\begin{align*} \beta ( C _ S X , Y ) = \beta ( X , C _ S Y ) , \quad \forall S \in \Gamma , \quad \forall X , Y \in \Gamma ^ \perp . \end{align*}


\begin{align*} v _ 1 ( x , t ) = v _ 0 ( x - \lambda t ) + \int _ { t _ { 1 , e x } ( x , t ) } ^ t v ( s , x - \lambda t + \lambda s ) \dd s + \int _ { 0 } ^ { t _ { 1 , e n } ( x , t ) } v ( s , x - \lambda t + \lambda s ) \dd s . \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta y + f ( \cdot , y ) & = u , \ ; \ ; \Omega , \\ y & = 0 , \ ; \ ; \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} ( \nabla u ( x _ { 0 } ) ) ^ { \top } = 0 . \end{align*}


\begin{align*} ( { \rm D } \mathbf { A } ) _ { i j l } = A _ { i j , l } = - \epsilon _ { i j k } \ , { \rm a x l } ( \mathbf { A } ) _ { k , l } = - \epsilon _ { i j k } ( { \rm D } { \rm a x l } ( \mathbf { A } ) ) _ { k l } . \end{align*}


\begin{align*} \Delta _ { \Psi } u = \div _ { \Psi } ( \nabla u ) . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ m \sum _ { h = 1 } ^ 2 C _ { j , h } \int _ { \Omega } \frac { \partial } { \partial z _ { j , h } } ( - \delta ^ 2 ( K ( z _ j ) \nabla V _ { \delta , Z , j } ) ) \frac { \partial V _ { \delta , Z , i } } { \partial z _ { i , \hbar } } = \int _ { \Omega } u \frac { \partial V _ { \delta , Z , i } } { \partial z _ { i , \hbar } } , \ \ \forall i = 1 , \cdots , m , \ \hbar = 1 , 2 . \end{align*}


\begin{align*} \begin{aligned} \min & \ \frac { 1 } { 2 } x ^ T A x + b ^ T x \\ \mbox { s . t . } & \ \frac { 1 } { 2 } x ^ T B _ i x + c _ i ^ T x + d _ i \leq 0 , i = 1 , \dots , m , \end{aligned} \end{align*}


\begin{align*} \Phi _ { m , e } ^ { ( 1 ) } ( \lambda _ 1 , \lambda _ 2 ) \Phi _ { m , e } ^ { ( - 1 ) } ( \lambda _ 1 , \lambda _ 2 ) = \Phi _ { m e } ( \lambda _ 1 , \lambda _ 2 ) . \end{align*}


\begin{align*} ( f _ 1 ( \mathbf { x } ) , f _ 2 ( \mathbf { x } ) , \cdots , f _ { n - 1 } ( \mathbf { x } ) ) = ( f _ 2 ( \mathbf { y } ) , f _ 3 ( \mathbf { y } ) , \cdots , f _ n ( \mathbf { y } ) ) = ( f _ 2 ( \mathbf { x } ) , f _ 3 ( \mathbf { x } ) , \cdots , f _ n ( \mathbf { x } ) ) . \end{align*}


\begin{align*} E _ n = \frac { - \varepsilon _ 1 } { \sqrt { \varepsilon _ 1 ^ 2 + | \hat { x } | ^ 2 } } e _ 1 + \frac { | \hat { x } | } { \sqrt { \varepsilon _ 1 ^ 2 + | \hat { x } | ^ 2 } } e _ 2 . \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*} L = \bigoplus _ { i = 1 } ^ \ell \mathfrak a _ i b _ i . \end{align*}


\begin{align*} f _ { i : n } ( x ) = \frac { f ( x ) } { \operatorname { B } ( i , n - i + 1 ) } F ( x ) ^ { i - 1 } \{ 1 - F ( x ) \} ^ { n - i } , \end{align*}


\begin{align*} ( F \circ G ) ^ 2 ( x ) & = ( f ^ i ( g ^ m ( x ) ) , f ^ i { } _ \alpha ( g ^ m ( x ) ) g ^ \alpha { } _ j ( x ) , \\ * & \hphantom { { } = { } } f ^ i { } _ { \alpha \beta } ( g ^ m ( x ) ) g ^ \alpha { } _ j ( x ) g ^ \beta { } _ k ( x ) + f ^ i { } _ \alpha ( g ^ m ( x ) ) g ^ \alpha { } _ { j k } ( x ) ) . \end{align*}


\begin{align*} { { \bf { F } } _ i } \left ( { { \bf { x } } , t } \right ) = f _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { u } } ^ { e q } } + \Delta { \bf { u } } } \right ) - f _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { u } } ^ { e q } } } \right ) , \end{align*}


\begin{align*} \mathcal { W } \left ( t \right ) = \ln \frac { K _ 1 \left ( t \right ) \cdot | J _ 1 ( \left ( \lambda - 1 \right ) ^ { 1 / 2 } t ) | ^ { 1 / \left ( \lambda - 1 \right ) } } { t ^ { \lambda / \left ( \lambda - 1 \right ) } } , t \ , \in \ , A . \end{align*}


\begin{align*} { p _ { E O S } } = \frac { { \rho R T } } { { 1 - b \rho } } - \frac { { a \xi \left ( T \right ) { \rho ^ 2 } } } { { 1 + 2 b \rho - { b ^ 2 } { \rho ^ 2 } } } , \end{align*}


\begin{align*} \begin{aligned} | x _ a ( t ) - x ^ \star | & \leq \beta ( | x _ a ( 0 ) | _ { { \cal A } _ \delta } , 0 ) + \delta \leq \beta ( | x _ a ( 0 ) - x ^ \star | , 0 ) + \delta \\ & \leq \beta ( r _ 0 , 0 ) + \delta = r - d - \delta \end{aligned} \end{align*}


\begin{align*} A _ n ( \ell ) a _ { \ell } = Q _ { n , 1 } ( \ell ) \beta _ { n , 1 } ^ { \ell } + \cdots + Q _ { n , s ( n ) } ( \ell ) \beta _ { n , s ( n ) } ^ { \ell } . \end{align*}


\begin{align*} \varphi ( y ) \geq \inf \left \{ \varphi ( y ) \ | \ | y | = R \right \} , | y | \leq R . \end{align*}


\begin{align*} \sum _ { \substack { p \leq x \\ ( p , N ) = 1 } } a _ f \left ( p ^ { 2 l } \right ) \ll ( l \log l ) \sqrt { x } \log x \log ( N ( k - 1 ) ) . \end{align*}


\begin{align*} \langle { \boldsymbol { \nu } } _ { x , t } , g ( A ( \lambda , x , t ) , x , t ) \rangle = \langle { \boldsymbol { \nu } } _ { x , t } , f ( \lambda , x , t ) \rangle = \langle { \boldsymbol { N } } _ { x , t } , g ( \rho , x , t ) \rangle \end{align*}


\begin{align*} \left \{ \begin{aligned} S ^ \prime _ t & = - \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t , & S _ 0 & = s _ 0 , \\ I ^ \prime _ t & = \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t - \gamma I _ t - I _ t \phi \left ( I _ t \right ) , & I _ 0 & = i _ 0 , \\ R ^ \prime _ t & = \gamma I _ t , & R _ 0 & = r _ 0 , \\ D ^ \prime _ t & = I _ t \phi \left ( I _ t \right ) , & D _ 0 & = d _ 0 . \\ \end{aligned} \right . \end{align*}


\begin{align*} & b _ k ^ T y = 0 \ \ k \in \{ 0 , . . . , m \} \\ & \begin{bmatrix} b ^ 0 _ 1 & \cdots & b ^ 0 _ n \\ \vdots & \ddots & \vdots \\ b ^ m _ 1 & \cdots & b ^ m _ n \end{bmatrix} \begin{bmatrix} y _ 1 \\ \vdots \\ y _ n \end{bmatrix} = \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} \end{align*}


\begin{align*} \begin{aligned} & \dot y _ { b e n } = A _ { b e n } y '' _ { b e n } + M _ { b e n } y _ { b e n } , y \in ( - R / 2 , R / 2 ) , \\ & \dot y _ { n b } = A _ { n b } y '' _ { n b } + M _ { n b } y _ { n b } , y \in ( R / 2 , r + R / 2 ) , \end{aligned} \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{cases} \Delta _ \Psi u = f ( u ) & A ( r _ { 1 } , r _ { 2 } ) \\ u \equiv c _ { 1 } & \{ r _ { 1 } \} \times N \\ u \equiv c _ { 2 } & \{ r _ { 2 } \} \times N . \end{cases} \end{align*}


\begin{align*} \mathcal { M } _ { \omega } : = \Bigg \{ u \in H _ { r } ^ { 1 } ( \R ^ { 2 } ) \setminus \{ 0 \} : \int _ { \R ^ { 2 } } ( \alpha | \nabla u | ^ { 2 } + ( \alpha - 1 ) \omega u ^ { 2 } ) d x + ( 3 \alpha - 2 ) B ( u ) = \frac { p \alpha - 1 } { p } \int _ { \R ^ { 2 } } u ^ { 2 p } d x \Bigg \} . \end{align*}


\begin{align*} 2 x ^ { \tilde \varepsilon x / 2 } h _ s ( i ) ^ { ( \tilde \varepsilon x ) ^ 2 / 6 4 } = 2 \big ( x h _ s ( i ) ^ { \tilde \varepsilon x / 3 2 } \big ) ^ { \tilde \varepsilon x / 2 } \underbrace { \le } _ { \mbox { b y \eqref { e q : P r o b h x 0 } } } 2 n ^ { - \tilde \varepsilon x / 2 } h _ s ( i ) ^ { ( \tilde \varepsilon x ) ^ 2 / 1 2 8 } . \end{align*}


\begin{align*} A '' _ n ( k ) = A _ n ( k _ V ) A ' _ n ( k _ { V ' } ) \end{align*}


\begin{align*} Q = ] a , b [ \times B ( x _ 0 , r ) , \mbox { w i t h } 0 < a < b < + \infty , \ ; \ ; x _ 0 \in \R \ ; \ ; \mbox { a n d } \ ; \ ; 0 < r < + \infty , \end{align*}


\begin{align*} \bigcup _ { k = 1 } ^ { m } B \left ( y _ { k } , 2 r \right ) = \bigcup _ { k = 1 } ^ { m } E \left ( \mathbf { y } \right ) _ { k } \end{align*}


\begin{align*} \tilde { \nu _ 1 } ( 0 ) = 0 \qquad \frac { d } { d \tilde { \nu _ 2 } } \tilde { \nu _ 1 } ( 0 ) = 0 . \end{align*}


\begin{align*} \frac { \partial \tilde y } { \partial y } = g ^ { \frac { 1 } { 4 } } + y \cdot \biggl ( \frac { 1 } { 4 } g ^ { - \frac { 3 } { 4 } } \cdot \frac { 1 } { 6 } \int \limits _ { 0 } ^ 1 s f _ { y ^ { 5 } } ^ { ( 5 ) } ( x , s y ) ( 1 - s ) ^ 3 \ , d s \biggr ) . \end{align*}


\begin{align*} & \max \Big \{ \ ! \left \lVert \mu ( s , x ) - \mu ( t , x ) \right \rVert ^ 2 , \left \lVert \sigma ( s , x ) - \sigma ( t , x ) \right \rVert ^ 2 \Big \} \leq c \Big [ \lvert t - s \rvert + \ ! \ ! \sup _ { \substack { u , v \in [ 0 , t ] \colon \\ \lvert u - v \rvert \leq \lvert t - s \rvert } } \left \lVert x ( u ) - x ( v ) \right \rVert ^ 2 \Big ] , \end{align*}


\begin{align*} \big | T _ { W _ \ell , \phi } ( \mu ) - T _ { W , \phi } ( \mu ) \big | \le \sum _ { i = 1 } ^ L | c _ i | d _ \square ( W _ \ell , W ) . \end{align*}


\begin{align*} A _ n & : = \big \{ M ^ * _ v ( \delta n ) \in ( - \delta , \delta ) \big \} , \\ B _ n & : = \big \{ | M _ v ( t ) - M _ v ( \delta n ) | < \delta e ^ { ( \lambda ^ * - \lambda ) \delta n } t \in [ \delta n , \delta ( n + 1 ) ] \big \} . \end{align*}


\begin{align*} \mu ( T ^ { 2 r _ n } A \cap A ) & \geq \mu ( A \cap T ^ { - r _ n } A \cap T ^ { r _ n } A ) \\ & > 2 \big ( \beta - \frac { 1 } { 2 } \big ) \mu ( A ) \\ & = \big ( 2 \beta - 1 \big ) \mu ( A ) > \mu ( A ) ^ 2 . \end{align*}


\begin{gather*} \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \ge \inf _ { Q \in \mathcal { U } } Q \Bigl ( \oplus _ { i = 1 } ^ n f _ i \Bigr ) \quad \quad f _ i \in B ( \Omega _ i , \mathcal { A } _ i ) i \in I . \end{gather*}


\begin{align*} m _ { n } ( x ) = \frac { \nu _ n ( A _ { h , j } ) } { \mu _ n ( A _ { h , j } ) } \end{align*}


\begin{align*} \P \left ( \Omega _ 0 = \left \{ \omega \in \Omega : \int _ 0 ^ \eta | y _ t | ^ 2 d t \le \sqrt { \delta _ 1 } \eta K ^ { 2 r } , \sup _ { 0 \le t \le \eta } | z _ t - z _ 0 | \le R K ^ { \frac { r - 1 } { 3 } } \right \} \right ) \ge 1 - \frac { \beta } { 2 } . \end{align*}


\begin{align*} \Sigma \cap T \cap \bigcap _ { j = 1 } ^ J B ( p _ j , s _ \Sigma ( p _ j ) ) \partial \Sigma \cap T \cap \bigcap _ { j = 1 } ^ J B ( p _ j , s _ \Sigma ( p _ j ) ) \end{align*}


\begin{align*} & \hphantom { { } = { } } \{ f , g ; F , G \} ( \xi , \eta ; \dot { \xi } , \dot { \eta } ) \\ * & = ( f ( \xi ) , D f ( \xi ) \eta + g ( \xi ) ; F ( \xi ) \dot { \xi } , ( D F ( \xi ) ^ i { } _ { \alpha j } \eta ^ \alpha + G ( \xi ) ) \dot { \xi } + F ( \xi ) \dot { \eta } ) , \end{align*}


\begin{align*} \phi _ A ( t _ 1 , \hdots , t _ n ) = \left ( \prod _ { i = 1 } ^ n t _ i ^ { a _ { i 1 } } , \prod _ { i = 1 } ^ n t _ i ^ { a _ { i 2 } } , \hdots , \prod _ { i = 1 } ^ n t _ i ^ { a _ { i m } } \right ) . \end{align*}


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


\begin{align*} \begin{aligned} \frac { 1 } { 2 } \sum _ { \beta = 1 } ^ { \alpha } b _ \beta \bar { x } _ \beta ^ 2 \leq C b _ \alpha \bar { x } _ n + b _ \alpha O ( | \bar { x } ' | ^ 3 ) \leq C b _ \alpha \bar { x } _ n + C \delta ^ 3 b _ \alpha ^ 2 . \end{aligned} \end{align*}


\begin{align*} W ( y | x ) = b _ k - a + \sum _ { n = 2 } ^ \infty ( a _ n - a ) \ , \end{align*}


\begin{align*} \hat M ^ { ( k ) } = G _ k G _ { k - 1 } \cdots G _ 1 ( M + \tilde E ^ { ( k ) } ) , \end{align*}


\begin{align*} Z g \left ( X , Y \right ) = g \left ( \nabla _ { Z } X , Y \right ) + g \left ( X , \nabla _ { Z } ^ { \ast } Y \right ) \end{align*}


\begin{align*} \begin{cases} \nabla w \cdot \nabla ^ \perp ( u - q \ln \frac { 1 } { \varepsilon } ) = 0 , \\ w = \mathcal { L } _ H u , \\ \nabla ^ \perp u \cdot \nu | _ { \partial \Omega } = 0 . \end{cases} \end{align*}


\begin{align*} h ^ 0 \left ( X , K _ { X _ s } ^ { \otimes 2 } ( - D _ s ) \right ) - h ^ 1 \left ( X , K _ { X _ s } ^ { \otimes 2 } ( - D _ s ) \right ) = 3 g - 3 - n \end{align*}


\begin{align*} ( F \circ G ) ^ 2 ( x ) & = ( f ^ i ( g ^ m ( x ) ) , f ^ i { } _ \alpha ( g ^ m ( x ) ) g ^ \alpha { } _ j ( x ) , \\ * & \hphantom { { } = { } } f ^ i { } _ { \alpha \beta } ( g ^ m ( x ) ) g ^ \alpha { } _ j ( x ) g ^ \beta { } _ k ( x ) + f ^ i { } _ \alpha ( g ^ m ( x ) ) g ^ \alpha { } _ { j k } ( x ) ) . \end{align*}


\begin{align*} n - m + f = 2 , \end{align*}


\begin{align*} W ( r ) = - f \frac { d } { d r } \log \psi _ 0 ( r ) - \frac { 1 } { 2 } \frac { d f } { d r } . \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \nabla ^ k [ P _ t ^ L ( x ' - y ' ) ] d y ' = 0 , \quad \forall t > 0 , \ , k \in \N . \end{align*}


\begin{align*} a _ c \left [ t + 1 \right ] = & a _ c \left [ t \right ] - 4 \mu a _ c \left [ t \right ] \sum _ { j = 1 } ^ { M } \lvert \phi ^ { \left ( j , c \right ) } \rvert ^ 2 - 2 \mu \sum _ { l = 1 } ^ { M } \Re \left \{ \phi ^ { \left ( l , c \right ) } \right \} + 2 \mu a _ c \left [ t \right ] f ^ { \left ( c \right ) } , \end{align*}


\begin{align*} \mathcal { S } _ 2 = \left ( 2 - \frac { \tau _ { \rm d } } { T _ s } \right ) \frac { \tau _ { \rm d } } { N _ { \rm a } T _ { \rm s } } \mathcal { U } , \end{align*}


\begin{align*} f ( x \vee y \vee z ) + f ( x ) - f ( x \vee y ) - f ( x \vee z ) = f ( a \wedge b ) + f ( a \vee b ) - f ( a ) - f ( b ) \ge 0 . \end{align*}


\begin{align*} \begin{aligned} \limsup _ { k \to \infty } \Bigg \{ & \left | | | b _ { n _ k } | | _ { L ^ { 2 } ( B ( z _ k e _ z , R _ 0 ) ) } ^ { 2 } - \alpha \right | + \left | | | b _ { n _ k } | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } \backslash B ( z _ k e _ z , R _ k ) ) } ^ { 2 } - ( l - \alpha ) \right | \\ & + | | b _ { n _ k } | | _ { L ^ { 2 } ( B ( z _ k e _ z , R _ k ) \backslash B ( z _ k e _ z , R _ 0 ) ) } ^ { 2 } \Bigg \} \leq \varepsilon . \end{aligned} \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*} \begin{aligned} L ^ { p } ( \Omega ) & = L ^ { p } _ { \sigma } ( \Omega ) \oplus G ^ { p } ( \Omega ) , 1 < p < \infty , \\ L ^ { p } _ { \sigma } ( \Omega ) & = \{ u \in L ^ { p } ( \Omega ) \ | \ \nabla \cdot u = 0 \ \textrm { i n } \ \Omega , \ u \cdot n = 0 \ \textrm { o n } \ \partial \Omega \} , \\ G ^ { p } ( \Omega ) & = \{ \nabla q \in L ^ { p } ( \Omega ) \ | \ q \in L ^ { 1 } _ { \textrm { l o c } } ( \Omega ) \} , \end{aligned} \end{align*}


\begin{align*} a = \frac { \left ( \beta - 3 \right ) n + 2 \beta } { 3 } \end{align*}


\begin{align*} \tau = e _ { i _ 1 } + \dots + e _ { i _ k } \in ( \mathbb Z / 2 \mathbb Z ) ^ n \leq B _ n \end{align*}


\begin{align*} \hat \Phi ^ \dagger ( 0 , 0 ) & = 0 , & \hat \Phi ^ \dagger _ a ( 0 , 0 ) & = 0 , \\ \hat \Phi ^ \dagger _ { a a } ( 0 , 0 ) & = 0 , & \hat \Phi ^ \dagger _ { a a a } ( 0 , 0 ) & = 6 \gamma _ 1 , \end{align*}


\begin{align*} S = \left ( \begin{array} { c c } 1 / 2 + \alpha & \beta \\ \beta & 1 / 2 - \alpha \end{array} \right ) \end{align*}


\begin{align*} f ^ \star : = \inf \limits _ { x \in S ( g ) } f ( x ) \ , , \end{align*}


\begin{align*} ( \mathbf { u } - \mathbf { v } ) ( \underbrace { G _ 2 , G _ 3 , \cdots , G _ { 2 k } } _ { \bar { G } } ) = ( 0 , 0 , \cdots , 0 ) . \end{align*}


\begin{align*} \psi = \sqrt { \frac { { 2 \Delta { t ^ 2 } \left ( { { p _ { E O S } } - \rho \hat c _ s ^ 2 } \right ) } } { { G \Delta { x ^ 2 } } } } . \end{align*}


\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \in \R ^ n _ + , \end{align*}


\begin{align*} h ( G ) & = h ( G ' ) + f ( y ) + \sum _ { j = 1 } ^ y f ( x _ j ) - \sum _ { j = 1 } ^ y f ( x _ j - 1 ) \\ & \le h ( K _ { 1 , m - y } ) + f ( y ) + y \left ( f \left ( \frac { m } { y } \right ) - f \left ( \frac { m } { y } - 1 \right ) \right ) . \end{align*}


\begin{align*} \tilde { \alpha } = D ^ { - 1 } \circ \alpha \circ ( \mathrm { i d } _ S \times d ) \colon S \times _ \R ( \tilde { X } _ n \setminus \pi ^ { - 1 } ( 0 ) ) \to \tilde { \Pi } ^ { - 1 } ( S \times _ \R B ^ \star ) \end{align*}


\begin{align*} \kappa ( t ) = h _ { 1 - \beta + \alpha } ( t ) \ , + \ , h _ { 1 - \beta } ( t ) , \ 0 < \alpha < \beta < 1 , \end{align*}


\begin{align*} [ \beta ^ t _ { a , b , n } \ , , \ , \gamma ^ t _ { a , b , m } ] = \delta _ { n + m , 0 } \end{align*}


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


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


\begin{align*} \begin{cases} \dim H ^ 1 ( S , T _ S ) = 6 8 ; \\ \dim H ^ 2 ( S , T _ S ) = 6 . \end{cases} \end{align*}


\begin{align*} \underline { \delta } ^ \star ( s ) : = 2 \int _ { 0 } ^ { 1 / 2 } \alpha ( s \sin ( 2 \pi t ) ) \ , d t \end{align*}


\begin{align*} F ^ q ( v , p ) = 0 \end{align*}


\begin{align*} \frac { g ( t ) - g ( r ) } { t - r } = \lambda \frac { g ( t ) - g ( s ) } { t - s } + ( 1 - \lambda ) \frac { g ( s ) - g ( r ) } { s - r } . \end{align*}


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


\begin{align*} \mathcal { L } ( u ^ h ( t , x ) ) = \mathcal { L } \left ( \sum _ i w _ i ( t ) \sigma _ i ( x ) \right ) = \sum _ i w _ i ( t ) \mathcal { L } ( \sigma _ i ( x ) ) . \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*} A _ \phi = \begin{pmatrix} \cos ( 2 \pi \ell \phi ) & \sin ( 2 \pi \ell \phi ) \\ - \sin ( 2 \pi \ell \phi ) & \cos ( 2 \pi \ell \phi ) \end{pmatrix} \end{align*}


\begin{align*} \langle A ( u ) , v \rangle = - \int _ { \mathbb { T } ^ d } \langle a ( u ( x ) ) \nabla u ( x ) + b ( u ( x ) ) , \nabla v ( x ) \rangle d x . \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*} R _ 2 = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 2 } } \Theta \begin{bmatrix} m _ 1 ' & X \\ 0 & m _ 2 ' \end{bmatrix} \overline { \psi _ A ( { m _ 1 ' } ^ { - 1 } X } ) = \theta ( a ) q ^ 8 ( 1 - q ) . \end{align*}


\begin{align*} h _ d ( q ) = \log ( c _ { d , 2 } ( q ) ^ q ) - \log ( c _ { d , \infty } ( q ) ^ q ) . \end{align*}


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


\begin{align*} - \sum _ { i \geq 3 } \sum _ { d \geq 2 } a _ { i , d } ( \underline { b } ) = - a ( \underline { b } ) + \sum _ { i \geq 1 } \left ( a _ { i , 0 } ( \underline { b } ) + a _ { i , 1 } ( \underline { b } ) \right ) + a _ { 2 , 2 } ( \underline { b } ) . \end{align*}


\begin{align*} \begin{aligned} \int _ G \partial _ j \zeta ( 0 , \omega ) \ ; \overline { f ( 0 , \omega ) } d \mu ( \omega ) & = \int _ { G ^ \wedge } 2 \pi i \ ; y _ j ( \xi ) \widehat { \zeta ( 0 , \xi ) } \ ; \overline { \widehat { f ( 0 , \xi ) } } d \nu ( \xi ) \\ [ 5 p t ] & = - \int _ { G ^ \wedge } \widehat { \zeta ( 0 , \xi ) } \ ; \overline { 2 \pi i \ ; y _ j ( \xi ) \widehat { f ( 0 , \xi ) } } d \nu ( \xi ) . \end{aligned} \end{align*}


\begin{align*} E ^ { \pm } ( \Q _ { p ^ g , \infty } ) = \bigcup _ { n \geq - 1 } E ^ { \pm } ( \Q _ { p ^ g , n } ) . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n { \binom n k F _ { j k + m } B _ { n - k } z ^ k } = \frac { 1 } { \sqrt 5 } \big ( \alpha ^ m B _ n ( \alpha ^ j z ) - \beta ^ m B _ n ( \beta ^ j z ) \big ) \ , , \end{align*}


\begin{align*} \mathcal { B } _ { A B C } \coloneqq \mathcal { B } ( e _ A , e _ B , e _ C ) = \langle [ e _ A , e _ B ] _ H , e _ C \rangle . \end{align*}


\begin{align*} \Phi ( x ; z ) = A ( x ; k ) \psi ( x ; k ) B ( x , k ) , \end{align*}


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


\begin{align*} \Theta ( \beta _ 2 ( t ) ) & = f ( y _ 2 ( t ) ) - f ^ \star \\ & = f ( y _ 1 ( \alpha ( t ) ) ) - f ^ \star = \Theta ( \beta _ 1 ( \alpha ( t ) ) ) \end{align*}


\begin{align*} \begin{aligned} \mathcal { H } _ m ( K _ m ) = C _ { m , n } \delta _ a \end{aligned} \end{align*}


\begin{align*} [ - T , T ] = \bigcup _ { j = - [ T / \tau ] - 1 } ^ { [ T / \tau ] } I _ j , I _ j = [ j \tau , ( j + 1 ) \tau ] \cap [ - T , T ] , \end{align*}


\begin{align*} & \sup _ { x \in \mathbb { R } } \left \| \mathcal { P } ^ + \left ( z ^ { - i } \bar { r } _ 1 ( z ) e ^ { - 2 i z x } \right ) \right \| _ { L _ z ^ \infty } \leq \frac { 1 } { \sqrt { 2 } } \left \| z ^ { - i } r _ 1 ( z ) \right \| _ { H ^ 1 } , i = 0 , 1 , \\ & \sup _ { x \in \mathbb { R } } \left \| \mathcal { P } ^ - \left ( r _ 2 ( z ) \mathrm { e } ^ { 2 i z x } \right ) \right \| _ { L _ { z } ^ { \infty } } \leq \frac { 1 } { \sqrt { 2 } } \left \| r _ 2 ( z ) \right \| _ { H ^ { 1 } } . \end{align*}


\begin{align*} \gamma \varphi ( h ) G ( \nabla h ) \sigma _ { k } ( x ) = f ( x ) \ { \rm f o r } \ \gamma = 1 . \end{align*}


\begin{align*} f ^ { - 1 } ( \overline { V } _ \bullet ) = \{ V _ \bullet ; V _ j = \overline { V } _ j j \in [ n ] \setminus \{ l \} , V _ { l - 1 } + X V _ i \subset V _ l \subset V _ { l + 1 } \} \end{align*}


\begin{align*} \frac { 1 } { N ( f ( x ) ) } : = - \frac { f ' ( x ) } { f ( x ) } \end{align*}


\begin{align*} \lim _ { r \to \infty } \int _ { | x | = r } \sum _ { i , j = 1 } ^ n \left ( \frac { \partial h _ { i j } } { \partial x _ i } - \frac { \partial h _ { i j } } { \partial x _ j } \right ) \frac { x _ j } { | x | } \ , d \sigma _ { \bar g } = 0 . \end{align*}


\begin{align*} \Gamma ( x _ 0 , y ) = d _ x T ( x _ 0 , y ) . \end{align*}


\begin{align*} \Delta u _ 0 + k ^ 2 q u _ 0 = 0 \mbox { i n } \mathbb R ^ n , \end{align*}


\begin{align*} \mathcal { Q } = \int _ { 0 } ^ t v _ 1 ( s , x - \lambda t + \lambda s ) \dd s - \int _ { t _ { 1 , e n } ( x , t ) } ^ { t _ { 1 , e x } ( x , t ) } v _ 1 ( s , x - \lambda t + \lambda s ) \dd s \end{align*}


\begin{align*} \mathbb { M } ( \theta ) _ J = \sum _ { w \in X _ J } \Bbbk { \bf U } \dot { w } \eta ( \theta ) _ J = \sum _ { w \in X _ J } \Bbbk { \bf U } _ { w _ J w ^ { - 1 } } \dot { w } \eta ( \theta ) _ J . \end{align*}


\begin{align*} Z ^ 2 ( v , s ) = \Psi ^ q + v + \frac 1 2 \psi _ { v v } ( 0 , p _ 0 ) [ v , v ] \end{align*}


\begin{align*} \sigma ( B ) \leq \sum _ { j = 1 } ^ N \sigma ( B ^ j ) \mbox { w h e n e v e r } N \in \mathbb { Z } ^ + , \{ B ^ j \} _ { j = 1 } ^ N \subset \mathcal { B } , B \subset \bigcup _ { j = 1 } ^ N B ^ j . \end{align*}


\begin{align*} \P \left \{ Y = 2 ^ { - i } \right \} = r 2 ^ i , \end{align*}


\begin{align*} \mathrm { E } ( Z ^ s ) = \sum _ { r = 0 } ^ \infty w _ { r } ( \alpha , \beta ) \ , \tau _ { s , r + a - 1 } \mathrm { E } ( Z ^ s ) = \sum _ { i , j = 0 } ^ \infty \sum _ { r = 0 } ^ j w _ { i , j , r } ( \alpha , \beta ) \ , \tau _ { s , r } , \end{align*}


\begin{align*} R _ 1 = \int ^ t _ 0 \tau \| \partial _ \tau f \| _ { L ^ 2 } ^ 2 d \tau \le C _ \delta B _ 0 ^ 2 \epsilon ^ 2 + \delta \| \tau \partial _ \tau f \| ^ 2 _ { L ^ 2 ( [ 0 , t ] ; L ^ 2 _ A ) } . \end{align*}


\begin{align*} \tau ( K ) : = \bigcap _ { K \subset U } \ker ( r : S H _ { M } ( M ; \Lambda ) \rightarrow S H _ { M } ( M - U ; \Lambda ) ) \end{align*}


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


\begin{align*} X ^ { \Delta } ( \beta ) = \left \{ \sum _ { \ell = 0 } ^ { m - 1 } a _ \ell \beta ^ { m - 1 - \ell } : m \in \N , \ a _ 0 , \ldots , a _ { m - 1 } \in \Delta \right \} \end{align*}


\begin{align*} L ( x ) = x ^ { 2 ^ r } + t x . \end{align*}


\begin{align*} \norm { \Theta _ { \norm { \cdot } } ( \nu ) } _ { ( C ( [ 0 , T ] ; ( U , \norm { \cdot } _ { U } ) ) , \norm { \cdot } _ \infty ) ' } = \norm { \nu } _ { v a r } ( \nu \in M ( [ 0 , T ] ; U ' ) ) \end{align*}


\begin{align*} \begin{aligned} K f = K ^ { 1 - \chi } f + K ^ \chi f \ , , \ K ^ \Lambda f = K _ 1 ^ \Lambda f - K _ 2 ^ \Lambda f \textrm { f o r } \Lambda = 1 - \chi \textrm { a n d } \chi \ , , \end{aligned} \end{align*}


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


\begin{align*} G = \begin{pmatrix} \mathbf { 1 } _ { n - 1 } & 1 \\ G _ { n - 1 } & \mathbf { 0 } _ { n - 1 } ^ T \end{pmatrix} , \end{align*}


\begin{align*} d g _ 1 ' = ( \log \sqrt { \tau _ 0 } ) _ z d z \ g ' _ 1 + \frac { 1 } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) \ J \widehat { g _ 1 ' } I , \end{align*}


\begin{align*} F ( t ) = \int _ { \mathcal { O } } f _ 1 ( t , x ) ^ { \frac { \alpha } { \alpha - 1 } } d x . \end{align*}


\begin{align*} a _ { n - i } ( \gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \end{align*}


\begin{align*} \frac D 2 = \{ q _ 1 \mid q _ 1 \overline { q p } q \in D \} . \end{align*}


\begin{align*} \mathcal { G } ^ { \Delta _ { T , E } } ( V , W ) = \mathcal { G } ^ { \Delta _ { T } } ( V , W ) + \frac { 1 } { 2 } \int _ { M } ( E ) \ , g ( V , W ) ~ v o l _ { g } . \end{align*}


\begin{align*} | s _ { h , q } - s _ h | \le \epsilon _ 0 = 1 / 2 ^ { b _ 0 } \end{align*}


\begin{align*} 0 \rightarrow E \rightarrow \bigoplus _ { j = 1 } ^ n E _ j \rightarrow \mathcal { T } _ E \rightarrow 0 . \end{align*}


\begin{gather*} \mathcal { A } ( x , \xi ) = D _ { \xi } F ( x , \xi ) . \end{gather*}


\begin{align*} \widehat { \tau } : x \mapsto \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \tau _ { 1 } ( x _ { i i } ) \forall x = ( x _ { i j } ) \in \mathcal { M } _ { n } ( M _ { 1 } ) . \end{align*}


\begin{align*} \chi ( x ) = \begin{cases} 1 & : \abs { x } \leq 1 \\ 0 & : \abs { x } \geq 2 \end{cases} \end{align*}


\begin{align*} \mathcal { T } ( x , \textbf { 1 } ) = \log \log x + C ( \textbf { 1 } ) + o ( 1 ) , \end{align*}


\begin{align*} F _ { \delta , Z } = \{ u \in L ^ p ( \Omega ) \mid \int _ { \Omega } \frac { \partial V _ { \delta , Z , j } } { \partial z _ { j , h } } u = 0 , \ \ \forall j = 1 , \cdots , m , \ h = 1 , 2 \} , \end{align*}


\begin{align*} \varepsilon \circ i ( B ) = A \varepsilon ( B ) A ^ { - 1 } . \end{align*}


\begin{align*} & M i n i m i z e _ { \mathbf { x } \in \mathbb { R } ^ { 3 n } } \ \ \sum \limits _ { i = 1 } ^ n ( a _ J , a _ J , b _ { J i } ) ^ T \mathbf { x } _ i \\ & \sum \limits _ { i = 1 } ^ n ( a _ k , a _ k , b _ { k i } ) ^ T \mathbf { x } _ i + c _ k \le 0 \ \ \forall \ , k \in { 1 , . . . , m } \\ & \mathbf { x } _ i \in \mathcal { L } _ i \end{align*}


\begin{align*} r = | x | , u _ r = \dfrac { x } { | x | } \cdot \nabla u \end{align*}


\begin{align*} \Psi _ { [ q ] } ( x ) = x ^ 2 - 2 \Re ( q ) x + I _ q . \end{align*}


\begin{align*} \varphi '' + \Big ( ( m - 1 ) \frac { \sigma ' } { \sigma } - \Phi ' \Big ) \varphi ' - f ' ( u ) \varphi \leq \varphi '' + \Big ( ( m - 1 ) \frac { \sigma ' } { \sigma } - \Phi ' \Big ) \varphi ' + B = 0 . \end{align*}


\begin{align*} c _ { b _ { k } } & = \frac { 1 } { 2 } \| ( \widetilde { u } _ { k } , \widetilde { v } _ { k } ) \| _ { E } ^ { 2 } + \frac { 1 } { 2 } \Big ( B ( \widetilde { u } _ { k } ) + B ( \widetilde { v } _ { k } ) \Big ) - \frac { 1 } { 2 p } F ( \widetilde { u } _ { k } , \widetilde { v } _ { k } ) \\ & \geq \Big ( \frac { 1 } { 2 } - \frac { \alpha } { 2 ( p \alpha - 1 ) } \Big ) \| ( \widetilde { u } _ { k } , \widetilde { v } _ { k } ) \| _ { E } ^ { 2 } , \end{align*}


\begin{align*} \partial ^ 2 _ { x _ i x _ j } \phi = \partial ^ 2 _ { x _ i x _ j } \psi + \partial ^ 2 _ { x _ i y } \psi \Gamma _ j + \partial ^ 2 _ { x _ j y } \psi \Gamma _ i + \partial ^ 2 _ y \psi \Gamma _ j \Gamma _ i + \partial _ y \psi \partial ^ 2 _ { x _ j x _ i } T - \partial _ y T \partial ^ 3 _ { x _ j x _ i y } T + \partial _ { y } \Gamma _ i \partial _ { y } \Gamma _ j \end{align*}


\begin{align*} \frac { \partial \tilde y } { \partial x } = y \cdot \frac { 1 } { 4 } g ^ { - \frac { 3 } { 4 } } \cdot \biggl ( \frac { 1 } { 6 } \int \limits _ { 0 } ^ 1 f _ { x y ^ { 4 } } ^ { ( 5 ) } ( x , s y ) ( 1 - s ) ^ 3 \ , d s \biggr ) , \end{align*}


\begin{align*} \frac { 1 } { A r } = h ^ q , \end{align*}


\begin{align*} G _ n / \langle \epsilon \rangle = G _ n / \{ 1 , \epsilon \} \cong ( \mathbb Z / 2 \mathbb Z ) ^ n \rtimes S _ n = B _ n . \end{align*}


\begin{align*} \left ( \alpha + 1 \right ) \kappa \dot { P } - \varpi \dot { P } = \alpha P . \end{align*}


\begin{align*} \pi ( x ) = \frac { f ( x , 1 ) } { g ( x , 1 ) } = \frac { s } { r } \end{align*}


\begin{align*} h = 0 , \nabla h = 0 , v = 0 , \nu ( v ) = 0 \mbox { o n } \hat \Sigma \end{align*}


\begin{align*} M ( n + 3 ) \coloneqq \begin{cases} ( F ^ + ) ^ n E _ 1 & \\ ( F ^ + ) ^ n E _ 2 & \end{cases} \quad M ( - n ) \coloneqq \begin{cases} ( F ^ - ) ^ n E _ 2 & \\ ( F ^ - ) ^ n E _ 1 & \end{cases} \end{align*}


\begin{align*} K = \{ k _ n : n \in \N \} \subset M ^ c . \end{align*}


\begin{align*} X _ { i ; j } V _ j & = \omega _ { i j } V _ j \\ \omega _ { i j ; \ell } V _ \ell & = - R _ { k \ell i j } X _ k V _ \ell + \tfrac { 1 } { 2 } \left ( h _ { i j , \ell } + h _ { i \ell , j } - h _ { j \ell , i } \right ) V _ \ell \end{align*}


\begin{align*} f \cdot g : = \int _ \S f ( x ) g ( x ) \ \d x + \int _ \S D f ( x ) D g ( x ) \ \d x \end{align*}


\begin{align*} 0 = \phi ^ \theta ( e ) = \tau ( u _ { l - 1 } , e ) \theta ( u _ { l - 1 } ) + \phi ( e ) + \tau ( u _ l , e ) \theta ( u _ l ) , \end{align*}


\begin{align*} h ( M ) = \inf \frac { _ { n - 1 } ( \Sigma ) } { \min \{ _ n ( A ) , _ n ( B ) \} } , \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { 1 } { \varepsilon ^ 2 \ln \frac { 1 } { \varepsilon } } \int _ \Omega \left ( v ^ \tau _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ p _ + v ^ \tau _ \varepsilon d x = q ^ { p + 1 } ( \bar { x } ) \int _ { \hat { B } _ \tau ( 0 ) } ( \hat { U } ( y ) + \ln \tau ) ^ { p } _ + d y . \end{align*}


\begin{align*} a ( - k ) = a ( k ) , k \in \overline { D } _ + , b ( - k ) = - b ( k ) , \operatorname { I m } k ^ { 2 } = 0 . \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*} \widehat { f } ( a , z ) = \rho _ k ^ \lambda ( f ) e _ a ( z , 0 ) , \ , \ , \ , \rho _ k ^ \lambda ( f ) = \int _ { G _ n } f ( g ) \rho _ k ^ \lambda ( g ) d g . \end{align*}


\begin{align*} \lambda ^ * = - 3 \sqrt 2 \frac { \sqrt { 1 + a ^ 2 } } { 2 + \frac { 1 9 7 } { 1 9 9 } a ^ 2 } \eta . \end{align*}


\begin{align*} \Delta _ + = \Delta _ - ^ 0 \cup \{ \alpha + n \delta , n \delta | n \in \mathbb Z _ { < 0 } , \alpha \in \Delta ^ 0 \} \end{align*}


\begin{align*} - \Delta ( v \psi _ { m } ) ( x ) + F ' ( \overline { y } ; v \psi _ { m } ) ( x ) = 0 , x \in \mathcal { A } , \end{align*}


\begin{align*} | F _ m ( n ) | \leq 1 + \sum _ { k = 2 } ^ m 1 + r _ m ( n ) < 2 m , n , m \in \N . \end{align*}


\begin{align*} d G = \sqrt { \tau _ 0 } / 4 \left \{ \left ( \left ( 1 + 4 Q _ 0 / \tau _ 0 \right ) d z + \left ( 1 + 4 \overline { Q _ 0 } / \tau _ 0 \right ) d \overline { z } \right ) u _ 1 \right . \\ \left . - i \left ( \left ( 1 - 4 Q _ 0 / \tau _ 0 \right ) d z - \left ( 1 - 4 \overline { Q _ 0 } / \tau _ 0 \right ) d \overline { z } \right ) u _ 2 \right \} \end{align*}


\begin{align*} \mathcal C '^ \kappa = \varprojlim _ n ( \mathcal C ^ n ) ^ \kappa = \varprojlim _ n \varinjlim _ i ( \mathcal C _ i ^ n ) ^ \kappa = \varinjlim _ i \varprojlim _ n ( \mathcal C _ i ^ n ) ^ \kappa = \varinjlim _ i \mathcal C ^ \kappa = \mathcal C ^ \kappa , \end{align*}


\begin{align*} \int _ M | | ( \eta + g ^ { - 1 } ) ^ { \frac { 1 } { 2 } } u _ 0 | | _ { \omega } + \int _ M | | \tau _ 0 | | _ { \omega } \le \int _ M \langle ( B + \lambda I ) ^ { - 1 } v , v \rangle _ { \omega , h } d V _ { \omega } . \end{align*}


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


\begin{align*} \rho ^ { \sigma } ( g ) : = \rho ( g ^ { \sigma } ) , \ \ g \in G . \end{align*}


\begin{align*} t _ n = \frac { \alpha ^ n - \beta ^ n } { \alpha - \beta } \end{align*}


\begin{align*} f ( x ) = \int _ \S W _ r ( x - y ) \prod _ { i = 1 } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( y ) - \tilde \Psi ( x ) ) \ \d y \end{align*}


\begin{align*} \lambda _ { f } ( n ) = \lambda _ t ( n ) = \lambda _ s ( n ) \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n { \binom n k F _ { j k + m } B _ { n - k } ( x ) z ^ k } = \frac { 1 } { \sqrt 5 } \big ( \alpha ^ m B _ n ( x + \alpha ^ j z ) - \beta ^ m B _ n ( x + \beta ^ j z ) \big ) \ , , \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*} \lim _ { i \rightarrow \infty } g _ { n _ i } ( t , \omega ) = 0 , ( t , \omega ) . \end{align*}


\begin{align*} \dot { s } \varepsilon ( x ) \dot { s } = \displaystyle \varepsilon ( - x ^ { - 1 } ) \dot { s } h ( - x ) \varepsilon ( - x ^ { - 1 } ) . \end{align*}


\begin{align*} F ^ { i j } v _ { i j } \geq \delta _ 1 \left ( \frac { 1 } { b _ { n - 1 } ^ { 1 / k } } + \sum _ { i = 1 } ^ n F ^ { i i } \right ) \mbox { i n } \omega . \end{align*}


\begin{align*} \mathbf z \cdot \theta = \sum _ { j = 2 } ^ { n } \alpha _ j \frac { \mathbf z \cdot \omega _ j - s \omega _ 1 \cdot \omega _ j } { \sqrt { 1 + s ^ 2 } } . \end{align*}


\begin{align*} \gamma ( s , \tau ) = \left ( r _ * \cos \left ( \frac { - s - a _ 1 \tau } { \sqrt { k ^ 2 + r _ * ^ 2 } } \right ) , r _ * \sin \left ( \frac { - s - a _ 1 \tau } { \sqrt { k ^ 2 + r _ * ^ 2 } } \right ) , \frac { k s - b _ 1 \tau } { \sqrt { k ^ 2 + r _ * ^ 2 } } \right ) ^ t , \end{align*}


\begin{align*} \left \{ \begin{aligned} \sigma _ k \big ( \lambda ( D ^ { 2 } u ) \big ) & = f \ ; \ ; \mbox { i n } ~ \Omega , \\ u & = \varphi \ ; \ ; \mbox { o n } ~ \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \varphi ( - t ) ^ 2 H ( - t ) + \varphi ( t ) ^ 2 H ( t ) = 2 \varphi ( t ^ 4 ) ^ 2 G ( t ^ 4 ) . \end{align*}


\begin{align*} g _ d ( \ell ) = \begin{cases} \ell , & d = 1 , \\ \log \ell , & d = 2 , \\ 1 , & d \geq 3 . \end{cases} \end{align*}


\begin{align*} \varphi ( \lambda _ 1 , \lambda _ 2 ) : = h ( \lambda _ 1 , \lambda _ 2 ) \cdot q _ { l , n } ( \lambda _ 1 , \lambda _ 2 ) . \end{align*}


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


\begin{align*} k _ \theta \cdot f _ { - } & = \frac { 1 } { \sqrt { 2 } } ( s ( \theta ) , c ( \theta ) , s ( \theta ) , c ( \theta ) ) , \\ k _ \theta \cdot f _ { + } & = \frac { 1 } { ( 1 0 8 ) ^ { \frac { 1 } { 4 } } } ( s ( 3 \theta ) , 3 c ( 3 \theta ) , - 3 s ( 3 \theta ) , - c ( 3 \theta ) ) , \end{align*}


\begin{align*} \mathcal D _ { \psi } : = \psi _ 0 \frac { \partial } { \partial x } + \psi _ 1 \frac { \partial } { \partial y } . \end{align*}


\begin{align*} \rho { c _ v } { \partial _ { { t } } } T = \nabla \cdot \left ( { \lambda \nabla T } \right ) + { \bar F } + \left ( { \vartheta - 1 } \right ) \frac { { \Delta t } } { 2 } \rho { c _ v } \partial _ { { t } } ^ 2 T , \end{align*}


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


\begin{align*} F ^ { i j } = \frac { \partial F ( D ^ 2 u ) } { \partial u _ { i j } } . \end{align*}


\begin{align*} F ^ { q , \dagger } ( 0 , p ) = 0 \end{align*}


\begin{align*} \mathcal { C } _ { w } f = \mathcal { P } ^ + \left ( f w _ - \right ) + \mathcal { P } ^ { - } \left ( f w _ + \right ) = \mathcal { P } ^ - ( f J ) . \end{align*}


\begin{align*} \Phi ( x ' ) : = \partial _ n K ^ L ( x ' , 1 ) , x ' \in \R ^ { n - 1 } . \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*} & \int _ M ( \eta + g ^ { - 1 } ) | D ^ { '' * } v | ^ 2 _ Q d V _ M + \int _ M \eta | D ^ { '' } v | ^ 2 _ Q d V _ M \\ \ge & \int _ M \langle [ \eta \sqrt { - 1 } \Theta _ Q - \sqrt { - 1 } \partial \bar { \partial } \eta - \sqrt { - 1 } g \partial \eta \wedge \bar { \partial } \eta , \Lambda _ { \omega } ] v , v \rangle _ Q d V _ M . \end{align*}


\begin{align*} a _ I : = \sum _ { w \in W _ I } w \textrm { a n d } b _ I : = \sum _ { w \in W _ I } \epsilon ( w ) \ , w \ , , \end{align*}


\begin{align*} \omega : = \{ x \in \Omega : | x _ \beta | < \delta \frac { b _ \alpha } { \sqrt { b _ \beta } } , \beta = \alpha + 1 , \ldots , n - 1 , x _ n < \delta ^ 2 b _ \alpha \} , \end{align*}


\begin{align*} \forall \theta > 0 \ \Rightarrow \sup _ { p \ge 1 } \left [ \ \frac { L ( p ^ { \theta } ) } { L ( p ) } \ \right ] = C ( \theta ) < \infty . \end{align*}


\begin{align*} \mathcal { W } _ { f _ { 1 , \infty } } ( n , \alpha , \epsilon , g , \psi ) : = \inf \bigg \{ \sum _ { i } c _ { i } ^ { - \alpha n _ { i } + S _ { 1 , n _ { i } } \psi ( x _ { i } , \epsilon ) } \bigg \} , \end{align*}


\begin{align*} F ' ( y ; h ) ( x ) = f _ { x } ' ( y ( x ) ; h ( x ) ) . \end{align*}


\begin{align*} [ i , j ] : = ( | \cdot | ^ i , | \cdot | ^ { i + 1 } , \ldots , | \cdot | ^ j ) \end{align*}


\begin{gather*} \inf _ { Q \in \Gamma } Q ( c ) = \sup _ { f _ 1 , \ldots , f _ n } \ , \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \end{gather*}


\begin{align*} p ^ { \ell } = p ^ { \ell - 1 } \ast \frac { A ^ { i } d } { A ^ { i } p ^ { \ell - 1 } } \end{align*}


\begin{align*} x _ { k } a _ { i j } ( x ) u _ { i k } u _ { j } = \frac { 1 } { 2 } \partial _ k \big ( a _ { i j } ( x ) u _ i u _ j x _ { k } \big ) - \frac { n } { 2 } a _ { i j } ( x ) u _ i u _ j - \frac { 1 } { 2 } x _ { k } \partial _ k a _ { i j } ( x ) u _ i u _ j , \end{align*}


\begin{align*} ( f , \dot \mu _ t ) = ( A _ { \mu _ t } f , \mu _ t ) , \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*} \widetilde { \varphi } ( g ) = \begin{cases} \varphi ( g ) & g \in H \\ 0 & g \notin H . \end{cases} \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*} v _ a ( u ) = \lim _ { r \to 0 } { \frac { \sigma ( a , r ) } { { r } ^ { \frac { 4 n ( m - 1 ) } { m } } } } \end{align*}


\begin{align*} M \left ( \begin{smallmatrix} 0 & t \\ 1 & 0 \end{smallmatrix} \right ) M = \left ( \begin{smallmatrix} 0 & t \\ 1 & 0 \end{smallmatrix} \right ) \textrm { a n d } A M = \left ( \begin{smallmatrix} 0 & t \\ 1 & 0 \end{smallmatrix} \right ) A \left ( \begin{smallmatrix} 0 & t \\ 1 & 0 \end{smallmatrix} \right ) . \end{align*}


\begin{align*} \epsilon _ { f \otimes \chi _ D } = \chi _ D ( - r ) \mu ( q ' ) q '^ { 1 / 2 } \lambda _ f ( q ' ) \epsilon _ f . \end{align*}


\begin{align*} f ( x ) _ { i j } = p _ { 1 } \pi ( t _ { i } ^ { - 1 } ) x \pi ( t _ { j } ) p _ { 1 } \forall x \in M , \ ; \forall i , j \in \{ 1 , \ldots , n \} \end{align*}


\begin{align*} P _ { \lambda } ( Z ) : = \prod _ { k = 1 } ^ m \left ( k ! \ , \det \left ( ( z _ { i , j } ) _ { i , j = 1 } ^ { k } \right ) \right ) ^ { \lambda _ { k } - \lambda _ { k + 1 } } . \end{align*}


\begin{align*} \abs { t _ n } _ p & = \abs { t _ \ell } _ p \abs { k } _ p , \intertext { w h i l e f o r $ p = 2 $ , } \abs { t _ n } _ 2 & = \begin{cases} \abs { t _ \ell } _ 2 & k \\ 2 \abs { t _ { 2 \ell } } _ 2 \abs { k } _ 2 & k \end{cases} \end{align*}


\begin{align*} & \sup _ { x \in \left ( x _ { 0 } , \infty \right ) } \left \| \langle x \rangle \mathcal { P } ^ { + } \left ( z ^ { - i } \bar { r } _ 1 ( z ) e ^ { - 2 i z x } \right ) \right \| _ { L _ z ^ 2 } \leq \left \| z ^ { - i } \bar r _ 1 ( z ) \right \| _ { H ^ 1 _ z } , i = 0 , 1 , \\ & \sup _ { x \in \left ( x _ 0 , \infty \right ) } \left \| \langle x \rangle \mathcal { P } ^ - \left ( r _ 2 ( z ) e ^ { 2 i z x } \right ) \right \| _ { L _ { z } ^ 2 } \leq \left \| r _ 2 ( z ) \right \| _ { H ^ 1 _ z } . \end{align*}


\begin{align*} \begin{array} { l l l } \tilde { F } ( \mathsf { g } _ { \mathbb { C } , \mathsf { S } } ( L ) ) \circ \Phi _ { S , S } & = & F ( \mu ) \circ F ( i d _ S \otimes _ C L ) \circ \Phi _ { S , S } \\ & = & F ( \mu ) \circ \Phi _ { S , S } \circ ( i d _ { F ( S ) } \otimes _ D F ( L ) ) \\ & = & \mathsf { g } _ { \mathbb { D } , \widetilde { \mathbb { F } } ( \mathsf { S } ) } ( \tilde { F } ( L ) ) . \end{array} \end{align*}


\begin{align*} u ( x , t ) = \lim _ { k \rightarrow 0 } ( k ^ { - 1 } \varphi ( x , t ; k ) ) _ { 1 2 } . \end{align*}


\begin{align*} p \ast q : = \sum _ { n \geq 1 } p _ n q _ n T ^ n . \end{align*}


\begin{align*} \sigma _ b \left ( Q ( z ) \right ) = 0 \quad | b ( z ) | \geq C _ 1 \ . \end{align*}


\begin{align*} R ( x ; z ) = \begin{pmatrix} \bar { r } _ 1 ( z ) r _ 2 ( z ) & \bar { r } _ 1 ( z ) \mathrm { e } ^ { - 2 i z x } \\ r _ 2 ( z ) \mathrm { e } ^ { 2 i z x } & 0 \end{pmatrix} , \end{align*}


\begin{align*} { \bf { u } } _ k ^ { \star } = { \left [ { \bf { \bar U } } ^ { \star } \right ] _ { : , k + 1 } } = { \left [ \begin{array} { l } { { \bf { I } } _ K } \\ { { \bf { 0 } } _ { \left ( { { M _ t } - K } \right ) \times K } } \end{array} \right ] _ { : , k + 1 } } , \ \forall k , \end{align*}


\begin{align*} D ^ k \varphi ( Z ) = \sum _ { q \geq k } \frac { b _ { q } } { ( q - k ) ! } H _ { q - k } ( Z ) D ^ k L \varphi ( Z ) = - \sum _ { q \geq k } q \ , \frac { b _ { q } } { ( q - k ) ! } H _ { q - k } ( Z ) , \end{align*}


\begin{align*} \overline { \mathfrak { D } _ { q ^ n } } \ \overline { U _ { q ^ n } } \dot { s } \varepsilon ( v ) \zeta = ( A + B + C ) \overline { U _ { q ^ n } } \dot { s } \overline { U ^ * _ { q ^ n } } { \bf 1 } _ { - } \in \Bbbk { \bf G } \zeta . \end{align*}


\begin{align*} F _ k ^ + ( f ) = ( f _ i ' ) _ { i \in I } \colon F _ k ^ + ( M ) \rightarrow F _ k ^ + ( M ' ) \end{align*}


\begin{align*} f ( z ) c _ { \psi } = \int _ { \partial \Omega } f ( w ) E _ { \psi } ( w , z ) d \sigma _ { \psi } ( w ) - \int _ \Omega E _ { \psi } ( w , z ) { \mathcal D } _ { \psi } f ( w ) d x d y , \end{align*}


\begin{align*} T x = \frac { 1 } { \lambda } x . \end{align*}


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


\begin{align*} I _ { \varepsilon } ( u ) = \frac { 1 } { 2 } \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x - \frac { 1 } { ( p + 1 ) \varepsilon ^ 2 } \int _ { \Omega } \left ( u - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p + 1 } _ + d x , \ \ \forall u \in \mathcal { H } ( \Omega ) . \end{align*}


\begin{align*} \frac { 1 } { \varepsilon ^ 2 \ln \frac { 1 } { \varepsilon } } \int _ \Omega \left ( v ^ \tau _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + q \ln \frac { 1 } { \varepsilon } d x \rightrightarrows q ^ { p + 1 } ( \bar { x } ) \int _ { \hat { B } _ \tau ( 0 ) } ( \hat { U } ( y ) + \ln \tau ) ^ { p } _ + d y . \end{align*}


\begin{align*} \mu ^ * & = \mu ^ { \rm N } - \varkappa ^ { \rm N } , \qquad \ , \varkappa = 2 \varkappa ^ { \rm N } , \qquad \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*} Y [ k ] = \frac { 1 } { \sqrt { K } } \sum _ { n = 0 } ^ { K - 1 } y [ n ] e ^ { - \frac { 2 \pi n k j } { K } } . \end{align*}


\begin{align*} \mu _ { n } = \int _ { \Pi } x _ { 2 } \omega _ { n } \mathrm { d } x \rightarrow \mu , ~ n \rightarrow \infty , \end{align*}


\begin{align*} \sum _ { Y < p \leq X } \frac { 1 - ( p ^ { 2 i D t } ) } { p } & = \frac { 1 } { 2 \pi } \left ( \int _ 0 ^ { 2 \pi } ( 1 - \cos \alpha ) d \alpha \right ) \cdot \log \left ( \frac { \log ( 2 D | t | \log X ) } { \max \{ 1 , D | t | \log N _ { X , \eta } \} } \right ) - O _ K ( 1 ) \\ & \geq \left ( \frac { 1 } { 3 } - \eta \right ) \min \{ \log \log X , \log ( 1 + | t | \log X ) \} - O _ K ( 1 ) . \end{align*}


\begin{align*} A = \left ( \begin{array} { c c } B & C \\ C ^ T & D \end{array} \right ) \end{align*}


\begin{align*} \left ( \sqrt { p ' _ j } - \sqrt { q ' _ j } \right ) ^ 2 & = q ' _ j \left ( \sqrt { 1 + \frac { p ' _ \gamma - q ' _ j } { q ' _ j } } - 1 \right ) ^ 2 \leq \frac { ( p ' _ i - q ' _ j ) ^ 2 } { q ' _ j } . \end{align*}


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


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


\begin{align*} \vert S \cap [ 1 , n ] \vert \leq \sum _ { \ell = 1 } ^ U \vert S _ { \ell } \vert = O ( U ^ 2 ) = O ( ( \log n ) ^ 2 ) = o ( n / \log n ) \end{align*}


\begin{align*} - \Delta z _ { n } + \frac { F ( y _ { n } ) - F ( y ) } { t _ { n } } = h _ { n } . \end{align*}


\begin{align*} \liminf _ { i \to \infty } \frac { w _ { n _ i } } { n _ i ^ { r - \frac { 1 } { 2 } } } = 0 \limsup _ { i \to \infty } \frac { w _ { n _ i } } { n _ i ^ { r - \frac { 1 } { 2 } } } = \infty \end{align*}


\begin{align*} A ^ U _ \xi : = I ^ * ( { \ge } \xi , { < } \nu ) \end{align*}


\begin{align*} \alpha _ y = \mu _ y , \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*} H ( d G , d G ) = Q _ 0 d z ^ 2 + \left ( \frac { \tau _ 0 } { 4 } + \frac { 4 | Q _ 0 | ^ 2 } { \tau _ 0 } \right ) d z d \overline { z } + \overline { Q _ 0 } d \overline { z } ^ 2 . \end{align*}


\begin{align*} { \rm I m } ( M ( w ) ) = \frac { { \rm I m } ( w ) } { | c w + d | ^ { 2 } } . \end{align*}


\begin{align*} [ M ] & = \frac { \prod \limits _ { i = 0 } ^ N v _ i ^ { x _ i } } { \prod \limits _ { n = 1 } ^ \infty ( 1 - z ^ n ) ^ { N } \prod \limits _ { n = 1 } ^ \infty \prod \limits _ { 1 \leq i < j \leq N } ( 1 - v _ j v _ i ^ { - 1 } z ^ { n - 1 } ) ( 1 - v _ i v _ j ^ { - 1 } z ^ n ) } . \end{align*}


\begin{align*} \begin{array} { l } \mathop { \max } \limits _ { \bf p } \eta = \frac { R _ T } { P _ T } \\ { \rm { s } } { \rm { . t } } { \rm { . } } \\ C 2 , \ , C 3 , \ , C 8 , \ , { \rm a n d } \ , \ , C 9 . \end{array} \end{align*}


\begin{align*} \Psi _ { a , b } ( x ' ) : = 4 \int _ a ^ b \Phi _ t * \Phi _ t ( x ' ) \ , \frac { d t } { t } , x \in \R ^ { n - 1 } . \end{align*}


\begin{align*} a _ \lambda v = \sum _ { j \in \mathbb { Z _ { + } } } ( a _ { ( j ) } v ) \frac { \lambda ^ { j } } { j ! } . \end{align*}


\begin{align*} \Phi \colon ( \C ^ { Z \times Z } ) ^ { G } \to \bigoplus _ { i = 1 } ^ k \C ^ { m _ i \times m _ i } \ , \ , \ , \ , A \mapsto \bigoplus _ { i = 1 } ^ k U _ i ^ * A U _ i . \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*} V _ 1 ( k ) : = \begin{pmatrix} 1 & 0 \\ 0 & 2 i k \end{pmatrix} , V _ 2 ( k ) : = \begin{pmatrix} ( 2 i k ) ^ { - 1 } & 0 \\ 0 & 1 \end{pmatrix} , \end{align*}


\begin{align*} T _ { \delta } ( x ; z ) : = \left ( M _ { \delta , + , 1 } ( x ; z ) - e _ 1 , M _ { \delta , - , 2 } ( x ; z ) - e _ 2 \right ) \begin{pmatrix} 1 & 0 \\ - r _ { \delta , 2 } ( z ) e ^ { 2 i z x } & 1 \end{pmatrix} \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \Big [ \sup _ { t \leq T } \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H ^ p \Big ] = 0 , \end{align*}


\begin{align*} \begin{aligned} N _ 3 ^ { ( m / 2 ) } [ \hat { f } ] < \infty \end{aligned} \end{align*}


\begin{align*} \operatorname { R } = \frac { 1 } { 2 } \mathcal { H } ^ 2 \end{align*}


\begin{align*} [ ( \lambda _ 1 , \mu _ 1 ) , \kappa _ 1 ] \cdot [ ( \lambda _ 2 , \mu _ 2 ) , \kappa _ 2 ] : = [ ( \lambda _ 1 + \lambda _ 2 , \mu _ 1 + \mu _ 2 ) , \kappa _ 1 + \kappa _ 2 + \lambda _ 1 \mu _ 2 ^ t - \mu _ 1 \lambda _ 2 ^ t ] . \end{align*}


\begin{align*} \lim _ { n \to \infty } \| T _ n p _ n f - p _ n T f \| = 0 f \in X . \end{align*}


\begin{align*} \underline { u } = \frac { 1 } { 2 } \sum _ { 1 \leq j < k \leq N } \langle u ( e _ j ) , e _ k \rangle \ e _ j \cdot e _ k . \end{align*}


\begin{align*} \dot { x } ( t ) & = - A x ( t ) + B u ( t ) \quad ( t > 0 ) , \\ x ( 0 ) & = x _ 0 \in X , \end{align*}


\begin{align*} \psi ( x , w ) = ( \gamma h + \nabla \gamma ) ( x ) + w . \end{align*}


\begin{align*} T _ \sigma ( f , g ) ( x ) = \langle K _ \sigma ( x , y , z ) , ( f \otimes g ) ( y , z ) \rangle , \end{align*}


\begin{align*} c _ s ^ 2 \partial _ { x 1 } T = - \frac { 1 } { \Delta t } { \varsigma } _ 1 { m } _ 1 ^ { ( 1 ) } , \end{align*}


\begin{gather*} 0 = P _ \alpha ( f ) = P _ \alpha \bigl \{ f \ , 1 ( \abs { f } \le k ) \bigr \} + P _ \alpha \bigl \{ f \ , 1 ( \abs { f } > k ) \bigr \} \\ \le P _ \alpha \bigl \{ f \ , 1 ( \abs { f } \le k ) \bigr \} + \sup _ { Q \in M } Q \bigl \{ \abs { f } \ , 1 ( \abs { f } > k ) \bigr \} \quad \quad k . \end{gather*}


\begin{align*} \beta _ y = \min \left \{ \frac { 1 } { 2 L _ x } , \frac { 1 } { 2 \eta _ y L _ { x y } ^ 2 } \right \} . \end{align*}


\begin{align*} \psi ^ { \pm } ( x , t ; k ) = I + k \int _ { \pm \infty } ^ { x } e ^ { - 2 i k ^ 2 ( x - y ) \widehat { \sigma } _ 3 } P _ y ( y ) \psi ^ { \pm } ( y , t ; k ) d y . \end{align*}


\begin{align*} H = 2 \int _ { \mathbb { R } ^ { 3 } } ( \phi - \phi _ { \infty } ) _ { + } \frac { G } { r ^ { 2 } } \dd x . \end{align*}


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


\begin{align*} \limsup _ { n \to \infty } \sum _ { m = 1 } ^ n \| z _ { n , m } - w _ { n , m } \| _ { L ^ 2 ( \mu ) } \leq \limsup _ { n \to \infty } \varepsilon \| g \| _ { L ^ 2 ( \mu ) } ^ 3 \sum _ { m = 1 } ^ n \mathbb { E } _ { n , m } \big [ \| \chi _ { n , m } \| _ { L ^ 2 ( \mu ) } ^ 2 \big ] \end{align*}


\begin{align*} h = 0 , \nabla h = 0 \mbox { o n } \hat \Sigma \end{align*}


\begin{align*} \langle [ e _ i ] , [ e _ j ] \rangle = d _ { i j } = 1 + \frac { \delta _ { i j } } { \varphi _ i } \quad \forall i , j \in I , \end{align*}


\begin{align*} \partial _ \theta \Phi = \partial _ \lambda \Phi = \partial _ t \Phi = 0 \end{align*}


\begin{align*} y ( q ) = \psi ( q ^ 3 ) \left ( \frac { \varphi _ 6 \varphi _ 9 ^ 2 } { \varphi _ 3 \varphi _ { 1 8 } } - 2 q \psi ( q ^ 9 ) \right ) = \frac { \varphi _ 6 ^ 2 } { \varphi _ 3 } \frac { \varphi _ 6 \varphi _ 9 ^ 2 } { \varphi _ 3 \varphi _ { 1 8 } } - 2 q \psi ( q ^ 3 ) \psi ( q ^ 9 ) = z ( q ^ 3 ) - 2 q \psi ( q ^ 3 ) \psi ( q ^ 9 ) \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*} \rho ( g ) & = g & g \in G \\ \rho ^ { \vee } ( g ) & = ( g ^ { - 1 } ) ^ { \vee } ( \varphi ) & g \in G . \end{align*}


\begin{align*} H _ 2 = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 3 } } \Theta \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = \theta ( a ) q ^ 8 ( 1 - q ) ( 1 - q ^ 2 ) . \end{align*}


\begin{align*} { \rm G a l } ( L ' / K ' ) = { \rm G a l } ( L ' / N _ 1 ) \times { \rm G a l } ( L ' / N _ 2 ) . \end{align*}


\begin{align*} S ( u ) = \frac { 1 } { 2 } \sum _ { n = 0 } ^ { + \infty } \xi _ { n } \cdot \nabla \left ( \xi _ { n } \cdot \nabla u \right ) . \end{align*}


\begin{align*} H [ b _ n ] = 4 \pi \int _ { D } ( \phi _ n - \phi _ \infty ) _ + G _ n \frac { 1 } { r } \dd z \dd r + 4 \pi \int _ { \mathbb { R } ^ { 2 } _ { + } \backslash D } ( \phi _ n - \phi _ \infty ) _ + G _ n \frac { 1 } { r } \dd z \dd r . \end{align*}


\begin{align*} ( A ^ q \eta ) ( x ) = \int _ \S W _ r ( x - y ) ( \eta ( y ) - \eta ( x ) ) \cos ( 2 \pi q ( y - x ) ) \ \d y - \int _ \S W _ r ( y ) \eta ( y ) \cos ( 2 \pi q y ) ) \ \d y . \end{align*}


\begin{align*} f ( x ) = \frac { 1 } { \sigma \mathrm { B } ( \alpha , \beta ) } \ , \left [ \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \alpha - 1 } \ , \left [ 1 - \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \beta - 1 } \ , \phi \left ( \frac { x - \mu } { \sigma } \right ) , x \in \mathbb { R } , \end{align*}


\begin{align*} X _ \beta = \{ ( x , y , w , z ) \in X : w - x = \beta \} \end{align*}


\begin{align*} \mathfrak g = \mathfrak g _ + \oplus \mathfrak h \oplus \mathfrak g _ - \end{align*}


\begin{align*} \Psi _ { a , b } ( x ' ) : = 4 \int _ a ^ b \Phi _ t * \Phi _ t ( x ' ) \ , \frac { d t } { t } , x \in \R ^ { n - 1 } . \end{align*}


\begin{align*} s _ i ( A ) = \min \limits _ { E : \ ; \dim ( E ) = n - i + 1 } \max \limits _ { x \in E , \ , \lVert x \rVert _ 2 = 1 } \lVert A x \rVert _ 2 , 1 \leq i \leq n , \end{align*}


\begin{align*} \int _ { \Omega } n \xi ( \cdot , t _ { 2 } ) - \int _ { t _ { 1 } } ^ { t _ { 2 } } \int _ { \Omega } n \xi _ { t } + \int _ { t _ { 1 } } ^ { t _ { 2 } } \int _ { \Omega } ( \nabla n - n S ( x , n , c ) \cdot \nabla c ) \cdot \nabla \xi = \int _ { \Omega } n \xi ( \cdot , t _ { 1 } ) , \end{align*}


\begin{align*} \mathbb { B } _ { k , S ; ( n , g ) } ( \tau , z ) = \mathbb { B } _ { k , S [ u ] ; ( n , g ) } ( \tau , ( u ^ t ) ^ { - 1 } z ) , \end{align*}


\begin{align*} \sum _ { \substack { \tau \in D ( \ell ) \\ \mathrm { s i g n } ( \tau ) = 1 } } \prod _ { j = 1 } ^ { \ell } \frac { 1 } { x _ { \tau ( j ) } - x _ j } = \sum _ { \substack { X = \bigsqcup _ { i = 1 } ^ { s } X _ i \\ | X _ i | \geqslant 2 \\ s \ } } \left ( \prod _ { j = 1 } ^ s f ( X _ i ) \right ) . \end{align*}


\begin{align*} \theta ^ h = 2 ~ { \rm a n d } ~ q \ge h \log _ 2 ( 1 + d 2 ^ { b _ 0 + 1 } ) . \end{align*}


\begin{align*} g _ I = \big ( g _ { \{ i \} } \cup g _ { I \setminus \{ i \} } \big ) \setminus \big ( g _ { \{ i \} } \cap g _ { I \setminus \{ i \} } \big ) \ , , \forall i \in I . \end{align*}


\begin{align*} \Lambda ( s , \rho ) = \epsilon ( \rho ) c ( \rho ) ^ { \frac { 1 } { 2 } - s } \Lambda ( 1 - s , \overline { \rho } ) , \end{align*}


\begin{gather*} m _ { n } ( t ) = \allowbreak \sqrt { \frac { n ! } { \left ( \beta \right ) ^ { \left ( n \right ) } } } \frac { 1 } { ( 1 - t ) ^ { \beta } } \frac { 1 } { \Gamma ( \beta ) } \times \\ \int _ { 0 } ^ { \infty } \left ( \sum _ { j = 0 } ^ { n } \left ( \frac { - t y } { 1 - t } \right ) ^ { j } L _ { n - j } ( x | ( \beta + j ) ) / j ! \right ) y ^ { \beta - 1 } \exp ( - y ) d y . \end{gather*}


\begin{align*} S ( k ) = \begin{pmatrix} a ( k ) & b ( k ) \\ [ 3 p t ] - \overline { b ( \bar { k } ) } & \overline { a ( \bar { k } ) } \end{pmatrix} , \ \ \det S ( k ) = 1 . \end{align*}


\begin{align*} \begin{aligned} x _ n ^ * - x _ n ^ 0 = \ , & x _ n - \sum _ { \beta = 1 } ^ \alpha x _ \beta \rho _ \beta ( 0 , \tilde { x } ) - \rho ( 0 , \tilde { x } ) \\ > \ , & \rho ( \hat { x } , \tilde { x } ) - \sum _ { \beta = 1 } ^ \alpha x _ \beta \rho _ \beta ( 0 , \tilde { x } ) - \rho ( 0 , \tilde { x } ) \geq 0 \end{aligned} \end{align*}


\begin{align*} \mathcal { M } ^ { r } ( t ) \geq \mathcal { M } ^ { r } ( \delta ) + ( r - 1 ) \theta _ { 1 } \varrho _ { 0 } ^ { 1 - \sigma } \int _ { \delta } ^ { t } ( \mathcal { M } ^ { r } ( s ) ) ^ { \sigma + 1 } d s , 0 < \delta < t < t _ { 0 } < T _ { \mathrm { g e l } } . \end{align*}


\begin{align*} F ( \eta ) : = T - \frac { 1 } { \eta } I , \eta \in \Omega . \end{align*}


\begin{align*} a ( u , v ) = \int _ D \nabla u \cdot \nabla \bar { v } ~ d x , ( f , v ) = \int _ D f \bar { v } ~ d x . \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 x = T ( x _ 1 , x _ 2 , \dots ) : = ( \alpha _ 1 T _ 1 x _ 1 , \alpha _ 2 T _ 2 x _ 2 , \dots ) . \end{align*}


\begin{align*} \Delta _ { \Psi } u = \div _ { \Psi } ( \nabla u ) . \end{align*}


\begin{align*} H = \begin{pmatrix} \psi _ { v v } ( 0 , p _ 0 ) [ v , v ] & 0 \\ 0 & 0 \end{pmatrix} \end{align*}


\begin{align*} \mathcal { L } _ 1 f = - \sum _ { i , j = 1 } ^ 3 \left \{ \partial _ i [ ( a _ { i j } * \mu ) \partial _ j f ] + ( a _ { i j } * \mu ) \frac { v _ i } { 2 } \frac { v _ j } { 2 } f - \partial _ i \left [ ( a _ { i j } * \mu ) \frac { v _ j } { 2 } \right ] f \right \} , \end{align*}


\begin{align*} [ T ] + \nu [ N ] = \frac { 2 } { \mu } h _ z J + \nu I , \end{align*}


\begin{align*} \mu _ m ^ { ( 1 2 ) } \big ( \mathfrak { s } \boldsymbol { J } \big ) = \mu _ m ^ { ( 1 2 3 ) } \big ( \boldsymbol { J } \big ) \mu _ m ^ { ( 1 2 3 ) } \big ( \mathfrak { s } \boldsymbol { J } \big ) = \mu _ m ^ { ( 1 2 ) } \big ( \boldsymbol { J } \big ) . \end{align*}


\begin{align*} \begin{aligned} \abs { c _ 3 ( S ^ 3 \setminus S _ i ) } & \le \abs { c _ 2 ( S ^ 2 \setminus S _ i ) } + \abs { c _ 3 ( P ^ 3 \setminus S _ i ) } \\ & \le ( 2 k - t ' - \abs { S \cap S _ i } - q _ i - s _ 1 ) + ( s _ 1 - t _ i ) = 2 k - t ' - \abs { S \cap S _ i } - p _ i . \end{aligned} \end{align*}


\begin{align*} \psi _ 1 = \mathcal { C } _ { \rm s } \delta , \rm { a n d } \ , \ , \ , \psi _ 2 = \mathcal { C } _ { \rm s } P _ { \rm b i a s } + \mathcal { C } _ { \rm n } . \end{align*}


\begin{align*} \| v \| _ { - m , p ' } : = \sup _ { 0 \not = u \in W ^ { p , m } _ 0 } \frac { | \langle u , v \rangle | } { \| u \| _ { m , p } } . \end{align*}


\begin{align*} 1 - n + m - t = 0 , \end{align*}


\begin{align*} [ E : F ] = [ F ( \alpha , \gamma ) : F ] = [ F ( \alpha , \gamma ) : F ( \gamma ) ] [ F ( \gamma ) : F ] \leq m ( q - 1 ) \end{align*}


\begin{align*} H _ 2 = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 3 } } \Theta \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = q ^ 8 ( 1 - q ) ( 1 - q ^ 2 ) . \end{align*}


\begin{align*} ( R _ \lambda ( s ) f ) ( \tau ) = \int _ { \tau } ^ 1 { E _ \lambda ( \tau - \xi , s ) C ^ { - 1 } ( s ) f ( \xi ) \ \dd \xi } . \end{align*}


\begin{align*} \sigma ( \sqrt { a } ) = - \sqrt { a } , \ & \sigma ( \sqrt { d } ) = \sqrt { d } \\ \tau ( \sqrt { a } ) = \sqrt { a } , \ & \tau ( \sqrt { d } ) = - \sqrt { d } . \end{align*}


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


\begin{align*} \sum _ { k = 0 } ^ n { \binom n k L _ { j k + m } B _ { n - k } z ^ k } = \alpha ^ m B _ n ( \alpha ^ j z ) + \beta ^ m B _ n ( \beta ^ j z ) . \end{align*}


\begin{align*} | h ( x + \delta u ) - h ( x ) | = | R ( x , \delta u ) | \le L _ r \delta . \end{align*}


\begin{align*} r _ { n , k } = \deg ( C _ { n , k } ) \leq n . \end{align*}


\begin{align*} \mathcal { P } ^ { \pm } ( h ) ( z ) = \pm \frac { 1 } { 2 } h ( z ) - \frac { i } { 2 } \mathcal { H } ( h ) ( z ) z \in \mathbb { R } . \end{align*}


\begin{align*} \partial _ i \circ \pi : = \partial _ { u _ i } \circ \pi = \pi _ { * } X _ i . \end{align*}


\begin{align*} F ' ( x , y ) = L _ 1 \circ F \circ L _ 2 ( x , y ) . \end{align*}


\begin{align*} \Sigma ^ { \pm } ( s - w , \phi , F _ { \pm } ) = & \lim _ { z \downarrow 0 } 2 \int _ 0 ^ \infty \lambda ^ { 4 ( 1 - s + w ) - 1 } \int _ { - \infty } ^ \infty \int _ 0 ^ \infty \sum _ { \ell , m = 1 } ^ \infty \rho _ \phi ( 3 \ell m ) K _ \nu ( 6 \pi \ell m t ^ 2 ) \\ & \times \cos \left ( \frac { 6 \pi \ell m t ^ 3 u } { \lambda } \right ) \hat { F } \left ( 0 , 0 , \frac { 3 m \lambda } { t } , u \right ) \frac { d \lambda } { \lambda } t d t d u . \end{align*}


\begin{align*} h _ { k l i j } = & \ h _ { i j k l } - h _ { m l } ( h _ { i m } h _ { k j } - h _ { i j } h _ { m k } ) - h _ { m j } ( h _ { m i } h _ { k l } - h _ { i l } h _ { m k } ) \\ & - h _ { m l } ( \delta _ { i j } \delta _ { k m } - \delta _ { i k } \delta _ { j m } ) - h _ { m j } ( \delta _ { i l } \delta _ { k m } - \delta _ { i k } \delta _ { l m } ) . \end{align*}


\begin{align*} \mu _ { ( \boldsymbol { x } , \boldsymbol { \Psi } ( t , \boldsymbol { x } ) ) } ^ { N , i } + \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } h \Big ( \mu _ { ( \boldsymbol { x } , \boldsymbol { \Psi } ( t , \boldsymbol { x } ) ) } ^ { N , j } \Big ) = - R ^ { - 1 } B \Big [ P _ t \nu ^ { N , i } _ { \boldsymbol { x } } + \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } \Psi ^ { N , j } ( t , \boldsymbol { x } ) \Big ] . \end{align*}


\begin{align*} \sum _ { i , j = 1 } ^ 3 \int _ { \mathbb R ^ 3 } \partial _ i [ ( a _ { i j } * \mu ) v _ j ] f ^ 2 d v & = \sum _ { i , j = 1 } ^ 3 \int _ { \mathbb R ^ 3 } \partial _ i \bigg ( \int _ { \mathbb R ^ 3 } a _ { i j } ( v - v ' ) v _ j ' \mu ( v ' ) d v ' \bigg ) f ^ 2 d v \\ & = \sum _ { i , j = 1 } ^ 3 \int _ { \mathbb R ^ 3 } \partial _ i [ a _ { i j } * ( v _ j \mu ) ] f ^ 2 d v . \end{align*}


\begin{align*} \begin{cases} - \delta ^ 2 ( K _ H ( x ) \nabla v ) = \left ( v - \left ( \frac { \alpha | x | ^ 2 } { 2 } + \beta \right ) \right ) ^ { p } _ + , \ \ & x \in B _ { R ^ * } ( 0 ) , \\ v = 0 , \ \ & x \in \partial B _ { R ^ * } ( 0 ) . \end{cases} \end{align*}


\begin{align*} \forall x \in E , \forall \lambda \in \mathbf { C } , \ \overline { x \otimes _ \gamma \lambda } : = x \otimes _ { _ { \overline { \gamma } } } \overline { \lambda } . \end{align*}


\begin{align*} P _ n P _ { n ' } = \begin{cases} P _ n , & n ' = n , \\ 0 , & n ' \ne n . \end{cases} \end{align*}


\begin{align*} P ^ \infty & = \varprojlim P ^ r = \left \{ u = ( u ^ r ) _ { r \in \mathbb { N } } \in \prod _ { r \in \mathbb { N } } P ^ r \ ; \middle | \ ; \pi ^ r _ s u ^ r = u ^ s \ \right \} , \\ * \widetilde { P } ^ \infty & = \varprojlim \widetilde { P } ^ r , \\ * G ^ \infty & = P ^ \infty ( \mathbb { R } ^ n , o ) , \\ * \widetilde { G } ^ \infty & = \widetilde { P } ^ \infty ( \mathbb { R } ^ n , o ) , \end{align*}


\begin{align*} L _ x v = B ( x , v ) + B ( v , x ) \end{align*}


\begin{align*} \partial _ X \langle \langle \varphi , \varphi ' \rangle \rangle = \langle \langle \nabla _ X \varphi , \varphi ' \rangle \rangle + \langle \langle \varphi , \nabla _ X \varphi ' \rangle \rangle \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n \binom n k \frac { 2 ^ k L _ { j k } } { L _ j ^ { k } } B _ { n - k } = n \Big ( \frac { \sqrt 5 F _ j } { L _ j } \Big ) ^ { n - 1 } , \qquad \mbox { $ n $ o d d } . \end{align*}


\begin{align*} \hat { f } _ i ( x _ i ) = \int _ { \R ^ { n - 1 } } f ( x _ 1 , \ldots , x _ n ) \ , \prod _ { j \neq i } Q _ j ( d x _ j ) \ge - c _ 1 e ^ { c _ 2 x _ i ^ 2 } \prod _ { j \neq i } \int _ { \R } e ^ { c _ 2 x _ j ^ 2 } Q _ j ( x _ j ) \ , d x _ j \end{align*}


\begin{align*} \partial _ y T ( x _ 0 , y ) = 1 . \end{align*}


\begin{align*} \overline { \mathrm { s p a n } } _ { H ^ 2 ( \mathbb { D } ) } \{ ( I - S ) h _ k \mid k \geq 2 \} = H ^ 2 ( \mathbb { D } ) . \end{align*}


\begin{align*} I : = & \mathbb { E } \int _ 0 ^ { T \wedge \tau _ { X } ^ M } f ( s ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s , \\ I I : = & \mathbb { E } \int _ 0 ^ { T \wedge \tau _ { X } ^ M } \rho ( X _ n ( s ) ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s , \\ I I I : = & \mathbb { E } \int _ 0 ^ { T \wedge \tau _ { X } ^ M } \eta ( X ( s ) ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s . \end{align*}


\begin{align*} \int _ { \{ z \in M : - \Psi ( z ) \in N \} } | \tilde { F } | ^ 2 _ h = 0 \end{align*}


\begin{align*} C _ { * } ( n ) : = ( C _ { * } ( R ^ { 2 n } ) , d ) = \cdots \rightarrow C _ { 2 } ( R ^ { 2 n } ) \rightarrow C _ { 1 } ( R ^ { 2 n } ) \xrightarrow { \varepsilon } \mathbb { Z } \rightarrow 0 \end{align*}


\begin{align*} B _ n ( - x ) = ( - 1 ) ^ n \big ( B _ n ( x ) + n x ^ { n - 1 } \big ) , \end{align*}


\begin{align*} { f _ i } \left ( { { \bf { x } } + { { \bf { c } } _ i } \Delta t , t + \Delta t } \right ) = { f _ i } ^ * \left ( { { \bf { x } } , t } \right ) . \end{align*}


\begin{align*} m _ { \mathrm A D M } ( g ) = \frac { 1 } { 2 ( n - 1 ) \omega _ { n - 1 } } \lim _ { r \to \infty } \int _ { | x | = r } \sum _ { i , j = 1 } ^ n \left ( \frac { \partial g _ { i j } } { \partial x _ i } - \frac { \partial g _ { i j } } { \partial x _ j } \right ) \frac { x _ j } { | x | } \ , d \sigma . \end{align*}


\begin{align*} & S ( k ) = I + \mathcal { O } ( k ^ { - 1 } ) , k \rightarrow \infty . \end{align*}


\begin{align*} [ J _ + ^ \alpha , J _ - ^ \alpha ] = 2 \omega ^ { - 2 p + 1 } ( \alpha ) J _ 0 ^ \alpha . \end{align*}


\begin{align*} \frac { 1 } { 2 } \underset { j = 1 } { \overset { N } { \sum } } \partial _ { x ^ j x ^ j } \phi ^ { N , i } ( t , \boldsymbol { x } ) \sigma ^ 2 + \frac { 1 } { 2 } \underset { j , \tau = 1 } { \overset { N } { \sum } } \partial _ { x ^ j } \partial _ { x ^ \tau } \phi ^ { N , i } ( t , \boldsymbol { x } ) \sigma _ 0 ^ 2 = \frac { 1 } { 2 } \partial _ { \nu \nu } \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) \sigma _ 0 ^ 2 + O ( \frac { 1 } { N } ) . \end{align*}


\begin{align*} \widetilde { H } _ { j } ( s ) & : = \frac { \widetilde { F } _ { j } ( s ) } { \widetilde { G } _ { j } ( s ) } \\ & = \prod _ { p } ^ { } \left ( \frac { 1 + \widetilde { A } } { 1 + \widetilde { B } } \right ) \\ & \ll _ { \epsilon } 1 , & & ( \mbox { i n $ \Re ( s ) > \frac { 5 } { 2 } $ } ) \end{align*}


\begin{align*} \xi = - L - 1 , \eta = \frac { Q } { 2 ( L + 1 ) } - \sqrt { \kappa B _ 4 } , \zeta = \kappa \left ( \frac { B _ 3 } { 2 \sqrt { \kappa B _ 4 } } + 1 \right ) , \sigma = \sqrt { \kappa B _ 4 } , \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*} J a = \frac { { { c _ p } \left ( { { T _ w } - { T _ { s a t } } } \right ) } } { { { h _ { l v } } } } , \end{align*}


\begin{align*} \sigma _ W ^ 2 = 3 2 / \alpha ^ 2 \sigma _ Z ^ 2 = 3 2 T ^ 2 / \alpha ^ 2 \end{align*}


\begin{align*} \omega ^ { - 1 } ( d x ^ i , d x ^ j ) = \omega ^ { - 1 } ( d \theta _ i , d \theta _ j ) = 0 , \omega ^ { - 1 } ( d \theta _ i , d x ^ j ) = \delta _ i ^ j , \end{align*}


\begin{align*} \int _ { \Omega } ( K _ H ( x ) \nabla v ^ \tau _ \varepsilon | \nabla v ^ \tau _ \varepsilon ) d x = & 2 \pi q ^ 2 \sqrt { ( K _ { H } ) _ { 1 1 } ( K _ { H } ) _ { 2 2 } } ( \bar { x } ) \ln \frac { 1 } { \varepsilon } + O ( 1 ) + C | \ln \tau | \\ = & 2 \pi q ^ 2 \sqrt { d e t ( K _ { H } ) } ( \bar { x } ) \ln \frac { 1 } { \varepsilon } + O ( 1 ) + C | \ln \tau | , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T \Vert B ( t , X ( t , x _ n ) ) - B ( t , X ( t , x ) ) \Vert _ { L _ 2 } ^ 2 d t = 0 . \end{align*}


\begin{align*} \begin{cases} \Delta _ { \Psi } w = A w & \O \\ w = 0 & \partial \O . \end{cases} \end{align*}


\begin{align*} P V _ { \delta , \hat { x } , \hat { q } , z } ( x ) = V _ { \delta , \hat { x } , \hat { q } , z } ( x ) - \frac { \hat { q } } { \ln \frac { R } { s _ \delta } } g _ { \hat { x } } ( T _ { \hat { x } } x , T _ { \hat { x } } z ) , \ \ \forall x \in \Omega , \end{align*}


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


\begin{align*} \lim \limits _ { \alpha \to \infty } \sup _ { \Vert f \Vert _ { L ^ 1 ( \nu ) } \leq 1 } \psi _ \alpha ( K ^ { * } f ) = 0 \iff \lim \limits _ { \alpha \to \infty } \sup _ { \omega = \frac { 1 } { H } \sum _ { h = 1 } ^ { H } \delta _ { a _ h } } \psi _ { \alpha } ( K ^ * \omega ) = 0 \end{align*}


\begin{align*} \phi ( x ) = \Psi \left ( \frac { \ln \left ( \frac { | x | } { \sqrt { R } } \right ) } { \ln \left ( \sqrt { R } \right ) } \right ) , \ , \ , x \in \mathbb { R } ^ { N } . \end{align*}


\begin{align*} \gamma _ { \rm d } = \frac { \alpha ^ 2 \frac { K } { K - 2 } } { \sigma _ { W _ { \rm d } } ^ 2 } , \mathrm { a n d } \gamma _ { \rm s } = \frac { \alpha ^ 2 \frac { K } { K - 2 } } { \sigma _ { W _ { \rm s } } ^ 2 } , \end{align*}


\begin{align*} F _ n ( \eta ) : = T _ n - \frac { 1 } { \eta } I _ n , \end{align*}


\begin{align*} [ g ' k g h , \psi k \varphi h ] = [ g ' g \underbrace { c _ g ( k ) h } _ { \in H } , \psi k \varphi h ] = [ g ' g , \psi k \varphi h ( c _ { g } ( k ) h ) ^ { - 1 } ] = [ g ' g , \psi k \varphi c _ { g ^ { } } ( k ^ { - 1 } ) ] = [ g ' g , \psi \varphi ] \end{align*}


\begin{align*} B _ n ( 1 - x ) = ( - 1 ) ^ n B _ n ( x ) , \end{align*}


\begin{align*} \mathcal { U } = \{ u \in \hat { \mathcal { N } } _ p ( q ) : u \} \subseteq \hat { \mathcal { N } } _ p ( q ) \cap \{ u = ( u _ i ) _ i : 1 + \sum _ i u _ i = 0 \} \cong \mathbb { R } ^ { p + 1 - k } . \end{align*}


\begin{align*} \norm { u _ j } _ { L ^ 2 } ^ 2 & = \int _ \S \left ( \int _ { x - r } ^ { x + r } \prod _ { \stackrel { i = 1 } { i \neq j } } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) ( - \partial \eta _ j ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( y ) - \tilde \Psi ( x ) ) \ \d y \right ) ^ 2 \ \d x \\ & \le \int _ \S \int _ \S \prod _ { \stackrel { i = 1 } { i \neq j } } ^ n \norm { \partial \eta _ i } _ { L ^ 2 } ^ 2 ( \partial \eta _ j ( x ) ) ^ 2 \ \d y \ \d x \\ & = \prod _ { i = 1 } ^ n \norm { \partial \eta _ i } _ { L ^ 2 } ^ 2 \end{align*}


\begin{align*} \Phi _ { t } + u \cdot \nabla \Phi & = \mu \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi . \end{align*}


\begin{align*} \lambda _ \mathrm { a } [ n ] = \mathcal { C } _ { \rm s } x _ { \rm t } [ n ] + \mathcal { C } _ { \rm n } . \end{align*}


\begin{align*} \lim _ { n \to \infty } \| T _ n p _ n f - p _ n T f \| = 0 f \in X . \end{align*}


\begin{align*} \partial _ r ( d _ { i s } ) + ( d _ { i s } - d _ { i r } ) \Gamma _ { r s } ^ s = \sum _ j \phi _ { i j } ( \partial _ r ) d _ { j s } \varphi _ j = \phi _ { i s } ( \partial _ r ) ( d _ { s s } - 1 ) \varphi _ s + \sum _ j \phi _ { i j } ( \partial _ r ) d _ j \varphi _ j = \phi _ { i s } ( \partial _ r ) , \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*} \mathcal { P } _ s = \{ V _ 1 \subset V _ 2 \subset \ldots V _ { l - 1 } \subset V _ { l + 1 } \subset \ldots \subset V _ { n } = \mathbb { C } ^ n ; \dim _ { \mathbb { C } } ( V _ i ) = i \} . \end{align*}


\begin{align*} \int _ { \Omega } ( g \Delta f - f \Delta g ) { \rm d } \mathbf x = \int _ { \partial \Omega } ( g \partial _ { \nu } f - f \partial _ \nu g ) { \rm d } \sigma , \end{align*}


\begin{align*} \hat U ( \xi , t ) = \exp ( E ( \xi ) t ) \hat w _ 0 ( \xi ) . \end{align*}


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


\begin{align*} | f _ \alpha ( U | \bar U ; x , t ) | & \le | f _ \alpha - \bar f _ \alpha - \nabla \bar f _ \alpha ( U - \bar U ) ) | \\ & + \left | \nabla \bar f _ \alpha \nabla \bar A ^ { - 1 } \big ( A ( U , x , t ) - \bar A - \nabla \bar A ( U - \bar U ) \big ) \right | \\ & \le k _ 4 | A ( U , x , t ) - \bar A | ^ 2 \ , , \end{align*}


\begin{align*} f ' ( x , y ) & = \alpha ( f \circ L ( x , y ) ) , \\ & = \alpha \left ( a ( t x + u y ) ^ { q + 1 } + b ( t x + u y ) ^ q ( v x + w y ) + c ( t x + u y ) ( v x + w y ) ^ q + d ( v x + w y ) ^ { q + 1 } \right ) , \end{align*}


\begin{align*} \bigcap _ { i = 1 } ^ { n } \left ( B \left ( X _ { i } , r \right ) ^ { c } \cap C _ { j } \right ) \neq \emptyset . \end{align*}


\begin{align*} N \varphi : = A ( x ) \nabla \varphi \cdot \frac { x } { | x | } . \end{align*}


\begin{align*} & \lim _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \int _ { \{ z \in M : - \Psi ( z ) \in I _ { i , n } \backslash U _ k \} } | \tilde F | ^ 2 _ h \\ \le & \frac { G ( T _ 1 ) } { \int _ { T _ 1 } ^ { + \infty } c ( s ) e ^ { - s } d s } \limsup _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \frac { 1 } { \inf _ { I _ { i , n } \backslash U _ k } c ( t ) } \int _ { I _ { i , n } \backslash U _ k } c ( s ) e ^ { - s } d s . \end{align*}


\begin{align*} \alpha ( g , Z ) = ( \xi + z _ 1 \psi _ 1 ) \alpha ( g , Z ) - \alpha ( g , Z ) \cdot \left ( \sum _ { \begin{smallmatrix} 1 \in I \subseteq [ 1 , n ] \\ | I | > 1 \end{smallmatrix} } ( z _ { I ' } + 1 ) \cdot \delta _ { I ' } \right ) \end{align*}


\begin{align*} - F v = \norm { v } _ { V } F \left ( \frac { - v } { \norm { v } _ { V } } \right ) = \norm { v } _ { V } ( G \widetilde { w } + z ) = G ( \norm { v } _ { V } \widetilde { w } ) + \norm { v } _ { V } z , \end{align*}


\begin{align*} \lim _ { n \to \infty } T ^ { c _ 1 ( n ) } ( x _ 0 ) = x _ 1 . \end{align*}


\begin{align*} L _ X g ( \nu , \cdot ) = h ( \nu , \cdot ) \mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \Sigma . \end{align*}


\begin{align*} S _ q = B _ q \circ \mathcal F \circ B _ q ^ { - 1 } . \end{align*}


\begin{align*} \lim _ { z \downarrow 0 } \int _ 0 ^ \infty w _ z ( \lambda ) \lambda ^ { 4 ( 1 - s + w ) } \frac { d \lambda } { \lambda } \int _ 0 ^ 1 d u \int _ 0 ^ \infty \frac { d t } { t ^ 3 } \phi ( n _ { u } ^ t a _ { \frac { 1 } { t } } ) \sum _ { m = 1 } ^ \infty \hat { F } _ { \pm } ( \lambda a _ { \frac { 1 } { t } } \cdot ( 0 , 0 , 0 , m ) ) \end{align*}


\begin{align*} \begin{aligned} x _ n ^ * - x _ n ^ 0 = \ , & x _ n - \sum _ { \beta = 1 } ^ \alpha x _ \beta \rho _ \beta ( 0 , \tilde { x } ) - \rho ( 0 , \tilde { x } ) \\ > \ , & \rho ( \hat { x } , \tilde { x } ) - \sum _ { \beta = 1 } ^ \alpha x _ \beta \rho _ \beta ( 0 , \tilde { x } ) - \rho ( 0 , \tilde { x } ) \geq 0 \end{aligned} \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \nabla ^ k [ P _ t ^ L ( x ' - y ' ) ] d y ' = 0 , \quad \forall t > 0 , \ , k \in \N . \end{align*}


\begin{align*} g _ i ^ A = i \wedge A , \end{align*}


\begin{align*} G ^ { \alpha \beta } _ 0 = \frac { \partial G } { \partial r _ { \alpha \beta } } ( \nabla _ { \alpha \beta } u ( x _ 0 ) ) , 1 \leq \alpha , \beta \leq n - 1 . \end{align*}


\begin{align*} f \vee g = \frac { 1 } { 2 } \big ( f + g + | f - g | \big ) , \end{align*}


\begin{align*} \bar { y } _ n : = y _ n - \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( x ' _ 0 ) y _ \beta . \end{align*}


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


\begin{align*} k < & \Big ( \Pi _ { i = j } ^ { n - 1 } \big ( r _ i + 1 \big ) \Big ) h _ j , \end{align*}


\begin{align*} \mathcal { G } ^ { \Delta } ( V , W ) : = \mathcal { W } \left ( V W \Delta ^ { - m } \right ) , \end{align*}


\begin{align*} \tilde H - ( k + 2 ) B & = K _ { S ^ { [ 2 ] } } - K _ { S ^ { [ 2 ] } } + \tilde H - ( k + 2 ) B \\ & = \widetilde { K _ S } + \widetilde { H - K _ S } - ( k + 2 ) B . \end{align*}


\begin{align*} F _ n ( \eta ) = T _ n - \frac { 1 } { \eta } I _ n . \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*} p ^ { C _ 5 } = e . \end{align*}


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


\begin{align*} & \gamma \beta ( \gamma ) = \frac { \gamma _ { t - 1 } ( 1 - \kappa _ t ) } { ( 1 - \kappa _ { t - 1 } ) } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 3 } { 2 } \gamma _ 0 , \\ & \alpha ( \gamma ) = \frac { \kappa _ { t - 1 } \gamma \beta ( \gamma ) } { \gamma _ { t - 1 } } = \frac { \kappa _ { t - 1 } ( 1 - \kappa _ { t } ) } { 1 - \kappa _ { t - 1 } } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 1 } { 2 } . \end{align*}


\begin{align*} A _ { k j } = \begin{cases} \sum _ { l = 1 } ^ { n _ 1 } \Psi _ { k l } \bar \Psi _ { j l } , & \lambda _ k = \lambda _ j , \\ 0 , & . \end{cases} \end{align*}


\begin{align*} [ X + \xi , Y + \eta ] _ H = \mathcal { L } _ X ( Y + \eta ) - \iota _ Y d \xi + H ( X , Y , \cdot ) , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { n } { n \choose k } L _ { j k } L _ { j ( n - k ) } = 2 ^ n L _ { j n } + 2 L _ j ^ n , \end{align*}


\begin{align*} F _ { i } ^ \pm ( M ) \coloneqq \begin{cases} F _ i ^ \pm ( M ) & , \\ x _ i & , \\ E _ i & , \\ x _ j & . \end{cases} \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*} \mathcal { I } ( x , t ) = \bigcup _ { i = 1 } ^ { n } [ t _ { i , e n } ( x , t ) , t ] . \end{align*}


\begin{align*} \langle C , C \rangle = \sum _ { i = 1 } ^ s | c _ { i } | ^ { 2 } \langle g _ { i } , g _ { i } \rangle . \end{align*}


\begin{align*} N _ j ( t ) - N _ j ( 0 ) = \sum _ { k = 1 } ^ d \int _ 0 ^ t \Psi _ { j k } ( s ) d \beta ^ c _ k ( s ) \forall \ , 0 \le t \le T , \ ; j = 1 , \dots , d . \end{align*}


\begin{align*} e ^ { - \phi ( x ) } = \int _ { \mathbb R } e ^ { - { \psi } ( x , y ) } d y . \end{align*}


\begin{align*} \Phi : = \sup \{ \zeta \in U S C ( \overline { \Omega } \times \mathbb R ^ m ) : \zeta | _ { \Omega \times \mathbb R ^ m } F \star \mathcal P \zeta | _ { \partial \Omega \times \mathbb R ^ m } \le \phi \} \end{align*}


\begin{align*} \dot { x } = - \nabla f ( x ) = - x ^ 3 , x ( 0 ) = 1 . \end{align*}


\begin{align*} & \sup \{ | \varphi _ { 2 k j _ 0 } ( z ) - \rho ( z ) | ; \ z \in \mathbb { B } _ n , \ | \varphi _ { k j _ 0 } ( z ) | \leq r \} \\ & = \sup \{ | \varphi _ { k j _ 0 } \circ \varphi _ { k j _ 0 } ( z ) - \rho \circ \varphi _ { k j _ 0 } ( z ) | ; \ z \in \mathbb { B } _ n , \ | \varphi _ { k j } ( z ) | \leq r \} \\ & \leq \sup \{ | \varphi _ { k j _ 0 } ( z ) - \rho ( z ) | ; \ z \in \mathbb { B } _ n , \ | z | \leq r \} \leq \varepsilon \end{align*}


\begin{align*} \Psi _ { \mathrm { C } } ( x ) = \begin{cases} \left ( A _ { \mathrm { C } } J _ { 1 } \left ( \lambda ^ { 1 / 2 } r \right ) + W r \right ) \sin \theta , & r \leq a , \\ \frac { a ^ { 2 } } { r } \sin \theta , & r > a , \end{cases} \end{align*}


\begin{align*} \begin{cases} - \Delta _ y \phi + \phi = \lambda ( \phi - W ) _ + , \quad & \mathrm { i n } \ \ \mathbb { R } ^ 4 , \\ \phi \rightarrow 0 , & \mathrm { a s } \ \ | y | \rightarrow \infty . \\ \end{cases} \end{align*}


\begin{align*} \nabla \hat X = \omega \quad \mbox { i n } \Omega . \end{align*}


\begin{align*} q _ 1 = \begin{pmatrix} | \xi | I _ { n - 1 } + \frac { s _ { 1 } } { | \xi | } ( g ^ { \alpha \beta } \xi _ \alpha \xi _ \gamma ) & i s _ { 1 } ( g ^ { \alpha \beta } \xi _ \beta ) & 0 \\ [ 1 m m ] i s _ { 1 } ( \xi _ \beta ) & ( 1 - s _ { 1 } ) | \xi | & 0 \\ [ 1 m m ] 0 & 0 & | \xi | \end{pmatrix} , \end{align*}


\begin{align*} \nabla _ { i j } u : = \nabla _ { e _ i } \nabla _ { e _ j } u = e _ i ^ k e _ j ^ l \partial _ { k } \partial _ l u = e _ i ^ k e _ j ^ l u _ { k l } . \end{align*}


\begin{align*} \langle z _ 1 ^ * , v \rangle = \hat z _ 1 \cdot v , \langle z _ 2 ^ * , v \rangle = \hat z _ 2 \cdot v , \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*} H ( v , w , t ^ { - 1 } ) = t ^ { ( _ { \bullet } v - w ) \cdot ( _ { \bullet } v - w ) - _ { \bullet } v \cdot _ { \bullet } v } \frac { \mathit { g l } ( _ { \bullet } v , t ) } { \mathit { g l } ( _ { \bullet } v - w , t ) } . \end{align*}


\begin{align*} | ( k + 1 ) ^ { 1 - s } - k ^ { 1 - s } | = | 1 - s | \left \lvert \int _ k ^ { k + 1 } y ^ { - s } \dd y \right \rvert \asymp | 1 - s | k ^ { - \Re ( s ) } \end{align*}


\begin{align*} \alpha _ { \rm i d } = \psi _ { 1 } T _ { \rm s } \left [ Q \left ( \kappa _ { \rm b } \right ) - Q \left ( \kappa _ { \rm t } \right ) \right ] . \end{align*}


\begin{align*} \mathbf { x } ^ { \left ( \right ) } = & \mathbf { P } ^ { \left ( \right ) } \mathbf { A } ^ { \left ( \right ) } \mathbf { s } ^ { \left ( \right ) } = a _ c s _ c \mathbf { p } _ c + \sum _ { m = 1 } ^ { M } a _ m s _ m \mathbf { p } _ m , \end{align*}


\begin{align*} \phi ( U | \bar U ; x , t ) & : = ( \nabla \bar A ) ^ { - 1 } ( A - \bar A ) - ( U - \bar U ) \ ; , \\ G _ 1 ( U | \bar U ; x , t ) & : = \nabla G - \nabla \bar G - \nabla ^ 2 \bar G \cdot ( \nabla \bar A ) ^ { - 1 } ( A - \bar A ) \ ; , \\ G _ 2 ( U | \bar U ; x , t ) & : = G _ { x _ \alpha } - \bar G _ { x _ \alpha } - \nabla \bar G _ { x _ \alpha } \cdot ( \nabla \bar A ) ^ { - 1 } ( A - \bar A ) \ ; , \end{align*}


\begin{align*} A \oplus _ 2 \left ( \sqrt { ( 1 - \lambda ) s + \lambda t } B \right ) = \left ( \sqrt { 1 - \lambda } ( A \oplus _ 2 \sqrt { s } B ) \right ) \oplus _ 2 \left ( \sqrt { \lambda } ( A \oplus _ 2 \sqrt { t } B ) \right ) . \end{align*}


\begin{align*} \widetilde { f } ( \lambda , b ) = \pi _ \lambda ( f ) Y _ 0 ( b ) , \ , \ , \ , \pi _ \lambda ( f ) = \int _ G f ( g ) \pi _ \lambda ( g ) d g . \end{align*}


\begin{align*} q ' _ j & = \sum _ { i \in A _ j } q _ i = \sum _ { i \in [ m ] : \tilde { \delta } _ i \in [ \nu _ j , \nu _ { j + 1 } ) } q _ i = \sum _ { i \in [ m ] : \tilde { \delta } _ i \in [ \nu _ j , \nu _ { j + 1 } ) } 0 . 5 \tilde { q } _ i = 0 . 5 \P \{ \tilde { X } \in [ \nu _ j , \nu _ { j + 1 } ) \} . \end{align*}


\begin{align*} \dot { { x } } _ a = \ , - \gamma c _ 1 \delta \left . \dfrac { \partial h } { \partial x } \right | _ { { x } _ a } \ , . \end{align*}


\begin{align*} S ( t ) : = \exp \left ( t \begin{pmatrix} 0 & x _ { 0 , n - 1 } \\ x _ { 0 , 2 } & 0 \end{pmatrix} \right ) . \end{align*}


\begin{align*} \lim _ { p \to \infty } \frac { \psi ( p ) } { p } = 0 . \end{align*}


\begin{align*} \textup { S t o C } : = \min \{ \alpha + \gamma - \frac { 1 } { 2 } - \frac { \alpha } { 2 \beta } ( \kappa - r ) , 1 - \varepsilon \} \leq \min \Big \{ \alpha + \frac { 1 } { 2 } , 1 - \varepsilon \Big \} . \end{align*}


\begin{align*} \begin{aligned} & \sum _ { \begin{subarray} { l } k _ 1 + \cdots + k _ n = k \\ k _ i \in \mathbb { Z } _ { \ge 1 } ( i = 1 , \ldots , n - 1 ) , \\ k _ n \in \mathbb { Z } _ { \ge 2 } \end{subarray} } Z ^ { \star } _ { I } ( \{ k _ i \} _ { i = 1 } ^ { n } ; ( \alpha , \beta ) ) \\ = & \binom { k - 2 } { n - 1 } Z ( n - 1 | k - n + 1 ; ( \alpha , \beta ) ) + \binom { k - 2 } { n - 2 } Z ( n | k - n ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \textup { S t o C } : = \min \Big \{ \alpha + \gamma - \frac { 1 } { 2 } - \frac { \alpha } { 2 \beta } ( \kappa - r ) , 1 - \varepsilon \Big \} = \min \Big \{ \frac { \alpha r } { 2 \beta } + ( \gamma - \frac { 1 } { 2 } ) ^ { + } , 1 - \varepsilon \Big \} . \end{align*}


\begin{align*} \begin{cases} - \delta ^ 2 ( K ( x ) \nabla w ) = ( w - q ) ^ { p } _ + , \ \ & x \in \Omega , \\ w = 0 , \ \ & x \in \partial \Omega . \end{cases} \end{align*}


\begin{align*} R _ { \beta } : = L _ { T ' } , \end{align*}


\begin{align*} \Omega _ \delta : = \{ x \in \Omega : | x - x _ 0 | < \delta \} \end{align*}


\begin{align*} \operatorname { E } ( X ^ r ) & = \frac { \theta ^ { r / 2 } } { \operatorname { B } ( a , b ) } \int _ 0 ^ \infty y ^ { - r / 2 } \exp ( - a y ) \left \{ 1 - \exp ( - y ) \right \} ^ { b - 1 } \mathrm { d } y . \end{align*}


\begin{align*} \nabla _ { e _ { i } } e _ { j } = \alpha _ { i j k } e _ { k } , \end{align*}


\begin{align*} \sigma ( \alpha + \beta ) = \sigma ( \alpha ) + \sigma ( \beta ) \end{align*}


\begin{align*} & | X ^ \beta ( 0 ; t , x ) - X ^ \beta ( 0 ; t , y ) | \le | x - y | + \int _ 0 ^ t \left | \frac { d } { d s } X ^ \beta ( s ; t , x ) - \frac { d } { d s } X ^ \beta ( s ; t , y ) \right | \dd s \\ & = | x - y | + \int _ 0 ^ t | u ( X ^ \beta ( s ; t , x ) , s ) - u ( X ^ \beta ( s ; t , y ) , s ) | \dd s \\ & \le | x - y | + \int _ 0 ^ t \varphi _ \Theta ( | X ^ \beta ( s ; t , x ) - X ^ \beta ( s ; t , y ) | ) B \dd s . \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } z \mathcal { C } ( h ) ( z ) = - \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } } h ( s ) d s . \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*} \tau = e _ { i _ 1 } + \dots + e _ { i _ k } \in ( \mathbb Z / 2 \mathbb Z ) ^ n \leq B _ n \end{align*}


\begin{align*} \mathcal { P } ' \left ( \partial _ Y \Phi ( X ) \right ) = - \sum _ { i = 1 , 2 } \sqrt { c _ i } \ \langle Y _ i , X \rangle \mathcal { P } ' ( N _ i ) + \Phi ( \mathcal { P } ( B ( Y , X _ T ) ) - \mathcal { P } ( B ^ * ( Y , X _ N ) ) ) ; \end{align*}


\begin{align*} \frac { \partial p } { \partial x } & = h ^ x ( x , 0 ) + \int _ 0 ^ y \frac { \partial h ^ y } { \partial x } ( x , s ) \ , d s \\ & = h ^ x ( x , 0 ) + \int _ 0 ^ y g ^ { x y } ( x , s ) \ , d s \end{align*}


\begin{align*} \beta _ { \infty } ( n ) = \frac { 4 \pi ^ { 2 } n } { \sqrt { \det ( A ) } } . \end{align*}


\begin{align*} \mu _ j = 1 & \mbox { a n d } j = i + 1 , \mbox { o r } \\ \mu _ j = 0 & \mbox { a n d } j \neq i + 1 . \end{align*}


\begin{align*} \left \vert \frac { \tau } { \eta \cdot \omega _ 1 } \right \vert = \left \vert \frac { 1 } { \mathbf d \cdot \omega _ 1 } \right \vert = \frac { \sqrt { 1 + s ^ 2 } } { s } > \frac { 1 } { s } . \end{align*}


\begin{align*} X - W = c _ i W ^ { ( i ) } + O ^ { 3 , \alpha } ( | x | ^ { - q } ) . \end{align*}


\begin{align*} \rho { c _ v } { \partial _ { { t } } } T = \nabla \cdot \left ( { \lambda \nabla T } \right ) + { \bar F } + \left ( { \vartheta - 1 } \right ) \frac { { \Delta t } } { 2 } \rho { c _ v } \partial _ { { t } } ^ 2 T , \end{align*}


\begin{align*} \begin{aligned} \langle \partial _ { t , x } ^ \beta f , \phi \rangle = \partial _ { t , x } ^ \beta \langle f , \phi \rangle - \langle f , \partial _ { t , x } ^ \beta \phi \rangle = - \langle f , \partial _ { t , x } ^ \beta \phi \rangle \ , , \end{aligned} \end{align*}


\begin{align*} v _ 1 \cdot v _ 2 = - \int _ X v _ 1 \wedge v _ 2 ^ \vee . \end{align*}


\begin{align*} \frac { u ^ k - u ^ { k - 1 } } { \tau _ k } + \mathcal { L } u ^ k = F ^ k , \end{align*}


\begin{align*} | A | ^ { m - 1 } | B _ 1 + \ldots + B _ m | \le \prod _ { i = 1 } ^ m | A + B _ i | . \end{align*}


\begin{align*} z ^ k & \mapsto f _ k ( s ) = - \frac { 1 } { s } \left ( ( k + 1 ) ^ { 1 - s } - k ^ { 1 - s } \right ) & & ( k \in \mathbb { N ^ * } ) \\ 1 & \mapsto - \frac { 1 } { s } \end{align*}


\begin{align*} c _ 1 ( q , k , p ) : = \frac { 1 } { 4 } \Big ( \hat W _ r ( q - k ) + \hat W _ r ( q + k ) - 2 \hat W _ r ( q ) - ( 4 \lambda + 2 \mu ) \hat W _ r ( q ) \Big ) , \end{align*}


\begin{align*} x = \sum _ { n = 0 } ^ \infty \frac { \varepsilon _ n } { \beta ^ { n + 1 } } = \sum _ { n = 0 } ^ { s - 1 } \frac { \varepsilon _ n } { \beta ^ { n + 1 } } + \left ( \sum _ { n = s } ^ { s + t - 1 } \frac { \varepsilon _ n } { \beta ^ { n + 1 } } \right ) \frac { \beta ^ t } { \beta ^ t - 1 } . \end{align*}


\begin{align*} \alpha _ * ^ 2 \| u _ * \| ^ 2 + 2 \alpha _ * \langle u _ * , \lambda x _ v + ( 1 - \lambda ) x _ w \rangle - \lambda ( 1 - \lambda ) \| x _ v - x _ w \| ^ 2 = 0 \end{align*}


\begin{align*} f = \sum _ { \mathcal { J } } \sum _ { \beta } f _ { \mathcal { J } , \beta } ( x ^ { \mu } ) \prod _ { j \in \mathcal { J } } \theta _ j ^ { \beta ^ j } , \end{align*}


\begin{align*} d _ \square ( W _ 1 , W _ 2 ) : = \sup _ { S , T \subset [ 0 , 1 ] } \bigg | \int _ { S \times T } \big ( W _ 1 ( u , v ) - W _ 2 ( u , v ) \big ) d u d v \bigg | , \end{align*}


\begin{align*} F ' _ m ( n ) = 1 + H ' ( n , m ) - r _ m ( n ) \sum _ { \begin{smallmatrix} k = 1 \\ k | m \end{smallmatrix} } ^ m \frac { \mu ( k ) } { k } , \end{align*}


\begin{align*} \alpha _ n ( 0 ) = h \psi _ 0 ( h n ) + ( - 1 ) ^ n h \phi _ 0 ( h n ) , \end{align*}


\begin{align*} g ' _ 2 = - \frac { 1 } { \nu } I \widehat { g _ 1 ' } I + \frac { 2 i } { \mu \nu } h _ z J g _ 1 ' \end{align*}


\begin{align*} \mu _ t ^ { * , \xi } = \mathbb E [ \alpha ^ { * , \xi } _ t | \mathcal { F } _ t ^ { W ^ 0 } ] \quad \nu _ t ^ { * , \xi } = \mathbb E [ x _ t ^ { \xi , \alpha ^ { * , \xi } } | \mathcal { F } _ t ^ { W ^ 0 } ] . \end{align*}


\begin{align*} N K B L ( m , n ) = \sum _ { { \scriptscriptstyle A \in M N A P P ( m , n ) } } N ( A ) \cdot ( m - 1 ) ! + \sum _ { { \scriptscriptstyle A \in M P A P P ( m , n ) } } N ( A ) \cdot 2 ^ { m - 1 } ( m - 1 ) ! \\ \ge \sum _ { i = 1 } ^ { 3 } N ( A _ i ) \cdot ( m - 1 ) ! + \sum _ { i = 7 } ^ { 1 0 } N ( A _ i ) \cdot 2 ^ { m - 1 } ( m - 1 ) ! \ge ( 5 \cdot 2 ^ m ) \cdot ( m - 1 ) ! . \end{align*}


\begin{gather*} \inf _ { Q \in M } Q ( c ) = \sup _ { f _ 1 , \ldots , f _ n } \ , \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \end{gather*}


\begin{align*} G _ { \iota } = ( \nabla G \cdot e _ { \iota } ) w _ { \iota \iota } , \end{align*}


\begin{align*} V ' _ i ( u ) [ v ] = 4 B _ i ( u ^ 2 , u v ) , \ \forall u , v \in X , \ i = 0 , 1 , 2 . \end{align*}


\begin{align*} \begin{aligned} & y _ \delta ( x _ a , t ) = \dfrac { a _ { 0 , \delta } ( x _ a ) } { 2 } + \\ & \sum _ { k = 1 } ^ { \infty } a _ { k , \delta } ( x _ a ) \cos \left ( k 2 \pi t \right ) + b _ { k , \delta } ( x _ a ) \sin \left ( k 2 \pi t \right ) \end{aligned} \end{align*}


\begin{align*} N \mapsto \mu _ N = \frac { 1 } { | \Phi _ N | } \sum _ { n \in \Phi _ N } \delta _ { T ^ n a } \end{align*}


\begin{align*} S \subset \bigcup _ { i = 1 } ^ { C _ { B C } } \bigcup _ { B \in \mathcal { B } ^ { ( i ) } } B . \end{align*}


\begin{align*} \lambda = a + b x \in \sigma _ l ( A ) , \ , v = ( 1 , x ) ^ T \in V _ l ( \lambda ) , \forall x \in S _ Q ( b , a - d , 0 ) . \end{align*}


\begin{align*} c _ s ^ 2 \partial _ { z 1 } T = - \frac { 1 } { \Delta t } { \varsigma } _ 3 { m } _ 3 ^ { ( 1 ) } , \end{align*}


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


\begin{align*} \tilde { h } _ d '' ( x ) = \sum _ { n = 0 } ^ \infty \left ( \frac { 1 } { ( x + n ) ^ 2 } - \frac { 1 } { \left ( x + n + \frac { 1 } { 2 } \right ) ^ 2 } - \frac { 1 } { \left ( x + n + \frac { d - 1 } { 2 } \right ) ^ 2 } \right ) . \end{align*}


\begin{align*} m _ k ' = - m _ k + \sum _ { j \in \Omega ( k , - ) } b _ { k j } m _ j . \end{align*}


\begin{align*} \mathcal { M } = \{ Z = ( z _ 1 , z _ 2 , \cdots , z _ m ) \in \mathbb { R } ^ { ( 2 m ) } \mid z _ i \in B _ { \bar { \rho } } ( x _ { 0 , i } ) , \ i = 1 , \cdots , m \} . \end{align*}


\begin{align*} \left \langle T _ a g _ t , T _ a g _ j \right \rangle _ { \alpha } = \frac { ( 1 + o ( 1 ) ) ( a + 1 ) ( \alpha - \frac { 1 } { 2 } ) } { ( a - \alpha + \frac { 3 } { 2 } ) } ( j + 1 ) ^ { a + \frac { 1 - \alpha } { 2 } } ( t + 1 ) ^ { - a + \frac { \alpha - 3 } { 2 } } . \end{align*}


\begin{align*} u ( t ) & \leq \int _ { 0 } ^ { t } \bigg ( 2 \sum _ { i = 1 } ^ { \infty } \sum _ { j = 1 } ^ { i } C _ { V } j ( i + j ) | u _ j | \ , \psi _ i + \sum _ { i = 1 } ^ { \infty } \sum _ { j = i } ^ { \infty } C _ V i ^ { \eta } | u _ j | \ , \psi _ i \bigg ) ( h ) d h . \\ \end{align*}


\begin{align*} L ( x ) : = \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( y ' ) ( x _ \beta - y _ \beta ) . \end{align*}


\begin{align*} I ( A ^ { 1 , \dots , \ell } ) = \langle p _ { ( u , s ) } - p _ { ( u , r ) } \mid u \in [ \beta ] , s , r \in \{ 0 , \dots , x _ u Y _ u - 1 \} \rangle + J . \end{align*}


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


\begin{align*} \sigma ( \gamma _ 1 ) & = \gamma _ 3 , & \sigma ( \gamma _ 2 ) & = \gamma _ 2 , & \sigma ( \gamma _ 3 ) & = \gamma _ 4 , & \sigma ( \gamma _ 4 ) & = \gamma _ 1 . \end{align*}


\begin{align*} \left | \Delta _ C \Delta _ B ( A ) \right | & = \left | \big ( ( A + B ) \setminus A \big ) + C \right | - \left | ( A + B ) \setminus A \right | \\ & \geq \left | ( A + B + C ) \setminus ( A + C ) \right | - \left | A + B \right | + \left | A \right | \\ & = | A + B + C | - | A + C | - | A + B | + | A | , \end{align*}


\begin{align*} d v ' = \left \{ ( \log \sqrt { \tau _ 0 } ) _ z \ d z + \frac { i } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) J \right \} v ' . \end{align*}


\begin{align*} L ^ { i } = \{ p \in \Delta _ { m - 1 } \vert \ , A ^ { i } p = A ^ { i } d \} \end{align*}


\begin{align*} \mathcal { W } \ , ( P ) : = \int _ { M } \left ( \int _ { | \xi | = 1 } t r \ , \sigma _ { - n } ( P ) ( x , \xi ) ~ { \mathcal V } _ { \xi } \right ) ~ d ^ { n } x , \end{align*}


\begin{align*} D _ x ( A ^ q \eta ) ( x ) & = D _ x \left ( \int _ { x - r } ^ { x + r } ( \eta ( y ) - \eta ( x ) ) \cos ( 2 \pi q ( y - x ) ) \ \d y \right ) \\ & = ( \eta ( x + r ) - \eta ( x ) ) \cos ( 2 \pi q r ) - ( \eta ( x - r ) - \eta ( x ) ) \cos ( 2 \pi q r ) \\ & + \int _ { x - r } ^ { x + r } - \partial \eta ( x ) \ \cos ( 2 \pi q ( y - x ) ) + 2 \pi q ( \eta ( y ) - \eta ( x ) ) \sin ( 2 \pi q ( y - x ) ) \ \d y . \end{align*}


\begin{align*} \eta ( t ) = \begin{cases} R , t \leq \delta \\ R ( 1 - ( \frac { t - \delta } { R - \delta } ) ^ 2 ) ^ 2 , t > \delta \end{cases} \end{align*}


\begin{align*} \kappa \stackrel { d e f } { = } \lim _ { n \to \infty } \kappa _ n . \end{align*}


\begin{align*} \Phi _ t ( x ' ) = t ^ { 1 - n } ( \partial _ n K ^ L ) ( x ' / t , 1 ) = t \ , ( \partial _ n K ^ L ) ( x ' , t ) = t \partial _ t K ^ L ( x ' , t ) = t \partial _ t P _ t ^ L ( x ' ) , \end{align*}


\begin{align*} \alpha _ m : = \min \left \{ \langle Q , X \rangle \ , | \ , X \in \R ^ { Z _ m \times Z _ m } _ { \geq 0 } , \ , \langle J , X \rangle = 1 , \ , X \succeq 0 \right \} . \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*} \omega _ { u v } = \begin{cases} x _ u y _ v & t ( u , \bullet ) = t ( \bullet , v ) \\ 0 & \end{cases} \end{align*}


\begin{align*} \gamma \cdot f ( x , y ) = \frac { f ( ( x , y ) \gamma ) } { | \det \gamma | } , \end{align*}


\begin{align*} X ^ v \circ X ^ w = q ^ { - \langle v , w \rangle } X ^ { v + w } \ , v , w \in \mathbb { N } ^ n . \end{align*}


\begin{align*} a _ { k - j , 0 } & = \frac { \Gamma \bigl ( - w _ 2 + k \bigr ) \Gamma \bigl ( 0 \bigr ) \Gamma \bigl ( w _ 2 + \frac { n } { 2 } \bigr ) } { \Gamma \bigl ( - w _ 2 + j \bigr ) \Gamma \bigl ( - j \bigr ) \Gamma \bigl ( w _ 2 + \frac { n - 2 k } { 2 } \bigr ) } \\ & = ( - 1 ) ^ j j ! \frac { \Gamma \bigl ( - w _ 2 + k \bigr ) \Gamma \bigl ( w _ 2 + \frac { n } { 2 } \bigr ) } { \Gamma \bigl ( - w _ 2 + j \bigr ) \Gamma \bigl ( w _ 2 + \frac { n - 2 k } { 2 } \bigr ) } . \end{align*}


\begin{align*} L ( v ) = \int _ U \left ( \langle W _ u ( u ^ { - 1 } ( y ) ) , v ( y ) \rangle + W _ \xi ( u ^ { - 1 } ( y ) ) [ \nabla _ x u ] ^ t ( u ^ { - 1 } ( y ) ) : [ \nabla _ y v ] ( y ) \right ) d y . \end{align*}


\begin{align*} W ( 0 ^ \infty | 1 ^ \infty ) = b - a + \sum _ { n = 2 } ^ \infty ( a _ n - a ) \ . \end{align*}


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


\begin{align*} \partial _ { t } u = \nabla \cdot ( | \nabla u | ^ { p - 2 } \nabla u ) - c | u | ^ { p - 2 } u , \end{align*}


\begin{align*} \Delta x _ k + \bar u ^ { - 1 } \nabla \bar u \cdot \nabla x _ k = - \Gamma ^ k + \bar u ^ { - 1 } \bar g ^ { i k } \frac { \partial \bar u } { \partial x _ i } . \end{align*}


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


\begin{align*} d x = b ( x ) d \tau + \sigma ( x ) d \beta ( \tau ) , \ ; \ ; \ ; x \in \mathbb { R } ^ l , \ ; \tau \geqslant 0 , \end{align*}


\begin{align*} \Delta x _ k + \bar u ^ { - 1 } \nabla \bar u \cdot \nabla x _ k = 0 \mbox { f o r a l l } k = 1 , \dots , n . \end{align*}