\begin{align*} \beta ^ t _ { a , b } ( z ) \gamma ^ t _ { a , b } ( w ) = ( z - w ) ^ { - 1 } \end{align*}


\begin{align*} \delta ^ { \frac { 2 } { p - 1 } } s _ \delta ^ { - \frac { 2 } { p - 1 } } \phi ' ( 1 ) = a / \ln \frac { s _ \delta } { R } . \end{align*}


\begin{align*} f _ i ( r ) : = \Big \lfloor \frac { r } { 1 + ( 1 + \tilde \varepsilon ) ^ { - i } } \Big \rfloor , r \in \N . \end{align*}


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


\begin{align*} \widetilde G _ { n } ( x , y ) & = \mathrm { d e t } \left ( f _ n ^ { ( 1 ) } ( y _ j - x _ i ) \right ) _ { i , j = 1 } ^ d \\ & = \mathrm { d e t } \left ( \mathrm { e } ^ { \lambda ( y _ j - x _ i ) } ( 1 + \lambda ) ^ { - n } f _ n ^ { ( 1 + \lambda ) } ( y _ j - x _ i ) \right ) _ { i , j = 1 } ^ d \\ & = \mathrm { e } ^ { \lambda \sum _ { i = 1 } ^ d ( y _ i - x _ i ) - d n \ln ( 1 + \lambda ) } \widetilde G _ { n } ^ { ( 1 + \lambda ) } ( x , y ) . \end{align*}


\begin{align*} N _ m ( x ) \leq | h _ m ( x ) | ^ { 3 - 2 \beta } = | h _ m ( x ) | ^ \rho . \end{align*}


\begin{align*} \lambda _ S ( G ) = \lim _ { n \to \infty } \frac { | B _ S ( n + 1 ) | } { | B _ S ( n ) | } . \end{align*}


\begin{align*} d _ 0 | q | ^ 2 \wedge d _ 1 | q | ^ 2 = 4 \sum _ { l = 0 } ^ { n - 1 } | q _ l | ^ 2 \omega ^ l \wedge \omega ^ { n + l } + \sum _ { | j - k | \not = n } a _ { j k } \omega ^ j \wedge \omega ^ k \end{align*}


\begin{align*} \omega \kappa \begin{pmatrix} 1 \\ \vdots \\ \theta ^ { n - 1 } \end{pmatrix} = v , \end{align*}


\begin{align*} & R e \left ( \omega ( s ) e ^ { i \theta } \right ) \overset { \eqref { c o n c a t e n a t e d _ o m e g a } , \eqref { c o n c a t e n a t e d _ s } } { = } - s R e \left ( e ^ { i ( \theta - \theta _ 2 / 2 - \theta _ 1 / 2 ) } \right ) = \\ & - s \cos ( \theta - \theta _ 2 / 2 - \theta _ 1 / 2 ) \overset { \eqref { c o n c a t e n a t e d _ s } , \eqref { c o n c a t e n a t e d _ t h e t a } , \eqref { a l p h a } } { \leq } - s \cos \left ( \alpha \right ) . \end{align*}


\begin{align*} \mu _ N ( E ) = \dfrac { | A \cap \Phi _ N | } { | \Phi _ N | } \end{align*}


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


\begin{align*} j _ { x , x + 1 } = \mathcal G _ t \frak f _ x - \frac 1 { 4 \gamma } \nabla \mathfrak F _ x , \end{align*}


\begin{align*} ( I - Q ) F ^ q ( v + \psi ( v , p ) , p ) = 0 \end{align*}


\begin{align*} \begin{pmatrix} \theta ^ 2 & \theta \cdot \sigma \\ \theta \cdot \sigma & \sigma ^ 2 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & - 2 \end{pmatrix} . \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) : = \sum _ { K \in { \mathcal T } _ n } \int _ K \nabla u _ n \cdot \nabla v _ n d K , u _ n , v _ n \in X _ n . \end{align*}


\begin{align*} \Sigma \big ( X , q \big ) = X \frac { \lambda ( q ) } { q } + \mathcal { O } \big ( \lambda ( q ) \big ) \ , . \end{align*}


\begin{align*} \Delta u _ 1 \wedge \dots \wedge \Delta u _ n = n ! \det ( A _ 1 , A _ 2 , \dots , A _ n ) \Omega _ { 2 n } . \end{align*}


\begin{align*} \begin{aligned} | h ( x + & \delta u ( \tau ) ) - h ( x _ a + \delta u ( \tau ) ) | \le L _ r | x - x _ a | \\ & \le L _ r ( | x - z _ 1 | + \gamma | \epsilon _ 1 ( \eta _ a , \tau ) | ) \\ & \le L _ r ( | x - z _ 1 | + \gamma \delta L _ r ( 3 + L _ r ) ) . \end{aligned} \end{align*}


\begin{align*} d X ^ i _ t = \alpha _ i ( t , X _ t ) d t + d B ^ i _ t , X ^ i _ 0 = 0 , \ \ i = 1 , \ldots , n , \end{align*}


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


\begin{align*} - L u = f ( u ) \Omega , \end{align*}


\begin{align*} \inf \Bigg \{ \int _ { \{ \Psi < - t \} } | \tilde { f } | ^ 2 _ h c ( - \Psi ) : \tilde { f } \in H ^ 0 ( \{ \Psi < - t \} , \mathcal { O } ( K _ M \otimes E ) ) \\ \& \ , ( \tilde { f } - f ) _ { z _ 0 } \in \mathcal { O } ( K _ M ) _ { z _ 0 } \otimes J _ { z _ 0 } , z _ 0 \in Z _ 0 \Bigg \} \end{align*}


\begin{align*} \mathbb { E } [ \langle Z _ { \lambda } ^ { S _ { n } } , f \rangle | \sigma ( \Phi ^ { S _ { n } , \tilde { S } _ { n , m } } ) ] = \langle Y _ { m } ^ { S _ { n } } , f \rangle . \end{align*}


\begin{align*} { A \choose B } : = \frac { B ( - 1 ) } { q } J ( A , \overline { B } ) , \end{align*}


\begin{align*} \ell ( x _ 0 ) \geq \ell ( x _ 1 ) = \ell ( x _ L x _ 0 x _ R ) \geq \ell ( x _ 0 x _ R ) - \ell ( x _ L ) = \ell ( x _ 0 ) + \ell ( x _ R ) - \ell ( x _ L ) . \end{align*}


\begin{align*} u ( \infty ) = \Re \ , \psi ( z _ 0 ) \end{align*}


\begin{align*} \zeta ( x ) : = \begin{cases} \frac { \xi ( x ) } { p ( x ) } , & \{ p ( x ) \ne 0 \} , \\ f _ { - } ' ( \overline { y } ( x ) ) , & \{ p ( x ) = 0 \} , \end{cases} \end{align*}


\begin{align*} H ^ 2 ( \mathbb { D } ) = \{ f = \sum _ { n = 0 } ^ { \infty } a _ n z ^ n \mid ( a _ n ) _ { n \in \mathbb { N } } \in \ell ^ 2 \} . \end{align*}


\begin{align*} & : , ~ \forall n \in \N , ~ \omega _ n = \alpha _ 1 \beta _ 1 \dots \alpha _ { n - 1 } \beta _ { n - 1 } \alpha _ n , \\ & : , ~ \forall n \in \N , ~ \omega _ n = \beta _ 1 \alpha _ 1 \dots \beta _ { n } \alpha _ { n } . \end{align*}


\begin{align*} h ( h e _ n h ) ( h g e _ n h g ) \cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) = ( h e _ n h ) ( h g e _ n h g ) \cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) , \end{align*}


\begin{align*} \widetilde { X } = \widehat { X } = \overline { X } , \widetilde { \mathbb { P } } \otimes d t . \end{align*}


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


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


\begin{align*} \frac { 1 } { \rho _ K } \| x ^ { K + 1 } - x ^ K \| \leq \frac { r _ 0 + \theta } { \rho _ 0 \zeta ^ K } \overset { \eqref { K - s p } } { \leq } \frac { \varepsilon } { 2 } , \eta _ K = \eta _ 0 \sigma ^ K \overset { \eqref { K - s p } } { \leq } \frac { \varepsilon } { 2 } , \end{align*}


\begin{align*} \deg ( ( g _ { m , ( m ' ; l ' ) } ^ \epsilon ) _ { i j } ) = ( - 1 ) ^ j i l + i j ( d - 2 ) + \frac { ( - 1 ) ^ i } { 2 } \left ( 1 - \left ( - 1 \right ) ^ j \epsilon \right ) \end{align*}


\begin{align*} \mathcal { F } _ { \geq p } C F ^ { k } ( H _ { t } ) : = \lbrace x \in C F ^ { k } ( H _ { t } ) \mid \mathcal { A } ( x ) \geq p \rbrace \end{align*}


\begin{align*} \begin{aligned} \nabla \times U = f U , \nabla \cdot U & = 0 \textrm { i n } \ \mathbb { R } ^ { 3 } , \\ U & \to B _ { \infty } \textrm { a s } \ | x | \to \infty . \end{aligned} \end{align*}


\begin{align*} r _ a ( b ) = \left ( s _ \alpha ( \beta ) , - \chi _ K ( \beta ) + \langle \alpha ^ \vee , \beta \rangle \chi _ K ( \alpha ) \right ) , \end{align*}


\begin{align*} k _ \theta \cdot f _ - & = \frac { 1 } { \sqrt { 2 } } \left ( s ( \theta ) , c ( \theta ) , s ( \theta ) , c ( \theta ) \right ) \\ k _ \theta \cdot f _ + & = \frac { 1 } { ( 1 0 8 ) ^ { \frac { 1 } { 4 } } } \left ( s ( 3 \theta ) , 3 c ( 3 \theta ) , - 3 s ( 3 \theta ) , - c ( 3 \theta ) \right ) . \end{align*}


\begin{align*} \gamma \cdot \alpha & = ( \alpha _ 1 , \ldots , \gamma \alpha _ i , \ldots , \alpha _ k ) , \\ \sigma \cdot \alpha & = ( \alpha _ { \sigma ^ { - 1 } ( 1 ) } , \alpha _ { \sigma ^ { - 1 } ( 2 ) } , \ldots , \alpha _ { \sigma ^ { - 1 } ( k ) } ) . \end{align*}


\begin{align*} \int _ { \Omega } A \cdot B \dd x = \sum _ { j } \frac { 1 } { \lambda _ j } c _ j ^ { 2 } = \sum _ { j } \frac { 1 } { f _ j ^ { + } } c _ j ^ { 2 } + \sum _ { j } \frac { 1 } { f _ j ^ { - } } c _ j ^ { 2 } . \end{align*}


\begin{align*} m ^ { \star } ( C _ \mu \cap ( \bigcup _ { k \in \mathbb { N } } I _ k ) ) & = m ^ { \star } ( \bigcup _ { k \in \mathbb { N } } ( C _ \mu \cap I _ k ) \leq \sum _ { k \in \mathbb { N } } m ^ { \star } ( C _ \mu \cap I _ k ) \leq \sum _ { k \in \mathbb { N } } m ( H \cap I _ k ) \\ & \leq ( 1 - \mu ) \sum _ { k \in \mathbb { N } } m ( I _ k ) < ( 1 - \mu ) m ( G ) < m ^ { \star } ( C _ { \mu } ) \end{align*}


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


\begin{align*} \mathbf { y } _ k = \mathbf { H } _ k \sum _ { i \in \mathcal { M } _ k } a _ i s _ i \mathbf { p } _ i + \mathbf { H } _ k \sum \limits _ { \substack { l = 1 \\ l \neq k } } ^ { K } \sum \limits _ { j \in \mathcal { M } _ l } a _ j s _ j \mathbf { p } _ j + \mathbf { n } _ k \end{align*}


\begin{align*} \lim _ { n \to \infty } T ^ { c _ 1 ( n ) } ( x _ 0 ) = x _ 1 . \end{align*}


\begin{align*} T _ { \textnormal { i n n e r } } = \left \lceil \left ( \tfrac { 2 ^ { p + 1 } ( 5 p - 2 ) } { p ! } \tfrac { L D ^ { p - 1 } } { \mu } \right ) ^ { \frac { 2 } { p + 1 } } \right \rceil , \end{align*}


\begin{align*} h ( p ) = \sum _ k \lambda _ k a _ k ^ p \end{align*}


\begin{align*} \langle \rho _ { \Sigma _ s } ( m ) , h _ 1 \otimes . . . \otimes h _ m \rangle _ { \mathcal H ^ m } = \sum _ { p \in \mathcal P ( m ) } \prod _ { ( x , y ) \in p } \langle \Sigma h _ x , h _ y \rangle , \end{align*}


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


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


\begin{align*} y ^ * = t \int _ { \R ^ n } \nabla f \ , d \gamma _ { y ^ * , t } , \end{align*}


\begin{align*} T _ { \delta } ( x ; z ) : = \left ( M _ { \delta , + , 1 } ( x ; z ) - e _ 1 , M _ { \delta , - , 2 } ( x ; z ) - e _ 2 \right ) \begin{pmatrix} 1 & 0 \\ - r _ { \delta , 2 } ( z ) e ^ { 2 i z x } & 1 \end{pmatrix} \end{align*}


\begin{align*} h ( s ) = h _ { a , \Delta } ( s ) = \left ( \dfrac { a } { a ^ 2 + s ^ 2 } \right ) \left ( \dfrac { e ^ { 2 \pi a \Delta } + e ^ { - 2 \pi a \Delta } - 2 \cos ( 2 \pi \Delta s ) } { \left ( e ^ { \pi a \Delta } - e ^ { - \pi a \Delta } \right ) ^ 2 } \right ) \end{align*}


\begin{align*} \hat \Phi ^ \dagger ( a , s ) & = \left [ \gamma _ 1 a ^ 3 + \mathcal O ( a ^ 4 ) \right ] + s \left [ \gamma _ 2 a + \mathcal O ( a ^ 2 ) \right ] + \mathcal O ( s ^ 2 ) \\ & = a ( \gamma _ 1 a ^ 2 + \gamma _ 2 s ) + \mathcal O ( a ^ 4 + \abs { s } a ^ 2 + s ^ 2 ) \end{align*}


\begin{align*} \mu \left ( x \right ) = D ^ { r } \left ( e ^ { a \left \vert x \right \vert } f \left ( x \right ) \right ) \end{align*}


\begin{align*} | \Pi _ { \mathrm { k e r } A ^ \perp } X _ t | & = \left | \Pi _ { \mathrm { k e r } A ^ \perp } X _ 0 + t \Pi _ { \mathrm { k e r } A ^ \perp } B ( X _ 0 , X _ 0 ) + \Pi _ { \mathrm { k e r } A ^ \perp } \int _ 0 ^ t ( t - s ) \frac { d } { d s } B ( X _ s , X _ s ) d s \right | \\ & \ge c t | \Pi _ { V _ 1 } X _ 0 | | \Pi _ { V _ 2 } X _ 0 | - C \sqrt { \delta } K ^ { r } - C K \tilde { \eta } \int _ 0 ^ t | B ( X _ s , X _ s ) | d s , \end{align*}


\begin{align*} \tilde Z _ n ^ N ( i ) : = & \Delta _ n ^ { - \frac 1 2 } \left ( ( p _ N \tilde { \Delta } _ i ^ n Y ) ^ { \otimes 2 } - \langle \langle p _ N \tilde { \Delta } _ i ^ n Y \rangle \rangle \right ) \\ = & \Delta _ n ^ { - \frac 1 2 } \left ( ( p _ N \tilde { \Delta } _ i ^ n Y ) ^ { \otimes 2 } - \int _ { t _ { i - 1 } } ^ { t _ i } p _ N \mathcal S ( t _ i - s ) \Sigma _ s \mathcal S ( t _ i - s ) ^ * p _ N d s \right ) . \end{align*}


\begin{align*} \partial ^ \square c _ { t \ , | t = 0 } = \partial ^ \square \big ( ( c + t w ) \circ \theta _ t ^ { - 1 } \big ) _ { | t = 0 } + \partial ^ \square m _ { t \ , | t = 0 } = w . \end{align*}


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


\begin{align*} H ^ 2 ( \mathbb { D } ) & = \{ \sum _ { n = 0 } ^ { \infty } a _ n z ^ n \mid ( a _ n ) \in \ell ^ 2 \} \\ Y & = \{ \sum _ { n = 0 } ^ { \infty } a _ n f _ n \mid ( a _ n ) \in \ell ^ 2 \} \end{align*}


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


\begin{align*} \lim _ { n \to \infty } \frac { 1 } { n } \log \int _ { \R ^ n } e ^ { f ( x ) } d x = \sup _ { Q \in \P ( \R ) } \left ( \int _ { \R } V \ , d Q + \frac 1 2 \int _ { \R } \int _ { \R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \right ) . \end{align*}


\begin{align*} k _ { i } ( \Delta ) = \sum _ { \Upsilon \in \mathcal { T } _ { i } ( \Delta ) } { | \mathbf { T } ( \tilde { H } _ { i - 1 } ( \Upsilon ) ) | ^ 2 } = \sum _ { \Upsilon \in \mathcal { T } _ { i } ( \Delta ) } { \mathbf { t } _ { i - 1 } ( \Upsilon ) ^ 2 } . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { | R _ q ( n ) | } { | B _ S ( n ) | } = 0 . \end{align*}


\begin{align*} \sigma = \sum \limits _ { d = 1 } ^ \infty \frac { \mu ( d ) \lambda ( d ^ 2 ) } { d ^ 2 } = \prod _ p \left ( 1 - \frac { \lambda ( p ^ 2 ) } { p ^ 2 } \right ) \ , . \end{align*}


\begin{align*} P ( \alpha ) = \big \{ ( z , w ( \cdot ) ) \in \mathbb R ^ n \times \mathrm { P L i p } \colon \| z \| \leq \alpha , \ , \| w ( \cdot ) \| _ \infty \leq \alpha \big \} . \end{align*}


\begin{align*} \hat { A } ^ + = - f \frac { d } { d r } - \frac { L + 1 } { r } f + \frac { Q } { 2 ( L + 1 ) } + \kappa \left ( L + \frac { 3 } { 2 } \right ) \frac { r } { f } . \end{align*}


\begin{align*} c _ { k + 1 } < 2 c _ k , \ \displaystyle \lim _ { k \to \infty } c _ k = \infty , \ \log c _ k = o ( \alpha ( 2 ^ { k - 1 } ) ) , \ \ \log c _ k = o ( k ) , \end{align*}


\begin{align*} \left | \Pi _ { \mathrm { k e r } A ^ \perp } \frac { d ^ { \ell - 1 } } { d t ^ { \ell - 1 } } X _ t | _ { t = t _ 0 } \right | \ge \frac { \gamma ^ 3 } { C _ J } K ^ \ell . \end{align*}


\begin{align*} F _ n - n ^ p \ \ = \ \ \mathcal { C } _ n ^ { ( p ) } \ - \ 2 \sum _ { j = 0 } ^ { p / 2 } \binom { p } { 2 j + 1 } \mathcal { C } _ n ^ { ( p - 2 j - 1 ) } \ + 2 \binom { p } { 2 \lfloor p / 2 \rfloor + 1 } F _ n \mbox { f o r $ p $ o d d } . \end{align*}


\begin{align*} Q ( X ) = \sum _ { 0 \le i \ne j \le n - 1 } B _ { i j } X ^ { 2 ^ i + 2 ^ j } + \sum _ { 0 \le i \le n - 1 } C _ i X ^ { 2 ^ i } + D . \end{align*}


\begin{align*} & \phi _ i ( v w ) = v w + ( v w | e _ i ) y = v w + ( v | w e _ i ) y \\ & = v w + ( v | w ) y = ( v + ( v | e _ i ) y ) * ( w + ( w | e _ i ) y ) = \phi _ i ( v ) * \phi _ i ( w ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r l } \div ( A ( x ) \nabla v ) + \dd ( x ) \cdot \nabla u = 0 & B _ { \rho } \\ v = u & \partial B _ { \rho } . \end{array} \right . \end{align*}


\begin{align*} f ( x ) = \begin{cases} 1 + x , & x \leq 0 , \\ 1 - x , & x > 0 . \end{cases} \end{align*}


\begin{align*} \mathcal B _ { \mathcal X } ^ { \mathfrak a } = \left \{ ( s , b ) | s \in S , b \in B _ { X _ s } ^ { \alpha _ s } \right \} \end{align*}


\begin{align*} & \abs { \varphi ( v \beta ' ) - \langle \alpha ^ \vee , \beta \rangle \varphi ( v \alpha ) - \varphi ( - v \beta ) } \\ \leq & \sum _ { k = 1 } ^ { \langle \alpha ^ \vee , \beta \rangle } \abs { \varphi ( v ( k \alpha - \beta ) ) - \varphi ( v \alpha ) - \varphi ( v ( k - 1 ) \alpha - \beta ) } \\ \leq & \langle \alpha ^ \vee , \beta \rangle . \end{align*}


\begin{align*} \varphi ( - t ) ^ 2 G ( - t ) - \varphi ( t ) ^ 2 G ( t ) = 2 t \varphi ( t ^ 4 ) ^ 2 H ( t ^ 4 ) \end{align*}


\begin{align*} \Phi ( X ) = \left [ \begin{array} { c c } X & 0 \\ 0 & X \end{array} \right ] \quad { \rm s o \ t h a t } \Phi ^ \dagger \left ( \left [ \begin{array} { c c } A & B \\ C & D \end{array} \right ] \right ) = A + D \ . \end{align*}


\begin{align*} h = 0 , \nabla h = 0 , \nabla ^ { k + 2 } h = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \hat \Phi ^ \dagger ( 0 , 0 ) & = 0 \end{align*}


\begin{align*} \langle \lambda ^ * - \tilde { \lambda } ^ k , \frac { 1 } { \beta } ( \tilde { \lambda } ^ k - \lambda ^ k ) \rangle _ { \mathcal { L } ^ 2 } = \langle \lambda ^ * - \tilde { \lambda } ^ k , \tilde { y } ^ k - \tilde { x } ^ k \rangle _ { \mathcal { L } ^ 2 } = \langle \lambda ^ * - \tilde { \lambda } ^ k , \tilde { y } ^ k - y ^ * - ( \tilde { x } ^ k - x ^ * ) \rangle _ { \mathcal { L } ^ 2 } . \end{align*}


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


\begin{align*} \widehat { a } = \chi _ E a . \end{align*}


\begin{align*} \mathcal { B } _ { A B C } \coloneqq \mathcal { B } ( e _ A , e _ B , e _ C ) = \langle [ e _ A , e _ B ] _ H , e _ C \rangle . \end{align*}


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


\begin{align*} \begin{alignedat} { 3 } F ( z _ l ) & = ( 0 , 0 ) , \phantom { x x x } & F ( z _ r ) & = ( 0 , 0 ) , \\ \nu _ \Sigma ( z _ l ) & = ( - 1 , 0 ) , \phantom { x x x } & \nu _ \Sigma ( z _ r ) & = ( + 1 , 0 ) . \end{alignedat} \end{align*}


\begin{align*} \mathbf { S ( R ) } = \sum \limits _ { i = 1 } ^ { N } \sum \limits _ { j = 1 } ^ { N } \mathbf { R } _ { i j } \end{align*}


\begin{align*} a = \Big ( \frac { 1 - q } { 2 } \Big ) ^ { k / p _ k } , b = \Big ( \frac { q } { 2 } \Big ) ^ { k / p _ k } . \end{align*}


\begin{align*} x _ { i j } : = x _ { i i - 1 } x _ { i - 1 i - 2 } \cdots x _ { j + 1 j } \end{align*}


\begin{align*} s _ 1 ( z ) : = \mathcal { C } ( s J ) ( z ) s _ 2 ( z ) : = \mathcal { C } ( s J ) ^ { H } ( z ) , \end{align*}


\begin{align*} & ( \nexists \ , x \in \mathbb { R } ^ n ) \ \ s . t . \\ & f ( x ) < 0 \\ & g ( x ) \le 0 \end{align*}


\begin{align*} O _ { n , n } ( R ) \subseteq O _ { n + 1 , n + 1 } ( R ) : A \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & A \end{pmatrix} . \end{align*}


\begin{align*} p _ 0 = 1 - 2 \exp \left ( \frac { a } { 2 } ( 2 \epsilon _ f + \epsilon _ g ^ { 3 / 2 } - r ) \right ) . \end{align*}


\begin{align*} V _ n ( \varphi ) = V ( \varphi ) - \tilde { V } _ n ( \varphi ) \leq - \eta _ 0 + \norm { \tilde { V } _ n ( \varphi ) } , \end{align*}


\begin{align*} ( I - F ) \Psi ^ - _ 1 = 0 \Longleftrightarrow F \Psi ^ - _ 1 = \Psi ^ - _ 1 . \end{align*}


\begin{align*} \partial _ t w + ( \mathbf { v } \cdot \nabla ) w = 0 . \end{align*}


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


\begin{align*} D = m \ , q _ 0 - \sum _ { j = 1 } ^ s m _ j q _ j \end{align*}


\begin{align*} W ( r ) = - f \frac { d } { d r } \log \psi _ 0 ( r ) - \frac { 1 } { 2 } \frac { d f } { d r } . \end{align*}


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


\begin{align*} \langle \mu _ 1 , \alpha \rangle : = \Phi ^ + ( - y ^ { - 1 } \alpha ) , \langle \mu _ 2 , \alpha \rangle : = \Phi ^ + ( x \alpha ) . \end{align*}


\begin{align*} F ^ L = \C [ z _ { s , k } | 1 \leq s \leq m , 1 \leq k \leq L ] \otimes C ^ { \C } ( r _ { t , k } , c _ k | 1 \leq t \leq n , 1 \leq k \leq L ) . \end{align*}


\begin{align*} \gamma _ { t - 1 } \alpha _ t ^ 2 = ( 1 - \alpha _ t ) \alpha _ { t - 1 } ^ 2 \gamma _ t + \mu \alpha _ t \gamma _ t \gamma _ { t - 1 } \leq ( 1 - \alpha _ t ) \alpha _ { t - 1 } ^ 2 \gamma _ t + \alpha _ t \gamma _ t \alpha _ { t - 1 } ^ 2 = \gamma _ t \alpha _ { t - 1 } ^ 2 , \end{align*}


\begin{align*} & \Delta _ j \Delta _ k f = 0 , ~ ~ ~ | j - k | \geq 2 , \\ & \Delta _ j ( S _ { k - 1 } f \Delta _ k f ) = 0 , ~ ~ ~ | j - k | \geq 5 . \end{align*}


\begin{align*} \tilde \Sigma = \{ x \in M \mid { \rm V o l } ( A \cap B _ x ( r ) ) = { \rm V o l } ( B \cap B _ x ( r ) ) \} , \end{align*}


\begin{align*} \exists \ r = r ( \theta , p ) \in [ 1 , p ) , \ \exists \ v = \ v ( \theta , p , r ) < \infty . \end{align*}


\begin{align*} a ( U ^ { - 1 } Z V ) = a ( Z ) \end{align*}


\begin{align*} \mathcal { T } : = \bigcup _ { n \in S ' } \{ n \} \times [ t ( n ) , t \left ( \mathrm { s u c c } \ , ( n ) \right ) ) \subset S \times \mathbb { R } _ { \geq 0 } , \end{align*}


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


\begin{align*} 2 \pi i f ( z ) = - \int _ { p - i e ^ { i \alpha } [ 0 , + \infty ) } \left ( \int _ { e ^ { - i \alpha } [ 0 , + \infty ) } f ( \zeta ) e ^ { \omega \zeta } d \zeta \right ) e ^ { - \omega z } d \omega + \\ + \int _ { p + i e ^ { - i \alpha } [ 0 , + \infty ) } \left ( \int _ { e ^ { i \alpha } [ 0 , + \infty ) } f ( \zeta ) e ^ { \omega \zeta } d \zeta \right ) e ^ { - \omega z } d \omega . \end{align*}


\begin{align*} G ( s ^ * , s ^ * ) = \dfrac { 1 } { \Gamma ( \alpha ) } \left ( \frac { ( b - a ) ( \alpha - 1 ) } { 2 \alpha - 2 - \beta } \right ) ^ { \alpha - 1 } \left ( \frac { \alpha - 1 - \beta } { 2 \alpha - 2 - \beta } \right ) ^ { \alpha - 1 - \beta } . \end{align*}


\begin{align*} \widetilde { f } ( \lambda , b ) = \pi _ \lambda ( f ) Y _ 0 ( b ) , \ , \ , \ , \pi _ \lambda ( f ) = \int _ G f ( g ) \pi _ \lambda ( g ) d g . \end{align*}


\begin{align*} ( \varinjlim _ { n } X _ { H ' , n } \widehat { \otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) ) ^ { R G - l a } = 0 . \end{align*}


\begin{align*} q ^ \perp = 1 \oplus 0 \oplus 1 \oplus \begin{pmatrix} b ^ 2 & - a b \\ - a b & a ^ 2 \end{pmatrix} . \end{align*}


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


\begin{align*} q _ { n , m } \cdot q _ { l , n } = r ^ { l } _ { n , m } \cdot q _ { l , m } \end{align*}


\begin{align*} \int _ { - 1 } ^ 1 d x T _ n ( x ) = \int _ 0 ^ \pi d \theta \sin \theta \cos ( n \theta ) = \end{align*}


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


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


\begin{align*} \begin{aligned} \Delta u _ 1 \wedge \Delta u _ 2 \wedge \dots \wedge \Delta u _ n & = d _ 0 ( d _ 1 u _ 1 \wedge \Delta u _ 2 \wedge \dots \wedge \Delta u _ n ) = - d _ 1 ( d _ 0 u _ 1 \wedge \Delta u _ 2 \wedge \dots \wedge \Delta u _ n ) \\ & = d _ 0 d _ 1 ( u _ 1 \Delta u _ 2 \wedge \dots \wedge \Delta u _ n ) = \Delta ( u _ 1 \Delta u _ 2 \wedge \dots \wedge \Delta u _ n ) . \end{aligned} \end{align*}


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


\begin{align*} \begin{aligned} 1 - \cos S _ c ( x , y ) = \frac { 2 | x - y | } { | x + y | + | x - y | } \ , . \end{aligned} \end{align*}


\begin{align*} \hat A = A [ [ T ] ] / ( T - x ) \end{align*}


\begin{align*} a s _ { \alpha } ( x , x , y ) = a s _ { \alpha } ( y , x , x ) = 0 , \end{align*}


\begin{align*} \mu _ t = j _ t - \frac { L } { 2 } \medspace ( 1 \leq t \leq n ) . \end{align*}


\begin{align*} \prescript L { } { } \ell { } ^ R ( x , w ^ { - 1 } \alpha ) = \prescript L { } { } \ell { } ^ R ( x , v ( w v ) ^ { - 1 } \alpha ) \leq 0 \end{align*}


\begin{align*} \phi ^ { N , i } ( t , \boldsymbol { x } ) = \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) ~ ~ ~ ~ \nu ^ { N , i } _ { \boldsymbol { x } } = \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } x ^ j , \end{align*}


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


\begin{align*} \begin{aligned} a _ { k , \delta } ( x _ a ) & : = 2 \int _ { 0 } ^ { 1 } y _ \delta ( x _ a , t ) \cos ( k 2 \pi t ) \ , d t \\ b _ { k , \delta } ( x _ a ) & : = 2 \int _ { 0 } ^ { 1 } y _ \delta ( x _ a , t ) \sin ( k 2 \pi t ) \ , d t \ , . \end{aligned} \end{align*}


\begin{align*} \overline { d ^ 2 } = \sum \limits _ { m = 1 } ^ { N } { d _ m ^ 2 } \end{align*}


\begin{align*} N \circ \alpha = \alpha \circ N , \end{align*}


\begin{align*} \norm { \theta } _ { \mathcal { L } ^ 2 , { G } } ^ 2 = \beta r \norm { x } _ { \mathcal { L } ^ 2 } ^ 2 + \beta \norm { y } _ { \mathcal { L } ^ 2 } ^ 2 + \frac { 1 } { \beta } \norm { \lambda } _ { \mathcal { L } ^ 2 } ^ 2 , \end{align*}


\begin{align*} L = a _ 0 x + a _ 1 x ^ q + a _ 2 x ^ { q ^ 2 } + \cdots + a _ n x ^ { q ^ n } \in F [ x ] . \end{align*}


\begin{align*} G _ n ( x , z ) d z : = \P _ x ( S ( n ) \in d z , \tau > n ) , x , z \in W ^ d . \end{align*}


\begin{align*} \Psi ( t , \cdot ) : = - \Delta ^ { - 1 } \omega ( t , \cdot ) \end{align*}


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


\begin{align*} \phi ^ q _ i = 2 \pi q i / M + \alpha , i = 1 , \dots , M , \end{align*}


\begin{align*} \Vert M _ { \widetilde { m } \vert _ { \Lambda _ N \times \Lambda _ N \times \Lambda _ N } ( \ : \cdot \ : , 0 \ : \cdot \ : ) } : S _ { 1 } ^ { N + 1 } \rightarrow S _ { 1 } ^ { N + 1 } \Vert \leq \Vert M _ { \widetilde { m } \vert _ { \Lambda _ N \times \Lambda _ N \times \Lambda _ N } } : S _ { p _ 1 } ^ { N + 1 } \times S _ { p _ 2 } ^ { N + 1 } \rightarrow S _ { p } ^ { N + 1 } \Vert . \end{align*}


\begin{align*} \overline { \beta } _ { L } = \sup \left \{ \eta : \lim \sup _ { r \rightarrow \infty } r ^ { \eta } \overline { h } \left ( r \right ) = 0 \right \} = 0 \end{align*}


\begin{align*} d ( p , g ) \leq L d ( p , \pi ^ { - 1 } ( \pi ( g ) ) ) + M = L d ( p , \pi ^ { - 1 } ( \pi ( q ) ) ) + M \leq L d ( p , q ) + M < r + M , \end{align*}


\begin{align*} ( \sigma , \sigma ( 1 ) , \ldots , \sigma ( r - 1 ) ) = [ \sigma , \sigma ( r - 1 ) ] = \Delta \end{align*}


\begin{align*} \left | \begin{array} { c c } x ^ 2 + x y + y ^ 2 = n \ ; \ ; \ , \\ x '^ 2 + x ' y ' + y '^ 2 = n \end{array} \right . \ , , \end{align*}


\begin{align*} P _ { \xi } \circ Q _ { \xi } = \sum _ { k \geq 0 } \frac { 1 } { k ! } \ , \frac { \partial ^ { k } P _ { \xi } } { \partial \xi ^ { k } } \ , \dd ^ { k } Q _ { \xi } \ ; , \end{align*}


\begin{align*} \begin{aligned} & F ( s ) \lesssim \frac { 1 } { s ^ { \tau } } , 0 < \tau \leq \frac { 3 } { 2 } , \\ & F ^ { ( k ) } ( s ) \lesssim \frac { 1 } { s ^ { k + \tau } } , 0 \leq \tau \leq \frac { 3 } { 2 } , \ k \in \mathbb { N } . \end{aligned} \end{align*}


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


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


\begin{align*} ( w \varepsilon ^ \mu ) ( \alpha , k ) : = ( w \alpha , k - \langle \mu , \alpha \rangle ) . \end{align*}


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


\begin{align*} X _ S = \boxtimes _ { i \in S } X _ i . \end{align*}


\begin{align*} | | R ^ { \nabla } | | ^ 2 _ \infty = \sup _ { x \in M } | | R ^ { \nabla } | | ^ 2 _ x \ , . \end{align*}


\begin{align*} \int _ { \mathbb { R } } s V s ^ { H } d z = 0 \Longrightarrow \int _ { \mathbb { R } } { \rm R e } ( s V s ^ { H } ) \mathrm { d } z = 0 , \end{align*}


\begin{align*} \pi ( s ) = \lim _ { n \to \infty } w _ n ^ { ( i ) } ( s ) = - s \mathcal { A } ' _ { \delta } ( q s ) . \end{align*}


\begin{align*} \tilde { I _ j } = \{ i \in [ n / \ell ] : \ : V _ i \subseteq I _ j \} . \end{align*}


\begin{align*} & b _ k ^ T y = 0 \ \ k \in \{ 0 , . . . , m \} \\ & \begin{bmatrix} b ^ 0 _ 1 & \cdots & b ^ 0 _ n \\ \vdots & \ddots & \vdots \\ b ^ m _ 1 & \cdots & b ^ m _ n \end{bmatrix} \begin{bmatrix} y _ 1 \\ \vdots \\ y _ n \end{bmatrix} = \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} \end{align*}


\begin{align*} 0 \leq \beta _ t = \mu \gamma _ t \alpha _ t ^ { - 1 } \leq \sqrt { \mu \gamma _ t } \leq 1 . \end{align*}


\begin{align*} & L _ 4 \colon \mathbf { Z } = 2 \eta \mathbf { X } - \iota \mathbf { W } = 0 , & L ' _ 4 \colon \mathbf { W } - \rho \mathbf { Z } = 2 \eta ' \mathbf { X } - \iota ' \mathbf { W } = 0 , \end{align*}


\begin{align*} \frac { \lVert y - x \rVert _ 2 } { \lVert x \rVert _ 2 } = O \big ( \delta \ , \kappa ( A ) \big ) \end{align*}


\begin{align*} \abs { \eta ( y ) - \eta ( x ) } ^ 2 & = \abs { \int _ x ^ y \partial \eta ( z ) \ \d z } ^ 2 \le \int _ \S \partial \eta ( z ) ^ 2 \ \d z = \norm { \partial \eta } _ { L ^ 2 } ^ 2 , \end{align*}


\begin{align*} A = & \left \lbrace \left | \langle B _ 1 , B _ 2 \rangle \sum _ { i = 1 } ^ n \int _ { ( i - 1 ) \Delta _ n } ^ { i \Delta _ n } \Delta _ n ^ { - \frac 1 2 } \langle ( \mathcal S ( i \Delta _ n - s ) - I ) B _ 1 , B _ 2 \rangle d s \right | > c \right \rbrace . \end{align*}


\begin{align*} \sum _ { \pi \in \mathcal S _ d } ( \pi ) \P _ x ( \widehat S _ i ( n + d - \pi ( i ) ) \in d z _ { \pi ( i ) } , i = 1 , \ldots , d , \widehat \tau < \infty ) = 0 . \end{align*}


\begin{align*} \left \langle \left ( x , C _ { \xi } \right ) , \left ( y , C _ { \eta } \right ) \right \rangle = \tfrac { 1 } { 4 } \left ( \left \langle x , \eta \right \rangle + \left \langle y , \xi \right \rangle \right ) \end{align*}


\begin{align*} \partial ^ \square \big ( ( c + t w ) \circ \theta _ t ^ { - 1 } \big ) _ { | t = 0 } = \frac { \mathrm { d } } { \mathrm { d } t } ( c + t w ) _ { | t = 0 } - \partial _ t \theta _ t \cdot \nabla _ { \Gamma _ t } \big ( ( c + t w ) \circ \theta _ t ^ { - 1 } \big ) _ { | t = 0 } = w - V \nu \cdot \nabla _ \Sigma c = w \end{align*}


\begin{align*} \Psi = \rho ( x ' ) - x _ n - \theta | x ' | ^ 2 + K x _ n ^ 2 . \end{align*}


\begin{align*} & [ P ( u _ t ( t ) ) ] _ t + A u ( t ) + B ( t , x , u _ t ( t ) ) = 0 , t > 0 , \ ; \ ; x \in \mathbb { R } ^ n , \end{align*}


\begin{align*} \mathcal { M } _ { f _ { 1 , \infty } } ( N , \alpha + \delta , 6 \epsilon , \bigcup _ { n = N } ^ { \infty } Z _ { n , n ^ { - 2 } t } , \psi ) \leq \sum _ { n = N } ^ { \infty } \mathcal { M } _ { f _ { 1 , \infty } } ( N , \alpha + \delta , 6 \epsilon , Z _ { n , n ^ { - 2 } t } , \psi ) . \end{align*}


\begin{align*} F ( s , x ) = \left | \frac { \zeta ' } { \zeta } ( s ) + \sum _ { n < x ^ 2 } \frac { \Lambda _ x ( n ) } { n ^ s } \right | \leq | Z _ 1 | + | Z _ 2 | + | Z _ 3 | \end{align*}


\begin{align*} \begin{gathered} ( x _ { 1 0 } , x _ { 1 1 } ) \in \omega ( ( x _ { 0 0 } , x _ { 0 1 } ) , T \times T ) \quad \bigl ( x _ { 0 1 } , x _ { 1 1 } \bigr ) \in \omega \bigl ( ( x _ { 0 0 } , x _ { 1 0 } ) , T \times T \bigr ) . \end{gathered} \end{align*}


\begin{align*} T _ { W , \widetilde { K } } ( \mu ^ t ) & = \frac 1 2 \int _ { [ 0 , 1 ] \times \R } \int _ { [ 0 , 1 ] \times \R } W ( u , v ) K ( x - y ) \mu ^ t ( d u , d x ) \mu ^ t ( d v , d y ) \\ & = \frac 1 2 \int _ { [ 0 , 1 ] \times \R } \int _ { [ 0 , 1 ] \times \R } W ( u , v ) K \big ( ( 1 - t ) ( x - y ) + t ( T _ u ( x ) - T _ u ( y ) ) \big ) \mu ^ 0 ( d u , d x ) \mu ^ 0 ( d v , d y ) . \end{align*}


\begin{align*} \xi [ X ] \stackrel { d e f } { = } \int _ X \xi ( x ) \ \mu ( d x ) \ - \end{align*}


\begin{align*} \mathcal { T } _ b ^ * ( f ) ( x ) = \sup _ { t > 0 } \Big | \int _ { \mathbb { R } ^ n } k _ t ( x , y ) ( b ( x ) - b ( y ) ) f ( y ) d y \Big | . \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x & \leq C \int _ { B _ { 8 \rho } \setminus B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x \\ & \rho \leq 1 / 8 \varepsilon \leq \varepsilon _ 0 . \end{align*}


\begin{align*} \left \{ \begin{aligned} d x _ t ^ { \xi , \alpha } = & ~ [ A x _ t ^ { \xi , \alpha } + B \alpha _ t + f ( \nu _ t ) + b ( \mu _ t ) ] d t + \sigma d W _ t + \sigma _ 0 d W ^ 0 _ t , \\ x _ 0 ^ { \xi , \alpha } = & ~ \xi , \end{aligned} \right . \end{align*}


\begin{align*} \frac { d } { d r } u ( \delta _ { r } g ) = \frac { 1 } { r } Z u ( \delta _ { r } g ) . \end{align*}


\begin{align*} \bar { \Gamma } _ { j k } = ( - 1 ) ^ { j + k } \bar { \Gamma } ^ \flat _ { j k } , \forall ( j , k ) \in \{ 0 , \dots , n - 1 \} ^ 2 . \end{align*}


\begin{align*} d _ n ^ { \mathrm { a v g } } ( \{ \xi _ 1 , \ldots , \xi _ N \} , L _ p ( \Omega ) ) _ p = N ^ { - 1 / p } d _ n ^ { \mathrm { a v g } } ( ( \xi _ 1 , \ldots , \xi _ N ) , \ell _ p ^ N ) _ p . \end{align*}


\begin{align*} \mu _ k ( G ) : = \frac { \deg ( G ) + k } { ( G ) } . \end{align*}


\begin{gather*} \phi ^ + ( x ) : = \lim _ { n \rightarrow \infty } \frac 1 n \sum ^ { n - 1 } _ { k = 0 } \phi ( f ^ k ( x ) ) , \\ \phi ^ - ( x ) : = \lim _ { n \rightarrow \infty } \frac 1 n \sum ^ { n - 1 } _ { k = 0 } \phi ( f ^ { - k } ( x ) ) , \\ \bar \phi ( x ) : = \lim _ { n \rightarrow \infty } \frac 1 { 2 n - 1 } \sum ^ { n - 1 } _ { k = - ( n - 1 ) } \phi ( f ^ k ( x ) ) . \end{gather*}


\begin{align*} I = \varinjlim _ { H \subset G } I ^ H . \end{align*}


\begin{align*} \vec { b } = & \sum _ { i = j } ^ { n - 1 } b _ i \vec { e } _ i \end{align*}


\begin{align*} E q _ k = \left [ \begin{array} { c c c } \ 1 _ { n - 1 } ^ { ' } u _ k \\ 2 ( J _ { n - 1 } - I _ { n - 1 } ) u _ k \\ S ^ { ' } u _ k \end{array} \right ] . \end{align*}


\begin{align*} \Gamma _ 1 ( X ) = \sigma X + \mathcal { O } \big ( z X ^ \varepsilon \big ) \ , , \end{align*}


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


\begin{align*} \pi ( x ) = \frac { f ( x , 1 ) } { g ( x , 1 ) } = \frac { s } { r } \end{align*}


\begin{align*} \beta _ 1 ( u ) = 0 & \Longrightarrow \beta _ 1 ( u _ { \varepsilon } ) = 0 , \\ \beta _ 1 ( u ) = \beta ' _ 3 ( u ; Q ) = 0 & \Longrightarrow \beta ' _ 3 ( u _ { \varepsilon } ; Q ) = 0 . \end{align*}


\begin{align*} m _ { n } ( x ) = \frac { \nu _ n ( A _ { h , j } ) } { \mu _ n ( A _ { h , j } ) } \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty b _ 3 \left ( \frac { m n - 1 } { 1 2 } \right ) q ^ n \in S _ { 3 m - 3 } ( \Gamma _ 0 ( 4 3 2 ) , \chi _ { 1 2 } ) _ m . \end{align*}


\begin{align*} p \Delta \left ( H - \frac { \mu } { 2 } \right ) ^ { p - 1 } + 2 p \left ( H - \frac { \mu } { 2 } \right ) ^ { p - 1 } \left ( 2 H ^ 2 - K \right ) - 4 H \left ( \left ( H - \frac { \mu } { 2 } \right ) ^ { p } + \varsigma \right ) = 0 , \end{align*}


\begin{align*} H ( t , x , y ) = \mathbb { P } ( t , x , y ) . \end{align*}


\begin{align*} X = N \oplus X _ 0 \qquad Z = R \oplus Z _ 0 , \end{align*}


\begin{align*} P V _ { \delta , \hat { x } , \hat { q } , z } ( x ) = V _ { \delta , \hat { x } , \hat { q } , z } ( x ) - \frac { \hat { q } } { \ln \frac { R } { s _ \delta } } g _ { \hat { x } } ( T _ { \hat { x } } x , T _ { \hat { x } } z ) , \ \ \forall x \in \Omega , \end{align*}


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


\begin{align*} k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } } , \lambda ^ { N , 1 } _ { \boldsymbol { \varphi } } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \varphi } } ) : = \rho _ i ^ N \big ( - R ^ { - 1 } B [ P _ t \nu ^ { N , 1 } _ { \boldsymbol { x } } + \lambda ^ { N , 1 } _ { \boldsymbol { \varphi } } ] , \ldots , - R ^ { - 1 } B [ P _ t \nu ^ { N , N } _ { \boldsymbol { x } } + \lambda ^ { N , N } _ { \boldsymbol { \varphi } } ] \big ) . \end{align*}


\begin{align*} ( D ^ t ) ^ N ( \psi ) = \sum _ { \substack { \alpha = ( \alpha _ 0 , . . . , \alpha _ k ) \\ | \alpha | = N } } C _ { N , \alpha } \frac { \partial ^ { \alpha _ 0 } \psi \partial ^ { \alpha _ 1 + 1 } \phi \partial ^ { \alpha _ 2 + 1 } \phi . . . \partial ^ { \alpha _ k + 1 } \phi } { \lambda ^ N ( \phi ' ) ^ { N + k } } . \end{align*}


\begin{align*} a _ { n - i } ( \gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \end{align*}


\begin{align*} L ( p , \pi ^ h ) = L ( p , \pi _ 1 ^ h ) = L ( p , \pi _ 2 ^ h ) = L ( p , \bar { \pi } ^ h ) . \end{align*}


\begin{align*} 2 \bar \pi = \sum _ { v \in V _ 1 } \pi _ v . \end{align*}


\begin{align*} \hat W _ r ( q + k ) - \hat W _ r ( 2 q ) & \le \frac { 2 } { \pi } \left ( \frac { 1 } { q + k } + \frac { 1 } { 2 q } \right ) \\ & \le \frac { 4 } { \pi q } \\ & \le 4 r - \frac { 2 } { \pi } \sin ( 2 \pi r ) \\ & = \hat W _ r ( 0 ) - \hat W _ r ( 1 ) . \end{align*}


\begin{align*} ( \nabla _ X \beta ) ( Y , Z ) = ( \nabla _ Y \beta ) ( X , Z ) . \end{align*}


\begin{align*} \begin{cases} \delta _ { + } ( z ) = ( 1 + \bar { r } _ 1 ( z ) r _ 2 ( z ) ) \delta _ { - } ( z ) , z \in \mathbb { R } \\ \delta _ { \pm } ( z ) \rightarrow 1 \quad | z | \rightarrow \infty . \end{cases} \end{align*}


\begin{align*} \overline { h } \left ( r \right ) > \int _ { \left \vert x \right \vert > r } \nu _ { \alpha } ^ { \prime } \left ( d x \right ) = \frac { 1 } { \alpha \log ^ { \alpha } \left ( 1 + r \right ) } \end{align*}


\begin{align*} M / ( x _ 1 ^ n , \dots , x _ k ^ n ) = ( ( M / x _ 1 ^ n ) / x _ 2 ^ n ) / \dots / x _ k ^ n \end{align*}


\begin{align*} S _ 0 = \{ s _ { | i } \mid s \in S , i \in \mathbb { Z } \} = \left \{ ( s _ j ) _ { | i } \mid 1 \le j \le d i \in \mathbb { Z } \right \} \subseteq H = H _ 0 , \end{align*}


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


\begin{align*} x ' : = w ' \varepsilon ^ { \mu ' } : = x r _ { v \alpha , \Phi ^ + ( - v \alpha ) } \in \widetilde W \end{align*}


\begin{align*} \sum _ { \gamma _ { \chi } } \frac { \frac { 1 } { 2 } } { \frac { 1 } { 4 } + \gamma _ { \chi } ^ 2 } & \leq \left ( \dfrac { 1 } { 2 } + \dfrac { 2 } { \log q - 2 } \right ) \log { \dfrac { q } { \pi } } + 2 \left ( 1 - \dfrac { 2 } { \log q } \right ) ^ { - 2 } \log \log q \leq \dfrac { 1 } { 2 } \log { \dfrac { q } { \pi } } + 2 . 6 \log \log q . \end{align*}


\begin{align*} L _ { q } [ f ( a t ) ] = \frac { 1 } { a } F _ { q } ( s / a ) . \end{align*}


\begin{align*} \varphi ( t ) : = \mathrm { K L } \left ( \rho _ { t } \mid \pi \right ) \end{align*}


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


\begin{align*} F ( x ) = \frac { 2 } { \pi } \arcsin \left \{ \exp \left ( - \frac { \theta } { 2 x ^ 2 } \right ) \right \} . \end{align*}


\begin{align*} J ( u ) : = \int _ { \R ^ N } \left ( I _ \alpha \ast | u | ^ { p } \right ) | u | ^ { p } d x \end{align*}


\begin{align*} X ^ \dagger = N ^ \dagger \oplus X _ 0 ^ \dagger Z ^ \dagger = R ^ \dagger \oplus Z _ 0 ^ \dagger , \end{align*}


\begin{align*} A R ( G ) = \begin{cases} A \rtimes \{ I \} & d \\ A \rtimes \langle - I \rangle & d \end{cases} \end{align*}


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


\begin{align*} \mu _ { \rm e f f } ^ 2 ( r , \theta ) = \mu _ r ^ 2 ( r ) \sin ^ { 2 q } \theta + \mu _ 0 ^ 2 ( r ) \end{align*}


\begin{align*} A = K \langle T _ 1 ^ { 1 / p ^ \infty } , \dots , T _ n ^ { 1 / p ^ \infty } \rangle , A ^ + = K ^ + + A ^ { \circ \circ } . \end{align*}


\begin{align*} \sigma ( \lambda \alpha ) = \sigma ( \lambda ) \sigma ( \alpha ) = \lambda \sigma ( \alpha ) . \end{align*}


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


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


\begin{align*} d G = \sqrt { \tau _ 0 } / 4 \left \{ \left ( \left ( 1 + 4 Q _ 0 / \tau _ 0 \right ) d z + \left ( 1 + 4 \overline { Q _ 0 } / \tau _ 0 \right ) d \overline { z } \right ) u _ 1 \right . \\ \left . - i \left ( \left ( 1 - 4 Q _ 0 / \tau _ 0 \right ) d z - \left ( 1 - 4 \overline { Q _ 0 } / \tau _ 0 \right ) d \overline { z } \right ) u _ 2 \right \} \end{align*}


\begin{align*} \begin{array} { l l l } \nu _ { \alpha } \left ( d x \right ) = \frac { 1 } { \left ( 1 + \left \vert x \right \vert \right ) \log ^ { 1 + \alpha } \left ( 1 + \left \vert x \right \vert \right ) } d x & & 0 < \alpha < 2 \end{array} \end{align*}


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


\begin{align*} \mathrm { D E } ( \Sigma , c ) ( V , w ) = \big \langle \mathrm { E } ( \Sigma , c ) , ( V , w ) \big \rangle \end{align*}


\begin{align*} ( x \cdot y ) \diamond \alpha ( z ) & = \varepsilon ( y , z ) ( x \diamond z ) \cdot \alpha ( y ) , \\ ( x \diamond y ) \cdot \alpha ( z ) - \alpha ( x ) \diamond ( y \cdot z ) & = \varepsilon ( x , y ) \big ( ( y \diamond x ) \cdot \alpha ( z ) - \alpha ( y ) \diamond ( x \cdot z ) \big ) . \end{align*}


\begin{align*} ( \sin S _ c ^ l - \sin S _ c ) \prod _ { m = 1 } ^ { 2 l - 1 } \partial _ { x } ^ { a _ m } S _ c , ( \cos S _ c ^ l - \cos S _ c ) \prod _ { m = 1 } ^ { 2 l } \partial _ { x } ^ { b _ m } S _ c , l \ge 0 , \end{align*}


\begin{align*} \int _ { \Omega } - \delta _ N ^ 2 \left ( ( K ( x ) - K ( z _ { N , i } ) ) \nabla \frac { \partial V _ { \delta _ N , Z _ N , i } } { \partial z _ { i , h } } \right ) u _ N = O \left ( \frac { \varepsilon _ N ^ 2 } { | \ln \varepsilon _ N | ^ { p - 1 } } \right ) . \end{align*}


\begin{align*} \nabla _ X \Psi _ 2 = - \frac { 1 } { 2 } \left ( X - \langle X , T _ 1 \rangle ( T _ 1 + f _ 1 ) - \langle X , T _ 2 \rangle ( T _ 2 + f _ 2 ) \right ) \cdot \Psi _ 1 + \frac { 1 } { 2 } S ( X ) \cdot \Psi _ 2 , \end{align*}


\begin{align*} N e w t o n ( F ) = C o n v ( \bigcup \limits _ { i = 0 } ^ l N e w t o n ( s _ { \lambda ^ i } ) ) . \end{align*}


\begin{align*} \phi ( p ) = \arctan ( p ^ { 2 } ) \qquad \mbox { o r } \phi ( p ) = ( 1 + p ^ { 2 } ) ^ { \alpha } \quad \alpha \in ( 0 , 1 / 2 ) , \end{align*}


\begin{align*} G _ { \xi _ { \Delta } } : \Omega \to [ \xi _ { \Delta } ] : = \{ \omega \in \Omega \colon \omega _ \Delta = \xi _ \Delta \} , \eta \mapsto \xi _ { \Delta } \eta _ { \Delta ^ c } . \end{align*}


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


\begin{align*} \mathbb { E } _ { \mathbb { P } } \left [ \left | \sum _ { j = 2 } ^ { n } \left ( b ( t , X ^ { 1 , \infty } , X ^ { j , \infty } ) - \left \langle b \left ( t , X ^ { 1 , \infty } , \cdot \right ) , \mu \right \rangle \right ) \right | ^ { 2 p } \right ] \leq p ! \left ( ( n - 1 ) \beta \right ) ^ { p } . \end{align*}


\begin{align*} \big ( D _ 0 ( t , x ) ^ { \ast } \big ) _ { i j } : = \int _ { Y ^ { \ast } } D _ 0 ( t , x , y ) \big [ \nabla _ y w _ i ( t , x , y ) + e _ i \big ] \cdot \big [ \nabla _ y w _ j ( t , x , y ) + e _ j \big ] d y , \end{align*}


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


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


\begin{align*} \| A _ j x \| ^ 2 - \big | \langle x | A _ j x \rangle \big | ^ 2 & = \| ( A _ j - z _ j I ) x \| ^ 2 - \big | \langle x | ( A _ j - z _ j I ) x \rangle \big | ^ 2 \\ & \leq \| ( A _ j - z _ j I ) x \| ^ 2 . \end{align*}


\begin{gather*} \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \ge \inf _ { Q \in \mathcal { U } } Q \Bigl ( \oplus _ { i = 1 } ^ n f _ i \Bigr ) \quad \quad f _ i \in B ( \Omega _ i , \mathcal { A } _ i ) i \in I . \end{gather*}


\begin{align*} \sum _ { l = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M \sum _ { q = 1 } ^ M a _ q ^ 2 \hat { \phi } ^ { \left ( l , q \right ) ^ * } \hat { \phi } ^ { \left ( j , q \right ) } = 2 \sum _ { l = 1 } ^ { M - 1 } \sum _ { j = i + 1 } ^ { M } \sum _ { q = 1 } ^ M a _ q ^ 2 \Re \left [ \hat { \phi } ^ { \left ( l , q \right ) ^ * } \hat { \phi } ^ { \left ( j , q \right ) } \right ] . \end{align*}


\begin{align*} A _ { k j } = \begin{cases} \sum _ { l = 1 } ^ { n _ 1 } \Psi _ { k l } \bar \Psi _ { j l } , & \lambda _ k = \lambda _ j , \\ 0 , & . \end{cases} \end{align*}


\begin{align*} \mathbf { H } _ { \mathrm { a l l } , k } = \left \{ \begin{array} { c c } \mathbf { H } _ { \mathrm { d } , k } + \mathbf { H } _ { k } \mathbf { \Phi } _ { \mathrm { r } } \mathbf { G } , & k \in \mathcal { K } _ { \mathrm { r } } , \\ \mathbf { H } _ { \mathrm { d } , k } + \mathbf { H } _ { k } \mathbf { \Phi } _ { \mathrm { t } } \mathbf { G } , & k \in \mathcal { K } _ { \mathrm { t } } . \end{array} \right . \end{align*}


\begin{align*} \big ( \bar \pi \bar p _ t \big ) \bar p _ T = \bar \pi \bar p _ t , \end{align*}


\begin{align*} \theta _ { \rho ^ \varepsilon } ( 0 , \cdot ) \circ F ^ { - 1 } : F ( U _ i ) \rightarrow \R ^ 2 , \phantom { x x } p \mapsto p + \rho _ 0 ^ \varepsilon \nu _ \Sigma \big ( F ^ { - 1 } ( p ) \big ) = p + \rho _ 0 ^ \varepsilon \nu _ { F ( U _ i ) } ( p ) \end{align*}


\begin{align*} v ( t ) = \sum _ { \mathbf { k } = 1 } ^ { \infty } \widehat { v } _ { \mathbf { k } } { \rm e } ^ { { \rm i } \mathbf { k } t } \sum _ { \mathbf { k } = 1 } ^ { \infty } | \widehat { v } _ { \mathbf { k } } | < \infty , \end{align*}


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


\begin{align*} c _ 1 ( q , 2 q , p ^ 0 ( 0 ) ) & = \frac 1 4 \left ( \frac { 1 } { - q } f ( - q r ) + \frac { 1 } { 3 q } f ( 3 q r ) - \frac { 2 } { q } f ( q r ) - H ( q , r ) \frac 1 q f ( q r ) \right ) \\ & = \frac 1 q g ( q r ) , \end{align*}


\begin{align*} G ( \beta , 1 ) = H ( \beta , 1 ) \circ [ g _ { \beta } ] _ { 1 } = \beta \circ \widetilde { \beta ( 1 ) } \simeq i d _ { X _ { U ' } ^ { I } } \end{align*}


\begin{align*} [ r ^ \beta ] C _ k ( q , r , t ) = \frac { C _ k ( t ) ^ { \beta + 1 } - q ^ { \beta + 1 } } { 1 - q + q ^ k t } = \left ( C _ k ( t ) ^ { \beta + 1 } - q ^ { \beta + 1 } \right ) \left ( 1 + ( q - q ^ k t ) + ( q - q ^ k t ) ^ 2 + \hdots \right ) . \end{align*}


\begin{align*} \sum _ { \xi _ { \Delta } \in \Omega _ { \Delta } } \int _ { \Omega } c _ { \Delta } ( \omega , \xi _ { \Delta } ) f ( \omega ) g ( \xi _ { \Delta } \omega _ { \Delta ^ c } ) \mu ( d \omega ) = \sum _ { \xi _ { \Delta } \in \Omega _ { \Delta } } \int _ { \Omega } \hat { c } _ { \Delta } ( \omega , \xi _ { \Delta } ) f ( \xi _ { \Delta } \omega _ { \Delta ^ c } ) g ( \omega ) \mu ( d \omega ) , \end{align*}


\begin{align*} \Phi _ { t } + u \cdot \nabla \Phi = 0 . \end{align*}


\begin{align*} 0 = ( \nabla _ T C _ S ( X _ i ) - C _ S C _ T ( X _ i ) - C _ { \nabla _ T S } ( X _ i ) ) - ( \nabla _ S C _ T ( X _ i ) - C _ T C _ S ( X _ i ) - C _ { \nabla _ S T } ( X _ i ) ) = d \lambda _ i ( T , S ) X _ i . \end{align*}


\begin{align*} h _ { n } = \rho _ p ^ { n - j _ p } h _ { j _ p } + \frac { \rho _ p ( \rho _ p ^ { n - j _ p } - 1 ) } { 2 } . \end{align*}


\begin{align*} \liminf _ { n \to \infty } M _ n ^ { \psi } / n \ge \lim _ { n \to \infty } M _ n ^ { \psi } ( Q ^ n ) / n = T _ { W , \psi } ( \mu ) - I ( \mu ) . \end{align*}


\begin{align*} \mathbf { x } \circ \mathbf { s } = \left ( \mathbf { x } ^ { ( 1 ) } \circ \mathbf { s } ^ { ( 1 ) } , \cdots , \mathbf { x } ^ { ( k ) } \circ \mathbf { s } ^ { ( k ) } \right ) , \forall \mathbf { x } , \mathbf { s } \in \mathcal { R } ^ { n } \end{align*}


\begin{align*} | \phi _ { \mathrm { r } , m } | ^ { 2 } + | \phi _ { \mathrm { t } , m } | ^ { 2 } = 1 , \forall m \in \mathcal { M } . \end{align*}


\begin{align*} \int _ { M } c ( t , p ) \ , \mathrm { d } \mathcal { H } ^ d ( p ) = m \end{align*}


\begin{align*} s _ { m } = - \sum _ { \emptyset \neq I \subset \{ 1 , \dots , m \} } ( - 1 ) ^ { | I | } \langle a _ { I } \rangle \in \mathbb { Z } [ R ^ { * } ] , \end{align*}


\begin{align*} \liminf _ { N \to \infty } \frac { 1 } { N } \sum _ { n = 0 } ^ { N - 1 } \max _ { 1 \leq i < j \leq k } { d } ( T ^ { ( n , [ n \beta + t ] ) } x _ i , T ^ { ( n , [ n \beta + t ] ) } x _ j ) = 0 \end{align*}


\begin{align*} \begin{aligned} b _ { 1 , \delta } ( x ^ \star ) = & \ , 2 \int _ { 0 } ^ { 1 } h ( x ^ \star + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \\ = & \ , 2 \int _ { 0 } ^ { 1 / 2 } h ( x ^ \star + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \\ & - 2 \int _ { 0 } ^ { 1 / 2 } h ( x ^ \star - \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \end{aligned} \end{align*}


\begin{align*} A _ { p , N } : = \sum _ { A = 0 } ^ { 2 n - 1 } \int _ S \left [ v _ p ( d _ 1 v _ { p + N } \wedge \alpha ) _ A \tau ( \mathbf { { n } } ) _ { A 0 } + v _ { p + N } ( d _ 0 v _ p \wedge \alpha ) _ A \tau ( \mathbf { { n } } ) _ { A 1 } \right ] d S , \end{align*}


\begin{align*} \int _ \Omega f ( \eta ) \mu ( d \eta ) \cdot \left ( \sum _ { m = 0 } ^ \infty \int _ \Omega ( \varphi _ n ( r _ m \eta ) - \varphi _ n ( r _ { m + 1 } \eta ) ) \mu ( d \eta ) + \varphi _ n ( \mathbf { 1 } ) \right ) = 0 . \end{align*}


\begin{align*} & \frac { 1 } { 2 } ( V + V ^ H ) = \left ( \begin{array} { c c } 1 + | r ( k ) | ^ 2 & \overline { r ( k ) } e ^ { - 2 i k ^ 2 x } \\ r ( k ) e ^ { 2 i k ^ 2 x } & 1 \end{array} \right ) , \ \ \ k \in \mathbb { R } , \\ & \frac { 1 } { 2 } \left ( V + V ^ H \right ) = \left ( \begin{array} { c c } 1 - | r ( k ) | ^ 2 & 0 \\ 0 & 1 \end{array} \right ) , \ \ \ k \in i \mathbb { R } , \end{align*}


\begin{align*} \mathrm { E } ( X ^ n ) = \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \frac { 1 } { \sigma } \int _ { - \infty } ^ \infty x ^ n \left [ \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \alpha - 1 } \left [ 1 - \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \beta - 1 } \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \mathrm { d } x . \end{align*}


\begin{align*} \left | \begin{array} { c c } x + y = x ' \ , \\ x ' + y ' = y \end{array} \right . . \end{align*}


\begin{align*} s ^ W = s ^ g - 3 \ , \delta ^ g \theta - \frac { 3 } { 2 } \| \theta \| ^ 2 , \end{align*}


\begin{align*} & f _ k ( \Tilde { x } ) \le \lambda f _ k ( x _ v ) + ( 1 - \lambda ) f _ k ( x _ w ) < \lambda v _ k + ( 1 - \lambda ) w _ k . \\ & a _ k \| \Tilde { x } \| ^ 2 + 2 b _ k ^ T \Tilde { x } + c _ k \le a _ k ( \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 ) + 2 b _ k ^ T ( \lambda x _ v + ( 1 - \lambda ) x _ w ) + c _ k . \end{align*}


\begin{align*} ( W \times T \times L ) / D = ( V \times U \times T \times L ) / D \cong V \times K \end{align*}


\begin{align*} \overline { W ^ { s } ( ( x , t ) , T _ { \vec { v } } ) \cap \operatorname { s u p p } \widetilde \mu _ { ( x , t ) } } = \operatorname { s u p p } \widetilde \mu _ { ( x , t ) } . \end{align*}


\begin{align*} \mathbb { E } | \Gamma _ { N } ^ { D } ( b ) \backslash \hat { \Gamma } _ { N } ^ { D , M } ( b ) | \leq \ & \mathbb { E } | \Gamma _ { N } ^ { D } ( b ) \backslash D _ { N } ^ { \varepsilon } | + \mathbb { E } | \Gamma _ { N } ^ { D } ( b ' ) | \\ & + \sum _ { x \in D _ { N } ^ { \varepsilon } } \sum _ { k = k _ { N } } ^ { n ( x ) } \mathbb { P } ( h ^ { D _ { N } } ( x ) - a _ { N } \in [ b , b ' ) , T _ { N , M } ^ { k } ( x ) ) , \end{align*}


\begin{align*} \left \lvert 1 - 4 \mu \left ( \sum _ { l = 1 } ^ { M } \lvert \phi ^ { \left ( l , j \right ) } \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*} 0 & = \int _ \Sigma \Big \langle Q ( h , v ) , \big ( k ^ \intercal , H ' ( k ) \big ) \Big \rangle d \sigma \\ & = \int _ \Sigma \Big \langle \big ( v A + \bar u A ' ( h ) - \nu ( v ) g ^ \intercal , 2 v \big ) , \big ( - 2 \zeta A ' ( h ) , \zeta A \cdot A ' ( h ) \big ) \Big \rangle \ , d \sigma \\ & = - \int _ \Sigma 2 \bar u \zeta | A ' ( h ) | ^ 2 \ , d \sigma \end{align*}


\begin{align*} \Phi ( p _ 0 ) = R { s ( p _ 0 ) } \leq \Phi ( p ) \leq C ( \delta , m _ \delta ) \end{align*}


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


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


\begin{align*} L _ { q } \left [ \int _ { 0 } ^ { \infty } f ( x ) d x \right ] = \frac { 2 - q } { s } L _ { \frac { 1 } { 2 - q } } \left [ f ( t ) \right ] \left ( s ( 2 - q ) \right ) . \end{align*}


\begin{align*} e _ 1 : = \alpha q ^ { d ^ 2 + d - 2 } , \end{align*}


\begin{align*} | \nabla _ { H } u | ^ 2 = \sum _ { i = 1 } ^ m ( X _ i u ) ^ 2 . \end{align*}


\begin{align*} \mathcal { A } _ { \varsigma } = \{ \mathcal { A } _ { \varsigma } ( \tau , \omega ) : \tau \in \R , \omega \in \Omega \} , \end{align*}


\begin{align*} | j _ 1 ( \alpha - \frac { p _ n } { q _ n } ) | = \frac { j _ 1 } { q _ n } \| q _ n \alpha \| \leq \frac { 1 } { 2 } \| q _ n \alpha \| . \end{align*}


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


\begin{align*} s _ { \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor } ( F ) \geq g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor \big ) , j = 1 , 2 , \dots , i , \end{align*}


\begin{align*} N _ { P ^ * } ( c _ 4 ) = \emptyset . \end{align*}


\begin{align*} | \phi ^ p _ j \cap \phi ^ s _ j | = | \phi ^ s _ j | - 1 . \end{align*}


\begin{align*} \begin{aligned} 0 = \ , & ( A \Psi \pm T _ \alpha ( u - \varphi ) ) _ n \\ = \ , & - A + 2 A K x _ n \\ & \pm \left [ ( u - \varphi ) _ { \alpha n } + \sum _ { \beta < n } B _ { \alpha \beta } ( x _ { \beta } ( u - \varphi ) _ { n n } - x _ n ( u - \varphi ) _ { \beta n } ) - \sum _ { \beta < n } B _ { \alpha \beta } ( u - \varphi ) _ { \beta } \right ] \\ < \ , & 0 , \end{aligned} \end{align*}


\begin{align*} I _ 2 ( x ) = - \int _ { \mathbb { R } } r _ 2 ( z ) M _ { + , 1 2 } ( x ; z ) \mathrm { e } ^ { 2 i s x } \mathcal { P } ^ + \left ( z ^ { - 1 } \bar { r } _ 1 ( z ) \mathrm { e } ^ { - 2 i z x } \right ) \mathrm { d } z . \end{align*}


\begin{align*} N = m _ 1 s _ 1 + \cdots + m _ L s _ L \end{align*}


\begin{align*} \mathcal D _ { \psi } f ( t z ) = \psi _ 0 \frac { \partial f } { \partial x } ( t z ) + \psi _ 1 \frac { \partial f } { \partial y } ( t z ) = 0 \end{align*}


\begin{align*} m _ n ^ { ( 1 ) } + \ell _ n ^ { ( 1 ) } q _ n = k _ n . \end{align*}


\begin{align*} R _ { \bar D } ( \theta ) = \begin{cases} & ( 0 \leq \theta \leq \frac { \pi } { 4 } ) \\ \theta & ( \frac { \pi } { 4 } < \theta < \frac { 3 } { 4 } \pi ) \\ & ( \frac { 3 } { 4 } \pi < \theta < \frac { 5 } { 4 } \pi ) \\ \theta & ( \frac { 5 } { 4 } \pi < \theta < \frac { 7 } { 4 } \pi ) \\ & ( \frac { 7 } { 4 } \pi \leq \theta < 2 \pi ) \end{cases} \end{align*}


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


\begin{align*} P _ u : = R ^ { 2 - \frac { 9 } { r } } \left ( R ^ { \frac { 3 } { r } } \left ( \frac { 1 } { R ^ 3 } \int _ { \mathcal { C } ( R / 2 , R ) } \vert u \vert ^ { p } d x \right ) ^ { \frac { 1 } { p } } \right ) . \end{align*}


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


\begin{align*} \begin{aligned} & \chi ( x , r , s ) = \chi ( y , r , s ) \quad \mbox { i f } | x | = | y | \quad \mbox { a n d } \\ & \chi ( x , r , s ) + | \partial _ { r } \chi ( x , r , s ) | \le \chi _ { 0 } ( s ) \quad \mbox { w i t h s o m e } \ , \ , \chi _ { 0 } \in \mathcal { C } ( \R _ { + } ) . \end{aligned} \end{align*}


\begin{align*} r ^ \vee ( \kappa + h ^ \vee ) ( \check { \kappa } + \check { h } ^ \vee ) = 1 , r ^ \vee ( \check { \kappa } + n + \check { h } ^ \vee ) ( \varkappa + h ^ \vee ) = 1 . \end{align*}


\begin{align*} \abs { \mathcal { T } } \le 2 \sum _ { i = 1 } ^ { C _ { B C } } \abs { \mathcal { B } ^ { ( i ) } } . \end{align*}


\begin{align*} \Gamma _ 3 = \mathrm { c o n e } ( \mathbf { e } _ j , j \in E ) & \triangleq \left \{ \sum _ { j \in E } \lambda _ j \mathbf { e } _ j : \lambda _ j \geq 0 , j \in E \right \} \end{align*}


\begin{align*} N = n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } = \left ( n + a \right ) ^ { 3 } - \left ( n + a - \beta \right ) ^ { 3 } \end{align*}


\begin{align*} ( \nabla _ V ( \nabla X ) ) ^ i _ j = V _ k \ , g ( \nabla _ { e _ k } \nabla _ { e _ j } X - \nabla _ { \nabla _ { e _ k } e _ j } X , e _ i ) : = V _ k X _ { i ; j k } . \end{align*}


\begin{align*} \int _ { \Omega } w \varphi \ , d x = \langle \mathcal A y + a y , \varphi \rangle = \langle \mathcal A ^ * \varphi + a \varphi , y \rangle = \langle h , y \rangle \ge 0 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H = 0 , \ \mathbb { P } . \end{align*}


\begin{align*} { R ^ * } = \mathop { \arg \max } \limits _ { { W _ s } \in \left [ { { W ^ { l o w } } , { W ^ { u p } } } \right ] } \ ; \left ( { W - { W _ s } } \right ) { \log _ 2 } \left ( { 1 + \frac { { \left [ { P - \overline { { p _ K } \left ( { \overline S , { W _ s } } \right ) } } \right ] { { \left | { { h _ b } } \right | } ^ 2 } } } { { \left ( { W - { W _ s } } \right ) { N _ 0 } } } } \right ) , \end{align*}


\begin{align*} & \frac { ( n - 2 ) ( 1 0 - n ) } { 4 } \int _ { B _ { \rho } } r ^ { 2 - n } u _ r ^ 2 \d x \\ & \leq C \int _ { B _ { 2 \rho } \setminus B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon \int _ { B _ { 2 \rho } } r ^ { 4 - n } | D ^ 2 u | | \nabla u | \d x \\ & + C \varepsilon \int _ { B _ { 2 \rho } } \big ( r ^ { 3 - n } + \varepsilon r ^ { 4 - n } \big ) | \nabla u | ^ 2 \d x . \end{align*}


\begin{align*} \det \big ( M ( \xi ) - \lambda I \big ) & = \det \left ( - ( \xi _ 1 ^ 2 + \xi _ 2 ^ 2 + \cdots + \xi _ d ^ 2 ) \left ( \begin{smallmatrix} 1 & 0 \\ 0 & \epsilon \end{smallmatrix} \right ) + i \xi _ 1 c I + \left ( \begin{smallmatrix} 0 & e ^ { - \kappa } \\ 0 & - \kappa e ^ { - \kappa } \end{smallmatrix} \right ) \right ) \\ & = 0 . \end{align*}


\begin{align*} \phi _ { i s } ( \partial _ r ) = 0 \quad \forall r \neq i \neq s \neq r , \end{align*}


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


\begin{align*} \frac { \partial \tilde y } { \partial y } = g ^ { \frac { 1 } { 4 } } + y \cdot \biggl ( \frac { 1 } { 4 } g ^ { - \frac { 3 } { 4 } } \cdot \frac { 1 } { 6 } \int \limits _ { 0 } ^ 1 s f _ { y ^ { 5 } } ^ { ( 5 ) } ( x , s y ) ( 1 - s ) ^ 3 \ , d s \biggr ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { r _ n ^ 2 } { h _ n } = 0 . \end{align*}


\begin{align*} \begin{array} { l l l } \mu \circ ( \eta \otimes ( f \circ \eta ) ) & = & \mu \circ ( \eta \otimes i d _ M ) \circ ( i d _ I \otimes ( f \circ \eta ) ) \\ & = & \lambda _ M \circ ( i d _ I \otimes ( f \circ \eta ) ) \\ & = & ( f \circ \eta ) \circ \lambda _ I . \\ & & \mbox { ( b y n a t u r a l i t y o f $ \lambda $ ) } \end{array} \end{align*}


\begin{align*} \phi ( t ) = \frac { d } { d t } \ , \frac { 1 } { \Gamma ( 1 - \alpha ) } \ , \int _ 0 ^ t ( t - \tau ) ^ { - \alpha } f ( \tau ) \ , d \tau , \ t > 0 . \end{align*}


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


\begin{align*} \langle { \boldsymbol { \nu } } _ { x , t } , g ( A ( \lambda , x , t ) , x , t ) \rangle = \langle { \boldsymbol { \nu } } _ { x , t } , f ( \lambda , x , t ) \rangle = \langle { \boldsymbol { N } } _ { x , t } , g ( \rho , x , t ) \rangle \end{align*}


\begin{align*} \pi ( [ X ] ) \wedge \pi ( [ Y ] ) = [ f _ \alpha '' X \cap ( ( \lambda \setminus A _ \alpha ) \cup f _ \alpha '' ( Y \cap A _ \alpha ) ] = [ f _ \alpha '' X \cap f '' _ \alpha ( Y \cap A _ \alpha ) ] = [ f _ \alpha '' ( X \cap Y ) ) ] . \end{align*}


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


\begin{align*} - \epsilon _ 1 b _ { n - 1 } u _ n ( x ) \leq u ( 0 ) - u ( x ) + \sum _ { \alpha = 1 } ^ { n - 1 } x _ \alpha u _ \alpha ( x ) \leq C b _ { n - 1 } ^ 2 . \end{align*}


\begin{align*} \Phi _ { n , e } ^ { ( i ) } ( x , y ) \Psi _ { n , e } ^ { ( i ) } ( x , y ) = x ^ n - \zeta ^ i _ { e } y ^ n . \end{align*}


\begin{align*} u \cdot n = 0 \end{align*}


\begin{align*} \gamma _ k ( t , x , \xi , y ) = \varphi ( t , x , \kappa _ t ( \cdot ) ) + k \| x - y \| ^ 2 + k ( t - \xi ) ^ 2 - \epsilon _ \circ ( \xi - \tau ) , \end{align*}


\begin{align*} \xi _ { t } = h ( \xi ) \end{align*}


\begin{align*} | \Phi _ S ( \sigma , \tau ) | & = \sum _ { i \in [ m ] } | \Phi _ S ( \sigma , \tau ) _ i | = | S | | \sigma | + ( m - | S | ) | \tau | . \end{align*}


\begin{align*} \begin{aligned} \left | \int _ { \Omega } d _ 0 u \wedge d _ 1 v \wedge \alpha \right | ^ 2 \leq \int _ { \Omega } d _ 0 u \wedge d _ 1 u \wedge \alpha \cdot \int _ { \Omega } d _ 0 v \wedge d _ 1 v \wedge \alpha . \end{aligned} \end{align*}


\begin{align*} \sum _ { i \geq 1 \atop { d \geq 0 } } a _ { i , d } ( \underline { b } ) = a ( \underline { b } ) . \end{align*}


\begin{align*} m _ k : = \left ( \frac { 2 ^ { \star } } { 2 } \right ) ^ k m _ 0 . \end{align*}


\begin{align*} \sigma _ { k , i } ( x ) = a _ { k , i } e _ { k } ( x ) , \end{align*}


\begin{align*} \begin{aligned} P \times V \times G & \to P \times V \\ ( p , v ) \cdot g & = ( p \cdot g , g ^ { - 1 } v ) \ , . \end{aligned} \end{align*}


\begin{align*} I : = & \mathbb { E } \int _ 0 ^ { T \wedge \tau _ { X } ^ M } f ( s ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s , \\ I I : = & \mathbb { E } \int _ 0 ^ { T \wedge \tau _ { X } ^ M } \rho ( X _ n ( s ) ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s , \\ I I I : = & \mathbb { E } \int _ 0 ^ { T \wedge \tau _ { X } ^ M } \eta ( X ( s ) ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s . \end{align*}


\begin{align*} d z _ t = \Pi _ { \mathrm { k e r } A } ( B ( y _ t , y _ t ) - A y _ t ) d t + \Pi _ { \mathrm { k e r } A } \sigma d W _ t . \end{align*}


\begin{align*} \sigma _ \mathfrak { s } ( \boldsymbol { J } ; n , p ) = \sigma _ \mathfrak { s } ( \boldsymbol { J } ; q , p ) \quad \ | n - q | \leq 1 \quad \sigma _ \mathfrak { s } ( \boldsymbol { J } ; n , p ) = \sigma _ \mathfrak { s } ( \boldsymbol { J } ; n , m ) \quad \ | p - m | \leq 1 \ , . \end{align*}


\begin{align*} \left ( 2 n - \beta \left ( 3 \beta ^ { 2 } + 4 \right ) + 1 \right ) \left ( n ^ { 2 } + \left ( 2 \beta + 1 \right ) n + \beta ^ { 2 } + \beta + 1 \right ) = 0 \end{align*}


\begin{align*} m ( x , t ) = \lim _ { k \rightarrow \infty } ( k \psi ( x , t ; k ) ) _ { 1 2 } . \end{align*}


\begin{align*} \gamma _ k ( t _ k , x _ k , \xi _ k , y _ k ) = \min \limits _ { ( t , x , \xi , y ) \in \Omega _ \delta \times \Omega _ \delta ^ * } \gamma _ k ( t , x , \xi , y ) . \end{align*}


\begin{align*} z _ i = \lambda _ i \cdot \left ( { 1 \over 1 + \sum _ { j \in \mathbb Z _ 0 } z _ j } \right ) ^ k , \ \ i \in \mathbb Z _ 0 . \end{align*}


\begin{align*} { { \bf { W } } ^ { \star } } = \left [ { { { \bf { W } } ^ { \star } } , { { \bf { 0 } } _ { \left ( { \sum { { { \bar r } _ { l ' } } } } \right ) \times \left ( { { M _ t } - K } \right ) } } } \right ] \left [ \begin{array} { l } { { \bf { I } } _ K } \\ { { \bf { 0 } } _ { \left ( { { M _ t } - K } \right ) \times K } } \end{array} \right ] . \end{align*}


\begin{align*} f _ { a } ( x ) = \dfrac { a } { a ^ 2 + x ^ 2 } . \end{align*}


\begin{align*} C ( H , K , M ^ \vee ) : = \bigoplus _ { n \geq 0 } \ , C _ n ( H , K , M ^ \vee ) \end{align*}


\begin{align*} \Phi ( t , x , y ) = ( x e ^ { t } , y e ^ { t } ) \end{align*}


\begin{align*} \mathbf { r } ^ { ( k ) } = \mathbf { x } ^ { ( k - 1 ) } - \rho \Phi ^ { \top } \left ( \Phi \mathbf { x } ^ { ( k - 1 ) } - \mathbf { y } \right ) , \end{align*}


\begin{align*} \Delta ( y _ m - y ) = & U ( t , 0 ) \Delta ( x _ m - x ) + \int _ 0 ^ t U ( t , s ) \{ 2 i ( D _ j \nabla \tilde { b } ^ j ( s ) + \nabla \tilde { b } ^ j ( s ) D _ j + \nabla \tilde { c } ( s ) ) \\ & \times \nabla ( y _ m - y ) + i ( D _ j \Delta \tilde { b } ^ j ( s ) + \Delta \tilde { b } ^ j ( s ) D _ j + \Delta \tilde { c } ( s ) ) ( y _ m - y ) \\ & - \lambda i \Delta [ e ^ { ( \alpha - 1 ) W } ( g ( y _ m ( s ) ) - g ( y ( s ) ) ) ] \} d s , \end{align*}


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


\begin{align*} \alpha \wedge \beta : = \{ \gamma \in R ^ + \mid \gamma = \alpha + \beta - \delta \delta \in \alpha \vee \beta \} \end{align*}


\begin{align*} M N A P P ( m , n ) = & \biggl \{ A \in N A P P ( m , n ) : A \mbox { i s t h e m i n i m a l e l e m e n t i n t h e s e t } \\ & \{ A , \rho ( A ) , \sigma ( A ) , ( \sigma \rho ) ( A ) \} \mbox { u n d e r t h e l e x i c o g r a p h i c o r d e r } \biggr \} \end{align*}


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


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


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


\begin{align*} \Bigl ( \frac { 1 } { \lambda ( t ) } \mathcal { G } ( t ) \Bigr ) ^ { \prime } \leq - \frac { k _ 3 } { k _ 2 } \frac { \alpha ( t ) } { \lambda ( t ) } \Bigl [ \frac { 1 } { \lambda ( t ) } \mathcal { G } ( t ) \Bigr ] ^ { \frac { m } { \ell } } . \end{align*}


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


\begin{align*} { \rm e x } ( n , H _ 3 ^ r ) & \ : \geq \ : \frac { 1 } { n } \binom { n } { r } \ : + \ : \frac { 1 } { r } \sum _ { d = 0 } ^ { \lfloor n / r \rfloor - 1 } \binom { n - 1 - r d } { r - 1 } \ , . \end{align*}


\begin{align*} \mu _ \infty ( \dd { \bf q } ' , \dd { \bf p } ' ) = \mu _ \infty { \cal P } _ { 0 , \theta } ( \dd { \bf q } ' , \dd { \bf p } ' ) \ge c _ * \mu _ \infty { \cal Q } _ { 0 , \theta } ( { \bf q } ' , { \bf p } ' ) \dd { \bf q } ' \dd { \bf p } ' . \end{align*}


\begin{align*} h _ { s , t } ^ { m , n } = \frac { ( m t - n s ) ^ 2 - ( s - t ) ^ 2 } { 4 s t } . \end{align*}


\begin{align*} - \Delta u = F \quad \Omega , u = g \quad \partial \Omega \end{align*}


\begin{align*} \alpha _ { k } ^ { \mathrm { o p t } } = 1 + \frac { \sum _ { l \in \mathcal { M } _ { k } } \overline { \mathbf { h } } _ { k l } ^ { \mathbf { H } } \mathbf { z } _ { k l } \mathbf { z } _ { k l } ^ { \mathbf { H } } \overline { \mathbf { h } } _ { k l } } { \sum _ { i \in \mathcal { K } _ { l } \setminus \{ k \} } \sum _ { l \in \mathcal { M } _ { i } } \overline { \mathbf { h } } _ { k l } ^ { \mathbf { H } } \mathbf { z } _ { i l } \mathbf { z } _ { i l } ^ { \mathbf { H } } \overline { \mathbf { h } } _ { k l } + \delta _ { } ^ 2 } . \end{align*}


\begin{align*} G ( \breve F ) = \bigsqcup _ { x \in \widetilde W } I x I , \end{align*}


\begin{align*} \mathbb { E } \left [ y _ i ^ * y _ i | \hat { \mathbf { H } } \right ] = & a _ c ^ 2 \left ( \lvert \hat { \phi } ^ { \left ( i , c \right ) } \rvert ^ 2 + \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ c \rVert ^ 2 \right ) + \sum _ { j = 1 } ^ { K } a _ j ^ 2 \left ( \lvert \hat { \phi } ^ { \left ( i , j \right ) } \rvert ^ 2 + \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 \right ) + \sigma _ n ^ 2 . \end{align*}


\begin{align*} P ^ g ( v _ 1 , \ldots , v _ n ) = P ( w _ 1 , \ldots , w _ n ) . \end{align*}


\begin{align*} I _ h \leq E [ b ] \leq \liminf _ { n \to \infty } E [ b _ n ] = I _ h , \end{align*}


\begin{align*} g _ { \theta } ( \omega ) : = \frac { 1 } { 2 \pi i } \int _ { e ^ { i \theta } [ 0 , + \infty ) } f ( \zeta ) e ^ { \omega \zeta } d \zeta , \omega \in \Omega _ { \theta } , \end{align*}


\begin{align*} \Delta _ n ^ { 1 - \frac m 2 } \langle S A M P V _ t ^ n ( m _ 1 , . . . , m _ k ) , & \bigotimes _ { i = 1 } ^ k \bigotimes _ { j = 1 } ^ { m _ j } e _ { p _ { i , j } } \rangle _ { \mathcal H ^ m } \\ & \stackrel { u . c . p . } { \longrightarrow } \langle \int _ 0 ^ t \rho _ { \Sigma _ s } ^ { \otimes k } ( m _ 1 , . . . , m _ k ) d s , \bigotimes _ { i = 1 } ^ k \bigotimes _ { j = 1 } ^ { m _ j } e _ { p _ { i , j } } \rangle _ { \mathcal H ^ m } , \end{align*}


\begin{align*} w ( t , \xi ) : = e ^ { 3 i \lambda _ 1 ( \Phi _ 1 ( t ) - \theta _ 1 ) } v _ 2 ( t ) . \end{align*}


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


\begin{align*} F ( P ) = B \ ! \left ( a , \frac { q } { 2 } \right ) \subset B \ ! \left ( F ( p ( s , k ) ) , \frac { 3 q } { 5 } \right ) \subset B \ ! \left ( F ( p ( s , k ) ) , \frac 2 3 | F ( p ( s , k ) ) | \right ) . \end{align*}


\begin{align*} \begin{aligned} \langle x _ k , \alpha _ 1 x _ k + \alpha _ 2 v \rangle & = y _ 1 y _ 2 \\ \| \alpha _ 1 x _ k + \alpha _ 2 v \| & = y _ 2 , \end{aligned} \end{align*}


\begin{align*} S _ i = { { w } _ i } \rho c _ v \frac { \Delta t } { 2 } \partial _ t ^ 2 T , \end{align*}


\begin{align*} \# \{ \beta \in \Phi ^ - \setminus ( - I ) \mid \varphi ( v ' \beta ) > 0 \} = & \# \{ \beta \in \Phi ^ - \setminus ( - I \cup \{ - \alpha \} ) \mid \varphi ( v ' \beta ) > 0 \} \\ = & \# \{ \beta \in \Phi ^ - \setminus ( - I \cup \{ - \alpha \} ) \mid \varphi ( v \beta ) > 0 \} \\ \leq & \# \{ \beta \in \Phi ^ - \setminus ( - I ) \mid \varphi ( v \beta ) > 0 \} . \end{align*}


\begin{align*} C R B = \frac { \lambda ^ 2 } { 8 { \pi } ^ 2 K S N R { c o s } ^ 2 \theta \overline { d ^ 2 } } \end{align*}


\begin{align*} a _ { j k } x _ k + x _ j b _ { j k } = \delta _ { j k } v _ j , x _ j a _ { j k } + b _ { j k } x _ k = - \delta _ { j k } v _ j , \end{align*}


\begin{align*} \alpha _ i ( t , x _ 1 , \ldots , x _ n ) = \widehat \alpha _ i ( t , x _ i ) , \end{align*}


\begin{align*} g _ b ( f ) ( x ) = \Big ( \int _ { 0 } ^ { \infty } \Big | \frac { d } { d t } e ^ { - t L } ( b ( x ) - b ( \cdot ) ) ( f ) ( x ) \Big | ^ 2 t d t \Big ) ^ { 1 / 2 } ; \end{align*}


\begin{align*} \phi ( \sigma , \tau ) = [ \widetilde \sigma , \widetilde \tau ] = [ \widetilde \tau , \widetilde \sigma ] ^ { - 1 } = [ \widetilde \tau , \widetilde \sigma ] = \phi ( \tau , \sigma ) . \end{align*}


\begin{align*} \bigg ( \frac { 1 } { | Q | } \iint _ { T _ Q } | \Theta f ( x ' , t ) | ^ 2 \frac { d x ' d t } { t } \bigg ) ^ { \frac 1 2 } = \bigg ( \frac { 1 } { | Q | } \iint _ { T _ Q } | \Theta ( f - f _ Q ) ( x ' , t ) | ^ 2 \frac { d x ' d t } { t } \bigg ) ^ { \frac 1 2 } \le { \rm I } + { \rm I I } , \end{align*}


\begin{align*} r s [ R , G ] = s r [ R , G ] r s r ' s ' [ R , G ] = r r ' s s ' [ R , G ] . \end{align*}


\begin{align*} \ln \left \{ 1 - \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right \} = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ n } { n } \ , \Phi \left ( \frac { x - \mu } { \sigma } \right ) ^ n . \end{align*}


\begin{align*} s ( x \cdot y ) \beta ( v ) = s ( \alpha ( x ) ) s ( y ) v . \end{align*}


\begin{align*} \gamma \alpha \omega - \gamma \omega = \dfrac { ( \alpha \omega - \omega ) \det ( \gamma ) } { ( c \omega + d ) ( c \alpha \omega + d ) } . \end{align*}


\begin{align*} \mbox { e i t h e r } T _ { \rm m a x } = \infty \mbox { o r } \limsup _ { t \nearrow T _ { \rm m a x } } \| n ( \cdot , t ) \| _ { L ^ { \infty } ( \Omega ) } = \infty . \end{align*}


\begin{align*} u = 0 , ( \nabla \times B ) \times n = 0 , B \cdot n = 0 , \end{align*}


\begin{align*} { U _ \theta } ^ \top U _ \theta = \begin{bmatrix} \cos ^ 2 \theta \dfrac { m } { k } V ^ \top V & \vline & \cos \theta \sin \theta \dfrac { m } { \sqrt { k l } } V ^ \top W \\ \hline \cos \theta \sin \theta \dfrac { m } { \sqrt { k l } } W ^ \top V & \vline & \sin ^ 2 \theta \dfrac { m } { l } W ^ \top W \end{bmatrix} . \end{align*}


\begin{align*} a = \left ( \beta - 1 \right ) n + \beta ^ { 2 } + \beta + 1 \end{align*}


\begin{align*} H \left ( P ^ { ( n , n ) } [ t ] \mid \mu ^ { \otimes n } [ t ] \right ) = \mathbb { E } _ \mathbb { P } [ Z _ t ^ n \log Z _ t ^ n ] . \end{align*}


\begin{align*} \mathcal S ( t ) h ( x ) : = \begin{cases} h ( x + t ) , & x + t \leq 1 , \\ h ( 1 ) , & x + t > 1 . \end{cases} \end{align*}


\begin{align*} \mathcal { C } _ { w } f = \mathcal { P } ^ + \left ( f w _ - \right ) + \mathcal { P } ^ { - } \left ( f w _ + \right ) = \mathcal { P } ^ - ( f J ) . \end{align*}


\begin{align*} \psi _ { 2 n } = \begin{pmatrix} \psi _ { 2 } \\ & \psi _ { 2 } \\ & & \ddots \\ & & & \psi _ { 2 } \end{pmatrix} = \bigoplus _ { 1 } ^ { n } \psi _ { 2 } , \psi _ { 2 } = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} . \end{align*}


\begin{align*} & \left \langle \frac 1 N v ^ { - 1 } \sum _ { k = 1 } ^ N ( \sigma \circ w ) ^ k \mu , \alpha \right \rangle \\ = & \frac 1 N \sum _ { k = 1 } ^ N \langle \mu , ( \sigma \circ w ) ^ k v \alpha \rangle \\ = & \frac 1 N \sum _ { k = 1 } ^ N \langle \mu , ( \sigma \circ w ) ^ k v \alpha \rangle + \Phi ^ + ( ( \sigma \circ w ) ^ k v \alpha ) - \Phi ^ + ( ( \sigma \circ w ) ^ { k + 1 } v \alpha ) \\ = & \frac 1 N \sum _ { k = 1 } ^ N \ell ( x , ( \sigma \circ w ) ^ k v \alpha ) . \end{align*}


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


\begin{align*} R _ { k , i } = \log _ { 2 } \bigg ( 1 + \frac { P _ { k , i } { \lvert \mathbf { d } _ { k , i } ^ { H } \mathbf { h } _ { k } \rvert } ^ { 2 } } { \sum _ { \pi _ { k ' , i ' } > \pi _ { k , i } } P _ { k ' , i ' } { \lvert \mathbf { d } _ { k ' , i ' } ^ { H } \mathbf { h } _ { k ' } \rvert } ^ { 2 } + \sigma _ { \textrm { u l } } ^ { 2 } } \bigg ) , \end{align*}


\begin{align*} \hat { c } _ \Delta ^ { B _ { n - 1 } ( x ) } ( \xi _ \Delta \eta _ { \Delta ^ c } , \eta _ \Delta ) = 0 \end{align*}


\begin{align*} \int _ W | \Delta \rho | \leq \int _ W \left ( | C - \Delta \rho | + C \right ) = \int _ W ( 2 C - \Delta \rho ) = 2 C \ , _ n ( W ) + _ { n - 1 } ( \partial W ) . \end{align*}


\begin{align*} ( g _ 1 , g _ 2 , g _ 3 ) \cdot \phi ( h ) = g _ 3 \cdot \phi ( g _ 1 ^ { - 1 } h g _ 2 ) . \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x & \leq C \int _ { B _ { 4 \rho } \setminus B _ { 2 \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon ( 1 + \varepsilon \rho ) \int _ { B _ { 8 \rho } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \\ & \rho \leq 1 / 8 \varepsilon \leq \varepsilon _ 0 . \end{align*}


\begin{align*} \tilde N ( \mathbf { v } ) = \mathbf { 0 } . \end{align*}


\begin{align*} \mathcal { U } _ \mu = \{ u \in \mathcal { U } : u \mu \} \subseteq \mathcal { U } , \end{align*}


\begin{align*} ( i _ X R ^ { \nabla } ) ( Y _ 1 ) = R ^ { \nabla } ( X , Y _ 1 ) \ , , \end{align*}


\begin{align*} \mu ^ { T a m a } : = \rho _ T ^ { - 1 } | \omega _ \infty | \prod _ p L _ p ( 1 , \sigma _ T ) | \omega | _ p , \end{align*}


\begin{align*} u ^ K _ { n , h } ( p ) + H ( p , u ^ K _ { n , h } ( p ) , \nabla _ H u ^ K _ { n , h } ( p ) ) = u ^ K _ { n - 1 , h } ( p ) - \omega _ f ( | h | _ G ) \end{align*}


\begin{align*} & \mathcal { \tilde { R } } _ { 1 , \epsilon } ( t ) = \int ^ { t } _ { 0 } \big ( b ( \varphi ( s ) ) - b ( X ^ { \epsilon } ( s ) ) ) \big ) d s , \\ & \mathcal { \tilde { R } } _ { 2 , \epsilon } ( t ) = \lambda ( \epsilon ) \int ^ { t } _ { 0 } \big ( \sigma ( \varphi _ { m } ( s ) ) - \sigma ( X ^ { \epsilon } ( s ) ) \big ) \theta _ { \epsilon } ( s ) d s . \end{align*}


\begin{align*} \phi = 2 \pi \frac { d } { \lambda } { s i n } \theta _ 0 = \frac { 2 \pi Z } { N } \end{align*}


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


\begin{align*} \begin{aligned} \sup _ { x \in \left ( - \infty , x _ 0 \right ) } \| \langle x \rangle m ( x ; z ) \| _ { L _ z ^ 2 ( \mathbb { R } ) } & \leq \sqrt { \pi } \left ( \left \| u _ { x x } \right \| _ { L ^ { 2 , 1 } } + \frac { 1 } { 2 } \left \| u _ x ^ 3 \right \| _ { L ^ { 2 , 1 } } \right ) \\ & \leq c \left ( \| u \| _ { H ^ { 2 , 1 } } + \| u \| _ { H ^ { 2 , 1 } } ^ 3 \right ) , \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \nabla \times U = f U , \nabla \cdot U & = 0 \textrm { i n } \ \Omega , \\ U \cdot n & = 0 \textrm { o n } \ \partial \Omega . \end{aligned} \end{align*}


\begin{align*} | \delta ( \zeta \theta ^ { - 1 } , \theta \gamma ^ + ) | = | f ( \zeta \gamma ^ + , \theta \gamma ^ - ) - f ( \theta \gamma ^ + , \theta \gamma ^ - ) | \leq 2 M , \end{align*}


\begin{align*} G '' ( c ) = - \frac { g ' ( c ) } { c } \end{align*}


\begin{align*} [ \sqrt { - 1 } \theta \wedge \bar { \theta } , \Lambda _ \omega ] \alpha = \bar { \theta } \wedge ( \alpha \llcorner ( \bar { \theta } ) ^ \sharp ) , \end{align*}


\begin{align*} ( \phi | _ { k , S } \gamma ) ( \tau , z ) : = J _ { k , S } ( \gamma , ( \tau , z ) ) \phi ( \gamma ( \tau , z ) ) , \end{align*}


\begin{align*} \tilde { m } _ n = \begin{cases} m _ n + q _ n - q _ { n - 1 } , \ell _ n = 2 \\ m _ n - q _ n + q _ { n - 1 } , \ell _ n = - 2 \end{cases} . \end{align*}


\begin{align*} \begin{aligned} \int u _ k ^ s \Delta u _ 1 ^ t \wedge \dots \wedge \Delta u _ { k - 1 } ^ t \wedge \beta _ n ^ { n - m } \wedge \Delta \omega = \int \Delta u _ 1 ^ t \wedge \dots \wedge \Delta u _ { k - 1 } ^ t \wedge \Delta u _ k ^ s \wedge \beta _ n ^ { n - m } \wedge \omega \geq 0 , \end{aligned} \end{align*}


\begin{align*} \Psi = \Psi _ 1 ^ { - 1 } \circ \Psi _ 2 ^ { - 1 } . \end{align*}


\begin{align*} S _ \beta ( x , y , z ) = ( x + \alpha , y + x , z + x + \beta ) \end{align*}


\begin{align*} \psi \left ( d _ - ^ { \ell + 1 } L ( 1 v , 0 w ) \right ) & = \psi \left ( t ^ { - \ell } d _ - ^ { \ell + 1 } d _ = L ( v 1 , w \bullet ) \right ) \end{align*}


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


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


\begin{align*} e ^ { \gamma x } f _ 1 \ast f _ 2 ( x ) & = \int _ { - \infty } ^ \infty e ^ { \gamma ( x - y ) } f _ 2 ( x - y ) e ^ { \gamma y } f _ 1 ( y ) d y \\ & \le o ( 1 ) \int _ { - \infty } ^ 0 e ^ { \gamma y } f _ 1 ( y ) d y . \end{align*}


\begin{align*} \phi _ { g ^ { - 1 } g _ 2 , g _ 1 } = \phi _ { g _ 2 , g g _ 1 } , \qquad \end{align*}


\begin{align*} C _ 1 ( t ) = \sum _ { u : \ , i _ u \geq 3 \atop { 0 \leq d _ u < i _ u } } i _ u = \sum _ { i \geq 3 } \sum _ { d = 0 } ^ { i - 1 } i x _ { i , d } , \end{align*}


\begin{align*} a ^ { D } ( x , y ) : = \tilde { G } ^ { D } ( x , y ) - \Gamma ( x , y ) ( x , y \in D , x \neq y ) , \end{align*}


\begin{align*} \sum _ { n } \sum _ { \sigma \subset \mu _ \ell } c _ { \ell } x _ { \ell , n } \pi ^ { n - \omega ( \mu _ \ell ) } + c _ { \kappa } \pi ^ { - \omega ( \kappa ) } & = 0 \sigma \subset \kappa \\ \sum _ { n } \sum _ { \sigma \subset \mu _ \ell } c _ { \ell } x _ { \ell , n } \pi ^ { n - \omega ( \mu _ \ell ) } & = 0 \sigma \not \subset \kappa . \end{align*}


\begin{align*} g _ { i j } = \left \langle e _ i , e _ j \right \rangle , h _ { i j } = \left \langle \overline { D } _ { e _ i } e _ j , \mathbf { n } \right \rangle , \end{align*}


\begin{align*} C _ { { 3 i + 1 } } ( \mathcal { H } _ { \ast } ) = \mathrm { I m } \ , \partial _ i \oplus s _ { _ { 3 i + 1 } } ( \mathrm { I m } \ , \partial '' _ i ) = \mathrm { I m } \ , \partial _ i . \end{align*}


\begin{align*} \xi ( t ) = \sigma _ { \xi ( u ) \xi ( v ) } ( ( t - u ) / ( v - u ) ) \in \Sigma _ { \Gamma ( u ) \Gamma ( v ) } ( ( t - u ) / ( v - u ) ) . \end{align*}


\begin{align*} g _ \sigma = \varphi _ { + } ^ { - 1 } \circ T _ \sigma \circ \varphi _ { - } \end{align*}


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


\begin{align*} { { \mathbf { A } } ^ { T } } \mathbf { d } _ { h y } = 0 . \end{align*}


\begin{align*} i A _ \ell ' = t ^ { - 1 } N _ \ell ( A _ 1 , A _ 2 , \dots , A _ L ) ( \ell = 1 , 2 , \dots , L ) . \end{align*}


\begin{align*} \| y \| \geq g \bigl ( \sum _ { j = 1 } ^ m \alpha _ j y _ j \bigr ) = \sum _ { j = 1 } ^ m \alpha _ j g ( y _ j ) = \sum _ { j = 1 } ^ m \alpha _ j > 1 - \frac { \varepsilon } { 5 } \end{align*}


\begin{align*} e ^ { 2 \varphi ( y ) } & = \int _ { T _ y N } e ^ { 2 w } \circ \exp _ y ( v ) \chi ( \Vert v \Vert ) \dd V _ y \\ & = \int _ { T _ x N } ( p ^ * e ^ { 2 w } ) \circ \exp _ x ( v ) \chi ( \Vert v \Vert ) p ^ * ( \dd V _ y ) \\ & = \int _ { T _ x N } \lambda ^ 2 e ^ { 2 w } \circ \exp _ x ( v ) \chi ( \Vert v \Vert ) \dd V _ x \\ & = \lambda ^ 2 e ^ { 2 \varphi } . \end{align*}


\begin{align*} h ^ { ( \lambda _ 1 , \ldots , \lambda _ d ) } ( x ) = \prod _ { i = 1 } ^ d \lambda _ i ^ { i - d } e ^ { \sum _ { i = 1 } ^ d \lambda _ i x _ i } \det \big ( ( - 1 ) ^ { d - j } \big ( \frac { d } { d x } \big ) ^ { d - j } e ^ { - \lambda _ i x _ j } \big ) _ { i , j = 1 } ^ d . \end{align*}


\begin{align*} T _ { \mathrm { g e l } } : = \inf \left \{ t \geq 0 \ \ \mbox { s u c h t h a t } \ \ \mathcal { M } ^ { 1 } ( t ) < \mathcal { M } ^ { 1 } ( 0 ) = \varrho _ { 0 } \right \} . \end{align*}


\begin{align*} \nu ( t , p ) = \frac { p } { R ( t ) } \phantom { x x } \phantom { x x } H ( t , p ) = - \frac { d } { R ( t ) } \end{align*}


\begin{align*} \lambda _ { i , j } = \begin{cases} \lambda _ j ~ ~ { \rm i f } ~ ~ i = 1 , \\ \\ \frac { \lambda _ { i } \lambda _ { j } } { \Lambda } ~ ~ { \rm i f } ~ ~ i > 1 . \\ \end{cases} \end{align*}


\begin{align*} r _ { \delta , j } ( z ) : = \bar { \delta } _ + ( z ) \bar { \delta } _ - ( z ) r _ { j } ( z ) , \ \ j = 1 , 2 . \end{align*}


\begin{align*} \begin{aligned} \mathbb E \exp \left ( \sum _ { i = 1 } ^ k \alpha _ i \int _ 0 ^ { \epsilon ^ { - 2 } t _ i } \xi ( s ) d s \right ) & = \mathbb E \exp \left ( \sum ^ { k } _ { \ell = 1 } \Delta _ \ell \right ) \\ & = \mathbb E \left [ \exp \left ( \sum ^ { k - 1 } _ { \ell = 1 } \Delta _ \ell \right ) \mathbb E \left ( e ^ { \Delta _ k } \Big | \mathcal { F } _ { \epsilon ^ { - 2 } t _ { k - 1 } } \right ) \right ] , \end{aligned} \end{align*}


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


\begin{align*} ( \Delta K _ { m , \epsilon } ) ^ p \wedge \beta _ n ^ { n - p } = ( p A \wedge B ^ { p - 1 } + B ^ p ) \wedge \beta _ n ^ { n - p } , p = 1 , \ldots , m , \end{align*}


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


\begin{align*} \begin{aligned} I _ 4 ^ { \prime } ( x ) & = - 4 \pi ^ { - 1 } \int _ { - \infty } ^ { \infty } r _ 2 ( z ) e ^ { - 2 i z x } \left ( M _ { + , 2 2 } ( x ; z ) - 1 \right ) d z \\ & - 2 i \pi ^ { - 1 } \int _ { - \infty } ^ { \infty } s ^ { - 1 } r _ 2 ( z ) e ^ { - 2 i z x } \partial _ { x } M _ { + , 2 2 } ( x ; z ) d z . \end{aligned} \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } ( \tau _ { n } ^ M < T ) = 0 . \end{align*}


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


\begin{align*} \hat { \mathbf { r } } _ p = \mathbf { A } \mathbf { x } , \hat { \mathbf { r } } _ d = \mathbf { A } ^ { T } \mathbf { y } + \mathbf { C } \mathbf { x } + \mathbf { s } . \end{align*}


\begin{align*} \mathcal { W } ^ 2 ( \mu _ 1 , \mu _ 2 ) = \inf \mathbb { E } [ | X - Y | ^ 2 ] . \end{align*}


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


\begin{align*} E _ n = \frac { - \varepsilon _ 1 } { \sqrt { \varepsilon _ 1 ^ 2 + | \hat { x } | ^ 2 } } e _ 1 + \frac { | \hat { x } | } { \sqrt { \varepsilon _ 1 ^ 2 + | \hat { x } | ^ 2 } } e _ 2 . \end{align*}


\begin{align*} w = \begin{pmatrix} 1 & 0 \end{pmatrix} P Q ( t ) P ^ { - 1 } \begin{pmatrix} W _ 2 \\ \overline { W _ 2 } \end{pmatrix} + O ( t ^ { - 1 / 4 + C _ 1 \varepsilon ^ 2 _ 1 + 2 \mu C _ 2 \varepsilon ^ 2 _ 1 + C _ 4 \varepsilon _ 1 ^ 2 } ) \end{align*}


\begin{align*} x _ 1 = x _ L x _ 0 x _ R = \underbrace { ( x _ L r _ a ) } _ { < x _ L } x _ 0 \underbrace { ( r _ { x _ 0 ^ { - 1 } ( a ) } x _ R ) } _ { < x _ R } , \end{align*}


\begin{align*} y _ n = \left [ g , x ^ { n } \right ] = \left [ g , x ^ { n - 1 } \right ] \left [ g , x \right ] ^ { x ^ { n - 1 } } = y _ { n - 1 } y \left [ y , x ^ { n - 1 } \right ] \end{align*}


\begin{align*} \begin{aligned} R ^ { \nabla ^ t } _ { X , Y } & = ( \nabla _ X + t A _ X ) ( \nabla _ Y + t A _ Y ) - ( \nabla _ Y + t A _ Y ) ( \nabla _ X + t A _ X ) \\ & = R ^ \nabla _ { X , Y } + t [ \nabla _ X , A _ Y ] - t [ \nabla _ Y , A _ X ] + t ^ 2 [ A _ X , A _ Y ] \\ & = R ^ \nabla _ { X , Y } + t \nabla _ X ( A _ Y ) - t \nabla _ Y ( A _ X ) + t ^ 2 [ A , A ] _ { X , Y } \\ & = R ^ \nabla _ { X , Y } + t ( d ^ \nabla A ) _ { X , Y } + t ^ 2 [ A , A ] _ { X , Y } \ , . \\ \end{aligned} \end{align*}


\begin{align*} \frac { 1 } { 1 - y } = \sum _ { n = 0 } ^ \infty y ^ n . \end{align*}


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


\begin{align*} Q ( t ) = \left \{ x \in K ( t ) \colon \frac { t } { 6 } \leq x _ 1 \leq \frac { t } { 4 } , \ ; \frac { t } { 6 } \leq x _ 2 \leq \frac { t } { 4 } , \ ; | x _ 3 - t | \leq \frac { t } { 4 } \right \} . \end{align*}


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


\begin{align*} \widehat { \mathcal D } _ { \hat { A } } ^ { \times } = \{ P \in \widehat { \mathcal D } _ { \mathcal { O } _ { \hat { A } } } \ ; | \ ; \pi ( P ) = 1 \} \ ; . \end{align*}


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


\begin{align*} { \partial _ t } = \sum \limits _ { n = 1 } ^ { + \infty } { { \varepsilon ^ n } \partial { t _ n } } , \ ; \ ; \ ; \nabla = \varepsilon { \nabla _ 1 } , \ ; \ ; \ ; { g _ i } = \sum \limits _ { n = 1 } ^ { + \infty } { { \varepsilon ^ n } g _ i ^ { \left ( n \right ) } } , \ ; \ ; \ ; \bar F = \varepsilon { \bar F ^ { \left ( 1 \right ) } } , \end{align*}


\begin{align*} s ^ { \pm } ( \rho ) : = \mp \frac { b ( \mp \rho ) } { a ( \rho ) } , \quad \rho \in \mathbb { R } . \end{align*}


\begin{align*} \lambda _ c = 2 \sum _ { j = 1 } ^ { M } \lvert \phi ^ { \left ( j , c \right ) } \rvert ^ 2 - f ^ { \left ( c \right ) } \end{align*}


\begin{align*} g ( r ) & = \lim _ { k \rightarrow { 0 } } \frac { G ( r + k ) - G ( r ) } { k } = \lim _ { n \rightarrow { \infty } } \frac { G ( r + \frac { 1 } { n } ) - G ( r ) } { \frac { 1 } { n } } = \lim _ { n \rightarrow { \infty } } n \{ G ( r + \frac { 1 } { n } ) - G ( r ) \} \end{align*}


\begin{align*} k _ 3 ( \langle H \rangle , a ) = k _ 3 ( \langle H \rangle , b ) . \end{align*}


\begin{gather*} m _ { n } ( t ) = \allowbreak \sqrt { \frac { n ! } { \left ( \beta \right ) ^ { \left ( n \right ) } } } \frac { 1 } { ( 1 - t ) ^ { \beta } } \frac { 1 } { \Gamma ( \beta ) } \times \\ \int _ { 0 } ^ { \infty } \left ( \sum _ { j = 0 } ^ { n } \left ( \frac { - t y } { 1 - t } \right ) ^ { j } L _ { n - j } ( x | ( \beta + j ) ) / j ! \right ) y ^ { \beta - 1 } \exp ( - y ) d y . \end{gather*}


\begin{align*} p ( x ) : = \sum _ { i = 0 } ^ d p _ i x ^ i = p _ d \prod _ { j = 1 } ^ d ( x - x _ j ) , ~ p _ d \neq 0 , \end{align*}


\begin{align*} p ( s , k ) = ( 2 s _ 1 , 2 s _ 2 , k \eta ( k ) ) . \end{align*}


\begin{align*} K ( Z , W ) = \det ( I _ n - Z W ^ * ) ^ { - 2 n } . \end{align*}


\begin{align*} \begin{cases} d x _ t = B ( x _ t , x _ t ) d t - A x _ t d t + \sigma d W _ t \\ x _ t | _ { t = 0 } = x _ 0 \in \R ^ n . \end{cases} \end{align*}


\begin{align*} B ( x , t , \lambda ) = - \lambda + y ( t ) h ' ( 0 ) . \end{align*}


\begin{align*} \boxed { \scriptstyle \sf H r a n k ~ 1 } \ , : = \ , \boxed { \aligned 0 & \ , \neq \ , F _ { x x } \\ 0 & \ , \equiv \ , F _ { x x } \ , F _ { y y } - F _ { x y } ^ 2 \endaligned } , \end{align*}


\begin{align*} \lambda \sum _ { i = 1 } ^ d ( y _ i - x _ i ) = \lambda d ( y _ 1 - x _ 2 ) + \lambda \sum _ { i = 2 } ^ d ( ( y _ i - y _ { 1 } ) - ( x _ i - x _ 1 ) ) \le \lambda d ( y _ 1 - x _ 1 ) + | \lambda | d ^ 2 \sqrt n . \end{align*}


\begin{align*} \| f \| = \sup \bigl \{ \frac { | f ( p ) - f ( q ) | } { d ( p , q ) } \colon p , q \in M , p \neq q \bigr \} , \end{align*}


\begin{align*} \int _ { \lambda I _ Q } \| R _ j \partial _ j g ( \cdot , t ) \| _ { L ^ 1 ( 2 \lambda Q _ 1 ) } d t & \lesssim \int _ { \lambda I _ Q } \ell ( Q ) ^ { N / p } \| R _ j \partial _ j g ( \cdot , t ) \| _ { L ^ q ( 2 \lambda Q _ 1 ) } d t \\ & \lesssim \int _ { \lambda I _ Q } \ell ( Q ) ^ { N / p } \| \partial _ j g ( \cdot , t ) \| _ { L ^ q ( 2 \lambda Q _ 1 ) } d t \lesssim \ell ( Q ) ^ { N / p } \ell ( Q ) ^ { N / q } = \ell ( Q ) ^ N . \end{align*}


\begin{align*} \alpha = \sum _ { i = 1 } ^ s a _ i w _ i + \sum _ { i = 1 } ^ t b _ i e _ { j _ i } = \sum _ { i = 1 } ^ s a _ i \left ( \sum _ { j = 1 } ^ k w _ { i j } e _ j \right ) + \sum _ { i = 1 } ^ t b _ i e _ { j _ i } \end{align*}


\begin{align*} \sqrt n \langle \Pi _ m Q \Pi _ m h - Q h , h \rangle \leq & \| ( I - \Pi _ m ) Q h \| + \| ( I - \Pi _ m ) Q h _ m \| \\ = & \sqrt n \sqrt { \Delta _ m } \left ( \| ( Q h ) ' \| + \| Q \| _ { L _ { } ( L ^ 2 ( 0 , 1 ) , \dot H ^ 1 ) } \| h _ m \| \right ) . \end{align*}


\begin{align*} V _ i ( n ) = \left \{ \begin{array} { l l } V ( n ) & n \in [ 0 , n _ i ] \\ V ( n _ i ) + V ' ( n _ i ) ( n - n _ i ) & n \in [ n _ i , L ] \end{array} \right . . \end{align*}


\begin{align*} \Pi ( \widetilde { \pi } ) : = \left \{ \bigotimes _ v ^ \prime \pi _ v \mid \pi _ v \in \Pi ( \widetilde { \pi } _ v ) , \pi _ v = \pi ^ 0 _ v \ , \ , v \right \} . \end{align*}


\begin{align*} E ^ L _ Y ( u ) = \varphi ^ L _ { \sigma , s } ( E ^ L _ X ( v ) ) \end{align*}


\begin{align*} \partial _ t u ( t , x ) = \nabla \cdot a \big ( t , x , u ( t , x ) , \nabla u ( t , x ) \big ) - a _ 0 \big ( t , x , , u ( t , x ) , \nabla u ( t , x ) \big ) , \end{align*}


\begin{align*} R _ { x } \left ( y \right ) = \cos \left ( \left \vert x \right \vert \right ) y + \sin \left ( \left \vert x \right \vert \right ) \left ( x / { \left \vert x \right \vert } \right ) \times y \end{align*}


\begin{align*} F _ n - n ^ 3 \ = \ \sum _ { i = 1 } ^ { n } \left ( i ^ 3 - 2 \left ( \binom { 3 } { 2 } i ^ 2 + \binom { 3 } { 0 } i ^ 0 \right ) \right ) \cdot F _ { n - i } . \end{align*}


\begin{align*} N K B L ( m , n ) = \sum _ { { \scriptscriptstyle A \in M N A P P ( m , n ) } } N ( A ) \cdot ( m - 1 ) ! + \sum _ { { \scriptscriptstyle A \in M P A P P ( m , n ) } } N ( A ) \cdot 2 ^ { m - 1 } ( m - 1 ) ! \\ \ge \sum _ { i = 1 } ^ { 3 } N ( A _ i ) \cdot ( m - 1 ) ! + \sum _ { i = 7 } ^ { 1 0 } N ( A _ i ) \cdot 2 ^ { m - 1 } ( m - 1 ) ! \ge ( 5 \cdot 2 ^ m ) \cdot ( m - 1 ) ! . \end{align*}


\begin{align*} | H ( x ) | \leq \max _ { y \in [ - 1 , 1 ] ^ 2 } | h ( y ) | = \sqrt { 2 } \quad \ x \in \R ^ 3 \setminus \Omega . \end{align*}


\begin{align*} I _ { \alpha } \left ( f \right ) = \int _ { 1 } ^ { \infty } \overline { \nu _ { \alpha } ^ { \prime } } \left ( f \left ( x \right ) \right ) d x \end{align*}


\begin{gather*} K _ 2 = \bigl \{ P \in M : ( \pi _ 1 , \ldots , \pi _ n ) P \bigr \} \quad \quad \\ K _ 3 = \bigl \{ P \in M : P \ll P ^ * ( \pi _ 1 , \ldots , \pi _ n ) P \bigr \} . \end{gather*}


\begin{align*} \hat { u } _ i = c _ 1 \frac { \partial \phi } { \partial x _ 1 } + c _ 2 \frac { \partial \phi } { \partial x _ 2 } . \end{align*}


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


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


\begin{align*} \{ | v _ 1 | > | v _ 2 | > \dots > | v _ N | \} \cap \{ | z v _ i | < | v _ { i + 1 } | | i = 1 , \dots N - 1 \} \subset \mathbb C ^ N \times \mathbb H \end{align*}


\begin{align*} u = ( Q _ { \delta } ( - \delta ^ 2 ( K ( x ) \nabla ) ) ) ^ { - 1 } p Q _ \delta ( V _ { \delta , Z } - q ) ^ { p - 1 } _ + u + ( Q _ { \delta } ( - \delta ^ 2 ( K ( x ) \nabla ) ) ) ^ { - 1 } h , \ \ u \in E _ { \delta , Z } . \end{align*}


\begin{align*} & F ( W ) c _ { ( \vartheta , \varphi ) } = \int _ { \gamma } F ( Z ) E _ { ( \vartheta , \varphi ) } ( Z , W ) d \sigma _ { ( \vartheta , \varphi ) } ( Z ) - \int _ { \Gamma } E _ { ( \vartheta , \varphi ) } ( Z , W ) \frac { \partial F } { \partial Z _ { ( \vartheta , \varphi ) } } d Z \wedge d Z ^ { * } , \end{align*}


\begin{align*} H \left ( P ^ { ( n , k ) } [ t ] \mid \mu ^ { \otimes k } [ t ] \right ) \leq \frac { k } { n } M , k = 1 , \cdots , n , t \in [ 0 , T _ M ] . \end{align*}


\begin{align*} W ' ( r ) = \frac { \xi ' } { r } f + \eta ' + \zeta ' \frac { r } { f } + \frac { \sigma ' } { f ^ 2 } , \xi ' \le 0 , \sigma ' > 0 , \end{align*}


\begin{align*} \zeta : = \zeta _ q : = \exp \Big ( \frac { 2 \pi \sqrt { - 1 } } { q } \Big ) . \end{align*}


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


\begin{align*} x _ n ( t ) : = x _ a ( t - n { \bar { t } } / { \gamma } , x ( n { \bar { t } } / { \gamma } ) ) \forall \ , t \in I _ n . \end{align*}


\begin{align*} R : ( x , y ) \mapsto R ( x , y ) = ( f ( x , y ) , g ( x , y ) ) \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) \| _ { \mathcal { A } } & \le \beta ( \| x ( \bar { t } / \gamma ) \| _ { \mathcal { A } } , 0 ) + \tilde { d } \\ & \le \beta ( \underline { \rho } , 0 ) + \tilde { d } \\ & \le d & & \forall \ , t \in I _ 1 \end{aligned} \end{align*}


\begin{align*} \Pi _ n Y _ { i \Delta _ n } = \sum _ { j = 1 } ^ n \alpha _ { j , i } k ( j \Delta _ n , \cdot ) , \end{align*}


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


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


\begin{align*} u = \int ^ x \frac { \dd x } { \sqrt { f _ 5 ( x ) } } + \int ^ { x ' } \frac { \dd x ' } { \sqrt { f _ 5 ( x ' ) } } , v = \int ^ x \frac { x \dd x } { \sqrt { f _ 5 ( x ) } } + \int ^ { x ' } \frac { x ' \dd x ' } { \sqrt { f _ 5 ( x ' ) } } , \end{align*}


\begin{align*} \epsilon ( x _ a , t ) = \int _ N ^ t h ( x _ a + \delta u ( \tau ) ) u ( \tau ) - \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \ , d \tau . \end{align*}


\begin{align*} \lambda | _ { E _ { k , [ p , q ] } } ^ * ( a ) = \lambda ( p + a ) \end{align*}


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


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