\begin{align*} \mathbf { D } _ { i l } = \begin{cases} \mathbf { I } _ { N _ { } } , & l \in \mathcal { M } _ { i } , \\ \mathbf { O } _ { N _ { } } , & l \notin \mathcal { M } _ { i } . \end{cases} \end{align*}


\begin{align*} \xi _ z = \frac { 1 } { r ! } \sum _ { j = 1 } ^ { r ! } T ^ j ( \xi _ { i _ z , z } ) \end{align*}


\begin{align*} & \rho _ L ( \theta ) = \sum _ { | l | \leq L } \widehat { \rho _ L } ( l ) e ( l \theta ) = \widehat { \rho _ L } ( 0 ) + \sum _ { 1 \leq l \leq L } \widehat { \rho _ L } ( l ) 2 \cos ( 2 \pi l \theta ) , \end{align*}


\begin{align*} \frac { d } { d s } \left ( \frac { \zeta ' ( s ) } { \zeta ( s ) } \right ) = \frac { 1 } { ( s - 1 ) ^ 2 } - \sum _ { n = 1 } ^ \infty \frac { 1 } { ( s + 2 n ) ^ 2 } - \sum _ \rho \frac { 1 } { ( s - \rho ) ^ 2 } . \end{align*}


\begin{align*} ( x = w \varepsilon ^ \mu , x ' = w ' \varepsilon ^ { \mu ' } , v , v ' , L , R , J ) \end{align*}


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


\begin{align*} H = \bigcup _ { n \in \mathbb { N } } Q _ n \end{align*}


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


\begin{align*} \partial ^ \square H = g ( c ) \Delta _ \Gamma H + 2 \nabla _ \Gamma g ( c ) \cdot \nabla _ \Gamma H + \big ( \Delta _ \Gamma g ( c ) + \big | \nabla _ \Gamma \nu \big | ^ 2 g ( c ) \big ) H \end{align*}


\begin{align*} A ^ { 1 } A ^ { 2 } : = \begin{bmatrix} A ^ { 1 } \\ \hline A ^ { 2 } \end{bmatrix} . \end{align*}


\begin{align*} 0 = W _ j + \gamma _ j ( W _ i ) - W _ i - \gamma _ i ( W _ j ) + ( 1 - R _ { H ' , n } ) ( Q _ 2 ) \end{align*}


\begin{align*} h ( \mu ) = \frac { 2 ^ { 2 - \alpha - \beta } } { \sqrt { 2 \pi } \ , \sigma \ , \mathrm { B } ( \alpha , \beta ) \left [ 1 - I _ { \frac { 1 } { 2 } } ( \alpha , \beta ) \right ] } . \end{align*}


\begin{align*} R \cdot R ( u , J u , x , J x ; x , J x ) = \frac { 1 } { 2 } \ , L ( p , \pi ^ h ) \ , Q ^ c ( g , R ) ( u , J u , x , J x ; x , J x ) . \end{align*}


\begin{align*} i x , x ' \in E _ i , ~ \frac { q ( x ) } { p ( x ) } = \frac { q ( x ' ) } { p ( x ' ) } , \end{align*}


\begin{align*} L : = \max \bigg ( \frac { 1 } { \tilde \varepsilon } , 8 0 p \ , \frac { C ( \tilde { \varepsilon } ) } { c ( \tilde { \varepsilon } ) } \bigg ) . \end{align*}


\begin{align*} [ a _ 1 , a _ 2 , \cdots , a _ m ] : = \cfrac { 1 } { a _ 1 - \cfrac { 1 } { a _ 2 - \cfrac { 1 } { \ddots - \cfrac { 1 } { a _ { m } } } } } \end{align*}


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


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


\begin{align*} \lambda _ 1 \alpha ^ { m - 1 } + \cdots + \lambda _ m \beta ^ { m - 1 } = 0 , \end{align*}


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


\begin{align*} \mu | _ { E _ { 2 , m + e _ i - e _ j } } = \lambda ^ j = \lambda _ x | _ { E _ { 2 , m + e _ i - e _ j } } \end{align*}


\begin{align*} \Phi ( [ a \cos ( 2 \pi \ell \phi ) + b \sin ( 2 \pi \ell \phi ) ] u _ \ell + [ - a \sin ( 2 \pi \ell \phi ) + b \cos ( 2 \pi \ell \phi ) ] w _ \ell , p ) \\ = [ c \cos ( 2 \pi \ell \phi ) + d \sin ( 2 \pi \ell \phi ) ] u _ \ell + [ - c \sin ( 2 \pi \ell \phi ) + d \cos ( 2 \pi \ell \phi ) ] w _ \ell . \end{align*}


\begin{align*} X = \nabla f , X ^ { \flat } = d f \nabla X ^ { \flat } = ( f ) \ , . \end{align*}


\begin{align*} V _ { \delta , Z } ( x ) - q ( x ) = V _ { \delta , z _ i , \hat { q } _ { \delta , i } , z _ i } ( x ) - \hat { q } _ { \delta , i } + O \left ( \frac { \ln | \ln \varepsilon | } { | \ln \varepsilon | ^ 2 } \right ) . \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*} \left \{ \begin{aligned} - d y _ t = & \big [ f ^ \alpha ( t , \zeta _ t , \mu _ t ) + f _ 0 ( t ) \big ] d t + \big [ g ^ \alpha ( t , \zeta _ t , \mu _ t ) + g _ 0 ( t ) \big ] d \overleftarrow B _ t - z _ t d W _ t , \\ d p _ t = & \big [ F ^ \alpha ( t , \zeta _ t , \mu _ t ) + F _ 0 ( t ) \big ] d t + \big [ G ^ \alpha ( t , \zeta _ t , \mu _ t ) + G _ 0 ( t ) \big ] d W _ t - q _ t d \overleftarrow B _ t , \\ y _ T = & \xi , \ , \ , p _ 0 = \Psi ^ \alpha ( y _ 0 , \mathcal L ( y _ 0 ) ) + \Psi _ 0 , \end{aligned} \right . \end{align*}


\begin{align*} r _ { \ell } \leq e ^ { 4 0 \varepsilon q _ n } \frac { e ^ { - q _ n L } } { \max ( | \ell | , 1 ) } \max ( r _ { \ell - 1 } , r _ { \ell + 1 } ) \times \begin{cases} \max ( | \ell | , e ^ { \delta _ n q _ n } , 1 ) , & \beta _ n \geq \delta _ n + 2 0 0 \varepsilon \\ e ^ { \beta _ n q _ n } , & \beta _ n < \delta _ n + 2 0 0 \varepsilon \end{cases} . \end{align*}


\begin{align*} - \int _ { \R } \left ( T _ i ' ( x _ i ) - 1 \right ) Q _ i ( x _ i ) d x _ i = \int _ \R \left ( T _ i ( x _ i ) - x _ i \right ) Q _ i ' ( x _ i ) \ , d x _ i . \end{align*}


\begin{align*} C _ { \ast } ( K ; \mathfrak { g } _ { \mathrm { A d } _ \rho } ) : = \displaystyle C _ { \ast } ( \widetilde { K } ; \mathbb { Z } ) \displaystyle \otimes \mathfrak { g } / \sim , \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*} { \rm I m } ( M ( w ) ) = \frac { { \rm I m } ( w ) } { | c w + d | ^ { 2 } } . \end{align*}


\begin{align*} d = d _ \alpha + b - a _ { \alpha + 1 } + \sum _ { j = 2 } ^ \infty ( a _ j - a _ { \alpha + j } ) \ . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \aligned & - \Delta u + V ( x ) u = \left ( I _ \alpha \ast | u | ^ p \right ) | u | ^ { p - 2 } u + \lambda u ~ ~ x \in \R ^ N , \\ & u ( x ) \to 0 ~ ~ ~ ~ | x | \to \infty , \endaligned \end{array} \right . \end{align*}


\begin{align*} \boldsymbol { \alpha } _ t ^ * : = ( \alpha _ t ^ { * , i } ) _ { 1 \leq i \leq N } = \left ( - R ^ { - 1 } B ( P _ t x _ t ^ { * , i } + \varphi _ t ^ { * , i } ) - h \Big ( k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } ^ * _ t } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } ^ * _ t } , \lambda ^ { N , 1 } _ { \boldsymbol { \varphi } ^ * _ t } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \varphi } ^ * _ t } ) \Big ) \right ) _ { 1 \leq i \leq N } . \end{align*}


\begin{align*} \P \left ( \frac { 1 } { m } \sum _ { i = 1 } ^ m V _ i \leq \left ( \frac { \pi } { 4 } + \frac { 1 9 9 \pi } { 3 2 0 0 k } \right ) \| \mathbf { x } \| \right ) \leq 2 \exp \left ( - \frac { c _ 5 m } { k } \right ) . \end{align*}


\begin{align*} S _ r ( a , b ) = \Gamma ( b ) \Gamma \left ( 1 - \frac { r } { 2 } \right ) \sum _ { n = 0 } ^ \infty \frac { ( - 1 ) ^ n } { ( a + n ) ^ { 1 - r / 2 } \Gamma ( b - n ) n ! } , r < 2 . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 3 \arctan \lambda _ i = \hat \theta _ 0 \in \left ( \pi , \frac { 3 \pi } { 2 } \right ) . \end{align*}


\begin{align*} \mu = \mu ( \mathbf { x } , \mathbf { s } ) = \mathbf { x } ^ T \mathbf { s } / k , \end{align*}


\begin{align*} \epsilon ( x , t ) = \int _ 0 ^ t f ( x , \tau ) - f _ a ( x ) \ , d \tau = \int _ { N T } ^ t f ( x , \tau ) - f _ a ( x ) \ , d \tau , \end{align*}


\begin{align*} \mathbb G = [ 0 , \vartheta ] \times \mathbb R ^ n \times \mathrm { P L i p } , \mathbb G _ * = \cup _ { i = 0 } ^ { I - 1 } ( i h , ( i + 1 ) h ) \times \mathbb R ^ n \times \mathrm { P L i p } _ * . \end{align*}


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


\begin{align*} S _ j ( n ) = \inf \{ t \geq 0 : N _ { d - j + 1 } ( t ) \geq n \} , n \geq 0 . \end{align*}


\begin{align*} \bigl ( | u _ t | ^ { \ell - 2 } u _ t \bigr ) _ t - a \nabla \cdot \bigl ( | \nabla u | ^ { q - 2 } \nabla u \bigr ) + b ( t , x ) | u _ t | ^ { m - 2 } u _ t = 0 , \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*} \alpha ( g , ( z _ 1 + 1 , z _ 2 , \ldots ) ) = ( \xi + ( z _ 1 + 1 ) \psi _ 1 ) \alpha ( g , Z ) - \sum _ { \overline { \Gamma } \in { \rm B i c } ( { g } , { Z } ) _ 1 } m ( \overline { \Gamma } ) \alpha ( \overline { \Gamma } ) . \end{align*}


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


\begin{align*} f ( t ) = \frac { | A \oplus _ 2 \sqrt { t } B | ^ { \frac { 2 } { k } } } { | V ( ( A \oplus _ 2 \sqrt { t } B ) [ n - k ] , Z _ 1 , \dots , Z _ k ) | ^ { \frac { 2 } { k } } _ { n - k } } \end{align*}


\begin{align*} R ( X , Y ) Z = c ( X \wedge _ g Y ) Z , \end{align*}


\begin{align*} B _ { x , y } ( \xi ) = \lim _ { t \rightarrow \infty } ( d _ X ( y , \gamma _ \xi ( t ) ) - t ) \end{align*}


\begin{align*} & \sum \limits _ { y \neq 0 , x } [ ( 2 + h \chi _ 4 ( y ) + \overline { h } \overline { \chi _ 4 } ( y ) ) ( 2 + h \chi _ 4 ( x - y ) + \overline { h } \overline { \chi _ 4 } ( x - y ) ) ] \\ & = 4 ( q - 3 ) - 4 h \chi _ 4 ( x ) - 4 \overline { h } \overline { \chi _ 4 } ( x ) - 2 \chi _ 4 ( g ) \varphi ( x ) J ( \chi _ 4 , \chi _ 4 ) + 2 \chi _ 4 ( g ) \varphi ( x ) \overline { J ( \chi _ 4 , \chi _ 4 ) } . \end{align*}


\begin{align*} \begin{aligned} \Delta u _ 1 ^ { t } \wedge \dots \wedge \Delta u _ k ^ { t } \wedge \beta _ n ^ { n - m } \rightarrow \Delta u _ 1 \wedge \dots \wedge \Delta u _ k \wedge \beta _ n ^ { n - m } . \end{aligned} \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { m \to \infty } \lim _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \sup _ { t \in [ 0 , 1 ] } | \hat { X } ^ { \epsilon } ( t ) - \varphi ( t ) | < \delta , \chi _ 2 < 1 \Big ) = - \infty . \end{align*}


\begin{align*} x '' : = w '' \varepsilon ^ { \mu '' } : = x ' r _ { a } = w ' s _ \alpha \varepsilon ^ { \mu ' - ( 1 + \langle \mu ' , \alpha \rangle ) \alpha ^ \vee } < x ' . \end{align*}


\begin{align*} \psi _ \alpha ( f ) : & = \nu ( \{ x \in X : | f ( x ) | > \alpha \} ) \end{align*}


\begin{align*} 0 = E ^ 0 \subset E ^ 1 \subset \cdots \subset E ^ { k - 1 } \subset E ^ { k } = E , \end{align*}


\begin{align*} I _ j = W _ j \setminus \bigcup _ { e \in G [ W _ j ] } e . \end{align*}


\begin{align*} \mathbb { P } ( S _ k = 0 ) ^ { 1 / p _ k } \leq \mathbb { P } ( S _ k = - k ) ^ { 1 / p _ k } + \mathbb { P } ( S _ k = k ) ^ { 1 / p _ k } \end{align*}


\begin{align*} \psi _ n ( x ) = \sum _ { j \leq n , x \geq \sup I ^ + _ j } \lambda ( I ^ + _ j ) \ ; \ ; - \sum _ { j \leq n , x \leq \inf I ^ - _ j } \lambda ( I ^ - _ j ) . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ l \beta _ k & = \lambda _ { n + 1 } \sum _ { k = 1 } ^ l \frac { d ( \gamma ( t _ { k - 1 } ) , \gamma ( t _ { k } ) ) } { d ( p _ { n + 1 } , q _ { n + 1 } ) } \leq \frac { \lambda _ { n + 1 } L ( \gamma ) } { d ( p _ { n + 1 } , q _ { n + 1 } ) } \\ & < \lambda _ { n + 1 } + \frac { \lambda _ { n + 1 } \varepsilon \delta } { d ( p _ { n + 1 } , q _ { n + 1 } ) } \leq \lambda _ { n + 1 } + \frac { \varepsilon } { 2 } . \end{align*}


\begin{align*} \overline { \Psi } ( A _ t ) = \left \{ \sum _ { \alpha \in { \mathbb { Z } } } a _ { \alpha } \ , \dd ^ { \alpha } \in \widehat { \Psi } ( A _ t ) \ ; | \ ; v a l _ t ( a _ \alpha ) \geq \alpha \right \} \end{align*}


\begin{align*} \epsilon ' = \sum _ { i \in I } \bar { \nu } _ { j ! } & \ & ( \delta _ k ^ n ) _ ! = \sum _ { J = \{ i _ 0 < \hdots < \widehat { i _ k } < \hdots < i _ n \} \subset K = \{ i _ 0 < \hdots < i _ n \} } ( \nu _ { K } ^ J ) _ ! \end{align*}


\begin{align*} \lim _ { \norm { \eta _ n } _ { H ^ 1 } \to 0 } \frac { 1 } { \norm { \eta _ n } _ { H ^ 1 } } \norm { A ^ { \tilde \Psi + \eta _ n } [ \eta _ 1 , \dots , \eta _ { n - 1 } ] - A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ { n - 1 } ] - A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ n ] } _ { H ^ 1 } = 0 . \end{align*}


\begin{align*} \left ( R ^ W _ { X , Y } \right ) ^ { J , - } = \frac { 1 } { 2 } \left ( D ^ W _ X \left ( D ^ W _ Y J \right ) - D ^ W _ Y \left ( D ^ W _ X J \right ) - D ^ W _ { [ X , Y ] } J \right ) . \end{align*}


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


\begin{align*} f _ \xi ( z ) = \begin{cases} \frac { f ( z ) } { \xi - z } & , \\ 0 & \end{cases} \end{align*}


\begin{align*} \dot { x } _ a = \gamma f _ a ( x _ a ) x _ a ( 0 ) = x _ 0 \end{align*}


\begin{align*} \Sigma _ 3 = \sum \limits _ { D \leq d < 2 D } \sum \limits _ { n \in \mathcal { N } ( d ) } \psi \left ( \frac { \sqrt { X } - n } { d } \right ) \ , . \end{align*}


\begin{align*} \begin{aligned} \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \geq \underline { \delta } ^ \star ( \delta ) & \forall \ , x \geq x ^ \star + \delta \\ \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \leq - \underline { \delta } ^ \star ( \delta ) & \forall \ , x \leq x ^ \star - \delta \ , . \end{aligned} \end{align*}


\begin{align*} C _ { B _ n } ( \sigma ) = \{ \tau \in B _ n : \sigma \tau = \tau \sigma \} \end{align*}


\begin{align*} F ( z ) = - F ( z + F ( z ) ) . \end{align*}


\begin{align*} \begin{cases} \partial _ { t } u _ i - d _ i \partial _ { x x } u _ i = f _ i ( x , t , u ) , & x \in \Omega , ~ t > 0 , \\ \partial _ x u _ i ( 0 , t ) = \partial _ x u _ i ( L , t ) = 0 , & t > 0 , \\ u _ i ( x , 0 ) = u _ { i , 0 } ( x ) , & x \in \Omega , \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*} \langle A _ q ^ w , A _ q ^ z \rangle _ { L ^ 2 ( \mathbb { R } ) } = K _ q ( w , z ) . \end{align*}


\begin{align*} \chi _ K ( 2 ) = \begin{cases} 1 & d \equiv 1 \mod 8 \\ - 1 & d \equiv 5 \mod 8 \\ 0 & \end{cases} . \end{align*}


\begin{align*} ( D _ { a ^ + } ^ { \alpha } 1 ) ( x ) = \frac { ( x - a ) ^ { - \alpha } } { \Gamma [ 1 - \alpha ] } , \ \forall x \in [ a , b ] . \end{align*}


\begin{align*} \left \| g ( x ) - h ( x ) \right \| & = \lim _ { n \rightarrow \infty } | 2 | ^ { n } \left \| g ( 2 ^ { n + 1 } x ) - h ( 2 ^ { n + 1 } x ) \right \| \\ & \leq \lim _ { k \rightarrow \infty } | 2 | ^ { n } \max \{ \left \| g ( 2 ^ { n + 1 } x ) - h ( 2 ^ { n + 1 } x ) \right \| , \left \| g ( 2 ^ { n + 1 } x ) - h ( 2 ^ { n + 1 } x ) \right \| \} \\ & \leq \lim _ { j \rightarrow \infty } \lim _ { n \rightarrow \infty } \max \{ | 2 | ^ { k } \mu ( 0 , 2 ^ { k + 1 } x ) ; j \leq k < n + j \} \\ & = 0 \end{align*}


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


\begin{align*} \lambda _ 0 = \sup \{ \lambda \geq 0 : p \cdot \lambda ( p ^ { - 1 } \cdot q ) \notin \Omega \} . \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } \sup _ { n \in \mathbb { N } } I I _ n = 0 . \end{align*}


\begin{align*} \Psi _ 1 ( \hat \alpha ) = 0 , \Psi _ 1 ' ( \hat \alpha ) > 0 \end{align*}


\begin{align*} g ( E ) = \displaystyle \lim _ { k \rightarrow \infty } \frac { ( - 1 ) ^ { k } } { \Gamma ( k + 1 ) } Z _ { q } ^ { ( k ) } ( \beta ) \beta ^ { k + 1 } Q _ { 1 } ( 2 - q ) \Biggr | _ { \beta = \frac { k \xi _ { m } } { E } } . \end{align*}


\begin{align*} \omega _ { u v } = \begin{cases} x _ u y _ v & t ( u , \bullet ) = t ( \bullet , v ) \\ 0 & \end{cases} \end{align*}


\begin{align*} \lim _ { n \to + \infty } | D z _ { n } | ( ( a , b ) ) = \lim _ { n \to + \infty } \sum _ { i = 1 } ^ { \infty } | J _ { n } ( i ) | = \sum _ { i = 1 } ^ { \infty } | J _ { \infty } ( i ) | . \end{align*}


\begin{align*} \limsup _ { n \to \infty } T _ { W _ { n J } , \psi } ( \mu _ n ( Q ^ n ) ) & \le \limsup _ { n \to \infty } T _ { W _ { n J } , \psi _ m } ( \mu _ n ( Q ^ n ) ) \le \limsup _ { n \to \infty } T _ { W , \psi _ m } ( \mu _ n ( Q ^ n ) ) = T _ { W , \psi _ m } ( \mu _ \infty ) , \end{align*}


\begin{align*} s _ { h , q } ( p , c , \rho ) : = \frac { \rho } { q } \sum _ { g = 0 } ^ { q - 1 } \zeta ^ { ( h + 1 ) g } ~ \frac { p ' ( c + \rho \zeta ^ g ) } { p ( c + \rho \zeta ^ g ) } ~ { \rm f o r } ~ \zeta ~ { \rm o f ~ ( \ref { e q z t } ) ~ a n d } ~ h = 0 , 1 , \dots , q - 1 , \end{align*}


\begin{align*} \phi _ t + u \cdot \nabla \phi = \mu \Delta \phi . \end{align*}


\begin{align*} A u t \left ( \mathcal { R P } ( G ) \right ) = \mathbb Z _ 2 \times { \prod _ { i = 1 } ^ { t } S _ { \widehat { u _ i } } } . \end{align*}


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


\begin{align*} J a = \frac { { { c _ p } \left ( { { T _ w } - { T _ { s a t } } } \right ) } } { { { h _ { l v } } } } , \end{align*}


\begin{align*} \lim _ { n \to \infty } \sup _ { y \in \R ^ N } \int _ { B _ r ( y ) } u _ n ^ 2 d x = 0 , \end{align*}


\begin{align*} \mathsf { M } ^ n = & \{ \mathsf { m } ^ n _ 1 , \cdots , \mathsf { m } ^ n _ n \} \\ = & \left \{ g \bigl ( \xi _ 0 \bigr ) , g \bigl ( \xi _ 1 ^ { \mathsf { M } ^ n } \bigr ) , \cdots , g \bigl ( \xi _ { n - 1 } ^ { \mathsf { M } ^ n } \bigr ) \right \} \in \Theta _ n , \end{align*}


\begin{align*} S H ^ { * } _ { M } ( K ; \Lambda _ { 0 } ) : = H ^ { * } ( \widehat { t e l } ( \mathcal { C } ; \Lambda _ { 0 } ) ) . \end{align*}


\begin{align*} f \circ \gamma ( t ) = \gamma ( t + b _ \gamma ( f ) ) . \end{align*}


\begin{align*} \delta ( z ) = \exp [ { \mathcal { C } \log \left ( 1 + \bar { r } _ 1 r _ 2 \right ) } ] , z \in \mathbb { C } ^ { \pm } , \end{align*}


\begin{align*} \int _ 0 ^ { ( i - 1 ) \Delta _ n } ( \mathcal S ( \Delta _ n ) - I ) \Sigma _ s ^ { \mathcal S _ n } ( \mathcal S ( \Delta _ n ) - I ) ^ * d s = & ( i - 1 ) \Delta _ n ( e \otimes ( \Delta _ i \mathcal S f ) ) ( e \otimes ( \Delta _ i \mathcal S f ) ) ^ * \\ = & ( i - 1 ) \Delta _ n ( \Delta _ i \mathcal S f ) ^ { \otimes 2 } . \end{align*}


\begin{align*} A _ 0 ( \Lambda ) = \frac { \Lambda - 2 \pm \sqrt { \Lambda ( \Lambda - 4 ) } } { 2 } , \ B _ 0 ( \Lambda ) = \frac { \Lambda - 2 \mp \sqrt { \Lambda ( \Lambda - 4 ) } } { 2 } . \end{align*}


\begin{align*} O _ { \gamma } ( f _ 0 ) = 1 + ( q + 1 ) \frac { q ^ { d _ \gamma } - 1 } { q - 1 } . \end{align*}


\begin{align*} h _ { p } ^ { \left ( 0 \right ) } = \frac { \left ( - 1 \right ) ^ { l } } { 2 \pi i } \frac { 1 } { \left ( p + 1 \right ) \left ( p + 2 \right ) \cdots \left ( p + l \right ) } \int _ { \mathbb { T } } \left ( \widetilde { \varphi } _ { 0 } ^ { - } \left ( u \right ) \right ) ^ { \left ( l \right ) } u ^ { p + l } d u . \end{align*}


\begin{align*} \gamma _ \Lambda ( \eta _ { \Lambda } \eta _ \Delta | \eta _ { \Lambda ^ c } ) = \gamma _ { \Lambda } ( \eta _ { \Lambda } | \eta _ { \Lambda ^ c } ) \mathbf { 1 } _ { \eta _ \Delta } ( \eta _ { \Lambda ^ c } ) . \end{align*}


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


\begin{align*} \sup _ { t _ N \in [ - T , T ] } \| P _ N u ( t _ N ) \| _ { L ^ 2 } ^ 2 & \leqslant \| P _ N u _ 0 \| _ { L ^ 2 } ^ 2 \\ & + \sup _ { t _ N \in [ - T , T ] } \Big | \int _ { [ 0 , t _ N ] \times \R ^ 2 } P _ N u P _ N ( u \partial _ x u ) ~ d t d x d y \Big | . \end{align*}


\begin{align*} [ \partial a _ \lambda b ] & = - \lambda [ a _ \lambda b ] , \ \ [ a _ \lambda \partial b ] = ( \partial + \lambda ) [ a _ \lambda b ] \ \ \mbox { ( c o n f o r m a l \ s e s q u i l i n e a r i t y ) } , \\ { [ a _ \lambda b ] } & = - [ b _ { - \lambda - \partial } a ] \ \ \mbox { ( s k e w - s y m m e t r y ) } , \\ { [ a _ \lambda [ b _ \mu c ] ] } & = [ [ a _ \lambda b ] _ { \lambda + \mu } c ] + [ b _ \mu [ a _ \lambda c ] ] \ \ \mbox { ( J a c o b i \ i d e n t i t y ) } . \end{align*}


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


\begin{align*} g \big | _ { U _ \alpha } = D ^ 2 u _ \alpha . \end{align*}


\begin{align*} \left ( \mathfrak { M } ^ \prime ( f ) \eta \right ) ( x , m ) = f \left ( O ^ m x \right ) \eta ( x , m ) , \ ; \left ( \mathcal { F } ' ( O ^ k ) \eta \right ) ( x , m ) = \eta ( x , m - k ) . \end{align*}


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


\begin{align*} x = - r , y = - q . \end{align*}


\begin{align*} & \sum _ q \sum _ p \sum _ { u \in \mathcal { O } ^ * _ F } \int _ { \mathbb { A } ^ n } f _ { B } ( x ) f _ { B } ( \gamma u x ) d \alpha ^ n \\ & = \sum _ q \sum _ p \sum _ { u \in \mathcal { O } ^ * _ F } \sum _ { 1 \leq i , j \leq h _ F } \frac { B ^ n } { w _ F ^ 2 } \int _ { \mathbb { A } ^ n } \phi ( a _ i x ) \phi ( a _ j \gamma u x ) d \alpha ^ n , \end{align*}


\begin{align*} \begin{aligned} & \Phi _ { j } [ ( v _ { 1 } , v _ { 2 } ) ] ( t ) \\ & = i \int _ { t } ^ { \infty } U ( t - \tau ) \left \{ \tilde { { \mathcal N } } _ { j } ( v _ { 1 } + u _ { \mathrm { a p } , 1 } , v _ { 2 } + u _ { \mathrm { a p } , 2 } ) - { \tilde { \mathcal N } } _ { j } ( u _ { \mathrm { a p } , 1 } , u _ { \mathrm { a p } , 2 } ) \right \} ( \tau ) d \tau \\ & + i \int _ { t } ^ { \infty } U ( t - \tau ) \mathcal { E } _ { j } ( \tau ) d \tau + \mathcal { R } _ j \end{aligned} \end{align*}


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


\begin{align*} \aligned e _ 1 & \ , : = \ , ( 1 - y ) \ , \partial _ x - z \ , \partial _ y + x \ , \partial _ u , \\ e _ 2 & \ , : = \ , ( 1 - y ) \ , \partial _ y - 2 z \ , \partial _ z + u \ , \partial _ u , \\ e _ 3 & \ , : = \ , u \ , \partial _ x - \tfrac { 4 } { 3 } \ , x \ , \partial _ y + ( 1 - y ) \ , \partial _ z , \\ e _ 4 & \ , : = \ , x \ , \partial _ x - z \ , \partial _ z + 2 \ , u \ , \partial _ u . \endaligned \end{align*}


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


\begin{align*} \sum _ { \xi _ { \Theta } } \int _ { \Omega } h _ { \Delta } ( \eta ) ( \hat { c } _ { \Theta } ( \eta , \xi _ { \Theta } ) - c _ { \Theta } ( \eta , \xi _ { \Theta } ) ) \mu ( d \eta ) = \sum _ { \xi _ { \Theta } } \int _ { \Omega } c _ { \Theta } ( \eta , \xi _ { \Theta } ) \left ( h _ { \Delta } ( \xi _ { \Theta } \eta _ { \Theta ^ c } ) - h _ { \Delta } ( \eta ) \right ) \mu ( d \eta ) . \end{align*}


\begin{align*} T ( u ) = T _ - - \frac { 4 \gamma J u } { D } , u \in [ 0 , 1 ] , \end{align*}


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


\begin{align*} M _ i ( t ) = Q ^ * _ i \circ ( ( 1 - t ) \mathrm { I d } + t T _ i ) ^ { - 1 } , \end{align*}


\begin{align*} \bar { y } _ t = - \int _ 0 ^ t \exp \left ( - \int _ s ^ t Z _ { t ' , 3 } d t ' \right ) \bar { z } _ { s , 3 } y _ s d s . \end{align*}


\begin{align*} \dfrac { p } { p + 1 } \displaystyle \sum _ { n = 1 } ^ \infty \dfrac { 2 ^ { \omega ( \gcd ( n , p ) ) } \cdot 2 r _ { Q ^ \ast } ( n ) ^ 2 } { n } \displaystyle \sum _ { d = 1 } ^ \infty \psi \left ( d \sqrt { \dfrac { n } { p } } \right ) \leq \frac { 1 } { \min Q ^ { * } } + 3 2 1 6 . 6 5 2 4 \frac { M ( 2 5 . 0 9 p ^ { 3 5 / 6 } ) } { p ^ { 1 / 4 } } . \end{align*}


\begin{align*} f ( t ) = ( \alpha t ) ^ { \delta } { } _ { 0 } F _ { 1 } \left ( \ ; ; \frac { 1 } { 2 } + \delta ; - \frac { 1 } { 4 } \alpha ^ { 2 } t ^ { 2 } \right ) = \left \{ \begin{array} { l l } \cos \alpha t & \mbox { f o r } \ ; \delta = 0 \\ \sin \alpha t & \mbox { f o r } \ ; \delta = 1 \\ \end{array} \right . \end{align*}


\begin{align*} \div ( A ( x ) \nabla v ) + h ( x ) = 0 B _ 1 . \end{align*}


\begin{align*} x y = y x = x \wedge y \end{align*}


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


\begin{align*} \lambda _ n ( A ) = \frac { \lambda ( A \cap A _ n ) } { \lambda ( A _ n ) } , \end{align*}


\begin{align*} G _ x ( \theta ) : = \sum _ { n \in \Z } g \left ( \pi _ N ( x ) ( \theta + n ) \right ) , \end{align*}


\begin{align*} \limsup _ { N \to \infty } \frac { 1 } { N } \sum _ { n = 0 } ^ { N - 1 } \min _ { 1 \leq i < j \leq k } \widetilde { d } ( T _ { \vec { v } } ^ n ( x _ i , t _ i ) , T _ { \vec { v } } ^ n ( x _ j , t _ j ) ) \geq \eta > 0 . \end{align*}


\begin{align*} ( \Delta u ) ^ { m } \wedge \beta _ n ^ { n - m } = m ! ( n - m ) ! \mathcal { H } _ m ( u ) \Omega _ { 2 n } . \end{align*}


\begin{align*} \sum _ { k \leq j < m } \delta _ { 0 , j } + \sum _ { 0 < i < k } \delta _ { i , k } - \sum _ { k < j \leq m } \delta _ { k , j } - \sum _ { 0 < i < k } \delta _ { i , m } = 1 + \sum _ { 0 \leq i < m } ( \delta _ { 0 , i } + \delta _ { i , k } - 1 ) , \end{align*}


\begin{align*} & \lambda _ x | _ { E _ { d ^ * ( y ) } } = \lambda _ y \\ & \lambda _ x | _ { E _ { [ w b ^ { - n } , w b ^ { - n + 1 } a ] } } = \phi _ n \end{align*}


\begin{align*} G _ { \epsilon } ( t ) = \int _ 0 ^ 1 K ( t , r ) \theta _ { \epsilon } ( r ) d r , 0 \leq t \leq 1 , \end{align*}


\begin{align*} h _ { i } = e _ { i } \cdot F ( x ) . \end{align*}


\begin{align*} X ^ { \epsilon } ( t ) & = x _ { 0 } + \int ^ { t } _ { 0 } b ( X ^ { \epsilon } ( s ) ) d s + \lambda ( \epsilon ) \int ^ { t } _ { 0 } \sigma ( X ^ { \epsilon } ( s ) ) \theta _ { \epsilon } ( s ) d s , \end{align*}


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


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


\begin{align*} H \left ( \rho _ n ^ { k } ( t , \cdot ) \mid \bar { \rho } ^ { \otimes k } ( \cdot ) \right ) = \sup _ { g \in L ^ \infty ( \mathbb { T } ^ { k d } ) } \mathbb { E } _ { \rho _ n ^ { k } } ( g ) - \mathbb { E } _ { \bar { \rho } ^ { \otimes k } } ( e ^ g ) + 1 . \end{align*}


\begin{align*} { \bar F _ i } = - { { w } _ i } { \bar F } , \end{align*}


\begin{align*} \begin{aligned} \det \left ( \frac { \partial ^ 2 u } { \partial \overline { q _ l } \partial { q _ k } } \right ) , \end{aligned} \end{align*}


\begin{align*} ( \widehat { h } ^ i , \widehat { h } ^ i { } _ j , \widehat { h } ^ i { } _ { j k } ) & = ( \phi ( h ^ a ) ^ i , D \phi ( h ^ a ) ^ i { } _ \alpha h ^ \alpha { } _ j , H \phi ( h ^ a ) ^ i { } _ { \alpha \beta } h ^ \alpha { } _ j h ^ \beta { } _ k + D \phi ( h ^ a ) ^ i { } _ \alpha h ^ \alpha { } _ { j k } ) \\ * & = ( \phi ( h ^ a ) ^ i , ( D \phi ( h ^ a ) ^ i { } _ j , H \phi ( h ^ a ) ^ i { } _ { j k } ) ( h ^ i { } _ j , h ^ i { } _ { j k } ) ) , \end{align*}


\begin{align*} \mathbf { \Phi } _ { \mathrm { r } } ^ { H } \mathbf { \Phi } _ { \mathrm { r } } + \mathbf { \Phi } _ { \mathrm { t } } ^ { H } \mathbf { \Phi } _ { \mathrm { t } } = \mathbf { I } _ { M } . \end{align*}


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


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


\begin{align*} z = ( \pi \otimes ( \chi \circ \det ) , \alpha _ z ) \end{align*}


\begin{gather*} \Omega = \Omega _ 1 \times \ldots \times \Omega _ n \quad \quad \mathcal { A } = \mathcal { A } _ 1 \otimes \ldots \otimes \mathcal { A } _ n \end{gather*}


\begin{align*} d u = \Delta _ p u \ , d t + \nabla u \circ d W , u ( 0 , \cdot ) = u _ 0 , \end{align*}


\begin{align*} s _ l ( k ) = \left \{ \begin{array} { l l } { k + l - 1 \choose l - 1 } & k \in [ 0 , N ] \\ { k + l - 1 \choose l - 1 } - l { k + l - N - 2 \choose l - 1 } & k \in [ N + 1 , 2 N ] \end{array} \right . . \end{align*}


\begin{align*} \dim H & \geq x - \lfloor \tilde \varepsilon x \rfloor + \ell _ { i + 1 } '' - \lfloor \tilde \varepsilon x \rfloor + \sum _ { j = 1 } ^ i \big ( \ell _ { j } - \lfloor 2 ^ { j - i - 1 } \tilde \varepsilon x \rfloor \big ) \\ & \geq r + x - \ell _ { i + 1 } ' - 3 \tilde \varepsilon x \\ & \geq r + x - ( 1 + \tilde \varepsilon ) ^ { - i - 1 } ( r + x ) . \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*} \cos ^ 2 \theta \delta _ { m , k } + \sin ^ 2 \theta \sqrt { m } = \cos ^ 2 \theta \sqrt { m } + \sin ^ 2 \theta \delta _ { m , l } = \gamma _ { m , k , l } . \end{align*}


\begin{align*} \beta _ { \bar { \mathbf { g } } } \mathbf { g } = \beta _ { \bar g } g + \bar u ^ { - 2 } u d u - \bar u ^ { - 1 } g ( \nabla _ { \bar g } \bar u , \cdot ) . \end{align*}


\begin{align*} \delta ( \eta , \theta \gamma ^ + ) = f ( \eta \theta \gamma ^ + , \theta \gamma ^ - ) - f ( \theta \gamma ^ + , \theta \gamma ^ - ) . \end{align*}


\begin{align*} \tilde p : = p _ \beta ( \beta ) , \end{align*}


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


\begin{align*} \sup _ { y \in \Z ^ d } \sum _ { \Delta \ni y } \sum _ { z \neq y } \sum _ { \xi _ { \Delta } } \sum _ { i = 1 } ^ q \norm { \nabla ^ i _ z \hat { c } _ \Delta ( \cdot , \xi _ \Delta ) } _ \infty < \infty . \end{align*}


\begin{align*} a _ { i } = \sum _ { j = 1 } ^ { n } k _ { i j } \theta ^ { j - 1 } . \end{align*}


\begin{align*} \partial _ t \left [ t ^ 2 \left [ \left ( \frac { \alpha } { t ^ 4 } + \beta + \gamma \frac { u ^ 2 } { t ^ 4 } \right ) ^ 2 + \delta ^ 2 \frac { u ^ 2 } { t ^ 4 } \right ] \right ] = 0 . \end{align*}


\begin{align*} c = c _ - ( x ) - c _ + ( x ) = \frac { 1 } { 2 } \int _ { - \infty } ^ \infty | u _ x | ^ 2 d x , \end{align*}


\begin{align*} \Omega ^ 1 _ { O ( \Delta ( \rho _ 1 , \ldots , \rho _ n ) ) } \cong \bigoplus _ { i = 1 } ^ n O ( \Delta ( \rho _ 1 , \ldots , \rho _ n ) ) . d x _ i , \end{align*}


\begin{align*} & \Phi _ S ( \sigma , \tau ) _ i = \left \{ \begin{aligned} & \sigma & & ( i \in S ) \\ & \tau & & ( i \notin S ) . \end{aligned} \right . \end{align*}


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


\begin{align*} x = \mu - \sigma \left [ F ^ { - 1 } _ \Gamma ( 1 - 2 u ) \right ] ^ { 1 / s } , \end{align*}


\begin{align*} X _ { \gamma } ' = \{ x ' \in X ' \ ; | \ ; x ' \ ; \tau \| \cdot \| \} \eqqcolon X ^ { \circ } \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*} \frac { \mathrm { d } } { \mathrm { d } \alpha } \ell = & \psi ( \alpha + \beta ) - \psi ( \alpha ) + \log \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \end{align*}


\begin{align*} D ( f \circ l _ g ) = ( D f ) \circ l _ g , \ ; \ ; \ ; f \in C ^ \infty ( G ) , \forall g \in G , \end{align*}


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


\begin{align*} M ^ 2 ( t ) = 2 a t . \end{align*}


\begin{align*} \chi _ K ( p ) = \begin{cases} 1 & p K \\ - 1 & p K \\ 0 & p K \end{cases} . \end{align*}


\begin{align*} K _ \lambda ( Z , W ) = \det ( I _ n - Z W ^ * ) ^ { - \lambda } . \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*} N ( f ) & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert ( 1 - m ) \det ( I - A ^ i M ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert ( 1 - m ) \det ( I - B ^ i C ^ i M ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert ( 1 - m ) \det ( I - C ^ i M ) \rvert . \end{align*}


\begin{align*} \eta ( U | \bar U ; x , t ) \ge \begin{cases} c _ 1 ' \big | U - \bar U \big | ^ 2 & \ ; | U | \le r _ 2 , \\ c _ 2 ' \ , \big | U - \bar U \big | ^ p & \ ; | U | \ge r _ 2 , \end{cases} \end{align*}


\begin{align*} E ( u ) = \int _ M \frac 1 2 \sum _ { i = 1 } ^ m h \big ( d u ( e _ i ) , d u ( e _ i ) \big ) \ , d v = \int _ M { \frac 1 2 } { \big ( \sigma _ 1 ( u ^ { \ast } ) \big ) } \ , d v . \end{align*}


\begin{align*} x ^ { ( k ) } _ { i } = x ^ { ( \hat { k } ) } _ { i } ( \forall i \in \{ 1 , \dots , D \} ) ( \forall k > \hat { k } ) . \end{align*}


\begin{align*} a ( Z ) = a \big ( ( Z Z ^ * ) ^ \frac { 1 } { 2 } \big ) \end{align*}


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


\begin{align*} \mathbf { P } _ n = \begin{bmatrix} U ( 0 ) & \mathbf { Q } _ n \\ \ast & \mathbf { T } _ n + \mathbf { I } _ n ^ { - 1 } \end{bmatrix} . \end{align*}


\begin{align*} D ( m , a ) = \det ( \Gamma ( a + i + j ) ) _ { i , j = 1 } ^ { m } . \end{align*}


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


\begin{align*} \frac { d } { d \rho } \int _ { B _ \rho ( x _ 0 ) } \big ( \exp ( \frac { | | R ^ \nabla | | ^ 2 } 2 ) - 1 \big ) \ , d v = \int _ { \partial B _ \rho ( x _ 0 ) } \big ( \exp ( \frac { | | R ^ \nabla | | ^ 2 } 2 ) - 1 \big ) \ , d s \ , . \end{align*}


\begin{align*} D ( F r o b _ v ) = q _ v S _ v \end{align*}


\begin{align*} \begin{aligned} & A y '' _ { b e n } + M _ { b e n } y _ { b e n } = E y _ { b e n } \\ & a A y '' _ { n b } + M _ { n b } y _ { n b } = E y _ { n b } , \end{aligned} \end{align*}


\begin{align*} \frac { \sqrt { p } } { \log p + 1 } > \frac { ( s + 1 ) 2 ^ { l + 1 } } { \left ( 1 - \sum _ { i = 1 } ^ s \frac 1 { p _ i } \right ) \prod _ { j = 1 } ^ l \left ( 1 - \frac 1 { q _ j } \right ) } . \end{align*}


\begin{align*} \int _ \Omega \left ( \eta x _ { n + 1 } ^ m + \lambda \right ) d V = \int _ \Sigma \langle Y , \nu \rangle \ , d A . \end{align*}


\begin{align*} d _ { C y } ( ( z , t , u ) , ( z ' , t ' , u ' ) ) = \big \lvert ( \lvert z - z ' \rvert ^ { 2 } + \lvert u - u ' \rvert ) ^ { 2 } + \lvert t - t ' + 2 \Im ( z \overline { z ' } ) \rvert ^ { 2 } \big \rvert ^ { 1 / 4 } . \end{align*}


\begin{align*} | T | = | T ' | \end{align*}


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


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


\begin{align*} R ( \rho ) = T _ { + } ( \rho ) G _ { - } ( \rho ) \end{align*}


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


\begin{align*} \bar K ^ { ( n , \ell ) } ( { x , y } ) : = \frac { 1 } { 4 ( n + 1 ) ^ 2 } \sum _ { j , j ' = - n - 1 } ^ n \Phi \left ( \frac { j } { n + 1 } , \frac { j ' } { n + 1 } \right ) \cos \left ( \frac { \pi j ( y - x ) } { n + 1 } \right ) \cos \left ( \frac { \pi j ' ( y - x - \ell ) } { n + 1 } \right ) . \end{align*}


\begin{align*} \ell _ \beta ( \langle \gamma \rangle ) = \beta ( \gamma , \gamma ^ + ) \end{align*}


\begin{align*} T ( \theta ) : = \begin{pmatrix} F ( x ) - \lambda \\ \lambda \\ x - y \end{pmatrix} , \forall \ \theta : = \begin{pmatrix} x \\ y \\ \lambda \end{pmatrix} \in \mathcal { K } . \end{align*}


\begin{align*} \lim _ { \sum m _ i \to \infty } | \lambda _ { m _ 1 , \ldots , m _ n } | ^ { 1 / \sum m _ i } = 0 . \end{align*}


\begin{gather*} \Phi ( r ) = \displaystyle \int _ { B _ r } | D u | ^ { p + 2 \beta } d x , \end{gather*}


\begin{align*} S _ k ^ { i j } ( \Delta u ) _ { i j } + \sum _ i S _ k ^ { p q , r s } u _ { p q i } u _ { r s i } = \Delta f , \end{align*}


\begin{align*} \omega ( X _ v , X _ w ) = 0 \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*} m = \max \{ q \in [ M - 1 ] : \ r _ q \neq \ell _ q \} , \end{align*}


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


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


\begin{align*} t = r _ 0 + r _ 1 q + \cdots + r _ { n - 1 } q ^ { n - 1 } , \end{align*}


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


\begin{align*} \prod _ { i = 1 } ^ j a _ f ( p _ i ^ { 2 m _ i } ) \prod _ { i = j + 1 } ^ r a _ f ( p _ i ^ { l _ i } ) \prod _ { i = 1 } ^ j a _ f ( q _ i ^ { m _ i ' } ) \prod _ { i = j + 1 } ^ r a _ f ( q _ i ^ { l ' _ i } ) = \prod _ { u = 1 } ^ t \left ( \prod _ { i \in \mathcal I ( s _ u ) } a _ f ( s _ u ^ { 2 i } ) \right ) , \end{align*}


\begin{align*} \phi ( Y ) = f ^ { - 1 } ( z _ 0 ) . \end{align*}


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


\begin{align*} | T \cap M ' | \le \begin{cases} \theta _ { d - 1 } q ^ { d ^ 2 - d + 1 } & P \in \tau _ 1 \cup \tau _ 2 , \\ ( q ^ { d - 1 } + \theta _ { d - 1 } ) q ^ { d ^ 2 - d } & \end{cases} \end{align*}


\begin{align*} \dim K = | \tau ^ 0 | - 1 = \dim L . \end{align*}


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


\begin{align*} \hat \Phi ^ \dagger _ s ( 0 , 0 ) = 0 , \qquad \hat \Phi ^ \dagger _ { a s } ( 0 , 0 ) = \gamma _ 2 , \end{align*}


\begin{align*} \iota _ { k } ^ \star = \gamma _ { k } , \forall k \in \mathcal { K } , \end{align*}


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


\begin{align*} \omega _ \alpha ( \xi , \eta ) = | \xi | ^ { \alpha } \xi + \frac { \eta ^ 2 } { \xi } , \xi \in \R \backslash \{ 0 \} , \ ; \eta \in \R , \end{align*}


\begin{align*} \rho _ { \prescript { \sigma ^ n } { } x } ( \prescript { \sigma ^ n } { } ( u ) ) = \prescript { \sigma ^ n } { } ( \rho _ x ( u ) ) . \end{align*}


\begin{align*} F ^ { q , \dagger } \colon X ^ \dagger \times \mathcal P \to Z ^ \dagger , F ^ { q , \dagger } ( v , p ) : = F ^ q ( v , p ) , \end{align*}


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


\begin{align*} k _ { r s } = \sum _ { j , h , l = 1 } ^ { n } p _ { r j h } k _ { j l } c _ { h l s } , \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*} | D z _ { \infty } | ( ( a , b ) ) = \sum _ { s \in S _ { z _ { \infty } } } | J _ { z _ { \infty } } ( s ) | \leq \sum _ { i = 1 } ^ { \infty } | J _ { \infty } ( i ) | , \end{align*}


\begin{align*} \| y \| & \leq \sum _ { j = 1 } ^ m \alpha _ j \| { y _ j } \| = \sum _ { j = 1 } ^ m \frac { \alpha _ j } { 2 } \| m _ { a _ j , b _ j } + y _ j - m _ { a _ j , b _ j } + y _ j \| \\ & \leq \sum _ { j = 1 } ^ m \frac { \alpha _ j } { 2 } \bigl ( \| m _ { a _ j , b _ j } + y _ j \| + \| m _ { a _ j , b _ j } - y _ j \| \bigr ) \\ & \leq \sum _ { j = 1 } ^ m \alpha _ j \bigl ( 1 + \frac { \varepsilon } { 5 m \alpha _ j } \bigr ) < 1 + \frac { 2 \varepsilon } { 5 } , \end{align*}


\begin{align*} \chi _ K ( \alpha ) + \chi _ K ( - \alpha ) = \begin{cases} 1 , & \alpha \in \Phi \setminus \Phi _ K , \\ 0 , & \alpha \in \Phi _ K . \end{cases} \end{align*}


\begin{align*} \rho ( t , \theta ) = ( t ( \cos \theta + 1 ) , t ( \cos \theta - 1 ) , t \sin \theta ) . \end{align*}


\begin{align*} \mathcal { B } ( u , v , w ) = \langle [ u , v ] _ H , w \rangle , u , v , w \in E . \end{align*}


\begin{align*} \frac { 1 } { \mu ( \eta _ { \Lambda _ n } ) } \int \mathbf { 1 } _ { \eta _ { \Lambda _ n } } ( \omega ) \left ( \sum _ { \Theta \Subset \Z ^ d } h _ { \Theta } ( \omega ) \right ) ^ 2 \mu ( d \omega ) = 0 . \end{align*}


\begin{align*} \hat \Phi _ s ( 0 , 0 ) = 0 . \end{align*}


\begin{align*} U \cdot V = U _ { i _ 1 \cdots i _ n } V _ { i _ 1 \cdots i _ n } , \quad | U | ^ 2 = U \cdot U . \end{align*}


\begin{align*} \abs { \mathcal { R } } \le C _ { V B } = C _ { V B } ( M , g ) . \end{align*}


\begin{align*} Q _ \delta L _ \delta \omega = Q _ \delta l _ { 1 , \delta } + Q _ \delta l _ { 2 , \delta } + Q _ \delta R _ \delta ( \omega ) , \end{align*}


\begin{align*} h _ \mu ( t ) = t - \log ( 1 + \mu t ) . \end{align*}


\begin{align*} \R : = \{ R _ i : i \in \{ 1 , \dots , j - \theta _ { d - 1 } - 1 \} R _ i \notin \tau \} . \end{align*}


\begin{align*} F _ k ( \theta _ y ) = F _ { k - q _ n } ( \theta _ { ( \ell + 1 ) q _ n + m _ n + 1 } ) F ( \theta _ { ( \ell + 1 ) q _ n + m _ n } ) F _ { q _ n - 1 } ( \theta _ { \ell q _ n + m _ n + 1 } ) \end{align*}


\begin{align*} \mathcal { U } ( E , \Omega ) : = \{ u \in Q S H _ m ( \Omega ) , u \vert _ { \Omega } \leq 0 , u \vert _ { E } \leq - 1 \} , \end{align*}


\begin{align*} m ^ a \ell ^ { t _ { \widetilde K } } = m ^ a ( \ell ' v ) ^ { t _ { \widetilde K } } = m ^ a { \ell ' } ^ { t _ { \widetilde K } } \cdot ( v + \ell ' ( v ) + \dots + \ell '^ { t _ { \widetilde K } - 1 } ( v ) ) . \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*} \ell ( r _ { ( \alpha , k ) } , \beta ) = \begin{cases} 1 , & \beta = \alpha , \\ - 1 , & \beta = - \alpha , \\ 0 , & \beta \neq \pm \alpha . \end{cases} \end{align*}


\begin{align*} H ( s , t ) = \frac { \kappa ( s ) } { n } . \end{align*}


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


\begin{align*} g ( x ) = \sum _ { i = 1 } ^ n \frac { f ( x + \sigma u _ i ) - f ( x ) } { \sigma } \tilde u _ i . \end{align*}


\begin{align*} u ^ K _ n ( p ) = \begin{cases} u _ n ( p ) & \\ K & \end{cases} \end{align*}


\begin{align*} \mathbf { v } \cdot \overrightarrow { \zeta } = 0 , \end{align*}


\begin{align*} \mathbf { v } ^ { 0 } = \mathbf { e } _ { k } = [ \cdots , 0 , 0 , \cdots , 1 , \cdots ] ^ { T } , k = 0 , \cdots , N - 1 . \end{align*}


\begin{align*} g ( \upsilon ) = \frac 1 4 \left ( f ( \upsilon ) + \frac 1 3 f ( 3 \upsilon ) - 2 f ( \upsilon ) - u ( \upsilon ) f ( \upsilon ) \right ) . \end{align*}


\begin{align*} \rho { u _ y } = \sum \limits _ { i = 0 } ^ { 1 8 } { { c _ { i y } } { f _ i } + \frac { { \Delta t } } { 2 } { F _ y } } , \end{align*}


\begin{align*} \sum _ { A = 0 } ^ { 2 n - 1 } \nabla _ { A \alpha } \varrho \left ( d _ 1 u _ k \wedge \Theta \wedge ( \Delta \varrho ) ^ { n - k } \right ) _ A \Omega _ { 2 n } = d _ \alpha \varrho \wedge d _ 1 u _ k \wedge \Theta \wedge ( \Delta \varrho ) ^ { n - k } , \end{align*}


\begin{align*} \mathbf { \bar { x } } = ( \Theta \mathbf { G } ) ^ T \mathbf { x } , \mathbf { \bar { s } } = { { ( \Theta \mathbf { G } ) } ^ { - 1 } } \mathbf { s } , \end{align*}


\begin{align*} d \geq 2 : 0 < s < s _ s \leq - \frac d 2 , d = 1 : s < \frac 1 6 s \ne - \frac 1 2 , \end{align*}


\begin{align*} a ( Z V ) = a ( Z ) , \end{align*}


\begin{align*} ( b ^ { [ j ] } \wedge a ) ( \alpha ) ( \xi ) = \begin{cases} b ( \alpha ) ( \xi ) & \alpha < \beta , \\ x _ j ( \beta ) ( \xi ) & \alpha = \beta \xi \le \zeta , \\ b ( \beta ) ( \xi ) & \alpha = \beta \xi > \zeta , \\ a ( \alpha ) ( \xi ) & \end{cases} \end{align*}


\begin{align*} m ( D _ { t _ 0 } \setminus P ) = 0 \end{align*}


\begin{align*} d ^ * ( \lceil x \rceil ) & = q ( c ( \lceil x \rceil ) ) \\ & = q ( c _ a c _ a c _ b c _ b c _ b c _ b c _ b c _ b c _ b c _ b ) \\ & = a ^ 2 b ^ 8 \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*} ( u _ \ast - \varphi ) ( p ) > ( u _ \ast - \varphi ) ( p _ 0 ) = 0 \end{align*}


\begin{align*} \lim _ { s \to 1 ^ + } ( s - 1 ) \zeta _ K ( s ) = \frac { 2 ^ { r + t } \pi ^ t R _ K h _ K } { w _ K | \Delta _ K | ^ { 1 / 2 } } , \end{align*}


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


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


\begin{align*} v \wedge \eta = ( v _ 2 \eta _ 3 - v _ 3 \eta _ 2 , v _ 3 \eta _ 1 - v _ 1 \eta _ 3 , v _ 1 \eta _ 2 - v _ 2 \eta _ 1 ) . \end{align*}


\begin{align*} F ( x ) = \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \int _ 0 ^ { \Phi _ s ( \frac { x - \mu } { \sigma } ) } t ^ { \alpha - 1 } \ , ( 1 - t ) ^ { \beta - 1 } \ , \mathrm { d } t = I _ { \Phi _ s ( \frac { x - \mu } { \sigma } ) } ( \alpha , \beta ) . \end{align*}


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


\begin{align*} L a _ { 0 } - a _ { 0 } ^ { \prime } = q , \end{align*}


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


\begin{align*} \mathbf { w } _ { k } ^ \star = \left ( \sum _ { p \in \mathcal { K } } \bar { \mathbf { h } } _ { p } \bar { \mathbf { h } } _ { p } ^ H + \lambda ^ \star \mathbf { I } _ N \right ) ^ { - 1 } \times \sqrt { 1 + \iota _ { k } } \bar { \mathbf { h } } _ { k } , \forall k \in \mathcal { K } , \end{align*}


\begin{align*} h ( \nu , \cdot ) = 0 , ( \nabla _ { \nu } h ) ( \nu , \cdot ) = 0 . \end{align*}


\begin{align*} u _ n = a ( \theta _ 1 ^ n - \zeta \theta _ 2 ^ n ) \end{align*}


\begin{align*} \iota ( \upsilon ) = - \frac { 1 } { 4 } f ( \upsilon ) + \frac { h ( \upsilon ) } { 4 g ( \upsilon ) } \left ( - 4 \upsilon + \frac 1 2 f ( 2 \upsilon ) \right ) . \end{align*}


\begin{align*} d \tilde { x } _ t = \lambda _ A \tilde { x } _ t d t + e ^ { \lambda _ A t } B ( x _ t , x _ t ) d t - A \tilde { x } _ t d t + e ^ { \lambda _ A t } \sigma d W _ t . \end{align*}


\begin{align*} \begin{cases} - ( K _ H ( x ) \nabla v ) = \frac { 1 } { \varepsilon ^ 2 } \left ( v - q _ { \hat { x } } \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ \ \ & \ \Omega _ { \hat { x } } . \\ v = 0 , \ \ \ \ & \ \partial \Omega _ { \hat { x } } . \end{cases} \end{align*}


\begin{align*} [ \mathbf { h } ' _ { 3 i } , \mathbf { h } _ { 3 i } ] = 1 . \end{align*}


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


\begin{align*} \hat M = ( M \setminus \Omega ) \cup \overline U \end{align*}


\begin{align*} [ P ( u ^ \prime ( t ) ) ] ^ \prime + A ( u ( t ) ) + Q ( t , u ^ \prime ( t ) ) + F ( u ( t ) ) = 0 , \end{align*}


\begin{align*} \frac { \partial \textbf { \textit { U } } } { \partial x _ n } \bigg | _ { \partial \Omega } = - Q \textbf { \textit { U } } | _ { \partial \Omega } + \mathcal { R } \textbf { \textit { V } } . \end{align*}


\begin{align*} \frac { \alpha } { z _ 0 } + \frac { \beta } { z _ 0 - 1 } + \frac { \gamma } { z _ 0 - t } = \frac { 1 } { \i y _ 0 } \ , . \end{align*}


\begin{align*} \mathcal { G } ( t ) = \mathcal { H } ( t ) + \nu \alpha ( t ) \delta ^ { \frac { 1 } { \ell } } ( t ) \mathcal { E } ^ { r + \frac { 1 } { q } + \frac { 1 } { \ell ^ { \prime } } } ( t ) , \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*} \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*} \begin{array} { l l l } \tilde { F } ( \mathsf { g } _ { \mathbb { C } , \mathsf { S } } ( L ) ) \circ \Phi _ { S , S } & = & F ( \mu ) \circ F ( i d _ S \otimes _ C L ) \circ \Phi _ { S , S } \\ & = & F ( \mu ) \circ \Phi _ { S , S } \circ ( i d _ { F ( S ) } \otimes _ D F ( L ) ) \\ & = & \mathsf { g } _ { \mathbb { D } , \widetilde { \mathbb { F } } ( \mathsf { S } ) } ( \tilde { F } ( L ) ) . \end{array} \end{align*}


\begin{align*} C _ 1 ( \mathcal { H } _ { \ast } ) = \mathrm { I m } \ , \partial ' _ 0 \oplus s _ { _ { 1 } } ( \mathrm { I m } \ , \partial _ 0 ) . \end{align*}


\begin{align*} \bar { x } _ n - x _ n ^ 0 = \frac { - \varepsilon _ 1 \bar { \xi } _ 1 + | \hat { x } | \bar { \xi } _ 2 } { \sqrt { \varepsilon _ 1 ^ 2 + | \hat { x } | ^ 2 } } \leq | D \rho ( 0 , \tilde { x } ) | | \bar { \xi } _ 1 | + | \bar { \xi } _ 2 | \leq C \epsilon _ 0 \delta ^ 2 b _ \alpha \end{align*}


\begin{align*} K _ n = [ x - \frac { 1 } { 2 n + 1 } , x + \frac { 1 } { 2 n + 1 } ] \ \textrm { f o r } \ n \neq m ^ 2 \ \textrm { w h e r e } \ m \in \mathbb { N } \\ K _ n = [ x - n , x + n ] \ \ \ \textrm { f o r } \ n = m ^ 2 \ \textrm { w h e r e } \ m \in \mathbb { N } \\ \end{align*}


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


\begin{align*} & \mathcal { Z } ^ * _ K ( \underline { \Delta ^ { J - 1 } } , \underline { \partial \Delta ^ { J - 1 } } ) ( \underline { \Delta ^ { J - 1 } } , \underline { \partial \Delta ^ { J - 1 } } ) = \{ ( \Delta ^ { j _ i - 1 } , \partial \Delta ^ { j _ i - 1 } ) \} _ { i \in \{ 1 , \ldots , m \} } . \end{align*}


\begin{align*} B _ q ( \phi ) ( z ) : = \displaystyle \int _ { \mathbb { R } } \overline { A _ q ( z , x ) } \phi ( x ) d x . \end{align*}


\begin{align*} \prescript L { } { } \ell { } ^ R ( x , \alpha ) + \prescript L { } { } \ell { } ^ R ( x , - \alpha ) = & \langle \mu , \alpha \rangle + \langle \mu , - \alpha \rangle + \underbrace { \chi _ R ( \alpha ) + \chi _ R ( - \alpha ) } _ { \in \{ 0 , 1 \} } \\ & \qquad - \underbrace { ( \chi _ L ( w \alpha ) + \chi _ L ( - w \alpha ) ) } _ { \in \{ 0 , 1 \} } \\ \in & \{ - 1 , 0 , 1 \} . \end{align*}


\begin{align*} \mathcal { Z } \left ( \hat { \mu } \right ) = \bigcup _ { n = 1 } ^ { \infty } \mathcal { Z } \left ( M _ { \mathcal { D } _ { n } } ( \rho ^ { n } \xi ) \right ) = \bigcup _ { n = 1 } ^ { \infty } \frac { \rho ^ { - n } a _ { n } } { N _ { n } } , \ a _ { n } \in \mathbb { Z } \backslash N _ { n } \mathbb { Z } . \end{align*}


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


\begin{align*} & \mid 2 ^ f h ^ { i ' } \overline { h } ^ { j ' } \sum \limits _ { a _ m \neq a _ i } \chi _ 4 ( g ( a _ m ) ) \mid \\ & = \mid 2 ^ f h ^ { i ' } \overline { h } ^ { j ' } \left \lbrace \sum \limits _ { a _ m } \chi _ 4 ( g ( a _ m ) ) - \chi _ 4 ( g ( a _ 1 ) ) - \cdots - \chi _ 4 ( g ( a _ { m - 1 } ) ) \right \rbrace \mid \\ & \leq 2 ^ { 3 m } \cdot 3 m \sqrt { q } + 2 ^ { 2 m - 3 } \\ & \leq 2 ^ { 2 m } ( 1 + 2 ^ m \cdot 3 m \sqrt { q } ) , \end{align*}


\begin{align*} \mathrm { M S E } = \sum _ { m = 1 } ^ { M } \left | P _ { \mathrm { d } } \left ( \theta _ { m } \right ) - \mathbf { a } ^ { H } \left ( \theta _ { m } \right ) \mathbf { R } _ { X } \mathbf { a } \left ( \theta _ { m } \right ) \right | ^ { 2 } , \end{align*}


\begin{align*} & \limsup _ { \delta \to 0 } \limsup _ { \epsilon \to 0 } \epsilon ^ { 2 } \log \mathbb P \Big ( \sup _ { | t - s | < \delta } \big | \epsilon \Theta _ { \epsilon } ( t ) - \epsilon \Theta _ { \epsilon } ( s ) \big | > a \Big ) = - \infty , \end{align*}


\begin{align*} & \Phi ( \tau ) ( \Lambda _ g - \tau I ) = I \mod O P S ^ { - \infty } , \\ & ( \Lambda _ g - \tau I ) \Phi ( \tau ) = I \mod O P S ^ { - \infty } . \end{align*}


\begin{align*} V _ g & = u _ g c H - G ( c ) H = \big ( G ' ( c ) c - G ( c ) \big ) H , \\ w _ g & = - \Delta _ \Sigma u _ g + c H V _ g = - \Delta _ \Sigma \big ( G ' ( c ) \big ) + c H V _ g . \end{align*}


\begin{align*} \begin{aligned} | \varepsilon _ \delta ( z _ 1 , \bar { y } & , t ) - \varepsilon _ \delta ( x _ a , \bar { y } _ a , t ) | \\ \le & L _ r \gamma | \epsilon _ 1 ( \eta _ a , t ) | + | \bar { y } ( t ) - \bar { y } _ a ( t ) | \\ \le & \gamma \delta L _ r \left ( 2 + L _ r \ ( 3 + L _ r ) \right ) + | \bar { y } - z _ 2 | . \end{aligned} \end{align*}


\begin{align*} [ a _ { ( m ) } , b _ { ( n ) } ] = \sum _ { j \in \mathbb { Z _ { + } } } \left ( \begin{array} { c c c } m \\ j \end{array} \right ) ( a _ { ( j ) } b ) _ { ( m + n - j ) } . \end{align*}


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


\begin{align*} F _ m = \begin{cases} H & \ k \ , \\ G & \ k \ , \end{cases} \end{align*}


\begin{align*} \left ( \psi ^ L _ { a ; S } \right ) _ { 0 j } \colon & f ( x , y , z ) \mapsto \eta ^ { - a \vert S \vert j } f ( x , \eta ^ { - a } x , z ) \\ \left ( \psi ^ R _ { S ; b } \right ) _ { i 0 } \colon & f ( x , y , z ) \mapsto f ( x , \eta ^ { b } z , z ) . \end{align*}


\begin{align*} \int _ { l _ 1 } ^ { l _ 2 } y ^ { \frac { i + j + 1 } { s } - 1 } \ , \exp ( - y ) & \left [ \sum _ { m = 0 } ^ \infty \frac { ( - y ) ^ m } { ( 1 / s + m ) m ! } \right ] ^ j \mathrm { d } y = \sum _ { m = 0 } ^ \infty c _ { m , j } \int _ { l _ 1 } ^ { l _ 2 } y ^ { m + \frac { i + j + 1 } { s } - 1 } \exp ( - y ) \mathrm { d } y \\ & = \sum _ { m = 0 } ^ \infty c _ { m , j } \left [ \Gamma \left ( m + \frac { i + j + 1 } { s } , l _ 1 \right ) - \Gamma \left ( m + \frac { i + j + 1 } { s } , l _ 2 \right ) \right ] , \end{align*}


\begin{align*} \alpha = \alpha _ { ( 0 ) } + \alpha _ { ( 1 ) } \end{align*}


\begin{align*} I _ 7 = 4 \beta \int \rho ^ { 2 \beta - 2 \alpha - 2 } Z v \ \Delta _ { H } v = 4 \beta \int \rho ^ { - Q + 2 } Z v \ ; \Delta _ { H } v . \end{align*}


\begin{align*} \hat { k } _ d ( \Delta ) = \sum _ { \Upsilon \in \mathcal { T } _ d ( \Delta ) } \prod _ { \tau \in \Upsilon _ d } \left ( \prod _ { v \in \tau } x _ { \tau } \right ) { \mathbf { t } _ { d - 1 } ( \Upsilon ) ^ 2 } = \sum _ { \Upsilon \in \mathcal { T } _ d ( \Delta ) } \left ( \prod _ { v \in V } x _ { v } ^ { \deg _ \Upsilon v } \right ) { \mathbf { t } _ { d - 1 } ( \Upsilon ) ^ 2 } . \end{align*}


\begin{align*} \mathbb E \left [ \| \Delta _ j \mathcal S B ^ { \mathfrak H } \| _ H ^ { 8 } \right ] \leq & \int _ 0 ^ 2 \mathbb E \left [ \left ( B ^ { \mathfrak H } _ { x + i \Delta _ n } - B ^ { \mathfrak H } _ { x + ( i - 1 ) \Delta _ n } \right ) ^ 8 \right ] d x = 2 1 0 \Delta _ n ^ { 8 H } = 2 1 0 \Delta _ n ^ 2 , \end{align*}


\begin{align*} r i \ , C : = \{ x \in \textit { a f f } \ , C \ , | \ , ( \exists \ , \epsilon > 0 ) \ , ( x + \epsilon B ) \cap ( \textit { a f f } \ , C ) \subset C \} \end{align*}


\begin{align*} \left \{ \begin{aligned} d x _ t ^ { * , i } = & ~ \Big [ A x _ t ^ { * , i } - B ^ 2 R ^ { - 1 } y _ t ^ { * , i } - B h \big ( \rho _ i ^ N ( \Delta ^ { N , 1 } _ { * , t } , \ldots , \Delta ^ { N , N } _ { * , t } ) \big ) \Big ] d t + \sigma d W _ t ^ i + \sigma _ 0 d W ^ 0 _ t , \\ x ^ { * , i } _ 0 = & ~ \xi ^ i . \end{aligned} \right . \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ H u = - ( K _ H ( x ) \nabla u ) = \frac { 1 } { \varepsilon ^ 2 } \left ( u - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ & x \in \Omega , \\ u = 0 , \ \ & x \in \partial \Omega , \end{cases} \end{align*}


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


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


\begin{align*} \alpha _ i + \bar { \alpha } _ j = 1 . \end{align*}


\begin{align*} { } _ a D _ g ^ { \alpha } \circ { } _ { a } I _ g ^ { \alpha } = I , \end{align*}


\begin{align*} \varepsilon ( h , x + y ) [ [ \alpha ( x ) , \alpha ( y ) ] , \alpha ( h \cdot z ) ] & + \varepsilon ( x + y , z ) [ [ h \cdot z , \alpha ( x ) ] , \alpha ^ 2 ( y ) ] \\ & + \varepsilon ( h , y ) \varepsilon ( x , y + z ) [ [ \alpha ( y ) , h \cdot z ] , \alpha ^ 2 ( x ) ] = 0 , \end{align*}


\begin{align*} F = \{ x \in [ 0 , 1 ) : E ( x ) \} . \end{align*}


\begin{align*} \phi ( t ) = f ( t ) + C ( \kappa \ , * \ , k _ 1 ) ( t ) , \ t > 0 . \end{align*}


\begin{align*} \ell ( x , \alpha ) = \prescript L { } { } \ell { } ^ R ( x , \alpha ) + \Phi ^ + _ R ( \alpha ) \geq 0 . \end{align*}


\begin{align*} \tilde { d } _ H ( \xi , p ) = | \xi \cdot p ^ { - 1 } | _ G = | \zeta \cdot w ^ { - 1 } | _ G \leq - \lambda , \end{align*}


\begin{align*} \begin{aligned} & \sum _ { \begin{subarray} { c } r _ 1 + \cdots + r _ { n + 1 } = r \\ r _ i \in \mathbb { Z } _ { \ge 0 } \end{subarray} } Z _ { I } ( 1 + r _ 1 , \{ 2 + r _ i \} _ { i = 2 } ^ { n + 1 } ; ( \alpha , \beta ) ) \\ = & \sum _ { \begin{subarray} { c } r _ 1 + \cdots + r _ { n } = r \\ r _ i \in \mathbb { Z } _ { \ge 0 } \end{subarray} } Z _ { I I } ( \{ 2 + r _ i \} _ { i = 1 } ^ { n - 1 } , 3 + r _ n ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} g _ 1 ( x ) = \frac { x ^ 2 } { 1 + \cos x } , g _ 2 ( x ) = \frac { x \sinh x } { 1 + \cosh x } . \end{align*}


\begin{align*} \left \{ \begin{aligned} - d y _ t = & f ( t , y _ t , y _ t , u _ t , \mathcal L ( y _ t , z _ t , u _ t ) ) d t + g ( t , y _ t , z _ t , u _ t , \mathcal L ( y _ t , z _ t , u _ t ) ) d \overleftarrow B _ t \\ & - z _ t d W _ t , \ , t \in [ 0 , T ] , \\ y _ T = & \xi , \end{aligned} \right . \end{align*}


\begin{align*} \psi _ { \tau } ( p ) = | | \tau | | _ p \le C _ 1 ( \beta , \gamma , L ) \ ( \beta - p ) ^ { - ( \gamma + 1 ) / \beta } \ L ^ { 1 / \beta } ( 1 / ( \beta - p ) ) , \end{align*}


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


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


\begin{align*} r _ { \ell } \leq & e ^ { 5 0 \varepsilon q _ n } \frac { e ^ { - q _ n L } } { \max ( | \ell | , 1 ) } \max ( r _ { \ell - 1 } , r _ { \ell + 1 } ) \times \begin{cases} \max ( | \ell | , e ^ { \delta _ n q _ n } ) , \beta _ n \geq \delta _ n + 2 0 0 \varepsilon , \\ e ^ { \beta _ n q _ n } , \beta _ n < \delta _ n + 2 0 0 \varepsilon \end{cases} \end{align*}


\begin{align*} \mathbb { x } _ c ^ { \mathcal { X } ' , \mathbb { F } } \circ \alpha ^ { m } ( \mathbb { C } ) _ c = \alpha ^ m ( \mathbb { D } ) _ { F c } \circ \mathbb { x } ^ { \mathcal { X } , \mathbb { F } } _ c . \end{align*}


\begin{align*} g _ { x _ 1 x _ 2 \ldots x _ d } = { ( - 1 ) } ^ d \prod _ { j = 1 } ^ d \lambda _ j g ( x _ 1 , \ldots , x _ d ) \end{align*}


\begin{align*} c = \sum _ { \tau \in \Delta _ i } c _ { \tau } [ \tau ] , \end{align*}


\begin{align*} j ' ( t ) = & - \frac { 1 } { t ^ 2 \varepsilon ^ 2 } \int _ { \Omega } \left ( p ( t u - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p - 1 } t u ^ 2 - \left ( t u - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p } u ) d x \\ = & \frac { 1 } { t ^ 2 \varepsilon ^ 2 } \int _ { \Omega } \left ( t u - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p - 1 } u [ ( 1 - p ) ( t u - q \ln \frac { 1 } { \varepsilon } ) - p q \ln \frac { 1 } { \varepsilon } ] d x \\ \leq & 0 . \end{align*}


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


\begin{align*} \widetilde { L } : = \bigcup _ i \widetilde { U _ i } , \end{align*}


\begin{align*} q ' _ x ( t ) : = q _ x ( t ) - \bar q _ x ( t ) \quad \mbox { a n d } p ' _ x ( t ) : = p _ x ( t ) - \bar p _ x ( t ) \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } l _ n = g . \end{align*}


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


\begin{align*} \ < \big ( \widehat { f } \ ; \big ) ^ * ( g ) , \delta ( x ) \ > = \ < g , \widehat { f } ( \delta ( x ) ) \ > = \ < g , \delta ( f ( x ) ) \ > = g \circ f ( x ) = \ < C _ f ( g ) , \delta ( x ) \ > . \end{align*}


\begin{align*} \ln \left \lbrace \prod _ { j \neq k } \frac { | \sin \pi ( \theta - \theta _ j ) | } { | \sin \pi ( \theta _ k - \theta _ j ) | } \right \rbrace = \sum _ { j \neq k } \ln | \sin \pi ( \theta - \theta _ j ) | - \sum _ { j \neq k } \ln | \sin \pi ( \theta _ k - \theta _ j ) | < ( 2 q _ n - 1 ) ( \frac { \ln { q _ { n + 1 } / | \ell | } } { 2 q _ n - 1 } + \epsilon ) . \end{align*}


\begin{align*} X _ { t } = X _ 0 + \int _ 0 ^ t a _ s \ , d s + \int _ 0 ^ t b _ s \ , d W _ s , \end{align*}


\begin{align*} l ( y _ p ) \sim r ( y _ p ) = p = p _ 1 + \cdots + p _ n = r ( y _ { p _ 1 } ) + \cdots + r ( y _ { p _ n } ) \succ l ( y _ { p _ 1 } ) \vee \cdots \vee l ( y _ { p _ n } ) . \end{align*}


\begin{align*} \beta _ i + \bar { \beta _ j } = 1 \ , \ , { \rm a n d } \ , \ , \beta _ u + \bar { \beta } _ 4 = 1 . \end{align*}


\begin{align*} M _ n & \ge \sup _ { Q \in \P ( \R ) } \bigg ( n \int _ { \R } V ( x ) \ , Q ( d x ) + \frac 1 2 \sum _ { i , j = 1 } ^ n J _ { i j } \int _ { \R } \int _ { \R } K ( x - y ) \ , Q ( d x ) Q ( d y ) - n H ( Q ) \bigg ) \\ & = n \sup _ { Q \in \P ( \R ) } \bigg ( \int _ { \R } V \ , d Q + \frac 1 2 \int _ { \R } \int _ { \R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \bigg ) . \end{align*}


\begin{align*} \begin{aligned} & \dot y _ { b e n } = A _ { b e n } y '' _ { b e n } + M _ { b e n } y _ { b e n } , y \in ( - R / 2 , R / 2 ) , \\ & \dot y _ { n b } = A _ { n b } y '' _ { n b } + M _ { n b } y _ { n b } , y \in ( R / 2 , r + R / 2 ) , \end{aligned} \end{align*}


\begin{align*} ( \nu ( u ) ) ' = \nu ' ( h ) ( \bar u ) + \nu ( v ) \end{align*}


\begin{align*} \lim _ { \lambda _ 1 , \ldots , \lambda _ d \rightarrow 1 } \frac { \det ( \varphi _ i ( \lambda _ j ) ) _ { i , j = 1 } ^ d } { \Delta ( \lambda _ 1 , \ldots , \lambda _ d ) } = \frac { \det ( \varphi _ i ^ { ( j - 1 ) } ( 1 ) ) _ { i , j = 1 } ^ d } { \prod _ { j = 1 } ^ { d - 1 } j ! } . \end{align*}


\begin{align*} \abs { f } = \inf _ { x \in X } d _ f ( x ) . \end{align*}


\begin{align*} g _ s ( x ) = \left \{ \begin{array} { l l } g ( x ) / \lambda _ s & \ x > c _ 1 \\ g ( c _ 1 ) / \lambda _ s & \ - c _ 1 \le x \le c _ 1 \\ g ( - x ) / \lambda _ s & \ x < - c _ 1 , \end{array} \right . \end{align*}


\begin{align*} \psi ^ - ( x , t ; k ) = \psi ^ + ( x , t ; k ) e ^ { - i k ^ 2 x \widehat { \sigma } _ 3 } S ( k ) , \end{align*}


\begin{align*} \frac { 1 } { 2 } B ( X ) : = \frac { 1 } { 2 } \sum _ { j = 1 } ^ p e _ j \cdot B ( X , e _ j ) \ \in C l ( T M \oplus E \oplus \mathcal { E } _ 2 ) \end{align*}


\begin{align*} y ^ q ( x ^ { q + 1 } + d ) = ( x + d ) . \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty x ^ n \ , h _ j ^ { ( s ) } \mathrm { d } x & = \sigma \ , \int _ { - \infty } ^ \infty ( \sigma z + \mu ) ^ n \ , [ \Phi _ s ( z ) ] ^ { j + \alpha - 1 } \ , \phi _ s ( z ) \mathrm { d } z \\ & = \sigma \mu ^ n \ , \sum _ { i = 0 } ^ n \ , \binom { n } { i } \ , \left ( \frac { \sigma } { \mu } \right ) ^ i \ , \int _ { - \infty } ^ \infty z ^ i \ , [ \Phi _ s ( z ) ] ^ { j + \alpha - 1 } \ , \phi _ s ( z ) \mathrm { d } z . \end{align*}


\begin{align*} \mathrm { v a r } [ S _ { k } ( x ) - S _ { m } ( x ) ] = ( m - k ) \gamma + o ( 1 ) \end{align*}


\begin{align*} \hat { Y } [ i , k ] = H [ k ] \bar { Y } [ i , k ] + W [ i , k ] , \end{align*}


\begin{align*} & M = \begin{pmatrix} D & * \\ 0 & * \end{pmatrix} , D = d i a g ( d _ 1 , d _ 2 , \ldots , d _ { l _ 0 - 1 } ) \\ & \deg ( \det ( D ) ) \geq \deg ( \det ( D _ 0 ) ) , \deg ( \det ( M ) ) = \deg ( \det ( M _ 0 ) ) \end{align*}


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


\begin{align*} \lambda _ c ^ { \left ( \right ) } = 2 \sum _ { j = 1 } ^ { M } \lvert \hat { \phi } ^ { \left ( j , c \right ) } \rvert ^ 2 + \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 - f ^ { \left ( c \right ) } \end{align*}


\begin{align*} \hat \Phi \left ( \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) : = \begin{pmatrix} \langle z _ 1 ^ * , \Phi ( a u _ \ell + b w _ \ell , p ( s ) ) \rangle \\ \langle z _ 2 ^ * , \Phi ( a u _ \ell + b w _ \ell , p ( s ) ) \rangle \end{pmatrix} . \end{align*}


\begin{align*} & \Delta x _ j - \bar u ^ { - 1 } \nabla \bar u \cdot \nabla x _ j = 0 \\ & x _ j ( p ) = 0 \ ; \mbox { a n d } \ ; \frac { \partial x _ j } { \partial y _ i } ( p ) = \delta _ { i j } . \end{align*}


\begin{align*} \chi _ J ( \beta ) = \Phi ^ + ( \beta ) - \Phi ^ + _ J ( \beta ) . \end{align*}


\begin{gather*} \delta _ 1 ( X ) = 2 \mu F ( \mu ) - 2 \mu + 2 \int _ { \mu } ^ { \infty } x f ( x ) \mathrm { d } x \delta _ 2 ( X ) = 2 \int _ { m } ^ { \infty } x f ( x ) \mathrm { d } x - \mu , \end{gather*}


\begin{align*} B ( \alpha , \beta ) = \left \{ \begin{array} { l l } \displaystyle { \int _ { 0 } ^ { 1 } t ^ { \alpha - 1 } ( 1 - t ) ^ { \beta - 1 } d t } , & ( \Re ( \alpha ) > 0 , \Re ( \beta ) > 0 ) \\ \\ \displaystyle { \frac { \Gamma ( \alpha ) \Gamma ( \beta ) } { \Gamma ( \alpha + \beta ) } } , & ( ( \alpha , \beta ) \notin { \bf Z } _ { 0 } ^ { - } ) . \end{array} \right . \end{align*}


\begin{align*} ( 1 - \alpha ) \pi , \ ; ( 1 - \beta ) \pi , \ ; ( 1 - \gamma ) \pi , \ ; ( 1 - \delta ) \pi , \alpha , \beta , \gamma , \delta \in ( 0 , 1 ) , \alpha + \beta + \gamma + \delta = 2 . \end{align*}


\begin{align*} f ( z ) = \int _ { \Gamma } g ( \omega ) e ^ { - \omega z } d \omega . \end{align*}


\begin{align*} f ( g ) = \sum _ k r _ k g \cdot v _ { k } . \end{align*}


\begin{align*} ( x , & y , z , \zeta ) ( x ' , y ' , z ' , \zeta ' ) \\ & = ( x + x ' , y + y ' , z + z ' + x y ' - x ' y , \zeta + \zeta ' + \xi ( x y ' - x ' y ) ) . \end{align*}


\begin{align*} \frac { d } { d t } E \big [ [ M _ v ] ( t ) \big ] = \sum _ { k = 1 } ^ N e ^ { - 2 \lambda t } \lambda ^ 2 v _ { k } ^ 2 E | Z _ k ( t ) | \le 4 | v | ^ 2 e ^ { - 2 \lambda t } E \bigg [ \sum _ { k = 1 } ^ N | Z _ k ( t ) | \bigg ] . \end{align*}


\begin{align*} \gamma _ { \textrm { c } , k } = \frac { { \lvert \mathbf { h } _ { k } ^ { H } \mathbf { p } _ { \textrm { c } } \rvert } ^ { 2 } } { \sum _ { i = 1 } ^ { K } { \lvert \mathbf { h } _ { k } ^ { H } \mathbf { p } _ { \textrm { p } , i } \rvert } ^ { 2 } + \sigma _ { z } ^ { 2 } } . \end{align*}


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


\begin{align*} d _ n ^ T ( \{ e ( k x ) \} _ { k = - N } ^ N , L _ p ) \le \log ^ { - c _ 1 ( p ) } N , \quad \mbox { i f $ n \ge N \exp ( - \log ^ { c _ 2 } N ) $ } . \end{align*}


\begin{align*} x ' = \sum _ { i = 1 } ^ n \lambda _ i \ , m _ { p _ i , q _ i } . \end{align*}


\begin{align*} x = \Phi \left ( - t , \Phi ( t , x ) \right ) = \Phi \left ( - t , \Phi ( s , y ) \right ) . \end{align*}


\begin{align*} a _ { n , k } & = \begin{cases} O ( 1 ) \quad & H < H ( k ) \\ O ( \log n ) \quad & H = H ( k ) \\ O ( n ^ { ( 2 H - 1 ) k - 1 } ) \quad & H > H ( k ) \end{cases} \end{align*}


\begin{align*} \varphi _ { c , d } ( u ) = \varphi _ { c , d } ( v ) , \end{align*}


\begin{align*} v = 0 , \nu ( v ) = 0 \mbox { o n } \hat \Sigma . \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*} E _ { \delta , Z } = \{ u \in W ^ { 2 , p } \cap H ^ 1 _ 0 ( \Omega ) \mid \int _ { \Omega } ( K ( x ) \nabla \frac { \partial V _ { \delta , Z , j } } { \partial z _ { j , h } } ) u = 0 , \ \ \forall j = 1 , \cdots , m , \ h = 1 , 2 \} . \end{align*}


\begin{align*} B _ n & \rightharpoonup B \textrm { i n } \ L ^ { 2 } ( \Omega ) , \\ A _ n = \textrm { c u r l } ^ { - 1 } B _ n & \to A = \textrm { c u r l } ^ { - 1 } B \textrm { i n } \ L ^ { 2 } ( \Omega ) . \end{align*}


\begin{align*} \hat { \theta } _ i = \arcsin \left ( \frac { \lambda \arg z _ i } { 2 \pi d } \right ) \end{align*}


\begin{align*} \chi = \begin{cases} \max \{ 1 + \beta , 1 + \lambda , 1 + \gamma + \frac { 2 \theta } { \alpha } \} , & \alpha \leq 2 , \\ \max \{ 1 + \beta , 3 + \lambda - \alpha , 3 + \gamma + \theta - \alpha \} , & \alpha > 2 . \end{cases} \end{align*}


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


\begin{align*} K = \frac 1 { | t | ^ { \frac 1 M ( M + 1 - \frac 1 { 2 s } ) } } * \overset { M ) } { \cdots } * \frac 1 { | t | ^ { \frac 1 M ( M + 1 - \frac 1 { 2 s } ) } } , \end{align*}


\begin{align*} \begin{aligned} \Bigg | & \dfrac { \partial \epsilon _ 1 ( \eta , t ) } { \partial \eta } \Bigg | _ { \eta = \eta _ a } \dot { \eta } _ a \Bigg | \le \ , \gamma | a _ { 0 , \delta } ( x _ a ) / 2 - \bar { y } _ a | \\ & + \gamma \left | \left . \dfrac { \partial \varepsilon _ \delta ( x , \bar { y } _ a , t ) - b _ { 1 , \delta } ( x ) / 2 } { \partial x } \right | _ { x = x _ a } \right | \dfrac { | b _ { 1 , \delta } ( x _ a ) | } { 2 } . \end{aligned} \end{align*}


\begin{align*} M ( - t ) = \frac { 1 } { \sqrt 2 \pi } \sum _ { r = 0 } ^ \infty \pi _ r \ , \ , \int _ { - \infty } ^ { \infty } \Phi ( x ) ^ r \ , \exp \left ( - t \ , x - \frac { x ^ 2 } { 2 } \right ) \mathrm { d } x . \end{align*}


\begin{align*} D '' u _ { m ' , \epsilon , j } + P _ { m ' } \big ( \sqrt { N _ 1 \tilde { \lambda } _ { m ' } } h _ { m ' , \epsilon , j } \big ) = D '' \left ( ( 1 - v ' _ { \epsilon } ( \Psi ) ) f F ^ { 1 + \delta } \right ) . \end{align*}


\begin{align*} \mathbf { v } ^ { 0 } = [ \cdots , 1 , 0 , 0 , 0 , \cdots , 1 , \cdots ] ^ { T } , m , n = 1 , \cdots , N . \end{align*}


\begin{align*} \mathbb { E } _ { \mathbb { P } } \left [ \left | \frac { 1 } { t ^ { \frac { 1 } { 2 } - H } } \sum _ { j = 2 } ^ n \triangle k _ t ^ { 1 , j , n } \right | ^ { 2 p } \right ] \leq p ! \left ( ( n - 1 ) \beta \right ) ^ { p } . \end{align*}


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


\begin{align*} \partial ^ { c i } _ { \tau , w } g _ * ( \tau , z , w ( \cdot ) ) = G ( \tau , w ( - h ) , \mathrm { d } ^ + w ( - h ) / \mathrm { d } \xi ) , \nabla _ z g _ * ( \tau , z , w ( \cdot ) ) = 0 \end{align*}


\begin{align*} G ( \overline { \Psi } ( A _ t ) ) = \{ P \in \overline { \Psi } ( A _ t ) \ ; | \ ; P \vert _ { t = 0 } \in G _ { A _ t } \} \end{align*}


\begin{align*} \Omega ( X , q , n ) = \sum \limits _ { m \leq X \atop { m \equiv n \ , ( q ) } } 1 \ , . \end{align*}


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


\begin{align*} f _ 1 & = \mathcal { F } ( \gamma ( N _ { \max } ^ { ( 5 - 2 \alpha ) + \varepsilon } t - n ) \mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ { N _ 1 } u ) , \\ \tilde { f } _ 2 & = \mathcal { F } ( \gamma ( N _ { \max } ^ { ( 5 - 2 \alpha ) + \varepsilon } t - n ) \mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ { N _ 2 } \partial _ x u ) , \\ f _ 3 & = \mathcal { F } ( \gamma ( N _ { \max } ^ { ( 5 - 2 \alpha ) + \varepsilon } t - n ) \mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ N u ) , \end{align*}


\begin{align*} b ( x _ 1 , x _ 2 ) = \nabla \times ( \phi ( x _ 1 , x _ 2 ) \nabla z ) + G ( x _ 1 , x _ 2 ) \nabla z . \end{align*}


\begin{align*} \bigvee _ { ( s , t ) \in \Lambda _ N ^ { \vec { v } } ( 1 ) } T ^ { - ( s , t ) } \{ A , A ^ c \} \bigvee _ { n = 0 } ^ { N - 1 } \mathcal { A } ^ n . \end{align*}


\begin{align*} n _ { i } & = 3 n _ { i - 1 } - 3 n _ { i - 2 } + n _ { i - 3 } + \kappa \\ \left ( n + a \right ) _ { i } & = 3 \left ( n + a \right ) _ { i - 1 } - 3 \left ( n + a \right ) _ { i - 2 } + \left ( n + a \right ) _ { i - 3 } + \lambda \\ \left ( n + b \right ) _ { i } & = 3 \left ( n + b \right ) _ { i - 1 } - 3 \left ( n + b \right ) _ { i - 2 } + \left ( n + b \right ) _ { i - 3 } - \lambda \end{align*}


\begin{align*} P _ n u : = \sum _ { i = 1 } ^ { n } \langle u , e _ i \rangle e _ i . \end{align*}


\begin{align*} \P _ x ( Z _ d ( n _ 1 ) \geq \xi _ 1 , \ldots , Z _ d ( n _ m ) \geq \xi _ m ) = \det ( I - \chi _ \xi K \chi _ { \xi } ) _ { l ^ 2 ( \{ n _ 1 , \ldots n _ k \} \times \mathbb { N } } \end{align*}


\begin{align*} \begin{aligned} | \mathcal { R } _ { 2 , \epsilon } ( t ) | & \leq \vartheta \lambda ^ { 2 } ( \epsilon ) \int ^ { t } _ { 0 } 1 + | \varphi ( s ) | + | \varphi ' _ m ( s ) | d s \\ & + \lambda ( \epsilon ) \epsilon ^ { 2 } \Big | \int ^ { t } _ { 0 } \big ( \sigma ( \varphi _ { m } ( s ) ) - \sigma ( \hat { X } ^ { \epsilon } ( s - ) ) \big ) \theta _ { \epsilon } ( s - ) \tilde { N } ^ { \epsilon ^ { - 2 } } ( d s ) \Big | , \end{aligned} \end{align*}


\begin{align*} f ( t ) = \displaystyle \sum _ { n = 0 } ^ { \infty } \frac { \left ( \ ; \alpha ^ { 2 } t ^ { 2 } \right ) ^ { n } } { ( 2 n ) ! } = \cosh \alpha t . \end{align*}


\begin{align*} \gamma : = \begin{cases} \max ( e ^ { \delta _ n q _ n } , | \ell | , 1 ) e ^ { - \beta _ n q _ n } , \beta _ n \geq \delta _ n + 2 0 0 \varepsilon \\ 1 , \end{cases} \end{align*}


\begin{align*} \gamma _ k ( t _ k , x _ k , \xi _ k , y _ k ) \leq \gamma _ k ( \tau , z , \tau , z ) = \varphi ( \tau , z , w ( \cdot ) ) . \end{align*}


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


\begin{align*} - \int _ 0 ^ \infty u \phi = \int _ 0 ^ \infty f \phi ' , \qquad \phi \in C _ 0 ^ \infty ( 0 , \infty ) , \end{align*}


\begin{align*} \tilde { m } _ n = \begin{cases} m _ n - q _ { n - 1 } , \ell _ n = 1 \\ m _ n + q _ { n - 1 } , \ell _ n = - 1 \end{cases} . \end{align*}


\begin{align*} R i c ^ B = \rho ^ \nabla + d J \theta , \end{align*}


\begin{align*} _ { \rm r e f } [ V ] ( y , z , q ) = _ { ^ F V } \left ( q ^ { L _ 0 - \frac { c } { 2 4 } } z ^ { h } y ^ { - 2 g } \right ) . \end{align*}


\begin{align*} \xi = - L - 1 , \eta = \frac { Q } { 2 ( L + 1 ) } , \zeta = \frac { \kappa ( L + 1 ) B _ 1 } { Q } . \end{align*}


\begin{align*} F ( s ) = \int _ { 0 } ^ { \infty } \exp ( - s t ) f ( t ) d t . \end{align*}


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


\begin{align*} \begin{aligned} | u _ { \tilde { \nu } ( y ' ) } ( y ) | \ , & \leq | u _ { \tilde { \nu } ( 0 ) } ( 0 ) | \\ & + \sum _ { \beta = \alpha + 1 } ^ { n - 1 } \sup _ { \xi \in \partial \omega \cap \partial \Omega } \left ( | \nabla _ { \tilde { \nu } \tau _ \beta } u ( \xi ) | + \sum _ { \gamma = 1 } ^ { n - 1 } | \rho _ { \gamma \beta } ( \xi ) u _ \gamma ( \xi ) | \right ) \cdot | y _ \beta | \leq C b _ \alpha \end{aligned} \end{align*}


\begin{align*} - T ( x ) + x = \varphi \varphi ( - T ( x ) + x ) = \varphi ( - T ( x ) - x ) . \end{align*}


\begin{align*} \sup _ { p \ge 1 } \left \{ \ \frac { | | \tau | | _ { p , \Omega } } { \psi _ { m , L } ( p ) } \ \right \} = C ( m , L ) < \infty \end{align*}


\begin{align*} & \partial _ { t } Q - \Theta \sigma ^ { i j } _ { k } \nabla _ { i j } Q \\ & = \frac { 1 } { P } ( \partial _ { t } P - \Theta \sigma ^ { i j } _ { k } \nabla _ { i j } P ) - A \left [ \partial _ { t } \left ( \frac { \rho ^ { 2 } } { 2 } \right ) - \Theta \sigma ^ { i j } _ { k } \nabla _ { i j } \left ( \frac { \rho ^ { 2 } } { 2 } \right ) \right ] + \frac { \Theta } { P ^ { 2 } } \sigma ^ { i j } _ { k } \nabla _ { i } P \nabla _ { j } P . \end{align*}


\begin{align*} \mathbf { v } ^ { 0 } = [ b _ 0 , b _ 1 , \cdots , b _ { N - 1 } ] ^ { T } , \end{align*}


\begin{align*} & H _ { x , x ' } ^ { ( n ) } = \frac { 1 } { ( n + 1 ) ^ 2 } \sum _ { j , j ' = 0 } ^ n \Phi \left ( \frac { j } { n + 1 } , \frac { j ' } { n + 1 } \right ) \frac { \exp \left \{ \frac { 2 i \pi x } { n + 1 } \right \} - 1 } { \exp \left \{ \frac { 2 i \pi x } { n + 1 } \right \} - 1 } \exp \left \{ \frac { 2 i \pi j x } { n + 1 } \right \} \exp \left \{ \frac { 2 i \pi j ' x ' } { n + 1 } \right \} \end{align*}


\begin{align*} \hat W _ r ( k ) = \frac 1 k f ( r k ) \end{align*}


\begin{align*} g \big ( c ( t , x ) \big ) \left ( \frac { \partial _ { x x } w ( t , x ) } { 1 + | \partial _ x w ( t , x ) | ^ 2 } - \frac { 1 } { w ( t , x ) } \right ) \begin{cases} \leq - g \big ( c _ 0 ( x ) \big ) \frac { 1 } { w _ 0 ( x ) } + \varepsilon , \\ \geq - g \big ( c _ 0 ( x ) \big ) \frac { 1 } { w _ 0 ( x ) } - \varepsilon \end{cases} \end{align*}


\begin{align*} f _ \alpha ^ 2 e _ \alpha & = f _ \alpha e _ \alpha f _ \alpha + f _ \alpha [ f _ \alpha , e _ \alpha ] \\ & = f _ \alpha e _ \alpha f _ \alpha + f _ \alpha ( - h _ \alpha ) \\ & = f _ \alpha e _ \alpha f _ \alpha + ( - h _ \alpha - 2 ) f _ \alpha . \end{align*}


\begin{align*} \boldsymbol { \alpha } _ t ^ * = ( \alpha _ t ^ { * , i } ) _ { 1 \leq i \leq N } = \left ( - R ^ { - 1 } B y _ t ^ { * , i } - h \big ( \rho _ i ^ N ( \Delta ^ { N , 1 } _ { * , t } , \ldots , \Delta ^ { N , N } _ { * , t } ) \big ) \right ) _ { 1 \leq i \leq N } . \end{align*}


\begin{align*} F : = \lim _ { n \to \infty } T ^ n [ f ] . \end{align*}


\begin{align*} \alpha _ { i _ k , j _ \ell } \oplus \alpha _ { i _ r , j _ t } = \emptyset \end{align*}


\begin{align*} f ( x ) ( x f ' ( x ) + x ^ 2 f '' ( x ) ) & = \left ( \sum _ { i = k } ^ l \frac { x ^ i } { i ! } \right ) \left ( \sum _ { i = k } ^ l \frac { x ^ i } { ( i - 1 ) ! } + \frac { x ^ i } { ( i - 2 ) ! } \right ) = \left ( \sum _ { i = k } ^ l \frac { x ^ i } { i ! } \right ) \left ( \sum _ { i = k } ^ l \frac { i x ^ i } { ( i - 1 ) ! } \right ) \\ & \geq \left ( \sum _ { i = k } ^ l \frac { x ^ i } { ( i - 1 ) ! } \right ) ^ 2 = ( x f ' ( x ) ) ^ 2 . \end{align*}


\begin{align*} v _ { t _ 0 } ( t ) : = \lim _ { j \to + \infty } v _ { t _ 0 , B _ j } ( t ) = \left \{ \begin{aligned} & - t _ 0 & & x < - t _ 0 , \\ & \ t & & x \ge t _ 0 . \end{aligned} \right . \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*} \prod _ { i = 1 } ^ { \pi ( a ) } q _ i \neq \prod _ { i = 1 } ^ { \pi ( a ) } p _ i ^ { \alpha _ i } . \end{align*}


\begin{align*} \hat { b } _ { 1 , \delta } ( x ) = \left \{ \begin{array} { c c } b _ { 1 , \delta } ( x ) & x \in \bar \Delta \\ 0 & x \not \in \hat \Delta \ , . \end{array} \right . \end{align*}


\begin{align*} O _ { \gamma } ( f _ 0 ) = q ^ { d _ \gamma } = | D ( \gamma ) | ^ { - 1 / 2 } \end{align*}


\begin{align*} b \in \mbox { I m } ( f ) & \mbox { i f f } b + I m ( f ) = 0 \\ & \mbox { i f f } \pi ( b ) = 0 \\ & \mbox { i f f } b \in \mbox { K e r } ( \pi ) \end{align*}


\begin{align*} \int _ { c - i \infty } ^ { c + i \infty } d s ^ { \prime } \frac { F _ { q } ( s ^ { \prime } ) } { ( s - s ^ { \prime } ) } = 2 \pi i F _ { q } ( s ) . \end{align*}


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


\begin{align*} m = \left \lceil \frac { \log C _ 0 + 1 } { \beta \tau } \right \rceil . \end{align*}


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


\begin{align*} H _ K \times H ^ n ( Y _ K , N ) \rightarrow H ^ n ( Y _ K , N ) , \ , \ , \ , ( \chi , c ) \mapsto \chi \cdot c : = \chi \cup c \end{align*}


\begin{align*} \hat { X } ^ { \epsilon } ( t ) & = x _ { 0 } + \int ^ { t } _ { 0 } b ( \varphi ( s ) ) d s + \lambda ( \epsilon ) \int ^ { t } _ { 0 } \sigma ( \varphi ( s ) ) \theta _ { \epsilon } ( s ) d s . \end{align*}


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


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


\begin{align*} I _ h \leq \inf \left \{ E [ b ] \ \middle | \ b \in \tilde { T } _ h \right \} \leq \inf \left \{ E [ b ] \ \middle | \ b \in T _ h \right \} = I _ h , \end{align*}


\begin{align*} \exists C _ 3 = C _ 3 ( m ) \in ( 0 , \infty ) \ \Rightarrow T [ M ^ * n ] ( t ) \le \exp ( - C _ 3 ( m ) \ t ^ m ) , \ t \ge 0 , \end{align*}


\begin{align*} ( R ( X , T ) S ) ^ h = 0 \quad \forall T , S \in \Delta , \ , X \in T M . \end{align*}


\begin{align*} \mathbf { z } _ { k l } ^ { \mathrm { o p t } } = { \alpha _ { k } { \mu } _ { k } { \mathbf { Z } } ^ { - 1 } \overline { \mathbf { h } } _ { k l } } , \end{align*}


\begin{align*} \sigma ( \delta ) = ( \det \sigma ) \ \delta . \end{align*}


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


\begin{gather*} \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) = P \Bigl ( \oplus _ { i = 1 } ^ n f _ i \Bigr ) \ge \inf _ { Q \in \mathcal { U } } Q \Bigl ( \oplus _ { i = 1 } ^ n f _ i \Bigr ) \end{gather*}


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


\begin{align*} \| u \| _ { 2 , h _ n } = \sum _ { K \in \mathcal { T } _ n } ( u , u ) _ { H ^ 2 ( K ) } , \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*} F ( x ) = \sum _ { { \bf k } \in \Lambda _ { 1 } } \begin{pmatrix} a _ { \bf k } & 0 \\ 0 & - a _ { \bf k } \end{pmatrix} { \rm e } ^ { { \rm i } \langle { \bf k } , x \rangle } + \sum _ { { \bf k } \in \Lambda _ { 2 } } \begin{pmatrix} 0 & b _ { \bf k } \\ c _ { \bf k } & 0 \end{pmatrix} { \rm e } ^ { { \rm i } \langle { \bf k } , x \rangle } , \end{align*}


\begin{align*} \{ 1 , \ldots , \ell \} \simeq \bigcup _ { i = 1 } ^ L \{ m _ 1 + \cdots + m _ { i - 1 } + 1 , \ldots , m _ 1 + \cdots + m _ i \} \end{align*}


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


\begin{align*} \int _ { \mathfrak U } | \omega | & = { \pi ' _ s } _ ! \int _ { \mathfrak U ' } | \omega | = { \pi ' _ s } _ ! \Big ( { i ' _ s } ^ * \int _ { \mathfrak X ' } | \omega | \Big ) \\ & = ( \mathfrak i _ 0 ^ * \circ { \pi _ s } _ ! ) \int _ { \mathfrak X ' } | \omega | \\ & = \mathfrak i _ 0 ^ * \int _ { \mathfrak X } | \omega | , \end{align*}


\begin{align*} \limsup _ { x \to \infty } \frac { g ^ { \ast 2 } ( x ) } { g ( x ) } & \le ( 2 / \lambda ^ 2 ) e ^ \lambda \Big ( \lambda - e ^ { - \lambda } \sum _ { n \neq 2 } ^ \infty \liminf _ { x \to \infty } \frac { \lambda ^ n } { n ! } \frac { g ^ { \ast n } ( x ) } { g ( x ) } \Big ) = 2 , \end{align*}


\begin{align*} & w _ 0 = e _ 1 , \ \ w _ { n + 1 } ( x ; z ) = F w _ n = \int _ { - \infty } ^ x F ( x , y ; z ) w _ n ( y ) d y . \end{align*}


\begin{align*} & \lim _ { N \to \infty } \sup _ { t \in [ 0 , T ] } \sup _ { n \in \mathbb N } \sum _ { m , k \geq N } \mathbb E \left [ \langle \tilde Z _ t ^ { 2 , n } , e _ k \otimes e _ m \rangle _ { \mathcal H } ^ 2 \right ] \\ = & \lim _ { N \to \infty } \sup _ { t \in [ 0 , T ] } \sup _ { n \in \mathbb N } \mathbb E \left [ \| p _ N \tilde Z _ t ^ { 2 , n } \| _ { \mathcal H } ^ 2 \right ] = 0 , \end{align*}


\begin{align*} P _ 1 ( f , g ) & { } = \left ( f - \frac { 1 } { 2 } ( ( f - g ) \vee 0 ) , g + \frac { 1 } { 2 } ( ( f - g ) \vee 0 ) \right ) , \\ P _ { 2 , \alpha } ( f , g ) & { } = \left ( g + \frac { 1 } { 2 } \varphi _ \alpha \circ ( f - g ) , f - \frac { 1 } { 2 } \varphi _ \alpha \circ ( f - g ) \right ) , \end{align*}


\begin{align*} \prescript J { } \pi ( s _ \beta w ) = ( w ' , \lambda - \lambda ' + \Phi ^ + ( - \beta ) w ^ { - 1 } \beta ^ \vee ) . \end{align*}


\begin{align*} \alpha _ { i _ k , j _ \ell } \oplus \alpha _ { i _ r , j _ t } = \{ \alpha _ { i _ k , j _ t } , \alpha _ { j _ \ell , i _ r } \} \end{align*}


\begin{align*} { \rm e } _ { q , \ell , \ell ' } = \delta _ { \ell , \ell ' } \quad \mbox { a n d } { \rm e } _ { p , \ell , \ell ' } = \delta _ { n + 1 + \ell , \ell ' } , \ell ' = 1 , \ldots , 2 n + 2 . \end{align*}


\begin{align*} \hat { \Theta } _ { R M } = \left \{ \hat { \theta } _ i , i \in \{ 1 , 2 , \cdots , 2 N - 2 \} \right \} , \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { 1 } { \abs { \Lambda _ { n } } } s _ { n } ( \nu | \mu ) = \lim _ { n \to \infty } \frac { G _ { n } } { \abs { \Lambda _ { n } } } s _ { n } ( \nu | \mu ) , \end{align*}


\begin{align*} \dot \phi _ i = \frac 1 M \sum _ { j = 1 } ^ M a _ { i - j } \sin ( \phi _ j - \phi _ i ) , i = 1 , \dots , M , \end{align*}


\begin{align*} \cosh { { d } } = \frac { - \langle ~ \mathbf { x } , ~ \mathbf { y } \rangle } { \sqrt { \langle ~ \mathbf { x } , ~ \mathbf { x } \rangle \langle ~ \mathbf { y } , ~ \mathbf { y } \rangle } } . \end{align*}


\begin{align*} w = \Psi ( \hat z ) \end{align*}


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


\begin{align*} \Phi _ 1 ( t ) & = | \alpha | ^ 2 \log t + \theta _ 1 + O ( t ^ { - 1 / 4 + C _ 1 \varepsilon ^ 2 _ 1 } ) \end{align*}


\begin{align*} z : = \frac { \frac { 1 } { 2 } + i \rho } { \frac { 1 } { 2 } - i \rho } . \end{align*}


\begin{align*} \rho _ { Z Z } ( L , L ) = a \rho _ w ^ 2 + b \rho _ z ^ 2 . \end{align*}


\begin{align*} \int _ { 4 I _ Q } \| R _ j \partial _ j h ( \cdot , t ) \| _ { L ^ 1 ( 8 Q _ 1 ) } d t \lesssim \ell ( Q ) ^ N , \ ; \ ; j = 1 , 2 , \cdots , N . \end{align*}


\begin{align*} \widetilde { \varphi } ^ { H } _ { n , p } & = \varphi ^ { H } _ { n , p } - \psi _ { n , p } \ , , \\ \widetilde { \varphi } ^ { V } _ { n , p } & = \varphi ^ { V } _ { n , p } - \rho _ { n , p } \ , . \end{align*}


\begin{align*} & \sqrt n \left ( \hat { \Sigma } _ t ^ { n , m } - t Q \right ) \\ = & \sqrt n \left ( R V _ t ^ n - t Q \right ) + \left ( \sqrt n \left ( \hat { \Sigma } _ t ^ { n , m } - \Pi _ m t Q \Pi _ m \right ) - \sqrt n \left ( R V _ t ^ n - t Q \right ) \right ) + \sqrt n \left ( \Pi _ m t Q \Pi _ m - t Q \right ) . \end{align*}


\begin{align*} \Phi _ { t } + u \cdot \nabla \Phi & = 0 \textrm { i n } \ \mathbb { R } ^ { 3 } \times ( 0 , T ) . \end{align*}


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


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


\begin{align*} \int _ { 0 } ^ { 1 } h ( s ) \sin ( 2 \pi t ) \ , d t = 0 \end{align*}


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


\begin{align*} \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*} M _ { I } = \bigoplus _ { J \subset I } R _ { J } , \end{align*}


\begin{align*} e _ { G } ( \tilde { \theta } ^ k , T , \mathcal { K } ) = \tilde { \theta } ^ k - \Pi _ { \mathcal { K } , G } ( \tilde { \theta } ^ k - G ^ { - 1 } T ( \tilde { \theta } ^ k ) ) . \end{align*}


\begin{align*} ( G _ \alpha T _ \alpha f \circ \psi _ M ( m ) ) ( s ) & = G _ \alpha T _ \alpha f ( s \otimes m ) \\ & = T _ \alpha f ( s \otimes m ) \\ & = s \otimes f ( m ) \\ & = \psi _ M ( f ( m ) ) ( s ) \\ & = ( \psi _ M \circ f ) ( m ) ( s ) . \end{align*}


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


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


\begin{align*} z _ \star ^ { ( i ) } = \begin{cases} 2 T - i + 1 , & \textnormal { i f } 1 \leq i \leq 2 T , \\ 0 & \textnormal { o t h e r w i s e } . \end{cases} y _ \star ^ { ( i ) } = \begin{cases} 1 , & \textnormal { i f } 1 \leq i \leq 2 T , \\ 0 & \textnormal { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} \xi _ p ( V ) = \left ( \sum _ { l \ge n } ( l + 2 ) ( l + n + 3 ) ( l - n + 1 ) a _ { l + 3 } z ^ l \right ) d z ^ { \otimes 2 } \end{align*}


\begin{align*} a _ 2 ( x ) = \frac { \Gamma ( n - 2 ) \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) } { ( 2 \pi ) ^ { n - 1 } } \bigg ( a _ { 2 , 1 } H ^ 2 + a _ { 2 , 2 } \sum _ { \alpha } \kappa _ \alpha ^ 2 + a _ { 2 , 3 } R + a _ { 2 , 4 } R _ { n n } \bigg ) \end{align*}


\begin{align*} A ( p , s ) = 2 ^ { - p / 2 } \left ( \frac { 3 s } { 2 } - p \right ) \frac { 1 2 s - \left ( \frac { p } { 2 } + 2 \right ) \left ( \frac { p } { 2 } + 3 \right ) } { 1 4 4 } \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } d _ { q } s \ , F _ { q } ( s ) = \int _ { s } ^ { \infty } d _ { q } s \int _ { 0 } ^ { \infty } \exp _ { q } ( - s t ) f ( t ) d t . \end{align*}


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


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


\begin{align*} \Omega ( n , m ) : = [ h ( l _ n \xi , l _ n \eta , l _ n \omega ) - h ( g \xi , g \eta , g \omega ) ] - [ h ( l _ m \xi , l _ m \eta , l _ m \omega ) - h ( g \xi , g \eta , g \omega ) ] . \end{align*}


\begin{align*} \int _ { \Omega } \bold { A } _ j \cdot \bold { B } _ k \dd x = \frac { 1 } { \lambda _ k } \int _ { \Omega } \bold { A } _ j \cdot \nabla \times \bold { B } _ k \dd x = \frac { 1 } { \lambda _ k } \int _ { \Omega } \bold { B } _ j \cdot \bold { B } _ k \dd x = \frac { 1 } { \lambda _ k } \delta _ { j , k } . \end{align*}


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


\begin{align*} \int _ { D ( 0 , 2 ) \backslash D ( 0 , 1 ) } | \phi _ n | ^ { 2 } \frac { 1 } { r } \dd z \dd r = 1 , \int _ { D ( 0 , 2 ) \backslash D ( 0 , 1 ) } | \nabla \phi _ n | ^ { 2 } \frac { 1 } { r } \dd z \dd r \to 0 . \end{align*}


\begin{align*} \varepsilon _ { f , n } = \sum _ { j = 1 } ^ { \infty } ( - 1 ) ^ { j - 1 } \zeta _ { ( f ) } ^ { \varphi ^ { - ( n + 1 + 2 j ) } } p ^ j \end{align*}


\begin{align*} ( \partial - \nabla ) _ X Y = - \sum _ { i = 1 , 2 } \sqrt { c _ i } \ \langle X _ i , Y \rangle \nu _ i + B ( X , Y _ T ) - B ^ * ( X , Y _ N ) \end{align*}


\begin{align*} A = \bigcup _ { r = 1 } ^ { \infty } \bigcup _ { k = 1 } ^ { \infty } A _ { k , r } . \end{align*}


\begin{align*} X _ { n + 1 } = L ( X _ n ) + F ( X _ n ) + G ( X _ n ) \xi _ { n + 1 } , \end{align*}


\begin{align*} \psi ( B _ \phi u , p ) = B _ \phi \psi ( u , p ) \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*} \vert S \cap [ 1 , n ] \vert = o ( \alpha ( n ) \cdot n / \log n ) \end{align*}


\begin{align*} H ^ { ( k ) } ( E ) = \min \limits _ { \substack { 0 \neq F \subset E \\ \dim F = k } } H _ r ( F ) \end{align*}


\begin{align*} 2 a = ( a + b ) + ( a - b ) \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*} M _ n ( Q ) / n & \le \int _ \R V \ , d \overline { Q } + \frac 1 2 \int _ \R \int _ \R K ( x - y ) \ , \overline { Q } ( d x ) \overline { Q } ( d y ) - H ( \overline { Q } ) = M _ n ( \overline { Q } ^ { \otimes n } ) / n . \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*}