\begin{align*} M = \left ( \begin{array} { c c } \varepsilon _ 1 \lambda & - \varepsilon _ 1 \mu \\ \varepsilon _ 3 \mu & - \varepsilon _ { 3 } \lambda \end{array} \right ) \end{align*}


\begin{align*} m _ k : = \left ( \frac { 2 ^ { \star } } { 2 } \right ) ^ k m _ 0 . \end{align*}


\begin{align*} \kappa _ 1 = a \kappa _ 2 + b , \end{align*}


\begin{align*} W ( y | x ) = \ , \sum _ { n = 1 } ^ \infty \left [ A \circ \hat { \sigma } ^ { - n } ( y | x ) - A \circ \hat { \sigma } ^ { - n } ( y | x ' ) \right ] , \end{align*}


\begin{align*} \widetilde { \mu } \ , ' ( n ) \ r _ n ( m ) = H ' _ { m } \ \widetilde { r } \ , ' _ { n } ( m + 1 ) + I ' _ m \ \widetilde { r } \ , ' _ { n } ( m ) + J ' _ m \ \widetilde { r } \ , ' _ { n } ( m - 1 ) \ , , \end{align*}


\begin{align*} \S = \{ v \in C ^ { 2 } ( \overline \O ) : v | _ { ( \partial \O ) _ { j } } = \phi _ { j } \} . \end{align*}


\begin{align*} ( 2 \pi ) ^ { - n } | \lambda | ^ n W _ \lambda ( g \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ) = W _ \lambda ( g ) P _ k ( \lambda ) . \ , \ , \end{align*}


\begin{align*} | r ^ { N , i } ( t , \boldsymbol { x } ) | \leq \frac { C } { N } \Big ( 1 + \frac { 1 } { N } \underset { j = 1 } { \overset { N } \sum } | x ^ i - x ^ j | \Big ) . \end{align*}


\begin{align*} V ( \check h ) = V ( h , x _ { n + 1 } - f ) = V ( h ^ G ) \ , . \end{align*}


\begin{align*} \begin{aligned} f & = f ( \mathbf 0 ) + \delta f , \vert \delta f \vert \leq \| f \| _ { C ^ \alpha } \vert \mathbf x \vert ^ \alpha , \\ v & = v ( \mathbf 0 ) + \delta v , \vert \delta v \vert \leq \| v \| _ { C ^ \alpha } \vert \mathbf x \vert ^ \alpha , \\ \eta & = \eta ( \mathbf 0 ) + \delta \eta , \vert \delta \eta \vert \leq \| \eta \| _ { C ^ \alpha } \vert \mathbf x \vert ^ \alpha . \end{aligned} \end{align*}


\begin{align*} \Delta _ C = \big ( \theta _ { j _ 1 } - \theta _ { i _ 1 } \big ) \big ( \theta _ { j _ 2 } - \theta _ { i _ 2 } \big ) \cdots \big ( \theta _ { j _ k } - \theta _ { i _ k } \big ) \ , . \end{align*}


\begin{align*} \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) = \int \eta ( \lambda | \bar { U } ; x , t ) d \boldsymbol { \nu } ( \lambda ) \end{align*}


\begin{align*} \mu \Big ( \bigcap _ { j = 0 } ^ { t _ n } T ^ { j h _ n } J \Big ) \geq ( 1 - \epsilon _ n ) \mu ( J ) . \end{align*}


\begin{align*} \eta ( \mathbf 0 ) \not = 0 ; \end{align*}


\begin{align*} \tilde U ^ { i j } D _ { i j } \tilde w = - \gamma ^ { - 1 } f , \end{align*}


\begin{align*} V ( r ) = \frac { L ( L + 1 ) } { r ^ 2 } - \frac { Q } { r } f + \kappa Q \frac { r } { f } + \kappa \frac { L ( L + 1 ) } { f ^ 2 } , \end{align*}


\begin{align*} { \bf { \hat \Lambda } } = d i a g \left [ { 1 , 1 , 1 , 1 , { { \hat s } _ e } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ \pi } , { { \hat s } _ \pi } , { { \hat s } _ \pi } } \right ] , \end{align*}


\begin{align*} \Im m ( a _ 0 \overline { a _ 1 } ) = \Im m ( a _ 2 \overline { a _ 3 } ) ; \end{align*}


\begin{align*} \abs { \eta ( y ) - \eta ( x ) } ^ 2 & = \abs { \int _ x ^ y \partial \eta ( z ) \ \d z } ^ 2 \le \int _ \S \partial \eta ( z ) ^ 2 \ \d z = \norm { \partial \eta } _ { L ^ 2 } ^ 2 , \end{align*}


\begin{align*} T _ \alpha = \partial _ { \alpha } + \sum _ { \beta < n } B _ { \alpha \beta } ( x _ { \beta } \partial _ { n } - x _ { n } \partial _ { \beta } ) , \ ; \ ; \mbox { f o r } \alpha < n , \end{align*}


\begin{align*} ( T _ 1 + f _ 1 ) \cdot ( T _ 2 - f _ 2 ) \cdot \Psi _ 2 = i \Psi _ 2 \end{align*}


\begin{align*} d _ x T ( x , y ) = - \frac { d _ x \Phi ( x , y , T ( x , y ) ) } { \partial _ z \Phi ( x , y , T ( x , y ) ) } . \end{align*}


\begin{align*} H _ \alpha ( X ) = \frac { 1 } { 1 - \alpha } \log \left ( \int ^ \infty _ 0 f ( x ) ^ { \alpha } \mathrm { d } x \right ) , \alpha > 0 , \ ; \alpha \neq 1 , \end{align*}


\begin{align*} V ( \xi _ 0 , n , x ) = \sup _ { \theta \in \Theta _ n } E _ { \xi _ 0 } \left [ h ^ \theta ( x , Z _ 1 ^ \theta ) \right ] , \end{align*}


\begin{align*} \begin{aligned} e ^ { - a } \int _ { | v | \leq 1 } | v | ^ k \d v \leq \int _ { | v | \leq 1 } | v | ^ k e ^ { - a | v | ^ 2 } \d v \leq \int _ { | v | \leq 1 } | v | ^ k \d v \ , . \end{aligned} \end{align*}


\begin{align*} \begin{pmatrix} v _ 1 \\ v _ 2 \end{pmatrix} = - \frac { 1 } { k ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 } \begin{pmatrix} x _ 1 x _ 2 & - k ^ 2 - x _ 1 ^ 2 \\ k ^ 2 + x _ 2 ^ 2 & - x _ 1 x _ 2 \end{pmatrix} \begin{pmatrix} \partial _ { x _ 1 } \varphi \\ \partial _ { x _ 2 } \varphi \end{pmatrix} . \end{align*}


\begin{align*} B _ { 2 } = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 2 } } \Theta _ { \theta } \begin{bmatrix} m _ { 1 } & X \\ 0 & m _ { 2 } \end{bmatrix} \overline { \psi _ { A } ( { m _ { 1 } } ^ { - 1 } X } ) = q ^ { 8 } ( 1 - q ) . \end{align*}


\begin{align*} \inf \Bigg \{ \int _ { \{ \Psi < - t \} } | \tilde { f } | ^ 2 _ h c ( - \Psi ) : \tilde { f } \in H ^ 0 ( \{ \Psi < - t \} , \mathcal { O } ( K _ M \otimes E ) ) \\ \& \ , ( \tilde { f } - f ) _ { z _ 0 } \in \mathcal { O } ( K _ M ) _ { z _ 0 } \otimes J _ { z _ 0 } , z _ 0 \in Z _ 0 \Bigg \} \end{align*}


\begin{align*} \nabla _ X \psi _ 1 = \frac { 1 } { 2 } X _ 1 \cdot \psi _ 1 - \frac { 1 } { 2 } S ( X ) \cdot N \cdot \psi _ 1 \end{align*}


\begin{align*} \Gamma ( Y ) & = - \Gamma ( Z ^ { ( i ) } ) \mbox { i n } M \setminus \Omega \\ Y & = - Z ^ { ( i ) } \mbox { o n } \Sigma . \end{align*}


\begin{align*} & \hphantom { { } = { } } ( x ^ i , ( a ^ i { } _ j , a ^ i { } _ { j \alpha } a ^ \alpha { } _ k ) , ( b ^ i { } _ j , b ^ i { } _ { j k } ) ) . ( g , X ) \\ * & = ( x ^ i , ( a ^ i { } _ \alpha g ^ \alpha { } _ j , a ^ i { } _ { \beta \alpha } g ^ \beta { } _ j a ^ \alpha { } _ \beta g ^ \beta { } _ k ) , ( h ^ i { } _ \alpha b ^ \alpha { } _ \beta g ^ \beta { } _ j + X ^ i { } _ j , h ^ i { } _ \alpha b ^ \alpha { } _ { \beta k } g ^ \beta { } _ j ) ) . \end{align*}


\begin{align*} p ( T ( t ) x ) = \lim _ { n \to \infty } \left ( \frac { n } { t } \right ) ^ { n } p \left ( R \left ( \frac { n } { t } , A \right ) ^ { n } x \right ) \underset { \eqref { e q : r e s _ c o n t } } { \leq } p ( x ) \end{align*}


\begin{align*} F \subseteq \bigcup _ { n = 0 } ^ \infty [ x _ n , y _ n ] \end{align*}


\begin{align*} h ( T ) & = x y \log ( x y ) - \sum _ { k = 1 } ^ { r } \left ( \Delta ( i _ k ) + \Delta ( j _ k ) \right ) \\ & \le x y \log ( x y ) - \sum _ { k = 1 } ^ { r } \left ( \Delta ( x ) + \Delta ( y + 1 - k ) \right ) \\ & = h ( T ' ) , \end{align*}


\begin{align*} \lim \limits _ { j \to + \infty } \frac { G ( t _ 0 ) - G ( t _ 0 + B _ j ) } { \int _ { t _ 0 } ^ { t _ 0 + B _ j } c ( t ) e ^ { - t } d t } = \liminf \limits _ { B \to 0 + 0 } \frac { G ( t _ 0 ) - G ( t _ 0 + B ) } { \int _ { t _ 0 } ^ { t _ 0 + B } c ( t ) e ^ { - t } d t } \end{align*}


\begin{align*} x ' _ 1 = \varphi _ { \sigma , s } ^ L ( x _ 1 ) , x ' _ 2 = \varphi _ { \sigma , s } ^ L ( x _ 2 ) , y ' _ 1 = \varphi _ { \sigma , p } ^ R ( y _ 1 ) y ' _ 2 = \varphi _ { \sigma , p } ^ R ( y _ 2 ) . \end{align*}


\begin{gather*} g ( t ) \allowbreak = \allowbreak \frac { 1 } { ( 1 - t ^ { 2 } ) ^ { \beta } } \left ( 1 + \frac { \rho t ^ { 2 } } { 1 - t ^ { 2 } } \right ) ^ { - \beta } \allowbreak = \\ = \allowbreak \frac { 1 } { ( 1 - t ^ { 2 } ( 1 - \rho ) ) ^ { \beta } } = \sum _ { n \geq 0 } t ^ { 2 n } ( 1 - \rho ) ^ { n } \left ( \beta \right ) ^ { \left ( n \right ) } / n ! . \end{gather*}


\begin{align*} \nabla \times ( a _ { t } ^ { \varepsilon } + ( B \times u ) ^ { \varepsilon } + \mu \nabla \times b ^ { \varepsilon } ) = 0 \textrm { o n } \ ( L ^ { 2 } _ { \sigma } \cap H ^ { 1 } ) ( \mathbb { R } ^ { 3 } ) ^ { * } , \end{align*}


\begin{align*} S _ m ( \lambda ) = \sum \limits _ { 1 \leq j _ 1 < \cdots < j _ m \leq n } \lambda _ { j _ 1 } \dots \lambda _ { j _ m } , \end{align*}


\begin{align*} J _ 3 \leq \begin{cases} C t _ m ^ { \alpha - 1 - \frac { \alpha \zeta } { 2 \beta } } h ^ { \min \{ \frac { 1 } { 2 } + \frac { \alpha } { 2 \beta } ( r + \lambda - \kappa ) , 1 \} } , & \mbox { i f } \alpha + \gamma = 1 , \\ C t _ m ^ { \alpha - 1 - \frac { \alpha \zeta } { 2 \beta } } h ^ { \min \{ \frac { \alpha } { 2 \beta } \min \{ \kappa , r + \lambda \} + ( \gamma - \frac { 1 } { 2 } ) ^ { + } , 1 - \varepsilon \} } , & \mbox { i f } \alpha + \gamma \neq 1 , \end{cases} \end{align*}


\begin{align*} \log Z = \log \int _ { \R ^ n } e ^ f \ , d \mu ^ { \otimes n } = \sup _ { Q \in \P ( \R ^ n ) } \left ( \int _ { \R ^ n } f \ , d Q - H ( Q \ , | \ , \mu ^ { \otimes n } ) \right ) , \end{align*}


\begin{align*} 2 \int _ 0 ^ { t } \int \bigl ( | \widetilde \psi ^ h | ^ 2 \widetilde \psi ^ h \bigr ) ( s , x ) \overline { \bigl [ e ^ { - i ( t - s ) \Delta } f \bigr ] } ( x ) \ , d x \ , d s = 2 \int _ 0 ^ { t } \bigl \langle e ^ { i ( t - s ) \Delta } \bigl ( | \widetilde \psi ^ h | ^ 2 \widetilde \psi ^ h \bigr ) ( s ) , f \bigr \rangle \ , d s . \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } & n ^ { p } ( \cdot , t ) \\ & \le \int _ { \Omega } n _ { 0 } ^ { p } + \frac { p ( p - 1 ) } { 4 } \| S _ { 0 } \| _ { \mathcal { C } ( [ 0 , \gamma ] ) } ^ { 2 } \| \nabla c \| _ { L ^ { \infty } ( 0 , T ; L ^ { \infty } ( \Omega ) ) } ^ { 2 } \int _ { 0 } ^ { t } \int _ { \Omega } n ^ { p } \quad \mbox { f o r } t \le T . \end{aligned} \end{align*}


\begin{align*} { \mathbf { A } } = \mathbf { A n t i } ( \vartheta _ 1 , \vartheta _ 2 , \vartheta _ 3 ) : = \begin{footnotesize} \begin{pmatrix} 0 & - \vartheta _ 3 & \vartheta _ 2 \\ \vartheta _ 3 & 0 & - \vartheta _ 1 \\ - \vartheta _ 2 & \vartheta _ 1 & 0 \end{pmatrix} \end{footnotesize} \in \mathfrak { s o } ( 3 ) , \end{align*}


\begin{align*} \begin{cases} \Delta _ \Psi u = f ( u ) & A ( r _ { 0 } , + \infty ) \\ u \equiv c _ { 0 } & \{ r _ { 0 } \} \times N . \end{cases} \end{align*}


\begin{align*} X _ S = \boxtimes _ { i \in S } X _ i . \end{align*}


\begin{align*} H _ \epsilon = \bigcap _ { b \in B _ 1 \times \cdots \times B _ k } T ^ { - \overline { \eta ( \epsilon ) } \cdot b } E \end{align*}


\begin{align*} S t ( e _ { 1 } ) = \left \{ \begin{pmatrix} 1 & c & ^ { t } u \\ 0 & 1 & 0 \\ 0 & x & B \end{pmatrix} \Bigg | \ , \ , u + \ , ^ { t } B \psi x = 0 ; \ , \ , 2 c + \langle x , x \rangle = 0 ; \ , \ , B \in O _ { n - 1 , n - 1 } . \right \} . \end{align*}


\begin{align*} \left ( \left ( e ^ { 1 - \binom { k } { \ell } } + o ( 1 ) \right ) m ^ { k - \ell } \right ) ^ { m ^ \ell } \end{align*}


\begin{align*} D _ { \uparrow } V ( u ) ( h ) = \liminf _ { \substack { t \downarrow 0 \\ h ' \to h } } \frac { V ( u + t h ' ) - V ( u ) } { t } . \end{align*}


\begin{align*} a _ { n + 1 } = a _ { \alpha + n + 1 } + d _ { \alpha + n + 1 } - d _ { \alpha + n } \iff a _ { \alpha + n + 1 } - a _ { n + 1 } = d _ { \alpha + n } - d _ { \alpha + n + 1 } \ , \ \forall \ n \in \N \ . \end{align*}


\begin{align*} N ( \tau ) & = \frac { \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) \operatorname { v o l } ( \partial \Omega ) } { n - 1 } \biggl [ \biggl ( \frac { \lambda + 3 \mu } { 2 \mu ( \lambda + \mu ) } \biggr ) ^ { n - 1 } + \frac { 1 } { ( 2 \mu ) ^ { n - 1 } } + \frac { n - 2 } { \mu ^ { n - 1 } } + \frac { 1 } { \alpha ^ { n - 1 } } \biggr ] \tau ^ { n - 1 } \\ & + o ( \tau ^ { n - 1 } ) \ \tau \to + \infty . \end{align*}


\begin{align*} \varphi ( x ' , \rho ( x ' ) ) = \frac { 1 } { 2 } \sum _ { \alpha \leq n - 1 } b _ \alpha x _ \alpha ^ 2 + \frac { 1 } { 6 } \sum _ { \alpha , \beta , \gamma \leq n - 1 } \varphi _ { \alpha \beta \gamma } ( 0 ) x _ \alpha x _ \beta x _ \gamma + O ( | x ' | ^ 4 ) , \end{align*}


\begin{align*} h ( x , u ) = E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 2 } ^ { n } Z _ i ^ { \theta [ u ] } \Bigr ) \right ] = E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \Bigr ) \right ] . \end{align*}


\begin{align*} \sum _ i \sigma _ { k } ^ { i i } \kappa _ i ^ 2 = \sigma _ 1 \sigma _ k - ( k + 1 ) \sigma _ { k + 1 } . \end{align*}


\begin{align*} \tilde { Y } _ t = e ^ { | z | J ^ \perp t } \tilde { Y } _ 0 + \int _ 0 ^ t e ^ { | z | J ^ \perp ( t - s ) } P ^ { - 1 } \Pi _ { \mathrm { k e r } A ^ \perp } \sigma d W _ s . \end{align*}


\begin{align*} K ( c _ 1 , \dots , c _ n , 2 m ) = & p _ j \circ H ( c _ 1 , \dots , c _ n , m ) \\ = & p _ j \circ i _ { U _ k } ( c 1 , \dots , c _ n ) = p _ j ( c _ 1 , \dots , c _ n ) . \end{align*}


\begin{align*} \mathcal { A } = \frac { 1 } { 2 } \sum _ { j = 1 } ^ p e _ j \cdot \big ( ( \widetilde { \nabla } _ Y B ) ( X , e _ j ) - ( \widetilde { \nabla } _ X B ) ( Y , e _ j ) \big ) , \end{align*}


\begin{align*} & \frac { d } { d r } \log _ { m + 1 } \frac { e } { r } = \frac { 1 } { \prod _ { k = 1 } ^ m \log _ k \frac { e } { r } } \left ( - \frac { 1 } { r } \right ) , \\ & \frac { d } { d r } \log _ { m + 1 } \frac { 1 } { C r ^ \alpha } = \frac { 1 } { \prod _ { k = 1 } ^ m \log _ k \frac { 1 } { C r ^ \alpha } } \left ( - \frac { \alpha } { r } \right ) . \end{align*}


\begin{align*} h _ k \dot { \alpha } ( t _ { k - 1 } ) \le \frac { r } { c } ( k = k _ 0 + 1 , k _ 0 + 2 , \ldots ) . \end{align*}


\begin{align*} r _ k ( n ) = k \left \{ \frac { n } { k } \right \} , \end{align*}


\begin{align*} \norm { \tfrac { 1 } { p _ n } - \tfrac { 1 } { f } } _ \infty = \norm { \tfrac { p _ n - f } { p _ n f } } _ \infty \leq \tfrac { 2 } { \delta ^ 2 } \| p _ n - f \| _ \infty \to 0 . \end{align*}


\begin{align*} & \ \dot { s } \epsilon ( z ) \dot { s } ( e - \varepsilon ( 1 ) ) \lambda _ i = \varepsilon ( - z ^ { - 1 } ) \dot { s } ( \varepsilon ( - z ) - \varepsilon ( z ^ 2 - z ) ) \lambda _ i \\ & \ = \sum _ { k , \nu } g ^ i _ { k , \nu } ( \varepsilon ( - z ^ { - 1 } ( 1 + x ^ i _ { k , \nu } ) ) - \varepsilon ( z ^ { - 1 } ( z - 1 ) ^ { - 1 } ( x ^ i _ { k , \nu } - z + 1 ) ) ) \lambda _ k . \end{align*}


\begin{align*} \psi ^ \pm ( x , t ; k ) = e ^ { - i c _ - ( x ) \widehat { \sigma } _ 3 } \mu ^ \pm ( x , t ; k ) e ^ { i c _ + ( x ) \sigma _ 3 } , \end{align*}


\begin{align*} \int _ { M \setminus \Omega } \bigg \langle S ( g , u ) , \ , \kappa _ 0 ( g , u , X ) \bigg \rangle _ g \ , d \mathrm { v o l } _ g & = 0 , \end{align*}


\begin{align*} \begin{cases} t _ { i , e n } ( x _ 0 , t _ 0 ) = \max \left ( 0 , t _ 0 - \dfrac { x _ 0 + R } { \lambda _ i } \right ) \forall \ : i \in [ 1 , p ] , \\ t _ { i , e n } ( x _ 0 , t _ 0 ) = \max \left ( 0 , t _ 0 - \dfrac { x _ 0 - R } { \lambda _ i } \right ) \forall \ : i \in [ p + 1 , n ] . \end{cases} \end{align*}


\begin{align*} D ^ { k } B ^ C _ n ( j _ { V _ C } ) = D ^ { k } \check { B } ^ C _ n ( j _ { V _ C } ) \end{align*}


\begin{align*} D _ g ^ 1 = D _ g ^ + \oplus D _ g ^ - . \end{align*}


\begin{align*} \alpha _ 2 = 0 \ , \Leftrightarrow \ , a _ 2 = 0 \ , \Leftrightarrow \ , \ell _ t = 2 \ , \ell _ b \ , \Leftrightarrow \ , \beta = \gamma 2 \ , \alpha _ 1 + 3 \ , \alpha _ 3 = 0 \ , \Leftrightarrow \ , a _ 3 = 0 \ , \Leftrightarrow \ , \Psi = \frac { 3 } { 2 } , \end{align*}


\begin{align*} \rho ^ { k _ 1 } = ( a / b ) ^ { x k _ 1 } ( \alpha / \beta ) ^ { y k _ 1 } = ( \alpha / \beta ) ^ { x l _ 1 } \zeta ^ x ( \alpha / \beta ) ^ { y k _ 1 } = ( \alpha / \beta ) \zeta ^ x . \end{align*}


\begin{align*} \beta _ 2 ( u ) = 0 \Longrightarrow \beta _ 2 ( u _ { \varepsilon } ) = 0 . \end{align*}


\begin{align*} \Delta _ { \Psi } u = e ^ { \Psi } \div ( e ^ { - \Psi } \nabla u ) = \Delta u - g ( \nabla \Psi , \nabla u ) . \end{align*}


\begin{align*} R ^ i _ { n _ 0 ; l _ 1 , \ldots , l _ { n _ 0 } } = \underset { n _ 0 \geq j \geq i } { \bigsqcup } Q ^ j _ { n _ 0 ; l _ 1 , \ldots , l _ { n _ 0 } } . \end{align*}


\begin{align*} \langle T ( f , g ) , h \rangle = \langle T ^ { * 1 } ( h , g ) , f \rangle = \langle T ^ { * 2 } ( f , h ) , g \rangle . \end{align*}


\begin{align*} & \int _ { X _ j } | u _ { m , m ' , \epsilon , j } | ^ 2 _ { h _ { m ' } } e ^ { v _ \epsilon ( \Psi _ m ) - \delta M _ { \eta _ m } } c ( - v _ \epsilon ( \Psi _ m ) ) + \int _ { X _ j } | h _ { m , m ' , \epsilon , j } | ^ 2 _ { h _ { m ' } } e ^ { - \phi - \delta M _ { \eta _ m } } \\ \le & \int _ { X _ j } v '' _ { \epsilon } ( \Psi _ m ) | f F ^ { 1 + \delta } | ^ 2 _ { h _ { m ' } } e ^ { - \phi - \delta M _ { \eta _ m } } < + \infty . \end{align*}


\begin{align*} \phi = \frac { 1 } { 2 } \left ( u _ { , p } \xi ^ p + v _ q x ^ q \right ) . \end{align*}


\begin{align*} \mathcal { W } ^ 2 ( \mu _ 1 , \mu _ 2 ) = \inf \mathbb { E } [ | X - Y | ^ 2 ] . \end{align*}


\begin{align*} a _ c \left [ t + 1 \right ] & = a _ c \left [ t \right ] - \mu \frac { \partial \mathbb { E } \left [ \varepsilon \right ] } { \partial a _ c } \\ a _ i \left [ t + 1 \right ] & = a _ i \left [ t \right ] - \mu \frac { \partial \mathbb { E } \left [ \varepsilon \right ] } { \partial a _ i } . \end{align*}


\begin{align*} \int _ { \partial \Omega } { \texttt f } \sigma ^ \psi _ x \mathfrak { f } = & \int _ { \Omega } \left ( { \texttt f } { } ^ \psi \mathcal D [ \mathfrak { f } ] + { } ^ { { \psi } } \mathcal D _ r [ { \texttt f } ] \mathfrak { f } \right ) d x , \end{align*}


\begin{align*} \frac { B ( z ) } { A ( z ) } = p z + q + \sum _ n \Big ( \frac { 1 } { t _ n - z } - \frac { 1 } { t _ n } \Big ) \mu _ n \end{align*}


\begin{align*} & \psi _ x + i k ^ 2 [ \sigma _ 3 , \psi ] = k P _ x \psi , \\ & \psi _ t + i \eta ^ 2 [ \sigma _ 3 , \psi ] = H \psi . \end{align*}


\begin{align*} \mathbf { g } = \pm u ^ 2 d t ^ 2 + g . \end{align*}


\begin{align*} \begin{pmatrix} v _ 1 \\ v _ 2 \end{pmatrix} = - \frac { 1 } { k ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 } \begin{pmatrix} x _ 1 x _ 2 & - k ^ 2 - x _ 1 ^ 2 \\ k ^ 2 + x _ 2 ^ 2 & - x _ 1 x _ 2 \end{pmatrix} \begin{pmatrix} \partial _ { x _ 1 } \varphi \\ \partial _ { x _ 2 } \varphi \end{pmatrix} . \end{align*}


\begin{align*} \ T _ { \tau } ( t ) \stackrel { d e f } { = } { \bf P } ( | \tau | \ge t ) , \ t \ge 0 . \end{align*}


\begin{align*} C \left ( X , Y , Z \right ) \equiv \left ( \nabla _ { Z } g \right ) \left ( X , Y \right ) = Z g \left ( X , Y \right ) - g \left ( \nabla _ { Z } X , Y \right ) - g \left ( X , \nabla _ { Z } Y \right ) \end{align*}


\begin{align*} & W _ 0 = \{ ( p , q ) \in \hat A \times \hat B \mid d ( q , w ) \geq d ( p , w ) \} ; \\ & W _ 1 = \{ ( p , q ) \in \hat A \times \hat B \mid d ( p , \tilde { w } ) \geq d ( q , \tilde { w } ) \} . \end{align*}


\begin{align*} h _ { \hat { \mu } } ( \hat { \sigma } ) = h _ { \hat { \mu } } ( \hat { \sigma } , \hat { \P } ) \ \ h _ \mu ( \sigma ) = h _ \mu ( \sigma , \P ) \ . \end{align*}


\begin{align*} \nabla _ V ( \nabla X ) = - R ( X , V ) + T _ h ( V , \cdot ) \end{align*}


\begin{align*} \sup \limits _ i \int _ K | \tilde { F } _ i - ( 1 - b _ { t _ 0 , B } ( \Psi ) ) f F ^ { 1 + \delta } | ^ 2 _ { h _ { j _ K , 1 } } < + \infty . \end{align*}


\begin{align*} \begin{aligned} y \left [ i \right ] & = \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { \bf { \bar q } } \left [ { i - { n _ l } } \right ] } + z \left [ i \right ] \\ & = \sum \limits _ { l = 1 } ^ L { \sum \limits _ { { l ' } = 1 } ^ { L ' } { { \bf { h } } _ l ^ H { { \bf { { F } } } _ { l ' } } { \bf { d } } \left [ { i - { n _ l } - { \kappa _ { l ' } } } \right ] } + z \left [ i \right ] } . \end{aligned} \end{align*}


\begin{align*} P _ b ( x ) = x ^ { q + 1 } + x + b , \end{align*}


\begin{align*} x \cdot x = x \cdot 1 = 1 , 1 \cdot x = x \end{align*}


\begin{align*} E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] = \xi _ 0 E _ 1 \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] + ( 1 - \xi _ 0 ) E _ 2 \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] . \end{align*}


\begin{align*} d u = \Delta _ p u \ , d t + \kappa \Delta u \ , d t + \sum _ { k \in \Z ^ d } \sum _ { i = 1 } ^ { d - 1 } \xi _ { k , i } ( x , t ) \cdot \nabla u \ , d B _ t ^ { k , i } , \end{align*}


\begin{align*} X = \{ \sum _ { n = 0 } ^ { \infty } a _ n z ^ n \mid \left ( a _ n / w _ n \right ) _ { n \in \mathbb { N } } \in \ell ^ 2 \} \end{align*}


\begin{align*} r ( a b ) = ( r a ) b = ( - 1 ) ^ { p ( r ) p ( a ) } a ( r b ) \end{align*}


\begin{align*} \exists \ r = r ( \theta , p ) \in [ 1 , p ) , \ \exists \ v = \ v ( \theta , p , r ) < \infty . \end{align*}


\begin{align*} p ( x ) \Longleftrightarrow t _ { c , \rho } ( y ) = p \Big ( \frac { x - c } { \rho } \Big ) , \end{align*}


\begin{align*} - \Delta v - p w ^ { p - 1 } _ + v = 0 , \ \ v \in L ^ { \infty } ( \mathbb { R } ^ 2 ) . \end{align*}


\begin{align*} A ( \tau _ y ( x ) ) + \sum _ { n = 1 } ^ N \left [ \hat { A } \circ \hat { \sigma } ^ { - ( n + 1 ) } ( y | x ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( \sigma ( y ) | x ' ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ) + \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ' ) \right ] & = \\ A ( \tau _ { y , N + 1 } ( x ) ) + \sum _ { n = 1 } ^ N \left [ \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ' ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( \sigma ( y ) | x ' ) \right ] \ . \end{align*}


\begin{align*} \mathcal { C } _ n ^ { ( p ) } \ = \ A _ p F _ n + B _ p F _ { n + 1 } - \sum _ { k = 0 } ^ p \binom { p } { k } B _ k n ^ { p - k } . \end{align*}


\begin{align*} u = \arg \min _ { w \in L _ 2 ( 0 , T ; \mathcal { V } ) } \bigg ( | | \partial _ t w + \mathcal { L } w - F | | ^ 2 _ { L _ 2 ( 0 , T ; \mathcal { H } ) } + | | \alpha w + \beta \mathbf { n } \cdot \nabla w - g | | ^ 2 _ { L _ 2 ( 0 , T ; \partial \mathcal { H } ) } + | | w ( 0 , \cdot ) - u _ 0 | | ^ 2 _ { \mathcal { H } } \bigg ) , \end{align*}


\begin{align*} \bigg ( \frac { 1 } { | Q | } \iint _ { T _ Q } | \Theta f ( x ' , t ) | ^ 2 \frac { d x ' d t } { t } \bigg ) ^ { \frac 1 2 } = \bigg ( \frac { 1 } { | Q | } \iint _ { T _ Q } | \Theta ( f - f _ Q ) ( x ' , t ) | ^ 2 \frac { d x ' d t } { t } \bigg ) ^ { \frac 1 2 } \le { \rm I } + { \rm I I } , \end{align*}


\begin{align*} \Omega _ w = \{ ( F _ \bullet , V _ \bullet ) ; \dim V _ i \cap F _ j \geq r _ { i , j } ( w ) \} . \end{align*}


\begin{align*} \left \{ \begin{aligned} \dot { x } & = \frac { p } { t } ( z - x ) , \\ \dot { z } & = - p t ^ { p - 1 } \nabla f ( x ) . \end{aligned} \right . \end{align*}


\begin{align*} a _ { P _ T } ( T ) = \delta ( T , T ) + O _ { \epsilon } ( k ^ { - 2 / 3 } \det ( T ) ^ { 1 + \epsilon } ) . \end{align*}


\begin{align*} \partial _ t \gamma = \frac { c } { 4 \pi } | \ln \varepsilon | ( \partial _ s \gamma \times \partial _ { s s } \gamma ) = \frac { c \bar { K } } { 4 \pi } | \ln \varepsilon | \mathbf { b } _ { \gamma ( t ) } , \end{align*}


\begin{align*} 0 \ = \ - { \rm c u r l } \left ( { \rm c u r l } \ , u \right ) + { \rm D } ( { \rm d i v } \ , u ) \ , - \ , \Delta u \end{align*}


\begin{align*} \norm { f } _ { H ^ 1 _ 0 } = \sqrt { f \cdot f } = \sqrt { \norm { f } _ { L ^ 2 } ^ 2 + \norm { D f } _ { L ^ 2 } ^ 2 } . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ m { w _ { i j } } = 1 i = 1 , \dots , n . \end{align*}


\begin{align*} \sum _ { j \le x } \tau _ z ( j ) = \sum _ { d < z } \left \lfloor \frac x d \right \rfloor = \sum _ { d < z } \left \lfloor \frac { \lfloor x \rfloor } d \right \rfloor < \lfloor x \rfloor ( \log z + 1 ) . \end{align*}


\begin{align*} \nabla _ X \psi _ 1 = \frac { 1 } { 2 } X _ 1 \cdot \psi _ 1 - \frac { 1 } { 2 } B ( X ) \cdot \psi _ 1 \end{align*}


\begin{align*} { } \hat { T } _ N : = \arg \min _ { T \in \mathcal { T } } M _ N ( T ) , M _ N ( T ) : = \frac { 1 } { 2 N } \sum _ { i = 1 } ^ N d ^ 2 _ { \mathcal { W } } ( T \# \mu _ i , \nu _ i ) . \end{align*}


\begin{align*} \gamma _ i ( s , \tau ) = \bar { Q } _ { \frac { 2 \pi ( i - 1 ) } { m } } \gamma ( s , \tau ) . \end{align*}


\begin{align*} \begin{cases} \chi _ 1 \leq w _ 1 \chi + r \\ \chi _ 1 \geq w _ 1 \chi . \end{cases} \end{align*}


\begin{align*} c _ \varepsilon = I _ \varepsilon ( u _ \varepsilon ) = \pi \min \limits _ { \overline { \Omega } } q ^ 2 \sqrt { d e t ( K _ H ) } \ln \frac { 1 } { \varepsilon } + O ( 1 ) . \end{align*}


\begin{align*} & M i n i m i z e _ { x , y \in \mathbb { R } ^ { n } } \ \ \alpha _ J ^ T y + 2 b _ J ^ T x \\ & s . t . \ \ \alpha _ k ^ T y + 2 b _ k ^ T x + c _ j \le 0 \ \ ( \forall \ , k \in \{ 1 , . . . , m \} ) \\ & s . t . \ \ x _ i ^ 2 - y _ i = 0 , \ \ \forall \ , i \end{align*}


\begin{align*} | | A _ q ^ z | | ^ 2 _ { L ^ 2 ( \mathbb R ) } = E _ q \left ( | z | ^ 2 \right ) . \end{align*}


\begin{align*} \mu ^ { N , i } _ { \boldsymbol { \alpha } } : = \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } \alpha ^ j , \nu ^ { N , i } _ { \boldsymbol { x } } : = \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } x ^ j . \end{align*}


\begin{gather*} \begin{aligned} & 2 \langle A ( t , u ) - A ( t , v ) , u - v \rangle + \Vert B ( t , u ) - B ( t , v ) \Vert _ { L _ 2 } ^ 2 \\ \leq & [ f ( t ) + \rho ( u ) + \eta ( v ) ] \Vert u - v \Vert _ { H } ^ 2 , \\ \end{aligned} \\ | \rho ( u ) | + | \eta ( u ) | \leq C ( 1 + \Vert u \Vert _ V ^ { \alpha } ) ( 1 + \Vert u \Vert _ H ^ { \gamma } ) , \end{gather*}


\begin{align*} R _ c = \pi \sqrt { 5 0 / 2 } \approx 1 5 . 7 \ , , \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I I _ 2 = 0 . \end{align*}


\begin{align*} \int _ { \Omega } ( U - f \textrm { c u r l } ^ { - 1 } U ) \cdot \tilde { U } \dd x = 0 , f = \frac { \int _ { \Omega } U _ 0 \cdot U \dd x } { \int _ { \Omega } \textrm { c u r l } ^ { - 1 } U _ 0 \cdot U \dd x } . \end{align*}


\begin{align*} I \coloneqq \langle \varepsilon _ 1 ^ { c _ 1 } , \ \varepsilon _ 2 ^ { c _ 2 } , \ \varepsilon _ 2 ^ { b / g } \alpha _ k - \alpha _ k \varepsilon _ 1 ^ { c / g } \mid k = 1 , 2 , \dots , g \rangle . \end{align*}


\begin{align*} W ( \xi _ 0 , k , x , \mathsf { M } ^ k ) = W ( 1 - \xi _ 0 , k , x , \mathsf { M } ^ k ) , ~ \forall \xi _ 0 \in [ 0 , 1 ] , \forall x \in \mathbb { R } . \end{align*}


\begin{align*} \int _ { \mathfrak U } | \omega | = { \phi _ s } _ ! \int _ { \mathfrak Y } | \omega | , \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { \ln \frac { ( A _ \varepsilon , \partial \Omega ) } { ( A _ \varepsilon ) } } { \ln \frac { 1 } { \varepsilon } } = 1 . \end{align*}


\begin{align*} \begin{aligned} \| u _ { 0 } \| _ { L ^ { 2 } ( \Lambda _ h ) } & \leq \| e ^ { \rho \cdot \mathbf x } \| _ { L ^ { 2 } ( \Lambda _ { h } ) } + \| e ^ { \rho \cdot \mathbf x } \| _ { L ^ { 4 } ( \Lambda _ { h } ) } \| \psi ( \mathbf x ) \| _ { L ^ { 4 } ( \Lambda _ { h } ) } \\ & \lesssim \left ( 1 + \tau ^ { - \frac { 2 } { 3 } } \right ) e ^ { - \zeta h \tau } , \mbox { a s $ \tau \rightarrow \infty $ . } \end{aligned} \end{align*}


\begin{align*} \begin{aligned} L [ \hat { f } ] = \nu \hat { f } - K [ \hat { f } ] = \hat { \textrm { g } } \ , , \ | \hat { f } | = \tfrac { 1 } { \nu } | \hat { \textrm { g } } + K [ \hat { f } ] | < \tfrac { 1 } { \nu _ 0 } ( | \hat { \textrm { g } } | + | K [ \hat { f } ] | ) \ , , \end{aligned} \end{align*}


\begin{align*} \Gamma _ { k } = \{ \lambda \in \mathbb { R } ^ { n } : \sigma _ { j } ( \lambda ) > 0 , j = 1 , \ldots , k \} . \end{align*}


\begin{align*} M i n i m i z e _ { x \in \mathbb { R } ^ n } \ \ J ( x ) : = 1 / 2 \| x - z \| ^ 2 \ \ s . t . \ x \in A \end{align*}


\begin{align*} \left \{ \begin{aligned} b x = \mu , \\ c + ( d - a ) x = \mu x . \end{aligned} \right . \end{align*}


\begin{align*} R _ 1 = \sum _ { X \in S ( 0 ) } \Theta \begin{bmatrix} m _ 1 ' & X \\ 0 & m _ 2 ' \end{bmatrix} \overline { \psi _ A ( { m _ 1 ' } ^ { - 1 } X } ) = - \theta ( a ) q ^ 8 ( q - 1 ) ^ 2 . \end{align*}


\begin{align*} x _ 2 \partial _ { x _ 1 } v _ 3 - x _ 1 \partial _ { x _ 2 } v _ 3 + k \partial _ { x _ 3 } v _ 3 = 0 . \end{align*}


\begin{align*} \Delta u _ 1 \wedge \dots \wedge \Delta u _ k \wedge \beta _ n ^ { n - m } ( \omega ) = \int u _ k \Delta u _ 1 \wedge \dots \wedge \Delta u _ { k - 1 } \wedge \beta _ n ^ { n - m } \wedge \Delta \omega , k = 1 , \dots , m , \end{align*}


\begin{align*} \lambda _ x | _ { E _ w } = \lambda _ y | _ { E _ w } , \lambda _ x ( w , w b ) = f \lambda _ x ( w b ) = s ( f ) \end{align*}


\begin{align*} \begin{aligned} \int _ { \mathcal C ^ h } \vert \mathbf x \vert ^ { \alpha } e ^ { - \zeta \tau \vert \mathbf x \vert } \mathrm { d } \mathbf x \lesssim \frac { 1 } { \tau ^ { \alpha + 3 } } + \frac { 1 } { \tau } e ^ { - \frac { 1 } { 2 } \zeta h \tau } , \end{aligned} \end{align*}


\begin{align*} \sum _ { p \leq x } a _ { \pi } ( p ^ 2 ) \log p = \sum _ { p \leq x } \sum _ { 1 \leq j \leq M } \alpha ^ 2 _ { \pi } ( p , j ) \log p = & - \delta ( \pi ) x + O ( x ^ { 1 / 2 } \log ^ { 2 } x ) . \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) : = \sum _ { K \in { \mathcal T } _ n } \int _ K \nabla u _ n \cdot \nabla v _ n d K , u _ n , v _ n \in X _ n . \end{align*}


\begin{align*} \begin{aligned} \Psi \leq & - \frac { \theta } { 2 } | x ' | ^ 2 , \mbox { o n } \partial _ 1 \omega _ \delta \\ \Psi \leq & - \frac { \delta ^ 2 } { 2 } , \mbox { o n } \partial _ 2 \omega _ \delta \\ \Psi \leq & - \frac { \theta \delta ^ 2 } { 2 } , \mbox { o n } \partial _ 3 \omega _ \delta . \end{aligned} \end{align*}


\begin{align*} g _ i ^ { ( e q ) } = { { w } _ i } T , \end{align*}


\begin{align*} S \cap [ q ] = ( U \cap [ q ] ) \cup ( T \cap [ q ] ) = ( T \cap F _ { \ell + 1 } ) \cup T = T . \end{align*}


\begin{align*} H = \frac { 1 } { 2 m } \left ( \sum _ i p _ i ^ 2 + \lambda \left ( \sum _ i x _ i p _ i \right ) ^ 2 \right ) + { \cal V } ( r ) = \frac { 1 } { 2 m } \left ( ( 1 + \lambda r ^ 2 ) \sum _ i p _ i ^ 2 - \lambda \sum _ { i < j } J _ { i j } ^ 2 \right ) + { \cal V } ( r ) , \end{align*}


\begin{align*} \begin{cases} \partial _ t u ( t ) = - \Delta ^ 2 u + \Delta \varphi ( u ) , \\ \nabla u \cdot \nu = \nabla ( \Delta u ) \cdot \nu = 0 \ \partial \mathcal { O } , \\ u ( 0 ) = u _ 0 , \end{cases} \end{align*}


\begin{align*} \frac { \partial z } { \partial \alpha } = & \frac { \phi ( z ) \ , [ 1 - \Phi ( z ) ] } { \Phi ( z ) \ , [ 1 - \Phi ( z ) ] + z \ , \phi ( z ) \left [ \alpha - ( \alpha + \beta ) \ , \Phi ( z ) \right ] - ( 2 - \alpha - \beta ) \ , \phi ^ 2 ( z ) } . \end{align*}


\begin{align*} E [ M ^ * _ v ( t ) ^ 2 ] \le e ^ { - 2 ( \lambda ^ * - \lambda ) t } E [ M _ v ( 0 ) ^ 2 ] + \left \{ \begin{aligned} & 4 | v | ^ 2 \| x \| t e ^ { - 2 ( \lambda ^ * - \lambda ) t } & & \lambda \ge 1 , \\ & 4 | v | ^ 2 \| x \| \frac { 1 } { 1 - \lambda } e ^ { - 2 ( \lambda ^ * - 1 ) t } & & \lambda < 1 . \end{aligned} \right . \end{align*}


\begin{align*} \partial _ t w + ( \mathbf { v } \cdot \nabla ) w = 0 . \end{align*}


\begin{align*} | \sigma | ^ 2 _ g = \frac { \i ^ { n ^ 2 } \sigma \wedge \bar { \sigma } } { d V _ g } = c _ n \frac { ( d y _ 1 \wedge \dots \wedge d y _ n ) \wedge ( d \theta _ 1 \wedge \dots \wedge d \theta _ n ) } { ( d x ^ 1 \wedge \dots \wedge d x ^ n ) \wedge ( d \theta _ 1 \wedge \dots \wedge d \theta _ n ) } = c _ n \frac { d y _ 1 \wedge \dots \wedge d y _ n } { d x ^ 1 \wedge \dots \wedge d x ^ n } , \end{align*}


\begin{align*} & = \sup _ { ( x ' , y ' ) \in X \times \mathbb R } \{ \psi ( x ' , y ) - \frac { 1 } { 2 \epsilon } \left ( \| x - x ' \| ^ 2 - \| x \| ^ 2 \right ) - \frac { 1 } { 2 } \left ( | y - y ' | ^ 2 - \| y \| ^ 2 \right ) \} \\ & = \sup _ { ( x ' , y ' ) \in X \times \mathbb R } \{ \psi ( x ' , y ) + \frac { 1 } { \epsilon } x \cdot x ' + \frac { 1 } { \epsilon } y \cdot y ' - \frac { 1 } { 2 \epsilon } \| x ' \| ^ 2 - \frac { 1 } { 2 \epsilon } \| y ' \| ^ 2 \} \end{align*}


\begin{align*} \alpha _ { i j k } = - \frac { 1 } { 2 } R _ { \ell i j k } x ^ { \ell } + \mathbf { o ( } \mathbf { x ) } , \mathbf { } \end{align*}


\begin{align*} \langle \overline { y } - g , S ' ( \overline { u } ; h ) \rangle = \langle \overline { y } - g , z \rangle = \langle \zeta z , p \rangle + \langle - \Delta z , p \rangle = \langle \zeta z , p \rangle - \langle F ' ( \overline { y } ; z ) , p \rangle + \langle h , p \rangle \ge - \alpha \langle \overline { u } , h \rangle . \end{align*}


\begin{align*} f ( p , s ) = s \log a + \log A _ k - \left ( \frac { p } { 2 } + 2 \right ) \log s + \log \Gamma \left ( \frac { p } { 2 } + 2 \right ) . \end{align*}


\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) = A ( 1 ^ \infty ) - A ( 1 ^ n 0 ^ \infty ) = \begin{cases} c - d \ , \\ c - c _ n \ , \end{cases} \ , \end{align*}


\begin{align*} \nu _ z = - \frac { 1 } { 2 } h _ z - ( \alpha - I \beta ) h _ { \overline { z } } . \end{align*}


\begin{align*} 2 H = \alpha \frac { \nu _ 3 } { z } + \varpi , \end{align*}


\begin{align*} \delta _ { + } \delta _ { - } = \mathrm { e } ^ { - \mathrm { i } \mathcal { H } \log \left ( 1 + \bar { r } _ 1 r _ 2 \right ) } . \end{align*}


\begin{align*} G ( z ) = A ( z ) \bigg ( 1 + \sum _ n a _ n \bar b _ n t _ n + \sum _ n \frac { a _ n \bar b _ n t ^ 2 _ n } { z - t _ n } \bigg ) . \end{align*}


\begin{align*} i \leq N _ v / 2 , \ \beta _ i ^ v = \beta _ i ^ { v _ l } \oplus \beta _ i ^ { v _ r } , i > N _ v / 2 , \ \beta _ i ^ v = \beta ^ { v _ r } _ { i - N _ v / 2 } . \end{align*}


\begin{align*} \gamma ( X _ i , X _ j ) = 0 \quad \forall i \neq j , \end{align*}


\begin{align*} A _ k \ = \ \sum _ { j = 0 } ^ { k - 1 } \binom { k } { j } B _ j . \end{align*}


\begin{align*} \mathbf { g } = \pm u ^ 2 d t ^ 2 + g . \end{align*}


\begin{align*} \int _ { \partial \Omega } \nabla c \cdot \nu c \psi ^ { 2 } = \int _ { \partial \Omega \cap B _ { \eta } } \nabla c \cdot \nu c \psi ^ { 2 } . \end{align*}


\begin{align*} 0 . 2 5 I _ f ( p , q ) & \leq \sum _ { i \in A _ { 1 + \kappa , \infty } } p _ i f \left ( \frac { q _ i } { p _ i } \right ) \leq \sum _ { i \in A _ { 1 + \kappa , \infty } } p _ i f ( \nu ) = p ' f ( \nu ) , \end{align*}


\begin{align*} c _ { 1 } : = \mathop { \inf } \limits _ { ( u , v ) \in E \setminus \{ ( 0 , 0 ) \} } \mathop { \max } \limits _ { t \geq 0 } I ( t u , t v ) . \end{align*}


\begin{align*} ( \sigma _ C \oplus \sigma _ D ) ( l ) = ( \delta _ C + \delta _ D ) ^ * ( l ) = \delta _ { C \cap D } ^ * ( l ) = \sigma _ { C \cap D } ( l ) = l ( x ) . \end{align*}


\begin{align*} h _ = ( \sum _ { j = 1 } ^ m \lambda _ j \nabla g _ j , \lambda _ 1 g _ 1 , \dots , \lambda _ m g _ m , \sum _ { j = 1 } ^ m \lambda _ j ^ 2 - 1 ) \ , , \end{align*}


\begin{align*} \lim _ { \ell \to \infty } \alpha _ \ell = + \infty . \end{align*}


\begin{align*} v ( x ) = \left | \sum _ { i = 1 } ^ k x _ i B _ i \right | \end{align*}


\begin{align*} | | \mathcal { A } _ q ^ p | | ^ 2 _ { L ^ 2 ( \mathbb R , \mathbb { H } ) } = E _ q \left ( | p | ^ 2 \right ) . \end{align*}


\begin{align*} z _ 0 ^ { p ^ r } x _ 0 + \cdots + z _ n ^ { p ^ r } x _ n = 0 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \Vert X _ n ( t , \omega ) - X ( t , \omega ) \Vert _ H = 0 , \forall \ , ( t , \omega ) \in \Gamma , \end{align*}


\begin{align*} f ^ \lambda \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ( z ) = ( 2 \pi ) ^ { - p - q } \lambda ^ { p + q } \ , P ( z ) \ , g ^ \lambda \ast _ \lambda \varphi _ { k - p , \lambda } ^ { n + p + q - 1 } ( z ) \end{align*}


\begin{align*} \varprojlim _ { n \in \Delta } \mathcal F ( Y _ n ) = \varprojlim _ { n \in \Delta } \varprojlim _ { k \in \Delta } \mathcal F ( \tilde Y _ { n , k } ) = \varprojlim _ { k \in \Delta } \left ( \varprojlim _ { n \in \Delta } \mathcal F ( \tilde Y _ { n , k } ) \right ) . \end{align*}


\begin{align*} I V = - \Vert \Delta ( n - \widetilde { n } ) \Vert _ { L ^ 2 } ^ 2 = - \Vert n - \widetilde { n } \Vert _ { H ^ 2 } ^ 2 + \Vert n - \widetilde { n } \Vert _ { H ^ 1 } ^ 2 . \end{align*}


\begin{align*} e ^ { i c _ + ( x ) } \partial _ { x } \left ( \bar { u } _ x ( x ) e ^ { - i c _ - ( x ) } \right ) = - \frac { 1 } { \pi } \int _ { \mathbb { R } } r _ { \delta , 2 } ( z ) \mathrm { e } ^ { 2 i z x } M _ { \delta , - , 2 2 } ( x ; z ) d z . \end{align*}


\begin{align*} \begin{cases} \Delta _ { \Psi } u = A u + B & \O \\ u = c _ { j } & ( \partial \O ) _ { j } \end{cases} \end{align*}


\begin{align*} \begin{aligned} \Bigg | & \left . \dfrac { \partial \epsilon _ 2 ( \eta , \tau ) } { \partial \eta } \right | _ { \eta = \eta _ a ( \tau ) } \dot { \eta } _ a ( \tau ) \Bigg | \le \\ & \gamma \Bigg | \left . \dfrac { \partial \epsilon _ 2 ( \eta , \tau ) } { \partial x _ a } \right | _ { \eta = \eta _ a ( \tau ) } \Bigg | \dfrac { | b _ { 1 , \delta } ( x _ a ) | } { 2 } \le \gamma 2 L _ r ^ 2 \delta ^ 2 . \end{aligned} \end{align*}


\begin{align*} \sigma ( ( u ^ i , u ^ i { } _ j ) . a ) & = \sigma ( u ^ i , u ^ i { } _ \alpha a ^ \alpha { } _ j ) \\ * & = ( u ^ i , u ^ i { } _ \alpha a ^ \alpha { } _ j , - \Gamma ^ i { } _ { \alpha \beta } u ^ \alpha { } _ \gamma u ^ \beta { } _ \delta a ^ \gamma { } _ j a ^ \delta { } _ k ) \\ * & = \sigma ( u ^ i , u ^ i { } _ j ) . a \end{align*}


\begin{align*} Q _ { \delta _ N } L _ { \delta _ N } u _ N = L _ { \delta _ N } u _ N - \sum _ { j = 1 } ^ m \sum _ { h = 1 } ^ 2 C _ { j , h , N } \frac { \partial } { \partial z _ { j , h } } ( - \delta _ N ^ 2 ( K ( z _ { N , j } ) \nabla V _ { \delta _ N , Z _ N , j } ) ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { \infty } \epsilon _ i < \infty . \end{align*}


\begin{align*} S _ i : = \sup _ { Q _ i \in \P ( \R ) } \left ( \int _ { \R } \hat { f } _ i \ , d Q _ i - H ( Q _ i ) \right ) . \end{align*}


\begin{align*} X ( t ) & = x + \int _ 0 ^ t A ( s , \overline { X } ( s ) ) d s + \int _ 0 ^ t B ( s , \overline { X } ( s ) ) d W ( s ) \\ & \quad \ + \int _ 0 ^ t \int _ { \{ | z | < 1 \} } H ( s , \overline { X } ( s ) , z ) \widetilde { N } ( d s , d z ) \\ & \quad \ + \int _ 0 ^ t \int _ { \{ | z | \ge 1 \} } J ( s , \overline { X } ( s ) , z ) N ( d s , d z ) . \end{align*}


\begin{align*} V _ { \mathrm { d e t } } = \sup _ { y \in \R ^ n } \left ( \int _ { \R ^ n } g \ , d \gamma _ { y , T } - \frac { 1 } { n } H ( \gamma _ { y , T } \ , | \ , \gamma _ T ) \right ) = \sup _ { y \in \R ^ n } \left ( \int _ { \R ^ n } g \ , d \gamma _ { y , T } - \frac { | y | ^ 2 } { 2 n T } \right ) , \end{align*}


\begin{align*} \frac { \partial ^ 2 f ( z ) } { \partial z ^ 2 } = & \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \ , \phi ( z ) \ , [ \Phi ( z ) ] ^ { \alpha - 2 } \ , [ 1 - \Phi ( z ) ] ^ { \beta - 2 } \ , \frac { \partial s ( z ) } { \partial z } . \end{align*}


\begin{align*} V ( \xi _ 0 , n , x ) : = \sup _ { \theta \in \Theta _ n } W ( \xi _ 0 , n , x , \theta ) = \sup _ { \theta \in \Theta _ n } E _ P \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] . \end{align*}


\begin{align*} \sum \limits _ { i = 1 } ^ j \chi _ i \leq \chi ( \sum \limits _ { i = 1 } ^ j w _ i ) + j r , \end{align*}


\begin{align*} F ^ { \perp } : = \left \lbrace \varphi \in E ^ { \vee } \mid \varphi ( F ) = 0 \right \rbrace . \end{align*}


\begin{align*} ( x ) _ k = \frac { \Gamma ( x + k ) } { \Gamma ( x ) } = x ( x + 1 ) \dots ( x + k - 1 ) \end{align*}


\begin{align*} \begin{aligned} \textrm { t h e i n t e g r a l } \int _ { \R ^ 3 } | v | ^ k e ^ { - a | v | ^ 2 } \d v \textrm { c o n v e r g e s , i f a n d o n l y i f } k > - 3 \ , . \end{aligned} \end{align*}


\begin{align*} u ( t ) = u _ 0 + \int _ 0 ^ t \Delta _ p u \ , \ , d s + \int _ 0 ^ t S ( u ) \ , \ , d s + \sum _ { n = 0 } ^ { + \infty } \int _ 0 ^ t \xi _ { n } ( x ) \cdot \nabla u ( s ) \ , d B _ s ^ { n } \end{align*}


\begin{align*} p \frac { d ^ 2 } { d s ^ 2 } \left ( \left ( \kappa - \mu \right ) ^ { p - 1 } \right ) + p \kappa ^ 2 \left ( \kappa - \mu \right ) ^ { p - 1 } - \kappa \left ( \left ( \kappa - \mu \right ) ^ p + \sigma \right ) = 0 . \end{align*}


\begin{align*} \psi _ { \tau } ( p ) \stackrel { d e f } { = } \left [ \ p \int _ 0 ^ { \infty } \ t ^ { p - 1 } \ T _ { \tau } ( t ) \ d t \ \right ] ^ { 1 / p } = | | \tau | | L _ p ( \Omega ) , \end{align*}


\begin{align*} E _ { \xi _ 1 ^ \theta ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 2 } ^ { n } Z _ i ^ { \theta [ u ] } \biggr ) \biggr ] = E _ { \xi _ 1 ^ \theta ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \biggr ) \biggr ] . \end{align*}


\begin{align*} \sigma _ { \rm a } ^ 2 ( x [ n ] ) = \lambda _ \mathrm { a } [ n ] T _ { \rm s } \exp \left ( - \frac { \lambda _ \mathrm { a } [ n ] \tau _ { \rm d } } { N _ { \rm a } } \right ) - \frac { \lambda _ \mathrm { a } ^ 2 [ n ] T _ { \rm s } \tau _ { \rm d } } { N _ { \rm a } } \exp \left ( - \frac { 2 \lambda _ \mathrm { a } [ n ] \tau _ { \rm d } } { N _ { \rm a } } \right ) \left ( 2 - \frac { \tau _ { \rm d } } { T _ { \rm s } } \right ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 3 ( I _ N ^ i ( p ) - J _ N ^ i ( p ) ) = O ( p ^ { \mu - \nu } ) \quad \mbox { f o r l a r g e } p > 0 , \end{align*}


\begin{align*} \begin{aligned} \beta ( \theta ) \sin \theta = \beta ( \tfrac { \pi } { 2 } - \theta ) \sin ( \tfrac { \pi } { 2 } - \theta ) \ , , \ \beta ( \theta ) \sin \theta = \beta ( \pi - \theta ) \sin ( \pi - \theta ) \ , , \end{aligned} \end{align*}


\begin{align*} f ^ \sigma \circ \tilde { \sigma } _ { Y ^ \sigma , \mathrm { t r } } ^ { * } = \sigma _ { X , \mathrm { t r } } ^ * \circ f ^ \sigma ; \end{align*}


\begin{align*} \sum _ { u = 1 } ^ t i _ u = 2 r , \end{align*}


\begin{align*} N = n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } = \left ( n + 3 \right ) ^ { 3 } + \left ( n + \alpha \right ) ^ { 3 } \end{align*}


\begin{align*} \begin{aligned} & \sum _ { i = 1 } ^ { n } \sum _ { j = 0 } ^ { k _ i - 2 } Z ^ { \star } _ { I } ( j + 1 , k _ { i + 1 } , \ldots , k _ n , k _ 1 , \ldots , k _ { i - 1 } , k _ i - j ; ( \alpha , \beta ) ) \\ = & ( k - n ) Z ( n | k - n + 1 ; ( \alpha , \beta ) ) + n Z ( n + 1 | k - n ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \int e ^ { W ( y | x ) } d \nu _ { A ^ { * } } ( y ) = \varphi _ A ( x ) \end{align*}


\begin{align*} M ^ d \times ( - 1 , 0 ) \ni ( x , z ) \mapsto \mathcal { S } ( x , z ) = ( x , \varrho ( x , z ) ) \in \Omega , \end{align*}


\begin{align*} \cos ^ 2 \theta \delta _ { m , k } + \sin ^ 2 \theta \sqrt { m } = \cos ^ 2 \theta \sqrt { m } + \sin ^ 2 \theta \delta _ { m , l } = \gamma _ { m , k , l } . \end{align*}


\begin{align*} l ( y _ p ) \leq \bigvee _ { 1 \leq i \leq n } l ( \Psi ( v _ i ) ) = \bigvee _ { 1 \leq i \leq n } \Phi ( l ( v _ i ) ) = \bigvee _ { 1 \leq i \leq n } \Phi ( p _ i ) = \Phi \left ( \bigvee _ { 1 \leq i \leq n } p _ i \right ) = \Phi ( p ) . \end{align*}


\begin{align*} u ( t ) : = \sum _ { i = 1 } ^ { \infty } \big | \psi _ i ( t ) - \rho _ i ( t ) \big | = \sum _ { i = 1 } ^ { \infty } | u _ i ( t ) | . \end{align*}


\begin{align*} \left | \sum _ { j = 0 } ^ { n - 1 } A ( \sigma ^ j ( x ) ) - \sum _ { j = 0 } ^ { n - 1 } A ( \sigma ^ j ( y ) ) \right | < \epsilon \ , \end{align*}


\begin{align*} \frac { \partial s _ { \delta , i } } { \partial z _ { i , h } } = O ( \delta | \ln \delta | ^ { \frac { p - 1 } { 2 } } ) . \end{align*}


\begin{align*} M _ 1 = & \bigcup _ { i = 1 } ^ { \infty } \Big ( [ \epsilon _ i \mathcal { H } _ { n _ i } , \mathcal { H } _ { n _ i } - L _ i ] \cup [ \mathcal { H } _ { n _ i } + L _ i , \mathcal { H } _ { n _ i + 1 } ] \cap \N \Big ) , \\ M _ 2 = & M \setminus M _ 1 . \end{align*}


\begin{align*} x l _ 1 + y k _ 1 = 1 \end{align*}


\begin{align*} \partial _ v \Gamma _ { u v } ^ v - \Gamma _ { v u } ^ u \Gamma _ { u v } ^ v + g _ { u v } = 0 . \end{align*}


\begin{align*} \dot y = A y '' + M y , y ( 0 , x ) = y _ 0 ( x ) , \end{align*}


\begin{align*} \lambda _ s = \sum _ { a = m - s + 1 } ^ { m } i _ a + \frac { L } { 2 } \medspace ( 1 \leq s \leq m ) . \end{align*}


\begin{align*} h ( G ) \le ( m - y ) \log ( m - y ) + y \log ( y ) + m \left ( \log ( m ) - \log ( y ) \right ) - ( m - y ) \left ( \log ( m - y ) - \log ( y ) \right ) = m \log m . \end{align*}


\begin{align*} \left \| \psi \left ( \frac { \mathbf y } { \tau } \right ) \right \| _ { L ^ 8 ( \mathcal K ) } = \tau ^ { \frac { 1 } { 4 } } \| \psi \left ( \mathbf x \right ) \| _ { L ^ 8 ( \mathcal K ) } = \mathcal O ( \tau ^ { - \frac { 5 } { 1 2 } } ) , \mbox { a s $ \tau \rightarrow \infty $ . } \end{align*}


\begin{align*} F _ k ( x ) = \max _ { i = 1 , \ldots , 2 ^ k } x _ i . \end{align*}


\begin{align*} L _ x = \{ l _ x \coloneqq l - l ( x ) : l \in L \} . \end{align*}


\begin{align*} \partial \Omega _ \varepsilon = \{ x + \varepsilon \rho ( x ) \mathbf { n } _ x \mid x \in \partial \Omega \} , \end{align*}


\begin{align*} \Sigma ^ 2 = \left ( \frac { 1 } { 2 L _ { x } } + \frac { \omega } { L _ { x y } } \right ) \frac { \sigma _ { F , r } ^ 2 } { n } + \left ( \frac { 1 } { 2 L _ { y } } + \frac { 1 } { L _ { x y } \omega } \right ) \frac { \sigma _ { G , r } ^ 2 } { n } . \end{align*}


\begin{align*} c _ { k + 1 } - c _ { k } = ( c _ { k + 1 } - c _ { k } ) \left ( \sum _ { i \in \Omega _ t ^ o } a _ i - \sum _ { i \in \Omega _ t ^ e } a _ i \right ) . \end{align*}


\begin{align*} \begin{cases} \gamma & \leq { ( \alpha - 1 ) } \min \{ 1 , { 1 } / { L _ 1 } \} \left [ { \alpha B ^ 2 \left ( C _ 0 + 1 \right ) } \right ] ^ { - 1 } ; \\ \gamma & \leq B ^ { - 2 } \Big [ \alpha ^ 2 + ( e - 1 ) \Big ( L _ { 0 } + L _ 1 C _ 0 \Big ) \Big ] ^ { - 1 } . \end{cases} \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n p _ { i } = 1 _ { Q } = 1 _ { M } p _ i p _ j = 0 \forall i \neq j \end{align*}


\begin{align*} \nabla _ X \psi _ 1 = \frac { 1 } { 2 } X _ 1 \cdot \psi _ 1 - \frac { 1 } { 2 } S ( X ) \cdot N \cdot \psi _ 1 \end{align*}


\begin{align*} I _ \varepsilon ( t ( u ) u ) = \max \{ I _ \varepsilon ( t u ) , t > 0 \} . \end{align*}


\begin{align*} \nu ^ * A = \mu ^ * h ^ * D + E . \end{align*}


\begin{align*} \bigcup _ { y = 1 } ^ z I _ y = \{ 1 , \dots , n \} \end{align*}


\begin{align*} P ^ g ( x _ 1 , \ldots , x _ n ) = P ( g x _ 1 , \ldots , g x _ n ) . \end{align*}


\begin{align*} R = - E _ 0 . \end{align*}


\begin{align*} \Phi _ n ( x , y ) = \prod ^ n _ { \substack { j = 1 \\ ( j , n ) = 1 } } ( x - \zeta ^ j _ n y ) . \end{align*}


\begin{align*} r _ { \delta , j } ( z ) : = \bar { \delta } _ + ( z ) \bar { \delta } _ - ( z ) r _ { j } ( z ) , \ \ j = 1 , 2 . \end{align*}


\begin{align*} \begin{aligned} \beta ( \arccos \tfrac { | v _ 1 - v | } { \sqrt { | v _ 1 - v | ^ 2 + | y | ^ 2 } } ) \leq \beta _ 0 | v _ 1 - v | \big ( | v _ 1 - v | ^ 2 + | y | ^ 2 \big ) ^ { - \frac { 1 } { 2 } } \ , . \end{aligned} \end{align*}


\begin{align*} B ( x , x ) = 0 , \lim _ { t \to \infty } \norm { e ^ { t L ^ \perp _ { x } } } = \infty . \end{align*}


\begin{align*} \begin{aligned} \beta ( \theta ) \sin \theta = \beta ( \tfrac { \pi } { 2 } - \theta ) \sin ( \tfrac { \pi } { 2 } - \theta ) \ , , \ \beta ( \theta ) \sin \theta = \beta ( \pi - \theta ) \sin ( \pi - \theta ) \ , , \end{aligned} \end{align*}


\begin{align*} K ^ L ( x ' , t ) : = P _ t ^ L ( x ' ) = t ^ { 1 - n } P ^ L ( x ' / t ) , \forall ( x ' , t ) \in \R ^ n _ + , \end{align*}


\begin{align*} \mu ( [ \gamma , u ] ) = n + C Z ( [ \gamma , u ] ) \end{align*}


\begin{align*} \psi _ \epsilon = \sup _ { \| x - x ' \| < \delta , \| y - y ' \| \le \delta } u ( x ' , y ' ) \end{align*}


\begin{align*} \mathcal { C } = \left \{ f ( x ) = \sum _ { k = 1 } ^ N c _ k \left \{ \frac { 1 } { k x } \right \} : \ , x \in ( 0 , 1 ) , \ ; \sum _ { k = 1 } ^ N \frac { c _ k } { k } = 0 \right \} . \end{align*}


\begin{align*} & \Tilde { x } = \sum \limits _ { i = 1 } ^ p \alpha _ i u _ i + \lambda x _ v + ( 1 - \lambda ) x _ w \in \mathcal { S } ^ n \\ & \| \sum \limits _ { i = 1 } ^ p \alpha _ i u _ i + \lambda x _ v + ( 1 - \lambda ) x _ w \| ^ 2 = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 \end{align*}


\begin{align*} \frac { 1 } { ( 1 + \varepsilon ) ^ { j - 1 } } \frac { 1 } { E ( t _ { i _ { j } } ) } ( i _ { j + 1 } - i _ j ) = \Omega \left ( \frac { 1 } { j ^ 2 } \right ) . \end{align*}


\begin{align*} N _ { c o m p } ^ i & = N ( r _ { i , f } + r _ { i , g } ) \\ & = 2 N + \mathcal { O } \left ( \frac { \max \{ \omega , \omega ^ { - 1 } \} } { n L _ { x y } ( 1 - \theta ) ^ 2 \epsilon } \left ( \left ( \frac { 1 } { L _ { x } } + \frac { \omega } { L _ { x y } } \right ) \sigma _ { f , i } ^ 2 + \left ( \frac { 1 } { L _ { y } } + \frac { 1 } { L _ { x y } \omega } \right ) \sigma _ { g , i } ^ 2 \right ) \log \left ( \frac { \Psi ^ 0 \nu } { \epsilon } \right ) \right ) . \end{align*}


\begin{gather*} g ( X , \nabla \Psi ) \equiv c o n s t \\ \Updownarrow \\ \div _ \Psi ( X ) = \div ( X ) - g ( X , \nabla \Psi ) \equiv c o n s t . \end{gather*}


\begin{align*} S _ \beta ( x , y , z ) = ( x + \alpha , y + x , z + x + \beta ) \end{align*}


\begin{align*} D _ n ( x , t _ X , t _ Y ) = \begin{cases} ( t _ X + t _ Y ) \Delta _ n ( x , \frac { t _ X } { t _ X + t _ Y } ) & \\ 0 & \end{cases} \end{align*}


\begin{align*} \left \{ e _ { m , q } ( z ) = \frac { z ^ m } { \sqrt { \Gamma ( m q + 1 ) } } \ ; ; \ ; m \geq 0 \right \} \end{align*}


\begin{align*} \sum _ { j = 1 } ^ d y _ { n , j } < \min _ { k = 1 , 2 , \ldots , d } \left ( x _ { n + 1 , k } + \sum _ { j \neq k } x _ { 0 , j } \right ) \forall n \geq N _ 0 . \end{align*}


\begin{align*} \begin{aligned} { \mathcal { G } } ( z , r , z ' , r ' ) & = \frac { \sqrt { r r ' } } { 2 \pi } F ( s ) , s = \frac { | z - z ' | ^ { 2 } + | r - r ' | ^ { 2 } } { r r ' } , \\ F ( s ) & = \int _ { 0 } ^ { \pi } \frac { \cos \theta } { \sqrt { 2 ( 1 - \cos \theta ) + s } } \dd \theta . \end{aligned} \end{align*}


\begin{align*} \Phi ( \mathcal { P } ( Z ) ) = \mathcal { P } ' ( \Phi ( Z ) ) \end{align*}


\begin{align*} T _ \ell ( x ) = \sqrt { \frac { \mu _ d } { n _ { \ell ; d } } } \sum _ { m = 1 } ^ { n _ { \ell ; d } } a _ { \ell , m } Y _ { \ell , m ; d } ( x ) , x \in \mathbb { S } ^ d , \end{align*}


\begin{align*} \Delta ^ 2 u = f \qquad & D , \\ [ 1 m m ] u = \frac { \partial u } { \partial \nu } = 0 & \partial D . \end{align*}


\begin{align*} u _ 2 ^ + = u _ 2 ^ - = u _ 1 ^ + , \ \partial u _ 2 ^ - = \partial u _ 2 ^ + + \eta _ 2 u _ 2 ^ + = \partial u _ 1 ^ + + \eta _ 2 u _ 1 ^ + \ \mathrm { o n } \ \Gamma _ h . \end{align*}


\begin{align*} & \mathcal { P } ^ + \left ( f ( z ) e ^ { - 2 i z x } \right ) = \int _ { 2 x } ^ { + \infty } \widehat { f } ( \xi ) e ^ { i z ( \xi - 2 x ) } \mathrm { d } \xi . \end{align*}


\begin{align*} \Bigg ( \sum _ { v \in \mathbb { N } ^ n } \frac { c ( v , q ) } { \mathit { g l } ( v , q ) } X ^ v \Bigg ) \ ! \circ \mathrm { E x p } \Bigg ( \sum _ { v \in \mathbb { N } ^ n \backslash \{ 0 \} } \ ! \frac { s ( v , q ) } { 1 - q } X ^ v \Bigg ) = 1 , \\ \end{align*}


\begin{align*} f ^ { \sigma } ( \mathrm { T } ( Y ^ \sigma , \widehat { \Z } ) ) = f ( \mathrm { T } ( Y , \widehat { \Z } ) ) . \end{align*}


\begin{align*} M i n i m i z e _ { x \in \mathbb { R } ^ n } \ \ J ( x ) : = 1 / 2 \| x - z \| ^ 2 \ \ s . t . \ x \in A \end{align*}


\begin{align*} \partial _ t \varrho + m \cdot \nabla J ( \varrho ) = 0 , \varrho ( 0 , \cdot ) = \varrho ^ { \rm i n i } , \end{align*}


\begin{align*} \begin{aligned} \Phi _ 1 ( \gamma ) : = \{ b _ 0 | \ , b _ 0 \in [ 0 , 1 - \gamma ] \textrm { i f } - 1 < \gamma \leq 1 , \textrm { a n d } b _ 0 \in ( - \gamma - 1 , 2 ) \textrm { i f } - 3 < \gamma \leq - 1 \} \ , . \end{aligned} \end{align*}


\begin{align*} y _ j ' \vert _ { y _ i \leftarrow \hat y _ i , i = 1 , \dots , n } = \hat y _ j ' , j \in I . \end{align*}


\begin{align*} \tilde { u } ^ { [ n ] } ( x ) = \sum _ i w ^ { [ n ] } _ i \sigma _ i ( x ) , \end{align*}


\begin{align*} b _ n - b _ { 1 , n } - b _ { 2 , n } = \nabla \times ( \phi _ n ( \chi _ { R _ n / 2 } - \chi _ { R _ 0 } ) \nabla \theta ) + G _ n ( \chi _ { R _ n / 2 } - \chi _ { R _ 0 } ) \nabla \theta , \end{align*}


\begin{align*} G _ { \iota \iota } = ( ( \nabla ^ { 2 } G \cdot e _ { \iota } ) \cdot e _ { \iota } ) w ^ { 2 } _ { \iota \iota } - ( \nabla G \cdot x ) w _ { \iota \iota } + ( \nabla G \cdot e _ { i } ) w _ { i \iota \iota } . \end{align*}


\begin{align*} \lim _ { k \to \infty } \deg _ p ( N _ k ' ) = \lim _ { k \to \infty } \deg _ p ( N _ k '' ) . \end{align*}


\begin{align*} \varphi ^ L _ { \sigma , s ' } ( D ) = \{ a , b \} \varphi ^ L _ { \sigma , s ' } ( E ) = \varphi ^ L _ { \sigma , s ' } ( C _ i ) \setminus \{ a \} = C \setminus \{ a \} . \end{align*}


\begin{align*} \mu _ t = j _ t - \frac { L } { 2 } \medspace ( 1 \leq t \leq n ) , \end{align*}


\begin{align*} & M i n i m i z e _ { x \in \mathbb { R } ^ n } \ , T r ( A _ J , X ) + 2 b _ J ^ T x \\ & s . t . \ \ T r ( A _ k , X ) + 2 b _ k ^ T x + c _ k \le 0 \ , k = 1 , . . . , m \\ & X = x x ^ T \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { m } \gamma _ { k } ( f _ { k } ( x ) + C _ { k } ) \ge 0 \ \ \forall \ x \in \mathbb { R } ^ { n } , \ \ C _ k \in \mathbb { R } _ + . \end{align*}


\begin{align*} v _ i = 1 \{ \theta _ i ^ \star = 0 \} - 1 \{ \theta _ i ^ \star = \pi \} , i = 1 , . . . , N . \end{align*}


\begin{align*} B _ q \circ T ( f ) = f . \end{align*}


\begin{align*} a _ { r , s } = \sum _ { \substack { A \subseteq \{ 1 , 2 , \ldots , r + s \} \\ | A | = r } } ( - 1 ) ^ { | \{ 1 \le b < a \le r + s \ , | \ , a \in A , b \not \in A \} | } \end{align*}


\begin{align*} \mu _ k ( G ) : = \frac { \deg ( G ) + k } { ( G ) } . \end{align*}


\begin{align*} \hat { L } _ 0 = \{ 0 \} \sqcup \bigsqcup _ { m = 1 } ^ \infty \bigsqcup _ { \gamma \in \Gamma / ( \Gamma \cap N ) } \{ \gamma \cdot ( 0 , 0 , 0 , m ) \} \sqcup \bigsqcup _ { m = 1 } ^ \infty \bigsqcup _ { n = 0 } ^ { 3 m - 1 } \bigsqcup _ { \gamma \in \Gamma } \{ \gamma \cdot ( 0 , 0 , 3 m , n ) \} . \end{align*}


\begin{align*} \delta ^ 2 \int _ { \Omega } \left ( K ( x ) \nabla u | \nabla \varphi \right ) = \int _ { \Omega } \hat { h } \varphi , \ \ \ \ \forall \varphi \in H ^ 1 _ 0 ( \Omega ) . \end{align*}


\begin{align*} \xi \left ( ( a ) \right ) = \left ( \frac { a } { | a | } \right ) ^ { k - 1 } \end{align*}


\begin{align*} u _ N = z + v _ N \end{align*}


\begin{align*} \mu \Big ( \bigcap _ { j = 0 } ^ { t _ n } T ^ { s _ { j , i } } A \Big ) \geq ( 1 - \epsilon _ n ) \mu ( A ) . \end{align*}


\begin{align*} \frac { i } { ( 2 \pi ) ^ { n } } \int _ { \mathbb { R } ^ { n - 1 } } \int _ { \mathcal { C } } e ^ { - t \tau } \operatorname { T r } \phi _ { - 2 } \ , d \tau \ , d \xi = a _ 1 ( x ) t ^ { 2 - n } , \end{align*}


\begin{align*} \begin{cases} - \Delta h _ { \hat { x } } ( x , y ) = 0 , \ \ & x \in T _ { \hat { x } } ( \Omega ) , \\ h _ { \hat { x } } ( x , y ) = \frac { 1 } { 2 \pi } \ln \frac { 1 } { | x - y | } , \ \ & x \in \partial T _ { \hat { x } } ( \Omega ) . \end{cases} \end{align*}


\begin{align*} \bigoplus _ { j = l } ^ { n - 1 } K _ j ' \subset ( \bigoplus _ { j = l } ^ { n - 1 } \sigma _ j \circ \rho _ j ' : \bigoplus _ { j = l } ^ { n - 1 } j _ { j * } E _ j \rightarrow \bigoplus _ { j = l } ^ { n - 1 } ( \sigma _ j ) ) \end{align*}


\begin{align*} \Delta v + \big ( \Delta ' ( h ) \big ) \bar u = 0 \mbox { i n } M \setminus \Omega . \end{align*}


\begin{align*} F ( W ) = \frac { 1 } { 2 \pi { \bf i } } \int _ { \gamma } \frac { F ( Z ) } { Z - W } d Z + \frac { 1 } { 2 \pi { \bf i } } \int _ { \Gamma } \frac { \dfrac { \partial F } { \partial Z ^ * } } { Z - W } d Z \wedge d Z ^ * , \end{align*}


\begin{align*} & \hphantom { { } = { } } \phi \circ ( \varphi \circ f ) ( x ) \\ * & = \widehat { \varphi } ( f ( o ) ) + D \phi ( p ) ^ i { } _ \alpha a ^ \alpha { } _ \beta x ^ \beta \\ * & \hphantom { { } = { } } + \frac 1 2 \left ( H \phi ( p ) ^ i { } _ { \alpha \beta } a ^ \alpha { } _ j a ^ \beta { } _ k + D \phi ( p ) ^ i { } _ \alpha b ^ \alpha { } _ { j k } \right ) x ^ j x ^ k + ( ) . \end{align*}


\begin{align*} \sum _ { l = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M \mathbb { E } \left [ y _ l ^ * y _ j | \hat { \mathbf { H } } \right ] = \sum _ { l = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M \left ( a _ c ^ 2 \hat { \phi } ^ { \left ( l , c \right ) ^ * } \hat { \phi } ^ { \left ( j , c \right ) } + \sum _ { q = 1 } ^ M a _ q ^ 2 \hat { \phi } ^ { \left ( l , q \right ) ^ * } \hat { \phi } ^ { \left ( j , q \right ) } \right ) \end{align*}


\begin{align*} Q _ { j , - } ( x ; k ) = \mathcal { P } ^ - \left ( Q _ { j , - } ( x ; k ) J + D _ j \right ) ( z ) , z \in \mathbb { R } , \end{align*}


\begin{align*} c _ k : = \frac { N _ k ( A ) } { r ( A ) ^ k } \ \end{align*}


\begin{align*} F ( x ) = \max _ { i = 1 , \ldots , 2 ^ k } x _ i . \end{align*}


\begin{align*} \Gamma _ { r } = \mathbb { Z } ^ { d } _ { * } \cap \bigg \{ \mathbf { k } : \sum _ { j } \langle j \rangle ^ { \eta } { \bf k } _ { j } w _ { j } \geq r | { \bf k } | _ { \eta } , \sum _ { j } w _ { j } = 1 , w _ j > 0 \bigg \} , \end{align*}


\begin{align*} d = 2 n = 4 l + 2 \equiv 2 \mod 3 \end{align*}


\begin{align*} G _ { \ell ; d } ( t ) ^ r = \sum _ { j = 0 } ^ { r \ell } \gamma _ { j , \ell ; r } G _ { j ; d } ( t ) \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 3 } } | b | ^ { 2 } \dd x = 2 \pi \int _ { \mathbb { R } ^ { 2 } _ { + } } \left ( | \nabla \phi | ^ { 2 } + | G | ^ { 2 } \right ) \frac { 1 } { r } \dd z \dd r . \end{align*}


\begin{align*} \frac { 2 } { m ( m - 1 ) } \sum _ { 1 \le i < j \le m } | P _ { u _ i ^ \bot } Z | | P _ { u _ j ^ \bot } Z | \le \left ( \frac { 1 } { m } \sum _ { i = 1 } ^ m | P _ { u _ i ^ \bot } Z | \right ) ^ 2 . \end{align*}


\begin{align*} B _ { a } : = \begin{pmatrix} 1 \\ & a \\ & & \ddots \\ & & & 1 \\ & & & & a \end{pmatrix} \end{align*}


\begin{align*} \varphi ( t ) = 1 + A ( t ) - B ( t ) \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } J _ { 1 } ( \overline { u } _ { k } ) = \widetilde { E } _ { 1 } , \ \ \ \lim _ { k \rightarrow \infty } J _ { \omega } \overline { v } _ { k } = \widetilde { E } _ { \omega } . \end{align*}


\begin{align*} 1 = \langle \varpi \otimes \varpi ( \cdot ) \xi ' _ i , \xi ' _ i \rangle i . \end{align*}


\begin{align*} \lim _ { j \to + \infty } \int _ { \{ \Psi < - t _ 0 \} } | f _ j | ^ 2 _ h c ( - \Psi ) = 0 , \end{align*}


\begin{align*} \delta _ 1 = 2 \big [ \nu F ( \nu ) - J ( q ) \big ] \quad \ , \ , \ , \quad \ , \ , \ , \delta _ 2 = \nu - 2 \ , J ( m ) , \end{align*}


\begin{align*} a _ { E } ( n ) = \prod _ { p \leq \infty } \beta _ { p } ( n ) \end{align*}


\begin{align*} ( a b ) ^ i { } _ { j k l } & = a ^ i { } _ { \alpha \beta \gamma } b ^ \alpha { } _ j b ^ \beta { } _ k b ^ \gamma { } _ l + a ^ i { } _ { \alpha \beta } b ^ \alpha { } _ { j l } b ^ \gamma { } _ k + a ^ i { } _ { \alpha \beta } b ^ \alpha { } _ j b ^ \gamma { } _ { k l } + a ^ i { } _ { \alpha \beta } b ^ \beta { } _ l b ^ \gamma { } _ { j k } + a ^ i { } _ \alpha b ^ \alpha { } _ { j k l } \end{align*}


\begin{align*} \mathcal { W } ( f V W \Delta ^ { - n } ) = \mathcal { G } ^ { \Delta } ( f V , W ) = \mathcal { G } ^ { \Delta } ( V , f W ) \end{align*}


\begin{align*} m : = b _ { n - 1 } \cdots b _ { \alpha + 1 } b _ \alpha ^ { k + \alpha - n } . \end{align*}


\begin{gather*} \sum _ { m = 0 } ^ { n } ( - 1 ) ^ { n - m } \binom { n } { m } 2 ^ { - m } \sum _ { j = 0 } ^ { m } \binom { m } { j } \left ( \beta \right ) ^ { \left ( n - j \right ) } \left ( \beta + m - j \right ) ^ { \left ( j \right ) } \\ = \left \{ \begin{array} { c c c } 0 & & n \\ \frac { n ! } { 2 ^ { n / 2 } ( n / 2 ) ! } \left ( \beta \right ) ^ { \left ( n / 2 \right ) } & & n \end{array} \right . . \end{gather*}


\begin{align*} x _ 2 \partial _ { x _ 1 } v _ 3 - x _ 1 \partial _ { x _ 2 } v _ 3 + k \partial _ { x _ 3 } v _ 3 = 0 . \end{align*}


\begin{align*} \partial _ t ^ k f & = \partial _ t ( \partial _ t ^ { k - 1 } f ) = - \mathcal L ( \partial _ t ^ { k - 1 } f ) + \partial _ t ^ { k - 1 } \Gamma ( f , f ) \\ & = - \mathcal L ( \partial _ t ^ { k - 1 } f ) + \sum _ { 0 \le j \le k - 1 } C ^ j _ { k - 1 } \Gamma ( \partial ^ j _ t f , \partial ^ { k - 1 - j } _ t f ) . \end{align*}


\begin{align*} \phi ^ { \zeta \theta } ( e _ { v w } ) : = ( \phi ^ \zeta ) ^ \theta ( e _ { v w } ) & = \tau ^ \zeta ( v , e ) \theta ( v ) + \phi ^ \zeta ( e _ { v w } ) + \tau ^ \zeta ( w , e ) \theta ( w ) \\ & = \zeta ( v ) \tau ( v , e ) \theta ( v ) + \phi ( e _ { v w } ) + \zeta ( w ) \tau ( w , e ) \theta ( w ) \\ & = \tau ( v , e ) \theta ^ \zeta ( v ) + \phi ( e _ { v w } ) + \tau ( w , e ) \theta ^ \zeta ( w ) \\ & = \phi ^ { \theta ^ \zeta } ( e _ { v w } ) = \phi ^ { \theta ^ \zeta \zeta } ( e _ { v w } ) \end{align*}


\begin{align*} \psi _ { 2 n } = \begin{pmatrix} \psi _ { 2 } \\ & \psi _ { 2 } \\ & & \ddots \\ & & & \psi _ { 2 } \end{pmatrix} = \bigoplus _ { 1 } ^ { n } \psi _ { 2 } , \psi _ { 2 } = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \end{align*}


\begin{align*} \begin{aligned} \iota ( C ' _ w ) & = C ' _ w , \\ q ^ { \frac { \ell ( w ) } { 2 } } C ' _ w & = \sum _ { z \leq w } P _ { z , w } ( q ) T _ z , \end{aligned} \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \theta ^ j _ { \alpha \beta } ( x ' , t ; y ' ) d y ' = \int _ { \R ^ { n - 1 } } t \ , \partial _ j K ^ L _ { \alpha \beta } ( x ' - y ' , t ) d y ' = t \ , \partial _ j \int _ { \R ^ { n - 1 } } K ^ L _ { \alpha \beta } ( y ' , t ) d y ' = 0 . \end{align*}


\begin{align*} \xi [ X ] \stackrel { d e f } { = } \int _ X \xi ( x ) \ \mu ( d x ) \ - \end{align*}


\begin{align*} \theta _ { \Psi } ( r , \xi ) = \theta _ { \Psi } ( r ) \end{align*}


\begin{align*} u ^ k = \arg \min _ { w \in H ^ 1 ( \Omega ) } \bigg ( \frac { | \Omega | } { 2 \mathrm { N } _ I } \sum _ { n = 1 } ^ { \mathrm { N } _ I } ( w - u ^ { k - 1 } ) ^ 2 ( x _ n ^ I ) + \tau _ k \tilde { N } ( w ; w ) \bigg ) . \end{align*}


\begin{align*} h _ { s , t } ^ { m , n } = \frac { ( m t - n s ) ^ 2 - ( s - t ) ^ 2 } { 4 s t } . \end{align*}


\begin{align*} \mathcal { H } _ m ( A ) : = S _ m ( \lambda ( A ) ) , \end{align*}


\begin{align*} T - \mathcal { P } ^ + \left ( T R _ + \right ) - \mathcal { P } ^ - \left ( T R _ - \right ) = F , \end{align*}


\begin{align*} \gamma ( x _ n ^ D ; w ) \ge \tau ( x _ n ^ D ) : = 8 \frac { | \Gamma _ D | \mathrm { N } _ I \lambda _ d ^ 2 } { | \Omega | \mathrm { N } _ D \lambda _ 1 } \frac { | \nabla w ( x _ n ^ D ) | ^ 2 } { | \nabla w ( y _ n ) | ^ 2 } , \end{align*}


\begin{align*} \gamma _ k = \frac { a _ k ^ 2 \lvert \mathbf { \hat { h } } _ k ^ { \textrm { T } } \mathbf { p } _ k \rvert ^ 2 } { \sum \limits _ { \substack { i = 1 \\ i \neq k } } ^ K a _ i ^ 2 \lvert \mathbf { h } _ k \mathbf { p } _ i \rvert ^ 2 + \sigma _ w ^ 2 } . \end{align*}


\begin{align*} 2 \widetilde y _ n - \widetilde x _ 0 & = 2 e ^ { r y _ n } - e ^ { r x _ 0 } \\ & \leq e ^ { r ( y _ n + a ) } - e ^ { r x _ 0 } \\ & \leq e ^ { r x _ { n + 1 } } - e ^ { r x _ 0 } \\ & = \widetilde x _ { n + 1 } - e ^ { r x _ 0 } \\ & < \widetilde { x } _ { n + 1 } , \end{align*}


\begin{align*} \Vert M _ { \widetilde { m } \vert _ { \Lambda _ N \times \Lambda _ N \times \Lambda _ N } ( \ : \cdot \ : , 0 \ : \cdot \ : ) } : S _ { 1 } ^ { N + 1 } \rightarrow S _ { 1 } ^ { N + 1 } \Vert \leq \Vert M _ { \widetilde { m } \vert _ { \Lambda _ N \times \Lambda _ N \times \Lambda _ N } } : S _ { p _ 1 } ^ { N + 1 } \times S _ { p _ 2 } ^ { N + 1 } \rightarrow S _ { p } ^ { N + 1 } \Vert . \end{align*}


\begin{align*} \int _ { \Omega \cap B _ { \eta } } n c ^ { 2 } \psi ^ { 2 } + \int _ { \Omega \cap B _ { \eta } } | \nabla c | ^ { 2 } \psi ^ { 2 } = \int _ { \partial \Omega } \nabla c \cdot \nu c \psi ^ { 2 } - 2 \int _ { \Omega \cap B _ { \eta } } \nabla c \cdot \nabla \psi c \psi . \end{align*}


\begin{align*} P _ i ^ j = P _ 0 ^ 0 + j ( q ^ 2 + 1 ) + i . \end{align*}


\begin{align*} \log \int _ { \R ^ n } e ^ { f } \ , d \gamma _ t - \sup _ { y \in \R ^ n } \left ( \int _ { \R ^ n } f \ , d \gamma _ { y , t } - H ( \gamma _ { y , t } \ , | \ , \gamma _ t ) \right ) = \inf _ { y \in \R ^ n } H ( \gamma _ { y , t } \ , | \ , P ) . \end{align*}


\begin{align*} \begin{cases} t _ { i , e x } ( x _ 0 , t _ 0 ) = \max \left ( 0 , t _ 0 - \dfrac { x _ 0 - R } { \lambda _ i } \right ) \forall \ : i \in [ 1 , p ] , \\ t _ { i , e x } ( x _ 0 , t _ 0 ) = \max \left ( 0 , t _ 0 - \dfrac { x _ 0 + R } { \lambda _ i } \right ) \forall \ : i \in [ p + 1 , n ] . \end{cases} \end{align*}


\begin{align*} h _ { 0 j ; k _ 1 k _ 2 } = 0 , h _ { i j ; k _ 1 a } = 0 , h _ { i j ; a k _ 1 } = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \sum _ { r _ { Q } ( n ) = 0 } n \ll p ^ { 1 + \epsilon } \frac { m } { \min Q ^ { * } } \ll \frac { p ^ { 3 + \epsilon } } { \left ( \min Q ^ { * } \right ) ^ { 2 } } . \end{align*}


\begin{align*} \Delta u ^ i ( \mathbf x ) + k ^ 2 u ^ i ( \mathbf x ) = 0 , \mathbf x \in \mathbb R ^ n , \end{align*}


\begin{align*} x = \sum _ { m = 0 } ^ \infty \frac { \psi ( \eta _ m ) } { \gamma ^ { m + 1 } } . \end{align*}


\begin{align*} T ( \lambda u ) = u . \end{align*}


\begin{align*} \begin{cases} \Delta _ { \Psi } v = f ' ( u ) \ , v & \O \\ v = 0 & \partial \O . \end{cases} \end{align*}


\begin{align*} T _ { ( a , 1 ) } \left ( \frac { h _ k } { ( 1 - z ) ^ { a - 1 } } \right ) = \Psi ( r _ k ) . \end{align*}


\begin{align*} \widehat { f } ( \lambda , \omega ) = ( 2 \pi ) ^ { - n / 2 } \int _ { \R ^ n } f ( x ) e ^ { - i \lambda x \cdot \omega } d x , \ , \ , \ , \lambda \in \R , \omega \in S ^ { n - 1 } . \end{align*}


\begin{align*} L = a _ 0 x + a _ 1 x ^ q + a _ 2 x ^ { q ^ 2 } + \cdots + a _ n x ^ { q ^ n } \in F [ x ] . \end{align*}


\begin{align*} \vec { r } = ( r _ 1 , r _ 2 , \ldots ) . \end{align*}


\begin{align*} \tilde { K _ i } : = K _ i \cap { K _ i } ' , \end{align*}


\begin{align*} X ^ \dagger = N ^ \dagger \oplus X _ 0 ^ \dagger Z ^ \dagger = R ^ \dagger \oplus Z _ 0 ^ \dagger , \end{align*}


\begin{align*} { \mathbf { A } } = \mathbf { A n t i } ( \vartheta _ 1 , \vartheta _ 2 , \vartheta _ 3 ) : = \begin{footnotesize} \begin{pmatrix} 0 & - \vartheta _ 3 & \vartheta _ 2 \\ \vartheta _ 3 & 0 & - \vartheta _ 1 \\ - \vartheta _ 2 & \vartheta _ 1 & 0 \end{pmatrix} \end{footnotesize} \in \mathfrak { s o } ( 3 ) , \end{align*}


\begin{align*} \epsilon _ n = \max _ { | \eta - \lambda | \le \delta } \max _ { v \in G ( \lambda ) } \| F _ n ( \eta ) p _ n v - q _ n F ( \eta ) v \| _ { X _ n } \end{align*}


\begin{align*} \begin{cases} \alpha a + ( \alpha - 1 ) b + ( 3 \alpha - 2 ) c - \frac { p \alpha - 1 } { p } d = 0 ; \\ ( 1 - 2 \alpha \mu ) a + ( 1 - 2 \mu ( \alpha - 1 ) ) b + 3 \big ( 1 - \mu ( 6 \alpha - 4 ) \big ) c - \big ( 1 - \mu ( 2 p \alpha - 2 ) \big ) d = 0 ; \\ \big ( 2 \mu ( \alpha - 1 ) - 1 \big ) b + 2 \big ( \mu ( 6 \alpha - 4 ) - 1 \big ) c - \frac { \mu ( 2 p \alpha - 2 ) - 1 } { p } d = 0 . \end{cases} \end{align*}


\begin{align*} \phi ( t ) = \frac { d } { d t } ( k \ , * \ , f ) ( t ) \ , = \ , \frac { d } { d t } \ , \int _ 0 ^ t k ( t - \tau ) f ( \tau ) \ , d \tau , \ t > 0 . \end{align*}


\begin{align*} | Z _ 3 | & = \left | \frac { 1 } { \log x } \sum _ { \rho } \frac { x ^ { \rho - s } - x ^ { 2 ( \rho - s ) } } { ( s - \rho ) ^ 2 } \right | \\ & \leq \frac { x ^ { \frac { 1 } { 2 } - \sigma } + x ^ { 1 - 2 \sigma } } { \log x } \sum _ { \gamma } \frac { 1 } { ( \sigma - \beta ) ^ 2 + ( t - \gamma ) ^ 2 } . \end{align*}


\begin{align*} c _ { q ; d } = \begin{cases} \Big ( 2 ^ { \frac d 2 - 1 } \Big ( \frac d 2 - 1 \Big ) ! \Big ) ^ q \int _ 0 ^ \infty J _ { \frac d 2 - 1 } ( u ) ^ q u ^ { - q ( \frac d 2 - 1 ) + d - 1 } d u & \mbox { i f } q \geq 3 , \cr \frac { ( d - 1 ) ! \mu _ d } { 4 \mu _ { d - 1 } } & \mbox { i f } q = 2 , \\ \end{cases} \end{align*}


\begin{align*} f _ k ( d ) = \lim _ { n \rightarrow \infty } \frac { \vartheta ( n , d , k ) } { \log n } , \end{align*}


\begin{align*} \sum _ { \tau \in D ( n ) } \prod _ { j = 1 } ^ { n } \dfrac { 1 } { 1 - \zeta ^ { j - \tau ( j ) } } = \dfrac { \left ( \left ( n - 1 \right ) ! ! \right ) ^ { 2 } } { 2 ^ n } = \dfrac { n ! } { 4 ^ n } \binom { n } { n / 2 } , \end{align*}


\begin{align*} D \phi ( F ) = \sum _ { r = 1 } ^ k \frac { \partial \phi } { \partial x _ r } ( F ) D F _ r . \end{align*}


\begin{align*} u \cdot \nabla \omega = 0 , & \mathrm { i n } \ , \ , \Pi , \\ u \rightarrow u _ { \infty } & \mathrm { a s } \ , \ , | x | \rightarrow \infty . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { M } \mathbb { E } \left [ y _ i ^ * y _ i \right ] = \sum _ { l = 1 } ^ M a _ c ^ 2 \lvert \mathbf { h } _ { l , * } \mathbf { p } _ c \rvert ^ 2 + \sum _ { i = 1 } ^ M \sum _ { j = 1 } ^ M a _ j ^ 2 \lvert \mathbf { h } _ { i , * } \mathbf { p } _ j \rvert ^ 2 + M \sigma _ n ^ 2 , \end{align*}


\begin{gather*} \left \{ \begin{array} { c c c } 0 & & n \\ \frac { n ! } { ( n / 2 ) ! } \left ( \beta \right ) ^ { \left ( n / 2 \right ) } ( 1 - \rho ) ^ { n / 2 } & & n \end{array} \right . = E ( - X + Y ) ^ { n } \allowbreak \\ = \sum _ { m = 0 } ^ { n } ( - 1 ) ^ { m } \binom { n } { m } E X ^ { m } Y ^ { n - m } . \allowbreak \end{gather*}


\begin{align*} b = 0 \mbox { o r } c = 0 ; b \neq 0 \mbox { a n d } c \neq 0 . \end{align*}


\begin{align*} q _ 1 ( z ) : = \mathcal { C } \left ( Q _ { j , - } J \right ) ( z ) q _ 2 ( z ) : = \mathcal { C } \left ( Q _ { j , - } J + D _ j \right ) ^ H ( z ) \end{align*}


\begin{align*} W ( a u + b v , x ) = a W ( u , x ) + b W ( v , x ) . \end{align*}


\begin{align*} & \frac { ( n - 2 ) ( 1 0 - n ) } { 4 } \int _ { B _ { \rho } } r ^ { 2 - n } u _ r ^ 2 \d x \\ & \leq C \int _ { B _ { 2 \rho } \setminus B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon \int _ { B _ { 4 \rho } } r ^ { 3 - n } | \nabla u | ^ 2 \d x , \rho \leq 1 / 4 \varepsilon \leq \varepsilon _ 0 , \end{align*}


\begin{align*} N _ { r s } \coloneqq \begin{cases} M _ { r s } & \\ N _ { k s } & . \end{cases} \end{align*}


\begin{align*} G _ { \alpha \beta } ^ T H ^ { \beta } _ i \overline { G _ { \alpha \beta } } & = H ^ { \alpha } _ i \\ & = C ^ { \alpha } _ { i } H ^ { \alpha } _ { 1 } ( \overline { C ^ { \alpha } _ { i } } ^ { T } ) = C ^ { \alpha } _ { i } G _ { \alpha \beta } ^ T H ^ { \beta } _ 1 \overline { G _ { \alpha \beta } } ( \overline { C ^ { \alpha } _ { i } } ^ { T } ) . \end{align*}


\begin{align*} & \frac { \varpi _ n } { \varpi _ { n + 1 } } x ( \varpi _ { n + 1 } ) = \frac { \varpi _ n } { \varpi _ { n + 1 } } \varphi _ M ^ { - n - 1 } ( \frac \varpi \pi x ' ) = \varphi _ M ^ { - n - 1 } ( \frac { \varpi ^ p } { \varpi } \frac \varpi \pi x ' ) = \varphi _ M ^ { - n } ( \varphi _ M ^ { - 1 } ( \frac { \varpi ^ p } \pi x ' ) ) = \\ & \qquad = \varphi _ M ^ { - n } ( \frac \varpi \pi x ' ) = x ( \varpi _ n ) . \end{align*}


\begin{align*} \dim A _ i = 4 i + 1 - m _ i - n _ i . \end{align*}


\begin{align*} x _ { i j } : = x _ { i i - 1 } x _ { i - 1 i - 2 } \cdots x _ { j + 1 j } \end{align*}


\begin{align*} \sigma _ { k - 1 } ( b ) u _ { n n } ( 0 ) - \sum _ { \alpha \leq n - 1 } u _ { n \alpha } ^ 2 ( 0 ) \sigma _ { k - 2 ; \alpha } ( b ) + \sigma _ k ( b ) = f ( 0 ) . \end{align*}


\begin{align*} M = \left ( \begin{array} { c c } - \varepsilon _ 1 \mu & - \varepsilon _ 1 \mu \\ \varepsilon _ 3 \mu & \varepsilon _ { 3 } \mu \end{array} \right ) \end{align*}


\begin{align*} \langle v , w \rangle : = \sum _ { i \in Q _ 0 } v _ i w _ i - \sum _ { h \in Q _ 1 } v _ { h ' } w _ { h '' } \ , \ , v , w \in \mathbb { Z } ^ n . \end{align*}


\begin{align*} 1 = - \sum _ { \emptyset \neq I \subset \{ 1 , \dots , m \} } ( - 1 ) ^ { | I | } , \end{align*}


\begin{align*} \begin{aligned} | x & ( t ) - z _ 1 ( t ) | \le \ , | x ( 0 ) - z _ 1 ( 0 ) | + t \gamma ^ 2 \bar { k } _ 3 ( L _ r , \delta ) \\ & + \gamma \int _ 0 ^ t L _ r | x ( \tau ) - z _ 1 ( \tau ) | + | \bar { y } ( \tau ) - z _ 2 ( \tau ) | d \tau \end{aligned} \end{align*}


\begin{align*} X _ \ell : = \int _ { \mathbb S ^ d } \varphi ( T _ \ell ( x ) ) d x , \end{align*}


\begin{align*} \beta = ( \beta _ 1 , \ldots , \beta _ k ) \textrm { w i t h } \sum _ i \beta _ i = m . \end{align*}


\begin{align*} S _ i = { { w } _ i } \rho c _ v \frac { \Delta t } { 2 } \partial _ t ^ 2 T , \end{align*}


\begin{align*} A \circ \widetilde \Phi \circ \widetilde { \mathcal { P } } \circ \widetilde \Phi ^ { - 1 } \circ A ^ { - 1 } = \mathcal { P } ' \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { r _ n ^ 2 } { h _ n } = 0 . \end{align*}


\begin{align*} ( N _ j ( \varphi ) - P ) & = S ( I - T _ 1 + P ) ( N _ j ( \varphi ) - P ) \\ & = S ( I - C _ \varphi + P ) ( M _ j ( \varphi ) - P ) \\ & = \dfrac { 1 } { j } S ( C _ \varphi - C _ { \varphi _ { j + 1 } } ) \rightarrow 0 , \end{align*}


\begin{align*} d \in \sigma _ l ( A ) , \ , v = ( x , 1 ) ^ T \in V _ l ( d ) , \forall x \in S _ L ( c , 0 ) \cap S _ L ( d - a , 0 ) ; \end{align*}


\begin{align*} \frac { \partial \textbf { \textit { W } } } { \partial t } - ( B - Q ) \textbf { \textit { W } } = - \textbf { \textit { Y } } . \end{align*}


\begin{align*} \int _ { \mathfrak X } | \omega | & = { \pi _ s } _ ! \int _ { \mathfrak U } | \omega | = { \pi _ s } _ ! \Big ( { \psi _ s } _ ! \int _ { \mathfrak Y } | \omega | \Big ) = { \phi _ s } _ ! \int _ { \mathfrak Y } | \omega | , \end{align*}


\begin{align*} P ( X _ a = c X _ b ) = \int \int & P ( X _ a = c t | \sum _ { i = 1 } ^ { a - 1 } X _ i = s X _ b = t ) \\ & \quad \times ( \sum _ { i = 1 } ^ { a - 1 } X _ i X _ b s t ) d s d t . \end{align*}


\begin{align*} P ( x _ 1 , \ldots , x _ n ) = \lambda _ 0 x _ 1 ^ s + \lambda _ 1 x _ 1 ^ { s - 1 } x _ 2 + \cdots + \lambda _ s x _ 2 ^ s , \end{align*}


\begin{align*} \mathcal { L } = \frac { 1 } { 2 } \sum _ { i , j = 1 } ^ 3 \Lambda _ { i j } \partial _ { x _ i x _ j } + x _ 1 x _ 3 \partial _ { x _ 1 } - x _ 2 x _ 3 \partial _ { x _ 2 } + ( x _ 2 ^ 2 - x _ 1 ^ 2 ) \partial _ { x _ 3 } - x _ 2 \sum _ { j = 1 } ^ 3 a _ j \partial _ { x _ j } . \end{align*}


\begin{align*} P ( \alpha , \alpha ^ q ) = \alpha ^ { q ( q + 1 ) } + a \alpha ^ { 2 q } + b \alpha ^ { q + 1 } = \alpha ^ q ( \alpha ^ { q ^ 2 } + a \alpha ^ { q } + b \alpha ) = 0 . \end{align*}


\begin{align*} - L u = f ( u ) \Omega , \end{align*}


\begin{align*} & { M i n i m i z e \ , } _ { x \in \mathbb { R } ^ n } \ \ J ( x ) : = a _ J x ^ T x + 2 b _ J ^ T x + c _ J \\ & s . t . \ \ f _ k ( x ) : = a _ k x ^ T x + 2 b _ k ^ T x + c _ k \le 0 , \ \ k = 1 , . . . , m \end{align*}


\begin{align*} \lim _ { j \to + \infty } \int _ { \{ \Psi < - t _ 0 \} } | f _ j | ^ 2 _ h c ( - \Psi ) = 0 , \end{align*}


\begin{align*} \sigma _ { n - 1 } ( \kappa ) = \sigma , \partial \Sigma = \Gamma . \end{align*}


\begin{align*} \left | v _ { i , n } \right | ^ { 2 } \prod _ { k = 1 ; k \neq i } ^ { n } \left ( - \lambda _ { k } ( A ) \right ) = \prod _ { k = 1 } ^ { n - 1 } \left ( - \lambda _ { k } \left ( M _ { n } \right ) \right ) . \end{align*}


\begin{align*} \begin{aligned} ( \Delta \varphi _ { p , N } ^ { + } ) ^ m \wedge \beta _ n ^ { n - m } = 2 ^ { - m } ( \Delta u _ p + \Delta u _ { p + N } ) ^ m \wedge \beta _ n ^ m = 2 ^ { - m } \sum \limits _ { k = 0 } ^ m \binom m k ( \Delta u _ p ) ^ k \wedge ( \Delta u _ { p + N } ) ^ { m - k } \wedge \beta _ n ^ m . \end{aligned} \end{align*}


\begin{align*} e _ { i } = \frac { \partial } { \partial x ^ { i } } - \frac { 1 } { 6 } R _ { i j k \ell } \ , x ^ { j } x ^ { k } \frac { \partial } { \partial x ^ { \ell } } + o ( \mathbf { x ^ { 2 } ) } \end{align*}


\begin{align*} K ^ { ( 1 ) } ( S _ t , I _ t ) & = \frac { w } { 2 \theta ( \partial _ i V ( S _ t , I _ t ) - \partial _ s V ( S _ t , I _ t ) ) } , \\ K ^ { ( 2 ) } ( S _ t , I _ t ) & = \frac { w } { 2 \theta ( 1 - \theta \bar L ) ( \partial _ i V ( S _ t , I _ t ) - \partial _ s V ( S _ t , I _ t ) ) } . \end{align*}


\begin{align*} \sigma ( t ) = \max ( t , 0 ) , t \in \mathbb { R } . \end{align*}


\begin{align*} \tilde { Q } _ { i j } ( H ) = \partial ^ 2 _ { i j } H - \tilde { \Gamma } ^ i _ { j i } \partial _ i H - \tilde { \Gamma } ^ j _ { i j } \partial _ j H = 0 , \quad \forall i < j , \end{align*}


\begin{align*} \int _ { \C ^ n } | g ( z ) | ^ 2 d z = ( 2 \pi ) ^ { - 2 n } \ , | \lambda | ^ { 2 n } \ , \sum _ { k = 0 } ^ \infty \int _ { \C ^ n } | g \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ( z ) | ^ 2 d z . \end{align*}


\begin{align*} | Z _ t | ^ 2 = | Z _ 0 | ^ 2 \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } \mu \xi ( \mu , t ) \ d \mu = \int _ { 0 } ^ { \infty } \mu \xi ^ { \mathrm { i n } } ( \mu ) \ d \mu = \varrho _ { 0 } , \forall ~ t \geq 0 . \end{align*}


\begin{align*} f _ s ( Z ^ n _ s ) = C ^ { ( n ) } _ s \end{align*}


\begin{align*} \frac { d } { d t } \theta _ i = \omega _ i + \frac { K } { N } \sum _ { j = 1 } ^ N \sin ( \theta _ j - \theta _ i ) , i = 1 , \ldots , N . \end{align*}


\begin{align*} P _ \delta ( z ) = \frac { \pi \delta ^ 2 } { \ln \frac { R } { \varepsilon } } q ^ 2 \sqrt { d e t ( K _ H ) } ( z ) + N _ \delta ( z ) , \end{align*}


\begin{align*} \mu \Lambda ( \mu , \nu ) \xi ( \mu , s ) \xi ( \nu , s ) \chi _ { ( 0 , \mu ) } ( \nu ) = 0 a . e . ~ \mbox { i n } ( 0 , t _ { 0 } ) \times ( R _ { 0 } , \infty ) \times ( 0 , \infty ) . \end{align*} % \end{align*}


\begin{align*} \varrho ( x , z ) = ( z + 1 ) f ( x ) - z b ( x ) \end{align*}


\begin{align*} \mu ^ * + \varkappa & = \mu ^ { \rm N } + \varkappa ^ { \rm N } , \mu ^ * = \mu ^ { \rm N } - \varkappa ^ { \rm N } , \lambda = \lambda ^ { \rm N } , \\ \beta & = \gamma ^ { \rm N } - \beta ^ { \rm N } , \ \ \gamma = \gamma ^ { \rm N } + \beta ^ { \rm N } , \qquad \alpha = \alpha ^ { \rm N } \ , , \end{align*}


\begin{align*} k _ 2 ( t ) = h _ { 1 - \alpha - \gamma } ( t ) \cdot \ , f _ 3 ( t ) , \ f _ 3 ( t ) = \sum _ { k = 0 } ^ { + \infty } \ , c _ k t ^ k \end{align*}


\begin{align*} W ' ( r ) = \frac { \xi ' } { r } f + \eta ' + \zeta ' \frac { r } { f } + \frac { \sigma ' } { f ^ 2 } , \xi ' \le 0 , \sigma ' > 0 , \end{align*}


\begin{align*} \bar \pi \bar p _ t = \bar \pi , ~ ~ ~ ~ \forall t > 0 . \end{align*}


\begin{align*} O _ { n , n } ( R ) \subseteq O _ { n + 1 , n + 1 } ( R ) : A \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & A \end{pmatrix} . \end{align*}


\begin{align*} \begin{aligned} \Delta u _ 1 \wedge \dots \wedge \Delta u _ k & \wedge \beta _ n ^ { n - m } ( \omega ) = \int u _ k \Delta u _ 1 \wedge \dots \wedge \Delta u _ { k - 1 } \wedge \beta _ n ^ { n - m } \wedge \Delta \omega \end{aligned} \end{align*}


\begin{align*} \varphi _ { * } : = - ( \varphi _ 0 + \ldots + \varphi _ p + 1 ) = 0 . \end{align*}


\begin{align*} i _ { \chi \theta } ( f ) ( I _ \bullet ) = \sum _ { J _ \bullet \in \partial _ { \chi \theta } ^ { - 1 } ( I _ \bullet ) } f ( J _ \bullet ) \end{align*}


\begin{align*} V _ { \lambda / \mu } : = L _ T , \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \theta ^ j _ { \alpha \beta } ( x ' , t ; y ' ) d y ' = \int _ { \R ^ { n - 1 } } t \ , \partial _ j K ^ L _ { \alpha \beta } ( x ' - y ' , t ) d y ' = t \ , \partial _ j \int _ { \R ^ { n - 1 } } K ^ L _ { \alpha \beta } ( y ' , t ) d y ' = 0 . \end{align*}


\begin{align*} \Big \| \sum \limits _ { k = 2 } ^ { n } \mu ( k ) ( I - S ) h _ k - ( 1 - z ) \Big \| _ { H ^ 2 ( \mathbb { D } ) } \longrightarrow 0 n \to \infty \end{align*}


\begin{align*} \begin{aligned} & k + \alpha _ 1 + i \beta _ 1 + \delta _ { 1 k } , i k + i \alpha _ 2 - \beta _ 2 + \delta _ { 2 k } , \\ - & k + \alpha _ 3 + i \beta _ 3 + \delta _ { 3 k } , - i k + i \alpha _ 4 - \beta _ 4 + \delta _ { 4 k } , \end{aligned} \end{align*}


\begin{align*} \langle u , v \rangle : = \ , \ , ^ { t } u \psi _ { 2 n } v \end{align*}


\begin{align*} \alpha = 1 + z \frac { \Phi ( z ) } { \phi ( z ) } \frac { 1 - \Phi ( z ) } { 1 - 2 \Phi ( z ) } , . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { n - 1 } } e ^ { - C | \xi | } \ , d \xi = \frac { \Gamma ( n - 1 ) \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) } { C ^ { n - 1 } } , C > 0 , \ n \geqslant 2 , \end{align*}


\begin{align*} \frac { F ( z ) L ( z ) } { \sinh ^ 2 \ ! \Big ( \frac { \sqrt { 5 } F _ j } { 2 } z \Big ) } = \frac { 4 } { \sqrt { 5 } } e ^ { L _ j z } \coth \ ! \Big ( \frac { \sqrt { 5 } F _ j } { 2 } z \Big ) . \end{align*}


\begin{align*} | F ^ { i j } ( \nabla _ \beta \varphi ) _ { i j } | \leq C \sqrt { b _ \alpha } \left ( \sum _ { i = 1 } ^ { \alpha } F ^ { i i } + \sum _ { i = \alpha + 1 } ^ { n - 1 } \frac { b _ i } { b _ \alpha } F ^ { i i } \right ) \mbox { i n } \omega _ \epsilon . \end{align*}


\begin{align*} \begin{cases} d X _ { t , 3 } = - X _ { t , 1 } X _ { t , 2 } d t \\ d X _ { t , j } = 0 & j \neq 3 \end{cases} \end{align*}


\begin{align*} & S ' ( h , v ) = 0 \mbox { i n a c o l l a r n e i g h b o r h o o d $ U $ o f $ \hat \Sigma $ i n } M \setminus \Omega \\ & \left \{ \begin{array} { l } h ^ \intercal = 0 \\ H ' ( h ) = 0 \end{array} \right . \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \rho _ L ( \theta ) : = \sum _ { n \in \Z } \rho ( L ( \theta + n ) ) L = L ( x ) \geq 1 , \end{align*}


\begin{align*} \phi _ { i s } ( \partial _ i ) = - \frac { \Gamma _ { i s } ^ s } { \varphi _ i } , \quad \forall s \neq i , \end{align*}


\begin{align*} \int e ^ { W ( y | x ) } d \nu _ { A } ( y ) = \varphi _ A ( x ) . \end{align*}


\begin{align*} 2 \bar \pi = \sum _ { v \in V _ 1 } \pi _ v . \end{align*}


\begin{align*} H _ N ( t ) = H ( f _ { N , t } ; \nu _ { N , t } ) : = \int _ { \Omega _ N } f _ { N , t } \log f _ { N , t } d \nu _ { N , t } . \end{align*}


\begin{align*} A = K \langle T _ 1 ^ { 1 / p ^ \infty } , \dots , T _ n ^ { 1 / p ^ \infty } \rangle , A ^ + = K ^ + + A ^ { \circ \circ } . \end{align*}


\begin{align*} 2 \widetilde y _ n - \widetilde x _ 0 & = 2 e ^ { r y _ n } - e ^ { r x _ 0 } \\ & \leq e ^ { r ( y _ n + a ) } - e ^ { r x _ 0 } \\ & \leq e ^ { r x _ { n + 1 } } - e ^ { r x _ 0 } \\ & = \widetilde x _ { n + 1 } - e ^ { r x _ 0 } \\ & < \widetilde { x } _ { n + 1 } , \end{align*}


\begin{align*} \prod _ { 1 \leq j \leq N , n _ j \mid n _ 1 } \vert \eta _ { j , ( 1 ) } \vert _ F = \frac { 1 } { p ^ { D } } \end{align*}


\begin{align*} V _ 1 \oplus V _ 2 = \mathrm { s p a n } ( \cos \ell x _ 1 , \sin \ell x _ 1 ) \oplus \mathrm { s p a n } ( \cos k x _ 2 , \sin k x _ 2 ) , \end{align*}


\begin{align*} \mathbf { c } ^ { ( 1 ) } = ( \underbrace { 1 , 1 , \cdots , 1 } _ { 2 k } , \ast , \ast , \cdots , \ast ) . \end{align*}


\begin{align*} \dot x _ a = - { \gamma \over 2 } b _ { 1 , \delta } ( x _ a ) \ , . \end{align*}


\begin{align*} \sup _ { M } v = \sup _ { \partial M } v . \end{align*}


\begin{align*} g ( x ) = G \bigg ( \frac { 1 } { n } \sum _ { i = 1 } ^ n \delta _ { x _ i } \bigg ) , \ \ G : \P ( \R ) \to \R , \end{align*}


\begin{align*} \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { y } ' \right ] = a _ c \sum _ { j = 1 } ^ { M } \mathbf { h } _ { j , * } \mathbf { p } _ c + \sum _ { l = 1 } ^ M a _ l \mathbf { h } _ { l , * } \mathbf { p } _ l . \end{align*}


\begin{align*} T _ 2 = \sum _ { \beta \in F ^ { \times } } \sum _ { X \in S ( \beta ) } \Theta \begin{bmatrix} m _ 1 ' & X \\ 0 & m _ 2 ' \end{bmatrix} \overline { \psi _ A ( { m _ 1 ' } ^ { - 1 } X } ) = \theta ( a ) q ^ 8 ( q - 1 ) ^ 2 . \end{align*}


\begin{align*} & S ( k ) = \sigma _ 2 \overline { S ( \bar { k } ) } \sigma _ 2 , \ \ \ S ( k ) = \sigma _ 3 S ( - k ) \sigma _ 3 \end{align*}


\begin{align*} d ( X ( \bar u ) ) - L _ X g ( \nabla \bar u , \cdot ) & = \nabla ^ 2 \bar u ( X , \cdot ) - \nabla X ( \nabla \bar u , \cdot ) . \end{align*}


\begin{align*} \mathcal B _ { \mathcal X } ^ { \mathfrak a } = \left \{ ( s , b ) | s \in S , b \in B _ { X _ s } ^ { \alpha _ s } \right \} \end{align*}


\begin{align*} \hat { A } ^ + = - f \frac { d } { d r } - \frac { L + 1 } { r } f + \frac { Q } { 2 ( L + 1 ) } + \kappa \left ( L + \frac { 3 } { 2 } \right ) \frac { r } { f } . \end{align*}


\begin{align*} f ( ( a + 1 ) y + b ) - f ( a y + b ) & = y + y \int _ { u = a + \frac b y } ^ { a + 1 + \frac b y } \left ( \log ( u ) + \log ( y ) \right ) \d u \\ \end{align*}


\begin{align*} \mathcal { G } = \left \lbrace u _ 0 ( x ) : \ u _ 0 ( x ) \in H ^ { 3 } ( \mathbb { R } ) \cap H ^ { 2 , 1 } ( \mathbb { R } ) , \ a ( k ) \neq 0 \mathbb { C } ^ - \right \rbrace , \end{align*}


\begin{gather*} \Gamma _ 0 = \bigl \{ P \in \Gamma : P ( \abs { f } ) < \infty P ( f ) = 0 f \in F \bigr \} \end{gather*}


\begin{align*} \varphi '' + \Big ( ( m - 1 ) \frac { \sigma ' } { \sigma } - \Phi ' \Big ) \varphi ' - f ' ( u ) \leq 0 \textnormal { i n } \ I = ( r _ 1 , r _ 2 ) . \end{align*}


\begin{align*} S _ 2 \wr S _ n = ( \mathbb Z / 2 \mathbb Z ) \wr S _ n = ( \mathbb Z / 2 \mathbb Z ) ^ n \rtimes S _ n = B _ n . \end{align*}


\begin{align*} \begin{aligned} \left ( C _ { \mathcal C ^ h } - j ^ s e ^ { - \frac { 1 } { 2 } \zeta h j ^ s } \right ) \vert \eta ( \mathbf 0 ) v _ j ( \mathbf 0 ) \vert & \lesssim j ^ { - \beta + \frac { 1 7 } { 2 0 } s } + j ^ { \gamma - ( \alpha + 1 ) s } + j ^ { - \beta + \frac { 1 1 } { 1 0 } s } \\ & + j ^ { - \beta + ( - \alpha + \frac { 1 1 } { 1 0 } ) s } + j ^ { \gamma - \alpha s } + j ^ { \gamma - 2 \alpha s } \end{aligned} \end{align*}


\begin{align*} 0 \ = \ - { \rm c u r l } \left ( { \rm c u r l } \ , u \right ) + { \rm D } ( { \rm d i v } \ , u ) \ , - \ , \Delta u \end{align*}


\begin{align*} \psi ( v , p ) = \psi ^ \dagger ( v , p ) \in O \end{align*}


\begin{align*} g \circ L _ f = f \bullet _ \delta g \ \mbox { a n d } \ g \circ R _ f = g \bullet _ \delta f . \end{align*}


\begin{align*} X ( t ) = x + \int _ 0 ^ t \mathcal { A } ( s ) d s + \int _ 0 ^ t \mathcal { B } ( s ) d W ( s ) , \mathbb { P } \otimes d t , \end{align*}


\begin{align*} \left \lvert 1 - 4 \mu \left ( \sum _ { l = 1 } ^ { M } \lvert \phi ^ { \left ( l , j \right ) } \rvert ^ 2 + \sum _ { q = 1 } ^ { M - 1 } \sum _ { r = q + 1 } ^ { M } f _ { q , r } ^ { \left ( j \right ) } \right ) \right \rvert < 1 \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x & \leq C \int _ { B _ { 4 \rho } \setminus B _ { 2 \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon ( 1 + \varepsilon \rho ) \int _ { B _ { 8 \rho } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \\ & \rho \leq 1 / 8 \varepsilon \leq \varepsilon _ 0 . \end{align*}


\begin{align*} P _ j = \sum \limits _ { \forall v } \sum \limits _ { \forall q } p _ { v , j } ^ q + \sum \limits _ { \forall k , k \ne 0 } \sum \limits _ { \forall m } p _ { k , j } ^ m + \sum \limits _ { \forall n } p _ { 0 , j } ^ n , \end{align*}


\begin{align*} \sigma _ 0 = \min \{ \frac { p _ 0 } { \nu _ 0 } , q _ 0 \} . \end{align*}


\begin{align*} \left | \int _ M f ( x ) \right | = \left | \int _ M ( f ( x ) - \tilde f ( x ) ) \right | \leq { \rm V o l } ( \tilde \Sigma ^ r ) \cdot { \rm V o l } ( M ) . \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } \Psi ^ \pm _ 2 ( x ; z ) = e ^ { - i c _ \pm } e _ 2 , \end{align*}


\begin{align*} \frac { \left ( 6 n - \beta ^ { 3 } + 3 \right ) \left ( n ^ { 2 } + n + 1 \right ) } { 3 } = 0 \end{align*}


\begin{align*} \forall j \in [ i ] , g ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor ) = g _ s ( j ) , \end{align*}


\begin{align*} S t _ { D N P } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( f , z ) - f ( z ) | | _ { \infty } = 0 \end{align*}


\begin{align*} x ^ { t + 1 } = ( I + \gamma _ t B ) ^ { - 1 } \left ( x ^ t + \alpha _ t ( x ^ t - x ^ { t - 1 } ) - \gamma _ t ( F ( x ^ t ) + \beta _ t ( F ( x ^ t ) - F ( x ^ { t - 1 } ) ) ) \right ) , \end{align*}


\begin{align*} L \cong M / N \cong \bigoplus _ { i = 1 } ^ l M _ i / N _ i \end{align*}


\begin{align*} A _ n : = B _ n + C _ n \longrightarrow B + C = : A . \end{align*}


\begin{align*} \int _ { 1 } ^ { \infty } \frac { d s } { \left ( \int _ { 0 } ^ { s } F ( t ) d t \right ) ^ { \frac { p } { 2 p - q + 1 } } } < \infty \quad \mbox { a n d } \int _ { 1 } ^ { \infty } \frac { s \ d s } { \left ( \int _ { 0 } ^ { s } F ( t ) d t \right ) ^ { \frac { p } { 2 p - q + 1 } } } = \infty . \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \bar { \theta } ) = & E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ n Z ^ { \bar { \theta } } \biggr ) \biggr ] \\ = & E _ { \xi _ 0 } \biggl [ V \biggl ( \frac { \xi _ 0 f _ 1 ( X _ 1 ) } { \xi _ 0 f _ 1 ( X _ 1 ) + ( 1 - \xi _ 0 ) f _ 2 ( X _ 1 ) } , n - 1 , x + X _ 1 \biggr ) \biggr ] . \end{align*}


\begin{align*} X _ { j } ( u ) ( x _ { 0 } ) = 0 , \forall j = 1 , \cdots , k , \ \ \ \forall x _ 0 \in \bar \O . \end{align*}


\begin{align*} ( H _ { \lambda v , \alpha , x } \psi ) _ { n } = \psi _ { n + 1 } + \psi _ { n - 1 } + \lambda v ( x + n \alpha ) \psi _ { n } , \end{align*}


\begin{align*} & { M i n i m i z e \ , } _ { x \in \mathbb { R } ^ n } \ \ J ( x ) : = x ^ T A _ J x + 2 b _ J ^ T x + c _ J \\ & s . t . \ \ f _ k ( x ) : = x ^ T A _ k x + 2 b _ k ^ T x + c _ k \le 0 , \ \ k = 1 , . . . , m \end{align*}


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


\begin{align*} \widehat { \partial _ j f ( 0 , \xi ) } = 2 \pi i \ ; y _ j ( \xi ) \widehat { f ( 0 , \xi ) } . \end{align*}


\begin{align*} v \left ( x , t \right ) & = u \left ( x + u _ { \infty } t \right ) - u _ { \infty } , \\ q \left ( x , t \right ) & = \omega \left ( x + u _ { \infty } t \right ) , \end{align*}


\begin{align*} \mathcal { J } ' ( \overline { u } ; h ) & = \partial _ { y } J ( \overline { y } , \overline { u } ) S ' ( \overline { u } ; h ) + \partial _ { u } J ( \overline { y } , \overline { u } ) h \\ & = \langle \overline { y } - g , S ' ( \overline { u } ; h ) \rangle + \alpha \langle \overline { u } , h \rangle , \end{align*}


\begin{align*} \nu _ j = \begin{cases} \gamma _ j - 1 , & j \geq j ^ * \\ 1 - \gamma _ j , & \end{cases} \end{align*}


\begin{align*} B ( x , x ) = 0 , \lim _ { t \to \infty } \norm { e ^ { t L ^ \perp _ { x } } } = \infty . \end{align*}


\begin{align*} \Gamma ( G \times H ) = \Gamma ( G ) \boxtimes \Gamma ( H ) . \end{align*}


\begin{align*} \Omega _ 0 \cap \ ( - \mathbb { R } _ + ^ { m + 1 } ) = \emptyset . \end{align*}


\begin{align*} & u _ { x t } + \alpha \beta ^ 2 u - 2 i \alpha \beta u _ x - \alpha u _ { x x } - i \alpha \beta ^ 2 | u | ^ 2 u _ x = 0 , \\ & u ( x , t = 0 ) = u _ 0 ( x ) , \end{align*}


\begin{align*} \psi \left ( d _ - ^ { \ell + 1 } L ( 1 v , 0 w ) \right ) & = \psi \left ( t ^ { - \ell } d _ - ^ { \ell + 1 } d _ = L ( v 1 , w \bullet ) \right ) \end{align*}


\begin{align*} V _ { a , b } = \mathbb { C } [ \partial ] v , L _ \lambda v = ( \partial + a \lambda + b ) v , { H _ i } _ \lambda v = 0 , \end{align*}


\begin{align*} & \int _ { - \infty } ^ { \infty } \bar { u } _ y ( y ) \left ( \psi ^ - _ { 1 1 } - 1 \right ) e ^ { - 2 i z y } \mathrm { d } y = - z \int _ { - \infty } ^ { \infty } \bar { u } _ y ( y ) e ^ { - 2 i z y } K \psi ^ - _ { 1 1 } \mathrm { d } y \\ & = - \frac { 1 } { 2 } z \widehat { \bar { u } _ x K \psi ^ - _ { 1 1 } } ( z ) . \end{align*}


\begin{align*} \mathbf { h } ( H _ { \rho } ( x ) ) = R _ \rho \mathbf { h } ( x ) \end{align*}


\begin{align*} B _ \gamma ( z ) = A ( z ) \bigg ( \gamma + \sum _ n \frac { \mu _ n } { t _ n - z } \bigg ) . \end{align*}


\begin{align*} \mathcal D _ \pi = \sum _ { \widetilde { K } \ \ \widetilde { L } } ( e _ { \widetilde { K } } - 1 ) [ \widetilde { K } ] \in H _ 1 ( \widetilde M , \mathbb Z / 2 \mathbb Z ) \end{align*}


\begin{align*} \lambda ^ { - m } \ , \int _ { S ^ { n - 1 } } \widehat { f } ( \lambda , \omega ) \ , Y _ m ( \omega ) \ , d \omega = c _ { n , m } \int _ 0 ^ \infty r ^ { - m } f _ m ( r ) \ , \frac { J _ { n / 2 + m - 1 } ( \lambda r ) } { ( \lambda r ) ^ { n / 2 + m - 1 } } r ^ { n + 2 m - 1 } d r \end{align*}


\begin{align*} Z ^ i ( v ^ \mathrm { a p p } ( s ) , s ) - Z ^ i ( v ( s ) , s ) = \mathcal O ( s ^ \frac 3 2 ) . \end{align*}


\begin{align*} \begin{array} { l l l } \mu \circ ( ( f \otimes \eta ) \otimes \eta ) & = & \mu \circ ( i d _ M \otimes \eta ) \circ ( ( f \circ \eta ) \otimes i d _ I ) \\ & = & \rho _ M \circ ( ( f \circ \eta ) \otimes i d _ I ) \\ & = & ( f \circ \eta ) \circ \rho _ I \\ & & \mbox { ( b y n a t u r a l i t y o f $ \rho $ ) } \end{array} \end{align*}


\begin{align*} & \lim _ { M \rightarrow \infty } \mathbb { E } \int _ 0 ^ T \psi ( t ) \Big [ \Vert X ( t \wedge \tau _ u ^ M ) \Vert _ H ^ 2 - \Vert x \Vert _ H ^ 2 \Big ] d t \\ = & \lim _ { M \rightarrow \infty } \liminf _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T \psi ( t ) \Big [ \Vert X _ n ( t \wedge \tau _ u ^ M ) \Vert _ H ^ 2 - \Vert P _ n x \Vert _ H ^ 2 \Big ] d t . \end{align*}


\begin{align*} \nabla _ { 1 1 } \Theta - \frac { k } { k + 1 } \frac { ( \nabla _ { 1 } \Theta ) ^ { 2 } } { \Theta } + ( k + 1 ) \Theta = ( k + 1 ) \Theta ^ { \frac { k } { k + 1 } } \left ( \Theta ^ { \frac { 1 } { k + 1 } } + ( \Theta ^ { \frac { 1 } { k + 1 } } ) _ { \iota \iota } \right ) = ( k + 1 ) \Theta ( 1 + \Theta ^ { - \frac { 1 } { k + 1 } } \Theta ^ { \frac { 1 } { k + 1 } } _ { \iota \iota } ) . \end{align*}


\begin{align*} B ( v _ 1 , v _ 2 ) + B ( v _ 2 , v _ 1 ) & = \frac { k } { \ell } ( - \alpha \sin \ell x _ 1 + \beta \cos \ell x _ 1 ) \cos k x _ 2 - \frac { \ell } { k } \cos k x _ 2 ( - \alpha \sin \ell x _ 1 + \beta \cos \ell x _ 1 ) \\ & = \left ( \frac { k } { \ell } - \frac { \ell } { k } \right ) ( - \alpha \sin \ell x _ 1 + \beta \cos \ell x _ 1 ) \cos k x _ 2 . \end{align*}


\begin{align*} \big ( \bar \pi \bar p _ t \big ) \bar p _ T = \bar \pi \bar p _ t , \end{align*}


\begin{align*} \tilde W \cap h _ r ( \tilde W ) = ( S _ 0 ^ - \cap E _ r ) \cap h _ r ( S _ 0 ^ - \cap E _ r ) = ( S _ 0 ^ - \cap h _ r ( S _ 0 ^ - ) ) \cap E _ r = \emptyset \cap E _ r = \emptyset , \end{align*}


\begin{align*} \epsilon x _ i = x _ i \epsilon , 1 \leq i \leq n . \end{align*}


\begin{align*} \kappa : = D _ { a , n } = \begin{pmatrix} a \\ & a ^ { - 1 } \\ & & \ddots \\ & & & a \\ & & & & a ^ { - 1 } \end{pmatrix} . \end{align*}


\begin{align*} - \Delta _ y \varphi = \mu ^ { 2 } \left ( \varphi - 1 - \frac { \gamma } { r ^ { 2 } } \right ) _ { + } \textrm { i n } \ \mathbb { R } ^ { 5 } . \end{align*}


\begin{align*} h ( x ) = h _ 0 + \left \{ \begin{array} { c c } ( x - \pi ) ^ 2 - 1 & x < \pi \\ \cos ( x - \pi ) - 2 & x \in [ \pi , \ , 2 \pi ) \\ ( x - 2 \pi ) ^ 2 - 3 & x \ge 2 \pi \end{array} \right . \end{align*}


\begin{align*} S _ { \ell } = \left \{ \ k \in ( p ^ { \ell - 1 } , p ^ { \ell } ] : \ k \equiv 0 \bmod p ^ { \ell - \lfloor C _ 5 \log \ell \rfloor } \right \} \end{align*}


\begin{align*} \lambda _ i \neq 0 , i = 1 , \dots , n . \end{align*}


\begin{align*} { \rm O } ( \epsilon ^ 0 ) : g _ i ^ { ( 0 ) } = g _ i ^ { ( e q ) } , \end{align*}


\begin{align*} \begin{aligned} \int _ { \mathbb { R } ^ { 2 } _ { + } } \nabla \phi \cdot \nabla \tilde { \phi } \frac { 2 \pi ^ { 2 } } { r } \dd z \dd r = \int _ { \mathbb { R } ^ { 5 } } \nabla _ y \varphi \cdot \nabla _ y \tilde { \varphi } \dd y , \varphi = \frac { \phi } { r ^ { 2 } } , \tilde { \varphi } = \frac { \tilde { \phi } } { r ^ { 2 } } . \end{aligned} \end{align*}


\begin{align*} \theta _ A = \theta _ 1 ^ { a _ 1 } \theta _ 2 ^ { a _ 2 } \cdots \theta _ n ^ { a _ n } : = \theta ^ { \alpha ( A ) } ~ . \end{align*}


\begin{align*} ( I ^ \gamma _ { T - } \psi ) ( t ) = \frac { \Gamma ( m + 1 ) } { \Gamma ( \gamma + m + 1 ) } T ^ { \gamma } \biggl ( 1 - \frac { t } { T } \biggr ) ^ { m + \gamma } , \ , \ , \ , t \in [ 0 , T ) , \end{align*}


\begin{align*} V ^ { - 1 } ( \rho \circ \varphi ) V = V ^ { - 1 } \circ \rho \circ V ( V ^ { - 1 } \circ \varphi \circ V ) = 0 _ s \oplus \begin{bmatrix} ( V ^ { - 1 } \varphi V ) ^ { s + 1 } ( z ) \\ \vdots \\ ( V ^ { - 1 } \varphi V ) ^ n ( z ) \\ \end{bmatrix} . \end{align*}


\begin{align*} \d X _ t ^ i = b ^ i ( X _ t ) \d t + \sigma \d W _ t ^ i , ~ ~ \tilde X _ { n + 1 } ^ i = \tilde X _ n ^ i + b ^ i ( \tilde X _ n ) \tau + \sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , ~ ~ i = 1 , \cdots , N . \end{align*}


\begin{align*} F _ m ( n ) = 1 - G ( n , m ) + m \left \lfloor \frac { n } { m } \right \rfloor \sum _ { k = 1 } ^ m \frac { \mu ( k ) } { k } . \end{align*}


\begin{align*} \widehat { \mathrm { S } ( V ) } _ I : = \varprojlim \mathrm { S } ( V ) / I ^ n \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Theta ( t , x ) = \int _ \S \int _ \S W _ r ( z + y - 2 x ) \sin ( \Theta ( t , z ) + \Theta ( t , y ) - 2 \Theta ( t , x ) ) \ \d y \d z , \end{align*}


\begin{align*} R _ { i j k l } = - ( \delta _ { i k } \delta _ { j l } - \delta _ { i l } \delta _ { j k } ) + h _ { i k } h _ { j l } - h _ { i l } h _ { j k } , \end{align*}


\begin{align*} F ( r ) = \sigma _ k ^ { 1 / k } ( \lambda ( r ) ) \mbox { w i t h } \lambda ( r ) \in \Gamma _ k \end{align*}


\begin{align*} \forall t > 0 , \ H _ { m i n } ( E [ t ] ) = t H _ { m i n } ( E ) . \end{align*}


\begin{align*} c x ^ 2 + ( d - a ) x - b = 0 . \end{align*}


\begin{align*} f ( x ) = f _ { s ( x ) } ( x ) \end{align*}


\begin{align*} | x _ t | ^ 2 - | x _ 0 | ^ 2 = 2 \int _ 0 ^ t x _ s \cdot \sigma d W _ s - 2 \int _ 0 ^ t A x _ s \cdot x _ s d s + t \sum _ { i , j = 1 } ^ n | \sigma _ { i j } | ^ 2 . \end{align*}


\begin{align*} \int _ \Omega \Phi ( x ) \ , | \operatorname { g r a d } f ( x ) | \ , d V ( x ) = \int _ 0 ^ \infty d s \int _ { \Omega \cap \{ | f | = s \} } \Phi ( x ) \ , d S ( x ) , \end{align*}


\begin{align*} \int _ 0 ^ \infty K _ \mu ( \alpha x ) K _ \nu ( \beta x ) x ^ { s - 1 } d x & = 2 ^ { s - 3 } \alpha ^ { - s - \nu } \beta ^ \nu \Gamma \left ( \frac { 1 } { 2 } ( s + \mu + \nu ) \right ) \Gamma \left ( \frac { 1 } { 2 } ( s - \mu - \nu ) \right ) \\ & \int _ 0 ^ 1 t ^ { \frac { s - \mu + \nu } { 2 } - 1 } ( 1 - t ) ^ { \frac { s + \mu - \nu } { 2 } - 1 } \left ( 1 - \left ( 1 - \frac { \beta ^ 2 } { \alpha ^ 2 } \right ) t \right ) ^ { - \frac { s + \mu + \nu } { 2 } } d t . \end{align*}


\begin{align*} \sigma = \alpha ( Y ) ( \mathrm { i d } _ { Y } ) \in \underline { Y } ( \overline { Y } ) = \mathrm { H o m } _ { \mathbb { C } } ( \overline { Y } , Y ) \end{align*}


\begin{align*} \begin{cases} a \mapsto a b , \\ c \mapsto c , & c \neq a , \end{cases} \begin{cases} a \mapsto b a , \\ c \mapsto c , & c \neq a . \end{cases} \end{align*}


\begin{align*} \widehat { f _ \delta } ( a , z ) = ( 2 \pi ) ^ { - p - q } | \lambda | ^ { p + q } \ , \Big ( \sum _ { j = 1 } ^ { d ( p , q ) } P _ { p , q } ^ j ( z ) \ , \mathcal { G } _ { n + p + q } ( g _ { \delta , j } ) ( a ( p , q ) ) \Big ) \ , \varphi _ { k - q , \lambda } ^ { n + p + q - 1 } ( z ) . \end{align*}


\begin{align*} \phi ( z ) = \sum _ { q \geq 2 } \frac { | b _ q | } { q ! } H _ { q } ( z ) , \end{align*}