\begin{align*} G _ { \mu \boxplus \nu } ( s _ { 0 } ) = \lim _ { y \rightarrow 0 ^ { + } } G _ { \nu } \left ( \omega ( s _ { 0 } + i y ) \right ) = G _ { \nu } ^ { * } ( \alpha ) \in \mathbb { R } \setminus \left \{ 0 \right \} . \end{align*}


\begin{gather*} \Gamma _ 0 ( N ) _ Q : = \left \{ \gamma \in \Gamma _ 0 ( N ) \mid Q | \gamma = Q \right \} . \end{gather*}


\begin{align*} h ( v ) = 1 + \max _ { w | } { h ( w ) } . \end{align*}


\begin{align*} S ^ { n , M , 1 } _ \tau = n ^ { \alpha + \frac 1 2 } \sum _ { i = 0 } ^ { M - 1 } \sigma ' ( X _ { \tau _ i } ) \int _ { \tau _ i } ^ { \tau _ { i + 1 } } ( t - s ) ^ \alpha \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) d s \end{align*}


\begin{align*} \psi ( u T ) & = \varphi ( e _ 1 u T e _ 1 ) = \varphi ( e _ 1 ( u J u J ) T ( J u ^ * J ) e _ 1 ) = \varphi ( e _ 1 T ( J u ^ * J ) e _ 1 ) \\ & = \varphi ( e _ 1 T ( J u ^ * J ) ( u J u J ) e _ 1 ) = \psi ( T u ) . \end{align*}


\begin{align*} G ^ n _ \tau & : = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s \left ( \sigma ' ( X _ s ) \right ) ^ 2 \ , \left ( \int ^ s _ { \eta _ n ( s ) } ( s - \eta _ n ( u ) ) ^ { \alpha } \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) \ , \\ & \times \left ( \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) d s , \end{align*}


\begin{align*} \mathcal { L } _ { \sigma p } ( \mathbb { R } ^ { n } ) : = \left \{ u \in L ^ { p - 1 } _ { l o c } ( \mathbb { R } ^ { n } ) \ , \Big | \ ; \int _ { \mathbb { R } ^ { n } } \frac { | u ( x ) | ^ { p - 1 } } { 1 + | x | ^ { n + \sigma p } } d x < \infty \right \} . \end{align*}


\begin{align*} \kappa _ { n } = ( m - 1 ) \frac { \sum \limits _ { i = 1 } ^ { m - 1 } \kappa _ i } { m - 1 } = \sum \limits _ { i = 1 } ^ { m - 1 } \kappa _ i = \sum \limits _ { i = 1 } ^ { m - 1 } \sum \limits _ { j = 1 } ^ { m - 1 } \sum \limits _ { k = 1 } ^ { m - 1 } \tau _ { i j } \tau _ { i k } l _ { j k } = \sum \limits _ { i = 1 } ^ { m - 1 } \sum \limits _ { j = 1 } ^ { m - 1 } h ^ { i j } l _ { i j } , \end{align*}


\begin{align*} \sum _ { X \in M '' } P _ 0 ( X ) = \sum _ { ( i _ n , i _ { n + 1 } , \ldots , i _ q ) \in I } P _ 0 \big ( D [ i _ 0 , i _ 1 , \ldots , i _ n , i _ { n + 1 } , \ldots , i _ q ] \big ) . \end{align*}


\begin{align*} P _ f = \begin{cases} P ^ { - 1 } , & f \in P ^ { - 1 } \\ P & f \in P . \end{cases} \end{align*}


\begin{align*} \theta ^ 2 = 1 . \end{align*}


\begin{align*} { f } _ { i } ^ 0 ( \alpha ) + \sum _ { j \in S _ { p } } { f } _ { j } ^ 0 ( \alpha ) = f ( \gamma _ p ) . \end{align*}


\begin{align*} \int \frac { 1 } { z - x } \ , d \mu ( x ) = \cfrac { 1 } { z - b _ { 0 } - \cfrac { a _ 0 } { z - b _ { 1 } - \cfrac { a _ { 1 } } { \ddots } } } \end{align*}


\begin{align*} a _ { n + 1 } = p a _ n + 1 , \ a _ 1 = 0 , \end{align*}


\begin{align*} \max \left \{ \sum _ { j = 1 } ^ m F ( x ^ j ) : y \in P ( \Pi ) \land Z ( y ) = h ^ 1 , h ^ 2 , \ldots , h ^ m \land x ^ j \in P ( h ^ j ) \right \} , \end{align*}


\begin{align*} y ^ d \mapsto \prod _ { i = 1 } ^ { \mathrm { r } } y _ i ^ { p _ i ^ + \cdot d } . \end{align*}


\begin{align*} | \pi ^ t _ i ( s ) | : = \lim _ { n \to \infty } n ^ { - 1 } \sum _ { j = 1 } ^ n 1 ( j \in \pi ^ t _ i ( s ) ) . \end{align*}


\begin{align*} \varphi _ j ( D ) = 2 ^ { - 2 \ell j } ( - \Delta ) ^ \ell \widetilde \varphi _ k ( D ) \end{align*}


\begin{align*} P ( T , x ) & = \prod _ { v \in V ( T ) } \mathcal { G } ( v , x ) . \end{align*}


\begin{align*} \langle v , K _ { 1 1 } u \rangle = \ ; & \frac { 1 } { 2 } \langle v \otimes u _ 1 , w \ , u _ 1 \otimes u \rangle \\ \langle v , K _ { 2 2 } u \rangle = \ ; & \frac { 1 } { 2 } \langle v \otimes u _ 2 , w \ , u _ 2 \otimes u \rangle . \end{align*}


\begin{align*} \varphi \circ { \rm A d } ( u ) ( T ) = \varphi \circ { \rm A d } ( J u ^ * J ) ( T ) = \varphi ( T ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : m c k e a n v l a s o v } \forall i \in \{ 1 , \ldots , N \} , \dd X ^ { i , N } _ t = b \big ( X ^ { i , N } _ t , \mu _ { \mathcal { X } ^ N _ t } \big ) \dd t + \sigma \big ( X ^ { i , N } _ t , \mu _ { \mathcal { X } _ t } \big ) \dd B ^ { i } _ t \end{align*}


\begin{align*} ( \alpha _ 1 , \ldots , \alpha _ n ) \cdot C = ( 1 , 0 , \ldots , 0 ) , \end{align*}


\begin{align*} \begin{array} { l l } e ^ { A } e ^ { B } = \exp \left [ A + B + \frac { 1 } { 2 } [ A , B ] + \frac { 1 } { 1 2 } [ A , [ A , B ] ] - \frac { 1 } { 1 2 } [ B , [ A , B ] ] + \cdots \right ] , \end{array} \end{align*}


\begin{align*} \sum \limits _ { i = j } ^ \infty \frac { 1 } { a _ i } \leq \frac { C _ a } { a _ j } , \ \ \ j \in \N . \end{align*}


\begin{align*} \| \mu ( E ) - \nu ( E ) \| _ { T V } : = \sup _ { E \in \Sigma } | \mu ( E ) - \nu ( E ) | , \end{align*}


\begin{align*} \partial _ x \partial _ x g ( x , y ) & = \gamma \left ( \frac { 2 b ' ( x ) ^ 2 } { b ( x ) ^ 3 } - \frac { b '' ( x ) } { b ( x ) ^ 2 } \right ) \| y \| ^ 2 _ d , \\ \partial _ { y _ i } \partial _ { y _ j } g ( x , y ) & = \frac { 2 \gamma } { b ( x ) } \ 1 { i = j } , \\ \partial _ x \partial _ { y _ i } g ( x , y ) & = - 2 \gamma \frac { b ' ( x ) } { b ( x ) ^ 2 } y _ i . \end{align*}


\begin{align*} H _ 1 & = - \Delta + c V _ 0 ( x ) \\ H _ x & = - \frac { d ^ 2 } { d x ^ 2 } + c V _ 0 ( x ) \end{align*}


\begin{align*} \mathcal { U } _ { [ n , j ] } : = \{ \gamma \in \mathcal { P } _ { [ n , 0 ] } : \mbox { t h e f i r s t s t e p i n $ \gamma $ i s } \ , \ , ( 0 , 0 ) \rightarrow ( 1 , - j ) \} , - 1 \leq j \leq p . \end{align*}


\begin{align*} \lambda ^ { - \frac { n - 2 } { 2 } } { \sf Q } ( y ) + \lambda ^ { - \frac { n - 2 } { 2 } } \sigma { \sf A } _ 1 & = \pm \eta ^ \frac { 2 } { 1 - q } { \sf U } ( \xi ) | y | \to \infty , \ | \xi | \to 0 , \\ \pm \eta ^ \frac { 2 } { 1 - q } { \sf U } ( \xi ) & = \pm { \sf U } _ \infty ( x ) + \Theta _ J ( x , t ) | \xi | \to \infty , \ | z | \to 0 . \end{align*}


\begin{align*} \mathbb { P } _ x ( X _ n = o d ( o , X _ m ) > r 0 \leq m \leq n ) & \leq \sum _ { m = 0 } ^ n \sum _ { z \in V } \mathbb { P } _ x ( \kappa = m , X _ \kappa = z ) \mathbb { P } _ z ( X _ { n - m } = o ) \\ & \leq \sqrt { \frac { c ( o ) + K ( o ) } { c ( z ) + K ( z ) } } \exp \left [ - \frac { r ^ 2 } { 2 n } \right ] \leq \sqrt { \frac { 2 c ( o ) } { c _ { \min } } } \exp \left [ - \frac { r ^ 2 } { 2 n } \right ] , \end{align*}


\begin{align*} u ' ( x ) + u ^ 2 ( x ) + p ( x ) u ( x ) = 0 , \end{align*}


\begin{align*} \Vert \mathfrak { R } _ { \Phi } ^ { f , \gamma } \Vert _ { \mathcal { U } } \leq \mathfrak { F } _ { \gamma } \left ( \Phi , f \right ) \lim _ { \gamma \rightarrow 0 ^ { + } } \Vert \mathfrak { R } _ { \Phi } ^ { f , \gamma } \Vert _ { \mathcal { U } } = 0 \ . \end{align*}


\begin{align*} ( ( 1 \otimes x _ 1 ) \cdot ( 1 \otimes x _ 2 ) ) _ { \varphi _ { \ast } \lambda } = 1 \otimes x _ 1 x _ 2 + \sum _ { i } \varphi ( m _ i ) \otimes y _ i . \end{align*}


\begin{align*} \bar { \phi } _ n ( y ) = \phi _ n ( y + h _ n t _ 1 ) . \end{align*}


\begin{align*} J ( \xi _ T ) = { { \cal L } ( \xi _ T , 0 , 0 , \dots ) } + C \end{align*}


\begin{align*} \begin{aligned} \dot x ( t ) & = A x ( t ) + B u ( t ) + w ( t ) , \\ y ( t ) & = C x ( t ) + v ( t ) , \end{aligned} \end{align*}


\begin{align*} \tau ( \alpha \beta ) = \sum _ { k = 0 } ^ { n - 1 } \alpha _ { k } \tau \left ( \theta ^ { k } \beta \right ) = \sum _ { k = 0 } ^ { n - 1 } \alpha _ { k } H ^ { k } \overline { \beta } = H ^ { * } ( \overline { \alpha } ) \overline { \beta } = \overline { \alpha } \otimes \overline { \beta } , \end{align*}


\begin{align*} \psi _ n ( f ) ( z ) = \begin{pmatrix} f ( 1 ) & 0 \\ 0 & f ( z ) \end{pmatrix} \end{align*}


\begin{align*} \Vert u - J z _ { \mathrm { n c } } \Vert _ { L ^ 2 ( \Omega ) } ^ 2 & = a ( u , { w } - J I { w } ) + a _ { \mathrm { p w } } ( { w } , z _ { \mathrm { n c } } - J z _ { \mathrm { n c } } ) + a ( u , J I { w } ) - a _ { \mathrm { p w } } ( { w } , z _ { \mathrm { n c } } ) . \end{align*}


\begin{align*} a \cdot b = a + b - a b \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } B ( t ) = + \infty . \end{align*}


\begin{align*} R ^ G _ { \varphi } \theta : = | W _ M | ^ { - 1 } \sum _ { w \in W _ M } \tilde { \varphi } ( w F ) R ^ G _ { T _ w } \theta _ { T _ w } , \end{align*}


\begin{align*} { \bar \Omega } _ { A B } + \Psi _ { A B } = \Omega _ { A B } , \mbox { f o r } 1 \le A , B \le N + 1 . \end{align*}


\begin{align*} | \omega _ k | \leq r _ k = r \max \left \{ \left | \frac { j } { n } \right | ^ { \frac { 1 } { n - j } } , \ , \ , j = 1 , 2 , \dots , k \right \} . \end{align*}


\begin{align*} a = M \cdot m . \end{align*}


\begin{align*} \epsilon _ n = \frac { a } { n ^ { \frac { 1 } { 2 } ( 1 - b ) } } \ ; , \ , \end{align*}


\begin{align*} V _ { U } = \left \lbrace v \in \mathcal { H } \left ( \begin{array} { c c } U & - I \\ 0 & 0 \end{array} \right ) ( D P ) v = 0 \right \rbrace . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \ge M } \sup _ { \tau \in [ 0 , t ] } \| S ^ { n , M , 2 } _ \tau \| _ 2 = 0 . \end{align*}


\begin{align*} \Lambda ^ { ( 4 ) } _ { n , \delta } = \exp \left ( i \mu n ^ { \alpha + \frac 1 2 } \int _ 0 ^ { t - \delta } ( t - s ) ^ \alpha \xi ^ n _ s d W _ s + i \lambda n ^ { \alpha + \frac 1 2 } \int _ { t - \delta } ^ { \eta _ n ( t ) } \psi _ { n , 1 } ( s , \eta _ n ( t ) ) d W _ s \right ) . \end{align*}


\begin{align*} \sum \limits _ { i } ^ { N } \alpha _ i = 0 . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \frac { \partial } { \partial t } \omega ( t ) = - \mathrm { R i c } ^ { T } ( \omega ( t ) ) , \omega ( 0 ) = \omega _ { 0 } . \end{array} \end{align*}


\begin{align*} p _ { \bar { X } , \bar { Y } } ( a , b ) = p _ { \bar { Y } | \bar { X } } ( b | a ) p _ { \bar { X } } ( a ) , \end{align*}


\begin{align*} \tilde { g } = 2 d \tau d u + g \end{align*}


\begin{align*} \gg = \oplus _ { \mu } \gg ^ { \mu } , \end{align*}


\begin{align*} \| B _ { k ; j , l } ^ { e s s ; m } ( t , x ) \| _ { L ^ 2 } \lesssim \sup _ { g \in L ^ 2 , \| g \| _ { L ^ 2 } = 1 } \big | \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } ( t - s ) \big | { ( \hat { v } \times \xi ) } \big | \big | \overline { \hat { g } ( \xi ) } \big | \big | \hat { f } ( s , \xi , v ) \big | \varphi _ { j , l } ( v , { \omega } ) \varphi _ { m , k , l } ^ { e s s } ( \omega , \xi ) \varphi _ { m ; - 1 0 M _ t } ( t - s ) \end{align*}


\begin{align*} l ( \sigma ) \left ( - \frac { 2 } { \sigma ^ 3 } \right ) = \frac { 5 } { \sigma ^ 4 } + G ( P _ X V , P _ X V ) . \end{align*}


\begin{align*} f _ { k - 2 } ( z , \overline \omega _ 1 ) = \int _ C \frac { e ^ { z s } \ , s ^ { k - 2 } } { e ^ { \overline \omega _ 1 s } - 1 } \ d s , \end{align*}


\begin{align*} \left ( \frac { \sum _ { m = 1 } ^ { M } ( T _ { { m } } - \alpha ) } { M } \right ) ^ 2 \le \frac { \sum _ { m = 1 } ^ { M } ( T _ { { m } } - \alpha ) ^ 2 } { M } \end{align*}


\begin{align*} ( \Psi _ { t , \eta _ i } ^ { q ( t ) } ) ^ * g _ { \mathrm { c a n } } = \frac { \mathrm { t r } _ g ( \Psi _ { t , \eta _ i } ^ { q ( t ) } ) ^ * g _ \mathrm { c a n } } { n } \cdot g + O ( t ^ l ) \end{align*}


\begin{align*} M _ { n , k } ( n ) = \dbinom { 2 n } { k } - \dbinom { 2 n } { k - 1 } . \end{align*}


\begin{align*} \tilde { g } _ { i j } ( \xi ) : = \int \partial _ i \tilde { l } ^ { ( \alpha ) } \partial _ j \tilde { l } ^ { ( - \alpha ) } \ , d x , \quad \tilde { \Gamma } ^ { ( \alpha ) } _ { i j , k } = \int \partial _ i \partial _ j \tilde { l } ^ { ( \alpha ) } \partial _ k \tilde { l } ^ { ( - \alpha ) } \ , d x . \end{align*}


\begin{align*} d \theta _ A = \sum \theta _ { A B } \wedge \theta _ B , 1 \le A , B \le N + 2 . \end{align*}


\begin{align*} \Lambda _ i ^ { \nu + 1 } = \Lambda _ i ^ { \nu } + \pi ^ { \nu } Z ^ { \nu } _ i \end{align*}


\begin{align*} e _ 1 = \frac { \sum a _ i - ( r - 2 ) } { r } \in \Z , ~ ~ e _ 2 = \frac { \sum b _ i - ( s - 2 ) } { s } \in \Z . \end{align*}


\begin{align*} \theta ( s ) = \left \{ \begin{aligned} 1 & 0 \leq s < t \\ \frac { t - s } { \delta } + 1 & t \leq s < t + \delta \\ 0 & s > t + \delta . \end{aligned} \right . \end{align*}


\begin{align*} \lim _ { n \to \infty } e _ n ( B _ { R _ \ell } ( x _ 0 ) ) = e ( B _ { R _ \ell } ( x _ 0 ) ) \quad \lim _ { n \to \infty } \int _ { B _ { r _ \ell } ( x _ 0 ) } u _ n = u ( B _ { r _ \ell } ( x _ 0 ) ) . \end{align*}


\begin{align*} ( h \rightharpoonup f ) ( m ) = \sum _ { ( h ) } h _ 1 \rightharpoonup f ( S h _ 2 \rightharpoonup m ) \end{align*}


\begin{align*} 0 \longrightarrow \mathbb { L } : = ( \beta ) \longrightarrow \mathbb { Z } ^ { m } \stackrel { \beta } { \longrightarrow } N \longrightarrow 0 , \end{align*}


\begin{align*} c _ b ^ { - 2 } \Vert u _ { \mathrm { n c } } \Vert _ { L ^ 2 ( T ) } ^ 2 \le \Vert u _ { \mathrm { n c } } \Vert _ { b _ T } ^ 2 = ( u _ { \mathrm { n c } } , u ) _ { b _ T } - ( u _ { \mathrm { n c } } , u - u _ { \mathrm { n c } } ) _ { b _ T } . \end{align*}


\begin{align*} \mathrm { d } \alpha _ { \lambda , \xi } ( x _ 1 , x _ 2 , x _ 3 ) = \sum _ { \circlearrowleft x _ 1 , x _ 2 , x _ 3 } \alpha _ { \lambda , \xi } ( x _ 1 , x _ 2 x _ 3 ) + x _ 1 \alpha _ { \lambda , \xi } ( x _ 2 , x _ 3 ) . \end{align*}


\begin{align*} & \psi _ i \psi _ j = - q _ { i j } \psi _ j \psi _ i , & & \sum _ { i = 1 } ^ r q _ i \psi _ i \psi _ { - i } = 0 . \end{align*}


\begin{align*} g = \frac { d r ^ 2 } { \alpha ( r ) } + \sin ^ 2 r \cdot h _ { \mathbb { S } ^ { n - 2 } } + e ^ { 2 \beta ( s , r ) } \cdot d s ^ 2 . \end{align*}


\begin{align*} & 2 f _ { \lambda } ( \lambda ) f _ { \lambda \lambda } ( \lambda ) - \nabla f _ { \lambda } \cdot \nabla f ( \lambda ) \\ & = - f _ { \lambda } ( \lambda ) \Delta f ( \lambda ) - f ( \lambda ) \Delta f _ { \lambda } ( \lambda ) + 2 \sum _ { \l - k > 1 } \dfrac { \partial } { \partial \lambda } \left ( \det \left ( ( \lambda I _ N - H ) _ { k | k } \right ) \det \left ( ( \lambda I _ N - H ) _ { \l | \l } \right ) \right ) . \end{align*}


\begin{align*} \delta _ j : = D \exp \left ( - \frac { \log n } { b ( n ) ^ { j / 4 } } \right ) = D n ^ { - 1 / b ( n ) ^ { j / 4 } } . \end{align*}


\begin{align*} \int _ { ( - \infty , - n ^ m ) } \frac { n s ^ { 2 } \ , d \sigma _ { \mu } ^ { - } ( s ) } { ( \widetilde { x } _ n - s ) ^ { 2 } } & = n \sum _ { k = n + 1 } ^ \infty \left ( \frac { k ^ m } { \widetilde { x } _ n + k ^ m } \right ) ^ 2 \sigma _ \mu ^ - ( \{ - k ^ m \} ) \\ & \leq 1 6 n ^ 3 \sigma _ \mu ^ - ( ( - \infty , - n ^ m ) ) = \frac { 1 6 \widetilde { \ell } ( n ^ m ) } { n ^ { a m - 3 } } \end{align*}


\begin{align*} \sum \limits _ { l = r } ^ k ( - 1 ) ^ { l - r } \binom { k } { l } \binom { l } { r } = 0 . \end{align*}


\begin{align*} \rho = \frac { 1 } { [ K : \Q ] } \sum _ { g \in \mathrm { G a l } ( K / \Q ) } g . \rho \ ; . \end{align*}


\begin{align*} \beta _ t = \sup _ { s \in [ 0 , t ] } \beta _ t ( s ) \leq 1 - 2 \epsilon , \alpha _ t = \sup _ { s \in [ 0 , t ] } \alpha _ t ( s ) \leq \alpha ^ { \star } - \epsilon . \end{align*}


\begin{align*} Y _ { n , s , m } = V \left ( \sum _ { i = 1 } ^ n z _ i ( F _ i \cdot \underline { y } ) \right ) . \end{align*}


\begin{align*} \partial _ \xi e ^ { i x \xi + i p ( \xi ) t } = i [ x + t p ' ( \xi ) ] e ^ { i x \xi + i p ( \xi ) t } . \end{align*}


\begin{align*} \bigcirc _ { w = 0 } ^ i ( \theta _ t + \gamma _ { r - i + w } ) ^ { - 1 } \circ ( \theta _ t + \xi _ { i , 1 } ) \circ \cdots \circ ( \theta _ t + \xi _ { i , \ell _ i } ) = \bigcirc _ { w ' = 0 } ^ j ( \theta _ t + \gamma _ { r - j + w ' } ) ^ { - 1 } \circ ( \theta _ t + \xi _ { j , 1 } ) \circ \cdots \circ ( \theta _ t + \xi _ { j , \ell _ j } ) \enspace , \end{align*}


\begin{align*} L ( f ) = \sum _ { i = 1 } ^ k X _ i [ f ] \Theta _ i , \forall f \in \mathcal { O } . \end{align*}


\begin{align*} v o l ( D ^ 2 _ x ( 1 ) ) : = \frac { v o l ( B ^ 2 ( 1 ) ) b ^ 2 A ^ { \epsilon \eta } z ^ 1 _ { \epsilon } z ^ 1 _ { \eta } } { 2 \sqrt { \det A } } . \end{align*}


\begin{align*} Z _ j = \begin{cases} \dot { x } = - y + d _ j x + l _ j \ , x ^ 2 + m _ j \ , x \ , y + n _ j \ , y ^ 2 , \\ \dot { y } = x ( 1 - y ) , \end{cases} \ \ ( x , y ) \in \Sigma ^ { \mathcal { V } } _ j , \end{align*}


\begin{align*} \mathcal A \rightarrow \mathcal A / \mathcal P \simeq R _ { \pi } \stackrel { c ^ * } { \longrightarrow } R , \ \ \ F \mapsto F _ { c } : = c ^ * ( F ( 1 ) ) \end{align*}


\begin{align*} \chi _ k = \min _ { S _ k \subset V \atop \dim ( S _ k ) = k } \max _ { 0 \not = u \in S _ k } \frac { \| u \| _ V } { \| u \| _ { L _ 2 ( D ) } } . \end{align*}


\begin{align*} \nu ( f ) = \beta _ b + b \gamma _ \sigma . \end{align*}


\begin{align*} \left . \begin{array} { l } \left \{ x _ { n } - y _ { n } \right \} _ { n = 0 } ^ { \infty } \ , \ , \mathrm { i s } \ , \mathrm { \ , b o u n d e d } \\ \left \Vert x _ { n } - y _ { n } \right \Vert - \left \Vert T x _ { n } - T y _ { n } \right \Vert \rightarrow 0 \end{array} \right \} \Longrightarrow x _ { n } - y _ { n } - \left ( T x _ { n } - T y _ { n } \right ) \rightarrow 0 . \end{align*}


\begin{align*} \sum _ { \substack { 0 \le p \le i , \ 0 \le q \le n - i \\ p + q = k } } ( y _ i - y _ { i + 1 } ) e _ { p } ( y _ 1 , \ldots , y _ i ) e _ { q } ( y _ { i + 1 } , \ldots , y _ n ) = 0 . \end{align*}


\begin{align*} u _ 1 u _ 2 = e ^ { i x \cdot \xi } e ^ { i ( \varphi _ 1 + \varphi _ 2 ) } + e ^ { i x \cdot \xi } \big ( e ^ { i \varphi _ { 2 } } r _ { 1 } + e ^ { i \varphi _ { 1 } } r _ { 2 } + r _ 1 r _ 2 \big ) . \end{align*}


\begin{align*} U = - S ( s ) \partial _ \nu ^ + U + D ( s ) \gamma ^ + U . \end{align*}


\begin{align*} | f _ { i } ( \alpha ) | = | f _ { i } ^ 0 ( \alpha ) | < \| f \| _ { \infty } + \epsilon . \end{align*}


\begin{align*} D _ { q } u ( x ) + u ( x ) u ( q x ) + D ( x ) u ( q x ) = 0 , \end{align*}


\begin{align*} Q ( p , 0 ) \otimes Q ( 0 , q ) = \bigoplus _ { n = 0 } ^ { \min \{ p , q \} } Q ( p - n , q - n ) \ , \end{align*}


\begin{align*} | A | ^ 2 _ { g _ X } = g _ X ^ { i l } g _ X ^ { j q } A _ { i j } A _ { l q } . \end{align*}


\begin{align*} - \Delta v _ e ^ i = 0 , \ v _ e ^ i ( x , 0 ) = - v _ p ^ { i - 1 } ( x , 0 ) , \ v _ e ^ i ( x , \infty ) = 0 , \ u _ e ^ i = \int _ x ^ \infty v _ { e Y } ^ i ( \theta , Y ) { \rm d } \theta , \end{align*}


\begin{align*} a ( w , z ; \alpha ) = \displaystyle \int _ \Omega \mu \nabla w \cdot \nabla z \ , d \Omega + \int _ \Omega \big ( { \bf b } \cdot \nabla w + \sigma w \big ) z \ , d \Omega , f ( z ; \alpha ) = \displaystyle \int _ \Omega f z \ , d \Omega , \end{align*}


\begin{align*} \sum _ { j = 1 } ^ { m } j c _ j = O \left ( m v \left ( \frac { 1 } { m } \right ) \right ) , \end{align*}


\begin{align*} \zeta _ { 3 } ( x ) = \left \{ \begin{array} { l l } 0 & 1 \leq x \leq \frac { 3 } { 2 } , \\ 1 & x \geq 2 . \end{array} \right . \end{align*}


\begin{align*} \left \langle \Phi _ 0 ( x ) ^ { ( 0 ) } ( \alpha ) , e \right \rangle = \varrho ( x ) [ \langle \alpha , e \rangle ] - \left \langle \alpha , \nabla _ x ( e ) \right \rangle , \quad \end{align*}


\begin{align*} \varrho ( r ) = \int _ 0 ^ r \ \inf _ { d _ X ( x , x ^ * ) \leq \rho } C ( J f ( x ) ) d \rho . \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } \| \chi _ { S ^ c _ { t ^ \beta } } J U _ 0 f \| = 0 \end{align*}


\begin{align*} \begin{array} [ c ] { c } \omega _ { T _ { 0 } } = \widehat { \omega } _ { T _ { 0 } } + \sqrt { - 1 } \partial _ { B } \overline { \partial } _ { B } \varphi _ { T _ { 0 } } \geq 0 . \end{array} \end{align*}


\begin{align*} ( z \partial _ z + \mu - \rho / z ) ( z ^ { - \mu } z ^ \rho \alpha ) = 0 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { t \in [ 0 , T ] } E [ | A ^ n _ t - \widetilde { A } ^ n _ t | ^ 2 ] = 0 , \end{align*}


\begin{align*} \label [ I e q ] { e q : p d e p d m p } \frac { \dd } { \dd t } \langle f _ t , \varphi \rangle = \langle f _ t , a \cdot \nabla _ x \varphi \rangle + \iint _ { E \times E } \lambda ( x , f _ t ) \{ \varphi ( y ) - \varphi ( x ) \} P _ { f _ t } ( x , \dd y ) f _ t ( \dd x ) , \end{align*}


\begin{align*} X = p \partial _ { 1 } + q \partial _ { 2 } . \end{align*}


\begin{align*} A x + e = F ^ { - 1 } G ( x ) + F ^ { - 1 } d , \end{align*}


\begin{align*} \left \{ \begin{array} { l } Y ' ( t ) = A ( t ) Y ( t ) , \\ Y ( 0 ) = , \end{array} \right . \end{align*}


\begin{align*} \Phi ^ \tau _ x ( \eta , \ , h ) \ = \ \big ( \tau ^ { x , h ( x ) } \eta , \ , h + \delta _ x \big ) \ , , \end{align*}


\begin{align*} r = r \cdot w _ 1 \cdots w _ s = h _ s \cdots h _ 1 \cdot r , \end{align*}


\begin{align*} R P P ( w , n ) = B _ w ( n \lambda ) \end{align*}


\begin{align*} \nu \left ( f - l ( h _ \sigma ) \right ) = \beta _ b + b \gamma _ \sigma . \end{align*}


\begin{align*} q _ A = \frac { 2 } { 1 + \frac { G ( x , y ) } { G ( x , x ) } } . \end{align*}


\begin{align*} \int f ( x ) d _ q x = F ( x ) , \end{align*}


\begin{align*} \operatorname { H e s s } _ { p } ( V , W ) = \operatorname { H e s s } _ { p } ( W , V ) , \operatorname { H e s s } _ { \| X \| ^ { 2 } } ( V , W ) = \operatorname { H e s s } _ { \| X \| ^ { 2 } } ( W , V ) . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ \infty | \beta ( k ) - \alpha ( k ) | ^ 2 w ' ( k ) < \infty . \end{align*}


\begin{align*} f _ i ( x ) = ( f \circ S ^ { 1 - i } ) ( A x + e ) = f ( A S ^ { ( 1 - i ) k } x + e ) . \end{align*}


\begin{align*} \Delta | \nabla u | = | k | ^ 2 | \nabla u | - \langle \div k , \nabla u \rangle \end{align*}


\begin{align*} { \rm t r } ( A D ^ 2 u ) + \langle \beta , \nabla u \rangle - \lambda u = - \beta ^ k , \lambda > 0 . \end{align*}


\begin{align*} \begin{aligned} d _ h ( a , b ) & \leq d _ h ( a , \tilde x ) + d _ h ( \tilde x , \tilde y ) + d _ h ( \tilde y , b ) \\ & \leq 3 \eta + \delta + 3 \eta \\ & \leq 2 \delta \end{aligned} \end{align*}


\begin{align*} & \| P _ { \delta } ( A _ { v t , m } ) e ^ { - i H _ 0 t } \psi \| = \| P _ { \delta } ( A _ { v t , m } ) e ^ { - i H _ 0 t _ n } \chi _ { S _ R } \psi \| \leq \| \chi _ { S _ R } e ^ { i H _ 0 t _ n } P _ { \delta } ( A _ { v t , m } ) \| _ \textrm { o p } \| \psi \| \xrightarrow { t \rightarrow \infty } 0 \end{align*}


\begin{align*} f = \pi _ { i _ 1 } \pi _ { i _ 2 } \cdots \pi _ { i _ d } \end{align*}


\begin{align*} { B } ^ { \prime } _ { { \rm S T } } = \{ t { t ^ { \prime } _ 1 } \ldots { t ^ { \prime } _ n } , \ n \in \mathbb { N } \} . \end{align*}


\begin{align*} H ( x , y , t ) \lesssim \frac { 1 } { \sqrt { \mathrm { v o l } ( B ( x , r ) ) \mathrm { v o l } ( B ( y , r ) ) } } e ^ { - \frac { \rho ^ 2 } { 4 t } - \lambda _ 1 ( M ) t } = \overline { H } ( \rho , t ) , \end{align*}


\begin{align*} \Gamma ( T ^ * \mathcal { U } ) = \Gamma ( T ^ * \mathcal { M } _ { \mathrm { S p } } ( 2 m , \alpha , L ) ) . \end{align*}


\begin{align*} n ^ { - q _ A } \leq n ^ { - k } n ^ { K \log ( D / \delta ) / f ( n ) } = n ^ { - k } ( D / \delta ) ^ { K ( \log n ) / f ( n ) } = n ^ { - k } ( D / \delta ) ^ { K / b ( n ) } . \end{align*}


\begin{align*} \frac { \partial } { \partial y } \Im H ( x + i y ) > 0 y = f ( x ) , \ : x \in ( \alpha - \varepsilon , \alpha + \varepsilon ) . \end{align*}


\begin{align*} S ( s ) \varphi ( x ) : = \int _ \Gamma \mathcal { K } ( | x - y | , s ) \varphi ( y ) d \Gamma _ y x \in \R ^ d \setminus \Gamma \end{align*}


\begin{align*} \sum _ { n \le x } \tau _ k ( n ) = x \ , P _ { k - 1 } ( \log x ) + O ( x ^ { \theta _ k + \varepsilon } ) \end{align*}


\begin{align*} \widetilde { C } ' ( m , p ) \begin{bmatrix} \widehat { m } \\ \widehat { p } \end{bmatrix} = - \widetilde { C } ( m , p ) \circ C ( \widehat { m } , \widehat { p } ) \circ \widetilde { C } ( m , p ) \end{align*}


\begin{align*} e _ { j , k , l } = \left ( \dfrac { p ! } { j ! \ , k ! \ , l ! } \right ) ^ { - 1 / 2 } \sum _ { f \in S _ p } e _ { i _ { f ( 1 ) } } \otimes . . . \otimes e _ { i _ { f ( p ) } } \in \mathcal H _ { p , 0 } \ . \end{align*}


\begin{align*} | \mathcal K _ { \kappa , 2 } | = \left | \mathcal { T } _ { \kappa , 2 } \right | + \frac 1 2 \left | \mathcal S _ { \kappa , 2 } \setminus \mathcal T _ { \kappa , 2 } \right | . \end{align*}


\begin{align*} k = k _ 1 - k _ 2 + \cdots + k _ p . \end{align*}


\begin{align*} \left | \sum _ { k = 1 } ^ N f ( n _ k \alpha ) \right | \ll ( N \log N ) ^ { 1 / 2 } ( \log \log N ) ^ { 3 / 2 + \varepsilon } \textrm { f o r a . e . } \alpha , \end{align*}


\begin{align*} b ( \hat \eta ) x \Omega _ \phi & = x \hat \eta \\ b ( \hat \eta ) \eta & = \eta \otimes _ \phi \hat \eta \end{align*}


\begin{align*} \Big ( x \frac { d } { d x } \Big ) _ { p , \lambda } e ^ { x } = \sum _ { n = 1 } ^ { \infty } \frac { ( n ) _ { p , \lambda } } { n ! } x ^ { n } = e ^ { x } \phi _ { p , \lambda } ( x ) . \end{align*}


\begin{align*} \Sigma : = ( X , U , F ) , \end{align*}


\begin{align*} \frac { y _ { n } } { x _ { n } } = \prod _ { k \leq n } \frac { p _ k ^ 2 } { p _ k ^ 2 - \chi ( p _ k ) } = \left ( \frac { p _ n ^ 2 } { p _ n ^ 2 - 1 } \right ) \prod _ { k \leq n - 1 } \frac { p _ k ^ 2 } { p _ k ^ 2 - 1 } , \end{align*}


\begin{align*} \mathrm { T r } _ { \perp , r } ( A ) : = \sum _ { \alpha \ge 1 } \langle u _ { r , \alpha } , A \ , u _ { r , \alpha } \rangle , \mathrm { T r } _ { \perp , \ell } ( A ) : = \sum _ { \alpha \ge 1 } \langle u _ { \ell , \alpha } , A \ , u _ { \ell , \alpha } \rangle \end{align*}


\begin{align*} \sup _ { \mathbb { R } ^ n } \left | \underline { u } ( x ) - \frac { 1 } { 2 } \sum _ { i = 1 } ^ n a _ i x _ i ^ 2 \right | \leq C . \end{align*}


\begin{align*} D _ 2 \approx \sum _ { k = - \infty } ^ { M - 1 } \bigg ( \int _ { x _ k } ^ { x _ { k + 1 } } u ( s ) \bigg ( \int _ { s } ^ { \infty } u \bigg ) ^ { \frac { q } { p - q } } d s \bigg ) \sum _ { i = - \infty } ^ k 2 ^ { - i \frac { q } { p - q } } V _ r ( x _ { i - 1 } , x _ i ) ^ { \frac { p q } { p - q } } . \end{align*}


\begin{align*} { \rm e r f } _ { \kappa } \ ! \left ( x \right ) \ ! = \ ! \left ( \ ! 1 + \frac { 1 } { 2 } \ , \kappa \ ! \right ) \ ! \sqrt { 2 \kappa } \ , \ , \frac { \Gamma \ ! \left ( \frac { 1 } { 2 \kappa } + \frac { 1 } { 4 } \right ) } { \Gamma \ ! \left ( \frac { 1 } { 2 \kappa } - \frac { 1 } { 4 } \right ) } \frac { 2 } { \sqrt { \pi } } \int _ 0 ^ x \ ! \exp _ { \kappa } ( - t ^ 2 ) \ , d t \ \ , \end{align*}


\begin{align*} T \mathcal { M } ( p , q ) = \R ( - \partial _ s ) \oplus T ( \mathcal { M } ( p , q ) / \R ) . \end{align*}


\begin{align*} t \mapsto \frac { p _ { d , t , \mu , m ^ 2 } ( \lambda ) } { t ^ 2 } \end{align*}


\begin{align*} u _ t + ( - \Delta ) ^ { d } u = | x | ^ { \alpha } | u | ^ { p } + \zeta ( t ) { \mathbf w } ( x ) \ \quad \mbox { f o r } ( x , t ) \in \mathbb { R } ^ { N } \times ( 0 , \infty ) , \end{align*}


\begin{align*} z ^ * = \sum _ { w \in \mathcal { W } \setminus \left ( \{ z \} \cup \{ x ^ t _ s : \ ; s \in \mathbb { N } , \ : t \in A \cup \{ k \} \} \right ) } y ^ * ( w ) \widetilde { w } . \end{align*}


\begin{align*} P ( u ) \cdot v = \begin{bmatrix} f & h \end{bmatrix} ^ T \end{align*}


\begin{align*} \sum _ i ( \alpha , h _ i ) _ { M , n } = \Tilde { \Phi } _ L ( \beta ) ( \sum _ j ( \Tilde { \sigma } _ j ^ { - 1 } ( \xi ) ) ) - \sum _ j ( \Tilde { \sigma } _ j ^ { - 1 } ( \xi ) ) . \end{align*}


\begin{align*} \lambda b ( u , b _ { T } u _ { \mathrm { n c } } ) & = a _ { \mathrm { p w } } ( u , b _ { T } u _ { \mathrm { n c } } ) = ( D ^ { m } u , D ^ { m } ( b _ T u _ { \mathrm { n c } } ) ) _ { L ^ 2 ( T ) } \le \vert u - I u \vert _ { H ^ m ( T ) } \vert b _ T u _ { \mathrm { n c } } \vert _ { H ^ m ( T ) } . \end{align*}


\begin{align*} \Bigl ( \rho \wedge \pi _ { { } _ S } = \pi \Bigr ) \ \Leftrightarrow \ \Bigl ( \begin{array} { c } S \cap V _ i = \emptyset \mbox { f o r e v e r y $ 1 \leq i \leq p $ , a n d } \\ S \cap W _ j \neq \emptyset \mbox { f o r e v e r y $ 1 \leq j \leq q $ } \end{array} \Bigr ) . \end{align*}


\begin{align*} \mathcal { R } ( \psi ) = R _ { \varphi _ 1 } R _ { \varphi _ 1 } \cdots R _ { \varphi _ n } , \end{align*}


\begin{align*} \Phi '' ( \vec p ) [ \widehat { \vec p } , \cdot ] = F '' ( V ( \vec p ) ) [ V ' ( \vec p ) \widehat { \vec p } , V ' ( \vec p ) \ , \cdot ] + F ' ( V ( \vec p ) ) V '' ( \vec p ) [ \widehat { \vec p } , \cdot ] . \end{align*}


\begin{align*} \begin{cases} Z _ 2 ( r ) = a _ 1 r ^ { - ( n - 2 ) } + o ( r ^ { - ( n - 2 ) } ) r \to 0 , \\ Z _ 2 ( r ) = a _ 2 + o ( 1 ) r \to \infty \end{cases} \end{align*}


\begin{align*} \prod _ { k \ge 1 } \frac { \mu _ k ^ 2 } { \lambda _ k ^ 2 } = 0 , \sum _ { k \ge 1 } \Bigg ( \frac { \lambda _ { k } ^ { 2 } } { \mu _ { k } ^ { 2 } } - 1 \Bigg ) = \infty ; \end{align*}


\begin{align*} f _ * = \min _ { x \in \mathbb { R } ^ d } f ( x ) = \min _ { i \in \mathcal { I } } \big \{ f _ i - \tfrac { 1 } { 2 L } \| g _ i \| ^ 2 \big \} \end{align*}


\begin{align*} h _ t ( x , y ) = p _ t ( x , y ) - \frac { 1 } { n } . \end{align*}


\begin{align*} t \in \tau : = [ 0 , 1 ] \setminus \{ t _ i \ , | \ , 1 \le i \le 2 ^ { | I | } - 1 \} , \end{align*}


\begin{align*} \max _ { v \in [ 0 , V _ F ] } \rho _ { } ( v ) \geq \frac { \int _ 0 ^ { V _ F } \rho _ { } ( v ) d v } { V _ F } = \frac { 1 } { V _ F } \geq \frac { 1 } { g _ 1 } . \end{align*}


\begin{align*} e _ k ( \tau _ 1 , \tau _ 2 , \ldots , \tau _ n ) = 0 ( 1 \le k \le n ) . \end{align*}


\begin{align*} - m \lambda \log ( \tfrac { \lambda } { 2 } ) & = \frac { \log ( L \log ( m ) ) } { L \log ( m ) } \log \left ( \frac { 2 L m \log ( m ) } { \log ( L \log ( m ) ) } \right ) \\ [ 3 p t ] & \leq \frac { ( \log ( 2 L \log ( m ) ) ^ 2 } { L \log ( m ) } + \frac { \log ( L \log ( m ) ) } { L } \leq C _ 4 + \frac { \log ( L \log ( m ) ) } { L } \end{align*}


\begin{align*} \mathcal { L } = \int _ 0 ^ T \left [ \bold { K - ( P + E ) + B + W } \right ] d t , \end{align*}


\begin{align*} & \langle y , ( - u _ i , 1 , 0 , 0 ) \rangle \leq 0 , \ i = 1 , \ldots , m , \ , \langle y , - e _ { s + 1 } + e _ { s + 3 } \rangle \leq 0 , \\ & \langle y , - e _ i \rangle \leq 0 , \ , i = 1 , \ldots , s , \ , \langle y , - e _ { s + 3 } \rangle \leq 0 , \\ & \langle y , ( \gamma _ j , 0 , 0 , 0 ) - d _ j e _ { s + 2 } + e _ { s + 3 } \rangle \leq 0 , \ , \langle y , e _ { s + 2 } \rangle \leq 1 , \ , \langle y , - e _ { s + 2 } \rangle \leq - 1 . \end{align*}


\begin{align*} \Psi ( & E _ { j + 1 } ( x , d - 1 ) ) - \Psi ( E _ j ( x , d - 1 ) ) = \\ & = 2 \sqrt { d - 1 } \left ( \csc \left ( \dfrac { \pi } { 2 j + 4 } \right ) - 1 \right ) - 2 \sqrt { d - 1 } \left ( \cot \left ( \dfrac { \pi } { 2 j + 2 } \right ) - 1 \right ) \\ & = 2 \sqrt { d - 1 } \left ( \csc \left ( \dfrac { \pi } { 2 j + 4 } \right ) - \cot \left ( \dfrac { \pi } { 2 j + 2 } \right ) \right ) \\ & = f _ j \ , \sqrt { d - 1 } \ , . \end{align*}


\begin{align*} \frac { 2 L \ , \big ( f ( x _ 0 ) - f _ * \big ) } { 1 + \sum \limits _ { i = 0 } ^ { N - 1 } \left ( 2 h _ i - h _ i ^ 2 \frac { - \kappa } { 1 - \kappa } \right ) } \end{align*}


\begin{align*} \times \varphi _ { j , l } ( v , { \omega } ) d { \omega } d v d s \big | \lesssim \sup _ { s \in [ 0 , t ] } \sup _ { g \in L ^ 2 , \| g \| _ { L ^ 2 } = 1 } 2 ^ { 2 m + k + 2 l + \tilde { c } ( m , k , l ) } \int _ { \R ^ 3 } \int _ { \R ^ 3 } \big | \hat { f } ( s , \xi , v ) \big | \varphi _ k ( \xi ) \psi _ { \leq \tilde { c } ( m , k , l ) + \epsilon M _ t } ( \tilde { v } \times \tilde { \xi } ) \big | \overline { \hat { g } ( \xi ) } \big | \varphi _ j ( v ) d v d \xi \end{align*}


\begin{align*} ( - 1 ) ^ k \sum _ { j = - k } ^ k ( - 1 ) ^ j p ( n - j ( 3 j + 1 ) / 2 ) = \widetilde P _ k ( n ) . \end{align*}


\begin{align*} \norm { f _ m } ^ 2 = \norm { g _ 0 } ^ 2 + \norm { g _ 1 } ^ 2 + \cdots + \norm { g _ m } ^ 2 \leq \norm { r h } ^ 2 \end{align*}


\begin{align*} \bigg ( \sum _ { k = - \infty } ^ { M - 1 } \bigg ( \int _ 0 ^ { x _ k } f ^ r v \bigg ) ^ { \frac { q } { r } } \int _ { x _ k } ^ { x _ { k + 1 } } u ( t ) d t \bigg ) ^ { \frac { 1 } { q } } \leq \mathfrak { C ' } \bigg ( \sum _ { k = - \infty } ^ { M - 1 } 2 ^ k \bigg ( \int _ { x _ k } ^ { x _ { k + 1 } } f \bigg ) ^ p \bigg ) ^ { \frac { 1 } { p } } \end{align*}


\begin{align*} \kappa ( z ) \triangleq \log \phi ( z ) = \sum _ { j = 1 } ^ \infty \kappa _ j \frac { z ^ j } { j ! } , \end{align*}


\begin{align*} \widehat { \mathsf { F D P } } ( t ) : = \frac { 1 + \# \{ j : W _ j \leq - t \} } { \# \{ j : W _ j \geq t \} \vee 1 } , \end{align*}


\begin{align*} \lim _ { z \to \infty } A ( z ) = \infty . \end{align*}


\begin{align*} Y = \exp ( \lambda _ 0 M _ 0 ) \exp ( \lambda _ 1 M _ 1 ) \exp ( \lambda _ 2 M _ 2 ) \ , . \end{align*}


\begin{align*} \mathfrak { X } _ L : = \{ y \in L ; ~ < x , y > _ L \in \mathcal { O } ~ \forall x \in \lambda _ { \rho } ( \mathfrak { p } _ L ) \} . \end{align*}


\begin{align*} \Omega ( \alpha , A , z + a f _ k ) = \{ z + a f _ k \} \cup \bigcup _ { n = 1 } ^ \infty \Omega ( \alpha _ n , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \end{align*}


\begin{align*} \frac { 1 } { \zeta ( s ) } \sum _ { n = 1 } ^ { \infty } \frac { c _ k ( n ) } { n ^ s } = \sum _ { d | k } \mu ( \frac { k } { d } ) d ^ { 1 - s } , \end{align*}


\begin{align*} \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix} , \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} . \end{align*}


\begin{align*} & ( c _ i ^ 2 - c _ i ) = 0 i > \ell \\ & c _ i c _ j = 0 i > \ell , j > i . \end{align*}


\begin{align*} \sum _ { k \ge 1 } ^ \infty \left ( \frac { \mu _ k ^ 2 } { \lambda _ { k + 1 } ^ 2 } - 1 \right ) = \infty . \end{align*}


\begin{align*} Q _ { 3 , \tilde { q } , \tilde { p } } = \underbrace { \sigma _ 2 ^ { \lambda ( 1 ) } \sigma _ 1 ^ { \lambda ( 2 ) } \sigma _ 2 ^ { \lambda ( \frac { \tilde { q } - 3 } { 2 } ) } \sigma _ 1 ^ { \lambda ( \frac { \tilde { q } - 1 } { 2 } ) } } _ { P } \sigma _ 2 ^ { - \lambda ( \frac { \tilde { q } - 1 } { 2 } ) } \sigma _ 1 ^ { - \lambda ( \frac { \tilde { q } - 3 } { 2 } ) } . . . \sigma _ 2 ^ { - \lambda ( 2 ) } \sigma _ 1 ^ { - \lambda ( 1 ) } \end{align*}


\begin{align*} \overline { \varepsilon } \ ! = \ ! \frac { 1 } { 2 ^ { K + 2 } } \left | a ^ { p / 2 } - b ^ { p / 2 } \right | = \frac { 1 } { 2 ^ { K + 2 } } \left | \frac { a ^ p - b ^ p } { a ^ { p / 2 } + b ^ { p / 2 } } \right | & \geq \frac { 1 } { 2 ^ { K + 2 } } \left | \frac { a ^ { \lfloor p \rfloor } - b ^ { \lfloor p \rfloor } } { a ^ { \lceil p / 2 \rceil } + b ^ { \lceil p / 2 \rceil } } \right | \triangleq \varepsilon > 0 . \end{align*}


\begin{align*} p ( n - 1 ) + p ( n - 2 ) - p ( n ) & = \widetilde P _ 1 ( n ) \\ p \left ( n - \frac { k ( 3 k - 1 ) } { 2 } \right ) + p \left ( n - \frac { k ( 3 k + 1 ) } { 2 } \right ) & = \widetilde P _ { k - 1 } ( n ) + \widetilde P _ k ( n ) , \ k \geqslant 2 . \end{align*}


\begin{align*} \tau ^ * ( p ^ m ) = 2 \end{align*}


\begin{align*} \phi _ { n } ^ { \pm } ( x ) = \frac { e ^ { i \left [ \left ( \varphi \pm 2 n \pi \right ) x / a \pm \theta \right ] } } { \sqrt { a } } ; E _ { n } ^ { \pm } = \frac { \hslash ^ { 2 } } { 2 m } \left ( \frac { 2 n \pi \pm \varphi } { a } \right ) ^ { 2 } + \frac { \beta \hslash ^ { 4 } } { 3 m } \left ( \frac { 2 n \pi \pm \varphi } { a } \right ) ^ { 4 } , \end{align*}


\begin{align*} \| u _ k ( t ) \| _ \infty = ( T - t ) ^ { - 3 k } ( k \in \N ) . \end{align*}


\begin{align*} \gamma ^ + D ( \partial _ t ) \psi = \frac 1 2 \psi + K ( \partial _ t ) \psi \end{align*}


\begin{align*} \mathcal { I } _ n = \underbrace { \iint _ { Q _ T } Z \frac { B _ n } { A _ n } ( A - A _ n ) [ \nabla \cdot ( m \nabla \varphi _ n ) ] } _ { \mathcal { I } _ { n , 1 } } + \underbrace { \iint _ { Q _ T } Z ( B _ n - B ) [ \nabla \cdot ( m \nabla \varphi _ n ) ] } _ { \mathcal { I } _ { n , 2 } } . \end{align*}


\begin{align*} \sum _ { | \alpha | \leq m } \langle a _ { 1 , \alpha } - a _ { 2 , \alpha } , ( D ^ \alpha v _ 1 ) v _ 2 \rangle = 0 \mbox { f o r a l l } v _ 1 , v _ 2 \in C ^ \infty _ c ( \Omega ) . \end{align*}


\begin{align*} f ( x ) = \sum _ { n \in \Z } f _ n e ^ { 2 \pi i n x } \ , . \end{align*}


\begin{align*} \varphi ( x ) = ( 1 + \lVert x \rVert ^ 2 ) ^ 8 , U ( x ) = 8 T \left ( 8 T e ^ { \alpha T } \bar { r } ^ 2 \bar { a } ^ 2 + 3 + \bar { r } \right ) + e ^ { - \alpha T } \lVert x \rVert ^ 2 , \bar { U } ( x ) = 0 , \end{align*}


\begin{align*} \int \dfrac { x ^ \alpha } { ( - x ; q ) _ \infty } L _ n ^ { \alpha } ( x ; q ) d _ q x = \dfrac { x ^ { \alpha + 1 } } { [ n ] _ q ( - x ; q ) _ \infty } L _ { n - 1 } ^ { \alpha + 1 } ( x ; q ) . \end{align*}


\begin{align*} \begin{aligned} \sum _ { i \in I } 2 r s d _ i + 2 \sum _ { i \in I ( \Gamma ) } s a _ i - 2 s k _ 1 ( \Gamma ) + 2 \sum _ { i \in I ( \Gamma ) } r b _ i - 2 r k _ 2 ( \Gamma ) & = 2 r s | I | - 2 r s . \end{aligned} \end{align*}


\begin{align*} \frac { \partial } { \partial \theta } \arg \left ( \Phi ( r e ^ { i \theta } ) \right ) > 0 r = r _ { 0 } \theta = h _ { \nu } ( r _ { 0 } ) . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l } \psi ( x ) = \sqrt { \frac { 2 } { a } } \cos ( \frac { n \pi x } { a } ) , $ n o d d $ \\ \psi ( x ) = \sqrt { \frac { 2 } { a } } \sin ( \frac { n \pi x } { a } ) , $ n e v e n $ \end{array} \right . . \end{align*}


\begin{align*} \rho _ \mathcal { A } ( x _ i , X _ \lambda ) = \varrho ( x _ i ) + X _ \lambda \end{align*}


\begin{align*} w _ m = \frac { [ T _ m - \alpha ] ^ + } { \sum _ { k = 1 } ^ { M } [ { T _ k } - \alpha ] ^ + } , m = 1 , 2 , \ldots , M \end{align*}


\begin{align*} \tau ^ \mathrm { a n } ( M ' / S , \mathbb { 1 } ) = \tau ^ \mathrm { a n } \big ( M / S , \mathrm { I n d } ^ G _ H \mathbb { 1 } \big ) \ ; . \end{align*}


\begin{align*} \sup _ { N \leq k \leq M } a _ k \bigg ( \sum _ { i = N } ^ k b _ i \bigg ) ^ \alpha \approx \sup _ { N \leq k \leq M } a _ k b _ k ^ \alpha . \end{align*}


\begin{align*} \widetilde { u } _ n & \in \left ( Q _ 1 ( \alpha - 1 , A _ n , u _ n + a _ n f _ { q ( n ) } ) \right ) ^ { ( \alpha ) } \\ & \subseteq \left ( \bigcap _ { j = 1 } ^ \infty K ( \alpha - 1 , A _ j , u _ j + a _ j f _ { q ( j ) } ) \right ) ^ { ( \alpha ) } = K ^ { ( \alpha ) } . \end{align*}


\begin{align*} \mathrm { T C M } _ { q } ( X ^ { n } ) = \mathrm { E } \left [ ( X - \mathrm { T C E } _ { q } ( X ) ) ^ { n } | X > x _ { q } \right ] , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { n - 1 } { \lambda _ i } ( A ) \det ( ( B v ) ^ * ( B v ) ) = \det ( B ^ * A B ) , \end{align*}


\begin{align*} ( [ \alpha ] _ { \vec s } , [ \beta ] _ { \vec s ^ \prime } ) = \begin{cases} \int _ { D _ { I _ { \vec s } } } \alpha \cup \beta , & \vec s = - \vec s ^ \prime ; \\ 0 , & \end{cases} \end{align*}


\begin{align*} \begin{cases} \dot u = R _ { \varphi ( u \wedge v ) } ( u ) \\ \dot v = R _ { \varphi ( u \wedge v ) } ( v ) \end{cases} \end{align*}


\begin{align*} \langle a , b \rangle _ { l ^ 2 _ n } = \frac { \pi } { n } \sum _ { i = 1 } ^ { n - 1 } a _ i b _ i , \qquad \| a \| _ { l ^ p _ n } = \begin{cases} \Big ( \frac { \pi } { n } \sum \limits _ { i = 1 } ^ { n - 1 } | a _ i | ^ p \Big ) ^ { \frac { 1 } { p } } , \quad & 1 \le p < \infty , \\ \sup \limits _ { 1 \le j \le n - 1 } | a _ i | , \quad & p = \infty , \end{cases} \end{align*}


\begin{align*} A \Phi _ { \nu , \nu ' } = ( 2 | \nu | + n ) \Phi _ { \nu , \nu ' } . \end{align*}


\begin{align*} \lvert g ( t - s , x - y ) - g ( t - s , x ' - y ) \rvert ^ 2 z ^ 2 & \leq C ( t - s ) ^ { - d } z ^ 2 \Bigl \lvert e ^ { - \frac { \lvert x - y \rvert ^ 2 } { 2 ( t - s ) } } - e ^ { - \frac { \lvert x ' - y \rvert ^ 2 } { 2 ( t - s ) } } \Bigr \rvert ^ 2 \\ & \leq C ( t - s ) ^ { - \frac d 2 q } z ^ q \Bigl \lvert e ^ { - \frac { \lvert x - y \rvert ^ 2 } { 2 ( t - s ) } } - e ^ { - \frac { \lvert x ' - y \rvert ^ 2 } { 2 ( t - s ) } } \Bigr \rvert ^ q \\ & = C \lvert g ( t - s , x - y ) - g ( t - s , x ' - y ) \rvert ^ q z ^ q , \end{align*}


\begin{align*} - \operatorname { D i v } \sigma & = u & & \Omega , \\ \nabla \cdot y & = 0 & & \Omega , \\ y & = 0 & & \Gamma , \end{align*}


\begin{align*} I _ { \vec s } : = \{ i : s _ i \neq 0 \} \subseteq \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} d V _ { B H } = ( 1 - \| \beta \| ^ 2 _ { \alpha } ) ^ { \frac { n + 1 } { 2 } } d V _ { \alpha } \end{align*}


\begin{align*} \lambda _ { k + 2 } ( ( 0 , \infty ) ) = \lambda ( ( z _ { k + 2 } , z _ { k + 1 } ] ) = \overline \lambda ( z _ { k + 2 } ) - \overline \lambda ( z _ { k + 1 } ) \leq 2 . \end{align*}


\begin{align*} \mathcal { N } ' = \phi ^ { - 1 } \big ( \mathbb { A } \big ( \rho ( \tfrac 1 5 \sigma ^ * ) \big ) \big ) . \end{align*}


\begin{align*} v ( x ) = g ( x ) \int _ { 0 } ^ { x } \frac { 1 } { g ( u ) } d _ q u , x \in I \end{align*}


\begin{align*} \mathrm { e } ^ { - z t } B R ( z , A ) & = \mathrm { e } ^ { - z t } \int _ 0 ^ t \mathrm { e } ^ { z s } \tilde { B } T ( s , A ) \ , \mathrm { d } s + B R ( z , A ) T ( t , A ) . \end{align*}


\begin{align*} w _ { a b } \stackrel { \mathrm { d } } { = } N ^ { - 1 / 2 } \times \begin{cases} \chi _ \mathrm { o d } , & a \ne b \\ \chi _ \mathrm { d } , & a = b , \end{cases} \end{align*}


\begin{align*} J '' ( \gamma _ 0 , y ) = \int \limits _ { t _ 0 } ^ { t _ 1 } 2 \omega ( t , y ( t ) , \dot { y } ( t ) ) d t , \end{align*}


\begin{align*} & \int _ 0 ^ r \left [ ( - \phi '' - \frac { z } { 2 } \phi ' + \frac { z } { 2 } e _ \sigma \psi ' ) \cdot \phi + ( - \psi '' - \frac { z } { 2 } \psi ' + \frac { z } { 2 } e _ \sigma \phi ' ) \cdot \psi \right ] { \rm { d } } z \\ & = \int _ 0 ^ r | ( \phi ' , \psi ' ) | ^ 2 + \frac { 1 } { 4 } \int _ 0 ^ r | ( \phi , \psi ) | ^ 2 - \frac { 1 } { 2 } \int _ 0 ^ r ( e _ \sigma + z e _ \sigma ' ) \psi \phi . \end{align*}


\begin{align*} z _ s = e ^ { - \pi t + \pi \sigma i } , \cos \frac { \pi s } { 2 } = \frac { e ^ { \frac { \pi } { 2 } \left ( t - \sigma i \right ) } } { 2 } \left ( 1 + z _ s \right ) , \quad \sin \frac { \pi s } { 2 } = \frac { e ^ { \frac { \pi } { 2 } \left ( t - \sigma i \right ) } i } { 2 } \left ( 1 - z _ s \right ) . \end{align*}


\begin{align*} \sum _ { k \ge 1 } \left ( 1 - \frac { \mu _ k ^ 2 } { \lambda _ k ^ 2 } \right ) = \infty , \end{align*}


\begin{align*} d u \left ( t \right ) & + \nu A u \left ( t \right ) d t + \lambda ( r ( t ) ) u \left ( t \right ) d t = \left ( { f \left ( { t , r ( t ) , u ( t ) } \right ) + g ( r ( t ) ) } \right ) d t \\ & + \varepsilon \sum \limits _ { k = 1 } ^ \infty { \left ( { h _ k ( r ( t ) ) + \sigma _ k \left ( { t , r ( t ) , u \left ( t \right ) } \right ) } \right ) } d W _ k \left ( t \right ) , t > s , \end{align*}


\begin{align*} g _ { r , x } ( z ) & = \left \lbrace \begin{matrix} g ( x ) & \mathrm { d i s t } ( x , z ) \leq \frac { r } { 2 } \\ \Phi _ x ^ { - 1 } \left ( \chi _ r ( \mathrm { d i s t } ( x , z ) ) \Phi _ x ( g ( z ) ) \right ) & \frac { r } { 2 } \leq \mathrm { d i s t } ( x , z ) \leq r \\ g ( z ) & \end{matrix} \right . . \end{align*}


\begin{align*} \hat \delta _ j \cdot ( I _ 3 , n _ j ) = 1 , n _ j \in \Z . \end{align*}


\begin{align*} \norm { g _ { \le \iota } ^ { ( 1 ) } } _ p ^ p \lesssim 2 ^ { - \varepsilon \iota } \sum _ { \ell = - 1 } ^ \iota ( 2 ^ { \ell ( d _ 1 - d _ 2 ) + \iota ( d _ 2 - d _ 1 ) } ) ^ { p / q } \norm { F ^ { ( \iota ) } } _ { L ^ 2 _ s } ^ p \norm { f } _ p ^ p \end{align*}


\begin{align*} W ^ { s , p } ( \Omega ) \vcentcolon = \{ \ , u \in W ^ { m , p } ( \Omega ) \ , ; \ , \partial ^ { \alpha } u \in W ^ { \sigma , p } ( \Omega ) \forall | \alpha | = m \} , \end{align*}


\begin{align*} ( x , t ) \in D _ { n } : = D _ { \rm I n t } \cap \{ x > 0 \} \cap \{ t - x > l _ n R \} , l _ n : = \sum _ { i = 0 } ^ n \left ( \frac { 1 } { 2 } \right ) ^ i . \end{align*}


\begin{align*} \begin{cases} a = p _ 1 + p _ 2 - u - v - 2 s \ , \ \ b = q _ 1 + q _ 2 - u - v + s \ , \ \ \mbox { f o r ( \ref { p s i - 1 c a s e } ) } \ , \\ a = p _ 1 + p _ 2 - u - v + s \ , \ \ b = q _ 1 + q _ 2 - u - v - 2 s \ , \ \ \mbox { f o r ( \ref { p s i - 2 c a s e } ) } \ , \end{cases} \end{align*}


\begin{align*} P _ { n _ { k + 1 } } = T _ { h \left ( n _ { k + 1 } \right ) } \cdots T _ { h \left ( n _ { k + 1 } + M + 1 \right ) } T _ { l _ { M } } \cdots T _ { l _ { 1 } } P _ { n _ { k } } . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb P \Big ( \sup _ { t , x } | u ^ n ( t , x ) - u ( t , x ) | \ge 1 \Big ) = \mathbb P \Big ( \sup _ { t , x } | u ^ n ( t , x ) - u ( t , x ) | ^ { 2 p } \ge 1 \Big ) = 0 . \end{align*}


\begin{align*} g ( x , Y , Z ) = u , g _ x ( x , Y , Z ) = p . \end{align*}


\begin{align*} P _ { W | X } ( 0 | x ) = \begin{cases} 1 & z _ x > \gamma \\ \tau & z _ x = \gamma \\ 0 & z _ x < \gamma \end{cases} , \end{align*}


\begin{align*} \bar A ( x , s ) & = A ( x _ 0 + \Theta x , s ) \\ \bar f ( x , s ) & = \Phi \Theta ^ 2 f ( x _ 0 + \Theta x ) \\ \bar Q ( x ) & = \Phi ^ 2 \Theta ^ 2 Q ( x _ 0 + \Theta x ) . \end{align*}


\begin{align*} - \widehat { H } ' _ { n H } ( t ) = - e ^ { - \alpha ( t ) } \widehat { H } _ 1 + \left ( \frac { d \alpha } { d t } ( t ) + 2 \beta ( t ) \right ) \widehat { H } _ 2 + e ^ { \alpha ( t ) } \left ( \frac { d \beta } { d t } ( t ) - \beta ^ 2 ( t ) - \omega ^ 2 ( t ) \right ) \widehat { H } _ 3 - e ^ { \frac { k \alpha ( t ) } 2 } c ( t ) \widehat { H } _ 4 , \end{align*}


\begin{align*} f = \lambda \tilde R _ \lambda f - \tilde R _ \lambda q f . \end{align*}


\begin{align*} a = p + q - n - m - 2 k \ , \ \ 0 \le n \le p - k \ , \ \ 0 \le m \le q - k \ , \end{align*}


\begin{align*} R ( \mu , A _ 2 ) & - R ( \mu , A _ 1 ) \\ & = \left ( 1 + ( w - x ) R ( \lambda , A _ 2 ) \right ) ^ { - 1 } \left ( R ( \lambda , A _ 2 ) - R ( \lambda , A _ 1 ) \right ) \left ( 1 + ( x - w ) R ( \mu , A _ 1 ) \right ) , \end{align*}


\begin{align*} c _ 1 ( \alpha , h ) = r s \ ; \ ; \ ; \ ; \ ; c _ 2 ( \alpha , h ) = t u \end{align*}


\begin{align*} A ^ { i } : = I - G ^ { i + 1 } d _ { V } ^ { i } . \end{align*}


\begin{align*} C ( \lambda w ) = C \left ( \lambda \sqrt { w ^ 2 } \right ) = \ln \left [ \delta \cdot \cosh \left ( z _ 0 \lambda \sqrt { w ^ 2 } \right ) + \frac { 1 - \delta } { 1 - \lambda ^ 2 w ^ 2 / q ^ 2 } \right ] , \end{align*}


\begin{align*} \bigg ( \sum _ { k = N } ^ M \bigg ( \sum _ { i = N } ^ k x _ i b _ i \bigg ) ^ q a _ k \bigg ) ^ { \frac { 1 } { q } } \leq C \bigg ( \sum _ { k = N } ^ M x _ k ^ p \bigg ) ^ { \frac { 1 } { p } } \end{align*}


\begin{align*} g ^ f _ { k \rho } ( k X , k X ) = k g ^ f _ \rho ( X , X ) . \end{align*}


\begin{align*} \sigma _ { B H } ( x ) = \frac { v o l ( B ^ n ( 1 ) ) } { v o l \left \lbrace ( y ^ i ) \in T _ x M : F ( x , y ) < 1 \right \rbrace } . \end{align*}


\begin{align*} \mathbb { Y } : = \{ ( I , R ) \ | \ 0 < I \leq 1 , 0 \leq R \leq 1 - I \} \times \mathbb { X } \times \mathbb { R } \end{align*}


\begin{align*} \left \{ \begin{array} { l l l } \tau \left ( \theta ^ { k } \right ) = \overline { e } _ { k + 1 } , \ \ 0 \leq k \leq n - 1 \\ H ^ { * } \left ( \overline { e } _ { k } \right ) = H ^ { k - 1 } , \ \ 1 \leqslant k \leqslant n . \end{array} \right . \end{align*}


\begin{align*} d ( \mu \boxplus \nu ) _ { } = p _ { \mu \boxplus \nu } \ , d \lambda . \end{align*}


\begin{align*} f _ \infty ( x , v ) = \frac { 1 } { Z } e ^ { - \Phi ( x ) } , \end{align*}


\begin{align*} \delta ^ { ( s ) } u = f ^ { ( 1 ) } ( u , u ^ { \phi ^ s } ) \end{align*}


\begin{align*} L _ { t } ( \dot { x } , x , t ] = L _ { n s } ( \dot { x } , x , t ) - \frac { \partial \Phi _ { n u l l } ( x , t ) } { \partial t } \ , \end{align*}


\begin{align*} u ( x _ 1 , x ' ) = u ( y _ 1 , x ' ) + \int _ { y _ 1 } ^ { x _ 1 } \frac { \partial u } { \partial x _ 1 } ( t , x ' ) \dd t . \end{align*}


\begin{align*} F l _ n = \{ V _ { \bullet } = ( V _ 1 \subset V _ 2 \subset \cdots \subset V _ n = \C ^ n ) \mid \dim _ { \C } V _ i = i \ ( 1 \le i \le n ) \} . \end{align*}


\begin{align*} d = \frac { a _ i ^ 2 - a _ i } { b _ i ^ 2 - b _ i } . \end{align*}


\begin{align*} E \left [ Z ^ n _ t - \widetilde { Z } ^ n _ t \right ] & = n ^ { 2 \alpha + 1 } E \left [ \left | \int ^ t _ 0 ( t - s ) ^ { \alpha } \left ( \sigma ' ( X _ s ) - \Psi ^ n _ s \right ) \left ( X ^ n _ { s } - X _ s \right ) \ , d W _ s \right | ^ 2 \right ] \\ & \leq n ^ { 2 \alpha + 1 } \int ^ t _ 0 ( t - s ) ^ { \alpha } \left ( E [ ( \sigma ' ( X _ s ) - \Psi ^ n _ s ) ^ 4 ] \right ) ^ { \frac 1 2 } \left ( E [ ( X ^ n _ { s } - X _ s ) ^ 4 ] \right ) ^ { \frac 1 2 } \ , d s \\ & \leq C n ^ { - \beta ( 2 \alpha + 1 ) } . \end{align*}


\begin{gather*} \# \mathcal { Q } _ N ^ 0 ( D ) / \Gamma _ 0 ( N ) = \left ( n _ D ( N ) + n _ { \frac { D } { N ^ 2 } } ( 1 ) \right ) \# \mathcal { Q } _ 1 ^ 0 ( D ) / \Gamma _ 0 ( 1 ) \end{gather*}


\begin{align*} \mathbf { c } + \left ( \Z ^ { 1 + n } \cap C \right ) = \Z ^ { 1 + n } \cap C ^ \circ , \end{align*}


\begin{align*} p ( 0 ) = 0 , p ' ( \rho ) > 0 \ \forall \ \rho > 0 , \liminf _ { \rho \rightarrow 0 } \frac { p ' ( \rho ) } { \rho } > 0 , p ( \rho ) \sim _ { \rho \rightarrow \overline { \rho } - } | \overline { \rho } - \rho | ^ { - \beta } , \mbox { f o r s o m e } \beta > 5 / 2 , \end{align*}


\begin{align*} F _ m ( \varphi ) ( ( x _ j , y _ j ) _ { 1 \leq j \leq K } ) = \left ( \sum _ { j = 1 } ^ K a _ { i , j } x _ j , \sum _ { j = 1 } ^ K b _ { i , j } y _ j \right ) _ { 1 \leq i \leq L } \end{align*}


\begin{align*} g _ { j + 1 } ( \lambda ) = { \mu _ j } g _ j ( \lambda ) \neq 0 . \end{align*}


\begin{align*} \psi ( t + s ) = \| \tilde { B } T ( t + s , A ) \| _ q \leq \| \tilde { B } T ( t , A ) \| _ q \| T ( s , A ) \| _ \infty = \psi ( t ) \phi ( s ) . \end{align*}


\begin{align*} P _ 0 ( D _ j ) = \sum \limits _ { k = 0 } ^ m P _ 0 ( D _ j \cap D ' _ k ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \| P _ k ( \alpha _ { \pi ( t ) } \phi _ n ( T ) \alpha _ { \pi ( t ^ { - 1 } ) } - \phi _ n ( \alpha _ { \pi ( t ) \oplus \pi ( t ) } T \alpha _ { \pi ( t ^ { - 1 } ) \oplus \pi ( t ^ { - 1 } ) } ) ) P _ k \| = 0 . \end{align*}


\begin{align*} \int _ 0 ^ { \infty } n s P _ s ( o , o ) e ^ { - \frac { s } { t _ { 1 } } } \mathrm { d } s \le t _ 1 ^ { 2 } + \sum _ { i = 2 } ^ { n } \beta _ i ^ { - 2 } . \end{align*}


\begin{align*} c _ i ( \alpha , g ) = c _ i ( \alpha , h ) \mbox { f o r } i = 1 , 2 , \end{align*}


\begin{align*} A _ { I , \mathrm { s y m } } : = \Q [ t _ { \alpha , \beta , d } \ , | \ , 0 \le \alpha \le r - 2 , 0 \le \beta \le s - 2 , d \in \Z _ { \ge 0 } ] / \mathrm { I d e a l } ( I ) , \end{align*}


\begin{align*} \mathcal { S } ^ { ( q ) } _ { [ n , j ] } & : = \{ \gamma + q : \gamma \in \mathcal { S } _ { [ n , j ] } \} , \\ \widehat { \mathcal { S } } ^ { ( q ) } _ { [ n , j ] } & : = \{ \gamma - q : \gamma \in \widehat { \mathcal { S } } _ { [ n , j ] } \} , \end{align*}


\begin{align*} E ( G / H ) = E ( ( G / G ^ 0 ) / ( H / A ) ) \neq 0 \mod p \end{align*}


\begin{align*} k = h = \dfrac { 1 } { p - 1 } \end{align*}


\begin{align*} J _ 2 = & \int _ 0 ^ r - \frac { z ^ 3 } { 2 } ( \bar \psi \bar \psi ' \cdot \phi + \bar \psi \bar \phi ' \cdot \psi ) { \rm { d } } z \\ \leq & \Vert z ( \phi , \psi ) \Vert _ { L ^ 2 } \cdot \Vert z ( \bar \phi ' , \bar \psi ' ) \Vert _ { L ^ 2 } \cdot \frac { 1 } { 2 } \Vert z \bar \psi \Vert _ { L ^ \infty } , \end{align*}


\begin{align*} \delta = \begin{cases} 0 , & , \\ 1 , & . \end{cases} \end{align*}


\begin{align*} D = P _ { \Upsilon } d _ { V } A = \left ( \begin{array} { c c c c c c } P _ { \Upsilon } d & 0 & 0 & 0 & \cdots & 0 \\ P _ { \Upsilon } d T d & P _ { \Upsilon } d & 0 & 0 & \cdots & 0 \\ P _ { \Upsilon } ( d T ) ^ { 2 } d & P _ { \Upsilon } d T d & P _ { \Upsilon } d & 0 & \cdots & 0 \\ P _ { \Upsilon } ( d T ) ^ { 3 } d & P _ { \Upsilon } ( d T ) ^ { 2 } d & P _ { \Upsilon } d T d & P _ { \Upsilon } d & \cdots & 0 \\ & \cdots & \cdots & \cdots & & \end{array} \right ) . \end{align*}


\begin{align*} \widetilde { \varphi } = { w _ 1 } ^ + \tilde { f } _ + + { w _ 1 } ^ - \tilde { f } _ - . \end{align*}


\begin{align*} \frac { 1 } { m } \nabla \cdot ( m \nabla \varphi ) = 1 \R ^ d \end{align*}


\begin{align*} Z ( \overline \delta , \overline \mu , \beta ) = \frac { \beta } { \sqrt { \overline \mu - \frac { 1 } { 2 } ( 1 - \beta ) } } \frac { _ 3 \big ( \frac { 1 } { 2 } ( \sqrt { \beta } + \frac { 1 } { \sqrt { \beta } } ) + \overline \delta \overline \mu \ , | \ , \frac { 1 } { \sqrt { \beta } } , \sqrt { \beta } , \overline \delta \big ) } { _ 3 \big ( \sqrt { \beta } \ , | \ , \frac { 1 } { \sqrt { \beta } } , \sqrt { \beta } , \overline \delta \big ) } , \end{align*}


\begin{align*} \dot { x } ( t ) = & A x ( t ) + B u ( t ) , \\ y ( t ) = & \ ; C x ( t ) + D u ( t ) , \end{align*}


\begin{align*} & \nu _ p ( \{ ( a , r ) \in \omega \times \R _ + : \ c _ d ' r | \nabla f ( a ) | + C A r ^ 2 > \kappa \} ) \\ & = \frac { c _ { d , p } } { \kappa ^ p } \int _ { \omega } \left ( \sqrt { \tfrac { C A } { c _ d '^ 2 } \kappa + \left ( \tfrac 1 2 | \nabla f ( a ) | \right ) ^ 2 } + \tfrac 1 2 | \nabla f ( a ) | \right ) ^ p d a \ , . \end{align*}


\begin{align*} w & : = v _ 1 \Uparrow u = v _ 2 \Uparrow u , \\ w ' & : = v ' _ 1 \Uparrow u ' = v ' _ 2 \Uparrow u ' , \\ y _ 1 & : = v _ 1 \Uparrow x _ 1 = v ' _ 1 \Uparrow x _ 1 , \\ y _ 2 & : = v _ 2 \Uparrow x _ 2 = v ' _ 2 \Uparrow x _ 2 . \end{align*}


\begin{align*} w \cdot \lambda : = w \left ( \lambda + \rho \right ) - \rho . \end{align*}


\begin{align*} \langle f , I ^ { \circ } ( w _ 1 , z _ 2 ) u _ { k + \frac { j _ 2 } { T } } w _ 2 \rangle = 0 . \end{align*}


\begin{align*} \int _ { 0 } ^ { x } f ( t ) h ( t / q ) \left ( - \frac { q ( 1 + q ) + t } { q ^ 2 ( 1 - q ) t } u ( t / q ) + \frac { 1 } { q ( 1 - q ) ^ 2 t } \right ) y ( t ) d _ q t = H ( x ) - \lim _ { n \rightarrow \infty } H ( q ^ n x ) . \end{align*}


\begin{align*} F _ m ( \varphi ) : \bigoplus _ { j = 1 } ^ K V _ { n _ j } ^ m \to \bigoplus _ { i = 1 } ^ L V _ { \ell _ i } ^ m \end{align*}


\begin{align*} \left ( \frac { 2 \beta \hslash ^ { 2 } } { 3 a ^ { 2 } } \right ) ^ { 2 } . \left ( \frac { \lambda ^ { 2 } - 1 } { \lambda } \right ) ^ { 2 } = 1 . \end{align*}


\begin{align*} 0 \leq \mathcal { X } \in C _ \textup { b } ^ \infty ( \R ^ N ) \qquad \mathcal { X } ( x ) = \begin{cases} 0 & | x | \leq \frac { 1 } { 2 } , \\ 1 & | x | \geq 1 . \end{cases} \end{align*}


\begin{align*} 1 = \| C _ { \sigma _ 2 } \psi \| _ { L ^ 2 ( \widehat { \R ^ 2 } \times \widehat { \R } ) } = \| r T \psi \| _ { L ^ 2 ( \widehat { \R ^ 2 } \times \widehat { \R } ) } = r \| T \psi \| _ { L ^ 2 ( \widehat { \R ^ 2 } \times \widehat { \R } ) } = r . \end{align*}


\begin{align*} u _ t = \Delta u - | u | ^ { q - 1 } u . \end{align*}


\begin{align*} \begin{aligned} \mathcal { N } ^ s _ { p ( x , . ) } u ( x ) = \int _ { \Omega } \dfrac { | u ( x ) - u ( y ) | ^ { p ( x , y ) - 2 } ( u ( x ) - u ( y ) ) } { | x - y | ^ { N + s p ( x , y ) } } d y , ~ ~ \forall x \in \R ^ N \setminus \Omega , \end{aligned} \end{align*}


\begin{align*} L = L ^ 0 + \hat L , \end{align*}


\begin{align*} \langle H _ N \rangle _ { \psi _ \mathrm { g s } } \ge \ ; & E _ { 2 - \mathrm { m o d e } } + \frac { c \lambda } { N } \big \langle ( \mathcal { N } _ 1 - \mathcal { N } _ 2 ) ^ 2 \big \rangle _ { \psi _ \mathrm { g s } } + E ^ \mathrm { B o g } \\ & - C _ \Lambda o _ N ( 1 ) - \frac { C } { ( \mu _ { 2 M _ \Lambda + 2 } - \mu _ + ) } ^ { 1 / 2 } , \end{align*}


\begin{align*} \frac { f ^ { 2 } } { 2 } \operatorname { g r a d } ( \| X \| ^ { 2 } ) = \operatorname { g r a d } p \end{align*}


\begin{align*} f ( t ) = \lambda u ( t ) + \Gamma ( \beta + 1 ) ( 1 + H ( t - r ) ) . \end{align*}


\begin{align*} G _ { ( t , \rho ) } \left ( \frac { \partial } { \partial z } , \frac { \partial } { \partial z } \right ) = t ^ 2 G _ { ( 1 , \rho ) } \left ( \frac { \partial } { \partial z } , \frac { \partial } { \partial z } \right ) . \end{align*}


\begin{align*} k \tan ( k a / 2 ) + k ' \tanh ( k ' a / 2 ) = 0 , \end{align*}


\begin{align*} H _ N = \sum _ { m , n \ge 1 } h _ { m n } a ^ * _ m a _ n + \frac { \lambda } { 2 ( N - 1 ) } \sum _ { m , n , p , q \ge 1 } w _ { m n p q } a ^ * _ m a ^ * _ n a _ p a _ q , \end{align*}


\begin{align*} [ x _ 0 , \dots , x _ n ] ( f g ) = \sum _ { k = 0 } ^ n [ x _ 0 , \dots , x _ k ] f [ x _ k , \dots , x _ n ] g . \end{align*}


\begin{align*} Z _ g & = \# ^ g S ^ n \times S ^ { n + 1 } , \\ Z _ { g , 1 } & = Z _ g \setminus \operatorname { i n t } D ^ { 2 n + 1 } , \end{align*}


\begin{align*} \int _ 0 ^ \infty \inf _ { d _ X ( x , x ^ * ) \leq \rho } C ( J f ( x ) ) d \rho = \infty . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \abs { \frac { H _ n ( 1 - s ) } { H _ n ( s ) } } = 1 . \end{align*}


\begin{align*} H ^ { S ; U ; m , i } _ { j , n , l , r } ( t , x , \zeta ) = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } ( t - s ) \big ( E ( s , x + ( t - s ) \omega ) + \hat { v } \times B ( s , x + ( t - s ) \omega ) \big ) \cdot \nabla _ v \big ( \frac { m _ { U } ( v , \omega ) \varphi _ { j , n } ^ { i ; r } ( v , \zeta ) \varphi _ { l ; r } ( \tilde { v } + \omega ) } { 1 + \hat { v } \cdot \omega } \big ) \end{align*}


\begin{align*} \Gamma _ p : = \{ \tau - \tau _ 1 + \cdots - \tau _ p = 0 , \ , \ , k - k _ 1 + \cdots - k _ p = 0 \} , \end{align*}


\begin{align*} \Gamma \left ( \Delta _ { \mathfrak { a } } \right ) \left ( \rho \right ) = \int _ { \mathbb { S } } \left \vert \rho \left ( \mathfrak { e } _ { \Psi } \right ) \right \vert ^ { 2 } \mathfrak { a } _ { + } \left ( \mathrm { d } \Psi \right ) , \rho \in E _ { 1 } \ . \end{align*}


\begin{align*} \mathbf { f } = \psi \circ f \circ \varphi ^ { - 1 } : \varphi ( W ) \rightarrow \psi ( V ) \end{align*}


\begin{align*} h ^ { - 1 } ( \mathcal { D } _ x ) \cap T _ { ( E _ * , \psi ) } ^ * \mathcal { M } _ { \textnormal { S p } } ( 2 m , \alpha , L ) = \bigcup _ { i = 1 } ^ { 2 m } \tilde { N } _ i ( E _ * ) \end{align*}


\begin{align*} \dim ( C ) + \dim ( C ^ \perp ) = n \omega . \end{align*}


\begin{align*} \mathcal { X } \cap [ \emptyset , N ] = \bigcup _ { j < \tilde { j } } ( \mathcal { X } \cap [ B _ j , N ] ^ * ) . \end{align*}


\begin{align*} \Lambda = ( \lambda _ 1 , \lambda _ 2 , \ldots , \lambda _ m ) \end{align*}


\begin{align*} \sum \limits _ { i = 1 } ^ { \infty } \frac { 1 } { \alpha _ i } \sum \limits _ { k : k \geq 0 , k \leq \alpha _ i } \mu ( k ) k \leq c _ { \alpha } , \end{align*}


\begin{align*} E : = c _ 1 ( T _ X ( - \log D ) ) + \sum _ { \vec s , k } \left ( 1 - \frac 1 2 \deg ( \tilde { T } _ { \vec s , k } ) - \# \{ i : s _ i < 0 \} \right ) t _ { \vec s , k } \tilde { T } _ { \vec s , k } \end{align*}


\begin{align*} S ( \delta _ { } ) = o ( 1 ) . \end{align*}


\begin{align*} \langle \chi , \Lambda ( \rho _ { i _ 1 } ) \rangle = 1 , \ , \langle \chi , \Lambda ( \rho _ { i _ 2 } ) \rangle = 0 , \ , \ldots , \ , \langle \chi , \Lambda ( \rho _ { i _ s } ) \rangle = 0 . \end{align*}


\begin{align*} & \lim _ { n \rightarrow \infty } n ^ { 2 \alpha + 1 } \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' ( X _ s ) ) ^ 2 \Psi ^ { n , 2 } _ s d s \\ & = \frac 1 { 2 \alpha + 2 } \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' \sigma ) ^ 2 ( X _ s ) d s , \end{align*}


\begin{align*} u _ { \l , o ' } ( \cdot , t ) = \widehat { u } ( \cdot , t ) . \end{align*}


\begin{align*} \frac { \displaystyle \sum _ { m = 0 } ^ \infty [ p ^ \# _ m ( o , o ) - \pi ( o ) ] } { \displaystyle \sum _ { m = 0 } ^ \infty [ \tilde p ^ \# _ m ( A , A ) - \pi ( A ) ] } \ge 1 - C \frac { \log ( D / \delta ) } { f ( n ) } \end{align*}


\begin{align*} \mathcal { J } _ { 2 , 1 } = - 2 \iint _ { Q _ T } \nabla \cdot ( p ( m \nabla p ) ) \Delta p = - 2 \iint _ { Q _ T } m | \nabla p | ^ 2 \Delta p + p \Delta p \left [ \nabla \cdot ( m \nabla p ) \right ] . \end{align*}


\begin{align*} \textup { c o l u m n $ i $ o f $ A $ } = S ^ { ( i - 1 ) k } b 1 \leq i \leq n , \end{align*}


\begin{align*} \mathcal L = \mathcal L [ u ] : = w ^ { i j } [ D _ { i j } - D _ { p _ k } A _ { i j } ( \cdot , u , D u ) D _ k ] , \end{align*}


\begin{align*} \begin{aligned} & 0 \le u \le \min \{ p _ 1 , q _ 2 \} \ , \ \ 0 \le v \le \min \{ p _ 2 , q _ 1 \} \ , \\ & \begin{cases} 0 \le s \le \min \{ p _ 1 - u , p _ 2 - v \} \ , \ \ \mbox { f o r ( \ref { p s i - 1 c a s e } ) } \ , \\ 0 \le s \le \min \{ q _ 1 - v , q _ 2 - u \} \ , \ \ \mbox { f o r ( \ref { p s i - 2 c a s e } ) } \ . \end{cases} \end{aligned} \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , T ] } \| L ^ { n , 2 } _ \tau + L ^ { n , 3 } _ \tau \| _ 2 = 0 . \end{align*}


\begin{align*} \begin{aligned} q _ { n } ( z ) & : = \det ( z I _ { n } - H _ { n } ) , \\ q _ { n , k } ( z ) & : = \det ( z I _ { n - k } - H _ { n } ^ { [ k ] } ) , 1 \leq k \leq p . \end{aligned} \end{align*}


\begin{align*} { \bf { a } } = ( a _ 1 = a , a _ 2 = a , \ldots , a _ k = a ) , \end{align*}


\begin{align*} \sigma _ 2 * \sigma _ 1 * \sigma _ 2 = \sigma _ 1 . \end{align*}


\begin{align*} \Lambda ^ { \nu } = ( \Lambda _ 1 ^ { \nu } , \ldots , \Lambda _ n ^ { \nu } ) , \ \ \nu \geq 1 \end{align*}


\begin{align*} f = \sum \nolimits _ { 0 \le s } a _ s g ^ s = \sum \nolimits _ { 0 \le s } b _ s \phi ^ s , b _ s = a _ s G ^ s . \end{align*}


\begin{align*} & \lim _ { k \to \infty } \left ( \left [ \prod _ { j = k } ^ { k + n - 1 } w ^ j \right ] \left ( x _ { k + n } - l ( x ) \right ) + \left [ \prod _ { j = 0 } ^ { n - 1 } w ^ j \right ] \left ( x _ { n } - l ( x ) \right ) \right ) \\ & = \left [ \prod _ { j = 0 } ^ { n - 1 } w ^ j \right ] \left ( x _ { n } - l ( x ) \right ) . \end{align*}


\begin{align*} \begin{array} { c c } \Lambda ( g ) ^ { - 1 } \tilde { f } ^ { | u \rangle } ( \Lambda ( g ) p ) = \tilde { f } ^ { | u ' \rangle } ( p ) = \frac { u ' } { u ' \cdot p } , \ , \ , \ , u ' = \Lambda ( g ) ^ { - 1 } u , \\ g \in S L ( 2 , \mathbb { C } ) , \end{array} \end{align*}


\begin{align*} m ( z ) = \sum _ { n = 0 } ^ { \infty } \frac { A _ { n } } { z ^ { n + 1 } } \end{align*}


\begin{align*} 2 g ( \nabla _ { X } X , W ) = - g ( \operatorname { g r a d } ( \| X \| ^ { 2 } ) , W ) \end{align*}


\begin{align*} \cos ^ 2 ( \alpha _ k ^ * ) | e _ k | + \sin ^ 2 ( \alpha _ k ^ * ) | e _ k | = | e _ k | . \end{align*}


\begin{align*} \Theta ^ n _ s : = \sigma ( X _ s ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) + X ^ n _ { \eta _ n ( s ) } - X ^ n _ s . \end{align*}


\begin{gather*} \left ( { \varrho _ m ( \gamma , \upsilon ) } \hat h | _ { k - \frac 1 2 } ( \gamma , \upsilon ) \right ) ( \tau ) = O ( 1 ) \end{gather*}


\begin{align*} l _ { \rho } \left ( \Gamma , \sum _ { n = 1 } ^ { \infty } a _ n \mu ^ n \right ) \leq \left ( 1 + \sum _ { n = 1 } ^ { \infty } n a _ n \right ) l _ { \rho } ( \Gamma , \mu ) . \end{align*}


\begin{align*} \mathbb K _ \pm : = \left \{ f \in \mathbb L \otimes \mathbb Q | \left \{ i \in \{ 1 , \ldots , m \} | D _ i \cdot f \in \mathbb Z \right \} \in \mathcal A _ \pm \right \} . \end{align*}


\begin{align*} S = A \times T _ 1 \times \dots \times T _ m \end{align*}


\begin{align*} \| u \| _ { k , \alpha } = \sup _ { x \in M } \left \| \left . \Psi ^ * u \right | _ { B _ \delta ( x ) } \right \| _ { k , \alpha , B _ \delta ( 0 ) \times [ 0 , T ] } . \end{align*}


\begin{align*} F ( k - 1 , 2 , 2 ) = 4 \left \lceil \frac { k } { 4 } \right \rceil . \end{align*}


\begin{align*} \begin{aligned} \mathcal { N } ^ s _ 2 u ( x ) = \int _ { \Omega } \dfrac { u ( x ) - u ( y ) } { | x - y | ^ { N + 2 s } } d y , ~ ~ \forall x \in \R ^ N \setminus \Omega . \end{aligned} \end{align*}


\begin{align*} T _ m ( d ) = \sum _ { Q \in \Gamma \backslash \mathcal Q _ { d ^ 2 } } \chi _ d ( Q ) \begin{dcases} \int _ { C _ Q } F _ { m , Q } ( z , s ) y ^ { - 1 } | d z | & d > 0 , \\ \int _ { C _ Q } i \partial _ z F _ { m , Q } ( z , s ) \ , d z & d < 0 . \end{dcases} \end{align*}


\begin{align*} E _ { x \leftrightarrow y } = E _ { x \rightarrow y } \land E _ { y \rightarrow x } , \end{align*}


\begin{align*} v ( \phi , w ) \leq v ( \phi ' , w ) & v ( \Delta ^ \neg ( \phi \rightarrow \phi ' ) , w ) = ( 1 , 0 ) , \\ v ( \phi , w ) > v ( \phi ' , w ) & v ( \Delta ^ \neg ( \phi ' \rightarrow \phi ) \wedge { \sim } \Delta ^ \neg ( \phi \rightarrow \phi ' ) , w ) = ( 1 , 0 ) . \end{align*}


\begin{align*} N ( \alpha R ) = | N ( \alpha ) | , \forall \alpha \in R . \end{align*}


\begin{align*} W ^ { n , p } ( \O ) : = \{ f \in L ^ p ( \O ) \mid D ^ \alpha f \in L ^ p ( \O ) , \ , 0 \leq | \alpha | \leq n \} , \end{align*}


\begin{align*} \# Z \left ( H _ 2 | _ { [ 0 , 1 ] \times \Phi _ p } \right ) = \# Z \left ( \hat H _ { 2 , p } \right ) . \end{align*}


\begin{align*} \lambda _ i = r \kappa _ i + s \end{align*}


\begin{align*} \mathbb { E } ( X _ i X _ j ) = \sum \limits _ { \substack { 0 \leq l \leq i \\ \gcd ( l , i ) \ { \rm i s } \ k - { \rm f r e e } } } C _ { \alpha } ( i , l ) T _ k ( l , i , j , \alpha ) , \end{align*}


\begin{align*} D _ f ( x ) = \left ( f ( x ) , f ' ( x ) , \ldots , f ^ { ( j ) } ( x ) , \ldots , f ^ { ( n ) } ( x ) \right ) \end{align*}


\begin{align*} w ( y , d ) : = \int _ D Q _ { - A , q } e ^ { i k z \cdot d } u _ { A , q } ( z , y ) d z . \end{align*}


\begin{align*} & \left \langle \gamma _ j ( w ) , \gamma _ i ( z ) \right \rangle \\ & \quad = \big ( K _ 2 \partial _ i \bar { \partial } _ j K _ 3 \big ) ( z , w ) + \big ( K _ 3 \partial _ i \bar { \partial } _ j K _ 2 \big ) ( z , w ) - \big ( \partial _ i K _ 2 \bar { \partial } _ j K _ 3 \big ) ( z , w ) - \big ( \partial _ i K _ 3 \bar { \partial } _ j K _ 2 \big ) ( z , w ) . \end{align*}


\begin{align*} \overline { G } _ { ( 1 ) } ( u ) = \frac { \exp ( - u ) } { 1 + \exp ( - u ) } \end{align*}


\begin{align*} \sum _ { j = 0 } ^ \infty \varrho ^ { 2 j } \left \langle \varphi _ j , K ( \partial _ t ^ { \Delta t } ) \varphi ( t _ j ) \right \rangle \geq C _ 2 ( \sigma ) \sum _ { j = 0 } ^ \infty \varrho ^ { 2 j } \| ( \partial _ t ^ { \Delta t } ) ^ \eta \varphi _ j \| ^ 2 _ X , \end{align*}


\begin{align*} ( \Psi \otimes \cdots \otimes \Psi ) \rho ( u ) & = \varepsilon ( u ) ( \Psi \otimes \cdots \otimes \Psi ) , \\ \rho ( u ) ( \Psi ^ * \otimes \cdots \otimes \Psi ^ * ) & = \varepsilon ( u ) ( \Psi ^ * \otimes \cdots \otimes \Psi ^ * ) \end{align*}


\begin{align*} P Q ^ k P ^ { - 1 } = P \begin{pmatrix} M ^ k & \vec { 0 } \\ e _ k ^ { \top } & 1 \end{pmatrix} P ^ { - 1 } = \begin{pmatrix} M ^ k & \vec { 0 } \\ f _ k ^ { \top } & 1 \end{pmatrix} \end{align*}


\begin{align*} y ( z x ) = - ( - 1 ) ^ { \vert x \vert ( \vert y \vert + \vert z \vert ) } ( x y ) z . \end{align*}


\begin{align*} \min _ { v \in V } J ( v ) \ \ J ( v ) : = \frac { 1 } { q } \int _ { \R ^ { d } } | v | ^ { q } d \rho - L ( v ) . \end{align*}


\begin{align*} \theta ( s ) = \begin{cases} 1 , & \ 0 \leq s < t , \\ \displaystyle { \frac { t - s } { \delta } + 1 } , & \ t \leq s < t + \delta , \\ 0 , & \ s > t + \delta , \end{cases} \end{align*}


\begin{align*} \sum \limits _ { 1 \leq n \leq N } f _ k ( n ) f _ k ( n + 1 ) = \sum \limits _ { 1 \leq n \leq N } \sum _ { w _ 1 \mid n } \frac { g _ k ( w _ 1 ) } { w _ 1 } \sum _ { w _ 2 \mid ( n + 1 ) } \frac { g _ k ( w _ 2 ) } { w _ 2 } . \end{align*}


\begin{align*} \Delta \left ( \sum _ { i = 1 } ^ 3 \frac { \partial U _ { 0 } ( y ) } { \partial y _ i } \right ) - \left ( \sum _ { i = 1 } ^ 3 \frac { \partial U _ { 0 } } { \partial y _ i } \right ) + p U _ 0 ( y ) ^ { p - 1 } \left ( \sum _ { i = 1 } ^ 3 \frac { \partial U _ 0 ( y ) } { \partial y _ i } \right ) = 0 \end{align*}


\begin{align*} N ( t ) = \int _ 0 ^ { + \infty } J _ v ( V _ F , g ) \ , p ( t , V _ F , g ) \ , d g = \int _ 0 ^ { + \infty } \bigl ( - g _ L V _ F + g ( V _ E - V _ F ) \bigr ) \ , p ( t , V _ F , g ) \ , d g . \end{align*}


\begin{align*} \label [ I e q ] { e q : T i n f t y t p h i k } T _ { \infty , t } \Phi _ k ( \nu ) = \big \langle \overline { S } _ t ( \nu ) ^ { \otimes k } , \varphi _ k \big \rangle , \end{align*}


\begin{align*} \left \{ ~ \begin{aligned} & \theta _ t + \mathbf { u } \cdot \nabla \theta = 0 , \\ & \mathbf { u } = \nabla ^ \perp \Psi , \\ & \mathbf { A } \Psi = \theta , \end{aligned} \right . \end{align*}


\begin{align*} \mathcal H _ 1 \to \mathcal H _ { p , 0 } : w = ( z _ 1 , z _ 2 , z _ 3 ) \mapsto w \otimes . . . \otimes w = \ ! \ ! \sum _ { j + k + l = 1 } \sqrt { \dfrac { p ! } { j ! \ , k ! \ , l ! } } z _ 1 ^ j z _ 2 ^ k z _ 3 ^ l \ , e _ { j , k , l } \ . \end{align*}


\begin{align*} ( x , t ) \in D : = D _ { \rm I n t } \cap \{ x > 0 \} \cap \{ t - x > R \} . \end{align*}


\begin{align*} \lim _ { n \to \infty } \phi _ n ^ * ( x ) = \phi ^ * ( x ) \end{align*}


\begin{align*} G ( z \ , | \ , \omega _ 1 , \widetilde \omega _ 1 , \omega _ 2 ) = \exp \Bigg ( \int _ C \frac { - e ^ { ( z + \overline \omega _ 1 ) s } } { ( e ^ { \omega _ 1 s } - 1 ) ( e ^ { \widetilde \omega _ 1 s } - 1 ) ( e ^ { \omega _ 2 s } - 1 ) } \frac { d s } { s } \Bigg ) , \end{align*}


\begin{align*} & \quad \| \{ \Sigma _ n ( \mathbb U ( t ) ) - \Sigma _ n ( U ( t ) ) \} e _ l \| ^ 2 = \sum _ { m = 1 } ^ { n - 1 } \left | \left ( \sigma ( \mathbb U _ m ( t ) ) - \sigma ( U _ m ( t ) ) \right ) e _ l ( m ) \right | ^ 2 \\ & \le \frac { C } { n } \sum _ { m = 1 } ^ { n - 1 } \left | \mathbb U _ m ( t ) - U _ m ( t ) \right | ^ 2 = \frac { C } { n } \left \| \mathbb U ( t ) - U ( t ) \right \| ^ 2 , \quad \forall ~ l \in \Z _ n . \end{align*}


\begin{align*} \| u \| _ { W ^ { s , p } ( \Omega ) } \vcentcolon = \left ( \| u \| _ { W ^ { m , p } ( \Omega ) } ^ p + \sum _ { | \alpha | = m } [ \partial ^ { \alpha } u ] _ { W ^ { \sigma , p } ( \Omega ) } ^ p \right ) ^ { 1 / p } . \end{align*}


\begin{align*} U _ t = { \mathcal { A } } _ m U , U ( 0 ) = U _ 0 , \end{align*}


\begin{align*} \frac { ( x + y + z ) ^ 3 } { x y z } = n \end{align*}


\begin{align*} \left | 1 - \frac { z - \mu _ k ^ 2 } { z - \lambda _ k ^ 2 } \right | = \left | \frac { \lambda _ k ^ 2 - \mu _ k ^ 2 } { z - \lambda _ k ^ 2 } \right | \le \frac 2 R ( \lambda _ k ^ 2 - \mu _ k ^ 2 ) \forall z : \ | z | > R . \end{align*}


\begin{align*} \left [ \sum _ { j = 1 } ^ n Q ^ j _ k \xi ^ k \frac { \partial } { \partial x _ j } , X \right ] = \xi ^ k \left [ \sum _ { j = 1 } ^ n Q ^ j _ k \frac { \partial } { \partial x _ j } , X \right ] . \end{align*}


\begin{gather*} ( \gamma , \upsilon ) ( \gamma ' , \upsilon ' ) = ( \gamma \gamma ' , ( \upsilon \circ \gamma ' ) \upsilon ' ) . \end{gather*}


\begin{align*} \kappa _ 5 = \sum _ { k = 1 } ^ { \infty } \int _ 0 ^ 1 \int _ { 0 } ^ { 1 } [ ( y + k ) ^ \alpha - ( y + k - z ) ^ \alpha ] [ ( y + k ) ^ \alpha - k ^ \alpha ] d z d y . \end{align*}


\begin{align*} \sum _ { m = 0 } ^ { \delta ^ 2 } p ^ \# _ m ( o , y ) & \le \int _ 0 ^ { 1 0 \delta ^ 2 } p _ t ( o , y ) \mathrm { d } t + \delta ^ 2 \P ( N \le 2 \delta ^ 2 ) \\ & \le O ( 1 / f ( n ) ) + \delta ^ 2 \exp ( - c \delta ^ 2 ) , \end{align*}


\begin{align*} \Psi ^ * ( \lambda _ * H ^ { ( 1 ) } ( \mu _ N ) - \lambda _ * \Psi ( - 1 ) ) \cong \Psi ^ * \Psi _ * \binom { N } { 2 } \cong * . \end{align*}


\begin{align*} ( \mathcal { O } _ C ( n ) ) = \beta - n \delta , \ ( \mathcal { O } _ x ) = - \delta \ . \end{align*}


\begin{align*} \varphi _ n = \varphi _ { n ; \mathrm { o u t } } + \varphi _ { n ; \mathrm { i n } } + \varphi _ { n ; \mathrm { s u r } } \end{align*}


\begin{align*} g ( \operatorname { g r a d } ( f ^ { 2 } ) , V ) g ( \operatorname { g r a d } ( \| X \| ^ { 2 } ) , W ) = g ( \operatorname { g r a d } ( f ^ { 2 } ) , W ) g ( \operatorname { g r a d } ( \| X \| ^ { 2 } ) , V ) \end{align*}


\begin{align*} \sum W ( U , f ) \int _ U Q \omega = \int _ { \Delta ^ \perp } f ^ * \alpha \end{align*}


\begin{align*} \widehat { H } _ { { \textrm { C K } } } ( t ) : = \exp { ( - 2 \gamma t ) } \frac { \widehat { p } ^ 2 } { 2 } + \omega ^ 2 \exp { ( 2 \gamma t ) } \frac { \widehat { x } ^ 2 } { 2 } , \gamma \in \mathbb { R } , \ , \ , \omega \in \mathbb { R } , \end{align*}


\begin{align*} \sup _ { \delta < | t | \leq \lambda } \left \lvert \frac { \phi ( s + \mathrm { i } t ) } { \phi ( s ) } \right \rvert & = \sup _ { \delta < | t | \leq \lambda } \left \lvert \prod _ { j = 1 } ^ { w _ n } \frac { \tilde { \phi } _ j ( s + \mathrm { i } t ) } { \tilde { \phi } _ j ( s ) } \right \rvert \cdot 1 \\ & \leq c _ 2 ^ { w _ n } \\ & = o ( n ^ { - 1 / 2 } ) , \end{align*}


\begin{align*} y _ i \pi _ { [ a , b ] } = y _ { b + 1 } \pi _ { [ a , b ] } ( a \le i \le b ) \end{align*}


\begin{align*} \Phi _ { \mu } ( \eta _ { \mu } ( z ) ) = z \end{align*}


\begin{align*} \Lambda ^ { ( 3 ) } _ { n , \delta } & = \exp \left ( i \mu n ^ { \alpha + \frac 1 2 } \int _ 0 ^ { t - \delta } ( t - s ) ^ \alpha \xi ^ n _ s d W _ s + i \lambda n ^ { \alpha + \frac 1 2 } \int _ { t - \delta } ^ { \eta _ n ( t ) } \psi _ { n , 1 } ( s , \eta _ n ( t ) ) d W _ s \right ) \\ & \times \left [ \exp \left ( i \mu n ^ { \alpha + \frac 1 2 } \int _ { t - \delta } ^ t ( t - s ) ^ \alpha \xi ^ n _ s d W _ s \right ) - 1 \right ] \end{align*}


\begin{align*} { } \Delta _ n \underset { \sim } { < } ( \frac { \log T _ n } { T _ n } ) ^ { \frac { \bar { \beta } _ 3 } { 2 \bar { \beta } _ 3 + d - 2 } ( \frac { 1 } { \beta _ 1 } + \frac { 1 } { \beta _ 2 } ) } \log T _ n \mbox { f o r } \beta _ 2 < \beta _ 3 , \\ \Delta _ n \underset { \sim } { < } ( \frac { 1 } { T _ n } ) ^ { \frac { \bar { \beta } _ 3 } { 2 \bar { \beta } _ 3 + d - 2 } ( \frac { 1 } { \beta _ 1 } + \frac { 1 } { \beta _ 2 } ) } \mbox { f o r } \beta _ 2 = \beta _ 3 . { } \end{align*}


\begin{align*} \lambda _ k = \left ( { b + c } \right ) \left ( { d + 1 } - 2 k \right ) \end{align*}


\begin{align*} \left | 1 - \frac { z - \mu _ k ^ 2 } { z - \lambda _ k ^ 2 } \right | = \left | \frac { \lambda _ k ^ 2 - \mu _ k ^ 2 } { z - \lambda _ k ^ 2 } \right | \le C ( K ) ( \lambda _ k ^ 2 - \mu _ k ^ 2 ) \forall z \in K . \end{align*}


\begin{align*} \sup \left \{ \int _ 0 ^ \infty f g \ , d r : \ g \geq 0 \ , , \ \underline \nu _ \beta ( g ) \leq 1 \right \} = \beta \ , \overline \mu _ \beta ( f ) \ , , \end{align*}


\begin{align*} \mathcal { U } = \{ ( x , g ( x , y , z ) , g _ x ( x , y , z ) ) | \ ( x , y , z ) \in \Gamma \} , \end{align*}


\begin{align*} D ^ s u : = \dfrac { u ( x ) - u ( y ) } { | x - y | ^ s } . \end{align*}


\begin{align*} D ^ * _ X U = D ^ * _ U X = \frac { 1 } { t } U - \frac { 1 } { t } U = 0 . \end{align*}


\begin{align*} H = V _ { \phi } ^ { - 1 } \operatorname { d i a g } \left \{ \theta _ { 0 } , \theta _ { 1 } , \cdots , \theta _ { n - 1 } \right \} V _ { \phi } \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 } \left \| \dfrac { G ( z + t h ) - G ( z ) } { t } - G ' ( z , h ) \right \| = 0 , \end{align*}


\begin{align*} \dfrac { \log _ 2 \left ( N _ { X _ { H , V } } ( p n , n ) \right ) } { p n ^ 2 } \geq \dfrac { \log _ 2 \left ( \max \limits _ { v \in \{ 0 , \dots , p - 1 \} ^ n } \Pi _ { j = 0 } ^ { p - 1 } N _ { v ^ j } \right ) } { p n } \end{align*}


\begin{align*} E ( X _ 3 ^ 2 ) = \varepsilon ^ 2 \big [ P ( \boldsymbol { X } = \mathbf { s } ( 3 ) ) + P ( \boldsymbol { X } = \mathbf { s } ( 6 ) ) \big ] . \end{align*}


\begin{align*} \frac { s \ , e ^ { z s } } { e ^ s - 1 } = \sum _ { n = 0 } ^ { \infty } B _ n ( z ) \frac { s ^ n } { n ! } . \end{align*}


\begin{align*} m _ 1 & = \left \lceil \dfrac { n } { 2 } \right \rceil \\ m _ 2 & = \left \lceil \dfrac { n - m _ 1 } { 3 } \right \rceil \\ m _ 3 & = \left \lceil \dfrac { n - ( m _ 1 + m _ 2 ) } { 4 } \right \rceil \\ & \ ; \ ; \vdots \\ m _ { \ell } & = \left \lceil \dfrac { n - ( m _ 1 + m _ 2 + \cdots + m _ { \ell - 1 } ) } { \ell + 1 } \right \rceil . \end{align*}


\begin{align*} \widehat { \varPhi } _ { x } ( t ) : = \widehat { \varPhi } ( x , t ) = \int _ { 0 } ^ { t } \widehat { \varphi } _ { x } ( \tau ) d \tau ~ ~ ~ ~ t \geqslant 0 . \end{align*}


\begin{align*} I _ { j , l } ^ { 1 ; 1 } ( t , x ) = \int _ { \R ^ 3 } \int _ { | y - x | \leq t } \frac { \varphi _ { m ; - 1 0 M _ t } ( | y - x | ) } { | y - x | } \frac { \hat { v } _ 2 \omega _ 3 - \hat { v } _ 3 \omega _ 2 } { 1 + \hat { v } \cdot \omega } S f ( t - | y - x | , y , v ) \varphi _ { j , l } ( v , \omega ) d y d v \end{align*}


\begin{align*} \Gamma ^ { k ( s ) } _ { i j } - \Gamma ^ { k ( s ) } _ { j i } = L ^ { k ( s ) } _ { i j } ( \Lambda ^ { ( s ) } ) . \end{align*}


\begin{align*} \nu _ Q ( q Q ) - ( \epsilon ( Q ) - \gamma ) \geq \nu ( f ) = \nu ( r ) . \end{align*}


\begin{align*} \rho ( I _ { d , s } ) = \rho _ { i c } ( I _ { d , s } ) = \rho _ { i c } ( ( I _ { d , s } ) ^ \vee ) = \frac { d ( s - d + 1 ) } { s } = \frac { d } { \widehat { \alpha } ( I _ { d , s } ) } = \frac { s - d + 1 } { \widehat { \alpha } ( ( I _ { d , s } ) ^ \vee ) } . \end{align*}


\begin{align*} \min _ { 0 { } \leq { } i { } \leq { } N } \| \nabla f ( x _ i ) \| ^ 2 { } \leq { } \frac { 2 L \ , \big ( f ( x _ 0 ) - f _ * \big ) } { 1 + \sum \limits _ { i = 0 } ^ { N - 1 } \left [ 2 h _ i - h _ i ^ 2 \frac { - \kappa } { 2 \min \left ( 1 , \frac { 1 } { h _ i } \right ) - ( 1 + \kappa ) } \right ] } \end{align*}


\begin{align*} \| N ^ { n , 5 } _ \tau \| _ 2 \le C \frac 1 n \sum _ { j = 0 } ^ { \lfloor n T \rfloor } \sum _ { k = j + 1 } ^ { \infty } k ^ { 2 \alpha - 2 } \to 0 , \end{align*}


\begin{align*} \langle \nabla _ i \nabla _ j \Psi _ t - \nabla _ l \Psi _ t \Gamma _ { i j } ^ l , \nabla _ i \nabla _ j \Psi _ t - \nabla _ l \Psi _ t \Gamma _ { i j } ^ l \rangle = \frac { 1 } { 2 t } \Big ( 1 + O ( t ) \Big ) + \sum _ l \Gamma _ { i j } ^ l \Gamma _ { i j } ^ l . \end{align*}


\begin{align*} & ( b _ i ^ 2 - b _ i ) = 0 k < i \\ & b _ i b _ j = 0 k < i < j . \end{align*}


\begin{align*} \left \vert \frac { g ' ( z ) } { f ' ( z ) } \right \vert \leq \beta + \frac { 1 - \beta ^ 2 } { 1 - r ^ 2 } \frac { r } { m ( r ) } = : h ( \beta , r ) . \end{align*}


\begin{align*} w ( y , d ) = - \int _ D \Phi ( z , y ) Q _ { - A , q } u _ { - A , q } ( z , d ) d z , \end{align*}


\begin{align*} \tau ^ \mathrm { a n / t o p } ( M / S , \rho ) = \tau ^ \mathrm { a n / t o p } ( M / S , F _ \rho ) \ ; . \end{align*}


\begin{align*} f _ h ( v _ I ) = x _ I , \end{align*}


\begin{align*} T _ { \Xi } ( \theta ) : = \left \{ h \in D : h = \lim _ { n \to \infty } \frac { \theta _ n - \theta } { t _ n } \ \right \} . \end{align*}


\begin{align*} \Gamma _ { \alpha , \beta } = \{ x + i y \in \mathbb { C } : y \ge \max \{ \beta , \alpha | x | \} \} . \end{align*}


\begin{align*} \pi = \bigoplus _ { i = 0 } ^ { n } c _ i \sigma _ i , \end{align*}


\begin{align*} \int _ 0 ^ \infty \ 1 _ { \{ \underline f > \tau \} } ( r ) g ( r ) \ , d r & = \int _ { a _ \tau } ^ \infty g ( r ) \ , d r \leq a _ \tau ^ { - p + 1 } \sup _ { \alpha > 0 } \alpha ^ { p - 1 } \int _ \alpha ^ \infty g ( r ) \ , d r \\ & = ( p - 1 ) \int _ 0 ^ \infty \ 1 _ { \{ \underline f > \tau \} } ( r ) \ , \frac { d r } { r ^ p } \ \overline \nu _ p ( g ) \ , . \end{align*}


\begin{align*} L _ { \mathcal { R } } ( f ) ( x ) = \int _ { \Theta } f ( \psi _ { \theta } ( x ) ) d p ( \theta ) , \end{align*}


\begin{align*} | \psi | ( x , u , p ) = \frac { f ( x ) } { f ^ * \circ Y ( x , u , p ) } , \end{align*}


\begin{align*} \phi ^ * ( x ) : = \inf \{ t \geq 0 \mid \mu ( \phi ^ { - 1 } ( \mathbb { R } _ { > t } ) ) \leq 2 | x | \} . \end{align*}


\begin{align*} \lim _ { k \to \infty } \lambda _ \delta ( \Omega _ k , \Sigma _ k ) = \lambda ( \Omega , \Sigma ) . \end{align*}


\begin{align*} g ( R ( U , V ) W , S ) = g ( R ^ \nu ( U , V ) W , S ) + g ( T _ U W , T _ V S ) - g ( T _ V W , T _ U S ) , \end{align*}


\begin{align*} m ^ 2 _ t : = m ^ 2 + \frac { 1 } { t } \quad \Rightarrow \lim _ { t \downarrow 0 } m _ t = + \infty . \end{align*}


\begin{align*} \tau x = x ^ q \tau , ~ ~ \forall x \in \Omega . \end{align*}


\begin{align*} \psi ( - a / 2 ) = \psi ( a / 2 ) = 0 , \end{align*}


\begin{align*} \begin{cases} \limsup _ { | x | \rightarrow \infty } | x | ^ { \min \{ \beta , n \} - 2 } \left | u ( x ) - \left ( \frac { 1 } { 2 } x ^ T A x + b \cdot x + c \right ) \right | < \infty , & \ \beta \neq n , \\ \limsup _ { | x | \rightarrow \infty } | x | ^ { n - 2 } ( \ln | x | ) ^ { - 1 } \left | u ( x ) - \left ( \frac { 1 } { 2 } x ^ T A x + b \cdot x + c \right ) \right | < \infty , & \ \beta = n . \end{cases} \end{align*}


\begin{align*} \sum _ k \norm { \omega _ k \circ \sigma ^ \phi _ t } \norm { V _ { - t } \zeta _ k } = \sum _ k \norm { \omega _ k } \norm { \zeta _ k } , \end{align*}


\begin{align*} \pi \wedge \pi _ { { } _ S } = \pi \wedge \pi _ { { } _ { \Phi ( S ) } } , \ \ \forall \ , S \subseteq \{ 1 , \ldots , k \} . \end{align*}


\begin{align*} n l _ h ( u , v ) : = ( \nabla \times u ) \times v , \end{align*}


\begin{align*} \hat { \pi } _ { h , T } ( x ) = \frac { 1 } { T \prod _ { l = 1 } ^ d h _ l } \int _ 0 ^ T \prod _ { m = 1 } ^ d K ( \frac { x _ m - X _ u ^ m } { h _ m } ) d u . \end{align*}


\begin{align*} \Upsilon = \{ ( g _ { i , j } ) \in \Gamma ( \sqrt { - 3 } ) : g _ { 1 , 1 } \equiv 1 \bmod 3 \} . \end{align*}


\begin{align*} \min \limits _ { h = 2 , \dots , k } \left | \xi _ 1 - \xi _ h \right | = \left | \xi _ 1 - \xi _ 2 \right | = 2 \sin { \pi \over k } \end{align*}


\begin{align*} ( - L z _ s , z _ s ) _ V & = \frac { 1 } { 2 } \sum _ { e \in E } \frac { \delta _ E ( e ) } { | e | ^ 2 } \sum _ { u , v \in e } ( z _ s ( u ) - z _ s ( v ) ) ^ 2 \\ & = \frac { 1 } { 2 } \sum _ { e \in \partial S } \frac { \delta _ E ( e ) } { | e | ^ 2 } 2 | e \setminus S | | S \cap e | | V | ^ 2 \\ & \le \frac { 1 } { 4 } \delta _ E ( \max ) | \partial S | | V | ^ 2 . \end{align*}


\begin{align*} \Lambda _ i ^ { \nu t } A _ i \Lambda _ i ^ { \nu } = B - \pi ^ { \nu } C ^ { \nu } _ i , \end{align*}


\begin{align*} \limsup _ { t \to \infty } \tilde p _ { d , t , \mu } ( \lambda ) \underset { \mu \rightarrow \infty } { = } o ( \mu ^ { - 1 } ) . \end{align*}


\begin{align*} \sum _ { e \in E _ i } t '' _ e = 2 n _ i \ , . \end{align*}


\begin{align*} a _ i = a _ { \theta 0 } \left ( \partial _ i L ( h _ \theta ) \right ) \end{align*}


\begin{align*} Y ^ n _ t = n ^ { \alpha + \frac 1 2 } ( X ^ n _ t - X _ t ) . \end{align*}


\begin{align*} F _ n ( - x ) & = Q ( x ) \exp \bigg \{ - a _ 0 \frac { x ^ 3 } { \sqrt { n } } + a _ 1 \frac { x ^ 4 } { n } \\ & - O \left ( \frac { x ^ 5 } { n ^ { 3 / 2 } } \right ) + O \left ( \frac { x } { \sqrt { n } } \right ) \bigg \} . \end{align*}


\begin{align*} \left | \left . \int u _ { 1 , t } d \nu ^ 2 _ { t } \right | _ { t = \bar s } ^ { t = s ' } \ , \right | & \le C \varepsilon ( 1 + \sqrt { s ' - s } ) \leq C \varepsilon ( 1 + s ' - s ) . \end{align*}


\begin{align*} ( - \Delta ) ^ s m - \frac { ( - \Delta ) ^ s m _ 1 } { \gamma _ 1 ^ { 1 / 2 } } m & = 0 \quad \Omega , \\ m & = 0 \quad \Omega _ e . \end{align*}


\begin{align*} I _ { D _ { Y , - } , d } = \frac { 1 } { \prod _ { i \in I _ - , D _ i \cdot d > 0 } ( \bar D _ i + ( D _ i \cdot d ) z ) } [ \textbf { 1 } ] _ { ( - D _ i \cdot d ) _ { i \in I _ - } } . \end{align*}


\begin{align*} \begin{array} { l l } \partial _ t \psi ( t ) = A _ { 1 } ( t ) \psi ( t ) , \end{array} \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } t ^ { \frac { d - 2 \delta } { \alpha } ( 1 - \frac { 1 } { q } ) } \| u ( t , \cdot ) - A \Psi _ t \| _ { q , h } & = \lim _ { t \to \infty } t ^ { \frac { d - 2 \delta } { \alpha } ( 1 - \frac { 1 } { q } ) } \| u ( t + 1 , \cdot ) - A \Psi _ { t + 1 } \| _ { q , h } \\ & = \lim _ { t \to \infty } t ^ { \frac { d - 2 \delta } { \alpha } ( 1 - \frac { 1 } { q } ) } \| \tilde P _ t \left ( \tilde P _ 1 f - A \Psi _ 1 \right ) \| _ { q , h } = 0 . \end{align*}


\begin{align*} c U + U ^ * b = 0 \ , . \end{align*}


\begin{align*} \prod _ { v \in { { \mathfrak { M } } } _ K } | \xi | _ v = 1 \ \ \ \ \xi \in K \setminus \{ 0 \} \enspace . \end{align*}


\begin{align*} \mathrm { q a d j } ( A ) A = A \mathrm { q a d j } ( A ) = \det ( A ) E . \end{align*}


\begin{align*} J _ { w , \sigma , \lambda } \circ \phi ( \sigma , \lambda ) = \phi ( w \sigma , w \lambda ) \circ J _ { w , \sigma , \lambda } \end{align*}


\begin{align*} \phi ( x , v ) = e ^ { \chi ( | x | ^ 2 + | v ^ 2 | ) } . \end{align*}


\begin{align*} \iota ( r ( e ) , z ) \ , e = e \ , \iota ( s ( e ) , z ) \quad \end{align*}


\begin{align*} \nu ( g ) = \left \{ \begin{array} { c c c } 1 & \mbox { i f } & g \ \mbox { i s o r i e n t a t i o n p r e s e r v i n g } \\ - 1 & \mbox { i f } & g \ \mbox { i s o r i e n t a t i o n r e v e r e s i n g } . \end{array} \right . \end{align*}


\begin{align*} \label [ I e q ] { e q : p o l y n o m i a l s o b o l e v } \| \mu - \nu \| _ { H ^ { - s } } ^ 2 & = \int _ { \R ^ { 2 d } } \Phi _ s ( x - y ) ( \mu ^ { \otimes 2 } - \mu \otimes \nu ) ( \dd x , \dd y ) \\ & \qquad \qquad + \int _ { \R ^ { 2 d } } \Phi _ s ( x - y ) ( \nu ^ { \otimes 2 } - \nu \otimes \mu ) ( \dd x , \dd y ) \\ & = \int _ { \R ^ { 2 d } } \Phi _ { s } ( x - y ) ( \mu - \nu ) ^ { \otimes 2 } ( \dd x , \dd y ) , \end{align*}


\begin{align*} | \varphi ( S ) - \varphi ( \alpha _ u S \alpha _ u ^ * ) | \leq 2 \| S \| \varphi ( \sum _ { t \not \in F } p _ t ) ^ { 1 / 2 } + \| S \| \varphi ( 1 - \sum _ { t \in F } \alpha _ { t ^ { - 1 } } ( p _ t ) ) ^ { 1 / 2 } = 3 \| S \| \varphi ( \sum _ { t \not \in F } p _ t ) ^ { 1 / 2 } . \end{align*}


\begin{align*} P ^ { \nu } _ { s + t } ( x , \cdot ) = \int _ E P ^ { \overline { S } _ t ( \nu ) } _ { s } ( y , \cdot ) P ^ { \nu } _ { t } ( x , \dd y ) . \end{align*}


\begin{align*} \left \{ \begin{array} { c l } i \partial _ t u + \Delta u & = \pm | u | ^ { \frac { 4 } { d - 2 } } u ( t , x ) \in \R \times \R ^ d , \\ u ( 0 ) & = u _ 0 \in M ^ { 1 + \varepsilon } _ { 4 , 2 } ( \R ^ d ) . \end{array} \right . \end{align*}


\begin{align*} t ^ { - 1 } ( f ( x ) g ( x ) ) = \overline { f } \otimes \overline { g } . \end{align*}


\begin{align*} \begin{array} { l l } \mbox { m i n i m i z e } & f ( x ) + h ( x ) \\ \mbox { s u b j e c t t o } & A x = b . \end{array} \end{align*}


\begin{align*} d i m _ x f ^ { - 1 } ( f ( x ) ) = d i m \mathcal { O } _ { X _ y , x } + t r . d e g \kappa ( x ) / \kappa ( y ) . \end{align*}


\begin{align*} \mathrm { d i s t } ( h _ d ( x ) , f ( x ) ) \leq 2 C ' \sup _ { x \in M } \abs { \Phi ( x ) } = D . \end{align*}


\begin{align*} f ( x ) = \sum \limits _ { y \leq x } g ( y ) \quad g ( x ) = \sum \limits _ { y \leq x } \mu ( y , x ) \cdot f ( y ) \end{align*}


\begin{align*} D _ { \frac { \partial } { \partial \sigma } } \frac { \partial } { \partial \sigma } = \frac { 1 } { \sigma } \frac { \partial } { \partial \sigma } \mbox { o r } - \frac { 3 } { \sigma } \frac { \partial } { \partial \sigma } . \end{align*}


\begin{align*} | A | \times | A _ { i j | k l } | = \begin{vmatrix} | A _ { i | k } | & | A _ { i | l } | \\ | A _ { j | k } | & | A _ { j | l } | \end{vmatrix} . \end{align*}


\begin{align*} z & = \chi ( t / \tilde { T } ) \left ( W _ t ^ \gamma T [ u _ 0 , \cdots , u _ 0 ] - \frac { f } { \bigtriangleup + i \gamma } \right ) \\ & \qquad - i \chi ( t / \tilde { T } ) \int _ 0 ^ t W _ { t - s } ^ \gamma \left ( \mathfrak { F } ( v ) + \mathcal { H L } _ B [ z , v , \cdots , v ] \right ) \ , d s . \end{align*}


\begin{align*} \widehat { \varphi } _ { i , j } ( z ) = [ \lambda \in \sigma ( w _ { i , j } ( z ) ) : | \lambda | = 1 ] . \end{align*}


\begin{align*} H ( z ) = a + b z + \int _ \mathbb { R } \frac { 1 + s z } { z - s } \ , d \rho ( s ) , z \in \mathbb { C } ^ + , \end{align*}


\begin{align*} \bigcup _ { m \in \N } & \phi _ m \big ( [ 0 , T ] \times \phi _ m ( s _ m '' , p _ m ) \big ) \\ & = \Big ( \bigcup _ { m < m _ \epsilon } \phi _ m \big ( [ 0 , T ] \times \phi _ m ( s _ m '' , p _ m ) \big ) \Big ) \cup \Big ( \bigcup _ { m \geq m _ \epsilon } \phi _ m \big ( [ 0 , T ] \times \phi _ m ( s _ m '' , p _ m ) \big ) \Big ) \end{align*}


\begin{align*} \mathcal { D } _ { [ n , j ] } ^ { ( q ) } & : = \{ \gamma + q : \gamma \in \mathcal { D } _ { [ n , j ] } \} q \in \mathbb { Z } _ { \geq 0 } , \\ \widehat { \mathcal { D } } _ { [ n , j ] } ^ { ( q ) } & : = \{ \gamma - q : \gamma \in \widehat { \mathcal { D } } _ { [ n , j ] } \} q \in \mathbb { Z } _ { \geq 0 } . \end{align*}


\begin{align*} - \tilde u '' _ i + \tilde \lambda \tilde u _ i = \tilde u _ i ^ { p - 1 } , \tilde u _ i \ge 0 \end{align*}


\begin{align*} c _ 1 ( \alpha , g ) = s t \ ; \ ; \ ; \ ; \ ; c _ 2 ( \alpha , g ) = u r \end{align*}


\begin{align*} \widehat { H } _ { n H } ( t ) : = \frac { 1 } { 2 } \sum _ { i = 1 } ^ n \left ( { \widehat { p } } _ i ^ 2 + \omega ^ 2 ( t ) \widehat { x } _ i ^ 2 \right ) + c ( t ) { U _ { n H } } ( { \widehat { x } } _ 1 , \ldots , { \widehat { x } } _ n ) , \end{align*}


\begin{align*} \lim _ { n \to + \infty } c _ n = 0 , \ \ \int _ { \mathbb { R } ^ N } \sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \frac { \partial U _ { h _ n , i } } { \partial h _ n } \left ( \psi _ n + c _ n \sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \frac { \partial U _ { h _ n , i } } { \partial h _ n } \right ) ~ d y = 0 . \end{align*}


\begin{align*} M = \left ( \mu _ 0 , \mu _ 1 , \ldots , \mu _ n \right ) \end{align*}


\begin{align*} \rho _ g ( a ) = \mathcal { U } _ g a \mathcal { U } _ g ^ { - 1 } . \end{align*}


\begin{align*} \begin{array} { l l } g ( ( \nabla _ U A ) _ { X _ j } X _ j , V ) & = g ( \nabla _ U g ( X _ j , X _ j ) H ' - g ( \mathcal { H } \nabla _ U X _ j , X _ j ) H ' \\ & - g ( X _ j , \mathcal { H } \nabla _ U X _ j ) H ' , V ) . \end{array} \end{align*}


\begin{align*} P ' ( f p ) = \sum _ { x \in N _ { s o r t e d } } \sum _ { y \in S _ x } f _ { f p } ( m _ { x } , \gamma _ { x } , k _ { x } ) f _ { f p } ( m _ { y } , \gamma _ { y } , k _ { y } ) . \end{align*}


\begin{align*} \frac { d } { d t } g ^ { T } ( x , t ) = - ( R i c ^ { T } ( x , t ) - \varkappa g ^ { T } ( x , t ) ) \end{align*}


\begin{align*} \hat g ( \zeta ) = \int _ { \R ^ { 2 n } } g ( z ) e ^ { - i \langle \zeta , z \rangle _ { \R ^ { 2 n } } } \ , d z , \zeta \in \R ^ { 2 n } . \end{align*}


\begin{align*} j ( a ^ { * } ) ( \sigma ) = j ( a ) ( \sigma ^ { - 1 } ) ^ { * } j ( a * b ) ( \sigma ) = \sum _ { q ( \tau ) \in G ^ { r ( \sigma ) } } j ( a ) ( \tau ) \ , \alpha _ { \tau } \bigl ( j ( b ) ( \tau ^ { - 1 } \sigma ) \bigr ) \end{align*}


\begin{align*} \mathcal { E } _ h ^ { ( n ) } = & ( \chi - 1 ) f _ h ^ { ( n ) } + \mathcal { \tilde E } _ h ^ { ( n ) } , \end{align*}


\begin{align*} \partial _ t m _ t ( z ) = - m _ t \partial _ z m _ t ( z ) , m _ 0 ( z ) = m ( z ) , \end{align*}


\begin{align*} \gamma : = & \gamma ( n , \sigma , p ) = \int _ { B _ { 1 } ^ { c } } \frac { d y } { | y | ^ { n + \sigma p } } = \frac { | \mathbb { S } ^ { n - 1 } | } { \sigma p } , \\ \tau : = & \tau ( n , \sigma , p , s ) = \int _ { B _ { 1 } ^ { c } } \frac { d y } { | y | ^ { n + \sigma p + s ( p - 1 ) } } = \frac { | \mathbb { S } ^ { n - 1 } | } { \sigma p + s ( p - 1 ) } . \end{align*}


\begin{align*} \dot { x } _ i & = v _ { i } \left ( \cos { \psi _ i } - \sin { \psi _ i } \tan { \beta _ i } \right ) \\ \dot { y } _ i & = v _ { i } \left ( \sin { \psi _ i } + \cos { \psi _ i } \tan { \beta _ i } \right ) \\ \dot { \psi } _ i & = \frac { v _ { i } } { l _ r } \tan { \beta _ i } \\ \dot { \beta } _ i & = \omega _ i \\ \dot { v } _ { i } & = a _ { i } , \end{align*}


\begin{align*} \frac { 1 } { p _ 2 } = \frac { \theta } { p _ 1 } + \frac { 1 } { r _ 2 } , \end{align*}


\begin{align*} \mathcal { F } ( x , z ) = \frac { v o l ( B ^ n ( 1 ) ) } { v o l ( D ^ n _ x ( 1 ) ) } . \end{align*}


\begin{align*} \phi _ { \sigma ' } = 0 \sigma ' \in \widehat { M } \vert \sigma ' \vert > \vert \sigma \vert . \end{align*}


\begin{align*} \dot { h } ( t ) = R \tilde { u } ( t ) , \dot { R } = 0 , \end{align*}


\begin{align*} A = \bigcup \limits _ { l = 0 } ^ m D ' _ l \cup D _ { n + 1 } . \end{align*}


\begin{align*} \hat { \beta } _ { \alpha } = \hat { \beta } _ { O L S } + \alpha ( Y ^ { \prime } H _ { 1 } Y , \ldots , Y ^ { \prime } H _ { k } Y ) ^ { \prime } \end{align*}


\begin{align*} P _ { i } = N _ { i } ^ { 1 / 9 - 2 \varepsilon } , Q _ { i } = N _ { i } ^ { 8 / 9 + \varepsilon } \end{align*}


\begin{align*} A = \left [ \begin{matrix} B & { \bf 0 } \\ { \bf 0 } & C \end{matrix} \right ] , B = \left [ \begin{matrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ \epsilon & 0 & 1 \end{matrix} \right ] , \end{align*}


\begin{align*} R ( w + i y , A ) = R ( x + i y , A ) \sum _ { k = 0 } ^ \infty R ( x + i y , A ) ^ k ( x - w ) ^ k , \end{align*}


\begin{align*} \hat { H } \phi _ { n } ^ { \pm } ( x ) = E _ { n } \phi _ { n } ^ { \pm } ( x ) , \end{align*}


\begin{align*} \frac { d } { d t } \varphi = \log \det ( g _ { \alpha \overline { \beta } } ^ { T } + \varphi _ { \alpha \overline { \beta } } ) - \log \det ( g _ { \alpha \overline { \beta } } ^ { T } ) + \kappa \varphi - F . \end{align*}


\begin{align*} \nu _ d = \mu _ d * N ( 0 , 1 / d ^ 2 ) , \end{align*}


\begin{align*} \log M ^ * ( n , \epsilon ) = n C - \sqrt { n V _ { \epsilon } } Q ^ { - 1 } ( \epsilon ) + O ( \log n ) . \end{align*}


\begin{align*} \widetilde { H } _ 0 ( \phi ) : = \nu ( \widetilde { H } ( \phi ) _ 0 ) \widetilde { H } ( \phi ) _ 0 , \end{align*}


\begin{align*} \int _ { t _ 1 } ^ { t _ 2 } \tilde { V } ( s ) \cdot K ( s , X ( s ) , V ( s ) ) d s , \tilde { V } ( s ) : = V ( s ) / | V ( s ) | . \end{align*}


\begin{align*} \dot \kappa _ t = \frac { 1 } { t } - \frac { \chi _ t } { t ^ 2 } . \end{align*}


\begin{align*} \big ( ( \hat { v } + \omega ) \times B \big ) _ 1 = ( \hat { v } + \omega ) _ 2 B _ 3 - ( \hat { v } + \omega ) _ 3 B _ 2 , \big ( ( \hat { v } + \omega ) \times B \big ) _ 2 = - ( \hat { v } + \omega ) _ 1 B _ 3 + ( \hat { v } + \omega ) _ 3 B _ 1 \end{align*}


\begin{align*} \tilde g = 2 d \tau d u + ( | \nabla u | ^ { - 2 } + a _ 0 ^ 2 + \rho ^ { - 2 } b _ 0 ^ 2 ) d u ^ 2 + 2 a _ 0 d u d \rho + 2 b _ 0 d u d \theta + d \rho ^ 2 + \rho ^ 2 d \theta ^ 2 \end{align*}


\begin{align*} \Tilde { \mathcal { N } } ( f ) ( v _ n ) = \operatorname { N } _ { n + 1 , n } ( f ( v _ { n + 1 } ) ) . \end{align*}


\begin{align*} \left \langle u , v \right \rangle _ { H ^ 2 _ { \frac { 1 } { a } } ( 0 , 1 ) } = \int _ { 0 } ^ { 1 } \frac { f v } { a } \ , d x \Longleftrightarrow \int _ { 0 } ^ { 1 } \biggl ( \frac { u v } { a } + u '' v '' \biggr ) d x = \int _ { 0 } ^ { 1 } \frac { f v } { a } \ , d x , \end{align*}


\begin{align*} - \Delta v _ e ^ 1 + \sigma \Delta g _ e ^ 1 = 0 . \end{align*}


\begin{align*} \dd X _ t & = b ( X _ t ) \dd t + \sigma \dd B _ t \\ \dd Y _ t & = b ( Y _ t ) \dd t + \sigma ( I _ d - 2 e _ t e _ t ^ \mathrm { T } ) \dd B _ t , \end{align*}


\begin{align*} \begin{aligned} Z ( x _ i ) = f _ i - \frac { \mu } { 2 } \| x _ i \| ^ 2 \nabla Z ( x _ i ) = g _ i - \mu x _ i . \end{aligned} \end{align*}


\begin{align*} \Vert u ^ { \infty } \Vert _ { \mathcal { F } } ^ 2 : = \sum _ { ( \ell _ 1 , m _ 1 ) \in \Gamma } \sum _ { ( \ell _ 2 , m _ 2 ) \in \Gamma } \Big ( \frac { 2 \ell _ 1 + 1 } { e k a } \Big ) ^ { 2 \ell _ 1 } \Big ( \frac { 2 \ell _ 2 + 1 } { e k a } \Big ) ^ { 2 \ell _ 2 } \left \vert \mu _ { ( \ell _ 1 , m _ 1 ; \ell _ 2 , m _ 2 ) } \right \vert ^ 2 . \end{align*}


\begin{align*} \widetilde a _ i = \widetilde W _ c ( \mu , \nu _ { i / I } ) - \widetilde W _ c ( \mu , \nu _ { ( i - 1 ) / I } ) \forall i \in \mathcal I , \end{align*}


\begin{align*} \sum _ { i = 3 , 4 } \big \| T _ { k , j ; n , l , r } ^ { \mu , m , i } ( \mathfrak { m } , E ) ( t , x , \zeta ) + \hat { \zeta } \times T _ { k , j ; n , l , r } ^ { \mu , m , i } ( \mathfrak { m } , B ) ( t , x , \zeta ) \big \| _ { L ^ \infty _ x } \lesssim \| \mathfrak { m } ( \cdot , \zeta ) \| _ { \mathcal { S } ^ \infty } \big [ 2 ^ { ( 1 - 1 9 . 5 \epsilon ) M _ { t ^ { \star } } } \end{align*}


\begin{align*} b _ 0 ( s ) = \widetilde { b } _ 0 ( s ) = V _ \alpha ( s ) ^ { - 1 } \zeta ( \Delta , s ) . \end{align*}


\begin{align*} \mathcal { I } _ { 2 , 1 } - \iint _ { Q _ T } [ \nabla m \cdot \nabla p ] | \nabla p | ^ 2 = \iint _ { Q _ T } m \Delta p | \nabla p | ^ 2 . \end{align*}


\begin{align*} s = - 1 - l , l \geq 0 s = - \frac { H } { 2 \pi i } - \frac { m } { d } - \bar { f } , m \geq 1 . \end{align*}


\begin{align*} Z = f ( p \partial _ { 1 } + q \partial _ { 2 } ) , \end{align*}


\begin{align*} \begin{aligned} \tfrac { \sigma _ 0 L } { h _ 0 B } = 1 - \tfrac { 1 - \kappa h _ 0 ^ 2 } { 2 \left [ 2 - ( 1 + \kappa ) h _ 0 \right ] } = \tfrac { \kappa h _ 0 ^ 2 - 2 ( 1 + \kappa ) h _ 0 + 3 } { 2 \left [ 2 - ( 1 + \kappa ) h _ 0 \right ] } \ , \ , , \tfrac { \alpha _ 0 } { B } = \tfrac { 1 - \kappa h _ 0 } { 2 - ( 1 + \kappa ) h _ 0 } \end{aligned} \end{align*}


\begin{align*} u _ t = \sum _ { h = 1 } ^ { H } M ^ { [ h ] } w _ { t - h } , \end{align*}


\begin{align*} \frac { d } { d t } E _ { m , 3 } ( t ) - \delta \int _ 0 ^ L u _ t \overline { w } _ x d x = c ( m - 1 ) \int _ 0 ^ L \abs { \omega _ x } ^ 2 d x + \frac { c m } { 2 } \int _ 0 ^ { L } \int _ 0 ^ \infty \sigma ' ( s ) \abs { \eta _ x } ^ 2 d s d x . \end{align*}


\begin{align*} \begin{aligned} - \kappa \left ( 1 + \kappa \right ) h ^ 3 + \left [ 3 \kappa + \left ( 1 + \kappa \right ) ^ 2 \right ] h ^ 2 - 4 \left ( 1 + \kappa \right ) h + 4 = 0 \end{aligned} \end{align*}


\begin{align*} n ( T , l ( T ) + 1 - j ) = c _ { l ( T ) - j } n ( T , l ( T ) - j ) , \end{align*}


\begin{align*} \mathbb { Z } _ { r } & \rightarrow \mathbb { C } ^ { 2 } \\ \gamma & \mapsto \begin{bmatrix} e ^ { 2 \pi i \frac { a _ { 1 } } { r _ { 1 } } } & 0 \\ 0 & e ^ { 2 \pi i \frac { a _ { 2 } } { r _ { 2 } } } , \end{bmatrix} \end{align*}


\begin{align*} k \left [ \tan ( k a / 2 ) + \cot ( k a / 2 ) \right ] = k ' \left [ \tanh ( k ' a / 2 ) - \coth ( k ' a / 2 ) \right ] , \end{align*}


\begin{align*} \mathbf { J \left ( \boldsymbol { \theta } \right ) = \mathbb { E } } \left \{ \left [ \frac { \partial \ln p \left ( \boldsymbol { y } \mid \boldsymbol { \theta } \right ) } { \partial \boldsymbol { \theta } } \right ] \left [ \frac { \partial \ln p \left ( \boldsymbol { y } \mid \boldsymbol { \theta } \right ) } { \partial \boldsymbol { \theta } } \right ] ^ { T } \right \} , \end{align*}


\begin{align*} \begin{array} { l l } \underset { x \in \Re ^ n } { \mbox { m i n i m i z e } } & f ( x ) \\ \mbox { s u b j e c t t o } & g ( x ) = 0 , \ : X ( x ) \in \S ^ d _ + , \end{array} \end{align*}


\begin{align*} M = \begin{pmatrix} f _ 1 ^ 1 & \ldots & f _ { n + m } ^ 1 \\ f _ 1 ^ 2 & \ldots & f _ { n + m } ^ 2 \\ \vdots & \ddots & \vdots \\ f _ 1 ^ n & \ldots & f _ { n + m } ^ n \\ \end{pmatrix} \end{align*}


\begin{align*} \lim _ { i \rightarrow \infty } ( - \Delta ) ^ { \sigma } _ { p } v _ { j } ( x ) = - 1 . \end{align*}


\begin{align*} G ^ 1 ( R _ { \pi } ) : = \textup { k e r } ( G ( R _ { \pi } ) \rightarrow G ( k ) ) \end{align*}