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


\begin{align*} \begin{aligned} \sqrt { a \cdot b } & \le \frac { a + b } { 2 } \ , , \operatorname { f o r } a , b > 0 \\ \operatorname { w i t h } \ , ` ` & = " \ , \operatorname { h o l d s } \ , \operatorname { i f } \ , \operatorname { a n d } \ , \operatorname { o n l y } \ , \operatorname { i f } \ , a = b \ , . \end{aligned} \end{align*}


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


\begin{align*} U ( 1 , 1 , \Delta ) = 1 . \end{align*}


\begin{align*} \int _ { 1 } ^ \infty \frac { d s } { \Big ( \displaystyle \int _ 0 ^ s F ( t ) d t \Big ) ^ { p / ( 2 p - q + 1 ) } } < \infty \quad \mbox { w h e r e } \ ; F ( t ) = \int _ 0 ^ t f ( k ) d k . \end{align*}


\begin{align*} \gamma ( x _ n ^ D ; w ) \ge \tau ( x _ n ^ D ) : = 8 \frac { | \Gamma _ D | \mathrm { N } _ I \lambda _ d ^ 2 } { | \Omega | \mathrm { N } _ D \lambda _ 1 } \frac { | \nabla w ( x _ n ^ D ) | ^ 2 } { | \nabla w ( y _ n ) | ^ 2 } , \end{align*}


\begin{align*} \sigma = ( y _ 1 + \i \theta _ 1 ) \wedge \dots \wedge ( y _ n + \i \theta _ n ) . \end{align*}


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


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


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


\begin{align*} F ( x ) = \frac { 1 } { \operatorname { B } ( a , b ) } \int ^ { x } _ { 0 } \frac { 2 \theta } { y ^ { 3 } } \exp \left ( - \frac { a \theta } { y ^ { 2 } } \right ) \left \{ 1 - \exp \left ( - \frac { \theta } { y ^ { 2 } } \right ) \right \} ^ { b - 1 } \mathrm { d } y . \end{align*}


\begin{align*} v = 0 , \nu ' ( h ) ( \bar u ) + \nu ( v ) = 0 , A ' ( h ) = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} k _ i ( x ) = \prod _ { \sigma \in \Sigma _ \infty } \sigma ( x ) ^ { e _ i ( \sigma ) } \end{align*}


\begin{align*} T ( x ) = x - g ( x ) = x - G ^ { - 1 } ( x ) . \end{align*}


\begin{align*} \prescript L { } { } \ell { } ^ R ( x , \alpha ) = & \langle \mu , \alpha \rangle + \chi _ R ( \alpha ) - \chi _ L ( w \alpha ) \\ = & \langle \mu , \alpha \rangle + \Phi ^ + ( \alpha ) - \Phi ^ + _ R ( \alpha ) - \Phi ^ + ( w \alpha ) + \Phi _ L ^ + ( w \alpha ) \\ = & \ell ( x , \alpha ) - \Phi ^ + _ R ( \alpha ) + \Phi _ L ^ + ( w \alpha ) \end{align*}


\begin{align*} \delta _ { j k } X _ { i j \ell } + \delta _ { i \ell } X _ { i j k } = \delta _ { i j } X _ { i k \ell } + \delta _ { k \ell } X _ { i j k } . \end{align*}


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


\begin{align*} \mathbf { w } _ { x _ w , s _ w } = \mathbf { T } _ { x _ w } \mathbf { s } _ w . \end{align*}


\begin{align*} \Lambda ^ { * } ( x ) = \left \{ \begin{array} { l l } 1 - \sqrt { 1 - x ^ { 2 } } , & ~ | x | \leq 1 ; \\ + \infty , & | x | > 1 . \\ \end{array} \right . \end{align*}


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


\begin{align*} { \bf E } \left ( { \bf A } , { \mathbb V } \right ) = \left \{ { \bf A } _ { \theta } : \theta \left ( x , y \right ) = \sum _ { i = 1 } ^ { s } \theta _ { i } \left ( x , y \right ) e _ { i } \ \ \ \ \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle \in { \bf T } _ { s } ( { \bf A } ) \right \} . \end{align*}


\begin{align*} D _ { \eta } = A ^ { - 1 } A _ { \eta } : \Gamma _ { \mathbb { C } } ^ \perp \rightarrow \Gamma _ { \mathbb { C } } ^ \perp , \end{align*}


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


\begin{align*} F ^ q ( B _ \phi \eta , p ) = B _ \phi F ^ q ( \eta , p ) \end{align*}


\begin{align*} X _ t \leq \int _ 0 ^ t \alpha _ s \sup _ { r \in [ 0 , s ] } X _ r \ , d s + M _ s + H _ s , \end{align*}


\begin{align*} x ^ { \ast , \sigma , N + n } = x ^ { \ast , \sigma , N } \cdot \prescript { \sigma ^ N } { } x _ \infty \cdots \prescript { \sigma ^ { N + n - 1 } } { } x _ \infty \end{align*}


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


\begin{align*} N _ 2 ( s ) \leq N _ 2 : = \frac { 4 ( \tau + C _ 2 ) } { \eta ( t ) } + 1 . \end{align*}


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


\begin{align*} y \cdot [ x , z ] = \varepsilon ( y , x ) [ x , y \cdot z ] - \varepsilon ( x + y , z ) [ z , y \cdot x ] + [ y , x ] \cdot z - \varepsilon ( x , z ) [ y , z ] \cdot x . \end{align*}


\begin{align*} ( \Delta \omega ^ * ( q , K , \Omega ) ) ^ m \wedge \beta _ n ^ { n - m } = 0 \Omega \setminus K . \end{align*}


\begin{align*} \frac { d } { d t } z _ t = B ( z _ t , z _ t ) \end{align*}


\begin{align*} u _ { n + k } + \alpha _ { 1 , n } u _ { n + k - 1 } + \alpha _ { 2 , n } u _ { n + k - 2 } + \cdots + \alpha _ { k , n } u _ { n } = 0 \end{align*}


\begin{align*} ( \mu + i \eta ) W _ 2 ( \xi ) - i \lambda _ 6 \tfrac { W _ 1 ( \xi ) ^ 2 } { | W _ 1 ( \xi ) | ^ 2 } \overline { W _ 2 ( \xi ) } = 0 \end{align*}


\begin{align*} K _ i = \langle \{ a _ i , b _ i \} , \{ b _ i , c _ i \} \rangle , \ L _ i = \langle \{ a _ i \} , \{ c _ i \} \rangle . \end{align*}


\begin{align*} k = \frac { z ^ 2 + z + 1 } { n } q - a z - a \ , . \end{align*}


\begin{align*} \begin{aligned} S _ { e , \mathcal { Y M } } ( X , Y ) & = \exp ( \frac { | | R ^ { \nabla } | | ^ 2 } 2 ) \big ( g ( X , Y ) - \langle i _ X R ^ { \nabla } , i _ Y R ^ { \nabla } \rangle \big ) \end{aligned} \end{align*}


\begin{align*} { \mathcal { Q } } _ { i j } ( n ) : = \textsf { P } \left \{ { W ( { n + k } ) = j } \bigm | { W ( k ) = i } \right \} = { \frac { j q ^ { j - i } } { i \beta ^ n } } P _ { i j } ( n ) i , j \in { \mathcal { E } } , \end{align*}


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


\begin{align*} \| ( \mathcal S ( t ) - I ) B ^ { \mathfrak H } ( \omega ) \| ^ 2 = & \int _ 0 ^ { 2 - t } ( B _ { t + x } ^ { \mathfrak H } ( \omega ) - B _ x ^ { \mathfrak H } ( \omega ) ) ^ 2 d x + \int _ { 2 - t } ^ 2 ( B _ x ^ { \mathfrak H } ( \omega ) ) ^ 2 d x \leq \bar C _ { \omega } t ^ { 2 \mathfrak H } . \end{align*}


\begin{align*} \hat { \Gamma } _ t ^ n : = \Delta _ n ^ { - 1 } \bigl ( S A M P V _ t ^ n ( 4 ) - S A M P V _ t ^ n ( 2 , 2 ) \bigr ) . \end{align*}


\begin{align*} z _ 0 ^ { p ^ r } x _ 0 + z _ 1 ^ { p ^ r } x _ 1 \cdots + z _ n ^ { p ^ r } x _ n = 0 . \end{align*}


\begin{align*} \{ v _ 1 , \ldots , v _ k \} \in G \bigcap \limits _ { i = 1 } ^ k S _ { v _ i } = \emptyset . \end{align*}


\begin{align*} K ^ L ( x ' , t ) : = P _ t ^ L ( x ' ) = t ^ { 1 - n } P ^ L ( x ' / t ) , \forall ( x ' , t ) \in \R ^ n _ + , \end{align*}


\begin{align*} Q ^ c ( g , R ) ( x , J x , J x , x ; u , J u ) = 2 \ , Q ^ c ( g , R ) ( u , J u , x , J x ; x , J x ) \neq 0 . \end{align*}


\begin{align*} I _ { l } ^ { \left ( 2 \right ) } ( s , \rho ) = \frac { 1 } { \left ( 2 i \right ) ^ { 2 } } \int _ { \mathbb { R } } \frac { e ^ { i S ( \tau , x - s ) } \tau ^ { l - 3 } } { \left ( \tau - \rho \right ) ^ { l + 1 } } d \tau . \end{align*}


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


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


\begin{align*} G \frac { d } { d t } w ( t ) = L w ( t ) + f ( t ) \end{align*}


\begin{align*} Y ^ n _ 0 = \xi _ 0 , \quad Y _ { k + 1 } ^ n = Y _ k ^ n + \mu ( \tfrac { k T } { n } , \mathcal { Y } ^ n ) \tfrac { T } { n } + \sigma ( \tfrac { k T } { n } , \mathcal { Y } ^ n ) ( W _ { \frac { ( k + 1 ) T } { n } } - W _ { \frac { k T } { n } } ) . \end{align*}


\begin{align*} \Delta \rho _ 1 ( x ) \leq \psi _ { K , - H } ( \rho _ 1 ( x ) ) = ( n - 1 ) \sqrt K \cdot \frac { ( n - 1 ) \sqrt K \sinh \sqrt K \rho _ 1 - H \cosh \sqrt K \rho _ 1 } { ( n - 1 ) \sqrt K \cosh \sqrt K \rho _ 1 - H \sinh \sqrt K \rho _ 1 } . \end{align*}


\begin{align*} p \frac { d ^ 2 } { d s ^ 2 } \left ( \left ( \kappa - \mu \right ) ^ { p - 1 } \right ) + p \kappa ^ 2 \left ( \kappa - \mu \right ) ^ { p - 1 } - \kappa \left ( \left ( \kappa - \mu \right ) ^ p + \sigma \right ) = 0 . \end{align*}


\begin{align*} \max _ { - N \leq j \leq N } \Vert M _ { H _ j } : S _ { 1 } ^ { 2 N + 1 } \rightarrow S _ { 1 } ^ { 2 N + 1 } \Vert \leq \Vert M _ { \widetilde { H } \vert _ { \mathbb { Z } _ N \times \mathbb { Z } _ N } } ^ { ( N ) } : S _ { p _ 1 } ^ { 2 N + 1 } \times S _ { p _ 2 } ^ { 2 N + 1 } \rightarrow S _ { 1 } ^ { 2 N + 1 } \Vert . \end{align*}


\begin{align*} E _ \lambda : = \{ p \in \Omega : \psi _ E ( p ) < \lambda \} \end{align*}


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


\begin{align*} \varphi ( \theta ^ { k } , \tilde { \theta } ^ { k } , \zeta ^ k ) = \langle \lambda ^ { k } - \tilde { \lambda } ^ { k } , \tilde { y } ^ { k } - y ^ { k } \rangle _ { \mathcal { L } ^ 2 } + \langle \theta ^ { k } - \tilde \theta ^ { k } , G d ( \theta ^ k , \tilde { \theta } ^ { k } , \zeta ^ k ) \rangle _ { \mathcal { L } ^ 2 } , \end{align*}


\begin{align*} F ^ + ( t , x ) : = \left ( \begin{array} { c } R ^ + ( t , x ) \cos \pi t , \\ R ^ + ( t , x ) \sin \pi t \end{array} \right ) , \end{align*}


\begin{align*} \nu ^ * A = \mu ^ * h ^ * D + E . \end{align*}


\begin{align*} \phi _ \mu ( g ) = \frac { J _ \lambda ^ { ( 2 ) } ( a _ 1 , \ldots , a _ N ) } { J _ \lambda ^ { ( 2 ) } ( 1 , \ldots , 1 ) } , \mu = 2 \lambda \end{align*}


\begin{align*} \partial _ { t } ^ n \sqrt { P _ { \pm , \pm } \left ( t , u + \left ( \frac { t _ 1 } { t } \right ) ^ 2 u _ 1 \right ) } \Bigg | _ { t = t _ 0 } \asymp _ n \frac { 1 } { t _ 0 ^ { n - 1 } } , \end{align*}


\begin{align*} u _ 2 ( t ) & = M ( t ) D ( t ) e ^ { - 3 i \lambda _ 1 | W _ 1 | ^ 2 \log t } \begin{pmatrix} 1 & 0 \end{pmatrix} P Q ( t ) P ^ { - 1 } \begin{pmatrix} W _ 2 \\ \overline { W _ 2 } \end{pmatrix} \\ & + O ( t ^ { - 3 / 4 + C _ 1 \varepsilon ^ 2 _ 1 + 2 \mu C _ 2 \varepsilon ^ 2 _ 1 + C _ 4 \varepsilon _ 1 ^ 2 } ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \langle A ( u _ n ) , u _ n \rangle = \langle A ( u ) , u \rangle . \end{align*}


\begin{align*} \lim _ { N \to \infty } \sup _ { n \in \mathbb N } \mathbb P \left [ \sum _ { k \geq N } \langle Y _ n , e _ k \rangle ^ 2 > \delta \right ] = 0 , \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } d _ { q } s \ , F _ { q } ( s ) = L _ { q } \left \{ \frac { f ( t ) } { t } \right \} . \end{align*}


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


\begin{align*} h _ 0 \ , * \ h _ 1 = h _ 1 . \end{align*}


\begin{align*} g ^ * f _ * \mathcal M = \varprojlim _ k g ^ * f _ * \tau _ { \le k } \mathcal M = \varprojlim _ k f ' _ * g '^ * \tau _ { \le k } \mathcal M = f ' _ * \varprojlim _ k g '^ * \tau _ { \le k } \mathcal M = f ' _ * g '^ * \mathcal M , \end{align*}


\begin{align*} \varphi _ { * } : = - ( \varphi _ 0 + \ldots + \varphi _ p + 1 ) = 0 . \end{align*}


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


\begin{align*} \mathbb { E } \Big [ \underset { t _ 0 \leq t \leq T } { \sup } | \phi ^ { N , i } ( t , \boldsymbol { x } _ t ^ * ) - \Psi ^ { N , i } ( t , \boldsymbol { x } _ t ^ * ) | ^ 2 \Big ] \leq \frac { C } { N ^ 2 } \Big ( 1 + \frac { 1 } { N } \underset { j = 1 } { \overset { N } \sum } \mathbb { E } [ | \xi ^ i - \xi ^ j | ^ 2 ] + \frac { 1 } { N ^ 2 } \underset { i , j = 1 } { \overset { N } { \sum } } \mathbb { E } [ | \xi ^ { i } - \xi ^ { j } | ^ 2 ] \Big ) . \end{align*}


\begin{align*} \tilde Y _ { n + 1 } ^ i = \tilde Y _ n ^ i + \bigg ( b ( \tilde Y _ n ^ i ) + \frac 1 { p - 1 } \sum _ { j \neq i , j \in \C } K ( \tilde Y _ n ^ i - \tilde Y _ n ^ j ) \bigg ) \tau + \sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , ~ ~ i \in \C . \end{align*}


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


\begin{align*} h ( a , x ) : = \min _ k \mu _ k ( a , x ) > 0 . \end{align*}


\begin{align*} \varphi ( - T ( x ) + x ) = \varphi ( z ) = z - 2 g ^ { - 1 } ( z ) = x - T ( x ) - 2 x = - T ( x ) - x . \end{align*}


\begin{align*} \frac { 1 } { ( M ) } \mathcal { Y M } _ p ( { \nabla } ) = \frac { | | R ^ \nabla | | ^ p } { p } = \frac 1 p \bigg ( \frac { 2 } { ( M ) } \mathcal { Y M } ( \nabla ) \bigg ) ^ { \frac p 2 } \end{align*}


\begin{align*} \Phi ^ \dagger ( v , p ) : = Q ^ \dagger F ^ { q , \dagger } ( v + \psi ^ \dagger ( v , p ) , p ) , \end{align*}


\begin{align*} \mu \big ( T ^ { ( r + 1 ) r _ n } A \cap A ^ c \big ) \leq & \sum _ { i = 1 } ^ { r + 1 } \mu \big ( T ^ { i r _ n } A \cap T ^ { ( i - 1 ) r _ n } A ^ c \big ) \\ < & \big ( r + 1 \big ) \big ( 1 - \beta \big ) \mu ( A ) \\ = & \Big ( \big ( r + 1 \big ) - \big ( r + 1 \big ) \beta \Big ) \mu ( A ) \\ = & \Big ( 1 - \big ( ( r + 1 ) \beta - r \big ) \Big ) \mu ( A ) \\ < & \Big ( 1 - \mu ( A ) \Big ) \mu ( A ) . \end{align*}


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


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


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


\begin{align*} h ^ \intercal = 0 , A ' ( h ) = 0 , ( \nabla ^ k _ \nu A ) ' ( h ) = 0 \mbox { o n } \hat \Sigma \end{align*}


\begin{align*} \theta _ { \rho ^ \varepsilon } ( t , z _ l ) & = F ( z _ l ) + \rho ^ \varepsilon ( t , z _ l ) \nu _ \Sigma ( z _ l ) = ( 0 , 0 ) + \rho ^ \varepsilon ( t , z _ l ) ( - 1 , 0 ) = \big ( - \rho ^ \varepsilon ( t , z _ l ) , 0 \big ) , \\ \theta _ { \rho ^ \varepsilon } ( t , z _ r ) & = F ( z _ r ) + \rho ^ \varepsilon ( t , z _ r ) \nu _ \Sigma ( z _ r ) = ( 0 , 0 ) + \rho ^ \varepsilon ( t , z _ r ) ( + 1 , 0 ) = \big ( \rho ^ \varepsilon ( t , z _ r ) , 0 \big ) \end{align*}


\begin{align*} A = \mathcal { N } ( A ) - \mathcal { R } ( A ) . \end{align*}


\begin{align*} I _ 2 & = | c | | \langle f , g \rangle | = \left | c \int ( f - m _ Q f ) \ ; g \ ; d m \right | \\ & \lesssim \int _ { 2 Q } | f - m _ Q f | \ ; | g | \ ; d m + \int _ { \R ^ { N + 1 } \setminus 2 Q } ( f - m _ Q f ) \ ; g \ ; d m \\ & = I _ { 2 1 } + I _ { 2 2 } . \end{align*}


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


\begin{align*} f ( t ) : = \frac { ( t - a ) ^ { \alpha - 1 } } { \Gamma ( \alpha + 1 ) } \left [ \frac { \alpha } { \alpha - \beta } ( b - a ) - ( t - a ) \right ] , t \in [ a , b ] . \end{align*}


\begin{align*} \pi _ L ( h ) ( u \otimes v ) = ( \pi ( h ) ( u ) ) \otimes v \end{align*}


\begin{align*} B ( p ) = \frac { 1 } { p \mu } \int _ { 0 } ^ { Q ( p ) } x f ( x ) \mathrm { d } x = \frac { 1 } { p \mu } J ( Q ( p ) ) L ( p ) = \frac { 1 } { \mu } \int _ { 0 } ^ { Q ( p ) } x f ( x ) \mathrm { d } x = \frac { 1 } { \mu } J ( Q ( p ) ) , \end{align*}


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


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


\begin{align*} \kappa ( s ) = \alpha \ , \frac { \nu _ { n + 1 } ( s ) } { x _ { n + 1 } ( s ) } + \varpi . \end{align*}


\begin{align*} s _ i ( A ) : = \sqrt { \lambda _ i ( A ^ * A ) } , i = 1 , 2 , \dots , n , \end{align*}


\begin{align*} G ( H ( x ) ) = H ( x ) \quad \ x \in \partial T ( s ) . \end{align*}


\begin{align*} R \cdot R ( x , J x , J x , x ; u , J u ) = L ( p ) \ , \ , Q ( g , R ) ( x , J x , J x , x ; u , J u ) , \end{align*}


\begin{align*} & \lambda _ p < \lambda _ q \equiv 0 < \lambda _ r , \\ & p = 1 , \dots , l , q = l + 1 , \dots , m , r = m + 1 , \dots , n , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ r \frac { t ^ { d _ i p ^ { j _ i } } - 1 } { t ^ { d _ i } - 1 } \in \Z [ t ] . \end{align*}


\begin{align*} \tilde { F } _ i ( x ) = \bar { F } _ i ( x ) \times \tilde { H } ( x ) , i = 1 , \dots , n , \end{align*}


\begin{align*} \tilde { r } = r + \left ( \Delta ^ g f + g ( \theta , d f ) \right ) F , \end{align*}


\begin{align*} T : L ^ 2 ( D ) \rightarrow L ^ 2 ( D ) T f = u . \end{align*}


\begin{align*} g ( \omega ) : = g _ { \theta } ( \omega ) , \omega \in \Omega _ { \theta } \subset \Omega . \end{align*}


\begin{align*} \phi _ 1 '' + c \phi _ 1 ' + \frac { \epsilon } { \kappa } \phi _ 2 '' + \frac { c } { \kappa } \phi _ 2 ' = 0 . \end{align*}


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


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


\begin{align*} p _ l ( s \cdot z ) = s ^ l p _ l ( z ) , \ ; \ ; \forall l = 0 , \dots , k . \end{align*}


\begin{align*} \begin{aligned} ( \Delta ( u + v ) ) ^ m \wedge \beta _ n ^ { n - m } & = ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } + ( \Delta v ) ^ m \wedge \beta _ n ^ { n - m } + \sum \limits _ { p = 1 } ^ { m - 1 } \binom { m } { p } ( \Delta u ) ^ p \wedge ( \Delta v ) ^ { m - p } \wedge \beta _ n ^ { n - m } \\ & \geq ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } + ( \Delta v ) ^ m \wedge \beta _ n ^ { n - m } . \\ \end{aligned} \end{align*}


\begin{align*} \begin{aligned} & \xi y \left [ n \right ] = \sqrt P s \left [ { n - { n _ { \max } } } \right ] + \\ & \sqrt P \sum \limits _ { l = 1 } ^ L { \sum \limits _ { l ' \ne l } ^ L { \frac { { { \bf { h } } _ l ^ H { { \bf { h } } _ { l ' } } } } { { \sum \nolimits _ { k = 1 } ^ L { { { \left \| { { { \bf { h } } _ k } } \right \| } ^ 2 } } } } s \left [ { n - { n _ { \max } } + { n _ { l ' } } - { n _ l } } \right ] } } + \xi z \left [ n \right ] . \end{aligned} \end{align*}


\begin{align*} R _ f \circ g = g \bullet _ \mu f . \end{align*}


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


\begin{align*} ( B _ \phi f ) ( x ) = f ( x + \phi ) - f ( \phi ) . \end{align*}


\begin{align*} \frac { d \nu _ x } { d \nu _ y } ( \xi ) = e ^ { h ( g ) B _ { x , y } ( \xi ) } , \end{align*}


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


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


\begin{align*} I ( \mu _ m ) & = H ( \mu ^ m \ , | \ , \overline { \mu } ) = \int _ { [ 0 , 1 ] \times \R } \log \frac { d \mu ^ m } { d \overline { \mu } } \ , d \mu ^ m \\ & = \int _ { [ 0 , 1 ] \times \R } \log \frac { d \mu ^ m } { d \mu } \ , d \mu ^ m + \int _ { [ 0 , 1 ] \times \R } \log \frac { d \mu } { d \overline { \mu } } \ , d \mu ^ m . \end{align*}


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


\begin{align*} \lim _ { k \to + \infty } \mathrm { d i s t } _ { \mathbb { U } } ( \mathcal { A } ( \tau _ { n _ k } , \theta _ { \tau _ { n _ k } } \omega ) , \mathcal { A } _ { \infty } ( \theta _ { \tau _ { n _ k } } \omega ) ) = 0 \ \emph { f o r a l l } \ \omega \in \Omega . \end{align*}


\begin{align*} \begin{aligned} | \langle \mu _ t - \nu , b ( t , x , \cdot ) \rangle \leq \gamma H \left ( \nu \mid \mu _ t \right ) \quad t \in [ 0 , T ] , x \in \mathbb { R } ^ d , \\ \nu \in \mathcal { P } ( \mathbb { R } ^ d ) b ( t , x , \cdot ) \in L ^ 1 ( \nu ) . \end{aligned} \end{align*}


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


\begin{align*} \mathbf { h } = & \lambda _ { 1 } \mathbf { e } _ 1 + \lambda _ { 2 } \mathbf { e } _ 2 + \lambda _ { 1 2 3 ' } \mathbf { e } _ { 1 2 3 ' } \\ = & [ \lambda _ 1 + \lambda _ { 1 2 3 ' } , \lambda _ 2 + \lambda _ { 1 2 3 ' } , \lambda _ { 1 2 3 ' } , \lambda _ 1 + \lambda _ 2 + 2 \lambda _ { 1 2 3 ' } , \\ & \lambda _ 1 + 2 \lambda _ { 1 2 3 ' } , \lambda _ 2 + 2 \lambda _ { 1 2 3 ' } , \lambda _ 1 + \lambda _ 2 + 2 \lambda _ { 1 2 3 ' } ] ^ \intercal \end{align*}


\begin{align*} \left ( \rho ^ \star \right ) ^ { s y m } - \left ( R i c ^ g \right ) ^ { J , + } = - \frac { 1 } { 4 } \left ( 2 \delta ^ g \theta + \| \theta \| ^ 2 - 2 \| N \| ^ 2 \right ) g . \end{align*}


\begin{align*} q \mathcal { A } ( u ) = \Vert \nabla u \Vert ^ q _ q = \langle A ( u ) , u \rangle , \\ \ell \mathcal { P } ( v ) = \Vert v \Vert ^ \ell _ { \ell } = \langle P ( v ) , v \rangle \end{align*}


\begin{align*} F ( \theta , E ) = \cos { \pi \theta } \cdot A ( \theta , E ) = \left ( \begin{matrix} E \cos { \pi \theta } - \lambda \sin { \pi \theta } \ & - \cos { \pi \theta } \\ \cos { \pi \theta } & 0 \end{matrix} \right ) . \end{align*}


\begin{align*} \bigoplus _ { j = l } ^ { n - 1 } K _ j ' \subset ( \bigoplus _ { j = l } ^ { n - 1 } \sigma _ j \circ \rho _ j ' : \bigoplus _ { j = l } ^ { n - 1 } j _ { j * } E _ j \rightarrow \bigoplus _ { j = l } ^ { n - 1 } ( \sigma _ j ) ) \end{align*}


\begin{align*} 2 ^ { - ( k + 2 ) } n \ge 2 \ \mbox { a n d } \binom { n } { k } 2 ^ k \cdot \exp \left ( - 2 ^ { - ( k + 4 ) } n \right ) < \frac { 1 } { ( 2 n ) ^ { \log ( 2 n ) } } \end{align*}


\begin{align*} \operatorname { E } ( X ^ { - 1 } ) = \frac { \sqrt { \pi / \theta } } { 2 \operatorname { B } ( a , b ) } \sum _ { n = 0 } ^ b ( - 1 ) ^ n \binom { b - 1 } { n } \frac { 1 } { \sqrt { ( a + n ) ^ 3 } } . \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } \mu } \ell = & - \frac { \alpha - 1 } { \sigma } \frac { \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) } { \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) } + \frac { \beta - 1 } { \sigma } \frac { \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) } { 1 - \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) } + \frac { s } { \sigma } \left | \frac { x - \mu } { \sigma } \right | ^ { s - 1 } \mathrm { s i g n } ( x - \mu ) \end{align*}


\begin{align*} \nabla _ X Y = D ^ g _ X Y - \frac { 1 } { 2 } \theta ( J X ) J Y - \frac { 1 } { 2 } \theta ( Y ) X + \frac { 1 } { 2 } g ( X , Y ) \theta ^ \sharp + g ( X , N ( \cdot , Y ) ) , \end{align*}


\begin{align*} g _ { | i } = \begin{cases} \dot g ( j ) _ { | i } \ , \dot g ( j ) _ { | \ , i + 3 q ( n ) + 4 C } & \\ 1 & \end{cases} \end{align*}


\begin{align*} L _ 1 \circ F \circ L _ 2 = F ' , \end{align*}


\begin{align*} | a ( z ) | \geq 1 / 2 , \operatorname { I m } z = 0 . \end{align*}


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


\begin{align*} h _ { i j k } = h _ { i k j } . \end{align*}


\begin{align*} x ^ { j } \allowbreak = \allowbreak j ! \sum _ { k = 0 } ^ { j } ( - 1 ) ^ { k } \frac { ( \beta ) ^ { ( j ) } } { ( j - k ) ! ( \beta ) ^ { ( k ) } } L _ { k } ^ { ( \beta ) } ( x ) , \end{align*}


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


\begin{align*} \frac { d L } { d t _ { k } } = \left [ ( L ^ { k } ) _ { D } , L \right ] \ ; , k \geq 1 \ ; , \end{align*}


\begin{align*} \left | \frac { z } { n } - \frac { a } { q } \right | < \frac { 1 } { q \sqrt { n } } \ , , \quad 1 \leq q \leq \sqrt { n } , \quad ( a , \ , q ) = 1 \ , . \end{align*}


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


\begin{align*} N e w t o n ( s _ { p ^ { \downarrow } } ) \cap \mathbb { Z } ^ m \subseteq N e w t o n ( s _ { \lambda ^ j } ) \cap \mathbb { Z } ^ m = S u p p ( s _ { \lambda ^ j } ) \subseteq S u p p ( F ) . \end{align*}


\begin{align*} f _ { * } \nabla _ X A D _ i Y & = \widetilde { \nabla } _ X f _ { * } A D _ i Y - \langle A X , A D _ i Y \rangle \rho = - \widetilde { \nabla } _ X h _ { * } \pi _ { * } D _ i Y - \langle h _ { * } \pi _ { * } X , h _ { * } \pi _ { * } D _ i Y \rangle h \circ \pi \\ & = - h _ { * } \nabla ' _ { \hat { X } } \hat { D } _ i \hat { Y } - \alpha ^ h ( \hat { X } , \hat { D } _ i \hat { Y } ) . \end{align*}


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


\begin{align*} \tau ( \widetilde K _ n ) \ge [ \Lambda ( g , 2 A , M ) ] ^ { - 1 } ( 1 + \varepsilon ) & \geq [ \Lambda ( g , A , M ) ] ^ { - 2 } ( 1 + \varepsilon ) \\ & > \left [ \frac { 1 + \varepsilon } { 1 + \frac { \varepsilon } { 2 } } \right ] ^ { - 1 } ( 1 + \varepsilon ) \\ & = 1 + \frac { \varepsilon } { 2 } , \end{align*}


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


\begin{align*} \left ( \left ( e ^ { 1 - \binom { k } { \ell } } + o ( 1 ) \right ) m ^ { k - \ell } \right ) ^ { m ^ \ell } \end{align*}


\begin{align*} G ( t ) = \int _ 0 ^ 1 K ( t , r ) d W ( r ) , \ \ \ 0 \le t \le 1 . \end{align*}


\begin{align*} \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { \infty } h _ { n } ( x ) h _ { m } ( x ) \exp ( - x ^ { 2 } / 2 ) d x = \delta _ { n m } , \end{align*}


\begin{align*} L ( x ) = \sum _ { i = 0 } ^ d b _ i x ^ { q ^ i } \end{align*}


\begin{align*} \left \{ \begin{array} { l } \lambda ^ { N _ { i - 1 } } + \sum _ { \ell = 1 } ^ { r } c _ \ell u _ { i - 1 } ^ { \gamma _ \ell } \lambda ^ { ( i - 1 ) \gamma _ \ell + N _ { i - 1 } - \ell } = 0 \\ \lambda = u _ { i - 1 } v _ { i } \end{array} \right . \end{align*}


\begin{align*} \phi _ i = \left \{ \begin{aligned} & \sigma & & ( i = k ) \\ & \tau & & ( i \neq k ) , \end{aligned} \right . \end{align*}


\begin{align*} \begin{pmatrix} 1 / d & \sqrt { 1 - 1 / d ^ 2 } \\ \sqrt { 1 - 1 / d ^ 2 } & - 1 / d \end{pmatrix} , \end{align*}


\begin{align*} \frac { L ' } { L } ( s , \chi ) = - \sum _ { n \leq x y } \frac { \Lambda _ { x , y } ( n ) \chi ( n ) } { n ^ s } + \frac { 1 } { \log { y } } \sum _ { \rho _ { \chi } } \frac { x ^ { \rho _ { \chi } - s } - ( x y ) ^ { \rho _ { \chi } - s } } { \left ( \rho _ { \chi } - s \right ) ^ { 2 } } + \frac { 1 } { \log { y } } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { - 2 n - \mathfrak { a } - s } - ( x y ) ^ { - 2 n - \mathfrak { a } - s } } { \left ( 2 n + \mathfrak { a } + s \right ) ^ { 2 } } \end{align*}


\begin{align*} | \phi | = \sum _ { i \in [ m ] } | \phi _ i | = m ( \dim K + 1 ) . \end{align*}


\begin{align*} F _ { q } ^ { ( k ) } ( s , q ^ { \prime } ) = \displaystyle \sum _ { n = 0 } ^ { \infty } \frac { \left ( - \frac { 1 } { 1 - q ^ { \prime } } \right ) _ { n } \left ( \mp ( 1 - q ^ { \prime } ) \sigma _ { \epsilon } \right ) ^ { n } } { Q _ { n + 1 } ( 2 - q ) \ ; \ ; n ! } \frac { ( - 1 ) ^ { k } \Gamma ( n + k + 1 ) } { s ^ { n + k + 1 } } . \end{align*}


\begin{align*} \alpha = \frac { c } { 4 \pi k \sqrt { k ^ 2 + r _ * ^ 2 } } , \ \ \beta = \frac { \alpha } { 2 } ( 3 r _ * ^ 2 + 4 k ^ 2 ) . \end{align*}


\begin{align*} | f _ i ( u ) | \lesssim 1 + | u | ^ p , \forall u \in \R _ + ^ m , \ ; \forall i = 1 , \ldots , m . \end{align*}


\begin{align*} & \Delta _ \Gamma V + V \big | \nabla _ \Gamma \nu \big | ^ 2 = A \colon D _ \Gamma ^ 2 H + B \cdot \nabla _ \Gamma H + C H \end{align*}


\begin{align*} c _ { \lambda _ n } \geq \kappa ( \lambda _ n ) \geq \left ( \frac { 1 } { 8 } - \frac { 1 } { 2 ^ { 2 p + 1 } } \right ) \left ( \frac { \beta _ { \lambda _ n } ^ p } { 2 C ( N , \alpha , p ) } \right ) ^ { \frac { 1 } { p - 1 } } = \left ( \frac { 1 } { 8 } - \frac { 1 } { 2 ^ { 2 p + 1 } } \right ) \left ( \frac { \beta _ { 0 } ^ p } { 2 C ( N , \alpha , p ) } \right ) ^ { \frac { 1 } { p - 1 } } > 0 . \end{align*}


\begin{align*} \tau = \beta \left ( s \right ) u , \quad \rho = \beta \left ( s \right ) \xi \quad \beta \left ( s \right ) = \left ( \frac { s - x } { 1 2 t } \right ) ^ { \frac { 1 } { 2 } } . \end{align*}


\begin{align*} I _ { \theta h } \leq E [ t ( \theta ) b ] = t ( \theta ) ^ { 2 } E [ b ] = t ( \theta ) ^ { 2 } I _ { h } < \theta I _ h , \end{align*}


\begin{align*} \pi _ 1 ( M ) = \cdots = \pi _ { 6 } ( M ) = 0 . \end{align*}


\begin{align*} { { \xi } _ { o } } = \left \| \left ( ( { { \mathbf { x } } _ { o } } + { { \mathbf { s } } _ { o } } ) - { { \psi } _ { o } } \mathbf { e } \right ) \right \| + \rho \left \| { { \mathbf { s } } _ { o } } \right \| - \gamma { { \psi } _ { o } } . \end{align*}


\begin{align*} h _ { \hat { \mu } } ( \hat { \sigma } ) = h _ { \hat { \mu } } ( \hat { \sigma } , \hat { \P } ) \ \ h _ \mu ( \sigma ) = h _ \mu ( \sigma , \P ) \ . \end{align*}


\begin{align*} E _ { u } = < \nabla _ { H } u , \nabla _ { H } \rho > - \frac { Z u } { \rho } | \nabla _ { H } \rho | ^ 2 . \end{align*}


\begin{align*} \mathcal { L } u _ { 1 } = 3 \lambda _ 1 | u _ 1 | ^ 2 u _ 1 . \end{align*}


\begin{align*} \kappa ' \begin{pmatrix} 1 \\ \vdots \\ \theta ^ { n - 1 } \end{pmatrix} = \begin{pmatrix} a _ { 1 } \\ \vdots \\ a _ { n } \end{pmatrix} . \end{align*}


\begin{align*} s \cdot ( x , y , z , \zeta ) = ( a x + b y , c x + d y , ( \det s ) z , ( \det s ) \zeta ) . \end{align*}


\begin{align*} \mathbf { R } \mathbf { v } ^ { 0 } & = \mathbf { R } \frac { ( \sum \limits _ { j = 1 } ^ { N } \mathbf { R } _ { . j } ) ^ { H } } { \mathbf { S ( R ) } } = \mathbf { R } ( \mathbf { 1 } ^ { H } _ { N \times 1 } \mathbf { R } ) ^ { H } / \mathbf { S ( R ) } \\ & = \mathbf { R } \mathbf { R } ^ { H } \mathbf { 1 } _ { N \times 1 } = \mathbf { 1 } _ { N \times 1 } \end{align*}


\begin{align*} \alpha = \omega \kappa \theta \kappa ^ { - 1 } \omega ^ { - 1 } , \end{align*}


\begin{align*} v = 0 , \nu ' ( h ) ( \bar u ) + \nu ( v ) = 0 , A ' ( h ) = 0 \mbox { o n } \hat \Sigma , \end{align*}


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


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


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I I _ 1 \leq C \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ { T } ( 1 + \Vert X _ n ( s ) \Vert _ H ^ { \lambda } ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s = 0 . \end{align*}


\begin{align*} \operatorname { I } _ { y } ( a , b ) = \frac { 1 } { \operatorname { B } ( a , b ) } \int ^ { y } _ { 0 } \omega ^ { a - 1 } ( 1 - \omega ) ^ { b - 1 } \mathrm { d } \omega , \end{align*}


\begin{align*} c _ { \N } : = \mathop { \inf } \limits _ { ( u , v ) \in \N } I ( u , v ) . \end{align*}


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


\begin{align*} U = ( - 1 ) ^ { p ( \eta , \ell _ { m } ) } ( - 1 ) ^ { | \pi ( \eta \cup \ell _ m ) | } \prod _ { r \notin [ m + 1 , M ] } U ^ { \eta \cup j } _ r . \end{align*}


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


\begin{align*} h ( \nu | \Phi ) : = \limsup _ { n \to \infty } \frac { 1 } { \abs { \Lambda _ n } } h _ { \Lambda _ n } ( \nu | \mu ) . \end{align*}


\begin{align*} \tilde { V } ( x ) = \frac { 1 } { T } \int _ 0 ^ T \mathcal { P } _ t V ( x ) d t \end{align*}


\begin{align*} \gamma ( x ) * ( \mu \circ \nu ) = \big ( ( \nu \cdot \gamma ) ( x ) * \mu \big ) \circ \big ( \gamma ( x ) * \nu \big ) \end{align*}


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


\begin{align*} f ( z ) e ^ { p z } = \frac { 1 } { 2 \pi i } \int _ { \partial \Delta } \frac { f ( \zeta ) e ^ { p \zeta } } { \zeta - z } d \zeta , z \in \Delta , \end{align*}


\begin{align*} \phi _ { z } ( { \bf { v } } , { \bf { u } } ) \triangleq \phi _ { z } ( d ) = \ss _ { z } - 1 0 \ , \alpha _ { z } \log _ { 1 0 } \left ( d \right ) , \end{align*}


\begin{align*} \mathcal { R } ( e _ 1 ) = e _ 1 + \frac { a _ { 4 } } { 2 } e _ 4 , \ \mathcal { R } ( e _ 2 ) = \lambda _ 1 e _ 3 , \ \mathcal { R } ( e _ 3 ) = \mathcal { R } ( e _ 4 ) = 0 , \end{align*}


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


\begin{align*} \log ( h _ { t } + h ) = \log \sigma _ { n - 1 } + \chi ( x , t ) , \end{align*}


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


\begin{align*} \begin{cases} x _ { 2 n + 1 } & = ( 1 - \alpha _ n ) T x _ { 2 n } + \alpha _ n u , \\ x _ { 2 n + 2 } & = ( 1 - \beta _ n ) U x _ { 2 n + 1 } + \beta _ n x _ { 2 n + 1 } . \end{cases} \end{align*}


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


\begin{align*} < n \times u , v > _ { \partial \Omega } = \int _ { \partial \Omega } u \times v \cdot n \dd H = \int _ { \partial \Omega } \alpha \wedge \beta = \int _ { \partial \Omega } d ( \phi \wedge \beta ) = 0 . \end{align*}


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


\begin{align*} \det \rho ( I ) = | \Delta _ F | ^ { \frac { 1 } { 2 } } N _ F ( I ) . \end{align*}


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


\begin{align*} _ { \rm m i n } \left ( a _ j ^ { \left ( o \right ) } \right ) = 2 a _ j ^ { \left ( o \right ) } \left ( a _ j ^ { \left ( o \right ) } \sum _ { i = 1 } ^ { M } \lvert \phi ^ { \left ( i , j \right ) } \rvert ^ 2 - \Re \left \{ \phi ^ { \left ( j , j \right ) } \right \} \right . \left . + \sum _ { q = 1 } ^ { M - 1 } \sum _ { r = q + 1 } ^ { M } f _ { q , r } ^ { \left ( j \right ) } \right ) . \end{align*}


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


\begin{align*} \psi _ * \mu \big ( B ( \psi ( y ) , r ) \cap \psi ( Y ) \big ) = \mu ( \psi ^ { - 1 } \big ( B ( \psi ( y ) , r ) \cap \psi ( Y ) \big ) ) = \mu ( \pi \big ( B ( \psi ( y ) , r ) \cap \psi ( Y ) \big ) ) , \end{align*}


\begin{align*} \partial _ t \Phi _ { + } ^ { 2 } + u \cdot \nabla \Phi _ { + } ^ { 2 } = \mu \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi _ { + } ^ { 2 } - 2 \mu | \nabla \Phi _ { + } | ^ { 2 } \textrm { o n } \ L ^ { 1 } _ { \textrm { l o c } } ( \mathbb { R } ^ { 3 } ) . \end{align*}


\begin{align*} M ^ d \times ( - 1 , 0 ) \ni ( x , z ) \mapsto \mathcal { S } ( x , z ) = ( x , \varrho ( x , z ) ) \in \Omega , \end{align*}


\begin{align*} u _ \lambda ( t , x ) & = \lambda u ( \lambda ^ 2 t , \lambda x ) \\ b _ \lambda ( t , x ) & = \lambda b ( \lambda ^ 2 t , \lambda x ) \\ \pi _ \lambda ( t , x ) & = \lambda ^ 2 \pi ( \lambda ^ 2 t , \lambda x ) , \end{align*}


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


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


\begin{align*} \nabla _ T D _ i = [ D _ i , C _ T ] = 0 \quad \forall T \in \Gamma \forall i . \end{align*}


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


\begin{align*} T _ { \mu } f ( x ) = \sum _ { n = 0 } ^ { \infty } \mu ( 2 n + d ) P _ { n } f ( x ) , \end{align*}


\begin{align*} d ( v _ 1 \Rightarrow w _ 2 v _ 2 ) = d ( v _ 1 \Rightarrow v _ 1 ' ) + d ( v _ 1 ' \Rightarrow w _ 2 v _ 2 ' ) + d ( w _ 2 v _ 2 ' \Rightarrow w _ 2 v _ 2 ) . \end{align*}


\begin{align*} \pi \circ \phi = \mbox { i d } _ Y , \end{align*}


\begin{align*} d ( e ^ { ( \alpha - 1 ) W ( t - s ) } ) = ( \alpha - 1 ) e ^ { ( \alpha - 1 ) W ( t - s ) } d W ( t ) + ( \alpha - 1 ) ^ 2 \mu e ^ { ( \alpha - 1 ) W ( t - s ) } d t . \end{align*}


\begin{align*} W _ r ( x ) = \frac 1 2 \hat W _ r ( 0 ) + \sum _ { k = 1 } ^ \infty \hat W _ r ( k ) \cos ( 2 \pi k x ) , \end{align*}


\begin{align*} F ( y ) ( t ) = U ( t , 0 ) x - \lambda i \int _ 0 ^ t U ( t , s ) ( e ^ { ( \alpha - 1 ) W ( s ) } g ( y ( s ) ) ) d s , \ t \in [ 0 , T ] . \end{align*}


\begin{align*} \frac { \partial ^ 2 v } { \partial n _ e ^ 2 } = n _ e \cdot ( \triangle v ) n _ e \end{align*}


\begin{align*} \{ d _ F ( \alpha , \beta ) = \max _ { a \in F } \| \alpha ( a ) - \beta ( a ) \| \mid F \subseteq A \} . \end{align*}


\begin{align*} \sum _ i \sigma _ { k } ^ { i i } \kappa _ i ^ 2 = \sigma _ 1 \sigma _ k - ( k + 1 ) \sigma _ { k + 1 } . \end{align*}


\begin{align*} T ( \lambda u ) = u . \end{align*}


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


\begin{align*} \sqrt { a \cdot b } = \overset { ` ` f ( a v e r a g e ) " \swarrow } { \exp ( \frac { A + B } { 2 } ) } \overset { ( \ , ) } \le \overset { ` ` a v e r a g e ( f ) " \swarrow } { \frac { \exp A + \exp B } { 2 } } = \frac { a + b } { 2 } \ , . \end{align*}


\begin{align*} x = x _ L \cdot \prescript L { } { } x { } ^ R \cdot x _ R & \ell ( x ) = \ell ( x _ L ) + \ell \left ( \prescript L { } { } x { } ^ R \right ) + \ell ( x _ R ) , \\ \prescript { - L } { } { } x { } ^ { - R } = x _ L ' \cdot x \cdot x _ R ' & \ell \left ( \prescript { - L } { } { } x { } ^ { - R } \right ) = \ell ( x _ L ' ) + \ell \left ( x \right ) + \ell ( x _ R ' ) . \end{align*}


\begin{align*} \mathbf { a } ( \theta _ { 0 } ) = [ e ^ { j 2 \pi d _ { 1 } { s i n } \theta _ 0 / \lambda } , e ^ { j 2 \pi d _ { 2 } { s i n } \theta _ 0 / \lambda } , \cdots , e ^ { j 2 \pi d _ { N } { s i n } \theta _ 0 / \lambda } ] ^ { T } , \end{align*}


\begin{align*} a ( \rho ^ \varepsilon ) ( 0 , z _ l ) = 1 \phantom { x x x } \phantom { x x x } a ( \rho ^ \varepsilon ) ( 0 , z _ r ) = 1 \end{align*}


\begin{align*} \L ^ n _ { \overline { A } } f \to \int f \ d \mu \ , \ \ \ \L ^ n _ { \overline { A } ^ * } f \to \int f \circ \theta ^ { - 1 } \ d \mu ^ * = \int f \ d \left ( \theta ^ { - 1 } _ * \mu ^ * \right ) \ , \end{align*}


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


\begin{align*} f ( \theta ) & = S ^ { - 1 } ( \theta ) = \mathbf { a } ^ H ( \theta ) \mathbf { U } _ V \mathbf { U } ^ H _ V \mathbf { a } ( \theta ) \\ & = \mathbf { a } ^ T ( \frac { 1 } { z } ) \mathbf { U } _ V \mathbf { U } ^ H _ V \mathbf { a } ( z ) \triangleq f ( z ) \end{align*}


\begin{align*} \left | \big ( A t ^ { - 2 \ell - 4 C } \cup A t ^ { 2 \ell + 4 C } \big ) \cap B _ S ( n + D ' ) \right | \ge \frac { 1 } { \ell + 1 } \binom { \ell + 1 } { 2 } \ , | R _ { q , \ell } ^ \flat ( n ) | > \frac { q ( n ) } { 2 } \ , | R _ { q , \ell } ^ \flat ( n ) | . \end{align*}


\begin{align*} \lim \limits _ { q \to \infty } { _ { 3 } } F _ { 2 } \left ( \begin{array} { c c c } \chi _ 4 , & \chi _ 4 , & \chi _ 4 ^ 3 \\ & \varepsilon , & \varepsilon \end{array} | 1 \right ) = 0 . \end{align*}


\begin{align*} \mathcal { H } _ m ( s I + A ) = \sum \limits _ { 1 \leq i _ 1 < \dots < i _ m \leq n } ( s + \lambda _ { i _ 1 } ) \dots ( s + \lambda _ { i _ m } ) = \sum \limits _ { p = 0 } ^ m \binom { n - p } { m - p } \mathcal { H } _ p ( A ) s ^ { m - p } . \end{align*}


\begin{align*} \lambda \Lambda _ 1 L _ { g _ 1 } ( z ) = - \sum _ { n = 1 } ^ \infty n ^ { - 1 } ( 1 - e ^ { \lambda \Lambda _ 1 } ) ^ n L ^ n _ { f _ 1 } ( z ) \end{align*}


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


\begin{align*} A _ 1 : = \left ( \begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 2 \\ 0 & 1 & - 1 \end{matrix} \right ) , \end{align*}


\begin{align*} \{ x \in E : i ( x ) < d ( x ) \} = \bigcup _ { 1 \le k \le n } \mathfrak { J } _ k \end{align*}


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


\begin{align*} { \cal V } ( r ) = - \frac { Q } { r } \sqrt { 1 + \lambda r ^ 2 } , Q > 0 . \end{align*}


\begin{align*} \Phi _ p ( x ) = \begin{cases} \phi _ p ( x ) , & x \geq 1 , \\ 2 \phi _ p ( 1 ) - \phi _ p ( 2 - x ) , & 0 \leq x \leq 1 . \end{cases} \end{align*}


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


\begin{align*} \begin{aligned} E _ A & = R _ { X \backslash A } , \\ R _ B & = E _ { X \backslash B } . \end{aligned} \end{align*}


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


\begin{align*} \Delta { { \mathbf { x } } ^ { T } } \Delta \mathbf { s } = \left ( 1 - \nu \right ) \left ( \Delta { { \mathbf { y } } ^ { T } } \mathbf { A x } - \Delta { { \mathbf { x } } ^ { T } } \left ( { { \mathbf { A } } ^ { T } } \mathbf { y } + \mathbf { C } \mathbf { x } + \mathbf { s } \right ) \right ) . \end{align*}


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


\begin{align*} j ^ 1 _ q ( F _ \varphi \circ f _ \varphi { } ^ { - 1 } ) & = ( q , F _ \varphi ( o ) , D F _ \varphi ( o ) ( D f _ \varphi ( o ) ) ^ { - 1 } ) \\ * & = ( q , F _ \varphi ( o ) , D F _ \varphi ( o ) ^ i { } _ { j \alpha } ( ( F _ \varphi ( o ) ) ^ { - 1 } ) ^ \alpha { } _ k ) \end{align*}


\begin{align*} & g : [ 0 , \infty ] \rightarrow \mathbb { R } , \\ & F ( z ) = g ( | z | ) \ , \ , \ , \ , z \in \mathbb { C } ^ n , \\ & f ( z ) = \frac { g ' ( | z | ) z } { | z | } . \end{align*}


\begin{align*} D ( \Omega ) : = \Big \{ f \in C ( \Omega ) : \sum _ { x \in \Z ^ d } \delta _ x ( f ) < \infty \Big \} . \end{align*}


\begin{align*} ( \widetilde { B } _ { \infty } ) ^ { \Gamma _ n } = \widetilde { B } _ n \mbox { a n d } ( \widetilde { B } _ { \infty } ) ^ { G ' } = B _ { G ' , \infty } . \end{align*}


\begin{align*} c : = ( \alpha + \beta , - \chi _ K ( \alpha ) - \chi _ K ( \beta ) ) \in R . \end{align*}


\begin{align*} A d ( P ) = P \times _ { A d } \mathcal { G } \ , , \end{align*}


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


\begin{align*} k = \dim ( \Delta ) = \dim ( \pi ( \Delta ) ) + \dim ( \ker ( \pi | _ \Delta ) ) = m + \dim ( \ker ( \pi | _ \Delta ) ) . \end{align*}


\begin{align*} \int _ \sigma ^ r d t \int _ { \varrho \leq t } ( \Delta \varrho ) ^ { n - k } \wedge \Delta u _ 1 \wedge \dots \wedge \Delta u _ k = - \int _ { \varrho \leq r } u _ k ( \Delta \varrho ) ^ { n - k + 1 } \wedge \Delta u _ 1 \wedge \dots \wedge \Delta u _ { k - 1 } . \end{align*}


\begin{align*} \begin{aligned} | h ( x ( t ) + \delta u ( t ) ) - h ( z ( t ) + \delta u ( t ) ) | \le & \ , L _ r | x ( t ) - z ( t ) | \\ | h ( z ( t ) + \delta u ( t ) ) - h ( x _ a ( t ) + \delta u ( t ) ) | \le & \ , \gamma L _ r | \epsilon ( x _ a ( t ) , t ) | . \end{aligned} \end{align*}


\begin{align*} \eta _ \sigma ( x ) = \prescript { \sigma ^ { - 1 } } { } ( v ) ^ { - 1 } w v \in W . \end{align*}


\begin{align*} \begin{aligned} & M _ { \delta , + , 1 } ( x ; z ) = e _ 1 + \mathcal { P } ^ + \left ( r _ { \delta , - , 2 } e ^ { 2 i z x } M _ { \delta , - , 2 } ( x ; \cdot ) \right ) , \\ & M _ { \delta , - , 2 } ( x ; z ) = e _ 2 + \mathcal { P } ^ { - } \left ( \bar { r } _ { \delta , - , 1 } e ^ { - 2 i z x } M _ { \delta , + , 1 } ( x ; \cdot ) \right ) . \end{aligned} \end{align*}


\begin{align*} \bigcup \limits _ { \mu } S u p p ( s _ \mu ) = \bigcup \limits _ { i = 0 } ^ l S u p p ( s _ { \lambda ^ i } ) . \end{align*}


\begin{align*} h ( \nu | \mu ) : = \sum _ { x \in E } \nu ( x ) \log \left ( \frac { \nu ( x ) } { \mu ( x ) } \right ) , \end{align*}


\begin{align*} \delta ^ \nabla \big ( \exp ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) R ^ \nabla \big ) = 0 \ , . \end{align*}


\begin{align*} J ( x , \delta ) = \int _ 1 ^ { x } | \theta ( y + \delta y ) - \theta ( y ) - \delta y | ^ 2 d y \end{align*}


\begin{align*} \hat \Phi ^ \dagger _ a ( 0 , 0 ) & = \langle z _ 1 ^ * , \Phi _ v ( 0 , p _ 0 ) u _ \ell \rangle = 0 . \end{align*}


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


\begin{align*} \ell ^ { \prime } \left ( r \right ) = \frac { c } { 3 } \frac { 1 - \cos r } { \left ( r - \sin r \right ) ^ { 2 / 3 } } \end{align*}


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


\begin{align*} B _ { x , \eta x } ( \xi ) = 0 . \end{align*}


\begin{align*} \widetilde G _ n ( x , y ) & = \left ( \frac { 1 } { 2 \pi } \right ) ^ { d / 2 } \int _ { \mathbb { R } ^ d } \mbox { d e t } \left ( e ^ { - i \theta _ i ( y _ { j } - x _ { i } ) } \right ) _ { i , j = 1 } ^ d \prod _ { i = 1 } ^ d ( \varphi ( \theta _ i ) ) ^ n d \theta _ i . \end{align*}


\begin{align*} \begin{aligned} \Bigg | & \left . \dfrac { \partial \epsilon _ 2 ( \eta , \tau ) } { \partial \eta } \right | _ { \eta = \eta _ a ( \tau ) } \dot { \eta } _ a ( \tau ) \Bigg | \le \\ & \gamma \Bigg | \left . \dfrac { \partial \epsilon _ 2 ( \eta , \tau ) } { \partial x _ a } \right | _ { \eta = \eta _ a ( \tau ) } \Bigg | \dfrac { | b _ { 1 , \delta } ( x _ a ) | } { 2 } \le \gamma 2 L _ r ^ 2 \delta ^ 2 . \end{aligned} \end{align*}


\begin{align*} \mu ( B _ p ( \bar x , r ) ) \leq C r ^ { N + 2 s - 1 } \quad \mbox { f o r a l l $ \bar x \in \R ^ { N + 1 } , \ , r > 0 $ . } \end{align*}


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


\begin{align*} k _ 4 ( P ^ \ast ( q ) ) & = \frac { q } { 4 } \times 0 \\ & = \frac { q } { 4 } \times k _ 3 ( \langle H \rangle ) . \end{align*}


\begin{align*} f ( s \xi , s \eta ) - f ( \xi , \eta ) = \hat \beta ( s , \xi ) + \hat \beta ( s , \eta ) . \end{align*}


\begin{align*} Y ^ { t , \Upsilon ; v } ( s ) = \Phi ( X _ T ^ { t , \Upsilon ; v } ) + \int _ { s } ^ { T } g ( r , X _ r ^ { t , \Upsilon ; v } , Y ^ { t , \Upsilon ; v } ( r ) , Z ^ { t , \Upsilon ; v } ( r ) , v ( r ) ) d r - \int _ { s } ^ { T } Z ^ { t , \Upsilon ; v } ( r ) d W ( r ) . \end{align*}


\begin{align*} \begin{cases} | b _ 1 | \leq C \max ( e ^ { \delta _ n q _ n } , | \ell | , 1 ) e ^ { ( \tilde { L } - \beta _ n + 4 \varepsilon ) q _ n } \\ | b _ 3 | \leq \max ( e ^ { \delta _ n q _ n } , | \ell | , 1 ) c _ { n , \ell + 1 } e ^ { ( \tilde { L } - \beta _ n + 4 \varepsilon ) q _ n } \\ | b _ 4 | \leq c _ { n , \ell + 1 } e ^ { ( \tilde { L } + \varepsilon ) q _ n } \leq C \max ( e ^ { \delta _ n q _ n } , | \ell | , 1 ) e ^ { ( \tilde { L } - \beta _ n + \varepsilon ) q _ n } \end{cases} \end{align*}


\begin{align*} \varphi ^ { V , U } \overset { \mathrm { l a w } } { = } \mathbb { E } [ h ^ { V } | \sigma ( h _ { z } ^ { V } : z \in V \backslash U ) ] . \end{align*}


\begin{align*} \begin{pmatrix} v _ 1 \\ v _ 2 \end{pmatrix} = - \frac { 1 } { k ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 } \begin{pmatrix} x _ 1 x _ 2 & - k ^ 2 - x _ 1 ^ 2 \\ k ^ 2 + x _ 2 ^ 2 & - x _ 1 x _ 2 \end{pmatrix} \begin{pmatrix} \partial _ { x _ 1 } \varphi \\ \partial _ { x _ 2 } \varphi \end{pmatrix} . \end{align*}


\begin{align*} J ( s ) = \lim _ { x \to s ^ { + } } u ( x ) - \lim _ { x \to s ^ { - } } u ( x ) \forall s \in S _ { u } . \end{align*}


\begin{align*} \begin{aligned} \mathcal { U } ^ * ( K , \Omega ) = \left \{ u \in Q S H _ m ( \Omega ) \cap C ( \Omega ) , u \vert _ { K } \leq - 1 , \varliminf _ { q \to \partial \Omega } u ( q ) \geq 0 \right \} . \end{aligned} \end{align*}


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


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


\begin{align*} \log \binom { C n } { 2 L } & < C n \log C n - \epsilon n \log \epsilon n - ( C - \epsilon ) n \log ( C - \epsilon ) n + O ( \log n ) \\ & = ( C \log C - \epsilon \log \epsilon - ( C - \epsilon ) \log ( C - \epsilon ) ) n + O ( \log n ) . \end{align*}


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


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


\begin{align*} P _ { n } \left ( I + \mathbf { H } ( \varphi ) \right ) P _ { n } \cdot P _ { n } A ^ { - 1 } P _ { n } & = P _ { n } \left ( I + \mathbf { H } ( \varphi ) \right ) A ^ { - 1 } P _ { n } - P _ { n } \left ( I + \mathbf { H } ( \varphi ) \right ) Q _ { n } A ^ { - 1 } P _ { n } \\ & = P _ { n } - P _ { n } \mathbf { H } ( \varphi ) Q _ { n } A ^ { - 1 } P _ { n } . \end{align*}


\begin{align*} w = ( \Psi _ 1 ^ { - 1 } \circ \Psi _ 2 ^ { - 1 } ) ( \hat z ) \end{align*}


\begin{align*} \widetilde { E } ( x , t ) = \log \lambda _ { m a x } ( \{ w ^ { i j } \} ) - d \log h + \frac { l } { 2 } \rho ^ { 2 } , \end{align*}


\begin{align*} h ( \delta ) = C _ 8 \min \{ 1 , \delta ^ { 3 } \} C _ 8 \stackrel { \rm d e f } { = } \frac { \eta _ 1 \eta _ 2 \kappa _ { \rm f o d } } { 2 } \min \left \{ \frac { \eta _ 2 } { \kappa _ { \rm b h m } } , 1 \right \} . \end{align*}


\begin{align*} \psi _ * \mu \big ( B ( \psi ( y ) , r ) \cap \psi ( Y ) \big ) = \mu ( \psi ^ { - 1 } \big ( B ( \psi ( y ) , r ) \cap \psi ( Y ) \big ) ) = \mu ( \pi \big ( B ( \psi ( y ) , r ) \cap \psi ( Y ) \big ) ) , \end{align*}


\begin{align*} L _ { o d d } : = A + H + \sum _ { k = 1 } ^ { \frac { n - 3 } { 2 } } B _ k . \end{align*}


\begin{align*} g ( u \times v , w ) = \mathrm { v o l } _ g ( u , v , w ) , \end{align*}


\begin{align*} \begin{aligned} \mathbf { V } _ i & \triangleq \sum _ { k \in \mathcal { K } _ i } | \tau _ { k } | ^ 2 \sum _ { p \in \mathcal { K } } \mathbf { v } _ { k , p } \mathbf { v } _ { k , p } ^ H , \\ \tilde { \mathbf { v } } _ i & \triangleq \sum _ { k \in \mathcal { K } _ i } \sqrt { 1 + \iota _ { k } } \mathbf { v } _ { k , k } \tau _ { k } , \forall i \in \{ \mathrm { t , r } \} . \end{aligned} \end{align*}


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


\begin{align*} | | A _ q ^ z | | ^ 2 _ { L ^ 2 ( \mathbb R ) } = E _ q \left ( | z | ^ 2 \right ) . \end{align*}


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


\begin{align*} \mathcal { H } _ m ( u ) : = \mathcal { H } _ m \left ( \frac { \partial ^ 2 u } { \partial \overline { q _ l } \partial { q _ k } } \right ) . \end{align*}


\begin{align*} ( 2 \pi ) ^ { - n } | \lambda | ^ n W _ \lambda ( g \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ) = W _ \lambda ( g ) P _ k ( \lambda ) . \ , \ , \end{align*}


\begin{align*} \mathbf { \Phi } _ { g } ^ \star = \arg \min _ { \mathbf { \Phi } _ { g } \in \mathcal { M } _ g } \tilde { f } _ g ( \mathbf { \Phi } _ { g } ) , \forall g \in \mathcal { G } . \end{align*}


\begin{align*} G _ { \iota \iota } = ( ( \nabla ^ { 2 } G \cdot e _ { \iota } ) \cdot e _ { \iota } ) w ^ { 2 } _ { \iota \iota } - ( \nabla G \cdot x ) w _ { \iota \iota } + ( \nabla G \cdot e _ { i } ) w _ { i \iota \iota } . \end{align*}


\begin{gather*} F _ { \textrm { B } i } ( x ) = \bar { F } _ i ( x ) \tilde { H } ( x ) + F _ { \textrm { B R } i } ( x ) , \deg ( F _ { \textrm { B R } i } ) < d , i = 1 , \dots , n , \\ F _ { \textrm { S } i } ( x ) = \bar { F } _ i ( x ) \tilde { H } ( x ) + F _ { \textrm { S R } i } ( x ) , \deg ( F _ { \textrm { S R } i } ) < d , i = 1 , \dots , n . \end{gather*}


\begin{align*} \begin{array} { l } \circlearrowleft _ { x , y , z } \varepsilon ( z , x ) \Big ( \varepsilon ( h , x + y ) [ [ \alpha ( x ) , \alpha ( y ) ] , \alpha ( h \cdot z ) ] \Big ) \\ + \circlearrowleft _ { x , y , z } \varepsilon ( z , x ) \Big ( 2 [ \alpha ( h ) \cdot [ x , y ] , \alpha ^ 2 ( x ) ] \Big ) = 0 . \end{array} \end{align*}


\begin{align*} \eta _ { N } ^ { D } : = \frac { 1 } { K _ { N } } \sum _ { x \in D _ { N } } \delta _ { x / N } \otimes \delta _ { h _ { x } ^ { D _ { N } } - a _ { N } } , \end{align*}


\begin{align*} \begin{aligned} \omega ( x , y , z ) & = - 2 x \ d H ^ * \wedge d Y ^ * + 2 y \ d X ^ * \wedge d Y ^ * - 2 z \ d X ^ * \wedge d H ^ * \\ & = - 2 ( x \ d H ^ * \wedge d Y ^ * - y \ d X ^ * \wedge d Y ^ * + z \ d X ^ * \wedge d H ^ * ) . \end{aligned} \end{align*}


\begin{align*} d = k ^ 2 + 2 n k = 4 m k + 2 k - k ^ 2 \equiv 2 k - k ^ 2 \mod 4 k \end{align*}


\begin{align*} I _ { n , l } = \int _ 0 ^ \pi d \theta \sin \theta \cos ( n \theta ) P _ l ( \cos \theta ) = \end{align*}


\begin{align*} \log \det u _ { , i j } = - v _ p x ^ p + u _ { , q } \xi ^ q + c . \end{align*}


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


\begin{gather*} P ( c ) = \inf _ { Q \in \Gamma } Q ( c ) ; \end{gather*}


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


\begin{align*} \mathcal { L } ( u ^ h ( t , x ) ) = \mathcal { L } \left ( \sum _ i w _ i ( t ) \sigma _ i ( x ) \right ) = \sum _ i w _ i ( t ) \mathcal { L } ( \sigma _ i ( x ) ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 3 } } | \tilde { b } | ^ { 2 } \dd x & = \left ( \frac { W } { 2 } \right ) ^ { - 2 } \int _ { \mathbb { R } ^ { 3 } } | b | ^ { 2 } \dd x , \\ \int _ { \mathbb { R } ^ { 3 } } \left ( \tilde { \phi } - r ^ { 2 } - \tilde { \gamma } \right ) _ + \frac { \tilde { G } } { r ^ { 2 } } \dd x & = \left ( \frac { W } { 2 } \right ) ^ { - 2 } \int _ { \mathbb { R } ^ { 3 } } \left ( \phi - \frac { W } { 2 } r ^ { 2 } - \gamma \right ) _ + \frac { G } { r ^ { 2 } } \dd x . \end{align*}


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


\begin{align*} q _ { t , \gamma } ( x , y ) = q _ t ( x , y ) ( 1 - \varphi ( \frac { \sqrt { t } } { \gamma } ) ) , \end{align*}


\begin{align*} & s _ { B S } ( d ( \lambda ) ) = \star = d ^ 0 ( s _ { \Lambda } ) \\ & r _ { B S } ( d ( \lambda ) ) = \star = d ^ 0 ( r _ { \Lambda } ) \end{align*}


\begin{align*} \left \{ \begin{array} { c l } \partial _ t v - D ^ { \alpha } _ x \partial _ { x } v - \partial _ { x } ^ { - 1 } \partial _ y ^ 2 v & = \partial _ x ( v ( u _ 1 + u _ 2 ) ) / 2 , ( t , x , y ) \in \R \times \R \times \R , \\ v ( 0 ) & = \phi _ 1 - \phi _ 2 = : \phi . \end{array} \right . \end{align*}


\begin{align*} F ( \lambda ) x _ j + \frac { 1 } { 1 ! } F ^ { ( 1 ) } ( \lambda ) x _ { j - 1 } + \ldots + \frac { 1 } { j ! } F ^ { ( j ) } ( \lambda ) x _ 0 = 0 , j = 0 , 1 , \ldots , k , \end{align*}


\begin{align*} \left ( \left ( P _ { n } + P _ { n } \mathbf { H } ( \varphi ) \right ) P _ { n } y \right ) \left ( \rho \right ) = - P _ { n } \mathbf { H } ( \varphi ) \left ( 1 \right ) \left ( \rho \right ) . \end{align*}


\begin{align*} D & : = P + Q \intertext { a n d } \overline { D } & : = \overline { P } + \overline { Q } \end{align*}


\begin{align*} g \big ( u _ 0 ( z _ l ) \big ) = g \big ( u _ 0 ( z _ r ) \big ) + K . \end{align*}


\begin{align*} x ( b ) = & ( \alpha , \Phi ^ + ( - w ^ { - 1 } \alpha ) - \langle \mu , w ^ { - 1 } \alpha \rangle ) \\ = & ( \alpha , \Phi ^ + ( - \alpha ) + \ell ( x , - w ^ { - 1 } \alpha ) \rangle ) = ( \alpha , \Phi ^ + ( - \alpha ) ) = a . \end{align*}


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


\begin{align*} | W _ i \cap C ^ { ( i ) } | = \prod _ { l = 1 } ^ { i } \frac { \beta _ { j + l - 1 } } { c _ { l } } . \end{align*}


\begin{align*} x _ { 2 i , j } = \tfrac { 1 } { 2 } S - x _ { 2 i , n + 1 - j } . \end{align*}


\begin{align*} c _ 1 ( q ) = \begin{cases} c _ { 1 , 2 } ( q ) , & 0 < q \leq q _ 1 ^ * , \\ c _ { 1 , \infty } ( q ) , & q _ 1 ^ * \leq q < 2 , \end{cases} \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } | \nabla v _ { l } | ^ { 2 } + \int _ { \partial \Omega } | v _ { l } | ^ { 2 } + \int _ { \Omega } u _ { l } v _ { l } ^ { 2 } & = \int _ { \Omega } f v _ { l } + \int _ { \partial \Omega } \eta v _ { l } \\ & \le C ( \| f \| _ { L ^ { p } ( \Omega ) } \| v _ { l } \| _ { H ^ { 1 } ( \Omega ) } ) + \int _ { \partial \Omega } \eta v _ { l } . \end{aligned} \end{align*}


\begin{align*} r ( k ) : = \frac { b ( k ) } { a ( k ) } , k \in \mathbb { R } \cup i \mathbb { R } , \end{align*}


\begin{align*} \varphi _ a ( z ) = \dfrac { a - P _ a ( z ) - s _ a Q _ a ( z ) } { 1 - \langle z , a \rangle } , z \in \mathbb { B } _ n , \end{align*}


\begin{align*} c ^ { k , j } _ { \ell , p , q } = 0 \ \textrm { i f } j > 2 k \textrm { o r a n y o n e e n t r y i n t h e i n d e x } ( j , \ell , p , q ) \textrm { i s } n e g a t i v e . \end{align*}


\begin{align*} \{ f , g \} = P ( d f \wedge d g ) , \end{align*}


\begin{align*} R ( J X , J Y ) Z = R ( X , Y ) Z \textrm { a n d } R ( X , Y ) J Z = J ( R ( X , Y ) Z ) , \end{align*}


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


\begin{align*} \dot { { x } } _ a & = - \gamma \int _ { 0 } ^ { 1 } \left ( y _ \delta ( { x _ a } , \tau ) - \bar { y } _ a \right ) \ , u ( \tau ) \ , d \tau \\ \dot { \bar { y } } _ a & = - \gamma \ , \bar { y } _ a + \gamma \int _ { 0 } ^ { 1 } y _ \delta ( { x } _ a , \tau ) \ , d \tau . \end{align*}


\begin{align*} \frac { d ^ 2 } { d t ^ 2 } X ( \gamma ( t ) ) = B ( t ) X ( \gamma ( t ) ) + D ( t ) \end{align*}


\begin{align*} \lim _ { A \rightarrow \infty } \int _ { | x | > A } | \mathcal { T } ^ * _ { b , \gamma } f ( x ) | ^ p w ( x ) d x = 0 , \end{align*}


\begin{align*} W ^ { s } ( ( x , t ) , T _ { \vec { v } } ) = W ^ s ( x , T _ { \vec { v } } ) \times \{ t \} . \end{align*}


\begin{align*} \hat { \mu } ( \xi ) = \hat { \mu } _ { \rho , \{ \mathcal { D } _ { n } \} } ( \xi ) = \prod _ { n = 1 } ^ { \infty } M _ { \mathcal { D } _ { n } } ( \rho ^ { n } \xi ) , \xi \in \mathbb { R } , \end{align*}


\begin{align*} \limsup _ { t \nearrow T _ { \max } } \| u _ i ( t ) \| _ { L ^ \infty ( \Omega ) } < \infty ~ i = 1 , 2 , \cdots , m , \end{align*}


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


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


\begin{align*} - \Delta v - p w ^ { p - 1 } _ + v = 0 , \ \ v \in L ^ { \infty } ( \mathbb { R } ^ 2 ) . \end{align*}


\begin{align*} \frac { e ^ { p ( \zeta - z ) } } { \zeta - z } = - \int _ { p + i e ^ { - i \alpha } [ 0 , + \infty ) } e ^ { \omega ( \zeta - z ) } d \omega , \zeta \in e ^ { i \alpha } [ 0 , + \infty ) , \ ; z \in \Delta . \end{align*}


\begin{align*} G ( x ) = \begin{cases} \frac { 1 } { 2 } \left \{ 1 - F _ \Gamma \left [ \left ( - \frac { x - \mu } { \sigma } \right ) ^ s \right ] \right \} , & x \leq \mu , \\ \frac { 1 } { 2 } \left \{ 1 + F _ \Gamma \left [ \left ( \frac { x - \mu } { \sigma } \right ) ^ s \right ] \right \} , & x > \mu , \end{cases} \end{align*}


\begin{align*} \sum \limits _ { y \neq 0 , x } \chi _ 4 ( y ( x - y ) ) = \sum \limits _ { y \neq 0 , 1 } \chi _ 4 ( x y ) \chi _ 4 ( x - x y ) = \varphi ( x ) J ( \chi _ 4 , \chi _ 4 ) . \end{align*}


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


\begin{align*} \partial ^ \square c _ { | ( t , p ) } = \partial ^ \circ c _ { | ( t , p ) } - V ^ { } _ { \Gamma \ , | ( t , p ) } \cdot \nabla _ \Gamma c _ { | ( t , p ) } = \partial ^ \circ c _ { | ( t , p ) } = \partial _ t c _ { | ( t , p ) } \end{align*}


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


\begin{align*} \left | \det \begin{pmatrix} x _ j & x _ k \\ y _ j & y _ k \end{pmatrix} \right | \le | x _ j y _ k | + | x _ k y _ j | . \end{align*}


\begin{align*} \delta \rho = \varepsilon \kappa ( \rho ) \ , { y _ { N ( k - 1 ) + i } \ , x _ { N ( \ell - 1 ) + j } } + O ( \varepsilon ^ 2 ) . \end{align*}


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


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


\begin{align*} y \left ( \rho \right ) : = y \left ( \rho , x \right ) = \sum _ { n = 0 } ^ { \infty } \frac { a _ { n } ( x ) z ^ { n } \left ( \rho \right ) } { \frac { 1 } { 2 } - i \rho } \end{align*}


\begin{align*} \begin{aligned} & \omega _ T = \frac { d ( x + \varphi y ) } { x + \varphi y } \wedge \frac { d ( x + \varphi ' y ) } { x + \varphi ' y } = \frac { ( d x + \varphi d y ) \wedge ( d x + \varphi ' d y ) } { x ^ 2 + x y - y ^ 2 } \\ & = \frac { \varphi ' - \varphi } { N _ { E / F } ( x + \varphi y ) } d x \wedge d y = \frac { - \sqrt { 5 } } { N _ { E / F } ( x + \varphi y ) } d x \wedge d y . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \psi ( Y ^ { \epsilon } ( t ) ) & = \psi ( y ^ { \epsilon } _ { 0 } ) + \int _ 0 ^ t \mathcal { B } ^ { \epsilon } ( s ) \tilde { N } ^ { \epsilon ^ { - 2 } } ( d s ) \\ & \quad + \int _ 0 ^ t \langle \nabla \psi ( Y ^ { \epsilon } ( s - ) ) , h ^ { \epsilon } ( s ) \rangle d s + \int _ 0 ^ t \mathcal { A } ^ { \epsilon } ( s ) d s , \end{aligned} \end{align*}


\begin{align*} \psi ( n ) : = n ^ { - 1 } \phi ( n H ) , \forall n \in N \end{align*}


\begin{align*} \bigg ( \sum _ { j = 0 } ^ { k } \dbinom { k } { j } ( 1 - q ) ^ { k - j } q ^ j \dbinom { k } { j } \Big ( \frac { 1 } { 2 } \Big ) ^ k \bigg ) ^ { 1 / p _ k } \leq \Big ( \frac { 1 - q } { 2 } \Big ) ^ { k / p _ k } + \Big ( \frac { q } { 2 } \Big ) ^ { k / p _ k } \end{align*}


\begin{align*} T _ n ( x ) = \lim \limits _ { \gamma \rightarrow 0 } \frac { n + 2 \gamma } { 2 \gamma } C ^ \gamma _ n ( x ) \ , , P _ l ( x ) = C ^ { 1 / 2 } _ l ( x ) \ , , \end{align*}


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


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


\begin{align*} \eta ( z ) = q ^ \frac { 1 } { 2 4 } \prod _ { n = 1 } ^ \infty ( 1 - q ^ n ) , \end{align*}


\begin{align*} \hat { \theta } = \arcsin ( \frac { \phi \lambda } { 2 \pi d } ) , \end{align*}


\begin{align*} \tilde { F } _ k ( x , y ) = \begin{pmatrix} \max _ { i = 1 , \ldots , 2 ^ k } x _ i \\ \max _ { i = 1 , \ldots , 2 ^ k } y _ i \end{pmatrix} . \end{align*}


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


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I I _ 2 = 0 . \end{align*}


\begin{align*} a ( A ^ { - 1 } Z B ) = a ( Z ) \end{align*}


\begin{align*} \int _ { B _ \rho ( x _ 0 ) - B _ 1 ( x _ 0 ) } \big ( \exp ( \frac { | | R ^ \nabla | | ^ 2 } 2 ) - 1 \big ) \ , d v = o ( \rho ^ { 1 + \delta } ) \rho \rightarrow \infty \end{align*}


\begin{align*} & \Tilde { x } = \sum \limits _ { i = 1 } ^ p \alpha _ i u _ i + \lambda x _ v + ( 1 - \lambda ) x _ w \in \mathcal { S } ^ n \\ & \| \sum \limits _ { i = 1 } ^ p \alpha _ i u _ i + \lambda x _ v + ( 1 - \lambda ) x _ w \| ^ 2 = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 \end{align*}


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


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


\begin{align*} ( x \wedge y ) \cdot z = ( x \cdot y ) x \cdot z = ( x \cdot y ) \cdot ( x \cdot z ) , \end{align*}


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


\begin{align*} \Theta & = d \theta ^ 0 + \theta ^ 1 \wedge \theta ^ 0 , \\ * \Omega & = d \theta ^ 1 + \theta ^ 1 \wedge \theta ^ 1 , \end{align*}


\begin{align*} \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { y } ' | \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*} F ' ( x , y ) = L _ 1 \circ F \circ L _ 2 ( x , y ) . \end{align*}


\begin{align*} \alpha ( A ) & = \inf \{ t > 0 : A \mbox { i s i n c l u d e d i n s o m e b a l l o f r a d i u s $ t $ } \} , \\ \beta ( A ) & = \inf \{ t > 0 : A \mbox { h a s a d i a m e t e r l e s s t h a n $ t $ } \} , \end{align*}


\begin{align*} H _ { m i n } ( E ) = \min \limits _ { 0 \neq F \subset E } H _ r ( F ) . \end{align*}


\begin{align*} R _ c ^ { s y m } = \pi / 0 . 8 6 3 \approx 3 . 6 4 . \end{align*}


\begin{align*} \tilde z _ k = { \tt t z e t ( ( k - 1 ) n / 4 + 1 ) } , k = 1 , 2 , 3 , 4 , \end{align*}


\begin{align*} T = \bigcup _ { i = 1 } ^ I A ( p _ i , s _ i , 2 s _ i ) \end{align*}


\begin{align*} \mu _ \varphi ( \{ \chi \in \widehat { H } \ : : \ : \chi ^ g = \chi \} ) > 0 . \end{align*}


\begin{align*} \int _ { B _ { 1 / 2 } } r ^ { 2 - n } | \nabla u | ^ 2 \d x \leq C \int _ { B _ { 1 } } r ^ { 2 - n } u _ r ^ 2 \d x + C \varepsilon ^ 2 \int _ { B _ { 1 } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \varepsilon \leq \varepsilon _ 0 , \end{align*}


\begin{align*} t < t ' = F _ E ( t ' ) \leq F _ E ( t ) , \end{align*}


\begin{align*} T = \bigcup _ { s \in S } T ( s ) . \end{align*}


\begin{align*} \dot { W } ( x _ t ) = - \begin{bsmallmatrix} x ( t ) \\ x ( t - h ) \end{bsmallmatrix} ^ \top \begin{bsmallmatrix} \eta _ 0 \mathcal { H } ( A ) + \frac { 1 } { 2 } I _ m & \eta _ 0 A _ d \\ \ast & \frac { 1 } { 2 } I _ m \end{bsmallmatrix} \begin{bsmallmatrix} x ( t ) \\ x ( t - h ) \end{bsmallmatrix} , \end{align*}


\begin{align*} T ^ v ( u , J u , J u , u ) = 0 \textrm { f o r e v e r y } u \in V . \end{align*}


\begin{align*} & = \frac { 1 } { q ^ { m - 1 } - 1 } \left ( q ^ { m - 1 } \zeta _ \nu ( n - 1 ) \cdots \zeta _ \nu ( n - m + 1 ) - \zeta _ \nu ( n ) \cdots \zeta _ \nu ( n - m + 2 ) \right ) \\ & = \frac { 1 } { q ^ { m - 1 } - 1 } \zeta _ \nu ( n - 1 ) \cdots \zeta _ \nu ( n - m + 2 ) \left ( q ^ { m - 1 } \zeta _ \nu ( n - m + 1 ) - \zeta _ \nu ( n ) \right ) , \end{align*}


\begin{align*} h _ \alpha ( t ) : = \frac { t ^ { \alpha - 1 } } { \Gamma ( \alpha ) } , \ \alpha > 0 . \end{align*}


\begin{align*} \int _ { \Gamma } J _ 0 d \sigma _ y = \int _ { \Gamma ( t , x ) } d \sigma = \vert \Gamma ( t , x ) \vert \end{align*}


\begin{align*} d _ { C y } ( ( z , t , u ) , ( z ' , t ' , u ' ) ) = \lvert 2 \langle \psi ( z , t , u ) , \psi ( z ' , t ' , u ' ) \rangle \rvert ^ { 1 / 2 } . \end{align*}


\begin{align*} \mathcal { M } _ g = \{ \mathbf { \Phi } _ { g } \in \mathbb { C } ^ { 2 \bar { M } \times \bar { M } } : \mathbf { \Phi } _ { g } ^ H \mathbf { \Phi } _ { g } = \mathbf { I } _ { \bar { M } } \} , \forall g \in \mathcal { G } , \end{align*}


\begin{align*} \Sigma _ 3 & = \sum \limits _ { D \leq d < 2 D } \sum \limits _ { n \in \mathcal { N } ( d ) } \Bigg ( - \sum \limits _ { 1 \leq | m | \leq M } \frac { e \left ( m \left ( \frac { \sqrt { X } - n } { d } \right ) \right ) } { 2 \pi i m } + \mathcal { O } \left ( f _ { M } \left ( \frac { \sqrt { X } - n } { d } \right ) \right ) \Bigg ) \\ & = \Sigma _ 4 + \Sigma _ 5 \ , , \end{align*}


\begin{align*} A ^ * ( y ) : = A ^ * ( y | x ) = A ( \tau _ y ( x ) ) + W \circ \hat { \sigma } ^ { - 1 } ( y | x ) - W ( y | x ) \ , \end{align*}


\begin{align*} \phi _ 3 + c \phi _ 1 + \frac { \epsilon } { \kappa } \phi _ 4 + \frac { c } { \kappa } \phi _ 2 = c o n s t a n t : = k . \end{align*}


\begin{align*} P _ { e , w } ( q ) = \sum _ { 1 \leq i \leq k } q ^ { \frac { \ell ( w ) - \ell ( w _ i ) } { 2 } } . \end{align*}


\begin{align*} \begin{cases} t _ { i , e x } ( x _ 0 , t _ 0 ) = \max \left ( 0 , t _ 0 - \dfrac { x _ 0 - R } { \lambda _ i } \right ) \forall \ : i \in [ 1 , p ] , \\ t _ { i , e x } ( x _ 0 , t _ 0 ) = \max \left ( 0 , t _ 0 - \dfrac { x _ 0 + R } { \lambda _ i } \right ) \forall \ : i \in [ p + 1 , n ] . \end{cases} \end{align*}


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


\begin{align*} \int _ { 4 I _ Q } \| R _ j \partial _ j h ( \cdot , t ) \| _ { L ^ 1 ( 8 Q _ 1 ) } d t & = \int _ { 4 I _ Q } \int _ { 8 Q _ 1 } | R _ j \partial _ j h ( x , t ) | d x d t \\ & \leq \int _ { 4 I _ Q } \ell ( Q ) ^ { N / p } \| R _ j ( \partial _ j h ) ( \cdot , t ) \| _ { L ^ q ( \R ^ N ) } d t \\ & \lesssim \ell ( Q ) ^ { N / p } \int _ { 4 I _ Q } \| \partial _ j h ( \cdot , t ) \| _ { L ^ q ( \R ^ N ) } d t . \end{align*}


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


\begin{align*} \operatorname { I } ^ { - 1 } _ { u } ( a , b ) = \sum _ { i = 1 } ^ { \infty } q _ i \ , [ a \operatorname { B } ( a , b ) u ] ^ { i / a } , \end{align*}


\begin{align*} \sup _ { n } \mathbb { E } _ { \mathbb { P } } \left [ \exp \left \{ \kappa \sum _ { i = 1 } ^ { n } \int _ { T _ { 0 } } ^ { T _ { 0 } + \delta } \triangle b _ { t } ^ { i , n , \infty } \cdot d W _ { t } ^ { i } - \frac { \kappa } { 2 } \sum _ { i = 1 } ^ n \int _ { T _ { 0 } } ^ { T _ { 0 } + \delta } \left | \triangle b _ { t } ^ { i , n , \infty } \right | ^ { 2 } d t \right \} \right ] \end{align*}


\begin{align*} \sum _ h a _ h v ( \sigma _ h ) \sigma _ h = \sum _ h \left [ \sum _ { \sigma _ h \subset \tau _ j } c _ { h , j } b _ j \frac { v ( \sigma _ h ) } { v ( \tau _ j ) } \right ] \sigma _ h , \end{align*}


\begin{align*} c _ k = \frac 1 2 \sum _ { i = 0 } ^ k ( 2 i - k ) ^ 2 = { k + 2 \choose 3 } , \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } \Big ( \sup _ { t \leq T } \Vert Y _ n ( t ) \Vert _ H > \sqrt { M } \Big ) \leq \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } ( \tau _ { n } ^ M < T ) = 0 , \end{align*}


\begin{align*} R _ { q , \ell } ^ \flat ( n ) = \{ g \in A \cap B _ S ( n ) \mid \sigma _ g ^ + - \sigma _ g ^ - = \ell \} , \ell > q ( n ) , \end{align*}


\begin{align*} \inf \left \{ | | b _ 0 - \tilde { b } | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } \ | \ \tilde { b } \in S _ { h _ C , W , 0 } \ \right \} & = \inf \left \{ | | b _ 0 - \tilde { b } | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } \ \middle | \ \tilde { b } = U _ C ( \cdot + z e _ z ) - B _ { \infty } , \ z \in \mathbb { R } \right \} \\ & = \inf \left \{ | | b _ 0 + B _ { \infty } - U _ C ( \cdot + z e _ z ) | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } \ \middle | \ z \in \mathbb { R } \right \} . \end{align*}


\begin{align*} X _ { r } ^ i = \xi _ r \quad { X } ^ { i } _ t = \xi _ 0 + \int _ { 0 } ^ { t } \mu ( { \delta } _ i ( s ) , \mathcal { X } ^ { i } ) \ , d s + \int _ { 0 } ^ { t } \sigma ( { \delta } _ i ( s ) , \mathcal { X } ^ { i } ) \ , d W _ s . \end{align*}


\begin{align*} ( d \circ \delta _ { \lambda } ) ( g ) = \lambda ^ { Q } d g , \end{align*}


\begin{align*} \Delta : = \lbrace z \in \mathbb { C } \setminus \lbrace 0 \rbrace \colon - \alpha < \arg ( z ) < \alpha \rbrace , \end{align*}


\begin{align*} \ 1 _ { n - 1 } ' u _ k = 0 . \end{align*}


\begin{align*} \chi _ K ( s _ \alpha ( \beta ) ) = \chi _ K ( \beta ) - \langle \alpha ^ \vee , \beta \rangle \chi _ K ( \alpha ) . \end{align*}


\begin{align*} s _ n ( \nu | \mu ) = 0 \end{align*}


\begin{align*} \tilde { K _ i } : = K _ i \cap { K _ i } ' , \end{align*}


\begin{align*} Z ^ i ( v ( s ) , s ) - Z ( v ( s ) , s ) = \mathcal O ( \norm { ( v ( s ) , s ) } ^ { i + 1 } ) = \mathcal O ( s ^ \frac { i + 1 } { 2 } ) . \end{align*}


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


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


\begin{align*} \tau = \underbrace { \min \ ! \big ( \mathcal { H } _ 2 - \rho ( h _ { j , 1 } ) + 3 q ( n ) + 4 C \big ) } _ { \ge \ , - q ( n ) - C + 3 q ( n ) + 4 C \ , = \ , 2 q ( n ) + 3 C } - \underbrace { \max \ ! \big ( \mathcal { H } _ 1 \big ) } _ { \le \ , q ( n ) + C } \ge q ( n ) + 2 C , \end{align*}


\begin{align*} \delta _ k \le \beta ( x _ k ) \stackrel { \rm d e f } { = } \max \left \{ C _ 4 \| \nabla \phi ( x _ k ) \| - C _ 5 \epsilon _ g , - C _ 6 \lambda _ { \rm m i n } ( \nabla ^ 2 \phi ( x _ k ) ) - C _ 7 \epsilon _ H \right \} , \end{align*}


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


\begin{align*} \sum _ { i = 1 } ^ m f _ i ( u ) \log u _ i \leq 0 , \end{align*}


\begin{align*} u ( t ) = e ^ { \kappa t \Delta } u _ 0 + \int _ 0 ^ t e ^ { \kappa ( t - s ) \Delta } \Delta _ p u ( s ) \ , \ , d s + \sum _ { k \in \Z ^ d } \sum _ { i = 1 } ^ { d - 1 } \int _ 0 ^ t e ^ { \kappa ( t - s ) \Delta } \xi _ { k , i } \cdot \nabla u ( s ) \ , d B _ s ^ { k , i } . \end{align*}


\begin{align*} \int _ \R ( | T _ i ( x _ i ) | + | x _ i | ) Q _ i ( x _ i ) d x _ i = \int _ \R | x _ i | Q ^ * _ i ( x _ i ) d x _ i + \int _ \R | x _ i | Q _ i ( x _ i ) d x _ i < \infty . \end{align*}


\begin{align*} \hat { \lambda } _ { 1 } = \frac { ( \mathbf { v } ^ { n + 1 } ) _ i } { ( \mathbf { v } ^ { n } ) _ i } . \end{align*}


\begin{align*} \mu \Lambda ( \mu , \nu ) \xi ( \mu , s ) \xi ( \nu , s ) \chi _ { ( 0 , \mu ) } ( \nu ) = 0 a . e . ~ \mbox { i n } ( 0 , t _ { 0 } ) \times ( R _ { 0 } , \infty ) \times ( 0 , \infty ) . \end{align*} % \end{align*}


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


\begin{align*} \mbox { $ ( ( X _ { 2 , 6 } \cap N ( c _ 2 ) ) \cup ( X _ { 2 , 7 } \cap N ( c _ 2 ) ) \cup \{ c _ 2 \} ) \cap { \cal N } = \emptyset $ . } \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sqrt [ n ] { \binom { n } { \lceil \alpha n \rceil } } \le \frac { 1 } { \alpha ^ { \alpha } ( 1 - \alpha ) ^ { 1 - \alpha } } . \end{align*}


\begin{align*} \mathbf x ^ { t } \gets \underset { \mathbf z : \operatorname { s u p p } ( \mathbf z ) = \hat { S } ^ { t } } { \arg \min } f ( \mathbf z ) , \end{align*}


\begin{align*} \pi _ { 2 } \circ \lambda _ { f \times i d _ { X } } [ ( f \times i d _ { X } ) ( r , x ) , \alpha ] ( t ) = \pi _ { 2 } ( f ( r ) , \alpha ( t ) ) = \alpha ( t ) . \end{align*}


\begin{align*} \Delta _ { B } \textbf { \textit { u } } = ( \nabla ^ j \nabla _ j u ^ i ) \frac { \partial } { \partial x _ i } , \textbf { \textit { u } } = u ^ i \frac { \partial } { \partial x _ i } \in \mathfrak { X } ( \Omega ) . \end{align*}


\begin{align*} \mathbf { S } \mathbf { u } + \mathbf { X } \mathbf { v } = \mathbf { h } , \mathbf { u } ^ T \mathbf { v } \ge 0 , \end{align*}


\begin{align*} \mu ( \mathbf { x } , \mathbf { s } , \kappa , \tau ) = ( \mathbf { x } ^ T \mathbf { s } + \kappa \tau ) / ( k + 1 ) . \end{align*}


\begin{align*} M _ { \delta , + } ( x ; z ) = M _ { \delta , - } ( x ; z ) ( I + \hat { R } _ { \delta } ( x ; z ) ) , z \in \mathbb { R } . \end{align*}


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


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


\begin{align*} X _ t ^ i - X _ { t _ n } ^ i = \int _ { t _ n } ^ t b ^ i ( X _ s ) \d s + \sigma ( W _ t ^ i - W _ { t _ n } ^ i ) . \end{align*}


\begin{align*} - \Delta u = f & D , \\ [ 1 m m ] u = 0 & \partial D . \end{align*}


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


\begin{align*} \Delta + \Gamma & = \{ x + y \mid x \in \Delta , y \in \Gamma \} , \\ \Delta - \Gamma & = \{ x - y \mid x \in \Delta , y \in \Gamma \} , \\ - \Delta & = \{ - x \mid x \in \Delta \} . \end{align*}


\begin{align*} T ^ e _ A ( x ) : = \inf \{ t > 0 \mid \Phi ( t , x ) \not \in A \} . \end{align*}


\begin{align*} \Im \mathcal { P } [ \varrho ] = a \Re ( \varrho _ z ) \Im ( \varrho _ z ) ( ( 5 a ^ 2 - 4 ) | w | ^ 4 - 4 ( | \varrho _ z | ^ 4 + 2 | w | ^ 2 | \varrho _ z | ^ 2 ) ) . \end{align*}


\begin{align*} d \log \varphi _ { \lambda , u } ( x ) = 0 . \end{align*}


\begin{align*} \left \lvert 1 - 4 \mu \left ( \sum _ { l = 1 } ^ { M } \lvert \hat { \phi } ^ { \left ( l , j \right ) } \rvert ^ 2 + \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 + \sum _ { q = 1 } ^ { M - 1 } \sum _ { r = q + 1 } ^ { M } f _ { q , r } ^ { \left ( j \right ) } \right ) \right \rvert < 1 \end{align*}


\begin{align*} \langle \mathcal { A } _ q ^ s , \mathcal { A } _ q ^ p \rangle _ { L ^ 2 ( \mathbb { R , \mathbb { H } } ) } = K _ q ( p , s ) . \end{align*}


\begin{align*} \partial _ n ^ v ( \sum _ \ell r _ \ell \mu _ \ell + r _ { m + 1 } \sigma _ { m + 1 } ) = 0 , \end{align*}


\begin{align*} [ \mathbf { h } ' _ 2 , \mathbf { h } _ 2 ] = 1 . \end{align*}


\begin{align*} \varphi \left ( \rho \right ) = R ( \rho ) e ^ { 8 i t \rho ^ { 3 } + 2 i x \rho } . \end{align*}


\begin{align*} \Lambda = \cup _ { m \in \mathbb { N } ^ 2 } \Lambda ^ m . \end{align*}


\begin{align*} \tilde G ( p , s ) = s ^ { - p / 2 } 2 ^ { 3 p / 2 - 1 } \Gamma ( p / 2 ) D ( p ) \end{align*}


\begin{align*} ( \mathbf { x } , - \mathbf { y } ) \underbrace { \begin{pmatrix} G _ { 4 } & G _ { 5 } & \cdots & G _ { 2 k + 2 } \\ G _ { 5 } & G _ { 6 } & \cdots & G _ { 2 k + 3 } \end{pmatrix} } _ M = 0 . \end{align*}


\begin{align*} \ell ( w s _ { - w ^ { - 1 } \gamma } ) = \ell ( w ) - \ell ( s _ { - w ^ { - 1 } \gamma } ) . \end{align*}


\begin{align*} \theta _ t \omega ( \cdot ) = \omega ( \cdot + t ) - \omega ( t ) , \ \ t \in \R , \ \omega \in \Omega . \end{align*}


\begin{align*} \psi ( q ) = \varphi ( p _ 0 ) + \frac { \tilde { d } _ H ( p _ 0 , q ) ^ \beta } { \delta ^ \beta } . \end{align*}


\begin{align*} & E _ { * } ( g _ 1 , w ) = \\ & \int _ { - \infty } ^ \infty \int _ 0 ^ 1 \int _ 0 ^ \infty \int _ 0 ^ \infty \Psi _ * \left ( \lambda , \theta , t , u + \left ( \frac { t _ 1 } { t } \right ) ^ 2 u _ 1 \right ) e ^ { i \Phi \left ( \lambda , \theta , t , u + \left ( \frac { t _ 1 } { t } \right ) ^ 2 u _ 1 \right ) } W ( \lambda , t , u ) d \lambda d \theta d t d u . \end{align*}


\begin{align*} & \ell ( x , \alpha ) + \ell ( x , - \alpha ) \\ = & \langle \mu , \alpha \rangle + \Phi ^ + ( \alpha ) - \Phi ^ + ( w \alpha ) + \langle \mu , - \alpha \rangle + \Phi ^ + ( - \alpha ) - \Phi ^ + ( - w \alpha ) \\ = & \Phi ^ + ( \alpha ) + \Phi ^ + ( - \alpha ) - ( \Phi ^ + ( w \alpha ) + \Phi ^ + ( - w \alpha ) ) = 1 - 1 = 0 . \end{align*}


\begin{align*} \rho _ { x _ 1 \ast x _ 2 } = \rho _ { x _ 2 } \circ \rho _ { x _ 1 } . \end{align*}


\begin{align*} \theta _ 0 ( x ) - x = - \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } \frac { x ^ { s } } { s } \bigg \{ \frac { \zeta ' ( s ) } { \zeta ( s ) } + g ( s ) \bigg \} d t . \end{align*}


\begin{align*} x ( \varpi _ n ) = \varphi _ M ^ { - n } ( \frac \varpi \pi x ' ) \end{align*}


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


\begin{align*} K _ 1 * \cdots * K _ m = \left \{ \sigma \subset \bigsqcup _ { i \in [ m ] } V ( K _ i ) \ \middle | \ \sigma \cap V ( K _ i ) \in K _ i \right \} . \end{align*}


\begin{align*} T ^ r _ B ( x ) : = \inf \{ t > 0 \mid \Phi ( t , x ) \in B \} . \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 } ) , & \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*} \sum _ { w \in E } w t ( w ) = \sum _ { w \in O } w t ( w ) . \end{align*}


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


\begin{align*} \prescript J { } \pi ( s _ \alpha w _ 2 ) = ( w ' _ 2 , \mu _ 2 + \Phi ^ + ( - \alpha ) w _ 2 ^ { - 1 } \alpha ^ \vee ) . \end{align*}


\begin{align*} w _ 0 w _ 1 = s _ 1 \cdots s _ q . \end{align*}


\begin{align*} \widehat { \iota } ( D ) = \frac { \beta m + \gamma } { m + b _ 1 + 1 } . \end{align*}


\begin{align*} \sum \limits _ { j = 1 } ^ { N } \mathbf { R } _ { . j } = \mathbf { 1 } ^ { H } _ { N \times 1 } \mathbf { R } \end{align*}


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


\begin{align*} \bigcup _ { \rho _ i > r _ i } K \ < \frac { x _ 1 } { \rho _ 1 } , \ldots , \frac { x _ 1 } { \rho _ n } \ > = \bigcup _ { \rho _ i > r _ i } T ( \rho _ 1 , \ldots , \rho _ n ) \end{align*}


\begin{align*} \partial _ s \rho ( s , u ) - m \cdot \nabla \big ( \rho ^ 2 ( s , u ) \big ) = 0 , \rho ( 0 , u ) = \rho ^ { \rm i n i } ( u ) , \end{align*}


\begin{align*} \mathcal { Y M } ( \nabla ) = \int _ M \frac 1 2 | | R ^ \nabla | | ^ 2 \ , d v \ , , \end{align*}


\begin{align*} \xi ( F ) = u _ \S ( F ) + 2 . \end{align*}


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


\begin{align*} \dfrac { \varepsilon } { 2 } \leq \beta ( \varphi _ { 2 k } ( a _ 1 ) , \rho ( a _ 1 ) ) = \beta ( \varphi _ { 2 k } ( a _ 1 ) , \rho \circ \varphi _ { 2 k } ( a _ 1 ) ) \leq \beta ( \varphi _ i ( a _ 1 ) , \rho \circ \varphi _ i ( a _ 1 ) ) . \end{align*}


\begin{align*} \mbox { $ ( ( X _ { 2 , 7 } \setminus N ( c _ 2 ) ) \cup ( X _ { 3 , 7 } \setminus N ( c _ 3 ) ) \cup \{ c _ 7 \} ) \cap { \cal N } = \emptyset $ . } \end{align*}


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


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


\begin{align*} J ( z ) = & \frac { \sqrt \theta } { B ( a , b ) } \sum _ { n = 0 } ^ \infty \frac { ( - 1 ) ^ n } { \Gamma ( b - n ) n ! } \int _ { \theta / z ^ 2 } ^ { \infty } y ^ { - 1 / 2 } \exp \{ - ( a + n ) y \} \mathrm { d } y \\ = & \sqrt { \pi \theta } \frac { \Gamma ( b ) } { B ( a , b ) } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ n } { \Gamma ( b - n ) n ! } \frac { 1 } { \sqrt { a + n } } \operatorname { e r f c } ( \sqrt { \theta / z ^ 2 } \sqrt { a + n } ) , \end{align*}


\begin{align*} \phi ( k ) = \sum _ { z \in \partial I ( k ) } G _ { I ( k ) } ( z , k ) \phi ( z ' ) . \end{align*}


\begin{align*} { \tt \hat z = 1 . / e v a l u ( P h i 1 , z , { ' d ' } ) } \end{align*}


\begin{align*} N : = \bigwedge ^ { p - 1 } ( J _ { p - 2 } \oplus J _ p ^ { \oplus k - 1 } ) \cong \bigoplus ^ { p - 1 } _ { i = 0 } \bigwedge ^ { p - 1 - i } J _ { p - 2 } \otimes \bigwedge ^ { i } ( J _ p ^ { \oplus k - 1 } ) . \end{align*}


\begin{align*} A _ t : = t A _ 1 + ( 1 - t ) A _ 0 t \in [ 0 , 1 ] \end{align*}


\begin{align*} a _ { n + 1 } = a _ { \alpha + n + 1 } + d _ { \alpha + n + 1 } - d _ { \alpha + n } \iff a _ { \alpha + n + 1 } - a _ { n + 1 } = d _ { \alpha + n } - d _ { \alpha + n + 1 } \ , \ \forall \ n \in \N \ . \end{align*}


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


\begin{align*} \exp ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) \delta ^ \nabla R ^ \nabla - i _ { \big ( \exp ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) \big ) } R ^ \nabla = 0 \ , , \end{align*}


\begin{align*} \lim _ { N \to \infty } \dfrac { | ( \Phi _ N - t ) \cap \Phi _ N | } { | \Phi _ N | } = 1 \end{align*}


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


\begin{align*} | \phi ( 0 ) | \geq 1 \end{align*}


\begin{align*} \Tilde { x } = \sum \limits _ { i = 1 } ^ p \alpha _ i u _ i + \lambda x _ v + ( 1 - \lambda ) x _ w \in \mathcal { S } ^ n , \end{align*}


\begin{align*} A _ j = \textrm { c u r l } ^ { - 1 } B _ j \to A = \textrm { c u r l } ^ { - 1 } B \textrm { i n } \ L ^ { 2 } _ { \textrm { l o c } } ( 0 , T ; L ^ { 2 } ( \Omega ) ) , \end{align*}


\begin{align*} x \cdot y : = \sum _ { g \in G } \underline g \left ( \sum _ { h _ 1 h _ 2 = g } x _ { h _ 1 } y _ { h _ 2 } \right ) . \end{align*}


\begin{align*} \mathcal { J } ' ( \overline { u } ; h ) = \langle \overline { y } - g , S ' ( \overline { u } ; h ) \rangle + \alpha \langle \overline { u } , h \rangle \ge 0 , h \in T _ { \mathcal { C } _ { a d } } ( \overline { u } ) . \end{align*}


\begin{align*} ( x , v ) \sim ( y , w ) \ : \Longleftrightarrow \ x = y \ \ w = \varphi _ { \beta \alpha } ( x ) v \ ( x \in U _ { \alpha } , y \in U _ { \beta } , v , w \in \mathbb R ^ n ) \ , . \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta z _ { h } + F ' ( y ; z _ { h } ) & = h , \ ; \ ; \Omega , \\ z _ { h } & = 0 , \ ; \ ; \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \hat { f } ( \chi ) = \int _ G f ( g ) \chi ( g ) d g \end{align*}


\begin{align*} \phi _ { i j } ( X ) = \langle \nabla ^ \perp _ X \eta _ i , \eta _ j \rangle . \end{align*}


\begin{align*} \det ( D ^ 2 ( a _ k \hat C _ { q _ k , N } ) ) = c _ k \delta _ { q _ k } . \end{align*}


\begin{align*} \log \lceil 2 ^ { \lambda _ { 1 2 3 ' } } \rceil & = \log \left \lceil \frac { \zeta } { 3 6 } \right \rceil = \log 3 \end{align*}


\begin{align*} \nabla G ( U , x , t ) ^ T \nabla A ( U , x , t ) & = \nabla A ( U , x , t ) ^ T \nabla G ( U , x , t ) \\ \nabla f _ \alpha ( U , x , t ) ^ T \nabla A ( U , x , t ) & = \nabla f _ \alpha ( U , x , t ) ^ T \nabla G ( U , x , t ) \qquad \alpha = 1 , \dots , d , \end{align*}


\begin{align*} a ( U _ 1 Z ) = a ( U _ 1 U D ( x ) V ) = a ( D ( x ) V ) = a ( U D ( x ) V ) = a ( Z ) . \end{align*}


\begin{align*} k : a \mapsto a ^ { k } : = \prod _ { \sigma \in \Sigma _ v } \sigma ( a ) ^ { k _ \sigma } \end{align*}


\begin{align*} A ( x , t ) = \sum _ { n = 0 } ^ { \infty } a _ { n } ( x ) L _ { n } ( t - x ) e ^ { \frac { x - t } { 2 } } \end{align*}


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


\begin{align*} \lim _ { x \to \infty } \breve g _ 1 ( x ) / f _ 1 ( x ) = ( 1 - e ^ { - \lambda \Lambda _ 1 } ) / ( \lambda \Lambda _ 1 ) , \end{align*}


\begin{align*} & \varphi _ x + i k ^ 2 [ \sigma _ 3 , \varphi ] = k P _ x \varphi , \\ & \varphi _ t + i \eta ^ 2 [ \sigma _ 3 , \varphi ] = H \varphi , \end{align*}


\begin{align*} T _ 1 ( u , J u , J u , u , v , J v ) = T _ 2 ( u , J u , J u , u , v , J v ) , \end{align*}


\begin{align*} & \chi ( \xi ) + \sum _ { j \geq 0 } \varphi ( 2 ^ { - j } \xi ) = 1 , ~ ~ \forall \xi \in \mathbb { R } ^ d , \\ & | j - j ' | \geq 2 \Rightarrow ~ \varphi ( 2 ^ { - j } \xi ) \cap ~ \varphi ( 2 ^ { - j ' } \xi ) = \emptyset . \end{align*}


\begin{gather*} \lim _ { N \to \infty } \dfrac { | \Phi _ N g \cap \Phi _ N | } { | \Phi _ N | } = 1 \\ \lim _ { N \to \infty } \dfrac { | \Phi _ N \cap g \Phi _ N | } { | \Phi _ N | } = 1 \end{gather*}


\begin{align*} Y _ t - y _ t = \int _ 0 ^ t e ^ { L _ { z _ 0 } ^ \perp ( t - s ) } \Pi _ { \mathrm { k e r } A ^ \perp } ( B ( y _ s , z _ 0 - z _ s ) + B ( z _ 0 - z _ s , y _ s ) - B ( y _ s , y _ s ) + A y _ s ) d s . \end{align*}


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


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


\begin{align*} \theta _ { i , ( 2 ) } ( z _ j ) = { \delta } _ { i , j } \mbox { a n d } \theta _ { i , ( 1 ) } ( z _ j ) = a _ { i , j } . \end{align*}


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


\begin{align*} \widetilde \theta _ 1 & = 0 , \\ \widetilde \theta _ i & = \pi ^ { [ n - 1 ] } _ { i - 1 } ( \xi ^ { \widetilde \theta } _ 1 ( Y _ 1 ) , x + Y _ 1 , X _ 2 , Y _ 2 , \dots , X _ { i - 1 } , Y _ { i - 1 } ) , \end{align*}


\begin{align*} S ( t ) = \begin{bmatrix} e ^ { t \nu \Delta } & 0 \\ 0 & e ^ { t \mu \Delta } \\ \end{bmatrix} . \end{align*}


\begin{align*} \tilde X _ { n + 1 } ^ i = \tilde X _ n ^ i + b ( \tilde X _ n ^ i ) \tau + \gamma ^ i ( \tilde X _ n ) \tau + \sigma W _ \tau ^ i , \end{align*}


\begin{align*} \begin{aligned} & \hat { \mathbf { r } } _ p ( \alpha ) = ( 1 - \alpha \delta / \sqrt { 2 k } ) \hat { \mathbf { r } } _ p , \\ & \hat { \mathbf { r } } _ d ( \alpha ) = ( 1 - \alpha \delta / \sqrt { 2 k } ) \hat { \mathbf { r } } _ d , \\ & \mu ( \alpha ) = ( 1 - \alpha \delta / \sqrt { 2 k } ) \mu . \end{aligned} \end{align*}


\begin{align*} F = \varinjlim _ n ( \varprojlim _ { k } X _ { H , n } \widehat { \otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V _ k ) \oplus \varprojlim _ k A _ { H ' , n } \widehat { \otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V _ k ) ) . \end{align*}


\begin{align*} F _ n ( z ) ( t ) = U ( \tau _ n + t , \tau _ n ) y _ n ( \tau _ n ) - \lambda i \int _ 0 ^ t U ( \tau _ n + t , \tau _ n + s ) ( e ^ { ( \alpha - 1 ) W ( \tau _ n + s ) } g ( z ( s ) ) ) d s . \end{align*}


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


\begin{align*} H ^ { s , m } & = \{ f \in \mathcal { S } ' \ ; \| f \| _ { H ^ { s , m } } < \infty \} , \\ \| f \| _ { H ^ { s , m } } & = \| \langle x \rangle ^ m \mathcal { F } ^ { - 1 } \langle \xi \rangle ^ s { { \mathcal F } } f \| _ { L _ { x } ^ { 2 } } , \end{align*}


\begin{align*} \frac { \partial \xi ( \mu , t ) } { \partial t } = & - \frac { \partial } { \partial \mu } \bigg ( \xi ( \mu , t ) \int _ 0 ^ { \mu } \nu \Lambda ( \mu , \nu ) \xi ( \nu , t ) \ d \nu \bigg ) \\ & - \int _ { \mu } ^ { \infty } \Lambda ( \mu , \nu ) \xi ( \mu , t ) \xi ( \nu , t ) \ d \nu , \ ( \mu , t ) \in \mathbb { R } _ { > 0 } ^ 2 , \ \end{align*}