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


\begin{align*} \begin{aligned} & \sum _ { 0 { \le } m _ 1 { \le } \cdots { \le } m _ n < \infty } \frac { ( \alpha ) _ { m _ 1 } } { { m _ 1 } ! } \frac { ( \beta ) _ { m _ 1 } } { ( \gamma ) _ { m _ 1 } } \frac { { m _ n } ! } { ( \alpha ) _ { m _ n } } \frac { ( \gamma ) _ { m _ { n } } } { ( \beta ) _ { m _ n } } \\ & \times \left \{ \prod _ { i = 1 } ^ { n } \frac { 1 } { ( m _ i + \alpha ) ^ { a _ i } ( m _ i + \beta ) ^ { b _ i } ( m _ i + \gamma ) ^ { c _ i } } \right \} , \end{aligned} \end{align*}


\begin{align*} & \frac { d } { d t } \int _ { V } \Re ( \alpha _ { t } + \sqrt { - 1 } \omega ) ^ { p } - \cot \Theta _ { 0 } \cdot \Im ( \alpha _ { t } + \sqrt { - 1 } \omega ) ^ { p } \\ = { } & p \int _ { V } \left ( \Re ( \alpha _ { t } + \sqrt { - 1 } \omega ) ^ { p - 1 } - \cot \Theta _ { 0 } \cdot \Im ( \alpha _ { t } + \sqrt { - 1 } \omega ) ^ { p - 1 } \right ) \wedge \omega > 0 . \end{align*}


\begin{align*} B _ 2 = L ( L + 1 ) , \zeta = \kappa ( L + 1 ) , \end{align*}


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


\begin{align*} l ( \rho \circ \Psi \circ \pi ^ { - 1 } ( 1 \otimes e _ 0 ) ) = l ( \rho \circ \Psi ( p r ) ) = \rho ( l ( \Psi ( p r ) ) ) = \rho ( \Phi ( p r ) ) . \end{align*}


\begin{align*} \omega : = - p _ 1 ^ * ( \log \sqrt { \tau _ 0 } ) _ z d z + p _ 2 ^ * \omega _ 0 . \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*} \tau \lim _ { n \to \infty } T ( t ) ( x _ n - x ) = 0 \end{align*}


\begin{align*} X = X _ 1 \boxtimes X _ 2 \boxtimes \cdots \boxtimes X _ k , \end{align*}


\begin{align*} \begin{cases} c _ { \alpha + n + 1 } - c _ { n + 1 } = b _ { \alpha + n } - b _ { \alpha + n + 1 } \\ d _ n = c _ { \alpha + 1 } + ( b _ { \alpha + 1 } - b _ \alpha ) = d \\ b _ { \alpha + n } = b _ \alpha + ( d _ 1 - c _ { \alpha + 1 } ) + \sum _ { j = 2 } ^ n ( c _ j - c _ { \alpha + j } ) \\ b = b _ \alpha + d - c _ { \alpha + 1 } + \sum _ { j = 2 } ^ \infty ( c _ j - c _ { \alpha + j } ) \end{cases} \ , \end{align*}


\begin{align*} \rho ( x ) = ( \rho _ k ( x ) : k \in \N ) \end{align*}


\begin{align*} [ S ] = \mu \left ( \frac { 1 } { 2 } d \overline { z } + ( \alpha - I \beta ) d z \right ) J \end{align*}


\begin{align*} \tau ( x ) \omega : = \omega + \Lambda x - \left \lfloor \omega + \Lambda x \right \rfloor \end{align*}


\begin{align*} q ( A ) = \sum _ i q ( E _ i ) p ( A | E _ i ) , A \in \bf { A } . \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*} \mathcal K _ 0 = { c _ 1 \omega _ 1 + c _ 2 \omega _ 2 + c _ 3 \omega _ 3 ; c _ 1 , c _ 2 , c _ 3 > 0 } . \end{align*}


\begin{align*} \kappa = \frac { \alpha } { f \sqrt { 1 + { f ' } ^ 2 } } + \varpi \end{align*}


\begin{align*} \frac { \alpha } { 2 \beta } \min \{ \kappa - r , \lambda \} \leq \frac { \alpha } { 2 \beta } \min \{ \kappa - r , r \} \leq \frac { 1 } { 2 } \frac { \alpha \kappa } { 2 \beta } = \frac { 1 } { 2 } \min \{ \alpha + \gamma - \frac { 1 } { 2 } , \alpha \} \leq \frac { \alpha } { 2 } \leq \min \{ \alpha - \varepsilon , \frac { 1 } { 2 } \} . \end{align*}


\begin{align*} s _ 1 ( \Omega ) = s _ 2 ( \Omega ) = \Omega ^ * = \{ ( 1 , 2 ) \} . \end{align*}


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


\begin{align*} [ q ^ \alpha r ^ \beta ] C _ k ( q , r , t ) = C _ k ( t ) ^ { \beta + 1 } \sum _ { i \geq 0 } ( - 1 ) ^ i \binom { \alpha - ( k - 1 ) i } { i } t ^ i - \sum _ { i \geq 0 } ( - 1 ) ^ i \binom { \alpha - \beta - 1 - ( k - 1 ) i } { i } t ^ i . \end{align*}


\begin{align*} \begin{aligned} \dfrac { \partial h ( s ) } { \partial s } & = \left . \dfrac { \partial h ( s ) } { \partial s } \right | _ x + \left . \dfrac { \partial ^ 2 h ( s ) } { \partial s ^ 2 } \right | _ x ( s - x ) + O ( ( s - x ) ^ 2 ) . \\ \end{aligned} \end{align*}


\begin{align*} [ X _ 1 ] = ( \frac { \mu } { 2 } d \overline { z } - \frac { 2 } { \mu } h _ z ^ 2 d z + \frac { \tau _ 0 } { 2 \mu } d \overline { z } ) J - \frac { 1 } { \mu } d h \sqrt { \tau _ 0 } I . \end{align*}


\begin{align*} \langle \delta _ \chi , \delta _ \pi \rangle _ { L ^ 2 ( G ^ \wedge ) } = \int _ { G ^ \wedge } \delta _ \chi ( \xi ) \ ; \delta _ \pi ( \xi ) \ , d \nu ( \xi ) = \left \{ \begin{array} { c c l } 1 , & & \chi = \pi , \\ 0 , & & \chi \not = \pi . \end{array} \right . \end{align*}


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


\begin{align*} \begin{aligned} \int _ { \Omega } | \nabla c | ^ { 2 } & \le \int _ { \Omega } | \nabla c | ^ { 2 } + \int _ { \Omega } n c ^ { 2 } + \frac { 1 } { 2 } \int _ { \partial \Omega } c ^ { 2 } \\ & = \int _ { \partial \Omega } \gamma c - \frac { 1 } { 2 } \int _ { \partial \Omega } c ^ { 2 } \\ & \le \frac { 1 } { 2 } \gamma ^ { 2 } | \partial \Omega | . \end{aligned} \end{align*}


\begin{align*} X ( q , r ) = \frac 1 q \iota ( q r ) \end{align*}


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


\begin{align*} L ( p \mid u ) = \prod _ { i = 1 } ^ m p _ j ^ { u _ j } . \end{align*}


\begin{align*} \frac { p } { p + 1 } \sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { \omega ( \gcd ( n , p ) ) } \cdot 2 r _ { Q ^ { \ast } } ( n ) ^ { 2 } } { n } \sum _ { d = 1 } ^ { \infty } \psi \left ( d \sqrt { \frac { n } { p } } \right ) \end{align*}


\begin{align*} \phi ( \pi \cdot e _ { \sigma } ) = \phi ( e _ { \pi \sigma \pi ^ { - 1 } } ) = \phi ( e _ { ( \pi \sigma _ 1 \ , \pi \sigma _ 2 \ , \ldots \ , \pi \sigma _ m ) } ) = ( \overline { \pi r } , 1 ) = \pi \cdot \phi ( e _ { \sigma } ) , \end{align*}


\begin{align*} & \prod _ { p } ^ { } \left ( 1 + \frac { b ( p ) } { p ^ { s } } + \cdots + \frac { b ( p ^ { m } ) } { p ^ { m s } } + \cdots \right ) \\ & = \zeta ( s - 2 ) L ( s , \chi ) \prod \limits _ { n = 1 } ^ { j } L ( s - 2 , s y m ^ { 2 n } f ) L ( s , s y m ^ { 2 n } f \otimes \chi ) \\ & = G _ { j } ( s ) , & & ( \mbox { s a y } ) \end{align*}


\begin{align*} ( A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ n ] ) ( x ) = f ( x ) - g ( x ) . \end{align*}


\begin{align*} H _ N ( t ) - \widetilde H _ N ( t ) = \int _ { \Omega _ N } g _ { N , t } \ , d \mu _ { N , t } , g _ { N , t } : = \log \left ( \frac { d \tilde \nu _ { N , t } } { d \nu _ { N , t } } \right ) . \end{align*}


\begin{align*} g _ 2 = - \frac { 1 } { \nu } I \widehat { g _ 1 } I + \frac { 2 i } { \sqrt { \tau _ 0 } } h _ z J g _ 1 . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ m b _ i ( x ^ { q ^ i } \ \ f ) = 0 . \end{align*}


\begin{align*} \lim _ { t \to \infty } \frac 1 { t ^ n | A | } \int _ { A _ t } f ( x ) \ , d x = M ( f ) , \end{align*}


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


\begin{align*} \partial _ { \theta _ i } ( \theta _ A ) = \begin{cases} ( - 1 ) ^ { \# \{ j \in A : j < i \} } \theta _ { A \backslash \{ i \} } & \hbox { i f } i \in A \\ 0 & \hbox { i f } i \notin A \end{cases} ~ . \end{align*}


\begin{align*} \mathcal { V } _ p = T _ p \Sigma _ t \ \ \ \ \forall p \in \Sigma _ t \end{align*}


\begin{align*} \begin{aligned} x ( t ) - z _ 1 ( t ) = & \ , x ( 0 ) - z _ 1 ( 0 ) + \int _ 0 ^ t \dot { x } ( \tau ) - \dot { x } _ a ( \tau ) d \tau \\ & - \gamma \int _ 0 ^ t \dfrac { \partial \epsilon _ 1 ( \eta , t ) } { \partial \eta } \Big | _ { \eta = \eta _ a ( \tau ) } \dot { \eta } _ a ( \tau ) d \tau \\ & - \gamma \int _ 0 ^ t \dfrac { \partial } { \partial \tau } \epsilon _ 1 ( \eta _ a ( \tau ) , \tau ) d \tau . \end{aligned} \end{align*}


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


\begin{align*} A _ { l , u } & = \Big \{ i \in [ k ] : \frac { p _ i } { q _ i } \in [ l , u ) \Big \} , \\ A _ { l , \infty } & = \Big \{ i \in [ k ] : \frac { p _ i } { q _ i } \in [ l , \infty ] \Big \} . \end{align*}


\begin{align*} I _ 1 = & \mathbb { E } \int _ 0 ^ { T \wedge \tau _ n ^ M } \big [ 2 \langle A ( r , Y _ n ( r ) ) , Y _ n ( r ) \rangle + \Vert P _ n B ( r , Y _ n ( r ) ) Q _ n \Vert _ { L _ 2 } ^ 2 \big ] d r \int _ { 0 \vee ( r - \delta ) } ^ { r } \mathbf { 1 } _ { \{ \tau _ n ^ M > t \} } d t \\ \leq & \delta \ , \mathbb { E } \int _ 0 ^ { T \wedge \tau _ n ^ M } f ( s ) ( 1 + \Vert Y _ n ( s ) \Vert _ H ^ 2 ) d s \\ \leq & C _ M \delta . \end{align*}


\begin{align*} \mathcal { H } _ m ( s I + A ) = \sum \limits _ { 1 \leq i _ 1 < \dots < i _ m \leq n } ( s + \lambda _ { i _ 1 } ) \dots ( s + \lambda _ { i _ m } ) = \sum \limits _ { p = 0 } ^ m \binom { n - p } { m - p } \mathcal { H } _ p ( A ) s ^ { m - p } . \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*} \beta _ { \bar { \mathbf { g } } } \mathbf { g } ( \partial _ t ) & = 0 \\ \beta _ { \bar { \mathbf { g } } } \mathbf { g } ( \partial _ a ) & = \beta _ { \bar g } g ( \partial _ a ) + \bar u ^ { - 2 } u \partial _ a u - \bar u ^ { - 1 } g ( \nabla _ { \bar g } \bar u , \partial _ a ) \mbox { f o r $ a = 1 , \dots , n $ } . \end{align*}


\begin{align*} R ^ { l } _ { i j k } = \frac { \partial \Gamma ^ { l } _ { j k } } { \partial x _ i } - \frac { \partial \Gamma ^ { l } _ { i k } } { \partial x _ j } + \Gamma ^ { h } _ { j k } \Gamma ^ { l } _ { i h } - \Gamma ^ { h } _ { i k } \Gamma ^ { l } _ { j h } , \end{align*}


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


\begin{align*} B _ n \Big ( \frac { 2 \beta ^ j } { L _ j } \Big ) - B _ n \Big ( \frac { F _ j \sqrt 5 } { L _ j } \Big ) = 0 , \qquad \mbox { $ n $ e v e n } . \end{align*}


\begin{align*} \{ u _ { \alpha \beta } ( 0 ) \} _ { 1 \leq \alpha , \beta \leq n - 1 } = \mathrm { d i a g } \{ b _ 1 , \ldots , b _ { n - 1 } \} . \end{align*}


\begin{align*} V : = U ( \mathfrak g ) \otimes _ { U ( \mathfrak g _ { \geq 0 } ) } \mathbb C _ x . \end{align*}


\begin{align*} \sum _ { J } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } q \ , \partial _ { x ^ \prime } ^ { J } q - \sum _ { J } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } b \ , \partial _ { x ^ \prime } ^ { J } q - \frac { \partial q } { \partial x _ n } + c = 0 . \end{align*}


\begin{align*} \int _ { \mathfrak U } | \omega | = \pi ^ * _ s \int _ { \mathfrak X } | \pi _ { \eta } ^ * \omega | \end{align*}


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


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


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


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


\begin{align*} \begin{aligned} \chi ( s ) \equiv 0 \textrm { f o r } 0 < s \leq r \ , , \ \chi ( s ) \equiv 1 \textrm { f o r } s \geq 2 r \ , , \ 0 \leq \chi ( s ) \leq 1 \textrm { f o r a l l } s > 0 \ , . \end{aligned} \end{align*}


\begin{align*} \mathrm { L i f t } ( f ) = \left \lbrace \prod _ { \alpha = 1 } ^ d z ^ { i _ { \alpha } } _ { j _ { \alpha } k _ { \alpha } } - \prod _ { \alpha = 1 } ^ d z ^ { i _ { \alpha } } _ { j ' _ { \alpha } k _ { \alpha } } \mid k _ { \alpha } \in [ t _ { i _ { \alpha } } ] \alpha \in [ d ] \right \rbrace . \end{align*}


\begin{align*} D = 3 \left ( \left ( a - b \right ) ^ { 4 } - \left ( a ^ { 4 } + b ^ { 4 } + \left ( a - 1 \right ) ^ { 4 } + \left ( b - 1 \right ) ^ { 4 } \right ) + 1 \right ) \end{align*}


\begin{align*} \lim _ { n \to \infty } \lim _ { N \to \infty } \sum _ { m = 1 } ^ { n } \sum _ { i = N } ^ { \infty } \mathbb { E } _ { n , m } [ \langle \chi _ { n , m } , e _ i \rangle ^ 2 ] = 0 \end{align*}


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


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T g _ n ( t ) d t = 0 . \end{align*}


\begin{align*} f ^ \lambda ( z ) = ( 2 \pi ) ^ { - n } \ , | \lambda | ^ n \ , \sum _ { k = 0 } ^ \infty R _ k ^ { n - 1 } ( \lambda , f ) \ , \varphi _ { k , \lambda } ^ { n - 1 } ( z ) . \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } \Psi ^ - _ 1 ( x ; z ) = e ^ { - i c _ - ( x ) } e _ 1 . \end{align*}


\begin{align*} \begin{aligned} & y '' _ { b e n } + N _ { b e n } ( E ) y _ { b e n } = 0 \\ & y '' _ { n b } + N _ { n b } ( E ) y _ { n b } = 0 . \end{aligned} \end{align*}


\begin{align*} E _ 0 = \kappa \left ( \frac { B _ 3 } { 2 \sqrt { \kappa B _ 4 } } + L + 2 \right ) ^ 2 - \left ( \frac { Q } { 2 ( L + 1 ) } - \sqrt { \kappa B _ 4 } \right ) ^ 2 , \end{align*}


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


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


\begin{align*} \mathbb { E } \left [ \varepsilon \right ] = & { \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { s } ^ { \left ( \right ) } \right ] } - { \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { y } ' \right ] } - { \mathbb { E } \left [ \mathbf { y ' } ^ H \mathbf { s } ^ { \left ( \right ) } \right ] } + { \mathbb { E } \left [ \mathbf { y ' } ^ H \mathbf { y } ' \right ] } . \end{align*}


\begin{align*} \frac { 1 + b ( z ) } { 1 - b ( z ) } = \int _ { \partial \mathbb { D } } \frac { t + z } { t - z } d \mu ( t ) \ , z \in \mathbb { D } \end{align*}


\begin{align*} H = \frac { \alpha } { n } \frac { \nu _ { n + 1 } } { x _ { n + 1 } } + \frac { \varpi } { n } , \end{align*}


\begin{align*} W ( t , x ) = \sqrt { C _ { d } \kappa } \ , \sum _ { k \in \Z ^ d _ 0 } \sum _ { i = 1 } ^ { d - 1 } \theta _ { k } \sigma _ { k , i } ( x ) W ^ { k , i } _ { t } , \end{align*}


\begin{align*} F ( X ) = \sum _ { i = 0 } ^ { 2 ^ n - 1 } A _ i X ^ i . \end{align*}


\begin{align*} \partial _ X \varphi = \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*} \begin{cases} \partial _ t w + \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \nabla ^ \perp \varphi \cdot \nu | _ { \partial \Omega } = v _ n | _ { \partial \Omega } . \end{cases} \end{align*}


\begin{align*} f ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \exp \left ( - \frac { x ^ 2 } { 2 } \right ) , \end{align*}


\begin{align*} d ^ 1 _ { p , q } = \begin{cases} 0 , & p \ , \ , \\ i _ { * } , & p \ , \ , , \end{cases} \end{align*}


\begin{align*} | | \zeta | | G \kappa \stackrel { d e f } { = } \sup _ { p \in U } \left \{ \ \frac { | \zeta | _ p } { \kappa ( p ) } \ \right \} . \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{array} { r l } h _ = & ( \lambda _ 1 \begin{bmatrix} 2 x _ 1 \\ - 1 \end{bmatrix} + \lambda _ 2 \begin{bmatrix} - 2 x _ 1 \\ 4 \end{bmatrix} + \lambda _ 3 \begin{bmatrix} 0 \\ - 1 \end{bmatrix} , \\ \\ & \lambda _ 1 ( x _ 1 ^ 2 - x _ 2 ) , \lambda _ 2 ( - x _ 1 ^ 2 + 4 x _ 2 ) , \lambda _ 3 ( - x _ 2 + 1 ) , 1 - \lambda _ 0 ^ 2 - \lambda _ 1 ^ 2 - \lambda _ 2 ^ 2 - \lambda _ 3 ^ 2 ) \ , . \end{array} \end{align*}


\begin{align*} R _ { i a } = \sum _ { j , b } \mathcal { B } _ { b i } ^ j \mathcal { B } _ { a j } ^ b = \sum _ { j , b } \mathcal { B } _ { b i j } \mathcal { B } _ { a j b } ( - \varepsilon _ { j ' } ) \varepsilon _ b = \sum _ { j , b } \mathcal { B } _ { b i j } \mathcal { B } _ { j a b } \varepsilon _ { j ' } \varepsilon _ b \end{align*}


\begin{align*} \langle i _ p ( T _ \phi ( x _ 1 , \ldots , x _ n ) ) , i _ { p ' } ( y ^ * ) \rangle _ { p , p ' } = \left \langle \left ( M _ { \widetilde { \phi } } ( i _ { p _ 1 , \alpha } ( x _ 1 ) , \ldots , i _ { p _ n , \alpha } ( x _ n ) ) \right ) _ { \alpha \in I } , i _ { p ' } ( y ^ * ) \right \rangle _ { p , p ' } , \end{align*}


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


\begin{align*} ( \mathbf { x } , - \mathbf { y } ) M _ { I J } = { \bf 0 } , \end{align*}


\begin{align*} J _ { 1 } ( u ) \leq \liminf _ { k \rightarrow \infty } J _ { 1 } ( \overline { u } _ { k } ) = B _ { 1 } \end{align*}


\begin{align*} \ell ( a , b , \theta ) = & n [ \log ( 2 \theta ) - \log \{ \operatorname { B } ( a , b ) \} ] - 3 \sum ^ n _ { i = 1 } \log ( x _ { i } ) - a \theta \sum ^ n _ { i = 1 } \frac { 1 } { x ^ 2 _ i } \\ & + ( b - 1 ) \sum ^ { n } _ { i = 1 } \log \left \{ 1 - \exp \left ( - \frac { \theta } { x _ i ^ 2 } \right ) \right \} . \end{align*}


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


\begin{align*} \mathcal { I } ( x , t ) = \bigcup _ { i = 1 } ^ p [ t _ { i , e n } ( x , t ) , t _ { i , e x } ( x , t ) ] . \end{align*}


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


\begin{align*} \begin{aligned} | \mathfrak { a } ( | v | ) | + | \mathfrak { b } ( | v | ) | \leq C ( 1 + | v | ) \ \forall v \in \R ^ 3 \ , . \end{aligned} \end{align*}


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


\begin{align*} \begin{aligned} & \int _ { B } ( u _ p - u _ { p + N } ) ( \Delta u _ p ) ^ k \wedge ( \Delta u _ { p + N } ) ^ { m - k } \wedge \beta _ n ^ m \\ \leq & \sigma ( v , p , N ) + \int _ B u _ p ( \Delta u _ p ) ^ m \wedge \beta _ n ^ { n - m } - \int _ B u _ { p + N } ( \Delta u _ { p + N } ) ^ m \wedge \beta _ n ^ { n - m } \end{aligned} \end{align*}


\begin{align*} B _ q ( \phi ) ( z ) & = \int _ { \mathbb { C } } f ( w ) | w | ^ { \frac { 2 } { q } - 2 } e ^ { - \frac { | w | ^ 2 } { q } } \overline { K _ q ( w , z ) } d A ( w ) \end{align*}


\begin{align*} \begin{aligned} \langle \L f , f \rangle \geq \lambda \| f \| ^ 2 _ { L ^ 2 ( \nu ) } \end{aligned} \end{align*}


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


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


\begin{align*} c _ \sigma \colon \Gamma = \{ \pm 1 \} \to \mathrm { A u t } _ { \mathbb { C } } ( X _ { \mathbb { C } } ) , \left \{ \begin{array} { r l } 1 \mapsto & \mathrm { i d } _ { X _ { \mathbb { C } } } \\ - 1 \mapsto & \tau _ * ( \sigma _ X ^ { - 1 } \circ \sigma ) = \Psi \end{array} \right . \end{align*}


\begin{align*} V _ { \alpha , \beta , \gamma } = \mathbb { C } [ \partial ] v , L _ \lambda v = ( \partial + \alpha \lambda + \beta ) v , H _ \lambda v = \gamma v . \end{align*}


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


\begin{align*} A ( 0 ) \nabla ( L u ) \cdot \nabla u = - f ' ( u ) | \nabla u | _ { A ( 0 ) } ^ 2 . \end{align*}


\begin{align*} \lim _ { k \to \infty } \sup _ { z _ 0 \in \mathbb { R } } \int _ { B ( z _ 0 e _ z , R ) } | b _ { n _ { k } } | ^ { 2 } \dd x = 0 . \end{align*}


\begin{align*} C \geq h ( x , t ) - h ( x , 0 ) = \int _ { 0 } ^ { t } ( P - h ) ( x , t ) d t . \end{align*}


\begin{align*} c = 4 + ( n + 3 ) \left ( n + 2 + \norm { \tilde \Psi } _ { H ^ 1 } ^ 2 \right ) . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow 0 } I I _ 3 = 0 . \end{align*}


\begin{align*} W = \frac { 1 } { 2 } \Big [ & ( \mu ^ * + \varkappa ) \ , { \rm e } ^ * _ { j i } { \rm e } ^ * _ { j i } + \mu ^ * \ , { \rm e } ^ * _ { j i } { \rm e } ^ * _ { i j } + \lambda \ , { \rm e } ^ * _ { i i } { \rm e } ^ * _ { j j } + \gamma \ , \mathfrak { K } _ { j i } \mathfrak { K } _ { j i } + \beta \ , \mathfrak { K } _ { j i } \mathfrak { K } _ { i j } + \alpha \ , \mathfrak { K } _ { i i } \mathfrak { K } _ { j j } \Big ] . \end{align*}


\begin{align*} a x ^ q - b x = 0 \end{align*}


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


\begin{align*} \max _ { v \in \Lambda } e ( \alpha ^ { ( v ) } , G ^ { ( v ) } ) = e ( 0 , G ) + 2 H - 3 / 2 \end{align*}


\begin{align*} f ( x ) = \sum _ { 1 \le i < j \le M } \left | \det \begin{pmatrix} x _ i & x _ j \\ z _ i & z _ j \end{pmatrix} \right | \quad \hbox { a n d } g ( x ) = \sum _ { 1 \le i < j < k \le M } \left | \det \begin{pmatrix} x _ i & x _ j & x _ k \\ y _ i & y _ j & y _ k \\ z _ i & z _ j & z _ k \end{pmatrix} \right | . \end{align*}


\begin{align*} E = \big \{ ( u , v ) \in V ^ 2 : \ ; \pi ( u , v ) > 0 \big \} . \end{align*}


\begin{align*} \widehat { f } ( a , z ) = R _ k ^ { n - 1 } ( - \lambda , f ) e _ { k , \lambda } ^ { n - 1 } ( z , 0 ) \end{align*}


\begin{align*} \mathcal W = \mathcal W ( e _ k , ( M ' _ { k , \mathrm { o u t } } ) ^ { - 1 } ( N _ { ( - , k ) } ) ) \coloneqq \{ N _ k \mid N _ k \subset ( M ' _ { k , \mathrm { o u t } } ) ^ { - 1 } ( N _ { ( - , k ) } ) \subset M _ k ' , \ \} , \end{align*}


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


\begin{align*} M ( x ^ 1 , \dots , x ^ m ) = \frac { 1 } { m ! } \prod _ k \left ( \sum \limits _ i x _ i ^ k \frac { \partial } { \partial x _ i } \right ) P ( x ) , \end{align*}


\begin{align*} \lim _ { \lambda \to \infty } \frac { 1 } { m ( B ( \mathbf 0 , \lambda ) \cap \Omega ) } \int _ { B ( \mathbf 0 , \lambda ) \cap \Omega } \vert v ( \mathbf x ) \vert \mathrm d \mathbf x = 0 , \end{align*}


\begin{align*} \Vert M _ { \widetilde { H } \vert _ { \mathbb { Z } _ N \times \mathbb { Z } _ N } } : S _ { p _ 1 } ^ { 2 N + 1 } \times S _ { p _ 2 } ^ { 2 N + 1 } \rightarrow S _ { 1 } ^ { 2 N + 1 } \Vert \leq A _ { p _ 1 , p _ 2 , 2 N + 1 } . \end{align*}


\begin{align*} W _ + ( r ) & = ( 2 L + 3 ) \left ( - \frac { f } { r } + \Q + \kappa \frac { r } { f } \right ) , \\ W _ - ( r ) & = - \frac { f } { r } - \Q + \kappa \frac { r } { f } , \end{align*}


\begin{align*} \widetilde { f } ( \lambda , b ) = \int _ { G / K } f ( x ) e ^ { ( - i \lambda + \rho ) A ( x , b ) } d x \end{align*}


\begin{align*} \widetilde { E } _ { \omega } : = \inf _ { u \in \mathcal { M } _ { \omega } } J _ { \omega } ( u ) \end{align*}


\begin{align*} \lim _ { n \to \infty } \mu ( T ^ { m _ n + j } A \cap B ) = \lim _ { n \to \infty } \mu ( T ^ { m _ n } A \cap T ^ { - j } B ) = \mu ( A ) \mu ( T ^ { - j } B ) = \mu ( A ) \mu ( B ) . \end{align*}


\begin{align*} Z = \prod _ { k \in \N } Z _ k , \end{align*}


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


\begin{align*} \sum _ s \phi _ { i s } ( \partial _ i ) \varphi _ s & = \frac { 1 } { 2 } \partial _ i ( \varphi ^ { - 1 } _ i ) \varphi _ i - \sum _ { s \neq i } \frac { \Gamma _ { i s } ^ s \varphi _ s } { \varphi _ i } = \frac { 1 } { 2 \varphi _ i } \partial _ i \Big ( \sum _ s \varphi _ s \Big ) = 0 . \end{align*}


\begin{align*} H = ( p \wedge q ^ { \perp } ) H \oplus ( p ^ { \perp } \wedge q ) H \oplus ( p \wedge q ) H \oplus ( p ^ { \perp } \wedge q ^ { \perp } ) H \oplus ( e _ 1 + e _ 2 ) H . \end{align*}


\begin{align*} D _ { x , 2 } & = \frac { 1 } { 2 L _ { x y } } \left ( \sqrt { n \sigma _ { G , r } ^ 2 } + L _ { y } \sqrt { n } M _ y + \| \nabla G ( Y ^ * ) \| \right ) \\ & + \frac { 1 } { 2 \mu _ { x } } \left ( \sqrt { n \sigma _ { F , r } ^ 2 } + L _ { x } \sqrt { n } M _ x + \| \nabla F ( X ^ * ) \| \right ) , \end{align*}


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


\begin{align*} \sum _ { j = 1 } ^ d F _ j = F _ 1 + \cdots + F _ d = \left \{ \sum _ { j = 1 } ^ d a _ j : a _ j \in F _ j \ \forall 1 \le j \le d \right \} . \end{align*}


\begin{align*} \tilde { \kappa } ( p ) \cdot \tilde { k _ 1 } ( p ) \cdot \tilde { k _ 2 } ( p ) \ , = \ , \frac { 1 } { p } , \ \Re ( p ) > c \end{align*}


\begin{align*} \langle a , b \rangle : = ( f | [ a , b ] ) . \end{align*}


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


\begin{align*} \sum _ { k = 1 } ^ { \infty } e ^ { - t \tau _ { k } } = t ^ { 1 - n } \int _ { \partial \Omega } a _ 0 ( x ) \ , d S + t ^ { 2 - n } \int _ { \partial \Omega } a _ 1 ( x ) \ , d S + \begin{cases} O ( t \log t ) , & n = 2 , \\ O ( t ^ { 3 - n } ) , & n \geqslant 3 , \end{cases} \ t \to 0 ^ + . \end{align*}


\begin{align*} \left ( K _ H ( x ) \nabla q | \nabla \left ( \frac { u ^ 2 } { q } \right ) \right ) = \left ( K _ H ( x ) \nabla u | \nabla u \right ) - q ^ 2 \left ( K _ H ( x ) \nabla \left ( \frac { u } { q } \right ) | \nabla \left ( \frac { u } { q } \right ) \right ) . \end{align*}


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


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


\begin{align*} \theta _ 2 ^ { l _ 1 + k _ 1 n } u _ n = b \beta ^ n \zeta ^ { y - x n } ( \theta _ 1 ^ { l _ 1 + k _ 1 n } - ( - \zeta ^ { x n - y } ) \theta _ 2 ^ { l _ 1 + k _ 1 n } ) \end{align*}


\begin{align*} \ell _ t = 2 \ , \ell _ b \qquad ( \beta = \gamma ) \Psi = \frac { 3 } { 2 } . \end{align*}


\begin{align*} | A | \ , | A + B + C | = & \sum _ { k + j + m = n } { n \choose k , j , m } | A | V ( A [ k ] , B [ j ] , C [ m ] ) \\ = & \sum _ { 0 \le j + m \le n } { n \choose j , m , n - j - m } | A | V ( A [ n - j - m ] , B [ j ] , C [ m ] ) , \end{align*}


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


\begin{align*} a _ { n , k } = \frac 1 n \sum _ { j \in [ n ] ^ k } \rho _ H ( j _ 1 - j _ 2 ) \rho _ H ( j _ 2 - j _ 3 ) \cdots \rho _ H ( j _ { k - 1 } - j _ k ) \rho _ H ( j _ k - j _ 1 ) . \end{align*}


\begin{align*} \langle A _ q ^ w , A _ q ^ z \rangle _ { L ^ 2 ( \mathbb { R } ) } = K _ q ( w , z ) . \end{align*}


\begin{align*} \sum _ { k , \ell = 1 } ^ n A _ { k \ell } ( y , \omega ) \ , \xi _ k \ , \xi _ \ell \geqslant a _ 0 { \vert \xi \vert } ^ 2 . \end{align*}


\begin{align*} [ T ] = \frac { 1 } { \mu } J ( \partial _ x h + I \partial _ y h ) = \frac { 2 } { \mu } h _ z J \end{align*}


\begin{align*} f ( z , t ) = ( 2 \pi ) ^ { - n - 1 } \ , \int _ { - \infty } ^ \infty e ^ { i \lambda t } \ , \big ( \sum _ { k = 0 } ^ \infty \rho _ k ^ \lambda ( f ) e ^ { n - 1 } _ { k , \lambda } ( z , 0 ) \big ) | \lambda | ^ n d \lambda . \end{align*}


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


\begin{align*} \left ( \begin{array} { c c } 0 & b \\ c & d - a \\ \end{array} \right ) \left ( \begin{array} { c } 1 \\ x \\ \end{array} \right ) = \mu \left ( \begin{array} { c } 1 \\ x \\ \end{array} \right ) . \end{align*}


\begin{align*} s _ \Sigma ( \bar x ) & \le s _ \Sigma ( x ) + \frac { 1 } { \lambda } d ( x , \bar x ) \le r _ m + \frac { \bar K } \lambda r _ { m ' } < \left ( \frac 1 { \bar K ^ 2 } + \frac 1 { \lambda \bar K } \right ) r _ { m ' - 2 } \le \frac { r _ { m ' - 2 } } { \bar K } \le \frac { r _ i } { \bar K } . \end{align*}


\begin{align*} \mathcal { M } _ { \Gamma } = \left \lbrace p R \ , \vert \ , p _ { i } = \dfrac { 1 } { Z ( \theta ) } \prod \limits _ { F \in \mathrm { f a c e t } ( \Gamma ) } \theta ^ { F } _ { i _ { F } } \right \rbrace \end{align*}


\begin{align*} k ( M ) = \inf \frac { { \rm V o l } _ { n - 1 } ( \Sigma ) } { { \rm V o l } _ n ( A ) | \log ( { \rm V o l } _ n ( A ) ) | } , \end{align*}


\begin{align*} p _ { \gamma _ Z } ( F v + G w ) = p _ { \gamma _ Z } ( - \norm { v } _ { V } z ) = \norm { v } _ { V } p _ { \gamma _ Z } ( z ) \leq \norm { v } _ { V } \frac { \varepsilon } { \norm { v } _ { V } } = \varepsilon , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T | g _ n ( t ) | d t = 0 . \end{align*}


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