\begin{align*} \textbf { M } _ { x } ( u , v ) = \sum m _ { k } < \pmb { u } _ { k } \mid \pmb { v } _ { k } > \\ \textbf { M } _ { x } ( u , v ) = \textbf { M } _ { g x } ( g u , g v ) \end{align*}


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


\begin{align*} N ' = y _ n C ( z _ n ) + y _ n z _ n C ' ( z _ n ) \delta _ n + c _ m S _ n + o ( C ' ( z _ n ) \delta _ n ) . \end{align*}


\begin{align*} \lim _ { s \to 1 } \left ( \zeta _ Q ( s ) - \frac { \pi } { s - 1 } \right ) = 2 \pi ( \gamma _ o - \log 2 - \log ( \sqrt y | \eta ( z ) | ^ 2 ) ) , \end{align*}


\begin{align*} ( H , T ) = \left ( \begin{bmatrix} 1 & c & 0 \\ \eta & ( 1 + d ) & 1 \\ 0 & \eta & ( 1 + 2 d ) c ^ { - 1 } \end{bmatrix} , \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & c ^ { - 1 } \end{bmatrix} \right ) \end{align*}


\begin{align*} X ( \omega ) = \lim _ { m \rightarrow \infty } X _ m ( \omega ) \forall \omega \in \Omega \end{align*}


\begin{align*} \tau _ N \left ( Q ( a _ 1 ^ N , \dots , a _ k ^ N ) \right ) = \tau \big ( Q ( a _ 1 , \dots , a _ k ) \big ) + \mathcal { O } \bigg ( \frac { N ^ \varepsilon } { N } \max \limits _ { i \neq j } | y _ i ^ N - y _ j ^ N | ^ 4 + \frac { 1 } { \min \limits _ { i \neq j } | y _ i ^ N - y _ j ^ N | ^ { \delta _ P } } \bigg ) . \end{align*}


\begin{align*} \P \left \{ H \in C ^ { \alpha , \beta } _ { \textit { l o c } } ( \R _ + \times \R ^ d ) \right \} = 1 \end{align*}


\begin{align*} x _ { j j } = 0 , x _ { i i } = 1 , \end{align*}


\begin{align*} \overline { D } _ t \mu _ t \ ; = \ ; \limsup _ { s \to t } \frac { \mu _ s \ , - \ , \mu _ t } { s - t } , \end{align*}


\begin{align*} ( ( g f ) ( \pi ) ^ I ) _ { i j } = \sum _ y f ( \pi ) _ { x _ i , y _ j } = \sum _ y ( \pi _ i ^ G ) _ { x y } \pi ^ I _ { i j } = \pi ^ I _ { i j } . \end{align*}


\begin{align*} \left ( \frac { d m _ 0 } { d \zeta } + \frac { i \zeta } { 2 } \sigma _ 3 m _ 0 \right ) m _ 0 ^ { - 1 } = \left [ \frac { d m ^ { p c } _ { - k _ 0 } } { d \zeta } + m ^ { p c } _ { - k _ 0 } \frac { i \nu } { \zeta } \sigma _ 3 \right ] \left ( m ^ { p c } _ { - k _ 0 } \right ) ^ { - 1 } + \frac { i \zeta } { 2 } \left [ \sigma _ 3 , m ^ { p c } _ { - k _ 0 } \right ] \left ( m ^ { p c } _ { - k _ 0 } \right ) ^ { - 1 } , \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u & = h ( x , u , \nabla u ) \ ; \ ; & & \mbox { i n } \ ; \ ; \R ^ N , \\ u & > 0 & & \mbox { i n } \ ; \ ; \R ^ N , \\ u & ( x ) \to 0 & & \mbox { a s } \ ; | x | \to \infty . \end{alignedat} \right . \end{align*}


\begin{align*} e ^ { \pi i \theta _ 1 s } F ( z + \pi i \theta _ 1 , s ) = e ^ { \pi \theta _ 1 i s } \frac { e ^ { - ( z + \pi i \theta _ 1 ) s } } { 1 - e ^ { - z - \pi i \theta _ 1 } } = \frac { e ^ { - z s } } { 1 + ( - 1 ) ^ { \theta _ 1 } e ^ { - z } } . \end{align*}


\begin{align*} \zeta ^ \epsilon ( t ) = \frac { 1 } { \epsilon } \int _ { 0 } ^ { t } m ^ \epsilon ( s ) d s , \end{align*}


\begin{align*} \varphi ( t ) : = \kappa _ n C _ \Theta ^ 2 \Theta ^ { - 1 } ( \Theta ( C _ \Theta ) \vee C ( n , R , \alpha ) ^ { \frac { \alpha } { \alpha - 1 } } ) \ , \Theta ^ { - 1 } \left ( \Theta ( C _ \Theta ) \vee \int _ { B ( o , 2 R ) } \Theta \left ( \| D b ( t , z ) \| \right ) \dd z \right ) \ , , \end{align*}


\begin{align*} \Phi = ( x _ 1 \to x _ 2 ) \land y \leq z \ , \& \ , ( x _ 2 \to x _ 1 ) \land y \leq z \Longrightarrow y \leq z \end{align*}


\begin{align*} i _ \omega \ = \ \lim _ { M \rightarrow \infty } \frac { \ln ( \varphi _ \omega ( M ) ) } { \ln M } \mbox { a n d } I _ \omega = \lim _ { M \rightarrow \infty } \frac { \ln ( \Phi _ \omega ( M ) ) } { \ln M } . \end{align*}


\begin{align*} & \tilde { z } _ { R ^ \prime } ( t ) - \tilde { z } _ R ( t ) = \\ & \int _ 0 ^ t S ( t - s ) ( i \nu ( \tilde { z } _ { R ^ \prime } ( s ) - \tilde { z } _ R ( s ) ) + \epsilon ( \gamma ( \tilde { z } _ { R ^ \prime } ( s ) - \tilde { z } _ R ( s ) ) - \mu ( \overline { \tilde { z } _ { R ^ \prime } } ( s ) - \overline { \tilde { z } _ R } ( s ) ) ) ) \d s + I ( t ) , \end{align*}


\begin{align*} G ( k ) : = \left \| \frac { ( h _ { \alpha } - e _ { \alpha } ) x u _ { \alpha } } { ( h _ { \alpha } - e _ { \alpha } + | k | ) ^ { 1 / 2 } } \right \| _ { L ^ 2 } ^ 2 , \end{align*}


\begin{align*} \sum _ { j , k = 0 } ^ n g _ { j k } ( x ) \frac { d x _ j } { d t } \frac { d x _ k } { d t } > 0 . \end{align*}


\begin{align*} E ^ 2 _ { p q } = H _ p ( G \ltimes A , K _ q ( C _ 0 ( B ) ) ) \Rightarrow K _ { p + q } ( G \ltimes C _ 0 ( B ) ) . \end{align*}


\begin{align*} x \to y : = \max \{ u \colon u \wedge x = x \wedge y \} , \end{align*}


\begin{align*} \frac { | f ( z ) | ^ p } { \int _ { \mathbb { D } ^ \ast } | f | ^ p } = \frac { | g ( z ) | ^ p } { \int _ { \mathbb { D } ^ \ast } | g ( w ) | ^ p / | w | ^ { p k _ p } } \frac { 1 } { | z | ^ { p k _ p } } . \end{align*}


\begin{gather*} r _ { i , j , k } = \left [ E _ { i , j } , E _ { i , k } \right ] , r _ { i , j , k } ' = \left [ E _ { i , j } , E _ { j , k } \right ] E _ { i , k } ^ { - 1 } \intertext { a r e t h e S t e i n b e r g r e l a t i o n s ( $ i , j $ a n d $ k $ a r e d i s t i n c t ) , a n d } r = \left ( E _ { 1 , 2 } E _ { 2 , 1 } ^ { - 1 } E _ { 1 , 2 } \right ) ^ 4 . \end{gather*}


\begin{align*} W ^ { \prime } \left \{ f _ { \lambda } , f _ { \nu } \right \} = \left ( \lambda ^ { 2 } - \nu ^ { 2 } \right ) f _ { \lambda } f _ { \nu } \lambda , \nu . \end{align*}


\begin{align*} ( h x ) ( t ) = x _ 0 + \int _ { a } ^ { t } f ^ 0 ( s , x ( s ) ) d s + \sum \limits _ { j = 1 } ^ m \int _ { a } ^ { t } f ^ j ( s , x ( s ) ) d W _ j ( s ) \end{align*}


\begin{align*} I _ \omega = \inf \{ \beta < 1 : \omega \in U R P ( \beta ) \} . \end{align*}


\begin{align*} \omega ( k ) = | k | . \end{align*}


\begin{align*} ( f _ 1 \circ g _ 1 ) \otimes ( f _ 2 \circ g _ 2 ) = ( f _ 1 \otimes f _ 2 ) \circ ( g _ 1 \otimes g _ 2 ) \end{align*}


\begin{align*} r = \frac { 4 - 4 { h ^ \prime } ^ 2 - 2 h h ^ { \prime \prime } } { h ^ 2 } E _ 1 ^ \flat \wedge E _ 2 ^ \flat - \left ( \frac { 5 h ^ \prime h ^ { \prime \prime } + h h ^ { \prime \prime \prime } } { h h ^ \prime } + \frac { 1 } { 8 } { \textbf { H } ^ \prime } ^ 2 \right ) E _ 3 ^ \flat \wedge E _ 4 ^ \flat + \frac { 1 } { 8 } { \textbf { H } ^ \prime } ^ 2 d \theta \wedge d \varphi . \end{align*}


\begin{align*} X _ { i , j } ( x , x _ { 1 } ) \left | J \right \rangle \simeq \sum _ { n = 0 } ^ { | J | } \sum _ { \substack { J ' \subseteq J \\ | { J ' } | = n } } \mathcal { G } ^ { n } ( i , J , { J ' } ) W _ { n + i + 1 , n + j + 1 } ( x ) \left | J \setminus { J ' } \right \rangle \ , , \end{align*}


\begin{align*} \kappa ( n ) = ( n + 1 ) \big [ ( n + 1 ) ! \big ] ^ { n } - n . \end{align*}


\begin{align*} \Omega _ j \equiv \Omega _ j ( \zeta ) = | j | _ { \bar g } \sqrt { | j | _ { \bar g } ^ 2 + \zeta } , j \in \Z ^ d \setminus \{ 0 \} \ , . \end{align*}


\begin{align*} { F _ N } = { F _ N } U _ N ' - ( U _ N ' ) ^ 2 \end{align*}


\begin{align*} T _ t f = \Psi ( i _ { \mathsf { b } } ( t ) ) \mbox { f o r a l l } t \in G \ , . \end{align*}


\begin{align*} n \mapsto \begin{cases} + 1 & \mbox { i f $ n \equiv 1 \bmod 4 $ } \\ - 1 & \mbox { i f $ n \equiv 3 \bmod 4 $ } \\ 0 & \mbox { i f $ n $ i s e v e n . } \end{cases} \end{align*}


\begin{align*} \Psi ( x , t ) : = t ^ { r ( x ) } + \mu ( x ) ^ { \frac { s ( x ) } { q ( x ) } } t ^ { s ( x ) } \quad ( x , t ) \in \overline { \Omega } \times [ 0 , \infty ) , \end{align*}


\begin{align*} M = \begin{pmatrix} W & X \\ Y & Z \end{pmatrix} \ ; . \end{align*}


\begin{align*} \delta _ 0 ( f ) = f ( 0 ) \end{align*}


\begin{align*} \mathring { R } _ { q + 1 } ^ B : = \mathring { R } _ { l i n } ^ B + \mathring { R } _ { o s c } ^ B + \mathring { R } _ { c o r } ^ B , \end{align*}


\begin{align*} \sup _ { w : \ , \| w \| _ { W ^ { s , p } ( \mathcal O ) } = 1 } \| \varphi \ , w \| _ { W ^ { s , p } ( \mathcal O ) } < \infty . \end{align*}


\begin{align*} & - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \xi w + \partial _ \tau w \leq \frac { ( \delta b ) ^ { \frac { 1 } { \alpha _ 0 } } } { K } ( 1 - \eta ) ^ { \alpha _ 0 } i n [ 0 , T ^ * ] \times [ 0 , X ] \times [ 0 , 1 ] , \\ & - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \tau w \leq b \delta ( 1 - \eta ) ^ { \alpha _ 0 } i n [ 0 , T ^ * ] \times [ 0 , X ] \times [ 0 , 1 ] . \end{align*}


\begin{align*} [ 2 c ] _ n = \begin{cases} 0 & \\ q ^ { n - 1 } t & \end{cases} \end{align*}


\begin{align*} M ( f ) = \int _ 0 ^ \infty f ( t , \omega ) \ , \omega ^ { 1 / 2 } \ , \dd \omega E ( f ) = \int _ 0 ^ \infty f ( t , \omega ) \ , \omega ^ { 3 / 2 } \ , \dd \omega . \end{align*}


\begin{align*} c ( \sigma , \alpha ) = c \big ( \sigma , \sigma ( \alpha ) \big ) . \end{align*}


\begin{align*} D _ s ( X ) = \min _ { \substack { Y \subset X \\ | Y | = n } } D _ s ( X \setminus Y , Y ) . \end{align*}


\begin{align*} \theta _ 0 : = \chi ^ { 1 + \delta } . \end{align*}


\begin{align*} \langle R ^ \perp ( e _ 1 , e _ 2 ) e _ 3 , e _ 5 \rangle = 0 \langle R ^ \perp ( e _ 1 , e _ 2 ) e _ 4 , e _ 6 \rangle = 0 , \end{align*}


\begin{align*} \mathcal { A } _ 0 : = \left \{ ( R _ 0 , r _ 0 , a _ 0 ) \in \widetilde { \mathcal { A } _ 0 } : \mathcal { E } _ 0 ( R _ 0 , r _ 0 , a _ 0 ) < 0 \right \} . \end{align*}


\begin{align*} 2 h ^ { \mu _ { \theta } } + V + \frac { 1 } { \theta } \log \mu _ { \theta } = c \end{align*}


\begin{align*} M \rvert _ { \chi \xi } ( \alpha , \beta ) = M ( ( \chi , \alpha ) , ( \xi , \beta ) ) \ ; , \end{align*}


\begin{align*} \left ( z \gamma _ { n } ^ { \prime } \right ) ^ { \prime } = \frac { z \left ( \gamma _ { n } ^ { \prime } \right ) ^ { 2 } } { \gamma _ { n } } + \frac { 4 } { z } \gamma _ { n } \left [ \gamma _ { n - 1 } \left ( \gamma _ { n - 2 } - \gamma _ { n } + 1 \right ) - \gamma _ { n + 1 } \left ( \gamma _ { n } - \gamma _ { n + 2 } + 1 \right ) \right ] . \end{align*}


\begin{align*} { B } _ { \mathcal { A } _ t } = \begin{pmatrix} A _ { t , 1 3 } & \frac { 1 } { 2 } I _ { d \times d } - A _ { t , 1 1 } \\ \frac { 1 } { 2 } I _ { d \times d } - A _ { t , 1 1 } ^ T & - A _ { t , 2 1 } \end{pmatrix} , \end{align*}


\begin{align*} Y ^ 2 Z = X ^ 3 + ( \lambda _ 1 + \lambda _ 2 ) X ^ 2 Z + \lambda _ 1 \lambda _ 2 X Z ^ 2 \ , . \end{align*}


\begin{align*} \left \{ \begin{aligned} \omega _ { y } + u v _ { y } - ( v + b ) u _ { y } & = p _ { x } , \\ - \omega _ { x } - u v _ { x } + ( v + b ) u _ { x } & = p _ { y } , \\ u _ { x } + v _ { y } & = 0 , \end{aligned} \right . \end{align*}


\begin{align*} \frac { \partial ^ n } { \partial \upsilon ^ n _ j } \prod _ { i = 1 } ^ d [ \phi _ \tau ( \upsilon ) ] _ { i , \sigma ( i ) } = \sum _ { \substack { m _ 1 , \dots , m _ d \in \N _ 0 , \\ m _ 1 + \ldots + m _ d = n } } \binom { n } { m _ 1 , \ldots , m _ d } \prod _ { i = 1 } ^ d \frac { \partial ^ { m _ i } } { \partial \upsilon _ j ^ { m _ i } } [ \phi _ \tau ( \upsilon ) ] _ { i , \sigma ( i ) } , \end{align*}


\begin{align*} \epsilon _ { j , k } \cdot e _ { j ' , k ' } = \begin{cases} 1 , & ( j , k ) = ( j ' , k ' ) , \\ 0 , & . \end{cases} \end{align*}


\begin{align*} \overline { \Phi } ( x ) : = \int _ x ^ \infty \frac { e ^ { - u ^ 2 / 2 } } { \sqrt { 2 \pi } } d u \ , , \end{align*}


\begin{align*} \forall t \geq 0 , \forall i = 1 , \ldots , d , \ ; f ^ i ( t ) = w ^ i ( t ) + \alpha _ i \max _ { 0 \leq s \leq t } f ^ i ( s ) + \beta \min _ { 0 \leq s \leq t } f ^ i ( s ) . \end{align*}


\begin{align*} h ( z ) = \frac { z + 1 } { z - 1 } + 1 , \ \ \ z \in \Omega . \end{align*}


\begin{align*} x \star y = \gamma _ x ( y ) - y = [ \gamma _ x , y ] , \end{align*}


\begin{align*} n = r + s , \ m = s . \end{align*}


\begin{align*} x ( t ) = \begin{cases} \varphi ( t ) , & - h \leq t \leq 0 , \\ \varphi ( 0 ) , & t \geq 0 . \end{cases} \end{align*}


\begin{align*} [ u _ B , v _ B ] = \Big [ f ' ( \zeta ) , \frac { 1 } { 2 \sqrt { x + x _ 0 } } \{ \zeta f ' ( \zeta ) - f ( \zeta ) \} \Big ] , \end{align*}


\begin{align*} \mathbf { A } = \left ( A ^ { i , j } _ { s , t } ( x ) , \mathbb A ^ { i , j } _ { s , t } ( x ) \right ) _ { \substack { 1 \leq i , j \leq n ; \\ s \le t \in [ 0 , T ] ; x \in D } } \mathbf { B } = \left ( B ^ { i , j } _ { s , t } ( x ) , \mathbb { B } ^ { i , j } _ { s , t } ( x ) \right ) _ { \substack { 1 \leq i , j \leq n ; \\ s \le t \in [ 0 , T ] ; x \in D } } \end{align*}


\begin{align*} \frac { d _ n } { r _ n } \to + \infty , r _ n \to 0 , \frac { r _ n ^ \alpha } { M _ n } \to 0 . \end{align*}


\begin{align*} \left \{ \begin{aligned} & L \partial _ { \tau , \xi } w = 0 ( \tau , \xi , \eta ) \in D , \\ & \partial _ \eta \partial _ { \tau , \xi } w \mid _ { \eta = 0 } = 0 , \displaystyle \lim _ { \eta \to 1 } \partial _ { \tau , \xi } w = 0 , \end{aligned} \right . \end{align*}


\begin{align*} d t = \frac { d l } { \sqrt { E - V } } \end{align*}


\begin{align*} ( F \otimes G ) ( H \otimes M ) = ( F H ) \otimes ( G M ) . \end{align*}


\begin{align*} \log \norm { \chi } = 4 \log ( 5 \cdot 4 3 ) , 2 \ , ( \overline { p } \cdot \overline { q } ) _ { \mathrm { f i n } } = 2 \log ( 5 \cdot 4 3 ) \ , . \end{align*}


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


\begin{align*} \begin{aligned} & P _ t - F _ 0 ( D ^ 2 P ) - P _ f = 0 , \\ & P ( x ' , 0 , t ) \equiv P _ g ( x ' , 0 , t ) , \end{aligned} \end{align*}


\begin{align*} 0 \le E [ V _ { 0 , a , a + b } ] = \varphi _ a + \varphi _ b - \varphi _ { a + b } & \le 2 E \sum _ { i = 1 } ^ a \sum _ { \ell = 1 } ^ b g _ { a , \alpha } ( i ) G ( S _ i , \tilde { S } _ { \ell } ) \tilde { g } _ { b , \alpha } ( \ell ) = 2 E Y _ { a , b } \ , . \end{align*}


\begin{align*} & \left | - \left \langle A B h \partial _ x \partial _ \tau ^ 3 \tilde { h } , \partial _ \tau ^ 3 u \right \rangle - \left \langle A B h \partial _ x \partial _ \tau ^ 3 u , \partial _ \tau ^ 3 \tilde { h } \right \rangle \right | \\ = & \left | \left \langle \partial _ x ( A B h ) \partial _ \tau ^ 3 \tilde { h } , \partial _ \tau ^ 3 u \right \rangle \right | \\ \le & C D ( t ) ^ { \frac 1 4 } E ( t ) ^ { \frac 5 4 } . \end{align*}


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


\begin{align*} ( A + \lambda B - \mu I ) \ , x & = 0 , \\ ( A + \lambda B - \mu I ) ^ 2 \ , y & = 0 , \end{align*}


\begin{align*} U ( t , s _ { 0 } ) = P ( t ) e ^ { Q ( t - s _ { 0 } ) } P ^ { - 1 } ( s _ { 0 } ) , \textnormal { f o r a n y $ t \geq s _ { 0 } \geq 0 $ } , \end{align*}


\begin{align*} \begin{aligned} r ^ { 2 } \left \| p \right \| _ { L ^ { \infty } ( B _ { \frac 7 4 r } ^ { + } \backslash B _ { \frac 5 4 r } ^ { + } ) } \leq & C ( 1 + r \left \| p \right \| _ { L ^ { 2 } ( B _ { 2 r } ^ { + } \backslash B _ { r } ^ { + } ) } + r ^ { 2 } \left \| \nabla p \right \| _ { L ^ { 2 } ( B _ { 2 r } ^ { + } \backslash B _ { r } ^ { + } ) } ) \\ & \cdot \sqrt { \log ( e + r ^ { 3 } \left \| \Delta p \right \| _ { L ^ { 2 } ( B _ { 2 r } ^ { + } \backslash B _ { r } ^ { + } ) } ) } . \end{aligned} \end{align*}


\begin{align*} c + o ( 1 ) \| u _ n \| & \geq I _ { \lambda } ( u _ n ) - \frac { 1 } { \alpha } \langle I ^ { ' } _ { \lambda } ( u _ n ) , u _ n \rangle \\ & = \left ( \frac { 1 } { 2 } - \frac { 1 } { \alpha } \right ) \left ( | \nabla _ { s _ 1 } u | _ 2 ^ 2 + | \nabla _ { s _ 2 } u | _ 2 ^ 2 + \lambda | u | _ 2 ^ 2 \right ) + \int _ { \R ^ d } \left ( \frac { 1 } { \alpha } g ( u ) u - G ( u ) \right ) d x \\ & \geq C \| u \| ^ 2 . \end{align*}


\begin{align*} \dim \mathsf { W } _ M ^ \lambda = s _ \lambda ( 1 ^ M ) \geq s _ \lambda ( 1 ^ N ) = \dim \mathsf { W } _ N ^ \lambda , \end{align*}


\begin{align*} \left \{ \begin{aligned} & g _ 1 ( x , t ) = i \kappa + R _ 1 ( \xi , t ) , \\ & g _ 2 ( x , t ) = i \kappa \left ( \frac { \mathfrak { B } ^ r _ { 2 1 } ( \xi , t ) } { k _ 0 + i \kappa } + \frac { \overline { \mathfrak { B } ^ r _ { 2 1 } ( \xi , t ) } } { k _ 0 - i \kappa } \right ) + R _ 1 ( \xi , t ) , \end{aligned} \right . \end{align*}


\begin{align*} & G _ { \mathcal R , \zeta _ o , 2 } ( x , y ) : = \\ & \frac { i } { 2 \pi } \int _ { \mathcal D _ { \zeta _ o , \frac \pi 2 + \theta _ { \hat { x } } - i \infty } } \frac { 2 i \sin \zeta \mathcal S ( \cos \zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \left ( - \vert y \vert \cos ( \zeta + \theta _ { \hat y } ) + \vert x \vert \cos ( \zeta - \theta _ { \hat x } ) \right ) } } d \zeta . \end{align*}


\begin{align*} \begin{cases} { \rm e n t } [ \mu | \nu ] = \int \frac { d \mu } { d \nu } \log \left ( \frac { d \mu } { d \nu } \right ) d \nu \ { \rm i f } \ \mu \ll \nu \\ { \rm e n t } [ \mu | \nu ] = \infty \ { \rm i f } \ { \rm n o t , } \end{cases} \end{align*}


\begin{align*} G _ { 2 , R } ( 0 , y ) = \frac { 1 } { ( 2 - n ) \omega _ n } \left ( | y | ^ { 2 - n } - R ^ { 2 - n } \right ) , \forall \ , y \in B _ R ( 0 ) \backslash \{ 0 \} , \end{align*}


\begin{align*} \begin{cases} i \displaystyle \frac { \partial u } { \partial t } + O p _ w ( a ) u = 0 \\ u ( 0 , x ) = u _ 0 ( x ) . \end{cases} \end{align*}


\begin{align*} \hat \omega _ 1 ( h _ 1 + h _ 2 ) \hat \omega _ 2 ( h _ 1 + \tilde \alpha _ { G } h _ 2 ) = \hat \omega _ 1 ( h _ 1 - h _ 2 ) \hat \omega _ 2 ( h _ 1 - \tilde \alpha _ { G } h _ 2 ) , \ \ h _ j \in H . \end{align*}


\begin{align*} \nabla ^ 2 v \ , t = 0 \qquad \textrm { o n } \partial A \ , , \end{align*}


\begin{align*} \eta _ 1 ( r ) = - k _ d ^ { - 1 } r V _ \Gamma \eta _ 0 , r \in [ g _ 0 , g _ 1 ] . \end{align*}


\begin{align*} L \left [ \partial _ { x } \left ( \phi p \right ) \right ] = L \left [ \psi p \right ] , \end{align*}


\begin{align*} \phi _ { \pm } ( x , t , k ) = N _ { \pm } ( k ) e ^ { - ( i k x + 4 i k ^ 3 t ) \sigma _ 3 } , \end{align*}


\begin{align*} \left \lfloor \frac { | C | } { | C | - 1 } n \right \rfloor = \left \lfloor \frac { 4 ^ k } { 4 ^ k - 1 } \left ( 2 ^ { k - 1 } ( 2 ^ k - 1 ) \right ) \right \rfloor = 2 ^ { k - 1 } ( 2 ^ k - 1 ) = d _ L ( C ) . \end{align*}


\begin{align*} & \bar f ( x ) = f \left ( U _ { d + 1 } ^ { r _ { d + 1 } } \ldots U _ n ^ { r _ n } x \right ) , \\ & \bar U _ j = U _ j ^ { l _ j } , d < j \le n . \end{align*}


\begin{align*} r _ { i } ^ { - t _ k } = & \sum \limits _ { j \in J ^ k } r _ { j } ^ { t _ k } \geq \sum \limits _ { j \in J ^ k } r _ { j } ^ { \dim _ H ( \mathcal { G } ( F ^ { \alpha } ) ) } = \sum \limits _ { j \in J ^ k } r _ { j } ^ { t _ * } r _ { j } ^ { \dim _ H ( \mathcal { G } ( F ^ { \alpha } ) ) - t _ * } \\ \geq & \sum \limits _ { j \in J ^ k } r _ { j } ^ { t _ * } r _ { m a x } ^ { k ( \dim _ H ( \mathcal { G } ( F ^ { \alpha } ) ) - t _ * ) } \\ = & r _ { m a x } ^ { k ( \dim _ H ( \mathcal { G } ( F ^ { \alpha } ) ) - t _ * ) } , \end{align*}


\begin{align*} \left \| \sum _ { i = 0 } ^ { n - 1 } g _ i ' ( 0 ) \right \| _ U \gtrsim \sqrt { n } A ^ { - 1 } , \end{align*}


\begin{align*} \widetilde \Sigma _ { \widetilde p , \widetilde q } = \Sigma _ { p , q } . \end{align*}


\begin{align*} \mathcal { C } ( t , \varphi ) : = u _ t ^ \star ( \varphi ) ( t ) , \forall ( t , \varphi ) \in X _ 0 , \end{align*}


\begin{align*} \nabla = C \frac { \beta ( \gamma , \alpha ) } { \Gamma ( \alpha ) } \left ( \Omega \sum _ { i = 1 } ^ { m } \omega _ { i } \left ( \frac { \xi _ { i } ^ { \rho } } { \rho } \right ) ^ { \alpha + \gamma - 1 } + \left ( \frac { T ^ { \rho } } { \rho } \right ) ^ { \alpha } \right ) < 1 . \end{align*}


\begin{align*} \mathbf { x } ^ * _ i = 0 , \quad \frac { \partial f } { \partial x _ i } ( \mathbf { x } ^ * ) = \lambda \ ; . \end{align*}


\begin{align*} \int _ 1 ^ \infty t ^ { - ( \beta - \gamma + 1 ) } P _ 1 ( t ) d t = \frac { \zeta ( \beta - \gamma ) } { \gamma - \beta } + \frac 1 { ( \gamma - \beta ) ( \gamma - \beta + 1 ) } - \frac { 1 } { 2 ( \gamma - \beta ) } . \end{align*}


\begin{align*} u ( a , t ) = \alpha _ { 1 } , u ( b , t ) = \alpha _ { 2 } , \end{align*}


\begin{align*} \tanh ( \ell _ \gamma ) = \tan ( \alpha ) \sinh ( d ) \leq \tan ( \vartheta ) \sinh ( d ) = \tanh ( \ell _ N ) . \end{align*}


\begin{align*} e ( \bar { \mu } / v ) = e ( \bar { \mu } ' / v ) = 2 , f ( \bar { \mu } / v ) = f ( \bar { \mu } ' / v ) = 1 . \end{align*}


\begin{align*} \theta _ c : = \begin{cases} \arccos ( n ) \in ( 0 , \pi / 2 ) , \ ; & k _ + > k _ - , \\ \arccos ( 1 / n ) \in ( 0 , \pi / 2 ) , \ ; & k _ + < k _ - . \end{cases} \end{align*}


\begin{align*} D f & = i \delta - i f ' \\ x g & = - i - i h , \end{align*}


\begin{align*} s _ p ( E , \Omega ) : = \sup \left \{ \frac { \int _ E | f | ^ p } { \int _ \Omega | f | ^ p } : f \in A ^ p ( \Omega ) \backslash \{ 0 \} \right \} . \end{align*}


\begin{align*} q ^ 0 ( t ) = q _ 0 ^ 0 + \int _ 0 ^ t f ( q ^ 0 ( s ) ) d s + \int _ 0 ^ t \sigma ( q ^ 0 ( s ) ) d \beta ( s ) \end{align*}


\begin{align*} G _ i ( y _ i ) \coloneqq \prod _ { k = 0 } ^ \infty \left ( 1 - a _ i ^ { R _ k } y _ i \right ) , H _ i ( y _ i ) \coloneqq \sum _ { k = 0 } ^ \infty \frac { a _ i ^ { R _ k } } { 1 - a _ i ^ { R _ k } y _ i } \end{align*}


\begin{align*} O _ { f _ 0 } ( a ) = \Delta ^ { - \frac { 1 } { 2 } } ( a ) \times \sum _ { \overline { m } } \left ( 1 - \frac { 1 } { p } \right ) ^ { \kappa ( \overline { m } ) } , \end{align*}


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


\begin{align*} H _ N ( \alpha , \beta ; q ) = \sum _ { r = 0 } ^ \infty \left ( - \frac { 1 } { N } \right ) ^ r \sum _ { s = 0 } ^ r q ^ s \vec { W } ^ r ( \alpha , \beta ; s ) \end{align*}


\begin{align*} ( H _ i ( \epsilon ) , T _ i ) = \left ( \begin{bmatrix} h _ { i - 1 , i - 1 } & h _ { i - 1 , i } \\ \epsilon & h _ { i , i } \\ \end{bmatrix} , \begin{bmatrix} t _ { i - 1 , i - 1 } & t _ { i - 1 , i } \\ 0 & t _ { i , i } \\ \end{bmatrix} \right ) . \end{align*}


\begin{align*} p z _ i = q z ' _ i \end{align*}


\begin{align*} \rho \ : = \ : \rho _ 1 \oplus \cdots \oplus \rho _ { \ell } . \end{align*}


\begin{align*} V ( z ) = \mathcal { L } v ( z ) : = \int _ 0 ^ \infty e ^ { - k z } v ( k ) \ , d k \end{align*}


\begin{align*} \pi _ i ( E \otimes _ R Q ) = \pi _ i ( E ) \otimes _ R Q \end{align*}


\begin{align*} d \mathbf { P } _ { N , \beta } ( X _ { N } ) = \frac { 1 } { \widetilde { K } _ { N , \beta } } \exp \left ( - N ^ { 2 } \beta { \rm F } _ { N } ( X _ { N } , \mu _ { V } ) + N \log \omega _ { N } \right ) \Pi _ { i = 1 } ^ { N } \rho _ { N } ( x _ { i } ) d X _ { N } . \end{align*}


\begin{align*} 1 \leq \left | T ( n + 1 , \mu ) \right | \ , \left | T ( n + 1 , \mu ) \right | _ p = \left | T ( n + 1 , \mu ) \right | \ , \left | B _ { n + 1 , \mu , 0 } ( 1 ) \Lambda _ p \right | _ p \leq \left | T ( n + 1 , \mu ) \right | \ , \left | \Lambda _ p \right | _ p . \end{align*}


\begin{align*} R ( x , t ) = \{ I _ n - \varepsilon r W ( y , t ) \} ^ { - 1 } = I _ n + \varepsilon r W ( y , t ) + O ( \varepsilon ^ 2 ) \end{align*}


\begin{align*} d l ^ { 2 } = \sum _ { \alpha , \beta = 1 } ^ { 3 N - 7 } N _ { \alpha \beta } d s ^ { \alpha } d s ^ { \beta } + \sum _ { i = 1 } ^ { N - 1 } \mid \frac { \pmb { r } _ { i } } { \lambda } \mid ^ { 2 } ( d \lambda ) ^ { 2 } \end{align*}


\begin{align*} \frac { 1 } { a + b z } = \frac { 1 } { a } \sum _ { k = 0 } ^ { \infty } \left ( - \frac { b } { a } z \right ) ^ k . \end{align*}


\begin{align*} \beta _ { i , i + j } ( I ) \ = \ \sum _ { u \in G ( I ) _ j } \binom { n - \min ( u ) - t ( j - 1 ) } { i } . \end{align*}


\begin{align*} m ( w ) = O ( \| \psi \| _ { W _ \zeta ( B ) } \kappa ) + O ( \psi ( 0 ) \kappa \| \zeta \| _ { W ' _ \zeta ( B ) } ) = O ( \| \psi \| _ { W _ \zeta ( B ) } ( 1 + C ) \kappa ) , \end{align*}


\begin{align*} Y ( t + 1 ) & = \alpha ( t ) ( Y ( t ) - \beta \nabla G ( Y ( t ) ) ) \\ & \quad { } + ( 1 - \alpha ( t ) ) ( ( 1 - \eta ) Y ( t ) + \eta W ( \omega ^ { * } ( t ) ) Y ( t ) ) \end{align*}


\begin{align*} \frac { 1 } { ( 1 + t ) ^ \beta } = \sum _ { n = 0 } ^ \infty ( - 1 ) ^ n \binom { \beta + n - 1 } { n } t ^ n \ , \end{align*}


\begin{gather*} \left ( \gamma _ { n - 1 } U _ { n - 1 } U _ { n } - x U _ { n } + 1 \right ) y = \gamma _ { n - 1 } A _ { n - 1 } A _ { n } y ^ { \prime \prime } \\ + \gamma _ { n - 1 } \left [ A _ { n - 1 } \left ( \partial _ { x } A _ { n } - B _ { n } \right ) - A _ { n } B _ { n - 1 } \right ] y ^ { \prime } - x A _ { n } y ^ { \prime } \\ + \gamma _ { n - 1 } \left ( B _ { n } B _ { n - 1 } - A _ { n - 1 } \partial _ { x } B _ { n } \right ) y + x B _ { n } y + y . \end{gather*}


\begin{align*} f _ 5 ( n ) = \sqrt { n } , f _ 6 ( n ) = \log n , f _ d ( n ) = 1 \forall d \ge 7 . \end{align*}


\begin{align*} \frac { e ^ \eta - 1 } { \eta } \cdot e ^ { | \xi - \eta | } = \frac { e ^ \eta - 1 } { \eta } \cdot e ^ { \xi - \eta } = \frac { e ^ \xi - e ^ { | \xi - \eta | } } { \eta } \geq \frac { e ^ \xi - 1 } { \xi } \ , . \end{align*}


\begin{align*} \partial ^ \circ ( g \eta _ 0 ) - g V _ \Gamma H \eta _ 0 - k _ d \ , \mathrm { d i v } _ \Gamma ( g \nabla _ \Gamma \eta _ 0 ) = g f . \end{align*}


\begin{align*} g _ H ( N ( \eta _ 2 ( t _ 2 ) ) , \nu _ 2 ( t _ 2 ) ) = \cos ( \Theta _ 2 ) . \end{align*}


\begin{align*} - 2 a \delta _ a \cdot ( w a ^ { w } + ( 1 - w ) c _ w a ^ { 1 - w } ) = - 2 \delta _ a \cdot ( 2 w - 1 ) a ^ { w + 1 } . \end{align*}


\begin{align*} \mathcal { F } ( \overline { \mathbf { P } } ) = \mathcal { G } ( \overline { \mathbf { P } } ) - \inf _ { \overline { \mathbf { P } } ^ * } \mathcal { G } ( \overline { \mathbf { P } } ^ * ) , \end{align*}


\begin{align*} S _ { q , r } ( m ) = p \left ( q - 1 \right ) q ^ { v - 1 } w + r \in \Omega _ p \end{align*}


\begin{align*} \langle \hat f , \varphi \rangle = \langle f , \mathcal { F } ^ { - 1 } \varphi \rangle , \ \ \ \ \ \forall \varphi \in \mathcal { S } ( \mathbb { R } ) . \end{align*}


\begin{align*} g = \beta \bar g , \beta \in { \cal B } : = ( \beta _ 1 , \beta _ 2 ) , 0 < \beta _ 1 < \beta _ 2 < + \infty \ , . \end{align*}


\begin{align*} S = \{ ( m _ { 1 } , m _ { 2 } ) \in \mathbb { Q } ( S , 2 ) \times \mathbb { Q } ( S , 2 ) \mid m _ { 1 } > 0 \} . \end{align*}


\begin{align*} R _ 0 = R _ 0 ( T ) : = 1 \vee ( 2 T \kappa ) ^ { 1 / \alpha } . \end{align*}


\begin{align*} \left \{ \begin{aligned} & - \Delta v = f _ { \epsilon } ( x , z ) & & \mbox { i n } \Omega \\ & v = 0 & & \mbox { o n } \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} k _ { D _ n } ( p , q ) = O ( 1 + R ) \end{align*}


\begin{align*} M = \bigoplus _ { i = 1 } ^ n \frac { \Z _ \ell } { \ell ^ { e _ i } \Z _ \ell } \ , \dd X _ i . \end{align*}


\begin{align*} \mathcal { S } _ { \ell } : = \{ r = ( r _ 1 , . . . , r _ n ) \in \N ^ n : \sum _ { j = 1 } ^ n r _ j = \ell . \} \end{align*}


\begin{align*} J _ n = \frac { 2 ^ n - ( - 1 ) ^ n } { 3 } . \end{align*}


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


\begin{align*} ( - \Delta ) ^ { \frac { n - 1 } { 2 } } h _ { R } ( x ) = \phi _ R ( x ) , \forall \ , x \in B _ { R } ( 0 ) , \ , \ , n \ , \ , \mbox { i s o d d } . \end{align*}


\begin{align*} \mathcal { E } _ { m , n , 1 , 1 } ^ { \epsilon , \Delta t } = \mathcal { E } _ { m , n , 1 , 1 , 1 } ^ { \epsilon , \Delta t } + \mathcal { E } _ { m , n , 1 , 1 , 2 } ^ { \epsilon , \Delta t } , \end{align*}


\begin{align*} G _ 0 ^ B ( ( P \times _ B M ) \times _ X N ) & = G _ 0 ^ { B \times B } ( ( P \times M ) \times _ X N ) \\ \stackrel { \sim } { \to } G _ 0 ^ { B \times B } ( M \times _ X ( P \times N ) ) & = G _ 0 ^ B ( M \times _ X ( P \times _ B N ) ) . \end{align*}


\begin{align*} \varphi ^ * \lambda = d t + \frac { 1 } { 2 } r ^ 2 \ , d \theta . \end{align*}


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


\begin{align*} t \alpha _ i = ( e - h ) ( \gamma _ i - \overline { \gamma _ i } ) = e \gamma _ i - \gamma _ i h + e \overline { \gamma _ i } + h \overline { \gamma _ i } \\ = e \gamma _ i - \gamma _ i h + e \overline { \gamma _ i } \end{align*}


\begin{align*} \Box ( a \land b ) = \Box a \land \Box b \ , \ , \ , \ , \Box 1 = 1 . \end{align*}


\begin{align*} \vert \Delta _ i \vert = \frac { \vert G _ \alpha \vert } { \vert G _ { \alpha \beta } \vert } = \frac { 2 ( m ! ) ^ 2 } { 2 ( ( m - i ) ! i ! ) ^ 2 } = \left ( \frac { m ! } { ( m - i ) ! i ! } \right ) ^ 2 = 2 ^ { - \lfloor \frac { 2 i } { m } \rfloor } \left ( \frac { m ! } { ( m - i ) ! i ! } \right ) ^ 2 . \end{align*}


\begin{align*} N _ { + } ( k ) = \begin{pmatrix} 1 & \frac { A } { 2 i k } \\ 0 & 1 \end{pmatrix} , N _ { - } ( k ) = \begin{pmatrix} 1 & 0 \\ \frac { \sigma A } { 2 i k } & 1 \end{pmatrix} . \end{align*}


\begin{align*} \Delta _ { n - 1 } ^ 2 = ( [ n \beta ] - [ ( n - 1 ) \beta ] ) - ( [ n \alpha ] - [ ( n - 1 ) \alpha ] ) . \end{align*}


\begin{align*} \frac { d } { d t } v + \nu A v + B ( v , v ) = h - \mu P _ N ( v - u ) , \end{align*}


\begin{align*} q _ { - \sigma } \left ( x , t \right ) = q \left ( x , t \right ) - 2 \partial _ { x } ^ { 2 } \log \det \left \{ I - \mathbf { K } \left ( x , t \right ) \right \} \end{align*}


\begin{align*} h \circ \Pi ( - r , \pi - \theta ) = h ( \overline { r e ^ { i \theta } } ) = \left ( R _ { \Gamma } \circ h \circ \Pi \right ) ( r , \theta ) . \end{align*}


\begin{align*} f ( \mu ) = \max _ { \mathbf { x } \in \Delta } f ( \mathbf { x } ) \ ; , \Delta = \left \{ \mathbf { x } : \sum _ { i = 1 } ^ { \binom { n } { 2 } } \mathbf { x } _ i = 1 \mathbf { x } _ 1 , \ldots , \mathbf { x } _ { \binom { n } { 2 } } \geq 0 \right \} \ ; . \end{align*}


\begin{align*} Z = \overline { Z _ 1 \oplus \ldots \oplus Z _ l } \end{align*}


\begin{align*} \partial _ t \rho ( x , t ) = - \varepsilon ^ { - 1 } V _ \Gamma ( y , t ) \partial _ r \eta ( y , t , r ) + \partial ^ \circ \eta ( y , t , r ) + O ( \varepsilon ) . \end{align*}


\begin{align*} \Theta [ \phi _ n , k _ n ] : = \Theta [ n ] , \end{align*}


\begin{align*} \langle H _ y \rangle _ { R _ y } = \langle H _ y \rangle _ { \kappa \Phi _ { \# } ^ y } + \langle H _ y \rangle _ { R _ y ^ { \# } } + 2 \mathrm { R e } \langle \kappa \Phi _ { \# } ^ y \ , | \ , H _ y R _ y ^ { \# } \rangle , \end{align*}


\begin{align*} G ^ { ( j ) } _ { \mathcal R } ( x , y ) = \frac { i e ^ { i k _ { + } \vert x \vert } } { 4 \pi } \int ^ { + \infty } _ { - \infty } v _ j ( s ) F ( s ) \frac { d \zeta ( s ) } { d s } e ^ { - \vert x \vert s ^ 2 } d s , \end{align*}


\begin{align*} S _ { H } ^ { \prime } \left ( t \right ) = - 2 t S _ { H } \left ( t \right ) + 2 \sqrt { \pi } = - 2 t S _ { H } \left ( t \right ) + 2 \mu _ { 0 } ^ { H } . \end{align*}


\begin{align*} E ( x , t , k ) = I + C ( \mu w ) = I + \frac { 1 } { 2 \pi i } \int _ { \Gamma _ E } \mu ( x , t , s ) w ( x , t , s ) \frac { d s } { s - k } , \end{align*}


\begin{align*} t = \frac { 1 } { \sqrt { p } } . \end{align*}


\begin{align*} \pi ^ j \gamma ( \widetilde { 0 } ) = \pi ^ { j ' } \gamma ' ( \widetilde { 0 } ) \ ; \Leftrightarrow \ ; ( \gamma ' ) ^ { - 1 } \pi ^ { j - j ' } \gamma ( \widetilde { 0 } ) = \widetilde { 0 } \ ; \Leftrightarrow \ ; ( \gamma ' ) ^ { - 1 } \pi ^ { j - j ' } \gamma \in \mathcal { G } ( X ) _ { \widetilde { 0 } } . \end{align*}


\begin{align*} \begin{aligned} \lim _ { r \to \infty } \int _ { 0 } ^ { \pi } | p ( r , \theta ) - \bar { p } ( r ) | ^ { 2 } d \theta = 0 . \end{aligned} \end{align*}


\begin{align*} h ( t \cdot x ) = \prod _ { v \leq \infty } h _ v ( t \cdot x ) = \prod _ { v \leq \infty } | t | _ v \cdot h _ v ( x ) = \prod _ { v \leq \infty } | t | _ v \cdot \prod _ { v \leq \infty } h _ v ( x ) = 1 \cdot h ( x ) \end{align*}


\begin{align*} \# C _ { i , i + 1 } = p ( p + 1 ) . \end{align*}


\begin{align*} f _ \varepsilon ( t ) : = \sup _ { x \in \R ^ d } \| u ( t + \varepsilon \ , , x ) - u ( t \ , , x ) \| _ k , 0 < t \le T , \end{align*}


\begin{align*} \kappa ( \mathsf { F } ^ { ( n ) } ) : = \sum _ { 1 \le k \le n } \mathsf { F } _ k ^ { ( n ) } \end{align*}


\begin{align*} & C _ { n , A , \infty } [ z ; \psi _ 1 , \psi _ 2 ] : = \lim _ { L \to \infty } C _ { n , A , L } [ z ; \psi _ 1 , \psi _ 2 ] . \end{align*}


\begin{align*} c _ k \left ( \coprod _ { i = 1 } ^ m ( Y _ i , \lambda _ i ) \right ) = \max _ { k _ 1 + \cdots + k _ m = k } \sum _ { i = 1 } ^ m c _ { k _ i } ( Y _ i , \lambda _ i ) . \end{align*}


\begin{align*} a _ n & = \frac { 2 } { \pi } \int _ { 0 } ^ { \infty } \bigg ( \frac { \sin x } { x } \bigg ) ^ { n + 1 } \cos x \ , d x . \\ & = \frac { 2 } { \pi } \bigg ( \frac { n + 1 } { n + 2 } \bigg ) \int _ { 0 } ^ { \infty } \bigg ( \frac { \sin x } { x } \bigg ) ^ { n + 2 } \ , d x . \end{align*}


\begin{align*} \mathcal F : = \mathcal L \bigl ( - \dfrac t \varepsilon \bigr ) \biggl ( \begin{array} { c } 0 \\ K + L _ 1 + L _ 2 \end{array} \biggr ) \rightarrow 0 , ~ ~ ~ ~ . \end{align*}


\begin{align*} \psi _ 1 ( x , t , k ) = \psi _ 2 ( x , t , k ) e ^ { - ( i k x + 4 i k ^ 3 t ) \sigma _ 3 } \sigma \Lambda \overline { S ^ { - 1 } ( - k ) } \Lambda e ^ { ( i k x + 4 i k ^ 3 t ) \sigma _ 3 } , k \in \mathbb { R } \backslash \{ 0 \} . \end{align*}


\begin{align*} B _ { \mathfrak a } \left ( \rho ( y ) f ( \alpha ( z ) , t ) , g ( x ) \right ) & = B _ { \mathfrak a } \left ( f ( \alpha ( z ) , t ) , \rho ( y ) g ( x ) \right ) \\ & = B _ { \mathfrak a } \left ( f \circ \alpha ( z , a ) , \rho ( y ) g ( x ) \right ) . \end{align*}


\begin{align*} \Theta _ 2 = \vartheta ( s _ 2 ) - \vartheta ( s _ 2 ^ * ) = \int _ { s _ 2 ^ * } ^ { s _ 2 } \dot \vartheta ( s ) \ , d s . \end{align*}


\begin{align*} p ( x , t ) = a ( t ) \delta _ 0 ( x ) + r ( x , t ) + b ( t ) \delta _ 1 ( x ) \ , \end{align*}


\begin{align*} - \Delta P = \mathrm { d i v } ( u \cdot \nabla u ) = \underset { i , j } { \sum } \partial _ { x _ { i } } u ^ { j } \partial _ { x _ { j } } u ^ { i } \coloneqq \mathfrak { U } . \end{align*}


\begin{align*} \left \vert a \right \vert _ { \mathfrak { Z } ^ { \circ } } = \mathbb { E } \max _ { 0 \leq j \leq N } \left \langle a , X ^ { ( j ) } \right \rangle = \delta ^ { - 1 } \int _ { 0 } ^ { 1 - F _ { a } \left ( 0 \right ) } F _ { a } ^ { - 1 } \left ( 1 - s \right ) \exp \left ( - \delta ^ { - 1 } s \right ) d s \end{align*}


\begin{align*} J ^ { \pm } ( u ) = \begin{pmatrix} 1 & 0 \\ f _ { 1 } ^ { \pm } ( u ) & 1 \end{pmatrix} \begin{pmatrix} k _ { 1 } ^ { \pm } ( u ) & 0 \\ 0 & k _ { 2 } ^ { \pm } ( u ) \end{pmatrix} \begin{pmatrix} 1 & e _ { 1 } ^ { \pm } ( u ) \\ 0 & 1 \end{pmatrix} . \end{align*}


\begin{align*} \| g r a d ^ g \textbf { H } \| ^ 2 = 2 s ^ H _ \Sigma , \end{align*}


\begin{align*} x ^ 2 + y ^ 2 + f _ { 2 r } ( t , s ) = 0 , \end{align*}


\begin{align*} \begin{aligned} \lim \limits _ { n \rightarrow \infty } \left | \delta ( n ) u ^ { \delta ( n ) , n } ( y ) + \mu ( h ) \right | = 0 , y , \end{aligned} \end{align*}


\begin{align*} \left ( \frac { \partial } { \partial \lambda } \right ) ^ d \omega _ \mu = \prod _ { i = 1 } ^ n \left ( \frac { \partial } { \partial \mu _ i } \right ) ^ { w _ i } \omega _ \mu , \\ \left ( \sum _ { i = 1 } ^ n a _ { i j } \mu _ i \frac { \partial } { \partial \mu _ i } + c _ j \right ) \omega _ \mu = 0 . \end{align*}


\begin{align*} X ^ { \mathfrak { s } } _ T : = C ( [ 0 , T ] ; H _ x ^ { \mathfrak { s } } ) \cap L ^ r ( 0 , T ; W _ x ^ { \mathfrak { s } , p } ) . \end{align*}


\begin{align*} \begin{aligned} & d ^ { - ( 2 ^ h + 1 ) } = \alpha ^ { q - 1 } & & \mbox { i f } h \mbox { i s o d d , } \\ & d ^ { - ( 2 ^ { n + h } + 1 ) } = \alpha ^ { q - 1 } & & \mbox { i f } h \mbox { i s e v e n . } \end{aligned} \end{align*}


\begin{align*} T _ 0 = \max ( k , k ' ) , \end{align*}


\begin{align*} P \ , ( A - \lambda B ) \ , Q = \left [ \begin{array} { c c } R ( \lambda ) & 0 \\ 0 & S ( \lambda ) \end{array} \right ] \quad \mbox { a n d } P U = \left [ \begin{array} { c } 0 \\ \widetilde U \end{array} \right ] , \end{align*}


\begin{align*} F _ k = \bigcup _ { j = 1 } ^ { k } F _ { k j } , \mbox { w h e r e } F _ { k j } = \psi _ j ^ { - 1 } \big \{ \big ( F _ k ' \cap Q _ j \big ) \times \big [ \R \setminus ( - m _ k , m _ k ) \big ] \big \} . \end{align*}


\begin{align*} v ( y , s ) = \frac { u ( x , t ) - a _ { m _ 0 } x _ n } { r ^ { 1 + \alpha } } . \end{align*}


\begin{align*} \delta \left ( s \right ) : = \operatorname * { s g n } \left ( s \right ) \sum _ { n } c _ { n } ^ { 2 } \left ( t \right ) \delta _ { \kappa _ { n } } \left ( \left \vert s \right \vert \right ) . \end{align*}


\begin{align*} \int _ { E } p ( t , x ) ( 1 - \rho ( t , x ) ) \ , d x \ , d t = \lim _ { k \to \infty } \int _ { E } p _ { \gamma _ k } ( t , x ) ( 1 - \rho _ { \gamma _ k } ( t , x ) ) \end{align*}


\begin{align*} Y ^ 2 Z + a _ 1 X Y Z + a _ 1 a _ 2 Y Z ^ 2 = X ^ 3 + a _ 2 X ^ 2 Z \ , . \end{align*}


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


\begin{align*} \int _ { Z _ { \overline { K } _ v } ^ { \mathrm { a n } } } \log \| s _ d \| _ v \prod _ { i = 0 } ^ { d - 1 } c _ 1 ( \overline { L } _ { i , v } ) = \frac { 1 } { e ^ d } c _ 1 ( \widetilde { L } _ 0 ) \dots c _ 1 ( \widetilde { L } _ { n - 1 } ) [ V _ v ] \log N ( v ) . \end{align*}


\begin{align*} \bar { \nu } ( x , t ) \cdot \nabla \rho _ \zeta ^ \varepsilon ( x , t ) = - k _ d ^ { - 1 } \Bigl ( \overline { V _ \Gamma \zeta } \Bigr ) ( x , t ) + d ( x , t ) \bar { \zeta } _ 2 ( x , t ) , ( x , t ) \in \overline { Q _ { \varepsilon , T } } \end{align*}


\begin{align*} \mathbf { P } _ n ^ { \alpha , \theta , T } ( \Gamma ( x _ i : i \in \mathcal { B } \sqcup \mathcal { O } \sqcup \mathcal { U } ) ) = \lim _ { m \to \infty } m ^ { \# \mathcal { B } } P ^ { 1 , 1 - \frac { \lambda } { m } , \frac { T } { m } } _ { m , n } \left ( \sigma ( \tilde { x } ^ 1 ) = \tilde { y } ^ 1 , \ldots , \sigma ( \tilde { x } ^ p ) = \tilde { y } ^ p \right ) , \end{align*}


\begin{align*} - \kappa _ \eta ( t ) = \dot \beta ( t ) - K _ 1 ( \eta ( t ) ) \cos ( \beta ( t ) ) , \end{align*}


\begin{align*} U [ x ] : = \{ y \in X \ , : \ , ( x , y ) \in U \} \ , . \end{align*}


\begin{align*} S = \left \{ ( s , t , r , \theta ) \in \R \times I \times D ^ 2 \ ; \big | \ ; ( s , \varphi ( t , r , \theta ) ) \in M _ { \lambda _ g } \right \} , \end{align*}


\begin{align*} [ 2 ] _ n = t [ 1 a ] _ { n - 1 } . \end{align*}


\begin{align*} \mathcal { C } ( f , g , h ) = \frac 1 6 \left ( \bar { \mathcal { C } } ( f , g , h ) + \bar { \mathcal { C } } ( f , h , g ) + \bar { \mathcal { C } } ( g , f , h ) + \bar { \mathcal { C } } ( g , h , f ) + \bar { \mathcal { C } } ( h , f , g ) + \bar { \mathcal { C } } ( h , g , f ) \right ) , \end{align*}


\begin{align*} x \sum _ { k = 0 } ^ 4 \frac { k ! } { \log ^ { k + 1 } x } + \frac { m _ a ( x ) x } { \log ^ 6 x } \leq \pi ( x ) \leq x \sum _ { k = 0 } ^ 4 \frac { k ! } { \log ^ { k + 1 } x } + \frac { M _ a ( x ) x } { \log ^ 6 x } \end{align*}


\begin{align*} \left \{ \begin{array} { c c } \dot { x _ { k } } ( t ) = \xi _ { k } ( t ) , & x _ { k } ( 0 ) = x _ { k } ^ { i n } , \\ \dot { \xi _ { k } } ( t ) = - \frac { 1 } { N } \underset { 1 \leq j \leq N , j \neq k } { \sum } \nabla V ( x _ { k } ( t ) - x _ { j } ( t ) ) , & \xi _ { k } ( 0 ) = \xi _ { k } ^ { i n } . \end{array} \right . 1 \leq k \leq N . \end{align*}


\begin{align*} \beta _ { 1 , n } ^ * ( P _ { \ell , m - \ell } ) \leq \beta _ { w ' , n - \ell } ^ * ( P _ { 0 , m - \ell } ) \cdot \prod _ { i = 1 } ^ { \ell } w _ i \ ; , \end{align*}


\begin{align*} \sum _ { i \neq j } \left | \left [ { \rm e m p } _ { N } ( K _ { i } ) - \rho ( K _ { i } ) \right ] \max _ { y \in K _ { i } } g ^ { \epsilon } ( x - y ) \right | & \leq { \rm D i s c } ( \eta , \delta , N ) \sum _ { i \neq j } \left | \max _ { y \in K _ { i } } g ^ { \epsilon } ( x - y ) \right | \\ & \leq \frac { c _ { g , \epsilon } } { \eta ^ { d } } { \rm D i s c } ( \eta , \delta , N ) , \end{align*}


\begin{align*} \| \nabla \hat { J } ( u _ { 0 } ) - \nabla J ( u _ { 0 } ) \| = O ( d \epsilon ) . \end{align*}


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


\begin{align*} ( \mathcal { A } \otimes \mathcal { F } ) \cap ( \mathcal { A } \otimes \mathcal { G } ) = \mathcal { A } \otimes \left ( \mathcal { F } \cap \mathcal { G } \right ) , \end{align*}


\begin{align*} \begin{array} { r @ { \ ; = \ ; } l } R _ A ( c ( R _ A f ^ * \times _ { A _ 0 } c ( g \times _ { A _ 0 } S _ A f ^ * ) ) ) ^ * & c ( R _ A g ^ * \times _ { A _ 0 } S _ A f ^ * ) \quad \\ S _ A ( c ( R _ A f ^ * \times _ { A _ 0 } c ( g \times _ { A _ 0 } S _ A f ^ * ) ) ) ^ * & c ( R _ A f ^ * \times _ { A _ 0 } S _ A g ^ * ) . \end{array} \end{align*}


\begin{align*} \partial _ t \mu _ t - \nabla _ M \cdot ( ( \nabla _ M ( W \ast \mu _ t ) ) \mu _ t ) = 0 \end{align*}


\begin{align*} \langle \cdot \ , | \ , \cdot \rangle _ { \# } : = \langle \cdot \ , | \ , ( h _ { \alpha } - e _ { \alpha } + H _ f ) \ , | \ , \cdot \rangle , \langle \cdot \ , | \ , \cdot \rangle _ { * } : = \langle \cdot \ , | \ , H _ f \ , | \ , \cdot \rangle , \end{align*}


\begin{align*} \biggl [ \frac { \partial } { \partial s _ { 1 , 0 , 1 } } , \ D \biggr ] = \biggl [ \frac { \partial } { \partial s _ { 1 , 0 , 1 } } , \ r s _ { 1 , 0 , 1 } \biggr ] = r , \end{align*}


\begin{align*} F ( p , q ) = \log \left | 2 \sin ( \pi ( p + q i ) \right | - \pi q . \end{align*}


\begin{align*} \ker ( t , d ) : = \{ g \in G : d ( g ) = 1 , \ t ( g h ) = t ( h ) \ \mbox { f o r a l l $ h \in G $ } \} \subset G . \end{align*}


\begin{align*} \frac { d \hat \eta _ \tau } { d \tau } = - \frac { 1 } { \frac { \partial \hat x _ \tau } { \partial \eta } } \frac { d \hat x _ \tau } { d \tau } . \end{align*}


\begin{align*} x * y = \phi ( \phi ^ { - 1 } ( x ) \cdot \phi ^ { - 1 } ( x ) ) , \ \ x , y \in { \mathfrak L } . \end{align*}


\begin{align*} & \sin ( \zeta ( s ) ) = 2 P ( s ) Q ( s ) \cos \theta _ { \hat x } + ( 2 P ^ 2 ( s ) - 1 ) \sin \theta _ { \hat x } , \\ & \cos ( \zeta ( s ) + \theta _ y ) = { ( 2 P ^ 2 ( s ) - 1 ) \cos ( \theta _ { \hat x } + \theta _ { \hat y } ) - 2 P ( s ) Q ( s ) \sin ( \theta _ { \hat x } + \theta _ { \hat y } ) } . \end{align*}


\begin{align*} R i c + \nabla ^ 2 \varphi = ( \lambda + \rho S ) g , \end{align*}


\begin{align*} \gamma : = \sup _ { 1 \leq j \leq n } \bigg \{ \sum _ { i = 1 } ^ n | a _ { i j } | \bigg \} \end{align*}


\begin{align*} x \to y = \max \{ u \colon ( u ] \cap ( x \vee y ] \cap [ y ) = \{ y \} \} . \end{align*}


\begin{align*} C _ 1 = \sum _ { x \in \Z } \sum _ { t = 0 } ^ \infty \Delta ( x , t ) ^ 4 , \ \ \ C _ 2 = \biggl ( \sum _ { x \in \Z } \sum _ { t = 0 } ^ \infty \Delta ( x , t ) ^ 2 \biggr ) ^ 2 - C _ 1 . \end{align*}


\begin{align*} V : = \{ g ( x ) h ( t ) \in k [ x , t ] \ , | \ , g ( x ) \in k [ x ] \setminus \{ 0 \} , h [ t ] \in k [ t ] \setminus \{ 0 \} \} . \end{align*}


\begin{align*} | j | _ g = \beta | j | _ { \overline g } \end{align*}


\begin{align*} X ^ * ( r e ^ { i \theta } ) = \left ( r \sin { \theta } , - r \cos { \theta } , \frac { 1 } { \pi } \log { r } \right ) , w = r e ^ { i \theta } \in \mathbb { H } , \end{align*}


\begin{align*} X _ k = g _ k ( S _ 1 , \ldots , S _ k , R ) \in C ( S _ k ) \forall k \in \mathbb { N } \end{align*}


\begin{align*} c ( k ) = \frac { k \ , d } { k + h ^ \vee } - 6 k + h ^ \vee - 4 , \end{align*}


\begin{align*} \partial _ t \mathcal { Z } = \frac { 1 } { 2 } \partial _ x ^ 2 \mathcal { Z } + \sqrt { 2 \mu _ 2 } \beta \mathcal { Z } \xi , \ \ \ \mathcal { Z } ( 0 , \cdot ) \equiv 1 , \end{align*}


\begin{align*} \frac { a - n } { 2 } \pm \delta ^ \pm _ { \mu } , \delta ^ \pm _ { \mu } = \pm \frac { 1 } { 2 } \sqrt { a ^ 2 + 4 \mu } , \end{align*}


\begin{align*} M _ t ( z , z ' ) & = \int _ { \R } e ^ { - i \tau \rho ' } \Big \{ e ^ { i \tau \rho } \sum _ { \mu \in \N ^ d } e ^ { - t ( \tau ^ 2 + 2 | \mu | + d ) } m ( \tau , | \mu | ) \Phi _ { \mu } ( x ' ) \Phi _ { \mu } ( x ) \Big \} \dd \tau . \end{align*}


\begin{align*} D _ { \Lambda ' } = \Lambda _ 1 + D _ { Z ' } = \{ \ , \Lambda _ 1 + \Lambda \mid \Lambda \in D _ { Z ' } \ , \} . \end{align*}


\begin{align*} h ^ { \mu } = g \ast \mu . \end{align*}


\begin{align*} \nabla \pi = \begin{pmatrix} \partial _ 1 \pi _ 1 & \cdots & \partial _ 1 \pi _ n \\ \vdots & \ddots & \vdots \\ \partial _ n \pi _ 1 & \cdots & \partial _ n \pi _ n \end{pmatrix} . \end{align*}


\begin{align*} \kappa ( \mathsf { F } ^ { ( n ) } ) : = \sum _ { 1 \le k \le n } \mathsf { F } ^ { ( n ) } _ k . \end{align*}


\begin{align*} \rho ^ \mathrm { p a i r } [ n ] ( x ) : = \Big ( \frac { k _ n \circ \overline { k _ n } } { \delta - k _ n \circ \overline { k _ n } } \Big ) ( x , x ) , \end{align*}


\begin{align*} G ( x , y ) = \sum _ { i = 0 } ^ \infty P ^ x ( S _ i = y ) \ , . \end{align*}


\begin{align*} \mathbb { E } [ u ^ x _ { s + t } | \mathcal { F } _ s ] = \mathbb { E } [ \phi ( u ^ x _ { s + t } ) ] = P _ t \phi ( u _ s ^ x ) \ , , \quad \mathbb { P } - \textrm { a . s . } \ , , \end{align*}


\begin{align*} y \left ( \alpha \right ) + \int K \left ( \alpha , s ; x , t \right ) y \left ( s \right ) \mathrm { d } \sigma _ { t } \left ( s \right ) & = \psi \left ( x , t ; \mathrm { i } \alpha \right ) , \ \alpha \in \operatorname * { S u p p } \sigma , \\ \mathrm { d } \sigma _ { t } \left ( s \right ) & : = \mathrm { e } ^ { 8 s ^ { 3 } t } \mathrm { d } \sigma \left ( s \right ) , \end{align*}


\begin{align*} p ^ i ( \tilde x ( \omega ) ) = p ^ i ( \tilde y ( \omega ) ) \ \ \mbox { a n d } \ \ ( p ^ i \circ h ) ( \tilde x ( \omega ) ) \ne ( p ^ i \circ y ) ( \tilde y ( \omega ) ) \ \ \ ( \omega \in B ) . \end{align*}


\begin{align*} b = \frac { c _ 0 } { 4 } e ^ { - \beta X } e ^ { - \frac { 8 } { 3 } } \end{align*}


\begin{align*} \frac { d ^ \ell _ x \lambda ^ c _ a ( \tau ( \ell ) + k ) } { d ^ \ell _ x \lambda ^ c _ { x ^ u } ( \tau ( \ell ) ) } = \frac { \lambda ^ c _ a ( \tau ( \ell ) ) } { \lambda ^ c _ { x ^ u } ( \tau ( \ell ) ) } \times \lambda ^ c _ { a _ { \tau ( \ell ) } } ( k ) . \end{align*}


\begin{align*} \tilde { H } _ 6 = i ( \kappa _ 1 - \tilde { H } _ 5 ) . \end{align*}


\begin{align*} \frac { d } { d t } v _ n + \nu A v _ n + B ( v _ n , v _ n ) = f _ { n - 1 } - \mu _ n P _ N ( v _ n - u ) , v _ n ( t _ { n - 1 } ) = v _ n ^ 0 , t \in I _ { n - 1 } . \end{align*}


\begin{align*} a g = g b \qquad b h = h a . \end{align*}


\begin{align*} \prod _ { 1 \le i \le n } \binom { C _ i + N _ i - 1 } { N _ i } , \end{align*}


\begin{align*} y \left ( \alpha , x \right ) + \int K \left ( \alpha , s , x \right ) y \left ( s , x \right ) \mathrm { d } \sigma \left ( s \right ) = \psi \left ( x , \mathrm { i } \alpha \right ) , \ \ \ \alpha \in \operatorname * { S u p p } \sigma , \end{align*}


\begin{align*} \lim _ { a \to 0 + } a ^ 2 c ( a ) = \infty . \end{align*}


\begin{align*} v _ 1 ( x , t ) = \frac { A } { 2 i } \frac { 1 } { 1 - \gamma _ 0 e ^ { - A x + A ^ 3 t } } \end{align*}


\begin{align*} | M _ k | _ * - \epsilon < \left | S \cap \cup _ { i = 1 } ^ N B _ i \right | _ * \leq \sum _ { i = 1 } ^ N | S \cap B _ i | _ * \leq \left ( 1 - \dfrac { 1 } { k } \right ) \sum _ { i = 1 } ^ N | B _ i | < ( 1 + \epsilon ) \left ( 1 - \dfrac { 1 } { k } \right ) | M _ k | _ * , \end{align*}


\begin{align*} \limsup _ { r \to 0 } \dfrac { | E \cap B _ r ( x ) | _ * } { | B _ r ( x ) | } = 1 \end{align*}


\begin{align*} \overline { r _ 1 ( - k ) } = r _ 1 ( k ) , \overline { r _ 2 ( - k ) } = r _ 2 ( k ) , k \in \mathbb { R } \backslash \{ 0 \} . \end{align*}


\begin{align*} \partial ^ { \alpha } | \tau | ^ 2 = \begin{cases} | \tau | ^ 2 , & \alpha = 0 , \\ 2 \tau _ j , & \alpha = e _ j j \in \underline { d } , \\ 2 , & \alpha = 2 e _ j j \in \underline { d } , \\ 0 , & , \end{cases} \end{align*}


\begin{align*} \Theta [ \phi , k ] ( x , y ) & : = \upsilon _ N ( x - y ) \Big \{ \phi ( x ) \phi ( y ) + \frac { 1 } { N } \big ( k \circ ( \delta - \overline { k } \circ k ) ^ { - 1 } \big ) ( x , y ) \Big \} , \end{align*}


\begin{align*} m _ 1 = m , m _ 2 = n \alpha = 1 , \beta = 1 - \frac { \lambda } { m } , \gamma = \frac { T } { m } . \end{align*}


\begin{align*} \min _ { \| \phi \| = 1 } \Big \{ E _ \mathrm { H } [ \phi ] + \frac { 1 } { N } \mathcal { E } \big [ k _ { n + 1 } ; \ , H _ 0 [ \phi , k _ { n + 1 } ] , \ , \Theta [ \phi , k _ { n + 1 } ] \big ] , \phi \in \mathfrak { h } ^ 1 _ V \Big \} . \end{align*}


\begin{align*} \gamma _ i = | y _ i ^ * B x _ i | \ , ( 1 + | \lambda _ i | ^ 2 ) ^ { - 1 / 2 } \end{align*}


\begin{align*} \psi ( u , v ) = \frac { u + v } { 2 } + ( u - v ) ^ 2 , \end{align*}


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


\begin{align*} u _ { y _ 0 } ( z ) = \begin{cases} \frac { 2 } { z + i y _ 0 } & z \in \Omega , \\ \frac { 2 } { \bar z + i y _ 0 } & z \in \mathbb C \setminus \Omega . \end{cases} \end{align*}


\begin{align*} \mathcal P _ { P K } ( x ) = \sum _ { g \in K } E _ { P } ( x u _ { g ^ { - 1 } } ) u _ g . \end{align*}


\begin{align*} \chi _ { \Lambda } ( k ) = \chi \Big ( \frac { | k | } { \Lambda } \Big ) , \chi ( r ) = \begin{cases} 1 , & r < 1 / 2 , \\ 0 , & r > 1 , \end{cases} \chi \in [ 0 , 1 ] , \end{align*}


\begin{align*} \ell _ { R } \left ( R / { I _ { q } } \right ) = \sum _ { h = 1 } ^ { [ \mathbb { K } _ { \vartheta } : \mathbb { K } ] } \# ( \vartheta ^ { h } ( R ) \cap q H ) - \sum _ { h = 1 } ^ { [ \mathbb { K } _ { \vartheta } : \mathbb { K } ] } \# ( \vartheta ^ { h } ( I _ { q } ) \cap q H ) \end{align*}


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


\begin{align*} v ^ * = V _ 0 , \theta ^ * = \frac { d [ V , S ] ^ { p r e d } } { d [ S ] ^ { p r e d } } , \end{align*}


\begin{align*} \mathbb { E } _ 1 ( t _ 0 , t _ 1 ) & : = H ^ 1 \left ( t _ 0 , t _ 1 ; \mathbb { L } ^ 2 \right ) \cap L ^ 2 \left ( t _ 0 , t _ 1 ; \mathbb { H } ^ 2 \right ) , \\ \mathbb { E } _ 1 & : = \mathbb { E } _ 1 ( 0 , T ) . \end{align*}


\begin{align*} \sum _ { \Omega _ { n , \mathrm { k } } } = \sum _ { ( N _ 1 , \dots , N _ n ) \in \Omega _ { n , \mathrm { k } } } , \Sigma _ { \mathrm { k } } : = k _ 1 + \cdots + k _ \ell \qquad [ n ] _ { \neq \mathrm { k } } : = \{ 1 , \dots , n \} \setminus \{ k _ 1 , \dots , k _ \ell \} . \end{align*}


\begin{align*} | I _ 3 | < \lceil \frac { k _ { a + 1 } - | I _ 2 | - 1 } { 2 } \rceil + 1 . \end{align*}


\begin{align*} Q = \begin{pmatrix} 1 _ { \ell } & * \\ 0 & 1 _ { r - \ell } \end{pmatrix} \begin{pmatrix} A & 0 \\ 0 & D \end{pmatrix} \cdot k \end{align*}


\begin{align*} \hat \sigma _ D ( t _ 1 ) : = \sup _ { 0 < d ( z , \partial D ) \leq t _ 1 \atop z \in D } \rho _ { D } ^ { - 2 } ( z ) | S f ( z ) | \leq \sigma ( t _ 0 ) . \end{align*}


\begin{align*} I ( e \cdot , f ) = I _ { f , \psi } = \omega _ e + \sum _ { i = 1 } ^ { 6 } K _ e ^ { w _ i } \star \omega _ { w _ i } + K _ e ^ { w _ { G _ 4 } } \star \omega _ { w _ { G _ 4 } } , \end{align*}


\begin{align*} O _ { g ^ \ast } ^ { M _ \nu } ( f ^ \ast ) = e ( J _ b ) \cdot O _ { g } ^ { J _ b } ( f ) , \end{align*}


\begin{align*} D ^ + V ( \phi ( t , x , u ) ) = \dot { V } _ { u ( \cdot + t ) } ( \phi ( t , x , u ) ) \leq a V ( \phi ( t , x , u ) ) . \end{align*}


\begin{align*} \begin{cases} n S ^ { m } - T ^ { m } = S ^ { m - k } U ^ { k } , \\ n S ^ { m } + T ^ { m } = S ^ { m - k } V ^ { k } , \end{cases} k \le m , \begin{cases} n S ^ { k } - S ^ { k - m } T ^ { m } = U ^ { k } , \\ n S ^ { k } + S ^ { k - m } T ^ { m } = V ^ { k } , \end{cases} k \ge m , \end{align*}


\begin{align*} B = \sum ^ { n - r } _ { i = 1 } b _ i Y _ i ^ 2 , \end{align*}


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


\begin{align*} x \circ ( y + z ) = ( x \circ y ) - x + ( x \circ z ) . \end{align*}


\begin{align*} & \int _ { [ X _ { 0 , n , 0 } ] ^ { \rm v i r t } } c _ 1 ( \mathcal { L } _ 1 ) ^ { i _ 1 } { \rm e v } _ 1 ^ * ( \phi _ { \alpha _ 1 } ) \cdots c _ 1 ( \mathcal { L } _ n ) ^ { i _ n } { \rm e v } _ n ^ * ( \phi _ { \alpha _ n } ) = \int _ { \overline { \mathcal { M } } _ { 0 , n } } \psi _ 1 ^ { i _ 1 } \cdots \psi _ n ^ { i _ n } \int _ X \phi _ { \alpha _ 1 } \cdots \phi _ { \alpha _ n } . \end{align*}


\begin{align*} \Lambda _ p ' : = \int _ I \int _ \Omega \left ( \frac { \ell } { \ell ( s , x ) } \right ) ^ { \frac { n } { p - n } } \exp \biggl ( \frac { \ell p ^ 2 } { p - n } \| D _ x b ( s , x ) \| \biggr ) \dd x \dd s < \infty . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { \Gamma ( \alpha - n ) \Gamma ( \beta + n + 1 ) } { \Gamma ( \beta + 1 ) } = \frac { \Gamma ( 1 + \alpha ) } { 1 + \alpha + \beta } \ , , \end{align*}


\begin{align*} \langle m _ i , m _ j \rangle = \begin{cases} [ 2 ] , & \ ; j \equiv i \bmod d ; \\ z , & \ ; i \equiv 0 \equiv j - 1 \bmod d ; \\ z ^ { - 1 } , & \ ; i - 1 \equiv 0 \equiv j \bmod d ; \\ 1 , & \ ; i \equiv j \pm 1 \bmod d , \ ; ; \\ 0 , & . \end{cases} \end{align*}


\begin{align*} \zeta _ y ( x ) = \begin{cases} 1 , & | x | \leq \frac { 1 } { 4 } y , \\ 0 , & | x | \geq \frac { 1 } { 3 } y , \end{cases} \end{align*}


\begin{align*} \langle a _ i , a _ j ^ { \star } \rangle = \delta _ { i , j } , \ ; i , j = 1 , \ldots , 4 d , \end{align*}


\begin{align*} ( P , P ' ) ^ * = ( P ' , P ) \ ; . \end{align*}


\begin{align*} w _ 1 ( V _ p f ) = p \ , w _ 1 ( f ) . \end{align*}


\begin{align*} 2 T ^ { d } + U ^ { d } = V ^ { d } \end{align*}


\begin{align*} \dd _ p ( X ^ { \epsilon } ( t ) , X ( t ) ) & = \dd _ p \bigl ( \varphi ( \zeta ^ \epsilon ( t ) , x _ 0 ^ \epsilon ) , \varphi ( \beta ( t ) , x _ 0 ) \bigr ) \\ & \le \| \zeta ^ \epsilon ( t ) - \beta ( t ) \| _ p + \| e ^ { C | \zeta ^ \epsilon ( t ) | + C | \beta ( t ) | } \| _ p \dd ( x _ 0 ^ \epsilon , x _ 0 ^ 0 ) . \end{align*}


\begin{align*} c e ^ { - C y _ 0 ^ 2 } \leq 1 - u ( y _ 0 ) = \int _ { y _ 0 } ^ { + \infty } \partial _ { y } u d y \leq C e ^ { - \frac { b } { 2 } y _ 0 } \end{align*}


\begin{align*} ( L N R ) _ { i j } = \sum _ k L _ { i k } N _ { k ( n + 2 - k ) } R _ { ( n + 2 - k ) j } = \sum _ k \sum _ { \delta , \partial } L _ { i k } ( \delta ) R _ { ( n + 2 - k ) j } ( \partial ) . \end{align*}


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


\begin{align*} x _ j ^ T z ^ * _ j = x _ { j 0 } z _ { j 0 } + \bar x ^ T \bar z ^ * _ j = x _ { j 0 } z _ { j 0 } + \bar x ^ T \bar z ^ * _ j \stackrel { r _ j ( z _ j ^ * ) < 0 } { > } x _ { j 0 } \| \bar z _ j ^ * \| + \bar x ^ T \bar z ^ * _ j = 0 \end{align*}


\begin{align*} E _ s ( \omega ) = \left ( \frac { \sqrt { 3 } } { 2 } \right ) ^ { s / 2 } \cdot \frac { \zeta ( s ) L ( s , \chi _ { - 3 } ) } { \zeta ( 2 s ) } . \end{align*}


\begin{align*} D h ( x - T x ) = \ ( h _ { x _ 1 } ( x - T x ) , \cdots , h _ { x _ n } ( x - T x ) \ ) \end{align*}


\begin{align*} M ( A ) ( ( \vec { \alpha } _ i , \vec { \alpha } _ o ) ) = A _ P \cdot \det ( A _ M \rvert _ { \vec { \alpha } _ i , \vec { \alpha } _ o } ) \ ; . \end{align*}


\begin{align*} M = W \cup U _ { C } \cup U _ { \Gamma } . \end{align*}


\begin{align*} n = b 2 ^ { v + 1 } + 1 \end{align*}


\begin{align*} \nu _ \varepsilon \cdot \nabla \lambda = \frac { ( - 1 ) ^ { i + 1 } k _ d ^ { - 1 } } { \sqrt { 1 + \varepsilon ^ 2 | \bar { \tau } _ \varepsilon ^ i | ^ 2 } } \Bigl ( - \overline { V _ \Gamma } + \varepsilon ^ 2 \bar { g } _ i \bar { \tau } _ \varepsilon ^ i \cdot \nabla \overline { V _ \Gamma } \Bigr ) \quad \Gamma _ \varepsilon ^ i ( t ) \end{align*}


\begin{align*} \norm { v _ \varepsilon - v _ p } { L ^ 2 ( 0 , T ; L ^ 6 _ { l o c } ( \mathbb R ^ 2 \times 2 \mathbb T ) ) } + \norm { \xi _ \varepsilon } { L ^ 2 ( 0 , T ; L ^ 6 ( \mathbb R ^ 2 \times 2 \mathbb T ) ) } \rightarrow 0 ; \end{align*}


\begin{align*} \Bigl ( d ^ k \bar { \zeta } \Bigr ) ( x , t ) = d ( x , t ) ^ k \bar { \zeta } ( x , t ) , ( x , t ) \in \overline { Q _ { \varepsilon , T } } , \ , k = 0 , 1 , 2 , \end{align*}


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


\begin{align*} \delta _ { b _ 1 , \ldots , b _ m } \preceq \delta _ { c _ 1 , \ldots , c _ m } \Leftrightarrow b _ i \leq c _ i \textrm { f o r } i = 1 , \ldots , m . \end{align*}


\begin{align*} | g | = | g y _ x y _ x ^ { - 1 } | \leq | g y _ x | | y _ x ^ { - 1 } | \leq | g y _ x | C = C | g x k _ x | = C | g x | \ , . \end{align*}


\begin{align*} 2 \delta \geq w \partial _ \eta ^ 2 w = & w ^ { - 1 } ( \eta \partial _ \xi w + \partial _ \tau w ) \\ = & w ^ { - 1 } [ \eta ( \partial _ \xi w + \partial _ \tau w ) + ( 1 - \eta ) \partial _ \tau w ] i n D _ { T _ * } , \end{align*}


\begin{align*} \left [ \alpha ( x ) , [ y , z ] \right ] + \left [ \alpha ( y ) , [ z , x ] \right ] + \left [ \alpha ( z ) , [ x , y ] \right ] = 0 \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } \int _ I \int _ { A ^ \epsilon _ { ( t , s ) } } \| D _ x X ( t , s , x ) \| ^ p \dd x \dd s = \int _ I \int _ { \Omega _ { ( t , s ) } } \| D _ x X ( t , s , x ) \| ^ p \dd x \dd s . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n | h _ { k } | ^ 2 \lesssim n ^ { 1 - \frac { 1 } { \alpha } } , n \geq 1 . \end{align*}


\begin{gather*} x ^ { n + 1 } + { \displaystyle \sum \limits _ { k = 1 } ^ { \infty } } x \alpha _ { n , k } \left ( x \right ) z ^ { 2 k } = x ^ { n + 1 } + { \displaystyle \sum \limits _ { k = 1 } ^ { \infty } } \alpha _ { n + 1 , k } \left ( x \right ) z ^ { 2 k } \\ + x ^ { n - 1 } \gamma _ { n } \left ( z \right ) + \gamma _ { n } \left ( z \right ) \left ( { \displaystyle \sum \limits _ { k = 1 } ^ { \infty } } \alpha _ { n - 1 , k } \left ( x \right ) z ^ { 2 k } \right ) , \end{gather*}


\begin{align*} \mathcal R _ i ( M _ i ) : = \{ d _ { g _ i } ( p , \cdot ) | _ { \overline { \mathcal O _ i } } : p \in M _ i \} \ , i = 1 , 2 \end{align*}


\begin{align*} \alpha ( q , a ) & = \begin{cases} ( q ' , b ) & ( q , a , q ' , b ) \in \Delta \\ ( q ' , a ) & ( q , \pm 1 , q ' ) \in \Delta \end{cases} \\ \beta _ { + 1 } & = \prod _ { q ' \mid \exists q , ( q , + 1 , q ' ) \in \Delta } \rho _ { q ' } \\ \beta _ { - 1 } & = \prod _ { q ' \mid \exists q , ( q , - 1 , q ' ) \in \Delta } { \rho _ { q ' } } ^ { - 1 } \end{align*}


\begin{align*} ( - i \gamma ^ \mu \partial _ \mu + M ) \psi = V _ b * ( \psi ^ \dagger \gamma ^ 0 \Gamma \psi ) \Gamma \psi . \end{align*}


\begin{align*} \operatorname { I m } \varphi \left ( \mathrm { i } \alpha - \varepsilon \right ) = - \pi \delta \left ( \alpha \right ) \psi \left ( \mathrm { i } \alpha \right ) . \end{align*}


\begin{align*} \rho _ { \lambda , L } ( f ) = \frac { 1 } { | \Lambda _ L | } { \bf E } _ L { \rm t r } ( f ( H _ { \lambda , L } ) ) , \end{align*}


\begin{align*} T _ 2 - T _ 1 \ge 2 \quad \mbox { a n d } \quad \Bigl ( r ( t ) = 0 \ , \mbox { o r } s ( t ) = 0 \ , \mbox { f o r } \ , T _ 1 \le t \le T _ 2 \Bigr ) , \end{align*}


\begin{align*} \frac { d } { d t } \frac { K ' } { \partial Y _ { k } } - \frac { \partial K ' } { \partial y _ { k } } + \gamma _ { k i j } \frac { \partial K ' } { \partial Y _ { i } } Y _ { j } = F ' _ { k } \end{align*}


\begin{align*} \begin{gathered} \begin{multlined} \Phi _ \cdot ^ { a , b } ( i , j ) : \{ 0 , \ldots , a - 1 \} \times \{ 0 , \ldots , b - 1 \} \\ \rightarrow \{ 0 , \ldots , a b - 1 \} \end{multlined} \\ \Phi _ \cdot ^ { a , b } ( i , j ) \coloneqq b i + j \ ; , \\ \bar { \Phi } _ \cdot ^ { a , b } ( i ) \coloneqq ( \bar { \Phi } _ { \cdot 0 } ^ { a , b } ( i ) , \bar { \Phi } _ { \cdot 1 } ^ { a , b } ( i ) ) \coloneqq ( \Phi _ \cdot ^ { a , b } ) ^ { - 1 } ( i ) \ ; . \end{gathered} \end{align*}


\begin{align*} g = g _ M + T ^ \flat \otimes T ^ \flat + ( J T ) ^ \flat \otimes ( J T ) ^ \flat . \end{align*}


\begin{align*} J _ { 2 n } = \begin{pmatrix} & w _ n \\ - w _ n & \end{pmatrix} , \quad { \rm w i t h } w _ n = \begin{pmatrix} & & & 1 \\ & & 1 & \\ & \cdots & & \\ 1 & & & \end{pmatrix} _ { n \times n } . \end{align*}


\begin{align*} \Delta ( d \bar { \eta } ) = ( \mathrm { d i v } \ , \bar { \nu } ) \bar { \eta } + 2 \bar { \nu } \cdot \nabla \bar { \eta } + d \Delta \bar { \eta } = ( \mathrm { d i v } \ , \bar { \nu } ) \bar { \eta } + d \Delta \bar { \eta } \quad \overline { N } \end{align*}


\begin{align*} X _ n = \alpha ^ { ( n ) } ( \vec { W } ) \end{align*}


\begin{align*} J : = \int _ { \mathbb { R } ^ 3 } \mathrm { d } k \ , \frac { f _ y ( k ) } { \alpha ^ 2 + | k | } , \end{align*}


\begin{align*} \widetilde { B _ s } = \left ( \begin{matrix} 0 & 1 & 0 \\ - 1 & 0 & - 1 \\ 0 & 1 & 1 \end{matrix} \right ) \end{align*}


\begin{align*} \left \{ \int _ { 0 } ^ { 1 } g _ s d h _ s = 0 , \forall h \in \mathcal { H } \right \} \Rightarrow \left \{ g \equiv 0 \right \} , \end{align*}


\begin{align*} { } _ 2 F _ 1 ( 1 , 1 ; 2 + \epsilon ; 1 ) = \frac { 1 + \epsilon } { \epsilon } \end{align*}


\begin{align*} \langle e _ i , e _ j \rangle = \langle f _ i , f _ j \rangle = 0 , \langle e _ i , f _ j \rangle = \delta _ { i j } . \end{align*}


\begin{align*} f ( A _ 2 , \theta _ 2 , \omega _ 2 , S ) = d ^ { \nu } \cdot f ( A _ 1 , \theta _ 1 , \omega _ 1 , S ) . \end{align*}


\begin{align*} \sigma _ { A , j } : \{ 1 , \ldots , 2 n \} \to \{ 0 , 1 \} \sigma _ { A , j } ( l ) = \begin{cases} 1 : \max { a ( l ) } > j , \\ 0 : . \end{cases} \end{align*}


\begin{align*} E _ s ( \tau _ { \mathfrak { a } } ) = \frac { \# K ^ \times } { 4 } \left ( \frac { \sqrt { | - D | } } { 2 } \right ) ^ s \frac { \zeta _ K ( s , \mathfrak { a } ) } { \zeta ( 2 s ) } , \end{align*}


\begin{align*} \Re \frac { \zeta ' } { \zeta } \big ( { \textstyle \frac 1 2 } + i t \big ) = - \ , { \textstyle \frac 1 2 } \log \frac { t } { 2 \pi } + O ( 1 / t ) \end{align*}


\begin{align*} u ( x , t ) = 2 i \lim _ { k \rightarrow \infty } { k \psi _ j ( x , t , k ) } _ { 1 2 } , - \sigma u ( - x , - t ) = 2 i \lim _ { k \rightarrow \infty } { k \psi _ j ( x , t , k ) } _ { 2 1 } . \end{align*}


\begin{align*} H = O p _ w ( a ) + O p _ w ( \sigma ) , \end{align*}


\begin{align*} \big \langle T _ { \kappa _ N , p } ^ Y g , g \big \rangle _ { L ^ 2 ( Y ) } = \big \langle \kappa _ N \cdot B _ { k , p } ^ Y g , B _ { k , p } ^ Y g \big \rangle _ { L ^ 2 ( Y ) } . \end{align*}


\begin{align*} \Phi ( t , s ) = e ^ { \int _ { s } ^ { t } a ( \tau ) \ , d \tau } \leq K _ { 0 } e ^ { - ( \omega - c ) ( t - s ) } e ^ { 2 c s } , \textnormal { f o r $ t \geq s \geq 0 $ } . \end{align*}


\begin{align*} | \Gamma _ 3 ^ h ( u ) | + | \Gamma _ 4 ^ h ( u ) | = & \mathfrak { s } | \langle | I _ h z | ^ 2 I _ h z - | z | ^ 2 z , I _ h ( z u ) \rangle _ { L _ x ^ 2 } | + \mathfrak { s } | \langle | z | ^ 2 z , ( I _ h - I ) ( z u ) \rangle _ { L _ x ^ 2 } | \\ = & \mathfrak { s } | \langle I _ h ( | I _ h z | ^ 2 I _ h z - | z | ^ 2 z ) , z u \rangle _ { L _ x ^ 2 } | + \mathfrak { s } | \langle ( I _ h - I ) | z | ^ 2 z , z u \rangle _ { L _ x ^ 2 } | \end{align*}


\begin{align*} u ( x , 0 ) = \sin ( \varPi x ) , x \in [ 0 , 1 ] \end{align*}


\begin{align*} \lim _ { \theta \to 0 } \theta \overline { \mathbb { W } } ( \overline { \mathbf { P } } , \mu _ { V } ) + \overline { \rm E n t } [ \overline { \mathbf { P } } | \overline { \mathbf { \Pi } } ^ { 1 } ] = \mathcal { G } ( \overline { \mathbf { P } } ) \end{align*}


\begin{align*} n ( t , k ) ^ { 2 k } = \bigg ( \sum _ { t \le \gamma _ d \le t + 2 \pi k / \log T } m ( \gamma _ d ) \bigg ) ^ { \ ! 2 k } \ge \sum _ { t \le \gamma _ d \le t + 2 \pi k / \log T } m ( \gamma _ d ) ^ { 2 k } . \end{align*}


\begin{align*} & \mathrm { R e } \{ \mathrm { t r } ( \mathbf { W } \mathbf { D } _ { \mathbf { W } } ^ H ) \} < 0 , \\ & \nabla \boldsymbol { \mu } _ i \mathbf { d } _ { \boldsymbol { \omega } } = \mathbf { 0 } , \forall i \in \{ 1 , 2 , 3 , 4 \} , \end{align*}


\begin{align*} \begin{aligned} \bar { p } ' ( r ) & = \frac { 1 } { \pi r } \int _ { 0 } ^ { \pi } [ \omega _ { \theta } + u v _ { \theta } - ( v + b ) u _ { \theta } ] d \theta \\ & = \frac { 1 } { \pi r } \int _ { 0 } ^ { \pi } [ 2 u v _ { \theta } - ( v u _ { \theta } + u v _ { \theta } ) ] d \theta \\ & = \frac { 2 } { \pi r } \int _ { 0 } ^ { \pi } ( u - \bar { u } ) v _ { \theta } d \theta . \end{aligned} \end{align*}


\begin{align*} \varphi _ { s , a , b } ( t ) = e ^ { i s t } \varphi _ { s , a , t + b } ( 0 ) , \end{align*}


\begin{gather*} \gamma _ 0 z = z _ { | _ \Gamma } { \rm f o r ~ a l l ~ } z \in C ^ \infty ( \overline { \Omega } ) \cap V , \\ \gamma _ 1 z = \partial _ { \boldsymbol { \nu } } z { \rm f o r ~ a l l ~ } z \in C ^ \infty ( \overline { \Omega } ) \cap W . \end{gather*}


\begin{align*} B _ \sigma ( i ) \cdot B _ S ( \cdot , i ) = \prod _ j O ( i , j ) \cdot A _ \sigma ( j ) \cdot A _ S ( \cdot , j ) \ ; . \end{align*}


\begin{align*} w _ K = 3 2 1 4 8 7 6 5 9 . \end{align*}


\begin{align*} \Phi _ k & \coloneqq \begin{bmatrix} \phi _ { k - n } & \dots & \phi _ k \end{bmatrix} = \begin{bmatrix} x _ { k - n } & \dots & x _ k \\ u _ { k - n } & \dots & u _ k \end{bmatrix} , \\ U _ k & \coloneqq \begin{bmatrix} u _ { k - n + 1 } & u _ { k - n + 2 } & \dots & u _ { k + 1 } \\ u _ { k - n + 2 } & u _ { k - n + 3 } & \dots & u _ { k + 2 } \\ \vdots & \vdots & \ddots & \vdots \\ u _ { k - n + L - 1 } & u _ { k - n + L } & \dots & u _ { k + L - 1 } \end{bmatrix} . \end{align*}


\begin{align*} G _ p ^ X = \Big ( \begin{matrix} 0 & T _ p ^ X \\ 0 & 0 \end{matrix} \Big ) . \end{align*}


\begin{align*} F ( ( x ^ 0 , { O } \overline { x } ) , ( y ^ 0 , { O } \overline { y } ) ) = F ( ( x ^ 0 , \overline { x } ) , ( y ^ 0 , \overline { y } ) ) , \end{align*}


\begin{align*} f ( a _ N ) = ( b _ N - a _ N ) - ( b _ k - a _ k ) = \sum _ { j = k + 1 } ^ N \widetilde { f } ( a _ j ) \geq f ( a _ { k + 1 } ) . \end{align*}


\begin{align*} H ^ S = & h ^ { \emptyset } + \sum _ { 1 \leq | I | \leq m } h ^ I \int _ 0 ^ T \langle \epsilon _ { I ' } , \hat { \mathbb { S } } ( \boldsymbol { \ell } ) _ { s ^ - } \rangle d S ( \boldsymbol { \ell } ) _ s , \end{align*}


\begin{align*} \bigcup _ { j \in J _ A } \left ( a ( j ) \setminus \{ j \} \right ) = \{ 1 , \ldots , n \} \setminus J _ A = I _ A . \end{align*}


\begin{align*} p _ { i } = \frac { \partial L } { \partial q _ { i } } = \frac { \partial K } { \partial \dot { q } _ { i } } = \sum _ { j = 1 } ^ { f } M _ { i j } \dot { q } _ { j } \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \frac { h _ { R , l } ( z _ n , v _ n ' ) } { h _ { R , r } ( z _ n , v _ n ' ) } \ = \ \infty . \end{align*}


\begin{align*} \lim _ { i \to \infty } u _ i = - 1 ( \theta _ 2 , \theta _ 3 ) \times M , \end{align*}


\begin{align*} \sum _ { n _ 1 < j \leq n _ 2 } a _ j b ^ j = \sum _ { n _ 1 < j \leq n _ 2 } b ^ j \sum _ { \ell \geq j } c _ j = \sum _ { n _ 1 < \ell \leq n _ 2 } c _ \ell \sum _ { n _ 1 < j \leq \ell } b ^ j . \end{align*}


\begin{align*} & \partial _ \tau w _ B = 0 , \quad \displaystyle \lim _ { \eta \to 1 } w _ B \partial _ \eta ^ 2 w _ B = - \frac { 1 } { 2 } \frac { 1 } { x + x _ 0 } , \quad \partial _ \xi w _ B | _ { \eta = 0 } \leq - a _ 0 , \partial _ { \eta } ^ 2 w _ B | _ { \eta = 0 } = 0 , \\ & c ( 1 - \eta ) \leq w _ B \leq c ^ { - 1 } ( 1 - \eta ) \sqrt { - \ln ( \mu ( 1 - \eta ) ) } , \partial _ \xi w _ B < 0 i n [ 0 , X ] \times [ 0 , 1 ) , \end{align*}


\begin{align*} r ( E _ 3 , E _ 4 ) - \rho ^ \nabla _ M ( E _ 3 , E _ 4 ) & = - 2 g \left ( N ( E _ 3 , T ) , J T \right ) g \left ( N ( E _ 4 , T ) , T \right ) , \\ & = - \frac { 1 } { 8 } { \textbf { H } ^ \prime } ^ 2 . \end{align*}


\begin{align*} a _ 1 y _ 1 - b _ 3 y _ 2 = c _ 2 - d _ 2 . \end{align*}


\begin{align*} \epsilon : = \min \left \{ \eta : = \frac { k _ 0 } { 2 } , \frac { 1 } { 2 } \vert i \kappa + k _ 0 \vert \right \} , \end{align*}


\begin{align*} I _ N = \sum _ { d = 1 } ^ \infty z ^ d \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } s _ \lambda ( a _ 1 , \dots , a _ N ) s _ \lambda ( b _ 1 , \dots , b _ N ) \Omega _ { \frac { q } { N } } ^ { - 1 } ( \lambda ) \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) . \end{align*}


\begin{align*} y '' _ i = \left \{ \begin{array} { l l } x _ i & \mbox { i f } i \in I \setminus F , \\ y _ i & \mbox { i f } i \in F . \end{array} \right . \end{align*}


\begin{align*} \lim _ { a \searrow 0 } g ( a ) = 1 . \end{align*}


\begin{align*} \frac d { d t } \sum \limits _ { \beta = 0 } ^ { 1 } \sum \limits _ { | \alpha | = 0 } ^ { 3 - \beta } ( \| \sqrt { B } \partial _ \tau ^ \alpha \partial _ y ^ \beta u \| _ { L ^ 2 } ^ 2 & + \| \sqrt { A } \partial _ \tau ^ \alpha \partial _ y ^ \beta \tilde { h } \| _ { L ^ 2 } ^ 2 ) + C _ 0 D ( t ) \\ & \le C D ( t ) ^ { \frac 1 4 } E ( t ) ^ { \frac 5 4 } + C D ( t ) ^ { \frac 1 2 } E ( t ) + C D ( t ) E ( t ) ^ { \frac 1 4 } . \end{align*}


\begin{align*} X = A C \quad Y = D B , \end{align*}


\begin{align*} \Phi _ N = \max \Bigl \{ \tfrac { 1 + m + x _ N } { k + \ell m + \ell x _ N } : m , k , \ell \ \mathfrak { P } 1 \mathfrak { P } 2 A = A _ N \delta = \delta _ N \Bigr \} , \end{align*}


\begin{align*} P ^ { \ell , r - \ell } = \left \{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} : a \in S L _ \ell ( k ) , b = \ell \times ( r - \ell ) , d \in S L _ { r - \ell } ( k ) \right \} , \end{align*}


\begin{align*} \psi _ 1 ( x , t , k ) = \psi _ 2 ( x , t , k ) e ^ { - ( i k x + 4 i k ^ 3 t ) \sigma _ 3 } S ( k ) e ^ { ( i k x + 4 i k ^ 3 t ) \sigma _ 3 } , k \in \mathbb { R } \backslash \{ 0 \} . \end{align*}


\begin{align*} B = \{ Q ^ 2 - x _ 0 x _ 1 x _ 2 x _ 3 = 0 \} \subset \mathbb P ^ 3 \end{align*}


\begin{align*} ( x , t ) = \left ( \bar x _ n + r _ n y , \bar t _ n + r _ n ^ 2 s \right ) , ( x ' , t ' ) = \left ( \bar x _ n + r _ n y ' , \bar t _ n + r _ n ^ 2 s ' \right ) \end{align*}


\begin{align*} ( B _ { \langle f \rangle } ) _ { ( p ) } = B _ { ( ( p ) ) } . \end{align*}


\begin{align*} H ( G , x ) = \sum _ { i \geq 0 } d i s ( G , i ) x ^ i , \ \ i \leq d i a m ( G ) = 2 . \end{align*}


\begin{align*} ( y ^ { - 1 } g y ) ^ { i n } = y ^ { - 1 } g ^ { i n } y = y ^ { - 1 } x ^ { j n } y = y ^ { - 1 } y ^ { j n } y = y ^ { j n } = x ^ { j n } \in \langle x \rangle , \end{align*}


\begin{align*} u ( t ) = U ( t , s ) u ( s ) + j ^ { - 1 } \int _ s ^ t U ^ { \odot \star } ( t , \tau ) R _ { \delta , s } ( \tau , u ( \tau ) ) d \tau , - \infty < s \leq t < \infty , \end{align*}


\begin{gather*} s _ 3 = \begin{cases} V _ 0 W _ 2 = W _ 3 V _ 4 & ~ V _ 0 < W _ 2 ( \Longleftrightarrow W _ 3 < V _ 4 ) , \\ W _ 2 V _ 0 = V _ 4 W _ 3 & ~ W _ 2 < V _ 0 ( \Longleftrightarrow V _ 4 < W _ 3 ) \end{cases} \end{gather*}


\begin{align*} \omega : = \{ x \in \mathbb { R } ^ { d } | \zeta ( x ) = 0 \} . \end{align*}


\begin{align*} \rho \ , \overline { m } _ j = \begin{cases} \lambda \overline { m } _ { j + 1 } , & \ , j = 1 , \ldots , d - 2 ; \\ - \lambda \sum _ { k = 1 } ^ { d - 1 } ( - q ) ^ { k } \overline { m } _ { d - k } , & \ , j = d - 1 . \end{cases} \end{align*}


\begin{align*} \langle \check { \alpha } _ 1 , u ^ * \alpha _ 2 \rangle _ { A _ 1 } = \langle \check { u } ^ * \check { \alpha } _ 1 , \alpha _ 2 \rangle _ { A _ 2 } . \end{align*}


\begin{align*} \tau ( g ) = \tau ( x + i y ) = \begin{cases} 1 & y > a ' \\ 0 & y < a '' . \end{cases} \end{align*}


\begin{align*} M _ { k } [ y ] \succeq 0 , ~ L _ { c _ i } ^ { ( k ) } [ y ] = 0 ~ ( i \in \mathcal { E } ) , ~ L _ { c _ j } ^ { ( k ) } [ y ] \succeq 0 ~ ( j \in \mathcal { I } ) . \end{align*}


\begin{align*} 4 N ( X , T ) & = [ J X , J T ] - [ X , T ] - J [ X , J T ] - J [ J X , T ] , \\ & = \frac { J X ( \textbf { H } ) } { 2 } \left ( T - J T \right ) - \frac { X ( \textbf { H } ) } { 2 } \left ( T - J T \right ) - \frac { X ( \textbf { H } ) } { 2 } \left ( J T + T \right ) - \frac { J X ( \textbf { H } ) } { 2 } \left ( J T + T \right ) , \\ & = - X ( \textbf { H } ) T - J X ( \textbf { H } ) J T . \end{align*}


\begin{align*} L ^ { - 1 } = \begin{pmatrix} A _ { 1 1 } & I _ { d \times d } - A _ { 1 1 } \\ I _ { d \times d } & - I _ { d \times d } \end{pmatrix} , \end{align*}


\begin{align*} \mathbb { E } \big [ R _ { 2 , \infty } ^ { ( d ) } ( s , . , N ) \big ] = \displaystyle \int _ { [ 0 , 1 ) ^ d } R _ { 2 , \infty } ^ { ( d ) } ( s , \boldsymbol { \alpha } , N ) d \boldsymbol { \alpha } = ( 2 s ) ^ d \frac { N - 1 } { N } . \end{align*}


\begin{align*} & \breve { M } ^ r ( x , t , 0 ) = I + \frac { \mathfrak { B } ^ r ( \xi , t ) } { k _ 0 } + \frac { \overline { \mathfrak { B } ^ r ( \xi , t ) } } { k _ 0 } + R ( \xi , t ) , \\ & \breve { M } ^ r ( x , t , i \kappa ) = I + \frac { \mathfrak { B } ^ r ( \xi , t ) } { k _ 0 + i \kappa } + \frac { \overline { \mathfrak { B } ^ r ( \xi , t ) } } { k _ 0 - i \kappa } + R ( \xi , t ) . \end{align*}


\begin{align*} 3 m + 1 = 2 ^ e m \end{align*}


\begin{align*} - d u ( t , x ) = \frac 1 2 \Delta u d t + u ( t , x ) W ( d t , x ) , \ u ( T , x ) = \phi ( x ) \ , . \end{align*}


\begin{align*} \Phi ( X _ 1 , X _ 2 ) & = \pi _ p ( X _ 1 , X _ 2 ) + \frac { 1 } { p ! } \sum _ { \varepsilon \in \{ 1 , 2 \} ^ { p + 1 } } X _ { \varepsilon _ 1 } \cdots X _ { \varepsilon _ { p + 1 } } \int _ 0 ^ 1 \partial _ { \varepsilon _ 1 } \cdots \partial _ { \varepsilon _ { p + 1 } } \Phi ( t X _ 1 , t X _ 2 ) \ ( 1 - t ) ^ p \ d t \\ & = \pi _ p ( X _ 1 , X _ 2 ) + g _ { 0 , 0 } ( X _ 1 , X _ 2 ) . \end{align*}


\begin{align*} 1 _ p = p * p . \end{align*}


\begin{align*} H [ \phi _ n , k _ n ; \mu _ n ] : = H [ n ] , \end{align*}


\begin{align*} \psi ( x ) = \frac { \sigma ^ 2 } { \sqrt { \mu ^ 2 + 2 q \sigma ^ 2 } } \mathrm e ^ { - ( \mu / \sigma ^ 2 ) x } \sinh \left ( ( x / \sigma ^ 2 ) \sqrt { \mu ^ 2 + 2 q \sigma ^ 2 } \right ) . \end{align*}


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


\begin{align*} \Lambda ( z , t ) = ( \sqrt \mu \ , z , \mu t ) . \end{align*}


\begin{align*} & W _ { ( k ) } = \phi _ { ( k ) } k _ 1 , k \in \Lambda _ u \cup \Lambda _ B , \\ & D _ { ( k ) } = \phi _ { ( k ) } k _ 2 , k \in \Lambda _ B . \end{align*}


\begin{align*} \sigma _ \beta ( \beta E ) = C _ n \beta ^ { n - 1 } \sigma _ 1 ( E ) C _ n > 0 \ , . \end{align*}


\begin{align*} \Phi ( t ) = t \left ( 1 + \max \{ 0 , \log ^ n t \} \right ) , n \ge 1 , \end{align*}


\begin{align*} \hat { \beta } ^ \varepsilon = \beta ^ \varepsilon - ( \nu ^ \varepsilon \cdot \bar { \nu } ) \omega ^ \varepsilon - \varepsilon \bar { g } _ i ( \nu ^ \varepsilon \cdot \nabla \omega ^ \varepsilon ) \quad \Gamma _ \varepsilon ^ i ( t ) \end{align*}


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


\begin{align*} \left | j _ 1 \right | \geq \frac { 2 ( r - 2 ) ^ { 1 / \delta } } { C } N ^ { \beta / \delta } = : N _ 2 \ , . \end{align*}


\begin{align*} P \ , ( \widehat A - \lambda \widehat B ) \ , \widehat Q = \left [ \begin{array} { c c c } R ( \lambda ) & 0 & 0 \\ 0 & S ( \lambda ) & 0 \end{array} \right ] \quad \mbox { a n d } P U = \left [ \begin{array} { c } 0 \\ \widetilde U \end{array} \right ] . \end{align*}


\begin{align*} L ^ { \pm } ( u ) _ { b _ { 1 } \cdots b _ { k } } ^ { a _ { 1 } \cdots a _ { k } } = \sum _ { \sigma \in \mathfrak { S } _ { k } } s g n ( \sigma ) l _ { a _ { \sigma ( 1 ) } b _ { 1 } } ^ { \pm } ( u ) \cdots l _ { a _ { \sigma ( k ) } b _ { k } } ^ { \pm } ( u + ( k - 1 ) h ) \end{align*}


\begin{align*} g _ i ( y _ i ) \coloneqq \prod _ { k = 0 } ^ \infty \left ( 1 - a _ k ^ { ( i ) } y _ i \right ) , h _ i ( y _ i ) \coloneqq \sum _ { k = 0 } ^ \infty \frac { a _ k ^ { ( i ) } } { 1 - a _ k ^ { ( i ) } y _ i } \quad \ ; ( 1 \leq i \leq s ) . \end{align*}


\begin{align*} { \lim \sup } _ { x \to \infty } \frac { \log \mu ( [ 0 , \ , \varphi ( x ) ] ) } { x } = A \end{align*}


\begin{align*} \psi _ { \sigma } \left ( x , t , \mathrm { i } s \right ) = y \left ( s , x , t \right ) ; \end{align*}


\begin{align*} y ^ 2 + \nu z ^ 2 = \beta ^ 2 + \nu \gamma ^ 2 , \end{align*}


\begin{align*} X _ i ^ \odot ( s ) = X _ i ^ \star ( s ) \cap X ^ \odot . \end{align*}


\begin{align*} K = a ^ 2 + b ^ 2 + c ^ 2 - 2 a b c - 1 , \end{align*}


\begin{align*} H _ N ( \alpha , \beta ) = \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } \frac { ( \dim \mathsf { V } ^ \lambda ) ^ 2 } { d ! } \omega _ \alpha ( \lambda ) \Omega _ { \frac { q } { N } } ^ { - 1 } ( \lambda ) \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) \omega _ \beta ( \lambda ) \end{align*}


\begin{align*} f ( u , v ) - f ( z , w ) = 0 \end{align*}


\begin{align*} \begin{aligned} & C \left ( \mathbf { p } , \mathbf { q } \right ) ( \tau ) + C \left ( \mathbf { u } , \mathbf { v } \right ) ( \tau ) \\ = & \sum _ { i = 0 } ^ { N - \tau } \left ( ( - 1 ) ^ { g _ i - g _ j } \left ( ( - 1 ) ^ { - j _ { \pi ( 0 ) } - j _ { \pi ( m - 3 ) } } + ( - 1 ) ^ { i _ { \pi ( m - 3 ) } - j _ { \pi ( 0 ) } } \right ) \right ) , \end{aligned} \end{align*}


\begin{align*} \gamma ( x ) = x \times \cdot = \left [ { \begin{array} { c c c } 0 & - x _ 3 & x _ 2 \\ x _ 3 & 0 & - x _ 1 \\ - x _ 2 & x _ 1 & 0 \\ \end{array} } \right ] \end{align*}


\begin{align*} u ' _ k u _ k = \displaystyle \prod _ { h ( \alpha ) = k } u _ { t ( \alpha ) } + \displaystyle \prod _ { t ( \alpha ) = k } u _ { h ( \alpha ) } , \end{align*}


\begin{align*} \psi _ j \circ \Phi = \Phi ^ + _ j + \sum _ { m = 1 } ^ M a _ { j , m } \cdot ( \varphi _ { j , m } \circ \Pi ) + \sum _ { m = 1 } ^ M b _ { j , m } \cdot ( \delta \circ \varphi _ { j , m } \circ \Pi ) , \end{align*}


\begin{align*} \hat { H } = \sum _ { i , j } H ( i , j ) c _ i ^ \dagger c _ j \ ; . \end{align*}


\begin{align*} \gamma . [ C ] = [ H ] . [ C ] > 0 . \end{align*}


\begin{align*} \lim _ { | \xi | \to \infty } \frac { \psi ( \xi ) } { | \xi | ^ { \alpha } } = 1 , \end{align*}


\begin{align*} X _ k = h _ k ( ( Y _ j ) _ { j \in \tilde { J } _ k } ) \end{align*}


\begin{align*} \left | \log \left ( \frac { \frac { z f _ 1 ' ( z ) } { f _ 1 ( z ) } } { 2 - \frac { z f _ 1 ' ( z ) } { f _ 1 ( z ) } } \right ) \right | = \left | \log \left ( \frac { 1 + \frac { e - 1 } { e + 1 } } { 2 - \left ( 1 + \frac { e - 1 } { e + 1 } \right ) } \right ) \right | = \left | \log ( e ) \right | = 1 . \end{align*}


\begin{align*} \Phi _ n = \varphi _ 1 \land y \leq z \ , \& \dots \& \ , \varphi _ { n + 1 } \land y \leq z \Longrightarrow y \leq z , \end{align*}


\begin{align*} \Pi _ p ( x , y ) \ ; = \ ; 0 \end{align*}


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


\begin{align*} f ( u ) = u ( u - 1 ) ^ 2 \cdots ( u - N ) ^ 2 \end{align*}


\begin{align*} \int _ { v _ 0 \in \mathbb { S } ^ { d - 1 } } v _ { 0 , i } v _ { 0 , j } d S = 0 . \end{align*}


\begin{align*} \frac { \partial } { \partial r } \Big ( \frac { \partial X } { \partial \mu } \Big ) & = A ' _ X ( X ) \frac { \partial X } { \partial \mu } , \frac { \partial X } { \partial \mu } ( 0 , \mu ) = ( X _ 0 ) ' _ \mu ( \mu ) , \\ \frac { \partial } { \partial r } \Big ( \frac { \partial Y } { \partial \mu } \Big ) & = B ' _ Y ( Y ) \frac { \partial Y } { \partial \mu } , \frac { \partial Y } { \partial \mu } ( 0 , \mu ) = ( Y _ 0 ) ' _ \mu ( \mu ) . \end{align*}


\begin{align*} \left \lbrace \begin{aligned} d \eta _ q ^ { \epsilon , { \bf h } } ( t ) & = \frac { \eta _ p ^ { \epsilon , { \bf h } } ( t ) } { \epsilon } d t \\ d \eta _ p ^ { \epsilon , { \bf h } } ( t ) & = - \frac { \eta _ p ^ { \epsilon , { \bf h } } ( t ) } { \epsilon ^ 2 } d t + \frac { 1 } { \epsilon } D f ( q ^ \epsilon ( t ) ) . \eta _ q ^ { \epsilon , { \bf h } } ( t ) d t + \frac { 1 } { \epsilon } D \sigma ( q ^ \epsilon ( t ) ) . \eta _ q ^ { \epsilon , { \bf h } } ( t ) d \beta ( t ) , \end{aligned} \right . \end{align*}


\begin{align*} \phi ' ( s ) = \frac { \Theta ( s ) ^ { - \frac { \alpha - 1 } { \alpha } } } { s ^ 2 } \left ( - \frac { \alpha - 1 } { \alpha } \frac { \Theta ' ( s ) } { s \Theta ( s ) } - 1 \right ) < 0 , \end{align*}


\begin{align*} P ^ * \left ( ( \Omega _ { i j } \times \phi _ { t _ i , \nu } ^ { - 1 } ( B _ { i j } ^ { t _ i , \nu } ) ) \triangle ( \Omega _ { i j } \times B _ { i j } ) \right ) < \frac { \varepsilon } { 2 s ^ 2 m } \end{align*}


\begin{align*} \big [ \sigma ^ { 2 } ( \alpha ) \big ] ^ { p ^ { 2 } } & = \big [ \sigma ( \sigma ( \alpha ) ) \big ] ^ { p ^ { 2 } } = \big [ \sigma \big ( \sigma ( \alpha ) \big ) ^ { p } \big ] ^ { p } = \big [ \sigma \big ( \sigma ( \alpha ) ^ { p } \big ) \big ] ^ { p } \\ & = \big [ \sigma \big ( \alpha ^ { q } \big ) \big ] ^ { p } = \big [ \sigma ( \alpha ) \big ] ^ { p q } = \big [ \sigma ( \alpha ) ^ { p } \big ] ^ { q } = \alpha ^ { q ^ { 2 } } . \end{align*}


\begin{align*} \L _ T h = \frac { h } { | \det ( D T ) | } \circ T ^ { - 1 } . \end{align*}


\begin{align*} T ^ 2 \log ^ 4 T \le \bigg ( \sum _ { \substack { T < \gamma \le 2 T \\ Z ' ( \gamma ) \ , Z ( \gamma + \frac { 2 \pi \kappa } { \log T } ) < 0 } } \ ! \ ! \ ! \ ! 1 \bigg ) \bigg ( \sum _ { T < \gamma \le 2 T } Z ' ( \gamma ) ^ 2 \ , Z \Big ( \gamma + \frac { 2 \pi \kappa } { \log T } \Big ) ^ 2 \ , \Big | M \Big ( \frac 1 2 + i \gamma + \frac { 2 \pi i \eta } { \log T } , P \Big ) \Big | ^ 4 \bigg ) , \end{align*}


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


\begin{align*} \mathbf { P r } \left ( \left | \sum _ { i = 1 } ^ { n } a _ i X _ i \right | \geq t \right ) \leq 2 \exp \left ( - \frac { c t ^ 2 } { K ^ 2 \| a \| ^ 2 } \right ) , \end{align*}


\begin{align*} \chi _ { w \cdot \mu + \nu } ( x ) = w \cdot \mu ( \pi _ R ( x ) ) + \nu ( \pi _ R ( x ) ) = w \cdot \mu ( \pi _ R ( x ) ) - \mu ( \pi _ R ( x ) ) = \chi _ { w \cdot \mu - \mu } ( x ) = 0 . \end{align*}


\begin{align*} X _ t = x + \int _ 0 ^ t V _ 0 ( X _ s ) \ , d s + \sum _ { i = 1 } ^ d \int _ 0 ^ t V _ i ( X _ s ) \ , d B _ s ^ i , \end{align*}


\begin{align*} Z _ n ( \lambda , T ) = \frac { 1 } { 2 } \sum _ { \theta \in \{ 0 , 1 \} ^ 2 } \prod _ { w ^ n = ( - 1 ) ^ { \theta _ 2 } } \left | e ^ { T w } - ( - 1 ) ^ { \theta _ 1 } e ^ { - \lambda } \right | . \end{align*}


\begin{align*} \begin{tabular} [ c ] { l } $ D _ { n } = \phi ^ { 2 } C _ { n } \partial _ { x } ^ { 2 } - 2 x \phi \left [ \left ( \phi - 1 \right ) C _ { n } + \phi \right ] \partial _ { x } $ \\ $ + \left ( n - 2 \gamma _ { n } \right ) \left [ 2 x ^ { 2 } \phi - \left ( \phi - 2 x ^ { 2 } \phi + n x ^ { 2 } - 2 x ^ { 2 } \gamma _ { n } \right ) C _ { n } \right ] + 4 \gamma _ { n } C _ { n - 1 } C _ { n } ^ { 2 } . $ \end{tabular} \ \end{align*}


\begin{align*} \left \{ \begin{aligned} & J ( S ) = 0 ( \tau , \xi , \eta ) \in D , \\ & \partial _ \eta S \mid _ { \eta = 0 } = 0 , \displaystyle \lim _ { \eta \to 1 } S = 0 . \end{aligned} \right . \end{align*}


\begin{align*} \begin{aligned} \sum _ { h = 0 } ^ { n - 1 } \| f ( s _ { h + 1 } ) - f ( s _ h ) \| _ E ^ q & \lesssim \sum _ { N } N ^ \epsilon \sum _ { h = 0 } ^ { n - 1 } \sum _ { I \in J _ h ^ N } \| f ( \max I ) - f ( \min I ) \| _ E ^ q \\ & \leq \sum _ { N } N ^ \epsilon \sum _ { k = 0 } ^ { N - 1 } \| f ( t _ { k + 1 } ^ N ) - f ( t _ k ^ N ) \| _ E ^ q . \end{aligned} \end{align*}


\begin{align*} d ( g ) t ( g ^ { - 1 } h ) + t ( g h ) = t ( g ) t ( h ) . \end{align*}


\begin{align*} P \bigg ( \sup _ { t \in I _ n } \{ | B _ t - B _ { s _ { n - 1 } } | \} \le r ( s _ n ) \bigg ) \ge c \exp ( - \log _ 2 s _ n ) = \frac { c } { n \log n } \ , . \end{align*}


\begin{align*} X _ t = x + \mu t + \sigma W _ t , \end{align*}


\begin{align*} \beta ( H ) = \sup _ { \mu } \beta ( \mu ; H ) \ ; , \end{align*}


\begin{align*} f ( a ) = \lim _ { k \rightarrow \infty } f _ k ( a ) = \lim _ { k \rightarrow \infty } \frac { 1 } { | \Omega | } \int _ \Omega { f _ k } = \frac { 1 } { | \Omega | } \int _ \Omega { f } = \frac { 1 } { | \Omega _ 1 | } \int _ { \Omega _ 1 } f , \end{align*}


\begin{align*} \breve { J } ^ { r } ( x , t , k ) \begin{pmatrix} 1 & 0 \\ 0 & \frac { k - i \kappa } { k } \end{pmatrix} \breve { J } ( x , t , k ) \begin{pmatrix} 1 & 0 \\ 0 & \frac { k } { k - i \kappa } \end{pmatrix} . \end{align*}


\begin{align*} u = \frac { 3 6 N - y } { 6 x } , v = \frac { 3 6 N + y } { 6 x } . \end{align*}


\begin{align*} R ( x ) = \Big ( \frac { \sin \pi x } { \pi x } \Big ) ^ 2 \frac { 1 } { 1 - x ^ 2 } \end{align*}


\begin{align*} \gamma _ { o i r 2 } ( C _ { p } ) = \lfloor p / 2 \rfloor + \lceil p / 4 \rceil - \lfloor p / 4 \rfloor \end{align*}


\begin{align*} D _ { \sigma , k - 1 } ( h ) & = r ^ 2 D _ { \sigma , k - 1 } \left ( \frac { ( z - \lambda ) ( g ( z ) - g ( r \lambda ) ) } { z - r \lambda } \right ) \\ & \leqslant \frac { 4 r ^ 2 } { ( 1 + r ) ^ 2 } D _ { \sigma , k - 1 } ( g ( z ) - g ( r \lambda ) ) \\ & = \frac { 4 r ^ 2 } { ( 1 + r ) ^ 2 } D _ { \sigma , k - 1 } ( g ) . \end{align*}


\begin{align*} e ^ { 2 \pi i B _ { C } ( m + a , b ) } f _ { - \frac { 1 } { N \tau } } ( m + a ) & = P _ { k - 1 } ( m + a ) e ^ { 2 \pi i \left ( B _ { C } ( m + a , b ) - \frac { Q _ { C } ( m + a ) } { N \tau } \right ) } \\ & = e ^ { 2 \pi i \left ( B _ { C } ( a , b ) - \frac { Q _ { C } ( a ) } { N \tau } \right ) } P _ { k - 1 } ( m + a ) e ^ { 2 \pi i \left ( B _ { C } \left ( m , \frac { b N \tau - a } { N \tau } \right ) - \frac { Q _ { C } ( m ) } { N \tau } \right ) } . \end{align*}


\begin{align*} \eta _ n ^ { ( a ) } = G ^ { - 1 } \Phi _ { n , a } \end{align*}


\begin{align*} \begin{aligned} & X ( e _ i ) X ' ( e _ i ) \\ & = v ^ { \varLambda ( e _ i , \sum \limits _ { 1 \leq l \leq m } [ b _ { l i } ] _ + e _ l ) } X ( { \sum \limits _ { 1 \leq l \leq m } [ b _ { l i } ] _ + e _ l } ) + v ^ { \varLambda ( e _ i , \sum \limits _ { 1 \leq l \leq m } [ - b _ { l i } ] _ + e _ l ) } X ( { \sum \limits _ { 1 \leq l \leq m } [ - b _ { l i } ] _ + e _ l } ) . \end{aligned} \end{align*}


\begin{align*} ( - \Delta _ x + V _ \mathrm { t r a p } ( x ) ) \phi ( x ) + ( \upsilon _ N \ast | \phi | ^ 2 ) ( x ) \phi ( x ) = \mu \phi ( x ) . \end{align*}


\begin{align*} \Delta : = \{ ( x , x ) \ , : \ , x \in X \} \subseteq U \end{align*}


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


\begin{align*} I _ h = \iint _ D | \nabla ( \Delta ^ h _ e u ) | ^ 2 \Psi ^ 2 \ , d X \leq C _ { E , u } , \end{align*}


\begin{align*} 2 \langle \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) , \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \rangle \cos 6 \theta = \left ( \Vert \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) \Vert ^ 2 - \Vert \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \Vert ^ 2 \right ) \sin 6 \theta . \end{align*}


\begin{align*} ( \tilde { H } _ b - i \sqrt { \lambda } ) S _ b f _ n = f _ n + T _ b f _ n , \end{align*}


\begin{align*} g = S + e ^ { - \beta _ 0 \tau } M ( 1 - \eta ) ^ { \alpha _ 0 } + \mu _ 1 \tau , \end{align*}


\begin{align*} 1 + r _ 1 ( k ) r _ 2 ( k ) = \frac { 1 } { a _ 1 ( k ) a _ 2 ( k ) } , k \in \mathbb { R } \backslash \{ 0 \} . \end{align*}


\begin{align*} A ' = \begin{pmatrix} R & S & T \\ U & W & X \\ V & Y & Z \end{pmatrix} \ ; . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ 2 \alpha _ i \ , \widehat { \Pi ^ { - 1 } S _ i } \end{align*}


\begin{align*} \tilde { F } _ { \kappa } ( t ) : = - t \arctan \left ( \tfrac { t } { \kappa } \right ) + \tfrac 1 2 \kappa \log \left ( 1 + t ^ 2 / \kappa ^ 2 \right ) . \end{align*}


\begin{align*} u ( x , 1 ) = e x p \left ( \frac { 1 } { 8 \lambda } \right ) , 0 \leq x \leq 1 , \end{align*}


\begin{align*} \begin{array} { l } \underset { \epsilon \rightarrow 0 } { \lim } \ , \underset { \eta \rightarrow 0 } { \lim } \ , \left \vert \sum _ { k = 1 } ^ { n } \left [ \breve { F } _ { \epsilon } ( y _ { k } ) - \breve { F } _ { \epsilon } ( y _ { k - 1 } ) \right ] - \sum _ { k = 1 } ^ { n } y _ { k } ^ { \prime } \left [ F _ { \epsilon } ( y _ { k } ) - F _ { \epsilon } ( y _ { k - 1 } ) \right ] \right \vert = 0 . \end{array} \end{align*}


\begin{align*} 0 = \widehat { Q } _ 0 ^ 1 + Q '^ 1 _ 1 \circ F ^ 1 - F ^ 1 \circ Q + \sum _ { n = 2 } ^ \infty \frac { 1 } { n ! } Q '^ 1 _ n \circ ( F ^ 1 ) ^ { \star n } . \end{align*}


\begin{align*} \int _ { \real ^ { 2 ( n - m ) } } | z _ N ^ { 2 \beta } | \cdot \exp ( - \pi | Z _ N | ^ 2 ) \cdot d Z _ { 2 m + 1 } \wedge \cdots \wedge d Z _ { 2 n } = \frac { \beta ! } { \pi ^ k } . \end{align*}


\begin{align*} \begin{aligned} \rho ^ \varepsilon ( x , t ) & = \sum _ { k = 0 } ^ \infty \varepsilon ^ k \eta _ k ( \pi ( x , t ) , t , \varepsilon ^ { - 1 } d ( x , t ) ) , \\ f ^ \varepsilon ( x , t ) & = f ( \pi ( x , t ) , t ) + O ( \varepsilon ) , \end{aligned} \end{align*}


\begin{align*} H _ { \infty } ^ { } = ( P \otimes \mathbb { 1 } - \alpha ^ { 1 / 2 } A _ { \infty } ( x ) ) ^ 2 + \mathbb { 1 } \otimes H _ f - \frac { \alpha } { | x | } \otimes \mathbb { 1 } , \end{align*}


\begin{align*} S ( v ) ( x ) : = \bigg ( \iint _ { \widehat { \Gamma } _ a ( x ) } | \nabla v ( y , s ) | ^ 2 \ , \frac { d y d s } { | s | ^ { n - 2 } } \bigg ) ^ { \frac { 1 } { 2 } } , \end{align*}


\begin{align*} \log ( k n + k ) = \log ( n + 1 ) + \log k < \log n + \log k + \frac { 1 } { n } , \end{align*}


\begin{align*} r _ 0 & : = \frac { 4 \pi } { T ( a _ 0 ) } , \\ R _ 0 & : = \operatorname { i d } \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \pi _ n ( x ) } { \pi _ n ( y ) } \ ; = \ ; a \in ( 0 , \infty ) \ ; . \end{align*}


\begin{align*} \int _ { \mathbb { \mathbb { R } } ^ { N } } \Psi ( x ) d x = \int _ { \mathbb { \mathbb { R } } ^ { N } } \Pi ( x ) d x = 1 . \end{align*}


\begin{align*} F _ { > s } ( x ) : = \begin{dcases*} \exp \bigg \{ \sum _ { k > s } c _ k x ^ k \bigg \} , & $ \mathsf { F } ^ { ( n ) } = \mathsf { S } ^ { ( n ) } $ \\ \exp \bigg \{ \sum _ { j \ge 1 } \sum _ { k > s } c _ k x ^ { j k } / j \bigg \} , & $ \mathsf { F } ^ { ( n ) } = \mathsf { G } ^ { ( n ) } $ \end{dcases*} . \end{align*}


\begin{align*} \forall v , w , u \in V , \ ; \ ; \left ( m \circ \left ( \dot { + } \otimes \dot { + } \right ) \right ) ( v \otimes u \otimes u \otimes w ) = \delta _ { v \dot { + } u , u \dot { + } w } ( v \dot { + } u ) = \dot { + } ( u \otimes \delta _ { v , w } v ) \end{align*}


\begin{align*} \Phi _ \cdot ^ \infty = ( \Phi _ { \cdot 0 } ^ \infty , \Phi _ { \cdot 1 } ^ \infty ) : \mathbb { N } _ 0 \rightarrow \mathbb { N } _ 0 ^ 2 \ ; , \end{align*}


\begin{align*} \varphi = \begin{pmatrix} u ^ { p - 1 } & v \\ v & - u \end{pmatrix} \psi = \begin{pmatrix} u & v \\ v & - u ^ { p - 1 } \end{pmatrix} \end{align*}


\begin{align*} V ( \ell ) = 0 \forall \ell : \ ; \pi ( \ell ) = 0 \ , , \mbox { o r } \sum _ { i = 1 } ^ d \ell _ i \ne 0 \ , . \end{align*}


\begin{align*} \int _ Y y \cdot t _ * ( s ^ * x \cdot \pi ^ * w \cdot \mu ) = \int _ X x \cdot s _ * ( t ^ * y \cdot \pi ^ * w \cdot \mu ) . \end{align*}


\begin{align*} \frac { z } { 2 } \gamma _ { n } ^ { \prime } \left ( z \right ) = { \displaystyle \sum \limits _ { k = 1 } ^ { \infty } } k \eta _ { n , k } z ^ { 2 k } , \end{align*}


\begin{align*} \Delta _ { ( I - \tilde \alpha ) k _ 3 } \Delta _ { 2 k _ 2 } \Delta _ { ( I + \tilde \alpha ) k _ 1 } P ( u ) = 0 , \ \ u \in Y . \end{align*}


\begin{align*} g ( y ) = f ( 2 ( I - \tilde \alpha ) ^ { - 1 } y ) , \ \ y \in Y , \end{align*}


\begin{align*} Q ( s ) = - \beta ^ 2 + ( 2 \nu \alpha - \alpha ^ 2 - \beta ^ 2 ) s + ( 1 - 2 \alpha + 2 \nu ) s ^ 2 . \end{align*}


\begin{align*} | - \partial _ s w _ n - \Delta w _ n | = | - \theta _ n h ( \bar x _ n + r _ n y , \bar t _ n + r _ n ^ 2 s ) | D w _ n | ^ \gamma + g _ n | \stackrel { \eqref { t h e t a b o u n d } } { \le } h _ 1 K ^ { \gamma - 1 } | D w _ n | ^ \gamma + 1 , \end{align*}


\begin{align*} 2 h ^ { \mu _ { \theta } } + V + \frac { 1 } { \theta } \log \mu _ { \theta } = c , \end{align*}


\begin{align*} - \Delta _ { p _ i } u _ i = f _ i ( x , u _ 1 , u _ 2 , \nabla u _ 1 , \nabla u _ 2 ) \quad \Omega , u _ i = 0 \quad \partial \Omega , \end{align*}


\begin{align*} \begin{pmatrix} H _ 0 \\ H _ 1 \end{pmatrix} Z \begin{pmatrix} G _ 0 & G _ 1 \end{pmatrix} = \begin{pmatrix} S & 0 \\ 0 & 0 \end{pmatrix} \end{align*}


\begin{align*} \tilde { D } ( x ) = D ( x ) - \frac { 1 } { 1 + 2 x } A ( x ) \ , , \end{align*}


\begin{align*} | \nabla _ s ( \theta ^ A u ( \theta ^ B x ) ) | _ 2 ^ 2 = \int _ { \R ^ d \times \R ^ d } \frac { \theta ^ { 2 A } | u ( \theta ^ B x ) - u ( \theta ^ B y ) | ^ 2 } { | x - y | ^ { d + 2 s } } d x d y = \theta ^ { 2 A + B ( 2 s - d ) } | \nabla _ s u | _ 2 ^ 2 , \end{align*}


\begin{align*} n ( t , \lambda ) = N \Big ( t + \frac { 2 \pi \lambda } { \log T } \Big ) - N ( t ) . \end{align*}


\begin{align*} \Lambda _ { p , h } ' : = \int _ I \int _ \Omega ( \ell _ h ( s , x ) ) ^ { n / ( n - p ) } \exp \biggl ( \frac { \ell p ^ 2 } { p - n } \| D _ x b _ h ( s , x ) \| \biggr ) \dd x \dd s \ , , \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty h _ { k } ( x ) W ( x ) \mathrm { d } x = \sum _ { x \in X _ n } \omega ( x ) h _ { k } ( x ) , k \leq 2 n - 1 . \end{align*}


\begin{align*} [ 2 c ] = \frac { q ^ 2 t x ^ 3 } { 1 - q x } . \end{align*}


\begin{align*} \left \{ \begin{aligned} & - \Delta \psi = \hat { \phi } & & \mbox { i n } \Omega \\ & \psi = 0 & & \mbox { o n } \partial \Omega \end{aligned} \right . \ , \ , \mbox { a n d } \ , \ , \left \{ \begin{aligned} & - \Delta \chi = 1 & & \mbox { i n } \Omega \\ & \chi = 0 & & \mbox { o n } \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} 2 \left ( n ( i + q ) - m \left ( j + \frac q 2 \right ) \right ) d ^ 1 _ { r , s } ( m + n , i + j ) & = \left ( n ( i + s + q ) - ( m + r ) \left ( j + \frac q 2 \right ) \right ) d ^ 0 _ { r , s } ( m , i ) \\ & \quad + \left ( ( n + r ) ( i + q ) - m \left ( j + s + \frac q 2 \right ) \right ) d ^ 1 _ { r , s } ( n , j ) , \\ 2 q d ^ 0 _ { r , s } ( m + n , i + j ) & = q ( d ^ 1 _ { r , s } ( m , i ) + d ^ 1 _ { r , s } ( n , j ) ) . \end{align*}


\begin{align*} \hat x _ \tau ( T , \hat y , \hat \eta _ \tau ) = x _ T \ \ \mbox { f o r } \ \ 0 \leq \tau \leq 1 . \end{align*}


\begin{align*} \mathfrak { B } ^ r ( \xi , t ) = - \frac { \mathfrak { B _ 0 } } { \sqrt { \tau } } = \begin{pmatrix} 0 & - i \eta \beta ^ r ( \xi ) e ^ { - t \varphi ( \xi , 0 ) } \tau ^ { - i \nu - \frac { 1 } { 2 } } \\ i \eta \gamma ^ r ( \xi ) e ^ { t \varphi ( \xi , 0 ) } \tau ^ { i \nu - \frac { 1 } { 2 } } & 0 \end{pmatrix} \end{align*}


\begin{align*} \dot { x } = \frac { 1 } { 1 + | u ( t ) | } x . \end{align*}


\begin{align*} L \left [ 2 x \left ( \phi - 1 \right ) P _ { n + 1 } P _ { n } \right ] & = \left [ \gamma _ { n } + \gamma _ { n + 1 } + \gamma _ { n + 2 } - \left ( z ^ { 2 } + 1 \right ) \right ] h _ { n + 1 } , \\ L \left [ 2 x \left ( \phi - 1 \right ) P _ { n + 1 } P _ { n - 2 } \right ] & = h _ { n + 1 } , \end{align*}


\begin{align*} \sum _ { z \in \Z } \sum _ { s = 1 } ^ t | \Delta ( x - z , t - s ) | ^ 2 \lesssim 1 . \end{align*}


\begin{align*} J ( u _ 0 ; U _ 0 ) = \left \| g ^ { T } ( U _ 0 + u _ 0 ) - g ^ { T } ( U _ 0 ) \right \| ^ 2 , \footnote { T h r o u g h o u t t h e p a p e r , t h e n o r m $ \| \cdot \| $ i s s p e c i f i c f o r t h e e n e r g y n o r m . W h e n w e i m p l e m e n t t h e n u m e r i c a l c o m p u t a t i o n o r s a y i n t h e d i s c r e t e c a s e , i t r e d u c e s t o t h e E u c l i d e a n n o r m a s \begin{align*} \| v \| = \sqrt { \sum _ { i = 1 } ^ { d } v _ i ^ 2 } . \end{align*} } \end{align*}


\begin{align*} S = \left \{ p \in \R ^ n : \right \} \end{align*}


\begin{align*} 1 = \sum _ { \ell = 1 } ^ j G ( x _ i , x _ \ell ) P ^ { x _ \ell } ( \tau _ X = \infty ) , \forall 1 \le i \le j \ , . \end{align*}


\begin{gather*} d ^ 3 \gamma ' = d ^ 3 \gamma - \frac { 1 } { 2 } d ^ 3 B ( \tau \wedge d ^ 1 \tau ) - d ^ 3 B \left ( \theta \wedge \tau \right ) . \end{gather*}


\begin{align*} \bar f _ h ( t ) = \sup _ { x \in \R ^ d } \| u ( t \ , , x + h ) - u ( t \ , , x ) \| _ k 0 < t \le T . \end{align*}


\begin{align*} & \left [ \Bigl ( \overline { V _ \Gamma } + \varepsilon \ , \overline { \partial ^ \circ g _ i } + \varepsilon ^ 2 \bar { g } _ i \bar { \tau } _ \varepsilon ^ i \cdot \overline { \nabla _ \Gamma V _ \Gamma } \Bigr ) \rho ^ \varepsilon \right ] ( x , t ) \\ & = V _ \Gamma \eta _ 0 ( r ) + \varepsilon \{ ( \partial ^ \circ g _ i ) \eta _ 0 ( r ) + V _ \Gamma \eta _ 1 ( r ) \} + O ( \varepsilon ^ 2 ) , \end{align*}


\begin{align*} G = \left \langle E _ { i , j } \ ; \big | \ ; r _ { i , j , k } , r _ { i , j , k } ' , r \right \rangle , \end{align*}


\begin{align*} x \to y = \max \{ u \colon ( u ] \cap ( ( x ] \cup ( y ] ) \cap [ y ) = \{ y \} \} . \end{align*}


\begin{align*} \Upsilon ( x , t ) = t ^ { \ell ( x ) } + \mu ( x ) ^ { \frac { m ( x ) } { q ( x ) } } t ^ { m ( x ) } \quad ( x , t ) \in \overline { \Omega } \times [ 0 , \infty ) , \end{align*}


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


\begin{align*} \| g r a d ^ g \textbf { H } \| ^ 2 = 2 s ^ H _ \Sigma , \end{align*}