\begin{align*} \Gamma ( s ) \ , ( s , \lambda ) _ { k } = \Gamma ( s ) ( - 1 ) ^ { k } + \Gamma ( s ) \ , k ^ { s } e ^ { \lambda } \ , \sum _ { \ell = 1 } ^ { k } \left ( \begin{array} { c } k \\ \ell \end{array} \right ) \left ( \frac { 2 \lambda } { k } \right ) ^ { \ell } ( - 1 ) ^ { k - \ell } \ , \intop _ { k } ^ { \infty } t ^ { - s } e ^ { - \frac { \lambda } { k } t } \ , \frac { ( t - k ) ^ { \ell - 1 } } { ( \ell - 1 ) ! } \ , d t , \end{align*}


\begin{align*} y ( q x ) = m ( x ) y ( x ) + r ( x ) , \end{align*}


\begin{align*} \Gamma = \bigcup _ { i = 1 } ^ \epsilon \left ( \gamma _ i + v K \right ) . \end{align*}


\begin{align*} v ( p ^ \alpha ) \ = \ p + \alpha \ \le \ p \alpha \ = \ \underbrace { p + \cdots + p } _ { \alpha } \ \le \ p ^ \alpha . \end{align*}


\begin{align*} \phi = \sum _ { j = 0 } ^ N c _ j Z _ j , \end{align*}


\begin{align*} L = L _ 1 \oplus \Z \zeta \oplus \Z z \end{align*}


\begin{align*} \mathbb { E } _ { \lambda } \left [ \tilde { f } ( g , \cdot ) | \mathcal { F } ^ { s } \right ] = \mathbb { E } _ { \lambda } \left [ \tilde { f } ( e , \cdot ) | \mathcal { F } ^ { s } \right ] \mbox { } \lambda \mbox { - a . e . } , \end{align*}


\begin{align*} \widehat { S } \equiv ( \widehat { Y } _ 1 , \widehat { Y } _ 2 , \widehat { X } ) = \varepsilon ( Y _ 1 , Y _ 2 , X ) + ( 1 - \varepsilon ) ( \widetilde { Y } _ 1 , \widetilde { Y } _ 2 , \widetilde { X } ) , \end{align*}


\begin{align*} \max _ { q \in S ^ { 2 } : \ , q \cdot v ' = 0 } ( q \cdot x ) = \max _ { q \in S ^ { 2 } : \ , q \cdot ( O v ) = 0 } ( q \cdot x ) = \max _ { q \in S ^ { 2 } : \ , ( O ^ \intercal q ) \cdot v = 0 } ( ( O ^ \intercal q ) \cdot ( O ^ \intercal x ) ) = \max _ { q \in S ^ { 2 } : \ , q \cdot v = 0 } ( q \cdot ( O ^ \intercal x ) ) . \end{align*}


\begin{align*} A \mapsto \langle h , A h \rangle = \sum _ { i = 1 } ^ n w _ i \lambda _ i ( A ) \end{align*}


\begin{align*} & = \frac { | h _ 1 | ^ 2 } { \sum _ { x _ i \in \Phi \backslash \{ x _ 1 \} \cap \mathcal { A } } | h _ i | ^ 2 } \overset { \Delta } { = } \frac { | h _ 1 | ^ 2 } { I } , \end{align*}


\begin{align*} \pi _ 0 ( \mathcal { M } ) = \dots = \pi _ j ( \mathcal { M } ) = 0 . \end{align*}


\begin{align*} \Pi _ \lambda = \chi \Pi _ \lambda \chi + \sum \limits _ j \chi _ j \Pi _ \lambda \chi _ j + \sum \limits _ { j \neq j ' } \chi _ { j } \Pi _ \lambda \chi _ { j ' } + \sum \limits _ { j } ( \chi \Pi _ \lambda \chi _ j + \chi _ j \Pi _ \lambda \chi ) \end{align*}


\begin{align*} b _ { \delta ' , \hat { T } } ( \hat { \ell } ) = b _ { \delta ' , \hat { T } } ( ( \hat { \ell } ) _ { \sf p } ) + b _ { \delta ' , \hat { T } } ( ( \hat { \ell } ) _ { \sf q } ) \end{align*}


\begin{align*} \Psi _ { \mathcal { B } ^ H } ( r ) \coloneqq \begin{cases} \sup \limits _ { x \in \partial \Omega } | | \mathcal { B } ^ H | | _ { L ^ { n - 1 , \infty } ( \partial \Omega \cap B _ r ( x ) ) } \quad n \geq 3 , \\ \\ \sup \limits _ { x \in \partial \Omega } | | \mathcal { B } ^ H | | _ { L ^ { 1 , \infty } \log L ( \partial \Omega \cap B _ r ( x ) ) } \quad n = 2 . \end{cases} \end{align*}


\begin{align*} V _ { \infty } : = \lim _ { \vert x \vert \rightarrow \infty } \ , V ( x ) > \inf _ { x \in \mathbb { R } ^ N } \ , V ( x ) = V _ { 0 } > - 1 . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ n } J _ \varepsilon ( | x | ) x _ l x _ m \ : x = 0 \end{align*}


\begin{align*} & \texttt { m p . p r e c = d } \\ & \texttt { x 1 = r o u n d ( x , d ) ; y 1 = r o u n d ( y , d ) } \\ & \texttt { z = x 1 + y 1 } . \end{align*}


\begin{align*} \tilde { p } : = p + q . \end{align*}


\begin{align*} 3 \ , d \Phi ( X , \xi _ i , \xi _ j ) = g ( [ \xi _ i , \xi _ j ] , f X ) , 2 \ , d \eta ^ k ( \xi _ j , \xi _ i ) = g ( [ \xi _ i , \xi _ j ] , \xi _ k ) \end{align*}


\begin{align*} \alpha ' _ n = ( 1 - a ) \left ( \frac { q ^ n \alpha _ n } { 1 - a q ^ { 2 n } } - \frac { q ^ { n - 1 } \alpha _ { n - 1 } } { 1 - a q ^ { 2 n - 2 } } \right ) , \beta ' _ n = q ^ n \beta _ n , \end{align*}


\begin{align*} F ( \underline { k } , t , \mathbf { p } _ { n + 1 } ) = \left ( \prod _ { j = 0 } ^ { k } p _ j ^ { } \right ) \sum _ { j = 0 } ^ { k } \frac { q _ j ^ { t - k } } { 1 - q _ j } \prod _ { \substack { i = 0 \\ i \neq j } } ^ { k } \frac { 1 } { q _ j ^ { } - q _ i ^ { } } \ , . \end{align*}


\begin{align*} \mathcal { I } ( \delta ) = \mathbb { E } [ Y _ 2 ] - \mathbb { E } [ Y _ 1 ] + \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \mathrm { R } _ { \mathrm { D } } ( \lambda , \delta ) = \mathcal { I } _ { \mathrm { D } } ( \delta ) . \end{align*}


\begin{align*} x _ { t } = x _ { 0 } + \int _ { 0 } ^ { t } b _ { r } \mathrm { d } r + \int _ { 0 } ^ { t } \sigma _ { r } \mathrm { d } W _ { r } , 0 \leq t \leq T , \end{align*}


\begin{align*} _ { \alpha } : = \{ \mathbf { W } \in M ( \mathbb { R } ^ { 3 \times 3 } ) : \mathbf { W } \geq \alpha ^ 3 , \ ; ( \mathbf { W } ) \geq 2 \alpha ^ 2 , \ ; \mathbf { W } \geq 3 \alpha \} \end{align*}


\begin{align*} \mathcal { I } ( 0 ) = \int _ { \mathcal { X } } \left [ \sup _ { \gamma ( \cdot | x ) \in \Pi ( \mu _ { 1 | 3 } , \mu _ { 2 | 3 } ) } \int _ { \mathcal { Y } _ 1 \times \mathcal { Y } _ 2 } f ( y _ 1 , y _ 2 , x ) \ , d \gamma ( y _ 1 , y _ 2 | x ) \right ] d \gamma _ X ( x ) , \end{align*}


\begin{align*} - ^ \delta : M & \to M \\ m & \mapsto m ^ \delta : = \delta \ast m . \end{align*}


\begin{align*} \Rightarrow \| f ^ { = t } \| _ 2 ^ 2 = ( p ( 1 - p ) ) ^ t \approx p ^ t . \end{align*}


\begin{align*} r ' _ d = \frac { \sqrt { d } } { \log ^ { 1 / 2 } ( \gamma _ 2 / \gamma _ 1 ) } \mbox { a n d } \gamma ' _ d = \left ( \frac { 3 3 r } { \sqrt { d } } \right ) ^ d \gamma _ 1 \log ^ { d / 2 } \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) . \end{align*}


\begin{align*} \widetilde { \rho } _ w ( - i ^ - , \kappa ( - i ) ) = - \widetilde { \rho } _ w ( - i ^ + , \kappa ( - i ) ) \end{align*}


\begin{align*} R _ N ^ 2 ( L , \alpha , \Delta ) = \sum _ { k \in \mathbb { Z } } b _ { k , N } ( L ) e ( k \alpha ) . \end{align*}


\begin{align*} \overset \cdot { \ ; \ ; R ^ 1 _ { i , 0 } } = a _ 0 \dots a _ r R _ { i , r + 1 } ^ 1 ; i = 0 , \dots , r - 1 . \end{align*}


\begin{align*} L i _ 2 ( 2 ) = - \frac { \pi ^ 2 } { 4 } - i \pi \log 2 , \end{align*}


\begin{align*} \int f \otimes \bigl ( \otimes _ { i = 1 } ^ n f _ i \bigr ) \otimes g \ , d \rho _ n = \int f \cdot \bigl ( \prod _ { i = 1 } ^ n V f _ i \bigr ) \cdot J g \ , d \mu = \int f \cdot \bigl ( \prod _ { i = 1 } ^ n V f _ i \bigr ) \otimes g \ , d \eta . \end{align*}


\begin{align*} \mathfrak { S } ( f _ k ) = k + r - 1 , \ , \ , \ , \ , \ , \overline { \mathfrak { S } } ( f _ k ) = n - k + 1 . \end{align*}


\begin{align*} \partial _ { t } \mu ( t , x ) - \partial _ { x } \left ( \mu ( t , x ) \left ( 2 \int ^ x _ { - \infty } \mu ( t , y ) d y - 1 \right ) \right ) = \mu ( t , x ) S \star \mu ( t , x ) , \mu ( 0 , x ) = \mu ^ { i n } . \end{align*}


\begin{align*} \left | \frac { a _ i } { r } - \frac { a ' _ i } { r ' } \right | = \left | \frac { a _ { 1 , i } } { r } - \frac { a _ { 2 , i } } { r ' } + \frac { d _ i - d _ i ' } { d } \right | \end{align*}


\begin{align*} \lim _ { p \rightarrow \infty } \zeta _ { p } ( s ) = \zeta ( s ) , \ , \ , \ , \ , \ , \lim _ { p \rightarrow 0 ^ { + } } \zeta _ { p } ( s ) = ( 2 ^ { s } - 1 ) \ , \zeta ( s ) . \end{align*}


\begin{align*} \Lambda _ { \alpha , i } = { \rm d i a g } ( g _ { \lambda _ i } ( \alpha \theta _ n ^ { k } ) ) _ { k = 0 } ^ { n - 1 } \textrm { a n d } g _ { \lambda _ i } ( z ) : = \lambda _ i - 2 z + \lambda _ i z ^ 2 . \end{align*}


\begin{align*} \sigma _ r w t ( p _ s ) . ( p \widetilde { \mathbb { S } } _ { \lambda ^ i } | _ { z = 0 } ) = { \bf { b } } w t ( p _ s ) . ( p \widetilde { \mathbb { S } } _ { \lambda ^ i } | _ { z = 0 } ) + \frac { { \bf { b } } ( p _ s ) } { b _ { 0 } } ( p \widetilde { \mathbb { S } } _ { \lambda ^ 1 } | _ { z = 0 } ) ( p \widetilde { \mathbb { S } } _ { \lambda ^ i } | _ { z = 0 } ) . \end{align*}


\begin{align*} W _ p ^ - ( \mu , \nu ) = \inf _ { \beta \in \Phi ^ - ( \nu , m ) } W _ p ( \mu , \beta ) , \\ W _ p ^ + ( \mu , \nu ) = \inf _ { \alpha \in \Phi ^ + ( \mu , n ) } W _ p ( \alpha , \nu ) . \end{align*}


\begin{align*} B ^ * B = \sum _ { i , j = 1 } ^ d \zeta _ i \overline { \zeta _ j } S _ i ^ * S _ j = 1 . \end{align*}


\begin{align*} k _ { n e w \left ( 1 \right ) } = \underset { k \in \mathcal { K } _ { c l r \left ( 1 \right ) } } { { \arg \max } } ~ \textbf { g } _ { k } ^ { H } \textbf { g } _ { k } \end{align*}


\begin{align*} H = \prod _ { \ell \in \mathcal { S } } \prod _ { i = 1 } ^ { r ( \ell ) } \Delta ^ { \mathcal { T } ( \ell ) _ i } ( \mathbf { P G } ^ 1 ( \mathbb { Q } _ p ) ) . \end{align*}


\begin{align*} \mu ( g _ z ) = - z ' + a g _ z z ^ \ast = - z ' + z ^ \ast + u - q ( u ) z . \end{align*}


\begin{align*} \mathcal { J } : = \left \{ j \in \Z : [ ( 1 + T ^ { - 1 } ) ^ { j } , ( 1 + T ^ { - 1 } ) ^ { j + 1 } ) \cap \mathcal { M } \not = \emptyset \right \} \end{align*}


\begin{align*} \tilde { \Lambda } ( z ) = p _ { + } ( z ) p _ { - } ( z ) . \end{align*}


\begin{align*} \begin{aligned} T _ 3 \ge & - C s ^ 2 \lambda ^ 2 \iint _ { Q } \xi ^ 3 \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 u ^ 2 d x d t - C s ^ 2 \lambda ^ 2 \int _ { 0 } ^ { T } \int _ { \omega } \xi ^ 3 u ^ 2 d x d t , \end{aligned} \end{align*}


\begin{align*} M _ { i + 1 } - ( k _ { i + 1 } - \overline { \lambda } _ { i + 1 } - 1 ) - 1 & = \sharp \{ j | - \overline { \lambda } ^ { i + 1 } _ { - j } \leq \overline { \lambda } _ { i + 1 } \} - 1 \leq \sharp \{ j | - \overline { \lambda } ^ { i } _ { - j } \leq \overline { \lambda } _ { i + 1 } \} \\ & < \sharp \{ j | - \overline { \lambda } ^ { i } _ { - j } \leq \overline { \lambda } _ { i } \} = M _ { i } - ( k _ { i } - \overline { \lambda } _ { i } - 1 ) . \end{align*}


\begin{align*} \theta _ { L , X ^ { \ast } } ( A ) ( [ \epsilon , 0 ] ) = \sum _ { l \in L \cap X } A ( \epsilon , l ) = \sum _ { n \in \Z } A ( \epsilon , n ) . \end{align*}


\begin{align*} ( x _ - , x _ + ) = \bigcup _ { 1 \leq q \leq q ( k ) } J ^ { ( k ) } ( q ) \cup \bigcup _ { l > k } \bigcup _ { \epsilon = \pm } \bigcup _ { 1 \leq q \leq q _ \epsilon ( l ) } J _ \epsilon ^ { ( l ) } ( q ) . \end{align*}


\begin{align*} Q _ { 1 , \vec { v } _ { s , t , f , 0 } } = \end{align*}


\begin{align*} \pi \psi ( G ) = \pi ( \pi ( G ) ) = G , \end{align*}


\begin{align*} \mu _ i = \Big ( \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big ) ^ { \frac { 2 i - 1 } { n - 2 } } d _ i , \quad \mbox { w i t h } d _ i > 0 , i = 1 , \cdots , k . \end{align*}


\begin{align*} \tau - \psi _ m = \sum _ { k = 1 } ^ m \frac { ( 2 r ) ^ { k - 1 } } { ( 2 r + 1 ) ^ k } ( T ( \phi ) \phi _ k - \phi _ k ) . \end{align*}


\begin{align*} G = \omega _ Q ^ { 1 / \bar { p } _ 2 - 1 / \bar { p } _ 1 } \int _ { \mathbb H ^ n } \frac { | y | _ h ^ { - Q / p } } { \max ( 1 , | y | _ h ^ Q ) } d y . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c u _ { j + 1 } ) ^ n = \psi ( \cdot , u _ j ) \ , \omega ^ n & \textnormal { o n } & \Omega \\ u _ { j + 1 } = 0 & \textnormal { o n } & \partial \Omega . \end{array} \right . \end{align*}


\begin{align*} l _ T ( u , v , x ) & = [ T u , T v , x ] , \\ m _ T ( u , x , v ) & = [ T u , x , T v ] - T \rho ( T u , x ) v , \\ r _ T ( x , u , v ) & = [ x , T u , T v ] - T \rho ( x , T u ) v , \end{align*}


\begin{align*} U ( z ) \nabla U ^ { - 1 } ( z ) = \partial _ z + Z . \end{align*}


\begin{align*} \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) u : = \sum _ { i = 1 } ^ { M ' } \tilde { \alpha } _ i ( - \Delta ) ^ { \tilde { s } _ i } u \tilde { s } = ( \tilde { s } _ 1 , \dots , \tilde { s } _ { M ' } ) , 0 < \tilde { s } _ 1 < \cdots < \tilde { s } _ { M ' } \leq s _ M , \tilde { \alpha } _ i \in L ^ \infty ( \mathbb { R } ^ n ) \end{align*}


\begin{align*} \langle \phi , \psi \rangle = \frac { 1 } { | G | } \sum _ { g \in G } \phi ( g ) \overline { \psi ( g ) } . \end{align*}


\begin{align*} \forall ( y , \xi ) \in B _ 1 \times \R ^ k , f _ 0 ( y , \xi ) \equiv f _ 0 ( \xi ) : = \abs { \xi } ^ { p _ 0 } . \end{align*}


\begin{align*} - 1 = \sum _ { k = 1 } ^ { N _ 1 } u ' _ k e _ k \end{align*}


\begin{align*} p _ m ( u , x ) : = \sum _ { k = 0 } ^ { m } \sum _ { i = 0 } ^ { m - k } p ( m , k , i ) u ^ k x ^ i \end{align*}


\begin{align*} u _ { + } = 0 , \mathbb { F } _ { + } = \mathbb { I } . \end{align*}


\begin{align*} A ( n ) \ = \ \sum _ { i = 1 } ^ { k } p _ i \cdot \alpha _ i . \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } S _ j ( t ) = \frac { \nu Y } { \lambda } \frac 1 { j ( j + 1 ) } . \end{align*}


\begin{align*} \rho _ t ( 0 , y ) : = \lim _ { \Gamma \ni x \to 0 } \rho _ t ( x , y ) \in ( 0 , \infty ) , t > 0 , \ , y \in \Gamma , \end{align*}


\begin{align*} p = \sum _ { k = 0 } ^ { n } A ^ { \ast } _ { k } , \end{align*}


\begin{align*} f ( k , n ) = \dfrac { \binom { n } { k } } { 2 ^ n } , 0 \le k \le n . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b ) = 1 \\ a , b \geq 1 } } \left ( \frac { 1 } { 1 - y ^ a z ^ b } \right ) ^ { \frac { b } { a ^ 2 } } = \exp \left \{ \frac { z L i _ 2 ( y ) } { ( 1 - z ) ^ 2 } \right \} \end{align*}


\begin{align*} S _ 1 & = \sum _ { k = 0 } ^ \infty \frac { 1 } { ( 2 k ) ! } \bigg ( \frac { 2 \beta } { \alpha } \bigg ) ^ { 2 k } \Gamma \bigg ( k + \frac { 1 } { 2 } \bigg ) \Gamma \bigg ( \nu + k + \frac { 1 } { 2 } \bigg ) , \\ S _ 2 & = - \sum _ { k = 0 } ^ \infty \frac { 1 } { ( 2 k + 1 ) ! } \bigg ( \frac { 2 \beta } { \alpha } \bigg ) ^ { 2 k + 1 } k ! \Gamma ( \nu + k + 1 ) . \end{align*}


\begin{align*} 0 = \int _ { \Omega } \left . \nabla ^ { \mathbb { N } ^ { n + 1 } ( K ) } _ p \left ( d _ K ^ 2 ( y , p ) \right ) \right | _ { p = O } \mathrm { d } \mu _ K ( y ) = \int _ { \Omega } \left . 2 d _ K ( y , O ) \nabla ^ { \mathbb { N } ^ { n + 1 } ( K ) } _ p [ d _ K ( y , p ) ] \right | _ { p = O } \mathrm { d } \mu _ K ( y ) . \end{align*}


\begin{align*} F ( \omega ) = f ( W ( h _ 1 ) , \ldots , W ( h _ { m _ 1 } ) , N ( g _ 1 ) , \ldots , N ( g _ { m _ 2 } ) ) , \end{align*}


\begin{align*} \langle z ^ n \otimes \lbrace 0 , \infty \rbrace , f \rangle = 0 \end{align*}


\begin{align*} \norm { ( \phi _ n , B _ n \phi _ n ) - ( \phi , B \phi ) } ^ 2 & = \norm { \phi _ n - \phi } ^ 2 + \norm { B _ n \phi _ n - B \phi } ^ 2 = \norm { \phi _ n - \phi } ^ 2 + \norm { i \phi - i \phi _ n } ^ 2 \\ & = 2 \norm { ( B _ n + i ) ^ { - 1 } \psi - ( B + i ) ^ { - 1 } \psi } ^ 2 \longrightarrow 0 \end{align*}


\begin{align*} \widetilde { \mathcal R } ( \gamma , \gamma _ t , \sigma _ t ) & = - \beta J \left ( \phi '' \left ( \cdot + \gamma ( \cdot , t ) + \frac { 1 } { N } \sigma ( t ) \right ) \left ( 2 \gamma _ x + \gamma _ x ^ 2 \right ) + \phi ' \left ( \cdot + \gamma ( \cdot , t ) + \frac { 1 } { N } \sigma ( t ) \right ) \gamma _ { x x } \right ) \\ & + \ , \phi ' \left ( \cdot + \gamma ( \cdot , t ) + \frac { 1 } { N } \sigma ( t ) \right ) \left ( \gamma _ t + \frac { 1 } { N } \sigma _ t \right ) . \end{align*}


\begin{align*} \langle T _ { ( 1 - p ) \rightarrow p } A h , \ , g \rangle = \sum _ { d = 0 } ^ m \left ( - \frac { p } { 1 - p } \right ) ^ d \langle h ^ { = d } , g \rangle \geq \mu _ { p } ( h ) \mu _ { p } ( g ) - \sum _ { d = 1 } ^ m \left ( \frac { p } { 1 - p } \right ) ^ d | \langle h ^ { = d } , g \rangle | . \end{align*}


\begin{align*} f * ( \psi * g ) = ( f * \psi ) * g = \mu _ ! ( f \ltimes \psi \rtimes g ) & & f \in C _ c ^ \infty ( G _ 1 ) , \psi \in C _ c ^ \infty ( M ) , g \in C _ c ^ \infty ( G _ 2 ) . \end{align*}


\begin{align*} e ^ t M _ \ell ( \boldsymbol { a } , t ) = a _ \ell + \int _ 0 ^ t e ^ u M _ \ell ( \boldsymbol { a } , u ) \left ( q M _ \ell ( \boldsymbol { a } , u ) + \phi ( u ) + 1 \right ) . \end{align*}


\begin{align*} \zeta _ { p ^ { \prime } } ( 1 - s ) = 2 \cos \left ( \frac { \pi s } { 2 } \right ) \intop _ { 0 } ^ { \infty } \frac { x ^ { s - 1 } } { \sigma ^ { \prime } ( x ) \ , e ^ { 2 \pi x } - 1 } \ , d x . \end{align*}


\begin{align*} R ( l _ 1 ) \otimes R ( l _ 2 ) = R ( l ) _ 1 \otimes R ( R ( l ) _ 2 ) + R ( R ( l ) _ 1 ) \otimes R ( l ) _ 2 + \lambda R ( l ) _ 1 \otimes R ( l ) _ 2 . \end{align*}


\begin{align*} w _ b = \Phi _ b + ( \widetilde { T } ( f ) ) _ b . \end{align*}


\begin{align*} \iota ( g ) ( f ) : = \int _ \Omega \ ! f g \ , \mathrm { d } \mu , f \in X , \end{align*}


\begin{align*} Q = \left ( \begin{array} { c c } 3 & 0 \\ 0 & 3 \end{array} \right ) , A = \left ( \begin{array} { c c c } 2 & 3 \\ 1 & 0 \end{array} \right ) , L = \left ( \begin{array} { c c c } 1 & - 3 \\ - 1 & 3 \end{array} \right ) . \end{align*}


\begin{align*} \Theta _ { \mathrm { D } } ( \delta ) : = \left \{ \int _ { \mathcal { Y } _ 1 \times \mathcal { Y } _ 2 } f ( y _ 1 , y _ 2 ) \ , d \gamma ( y _ 1 , y _ 2 ) : \gamma \in \Sigma _ { \mathrm { D } } ( \delta ) \right \} , \end{align*}


\begin{align*} \vec U ^ T ( I _ n \otimes E ^ T P _ { \partial \Omega } N _ { x _ i } E ) { \bf A _ i } \vec U = \sum _ { j = 1 , N } \lbrack ( E \vec U ) ^ T ( N _ { i } { \bf A _ i } ) ( E \vec U ) \rbrack _ j d s _ j = \sum _ { j = 1 , N } \lbrack \vec W ^ T \Lambda \vec W \rbrack _ j d s _ j . \end{align*}


\begin{align*} \operatorname { O } _ { V ^ { \perp } } ( V , q ) : = \big \{ \varphi \in \operatorname { O } ( V , q ) \big | \varphi | _ { V ^ { \perp } } = \operatorname { I d } _ { V ^ { \perp } } \big \} = \{ \varphi \in \operatorname { O } ( V , q ) | \varphi ( v ) = v \ \forall v \in V ^ { \perp } \} . \end{align*}


\begin{align*} ( q _ 1 ( x ) - q _ 2 ( x ) ) u _ 1 ( x ) = 0 \quad E , \end{align*}


\begin{align*} \varphi ( p ( \mathcal J ) ) _ { k + 1 + i } = \varphi ( p ( \mathcal J ) ) _ { k + i } \oplus z ^ { n + i + 2 - x } , \end{align*}


\begin{align*} \Omega _ t ^ p : = \{ \phi _ t ( p , q ) : q \in \Omega \} . \end{align*}


\begin{align*} \begin{aligned} \langle S _ N ( \ell ) ^ 2 \rangle & = \sum _ { i , j = 1 } ^ N \sum _ { m , n \in \mathbb { Z } } \int _ 0 ^ 1 \chi \left ( \frac { x _ i - x _ 0 + m } { \ell } \right ) \chi \left ( \frac { x _ j - x _ 0 + n } { \ell } \right ) \ , d x _ 0 \\ & = \sum _ { i , j = 1 } ^ N \sum _ { m \in \mathbb { Z } } \int _ \mathbb { R } \chi \left ( \frac { x _ i - x _ 0 + m } { \ell } \right ) \chi \left ( \frac { x _ j - x _ 0 } { \ell } \right ) \ , d x _ 0 . \end{aligned} \end{align*}


\begin{align*} \beta ( M , C ) = \min _ { \substack { P \subset \mathbb { Z } _ M \\ P \ne \emptyset } } \left \lbrace \vert P \vert \colon \Vert P \Vert _ u \leq \frac { C - 3 } { C - 1 } \mathbf { P } ( P ) \right \rbrace . \end{align*}


\begin{align*} a _ { i } : = \frac { b _ { i } } { a _ { i + 1 } \cdots a _ { i + p - 1 } } , i \le I - 1 ; \end{align*}


\begin{align*} \vartheta _ L ( f ; h _ \kappa ^ 3 ) ( g _ K ) = \bigl ( \Theta _ { ( L , K ) } ( f ; \tau ) , \Theta _ K ( \tau , g _ K ; h _ \kappa ^ 3 ) \bigr ) _ \tau \end{align*}


\begin{align*} ( q , p ) = ( q ^ 1 , \ldots , q ^ n , p ^ 1 , \ldots , p ^ n ) = q + i p . \end{align*}


\begin{align*} \left \{ \begin{array} { l } x _ 0 y _ 0 + x _ 1 y _ 1 + x _ 2 y _ 2 = 0 , \\ \\ \displaystyle \sum _ { \substack { 0 \le i , j \le 2 } } a _ { i , j } x _ i y _ j = 0 . \end{array} \right . \end{align*}


\begin{align*} ( p ( u ) - p ( v ) ) ^ \top ( A ^ \top A - I _ d ) ( p ( u ) - p ( v ) ) = 0 . \end{align*}


\begin{align*} \langle D ^ { k - 1 } ( f ) , g \rangle = 1 . \end{align*}


\begin{align*} \delta _ { t } ^ I \delta _ { r } ^ { \Lambda } x = \delta _ { r } ^ { \Lambda } \delta _ { t } ^ I x , r , t > 0 . \end{align*}


\begin{align*} n _ t : = \left \lceil \frac { k t } { \rho ( t ) / ( 2 \sqrt { d } ) } \right \rceil ^ d \end{align*}


\begin{align*} ( D v ) _ { \# } ( g + f \chi _ { _ { \Omega \backslash \overline { U } } } ) ) = f . \end{align*}


\begin{align*} \xi _ { \ast } \left ( p _ { \ast } \left ( m _ { K \cap Q } \right ) \times \lambda \right ) = \int _ { K \cap Q } \vartheta ( k P ) . \lambda d m _ { K \cap Q } ( k ) = \int _ { K \cap Q } k . \lambda d m _ { K \cap Q } ( k ) = m _ { K \cap Q } \ast \lambda = \eta . \end{align*}


\begin{align*} T _ { \mu \nu } = ( \rho + p ) u _ { \mu } u _ \nu + p g _ { \mu \nu } , \end{align*}


\begin{align*} A ^ * _ { h _ 0 } \mathcal { C } _ { K , r } = A ^ * _ { h _ 0 } T _ { h _ 0 } ^ { - 1 } T _ { h _ 0 } \mathcal { C } _ { K , r } \rightarrow \{ x \in \mathbb { R } ^ n : x \cdot e _ 0 \geq 0 \} \end{align*}


\begin{align*} c _ { F , j } = ( - 1 ) ^ { n - j } \frac { \partial V } { \partial \bar { c } _ { n - ( j + 1 ) } } \end{align*}


\begin{align*} E [ Z _ 0 ( s ) ^ 2 ] = c _ 1 e ^ { 2 \lambda s } - c _ 2 e ^ { \lambda s } . \end{align*}


\begin{align*} u ( t ) = u _ 0 + \int _ 0 ^ t b ( s ) \ , d s + \int _ 0 ^ t G ( s ) \ , d W ( s ) , \end{align*}


\begin{align*} [ \xi _ i , \xi _ j ] & = 0 , 1 \le i , j \le p . \end{align*}


\begin{align*} \mathcal { G } _ { \overline { \psi } _ 0 } ^ { \mathbf { K } } : = \mathcal { G } ( \mathbb { Q } ) \bigcap \mathbf { K } _ { { \overline { \psi } _ 0 } } ^ p \end{align*}


\begin{align*} \left \| \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} \right \| \le \| A \| + \| B \| . \end{align*}


\begin{align*} \tfrac { d } { d \tau } \bigl [ e ^ \tau F ( t _ 0 , x _ 0 , e ^ \tau ) - e ^ \tau \tilde F ( t _ 0 , x _ 0 , e ^ \tau ) \bigr ] = 0 , \end{align*}


\begin{align*} \psi \left ( \begin{pmatrix} b _ 1 & b _ 2 \\ b _ 3 & b _ 4 \end{pmatrix} \right ) = \begin{pmatrix} b _ 1 ^ 2 & b _ 1 b _ 2 & b _ 2 ^ 2 \\ 2 b _ 1 b _ 3 & b _ 1 b _ 4 + b _ 2 b _ 3 & 2 b _ 2 b _ 4 \\ b _ 3 ^ 2 & b _ 3 b _ 4 & b _ 4 ^ 2 \end{pmatrix} . \end{align*}


\begin{align*} \sum _ { n \geq 0 } \overline { p } _ { - t } ( n ) q ^ n = \frac { f _ 2 ^ t } { f _ 1 ^ { 2 t } } = \left ( \frac { f _ 2 } { f _ 1 ^ { 2 } } \right ) ^ t . \end{align*}


\begin{align*} \phi ' ( \rho ( 1 + u ) ) = \phi ' ( \rho ) - K \phi ( \rho ) \rho u - \dfrac { 1 } { 2 } K \phi ' ( \rho ) \rho ^ 2 u ^ 2 + o ( u ^ 2 ) , \end{align*}


\begin{align*} f \circ _ { } g \coloneqq \sum _ { i = 0 } ^ { \infty } \left ( \sum _ { k = 0 } ^ { i } \binom { i } { k } a ( k ) b ( i - k ) \right ) t ^ { i } \in k [ [ t ] ] \ , , \end{align*}


\begin{align*} \| u \| : = \left ( \int _ { \Omega } \left | X u \right | ^ { p } d x \right ) ^ { \frac { 1 } { p } } . \end{align*}


\begin{align*} M _ b ( \theta ) = \mathbb { E } _ { \Phi | \Phi ( \mathcal { A } ) > 0 } \{ P _ s ^ b ( \theta ) \} \Pr ( \Phi ( \mathcal { A } ) > 0 ) . \end{align*}


\begin{align*} V ^ { \sigma } ( \hat { \nu } ^ * , \mu ^ * ) - V ^ { \sigma } ( \nu ^ * , \mu ^ * ) & \geq \int _ { \mathcal { X } } \left ( \frac { \delta F } { \delta \nu } ( \nu ^ * , \mu ^ * , x ) + \frac { \sigma ^ 2 } { 2 } \log \left ( \frac { \nu ^ * ( x ) } { \pi ( x ) } \right ) \right ) ( \hat { \nu } ^ * - \nu ^ * ) ( \mathrm { d } x ) \\ & + \frac { \sigma ^ 2 } { 2 } \operatorname { D _ { K L } } ( \hat { \nu } ^ * | \nu ^ * ) = \frac { \sigma ^ 2 } { 2 } \operatorname { D _ { K L } } ( \hat { \nu } ^ * | \nu ^ * ) , \end{align*}


\begin{align*} \langle \sum _ { n \gg - \infty } a _ { n } q ^ { n } , \sum _ { n > 0 } b _ { n } q ^ { n } \rangle : = \displaystyle \sum _ { n < 0 } \dfrac { a _ { n } b _ { - n } } { n ^ { k - 1 } } . \end{align*}


\begin{align*} c _ 3 = \sum _ r \frac { \mu ( r ) } { r ^ 2 } f _ 1 ( r ) f _ 2 ( r ) \end{align*}


\begin{align*} \| R _ q ( \lambda ) f \| _ { H ^ j ( \mathrm { M } ) } \le c _ 1 \lambda ^ { j / 2 } \mathbf { e } _ \lambda \| f \| _ { L ^ 2 ( \mathrm { M } ) } , j = 1 , 2 . \end{align*}


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


\begin{align*} \sum _ { i = 1 } ^ n \beta _ i z _ { i p } = \sum _ { i = 1 } ^ n \gamma _ i z _ { i p } < \sum _ { i = 1 } ^ n \alpha _ i z _ { i p } . \end{align*}


\begin{align*} \frac { d } { d \lambda _ n } g _ n ' ( 0 , 0 ) = \rho - | \alpha | \overline { P \left ( \frac { d \mu _ { \lambda , n } } { d \lambda _ n } \right ) ( 1 ) } . \end{align*}


\begin{align*} \mathrm { I } \left ( \xi _ { 1 } , \mathcal { T } | X , \eta \right ) & : = \int _ { X } \mathrm { I } \left ( \xi _ { 1 } ^ { x } , \mathcal { T } _ { x } \right ) d \eta ( x ) . \end{align*}


\begin{align*} I _ { \omega , \mathbf { c } } ( U ( t ) ) \le 2 ( S _ { \omega , \mathbf { c } } ( U ( t ) ) - S _ { \omega , \mathbf { c } } ( \Phi _ 0 ) ) = 2 ( S _ { \omega , \mathbf { c } } ( U ( t ) ) - \mu _ { \omega , \mathbf { c } } ) \end{align*}


\begin{align*} \textbf { Q } ^ { \lambda _ j } = Q _ { i _ 1 } ^ { \lambda _ j ( Q _ { i _ 1 } ) } \cdots Q _ { i _ r } ^ { \lambda _ j ( Q _ { i _ r } ) } . \end{align*}


\begin{align*} \mathbb { E } ( X ) = \sum _ { \substack { 0 \leq i < j \leq h \\ j - i \leq t } } \mathbb { P } ( s _ i = s _ j ) + \sum _ { \substack { 0 \leq i \leq h < j \leq k \\ j - i \leq t } } \mathbb { P } ( s _ i = s _ j ) + \sum _ { \substack { h < i < j \leq k \\ j - i \leq t } } \mathbb { P } ( s _ i = s _ j ) . \end{align*}


\begin{align*} \left ( \int _ s ^ t \pi _ { r } ^ N ( A \phi _ j ) d r \right ) ^ 2 = \frac { 1 } { N ^ 2 } \left [ \int _ s ^ t \sum _ { ( x , \sigma ) \in V } \eta _ r ^ N ( x , \sigma ) \cdot ( A \phi _ j ) ( \tfrac { x } { N } , \sigma ) d r \right ] ^ 2 . \end{align*}


\begin{align*} \eta ( \sigma ) ^ * ( y _ { \tau , \mathfrak { u } } ) = \left ( \prod ^ k _ { \ell = 1 } \frac { c _ { i _ { \ell } } ^ { \sigma } } { c _ { i _ { \ell } } ^ { \tau } } \right ) x _ { \tau , \mathfrak { u } } . \end{align*}


\begin{align*} I _ { 3 } = \mathbb { P } \left ( \sum _ { i = 1 } ^ { n } \eta _ { n , i } \geq \frac { x } { 2 } \right ) \ \ \ \ \ \ I _ { 4 } = \mathbb { P } \left ( \frac { \log W _ { n } } { \sigma \sqrt { n } } \geq \frac { x } { 2 } \right ) . \end{align*}


\begin{align*} \partial _ t v + v \cdot \nabla v - \Delta v = - \nabla q , { \rm d i v } \ , v = 0 \end{align*}


\begin{align*} \hat { x } _ z ( 0 ) = x ( \min \{ k \geq 0 : z ( k ) = 1 \} ) . \end{align*}


\begin{align*} l ( U ^ { n , \sigma } ) - d \ , = \ , 0 , \end{align*}


\begin{align*} \mathcal { B } _ a : = \left \{ y + a : y \in \mathcal { B } \right \} \mbox { a n d } \mathcal { X } _ a : = \left \{ y + a : y \in \mathcal { X } \right \} \end{align*}


\begin{align*} \mathbb { R } ^ { n } \setminus \Omega ^ { * } & \subset \mathbb { R } ^ { n } \setminus \big ( Q _ { 1 , k _ { 1 } } ^ { * } \bigcup Q _ { 2 , k _ { 2 } } ^ { * } \big ) \subset \bigcup _ { h = 1 } ^ { \infty } \bigcup _ { i = 1 } ^ { \infty } \big ( \mathcal { Q } _ { 1 , k _ { 1 } } ^ { i } \bigcap \mathcal { Q } _ { 2 , k _ { 2 } } ^ { h } \big ) . \end{align*}


\begin{align*} \frac { \partial g } { \partial \theta } = \left ( e ^ { - i ( d + i H d ) } \right ) \left ( e ^ { i ( d + i H d ) } \frac { \partial g } { \partial \theta } \right ) . \end{align*}


\begin{align*} ( \prod \limits _ { i = 1 } ^ m a ( i ) \ast f ( t ( i ) ) ) \ast a ( m + 1 ) \in A \cap \sigma ( L ) , \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle G ( t ) = \int _ { \mathbb { R } ^ N } | x | ^ 2 \left ( a _ 2 | \phi | ^ 2 + a _ 1 | \psi | ^ 2 \right ) d x , \end{array} \right . \end{align*}


\begin{align*} \mathcal { H } ^ 1 ( J \cap \partial B _ R ) = 0 . \end{align*}


\begin{align*} ( x ) _ 0 = 1 \quad ( x ) _ m = x ( x + 1 ) \cdots ( x + m - 1 ) m \in \mathbb { Z } ^ { + } . \end{align*}


\begin{align*} \int \otimes _ { i = 1 } ^ n f _ i \otimes g \ , d \tilde \rho = \int \bigl ( \prod _ { i = 1 } ^ n V f _ i \bigr ) \ , J ^ { \ast } g \ , d \mu , \end{align*}


\begin{align*} \lim _ { r \nearrow 1 } \int _ 0 ^ { 2 \pi } | T ( f ) ( e ^ { i \theta } ) - T ( f ) ( r e ^ { i \theta } ) | ^ \gamma \ , d \theta = 0 , \end{align*}


\begin{align*} \mathcal { W A } _ \lambda [ u ] = \frac { 1 } { 2 } \norm { u } _ { L ^ 2 _ \lambda } ^ 2 ; \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c , d , e ) = 1 \\ a , b , c , d , e \geq 1 } } \left ( \frac { 1 } { 1 - v ^ a w ^ b x ^ c y ^ d z ^ e } \right ) ^ { \frac { 1 } { a ^ q b ^ r c ^ s d ^ t e ^ u } } \end{align*}


\begin{align*} X _ T \prec X _ F = X _ { F T } , \end{align*}


\begin{align*} \alpha _ n = ( - 1 ) ^ n q ^ { \left ( { n \atop 2 } \right ) } \frac { 1 - a q ^ { 2 n } } { 1 - a } \frac { ( a ) _ n } { ( q ) _ n } , \beta _ n = \delta _ { n , 0 } . \end{align*}


\begin{align*} x _ i = b _ i + \sum _ { j > i } t _ { i j } b _ j \end{align*}


\begin{align*} C = \int _ { \mathbb { R } ^ n } K \left ( x , \hat { e } _ 1 \right ) | x | ^ { - n / p ^ { \prime } } d x \end{align*}


\begin{align*} \lambda _ k [ f ( A _ { \mathcal { S ' } } ) ] & \le \min _ { h \in { \mathcal { F } } ; \ \Vert h \Vert = 1 } \langle h , f ( A ) h \rangle \\ & \le \lambda _ k [ f ( A ) _ { \mathcal { S ' } } ] . \end{align*}


\begin{align*} I : = \int _ { [ 0 , 1 ] ^ 3 } \ ! \ ! f ( x , y ) f \left ( x + P _ 1 ( t ) , y \right ) f \left ( x , y + P _ 2 ( t ) \right ) \ , \mathrm { d } t \mathrm { d } x \mathrm { d } y > \delta \end{align*}


\begin{align*} H ^ { n - 1 } ( \partial B _ 1 ) = \int _ { \partial B _ 1 } | e - \omega | ^ { 2 s - n } \ , d H ^ { n - 1 } _ \omega . \end{align*}


\begin{align*} ( \pounds _ { \xi _ i } \ , d \eta ^ j ) ( X , Y ) = ( \pounds _ { \xi _ i } \ , g ) ( X , { f } Y ) + g ( X , ( \pounds _ { \xi _ i } { f } ) Y ) . \end{align*}


\begin{align*} M _ q ( \Gamma ^ { \ , * } _ m ) \geqslant C _ 0 \cdot m ^ { \frac { q - n } { n } } ( D ) \cdot m ^ { - \frac { q - n } { n } } ( D ) = C _ 0 . \end{align*}


\begin{align*} \Delta _ { 0 } = 0 \mu _ { 1 } = \mu _ { 2 } \textup { t r } ( \Sigma _ 1 ) = \textup { t r } ( \Sigma _ 2 ) . \end{align*}


\begin{align*} K _ { \omega , \mathbf { c } } ( \Phi ) = 2 L ( \Phi ) + 3 N ( \Phi ) + 2 \omega Q ( \Phi ) + 2 \mathbf { c } \cdot \mathbf { P } ( \Phi ) = 0 . \end{align*}


\begin{align*} \mathfrak { d } ^ k _ { \sigma , j } ( f ) = \sum _ { 0 \leq n \leq \frac { k - j } { m _ \sigma } } \frac { \binom { k } { j + n m _ \sigma } \binom { \frac { ( m _ \sigma - 1 ) - j } { m _ \sigma } - 1 } { n } } { \binom { \frac { ( k - j ) - ( m _ \sigma - 1 ) } { m _ \sigma } } { n } } \frac { \partial ^ k ( f \cdot V ) } { \partial z ^ { j + n m _ \sigma } \partial \overline { z } ^ { k - j - n m _ \sigma } } ( 0 , 0 ) . \end{align*}


\begin{align*} \nabla _ g u _ \delta & = ( \phi ' ( \rho ) - s \phi ( \rho ) ) e ^ { - s \rho } \nabla _ g \rho , \\ \Delta _ g u _ \delta & = \left [ - 2 s \phi ' ( \rho ) + s ^ 2 \phi ( \rho ) + ( \phi ' ( \rho ) - s \phi ( \rho ) ) p s \coth ( \kappa \rho ) \right ] e ^ { - s \rho } \\ & = \left [ s ( p \coth ( \kappa \rho ) - 2 ) \phi ' ( \rho ) + s ^ 2 ( 1 - p \coth ( \kappa \rho ) ) \phi ( \rho ) \right ] e ^ { - s \rho } . \end{align*}


\begin{align*} \lim _ { m \to \infty } A ^ m = x \cdot y ^ T = \frac { 1 } { n } J = \frac { 1 } { n } P _ { \sigma _ 1 } + \cdots + \frac { 1 } { n } P _ { \sigma _ n } , \end{align*}


\begin{align*} J _ K \begin{bmatrix} I \\ S \end{bmatrix} = J _ K \begin{bmatrix} I _ { k \times k } & 0 \\ 0 & I _ { ( n - k ) \times ( n - k ) } \\ S _ 1 & S _ 2 \\ S _ 3 & S _ 4 \end{bmatrix} = \begin{bmatrix} - S _ 1 & - S _ 2 \\ 0 & I \\ I & 0 \\ S _ 3 & S _ 4 \\ \end{bmatrix} . \end{align*}


\begin{align*} ( \lambda + \mu ) ^ { - 1 } - p _ k ( \lambda , \mu ) = \frac { \mu ^ { 2 n + 2 } } { \lambda ^ { 2 n + 2 } ( \lambda + \mu ) } \end{align*}


\begin{align*} \sum _ { k \in \hbar \mathbb { Z } ^ { n } } | f ( k ) | ^ { 2 } = \left ( \sum _ { \xi \in \mathcal { I } _ { \hbar } } \widehat { f } ( \xi ) u _ { \xi } , \sum _ { \eta \in \mathcal { I } _ { \hbar } } \widehat { f } ( \eta ) u _ { \eta } \right ) = \sum \limits _ { \xi , \eta \in \mathcal { I } _ { \hbar } } \widehat { f } ( \xi ) \overline { \widehat { f } ( \eta ) } \left ( u _ { \xi } , u _ { \eta } \right ) = \sum _ { \xi \in \mathcal { I } _ { \hbar } } | \widehat { f } ( \xi ) | ^ { 2 } . \end{align*}


\begin{gather*} 0 = \int _ a ^ t \langle L _ { \lambda } [ X ] , X \rangle d s = - \int _ a ^ t \langle X '' , X \rangle d s + \int _ a ^ t \langle ( R - \lambda I ) X , X \rangle d s \\ = \int _ a ^ t \| X ' \| ^ 2 d s - \langle X ' , X \rangle | _ a ^ t + \int _ a ^ t \langle ( R - \lambda I ) X , X \rangle d s \\ = \int _ a ^ t \| X ' \| ^ 2 + \langle ( R - \lambda I ) X , X \rangle d s . \end{gather*}


\begin{align*} \alpha _ i ^ { \prime } = a _ 1 \cdots a _ p a _ { p + 1 } \cdots a _ q x a _ { q + 1 } \cdots a _ l \quad \quad \tilde { \alpha } _ i ^ { \prime } = ( a _ l , \dots , a _ { q + 1 } , x , a _ q , \dots , a _ { p + 1 } , a _ p , \dots , a _ 1 ) ; \end{align*}


\begin{align*} M _ q ( \Gamma ) \leqslant \frac { 1 } { \varepsilon _ 0 ^ q } \int \limits _ { { \mathbb R } ^ n } Q ( z ) \ , d m ( z ) = \frac { \Vert Q \Vert _ 1 } { { | f ( x ) - f ( y ) | } ^ { q } } \ , . \end{align*}


\begin{align*} \frac { n } { n - k } \Big ( F ( k , t , n ) - F ( k , t + 1 , n ) \Big ) = F ( k , t , n ) - F ( k - 1 , t , n ) , \end{align*}


\begin{align*} \mathcal { A } = \left \{ \mathcal { K } \times I \in \overline { \mathcal { A } } \ , : \mathcal { K } \times I \not \subset \mathcal { Q } _ { 5 \rho _ { { i } } } \left ( z _ { i } \right ) i \right \} . \end{align*}


\begin{align*} \int _ { B _ { \delta } ( \hat { x } ^ \prime ) \times ( 0 , \delta ) } J _ \varepsilon ( | A _ \gamma ( \hat { x } , \hat { x } ) ( \hat { x } - \bar { y } ) | ) \nabla _ { x ^ \prime } u ( \hat { x } ) \cdot ( \hat { x } ^ \prime - \bar { y } ^ \prime ) \big | \det F _ \gamma ( \hat { x } ) \big | \ ; y = 0 . \end{align*}


\begin{align*} \sum \limits _ { i = 1 } ^ 3 \mathrm { v } _ i \ , h _ i ^ 2 \ , w _ { p \alpha + k } ^ { ( i ) } ( 0 , t ) = \boldsymbol { d _ { p \alpha + k } } ( t ) , \end{align*}


\begin{align*} d ( x y ) = - d ( x ) y - d ( y ) x , \forall x , y \in X . \end{align*}


\begin{align*} \lim \limits _ { \varepsilon \searrow 0 } \mathcal { E } _ \varepsilon ( c ) = \frac { 1 } { 2 } \int _ { \Omega } \big | \nabla c ( x ) \big | ^ 2 \ : x \end{align*}


\begin{align*} \overline { \Omega } \subseteq \bigcup _ { j = 0 } ^ N U _ j \ ; \ ; \ ; \ ; \ ; \ ; \Omega \cap U _ j = \mathbb { R } ^ n _ { \gamma _ j } \cap U _ j \ ; \ ; \ ; j = 0 , \ldots , N . \end{align*}


\begin{align*} \mathbb { F } ( x ( X , t ) , t ) = \widetilde { \mathbb { F } } ( X , t ) . \end{align*}


\begin{align*} t _ \delta ( x ) = \delta x \end{align*}


\begin{align*} \rho _ w ( \xi , w ) = \rho _ { \overline { w } } ( \overline { \xi } , \overline { w } ) = \overline { \rho _ w ( \overline { \xi } , \overline { w } ) } { \rm \ f o r \ } \xi \ne \pm 1 \end{align*}


\begin{align*} \Vert g \Vert _ { C ^ { 1 , \bar 1 } ( \bar \Omega \times I ) } : = \max \left \{ \Vert g \Vert , \Vert \nabla g \Vert , \Vert \Delta g \Vert , \Vert g _ t \Vert , \Vert g _ { t t } \Vert \right \} \leq A , \end{align*}


\begin{align*} { \rm R e } \left ( r e ^ { i ( v + i H v ) } \frac { \partial \kappa } { \partial \theta } \right ) = - e ^ { - u } e ^ { - ( H v ) } \widetilde { \rho } _ { \theta } ( \theta , \kappa ) , \end{align*}


\begin{align*} \frac { \partial x } { \partial \omega } = O ( t ^ 2 ) \qquad \qquad \frac { \partial y } { \partial \omega } = O ( t ^ 2 ) , \end{align*}


\begin{align*} \nu _ i \left ( g ^ { ( i ) } ( x ) \right ) = 0 \end{align*}


\begin{align*} \det \ , D ^ 2 v _ 0 = c _ 0 \chi _ { _ { \Omega ^ * _ 0 } } \quad \R ^ n \end{align*}


\begin{align*} \begin{aligned} \mathcal { I } ( \lambda \delta + ( 1 - \lambda ) \delta ^ \prime ) & = \sup _ { \nu \in \Sigma \left ( \lambda \delta + ( 1 - \lambda ) \delta ^ \prime \right ) } \int _ { \mathcal { S } } f ( s ) d \nu ( s ) \\ & \geq \int _ { \mathcal { S } } f d \gamma ^ { \prime \prime } = \lambda \int _ { \mathcal { S } } f d \gamma + ( 1 - \lambda ) \int _ { \mathcal { S } } f d \gamma ^ { \prime } . \end{aligned} \end{align*}


\begin{align*} \mathcal { R } ( I ) = \bigcup _ { s \in I } ( Z _ s ) . \end{align*}


\begin{align*} \begin{aligned} U _ j & = y + \alpha j - 1 0 \log ( j \wedge ( n - j ) ) , \\ L _ j & = - 1 0 + ( \alpha + \frac { y } { n _ \mathcal L } ) j - ( j \wedge ( n - j ) ) ^ { 3 / 4 } . \end{aligned} \end{align*}


\begin{align*} \left [ { N + 1 \atop j } \right ] = q ^ j \left [ { N \atop j } \right ] + \left [ { N \atop j - 1 } \right ] , \end{align*}


\begin{align*} e ^ { \rho ( u ) } = X ( T ( u ) ) . \end{align*}


\begin{align*} A ^ j _ s [ T _ m ] ^ i _ j ( A ) = \delta ^ i _ s \sigma _ { m + 1 } ( A ) - [ T _ { m + 1 } ] ^ i _ s ( A ) . \end{align*}


\begin{align*} E \big | \bar { S } ^ k _ { j , + } ( t ) - \hat { S } ^ k _ { j , + } ( t ) \big | = O ( \theta ^ k e ^ { \lambda t / 2 } ) . \end{align*}


\begin{align*} \bar { G } _ h ( s _ 1 , \xi ) = & \frac { \alpha _ n } { 2 } a _ 2 \partial _ { \xi _ h } \varphi ( \xi ) s _ 1 ^ { n - 2 } , \quad \mbox { f o r } h = 1 , \cdots , n , \\ \bar { G } _ 0 ( \bar { s } , \bar { \sigma } , \xi ) = & \alpha _ n a _ 1 s _ 1 ^ { n - 2 } \varphi ( \xi ) + a _ 3 \sum \limits _ { i = 2 } ^ { k } s _ i ^ { \frac { n - 2 } { 2 } } g ( \sigma _ i ) - a _ 4 \sum \limits _ { i = 1 } ^ k \frac { 2 } { 2 i - 1 } | \ln s _ i | . \end{align*}


\begin{align*} \mathfrak { C } = \{ \mathfrak { c } \in C ^ \infty ( \mathrm { M } , \mathbb { R } ) ; \ ; \mathfrak { c } > 0 \} . \end{align*}


\begin{align*} \| f - s _ \alpha \| _ { L ^ \infty [ a , b ] } = \| \cosh ( \alpha \cdot ) \left ( g - t _ \alpha \right ) \| _ { L ^ { \infty } [ a , b ] } \leq C \overline { h } ^ 2 \end{align*}


\begin{align*} A ( \alpha ) = \left [ \begin{matrix} B & 0 \\ 0 & C \end{matrix} \right ] . \end{align*}


\begin{align*} \sigma = \alpha _ 1 ^ { \prime } \alpha _ 2 ^ { \prime } \cdots \alpha _ m ^ { \prime } \beta ^ { \prime } \end{align*}


\begin{align*} N _ { d r } = \min \left \{ O _ { B } ^ { H } ( m a + r ) \cdot a + ( m a + r ) d \mid m \in \mathbb { N } \right \} . \end{align*}


\begin{align*} \alpha _ { \rho } ^ { x } = \left ( \bar { \psi } _ { H } \right ) _ { \ast } \left ( r ^ { - 1 } ( \bar { p } _ { k } ) _ { \ast } \left ( \tau ( y ) ^ { - 1 } . \nu _ { P } ^ { y } \right ) \right ) \mbox { w h e r e } x = ( y , r , H ) . \end{align*}


\begin{align*} \sum \limits _ { k = 1 } ^ { + \infty } \| r ^ { k } \| ^ 2 < \infty \end{align*}


\begin{align*} C ^ \varphi _ { i j k , k } R ^ \varphi _ { i j } = ( C ^ \varphi _ { i j k } R ^ \varphi _ { i j } ) _ k - \frac { 1 } { 2 } | C ^ \varphi | ^ 2 \ , . \end{align*}


\begin{align*} d u _ \lambda + A u _ \lambda \ , d t = f _ \lambda ( u _ \lambda ) \ , d t + \sigma ( u _ \lambda ) B \ , d W , u _ \lambda ( 0 ) = u _ 0 . \end{align*}


\begin{align*} I _ { m , p } ^ { \star } ( s , x ) : = \frac { 1 } { 2 \pi i } \ , \intop _ { \sigma - i \infty } ^ { \sigma + i \infty } \Gamma ( z ) \ , \Gamma ( s - z ) \ , ( 2 z , 2 \pi p m ) _ { m } \ , \left ( \frac { x } { m } \right ) ^ { 2 z } \ , d z = I _ { m , p } ^ { \star ( 1 ) } ( s , x ) + I _ { m , p } ^ { \star ( 2 ) } ( s , x ) , \end{align*}


\begin{align*} \ell = \ell _ \gamma : Z = \N _ \star ^ + \ni \phi \mapsto \phi ^ \gamma \in ( L _ { 1 / \gamma } ( \N ) ) ^ + \end{align*}


\begin{align*} 0 = \sum _ { i = 1 } ^ D \sum _ { j = 1 } ^ D \alpha _ i \bar { \alpha } _ j \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( i ) \lambda _ f ( j ) \big | L ( f , n + 1 ) \big | ^ 2 . \end{align*}


\begin{align*} \pi _ 1 ( X ^ { y , \Gamma } _ n ) & \leqslant \pi _ 1 ( Z _ { j _ 0 - 1 } ) + \frac { \delta H } { 5 } \\ & < \pi _ 1 ( w _ { j _ 0 } ) + \frac { 4 \delta H } { 5 } + \frac { \delta H } { 5 } & \mbox { u s i n g \eqref { e : l o c a l i z a t i o n _ x _ 2 } } \\ & = \pi _ 1 ( w _ { j _ 0 } ) + \delta H . \end{align*}


\begin{align*} \det \big ( M _ 2 ( \phi , \omega , r ; t ) \big ) , \ , \det \big ( M _ 3 ( \phi , \omega , r ; t ) \big ) = O ( t ^ 3 ) , t \to 0 . \end{align*}


\begin{align*} ( 1 - z ^ 2 ) \frac { \dd ^ 2 w } { \dd z ^ 2 } + \left ( \beta - \alpha - z ( \alpha + \beta + 2 ) \right ) \frac { \dd w } { \dd z } + \gamma ( \alpha + \beta + \gamma + 1 ) w = 0 , \end{align*}


\begin{align*} p ( x , y ) = \frac { D ( x , y ) } { \sum _ { z \neq x } D ( x , z ) } . \end{align*}


\begin{align*} \begin{aligned} f ( x ) = \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } \omega ( Q ) \psi _ { j } ( x , x _ { Q } ) q _ { j } h ( x _ { Q } ) , \end{aligned} \end{align*}


\begin{align*} w _ 1 = z _ 1 + \mathcal { O } ( \| H \| ) = x _ 1 + \iota y _ 1 + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} \mu _ T ( \overline { T _ { } x } ) = \{ \phi _ { i , j } = \varepsilon _ i + \varepsilon _ j \ : : \ : \{ i , j \} \in { \mathcal B } ( x ) \} . \end{align*}


\begin{align*} \zeta ( s , t ) + \zeta ( t , s ) = \zeta ( s ) \zeta ( t ) - \zeta ( s + t ) \Re ( s ) > 1 , \ ; \Re ( t ) > 1 , \end{align*}


\begin{align*} \mathrm { d i m } _ M ( \partial X ) = 4 - \liminf _ { d \to 0 } \frac { \ln V _ d ( U ) } { \ln ( d ) } . \end{align*}


\begin{align*} M ^ \delta = M ^ \delta _ 0 + M ^ \delta _ 1 \end{align*}


\begin{align*} n _ + = \min \left \{ n ' \in \N \ \left | \ n ' > n , \textbf { Q } _ { n ' } \ne \emptyset \right . \right \} . \end{align*}


\begin{align*} \langle L \rangle = \sum _ { C \subset \# L } \langle L , C \rangle \in \Bbb Z [ A , A ^ { - 1 } ] \end{align*}


\begin{align*} \varphi \circ \xi \left ( g , x _ { 0 } \right ) = g Q \mbox { f o r a l l } g \in G . \end{align*}


\begin{align*} f _ \Theta ( a ) \coloneqq \sum _ { i = 1 } ^ n y _ i [ \Theta ^ { - 1 } U ( a ) ] _ i = \sum _ { i = 1 } ^ n [ \Theta ^ { - 1 } y ] _ i U ( a ) _ i , \end{align*}


\begin{align*} \| \Tilde { \theta } \| _ { \C ^ k } \ll \rho ^ { - ( 2 k + 2 N ) } , N = n ^ 2 + m ^ 2 + n m - 1 \end{align*}


\begin{align*} \mathcal { L } _ { \hbar } u ( k ) : = \sum \limits _ { j = 1 } ^ { n } \left ( u \left ( k + \hbar v _ { j } \right ) + u \left ( k - \hbar v _ { j } \right ) \right ) - 2 n u ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \end{align*}


\begin{align*} \frac { \dd ^ 2 Z _ { t } ^ { ( \varepsilon ) } } { \dd t ^ 2 } = \lambda ^ 2 \partial _ { x x } Z _ { t } ^ { ( \varepsilon ) } + 2 \gamma \frac { \dd Z _ { t } ^ { ( \varepsilon ) } } { \dd t } + \varepsilon 2 \lambda \sqrt { \gamma \rho } \partial _ x \frac { \dd \tilde { W } _ { t } } { \dd t } . \end{align*}


\begin{align*} \| \partial _ k \varphi _ 3 \| _ { H ^ m } & = \| \partial _ k ( - \gamma \Delta + \omega + i \mathbf { c } \cdot \nabla ) ^ { - 1 } \nabla ( \varphi _ 1 \cdot \overline { \varphi _ 2 } ) \| _ { H ^ m } \lesssim \| \varphi _ 1 \cdot \varphi _ 2 \| _ { H ^ m } \lesssim \| \varphi _ 1 \| _ { H ^ m } \| \varphi _ 2 \| _ { H ^ m } < \infty \end{align*}


\begin{align*} F ( n ) & \ = \ \sum _ { i = 1 } ^ { k } \left ( p _ i + \alpha _ i \right ) , \\ G ( n ) & \ = \ \prod _ { i = 1 } ^ k p _ i \cdot \alpha _ i , \end{align*}


\begin{align*} \widetilde { M } _ k ^ { ( 1 ) } = \Big ( ( \alpha + j + h - 1 + t _ { k - j } ) a _ { \alpha + i + j + h - 1 + t _ { k - j } } \Big ) _ { i , j = 0 , \ldots , k - 1 } , \end{align*}


\begin{align*} \Delta _ { \mathfrak { c } g } u = 0 \ ; \mathrm { i n } \ ; \mathrm { M } , u _ { | \partial \mathrm { M } } = \varphi . \end{align*}


\begin{align*} \int \chi ( x ) e ^ { - i x \cdot \xi _ 0 / h \Tilde { h } } ( - i h X - z - i ( \tilde { h } ^ { - 1 } Q _ { \infty , \tilde { h } } + \tilde { h } ^ { - 1 } q _ { 1 } + W ) ) ^ { - 1 } ( \psi ( x ) e ^ { i x \cdot r \eta _ 0 / h \Tilde { h } } ) d x = \mathcal { O } ( h ^ \infty \Tilde { h } ^ \infty ) . \end{align*}


\begin{align*} f ( k , s , t ) = \binom { k } { 2 } + k ( t - k ) + \left \lfloor \frac { t - k } { s } \right \rfloor \binom { s } { 2 } + \binom { t - k - \left \lfloor \frac { t - k } { s } \right \rfloor s } { 2 } \end{align*}


\begin{align*} \| f ^ { = d } \| _ q \le \sqrt { \gamma } ( 1 2 r q ) ^ d \le \gamma ( 1 2 e ^ 2 r q ) ^ { d } \end{align*}


\begin{align*} \breve { v } ( x , t ) : = \psi ( x , t ) - \phi \left ( x + \sigma ( t ) \right ) , \end{align*}


\begin{align*} \partial _ { t } F _ { N } + \partial _ { x } ( A _ { N } ( t , F _ { N } ) ) = \mathbf { S } [ F _ { N } ] ( t , x ) . \end{align*}


\begin{align*} i \partial _ t u = - \Delta u + | u | ^ 2 u \end{align*}


\begin{align*} \left \lvert \sum _ { l \equiv v ( \bmod a ) } \binom n l \alpha ^ l ( 1 - \alpha ) ^ { n - l } - \frac 1 a \right \rvert = O \left ( n ^ { - \frac 1 2 } \right ) . \end{align*}


\begin{align*} V \left ( t , \xi \right ) = \inf _ { u \in \Lambda _ { t , \xi } } J \left ( t , \xi ; u \right ) , \end{align*}


\begin{align*} n _ \alpha : = \frac { n } { \gcd ( n , Q ( \alpha ^ \vee ) ) } . \end{align*}


\begin{align*} \max _ { i , j } \vert a _ { i j } \vert = ( \frac { M n } { \log ( 2 N ) } ) ^ { \frac { 1 } { 2 + \varepsilon / 4 } } . \end{align*}


\begin{align*} \limsup _ { n \rightarrow \infty } ( n \log n ) ^ { - 1 / p } \big | S _ n - \sum _ { k = 1 } ^ n g _ k \big | > 0 \mbox { a . s . } \end{align*}


\begin{align*} H _ m : = \sum _ { k = m } ^ { J - 1 } 2 ^ k | c _ k | . \end{align*}


\begin{align*} \alpha _ { l , { \hat { n } _ t } } e ^ { - j \omega } = \alpha _ { l , { \hat { n } _ t } } \cos \omega - j \alpha _ { l , { \hat { n } _ t } } \sin \omega . \end{align*}


\begin{align*} \sum _ { d _ 1 , d _ 2 < D } \frac { 1 } { \sqrt { d _ 1 d _ 2 } } \sum _ { I d _ 1 , J d _ 2 < D } \alpha _ { I d _ 1 } \bar { \alpha } _ { J d _ 2 } \sum _ { L , M \geq 1 } \frac { G _ k ( L M d _ 1 d _ 2 \slash p ) } { \sqrt { L M } } \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( I L ) \lambda _ f ( J M ) = 0 . \end{align*}


\begin{align*} \Omega = \{ \langle \xi \rangle ^ { - 1 } p = 0 \} \setminus { \rm C o n } _ p ( { \rm e l l } _ h ( B ) ; { \rm e l l } _ h ( B _ 1 ) ) \end{align*}


\begin{align*} X ( K ) \cap \Gamma = \pi _ 0 ^ { - 1 } \left ( X _ 0 ( K ) \cap \Gamma _ 0 \right ) \cap \Gamma = \left ( \pi _ 0 | _ { \Gamma } \right ) ^ { - 1 } \left ( X _ 0 ( K ) \cap \Gamma _ 0 \right ) . \end{align*}


\begin{align*} f _ { \lambda , [ L ] } ( s ) \ge \sum _ { \ell = 1 } ^ { L } a _ { \lambda , \ell } ( s _ \ell ) , \forall s = ( s _ 1 , \ldots , s _ L ) \in \prod _ { \ell \in [ L ] } \mathcal { S } _ \ell . \end{align*}


\begin{align*} \left \{ \begin{aligned} & - \Delta u = \lambda u + u \log u ^ 2 , \hbox { i n } \ , \ , \mathbb { R } ^ N , \\ & \int _ { \mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } , \end{aligned} \right . \end{align*}


\begin{align*} \widehat { f _ j } = ( f _ j ( 1 ) , f _ j ( 2 ) , \cdots , f _ j ( r ) , s ^ 0 _ j , s ^ 1 _ j , \cdots ) \mbox { i n } S , \end{align*}


\begin{align*} M = G _ m ^ \Delta q ^ { n - m \Delta } > \left ( q ^ m ( 1 - 2 ( 0 . 9 9 ) ^ m ) \right ) ^ \Delta q ^ { n - m \Delta } = q ^ { n } \left ( 1 - 2 ( 0 . 9 9 ) ^ m \right ) ^ \Delta . \end{align*}


\begin{align*} M _ { i , \ell } = ( ( \lambda v _ i ) _ { \vec { v } } ) _ \ell \end{align*}


\begin{align*} \left ( \sum _ { j = 1 } ^ d T _ j ^ t \otimes T _ j ^ * \right ) \mathrm { v e c } ( X ) = \mathrm { v e c } \left ( \sum _ { j = 1 } ^ d T _ j ^ * X T _ j \right ) . \end{align*}


\begin{align*} I _ \mu ( u ) = & \frac { 1 } { 2 } \int _ { \mathbb { R } ^ N } \left ( | \nabla u | ^ 2 + ( \mu + 1 ) u ^ 2 \right ) d x + \int _ { \mathbb { R } ^ N } F _ 1 ( u ) d x - \int _ { \mathbb { R } ^ N } F _ 2 ( u ) d x \\ \geq & \frac { 1 } { 2 } \int _ { \mathbb { R } ^ N } | \nabla u | ^ 2 d x + \int _ { \mathbb { R } ^ N } F _ 1 ( u ) d x - C C _ 1 a ^ { \left ( 1 - \beta _ p \right ) p } \left ( \int _ { \mathbb { R } ^ N } | \nabla u | ^ 2 d x \right ) ^ { \frac { p \beta _ p } { 2 } } . \end{align*}


\begin{align*} E _ 1 = \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} \end{align*}


\begin{align*} \widetilde { g } _ { u , v } ( z ) : = u | A | z ^ { n ' + m ' } + v | B | \overline { z } ^ { m ' } + | C | , z \in \mathbb { C } , \end{align*}


\begin{align*} W _ { 1 } = \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle , W _ { 2 } = \left \langle \left [ \vartheta _ { 1 } \right ] , \left [ \vartheta _ { 2 } \right ] , \dots , \left [ \vartheta _ { s } \right ] \right \rangle \in G _ { s } \left ( { \rm H ^ { 2 } } \left ( { \bf A } , \mathbb C \right ) \right ) , \end{align*}


\begin{align*} s _ j ( x ) = \dfrac { \sinh ( \alpha ( x _ { j } - x ) ) } { \sinh ( \alpha h _ j ) } y _ { j - 1 } + \dfrac { \sinh ( \alpha ( x - x _ { j - 1 } ) ) } { \sinh ( \alpha h _ j ) } y _ j . \end{align*}


\begin{align*} B _ j \rho ^ j & = { \rm m a x } _ { j ' \leq j } \{ b _ { j ' } \rho ^ j \} \\ & = { \rm m a x } \{ { \rm m a x } _ { j ' < j _ { \varepsilon , 0 } } \{ b _ { j ' } \rho ^ { j ' } \rho ^ { j - j ' } \} , { \rm m a x } _ { j _ { \varepsilon , 0 } \leq j ' \leq j } \{ b _ { j ' } \rho ^ { j ' } \rho ^ { j - j ' } \} \} < \varepsilon \end{align*}


\begin{align*} \left ( { D } ^ { \alpha } _ { N } U ^ { n - \sigma } , v _ h \right ) \ , + \ , a \big ( l ( U ^ { n , \sigma } ) \big ) \ , ( \nabla U ^ { n , \sigma } , \nabla v _ h ) = & \ , ( f ^ { n - \sigma } , v _ h ) , \\ U ^ 0 = & \ , u ^ 0 _ h , \end{align*}


\begin{align*} L = \{ z \in K \ , ; \ , \lim _ { m \to \infty } d _ { K , q , m } ^ { S } ( f ) e ^ { m q ( z ) } = 0 \} \neq \varnothing \end{align*}


\begin{align*} s _ { i j } + t _ { i j } + \sum _ { i < k < j } s _ { i k } t _ { k j } = 0 \end{align*}


\begin{align*} L i _ n ( z ) = \int _ { 0 } ^ { z } \frac { L i _ { n - 1 } ( t ) } { t } d t . \end{align*}


\begin{align*} \widehat { p } = \widehat { p } ( x ) : = \varphi _ 1 ^ { - 1 } ( x ) . \end{align*}


\begin{align*} L ( H ) = u ^ { j \bar k } H _ { j \bar k } \leq - \alpha ( 1 - \alpha \ ; \textnormal { { d i a m } } ^ 2 ( \Omega ) ) \sum _ { j = 1 } ^ { n } u ^ { j \bar j } H . \end{align*}


\begin{align*} H _ \delta = \left ( \bigcap _ { g \in G ; n _ g \neq 0 } g H g ^ { - 1 } \right ) \cap H . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ p a _ i f _ i + a _ { p + 1 } s + g \end{align*}


\begin{align*} & = \frac { | h _ 1 | ^ 2 } { \sum _ { x _ i \in \Phi \backslash \{ x _ 1 \} \cap \mathcal { A } } | h _ i | ^ 2 + 1 / \rho } \overset { \Delta } { = } \frac { | h _ 1 | ^ 2 } { I + 1 / \rho } , \end{align*}


\begin{align*} \gamma > \max \left \{ 2 - \frac { 2 } { n } , 1 \right \} \ , \mbox { f o r } \ , n = 1 , 2 , 3 . \end{align*}


\begin{align*} x _ j + \varphi \left ( \sum _ { k = 1 } ^ n a _ { j k } x _ k \right ) , j = 1 , \ldots , n \end{align*}


\begin{align*} C _ { p } ^ { ( 2 ) } : = \lim _ { n \rightarrow \infty } \left \{ \sum _ { k = 1 } ^ { n - 1 } \frac { ( 1 , 2 \pi p k ) _ { k } } { k } - \frac { \log ( n ) } { 1 + \frac { 1 } { \pi p } } \right \} \end{align*}


\begin{align*} \sum _ { v \in X } \deg _ G ( v ) & = \sum _ { v \in X } | N ( v ) \cap X | + \sum _ { v \in X } | N ( v ) \setminus X | \\ & = \left ( \frac { 1 } { 2 } \sum _ { v \in X } | N ( v ) \cap X | + \sum _ { v \in X } | N ( v ) \setminus X | \right ) + \frac { 1 } { 2 } \sum _ { v \in X } | N ( v ) \cap X | \\ & = M + \frac { 1 } { 2 } \sum _ { v \in X } | N ( v ) \cap X | \\ & > \frac { t | X | } { 2 } + \frac { | X | \delta ( G [ X ] ) } { 2 } \\ & \geq \frac { 3 t | X | } { 4 } . \end{align*}


\begin{align*} u = \zeta u _ 1 ^ { m _ 1 } \dots u _ r ^ { m _ r } , \end{align*}


\begin{align*} \partial ^ 2 _ v \psi _ { \mu _ N } = \frac 1 { N + 1 } \sum _ { i = 2 } ^ { N + 1 } \partial ^ 2 _ v \psi _ { \delta _ { x _ i } } ( x _ 1 ) \geq \frac { ( \alpha - \beta ) K _ N } { N + 1 } \geq \frac { \alpha - \beta } { 2 ( N + 1 ) } \ , , \end{align*}


\begin{align*} A \circ B = V ^ * ( A \otimes B ) V . \end{align*}


\begin{align*} \overline { a } _ { i j } : = a _ { i j } \mathbb { I } _ { ( \vert a _ { i j } \vert \le a ) } , \tilde { a } _ { i j } : = a _ { i j } \mathbb { I } _ { ( \vert a _ { i j } \vert > a ) } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { n } \Big ( q _ i ( p - x _ { i } ) \prod _ { \substack { 1 \le j \le n \\ j \neq i } } ( \sum _ { 1 \le k \le d } ( p _ k - x _ { j k } ) ^ 2 ) ^ { \frac { m + 2 } { 2 } } \Big ) = 0 . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ r \sum _ { { \bf j } \in \mathcal S _ i } c _ { i , { \bf j } , \ell } a _ { i , { \bf k } - { \bf j } } = 0 \end{align*}


\begin{align*} e _ r \ , : = \ , \sum _ { \lambda \in \Lambda \smallsetminus \{ 0 \} } \frac { 1 } { \lambda ^ r } \ , \in \mathbb C \ , , \end{align*}


\begin{align*} y _ 0 = \exp \biggl ( \frac { c e ^ l } { 6 l } \biggr ) , \ ; \ ; y _ j = y _ { j - 1 } ^ { e ^ { \alpha } } , , \end{align*}


\begin{align*} \int _ { \tilde { X } } f d \tilde { \nu } & = \int _ { G / \Gamma } \int _ { X } f _ { z } d \phi _ { \tau ( z ) } d m _ { G / \Gamma } ( z ) \\ & = \int _ { G / \Gamma } \int _ { G / P } \int _ { X } f _ { z } d \boldsymbol { \beta } _ { \nu } ( w ) d \nu _ { P } ( \tau ( z ) . w ) d m _ { G / \Gamma } ( z ) . \end{align*}


\begin{align*} \Theta = \sum _ { i = 1 } ^ n \arctan \lambda _ i \end{align*}


\begin{align*} y ( q x ) = ( 1 - x ^ 3 ) y ( x ) , \end{align*}


\begin{align*} \sum _ { \abs { u } \leq \frac { \abs { x } } { 3 } < \abs { v } } = \sum _ { \abs { u } \leq \frac { \abs { x } } { 3 } , \ : \ : \frac { \abs { x } } { 2 } < \abs { v } } + \sum _ { \abs { u } \leq \frac { \abs { x } } { 3 } < \abs { v } \leq \frac { \abs { x } } { 2 } } . \end{align*}


\begin{align*} b _ 1 b _ 2 = e _ 1 \triangleright e _ 2 \triangleright \cdots \triangleright e _ { \ell - 2 } \ , . \end{align*}


\begin{align*} \mathcal E _ 1 \coloneqq \left \{ ( a _ 1 , \ldots , a _ n , b ) \in \mathbb N ^ { n + 1 } : b \leq d , \forall i , a _ i \leq h _ i , \exists i , a _ i = 0 \right \} \end{align*}


\begin{align*} \varphi = V _ n ( \varphi ) \circ T - V _ n ( \varphi ) + \Gamma _ n ( \varphi ) . \end{align*}


\begin{align*} \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \cdot ( \partial _ \nu w ) \nu = H ^ 2 ( \nabla w ) \ , . \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( \frac { 1 } { 1 - z ^ n } \right ) ^ { \frac { m ^ 3 } { n ^ 4 } } = \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( \frac { n ^ 2 } { 4 } + \frac { n ^ 3 } { 3 } + \frac { n ^ 4 } { 4 } \right ) \frac { z ^ n } { n ^ 4 } \right \} \end{align*}


\begin{align*} \Omega _ s = \{ \omega : \exists n _ 1 = n _ 1 ( \omega ) \ : \ : \forall n \geq n _ 1 , \ : \ : \ : \ : C _ { 1 , n } , \ldots , C _ { f ( n + 1 ) , n } \ : \ : \} . \end{align*}


\begin{align*} \begin{bmatrix} | Z ^ * | & Z \\ Z ^ * & | Z | \end{bmatrix} \end{align*}


\begin{align*} \mathcal { A } ( f , f ) = T ( f ^ { 2 } ) - 2 f T ( f ) - 2 B ( A ( f ) , A ( f ) ) \left ( f \in P \right ) \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle \frac { 1 } { 2 } \Lambda \left ( \dfrac { a _ 2 } { m _ 1 } \int _ { \mathbb { R } ^ N } | \nabla \xi | ^ 2 d x + \dfrac { a _ 1 } { 2 m _ 2 } \int _ { \mathbb { R } ^ N } | \nabla \eta | ^ 2 d x \right ) = 0 , \end{array} \right . \end{align*}


\begin{align*} \P [ A _ u ] = \int _ { A _ u } D ( u ) d \widehat { \P } , \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } 0 & = & \varphi ^ { \prime } \left ( t \right ) + \left [ A + \bar { A } + \beta \left ( C + \bar { C } \right ) \right ] ^ { \top } \varphi \left ( t \right ) \\ & & - \Pi _ { 5 } ^ { \top } \left ( t \right ) \Pi _ { 2 } ^ { - 1 } \left ( t \right ) \Pi _ { 4 } ^ { \top } \left ( t \right ) , \\ \varphi \left ( T \right ) & = & 0 \end{array} \right . \end{align*}


\begin{align*} p _ s = \exp \left [ - \frac { \lambda _ d s } { ( \hat { r } ( s ) ) ^ 2 } ( 1 + o ( 1 ) ) \right ] . \end{align*}


\begin{align*} \frac 1 { \omega ( B ) } \| ( b - b _ B ) f _ 1 \| ^ s _ { L ^ s _ \omega } & = \frac 1 { \omega ( B ) } \int _ { 5 B } | b ( y ) - b _ B | ^ s | f | ^ s d \omega ( y ) \\ & \le { \Big ( \frac 1 { \omega ( B ) } \int _ { 5 B } | b ( y ) - b _ B | ^ { s s _ 1 ' } d \omega ( y ) \Big ) ^ { \frac 1 { s _ 1 ' } } } \Big ( \frac 1 { \omega ( B ) } \int _ { 5 B } | f | ^ { s s _ 1 } d \omega ( y ) \Big ) ^ { \frac 1 { s _ 1 } } \\ & \lesssim \ell ( B ) ^ { \beta s } \| b \| ^ s _ { \Lambda ^ \beta } M ^ s _ { s s _ 1 } ( f ) ( x ) \end{align*}


\begin{align*} \partial _ { x _ n } u ( \hat { x } ) = \partial _ { x _ n } u ( \hat { x } ) - \partial _ { x _ n } u ( \hat { x } ^ \prime , 0 ) = \Big ( \int _ { 0 } ^ 1 \partial _ { x _ n } ^ 2 u ( \hat { x } ^ \prime , t x _ n ) \ : t \Big ) x _ n . \end{align*}


\begin{align*} \left . \begin{array} { c } Y \left ( x , y \right ) = \delta \left ( x _ { 1 } - y _ { 1 } \right ) \overline { Y } \left ( \overline { x } , \overline { y } \right ) , \\ \overline { Y } \in L _ { \infty } \left ( \Omega _ { 1 } \times \Omega _ { 1 } \right ) , \left \Vert \overline { Y } \right \Vert _ { L _ { \infty } \left ( \Omega _ { 1 } \times \Omega _ { 1 } \right ) } \leq N _ { 1 } . \end{array} \right . \end{align*}


\begin{align*} T _ { m i x } ( P _ { \phi } ) = \Big ( \frac { e ^ 2 \Delta } { b } \Big ) ^ { 9 + 4 \lceil \frac { 2 \eta } { b } \rceil } \cdot O ( \log n ) . \end{align*}


\begin{align*} A \simeq \begin{pmatrix} X & \ast \\ \ast & \ast \end{pmatrix} . \end{align*}


\begin{align*} \nabla d _ { S _ { i } } \left ( x \right ) = \left ( x - P _ { i } ( x ) \right ) / \left \Vert x - P _ { i } ( x ) \right \Vert . \end{align*}


\begin{align*} \delta \in \left ( 0 , \delta _ { 0 } \right ) , \delta _ { 0 } = \delta _ { 0 } \left ( N , \varepsilon , \Omega , T , c , \rho \right ) = \exp \left ( - \frac { \lambda _ { 1 } d } { \rho } \right ) \in \left ( 0 , 1 \right ) . \end{align*}


\begin{align*} \tilde { S } _ { \gamma _ i \gamma _ i } = M _ { [ 1 ] } \diamond \cdots \diamond M _ { [ | \alpha _ i | ] } . \end{align*}


\begin{align*} \texttt { L H S } \eqref { T 7 . 1 } \le \int _ { t - l } ^ { t } ( t - z ) ^ { \alpha + \beta - 1 / 2 } \ , d z = \int _ 0 ^ { l } z ^ { \alpha + \beta - 1 / 2 } \ , d z \simeq l ^ { \alpha + \beta + 1 / 2 } = \texttt { R H S } \eqref { T 7 . 1 } . \end{align*}


\begin{align*} p _ m ( u , x + 1 ) - p _ m ( u , x ) = \biggl [ m + ( 1 + u ) \frac { \partial } { \partial { u } } \biggr ] p _ { m - 1 } ( u , x ) . \end{align*}


\begin{align*} \| A \chi e ^ { - i t _ 0 h ^ { - 1 } \tilde { P } _ h ( z ) } \tilde { R } _ h ( z ) \chi B \| _ { L ^ 2 \to L ^ 2 } = \mathcal { O } ( h ^ { - 2 } ) . \end{align*}


\begin{align*} \vec U ^ T ( P \otimes P _ { \Omega } ) \vec U _ t + ( \vec U , { \bf D _ { x _ i } } { \bf A _ i } \vec U ) + ( { \bf A _ i } \vec U , { \bf D _ { x _ i } } \vec U ) + ( \vec U , { \vec L _ D ) } = 0 , \end{align*}


\begin{align*} \mathfrak { n } ^ { I } = \oplus _ { \alpha > 0 , \alpha \notin [ I ] } \mathfrak { g } _ { \alpha } , \ \mathfrak { n } ^ { - I } = \oplus _ { \alpha < 0 , \alpha \notin [ I ] } \mathfrak { g } _ { \alpha } . \end{align*}


\begin{align*} \norm { Q _ \psi + s v } _ * = 1 + C s ^ 2 + o ( s ^ 2 ) , s \to 0 . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \hat f ( \{ i \} ) ^ 2 \geq m \frac { p } { 1 - p } \alpha ^ 2 . \end{align*}


\begin{align*} \pi ( x ) = \sum _ { t \in G } \pi _ t ( x ( t ) ) . \end{align*}


\begin{align*} \phi ( _ g F ) = \phi ( F ) \end{align*}


\begin{align*} \tilde \beta _ i = \min \ , \tilde \beta = \min \ , \alpha . \end{align*}


\begin{align*} \mathrm { o s c } ( v , I ^ { ( k ) } ) = O ( e ^ { ( - \lambda _ 1 ( 1 - a ) + \tau ) k } ) c _ 1 ( \varphi ) . \end{align*}


\begin{align*} \Lambda _ { \alpha _ 1 \alpha _ 2 } : = \mathbb Z + \mathbb Z \alpha _ 1 + \mathbb Z \alpha _ 2 + \mathbb Z \alpha _ 1 \alpha _ 2 . \end{align*}


\begin{align*} \begin{aligned} T _ 4 = & s \iint _ { Q } ( A \nabla \sigma ) _ i \frac { \partial A } { \partial x _ i } \nabla u \cdot \nabla u d x d t \ge - C s \lambda \iint _ Q \xi A \nabla u \cdot \nabla u d x d t . \end{aligned} \end{align*}


\begin{align*} H ^ p _ A = \{ f \in \mathcal { S } ' \colon M ^ 0 _ { \phi , A } f \in L ^ p \} , \end{align*}


\begin{align*} \overline { _ N } = \overline { G _ M \cap { \mathcal H } } . \end{align*}


\begin{align*} \Psi _ \Omega ^ H ( r ) = \begin{cases} \sup \limits _ { x \in \partial \Omega } | | \mathcal B ^ H | | _ { L ^ { n - 1 , \infty } ( \partial \Omega \cap B _ r ( x ) ) } \quad n \geq 3 , \\ \\ \sup \limits _ { x \in \partial \Omega } | | \mathcal B ^ H | | _ { L ^ { 1 , \infty } \log L ( \partial \Omega \cap B _ r ( x ) ) } \quad n = 2 \end{cases} \end{align*}


\begin{align*} { { \boldsymbol { \hat x } } _ { k ; t + 1 } } = { { \boldsymbol { \hat x } } _ { k | k - 1 } } + { { \boldsymbol { K } } _ { k ; t } } \left ( { { { \boldsymbol { y } } _ k } - { { { \boldsymbol { \hat y } } } _ { k | k - 1 } } } \right ) \end{align*}


\begin{align*} \begin{aligned} F ( k , t + 1 , \mathbf { p } _ { n + 1 } ) = F ( k , t , \mathbf { p } _ { n + 1 } ) - p _ { k } ^ { } f ( k , t , \mathbf { p } _ { n + 1 } ) , \end{aligned} \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \alpha _ i z _ { i p } = \sum _ { i = 1 } ^ n \gamma _ i z _ { i p } < \sum _ { i = 1 } ^ n \beta _ i z _ { i p } , \end{align*}


\begin{align*} \mathbf { G } ( R ) : = \{ g \in \mathrm { G L } _ { 2 n } ( R ) \mid g ^ { \mathrm { t } } J _ n g = \nu ( g ) J _ n \ , \nu ( g ) \in R ^ \times \} . \end{align*}


\begin{align*} f \sharp g ( \xi ) \sim \sum _ \alpha \frac { ( - i ) ^ { | \alpha | } } { \alpha ! } ( \partial _ \xi ^ \alpha f ) ( \xi ) \partial _ x ^ \alpha | _ { x = 0 } \Big ( \alpha _ { - x } \big ( g ( \xi + B x ) \big ) \Big ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \sup _ { x \in M _ + , x \neq 0 } \abs { ( S _ n ( \nu _ k ) - \bar { \nu } _ k ) ( x ) } / \rho ( x ) & = \lim _ { n \to \infty } \sup _ { x \in M _ + , x \neq 0 } \abs { ( S _ n ( \nu _ k ) - \bar { \nu } ) ( x ) } / \rho ( x ) = 0 . \end{align*}


\begin{align*} W _ { h , q } ( i ) = \begin{cases} h & i = q ; \\ 1 & k - j + h \leq i \leq k - 2 i \neq q \\ l - j + 1 & i = k - 1 ; \\ 0 & . \end{cases} \end{align*}


\begin{align*} \mathcal { A } ( f , f ) = \mathcal { Q } ( f ) \left ( f \in P \right ) \end{align*}


\begin{align*} \begin{pmatrix} 2 & - 1 \\ - 3 & 2 \end{pmatrix} . \end{align*}


\begin{align*} f ' _ \psi ( 0 ) = 0 f '' _ \psi ( 0 ) = C ' _ \circ ( \psi ) > 0 . \end{align*}


\begin{align*} 2 ( \log 2 ) ^ 3 - 7 \zeta ( 3 ) = - 2 4 L i _ 3 \left ( \frac { 1 } { 2 } \right ) + 1 8 L i _ 3 \left ( \frac { 1 } { 4 } \right ) + 4 L i _ 3 \left ( \frac { 1 } { 8 } \right ) - L i _ 3 \left ( \frac { 1 } { 6 4 } \right ) , \end{align*}


\begin{align*} u ^ n - U ^ n = u ^ n - R _ h u ^ n + R _ h u ^ n - U ^ n = \rho ^ n + \theta ^ n , \end{align*}


\begin{align*} \widetilde { \rho } ( t , \kappa ( t ) ) = 0 { \rm \ f o r \ } t \ge 0 , \end{align*}


\begin{align*} | X Y | ^ 2 = \mathrm { t r } \big ( X Y ( X Y ) ^ t \big ) = \mathrm { t r } ( X Y Y X ) = \mathrm { t r } ( X ^ 2 Y ^ 2 ) \leq \l _ { { \rm m a x } } ^ 2 \mathrm { t r } ( Y ^ 2 ) = \l _ { { \rm m a x } } ^ 2 | Y | ^ 2 , \end{align*}


\begin{align*} G ( \Psi ) : = ( 4 - 2 d ) \omega Q ( \Psi ) + ( 3 - d ) \mathbf { c } \cdot \mathbf { P } ( \Psi ) . \end{align*}


\begin{align*} \lambda _ { 1 + j } \left ( \oplus ^ m A \right ) = \lambda _ { \langle ( 1 + j ) / m \rangle } ( A ) \end{align*}


\begin{align*} \overline { c } _ { X ^ { \ast } } ( g , g ' ) = \overline { c } _ { X ^ { \ast } } ( g ^ { \alpha } , g ^ { ' \alpha } ) \nu _ 2 ( \alpha , g ) \nu _ 2 ( \alpha , g ' ) \nu _ 2 ( \alpha , g g ' ) ^ { - 1 } . \end{align*}


\begin{align*} [ \partial x _ \lambda y ] & = - \lambda [ x _ \lambda y ] , [ x _ \lambda \partial y ] = ( \partial + \lambda ) [ x _ \lambda y ] ( c o n f o r m a l \ s e s q u i l i n e a r i t y ) , \\ { } [ x _ \lambda y ] & = - ( - 1 ) ^ { | x | | y | } [ y _ { - \lambda - \partial } x ] ( s k e w s y m m e t r y ) , \\ { } [ x _ \lambda [ y _ \mu z ] ] & = [ [ x _ \lambda y ] _ { \lambda + \mu } z ] + ( - 1 ) ^ { | x | | y | } [ y _ \mu [ x _ \lambda z ] ] ( J a c o b i \ i d e n t i t y ) , \end{align*}


\begin{align*} \varphi & = g _ 1 * \Phi _ 1 ( \varphi ) + \ldots + g _ N * \Phi ( \varphi ) \\ & = g _ i * ( \rho \cdot \Psi _ 1 ( \varphi ) ) + \ldots + g _ N * ( \rho \cdot \Psi _ N ( \varphi ) ) \\ & = f _ 1 * \Psi _ 1 ( \varphi ) + \ldots + f _ N * \Psi ( \varphi ) \end{align*}


\begin{align*} F _ i ( U ^ n , \ , d ) \ , = \ , 0 , \mbox { f o r } \ ; 1 \le i \le M + 1 , \end{align*}


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


\begin{align*} W _ i = \begin{cases} 1 , & \ ~ \alpha , \\ 0 , & \ ~ 1 - \alpha . \end{cases} \end{align*}


\begin{align*} H _ { T ^ { n + 1 } } ^ { * } ( P l ( k , n ) ; \mathbb { Q } ) = \Big \{ \alpha \in \bigoplus _ { i = 0 } ^ m \mathbb { Q } [ y _ 1 , \dots , y _ n , z ] / ( Y _ { \lambda ^ i } + z ) ~ \big { | } ~ & ( \alpha _ { i } - \alpha _ { j } ) \in ( Y _ { \lambda ^ i } + z , Y _ { \lambda ^ j } + z ) \\ & | \lambda ^ i \cap \lambda ^ j | = k - 1 \Big \} . \end{align*}


\begin{align*} T _ 5 \geq & - C s ^ 2 \lambda ^ 4 \iint _ Q \xi \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 | u | ^ 2 d x d t - C \lambda ^ 2 \iint _ { Q } \xi \left | A \nabla u \cdot \nabla \eta \right | ^ 2 d x d t \\ & - C s ^ 2 \lambda ^ 3 \int _ 0 ^ T \int _ { \omega } \xi | u | ^ 2 d x d t - C \lambda \iint _ Q \xi A \nabla u \cdot \nabla u d x d t . \end{align*}


\begin{align*} \frac { 1 } { q } : = \frac { 1 } { p } - \frac { 1 } { 2 } . \end{align*}


\begin{align*} \bar { S } _ { j , + , \Delta } ^ k ( t ) : = \nu \Delta \sum _ { \ell = 0 } ^ { \lfloor t / \Delta \rfloor } Z _ 0 ( \ell \Delta ) p _ { j , + } ^ k ( t - \ell \Delta ) . \end{align*}


\begin{align*} \mathcal { C } _ { \omega , \mathbf { c } } : = \left \{ \left . \left ( ( \sqrt { \omega } - \tau ) ^ 2 , \frac { \mathbf { c } } { \sqrt { \omega } } ( \sqrt { \omega } - \tau ) \right ) \ \right | \ \tau < \sqrt { \omega } \ \right \} \end{align*}


\begin{align*} \ K ( u ^ * , v ^ * ) \leq K ( u , v ) , \ L ( u ^ * , v ^ * ) \leq L ( u , v ) = 0 . \end{align*}


\begin{align*} \log \log \Pr [ \hat { M } = M ] ^ { - 1 } \geq ( 1 - \beta ) \log n + \log \log n + \omega ( 1 ) , \end{align*}


\begin{align*} \Q ( \alpha ) = \{ a _ 0 + a _ 1 \alpha + \dots + a _ { n - 1 } \alpha ^ { n - 1 } | a _ i \in \Q \} . \end{align*}


\begin{align*} \tilde { \mathcal { X } } = S ( \mathcal { U } ) - \mathcal { K } _ { K , \zeta ^ * } . \end{align*}


\begin{align*} U ^ T ( n _ 1 A + n _ 2 B ) U = U ^ T _ n \tilde A U _ n = W ^ T \Lambda W = ( W ^ + ) ^ T \Lambda ^ + W ^ + + ( W ^ - ) ^ T \Lambda ^ - W ^ - \end{align*}


\begin{align*} A _ N \begin{pmatrix} D _ { N } \\ W _ N \end{pmatrix} = \begin{pmatrix} S ^ 1 _ { N } \\ S ^ 2 _ N \end{pmatrix} , \ \ N \ge 2 , \end{align*}


\begin{align*} \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} = U \begin{bmatrix} A & 0 \\ 0 & 0 \end{bmatrix} U ^ * + V \begin{bmatrix} 0 & 0 \\ 0 & B \end{bmatrix} V ^ * , \end{align*}


\begin{align*} \mathcal { W } = \bigl \{ h \colon \ , [ 0 , 1 ] ^ 2 \to [ 0 , 1 ] \colon \ , h ( x , y ) = h ( y , x ) \ , \ , \forall \ , x , y \in [ 0 , 1 ] \bigr \} \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( 1 - z ^ n \right ) ^ { \frac { m ^ 1 } { n ^ 2 } } = \sqrt { 1 - z } \ ; \exp \left \{ \frac { - 1 } { 2 } \frac { z } { 1 - z } \right \} . \end{align*}


\begin{align*} \mathfrak { h } ^ * _ j ( \varphi ) = \lim _ { l \to \infty } Q ( 0 , l ) ^ { - 1 } \circ P _ { U ^ { ( l ) } _ { - j } } \circ \mathcal { M } ^ { ( l ) } \circ S ( l ) ( \varphi ) \in U _ { - j } \end{align*}


\begin{align*} \nu _ q ( f ) = \min \left \{ \nu \left ( f _ i q ^ i \right ) \right \} , \end{align*}


\begin{align*} \alpha ' _ n = ( 1 - a ) \left ( \frac { \alpha _ n } { 1 - a q ^ { 2 n } } - \frac { a q ^ { 2 n - 2 } \alpha _ { n - 1 } } { 1 - a q ^ { 2 n - 2 } } \right ) , \beta ' _ n = \beta _ n , \end{align*}


\begin{align*} Q _ { { \mathcal { H } } _ { n s } } \simeq \begin{pmatrix} I & 0 \\ R & 0 \end{pmatrix} \end{align*}


\begin{align*} \tau \circ ( 1 \otimes T ) \Delta ( l ) = \tau \circ ( l _ 1 \otimes T ( l _ 2 ) ) = T ( l _ 2 ) \otimes l _ 1 = ( T \otimes 1 ) ( l _ 2 \otimes l _ 1 ) = ( T \otimes 1 ) \tau \circ \Delta ( l ) . \end{align*}


\begin{align*} & \vec L _ D = ( I _ n \otimes P _ { \Omega } ) ^ { - 1 } ) ( D C ) \vec L _ C & & \ \ \ D C = ( I _ n \otimes E ^ T ) ( P _ { e r m } ) ^ T ( P _ { \partial \Omega } \otimes I _ n ) , & \\ & \vec L _ C = ( ( L _ C ) _ 1 , ( L _ C ) _ 2 , . . . , ( L _ C ) _ N ) ^ T & & ( L _ C ) _ j = ( 2 ( J ^ - T ^ { - 1 } ) ^ T \Sigma ( \sqrt { | \Lambda ^ - | } \vec W ^ - - R \sqrt { \Lambda ^ + } \vec W ^ + - S \vec G ) ) _ j . & \end{align*}


\begin{align*} \tilde { U } _ { \epsilon , \xi } ( x ) = S ^ { \frac { ( N - \mu ) ( 2 - N ) } { 4 ( N - \mu + 2 ) } } \bigl ( C ( N , \mu ) \bigr ) ^ { \frac { 2 - \mu } { 2 ( N - \mu + 2 ) } } U _ { \epsilon , \xi } ( x ) , \end{align*}


\begin{align*} \int _ 0 ^ 1 ( 1 - y ) y ^ { j - 1 } d y = \frac 1 { j ( j + 1 ) } . \end{align*}


\begin{align*} \widetilde { Q } _ { \alpha , z } ( \psi _ 1 \bar { \otimes } \psi _ 2 | | \varphi _ 1 \bar { \otimes } \varphi _ 2 ) = \widetilde { Q } _ { \alpha , z } ( \psi _ 1 | | \varphi _ 1 ) \widetilde { Q } _ { \alpha , z } ( \psi _ 2 | | \varphi _ 2 ) \end{align*}


\begin{align*} ( \widetilde { \alpha } \otimes \widetilde { \varphi } ) \circ \widetilde { \Delta } ( x , m ) = ( \alpha \otimes \varphi _ L ) \Delta ( x ) + ( \beta \otimes \varphi _ L ) ( \rho _ { \beta } ( m ) ) - ( \alpha _ M \otimes \varphi _ L ) ( \tau \circ ( \rho _ { \beta } ( m ) ) ) . \end{align*}


\begin{align*} 0 > \mathcal U ' ( r ) \cdot \frac { u _ 0 ( r ) } { u _ 0 ' ( r ) } = \frac { u ' ( r ) u _ 0 ( r ) - u _ 0 ' ( r ) u ( r ) } { u _ 0 ( r ) u _ 0 ' ( r ) } = \mathcal W ( r ) - \mathcal U ( r ) , \end{align*}


\begin{align*} f ' = f _ 0 ' + f _ 1 Q _ i ' + f _ 1 ' Q _ i + \ldots + f _ r ' Q ^ r + r f _ r Q _ i ' Q _ i ^ { r - 1 } . \end{align*}


\begin{align*} V _ A ^ { } ( \mathbf { h } _ t , a _ t ) & = ( 1 - \eta ) V _ A ( \mathbf { h } _ t , a _ t ) + \eta \big ( r _ t + \gamma _ { _ { } } \max _ { a _ { t + 1 } \in \mathcal { A } } V _ B ( \mathbf { h } _ { t + 1 } , a _ { t + 1 } ) \big ) , \end{align*}


\begin{align*} Y _ { n + 1 } = ( A \otimes I ) Y _ n + h \Phi ( t _ n , Y _ n , h ) \quad ( n \in \mathbb { N } ) , \end{align*}


\begin{align*} T ( f \cdot g ) = T ( f ) \cdot g + f \cdot T ( g ) + 2 A ( f ) \cdot A ( g ) \end{align*}


\begin{align*} \frac { 2 \pi \delta } { m } \le \frac { t \delta } { M + ( j - 1 ) m } \le \frac { 2 \pi \delta } { \beta m } = \frac { 2 \pi \delta } { m } + \frac { 2 \pi \delta } { 8 m ^ { 2 } } , \end{align*}


\begin{align*} \overline { S } ( k , n ) \overset { } { = } \sum _ { j = k + 1 } ^ n \binom { n } { j } = 2 ^ n - \sum _ { j = 0 } ^ { k } \binom { n } { j } \end{align*}


\begin{align*} \gamma _ R ( x ) : = \gamma \left ( \frac { | x | } { R } \right ) \textrm { f o r a l l } \ ; \ ; x \in \R ^ N \ , . \end{align*}


\begin{align*} \Pi : = \big \{ \psi _ t ( \Sigma _ t ) \ , | \ , \{ \psi _ t \} _ { t \in I ^ n } \ , \textrm { i s a n i s o t o p y w i t h } \ , \psi _ t = \mathrm { i d } \ , \mathrm { f o r } \ , t \in \partial I ^ n \big \} . \end{align*}


\begin{align*} E ( U ^ \top A U ) = \sum _ { i , j } a _ { i j } E ( u _ i u _ j ) = ( A ) , \end{align*}


\begin{align*} - \Delta _ p v = 0 \ , \ , B _ r ( x _ 0 ) . \end{align*}


\begin{align*} \mathcal { M } _ { q _ 1 , \alpha } f = \mathcal { M } _ { q _ 2 , \alpha } f , f \in H ^ { s _ M } ( \Omega ^ c ) \end{align*}


\begin{align*} \begin{cases} \partial _ { t } z - \sum _ { i , j = 1 } ^ n \partial _ { x _ i } ( A _ { i j } \partial _ { x _ j } z ) + b z = \chi _ { \omega } g , & \mbox { i n } \ Q , \\ z = 0 \ \mbox { o r } \ A \nabla z \cdot \nu = 0 , & \mbox { o n } \ \Sigma , \\ z ( 0 ) = z _ { 0 } , & \mbox { i n } \ \Omega . \end{cases} \end{align*}


\begin{align*} \beta ( \xi , 0 ) + \sum _ \mathbf { n } \beta ( \mathbf { n } ) = 1 , \end{align*}


\begin{align*} { \mathrm { T r \ , } } & \left \{ g ( X ^ * A X + Y ^ * B Y ) + g ( Y ^ * A Y + X ^ * B X ) \right \} \\ & \le { \mathrm { T r \ , } } \left \{ X ^ * g ( A ) X + Y ^ * g ( B ) Y ) + Y ^ * g ( A ) Y + X ^ * g ( B ) X \right \} \\ & = { \mathrm { T r \ , } } \left \{ ( g ( A ) + g ( B ) ) ( X X ^ * + Y Y ^ * ) \right \} = { \mathrm { T r \ , } } \left \{ g ( A ) + g ( B ) \right \} \end{align*}


\begin{align*} w _ b = \Phi _ b + T ( f ) _ b , \end{align*}


\begin{align*} T : H _ \Gamma ^ { 3 / 2 } ( \partial \mathrm { M } ) \rightarrow H : \varphi \mapsto T \varphi : = u _ { | \mathrm { M } _ 0 } , \end{align*}


\begin{align*} Q ( x , y , z ) = p d d _ 2 ^ { 2 } x ^ 2 + p d d _ 1 ^ { 2 } y ^ 2 + z ^ 2 - t p x y , \end{align*}


\begin{align*} \chi _ + : U _ + \to \R & & \chi _ + ( x , y ) & = x \\ \chi _ - : U _ - \to \R & & \chi _ - ( x , y ) & = \begin{cases} x & x \leq 0 \\ \theta _ + ( x ) & x > 0 . \end{cases} \end{align*}


\begin{align*} \textbf { R } = \rho _ { f } \hat { \textbf { G } } ^ { T } \textbf { P } \textbf { P } ^ { H } \hat { \textbf { G } } ^ { \ast } \left ( \rho _ { f } \tilde { \textbf { G } } ^ { T } \textbf { P } \textbf { P } ^ { H } \tilde { \textbf { G } } ^ { \ast } + \sigma _ { w } ^ { 2 } \textbf { I } _ K \right ) ^ { - 1 } \end{align*}


\begin{align*} \| f ^ { = d } \| _ 2 ^ 2 \le \left ( \frac { 3 3 r ' _ d } { d } \right ) ^ d \gamma _ 1 \| f ^ { = d } \| _ 2 \log ^ d \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) . \end{align*}


\begin{align*} u _ m = \phi _ m - h _ m - \omega _ m , \ \| u _ m \| \rightarrow 1 . \end{align*}


\begin{align*} E _ { + } ^ * : = \lim \limits _ { t \to - \infty } ( d \varphi ^ t ( x ) ) ^ { T } \tau _ { x _ 0 \to \varphi ^ t ( x ) } E _ u ^ * ( x _ 0 ) \end{align*}


\begin{align*} \frac { \sum _ { v \in \mathcal { L } } f ( v ) ^ 2 } { \sum _ { n \in \mathcal { M } } f ( n ) ^ 2 } = 1 - o ( 1 ) , \textrm { a s } N \to \infty . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c ) = 1 \\ a , b , c \geq 1 } } \left ( \frac { 1 } { 1 - x ^ a y ^ b z ^ c } \right ) ^ { \frac { c ^ 2 } { a b ^ 2 } } = \exp \left \{ L i _ 1 ( x ) L i _ 2 ( y ) L i _ { - 2 } ( z ) \right \} \end{align*}


\begin{align*} X = \prod _ { j = 1 } ^ { \infty } [ a _ { j } , b _ { j } ] = \bigl \{ \mathcal { N } = ( n _ { j } ) _ { j \ge 1 } : a _ { j } \le n _ { j } \le b _ { j } \bigr \} . \end{align*}


\begin{align*} L i _ 2 \left ( - \frac { 1 } { 8 } \right ) + L i _ 2 \left ( \frac { 1 } { 9 } \right ) = - \frac { 1 } { 2 } \left ( \log \left ( \frac { 9 } { 8 } \right ) \right ) ^ 2 . \end{align*}


\begin{align*} \varGamma _ L = \{ g \in \mathrm { S O } ^ + ( L ) \mid g \lambda - \lambda \in L \hbox { f o r a l l } \lambda \in L ' \} . \end{align*}


\begin{align*} \det \big ( d _ z \phi _ t ( p , \cdot ) \big ) = \sum _ { i = 0 } ^ N a _ i ( z ) t ^ i + t ^ { N + 1 } R _ N ( t , z ) , \forall \ , z \in V , \end{align*}


\begin{align*} \sum _ { i = 0 } ^ { \mathrm { d e g } ( h ) } h _ i t ^ i & = \sum _ { i = 0 } ^ { d _ 0 } h _ i t ^ i = \sum _ { i = 0 } ^ { p d + p - 1 } h _ i t ^ i \ , h _ i = 0 i > \mathrm { d e g } ( h ) \\ & = \sum _ { \theta = 0 } ^ { d } \left ( h _ { p \theta } t ^ { p \theta } + h _ { p \theta + 1 } t ^ { p \theta + 1 } + \cdots + h _ { p \theta + p - 1 } t ^ { p \theta + p - 1 } \right ) \\ & = \sum _ { \ell = 0 } ^ { p - 1 } \sum _ { \theta = 0 } ^ { d } h _ { p \theta + \ell } t ^ { p \theta + \ell } . \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( \frac { 1 } { 1 - y ^ m z ^ n } \right ) ^ { \frac { m ^ 4 } { n ^ 5 } } \end{align*}


\begin{align*} \lim _ { \rho \to 0 } \frac { 1 } { \rho } F ( u , B _ { \rho } ( x ) ) = 0 , \end{align*}


\begin{align*} \begin{cases} \displaystyle \frac { d } { d t } x _ \pm ( X _ \pm , t ) = u _ \pm ( x _ \pm ( X , t ) , t ) , t > 0 , \\ x _ \pm ( X _ \pm , 0 ) = X , \end{cases} \end{align*}


\begin{align*} Z _ 3 ( H ) = \overline { \{ ( p , q ) \in X \times X \ , | \ , \exists \ , C \in | H | \mbox { s m o o t h s . t . } p , q \in C , \ , 3 [ p - q ] = 0 \in J C \} } \subseteq X \times X \end{align*}


\begin{align*} s = \sum _ { i = 1 } ^ n x _ { i , i ^ * } , \end{align*}


\begin{align*} \hat F = \frac { 1 } { 2 } - \frac { 1 } { 2 } \tanh \left ( \frac { 2 \ , | k | \ , \pi / N _ S - \xi _ 0 } { d } \right ) \ , . \end{align*}


\begin{align*} L _ \R = \Bigl ( \bigoplus _ { j = 0 } ^ r \langle z _ j , z _ j ' \rangle \Bigr ) \oplus ( L _ { r + 1 } ) _ \R \end{align*}


\begin{align*} \Delta ^ \delta ( \tau ) = \prod _ { D \mid N } \Delta ( D \tau ) ^ { n _ D } . \end{align*}


\begin{align*} M _ a ( \cdot ) : = \begin{cases} \frac { 1 } { a ^ d } \sum _ { 0 \leq i _ 1 < a } \cdots \sum _ { 0 \leq i _ d < a } T _ 1 ^ { i _ 1 } \cdots T _ d ^ { i _ d } ( \cdot ) & G = \Z _ + ^ d , ~ a \in \N , \\ \\ \frac { 1 } { a ^ d } \int _ { Q _ a } T _ t ( \cdot ) d t & G = \R _ + ^ d , ~ a \in \R _ + . \end{cases} \end{align*}


\begin{align*} C _ { p } ^ { ( 1 ) } : = \lim _ { n \rightarrow \infty } \left \{ \sum _ { j = 1 } ^ { n - 1 } \frac { p ^ { 2 } + \lambda _ { j } ^ { 2 } } { p \left ( p + \frac { 1 } { \pi } \right ) + \lambda _ { j } ^ { 2 } } \cdot \frac { 1 } { \lambda _ { j } } - \log \left ( \lambda _ { n } \right ) \right \} . \end{align*}


\begin{align*} \left | \begin{matrix} x & y \\ z & w \end{matrix} \right | = x w - y z = x ^ 3 - \frac { a } { p ' x } \ , p ' \left ( \frac { a } { p ' x } \right ) ^ 2 = x ^ 3 - \frac { a ^ 3 } { p '^ 2 x ^ 3 } = x ^ 3 - x ^ 3 = 0 , \end{align*}


\begin{align*} ( \partial _ t - \partial _ x ^ 2 ) u = u \xi + c _ { e _ { ( \xi , 0 ) } + 2 e _ { ( \xi , e _ 0 ) } } u . \end{align*}


\begin{align*} \sup _ { \pi \in \mathcal { G } _ { \lambda } } \int _ { \mathcal { V } \times \mathcal { S } } \phi _ \lambda \ , d \pi = \sup _ { \pi \in \Gamma \left ( \Pi ( \mu _ 1 , \mu _ 2 ) , \phi _ \lambda \right ) } \int _ { \mathcal { V } \times \mathcal { S } } \phi _ { \lambda } \ , d \pi \end{align*}


\begin{align*} \sum _ { n \geq 0 } g _ n \ , \alpha ^ { n - 1 } \ , : = \ , \frac { 1 } { \alpha } \ , \exp \Big ( - \sum _ { r \geq 1 } \frac { ( - \alpha ) ^ r } { r } \ , ( E _ r - e _ r ) \Big ) \ , . \end{align*}


\begin{align*} & \left | \frac { \delta F } { \delta \mu } ( \nu _ t , \mu _ t , y ) - \frac { \sigma ^ 2 } { 2 } \log \left ( \frac { \mu _ t ( y ) } { \rho ( y ) } \right ) + \frac { \sigma ^ 2 } { 2 } \operatorname { D _ { K L } } ( \mu _ t | \rho ) \right | \\ & \leq 3 C _ { \mu } + \frac { \sigma ^ 2 } { 2 } \left ( \max \{ | \log r _ { 1 , \mu } | , \log R _ { 1 , \mu } \} + 2 \log R _ { \mu } \right ) = : C _ { V , \mu } . \end{align*}


\begin{align*} d a = \rho ( a ) ( d ) \end{align*}


\begin{align*} R _ { i j k t , i } = R ^ \varphi _ { j t , k } - R ^ \varphi _ { j k , t } + \alpha ( \varphi ^ a _ { j k } \varphi ^ a _ t - \varphi ^ a _ { j t } \varphi ^ a _ k ) \ , . \end{align*}


\begin{align*} v \left ( x , t \right ) = \widetilde { u } _ { t } \left ( x , t \right ) , q \left ( x , t \right ) = \widetilde { m } _ { t } \left ( x , t \right ) . \end{align*}


\begin{align*} \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} = U \begin{bmatrix} A & 0 \\ 0 & 0 \end{bmatrix} U ^ * + V \begin{bmatrix} 0 & 0 \\ 0 & B \end{bmatrix} V ^ * \end{align*}


\begin{align*} E _ t + ( u E + u p - u S + q ) _ x = 0 , \end{align*}


\begin{align*} E \big | S _ { j , + } ( t ) - \hat { S } _ { j , + } ( t ) \big | & \leq \sum _ { k = 1 } ^ \infty E \big | S ^ k _ { j , + } ( t ) - \hat { S } ^ k _ { j , + } ( t ) \big | = O \big ( t e ^ { \lambda t / 2 } \big ) , \end{align*}


\begin{align*} b _ { I , J } ^ G = \sum _ { i \in I } b _ { i , j } \end{align*}


\begin{align*} ( \widetilde { \Delta } _ { 2 } ) = O ( N ^ { - 2 } p ^ { - 2 } ) , \end{align*}


\begin{align*} \mathcal { I } ( \delta ) = \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \left [ \langle \lambda , \delta \rangle + \sup _ { \varpi \in \Pi ( \mu _ { 1 3 } , \mu _ { 2 3 } ) } \int _ { \mathcal { V } } f _ { \theta , \lambda } \ , d \varpi \right ] , \end{align*}


\begin{align*} \Omega _ 1 ( \mathcal { G } _ 1 ( \mathbb { Q } ) ) = \Omega _ 2 ( \mathcal { G } _ 2 ( \mathbb { Q } ) ) . \end{align*}


\begin{align*} \lim \limits _ { t \rightarrow T ^ * } G ( t ) = \lim \limits _ { t \rightarrow T ^ * } \int _ { \mathbb { R } ^ N } | x | ^ 2 \left ( a _ 2 | \phi | ^ 2 + a _ 1 | \psi | ^ 2 \right ) d x = 0 . \end{align*}


\begin{align*} \P _ { N , r _ N } ( U _ \varepsilon ^ x ) = \P _ { N , h _ N } ( U _ \varepsilon ^ x ) \ , \frac { 1 } { \P _ { N , h _ N } ( U _ \varepsilon ^ x ) } \int _ { U _ \varepsilon ^ x } \exp \left ( - \log \frac { \d \P _ { N , h _ N } } { \d \P _ { N , r _ N } } \right ) \ , \d \P _ { N , h _ N } . \end{align*}


\begin{align*} g _ i \triangleright g _ j & = g _ { ( 1 - \alpha ) i + \alpha j } = g _ { j + ( 1 - \alpha ) ( i - j ) } , \\ g _ i ^ { - 1 } \triangleright g _ j & = g _ { j + ( 1 - \alpha ^ { - 1 } ) ( i - j ) } \end{align*}


\begin{align*} \pi _ n ( \cdot ) = \frac { \pi \left ( \cdot \cap ( A _ { 1 n } \cap A _ { 2 n } ) \right ) } { \pi ( A _ { 1 n } \cap A _ { 2 n } ) } , \end{align*}


\begin{align*} u = F _ { u } \circ \phi ( x , v _ { [ 0 , R - \kappa ] } ) \ , , \end{align*}


\begin{align*} \mathcal { I } = \{ I _ 1 , I _ 2 , \dots \} \end{align*}


\begin{align*} \frac { 1 } { T _ i ^ { \infty } } + \frac { 1 } { \theta _ i ^ 2 T ^ * _ i ( u ) } = \frac { 1 } { T _ i ( u ) } . \end{align*}


\begin{align*} \mathfrak { S } ( f _ k ) + \overline { \mathfrak { S } } ( f _ k ) = n + 1 . \end{align*}


\begin{align*} \gamma _ { n } : = \inf _ { \mathcal { A } \in \mathbb { F } _ n } \sup _ { u \in \mathcal { A } } G ( u ) . \end{align*}


\begin{align*} P _ { \ell , M } = \Delta _ \ell ^ { \ , \vee } . \end{align*}


\begin{align*} \partial _ { x _ n } u ( \hat { x } ) _ { \big | \{ x _ n = 0 \} } = \partial _ { x _ n } \big ( c \circ F _ \gamma \big ) ( \hat { x } ) _ { \big | \{ x _ n = 0 \} } = \nabla c ( x ^ \prime , \gamma ( x ^ \prime ) ) \cdot ( - \mathbf { n } ( x ^ \prime , \gamma ( x ^ \prime ) ) ) = 0 , \end{align*}


\begin{align*} L _ \R = W _ \R \oplus \langle z , z ^ \ast \rangle , W _ \R = \langle z ^ + \rangle ^ \perp \cap \langle z ^ - \rangle ^ \perp . \end{align*}


\begin{align*} \int _ { S _ { y } } D \left ( \alpha _ { x , g } \parallel \alpha _ { x , e } \right ) d \eta ^ { y } ( x ) = \sup _ { n } \left \{ \int _ { S _ { y } } H _ { q _ { x , g } ^ { n } \parallel q _ { x , e } ^ { n } } \left ( \mathcal { P } _ { x , n } \right ) d \eta ^ { y } - \delta _ { n } \right \} . \end{align*}


\begin{align*} | \nabla _ g \rho | = 1 \quad \mbox { a n d } \quad \Delta _ g \rho = ( n - 1 ) \kappa \coth ( \kappa \rho ) = p s \coth ( \kappa \rho ) . \end{align*}


\begin{align*} I _ \gamma : = \{ y \in L \ | \ v ( y ) \ge a a \in \gamma \} . \end{align*}


\begin{align*} \phi ( X E _ { i i } X ^ * \otimes B ) = W _ X ( E _ { i i } \otimes \varphi _ { i , X } ( B ) ) { W } _ X ^ * \hbox { f o r a l l \quad } B \in M _ n , \end{align*}


\begin{align*} \mu _ { p } ( h ) \mu _ { p } ( g ) \leq e \mu _ { p } ( h ) \mu _ { p } ( g ) \sum _ { d = 1 } ^ n \left ( \frac { p } { 1 - p } \ , 4 0 0 \ , r \ , \sqrt { q \cdot A } \right ) ^ d . \end{align*}


\begin{align*} \tfrac { 1 } { \sqrt { c ^ 2 + d ^ 2 } } \tfrac { d \det g } { \sqrt { c ^ 2 + d ^ 2 } } + ( \tfrac { b d + a c } { \sqrt { c ^ 2 + d ^ 2 } } ) \tfrac { c } { \sqrt { c ^ 2 + d ^ 2 } } & = \tfrac { d \det g } { \sqrt { c ^ 2 + d ^ 2 } } + \tfrac { c b d + a c ^ 2 } { c ^ 2 + d ^ 2 } \\ & = \tfrac { d a d - d b c + c b d + a c ^ 2 } { c ^ 2 + d ^ 2 } = a ; \end{align*}


\begin{align*} W _ { K _ 1 K _ 2 K _ 3 K } ( s ) & = \frac { 1 } { ( 2 \pi ) ^ 8 } | \beta _ h ( s ) | ^ 2 \beta _ h ( s ) \overline { \beta _ { \sigma } ( s ) } \int _ { \mathbb { R } ^ 2 } F _ { K _ 1 K _ 2 K _ 3 K } ( s , x ) \dd x \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( 1 - y ^ m z ^ n \right ) ^ { \frac { m ^ 1 } { n ^ 2 } } = \left ( \frac { 1 } { 1 - y z } \right ) ^ { \frac { y } { 1 - y } } \exp \left \{ - \frac { y } { ( 1 - y ) ^ 2 } \left ( L i _ 2 ( z ) - L i _ 2 ( y z ) \right ) \right \} . \end{align*}


\begin{align*} \big [ D \widetilde { \varphi _ 1 \bar { \otimes } \varphi _ 2 } : D \tau _ { M _ 1 \bar { \otimes } M _ 2 } \big ] _ t = \lambda ^ { \varphi _ 1 } ( t ) \otimes \lambda ^ { \varphi _ 2 } ( t ) , t \in \mathbb { R } \end{align*}


\begin{align*} & M ( \lambda ) : = \left \{ \left ( \phi _ { 2 } ( \lambda ) - \phi _ { 1 } ( \lambda ) , V \right ) \mid V \geq 0 \right \} ; \\ & M _ { \Phi } ( \lambda ) : = \left \{ \left ( \phi _ { 2 } ( \lambda ) - \phi _ { 1 } ( \lambda ) , \Phi \left ( \phi _ { 2 } ( \lambda ) - \phi _ { 1 } ( \lambda ) ; \lambda , \alpha \right ) \right ) \mid \alpha \geq 0 \right \} . \end{align*}


\begin{align*} s _ 1 = \sum _ { j , k = 1 } ^ n R ( e _ j , \bar e _ j , e _ k , \bar e _ k ) , s _ 2 = \sum _ { j , k = 1 } ^ n R ( e _ j , e _ k , \bar e _ k , \bar e _ j ) . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta ) = \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \sup _ { \pi \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \left [ \langle \lambda , \delta \rangle + \int _ { \mathcal { V } } g _ \lambda \ , d \pi \right ] = \sup _ { \pi \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \left [ \langle \lambda , \delta \rangle + \int _ { \mathcal { V } } g _ \lambda \ , d \pi \right ] . \end{align*}


\begin{align*} u _ \delta = \phi ( \rho ) e ^ { - s \rho } \end{align*}


\begin{align*} \begin{array} { l l l l } x \partial & = \theta , & \partial x & = \theta + 1 , \\ x ^ 2 \partial ^ 2 & = \theta ( \theta - 1 ) , & \partial ^ 2 x ^ 2 & = ( \theta + 1 ) ( \theta + 2 ) , \\ x ^ 3 \partial ^ 3 & = \theta ( \theta - 1 ) ( \theta - 2 ) , \quad & \partial ^ 3 x ^ 3 & = ( \theta + 1 ) ( \theta + 2 ) ( \theta + 3 ) , \end{array} \end{align*}


\begin{align*} k ! S ( n , k ) = \sum _ { \ell = 1 } ^ k A _ { n , \ell - 1 } \binom { n - \ell } { k - \ell } \end{align*}


\begin{align*} \mathcal J = \mathcal J _ 1 + \mathcal J _ 2 , \end{align*}


\begin{align*} T ( f ^ { 2 } ) = 2 f T ( f ) + 2 B ( A ( f ) , A ( f ) ) \end{align*}


\begin{align*} \nabla d _ { S _ { i } } ^ { p } \left ( x \right ) = p \left \Vert x - P _ { i } ( x ) \right \Vert ^ { p - 2 } \left ( x - P _ { i } \left ( x \right ) \right ) \end{align*}


\begin{align*} \sigma ^ * u : = \left ( f _ { \sigma } : \sigma ^ { - 1 } U \to X , \ , \sigma _ U ^ { - 1 } M , \ , \phi _ { \sigma } , \ , \sigma _ U ^ * ( t ) \right ) \end{align*}


\begin{align*} D ^ { k - 1 } ( f ) = c ^ { \sigma } F + D ^ { k - 1 } ( h ^ { \sigma } ) + d ^ { \sigma } g \end{align*}


\begin{align*} h _ i \sum \limits _ { j = 1 } ^ { s _ { { i _ { } i } } } b _ { i _ { } i j } { \textbf { X } } ^ { \lambda _ j } = g _ i ^ { ( i ) } . \end{align*}


\begin{align*} \begin{aligned} G _ 0 & = [ - \tfrac 1 2 , \tfrac 1 2 ] \cap e ^ { - ( n _ \mathcal L - n _ 0 ) } \mathbb Z , \\ G _ { \ell } & = \{ h \in G _ 0 : S _ { k } ( h ) \in [ L _ k , U _ k ] k \leq \ell \} . \end{aligned} \end{align*}


\begin{align*} \circ ( x , d ) = x ( 1 + \eta _ 1 ) | \eta _ 1 | \le 2 ^ { - d } . \end{align*}


\begin{align*} z _ \mathbf { n } = \tfrac { 1 } { \mathbf { n } ! } \partial ^ \mathbf { n } p ( x ) . \end{align*}


\begin{align*} \frac { 1 } { 2 n } \int _ { \mathbb { R } ^ n } \rho _ \varepsilon ( | x | ) \ : x = \frac { 1 } { 2 n } \ : \omega _ n \int _ { 0 } ^ \infty \rho ( r ) r ^ { n - 1 } \ : r = \frac { \omega _ n } { n } \frac { 1 } { C _ n } = 1 , \end{align*}


\begin{align*} \mp _ s = \begin{bmatrix} \mp _ { s , 1 } & \mp _ { s , 2 } & \cdots & \mp _ { s , 2 ^ { t - s } } \end{bmatrix} . \end{align*}


\begin{align*} q _ i ( t _ 1 , \dots , t _ n ) = \det \begin{pmatrix} F ( t _ 1 ) t _ 1 ^ i & F ( t _ 2 ) t _ 2 ^ i & \dots & F ( t _ n ) t _ n ^ i \\ t _ 1 ^ { n - 2 } & t _ 2 ^ { n - 2 } & \dots & t _ n ^ { n - 2 } \\ \vdots & \vdots & \ddots & \vdots \\ t _ 1 & t _ 2 & \dots & t _ n \\ 1 & 1 & \dots & 1 \end{pmatrix} \end{align*}


\begin{align*} ( \Delta + \lambda - q ) u = 0 \ ; \mathrm { i n } \ ; \mathrm { M } , u _ { | \partial \mathrm { M } } = \overline { \varphi } . \end{align*}


\begin{align*} f _ I : = \Delta ^ { \bar { k } t } f f _ N : = f - f _ I . \end{align*}


\begin{align*} x ^ 2 ( x ^ q y - x y ^ q ) + y ^ 2 ( y ^ q z - y z ^ q ) + ( z ^ 2 + k x y ) ( z ^ q x - z x ^ q ) = 0 . \end{align*}


\begin{align*} U _ t + ( A U ) _ x + A ^ T U _ x + ( B U ) _ y + B ^ T U _ y + C U = 0 , \end{align*}


\begin{align*} \begin{bmatrix} 1 & 1 & 1 \\ 1 & r ^ { p _ 1 } & r ^ { p _ 2 } \\ 1 & r ^ { 2 p _ 1 } & r ^ { 2 p _ 2 } \end{bmatrix} \begin{bmatrix} \Tilde { \phi } _ { e } \\ C _ { p _ 1 } h ^ { p _ 1 } \\ C _ { p _ 2 } h ^ { p _ 2 } \end{bmatrix} & = \begin{bmatrix} \phi _ 1 \\ \phi _ 2 \\ \phi _ 3 \end{bmatrix} . \end{align*}


\begin{align*} L _ k = \frac { 1 } { 2 \pi i } \log \left ( \phi ( \tau _ k + 0 ) ^ { - 1 } \phi ( \tau _ k - 0 ) \right ) \end{align*}


\begin{align*} R _ q ( \mu ) = ( A _ q - \mu ) ^ { - 1 } , \mu \in \rho ( A _ q ) . \end{align*}


\begin{align*} B ( F ) = \{ b \in \{ 1 , \ldots , D - 1 \} \ | \ \nu ( a _ { Q b } Q ^ b ) \in \delta ^ L \mbox { f o r e v e r y } Q \mbox { w i t h l a r g e e n o u g h v a l u e } \} . \end{align*}


\begin{align*} \varphi ( b ) ( \varphi ( b ' ) ( x ) ) = 1 _ X ( 1 _ X ( x ) ) = x \neq 0 . \end{align*}


\begin{align*} \eta _ t ( x , t ) & = v ( x , t ) \\ \eta ( x , 0 ) & = x \end{align*}


\begin{align*} \Xi _ 2 ( x ) = \sum _ { x ' \le n \le x } \beta _ n \Omega ( n ) \end{align*}


\begin{align*} R _ { [ 1 ] } & = \begin{pmatrix} 1 & 0 \\ 0 & u _ 1 ^ T J _ { 2 | \alpha _ i | } v _ 1 \end{pmatrix} = I _ 2 + \mathcal { O } ( \| H \| ^ 2 ) . \end{align*}


\begin{align*} N _ { S } \left ( x \right ) : = \left \{ x ^ { \ast } \in X ^ { \ast } \mid \left \langle s - x , x ^ { \ast } \right \rangle \leq 0 , s \in S \right \} , \end{align*}


\begin{align*} M = \pm \left ( \left ( { T _ { \varepsilon _ 1 } ^ { \vec { v } } } ^ T \right ) ^ { m _ 1 } \left ( { T _ { \varepsilon _ 2 } ^ { \vec { v } } } ^ T \right ) ^ { m _ 2 } \right ) \end{align*}


\begin{align*} \Big | \int _ { B _ { \delta } ( x ^ \prime ) \times ( 0 , \delta ) } J _ \varepsilon ( | x - \hat { y } | ) \partial _ { x _ n } c ( x ) ( x _ n + y _ n ) y \Big | & \leq \int _ 0 ^ 1 K | \partial _ { x _ n } ^ 2 c ( x ^ \prime , t x _ n ) | t \\ & = K \frac { 1 } { x _ n } \int _ 0 ^ { x _ n } | \partial _ { x _ n } ^ 2 c ( x ^ \prime , z _ n ) | z _ n , \end{align*}


\begin{align*} \begin{aligned} \mathcal { R } _ { \beta _ 0 , \beta _ 1 } ( z ) : = & ( \mathcal { A } _ 0 - z I ) ^ { - 1 } - \left ( I - z \mathcal { A } _ 0 ^ { - 1 } \right ) ^ { - 1 } \Pi ( \overline { \beta _ 0 + \beta _ 1 M ( z ) } ) ^ { - 1 } \beta _ 1 \bigl ( \Pi ^ * \left ( I - z \mathcal { A } _ 0 ^ { - 1 } \right ) ^ { - 1 } \bigr ) \\ [ 0 . 3 e m ] = & ( \mathcal { A } _ 0 - z I ) ^ { - 1 } - S ( z ) ( \overline { \beta _ 0 + \beta _ 1 M ( z ) } ) ^ { - 1 } \beta _ 1 S ^ * ( \overline { z } ) . \end{aligned} \end{align*}


\begin{align*} T ( x _ 0 ) = \left \{ \begin{array} { l l } \sup \ , \{ t \leqslant \tau ^ { x _ 0 } _ { a + H } , \ , \pi _ 2 ( X ^ { x _ 0 } _ t ) = \pi _ 2 ( x _ 0 ' ) \} & \\ \infty & \end{array} \right . \\ T ( x _ 0 ' ) = \left \{ \begin{array} { l l } \sup \ , \{ t \leqslant \tau ^ { x _ 0 ' } _ { H } , \ , \pi _ 2 ( X ^ { x _ 0 ' } _ t ) = \pi _ 2 ( x _ 0 ) \} & \\ \infty & \end{array} \right . \end{align*}


\begin{align*} \mathcal { M } = \left \{ m \in \N : \begin{matrix} m \textrm { i s s q u a r e - f r e e , a l l p r i m e f a c t o r s o f } m \textrm { a r e i n } P \\ \textrm { a n d } m \textrm { h a s a t m o s t } \frac { a \log { N } } { | \log ( 2 \sigma - 1 ) | } \textrm { p r i m e f a c t o r s } \end{matrix} \right \} . \end{align*}


\begin{align*} l ' ( \varphi ( b ) , x ) = l ( b , x ) , r ' ( x , \varphi ( b ) ) = r ( x , b ) , \end{align*}


\begin{align*} P ^ { ( \alpha , \beta ) } _ k ( x ) = \frac { ( \alpha + 1 ) _ k } { k ! } { } _ 2 F _ 1 \left ( \begin{matrix} - k , \ , k + \alpha + \beta + 1 \\ \alpha + 1 \end{matrix} \ , ; \ , \frac { 1 - x } { 2 } \right ) . \end{align*}


\begin{align*} V _ A ( \beta ) ^ { n _ 1 } V _ A ( 2 \beta ) ^ { n _ 2 } V _ A ( 0 ) ^ { 2 k - n _ 1 - n _ 2 } = e ^ { - \i N \frac { \beta } { 2 } n _ 1 } e ^ { - \i N \beta n _ 2 } \mathcal { Z } _ A ( 0 ) ^ k \overline { \mathcal { Z } _ A ( 0 ) } ^ { k - n _ 1 - n _ 2 } \overline { \mathcal { Z } _ A ( \beta ) } ^ { n _ 1 } \overline { \mathcal { Z } _ A ( 2 \beta ) } ^ { n _ 2 } . \end{align*}


\begin{align*} \rho _ F ( a ) \rho _ F ( a c d ) + \rho _ F ( c ) \rho _ F ( d ) & = \rho _ F ( a ) \rho _ F ( a c d ) + \rho _ F ( c ) \rho _ F ( d ) + \rho _ F ( a c ) \rho _ F ( a d ) \\ & = \rho _ F ( a ) \rho _ F ( a c ) + \rho _ F ( a ) \rho _ F ( d ) + \rho _ F ( c ) \rho _ F ( d ) + \rho _ F ( a c ) \rho _ F ( a d ) \\ & = [ \rho _ F ( a ) \rho _ F ( a c ) + \rho _ F ( a c ) \rho _ F ( a d ) ] + [ \rho _ F ( a ) \rho _ F ( d ) + \rho _ F ( c ) \rho _ F ( d ) ] \\ & = \rho _ F ( d ) \rho _ F ( a c ) + \rho _ F ( d ) \rho _ F ( a c ) = 0 , \end{align*}


\begin{align*} \phi ( z ) \left ( 1 - \rho + \rho \zeta _ { k } ( g ^ { - 1 } z ) \right ) , \end{align*}


\begin{align*} \left \lVert \widetilde { \Pi } _ { \chi , k } \right \rVert _ { H ^ { 1 / 2 } ( \Gamma ; \C ^ 3 ) \to H ^ 1 ( Y _ { \rm s t i f f } ; \C ^ 3 ) } \leq C | \chi | ^ k , k = 1 , \dots , n , \left \lVert \widetilde { \Pi } _ { \chi , n } ^ { \rm e r r o r } \right \rVert _ { H ^ { 1 / 2 } ( \Gamma ; \C ^ 3 ) \to H ^ 1 ( Y _ { \rm s t i f f } ; \C ^ 3 ) } \leq C | \chi | ^ { n + 1 } , \end{align*}


\begin{align*} \mathcal { H } ^ 1 \left ( \Omega \cap \left ( \overline { J _ u } \setminus J _ u \right ) \right ) = 0 . \end{align*}


\begin{align*} t \deg \phi ( x ) \leq \deg h ( x ) = \deg h _ t ( x ) + t \deg \phi ( x ) \leq ( t + 1 ) \deg \phi ( x ) - 1 . \end{align*}


\begin{align*} { \rm I m } ( e ^ { i \pi ( 1 - \beta _ 1 ) } \kappa ( t ) ) = 0 { \rm \ f o r \ } t \le 0 \end{align*}


\begin{align*} \sum _ { i = 1 } ^ k \lambda _ i ( A ) = \max \limits _ { U ^ * U = I _ k } \mathrm { t r } ( U ^ * A U ) \hbox { a n d } \sum _ { i = 1 } ^ k \lambda _ { n - i + 1 } ( A ) = \min \limits _ { U ^ * U = I _ k } \mathrm { t r } ( U ^ * A U ) , \end{align*}


\begin{align*} i _ - \chi _ { I } ( H _ \omega ) i _ + = \int _ { I } P _ \omega ( E ) d \mu _ \omega ( E ) \end{align*}


\begin{align*} A _ n & = P _ n + P _ n L _ 1 P _ n + W _ n L _ 2 W _ n + C _ n \end{align*}


\begin{align*} E : Y ^ 2 = X ^ 3 - 1 1 2 5 . \end{align*}


\begin{align*} [ P : Q ] & : = \mathrm { I n d } ( Q P | _ { R ( P ) } : R ( P ) \to R ( Q ) ) \\ & = \dim ( N ( Q ) \cap R ( P ) ) - \dim ( R ( Q ) \cap N ( P ) ) \end{align*}


\begin{align*} f _ 1 ( x _ 1 , \dots , x _ m ) = \dots = f _ p ( x _ 1 , \dots , x _ m ) = 0 . \end{align*}


\begin{align*} \lim _ { \lambda \to 1 ^ + } \limsup _ { m , n \to \infty } \max _ { P _ m \leq P _ i \leq \lambda P _ { m } } \left | u _ { i n } - u _ { m n } \right | = 0 , \end{align*}


\begin{align*} \left \langle A _ r ^ n P _ 0 B _ s ^ n 1 , 1 \right \rangle & = \sum _ { Q \in \Gamma } c _ Q \int _ Q \{ h _ { \theta } \} _ { \theta = 1 } ^ { r + s } , \end{align*}


\begin{align*} \frac { \delta ^ 2 F } { \delta \nu \delta \mu } ( \nu , \mu , y , x ) = \frac { \delta ^ 2 F } { \delta \mu \delta \nu } ( \nu , \mu , x , y ) , \end{align*}


\begin{align*} P ^ \top & = [ e _ { i _ 1 } , \ldots , e _ { i _ k } , e _ { i _ { k + 1 } } , \ldots , e _ { i _ \ell } ] \intertext { a n c o n s i d e r } P @ @ A _ { I ( u ) } P ^ \top & = \begin{bmatrix} A _ { I ( v ) } & A _ { I ( v ) I ( w ) } \\ A _ { I ( w ) I ( v ) } & A _ { I ( w ) } \\ \end{bmatrix} . \end{align*}


\begin{align*} B _ { \ell n } = \left \{ ( ( s _ 1 , s _ 2 ) , ( y _ 1 , y _ 2 , x ) ) \in \mathcal { V } \times \mathcal { S } : c _ \ell ( s _ \ell , ( y _ \ell , x ) ) < n \right \} , \end{align*}


\begin{align*} L _ \Q = ( L _ 1 ) _ \Q \oplus \langle z , z ' \rangle _ \Q , \end{align*}


\begin{align*} \mu N _ { \beta } = N _ { \tilde \beta } \prod _ { i = 1 } ^ n { \mu \tilde w _ i \over w _ i } . \end{align*}


\begin{align*} d ( \uparrow _ x ^ y , \eta ) = d ( \uparrow _ x ^ y , \Sigma _ x ( M \cap \partial W _ 0 ) ) \le \frac { \beta } { 2 } = \frac { \pi } { 2 } - \frac 1 2 \Theta ^ M ( x , \partial W _ 0 ) . \end{align*}


\begin{align*} N = \dfrac { - \rho s ^ { i j } u _ i \frac { \partial } { \partial \theta _ j } + \phi ^ 2 \frac { \partial } { \partial r } } { \phi D } . \end{align*}


\begin{align*} \begin{bmatrix} I \\ S ( t ) \end{bmatrix} , \begin{bmatrix} \cos t I & - \sin t I \\ \sin t I & \cos t I \end{bmatrix} \begin{bmatrix} I \\ S ( 0 ) \end{bmatrix} \end{align*}


\begin{align*} \lim _ { t \to 0 } \left \| e ^ { - t H ^ { \beta } } g - g \right \| _ { L ^ q } = 0 \end{align*}


\begin{align*} { \mathbf { R } } ^ * _ l { \mathbf { R } } _ l = \left ( A ^ * _ i A _ j \right ) _ { \alpha _ { l } \le i , j < \alpha _ { l + 1 } } \end{align*}


\begin{align*} S _ { [ 1 ] } \diamond \cdots \diamond S _ { [ j + 1 ] } & = \left ( S _ { [ 1 ] } \diamond \cdots \diamond S _ { [ j ] } \right ) \diamond S _ { [ j + 1 ] } . \end{align*}


\begin{align*} \widetilde { T } ( f ) ( z ) = - \frac { 1 } { \pi } \iint _ D \left ( \frac { f ( \zeta ) } { \zeta - z } + \frac { z \overline { f ( \zeta ) } } { 1 - \overline { \zeta } z } \right ) \ , d \xi \ , d \eta , \end{align*}


\begin{align*} \begin{cases} \dot { \varrho } _ t ( x ) = \frac { \kappa } { 2 } \partial _ { x x } \varrho _ t ( x ) - \lambda \partial _ x \Delta _ t ( x ) , \\ \\ \dot { \Delta } _ t ( x ) = \frac { \kappa } { 2 } \partial _ { x x } \Delta _ t ( x ) - \lambda \partial _ x \varrho _ t ( x ) - 2 \gamma \Delta _ t ( x ) . \end{cases} \end{align*}


\begin{align*} X _ { \gamma , 2 } ( x ) = ( \log 2 + \log ( \tfrac 1 2 x ^ { \frac { 1 } { \beta } } ) ) x = \tfrac { 1 } { \beta } \log ( x ) x \forall x \in ( 0 , 1 ] . \end{align*}


\begin{align*} \underset { t \to \infty } { \lim } \ : P ^ \omega \left ( \mathcal { R } ( t ) \cap \widehat { \Phi } _ t ^ \omega = \emptyset \ : \big \vert \ : S _ t \right ) = 0 . \end{align*}


\begin{align*} \exists \ , \ , ( - \Delta ) ^ s u ( x ) = \mathfrak F ^ { - 1 } \big ( | \xi | ^ { 2 s } \mathfrak F u \big ) ( x ) . \end{align*}


\begin{align*} \textrm { a n d } s = \left ( \begin{array} { c c c } e ^ { - t _ 1 } & 0 & 0 \\ 0 & e ^ { - t _ 1 } & 0 \\ 0 & 0 & e ^ { - t _ 2 } \\ \end{array} \right ) , \textrm { f o r } t _ 1 < t _ 2 . \end{align*}


\begin{align*} x \ast _ { n , n } x & = ( y + z ) \ast _ { n , n } ( y + z ) = y \ast _ { n , n } y + y \ast _ { n , n } z + z \ast _ { n , n } y + z \ast _ { n , n } z \\ & = y \ast _ { n , n } y + z \ast _ { n , n } z = \top _ n \ast _ { n , n } y + \top _ n \ast _ { n , n } z \\ & = \top _ n \ast _ { n , n } ( y + z ) = \top _ n \ast _ { n , n } x \end{align*}


\begin{align*} { \rm p v } \int \cot \Big ( \frac { z ( e ) - z ( e ' ) } { L / \pi } \Big ) f ( e ' ) \d e ' = \int \cot \Big ( \frac { z ( e ) - z ( e ' ) } { L / \pi } \Big ) \frac { f ( e ' ) z _ { e } ( e ) - f ( e ) z _ { e } ( e ' ) } { z _ { e } ( e ) } \d e ' \ , , \end{align*}