\begin{align*} \Gamma _ { 1 , y , s } = \bigcup _ { \delta = 1 } ^ d \Gamma _ { 1 , y , s } ^ { \delta } , \Gamma _ { 1 , y , s } ^ { \delta } = \{ n \in \Gamma _ { 1 , y , s } : J _ { n } ^ { \delta } \subset ( L ^ { \delta } ) ^ c \} . \end{align*}


\begin{align*} S ( H , f ) = \frac { f - 1 } { 1 2 } \hbox { a n d } S ( H _ 2 , 2 f ) = \frac { 2 f + c _ a ' } { 1 2 } \hbox { w h e r e } c _ a ' : = \begin{cases} - 3 a - 2 & \hbox { i f $ a \equiv 0 \pmod 2 $ } , \\ 3 a + 1 & \hbox { i f $ a \equiv 1 \pmod 2 $ } \end{cases} \end{align*}


\begin{align*} \gamma _ p = 1 + v ^ m u ^ { m - 1 } \beta _ m t ^ m + \cdots + v ^ { n - 1 } u ^ { n - 2 } \beta _ { n - 1 } t ^ { n - 1 } \quad w _ p = v ^ m u ^ m \end{align*}


\begin{align*} g ( z + t ) - Q _ z ( t ) = \int _ z ^ { z + t } [ A + g '' ( u ) ] ( z + t - u ) \ , d u , \ \ ( z , t ) \in E . \end{align*}


\begin{align*} \left | \left | \frac { w _ 1 + w _ 2 } { 2 } \right | \right | = r \sqrt { \frac { 1 + \cos ( \theta ) } { 2 } } . \end{align*}


\begin{align*} \mathfrak { u } _ { k n } ^ I : = \sum _ { i = 0 } ^ { | \bar u _ { k n } | - 1 } \chi _ { I } ( T _ * ^ i ( p ) ) \end{align*}


\begin{align*} I ( \alpha , \beta , Z ) = \epsilon ( \alpha ) - \epsilon ( \beta ) \ , \ , \ , \mathrm { m o d } \ , \ , 2 . \end{align*}


\begin{align*} \left ( - \Delta + 1 + \frac { N - 1 } { r ^ 2 } - \left ( p - 1 \right ) \left ( | \cdot | ^ { N - \gamma } * Q ^ p \right ) Q ^ { p - 2 } \right ) \left ( - \partial _ r Q \right ) = - \partial _ r \left [ | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ] Q ^ { p - 1 } . \end{align*}


\begin{align*} \sum _ { \substack { n \geq i \geq 0 \\ m \geq j \geq 0 } } g _ { i j } ( t ) \bigl ( \sum _ { r _ { i j } \geq h \geq 1 } \mathcal { L } _ { K , \mathfrak { s } _ { i j , h } } ( { \boldsymbol \alpha _ { i j , h } } ) \bigr ) = 0 \end{align*}


\begin{align*} \left \{ \Psi _ { r n 1 } \left ( x \right ) \right \} ^ { 2 } & = p ^ { - r } \chi _ { p } \left ( 2 p ^ { - 1 } \left ( p ^ { r } x - n \right ) \right ) \Omega \left ( \left \vert p ^ { r } x - n \right \vert _ { p } \right ) \\ & = p ^ { r } \left \{ \Psi _ { r n 1 } \left ( x \right ) \right \} ^ { 2 } = p ^ { \frac { r } { 2 } } \Psi _ { r n 2 } \left ( x \right ) , \end{align*}


\begin{align*} \begin{aligned} \| V _ l - \hat { V } \| _ F \leq \| V _ l - Z \| _ F + \| Z - \hat { V } \| _ F \leq \sqrt { \frac { 4 0 k \epsilon } { \eta } } \| C \| _ F + \xi . \end{aligned} \end{align*}


\begin{align*} \dim ( ( X + V ) / V ) + \dim ( ( Y + W ) / W ) = \\ \dim ( X + Y + V + W ) + \dim ( ( X + V ) \cap ( Y + W ) ) - \dim ( V + W ) - \dim ( V \cap W ) = \\ \dim ( ( X + Y + V + W ) / ( V + W ) ) + \dim ( ( X + V ) \cap ( Y + W ) ) - \dim ( V \cap W ) \geq \\ \dim ( ( X + Y + V + W ) / ( V + W ) ) + \dim ( ( ( X \cap Y ) + ( V \cap W ) ) / ( V \cap W ) ) , \end{align*}


\begin{align*} \frac { \mathbf { P } \Big ( a _ n S _ n \leq - x \sqrt { v _ n + a _ n ^ 2 \sum _ { i = 1 } ^ n ( Z _ i - 1 ) ^ 2 } \ , \Big ) } { \Phi \left ( - x \right ) } = 1 + o ( 1 ) \end{align*}


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


\begin{align*} \Delta ( a _ 1 \dotsm a _ n ) : = \sum _ { S \subseteq [ n ] } w _ S \otimes w _ { J ^ S _ 1 } \vert \dotsm \vert w _ { J ^ S _ m } , \end{align*}


\begin{align*} \mathcal { \psi } _ { + } ( \zeta ) = \mathcal { \psi } _ { - } ( \zeta ) J ^ { \mathcal { \psi } } , \zeta \in \mathbb { R } , \end{align*}


\begin{align*} P _ 0 ( \lambda ) = & \begin{vmatrix} \lambda & - 1 & - 1 \\ - 1 & \lambda & - 2 \\ - 1 & - 2 & \lambda \end{vmatrix} \\ P _ 0 ( \lambda ) = & \lambda ^ 3 - 6 \lambda - 4 \end{align*}


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


\begin{align*} n _ i = a _ i + n _ { \min } - 1 , i = 1 , 2 , \cdots , \left | \mathcal { A } \right | . \end{align*}


\begin{align*} X _ { n + 1 } & = \begin{cases} X _ { U _ n } & \mbox { w i t h p r o b a b i l i t y $ p $ } , \\ - X _ { U _ n } & \mbox { w i t h p r o b a b i l i t y $ 1 - p $ } . \\ \end{cases} \end{align*}


\begin{align*} \operatorname { V a r } _ { \mu } ( f _ j ) = \operatorname { C o v } _ { \mu } ( g _ j + \varepsilon f , g _ j + \varepsilon f ) = \operatorname { V a r } _ { \mu } ( g _ j ) + 2 \varepsilon \operatorname { C o v } _ { \mu } ( g _ j , f ) + \varepsilon ^ 2 \operatorname { V a r } _ { \mu } ( f ) , \end{align*}


\begin{align*} & \textbf { s } _ m ( t ) = \\ & \biggl [ \Re \left [ s _ m [ t ] e ^ { j 2 \pi f _ 0 t } e ^ { j \psi _ { 1 m } } \right ] , \ldots , \Re \left [ s _ m [ t ] e ^ { j 2 \pi f _ 0 t } e ^ { j \psi _ { N _ s m } } \right ] \biggr ] ^ T . \end{align*}


\begin{align*} \phi = \phi ^ R + \phi _ 0 + \phi _ 7 , \end{align*}


\begin{align*} \Bigl \{ ( \phi , s , t , \lambda , y ) \in \R ^ { m + d ( n + 1 ) + 2 ^ d + | V | + | E | } \Bigm | \phi = A ( t ) , \ t = L ( y ) , \ s \in S ( \lambda ) , \ ( \lambda , y ) \in ( \Lambda ) \Bigr \} \end{align*}


\begin{align*} S ( \underline { P } , B ) = \{ ( \omega , x ) \in \Omega \times B \ ; \vert \ ; \omega \underline { P } ( x ) \leq 0 \} . \end{align*}


\begin{align*} \norm { \Gamma ' g _ { \ell } } \leq \norm { g _ \ell } \sum \limits _ { s = 2 } ^ { j } \gamma ^ { i _ s + \ell - 2 s } _ { i _ s - 2 s } \prod _ { t = 2 } ^ { s - 1 } \left ( 1 - \delta ^ { i _ t + \ell - 2 t } _ { i _ t - 2 t } \right ) . \end{align*}


\begin{align*} w ( r ) = \begin{cases} A \ , r ^ { 1 - N / 2 } J _ { N / 2 - 1 } \left ( \sqrt { \tilde { \lambda } _ 0 \overline { m } } r \right ) & 0 \le r < 1 \ ; , \\ B \ , r ^ { 1 - N / 2 } K _ { N / 2 - 1 } \left ( \sqrt { \tilde { \lambda } _ 0 \underline { m } } r \right ) & r \ge 1 \ ; , \\ \end{cases} \end{align*}


\begin{align*} g _ j ^ Q = \frac { 1 } { | Q | } \int _ { Q } g _ j ( x ) \ , d x . \end{align*}


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


\begin{align*} \mathrm { e } _ { \pi _ { \lambda , \beta } } ( z ; x ) : = \frac { \mathrm { e } ^ { z x } } { l _ { \pi _ { \lambda , \beta } } ( z ) } = \frac { \mathrm { e } ^ { z x } } { E _ { \beta } ( \lambda ( e ^ { z } - 1 ) ) } , z \in \mathbb { C } , \ ; x \in \mathbb { R } , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ q ( a _ i + b _ i ) = \sum _ { A \subset S _ q } \prod _ { j \in A } a _ j \prod _ { j \notin A } b _ j , \end{align*}


\begin{align*} X = \{ M _ \omega T _ x g _ 0 \mid ( x , \omega ) \in \R ^ { 2 d } \} \end{align*}


\begin{align*} = \int _ { \R ^ 3 } | \nabla \mathbf { v } | ^ 2 \varphi _ R \ , d x + \sum _ { i , j = 1 } ^ { 3 } \int _ { \R ^ 3 } \partial _ { j } v _ i \cdot v _ i \partial _ { j } \varphi _ R \ , d x . \end{align*}


\begin{align*} x \geq 3 2 7 5 , \end{align*}


\begin{align*} \int _ { 0 } ^ { 1 } u ^ { \epsilon } _ x \bar { v } _ x + \int _ { 0 } ^ { 1 } u ^ { \epsilon } _ x \bar { v } + \int _ { 0 } ^ { 1 } \rho ^ { \epsilon } _ x \bar { v } + \lambda \int _ { 0 } ^ { 1 } u ^ { \epsilon } \bar { v } = \int _ { 0 } ^ { 1 } g \bar { v } . \end{align*}


\begin{align*} g = \frac { f ' _ { s ' _ { n } } ( \tilde { \infty } ) - f ' _ { s _ { n } } ( \tilde { \infty } ) } { \delta \kappa } \end{align*}


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


\begin{align*} \xi ( x , t , y , s ; 1 , v ) = \zeta ( X , t , Y , s ; 1 , v ) , \end{align*}


\begin{align*} a | z \rangle = \sum _ { n = 0 } ^ { \infty } A _ { n } a | n \rangle = \sum _ { n = 1 } ^ { \infty } A _ { n } \sqrt { n } | n - 1 \rangle , \end{align*}


\begin{align*} \Theta ^ t W _ s = W _ { t + s } - W _ t , s \ge 0 , \end{align*}


\begin{align*} \left \{ \begin{aligned} & u _ 0 \in L ^ 2 ( \Omega ) , \rho _ 0 \in W ^ { 1 , 2 } ( \Omega ) , \\ & 0 < \underline { \rho } \leq \rho _ 0 ( x ) \leq \bar { \rho } < \infty , x \in \Omega , \\ & 0 < \underline { \rho } \leq \rho _ B ( x ) \leq \bar { \rho } < \infty , x \in \Gamma _ { \rm { i n } } . \end{aligned} \right . \end{align*}


\begin{align*} K _ * = 0 , C _ b = 0 , C _ W = \sigma _ 1 ^ { - 2 } . \end{align*}


\begin{align*} h _ B ( \tfrac { \eta } { n } ) = h _ B ( \tfrac { 1 } { \eta } ) \Longleftrightarrow \frac { \eta } { n } = \frac { 1 } { \eta } \Longleftrightarrow \eta = \sqrt { n } . \end{align*}


\begin{align*} A f ( x , \omega ) = \frac { e ^ { - \pi i ( \omega - i ) | x | } } { 2 \pi i ( \omega - i ) } . \end{align*}


\begin{align*} C ( X , Y ) = C ( Y , X ) + [ R ( X , Y ) , A ] . \end{align*}


\begin{align*} { \rm G e n } ( x , t ) = { { \rm G e n } ( a ^ { - 1 } x , t ) \over { \rm G e n } ( a ^ { - 1 } , t ) } , \end{align*}


\begin{align*} f ^ \star : = \inf _ { x \in \mathbb { R } ^ n } f ( x ) , \end{align*}


\begin{align*} G = \{ u \rightarrow 2 u v , \ ; \ ; v \rightarrow u \} , \end{align*}


\begin{align*} \frac { \log _ { \lambda } ( 1 + t ) } { t } ( 1 + t ) ^ { x } & = \sum _ { k = 0 } ^ { \infty } \beta _ { k , \lambda } ( x ) \frac { 1 } { k ! } \big ( \log _ { \lambda } ( 1 + t ) \big ) ^ { k } \\ & = \sum _ { k = 0 } ^ { \infty } \beta _ { k , \lambda } ( x ) \sum _ { n = k } ^ { \infty } S _ { 1 , \lambda } ( n , k ) \frac { t ^ { n } } { n ! } \\ & = \sum _ { n = 0 } ^ { \infty } \bigg ( \sum _ { k = 0 } ^ { n } \beta _ { k , \lambda } ( x ) S _ { 1 , \lambda } ( n , k ) \bigg ) \frac { t ^ { n } } { n ! } . \end{align*}


\begin{align*} \theta \varphi \theta ^ { - 1 } & = ( \tau , \boldsymbol \varepsilon ) ( \sigma , R ) ( \tau , \boldsymbol \varepsilon ) ^ { - 1 } \\ & = ( \tau \sigma \tau ^ { - 1 } , R ^ { \tau ^ { - 1 } } ) \\ & = ( \tau \sigma \tau ^ { - 1 } , ( \varphi _ { \tau 1 } , \varphi _ { \tau 2 } , \varphi _ { \tau 3 } ) ) \\ & = ( \tau \sigma \tau ^ { - 1 } , ( \alpha _ 1 , \alpha _ 2 , \alpha _ 3 ) ) . \end{align*}


\begin{align*} m ^ { P C } _ { \eta } = \mathcal { \psi } ( \zeta ) \mathcal { P } \zeta ^ { - i \nu \sigma _ 3 } e ^ { \frac { i } { 4 } \zeta ^ 2 \sigma _ 3 } , \end{align*}


\begin{align*} ( K * \tilde f ) '' ( x ) = \int _ { \R } \big [ \tilde f ( x + z ) - \tilde f ( x ) - z \tilde f ' ( x ) \big ] \ , K '' ( z ) d z , \end{align*}


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


\begin{align*} n _ 1 , \ldots , n _ L = n \gg 1 , \end{align*}


\begin{align*} \| \hat { V } - Z \| _ F = \| R - I \| _ F < \xi . \end{align*}


\begin{align*} ( f , g ) = \int _ { \mathbb { R } } f ( x ) g ( x ) d x . \end{align*}


\begin{align*} I _ f ( u ) : = \int _ { \overline { \Omega } } g _ u ^ p \ , d \mu - p \int _ { \partial \Omega } u f \ , d \nu . \end{align*}


\begin{align*} 0 = \omega ( v _ i \# u ) = \omega ( v _ i ) + \omega ( u ) . \end{align*}


\begin{align*} \begin{aligned} & f ^ L _ i \leq a _ { i 0 } ( x ) \leq \ldots \leq a _ { i n _ i } ( x ) \leq f ^ U _ i , \\ & u _ { i j } ( x ) \leq \min \bigl \{ f _ i ( x ) , a _ { i j } ( x ) \bigr \} j \in \{ 0 , \ldots , n _ i \} , \\ & u _ { i 0 } ( x ) = a _ { i 0 } ( x ) , u _ { i n _ i } ( x ) = f _ { i } ( x ) . \end{aligned} \end{align*}


\begin{align*} \mathcal { R } _ L ( \Q ( \sqrt { - 1 } ) , 4 ) & = \mathcal { R } _ L ( F , 4 ) _ I \coprod \mathcal { R } _ L ( F , 4 ) _ { I I } , \\ \mathcal { R } _ L ( \Q ( \sqrt { - 3 } ) , 2 ) & = \mathcal { R } _ L ( \Q ( \sqrt { - 3 } ) , 2 ) _ { I I } , \\ \mathcal { R } _ L ( F , 2 ) & = \mathcal { R } _ L ( F , 2 ) _ I \coprod \mathcal { R } _ L ( F , 2 ) _ { I I } \coprod \mathcal { R } _ L ( F , 2 ) _ { I I I } \coprod \mathcal { R } _ L ( F , 2 ) _ { I V } \coprod \mathcal { R } _ L ( F , 2 ) _ { V } , \end{align*}


\begin{align*} \langle \Phi ^ P _ { e } , \operatorname { I n d } _ { C ^ * _ r ( G ) } ( D ) \rangle = \frac { 1 } { | W _ { M \cap K } | } \cdot \sum _ { w \in W _ K } m \left ( \sigma ^ { M } ( w \cdot \mu ) \right ) , \end{align*}


\begin{align*} g ( x ) - y = g ( z + t ) - Q _ z ( t ) = \int _ z ^ { z + t } ( g '' ( u ) + A ) ( z + t - u ) \ , d u , \end{align*}


\begin{align*} u ( z ) & = \sum _ { n = 0 } ^ { \infty } u _ { n } z ^ { n } \end{align*}


\begin{align*} \begin{aligned} & u _ n ( x ) = p ( x ) u _ j ( x ) \leq p ( x ) u ( x ) \leq a ( x ) & & j \in \{ 0 , \ldots , n \} , \\ & p ^ L = u _ { 0 } ( x ) = a _ 0 ( x ) \leq \cdots \leq a _ n ( x ) = p ^ U , \\ & \deg ( u _ j ) \leq c _ 1 \deg ( a _ j ) \leq c _ 2 & & j \in \{ 0 , \ldots , n \} . \end{aligned} \end{align*}


\begin{align*} { \sf F i x } _ X ( G ) = \{ u \in X : \sigma u = u , \ \forall \sigma \in G \} \end{align*}


\begin{align*} \int _ 0 ^ 1 ( T _ { \gamma ( t ) } G ) \ 1 { \gamma ( t ) \in W ' ( \omega ) } \gamma ' ( t ) d t = 0 , \end{align*}


\begin{align*} \norm { \left ( 1 - \delta ^ { i _ 1 + \ell - 2 } _ { i _ 1 - 2 } \right ) \Gamma ' g _ { \ell } + \Gamma g _ \ell } & \leq \left ( 1 - \delta ^ { i _ 1 + \ell - 2 } _ { i _ 1 - 2 } \right ) \norm { \Gamma ' } \norm { g _ { \ell } } + \norm { \Gamma } \norm { g _ \ell } \\ & \leq \sum \limits _ { s = 1 } ^ j \gamma _ { i _ s - 2 s } ^ { i _ s + \ell - 2 s } \prod \limits _ { t = 1 } ^ { s - 1 } \left ( 1 - \delta ^ { i _ t + \ell - 2 t } _ { i _ t - 2 t } \right ) \norm { g _ \ell } , \end{align*}


\begin{align*} \forall k \geq 1 , 0 = - \int _ 0 ^ T \int _ { \mathbb { R } } \rho _ { 3 } ( s , u ) \partial _ s G _ { k } ( s , u ) d u d s - & \int _ 0 ^ t \int _ { \mathbb { R } } \rho _ { 5 } ( s , u ) [ - ( - \Delta ) ^ { \gamma / 2 } G _ { k } ] ( s , u ) d u d s . \end{align*}


\begin{align*} - \Delta g + \tilde { \lambda } _ 0 \underline { m } \ , g = f _ 0 ( g ) \R ^ n . \end{align*}


\begin{align*} h _ n : X \ast I & \to X \ast I \\ ( x , ( s , t ) ) & \mapsto \begin{cases} ( x , ( s + n , t - n ) ) & n < t \\ ( x , ( s + t , 0 ) ) & n \ge t . \end{cases} \end{align*}


\begin{align*} \psi _ { \eta } ' ( 0 ) = 1 . \end{align*}


\begin{align*} \sum _ a \Gamma _ a \lrcorner \omega _ { Q , L } ^ a = d E _ L , \end{align*}


\begin{align*} \Delta _ 5 ( \lambda _ 1 , \lambda _ 2 , \lambda _ 3 ) = 0 \ , , \end{align*}


\begin{align*} \mu _ { R , 2 } ^ { ( 0 ) } : = \int \overline { \chi _ + ( x ) \mathcal { K } _ + ( x , k ) } \chi _ - ( x ) \mathcal { K } _ - ( x , \ell ) \overline { \mathcal { K } _ S ( x , m ) } \mathcal { K } _ S ( x , n ) \ , d x ; \end{align*}


\begin{align*} \alpha U ( z ) = U ( z ' ) \lambda ( \alpha , z ) . \end{align*}


\begin{align*} { \bf T } _ { s } ( { \bf A } ) = \left \{ W = \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle \in G _ { s } \left ( { \rm H ^ { 2 } } \left ( { \bf A } , \mathbb C \right ) \right ) : \bigcap \limits _ { i = 1 } ^ { s } \operatorname { A n n } ( \theta _ { i } ) \cap \operatorname { A n n } ( { \bf A } ) = 0 \right \} , \end{align*}


\begin{align*} g _ c ( x ) = \frac { 1 } { \sqrt { 2 \pi c } } e ^ { - \frac { x ^ 2 } { 2 c } } , x \in \R . \end{align*}


\begin{align*} \varepsilon ( \tau ) = \sum _ { j = 0 } ^ { \ell } \hat { \varepsilon } _ j ( \tau ) \phi _ { j , \infty } + \hat { \varepsilon } _ - ( \tau ) : = \hat { \varepsilon } _ + + \hat { \varepsilon } _ - , \end{align*}


\begin{align*} ( \tilde f _ k ) ^ * ( g _ { \textnormal { e u c } } ) = e ^ { 2 \tilde u _ k } \ , g _ { \textnormal { p o i n } , j _ k } \textnormal { o n } \ , \ , \Sigma \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { \# \{ A \in \mathcal A ( H ) \colon \max ( A ) \le x \} } { \# \{ A \in H \colon \max ( A ) \le x \} } = 1 \ , . \end{align*}


\begin{align*} a * b = \lambda _ a ( b ) - b = - a + a \circ b - b . \end{align*}


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


\begin{align*} \delta _ j ^ k = \prod \limits _ { i = k - j } ^ k \delta _ i , ~ \gamma ^ k _ j = \gamma \sum \limits ^ { j - 1 } _ { i = - 1 } \delta ^ k _ i , \end{align*}


\begin{align*} \int _ 2 ^ T \abs { E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 & \ll \begin{cases} T ^ { 4 - 2 \sigma _ 1 - 2 \sigma _ 2 - 2 \sigma _ 3 } & ( \frac { 3 } { 2 } < \sigma _ 1 + \sigma _ 2 + \sigma _ 3 < 2 ) \\ \log T & ( \sigma _ 1 + \sigma _ 2 + \sigma _ 3 = 2 ) , \\ \end{cases} \end{align*}


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


\begin{align*} \sum _ { \substack { | \Im ( \rho ) | \leq T \\ \Re ( \rho ) > \sigma } } \frac { x ^ { \Re ( \rho ) - 1 } } { | \Im ( \rho ) | } & \leq 2 \sum _ { k = 0 } ^ { K - 1 } \frac { x ^ { - \nu _ 1 \left ( t _ k \right ) } } { t _ k } N _ 0 ( \sigma , t _ { k + 1 } ) \\ & = s _ 2 ( x , \sigma , K ) , \ . \end{align*}


\begin{align*} \omega = d \theta ^ c - \theta \wedge \theta ^ c . \end{align*}


\begin{align*} & \frac 1 { \lambda _ 1 } + \frac 1 { \lambda _ 2 } + \frac 1 { \Lambda _ 2 } = 2 , \\ & \frac 1 { \lambda _ k } + \frac 1 { \Lambda _ k } = 1 \mbox { i f } \ ; 3 \le k \le p . \end{align*}


\begin{align*} \sum ^ { \mathcal { J } } _ { j = 1 } 2 \alpha _ { j } \lceil e ^ 2 \alpha ^ { - 3 / 4 } _ j \rceil \leq \lceil 4 e ^ 2 k 1 0 ^ { - M / 4 } \rceil + 1 . \end{align*}


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


\begin{align*} A + D + E = [ \mathsf F ( S ) + 1 + m , \mathsf F ( S ) + 1 + 2 M - m ] = C + B + E \ , , \end{align*}


\begin{align*} \sigma ( S \l ' , S \l ) = \sigma ( \l ' , \l ) . \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { | \Lambda ( M ) | } { M ^ { 2 } } = \lim _ { M \to \infty } \sum _ { \Gamma \in H _ { 1 } ( Y ) } \frac { | \Lambda ( M , \Gamma ) | } { M ^ 2 } = \frac { | H _ { 1 } ( Y ) | } { 2 \mathrm { V o l } ( Y , \lambda ) } . \end{align*}


\begin{align*} e _ A ( t ) : = \exp ( A _ t ) , t \ge 0 . \end{align*}


\begin{align*} \xi ( s ) = \frac { 1 } { 2 } s ( s - 1 ) \pi ^ { - s / 2 } \Gamma \left ( \frac { s } { 2 } \right ) \zeta ( s ) \end{align*}


\begin{align*} \| v ^ { n } ( t ) \| _ { L ^ p ( \Omega ; H ^ \gamma ) } & \leq \| E ( t ) P _ n P X _ 0 \| _ { L ^ p ( \Omega ; H ^ \gamma ) } + \Big \| \int _ 0 ^ t E ( t - s ) P _ n A P F ( X ^ { n } ( s ) ) \ , \dd s \Big \| _ { L ^ p ( \Omega ; H ^ \gamma ) } \\ & \leq \| X _ 0 \| _ { L ^ p ( \Omega ; H ^ \gamma ) } + C \leq C ( T , \gamma , X _ 0 , p ) \end{align*}


\begin{align*} Q ( x _ 0 , r _ n ) = \lbrace y \in \R ^ d : 0 \le \min _ i ( y _ i - ( x _ 0 ) _ i ) \le \max _ i ( y _ i - ( x _ 0 ) _ i ) < r _ n \rbrace . \end{align*}


\begin{align*} { \mathcal P } _ j ( p ) = & \sum _ { \substack { X ^ { \alpha _ { j - 1 } } < q \leq X ^ { \alpha _ { j } } \\ q } } \frac { \chi _ { 8 p } ( q ) \lambda _ f ( q ) } { \sqrt { q } } , { \mathcal M } _ j ( p , \alpha ) = E _ { e ^ 2 \alpha ^ { - 3 / 4 } _ j } \Big ( \alpha { \mathcal P } _ j ( p ) \Big ) , { \mathcal M } ( p , \alpha ) = ( \log X ) ^ { 1 / 2 } \prod ^ { \mathcal { J } } _ { j = 1 } { \mathcal M } _ j ( p , \alpha ) . \end{align*}


\begin{align*} U _ { p , q , \delta } = F _ { T } ( \Omega _ { N _ { p , q , \delta } } ) . \end{align*}


\begin{align*} \delta _ H ( f _ i ( L _ i ) , f _ 0 ( L _ 0 ) ) = \delta _ H ( f _ i ( \phi _ i ( ( L _ 0 ) ) , f _ 0 ( L _ 0 ) ) \leq d _ { C ^ 0 } ( f _ i \circ \phi _ i , f _ 0 ) , \end{align*}


\begin{align*} \partial _ { x _ i x _ j } f ( \mathbf { x } ) = o ( 1 ) \Omega \to 0 ^ { + } , \ ; n \to + \infty \ ; , \end{align*}


\begin{align*} \mathfrak { P } _ { \ell } \rho ^ { ( t ) } \Psi _ { \Sigma } ( f ) = \Psi _ { \Sigma } ( \rho ^ { ( t ) } _ { * } f ) = \Psi _ { \Sigma } ( f ) - t \Psi _ { \Sigma } ( \mathcal { L } _ X f ) + \mathcal { O } ( t ^ 2 ) . \end{align*}


\begin{align*} m ( D ) = \big ( \mathcal { F } ^ \# \big ) ^ { - 1 } m ( k ) \mathcal { F } ^ \# , \end{align*}


\begin{align*} s _ \mu ( B , U ) = \sup \left \{ s \geq 0 \ : \ \mathcal { H } ^ { \mu , s } _ { \infty } \left ( U \right ) \geq \mu ( B ) \right \} . \end{align*}


\begin{align*} \{ 1 \} \cup \{ e ^ { \lambda ^ h _ k t } \} _ { | k | \geq k _ 0 } \cup \{ e ^ { \widehat \lambda _ { n _ l } t } \} _ { l = 1 } ^ { l _ 0 } \end{align*}


\begin{align*} \phi ( x , y ) = \begin{dcases} 1 , & x = 0 , 0 \leq y \leq 1 , \\ - 1 , & x = 1 , 0 \leq y \leq 1 , \\ - 2 x + 1 , & 0 \leq x \leq 1 , y = 0 , \\ - 2 x + 1 , & 0 \leq x \leq 1 , y = 1 . \end{dcases} \end{align*}


\begin{align*} \mathcal { L } \mathfrak { A } _ { i j } = \sum ^ n _ { k = 1 } \mu _ k E _ { i j k } + F _ { i j } , \end{align*}


\begin{align*} ( \mathcal { F } ( \mathcal { F } \varphi ) ) ( \xi ) = \varphi ( - \xi ) , \end{align*}


\begin{align*} a = \frac { \log \left ( n + \sqrt { n ^ 2 + 4 } \right ) - \log ( 2 ) } { \gamma \ , n } . \end{align*}


\begin{align*} \tau ' ( a _ n \cdots a _ 1 v b _ 1 \cdots b _ m ) = \tau ( a _ n \cdots a _ 1 \tau ' ( v ) b _ 1 \cdots b _ m ) . \end{align*}


\begin{align*} \boxed { \log \Bigl ( \prod ( 1 + \alpha _ k ) \Bigr ) = \sum \log ( 1 + \alpha _ k ) . } \end{align*}


\begin{align*} & \forall \gamma ^ 1 , x ^ X \big ( \gamma > _ \mathbb { R } 0 \land \exists y ^ X , z ^ X ( z \in A y \land x = _ X y + _ X \gamma z ) \\ & \qquad \qquad \qquad \qquad \qquad \rightarrow \gamma ^ { - 1 } ( x - _ X J ^ A _ { \gamma } x ) \in A ( J ^ A _ { \gamma } x ) \big ) , \end{align*}


\begin{align*} C _ { 1 , p , \delta , r } : = \int _ 0 ^ { + \infty } p _ \delta ( x ) ( G _ \delta ( x ) ) ^ r d x < + \infty , \end{align*}


\begin{align*} R _ 0 = R _ 1 + R _ 2 + R _ 3 + R _ 4 , \end{align*}


\begin{align*} - k _ { r } \leqslant \sum _ { j = 1 } ^ { r } i _ { k _ { j } } \leqslant 0 f o r 1 \leqslant r \leqslant m . \end{align*}


\begin{align*} \partial _ t X _ t \cdot \nu ( X _ t ) : = - H _ { E ( t ) } ^ \alpha ( X _ t ) , \ ; \ ; \ ; \mbox { f o r a l l $ X _ t \in \partial E ( t ) $ a n d } t \in [ 0 , T ] , \end{align*}


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


\begin{align*} \mathcal { L } \Phi = F ^ { i j } ( \lambda ' \delta _ { i j } - u h _ { i j } ) = \lambda ' \sum _ { i } F ^ { i i } - u F = \lambda ' \sum _ { i } F ^ { i i } - u ^ { \frac { \alpha + 1 } { \alpha } } F ^ { 2 } . \end{align*}


\begin{align*} X _ 0 ( T ) & = \big \{ \ , \lambda \in X ^ \star ( T ) \langle \lambda , \alpha ^ \vee \rangle = 0 \alpha \in R ^ + \ , \big \} \\ X _ 1 ( T ) & = \big \{ \ , \lambda \in X ^ \star ( T ) 0 \leq \langle \lambda , \alpha ^ \vee \rangle \leq p - 1 \alpha \in R ^ + \ , \big \} \end{align*}


\begin{align*} & Y ^ { ( \ell ) } [ ( \partial _ \mu z ) ^ { \gamma _ 1 } , \ldots , ( \partial _ \mu z ) ^ { \gamma _ m } ; f _ 1 , \ldots , f _ m ] : = \sum _ { \mu \leq \ell } \lambda _ \mu \prod _ { t = 1 } ^ m \left \{ \frac { C _ W } { n _ { \ell } } \sum _ { k = 1 } ^ { n _ \ell } f _ { t ; k } ^ { ( \ell ) } ( \partial _ \mu z _ k ^ { ( \ell ) } ) ^ { \gamma _ t } \right \} \end{align*}


\begin{align*} G = \{ a \rightarrow a x , \ ; \ ; x \rightarrow 1 + x ^ 2 \} , \end{align*}


\begin{align*} \tilde M _ { n , \mathbf { p } _ { k , 1 } } \cdot { \tilde M } _ { n , \mathbf { p } _ { k , 2 } } & = \frac { 4 } { n ^ 2 } \sum _ { \mathbf { i } \in \mathcal { J } } \prod _ { \ell = 1 } ^ { k } \tilde I ^ { ( p _ { \ell , 1 } ) } _ { i _ 1 , i _ 2 } \tilde I ^ { ( p _ { \ell , 2 } ) } _ { i _ 3 , i _ 4 } , \end{align*}


\begin{gather*} \sum _ { i = 1 } ^ { N - 1 } \sup _ { h \in \mathrm { L i p } ( 1 ) } \left | \tilde { Q } _ { i - 1 } \big ( P _ { \eta } - \tilde { Q } _ { 1 } \big ) P _ { ( N - i ) \eta } h ( x ) \right | \leq C ( 1 + | x | ) \ , \eta ^ { 2 / \alpha - 1 } . \end{gather*}


\begin{align*} \abs { f ( x ) - g ( x ) } \ = \ h ( X , x ) , \end{align*}


\begin{align*} R ^ t \pi _ { 1 * } ( \Q _ S ) ) _ \lambda = 0 . \end{align*}


\begin{align*} D _ { 2 , m , n } = \tau _ { A \smallsetminus \{ \alpha \} } ( p ^ m ) \tau _ { B \smallsetminus \{ \beta \} \cup \{ - \alpha \} } ( p ^ n ) . \end{align*}


\begin{align*} \bigg ( \sum _ { k = 0 } ^ { \infty } \bigg ( \int _ Y | F _ k ( y ) | d \nu ( y ) \bigg ) ^ r \bigg ) ^ { 1 / r } ~ \leq ~ \int _ Y \bigg ( \sum _ { k = 0 } ^ { \infty } | F _ k ( y ) | ^ r \bigg ) ^ { 1 / r } d \nu ( y ) . \end{align*}


\begin{align*} { \sf F i x } _ M ( G ) = \{ x \in M : \sigma ( x ) = x , \forall \sigma \in G \} \end{align*}


\begin{align*} \sigma _ { } ( ( I + \tilde { F } \tilde { F } ^ \top ) ^ { - \frac { 1 } { 2 } } ) & = \frac { 1 } { \sigma _ { } ( ( I + \tilde { F } \tilde { F } ^ \top ) ^ { \frac { 1 } { 2 } } ) } \\ & = \frac { 1 } { \sqrt { \sigma _ { } ( I + \tilde { F } \tilde { F } ^ \top ) } } \\ & = \frac { 1 } { \sqrt { \lVert ( I + \tilde { F } \tilde { F } ^ \top ) \rVert _ 2 } } , \end{align*}


\begin{align*} \mathcal { L } ( q , \lambda ( q , t ) ) = \mathcal { E } ( q , \lambda ( q , t ) ) \end{align*}


\begin{align*} f _ \varepsilon ( x ) = \int _ { B _ \varepsilon ^ + ( x ) } \bar f ( y ) \big ( \eta _ \varepsilon ( x - y ) - \eta _ \varepsilon ( x _ \ast - y ) \big ) \dd y \geqslant - ( M + \varepsilon ) \int _ { B _ \varepsilon ^ + ( x ) } y _ 1 \big ( \eta _ \varepsilon ( x - y ) - \eta _ \varepsilon ( x _ \ast - y ) \big ) \dd y . \end{align*}


\begin{align*} R ^ 2 + R ^ \# = \theta R \end{align*}


\begin{align*} \alpha = \sqrt { \left ( L - \mu \right ) \left ( \mu _ p t _ 1 \left ( \mu _ p - \mu \right ) \left ( L t _ 1 - 2 \right ) + \left ( L - \mu \right ) \right ) } . \end{align*}


\begin{align*} \begin{aligned} & h _ q + G _ \beta ^ \ast ( v ( y _ q , t _ q ) , 0 , Y _ q , K \cap \{ v ( \cdot , t _ q ) \leq v ( y _ q , t _ q ) \} ) \geq 0 , \\ & k _ q + G _ \beta ^ \ast ( v ( z _ q , t _ q ) , 0 , Z _ q , K \cap \{ v ( \cdot , t _ q ) \leq v ( z _ q , t _ q ) \} ) \geq 0 , \end{aligned} \end{align*}


\begin{align*} \hat q ^ * = q ^ * + q _ d ^ * \epsilon . \end{align*}


\begin{align*} S _ n Q _ { n + 1 } - S _ { n + 1 } Q _ n = ( - 1 ) ^ n w \end{align*}


\begin{align*} \delta ( t _ n ' ) = \big | \| \nabla u ( t _ n ' ) \| _ 2 ^ 2 - \| \nabla Q \| _ 2 ^ 2 \big | \to 0 , \end{align*}


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


\begin{align*} f ( \xi , k ) = - 2 \cdot 6 ^ { \frac { 2 } { 3 } } \eta \left ( k - \eta \right ) \left ( 1 + \frac { k - \eta } { 2 \eta } + \mathcal { O } \left ( \left ( k - \eta \right ) ^ 2 \right ) \right ) . \end{align*}


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


\begin{align*} [ a ( z ) , b ( w ) ] = \sum _ { j = 0 } ^ { N - 1 } \partial _ w ^ { ( j ) } \delta ( z - w ) c ^ j ( w ) , \end{align*}


\begin{align*} g = a \ , w a \in \R . \end{align*}


\begin{align*} \forall n , m \in \mathbb { N } \forall p \in F \left ( H ( d ( x _ { n + m } , p ) ) \leq G ( d ( x _ n , p ) ) + \sum _ { i = n } ^ { n + m - 1 } \varepsilon _ i \right ) , \end{align*}


\begin{align*} \mu _ { T } \{ \lambda \in \mathbb { C } : | \lambda | \leq r \} = \begin{cases} 0 , & r < \lambda _ 1 ( \mu _ { | T | } ) ; \\ 1 + S ^ { \langle - 1 \rangle } _ { \mu _ { T ^ * T } } ( r ^ { - 2 } ) , & \lambda _ 1 ( \mu _ { | T | } ) < r < \lambda _ 2 ( \mu _ { | T | } ) ; \\ 1 , & r \geq \lambda _ 2 ( \mu _ { | T | } ) . \end{cases} \end{align*}


\begin{align*} Y _ { \alpha \beta } ^ { i j } = \frac { h ( \mathcal { I } ) \cdot h ( \mathcal { I } _ { \alpha \beta } ^ { i j } ) } { h ( \mathcal { I } _ { \beta } ^ { i } ) \cdot h ( \mathcal { I } _ { \alpha } ^ { j } ) } - 1 . \end{align*}


\begin{align*} Z _ { t _ 1 } \subseteq \mathring { Z } _ { t _ 1 ' } = T ( Z _ { t _ 1 ' } ) + C \subseteq 0 + C = \mathring { N } _ t \end{align*}


\begin{align*} G \cup H \cup I & = ( \mathbb { N } \setminus \{ \bar { b } _ n \} ) \cup \{ \bar { b } _ n \} = \mathbb { N } , \end{align*}


\begin{align*} z ( z - 1 ) y '' ( z ) = n ( n - 1 ) y ( z ) ; \end{align*}


\begin{align*} b _ j = \sum \limits _ { i = 1 } ^ { j } ( - 1 ) ^ i q ^ { { j - i \choose 2 } } { j \choose i } _ q a _ i \end{align*}


\begin{align*} R f ( x , \omega ) = \widehat { V _ f f } ( \omega , - x ) . \end{align*}


\begin{align*} \lhd \ , T ( x ) , T ( y ) \ , \rhd = \lhd \ , x , y \ , \rhd , \quad x , y \in Z _ 2 . \end{align*}


\begin{align*} \int _ { \R ^ N } | \tilde { u } _ n - w | ^ 2 = & \int _ { B _ R ( \mathbf { 0 } ) } | \tilde { u } _ n - w | ^ 2 + \int _ { B _ R ^ c ( \mathbf { 0 } ) } | \tilde { u } _ n - w | ^ 2 \le \\ & \int _ { B _ R ( \mathbf { 0 } ) } | \tilde { u } _ n - w | ^ 2 + 2 \varepsilon \le 3 \varepsilon \ ; , \end{align*}


\begin{align*} H ' = \{ ( \rho ( x ) , \varphi ( x ) ) \mid x \in A \} \end{align*}


\begin{align*} \Lambda _ n = \Lambda _ { n , 1 } + \Lambda _ { n , 2 } , \end{align*}


\begin{align*} [ L _ { i j } , L _ { h k } ] = a _ { j h } L _ { i k } - a _ { i h } L _ { j k } - a _ { j k } L _ { i h } + a _ { i k } L _ { j h } , \end{align*}


\begin{align*} \int ( w ( \tau _ 0 ) - Q _ b ) \phi _ { 1 , b , \beta } \frac { y ^ 2 } { 2 } \rho _ \beta d y = \left ( \frac { 2 } { \beta } \left ( \frac { d } { 2 } - \gamma + 2 \right ) - \frac { 1 } { \beta } \left ( \frac { d } { 2 } - \gamma + 1 \right ) \right ) \| \phi _ { 1 , b , \beta } \| _ { L ^ 2 _ { \rho _ \beta } } ^ { 2 } + O ( b ^ { 1 - \frac { \epsilon } { 2 } } ) . \end{align*}


\begin{align*} \alpha ( t , x ) = a _ { \pi ( j ) } ( x ) , ( t , x ) \in [ s _ { j - 1 } , s _ j ) \times X . \end{align*}


\begin{align*} V ( \alpha ) & = \Big \{ { \sum } _ { i = 1 } ^ m ( 1 - p ^ { - d _ i } ) \colon \alpha \le ( d _ 1 , \ldots , d _ m ) , \ d _ 1 = z \Big \} , \\ \Gamma _ z & = \Big \{ { \sum } _ { i = 0 } ^ { z - 1 } a _ i ( p ^ z - p ^ i ) \colon 0 \le a _ i < p \Big \} . \end{align*}


\begin{align*} B _ { k n } ( x , y ) = \begin{bmatrix} \xi _ { k n } ( x ) + E _ { k n } ( x , y ) \\ x \end{bmatrix} \end{align*}


\begin{align*} = \left [ \begin{array} { c } \rho ( g _ 1 , g _ 2 ) \iota ( z _ 1 , z _ 2 ) \\ 1 \end{array} \right ] \lambda ( \rho ( g _ 1 , g _ 2 ) , \iota ( z _ 1 , z _ 2 ) ) \times \end{align*}


\begin{align*} \pi ( \l ) = M _ \omega T _ x , \l = ( x , \omega ) \in \R ^ { 2 d } \ , . \end{align*}


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


\begin{align*} R ^ 2 + R ^ \# = \theta R , \end{align*}


\begin{align*} G ( X _ t ) & = G ( x _ 0 ) + \int _ 0 ^ t ( G ' ( X _ s ) \cdot \mu ( X _ s ) + \tfrac { 1 } { 2 } G '' ( X _ s ) \cdot \sigma ^ 2 ( X _ s ) ) \ , d s + \int _ 0 ^ t G ' ( X _ s ) \cdot \sigma ( X _ s ) \ , d W _ s \\ & = G ( x _ 0 ) + \int _ 0 ^ t \widetilde \mu ( G ( X _ s ) ) \ , d s + \int _ 0 ^ t \widetilde \sigma ( G ( X _ s ) ) \ , d W _ s . \end{align*}


\begin{align*} F _ k ^ { \mu , \sigma } ( \mathbf { x } , \mathbf { y } , \mathbf { s } ) : = \begin{bmatrix} ( H + \rho I ) \mathbf { x } - A ^ T \mathbf { y } + \mathbf { g } - \rho { \mathbf { x } _ k } - \begin{bmatrix} 0 \\ \mathbf { s } \end{bmatrix} \\ A \mathbf { x } + \delta ( \mathbf { y } - \mathbf { y } _ k ) - \mathbf { b } \\ S X _ { \mathcal { C } } \mathbf { e } - \sigma \mu \mathbf { e } \end{bmatrix} . \end{align*}


\begin{align*} D _ h ( r u + z , r v + z ) = D _ { h ( r \cdot + z ) + Q \log r } ( u , v ) , \forall u , v \in \mathbb C . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ { \infty } \sup _ { \tau \in [ \tau _ 0 , 0 ] } \left \| z ^ { ( k ) } ( \tau , \cdot ) \right \| _ { H ^ 2 ( \R \backslash \{ t , 0 \} ) } ~ < ~ \sum ^ { \infty } _ { k = 1 } \beta _ k ( \tau ) ~ < ~ \infty . \end{align*}


\begin{align*} G = \{ u \rightarrow 2 u v , \ ; \ ; v \rightarrow u \} . \end{align*}


\begin{align*} \frac { \zeta ' ( s ) } { \zeta ( s ) } = B - \frac { 1 } { s - 1 } + \frac { 1 } { 2 } \log \pi - \frac { \Gamma ' ( s / 2 + 1 ) } { 2 \ \Gamma ( s / 2 + 1 ) } + \sum _ { \rho } \left ( \frac { 1 } { s - \rho } + \frac { 1 } { \rho } \right ) \end{align*}


\begin{align*} \mathbf { g } _ \star ( X , Y ) = \langle X \star \langle Y , \mathbf { g } ^ A \rangle _ \star , \mathbf { g } _ A \rangle _ \star . \end{align*}


\begin{align*} h = \frac { 4 r ^ 2 } { 8 m - 4 r } . \end{align*}


\begin{align*} \| 1 _ A \| & \ = \ \| z - 1 _ { B \backslash E } - P _ E ( z ) \| \\ & \ \leqslant \ \mathbf C _ \ell \| z - P _ E ( z ) \| \ \leqslant \ \mathbf C _ \ell \mathbf C _ { \lambda , d k p } \sigma ^ { \mathcal { U } _ X , L } _ m ( z ) \ \leqslant \ \mathbf C _ \ell \mathbf C _ { \lambda , d k p } \| 1 _ B \| . \end{align*}


\begin{align*} f _ i = g _ i + f _ { i - 1 } \end{align*}


\begin{align*} u _ \varepsilon ( x ) = \int _ { \R ^ n _ + } \bar u ( y ) \big ( \eta _ \varepsilon ( x - y ) - \eta _ \varepsilon ( x _ \ast - y ) \big ) \dd y \geqslant 0 x \in \R ^ n _ + \end{align*}


\begin{align*} \frac { \mathbf { P } \Big ( \frac { 1 } { \sigma } ( \hat { \theta } _ n - \theta ) \sqrt { \Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } > x \Big ) } { 1 - \Phi \left ( x \right ) } = \exp \bigg \{ \theta _ 1 C \frac { \ln n + x ^ { 3 } } { \sqrt { n } } \bigg \} \end{align*}


\begin{align*} \theta _ { P _ H } = \frac { \mathbb { E } [ Y ^ 2 ] - \mathbb { E } [ Y ] ^ 2 } { \mathbb { E } [ Y ] - \frac { 1 } { 1 + e ^ { a b } } } . \end{align*}


\begin{align*} H \left ( u \right ) : = \int \frac { 1 } { 2 } \left | \partial _ x u \right | ^ { 2 } + \frac { 1 } { 2 } V \left | u \right | ^ { 2 } \pm \frac { 1 } { 4 } \left | u \right | ^ { 4 } \ , d x . \end{align*}


\begin{align*} j _ 0 ( \delta ) & : = \begin{cases} k & \mbox { i f $ \delta = - k $ f o r s o m e $ k \in \mathbb { N } $ , } \\ 0 & \mbox { o t h e r w i s e } , \\ \end{cases} \end{align*}


\begin{align*} M _ 2 ( p , H ) = \frac { \pi ^ 2 } { 8 } \times \begin{cases} \frac { a + 1 } { a - 1 } \times \left ( 1 - \frac { d } { p } \right ) & \hbox { i f $ a $ i s o d d , } \\ 1 - \frac { ( d - 1 ) a + 1 } { p } & \hbox { i f $ a $ i s e v e n . } \\ \end{cases} \end{align*}


\begin{align*} A _ \alpha ( p _ \alpha ) ( x ) = 0 . \end{align*}


\begin{align*} p ( x ) = \phi \bigl ( f _ { 1 } ( x ) , \ldots , f _ d ( x ) \bigr ) , \end{align*}


\begin{align*} \pi ( \Phi ) \pi ( \xi , \eta , 0 ) \pi ( \Phi ) = \pi ( \Phi ) \pi ( \Phi ^ { \xi , \eta } ) = \pi ( \Phi \natural \Phi ^ { \xi , \eta } ) = e ^ { - \tfrac { \pi } { 2 } ( \xi ^ 2 + \eta ^ 2 ) } \pi ( \Phi ) . \end{align*}


\begin{align*} \norm { A - A \mathbb { P } } & = \norm { A ( I - \mathbb { P } ) } = \norm { A ( I - Q _ { W _ k } \ ! Q _ { W _ k } ^ T ) ( I - \mathbb { P } ) } \\ & \leq \norm { A ( I - Q _ { W _ k } \ ! Q _ { W _ k } ^ T ) } ~ \norm { I - \mathbb { P } } . \end{align*}


\begin{gather*} \phi [ T ] = ( p , 0 ) . \end{gather*}


\begin{align*} Q ( \xi , k ) : = \left \{ \begin{aligned} & Q _ 1 = \sigma _ 1 r ^ { - \frac { \sigma _ 3 } { 2 } } D ^ { \sigma _ 3 } \sigma _ 1 \sigma _ 3 , & \mathbb { C } _ { + } \cap U _ \delta ( \eta ) , \\ & Q _ 2 = \sigma _ 1 ( r ^ * ) ^ { - \frac { \sigma _ 3 } { 2 } } D ^ { - \sigma _ 3 } , & \mathbb { C } _ { - } \cap U _ \delta ( \eta ) , \\ \end{aligned} \right . \end{align*}


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


\begin{align*} \sum _ { \substack { \log x < | x - n | \leq x \\ \Lambda ( n ) \neq 0 } } \frac { 1 } { | x - n | } = \int _ { \log x } ^ { x } \frac { d p ( u ) } { u } = \left [ \frac { p ( u ) } { u } \right ] _ { \log x } ^ { x } + \int _ { \log x } ^ { x } \frac { p ( u ) } { u ^ 2 } \ , d u . \end{align*}


\begin{align*} C _ { \mathbf { F } _ N } ^ T C _ { \mathbf { F } _ N } = \begin{pmatrix} \frac { N } { 2 } & 0 \\ 0 & \frac { N } { 2 } \end{pmatrix} , \end{align*}


\begin{align*} \frac { 1 } { t } d _ K \big ( x + t ( A x + v ) \big ) & = \frac { 1 } { t } d _ K \bigg ( S _ t x + t v + t \bigg ( A x - \frac { S _ t x - x } { t } \bigg ) \bigg ) \\ & \leq \frac { 1 } { t } d _ K ( S _ t x + t v ) + \bigg \| A x - \frac { S _ t x - x } { t } \bigg \| , \end{align*}


\begin{align*} \mu _ \omega ( A ) = \int _ A \omega \ , d \mu _ X \end{align*}


\begin{align*} D ^ n ( y ) = M _ n ( x , y ) . \end{align*}


\begin{align*} \tilde \phi _ { i , \beta } ( y ) = \left \{ \begin{array} { r c l } & & K _ 0 y ^ { - \tilde \gamma } ( 1 + O ( y ^ 2 ) ) y \to 0 , \\ & & K _ { \infty } y ^ { 2 i - \gamma } ( \ln y + O ( 1 ) ) y \to + \infty , \end{array} \right . \end{align*}


\begin{align*} \phi _ { i j k } ^ { - 1 } \tilde { F } _ i \phi _ { i j k } ( y ) & = ( \mathrm { i d } - p \psi _ { i j k } ) \tilde { F } _ i ( \mathrm { i d } + p \psi _ { i j k } ) ( y ) \\ & = ( \mathrm { i d } - p \psi _ { i j k } ) ( \tilde { F } _ i ( y ) + p \psi _ { i j k } ( y ) ^ p ) \\ & = \tilde { F } _ i ( y ) - p \psi _ { i j k } ( y ^ p ) + p \psi _ { i j k } ( y ) ^ p \\ & = \tilde { F } _ i ( y ) + p \psi _ { i j k } ( y ) ^ p , \end{align*}


\begin{align*} \mathcal { L } A _ { i j , 0 } = p \kappa _ 0 \sigma _ j Q ^ { p - 1 } , \end{align*}


\begin{align*} \partial _ t \gamma _ T ^ { s \to t } \circ \alpha _ { V ( t ) } ^ { ( t - s ) T } ( A ) & = \gamma _ T ^ { s \to t } ( - \i [ \widetilde { H } _ t , \alpha _ { V ( t ) } ^ { ( t - s ) T } ( A ) ] ) + \gamma _ T ^ { s \to t } ( T \alpha _ { V ( t ) } ^ { ( t - s ) T } \circ \delta _ { V ( t ) } ( A ) + \i [ \widetilde { H } _ t , \alpha _ { V ( t ) } ^ { ( t - s ) T } ( A ) ] ) \\ & = T \gamma _ T ^ { s \to t } \circ \alpha _ { V ( t ) } ^ { ( t - s ) T } \circ \delta _ { V ( t ) } ( A ) . \end{align*}


\begin{align*} \widetilde { c } _ k ( s ) = \alpha ( s ) \pi ^ { \frac { n ( n - 1 ) } { 2 } } \frac { \Gamma ( 2 \lambda + 1 ) \Gamma ( 2 \lambda + 3 ) \ldots \Gamma ( 2 \lambda + 2 n - 3 ) } { \Gamma ( 2 \lambda + n ) \Gamma ( 2 \lambda + n + 1 ) \ldots \Gamma ( 2 \lambda + 2 n - 2 ) } , \end{align*}


\begin{align*} \mathfrak { i } : M _ n ( \mathbb { B } ) \stackrel { \sim } \longrightarrow \{ x \in M _ { 2 n } ( \mathbb { K } ) : \overline { x } I _ n ' J _ n ' = I _ n ' J _ n ' x \} . \end{align*}


\begin{align*} \left \langle f , ( \lambda - \mathcal { L } ^ { ( p ) } ) \varphi \right \rangle = 0 , \varphi \in \mathcal { A } ( M \setminus K ) , \end{align*}


\begin{align*} \Lambda _ { ( n , \infty ) } ( M , \Gamma ) : = \bigcup _ { m = 0 } ^ { \infty } \Lambda _ { ( n , m ) } ( M , \Gamma ) \end{align*}


\begin{align*} \prod _ \ast b _ j ( x , k _ j ) : = \overline { b _ 1 ( x , k _ 1 ) } b _ 2 ( x , k _ 2 ) \overline { b _ 3 ( x , k _ 3 ) } b _ 4 ( x , k _ 4 ) . \end{align*}


\begin{align*} M ^ { ( 2 ) } _ n = \frac { M ^ { ( 2 ) } _ 1 } { n } \prod ^ { n - 1 } _ { j = 1 } \left ( 1 + \frac { 2 \alpha } { j } \right ) = - \frac { \Gamma ( n + 2 \alpha ) } { \Gamma ( n + 1 ) \Gamma ( 2 \alpha ) } \sim - \frac { n ^ { - ( 1 - 2 \alpha ) } } { \Gamma ( 2 \alpha ) } , \end{align*}


\begin{align*} G _ \beta ( r , p , X , A ) = F ( p , X , A ) \end{align*}


\begin{align*} \begin{aligned} & \lim _ { n \to \infty } P _ { n , k } ( W _ { \mathcal { S } ( n , k ; M _ n ) } ) = - ( 1 - c ) ^ k \sum _ { l = 1 } ^ { k - 1 } \binom k l ( \frac c { 1 - c } ) ^ l \frac l { k - l } + \\ & k \sum _ { l = 1 } ^ { k - 1 } \binom k l c ^ l \int _ 0 ^ { 1 - c } \frac { y ^ { k - l - 1 } } { 1 - y ^ k } d y - c ^ k \log ( 1 - ( 1 - c ) ^ k ) . \end{aligned} \end{align*}


\begin{align*} ( x , y ) \ = \ \abs { x - y } \end{align*}


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


\begin{align*} \nabla l = \nabla f _ { 1 : m } + P ^ T \nabla f _ { m + 1 : n } . \end{align*}


\begin{align*} \delta _ f ( z ) = \lim _ { x \to \infty } \frac { \pi _ { f } ( x , z ) } { \pi ( x ) } = c _ f ( z ) a _ 2 , \end{align*}


\begin{align*} z = u _ 1 \cdot \ldots \cdot u _ k v _ 1 \cdot \ldots \cdot v _ { \ell } z ' = u _ 1 \cdot \ldots \cdot u _ k w _ 1 \cdot \ldots \cdot w _ { m } \ , , \end{align*}


\begin{align*} ( a + 1 ) a ' & = a ' + a a ' = 1 , \\ ( b + 1 ) b ' & = b ' + b b ' = \tfrac { \sin \alpha } { \sin \gamma } , \\ ( c + 1 ) c ' & = c ' + c c ' = \tfrac { \sin \beta } { \sin \gamma } . \end{align*}


\begin{align*} M [ u ] = M [ Q ] , E [ u ] = E [ Q ] , \| \nabla u \| _ 2 < \| \nabla Q \| _ 2 , \end{align*}


\begin{align*} \widetilde { h } ( t ) = h ( t ) - A e ^ { - e _ 0 t } \mathcal { Y } _ + \end{align*}


\begin{align*} \begin{aligned} \xi ^ { \prime } ( x ) + \eta ^ { \prime } ( x ) & = \lambda \xi ( x ) , x \in ( 0 , 1 ) , \\ \eta ^ { \prime \prime } ( x ) + \eta ^ { \prime } ( x ) + \xi ^ { \prime } ( x ) & = \lambda \eta ( x ) , x \in ( 0 , 1 ) , \\ \xi ( 0 ) & = \xi ( 1 ) , \\ \eta ( 0 ) = 0 , \ \ \eta ( 1 ) & = 0 . \end{aligned} \end{align*}


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


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


\begin{align*} T _ { i + 1 } = H ^ { - 1 } T _ i , T _ 0 = c \Lambda _ \xi Q \end{align*}


\begin{align*} \begin{aligned} \min _ { \mathbf { x } \in \mathbb { R } ^ { d _ 1 } } \ ; & \frac { 1 } { 2 } \mathbf { x } ^ T H \mathbf { x } + \mathbf { g } ^ T \mathbf { x } \\ \hbox { s . t . } \ ; & A \mathbf { x } = \mathbf { b } , \ ; \mathbf { x } - \mathbf { z } = 0 \\ & \mathbf { z } \geq 0 \\ \end{aligned} \end{align*}


\begin{gather*} ( X , \alpha , \beta ) \otimes ( X ' , \alpha ' , \beta ' ) = ( X \otimes X ' , \alpha '' , \beta '' ) , \end{gather*}


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


\begin{align*} f ( y ) & = d ( y , K ) \\ & \leq d ( y , z ) \\ & = d ( x , z ) - d ( x , y ) \\ & \leq d ( x , K ) + \epsilon r - d ( x , y ) \\ & = f ( x ) + \epsilon r - d ( x , y ) . \end{align*}


\begin{align*} \begin{aligned} \Big ( \frac { d } { d x } + x \Big ) ^ \alpha u ( x ) & = e ^ { - x ^ 2 / 2 } \Big ( \frac { d } { d x } \Big ) ^ \alpha \big [ e ^ { ( \cdot ) ^ 2 / 2 } u ( \cdot ) \big ] ( x ) \\ \Big ( - \frac { d } { d x } + x \Big ) ^ \alpha v ( x ) & = e ^ { x ^ 2 / 2 } \Big ( - \frac { d } { d x } \Big ) ^ \alpha \big [ e ^ { - ( \cdot ) ^ 2 / 2 } v ( \cdot ) \big ] ( x ) . \end{aligned} \end{align*}


\begin{align*} L _ A ( t , y ) = & - \sum _ { \substack { i , j = 1 \\ j \not = i } } ^ n \bigg ( \frac { \dot { x } _ i ( t ) \partial _ y A _ { i j } ( t , y - x _ i ( t ) ) } { x ^ 2 _ { i j } ( t ) } + \frac { 2 \dot { x } _ { i j } ( t ) A _ { i j } ( t , y - x _ { i } ( t ) ) } { x ^ 3 _ { i j } ( t ) } \bigg ) , \\ & + \sum _ { \substack { i , j = 1 \\ j \not = i } } ^ n \frac { ( \partial _ t \vec { x } ( t ) , \partial _ t \vec { \mu } ( t ) ) \cdot \nabla _ { \vec { x } , \vec { \mu } } \mathfrak { A } _ { i j } ( \vec { x } ( t ) , \vec { \mu } ( t ) , y - x _ i ( t ) ) } { x ^ 2 _ { i j } ( t ) } , \end{align*}


\begin{align*} C _ k = V ^ 0 _ k \oplus \ldots \oplus V ^ k _ k . \end{align*}


\begin{align*} ( \xi _ 0 , \cdots , \xi _ n ) \bullet [ x _ 0 : \dots : x _ n ] = \left [ \xi _ 0 x _ 0 : \dots : \xi _ n x _ n \right ] . \end{align*}


\begin{align*} \norm { f } _ { \mathcal { M } } = \inf \sum _ { \gamma \in \Gamma } | c _ \gamma | , \end{align*}


\begin{align*} u ( r , t ) = \frac { 1 } { \mu ( t ) } w ( y , \tau ) , y = \frac { r } { \sqrt { \mu ( t ) } } \frac { d \tau } { d t } = \frac { 1 } { \mu ( t ) } . \end{align*}


\begin{align*} \mathcal { R } ( Z ) = \Pi \tilde { \mathcal { R } } ( Z ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \dot { \eta } _ s ( t ) & = & A \eta _ s ( t ) + a ( \eta _ s ( t ) ) \\ \eta _ s ( 0 ) & = & x \end{array} \right . \end{align*}


\begin{align*} \varphi _ R ( A , b , \lbrace x _ j : j \leq k \rbrace , \lbrace W _ j : j < k \rbrace , \lbrace \zeta _ j : j < k \rbrace ) = ( e _ { \mathrm { r e m } ( \zeta _ { k - 1 } , n ) + 1 } , \zeta _ { k - 1 } + 1 ) . \end{align*}


\begin{align*} J ^ r ( x \cdot y ) \cdot J ^ s x = J ^ t y \end{align*}


\begin{align*} E _ { X / A N } = M \times _ { M \cap K } \big ( S _ { \mathfrak { p } \cap \mathfrak { m } } \otimes S _ { \mathfrak { k } / ( \mathfrak { k } \cap \mathfrak { m } ) } \otimes W | _ S \big ) \cong M \times _ { M \cap K } \big ( S _ { \mathfrak { p } \cap \mathfrak { m } } \otimes \widetilde { W } \big ) , \end{align*}


\begin{align*} \tilde { \phi } ^ { n + 1 } = \phi ^ n - \frac { 2 \tau ^ 2 } { 2 + \tau \eta ^ n } \Big ( G ( \phi ^ n ) - \lambda ^ n \phi ^ n \Big ) + \frac { 2 - \tau \eta ^ n } { 2 + \tau \eta ^ n } \Big ( \phi ^ n - \phi ^ { n - 1 } \Big ) . \end{align*}


\begin{align*} \sum _ { a \in S } B _ \ell ( a ) & = \left ( B _ \ell ( n + 1 ) - \frac { \ell } { \ell + 1 } \frac { B _ { \ell + 1 } ( n + 1 ) } { n } \right ) t + \frac { \ell t } { ( \ell + 1 ) n } B _ { \ell + 1 } - \sum _ { k = 1 } ^ { n } k ^ { \ell - 1 } E _ k . \end{align*}


\begin{align*} k n ( 1 - \epsilon ) < d ( z ) + e ( S _ 1 , L _ 1 \cup L _ 2 ) + \frac { \epsilon ^ 2 n } { 2 } = e ( S _ 1 , L ) + \frac { \epsilon ^ 2 n } { 2 } . \end{align*}


\begin{align*} \left \| \frac { \ 6 } { \ 6 x } \right \| = \left \| \frac { \ 6 } { \ 6 y } \right \| = 1 , \end{align*}


\begin{gather*} \lim _ { x \downarrow 0 } \frac { \log \left ( 1 - x + \frac { c \alpha } { ( 2 - \alpha ) \sigma ^ 2 } \ , x ^ { 2 / \alpha } \right ) + x } { x ^ { 2 / \alpha } } = \frac { c \alpha } { ( 2 - \alpha ) \sigma ^ 2 } . \end{gather*}


\begin{align*} { m ( E _ { m } ^ { [ \alpha , \beta ] , \varepsilon } ) = m ( \left \{ x : \underline { \dim } _ { \mathrm { l o c } } ( m , x ) \in [ \alpha , \beta ] \right \} ) } . \end{align*}


\begin{align*} T _ n ( k ) = \sum \limits _ { \substack { A \subset \{ 1 , \dots , d \} \\ | A | = k } } S _ { n , A } ^ { \rm M } . \end{align*}


\begin{align*} \sum _ { t = 1 } ^ { T } Z _ t \leq \frac { 3 } { 4 } \lambda \sum _ { t = 1 } ^ { T } Y _ t ^ 2 + \frac { 1 } { \lambda } \log ( 1 / \delta ) \end{align*}


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


\begin{align*} \Gamma = \left \lbrace 1 - \sup _ { \lbrace \cup _ { j = 1 } ^ s U _ j : U _ j \in \mathfrak { U } _ { i _ j } \rbrace } \min _ { H \in \mathcal { H } } \det ( H ^ \intercal H ) : \lbrace \mathfrak { U } _ { i _ 1 } , \ldots , \mathfrak { U } _ { i _ s } \rbrace \in \mathfrak { P } , ~ s = 1 , \ldots , r \right \rbrace . \end{align*}


\begin{align*} \Phi ( h ) = \frac { 1 } { 2 } \int ( L _ + h _ 1 ) h _ 1 d x + \frac { 1 } { 2 } \int ( L _ - h _ 2 ) h _ 2 d x . \end{align*}


\begin{align*} ( \chi _ A * \chi _ B ) ( g ) = | A \cap ( g - B ) | . \end{align*}


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


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


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


\begin{align*} r _ \nu ( x , y ) = 2 ^ { 1 / 2 } \sin ( 2 \pi \nu y ) \ , e ^ { 2 \pi i n _ \nu x } , \end{align*}


\begin{align*} \frac { 1 } { p } \cdot \frac { d } { d t } \int _ { \Omega } u ^ { - p } ( t , x ) = & - ( p + 1 ) \int _ { \Omega } u ^ { - p - 2 } ( t , x ) | \nabla u ( t , x ) | ^ 2 + ( p + 1 ) \chi \int _ { \Omega } \frac { u ^ { - p - 1 } ( t , x ) } { v ( t , x ) } \nabla u ( t , x ) \cdot \nabla v ( t , x ) \\ & - \int _ { \Omega } a ( t , x ) u ^ { - p } ( t , x ) + \int _ { \Omega } b ( t , x ) u ^ { - p + 1 } ( t , x ) \forall \ , t > s . \end{align*}


\begin{align*} G = \{ a \rightarrow a v , \ ; \ ; v \rightarrow u , \ ; \ ; u \rightarrow 2 u v \} . \end{align*}


\begin{align*} \int _ { K _ 1 } ( g \circ \varphi ) d \mu = \int _ { K _ 2 } g d ( \Phi ^ * \mu ) . \end{align*}


\begin{align*} \mathbf { M } _ { u } : = \sum _ { u \leq v } \mu _ { \mathfrak { B } _ n } ( u , v ) \mathbf { F } _ v , \mathbf { F } _ { u } = \sum _ { u \leq v } \mathbf { M } _ { v } . \end{align*}


\begin{align*} \pi ( p ) = \sum _ { i = 1 } ^ d c _ i \pi ( p _ i ) h ( p ) > \sum _ { i = 1 } ^ d c _ i h ( p _ i ) \end{align*}


\begin{align*} \begin{aligned} & \sum _ { t \notin T _ { i k } } \lambda _ { i t } \leq \delta _ { i k } , \ \sum _ { t \notin \bar { T } _ { i k } } \lambda _ { i t } \leq 1 - \delta _ { i k } i \in \{ 1 , \ldots , d \} \ , k \in \{ 1 , \ldots , \lceil \log _ 2 l _ i \rceil \} , \\ & \lambda \in \Lambda , \ \delta \in \{ 0 , 1 \} ^ { \sum _ { i = 1 } ^ d \lceil \log _ 2 l _ i \rceil } , \end{aligned} \end{align*}


\begin{align*} s _ k = \min \left \{ s ( \mu , \mathcal { B } , \mathcal { U } ) , \alpha \right \} - \varepsilon _ { k } . \end{align*}


\begin{align*} \sigma _ \gamma ( p \otimes q ) = v ^ * ( \varphi _ 1 ( p ) \otimes \varphi _ 2 ( q ) ) v \end{align*}


\begin{align*} \mathcal { S } ( h , k ) = \mathcal { L } ( h , k ) + \mathcal { D } ( h , k ) + \mathcal { U } ( h , k ) , \end{align*}


\begin{align*} e _ { A } ( t ) : = \exp ( A _ t ) , t \ge 0 . \end{align*}


\begin{align*} f _ 1 L i _ { K , \mathfrak { s } } ( \boldsymbol { \alpha } ) + f _ 2 L i _ { K , w } ( \beta ) = 0 . \end{align*}


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


\begin{align*} \lim _ { k \to \infty } \frac { c _ { \sigma _ { k } } ( Y , \lambda ) ^ { 2 } } { I ( \sigma _ { k } ) } = \mathrm { V o l } ( Y , \lambda ) . \end{align*}


\begin{align*} \varphi = ( ( 1 \ , 2 ) , ( \alpha _ 2 ^ { - 1 } , \varepsilon , \alpha _ 3 ^ { - 1 } ) ) \in \prescript { ( 1 \ , 2 ) } { } N _ l ^ A . \end{align*}


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


\begin{align*} L ( b _ { d _ c } ) & = \left ( \prod _ { i = 1 } ^ { d _ c - 1 } \operatorname { s g n } \psi _ i \right ) 2 \tanh ^ { - 1 } \exp \left ( \sum _ { i = 1 } ^ { d _ c - 1 } \log | \psi _ i | \right ) . \end{align*}


\begin{align*} & \int \mathbf { 1 } _ { \{ k \geq 0 \} } e ^ { i k x } \partial _ { x } \mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \ , d k = I _ { 1 } ( t , x ) + I _ { 2 } ( t , x ) , \\ & I _ 1 ( t , x ) : = \int \mathbf { 1 } _ { \{ 0 \leq k \leq \frac { 1 } { \sqrt { t } } \} } e ^ { i k x } \partial _ { x } \mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \ , d k , \\ & I _ 2 ( t , x ) : = \int \mathbf { 1 } _ { \{ k \geq \frac { 1 } { \sqrt { t } } \} } e ^ { i k x } \partial _ { x } \mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \ , d k . \end{align*}


\begin{align*} \begin{aligned} \sup _ { Q _ s \in \mathfrak { Q } _ s , s \in \lbrace 1 , \ldots , k \rbrace } \min _ { G \in \mathcal { G } ( Q _ 0 , \ldots , Q _ k ) } \det ( G ^ \intercal G ) \geq \sup _ { Q _ s \in \mathfrak { Q } _ s ' , s \in \lbrace 1 , \ldots , k \rbrace } \min _ { G \in \mathcal { G } ( Q _ 0 , \ldots , Q _ k ) } \det ( G ^ \intercal G ) , \end{aligned} \end{align*}


\begin{align*} Z _ { \ell i } ^ \pi = w _ { \ell i } ^ \pi + d _ { \ell i } , \end{align*}


\begin{align*} \| p \| : = \max _ { 0 \leq j \leq n } | p _ j | \end{align*}


\begin{align*} - k _ { p } \leqslant \sum _ { j = 1 } ^ { p } i _ { k _ { j } } \leqslant 0 f o r 1 \leqslant p \leqslant m - 1 . \end{align*}


\begin{align*} A _ { \delta } & : = \begin{cases} k ! & \mbox { i f $ \delta = - k $ f o r s o m e $ k \in \mathbb { N } $ , } \\ [ 2 m m ] \dfrac { 1 } { \Gamma ( 1 + \delta ) } & \mbox { o t h e r w i s e . } \\ \end{cases} \end{align*}


\begin{align*} E _ 2 ( f , p ) = \sum _ { z \leq p } \frac { 1 } { p } \sum _ { \substack { 0 < a < p \\ g c d ( n , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi a ( \tau ^ { n } - u ) } { p } } \cdot \frac { 1 } { p } \sum _ { \substack { 0 < b < p \\ \gcd ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { e m } - v ) } { p } } \ll p ^ { 1 - 2 \varepsilon } . \end{align*}


\begin{align*} A _ { \lambda } ( x , t ) = \lambda A \left ( \lambda x , { \lambda } ^ 2 t \right ) , \lambda > 0 . \end{align*}


\begin{align*} \lim _ { x \to \infty } E _ { \theta } ( x ) = 0 , \ \ \lim _ { x \to \infty } E _ { \pi } ( x ) = 0 . \end{align*}


\begin{align*} - c _ \alpha \lim _ { y \to 0 ^ + } y ^ { 1 - 2 \alpha } U _ y ( x , y ) = ( \mathfrak { D } _ { { \rm l e f t } , a } ) ^ \alpha u ( x ) \end{align*}


\begin{align*} l ( \gamma ) = \left \{ \begin{array} { l l } 2 , & f o r \ ; \gamma \in ( \frac d 2 , 2 ] , \\ 3 , & f o r \ ; \gamma \in ( 2 , 3 ] , \\ 4 , & f o r \ ; \gamma \in ( 3 , 4 ] , \end{array} \right . \ ; a n d \ell ( \gamma ) = \left \{ \begin{array} { l l } 0 , & f o r \ ; \gamma \in ( \frac d 2 , 2 ] , \\ 2 , & f o r \ ; \gamma \in ( 2 , 4 ] . \end{array} \right . \end{align*}


\begin{align*} \forall x \in \R \colon \varphi ( x ) : = - x , \Phi ( x ) : = \begin{cases} \R & x \leq 0 , \\ [ x ^ 2 , \infty ) & x = \frac 1 k k \in \N , \\ \varnothing & \end{cases} \end{align*}


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


\begin{align*} \mathcal V _ { p } ^ p ( \P , \Q ) : = \max \big \{ V _ p ( \P , \Q ) , V _ p ( \Q , \P ) \big \} , \end{align*}


\begin{align*} r _ k ( \mathbf { x } , \mathbf { y } ) : = \begin{bmatrix} \mathbf { x } \\ \mathbf { y } \end{bmatrix} - \Pi _ { D } ( \begin{bmatrix} \mathbf { x } \\ \mathbf { y } \end{bmatrix} - \begin{bmatrix} H \mathbf { x } + \mathbf { g } - A ^ T \mathbf { y } + \rho ( \mathbf { x } - \mathbf { x } _ k ) \\ A \mathbf { x } - \mathbf { b } + \delta ( \mathbf { y } - \mathbf { y } _ k ) \end{bmatrix} ) . \end{align*}


\begin{align*} A _ { i j } ( t , y ) = A _ { i j , 0 } ( y ) + \mathfrak { A } _ { i j } ( \vec { x } ( t ) , \vec { \mu } ( t ) , y ) . \end{align*}


\begin{align*} \check H _ { c t } ^ q ( \Z _ { \ge 0 } ; A ) = \begin{cases} A & q = 0 \\ 0 & \mbox { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} q q _ d ^ * + q _ d q ^ * = q ^ * q _ d + q _ d ^ * q = 0 . \end{align*}


\begin{align*} h ^ i _ { \mathbb { Q } _ p } : H ^ i ( K _ n , V ) \xrightarrow { \sim } H ^ i _ { \Phi \Gamma } ( \mathbf { D } ( V ) ) : = H ^ i ( C ^ { \bullet } ( \mathbf { D } ( V ) ) ) . \end{align*}


\begin{align*} \frac { 1 } { q } \int _ \Omega u _ t ^ q ( t , x ) d x & = - ( q - 1 ) \int _ \Omega u ^ { q - 2 } ( t , x ) | \nabla u ( t , x ) | ^ 2 d x - \chi ( q - 1 ) \int _ \Omega \frac { u ^ { q - 1 } ( t , x ) } { v ( t , x ) } \nabla u ( t , x ) \cdot \nabla v ( t , x ) d x \\ & \ , \ , \ , \ , + \int _ \Omega a ( t , x ) u ^ q ( t , x ) - \int _ \Omega b ( t , x ) u ^ { q + 1 } ( t , x ) d x \forall \ , t > s , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \mathrm { d } X _ { t } = f ( X _ { t } , \theta ) \mathrm { d } t + \sigma \mathrm { d } W _ { t } + \mathrm { d } L _ { t } , \\ & X _ { 0 } = x \in [ a , \infty ) . \end{aligned} \right . \end{align*}


\begin{align*} f ( x , y ) = \begin{dcases} 3 2 , & \frac { 5 } { 8 } \leq x \leq \frac { 7 } { 8 } , \frac { 5 } { 8 } \leq y \leq \frac { 7 } { 8 } , \\ 0 , & e l s e . \\ \end{dcases} \end{align*}


\begin{align*} \phi _ t ( x , t ) - \Delta \phi ( x , t ) + a | \nabla \phi ( x , t ) | ^ \alpha = - a k ^ \alpha - { n - 1 \over | x | } + a k ^ \alpha \leq 0 . \end{align*}


\begin{align*} \tilde { g } _ 2 ( s , y ) & = s ^ { 2 H - 1 } y ^ { - \tilde { \theta } \rho + \tilde { \theta } + 1 } \Big ( \alpha _ 2 \frac { 1 } { \tilde { \theta } ^ { \rho } } - ( H + \epsilon ) \tilde { \sigma } ( \tilde { \theta } + 1 ) y ^ { \tilde { \theta } \rho - \tilde { \theta } - 2 } \Big ) \\ & \ge h _ 1 s ^ { 2 H - 1 } y ^ { - \alpha } \end{align*}


\begin{align*} ( \alpha 1 + x ) ( \beta 1 + y ) = ( \alpha \beta + f ( x , y ) ) 1 + \alpha y + \beta x , \end{align*}


\begin{align*} A ( y , u ) : = \{ t \in [ 0 , T ) \colon | y ( t ) | = r \nexists \varepsilon > 0 y - u = \mathrm { c o n s t } [ t , t + \varepsilon ) \} . \end{align*}


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


\begin{align*} \lambda ^ { - 1 } ( x - J ^ A _ \lambda x ) = _ X \frac { \gamma } { \lambda \gamma } ( x - J ^ A _ \gamma x ) = _ X \frac { 1 } { \gamma } \left ( \frac { \gamma } { \lambda } x + \left ( 1 - \frac { \gamma } { \lambda } \right ) J ^ A _ \lambda x - J ^ A _ \lambda x \right ) . \end{align*}


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


\begin{align*} \widetilde { A } _ 1 = \mathrm { p r o j } _ { \subseteq U } ( \varphi ( A _ 1 ) ) \quad \widetilde { A } _ 2 = \mathrm { p r o j } _ { \subseteq V } ( \varphi ( A _ 2 ) ) \end{align*}


\begin{align*} \lim \limits _ { r \to 0 } \frac { \log r } { q } = \lim \limits _ { q \to \infty } \frac { 1 } { q } \log \prod \limits _ { i = 1 } ^ { q } C _ { w _ i } = \sum \limits _ { k \in T } p _ k \log C _ k . \end{align*}


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


\begin{align*} \prod _ { k = 0 } ^ \infty ( 1 + \alpha X ^ k ) & = \sum _ { k = 0 } ^ \infty \frac { \alpha ^ k X ^ { \binom { k } { 2 } } } { X ^ k ! } , \\ \prod _ { k = 1 } ^ \infty \frac { 1 } { 1 - \alpha X ^ k } & = \sum _ { k = 0 } ^ \infty \frac { \alpha ^ k X ^ k } { X ^ k ! } . \end{align*}


\begin{align*} \lim _ { N \to \infty } \norm { f - \sum _ { k = 1 } ^ N e _ { \pi ( k ) } } _ { \mathcal { B } } = 0 . \end{align*}


\begin{align*} 2 e ( x y ) + 4 ( e x ) ( e y ) = x y . \end{align*}


\begin{align*} \beta _ { n , \lambda } ( x ) & = \sum _ { k = 0 } ^ { n } \binom { n } { k } \beta _ { k , \lambda } ( x ) _ { n - k , \lambda } \\ & = \sum _ { k = 0 } ^ { n } \binom { n } { k } ( x ) _ { k , \lambda } \beta _ { n - k , \lambda } , ( n \ge 0 ) . \end{align*}


\begin{align*} \left ( \frac { x } { n } \right ) ^ c \Lambda ( n ) \leq \left ( \frac { x } { n } \right ) ^ c \log n = \frac { x } { n } \cdot \frac { e \log n } { e ^ { \log n / \log x } } \leq \frac { x } { n } \log x , \end{align*}


\begin{align*} L _ n ( x ) = \sum _ { k = 0 } ^ { \lfloor n / 2 \rfloor } L ( n , k ) x ^ { k } , \end{align*}


\begin{align*} \varphi _ k ( P ) = \sum _ { G \in \mathcal { G } _ k } \varphi ( P , G ) . \end{align*}


\begin{align*} \tilde { h } _ i = \mathbf { v } ^ H \mathbf { w } _ i ' / ( N _ 1 \sqrt { p } ) \sim \mathcal { C N } \Big ( 0 , \frac { 1 } { N _ 1 \varrho } \Big ) . \end{align*}


\begin{align*} M _ { ( k , \tau ) } = \Bigl \{ z \in \R ^ { \binom { k + \tau } { \tau } } \Bigm | z = ( 1 , x _ 1 , x _ 2 , \ldots , x ^ \tau _ n ) , \ x \in [ 0 , 1 ] ^ k \Bigr \} , \end{align*}


\begin{align*} M _ 2 ( f , \{ 1 \} ) = { \pi ^ 2 \over 8 } \times \left \{ \prod _ { q \mid f } \left ( 1 - { 1 \over q ^ 2 } \right ) - { 1 \over f } \prod _ { q \mid f } \left ( 1 - { 1 \over q } \right ) \right \} \ \ \ \ \ \hbox { ( $ f > 2 $ o d d ) } . \end{align*}


\begin{align*} V _ j = E _ j \setminus \bigcup _ { i < j } E _ i . \end{align*}


\begin{align*} \mathrm { v a r } [ \widehat { K } _ { \mathrm { a - c p t - d } } ^ \mathbb { R } ] = \frac { \psi } { 2 } \Big ( \sum _ { i \in \mathcal { K } ' } \frac { \mathrm { v a r } [ Z _ i ] } { N _ 1 \varrho \vartheta _ i } + \frac { 1 } { N _ 2 \bar { \gamma } _ c ' } \Big ) . \end{align*}


\begin{align*} \prod _ { 1 \leq i < j \leq 2 n } ( t _ i - t _ j ) ^ 2 = \prod _ { \alpha \in \Phi _ H } \alpha ( t ) ^ 2 \prod _ { i = 1 } ^ n t _ i ^ 2 . \end{align*}


\begin{align*} L ( x ) : = \left \{ \begin{aligned} & K ( \sigma ) + ( x - \sigma ) K ' ( \sigma ) & & 0 \leq x \leq \sigma \\ & K ( x ) & & x > \sigma , \end{aligned} \right . M : = K - L \end{align*}


\begin{align*} d _ k = ( - 1 ) ^ k \sum _ { \nu = 0 } ^ m ( - 1 ) ^ { \nu } { n + \nu \choose \nu } { n + 1 \choose k - \nu } . \end{align*}


\begin{align*} \max _ { 0 \leq | \alpha | \leq 3 } \sup _ { x \in C _ { R + 1 } } | D _ x ^ { \alpha } D _ y ^ { \alpha } K _ { i _ 1 i _ 2 } ( x , y ) | _ { y = x } | \leq M , \ i _ 1 , i _ 2 = 1 , 2 , \dots , m . \end{align*}


\begin{align*} P _ { \mathbf { c } } = \Big \{ \mathbf { v } \in ( \mathbb { Z } / p ^ { 2 k } \mathbb { Z } ) ^ n : \mathbf { D } _ { \mathbf { c } } \cdot \frac { \mathbf { v } - \mathbf { c } } { p ^ k } \equiv 0 \mod p ^ k \Big \} . \end{align*}


\begin{align*} \tilde { \lambda } _ 0 \le \frac { 1 } { \int _ { \R ^ N } m ' u ^ 2 } \le \frac { 1 } { \int _ { \R ^ N } m u ^ 2 } = \tilde { \lambda } _ 0 \ ; , \end{align*}


\begin{align*} \Lambda _ 2 ( f , g ) = ( \sigma ( 0 , \cdot ) , v ( 0 , \cdot ) ) , \end{align*}


\begin{align*} X ( t ; x ) = S _ t x + \int _ 0 ^ t S _ { t - s } \alpha ( X ( s ; x ) ) d s + \int _ 0 ^ t S _ { t - s } \sigma ( X ( s ; x ) ) d W ( s ) , t \in [ 0 , T ] . \end{align*}


\begin{align*} U ^ k _ i f ( x _ k ) = \frac { 1 } { R ( k , i ) } \sum \limits _ { x _ i < x _ k } f ( x _ i ) \end{align*}


\begin{align*} [ a , b ] ( [ c , d ] [ e , f ] ) = [ a , b ] [ p c , q f ] = [ ( i u ) a , j ( q f ) ] . \end{align*}


\begin{align*} \partial _ t p _ t = \nabla \left ( p _ t \nabla \log \frac { d p _ t } { d \gamma ^ { \epsilon , L } } \right ) , p _ 0 = \nu ^ { \epsilon , L } , \end{align*}


\begin{align*} \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } F ( v , v _ * , v ' , v _ * ' ) B ( | v - v _ * | , \cos \theta ) d v d v _ * d \sigma = \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } F ( v ' , v _ * ' , v , v _ * ) B ( | v - v _ * | , \cos \theta ) d v d v _ * d \sigma \end{align*}


\begin{align*} K _ { \widetilde { X } } = \tau ^ \ast ( K _ X ) + d \cdot F + \sum _ { i = 1 } ^ 4 \sum _ { j = 1 } ^ k d _ j \cdot C _ { i j } \end{align*}


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


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


\begin{align*} u ( t , \cdot ; s , u _ 0 ) & = e ^ { - A ( t - s ) } u _ 0 - \chi \int _ s ^ t e ^ { - A ( t - \tau ) } \nabla \cdot \left ( \frac { u ( \tau , \cdot ; s , u _ 0 ) } { v ( \tau , \cdot ; s , u _ 0 ) } \nabla v ( \tau , \cdot ; s , u _ 0 ) \right ) d \tau \\ & + \int _ s ^ t e ^ { - A ( t - \tau ) } u ( \tau , \cdot ; s , u _ 0 ) \big [ { \mu } + a ( \tau , \cdot ) - b ( \tau , \cdot ) u ( \tau , \cdot ; s , u _ 0 ) \big ] d \tau \end{align*}


\begin{align*} ~ \begin{aligned} u _ { i j } ( x ) & \leq \min \{ a _ { i j } , f _ i ( x ) \} & & f _ { i } ( x ) \in [ a _ { i \tau ( i , t - 1 ) } , a _ { i \tau ( i , t ) } ] , \\ u _ { i j } ( x ) & = - \infty & & . \\ \end{aligned} \end{align*}


\begin{align*} \Delta ( k ) = \frac { 1 } { 2 } \left ( \begin{array} { c c } \chi _ { R , L } ( k ) + \chi _ { R , L } ^ { - 1 } ( k ) & i \left ( \chi _ { R , L } ( k ) - \chi _ { R , L } ^ { - 1 } ( k ) \right ) \\ - i \left ( \chi _ { R , L } ( k ) - \chi _ { R , L } ^ { - 1 } ( k ) \right ) & \chi _ { R , L } ( k ) + \chi _ { R , L } ^ { - 1 } ( k ) \end{array} \right ) , \end{align*}


\begin{align*} | \epsilon \int _ U u u _ t ( x , t ) \ , d x | & \leq \frac { \epsilon } { 2 } ( \int _ U u ( x , t ) ^ 2 \ , d x ) + \frac { \epsilon } { 2 } \int _ U ( u _ t ( x , t ) ) ^ 2 \ , d x \\ & \leq \frac { \epsilon B ^ 2 } { 2 } \int _ U ( u _ { x x } ( x , t ) ) ^ 2 \ , d x + \frac { \epsilon B ^ 2 } { 2 } \int _ U ( u _ t ( x , t ) ) ^ 2 \ , d x \\ & \leq \epsilon B ^ 2 E ( t ) . \end{align*}


\begin{align*} \Lambda ( Z ) = ( \Lambda ^ { - 1 } \circ A \circ \Lambda , \Lambda ^ { - 1 } \circ B \circ \Lambda ) \end{align*}


\begin{align*} & - \nabla \cdot ( \epsilon ( x ) \phi ) = f ( x ) + \sum _ { i = 1 } ^ s z _ i \rho _ i , \quad , x \in \Omega , \\ & \alpha \phi = \phi _ b , x \in \partial \Omega , \end{align*}


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


\begin{align*} S _ j = L ( S _ j ) = L ( 0 ) + D L ( S _ j ) , \end{align*}


\begin{align*} \phi _ 1 ( s , t ) = M ( s , t ) \left ( t ^ { 2 l } + \sum _ { j = 1 } ^ { 2 l } A _ j ( s ) t ^ { 2 l - j } \right ) \end{align*}


\begin{align*} \mathcal { U } ( h , k ) = - \mathcal { L } ^ 0 ( h , k ) + \mathcal { U } ^ 2 ( h , k ) + \mathcal { U } ^ r ( h , k ) + O \big ( ( X h k ) ^ \varepsilon X C \big ) + O \bigg ( Q \frac { ( X C H K ) ^ { \varepsilon } } { \sqrt { H K } } \bigg ) . \end{align*}


\begin{align*} \zeta ^ 2 = \alpha , \xi ^ 2 = \beta , \zeta \xi = - \xi \zeta . \end{align*}


\begin{align*} \langle f , h \rangle = \int _ { \Gamma \backslash \mathfrak { Z } } f ( z ) \overline { h ( z ) } \delta ( z ) ^ k \mathbf { d } z , \end{align*}


\begin{align*} x = 1 2 6 - \Theta \ , , \alpha = 6 0 - 2 \Theta \ , , g = 5 3 1 - \frac { 6 5 } { 2 } \Theta + \frac { \Theta ^ 2 } { 2 } - \Delta \ , . \end{align*}


\begin{align*} \nabla P = \lambda \partial _ { r } ^ { T } - u ^ { \frac { \alpha + 1 } { \alpha } } \nabla \log u , \end{align*}


\begin{align*} \int _ { \mathbb S ^ 1 } B _ { x x } J B _ x \ , d x = - \frac 1 2 \int _ { \mathbb S ^ 1 } B _ x J _ { x } B _ x \ , d x . \end{align*}


\begin{align*} V _ n ( I ) = \mathrm { Z e r } ( \phi _ n ( f _ 1 ) , \ldots , \phi _ n ( f _ k ) ) \subset \mathrm { R } ^ n , n > 0 . \end{align*}


\begin{align*} \begin{cases} \Psi _ 0 ( 0 , k , g , \Phi , \chi , \tilde \xi ) = 0 \\ \Psi _ 0 ( n + 1 , k , g , \Phi , \chi , \tilde \xi ) = \Phi ( \chi ^ M _ g ( \Psi _ 0 ( n , k , g , \Phi , \chi , \tilde \xi ) , 8 k + 7 , \tilde \xi ( 8 k + 7 ) ) \end{cases} \end{align*}


\begin{align*} P ( x ) = \sum _ { k = 0 } ^ n { n + m + 1 \choose k } x ^ { n - k } ( 1 - x ) ^ k = \sum _ { k = 0 } ^ n ( - 1 ) ^ k \frac { n + m + 1 } { k + m + 1 } { n + m \choose m } { n \choose k } x ^ k , \end{align*}


\begin{align*} p = \Lambda ( e ^ r - 1 ) \sim b \ , ( e ^ r - 1 ) ^ { 1 - \alpha } \end{align*}


\begin{align*} \widetilde { T } ( x ) ( t ) : = [ ( T ( x ) ) ( t ) ] z _ 0 ( x \in X , \ , t \in K ) . \end{align*}


\begin{align*} G _ t = & \{ ( x , r ) : x \in \mathbb { R } ^ k , 0 < r < d ( x , K ) , z _ 1 , z _ 2 \in K \\ & d ( x , z _ 1 ) , d ( x , z _ 2 ) < d ( x , K ) + \epsilon r + t \\ & | \theta | > \cos ^ { - 1 } \left ( 2 \left ( \frac { 1 - ( 2 \delta + \epsilon ) } { 1 + 2 \delta } \right ) ^ 2 - 1 \right ) \} , \end{align*}


\begin{align*} J ( x ) : = \left \{ j \in X ^ * \mid \langle x , j \rangle = \norm { x } ^ 2 = \norm { j } ^ 2 \right \} \end{align*}


\begin{align*} \Psi ^ i ( { t _ c } _ { * } f ) = \mathfrak { P } _ { \ell } t _ c \Psi ^ i ( f ) . \end{align*}


\begin{align*} \mathrm { d o m } J ^ A _ \gamma = \mathrm { r a n } ( I d + \gamma A ) \end{align*}


\begin{align*} \ker ( \pi ^ { M p } ) = \{ \pm I \} . \end{align*}


\begin{align*} | x ( \tau ) - x ( \sigma ) | = | X ( \tau ) - X ( \sigma ) | \le \int _ \sigma ^ \tau | \dot { X } ( t ) | d t \lesssim \int _ { \sigma } ^ \tau \delta ( t ) d t . \end{align*}


\begin{align*} \frac { t } { e _ { \lambda } ( t ) - 1 } e _ { \lambda } ^ { x } ( t ) = \sum _ { n = 0 } ^ { \infty } \beta _ { n , \lambda } ( x ) \frac { t ^ { n } } { n ! } , ( \mathrm { s e e } \ [ 2 , 3 ] ) . \end{align*}


\begin{align*} W _ 0 ^ 2 ( M ) = \left \lbrace u \in W _ 0 ^ 1 ( M ) : \Delta _ \mu u \in L ^ 2 ( M ) \right \rbrace , \end{align*}


\begin{align*} f & = \sum _ { \l \in \L } \langle f , \pi ( \l ) g \rangle \ \pi ( \l ) \widetilde { g } \\ & = \sum _ { \l \in \L } \langle f , \pi ( \l ) \widetilde { g } \rangle \ \pi ( \l ) g , \end{align*}


\begin{align*} J ( \xi , t ) : = \left | \exp \left ( \frac { i \gamma t \xi ^ { 3 } } { 1 + \xi ^ { 2 } } \right ) - \exp \left ( \frac { - t \xi ^ { 4 } } { 1 + \xi ^ { 2 } } \right ) \right | ^ { 2 } . \end{align*}


\begin{align*} { \mathcal N } _ 1 u = ( u \cdot \nabla ) u \ , b + \frac { ( 1 - b ) \ , \nabla | u | ^ 2 + ( \mathrm { d i v } u ) u } { 2 } = \star ( \star d _ 1 u \wedge u ) \ , b + \frac { d _ 0 | u | ^ 2 - ( d _ 0 ^ * u ) u } { 2 } \end{align*}


\begin{align*} J _ { 0 } ( \hat { \alpha } \cup { ( \gamma , N ) } \cup { ( \delta _ { 1 } , 1 ) } , \hat { \alpha } \cup { ( \gamma , N - p _ { i } ) } \cup { ( \delta _ { 2 } , 1 ) } ) = 1 . \end{align*}


\begin{align*} \begin{aligned} A ^ { 0 } ( V ^ { 2 } ) W _ { t } + A ^ { i } ( V ^ { 2 } ) \partial _ { i } W - & B ^ { i j } ( V ^ { 2 } ) \partial _ { i } \partial _ { j } W + D ( V ^ { 2 } ) W = \widehat { R } , \\ \left . W \right \rvert _ { t = 0 } & = 0 , \end{aligned} \end{align*}


\begin{align*} \partial ^ h _ 1 v ( x ) : = \frac { v ( x + h e _ 1 ) - v ( x ) } h \end{align*}


\begin{align*} & d \rho _ i ( t ) + \sum _ { j \in N ( i ) } \theta ( \rho _ i , \rho _ j ) ( S _ j - S _ i ) d t + \sum _ { j \in N ( i ) } \theta ( \rho _ i , \rho _ j ) ( \Sigma _ j - \Sigma _ i ) \circ d W _ t = 0 , \\ & d S _ i ( t ) + \sum _ { j \in N ( i ) } \frac 1 2 ( S _ j - S _ i ) ^ 2 \frac { \partial \theta } { \partial \rho _ i } ( \rho _ i , \rho _ j ) d t + \sum _ { j \in N ( i ) } ( S _ i - S _ j ) ( \Sigma _ i - \Sigma _ j ) \frac { \partial \theta } { \partial \rho _ i } ( \rho _ i , \rho _ j ) \circ d W _ t = 0 . \end{align*}


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


\begin{align*} X f ( x ) = x f ( x ) P f ( x ) = \tfrac { 1 } { 2 \pi i } f ' ( x ) . \end{align*}


\begin{align*} w ( 0 ) = \gamma \geq 0 , \ , w ' ( 0 ) = 0 , \ , w ( y ) = y ^ { - 2 } \mbox { a s } y \to + \infty , \end{align*}


\begin{gather*} \kappa : = \left ( \sum S ( R '' ) R ' \right ) \mathbf { r } ^ { - 1 } = K ^ { - 1 } . \end{gather*}


\begin{align*} p : = \iota _ 0 \circ \pi : X \ast I & \to X \ast I \\ ( x , ( s , t ) ) & \mapsto ( x , ( s + t , 0 ) ) \end{align*}


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


\begin{align*} ( f \circ _ i g ) \circ _ j h = \begin{cases} ( f \circ _ j h ) \circ _ { i + l - 1 } h & 0 \leq j \leq i - 1 \\ f \circ _ i ( g \circ _ { j - i } h ) & i \leq j \leq n - 1 , \end{cases} \end{align*}


\begin{align*} \mathbb E [ Y _ 1 ^ 2 ] = \mathbb E [ Y _ 2 ^ 2 ] = \mathbb E [ Y _ 3 ^ 2 ] . \end{align*}


\begin{align*} n _ { i x } = \frac { \tilde { E } _ i n _ i ^ 3 } { n _ i ^ 2 - 1 } , i = 1 , 2 . \end{align*}


\begin{align*} ( \gamma \delta ) ^ * = ( \gamma ^ * \delta ) ^ * = ( \alpha ^ * \beta ^ * \delta ) ^ * . \end{align*}


\begin{align*} \Phi = \varepsilon _ { \mathbb C } + \Phi \succ \beta , \end{align*}


\begin{align*} E ( T , X ( T ) ) & = - ( \nabla \phi \ast ( G \ast \mathcal S + \mathcal S ) ) ( t , X ( t ) ) , \\ \lim _ { s \rightarrow \infty } \nabla \phi * \rho [ h _ { V ( T ) } ] ) ( s , 0 ) & = \lim _ { R \rightarrow \infty } ( \nabla \phi \ast ( G \ast S _ { R , X ( T ) , V ( T ) } + S _ { R , X ( T ) , V ( T ) } ) ( R , X ( T ) ) . \end{align*}


\begin{align*} \mathcal { L } ( \mathbf { x } , \mathbf { y } ) = \frac { 1 } { 2 } \mathbf { x } ^ T H \mathbf { x } + \mathbf { g } ^ T \mathbf { x } - \mathbf { y } ^ T ( A \mathbf { x } - \mathbf { b } ) + I _ D ( \mathbf { x } , \mathbf { y } ) , \end{align*}


\begin{align*} D = ( - R , R ) \times ( - L ' , L ' ) , \end{align*}


\begin{align*} E \big [ ( s \mathcal { T } ( \nu ) ) ^ 2 \big ] & = \frac { s ^ 2 \nu } { \nu - 2 } , \\ \big ( s \mathcal { T } ( \nu ) \big ) & = \frac { K _ { \nu / 2 } ( \sqrt { \nu } t s ) ( \sqrt { \nu } t s ) ^ { \nu / 2 } } { 2 ^ { \nu / 2 - 1 } \Gamma ( \nu / 2 ) } , \ \forall t \ge 0 . \end{align*}


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


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


\begin{align*} \operatorname { t r } _ g ( f ) : = \int _ { G / Z _ g } f ( h g h ^ { - 1 } ) d ( h Z _ g ) . \end{align*}


\begin{align*} j ( \tilde { \tau } _ m ( g _ { \infty } \times h _ { \infty } ) , z _ 0 \times z _ 0 ) = j ( \tilde { \tau } _ m , g _ { \infty } z _ 0 \times h _ { \infty } z _ 0 ) j ( g _ { \infty } , z _ 0 ) \overline { j ( h _ { \infty } , z _ 0 ) } . \end{align*}


\begin{align*} \| \nabla Q \| _ 2 ^ 2 + \| Q \| _ 2 ^ 2 - \int \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ p d x = 0 . \end{align*}


\begin{align*} B f ( x ) = \sum \limits _ { j = k } ^ { \infty } q ^ { - ( j + 1 ) } g _ j \ast f ( x ) . \end{align*}


\begin{align*} \prod _ { k = 0 } ^ \infty \frac { 1 } { 1 - \alpha X ^ k } = \sum _ { k = 0 } ^ \infty \frac { \alpha ^ k } { X ^ k ! } . \end{align*}


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


\begin{align*} \phi _ n ^ \star = \mod \Bigl [ \phi _ { d } - ( \phi _ { h , n } + \phi _ { g , n } ) , 2 \pi \Bigr ] , \end{align*}


\begin{align*} \lim _ { K \rightarrow + \infty } p ( x _ K ) \sum _ { j = 0 } ^ { { 1 \over \delta _ K } - 1 } h _ K G \left ( \overline { ( i _ K - j ) \delta _ K } \ , \log K \right ) & = \lim _ { K \rightarrow + \infty } p ( x _ K ) \sum _ { \ell = - \lfloor 1 / 2 \delta _ K \rfloor } ^ { 1 / \delta _ K - 1 - \lfloor 1 / 2 \delta _ K \rfloor } h _ K G \left ( h _ K \ell \right ) \\ & = p ( x ) \int _ { \mathbb { R } } G ( y ) \ , d y = p ( x ) , \end{align*}


\begin{align*} \begin{cases} L u : = - \nabla \cdot ( \mu \nabla u ) + \boldsymbol { \beta } \cdot \nabla u + \sigma u = f & \Omega \ , , \\ u = 0 & \Gamma \ , , \end{cases} \end{align*}


\begin{align*} \widehat { K } _ { \mathrm { a - c p t - f } } ^ \mathbb { R } = 2 \Re \{ \zeta _ \mathcal { K } \} / \sqrt { \pi \bar { p } \big ( 1 + \frac { 1 } { N _ 1 \bar { \gamma } ' } \big ) } , \end{align*}


\begin{align*} \bar \chi = \sum _ { i \notin J } \frac { \bar \alpha _ { i - 1 } } { \lambda _ i } + \sum _ { i \in J } \frac { \bar \alpha _ i } { \mu _ i } . \end{align*}


\begin{align*} K _ { \alpha \alpha } ^ { ( \ell ) } = K _ { \beta \beta } ^ { ( \ell ) } = K , \end{align*}


\begin{align*} \frac { \mathbf { P } \Big ( a _ n S _ n \geq x \sqrt { v _ n + n a _ n ^ 2 \sigma ^ 2 } \ , \Big ) } { 1 - \Phi \left ( x \right ) } = 1 + o ( 1 ) \ \ \ a n d \ \ \ \frac { \mathbf { P } \Big ( a _ n S _ n \leq - x \sqrt { v _ n + n a _ n ^ 2 \sigma ^ 2 } \ , \Big ) } { \Phi \left ( - x \right ) } = 1 + o ( 1 ) \end{align*}


\begin{align*} F ( X ) = X ^ k - a _ 1 X ^ { k - 1 } - \cdots - a _ k , \end{align*}


\begin{align*} D _ i = \pm \prod _ { j \neq i } ( c _ j - c _ i ) ^ { - 1 } . \end{align*}


\begin{align*} \lefteqn { \beta _ j ^ K ( t _ K ) - \beta _ { i _ K } ^ K ( t _ K ) } \\ & = \widetilde { \beta } ^ K ( t _ K , j \delta _ K ) - \widetilde { \beta } ^ K ( t _ K , x _ K ) - \left ( \beta _ { i _ K } ^ K ( t _ K ) - \widetilde { \beta } ^ K ( t _ K , x _ K ) \right ) \\ & \leq \varphi ( t _ K , j \delta _ K ) - \varphi ( t _ K , x _ K ) + \widetilde { M } ^ K ( t _ K , j \delta _ K ) - \widetilde { M } ^ K ( t _ K , x _ K ) - \left ( \beta _ { i _ K } ^ K ( t _ K ) - \widetilde { \beta } ^ K ( t _ K , x _ K ) \right ) \end{align*}


\begin{align*} ( f \smile g ) \circ _ j h = \begin{cases} ( f \circ _ i h ) \smile g & 0 \leq i \leq m - 1 \\ f \smile ( g \circ _ { i - m } h ) & m \leq i \leq m + n - 1 . \end{cases} \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \dot { \gamma } ^ \ast ( s ) = - \pi R ^ 2 & \ s > 0 , \\ \gamma ^ \ast ( 0 ) = r , & \end{array} \right . \end{align*}


\begin{align*} ( - 1 ) \cdot * _ \Sigma i _ \Sigma ^ * \Pi _ \ell d , = : \partial _ \Sigma : \mathfrak { E } ( \mathcal { M } ) \longrightarrow \mathfrak { E } ( \Sigma ) \end{align*}


\begin{align*} \pi ( \xi , \eta , 0 ) \pi ( \Phi ) = \pi ( \Phi ^ { \xi , \eta } ) . \end{align*}


\begin{align*} \frac { d } { d t } K ( t ) = - a K ( t ) ^ 2 , \end{align*}


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


\begin{align*} \lim _ { t \rightarrow + \infty } x _ 1 ( t ) - x _ { 2 } ( t ) = + \infty , \end{align*}


\begin{align*} \Lambda ( \phi \succ \gamma ) = \sum _ { u , v \in \N ^ * } f _ v g _ u x _ u x _ v = g ( x ) f ( x ) . \end{align*}


\begin{align*} \norm { L ( m ) } _ { o p } = \norm { m } _ \infty \norm { L ( m ) ^ { - 1 } } _ { o p } \leq \norm { L ( m ^ { - 1 } ) } _ { o p } = \norm { m ^ { - 1 } } _ \infty \end{align*}


\begin{align*} \begin{aligned} \left \| w ^ j \right \| _ { C ^ { 2 , \alpha } \left ( B _ { 3 r / 5 } ^ { m } ( 0 ) \right ) } \leq C ( r ) \left \| \Delta _ { h } z ^ j \right \| _ { C ^ { 0 , \alpha } \left ( B _ { 3 r / 5 } ^ { m } ( 0 ) \right ) } \leq C ( r ) \Psi ( \tau \ , | \ , n , \alpha , Q ) . \end{aligned} \end{align*}


\begin{align*} \Phi _ { \delta } ^ { x _ 0 } ( d , e , t ) = e ^ { ( \beta + L ) t } d + \varphi _ { \beta + L } ( t ) e \end{align*}


\begin{align*} \partial _ { x _ { 0 ; \alpha } } z _ { i ; \alpha } ^ { ( \ell ) } : = z _ { i ; \alpha } ^ { ( \ell ) } . \end{align*}


\begin{align*} W - V = a ^ j Y _ j - b ^ k \overline { Y } _ k = 0 , \end{align*}


\begin{align*} \beta ^ 3 \gamma ^ 3 = \beta \gamma l = k \ , ; \gamma = \frac { k } { \beta l } \ , ; k ^ 2 = l ^ 3 \ , . \end{align*}


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


\begin{align*} \Omega _ { \Sigma } ^ * = \{ ( z _ 1 , \dots , z _ { n } ) \in \mathbb D ^ n \ , : ( z _ 1 , \dots , z _ { n } ) \in \sigma _ T ( P _ 1 ^ { \perp } U _ 1 ^ * + z P _ 1 U _ 1 ^ * , \dots , P _ n ^ { \perp } U _ n ^ * + z P _ n U _ n ^ * ) \} . \end{align*}


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


\begin{align*} S _ { 2 , \lambda } ( k , p ) = & \sum _ { m = p } ^ { k } \bigg \{ \binom { k } { m } ( - 1 ) ^ { k - m } \langle 1 \rangle _ { k - m , \lambda } S _ { 2 , \lambda } ( m + 1 , p + 1 ) \\ & + \lambda k \binom { k - 1 } { m } ( - 1 ) ^ { k - m - 1 } \langle 1 \rangle _ { k - m - 1 , \lambda } S _ { 2 , \lambda } ( m + 1 , p + 1 ) \bigg \} . \end{align*}


\begin{align*} P _ { N \eta } h ( x ) - \tilde { Q } _ N h ( x ) = \sum _ { i = 1 } ^ { N } \tilde { Q } _ { i - 1 } \big ( P _ { \eta } - \tilde { Q } _ { 1 } \big ) P _ { ( N - i ) \eta } h ( x ) . \end{align*}


\begin{align*} \beta _ p \colon N Y ^ { o p } _ p \times _ { Y _ 0 } B ( Y _ 0 \sl f ) \to N Y ^ { o p } _ p = N Y ^ { o p } _ p \times _ { Y _ 0 } Y _ 0 . \end{align*}


\begin{align*} \norm { \xi _ { \lambda ^ p _ k } - \phi _ { k } } ^ 2 _ { L ^ 2 } = \norm { \xi _ { \lambda ^ p _ k } } ^ 2 _ { L ^ 2 } \leq C k ^ { - 2 } , \ \ \forall k \geq k _ 0 . \end{align*}


\begin{align*} \Delta _ { \mathbf { r } _ { 0 } ( \tau ) } ( [ k ] _ { \alpha } ^ { i } ) ^ { - 1 } \cdot \Delta _ { \mathbf { r } _ { 0 } ( \tau ) } ( \mathcal { I } ) = \det ( \mathbf { r } _ { ( \alpha ) } ( \tau ) ) \end{align*}


\begin{align*} g ( p ) = r = \log \left ( 1 + c \ , | \log p | \right ) \sim \log | \log p | , \textrm { a s } p \to 0 . \end{align*}


\begin{align*} \zeta _ 2 ( z \pm 2 \ell ) = \zeta _ 2 ( z ) \pm \frac { \pi } { \delta } , \wp _ 2 ( z \pm 2 \ell ) = \wp ( z ) , f _ 2 ( z \pm 2 \ell ) = f _ 2 ( z ) \pm \frac { 2 \pi } { \delta } \zeta _ 2 ( z ) + \bigg ( \frac { \pi } { \delta } \bigg ) ^ 2 \end{align*}


\begin{align*} \sum _ { l \in S _ { \nu } \cup \{ 0 \} } w _ { \nu , \nu ' , l } = 1 , \ , \forall \nu ' \in { \cal P } . \end{align*}


\begin{align*} \Xi _ { \mathcal { C } ^ c } : = \{ X \in \Xi ~ | ~ X ( \mathcal { C } ^ c ) \subseteq \mathcal { C } ^ c \} ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \Xi _ { \mathcal { C } \mathcal { C } ^ c } : = \{ X \in \Xi ~ | ~ X ( \mathcal { X } ) \subseteq \mathcal { C } ^ c \} , \end{align*}


\begin{align*} \partial _ { \Sigma } \chi ^ { * } \phi _ 0 = \rho ^ { * } _ { ( 1 ) } \partial _ { \Sigma _ 0 } \phi _ 0 = \psi . \end{align*}


\begin{align*} \begin{cases} [ z _ 1 , [ \mathfrak { X } , z _ 2 ] , [ \mathfrak { X } , z _ 3 ] ] + [ [ \mathfrak { X } , z _ 1 ] , z _ 2 , [ \mathfrak { X } , z _ 3 ] ] + [ [ \mathfrak { X } , z _ 1 ] , [ \mathfrak { X } , z _ 2 ] , z _ 3 ] = 0 , \\ [ [ \mathfrak { X } , z _ 1 ] , [ \mathfrak { X } , z _ 2 ] , [ \mathfrak { X } , z _ 3 ] ] = 0 , \ ; \ ; z _ 1 , z _ 2 , z _ 3 \in L . \end{cases} \end{align*}


\begin{align*} F _ { s , t , x } : = \{ v \in \R ^ 3 : s < { \mathcal T } _ { t , x _ 1 , v _ 1 } - 1 , | v | < \delta ^ { - \beta } , | v _ * ^ \perp ( t , x , v _ 1 ) - v ^ \perp | \check \tau _ { t , x } > \sqrt { \mathcal T _ { t , x _ 1 , v _ 1 } - s } \} . \end{align*}


\begin{align*} \lim _ { k \to \infty } \sup _ { t \in [ 0 , T ] } \left \| w ^ { ( k ) } ( t , \cdot ) - w ( t , \cdot ) \right \| _ { H ^ 2 ( \R \backslash \{ 0 \} ) } ~ = ~ 0 . \end{align*}


\begin{align*} \omega _ { p } ( A ) & = \underset { \alpha \in \mathbb { R } } { s u p } \norm { R e ( e ^ { i \alpha } A ) } _ { p } \\ & = \underset { \theta \in \mathbb { R } } { s u p } \norm { R e ( e ^ { i ( \theta - \frac { \pi } { 2 } ) } A ) } _ { p } \\ & = \underset { \theta \in \mathbb { R } } { s u p } \norm { I m ( e ^ { i \theta } A ) } _ { p } . \end{align*}


\begin{align*} v + w + x + y + z & = 5 \psi u \\ v w x y z & = u ^ 5 , \end{align*}


\begin{align*} G = \{ x \rightarrow x y , y \rightarrow x ^ 2 \} , \end{align*}


\begin{align*} f ( t ) = \begin{cases} e ^ { 4 \pi i t } + e ^ { 8 \pi i t } , & - 3 \leq t \leq - 1 \\ e ^ { 2 \pi i t } , & - 1 \leq t \leq 1 \\ e ^ { 2 \pi i t ^ 2 } , & 1 \leq t \leq 4 \\ 0 , & \end{cases} . \end{align*}


\begin{align*} \alpha _ k = ( \alpha _ { k - 1 } \rho _ k ) \wedge 1 . \end{align*}


\begin{align*} \nabla _ { d f _ j ( X ) } \left ( \nabla ^ 2 \beta ( Y , Y ) \right ) & = d f _ j ( X ) \left ( Y Y \beta \right ) - d f _ j ( X ) \left ( ( \nabla _ Y Y ) \beta \right ) \\ & = Y Y \left < d f _ j ( X ) , \nabla \beta \right > - ( \nabla _ Y Y ) d f _ j ( X ) \beta . \end{align*}


\begin{align*} p \odot ^ \flat p ' & = ( \underline { - 2 \delta ^ { - 1 } } \wedge - 3 \delta ^ { - 1 } \wedge - 5 \delta ^ { - 1 } ) \oplus ( \underline { - 1 \delta ^ { 2 } } \wedge - 3 \delta ^ { 2 } ) \oplus ( \underline { 1 \delta ^ { 4 } } ) \\ & = - 2 \delta ^ { - 1 } \oplus - 1 \delta ^ { 2 } \oplus 1 \delta ^ { 4 } . \end{align*}


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


\begin{align*} \phi _ { i , , \beta } ( \xi ) = \sum _ { j = 0 } ^ i c _ { i , j } ( 2 \beta ) ^ j b ^ j T _ j ( \xi ) + \tilde \lambda \sum _ { j = 0 } ^ i b ^ { j + 1 } \left ( c _ { i , j } ( 2 \beta ) ^ { j + 1 } T _ { j + 1 } ( \xi ) + S _ j ( \xi ) \right ) + b R _ i ( \xi ) , \end{align*}


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


\begin{align*} \sum \limits _ { n = 0 } ^ \infty d _ k ( n ) q ^ n = \frac { ( q ^ 2 ; q ^ 2 ) _ \infty ^ k } { ( q ; q ) _ \infty ^ { 3 k + 1 } } , \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } d Z ( t ) & = & \ell ( Z ( t ) ) d t + a ( Z ( t ) ) d W ( t ) \medskip \\ Z ( 0 ) & = & z , \end{array} \right . \end{align*}


\begin{align*} Z _ * ( x , y ) = \Lambda \left ( \begin{pmatrix} ( \eta _ * ^ { r _ 0 } \circ \xi _ * ) ^ { r _ 1 } \circ \eta _ * ( x ) \\ ( \eta _ * ^ { r _ 0 } \circ \xi _ * ) ^ { r _ 1 - 1 } \circ \eta _ * ( x ) \end{pmatrix} , \begin{pmatrix} \eta _ * ^ { r _ 0 } \circ \xi _ * ( x ) \\ x \end{pmatrix} \right ) . \end{align*}


\begin{align*} S _ { 1 , \lambda } ( k + 1 , l ) = S _ { 1 , \lambda } ( k , l - 1 ) - ( k - l \lambda ) S _ { 1 , \lambda } ( k , l ) , \end{align*}


\begin{align*} \frac { b _ { i j k } \mu _ k ( t ) } { x ^ 3 _ { i j } ( t ) } & = \frac { b _ { i j k } \alpha _ k } { ( \alpha _ i - \alpha _ j ) ^ 3 t ^ { 2 } } \frac { 1 + [ \mu _ k ( t ) - \alpha _ k / \sqrt { t } ] / ( \alpha _ { k } \sqrt { t } ) } { ( 1 + [ x _ { i j } ( t ) - ( \alpha _ i - \alpha _ j ) \sqrt { t } ] / [ ( \alpha _ i - \alpha _ j ) \sqrt { t } ] ) ^ 3 } \\ & = \frac { b _ { i j k } \alpha _ k } { ( \alpha _ i - \alpha _ j ) ^ 3 t ^ { 2 } } + O \bigg ( \frac { 1 } { t ^ { 9 / 4 } } \bigg ) . \end{align*}


\begin{align*} \partial _ t \rho = & \nabla \cdot \left ( \nabla \rho + \rho \nabla \phi \right ) , \\ - \Delta \phi = & \rho , \end{align*}


\begin{align*} \int _ Q { \rm d i v } ( \phi ) \ , d x = 0 \quad \phi \in C ^ 1 _ c ( \Omega \cap B _ 1 ; \R ^ d ) \ , . \end{align*}


\begin{align*} I ( \omega ) = \int _ { \R } e ^ { - \pi M x ^ 2 } e ^ { - 2 \pi i \omega x } \ , d x . \end{align*}


\begin{align*} & v ( x , { \bf s } _ { \rm e v e n } ) = 0 , \\ & e ^ { u ( x , { \bf s } _ { \rm e v e n } ) } = \sum _ { k = 1 } \frac 1 k \sum _ { j _ 1 , \dots , j _ k \in \mathbb { Z } ^ { \rm e v e n } _ { \geq 0 } , \atop j _ 1 + \cdots + j _ k = 2 k - 2 } { \rm w t } ( j _ 1 ) \cdots { \rm w t } ( j _ k ) \binom { j _ 1 } { j _ 1 / 2 } \cdots \binom { j _ k } { j _ k / 2 } s _ { j _ 1 } \cdots s _ { j _ k } , \end{align*}


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


\begin{align*} \langle \pi ( F ) f , g \rangle = \int _ { \mathbf { H } } F ( \mathbf { h } ) \langle \pi ( \mathbf { h } ) f , g \rangle \ , d \mathbf { h } , \end{align*}


\begin{align*} \sum _ { \mathbf { i } \in \mathcal { J } ^ 2 } \varphi _ 2 ( \mathbf i _ { 1 : 4 } ) ^ 3 \varphi _ 2 ( \mathbf i _ { 5 : 8 } ) ^ 2 = \sum _ { \mathbf { i } \in \mathcal { I } _ 2 ^ 2 } \varphi _ 2 ( \mathbf i _ { 1 : 4 } ) ^ 3 \varphi _ 2 ( \mathbf i _ { 5 : 8 } ) ^ 2 + o ( n ^ 4 ) = \frac { n ^ 4 } { 4 } \cdot \frac { 1 } { 9 0 ^ 5 } ( 1 + o ( 1 ) ) , \end{align*}


\begin{align*} \inf _ { ( \mathbf { h } , \mathbf { g } ) \in E ^ m } \mathcal { F } _ N ( \mathbf { h } , \mathbf { g } ) = \liminf _ { j \to \infty } \mathcal { F } _ N ( \mathbf { h } _ j , \mathbf { g } _ j ) \ge \mathcal { F } _ N ( \mathbf { h } _ * , \mathbf { g } _ * ) . \end{align*}


\begin{align*} \left \Vert u _ { 0 } \right \Vert _ { H ^ { 1 } } + \left \Vert x u _ { 0 } \right \Vert _ { L ^ { 2 } } \leq \varepsilon \end{align*}


\begin{align*} 1 \geq \mu _ { \mathrm { d i s c } } ( \gamma _ { 1 } \cup { \gamma _ { 2 } } ) - \mu _ { \mathrm { d i s c } } ( \gamma _ { 4 } ) = 1 2 - \mu _ { \mathrm { d i s c } } ( \gamma _ { 4 } ) \end{align*}


\begin{align*} \mathbb { M } ^ 2 _ { \ell } \ni \gamma = \left \{ ( u _ 0 , v ) \in \mathbb { M } ^ 2 \ , | \ , v \in \mathbb { R } \right \} \end{align*}


\begin{align*} s ( t _ { m ( k ) } ) = s ' ( t _ { k ' } ) . \end{align*}


\begin{align*} & d \rho _ i = \sum _ { j \in N ( i ) } \omega _ { i j } ( S _ i - S _ j ) \theta _ { i j } ( \rho ) d t ; \\ & d S _ i + ( \sum _ { j \in N ( i ) } \frac 1 2 \omega _ { i j } ( S _ i - S _ j ) ^ 2 \frac { \partial \theta _ { i j } } { \partial \rho _ i } + \frac 1 8 \frac { \partial } { \partial \rho _ i } I ( \rho ) + \mathbb V _ i + \sum _ { j \in N ( i ) } \mathbb W _ { i j } \rho _ j ) d t + \sigma _ i d W _ t = 0 . \end{align*}


\begin{align*} U : = \bigg \{ x \in B _ \rho ^ + \frac { u ( x ) } { \zeta ( x ) } > \frac { u ( a ) } { 2 \zeta ( a ) } \bigg \} ; \end{align*}


\begin{align*} \int _ \sigma ^ \tau \delta ( t ) d t \lesssim \int _ \sigma ^ \tau \ddot { y } _ R ( t ) d t = \dot { y } _ R ( \tau ) - \dot { y } _ R ( \sigma ) . \end{align*}


\begin{align*} \lim \limits _ { x \rightarrow 0 ^ + } x \ , \coth ( x ) & = \lim \limits _ { x \rightarrow 0 ^ + } x + \lim \limits _ { x \rightarrow 0 ^ + } \Big ( \frac { e ^ { 2 x } - e ^ 0 } { 2 x - 0 } \Big ) ^ { - 1 } = 0 + \frac { 1 } { \exp ' ( 0 ) } = 1 . \end{align*}


\begin{align*} d _ n ( W _ { t , m } , \ , l _ q ^ { \nu _ { t , m } } ) \underset { \mathfrak { Z } _ 0 } { \asymp } \varphi _ i ( t , \ , m , \ , n ) : = 2 ^ { \kappa _ { 1 , i } t + \kappa _ { 2 , i } m } n ^ { \sigma _ i } \end{align*}


\begin{align*} J \cap K & = ( ( A \setminus \{ \bar { a } _ n \} ) \cap ( B \setminus \{ \bar { a } _ n \} ) ) \cup \\ & \qquad \qquad \cup ( ( A \setminus \{ \bar { a } _ n \} ) \cap [ \bar { b } _ n \mapsto \bar { a } _ n ] ( H ) ) \\ & \qquad \qquad \cup ( ( B \setminus \{ \bar { a } _ n \} ) \cap [ \bar { b } _ n \mapsto \bar { a } _ n ] ( G ) ) \\ & \qquad \qquad \cup ( [ \bar { b } _ n \mapsto \bar { a } _ n ] ( G ) \cap [ \bar { b } _ n \mapsto \bar { a } _ n ] ( H ) ) \end{align*}


\begin{align*} _ p \left ( \frac { B _ { k _ m - n / 2 , \chi _ T } } { k _ m - n / 2 } \right ) & = _ p \left ( B _ { k _ m - n / 2 , \chi _ T } \right ) - _ p \left ( k _ m - n / 2 \right ) \\ & \geq - 1 - _ p \left ( 2 - n / 2 \right ) , \end{align*}


\begin{align*} \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \Im \left \{ \log \zeta ( \sigma + i T ) \right \} \ d \sigma = 2 \epsilon \ \mathcal { O } ( \log T ) \end{align*}


\begin{align*} \theta _ z & = \big ( ( z - q _ c ) ^ { s _ c } , \ldots , ( z - q _ { b _ z } ) ^ { s _ { b _ z } } \big ) , \\ \Omega _ z & = \Big \{ 1 + p ^ z \delta _ { n - r } + { \sum } _ { i = 1 } ^ r p ^ { d ' _ i } \colon ( d ' _ 1 , \ldots , d ' _ r ) \le \theta _ z , \ d ' _ r = 0 \Big \} . \end{align*}


\begin{align*} - \Delta Q + Q - \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ { p - 1 } = 0 , \end{align*}


\begin{align*} 0 \geq \langle x - y , - ( u - v ) \rangle = - \langle x - y , u - v \rangle \end{align*}


\begin{align*} E ( \rho , \psi ) = \int _ { \Omega } \bigg ( \sum _ { i = 1 } ^ s \rho _ i \log \rho _ i + \frac { 1 } { 2 } \epsilon ( x ) | \nabla \psi | ^ 2 \bigg ) d x , \end{align*}


\begin{align*} \frac { \tilde M _ n ( 2 ) } { \delta _ n ( 2 ) } + \frac { \tilde M _ n ( 3 ) } { \delta _ n ( 3 ) } = \delta ^ { - 1 } _ n ( 3 ) \cdot \sum _ { \ell = 1 } ^ { d } \sum _ { q = 1 } ^ { \ell - 1 } \left [ \frac { \delta _ n ( 3 ) } { \delta _ n ( 2 ) } \cdot \tilde M _ { n , \{ q , \ell \} } + \sum _ { p = 1 } ^ { q - 1 } \tilde M _ { n , \{ p , q , \ell \} } \right ] . \end{align*}


\begin{align*} u _ { i 0 } ( x ) = p _ i ^ L , u _ { i j } ( x ) = \sum _ { k \in C _ j } g _ { k } ( x ) j = 1 , \ldots , n . \end{align*}


\begin{align*} m _ 0 : = \max \{ m : \lim _ { n \to \infty } a _ n ^ { m , j , k } = \infty \} . \end{align*}


\begin{align*} L ( z ) = \frac { f ' ( z ) } { f ( z ) } = \frac { m } z + O ( | z | ^ { - 2 } ) \hbox { a s $ z \to \infty $ , } \end{align*}


\begin{align*} & ( h + 2 \beta _ a \epsilon ( a ) ) \tilde \phi _ 0 + ( h - 2 \beta _ a \epsilon ( a ) ) \tilde \phi _ 1 = 0 , \\ & - \epsilon _ { j - 1 / 2 } \tilde \phi _ { j - 1 } + 2 \hat \epsilon _ j \tilde \phi _ j - \epsilon _ { j + 1 / 2 } \tilde \phi _ { j + 1 } = h ^ 2 z _ i \tilde \rho _ { i , j } j = 1 , \cdots , N , \\ & ( h - 2 \beta _ b \epsilon ( b ) ) \tilde \phi _ N + ( h + 2 \beta _ b \epsilon ( b ) ) \tilde \phi _ { N + 1 } = 0 . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \dot { \xi } ( t ) & = & A \xi ( t ) + \alpha ( t , \xi ( t ) ) \\ \xi ( 0 ) & = & x \end{array} \right . \end{align*}


\begin{align*} u _ { \star , \lambda } ( x , t ) = u ( x , t ) ( x , t ) \in \R ^ n \times [ 0 , \infty ) . \end{align*}


\begin{align*} \mathbb { E } [ ( x _ 1 ^ n - \mathbb { E } [ x ^ n _ 1 ] ) ^ 2 ] & = \mathbb { E } \Big [ \int _ 0 ^ 1 \Big ( \mathbb { E } [ D _ s x ^ n _ 1 \vert \mathcal { F } _ s ] \Big ) ^ 2 d s \Big ] \\ & \le \mathbb { E } \Big [ \int _ 0 ^ 1 \mathbb { E } [ ( D _ s x ^ n _ 1 ) ^ 2 \vert \mathcal { F } _ s ] d s \Big ] = \int _ 0 ^ 1 \mathbb { E } [ ( D _ s x ^ n _ 1 ) ^ 2 ] d s . \end{align*}


\begin{align*} h _ K : = \delta _ K \log K \ll 1 . \end{align*}


\begin{align*} \bigg \vert \frac { ( - \Delta ) ^ s v ( h e _ 1 ) } h - ( - \Delta ) ^ s \partial _ 1 v ( 0 ) \bigg \vert & = \big \vert ( - \Delta ) ^ s ( \partial ^ h _ 1 v - \partial _ 1 v ) ( 0 ) \big \vert \\ & \leqslant C \Big ( \| \partial ^ h _ 1 v - \partial _ 1 v \| _ { L ^ \infty ( \R ^ n ) } + \| D ^ 2 \partial ^ h _ 1 v - D ^ 2 \partial _ 1 v \| _ { L ^ \infty ( \R ^ n ) } \Big ) . \end{align*}


\begin{align*} \mathfrak { A } _ { \ell } ( \Sigma , \mathcal { M } ) = \left \{ ( F _ H ) _ { H \in \mathrm { H a d } ( \mathcal { M } ) } \subset \mathfrak { F } _ c ( \Sigma ) [ [ \hbar ] ] \ , | \ , \beta _ { H ' - H } F _ H = F _ { H ' } \right \} \end{align*}


\begin{align*} Q ^ { [ k ] } _ n : = \phi _ n ( k , F ) , \forall n \in 1 : N , \end{align*}


\begin{align*} \mathcal B : = \{ A _ 1 ^ { \varepsilon _ 1 } \cap \cdots \cap A _ n ^ { \varepsilon _ n } : \epsilon _ i = 1 , 2 \} . \end{align*}


\begin{align*} & g ( k , \xi ) : = 4 X ^ 3 _ \eta ( k ) . \end{align*}


\begin{align*} \theta \prescript { \sigma } { } N _ l ^ \circ \theta ^ { - 1 } \subseteq \prescript { \tau \sigma \tau ^ { - 1 } } { } N _ v ^ \ast \iff \sigma = \varepsilon \ ; \ ; ( 1 \ , 3 ) . \end{align*}


\begin{align*} x _ { w \cdot a } ( t ) : = { } ^ { s _ w } x _ a ( t ) \end{align*}


\begin{align*} M ( n , \alpha , \varepsilon , Z , T , f ) = \inf \{ \sum _ { i } e ^ { - \alpha n _ i + f _ { n _ i } ( x _ i ) } : Z \subset \cup _ { i } B _ { n _ i } ( x _ i , \varepsilon ) \} , \end{align*}


\begin{align*} \tilde { \mathcal { R } } Z = \Lambda ( \tilde { p } Z ) . \end{align*}


\begin{align*} \partial _ t \left ( e ^ { - e _ 0 t } \alpha _ - \right ) = e ^ { - e _ 0 t } B ( R , \mathcal { Y } _ + ) \Rightarrow e ^ { - e _ 0 t } \alpha _ - ( t ) = \int _ t ^ \infty e ^ { - e _ 0 s } B ( R , \mathcal { Y } _ + ) d s . \end{align*}


\begin{align*} \mathcal { E } _ m ^ \sigma ( \tilde { f } _ m ^ \sigma , g ) = \langle f ; g \rangle _ { L ^ 2 ( \mu _ m ) } , \tilde { f } _ m ^ \sigma = \int _ 0 ^ { + \infty } \mathcal { P } ^ { m , \sigma } _ t ( f ) d t . \end{align*}


\begin{align*} \begin{matrix} \Xi _ { k + 1 } = \mathcal { A } \Xi _ k + \mathcal { B } \mathbf { v } _ k , & \mathbf { y } _ k = \mathcal { C } \Xi _ k , \end{matrix} \end{align*}


\begin{align*} \| \sqrt \rho u \| _ { L ^ \infty ( 0 , T ; L ^ 2 ( \Omega _ R ) ) } & \leq c , \\ \left \| \frac { \rho - \varrho } { \varepsilon } \right \| _ { L ^ \infty ( 0 , T ; L ^ 2 ( \Omega _ R ) ) } & \leq c , \\ \nu ^ \frac { 1 } { 2 } \| \nabla u \| _ { L ^ 2 ( 0 , T ; L ^ 2 ( \Omega _ R ) ) } & \leq c \end{align*}


\begin{align*} \mathcal { L } _ 0 = - ( \widehat { A } \nabla ) , \end{align*}


\begin{align*} E _ { \mu } ^ { [ \alpha , \gamma ] , \varepsilon } = \bigcup _ { n \geq 1 } E _ { \mu } ^ { [ \alpha , \gamma ] , \frac { 1 } { n } , \varepsilon } \ \ \ \mbox { a n d } \ \ \ \ F _ { \mu } ^ { [ \alpha , \gamma ] , \varepsilon } = \bigcup _ { n \geq 1 } F _ { \mu } ^ { [ \alpha , \gamma ] , \frac { 1 } { n } , \varepsilon } . \end{align*}


\begin{align*} \begin{cases} \ 0 \ \leq x \ \leq \ 1 \\ \ 0 \ \leq y \ \leq \ 1 \end{cases} . \end{align*}


\begin{align*} R _ { H } ( t , s ) = \int _ { 0 } ^ { t \wedge s } K _ { H } ( t , u ) K _ { H } ( s , u ) d u , \end{align*}


\begin{align*} M ( \alpha , \varepsilon , Z , T , f ) & = \lim _ { n \to \infty } M ( n , \alpha , \varepsilon , Z , T , f ) , \\ \underline { R } ( \alpha , \varepsilon , Z , T , f ) & = \liminf _ { n \to \infty } R ( n , \alpha , \varepsilon , Z , T , f ) , \\ \overline { R } ( \alpha , \varepsilon , Z , T , f ) & = \limsup _ { n \to \infty } R ( n , \alpha , \varepsilon , Z , T , f ) , \\ M ^ P ( \alpha , \varepsilon , Z , T , f ) & = \lim _ { n \to \infty } M ^ P ( n , \alpha , \varepsilon , Z , T , f ) . \end{align*}


\begin{align*} \mathcal { L } _ \delta ( f ) ( x ) = f '' ( x ) - \delta | x | ^ { \delta - 1 } \operatorname { s g n } ( x ) f ' ( x ) . \end{align*}


\begin{align*} \int \left ( \partial _ { x _ 1 } Q \right ) h _ 1 = . . . = \int \left ( \partial _ { x _ N } Q \right ) h _ 1 = \int Q h _ 2 , \end{align*}


\begin{align*} u _ { t } = D u _ { x x } - u \left ( u - \kappa \right ) \left ( u - 1 \right ) - \varepsilon u _ { x } ^ { m } , \end{align*}


\begin{align*} \tilde \varphi : V ^ { q + 1 } & \to A \\ ( x _ 0 , \ldots , x _ q ) & \mapsto \begin{cases} 0 & \exists i \in \{ 0 , \ldots , q \} : x _ i \not \in U \\ \varphi ( x _ 0 , \ldots , x _ q ) & \mbox { o t h e r w i s e } . \end{cases} \end{align*}