\begin{align*} e _ 0 ( t , x ) & = \phi _ 0 \left ( \langle 2 \phi _ { 0 , x } + \varphi _ { 0 , x } , \varphi _ { 0 , x } \rangle - \langle 2 \phi _ { 0 , t } + \varphi _ { 0 , t } , \varphi _ { 0 , t } \rangle \right ) \\ & \ ; \ ; \ ; \ ; \ ; \ ; - \varphi _ 0 \left ( | \tilde \phi _ { 1 , t } | ^ 2 - | \tilde \phi _ { 1 , x } | ^ 2 \right ) + \mathbf { 1 } _ { \omega } \left ( f _ 0 ^ { p ^ { \perp } } - f _ { 0 } ^ { \tilde \phi ^ { \perp } _ 1 } \right ) . \end{align*}


\begin{align*} p ( \theta ) = P \theta + q , \end{align*}


\begin{align*} \limsup _ { \varepsilon \rightarrow 0 ^ { + } } \limsup _ { n \rightarrow \infty } \mathbb { E } _ { \mu _ n } \Big [ \Big | \int _ { 0 } ^ { t } \frac { 1 } { n } \sum _ { x } \Phi _ { n } ( s , \tfrac { x } { n } ) [ \overleftarrow { \eta } _ { s } ^ { \ell } ( x ) - \overleftarrow { \eta } _ { s } ^ { \varepsilon n } ( x ) ] \overrightarrow { \eta } _ { s } ^ { \varepsilon n } ( x + 1 ) d s \Big | \Big ] = 0 . \end{align*}


\begin{gather*} \| \psi \| _ { L ^ { \infty } _ x L ^ 2 _ t ( P _ { \alpha , \beta } ^ l ( y ) ) } : = \sup _ { x \in [ y , y + l ] } \| \psi ( t , x ) \| _ { L ^ 2 _ t ( \alpha + x - y , \beta - x + y ) } . \end{gather*}


\begin{align*} \eta ^ { \prime \prime } ( 0 ) = \eta ^ { \prime \prime } ( 1 ) = 0 . \end{align*}


\begin{align*} P ( t , T ) = \exp \bigg ( - \int _ t ^ T f ( t , s ) d s \bigg ) , \end{align*}


\begin{align*} g _ k ( B ) = g ( B ) - ( \dim M - \dim B ) + k . \end{align*}


\begin{align*} E ( t , X ( t ) ) & = - \nabla ( \phi * \rho ( t , \cdot ) ) ( X ( t ) ) \\ & = - \nabla ( \phi * ( G * _ { s , x } \mathcal S + \mathcal S ) ) ( X ( t ) ) \\ & = \mathcal { F } ^ R ( t ) + \mathcal { E } ^ R _ 1 ( t ) + \mathcal { E } _ 2 ( t ) + \mathcal { E } _ 3 ( t ) , \end{align*}


\begin{align*} \rho ( x ) = 2 \int _ { x ' = 0 } ^ x \left ( \frac { 1 } { t _ { + } ( x ) } + \frac { 1 } { t _ { - } ( x ) } \right ) \mathrm { d } x ' \end{align*}


\begin{align*} \widetilde { \pi } ( \l ) \widetilde { \pi } ( \l ' ) = e ^ { 2 \pi i x ' \cdot \omega } \widetilde { \pi } ( \l + \l ' ) . \end{align*}


\begin{align*} \omega ( a S ( b _ { ( 1 ) } ) c ) b _ { ( 2 ) } = 0 . \end{align*}


\begin{align*} J = ( j _ 0 , \dots , j _ n ) , \ | J | = \sum _ { s = 0 } ^ n j _ s , \ \binom { d } { J } = \frac { d ! } { j _ 0 ! \cdot \ldots \cdot j _ n ! } \end{align*}


\begin{align*} \delta _ { \varphi } ^ * = \delta _ { \varphi } . \end{align*}


\begin{align*} & \forall n ^ 0 , x ^ X \preceq _ X L 1 _ X , y ^ X \exists z ^ X \preceq _ X L 1 _ X \\ & \qquad \qquad \qquad \exists w ^ X \preceq _ X L 2 ^ { \alpha _ n + 1 } 1 _ X \left ( y \in A x \rightarrow ( w \in A z \land x = _ X z + _ X \gamma _ n w ) \right ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l } i \partial _ t u - \Delta u = \rho ^ 2 | u | ^ 2 + C _ { \rho } + \langle \nabla \rangle ^ { - \alpha } \dot { W } \ , , t \in [ 0 , T ] \ , , \ , x \in \R ^ d \ , , \\ u _ 0 = \phi \ , , \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} & u ' + 2 u ' u '' - 1 = 0 \\ \Longrightarrow & u '' = \frac { 1 - u ' } { 2 u ' } = \frac { 1 } { 2 u ' } - \frac { 1 } { 2 } \to + \infty \end{aligned} \end{align*}


\begin{align*} [ z ^ n ] f ( A ) = \frac { 1 } { n } [ t ^ { n - 1 } ] f ' ( t ) \Phi ( t ) ^ n . \end{align*}


\begin{align*} \mathcal S f ( x ) & = \Big ( \sum \limits _ { Q } | q _ { Q } f ( x _ { Q } ) | ^ 2 \chi _ Q ( x ) \Big ) ^ { \frac { 1 } { 2 } } \\ & = \Big ( \sum \limits _ { Q } | \int _ { \R ^ N } \chi _ Q ( x ) q _ { Q } ( x _ Q , y ) f ( y ) d \omega ( y ) | ^ 2 \Big ) ^ { \frac { 1 } { 2 } } . \end{align*}


\begin{align*} q _ \alpha ( t ) = \dfrac { e ^ { - t } } { ( 1 - e ^ { - \alpha t } ) ^ { 1 - \frac { 1 } { \alpha } } } \left ( ( 1 - e ^ { - \alpha t } ) ^ { \alpha - 1 } + e ^ { - \alpha t } \right ) ^ { \frac { 1 } { \alpha } } . \end{align*}


\begin{align*} \begin{cases} \forall x ^ X , y ^ X , \gamma ^ 1 \left ( \gamma > _ \mathbb { R } 0 \rightarrow \norm { x - _ X y } _ X \geq _ \mathbb { R } \norm { J ^ A _ { \gamma } x - _ X J ^ A _ { \gamma } y } _ X \right ) , \\ \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{cases} \end{align*}


\begin{align*} C ^ { ( w ) } ( t , x ) ~ = ~ E ^ { ( w ) } _ 1 ( t ) - ( c _ 1 - 1 ) \cdot \left [ { \dot { \sigma } ^ { ( w ) } ( t ) t \over 2 } - ( w ( t , x ) ) - w ( t , 0 - ) ) \right ] \cdot \phi _ x \left ( x , { \sigma ^ { ( w ) } ( t ) t \over 2 } \right ) , \end{align*}


\begin{align*} \lambda \frac { d Q } { d P } & \in \partial V ( u - \sum _ { t = 0 } ^ { T - 1 } x _ t \Delta s _ { t + 1 } - \bar c \cdot \bar x + S _ 0 ( \bar x ) ) , \\ E ^ Q _ t [ \Delta s _ { t + 1 } ] & \in N _ { D _ t } ( x _ t ) t = 0 , \dots , T , \\ E ^ Q [ \bar c ] & \in \partial S _ 0 ( \bar x ) , \end{align*}


\begin{align*} M ^ { ( 2 ) } _ n & = \dfrac { L ^ { ( 2 ) } _ n } { t ^ { ( 2 ) } _ n } - 1 = \dfrac { 1 } { \log n } \left ( \sum _ { j = 1 } ^ n \dfrac { 1 } { j } - \log n \right ) \sim \gamma ( \log n ) ^ { - 1 } ( n \to \infty ) . \end{align*}


\begin{align*} \nabla _ { v } X _ { t } ^ { x } : = \lim _ { \epsilon \to 0 } \frac { X _ { t } ^ { x + \epsilon v } - X _ { t } ^ { x } } { \epsilon } , t \geq 0 . \end{align*}


\begin{align*} \sum _ { \ell = 1 } ^ k ( 2 + d _ \ell ) \leq k ^ 2 + n . \end{align*}


\begin{align*} \| C X \| _ F ^ 2 \geq ( { 1 - \xi } ) \left [ \sum ^ { l } _ { t = 1 } \bar { \sigma } ^ 2 _ { t } - \frac { 2 \theta \| S \| ^ 2 _ F } { \sqrt { \alpha } } - \theta ( k + \sqrt { k } \xi ) \| C \| ^ 2 _ F \right ] . \end{align*}


\begin{align*} \beta _ { i } ^ K ( t ) = M ^ K _ { i } ( t ) + A ^ K _ { i } ( t ) \end{align*}


\begin{align*} \partial _ t ^ 2 \zeta ( x , t , y , s ; v ) = - v _ 1 \frac { \sqrt { s } } { 4 \sqrt { t } ^ 3 } \ , \Psi + v _ 2 \frac { \sqrt { 1 - s } } { 4 \sqrt { 1 - t } ^ 3 } . \end{align*}


\begin{align*} \left . \begin{array} { l } ( r _ { k , s } + r _ { s , k } ) ( 0 ) = 0 , k = \overline { 0 , m - 1 } , s = \overline { k , k + i _ { 2 k } - 1 } \\ ( r _ { k , s } - r _ { s , k } ) ( 0 ) = 0 , k = \overline { 0 , m + \tau - 2 } , s = \overline { k + 1 , k + i _ { 2 k + 1 } } \end{array} \right \} \end{align*}


\begin{align*} \begin{cases} e ^ { - Q } \delta _ { s t } \leq g _ { s t } \leq e ^ { Q } \delta _ { s t } ; \\ r ^ { 1 + \alpha } \| \partial _ { j } g _ { s t } \| _ { C ^ { 0 , \alpha } ( B _ { r } ( p ) ) } \leq e ^ { Q } , \end{cases} \end{align*}


\begin{align*} \tilde { Y } _ 0 = x , \tilde { Y } _ { k + 1 } = \tilde { Y } _ { k } + \eta b ( \tilde { Y } _ { k } ) + \frac { \eta ^ { 1 / \alpha } } { \sigma } \ , \widetilde { Z } _ { k + 1 } , k = 0 , 1 , 2 , \dots , \end{align*}


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


\begin{align*} | z _ i | = \left | ( P _ i ^ { \perp } + z P _ i ) \xi , U _ i ^ * \xi \rangle \right | \leq \| ( P _ i ^ { \perp } + z P _ i ) \xi \| \| U _ i ^ * \xi \| \leq \| P _ i ^ { \perp } + z P _ i \| < 1 . \end{align*}


\begin{align*} \mu _ { n , m } = ( n + 1 ) { n + m + 1 \choose m } . \end{align*}


\begin{align*} P _ i = \left \{ u _ i \in \R ^ { n + 1 } \ , \middle | \ , \begin{aligned} a _ { i 0 } \leq u _ { i j } \leq \min \{ a _ { i j } , u _ { i n } \} , \ u _ { i 0 } = a _ { i 0 } , \ ; a _ { i 0 } \leq u _ { i n } \leq a _ { i n } \end{aligned} \right \} . \end{align*}


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


\begin{align*} \alpha _ { s _ m } ( f ) = \alpha _ { s ' _ m } ( f _ { U ' } ) \end{align*}


\begin{align*} \varphi ( \theta ) = \sum _ { n \not = 0 } \varphi _ n e ^ { i n \theta } \end{align*}


\begin{align*} S ( H _ 3 , 3 f ) = \frac { 5 f - 6 d + 1 } { 6 } \hbox { a n d } N _ 3 ' ( f , H ) = - d , \end{align*}


\begin{align*} \prod _ { j = j _ 0 ( \delta ) + 1 } ^ { n - 1 } \left ( 1 + \dfrac { \delta } { j } \right ) \sim A _ { \delta } \cdot n ^ { \delta } ( n \to \infty ) . \end{align*}


\begin{align*} \varphi ( S ^ 2 ( a ) ) = \varphi ( S ( a ) \delta _ { \varphi } ) = \varphi ( S ( \delta _ { \varphi } ^ { - 1 } a ) ) = \varphi ( \delta _ { \varphi } ^ { - 1 } a \delta _ { \varphi } ) , a \in A , \end{align*}


\begin{align*} y [ n ] = \sum _ { i \in \mathcal { K } ' } \sqrt { \rho } \hat { h } _ i ^ * h _ i s [ n ] / | \hat { h } _ i | ^ 2 + w [ n ] = \sqrt { \rho } K ' s [ n ] \ ! + \ ! \varphi _ { \mathcal { K } ' } s [ n ] \ ! + \ ! w [ n ] , \ n \ ! = \ ! 1 , 2 , \cdots , N _ 2 , \end{align*}


\begin{align*} \langle \mathbf { T } ( g , h ) , \mathbf { f } ( h ) \rangle = \langle \mathfrak { E } ( g \times h , \mu ) , \mathbf { f } ( h ) \rangle . \end{align*}


\begin{align*} R _ M f ( x ) & = - \ln r \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } \int _ { Q } \big ( \psi _ { Q } ( x , y ) - \psi _ { Q } ( x , x _ { Q } ) \big ) q _ { Q } f ( y ) d \omega ( y ) \\ & + \ln r \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } \int _ { Q } \psi _ { Q } ( x , x _ { Q } ) \big ( q _ { Q } f ( x _ Q ) - q _ { Q } f ( y ) \big ) d \omega ( y ) \\ & = R ^ 1 _ M f ( x ) + R ^ 2 _ M f ( x ) , \end{align*}


\begin{align*} \frac { 1 } { ( 2 m - 1 ) ! ! } E \left [ \left ( \dfrac { S _ n } { \sqrt { n \log n } } \right ) ^ { 2 m } \right ] - 1 \sim m \gamma \cdot ( \log n ) ^ { - 1 } \end{align*}


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


\begin{align*} f \ = \ g \circ \big ( P _ 2 ^ 2 , P _ 1 ^ 2 \big ) \end{align*}


\begin{align*} \Big ( \lambda - c _ \alpha \ , \lambda ^ 2 \epsilon _ n \Big ) \Big ( 1 - \delta _ n ( \lambda ) \Big ) = x \sqrt { 1 + \delta _ n ( \lambda ) } . \end{align*}


\begin{align*} \mathsf c ( H ) = \sup \{ \mathsf c ( a ) \colon a \in H \} \ , . \end{align*}


\begin{align*} E _ { \psi } ( x ) \le \varepsilon _ { } ( x _ 0 , x ) = A ( x _ 0 ) \Big ( \frac { \log x } { R } \Big ) ^ { 3 / 2 } \exp \Big ( - 2 \sqrt { \frac { \log x } { R } } \Big ) \ \ x \ge x _ 0 . \end{align*}


\begin{align*} K ( \beta ) = ( g ( x _ 0 ) - \underline { g } ) \ ! \left [ \min \ ! \left ( c _ 1 \tau _ 1 \min ( \underline { \alpha _ 1 } , \underline { t _ 1 } ) \varepsilon _ 1 ^ 2 , c _ 2 \tau _ 2 ^ 2 \min \ ! \left ( \min \ ! \left ( \alpha _ { 0 2 } , \dfrac { 3 | 2 c _ 2 - 1 | \varepsilon _ 2 } { M _ g } \right ) \ ! , \underline { t _ 2 } \right ) ^ 2 \ ! \varepsilon _ 2 \right ) \ ! \right ] ^ { - 1 } \ ! . \end{align*}


\begin{align*} T = \langle \pi ( \l _ 1 ^ \circ ) g , \pi ( \l _ 2 ^ \circ ) g \rangle , \l _ 1 ^ \circ , \ , \l _ 2 ^ \circ \in \L ^ \circ _ { a , b } . \end{align*}


\begin{align*} { \rm W F } _ h ' ( e ^ { - i t _ 0 h ^ { - 1 } \widetilde { P } _ h ( z ) } \widetilde { R } _ h ( z ) ) \cap S ^ * ( X \times X ) & \subset \\ \kappa ( \{ ( x , \xi , y , \eta ) \ , : \ , ( e ^ { - t _ 0 H _ p } ( x , \xi ) , y , \eta ) & \in \Delta ( T ^ * X ) \cup \Omega _ + \cup ( E ^ * _ u \times E ^ * _ s ) \setminus \{ 0 \} \xi = 0 , \eta \neq 0 \} ) . \end{align*}


\begin{align*} F ( k ) = \psi ^ { - 1 } ( \mathrm { S u c c } ^ { \sigma } ( \psi ( k ) ) ) . \end{align*}


\begin{align*} p _ c ^ { ( k ) } ( i ) = \begin{cases} \frac 1 { \lim _ { n \to \infty } P _ { n , k } ( W _ { \mathcal { S } ( n , k ; M _ n ) } ) } ( 1 - c ) ^ { k - i + 1 } c ^ { i - 1 } \binom k { i - 1 } , \ i \in \{ 2 , \cdots , k \} ; \\ 1 - \frac 1 { \lim _ { n \to \infty } P _ { n , k } ( W _ { \mathcal { S } ( n , k ; M _ n ) } ) } \big ( 1 - c ^ k - ( 1 - c ) ^ k \big ) , \ i = 1 . \end{cases} \end{align*}


\begin{align*} G _ { k n } ( x , y ) = \left ( { g _ { k n } ( x ) + \delta _ 1 ( x , y ) \atop g _ { k n } ( y ) + \delta _ 2 ( x , y ) } \right ) , \end{align*}


\begin{align*} d _ K ( h ) \leq \| h ^ - \| = \sqrt { \int _ 0 ^ { x _ 0 } | h ' ( x ) | ^ 2 e ^ { \gamma x } d x } . \end{align*}


\begin{align*} \left \langle ( \partial _ \Sigma \otimes \partial _ \Sigma ) E , f \otimes g \right \rangle = - \frac { 1 } { 2 } \int _ \Sigma f ( * d _ \Sigma g ) \ , \mathrm { d } V _ \Sigma . \end{align*}


\begin{align*} D ^ n ( f g ) = \sum _ { k = 0 } ^ n { n \choose k } D ^ k ( f ) D ^ { n - k } ( g ) , \end{align*}


\begin{align*} \mathbf { L } _ n = \mathbf { L } _ n ^ 0 + \frac { \mu } { 4 \pi } \int _ 0 ^ { 2 \pi } e ^ { \gamma \varphi ( \theta ) } e ^ { i n \theta } d \theta , \widetilde { \mathbf { L } } _ n = \widetilde { \mathbf { L } } _ n ^ 0 + \frac { \mu } { 4 \pi } \int _ 0 ^ { 2 \pi } e ^ { \gamma \varphi ( \theta ) } e ^ { - i n \theta } d \theta . \end{align*}


\begin{align*} \frac { \partial w } { \partial x } + 2 t w = 0 . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \varepsilon _ 1 ( \tau ) & = & c _ { 1 , 0 } \| \phi _ { \ell , b , \beta } \| ^ { - 2 } _ { L ^ 2 _ { \rho _ \beta } } \langle \varepsilon , \phi _ { 1 , b , \beta } \rangle _ { L ^ 2 _ { \rho _ \beta } } , \\ \varepsilon _ 1 ( \tau ) & = & - \| \phi _ { 0 , b , \beta } \| ^ { - 2 } _ { L ^ 2 _ { \rho _ \beta } } \langle \varepsilon , \phi _ { 0 , b , \beta } \rangle _ { L ^ 2 _ { \rho _ \beta } } . \end{array} \right . \end{align*}


\begin{align*} \ker d _ 0 & = \{ \varphi : X \to A : d _ 0 \varphi = 0 \} \\ & = \{ \varphi : X \to A : \varphi ( x _ 1 ) - \varphi ( x _ 0 ) = 0 \forall x _ 1 , x _ 0 \in X \} \\ & = \{ \varphi : X \to A \mbox { c o n s t a n t } \} \\ & = A \end{align*}


\begin{align*} \langle \mathbf { f , h } \rangle = \sum _ { j } \langle f _ j , h _ j \rangle . \end{align*}


\begin{align*} ( 1 - x ^ 2 ) y '' + [ - \beta + \alpha + ( \alpha + \beta - 2 ) x ] \ , y ' + n ( n - \alpha - \beta + 1 ) y = 0 \end{align*}


\begin{align*} \frac { \mathbf { P } ( U _ n > x ) } { 1 - \Phi \left ( x \right ) } = 1 + o ( 1 ) , \ \ \ \ \ \ \ n \rightarrow \infty . \end{align*}


\begin{align*} \mu & = C \cos ( \omega \Delta ) \\ \omega & = - C \sin ( \omega \Delta ) \end{align*}


\begin{align*} H ^ 0 ( Y , \mathcal { O } _ Y ) \to H ^ 0 ( F _ i , \mathcal { O } _ { F _ i } ) \to H ^ 1 ( Y , \mathcal { O } _ Y ( - F _ i ) ) \to H ^ 1 ( Y , \mathcal { O } _ Y ) = 0 . \end{align*}


\begin{align*} \langle w \otimes b , v \otimes a \rangle _ A = \langle w , v \rangle b ^ * a , v , w \in V , a , b \in A , \end{align*}


\begin{align*} \mathcal { V } = \{ \mathbf { f } \in \mathcal { S } _ { k } ( K _ 1 ( \mathfrak { n } ) ) : \mathbf { f } | T _ { \xi } = \lambda ( \xi ) \mathbf { f } \} , \mathcal { V } ( \overline { \Q } ) = \mathcal { V } \cap \mathcal { S } _ k ( K _ 1 ( \mathfrak { n } ) ) \end{align*}


\begin{align*} t _ { i j } = \begin{cases} s _ { i j } & j \in K _ i \backslash \{ 0 , n \} \\ a _ { i j } & j = 0 \\ s _ { i k _ i ^ * } & j = n \\ ( 1 - \gamma _ { i j } ) s _ { i l ( i , j ) } + \gamma _ { i j } s _ { i r ( i , j ) } & j \notin K _ i \cup \{ 0 \} \cup \{ n \} , \end{cases} \end{align*}


\begin{align*} L i _ { K , s } ( z ) : = \sum _ { i \geq 1 } \frac { z ^ { q ^ i } } { ( \theta ^ { q ^ i } - \theta ) ^ s } \in \mathbb { C } _ { \infty } \end{align*}


\begin{align*} { \vec y } \ , ' = ( F ( x ) + \Lambda ) \vec y , \end{align*}


\begin{align*} \int _ { \R ^ n } \nabla \bar { u } \cdot \nabla v = \tilde { \lambda } _ 0 \int _ { \R ^ n } \tilde { m } _ 0 \bar { u } v \forall v \in H ^ 1 ( \R ^ N ) \ ; , \end{align*}


\begin{align*} \mu ( \phi ) = \int _ { \mathbb { R } ^ d } \left ( \frac 1 2 | \nabla \phi | ^ 2 + V | \phi | ^ 2 + \beta | \phi | ^ 4 - \Omega \overline { \phi } L _ z \phi \right ) \mathrm { d } \mathbf { x } = E ( \phi ) + \frac { \beta } { 2 } \int _ { \mathbb { R } ^ d } | \phi ( \mathbf { x } ) | ^ 4 \mathrm { d } \mathbf { x } . \end{align*}


\begin{align*} S _ c = \left \lbrace \sum _ { i = 1 } ^ { n - 1 } \lambda _ i c ^ { ( i ) } \middle | \lambda _ i \in \mathbb { R } \right \rbrace . \end{align*}


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


\begin{align*} \forall \ , ( h , h ' , i ) \ne ( 2 , 2 , 2 ) : \lim _ { n \to \infty } \Lambda ^ { ( h : h ' ) } _ { n , i } = 0 , \lim _ { n \to \infty } \Lambda ^ { ( 2 : 2 ) } _ { n , 2 } = \frac 1 2 . \end{align*}


\begin{align*} c _ 1 ( \widetilde { M } _ l ) \cdot E _ t = \delta _ { l , i _ t } . \end{align*}


\begin{align*} F _ n ^ D : = I _ n ^ D ( \overline { \mathbf { a } } ) - \beta \sum _ { i = 1 } ^ n Q ( \overline { a } _ i ) , F _ n ^ C : = I _ n ^ C ( \overline { \mu } ) - \beta \int _ \R Q ( x ) \ , d \overline { \mu } ( x ) \end{align*}


\begin{align*} \Gamma ( H G ) = \bigcup _ { k = 3 } ^ { 7 } \Gamma _ { 2 k } ( H G ) . \end{align*}


\begin{align*} & \ll \sum _ { q \ll Q } \sum _ { \ell \ll X } \frac { ( H K \ell ) ^ \varepsilon } { \ell \sqrt { H K } } \sum _ { \substack { c > C , d \geq 1 \\ c d = q } } \phi ( d ) \ \ll \frac { Q ^ 2 } { C } \frac { ( X H K ) ^ { \varepsilon } } { \sqrt { H K } } . \end{align*}


\begin{align*} \left \{ \begin{aligned} u ( x ) + | u ' ( x ) | ^ p - f ( x ) \leq 0 & ( a , b ) , \\ u ( x ) + | u ' ( x ) | ^ p - f ( x ) \geq 0 & [ a , b ] . \end{aligned} \right . \end{align*}


\begin{align*} v ^ + ( \xi , \tau ) = \theta ^ + ( \tau ) H _ 1 ( \xi ) - M ( \eta ) b ^ \frac { \eta } { 4 } ( \tau ) H _ 0 \left ( \xi \right ) v ^ { - } = \theta ^ - ( \tau ) H _ 1 ( \xi ) + M ( \eta ) b ^ { \frac { \eta } { 4 } } ( \tau ) H _ 0 ( \xi ) , \end{align*}


\begin{align*} \vert x - y \vert ^ { n + 2 s + 2 } = R ^ { n + 2 s + 2 } \vert \tilde x - \tilde y \vert ^ { n + 2 s + 2 } \geqslant C R ^ { n + 2 s + 2 } \big ( 1 + R ^ { - n - 2 s - 2 } \vert y - a \vert ^ { n + 2 s + 2 } \big ) . \end{align*}


\begin{align*} \Delta u = f ~ ~ ~ ~ \mbox { i n } ~ B _ 1 , \end{align*}


\begin{align*} = \int _ { P ^ t _ n \backslash G ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \sum _ { \xi \in G _ { 2 t + r } } \sum _ { \gamma \in P ^ t _ n \backslash G _ n } \phi ( \tilde { \tau } _ t ( ( \xi \times 1 _ { 2 n - 2 t } ) g \times \gamma h ) , s ) \mathbf { f } ( h ) \mathbf { d } h \end{align*}


\begin{align*} 1 = \norm { e _ { \gamma ' } } _ \mathcal { H } ^ 2 = \sum _ { \gamma \in \Gamma } | \langle e _ { \gamma ' } , e _ \gamma \rangle | ^ 2 = 1 + \sum _ { \gamma \neq \gamma ' } | \langle e _ { \gamma ' } , e _ \gamma \rangle | ^ 2 . \end{align*}


\begin{align*} \biggl [ { a \choose c } + { b \choose c } \biggr ] - \biggl [ { a - 1 \choose c } + { b + 1 \choose c } \biggr ] = { a - 1 \choose c - 1 } - { b \choose c - 1 } \geq 0 . \end{align*}


\begin{align*} \tilde S _ 0 ( \bar x ) - \bar x \cdot \bar c - \sum _ { t = 0 } ^ { T - 1 } x _ t \Delta s _ { t + 1 } \le - S _ 0 ^ \infty ( - \bar x ) - \bar x \cdot \bar c - \sum _ { t = 0 } ^ { T - 1 } x _ t \Delta s _ { t + 1 } < 0 \end{align*}


\begin{align*} \left \{ \partial _ \Sigma ^ * \Psi ( f ) , \partial _ \Sigma ^ * \Psi ( g ) \right \} = \left \langle ( \partial _ \Sigma \otimes \partial _ \Sigma ) E , f \otimes g \right \rangle _ { \Sigma ^ 2 } . \end{align*}


\begin{align*} R ( s ; u , v ) = \prod _ { p | v } \left ( 1 - \frac { 1 } { p ^ { s + 1 } } \right ) \prod _ { p \nmid u v } \left ( 1 + \frac { 1 } { p ^ { s + 1 } ( p - 1 ) } \right ) \end{align*}


\begin{align*} \| V \partial _ y \Phi _ 2 \| _ { L ^ \infty } \lesssim \frac { 1 } { t ^ 2 } , \quad \bigg \| \Phi _ 2 \partial _ y V - \sum _ { i = 1 } ^ n \sigma _ i \mu _ i \partial _ y \R _ i \bigg \| \lesssim \frac { 1 } { t ^ { \frac { 3 } { 2 } } } . \end{align*}


\begin{align*} \begin{aligned} \phi ( f ^ L ) & + \sum _ { i = 1 } ^ d \Biggl [ \phi \Biggl ( \sum _ { j = 1 } ^ { i - 1 } e _ { \omega _ j } f _ { \omega _ j } ^ U + e _ { \omega _ i } f _ { \omega _ i } + \sum _ { j = i + 1 } ^ d e _ { \omega _ j } f _ { \omega _ j } ^ L \Biggr ) \\ & - \phi \Biggl ( \sum _ { j = 1 } ^ { i - 1 } e _ { \omega _ j } f _ { \omega _ j } ^ U + \sum _ { j = i } ^ d e _ { \omega _ j } f _ { \omega _ j } ^ L \Biggr ) \Biggr ] . \end{aligned} \end{align*}


\begin{align*} s '' ( t + k \bar { \tau } ) & = \bigoplus _ { i = t + k \bar { \tau } } ^ { t + ( k + 1 ) \bar { \tau } - 1 } \bar { s } ( i ) = \bar { \nu } ^ { k } \left ( \bigoplus _ { i = t } ^ { t + \bar { \tau } - 1 } \bar { s } ( i ) \right ) = \bar { \nu } ^ { k } s '' ( t ) , \end{align*}


\begin{align*} \left \| f \right \| _ { \mathcal { D } ( \mathcal { L } ^ { ( p ) } ) } : = \left \| ( \lambda - \mathcal { L } ^ { ( p ) } ) f \right \| _ { L ^ p ( M ) } . \end{align*}


\begin{align*} w ( x , t ) = U [ \psi _ { 0 } ] ( x , t , 0 ) + N [ u , \psi ] ( x , t ) , \end{align*}


\begin{align*} \sup _ { s \in [ 0 , T ] } \left | z ( s ) - z _ j ( s ) \right | \leq \sup _ { s \in [ 0 , T ] } \left ( \sum _ { k \colon z ( t _ k ) = z ( t _ k - ) } \frac { \psi _ k ( s ) } { j } + \sum _ { k \colon z ( t _ k ) \neq z ( t _ k - ) } \frac { \psi _ k ( s ) } { j } \right ) = \frac { 1 } { j } . \end{align*}


\begin{align*} | \nabla u | _ w = | \nabla u | \end{align*}


\begin{align*} \partial _ t ( n _ 0 u ^ 0 ) + \nabla _ x ( n _ 0 u ) = 0 , \end{align*}


\begin{align*} \Lambda ( M , \Gamma ) = \{ \ , \alpha \ , | \ , \alpha \ , \ , \mathrm { i s \ , \ , a n \ , \ , a d m i s s i b l e \ , \ , o r b i t \ , \ , s e t \ , \ , s u c h \ , \ , t h a t \ , \ , } [ \alpha ] = \Gamma \mathrm { \ , \ , a n d } \ , \ , A ( \alpha ) < M \ , \} . \end{align*}


\begin{align*} L ( A ) ( U \oplus U ^ 2 ) = 0 , \ ; V ^ 2 \subseteq L ( A ) , \ ; v ( v u ) \in L ( A ) \mbox { f o r a l l } u \in U \mbox { a n d } v \in V \end{align*}


\begin{align*} F _ { \alpha } = \sum _ { \alpha \leq \beta } M _ { \beta } M _ { \alpha } = \sum _ { \alpha \leq \beta } ( - 1 ) ^ { \ell ( \beta ) - \ell ( \alpha ) } M _ { \beta } . \end{align*}


\begin{align*} | A | ^ 2 - 4 f ^ 2 - 8 f \langle \xi , E _ 3 ^ { \perp } \rangle ( \langle \xi , E _ 1 ^ { \perp } \rangle ^ 2 - \langle \xi , E _ 2 ^ { \perp } \rangle ^ 2 ) = 0 . \end{align*}


\begin{align*} \widehat { \Phi } ( x ) = 1 + \sum _ { w \in \N ^ * } m ( w ) x _ w , \widehat { \kappa } ( x ) = \sum _ { w \in \N ^ * } \kappa ( w ) x _ w , \end{align*}


\begin{align*} \phi _ { \Delta } ( t ) { = } \operatorname { s g n } ( \phi ( t ) ) \operatorname { m i n } \left ( \left \lfloor \frac { 1 } { \Delta } | \phi ( t ) | { + } \frac { 1 } { 2 } \right \rfloor { , } 2 ^ { w _ { \phi } { - } 1 } { - } 1 \right ) \end{align*}


\begin{align*} x _ { k + 1 } = x _ k + e _ i \frac { e _ i ^ \intercal A ^ \intercal ( b - A x _ k ) } { \norm { A e _ i } _ 2 ^ 2 } . \end{align*}


\begin{align*} \Phi ( S ) \geq 1 - \alpha - ( 1 - \alpha ) \eta - \sum \limits _ { i = 1 } ^ { r } ( \lambda _ i ( M ) - \eta ) R ( k , i ) \varepsilon _ i - c \gamma \end{align*}


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


\begin{align*} 1 . & \textrm { t h e i n t e r i o r s o f $ T _ i $ a r e p a i r w i s e d i s j o i n t a n d $ X = \cup _ i T _ i $ , } \\ 2 . & \textrm { t h e i n t e r i o r o f e a c h t r i a n g l e a d m i t s a b i j e c t i v e l o c a l h a l f - d i l a t i o n t o a t r i a n g l e i n $ \R ^ 2 $ , } \\ 3 . & \textrm { a l l v e r t i c e s a r e p o i n t s o f s i n g u l a r i t y , } \\ 4 . & \textrm { t h e c o l l e c t i o n o f p o i n t s o f s i n g u l a r i t y d o n o t m e e t a t r i a n g l e $ T _ i $ o f f i t s v e r t i c e s . } \end{align*}


\begin{align*} \Vert u \Vert _ { N ^ { 1 , p } ( X ) } : = \Vert u \Vert _ { L ^ p ( X ) } + \mathcal { E } _ p ( u ) . \end{align*}


\begin{align*} \rho _ 1 = \sigma _ 1 , & & \rho _ 2 = \sigma _ 1 ^ 2 - 2 \sigma _ 2 , & & \rho _ 3 = \sigma _ 1 ^ 3 - 3 \sigma _ 1 \sigma _ 2 + 3 \sigma _ 3 . \end{align*}


\begin{align*} & W _ \star [ x , t ] : = K \cap \{ w _ \star ( \cdot , t ) \leq w _ \star ( x , t ) \} , \\ & U [ x , t ] : = K \cap \{ u ( \cdot , t ) \leq u ( x , t ) \} = K \cap \{ v ( \cdot , t ) \leq v ( x , t ) \} . \end{align*}


\begin{align*} L _ n \left ( \sqrt { \bar { x } \bar { y } } , \frac { \bar { x } + \bar { y } } { 2 } \right ) = \sqrt { \bar { x } \bar { y } ^ { - 1 } } \sum _ { k = 0 } ^ n { n \choose k } A _ k ( \bar { x } , \bar { y } ) \frac { ( \bar { y } - \bar { x } ) ^ { n - k } } { 2 ^ { n - k } } . \end{align*}


\begin{align*} \mu ( E _ { \mu } ^ { [ \alpha , \gamma ] , \varepsilon } ) & = \mu ( \left \{ x : \underline { \dim } ( \mu , x ) \in [ \alpha , \gamma ] \right \} ) \\ \mu ( F _ { \mu } ^ { [ \alpha , \gamma ] , \varepsilon } ) & = \mu ( \left \{ x : \overline { \dim } ( \mu , x ) \in [ \alpha , \gamma ] \right \} ) \end{align*}


\begin{align*} D = \left ( \begin{array} { c c c } 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & D _ { 0 } \\ \end{array} \right ) \in \mathbb { M } _ { N \times N } . \end{align*}


\begin{align*} ( f _ j \circ \psi _ j ) ^ * ( g _ { \textnormal { e u c } } ) = \psi _ j ^ * ( e ^ { 2 \tilde u _ j } \ , g _ { \textnormal { p o i n } , j } ) = e ^ { 2 \tilde u _ j \circ \psi _ j } \ , \psi _ j ^ * g _ { \textnormal { p o i n } , j } = e ^ { 2 \tilde u _ j \circ \psi _ j + 2 v _ j } \ , g _ { \textnormal { e u c } } \textnormal { o n } \ , \ , \ , B _ 1 ^ 2 ( 0 ) . \end{align*}


\begin{align*} \cfrac { n ( \widetilde { \chi } _ 0 \pi _ E , \psi _ E ) } { e } = n ( \chi r _ l ( \pi _ F ) ^ { ( l ) } , \overline { \psi } _ F ^ l ) \end{align*}


\begin{align*} & \left | \partial _ { k } \left ( m _ { \pm } ( x , k ) - 1 \right ) \right | \lesssim \frac { 1 } { | k | } \mathcal { W } _ { \pm } ^ { 1 } ( x ) , \pm x \geq - 1 . \end{align*}


\begin{align*} \begin{cases} u _ { t } + D ^ { a + 1 } _ x \partial _ { x } u + u u _ { x } = 0 , \ ; \ ; x , t \in \R , a \in ( 0 , 1 ) , \\ u ( x , 0 ) = \phi ( x ) , \end{cases} \end{align*}


\begin{align*} \lim _ { k \to \infty } \frac { 2 | \Lambda ( A ( \alpha _ { k } ^ { \Gamma } ) , \Gamma ) | } { I ( \alpha _ { k } ^ { \Gamma } ) } = 1 . \end{align*}


\begin{align*} w = P _ { \Omega _ { \varepsilon } } w + e ^ { - \beta _ { \varepsilon } \Psi _ { \varepsilon } ( \mathbf { q } ) } V _ { \varepsilon } \Omega _ { \varepsilon } \ ; , \end{align*}


\begin{align*} ( x \cdot y ) \cdot J x = y \end{align*}


\begin{align*} & \omega ( x , y , z ) = \varphi ( [ x , y ] , z ) - \rho ( z ) \varphi ( x , y ) \\ & \ ; \ ; \ ; \ ; \ ; \ ; \ ; \quad = - \Big ( \varphi ( [ y , x ] , z ) - \rho ( z ) \varphi ( y , z ) \Big ) = - \omega ( y , x , z ) . \end{align*}


\begin{align*} m ( t ) = \sum _ { k \in \Z } m _ k e ^ { 2 \pi i k t } , \end{align*}


\begin{align*} \ell ( h ) = \iint _ { \R ^ { 2 d } } F ( x , \omega ) \overline { \langle h , M _ \omega T _ x g \rangle } \ , d ( x , \omega ) \end{align*}


\begin{align*} u ( x _ \ast ) = - u ( x ) x \in \R ^ n . \end{align*}


\begin{align*} \prod _ { j = j _ 0 ( \delta ) + 1 } ^ { n - 1 } \left ( 1 + \dfrac { \delta } { j } \right ) = \prod _ { j = 1 } ^ { n - 1 } \dfrac { j + \delta } { j } = \dfrac { \Gamma ( n + \delta ) } { \Gamma ( n ) \Gamma ( 1 + \delta ) } . \end{align*}


\begin{align*} \int _ { t } ^ { t + \varepsilon } z \dd ( y - u ) \geq c \int _ { t } ^ { t + \varepsilon } \dd ( u - y ) = c \left ( ( u - y ) ( t + \varepsilon ) - ( u - y ) ( t ) \right ) > 0 . \end{align*}


\begin{align*} \Phi _ { R , L } ( x , t , k ) = \Phi ^ { \pm \infty } ( x , t , k ) + \int _ { \pm \infty } ^ { x } \Phi ^ { \pm \infty } ( x , t , k ) \left ( \Phi ^ { \pm \infty } \right ) ^ { - 1 } ( y , t , k ) \left ( Q - Q ^ { \pm \infty } \right ) ( y , t , k ) \Phi _ { R , L } ( y , t , k ) d y . \end{align*}


\begin{align*} L ( x _ { \rm t h } ) = N \sqrt { 2 \pi } f _ D \frac { x _ { \rm t h } } { \sigma } e ^ { - \frac { x _ { \rm t h } ^ 2 } { \sigma ^ 2 } } \left ( 1 - e ^ { - \frac { x _ { \rm t h } ^ 2 } { \sigma ^ 2 } } \right ) ^ { N - 1 } . \end{align*}


\begin{align*} X g ( x ) = x g ( x ) P g ( x ) = \frac { 1 } { 2 \pi i } \ , g ' ( x ) . \end{align*}


\begin{align*} S = \left \{ R ( s , t ) = \left ( x ( s , t ) , y ( s , t ) , z ( s , t ) \right ) ; \ ( s , t ) \in \Omega \right \} , \end{align*}


\begin{align*} \tilde { f } ( s , y ) & = \alpha _ { - 1 } ( - \tilde { \theta } y ^ { 2 \tilde { \theta } + 1 } ) + \alpha _ 0 y ^ { \tilde { \theta } + 1 } - \alpha _ 1 \frac { y } { \tilde { \theta } } + \alpha _ 2 s ^ { 2 H - 1 } \frac { 1 } { \tilde { \theta } ^ { \rho } } y ^ { - \tilde { \theta } \rho + \tilde { \theta } + 1 } - \tilde { \sigma } H s ^ { 2 H - 1 } y ^ { - 1 } ( \tilde { \theta } + 1 ) , \end{align*}


\begin{align*} \sum _ { n = 0 } ^ { \infty } \phi _ { n + 1 , \lambda } ( x ) \frac { t ^ { n } } { n ! } & = \frac { d } { d t } \sum _ { n = 0 } ^ { \infty } \phi _ { n , \lambda } ( x ) \frac { t ^ { n } } { n ! } \\ & = \frac { d } { d t } e ^ { x ( e _ { \lambda } ( t ) - 1 ) } = x e _ { \lambda } ^ { 1 - \lambda } ( t ) e ^ { x ( e _ { \lambda } ( t ) - 1 ) } \\ & = \sum _ { n = 0 } ^ { \infty } \bigg ( x \sum _ { k = 0 } ^ { n } \binom { n } { k } ( 1 - \lambda ) _ { n - k , \lambda } \phi _ { k , \lambda } ( x ) \bigg ) \frac { t ^ { n } } { n ! } . \end{align*}


\begin{align*} \Omega _ 2 ^ j ( x ) : = \Big \{ \eta \in \Omega : \overleftarrow { \eta } ^ { \ell } ( x ) \geq \frac { 2 } { \ell } \Big \} \cup \Big \{ \eta \in \Omega : \overleftarrow { \eta } ^ { \ell } ( x - j \ell ) \geq \frac { 2 } { \ell } \Big \} . \end{align*}


\begin{align*} \R [ \mathcal { B } _ n ] = \R [ X _ 1 , \ldots , X _ n ] / ( X _ 1 ( X _ 1 - 1 ) , \ldots , X _ n ( X _ n - 1 ) ) , \end{align*}


\begin{align*} \phi _ k ^ n ( x ^ n , z ^ n ) \approx \sum _ { m = 1 } ^ { N _ { e p } } \hat { \phi } ^ n _ { k , m } \psi _ { m } ( x ^ n , z ^ n ) = \sum _ { i = 1 } ^ { N _ { e p } } \phi _ k ^ n ( x _ i ^ n , z _ i ^ n ) h _ { i } ( x ^ n , z ^ n ) , ( x ^ n , z ^ n ) \in E ^ n , \end{align*}


\begin{align*} \sigma : = \left ( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right ) \qquad \sigma ' : = \mathbf { d ' } ( F ) \ , ; \end{align*}


\begin{align*} t ( \mu , \delta , \mathcal { B } ) = \lim _ { \varepsilon \to 0 } t ( \mu , \delta , \varepsilon , \mathcal { B } ) . \end{align*}


\begin{align*} \chi _ k ( r , n , m ) : = \max \{ \delta ( k ) , \chi ( r , n , m ) \} \delta ( k ) : = 2 k + 1 \end{align*}


\begin{align*} \mathrm { W F } ( u ) = \left \{ ( s , s ; \xi _ s , - \xi _ s ) \in \dot { T } ^ { * } \Sigma _ 0 ^ 2 \ , | \ , \xi _ s > 0 \right \} . \end{align*}


\begin{align*} \widehat { F } ( x , \omega ) = F ( - \omega , x ) . \end{align*}


\begin{align*} \begin{aligned} & \lim _ { n \to \infty } P _ { n , k } ( W _ { \mathcal { S } ^ + ( n , k ; M _ n ) } ) = - \big ( 1 - ( 1 - c ) ^ k \big ) \log ( 1 - ( 1 - c ) ^ k ) . \end{aligned} \end{align*}


\begin{align*} \| \rho ^ { \epsilon } \| _ { L ^ { \infty } ( \mathbb { R } ^ + ; L ^ { \infty } ) } = \| 1 + \epsilon \phi ^ { \epsilon } \| _ { L ^ { \infty } ( \mathbb { R } ^ + ; L ^ { \infty } ) } \approx 1 , \end{align*}


\begin{align*} \sum _ { r \in \mathbb { Z } } c ^ F _ { - r } \big ( r _ l ( \pi _ F ) ^ { ( l ) } ( w _ n ) W , \sigma _ F ^ { ( l ) } ( w _ { n - 1 } ) W ' \big ) q _ F ^ { - \frac { r } { 2 } } X ^ { - l r } = \gamma \big ( X ^ l , r _ l ( \pi _ F ) ^ { ( l ) } , \sigma _ F ^ { ( l ) } , \overline { \psi } _ F ^ l \big ) \sum _ { r \in \mathbb { Z } } c ^ F _ r ( W , W ' ) q _ F ^ { \frac { r } { 2 } } X ^ { l r } . \end{align*}


\begin{align*} \Delta \circ S ^ 2 = ( \sigma ^ { \varphi } \otimes ( \sigma ^ { \varphi _ S } ) ^ { - 1 } ) \circ \Delta , \end{align*}


\begin{align*} \sigma _ k = \begin{cases} 1 & \mbox { f o r } k < k _ c , \\ 2 & \mbox { f o r } k \ge k _ c , \end{cases} \end{align*}


\begin{align*} \begin{aligned} \rho ( \mathcal { L } ) = \| \mathcal { L } \| \leq 1 - \frac { 2 \sigma _ { \min } ^ 2 } { \| A \| _ F ^ 2 } , \end{aligned} \end{align*}


\begin{align*} \begin{aligned} | N _ { G } ( h ) \backslash V ( T ' \cup B ) | & \geq d _ { G } ( h ) - | N _ { G } ( h ) \cap V ( B ) | \\ & - | N _ { G } ( h ) \cap ( V ( T ' ) \backslash \{ u _ { l + 1 } , \cdots , u _ { j } \} ) | \\ & \geq ( t + 3 ) - 2 - ( t - \lfloor \frac { j - l } { 2 } \rfloor ) \\ & \geq 1 + \lfloor \frac { j - l } { 2 } \rfloor , \end{aligned} \end{align*}


\begin{align*} n = q , \theta = \hat \theta = 0 . \end{align*}


\begin{align*} \varphi _ { \sigma ^ { - 1 } i } = \alpha ^ { - 1 } _ i , i = 1 , 2 , 3 . \end{align*}


\begin{align*} \phi * \psi : = m _ A ( \phi \otimes \psi ) \Delta , \end{align*}


\begin{align*} R ( \lambda ) f = \sigma \lim _ { \mathclap { n \to \infty } } R _ n ( \lambda ) f . \end{align*}


\begin{gather*} \Box \phi = \left ( | \phi _ { t } | ^ 2 - | \phi _ { x } | ^ 2 \right ) \phi + \mathbf { 1 } _ { \omega } f ^ { \phi ^ { \perp } } , \ ; \phi [ 0 ] = u [ 0 ] , \end{gather*}


\begin{align*} \int _ { G ( \Q ) \backslash G ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \mathbf { E } ( g \times h , s ) \mathbf { f } ( h ) \mathbf { d } h = c _ k ( s ) D ( s , \mathbf { f } , \chi ) \mathbf { f } ( g ) . \end{align*}


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


\begin{align*} \int _ 0 ^ 1 \left ( A ( n _ { 2 j } , n _ { 2 j x } ) - A ( n _ { 1 j } , n _ { 1 j x } ) \right ) \varphi _ x d x + \int _ 0 ^ 1 \left ( n _ { 2 j } - n _ { 1 j } \right ) \varphi d x = \int _ 0 ^ 1 \left ( b _ 2 - b _ 1 \right ) \varphi d x \le 0 , \end{align*}


\begin{align*} L \otimes \Z _ v & = \begin{cases} \displaystyle { \bigoplus _ { j = 0 } ^ 1 L _ { p _ i , j } } ( \pi ^ j ) & ( v = p _ i \ \mathrm { f o r } \ i = 1 , \cdots , k ) , \\ L _ { v , 0 } & ( \mathrm { o t h e r w i s e } ) , \end{cases} \end{align*}


\begin{align*} \Big ( \phi _ { t } = I d _ L + t \mathcal { L } ( \mathfrak { X } ) - = I d _ L + t [ \mathfrak { X } , - ] , \ ; \psi _ { t } = I d _ V + t D ( \mathfrak { X } ) ( - ) \Big ) , \end{align*}


\begin{align*} [ J ^ { \chi _ A } ] _ \mathcal { M } : = \lambda \alpha \in \mathbb { N } ^ \mathbb { N } , x \in X . \begin{cases} J ^ A _ { r _ \alpha } x & r _ \alpha > 0 \rho > - r _ \alpha / 2 , \\ 0 & , \end{cases} \end{align*}


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


\begin{align*} G '' ( \xi _ i ) = 2 \alpha _ i + 2 \ , \frac { \mu ( \xi _ i + ) - \mu ( \xi _ i ) } { \sigma ^ 2 ( \xi _ i ) } , \end{align*}


\begin{align*} G ( t ) = \int _ { 0 } ^ { t } g ( s ) \ d s , \end{align*}


\begin{align*} g _ 1 ( t ) + g _ 2 ( t ) _ { s + 1 } \mathcal { F } _ s ( a _ 1 , \ldots , a _ { s + 1 } ; b _ 1 , \ldots , b _ s ) ( \alpha ) ^ { q ^ d } = 0 . \end{align*}


\begin{align*} \pi ( \l ) \pi ( \l ^ \circ ) = \pi ( \l ^ \circ ) \pi ( \l ) \end{align*}


\begin{align*} \overline { \frac { \partial u } { \partial Z _ { j } } } = \sum _ { i = 1 } ^ { n } \overline { g _ { i } } \overline { \frac { \partial h _ { i } } { \partial Z _ { j } } } , j = 1 , \cdots , n . \end{align*}


\begin{align*} \tilde \Omega \cap \{ x _ 1 = 0 \} \subset \bigcup _ { k = 1 } ^ M B ^ { ( k ) } \Subset \Omega . \end{align*}


\begin{align*} \omega = \check x _ { t , x , v } = x - \check { \mathcal T } _ { t , x _ 1 , v _ 1 } v \Leftrightarrow v = \tfrac { x - \omega } { t - \tau _ \omega } , \end{align*}


\begin{align*} \mathfrak { T } ( r , L ) = O ( \log r ) \hbox { a s $ r \to + \infty $ . } \end{align*}


\begin{align*} M _ { q _ 1 , q _ 2 } ( p ) & = \frac { \pi ^ 2 } { 4 p ^ 2 } \sum _ { a = 1 } ^ { p - 1 } \sum _ { b = 1 } ^ { p - 1 } \epsilon ( q _ 1 a , q _ 2 b ) \cot \left ( \frac { \pi a } { p } \right ) \cot \left ( \frac { \pi b } { p } \right ) \\ & = \frac { \pi ^ 2 } { 2 p ^ 2 } \sum _ { a = 1 } ^ { p - 1 } \cot \left ( \frac { \pi q _ 1 a } { p } \right ) \cot \left ( \frac { \pi q _ 2 a } { p } \right ) = \frac { 2 \pi ^ 2 } { p } s ( q _ 1 , q _ 2 , p ) . \end{align*}


\begin{align*} S ^ T J S = J . \end{align*}


\begin{align*} \| x \| \ = \ \left \| x - P _ A ( x ) + \sum _ { n \in A } e _ n ^ * ( x ) e _ n \right \| & \ \leqslant \ \sup _ { \delta } \| x - P _ A ( x ) + 1 _ { \delta A } \| \\ & \ \leqslant \ \mathbf C _ { \lambda , p l } \| x - P _ A ( x ) + 1 _ { \varepsilon B } \| . \end{align*}


\begin{align*} \Vert u \Vert _ { \theta , p } ^ p : = \int _ { Z } \int _ { Z } \frac { | u ( y ) - u ( x ) | ^ p } { d ( x , y ) ^ { \theta p } \ , \nu ( B ( x , d ( x , y ) ) ) } \ , d \nu ( y ) \ , d \nu ( x ) , \end{align*}


\begin{align*} \dim ( X \cap Y ) + \dim ( Y \cap V ) = \dim ( X \cap Y + Y \cap V ) - \dim ( X \cap Y \cap V ) \leq \dim ( Y ) - \dim ( X \cap Y \cap V ) . \end{align*}


\begin{align*} \log \Gamma \left ( \frac { \sigma + i T } { 2 } \right ) & = \left ( \frac { \sigma - 1 } { 2 } + \frac { i T } { 2 } \right ) \left [ \frac { 1 } { 2 } \log \left ( \frac { \sigma ^ 2 + T ^ 2 } { 4 } \right ) + i \arg \left ( \frac { \sigma + i T } { 2 } \right ) \right ] - \\ & \frac { \sigma + i T } { 2 } - \frac { 1 } { 2 } \log ( 2 \pi ) + o ( 1 ) \end{align*}


\begin{align*} h ( s ) : = s + A s ^ { 1 + \delta } . \end{align*}


\begin{align*} P ( x ) = \sum _ { k = 0 } ^ n p _ k x ^ k \end{align*}


\begin{align*} \mathcal { P } f ( t ) = \sum _ { k \in \Z ^ d } f ( t + k ) = \sum _ { l \in \Z ^ d } \widehat { f } ( l ) e ^ { 2 \pi i l \cdot t } . \end{align*}


\begin{align*} m _ 1 = \inf _ { \mathcal { M } } J . \end{align*}


\begin{align*} \mathbf e \approx \sum _ { j = m + 1 } ^ M r _ j \begin{bmatrix} \mathbf 0 \\ [ \mathbf L _ { 2 2 } ^ { - 1 } ] _ { : , j } \end{bmatrix} \approx \sum _ { j = m + 1 } ^ M r _ j \frac { h } { 2 } \begin{bmatrix} \mathbf 0 \\ G ( \mathbf S _ 2 , S _ j ) \end{bmatrix} , \end{align*}


\begin{align*} \frac { d } { d s } \frac { s } { h ( s ) } = \frac { d } { d s } \frac { 1 } { \int _ \mathbb { R } \frac { 1 } { s ^ 2 + u ^ 2 } d \mu ( u ) } = 2 s \cdot \frac { \int _ \mathbb { R } \frac { 1 } { ( s ^ 2 + u ^ 2 ) ^ 2 } d \mu ( u ) } { \left ( \int _ \mathbb { R } \frac { 1 } { s ^ 2 + u ^ 2 } d \mu ( u ) \right ) ^ 2 } . \end{align*}


\begin{align*} \| \phi _ { 1 , b , \beta } \| _ { L ^ 2 _ { \rho _ \beta } } ^ 2 = \frac { \left ( \frac { 2 } { \sqrt { 2 \beta } } \right ) ^ { - 2 \gamma } } { 2 \pi ^ { \frac { d + 2 } { 2 } } } \Gamma \left ( \frac { d } { 2 } - \gamma + 1 \right ) \left [ \left ( \frac { 4 } { \beta ^ 2 } - \frac { 1 6 } { \beta } + 1 6 \right ) \left ( \frac { d } { 2 } - \gamma + 1 \right ) ^ 2 + \frac { 4 } { \beta ^ 2 } \left ( \frac { d } { 2 } - \gamma + 1 \right ) \right ] \end{align*}


\begin{align*} \varphi ( t , \ , m , \ , n ) = \varphi _ i ( t , \ , m , \ , n ) ( t , \ , m ) \in A _ i . \end{align*}


\begin{align*} \varepsilon ^ + = e ^ { \int _ { \tau _ 0 } ^ \tau \lambda _ { M , b } d \tau ' } \phi _ { M , b } f ( y ) \end{align*}


\begin{align*} \begin{array} { l l } R e ( ( d ^ i ) ^ * A w ) = c _ 2 , \ ; \ ; \forall i \in \mathcal { I } \\ \displaystyle \sum _ { j = 1 } ^ { n } w _ j = 1 \\ \lvert a r g w _ j \rvert \leq \beta , \ ; \ ; j = 1 , 2 , . . . , n \end{array} \end{align*}


\begin{align*} u = x y , \ ; \ ; 2 v = x + y . \end{align*}


\begin{align*} 0 = \omega ( u ^ k ) = k \omega ( u ) , \end{align*}


\begin{align*} \{ x _ { \rm T L S } , E _ { \rm T L S } , f _ { \rm T L S } \} : = \mathop { \rm a r g m a x } \limits _ { x , E , f } \| E , F \| _ { F } \ \ \ \ s . t . \ \ ( A + E ) x = b + f , \end{align*}


\begin{align*} \mathcal { H } ^ { \mu , s } _ { t } ( A ) = \inf \left \{ \mathcal { H } ^ { s } _ { t } ( E ) : \ E \subset A , \ \mu ( E ) = \mu ( A ) \right \} . \end{align*}


\begin{align*} B = \partial _ k ( p _ \alpha ) ( x ) D _ { k } ^ { \alpha - 1 } ( f ) ( x ) + p _ \alpha ( x ) \partial _ k D _ k ^ { \alpha - 1 } ( f ) ( x ) . \end{align*}


\begin{align*} \rho _ 3 ( \mathbf { c } ) \Sigma \rho _ 2 ( A ) \Sigma ^ { - 1 } \rho _ 1 ( a ) = \left ( \begin{array} { c c c } a \iota ( \alpha _ { 1 1 } ) & \iota ( \alpha _ { 1 2 } ) \sigma ^ { - 1 } & a C _ 1 \\ a \sigma \iota ( \alpha _ { 2 1 } ) & \sigma \iota ( \alpha _ { 2 2 } ) \sigma ^ { - 1 } & a C _ 2 \\ 0 & 0 & a I _ 2 \end{array} \right ) , \end{align*}


\begin{align*} a ^ { n + 1 } = \gamma _ 2 a ^ 2 + \dots + \gamma _ { n } a ^ { n } , \end{align*}


\begin{align*} t ^ v _ j = Q ^ v \left ( \phi ^ { c h } ( t ^ { c h } ) + \sum _ { i \in \mathcal { N } _ v \setminus \{ j \} } \phi ^ c _ i ( t ^ c _ i ) \right ) , \end{align*}


\begin{align*} x = \frac { b _ i \tilde a _ i } { \| \tilde a _ i \| ^ 2 } \in H _ i . \end{align*}


\begin{align*} \mathcal { M } _ s \nu ( x ) : = \sup _ { r > 0 } r ^ { - s } \nu ( B ( x , r ) ) , x \in M , \end{align*}


\begin{align*} c _ m = \left \langle \psi ( \cdot ) , e ^ { \frac { 2 \pi i m \cdot } { | I | } } \right \rangle _ { H ^ { - s } _ { p e r } ( I ) , H ^ s _ { p e r } ( I ) } , \ \ \ \forall m \in \mathbb { Z } . \end{align*}


\begin{align*} \int _ 0 ^ 1 Q _ \alpha ( x ) \ , d x = \frac 1 { P ( \alpha ) } \int _ 0 ^ \alpha Q ( z ) \ , d z = 1 \alpha > q _ 2 , \end{align*}


\begin{align*} | \lambda ^ 2 _ f ( p ) - 2 | \leq | \lambda ^ 2 _ f ( p ) | + 2 \leq d ( p ^ 2 ) + 2 \ll 1 . \end{align*}


\begin{align*} B _ { 1 } = 2 \left [ 1 - \frac { k } { m } + \left ( 1 - \frac { k } { m } \right ) \delta + \frac { \delta ^ { 2 } } { 4 } \right ] \end{align*}


\begin{align*} \sum _ { m = 2 } ^ { \infty } \frac { \tau _ { A _ { s _ 1 } \smallsetminus \{ \alpha + s _ 1 \} \cup \{ - \beta - s _ 2 \} } ( p ^ m ) \tau _ { B _ { s _ 2 } \smallsetminus \{ \beta + s _ 2 \} \cup \{ - \alpha - s _ 1 \} } ( p ^ m ) } { p ^ m } \ll \sum _ { m = 2 } ^ { \infty } \frac { p ^ { m ( \frac { 1 } { 2 } - \epsilon + \varepsilon ) } } { p ^ m } \ll \frac { 1 } { p ^ { 1 + \varepsilon } } \end{align*}


\begin{align*} p _ { \mathcal { S } } ( \psi _ i , \Phi ) = \prod _ { j = 1 } ^ n p _ { \mathcal { S } _ j } ( \psi _ { i j } , \Phi _ j ) = \prod _ { j = 1 } ^ n p _ { K _ j } ( _ { \mathcal { S } _ j / K _ j } ( \psi _ { i j } ) , \Phi ' _ j ) = p _ Y ( _ { \mathcal { S } / Y } ( \psi _ i ) , \Phi ' ) , \end{align*}


\begin{align*} & \lim _ { \varepsilon \to 0 } \Big ( \int _ { \{ \vert f \vert \geq \varepsilon \} } \vert f \vert ^ { 2 \alpha } f ^ { - N } \xi + 2 i \pi \sum _ { m = m ' + N } ^ { \alpha + r + m ' < 0 } \tilde { T } _ { m , m ' } ^ { r , j } ( \xi ) \frac { \varepsilon ^ { 2 ( \alpha + r + m ' ) } } { 2 ( \alpha + r + m ' ) } ( - L o g \varepsilon ) ^ j \Big ) \end{align*}


\begin{align*} \omega \underline { P } = \omega _ 1 P _ 1 + \cdots + \omega _ q P _ q , \end{align*}


\begin{align*} n = m \ell , \theta = \ell - 1 , \hat \theta = \frac { 1 } { \ell } \sqrt { \frac { ( \ell - 1 ) ( m \ell - 1 ) } { m ( m - 1 ) } } . \end{align*}


\begin{align*} h ( \mathcal { A } _ { \beta } ^ { i } ) = 0 , h ( \mathcal { A } _ { \omega \beta } ^ { i v } ) = h ( \mathcal { B } _ { \beta } ^ { i } ) = 0 \end{align*}


\begin{align*} a _ { f _ { \lambda + 1 } } \ge ( n - 1 ) f _ { \lambda + 1 } + g _ { \lambda + 1 } = f _ { \lambda + 1 } + f _ { \lambda + 2 } \end{align*}


\begin{align*} \langle J ^ { ' } ( u ) , u \rangle = 0 \end{align*}


\begin{align*} \sum _ { r = 1 } ^ { n + m + 1 } ( - 1 ) ^ { r + 1 } \sum _ { 1 \leq k _ 1 < \ldots < k _ r \leq n + m + 1 } \left ( \prod _ { i = 1 } ^ { r } \tau _ t ( c _ { k _ i } ) \right ) \tau _ t ( c _ 1 \cdots \hat { c } _ { k _ 1 } \cdots \hat { c } _ { k _ r } \cdots c _ { n + m + 1 } ) + o ( t ) , \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { E \tau ^ { ( M A , A ) } } { M A } = \frac 1 { A + 1 } . \end{align*}


\begin{align*} M _ n = a _ n S _ n . \end{align*}


\begin{align*} \frac { \mathbf { P } ( X _ n / \sqrt { \langle X \rangle _ n } > x ) } { 1 - \Phi \left ( x \right ) } = 1 + O \bigg ( x ^ 3 ( \epsilon _ n + \delta _ n ) + ( 1 + x ) \big ( \delta _ n | \ln \delta _ n | + \epsilon _ n | \ln \epsilon _ n | \big ) \bigg ) . \end{align*}


\begin{align*} \Gamma _ { 2 , y , ( r , s ) } ^ { \delta } = \{ n \in \Gamma _ { 2 , y , ( r , s ) } : A ^ { \delta } \subset ( H ^ { \delta } ) ^ c \} , \delta \in \{ 1 , \ldots , d \} . \end{align*}


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


\begin{align*} M ( f , H ) = { 2 \pi ^ 2 \over f } \sum _ { \delta \mid f } \frac { \mu ( \delta ) } { \delta } \sum _ { h \in H } s ( h , f / \delta ) . \end{align*}


\begin{align*} D ^ * \Phi ( x , y ) ( y ^ * ) \subset \begin{cases} D ^ * g ( x ) ( y ^ * ) & y ^ * \in \mathcal N _ D ( g ( x ) - y ) , \\ \varnothing & . \end{cases} \end{align*}


\begin{align*} \partial _ { t } ^ 2 u - \partial _ x ^ 2 u + u + V ( x ) u + a ( x ) u ^ 2 + \pm u ^ 3 = 0 , \ , u ( 0 ) = u _ 0 , \ , \partial _ t u ( 0 ) = u _ 1 . \end{align*}


\begin{align*} \mathcal { L } f = | D | f + f - p Q ^ { p - 1 } f . \end{align*}


\begin{align*} | \det { \mathfrak { S } } | = | A ( \beta _ 1 ) \cdots A ( \beta _ k ) | \geqslant \frac { 1 } { 2 ^ K } | \beta _ 1 | ^ N \dots | \beta _ k | ^ N . \end{align*}


\begin{align*} & \lim _ { m \to \infty } F _ p ( T , p ^ { k _ m - n - 1 } ) = 1 \end{align*}


\begin{align*} L ^ { p } ( \mathbb { R } ^ 3 ) = L ^ { p , p } ( \mathbb { R } ^ 3 ) \hookrightarrow L ^ { p , q } ( \mathbb { R } ^ 3 ) \hookrightarrow L ^ { p , \infty } ( \mathbb { R } ^ 3 ) \ , \ p \leq q < \infty . \end{align*}


\begin{align*} z ^ 3 + z ^ 2 f _ 2 ( \lambda , 1 ) + z f _ 4 ( \lambda , 1 ) + f _ 6 ( \lambda , 1 ) = 0 = \varphi _ 3 ( z ) \ , , \end{align*}


\begin{align*} v _ k ( x , y ) = u _ k ( x ) - u _ k ( y ) . \end{align*}


\begin{align*} \alpha ^ 2 \norm { x } _ X ^ 2 - \norm { ( \alpha - 1 ) x + _ X y } _ X ^ 2 = 2 \alpha \langle x - _ X y , y \rangle _ X - ( 1 - 2 \alpha ) \norm { x - _ X y } _ X ^ 2 . \end{align*}


\begin{align*} V _ { \widetilde { g } } ( V _ g ^ * F ) ( \xi , \eta ) & = \langle V _ g ^ * F , M _ \eta T _ \xi \widetilde { g } \rangle \\ & = \iint _ { \R ^ { 2 d } } F ( x , \omega ) \overline { V _ g ( M _ \eta T _ \xi \widetilde { g } ) ( x , \omega ) } \ , d ( x , \omega ) \\ & = \iint _ { \R ^ { 2 d } } F ( x , \omega ) V _ { \widetilde { g } } g ( \xi - x , \eta - \omega ) e ^ { - 2 \pi i x \cdot ( \eta - \omega ) } \ , d ( x , \omega ) . \end{align*}


\begin{align*} \varepsilon _ { e x t , e } ( \tau ) = \mathcal { K } _ \beta ( \tau , \tau ' ) \varepsilon _ { e x t , e } ( \tau ' ) + \int _ { \tau ' } ^ \tau \mathcal { K } _ \beta ( \tau , \tilde \tau ) \left [ \mathcal { C } ( \varepsilon _ { e x t } ) + \mathcal { N } ( \varepsilon _ { e x t } ) \right ] ( \tilde \tau ) d \tilde \tau . \end{align*}


\begin{align*} m _ t ' = \overline { m } _ t , \ ; \ ; m _ t ' = m _ t ; m _ t ' = \hat m _ t , \ ; \ ; m _ t ' = \tilde { m } _ t ; \end{align*}


\begin{align*} d _ K ( x ) : = \inf _ { y \in K } \| x - y \| , x \in X . \end{align*}


\begin{align*} \limsup _ { r \to 0 } \limsup _ { j \to \infty } \theta _ { | \Sigma _ j | } ( p _ j , r ) - \theta _ { | \Sigma _ j | } ( p _ j ) = 0 \ , . \end{align*}


\begin{align*} [ n ] = \{ 0 \leq 1 \leq \ldots \leq n \} n \geq 0 , \end{align*}


\begin{align*} F \star _ { H , \ell } G [ \psi ] = \sum _ { n = 0 } ^ { \infty } \frac { \hbar ^ { n } } { n ! } \left \langle W _ { \Sigma } ^ { \otimes n } , F ^ { ( n ) } [ \psi ] \otimes G ^ { ( n ) } [ \psi ] \right \rangle \end{align*}


\begin{align*} \int _ { \R ^ N } f ( x ) d \omega ( x ) = 0 . \end{align*}


\begin{align*} \left \{ \begin{aligned} \dot { \xi } ( s ) & = e ^ \frac { s } { q - 1 } p \left | \eta ( s ) \right | ^ { p - 2 } \eta ( s ) \\ \dot { \eta } ( s ) & = e ^ { - s } D f \left ( \xi ( s ) \right ) \end{aligned} \right . \end{align*}


\begin{align*} \left \{ \begin{aligned} & D _ { + } ( k ) D _ { - } ( k ) = r _ { + } ( k ) , & k \in I _ \eta \backslash I _ R , \\ & D _ { + } D _ { - } = 1 , & k \in I _ R , \\ & D _ { + } = D _ { - } , & \textnormal { e l s e w h e r e } . \end{aligned} \right . \end{align*}


\begin{align*} f ( x + 1 , x _ 2 , \ldots , x _ m ) \ = \ h ( x , f ( x , x _ 2 , \ldots , x _ m ) , x _ 2 , \ldots , x _ m ) \end{align*}


\begin{align*} x ^ i x ^ j = \frac 1 2 \left [ y ^ i x ^ j + y ^ j x ^ i \right ] , \end{align*}


\begin{align*} \P ( \exists S \in \mathcal S : \eqref { e q - e n c l o s e - L a m b d a - 2 r } ~ h o l d s ) \leq \sum _ { t \geq 1 } \sum _ { t ' \geq t } n ^ { - 8 ^ { - d } t ' } ( 1 0 0 t ) ^ { d } = o ( 1 ) \ , . \end{align*}


\begin{align*} \begin{aligned} & \sum \limits _ { i = 1 } ^ { \infty } { \left ( { { n _ { i + 1 } } - { n _ i } } \right ) { \epsilon _ i } } , \\ & \sum \limits _ { i = 1 } ^ { \infty } { \left ( { { n _ { i + 1 } } - { n _ i } } \right ) \left ( { 2 { \tau _ { \rm c } } + 2 \mathcal { T } + { n _ { i + 1 } } + { n _ i } } \right ) { \epsilon _ i } } . \end{aligned} \end{align*}


\begin{align*} y & \in \partial V ( u - \sum _ { t = 0 } ^ { T - 1 } x _ t \Delta s _ { t + 1 } - \bar c \cdot \bar x + S _ 0 ( \bar x ) ) , \\ E _ t [ y \Delta s _ { t + 1 } ] & \in N _ { D _ t } ( x _ t ) t = 0 , \dots , T , \\ \frac { E [ y \bar c ] } { E [ y ] } & \in \partial S _ 0 ( \bar x ) \end{align*}


\begin{align*} \sum _ { k = 1 } ^ { t } \frac { \alpha _ k } { \Gamma _ k } = \frac { 1 } { \Gamma _ t } ~ ~ ~ ~ ~ \implies ~ ~ ~ ~ ~ \Gamma _ t \sum _ { k = 1 } ^ t \frac { \alpha _ k } { \Gamma _ k } = 1 \end{align*}


\begin{align*} A ( z ) = \prod _ { j = 1 } ^ { n } ( z - \alpha _ j ) ^ { m _ j } , \end{align*}


\begin{align*} D T \left [ f ( \eta ) f '' ( \eta ) \right ] = \sum _ { l = 0 } ^ { k } ( l + 1 ) ( l + 2 ) F ( l ) F ( k - l ) \end{align*}


\begin{align*} | A ( x ) - A ( x + r v ) | & = | A ( x ) - f ( x ) | + | f ( x ) - f ( x + r v ) | + | f ( x + r v ) - A ( x + r v ) | \\ & \leq \delta r + r + \delta r \\ & = r ( 1 + 2 \delta ) . \end{align*}


\begin{align*} E ( \rho ( \theta ) , \phi ( \theta ) ) - \theta E ( \rho ^ 0 , \phi ^ 0 ) - ( 1 - \theta ) E ( \rho ^ 1 , \phi ^ 1 ) = - \theta ( 1 - \theta ) I \end{align*}


\begin{align*} \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } b ( \cos \theta ) | v - v _ * | ^ \gamma f ( v ' ) d \sigma d v _ * = \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } b ( \cos \theta ) \frac 1 { \sin ^ { 3 + \gamma } ( \theta / 2 ) } | v - v _ * | ^ \gamma f ( v _ * ) d \sigma d v _ * . \end{align*}


\begin{align*} \bar \alpha _ i = \begin{cases} 1 , & i \in H , \\ \bar \alpha _ { i + 1 } / \rho _ { i + 1 } , & i \notin H . \end{cases} \end{align*}


\begin{align*} \tilde \sigma _ 0 \in A C [ 0 , \infty ) , \tilde \sigma _ 0 ( x ) = \tilde \sigma _ 0 ( 0 ) - \int _ 0 ^ x \sigma _ 0 ( t ) \ , d t . \end{align*}


\begin{align*} \begin{dcases} - \int _ 0 ^ T e ^ { \lambda _ 0 ( T - t ) } p ( t ) \ , d t & = m _ { 2 , 0 } , \ \ \ - \int _ 0 ^ T p ( t ) \ , d t = m _ { 2 , 0 } , \\ - \int _ { 0 } ^ { T } \overline { e ^ { \lambda ^ h _ k ( T - t ) } } p ( t ) \ , d t & = m _ { 2 , k } , \forall | k | \geq k _ 0 , \\ - \int _ { 0 } ^ { T } \overline { e ^ { \widehat \lambda _ { n _ l } ( T - t ) } } p ( t ) \ , d t & = m _ { 1 , l } , \forall \ , 1 \leq l \leq l _ 0 , \end{dcases} \end{align*}


\begin{align*} \frac { \mathrm d ^ k } { \mathrm d t ^ k } \ , \Phi _ { a , b } ( t ) = \sum _ { j = 0 } ^ k C _ { a , j , k } t ^ { a - ( k - j ) } A ^ { b + j } S ( t ) Q ^ \frac { 1 } { 2 } = \sum _ { j = 0 } ^ k C _ { a , j , k } \Phi _ { a - ( k - j ) , b + j } ( t ) , \end{align*}


\begin{align*} z \cdot ( x _ 1 , x _ 2 ) = ( z x _ 1 , z x _ 2 ) , \ z \in S ^ 1 , ( x _ 1 , x _ 2 ) \in \mathbb { C } ^ 2 . \end{align*}


\begin{align*} \limsup _ { n \rightarrow \infty } \log ( a _ n + b _ n ) = \max \Big \{ \limsup _ { n \rightarrow \infty } \log ( a _ n ) , \limsup _ { n \rightarrow \infty } \log ( b _ n ) \Big \} . \end{align*}


\begin{align*} \kappa _ { i z } = \sqrt { \kappa _ i ^ 2 - \kappa _ x ^ 2 - \kappa _ y ^ 2 } \end{align*}


\begin{align*} \langle W f , W ( M _ { - \omega } T _ x g _ 0 ) \rangle & = | \langle f , M _ { - \omega } T _ x g _ 0 \rangle | ^ 2 \\ & = | V _ { g _ 0 } f ( x , - \omega ) | ^ 2 \\ & = | B f ( z ) | ^ 2 e ^ { - \pi | z | ^ 2 } . \end{align*}


\begin{align*} S _ { \tau ( \xi ) } : = S _ { \tau } ^ { S _ { \xi } } . \end{align*}


\begin{align*} \hat { \phi } ( \xi ) = \frac { 1 } { 1 + | \xi | ^ 2 } . \end{align*}


\begin{align*} w ( y , \tau _ 0 ) = Q _ { b ( \tau _ 0 ) } ( y ) + \varepsilon ( \tau _ 0 ) . \end{align*}


\begin{align*} \partial _ { x _ i x _ j } f ( \mathbf { x } ) = 0 \end{align*}


\begin{align*} E \left [ \left ( \dfrac { S _ n } { \sqrt { n / 2 } } \right ) ^ 2 \right ] - 1 = 0 \end{align*}


\begin{align*} m : A \times A \rightarrow A , ( a , b ) \mapsto a \cdot b = a b . \end{align*}


\begin{align*} f = a _ 1 \widetilde { f } + p , \mbox { w h e r e $ \widetilde { f } \in H ^ 2 $ a n d $ p \in \P _ { N - 1 } $ } , \end{align*}


\begin{align*} \phi _ 0 ( \xi ) & = \Phi _ { \Gamma ^ 0 } ( \xi ) , \\ \phi _ j ( \xi ) & = \Phi _ { \Gamma ^ j \setminus \Gamma ^ { j - 1 } } ( \xi ) , ~ ~ ~ j \in \mathbb { N } , \end{align*}


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


\begin{align*} \psi ^ H _ t ( x _ i , y _ i , z ) = \left ( x _ i - t x ^ 0 _ i , y _ i - t y ^ 0 _ i , z + \sum _ i \left ( \frac { x ^ 0 _ i y ^ 0 _ i } { 2 } t - y ^ 0 _ i x _ i \right ) t \right ) \end{align*}


\begin{align*} H _ L ( \hat { \Psi } ( \mathbf { u } , \tilde { \Xi } ) ) \alpha ^ * = \hat { \Psi } ( \mathbf { \bar { u } } , \hat { \Xi } ) + c . \end{align*}


\begin{align*} \alpha \triangledown _ { \mathcal { H } } \beta \overset { { \scriptstyle \mathrm { d e f } } } { \Leftrightarrow } \mathcal { H } _ { \alpha } ^ { \beta } \in \mathfrak { G } ( \mathbf { L } ) \mathcal { H } _ { \beta } ^ { \alpha } \in \mathfrak { G } ( \mathbf { L } ) , \triangledown : = \bigcup _ { \mathcal { H } \in \mathfrak { G } ( \mathbf { L } ) } \triangledown _ { \mathcal { H } } . \end{align*}


\begin{align*} x ^ n _ { t \wedge \tau _ n } - x _ { t \wedge \tau _ n } & = \int _ 0 ^ { t \wedge \tau _ n } \Big \{ \tilde { f } _ n ( s , x _ s ^ n ) - \tilde { f } ( s , x _ s ) \Big \} d s \\ & = \int _ 0 ^ t \chi _ { _ { [ 0 , \tau _ n ) } } ( s ) \Big \{ \tilde { f } _ n ( s , x _ { s \wedge \tau _ n } ^ n ) - \tilde { f } _ n ( s , x _ { s \wedge \tau _ n } ) \Big \} d s . \end{align*}


\begin{align*} \lambda _ j = \frac { q ^ { k - j } - 1 } { q ^ { k } - 1 } = q ^ { - j } - O ( q ^ { - k } ) \end{align*}


\begin{align*} \xi ( t ; r , x ) = \xi ( t ; s , \xi ( s ; r , x ) ) . \end{align*}


\begin{align*} \rho ( S z , \tau ) = \rho _ S ( z , \tau ) = \mu ( S ) \rho ( z , \tau ) \mu ( S ) ^ { - 1 } . \end{align*}


\begin{align*} \Big \{ \alpha = Q + i p \in \C \ , \Big | \ , { \rm R e } ( \alpha ) < Q , { \rm I m } \sqrt { p ^ 2 - 2 \ell } < \beta \Big \} \cup \Big \{ Q + i p \in Q + i \R \ , \Big | \ , | p | \in \bigcup _ { j \geq \ell } ( \sqrt { 2 j } , \sqrt { 2 ( j + 1 ) } ) \Big \} , \end{align*}


\begin{align*} C ^ { \gamma } ( \mathbb { R } ^ { N - 1 } ) = \bigg \{ u \in C _ b ( \mathbb { R } ^ { N - 1 } ) : \ , [ u ] _ { C ^ { \gamma } } : = \sup _ { \stackrel { x , y \in \mathbb { R } ^ { N - 1 } } { x \neq y } } \frac { | u ( x ) - u ( y ) | } { | x - y | ^ { \gamma } } < \infty \bigg \} \end{align*}


\begin{align*} \sqrt { 2 } \widehat { K } _ \mathrm { a - c p t - f } ^ \mathbb { R } / \upsilon = \sum \nolimits _ { i \in \mathcal { K } } U _ i + R , \end{align*}


\begin{align*} B _ { k n } ( x , y ) = \begin{bmatrix} \xi _ { k n } ( x ) + b ^ { ( 1 + O ( \alpha ^ { n } ) ) q _ { k n } } f ( x ) y ( 1 + O ( \alpha ^ { k } ) ) \\ x \end{bmatrix} \end{align*}


\begin{align*} \rho ( \Phi ) & = \iint _ { \R ^ { 2 d } } \int _ 0 ^ 1 \sum _ { k \in \Z } \Phi _ k ( x , \omega ) e ^ { 2 \pi i k \tau } e ^ { 2 \pi i \tau } \rho ( x , \omega ) \ , d \tau \ , d ( x , \omega ) \\ & = \iint _ { \R ^ { 2 d } } \Phi _ { - 1 } ( x , \omega ) \rho ( x , \omega ) \ , d ( x , \omega ) . \end{align*}


\begin{align*} v _ * ^ \perp ( t , x , v _ 1 ) = \frac { x ^ \perp } { \check { \mathcal T } _ { t , x _ 1 , v _ 1 } } . \end{align*}


\begin{align*} \sum _ { n \geq 0 } \vert B _ n \vert ^ { \varepsilon } \mu ( B _ n ) = \sum _ { k \geq 0 } \sum _ { B \in \mathcal { T } _ k ( \mathcal { B } ) } \vert B \vert ^ { \varepsilon } \mu ( B ) \leq \sum _ { k \geq 0 } \sum _ { 1 \leq j \leq J _ k } 2 ^ { - k \varepsilon } \sum _ { B \in T _ { k , j } ( \mathcal { B } ) } \mu ( B ) . \end{align*}


\begin{align*} \partial _ t ^ 2 \phi - \partial _ x ^ 2 \phi + \phi + V ( x ) \phi = a ( x ) \phi ^ 2 + b ( x ) \phi ^ 3 , ( \phi , \phi _ t ) ( 0 ) = ( \phi _ 0 , \phi _ 1 ) , \end{align*}


\begin{align*} R ^ j _ t & = R ^ j _ 0 + M ^ j _ t + A ^ j _ t + B ^ j _ t \\ & = R ^ j _ 0 + \int _ 0 ^ t u ^ j _ l ( s ) d W ^ l _ s + \int _ 0 ^ t a ^ j ( s ) d s + \int _ 0 ^ t b ^ j ( s ) d s , j = 1 , 2 , \dots , d , \end{align*}


\begin{align*} P \cap \tau _ v P = P ( x _ 1 , \dots , x _ k , x _ 1 - v , \dots , x _ k - v ; B \times B ) . \end{align*}


\begin{align*} \Psi _ \xi ( u ) : = \varphi ( u ) - \psi _ \xi ( u ) \end{align*}


\begin{align*} \widetilde { O } \widetilde { H } _ n = 2 n \widetilde { H } _ n . \end{align*}


\begin{align*} \varphi _ q ( \xi ) = \begin{cases} \varphi ( \lambda _ q ^ { - 1 } \xi ) , \ \ \ \mbox { f o r } \ \ q \geq 0 \\ \chi ( \xi ) , \ \ \ \ \ \ \ \ \mbox { f o r } \ \ q = - 1 . \end{cases} \end{align*}


\begin{align*} \alpha \ = \ \sum \limits _ { n \in A } 2 ^ { - n } \end{align*}


\begin{align*} m ^ 3 - \lambda m ^ 2 - 2 \lambda m + \lambda ^ 2 = 0 . \end{align*}


\begin{align*} d X _ i ( t ) = b ( t , X _ i ( t ) , \mu ( t ) , \alpha _ i ) d t + \sigma ( t , X _ i ( t ) , \mu ( t ) ) d B _ i ( t ) + \sigma _ 0 ( t , X _ i ( t ) , \mu ( t ) ) d W ( t ) , \end{align*}


\begin{align*} v ( r , \omega ) = \sum _ { j \geq 1 } ( v _ j ^ + ( r ) + v _ j ^ - ( r ) ) \varphi _ j ( \omega ) , \end{align*}


\begin{align*} \int _ { \R ^ N } | \nabla u _ { \ast } | ^ 2 = \int _ { \R ^ N } | \nabla u | ^ 2 \ ; . \end{align*}


\begin{align*} y = \mu + X \beta + e , \end{align*}


\begin{align*} X _ { n + 1 } = \theta X _ n + \varepsilon _ { n + 1 } , \end{align*}


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


\begin{align*} \big \{ ( x , x ^ \prime , y , y ^ \prime ) \ \ y = f ( x ) , \ y ^ \prime = f ^ \prime ( x ^ \prime ) \big \} \end{align*}


\begin{align*} W e _ i - W e _ j & = ( e _ { v } + w _ j e _ j ) - ( e _ { v } + w _ i e _ i ) = w _ j e _ j - w _ i e _ i , \end{align*}


\begin{align*} \sum _ { r \in \mathbb { Z } } c ^ F _ r \big ( \widetilde { U } , \widetilde { W ' } \big ) q _ F ^ { - \frac { r } { 2 } } X ^ { - r } = \gamma \big ( X , r _ l ( \pi _ F ) ^ { ( l ) } , \sigma _ F ^ { ( l ) } , \overline { \psi } _ F ^ l \big ) \sum _ { r \in \mathbb { Z } } c ^ F _ r ( U , W ' ) q _ F ^ { \frac { r } { 2 } } X ^ { r } , \end{align*}


\begin{align*} i U ( \alpha z _ 1 ) ^ { \ast } H U ( \alpha z _ 2 ) = \left [ \begin{array} { c c } \eta ( \alpha z _ 1 , \alpha z _ 2 ) & \ast \\ \ast & \ast \end{array} \right ] , i U ( z _ 1 ) ^ { \ast } H U ( z _ 2 ) = \left [ \begin{array} { c c } \eta ( z _ 1 , z _ 2 ) & \ast \\ \ast & \ast \end{array} \right ] . \end{align*}


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


\begin{align*} A _ \rho \coloneqq \left [ 0 \to \tau ( M _ \rho ) \to N _ \rho \to M _ \rho \to 0 \right ] , \end{align*}


\begin{align*} X _ { n , r } & : = \begin{dcases} \delta ^ { - 1 } _ n ( 3 ) \cdot \sum _ { q = 1 } ^ { r - 1 } \left [ \frac { \delta _ n ( 3 ) } { \delta _ n ( 2 ) } \cdot \tilde M _ { n , \{ q , r \} } + \sum _ { p = 1 } ^ { q - 1 } \tilde M _ { n , \{ p , q , r \} } \right ] & , r \ge 2 \\ 0 & , r = 1 , \end{dcases} \end{align*}


\begin{align*} \Delta ( K ; \alpha , \beta ) : = \Big [ \frac { ( K ^ { \alpha + \beta } - 1 ) ^ 3 } { \alpha + \beta } \Big ] ^ 2 - \frac { ( K ^ { 2 \alpha } - 1 ) ^ 3 } { 2 \alpha } \cdot \frac { ( K ^ { 2 \beta } - 1 ) ^ 3 } { 2 \beta } < 0 . \end{align*}


\begin{align*} t ^ \star = \begin{cases} \bar t & \mbox { \ i f \ } \bar t \mbox { \ e x i s t s } \\ \tfrac { 3 } { \mu + L + \sqrt { \mu ^ 2 - L \mu + L ^ 2 } } & \mbox { o t h e r w i s e } , \end{cases} \end{align*}


\begin{align*} \partial _ \xi ^ k \Lambda Q = \partial _ \xi ^ k \left ( a _ 0 \xi ^ { - \gamma } \right ) + O ( \xi ^ { - \gamma - g - k } ) \xi \to + \infty . \end{align*}


\begin{align*} b , b ' [ \bar \rho - \alpha _ 0 , \bar \rho ) \alpha _ 0 \in ( 0 , \bar \rho ) , \\ | b ' | ^ \frac { 5 } { 2 } + | b | ^ \frac { 5 } { 2 } \leq c ( 1 + p ) [ 0 , \bar \rho ) c > 0 . \end{align*}


\begin{align*} \mu _ k = J _ 0 \left ( \frac { 2 \pi ( k - 1 ) } { N - 1 } W \right ) , k = 2 , { \dots } , N , \end{align*}


\begin{align*} \hat { b } ( x ) = \int _ { - \infty } ^ { \infty } K _ { h } ( y - x ) b ( y ) \mathrm { d } y . \end{align*}


\begin{align*} \mathbb { T } _ t + \mathrm { d i v } _ x ( u \mathbb { T } ) - ( \nabla _ x u \mathbb { T } + \mathbb { T } \nabla _ x ^ T u ) = \nu \Delta _ x \mathbb { T } + \frac { k A _ 0 } { 2 } \eta \mathbb { I } - \frac { A _ 0 } { 2 } \mathbb { T } . \end{align*}


\begin{align*} | 1 + r | ^ { - \omega } + | 1 - r | ^ { - \omega } = \frac { 1 } { 2 \pi i } \int _ { ( c ) } \mathcal { H } ( z , \omega ) r ^ { - z } \ , d z . \end{align*}


\begin{align*} D _ { T ^ * } T _ i ^ * = P _ i ^ { \perp } U _ i ^ * D _ { T ^ * } + P _ i U _ i ^ * D _ { T ^ * } T ^ * 1 \leq i \leq n , \end{align*}


\begin{align*} a \otimes b = \sum _ { i j } x _ { i j } y _ { i j } . \end{align*}


\begin{align*} \gamma ( X , \pi _ E , Q ( \widehat { D _ i } ) , \psi _ E ) = \prod _ { j = 0 } ^ { m _ i - 1 } \gamma ( X , \pi _ E , \widehat { \sigma _ i } \nu _ E ^ j , \psi _ E ) . \end{align*}


\begin{align*} \sum _ { q \le Q } \ , \sideset { } { ^ * } \sum _ { \chi \bmod q } \left | L \left ( \tfrac { 1 } { 2 } , \chi \right ) \right | ^ { 2 k } \sim c _ k \sum _ { q \le Q } \ , \sideset { } { ^ * } \sum _ { \chi \bmod q } \prod _ { p | q } \Bigg ( \sum _ { m = 0 } ^ { \infty } \frac { \binom { m + k - 1 } { k - 1 } ^ 2 } { p ^ m } \Bigg ) ^ { - 1 } ( \log q ) ^ { k ^ 2 } , Q \to \infty \end{align*}


\begin{align*} R _ { 1 } & = ( \underbrace { 0 , 1 , \cdots , 1 } _ { k + 2 } , \underbrace { 0 , \cdots , 0 } _ { r } , \underbrace { 0 , \cdots , 0 } _ { t } ) \\ R _ { 2 } & = ( \underbrace { 1 , 0 , \cdots , 0 } _ { k + 2 } , \underbrace { 1 , \cdots , 1 } _ { r } , \underbrace { 0 , \cdots , 0 } _ { t } ) . \end{align*}


\begin{align*} U ( t ) = U ^ p ( t ) + U ^ h ( t ) , \ \ t \geq 0 . \end{align*}


\begin{align*} \abs { A ( c x + c ) - \alpha } \ \leq \ \frac { c } { c x + c } \ = \ \frac { 1 } { x + 1 } . \end{align*}


\begin{align*} \phi ( t ) = g ( t ) \cdot \overline { \phi } _ h ( t ) . \end{align*}


\begin{align*} \Delta ( a ) = a _ { ( 1 ) } \otimes a _ { ( 2 ) } \in \widetilde { M } ( A \otimes ^ I A ) , \end{align*}


\begin{align*} H _ t ^ { \beta } : = \lim _ { \varepsilon \to 0 } \int _ 0 ^ t | X _ s | ^ { \beta } ( ( X _ s ) ) \ 1 _ { \{ | X _ s | \geq \varepsilon \} } { \rm d } s . \end{align*}


\begin{align*} \mathbb { E } [ X | X < a ] & = \alpha - \sigma h \left ( \frac { a - \alpha } { \sigma } \right ) \\ \mathbb { E } [ X | X > a ] & = \alpha + \sigma h \left ( \frac { a - \alpha } { \sigma } \right ) \end{align*}


\begin{align*} D _ { 1 , m } = \tau _ { A \smallsetminus \{ \alpha \} \cup \{ - \beta \} } ( p ^ m ) \tau _ { B \smallsetminus \{ \beta \} } ( p ^ { m } ) . \end{align*}


\begin{align*} \sqrt { b } \ , \vartheta ' _ 1 ( 0 , i b ) = \sqrt { b } \ , \vartheta _ 2 ( 0 , i b ) \vartheta _ 3 ( 0 , i b ) \vartheta _ 4 ( 0 , i b ) = \vartheta _ 4 ( 0 , i / b ) \vartheta _ 3 ( 0 , i b ) \vartheta _ 4 ( 0 , i b ) , \end{align*}


\begin{align*} z L ( z ) \to 0 \hbox { a n d } \frac { f _ 1 ' ( z ) } { f _ 1 ( z ) } = O \left ( | z | \right ) \hbox { a s $ z \to \infty $ w i t h $ \delta < \arg z < \pi - \delta $ } . \end{align*}


\begin{align*} \nabla _ x \int f _ 0 ( x - t v ) g ( v ) \dd v & = \frac { 1 } { t } \nabla _ x \frac 1 { t ^ 3 } \int f _ 0 ( w ) g ( \frac { x - w } { t } ) \dd w \\ & = \frac 1 { t ^ 4 } \int f _ 0 ( w ) \nabla g ( \frac { x - w } { t } ) \dd w = \frac 1 { t } \int f _ 0 ( x - t v ) \nabla g ( v ) \dd v . \end{align*}


\begin{align*} \mathrm { W F } ( W ) = \Gamma _ { \ell } \sqcup \Gamma _ r , \end{align*}


\begin{align*} \int _ \mathbb { R } \frac { s } { s ^ 2 + u ^ 2 } d \mu ( u ) = \int _ 0 ^ \infty \frac { s } { s ^ 2 + u ^ 2 } d \mu _ { | T | } ( u ) . \end{align*}


\begin{align*} p _ { m + 1 , j } = \tilde f _ { m , j } - f _ { m , j } , j = \overline { 1 , m } . \end{align*}


\begin{align*} \frac { T ^ 2 } { 2 } \arctan \left ( \frac { \frac { 1 } { 2 } + \epsilon } { T } \right ) - \frac { T ^ 2 } { 2 } \arctan \left ( \frac { \frac { 1 } { 2 } - \epsilon } { T } \right ) = \epsilon T + \epsilon \ \mathcal { O } \left ( \frac { 1 } { T } \right ) \end{align*}


\begin{align*} \Psi ( a , b , c , d ) & : = \sum _ { \pi \in D } \omega _ \pi ( a , b , c , d ) = \frac { ( - a , - a b c ; Q ) _ \infty } { ( a b ; Q ) _ \infty } , \\ \Phi ( a , b , c , d ) & : = \sum _ { \pi \in U } \omega _ \pi ( a , b , c , d ) = \frac { ( - a , - a b c ; Q ) _ \infty } { ( a b , a c , Q ; Q ) _ \infty } . \end{align*}


\begin{align*} \div ( \beta ( \rho ) u ) = T \end{align*}


\begin{align*} \nu _ i = \lambda _ i \ \ 1 \leq i \leq n - s . \end{align*}


\begin{align*} \Bigl ( \sum \alpha _ k \Bigr ) \circ \beta = \sum _ { n = 0 } ^ \infty \Bigl ( \sum _ { k = 1 } ^ \infty a _ { k , n } \Bigr ) \beta ^ n = \sum _ { k = 1 } ^ \infty \Bigl ( \sum _ { n = 0 } ^ \infty a _ { k , n } \beta ^ n \Bigr ) = \sum \alpha _ k ( \beta ) . \end{align*}


\begin{align*} A ( t ) = L ( t ) + C , \end{align*}


\begin{align*} \zeta ( k ) = t ^ { 1 / 2 } \sqrt { - 2 g '' ( - \eta ) } ( k + \eta ) . \end{align*}


\begin{align*} D _ { \infty } ( \xi ) : = \exp \left \{ - \frac { 1 } { 2 \pi i } \left ( \int _ { - \eta } ^ { - C _ R } + \int _ { C _ R } ^ { \eta } \right ) \frac { \log r _ { + } ( s ) } { X _ { \eta + } ( s ) ( s - k ) } d s \right \} . \end{align*}


\begin{align*} \begin{aligned} | \tilde Y _ { s , t } ( x , v ) | & \lesssim \frac { \delta ( t - s ) } { \langle d _ { t , x } \rangle ^ 2 + \langle x ^ \perp \rangle ^ 2 } , \\ | \nabla _ v \tilde Y _ { s , t } ( x , v ) | & \lesssim \frac { \delta ( t - s ) } { \langle d _ { t , x } \rangle + \langle x ^ \perp \rangle } . \end{aligned} \end{align*}


\begin{align*} \mu _ * ( B ( ( t , h ( t ) ) , r ) ) \geq \frac { \prod \limits _ { i = 1 } ^ { q } p _ { \sigma _ i } } { \prod \limits _ { i = 1 } ^ { q } C _ { \sigma _ i } } r C _ { \min } { \beta } ^ { - 1 } = C _ * \cdot r \frac { \prod \limits _ { i = 1 } ^ { q } p _ { \sigma _ i } } { \prod \limits _ { i = 1 } ^ { q } C _ { \sigma _ i } } , \end{align*}


\begin{align*} \int _ N \langle \nabla ^ * \nabla \xi , \xi \rangle = \int _ N \| \nabla \xi \| ^ 2 + \int _ { \partial N } \langle \nabla _ \nu \xi , \xi \rangle , \end{align*}


\begin{align*} \theta _ { t } ^ { c } \otimes \theta _ { t } ^ { c } = \theta ^ { c } \otimes \theta ^ { c } + \zeta _ { t } \otimes \theta ^ { c } + \theta ^ { c } \otimes \zeta _ { t } + \zeta _ { t } \otimes \zeta _ { t } . \end{align*}


\begin{align*} \chi _ 0 \in C ^ \infty , \chi _ 0 ( x ) = 1 , \forall x \in [ 0 , 1 ] , \chi _ 0 ( x ) = 0 , \forall x \ge 2 . \end{align*}


\begin{align*} T ( z ) = \mathcal { O } ( h ^ { - 2 n - 2 } ) . \end{align*}


\begin{align*} 0 = x _ { \tau _ 0 } = x _ t + \int _ t ^ { \tau _ 0 } b ( s , x _ s ) d s + B ^ H _ { \tau _ 0 } - B ^ H _ t , t \in ( \hat { \tau } _ 0 , \tau _ 0 ] . \end{align*}


\begin{align*} 0 = \sum _ { x \leq p \leq 2 x } \Psi ( z ) \Psi ( f ( z ) ) . \end{align*}


\begin{align*} \sup _ { J _ j } | \psi _ \lambda | \leq C _ 7 \left ( 2 ^ { - 4 / 3 \ell } \right ) ^ { \frac { c _ 0 } { 2 } m \sqrt { \lambda } } e ^ { \sqrt { \lambda } } = C _ 7 e ^ { ( 1 - \frac { 2 } { 3 } ( \log 2 ) { c _ 0 } m \ell ) \sqrt { \lambda } } \leq C _ 7 e ^ { ( 1 - { c _ 0 } m \ell ) \sqrt { \lambda } } . \end{align*}


\begin{align*} P _ { \gamma } ^ { \mathrm { i n } } ( M ) = P _ { \gamma } ^ { \mathrm { o u t } } : = ( 1 , . . . , \ , 1 ) \end{align*}


\begin{align*} & \exists { a ^ * } ^ { 0 ( 0 ) } \forall a ^ { X ( X ) } \Big ( \forall x ^ X ( x \in \mathrm { d o m } A \rightarrow a x \in A x ) \\ & \qquad \qquad \qquad \land \forall x ^ X ( x \not \in \mathrm { d o m } A \rightarrow \norm { a x } = 0 ) \rightarrow a ^ * \gtrsim a \Big ) . \end{align*}


\begin{align*} | W f ( x , \omega ) | = 2 ^ d | V _ { f ^ \vee } f ( 2 x , 2 \omega ) | = 2 ^ d | \widehat { f } * M _ { - 2 x } \widehat { \overline { f } } ^ \vee ( 2 \omega ) | . \end{align*}


\begin{align*} ( \mathcal { L } f ) ' = g , \lim _ { y \rightarrow + \infty } f ( y ) = 0 , \lim _ { y \rightarrow - \infty } f ( y ) = - \int g . \end{align*}


\begin{align*} \int _ { \Omega } \varepsilon \nabla \mathbf { v } _ { \varepsilon } \nabla c _ { \varepsilon } \otimes \nabla c _ { \varepsilon } \ , \mathrm { d } x = \int _ { \Omega } \Big ( \frac { 1 } { 2 } \partial _ { t } ( \mathbf { v } _ { \varepsilon } ) ^ 2 + ( \nabla \mathbf { v } _ { \varepsilon } ) ^ 2 \Big ) \mathrm { d } x . \end{align*}


\begin{align*} T P ( S ) : = \biggl \{ ( \lambda , y ) \in \R _ + ^ { | V | + | E | } \biggm | \lambda \in S , \ \sum _ { e \in E : v \in e } y _ e = \lambda _ v , \ v \in V \biggr \} . \end{align*}


\begin{align*} A _ n ( y + \sqrt { y ^ 2 - x ^ 2 } , y - \sqrt { y ^ 2 - x ^ 2 } ) = D _ n ( x ^ 2 , y ) . \end{align*}


\begin{align*} T ( u , p ) = \sum _ { \substack { 0 < a < p \\ g c d ( n , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi a ( \tau ^ { n } - u ) } { p } } . \end{align*}


\begin{align*} S ( z ) = I _ E + i \alpha ( { L ^ { \phantom { s } } \ ! \ ! \ ! } ^ { \scriptscriptstyle { - | | } } - z I ) ^ { - 1 } \alpha , z \in \rho ( { L ^ { \phantom { s } } \ ! \ ! \ ! } ^ { \scriptscriptstyle { - | | } } ) , { L ^ { \phantom { s } } \ ! \ ! \ ! } ^ { \scriptscriptstyle { - | | } } : = ( { L ^ { \phantom { s } } \ ! \ ! } ^ { \scriptscriptstyle { | | } } ) ^ * . \end{align*}


\begin{align*} ( g \pi ^ { - 1 } , h k ^ { - 1 } ) = ( \gamma , \tau ( \gamma ) ) \in ( g \Pi ^ { - 1 } \times h K ^ { - 1 } ) \cap \Gamma \subseteq ( g \Pi ^ { - 1 } \times I K ^ { - 1 } ) \cap \Gamma \end{align*}


\begin{align*} | x | = \sqrt { x \cdot x } \ . \end{align*}


\begin{align*} N _ 0 ^ i ( L ) : = K \ominus \{ N _ - ( L ^ * ) \vee N _ + ( L ^ * ) \} \subset N _ i ( L ) , \end{align*}


\begin{align*} E \left [ \left ( \frac { S _ { n + 1 } } { \sqrt { ( n + 1 ) / 2 } } \right ) ^ 2 \right ] - 1 = \frac { n - 1 } { n + 1 } \left ( E \left [ \left ( \frac { S _ { n } } { \sqrt { n / 2 } } \right ) ^ 2 \right ] - 1 \right ) , \end{align*}


\begin{align*} \begin{aligned} & \frac { | \tilde { x } - \tilde { y } | ^ 4 } { \varepsilon ^ 4 } + \alpha ( | \tilde { x } | ^ 2 + | \tilde { y } | ^ 2 ) \leq L ( | \tilde { x } - \tilde { y } | + 1 ) + M ( \tilde { t } + 1 ) \\ & - u ( x _ 1 , t _ 1 ) + v ( x _ 1 , t _ 1 ) + 2 \alpha | x _ 1 | ^ 2 + { \lambda \over T - t _ 1 } - { \lambda \over T - \tilde { t } } . \end{aligned} \end{align*}


\begin{align*} x : = B h + \alpha ( h ) - \rho ( h ) \eta : = \sigma ( h ) u . \end{align*}


\begin{align*} \big ( \mathcal { Z } _ 1 ( t ) , \epsilon ( t ) \big ) = \big ( \mathcal { Z } _ 2 ( t ) , \epsilon ( t ) \big ) = 0 , \end{align*}


\begin{align*} u ( x ) & = \frac 1 2 \gamma _ { n , s } \int _ { \R ^ n \setminus B _ r } \bigg ( \frac { r ^ 2 - \vert x \vert ^ 2 } { \vert y \vert ^ 2 - r ^ 2 } \bigg ) ^ s \bigg ( \frac 1 { \vert x - y \vert ^ n } - \frac 1 { \vert x _ \ast - y \vert ^ n } \bigg ) u ( y ) \dd y . \end{align*}


\begin{align*} d _ 0 \tilde \varphi ( s , t ) - \varphi ( s , t ) = \varphi ( a _ k , c _ 0 ) + \sum _ { i = 0 } ^ { m - 1 } \varphi ( c _ i , c _ { i + 1 } ) - \varphi ( a _ k , b _ 0 ) - \sum _ { i = 0 } ^ { l - 1 } \varphi ( b _ i , b _ { i + 1 } ) - \varphi ( s , t ) \end{align*}


\begin{align*} & \mathbb { G } _ \delta = \{ G ( z , \delta ) \mid z \in \mathbb { Q } ^ d \} , \ ; \ ; \mathbb { G } = \bigcup _ { \delta \in ( 0 , 1 ) \cap \mathbb { Q } } \mathbb { G } _ \delta , \ ; \ ; \mathcal { Q } _ = \left \{ \bigcup _ { i = 1 } ^ N Q _ i \mid Q _ i \in \mathbb { G } \right \} . \end{align*}


\begin{align*} W _ n ( x ) = \sum _ { k = 1 } ^ { \lfloor ( n + 1 ) / 2 \rfloor } W ( n , k ) x ^ { k } . \end{align*}


\begin{align*} Y _ k ( \lambda ) = \sum _ { i = 1 } ^ k \eta _ i ( \lambda ) \quad \quad \textrm { a n d } \quad \quad B _ k ( \lambda ) = \sum _ { i = 1 } ^ k b _ i ( \lambda ) \end{align*}


\begin{align*} \left \| \mathcal { U } \left ( t \right ) x _ 0 \right \| ^ 2 & \leq M _ 2 \sum ^ \infty _ { n = 1 } \bigl | e ^ { \lambda _ n t } \left < x _ 0 , z _ n \right > \bigr | ^ 2 \\ & \leq M _ 2 e ^ { - 2 \varepsilon t } \sum ^ \infty _ { n = 1 } \left | \left < x _ 0 , z _ n \right > \right | ^ 2 \leq \frac { M _ 2 } { M _ 1 } e ^ { - 2 \varepsilon t } \left \| x _ 0 \right \| ^ 2 , \end{align*}


\begin{align*} r _ { i j } = \begin{pmatrix} 0 & 0 & 0 \\ 0 & \frac { 1 } { r } \cdot \mathrm { I d } _ { k - 1 } & 0 \\ 0 & 0 & h _ { i j } \end{pmatrix} + o ( 1 ) , \end{align*}


\begin{align*} \rho ( d , k ) = \frac { 1 } { 2 } \biggl [ { \lceil \frac { d } { 2 } \rceil \choose k } + { \lfloor \frac { d } { 2 } \rfloor \choose k } \biggr ] . \end{align*}


\begin{align*} ( h \rhd s ) ^ * = S ( h ) ^ * \rhd s ^ * \end{align*}


\begin{align*} f = \norm { g } _ 2 ^ { - 2 } \iint _ { \R ^ { 2 d } } V _ g f ( x , \omega ) M _ \omega T _ x g \ , d ( x , \omega ) . \end{align*}


\begin{align*} \int _ { \R ^ N } \nabla \bar { u } \cdot \nabla v = \tilde { \lambda } _ 0 \int _ { \R ^ N } \tilde { m } _ 0 \bar { u } v \ ; . \end{align*}


\begin{align*} \partial _ t V _ t = \frac { 1 } { 2 } \Delta _ { \dot { C } _ t } V _ t - \frac { 1 } { 2 } ( \nabla V _ t ) _ { \dot { C } _ t } ^ 2 , \end{align*}


\begin{align*} \frac { 1 } { k ! } \big ( e _ { \lambda } ( t ) - 1 \big ) ^ { k } = \sum _ { n = k } ^ { \infty } S _ { 2 , \lambda } ( n , k ) \frac { t ^ { n } } { n ! } , ( k \ge 0 ) , ( \mathrm { s e e } \ [ 4 ] ) . \end{align*}


\begin{align*} \big ( D _ E | _ { S ^ + ( T N ) \otimes E } \big ) & = \left < \hat { A } ( T N ) \cdot f ^ * \mathrm { c h } ( S ^ + ( T M ) ) , [ N ] \right > \\ & = \left < - \tfrac { 1 } { 1 2 } p _ 1 ( T N ) + \tfrac 1 4 f ^ * p _ 1 ( T M ) + \tfrac 1 2 f ^ * e ( T M ) , [ N ] \right > \\ & = - \tfrac 1 4 \sigma ( N ) + ( f ) \left ( \tfrac 3 4 \sigma ( M ) + \tfrac 1 2 \chi ( M ) \right ) > 0 , \end{align*}


\begin{align*} E _ n ( u , v ) = v ^ { n + 1 } E _ n \left ( { u \over v ^ 2 } \right ) . \end{align*}


\begin{align*} \check H ^ q _ { c t } ( X , \mathcal F ) = H ^ q ( 0 \to \mathcal F _ 0 ( X ) \to \mathcal F _ 1 ( X ) \to \cdots ) , \end{align*}


\begin{align*} E _ { \psi } ( x ) = \mathcal { O } \big ( \exp \big ( - c \sqrt { \log x } \big ) \big ) , \end{align*}


\begin{align*} \mathrm { W F } ( K ) = \left \{ ( - u , u , v ; \xi , \xi , 0 ) \in \dot { T } ^ { * } ( \Sigma _ 0 \times \mathbb { M } ^ 2 ) \right \} . \end{align*}


\begin{align*} \int _ \mathcal { Q } \left ( \int _ \mathcal { Q } | Z f ( x , \omega ) | ^ 2 \ , d \omega \right ) \ , d x = \sum _ { k \in \Z ^ d } \int _ \mathcal { Q } | f ( x + k ) | ^ 2 \ , d x = \norm { f } _ 2 ^ 2 . \end{align*}


\begin{align*} D ^ * : \mathcal { H } \to \ell ^ 2 ( \Gamma ) , D ^ * f = ( \langle f , e _ \gamma \rangle ) _ { \gamma \in \Gamma } \end{align*}


\begin{align*} X = \frac { 3 \sigma { \log ( 1 / \delta ) } \pm \sqrt { ( 3 \sigma { \log ( 1 / \delta ) } ) ^ 2 + 4 ( \kappa + { L } ) \sqrt { \sum _ { t = 1 } ^ T \| g _ t \| ^ 2 } + 1 2 ( G ^ 2 + G \tilde G ) { \log ( 1 / \delta ) } } } { 2 } \end{align*}


\begin{align*} S = \begin{pmatrix} A & B \\ C & D \end{pmatrix} , \end{align*}


\begin{align*} a _ t = \frac { - R ( z _ 1 , \dots , \widehat { z _ i } , \dots , z _ { j - 1 } , t , z _ { j + 1 } , \dots z _ n ) } { Q ( z _ 1 , \dots , \widehat { z _ i } , \dots , z _ { j - 1 } , t , z _ { j + 1 } , \dots , z _ n ) } \end{align*}


\begin{align*} \varphi ^ { ( 1 ) } ( x ) : = \zeta ( x ) - \zeta \big ( Q _ \rho ( x ) \big ) x \in \R ^ n . \end{align*}


\begin{align*} P _ { n } ^ t \times P _ { n } ^ t = \bigcup _ { \xi \in G _ { 2 t + r } } \tilde { V } _ t ( ( \xi \times 1 _ { 2 m - 2 t } ) \times 1 _ { 2 n } ) = \bigcup _ { \xi \in G _ { 2 t + r } } \tilde { V } _ t ( 1 _ { 2 n } \times ( \xi \times 1 _ { 2 m - 2 t } ) ) , \end{align*}


\begin{align*} \langle \phi , \psi \rangle = \sum _ { g \in G } \phi ( g ) \psi ( g ) \end{align*}


\begin{align*} M _ 0 ( x ) = x ^ { - 1 } \mbox { a n d } W _ 0 ( x ) = 1 . \end{align*}


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


\begin{align*} C ( \rho ) : = \frac { 1 - ( 1 + \delta ) ^ { \rho } } { \rho } . \end{align*}


\begin{align*} \int _ M | \nabla _ g u ( x ) | ^ 2 { \rm d } v _ g - \mu \int _ M \frac { u ^ 2 ( x ) } { d ^ 2 _ g ( x _ 0 , x ) } { \rm d } v _ g + \int _ M u ^ 2 ( x ) { \rm d } v _ g \leq | \lambda | C _ 0 \| \alpha \| _ { L ^ \infty } \int _ M u ^ 2 ( x ) { \rm d } v _ g . \end{align*}


\begin{align*} L = \begin{pmatrix} \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ \ddots & m _ 1 & m _ 0 & m _ { - 1 } & m _ { - 2 } & m _ { - 3 } & \ddots \\ \ddots & m _ 2 & m _ 1 & m _ 0 & m _ { - 1 } & m _ { - 2 } & \ddots \\ \ddots & m _ 3 & m _ 2 & m _ 1 & m _ 0 & m _ { - 1 } & \ddots \\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \end{pmatrix} \end{align*}


\begin{align*} - \left ( F _ h ( x ) [ \dot x ] \right ) [ \lambda ( x ) - 2 \beta h ( x ) ] = - \sum _ { i = 1 } ^ m ( \lambda _ i ( x ) - 2 \beta h _ i ( x ) ) \nabla ^ 2 h _ i ( x ) [ \dot x ] , \end{align*}


\begin{align*} P _ R = \{ z \in \C ^ d \mid | z _ k | \leq R , \ , k = 1 , \ldots d \} : \end{align*}


\begin{align*} a _ j = 0 \ \textrm { i f } \ j \ne q ( m + 2 ) \ \textrm { a n d } \ j \ne q ( m + 2 ) + 1 \ \textrm { w i t h } \ q \in \Z \ , . \end{align*}


\begin{align*} { d \over d b } g _ b ( x ) ~ = ~ { 2 \eta ( x ) \over \pi } [ \ln ( x + b ) - \ln ( b ) ] \forall x > 0 . \end{align*}


\begin{align*} \mu _ i \coloneqq \frac { \nu ( \eta _ i ) } { \sum _ { j = 1 } ^ K \nu ( \eta _ j ) } , i = 1 , \dots , K . \end{align*}


\begin{align*} T ^ { \rm P r o a c } _ j = \tau ^ { \rm P r o a c } _ { m } R _ j + \tau ^ { \rm P r o a c } _ { V _ j } + \tau _ { \rm f } . \end{align*}


\begin{align*} k _ 0 : = \max k \left [ \frac { k } { 2 ^ { n + 1 } } \leq r \right ] . \end{align*}


\begin{align*} \rho ' _ \kappa ( x , \omega , \tau ) = \rho _ \kappa ( \iota ^ { - 1 } ( x , \omega , \tau ) ) = e ^ { 2 \pi i \kappa \tau } T _ { \kappa x } M _ \omega , \end{align*}


\begin{align*} x \in \N , \ \sum _ { t = 0 } ^ { T - 1 } x _ t \cdot \Delta s _ { t + 1 } \ge 0 \ a . s . \implies \sum _ { t = 0 } ^ { T - 1 } x _ t \cdot \Delta s _ { t + 1 } = 0 \ a . s . ; \end{align*}


\begin{align*} \begin{aligned} | f ( x ) - f ( x ' ) | & \leqslant C \Big ( \frac { \| x - x ' \| } { r } \Big ) ^ \beta \Big \{ \frac { 1 } { V ( x , x _ 0 , r + d ( x , x _ 0 ) ) } \Big ( \frac { r } { r + \boldsymbol { d ( x , x _ 0 ) } } \Big ) ^ \gamma \\ & \qquad + \frac { 1 } { V ( x ' , x _ 0 , r + d ( x ' , x _ 0 ) ) } \Big ( \frac { r } { r + \boldsymbol { d ( x ' , x _ 0 ) } } \Big ) ^ \gamma \Big \} ; \end{aligned} \end{align*}


\begin{align*} x _ t = x _ 0 + \int ^ t _ 0 b ( s , x _ s ) d s + \sigma B ^ H _ t , 0 \le t \le 1 , H \in ( 0 , \frac { 1 } { 2 } ) , \sigma > 0 . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ d } U _ i ( \tau _ x \omega ) \partial _ j \eta ( x ) d x = \int _ { \mathbb { R } ^ d } U _ j ( \tau _ x \omega ) \partial _ i \eta ( x ) d x \end{align*}


\begin{align*} \tilde { f } ' ( s , y ) \le f _ 1 ( s , y ) \le K \le s ^ { 2 H - 1 } K = K _ s \end{align*}


\begin{align*} E ( x ) \ = \ \frac { 1 } { f ( x ) + 1 } . \end{align*}


\begin{align*} \mathbf { r } _ { ( \alpha ) } ( \tau ) = \tau ^ { - 1 } \cdot \mathbf { s } \cdot \mathbf { s } ^ { \mathtt { T } } + \mathbf { r } _ { 0 } ( 0 ) _ { \kappa \left ( \mathcal { I } \setminus \{ \alpha \} \right ) } ^ { \kappa \left ( \mathcal { I } \setminus \{ \alpha \} \right ) } + ( \mathbf { 1 } \cdot \mathbf { s } ^ { \mathtt { T } } - \mathbf { s } \cdot \mathbf { 1 } ^ { \mathtt { T } } ) . \end{align*}


\begin{align*} J _ 2 : = \sum _ { \substack { \alpha \in A \\ \beta \in B } } \frac { 1 } { 4 \pi i } \int _ { ( \epsilon ) } \underset { s _ 2 = - s _ 1 - \alpha - \beta } { } \ \mathcal { J } \ , d s _ 1 . \end{align*}


\begin{align*} ( - \Delta ) ^ s u _ \varepsilon + k u _ \varepsilon & = f _ \varepsilon B _ { 7 / 8 } . \end{align*}


\begin{align*} A = \begin{pmatrix} 9 & 6 & \overline { 4 } & 2 & 1 \\ \overline { 1 3 } & 8 & \overline { 7 } & 3 & \overline { 1 } \\ 1 1 & \overline { 7 } & 5 & \overline { 2 } & 0 \\ \end{pmatrix} . \end{align*}


\begin{align*} \mathbf { E } _ l ^ { \ast } ( g , s ) = j ( k _ { \infty } , z _ 0 ) ^ { - l } \sum _ { h } c ( h , q , s ) e _ { \mathbb { A } } ( \lambda ( h \sigma ) ) , \end{align*}


\begin{align*} F ^ { i j } \nabla _ { i } \log u \nabla _ { j } P = \lambda F ^ { i j } r _ { j } \nabla _ { i } \log u - u ^ { \frac { \alpha + 1 } { \alpha } } F ^ { i j } \nabla _ { i } \log u \nabla _ { j } \log u . \end{align*}


\begin{align*} \zeta ^ { \bar { s } } = \xi ^ { b _ { k } } \circ \eta ^ { a _ { k } } \circ \dots \circ \xi ^ { b _ { 1 } } \circ \eta ^ { a _ { 1 } } . \end{align*}


\begin{align*} g _ { \alpha } ^ { ( \beta ) } ( o ) = \int _ D g _ { \alpha } \overline { { g } _ { \beta } } e ^ { - \varphi } = \overline { \int _ D g _ { \beta } \overline { { g } _ { \alpha } } e ^ { - \varphi } } = 0 . \end{align*}


\begin{align*} \int \psi ( x ) e ^ { - i x \cdot \rho \eta _ 0 / h \Tilde { h } } \widetilde { R } _ h ( z ) ^ * ( \chi ( x ) e ^ { i x \cdot \rho \xi _ 0 / h \Tilde { h } } ) d x = \mathcal { O } ( h ^ \infty \Tilde { h } ^ \infty ) , \end{align*}


\begin{align*} A ^ { 0 } ( U ) V _ { t } + A ^ { i } ( U ) \partial _ { i } V + D ( U ) V = 0 \end{align*}


\begin{align*} \mathcal { Q } ^ \bullet _ { M _ c } = \bigoplus _ { p = 0 } ^ { n - 1 } \biguplus _ { q = 0 } ^ \infty \biguplus _ { r = 0 } ^ \infty \mathcal { Q } ^ { p q r } _ { M _ c } \end{align*}


\begin{align*} \Bigl ( ( \theta ^ { q ^ N } - t ) ^ { c ( N - n _ 1 ) q ^ { n _ 1 + d - N } } \bigl ( ~ _ r \mathcal { F } _ s ( \alpha ) \bigr ) ^ { q ^ d } \Bigr ) | _ { t = \theta ^ { q ^ N } } = \sum _ { i = 1 } ^ { l } \prod _ { \substack { m = 1 \\ m \neq N } } ^ { n _ i + d - 1 } ( \theta ^ { q ^ m } - \theta ^ { q ^ N } ) ^ { c ( m - n _ i ) q ^ { n _ i + d - m } } \alpha ^ { q ^ { n _ i + d } } \end{align*}


\begin{align*} \theta _ \mu ( t + 1 ) = \theta _ \mu ( t ) - \eta \partial _ \mu \mathcal L ( \theta ( t ) ) , \mu \in [ \# ] \end{align*}


\begin{align*} \zeta _ k ( s ) = \prod _ { \pi } L ( \C [ G ] ( \pi ) , s ) = \prod _ { \pi } L ( V _ { \pi } , s ) ^ { \dim V _ { \pi } } \end{align*}


\begin{align*} h ' ( t ) = \frac { A _ 1 \exp ( - C u ( t ) ) } { \log ^ \alpha t } \left ( \frac { \log t - \alpha } { \log t } - C t u ' ( t ) \right ) \ge A _ 1 \log ^ { B - 2 } t \exp ( - C u ( t ) ) \end{align*}


\begin{align*} P ^ N \tilde { \tau } _ t ( G _ { n } \times G _ { n } ) = \bigcup _ { \xi , \beta , \gamma } P ^ N \tilde { \tau } _ t ( ( \xi \times 1 _ { 2 m - 2 t } ) \beta \times \gamma ) = \bigcup _ { \xi , \beta , \gamma } P ^ N \tilde { \tau } _ t ( \beta \times ( \xi \times 1 _ { 2 m - 2 t } ) \gamma ) . \end{align*}


\begin{align*} P _ 1 = \frac { x _ 1 A } { F _ { k , 1 } ( A ) } \end{align*}


\begin{align*} x = ( x _ 1 , \ \ldots , \ x _ d ) . \end{align*}


\begin{align*} \min _ \R Q = Q ( 0 ) = 0 . \end{align*}


\begin{align*} C ^ k ( A ) \colon = \mathrm { H o m } _ { \mathbb { C } } \big ( A ^ { \otimes ( k + 1 ) } , \mathbb { C } \big ) \end{align*}


\begin{align*} \mathcal { L } f ( g ) : = - \mathcal { E } ( f , g ) , g \in \mathcal { D } ( \mathcal { E } ) . \end{align*}


\begin{align*} \tau = ( 2 \beta _ \infty ) ^ { - 1 } \tilde \tau ( 1 + O ( I ^ \eta ( \tilde \tau ) ) ) , \tilde \tau \to + \infty . \end{align*}


\begin{align*} b _ i = { k + i \choose k } _ q \left ( N _ k ^ i ( V , V ' ) - \frac { 1 } { { k + i \choose k } _ q } I ( V , V ' ) \right ) , \end{align*}


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


\begin{align*} f ( \eta ) = \sum _ { k = 0 } ^ { \infty } \frac { \left . f ^ { k } ( \eta ) \right | _ { \eta = \eta _ i } } { k ! } ( \eta - \eta _ i ) ^ k = \sum _ { k = 0 } ^ { \infty } F ( k ) ( \eta - \eta _ i ) ^ k \end{align*}


\begin{align*} \sqrt { \sum _ { i = 1 } ^ { n } a _ i } \leq \sum _ { i = 1 } ^ { n } \frac { a _ i } { \sqrt { \sum _ { k = 1 } ^ { i } a _ k } } \leq 2 \sqrt { \sum _ { i = 1 } ^ { n } a _ i } \end{align*}


\begin{align*} \epsilon = \frac { 1 } { z _ 6 } \| e ^ { - 2 z _ 6 \cdot } - e ^ { - z _ 7 \cdot } \| . \end{align*}


\begin{align*} \langle ( R \# T ) ( x \wedge y ) , u \wedge v \rangle = \langle ( T \# R ) ( u \wedge v ) , x \wedge y \rangle = \langle ( R \# T ) ( u \wedge v ) , x \wedge y \rangle . \end{align*}


\begin{align*} \begin{aligned} U ^ 0 ( x ) \preceq \cdots \preceq U ^ n ( x ) = F ( x ) & V ^ 0 ( x ) \preceq \cdots \preceq V ^ n ( x ) = G ( x ) , \\ U ^ 0 ( x ) = A ^ 0 ( x ) \preceq \cdots \preceq A ^ n ( x ) & V ^ 0 ( x ) = B ^ 0 ( x ) \preceq \cdots \preceq B ^ n ( x ) , \\ U ^ i ( x ) \preceq A ^ i ( x ) & U ^ i ( x ) \preceq B ^ i ( x ) i = 0 , \ldots , n . \end{aligned} \end{align*}


\begin{align*} \partial _ { J } \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 } \# ( \mathcal { M } _ { 1 } ^ { J } ( \alpha , \beta ) / \mathbb { R } ) \cdot \langle \beta \rangle . \end{align*}


\begin{align*} \| u ( \cdot , t ) - \chi ( \cdot , t ) \| _ { L ^ { \infty } } \sim \begin{cases} ( 1 + t ) ^ { - \frac { \alpha } { 2 } } , & 1 < \alpha < 2 , \\ ( 1 + t ) ^ { - 1 } \log ( 1 + t ) , & \alpha \ge 2 \end{cases} \end{align*}


\begin{align*} \begin{aligned} & ( h + 2 \beta _ a \epsilon ( a ) ) \phi _ 0 + ( h - 2 \beta _ a \epsilon ( a ) ) \phi _ 1 - 2 h \phi ^ b ( a ) = 0 , \\ & - d _ h ( \epsilon D _ h \phi ) _ j - f _ j - \sum _ { i = 1 } ^ s z _ i \rho _ { i j } = 0 , j = 1 , \cdots , N , \\ & ( h - 2 \beta _ b \epsilon ( b ) ) \phi _ N + ( h + 2 \beta _ b \epsilon ( b ) ) \phi _ { N + 1 } - 2 h \phi ^ b ( b ) = 0 . \end{aligned} \end{align*}


\begin{align*} \log \left ( \frac { \sigma + i T } { 2 } \right ) = \frac { 1 } { 2 } \log \left ( \frac { \sigma ^ 2 + T ^ 2 } { 4 } \right ) + i \arg \left ( \frac { \sigma + i T } { 2 } \right ) \end{align*}


\begin{align*} C : = \phi _ { \delta } ^ { x _ 0 } ( T ) \eta _ 0 : = \frac { \delta } { 4 C } . \end{align*}


\begin{align*} \vartheta ( x ) = \begin{cases} \eta ( x ) , & \ \ x \in ( 0 , 1 ) , \\ 0 , & \ \ x \in \mathbb { R } \setminus ( 0 , 1 ) . \end{cases} \end{align*}


\begin{align*} \rho ( d , k ) = \frac { 1 } { 2 } \biggl [ { \lceil \frac { d } { 2 } \rceil \choose k } + { \lfloor \frac { d } { 2 } \rfloor \choose k } \biggr ] . \end{align*}


\begin{align*} T _ h ( \lambda ) ( u ) ( { x } ) = \sum _ { k = 1 } ^ { k = n } \frac { \eta _ 0 ^ k } { 4 \pi } \int _ { B _ k } \frac { \exp ( { i \sqrt { \lambda } h | x - y | ) } } { | x - y | } u ( y ) d y \end{align*}


\begin{align*} F _ 4 ( z ) = H _ n \left ( 1 - \frac 2 { z } \right ) , \end{align*}


\begin{align*} | A f ( x , \omega ) | = | \langle f , M _ \omega T _ x f \rangle | \leq \norm { f } _ 2 ^ 2 \end{align*}


\begin{align*} f _ r = \frac { \omega _ r } { 2 \pi } , \omega _ r = \sqrt { g k _ { r } \tanh \left ( k _ { r } h \right ) } , k _ r = \frac { 2 \pi } { L _ r } , L _ r = \alpha ~ \Delta x _ { \max } , \end{align*}


\begin{align*} A ( x , n ) \ = \ A ( n ) \end{align*}


\begin{align*} _ { s + 1 } \mathcal { F } _ s ( 1 , \ldots , 1 ; 2 , \ldots , 2 ) ( \alpha ) ^ q = \mathcal { L i } _ { K , s } ( \alpha ) ( \alpha \in \overline { k } \ \ | \alpha | _ { \infty } < q ^ s ) . \end{align*}


\begin{align*} \Psi ^ i _ { \Sigma _ 0 } ( \Lambda s ) = \Lambda ^ { \mu - 1 } \mathfrak { A } _ { \ell } m _ { \Lambda } ( \Psi ^ i _ { \Sigma _ 0 } ( s ) ) , \\ \Psi ^ i _ { \Sigma _ 0 } ( s + c ) = \mathfrak { A } _ { \ell } t _ c ( \Psi ^ i _ { \Sigma _ 0 } ( s ) ) . \end{align*}


\begin{align*} E ( x ) = \alpha + \int _ 0 ^ x \left ( n ( y ) - b ( y ) \right ) d y . \end{align*}


\begin{align*} D ^ { \alpha - 1 } ( p _ \alpha ) ( x ) = - x p _ \alpha ( x ) . \end{align*}


\begin{align*} \tilde { \lambda } _ { n } = \inf _ { B ^ 1 ( \mathbf { x } ) \subset \tilde { \Omega } _ n } \lambda ^ 1 ( B ^ 1 ( \mathbf { x } ) , \tilde { \Omega } _ n ) \ ; , \end{align*}


\begin{align*} \partial _ t h + \mathcal { L } h = i R ( h ) , \mathcal { L } = \left ( \begin{matrix} 0 & - L _ - \\ L _ + & 0 \end{matrix} \right ) , \end{align*}


\begin{align*} \Psi _ \tau = \Phi _ 0 + \lambda ( \tau ) ( \Phi _ 1 - \Phi _ 0 ) , \end{align*}


\begin{align*} \partial _ i \Phi ^ u ( x ) & = ( \partial _ i + u _ i ( x ) \partial _ z ) | _ { ( x , u ( x ) ) } , \ \ \ \forall 1 \leq i \leq n \ , ; \\ \nu ^ { u , f } ( x , u ( x ) ) & = \frac { 1 } { \sqrt { ( 1 + f ) ( 1 + g _ u ^ { i j } u _ i u _ j ) } } \cdot ( \partial _ z - g _ u ^ i j u _ i ( x ) \partial _ j ) \ , . \end{align*}


\begin{align*} \alpha & = \beta \gamma \Bigl ( 1 + 8 \sum _ { k = 1 } ^ \infty \Bigl ( \frac { 2 k X ^ { 2 k } } { 1 + X ^ { 2 k } } - \frac { ( 2 k - 1 ) X ^ { 2 k - 1 } } { 1 + X ^ { 2 k - 1 } } \Bigr ) \Bigr ) . \end{align*}


\begin{align*} C ^ \infty ( \Omega ) = \bigcap _ { k = 0 } ^ \infty C ^ k ( \Omega ) . \end{align*}


\begin{align*} G _ \tau ( \rho , \phi ) = \frac { 1 } { 2 \tau } d ^ 2 ( \rho ^ * , \rho ) + E ( \rho , \phi ) , ( \rho , \phi ) \in \mathcal { A } . \end{align*}


\begin{align*} D : \ell ^ 2 ( \Gamma ) \to \mathcal { H } , D c = \sum _ { \gamma \in \Gamma } c _ \gamma e _ \gamma \in \mathcal { H } . \end{align*}


\begin{align*} X e ^ { - t } - \frac { e ^ { - \alpha t } Y } { ( 1 - e ^ { - \alpha t } ) ^ { 1 - \frac { 1 } { \alpha } } } = _ { \mathcal { L } } \left ( e ^ { - \alpha t } \dfrac { ( 1 - e ^ { - \alpha t } ) ^ { \alpha - 1 } + e ^ { - \alpha t } } { ( 1 - e ^ { - \alpha t } ) ^ { \alpha - 1 } } \right ) ^ { \frac { 1 } { \alpha } } X . \end{align*}


\begin{align*} { \bf S } _ \ell ( Q + i p ) \sum _ { j = 0 } ^ { \ell } F _ j : = \sum _ { j = 0 } ^ { \ell } F _ j ^ + ( Q + i p ) \end{align*}


\begin{align*} E ^ { [ 1 ] } ( f , G \rtimes S ) ( t , \overline { t } ) = ( - 1 ) ^ n E ^ { [ 1 ] } ( \widetilde { f } , \widetilde { G } \rtimes S ) ( t ^ { - 1 } , \overline { t } ) \ , . \end{align*}


\begin{align*} a _ h ( u ^ { \cal N \ ! N } , v _ h ) = F _ h ( v _ h ) \forall v _ h \in V _ h \ , . \end{align*}


\begin{align*} \vert \norm { a } - \norm { T ^ \circ p } \vert & \leq \vert \norm { a } - \norm { T ^ \circ q } \vert + \vert \norm { T ^ \circ p } - \norm { T ^ \circ q } \vert \\ & \leq \frac { 1 } { \delta ( k ) + 1 } + \norm { T ^ \circ p - T ^ \circ q } \\ & \leq \frac { 1 } { 2 ( k + 1 ) } + \frac { 1 } { 2 ( k + 1 ) } \\ & = \frac { 1 } { k + 1 } \end{align*}


\begin{align*} p _ d & \triangleq N ( p ' _ 0 + p _ 0 ) \\ D & = p _ d + \frac { N } { N - 1 } ( 1 - p _ d ) , \end{align*}


\begin{align*} \rho \mu ( u ) = \rho \mu ( v _ i \# u ) = \omega ( v _ i \# u ) = \omega ( v _ i ) + \omega ( u ) . \end{align*}


\begin{align*} \partial _ { t } ^ 2 u - \partial _ x ^ 2 u + u + V ( x ) u \pm u ^ 3 = 0 , \ , u ( 0 ) = u _ 0 , \ , \partial _ t u ( 0 ) = u _ 1 . \end{align*}


\begin{align*} \begin{aligned} & \mathbb { E } T ^ { \rm P r o a c } _ j = \tau ^ { \rm P r o a c } _ { m } \mathbb { E } R _ j + \mathbb { E } \tau ^ { \rm P r o a c } _ { V _ j } + \tau _ { \rm f } , \\ & \mathbb { E } \left ( T ^ { \rm P r o a c } _ j \right ) ^ 2 = \mathbb { E } \left ( \tau ^ { \rm P r o a c } _ { m } R _ j + \tau ^ { \rm P r o a c } _ { V _ j } + \tau _ { \rm f } \right ) ^ 2 . \\ \end{aligned} \end{align*}


\begin{align*} \norm { \psi } _ { { H } ^ { - s } _ { { p e r } } ( I ) } = \left ( \sum _ { m \in \mathbb Z } \left ( 1 + \frac { 4 \pi ^ 2 m ^ 2 } { \mod { I } ^ 2 } \right ) ^ { - s } \mod { c _ m } ^ 2 \right ) ^ { \frac { 1 } { 2 } } , \end{align*}


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


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


\begin{align*} \Sigma _ X = \Phi \Delta \Phi ' + \Phi _ 0 \Delta _ 0 \Phi _ 0 ' . \end{align*}


\begin{align*} s _ { i j } : = \min \bigl \{ f _ i ( x ) , m _ { i j t _ i ( H ) } \bigr \} i \in \{ 1 , \ldots , d \} \ ; j \in \{ 0 , \ldots , n \} . \end{align*}


\begin{align*} \begin{aligned} \lambda _ i \in \Lambda _ i , \ \delta _ { i } \in \{ 0 , 1 \} ^ { \lceil \log _ 2 ( n ) \rceil } , \ & \sum _ { j \notin L _ { i k } } \lambda _ { i j } \leq \delta _ { i k } \leq 1 - \sum _ { j \notin R _ { i k } } \lambda _ { i j } , \ \\ & k = 1 , \ldots , \lceil \log _ 2 ( n ) \rceil , \end{aligned} \end{align*}


\begin{align*} \sum _ { s , t = 1 } ^ { Q } \mu ( L _ { B , s } \cap L _ { B , t } ) \leq \frac { C } { \mu ( B ) } \left ( \sum _ { n = 1 } ^ { Q } \mu ( L _ { B , n } ) \right ) ^ 2 . \end{align*}


\begin{align*} \widehat { V } _ Q f ( x ) = \mu ( V _ Q ) f ( x ) = e ^ { \pi i Q x ^ 2 } f ( x ) , Q = Q ^ T . \end{align*}


\begin{align*} \prod _ { \mu = 0 } ^ \infty c ^ { \chi ^ { - \mu } } \chi ^ { \mu \chi ^ { - \mu } } \leq c ( d , \chi ) , \end{align*}


\begin{align*} \mathrm { e x } ( n , F ) = \left ( 1 - \frac { 1 } { \chi ( F ) - 1 } + o ( 1 ) \right ) \frac { n ^ 2 } { 2 } , \end{align*}


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


\begin{align*} ( d _ q v ( \cdot , t ) , d _ q h ) _ { L ^ 2 ( { \mathbb R } ^ n , \varLambda ^ { q + 1 } ) } = ( v ( \cdot , t ) , ( d _ q ^ * d _ q h ) _ { L ^ 2 ( { \mathbb R } ^ n , \varLambda ^ { q + 1 } ) } = 0 \end{align*}


\begin{align*} J _ q ( n , k , t ) = \frac { 1 } { q ^ { ( k - t ) ^ 2 } { k \choose t } _ q } \sum \limits _ { i = 0 } ^ { k - t } ( - 1 ) ^ { k - t - i } q ^ { { k - t - i \choose 2 } } { k - t \choose i } _ q { k + i \choose i } _ q N _ k ^ { i } . \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*} \hat { G } ( t , \xi ) = \frac { \hat { \phi } ( \xi ) } { t } \zeta _ \xi ( t | \xi | ) , \zeta _ \xi ( p ) = | p | \hat { \psi } _ \xi ( | p | ) . \end{align*}


\begin{align*} \int _ j : = \frac { 1 } { 2 \pi i } \int _ { C _ j } L ( z , \chi ) L ( 2 z , \chi ^ { 2 } ) G ( z , \chi ) F _ { u } ( z , \chi ) \frac { X ^ { z } } { z } d z . \end{align*}


\begin{align*} u \rhd _ { n } v = v ^ { - n } u v ^ n \in G . \end{align*}


\begin{align*} \mu _ \omega ( 2 B _ \rho ) \le \mu _ \omega ( B _ \rho ( \infty , 8 r ) ) = \mu ( Z ) \ , C _ { \beta , a } \ r ^ { ( 2 \beta - 1 - a ) / ( \beta - 1 ) } . \end{align*}


\begin{align*} \nabla _ { v } Y _ { t } ^ { x ; l ^ { \epsilon } } = J _ { l _ { 0 } ^ { \epsilon } , t } ^ { x ; l ^ { \epsilon } } v , \end{align*}


\begin{align*} \sup _ { 1 \leq m \leq M } \| X _ m ^ { M , N } \| _ { L ^ p ( \Omega ; H ^ \kappa ) } < \infty , \ ; \kappa = \min \big \{ \gamma , \tfrac d 2 + \tfrac 1 4 \big \} . \end{align*}


\begin{align*} \psi _ { 2 } ( \alpha ; \beta ) : = \Psi \left ( h ( \mathcal { J } ) ^ { - 1 } \cdot h ( \mathcal { J } _ { \beta } ^ { \alpha } ) \right ) \end{align*}


\begin{align*} Z : = \left \{ v \in H _ 0 ^ 1 ( \Omega ) \colon \psi _ 1 \leq v \leq \psi _ 2 \Omega \right \} \end{align*}


\begin{align*} Z f ( x , \omega + l ) = Z f ( x , \omega ) \end{align*}


\begin{align*} \int \chi ( x ) e ^ { - i x \cdot \rho \xi _ 0 / h \Tilde { h } } \widetilde { R } _ h ( z ) ( \psi ( x ) e ^ { i x \cdot \rho \eta _ 0 / h \Tilde { h } } ) d x = \mathcal { O } ( h ^ \infty \Tilde { h } ^ \infty ) \end{align*}


\begin{align*} \sum _ { n = 0 } \alpha _ n | f _ n | ^ 2 \leq C \| f \| ^ 2 _ \omega \end{align*}


\begin{align*} l _ { \gamma ^ { \epsilon } _ { t } } ^ { \epsilon } = t , t \geq l _ { 0 } ^ { \epsilon } \quad \gamma _ { l _ { t } ^ { \epsilon } } ^ { \epsilon } = t , t \geq 0 . \end{align*}


\begin{align*} \log \log t = \log B _ 0 + \frac { 3 } { 5 } \log \log x - \frac { 1 } { 5 } \log \log \log x \le \frac { 3 } { 5 } \log \log x . \end{align*}


\begin{align*} & \partial _ y ( \mathcal { L } \mathfrak { B } _ { i j k } ) = - b _ { i j k } \sigma _ i \Lambda Q + G _ { i j k } , \end{align*}


\begin{align*} \frac { \partial } { \partial t } \lambda = \frac { 1 - q } { e ^ { q \lambda } - q e ^ { \lambda } } . \end{align*}


\begin{align*} \beta _ { n , \lambda } = \sum _ { k = 0 } ^ { n } \frac { \lambda ^ { k } ( 1 ) _ { k + 1 , \frac { 1 } { \lambda } } } { k + 1 } S _ { 2 , \lambda } ( n , k ) , ( n \ge 0 ) . \end{align*}


\begin{align*} \Theta ( z ) = I _ E + i J \alpha ( L ^ * - z I ) ^ { - 1 } \alpha : E \to E , z \in \rho ( L ^ * ) . \end{align*}


\begin{align*} \frac { 1 } { 1 - x } F _ { n , \lambda } \bigg ( \frac { x } { 1 - x } \bigg ) = \sum _ { k = 0 } ^ { \infty } x ^ { k } ( k ) _ { n , \lambda } , ( n \ge 0 ) . \end{align*}


\begin{align*} \Box \phi = f , \ ; \phi [ 0 ] = ( 0 , 0 ) , \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d \tau } \| \varepsilon _ - \| _ { L ^ 2 _ { \rho } } ^ 2 = ( \partial _ \tau \varepsilon _ - , \varepsilon _ - ) \end{align*}


\begin{align*} e ^ { 2 \pi i \tau } \rho ( \l ) e ^ { 2 \pi i \tau ' } \rho ( \l ' ) = e ^ { 2 \pi i ( \tau + \tau ' ) } e ^ { - \pi i \sigma ( \l , \l ' ) } \rho ( \l + \l ' ) . \end{align*}


\begin{align*} E ^ \alpha ( \rho ) = \frac 1 2 \int _ \R ( K _ \alpha * \rho ) \rho + \int _ \R Q _ \alpha \rho . \end{align*}