\begin{align*} { \lim \sup } _ { n \to \infty } \frac { \max _ { 0 \leq k \leq n } \ , S _ k } { ( n \log \log n ) ^ { 1 / 2 } } = 2 ^ { 1 / 2 } \sigma \quad , \end{align*}


\begin{align*} \partial _ x u + \partial _ y v = - \frac { \partial _ t p + ( u \partial _ x + v \partial _ y ) p } { \gamma p } = \frac { h ( \partial _ t + u \partial _ x + v \partial _ y ) \tilde { h } } { \gamma ( \frac 3 2 - \frac 1 2 h ^ 2 ) } . \end{align*}


\begin{align*} c _ n = \frac { \hat { \nu } ^ c _ p ( \psi _ n ^ { - 1 } ( K ) ) } { \hat { \nu } ^ c _ p ( K ) } , \end{align*}


\begin{align*} \Delta _ w : = \{ \underline { \lambda } _ { i , i + 1 } : w ( i + 1 ) < w ( i ) \} . \end{align*}


\begin{align*} \langle H _ y \rangle _ { u _ { \alpha } \otimes \Phi _ y ^ { ( 0 ) } } = \Big ( e _ { \alpha } + \alpha \| \lambda _ y \| ^ 2 - \frac { \alpha ^ 2 } { L ^ 3 } + O \Big ( \frac { \alpha ^ 2 } { L ^ 5 } \Big ) + O \big ( \alpha ^ 2 L e ^ { - L / 2 } \big ) \Big ) | \Phi _ y ^ { ( 0 ) } | ^ 2 , \end{align*}


\begin{align*} ( \mathcal H _ { \mu , n } ) : = \{ \varphi \in \mathcal O ( { \mathbb D } ) : \varphi f \in \mathcal H _ { \mu , n } \ , \ , \ , f \in \mathcal H _ { \mu , n } \} . \end{align*}


\begin{align*} ( f \otimes f ) \circ \Delta _ C = \Delta _ { C ' } \circ f , \epsilon _ { C ' } \circ f = \epsilon _ C . \end{align*}


\begin{align*} Z _ k & = f ^ { ( k ) } \left ( ( \phi _ k ( Y _ j ) ) _ { j \in \tilde { J } _ k } \right ) \\ & = \alpha ^ { ( k ) } \left ( ( Y _ j ) _ { j \in \tilde { J } _ k } \right ) \end{align*}


\begin{align*} \frac { d R ( g _ \tau , T , \hat y , \hat \eta _ \tau ) } { d \tau } \Big | _ { \tau = 0 } = R _ 1 + R _ 2 , \end{align*}


\begin{align*} \lambda _ a = \sum _ { \beta \in \check { \Phi } ^ { + } } m _ { \beta } \beta = \sum _ { 1 \leq i < j \leq n } m _ { i , j } \check { \alpha } _ { i , j } , \end{align*}


\begin{align*} \eta _ a ( u _ w ) = \theta ( v _ w ) = \frac { \theta ( E _ w ) a ^ { 1 - w } } { 1 - 2 w } = \frac { \zeta _ k ( w ) a ^ { 1 - w } } { \zeta ( 2 w ) ( 1 - 2 w ) } . \end{align*}


\begin{align*} a \circ b = a + ( - 1 ) ^ a b = \begin{cases} a + b , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; e v e n , } \\ a - b , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; o d d . } \end{cases} \end{align*}


\begin{gather*} f ( s ( x ) ) = i ( x ) f ( s ' ( v ) ) = i ' ( v ) . \end{gather*}


\begin{align*} \partial _ t \rho _ { i , m + 1 } - \nabla \cdot ( \rho _ { i , m + 1 } \nabla p _ { m + 1 } ) = \rho _ { i , m + 1 } G _ i ( p _ { m + 1 } , n _ m ) \end{align*}


\begin{align*} v ( y ) = \int _ { \partial \Omega } \ ( v ( x ) \ , \dfrac { \partial \Gamma } { \partial \nu } ( x - y ) - \Gamma ( x - y ) \ , \dfrac { \partial v } { \partial \nu } ( x ) \ ) \ , d \sigma ( x ) + \int _ \Omega \Gamma ( x - y ) \ , \Delta v ( x ) \ , d x \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { s _ 3 ( x ) } { \pi ( x ) } = \frac { 1 } { p - 1 } . \end{align*}


\begin{align*} \Omega _ i : = \{ u _ i \geq 0 \} \cap [ \theta _ 1 + \delta , \theta _ 3 - \delta ] \times M \subset ( \theta _ 2 - \delta , \theta _ 2 + \delta ) \times M \end{align*}


\begin{align*} \frac { 1 } { s _ 1 \ldots s _ { d } } \sum _ { j _ 1 = 0 } ^ { s _ 1 - 1 } \ldots \sum _ { j _ d = 0 } ^ { s _ d - 1 } f \left ( U _ 1 ^ { j _ 1 } \ldots U _ d ^ { j _ d } x \right ) \end{align*}


\begin{align*} & - \sqrt { \omega } \mathcal C ( f ) = c _ \infty ^ 3 j _ M ^ * \delta _ \infty - c _ 0 ^ 3 j _ M ^ * \delta _ 0 + \omega ^ { 1 / 2 } \phi , \\ & - \omega ^ { 3 / 2 } \mathcal C ( f ) = \omega ^ { 3 / 2 } \phi - E ( \phi ) \delta _ \infty \end{align*}


\begin{align*} w _ 1 x _ 1 + \cdots + w _ n x _ n - d \lambda x _ 1 ^ { w _ 1 } \cdots x _ n ^ { w _ n } = 0 \end{align*}


\begin{align*} L _ { 1 } ^ { \pm } ( u ) & = L ^ { \pm } ( u ) \otimes i d , \\ L _ { 2 } ^ { \pm } ( u ) & = i d \otimes L ^ { \pm } ( u ) . \end{align*}


\begin{align*} P _ A = \sum _ { \theta \in \{ 0 , 1 \} ^ 2 } \mu _ { A , \theta } P _ { A , \theta } . \end{align*}


\begin{align*} \begin{cases} \Delta _ 0 ^ \ast F ( r ) = F ( x ) - F ( 2 r ) , \\ \Delta _ { k + 1 } ^ \ast F ( r ) = \Delta _ k ^ \ast F ( r ) - 2 ^ { k + 1 } \Delta _ k ^ \ast F ( 2 r ) . \end{cases} \end{align*}


\begin{align*} \psi _ 1 ^ T A = 0 . \end{align*}


\begin{align*} \chi ( \sigma ) x _ { \sigma ( 1 ) } \wedge \cdots \wedge x _ { \sigma ( n ) } = x _ 1 \wedge \cdots \wedge x _ n . \end{align*}


\begin{align*} \gamma _ 0 ( u x u ^ * ) u ^ * \psi = u ^ * \gamma _ 0 ( u x u ^ * ) \psi = u ^ * \gamma _ 0 ( u x u ^ * ) e _ N \psi = u ^ * ( u x u ^ * ) \psi = x u ^ * \psi , \ \ \ x \in N ' \cap M . \end{align*}


\begin{align*} \Pi ( b ) = - b ^ \perp \ , . \end{align*}


\begin{align*} \Phi ( u ) : = \mathcal { A } ( u ) + \lambda \mathcal { B } ( u ) - \mathcal { F } ( u ) - \mathcal { G } ( u ) . \end{align*}


\begin{align*} { \rm T r } ( S _ P ) \simeq U \oplus { \rm N S } ( S _ { P ^ \circ } ) = U \oplus L _ { P ^ \circ } \end{align*}


\begin{align*} \sum ^ n _ { i = 1 } \| E _ { M \overline \otimes L ( K ) } ( ( 1 \otimes v _ { g _ i } ) \Delta ^ \pi ( x ) ( 1 \otimes v _ { h _ i } ) ) \| _ 2 \geq C . \end{align*}


\begin{align*} g : = h + 2 \ , T ^ \flat \otimes T ^ \flat , \end{align*}


\begin{align*} V _ { L , \omega } ( x ) = \sum _ { \gamma = 1 } ^ M V _ { L , \gamma } ( x ) V _ { L , \gamma } ( x ) : = v _ \gamma B _ \# ( x - y _ { L , \gamma } ) . \end{align*}


\begin{align*} \begin{gathered} M ^ S : ( i \sqcup ( c \sqcup A _ i ) ) \otimes ( o \sqcup ( c \sqcup A _ o ) ) \rightarrow K \ ; , \\ M ^ S = \begin{pmatrix} W & X & I _ 0 \\ Y & Z & I _ 1 \\ O _ 0 & O _ 1 & 0 \end{pmatrix} \end{gathered} \end{align*}


\begin{align*} \| 1 _ { \varepsilon A } \| ^ p & \ = \ \left \| \frac { 1 } { 2 } ( 1 _ { \varepsilon A } + 1 _ { B \backslash A } ) + \frac { 1 } { 2 } ( 1 _ { \varepsilon A } - 1 _ { B \backslash A } ) \right \| ^ p \\ & \ \le \ \frac { 1 } { 2 ^ p } \| 1 _ { \varepsilon A } + 1 _ { B \backslash A } \| ^ p + \frac { 1 } { 2 ^ p } \| ( 1 _ { \varepsilon A } - 1 _ { B \backslash A } ) \| ^ p \ \le \ 2 ^ { 1 - p } ( \mathbf h _ r ( N ) ) ^ p . \end{align*}


\begin{align*} C _ { 1 - \gamma , \rho } ^ { \gamma } [ a , b ] = \left \{ g \in C _ { 1 - \gamma , \rho } [ a , b ] , \ ^ { \rho } D _ { a ^ + } ^ { \gamma } g \in C _ { 1 - \gamma , \rho } [ a , b ] \right \} \end{align*}


\begin{align*} \beta _ 4 ( V _ t ) = \beta _ 2 ( V _ t ) = \beta _ 2 ( V ) = \beta _ 4 ( V ) - d . \end{align*}


\begin{align*} \left \{ \begin{aligned} ( \gamma - 1 ) ( p - 1 ) - 1 & = \alpha _ 1 \gamma + \beta _ 1 \theta \\ ( \theta - 1 ) ( q - 1 ) - 1 & = \beta _ 2 \gamma + \alpha _ 2 \theta . \end{aligned} \right . \end{align*}


\begin{align*} A _ { \infty } ^ { \pm } ( x ) \longmapsto A _ { \infty } ^ { \pm } ( 0 ) = : A _ { \infty } ^ { \pm } \end{align*}


\begin{align*} 2 \sum _ { u = 1 } ^ p \underline { Z } _ u - \frac { \pi ^ 2 } { 8 } \bar h _ 4 ( n ) \ , , \end{align*}


\begin{align*} \psi _ { \rho } \left ( x , t ; k \right ) = \psi \left ( x , t ; k \right ) \left [ 1 + o \left ( 1 \right ) \right ] \rightarrow 0 , \ \ \ x \rightarrow \infty , \ \ \ \operatorname { I m } k \geq 0 . \end{align*}


\begin{align*} \| H \| _ { \mathcal { V } _ { 2 } ^ { p } ( J ; E ) } : = \sup _ { \mathcal { P } } \Big ( \sum \nolimits _ { [ s , t ] \in \pi } \| H _ { s , t } \| _ E ^ p \Big ) ^ \frac { 1 } { p } < + \infty \ , , \end{align*}


\begin{align*} \frac { 1 } { N ^ { 1 - \lambda d } } \sum _ { i = 1 } ^ { N } \delta _ { N ^ { \lambda } x _ { i } } | _ { \square _ { R } } , \end{align*}


\begin{align*} m _ i > 0 \quad i = 1 , \ldots , s . \end{align*}


\begin{align*} \alpha _ i = \sum \limits _ { j = 1 } ^ s a _ { i j } I _ { \Omega _ { i j } \times ( \phi _ { u _ i , \nu } ) ^ { - 1 } ( C _ { i j } ) } , \ \ \alpha _ i | _ { \Omega _ i ^ { * } } = 0 , \ \ \Omega _ i ^ { * } \in \mathcal F _ i \otimes ( \phi _ { u _ i , \nu } ) ^ { - 1 } ( \mbox { B o r } ( Z ^ \nu _ { u _ i } ) ) \end{align*}


\begin{align*} S ( m , m ' , c ) = \underset { d \ , ( c ) } { \left . \sum \right . ^ { \ast } } e \Big ( \frac { m d + m ' \bar { d } } { c } \Big ) . \end{align*}


\begin{align*} \omega = e ^ x , g ( k ) = G ( x ) , \phi ( \omega ) = \Phi ( x ) , \mathcal { K } ( x ) = e ^ { - x } k ( e ^ { - x } ) \end{align*}


\begin{align*} \hat { \psi } _ { a , q } ( k ) = \pi ^ { - d / 4 } \exp \left ( - \frac { 1 } { 2 } ( k - q ) ^ 2 + 2 \pi i x \cdot a \right ) . \end{align*}


\begin{align*} F \xmapsto { \tau ^ * } F s \xmapsto { \tau ^ * } t _ 1 \xmapsto { \tau ^ * } t _ 2 \xmapsto { \tau ^ * } t _ 3 \xmapsto { \tau ^ * } s \\ \tau ^ * ( C _ j ) = C _ { [ j + 4 ] _ { 1 6 } } \tau ^ * ( B _ i ) = B _ { [ i + 2 ] _ 4 } . \end{align*}


\begin{align*} \rho _ s ^ { ( n ) } ( x ) = \rho _ s ( x ) \log ^ n ( 1 + s / x ) \left ( 1 + O _ n ( x ^ { - 1 } ) \right ) . \end{align*}


\begin{align*} h ( a ) = \frac { 1 } { 2 \gamma } \log ( 1 + \frac { 1 } { a } ) ^ { 2 } + \frac { 1 } { 2 \gamma } \tilde { h } ( a ) \end{align*}


\begin{align*} m _ i = \sum _ a t _ { a i a } = ( 7 , 1 1 , 1 5 ) \ ; , \end{align*}


\begin{align*} S _ w ( \theta ; \ell ) : = \sum _ { v \in V _ w ( \ell ) } \theta ( v ) . \end{align*}


\begin{align*} \theta _ 1 + m _ 1 = \theta _ 2 + m _ 2 = 0 \mod 2 . \end{align*}


\begin{align*} \begin{aligned} & | P ( x , t ) | = | v ( x , t ) | \leq C x _ n ^ { 2 + \alpha } , \\ & | D P ( x , t ) | \leq \frac { C } { x _ n } \left ( \| v \| _ { L ^ { \infty } ( Q _ { x _ n } ( p ) ) } + x _ n ^ 2 | \tilde { F } _ { p } ( 0 ) | \right ) \leq C x _ n ^ { 1 + \alpha } , \\ & | D ^ 2 P ( x , t ) | \leq \frac { C } { x _ n ^ 2 } \left ( \| v \| _ { L ^ { \infty } ( Q _ { x _ n } ( p ) ) } + x _ n ^ 2 | \tilde { F } _ { p } ( 0 ) | \right ) \leq C x _ n ^ { \alpha } . \end{aligned} \end{align*}


\begin{align*} \underset { k = i \kappa } { \rm R e s } \tilde { M } ^ { ( 1 ) } ( x , t , k ) = \frac { \gamma _ 0 } { a _ { 1 } ' ( i \kappa ) \delta ^ 2 ( i \kappa , \xi ) } e ^ { - 2 \kappa x + 8 \kappa ^ 3 t } \tilde { M } ^ { ( 2 ) } ( x , t , i \kappa ) , \gamma _ 0 ^ 2 = 1 , \end{align*}


\begin{align*} w \partial _ \eta ^ 2 w \leq & w ^ { - 1 } ( \eta \partial _ \xi w + \partial _ \tau w ) \\ \leq & w ^ { - 1 } [ \eta ( \partial _ \xi w + \partial _ \tau w ) + ( 1 - \eta ) \partial _ \tau w ] \\ \leq & \frac { b \delta ( 1 - \eta ) ^ { \alpha _ 0 + 1 } } { c _ 0 ( 1 - \eta ) } + w ^ { - 1 } \eta \delta b ( 1 - \eta ) \\ \leq & \frac { 5 \delta } { 4 } , o n \ , \ , \{ \xi = 0 \} \cup \{ \tau = 0 \} , \end{align*}


\begin{align*} x ( t , s _ 0 , x _ 0 ) = U ( t , s _ 0 ) x _ 0 , t \in J , \end{align*}


\begin{align*} \mu ^ \omega _ { s , t } ( \cdot ) = \int _ s ^ t \ 1 _ { \omega _ r } ( \cdot ) \ , d r , \end{align*}


\begin{align*} w ( x , 0 ) = \inf _ { b _ s } \ , \mathbb E \int _ 0 ^ \tau \ell | b _ s | ^ { \gamma ' } + f ( X _ s , s ) d s + \mathbb E w ( X _ \tau , \tau ) , \end{align*}


\begin{align*} \flat _ { i j } ( \gamma ) = \# \Big \{ e = \vec { i j } \in \overline { E } \ , \Big | \ , t _ e = \gamma \Big \} \ , . \end{align*}


\begin{align*} & P _ A \left ( \sigma ( x ^ 1 ) = y ^ 1 , \ldots , \sigma ( x ^ p ) = y ^ p \right ) = \prod _ { i = 1 } ^ k w _ A ( x ^ i , y ^ i ) \sum _ { \theta } \mu _ { A , \theta } ( - 1 ) ^ { \theta _ 1 g _ 1 + \theta _ 2 g _ 2 } \det \limits _ { i , j = 1 } ^ k \left ( K ^ { - 1 } _ { A , \theta } ( y ^ i , x ^ j ) \right ) , \end{align*}


\begin{align*} u ( - L ) h _ x u ( L ) = u ( r e ^ { 2 d _ 0 \ell } ) h _ { x ' } , \end{align*}


\begin{align*} \chi _ G ( g ) : = \det ( g ) ^ { \frac { q - \varepsilon } { 2 } } , g \in G ^ F , \end{align*}


\begin{align*} { \downarrow } ^ { \mathbb { X } } p ^ { - 1 } [ Y \smallsetminus ( U \cup V ) ] = { \downarrow } ^ { \mathbb { X } } p ^ { - 1 } [ Y \smallsetminus U ] \cap { \downarrow } ^ { \mathbb { X } } p ^ { - 1 } [ Y \smallsetminus V ] . \end{align*}


\begin{align*} \Xi ( \xi , t ) = e ^ { - \frac { t } { 2 } \varphi ( \xi , 0 ) \sigma _ 3 } \cdot \tau ^ { - \frac { i \nu \sigma _ 3 } { 2 } } . \end{align*}


\begin{align*} C ( \mathbf { a } , \mathbf { c } ) ( \tau ) + C ( \mathbf { b } , \mathbf { d } ) ( \tau ) = 0 , \tau , \end{align*}


\begin{align*} u ( t ) = T _ 0 ( t - s ) \varphi + j ^ { - 1 } \int _ s ^ t T _ 0 ^ { \odot \star } ( t - \tau ) B ( \tau ) u ( \tau ) d \tau , \varphi \in X , \end{align*}


\begin{align*} \mathcal { K } _ { \mathtt { m } } ( z , \bar z ) : = \tilde { \mathcal { H } } ( \alpha , z , \bar z ) _ { | \alpha = \sqrt { \mathtt { m } - \sum _ { j \not = 0 } | z _ j | ^ 2 } } \ , . \end{align*}


\begin{align*} B _ t ^ E \left [ B _ { 2 / 3 } ^ E \right ] ^ { - 1 } & = \begin{bmatrix} E - q ( \omega ) & - [ ( 1 - \eta ( t ) ) + \eta ( t ) p ( T ^ { - 1 } \omega ) ] \\ ( 1 - \eta ( t ) ) + \eta ( t ) p ( \omega ) & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ - 1 & E - q ( \omega ) \end{bmatrix} \\ & = \begin{bmatrix} ( 1 - \eta ( t ) ) + \eta ( t ) p ( T ^ { - 1 } \omega ) & * \\ 0 & [ 1 - \eta ( t ) ] + \eta ( t ) p ( \omega ) \end{bmatrix} , \end{align*}


\begin{align*} \frac { 1 - 2 t } { 1 - t } \leq c _ { \nu } \leq \frac { t } { 1 - t } \nu = 1 , 2 , \end{align*}


\begin{align*} \sigma _ { \Gamma } ( D _ 1 + D _ 2 ) = \lim _ { \epsilon \to 0 ^ + } \sigma _ { \Gamma } ( D _ 1 + \frac { \epsilon } { 2 } A + D _ 2 + \frac { \epsilon } { 2 } A ) . \end{align*}


\begin{align*} \phi _ \lambda ( n ) : = \sum _ { k = n + 1 } ^ \infty \lambda _ k ^ { - 1 } \| u _ k \| ^ 2 _ n \in [ 0 , \infty ] . \end{align*}


\begin{align*} | ( \vec \alpha _ i , \vec \alpha _ o ) | = | \vec \alpha _ i | + | \vec \alpha _ o | = \sum _ { x \in a _ i } \vec \alpha _ i ( x ) + \sum _ { y \in a _ o } \vec \alpha _ o ( y ) \ ; . \end{align*}


\begin{align*} B _ j ^ * = \{ x \in ( T ) : T ( x ) \cap B _ j \neq \emptyset \} , \end{align*}


\begin{align*} Z ( X , \theta ) : = \{ ( z , w ) \in X \times \C \colon \Re ( w e ^ { - i \theta ( z ) } ) > 0 \} , \end{align*}


\begin{align*} e x p ( q _ { c ^ { i j } } ) \cdot u ^ { p _ { c ^ { i j } } } = u ^ { p _ { c ^ { i j } } } \cdot e x p ( q _ { c ^ { i j } } ) \cdot u ^ { - 1 } . \end{align*}


\begin{align*} & \int _ Q | \nabla u _ { \delta _ k } | ^ 2 \ , d x d t \ , - \int _ \Sigma \partial _ { \boldsymbol { \nu } } u _ { \delta _ k } v _ { \delta _ k } \ , d \Gamma d t \\ & { } + \int _ Q \xi _ { \delta _ k } u _ { \delta _ k } \ , d x d t = \int _ Q \bigl ( \mu _ { \delta _ k } - \pi ( u _ { \delta _ k } ) + f \bigr ) u _ { \delta _ k } \ , d x d t . \end{align*}


\begin{align*} \frac { \partial p } { \partial \epsilon } ( 0 , h _ { i i } t _ { i i } ^ { - 1 } ) = & - t _ { i i } ^ { - 1 } ( h _ { i - 1 , i } t _ { i i } - h _ { i i } t _ { i - 1 , i } ) \\ \end{align*}


\begin{align*} B ^ - : = \left \lbrace \begin{pmatrix} a & 0 \\ c & d \end{pmatrix} : a , c , d \in \C , a d \neq 0 \right \rbrace \end{align*}


\begin{align*} Q ( x ) | \psi _ m \rangle = \frac { 1 } { 2 m - p - q - 2 N s } \prod _ { k = 1 } ^ m ( x - x _ k ) ( x + x _ k + 1 ) | \psi _ m \rangle + \ , . \end{align*}


\begin{align*} \partial _ r ^ 2 \eta _ 2 ( r ) = \zeta _ 1 - \zeta _ 0 , r \in ( g _ 0 , g _ 1 ) , \partial _ r \eta _ 2 ( g _ i ) = g \zeta _ i , i = 0 , 1 . \end{align*}


\begin{align*} f _ n ( t ) = \frac { d } { d t } u _ 1 + \nu A u _ 1 + B ( u _ 1 , u _ 1 ) . \end{align*}


\begin{align*} ( \Delta - \lambda ) u = f \end{align*}


\begin{align*} W ^ { 2 , 1 } _ q ( X ) = \{ : \partial ^ r _ t D ^ \beta _ x u \in L ^ q ( X ) | \beta | + 2 r \le 2 \} . \end{align*}


\begin{align*} \left ( \partial _ { x } ^ { \ast } \mathfrak { L } \right ) \left [ p \right ] = - \mathfrak { L } \left [ \partial _ { x } p \right ] , \quad \left ( x \mathfrak { L } \right ) \left [ p \right ] = \mathfrak { L } \left [ x p \right ] . \end{align*}


\begin{align*} e ^ { - k z } = - \tfrac { 1 } { z } \tfrac { d } { d k } e ^ { - k z } \end{align*}


\begin{align*} \frac { 1 } { n } T _ { 2 n } \stackrel { } \to \frac { 1 } { \lambda + } + \frac { 1 } { \lambda - } = : c _ 2 , \ ; n \to \infty , \end{align*}


\begin{align*} E ( t ) \le E ( 0 ) \exp \left \{ \int _ 0 ^ { T _ \varepsilon } C E ( t ) ^ { \frac 2 3 } d t \right \} \le 2 \exp \{ 4 C \varepsilon ^ { \frac 4 3 } T _ \varepsilon \} \varepsilon ^ 2 = 4 \varepsilon ^ 2 . \end{align*}


\begin{align*} & \sum _ { z \in \Z } \sum _ { t - N ^ { \epsilon } \le s \le t } \Delta ( x - z , t - s ) \xi ( z , s ) \frac { \Gamma ( z , s ) } { \Gamma ( x , t + 1 ) } \\ & = \sum _ { z \in \Z } \sum _ { t - N ^ { \epsilon } \le s \le t } \Delta ( x - z , t - s ) \xi ( z , s ) + O ( N ^ { - 1 / 2 + 5 \epsilon } ) . \end{align*}


\begin{align*} \int _ 0 ^ \infty e ^ { - k z } ( e ^ k - 1 ) ^ \alpha \ , d k & = \int _ 0 ^ 1 u ^ { z - \alpha - 1 } ( 1 - u ) ^ \alpha \ , d u = \frac { \Gamma ( z - \alpha ) \Gamma ( 1 + \alpha ) } { \Gamma ( z + 1 ) } . \end{align*}


\begin{align*} \beta _ { a } ( I ) = \beta _ { a , b } ( I ) & = \big | \big \{ w \in \mathcal { L } _ t ^ f ( u ) : \min ( u ) = 1 \big \} \big | - \big | \big \{ w \in \mathcal { L } _ t ^ f ( v ) \setminus \{ v \} : \max ( w ) = n \big \} \big | \\ & \ge \big | \big \{ u > u _ { \ell } > \dots > u _ { d - 1 } \} \big | - ( d - \ell ) = d + 1 - \ell - ( d - \ell ) = 1 . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ n [ M _ A ( v ) ] _ j = 0 \end{align*}


\begin{align*} ( 1 - x ^ 2 ) f '' ( x ) - ( 2 \alpha + 1 ) x f ' ( x ) + \lambda _ j f ( x ) = 0 . \end{align*}


\begin{align*} \sum _ { j , k = 0 } ^ n g ^ { j k } ( y ) \eta _ j \eta _ k = 0 , \end{align*}


\begin{align*} \theta ( k , \xi ) : = 4 k ^ 3 + 1 2 k \xi , \end{align*}


\begin{align*} I ' _ \alpha ( i ) = ( - 1 ) ^ { [ i \in J _ \alpha ] } \ , I _ \alpha ( i ) \ , , \end{align*}


\begin{align*} \omega : = r \ , d r \ , d \theta ^ 0 + d \theta ^ { 1 2 } + \omega _ R + d h \ , d \theta ^ S . \end{align*}


\begin{align*} \displaystyle \lim _ { \eta \to 1 } w \partial _ \eta ^ 2 w = l ( t , x ) . \end{align*}


\begin{align*} 2 z \cdot [ x , y ] = [ z \cdot x , y ] + [ x , z \cdot y ] . \end{align*}


\begin{align*} g ( y ) : = \max \Big \{ | y - A y - b | , | u ( y ) | \Big \} . \end{align*}


\begin{align*} \Psi '' _ { u } ( 1 ) & = ( 2 s _ { 1 } - 1 ) s _ 1 | \nabla _ { s _ 1 } u | _ { 2 } ^ { 2 } + ( 2 s _ { 2 } - 1 ) s _ 2 | \nabla _ { s _ 2 } u | _ { 2 } ^ { 2 } + \displaystyle \int _ { \mathbb { R } ^ { d } } W ( x ) u ^ { 2 } d x + \displaystyle \int _ { \mathbb { R } ^ { d } } \langle \nabla W ( x ) , x \rangle u ^ { 2 } d x \\ & + d ( d + 1 ) \displaystyle \int _ { \mathbb { R } ^ { d } } \widetilde { G } ( u ) d x - \frac { d ^ { 2 } } { 2 } \displaystyle \int _ { \mathbb { R } ^ { d } } \widetilde { G } ^ { ' } ( u ) u d x . \end{align*}


\begin{align*} f ( n , 4 , 5 ) = \frac 5 6 n + o ( n ) . \end{align*}


\begin{align*} [ J _ \omega \psi ] ( n ) = \begin{cases} \overline { p ( T ^ { n - 1 } \omega ) } \psi ( n - 1 ) + q ( T ^ n \omega ) \psi ( n ) + p ( T ^ n \omega ) \psi ( n + 1 ) & n \ge 1 \\ q ( \omega ) \psi ( 0 ) + p ( \omega ) \psi ( 1 ) & n = 0 . \end{cases} \end{align*}


\begin{align*} \sigma ( e _ { i _ { 1 } } \otimes \cdots \otimes e _ { i _ { k } } ) = e _ { i _ { \sigma ( 1 ) } } \otimes \cdots \otimes e _ { i _ { \sigma ( k ) } } , \end{align*}


\begin{align*} 0 & = \langle A _ 2 ( u _ 2 ) + \lambda _ 2 B _ 2 ( u _ 2 ) , ( u _ 2 - \overline { u } _ 2 ) _ + \rangle _ { \mathcal { H } _ 2 } \\ & - \langle F _ 2 ( T _ 1 u _ 1 , \overline { u } _ 2 , \nabla ( T _ 1 u _ 1 ) , \nabla \overline { u } _ 2 ) + G _ 2 ( T _ 1 u _ 1 , \overline { u } _ 2 ) , ( u _ 2 - \overline { u } _ 2 ) _ + \rangle _ { \mathcal { H } _ 2 } . \end{align*}


\begin{align*} \gamma _ { n } = \frac { h _ { n } } { h _ { n - 1 } } , n \geq 1 . \end{align*}


\begin{align*} \sum _ i \sum _ j A _ { a j i x i j } = \sum _ j \sum _ i A _ { a j i x i j } \ ; . \end{align*}


\begin{align*} \int _ { \R ^ d } ( G ( t _ h * u _ h ) - G ( t _ h * u ) ) d x = o ( 1 ) , \ \ \ h \rightarrow 0 + . \end{align*}


\begin{align*} c _ P u _ w ( i a ) = \int _ 0 ^ 1 u _ w ( x + i a ) d x = a ^ { 1 - w } \cdot \frac { E _ w ( z ) } { 1 - 2 w } , \end{align*}


\begin{align*} \dim G _ { A _ I } = \dim G _ A - \dim ( G _ A ) _ { V _ I } . \end{align*}


\begin{align*} \mathcal { L } _ N ^ { L ^ p \otimes F } \tilde { g } = 0 . \end{align*}


\begin{align*} \mathfrak { d } ( \mathfrak { S } _ { 1 , \delta = 0 } ) = \frac { ( p - 3 ) } { ( p - 1 ) ^ 2 } . \end{align*}


\begin{align*} \partial _ { t _ 1 } \partial _ { t _ 2 } \delta \Phi ( t _ 1 , t _ 2 , x ) & = \partial _ { t _ 1 } \partial _ { t _ 2 } \bigl ( \Phi ( t _ 1 + t _ 2 , x ) - \Phi ( t _ 1 , \Phi ( t _ 2 , x ) ) \bigr ) \\ & = \partial _ { t _ 1 } \bigl ( \partial _ t \Phi ( t _ 1 + t _ 2 , x ) - \partial _ x \Phi ( t _ 1 , \Phi ( t _ 2 , x ) ) \partial _ t \Phi ( t _ 2 , x ) \bigr ) \\ & = \partial _ t ^ 2 \Phi ( t _ 1 + t _ 2 , x ) - \partial _ t \partial _ x \Phi ( t _ 1 , \Phi ( t _ 2 , x ) ) \partial _ t \Phi ( t _ 2 , x ) . \end{align*}


\begin{align*} G = \begin{pmatrix} I _ { k _ 1 } & A & B _ 1 + 2 B _ 2 \\ 0 & 2 I _ { k _ 2 } & 2 D \end{pmatrix} , \end{align*}


\begin{align*} \frac { \sum _ { \xi _ { k + 1 } = + } \tau _ \xi p ^ { \xi } _ i } { \sum _ { \xi _ { k + 1 } = + } \tau _ \xi q ^ { \xi } _ i } = \frac { p _ i } { q _ i } 1 \leq i \leq k . \end{align*}


\begin{align*} & G ^ { ( 1 ) } _ { \mathcal R } ( x , y ) = \\ & \frac { e ^ { i k _ { + } \vert x \vert } } { \sqrt { \vert x \vert } } \frac { e ^ { i \frac { \pi } 4 } } { \sqrt { 8 \pi k _ { + } } } \left ( \frac { 2 \cos ^ 2 \theta _ { \hat x } - 1 - n ^ 2 } { n ^ 2 - 1 } \right ) e ^ { - i k _ { + } \vert y \vert \cos ( \theta _ { \hat x } { + } \theta _ { \hat y } ) } + G ^ { ( 1 ) } _ { \mathcal R , R e s } ( x , y ) \end{align*}


\begin{align*} J ( g ) = - \mu _ 1 + \big ( ( w + \bar { w } ) \partial _ { \eta } ^ 2 \bar { w } \big ) ( \varepsilon + \mu _ 1 \tau + \mu _ 1 ( 1 - \eta ) ) \leq - \mu _ 1 < 0 . \end{align*}


\begin{align*} X ^ \odot = X ^ { \odot } _ { - } ( s ) \oplus X ^ { \odot } _ { 0 } ( s ) \oplus X ^ { \odot } _ { + } ( s ) , \forall s \in \mathbb { R } , \end{align*}


\begin{align*} \xi _ { \Delta _ { g _ s } } ( \zeta ) = \zeta ^ 2 - a \zeta + \Delta _ { g _ F } \end{align*}


\begin{align*} q _ t ( s _ j , y ) : = P \Big ( \big ( S _ { s _ j } ^ 1 \big ) _ + \ge \frac { \psi ( t ) s _ j } { ( 1 + \delta ) t } - y \Big ) \ , , p _ t ( r , y ) : = P \Big ( S _ { r } ^ 1 \ge \frac { \psi ( t ) ( \delta t + r ) } { ( 1 + \delta ) t } + y \Big ) \ , \end{align*}


\begin{align*} g ^ { - 1 } d g = : \sum _ { a = 1 } ^ { 3 } \theta ^ { a } R ( \textbf { e } _ { a } ) \end{align*}


\begin{align*} \left ( 1 - \frac { \lambda } { m } + \exp \left \{ 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \right \} + \frac { T w } { m } \right ) = e ^ { - \lambda / m } \left ( 1 + \exp \left \{ 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \right \} + \frac { T w + \lambda } { m } + O ( 1 / m ^ 2 ) \right ) . \end{align*}


\begin{align*} \beta _ j \in \{ 0 , \beta , 2 \beta \} , \beta _ 1 + \dots + \beta _ { 2 m - 1 } = 2 m \beta , \end{align*}


\begin{align*} \begin{aligned} \mathcal { G } _ { 0 } & : = \left \{ \left ( g _ { i j } \right ) _ { i \leq j } \in \R ^ { \frac { d ( d + 1 ) } { 2 } } \ ; : \ ; \inf _ { x \neq 0 } \frac { g ( x , x ) } { | x | ^ 2 } > 0 \right \} \ , . \end{aligned} \end{align*}


\begin{align*} \delta b _ { s , t } = \mathcal { D } ( b ) _ { s , t } + B _ { s , t } b ^ 1 _ s + \mathbb { B } _ { s , t } b ^ { 2 } _ { s } + b ^ { \natural } _ { s , t } \ , . \end{align*}


\begin{align*} H ^ { \varepsilon } ( t , x , y , p , M , Z ) : = \min \limits _ { u \in U } \left \{ - ( \sigma ^ { \varepsilon } \sigma ^ { \varepsilon \top } M ) - f \cdot p - 2 ( \sigma ^ { \varepsilon } \varrho ^ { \top } Z ^ { \top } ) - \ell \right \} , \end{align*}


\begin{align*} \ell _ m ( v ) \ell _ k ( u ) \textbf { 1 } = \bar { \ell } _ k ( u ) \bar { \ell } _ m ( v ) \textbf { 1 } . \end{align*}


\begin{align*} \lim _ { i \to \infty } \int _ X \frac { | f ( x ) - f ( y ) | } { | x - y | } \rho _ i ( x , y ) \ , d \mathcal L ^ 1 ( x ) = | f ' ( y ) | . \end{align*}


\begin{align*} \frac { ( \log m ) ^ 2 } { m } \max _ { ( k ' - 1 ) m < \ell \le k ' m } \{ \overline { R } _ { \ell } \} \le \sum _ { j = 1 } ^ { k ' } D _ j ^ { ( m ) } + \frac { ( \log m ) ^ 2 } { m } \Big ( k ' \varphi _ m - \varphi _ { k ' m } \Big ) + c ' \ , , \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } \int _ \Omega { f _ k } = \int _ \Omega { f } . \end{align*}


\begin{align*} \begin{gathered} \alpha _ { a , b , c } : a \times ( b \times c ) \rightarrow ( a \times b ) \times c \ ; , \\ \alpha _ { a , b , c } ( ( i , ( j , k ) ) ) = ( ( i , j ) , k ) \ ; . \end{gathered} \end{align*}


\begin{align*} s ( m ) = \sum _ { \substack { d | m \\ 1 \leq d < m } } d \end{align*}


\begin{align*} m _ { n + 1 } ^ \epsilon = \frac { 1 } { 1 + \frac { \Delta t } { \epsilon ^ 2 } } \bigl ( m _ n ^ \epsilon + \frac { \Delta \beta _ n } { \epsilon } \bigr ) . \end{align*}


\begin{align*} \frac { \partial { K _ p } } { \partial { x } _ j } ( z ) = p \mathrm { R e } \ , \frac { \partial { K _ p ( \cdot , z ) } } { \partial { x _ j } } \bigg | _ z , \end{align*}


\begin{align*} Y _ r - Y _ { t _ i } = \int _ r ^ { t _ i } Y _ s W ( d s , B _ s ) - \int _ r ^ { t _ i } Z _ s d B _ s , r \in [ { t _ i } , t _ { t + 1 } ] \ , . \end{align*}


\begin{align*} | e ^ { - i \alpha } \lambda - E - i \eta | & = | \lambda - e ^ { i \alpha } ( E + i \eta ) | \\ & = \left ( ( \lambda - \cos ( \alpha ) E + \sin ( \alpha ) \eta ) ^ 2 + ( \sin ( \alpha ) E + \cos ( \alpha ) \eta ) ^ 2 \right ) ^ { 1 / 2 } \end{align*}


\begin{align*} \Psi ( \alpha , \beta ) = \sup _ { | x - y | < \beta , | x | > \alpha , | y | > \alpha } \left | g ( x ) - g ( y ) \right | . \end{align*}


\begin{align*} \Xi ( \alpha , 0 ) + \Xi ( \alpha , \alpha ) = o \left ( \Xi ( 0 , 0 ) \right ) \end{align*}


\begin{align*} \partial _ x | u _ t | ^ 2 = 2 \partial _ x u _ t \cdot u _ t \ , . \end{align*}


\begin{align*} h _ i x = ( p ^ i \circ h ) ( \tilde x ) | _ { \Omega _ i } \ \ \mbox { w h e r e } \ \ x = p ^ i ( \tilde x ) | _ { \Omega _ i } \ \ \mbox { a n d } \ \ x \in \mathcal P a ( E ) . \end{align*}


\begin{align*} h ( x _ \alpha f ) = [ h , x _ \alpha ] f + x _ \alpha h f = \alpha ( h ) x _ \alpha f + \mu ( h ) x _ \alpha f = ( \alpha + \mu ) ( h ) x _ \alpha f , \ \forall h \in \mathfrak { h } _ R . \end{align*}


\begin{align*} \begin{array} { l } \lbrace \alpha ( A _ { 0 0 } , B ) + ( \dfrac { 1 } { q } - \dfrac { 1 } { q ^ 2 } ) \alpha ( A _ { 1 0 } , B ) - \dfrac { 1 } { q ^ 3 } \alpha ( A _ { 1 1 } , B ) \rbrace \\ \\ - \dfrac { 1 } { q ^ 4 } \lbrace \alpha ( A _ { 1 1 } , B ) + ( \dfrac { 1 } { q } - \dfrac { 1 } { q ^ 2 } ) \alpha ( A _ { 2 1 } , B ) - \dfrac { 1 } { q ^ 3 } \alpha ( A _ { 2 2 } , B ) \rbrace = 0 \end{array} \end{align*}


\begin{align*} X = \frac { u + C v + B } { \gamma } . \end{align*}


\begin{align*} S = \bigsqcup _ { \iota \in I ( S ) } S _ \iota . \end{align*}


\begin{align*} x \circ y = x + x \star y + y \end{align*}


\begin{align*} \lim _ { n \to \omega } \| u _ n \phi ( x ) - x u _ n \| _ 2 = 0 , x \in S . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ k ( z _ { i , 1 } t _ 1 + \dots + z _ { i , N } t _ N ) . \end{align*}


\begin{align*} \int _ 0 ^ { T } \int _ { \R ^ n } \left ( \partial _ t \varphi ( t , x ) + \langle b ( t , x ) , D _ x \varphi ( t , x ) \rangle \right ) \ , d \rho _ t ( x ) d t = \ , - \int _ { \R ^ n } \varphi ( 0 , x ) \ , d \rho _ 0 ( x ) \end{align*}


\begin{align*} ( \beta _ 1 , \beta _ 2 ) \to ( \zeta _ 1 , \zeta _ 2 ) : = \Big ( \frac { m } { \beta _ 2 ^ 4 } , \frac { m } { \beta _ 1 ^ 4 } \Big ) \ , , \beta \mapsto \zeta : = \frac { m } { \beta ^ 4 } \end{align*}


\begin{align*} \mathcal { H } _ 1 ( x , t ) = \begin{cases} t \mathcal { H } ( x , 1 ) & 0 \leq t \leq 1 , \\ \mathcal { H } ( x , t ) & t > 1 . \end{cases} \end{align*}


\begin{align*} ( - \alpha ^ { i - h } ( g ) , - h ) \circ ( b , j ) = ( \alpha ^ { - h } ( b ) - \alpha ^ { i - h } ( g ) , j - h ) , \end{align*}


\begin{align*} i _ { R _ 1 , n } ( f ) = i _ { \textrm { T } , n } ( f ) \ , \left ( \frac { \textrm { j } 2 \pi f a _ { \textrm { T } } } { c + \textrm { j } 2 \pi f a _ { \textrm { T } } } \right ) \ , [ ] . \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { N \to \infty } { \rm D i s c } ( \eta , \delta , N ) = 0 , \end{align*}


\begin{align*} \pi ^ { - s } \Gamma ( s ) Z _ r ( g g ^ { \top } , s ) = \pi ^ { - \left ( \frac { r } { 2 } - s \right ) } \Gamma \left ( \frac { r } { 2 } - s \right ) { Z _ r \left ( ( g g ^ \top ) ^ { - 1 } , \frac { r } { 2 } - s \right ) . } \end{align*}


\begin{align*} \delta \int _ { D } ( u - B ) \cdot ( u - B ) _ { s , t } \dd x & = \int _ { s } ^ { t } \int _ { D } b ( u _ r ) \cdot ( u _ r - B _ r ) \dd x \dd r \ , . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ { n - 1 } \sup _ { 0 \le t \le 1 } \| g _ i '' ( t ) \| _ U \lesssim n A ^ { - 3 } . \end{align*}


\begin{align*} p _ \alpha ( x _ 1 , \dots , x _ N ) = \prod _ { i = 1 } ^ { \ell ( \alpha ) } \sum _ { j = 1 } ^ N x _ j ^ { \alpha _ i } , \end{align*}


\begin{align*} \varphi _ i ( \vec { a } ) \land 1 = \varphi _ i ( \vec { a } ) \leq \varphi _ 1 ( \vec { a } ) \lor \dots \lor \varphi _ n ( \vec { a } ) . \end{align*}


\begin{align*} f _ X : = f ( A _ X , \theta _ X , \omega _ X , \mathcal O ^ { \infty } ( X ) ) \in \mathcal O ( J ^ r ( X ) ) \simeq \mathcal O ( J ^ r ( X ) ) \cdot z _ L ^ w , \end{align*}


\begin{align*} L = \frac { 1 6 } { 3 } \alpha ^ { - 1 + \eta } , \eta \in ( 0 , 1 ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \omega } \| v _ n - c _ n ( g ) v _ n u _ { g ^ { - 1 } } \| _ 2 = 0 . \end{align*}


\begin{align*} \left \lbrace \begin{aligned} d Q ^ \epsilon ( t ) & = f \bigl ( Q ^ \epsilon ( t ) - P ^ \epsilon ( t ) \bigr ) d t + \sigma \bigl ( Q ^ \epsilon ( t ) - P ^ \epsilon ( t ) \bigr ) d \beta ( t ) , \\ d P ^ \epsilon ( t ) & = - \frac { P ^ \epsilon ( t ) } { \epsilon ^ 2 } d t + f \bigl ( Q ^ \epsilon ( t ) - P ^ \epsilon ( t ) \bigr ) d t + \sigma \bigl ( Q ^ \epsilon ( t ) - P ^ \epsilon ( t ) \bigr ) d \beta ( t ) . \end{aligned} \right . \end{align*}


\begin{align*} \bar { \nu } \cdot \nabla d = | \bar { \nu } | ^ 2 = 1 , \bar { \nu } \cdot \nabla \bar { \xi } = 0 \quad \overline { N _ T } \end{align*}


\begin{align*} \widetilde { K } _ { N , \beta } = \frac { Z _ { N , \beta } } { \exp \left ( - N ^ { 2 } \beta \mathcal { E } _ { V } ( \mu _ { V } ) \right ) } . \end{align*}


\begin{align*} z _ j ^ + ( n + 1 ) = \frac { \Theta [ n ] ( e _ j , e _ j ) } { H [ n ] ( e _ j , \overline { e _ j } ) + \sqrt { H [ n ] ( e _ j , \overline { e _ j } ) ^ 2 - | \Theta [ n ] ( e _ j , e _ j ) | ^ 2 } } . \end{align*}


\begin{align*} u ^ { o u t } ( t ) = h ( t , u ^ { i n } ( t ) ) , t \ge 0 , \end{align*}


\begin{align*} ( x y z ) ^ U : = \prod _ i x _ i ^ { n ' _ i } \prod _ i y _ i ^ { n '' _ i } \prod _ j z _ j ^ { n _ j } \end{align*}


\begin{align*} \begin{aligned} & \sqrt { \frac { \hbar ^ 2 } { 4 } | j | _ { g } ^ 4 + \mathtt { m } p ' ( \mathtt { m } ) | j | _ { g } ^ 2 } = \frac { \hbar } { 2 } \omega _ j \ , , j \in \Z ^ d \setminus \{ 0 \} \\ & \omega _ j : = \sqrt { | j _ g | ^ 4 + \delta | j _ g | ^ 2 } , \delta : = \frac { 4 \mathtt { m } p ' ( \mathtt { m } ) } { \hbar ^ 2 } \ , . \end{aligned} \end{align*}


\begin{align*} i \Delta I _ h z ( t ) = & i \Delta S ( t ) I _ h z _ 0 + \int _ 0 ^ t \Delta S ( t - s ) I _ h ( \nu z ( s ) - i \epsilon ( \gamma z ( s ) - \mu \overline { z } ( s ) ) ) \d s \\ & - \tfrac { i } { 2 } \int _ 0 ^ t \Delta S ( t - s ) I _ h ( z ( s ) F _ \Phi ) \d s - \kappa \int _ 0 ^ t \Delta S ( t - s ) I _ h ( \theta _ R ( | z | _ { X ^ { \mathfrak { s } } _ s } ) | z ( s ) | ^ 2 z ( s ) ) \d s \\ & + \int _ 0 ^ t \Delta S ( t - s ) I _ h ( z ( s ) \d W ( s ) ) \end{align*}


\begin{align*} \Gamma _ \varepsilon ^ i ( t ) = \{ y + \varepsilon g _ i ( y , t ) \nu ( y , t ) \mid y \in \Gamma ( t ) \} , t \in [ 0 , T ] , \ , i = 0 , 1 . \end{align*}


\begin{align*} p _ n ^ { \epsilon , \Delta t } = \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ n } p _ 0 ^ { \epsilon } + \frac { \Delta t } { \epsilon } \sum _ { k = 0 } ^ { n - 1 } \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ { n - k } } f ( q _ k ^ { \epsilon , \Delta t } ) + \frac { 1 } { \epsilon } \sum _ { k = 0 } ^ { n - 1 } \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ { n - k } } \sigma ( q _ k ^ { \epsilon , \Delta t } ) \Delta \beta _ k . \end{align*}


\begin{align*} f ( z ) = \int _ \Omega | m _ p ( \cdot , z ) | ^ { p - 2 } \overline { K _ { 2 , p , z } ( \cdot , z ) } f , \ \ \ \forall \ , f \in A ^ 2 _ { p , z } ( \Omega ) . \end{align*}


\begin{align*} \nabla ^ * \nabla + A = 0 , \end{align*}


\begin{align*} P _ { A , b } ( x , y ) = h ( y - A x - b ) - h ( y - A y - b ) + h ( x - A y - b ) - h ( x - A x - b ) . \end{align*}


\begin{align*} \chi ( z ) = \left \{ \begin{alignedat} { - 1 } & 1 , 0 \le z \le 1 , \\ & z , z \ge 2 , \end{alignedat} \right . \end{align*}


\begin{align*} \lim _ { \substack { r \to \infty , \\ r e ^ { i t } \notin E } } \frac { \log | W ( r e ^ { i t } ) | } { r ^ { \rho ( r ) } } = \frac a \rho < 1 . \end{align*}


\begin{align*} J _ M ( f ) ( \omega ) = - c _ 0 ^ 3 j _ M ^ * + \int _ 0 ^ \omega \tilde \omega ^ { 1 / 2 } \phi ( \tilde \omega ) \ , \dd \tilde \omega , J _ E ( f ) ( \omega ) = \int _ 0 ^ \omega \tilde \omega ^ { 3 / 2 } \phi ( \tilde \omega ) \ , \dd \tilde \omega . \end{align*}


\begin{align*} c d = d c = 1 . \end{align*}


\begin{align*} | \mathcal { M } _ { \ell } | = | \mathcal { S } _ { \ell } | \cdot | \mathcal { T } _ r | \leq & ( 2 d e ) ^ n \left ( \frac { 2 \ell + n } { n } \right ) ^ { d n } . \end{align*}


\begin{align*} \overline { \mathrm { m o } } \ , ( Y ) = \limsup _ { \varepsilon \ , \to \ , 0 ^ + } \ , \frac { \log \log S _ Y ( \varepsilon ) } { - \log \varepsilon } \qquad \underline { \mathrm { m o } } \ , ( Y ) = \liminf _ { \varepsilon \ , \to \ , 0 ^ + } \ , \frac { \log \log S _ Y ( \varepsilon ) } { - \log \varepsilon } \end{align*}


\begin{align*} \varphi ( v ) = e ^ { - \pi | v | ^ 2 } . \end{align*}


\begin{align*} \int _ { g _ 0 } ^ { g _ 1 } \partial _ r ^ 2 \eta _ 2 ( r ) \ , d r = \partial _ r \eta _ 2 ( g _ 1 ) - \partial _ r \eta _ 2 ( g _ 0 ) . \end{align*}


\begin{align*} \imath _ L \circ \Phi = \imath _ L : E C H ^ L ( Y , \lambda ) \longrightarrow E C H ( Y , \xi ) . \end{align*}


\begin{align*} B ^ { \epsilon } _ { s , t } : = \Gamma ^ { \epsilon } _ { t } - \Gamma ^ \epsilon _ s \ , \mathbb { B } ^ { \epsilon } _ { s , t } : = \int _ { s } ^ t \dd \Gamma ^ \epsilon _ r ( \Gamma ^ \epsilon _ { r } - \Gamma ^ \epsilon _ s ) \ , , \end{align*}


\begin{align*} H = - L - V , V ( x ) = \frac { \kappa } { | x | ^ { \alpha } } , 0 < \kappa < \kappa ^ * = \frac { 2 ^ \alpha \Gamma \left ( \frac { d + \alpha } { 4 } \right ) ^ 2 } { \Gamma \left ( \frac { d - \alpha } { 4 } \right ) ^ 2 } . \end{align*}


\begin{align*} { Z } _ { N , \beta } = z ^ { N } . \end{align*}


\begin{align*} \int _ { Q _ T } \Delta p | \nabla p | ^ 2 = \int _ { Q _ T } 2 p | D ^ 2 p | ^ 2 + 2 p \nabla \Delta p \cdot \nabla p = \int _ { Q _ T } 2 p | D ^ 2 p | ^ 2 - 2 \Delta p | \nabla p | ^ 2 - 2 p | \Delta p | ^ 2 . \end{align*}


\begin{align*} e ^ { - v } r = | e ^ { - \frac { F } { 2 } } w | ^ 2 - 2 \Re ( w e ^ { - F } ) + e ^ { - v } \eta \circ \theta , \end{align*}


\begin{align*} \mathcal { R } ( \epsilon , \Delta t ) = \int _ { 0 } ^ { \Delta t } \frac { \epsilon } { \Delta t } \bigl ( 1 - e ^ { - \frac { t } { \epsilon ^ 2 } } \bigr ) d t \ge 0 \end{align*}


\begin{align*} d X ^ 0 ( t ) = \sigma ( X ^ 0 ( t ) ) \circ d \beta ( t ) \end{align*}


\begin{align*} W \ast \mu ( x ) = \int _ { \{ y : d ( x , y ) \leq \sqrt { r _ h } \} } W ( x , y ) d \mu ( y ) + \int _ { \{ y : d ( x , y ) > \sqrt { r _ h } \} } W ( x , y ) d \mu ( y ) . \end{align*}


\begin{align*} \frac { \dd ^ N ( \sinh t ) ^ { - d / 2 } } { \dd t ^ N } = \sum C _ { N , m _ 1 , \dots m _ N } ( \sinh t ) ^ { - d / 2 - ( m _ 1 + \dots + m _ N ) } \prod _ { j = 1 } ^ N \Big ( \frac { \dd ^ { j } \sinh t } { \dd t ^ { j } } \Big ) ^ { m _ j } , \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta u = K _ 1 ( x ) u ^ { \alpha _ 1 } v ^ { \beta _ 1 } \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; u > 0 \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; & & u \lfloor _ { \partial \Omega } = 0 , \\ - \Delta v = K _ 2 ( x ) v ^ { \alpha _ 2 } u ^ { \beta _ 2 } \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; v > 0 \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; & & v \lfloor _ { \partial \Omega } = 0 , \end{alignedat} \right . \end{align*}


\begin{align*} \Delta _ k g = \Delta _ k f . \end{align*}


\begin{align*} \begin{multlined} \psi ^ { a b , c } ( \psi ^ { a , b } ( i , j ) , k ) = c ( b i + j ) + k \\ = b c i + ( c j + k ) = \psi ^ { a , b c } ( i , \psi ^ { b , c } ( j , k ) ) \ ; . \end{multlined} \end{align*}


\begin{align*} P _ { n } ( x ; z ) = x ^ { n } - c _ { n } \left ( z \right ) x ^ { n - 2 } + d _ { n } \left ( z \right ) x ^ { n - 4 } + O \left ( x ^ { n - 6 } \right ) , \end{align*}


\begin{align*} \bigl \langle \partial _ t \bigl ( v _ { \delta , \lambda } ( t ) - v _ 0 \bigr ) , F ^ { - 1 } _ \Gamma \bigl ( v _ { \delta , \lambda } ( t ) - v _ 0 \bigr ) \bigr \rangle _ { V _ { \Gamma , 0 } ^ * , V _ { \Gamma , 0 } } \ ! + \ ! \int _ \Gamma \nabla _ \Gamma w _ { \delta , \lambda } ( t ) \cdot \nabla _ \Gamma F ^ { - 1 } _ \Gamma \bigl ( v _ { \delta , \lambda } ( t ) - v _ 0 \bigr ) \ , d \Gamma = 0 \end{align*}


\begin{align*} \begin{cases} h ^ { \mu _ { V } } + \frac { V } { 2 } + c \geq 0 \\ h ^ { \mu _ { V } } ( x ) + \frac { V } { 2 } ( x ) + c = 0 \ { \rm f o r } \ x \in \Sigma \end{cases} \end{align*}


\begin{align*} \left \vert \left \vert \left \vert g \right \vert \right \vert \right \vert ^ { 2 } : = \int _ { S } \int g \left ( x , s \right ) ^ { 2 } \mathrm { d } \mu \left ( x \right ) \mathrm { d } s < \infty . \end{align*}


\begin{align*} \tilde H _ 1 ' ( s ) = h '' g _ H ( T _ * N , N ^ \perp ) g _ H ( T _ * N , N ) - h ' g _ H ( T _ * N , N ^ \perp ) ( - \kappa _ \eta + g _ H ( T _ * N , N ) \kappa _ 1 ) < 0 , \end{align*}


\begin{align*} \mathfrak { L } \left [ x ^ { n } \right ] = \mu _ { n } . \end{align*}


\begin{align*} \begin{pmatrix} h [ n + 1 ] & \Theta ^ \mathrm { p a i r } [ n + 1 ] \\ \overline { \Theta ^ \mathrm { p a i r } [ n + 1 ] } & h [ n + 1 ] \end{pmatrix} \circ \begin{pmatrix} e \\ \overline { e } \end{pmatrix} = \lambda ( e ) \begin{pmatrix} e \\ \overline { e } \end{pmatrix} , \lambda ( e ) \ge \mu _ { n + 1 } , e ( x ) \in ( \phi _ { n + 1 } ) _ \perp , \end{align*}


\begin{align*} t ( \xi + i \eta ) = P _ { \hat { t } } ( \xi + i \eta ) = \frac { 1 } { \pi } \int _ { - \infty } ^ { \infty } \frac { \eta } { ( \xi - s ) ^ 2 + \eta ^ 2 } \hat { t } ( s ) d s . \end{align*}


\begin{align*} d _ 0 & = \begin{bmatrix} 1 - s _ 1 \\ \dots \\ 1 - s _ n \end{bmatrix} , s _ i \in S , & & & d _ 1 & = \left [ \dfrac { \partial r _ i } { \partial s _ j } \right ] _ { r _ i \in R , s _ j \in S } . \end{align*}


\begin{align*} p = \rho ^ \gamma , \gamma \ge 1 . \end{align*}


\begin{align*} \aleph _ { \alpha , L } : = \frac { 1 } { 6 \pi } \left \langle \alpha L \arctan \left ( \frac { 1 } { \alpha L ( h _ 1 - e _ 1 ) } \right ) \right \rangle _ { x u _ 1 } , \end{align*}


\begin{align*} P ( \mathbf { o } ) = \frac { Z ( \mathbf { o } ) } { Z ( \{ \} ) } \ ; . \end{align*}


\begin{align*} ( T _ t f ) \oplus ( T _ s f ) = T _ { s + t } f \ , . \end{align*}


\begin{align*} \int _ 0 ^ T \int _ { \R ^ n } v \ , \left ( \partial _ t \varphi + \div ( b \ , \varphi ) \ , \right ) d t d x = \ , - \int _ { \R ^ n } \bar u \ , \varphi ( 0 , \cdot ) \dd x \end{align*}


\begin{align*} \mathcal U _ \varepsilon ( t ) = \mathcal L ( \dfrac { t } { \varepsilon } ) \mathcal U _ 0 + \int _ 0 ^ t \mathcal L ( \dfrac { t - s } { \varepsilon } ) \mathcal G ( s ) \ , d s . \end{align*}


\begin{align*} b _ j \Psi _ { k , i , t } = \begin{cases} \Psi _ { k , i - 1 , t } b _ j , & , \\ \Psi _ { k , i , t } b _ j , & , \\ \end{cases} \end{align*}


\begin{align*} \tilde H '' _ 1 ( s ) & = h ''' g _ H ( T _ * N , \dot \eta ) ^ 2 g _ H ( \nu , T _ * N ) + h '' \frac { d } { d s } g _ H ( T _ * N , \dot \eta ) g _ H ( \nu , T _ * N ) \\ & \quad + h '' g _ H ( T _ * N , \dot \eta ) \frac { d } { d s } g _ H ( \nu , T _ * N ) + h '' g _ H ( T _ * N , \dot \eta ) \frac { d } { d s } g _ H ( \nu , T _ * N ) \\ & \quad + h ' \frac { d ^ 2 } { d s ^ 2 } g _ H ( \nu , T _ * N ) . \end{align*}


\begin{align*} \lim _ { z \to 0 } \frac { \Phi _ g ( \tau , z , w ) } { ( 2 \pi i z ) ^ 2 } = \eta _ g ( \tau ) \eta _ g ( \omega ) . \end{align*}


\begin{align*} | U ( t , s ) | \leq K e ^ { - ( \omega - c ) ( t - s ) + 2 c s } = K ( s ) e ^ { - ( \omega - c ) ( t - s ) } , \textnormal { f o r $ t \geq s \geq 0 $ } . \end{align*}


\begin{align*} | K l _ p ( \psi _ p , \psi _ p ' ; \tilde { c } , w _ { G _ 4 } ) | & \leq D _ 8 \cdot \min ( p ^ { r + \sigma + \varrho / 2 + 3 m } , p ^ { \varrho + 3 \sigma / 2 + r / 2 + 3 m } ) . \end{align*}


\begin{align*} L ( t ) = \frac { 1 } { 2 } : a a : + t T a \in V ^ 1 ( \C a ) . \end{align*}


\begin{align*} I _ A = \{ ( i , j ) : i + 1 < j < n , \ a _ { i j } \neq 0 \} . \end{align*}


\begin{align*} M _ 1 [ n + 1 ] & : = \mathrm { d i a g } \Big ( \{ ( \upsilon _ N \ast | \phi _ { n + 1 } | ^ 2 ) ( x ) + \frac { 1 } { N } ( \upsilon _ N \ast \rho ^ \mathrm { p a i r } ) ( x ) \} \delta ( x - y ) \\ & + \frac { 1 } { N } ( \upsilon _ N \Big ( \frac { k _ { n + 1 } \circ \overline { k _ { n + 1 } } } { \delta - k _ { n + 1 } \circ \overline { k _ { n + 1 } } } \Big ) ) ( x , y ) \Big ) , \end{align*}


\begin{align*} \Omega : = & \phi - s \phi _ s - z \phi _ z \\ \Lambda : = & \Omega \phi _ { z z } + ( r ^ 2 - s ^ 2 ) ( \phi _ { s s } \phi _ { z z } - \phi ^ 2 _ { s z } ) , \end{align*}


\begin{align*} \Psi ( f ) = \left ( \omega ^ { f _ 0 } , \omega ^ { f _ 1 } , \hdots , \omega ^ { f _ { 2 ^ m - 1 } } \right ) , \end{align*}


\begin{align*} \frak { d } ( \mathcal { R } _ 1 ) = \mathfrak { d } ( \mathfrak { S } _ { 1 , \delta = 0 } ) & = \frac { p - 3 } { ( p - 1 ) ^ 2 } \\ \frak { d } ( \mathcal { R } _ 2 ) = \mathfrak { d } ( \mathfrak { S } _ { 3 , \delta = 0 } ) & = \frac { p ^ 2 - p - 1 } { ( p - 1 ) ( p ^ 2 - 1 ) } . \end{align*}


\begin{align*} F ( f \circ g ) = F ( f ) \circ F ( g ) \ ; . \end{align*}


\begin{align*} G ( x , y ) & = \frac { e ^ { i k _ { - } \vert x \vert } } { \sqrt { \vert x \vert } } G ^ { \infty } ( \hat x , y ) + G _ { R e s } ( x , y ) , \\ \nabla _ y G ( x , y ) & = \frac { e ^ { i k _ { - } \vert x \vert } } { \sqrt { \vert x \vert } } H ^ { \infty } ( \hat x , y ) + H _ { R e s } ( x , y ) , \end{align*}


\begin{align*} & W _ { ( k ) } ^ c : = \frac { 1 } { \lambda ^ 2 N _ { \Lambda } ^ 2 } \Phi _ { ( k ) } k _ 1 , \ \ k \in \Lambda _ u \cup \Lambda _ B , \\ & D _ { ( k ) } ^ c : = \frac { 1 } { \lambda ^ 2 N _ { \Lambda } ^ 2 } \Phi _ { ( k ) } k _ 2 , \ \ k \in \Lambda _ B . \end{align*}


\begin{align*} Q _ x ( v ) = \Vert \ , v \ , \Vert _ x ^ 2 / 2 , x \in M . \end{align*}


\begin{align*} Z _ { q } ( x ) : = 1 + q \int _ 0 ^ x W _ { q } ( y ) d y , x \in \mathbb { R } , \end{align*}


\begin{align*} \mathrm { E } [ e ^ { - \theta X _ t } ] = : e ^ { \psi ( \theta ) t } , t , \theta \geq 0 , \end{align*}


\begin{align*} 0 < \lambda _ 1 : = \min \{ \delta _ 1 , \bar { \delta } _ 1 \} \leq \lambda _ 2 : = \max \{ \delta _ 2 , \bar { \delta } _ 2 \} . \end{align*}


\begin{align*} \left [ - \frac { 1 } { 2 ( 2 \pi ) ^ 2 } \Delta _ L f \right ] ^ \wedge ( p ) = \nu ( p ) \hat { f } ( p ) . \end{align*}


\begin{align*} \delta ' : = \| z \| _ { s } \ , . \end{align*}


\begin{align*} J _ A : = \{ \max a : a \in A \} \subset \{ 1 , \ldots 2 n \} , \end{align*}


\begin{align*} a _ i \mapsto \begin{cases} a _ i , & ; \\ d _ i , & , \end{cases} b _ j \mapsto \begin{cases} b _ j , & ; \\ c _ { j + 1 } , & . \end{cases} \end{align*}


\begin{align*} | X ( n , 1 ) | = 4 \sum _ { c \mid n } \left ( \frac { - 4 } { c } \right ) | X ( n , 2 ) | = 2 \sum _ { c \mid n } \left ( \frac { - 2 } { c } \right ) . \end{align*}


\begin{align*} V _ \varepsilon ( x , t ) = \frac { ( - 1 ) ^ { i + 1 } } { \sqrt { 1 + \varepsilon ^ 2 | \bar { \tau } _ \varepsilon ^ i ( x , t ) | ^ 2 } } \Bigl ( \overline { V _ \Gamma } + \varepsilon \ , \overline { \partial ^ \circ g _ i } + \varepsilon ^ 2 \bar { g } _ i \bar { \tau } _ \varepsilon ^ i \cdot \overline { \nabla _ \Gamma V _ \Gamma } \Bigr ) ( x , t ) . \end{align*}


\begin{align*} h _ 1 \neq 0 , \partial _ x h _ 1 = 0 \ , , \end{align*}


\begin{align*} \phi ( x ) v = v x x \in a A a . \end{align*}


\begin{align*} \exp \Bigl ( a \ , \frac { \partial } { \partial x } \Bigr ) \ , \sigma ( x ) = \sigma ( x + a ) . \end{align*}


\begin{align*} x _ 4 ^ 2 = \lambda \end{align*}


\begin{align*} e x p ( q _ { b ^ { i j } } ) \cdot u ^ { p _ { b ^ { i j } } } = u ^ { p _ { b ^ { i j } } } \cdot e x p ( q _ { b ^ { i j } } ) \cdot u \end{align*}


\begin{align*} & \lim _ { k \rightarrow \infty } k ( E ( x , t , k ) - I ) = - \frac { 1 } { 2 \pi i } \int _ { \Gamma _ E } \mu ( x , t , s ) w ( x , t , s ) d s \\ & = - \frac { 1 } { 2 \pi i } \left ( \oint _ { | s + k _ 0 | = \epsilon } + \oint _ { | s - k _ 0 | = \epsilon } \right ) \mu ( x , t , s ) w ( x , t , s ) d s - \frac { 1 } { 2 \pi i } \int _ { \Gamma } \mu ( x , t , s ) w ( x , t , s ) d s \end{align*}


\begin{align*} [ \psi _ i ^ \epsilon ( z ) , \psi _ j ^ { \epsilon ' } ( w ) ] = 0 \ , , \psi ^ \pm _ { i , \mp b ^ \pm _ i } \cdot ( \psi ^ \pm _ { i , \mp b ^ \pm _ i } ) ^ { - 1 } = ( \psi ^ \pm _ { i , \mp b ^ \pm _ i } ) ^ { - 1 } \cdot \psi ^ \pm _ { i , \mp b ^ \pm _ i } = 1 \ , , \end{align*}


\begin{align*} f _ r ( z ) = f _ r ^ * ( \lambda ) + ( z - \lambda ) h _ r ( z ) , \end{align*}


\begin{align*} \langle u , v \rangle = ( e | [ u , v ] ) . \end{align*}


\begin{align*} u ^ { o u t } ( t ) = P u ^ { i n } ( t ) , t \ge 0 , \end{align*}


\begin{align*} Z ' ( \gamma ) \ , Z \Big ( \gamma + \frac { 2 \pi \kappa } { \log T } \Big ) = - i \ , \chi ( \rho ) ^ { 1 / 2 } \ , \chi \Big ( 1 - \rho - \frac { 2 \pi i \kappa } { \log T } \Big ) ^ { 1 / 2 } \ , \zeta ' ( 1 - \rho ) \ , \zeta \Big ( \rho + \frac { 2 \pi i \kappa } { \log T } \Big ) . \end{align*}


\begin{align*} \lim \limits _ { \varepsilon \rightarrow 0 } E \Big [ \big \| X _ { t \wedge \tau ^ \varepsilon } ^ { h _ \varepsilon } - Y _ { t \wedge \tau ^ \varepsilon } ^ { h _ \varepsilon } \big \| ^ 2 \Big ] + E \Big [ \int _ { 0 } ^ { t \wedge \tau ^ \varepsilon } \big \| X _ s ^ { h _ \varepsilon } - Y _ s ^ { h _ \varepsilon } \big \| _ V ^ 2 d s \Big ] = 0 . \end{align*}


\begin{align*} \lim _ { m \to \infty , } \prod _ { 0 \leq j \leq m - 1 } \left | 1 + e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 } { m } } + \frac { z } { m } \right | = \left | e ^ z - ( - 1 ) ^ { \theta _ 1 } \right | . \end{align*}


\begin{align*} X & \overset { ( a ) } { = } g ( ( X _ { i , m } ) _ { ( i , m ) \in L } ) \\ & \overset { ( b ) } { = } g ( \alpha ( \vec { W } ) ) \end{align*}


\begin{align*} \left ( C f \right ) ( k ' ) : = \frac { 1 } { 2 \pi i } \int _ { \Gamma _ E } \frac { f ( s ) } { s - k ' } d s , k ' \in \mathbb { C } \backslash \Gamma _ E . \end{align*}


\begin{align*} B _ { n , \mu , 0 } ( 1 ) \Lambda _ p = \sum _ { j = 0 } ^ k \lambda _ j B _ { n , \mu , j } ( 1 ) + \sum _ { j = 1 } ^ k \lambda _ j S _ { n , \mu , j } ( 1 ) \coloneqq T ( n , \mu ) + \sum _ { j = 1 } ^ k \lambda _ j S _ { n , \mu , j } ( 1 ) . \end{align*}


\begin{align*} \Phi _ { N k } = E _ { N \overline { k } } I _ N , k \in \N _ 0 , \end{align*}


\begin{align*} A = R R ^ \dagger \ ; , \end{align*}


\begin{align*} A _ m ( x + i y ) = m \int _ 0 ^ 1 \log \left | 1 + e ^ { 2 \pi i \phi } + \frac { x + i y } { m } \right | \mathrm { d } \phi \end{align*}


\begin{align*} d _ \alpha ( n ) = g _ { 1 / \{ \beta \} } ( n ) - g _ { 1 / \{ \alpha \} } ( n ) \end{align*}


\begin{align*} A ( \Gamma ) : = \mathbb R [ A _ { n + 1 } , \dots , A _ { n + m } ] [ \mathcal V ] \subset \mathcal F , \end{align*}


\begin{align*} \gamma _ { A , \ell } \geq \max _ { \emptyset \neq I \subseteq \{ 1 , \ldots , n \} } \frac { 2 \dim A _ I } { \dim G _ { A _ I } } = \gamma _ A . \end{align*}


\begin{align*} L _ \varepsilon ( x , v ) = Q _ { x } ( v ) - \varepsilon ^ { - 2 } U ( x ) , ( x , v ) \in T M . \end{align*}


\begin{align*} n _ { s - k } \ge n _ { s - \ell + 1 } + ( \ell - 1 - k ) \ge s - \ell + 1 + ( \ell - 1 - k ) = s - k . \end{align*}


\begin{align*} \lim _ { r \to \infty } \frac { n ( r ) } { \varphi ( r ) } = a < \rho . \end{align*}


\begin{align*} \lim \limits _ { n \to + \infty } \bigg | \ , u ^ { T ( n ) , n } ( 0 , y ) \ , - \ , \overline { g } \ , \bigg | = 0 , y , \end{align*}


\begin{align*} f ( x ) = I ^ { - 1 } _ { c m } \end{align*}


\begin{align*} \chi ( q ) = \begin{cases} \chi ^ 1 ( q ) \quad & q \equiv 1 \ m o d \ \rho , \\ \chi ^ 2 ( q ) \quad & q \equiv 2 \ m o d \ \rho , \\ \quad \vdots & \quad \vdots \\ \chi ^ { \rho } ( q ) \quad & q \equiv \rho \ m o d \ \rho . \\ \end{cases} \end{align*}


\begin{align*} f ( p ) & = ( 1 - \varepsilon _ p ) f ( a _ p ) + \varepsilon _ p f ( b _ p ) = ( 1 - \varphi _ p ) f ( c _ p ) + \varphi _ p f ( d _ p ) , \\ f ( q ) & = ( 1 - \varepsilon _ q ) f ( a _ q ) + \varepsilon _ q f ( b _ q ) = ( 1 - \varphi _ q ) f ( c _ q ) + \varphi _ q f ( d _ q ) . \end{align*}


\begin{align*} \Phi ( ( b , j ) ) = \begin{cases} 2 b + 1 \quad { \rm i f } \ ; j = 0 , \\ 6 b + 6 \quad { \rm i f } \ ; j = 1 . \end{cases} \end{align*}


\begin{align*} g _ N ( x , t ) = f _ N ( x , t ) - \psi ( 0 , 0 ) t . \end{align*}


\begin{align*} \left ( \frac { d m _ 0 } { d \zeta } + \frac { i \zeta } { 2 } \sigma _ 3 m _ 0 \right ) = \beta ^ { m a t } m _ 0 . \end{align*}


\begin{align*} d _ { r , s } ( 0 , i ) = 0 , \mbox { i f } r \ne 0 \mbox { a n d } s \ne 0 . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { s } \frac { 1 } { 1 - ( \sum _ { j = 1 } ^ N z _ { i , j } t _ j ) } = \prod _ { i = 1 } ^ { s } \left ( 1 + \left ( \sum _ { j = 1 } ^ N z _ { i , j } t _ j \right ) + \left ( \sum _ { j = 1 } ^ N z _ { i , j } t _ j \right ) ^ 2 + \cdots \right ) . \end{align*}


\begin{align*} \omega _ { 3 5 } = \frac { \kappa _ 2 } { \kappa _ 1 } \omega _ 1 , \ \omega _ { 4 5 } = - \frac { \kappa _ 2 } { \kappa _ 1 } \omega _ 2 , \ \omega _ { 3 6 } = \frac { \mu _ 2 } { \kappa _ 1 } \omega _ 2 \omega _ { 4 6 } = \frac { \mu _ 2 } { \kappa _ 1 } \omega _ 1 . \end{align*}


\begin{align*} \partial _ t \nu - \nabla \cdot ( \nu \nabla p ) = 0 , \end{align*}


\begin{align*} f ( y ) = \sum _ { i = 1 } ^ n f _ i ( y _ i ) , \ \ ( A x ) _ i = A _ i x . \end{align*}


\begin{align*} \left \{ H _ 0 ; G \right \} + Z = F \end{align*}


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


\begin{align*} & \mathcal S _ A ^ { \ne 0 } ( D ; \ell ) \\ & = \frac { D } { 2 \pi i } \int _ { ( \frac 1 2 ) } \sum _ { ( n , 2 ) = 1 } n ^ { - w } \frac { \tau _ A ( n ) } { \sqrt { n } } \frac { 1 } { 2 n \ell } \sum _ { ( c , 2 n \ell ) = 1 \atop c \le Y } \frac { \mu ( c ) } { c ^ 2 } \sum _ { k \ne 0 } ( - 1 ) ^ k G _ k ( n \ell ) \int _ 0 ^ \infty W \left ( \frac t N \right ) \tilde \Psi \left ( \frac { k D } { 2 c ^ 2 t \ell } \right ) t ^ { w - 1 } ~ d t ~ d w \end{align*}


\begin{align*} \frac { d } { d a } \left [ ( a ) _ m \right ] ^ p = p \left [ ( a ) _ m \right ] ^ { p - 1 } \frac { d } { d a } ( a ) _ m \end{align*}


\begin{align*} | \partial _ { \tau , \xi } w | \leq ( b \delta + C _ 1 ) ( 1 - \eta ) ^ { \alpha _ 0 } o n \tau = 0 a n d \xi = 0 , \end{align*}


\begin{align*} g = \sum _ { \alpha = 0 } ^ { m - 4 } x _ { \pi ( \alpha ) } x _ { \pi ( \alpha + 1 ) } , \end{align*}


\begin{align*} ( 2 A - \lambda \nabla t D ) \delta ^ { n + 1 } = ( 2 A + \lambda \nabla t D ) \delta ^ { n + 1 / 2 } \end{align*}


\begin{align*} \left | \frac { p ( z , t ) - 1 } { p ( z , t ) + 1 } \right | = 2 t ^ 2 | S h ( z + t ) | \leq k < 1 \end{align*}


\begin{align*} \lim _ { n \to \infty } n \mathbb { P } ( | X | > b _ n ) = 0 , \end{align*}


\begin{align*} \mathcal { A } ^ { - 1 } = \begin{pmatrix} I _ { d \times d } & 0 _ { d \times d } & - A _ { 1 3 } & A _ { 1 1 } - I _ { d \times d } \\ I _ { d \times d } & 0 _ { d \times d } & - A _ { 1 3 } & A _ { 1 1 } \\ 0 _ { d \times d } & I _ { d \times d } & A _ { 1 1 } ^ T & A _ { 2 1 } \\ 0 _ { d \times d } & - I _ { d \times d } & I _ { d \times d } - A _ { 1 1 } ^ T & - A _ { 2 1 } \end{pmatrix} . \end{align*}


\begin{align*} y = L \alpha ^ { - 1 } , L > 1 . \end{align*}


\begin{align*} \dfrac { \partial T } { \partial x } ( x ) = \ ( \begin{matrix} \dfrac { \partial T _ 1 x } { \partial x _ 1 } & \cdots & \dfrac { \partial T _ 1 x } { \partial x _ n } \\ \vdots & \vdots & \vdots \\ \dfrac { \partial T _ n x } { \partial x _ 1 } & \cdots & \dfrac { \partial T _ n x } { \partial x _ n } \end{matrix} \ ) , \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } f ( \lambda ( P _ t ) ) = f \left ( \lim _ { t \rightarrow \infty } \lambda ( P _ t ) \right ) = f ( \lambda _ 0 ) . \end{align*}


\begin{align*} \nabla _ + A = c B \nabla B = 0 , \end{align*}


\begin{align*} P _ A ( \sigma ) : = \frac { 1 } { Y _ { m _ 1 , m _ 2 } ( A ) } \prod _ { x \in \mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x , \sigma ( x ) ) , \end{align*}


\begin{align*} & \underline { W } _ j : = \overline { g } _ { j m , \alpha } \overline { g } _ { m , \alpha } \sum _ { i = 0 } ^ { m - 1 } \sum _ { \ell = 1 } ^ { m } G ( S _ { j m - i } , S _ { j m + \ell } ) , \\ & \underline { \hat { W } } _ j : = \overline { g } _ { j m , \alpha } \overline { g } _ { m , \alpha } \sum _ { i = m } ^ { j m - 1 } \sum _ { \ell = 1 } ^ { m } G ( S _ { j m - i } , S _ { j m + \ell } ) , \end{align*}


\begin{align*} g _ { 1 / \{ \alpha \} } ( n ) = \begin{cases} 1 n \in X \\ 0 n \notin X \end{cases} g _ { 1 / \{ \beta \} } ( n ) = \begin{cases} 1 n \in Y \\ 0 n \notin Y . \end{cases} \end{align*}


\begin{align*} ( \alpha _ 1 , \beta _ 1 ) \circ ( \alpha _ 2 , \beta _ 2 ) = ( \alpha _ 1 + ( - 1 ) ^ { \Re ( \alpha _ 1 ) } \alpha _ 2 , \beta _ 1 + ( - 1 ) ^ { \Re ( \alpha _ 1 ) } \beta _ 2 ) \end{align*}


\begin{align*} G ( \alpha ) _ i = \sum _ { j = 0 } ^ N \pi _ * ( \alpha _ j \cdot \zeta ^ { j + i } ) = \sum _ { j = 0 } ^ N \alpha _ j \cdot \underbrace { \pi _ * ( \zeta ^ { j + i } ) } _ { = 0 j < N - i } \ , . \end{align*}


\begin{align*} \mathcal { C } ( f ) ( \omega _ 1 ) = \iint _ { \omega _ 2 , \omega _ 3 , \omega _ 4 \geq 0 } W [ ( f _ 1 + f _ 2 ) f _ 3 f _ 4 - ( f _ 3 + f _ 4 ) f _ 1 f _ 2 ] \ , \dd \omega _ 3 \ , \dd \omega _ 4 . \end{align*}


\begin{align*} g ( s ) : = \sqrt { 2 / k _ { + } } e ^ { - i \frac { \pi } { 4 } } H _ { \theta _ c } ( s ) H _ { \pi - \theta _ c } ( s ) \sqrt { s - s _ b ^ { * } } F ( s ) \frac { d \zeta ( s ) } { d s } \frac { 2 i \sin \zeta ( s ) } { n ^ 2 - 1 } \end{align*}


\begin{align*} \xi _ { j , i } ( e _ f ) = \sum _ { g \in I ( n , d ) } p _ { i , j } ^ { f , g } ( u ) e _ g , \forall f \in I ( n , d ) , \end{align*}


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


\begin{align*} 0 \leq \kappa _ { \eta _ 1 } ( l ) - \kappa _ { \eta _ 2 } ( l ) = \dot \alpha _ 2 ( l ) \sin ( \alpha _ 2 ( l ) ) - \dot \alpha _ 1 ( l ) \sin ( \alpha _ 1 ( l ) ) - \frac { 2 l } { 1 + l ^ 2 } \bigl ( \cos ( \alpha _ 2 ( l ) ) - \cos ( \alpha _ 1 ( l ) ) \bigr ) . \end{align*}


\begin{align*} \omega _ 0 ( z ) = \omega ( z ) = \rho ( z , 0 ) = \frac { 1 } { 2 } \log \frac { 1 + \norm { z } } { 1 - \norm { z } } . \end{align*}


\begin{align*} \bar { t } = t , \bar { x } = x , \bar { y } = \psi ( t , x , y ) , \end{align*}


\begin{align*} \sigma _ 1 \sigma _ 3 - \sigma _ 4 = 0 . \end{align*}


\begin{align*} 3 \omega _ { 1 2 } ( e _ 1 ) = \frac { \mu _ 2 } { \kappa _ 2 } \omega _ { 5 6 } ( e _ 1 ) + e _ 2 \big ( \log \frac { \kappa _ 2 } { \kappa _ 1 } \big ) - * d \log \kappa _ 1 ( e _ 1 ) \end{align*}


\begin{align*} \sum _ { i + j = \ell } e _ i ( z ) e _ j ( \hat { z } ) - \hat { g } _ { \ell } ( z , \zeta , q ) \ / ; 1 \leq \ell \leq n \ / . \end{align*}


\begin{align*} S \left ( t \right ) = \mathfrak { L } \left [ \frac { 1 } { t - x } \right ] = { \displaystyle \sum \limits _ { n \geq 0 } } \frac { \mu _ { n } } { t ^ { n + 1 } } , \end{align*}


\begin{align*} b ( y , s ) = h _ 1 \gamma | D w ( y , s ) | ^ { \gamma - 2 } D w ( y , s ) , \end{align*}


\begin{align*} \bigl [ ( \bar { \nu } - \varepsilon \bar { \tau } _ \varepsilon ^ i ) \cdot \nabla \rho ^ \varepsilon \bigr ] ( x , t ) & = \varepsilon ^ { - 1 } \partial _ r \eta _ 0 ( r ) + \partial _ r \eta _ 1 ( r ) \\ & + \varepsilon \{ \partial _ r \eta _ 2 ( r ) - \nabla _ \Gamma g _ i \cdot \nabla _ \Gamma \eta _ 0 ( r ) \} + O ( \varepsilon ^ 2 ) . \end{align*}


\begin{align*} \Omega = \displaystyle \frac { \rho ^ { \gamma - 1 } } { \Gamma ( \gamma ) \rho ^ { \gamma - 1 } - \sum ^ { m } _ { i = 1 } \omega _ { i } ( \xi _ { i } ^ { \rho } ) ^ { \gamma - 1 } } . \end{align*}


\begin{align*} u : = - \gamma ( \Delta p + G ) . \end{align*}


\begin{align*} ( T _ t g ) ( x ) = g ( x - t ) \ , , \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ 2 } | \nabla u _ { ( x , 0 ) , r _ i } | ^ 2 \eta d \mathcal { H } ^ 2 & = - \int _ { \mathbb { R } ^ 2 } u _ { ( x , 0 ) , r _ i } \nabla u _ { ( x , 0 ) , r _ i } \cdot \nabla \eta d \mathcal { H } ^ 2 \\ & \rightarrow - \int _ { \mathbb { R } ^ 2 } u _ { \infty } \nabla u _ { \infty } \cdot \nabla \eta d \mathcal { H } ^ 2 = \int _ { \mathbb { R } ^ 2 } | \nabla u _ { \infty } | ^ 2 \eta d \mathcal { H } ^ 2 . \end{align*}


\begin{align*} \beta _ { x } ( s ) = \prod _ { \ell = 1 } ^ { \infty } \frac { \lambda ^ u _ { \psi ^ x _ { - \ell } ( s ) } } { \lambda ^ u _ { x _ { - \ell } } } . \end{align*}


\begin{align*} d ( a \otimes b ) = ( a \otimes g + f \otimes b ) d t + d \boldsymbol \Gamma ^ { \mathbf { A } , \mathbf { B } } [ a \otimes b ] \ , . \end{align*}


\begin{align*} \acute { \chi } \bigl ( ( \alpha , e ) , ( \alpha ' , e ' ) \bigr ) = \sum _ { \substack { ( j , k ) , ( j ' , k ' ) \in \acute { J } \\ k , k ' \textnormal { e v e n } } } ( - 1 ) ^ { k / 2 } \ , \acute { M } _ { j , k } ^ { j ' , k ' } \ , \acute { S } _ { \alpha , e ; j , k , k / 2 } \ , \acute { S } _ { \alpha ' , e ' ; j ' , k ' , k ' / 2 } \ , . \end{align*}


\begin{align*} \mathcal { P } _ { A , \infty } ( k ) & : = \prod _ { a \in A } \left \{ m _ { | a | } \delta \left ( \sum _ { l \in a } ( k _ l - k _ { l + 1 } ) \right ) \prod _ { l \in a } \widehat { B } ( k _ l - k _ { l + 1 } ) \right \} . \end{align*}


\begin{align*} m ( ( a _ i , a _ o ) ) = a _ i \sqcup a _ o \ ; . \end{align*}


\begin{align*} \Phi = \varphi _ 1 ( x _ 1 , \dots , x _ n ) \land y \leq z \ , \& \dots \& \ , \varphi _ m ( x _ 1 , \dots , x _ n ) \land y \leq z \Longrightarrow y \leq z . \end{align*}


\begin{align*} \frac { \pi ^ 2 } { 4 } E \Theta _ n = ( 1 + o ( 1 ) ) \log _ 2 n \ , . \end{align*}


\begin{align*} & u _ 0 ( x ) = A + o ( 1 ) , x \rightarrow + \infty , \\ & u _ 0 ( x ) = o ( 1 ) , x \rightarrow - \infty . \end{align*}


\begin{align*} c _ k ( Y , \lambda ) = c _ k \left ( Y , e ^ { f _ 1 } \lambda \right ) . \end{align*}


\begin{align*} \varphi ( [ x , y ] ) = \frac { 1 } { 2 } \left ( [ \varphi ( x ) , y ] + ( - 1 ) ^ { | \varphi | | x | } [ x , \varphi ( y ) ] \right ) , \ \ x , y \in { \mathfrak L } _ 0 \cup { \mathfrak L } _ 1 . \end{align*}


\begin{align*} \kappa _ G = \begin{cases} 1 , & \textrm { i f } G \textrm { i s o f t y p e A } , \\ 2 , & \textrm { i f } G \textrm { i s o f t y p e C } . \end{cases} \end{align*}


\begin{align*} D ' _ { \nu } ( X ) = ( \nu 2 X / c _ K , ( \nu + 1 ) 2 X / c _ K ] . \end{align*}


\begin{align*} 1 & = \| x ^ * \| = \Bigl \| \sum _ { i = 1 } ^ \infty x ^ * _ i \Bigr \| \geqslant \Bigl \| \sum _ { i = 1 } ^ \infty a _ i y ^ * _ i \Bigr \| - \| x _ 1 ^ * \| - \Delta \\ & \geqslant \frac { 1 } { C } \Bigl \| \sum _ { i = 1 } ^ \infty a _ i e ^ * _ { t _ i } \Bigr \| _ { T ^ * } - 2 - \Delta \geqslant \frac { 1 } { C } \Bigl \| \sum _ { i = 1 } ^ \infty \| x ^ * _ { i + 1 } \| e ^ * _ { t _ i } \Bigr \| _ { T ^ * } - 2 - 2 \Delta . \end{align*}


\begin{align*} \eta _ p ^ { \epsilon , { \bf h } } ( t ) = e ^ { - \frac { t } { \epsilon ^ 2 } } h _ p + \frac { 1 } { \epsilon } \int _ 0 ^ t e ^ { - \frac { t - s } { \epsilon ^ 2 } } D f ( q ^ \epsilon ( s ) ) . \eta _ q ^ { \epsilon , { \bf h } } ( s ) d s + \frac { 1 } { \epsilon } \int _ 0 ^ t e ^ { - \frac { t - s } { \epsilon ^ 2 } } D \sigma ( q ^ \epsilon ( s ) ) . \eta _ q ^ { \epsilon , { \bf h } } ( s ) d \beta ( s ) . \end{align*}


\begin{align*} \begin{array} { c c } \left [ Q ^ q u \right ] _ j ( x , t ) = \widetilde { F } ( \bar v _ { 1 } ^ u , \dots , \bar v _ { n } ^ u ) , \end{array} \end{align*}


\begin{align*} \phi ( k ) = \begin{cases} 2 k + 2 , & , \cr 2 k + 3 , & . \cr \end{cases} \end{align*}


\begin{align*} \square [ \underline { n } ] = \square [ n _ 1 ] * \dots * \square [ n _ p ] \end{align*}


\begin{align*} \phi ( b , s ) = s h ( b ) - s \int _ { s _ 0 } ^ s t ^ { - 2 } g ( b ^ 2 - t ^ 2 ) \ , d t . \end{align*}


\begin{align*} X ^ \odot = \overline { \mathcal { D } ( A _ 0 ^ \star ) } , \end{align*}


\begin{align*} A ^ * ( S , \mathbb { Q } ) [ \zeta ] / ( Q ) = \bigoplus _ { i = 0 } ^ N A ^ * ( S , \mathbb { Q } ) \cdot \zeta ^ i \end{align*}


\begin{align*} h ^ 0 ( X , k H ) = h ^ 0 ( X , ( k - 1 ) H ) + h ^ 0 ( H , k H | _ H ) \end{align*}


\begin{align*} ( A \otimes B ) ( x ) = A ( \Phi _ { \cdot 0 } ^ \infty ( x ) ) \cdot B ( \Phi _ { \cdot 1 } ^ \infty ( x ) ) \ ; . \end{align*}


\begin{align*} ( \det ^ { \mathrm { p l } } ) ^ * \ , \Xi ( S _ { j , k , l } ) = \frac { 1 } { l ! } \biggl ( \sum _ { \substack { ( j ' , k ' ) \in J \\ ( j ' , k ' ) \neq ( 1 , 0 ) } } \epsilon _ { j ' , k ' } \boxtimes \Xi ( S _ { j ' , k ' , 1 } ) \biggr ) ^ l \setminus \epsilon _ { j , k } \ , . \end{align*}


\begin{align*} \frac { \partial u _ \epsilon } { \partial t } \ , - \ , \triangle u _ \epsilon \ , + \ , \frac 2 { \epsilon ^ 2 } ( u _ \epsilon ^ 2 \ , - \ , 1 ) u _ \epsilon \ ; = \ ; 0 . \end{align*}


\begin{align*} X = \Gamma \left ( x _ 0 + \int _ 0 ^ { \cdot } f ( s , X _ s ) d s + W \right ) , \end{align*}


\begin{align*} \nu _ z ( v ) & = \gamma _ z ( v ' ) - \gamma _ z ( v ) \\ \nu _ { x x } ( v ) & = \gamma _ { x x } ( v ' ) - 2 t \gamma _ { x z } ( v ' ) + t ^ 2 \gamma _ { z z } ( v ' ) - \gamma _ { x x } ( v ) , \end{align*}


\begin{align*} ( M \oplus N ) = \begin{pmatrix} M & 0 \\ 0 & N \end{pmatrix} \ ; . \end{align*}


\begin{align*} B ( \alpha , b ) + Q ( \alpha ) \tau + B ( \alpha , a ) \tau = B ( \alpha , a \tau + b ) + Q ( \alpha ) \tau = B ( \alpha , z ) + Q ( \alpha ) \tau \end{align*}


\begin{align*} [ A ] ( \vec { x } ) = \sum _ { \vec { y } \in \mathbb { N } ^ b } A ( \vec { x } \sqcup ( \vec { y } \sqcup \vec { y } ) ) \ ; . \end{align*}


\begin{align*} 0 = \sigma _ 0 ( \Omega , g , \beta ) < \sigma _ 1 ( \Omega , g , \beta ) \le \sigma _ 2 ( \Omega , g , \beta ) \le \dotso \nearrow \infty . \end{align*}


\begin{align*} \int _ { N ( F ) } ( R ( \varphi _ \pi ) ( f ) ) ( n g ) d n = 0 \forall g \in G ( F ) . \end{align*}


\begin{align*} { B } _ \le : = \{ \mathrm { k } \in { B } : \forall 1 \le i \le \ell : k _ i \le s _ n + \nu \} , { B } _ > : = \{ \mathrm { k } \in { B } : \exists 1 \le i \le \ell : k _ i > s _ n + \nu \} . \end{align*}


\begin{align*} \mathcal S _ \nu = ( \Omega ^ * , \mathcal F ^ * , P \alpha _ \nu ^ { - 1 } ) \ \ \ \mbox { a n d } \ \ \ \mathcal B _ \nu = ( \Omega ^ * , \mathcal F ^ * , ( \mathcal F ^ * _ t ) _ { t \in T } , P \alpha _ \nu ^ { - 1 } ) . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ { k - 1 } W _ j \le 3 \varepsilon \frac { n \log _ 3 n } { ( \log m ) ^ 2 } \ , . \end{align*}


\begin{align*} \mathbb { E } \left [ \prod _ { i = 1 } ^ l X ( x _ { k _ i } , s _ { k _ i } , x _ { k ' _ i } , s _ { k ' _ i } ) \prod _ { j = 1 } ^ l | X ( x _ { \hat { k } _ j } , s _ { \hat { k } _ j } ) \right ] . \end{align*}


\begin{align*} \lambda ( g ) = \lambda ( f ) + \sum _ { \ell \mid M } [ \delta ( f , \ell ) - \delta ( g , \ell ) ] . \end{align*}


\begin{align*} \begin{aligned} \partial _ t u = [ \lambda _ 1 u \times \partial ^ 2 _ x u - \lambda _ 2 u \times ( u \times \partial _ x ^ 2 u ) + \lambda _ 1 u \times g ' ( u ) - \lambda _ 2 u \times ( u \times g ' ( u ) ) ] \dd t + u \times \circ d W \ , , \end{aligned} \end{align*}


\begin{align*} \widetilde S : = { \rm I m } ( S \hookrightarrow A [ \underline { x } ] \to A [ \underline { x } , \underline { y } ] ) , \end{align*}


\begin{align*} P ( t ) U ( t + T , s + j T ) = U ( t + T , s + j T ) P ( s ) \end{align*}


\begin{align*} H \ & \subseteq \ \big \{ j _ 1 , \dots , j _ 1 + ( t - 1 ) , \ldots , j _ { s - 1 } , \dots , j _ { s - 1 } + ( t - 1 ) , \ j _ s + 1 , \dots , n \big \} \\ & = \ [ n ] \setminus F _ s = F . \end{align*}


\begin{align*} z = \left \{ \begin{aligned} & \cos \eta , & & 0 < \eta < \pi , & & \textrm { f o r } ~ - 1 < z < 1 , \\ & \cos i \eta , & & \eta \ge 0 , & & \textrm { f o r } ~ z \ge 1 , \\ & \cos ( \pi + i \eta ) , & & \eta \le 0 , & & \textrm { f o r } ~ z \le - 1 . \end{aligned} \right . \end{align*}


\begin{align*} M _ { k , n } \ \mapsto \sum _ { J \subset \{ 1 , \ldots , r + 1 \} } ^ { | J | = k } \prod _ { i \in J } x _ i ^ n \cdot \prod _ { i \in J } ^ { j \notin J } \frac { x _ i } { x _ i - x _ j } \cdot \prod _ { i \in J } \Gamma _ i \ , . \end{align*}


\begin{align*} \alpha _ 0 ( A ) _ M ( i , j ) = A _ M ( \Phi _ \sqcup ^ \alpha ( i ) , \Phi _ \sqcup ^ \alpha ( j ) ) \ ; , \end{align*}


\begin{align*} \omega ( \delta ) : = \delta \Theta ^ { - 1 } \left ( \Theta ( C _ \Theta ) \vee \left ( \frac 1 { \delta } \right ) ^ { \frac { \alpha } { \alpha - 1 } } \right ) \end{align*}


\begin{align*} \lambda \partial _ i Q + \partial R _ i = 0 \end{align*}


\begin{align*} F | _ { x _ { i , 2 } = q x _ { i , 1 } } \mathrm { i s \ d i v i s i b l e \ b y } ( x _ { j , 1 } - \gamma x _ { i , 1 } ) ^ { \flat _ { i j } ( \gamma ) } \\ \end{align*}


\begin{align*} \mathbb { P } \left \{ \int _ { 0 } ^ { 1 } g _ s d D X _ s = 0 , g _ t \neq 0 \right \} = 0 . \end{align*}


\begin{align*} v _ \ell = \Big ( \prod _ { s = \ell } ^ { d - 1 } x _ { n - s t - 1 } \Big ) \Big ( \prod _ { s = 0 } ^ { \ell - 1 } x _ { n - s t } \Big ) , \ell \in \{ 0 , \dots , d \} . \end{align*}


\begin{align*} \lambda _ { k } = \frac { 1 } { \omega } [ \ln | \rho _ { k } | + i \ , ( \arg \rho _ { k } + 2 m \pi ) ] \textnormal { w i t h $ m \in \mathbb { Z } $ } , \end{align*}


\begin{align*} \int _ { \R ^ n } \rho _ i ( x ) \ , d x = 1 \ \textrm { f o r a l l } i \in \N \quad \textrm { a n d } \lim _ { i \to \infty } \int _ { | x | > \delta } \rho _ i ( x ) \ , d x = 0 \quad \textrm { f o r a l l } \delta > 0 . \end{align*}


\begin{align*} M ( 1 ) : = \langle R ^ G _ { T , \chi } \otimes \omega _ \psi ^ \vee , R ^ G _ { S , \eta } \rangle _ { G ^ F } \end{align*}


\begin{align*} v ^ u ( \phi ^ t ( \Phi ^ c _ x ( s ) ) ) = D \phi ^ t ( v ^ u ( \Phi ^ c _ x ( s ) ) ) \ : \ : \ : \textrm { a n d } \ : \ : \ : D \phi ^ t ( \Phi ^ c _ x ( s ) ) \frac { d } { d s } \Phi ^ c _ x ( s ) . \end{align*}


\begin{align*} [ e _ i ( z ) , f _ j ( w ) ] = \frac { \delta _ { i j } } { q - q ^ { - 1 } } \delta \left ( \frac { z } { w } \right ) \left ( \psi ^ + _ i ( z ) - \psi ^ - _ i ( z ) \right ) \ , , \end{align*}


\begin{align*} G = \frac { q } { 2 } \left ( \bar { x } _ { m - 1 } x _ { m - 2 } g + x _ { m - 1 } \bar { x } _ { m - 2 } \left ( g + x _ { \pi ( 0 ) } + m - 2 \right ) \right ) + c , \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty \sum _ { | m | \leq n / 2 } ( - 1 ) ^ m q ^ { n ^ 2 - 2 m ^ 2 } ( 1 + q ^ { \alpha m + \beta n + k } ) . \end{align*}


\begin{align*} \{ a _ n \} _ { n = 0 } ^ \infty = \left \{ [ n \alpha ] \right \} _ { n = 0 } ^ \infty \end{align*}


\begin{align*} G ( F ) _ { \bar \nu } \coloneqq \{ \gamma \in G ( F ) \mid \bar \nu ^ { K V } _ G ( \gamma ) = \bar \nu \} \end{align*}


\begin{align*} E : = \{ ( z , w ) \in \partial W \colon \ \partial _ z \theta ( z ) = 0 , \ w \neq 0 \} , \end{align*}


\begin{align*} \Delta ( \hat { \boldsymbol { w } _ \varepsilon } \phi ) = \nabla ^ { \bot } ( \hat { \omega _ \varepsilon } \phi + \hat { \boldsymbol { w } _ \varepsilon } \cdot \nabla ^ \bot \phi ) + \nabla [ \nabla \cdot ( \hat { \boldsymbol { w } _ { \varepsilon } } \phi ) ] . \end{align*}


\begin{align*} \beta _ { a } ( I ) = \beta _ { a , b } ( I ) & = \big | \big \{ w \in \mathcal { L } _ t ^ f ( u ) : \min ( u ) = 1 \big \} \big | - \big | \big \{ w \in \mathcal { L } _ t ^ f ( v ) \setminus \{ v \} : \max ( w ) = n \big \} \big | \\ & = \big | \big \{ w \in \mathcal { L } _ t ^ f ( u ) : \min ( w ) = 1 \big \} \big | - \big | \big \{ v _ { \ell + 1 } > \dots > v _ d \big \} \big | \\ & = \big | \big \{ w \in \mathcal { L } _ t ^ f ( u ) : \min ( w ) = 1 \big \} \big | - ( d - \ell ) = 0 , \end{align*}


\begin{align*} \widehat r ( \alpha ) = \int _ { \mathbb R } r ( u ) \ , e ( - \alpha u ) \ , d u , e ( x ) : = e ^ { 2 \pi i x } . \end{align*}


\begin{align*} & F ( \rho + \epsilon \nu ) = \int _ M \int _ M W ( x , y ) d ( \rho + \epsilon \nu ) ( x ) d ( \rho + \epsilon \nu ) ( y ) \\ & = F ( \rho ) + \epsilon ^ 2 F ( \nu ) + \epsilon \left ( \int _ M \int _ M W ( x , y ) d \rho ( x ) d \nu ( y ) + \int _ M \int _ M W ( x , y ) d \rho ( y ) d \nu ( x ) \right ) . \end{align*}


\begin{align*} f ( \alpha ) & = \beta _ \alpha ( a _ 1 ^ 1 , \dots , a _ 1 ^ k , \dots , a _ n ^ 1 , \dots , a _ n ^ k ) \\ & = \beta _ \alpha ( f ( z _ 1 ^ 1 ) , \dots , f ( z _ 1 ^ k ) , \dots , f ( z _ n ^ 1 ) , \dots , f ( z _ n ^ k ) ) \\ & = f ( \beta _ \alpha ) \end{align*}


\begin{align*} S ^ { ( k ) } ( n ) \equiv \sum _ { i = 0 } ^ { n } F ^ { ( k ) } ( i ) \end{align*}


\begin{align*} M = \begin{pmatrix} M \rvert _ { 0 0 } & M \rvert _ { 0 1 } \\ M \rvert _ { 1 0 } & M \rvert _ { 1 1 } \end{pmatrix} \ ; . \end{align*}


\begin{align*} \breve { A } - \lambda \breve { B } : = \left [ \begin{array} { c c } A & U \\ V ^ * & 0 \end{array} \right ] \left [ \begin{array} { c c } x \\ z \end{array} \right ] = \lambda \ , \left [ \begin{array} { c c } B & 0 \\ 0 & 0 \end{array} \right ] \left [ \begin{array} { c c } x \\ z \end{array} \right ] . \end{align*}


\begin{align*} \phi _ x + i k \sigma _ 3 \phi = U _ { + } \phi , \phi _ t + 4 i k ^ 3 \sigma _ 3 \phi = V _ { + } ( k ) \phi , \end{align*}


\begin{align*} \widetilde { \mathcal { A } _ 0 } & : = \left \{ R _ 0 \in S O ( 3 ) : R _ 0 e _ 3 = e _ 3 \right \} \times ( 0 , \infty ) \times \left ( - \infty , 0 \right ) \end{align*}


\begin{align*} a \otimes b = b \otimes a \ ; . \end{align*}


\begin{align*} a ( \exp ( \pi \vee ) X ) = \exp ( X \vee ) \otimes \exp ( \pi \vee ) ( a X ) . \end{align*}


\begin{align*} j = \begin{pmatrix} 0 & 1 \\ - 1 & 0 \end{pmatrix} s = \begin{pmatrix} 0 & 1 _ p \\ 1 _ p & 0 \end{pmatrix} \ , , \end{align*}


\begin{align*} u ( x , t ) = O \left ( ( - t ) ^ { - \frac { 1 } { 2 } } e ^ { 8 t \kappa _ { \delta } ( 3 \xi - \kappa _ { \delta } ^ 2 ) } \right ) . \end{align*}


\begin{align*} \mathcal L G = \Psi \end{align*}


\begin{align*} \sum _ i ( A _ { k l } B _ { f i i } ) = A _ { k l } ( \sum _ i B _ { f i i } ) \ ; . \end{align*}


\begin{align*} \begin{aligned} \int _ I \int _ { A _ { ( t , s ) } } \| D _ x X ( t , s , x ) \| ^ p \dd x \dd s & \le \int _ I \int _ { A _ { ( t , s ) } } \exp \left ( p \left | \int _ s ^ t \| D _ x b ( v , X ( v , s , x ) ) \| \dd v \right | \right ) \dd x \dd s \\ & \le \int _ I \int _ { A } \int _ { I } \frac { \chi _ { A ( v , s ) } ( x ) } { \ell _ A ( s , x ) } \exp \left ( \ell p \| D _ x b ( v , X ( v , s , x ) ) \| \right ) \dd v \dd x \dd s , \end{aligned} \end{align*}


\begin{align*} \alpha \int _ { \mathbb { R } ^ 3 } \mathrm { d } k \ ; f _ y ( k ) \ , \left \langle x u _ { \alpha } \ , \bigg | \ , \frac { ( h _ { \alpha } - e _ { \alpha } ) | k | } { 3 ( h _ { \alpha } - e _ { \alpha } + | k | ) } \ , \bigg | \ , x u _ { \alpha } \right \rangle = \frac { \alpha ^ 2 } { L ^ 3 } - \aleph _ { \alpha , L } \frac { \alpha } { L ^ 4 } + O \Big ( \frac { \alpha ^ 3 } { L ^ 2 } \log ( \alpha ^ { - 1 } ) \Big ) , \end{align*}


\begin{align*} E ( M ) = \bigcup _ { x \in G } x ^ { - 1 } E ( \overline { T } ) x . \end{align*}


\begin{align*} u _ { \max a } = - \sum _ { j \in a \setminus \{ \max a \} } u _ j . \end{align*}


\begin{align*} ( F ^ { \diamondsuit } ) ^ { \phi } = \det ( f ^ { \partial } ) ^ { - 2 s } \cdot ( F ^ { \sigma \tau } ) ^ { \diamondsuit } . \end{align*}


\begin{align*} a _ n = \int _ { - \sqrt { 1 2 / n } } ^ 0 F _ n ( x ) \ , d x . \end{align*}


\begin{align*} \omega _ \psi ( s ) = \vartheta _ T ( s ) q ^ { \frac { 1 } { 2 } \dim V ^ s } , s \in T ^ F . \end{align*}


\begin{align*} G _ \jmath : = C _ G ( S _ \jmath ) = Z _ \jmath \times G _ { \jmath , [ 1 ] } , \end{align*}


\begin{align*} \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 6 , v _ j \rangle = 0 , \end{align*}


\begin{align*} x _ n \sim \rho , \limsup _ { n \to \infty } y _ n < \rho ^ { - m } \quad S _ n : = \frac { x _ n ^ m y _ n } { 1 - x _ n ^ m y _ n } = \Theta ( 1 ) . \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { N \to \infty } \left | \mathcal { E } ^ { \neq } ( { \rm e m p } _ { N } - \mu _ { \theta } ) - \mathcal { E } ( \rho - \mu _ { \theta } ) \right | = 0 , \end{align*}


\begin{align*} \mathcal N _ { c , \phi } ( x ) = \left \{ m \in D ^ * : \phi ( x ) + \phi ^ c ( m ) = c ( x , m ) \right \} \end{align*}


\begin{align*} \mathcal { P } _ { A , L } : ( \Lambda _ L ^ * ) ^ { n + 1 } \to \R , \ k \mapsto \mathcal { P } _ { A , L } ( k ) = \prod _ { a \in A } \left \{ m _ { | a | } \delta _ { * , L } \left ( \sum _ { l \in a } ( k _ l - k _ { l + 1 } ) \right ) \prod _ { l \in a } \widehat { B } _ \# ( k _ l - k _ { l + 1 } ) \right \} . \end{align*}


\begin{align*} \begin{aligned} & \underset { y _ { 1 } , \hdots , y _ { m } } { } & & \sum _ { i = 1 } ^ { m } h _ { i } ^ { * } ( y _ { i } ) \\ & & & \sum _ { i = 1 } ^ { m } y _ { i } = \textbf { 0 } _ { n } \end{aligned} \end{align*}


\begin{align*} P : = x ^ { 2 s } \Delta _ { g _ s } | _ { U ' } = \mathsf D ^ 2 + a \mathsf D + \Delta _ { g _ F } + o ( 1 ) , \end{align*}


\begin{align*} T = O p _ w ( \sigma _ 1 ) \mu ( \chi ) \mbox { a n d } T = \mu ( \chi ) O p _ w ( \sigma _ 2 ) . \end{align*}


\begin{align*} S ^ i ( n ) = 2 \cdot 3 ^ i 4 ^ { v _ 2 - i } q _ 2 + 1 \equiv 1 \pmod { 4 } \end{align*}


\begin{align*} \Psi _ { \epsilon } ( x ) = \frac { 1 } { \epsilon ^ { N } } \Psi \left ( \frac { x } { \epsilon } \right ) \ , \ , \mbox { a n d } \ , \ , \Pi _ { \epsilon } ( x ) = \frac { 1 } { \epsilon ^ { N } } \Pi \left ( \frac { x } { \epsilon } \right ) , \end{align*}


\begin{align*} [ a _ { \beta } ( k ) , a _ { \gamma } ( h ) ] = 0 = [ a _ { \beta } ^ { \dagger } ( k ) , a _ { \gamma } ^ { \dagger } ( h ) ] , [ a _ { \beta } ( k ) , a ^ { \dagger } _ { \gamma } ( h ) ] = \delta _ { \beta , \gamma } \delta ( k - h ) . \end{align*}


\begin{align*} K _ { k , t } c : = \int _ t ^ 1 H _ { k , \frac s 2 } \big ( \partial _ s \Psi _ { k , s } c \big ) d s \ : . \end{align*}


\begin{align*} \frac { z f ' ( z ) } { f ( z ) } = 1 + \frac { z h ' ( z ) } { h ( z ) } + \frac { z k ' ( z ) } { k ( z ) } + \frac { z p ' ( z ) } { p ( z ) } \end{align*}


\begin{align*} \hat { f } ( \xi ) = \int _ { \mathbb { R } ^ d } f ( x ) e ^ { - 2 \pi i \xi \cdot t } d t \ \ \ \ t \in \mathbb { R } ^ d , \end{align*}


\begin{align*} \Theta ( \rho _ x ) = \Theta ( R _ x ) . \end{align*}


\begin{align*} \frac d { d t } \sum \limits _ { \beta = 0 } ^ { 1 } \sum \limits _ { | \alpha | = 0 } ^ { 3 - \beta } \left ( \| \sqrt { B } \partial _ \tau ^ \alpha \partial _ y ^ \beta u \| _ { L ^ 2 } ^ 2 + \| \sqrt { A } \partial _ \tau ^ \alpha \partial _ y ^ \beta \tilde { h } \| _ { L ^ 2 } ^ 2 \right ) + \frac { C _ 0 } { 2 } D ( t ) \le C E ( t ) ^ { \frac 5 3 } . \end{align*}


\begin{align*} [ E _ { i j } [ r ] , E _ { k l } [ s ] ] = \delta _ { k j } E _ { i l } [ r + s ] - \delta _ { i l } E _ { k j } [ r + s ] + r \delta _ { r , - s } K ( \delta _ { k j } \delta _ { i l } - \frac { \delta _ { i j } \delta _ { k l } } { n } ) . \end{align*}


\begin{align*} ( H _ b ^ E - z ) U _ L ( z ) & = 1 + W _ L ( z ) , \end{align*}


\begin{align*} [ Q u ] _ j ( x , t ) = \left \{ \begin{array} { l c l } \displaystyle \left [ S R u \right ] _ j ( x , t ) & \mbox { i f } & x _ j ( x , t ) = 0 \mbox { o r } x _ j ( x , t ) = 1 \\ c _ j ( x _ j ( x , t ) , x , t ) \varphi _ j ( x _ j ( x , t ) ) & \mbox { i f } & x _ j ( x , t ) \in ( 0 , 1 ) , \end{array} \right . \end{align*}


\begin{align*} \partial _ r \eta _ 2 ( r ) - \nabla _ \Gamma g _ i \cdot \nabla _ \Gamma \eta _ 0 ( r ) + k _ d ^ { - 1 } ( \partial ^ \circ g _ i ) \eta _ 0 ( r ) + k _ d ^ { - 1 } V _ \Gamma \eta _ 1 ( r ) = 0 \end{align*}


\begin{align*} \left \{ \lambda _ \tau = e ^ { f _ \tau } \lambda \right \} _ { \tau \in [ 0 , 1 ] } \end{align*}


\begin{align*} X ( x , t ) : = \frac { 1 } { 2 ^ t } \sum _ { S \in R W ( x , t ) } \prod _ { s = 1 } ^ t [ 1 + \xi ( S ( s ) , s ) ] \end{align*}


\begin{align*} m ( \pi , \sigma ) : = \langle \pi \otimes \omega _ \psi ^ \vee , \sigma \rangle _ { G ^ F } = \dim \textrm { H o m } _ { G ^ F } ( \pi \otimes \omega _ { \psi } ^ \vee , \sigma ) . \end{align*}


\begin{align*} \lim \limits _ { \epsilon \to 0 ^ + } \phi _ { \epsilon } \ast u ( x ) = u ( x ) \end{align*}


\begin{align*} \phi ^ \pi _ n ( \gamma _ 1 \vee \cdots \vee \gamma _ n \otimes m ) = \sum _ { k = 0 } ^ \infty \frac { 1 } { k ! } \phi _ { n + k } ( \pi \vee \cdots \vee \pi \vee \gamma _ 1 \vee \cdots \vee \gamma _ n \otimes m ) \end{align*}


\begin{align*} \partial _ t c _ i - \nabla c _ i \cdot \nabla p = c _ i ( G _ i - G ) . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u & = f ( x , u , \nabla u ) + g ( x , u ) \ ; \ ; & & \mbox { i n } \ ; \ ; \Omega , \\ u & > 0 & & \mbox { i n } \ ; \ ; \Omega , \\ u & = 0 & & \mbox { o n } \ ; \ ; \partial \Omega , \end{alignedat} \right . \end{align*}


\begin{align*} a _ 1 : = \sup I _ L . \end{align*}


\begin{align*} [ 1 * ] _ n = q t ^ { n - 1 } . \end{align*}


\begin{align*} \begin{dcases} d ^ \star ( j \circ u ) ( t ) = A _ 0 ^ { \odot \star } j u ( t ) + B ( t ) u ( t ) , & t \geq s , \\ u ( s ) = \varphi , & \varphi \in X , \end{dcases} \end{align*}


\begin{align*} \langle H _ y \rangle _ { u _ { \alpha } \otimes \Omega } & = e _ { \alpha } + \alpha \| \lambda _ y \| ^ 2 + \langle \alpha V \rangle _ { u _ { \alpha } } \\ & = e _ { \alpha } + \alpha \| \lambda _ y \| ^ 2 - \frac { \alpha ^ 2 } { L ^ 3 } + O \Big ( \frac { \alpha ^ 2 } { L ^ 5 } \Big ) + O \big ( \alpha ^ 2 L e ^ { - L / 2 } \big ) . \end{align*}


\begin{align*} F _ r = R _ r \ , . \end{align*}


\begin{align*} \| s _ \lambda ( x _ 1 , \dots , x _ N ) \| = s _ \lambda ( 1 , \dots , 1 ) = \dim \mathsf { W } _ N ^ \lambda . \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes \Phi _ y \ , | \ , H _ y R _ y \rangle & = 2 \mathrm { R e } \langle u _ { \alpha } \otimes \Phi _ y ^ { ( 0 ) } \ , | \ , H _ y R _ y \rangle + 2 \mathrm { R e } \langle u _ { \alpha } \otimes 2 \alpha ^ { 3 / 2 } \eta \Phi _ * ^ 1 \ , | \ , H _ y R _ y \rangle \\ & + 2 \mathrm { R e } \langle u _ { \alpha } \otimes R _ * \ , | \ , H _ y R _ y \rangle . \end{align*}


\begin{align*} \sum _ { \substack { \alpha \in P _ r \\ \alpha ^ { - 1 } ( 0 ) = i } } c _ \alpha = c _ \Delta . \end{align*}


\begin{align*} w _ { \ast } = s _ { \alpha _ 1 } s _ { \alpha _ 2 } \ldots s _ { \alpha _ { n - 1 } } s _ { \alpha _ n } s _ { \alpha _ { n - 1 } } \ldots s _ { \alpha _ 2 } s _ { \alpha _ 1 } . \end{align*}


\begin{align*} \widetilde { t } _ { i j k } = t _ { j i k } \ ; . \end{align*}


\begin{align*} { \rm E n t } [ \mathbf { P } _ { 1 } | \mathbf { P } _ { 2 } ] = \begin{cases} \int \frac { d \mathbf { P } _ { 1 } } { d \mathbf { P } _ { 2 } } \log \left ( \frac { d \mathbf { P } _ { 1 } } { d \mathbf { P } _ { 2 } } \right ) d \mathbf { P } _ { 2 } \ { \rm i f } \ \mathbf { P } _ { 1 } \ll \mathbf { P } _ { 2 } \\ \infty \ { \rm o . w , } \end{cases} \end{align*}


\begin{align*} 4 \alpha ^ { 5 / 2 } | \mathrm { R e } \langle u _ { \alpha } \tilde { \Phi } _ * ^ 1 \ , | \ , V _ y \Phi _ { \# } ^ y \rangle | \leq C \alpha ^ { 5 / 2 } \| \tilde { \Phi } _ * ^ 1 \| \| V _ y u _ { \alpha } \| \| \Phi _ { \# } ^ y \| = O ( \alpha ^ 4 ) , \end{align*}


\begin{align*} \omega = e ^ x , f ( \omega ) = F ( x ) ; \end{align*}


\begin{align*} \alpha ^ { \vee } : = \xi _ 1 - \xi _ 2 \end{align*}


\begin{align*} d x ( t ) = f ^ 0 ( t , x ( t ) ) d t + \sum \limits _ { j = 1 } ^ m f ^ j ( t , x ( t ) ) d W _ j ( t ) \ ( t \in [ a , b ] ) \ \ \ \mbox { a n d } \ \ \ x ( a ) = x _ 0 , \end{align*}


\begin{align*} \sum _ { x \in X _ n } \tau ( x ) = \sum _ { x \in X _ n } \tau ( x ) \Lambda ^ { - 1 } _ n ( x ) \Lambda _ n ( x ) \leq b _ { n } ( n + 1 ) \sup _ { x \in X _ n } \Lambda _ n ( x ) . \end{align*}


\begin{align*} \exp \bigl ( - \log ( 1 - x ) \bigr ) = \frac { 1 } { 1 - x } , \end{align*}


\begin{align*} ( q _ i ^ { c _ { i j } } z - w ) f _ i ( z ) f _ j ( w ) = ( z - q _ i ^ { c _ { i j } } w ) f _ j ( w ) f _ i ( z ) \ , , \end{align*}


\begin{align*} | t ( f ^ n ( p ) ) | = \lambda ^ c _ p ( n ) \cdot | t ( p ) | . \end{align*}


\begin{align*} X & = \Omega \times [ 0 , 1 ] / ( ( \omega , 1 ) \sim ( T \omega , 0 ) ) \\ \int _ X f \ , d \nu & = \int _ 0 ^ 1 \int _ \Omega f ( [ \omega , t ] ) \ , d \mu ( \omega ) \ , d t , \end{align*}


\begin{align*} \left \lbrace \begin{aligned} q _ { n + 1 } ^ { \epsilon , \Delta t } & = q _ n ^ { \epsilon , \Delta t } + \frac { \Delta t } { \epsilon } p _ { n + 1 } ^ { \epsilon , \Delta t } \\ p _ { n + 1 } ^ { \epsilon , \Delta t } & = \frac { 1 } { 1 + \frac { \Delta t } { \epsilon ^ 2 } } \Bigl ( p _ n ^ { \epsilon , \Delta t } + \frac { \Delta t f ( q _ n ^ { \epsilon , \Delta t } ) } { \epsilon } + \frac { \sigma ( q _ n ^ { \epsilon , \Delta t } ) } { \epsilon } \Delta \beta _ n \Bigr ) . \end{aligned} \right . \end{align*}


\begin{align*} \gamma ^ 0 = \begin{bmatrix} I _ { 2 \times 2 } & \mathbf 0 \\ \mathbf 0 & - I _ { 2 \times 2 } \end{bmatrix} , \ \gamma ^ j = \begin{bmatrix} \mathbf 0 & \sigma ^ j \\ - \sigma ^ j & \mathbf 0 \end{bmatrix} , \end{align*}


\begin{align*} \gamma _ { A , \ell } = \frac { \dim W } { \dim G _ { A , \ell } - \dim ( G _ { A , \ell } ) _ W } \end{align*}


\begin{align*} \langle \epsilon _ I , Z _ t \rangle = \int _ 0 ^ t \langle \epsilon _ { I ' } , Y _ { s ^ - } \rangle \ d { X } ^ 1 _ s , \end{align*}


\begin{align*} \mathbb { X } ^ { ( 2 ) } _ { s , t } : = \int _ s ^ t X _ { s , r ^ - } \otimes d X _ r + \frac { 1 } { 2 } [ X , X ] ^ c _ { s , t } + \sum _ { s < u \leq t } \Delta X _ u \otimes \Delta X _ u , \end{align*}


\begin{align*} h _ { \beta } ( v ) ( x ) : = \inf \Big \{ r > 0 , \sup _ { ( y , s ) \in \Gamma ( x , r ) } | v ( y , s ) | < \beta \Big \} , \end{align*}


\begin{align*} \gamma _ A : = \max _ { \emptyset \neq I \subseteq \{ 1 , \ldots , n \} } \frac { 2 \dim A _ I } { \dim G _ { A _ I } } . \end{align*}


\begin{align*} \underset { \epsilon \to 0 } \lim ~ \underset { \Delta t \to 0 } \lim ~ \dd _ p ( X _ N ^ { \epsilon , \Delta t } , X ^ \epsilon ( T ) ) = \underset { \Delta t \to 0 } \lim ~ \underset { \epsilon \to 0 } \lim ~ \dd _ p ( X _ N ^ { \epsilon , \Delta t } , X ^ \epsilon ( T ) ) = 0 . \end{align*}


\begin{align*} \gamma ( x ) = \frac { c ^ T x - c ^ T x ^ * } { c ^ T x ^ * } \end{align*}


\begin{align*} \left \{ \begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = f & & ~ ~ \mbox { i n } ~ ~ \Omega \cap Q _ 1 ; \\ & u = g & & ~ ~ \mbox { o n } ~ ~ \partial \Omega \cap Q _ 1 . \end{aligned} \right . \end{align*}


\begin{align*} | f | ^ 2 _ { V ^ 1 } = \langle S f , f \rangle _ { L ^ 2 } . \end{align*}


\begin{align*} \begin{aligned} & \underset { x _ { 1 } , \hdots , x _ { m } } { } & & \sum _ { i = 1 } ^ { m } h _ { i } ( x _ { i } ) \\ & & & x _ { 1 } = x _ { 2 } = \hdots = x _ { m } \end{aligned} \end{align*}


\begin{align*} B \left ( \sum _ { i j } c _ { i j } v _ i \otimes w _ j \right ) = \sum _ { i j } c _ { i j } \beta ( S v _ i , T w _ j ) , \end{align*}


\begin{align*} \begin{array} { l } \underset { \epsilon \rightarrow 0 } { \lim } \left \vert \left ( \breve { F } _ { \epsilon } ( M ) - \breve { F } _ { \epsilon } ( m ) \right ) - \left ( M F _ { \epsilon } ( M ) - m F _ { \epsilon } ( m ) - \int _ { m } ^ { M } F _ { \epsilon } ( y ) d y \right ) \right \vert = 0 . \end{array} \end{align*}


\begin{align*} \int _ X f ^ * y \cdot \mu = \int _ Y y \cdot f _ * \mu . \end{align*}


\begin{align*} \lambda ^ * _ x ( n + m ) = \lambda ^ * _ x ( n ) \lambda ^ * _ { f ^ n ( x ) } ( m ) , \ : \ : \ : * \in \{ s , c , u \} . \end{align*}


\begin{align*} \left \lbrace \begin{aligned} X _ { n + 1 } ^ { \epsilon , \Delta t } & = \Phi ( \frac { \Delta t m _ { n + 1 } ^ { \epsilon , \Delta t } } { \epsilon } , X _ n ^ { \epsilon , \Delta t } ) \\ m _ { n + 1 } ^ { \epsilon , \Delta t } & = m _ n ^ { \epsilon , \Delta t } - \frac { \Delta t } { \epsilon ^ 2 } m _ { n + 1 } ^ { \epsilon , \Delta t } + \frac { \Delta \beta _ n } { \epsilon } \end{aligned} \right . \end{align*}


\begin{align*} Q ^ \epsilon ( s _ 2 ) - Q ^ \epsilon ( s _ 1 ) = \int _ { s _ 1 } ^ { s _ 2 } f ( q ^ \epsilon ( s ) ) d s + \int _ { s _ 1 } ^ { s _ 2 } \sigma ( q ^ \epsilon ( s ) ) d \beta ( s ) , \end{align*}


\begin{align*} \phi ( a ( s \otimes m ) ) = \phi ( s \otimes a m ) = \phi ( ( 1 _ S \otimes a ) ( s \otimes m ) ) = ( 1 _ S \otimes a ) \phi ( s \otimes m ) = a \phi ( s \otimes a ) . \end{align*}


\begin{align*} ( Y _ I ^ J ) ^ S = \{ w _ K \in W \mid K \subset \Delta , J \subset K \subset I \} . \end{align*}


\begin{align*} \begin{cases} \Delta ^ 2 w = 0 & \\ w = - \bar v _ h & \\ \partial _ n w = - \partial _ n \bar v _ h & \end{cases} \end{align*}


\begin{align*} \left \{ \begin{aligned} & \dot u = X _ { H } ( u ) \\ & u ( 0 ) = u _ 0 \end{aligned} \right . \ . \end{align*}


\begin{align*} c _ i ( x ) = \sum _ { k = 1 } ^ { 3 } \sum _ { j = 1 } ^ { 3 } \frac { \partial \gamma _ { i , j } } { \partial x _ j } ( x ) \gamma _ { j , k } \ , , \end{align*}


\begin{align*} M ( \nu ) = \sum _ \alpha \Psi _ \alpha ( q ^ \nu ) \Theta _ \alpha ( \nu ) , \end{align*}


\begin{align*} x y z - z ^ 2 = \alpha \beta \gamma - \gamma ^ 2 . \end{align*}


\begin{align*} C & \le 2 C _ n '' \varrho ^ p = 2 \varrho ^ p / C _ n ' = 2 \varrho ^ p / \prod _ { n = 1 } ^ \infty \mu ( q \le \beta ^ n ) \\ & = 2 \varrho ^ p / \prod _ { n = 1 } ^ \infty ( 1 - \mu ( q > \beta ^ n ) ) \le 2 \varrho ^ p \exp \Bigl ( 2 \sum _ { n = 1 } ^ \infty \mu ( q > \beta ^ n ) \Bigr ) \le 2 \varrho ^ p \exp ( 8 k ) . \end{align*}


\begin{align*} \frac { \Phi _ 1 } { \Phi _ 3 } = \frac { \Phi _ 2 } { \Phi _ 4 } \end{align*}


\begin{gather*} \langle \partial _ t \bar { v } , z _ \Gamma \rangle _ { V _ \Gamma ^ * , V _ \Gamma } + \int _ { \Gamma } \nabla _ \Gamma \bar { w } \cdot \nabla _ \Gamma z _ \Gamma \ , d \Gamma = 0 , \end{gather*}


\begin{align*} \alpha ( G ) = \min _ { \emptyset \neq I \subseteq \{ 1 , \ldots , n \} } \frac { \dim G - \dim G _ { V _ I } } { \dim V _ I } . \end{align*}


\begin{align*} \delta ( Q ) = C \delta ( C ) = 0 . \end{align*}


\begin{align*} { \bf E } _ M { \bf E } _ v ^ { \otimes M } \ { \bf E } _ { y _ L } ^ { \otimes M } \sum _ { \gamma _ 1 , . . . , \gamma _ n = 1 } ^ M \prod _ { j = 1 } ^ n v _ { \gamma _ j } \exp ( 2 \pi i q _ j \cdot y _ { L , \gamma _ j } ) = \sum _ { A \in \mathcal { A } _ n } \prod _ { a \in A } \left \{ m _ { | a | } \delta _ { * , L } \left ( \sum _ { l \in a } q _ l \right ) \right \} \end{align*}


\begin{align*} \delta \langle u \rangle _ { s , t } + \int _ s ^ t \langle - \lambda _ 2 [ \partial _ x ^ 2 u _ r + u _ r | \partial _ x u _ r | ^ 2 ] - \lambda _ 1 u _ r \times \partial _ x ^ 2 u _ r \rangle \dd r = h _ 2 W _ { s , t } \langle u _ s \rangle + h _ 2 ^ 2 \mathbb W _ { s , t } \langle u _ s \rangle + \langle u ^ \natural _ { s , t } \rangle \ , . \end{align*}


\begin{align*} E _ M ( \alpha _ t ( u _ g ) ) = \rho ( t ) ^ { | g | } u _ g , \ ; \ ; \ ; t \in \mathbb R g \in \mathbb F _ n . \end{align*}


\begin{align*} \psi ( x , y ) : = \Big ( \frac { k _ n } { ( \delta - k _ n \circ \overline { k _ n } ) ^ { 1 / 2 } } \Big ) ( x , y ) , \end{align*}


\begin{align*} & \int _ { \R ^ n } \left \langle D ^ 2 h \ ( x - T x \ ) \xi , \xi \right \rangle \ , \phi ( x ) \ , d x \\ & = \sum _ { i , j = 1 } ^ n \int _ { \R ^ n } h _ { x _ i x _ j } ( x - T x ) \ , \phi ( x ) \ , d x \ , \xi _ i \ , \xi _ j = \sum _ { i , j = 1 } ^ n \left \langle h _ { x _ i x _ j } ( x - T x ) , \phi \right \rangle \ , \xi _ i \ , \xi _ j . \end{align*}


\begin{align*} L _ * \hat { \partial } _ i = L _ * ( A r \nabla _ { f _ i } ) = A r r ^ { - 1 } \nabla _ \alpha = \hat { \partial } . \end{align*}


\begin{align*} g : = g _ 0 - g _ 1 - g _ 2 + \overline { g } _ { n , \alpha } ^ 2 \overline { g } _ { m , \alpha } ^ 2 = \Big ( g _ { n , \alpha } ( i _ 1 ) \tilde { g } _ { m , \alpha } ( \ell _ { \pi _ 1 } ) - \overline { g } _ { n , \alpha } \overline { g } _ { m , \alpha } \Big ) \Big ( g _ { n , \alpha } ( i _ 2 ) \tilde { g } _ { m , \alpha } ( \ell _ { \pi _ 2 } ) - \overline { g } _ { n , \alpha } \overline { g } _ { m , \alpha } \Big ) \ , , \end{align*}


\begin{align*} K _ { p , \varphi _ p } ( z ) : = \sup \left \{ \frac { | g ( z ) | ^ p } { \| g \| _ { p , \varphi _ p } ^ p } : g \in { A } ^ p ( \mathbb { D } ) , g \neq 0 \right \} . \end{align*}


\begin{align*} J ^ B _ w ( Z ; X , Y ) & : = \left \langle A ' ( Y ) B ( Z ) A ( X ) , w \right \rangle , \\ J ^ C _ w ( Z ; X , Y ) & : = \left \langle A ' ( Y ) C ( Z ) A ( X ) , w \right \rangle , \ \ \mathrm { a n d } \\ I ^ D _ w ( Z ; X , Y ) & : = \left \langle A ' ( Y ) D ( Z ) A ( X ) , w \right \rangle . \end{align*}


\begin{align*} r _ 0 = { 4 \pi \over \min _ { a _ 0 \in \Omega } T ( a _ 0 ) + 8 \pi \lambda } . \end{align*}


\begin{align*} \sum _ { \substack { 0 < \gamma , \gamma ' \le T \\ | \gamma - \gamma ' | \le \frac { 2 \pi \lambda } { \log T } } } \ ! \ ! \ ! \ ! \ ! \ ! 1 \ \ \ge \ , \sum _ { 0 < \gamma , \gamma ' \le T } r \Big ( ( \gamma - \gamma ' ) \frac { \log T } { 2 \pi } \Big ) \ , w ( \gamma - \gamma ' ) = N ( T ) \int _ { \mathbb R } \widehat r ( \alpha ) \ , F ( \alpha ) \ , d \alpha . \end{align*}


\begin{align*} I _ { 1 } = \pi \delta \left ( \alpha \right ) \int K \left ( \alpha , s \right ) \mathrm { d } \mu \left ( s \right ) . \end{align*}


\begin{align*} \lambda ( P \oplus _ m P _ t ) \geq f ( \lambda _ 0 + \epsilon ( t ) ) \geq \phi ( x , \lambda _ 0 + \epsilon ( t ) ) = f ( \lambda _ 0 ) + \epsilon ( t ) x _ { m } ^ r . \end{align*}


\begin{align*} c _ { \omega , \Sigma } = \log | \omega | - | \Sigma | + 1 , \end{align*}


\begin{align*} \lim _ { \| \xi \| \to \infty } \psi ( \xi ) = \infty , \end{align*}


\begin{align*} h ^ 1 ( \mathbb P ^ r , \mathcal I _ S ( i - 1 ) ) = i + \binom { i } { 2 } s - \binom { r + i - 1 } { i - 1 } . \end{align*}


\begin{align*} ( \alpha ( - s - t , \alpha ( t , x ) ) , \alpha ( - s - t , \alpha ( s , x ) ) ) \in V = U ^ { - 1 } \end{align*}


\begin{align*} \left ( Q \circ \gamma \right ) ( \tau , 1 ) = j ( \gamma , \tau ) ^ 2 Q ( \gamma \tau , 1 ) , j \left ( \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right ) , \tau \right ) \coloneqq c \tau + d \end{align*}


\begin{align*} \mathbf { P } ( A ) = \mathbf { P } ( \theta _ { \tau } A ) . \end{align*}


\begin{align*} d _ { \rm C o n f i g } ( \mathcal { C } _ { 1 } , C _ { 2 } ) = \sum _ { k = 1 } ^ { \infty } \frac { 1 } { 2 ^ { k } } \sup _ { f \in { \rm L i p } _ { 1 } ( M ) } \frac { \int _ { \square _ { K } } f d \left ( \mathcal { C } _ { 1 } - \mathcal { C } _ { 2 } \right ) } { | \mathcal { C } _ { 1 } | ( \square _ { k } ) + | \mathcal { C } _ { 2 } | ( \square _ { k } ) } , \end{align*}


\begin{align*} [ k * ] _ n = q ^ k t ^ { n - k } + \sum _ { j = k } ^ \infty q ^ { k - 1 } t [ j * ] _ { n - 1 } . \end{align*}


\begin{align*} | K l _ p ( \psi _ p , \psi _ p ' ; c , w _ { G _ n } ) | & \leq D _ n \cdot \min ( p ^ { \sigma + a _ 2 + \cdots + \varrho / 2 + \frac { n ( n - 1 ) } { 2 } m } , p ^ { \ell / 2 + 2 \sigma + ( n - 3 ) \varrho + a _ 2 + \cdots + a _ { n - 2 } - \ell + \frac { n ( n - 1 ) } { 2 } m } ) . \end{align*}


\begin{align*} n \mapsto \begin{cases} + 1 & \mbox { i f $ n \equiv 1 , 3 \bmod 8 $ } \\ - 1 & \mbox { i f $ n \equiv 5 , 7 \bmod 8 $ } \\ 0 & \mbox { i f $ n $ i s e v e n . } \end{cases} \end{align*}


\begin{align*} A ( \varphi ) = \omega _ { n - 1 } \int \limits _ 0 ^ \varphi \sin ^ { n - 2 } \theta \ , d \theta \ , . \end{align*}


\begin{align*} A _ { \infty } ^ + ( x ) = a ^ { \dagger } ( \lambda _ { \infty } ( x ) ) , A _ { \infty } ^ - ( x ) = a ( \lambda _ { \infty } ( x ) ) , \end{align*}


\begin{align*} \rho _ \eta ^ \varepsilon ( x , t ) = \sum _ { k = 0 } ^ 2 \varepsilon ^ k \eta _ k ( \pi ( x , t ) , r , \varepsilon ^ { - 1 } d ( x , t ) ) , ( x , t ) \in \overline { Q _ { \varepsilon , T } } , \end{align*}


\begin{align*} ( \sigma _ { m } \overline { \mathbf { P } } ) ^ { \sigma _ { m } x } = \sigma _ { m } \overline { \mathbf { P } } ^ { x } . \end{align*}


\begin{align*} \kappa _ n : = \kappa _ * \left ( 2 - \frac { 1 } { 2 ^ n } \right ) , n \in \mathbb { N } _ 0 \end{align*}


\begin{align*} g = g h ^ { - 1 } \cdot h \in N ( K ) T ( O ) Z _ G ( K ) \check { \lambda } ( t ) = \check { \lambda } ( t ) N ( K ) T ( O ) Z _ G ( K ) . \end{align*}


\begin{align*} j = \begin{pmatrix} 0 & 1 \\ - 1 & 0 \end{pmatrix} s = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \end{align*}


\begin{align*} \delta ( \langle u \rangle - & B ) _ { s , t } + \int _ s ^ t \left [ \langle \lambda _ 2 u _ r \times [ u _ r \times \partial _ x ^ 2 u _ r ] - \lambda _ 1 u _ r \times \partial _ x ^ 2 u _ r \rangle \right ] \dd r \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad = h _ 2 W _ { s , t } [ \langle u _ s \rangle - B _ s ] + h _ 2 ^ 2 \mathbb W _ { s , t } [ \langle u _ s \rangle - B _ s ] + \langle u ^ \natural _ { s , t } - B ^ \natural _ { s , t } \rangle \ , . \end{align*}


\begin{align*} K = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { N } m _ { i } \dot { \textbf { x } } _ { i } ^ { 2 } = \frac { 1 } { 2 } [ \dot { \textbf { x } } _ { 1 } , . . . , \dot { \textbf { x } } _ { N } ] \textbf { M } \left [ \begin{array} { c } \dot { \textbf { r } } _ { 1 } \\ . . . \\ \dot { \textbf { x } } _ { N } \end{array} \right ] \end{align*}


\begin{align*} D _ x b ( x ) = \begin{cases} 0 & x < 0 x > e ^ { - e } , \\ \beta \log \frac { 1 } { x } ( \log \log \frac { 1 } { x } ) ^ \alpha ( 1 + R ( x ) ) & 0 < x < e ^ { - e } , \end{cases} \end{align*}


\begin{align*} W _ y ^ { Q F T } & = E _ y - E _ { \infty } \\ & = - \aleph _ { \alpha , L } \frac { \alpha } { L ^ 4 } + O ( \alpha ^ 4 \log ( \alpha ^ { - 1 } ) ) + O \big ( \alpha ^ 2 L e ^ { - L / 2 } \big ) + O \Big ( \frac { \alpha ^ 2 } { L ^ 5 } \Big ) + O \Big ( \frac { \alpha ^ 3 } { L ^ 2 } \log ( \alpha ^ { - 1 } ) \Big ) , \end{align*}


\begin{align*} G = \langle a , b | a ^ m = b ^ n = e , b a b ^ { - 1 } = a ^ r , \gcd ( ( r - 1 ) n , m ) = 1 , r ^ n \equiv 1 \ ; \pmod m \rangle . \end{align*}


\begin{align*} \left \langle x u _ { \alpha } \ , \bigg | \ , \frac { ( h _ { \alpha } - e _ { \alpha } ) ^ 2 } { ( h _ { \alpha } - e _ { \alpha } + | k | ) } \ , \bigg | \ , x u _ { \alpha } \right \rangle = \alpha ^ 2 \left \langle x u _ { 1 } \ , \bigg | \ , \frac { ( h _ { 1 } - e _ { 1 } ) ^ 2 } { ( \alpha ^ 2 ( h _ { 1 } - e _ { 1 } ) + | k | ) } \ , \bigg | \ , x u _ { 1 } \right \rangle , \end{align*}


\begin{align*} \mu ( E ) = | T ( E ) | \end{align*}


\begin{align*} & a _ 1 ( k ) = \frac { 1 } { k ^ 2 } \left ( \sigma \vert v _ 2 ( 0 , 0 ) \vert ^ 2 - \vert v _ 1 ( 0 , 0 ) \vert ^ 2 \right ) + O \left ( \frac { 1 } { k } \right ) , \\ & a _ 2 ( k ) = \frac { 4 \sigma } { A ^ 2 } \left ( \sigma \vert v _ 2 ( 0 , 0 ) \vert ^ 2 - \vert v _ 1 ( 0 , 0 ) \vert ^ 2 \right ) + O ( k ) , \\ & b ( k ) = - \frac { 2 i } { A k } \left ( \sigma \vert v _ 2 ( 0 , 0 ) \vert ^ 2 - \vert v _ 1 ( 0 , 0 ) \vert ^ 2 \right ) + O ( 1 ) , \end{align*}


\begin{align*} \mathfrak { d } ( S ) = \lim _ { x \rightarrow \infty } \frac { \# \mathcal { S } \cap [ 1 , x ] } { \pi ( x ) } \end{align*}


\begin{align*} \int _ { N ( F ) } \varphi _ { \pi } ( g n h ) d n = 0 \forall g , h \in G ( F ) . \end{align*}


\begin{align*} g _ { \mathcal C , F } = d r ^ 2 + r ^ 2 g _ F , r \in \mathbb R _ { > 0 } . \end{align*}


\begin{align*} \frac { d } { d t } \frac { \partial K } { \partial \dot { q } _ { i } } - \frac { \partial K } { \partial q _ { i } } = F _ { i } \end{align*}


\begin{align*} \dd _ p ( X _ 0 ^ { \epsilon , \Delta t } , X _ 0 ^ { 0 , \Delta t } ) = \dd ( x _ 0 ^ \epsilon , x _ 0 ^ 0 ) \underset { \epsilon \to 0 } \to 0 \end{align*}


\begin{align*} & \Vert w ( x , t , \cdot ) \Vert _ { L ^ p ( \Gamma \backslash \Gamma _ { \epsilon } ) } = O ( \epsilon ^ { \frac { 1 } { p } } \tau ^ { - 1 } ) , p = 1 , 2 , \\ & \Vert w ( x , t , \cdot ) \Vert _ { L ^ \infty ( \Gamma \backslash \Gamma _ { \epsilon } ) } = O ( \tau ^ { - 1 } ) . \end{align*}


\begin{align*} \min _ { \tilde { \boldsymbol { \theta } } _ t , \tilde { \boldsymbol { \theta } } _ r } & \sum _ { i \in \{ t , r \} } \| \tilde { \boldsymbol { \theta } } _ i + \boldsymbol { \vartheta } _ i \| ^ 2 \\ \mathrm { s . t . } & \tilde { \beta } _ { t , n } ^ 2 + \tilde { \beta } _ { r , n } ^ 2 = 1 , \forall n \in \mathcal { N } , \\ & \cos ( \tilde { \phi } _ { t , n } - \tilde { \phi } _ { r , n } ) = 0 , \forall n \in \mathcal { N } , \end{align*}


\begin{align*} f _ { \tau _ \mathrm { c b } , \rightarrow t } & = \theta ^ { - t } \circ f _ { \tau _ \mathrm { c b } } \circ \theta ^ t \\ f _ { \tau _ \mathrm { c b } , t \leftarrow } & = \theta ^ { t } \circ f _ { \tau _ \mathrm { c b } } \circ \theta ^ { - t } \\ f _ { \tau _ \mathrm { c b } , t \leftrightarrow t } & = f _ { \tau _ \mathrm { c b } , \rightarrow t } \circ f _ { \tau _ \mathrm { c b } , t \leftarrow } \end{align*}


\begin{align*} \left \{ \begin{aligned} & v _ 1 ( x , t ) = \int _ { - \infty } ^ { x } u ( y , t ) v _ 2 ( y , t ) d y , \\ & v _ 2 ( x , t ) = - i \sigma A - \sigma \int _ { - \infty } ^ { x } \left ( u ( - y , - t ) - A \right ) v _ 1 \left ( y , t \right ) d y , \\ \end{aligned} \right . \end{align*}


\begin{align*} A _ k = \frac { 1 } { k ! } \sum _ { \sigma \in \mathfrak { S } _ k } ( s g n \ , \sigma ) \sigma , \end{align*}


\begin{align*} \tilde { g } _ { \bold { a } , { \pmb { \theta } } } = a _ 1 \tilde { g } _ { \theta _ { 1 } } \oplus \cdots \oplus a _ m \tilde { g } _ { \theta _ m } , \end{align*}


\begin{align*} g _ \tau ( t ) : = \tau ^ { \frac 1 2 } g ( \tau t ) , \end{align*}


\begin{align*} A = \bigoplus _ { i } M _ { n _ i } \ ; . \end{align*}