\begin{align*} \varrho _ N ^ { \rm i n i } ( u ) : = \varrho _ * + N ^ { - \alpha } \rho ^ { \rm i n i } ( u ) , \end{align*}


\begin{align*} ( \theta _ 1 , \theta _ 2 ) = ( b ^ { - x } \alpha ^ y , a ^ { - x } \beta ^ y ) . \end{align*}


\begin{align*} T & = \left ( [ x ] \times [ y ] \right ) \backslash \left ( [ x - r + 1 . . x ] \times y \right ) \backslash \left ( x \times [ y - s + 1 . . y - 1 ] \right ) \mbox { a n d } \\ T ' & = \left ( [ x - w ] \times [ y ] \right ) \backslash \left ( ( x - w ) \times [ y - r ' + 1 . . y ] \right ) , \end{align*}


\begin{align*} \# J _ i \sim \frac n { e ^ \gamma \alpha ^ { i - 1 } } - \frac n { e ^ \gamma \alpha ^ i } = \frac { n ( \alpha - 1 ) } { e ^ \gamma \alpha ^ i } \end{align*}


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


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


\begin{align*} x _ { n + 1 } = \begin{cases} x _ n - 1 , & x _ n > 1 \\ n + 1 , & x _ n = 1 . \end{cases} \end{align*}


\begin{align*} \begin{aligned} G ^ { \alpha \beta } _ 0 \sigma _ { \alpha \beta } \geq \ , & G ( \sigma _ { \alpha \beta } - \theta \delta _ { \alpha \beta } ) + \theta \sum _ { \alpha = 1 } ^ { n - 1 } G ^ { \alpha \alpha } _ 0 \\ \geq \ , & \gamma + \theta \sum _ { \alpha = 1 } ^ { n - 1 } G ^ { \alpha \alpha } _ 0 \end{aligned} \end{align*}


\begin{align*} ( \alpha + \beta ) ^ { q ^ i } = \alpha ^ { q ^ i } + \beta ^ { q ^ i } \end{align*}


\begin{align*} 2 m + n = \sum _ { x \in G } { \frac { | G | } { | x ^ G | } } = | G | \cdot \sum _ { x \in G } { \frac { 1 } { | x ^ G | } } = n \cdot k . \end{align*}


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


\begin{align*} \Pi _ { \mathrm { k e r } A } ( B ( y , z ) + B ( z , y ) ) = 0 \forall z \in \mathrm { k e r } A , y \in \mathrm { k e r } A ^ \perp \end{align*}


\begin{align*} R i c ^ g + \nabla \tau = 0 , \end{align*}


\begin{align*} \begin{aligned} \eta ( U | \bar U ; x , t ) & \doteq \eta ( U , x , t ) - \eta ( \bar U , x , t ) - G ( \bar U , x , t ) \cdot ( A ( U , x , t ) - A ( \bar U , x , t ) ) \\ & = \tilde { \eta } ( V , x , t ) - \tilde { \eta } ( \bar V , x , t ) - ( \nabla _ V \tilde { \eta } ) ( \bar V , x , t ) \ ; ( A ( U , x , t ) - A ( \bar U , x , t ) ) \ , \\ & \doteq \tilde { \eta } \big ( V | \bar V ; x , t \big ) \ , , \end{aligned} \end{align*}


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


\begin{align*} \tilde { h } _ d ( x ) = x \log d + \log \left ( \frac { \Gamma ( x ) } { \Gamma \left ( x + \frac { 1 } { 2 } \right ) \Gamma \left ( x + \frac { d - 1 } { 2 } \right ) } \right ) + \log \left ( \frac { 2 ^ { d - 2 } \Gamma \left ( \frac { d } { 2 } \right ) ^ 2 } { \sqrt { \pi } d ^ { \frac { d - 1 } { 2 } } } \right ) . \end{align*}


\begin{align*} f ( z , t ) = \int _ \Omega \int _ { \C ^ n } \widehat { f } ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \ , d \nu ( a ) . \end{align*}


\begin{align*} \begin{aligned} \sup _ { x \in \left ( x _ 0 , \infty \right ) } \left | \langle x \rangle I _ 4 ^ { \prime } ( x ) \right | \leq c \left \| r _ 2 \right \| _ { H ^ { 1 } \cap L ^ { 2 , 1 } } \left \| r _ 1 \right \| _ { H ^ { 1 } \cap L ^ { 2 , 1 } } \left ( \left \| z ^ { - 1 } r _ 1 \right \| _ { H ^ { 1 } \cap L ^ { 2 , 1 } } + \left \| r _ 2 \right \| _ { H ^ { 1 } \cap L ^ { 2 , 1 } } \right ) , \end{aligned} \end{align*}


\begin{align*} { \rm V o l } ( B ) \geq \min \{ { \rm V o l } ( A ) , { \rm V o l } ( B ) \} = \mathfrak h ^ { - 1 } { \rm V o l } ( \Sigma ) . \end{align*}


\begin{align*} \begin{aligned} | \L ^ { - 1 } g ( v ) | \lesssim ( 1 + | v | ) ^ { - m } \| ( 1 + | v | ) ^ m g \| _ { L ^ \infty } \ ( \forall v \in \R ^ 3 ) \end{aligned} \end{align*}


\begin{align*} \mathcal S ( f ) = \left ( \frac { f '' } { f ' } \right ) - \frac 1 2 \left ( \frac { f '' } { f ' } \right ) ^ 2 \end{align*}


\begin{align*} D _ j ( x ; k ) : = V _ j ( k ) J = R ( x ; z ) V _ j ( k ) , \ j = 1 , 2 . \end{align*}


\begin{align*} \mathcal { M } ^ { r } ( t ) - \mathcal { M } ^ { r } ( \delta ) = \int _ { \delta } ^ { t } \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \mu } \omega ( \mu , \nu ) \xi ( \mu , s ) \xi ( \nu , s ) \Lambda ( \mu , \nu ) \ d \nu d \mu d s , \end{align*}


\begin{align*} \begin{aligned} F _ \kappa = \sum _ { n = 0 } ^ \infty \kappa ^ n F _ n \ , , \end{aligned} \end{align*}


\begin{align*} { \bf { \bar q } } \left [ i \right ] = \sum \limits _ { l = 1 } ^ { L ' } { { { { \bf { F } } } _ l } { \bf { d } } \left [ { i - { \kappa _ l } } \right ] } , \end{align*}


\begin{align*} \chi _ P ( P _ a , P _ a ) & = \sum _ { i = 1 } ^ k k P _ a ^ 2 ( i ) - 1 = \sum _ { i = 1 } ^ k k \frac { 1 } { k ^ 2 } \left ( 1 + \epsilon v _ { a , i } \right ) ^ 2 - 1 = \sum _ { i = 1 } ^ k \frac { 1 } { k } \left ( 1 + 2 \epsilon v _ { a , i } + \epsilon ^ 2 v _ { a , i } ^ 2 \right ) - 1 = \epsilon ^ 2 , \end{align*}


\begin{align*} \begin{cases} - \varepsilon ^ 2 ( K _ H ( x ) \nabla u ) = \left ( u - \left ( \frac { \alpha | x | ^ 2 } { 2 } + \beta \right ) \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ & x \in B _ { R ^ * } ( 0 ) , \\ u = 0 , \ \ & x \in \partial B _ { R ^ * } ( 0 ) , \end{cases} \end{align*}


\begin{align*} S _ k [ r ] = \sigma _ k ( \lambda ( r ) ) \end{align*}


\begin{align*} \frac { 1 - e ^ { a _ { \alpha + n + 1 } + a _ { \alpha + n + 2 } } } { \left ( e ^ { - a _ { \alpha + 1 } } - 1 \right ) \left ( 1 - e ^ { a _ { n + 1 } } \right ) \exp \left [ \sum _ { j = 2 } ^ n ( a _ j - a _ { \alpha + j } ) \right ] } \end{align*}


\begin{align*} \mathcal D _ { E / F } = \prod _ { \mathfrak p \in \mathcal R _ { E / F } } \mathfrak p ^ { e _ { \mathfrak p } - 1 } . \end{align*}


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


\begin{align*} w _ i ( \alpha , \beta ) = \frac { ( - 1 ) ^ i \ , \binom { \beta - 1 } { i } } { \mathrm { B } ( \alpha , \beta ) } w _ { i , j , r } ( \alpha , \beta ) = \frac { ( - 1 ) ^ { i + j + r } \ , \binom { \alpha + i - 1 } { j } \ , \binom { \beta - 1 } { i } \ , \binom { j } { r } } { \mathrm { B } ( \alpha , \beta ) } . \end{align*}


\begin{align*} I & = - \Vert u - \widetilde { u } \Vert _ { V } ^ 2 , \\ I I & \leq \varepsilon \Vert u - \widetilde { u } \Vert _ { V } ^ 2 + C _ { \varepsilon } \Vert u \Vert _ { V } ^ 2 \Vert u - \widetilde { u } \Vert _ { H } ^ 2 . \end{align*}


\begin{align*} u _ + = \max \{ u , 0 \} \end{align*}


\begin{align*} \phi ( \sigma _ 1 \sigma _ 2 , \tau ) = \phi ( \tau , \sigma _ 1 \sigma _ 2 ) = \phi ( \tau , \sigma _ 1 ) \phi ( \tau , \sigma _ 2 ) = \phi ( \sigma _ 1 , \tau ) \phi ( \sigma _ 2 , \tau ) . \end{align*}


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


\begin{align*} g ( t ) = \frac { | A \oplus _ 2 \sqrt { t } B | ^ 2 } { | \partial ( A \oplus _ 2 \sqrt { t } B ) | ^ 2 _ { n - 1 } } \end{align*}


\begin{align*} \partial _ t a _ j + \mathbb { P } ( B _ j \times u _ j ) = - \mu _ j \nabla \times b _ j \textrm { o n } \ L ^ { 2 } _ { \sigma } ( \mathbb { R } ^ { 3 } ) , \end{align*}


\begin{align*} \mathbf z \cdot \omega _ j - s \omega _ 1 \cdot \omega _ j \leq - \frac { 2 } { 3 } , j = 2 , \cdots , n . \end{align*}


\begin{align*} \phi _ { t } ^ { \ast } g = g . \end{align*}


\begin{align*} _ { \rm r e f } [ V _ \lambda ] = q ^ { - \frac { c } { 2 4 } } q ^ { \frac { ( \lambda | \lambda + 2 \rho ) } { 2 ( k + h ^ \vee ) } - \lambda ( x + y ) } \ \prod _ { i = 1 } ^ L X _ i \ \prod _ { 1 \leq i < j \leq L } X _ { i , j } \end{align*}


\begin{align*} \begin{cases} \partial _ t \mathbf { v } + ( \mathbf { v } \cdot \nabla ) \mathbf { v } = - \nabla P , \ \ & D \times ( 0 , T ) , \\ \nabla \cdot \mathbf { v } = 0 , \ \ & D \times ( 0 , T ) , \\ \mathbf { v } \cdot \mathbf { n } = v _ n , \ \ & \partial D \times ( 0 , T ) , \\ \mathbf { v } ( x , 0 ) = \mathbf { v } _ 0 ( x ) , \ \ & D , \end{cases} \end{align*}


\begin{align*} ( \nabla ^ { \ell + 2 } _ \nu h ) ( e _ a , e _ b ) = 0 , \mbox { o r i n t h e l o c a l f r a m e } h _ { a b ; \tiny \underbrace { 0 \cdots 0 } _ { \ell + 2 - \mbox { t i m e s } } } = 0 . \end{align*}


\begin{align*} X _ n : = \{ v \in L ^ 2 ( D ) : v | _ { K } \in P ^ \ell ( K ) , K \in { \mathcal T } _ n \} , \end{align*}


\begin{align*} D _ { 2 k ; w _ 1 , w _ 2 } ^ { e ^ { 2 \Upsilon } g } \bigl ( u \otimes v \bigr ) = e ^ { ( w _ 1 + w _ 2 - 2 k ) \Upsilon } D _ { 2 k ; w _ 1 , w _ 2 } ^ { g } \bigl ( e ^ { - w _ 1 \Upsilon } u \otimes e ^ { - w _ 2 \Upsilon } v \bigr ) \end{align*}


\begin{align*} \langle x \rangle \vee \langle y \rangle = \langle x \wedge y \rangle . \end{align*}


\begin{align*} \lambda = ( \lambda _ 1 , \ldots , \lambda _ k ) , \ \lambda _ 1 \geq \ldots \geq \lambda _ k > 0 , \ \sum \lambda _ i = m . \end{align*}


\begin{align*} S _ 1 ( \chi _ { 8 d } ; \pi , X , \phi ) = \frac { 2 } { M \log X } \sum _ { p } \frac { a _ { \pi } ( p ) \log p } { \sqrt { p } } \chi _ { 8 d } ( p ) \hat { \phi } \left ( \frac { \log p } { M \log X } \right ) . \end{align*}


\begin{align*} \kappa ( t ) = h _ { \alpha , \rho } ( t ) , \ 0 < \alpha < 1 , \ \rho \ge 0 , \end{align*}


\begin{align*} \dot { v } ( s ) = \begin{cases} \frac { 1 } { | G / G _ 0 | } \sum _ { t G _ 0 \in G / G _ 0 } u ( t ^ { - 1 } s t ) & s \in G _ 0 \\ 0 & \end{cases} \end{align*}


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


\begin{align*} d _ { \widetilde K } = \# \{ 1 \leq r \leq j : C _ r \in O _ { \widetilde K } , \ \lambda _ r = 1 \} . \end{align*}


\begin{align*} I ( \lambda ) = \int _ a ^ b e ^ { i \lambda \phi ( x ) } \psi ( x ) d x \end{align*}


\begin{align*} & G _ x ( \theta ) : = \sum _ { n \in \Z } g \left ( \pi _ N ( x ) ( \theta + n ) \right ) \\ & = \sum _ { | n | \leq \pi _ N ( x ) } \widehat { G _ { x } } ( n ) e ( n \theta ) \\ & = \widehat { G _ { x } } ( 0 ) + \sum _ { 1 \leq n \leq \pi _ N ( x ) } \widehat { G _ { x } } ( n ) 2 \cos ( 2 \pi n \theta ) , \end{align*}


\begin{align*} ( d _ i y ) ( [ \mathbf { a } ] \otimes [ \mathbf { b } ] ) = 0 , \end{align*}


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


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


\begin{align*} B \cdot n = 0 . \end{align*}


\begin{align*} n _ t = \nabla \cdot ( \nabla n - \chi ( c ) n \nabla c ) , \quad c _ { t } = \Delta c - n c , \qquad x \in \Omega , \ , t > 0 , \end{align*}


\begin{align*} E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] = E _ { \xi _ 0 } \left [ h \bigl ( x , Z _ 1 ^ \theta \bigr ) \right ] , \forall x \in \mathbb R , \end{align*}


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


\begin{align*} f ( x ) = \left ( \log \left ( 1 + \frac { 1 } { 2 x } \right ) + \frac { 1 } { 4 x } \right ) - \left ( \psi \left ( x + \frac 1 2 \right ) - \psi ( x ) \right ) . \end{align*}


\begin{align*} v : = - d + \frac { 1 } { 8 b _ { n - 1 } } d ^ 2 . \end{align*}


\begin{align*} s ( G ^ { \sigma , \tau } ) & = \sum _ { \gamma \in \Gamma } s ( G ^ { ( \gamma ) } ) + \sum _ { v \in V } \sigma ( v ) - 2 ( \abs { \Gamma } - 1 ) . \end{align*}


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


\begin{align*} { \rm D } ^ 2 \widetilde { W } ( { \rm D } \phi ) . ( \xi \otimes \eta , \xi \otimes \eta ) > 0 \forall \ \eta , \xi \in \mathbb { R } ^ 3 , \ \ \lVert \eta \rVert = \lVert \xi \rVert = 1 , \end{align*}


\begin{align*} \min \{ c _ { d , 2 } ( q ) , c _ { d , \infty } ( q ) \} = \begin{cases} c _ { d , 2 } ( q ) , & - ( d - 1 ) < q \leq q _ d ^ * , \\ c _ { d , \infty } ( q ) , & q _ d ^ * \leq q \leq 2 , \end{cases} \end{align*}


\begin{align*} Q _ h : = { P } _ { k - 1 } ^ { \mathrm { d i s c } } ( \mathcal { T } ) \cap Q . \end{align*}


\begin{align*} Z _ n = \delta ( u _ n ) + r _ n N _ n , \end{align*}


\begin{align*} \hat { A } ^ * ( y | x ) = A ( y _ 2 y _ 1 x ) + \left [ A ( y _ 1 x ' ) - A ( y _ 2 x ' ) \right ] . \end{align*}


\begin{align*} \lim _ { y \to \pm \infty } e ^ { - \psi ( x _ 0 , y ) } d _ x T ( x _ 0 , y ) = 0 . \end{align*}


\begin{align*} \prod _ { { \substack { p < T \\ p \nmid 2 \alpha \beta } } } | t _ l | ^ { - s } _ p = A B \end{align*}


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


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


\begin{align*} h ^ 2 _ { [ - u _ 1 , u _ 1 ] \oplus _ 2 \cdots \oplus _ 2 [ - u _ m , u _ m ] } ( x ) & = \sum _ { i = 1 } ^ m \langle u _ i , x \rangle ^ 2 = \sum _ { i = 1 } ^ m \langle U e _ i , x \rangle ^ 2 = \sum _ { i = 1 } ^ m \langle e _ i , U ^ * x \rangle ^ 2 = | U ^ * x | ^ 2 = h ^ 2 _ { U B _ 2 ^ m } ( x ) \\ & = \langle U ^ * x , U ^ * x \rangle = \langle \sqrt { U U ^ * } x , \sqrt { U U ^ * } x \rangle = | \sqrt { U U ^ * } x | ^ 2 = h ^ 2 _ { \sqrt { U U ^ * } B _ 2 ^ n } ( x ) . \end{align*}


\begin{align*} \frac { i } { 2 \pi } \int _ { \mathcal { C } } e ^ { - t \tau } \operatorname { T r } \phi _ { - 1 } \ , d \tau = e ^ { - t \alpha | \xi | } + ( n - 2 ) e ^ { - t \mu | \xi | } + e ^ { - 2 t \mu | \xi | } + e ^ { - 2 t \mu s _ 1 | \xi | } . \end{align*}


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


\begin{align*} \sum _ { k = 1 } ^ \infty \frac { \mu ( k ) } { k } = 0 \ \ \mathrm { a n d } \ \ \sum _ { k = 1 } ^ \infty \frac { \mu ( k ) \log k } { k } = - 1 \end{align*}


\begin{align*} [ u , v ] = L ( u \times v ) , \forall u , v \in \mathfrak { g } , \end{align*}


\begin{align*} Q _ { i - 1 } ( x , w x ) = \frac { Q _ i ( x , w x ) - Q _ i ( x , s _ { \alpha _ { i - 1 } } w x ) } { \alpha _ i ( w x ) } . \end{align*}


\begin{align*} F ( x ) = \operatorname { I } _ { \exp \left ( - \frac { \theta } { x ^ 2 } \right ) } ( a , b ) = \frac { 1 } { \operatorname { B } ( a , b ) } \int _ 0 ^ { \exp \left ( - \frac { \theta } { x ^ 2 } \right ) } \omega ^ { a - 1 } ( 1 - \omega ) ^ { b - 1 } \mathrm { d } \omega , \end{align*}


\begin{align*} \mathcal { S } _ n ^ { ( p ) } \ & = \ \sum _ { i = 0 } ^ { n } \ , i ^ p \cdot F _ { i } , \\ \intertext { a n d r e p l a c i n g $ i $ w i t h $ n - i $ w e h a v e } \mathcal { S } _ n ^ { ( p ) } \ & = \ \sum _ { i = 0 } ^ { n } \ , ( n - i ) ^ p \cdot F _ { n - i } \ = \ \sum _ { i = 0 } ^ { n } \left ( \sum _ { k = 0 } ^ p \binom { p } { k } n ^ { p - k } ( - 1 ) ^ k i ^ k \right ) \cdot F _ { n - i } . \end{align*}


\begin{align*} \varrho \ , { u } _ { i , t t } & = \sigma _ { j i , j } , \ \ , i = 1 , 2 , 3 , \\ \varrho \ , j \ , { \vartheta } _ { i , t t } & = m _ { j i , j } ^ * + \epsilon _ { i j k } \ , \sigma _ { j k } , i = 1 , 2 , 3 , \end{align*}


\begin{align*} \int _ 0 ^ 1 ( 1 + f ( x ) ) x ^ { s - 1 } d x = \frac { 1 } { s } \left ( 1 - \zeta ( s ) \sum _ { k = 2 } ^ N c _ k k ^ { - s } \right ) . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I \leq \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ { T } f ( s ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s = 0 . \end{align*}


\begin{align*} \beta _ 1 + \bar { \beta _ j } = 1 \ , \ , { \rm a n d } \ , \ , \beta _ u + \bar { \beta } _ v = 1 . \end{align*}


\begin{align*} \begin{aligned} \tfrac { \partial \nu ( v ) } { \partial | v | } \geq 0 \ \textrm { f o r } 0 \leq \gamma \leq 1 \ , , \textrm { a n d } \tfrac { \partial \nu ( v ) } { \partial | v | } < 0 \ \textrm { f o r } - 3 < \gamma < 0 \ , . \end{aligned} \end{align*}


\begin{align*} S _ { E , a , n } ' ( d ) = \emptyset , \end{align*}


\begin{align*} b _ k : = \begin{cases} 1 & \min ( \abs { k } , M - \abs { k } ) \le k _ 0 \\ r M - k _ 0 & \min ( \abs { k } , M - \abs { k } ) = k _ 0 + 1 \\ 0 & \end{cases} \end{align*}


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


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


\begin{align*} \rho _ \phi ( m ) \rho _ \phi ( n ) = \sum _ { d | ( m , n ) } \rho _ \phi \left ( \frac { m n } { d ^ 2 } \right ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } } \frac { r ' ( r r ' ) ^ { 2 \tilde { \tau } + 1 } } { | r - r ' | ^ { 4 \tilde { \tau } - 2 / q } r ' } \dd r ' = C r ^ { 2 / q } . \end{align*}


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


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


\begin{align*} h ( \lambda ) = \frac { ( \lambda , \lambda + 2 \rho ) } { 2 ( \kappa + h ^ \vee ) } + \frac { ( \lambda , \lambda + 2 \rho ) } { 2 ( \kappa ^ * + h ^ \vee ) } - ( \lambda , \rho ^ \vee ) = \frac { ( \lambda , \lambda ) r ^ \vee n } { 2 } + ( \lambda , n r ^ \vee \rho - \rho ^ \vee ) . \end{align*}


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


\begin{align*} \Phi ^ * D = \Pi , \end{align*}


\begin{align*} \frac { \partial \hat { q } _ { \delta , i } } { \partial z _ { i , h } } = O ( 1 ) , \end{align*}


\begin{align*} u _ { \varepsilon } ( x ' , t ) : = u ( x ' , t + \varepsilon ) \quad f _ { \varepsilon } ( x ' ) : = u ( x ' , \varepsilon ) , x ' \in \R ^ n , \ , t > 0 . \end{align*}


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


\begin{align*} a ( u , v ) = \lambda ( u , v ) v \in H _ 0 ^ 2 ( D ) . \end{align*}


\begin{align*} d x _ { t , j } = - a _ j x _ { t , j } d t + B _ j ( x _ t , x _ t ) d t + \sigma _ j d W _ t ^ { ( j ) } , \end{align*}


\begin{align*} \left ( - ( 1 - \kappa r ^ 2 ) \frac { d ^ 2 } { d r ^ 2 } - ( d - 1 - d \kappa r ^ 2 ) \frac { 1 } { r } \frac { d } { d r } + \frac { l ( l + d - 2 ) } { r ^ 2 } - \frac { Q } { r } \sqrt { 1 - \kappa r ^ 2 } - { \cal E } \right ) R ( r ) = 0 , \end{align*}


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


\begin{align*} \mathbb { S } ^ { 1 } = \mathrm { S p e c } ( \mathbb { R } [ x , y ] / ( x ^ { 2 } + y ^ { 2 } - 1 ) ) . \end{align*}


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


\begin{align*} c = \oint _ l \mathbf { v } \cdot \mathbf { t } d l = \iint _ { \sigma } \mathbf { w } \cdot \mathbf { n } d \sigma , \end{align*}


\begin{align*} P [ \Delta _ { M _ n } < s _ n ] = O ( r _ n ^ { 1 + \kappa } ) \end{align*}


\begin{align*} & M ^ { \pm } _ 2 ( x ; z ) - e _ { 2 } = \mathcal { P } ^ { \pm } \left ( M _ { - } ( x ; \cdot ) R ( x ; \cdot ) \right ) _ { 2 1 } , z \in \mathbb { R } , \\ & 2 i k ( M ^ { \pm } _ 2 ( x ; z ) - e _ { 2 } ) = \mathcal { P } ^ { \pm } \left ( 2 i k M _ - ( x ; \cdot ) R ( x ; \cdot ) \right ) _ { 2 1 } , z \in \mathbb { R } , \end{align*}


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


\begin{align*} \sum _ { j = 0 } ^ { k } \dbinom { k } { j } ^ 2 a ^ { p _ k ( k - j ) / k } b ^ { p _ k j / k } \leq ( a + b ) ^ { p _ k } , \end{align*}


\begin{align*} n = \frac { - 3 \left ( \alpha ^ { 2 } + 8 \right ) \pm \sqrt { - 3 \left ( \alpha ^ { 4 } + 8 \left ( \alpha ^ { 3 } - 6 \alpha ^ { 2 } + 1 3 \alpha + 2 \right ) \right ) } } { 6 \left ( \alpha + 2 \right ) } \end{align*}


\begin{align*} E _ a ( x ) \ll \begin{cases} \pi _ N ( x ) ^ { j - a } L ^ { 2 r } & a \leq j \\ L ^ { 2 r - ( a - j ) } & a > j . \\ \end{cases} \end{align*}


\begin{align*} A _ t + \mathbb { P } ( B \times u ) = - \mu \nabla \times B \textrm { o n } \ L ^ { 2 } _ { \sigma } ( \mathbb { R } ^ { 3 } ) . \end{align*}


\begin{align*} \bar { N } ( x ) = \{ u \in G \mid u = v ( u , v ) \in E \} . \end{align*}


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


\begin{align*} K _ q ( z , w ) = E _ q ( z \overline { w } ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { ( z \overline { w } ) ^ n } { \Gamma ( q n + 1 ) } , \end{align*}


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


\begin{align*} \dot { x } _ 1 = - \gamma h ( x _ 1 ) u x ( 0 ) = x _ { 1 0 } \end{align*}


\begin{align*} M _ x ^ 2 & = \frac { \omega } { 3 L _ { x y } } \left ( \Psi ^ 0 + \frac { 4 } { ( 1 - \theta ) ^ 2 } ( \delta _ x + \delta _ y ) + \frac { \Sigma ^ 2 } { 2 ( 1 - \theta ) } \right ) , \\ M _ y ^ 2 & = \frac { 1 } { 4 L _ { x y } \omega } \left ( \Psi ^ 0 + \frac { 4 } { ( 1 - \theta ) ^ 2 } ( \delta _ x + \delta _ y ) + \frac { \Sigma ^ 2 } { 2 ( 1 - \theta ) } \right ) , \end{align*}


\begin{align*} \delta = \lim _ { N \to \infty } \dfrac { | A \cap \Phi _ N | } { | \Phi _ N | } \end{align*}


\begin{align*} M _ { f _ { 1 , \infty } } ( n , \alpha , \epsilon , Z , \psi ) : = \inf \bigg \{ \sum _ { i } ^ { - \alpha n _ { i } + S _ { 1 , n _ { i } } \psi ( x _ { i } , \epsilon ) } : Z \subseteq \bigcup _ { i } B _ { n _ { i } } ( x _ { i } , \epsilon ) \bigg \} , \end{align*}


\begin{align*} \frac { \psi ^ \pm _ 2 ( x ; k ) } { 2 i k } = \Psi ^ \pm _ 2 ( x ; z ) , \end{align*}


\begin{align*} \gamma ( s , \tau ) = \left ( r ^ * \cos \left ( \frac { s } { r ^ * } \right ) , r ^ * \sin \left ( \frac { s } { r ^ * } \right ) , \frac { c } { 4 \pi r ^ * } \tau \right ) ^ t , \end{align*}


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


\begin{align*} \zeta ' & \ = \sum _ { z \in \mathbb { F } ^ * _ { q ^ n } } \overline { U _ { q ^ n } } \Lambda ( z ) { \bf 1 } _ { - } - ( q ^ n - 1 ) \overline { U _ { q ^ n } } { \bf 1 } _ { - } \\ & \ = \sum _ { x , y \in \mathbb { F } ^ * _ { q ^ n } , x y \ne 1 } \varepsilon ( x ) \Lambda ( y ) { \bf 1 } _ { - } - ( q ^ n - 1 ) \overline { U _ { q ^ n } } { \bf 1 } _ { - } \end{align*}


\begin{align*} E Q H _ n = \Big \{ q \in R _ n : \quad \sum _ { 1 \le i \le n } \partial _ { \theta _ i } q = 0 \quad \hbox { a n d } \sum _ { 1 \le i < j \le n } \partial _ { \theta _ j } \partial _ { \theta _ i } q = 0 \Big \} ~ . \end{align*}


\begin{align*} P ( d x ) = Z ^ { - 1 } e ^ { f ( x ) } \mu ^ { \otimes n } ( d x ) , \end{align*}


\begin{align*} L ( x ) = \sum _ { i = 0 } ^ r a _ i x ^ { 2 ^ i } \end{align*}


\begin{align*} B _ { \sigma , \lambda } = \dfrac { ( 2 \sigma - 1 ) \left ( 1 - \exp \left ( - \frac { 2 \lambda ( 1 - \sigma ) } { 2 \sigma - 1 } \right ) \right ) } { 2 \lambda ( 1 - \sigma ) ^ 2 } . \end{align*}


\begin{align*} \sigma _ { \rm s } ^ 2 ( x [ n ] ) = \mu _ { \rm s } ( x [ n ] ) - \mu _ { \rm s } ^ 2 ( x [ n ] ) \left [ 1 - \left ( 1 - \frac { \tau _ { \rm d } } { T _ { \rm s } } \right ) ^ 2 \right ] , \end{align*}


\begin{align*} \partial _ i ( e ( T ) \otimes 1 ) : = \sum _ { j = 1 } ^ { i + 3 } \sum _ { \sigma \in H } ( - 1 ) ^ { j - 1 } e ( \sigma ( T _ j ) ) \otimes x _ { a _ j } \in V _ { ( d - 1 , d - i + 2 , 1 ^ { i - 1 } ) } \otimes _ K R ( - d - i ) = F _ { i - 1 } , \end{align*}


\begin{align*} \Sigma ( k ) = \theta \big ( \chi ( 2 k + 1 ) + 1 + \lceil \ln ( 2 L ( k + 1 ) ) \rceil \big ) + 1 . \end{align*}


\begin{align*} \int _ K | | \tau _ 0 | | _ { \omega _ { \epsilon _ 1 } } \le & \liminf _ { \epsilon \to 0 } \int _ K | | \tau _ { \epsilon } | | _ { \omega _ { \epsilon _ 1 } } \\ \le & \liminf _ { \epsilon \to 0 } \int _ K | | \tau _ { \epsilon } | | _ { \omega _ { \epsilon } } \\ \le & \liminf _ { \epsilon \to 0 } \int _ M | | \tau _ { \epsilon } | | _ { \omega _ { \epsilon } } . \end{align*}


\begin{align*} \Omega _ { i j } ( \partial _ j , \partial _ r ) = \frac { - \Gamma _ { j i } ^ i \Gamma _ { r j } ^ j + \Gamma _ { r i } ^ i \Gamma _ { j r } ^ r - \Gamma _ { r i } ^ i \Gamma _ { j r } ^ r } { \varphi _ j } . \end{align*}


\begin{align*} ( S \textbf { \textit { u } } ) ^ i _ j = \nabla ^ i u _ j + \nabla _ j u ^ i , \end{align*}


\begin{align*} ( S _ \bullet \to S ) = \varprojlim _ i ( T _ { i , \bullet } \to T _ i ) . \end{align*}


\begin{align*} ( 1 . 4 4 ) ^ { 2 } \cdot \frac { 3 2 \pi ^ { 8 } } { 9 p ^ { 3 } } \sum _ { n = a p } ^ { b p } n \log ( n + 2 ) \sum _ { d = 1 } ^ { \infty } \psi \left ( d \sqrt { \frac { n } { p } } \right ) \leq \frac { 6 9 9 5 8 } { p ^ { 3 } } \cdot \left [ b p \log ( b p + 2 ) \right ] ( b - a ) p \sum _ { d = 1 } ^ { \infty } \psi ( d \sqrt { a } ) . \end{align*}


\begin{align*} S ( \zeta ) = \mathrm { s t a b } _ N ( \psi _ \zeta ) \end{align*}


\begin{align*} \partial _ { L - u } f _ u ( x ) & = \{ l _ u \in L - u : \forall z \in X f _ u ( z ) \geq f _ u ( x ) + l _ u ( z ) - l _ u ( x ) \} \\ & = \{ \begin{multlined} [ t ] l - u : \forall z \in X , l \in L , \\ f ( z ) - u ( z ) \geq f ( x ) - u ( x ) + l ( z ) - u ( z ) - l ( x ) + u ( x ) \} \end{multlined} \\ & = \{ l - u : l \in \partial _ L f ( x ) \} \end{align*}


\begin{align*} A ^ { - 1 } \mathcal { L } _ g = I _ { n + 1 } \frac { \partial ^ 2 } { \partial x _ n ^ 2 } + B \frac { \partial } { \partial x _ n } + C , \end{align*}


\begin{align*} \begin{aligned} \int _ { 0 } ^ { 1 / 2 } \left [ m ( x + \delta \sin ( 2 \pi t ) ) - m ( x ) \right ] \sin ( 2 \pi t ) \ , d t \\ \int _ { 0 } ^ { 1 / 2 } \left [ m ( x ) - m ( x - \delta \sin ( 2 \pi t ) ) \right ] \sin ( 2 \pi t ) \ , d t . \end{aligned} \end{align*}


\begin{align*} \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + Q \Bigr ) \textbf { \textit { U } } & = \textbf { \textit { W } } , \textbf { \textit { U } } \big | _ { x _ n = 0 } = \textbf { \textit { V } } , \\ \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + B - Q \Bigr ) \textbf { \textit { W } } & = \textbf { \textit { Y } } \in ( C ^ \infty ( [ 0 , T ] ; \mathfrak { D } ^ \prime ( \mathbb { R } ^ { n } ) ) ) ^ { n + 1 } . \end{align*}


\begin{align*} d _ { \widetilde K } = \# \{ 1 \leq r \leq j : C _ r \in O _ { \widetilde K } , \ \lambda _ r = 1 \} . \end{align*}


\begin{align*} f ( z , t ) = \int _ \Omega \int _ { \C ^ n } F ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \ , d \nu ( a ) . \end{align*}


\begin{align*} [ M ] & = \frac { [ \mathbb C _ x ] } { D } \\ [ \mathbb C _ x ] & = \prod _ { i = 0 } ^ N e ^ { x _ i \alpha _ i } \\ D & = \prod _ { n = 1 } ^ \infty ( 1 - e ^ { - n \delta } ) ^ { N } \prod _ { n = 1 } ^ \infty \prod _ { 1 \leq i < j \leq r } ( 1 - e ^ { \alpha _ j - \alpha _ i } e ^ { ( 1 - n ) \delta } ) ( 1 - e ^ { \alpha _ i - \alpha _ j } e ^ { - n \delta } ) . \end{align*}


\begin{align*} { \mathcal { E } } = \frac { 1 } { 2 } \int _ { \mathbb { R } ^ { 3 } } \left ( | v | ^ { 2 } + | b | ^ { 2 } \right ) \dd x , { \mathcal { H } } = 2 \int _ { \mathbb { R } ^ { 3 } } ( \phi - \phi _ { \infty } ) \frac { G } { r ^ { 2 } } \dd x . \end{align*}


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


\begin{align*} g ^ * f _ ! = f ' _ ! g '^ * \end{align*}


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


\begin{align*} g _ \xi ( z ) = \begin{cases} 0 & z \in U ^ c , \\ \frac { g ( z ) } { \xi - z } & z \in U \end{cases} \end{align*}


\begin{align*} T _ w : = T _ { s _ 1 } T _ { s _ 2 } \ldots T _ { s _ { \ell ( w ) } } , \end{align*}


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


\begin{align*} \begin{aligned} | x ( t ) - z ( t ) | \le & | x ( 0 ) - z ( 0 ) | e ^ { \gamma L _ r t } \\ & + \left ( e ^ { \gamma L _ r t } - 1 \right ) \dfrac { \gamma \bar k ( L _ r , M _ r , \delta ) } { L _ r } . \end{aligned} \end{align*}


\begin{align*} Z _ j \leq \frac { D ' } { Q _ 1 } + 1 0 \sqrt { \frac { D ' } { Q _ 1 ^ 2 Q _ 2 ^ 2 } } \sqrt { \log ( k ) } = \frac { 1 } { Q _ 1 } \left ( D ' + 1 0 \frac { \sqrt { D ' \log k } } { Q _ 2 } \right ) . \end{align*}


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


\begin{align*} M ( A _ 1 , \dots , A _ m ) = \binom { n } { m } \det ( A _ 1 , \dots , A _ m , I , \dots , I ) . \end{align*}


\begin{align*} & \int _ { X _ j } | F _ { \epsilon , j } - ( 1 - v ' _ { \epsilon } ( \Psi ) ) f F ^ { 1 + \delta } | ^ 2 _ h e ^ { v _ \epsilon ( \Psi ) - \delta \max \{ \psi + T , 2 \log | F | \} } c ( - v _ \epsilon ( \Psi ) ) \\ \le & \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } < + \infty . \end{align*}


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


\begin{align*} \gamma ^ i ( x ) = \frac 1 { N - 1 } \sum _ { j \neq i } K ( x ^ i - x ^ j ) . \end{align*}


\begin{align*} X _ \ell = m _ { \ell ; d } + \sum _ { q \geq 2 } \frac { b _ q } { q ! } \int _ { \mathbb S ^ d } H _ q ( T _ \ell ( x ) ) d x , \end{align*}


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


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


\begin{align*} 2 \widetilde y _ n & = 2 e ^ { r n ( A + a ) + r A } \\ & > ( e ^ { - r A } + e ^ { r a } ) e ^ { r n ( A + a ) + r A } \\ & = e ^ { r n ( A + a ) } + e ^ { r ( n + 1 ) ( A + a ) } \\ & = \widetilde x _ n + \widetilde x _ { n + 1 } . \end{align*}


\begin{align*} u = X \cdot \mathbf { e } , \nu ^ { n + 1 } = \mathbf { e } \cdot \nu , \end{align*}


\begin{align*} { g _ i } \left ( { { { \bf { x } } _ b } , t } \right ) = g _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { x } } _ b } , t } \right ) + g _ i ^ { \left ( { n e q } \right ) } \left ( { { { \bf { x } } _ b } , t } \right ) , \end{align*}


\begin{align*} \Lambda _ { g } = A \Bigl ( - \frac { \partial } { \partial x _ n } \Bigr ) - D , \end{align*}


\begin{align*} F _ 1 = \frac { 1 } { \sqrt { | c _ 1 | } } \ \langle \langle \nu _ 1 \cdot \varphi , \varphi \rangle \rangle . \end{align*}


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


\begin{align*} \hat { \mathbf x } _ 1 ( \theta _ \xi ) = \begin{bmatrix} 0 \\ \sin \theta _ \xi \\ \cos \theta _ \xi \end{bmatrix} , \hat { \mathbf x } _ 2 ( \theta _ \xi ) = \begin{bmatrix} \sin \theta _ \xi \\ 0 \\ \cos \theta _ \xi \end{bmatrix} , \hat { \mathbf x } _ 3 ( \theta _ \xi ) = \begin{bmatrix} \cos \theta _ \xi \\ \sin \theta _ \xi \\ 0 \end{bmatrix} \end{align*}


\begin{align*} u _ E = U _ C ( x + B _ { \infty } t ) - B _ { \infty } , \end{align*}


\begin{align*} \nabla _ X \varphi = \frac { 1 } { 2 } \sqrt { c _ 1 } \ X _ 1 \cdot \nu _ 1 \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi \end{align*}


\begin{align*} H ( Q \ , | \ , P ) \le \inf \Big \{ H ( R \ , | \ , P ) : R \in \P ( \R ^ n ) , \ H ( R ) < \infty \Big \} = 0 , \end{align*}


\begin{align*} \begin{aligned} 0 < 2 \sigma _ 2 ( \lambda ( D ^ 2 u ) ) = \ , & 2 \sum _ { 1 \leq i < j \leq n } ( u _ { i i } u _ { j j } - u _ { i j } ^ 2 ) \\ = \ , & ( \Delta u ) ^ 2 - \sum _ i u _ { i i } ^ 2 - \sum _ { i \neq j } u _ { i j } ^ 2 . \end{aligned} \end{align*}


\begin{align*} \mathcal { M } W ( s ) \ll ( 1 + | s | ) ^ { - E } . \end{align*}


\begin{align*} \dot x = & \ , - \gamma \left ( y _ \delta ( { x } , t ) - \bar { y } \right ) \ , u ( t ) & & { x } ( 0 ) = { x } _ 0 \\ \dot { \bar { y } } = & \ , \gamma \left ( y _ \delta ( { x } , t ) - \bar { y } \right ) & & \bar { y } ( 0 ) = \bar { y } _ 0 \end{align*}


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


\begin{align*} \epsilon _ 2 ( \eta _ a , t ) = \int _ N ^ t h ( x _ a + \delta u ( \tau ) ) - \dfrac { a _ { 0 , \delta } ( x _ a ) } { 2 } \ , d \tau \end{align*}


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


\begin{align*} f ( A ) = \begin{cases} 1 , & s < - \frac d 2 , \\ ( \log A ) ^ \frac 1 2 , & s = - \frac d 2 , \\ A ^ { \frac { d } { 2 } + s } , & s > - \frac d 2 . \end{cases} \end{align*}


\begin{align*} p & : = \left ( 1 / 2 - 2 \epsilon - \epsilon ^ { 1 + \alpha } + \epsilon ^ { 1 + \beta } - \epsilon ^ { 1 + \delta } , 1 / 2 + 2 \epsilon , \epsilon ^ { 1 + \alpha } - \epsilon ^ { 1 + \beta } , \epsilon ^ { 1 + \delta } \right ) , \\ q & : = \left ( 1 / 2 , 1 / 2 - \epsilon ^ { 1 + \alpha } , \epsilon ^ { 1 + \alpha } , 0 \right ) , \end{align*}


\begin{align*} | t _ n | _ p = 1 . \end{align*}


\begin{align*} M ( S ) = \varinjlim _ i M ( S _ i ) . \end{align*}


\begin{align*} \widehat { A } ^ \Gamma = \{ \chi \in \widehat { A } \ : : \ : \gamma \chi = \chi \ , \forall \gamma \in \Gamma \} \end{align*}


\begin{align*} \forall \varepsilon > 0 , \lim _ { n \to \infty } \sum _ { m = 1 } ^ n \mathbb { E } _ { n , m } ' [ \| \chi _ { n , m } ' \| _ { L ^ 2 ( \mu ) } ^ 2 : \| \chi _ { n , m } ' \| _ { L ^ 2 ( \mu ) } > \varepsilon ] = 0 \end{align*}


\begin{align*} ( i _ 1 \alpha _ 1 + i _ 2 \alpha _ 2 + \cdots + i _ n \alpha _ n ) ^ { p - 1 } = \sum _ { k _ 1 + \cdots + k _ n = p - 1 } \binom { p - 1 } { k _ 1 , \ldots , k _ n } i _ 1 ^ { k _ 1 } \cdots i _ n ^ { k _ n } \alpha _ 1 ^ { k _ 1 } \cdots \alpha _ n ^ { k _ n } \end{align*}


\begin{align*} E ( \theta ) _ J = \sum _ { w \in Z _ J } \Bbbk { \bf U } _ { w _ J w ^ { - 1 } } \dot { w } C ( \theta ) _ J . \end{align*}


\begin{align*} a ^ { \varepsilon } _ { t } + ( B \times u ) ^ { \varepsilon } + \nabla Q ^ { \varepsilon } = - \mu \nabla \times b ^ { \varepsilon } \textrm { o n } \ L ^ { 2 } ( \mathbb { R } ^ { 3 } ) . \end{align*}


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


\begin{align*} \frac { \partial } { \partial t } \Theta ( t , x ) = \int _ \S \int _ \S W _ r ( m _ 1 z + m _ 2 y + m _ 3 x ) ~ \sin ( n _ 1 \Theta ( t , z ) + n _ 2 \Theta ( t , y ) + n _ 3 \Theta ( t , x ) ) \ \d y \d z , \end{align*}


\begin{align*} \rho ^ { \lambda } _ { k } ( f ) e ^ { n - 1 } _ { k , \lambda } ( z , t ) = e ^ { i \lambda t } f ^ { - \lambda } \ast _ { - \lambda } \varphi _ { k , \lambda } ^ { n - 1 } ( z ) . \end{align*}


\begin{align*} h _ { K \oplus _ 2 L } ( x ) = \sqrt { h _ K ( x ) ^ 2 + h _ L ( x ) ^ 2 } \ge \sqrt { \rho _ K ( u ) ^ 2 + \rho _ L ( u ) ^ 2 } | \langle x , u \rangle | = \sqrt { \rho _ K ( u ) ^ 2 + \rho _ L ( u ) ^ 2 } h _ { [ - u , u ] } ( x ) . \end{align*}


\begin{align*} \frac { \partial h } { \partial t } \geq \frac { 1 } { f _ { \max } } h [ h ^ { - \varepsilon } \beta _ { 1 } - f _ { \max } ] . \end{align*}


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


\begin{align*} \left | \Pi _ V e ^ { | z | J ^ \perp t } \tilde { Y } _ 0 \right | ^ 2 & = \left | e ^ { | z | J ^ \perp t } \Pi _ { V } \tilde { Y } _ 0 \right | ^ 2 = e ^ { 2 | z | \lambda _ R t } \left | \Pi _ { V } \tilde { Y } _ 0 \right | ^ 2 . \end{align*}


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


\begin{align*} t / v = \mu ( \infty ) \textrm { i s a z e r o o f $ f $ } \iff \infty \textrm { i s a z e r o o f } f ' \end{align*}


\begin{align*} ( f \oplus g ) ( x ) = \inf _ { \substack { x _ 1 + x _ 2 = x \\ x _ 1 \in X _ 2 \\ x _ 2 \in X _ 2 } } \big ( f ( x _ 1 ) + g ( x _ 2 ) \big ) . \end{align*}


\begin{align*} | | B _ q ( \psi _ m ) | | _ { M L _ q ( \mathbb { C } ) } = 1 = | | \psi _ m | | _ { L ^ 2 ( \mathbb { R } ) } . \end{align*}


\begin{align*} X ( q , r ) & = - \frac { 1 } { 4 q } f ( q r ) + \frac { h ( q r ) } { 4 g ( q r ) } \left ( - 4 r + \frac { 1 } { 2 q } f ( 2 q r ) \right ) \\ & = \frac { 1 } { q } \iota ( q r ) , \end{align*}


\begin{align*} [ V _ f ] = \prod _ { i = 0 } ^ r e ^ { f ( x _ i ) \omega _ i } \frac { 1 } { D } . \end{align*}


\begin{align*} \overline { T } _ { n } [ { \bf \Lambda } ] = \lambda S _ n [ 1 ] , \end{align*}


\begin{align*} { \rm E E } _ { v , j } = \frac { { \overline { \rho } _ { v , j } B _ { \rm V L C } \log _ 2 \left ( { 1 + { \frac { \exp ( 1 ) } { 2 \pi } } \frac { { \overline P } _ v { \left ( { { R _ { \rm { P D } } } { \overline { G } _ { v , j } } } \right ) } ^ 2 } { { \sum \limits _ { v ' \ne v } { \overline P } _ { v ' } { \left ( { { R _ { \rm { P D } } } { \overline { G } _ { v ' , j } } } \right ) } ^ 2 } + N _ { \rm V L C } B _ { \rm V L C } } } \right ) } } { P _ { \rm V L C } + P _ v } . \end{align*}


\begin{align*} \d | X _ t ^ i | ^ 2 = 2 X _ t ^ i \cdot ( b ^ i ( X _ t ^ i ) + \sigma \d W _ t ^ i ) + d \sigma ^ 2 \d t . \end{align*}


\begin{align*} D ( A ' ) \coloneqq \{ x ' \in ( X , \upsilon ) ' \ ; | \ ; \exists \ ; y ' \in ( X , \upsilon ) ' \ ; \forall \ ; x \in D ( A ) : \ ; \langle A x , x ' \rangle = \langle x , y ' \rangle \} \end{align*}


\begin{align*} \chi ( \tilde Z _ { \mathbf e ; r } ( M ) ) = \sum _ { j \in J } \chi ( \pi ^ { - 1 } ( Z _ j ) ) = \sum _ { j \in J } \binom { m _ k - r } { e _ k - r } \chi ( Z _ j ) = \binom { m _ k - r } { e _ k - r } \chi ( Z _ { \mathbf e ' ; r } ( M ) ) . \end{align*}


\begin{align*} \gamma _ 0 J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \ge \gamma _ 0 J ( x ^ * ) \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 3 } } \left ( \nabla \phi \cdot \nabla \tilde { \phi } + G \tilde { G } \right ) \frac { 1 } { r ^ { 2 } } \dd x = \mu \int _ { \mathbb { R } ^ { 3 } } \left ( \tilde { \phi } 1 _ { ( 0 , \infty ) } ( \phi - \phi _ { \infty } ) G + ( \phi - \phi _ { \infty } ) _ { + } \tilde { G } \right ) \frac { 1 } { r ^ { 2 } } \dd x . \end{align*}


\begin{align*} \chi ( \tilde X _ { \mathbf e ; s } ( M ' ) ) = \binom { s } { e _ k } \chi ( X _ { \mathbf e ' ; s } ( M ' ) ) . \end{align*}


\begin{align*} v = X ( u ) = g ( \nabla u , X ) . \end{align*}


\begin{align*} r _ 1 ( z ) : = - \frac { b ( k ) } { 2 i k a ( k ) } , r _ 2 ( z ) : = \frac { 2 i k b ( k ) } { a ( k ) } , z \in \mathbb { R } , \end{align*}


\begin{align*} S _ 2 : 2 + 3 X ^ { p - 1 } Y = 0 \end{align*}


\begin{align*} F ^ * x = ( x _ \epsilon : \epsilon \ne \vec 0 ) \end{align*}


\begin{align*} \underline { h } _ { \mu } ( f _ { 1 , \infty } , x ) : = \lim _ { \epsilon \to 0 } \liminf _ { n \to \infty } \dfrac { - \log \mu ( B _ { n } ( x , \epsilon ) ) } { n } . \end{align*}


\begin{align*} X _ i \left ( t , x _ { 0 } , t _ { 0 } \right ) : = \lambda _ i ( t - t _ { 0 } ) + x _ { 0 } , t \in [ 0 , T ] . \end{align*}


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


\begin{align*} \kappa ^ { n } _ m ( t ) : = \sum _ { i = m } ^ { 2 m } i \psi ^ { n } _ i + 2 m \sum _ { i = 2 m + 1 } ^ { n } \psi ^ { n } _ i , \end{align*}


\begin{align*} d F ( X ) = \langle \langle X \cdot \varphi , \varphi \rangle \rangle , \end{align*}


\begin{align*} G ( x ; \theta ) = \exp \left ( - \frac { \theta } { x ^ 2 } \right ) , x > 0 , \theta > 0 . \end{align*}


\begin{align*} \begin{aligned} I _ 3 ' ( x ) = 4 \pi ^ { - 1 } \int _ { \mathbb { R } } z r _ 2 ( z ) e ^ { 2 i z x } d z = 4 \pi ^ { - 2 } \widehat { z r _ 2 ( z ) } ( - 2 x ) , \end{aligned} \end{align*}


\begin{align*} \mathbb { E } _ { \xi _ x ^ k } \langle \nabla f _ { \delta } ( x _ g ^ k , \xi _ x ^ k ) - \nabla f ( x ^ * ) , x ^ { k + 1 } - x ^ * \rangle & = \mathbb { E } _ { \xi _ x ^ k } \langle \nabla f _ { \delta } ( x _ g ^ k , \xi _ x ^ k ) - \nabla f _ { \delta } ( x _ g ^ k ) , x ^ { k + 1 } - x ^ * \rangle \\ & + \langle \nabla f _ { \delta } ( x _ g ^ k ) - \nabla f ( x ^ * ) , x ^ { k + 1 } - x ^ * \rangle . \end{align*}


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


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


\begin{align*} { \rm V o l } ( \tilde A \setminus { \tilde \Sigma } ^ { r } ) & \geq { \rm V o l } ( A \cap ( \tilde A \setminus { \tilde \Sigma } ^ { r } ) \ , ) = { \rm V o l } ( A ) - { \rm V o l } ( A \cap ( \tilde B \cup { \tilde \Sigma } ^ { r } ) ) \\ & \geq { \rm V o l } ( A ) - { \rm V o l } ( { \tilde \Sigma } ^ { r } ) - \sum _ { i = s + 1 } ^ m { \rm V o l } ( A \cap B _ { p _ i } ( r ) ) . \\ \end{align*}


\begin{align*} \left \{ \begin{aligned} d \varphi _ t ^ \xi = & - [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \varphi _ t ^ \xi + P _ t f ( \nu _ t ) + P _ t b ( \mu _ t ) \\ & + Q l ( \nu _ t ) - P _ t B h ( \mu _ t ) ] d t + \Lambda _ t ^ { 0 , \xi } d W ^ 0 _ t , \\ \varphi _ T ^ \xi = & ~ G g ( \nu _ T ) . \end{aligned} \right . \end{align*}


\begin{align*} u \ , = \ , F \big ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , \dots , x _ n \big ) , \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } l _ { n } ( x | \beta ) l _ { m } ( x | \beta ) f _ { g } ( x | \beta ) d x = \delta _ { n , m } . \end{align*}


\begin{align*} \begin{cases} \Delta _ \Psi u = f ( u ) & A ( r _ { 1 } , r _ { 2 } ) \\ u \equiv c _ { 1 } & \{ r _ { 1 } \} \times N \\ u \equiv c _ { 2 } & \{ r _ { 2 } \} \times N . \end{cases} \end{align*}


\begin{align*} v _ \epsilon { '' } ( t ) = \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ( t ) , \end{align*}


\begin{align*} \lambda _ { m - l + 1 } + \mu _ { 1 } = l - 1 . \end{align*}


\begin{align*} \Bigg | \dfrac { \partial \epsilon _ 1 ( \eta , t ) } { \partial \eta } \Bigg | _ { \eta = \eta _ a } \dot { \eta } _ a \Bigg | \le \gamma \delta L _ r ^ 2 3 \end{align*}


\begin{align*} \begin{aligned} \Xi = \{ \gamma | \ , \Phi _ 1 ( \gamma ) \cap \Phi _ 2 ( \gamma ) \neq \emptyset , - 3 < \gamma \leq 1 \} \ , , \end{aligned} \end{align*}


\begin{align*} \sigma ( V W ) = \mathfrak v _ { 2 } + \mathfrak v _ { 1 } = - V ^ { a } W ^ { b } \xi _ { a } \xi _ { b } + i V ^ { a } \delta _ { a } ( W ^ { b } ) \xi _ { b } . \end{align*}


\begin{align*} \Psi _ x + i z \left [ \sigma _ 3 , \Psi \right ] = \widetilde { Q } \Psi , \end{align*}


\begin{align*} \mathcal { R } = \left ( N \left ( \frac { b } { c t s } + \frac { b } { a s d } + \frac { 1 } { t d } \right ) \right ) ^ { - 1 } . \end{align*}


\begin{align*} h ^ { q - 1 } = \frac { r + 1 } { r } = 1 + \frac { 1 } { r } . \end{align*}


\begin{align*} \rho _ - \left \| Q _ { j , - } \right \| _ { L ^ 2 } ^ 2 \leq \operatorname { R e } \int _ { \mathbb { R } } Q _ { j , - } ( I + J ) Q _ { j , - } ^ H d z = \operatorname { R e } \int _ { \mathbb { R } } \mathcal { P } ^ - ( D _ j ) Q _ { j , - } ^ H d z \leq \| D _ j \| _ { L ^ 2 } \left \| Q _ { j , + } \right \| _ { L ^ 2 } . \end{align*}


\begin{align*} V i r = \mathbb { C } [ \partial ] L , [ L _ \lambda L ] = ( \partial + 2 \lambda ) L . \end{align*}


\begin{align*} a _ k ( x ) f _ k '' ( x ) + b _ k ( x ) f _ k ' ( x ) + p _ k c _ k ( x ) f _ k ( x ) = 0 , \end{align*}


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


\begin{align*} S _ \sigma D = \sigma ( D ) , \end{align*}


\begin{align*} T x : = [ ( f _ n ( x ) ) _ n ] \ \ \ \ ( \forall x \in E ) . \end{align*}


\begin{align*} \pi ^ G _ k ( X ) = \begin{cases} \displaystyle \operatorname * { c o l i m } _ { V \in s ( \mathcal { U } _ G ) } \ ; \ ; [ S ^ { k + V } , X ( V ) ] _ * ^ G & \mathrm { f o r } \ ; \ ; k \geq 0 \\ \displaystyle \operatorname * { c o l i m } _ { { V \in s ( \mathcal { U } _ G ) } } \ ; \ ; [ S ^ { k + V } , X ( \mathbb { R } ^ { - k } \oplus V ) ] _ * ^ G & \mathrm { f o r } \ ; \ ; k \leq 0 \end{cases} \end{align*}


\begin{align*} ( 1 . 4 4 ) ^ { 2 } \cdot \frac { 3 2 \pi ^ { 8 } } { 9 p ^ { 3 } } \sum _ { n = a p } ^ { b p } n \log ( n + 2 ) \sum _ { d = 1 } ^ { \infty } \psi \left ( d \sqrt { \frac { n } { p } } \right ) \leq \frac { 6 9 9 5 8 } { p ^ { 3 } } \cdot \left [ b p \log ( b p + 2 ) \right ] ( b - a ) p \sum _ { d = 1 } ^ { \infty } \psi ( d \sqrt { a } ) . \end{align*}


\begin{align*} W ( 1 ^ \infty | 1 ^ \infty ) = c - d + \sum _ { n = 2 } ^ { \infty } ( c - c _ n ) \end{align*}


\begin{align*} \frac { d ^ 2 \ , } { d \ , \tau ^ 2 } W ( \mathbf { e } + \tau \ , \xi \otimes \eta , \boldsymbol { \mathfrak { K } } + \tau \ , \zeta \otimes \eta ) \bigg | _ { \tau = 0 } > 0 \forall \ \eta , \xi , \zeta \in \mathbb { R } ^ 3 , \ \ \lVert \eta \rVert = \lVert \xi \rVert = \lVert \zeta \rVert = 1 . \end{align*}


\begin{align*} \begin{aligned} L ^ \Phi ( \theta ) [ \cdot ] : = & - \big ( { \rm d i v } _ { \ ! z } + 2 i \pi \theta \big ) \big ( A { ( \Phi ^ { - 1 } ( z , \omega ) , \omega ) } { \big ( \nabla _ { \ ! \ ! z } + 2 i \pi \theta \big ) } [ \cdot ] \big ) \\ & + V { \big ( \Phi ^ { - 1 } \left ( z , \omega \right ) , \omega \big ) } [ \cdot ] . \end{aligned} \end{align*}


\begin{align*} D ( M _ { q } ) = \{ f \in \ell ^ { \infty } \ ; | \ ; q f \in \ell ^ { \infty } \} \end{align*}


\begin{align*} p _ i = \begin{cases} 0 . 5 \tilde { q } _ i ( 1 + \tilde { \delta } _ i ) , & i \in [ m ] \\ 0 . 5 \tilde { q } _ { i - m } ( 1 - \tilde { \delta } _ { i - m } ) , & i \in [ k ] \setminus [ m ] . \end{cases} \end{align*}


\begin{align*} \mathcal { E } _ { | _ { x \times C } } = \varphi ( x ) , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { m } \gamma _ { k } y _ { k } \ge 0 \ \ \forall \ y \in \Omega _ { 0 } \end{align*}


\begin{align*} \bar \pi \bar p _ T = \bar \pi . \end{align*}


\begin{align*} \rho c _ v \partial _ { t _ 1 } T = { \bar F } ^ { ( 1 ) } , \end{align*}


\begin{align*} \rho _ N = \sum _ { j = 1 } ^ { N } w _ { N , j } \delta _ { \theta _ { N , j } } \end{align*}


\begin{align*} ( r _ 0 , \ldots , r _ { \ell - 1 } ) = \big ( \ , \underbrace { s _ L , \ldots , s _ L } _ { m _ L } , \ldots , \underbrace { s _ 1 , \ldots , s _ 1 } _ { m _ 1 } \ , \big ) , 0 < s _ 1 < \cdots s _ L , m _ i > 0 , \ 1 \leq i \leq L . \end{align*}


\begin{align*} \eta ( K ) = 4 K ^ { \frac { 2 r - 2 } { 3 } } . \end{align*}


\begin{align*} \lceil x \rceil = & \lambda ( e , b ) \lambda ( b , b ^ 2 ) . . . \lambda ( b ^ { m - 1 } , b ^ m ) \\ \lceil x \rceil = & \lambda ( e , a ) \lambda ( a , a ^ 2 ) . . . \lambda ( a ^ { n - 1 } , a ^ n ) \\ \lceil x \rceil = & \lambda ( e , a ) \lambda ( a , a ^ 2 ) . . . \lambda ( a ^ n , a ^ n b ) . . . \lambda ( b ^ { m - 1 } , b ^ m ) \end{align*}


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


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


\begin{align*} \phi ( \alpha , \tau ) = \widetilde \alpha \widetilde \tau { \widetilde \alpha } ^ { - 1 } \cdot { \widetilde \tau } ^ { - 1 } = \alpha ( x _ { i _ 1 } \cdots x _ { i _ k } ) \cdot ( x _ { i _ 1 } \cdots x _ { i _ k } ) ^ { - 1 } = x _ { \alpha ( i _ 1 ) } \cdots x _ { \alpha ( i _ k ) } \cdot ( x _ { i _ 1 } \cdots x _ { i _ k } ) ^ { - 1 } = 1 . \end{align*}


\begin{align*} V _ p ( A [ n - 1 ] , B ) = \frac { 1 } { n } \int \limits _ { S ^ { n - 1 } } h ^ p _ B ( u ) h _ A ^ { 1 - p } d S _ A ( u ) . \end{align*}


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


\begin{align*} \nabla \times U = \mu 1 _ { ( 0 , \infty ) } ( \phi - \phi _ { \infty } ) U . \end{align*}


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


\begin{align*} \partial _ u \Gamma _ { v u } ^ u = 2 \Gamma _ { u v } ^ u \Gamma _ { v u } ^ v = \partial _ v \Gamma _ { u v } ^ v . \end{align*}


\begin{align*} \mathcal { I } & = \bigg \{ E \in \mathbb { C } : \log | \lambda | > \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | \bigg \} , \\ \mathcal { O } & = \bigg \{ E \in \mathbb { C } : \log | \lambda | < \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | \bigg \} . \end{align*}


\begin{align*} \begin{cases} \nabla W \cdot \nabla ^ \perp \left ( \varPhi - \frac { \alpha } { 2 } | x ' | ^ 2 | \ln \varepsilon | \right ) = 0 , \\ W = \mathcal { L } _ H \varPhi , \\ \varPhi | _ { \partial B _ { R ^ * } ( 0 ) } = 0 . \end{cases} \end{align*}


\begin{align*} \widehat { f } ( \xi ) : = \int _ G f ( x ) \ , \overline { \xi ( x ) } \ , d \mu ( x ) \end{align*}


\begin{align*} \sum _ { 1 \le i \le n } \partial _ { \theta _ i } \Delta _ C & = \sum _ { 1 \le r \le k } ( \partial _ { \theta _ { i _ r } } + \partial _ { \theta _ { j _ r } } ) \Delta _ C + \sum _ { i \notin \bigcup _ { r = 1 } ^ k C _ r } \partial _ { \theta _ i } \Delta _ C \\ & = \sum _ { 1 \le r \le k } ( \partial _ { \theta _ { i _ r } } + \partial _ { \theta _ { j _ r } } ) \Delta _ C = 0 \ , . \end{align*}


\begin{align*} X _ { i ; j k } = - R _ { \ell k j i } X ^ \ell + \tfrac { 1 } { 2 } ( h _ { i j ; k } + h _ { i k ; j } - h _ { j k ; i } ) . \end{align*}


\begin{align*} \tau = \tau ' _ 1 v _ 1 \cdots \tau ' _ z v _ z \end{align*}


\begin{align*} \beta = 2 i - 1 \end{align*}


\begin{align*} \pi ( u ) : = \sum _ { v \in V } \pi ( u , v ) > 0 ; A : = \sup _ { u \in V } \pi ( u ) < \infty . \end{align*}


\begin{align*} a ( u _ n , v _ n ) = ( p _ n f , v _ n ) v _ n \in X _ n . \end{align*}


\begin{align*} E = ( C _ i \setminus D ) \cup \{ c \} . \end{align*}


\begin{align*} \operatorname { r a n k } { F ( ( M _ { k , \mathrm { o u t } } ' ) ^ { - 1 } ( N _ { ( - , k ) } ) ) } = s , \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { h ( x ) } { 1 / x } = & \lim _ { x \to \infty } \frac { 2 \theta } { \operatorname { B } ( a , b ) } \frac { 1 } { x ^ 3 } \frac { \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } \exp \left ( - \frac { a \theta } { x ^ 2 } \right ) } { \operatorname { I } _ { 1 - \exp ( - \theta / x ^ 2 ) } ( b , a ) } x . \end{align*}


\begin{align*} ( M N T _ { 2 } ) _ { i j } = \left \{ \begin{array} { c c } { i + 2 j \choose 3 j + 1 } & i \geq j \geq 0 ; \\ 0 & i < j , \end{array} \right . \end{align*}


\begin{gather*} H = \{ \pi _ 1 \} \cup \bigl \{ \bigl ( \pi _ { i + 1 } - \pi _ i \bigr ) \ , g ( \pi _ 1 , \ldots , \pi _ i ) : 1 \le i < n , \ , \ , g \in B ( \mathbb { R } ^ i , \mathcal { B } ( \mathbb { R } ^ i ) ) \bigr \} ; \end{gather*}


\begin{align*} A _ t \cdot r \nabla \theta & = \frac { 1 } { r } \Phi _ t , \\ ( B \times u ) \cdot r \nabla \theta & = \frac { 1 } { r } u \cdot \nabla \Phi , \\ \nabla \times B \cdot r \nabla \theta & = - \frac { 1 } { r } \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi . \end{align*}


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


\begin{align*} f ( \kappa ) = \sigma , \partial \Sigma = \Gamma , \end{align*}


\begin{align*} \sigma _ { k } ^ { i i } = \frac { \partial \sigma _ k } { \partial \kappa _ i } , \sigma _ { k } ^ { i i , j j } = \frac { \partial ^ 2 \sigma _ { k } } { \partial \kappa _ i \partial \kappa _ j } . \end{align*}


\begin{align*} \overline { \pi } _ 1 ( q _ 1 ) \overline { \pi } _ 1 ( q _ 2 ) = c _ { \chi _ 1 } ( q _ 1 , q _ 2 ) \overline { \pi } _ 1 ( q _ 1 q _ 2 ) \forall q _ 1 , q _ 2 \in Q \end{align*}


\begin{align*} \mathcal { W } ' ( X ) = \sum _ { m = 0 } ^ n V _ m ' ( X ) \end{align*}


\begin{align*} \left | \sum _ { j = 1 } ^ 3 \frac { 1 } { z _ j ^ 2 } \right | \geq \frac { \sin ^ 3 \theta _ \xi } { 3 0 } > 0 . \end{align*}


\begin{align*} \xi _ { k , i } ( x ) = \left \{ \begin{array} { l l } 2 \sqrt { C _ { d } \kappa } \ , \theta _ { k } { \rm R e } ( \sigma _ { k , i } ( x ) ) , & k \in \Z _ { + } ^ d , \\ 2 \sqrt { C _ { d } \kappa } \ , \theta _ { k } { \rm I m } ( \sigma _ { k , i } ( x ) ) , & k \in \Z _ { - } ^ d , \end{array} \right . \end{align*}


\begin{align*} \overline { \{ T ^ n \pi ( g ) : n \in \Z \} } = g Y . \end{align*}


\begin{align*} A _ { \alpha , \beta , u , v } = \begin{pmatrix} \alpha & u \alpha \\ \beta & v \beta \end{pmatrix} \end{align*}


\begin{align*} B _ q ( \psi _ m ) ( z ) = \displaystyle \frac { z ^ m } { \sqrt { \Gamma ( q m + 1 ) } } , \end{align*}


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


\begin{gather*} \lim _ k \sup _ { Q \in \mathcal { U } } Q \bigl \{ \abs { f } \ , 1 ( \abs { f } > k ) \bigr \} = 0 \quad f \in F . \end{gather*}


\begin{align*} \int _ { T } \mathrm { d i v } ( { \boldsymbol \psi } ^ { \mathrm { R T } _ 1 } _ { j } ) \varphi _ k \ , d \boldsymbol x = \frac { 3 \delta _ { j k } - 1 } { 2 4 } j , k = 1 , 2 , 3 . \end{align*}


\begin{align*} M _ { I J } = \begin{pmatrix} G _ { I _ 1 } & G _ { I _ 2 } & \cdots & G _ { I _ s } \\ G _ { J _ 1 } & G _ { J _ 2 } & \cdots & G _ { J _ s } \end{pmatrix} . \end{align*}


\begin{align*} \hat { \mu } _ { A ^ * } = \hat { \mu } _ A \ . \end{align*}


\begin{align*} f ( n , \phi , \tilde { p } , \tilde { q } ) : = \P _ { ( x _ 1 , \dots , x _ n ) \sim \tilde { p } ^ { \otimes n } } ( \phi ( y _ 1 , \dots , y _ n ) \ne p ) + \P _ { ( x _ 1 , \dots , x _ n ) \sim \tilde { q } ^ { \otimes n } } ( \phi ( y _ 1 , \dots , y _ n ) \ne q ) . \end{align*}


\begin{align*} V ^ { - 1 } \rho \circ \varphi V = V ^ { - 1 } \varphi \circ \rho V = V ^ { - 1 } \rho V . \end{align*}


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


\begin{align*} 0 = H _ i = ( \Delta u ) _ i + B x _ i \end{align*}


\begin{align*} - \Phi ^ * ( x , u ) = { \sup } \{ f \in U S C ( \overline { \Omega } ) : f \in F ( \Omega ) f ( \tau ) \le - { \phi ^ * _ \tau } ( u ) \tau \in \partial \Omega \} \end{align*}


\begin{align*} \frac { i } { ( 2 \pi ) ^ { n } } \int _ { \mathbb { R } ^ { n - 1 } } \int _ { \mathcal { C } } e ^ { - t \tau } \operatorname { T r } \phi _ { - 1 } \ , d \tau \ , d \xi = a _ 0 ( x ) t ^ { 1 - n } , \end{align*}


\begin{align*} \mathcal { W } \ , ( \Delta ^ { - m } ) = v _ { n - 1 } \ , v o l ( M ) , \end{align*}


\begin{align*} \phi \left ( \begin{pmatrix} 1 & u \\ 0 & 1 \end{pmatrix} \begin{pmatrix} t & 0 \\ 0 & \frac { 1 } { t } \end{pmatrix} \right ) = 2 t \sum _ { n = 1 } ^ \infty \rho _ \phi ( n ) K _ { \nu } ( 2 \pi n t ^ 2 ) \cos ( 2 \pi n u ) . \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) - x _ a ( t ) \| & \ , \le \| x ( t ) - z ( t ) \| + \| z ( t ) - x _ a ( t ) \| \\ & \ , \le \| x ( t ) - z ( t ) \| + \gamma \| \epsilon ( x _ a ( t ) , t ) \| \\ & \ , \le \| x ( t ) - z ( t ) \| + \gamma 2 M _ \mathcal { D } T . \end{aligned} \end{align*}


\begin{align*} \frac { \partial \tilde x } { \partial y } = x \cdot \frac { 1 } { 3 } h ^ { - \frac { 2 } { 3 } } \cdot h _ y ' , \end{align*}


\begin{align*} { { \bf { F } } _ { { \mathop { \rm i n t } } } } = - G \psi \left ( { \bf { x } } \right ) \sum \limits _ { i = 1 } ^ { 1 8 } { \varpi \left ( { { { \left | { { { \bf { c } } _ i } } \right | } ^ 2 } } \right ) } \psi \left ( { { \bf { x } } + { { \bf { c } } _ i } \Delta t } \right ) { { \bf { c } } _ i } , \end{align*}


\begin{align*} \phi ( m , \ell ) = \sum _ { \widetilde K \ \ \pi ^ { - 1 } ( K ) } \phi ( m , \ell _ { \widetilde K } ) \equiv \sum _ { \widetilde K \ \ \pi ^ { - 1 } ( K ) } ( e _ { \widetilde K } - 1 ) d _ { \widetilde K } \mod 2 . \end{align*}


\begin{align*} \mathcal { D } + \mathcal { E } = 0 . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \partial _ t f + \mathcal { L } ( f ) = \Gamma ( f , f ) , \\ & f | _ { t = 0 } = f _ 0 , \end{aligned} \right . \end{align*}


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


\begin{align*} \begin{aligned} & \int _ { \Omega } \Delta u _ { \epsilon } ( x ) \wedge \Delta v _ 1 ( x ) \wedge \dots \wedge \Delta v _ { m - 1 } ( x ) \wedge \beta _ n ^ { n - m } \wedge \omega ( x ) \\ = & \int _ { B ( 0 , \epsilon ) } \chi _ { \epsilon } ( y ) d V ( y ) \int _ { \Omega } u ( z ) \Delta v _ 1 ( z + y ) \wedge \dots \wedge \Delta v _ { m - 1 } ( z + y ) \wedge \beta _ n ^ { n - m } \wedge \Delta \omega ( z + y ) \geq 0 , \end{aligned} \end{align*}


\begin{align*} \int _ \Omega h d _ \alpha T = - \int _ \Omega d _ \alpha h \wedge T + \int _ { \partial \Omega } \sum _ { A = 0 } ^ { 2 n - 1 } h \ , T _ A \ , \tau ( \mathbf { { n } } ) _ { A \alpha } \ , d S , \alpha = 0 , 1 , \end{align*}


\begin{align*} \Delta _ c ^ + = \{ e _ i - e _ j | 1 \leq i < j \leq m \} \cup \{ f _ i \pm f _ j | 1 \leq i < j \leq n \} \cup \{ f _ j | 1 \leq j \leq n \} , \end{align*}


\begin{align*} ( \ell ^ { \infty } , \beta _ { 0 } ) ' = ( \ell ^ { \infty } , \mu ( \ell ^ { \infty } , \ell ^ { 1 } ) ) ' = \ell ^ { 1 } . \end{align*}


\begin{align*} \lim _ { k \to \infty } \sup _ { z _ 0 \in \mathbb { R } } \int _ { D ( z _ 0 , R ) } \rho _ { n _ { k } } \dd z \dd r = 0 . \end{align*}


\begin{align*} ( \mathbf { u } G _ 2 , \mathbf { u } G _ 3 , \cdots , \mathbf { u } G _ { 2 k } ) = ( \mathbf { v } G _ 2 , \mathbf { v } G _ 3 , \cdots , \mathbf { v } G _ { 2 k } ) = ( 1 , 1 , \cdots , 1 ) , \end{align*}


\begin{align*} P ( x _ 1 , \ldots , x _ n ) = x _ 2 ^ { q + 1 } - x _ 1 ^ q x _ 3 . \end{align*}


\begin{align*} \beta = 3 \left ( 2 i - 1 \right ) \end{align*}


\begin{align*} { \bf h } | _ { \partial \bf N } & = \pm 2 \bar u v d t ^ 2 + h ^ \intercal \\ A ' | _ { \bar { \bf g } } ( \bf h ) & = \pm \big ( v \nu ( \bar u ) + \bar u ( \nu ( u ) ) ' \big ) d t ^ 2 + A ' | _ g ( h ) . \end{align*}


\begin{align*} \alpha : = \Delta v _ 1 \wedge \dots \wedge \Delta v _ { m - 1 } \wedge \beta _ n ^ { n - m } . \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \mathsf { U } ^ n ) = W ( \xi _ 0 , n , x , \mathsf { V } ^ n ) . \end{align*}


\begin{align*} \partial _ L f ( x ) & = \{ l \in L : \forall y \in X , f ( y ) \geq f ( x ) + l ( y ) - l ( x ) \} \\ & = \{ l \in L : \forall y \in X , f ( y ) \geq f ( x ) + l _ x ( y ) - l _ x ( x ) \} \\ & = \{ l \in L : l - l ( x ) \in \partial _ { L _ x } f ( x ) \} \end{align*}


\begin{align*} N ( x ; k ) : = \begin{cases} e ^ { i c _ + ( x ) \sigma _ 3 } \left ( \frac { \psi _ 1 ^ - ( x ; k ) } { a ( k ) } , \psi _ 2 ^ + ( x ; k ) \right ) , \ ; k \in D ^ + , \\ [ 5 p t ] e ^ { i c _ + ( x ) \sigma _ 3 } \left ( \psi _ 1 ^ + ( x ; k ) , \frac { \psi _ 2 ^ - ( x ; k ) } { \overline { a ( \bar { k } ) } } \right ) , \ ; k \in D ^ - , \end{cases} \end{align*}


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


\begin{align*} g _ \mu : = \arg \max _ { \psi \in \mathcal { H } } \left \{ - \frac { \mathrm { d } } { \mathrm { d } \gamma } \mathrm { K L } \left ( T _ { \gamma } { \# } \mu \| \pi \right ) \Big | _ { \gamma = 0 } \quad \| \psi \| _ { \mathcal { H } } \leq 1 \right \} . \end{align*}


\begin{align*} x = q _ i + p _ i \end{align*}


\begin{align*} A = \begin{pmatrix} \sigma & 0 \\ 0 & 1 _ { 2 ( n - p ) } \end{pmatrix} \end{align*}


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


\begin{align*} \tau ( x ) \omega : = \varphi ( x ) \ , \omega , \end{align*}


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


\begin{align*} E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] = E _ { \xi _ 0 } \left [ h \bigl ( x , Z _ 1 ^ \theta \bigr ) \right ] , \forall x \in \mathbb R , \end{align*}


\begin{align*} \# R ^ i _ { n _ 0 ; l _ 1 , \cdots , l _ { n _ 0 } } = \underset { \tilde { l } = ( \tilde { l } _ { i } , \tilde { l } _ { i + 1 } , \cdots , \tilde { l } _ { n _ 0 } ) \in L } { \sum } c _ { \tilde { l } } \# P _ { l _ 1 , l _ 2 , \cdots , l _ { i - 1 } , \tilde { l } _ i , \tilde { l } _ { i + 1 } , \cdots , \tilde { l } _ { n _ 0 } } \end{align*}


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


\begin{align*} H _ { m i n } ( E \otimes ( F _ 1 [ t ] \times F _ 1 [ t ] ^ { \vee } ) ) & = H _ { m i n } ( E \otimes F _ 1 [ t ] \times E \otimes F _ 1 [ t ] ^ { \vee } ) \\ & = \min H _ { m i n } ( E \otimes F _ 1 [ t ] , E \otimes F _ 1 [ t ] ^ { \vee } ) \\ & \leq H _ { m i n } ( E \otimes F _ 1 [ t ] ) = t H _ { m i n } ( E \otimes F _ 1 ) \leq t H _ { m i n } ( E ) H _ { m i n } ( F _ 1 ) \end{align*}


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


\begin{align*} \tau ' _ y ( i ) = \begin{cases} \tau ' ( i ) & i \in I _ y \\ i & i \notin I _ y . \end{cases} \end{align*}


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


\begin{align*} | A | \ , | B _ 1 + \dots + B _ m | \le \prod \limits _ { i = 1 } ^ m | A + B _ i | , \end{align*}


\begin{align*} V ( r ) = \frac { L ( L + 1 ) } { r ^ 2 } - \frac { Q } { r } f + \kappa B _ 1 \frac { r } { f } + \kappa B _ 2 \frac { 1 } { f ^ 2 } + \kappa B _ 3 \frac { r } { f ^ 3 } + \kappa B _ 4 \frac { 1 } { f ^ 4 } , B _ 4 > 0 , \end{align*}


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


\begin{align*} N : = \mathcal N ( F ^ q _ v ( 0 , p _ 0 ) ) = \operatorname { s p a n } \{ u _ \ell , w _ \ell \} . \end{align*}


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


\begin{align*} \left [ A ( y _ 1 x ) - A ( y _ 1 x ' ) \right ] + \sum _ { n = 2 } ^ \infty \left [ A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) \right ] \end{align*}


\begin{align*} \frac { \partial } { \partial t } u ^ h ( t , x ) = \frac { \partial } { \partial t } \left ( \sum _ i w _ i ( t ) \sigma _ i ( x ) \right ) = \sum _ i \sigma _ i ( x ) \frac { d } { d t } w _ i ( t ) , \end{align*}


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


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


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


\begin{align*} \lim _ { j \to \infty } \int _ { \mathbb { R } ^ { 3 } } \Phi _ { j , + } \frac { G _ j } { r ^ { 2 } } \dd x = \int _ { \mathbb { R } ^ { 3 } } \Phi _ { 0 , + } \frac { G _ 0 } { r ^ { 2 } } \dd x \textrm { u n i f o r m l y i n } \ [ 0 , T ] . \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { h ( x ) } { 1 / x } = & \frac { 2 \theta } { \operatorname { B } ( a , b ) } \lim _ { x \to \infty } \frac { \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } / x ^ 2 } { \operatorname { I } _ { 1 - \exp ( - \theta / x ^ 2 ) } ( b , a ) } . \end{align*}


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


\begin{align*} \begin{aligned} { \mathfrak a } _ { 2 } = & \bigl ( \delta _ { a b } + \frac { 1 } { 3 } R _ { a c b d } x ^ { c } x ^ { d } \bigr ) \xi _ { a } \xi _ { b } + o ( { \mathbf { x ^ { 2 } } } ) , \\ { \mathfrak a } _ { 1 } = & \frac { 2 i } { 3 } \mathrm { R i c } _ { a b } x ^ { a } \xi _ { b } + o ( { \mathbf { x ^ { 2 } } } ) . \end{aligned} \end{align*}


\begin{align*} g _ s ( i ) : = \frac { c ' } { 2 \sqrt { m _ s } } 1 6 ^ { - i } m _ s n ^ { - \alpha } , \end{align*}


\begin{align*} f ( z ) - f ( 0 ) & = \int _ { 0 } ^ 1 \left ( \frac { \partial f } { \partial x } ( t z ) x - \frac { \psi _ 0 } { \psi _ 1 } y \frac { \partial f } { \partial x } ( t z ) \right ) d t \\ & = \int _ { 0 } ^ 1 \left ( x - \frac { \psi _ 0 } { \psi _ 1 } y \right ) \frac { \partial f } { \partial x } ( t z ) d t = \left ( x - \frac { \psi _ 0 } { \psi _ 1 } y \right ) h _ 1 ( z ) , \\ \end{align*}


\begin{align*} \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ j \cdot B _ j \dd x + 2 \mu _ j \int _ { 0 } ^ { t } \int _ { \Omega } \nabla \times B _ j \cdot B _ j \dd x \dd s = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ { 0 } \cdot B _ 0 \dd x , \end{align*}


\begin{align*} v _ \epsilon ( t ) : = & \int _ { - \infty } ^ { t } ( \int _ { - \infty } ^ { t _ 1 } ( \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ) ( s ) d s ) d t _ 1 \\ & - \int _ { - \infty } ^ { - t _ 0 } ( \int _ { - \infty } ^ { t _ 1 } ( \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ) ( s ) d s ) d t _ 1 - t _ 0 , \end{align*}


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


\begin{align*} p ^ { C _ 5 } = e . \end{align*}


\begin{align*} D ( A ' ) \coloneqq \{ x ' \in ( X , \upsilon ) ' \ ; | \ ; \exists \ ; y ' \in ( X , \upsilon ) ' \ ; \forall \ ; x \in D ( A ) : \ ; \langle A x , x ' \rangle = \langle x , y ' \rangle \} \end{align*}


\begin{align*} W _ { \rm { m p } } ( \overline { \mathbf { U } } ) = 0 \Rightarrow \boldsymbol { \alpha } = 0 ( \boldsymbol { \mathfrak { K } } = 0 ) . \end{align*}


\begin{align*} \frac { 1 } { 2 } \int _ { \mathbb { R } ^ { 3 } } \left ( | v | ^ { 2 } + | b | ^ { 2 } \right ) \dd x + \int _ { \delta } ^ { t } \int _ { \mathbb { R } ^ { 3 } } \left ( \nu | \nabla v | ^ { 2 } + \mu | \nabla b | ^ { 2 } \right ) \dd x \dd s = \frac { 1 } { 2 } \int _ { \mathbb { R } ^ { 3 } } \left ( | v | ^ { 2 } + | b | ^ { 2 } \right ) ( \delta ) \dd x . \end{align*}


\begin{align*} \nabla \mathcal { N } ( x ) = W _ { L } \cdot \sigma ' ( N ^ { ( L - 1 ) } ( x ) ) \cdot W _ { L - 1 } \cdot \ldots \cdot \sigma ' ( N ^ { ( 1 ) } ( x ) ) \cdot W _ { 1 } , \end{align*}


\begin{align*} \begin{aligned} | x ( 2 \bar { t } / \gamma ) | _ { \mathcal { A } _ \delta } & \le \kappa \bar c _ 0 e ^ { - \gamma \lambda \bar { t } / \gamma } + \bar c \\ & \le \kappa \bar c _ 0 e ^ { - \lambda \bar { t } _ 2 } + \bar c < \bar c _ 0 . \end{aligned} \end{align*}


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


\begin{align*} \begin{aligned} \Big | \langle \boldsymbol { \nu } , \eta _ t \rangle - \bar \eta _ t + \langle \boldsymbol { \nu } , q _ { \alpha , x _ \alpha } \rangle - \bar q _ { \alpha , x _ \alpha } \Big | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; , \end{aligned} \end{align*}


\begin{align*} | s _ { h , q } - s _ h | \le \frac { d \theta ^ { h } } { \theta ^ { q } - 1 } ~ { \rm f o r } ~ h = 0 , 1 , \dots , q - 1 , \end{align*}


\begin{align*} E [ M _ v ( t ) ^ 2 ] = E \big [ [ M _ v ] ( t ) \big ] . \end{align*}


\begin{align*} W _ { 1 } \left ( \mu , \nu \right ) = \sup _ { \left \Vert f \right \Vert _ { L i p } = 1 } \int f \left ( d \mu - d \nu \right ) , \end{align*}


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


\begin{align*} ( x _ 1 , x _ 2 , \cdots , x _ k ) \bar { G } = { \bf 0 } , \end{align*}


\begin{align*} \overline { p ( z ) } = \sum _ { n = 0 } ^ k \overline { a } _ n \overline { z } ^ n = \sum _ { n = 0 } ^ k \overline { a } _ n z ^ { - n } = z ^ { - k } \sum _ { n = 0 } ^ \infty \overline { a } _ n z ^ { k - n } = z ^ { - k } q ( z ) . \end{align*}


\begin{align*} \begin{aligned} | \langle \boldsymbol { \nu } , G ( \lambda | \bar U , x , \tau ) \rangle | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; , \end{aligned} \end{align*}


\begin{align*} \max \bigg \lbrace \sum _ { i , j = 1 } ^ n & t _ i t _ j | U ^ \top U | _ { i j } : t \in \mathbb { R } ^ n , \ ; \| t \| _ 2 = 1 , U \in \mathbb { R } ^ { m \times n } , \ ; U U ^ \top = { \rm I } _ m \bigg \rbrace \\ & \le \frac { m } { n } \left ( 1 + \sqrt { \frac { ( n - 1 ) ( n - m ) } { m } } \right ) , \end{align*}


\begin{align*} h ( x , y ) \ge \frac { 1 } { 2 \pi } \ln \frac { 1 } { | x - y | + 2 \max \{ d i s t ( x , \partial \Omega ) , d i s t ( y , \partial \Omega ) \} } . \end{align*}


\begin{align*} \mathbb { E } _ { n , m } [ \chi _ { n , m } ( w ' , \cdot ) | O ( \chi _ { n , m } ' ( w ' , M , \cdot ) ) = O ( \chi _ { n , m } ' ( w , M , \cdot ) ) ] \\ = \mathbb { E } _ { n , m } [ \chi _ { n , m } ( w ' , \cdot ) | w , M ] \end{align*}


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


\begin{align*} U & = b + B _ { \infty } = \nabla \times ( ( \phi - \phi _ { \infty } ) \nabla \theta ) + G \nabla \theta , \\ \nabla \times U & = \nabla \times ( G \nabla \theta ) - L \phi \nabla \theta . \end{align*}


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


\begin{align*} \begin{aligned} \| \eta ( t ) & - \eta _ a ( t ) \| = \| \eta ( t ) - \eta _ a ( t ) \pm z ( t ) \| \\ \le & \| \eta ( t ) - z ( t ) \| + \| z ( t ) - \eta _ a ( t ) \| \\ \le & \| \eta ( t ) - z ( t ) \| + \gamma \| \epsilon ( \eta _ a ( t ) , t ) \| \\ \le & \| \eta ( t ) - z ( t ) \| + \gamma \bar { k } _ 3 ( L _ r , \delta ) . \end{aligned} \end{align*}


\begin{align*} r _ d \le d \Big | \frac { p _ 0 } { p _ 1 } \Big | = d \Big | \frac { p ( 0 ) } { p ' ( 0 ) } \Big | . \end{align*}


\begin{align*} f ( x ) = \sum _ { m = 0 } ^ \infty \frac { x ^ m } { \Gamma \bigl ( 1 + \frac { m } { 2 } \bigr ) m ! } , \end{align*}


\begin{align*} \beta ^ { q ^ n } + a _ { n - k } \beta ^ { q ^ { n - k } } + \cdots + a _ 0 \beta = 0 \end{align*}


\begin{align*} T _ \ell ( x ) = \int _ { \mathbb { S } ^ d } f _ { x ; \ell } ( y ) W ( d y ) , x \in \mathbb { S } ^ d , \end{align*}


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


\begin{align*} \gamma = \frac { \alpha ^ 2 \frac { K } { K - 2 } } { \sigma _ { W _ { \rm d } } ^ 2 + \sigma _ { W _ { \rm s } } ^ 2 } = \frac { 1 } { \frac { 1 } { \gamma _ { \rm d } } + \frac { 1 } { \gamma _ { \rm s } } } . \end{align*}


\begin{align*} g ^ \iota = \begin{pmatrix} 0 & 1 \\ - 1 & 0 \end{pmatrix} ( g ^ { - 1 } ) ^ t \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix} . \end{align*}


\begin{align*} \Gamma _ k : = \bigg \{ f \in L ^ p ( [ 0 , T ] , H ) : \int _ 0 ^ { T - \delta _ k } \Vert f ( t + \delta _ k ) - f ( t ) \Vert _ Y ^ p d t \leq \frac { 1 } { k } \bigg \} . \end{align*}


\begin{align*} Y _ \infty = \left ( \frac { m } { \textrm { G e V } } \right ) ^ { - 3 } \left [ 1 + \frac { 3 \ln ( m / \textrm { G e V } ) } { 1 5 } + \frac { \ln ( c _ 2 / 5 ) } { 1 5 } \right ] \end{align*}


\begin{align*} \vartheta : = \ , \mathbf { A } \ , , \qquad \textbf { A n t i } \ , \vartheta = \mathbf { A } \ , , \end{align*}


\begin{align*} \mathcal { B } _ q ( \psi _ m ) ( p ) = \displaystyle \frac { p ^ m } { \sqrt { \Gamma ( q m + 1 ) } } , \end{align*}


\begin{align*} H _ { m i n } \left ( ( E _ 1 [ t ] \times E _ 1 [ t ] ^ { \vee } ) \otimes F \right ) & = H _ { m i n } \left ( E _ 1 [ t ] \otimes F \times E _ 1 [ t ] ^ { \vee } \otimes F \right ) \\ & = \min \left ( H _ { m i n } ( E _ 1 [ t ] \otimes F ) , H _ { m i n } ( E _ 1 [ t ] ^ { \vee } \otimes F ) \right ) \\ & \leq H _ { m i n } ( E _ 1 [ t ] \otimes F ) = t H _ { m i n } ( E _ 1 \otimes F ) \leq t H _ { m i n } ( E _ 1 ) H _ { m i n } ( F ) \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( \int _ 0 ^ T \Vert X _ n ( t ) \Vert _ V ^ p d t > M \right ) = 0 , \end{align*}


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


\begin{align*} e m + \gamma = \widehat { \iota } ( D ) c \geq \widehat { \iota } ( D ) ( b _ 1 d + 1 ) . \end{align*}


\begin{align*} U ( t , x , \nu ) : = P _ t x + \Phi ( t , \nu ) \end{align*}


\begin{align*} \sigma _ \hbar \big ( \Delta ^ { ( k ) } \big ) = \hbar Q + \Pi _ \hbar . \end{align*}


\begin{align*} \sigma \cdot X : = \{ S ^ k \sigma ( x ) \mid x \in X , 0 \leq k < | \sigma ( x _ 0 ) | \} . \end{align*}


\begin{align*} \ddot { \kappa } + \frac { 1 } { \rho } \dot { \kappa } + \left ( 1 - \frac { ( 3 / 2 ) ^ { 2 } } { \rho ^ { 2 } } \right ) \kappa = 0 , \kappa & > 0 , 0 < \rho < R _ 0 , \\ \kappa ( R _ 0 ) & = 0 . \end{align*}


\begin{align*} \chi _ 2 = \chi - \chi _ 1 - \sum \limits _ { j \geq 3 } \chi _ j + ( n - 1 ) r \end{align*}


\begin{align*} v _ 0 = c X _ { - \beta _ m } X _ { - \beta _ { m - 1 } } \cdots X _ { - \beta _ { m - k + 1 } } v _ k . \end{align*}


\begin{align*} \psi _ { \Delta , \beta } ( p ) = \psi _ { \Delta , \ \beta } [ \psi ] ( p ) \stackrel { d e f } { = } \frac { p } { p - \Delta } \ \beta ^ p \ \psi ( p ) , \ \Delta < p \le b , \ \beta > 1 . \end{align*}


\begin{align*} f '' ( x ) = a + b ' x + c ' x ^ q , \end{align*}


\begin{align*} \left \{ \begin{array} { l } \partial _ { t } q + v \cdot \nabla q = 0 , ( t , x ) \in \mathbb { R } _ + \times \mathbb { R } ^ 2 , \\ v = \nabla ^ { \bot } \psi , \nabla ^ { \perp } = ( \partial _ { 2 } , - \partial _ { 1 } ) , \\ \psi = \left ( - \Delta + \varepsilon ^ 2 \right ) ^ { - 1 } q , \\ q _ { \mid { t } = 0 } = q _ 0 , \\ \end{array} \right . \end{align*}


\begin{align*} a ( x ) = \nabla \times ( \eta ( z , r ) \nabla \theta ) + \phi ( z , r ) \nabla \theta , \end{align*}


\begin{align*} \kappa ( t ) = h _ \alpha ( t ) , \ k _ 1 ( t ) = h _ \gamma ( t ) , \ k _ 2 ( t ) = h _ { 1 - \alpha - \gamma } ( t ) , \ t > 0 \end{align*}


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


\begin{align*} Z ^ { ( i ) } = \partial _ i \mbox { o r } Z ^ { ( i , j ) } = x _ i \partial _ j - x _ j \partial _ i \mbox { f o r } i , j = 1 , \dots , n . \end{align*}


\begin{align*} \int ^ \infty _ 0 x ^ { - 3 \alpha } \exp \left \{ - ( a \alpha + n ) \frac { \theta } { x ^ 2 } \right \} \mathrm { d } x = & \int ^ \infty _ 0 u ^ { \frac { 3 ( \alpha - 1 ) } { 2 } } \exp \left \{ - ( a \alpha + j ) u \right \} \mathrm { d } u \\ = & \left ( \frac { 1 } { a \alpha + n } \right ) ^ { \frac { 3 \alpha - 1 } { 2 } } \Gamma \left ( \frac { 3 \alpha - 1 } { 2 } \right ) . \end{align*}


\begin{align*} \varpi \partial _ i ( \partial _ j ( u _ k w _ \ell ) ) = \partial _ i ( \partial _ j ( \varpi u _ k w _ \ell ) ) - \partial _ i ( ( \partial _ j \varpi ) u _ k w _ \ell ) - \partial _ j ( ( \partial _ i \varpi ) u _ k w _ \ell ) - ( \partial _ i \partial _ j \varpi ) u _ k w _ \ell . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ N ( \alpha _ N ^ i ) ^ s = \begin{cases} 0 , & N \nmid s \\ N , & N \mid s \end{cases} , \end{align*}


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


\begin{align*} \sum _ { j = 1 } ^ { D - 1 } \left ( \nu _ { j } - \nu _ { j - 1 } \right ) \P \left \{ Y \geq \nu _ j \right \} , \end{align*}


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


\begin{align*} \varphi _ 0 \eta _ 0 + \ldots + \varphi _ p \eta _ p = 0 , \end{align*}


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


\begin{align*} \lim _ { \delta \rightarrow 0 + } \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( \int _ 0 ^ { T - \delta } \Vert Y _ n ( t + \delta ) - Y _ n ( t ) \Vert _ { H } ^ { \alpha } d t > \epsilon \right ) = 0 . \end{align*}


\begin{align*} { \tilde \Sigma } ^ t = \{ x \in M \mid { \rm d i s t } ( x , \tilde \Sigma ) \leq t \} . \end{align*}


\begin{align*} ( K * \mu ) ( x ) = \int _ { \mathbb R ^ d } K ( x - z ) \mu ( \d z ) . \end{align*}


\begin{align*} u ( \upsilon ) = \frac { 4 \upsilon + \frac 1 2 f ( 2 \upsilon ) - 2 f ( \upsilon ) } { f ( \upsilon ) } . \end{align*}


\begin{align*} L ( E ) = \max \{ 0 , \log | \lambda | \} , \forall \ E \in \Sigma _ { \lambda \exp , \alpha } . \end{align*}


\begin{align*} ( \bigoplus _ { i \in I _ j } V _ i \to W _ j , \bigoplus _ { j \in J } W _ j \to U ) \mapsto ( \bigoplus _ { i \in I } V _ i = \bigoplus _ { j \in J } \bigoplus _ { i \in I _ j } V _ i \to \bigoplus _ { j \in J } W _ j \to U ) \end{align*}


\begin{align*} \int _ { t _ 0 } ^ { t _ 1 } \int _ { \Omega } \omega ( t , x ) f _ h ' ( \phi ( x ) ) \ , u ( t , x ) \cdot \psi ( x ) d x d t + \int _ { \Omega } \omega ( t _ 0 , x ) f _ h ( \phi ( x ) ) d x - \int _ { \Omega } \omega ( t _ 1 , x ) f _ h ( \phi ( x ) ) d x = 0 , \end{align*}


\begin{align*} \sigma _ { W _ { \rm s } } ^ 2 = \sigma _ { \rm a } ^ 2 ( \kappa _ { \rm b } ) Q ( - \kappa _ { \rm b } ) + \sigma _ { \rm a } ^ 2 ( \kappa _ { \rm t } ) Q ( \kappa _ { \rm t } ) + \ ! \ ! \underbrace { \int _ { \kappa _ { \rm b } } ^ { \kappa _ { \rm t } } f ( x ) \sigma _ { \rm a } ^ 2 ( x ) \ , \mathrm { d } x } _ { \mathcal { S } } . \end{align*}


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


\begin{align*} & r _ 2 ( z ) - \widetilde { r } _ 2 ( z ) = \frac { 2 i k b } { a } - \frac { 2 i k \tilde { b } } { \tilde { a } } , \\ & z ^ { - 2 } ( r _ 2 ( z ) - \widetilde { r } _ 2 ( z ) ) = \frac { 2 i k z ^ { - 2 } b } { a } - \frac { 2 i k z ^ { - 2 } \tilde { b } } { \tilde { a } } . \end{align*}


\begin{align*} \widetilde { X } _ n ( t ) = & P _ n x + \int _ 0 ^ t P _ n A ( s , \widetilde { X } _ n ( s ) ) d s + \int _ 0 ^ t P _ n B ( s , \widetilde { X } _ n ( s ) ) Q _ n d \widetilde { W } ( s ) , t \in [ 0 , T ] . \end{align*}


\begin{align*} x _ n = \rho ( x ' ) = \frac { 1 } { 2 } \sum _ { \alpha , \beta < n } B _ { \alpha \beta } x _ \alpha x _ \beta + O ( | x ' | ^ 3 ) , \end{align*}


\begin{align*} \alpha _ m : = \min \left \{ \langle Q , X \rangle \ , | \ , X \in \R ^ { Z _ m \times Z _ m } _ { \geq 0 } , \ , \langle J , X \rangle = 1 , \ , X \succeq 0 \right \} . \end{align*}


\begin{align*} Z _ { I } ^ { \star } ( 1 , \{ 2 \} ^ { s - 1 } ; ( \alpha , \beta , \gamma ) ) = Z ( s - 1 | s - 1 | 1 ; ( \alpha , \beta , \gamma ) ) + Z ( s - 1 | s - 1 | 1 ; ( \alpha , \beta , \alpha + \beta - \gamma ) ) \end{align*}


\begin{align*} J ( x ) \ge J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \end{align*}


\begin{align*} b ( x ) = \nabla \times ( \phi ( z , r ) \nabla \theta ) + G ( z , r ) \nabla \theta . \end{align*}


\begin{align*} \dot { s } ( 0 ) = - \frac { \partial _ { \tau } j ( 0 , 0 ) } { \partial _ { s } j ( 0 , 0 ) } . \end{align*}


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


\begin{align*} \mathcal { A } _ { H _ { t } } ( [ \gamma , u ] ) = \int _ { \gamma } H _ { t } + \int u ^ { * } \omega . \end{align*}


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


\begin{align*} A ^ { - 1 } \mathcal { L } _ g = I _ { n + 1 } \frac { \partial ^ 2 } { \partial x _ n ^ 2 } + B \frac { \partial } { \partial x _ n } + C = \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + B - Q \Bigr ) \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + Q \Bigr ) , \end{align*}


\begin{align*} g ^ { a b } = \delta _ { a b } + \frac { 1 } { 3 } R _ { a c b d } x ^ { c } x ^ { d } + o ( { \mathbf { x ^ { 2 } } } ) , \end{align*}


\begin{align*} \begin{aligned} \partial _ t F _ \kappa + v \cdot \nabla _ x F _ \kappa = \tfrac { 1 } { \kappa } \mathcal { B } ( F _ \kappa , F _ \kappa ) \ , , \end{aligned} \end{align*}


\begin{align*} W ( F ) ( \tau , \tau ' ) : = F _ { z = 0 } . \end{align*}


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


\begin{align*} - \sum _ { K = 0 } ^ { p } \binom { p } { K } \ , B _ K \ , n ^ { p - K } \end{align*}


\begin{align*} B ( x , x ) = \begin{pmatrix} x _ 2 x _ 3 \\ x _ 1 x _ 3 \\ - 2 x _ 1 x _ 2 \end{pmatrix} . \end{align*}


\begin{align*} F \colon ( x , B , M ) \mapsto \frac { 1 } { p } \sum _ { i = 1 } ^ p | [ U B ^ T M x ] _ i | . \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { 1 } { \ln \frac { 1 } { \varepsilon } } \int _ { \Omega } ( K _ H ( x ) \nabla v ^ \tau _ \varepsilon | \nabla v ^ \tau _ \varepsilon ) d x = 2 \pi q ^ 2 \sqrt { d e t ( K _ { H } ) } ( \bar { x } ) \end{align*}


\begin{align*} W ( v , x ) \stackrel { d } { = } e ^ { - \lambda ^ * t } W ( v , z ) . \end{align*}


\begin{align*} \begin{cases} - \delta ^ 2 \Delta w = ( w - a ) ^ { p } _ + , \ \ & \ B _ R ( 0 ) , \\ w = 0 , \ \ & \ \partial B _ R ( 0 ) , \end{cases} \end{align*}


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


\begin{align*} \begin{aligned} \int _ { \Omega } f ^ { p + 1 } \le C \frac { ( p + 1 ) ^ { 2 } } { 2 \log s } \int _ { \Omega } ( f \log f + e ^ { - 1 } ) \int _ { \Omega } f ^ { p - 2 } | \nabla f | ^ { 2 } + 2 C \| w \| _ { L ^ { 1 } ( \Omega ) } ^ { 2 } + 3 s ^ { p + 1 } | \Omega | , \end{aligned} \end{align*}


\begin{align*} \dot { e } _ { y _ a } = - \gamma e _ { y _ a } - \dfrac { \gamma } { 4 } \dfrac { \partial a _ { 0 , \delta } ( x _ a ) } { \partial x _ a } b _ { 1 , \delta } ( x _ a ) e _ { y _ a } ( 0 ) = 0 \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) \| _ { \mathcal { A } } & \le \| x ( t ) - x _ a ( t ) \| + \| x _ a ( t ) \| _ \mathcal { A } \\ & \le \tilde { d } + \beta ( \| x _ 0 \| _ { \mathcal { A } } , \gamma t ) & & \forall \ , t \in [ 0 , \bar { t } / \gamma ] \end{aligned} \end{align*}


\begin{align*} A _ k ( a _ j ) = A _ k ( b _ j ) = 0 . \end{align*}


\begin{align*} \Upsilon ( \cdot ) = \Phi ( \cdot ) + \Psi ( \cdot ) \end{align*}


\begin{align*} \frac { \partial j } { \partial s } ( 0 , 0 ) = 2 \int _ { \mathbb { R } ^ { 3 } } ( \phi _ 0 1 _ { ( 0 , \infty ) } ( \phi - \phi _ { \infty } ) G + ( \phi - \phi _ { \infty } ) _ { + } G _ 0 ) \frac { 1 } { r ^ { 2 } } \dd x . \end{align*}


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


\begin{align*} m = \left \lfloor { \frac { { i + { N _ { { \rm { C P } } } } } } { { K + { N _ { { \rm { C P } } } } } } } \right \rfloor , \end{align*}


\begin{align*} I = \sum _ { m = 1 } ^ 9 I _ m + J _ 1 + J _ 2 , \end{align*}


\begin{align*} ( \mathbf { u } G _ 2 , \mathbf { u } G _ 3 , \cdots , \mathbf { u } G _ { 2 k } ) = ( \mathbf { v } G _ 2 , \mathbf { v } G _ 3 , \cdots , \mathbf { v } G _ { 2 k } ) = ( 1 , 1 , \cdots , 1 ) , \end{align*}


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


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


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \eta = \varepsilon ( - z ^ { - 1 } ) \dot { s } h ( - z ) \varepsilon ( - z ^ { - 1 } ) \varepsilon ( 1 ) \eta = \varepsilon ( - z ^ { - 1 } ) \dot { s } \varepsilon ( z ^ 2 - z ) \eta . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ k | V _ i \setminus X _ i | - \frac { m ^ k } { 4 } \geq m ^ k \left ( \left ( 1 - \frac { 1 } { 2 k ^ 2 } \right ) ^ k - \frac { 1 } { 4 } \right ) \geq m ^ k \left ( 1 - \frac { 1 } { 2 k } - \frac { 1 } { 4 } \right ) \geq \frac { m ^ k } { 2 } \end{align*}


\begin{align*} H ( E ) = \left ( \N \left ( \prod _ { i = 1 } ^ { \ell } \mathfrak a _ i \right ) \prod _ { v \in V _ \infty } \det ( h _ { v } ( b _ i , b _ j ) ) ^ { e _ { v } / 2 } \right ) ^ { 1 / [ K : \mathbb { Q } ] } \end{align*}


\begin{gather*} \Lambda ( z ; \alpha ) : = \alpha ( S - z ^ { - 1 } ) ^ { - 1 } , \Gamma ( z ; \alpha ) : = \beta ( z - S ) ^ { - 1 } , \end{gather*}


\begin{align*} c _ \varepsilon = I _ \varepsilon ( u _ \varepsilon ) \geq \inf \limits _ { w \in \mathcal { N } _ \varepsilon } I _ \varepsilon ( w ) . \end{align*}


\begin{align*} \begin{aligned} F ^ { i j } \left ( \sqrt { b _ { n - 1 } } ( A _ 1 v - A _ 2 w ) \pm \nabla _ \alpha ( u - \varphi ) \right ) _ { i j } \geq \ , & 0 \mbox { i n } \omega _ 1 \\ A \sqrt { b _ { n - 1 } } ( A _ 1 v - A _ 2 w ) \pm \nabla _ \alpha ( u - \varphi ) \leq \ , & 0 \mbox { o n } \partial \omega _ 1 \end{aligned} \end{align*}


\begin{align*} \nabla ^ { ( s ) } _ { e _ { i } } = e _ { i } - \frac { 1 } { 4 } \alpha _ { i j k } \gamma ^ { j } \gamma ^ { k } , \end{align*}


\begin{align*} | | \tau | | G \psi \stackrel { d e f } { = } \sup _ { p \in ( a , b ) } \left \{ \ \frac { | | \tau | | L _ p ( \Omega ) } { \psi ( p ) } \ \right \} = \sup _ { p \in ( a , b ) } \left \{ \ \frac { | | \tau | | _ p } { \psi ( p ) } \ \right \} . \end{align*}


\begin{align*} ( f , g ) _ { \mathcal { H } ( T , A , \mu ) } = \big ( ( c _ n ) , ( d _ n ) \big ) _ { \ell ^ 2 } = \sum _ n \frac { f ( t _ n ) \overline { g ( t _ n ) } } { | A ' ( t _ n ) | ^ 2 \mu _ n } \end{align*}


\begin{align*} y \left [ i \right ] = \sum \limits _ { l ' = 1 } ^ { L ' } { \sum \limits _ { l \in { { \bar { \cal L } } _ { l ' } } } { { \bf { h } } _ l ^ H { \bf { \bar H } } _ { l ' } ^ \bot { \bf { \bar X } } _ { l ' } { \bf { d } } \left [ { i - { n _ l } - { \kappa _ { l ' } } } \right ] } } + z \left [ i \right ] , \end{align*}


\begin{align*} \sqrt { ( g ) } = 1 - \frac { 1 } { 6 } \mathrm { R i c } _ { a b } x ^ { a } x ^ { b } + o ( { \mathbf { x ^ { 2 } } } ) , \end{align*}


\begin{align*} \delta _ i = \begin{cases} \tilde { \delta } _ i , & i \in [ m ] \\ - \tilde { \delta } _ { i - m } , & i \in [ k ] \setminus [ m ] . \end{cases} \end{align*}


\begin{align*} H _ 0 ^ 2 ( D ) : = \left \{ v \in H ^ 2 ( D ) : v = \frac { \partial v } { \partial \nu } = 0 \partial D \right \} , \end{align*}


\begin{align*} w ( x ) : = x _ n - y _ n - L ( x ) + \kappa | x ' - y ' | ^ 2 \geq 0 \mbox { o n } \partial \omega \cap \partial \Omega . \end{align*}


\begin{align*} \nabla _ X \varphi = \frac { 1 } { 2 } ( \sqrt { c _ 1 } \ X _ 1 \cdot \nu _ 1 + \sqrt { c _ 2 } \ X _ 2 \cdot \nu _ 2 ) \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi \end{align*}


\begin{align*} & \int _ { \{ z \in M : - \Psi ( z ) \in S _ { i , n } \backslash U _ k \} } | F _ { t _ 0 , \tilde { c } } | ^ 2 _ h c ( - \Psi ) \\ = & \int _ { \{ z \in M : - \Psi ( z ) \in S _ { i , n } \backslash U _ k \} } | \tilde F | ^ 2 _ h c ( - \Psi ) + \int _ { \{ z \in M : - \Psi ( z ) \in S _ { i , n } \backslash U _ k \} } | F _ { t _ 0 , \tilde { c } } - \tilde F | ^ 2 _ h c ( - \Psi ) . \end{align*}


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


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


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


\begin{align*} \begin{aligned} \gamma _ 2 ^ \star : = \dfrac { \bar c } { 3 e ^ { L _ r \bar { t } } } \gamma _ 3 ^ \star : = \dfrac { \bar c L _ r } { 3 \bar k ( L _ r , M _ r , \delta ) ( e ^ { L _ r \bar { t } } - 1 ) } \end{aligned} \end{align*}


\begin{align*} \sum _ { ( d , 2 ) = 1 } M _ Z ( d ) \left ( \frac { d } { p } \right ) W \left ( \frac { d } { X } \right ) = \sum _ { \substack { \alpha \leq Z \\ ( \alpha , 2 p ) = 1 } } \mu ( \alpha ) \sum _ { ( d , 2 ) = 1 } \left ( \frac { d } { p } \right ) W \left ( \frac { d \alpha ^ 2 } { X } \right ) . \end{align*}


\begin{align*} \lim _ { \substack { n \rightarrow \infty \\ M \rightarrow \infty } } \Vert X _ n ( t \wedge \tau _ u ^ M ) - X ( t \wedge \tau _ u ^ M ) \Vert _ H = 0 , \mathbb { P } \otimes d t . \end{align*}


\begin{align*} F ( x ) = \frac { 1 } { \operatorname { B } ( a , b ) } \sum ^ { \infty } _ { n = 0 } \frac { ( - 1 ) ^ n \ , \Gamma ( b ) } { ( a + n ) \ , \Gamma ( b - n ) \ , n ! } \exp \left \{ - \frac { ( a + n ) \theta } { x ^ 2 } \right \} . \end{align*}


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


\begin{align*} \frac { 1 } { \varepsilon ^ 2 } \int _ { A _ \varepsilon } \left ( u _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } q d x = \frac { 2 c _ \varepsilon } { \ln \frac { 1 } { \varepsilon } } + O \left ( \frac { 1 } { \ln \frac { 1 } { \varepsilon } } \right ) \leq C . \end{align*}


\begin{align*} ^ F V = \bigoplus _ { p \in \frac { 1 } { 2 } \mathbb Z } F _ p V / F _ { p - \frac { 1 } { 2 } } V \end{align*}


\begin{align*} \tilde { { \mathcal { I } } } ( B _ 0 ) = \inf \left \{ \frac { 1 } { 2 } \int _ { \Omega } | B | ^ { 2 } \dd x \ \middle | \ B \in \overline { M } ^ { w } ( \Omega , B _ 0 ) \right \} , \end{align*}


\begin{align*} d = 2 n + 1 = 4 l + 3 \equiv 3 \mod 4 \end{align*}


\begin{align*} \partial _ { t } P = \Theta \partial _ { t } \sigma _ { k } + \sigma _ { k } \partial _ { t } \Theta , \end{align*}


\begin{align*} H _ 2 ^ { [ 2 ] } \bigl ( \alpha ( t ) \bigr ) = \pm \int _ { - \pi } ^ \pi 4 \sin ^ 2 ( \theta ) | \widehat { \alpha } ( t , \theta ) | ^ 2 \ , \tfrac { d \theta } { 2 \pi } , \end{align*}


\begin{align*} ( p _ n f , x _ n ) = ( f , x _ n ) x _ n \in X _ n . \end{align*}


\begin{align*} f ( x + k , \omega ) = f ( x , \tau ( k ) \omega ) . \end{align*}


\begin{align*} V _ 1 ( r ; l , Q ) = \frac { L ( L + 1 ) } { r ^ 2 } - \frac { Q } { r } f ( r ) - E _ 0 \end{align*}


\begin{align*} \begin{cases} \partial _ t w + \nabla ^ \perp \varphi \cdot \nabla w = 0 , \ \ & \ B _ { R ^ * } ( 0 ) , \\ w = \mathcal { L } _ H \varphi , \ \ & \ B _ { R ^ * } ( 0 ) , \\ \varphi = 0 , \ \ & \ \partial B _ { R ^ * } ( 0 ) . \end{cases} \end{align*}


\begin{align*} \sigma _ n ( \Delta ^ + ) = ( - 1 ) ^ n \sigma _ n ( \Delta ) . \end{align*}


\begin{align*} \Theta ^ { 2 ( 1 ) } & = d \theta ^ i { } _ j + \theta ^ i { } _ { \alpha j } \wedge \theta ^ \alpha + \theta ^ i { } _ \alpha \wedge \theta ^ \alpha _ j \\ * & = \Omega ^ i { } _ j + \theta ^ i { } _ { \alpha j } \wedge \theta ^ \alpha , \\ * \Theta ^ { 2 ( 2 ) } & = d \theta ^ i { } _ j + \theta ^ i { } _ { j \alpha } \wedge \theta ^ \alpha + \theta ^ i { } _ \alpha \wedge \theta ^ \alpha { } _ j \\ * & = \Omega ^ i { } _ j + \theta ^ i { } _ { j \alpha } \wedge \theta ^ \alpha . \end{align*}