\begin{align*} & P _ { i , 1 } = ( \dot { x } _ i - \mu _ i ) \sigma _ i \partial _ y \widetilde { R } _ i \\ & P _ { i , 2 } = - \bigg ( \dot { \mu } _ i + \sum ^ n _ { \substack { j = 1 , \\ j \not = i } } \frac { a _ { i j } } { x ^ 3 _ { i j } } + \sum ^ n _ { \substack { k , j = 1 , \\ j \not = i } } \frac { b _ { i j k } \mu _ k } { x ^ 3 _ { i j } } \bigg ) \frac { \sigma _ i \Lambda \widetilde { R } _ i } { 1 + \mu _ i } . \end{align*}


\begin{align*} f ( t ) = e ^ { - \pi t \cdot A t + 2 \pi b \cdot t + c } , \end{align*}


\begin{align*} ( x \cdot t ( t + 1 ) ) ' = x ^ 2 + 1 \end{align*}


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


\begin{align*} d \tilde \Phi _ t = - \dot { C } _ { \tau - t } \nabla V _ { \tau - t } ( \tilde \Phi _ t ) d t + \dot { C } _ { \tau - t } ^ { 1 / 2 } d B _ t , t \in [ 0 , \tau ] , \end{align*}


\begin{align*} C _ { J ^ E } ( f ) ( k ) = C _ { - } { f ( J ^ E - I ) } = \lim _ { s \rightarrow k } \int _ { \Gamma _ E } \frac { f ( s ) ( J ^ E ( s ) - I ) } { s - k } d s . \end{align*}


\begin{align*} \begin{pmatrix} I & 0 \\ Q & I \end{pmatrix} \begin{pmatrix} L & 0 \\ 0 & L ^ { - T } \end{pmatrix} \begin{pmatrix} I & P \\ 0 & I \end{pmatrix} = \begin{pmatrix} L & L P \\ Q L & Q L P + L ^ { - T } \end{pmatrix} \end{align*}


\begin{align*} \overline { \dim } _ B \big ( G ( h _ i ) \big ) = \varlimsup _ { \delta \rightarrow 0 } \frac { \log N _ { \delta } ( G ( h _ i ) ) } { - \log \delta } \le 2 - \sigma ~ ~ ~ ~ ~ \forall ~ ~ ~ i = 1 , 2 , \cdots , M . \end{align*}


\begin{align*} \check { A } = \{ \varphi ( g - h ) \mid g , h \in A \} \end{align*}


\begin{align*} P - \alpha Q & = ( R + \alpha S + \alpha ^ 2 U ) ^ 2 \\ & = ( R ^ 2 - 2 m S U ) + \alpha ( 2 R S - 2 k S U - m U ^ 2 ) + \alpha ^ 2 ( S ^ 2 + 2 R U - k U ^ 2 ) \end{align*}


\begin{align*} \Big | [ - ( - \Delta ) ^ { \gamma / 2 } G ] ( u ) \Big | \leq & c _ { \gamma } \int _ { 0 } ^ { 3 b _ G } \frac { \| \Delta G \| _ { \infty } } { w ^ { \gamma - 1 } } d w = H ^ G ( u ) , \end{align*}


\begin{align*} \partial _ z t > 0 , | \partial _ z \eta ^ L | : = | \partial _ z \eta ^ L ( t ( z ) , x _ 0 ) | = 1 \quad t ( 0 ) = 0 . \end{align*}


\begin{align*} B f ( i \omega ) e ^ { - \frac { \pi } { 2 } \omega ^ 2 } = e ^ { q ( i \omega ) - \frac { \pi } { 2 } \omega ^ 2 } = e ^ { - \pi ( \frac { 1 } { 2 } + A ' ) \omega ^ 2 + 2 \pi i b ' \cdot \omega + c ' } , \end{align*}


\begin{align*} \left ( \sum _ { i = 1 } ^ { l } | S _ i | \right ) ^ 2 \leq \left ( \sum _ { i = 1 } ^ { l } | S _ i | ^ 2 \right ) l . \end{align*}


\begin{align*} \sum _ { \gamma \in \Gamma _ { 8 } ( H G ) } c ( g ( \gamma ) ) = 2 \cdot 7 + 3 \cdot 7 = 3 5 \equiv 1 \pmod 2 \end{align*}


\begin{align*} \lambda _ \alpha ^ \gamma ( t ^ 1 , \dots , t ^ s ) = \lambda _ \alpha ^ \gamma ( t ^ 1 , \dots , t ^ \ell ) \end{align*}


\begin{align*} i \circ { \mathcal { R } } \circ i ^ { - 1 } = \sigma \end{align*}


\begin{align*} I ^ { ( s + n h ) } \subseteq \prod _ { i = 1 } ^ n I ^ { ( s _ i + 1 ) } . \end{align*}


\begin{align*} \lim _ { N \to \infty } \sum _ { k \in F } s _ { k , N } = \sum _ { k \in F } g _ k . \end{align*}


\begin{align*} f _ j ( x _ k ) = ( A x _ k - b ) ^ \intercal S _ j ( S _ j ^ \intercal A A ^ \intercal S _ { i _ k } ) ^ \dagger S _ { i _ k } ^ \intercal ( A x _ k - b ) , ~ j = 1 , \ldots . \epsilon . \end{align*}


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


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


\begin{align*} a _ \pi ( w , v ) = \sum _ { E \in { \cal T } _ h } \int _ E \Pi _ { E , q } \left ( \mu \nabla w \right ) \cdot \nabla v + \Pi _ { E , q - 1 } \left ( \boldsymbol { \beta } \cdot \nabla w + \sigma w \right ) v \ , , \end{align*}


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


\begin{align*} X _ { n + 1 } + 1 = \varepsilon ^ 4 _ n u _ n ^ 4 ( X _ n + 1 ) + \varepsilon _ n ^ 3 u _ n ^ 4 ( a + b ) ( 1 + a b u _ n ^ 2 ) . \end{align*}


\begin{align*} \rho \in H ^ { 1 } ( [ 0 , 1 ] ; \mathcal P ( G ) ) , m \in L ^ 2 ( [ 0 , 1 ] ; \mathcal S ^ { n \times n } ) , \ ; ( \rho ( 0 ) , \rho ( 1 ) ) = ( \rho ^ a , \rho ^ b ) \end{align*}


\begin{align*} C _ { J ^ E } ( f ) ( k ) = C _ { - } { f ( J ^ E - I ) } = \lim _ { s \rightarrow k } \int _ { \Gamma _ E } \frac { f ( s ) ( J ^ E ( s ) - I ) } { s - k } d s . \end{align*}


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


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


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


\begin{align*} \mu _ \delta ( d x ) = C _ \delta \exp ( - | x | ^ \delta ) d x = p _ \delta ( x ) d x , \end{align*}


\begin{align*} \forall x ^ X , y ^ X \left ( x = _ X y \rightarrow \forall k ^ 0 H ^ * \left [ A x , A y , \frac { 1 } { k + 1 } \right ] \right ) \end{align*}


\begin{align*} \partial _ { x _ { 0 ; \alpha } } = . \end{align*}


\begin{align*} \phi _ k = 0 , \partial _ n \phi _ k = 0 , \Gamma ^ { s y m } , \end{align*}


\begin{align*} \Bigl ( \sum \alpha _ k \Bigr ) \circ \beta & = \sum \alpha _ k ( \beta ) , \\ \Bigl ( \prod ( 1 + \alpha _ k ) \Bigr ) \circ \beta & = \prod ( 1 + \alpha _ k ( \beta ) ) , \\ \alpha \circ ( \beta \circ \gamma ) & = ( \alpha \circ \beta ) \circ \gamma . \end{align*}


\begin{align*} \sum _ { \xi , \eta } \langle X _ \alpha , [ X _ \xi , X _ \eta ] \rangle \langle X _ \alpha , [ \tilde R X _ \xi , \tilde T X \eta ] \rangle = \sum _ { \xi \triangle \eta = \alpha } r _ \xi t _ \eta \langle X _ \alpha , [ X _ \xi , X _ \eta ] \rangle ^ 2 = 4 \sum _ { \xi \triangle \eta = \alpha } r _ \xi t _ \eta . \end{align*}


\begin{align*} N _ { d _ 0 } ' ( f , H ) : = - f + \frac { 4 \mu ( d _ 0 ) } { \prod _ { q \mid d _ 0 } ( q + 1 ) } \sum _ { \delta \mid d _ 0 } \frac { \delta \mu ( \delta ) } { \phi ( \delta ) } S ( H _ \delta , \delta f ) . \end{align*}


\begin{align*} \frac { J _ \infty ^ { \mathsf { M P C } } - J _ \infty ^ \star } { J _ \infty ^ \star } J _ \infty ^ { \mathsf { M P C } } = \sum _ { k = 0 } ^ \infty \ell \left ( x _ k ^ \mathsf { M P C } , \tilde \mu \left ( x _ k ^ \mathsf { M P C } \right ) \right ) , \end{align*}


\begin{align*} T _ { \pm \eta , j } : = f ^ { - 3 j / 2 } _ { \pm \eta } \Delta _ { \eta } Q ^ { - 1 } \mathcal { K } _ j Q \Delta _ { \eta } ^ { - 1 } , j \geqslant 1 , \end{align*}


\begin{align*} \mathcal { P } _ { m } ( T ) : = \left \lbrace u \in L ^ { \infty } ( 0 , T ; H ^ { m } ) : u _ { t } \in L ^ { 2 } ( 0 , T ; H ^ { m - 1 } ) , ~ \nabla u \in L ^ { 2 } ( 0 , T ; H ^ { m } ) \right \rbrace , \end{align*}


\begin{align*} X ( \theta ) = \theta X ^ 0 + ( 1 - \theta ) X ^ 1 , \\ Y ( \theta ) = \theta Y ^ 0 + ( 1 - \theta ) Y ^ 1 , \\ Z ( \theta ) = \theta Z ^ 0 + ( 1 - \theta ) Z ^ 1 , \end{align*}


\begin{align*} 1 - 4 | \lambda | ^ 2 h ( s ) ^ 2 = \Big ( 1 - 2 h ( s ) ( s - t ) \Big ) ^ 2 \geq 0 . \end{align*}


\begin{align*} \check \tau _ { t , x } = \check \tau _ { t , x _ 1 } : = [ t - \tau _ { x _ 1 } ] _ + . \end{align*}


\begin{align*} \left | \dfrac { \sqrt { s _ n ^ 2 } } { \sqrt { \sigma _ n ^ 2 } } - 1 \right | \leq \begin{cases} \dfrac { C ' _ { \alpha } } { n } & ( - 1 < \alpha < 0 ) , \\ [ 4 m m ] \dfrac { C ' _ { \alpha } } { n ^ { 1 - 2 \alpha } } & ( 0 < \alpha < 1 / 2 ) , \\ [ 4 m m ] \dfrac { C ' _ { 1 / 2 } } { \log n } & ( \alpha = 1 / 2 ) . \end{cases} \end{align*}


\begin{align*} \iota ( \mathbf { h } \bullet \mathbf { h } ' ) & = \iota \left ( x + x ' , \omega + \omega ' , \tau + \tau ' + \tfrac { 1 } { 2 } ( x ' \cdot \omega - x \cdot \omega ' ) \right ) \\ & = \left ( x + x ' , \omega + \omega ' , \tau + \tau ' + \tfrac { 1 } { 2 } ( x ' \cdot \omega - x \cdot \omega ' ) + \tfrac { 1 } { 2 } ( x + x ' ) \cdot ( \omega + \omega ' ) \right ) . \end{align*}


\begin{align*} \tau _ t ( a _ 1 \cdots a _ n ) = \sum _ { j = 1 } ^ n t \cdot \tau ( a _ 1 \cdots a _ { j - 1 } \tau ' ( a _ j ) a _ { j + 1 } \cdots a _ n ) + o ( t ) \end{align*}


\begin{align*} E _ { \Sigma _ 0 } ( s , s ' ) = \frac { 1 } { 2 } \delta ' ( s - s ' ) . \end{align*}


\begin{align*} f _ { r 1 } ' + \frac { i \beta \eta } { 2 } \omega ^ 2 _ r f _ { r 1 } - \frac { i \beta \eta } { 4 } \omega _ r f ' _ r - \frac { i ( \gamma - \eta ^ 2 ) } { 2 \beta \eta } \omega _ r f ' _ r = 0 \end{align*}


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


\begin{align*} \chi ( u , v ) = ( - \rho ( - u ) , \widetilde { \gamma } \rho \gamma ^ { - 1 } ( v ) ) , \end{align*}


\begin{align*} x = \sum _ { i = 0 } ^ { 3 j } \alpha _ { i } e _ { i } , \quad \alpha _ { 3 j } \neq 0 \quad ~ \langle x \rangle _ { T ^ { * 3 } } = j + 1 . \end{align*}


\begin{align*} 1 + \frac { p ^ { w - 1 } - 1 } { p ( p - 1 ) } = 1 + O \bigg ( \frac { 1 } { p ^ { 1 + \varepsilon } } \bigg ) . \end{align*}


\begin{align*} V \left ( \frac { g _ 2 g _ 3 u } { H X } \right ) = \Psi \Big ( \frac { u } { X Q ^ { \vartheta } } \Big ) V \left ( \frac { g _ 2 g _ 3 u } { H X } \right ) \end{align*}


\begin{align*} \Phi ( h ) = B ( h , h ) = 2 \alpha _ + \alpha _ - + B ( h _ \perp , h _ \perp ) , \end{align*}


\begin{align*} z _ { \alpha } ^ { ( L + 1 ) } = J _ { x _ \alpha } z _ { \alpha } ^ { ( L + 1 ) } x _ \alpha . \end{align*}


\begin{align*} \delta ( \L ) = D ^ - ( \L ) = D ^ + ( \L ) . \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { | \Lambda _ { ( \infty , m ) } ( \frac { M } { 2 } , \Gamma ) | } { | \Lambda ( \frac { M } { 2 } ) | } = & \lim _ { M \to \infty } \frac { | \Lambda _ { ( \infty , m ) } ( \frac { M } { 2 } ) | } { | \Lambda ( M ) | } \frac { | \Lambda ( M ) | } { | \Lambda ( \frac { M } { 2 } ) | } \\ = & \lim _ { M \to \infty } \frac { | \Lambda _ { ( \infty , m ) } ( \frac { M } { 2 } , \Gamma ) | } { | \Lambda ( M ) | } \frac { M ^ 2 } { ( \frac { M } { 2 } ) ^ 2 } = 0 \end{align*}


\begin{align*} \tilde { x } _ { m , n } = \sqrt { \eta _ { m k } P _ d } \hat { g } _ { m k , n } ^ * s _ { k , n } , \end{align*}


\begin{align*} H ( \eta ) = h ( \eta ) h '' ( \eta ) - h ' ( \eta ) ^ 2 , \end{align*}


\begin{align*} \norm { \widehat { f } } _ { M ^ { p } } ^ p & = \norm { V _ { g _ 0 } \widehat { f } } _ p ^ p \leq C \norm { V _ { \widehat { g _ 0 } } \widehat { f } } _ p ^ p \\ & = C \iint _ { \R ^ { 2 d } } | V _ { \widehat { g _ 0 } } \widehat { f } ( x , \omega ) | ^ p \ , d ( x , \omega ) \\ & = C \iint _ { \R ^ { 2 d } } | V _ { g _ 0 } f ( - \omega , x ) | ^ p \ , d ( x , \omega ) \\ & = C \iint _ { \R ^ { 2 d } } | V _ { g _ 0 } f ( x , \omega ) | ^ p \ , d ( x , \omega ) \leq C ' \norm { f } _ { M ^ p } ^ p . \end{align*}


\begin{align*} \ell : = \bigcup _ { t \in ( - \epsilon , \epsilon ) } \eta ^ L ( t , x _ * ) \quad x _ * \in \mathbb { R } ^ 3 . \end{align*}


\begin{align*} \varphi = \varphi ^ { \vee \vee } , \psi = \psi ^ { \vee \vee } . \end{align*}


\begin{align*} \frac { \lambda } { T ^ 2 } \le \frac { \lambda } { ( T - \tilde { t } ) ^ 2 } = h - k \le F ^ \ast ( v ( \tilde { y } , \tilde { t } ) , p _ 2 , Y , K ) - F _ \ast ( u ( \tilde { x } , \tilde { t } ) , p _ 1 , X , \emptyset ) . \end{align*}


\begin{align*} \dfrac { S _ n } { \sqrt { n \log n } } \stackrel { } { \to } N ( 0 , 1 ) ( n \to \infty ) . \end{align*}


\begin{align*} m _ { i , \lambda } ( \pi _ 1 ^ { - 1 } ( \omega ) ) = 0 . \end{align*}


\begin{align*} \mathcal { M } ( \tilde u _ 2 , a _ 2 , b _ 2 ) u - \mathcal { M } ( \tilde u _ 1 , a _ 1 , b _ 1 ) u = & [ \mathcal { M } ( \tilde u _ 2 , a _ 2 , b _ 2 ) u - \mathcal { M } ( \tilde u _ 1 , a _ 2 , b _ 2 ) u ] \\ & + [ \mathcal { M } ( \tilde u _ 1 , a _ 2 , b _ 2 ) u - \mathcal { M } ( \tilde u _ 1 , a _ 1 , b _ 2 ) u ] \\ & + [ \mathcal { M } ( \tilde u _ 1 , a _ 1 , b _ 2 ) u - \mathcal { M } ( \tilde u _ 1 , a _ 1 , b _ 1 ) u ] . \end{align*}


\begin{align*} \| u \| _ { L _ t ^ q L _ x ^ r } = \big ( \int \| u ( t , \cdot ) \| _ { L _ x ^ r } ^ q d t \big ) ^ { \frac { 1 } { q } } . \end{align*}


\begin{align*} \| S _ r \Sigma - \Sigma \| _ { L _ 2 ^ 0 ( H ) } ^ 2 = \sum _ { j > r } \| \Sigma ^ j \| ^ 2 \to \infty \end{align*}


\begin{align*} g _ 1 ( t ) \mathcal { L } _ { K , \mathfrak { s } _ 1 } ( \alpha ) + \cdots + g _ r ( t ) \mathcal { L } _ { K , \mathfrak { s } _ r } ( \alpha ) = 0 \end{align*}


\begin{align*} \left ( \psi ^ { \beta } _ { z _ 0 + \frac { 1 } { 2 } \sum _ i x ^ 0 _ i y ^ 0 _ i } \circ \psi ^ H _ 1 \right ) ( x ^ 0 _ i , y ^ 0 _ i , z ^ 0 ) = 0 \end{align*}


\begin{align*} \frac { d } { d t } E ( \rho , \phi ) ( t ) & = - \int _ { \Omega } \sum _ { i = 1 } ^ s D _ i ( x ) \rho _ i | \nabla ( \log \rho _ i + z _ i \phi ) | ^ 2 d x \\ & + \frac { 1 } { 2 } \int _ { \partial \Omega } \epsilon ( x ) \left [ \phi ( \partial _ n \phi ) _ t - ( \partial _ n \phi ) \phi _ t \right ] d s + \frac { d } { d t } B ( \phi ) . \end{align*}


\begin{align*} f _ 0 & = 1 \quad \mbox { a n d } f _ k = z f _ { k - 1 } + z g _ { k - 1 } + z h _ { k - 1 } , k \geq 1 , \\ g _ k & = z \sum \limits _ { \ell \geq k + 1 } f _ \ell + z \sum \limits _ { \ell \geq k + 1 } g _ \ell + z \sum \limits _ { \ell \geq k + 1 } h _ \ell , k \geq 0 , \\ h _ k & = z f _ k + z g _ k + z h _ k , k \geq 0 . \end{align*}


\begin{align*} 0 & = \lim _ { y \to 0 } \left ( \gamma + \ln \left ( \frac { 4 \lvert y \rvert } { \epsilon _ 2 } \right ) + \psi \left ( \frac { 1 + \lvert y \rvert } { 2 } \right ) + \ln { \frac { 2 } { \lvert y \rvert } } \right ) , \end{align*}


\begin{align*} \begin{cases} - k _ { \varepsilon } \Delta \tilde { \Psi } _ { \varepsilon } + | \nabla \tilde { \Psi } _ { \varepsilon } | ^ 2 - \tilde { \lambda } _ 0 \underline { m } = 0 & \Omega \\ \tilde { \Psi } _ { \varepsilon } = - k _ { \varepsilon } \log ( w ) ( \frac { \mathbf { x } - \mathbf { x } _ { \varepsilon } } { k _ { \varepsilon } } ) & \partial \Omega \end{cases} \end{align*}


\begin{align*} P = \sum _ { i , j = 1 } ^ 3 \Big ( \mathcal { R } _ i \mathcal { R } _ j ( u _ i u _ j ) + \mathcal { R } _ i \mathcal { R } _ j ( b _ i b _ j ) \Big ) , \end{align*}


\begin{align*} \lim \limits _ { n \to \infty } \frac { 1 } { n } \sum \limits _ { i = 0 } ^ { n - 1 } g _ 1 ( S _ i ( w ) ) & = \int _ { \Omega } g _ 1 ( w ) d \mu , \\ \lim \limits _ { n \to \infty } \frac { 1 } { n } \sum \limits _ { i = 0 } ^ { n - 1 } g _ 2 ( S _ i ( w ) ) & = \int _ { \Omega } g _ 2 ( w ) d \mu , \end{align*}


\begin{align*} S _ 3 \leq \sum _ { \gamma \in \mathcal { I } } | \gamma | \sum \limits _ { k = 1 } ^ { | \gamma | } \tilde { M } ( T ) ^ { 2 k } \bigg ( \sum _ { \substack { 0 \leq | \beta | \leq | \gamma | - k \\ \beta < \gamma } } a _ \beta ( T ) \bigg ( \sum _ { \substack { \theta _ 1 + \dots + \theta _ k = \gamma - \beta \\ \theta _ i \not = \mathbf { 0 } } } \prod _ { i = 1 } ^ { k } \| q _ { \theta _ i } \| _ { L ^ \infty } \bigg ) \bigg ) ^ 2 ( 2 \N ) ^ { - p \gamma } . \end{align*}


\begin{align*} 1 = f ( 1 , . . . , 1 ) \leq f ( \kappa ) + \frac { \partial f } { \partial \kappa _ { i } } ( \kappa ) ( 1 - \kappa _ { i } ) = \sum _ { i } \frac { \partial f } { \partial \kappa _ { i } } ( \kappa ) . \end{align*}


\begin{align*} P _ { k + 1 - h } = \frac { x _ { h + 1 } z } { P _ { h + 1 } } - \frac { x _ h z } { P _ h } . \end{align*}


\begin{align*} I _ { B \cup \{ - \alpha \} , \{ \beta \} } ( n ) = { \tau } _ { B \smallsetminus \{ \beta \} \cup \{ - \alpha \} } ( n ) \end{align*}


\begin{align*} f \bullet ( g \bullet h ) ( x ) & = ( g \bullet h ) ( x ) f ( x ( { g \bullet h } ) ( x ) ) = h ( x ) g ( x h ( x ) ) f ( x ( g \bullet h ) ( x ) ) . \end{align*}


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


\begin{align*} \mbox { f o r $ k \geq 0 $ , } \sqrt { 2 \pi } \mathcal { K } ( x , k ) & = \chi _ + ( x ) T ( k ) \psi _ + ( x , k ) + \chi _ - ( x ) \left [ \psi _ - ( x , - k ) + R _ - ( k ) \psi _ - ( x , k ) \right ] , \\ \mbox { f o r $ k < 0 $ , } \sqrt { 2 \pi } \mathcal { K } ( x , k ) & = \chi _ - ( x ) T ( - k ) \psi _ - ( x , - k ) + \chi _ + ( x ) \left [ \psi _ + ( x , k ) + R _ + ( - k ) \psi _ + ( x , - k ) \right ] . \end{align*}


\begin{align*} f _ j ( x ) = g _ j ( x ) + \varepsilon f ( x ) , \end{align*}


\begin{align*} \nabla ^ { 2 } \phi _ k & = 0 , \Omega , \\ \phi _ k & = \phi _ k , \Gamma ^ { F S } , \\ \partial _ { n } \phi _ k & = 0 , \Gamma ^ b , \\ \partial _ { n } \phi _ k & = \dot { x } _ k n _ k , \Gamma ^ { b o d y } , \end{align*}


\begin{align*} \| B _ { \tilde { h } } Q u \| _ { H ^ N _ { h \tilde { h } } } = \mathcal { O } ( h ^ \infty \tilde { h } ^ \infty ) \| u \| _ { H ^ { - N } _ { h \tilde { h } } } . \end{align*}


\begin{align*} \delta ( h ( S ) ) = f ( \delta ( S ) ) , \end{align*}


\begin{gather*} \gamma _ { X \otimes Y } = \gamma _ X \otimes \gamma _ Y \end{gather*}


\begin{align*} \lim _ { \lambda \rightarrow + \infty } \| f \| ^ 2 _ { q , \lambda , t } = \int _ { \{ q \Psi < - t \} } | f | ^ 2 e ^ { - \varphi _ 0 } , \ \forall t \in [ 0 , + \infty ) . \end{align*}


\begin{align*} \Theta \mapsto S = - ( \chi ^ + - \Theta \chi ^ - ) ^ { - 1 } ( \chi ^ - - \Theta \chi ^ + ) , S \mapsto \Theta = ( \chi ^ - + \chi ^ + S ) ( \chi ^ + + \chi ^ - S ) ^ { - 1 } , \end{align*}


\begin{align*} H P ^ { k } ( A ) : = \lim _ { n \to \infty } H C _ \lambda ^ { k + 2 n } ( A ) , \ k = 0 , 1 . \end{align*}


\begin{align*} G _ { \alpha } ( \xi , w ) : = \partial _ { x } ^ { \alpha } ( \xi w ) - \xi \partial _ { x } ^ { \alpha } w , \end{align*}


\begin{align*} u \leqslant v & \ , : \Longleftrightarrow \ , u _ i \leq v _ i , \ ; i = 1 , \ldots , K \ ; \ ; \ ; \ ; u \neq v , \end{align*}


\begin{align*} K _ { s , t , x } ' & : = \{ v : s < { \mathcal T } _ { t , x _ 1 , v _ 1 } - 2 , | v | < \frac { \delta ^ { - \beta } } 2 , \check \tau _ { t , x } \langle v ^ \perp \rangle < \langle x ^ \perp \rangle / 5 \} , \\ K '' _ { s , t , x } & : = \{ v : s < { \mathcal T } _ { t , x _ 1 , v _ 1 } - 3 , | v | < \frac { \delta ^ { - \beta } } 3 , \check \tau _ { t , x } \langle v ^ \perp \rangle < \langle x ^ \perp \rangle / 6 \} . \end{align*}


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


\begin{align*} H P ^ \bullet ( \mathbb { C } \Gamma ) = \prod _ { \hat { \gamma } \in \langle \Gamma \rangle ' } \big ( H ^ \bullet ( N _ \gamma , \mathbb { C } ) \otimes H P ^ \bullet ( \mathbb { C } ) \big ) \times \prod _ { \hat { \gamma } \in \langle \Gamma \rangle '' } H ^ \bullet ( N _ \gamma , \mathbb { C } ) , \end{align*}


\begin{align*} \lim _ { x \to 0 ^ + } n _ { i x } ( x ) = \frac { 1 } { 4 } \left ( \alpha - \sqrt { \alpha ^ 2 - 8 \left ( b _ i ( 0 ) - 1 \right ) } \right ) = : A _ i > 0 , i = 1 , 2 . \end{align*}


\begin{align*} \log t - C t \log t u ' ( t ) & \ge ( 1 - 4 \cdot 1 0 ^ { - 6 } ) \log t \ge \log ^ { 0 . 9 9 1 } t + 0 . 1 9 = \log ^ { B + \alpha - 1 } t + \alpha . \end{align*}


\begin{align*} m ^ { \star } & = \max \left \{ m _ { i } , \ i = 1 , 2 , \ldots , n \right \} , \\ \mu ^ { \star } & = \max \left \{ \mu _ { i } , \ i = 1 , 2 , \ldots , n \right \} , \\ \mu _ \star & = \min \left \{ \mu _ { i } , \ i = 1 , 2 , \ldots , n \right \} . \end{align*}


\begin{align*} M _ k ( T ) : = \int _ { 0 } ^ { T } \left | \zeta \left ( \tfrac { 1 } { 2 } + i t \right ) \right | ^ { 2 k } \ , d t \end{align*}


\begin{align*} \widehat h ( t ) = \rho ( t , 1 ) + \widetilde p ( t ) , \forall t \in ( 0 , T ) , \end{align*}


\begin{align*} u _ { \lambda } ( x , t ) = \frac { 1 } { \lambda } u \left ( \frac { x } { \sqrt { \lambda } } , \frac { t } { \lambda } \right ) \end{align*}


\begin{align*} \left \{ \Phi ( f ) , \Pi ( g ) \right \} ^ \Sigma _ \mathrm { c a n } = \int _ \Sigma f g \ , \mathrm { d } V _ \Sigma . \end{align*}


\begin{align*} \left ( \frac { \partial } { \partial t } - { \mathcal L } \right ) u ( t , x ) + { q ( x ) } \cdot u ( t , x ) = f ( t , x ) , u ( 0 , x ) = g ( x ) , \end{align*}


\begin{align*} \mathcal { J } ( \phi ( \theta ) ) - \theta \mathcal { J } ( \phi ^ 0 ) - ( 1 - \theta ) \mathcal { J } ( \phi ^ 1 ) = & - \frac { \theta ( 1 - \theta ) \beta _ a } { 8 h ^ 2 } ( \xi _ 0 - \xi _ 1 ) ^ 2 \\ & - \frac { \theta ( 1 - \theta ) } { 8 h } \sum _ { j = 1 } ^ N \epsilon _ j ( \xi _ { j - 1 } - \xi _ { j + 1 } ) ^ 2 \\ & - \frac { \theta ( 1 - \theta ) } { 8 \beta _ b } ( \xi _ N + \xi _ { N + 1 } ) ^ 2 = 0 . \end{align*}


\begin{align*} \widetilde { \pi } _ { \ell / r } \circ \chi = \chi _ { \ell / r } \circ \pi _ { \ell / r } . \end{align*}


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


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


\begin{align*} \big ( g \curvearrowright f \big ) ( x ) : = f ( x g ( x ) ) = \sum _ { w \in \N ^ * } f _ w ( x g ( x ) ) _ w \in G ^ 0 \end{align*}


\begin{align*} \binom { c _ n k n } l \sum _ { j = 1 } ^ { n - N } \frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \frac 1 { k ( n - j + 1 ) - l } \le C \sum _ { j = 1 } ^ { n - N } ( \frac j n ) ^ { c ' k n } . \end{align*}


\begin{align*} f = \sum _ { \gamma \in \Gamma } c _ \gamma \pi ( \gamma ) g , \end{align*}


\begin{align*} h ( t ) = \alpha _ + ( t ) \mathcal { Y } _ + + \alpha _ - ( t ) \mathcal { Y } _ - + \sum _ { j = 0 } ^ N \beta _ j ( t ) Q _ j + h _ \perp ( t ) , h _ \perp \in G _ \perp ' , \end{align*}


\begin{align*} w ( \alpha ) = w ( \beta ) + \# ( \mathbb { R } \times \gamma _ { 0 } \cap { v } ) \end{align*}


\begin{align*} & \dot { \widetilde \rho _ 2 } + \widetilde m _ { 2 3 } = ( \Sigma _ 2 - \Sigma _ 3 ) \dot W ^ { \delta } , \\ & \dot { \widetilde \rho _ 3 } + \widetilde m _ { 3 2 } = ( \Sigma _ 3 - \Sigma _ 2 ) \dot W ^ { \delta } . \end{align*}


\begin{align*} A = Z \ , D _ A \ , W ^ T , B = Z \ , D _ B \ , U ^ T , G = V \ , D _ G \ , W ^ T , \end{align*}


\begin{align*} \begin{aligned} u ( x + h ) + u ( x - h ) - 2 u ( x ) & \leq C | h | ^ 2 \int _ 0 ^ T e ^ { - s } \left ( 1 - \frac { s } { T } \right ) ^ 2 d s + C | h | ^ 2 \int _ 0 ^ T \frac { e ^ { - s } } { T } d s + C | h | ^ 2 \int _ 0 ^ T \frac { e ^ { - s } } { T ^ 2 } d s \\ & \leq C | h | ^ 2 \int _ 0 ^ T e ^ { - s } d s + C \frac { | h | ^ 2 } { T } + C \frac { | h | ^ 2 } { T } \int _ 0 ^ 1 e ^ { - s T } d s \\ & \leq C \left ( 1 + \frac { 1 } { T } \right ) | h | ^ 2 , \\ \end{aligned} \end{align*}


\begin{align*} ( A . R ) ( x , y ) = [ A , R ( x , y ) ] - R ( A ( x ) , y ) - R ( x , A ( y ) = 0 , \ ; \forall ( x , y ) \in V ^ 2 , \end{align*}


\begin{align*} \langle f , g \rangle = \langle \widehat { f } , \widehat { g } \rangle . \end{align*}


\begin{align*} Y ^ { ( n ) } = \Bigl ( \int _ 0 ^ { v \wedge \tau _ n } X _ u | X _ u | ^ { p _ 0 - 2 } \sigma ( X _ u ) \ , d W _ u \Bigr ) _ { v \ge 0 } , n \in \N , \end{align*}


\begin{align*} x ( t ) = X ( t ) , \forall t \in D _ { \delta _ 0 } , \end{align*}


\begin{align*} N _ 0 ( T ) = \frac { T } { 2 \pi } \log \left ( \frac { T } { 2 \pi } \right ) - \frac { T } { 2 \pi } + \mathcal { O } ( \log T ) \end{align*}


\begin{align*} \left | \int _ { y _ 0 } ^ y y ^ { - \beta } d y \right | = \frac { 1 } { \beta - 1 } \left | y _ 0 ^ { 1 - \beta } - y ^ { 1 - \beta } \right | < R . \end{align*}


\begin{align*} f _ \delta ( x ) = \frac { 1 } { p _ \delta ( x ) } \int _ x ^ { + \infty } g ( y ) p _ \delta ( y ) d y = - \frac { 1 } { p _ \delta ( x ) } \int _ { - \infty } ^ x g ( y ) p _ \delta ( y ) d y . \end{align*}


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


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


\begin{align*} \sum _ { i = 1 } ^ { j } w _ H ( e _ i ) \le w _ G ( \phi ( e _ j ) ) . \end{align*}


\begin{align*} \mathcal { H } ( z , w ) = \sqrt { \pi } \frac { \Gamma ( \tfrac { 1 - w } { 2 } ) \Gamma ( \tfrac { z } { 2 } ) \Gamma ( \tfrac { w - z } { 2 } ) } { \Gamma ( \tfrac { w } { 2 } ) \Gamma ( \tfrac { 1 - z } { 2 } ) \Gamma ( \tfrac { 1 - w + z } { 2 } ) } . \end{align*}


\begin{align*} \widehat { y _ 0 } = \widehat { \mu } + \widehat { \beta } ' x _ 0 . \end{align*}


\begin{align*} \bar { \psi } _ { 2 } ( \alpha ; \omega ) : = \sum _ { i = 1 } ^ { \kappa - 1 } \psi _ { 2 } ( \delta _ { i } ; \delta _ { i + 1 } ) . \end{align*}


\begin{align*} \chi _ { t } + \beta \chi \chi _ { x } - \chi _ { x x } = 0 , \int _ { \R } \chi ( x , t ) d x = M . \end{align*}


\begin{align*} h ^ { j } = L R \cdot U ^ { j } \end{align*}


\begin{align*} \Re \left ( \frac { 1 } { \rho } \right ) = \frac { \beta } { \beta ^ 2 + \gamma ^ 2 } > 0 \end{align*}


\begin{align*} \mu ^ { \# } _ { R , 1 } ( k , \ell , m , n ) : = \sum _ { ( A , B , C , D ) \in \mathcal { X } _ { R } } \int \overline { \mathcal { K } ^ { \# } _ { A } ( x , k ) } \mathcal { K } ^ { \# } _ { B } ( x , \ell ) \overline { \mathcal { K } ^ { \# } _ { C } ( x , m ) } \mathcal { K } ^ { \# } _ { D } ( x , n ) \ , d x , \\ \mathcal { X } _ { R } = : = \{ S , R \} ^ 4 \smallsetminus \{ ( S , S , S , S ) \} , \end{align*}


\begin{align*} \mathfrak { F } _ { ( \alpha , \beta ) } ( g ) = \{ \alpha \Z ^ d \times \beta \Z ^ d \mid G ( g , \alpha \Z ^ d \times \beta \Z ^ d ) \} \end{align*}


\begin{align*} \tfrac { \partial } { \partial x } \left [ \sinh ( x ) - x - x ^ 3 \right ] & = \cosh ( x ) - 1 - 3 x ^ 2 , \\ \tfrac { \partial ^ 2 } { \partial x ^ 2 } \left [ \sinh ( x ) - x - x ^ 3 \right ] & = \sinh ( x ) - 6 x = x \left ( \tfrac { \sinh ( x ) } { x } - 6 \right ) . \\ \end{align*}


\begin{align*} \begin{aligned} \partial _ t \rho _ 1 = & \partial _ x \left ( \partial _ x \rho _ 1 + \rho _ 1 \partial _ x \phi \right ) , \\ \partial _ t \rho _ 2 = & \partial _ x \left ( \partial _ x \rho _ 2 - \rho _ 2 \partial _ x \phi \right ) , \\ - \partial _ x ^ 2 \phi = & \rho _ 1 - \rho _ 2 , \end{aligned} \end{align*}


\begin{align*} f _ i f _ j E _ n ^ { \sigma } \Pi _ n ^ { \sigma } \big ( f _ i ( ( 0 , 1 ) ^ d ) \big ) \cap \Pi _ n ^ { \sigma } \big ( f _ j ( ( 0 , 1 ) ^ d ) \big ) = \emptyset . \end{align*}


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


\begin{align*} C C ^ k ( A , B ) : = C C ^ k ( A ) \oplus C C ^ { k + 1 } ( B ) , \end{align*}


\begin{align*} \left \{ \aligned & \partial _ { t } \omega + ( u \cdot \nabla ) \omega = ( b \cdot \nabla ) j , \\ & \partial _ { t } j + ( u \cdot \nabla ) j + \mathcal { L } j = ( b \cdot \nabla ) \omega + T ( \nabla u , \nabla b ) , \endaligned \right . \end{align*}


\begin{align*} \langle T \phi , \psi \rangle & = \langle K ( x , y ) , \phi ( y ) \psi ( x ) \rangle = \langle K ( x , y ) , \chi _ 0 ( y ) \phi ( y ) \psi ( x ) \rangle \\ & = \langle K ( x , y ) , \chi _ 0 ( y ) [ \phi ( y ) - \phi ( x ) ] \psi ( x ) \rangle + \langle K ( x , y ) , \chi _ 0 ( y ) \phi ( x ) \psi ( x ) \rangle \\ & = : p + q , \end{align*}


\begin{align*} | v | _ \alpha : = \big ( \sum _ { j = 1 } ^ \infty \lambda _ j ^ \alpha | \big < v , e _ j \big > | ^ 2 \big ) ^ { \frac 1 2 } , \ ; \| v \| _ \alpha : = \big ( | v | _ \alpha ^ 2 + | \big < v , e _ 0 \big > | ^ 2 \big ) ^ { \frac 1 2 } . \end{align*}


\begin{align*} \left . \begin{aligned} M & = { } [ \Delta ( { \mathbb K } ) , \Delta ( F ) ] \mbox { a n d } \\ s & \equiv 0 \pmod { e ( G ) } \mbox { w i t h } s \ge N . \end{aligned} \right \} \end{align*}


\begin{align*} \begin{aligned} \bar { h } _ { i j } = h \left ( \frac { \partial } { \partial x ^ { i } } , \frac { \partial } { \partial x ^ { j } } \right ) = h \left ( \frac { \partial z ^ { k } } { \partial x ^ i } \frac { \partial } { \partial z ^ { k } } , \frac { \partial z ^ { l } } { \partial x ^ j } \frac { \partial } { \partial z ^ { l } } \right ) = \frac { \partial z ^ { k } } { \partial x ^ i } \frac { \partial z ^ { l } } { \partial x ^ j } h _ { k l } . \end{aligned} \end{align*}


\begin{align*} \mathcal { N } _ { C } ( \Theta ) : = \begin{bmatrix} H + \rho I + \Theta ^ { - 1 } & A ^ T \\ A & - \delta I \end{bmatrix} , \end{align*}


\begin{align*} \operatorname { R e } { \lambda } \left \| x \right \| ^ 2 - \operatorname { R e } \left < T x , x \right > = 0 . \end{align*}


\begin{align*} \rho _ { i j } - \rho ^ n _ { i j } + d _ h ( m _ i ) _ j = 0 , \end{align*}


\begin{align*} \langle T ^ \circ x - T ^ \circ x ' , - T ^ \circ x ' \rangle & \leq \langle T ^ \circ x - y ' , - T ^ \circ x ' \rangle + \langle y ' - T ^ \circ x ' , - T ^ \circ x ' \rangle \\ & = \langle T ^ \circ x - y ' , - T ^ \circ x ' \rangle \\ & \leq \norm { T ^ \circ x - y ' } \norm { T ^ \circ x ' } \\ & \leq \frac { 1 } { ( k + 1 ) ^ 2 } . \end{align*}


\begin{align*} D _ { d _ 0 } ( p , H ) = 1 + o ( 1 ) \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ d } K ( x ) f ( x , w ^ - ) w ^ - d x & = A ^ - ( w ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { k } | \{ \lambda \in \Lambda \mid \lambda \supset e \cup e _ { i } \} | \equiv 0 \pmod m , \end{align*}


\begin{align*} N ^ 0 _ \beta = \sum _ { k | \beta } \frac { n ^ 0 _ { \beta / k } } { k ^ 3 } \end{align*}


\begin{align*} \frac { \left \langle \tilde a _ i , g \right \rangle - b _ i } { \left \langle \tilde a _ i , c \right \rangle } = \min \left \lbrace \sigma \in \mathbb { R } _ { \geqslant 0 } \ ; \middle | \ ; x - \sigma c \in H _ i ^ + \right \rbrace . \end{align*}


\begin{align*} A - A _ k = \widehat { Z } \ , \widehat { D } _ A \ , \widehat { W } ^ T , B - B _ k = \widehat { Z } \ , \widehat { D } _ B \ , \widehat { U } ^ T , G - G _ k = \widehat { V } \ , \widehat { D } _ G \ , \widehat { W } ^ T . \end{align*}


\begin{align*} & \phi \circ T = T ^ { ' } \circ \psi , \\ & \psi \theta ( x , y ) = \theta ( \phi ( x ) , \phi ( y ) ) \psi , \forall x , y \in L . \end{align*}


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


\begin{align*} \frac { \sinh ( x ) } { x } = \frac { \sum \limits _ { k = 0 } ^ \infty \tfrac { x ^ { 2 k + 1 } } { ( 2 k + 1 ) ! } } { x } = \sum \limits _ { k = 0 } ^ \infty \underbrace { \tfrac { x ^ { 2 k } } { ( 2 k + 1 ) ! } } _ { \nearrow } . \end{align*}


\begin{align*} \bar { \Delta } ' _ i : = \bigl \{ ( z _ i , \delta _ i , w ) \bigm | ( z _ i , \delta _ i ) \in \bar { \Delta } _ i , \ z _ { i ' \tau ( i ' , t ) - 1 } \geq w _ { i ' j ' } ^ t \geq z _ { i ' \tau ( i ' , t ) } t < t ' \bigr \} . \end{align*}


\begin{align*} I _ { H _ s } = ( p _ 1 ( { \bf \underline { t } } ) , \ldots , p _ s ( { \bf \underline { t } } ) ) , \end{align*}


\begin{align*} \kappa ^ n ( a ) \otimes b ^ * \rho ^ { n } ( b ) = \sum _ { i = 1 } ^ n ( ( 1 \otimes b ^ * ) \Delta ( p _ i ) ) ( 1 \otimes \rho ^ n ( q _ i ) ) . \end{align*}


\begin{align*} F ( x , t ) = ( x , ( a + t ) v ) \end{align*}


\begin{align*} U ^ i V ^ j = 0 \mbox { f o r a l l } i \geq 1 \mbox { a n d } j \geq 2 \end{align*}


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


\begin{align*} \begin{aligned} \min \ & f ( x ) \\ \enspace & A x = b . \end{aligned} \end{align*}


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


\begin{align*} \nabla _ v \mu ( v ) = - v \psi ( v ) , \end{align*}


\begin{align*} a _ i & = \lfloor i \overline { s } \rfloor - \lceil i \overline { u } \rceil + 1 \\ b _ { - i } & = \lfloor - i \overline { t } \rfloor - \lceil - i \overline { u } \rceil + 1 \end{align*}


\begin{align*} ( n , m ) \ = \ n ^ m \end{align*}


\begin{align*} \psi _ { Q _ { J - 2 A } } ( x ) = - \frac { x } { 2 } \ , \Big ( \det M _ 1 - x \det M _ 2 \Big ) , \end{align*}


\begin{align*} H X ^ q ( \R ^ n , \R ) = \begin{cases} \R & q = n \\ 0 & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


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


\begin{align*} \begin{aligned} | N _ { G } ( v ) \cap ( V ( G ) \setminus ( V ( B ) \cup \{ u _ { l } \} ) ) | = d _ { G } ( v ) - | N _ { G } ( v ) \cap V ( B ) | - | N _ { G } ( v ) \cap \{ u _ { l } \} | \geq t \\ \end{aligned} \end{align*}


\begin{align*} m ^ { P C } _ { \eta } = I + \frac { 1 } { \zeta } \left ( \begin{array} { c c } 0 & - i \beta ^ { ( \eta ) } _ { 1 2 } \\ i \beta ^ { ( \eta ) } _ { 2 1 } & 0 \end{array} \right ) + \mathcal { O } ( \zeta ^ { - 2 } ) . \end{align*}


\begin{align*} \Gamma _ t ( \delta ) \triangleq \{ y \in \Omega : \hbox { d i s t } ( y , \Gamma _ t ) < \delta \} , \Gamma ( \delta ) = \bigcup _ { t \in [ 0 , T _ 0 ] } \Gamma _ t ( \delta ) \times \{ t \} , \end{align*}


\begin{align*} E _ 1 ( H , r , w ) : = & \left \{ \exists \ , i \in \left \{ 0 , \dots , m \right \} : \ y _ i \big ( \tfrac { H } { r } , r \big ) \right \} , \\ E _ 2 ( H , r , w ) : = & \bigcup _ { \substack { 0 \leq i \leq m \\ 0 \leq j \leq r - 1 } } \left \{ X _ { ( j + 1 ) \frac { H } { r } } ^ { y _ i } - X _ { j \frac { H } { r } } ^ { y _ i } \geq \left ( v _ + + \frac { \theta } { 2 r } \right ) \frac { H } { r } \right \} . \end{align*}


\begin{align*} \iota ( x , \omega , \tau ) = ( x , \omega , \tau + \tfrac { 1 } { 2 } x \cdot \omega ) \end{align*}


\begin{align*} \mathrm { P C } ( \tilde T ) : = \bigotimes _ { \tilde u } \mathrm { P C } _ { \tilde u } \otimes \bigotimes _ { \tilde w } \mathrm { P C } _ { \tilde w } \ , . \end{align*}


\begin{align*} \mathrm { E C H } ( Y , \lambda , \Gamma ) : = \bigoplus _ { * : \ , \ , \mathbb { Z } \ , \ , \mathrm { g r a d i n g } } \mathrm { E C H } _ { * } ( Y , \lambda , \Gamma ) . \end{align*}


\begin{align*} P _ 1 + P _ 2 + \cdots + P _ k = 1 - \frac { z F _ { k , 1 } ( A ) } { A } \end{align*}


\begin{align*} J ^ A _ \gamma : = ( I d + \gamma A ) ^ { - 1 } . \end{align*}


\begin{align*} \lambda _ { i j } = z _ { i j } - z _ { i j + 1 } j = 0 , \ldots , n _ i - 1 \lambda _ { i n _ i } = z _ { i n _ i } , \end{align*}


\begin{align*} \mathfrak { e } : = \{ Z \in \Xi ~ | ~ \mathcal { L } _ Z \nabla _ X Y = \nabla _ { [ Z , X ] } Y + \nabla _ X [ Z , Y ] X , Y \in \Xi \} \end{align*}


\begin{align*} \phi ^ 2 = s ^ 2 \chi _ A + t ^ 2 \chi _ B + \chi _ C . \end{align*}


\begin{align*} z ^ 3 y ^ 2 + a z y ^ 4 + x ^ 5 + b y ^ 5 = 0 \ , . \end{align*}


\begin{align*} u \left ( t , x \right ) = \frac { e ^ { i \frac { x ^ { 2 } } { 4 t } } } { \sqrt { - 2 i t } } \exp \left ( - \frac { i } { 2 } \left | W _ { + \infty } \left ( - \frac { x } { 2 t } \right ) \right | ^ { 2 } \log t \right ) W _ { + \infty } \left ( - \frac { x } { 2 t } \right ) + \mathcal { O } \left ( \varepsilon t ^ { - \frac { 1 } { 2 } - \delta } \right ) , t \geq 1 . \end{align*}


\begin{align*} \frac { \partial } { \partial t } K _ H ( t , s ) = c _ H \Big ( H - \frac { 1 } { 2 } \Big ) \Big ( \frac { 1 } { 2 } \Big ) ^ { H - \frac { 1 } { 2 } } \Big ( t - s \Big ) ^ { H - \frac { 3 } { 2 } } . \end{align*}


\begin{align*} \lim _ { n \to \infty } E \left [ \left ( \dfrac { S _ n } { \sqrt { n / ( 1 - 2 \alpha ) } } \right ) ^ k \right ] = \mu _ k ( k \in \mathbb { N } ) , \end{align*}


\begin{align*} \lim _ { t \to \infty } f ( t ) = \lim _ { z \downarrow 0 } z \tilde f ( - i z ) , \end{align*}


\begin{align*} \forall \ell \in \{ 1 , \ldots , k - 1 \} ~ \exists \ell ' \neq \ell : p _ { \ell , 1 } = p _ { \ell ' , 2 } , \end{align*}


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


\begin{align*} H _ i ( x ) : = \begin{cases} | \phi ( x _ i ) | H ( R _ i \phi ( x ) ) & ; \\ 0 & . \end{cases} \end{align*}


\begin{align*} \| u \| = \sup \left \{ \left | \sum _ { i = 1 } ^ n x ^ * ( x _ i ) y ^ * ( y _ i ) \right | : x ^ * \in B _ { X ^ * } , y ^ * \in B _ { Y ^ * } \right \} , \end{align*}


\begin{align*} \begin{aligned} | N _ { G } ( v ) \cap ( V ( T ' ) \backslash \{ u _ { 1 } , \cdots , u _ { s } \} ) | & = | N _ { G } ( v ) \cap ( Y \backslash \{ u _ { 1 } , \cdots , u _ { s } \} ) | \\ & \leq | Y | - \lfloor \frac { s } { 2 } \rfloor \\ & \leq t - \lfloor \frac { s } { 2 } \rfloor \end{aligned} \end{align*}


\begin{align*} ( J ) = ( S ^ T J S ) = \det ( S ) ( J ) \Longrightarrow \det ( S ) = 1 . \end{align*}


\begin{align*} \mu ( E _ { \mu } ^ { [ \alpha _ 1 , \gamma _ 1 ] , \varepsilon } ) = 1 . \end{align*}


\begin{align*} \Delta _ f x _ k \Delta _ f p _ k \geq \frac { \hbar } { 2 } , k = 1 , \ldots , d . \end{align*}


\begin{align*} \int _ { B ( x , r ) } | \nabla v | ^ { p - 2 } \nabla v \cdot \nabla ( u - v ) \ , d \mu = 0 . \end{align*}


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


\begin{align*} \langle f , g \rangle & : = \langle - \ln r \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } w ( Q ) \psi _ Q ( \cdot , x _ { Q } ) q _ { Q } h ( x _ { Q } ) , g \rangle \\ & = - \ln r \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } w ( Q ) \psi _ Q g ( x _ { Q } ) q _ { Q } h ( x _ { Q } ) , \end{align*}


\begin{align*} G _ { k , l } ( x ) = \begin{cases} - \frac { x ^ k } { k - l } + x ^ l \int _ 0 ^ x \frac { y ^ { k - l - 1 } } { 1 - y ^ k } d y , \ l = 1 , \cdots , k - 1 ; \\ - \frac { x ^ k } k \log ( 1 - x ^ k ) , \ l = k . \end{cases} \end{align*}


\begin{align*} q _ { 0 } ( x ) = \left \{ \begin{aligned} & C _ { L } , x < 0 , \\ & C _ { R } , x > 0 , \end{aligned} \right . \end{align*}


\begin{align*} A = I _ 0 \supseteq I _ 1 \supseteq I _ 2 \supseteq \cdots \supseteq I _ n = 0 \end{align*}


\begin{gather*} a * ( \lambda _ a ^ { - 1 } ( b ) * x ) + \lambda _ a ^ { - 1 } ( b ) * x = 0 \shortintertext { a n d h e n c e } ( a + b ) * ( x + y ) = ( a * x ) + x + ( a * y ) - x . \end{gather*}


\begin{align*} \varphi _ { i , \delta } = \delta ^ { - 1 / 2 } \varphi _ i : B _ { \delta ^ { - 1 / 2 } r _ 1 } ( \hat x _ i , \hat g _ { i , \delta } ) \to \mathbb R ^ n \end{align*}


\begin{align*} G ^ { \mu } _ \delta ( f ) = \int _ 0 ^ { + \infty } e ^ { - \delta t } P ^ { \mu } _ t ( f ) d t . \end{align*}


\begin{align*} A = \begin{pmatrix} - \partial _ x & - \partial _ x \\ [ 4 p t ] - \partial _ x & \partial _ { x x } - \partial _ x \end{pmatrix} , \end{align*}


\begin{align*} f g ( x ) : = \sum _ { w \in \N ^ * \cup \{ \mathbf { 1 } \} } ( f g ) _ w x _ w . \end{align*}


\begin{align*} \| \mathbf { m } _ n \| _ \infty = \Theta \left ( \frac { 1 } { \sqrt { n } } \| \mathbf { m } _ n \| _ 2 \right ) . \end{align*}


\begin{align*} \tilde T _ { j } ( \xi ) = T _ j ( \xi ) - C _ j \xi ^ { 2 j - \gamma } , \end{align*}


\begin{align*} \partial ^ \alpha ( f g ) = \sum _ { \beta \leq \alpha } \frac { \alpha ! } { \beta ! ( \alpha - \beta ) ! } ( \partial ^ \beta f ) ( \partial ^ { \alpha - \beta } g ) . \end{align*}


\begin{align*} U \mapsto A _ \ast ( \rho ( U ) ) = \begin{cases} A & U \mbox { n o t b o u n d e d } \\ 0 & U \mbox { b o u n d e d } \end{cases} . \end{align*}


\begin{align*} T _ \Sigma ( f ) [ \psi ] : = \frac { 1 } { 2 } \int _ \Sigma f ( s ) \psi { ( s ) } ^ 2 \sqrt { \gamma ' ( s ) } \ , \mathrm { d } s . \end{align*}


\begin{align*} \begin{aligned} & k \sum _ { l = 1 } ^ k \binom { c _ n k n } l ( k ) _ l \sum _ { j = n - N + 1 } ^ { n - 1 } \frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \frac 1 { k ( n - j + 1 ) - l } = \\ & k \sum _ { l = 1 } ^ k \binom { c _ n k n } l ( k ) _ l \sum _ { s = 1 } ^ { N - 1 } \frac { ( k ( n - s - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \frac 1 { k ( s + 1 ) - l } . \end{aligned} \end{align*}


\begin{align*} d _ { \alpha } T _ { \psi } = T _ { ( d _ { - \alpha } \psi ) } , \ \ d _ { - \alpha } T _ { \psi } = - T _ { ( d _ { \alpha } \psi ) } . \end{align*}


\begin{align*} T B _ { 8 } ( f ) = 2 T B _ { 5 } ( f ) , \end{align*}


\begin{align*} R ( e _ i \wedge e _ j ) = r _ { i , j } e _ i \wedge e _ j . \end{align*}


\begin{align*} \mathrm { C h } ( u , v ) = D ( u , v ) , u \in C _ c ^ \infty ( \mathcal { R } ) , v \in C _ c ( M ) \cap W ^ { 1 } ( M ) . \end{align*}


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


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


\begin{align*} & | \sigma _ n ^ 2 ( y ) \cdot ( G '' ( x ) - G '' ( y ) ) | \leq \begin{cases} c \cdot ( 1 + | y | ^ { 2 \ell _ \sigma + 2 } ) \cdot | x - y | , & ( x , y ) \in B ^ c , \\ c \cdot ( 1 + | y | ^ { 2 \ell _ \sigma + 2 } ) , & ( x , y ) \in B . \end{cases} \end{align*}


\begin{align*} M ( v , X ( k ) \setminus S ) = \sum \limits _ { y \in X ( k ) \setminus S } M ( v , y ) \end{align*}


\begin{align*} \left ( \begin{array} { c } u _ 1 ( t + 2 \bar { T } ) \\ u _ 2 ( t + 2 \bar { T } ) \\ u _ 3 ( t + 2 \bar { T } ) \end{array} \right ) = \left ( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \cos ( \frac { 4 \pi p } { n } ) & - \sin ( \frac { 4 \pi p } { n } ) \\ 0 & \sin ( \frac { 4 \pi p } { n } ) & \cos ( \frac { 4 \pi p } { n } ) \end{array} \right ) \left ( \begin{array} { c } u _ 1 ( t ) \\ u _ 2 ( t ) \\ u _ 3 ( t ) \end{array} \right ) , \end{align*}


\begin{align*} z = [ g _ 1 \cdot ( g ^ + ) ^ { - 1 } \cdot g _ 2 ^ { - 1 } ] \cdot g _ 0 ^ { - 1 } \cdot \phi ( g ^ + ) \cdot g _ 0 \end{align*}


\begin{align*} \forall k \in \N \colon \lambda _ k : = y _ k ^ * \norm { x _ k - \bar x } / \norm { y _ k - \bar y } \end{align*}


\begin{align*} \Theta ( I , x , y ) = \sum _ { \tilde y \in \phi ^ { - 1 } ( y ) } \Theta ( \tilde I , \tilde x , \tilde y ) ; \end{align*}


\begin{align*} A ( d _ 0 , f ) = \sum _ { a \mod d _ 0 \atop \gcd ( d _ 0 , a ) = 1 } \sum _ { b \mod d _ 0 \atop { \gcd ( d _ 0 , b ) = 1 \atop b \neq a } } \cot \left ( \frac { \pi ( b - a ) } { d _ 0 } \right ) \left ( \cot \left ( \frac { \pi f a } { d _ 0 } \right ) - \cot \left ( \frac { \pi f b } { d _ 0 } \right ) \right ) , \end{align*}


\begin{align*} \norm { w } \leq \frac { 2 L } { 2 ^ { - \alpha _ n } } = L \cdot 2 ^ { \alpha _ n + 1 } . \end{align*}


\begin{align*} P [ u ] = \Im \int \bar { u } \nabla u d x = 0 , \end{align*}


\begin{align*} M _ { d _ 0 } ( p , H ) = \kappa _ { d _ 0 } \times \left ( 1 + \frac { N _ { d _ 0 } ( p , H ) } { p } \right ) , \hbox { w h e r e } \kappa _ { d _ 0 } : = \frac { \pi ^ 2 } { 6 } \prod _ { q \mid d _ 0 } \left ( 1 - \frac { 1 } { q ^ 2 } \right ) . \end{align*}


\begin{align*} \Delta ( D _ { n n } ) = \sum _ { 1 \leq k < l \leq 6 } a _ { n k } a _ { n l } \otimes D _ { k l } + \sum _ { 5 \leq k \leq 6 } a _ { n k } ^ { 2 } \otimes D _ { k k } \ , . \end{align*}


\begin{align*} { S } ( x ) ( t ) : = z _ 1 ^ * ( ( \widetilde { S } ( x ) ) ( t ) ) ( x \in X , \ , t \in K ) . \end{align*}


\begin{align*} \langle M \rangle _ t = \int _ 0 ^ t f ( X _ s ) d s , \ ; \ ; t > 0 , \ ; \ ; P _ { x _ 0 } \end{align*}


\begin{align*} \begin{matrix} \mathbf { x } _ { k + 1 } = \boldsymbol { f } ( \mathbf { x } _ k , \mathbf { u } _ k ) , & \quad & \mathbf { y } _ k = \boldsymbol { h } ( \mathbf { x } _ k ) , \end{matrix} \end{align*}


\begin{align*} 0 = E _ 0 \subset E _ 1 \subset \ldots \subset E _ m = E \end{align*}


\begin{align*} & \frac { 1 } { 2 \pi } \int _ { 0 } ^ { T } \left [ \log \left | \frac { \frac { 1 } { 2 } - \epsilon + i t } { 2 } \right | - \log \left | \frac { \frac { 1 } { 2 } + \epsilon + i t } { 2 } \right | \right ] \ d t + \\ & \frac { 1 } { 2 \pi } \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \left [ \arg \left ( \frac { \sigma + i T } { 2 } \right ) - \arg \frac { \sigma } { 2 } \right ] \ d \sigma = 0 \end{align*}


\begin{align*} \frac { \mathbf { P } \Big ( \frac { 1 } { \sigma } ( \hat { \theta } _ n - \theta ) \sqrt { \Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } > x \Big ) } { 1 - \Phi \left ( x \right ) } = 1 + o ( 1 ) \ \ \ \textrm { a n d } \ \ \ \frac { \mathbf { P } \Big ( \frac { 1 } { \sigma } ( \hat { \theta } _ n - \theta ) \sqrt { \Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } < - x \Big ) } { \Phi \left ( - x \right ) } = 1 + o ( 1 ) . \end{align*}


\begin{align*} J ^ { G P } = \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix} , k \in ( - \eta , \eta ) \end{align*}


\begin{align*} { } ^ b { \rm T r } _ { c _ Y } \left ( [ \Phi _ { A _ { 1 } } , \Phi _ { A _ { 2 } } ] \right ) = \frac { i } { 2 \pi } \int _ { \mathbb { R } } \int _ G { \rm T r } _ { \partial Z } \left ( \frac { \partial I ( \Phi _ { A _ 1 } , h ^ { - 1 } , \lambda ) } { \partial \lambda } \circ I ( \Phi _ { A _ 2 } , h , \lambda ) \right ) d h d \lambda . \end{align*}


\begin{align*} \mathbf L \tilde { \mathbf V } + \mathbf b + \rho \mathcal I _ { \tilde { \mathbf V } } ( \mathbf V ^ * - \tilde { \mathbf V } ) + \mathbf a _ 2 + \mathbf a _ 1 = \mathbf 0 , \end{align*}


\begin{align*} \lambda _ { \phi } ( t ) = \frac { E ( \phi ) + \frac { \beta } { 2 } \int _ { \mathbb { R } ^ d } | \phi ( \mathbf { x } , t ) | ^ 4 \mathrm { d } \mathbf { x } - \| \dot { \phi } ( \cdot , t ) \| ^ 2 } { \| \phi ( \cdot , t ) \| ^ 2 } = \mu _ { \phi } ( t ) - \frac { \| \dot { \phi } ( \cdot , t ) \| ^ 2 } { \| \phi ( \cdot , t ) \| ^ 2 } , \end{align*}


\begin{align*} \mathcal N _ Q ( \bar x ; u ) = \left \{ \eta \in \mathbb X \ , \middle | \ , \begin{aligned} & \exists \{ u _ k \} _ { k \in \N } \subset \mathbb X , \ , \exists \{ t _ k \} _ { k \in \N } \subset \R _ + , \ , \exists \{ \eta _ k \} _ { k \in \N } \subset \mathbb X \colon \\ & u _ k \to u , \ , t _ k \searrow 0 , \ , \eta _ k \to \eta , \ , \eta _ k \in \mathcal N _ Q ( \bar x + t _ k u _ k ) \ , \forall k \in \N \end{aligned} \right \} . \end{align*}


\begin{align*} [ \chi _ A ] _ \mathcal { M } : = \lambda x , y \in X . \begin{cases} 0 ^ 0 & y \in A x , \\ 1 ^ 0 & y \not \in A x , \end{cases} \end{align*}


\begin{align*} P ( u ) = w . \end{align*}


\begin{align*} \mathrm { E C H } ( S ^ { 3 } , \lambda , 0 ) = \mathbb { F } [ U ^ { - 1 } , U ] / U \mathbb { F } [ U ] \end{align*}


\begin{align*} L _ { t } ( w , l ) : = w _ { l _ { t } } . \end{align*}


\begin{align*} \min _ { x \in \mathbb R ^ d } f ( x ) : = \mathbb E _ { z \sim \mathcal D } \left [ f ( x ; z ) \right ] , \end{align*}


\begin{align*} S ( z _ 1 , \dots , \widehat { z _ i } , \dots , \widehat { z _ j } , \dots , z _ n ) = 0 , \end{align*}


\begin{align*} C ( \Psi , \varphi _ 0 , J , f ) = \sup _ { \substack { \xi \in A ^ 2 ( \{ \Psi < 0 \} , e ^ { - \varphi _ 0 } ) ^ * \setminus \{ 0 \} \\ \xi | _ { A ^ 2 ( \{ \Psi < 0 \} , e ^ { - \varphi _ 0 } ) \cap J } \equiv 0 } } \frac { | \xi \cdot f | ^ 2 } { K ^ { \varphi _ 0 } _ { \xi , \Psi , \lambda } ( 0 ) } . \end{align*}


\begin{align*} \| A ^ \mu E _ { k , N } ^ m v \| & \leq C t _ m ^ { - \frac \mu 2 } \| v \| , \quad \forall v \in H , \ ; m = 1 , 2 , 3 , \cdots , M . \end{align*}


\begin{align*} \mathbf { L } ( \mathbf { t } ) \in \mathbb { C } ^ { k \times n } , \mathbf { R } ( \mathbf { t } ) = \mathrm { d i a g } \left ( \mathbf { d } ( \mathbf { t } ) \right ) \cdot \mathbf { R } ( \mathbf { 1 } ) \end{align*}


\begin{align*} ( \mathfrak { A } _ { \ell } ( \rho , \chi ) F ) _ { \widetilde { H } } = F _ { \chi ^ { * } \widetilde { H } } \circ \rho ^ { * } _ { ( 1 ) } . \end{align*}


\begin{align*} \sqrt { \epsilon } = \begin{cases} T ^ { - 1 / 2 } & \\ ( \log ( T ) / T ) ^ { 1 / 2 } & \\ T ^ { - \frac { 1 } { d - 1 / H } } & \end{cases} \end{align*}


\begin{align*} \begin{pmatrix} X ^ \varepsilon & 0 \\ 0 & - Y ^ \varepsilon \end{pmatrix} \le 0 , \end{align*}


\begin{align*} n = | G | , \quad \hat \theta = \frac { | \theta | } { \sqrt { n } | | \phi | | _ 2 } . \end{align*}


\begin{align*} T f ( x ) & = \langle K ( x , y ) , ( \xi ( y ) + \eta ( y ) ) f ( y ) \rangle \\ & = \int _ { \R ^ N } K ( x , y ) \xi ( y ) ( f ( y ) - f ( x ) ) d \omega ( y ) + f ( x ) \langle K ( x , y ) , \xi ( y ) \rangle \\ & + \int _ { \R ^ N } K ( x , y ) \eta ( y ) f ( y ) d \omega ( y ) \\ & = : I _ 1 + I _ 2 + I _ 3 . \end{align*}


\begin{align*} \mathcal { C } ^ { 1 } ( [ 0 , T ] ; X ) : = \left \lbrace u \in \mathcal { C } ( [ 0 , T ] ; X ) : u _ { t } \in \mathcal { C } ( [ 0 , T ] ; X ) \right \rbrace . \end{align*}


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


\begin{align*} \sigma = \frac \alpha { q _ \alpha n } , \end{align*}


\begin{align*} f _ n ( x ) = \begin{cases} x - \frac { n - 1 } { n } & x < - \frac { 1 } { n } \\ n x & x \in [ - \frac { 1 } { n } , \frac { 1 } { n } ) \\ x + \frac { n - 1 } { n } & x \geq \frac { 1 } { n } , \end{cases} \end{align*}


\begin{align*} \int _ a ^ b f \dd g = \int _ a ^ c f \dd g + \int _ c ^ b f \dd g \end{align*}


\begin{align*} F _ r * _ { \mathbf { H } _ r } G _ r ( x , \omega ) = ( F _ r * G _ r ) ( x , \omega , e ^ { 2 \pi i \tau } ) \end{align*}


\begin{align*} ( \pi ^ { s + 1 , s } ) ^ { * } \rho = \Theta _ { \lambda } + d \mu + \eta , \end{align*}


\begin{align*} \tilde \Gamma ( t ) : = \frac { \Gamma ( t ) } { | t | ^ { 1 / 2 } } \end{align*}


\begin{align*} ( \phi ^ 2 + \phi * \phi ) ( 0 ) = c ^ 2 ( \alpha * \alpha ) ( 0 ) + \bar c ^ 2 ( \bar \alpha * \bar \alpha ) ( 0 ) + 2 ( \bar \alpha * \alpha ) = 2 ( q - 1 ) . \end{align*}


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


\begin{align*} \Gamma _ { \Sigma } ( f , f ) ( x ) = \langle \nabla ( f ) ( x ) ; \Sigma ( \nabla ( f ) ( x ) ) \rangle = \left \| \sqrt { \Sigma } ( \nabla ( f ) ( x ) ) \right \| ^ 2 . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ d c _ i p _ i = 0 \sum _ { i = 0 } ^ d c _ i = 0 c _ 0 = \dotsb = c _ d = 0 \end{align*}


\begin{align*} \pi _ 0 ( \hat x ^ 1 , \dots , \hat x ^ m , \dots , \hat x ^ n ) = ( \hat x ^ 1 , \dots , \hat x ^ m ) , \end{align*}


\begin{align*} a _ i = n _ i - n _ { \min } + 1 , i = 1 , 2 , \cdots , m . \end{align*}


\begin{align*} \tilde \psi \left ( \begin{pmatrix} x _ 0 \\ y _ 0 \end{pmatrix} , \ldots , \begin{pmatrix} x _ { q - 1 } \\ y _ { q - 1 } \end{pmatrix} \right ) = \sum _ { t = 0 } ^ { \max ( y _ j ) - 1 } ( h _ t ^ * \psi _ 1 ) \left ( \begin{pmatrix} x _ 0 \\ y _ 0 \end{pmatrix} , \ldots , \begin{pmatrix} x _ { q - 1 } \\ y _ { q - 1 } \end{pmatrix} \right ) . \end{align*}


\begin{align*} S E ( G ) \leq E ( J - I ) + 2 E ( A ( G ) ) = 2 ( n - 1 ) + 2 E ( A ( G ) ) , \end{align*}


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


\begin{align*} y _ { j _ { 1 } \ldots j _ { k } } ^ { \sigma } ( J _ { x } ^ { r } \gamma ) = D _ { j _ { 1 } } \ldots D _ { j _ { k } } ( y ^ { \sigma } \gamma \varphi ^ { - 1 } ) ( \varphi ( x ) ) , \quad 0 \leq k \leq r . \end{align*}


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


\begin{align*} d \Phi _ t = \nabla \log \left ( \frac { d \mu ^ { \epsilon , L } } { d \varphi } \right ) ( \Phi _ t ) d t + \sqrt { 2 } d B _ t , \Phi _ 0 \sim \nu ^ { \epsilon , L } , \end{align*}


\begin{align*} { k \choose i } _ q \alpha _ i = 1 + o _ q ( 1 ) \end{align*}


\begin{align*} \widehat \psi _ { d ^ 2 } ( \mathbf { u } ) = \frac { 1 } { d ^ { 2 n } } \sum _ { f \in ( \mathbb { Z } / d ^ 2 \mathbb { Z } ) ^ { n } } \psi _ d ( f ) e ^ { \frac { 2 \pi i \langle f , \mathbf { u } \rangle } { d ^ 2 } } . \end{align*}


\begin{align*} S ( a , b , 3 \delta f ) = s ( a , b , 3 \delta f ) + s ( a + 2 f , b , 3 \delta f ) . \end{align*}


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


\begin{align*} \beta ( z , \tau ) = \beta ( x , \omega , \tau ) = B \rho ( x , \omega , \tau ) B ^ { - 1 } . \end{align*}


\begin{align*} \Lambda ^ { ( 2 : 2 ) } _ { n , 2 , 0 } ( \mathcal { A } ) & = \frac { 1 6 \cdot \left ( 1 + o ( 1 ) \right ) } { 9 0 ^ 2 n ^ 4 \delta ^ 4 _ n ( k ) } \cdot \sum \limits _ { \mathbf { P } _ { k } \in \mathcal { A } } \sum \limits _ { \mathbf { i } \in \mathcal I _ 2 \times \mathcal I _ 2 } \varphi _ { k - 1 } \left ( \mathbf { P } _ { k } , \mathbf { i } \right ) . \end{align*}


\begin{align*} \frac { \zeta ' ( s ) } { \zeta ( s ) } = \sum _ { \rho } \left ( \frac { 1 } { s - \rho } - \frac { 1 } { 2 + i t - \rho } \right ) + \mathcal { O } ( \log t ) \end{align*}


\begin{align*} | \langle \partial _ { 3 } u \nabla u , \partial _ { 3 } u \rangle | & \leq \sum _ { i = 1 } ^ { 3 } \int _ { \R ^ 3 } | \partial _ { 3 } u _ { i } \partial _ { i } u \partial _ { 3 } u | \end{align*}


\begin{align*} \int _ { t ^ k } ^ { t _ { k + 1 } } d N ( \sigma , t ) = N ( \sigma , t _ { k + 1 } ) - N ( \sigma , t _ k ) . \end{align*}


\begin{align*} \mu _ 1 y \cdot \mu _ 2 x = \mu _ 3 ( x \cdot y ) \end{align*}


\begin{align*} - \int _ \R \log | x | \overline { f ' ( x ) } \ , d x = \langle \log ( | x | ) , - f ' \rangle = \langle \log ( | x | ) ' , f \rangle , \end{align*}


\begin{align*} f ( t , x , v ) = & \int _ 0 ^ t ( \nabla \phi \ast \rho [ f ] ) ( s , x - ( t - s ) v ) \cdot \nabla \mu ( v ) \dd s \\ + & \int _ 0 ^ t E ( s , x - ( t - s ) v ) \cdot \nabla \mu ( v ) \dd s - \int _ 0 ^ t E ( s , X _ { s , t } ( x , v ) ) \cdot \nabla _ v \mu ( V _ { s , t } ( x , v ) ) \\ - & \int _ 0 ^ t e _ 0 \nabla \Phi ( X _ { s , t } ( x , v ) - X ( s ) ) \cdot \nabla _ v \mu ( V _ { s , t } ( x , v ) ) , \end{align*}


\begin{align*} J _ { 0 } ( u ) = - \chi ( u _ { 1 } ) \end{align*}


\begin{align*} \mathcal { Q } _ H : = \left \{ s \left | \begin{aligned} & s _ { i n } = s _ { i j } & & j \geq \tau ( i , t _ i ( H ) ) \\ & \frac { s _ { i j + 1 } - s _ { i j } } { m _ { i ( j + 1 ) t _ i ( H ) } - m _ { i j t _ i ( H ) } } \leq \frac { s _ { i j } - s _ { i j - 1 } } { m _ { i j t _ i ( H ) } - m _ { i ( j - 1 ) t _ i ( H ) } } & & \tau ( i , t _ i ( H ) - 1 ) \leq j \leq \tau ( i , t _ i ( H ) ) \\ & s _ { i j } = m _ { i j t _ i ( H ) } & & j \leq \tau ( i , t _ i ( H ) - 1 ) \end{aligned} \right . \right \} \end{align*}


\begin{align*} \mathcal A ( \rho ^ * , m ^ * ) = \sup _ { S \in H _ R ^ 1 } \mathcal L ( \rho ^ * , m ^ * , S ) \ge \inf _ { ( \rho , m ) } \sup _ { S \in H _ R ^ 1 } \mathcal L ( \rho , m , S ) . \end{align*}


\begin{align*} m _ 0 = \frac { \rho _ 1 \rho _ 2 \rho _ 3 } { \lambda _ 0 Z } , m _ 1 = \frac { 1 } { \lambda _ 1 Z } , m _ 2 = \frac { \rho _ 1 } { \lambda _ 2 Z } , m _ 3 = \frac { \rho _ 1 \rho _ 2 } { \lambda _ 3 Z } , \end{align*}


\begin{align*} \left ( E - \mathcal { A } \right ) ^ { - \frac { r } { 2 } } f = \frac { 1 } { \Gamma ( \frac { r } { 2 } ) } \int _ 0 ^ { + \infty } \dfrac { e ^ { - t } } { t ^ { 1 - \frac { r } { 2 } } } P _ t ( f ) d t , \end{align*}


\begin{align*} ( \phi ^ 2 + \phi * \phi ) ( a ) = ( 1 + J ' ( \alpha ^ 2 , \alpha ^ 2 ) ) \alpha ^ 4 ( a ) . \end{align*}


\begin{align*} S ^ * \pi ( \mathbf { h } ) = \left ( \pi ( \mathbf { h } ^ { - 1 } ) S \right ) ^ * = \left ( S \pi ( \mathbf { h } ^ { - 1 } ) \right ) ^ * = \pi ( \mathbf { h } ) S ^ * , \end{align*}


\begin{align*} \Phi = \sum _ { k = 0 } ^ { \infty } \Phi _ { k } Q _ { k } ^ { \pi _ { \lambda , \beta } } . \end{align*}


\begin{align*} \theta ( \tau ) = \beta ( \tau ) \left ( b ' ( \tau ) - b ( \tau ) \right ) . \end{align*}


\begin{align*} D f ( x ) b ( x ) = A f ( x ) . \end{align*}


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


\begin{align*} [ W ] = [ w _ 1 , \ldots , w _ n ] = w _ 1 + \cfrac { 1 } { w _ 2 + \cfrac { 1 } { w _ 3 + \cfrac { 1 } { \ddots + \cfrac { 1 } { w _ n } } } } \end{align*}


\begin{align*} \mathcal { H } ^ n = \ ! \left ( - 1 + \frac { \tau } { 2 } \eta ^ n - \frac { \tau ^ 2 } { 2 } \vartheta ^ n \ ! \right ) \ ! \phi ^ { n - 1 } + \frac { \tau ^ 2 } { 4 } \Delta \phi ^ { n - 1 } + 2 \phi ^ n + \tau ^ 2 \left ( \ ! ( \vartheta ^ n + \lambda ^ n ) \phi ^ n - \frac { 1 } { 2 } \Delta \phi ^ n - G ( \phi ^ n ) \ ! \right ) \end{align*}


\begin{align*} \begin{cases} \Delta _ { g _ { i , \delta } } f _ { i } ^ { j } = 0 , & \ B _ { 4 R } ( x _ { i } , g _ { i , \delta } ) ; \\ f _ { i } ^ { j } = b _ { i } ^ { j } , & \ \partial B _ { 4 R } ( x _ { i } , g _ { i , \delta } ) , \end{cases} \end{align*}


\begin{align*} \lim _ { n \to \infty } \int _ a ^ b f _ n \dd g = \int _ a ^ b f \dd g . \end{align*}


\begin{align*} d _ j : = \sum _ { i = 1 } ^ { n _ 0 } c _ { i , j } \partial _ { x _ { j } } \bigg | _ { x = x _ { \alpha } } , j = 1 , 2 . \end{align*}


\begin{align*} \mathcal { L } _ { \alpha , \beta } ( f ) ( x ) = ( 1 - x ^ 2 ) f '' ( x ) + ( ( \beta - \alpha ) - ( \alpha + \beta ) x ) f ' ( x ) . \end{align*}


\begin{align*} \mbox { m i n i m i z e } \ ; \ ; f ( x ) : = ( g \circ F ) ( x ) \mbox { s u b j e c t t o } \ ; \ ; G ( x ) \in X \end{align*}


\begin{align*} v _ i \big [ u ^ { i - 1 } \beta _ i ( x y ) - u ^ { i - 1 } x \beta _ i ( y ) - u ^ { i - 2 } \beta _ m ( x ) \beta _ { i - m } ( y ) - \dots - u ^ { i - 1 } \beta _ i ( x ) y \big ] = 0 \end{align*}


\begin{align*} Z : = \frac { 4 | \tilde { x } - \tilde { y } | ^ 2 } { \varepsilon ^ 4 } ( I + 2 ( \tilde { x } - \tilde { y } ) \otimes ( \tilde { x } - \tilde { y } ) ) + 2 \alpha \begin{pmatrix} I & I \\ I & I \end{pmatrix} . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ d } d x \ , \mathbb { E } \bigg [ \Big | \mathcal { F } ^ { - 1 } \Big ( \big ( 1 + | . | ^ 2 \big ) ^ { - s } \mathcal { F } \Big ( \rho [ \ < P s i 2 > _ { n } ( t , . ) - \ < P s i 2 > _ { n } ( s , . ) ] \Big ) \Big ) ( x ) \Big | ^ { 2 p } \bigg ] \lesssim \Bigg ( \sum _ { k = 1 } ^ { 8 } I _ k \Bigg ) ^ p , \end{align*}


\begin{align*} \theta _ { \Sigma , \epsilon } \left ( F _ H \right ) _ { H \in \mathrm { H a d } ( \mathcal { M } ) } : = \left ( \partial ^ { * } _ { \Sigma , \epsilon } F _ { H } \right ) _ { H \in \mathrm { H a d } ( \mathcal { M } ) } \end{align*}


\begin{align*} Z f ( x + k , \omega + l ) = e ^ { 2 \pi i k \cdot \omega } Z f ( x , \omega ) \end{align*}


\begin{align*} \displaystyle \lambda _ 1 \leq & \sqrt { \left ( \frac { 1 } { 6 } \sum _ { i = 1 } ^ { m } 2 ^ 2 ( 2 ^ 2 - 1 ) - 0 \right ) \left ( \frac { n - 1 } { n } \right ) } \\ \displaystyle \lambda _ 1 \leq & \sqrt { 2 m \left ( \frac { n - 1 } { n } \right ) } \end{align*}


\begin{align*} \norm { u } _ { W ( \sigma ' R ) , \infty } \leq C \left ( \frac { 1 \vee \norm { \lambda ^ { - 1 } } _ { W ( R ) , q } \norm { \Lambda } _ { W ( R ) , p } } { ( \sigma - \sigma ' ) ^ 2 } \right ) ^ \kappa \norm { u } _ { W ( \sigma R ) , \rho } ^ { \prod _ { k = 1 } ^ \infty \gamma _ k } , \end{align*}


\begin{align*} S _ t ( ( Y _ 0 , 0 , 0 ) , d W _ \cdot ) = S _ t ( Y _ 0 , d W _ \cdot ) , t \ge 0 . \end{align*}


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


\begin{align*} \pi _ i ( x ) = \frac { 1 } { R ( i + 1 , i ) } \sum \limits _ { y \gtrdot x } \pi _ { i + 1 } ( y ) . \end{align*}


\begin{align*} b ( \tau ) = \Psi _ \infty \exp { \left ( \left ( 1 - \frac { 2 } { \alpha } \right ) \left ( \int _ { \tau _ 0 } ^ \tau 2 \beta ( \zeta ) d \zeta + \tau _ 0 \right ) \right ) } \left [ 1 + O \left ( I ^ \eta ( \tau ) \right ) \right ] \tau \to + \infty . \end{align*}


\begin{align*} \varphi ^ { ( 3 ) } ( x ) & = C \int _ { \R ^ n \setminus B _ 1 } \bigg ( \frac { 1 - \vert x \vert ^ 2 } { \vert y \vert ^ 2 - 1 } \bigg ) ^ s \frac { \varphi ( y _ 1 ) } { \vert x - y \vert ^ n } \dd y \\ & = C \int _ { \R ^ n _ + \setminus B _ 1 ^ + } \bigg ( \frac { 1 - \vert x \vert ^ 2 } { \vert y \vert ^ 2 - 1 } \bigg ) ^ s \bigg ( \frac 1 { \vert x - y \vert ^ n } - \frac 1 { \vert x _ \ast - y \vert ^ n } \bigg ) \varphi ( y _ 1 ) \dd y . \end{align*}


\begin{align*} \mathbf { D } ^ { \alpha - 1 } ( f ) ( x ) : = \frac { 1 } { 2 } \left ( D ^ { \alpha - 1 } \left ( f \right ) ( x ) - ( D ^ { \alpha - 1 } ) ^ * \left ( f \right ) ( x ) \right ) . \end{align*}


\begin{align*} H X _ b ^ 1 ( X , A ) = \begin{cases} 0 & e ( X ) = 0 \\ A ^ { e ( X ) - 1 } & 0 < e ( X ) < \infty \\ \bigoplus _ \N A & e ( X ) = \infty \end{cases} \end{align*}


\begin{align*} \int _ { \bar x } ^ { x _ 1 } \int _ { \R } \phi _ { x x } \cdot \phi _ x \psi ( t ) \ , d t d x & = \int _ { \bar x } ^ { x _ 1 } \int _ { \R } ( \phi _ { t t } \cdot \phi _ x \psi ( t ) + a ( x ) \phi _ t \cdot \phi _ x \psi ( t ) ) \ , d t d x \\ & = \int _ { \bar x } ^ { x _ 1 } \int _ { \R } ( - \phi _ { t } \cdot \phi _ x \psi ' ( t ) - \phi _ t \cdot \phi _ { x t } \psi ( t ) + a ( x ) \phi _ t \cdot \phi _ x \psi ( t ) ) \ , d t d x . \end{align*}


\begin{align*} \log \zeta ( s ) - \log \zeta ( 2 + i t ) = \sum _ { | \gamma - t | \leq 1 } \left [ \log ( s - \rho ) - \log ( 2 + i t - \rho ) \right ] + \mathcal { O } ( \log t ) \end{align*}


\begin{align*} ( - \triangle _ { g } ) ^ { \alpha } u + g ( u ) = K ( x ) f ( x , u ) , \ \ \ \mathbb { R } ^ { d } , \end{align*}


\begin{align*} A ( N ( x ) , x ) = A ( x , N ( x ) ) = 0 , \ \forall x \in [ 0 , 1 ] . \end{align*}


\begin{align*} a \cdot b = \sum _ { i = 1 } ^ m a ^ i b ^ i . \end{align*}


\begin{align*} \mathcal { D } ( \mathcal { L } ) = \left \lbrace f \in \mathcal { D } ( \mathcal { E } ) : \mathcal { L } f \in L ^ 2 ( M ) \right \rbrace . \end{align*}


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


\begin{align*} V ( \mathbf { x } ) = V _ { \mathrm { h o } } ( \mathbf { x } ) + \frac { \kappa } { 2 } \begin{cases} \sin ^ { 2 } \left ( q _ { x } x \right ) , & d = 1 , \\ \sin ^ { 2 } \left ( q _ { x } x \right ) + \sin ^ { 2 } \left ( q _ { y } y \right ) , & d = 2 , \\ \end{cases} \end{align*}


\begin{align*} \varphi ( b _ 0 a _ 1 c _ 1 a _ 2 \cdots c _ { n - 1 } a _ n b _ n ) = \varphi ( b _ 0 ) \varphi ( a _ 1 ) \varphi ( c _ 1 ) \varphi ( a _ 2 ) \cdots \varphi ( c _ { n - 1 } ) \varphi ( a _ n ) \varphi ( b _ n ) = 0 . \end{align*}


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


\begin{align*} V _ { n , m } ( x ) = 1 + \sum _ { k = m + 1 } ^ { n + m + 1 } d _ k x ^ k \end{align*}


\begin{gather*} \Delta ( H ) = H \otimes 1 + 1 \otimes H , \Delta ( E ) = E \otimes 1 + K \otimes E , \Delta ( F ) = F \otimes K ^ { - 1 } + 1 \otimes F , \\ \epsilon ( H ) = \epsilon ( E ) = \epsilon ( F ) = 0 , \\ S ( H ) = - H , S ( E ) = - K ^ { - 1 } E , S ( F ) = - F K . \end{gather*}


\begin{align*} \iota ( \phi ) = \sum _ { g \in G } \phi ( g ) g . \end{align*}


\begin{align*} \phi ( x ) = \min \bigg \{ j \in \{ 0 , 1 , 2 , \ldots , x \} : \abs { P _ x ( a + ( b - a ) \frac { 2 j + 1 } { 2 x + 2 } } - \frac { q ^ \prime } { x + 1 } \leq 0 \bigg \} , \ \bigg ( q ^ \prime = \frac { d ( b - a ) } { 2 } + q \bigg ) . \end{align*}


\begin{align*} J _ 2 ^ n ( t , u ) & : = \int _ u ^ t \exp \Big ( \int _ s ^ t \tilde { f } ' ( r , x _ r ) d r \Big ) K _ s \cdot K _ H ( s , u ) d s . \end{align*}


\begin{align*} \lambda _ { \phi } ( 0 ) = \int _ { \mathbb { R } ^ d } \left ( \frac 1 2 | \nabla \phi _ g | ^ 2 + V | \phi _ g | ^ 2 + \beta | \phi _ g | ^ 4 - \Omega \ , \overline { \phi } _ g L _ z \phi _ g \right ) \mathrm { d } \mathbf { x } = \mu ( \phi _ g ) = : \mu _ g , \end{align*}


\begin{align*} z \bigl ( ~ _ { s + 1 } F _ s ( 1 , \ldots , 1 ; 2 , \ldots , 2 ) ( z ) \bigr ) = L i _ s ( z ) . \end{align*}


\begin{align*} \Bigg | \int _ { \mathbb { R } ^ 5 } \left ( \Delta ^ 2 \varphi _ R \right ) | u | ^ 2 d x \Bigg | & = \Bigg | \frac { 1 } { R ^ 2 } \int \left ( \Delta ^ 2 \varphi \right ) \left ( \frac { x } { R } \right ) u ( t , x ) ^ 2 d x \Bigg | \\ & \le \frac { C } { R ^ 2 } \| u \| _ 2 ^ 2 = \frac { C } { R ^ 2 } \| Q \| _ 2 ^ 2 \le 2 s _ c ( p - 1 ) \delta _ 1 \le 2 s _ c ( p - 1 ) \delta ( t ) \end{align*}


\begin{align*} \phi ( x ) \ = \ \min \{ j \in \ \N : \Phi ( x , j ) = 0 \vee \ j = x \} , \end{align*}


\begin{align*} \begin{aligned} \iota _ 0 : X & \to X \ast I \\ x & \mapsto ( x , ( d ( x , x _ 0 ) , 0 ) ) \end{aligned} & \begin{aligned} \iota _ 1 : X & \to X \ast I \\ x & \mapsto ( x , ( 0 , d ( x , x _ 0 ) ) ) \end{aligned} \end{align*}


\begin{align*} \langle v , a w \rangle = \langle a ^ * v , w \rangle , \forall a \in A , v , w \in V . \end{align*}


\begin{align*} \sum _ { \substack { 1 \le d < \infty \\ ( d , m n h k ) = 1 } } W \left ( \frac { c d } { Q } \right ) = W \left ( \frac { c } { Q } \right ) + \frac { \phi ( m n h k ) } { m n h k } \int _ { 1 } ^ { \infty } W \left ( \frac { c x } { Q } \right ) \ , d x + \int _ { 1 } ^ { \infty } W \left ( \frac { c x } { Q } \right ) \ , d E . \end{align*}


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


\begin{align*} \frac 1 { n + 2 } \frac { \partial u } { \partial x _ 1 } ( 0 ) = \int _ 0 ^ 1 r ^ { n + 1 } \frac { \partial u } { \partial x _ 1 } ( 0 ) \dd r = n \gamma _ { n , s } \int _ 0 ^ 1 \int _ { \R ^ n \setminus B _ r } \frac { r ^ { 2 s + n + 1 } y _ 1 u ( y ) } { ( \vert y \vert ^ 2 - r ^ 2 ) ^ s \vert y \vert ^ { n + 2 } } \dd y \dd r . \end{align*}


\begin{align*} \varsigma _ { j , i } = s i g n \left ( \mathcal { X } _ j - \epsilon _ i \right ) i \in [ m ] , \end{align*}


\begin{align*} \langle \nabla ^ 2 T , U \otimes V \otimes S \rangle \coloneqq \langle \nabla \nabla T , U \otimes V \otimes S \rangle = \langle \nabla _ U \nabla T , V \otimes S \rangle \end{align*}


\begin{align*} E _ r = \big \{ x \in \R ^ d : \ 2 \langle ( \widehat { A } + ( \widehat { A } ) ^ T ) ^ { - 1 } x , x \rangle < r ^ 2 \big \} \end{align*}


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


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


\begin{align*} V _ P D _ L J V _ { - Q } J ^ { - 1 } = \begin{pmatrix} L & L Q \\ P L & P L Q + L ^ { - T } \end{pmatrix} \end{align*}


\begin{align*} \alpha = \alpha ^ \sigma , \epsilon ( \alpha ) = 0 , \alpha ^ 2 = \alpha , \alpha * \alpha = ( \lambda - \mu ) \alpha + \mu + ( r - \mu ) \delta _ e . \end{align*}


\begin{align*} \rho _ 1 ( x \cdot y ) \cdot \rho _ 2 y = \rho _ 3 x \end{align*}


\begin{align*} \mathcal { R } ( R , L ) & : = - 2 \sum _ { i } \beta _ i \otimes R ( L ( \beta _ i ) ) , \\ \mathcal { T } ( R , L ) & : = - \sum _ i \big ( L ^ * ( \alpha _ i ) \otimes 1 + 1 \otimes \alpha _ i \big ) \circ \big ( L ^ * ( R ( \alpha _ i ) ) \otimes 1 + 1 \otimes R ( \alpha _ i ) \big ) , \end{align*}


\begin{align*} B _ t + \nabla \times ( ( \nabla \times B ) \times B ) = & \ \mu \Delta B , \\ \nabla \cdot B = & \ 0 \end{align*}


\begin{align*} m = 6 \ , , p = 4 \end{align*}


\begin{align*} x _ 1 ^ \prime = \varepsilon x _ 1 \ , , x _ 2 ^ \prime = \varepsilon ^ 4 x _ 2 \ , , y _ 1 ^ \prime = \varepsilon ^ 2 y _ 1 \ , , y _ 2 ^ \prime = \varepsilon ^ 3 y _ 2 \ , . \end{align*}


\begin{align*} \dim W > 0 , \Omega = G \cap W \overline { \Omega } \cap \partial \overline { G } = \overline { \Omega } \cap b \overline { G } , \end{align*}


\begin{align*} \tilde { r } _ { h , i } ( w ) = F _ h ( \varphi _ i ) - a _ h ( w _ H , \varphi _ i ) \ , , i \in I _ h \ , ; \end{align*}


\begin{align*} \Phi _ { \gamma , \delta } ( \Phi _ { \gamma , \delta } ( d , e , s - r ) , e , t - s ) = \Phi _ { \gamma , \delta } ( d , e , t - r ) . \end{align*}


\begin{align*} W _ 1 T = \zeta _ { A V , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \sigma _ 3 ) T + O \left ( T ^ { 2 - 2 \sigma _ 1 - 2 \sigma _ 3 } \right ) . \end{align*}


\begin{align*} q \cdot \mathcal { E } ( q , \lambda ( q , t ) ) = \frac { 1 - q } { 1 - q e ^ { \lambda ( q , t ) \cdot ( 1 - q ) } } - 1 \end{align*}


\begin{align*} X ^ T ( t ) : = \begin{cases} X ( t ) & [ 0 , T ] , \\ X ( T ) + ( t - T ) V ( T ) & [ T , \infty ) , \\ t V _ 0 & ( - \infty , 0 ] . \end{cases} \end{align*}


\begin{align*} U ^ { 1 } ( r , \theta ) = \left ( r ^ { \lambda _ { p } } a ^ { 1 } ( \theta ) , r ^ { \lambda _ { p } } b ^ { 1 } ( \theta ) , r ^ { \lambda _ { p } + 1 - m } c ^ { 1 } ( \theta ) \right ) , \end{align*}


\begin{align*} H ( v ) : = \beta ( v ) ( d \alpha _ 0 ) ( v , u ) , \end{align*}


\begin{align*} \frac 1 q = \frac { 1 ‐ \lambda } { p _ 1 } + \frac { \lambda } { p _ 0 } , \frac 1 2 = \frac { 1 ‐ \tilde \lambda } { p _ 1 } + \frac { \tilde \lambda } { p _ 0 } . \end{align*}


\begin{align*} \left \langle \tilde a _ i , x \right \rangle \leqslant b _ i , i = 1 , \ldots , m . \end{align*}


\begin{align*} \| g - h \| ^ 2 = | g ( \infty ) + \eta | ^ 2 + \int _ 0 ^ { \infty } | g ' ( x ) | ^ 2 e ^ { \gamma x } d x \end{align*}


\begin{align*} & \theta ( z , t ) \theta ( x , y ) - \theta ( y , t ) \theta ( x , z ) - \theta ( x , [ y , z , t ] ) + D ( y , z ) \theta ( x , t ) = 0 , \\ & \theta ( z , t ) D ( x , y ) - D ( x , y ) \theta ( z , t ) + \theta ( [ x , y , z ] , t ) + \theta ( z , [ x , y , t ] ) = 0 , \end{align*}


\begin{align*} d ^ 2 ( \rho ^ 0 , \rho ^ 1 ) : = \min _ { ( \rho , u ) \in K } \sum _ { i = 1 } ^ s \int _ 0 ^ 1 \int _ { \Omega } D _ i ^ { - 1 } | u _ i | ^ 2 \rho _ i d x d t . \end{align*}


\begin{align*} \Phi _ p : \ , D _ \rho \ , \longrightarrow \ , U ^ p _ \rho = \Phi _ p ( D _ \rho ) \subset \mathcal { O } ; \Phi _ p ( 0 ) = p \end{align*}


\begin{align*} \mathfrak { F } _ \alpha ( g ) = \{ \alpha \Z ^ { 2 d } \mid G ( g , \alpha \Z ^ { 2 d } ) \} \end{align*}


\begin{align*} u _ N ( t , x ) = \sum ^ N _ { i = 1 } \zeta _ i ( t ) e _ i ( x ) , \end{align*}


\begin{align*} \Theta _ 2 = \nabla c _ 0 ^ { i n } \otimes \nabla g + \nabla g \otimes \nabla c _ 0 ^ { i n } \| ( \Theta _ 3 , \Theta _ 4 ) \| _ { L ^ \infty ( 0 , T ; L ^ 2 ( \Omega ) ) } \le C \varepsilon ^ 2 , \end{align*}


\begin{align*} \partial ^ 2 _ { t _ i t _ j } ( \Phi \circ \varphi ) = \sum _ { l , k } ( \partial ^ 2 _ { x _ l x _ k } \Phi ) ( \partial _ { t _ j } \varphi ^ l ) ( \partial _ { t _ i } \varphi ^ k ) + \sum _ l ( \partial _ { x _ l } \Phi ) ( \partial ^ 2 _ { t _ i t _ j } \varphi ^ l ) . \end{align*}


\begin{align*} \mathbf { F } _ d = \left ( \R [ X _ 1 , \ldots , X _ n ] _ { \leq d } \right ) _ { n > 0 } . \end{align*}


\begin{align*} \kappa ( g ) = \sum _ { ( d , e ) \in D _ { 1 } ( P G ) } | g ( d ) \cap g ( e ) | = 5 \equiv 1 \pmod 2 . \end{align*}


\begin{align*} p ( x ) = \phi \bigl ( g _ { j ( 1 ) } ( x ) , \ldots , g _ { j ( d ) } ( x ) \bigr ) j ( 1 ) , \ldots , j ( d ) \in \{ 1 , \ldots , m \} , \deg ( p ) = \delta . \end{align*}


\begin{align*} d _ \rho ( ( x _ 0 , y _ 0 ) , ( x , y ) ) = \int _ 0 ^ L \gamma _ { \R } ( t ) ^ { - \beta } \ , \max \{ | \gamma _ Z ^ \prime | ( t ) , | \gamma _ { \R } ^ \prime | ( t ) \} \ , d t & \ge \int _ 0 ^ L \gamma _ { \R } ( t ) ^ { - \beta } | \gamma _ { \R } ^ \prime | ( t ) \ , d t \\ & \ge \frac { 1 } { \beta - 1 } \bigg \vert \frac { 1 } { y _ 0 ^ { \beta - 1 } } - \frac { 1 } { y ^ { \beta - 1 } } \bigg \vert . \end{align*}


\begin{align*} \prod _ { i \in I ^ * } c _ i = 1 - 1 ) \Longleftrightarrow \bigg ( | I _ \kappa ^ * | - \sum _ { i \in I _ \kappa ^ * } c _ i \bigg ) \equiv 0 2 ) \mod 4 . \end{align*}


\begin{align*} & M _ t = \exp ^ { \triangleleft } ( t h ) = 1 + \sum _ { \tau \in \mathcal T } \operatorname { c m } ( \tau ) P _ h ( \tau ) \frac { t ^ { | \tau | } } { | \tau | ! } = 1 + \sum _ { \tau \in \mathcal T } \frac { 1 } { \tau ! \sigma ( \tau ) } P _ h ( \tau ) t ^ { | \tau | } , \end{align*}


\begin{align*} \Lambda ^ { ( h : h ' ) } _ { n , 2 , 0 } ( \mathcal { W } _ { k } ( 0 ) \setminus \mathcal S _ k ) = \frac { c \left ( 1 + o ( 1 ) \right ) } { n ^ { h + h ' } \delta ^ 4 _ n ( k ) } \cdot | \mathcal { W } _ { k } ( 0 ) \setminus \mathcal S _ k | \cdot | \mathcal { I } _ h \times \mathcal { I } _ { h ' } | \cdot O ( n ^ { - 1 } ) = O ( n ^ { - 1 } ) \end{align*}


\begin{align*} B ^ { k } _ t + a B ^ { k - 1 } J _ x ^ { k } + b J ^ { k - 1 } B _ x ^ { k } + \mu \Lambda ^ \alpha B ^ { k } = & \ 0 , \\ B _ x ^ k = & \ \mathcal H J ^ k \end{align*}


\begin{align*} ( F _ 1 * F _ 2 ) ( \mathbf { h } _ 0 ) = \int _ { \mathbf { H } } F _ 1 ( \mathbf { h } ) F _ 2 ( \mathbf { h } ^ { - 1 } \mathbf { h } _ 0 ) \ , d \mathbf { h } . \end{align*}


\begin{align*} & \Gamma ^ { A i } _ 1 = \left \{ \zeta \in \mathbb { C } \vert \textnormal { a r g } \zeta = 2 \pi / 3 \right \} , \Gamma ^ { A i } _ 2 = \left \{ \zeta \in \mathbb { C } \vert \textnormal { a r g } \zeta = \pi \right \} , \\ & \Gamma ^ { A i } _ 3 = \left \{ \zeta \in \mathbb { C } \vert \textnormal { a r g } \zeta = - 2 \pi / 3 \right \} , \Gamma ^ { A i } _ 4 = \left \{ \zeta \in \mathbb { C } \vert \textnormal { a r g } \zeta = 0 \right \} , \end{align*}


\begin{align*} \begin{cases} \ i = ( x + 1 ) ( f _ 0 ( x , n ) - g _ 0 ( x , n ) ) \\ [ 8 p t ] \ j = h _ 0 ( x , n ) \end{cases} . \end{align*}


\begin{align*} ( \lambda - \overline { \lambda } ) g \varphi ^ { j } & = \lambda \overline { h } ^ { j } - \overline { \lambda } h ^ { j } , \\ ( \lambda - \overline { \lambda } ) \psi ^ { j } & = h ^ { j } - \overline { h } ^ { j } . \end{align*}


\begin{align*} | V _ g f ( x , \omega ) | = \mathcal { O } ( e ^ { - \frac { \pi } { 2 } ( a x ^ 2 + b \omega ^ 2 ) } ) , \end{align*}


\begin{align*} \theta _ T ( u , v ) x = [ x , T u , T v ] + T \Big ( \theta ( x , T v ) u - D ( x , T u ) v \Big ) , \ \forall x \in L , \ u , v \in V . \end{align*}


\begin{align*} \frac { \partial u } { \partial t } = \hat { L } u , \ , \hat { L } = \frac { \partial } { \partial x } + \frac { \partial ^ { 2 } } { \partial x ^ { 2 } } , \ ; u ( x , 0 ) = x \end{align*}


\begin{align*} \int _ { R _ k \le | \mathbf { x } | \le R _ k + 1 } \tilde { u } _ n ^ 2 \eta _ k ^ 2 \le C e ^ { - \sqrt { \tilde { \lambda } _ 0 \underline { m } } R _ k } R _ k = o ( 1 ) k \to + \infty \ ; , \end{align*}


\begin{align*} k _ { 0 } ( u , \omega ) : = \mathcal { J } \setminus \{ u \} \cup \{ \omega \} , k _ { 0 } ( u , \mathfrak { g } _ { c } ) = k _ { 0 } ( \mathfrak { g } _ { c } , u ) : = \mathcal { J } \setminus \mathfrak { g } _ { r } \cup \{ \alpha _ { 1 } , i _ { 3 } , u \} \end{align*}


\begin{align*} \lim _ { t \downarrow 0 } \bigg \| \alpha ( x ) - \frac { 1 } { t } \int _ 0 ^ t S _ { t - s } \alpha ( \xi ( s ; x ) ) d s \bigg \| = 0 . \end{align*}


\begin{align*} \ell = \frac { | m h \pm n k | } { d } . \end{align*}


\begin{align*} \dot M _ t ( x ) = M _ t ( x ) h ( x M _ t ( x ) ) = h ( x ) + ( ( M _ t - 1 ) \triangleleft h ) ( x ) . \end{align*}


\begin{align*} \int _ 0 ^ 1 A ( n _ { i j } , n _ { i j x } ) \varphi _ x d x + \int _ 0 ^ 1 ( n _ { i j } - b _ i ) \varphi d x = 0 , \quad \forall \varphi \in H _ 0 ^ 1 ( 0 , 1 ) , \end{align*}


\begin{align*} j ^ * [ \mathfrak { h } _ \omega ] & = 0 \in H H ^ { p + 1 } ( \mathfrak { A } _ 0 ) \\ j ^ * [ \mathfrak { c } _ \omega ] & = 0 \in H C _ \lambda ^ { p } ( \mathfrak { A } _ 0 ) . \end{align*}


\begin{align*} d Z _ { \lambda } = E _ { \lambda } + F _ { 0 } + F _ { 1 } + F _ { 2 } , \end{align*}


\begin{align*} f = \sum _ { k , l \in \Z ^ d } c _ { k , l } M _ { \beta l } T _ { \alpha k } g \end{align*}


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


\begin{align*} J ( Y _ m ^ { M , N } ) - J ( Y _ { m - 1 } ^ { M , N } ) = \big < A Y _ m ^ { M , N } , \overline { \partial } Y _ m ^ { M , N } \big > + \big < F ( Y _ m ^ { M , N } ) , \overline { \partial } Y _ m ^ { M , N } \big > + \big < Y _ m ^ { M , N } - Y _ { m - 1 } ^ { M , N } , \overline { \partial } Y _ m ^ { M , N } \big > . \end{align*}


\begin{align*} 1 - F _ n ( x s _ n ) = \exp \left ( - \frac { 1 } { 2 } x ^ 2 Q _ N ( x ) \right ) \left ( 1 - \Phi ( x ) + \vartheta \lambda _ N e ^ { - x ^ 2 / 2 } \right ) . \end{align*}


\begin{align*} S \ = \ \bigcup \limits _ { i = 1 } ^ { m _ + } \ L _ i ^ + , T \ = \ \bigcup \limits _ { i = 1 } ^ { m _ - } \ L _ i ^ - . \end{align*}


\begin{align*} \mathcal { Y } _ 1 = \int _ { \mathbb { R } ^ N } G _ + ( x - z ) \left ( \int _ { \mathbb { R } ^ N } G _ - ( z - y ) F ( y ) d y \right ) d z . \end{align*}


\begin{align*} Z f ( x , \omega ) = e ^ { 2 \pi i x \cdot \omega } Z \widehat { f } ( \omega , - x ) , \end{align*}


\begin{gather*} \lim _ { k \to \infty } W _ { 1 } \big ( \mu , \mathrm { l a w } ( X _ { \eta k } ) \big ) = \lim _ { k \to \infty } W _ { 1 } \big ( \mathrm { l a w } ( \tilde { Y } _ { k } ) , \tilde { \mu } _ { \eta } \big ) = 0 . \end{gather*}


\begin{align*} \lim \limits _ { n \rightarrow \infty } H _ n ( s _ 1 , \ldots , s _ r ) = \zeta ( s _ 1 , \ldots , s _ r ) \quad \lim \limits _ { n \rightarrow \infty } \overline { H } _ n ( s _ 1 , \ldots , s _ r ) = t ( s _ 1 , \ldots , s _ r ) , \end{align*}


\begin{align*} \int _ 0 ^ 1 \overline { \xi ( x ) } \xi ^ \prime ( x ) d x = \frac { 1 } { 2 } \int _ 0 ^ 1 \frac { d } { d x } | \xi ( x ) | ^ 2 d x + i \int _ 0 ^ 1 \Im ( \overline { \xi ( x ) } \xi ^ \prime ( x ) ) d x = i \int _ 0 ^ 1 \Im ( \overline { \xi ( x ) } \xi ^ \prime ( x ) ) d x , \end{align*}


\begin{align*} J \cup K \cup L & = ( \mathbb { N } \setminus \{ \bar { a } _ n \} ) \cup \{ \bar { a } _ n \} = \mathbb { N } , \end{align*}


\begin{align*} \mathbf L = \begin{bmatrix} \mathbf L _ { 1 1 } & \mathbf L _ { 1 2 } \\ \mathbf L _ { 2 1 } & \mathbf L _ { 2 2 } \end{bmatrix} , \end{align*}


\begin{align*} 0 & = \norm { J ^ A _ \gamma x - J ^ A _ { \gamma ' } x + ( ( x - J ^ A _ \gamma x ) - ( x - J ^ A _ { \gamma ' } x ) ) } \\ & = \norm { J ^ A _ \gamma x - J ^ A _ { \gamma ' } x + \vert \gamma \vert ( \gamma ^ { - 1 } ( x - J ^ A _ \gamma x ) - \gamma '^ { - 1 } ( x - J ^ A _ { \gamma ' } x ) ) } \\ & \geq \norm { J ^ A _ \gamma x - J ^ A _ { \gamma ' } x } . \end{align*}


\begin{align*} \left \{ \begin{array} [ c ] { l l } u _ { t } = - \gamma \boldsymbol { D } _ { x } ^ { \alpha } u + F ( u ) + \boldsymbol { D } _ { x } ^ { \alpha _ { 1 } } u ^ { 3 } , & x \in \mathbb { Q } _ { p } ^ { n } , \ t \in \left [ 0 , T \right ] ; \\ & \\ u ( 0 ) = f _ { 0 } \in \mathcal { H } _ { \infty } , & \end{array} \right . \end{align*}


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


\begin{align*} \Big | \frac { 1 } { 2 \varepsilon n ^ 2 } \sum _ { x } \Phi _ n ( s , \tfrac { x } { n } ) \sum _ { r = 1 } ^ { \varepsilon n } \int _ { \Omega _ 1 ( x ) } \overleftarrow { \eta } ^ { \ell } ( x ) [ \eta ( x + 1 ) - \eta ( x + 1 + r ) ] [ f ( \eta _ { 2 , x , r } ) - f ( \eta _ { 3 , x , r } ) ] d \nu _ { b } \Big | , \end{align*}


\begin{align*} 0 \ge \Phi ( f ) = B ( f , f ) = \frac { 1 } { 2 } \int \left ( L _ + f _ 1 \right ) f _ 1 + \frac { 1 } { 2 } \int \left ( L _ - f _ 2 \right ) f _ 2 . \end{align*}


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


\begin{align*} \int _ 0 ^ { + \infty } ( \mathcal { S } _ L ( \tau ) L f , \mathcal { S } _ L ( \tau ) f ) _ { L ^ 2 _ v } d \tau = - \int _ 0 ^ { + \infty } \frac d { d \tau } \Vert \mathcal { S } _ L ( \tau ) f \Vert _ { L ^ 2 } ^ 2 d \tau = \Vert f \Vert _ { L ^ 2 } ^ 2 - \lim _ { \tau \to \infty } \Vert \mathcal { S } _ L ( \tau ) f \Vert _ { L ^ 2 } ^ 2 = \Vert f \Vert _ { L ^ 2 } ^ 2 , \end{align*}


\begin{align*} \{ ( \beta , j ) < ( \gamma , k ) \} : = \{ \beta + j < \gamma + k \ ; \ ; \ ; \ ; \beta + j = \gamma + k \ ; \ ; \ ; \ ; \beta < \gamma \} , \end{align*}


\begin{align*} \frac { 1 } { t } d _ K ( a + t \beta a ) = 0 , t > 0 , \end{align*}


\begin{align*} \lim _ { y \to 0 ^ + } y ^ a \partial _ y u = f . \end{align*}


\begin{align*} \begin{aligned} & s _ { i j } = a _ { i j } j \leq \min ( J _ i ) , \\ & 0 \leq \frac { s _ { i \max ( J _ i ) } - s _ { i \max ( J _ i ) - 1 } } { a _ { i \max ( J _ i ) } - a _ { i \max ( J _ i ) - 1 } } \leq \cdots \frac { s _ { i \min ( J _ i ) + 1 } - s _ { i \min ( J _ i ) } } { a _ { i \min ( J _ i ) + 1 } - a _ { i \min ( J _ i ) } } \leq 1 , \\ & s _ { i j } = s _ { i n } j \geq \max ( J _ i ) , \end{aligned} \end{align*}


\begin{align*} \left \{ \begin{array} { l l l l } H _ \mu u \ ! + \ ! \varPhi _ q d _ q B _ q ( w , u ) & = & \varPhi _ q d _ q f & ( x , t ) \in { \mathbb R } ^ n \times ( 0 , T ) , \\ u & = & u _ 0 , & ( x , t ) \in \mathbb { R } ^ n \times \{ 0 \} . \end{array} \right . \end{align*}


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


\begin{align*} \mathcal { L } ( f ) = \frac { 1 } { \alpha } \left ( \mathcal { L } ^ \alpha + ( \mathcal { L } ^ { \alpha } ) ^ * \right ) ( f ) , \end{align*}


\begin{align*} & ( 1 - t ) ^ 2 \frac { \partial w } { \partial t } + \big ( ( 1 - t ) ( \alpha - 1 ) - x \big ) w = 0 , \\ & ( 1 - t ) \frac { \partial w } { \partial x } - t w = 0 . \end{align*}


\begin{align*} \Delta ( a ) ( b \chi \otimes c \theta ) = ( \Delta ( a ) ( b \otimes c ) ) ( \chi \otimes \theta ) , ( \chi b \otimes \theta c ) \Delta ( a ) = ( \chi \otimes \theta ) ( ( b \otimes c ) \Delta ( a ) ) . \end{align*}


\begin{align*} Y _ m ^ j = \sup _ { t \in [ 0 , T ] } | \dot { B } _ m ^ j ( t ) | \leq \frac { 1 } { \delta _ m } \sum _ { k = 1 } ^ m | ( B ^ j ( k \delta _ m ) - B ^ j ( ( k - 1 ) \delta _ m ) ) | = \sum _ { k = 1 } ^ m | \eta _ m ^ { j , k } | , \end{align*}


\begin{align*} h ^ { - \frac { 1 } { 2 } + \alpha } k ^ { - \frac { 1 } { 2 } - \alpha } \mathcal { G } ( 2 , \alpha , \beta ) = \mathcal { K } ( 0 , - \alpha - \beta , 2 ) . \end{align*}


\begin{align*} h _ A ( x ) = h _ A ( y ) \Longleftrightarrow x = y \ ; \vee \ ; x = y ^ { - 1 } . \end{align*}


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


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


\begin{align*} B ( b ) = \left | Z _ { b ^ { - 1 } } \phi ( 0 , 0 ) \right | ^ 2 = \frac { \pi } { 2 } \left | b \ , \vartheta ' _ 1 ( 0 , i b ) \frac { \vartheta _ 3 ( 0 , i b ) } { \vartheta _ 4 ( 0 , i b ) \vartheta _ 4 ( 0 , i b ^ { - 1 } ) } \right | ^ 2 . \end{align*}


\begin{align*} V = \{ ( z , w ) \in \mathbb D ^ 2 \ , : \ , \det ( \psi ( z ) - w I ) = 0 \} , \end{align*}


\begin{align*} \rho ^ { * } _ { ( \mu ) } \psi : = \omega ^ { \mu } \rho ^ { * } \psi \end{align*}


\begin{align*} \lim _ { n \to \infty } \langle ( P X - X P ) g _ n , g _ n \rangle = \frac { 1 } { 2 \pi i } \lim _ { n \to \infty } \norm { g _ n } _ 2 ^ 2 = \frac { 1 } { 2 \pi i } \norm { g } _ 2 ^ 2 . \end{align*}


\begin{align*} & \begin{aligned} z _ i \in \Delta _ i , \ & \delta _ { i t } \in \{ 0 , 1 \} , \\ & z _ { i \tau ( i , t ) } \geq \delta _ { i t } \geq z _ { i \tau ( i , t ) + 1 } i = 1 , \ldots , d , \ t = 1 , \ldots , l _ i - 1 , \end{aligned} \\ & f _ i = a _ { i 0 } z _ { i 0 } + \sum _ { j = 1 } ^ { n } ( a _ { i j } - a _ { i j - 1 } ) z _ { i j } = : F _ i ( z _ i ) i = 1 , \ldots , d . \end{align*}


\begin{align*} \iint _ { \R ^ { 2 d } } F ( x , \omega ) \langle \pi ( x , \omega ) f , g \rangle e ^ { - 2 \pi i ( x \cdot \eta - \xi \cdot \omega ) } \ , d ( x , \omega ) = 0 . \end{align*}


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


\begin{align*} \phi * \phi = s ^ 2 \chi _ A * \chi _ A + t ^ 2 \chi _ B * \chi _ B + \chi _ C * \chi _ C + 2 s t \chi _ A * \chi _ B + 2 t \chi _ B * \chi _ C + 2 s \chi _ C * \chi _ A . \end{align*}


\begin{align*} G _ { z , \alpha , \nu } ( x ) = x + \sum _ { i = 1 } ^ k \alpha _ i \cdot ( x - z _ i ) \cdot | x - z _ i | \cdot \phi \Bigl ( \frac { x - z _ i } { \nu } \Bigr ) . \end{align*}


\begin{align*} L \chi = L \phi , \end{align*}


\begin{align*} \rho = \mu / \lambda . \end{align*}


\begin{align*} q | _ { \mathcal { R } _ { \xi , I I I } } ( x , t ) = C _ R + \mathcal { O } ( e ^ { - \gamma t } ) . \end{align*}


\begin{align*} I ( \alpha , \beta , Z ) - I ( \alpha , \beta , Z ' ) = < c _ { 1 } ( \xi ) + 2 \mathrm { P D } ( \Gamma ) , Z - Z ' > . \end{align*}


\begin{align*} ( D ^ m B ) _ t + [ D ^ m , B ] J _ x + B D ^ m J _ x - [ D ^ m , J ] B _ x - J D ^ m B _ x + \mu \Lambda ^ \alpha D ^ m B = 0 \end{align*}


\begin{align*} \mu \left ( F _ { \mu } ^ { [ \alpha _ 2 , \gamma _ 2 ] , \varepsilon } \right ) = 1 . \end{align*}


\begin{align*} f ( x , n ) \ = \ \bigg [ \frac { x } { n ! } \bigg ] . \end{align*}


\begin{align*} | \nabla u ( x , y ) | _ { \rho , ( x , y ) } = \rho ( x , y ) ^ { - 1 } | \nabla u ( x , y ) | _ { ( x , y ) } . \end{align*}


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


\begin{align*} r C _ { \min } { \beta } ^ { - 1 } & \leq \prod _ { i = 1 } ^ { q } C _ { \sigma _ i } < r { \beta } ^ { - 1 } \end{align*}


\begin{align*} \iota ( \mathbf { h } ) \odot \iota ( \mathbf { h } ' ) & = \left ( x , \omega , \tau + \tfrac { 1 } { 2 } x \cdot \omega \right ) \odot \left ( x ' , \omega ' , \tau ' + \tfrac { 1 } { 2 } x ' \cdot \omega ' \right ) \\ & = \left ( x + x ' , \omega + \omega ' , \tau + \tau ' + \tfrac { 1 } { 2 } ( x \cdot \omega + x ' \cdot \omega ' ) + x ' \cdot \omega \right ) \\ & = \iota ( \mathbf { h } \bullet \mathbf { h } ' ) . \end{align*}


\begin{align*} \langle X g , P g \rangle = \sum _ { k , l \in \Z } \langle X g , M _ l T _ k g \rangle \langle M _ l T _ k g , P g \rangle . \end{align*}


\begin{align*} U ^ p ( t ) = \sum _ { k \geq N } a _ k e ^ { \tilde { \lambda } _ k ( T - t ) } , \ \ U ^ h ( t ) = \sum _ { \mod { k } \geq N } b _ k e ^ { \tilde { \gamma } _ k ( T - t ) } , \ \ t \geq 0 , \end{align*}


\begin{align*} \norm { Y ^ b _ { j } g _ \ell - \prod \limits _ { s = 1 } ^ { j } \left ( 1 - \delta ^ { i _ s + \ell - 2 s } _ { i _ s - 2 s } \right ) Y _ 0 ^ b g _ \ell } \leq \norm { g _ \ell } \sum \limits _ { s = 1 } ^ j \gamma _ { i _ s - 2 s } ^ { i _ s + \ell - 2 s } \prod \limits _ { t = 1 } ^ { s - 1 } \left ( 1 - \delta ^ { i _ t + \ell - 2 t } _ { i _ t - 2 t } \right ) , \end{align*}


\begin{align*} \delta = \Im \omega _ 2 ( i \varepsilon ) = \frac { | \lambda | ^ 2 } { s ( | \lambda | , \varepsilon ) - \varepsilon } \end{align*}


\begin{align*} H _ \Lambda ( \Phi ) = \sum _ { X \subset \Lambda } \Phi ( X ) . \end{align*}


\begin{align*} \tilde { W } _ { s , t } ( x , v ) & = W _ { s , t } ( x - \check { \mathcal { T } } _ { t , x , v } v , v ) , \\ \tilde { Y } _ { s , t } ( x , v ) & = Y _ { s , t } ( x - \check { \mathcal { T } } _ { t , x , v } v , v ) . \end{align*}


\begin{align*} \frac { p } { 2 } - \frac { 1 } { s ' } + \frac { N p } { 2 } - N = \frac { 1 } { s } + \left ( \frac { p } { 2 } - 1 \right ) \left ( N + 1 \right ) \in \Bigg [ \left ( \frac { p } { 2 } - 1 \right ) \left ( N + 1 \right ) , \frac { N - \gamma } { N } + \left ( \frac { p } { 2 } - 1 \right ) \left ( N + 1 \right ) \Bigg ) , \end{align*}


\begin{align*} \langle \phi , 1 \rangle = s ( \ell - 1 ) + t ( m - 1 ) + ( \ell - 1 ) ( m - 1 ) . \end{align*}


\begin{align*} \begin{aligned} s _ { i k } & = G _ { i k } ( z , \delta ) : = \sum _ { t = 1 } ^ { l _ i } \Bigl ( m ^ t _ { i 0 } ( \delta _ { i t - 1 } - \delta _ { i t } ) + \sum _ { j = 1 } ^ k ( m ^ t _ { i j } - m ^ t _ { i j - 1 } ) ( z _ { i j } - \delta _ { i t } ) \Bigr ) , \end{aligned} \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { n - 1 } ( j + 2 ) ( j + 1 ) x ^ m \alpha _ { j + 2 } ( x ) y ^ j + \sum _ { j = n } ^ { \infty } \left [ ( j + 2 ) ( j + 1 ) x ^ m \alpha _ { j + 2 } ( x ) + \varepsilon \alpha '' _ { j - n } ( x ) \right ] y ^ j = 0 . \end{align*}


\begin{align*} \sum _ { v \in V _ 1 } b _ v = \sum _ { v \in V _ 2 } b _ v = \cdots = \sum _ { v \in V _ d } b _ v . \end{align*}


\begin{align*} f ( w ) = j ( h _ { \infty } , z _ 0 ) ^ { k } \mathbf { f } ( h _ { \infty } ) , j ( \tilde { \tau } _ m , z \times w ) = \delta ( w , z ) , \end{align*}


\begin{align*} 2 \sqrt { e ^ r - 1 } = | \log p | ( 1 + o ( 1 ) ) \end{align*}


\begin{align*} N _ { d _ 0 } ( f , H ) = - f + \frac { 1 2 \mu ( d _ 0 ) } { \prod _ { q \mid d _ 0 } ( q ^ 2 - 1 ) } \sum _ { \delta \mid d _ 0 } \delta \mu ( \delta ) \sum _ { h \in H _ { d _ 0 } } s ( h , \delta f ) . \end{align*}


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


\begin{align*} \begin{cases} - \Delta \tilde { h } _ { \varepsilon } + \tilde { \lambda } _ 0 \underline { m } \ , \tilde { h } _ { \varepsilon } = 0 & \tilde { \Omega } _ { \varepsilon } \ ; , \\ \tilde { h } _ { \varepsilon } = w & \partial \tilde { \Omega } _ { \varepsilon } \ ; . \end{cases} \end{align*}


\begin{align*} g _ { t } = g + d \zeta _ { t } \circ ( J \otimes \mathrm { I d } ) + \zeta _ { t } \otimes \theta ^ { c } + \theta ^ { c } \otimes \zeta _ { t } + \zeta _ { t } \otimes \zeta _ { t } . \end{align*}


\begin{align*} B = \sum _ { i = 2 } ^ 4 e _ 1 e _ i \nabla ^ \partial _ { e _ i } , A = \sum _ { i = 2 } ^ 4 e _ 1 e _ i \left ( \nabla _ { e _ i } - \nabla ^ \partial _ { e _ i } \right ) . \end{align*}


\begin{align*} & \partial _ t \rho _ i = \nabla \cdot \left [ D _ i ( x ) \left ( \nabla \rho _ i + z _ i \rho _ i \nabla \phi \right ) \right ] , x \in \Omega \subset \R ^ d , t > 0 , \\ & - \nabla \cdot ( \epsilon ( x ) \nabla \phi ) = f ( x ) + \sum _ { i = 1 } ^ { s } z _ i \rho _ i , \end{align*}


\begin{align*} \lim _ { s \rightarrow 0 ^ + } k ( s , 0 ) = \frac { 1 } { \int _ \mathbb { R } \frac { 1 } { u ^ 2 } d \mu ( u ) } \qquad \lim _ { s \rightarrow \infty } k ( s , 0 ) = \int _ \mathbb { R } u ^ 2 d \mu ( u ) . \end{align*}


\begin{align*} I ( \alpha _ { l } ^ { \Gamma } ) - I ( \alpha _ { k } ^ { \Gamma } ) = 2 ( l - k ) . \end{align*}


\begin{align*} P _ { \rm o u t } ^ { D _ 2 } = 1 - e ^ { - \frac { 1 } { \beta _ { s s } \beta _ { s 2 } N } \left ( \frac { \sigma ^ 2 \gamma _ { t h 2 } } { P _ t ( \alpha _ 2 - \alpha _ 1 \gamma _ { t h 2 } ) } \right ) } . \end{align*}


\begin{align*} B _ 1 ( U , V ) = \Big ( \frac { 1 } { 2 \pi } + \frac { b _ 1 \log U + b _ 2 } { U \log U \log ( U / 2 \pi ) } \Big ) \big ( \log ( V / U ) \ , \log ( \sqrt { V U } / ( 2 \pi ) ) \big ) + \frac { 2 R ( U ) } { U } . \end{align*}


\begin{align*} \Xi _ { k + 1 } = \mathcal { A } \Xi _ k + \mathcal { B } \mathcal { G } \hat { \Psi } _ k ( \mathbf { u } , \Xi ) , \mathbf { y } _ k = \mathcal { C } \Xi _ k . \end{align*}


\begin{align*} \big ( ( \mathcal { F } ^ \# ) ^ { - 1 } \circ \mathcal { F } ^ \# g \big ) ( x ) & = \int _ { \R _ k } \mathcal { K } ^ \# ( x , k ) \Big ( \int _ { \R _ y } \overline { \mathcal { K } ^ \# ( y , k ) } g ( y ) \ , d y \Big ) d k \\ & = \int _ { \R _ k } \mathcal { K } ( x , k ) \Big ( \int _ { \R _ y } \overline { \mathcal { K } ( y , k ) } g ( y ) \ , d y \Big ) d k = g ( x ) \end{align*}


\begin{align*} \tau \leqslant \inf _ { B _ { 1 / 2 } } \tilde u = \inf _ { B _ { \rho / 2 } ( e _ 1 ) } u . \end{align*}


\begin{align*} [ \varphi ( a ) , a ^ 2 \cdot v _ 0 ] = [ \varphi ( a ^ 2 \cdot v _ 0 ) , a ] . \end{align*}


\begin{align*} \widehat { K } _ \mathrm { a - c p t - d } ^ \mathbb { R } & = \widehat { K ' } _ \mathrm { a - c p t - d } ^ \mathbb { R } + \widehat { K '' } _ \mathrm { a - c p t - d } ^ \mathbb { R } , \end{align*}


\begin{align*} \widehat { S } _ { W , m } = \widehat { V } _ P \widehat { D } _ { L , m } \widehat { J } \widehat { V } _ Q . \end{align*}


\begin{align*} \int x w ( x , t ) d x = 0 , \end{align*}


\begin{align*} ( - \Delta ) ^ s u + c u = 0 \qquad B _ { \rho } ( e _ 1 ) . \end{align*}


\begin{align*} | z \rangle = e ^ { - \frac { | z | ^ { 2 } } { 2 } } \sum _ { n = 0 } ^ { \infty } \frac { z ^ { n } } { \sqrt { n ! } } | n \rangle . \end{align*}


\begin{align*} \| \left \langle \nabla \right \rangle R ( h ) \| _ { S ' ( ( t , + \infty ) , L ^ 2 ) } \lesssim \sum _ { n = 0 } ^ \infty \| \left \langle \nabla \right \rangle R ( h ) \| _ { S ' ( ( t + n , t + n + 1 ) , L ^ 2 ) } \lesssim \begin{cases} e ^ { - ( p - 1 ) c t } , & p \in ( 2 , 3 ) , \\ e ^ { - 2 c t } , & p = 2 p \ge 3 . \end{cases} \end{align*}


\begin{align*} f = \mathop { \sum \sum } _ { k , l \in \Z ^ d } \langle f , M _ { \beta l } T _ { \alpha k } \ , g \rangle M _ { \beta l } T _ { \alpha k } \ , \widetilde { g } , \end{align*}


\begin{align*} b ( S ) = \sum _ { i \geq 0 } b _ i ( S ) . \end{align*}


\begin{align*} a ( \widetilde E _ 2 ^ { ( 3 ) } , T ) = a ( \mathrm { g e n u s } \ \Theta ^ { ( 3 ) } ( S ^ { ( p ) } ) , T ) . \end{align*}


\begin{align*} L ( s , \operatorname { s y m } ^ 2 f ) \ll & \left ( 1 + | s | \right ) ^ { ( 3 ( 1 - \Re ( s ) ) ) / 2 + \varepsilon } , 0 \leq \Re ( s ) \leq 1 . \end{align*}


\begin{align*} \bullet \ \ & \mbox { $ { \frak p } $ i s n o t c o m p l e t e i n t e r s e c t i o n , a n d } \\ \bullet \ \ & \mbox { $ z ^ u - x ^ { s _ 3 } y ^ { t _ 3 } $ i s t h e n e g a t i v e c u r v e , i . e . , $ u c < \sqrt { a b c } $ . } \end{align*}


\begin{align*} E ( x , t , k ) : = \left \{ \begin{aligned} & m ^ { ( 3 ) } ( x , t , k ) \left ( m _ { \pm \eta } \left ( x , t , k \right ) \right ) ^ { - 1 } , & k \in U _ { \delta } ( \pm \eta ) , \\ & m ^ { ( 3 ) } ( x , t , k ) \left ( \Delta _ \eta \left ( x , t , k \right ) \right ) ^ { - 1 } , & e l s e w h e r e . \end{aligned} \right . \end{align*}


\begin{align*} \partial _ { t } \eta _ k & = \partial _ z \phi _ k , \partial _ { t } \phi _ k = - g \eta _ k , \Gamma ^ { F S } \times \mathcal { T } , \end{align*}


\begin{align*} [ a + t a \cdot v _ 0 , b + t b \cdot v _ 0 ] = 0 \end{align*}


\begin{align*} Y ^ b _ j g _ \ell = \left ( 1 - \delta ^ { i _ 1 + \ell - 2 } _ { i _ 1 - 2 } \right ) Y ^ { b } _ { j - 1 } g _ { \ell } + \Gamma g _ \ell , \end{align*}