ \mathcal { H } ^ 2 . \end{align*}


\begin{align*} \rho ( x , \omega , e ^ { 2 \pi i \tau } ) = e ^ { 2 \pi i \tau } \rho ( x , \omega ) . \end{align*}


\begin{align*} \frac { 1 } { n } \sum _ { x } \sup _ { s \in [ 0 , T ] } | \Phi _ { n } ( s , \tfrac { x } { n } ) | \leq M _ 1 \quad \textrm { a n d } \quad \| \Phi _ { n } \| _ { \infty } : = \sup _ { ( s , u ) \in [ 0 , T ] \times \mathbb { R } } | \Phi _ n ( s , u ) | \leq M _ 2 , \end{align*}


\begin{align*} H Y ^ 0 _ b ( U , A ) = \ker d _ 0 = A ( U ) \end{align*}


\begin{align*} q ( x , t ) = 2 i \lim _ { k \rightarrow \infty } [ k m ( x , t , k ) ] _ { 1 2 } , ( x , t ) \in \mathbb { R } \times [ 0 , + \infty ) . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n \binom { n } { k } a _ k c _ k = \sum _ { m = 0 } ^ n \binom { n } { m } d _ m \nabla ^ m b _ n . \end{align*}


\begin{align*} \forall R \ge R _ 1 + | x ( t ) | \Rightarrow | A _ R ( u ( t ) ) | \le C \varepsilon = 4 s _ c ( p - 1 ) \delta _ 0 \le 4 s _ c ( p - 1 ) \delta ( t ) , \forall t \notin D _ { \delta _ 0 } . \end{align*}


\begin{align*} E ( s _ 1 , s _ 3 ; y , n , M ) = O \left ( \frac { 1 } { y ^ { \sigma _ 1 } ( y + n ) ^ { \sigma _ 3 } } \right ) \end{align*}


\begin{align*} \left \{ \begin{array} [ c ] { l } u \in C \left ( \left [ 0 , T \right ] , D o m ( \boldsymbol { A } ) \right ) \cap C ^ { 1 } \left ( \left [ 0 , T \right ] , \mathcal { X } \right ) ; \\ \\ u _ { t } = \boldsymbol { A } u + f ( t ) , \ \ t \in \left [ 0 , T \right ] ; \\ \\ u ( 0 ) = u _ { 0 } \in \mathcal { X } . \end{array} \right . \end{align*}


\begin{align*} \frac { \lambda ( r ) '' } { \lambda ( r ) } + \frac { c - \lambda ( r ) '^ { 2 } } { \lambda ( r ) ^ { 2 } } \geq 0 . \end{align*}


\begin{align*} f _ k ( z ) = a _ { k } ^ { \# } ( z ) \widetilde { f } ( z ) + \frac { p ( z ) - p ( \xi ) } { z - \xi _ k } \in H ^ 2 . \end{align*}


\begin{align*} \left ( \widehat { w } ( x ) - \widehat { w } ( y ) \right ) = \left ( \widehat { w } ^ \pm ( x ) - \widehat { w } ^ \pm ( y ) \right ) , \ \ \ x , y \in \mathbb { R } ^ d . \end{align*}


\begin{align*} m _ t ^ * = \max \{ \hat m _ t ‐ \varepsilon | t ‐ t _ * ( n ) | , \ , 0 \} . \end{align*}


\begin{align*} \mbox { f o r e v e r y $ B \in \mathcal { G } ^ { U } $ , } \ \ \ \eta ( B ) = \eta ( U ) \times \frac { m _ { E _ U } ^ { s _ 2 } ( B ) } { \sum _ { B ' \in \mathcal { G } ^ { U } } m _ { E _ U } ^ { s _ 2 } ( B ' ) } . \end{align*}


\begin{align*} A ^ { 0 } U _ { t } ^ { \delta } + A ^ { i } \partial _ { i } U ^ { \delta } + D U ^ { \delta } - B ^ { i j } \partial _ { i } \partial _ { j } U ^ { \delta } = f + \delta \Lambda \Delta U ^ { \delta } , \end{align*}


\begin{align*} \Gamma _ 1 = \{ n > n ( E ) : J _ n \subset L ^ c \} , \Gamma _ 2 = \{ n > n ( E ) : J _ n \subset L \} , \end{align*}


\begin{align*} \int _ { n } ^ { n + 1 } B ( h , R ) d s & = \frac { 1 } { 2 } \int _ n ^ { n + 1 } \int _ { \mathbb { R } ^ N } ( L _ + h _ 1 ) R _ 1 d x d s + \frac { 1 } { 2 } \int _ n ^ { n + 1 } \int _ { \mathbb { R } ^ N } ( L _ - h _ 2 ) R _ 2 d x d s \\ & \lesssim \| \left \langle \nabla \right \rangle h \| _ { S ( ( n , + \infty ) , L ^ 2 ) } \| \left \langle \nabla \right \rangle R \| _ { S ' ( ( n , + \infty ) , L ^ 2 ) } \\ & \lesssim e ^ { - ( c _ 0 + c _ 1 ) n } , \end{align*}


\begin{align*} \lambda = h d \mu . \end{align*}


\begin{align*} \tilde \varphi : S & \to A \\ s & \mapsto \sum _ { i = 0 } ^ { n - 1 } \varphi ( a _ i , a _ { i + 1 } ) \end{align*}


\begin{align*} \Delta \left ( \mathbf { P } _ k \right ) = ( \mathbf p _ { k - 1 , 1 } \cup \mathbf p _ { k - 1 , 2 } ) \cap ( \mathbf p _ { k - 1 , 3 } \cup \mathbf p _ { k - 1 , 4 } ) \end{align*}


\begin{align*} U _ { J , z } \langle \alpha \rangle = \sum _ { \beta : \mathrm { a d m i s s i b l e \ , \ , o r b i t \ , \ , s e t \ , \ , w i t h \ , \ , } [ \beta ] = \Gamma } \# \{ \ , u \in \mathcal { M } _ { 2 } ^ { J } ( \alpha , \beta ) / \mathbb { R } ) \ , | \ , ( 0 , z ) \in u \ , \} \cdot \langle \beta \rangle . \end{align*}


\begin{align*} A ( \Gamma ) = \left ( \begin{array} { c c c c } 0 & a _ { 1 , 2 } & \cdots & a _ { 1 , n } \\ a _ { 2 , 1 } & 0 & \cdots & a _ { 2 , n } \\ \vdots & \vdots & \ddots & \vdots \\ a _ { n , 1 } & a _ { n , 2 } & \cdots & 0 \end{array} \right ) \end{align*}


\begin{align*} { \rm d e t } \left ( \begin{array} { c c } f _ i & f _ { i - 1 } \\ g _ i & g _ { i - 1 } \end{array} \right ) = 1 . \end{align*}


\begin{align*} \alpha = \frac { 1 } { \tau } , ~ ~ \mbox { t h e r e c i p r o c a l o f t h e m o m e n t u m r e l a x a t i o n t i m e } . \end{align*}


\begin{align*} f ( 0 ) = 1 , f ( k ) = k ( k > 0 ) . \end{align*}


\begin{align*} ~ \begin{aligned} \phi \Bigl ( \Pi \bigl ( ( a _ { - i } , s _ i ) ; p ^ { \tau _ { i j } } \bigr ) \Bigr ) & - \phi \Bigl ( \Pi \bigl ( ( a _ { - i } , u _ i ) ; p ^ { \tau _ { i j } } \bigr ) \Bigr ) \leq \phi \Bigl ( \Pi \bigl ( ( a _ { - i } , s _ i ) ; p ^ { \tau _ { i j + 1 } - 1 } \bigr ) \Bigr ) \\ & - \phi \Bigl ( \Pi \bigl ( ( a _ { - i } , u _ i ) ; p ^ { \tau _ { i j + 1 } - 1 } \bigr ) \Bigr ) , \end{aligned} \end{align*}


\begin{align*} \left ( \rho ( x , \omega , \tau ) f \right ) ^ \vee ( t ) = e ^ { 2 \pi i \tau } e ^ { \pi i x \cdot \omega } T _ { - x } M _ { - \omega } f ( - t ) = \rho ( - x , - \omega , \tau ) f ^ \vee ( t ) . \end{align*}


\begin{align*} I \ ! I _ 2 = \int _ { \| u - y \| > t } \frac 1 { V ( x , y , t + d ( x , y ) ) } \Big ( \frac { t } { t + d ( x , y ) } \Big ) ^ { \varepsilon _ 0 } \frac 1 { V ( u , y , s + d ( u , y ) ) } \Big ( \frac { s } { s + \| u - y \| } \Big ) ^ { \varepsilon _ 0 } d \omega ( u ) \end{align*}


\begin{align*} \sum _ { t = 1 } ^ { k } \bar { \sigma } ^ 2 _ { t } \geq \sum _ { t = 1 } ^ { k } \sigma ^ 2 _ { t } - 2 \sqrt { k } \theta \| S \| ^ 2 _ F . \end{align*}


\begin{align*} I _ { \chi , h } ( \xi , \eta ) = \int \chi ( x ) e ^ { i x \cdot ( \xi + \eta ) / h } d x . \end{align*}


\begin{align*} S \underset { \mathfrak { Z } _ 0 } { \asymp } n ^ { ‐ \beta _ { j _ * } } = \varphi ( t _ n , \ , m _ n , \ , n ) ; \end{align*}


\begin{align*} \Big ( k \sum _ { j = 1 } ^ m \| A E _ { k , N } ^ j v \| ^ 2 \Big ) ^ { \frac 1 2 } \leq C \| v \| , \forall v \in H . \end{align*}


\begin{align*} & \underline \dim ( \mu , x ) = \liminf _ { r \rightarrow 0 ^ { + } } \frac { \log \mu ( B ( x , r ) ) } { \log r } \\ \mbox { a n d } \ \ \ \ & \overline \dim ( \mu , x ) = \limsup _ { r \rightarrow 0 ^ { + } } \frac { \log \mu ( B ( x , r ) ) } { \log r } . \end{align*}


\begin{align*} A ^ { 0 } _ { 1 } ( U ) = \rho , A ^ { 0 } _ { 3 } ( U ) = \left ( \begin{array} { c c } \frac { \rho e _ { \theta } } { \theta } & \\ & \frac { \tau } { \kappa \theta } \mathbb { I } _ { 3 \times 3 } \end{array} \right ) \quad \mbox { a n d } A ^ { 0 } _ { 2 } ( U ) = \rho \mathbb { I } _ { 3 \times 3 } . \end{align*}


\begin{align*} { \bar { \Delta } _ { \rm C l a s s i c a l - A R Q } } = - \frac { 1 } { 2 } + { n _ 1 } \left ( { \frac { 2 } { { 1 - { \epsilon _ 1 } } } - \frac { 1 } { 2 } } \right ) . \end{align*}


\begin{align*} f = \frac { 1 } { \langle \widetilde { g } , g \rangle } \iint _ { \R ^ { 2 d } } V _ g f ( x , \omega ) M _ \omega T _ x \widetilde { g } \ , d ( x , \omega ) . \end{align*}


\begin{align*} \Psi _ { V _ 0 } = \mathcal { E } _ { V _ 0 } - \sum _ { i = 1 } ^ 2 \dot { \mathfrak { q } } _ i \sigma _ i \partial _ y R _ i + \sum _ { \substack { i , j = 1 , \\ j \not = i } } ^ 2 \frac { a _ { i j } \sigma _ i } { ( \mathfrak { q } _ i - \mathfrak { q } _ j ) ^ 3 } \Lambda R _ i + \sum _ { i = 1 } ^ 2 \dot { \mathfrak { q } } _ i ( \mathcal { P } _ i + \mathcal { Q } _ i ) \end{align*}


\begin{align*} \left \{ \begin{array} { l l } u _ t + \big \{ m ( K \cap \{ u ( \cdot , t ) < u ( x , t ) \} ) - 1 \big \} | \nabla u | = 0 & \\ u ( x , 0 ) = \varphi _ 0 ( | x | ) & \ \R ^ n , \end{array} \right . \end{align*}


\begin{align*} \omega _ n \min \{ R , r ^ \ast \} ^ n = 1 , \end{align*}


\begin{align*} \frac { d } { d t } \int x u ^ 2 ( x , t ) d x = 2 I _ 3 ( u ) \end{align*}


\begin{align*} i _ X \phi = i _ X d \phi = 0 , \end{align*}


\begin{align*} \lim \limits _ { T \to \infty } \frac { 1 } { T } \int _ { 0 } ^ { T } \delta ( t ) d t = 0 . \end{align*}


\begin{align*} M ( p , H ) = \frac { \pi ^ 2 } { 6 } \times \frac { a + 1 } { a - 1 } \times \left ( 1 - \frac { ( d - 1 ) a + 1 } { p } \right ) . \end{align*}


\begin{align*} \Im \int Q v = 0 , \Re \int ( \partial _ { x _ k } Q ) v = 0 , \ ; k = 1 , . . . , N , \end{align*}


\begin{align*} | \alpha _ \pm | = | B ( h , \mathcal { Y } _ \mp ) | \lesssim \Big | \frac { 1 } { 2 } \int ( L _ + h _ 1 ) \mathcal { Y } _ 1 d x \Big | + \Big | \frac { 1 } { 2 } \int ( L _ - h _ 2 ) \mathcal { Y } _ 2 d x \Big | \lesssim \| h ( t ) \| _ { L _ x ^ 2 } \lesssim e ^ { - c _ 0 t } . \end{align*}


\begin{align*} f ( x ) = \sum \limits _ { k = - \infty } ^ { \infty } \sum \limits _ { Q } | Q | \phi _ { Q } ( x - x _ Q ) \psi _ { Q } \ast f ( x _ Q ) , \end{align*}


\begin{align*} & P ( X _ { n + 1 } = \pm 1 \mid \mathcal { F } _ n ) \\ & = \dfrac { \# \{ i = 1 , \ldots , n : X _ i = \pm 1 \} } { n } \cdot p + \dfrac { \# \{ i = 1 , \ldots , n : X _ i = \mp 1 \} } { n } \cdot ( 1 - p ) \\ & = \dfrac { n \pm S _ n } { 2 n } \cdot p + \dfrac { n \mp S _ n } { 2 n } \cdot ( 1 - p ) = \dfrac { 1 } { 2 } \left ( 1 \pm \alpha \cdot \dfrac { S _ n } { n } \right ) . \end{align*}


\begin{align*} \sum _ i \omega _ i ( b _ { ( 3 ) } c S ^ { - 1 } ( b _ { ( 1 ) } ) ) a _ i b _ { ( 2 ) } d = 0 , \forall b , c , d \in A . \end{align*}


\begin{align*} \mathfrak { A } \chi \theta _ { \Sigma , \epsilon } ( F _ H ) = \theta _ { \widetilde { \Sigma } , \widetilde { \epsilon } } \mathfrak { A } _ { \ell } ( \rho , \chi ) ( F _ H ) + \mathfrak { I } _ S ( \widetilde { \mathcal { M } } ) , \end{align*}


\begin{align*} \left [ \nabla ^ { E } _ V , \mathbf { c } ( V ' ) \right ] = \mathbf { c } ( \nabla ^ { T X } _ V V ' ) , V , V ' \in C ^ \infty ( X , T X ) \end{align*}


\begin{align*} \mu _ { \tau , T } : = \sum _ { t = 1 } ^ { \lfloor T / \tau \rfloor } \delta _ { B _ { t \tau } ^ H } \tau , \end{align*}


\begin{align*} w _ t + a _ n ( t , x ) \cdot w _ x ~ = ~ F ^ { ( k ) } ( t , x ) , w ( \tau _ 0 , \cdot ) ~ = ~ \overline { w } ( \cdot ) \end{align*}


\begin{align*} B ( x ) : = \max \left \lbrace \beta _ 1 ( x ) , \beta _ 2 ( x ) , \beta _ 3 ( x ) \right \rbrace , \end{align*}


\begin{align*} x _ { \kappa , c } ( \sum _ { i \in I } w _ i ) = \sum _ { i \in I } u _ i w _ i , w _ i \in F _ i . \end{align*}


\begin{align*} ( a b ( a + b ) x ) ' = ( a b ) ' ( 1 + x ^ 2 ) . \end{align*}


\begin{align*} B = \nabla \times ( - \Delta ) ^ { - 1 } J . \end{align*}


\begin{align*} a _ { n + 2 } ( x ) = \frac { - \varepsilon } { A ^ n _ 1 } \sum _ { k = m + 2 } ^ \infty \left [ k + ( m + 2 ) \right ] \left [ k + ( m + 1 ) \right ] a _ { k + ( m + 2 ) } x ^ { k } \ , . \end{align*}


\begin{align*} \int _ { t } ^ { T } C _ p q \left | \dot { \xi } ( s ) \right | ^ q d s = C _ p ( q - 1 ) p ^ q \left [ f ( \xi ( T ) ) - u ( \xi ( T ) ) - \left ( f ( \xi ( t ) ) - u ( \xi ( t ) ) \right ) \right ] + f ( \xi ( t ) ) - f ( \xi ( T ) ) . \end{align*}


\begin{align*} R ( \beta _ j ) A ( \beta _ j ) = 1 , ( R A ) ^ { ( k ) } ( \beta _ j ) = 0 \quad 1 \le k \le \nu _ j - 1 ; \end{align*}


\begin{align*} \epsilon _ k = \frac { 1 } { 2 } \max _ { j \in \lbrace 1 , \ldots , n \rbrace } \frac { | e _ j ^ \intercal ( A x _ k - b ) | ^ 2 } { \norm { A x _ k - b } _ 2 ^ 2 \norm { A ^ \intercal e _ j } _ 2 ^ 2 } + \frac { 1 } { 2 \norm { A } _ F ^ 2 } ; \end{align*}


\begin{align*} s - \mathcal T _ { t , x _ 1 , v _ 1 } = \check { \mathcal T } _ { t , x _ 1 , v _ 1 } - ( t - s ) \geq \frac 1 2 \check { \mathcal T } _ { t , x _ 1 , v _ 1 } \geq \frac 1 4 \check \tau _ { t , x } s \geq t - \frac { \check { \mathcal T } _ { t , x _ 1 , v _ 1 } } 2 = : s ^ \ast _ { t , x , v _ 1 } . \end{align*}


\begin{align*} \langle f _ 1 \otimes f _ 2 , g _ 1 \otimes g _ 2 \rangle = \langle f _ 1 , g _ 1 \rangle _ 1 \langle f _ 2 , g _ 2 \rangle _ 2 , \forall f _ 1 , g _ 1 \in \mathcal { H } _ 1 , \ ; \forall f _ 2 , g _ 2 \in \mathcal { H } _ 2 . \end{align*}


\begin{align*} \rho ( x , \omega , e ^ { 2 \pi i \tau } ) = e ^ { 2 \pi i \tau } e ^ { \pi i x \cdot \omega } T _ x M _ \omega . \end{align*}


\begin{align*} a p ( x , y ) ^ 2 + b q ( x , y ) ^ 2 + c r ( x , y ) ^ 2 = 0 \end{align*}


\begin{align*} \tau ( a _ 1 \cdots a _ n ) & = 0 , \\ \varphi ( a _ 1 \cdots a _ n ) & = \varphi ( a _ 1 ) \cdots \varphi ( a _ n ) . \end{align*}


\begin{align*} \Delta v _ { \varepsilon } ( \mathbf { x } ) = \sigma _ 3 \sqrt { \tilde { \lambda } _ 0 \underline { m } } \frac { ( N - 1 ) } { | \mathbf { x } - \mathbf { y } _ { \varepsilon } | } , \end{align*}


\begin{align*} \left \langle ( \rho _ 0 ^ { * } \partial _ { \Sigma _ 0 } ) ^ { \otimes 2 } E _ { \mathcal { M } _ 0 } , f \otimes g \right \rangle = \int _ { \mathcal { I } } f \wedge d g \end{align*}


\begin{align*} g ( w ) = \det \left ( U ( w ) ^ { - 1 } \mathfrak { p } ( w ) \right ) ^ k \det ( \mathfrak { p } ( w ) ) ^ { - k } f ( w ) . \end{align*}


\begin{align*} \frac { \partial f _ { 0 } } { \partial y _ { j _ { 1 } \ldots j _ { k } } ^ { \sigma } } - d _ { i } f _ { \sigma } ^ { i , j _ { 1 } \ldots j _ { k } } - f _ { \sigma } ^ { j _ { k } , j _ { 1 } \ldots j _ { k - 1 } } & = 0 \qquad \mathrm { S y m } ( j _ { 1 } \ldots j _ { k } ) , \ , \ , k \leq s , \\ \frac { \partial f _ { 0 } } { \partial y _ { j _ { 1 } \ldots j _ { s + 1 } } ^ { \sigma } } - f _ { \sigma } ^ { j _ { s + 1 } , j _ { 1 } \ldots j _ { s } } & = 0 \qquad \mathrm { S y m } ( j _ { 1 } \ldots j _ { s + 1 } ) . \end{align*}


\begin{align*} \sum \limits _ { \ell = 0 } ^ { 2 } \frac { d } { d t } \langle \nabla ^ { \ell } u ^ { \epsilon } , \epsilon \nabla ^ { \ell + 1 } \phi ^ { \epsilon } \rangle + \frac { 3 } { 4 } \| \nabla \phi ^ { \epsilon } \| _ { H ^ 2 _ { \frac { P ' ( \rho ^ { \epsilon } ) } { \rho ^ { \epsilon } } } } ^ 2 \lesssim D ( u ^ { \epsilon } , \eta ^ { \epsilon } , \tau ^ { \epsilon } ) + \| \mathrm { d i v } u ^ { \epsilon } \| _ { H ^ 2 } ^ 2 , \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } \frac { c | m h \pm e ^ { \xi } n k | } { g x Q } W \left ( \frac { c | m h \pm e ^ { \xi } n k | } { g x Q } \right ) x ^ { w - 1 } \ , d x = \left ( \frac { c | m h \pm e ^ { \xi } n k | } { g Q } \right ) ^ { w } \widetilde { W } ( 1 - w ) . \end{align*}


\begin{align*} H ^ n ( \mathcal { Z } _ K ; k ) \cong \bigoplus _ { n = 2 | J | - i } \operatorname { T o r } ^ S _ i ( k [ K ] , k ) _ J . \end{align*}


\begin{align*} \mathbb { S } ( \nabla u ) = \mu _ 1 ( \nabla _ x u + \nabla _ x ^ \top u ) + \mu _ 2 { \rm { d i v } } u \mathbb { I } , \end{align*}


\begin{align*} r = h _ 1 B _ 1 ( \theta _ 1 ) s _ 1 + h _ 2 B _ 2 ( \theta _ 2 ) s _ 2 + n , \end{align*}


\begin{align*} \lim _ { k \to \infty } \int _ \Omega | \nabla u | ^ { p - 2 } \nabla u \cdot \nabla \phi _ k \ , d \mu = \int _ \Omega | \nabla u | ^ { p - 2 } \nabla u \cdot \nabla v \ , d \mu . \end{align*}


\begin{align*} W ( \widehat { f } , \widehat { g } ) ( x , \omega ) & = 2 ^ d e ^ { 4 \pi i x \cdot \omega } \langle \widehat { f } , M _ { 2 \omega } T _ { 2 x } \widehat { g } ^ \vee \rangle \\ & = 2 ^ d e ^ { 4 \pi i x \cdot \omega } \langle f , T _ { - 2 \omega } M _ { 2 x } g ^ \vee \rangle \\ & = 2 ^ d e ^ { - 4 \pi i x \cdot \omega } \langle f , M _ { 2 x } T _ { - 2 \omega } g ^ \vee \rangle \\ & = W ( f , g ) ( - \omega , x ) . \end{align*}


\begin{align*} { d \widetilde A ^ K _ { i } ( t ) \over d t } < \bar p e ^ { \varepsilon _ K \log K } \sum _ { \ell = - \lfloor 1 / 2 \delta _ K \rfloor } ^ { 1 / \delta _ K - 1 - \lfloor 1 / 2 \delta _ K \rfloor } h _ { K } G ( h _ { K } \ell ) \exp ( \log K ( \widetilde A ^ K _ { i + \ell } ( t ) - \widetilde A ^ K _ { i } ( t ) ) - 2 \bar p . \end{align*}


\begin{align*} \psi ( x ) > q _ 1 ( \sqrt { 0 . 5 } ) + 0 = 0 . \end{align*}


\begin{align*} { H } ^ { - 1 } f ( \xi ) = \Lambda Q ( \xi ) \int _ 0 ^ \xi \frac { { L } f ( \xi ' ) } { \Lambda Q ( \xi ' ) } d \xi ' , \end{align*}


\begin{align*} \left ( \frac { \sum \limits ^ { 2 k } _ { i = 1 } \ , | \lambda _ i | } { 2 k } \right ) ^ 2 \geq \frac { 1 } { \displaystyle { \frac { 2 k ( 2 k - 1 ) } { 2 } } } \ , \sum \limits _ { i < j } \ , | \lambda _ i | \ , | \lambda _ j | , \end{align*}


\begin{align*} f = \sum \limits _ { k = - \infty } ^ { \infty } \widetilde { D } _ { k } D _ { k } ( f ) = \sum \limits _ { k = - \infty } ^ { \infty } D _ { k } \bar { D } _ { k } ( f ) , \end{align*}


\begin{align*} L ( x , p , y ) & = \begin{cases} + \infty & , \\ E l ( x , y ) - \langle x , p \rangle & , \\ - \infty & . \end{cases} \end{align*}


\begin{align*} H ^ { \theta \sim \alpha } = \{ h \in H \mid ( \theta - \alpha ) ^ n ( h ) = 0 \ ; n \ge 1 \} \end{align*}


\begin{align*} \alpha ' = \alpha \rho > \alpha . \end{align*}


\begin{align*} P _ M ( \bar f ^ { \lambda + M } ) = b ( \lambda ) \dots b ( \lambda + M - 1 ) \bar f ^ { \lambda } . \end{align*}


\begin{align*} \varepsilon ( y , \tau ) = w ( y , \tau ) - Q _ { b ( \tau ) } ( y ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } d X ( t ) & = & \big ( \frac { \kappa } { 2 } \frac { d ^ 2 } { d x ^ 2 } X ( t ) + \frac { d } { d x } X ( t ) + \alpha \big ) d t + \sigma d W ( t ) \medskip \\ X ( 0 ) & = & h \end{array} \right . \end{align*}


\begin{align*} \sum _ { r \in \mathbb { Z } } c ^ E _ r \big ( \widetilde { V } , \widetilde { \mathcal { S } } \big ) q _ F ^ { - \frac { r } { 2 } f } X ^ { - f r } = \gamma ( X , \pi _ E , \widehat { \sigma _ E } , \psi _ E ) \sum _ { r \in \mathbb { Z } } c ^ E _ r ( V , \mathcal { S } ) q _ F ^ { \frac { r } { 2 } f } X ^ { f r } \end{align*}


\begin{align*} \mathcal { F } = 1 \otimes 1 + \mathcal { O } ( \nu ) \in ( U \Xi \otimes U \Xi ) [ [ \nu ] ] \end{align*}


\begin{align*} M ^ { ( n - k , k ) } \cong _ { \mathfrak { S } _ { n } } \bigoplus _ { j = 0 } ^ { k } \mathbb { S } ^ { n - j , j } \end{align*}


\begin{align*} \Psi ( \tau ) = b ( \tau ) \exp \left ( \left ( \frac { 2 } { \alpha } - 1 \right ) \left ( \int _ { \tau _ 0 } ^ \tau 2 \beta ( \tau ' ) d \tau ' + \tau _ 0 \right ) \right ) \Psi ( \tau _ 0 ) = 1 , \end{align*}


\begin{align*} d Z _ { \lambda } = 0 \quad \mathrm { i f \ , a n d \ , o n l y \ , i f } E _ { \lambda } = 0 , \end{align*}


\begin{align*} H ( t ) = E ( t ) + \epsilon \int _ U u u _ t ( x , t ) \ , d x \end{align*}


\begin{align*} X = \left [ X _ 1 , \left [ X _ 2 , \ , \cdots \ , \left [ X _ { n - 1 } , X _ { n } \right ] \right ] \right ] \end{align*}


\begin{align*} \begin{aligned} A ^ { 0 } U _ { t } + A ^ { i } \partial _ { i } U - D U & = F , \\ \left . V \right \rvert _ { t = 0 } & = U _ { 0 } , \end{aligned} \end{align*}


\begin{align*} \sum _ { 0 \leq m < n < \infty } \frac { \tau _ A ( p ^ { m } ) \tau _ B ( p ^ { n } ) } { p ^ { m \beta } p ^ { n ( 1 - \beta ) } } = \sum _ { m = 0 } ^ { \infty } \frac { \tau _ { A \cup \{ - \beta \} } ( p ^ m ) \tau _ B ( p ^ m ) } { p ^ { m } } - \sum _ { m = 0 } ^ { \infty } \frac { \tau _ { A } ( p ^ m ) \tau _ B ( p ^ m ) } { p ^ { m } } . \end{align*}


\begin{align*} \forall \gamma ^ 1 , p ^ X , x ^ X \left ( \gamma > _ \mathbb { R } 0 \land p \in A ( x - _ X \gamma p ) \rightarrow p = _ X A _ \gamma x \right ) . \end{align*}


\begin{align*} \epsilon ' = b _ 1 ^ * - \frac { k - r } { r } \epsilon = \frac { k } { r } - b _ 2 ^ * \ , . \end{align*}


\begin{align*} \min \{ v _ p ( u _ i ) , k \} = \min \{ v _ p ( u _ 1 ) + ( i - 1 ) b , k \} \end{align*}


\begin{align*} \ker D ^ * \Phi ( \bar x , \bar y ) = \{ 0 \} \end{align*}


\begin{align*} x _ 1 ^ \prime = j ^ 2 x _ 1 \ , , x _ 2 ^ \prime = x _ 2 \ , , y _ 1 ^ \prime = j y _ 1 \ , , y _ 2 ^ \prime = y _ 2 \ , , \left ( j = e ^ { \frac { 2 \pi i } { 3 } } \right ) \end{align*}


\begin{align*} f _ { \alpha } ( z ) = \sum _ { \beta \leq \alpha } a _ { \beta } g _ { \beta } ( z ) . \end{align*}


\begin{align*} \left | \int _ { \mathbb S ^ 1 } B _ x D ^ m J D ^ m B \ , d x \right | \leq & \ \| B _ x \| _ { L ^ \infty } \| D ^ m B \| _ { L ^ 2 } \| D ^ m J \| _ { L ^ 2 } \\ \left | \int _ { \mathbb S ^ 1 } B D ^ m J D ^ m B _ x \ , d x \right | \leq & \ \| B \| _ { L ^ \infty } \| D ^ m B _ x \| _ { L ^ 2 } \| D ^ m J \| _ { L ^ 2 } \\ \left | \int _ { \mathbb S ^ 1 } J D ^ m B _ x D ^ m B \ , d x \right | \leq & \ \| J _ x \| _ { L ^ \infty } \| D ^ m B \| _ { L ^ 2 } ^ 2 . \end{align*}


\begin{align*} \widetilde \mu ( x ) = \mu ( x ) , \ , \widetilde \sigma ( x ) = \sigma ( x ) . \end{align*}


\begin{align*} \Omega ( \mathfrak { p } ^ j x ) & = \Omega ( x ) ~ ~ j \in \mathbb { Z } , \\ \int \limits _ { \mathfrak { D ^ { * } } } \Omega ( x ) d x & = 0 . \end{align*}


\begin{align*} \Omega \coloneqq \{ w \in W _ H \mid w ( S _ G ) = S _ G \} . \end{align*}


\begin{align*} _ { q _ 1 , q _ 2 , p } : = \sum _ { x \bmod p } \max ( q _ 1 x , q _ 2 x ) \end{align*}


\begin{align*} \widetilde { f } ( s ) : = \int _ 0 ^ { \infty } f ( x ) x ^ { s - 1 } \ , d x . \end{align*}


\begin{align*} y _ n = \sum _ { i = 1 } ^ n h _ i \prod _ { j = i + 1 } ^ n g _ j + y _ 0 \prod _ { j = 1 } ^ n g _ j . \end{align*}


\begin{align*} \tau ( a _ i ) = - 4 a _ i c _ 1 + a _ i + 4 ( a _ i , c _ 1 ) c _ 1 . \end{align*}


\begin{align*} \chi _ { R , L } ( k ) : \mathbb { C } \backslash [ - C _ { R , L } , C _ { R , L } ] \rightarrow \mathbb { C } , \chi _ { R , L } ( k ) = \left ( \frac { k - C _ { R , L } } { k + C _ { R , L } } \right ) ^ { \frac { 1 } { 4 } } . \end{align*}


\begin{align*} x ^ { k } \bigg ( \frac { d } { d x } \bigg ) ^ { k } f ( x ) = \sum _ { m = 0 } ^ { k } S _ { 1 , \lambda } ( k , m ) \bigg ( x \frac { d } { d x } \bigg ) _ { m , \lambda } f ( x ) . \end{align*}


\begin{align*} d _ \rho ( ( x _ 1 , y _ 1 ) , ( x _ 2 , y _ 2 ) ) = \inf _ \gamma \int _ \gamma \rho \ , d s , \end{align*}


\begin{align*} \Phi ( f ) [ \psi , \pi ] = \int _ \Sigma f \psi \ , \mathrm { d } V _ \Sigma , \Pi ( f ) [ \psi , \pi ] = \int _ \Sigma f \pi \ , \mathrm { d } V _ \Sigma . \end{align*}


\begin{align*} \int w ( x , t ) d x = \int \sigma ( x ) d x \end{align*}


\begin{align*} \mathcal { U } ^ 2 ( h , k ) = I _ 1 + I _ 2 + O \left ( X ^ { \varepsilon } Q ^ { 1 + \varepsilon } \frac { ( h k ) ^ \varepsilon ( h , k ) } { ( h k ) ^ { 1 / 2 } } + ( X C h k ) ^ { \varepsilon } k X ^ 2 Q ^ { - 9 7 } \right ) , \end{align*}


\begin{align*} v _ i = v _ { i - 1 } + v i = 1 , \dots , k - 1 \rho _ 1 = v _ { k - 1 } + v \end{align*}


\begin{align*} F \cdot K _ { \widetilde { X } } = 4 k - 1 C _ { i , 0 } \cdot K _ { \widetilde { X } } = 1 C _ { i , j } \cdot K _ { \widetilde { X } } = 0 C _ { i , k } \cdot K _ { \widetilde { X } } = - 1 \end{align*}


\begin{align*} G _ \lambda f : = \int _ 0 ^ \infty e ^ { - \lambda t } P _ t f d t \end{align*}


\begin{align*} f _ { \alpha } ^ { ( \beta ) } ( o ) = f ^ { ( \beta ) } ( o ) . \end{align*}


\begin{align*} \epsilon \int _ { 0 } ^ { 1 } \rho ^ { \epsilon } _ x \bar { \sigma } _ x + \int _ { 0 } ^ { 1 } u ^ { \epsilon } _ x \bar { \sigma } + \int _ { 0 } ^ { 1 } \rho ^ { \epsilon } _ x \bar { \sigma } + \lambda \int _ { 0 } ^ { 1 } \rho ^ { \epsilon } \bar { \sigma } = \int _ { 0 } ^ { 1 } f \bar { \sigma } . \end{align*}


\begin{align*} \mathcal H = \mathcal D _ - \oplus K \oplus \mathcal D _ + . \end{align*}


\begin{align*} \frac { d ^ 2 } { d t ^ 2 } \int x ^ 2 w ( x , t ) d x = 2 t ( I _ 3 ( \phi ) - I _ 3 ( \varphi ) ) . \end{align*}


\begin{align*} | \frac { | \phi ( x , y ) \cap A \times B | } { | \phi ( x , y ) \cap X _ i \times Y _ j | } - \frac { | A \times B | } { | X _ i \times Y _ j | } | = O ( q ^ { - 1 / 4 } ) . \end{align*}


\begin{align*} p _ \ell ( G _ 1 / T ( G _ 1 ) ) = p _ \ell ( E | _ { D _ 1 } ) , \end{align*}


\begin{align*} \widetilde { c } _ k ( s ) = \alpha ( s ) \int _ { \mathfrak { B } } \det ( I + z \bar { z } ) ^ { s + k } \textbf { d } z , \end{align*}


\begin{align*} ( p _ k + i q _ k ) ^ 2 = ( p ^ 2 _ k - q ^ 2 _ k ) + i 2 p _ k q _ k = a _ k + i b _ k , \end{align*}


\begin{align*} \mathcal { P } \left ( \lambda \right ) = \lambda I + T , \lambda \in \mathbf { C } , \end{align*}


\begin{align*} \mathrm { T r } _ \omega ( P \mathfrak { W } _ { 1 / 2 } ( c _ 1 ) ( 1 - P ) [ \mathfrak { W } _ { 1 / 2 } ( c _ 2 ) ] ^ * P ) = \omega \circ \log _ 2 ( c _ 1 c _ 2 ^ * ) . \end{align*}


\begin{align*} f ' = f _ 0 ' + f _ i + \ldots + f _ k , \end{align*}


\begin{align*} Y = \sqrt { P } X + Z , \end{align*}


\begin{align*} \mathcal { J } \leq \log \log \log X , \alpha _ { 1 } = \frac { 1 } { ( \log \log X ) ^ { 2 } } , \mbox { a n d } \sum _ { p \leq X ^ { 1 / ( \log \log X ) ^ { 2 } } } \frac { 1 } { p } \leq \log \log X . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \dd u - \Delta w \dd t = \dd W ( t ) , & \ ; \ ; D \times ( 0 , T ] , \\ w = - \Delta u + f ( u ) , & \ ; \ ; D \times ( 0 , T ] , \\ \frac { \partial u } { \partial n } = \frac { \partial w } { \partial n } = 0 , & \ ; \ ; \partial D \times ( 0 , T ] , \\ u ( 0 , x ) = u _ 0 , & \ ; \ ; D , \end{array} \right . \end{align*}


\begin{align*} e ^ { - t } u ( y ) = e ^ { - t } u \left ( \xi ( t ) \right ) = \int _ t ^ { t + b } e ^ { - s } \left ( C _ p \left | \dot { \xi } ( s ) \right | ^ q + f \left ( \xi ( s ) \right ) \right ) d s + e ^ { - ( b + t ) } u ( \xi ( b + t ) ) . \end{align*}


\begin{align*} & \int _ 0 ^ t t ^ { \frac { \beta } { \alpha } } ( t - \tau ) ^ { - 1 + \frac { \beta } { \alpha } } \tau ^ { - \frac { 2 \beta } { \alpha } } \ , d \tau \\ = & \int _ 0 ^ 1 ( 1 - \tau ' ) ^ { - 1 + \frac { \beta } { \alpha } } ( \tau ' ) ^ { - 1 + ( 1 - \frac { 2 \beta } { \alpha } ) } \ , d \tau ' \\ \lesssim & \ 1 \end{align*}


\begin{align*} A _ \ast ( U ) = \begin{cases} A & U \mbox { i n f i n i t e } \\ 0 & U \mbox { f i n i t e } \end{cases} \end{align*}


\begin{align*} v ( x ) : = \left \{ \begin{aligned} & \max \{ u ( x ) , \phi ( x ) + \epsilon ^ 2 \} , x \in \mathrm { B ( x _ 0 , 2 \epsilon ) } \cap \overline { \Omega } , \\ & u ( x ) , x \in \overline { \Omega } \setminus \mathrm { B ( x _ 0 , 2 \epsilon ) } . \end{aligned} \right . \end{align*}


\begin{align*} R ^ A g _ x ( z ) \leq \sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 ^ n } = 1 . \end{align*}


\begin{align*} { \rm A S } _ g ( D ) : = \frac { \widehat { A } \big ( \frac { R _ { X ^ g } } { 2 \pi i } \big ) \operatorname { t r } \big ( g \exp ( \frac { R ^ F } { 2 \pi i } ) \big ) \exp ( \operatorname { t r } ( \frac { R ^ L } { 2 \pi i } ) ) } { \operatorname { d e t } \big ( 1 - g \exp ( - \frac { R ^ { \mathcal { N } } } { 2 \pi i } ) \big ) ^ { \frac { 1 } { 2 } } } . \end{align*}


\begin{align*} \partial _ 1 u ( a ) = \tau \partial _ 1 \big \vert _ { x = a } \big ( \zeta ( x ) ( \rho - \vert x \vert ) ^ { - n - 2 } \big ) = \tau ( \partial _ 1 \zeta ( a ) ) ( \rho - \vert a \vert ) ^ { - n - 2 } . \end{align*}


\begin{gather*} \psi _ { X \otimes Y , Z } = ( \psi _ { X , Z } \otimes Y ) ( X \otimes \psi _ { Y , Z } ) , \psi _ { X , Y \otimes Z } = ( Y \otimes \psi _ { X , Z } ) ( \psi _ { X , Y } \otimes Z ) \end{gather*}


\begin{align*} G \cap I & = ( ( F \setminus \{ \bar { b } _ n \} ) \cup [ \bar { a } _ n \mapsto \bar { b } _ n ] ( L ) ) \cap ( D _ 1 \cup [ \bar { a } _ n \mapsto \bar { b } _ n ] ( J ) ) \\ & \subseteq ( ( F \setminus \{ \bar { b } _ n \} ) \cup [ \bar { a } _ n \mapsto \bar { b } _ n ] ( L ) ) \cap ( ( D \setminus \{ \bar { b } _ n \} ) \cup [ \bar { a } _ n \mapsto \bar { b } _ n ] ( J ) ) \end{align*}


\begin{align*} \mathfrak { B } _ 1 \times \mathfrak { B } _ { 2 , \emptyset } = [ 1 2 3 , \bar 1 \ , 3 2 ] , \mathfrak { B } _ 1 \times \mathfrak { B } _ { 2 , \{ 1 \} } = [ 1 \bar { 2 } 3 , \bar 1 \ , \bar 3 2 ] , \mathfrak { B } _ 1 \times \mathfrak { B } _ { 2 , \{ 2 \} } = [ 1 3 \bar { 2 } , \bar 1 2 \bar { 3 } ] \end{align*}


\begin{align*} \sum _ { a < n \leq b } g ( n ) e ^ { 2 \pi i f ( n ) } & = \sum _ { \substack { \nu \in \mathbb { Z } \\ \alpha - \eta < \nu < \beta + \eta } } \int _ a ^ b g ( x ) e ^ { 2 \pi i ( f ( x ) - \nu x ) } d x \\ & \quad + O \left ( g ( a ) \log ( \beta - \alpha + 2 ) \right ) + O \left ( \abs { g ^ \prime ( a ) } \right ) \end{align*}


\begin{align*} I ( \rho ) = \frac 1 2 \sum _ { i = 1 } ^ N \sum _ { j \in N ( i ) } \widetilde \omega _ { i j } | \log ( \rho _ i ) - \log ( \rho _ j ) | ^ 2 \widetilde \theta _ { i j } ( \rho ) , \end{align*}


\begin{align*} \theta \varphi \theta ^ { - 1 } & = ( \tau \sigma \tau ^ { - 1 } , ( \alpha _ { ( \tau \sigma ^ { - 1 } ) 1 } \varphi _ { \tau 1 } \alpha ^ { - 1 } _ { \tau 1 } , \alpha _ { ( \tau \sigma ^ { - 1 } ) 2 } \varphi _ { \tau 2 } \alpha ^ { - 1 } _ { \tau 2 } , \alpha _ { ( \tau \sigma ^ { - 1 } ) 3 } \varphi _ { \tau 3 } \alpha ^ { - 1 } _ { \tau 3 } ) ) \\ & = ( \tau \sigma \tau ^ { - 1 } , ( \beta _ 1 , \beta _ 2 , \beta _ 3 ) ) . \end{align*}


\begin{align*} \widetilde \alpha _ q ( 2 T , 2 S ) = 2 ^ { m \delta _ { 2 , p } } \alpha _ q ( S , T ) \end{align*}


\begin{align*} H \left ( T _ { i + 1 } \right ) = T _ i . \end{align*}


\begin{align*} \Lambda f ( y ) = \frac { \partial f _ c ( y ) } { \partial c } \bigg | _ { c = 1 } = \frac { 1 } { p - 1 } f ( y ) + y f ' ( y ) , \end{align*}


\begin{align*} K _ E = g '' ( 0 ) = \frac { 2 a ^ 3 } { \sqrt { D } } , \ \partial _ s K _ E = g ''' ( 0 ) = \frac { 6 a ^ 4 b } { D } . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ k \hat q _ { j _ i j _ { i + 1 } } = 1 . \end{align*}


\begin{align*} \lim _ { n \to \infty } ( | x - x _ n | , | \omega - \omega _ n | , | \tau - \tau _ n | ) = ( 0 , 0 , \Z ) \Leftrightarrow \lim _ { n \to \infty } | \mathbf { h } _ r - { \mathbf { h } _ r } _ n | = ( 0 , 0 , 1 ) = \mathbf { 1 } _ r . \end{align*}


\begin{align*} \Sigma _ { a } ^ { b } = 2 \sum _ { \substack { 0 < \gamma < T \\ a \le \beta < b } } \frac { x ^ { \beta - 1 } } { \gamma } , \end{align*}


\begin{align*} 1 + \frac { 1 } { 2 \log \log t } \leq 1 . 0 4 4 2 5 = \gamma , . \end{align*}


\begin{align*} f ^ 1 ( \phi ( x _ 0 ) ) = 0 , \quad g ( 0 ) = 0 \end{align*}


\begin{align*} M ^ T = M M > 0 , \end{align*}


\begin{align*} \tau _ t ( a _ 1 \cdots a _ n ) = o ( 1 ) \end{align*}


\begin{align*} \mathbf { M } _ { u } = \sum _ { u \leq v } \mu _ { \mathfrak { B } _ n } ( u , v ) \mathbf { F } _ { v } , \mathbf { F } _ { u } = \sum _ { u \leq v } \mathbf { M } _ { v } . \end{align*}


\begin{align*} & \frac { 1 } { n } { \bf E } \Big \{ \sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \mathbb R } ^ { n _ x } } ^ 2 \Big \} = \frac { 1 } { n } \sum _ { t = 1 } ^ { n } t r \Big ( \Gamma _ t P _ t \Gamma _ t ^ T + D _ t K _ { Z _ t } D _ t ^ T \Big ) . \end{align*}


\begin{align*} A F _ k \supseteq A F _ { k + 1 } F = \bigcap _ { k \in \mathbb { N } } A F _ k . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \mathrm { d } X _ { t } = \theta f ( X _ { t } ) \mathrm { d } t + \sigma \mathrm { d } W _ { t } + \mathrm { d } L _ { t } , \\ & X _ { 0 } = x \in [ l , \infty ) . \end{aligned} \right . \end{align*}


\begin{align*} \frac { L } { 2 \eta _ T } \sum _ { t = 1 } ^ { T } \frac { \eta _ t ^ 2 } { \alpha _ t ^ 4 } \norm { g _ t } ^ 2 , \end{align*}


\begin{align*} \Delta _ { G } & = \{ \widehat { x } _ i - \widehat { x } _ { i + 1 } \ , , \ ; 1 \leq i \leq p + q - 1 \ , , \widehat { x } _ { p + q } \} \\ \intertext { a n d } \Delta _ { K } & = \left \{ \begin{cases} \widehat { x } _ i - \widehat { x } _ { i + 1 } \ , , \ ; 1 \leq i \leq p - 1 \ , , \ ; \widehat { x } _ { p - 1 } + \widehat { x } _ { p } \ , , \\ \widehat { x } _ i - \widehat { x } _ { i + 1 } \ , , \ ; p + 1 \leq i \leq p + q - 1 \ , , \ ; \widehat { x } _ { p + q } \ , , \end{cases} \right \} \ , , \end{align*}


\begin{align*} ( x ) _ { 0 , \lambda } = 1 , ( x ) _ { k , \lambda } = x ( x - \lambda ) ( x - 2 \lambda ) \cdots ( x - ( k - 1 ) \lambda ) , ( k \ge 1 ) . \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { | \Lambda ( M ) | } { | \Lambda ( M , \Gamma ) | } = | H _ { 1 } ( Y ) | \end{align*}


\begin{align*} \hat { K } & = \arg \max _ { k } \Big ( f \big ( \Re \{ \mathbf { y } \} ; \mathcal { H } _ k \big ) ^ 2 + f \big ( \Im \{ \mathbf { y } \} ; \mathcal { H } _ k \big ) ^ 2 \Big ) , \end{align*}


\begin{align*} \mathcal { E } _ { n , \lambda } ( x ) = \sum _ { k = 0 } ^ { n } k ! S _ { 2 , \lambda } ( n , k ) \sum _ { j = 0 } ^ { k } \binom { x } { j } \bigg ( - \frac { 1 } { 2 } \bigg ) ^ { k - j } , ( n \ge 0 ) . \end{align*}


\begin{align*} \mathcal N _ { \mathbf T ( u ) } ( w _ s ( u , 0 ) ) & = \mathcal N _ { \mathcal T _ D ( g ( \bar x ) ) } ( \nabla g ( \bar x ) u ; w _ s ( u , 0 ) ) \\ & \subset \mathcal N _ { \mathcal T _ D ( g ( \bar x ) ) } ( \nabla g ( \bar x ) u ) \cap [ w _ s ( u , 0 ) ] ^ { \perp } = \mathcal N _ D ( g ( \bar x ) ; \nabla g ( \bar x ) u ) \cap [ w _ s ( u , 0 ) ] ^ \perp \end{align*}


\begin{align*} \hat { \phi } ^ \omega ( f _ 1 , \ldots , f _ d ) : = \phi ( 0 ) + \sum _ { i = 1 } ^ d \Biggl ( \phi \biggl ( \sum _ { j = 1 } ^ i e _ { \omega _ j } \biggr ) - \phi \biggl ( \sum _ { j = 1 } ^ { i - 1 } e _ { \omega _ j } \biggr ) \Biggr ) f _ { \omega _ i } . \end{align*}


\begin{align*} E _ 1 ( x ) = \sum _ { x \leq p \leq 2 x } \frac { 1 } { p } \sum _ { \substack { 0 < b < p \\ g c d ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { m } - v ) } { p } } \cdot \frac { 1 } { p } \sum _ { \substack { 0 \leq b < p \\ g c d ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { e m } - v ) } { p } } = 0 . \end{align*}


\begin{align*} N ^ { 1 , p } _ * ( \Omega ) = \left \{ u \in N ^ { 1 , p } ( \Omega ) : \int _ \Omega u \ , d \mu = 0 \right \} , \end{align*}


\begin{align*} F \ ! = \ ! ( f , u _ 0 ) \in \mathcal { F } ^ { k , \mathbf { s } ( s , \lambda , \lambda ' , \delta ) } _ { T , \varLambda ^ q } \times C ^ { 2 s + k + 1 , \lambda , \delta } _ { \varLambda ^ q } \end{align*}


\begin{align*} \begin{aligned} U ^ { j } \overset { \ast } { \rightharpoonup } U ^ { 0 } ~ \mbox { i n } ~ L ^ { \infty } ( 0 , T _ { 0 } ; H ^ { s } ) , & v ^ { j } \rightharpoonup v ^ { 0 } ~ \mbox { i n } ~ L ^ { 2 } ( 0 , T _ { 0 } ; H ^ { s + 1 } ) , \\ U _ { t } ^ { j } \rightharpoonup U _ { 1 } ~ \mbox { i n } ~ & L ^ { 2 } ( 0 , T _ { 0 } ; H ^ { s - 1 } ) . \end{aligned} \end{align*}


\begin{align*} \kappa ( \tfrac { \eta } { n } , \tfrac { 1 } { \eta } ) = \big ( \coth ( \tfrac { \eta } { 2 } ) \ , e ^ { \frac { \eta } { n } } \big ) ^ 2 , \end{align*}


\begin{align*} I _ 1 & : = c _ { n , s } \lim _ { \varepsilon \to 0 ^ + } \int _ { B \setminus B _ \varepsilon ( x ) } \left ( \frac 1 { \vert x - y \vert ^ { n + 2 s } } - \frac 1 { \vert x _ \ast - y \vert ^ { n + 2 s } } \right ) \big ( u ( x ) - u ( y ) \big ) \dd y \\ I _ 2 & : = c _ { n , s } \int _ { \R ^ n _ + \setminus B } \left ( \frac 1 { \vert x - y \vert ^ { n + 2 s } } - \frac 1 { \vert x _ \ast - y \vert ^ { n + 2 s } } \right ) \left ( u ( x ) - u ( y ) \right ) \dd y . \end{align*}


\begin{align*} \frac { T _ n ( 2 ) - \nu _ n ( 2 ) } { \delta _ n ( 2 ) } = \frac { \tilde T _ n ( 2 ) } { \delta _ n ( 2 ) } . \end{align*}


\begin{align*} & \mathcal A ( \rho ^ * , m ^ * ) = \mathcal A ( \rho ^ { * , \infty } , m ^ { * , \infty } ) , \limsup _ { R \to + \infty } R \mathcal E ( \rho ^ { * , R } , m ^ { * , R } ) = 0 . \end{align*}


\begin{align*} G = G _ { \xi , \alpha , \nu } . \end{align*}


\begin{align*} 2 v _ 0 + \rho _ 2 + \rho _ 3 + \rho _ 4 = 0 \end{align*}


\begin{align*} S ^ T J S = J \ ; \Leftrightarrow \ ; S ^ T J = J S ^ { - 1 } \ ; \Leftrightarrow \ ; J = S ^ { - T } J S ^ { - 1 } . \end{align*}


\begin{align*} \left \{ \begin{aligned} \dot { x } ( s ) & = y ( s ) , \\ \dot { y } ( s ) & = y ( s ) + f ^ \prime ( x ( s ) ) , \\ x ( 0 ) & = x _ 0 , \\ y ( 0 ) & = y _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} & | \dot { x } _ i ( t ) - \mu _ i ( t ) | \lesssim \frac { 1 } { t ^ { 5 / 4 - \delta _ 0 } } , \\ & \bigg | \dot { \mu } _ i ( t ) + \sum ^ n _ { \substack { j = 1 , \\ j \not = i } } \frac { a _ { i j } } { x ^ 3 _ { i j } ( t ) } + \sum ^ n _ { \substack { k , j = 1 , \\ j \not = i } } \frac { b _ { i j k } \mu _ k ( t ) } { x ^ 3 _ { i j } ( t ) } \bigg | \lesssim \frac { 1 } { t ^ { 9 / 4 - \delta _ 0 } } . \end{align*}


\begin{gather*} \phi [ T ] = u [ T ] . \end{gather*}


\begin{align*} L ( x , t ) = \frac { \sqrt { 1 - x } } { \sqrt { 1 - x } \cosh { ( t \sqrt { 1 - x } ) } - \sinh { t \sqrt { 1 - x } } } . \end{align*}


\begin{align*} \Phi _ { \lambda ^ p _ k } ( x ) = \begin{pmatrix} \xi _ { \lambda ^ p _ k } ( x ) \\ \eta _ { \lambda ^ p _ k } ( x ) \end{pmatrix} , \forall k \geq k _ 0 . \end{align*}


\begin{align*} A ' ( \bar { x } , \bar { y } , t ) = L ^ 2 ( x , y , t ) , \end{align*}


\begin{align*} & e _ { B + a } x ^ h - x ^ h e _ { B + a } - \frac { \partial f ^ a } { \partial x ^ h } = 0 , a = 1 , \ldots , k \\ & e _ \alpha x ^ h - x ^ h e _ \alpha - x ^ \mu \tau ^ { \mu h } ( e _ \alpha ) = 0 , \alpha = 1 , \ldots , B \\ & t ^ \alpha _ \ell e _ \alpha = 0 , \ell = 1 , \ldots , B + k - n \\ & e _ \alpha e _ \beta - e _ \beta e _ \alpha - C ^ \gamma _ { \alpha \beta } e _ \gamma = 0 , \\ & e _ \alpha \xi ^ i - \xi ^ i e _ \alpha = 0 \end{align*}


\begin{align*} \norm { \mathcal { R } ^ { n } Z } _ { y } = O ( \epsilon ^ { 2 ^ { n } } ) \end{align*}


\begin{align*} \langle \mathbf { f , h } \rangle = \int _ { G ( \Q ) \backslash ( G ( \mathbb { A } _ { \mathbf { h } } ) / K _ 1 ( \mathfrak { n } ) \times \mathfrak { Z } ) } \mathbf { f } ( g , z ) \overline { \mathbf { h } ( g , z ) } \delta ( z ) ^ k \mathbf { d } g _ { \mathbf { h } } \mathbf { d } z . \end{align*}


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


\begin{align*} \overrightarrow { \eta } ^ { \ell } ( x ) : = \frac { 1 } { \ell } \sum _ { y = 1 } ^ { \ell } \eta ( x + y ) \ ; \ ; \ ; \ ; \overleftarrow { \eta } ^ { \ell } ( x ) : = \frac { 1 } { \ell } \sum _ { y = - \ell } ^ { - 1 } \eta ( x + y ) . \end{align*}


\begin{align*} L ( d ) = \frac { L _ 0 } { d _ 0 ^ { - \alpha } } , \end{align*}


\begin{align*} & [ T u , T v , T _ 1 ( w ) ] + [ T u , T _ 1 ( v ) , T w ] + [ T _ 1 ( u ) , T v , T w ] \\ & = T \Big ( D ( T u , T _ 1 ( v ) ) w + D ( T _ 1 ( u ) , T v ) w + \theta ( T v , T _ 1 ( w ) ) u + \theta ( T _ 1 ( v ) , T w ) u \\ & - \theta ( T u , T _ 1 ( w ) ) v - \theta ( T _ 1 ( u ) , T w ) v \Big ) + T _ 1 \Big ( D ( T u , T v ) w + \theta ( T v , T w ) u - \theta ( T u , T w ) v \Big ) , \end{align*}


\begin{align*} \sum _ { t = l + 1 } ^ { k } \bar { \sigma } ^ 2 _ { t } < ( k - l ) \alpha \| W \| _ F ^ 2 . \end{align*}


\begin{align*} \sum _ { j = 0 } ^ k \| f _ j \| \leq O ( \sqrt { k } \| f \| ) . \end{align*}


\begin{align*} \forall n ^ 0 , x ^ X , y ^ X \exists z ^ X , w ^ X \left ( y \in A x \rightarrow \left ( w \in A z \land x = _ X z + _ X \gamma _ n w \right ) \right ) . \end{align*}


\begin{align*} \mathcal { L } ( h , k ) = \mathcal { L } ^ 0 ( h , k ) + \mathcal { L } ^ r ( h , k ) , \end{align*}


\begin{align*} M ( p , H ) = \frac { \pi ^ 2 } { 6 } \left ( 1 + \frac { N ( p , H ) } { p } \right ) = \frac { \pi ^ 2 } { 6 } \left ( \left ( 1 - \frac { 1 } { p } \right ) \left ( 1 - \frac { 2 } { p } \right ) + \frac { 1 2 S ' ( H , p ) } { p } \right ) . \end{align*}


\begin{align*} \underset { R \rightarrow + \infty } { \lim } G _ R ( y ) = y . \end{align*}


\begin{align*} \widehat { \psi _ { p ^ { 2 k } } } ( \mathbf { u } ) : = \frac { 1 } { p ^ { 2 k n } } \sum _ { \mathbf { c } \in ( \mathbb { Z } / p ^ { 2 k } \mathbb { Z } ) ^ n } \psi _ { p ^ { 2 k } } ( f _ \mathbf { c } ) \exp \left ( \frac { 2 \pi i \langle \mathbf { c } , \mathbf { u } \rangle } { p ^ { 2 k } } \right ) , \end{align*}


\begin{align*} g _ \gamma ( x ) = \begin{cases} e ^ { - \frac { 1 } { ( x ( 1 - x ) ) ^ \gamma } } & x \in ( 0 , 1 ) , \\ 0 & x \notin ( 0 , 1 ) . \end{cases} \end{align*}


\begin{align*} \chi _ { \ell } = \begin{cases} 1 & x _ { \ell + 1 } \neq x _ { \ell } , \\ 0 & \mathrm { o t h e r w i s e } , \end{cases} \end{align*}


\begin{align*} & [ T u , T v , T _ 1 ( w ) ] + [ T u , T _ 1 ( v ) , T w ] + [ T _ 1 ( u ) , T v , T w ] \\ = & T \Big ( D ( T u , T _ 1 ( v ) ) w + D ( T _ 1 ( u ) , T v ) w + \theta ( T v , T _ 1 ( w ) ) u + \theta ( T _ 1 ( v ) , T w ) u \\ & - \theta ( T u , T _ 1 ( w ) ) v - \theta ( T _ 1 ( u ) , T w ) v \Big ) + T _ 1 \Big ( D ( T u , T v ) w + \theta ( T v , T w ) u - \theta ( T u , T w ) v \Big ) , \end{align*}


\begin{align*} \operatorname { h r m } ( \Delta ) = \sum _ { T _ i \in \Delta } \operatorname { h r m } ( T _ i ) \end{align*}


\begin{align*} \mathcal { Y } ( T ) = \{ a ( \cdot , \cdot ) \in \mathcal { X } _ T \ , : \ , | a ( t , x ) - a ( t ^ { ' } , x ) | \le B _ 1 | t - t ^ { ' } | ^ \gamma , \ , \ , \alpha \le a ( t , x ) \le B _ 2 \ , \ , \forall \ , t , t ^ { ' } \in [ s , s + T ] , \ , \ , x \in \bar \Omega \} , \end{align*}


\begin{align*} \sigma _ j : = \frac { 1 } { j ! } \frac { d ^ j } { d x ^ j } \bigg | _ { x = 0 } \sigma ( x ) \end{align*}


\begin{align*} { \rm A d } _ q v = q v q ^ * . \end{align*}


\begin{align*} & \partial _ t \epsilon - \partial _ y ( | D | \epsilon + \epsilon - p | V _ 0 | ^ { p - 1 } \epsilon ) \\ & \qquad \qquad + \partial _ y \big [ | V _ 0 + \epsilon | ^ { p - 1 } ( V + \epsilon ) - | V _ 0 | ^ { p - 1 } V _ 0 - p | V _ 0 | ^ { p - 1 } \epsilon \big ] + \Psi _ { V _ 0 } = 0 , \end{align*}


\begin{align*} \int _ { \exp ( 5 8 ) } ^ x \frac { | \theta ( t ) - t | } { t \log ^ 2 t } t & \le \int _ { \exp ( 5 8 ) } ^ { x } \frac { A _ 1 \log ^ B t \exp ( - C u ( t ) ) } { \log ^ 2 t } t \\ & \le \int _ { \exp ( 5 8 ) } ^ { x } h ' ( t ) t \\ & \leq A _ 1 x \log ^ { - \alpha } x \exp ( - C u ( x ) ) \\ & = \log ^ { 1 - B - \alpha } x \cdot A _ 1 x \log ^ { B - 1 } x \exp ( - C u ( x ) ) \\ & \le \log ^ { 1 - B - \alpha } x _ 0 \cdot A _ 1 x \log ^ { B - 1 } x \exp ( - C u ( x ) ) , \end{align*}


\begin{align*} U \Big [ \Big ( - \frac { d } { d x } + 2 x \Big ) f \Big ] & = \Big ( - \frac { d } { d x } + x \Big ) U f \\ U \Big [ \frac { d } { d x } f \Big ] & = \Big ( \frac { d } { d x } + x \Big ) U f . \end{align*}


\begin{align*} f ( z ) - 1 \sim c z ^ { - s } , L ( z ) = \frac { f ' ( z ) } { f ( z ) } \sim f ' ( z ) \sim - c s z ^ { - 1 - s } , F ( z ) \sim \frac { z ^ { 1 + s } } { c s } , \end{align*}


\begin{align*} | \Omega | & = \int _ \Omega u ^ { \frac { 1 } { r } } ( t , x ; s , u _ 0 ) u ^ { - \frac { 1 } { r } } ( t , x ; s , u _ 0 ) d x \\ & \le \left ( \int _ \Omega u ( t , x ; s , u _ 0 ) d x \right ) ^ { \frac { p } { p + 1 } } \left ( \int _ \Omega u ^ { - p } ( t , x ; s , u _ 0 ) d x \right ) ^ { \frac { 1 } { p + 1 } } \forall \ , t > s . \end{align*}


\begin{align*} \int ^ { T } _ { 0 } & \int _ { \Omega } \big ( \rho u \cdot \partial _ t \phi + ( \rho u \otimes u ) : \nabla \phi + \rho ^ \gamma { \rm { d i v } } \phi + \delta \rho ^ \beta { \rm { d i v } } \phi - \varepsilon \nabla \rho \cdot \nabla u \cdot \phi \\ & - Z _ \varepsilon : \nabla \phi - \mathbb { S } ( \nabla u ) : \nabla \phi + ( j - n u ) \cdot \phi \big ) \ , d x d t + \int _ { \Omega } \rho _ 0 u _ 0 \cdot \phi ( 0 , x ) \ , d x = 0 , \end{align*}


\begin{align*} \begin{cases} T _ 1 ( u ) + [ \mathfrak { X } , T u ] = T D ( \mathfrak { X } ) u + T ^ { ' } _ 1 ( u ) , \\ [ \mathfrak { X } , T _ 1 ( u ) ] = T ^ { ' } _ 1 D ( \mathfrak { X } ) ( u ) . \end{cases} \end{align*}


\begin{align*} \mathrm { E C H } ( Y , \lambda , 0 ) \cong \bigoplus _ { k = 0 } ^ { \infty } \mathbb { F } \langle \alpha _ { k } \rangle \bigoplus \bigoplus _ { j = 1 } ^ { m } \mathbb { F } \langle \beta _ { j } \rangle \end{align*}


\begin{align*} \bigwedge _ { i = 1 } ^ m \left ( \nu \delta ^ { \tau } \odot ^ \flat n _ i \delta ^ { t _ i } \right ) = \bigwedge _ { i = j } ^ m \left ( \nu \delta ^ { \tau } \odot ^ \flat n _ i \delta ^ { t _ i } \right ) = \bigwedge _ { i = j } ^ m \left ( \nu - n _ i \right ) \delta ^ \tau = ( \nu - n _ j ) \delta ^ \tau . \end{align*}


\begin{align*} \forall \gamma ^ 1 , p ^ X , x ^ X \left ( \gamma > _ \mathbb { R } 0 \land \gamma ^ { - 1 } ( x - _ X p ) \in A p \rightarrow p = _ X J ^ A _ { \gamma } x \right ) . \end{align*}


\begin{align*} \partial _ t \mathcal { V } _ k ^ A + \mathcal { L } \mathcal { V } _ k ^ A = i R ( \mathcal { V } _ k ^ A ) + O ( e ^ { - ( k + 1 ) e _ 0 t } ) \mathcal { S } ( \mathbb { R } ^ N ) . \end{align*}


\begin{align*} \partial _ \tau w = \partial _ y ^ 2 w + \frac { d + 1 } { y } \partial _ y w - \frac { 1 } { 2 } \Lambda _ y w - 3 ( d - 2 ) w ^ 2 - ( d - 2 ) y ^ 2 w ^ 3 . \end{align*}


\begin{align*} M _ \omega T _ x f ( t ) = f ( t - x ) \ , e ^ { 2 \pi i \omega \cdot t } = f ( t - x ) \ , e ^ { 2 \pi i \omega \cdot ( t - x ) } e ^ { 2 \pi i \omega \cdot x } = e ^ { 2 \pi i \omega \cdot x } \ T _ x M _ \omega f ( t ) . \end{align*}


\begin{align*} m _ 0 = \lim _ { n \rightarrow + \infty } J ( u _ n ) & \geq \liminf _ { n \rightarrow + \infty } J ( t _ { \widehat { u } } u _ n ) \geq J ( t _ { \widehat { u } } \widehat { u } ) \\ & \geq m _ 0 . \end{align*}


\begin{align*} \left \{ \begin{array} [ c ] { l } \partial _ { t } u ( t ) = F \left ( t , u \left ( t \right ) \right ) \\ \\ u ( 0 ) = \phi \in Y \end{array} \right . \end{align*}


\begin{align*} & m _ { i j } = - \frac { 3 a _ { i j } } { ( \alpha _ i - \alpha _ j ) ^ 4 } , i \not = j , \\ & m _ { i i } = \sum ^ n _ { \substack { j = 1 \\ j \not = i } } \frac { 3 a _ { i j } } { ( \alpha _ i - \alpha _ j ) ^ 4 } , \forall i = 1 , \ldots , n . \end{align*}


\begin{align*} u ( x ) = \inf \left \lbrace \int _ 0 ^ \infty e ^ { - s } \left ( C _ p | \dot { \gamma } ( s ) | ^ { q } + f ( \gamma ( s ) ) \right ) d s : \gamma \in \mathrm { A C } ( [ 0 , \infty ) ; \overline { \Omega } ) , \gamma ( 0 ) = x \right \rbrace , \end{align*}


\begin{align*} T _ { \ell , \pi ^ \star _ { O F } } ^ { a v } = \lim _ { t \to \infty } \frac { t } { R _ \ell ^ \star ( t ) } = \lim _ { t \to \infty } \frac { t } { h _ \ell ( t ) } \frac { h _ \ell ( t ) } { R _ { \ell } ^ \star ( t ) } \stackrel { ( a ) } { = } \frac { \mu _ { \ell } } { f _ \ell ^ \star } , \end{align*}


\begin{align*} \mathbf { M } _ { u } \mathbf { M } _ { v } = \sum _ { w \in \mathfrak { B } _ { p + q } } \sum _ { v \leq v ' } \mu ( v , v ' ) b _ { u , v ' } ^ w \mathbf { M } _ { w } . \end{align*}


\begin{align*} \int _ \R \left ( x f ( x ) \right ) \overline { f ' ( x ) } \ , d x = x | f ( x ) | ^ 2 \Bigg | _ { x = - \infty } ^ \infty - \int _ \R \left ( | f ( x ) | ^ 2 + x f ' ( x ) \overline { f ( x ) } \right ) \ , d x . \end{align*}


\begin{align*} ( \pi ^ { r + 1 , r } ) ^ { * } \rho = h \rho + \sum _ { k = 1 } ^ { q } p _ { k } \rho , \end{align*}


\begin{align*} \mathcal { D } ( h , k ) : = \frac { 1 } { 2 } \sum _ { \substack { 1 \leq q < \infty \\ ( q , h k ) = 1 } } W \left ( \frac { q } { Q } \right ) \sum _ { \substack { 1 \leq m , n < \infty \\ ( m n , q ) = 1 \\ m h = n k } } \frac { \tau _ A ( m ) \tau _ B ( n ) } { \sqrt { m n } } V \left ( \frac { m } { X } \right ) V \left ( \frac { n } { X } \right ) \sum _ { \substack { 1 \leq c \leq C , d \geq 1 \\ c d = q \\ d | m h \pm n k } } \phi ( d ) \mu ( c ) , \end{align*}


\begin{align*} x _ { t _ { m ( k ) } } \big ( s ( t _ { m ( k ) } ) \big ) = \varepsilon _ { t _ { m ( k ) } } = 1 . \end{align*}


\begin{align*} \int _ { \R ^ n _ + } \frac { \dd y } { \vert x _ \ast - y \vert ^ { n + 2 s } } & = x _ 1 ^ { - 2 s } \int _ { \R ^ n _ + } \frac { \dd z } { \vert e _ 1 + z \vert ^ { n + 2 s } } . \end{align*}


\begin{gather*} c ^ L _ X = ( c ^ R _ X ) ^ { - 1 } \end{gather*}


\begin{align*} C = \left ( a _ 1 , a _ 2 , \ldots , a _ k , \overline { a _ 1 } , \overline { a _ 2 } , \ldots , \overline { a _ k } \right ) , \end{align*}


\begin{align*} \sum _ { \substack { c > C , d \geq 1 \\ c d = q } } \mu ( c ) = \begin{cases} \displaystyle - \sum _ { \substack { c \leq C , d \geq 1 \\ c d = q } } \mu ( c ) & q > 1 \\ \\ \hphantom { - - - } 0 & q = 1 . \end{cases} \end{align*}


\begin{align*} \sum _ { 0 \leq n < m < \infty } \frac { \tau _ A ( p ^ { m } ) \tau _ B ( p ^ { n } ) \left ( 1 - \frac { p ^ w } { p ^ { 2 } } \right ) } { p ^ { m ( 1 - \alpha ) } p ^ { n ( 2 - \beta - w ) } } = \frac { \tau _ A ( p ) } { p ^ { 1 - \alpha } } - \frac { \tau _ A ( p ) } { p ^ { 3 - \alpha - w } } + O \left ( \frac { 1 } { p ^ { 1 + \varepsilon } } \right ) . \end{align*}


\begin{align*} \Phi ( h _ n ) \to 0 , \| h _ n \| _ 2 ^ 2 = 1 , \end{align*}


\begin{align*} ( - \mathcal { L } _ \delta ) ( F _ \delta ) = g , \end{align*}


\begin{align*} \tfrac { \partial } { \partial \eta } A ( \tfrac { \eta } { n } , \tfrac { 1 } { \eta } ) \Bigr | _ { \eta = \eta _ n } > 0 \end{align*}


\begin{align*} T : = \frac { 1 } { \max _ { t \ge H } \left ( \frac { x ^ { - \nu _ 3 ( t ) } } { t } \right ) } = \min _ { t \ge H } \left ( x ^ { \nu _ 3 ( t ) } t \right ) . \end{align*}


\begin{align*} \tilde { \lambda } _ 0 = \frac { \int _ { T } | \nabla \bar { u } | ^ 2 } { \int _ { T } \tilde { m } _ 0 \bar { u } ^ 2 } = \frac { \int _ { \R ^ N } | \nabla \bar { u } | ^ 2 } { \int _ { \R ^ N } \tilde { m } _ 0 \bar { u } ^ 2 } \ ; . \end{align*}


\begin{align*} ( \partial _ t - \mathcal { L } ) U + Q \lozenge U = F , U | _ { t = 0 } = G , \end{align*}


\begin{align*} \zeta ( s ) = \sum _ { n \ge 1 } n ^ { - s } \ \ \Re s > 1 . \end{align*}


\begin{align*} ( a ( x ) - a ( y ) ) \langle x , x - y \rangle + a ( y ) | x - y | ^ { 2 } & = a ( y ) ( | x - y | ^ { 2 } + \langle x , x - y \rangle ) \\ & = a ( y ) ( \langle x - y , x - y \rangle - \langle x , x - y \rangle \\ & a ( y ) \langle x - y , - y \rangle \\ & = a ( y ) ( | y | ^ 2 - \langle x , y \rangle ) \\ & \geq a ( y ) ( | y | ^ 2 - | x | | y | ) \geq 0 \end{align*}


\begin{align*} \ell _ { \mathfrak { B } } ( \Psi ^ k ( t ) ) \leq 1 + \sum _ { i = 0 } ^ { k - 1 } \ell _ { \mathfrak { B } } ( \Psi ^ i ( a _ { \Psi } ) ) \leq 1 + s ( k + 1 ) ^ { n + 1 } . \end{align*}


\begin{align*} \psi = \varphi _ S = \varphi _ { S ^ { - 1 } } . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \frac { \delta \mathcal { L } _ { \sigma } } { \delta \overline { \phi } } ( \phi _ { \star } , \chi _ { \star } ) = \frac { \delta E } { \delta \overline { \phi } } ( \phi _ { \star } ) - \chi _ { \star } \phi _ { \star } + \sigma R ^ \prime ( \| \phi _ { \star } \| ^ 2 - 1 ) \phi _ { \star } = 0 , \\ & \frac { \partial \mathcal { L } _ { \sigma } } { \partial \chi } ( \phi _ { \star } , \chi _ { \star } ) = \| \phi _ { \star } \| ^ { 2 } - 1 = 0 , \end{aligned} \right . \end{align*}


\begin{align*} \lambda _ 1 + \lambda _ 2 + \lambda _ 3 & = n - 3 , \\ \lambda _ 1 \ , \lambda _ 2 + \lambda _ 2 \ , \lambda _ 3 + \lambda _ 3 \ , \lambda _ 1 & = - ( 2 n - 3 ) , \\ \lambda _ 1 \ , \lambda _ 2 \ , \lambda _ 3 & = n - 1 - 4 s _ 2 t _ 1 ( s _ 1 + t _ 2 ) . \end{align*}


\begin{align*} f = S ( S ^ { - 1 } f ) = \sum _ { \gamma \in \Gamma } \langle S ^ { - 1 } f , e _ \gamma \rangle e _ \gamma = \sum _ { \gamma \in \Gamma } \langle f , S ^ { - 1 } e _ \gamma \rangle e _ \gamma \end{align*}


\begin{align*} \Delta _ \eta m ^ { - 1 } _ \eta = I + \mathcal { O } ( t ^ { - 1 } ) , k \in \partial U _ { \delta } ( \eta ) , t \rightarrow \infty , \end{align*}


\begin{align*} \pi _ j = \prod _ { i = 1 } ^ r \nu _ i ^ { n _ { j i } } \end{align*}


\begin{align*} \| D \| _ { H ^ 1 } \lesssim & \sup _ { \substack { i \not = j , k \geq 1 \\ 2 \leq k + \ell \leq 3 } } \frac { 1 } { d ^ 3 } \Big ( \big \| ( \partial _ y ^ k \varphi _ { i j } ) ( \partial _ y ^ \ell B _ { i j , 0 } ) \big \| _ { L ^ 2 } \Big ) + \frac { 1 } { d ^ 5 } \lesssim \frac { 1 } { d ^ 5 } . \end{align*}


\begin{align*} & V : = \{ \delta _ t \otimes z ^ * : t \in B _ 2 , z ^ * \in V _ Z \} \subseteq W \subseteq U , \\ & x _ 1 ^ * : = \delta _ { s _ 0 } \otimes z _ 1 ^ * , e ( t ) : = e _ A ( t ) e _ Z \ , \ , t \in \partial A \end{align*}


\begin{align*} P ' ( x ) + P ( x ) \tilde F ( x ) = F ( x ) P ( x ) , x \in \mathbb R _ + . \end{align*}


\begin{align*} p ( x ) = y _ 1 \delta _ { 1 } ( x ) + y _ 2 \delta _ 2 ( x ) + \cdots + y _ t \delta _ { t } ( x ) , \end{align*}


\begin{align*} F _ 1 ( x ) : = \norm { x - J ^ S _ { \mu _ 0 } ( x + \mu _ 0 T ^ \circ x ) } \end{align*}


\begin{align*} X ^ 2 - 3 \sigma { \log ( 1 / \delta ) } X - \left ( ( \kappa + L ) \sqrt { \sum _ { t = 1 } ^ T \| g _ t \| ^ 2 } + 3 ( G ^ 2 + G \tilde G ) { \log ( 1 / \delta ) } \right ) \leq 0 , \end{align*}


\begin{align*} \biggl \{ ( s , m ) \biggm | s \in Q , \ m _ { ( I , j ) } = \prod _ { i \in I } s _ { i j _ i } , \ ; I \subseteq \{ 1 , \ldots , d \} | I | \geq 2 , j \in \{ 1 , \ldots , n \} ^ d \biggr \} . \end{align*}


\begin{align*} H ^ 1 \big ( \Gamma , C _ c ^ \infty ( X _ E \setminus X _ F , \mathbb { \overline { F } } _ l ) \big ) = 0 . \end{align*}


\begin{align*} \begin{aligned} & w _ { ( D , j ) } \geq 0 w _ { ( I , j ) } = 0 & & I \subset D , \ ; j \in E , \\ & y _ { ( I , j ) } = \sum _ { I ' \supseteq I , j ' \in E ( I , j ) } w _ { ( I ' , j ' ) } & & I \subseteq D , \ ; j \in E . \end{aligned} \end{align*}


\begin{align*} I _ { \min } ( f ) = \inf _ { v \in N ^ { 1 , p } _ * ( \Omega ) } I _ f ( v ) . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ { \infty } \sup _ { \tau \in ( 0 , T ] } \left \| z ^ { ( k ) } ( \tau , \cdot ) \right \| _ { H ^ 2 ( \R \backslash \{ 0 \} ) } ~ < ~ \infty . \end{align*}


\begin{align*} \mathbf { E } _ l ( x , s ) = \mathbf { E } ^ m _ l ( x , s ; \chi ) = \sum _ { \gamma \in P _ m \backslash G _ { 2 m } } \varphi ( \gamma x , s ) , \end{align*}


\begin{align*} \left \lbrace \begin{aligned} \dfrac { \mathrm { d } } { \mathrm { d } t } x ( t ) & = - \nabla \varphi ( x ( t ) ) \\ x ( 0 ) & = x _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} E _ i ( x ) = \alpha + \int _ 0 ^ x ( n _ i - b _ i ) ( y ) d y , \quad \forall x \in [ 0 , 1 ] , i = 1 , 2 . \end{align*}


\begin{align*} \tilde { \xi } ( s ) = \left \{ \begin{aligned} & \xi ( s ) 0 \leq s < t _ 0 \\ & \xi ( t _ 0 ) s \geq t _ 0 \end{aligned} \right . \end{align*}


\begin{align*} \mathcal { M } _ K : = \biggl \{ ( \lambda , y ) \in \R _ + ^ { | V | + | E | } \biggm | \lambda \in \Lambda , \ y _ e = \prod _ { v \in e } \lambda _ v , \ e \in E \biggr \} . \end{align*}


\begin{align*} J _ { a , b } ( T ) = \int _ { T } ^ { \infty } \frac { \log ^ a y } { y ^ b } d y . \end{align*}


\begin{align*} K _ \alpha ( x ) : = \alpha K ( \alpha x ) \end{align*}


\begin{align*} \partial _ u \partial _ v \phi = : P \phi = 0 . \end{align*}


\begin{align*} \partial _ r \left ( \varphi _ \mu \right ) ( r \theta ) = - r \langle \theta ; \Sigma ( \theta ) \rangle \varphi _ \mu ( r \theta ) . \end{align*}


\begin{align*} \frac { c _ k ( \mu ) L ( \mu , \mathbf { f } , \chi ) } { \pi ^ { \beta } \langle \mathbf { f , f } \rangle } = \frac { \langle \mathbf { h } _ w , \mathbf { f } \rangle } { \langle \mathbf { f , f } \rangle } \left ( \frac { \mathbf { f } ( g _ { \mathbf { h } } \cdot g _ { \infty } ) } { \mathfrak { P } _ k ( w ) } \right ) ^ { - 1 } \in \overline { \Q } , \end{align*}


\begin{align*} \sigma = [ b _ 1 , \ldots , b _ { m - s } , a _ { i + 1 } , a _ 1 , \ldots , a _ { i } , a _ { i + 2 } , \ldots , a _ s ] ; \end{align*}


\begin{align*} b ^ * \delta _ { \varphi } ^ { n } ( ( ( \sigma ^ { \varphi } ) ^ n ) ( S ^ { 2 n } ( b ) ) ) = 0 . \end{align*}


\begin{align*} J ( u _ n ) & = J ( \Vert u _ n \Vert v _ n ) \\ & \leq g ^ + \max \left \lbrace \Vert u _ n \Vert ^ { g ^ - } \Vert v _ n \Vert ^ { g ^ - } , \Vert u _ n \Vert ^ { g ^ + } \Vert v _ n \Vert ^ { g ^ + } \right \rbrace - \int _ { \mathbb { R } ^ { d } } K ( x ) F ( x , \Vert u _ n \Vert v _ n ) d x \\ & \leq g ^ + \max \left \lbrace \Vert u _ n \Vert ^ { g ^ - } , \Vert u _ n \Vert ^ { g ^ + } \right \rbrace - \int _ { \mathbb { R } ^ { d } } K ( x ) F ( x , \Vert u _ n \Vert v _ n ) d x , \end{align*}


\begin{align*} 2 \langle A ^ { 0 } \partial _ { x } ^ { \alpha } u _ { t } , \partial _ { x } ^ { \alpha } u \rangle = \frac { d } { d t } \langle A ^ { 0 } \partial _ { x } ^ { \alpha } u , \partial _ { x } ^ { \alpha } u \rangle - \langle \partial _ { x } ^ { \alpha } u , ( \partial _ { t } A ^ { 0 } ) \partial _ { x } ^ { \alpha } u \rangle , \end{align*}


\begin{align*} \nabla _ Y E _ 3 ^ { \top } = \langle \xi , E _ 3 ^ { \perp } \rangle A Y + \langle Y , E _ 1 ^ { \top } \rangle E _ 1 ^ { \top } - \langle Y , E _ 2 ^ { \top } \rangle E _ 2 ^ { \top } . \end{align*}


\begin{align*} a _ { i j } = \frac { 4 \sigma _ i \sigma _ j \kappa _ 0 ( p - 1 ) \int Q ^ p } { ( p - 3 ) \int Q ^ 2 } . \end{align*}


\begin{align*} \norm { T } _ { p } ^ { p } \leq \overset { n } { \underset { i , j = 1 } { \sum } } \norm { T _ { i j } } _ { p } ^ { p } \leq n ^ { 2 - p } \norm { T } _ { p } ^ { p } , \end{align*}


\begin{align*} \mathbf { g } _ { m k } [ t ] & = \Bigl [ g _ { m k , 0 } [ t ] , \ldots , g _ { m k , L _ { m k } - 1 } [ t ] \Bigr ] ^ T \\ & = \sqrt { \beta _ { m k } } \mathbf { h } _ { m k } [ t ] , \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 ^ + } \frac { t } { s ( r , t ) } = \frac { \lambda _ 1 ( \mu ) ^ 2 - r ^ 2 } { \lambda _ 1 ( \mu ) ^ 2 } . \end{align*}


\begin{align*} \begin{cases} - \Delta u + \tilde { \lambda } _ 0 \underline { m } \ , u = 0 & \R ^ n \ ; , \\ u ( \mathbf { 0 } ) = 1 \ ; u > 0 & \R ^ n \ ; . \end{cases} \end{align*}


\begin{align*} G ^ { \Sigma ^ 1 } _ { \Sigma ^ 0 , g } ( x ) : = \begin{cases} \sup \{ t \geq 0 : \exp ^ g _ x ( s \nu ^ 0 ( x ) ) \in U ^ 1 s \in [ 0 , t ] \} , & \ x \in U ^ 1 ; \\ \inf \{ t \leq 0 : \exp ^ g _ x ( s \nu ^ 0 ( x ) ) \in N \setminus U ^ 1 s \in [ t , 0 ] \} , & \ x \notin U ^ 1 . \end{cases} \end{align*}


\begin{align*} I ( \hat { \alpha } \cup { ( \gamma , M - 1 ) } , \hat { \alpha } \cup { ( \delta , 1 ) } \cup { ( \gamma , M - p _ { i } - 1 ) } ) = 2 \end{align*}


\begin{align*} \mathcal { A } : = \left \{ ( \rho , \phi ) : - \nabla \cdot ( \epsilon ( x ) \nabla \phi ) = f ( x ) + \sum _ { i = 1 } ^ { s } z _ i \rho _ i , \alpha \phi + \beta \frac { \partial \phi } { \partial \mathbf { n } } = \phi ^ b , x \in \partial \Omega , \rho \in [ \mathcal { P } ( \Omega ) ] ^ s \right \} . \end{align*}


\begin{align*} \iint _ { \mathcal { Q } \times \mathcal { Q } } | Z f ( x , \omega ) | ^ 2 \ , d ( x , \omega ) = \norm { f } _ 2 ^ 2 . \end{align*}


\begin{align*} x ' = _ X x = _ X y + \gamma \gamma ^ { - 1 } ( x - y ) = _ X y + \gamma ' \gamma ^ { - 1 } ( x - y ) . \end{align*}


\begin{align*} \chi ( e ' ) _ v = \begin{cases} 1 & v \in e ' \\ 0 & v \notin e ' , \end{cases} \chi ( e ' ) _ e = \begin{cases} 1 & e = e ' \\ 0 & . \end{cases} \end{align*}


\begin{align*} \mathcal { I } _ { \ell } ( h , k ) = \mathcal { I } _ { \ell } ^ * ( h , k ) + O ( X ^ { \varepsilon } Q ^ { 1 + \varepsilon } ( h k ) ^ { \varepsilon } ) , \end{align*}


\begin{align*} \mu ( d x ) = \frac { 1 } { Z ( \mu ) } \exp \left ( - V ( x ) \right ) d x , \end{align*}


\begin{align*} \mathrm { S p e c } ( \Delta _ 0 ) & = \{ c _ { \gamma } \ ; ; \ ; \gamma \in \widehat { G } \ ; \mathrm { s . t . } \ ; \mathrm { m u l t } ( \mathrm { t r i v . r e p r . } , \mathrm { R e s } ^ { G } _ { K } ( \rho _ { \gamma } ) ) \neq 0 \} \ , , \\ \intertext { a n d } \mathrm { S p e c } ( \Delta _ 1 ) & = \{ c _ { \gamma } \ ; ; \ ; \gamma \in \widehat { G } \ ; \mathrm { s . t . } \ ; \mathrm { m u l t } ( \rho , \mathrm { R e s } ^ { G } _ { K } ( \rho _ { \gamma } ) ) \neq 0 \} \ , . \end{align*}


\begin{align*} \| \varepsilon ( \cdot , \tau ) \| _ { L ^ \infty } = o ( b ^ { - 1 } ) , \tau \to + \infty . \end{align*}


\begin{align*} \phi \prec \varepsilon _ A : = \phi , \varepsilon _ A \succ \phi : = \phi , \phi \succ \varepsilon _ A : = 0 , \varepsilon _ A \prec \phi : = 0 . \end{align*}


\begin{align*} ( J _ L ) _ b ^ \xi = \mathcal { G } ( v _ b , \xi _ Q ) = g _ { i j } v ^ i _ b \Lambda ^ j . \end{align*}


\begin{align*} \mathcal { D } ^ { ( k ) } \mathcal { X } _ t = \left ( \mathcal { D } ^ { j _ 1 , j _ 2 , \dots , j _ k } _ { r _ 1 , r _ 2 , \dots , r _ k } \mathcal { X } _ t \right ) _ { \substack { j _ 1 , j _ 2 , \dots , j _ k \in \{ 1 , 2 , \dots , d \} \\ r _ 1 , r _ 2 , \dots , r _ k \in [ 0 , T ] } } . \end{align*}


\begin{align*} g _ 1 ( t ) \mathcal { L } _ { K , \mathfrak { s } } ( \boldsymbol { \alpha } ) + g _ 2 ( t ) \mathcal { L } _ { K , w } ( \beta ) = 0 . \end{align*}


\begin{align*} v _ t + G _ \beta ( v , \nabla v , \nabla ^ 2 v , K \cap \{ v ( \cdot , t ) \leq v ( x , t ) \} ) = 0 \end{align*}


\begin{align*} \mathcal { H } ^ { \zeta } _ t ( E ) = \inf \left \{ \sum _ { n \in \mathbb { N } } \zeta ( \vert B _ n \vert ) : \ , \vert B _ n \vert \leq t , \ B _ n E \subset \bigcup _ { n \in \mathbb { N } } B _ n \right \} . \end{align*}


\begin{align*} \| { S } ( \widehat { \Theta } ) - S ( \Theta ) \| _ 2 = \| D _ A \| _ 2 < \| D _ C \| _ 2 = \| { \mathcal { N } } _ { C } ( \widehat { \Theta } ) - { \mathcal { N } } _ { C } ( \Theta ) \| _ 2 . \end{align*}


\begin{align*} F _ 3 ( z ) = ( z - K ) H _ n \left ( \frac { K + 1 } { K - 1 } - \frac { 2 K } { ( K - 1 ) z } \right ) \end{align*}


\begin{align*} L _ n \left ( { 4 x \over ( 1 + x ) ^ 2 } \right ) = { 1 \over ( 1 + x ) ^ n } \sum _ { k = 0 } ^ n { n \choose k } ( 1 - x ) ^ { n - k } 2 ^ k A _ k ( x ) . \end{align*}


\begin{align*} \lim _ { T \to \infty } s ( \nu _ 0 \circ \alpha _ { T \Psi } ^ { 0 \to 1 } | \nu _ 1 ) = 0 , \end{align*}


\begin{align*} Z f \overline { Z g } = \sum _ { k , l } \langle Z f \overline { Z g } , e _ { k , l } \rangle \ , e _ { k , l } \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d } } \int _ { \mathbb { R } ^ { d } } g \left ( \frac { u _ n ( x ) - u _ n ( y ) } { \vert x - y \vert ^ { \alpha } } \right ) \frac { u _ n ( x ) - u _ n ( y ) } { \vert x - y \vert ^ { \alpha + d } } d x d y + \int _ { \mathbb { R } ^ { d } } g ( u _ n ) u _ n d x = \int _ { \mathbb { R } ^ { d } } K ( x ) f ( x , u _ n ) u _ n d x , \ \ n \in \mathbb { N } . \end{align*}


\begin{align*} \mathcal { B } ( u _ 0 , r ) = \{ u \in C ^ 0 ( \bar \Omega ) \ , | \ , \| u - u _ 0 \| _ \infty \le r \} . \end{align*}


\begin{align*} \sup _ { f \in M } \| f \| _ { Y _ q ( \tilde \Omega _ { [ \hat t ( n ) ] } ) } \stackrel { ( \ref { n u 2 } ) , ( \ref { e m b _ n u } ) } { \underset { \mathfrak { Z } _ 0 } { \lesssim } } 2 ^ { ( ( 1 ‐ \lambda ) \mu _ * ‐ \lambda \alpha _ * ) k _ * \hat t ( n ) } \stackrel { ( \ref { h a t _ m t } ) } { = } n ^ { ( ( 1 ‐ \lambda ) \mu _ * ‐ \lambda \alpha _ * ) / \gamma _ * } = n ^ { ‐ \theta _ 2 } \end{align*}


\begin{align*} C _ { \ell } = C _ { \ell } ( \alpha , \beta ) : = \dfrac { ( 2 \ell - 1 ) ! ! } { ( 1 - 2 \alpha ) ^ { \ell - 1 } } \cdot { \dfrac { \beta } { \Gamma ( 1 + \alpha ) } } ( \ell \in \mathbb { N } ) . \end{align*}


\begin{align*} h _ 1 : U _ 2 \ast I & \to U _ 2 \\ ( ( x , t ) , ( i , j ) ) & \mapsto \begin{cases} ( x , \hat i t ) & t < 0 \\ ( x , t ) & t \ge 0 . \end{cases} \end{align*}


\begin{align*} M [ u ] = M [ Q ] , E [ u ] = E [ Q ] , \end{align*}


\begin{align*} \mathcal A ( \rho , m ) : = \int _ 0 ^ 1 \frac 1 4 \sum _ { ( i , j ) \in E } L ( \theta _ { i j } ( \rho ) , m _ { i j } ) d t , \end{align*}


\begin{align*} M ( f , \{ 1 \} ) = { \pi ^ 2 \over 6 } \times \left \{ \prod _ { q \mid f } \left ( 1 - { 1 \over q ^ 2 } \right ) - { 3 \over f } \prod _ { q \mid f } \left ( 1 - { 1 \over q } \right ) \right \} \ \ \ \ \ \hbox { ( $ f > 2 $ ) . } \end{align*}


\begin{align*} ( L _ i ) _ K = \begin{pmatrix} K & e ^ { ( i ) } _ 1 \times A \\ e _ 2 ^ { ( i ) } \times A & K \end{pmatrix} , \end{align*}


\begin{align*} \bigsqcup _ { i = 1 } ^ { \lfloor T \rfloor } B ^ M _ \epsilon ( x _ i ) & \subseteq B ^ M _ \epsilon ( K ) , \intertext { a n d t h u s } \sum _ { i = 1 } ^ { \lfloor T \rfloor } \mathrm { V o l } \left ( B ^ M _ \epsilon ( x _ i ) \right ) & \leq \mathrm { V o l } \left ( B ^ M _ \epsilon ( K ) \right ) . \end{align*}


\begin{align*} \frac { ( - \Delta ) ^ s v ( h e _ 1 ) } h & = \frac { c _ { n , s } } 2 \int _ { \R ^ n } \bigg ( 2 \partial ^ h _ 1 v ( 0 ) - \partial ^ h _ 1 v ( y ) - \partial ^ h _ 1 v ( - y ) - \frac { v ( y ) + v ( - y ) } h \bigg ) \frac { \dd y } { \vert y \vert ^ { n + 2 s } } \\ & = ( - \Delta ) ^ s \partial ^ h _ 1 v ( 0 ) . \end{align*}


\begin{align*} \mathrm { Q F } \mathrm { E R } : \quad \frac { A _ 0 \rightarrow s = _ \rho t } { A _ 0 \rightarrow r [ s / x ^ \rho ] = _ \tau r [ t / x ^ \rho ] } \end{align*}


\begin{align*} \widetilde { U } \Big [ \Big ( \frac { d } { d x } + 2 x \Big ) f \Big ] & = \Big ( \frac { d } { d x } + x \Big ) \widetilde { U } f \\ \widetilde { U } \Big [ \frac { d } { d x } f \Big ] & = \Big ( \frac { d } { d x } - x \Big ) \widetilde { U } f . \end{align*}


\begin{align*} K _ v = \left \{ \gamma = \left [ \begin{array} { c c c } a & b & c \\ g & e & f \\ h & l & d \end{array} \right ] \in G ( \mathcal { O } _ v ) : \gamma \equiv \left [ \begin{array} { c c c } 1 _ m & \ast & \ast \\ 0 & 1 _ r & \ast \\ 0 & 0 & 1 _ m \end{array} \right ] p _ v ^ { n _ v } \right \} . \end{align*}


\begin{align*} H _ 0 = \mathbb { N } , H _ i = \langle p ^ { 2 i - 1 } + 2 , p H _ { i - 1 } \rangle , i = 1 , \ldots , s . \end{align*}


\begin{align*} \mathbb { E } _ { U , \xi } \left [ \left \| \nabla \phi _ { \mu } ^ { 1 / \bar { \rho } } ( x _ { t ^ * } ) \right \| _ 2 ^ 2 \right ] \leq \frac { \bar { \rho } } { \bar { \rho } - \rho } \frac { \left ( \phi _ { \mu } ^ { 1 / \bar { \rho } } ( x _ 0 ) - \underset { x } { \min } \ \phi _ { \mu } ( x ) \right ) + 2 ( n ^ 2 + 2 n ) \bar { \rho } L _ { f , 0 } ^ 2 \sum _ { t = 0 } ^ { T } \alpha _ t ^ 2 } { \sum _ { t = 0 } ^ T \alpha _ t } . \end{align*}


\begin{align*} \deg ( \pi ^ n ) \ = \ n + 1 . \end{align*}


\begin{align*} \int _ { 0 } ^ { 1 } u _ x \bar { \sigma } - \int _ { 0 } ^ { 1 } \rho \bar { \sigma } _ x + \lambda \int _ { 0 } ^ { 1 } \rho \bar { \sigma } = \int _ { 0 } ^ { 1 } f \bar { \sigma } . \end{align*}


\begin{align*} | R _ j ( x , y ) - R _ j ( x , y ' ) | & \leqslant c | y _ j - y _ j ' | \int _ 0 ^ \infty | h _ t ( x , y ) - h _ t ( x , y ' ) | \frac { d t } { t \sqrt t } \\ & \leqslant c | y _ j - y _ j ' | \bigg ( \int _ 0 ^ { \| x - y \| ^ 2 } + \int _ { \| x - y \| ^ 2 } ^ \infty \bigg ) | h _ t ( x , y ) - h _ t ( x , y ' ) | \frac { d t } { t \sqrt t } \\ & = : I \ ! I _ 1 + I \ ! I _ 2 . \end{align*}


\begin{align*} \pi ( p ) = \sum _ { i = 1 } ^ k ( \beta ^ { - 1 } \alpha _ i ) \pi ( p _ i ) + \sum _ { i = k + 1 } ^ d ( - \beta ^ { - 1 } \beta _ i ) \pi ( p _ i ) . \end{align*}


\begin{align*} \lim _ { j \to \infty } \max _ { t \in [ 0 , T ] } R _ j ( t ) = 0 . \end{align*}


\begin{align*} \sum z _ i = 0 \ , , \end{align*}


\begin{align*} m _ { i , \lambda } ( \Z ( P , B ) ) = 0 \end{align*}


\begin{align*} \frac { 1 } { \phi ( d ) } \sum _ { \psi \bmod d } \psi ( m h ) \overline { \psi } ( n k ) = \left \{ \begin{array} { c l } 1 & d | m h - n k \\ \\ 0 & \end{array} \right . \end{align*}


\begin{align*} \min _ { | z | = 1 } \norm { f - z g } _ { L ^ p } \le C \big \| \ , | f | - | g | \ , \big \| _ { L ^ p } ^ \gamma \cdot ( \norm { f } _ { L ^ p } + \norm { g } _ { L ^ p } ) ^ { 1 - \gamma } \ \ \forall \ , f , g \in V . \end{align*}


\begin{align*} \begin{aligned} \norm { \bar x _ t - x _ t } ^ 2 & = \norm { ( 1 - \alpha _ t ) \Gamma _ { t - 1 } \sum _ { k = 1 } ^ { t - 1 } \frac { \alpha _ k } { \Gamma _ k } \frac { ( \eta _ k - \gamma _ k ) } { \alpha _ k } \nabla f ( \bar x _ k ) } ^ 2 \\ & \leq ( 1 - \alpha _ t ) ^ 2 \Gamma _ { t - 1 } \sum _ { k = 1 } ^ { t - 1 } \frac { \alpha _ k } { \Gamma _ k } \frac { ( \eta _ k - \gamma _ k ) ^ 2 } { \alpha _ k ^ 2 } \norm { \nabla f ( \bar x _ k ) } ^ 2 \end{aligned} \end{align*}


\begin{align*} \frac { \widetilde { f } ' ( \gamma ) } { \widetilde { f } ( \gamma ) } = - \frac { \widetilde { h } ' ( \gamma ) } { \widetilde { h } ( \gamma ) } . \end{align*}


\begin{align*} \left \langle ( \partial _ { \Sigma _ 0 , \epsilon } ^ { * } F ) ^ { ( n ) } [ \phi ] , h \right \rangle & = \left \langle K ^ { \otimes n } , h \otimes F ^ { ( n ) } [ \partial _ { \Sigma _ 0 , \epsilon } \phi ] \right \rangle \end{align*}


\begin{align*} \inf _ { ( \theta _ 0 , x _ 0 ) \in \mathbb { R } \times \mathbb { R } ^ N } \| u _ n - e ^ { i \theta _ 0 } Q ( \cdot - x _ 0 ) \| _ { H ^ 1 } \le \| r _ n ^ 1 \| _ { H ^ 1 } = o _ n ( 1 ) , \end{align*}


\begin{align*} \omega ( \pi ) : = \lambda _ { \# ( \pi ) } \cdot \prod _ { i = 1 } ^ { \# ( \pi ) - 1 } ( \lambda _ { i } - \lambda _ { i + 1 } - 1 ) . \end{align*}


\begin{align*} y = A x _ 0 + z , \end{align*}


\begin{align*} \frac { \partial } { \partial t } \bar { \sigma } _ j & = - \sigma _ j \\ \frac { \partial } { \partial a _ i } \bar { \sigma } _ j & = ( 1 - t ) \frac { \partial } { \partial a _ i } \sigma _ j \end{align*}


\begin{align*} R ( \varphi _ t ) ( x ) = \frac { 1 } { 2 } \sum _ { p = 1 } ^ { q _ t } \bigg ( e ^ { - i \langle x , \xi _ p ^ t \rangle } \phi _ t ( \xi _ p ^ t ) | V _ p ^ t | + e ^ { i \langle x , \xi _ p ^ t \rangle } \overline { \phi _ t ( \xi _ p ^ t ) } | V _ p ^ t | \bigg ) , \end{align*}


\begin{align*} d R _ { \varepsilon } ^ { 2 } = d R ^ { 2 } + o ( \varepsilon ^ m ) , \ \ \varepsilon \to 0 . \end{align*}


\begin{align*} n A = F \cup ( r , a n - q ) \cup a n - G ( n + 1 ) A = F \cup ( r , a ( n + 1 ) - q ) \cup a ( n + 1 ) - G \ , . \end{align*}


\begin{align*} \mathcal { P } _ { k } = \{ \zeta ^ { \bar { w } } ( I _ k ) \bar { w } \prec \bar { s } _ { k } \zeta ^ { \bar { w } } ( J _ { k } ) \bar { w } \prec \bar { t } _ { k } \} . \end{align*}


\begin{align*} \Phi ( x , y , t ) : = u ( x , t ) - \varphi ( y , t ) - { | x - y | ^ 4 \over \varepsilon } . \end{align*}


\begin{align*} \widetilde { A } ^ { 0 } ( U ) U _ { t } + \widetilde { A } ^ { i } ( U ) \partial _ { i } U + \widetilde { D } ( U ) U - \widetilde { B } ^ { i j } ( U ) \partial _ { i } \partial _ { j } U = \widetilde { F } ( U , D _ { x } U ) , \end{align*}


\begin{align*} \sum _ { i = \xi ( \delta ) } ^ \infty \varepsilon _ i < \delta \end{align*}


\begin{align*} & d \rho _ i = \sum _ { j \in N ( i ) } \omega _ { i j } ( S _ i - S _ j ) \theta _ { i j } ( \rho ) \circ d W _ t ; \\ & d S _ i + ( \frac 1 2 \sum _ { j \in N ( i ) } \omega _ { i j } ( S _ i - S _ j ) ^ 2 \frac { \partial \theta _ { i j } } { \partial \rho _ i } + \frac 1 8 \frac { \partial } { \partial \rho _ i } I ( \rho ) ) \circ d W _ t + ( \mathbb V _ i + \sum _ { j \in N ( i ) } \mathbb W _ { i j } \rho _ j ) d t = 0 , \end{align*}


\begin{align*} G r ( ( \phi , \psi ) ) = \Big \{ \Big ( ( x , u ) , ( \phi ( x ) , \psi ( u ) \Big ) \ ; | \ ; x \in L , u \in V \Big \} \subset ( L \oplus V ) \oplus ( L ^ { ' } \oplus V ^ { ' } ) , \end{align*}


\begin{align*} s ( \mu , \mathcal { B } , \mathcal { U } ) & = \sup \left \{ s \geq 0 : \ ( B _ n ) _ { n \in \mathcal { N } _ \mu ( \mathcal { B } , \mathcal { U } , s ) } \mbox { i s } \mu \mbox { - a . c . } \right \} . \end{align*}


\begin{align*} & \mathcal B ^ * _ u \Phi _ { \lambda ^ p _ k } = \xi _ { \lambda ^ p _ k } ( 1 ) + \eta ^ \prime _ { \lambda ^ p _ k } ( 1 ) , \forall k \geq k _ 0 , \\ & \mathcal B ^ * _ u \Phi _ { \lambda ^ h _ k } = \xi _ { \lambda ^ h _ k } ( 1 ) + \eta ^ \prime _ { \lambda ^ h _ k } ( 1 ) , \forall | k | \geq k _ 0 . \end{align*}


\begin{align*} J ^ A _ \gamma x = J ^ A _ { \lambda \gamma } ( \lambda x + ( 1 - \lambda ) J ^ A _ \gamma x ) . \end{align*}


\begin{align*} Z _ \alpha f ( x , \omega ) = \alpha ^ { - 1 / 2 } \sum _ { k \in \Z } f \left ( \frac { x + k } { \alpha } \right ) e ^ { 2 \pi i k \omega } , \alpha > 0 . \end{align*}


\begin{align*} \gamma ^ { \epsilon , L } ( d \varphi ) \propto \exp \left [ - \frac { \epsilon ^ d } { 2 } \sum _ { x \in \Lambda _ { \epsilon , L } } \left ( \varphi _ x ( - \Delta ^ { \epsilon } + m ) \varphi _ x \right ) \right ] d \varphi = \exp \left [ - \frac { 1 } { 2 } ( \varphi , A _ { \epsilon } \varphi ) \right ] d \varphi , \end{align*}