\begin{align*} & d ( \theta _ n u ( t ) ) = - \theta _ n \nu A u ( t ) d t - \theta _ n \lambda ( r ( t ) ) u ( t ) d t + \theta _ n f ( t , u ( t ) ) d t \\ & \quad + \theta _ n g ( t ) d t + \sum _ { k = 1 } ^ \infty ( \theta _ n h _ k ( t ) + \theta _ n \sigma _ k ( t , u ( t ) ) ) d W _ k ( t ) , t > s . \end{align*}


\begin{align*} \psi = \eta _ Y \widetilde { H } _ 0 ( \phi ) \eta _ X ^ { - 1 } . \end{align*}


\begin{align*} B ( \widetilde { u } - { u } _ 2 ) ^ \prime + ( A + B Q ) ( \widetilde { u } - { u } _ 2 ) = - H _ 1 ( \widetilde { u } \ , ^ \prime + Q \widetilde { u } ) - H _ 2 \big ( ( u _ + - u ) ' + Q ( u _ + - u ) \big ) \end{align*}


\begin{align*} \varphi ( x , v ) = H ( x , v ) + \frac { x \cdot v } { 4 } + \frac { | x | ^ 2 } { 8 } , \end{align*}


\begin{align*} g ( w | \rho , \lambda ) = \sigma _ Z ^ 2 \lambda w + \left ( \frac { \rho } { \lambda } + \sigma _ N ^ 2 \lambda \right ) \cdot w = \left [ ( \sigma _ N ^ 2 + \sigma _ Z ^ 2 ) \lambda + \frac { \rho } { \lambda } \right ] \cdot w , \end{align*}


\begin{align*} b _ n = \frac { N ! } { \pi n } \sum _ { k = 0 } ^ { N - 1 } \frac { ( - 1 ) ^ k F _ k ( n ) } { ( N - k ) ! } \sin \left ( \frac { \pi } { 2 } k \right ) . \end{align*}


\begin{align*} P ( z ) = a _ n ( z - \lambda _ 1 ) ( z - \lambda _ 2 ) \cdots ( z - \lambda _ n ) . \end{align*}


\begin{align*} I _ { j ; l ; 0 } ^ { m ; p , q } ( t , x ) : = \int _ { 0 } ^ { t } \int _ { \R ^ 3 } \int _ { 0 } ^ { 2 \pi } \int _ 0 ^ \pi 2 ^ { m - j - l + \max \{ l , p \} } \big [ \sum _ { \begin{subarray} { c } \tilde { j } \in [ ( 1 + 2 \epsilon ) M _ t , \infty ) \cap \Z _ + , \tilde { l } \in [ - \tilde { j } , 2 ] \cap \Z , \tilde { m } \in [ - 1 0 M _ t , \epsilon M _ t ] \cap \Z \end{subarray} } | B _ { \tilde { j } , \tilde { l } } ^ { \tilde { m } } ( s , x + ( t - s ) \omega ) \end{align*}


\begin{align*} \norm { p _ { X } p _ { Y | X } - q _ { X } p _ { Y | X } } _ { T V } = \norm { p _ { X } - q _ { X } } _ { T V } . \end{align*}


\begin{align*} D _ { \beta } ( \varphi , x _ 0 ) = \left \{ x \in [ 0 , 1 ] : | T _ { \beta } ^ n x - x _ 0 | < \varphi ( n ) n \in \N \right \} \end{align*}


\begin{align*} E _ { x \leftrightarrow y } = E _ { x \rightarrow y } \land E _ { y \rightarrow x } , \end{align*}


\begin{align*} q _ A = \frac { 2 } { 1 + \frac { G ( x , y ) } { G ( o , o ) } } . \end{align*}


\begin{align*} M _ 2 = \sum _ { \substack { 0 \leq l \leq i \\ \gcd ( l , i ) \ { \rm { i s } } \ k - { \rm { f r e e } } \\ \gcd ( l + 1 , i + 1 ) \ { \rm { i s } } \ k - { \rm { f r e e } } } } C _ { \alpha } ( i , l ) . \end{align*}


\begin{align*} 0 & = c ( c \gamma + \gamma + 1 ) ^ 2 + c \gamma ( c \gamma + c + 1 ) ( c \gamma + \gamma + 1 ) + \gamma ( c \gamma + c + 1 ) ^ 2 \\ & = \gamma ^ 3 c ^ 3 + \gamma ^ 2 c ^ 2 + ( \gamma + 1 ) c + \gamma \end{align*}


\begin{align*} \left ( u , \left [ x , z \right ] \right ) = \left ( \left [ u , x \right ] , z \right ) . \end{align*}


\begin{align*} \Lambda _ 2 = ( 0 , 2 , 3 , - 6 ) \end{align*}


\begin{align*} \sigma ( \widetilde \Delta _ n ) = \Bigg \{ & \frac { n ^ 2 } { \pi ^ 2 } \sin ^ 2 \Big ( \frac { \pi k _ 1 } { n } \Big ) + \frac { n ^ 2 } { \pi ^ 2 } \sin ^ 2 \Big ( \frac { \pi k _ 2 } { n } \Big ) \\ & \left . - \frac { 2 n ^ 2 } { 3 \pi ^ 2 } \sin ^ 2 \Big ( \frac { \pi k _ 1 } { n } \Big ) \sin ^ 2 \Big ( \frac { \pi k _ 2 } { n } \Big ) \right | \forall _ { j = 1 , 2 } : k _ j = 0 \ , , \cdots , n - 1 \Bigg \} . \end{align*}


\begin{align*} \left | { { \lambda _ 0 } } + \sum _ { i = 1 } ^ m \sum _ { s = 0 } ^ { r - 1 } { { \lambda _ { i , s } } } F _ { s } ( x , \alpha _ i / \beta ) \right | _ { v _ 0 } > C ( \boldsymbol { \eta } , \boldsymbol { \zeta } , \boldsymbol { \alpha } , \beta , \varepsilon ) { \mathrm { H } } _ { v _ 0 } ( { { \boldsymbol { \lambda } } } ) { \mathrm { H } } ( { { \boldsymbol { \lambda } } } ) ^ { - \mu ( \boldsymbol { \eta } , \boldsymbol { \zeta } , \boldsymbol { \alpha } , \beta , \varepsilon ) } \enspace , \end{align*}


\begin{align*} \delta \lVert \alpha \rVert _ { 1 } \leq & \rho _ { M } ( T ( \alpha ) ) = \sup _ { \theta \in M } \lvert \theta ( T ( \alpha ) ) \rvert \\ = & \sup _ { \theta \in M } \lvert ( T ^ { * } ( \theta ) ) ( \alpha ) \rvert < \lvert \varphi ( \alpha ) \rvert \\ \leq & \lVert \varphi \rVert _ { 1 } \lVert \alpha \rVert _ { 1 } \leq \delta \lVert \alpha \rVert _ { 1 } , \end{align*}


\begin{align*} \sum _ { k = N } ^ { \infty } | b _ k | r ^ k \leq \sum _ { n = N } ^ { \infty } | a _ n | r ^ n = \sum _ { n = N } ^ { \infty } \frac { | \tilde { b } _ n | } { n } r ^ n , \end{align*}


\begin{align*} \int _ { \Xi } x ^ { n r + k } e ^ { x ^ r / \hbar } d x = { } & ( - 1 ) ^ n \hbar ^ n \left ( \prod _ { i = 1 } ^ n ( i - 1 + \frac { k + 1 } { r } ) \right ) \int _ { \Xi } x ^ { k } e ^ { x ^ r / \hbar } d x \\ = { } & ( - 1 ) ^ n \hbar ^ n \frac { \Gamma \left ( n + \frac { k + 1 } { r } \right ) } { \Gamma \left ( \frac { k + 1 } { r } \right ) } \int _ { \Xi } x ^ { k } e ^ { x ^ r / \hbar } d x . \end{align*}


\begin{align*} \begin{aligned} x _ 1 = \exp ( 2 \pi i z / \overline \omega _ 1 ) , & x _ 2 = \exp ( 2 \pi i z / \omega _ 2 ) , \\ q _ 1 = \exp ( 2 \pi i { \omega _ 2 } / { \overline \omega _ 1 } ) , & q _ 2 = \exp ( 2 \pi i { \omega _ 1 } / { \omega _ 2 } ) , \widetilde { q } _ 2 = \exp ( 2 \pi i { \widetilde \omega _ 1 } / { \omega _ 2 } ) . \end{aligned} \end{align*}


\begin{align*} \underline B ' : = \sup _ { s > 0 } \left ( \int _ s ^ \infty V ( t ) ^ { - \frac { 1 } { p - 1 } } d t \right ) ^ { - 1 } \int _ s ^ \infty W ( t ) \left ( \int _ t ^ \infty V ( t ' ) ^ { - \frac { 1 } { p - 1 } } d t ' \right ) ^ p d t \ , . \end{align*}


\begin{align*} V = - \operatorname { g r a d } p \end{align*}


\begin{align*} \Psi ( t _ z , n ) = \log ( z ) - 2 \pi i \cdot n . \end{align*}


\begin{align*} \rho = \frac { 1 } { 2 } \sum _ { \alpha \in \Delta _ { 0 } ^ { + } } \alpha - \frac { 1 } { 2 } \sum _ { \alpha \in \Delta _ { 1 } ^ { + } } \alpha . \end{align*}


\begin{align*} \bigcirc _ { w = 0 } ^ i ( \theta _ t + \gamma _ { r - i + w } ) ^ { - 1 } \circ ( \theta _ t + \xi _ { i , 1 } ) \circ \cdots \circ ( \theta _ t + \xi _ { i , \ell _ i } ) = \bigcirc _ { w ' = 0 } ^ j ( \theta _ t + \gamma _ { r - j + w ' } ) ^ { - 1 } \circ ( \theta _ t + \xi ' _ { j , 1 } ) \circ \cdots \circ ( \theta _ t + \xi ' _ { j , \ell ' _ j } ) \enspace , \end{align*}


\begin{align*} \partial \Omega \cap \mathbb { R } = A \cup B \cup C . \end{align*}


\begin{align*} T u ( \Omega ) : = Y ( \cdot , u , D u ) ( \Omega ) = \Omega ^ * , \end{align*}


\begin{align*} Q _ { n e w } = \alpha Q _ { o l d } \end{align*}


\begin{align*} \sum _ { l = 0 } ^ m h _ l ( \lambda ^ 2 + k ^ 2 ) ( k \lambda ) ^ l = 0 , \ ; \forall \vert k \vert \leq m , \lambda \in \mathfrak { a } ^ * _ \mathbb { C } , h _ l = 0 , l = 0 , \dots , m . \end{align*}


\begin{align*} C ^ * ( \mathbf { T } ) = C ( \mathbf { T } ) = \| \mathbf { T } ^ { - 1 } \| ^ { - 1 } \end{align*}


\begin{align*} \mathcal { E } : = N ^ { n \xi } \left ( \frac { N ^ \epsilon \ell } { K } + \frac { N T _ 1 } { \ell } + \frac { N \eta } { \ell } + \frac { N ^ \epsilon } { \sqrt { N \eta } } + \frac { 1 } { \sqrt { K } } \right ) , \end{align*}


\begin{align*} \mathcal { L } ( D _ { \beta } ( \varphi , x _ 0 ) ) = \begin{cases} 0 & \sum _ { n = 1 } ^ { \infty } \varphi ( n ) < \infty , \\ 1 & \sum _ { n = 1 } ^ { \infty } \varphi ( n ) = \infty . \end{cases} \end{align*}


\begin{align*} ( F * G ) ( n _ 1 , \ldots , n _ k ) = \sum _ { d _ 1 \mid n _ 1 , \ldots , d _ k \mid n _ k } F ( d _ 1 , \ldots , d _ k ) G ( n _ 1 / d _ 1 , \ldots , n _ k / d _ k ) , \end{align*}


\begin{align*} \vec s = ( s _ 1 , \ldots , s _ n ) \in \mathbb Z ^ n . \end{align*}


\begin{align*} \lim _ { z \to \infty } \frac { F _ { \mu } ( z ) } { z } = 1 , \end{align*}


\begin{align*} w _ m ( x ) = r _ { k _ 1 } ( x ) \ldots r _ { k _ l } ( x ) \end{align*}


\begin{align*} T _ { i j } ^ k = \delta _ i ^ k \rho _ j - \delta _ j ^ k \rho _ i . \end{align*}


\begin{align*} \varphi _ t ( k , v ) = k \exp _ G ( t v ) , \ ( k _ 1 , v _ 1 , t ) \cdot _ t ( k _ 2 , v _ 2 , t ) = \varphi ^ { - 1 } _ t \big ( \varphi _ t ( k _ 1 , v _ 1 ) \cdot \varphi _ t ( k _ 2 , v _ 2 ) \big ) . \end{align*}


\begin{align*} \mathcal { H } _ { \varphi } ( v ) = \frac { 1 } { \mathcal { F } } \left \lbrace \frac { \partial ^ 2 \mathcal { F } } { \partial z ^ i _ { \epsilon } \partial z ^ j _ { \eta } } \frac { \partial ^ 2 \mathcal { \varphi } ^ j } { \partial x ^ { \epsilon } \partial x ^ { \eta } } \right \rbrace v ^ i . \end{align*}


\begin{align*} g ( g r a d ~ f , X ) = X ( f ) , ~ f o r ~ X \in \Gamma ( T M ) . \end{align*}


\begin{align*} ( [ x , y ] , z , w ) = 0 \mbox { i f } \mathsf { c h a r } ( F ) \neq 2 , 3 . \end{align*}


\begin{align*} \hat { s } _ { g } \left ( q , r , \tau _ t \right ) & = \\ & \frac { q \exp \left ( - \frac { \left ( r - d _ { g } \right ) ^ 2 } { 2 \tau _ t ^ 2 } \right ) } { q \exp \left ( - \frac { \left ( r - d _ { g } \right ) ^ 2 } { 2 \tau _ t ^ 2 } \right ) + ( 1 - q ) \exp \left ( - \frac { r ^ 2 } { 2 \tau _ t ^ 2 } \right ) } , \\ \end{align*}


\begin{align*} j _ \delta ' ( u _ \delta ) h & = J _ y ( y _ \delta , u _ \delta ) S _ \delta ' ( u _ \delta ; h ) + J _ u ( y _ \delta , u _ \delta ) h + ( u - \overline { u } , h ) \\ & = ( y _ \delta - z _ d , S _ \delta ' ( u _ \delta ; h ) ) + \alpha ( u , h ) + ( u - \overline { u } , h ) \end{align*}


\begin{align*} \dot { \hat { \tau } } ^ * = - \frac { \alpha _ x ( 2 \nu _ x \tau ^ * + \xi _ x ) + \alpha _ y ( 2 \nu _ y \tau ^ * + \xi _ y ) + \nu _ x ^ 2 + \nu _ y ^ 2 } { \nu _ x ^ 2 + \nu _ y ^ 2 + \varepsilon } \end{align*}


\begin{align*} \begin{aligned} h _ * = \arg \max _ { 1 { } \leq { } h { } \leq { } \bar { h } ( \kappa ) } 2 h + \frac { \kappa h ^ 3 } { 2 - h \left ( 1 + \kappa \right ) } \end{aligned} \end{align*}


\begin{align*} T = \left \{ t _ z : z \in \C ^ \times \right \} , t _ z = \begin{pmatrix} \frac { 1 } { \bar z } \\ & \frac { \bar z } { z } \\ & & z \end{pmatrix} . \end{align*}


\begin{align*} ( \beta + 3 ) d ' - \beta d + 1 = 2 i \leq 2 d . \end{align*}


\begin{align*} \rho _ { s t } ( t ^ { 1 / \alpha } x , t ^ { 1 / \alpha } y ) = t ^ { \frac { 2 \delta - d } { \alpha } } \rho _ s ( x , y ) , s > 0 . \end{align*}


\begin{align*} \lim _ { x \to \infty } \sup _ { y : \| y \| _ d \leq b ( x ) } \bigl | \mu ( x , y ) - \gamma \sigma ^ 2 \bigr | = 0 . \end{align*}


\begin{align*} \bigl ( ( y _ 6 - y _ 5 - 2 z _ 6 , y _ 7 - y _ 5 - 2 z _ 7 , y _ 8 - y _ 5 - 2 z _ 8 ) \cdot ( d ) \cdot D _ 1 \cdots D _ 4 \bigr ) ^ r \subseteq J \ k = 5 , \end{align*}


\begin{align*} M ^ { n , 3 } _ s = \int _ { \eta _ n ( s ) - \delta } ^ { \eta _ n ( s ) } \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u . \end{align*}


\begin{align*} \left | \frac { 1 } { \sqrt { m } } - y _ m \right | & = \frac { 1 } { | \frac 1 { \sqrt { m } } + y _ m | } \left | \frac { 1 } { m } - y _ m ^ 2 \right | \leq \sqrt { m } \left | \frac { 1 } { m } - \left ( 1 - \sum _ { j \le { m - 1 } } y _ j ^ 2 \right ) \right | \cr & \leq 3 \sum _ { j \le m - 1 } \left | y _ j - \frac { 1 } { \sqrt { m } } \right | < 3 ( m - 1 ) \frac { 1 } { 3 0 0 m \cdot x \sqrt { m } } . \cr & < \frac { 1 } { 1 0 0 x \sqrt { m } } . \end{align*}


\begin{align*} E _ { \mbox { \tiny M D } } ( \theta ) = \sup _ { \lambda \ge 0 } \sup _ { P \le P _ w } \left \{ \lambda ( \sqrt { P _ s P } - \theta ) - C \left ( \lambda \sqrt { P } \right ) - \frac { \lambda ^ 2 \sigma _ N ^ 2 P } { 2 } \right \} . \end{align*}


\begin{align*} D ^ { 1 } \left ( \begin{array} { c } u \\ 0 \\ v \end{array} \right ) = \left ( \begin{array} { c } 0 \\ 0 \\ ( d T ) ^ { 2 } d u + d { v } \end{array} \right ) , \end{align*}


\begin{align*} \frac { d } { d x } e ^ { x ( e _ { \lambda } ( t ) - 1 ) } & = ( e _ { \lambda } ( t ) - 1 ) e ^ { x ( e _ { \lambda } ( t ) - 1 ) } \\ & = \sum _ { n = 0 } ^ { \infty } \bigg ( \sum _ { k = 0 } ^ { n } \binom { n } { k } ( 1 ) _ { n - k , \lambda } \phi _ { k , \lambda } ( x ) - \phi _ { n , \lambda } ( x ) \bigg ) \frac { t ^ { n } } { n ! } . \end{align*}


\begin{align*} d \mu _ t : = u ( \cdot , t ) d g _ t : = ( 4 \pi | t | ) ^ { - \frac { n } { 2 } } e ^ { - f _ t } d g _ t , t \in I \end{align*}


\begin{align*} Y _ R ( P , G ) \geq C ( P ) > 1 = \tilde { Y } _ R ( P , G ) \end{align*}


\begin{align*} x ' = \Sigma x _ { \beta _ i } ' = \Sigma c _ { i } x _ { \beta _ i } \end{align*}


\begin{align*} u _ { A , q } ^ s ( x , d ) = \frac { e ^ { i k \vert x \vert } } { \vert x \vert } \Big ( u _ { A , q } ^ \infty ( \hat { x } , d ) + O \Big ( \frac { 1 } { \vert x \vert } \Big ) \Big ) , \end{align*}


\begin{align*} \mathcal { G } ( n + 1 ) = \mathcal { G } ( n ) \oplus 1 \ ; \ n \neq 0 \bmod 2 c \ ; \ \ ; \ n \neq 4 c + 1 \bmod 1 2 c \end{align*}


\begin{align*} F _ { \phi _ 1 , \dots , \phi _ n } ( \mu ) = \prod _ { i = 1 } ^ n \langle \phi _ i , \mu \rangle . \end{align*}


\begin{align*} \frac { \mathrm { d } r ^ \pm } { \mathrm { d } \theta } \ ! = \ ! \frac { \sum _ { k = 2 } ^ { n } R ^ \pm _ k ( \theta ) r ^ k } { 1 + \sum _ { k = 2 } ^ { n } \Theta ^ \pm _ k ( \theta ) r ^ { k - 1 } } = \sum _ { k = 2 } ^ { \infty } S ^ \pm _ k ( \theta ) r ^ k , \end{align*}


\begin{align*} G ( V , V ) G ( \mbox { g r a d } f , P _ U U ) = G ( U , U ) G ( \mbox { g r a d } f , P _ V V ) , \end{align*}


\begin{align*} \sum _ { j = m } ^ { \infty } c _ j = O \left ( v \left ( \frac { 1 } { m } \right ) \right ) , \end{align*}


\begin{align*} \left \{ \small { \begin{pmatrix} 1 & 0 \\ 0 & U \end{pmatrix} } : \ U \in S U ( 2 ) \right \} \subset S U ( 3 ) \end{align*}


\begin{align*} g ^ { i j } = \frac { 1 } { f ( u ) ^ 2 } \widetilde { g } ^ { i j } + \frac { 1 } { f ( u ) ^ 2 } \frac { \widetilde { g } ^ { j l } u _ l \widetilde { g } ^ { i m } u _ m } { f ( u ) ^ 2 - | \widetilde { \nabla } u | ^ 2 } . \end{align*}


\begin{align*} c _ 2 | t | ^ { q ( x ) } \leqslant F ( x , t ) : = \int _ { 0 } ^ { t } f ( x , \tau ) d \tau , \end{align*}


\begin{align*} \frac { d \big ( \phi ^ c _ { - t } \big ) _ * \nu ^ { \ss } _ p } { d \nu ^ { \ss } _ { \phi ^ c _ t p } } = e ^ { - \delta t } . \end{align*}


\begin{align*} r ( g ) = d - 1 , \end{align*}


\begin{align*} \bar { \Omega } ( \theta ) = \bar { \psi } _ \theta ( \mathbb { H } \setminus \Sigma _ B ) \cup \bigcup _ { \gamma \in \bar { \Gamma } ( \theta ) } \gamma G _ \theta . \end{align*}


\begin{align*} \widetilde { \nabla } ( \theta , \omega _ { \theta } ) = 0 . \end{align*}


\begin{align*} D \stackrel { \eqref { l o w e r } } { > } C F - D E \geq ( E + 1 ) F - F E = F \geq D , \end{align*}


\begin{align*} \int _ 1 ^ x I _ 2 { \rm d } x \leq \delta _ 0 ( \Vert ( \varphi , \psi ) x ^ { - \frac { 3 } { 4 } - \sigma _ i } \Vert _ { L _ x ^ 2 L _ y ^ 2 } ^ 2 + \Vert ( u , h ) x ^ { - \frac { 3 } { 4 } - \sigma _ i } \Vert _ { L _ x ^ 2 L _ y ^ 2 } ^ 2 ) + C . \end{align*}


\begin{align*} \nabla K _ { j } ( 0 ) = 0 , \end{align*}


\begin{align*} w _ 2 ( \pi ) = \begin{cases} \dfrac { m _ { \pi } } { 2 } a _ 2 , & q \equiv 1 \mod 4 , \\ \\ \dbinom { m _ \pi } { 2 } ( \sum v _ i ^ 2 ) , & q \equiv 3 \mod 4 , \end{cases} \end{align*}


\begin{align*} C _ a ^ v \ : = \ \big \{ c \in \N \ ; : \ ; p \mid c \implies p \in [ P ( a ^ * ) , P ( a ^ * ) ^ { 1 / \sqrt { v } } ) \big \} . \end{align*}


\begin{align*} g ( \underline { X } , \underline { Y } ) = g ( \alpha , \beta , \underline { X } , \underline { Y } ) = \prod _ { { { 0 } } \leq i , j \leq { { r - 1 } } } ( X _ i - Y _ j ) \prod _ { { { 0 } } \leq i < j \leq { { r - 1 } } } [ ( X _ j - X _ i ) ( Y _ j - Y _ i ) ] \enspace . \end{align*}


\begin{align*} ( - \Delta ) ^ s u ^ * + \sum _ { | \alpha | \leq m } ( - 1 ) ^ { | \alpha | } D ^ \alpha ( a _ \alpha u ^ * ) & = 0 \mbox { i n } \ ; \ ; \Omega , u ^ * - f \in \widetilde H ^ s ( \Omega ) \end{align*}


\begin{align*} \left | \underset { \Gamma _ { 2 p - 1 } } { \int \sum } \overline { \widehat { \eta } } _ k \widehat { ( W _ t u _ 0 ) } _ { k _ 1 } \prod _ { \substack { j = 3 \\ o d d } } ^ { 2 p - 1 } \widehat { v } _ { k _ j } \prod _ { \substack { l = 2 \\ e v e n } } ^ { 2 p - 1 } \overline { \widehat { v } } _ { k _ l } \ , d \Gamma \right | \lesssim \| \eta \| _ { X ^ { - s - \varepsilon , 1 / 2 - } } \| u _ 0 \| _ { H ^ s _ x } \| v \| ^ { 2 p - 2 } _ { X ^ { s , 1 / 2 + } _ T } . \end{align*}


\begin{align*} \sum _ { h = 1 } ^ H \frac { 1 } { h \| h \alpha \| } \ll \left \{ \begin{array} { l l } \log ^ 2 H & \mathrm { i f } \ , \ , \ , \gamma = 1 , \\ H ^ { \gamma - 1 } & \mathrm { i f } \ , \ , \ , \gamma > 1 \end{array} \right . \end{align*}


\begin{align*} \| u \| _ { X ^ p _ \Delta ( I ; E ) } = \| e ^ { - i t \Delta } u \| _ { X ^ p ( I ; E ) } \end{align*}


\begin{align*} | R _ N ^ 1 ( s , n ) | & \leqslant C \log ( n ) \int _ 0 ^ n z ^ { 2 \alpha - 2 s - 1 } \leq C , \ \textup { f o r } \ N = M + 1 , \\ | R _ { N ' } ^ 2 ( s , n ) | & \leqslant C \log ( n ) \int _ 0 ^ n z ^ { 2 \alpha - 2 s - 1 } \leq C , \ \textup { f o r } \ N ' = 2 ( M + 1 ) . \end{align*}


\begin{align*} f ^ \wedge \colon U \times E _ 1 \times \cdots \times E _ k \to F \ , , f ^ \wedge ( x , v ) : = f ( x ) ( v ) \quad \mbox { f o r $ x \in U $ , $ v \in E _ 1 \times \cdots \times E _ k $ } \end{align*}


\begin{align*} h ( q ^ n x ) = \frac { ( q x ; q ) _ n } { x ^ n } h ( x ) ( n \in \mathbb { N } _ 0 ) . \end{align*}


\begin{align*} f ( A _ n ) = \sum _ { a \in A _ n } \frac { 1 } { a \log a } \ & \le \ \frac { e ^ { \gamma } } { m _ q } \ , \frac { { \rm d } ( { \rm L } _ { A _ n } ) } { 1 + v } . \end{align*}


\begin{align*} \prod _ { i \in J } \varpi _ i = \prod _ { k = 1 } ^ m \Big ( \prod _ { i \in J _ k } \varpi _ i \Big ) = \prod _ { k = 1 } ^ m \Big ( | J _ k | ! e _ { | J _ k | } ( x _ 1 , x _ 2 , \ldots , x _ { \max J _ k } ) \Big ) \end{align*}


\begin{align*} \beta _ 1 ( z ) & : = \displaystyle \sum _ { m = 1 } ^ { s _ 1 } \frac { z ^ m } m , \\ \beta _ i ( z ) & : = \displaystyle \beta _ { i - 1 } ( z ) + \sum _ { m = 1 } ^ { s _ i } \frac { ( 1 + ( z - 1 ) \exp ( \beta _ { i - 1 } ( z ) ) ) ^ m } m , 2 \le i \le q . \end{align*}


\begin{align*} J ( \gamma , z ) = \frac { \theta ^ * ( \gamma z ) } { \theta ^ * ( z ) } , \theta ^ * ( z ) = y ^ { 1 / 4 } \sum _ { n \in \Z } e ( n ^ 2 z ) , \end{align*}


\begin{align*} { \bf U } ^ k ( \alpha _ i ) = \displaystyle \sum _ { j = 1 } ^ { N _ h } T _ j ^ k ( \alpha _ i ) { \boldsymbol \xi } _ j k = 1 , \ldots , m , \ i = 1 , \ldots , p . \end{align*}


\begin{align*} I _ 4 & = \int _ 0 ^ r - \frac { z } { 2 } ( e _ \sigma + \bar \psi ) \cdot ( e ' _ \sigma \phi + e ' _ \delta \psi ) { \rm { d } } z \\ & \leq \frac { 1 } { 2 } \Vert ( \phi , \psi ) \Vert _ { L ^ 2 } \cdot \Vert z ( e ' _ \delta , e ' _ \sigma ) \Vert _ { L ^ 2 } \cdot \left ( \Vert e _ \sigma \Vert _ { L ^ \infty } + \Vert \bar \psi \Vert _ { L ^ \infty } \right ) , \end{align*}


\begin{align*} \sigma _ k ( \lambda ( D ^ 2 u ) ) = 1 \end{align*}


\begin{align*} S \not \subseteq \bigcup _ { i = 1 } ^ n B ( x _ i , \sin \theta ) . \end{align*}


\begin{align*} \binom { j } { k _ 1 , k _ 2 , \ldots , k _ { m } } = \frac { j ! } { k _ 1 ! k _ 2 ! \cdots k _ { m } ! } . \end{align*}


\begin{align*} h ( \vec x , \vec p ) = \sum _ { i = 1 } ^ 3 \left ( \dfrac { p _ i ^ 2 } { 2 m } + \dfrac { 1 } { 2 } m \omega x _ i ^ 2 \right ) \ , \end{align*}


\begin{align*} - \gamma \leq J _ { 2 } \leq - \int _ { B _ { R - 2 } \setminus B _ { 6 } } \frac { d y } { | y | ^ { n + \sigma p } } = - ( 6 ^ { - \sigma p } - ( R - 2 ) ^ { - \sigma p } ) \gamma . \end{align*}


\begin{align*} Z = f ( p \partial _ { 1 } + q \partial _ { 2 } ) . \end{align*}


\begin{align*} { H } ^ \Lambda = F _ \Lambda ^ * \big ( \boxplus _ { i = 1 } ^ v H ^ { \Xi _ i } \big ) , \end{align*}


\begin{align*} g ( ( \nabla _ X A ) _ W X , W ) = g ( \nabla _ X A _ W X , W ) - g ( A _ { \mathcal { H } \nabla _ X W } X , W ) - g ( A _ W A _ X X , W ) . \end{align*}


\begin{align*} s x y & = \pi ( s ^ \prime x ^ \prime y ^ \prime ) & ( \ ; \pi \ ; ) \\ & = \pi ( x ^ \prime y ^ \prime t ^ \prime ) & ( \ ; s ^ \prime x ^ \prime y ^ \prime = x ^ \prime y ^ \prime t ^ \prime ) \\ & = x y t , & \end{align*}


\begin{align*} \sum _ { \gamma \in \mathcal { B } _ { n } } w ( \gamma ) = \sum _ { \gamma \in \mathcal { B } _ { n } } w ( \gamma _ { 1 } ) w ( \gamma _ { 2 } ) w ( \gamma _ { 3 } ) w ( \gamma _ { 4 } ) = a _ { 0 } \sum _ { \gamma \in \mathcal { B } _ { n } } w ( \gamma _ 2 ) w ( \gamma _ 4 ) = a _ { 0 } \sum _ { k = 0 } ^ { n - 2 } A _ { k } ^ { ( 1 ) } A _ { n - k - 2 } , \end{align*}


\begin{align*} d ^ { ( \infty ) } ( t ) = n p _ t ( o , o ) - 1 . \end{align*}


\begin{align*} h \left ( \left \{ n , \dots , n + M - 1 \right \} \right ) = h \left ( \mathbb { N } \right ) . \end{align*}


\begin{align*} M _ 4 = f _ k ( j ) f _ k ( j + 1 ) + O _ { \alpha } \Big ( \frac { \tau _ 3 ( j ) \tau _ 3 ( j + 1 ) } { \sqrt { j - i - 1 } } \Big ) . \end{align*}


\begin{align*} \boldsymbol { \mu } = \left ( \begin{array} { c c c c c c c c c c c } 3 \\ 4 \end{array} \right ) , \mathbf { \Sigma } = \left ( \begin{array} { c c c c c c c c c c c } 0 . 9 & 0 . 5 \\ 0 . 5 & 0 . 7 \end{array} \right ) , ~ \boldsymbol { \gamma } = \left ( \begin{array} { c c c c c c c c c c c } 1 \\ 2 \end{array} \right ) . \end{align*}


\begin{align*} Z = \left \{ P \in G _ { n , s , m } \ , \colon \ , \omega \big | _ { P } \equiv 0 \right \} \end{align*}


\begin{align*} \varphi \in \prescript { } { m } { } _ { m } h _ l \in ( \mathbb { C } ) , \ ; \ ; \ ; l = 0 , \dots , m . \end{align*}


\begin{align*} \beta : = \limsup _ { x \to \infty } \frac { x b ' ( x ) } { b ( x ) } . \end{align*}


\begin{align*} R _ { i j k l } = \bar { R } _ { \alpha \beta \gamma \delta } X _ { ; i } ^ \alpha X _ { ; j } ^ \beta X _ { ; k } ^ \gamma X _ { ; l } ^ \delta + h _ { i l } h _ { j k } - h _ { i k } h _ { j l } , \end{align*}


\begin{align*} \sqrt { n } ( \hat { \mu } _ n - \mu ) * \gamma _ { \sigma } = \tau _ { a } \big ( \sqrt { n } ( \hat { \mu } _ n ^ { - a } - \mu ^ { - a } ) * \gamma _ { \sigma } \big ) \stackrel { d } { \to } \tau _ a G ^ { \circ } _ { \mu ^ { - a } } = : G _ { \mu } \ \dot { H } ^ { - 1 , p } ( \mu * \gamma _ { \sigma } ) . \end{align*}


\begin{align*} \zeta _ 1 = \sqrt { \frac { 1 } { \mathfrak { A } t } } x _ 1 , \zeta _ 2 = \sqrt { \frac { 1 } { \mathfrak { A } t } } x _ 2 , \end{align*}


\begin{align*} \int _ 0 ^ \infty \left ( \frac { 1 } { y } \int _ 0 ^ y f ( t ) { \rm d } t \right ) ^ p y ^ \alpha { \rm d } y \leq \left ( \frac { p } { p - \alpha - 1 } \right ) ^ p \int _ 0 ^ \infty f ( y ) ^ p y ^ \alpha { \rm d } y . \end{align*}


\begin{align*} F _ 1 & = \frac { F } { 1 - F } \intertext { s o } \frac { F } { F _ 1 } & = 1 - F . \end{align*}


\begin{align*} \tau _ k = \begin{cases} t - \delta _ { n _ k + 1 } & n _ k < \infty , \\ t & n _ k = \infty . \end{cases} \end{align*}


\begin{align*} \lambda _ n = \frac { 1 } { n } \delta _ { n - 1 } + \frac { n - 1 } { n } \lambda _ { n - 1 } , n \geq 2 . \end{align*}


\begin{align*} P _ F ( t ) \cdot \hat { L } _ { x y } = \frac { 1 } { n } \left ( \frac { 3 } { 4 ( 4 n + 1 ) } - t \right ) , \ \ P _ F ( t ) \cdot \hat { R } _ 0 = P _ F ( t ) \cdot \hat { R } _ 1 = \frac { 3 } { 4 ( 4 n + 1 ) } - t . \end{align*}


\begin{align*} \Lambda = \| \mathcal G \| ^ \frac { 1 } { p - 1 } _ { q _ 0 , B _ 2 ( 0 ) } < + \infty \end{align*}


\begin{align*} R _ \mathrm { r a t } ( G ) = \big \{ \rho \in R ( G ) \ ; : \ ; \chi _ \rho ( g ) \in \Q g \in G \big \} \ ; . \end{align*}


\begin{align*} m = n _ l ( m ) - n _ { l - 1 } ( m ) - \ldots - n _ { 1 } ( m ) \end{align*}


\begin{align*} y ^ * = b \widetilde { z } + \sum _ { j = 1 } ^ m \left ( c _ j \widetilde { x } ^ k _ j + v _ j ^ * \right ) , \end{align*}


\begin{align*} \begin{aligned} \sum _ { i \in I } 2 d _ i + \frac { 2 \sum _ { i \in I ( \Gamma ) } a _ i + ( k _ 1 ( \Gamma ) + k _ { 1 2 } ( \Gamma ) - 1 ) ( r - 2 ) } { r } & + \frac { 2 \sum _ { i \in I ( \Gamma ) } b _ i + ( k _ 2 ( \Gamma ) + k _ { 1 2 } ( \Gamma ) - 1 ) ( s - 2 ) } { s } \\ & = 2 | I | + k _ 1 ( \Gamma ) + k _ 2 ( \Gamma ) + k _ { 1 2 } ( \Gamma ) - 3 . \end{aligned} \end{align*}


\begin{align*} \alpha _ 1 & \lesssim \int _ 0 ^ \infty e ^ { \mu _ 1 s _ 1 } \{ ( \lambda \dot \lambda - \sigma ) + \lambda ^ \frac { n - 2 } { 2 } \eta ^ \frac { 2 } { 1 - q } ( T - t ) ^ { { \sf d } _ 1 } + ( T - t ) ^ { { \sf d } _ 1 } \sigma \} d s _ 1 \\ & \lesssim ( T - t ) ^ { { \sf d } _ 1 } \sigma | _ { t = 0 } \int _ 0 ^ \infty e ^ { \mu _ 1 s _ 1 } d s _ 1 \\ & \lesssim ( T - t ) ^ { { \sf d } _ 1 } \sigma | _ { t = 0 } . \end{align*}


\begin{align*} ( X , { \delta _ X } \dot { D f } \wedge \delta _ X D g ) & = ( X , { \delta _ X } { D ^ 2 f } ( \dot X , \cdot ) \wedge \delta _ X D g ) = - ( X , \delta _ X D g \wedge { \delta _ X } { D ^ 2 f } ( \dot X , \cdot ) ) \\ & = - ( d _ X \rho _ X \delta _ X d D g , { D ^ 2 f } ( \dot X , \cdot ) ) = - ( \nu _ { X } D g , { D ^ 2 f } ( \dot X , \cdot ) ) \\ & = - ( \dot X , { D ^ 2 f } ( \nu _ { X } D g , \cdot ) ) \end{align*}


\begin{align*} \beta ( s ) = \frac { 3 } { 4 } \zeta ( s ) - 2 \sum _ { n \geq 1 } \frac { 1 } { ( 4 n + 3 ) ^ s } , \end{align*}


\begin{align*} v = g x ^ a y ^ b ( ( b + 1 ) x \partial _ x - ( a + 1 ) y \partial _ y ) \in \mathfrak { g } _ { A , ( a , b ) } , \end{align*}


\begin{align*} d _ { \mathrm { C C } } ( a g , a h ) = d _ { \mathrm { C C } } ( g , h ) \quad a , g , h \in G . \end{align*}


\begin{align*} F _ { \mu } ( z ) + \varphi _ { \mu } ( F _ { \mu } ( z ) ) = z \end{align*}


\begin{align*} F ( x ; \sigma , y , Z ) : = f ( x ) + \frac { 1 } { 2 \sigma } \norm { \sigma y - g ( x ) } ^ 2 + \frac { 1 } { 2 \sigma } \norm { \left [ \sigma Z - X ( x ) \right ] _ { + } } _ { \mathrm { F } } ^ 2 , \end{align*}


\begin{align*} S _ i ( t ) = \{ S _ { s , i } ( t ) \cap S _ { 0 , i } ( t ) \} , \end{align*}


\begin{align*} x = [ 0 ; a _ 1 , a _ 2 , \dots ] = \smash { \frac 1 { a _ 1 + \frac 1 { a _ 2 + \frac 1 { a _ 3 + \cdots } } } } \phantom { \frac 9 { 9 _ y } } \end{align*}


\begin{align*} g _ { i j } = - u _ i u _ j + f ( u ) ^ 2 \widetilde { g } _ { i j } . \end{align*}


\begin{align*} ( n I _ n - J _ n ) \cdot \begin{bmatrix} h _ { 1 , 1 1 } \\ h _ { 1 , 2 2 } \\ \cdots \\ h _ { 1 , n n } \end{bmatrix} = ( n I _ n - J _ n ) \cdot \begin{bmatrix} a _ { 1 1 } \\ a _ { 2 2 } \\ \cdots \\ a _ { n n } \end{bmatrix} . \end{align*}


\begin{align*} u ^ 0 ( x , t ) = \frac { 1 } { 2 } \{ f ( x + t ) + f ( x - t ) \} \ge \frac { 1 } { 2 } f ( x - t ) \mbox { f o r } \ ( x , t ) \in \R \times [ 0 , \infty ) . \end{align*}


\begin{align*} \frac { d g } { d t } = \sum _ { \alpha = 1 } ^ r b _ \alpha ( t ) X _ \alpha ^ R ( g ) , g \in G , t \in \mathbb { R } , \end{align*}


\begin{align*} u _ { m } ( x , { \bf y } ; \alpha ) = \displaystyle \sum _ { k = 1 } ^ { m } \sum _ { j = 1 } ^ { N _ h } { \tilde u } _ { k , j } ^ \alpha \vartheta _ j ( x ) \varphi _ k ( \psi _ x ( { \bf y } ) ) \end{align*}


\begin{align*} Z = f ( r , \theta ) \partial _ { \theta } . \end{align*}


\begin{align*} S _ j = R _ j D _ j P _ j ^ T , \end{align*}


\begin{align*} \frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) + p ( x ) D _ q y ( x ) + r ( x ) y ( x ) = 0 , \end{align*}


\begin{align*} s ( g ) = d - 2 , \end{align*}


\begin{align*} \label [ I e q ] { e q : m a i n j u m p g e n e r a t o r } L _ \mu \varphi ( x ) = a \cdot \nabla \varphi ( x ) + \lambda ( x , \mu ) \int _ { E } \{ \varphi ( y ) - \varphi ( x ) \} P _ \mu ( x , \dd y ) , \end{align*}


\begin{align*} \sum _ { \gamma \in \mathcal { P } _ { [ n , j , k ] } } w ( \gamma ) = W _ { [ k , i ] } A _ { [ n - k - 1 , j - i - 1 ] } ^ { ( i + 1 ) } . \end{align*}


\begin{align*} T ^ { i - 1 , j + 1 } \circ T ^ { i , j } = 0 , \end{align*}


\begin{align*} F ( t ) & = P f ( \cdot , t ) , \\ F _ n ( t ) & = P _ n f ( \cdot , t ) . \end{align*}


\begin{align*} a _ i ^ { - 1 } = \frac { \left ( \AA ( i ) \right ) ^ i } { i ! } \geq \exp \left \{ - \AA ( m - i , m ) \right \} \frac { \left ( \AA ( m - i , m ) \right ) ^ { i } } { i ! } . \end{align*}


\begin{align*} \left \lbrace \begin{array} { c } \phi ( + a / 2 ) = e ^ { i \varphi } . \phi ( - a / 2 ) \\ \partial _ { x } \phi ( + a / 2 ) = e ^ { i \varphi } . \partial _ { x } \phi ( - a / 2 ) \\ \partial _ { x } ^ { 2 } \phi ( + a / 2 ) = e ^ { i \varphi } . \partial _ { x } ^ { 2 } \phi ( - a / 2 ) \\ \partial _ { x } ^ { 3 } \phi ( + a / 2 ) = e ^ { i \varphi } . \partial _ { x } ^ { 3 } \phi ( - a / 2 ) \end{array} \right . . \end{align*}


\begin{align*} ( J _ m ^ k f ) ( x , r \xi ) & = \frac { r ^ { m - k } } { | r | } ( J _ m ^ k f ) ( x , \xi ) \mbox { f o r } r \neq 0 \\ ( J _ m ^ k f ) ( x + s \xi , \xi ) & = \sum \limits _ { l = 0 } ^ k \binom { k } { l } ( - s ) ^ { k - l } ( J _ m ^ l f ) ( x , \xi ) \mbox { f o r } s \in \mathbb { R } . \end{align*}


\begin{align*} { \sf t } _ 9 & = \{ f ( { \sf U } _ \infty + \Theta _ { \sf a } + \theta ) - f _ 2 ( { \sf U } _ \infty + \Theta _ { \sf a } + \theta ) + f _ 2 ( { \sf U } _ \infty ) + { \sf U } _ \infty ^ { q - 1 } ( \Theta _ { \sf a } + \theta ) + P _ 1 \theta \} \\ & \times ( 1 - \chi _ { 2 , { \sf a } } ) \chi _ { 3 , { \sf e } } . \end{align*}


\begin{align*} \phi ( g ) = \int _ { 0 } ^ { \infty } 1 _ { \phi ^ { - 1 } ( \mathbb { R } _ { > t } ) } ( g ) d t \end{align*}


\begin{align*} \displaystyle \frac { \displaystyle \ \ \sum _ { j = 1 } ^ L \lambda _ j ^ 2 \ \ } { \displaystyle \sum _ { j = 1 } ^ { N _ h } \lambda _ j ^ 2 } \ge \epsilon _ 1 , \frac { \ \ \displaystyle \sum _ { j = 1 } ^ { \mu _ j } d _ { j , k } ^ 2 \ \ } { \displaystyle \sum _ { j = 1 } ^ { m } d _ { j , k } ^ 2 } \ge \epsilon _ 2 , \end{align*}


\begin{align*} \theta = \log W + p ( u ) + N \rho , \end{align*}


\begin{align*} h _ e ^ 1 = \sigma u _ e ^ 1 . \end{align*}


\begin{align*} \int _ 0 ^ L \abs { u _ x } ^ 2 d x = o ( 1 ) . \end{align*}


\begin{align*} V _ \alpha ( s ) = 2 \ , \frac { \sin ( \pi s ) } { \pi } \frac { \Gamma ( 1 - s ) \Gamma ( \alpha ) } { \Gamma ( \alpha - s ) } . \end{align*}


\begin{align*} \omega > \tau \ \ \mbox { i f } \ \ \omega - \tau = c _ 1 \alpha _ 1 + c _ 2 \alpha _ 2 \ \ \mbox { f o r n o n n e g a t i v e i n t e g e r s } \ ( c _ 1 , c _ 2 ) \ne ( 0 , 0 ) . \end{align*}


\begin{align*} y _ n ^ * = y ^ * - \frac { 1 } { a } ( z + a f _ k ) ( y ^ * ) \widetilde { x } ^ k _ n , \end{align*}


\begin{align*} \frac { 1 } { N } \sum _ { k = 1 } ^ N f ( Z _ k ) \to \int _ 0 ^ 1 f ( x ) \ , \mathrm { d } x \textrm { a . s . } \end{align*}


\begin{align*} \int _ { G _ 1 } f \ , d \mu _ { G _ 1 } = \int _ { - \infty } ^ \infty \int _ { - \infty } ^ \infty f [ x , a ] \ , \frac { d x \ , d a } { a ^ 2 } , \quad f \in C _ c ( G _ 1 ) . \end{align*}


\begin{align*} a ( u , v ) = \lambda \ , b ( u , v ) v \in V \end{align*}


\begin{align*} \Phi ( E _ { \lambda , c } [ f ] ) \Delta = \Delta E _ { \lambda , c - \hbar } [ f ' ] , \ \Phi ( F _ { \lambda , c } [ f ] ) \Delta = \Delta E _ { \lambda , c - \hbar } [ f ' ] \end{align*}


\begin{align*} J ^ r ( R _ { \pi } [ y ] ) : = R _ { \pi } [ ( \delta ^ { ( s ) } ) _ \mu y _ j \ | \ \mu \in { \mathbb M } _ n ^ r , j \in \{ 1 , \ldots , N \} ] ^ { \widehat { \ } } \end{align*}


\begin{align*} \int _ { ( 0 , b _ 1 + \varepsilon ) \cup ( b _ 2 - \varepsilon , c _ 1 ) } \abs { v } ^ 2 d x + \int _ { ( 0 , b _ 1 ) \cup ( b _ 2 , c _ 1 ) } | u _ x | ^ 2 d x = o ( 1 ) . \end{align*}


\begin{align*} a _ j = \frac { \bar { \beta } _ 3 } { \beta _ j ( 2 \bar { \beta } _ 3 + d - 2 ) } \mbox { f o r a n y } j \in \{ 1 , . . . , d \} . \end{align*}


\begin{align*} \phi _ { \lambda C _ 1 , \lambda C _ 2 } = \lambda \circ \phi _ { C _ 1 , C _ 2 } \circ \lambda ^ { - 1 } \end{align*}


\begin{align*} ( x , x ) _ { \rho , L , n } = 0 . \end{align*}


\begin{align*} \sum _ { n \ge 0 } A _ n ( t ) \frac { z ^ n } { n ! } = \frac { t - 1 } { t - e ^ { ( t - 1 ) z } } . \end{align*}


\begin{align*} T _ n : = \left \{ t \in \R ^ { d _ 0 } : \abs { t } \leq C ^ { - 1 } ( 4 \big \| \mathbb { G } _ n ^ { ( \sigma ) } \big \| _ { \dot H ^ { - 1 , p } ( \mu * \gamma _ { \sigma } ) } + 2 \lambda _ n ) \right \} . \end{align*}


\begin{align*} \deg \mathcal { F } _ 1 ( v , x ) & = \deg \mathcal { F } _ 2 ( v , x ) + 1 . \end{align*}


\begin{align*} \Psi ^ { \ast } ( d i s t ^ { 2 } ( \Psi ( x ) , \mathbf { S } ^ { 3 } ) ) = d i s t ^ { 2 } ( x , \mathbf { S } _ { p } ^ { 1 } ) . \end{align*}


\begin{align*} \sum _ \xi m _ \xi [ \mathfrak { M } ^ { e v } _ { \xi } ] = \sum _ \tau \frac { m _ { \tau } } { | A u t ( \tau ) | } i _ * [ \mathfrak { M } ^ { e v } ( \mathcal { X } , \tau ) _ { s , r e d } ] . \end{align*}


\begin{align*} R + 1 > r _ { i + 1 } & = r _ 1 - \Lambda _ 0 \sqrt { a _ 0 } \cdot \sum _ { k = 1 } ^ i \sqrt { t _ k } \\ & \geq R + 3 - \Lambda _ 0 \sqrt { a _ 0 t _ i } \cdot \sum _ { k = 0 } ^ { \infty } ( 1 + \mu ) ^ { - k } \\ & = R + 3 - \sqrt { t _ i } \cdot \frac { \Lambda _ 0 \sqrt { a _ 0 } ( 1 + \mu ) } { \mu } . \end{align*}


\begin{align*} \delta _ 0 = \max _ { t _ 0 \in \mathbb { C } } \left \{ \max \left \{ \min _ { \lambda \in S ( \Lambda _ 1 ) } | \lambda - t _ 0 | - \max _ { \mu \in S ( \Lambda _ 2 ) } | \mu - t _ 0 | , \min _ { \lambda \in S ( \Lambda _ 2 ) } | \lambda - t _ 0 | - \max _ { \mu \in S ( \Lambda _ 1 ) } | \mu - t _ 0 | \right \} \right \} , \end{align*}


\begin{align*} \left ( e ^ { - i \hat { \theta } } \int _ { \mathcal { E } _ k } \Omega \right ) \leq 0 , \forall k = 1 , 2 , \ldots N - 1 . \end{align*}


\begin{align*} f ^ { b - a } + ( a - b ) q f ' f ^ { b - a - 1 } = 1 . \end{align*}


\begin{align*} \Psi \circ L _ t = \Psi \circ L _ 0 \quad . \end{align*}


\begin{align*} \frac { d x } { d t } = 2 t - \frac { x } { t } + \frac { x ^ 2 } { t ^ 3 } \ , , t \geq 1 . \end{align*}


\begin{align*} f ( z \cdot e ) = z f ( e ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : m c k e a n l i n e a r i n t r o } \partial _ t f _ t + \nabla \cdot ( f _ t K \star f _ t ) = \sigma \Delta f _ t , \end{align*}


\begin{align*} \mathcal { R } _ 2 ^ { k \times \infty } : = \{ A = ( A _ { i j } ) _ { i , j = 1 } ^ { k , \infty } \colon ( A _ { i j } ) _ { j \in \mathbb { N } } \in \ell _ 2 , i = 1 , \ldots , k \} \end{align*}


\begin{align*} B ^ { * } = \operatorname { d i a g } \left \{ b _ { 1 1 } , b _ { 2 2 } , \cdots , b _ { n n } \right \} . \end{align*}


\begin{align*} \sum _ { s = 0 } ^ { r - 1 } \Psi _ { \alpha } \circ \theta _ { X _ s } ( X ^ { i _ 0 } _ 0 \cdots X ^ { i _ { r - 1 } } _ { r - 1 } ) = \sum _ { s = 0 } ^ { r - 1 } \frac { i _ s \alpha ^ { \vert \underline i \vert } } { \prod _ { s ' = 0 } ^ { r - 1 } ( i _ { s ' } + \gamma _ { r - s ' } ) \cdots ( i _ { s ' } + \gamma _ r ) } = \vert \underline { i } \vert \frac { \alpha ^ { \vert \underline { i } \vert } } { \prod _ { s ' = 0 } ^ { r - 1 } ( i _ { s ' } + \gamma _ { r - s ' } ) \cdots ( i _ { s ' } + \gamma _ r ) } \enspace . \end{align*}


\begin{gather*} S _ t ( x ) : = \big < \varphi _ 0 \varphi _ x \big > ^ { \epsilon , L } _ { \lambda , \mu + 1 / t , m ^ 2 } , x \in \Lambda _ { \epsilon , L } , \\ C _ t ( x ) : = \Big ( - \Delta ^ \epsilon + m ^ 2 + \frac { 1 } { t } \Big ) ^ { - 1 } ( 0 , x ) , x \in \Lambda _ { \epsilon , L } , \end{gather*}


\begin{align*} \nu _ c ( g ) = \inf \{ \nu \in \R : \sup _ \Lambda \chi ^ \Lambda ( g , \nu ) < + \infty \} , \end{align*}


\begin{align*} \max _ { S \in \mathcal { B } ( \bigcap _ { i = 1 } ^ m \mathcal { F } _ i ) } | S | \geqslant \max _ { A \subseteq E } \biggl ( 2 \cdot \sum _ { i = 1 } ^ m r _ i ( E ) - ( m - 1 ) | A | - \sum _ { i = 1 } ^ m r _ i ( E - A \cap T _ i ^ c ) \biggr ) \end{align*}


\begin{align*} \Pr \big ( \Delta ( C _ M ) > \delta \big ) & = \Pr \big ( \Delta ( C _ M ) > \delta \mid { \rm r a n k } ( M ) = k \big ) \Pr \big ( { \rm r a n k } ( M ) = k \big ) \\ & ~ ~ + \Pr \big ( \Delta ( C _ M ) > \delta \mid { \rm r a n k } ( M ) < k \big ) \Pr \big ( { \rm r a n k } ( M ) < k \big ) . \end{align*}


\begin{align*} f ^ { ( j ) } ( x ) = g _ j ( x ) \prod _ { i = 1 } ^ { m } ( x - \lambda _ i ) ^ { \mu _ { i , j } } \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } t ^ { \frac { d - 2 \delta } { \alpha } ( 1 - \frac { 1 } { q } ) } \| u ( t , \cdot ) - A \Psi _ t \| _ { q , h } = 0 . \end{align*}


\begin{align*} \mathbb { E } \Big \lVert \sum _ { j = 1 } ^ n Z _ j V _ { \bullet , j } \Big \lVert _ 2 ^ 2 = \mathbb { E } \Big [ \sum _ { i , j = 1 } ^ n Z _ i Z _ j \langle V _ { \bullet , i } , V _ { \bullet , j } \rangle _ 2 \Big ] = \mathbb { E } [ Z _ 1 ^ 2 ] \sum _ { j = 1 } ^ n \langle V _ { \bullet , i } , V _ { \bullet , j } \rangle _ 2 = k \mathbb { E } [ Z _ 1 ^ 2 ] . \end{align*}


\begin{align*} \lim _ { i \rightarrow \infty } ( - \Delta ) ^ { \sigma } _ { p } u _ { i } ( x ) = ( - \Delta ) ^ { \sigma } _ { p } u ( x ) - \theta , \end{align*}


\begin{align*} b _ i \le b ' _ { i + \rho _ 1 - \rho _ 2 + ( \rho _ 2 - \rho _ 1 + \eta ) } = b ' _ { i + \eta } ( i \in \Z ) . \end{align*}


\begin{align*} \phi ( n ) \geq \phi \left ( \frac { 1 } { 8 } \psi ^ { - 1 } ( a ( \lfloor k / 4 \rfloor ) ) \right ) = a \left ( 8 \lfloor k / 4 \rfloor \right ) \geq a ( 2 k ) , \end{align*}


\begin{align*} f ( A ) = \sum _ { a \in A } \frac { 1 } { a \log a } = \sum _ { a \in A } \int _ 1 ^ \infty a ^ { - t } \dd { t } = \int _ 1 ^ \infty f _ t ( A ) \dd { t } , \end{align*}


\begin{align*} \int f ( x ) \left ( \frac { d ^ 2 h } { d x ^ 2 } + p ( x ) \frac { d h } { d x } + r ( x ) h ( x ) \right ) y ( x ) d x = f ( x ) \left ( \frac { d h } { d x } y ( x ) - h ( x ) \frac { d y } { d x } \right ) , \end{align*}


\begin{align*} f ( t , x , w ) = \sin ( \lVert x \rVert + w ) , g ( x ) = \lVert x \rVert ^ 2 , \end{align*}


\begin{align*} \zeta ( - n , x ) - \zeta \left ( - n , x + \frac { 1 } { 2 } \right ) = \frac { 1 } { n + 1 } B _ { n + 1 } \left ( x + \frac { 1 } { 2 } \right ) - \frac { 1 } { n + 1 } B _ { n + 1 } ( x ) , \end{align*}


\begin{align*} \nu _ \sigma ( l ( h _ \sigma ) ) = \nu _ \sigma ( f ) . \end{align*}


\begin{align*} B _ n = \begin{pmatrix} 1 , & * ^ n , & 1 \\ 1 , & 0 ^ n , & 1 \end{pmatrix} \end{align*}


\begin{align*} \begin{aligned} & \big ( \sum _ { - \delta N \le j \le 0 } 2 ^ { - \frac { 1 } { 2 } j q } I ^ q _ j \big ) ^ { \frac { 1 } { q } } \ge \frac { C } { ( \ln N ) ^ 3 N ^ { \frac { 1 } { 2 } } } ( \sum _ { - \delta N \le j \le 0 } 1 ) ^ { \frac { 1 } { q } } = \frac { C N ^ { \frac { 1 } { q } - \frac { 1 } { 2 } } } { ( \ln N ) ^ 3 } . \end{aligned} \end{align*}


\begin{align*} F ( \Delta ) = \int _ \Delta Q \omega \end{align*}


\begin{align*} b _ { r , \alpha } : = \Theta \ , a ( u _ { r , \alpha } ) b _ { r , \alpha } ^ * : = \Theta ^ { - 1 } a ^ * ( u _ { r , \alpha } ) c _ { \ell , \alpha } : = \Theta ^ { - 1 } a ( u _ { \ell , \alpha } ) c ^ * _ { \ell , \alpha } : = \Theta a ^ * ( u _ { \ell , \alpha } ) . \end{align*}


\begin{align*} \gamma ( \frac { \lfloor n \tau \rfloor } n ) \left ( \tau - \frac { \lfloor n \tau \rfloor } n \right ) ^ { 2 \alpha + 2 } = \mathbf { 1 } _ { \{ \frac { \lfloor n \tau \rfloor } n \le t _ 1 \wedge t _ 2 \} } \left ( t _ 1 - \frac { \lfloor n \tau \rfloor } n \right ) ^ { \alpha } \left ( t _ 2 - \frac { \lfloor n \tau \rfloor } n \right ) ^ { \alpha } \left ( \tau - \frac { \lfloor n \tau \rfloor } n \right ) ^ { 2 \alpha + 2 } . \end{align*}


\begin{align*} \forall x , y \in \Lambda , 0 \leq \big < \varphi _ x ; \varphi _ y \big > ^ { h } _ { A , g , \nu } \leq \big < \varphi _ x ; \varphi _ y \big > ^ { 0 } _ { A , g , \nu } = \big < \varphi _ x \varphi _ y \big > ^ { 0 } _ { A , g , \nu } . \end{align*}


\begin{align*} D _ 1 = H , D _ 2 = H - P , D _ 3 = D _ 4 = P . \end{align*}


\begin{align*} \left | \underset { \Gamma _ { 2 p - 1 } } { \int \sum } \overline { \widehat { \eta } } _ k \prod _ { \substack { j = 1 \\ o d d } } ^ { 2 p - 1 } \widehat { v } _ { k _ j } \prod _ { \substack { j = 2 \\ e v e n } } ^ { 2 p - 1 } \overline { \widehat { v } } _ { k _ j } \ , d \Gamma \right | \lesssim \| \eta \| _ { X ^ { - s , 1 / 2 - } } \| v \| _ { X ^ { s , 1 / 2 - } _ T } \| v \| ^ { 2 p - 2 } _ { X ^ { s , 1 / 2 + } _ T } . \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } \| e ^ { - i t H } \phi - J U _ 0 ( t ) f \| = 0 \end{align*}


\begin{align*} u ^ { \dagger } . A . v = \left ( \begin{array} { c } u _ { + } \\ u _ { - } \end{array} \right ) ^ { \dagger } . ( D P ) ^ { \dagger } . \left ( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right ) . ( D P ) . \left ( \begin{array} { c } v _ { + } \\ v _ { - } \end{array} \right ) = 0 , \end{align*}


\begin{align*} P _ h ^ 2 \colon H ^ { - 1 / 2 } ( \Gamma ) \to X _ h ^ { - \frac 1 2 } P _ h ^ 3 \colon H ^ { 1 / 2 } ( \Gamma ) \to X _ h ^ { \frac 1 2 } \end{align*}


\begin{align*} \left [ Q , \Phi _ 1 ( [ [ x , y ] _ \mathfrak { g } , z ] _ \mathfrak { g } ) \right ] + \circlearrowleft ( x , y , z ) & = - \left [ \Phi _ 0 \left ( [ x , y ] _ \mathfrak { g } \right ) , \Phi _ 0 ( z ) \right ] + \circlearrowleft ( x , y , z ) \\ & = [ [ \Phi _ 0 ( z ) , Q ] , \Phi _ 1 ( x , y ) ] - [ [ \Phi _ 1 ( x , y ) , \Phi _ 0 ( z ) ] , Q ] + \circlearrowleft ( x , y , z ) \\ & = [ Q , [ \Phi _ 0 ( x ) , \Phi _ 1 ( y , z ) ] ] + \circlearrowleft ( x , y , z ) . \end{align*}


\begin{align*} T ^ { ( a b ) } = \sum _ { i , j , l , m } T ^ { i j } _ { l m } ( 1 \otimes \cdots \otimes E _ i ^ l \otimes \cdots \otimes E _ j ^ m \otimes \cdots \otimes 1 ) , \end{align*}


\begin{align*} E e e ^ { \prime } = \sigma ^ { 2 } \Sigma , \end{align*}


\begin{align*} p ^ \mu \tilde { f } _ \mu = 0 , \end{align*}


\begin{align*} \int _ 0 ^ L \abs { y _ x } ^ 2 d x = o ( 1 ) . \end{align*}


\begin{align*} \sigma ^ { * } ( \mathcal { A } _ k ) & = \bigcup _ { j = 2 } ^ { k } \{ x \in \mathbb { R } \colon H e _ j ( x ) = 0 \} . \end{align*}


\begin{align*} \ell _ A = \max \{ | t | : t \in A \} \mathrm { \ \ a n d \ \ } \max ( A ) = \{ s \in A : | s | = \ell _ A \} . \end{align*}


\begin{align*} w ^ * = b \widetilde { z } + \sum _ { j = 1 } ^ m w _ j ^ * , \end{align*}


\begin{align*} \lim _ { | y | \to \infty } \| R ( x + i y , A _ j ) \| _ { \infty } = 0 \qquad \forall x < - \omega _ 0 ( A _ j ) . \end{align*}


\begin{align*} ( - \Delta ) ^ s m - \frac { ( - \Delta ) ^ s m _ 1 } { \gamma _ 1 ^ { 1 / 2 } } m & = 0 \quad \Omega , \\ m & = m _ 0 \quad \Omega _ e . \end{align*}


\begin{align*} \chi _ { \pi } ( h _ i ) = \sum _ { j = 0 } ^ n c _ j \cdot \chi _ { \sigma _ j } ( h _ i ) \end{align*}


\begin{align*} \hat { V } ( y ) = V _ 0 + \frac { a } { | y | ^ m } + O \left ( \frac { 1 } { | y | ^ { m + \theta } } \right ) \end{align*}


\begin{align*} \lvert v _ { i , j } \rvert ^ 2 \prod _ { k = 1 ; k \ne i } ^ { n - 1 } ( { \lambda _ i } ( A ) - { \lambda _ k } ( A ) ) = \prod _ { k = 1 } ^ { n - 1 } ( { \lambda _ i } ( A ) - { \lambda _ k } ( M _ j ) ) , \end{align*}


\begin{align*} T _ { \alpha } f \geq T f - ( c - \epsilon ) u \geq c u - ( c - \epsilon ) u = \epsilon u \end{align*}


\begin{align*} \nabla _ { Z } Z = - \operatorname { g r a d } p \end{align*}


\begin{align*} \sum _ { n = r + 1 } ^ \infty \| S _ n ( t ) \| _ 1 < \infty . \end{align*}


\begin{align*} Q ( A , B ) = \sum _ { a \in A , b \in B } \pi ( a ) P ( a , b ) . \end{align*}


\begin{align*} \Lambda = \left ( \lambda _ 1 , \lambda _ 2 , \ldots , \lambda _ m \right ) \end{align*}


\begin{align*} | g _ k ( n ) | \leq \tau _ 2 ( n ) = O _ { \varepsilon } ( n ^ { \varepsilon } ) . \end{align*}


\begin{align*} G _ { f , k } ( n ) = \sum _ { d ^ k \delta = n } ( \mu * f ) ( d ) \tau _ k ( \delta ) , \end{align*}


\begin{align*} \| \delta _ n ( a ) \| _ { M , N } = \left \| \int _ 0 ^ 1 e ^ { 2 \pi i n \theta } \rho _ \theta ^ { - 1 } \delta \rho _ \theta ( a ) \ , d \theta \right \| _ { M , N } \le \int _ 0 ^ 1 \| \rho _ \theta ^ { - 1 } \delta \rho _ \theta ( a ) \| _ { M , N } \ , d \theta \le C \| a \| _ { M ' , N ' } \ , . \end{align*}


\begin{align*} p ( t , v , g ) = \frac { 1 } { V _ F } \sum _ { k = - \infty } ^ { + \infty } \exp \left ( - k ^ 2 ( \frac { 2 \pi } { V _ F } ) ^ 2 D ( t ) + i \frac { 2 \pi } { V _ F } k v + i \frac { 2 \pi } { V _ F } k g \right ) ( p _ { t , k } * \bar { G } _ { t , k } ) ( g ) . \end{align*}


\begin{align*} \begin{array} { l l } \frac { 1 } { 2 \lambda ^ 2 } \{ h ( \nabla _ { \tilde { X } } ^ N \tilde { Z } , \tilde { Y } ) + h ( \nabla _ { \tilde { Y } } ^ N \tilde { Z } , \tilde { X } ) \} + R i c ( X , Y ) + \frac { \mu } { \lambda ^ 2 } h ( \tilde { X } , \tilde { Y } ) = 0 . \end{array} \end{align*}


\begin{align*} | f ( z ^ m ) | + \sum _ { n = N } ^ { \infty } | a _ n | | z | ^ n \leq d ( 0 , \partial { \Omega } ) \end{align*}


\begin{align*} \begin{cases} - U '' + \lambda U = \rho U ^ { p - 1 } & , \\ U > 0 & , \\ \sum _ { i = 1 } ^ m U _ i ' ( 0 ) = 0 , \end{cases} \end{align*}


\begin{align*} { \mathcal U } _ n ( t , s ) = \exp \left ( t A _ n ^ { 1 / n } \right ) = \exp \left ( t ~ \left \{ \Pi _ { k = 1 } ^ n \left [ ( I - \kappa e ^ { - a _ k ( t , s ) } ) ^ { - 1 } \partial _ t a _ k ( t , s ) \right ] \right \} ^ { 1 / n } \right ) , \end{align*}


\begin{align*} K ^ { n , 2 } _ \tau & = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ { ( \delta + \frac 1 n ) \wedge \tau } \gamma _ s \left ( \sigma ' ( X _ s ) \right ) ^ 2 \\ & \times \left [ \left ( \int _ { \eta _ n ( s ) - \delta } ^ { \eta _ n ( s ) } \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) ^ 2 - \left ( \int _ { \eta _ n ( s ) - \delta } ^ { \eta _ n ( s ) } \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( s ) - \delta } ) \ , d W _ u \right ) ^ 2 \right ] \ , d s . \end{align*}


\begin{align*} E ( ( A R ) / A ) = E ( R / ( R \cap A ) ) \neq 0 . \end{align*}


\begin{align*} \| E \| _ { L ^ { p } ( \rho ; \R ^ d ) } = \sup \left \{ \int _ { \R ^ { d } } \langle \nabla \varphi , E \rangle d \rho : \varphi \in C _ { 0 } ^ { \infty } , \| \nabla \varphi \| _ { L ^ { q } ( \rho ) } \le 1 \right \} , \end{align*}


\begin{align*} Z _ t : = \lim _ { n \to \infty } n ^ { - 1 } Z ^ { ( n ) } _ t . \end{align*}


\begin{align*} \label [ I e q ] { e q : c o l l i s i o n k e r n e l B } B ( u , \sigma ) = \Phi ( | u | ) \Sigma ( \theta ) , \end{align*}


\begin{align*} & \hat { \eta } _ { x , p ; \delta } ( \xi ) = e ^ { - i x \xi } \hat { \eta } _ \delta ( \xi - p ) \end{align*}


\begin{align*} \vec { H } = J \nabla \theta _ t , \end{align*}


\begin{align*} H _ { ( Q _ 4 , K 3 ) } ( y ) = e ^ { \frac { H \log y } { 2 \pi i } } \sum _ { d \geq 0 } y ^ d \frac { \Gamma ( 1 + \frac { 4 H } { 2 \pi i } + 4 d ) } { \Gamma ( 1 + \frac { H } { 2 \pi i } + d ) ^ 4 } [ \textbf { 1 } ] _ { d } . \end{align*}


\begin{align*} \| h \partial _ \nu u _ { h } \| _ { L ^ 2 ( \Gamma ) } = O ( 1 ) . \end{align*}


\begin{align*} \begin{cases} - u _ n '' + \lambda _ n u _ n = \rho _ n u _ n ^ { p - 1 } & , \\ u _ n > 0 & , \\ \sum _ { \rm { e } \succ \rm { v } } u _ { e , n } ' ( \rm { v } ) = 0 & \forall \rm { v } \in \mathcal { V } , \end{cases} \end{align*}


\begin{align*} \Phi ( \tilde { a } , \tilde { m } ) = \int _ { 0 } ^ { t } \mathcal { G } ( t - s ) \left ( \begin{array} { c c c } 0 \\ { g ( a _ { L } + \tilde { a } , m _ { L } + \tilde { m } ) } \\ \end{array} \right ) d s . \end{align*}


\begin{align*} c _ 1 \cos t _ 1 + c _ 2 \sin t _ 1 + \frac { \mu a ^ 2 } { \lambda } = 0 , \end{align*}


\begin{align*} s i g n ( \sigma _ i \beta \sigma _ j \beta ^ { - 1 } ) = M e y e r ( \sigma _ i , \beta \sigma _ j \beta ^ { - 1 } ) \end{align*}


\begin{align*} Q ^ { n , 1 } _ \tau = : Q ^ { n , 3 } _ \tau + Q ^ { n , 4 } _ \tau , \end{align*}


\begin{align*} \Phi _ { \ell _ 2 } ( t ) = \Phi _ { \ell _ 1 } ( t ) \circ \mathbb { S } _ q ( \Delta ) = \Phi _ { \ell _ 1 } ( t ) \circ \mathbb { S } _ q ( \tilde { \ell } _ 1 ) \circ \cdots \circ \mathbb { S } _ q ( \tilde { \ell } _ n ) , \end{align*}


\begin{align*} D _ { - \Delta } F _ c ( \frac { | x - 2 P t | } { ( t + 1 ) ^ { \alpha } } \leq 1 ) = - \alpha \frac { | x - 2 P t | } { ( t + 1 ) ^ { 1 + \alpha } } F _ c ' \geq 0 . \end{align*}


\begin{align*} ( 1 - \tau _ i ) \phi ( f ) & = \phi ( f - f \circ \tau _ i ) \\ & = \langle M _ f v , v \rangle - \langle M _ { f \circ \tau _ i } v , v \rangle \\ & = \langle M _ f v , v \rangle - \langle M _ f \tau _ i v , \tau _ i v \rangle \\ & = \langle M _ f v , v - \tau _ i v \rangle + \langle M _ f ( v - \tau _ i v ) , \tau _ i v \rangle . \end{align*}


\begin{align*} X _ j f ( x , u ) & = \frac { d } { d t } f ( ( x , u ) ( t X _ j , 0 ) ) \big | _ { t = 0 } \\ & = \partial _ { x _ j } f ( x , u ) + \frac 1 2 \sum _ { k = 1 } ^ { d _ 2 } \langle U _ k , [ x , X _ j ] \rangle \partial _ { u _ k } f ( x , u ) , \\ U _ k f ( x , u ) & = \partial _ { u _ k } f ( x , u ) . \end{align*}


\begin{align*} h \colon ( U \times E ^ k ) \times W \to F \ , , h ( ( x , y ) , v ) : = d ( f ^ \wedge ) ( ( x , v ) , ( y , 0 ) ) \ , . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \exp \left ( - \frac { \lambda ^ 2 } 2 n ^ { 2 \alpha + 1 } \int _ { t - \delta } ^ { \eta _ n ( t ) } \psi _ { n , 1 } ^ 2 ( s , \eta _ n ( t ) ) d s \right ) = \exp \left ( - \frac { \lambda ^ 2 } 2 \kappa _ 1 ^ 2 \right ) \end{align*}


\begin{align*} \frac { \abs { f ' ( z ) } } { \abs { f ' ( z _ 0 ) } } = \left . \abs { \frac { \textbf { z } _ 0 - \alpha } { \textbf { z } _ 0 + \lambda i } } ^ 2 \middle / \abs { \frac { \textbf { z } - \alpha } { \textbf { z } + \lambda i } } ^ 2 \right . . \end{align*}


\begin{align*} \| k _ n u \| ^ q _ { L ^ q ( \Omega ) } + b \| \gamma ( k _ n u ) \| ^ r _ { L ^ r ( \Sigma ) } = k _ n ^ q \| u \| ^ q _ { L ^ q ( \Omega ) } + k _ n ^ r b \| u \| ^ r _ { L ^ r ( \Sigma ) } = 1 . \end{align*}


\begin{align*} \| E \| _ { L ^ 1 } \leq \tilde G _ { d , t , \mu } ( \lambda ) + \sum _ { i = 1 } ^ 3 \lambda ^ i P ^ { ( i ) } _ { d , t , \mu , \mu , c ( \mu ) \lambda } \big ( \| E \| _ { L ^ 1 } \big ) . \end{align*}


\begin{align*} F \supset \bigcap _ { N = 1 } ^ { \infty } \bigcup _ { n = N } ^ { \infty } \bigcup _ { w \in \Sigma _ { \beta _ 1 } ^ n , v \in \Sigma _ { \beta _ 2 } ^ n } \tilde { J } _ { n , \beta _ 1 } ( w ) \times \tilde { J } _ { n , \beta _ 2 } ( v ) : = \widetilde { F } . \end{align*}


\begin{align*} ( \tilde { f } ^ { | u \rangle } , \tilde { f } ^ { | v \rangle } ) _ { { } _ { \mathfrak { J } } } = - 4 \pi \lambda \textrm { c o t h } \lambda , \end{align*}


\begin{align*} \Vert u _ 1 u _ 2 \Vert _ { L ^ 1 ( B ) } \leq C , \Vert u _ j \Vert _ { H ^ 2 ( B ) } \leq C s ^ 2 e ^ { \Lambda s } , \ , j = 1 , 2 . \end{align*}


\begin{align*} \int \frac { 1 } { z - x } d \mu ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { s _ { n } } { z ^ { n + 1 } } \end{align*}


\begin{align*} d ^ k f ( x , y _ 1 , \ldots , y _ k ) : = ( D _ { y _ k } \cdots D _ { y _ 1 } ) ( f ) ( x ) \end{align*}


\begin{align*} \prod _ { b \in \lambda } \dfrac { 1 - q ^ { c o a r m _ \lambda ( b ) } t ^ { n - c o l e g _ \lambda ( b ) } } { 1 - q ^ { a r m _ \lambda ( b ) } t ^ { l e g _ \lambda ( b ) + 1 } } = \prod _ { 1 \leq i < j \leq n } \prod _ { r = 0 } ^ { \lambda _ i - \lambda _ j - 1 } \dfrac { 1 - q ^ r t ^ { j - i + 1 } } { 1 - q ^ r t ^ { j - i } } . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \frac { \partial } { \partial t } \omega ( t ) = - \mathrm { R i c } _ { \omega ( t ) } ^ { T } - \omega ( t ) , \omega ( 0 ) = \omega _ { 0 } . \end{array} \end{align*}


\begin{align*} F ( p , s ) = ( u ( p , s ) , p ) . \end{align*}


\begin{align*} B ( x _ i , \frac { 1 } { 2 } \sin \theta ) \cap B ( x _ j , \frac { 1 } { 2 } \sin \theta ) = \emptyset . \end{align*}


\begin{align*} L _ { B } ( x ^ { n } , b _ { \mathcal { T } } ) = \sum _ { \pi _ { \bar { Y } _ { \mathcal { T } } | \bar { X } } } Z _ { \pi _ { \bar { Y } _ { \mathcal { T } } | \bar { X } } } ( x ^ n , b _ { \mathcal { T } } ) , \end{align*}


\begin{align*} H _ { k , j ; n , l , r } ^ { \mu , m , i ; n o n , a } ( t , x , \zeta ) = \int _ { 0 } ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \varphi _ { m ; - 1 0 M _ t } ( t - s ) E B ^ a ( t , s , x - y + ( t - s ) \omega , \omega , v ) \cdot \nabla _ v \big ( ( t - s ) \mathfrak { H } ^ { \mu , E , i } _ { k , j ; n , l , r } ( y , \omega , v , \zeta ) \end{align*}


\begin{align*} & ( - 1 ) ^ { k } \left ( \bigg ( \sum _ { n = 1 } ^ \infty u _ m ( n ) \ , q ^ n \bigg ) \bigg ( \sum _ { n = - k } ^ { k } ( - 1 ) ^ { n } \ , q ^ { n ( 3 n - 1 ) / 2 } \bigg ) - \sum _ { n = 0 } ^ \infty C _ m ( n ) \ , q ^ n \right ) \\ & = \left ( \sum _ { n = 0 } ^ \infty C _ m ( n ) \ , q ^ n \right ) \left ( \sum _ { n = 0 } ^ \infty \widetilde { P } _ k ( n ) \ , q ^ n \right ) , \end{align*}


\begin{align*} \kappa _ 1 = \sqrt { \sum _ { k = 1 } ^ \infty \int _ 0 ^ 1 [ k ^ \alpha - ( k - x ) ^ \alpha ] ^ 2 d x } \end{align*}


\begin{align*} \sum _ { 1 \le j _ 1 < \cdots < j _ k \le N } \ \prod _ { \ell = 1 } ^ k \lambda _ { j _ \ell } = \sum _ { 1 \le j _ 1 < \cdots < j _ k \le N } \underset { 1 \le \ell , m \le k } { { \rm d e t } } ( A _ { j _ \ell j _ m } ) , \end{align*}


\begin{align*} \mathbf { P } _ x ( \tau _ \dagger = n \mid X ) = K ( X _ { n - 1 } ) \prod _ { i = 0 } ^ { n - 2 } ( 1 - K ( X _ i ) ) . \end{align*}


\begin{align*} \tilde g _ { m } ^ { ( \ell ) } : = \chi _ { \tilde B _ m ^ { ( \ell ) } \cap B _ { 3 R } ^ { d _ { \mathrm { C C } } } ( 0 ) } F _ \ell ^ { ( \iota ) } ( L , U ) f _ { m } ^ { ( \ell ) } \end{align*}


\begin{align*} m ! \ , & N _ 0 ( ( R f ^ { i _ 1 \dots i _ k } ) _ { p _ 1 q _ 1 \dots p _ { m - k } q _ { m - k } } ) \\ & = \sigma ( i _ 1 \dots i _ k ) \ , \sum \limits _ { r = 0 } ^ k ( - 1 ) ^ r \binom { k } { r } \frac { \partial ^ { r } } { \partial x _ { i _ 1 } \dots \partial x _ { i _ r } } ( R ^ k ( G _ { m - r } ) ) _ { p _ 1 q _ 1 \dots p _ { m - k } q _ { m - k } i _ { r + 1 } \dots i _ k } , \end{align*}


\begin{align*} \underset { 1 \le \ell , m \le k } { { \rm d e t } } ( A _ { j _ \ell j _ m } ) = \det \begin{pmatrix} a _ { j _ 1 j _ 1 } & a _ { j _ 1 j _ 2 } & \cdots & a _ { j _ 1 j _ k } \\ a _ { j _ 2 j _ 1 } & a _ { j _ 2 j _ 2 } & \cdots & a _ { j _ 2 j _ k } \\ \vdots & & \ddots & \vdots \\ a _ { j _ k j _ 1 } & a _ { j _ k j _ 2 } & \cdots & a _ { j _ k j _ k } \end{pmatrix} . \end{align*}


\begin{align*} \sup _ { \mathbb { R } ^ n } \left | u ^ 0 _ \infty ( x ) - \frac { 1 } { 2 } \sum _ { i = 1 } ^ n a _ i x _ i ^ 2 \right | \leq C _ 1 . \end{align*}


\begin{align*} \Psi = e _ i ^ { \varepsilon _ i ( \Phi ) } ( \Phi ) . \end{align*}


\begin{align*} f ( x ^ 1 ) = \frac { x ^ 1 } { \sqrt { 2 } } + c , \end{align*}


\begin{align*} \textsf { F A C } ( M ) = \bigsqcup _ { A \subsetneqq M } \textsf { F A C } ( A / M ) . \end{align*}


\begin{align*} g ( \operatorname { g r a d } ( \| Z \| ^ { 2 } ) + 2 \nabla _ { Z } Z , 2 \nabla _ { Z } Z ) = - F f ^ { 2 } \| X \| ^ { 2 } \| \operatorname { g r a d } ( \| X \| ^ { 2 } ) \| ^ { 2 } \end{align*}


\begin{align*} \begin{aligned} A ^ { ( 1 ) } _ { N , d , D } & = \left \{ R _ { N , d , D } > K _ { 0 } \Big ( \frac { \beta + \gamma } { \beta } \Big ) ^ { 1 / 2 } N ^ { 4 / 3 } ( \log N ) ^ { 4 / 3 } \right \} , \\ A ^ { ( 2 ) } _ { N , d , D } & = \left \{ R _ { N , d , D } < \varepsilon _ 0 \frac { \gamma } { \beta + \gamma } N ^ { 4 / 3 } ( \log N ) ^ { - 2 / 3 } \right \} . \end{aligned} \end{align*}


\begin{align*} \tau _ x ( f ) : = \int _ { G / G _ x } f ( g \theta g ^ { - 1 } ) d g . \end{align*}


\begin{align*} \langle \otimes _ { l = 1 } ^ { m } \mathfrak { a } \rangle _ { t } = \exp \biggl ( - \frac { t } { 2 } \sum \limits _ { j = 1 } ^ N \sum \limits _ { i , p = 1 } ^ m \otimes _ { l = 1 } ^ m ( J K _ { j } ) ^ { \delta _ { i l } + \delta _ { p l } } \biggr ) \langle \otimes _ { l = 1 } ^ { m } \mathfrak { a } \rangle _ { 0 } , \end{align*}


\begin{align*} f _ 1 \ast f _ 2 = \circledast \circ f _ 1 \tilde { \otimes } f _ 2 \circ \Delta \end{align*}


\begin{align*} \frac { d x } { d t } = b _ 0 ( t ) + b _ 1 ( t ) x + b _ 2 ( t ) x ^ 2 \ , , \end{align*}


\begin{align*} h _ { j } ( p ) = 0 , h _ { j \overline { l } } ( p ) = \delta _ { j } ^ { l } , d h _ { j \overline { l } } ( p ) = 0 . \end{align*}


\begin{align*} \| L _ { t } ^ { - 1 } [ S _ t u _ 0 ] W ^ \gamma _ t u _ 0 \| _ { H ^ 1 _ x ( \mathbb { T } ) } = \| W ^ \gamma _ t u _ 0 \| _ { H ^ 1 _ x ( \mathbb { T } ) } \lesssim e ^ { - \gamma t } \to 0 . \end{align*}


\begin{align*} \begin{aligned} \mathcal { L } ( x ) = - \tfrac { 1 } { N } \sum _ { i = 1 } ^ N y _ i \log \big ( \sigma ( A _ i x ) \big ) + ( 1 - y _ i ) \log \big ( 1 - \sigma ( A _ i x ) \big ) . \end{aligned} \end{align*}


\begin{align*} c _ { n - 1 , \xi _ { \boldsymbol { s } k } } ^ { \ast \ast } = & \frac { \Gamma \left ( \frac { n - 1 } { 2 } \right ) } { ( 2 \pi ) ^ { ( n - 1 ) / 2 } } \bigg \{ \int _ { 0 } ^ { \infty } t ^ { \frac { n - 3 } { 2 } } \left [ ( 2 + 3 \sqrt { 2 } ) \left ( t + \frac { \xi _ { \boldsymbol { s } , k } ^ { 2 } } { 2 } \right ) + 3 \right ] \exp \left ( - \sqrt { 2 t + \xi _ { \boldsymbol { s } , k } ^ { 2 } } \right ) \mathrm { d } t \bigg \} ^ { - 1 } \end{align*}


\begin{align*} \delta _ n ( \beta ) : = \dfrac { f ' ( v _ n ) } { \beta } \mod \mathcal { D } _ n . \end{align*}


\begin{align*} \partial _ t u _ R + \Delta ^ 2 u _ R = \Delta f _ R ( u _ R ) + \sigma ( u _ R ) \dot { W } , R \ge 1 \end{align*}


\begin{align*} \sigma _ 1 \leqslant a _ 2 = \left ( a _ 1 ^ 2 - \left | \operatorname { d e t } \left ( a _ 1 ^ 2 - A ^ { H } A \right ) \right | \left ( \frac { n - 1 } { ( n + 1 ) a _ 1 ^ 2 - \| A \| _ { F } ^ { 2 } } \right ) ^ { n - 1 } \right ) ^ { 1 / 2 } \leqslant a _ 1 \end{align*}


\begin{align*} ( X : B ) = \bigcap _ { b \in B } ( X + b ) = \bigcap _ { i = 0 } ^ { r } ( X + b _ i ) \end{align*}


\begin{align*} \langle T ( z ) , v \rangle _ { Y ' , Y } = \langle l , v \rangle _ { Y ' , Y } , \forall v \in Y , \end{align*}


\begin{align*} \langle \cdot | \cdot \rangle _ { \lambda } ^ i \colon \left ( { \sf C } _ \lambda ^ i \otimes L _ \lambda \right ) \otimes \left ( { \sf C } _ { \lambda ^ \dagger } ^ i \otimes L _ { \lambda ^ \dagger } \right ) \rightarrow \C , ( i = 1 , 2 ) \end{align*}


\begin{align*} N _ n ( t , t ) = \sum _ { r = 1 } ^ n \frac { 1 } { n } \binom { n } { r } \binom { n } { r - 1 } t ^ { r } . \end{align*}


\begin{align*} \Phi '' ( \vec p ) [ \widehat { \vec p } _ 1 , \widehat { \vec p } _ 2 ] = F '' ( V ( \vec p ) ) [ V ' ( \vec p ) \widehat { \vec p } _ 1 , V ' ( \vec p ) \widehat { \vec p } _ 2 ] + F ' ( V ( \vec p ) ) V '' ( \vec p ) [ \widehat { \vec p } _ 1 , \widehat { \vec p } _ 2 ] \end{align*}


\begin{align*} w _ 4 ( \pi ) = \begin{cases} \sum \limits _ { i = 1 } ^ 4 t _ i ^ 2 , & , \\ 0 , & . \end{cases} \end{align*}


\begin{align*} m _ { i j } = \chi _ { \sigma _ j } ( h _ i ) . \end{align*}


\begin{align*} a _ j h _ \rho ^ j = a _ { \rho 0 j } + a _ { \rho 1 j } Q _ \rho + \ldots + a _ { \rho l j } Q _ \rho ^ l \end{align*}


\begin{align*} \bar R _ { \alpha \beta \alpha 3 } & = R _ { \alpha \beta \alpha 3 } + k _ { \alpha 3 } k _ { \beta \alpha } - k _ { \alpha \alpha } k _ { \beta 3 } \\ & = \nabla _ \alpha k _ { \alpha \beta } - \nabla _ \beta k _ { \alpha \alpha } = 0 , \end{align*}


\begin{align*} F _ \ell ^ { ( \iota ) } ( \lambda , \rho ) = ( F ^ { ( \iota ) } \psi ) ( \sqrt \lambda ) \chi _ \ell ( \lambda / \rho ) \quad \rho \neq 0 \end{align*}


\begin{align*} \int _ 0 ^ a f ( t ) d _ q t : = ( 1 - q ) a \sum _ { n = 0 } ^ \infty q ^ n f ( a q ^ n ) , \ , \ , a \in \mathbb R , \end{align*}


\begin{align*} \left ( f _ { r , s } \right ) _ { r ' , s ' } ( x ) = f ( x ) \end{align*}


\begin{align*} E ( G / H ^ 0 ) = E ( G / H ) E ( H / H ^ 0 ) = \pm | H / H ^ 0 | = \pm 1 . \end{align*}


\begin{align*} \lim _ { h \to 0 } \lim _ { \alpha \to 0 } \frac { i } h \int _ { \Omega _ \Gamma } [ - h ^ 2 \Delta - 1 , \chi _ \alpha ( x _ n ) h D _ n ] u _ h \overline { u _ h } d x = \int _ { S ^ * _ \Gamma M \backslash S ^ * \Gamma } | \xi _ n | d \mu ^ \perp , \end{align*}


\begin{align*} \Phi & = \{ l ( T ) \} \cup \{ j \in \mathbb { N } \colon 1 \le j \le l ( T ) - 1 , c _ { l ( T ) - j } > 1 \} . \end{align*}


\begin{align*} \begin{array} { c c l } \left \vert \int _ \Omega R u ^ 2 d V _ g \right \vert & = & \left \vert \int _ \Omega R ( ( \xi _ { \tilde K } + \xi _ E ) u ) ^ 2 d V _ g \right \vert \\ & \leq & \left \vert \int _ \Omega R \xi _ { \tilde K } ^ 2 u ^ 2 d V _ g \right \vert + \left \vert 2 \int _ \Omega R \xi _ E \xi _ { \tilde K } u ^ 2 d V _ g \right \vert + \left \vert \int _ \Omega R \xi _ E ^ 2 u ^ 2 d V _ g \right \vert . \end{array} \end{align*}


\begin{align*} \varphi ( x ) = \langle \pi ( x ) \eta _ 0 , \xi _ 0 \rangle = \langle P _ { \mathcal L _ 1 } \pi ( x ) \eta _ 0 , \xi _ 0 \rangle = \langle \tilde \pi ( x ) \eta _ 0 , \xi \rangle . \end{align*}


\begin{align*} & ( \underbrace { + } _ { a _ { u j } } , \ \underbrace { - \dots - } _ { a _ { u + 1 \ , j + 1 } } , \ \cdots \ , \ \underbrace { + \dots + } _ { a _ { w - 1 \ , j } } , \ \underbrace { - \dots - } _ { a _ { w \ , j + 1 } } ) . \end{align*}


\begin{align*} \begin{cases} ( u , v , h , g ) | _ { y = 0 } = ( u , v , h , g ) | _ { y \rightarrow \infty } = ( 0 , 0 , 0 , 0 ) , \\ ( u , v , h , g ) | _ { x = 1 } = ( u , v , h , g ) | _ { x \rightarrow \infty } = ( 0 , 0 , 0 , 0 ) . \end{cases} \end{align*}


\begin{align*} \Phi _ { n } ( x ) = \frac { a _ 1 } { a _ 2 } \ln \vert a _ 2 x + a _ 4 \vert \ . \end{align*}


\begin{align*} \sup _ { B _ 1 ( 0 ) } u \leq \lim _ { j \to + \infty } \Phi _ \rho ( \mu _ { j + 1 } , h ^ j _ 1 ) \leq C _ 1 \Phi _ \rho ( p _ 1 , 2 ) , C _ 1 = C ^ \frac { \kappa } { p _ 1 ( \kappa - 1 ) } ( 2 ^ { \beta } \kappa ^ { \nu } ) ^ { \frac { 1 } { p _ 1 } \sum _ j \frac { j } { \kappa ^ j } } \end{align*}


\begin{align*} \hat \tau \leq - 2 \frac { \xi _ x \nu _ x + \xi _ y \nu _ y } { \nu _ x ^ 2 + \nu _ y ^ 2 } = 2 \tau ^ * . \end{align*}


\begin{align*} \mathrm { E h r } _ P ( x ) = \sum _ { t \geq 0 } | t P \cap \Z ^ { n } | x ^ t = \frac { \sum _ { i = 0 } ^ d h _ i ^ * x ^ i } { ( 1 - x ) ^ { d + 1 } } \ , . \end{align*}


\begin{align*} \mathcal { U } _ { [ n , j ] } = \bigcup _ { k = j } ^ { n - 1 } \mathcal { U } _ { [ n , j , k ] } \end{align*}


\begin{align*} x y z = a b ^ 2 , x + y + z = a b c . \end{align*}


\begin{align*} \begin{cases} \frac { \dd \Theta _ t } { \dd t } & = \Bar { Q } \circ H _ t + H _ t \circ Q _ \mathfrak g \\ \Xi _ 0 & = \Bar \Xi _ 1 \end{cases} \end{align*}


\begin{align*} \| u \| _ { W ^ { s , p } ( \Omega ) } \vcentcolon = \left ( \| u \| _ { L ^ p ( \Omega ) } ^ p + [ u ] _ { W ^ { s , p } ( \Omega ) } ^ p \right ) ^ { 1 / p } , \end{align*}


\begin{align*} g _ 1 = \begin{vmatrix} x _ 0 & x _ 1 \\ a _ 0 & a _ 1 \end{vmatrix} \ , \ g _ 2 = \begin{vmatrix} y _ 0 & y _ 1 \\ b _ 0 & b _ 1 \end{vmatrix} \ , \ g _ 3 = \begin{vmatrix} z _ 0 & z _ 1 \\ c _ 0 & c _ 1 \end{vmatrix} \ . \end{align*}


\begin{align*} \sup \left \{ \int _ 0 ^ \infty f g \ , d r : \ f \geq 0 \ , , \ \underline \mu _ \alpha ( f ) \leq 1 \right \} = \alpha \ , \overline \nu _ \alpha ( g ) \end{align*}


\begin{align*} f _ { i j } ( v ) \ , \partial _ k F ( v ) + f _ { j k } ( v ) \ , \partial _ i F ( v ) + f _ { k i } ( v ) \ , \partial _ j F ( v ) = 0 \end{align*}


\begin{align*} & R ^ N ( \partial _ t , \partial _ i , \partial _ j , \partial _ k ) = R ^ N _ { 0 i j k } = 0 , \\ & R ^ N ( \partial _ t , \partial _ i , \partial _ j , \partial _ t ) = R ^ N _ { 0 i j 0 } = - f ( t ) f '' ( t ) \widetilde { g } _ { i j } . \end{align*}


\begin{align*} x _ i x _ j = q ^ { h _ { i j } } x _ j x _ i , i , j = 1 , 2 , \cdots , n , \end{align*}


\begin{align*} f _ t \left ( \int _ { \Omega } ^ { } \alpha ( \omega ) d \omega \right ) = \int _ { \Omega } ^ { } \alpha ( \omega ) d \omega - t = \int _ { \Omega } ^ { } ( \alpha ( \omega ) - T ( \omega ) ) d \omega \leq \int _ { \Omega } ^ { } f _ { T ( \omega ) } \circ \alpha ( \omega ) d \omega . \end{align*}


\begin{align*} \Lambda ( G ) = \mathrm { s p a n } _ K \{ \varepsilon _ 1 , \varepsilon _ 2 , \varepsilon _ 3 , \alpha _ 1 , \alpha _ 2 , \alpha _ 3 , \alpha _ 1 \alpha _ 2 \} \ , , \end{align*}


\begin{align*} \int _ { \Omega _ e } ( q _ 1 - q _ 2 ) f g \ , d x = 0 \end{align*}


\begin{align*} \hat { P } \phi _ { n } ^ { \pm } ( x ) = \hat { T } \phi _ { n } ^ { \pm } ( x ) = \phi _ { n \pm 1 } ^ { \mp } ( x ) . \end{align*}


\begin{align*} s _ { \Tilde { \theta } } = ( 1 - \Tilde { \theta } ) s _ 0 + \Tilde { \theta } s _ 1 = \frac { t - s } { t - r } r + \frac { s - r } { t - r } t = s . \end{align*}


\begin{align*} \left \{ \begin{array} { r l } u _ t ( t , r ) = \Delta u ( t , r ) - \lambda ( t , x ) | \nabla u ( t , r ) | ^ { \alpha } + a ( t , x ) v ^ { p } ( t , r ) , & t > 0 , \ 0 < r < h ( t ) , \\ v _ t ( t , r ) = \Delta v ( t , r ) - \lambda ( t , x ) | \nabla v ( t , r ) | ^ { \alpha } + a ( t , x ) u ^ { p } ( t , r ) , & t > 0 , \ 0 < r < g ( t ) , \end{array} \right . \end{align*}


\begin{align*} \frac { 1 } { \gamma } \leq \| \dot C _ 0 \| \int _ { 0 } ^ { \beta } e ^ { 2 \int _ 0 ^ t \chi _ s \ , d s } \ , d t = \frac { 1 } { \alpha ^ 2 } \int _ { 0 } ^ { \beta } e ^ { 2 \int _ 0 ^ t \chi _ s \ , d s } \ , d t . \end{align*}


\begin{align*} \left ( r \tilde { A } \right ) '' - \dfrac { 2 } { r } f ^ { 2 } \tilde { A } = - \dfrac { 1 } { 4 } g ^ { 2 } \tilde { \rho } ^ { 2 } r \left ( B - \tilde { A } \right ) \end{align*}


\begin{align*} \hbox { N } _ h ( N , w ; \xi ) = c _ h h ^ { - 2 } \sum _ { c \in \mathbb { Z } ^ 4 } \sum _ { k = 1 } ^ \infty ( k \ell ) ^ { - 4 } S _ { k } ( c ; \xi ) I _ { k , \ell } ( c ) , \end{align*}


\begin{align*} \int \limits _ { a } ^ { c } U \omega d t = 0 , \end{align*}


\begin{align*} \operatorname { i n t } ( F , G ) = \operatorname { i n t } ( F ) \cdot ( A , B ) = ( A , B ) \cdot \operatorname { i n t } ( G ) = \operatorname { i n t } ( F ) \cdot ( A , B ) \cdot \operatorname { i n t } ( G ) . \end{align*}


\begin{align*} \sup _ { D _ s } \left | u _ s ( x ) - \frac { 1 } { 2 } \sum _ { i = 1 } ^ n a _ i x _ i ^ 2 \right | < C _ 1 , \end{align*}


\begin{align*} \lim _ { T \rightarrow \infty } \frac { 1 } { T } \int _ 0 ^ T \| \chi _ K \psi _ t \| \ , d t = 0 \end{align*}


\begin{align*} \L I _ m f , g \R = \L f , I _ m ^ * g \R \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } \limsup _ { n \rightarrow \infty } \left | E \left [ \exp \left ( i \mu n ^ { \alpha + \frac 1 2 } \int _ 0 ^ { t - \delta } ( t - s ) ^ \alpha \xi ^ n _ s d W _ s \right ) \right ] - E \left [ \exp \left ( i \mu Y ^ { n , 1 } _ t \right ) \right ] \right | = 0 . \end{align*}


\begin{align*} e _ { ( \cdot ) } \left ( \omega \right ) \doteq \omega \left ( \mathfrak { e } _ { ( \cdot ) } \right ) = d _ { + } \mathfrak { a } _ { + } \end{align*}


\begin{align*} \widetilde { T } _ { k , j ; n } ^ { b i l ; \mu , i } ( \mathfrak { m } , U ) ( t , x , \zeta ) = \int _ { 0 } ^ { t } \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \big ( E ( s , x - y + ( t - s ) \omega ) + \hat { v } \times B ( s , x - y + ( t - s ) \omega ) \big ) \cdot \nabla _ v \big ( ( t - s ) \mathcal { K } ^ { U , i } _ { k , j , n } ( \mathfrak { m } ) ( y , v , \omega , \zeta ) \end{align*}


\begin{align*} ( \chi _ { ( 1 , 0 ) } \otimes \rho ) \left [ \begin{pmatrix} x _ 1 \\ x _ 2 \end{pmatrix} , \begin{pmatrix} 1 & 0 \\ u & v \end{pmatrix} \right ] f ( t ) = | v | ^ { - 1 / 2 } e ^ { 2 \pi i x _ 1 } f \left ( \frac { t - u } { v } \right ) , t \in \R . \end{align*}


\begin{align*} \beta | f ' ( z ^ m ) | + ( 1 - \beta ) | f ( z ^ m ) | + \sum _ { n = 1 } ^ { \infty } | a _ n | \phi _ { n } ( r ) \leq d ( 0 , \partial { \Omega } ) \end{align*}


\begin{align*} l _ 0 ( x ) = | x | _ 0 = \left \{ \begin{array} { l l } 1 & x \neq 0 \\ 0 & x = 0 \end{array} \right . . \end{align*}


\begin{align*} \lim _ { N \to \infty } \bigg | \Big \langle \Phi , \Big ( { U } _ N ( & H _ N - H _ { 2 \mathrm { - m o d e } } ) { U } _ N ^ * - \mathbb { H } - \mu _ + \mathcal { N } _ \perp \Big ) \Phi \Big \rangle \\ & - \frac { \lambda } { \sqrt { 2 ( N - 1 ) } } \sum _ { m \ge 3 } \Big \langle \Phi , \Big ( \big ( w _ { + 1 - m } \ , \Theta + w _ { + 2 - m } \ , \Theta ^ { - 1 } \big ) a _ m \mathfrak { D } + \mathrm { h . c . } \Big ) \Phi \big \rangle \bigg | = 0 . \end{align*}


\begin{align*} \frac { 1 } { 2 } \{ g ( \nabla _ X \xi , Y ) + g ( \nabla _ Y \xi , X ) \} + R i c ( X , Y ) + \mu g ( X , Y ) = 0 . \end{align*}


\begin{align*} c \Big ( T _ 1 , T _ 2 , T _ 3 \Big ) : = \Big \langle \Delta [ T _ 1 ] , T _ 2 \times ^ { H ^ { T _ 1 } } T _ 3 \Big \rangle . \end{align*}


\begin{align*} [ A , B ] = \bigcup _ { n \geq 1 } [ A , B ] _ n \end{align*}


\begin{align*} D _ i ( u - u _ 0 ) ( x _ 0 ) = \kappa ( u - u _ 0 ) ( x _ 0 ) D _ i \phi ( x _ 0 ) , { \rm f o r } \ i = 1 , \cdots , n . \end{align*}


\begin{align*} \frac { 1 } { a ( k + h ^ \vee ) } + \frac { 1 } { b ( \ell + h ^ \vee ) } = c n \end{align*}


\begin{align*} \begin{cases} h _ { j - 1 , j } = 1 , & j \geq 1 , \\ h _ { j + k , j } = a _ { j } ^ { ( k ) } , & 0 \leq k \leq p , j \geq 0 , \\ h _ { i , j } = 0 , & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} \left . { D _ { u , m } } \right | _ { \alpha _ m = 0 } & = \displaystyle c _ { u , m } \prod _ { i = 1 } ^ { m - 1 } ( - \alpha _ i ) ^ { ( 2 n + 1 ) r ^ 2 } \prod _ { 1 \le i < j \le m - 1 } ( \alpha _ { j } - \alpha _ { i } ) ^ { ( 2 n + 1 ) r ^ 2 } \enspace \\ & = \Psi _ { \boldsymbol { 1 } } \left ( Q _ { u , m - 1 } ( \boldsymbol { t } ) { { \mathfrak { A } } } ( \boldsymbol { t } ) \left . { { \mathfrak { B } } } ( \boldsymbol { t } ) \right | _ { \alpha _ m = 0 } \right ) \enspace . \end{align*}


\begin{align*} \widetilde c _ n = 1 / c _ n \ . \end{align*}


\begin{align*} w ( ( n , m ) \rightarrow ( n + 1 , m + 1 ) ) & = 1 , \\ w ( ( n , m ) \rightarrow ( n + 1 , m - p ) ) & = a _ { m - p } . \end{align*}


\begin{align*} \Phi _ 0 ( t ) = \sum _ { m \geq 3 } \sum _ { \beta } \frac { 1 } { m ! } \langle t , \cdots , t \rangle _ { 0 , m , \beta } ^ { X _ { D , \infty } } q ^ { \beta } . \end{align*}


\begin{align*} \int F ( x ) k ( q x ) T _ q ( x ) y ( x ) d _ q x = F ( x ) k ( x ) \left ( y ( x ) u ( x ) - D _ { q ^ { - 1 } } y ( x ) \right ) . \end{align*}


\begin{align*} B _ { \gamma } \left ( a \right ) = \int _ { \Theta _ { d } } \sum _ { z \in \mathfrak { L } } \gamma ^ { d } f \left ( \gamma a + \gamma z \right ) \mathrm { e } ^ { i \theta \cdot z } \mathrm { d } \left ( \mathrm { F } _ { \ast } \mathrm { P } \right ) \left ( \theta \right ) \ . \end{align*}


\begin{align*} 2 A \sum _ 1 ^ N | F ( p _ j ) | \leq 2 A N \frac { ( F ( p ) ) } { \sin \epsilon } = \frac { 2 A N } { \sin \epsilon } ( \tilde { \Omega } _ { L _ 0 } ) . \end{align*}


\begin{align*} K _ r ( \tau ( x , y ) ) = \left ( \sum _ { i = 1 } ^ { n - 2 } \frac { 3 } { \varepsilon } ( \varphi _ i ^ 1 ( x , y ) - \Phi _ i ( x , y ) ) ^ 2 \right ) ^ r < \left ( \frac 1 3 \right ) ^ r . \end{align*}


\begin{align*} 2 A _ Y X = - \left \{ ( \nu [ X , Y ] - \lambda ^ 2 g ( X , Y ) \left ( \nabla _ \nu \frac { 1 } { \lambda ^ 2 } \right ) \right \} - 2 \lambda ^ 2 g ( X , Y ) \left ( \nabla _ \nu \frac { 1 } { \lambda ^ 2 } \right ) . \end{align*}


\begin{align*} r ^ C _ A ( b ) \stackrel { ( \ref { s p l . c o m . 1 } ) } = & l ^ B _ A ( r ^ C _ A ( b ) ) r ^ B _ A ( r ^ C _ A ( b ) ) \\ = & \ ; l ^ B _ A ( r ^ C _ A ( b ) ) b \stackrel { ( \ref { s p l . c o m } ) } = l ^ B _ A ( r ^ C _ A ( b ) ) l ^ C _ A ( b ) r ^ C _ A ( b ) . \end{align*}


\begin{align*} P _ { o u t } ( i ) = \textrm { P r } \bigg [ \gamma _ i < \gamma _ { t h } ( i ) \bigg ] . \end{align*}


\begin{align*} G ( t , x ) e ^ { z t } = \sum _ { n = 0 } ^ { \infty } \bar { P } _ n ( x , z ) \frac { t ^ n } { n ! } . \end{align*}


\begin{align*} \nabla v _ { 2 , t } = - v _ { 2 , t } \nabla f _ t , \nabla ^ 2 v _ { 2 , t } = v _ { 2 , t } \left ( \nabla f _ t \otimes \nabla f _ t - \nabla ^ 2 f _ t \right ) . \end{align*}


\begin{align*} & ( v _ e ^ i , g _ e ^ i ) ( x , 0 ) = - ( v _ p ^ { i - 1 } , g _ p ^ { i - 1 } ) ( x , 0 ) , \\ & ( v _ e ^ i , g _ e ^ i ) \rightarrow ( 0 , 0 ) { \rm ~ a s ~ } Y \rightarrow \infty . \end{align*}


\begin{align*} f _ { ( n ) + } ' ( 0 ) = n f \left ( \frac { 1 } { n } \right ) > - \infty . \end{align*}


\begin{align*} \overline { f } \otimes \overline { g } = H ^ { * } ( \overline { f } ) \overline { g } \end{align*}


\begin{align*} S _ { \psi } f ( x ) : = \frac { 1 } { \Psi ( x ) } \int _ 0 ^ { x } f ( t ) \psi ( t ) d t \end{align*}


\begin{align*} F ( B ^ { * } ) = \left \{ \sum _ { i = 1 } ^ { n } x _ { i } \overline { b } _ { i } ^ { * } \mid 0 \leq x _ { i } < 1 \ \ x _ { i } \in \mathbb { R } \right \} . \end{align*}


\begin{align*} \Theta _ k : = \bigoplus _ { i = 1 } ^ n R _ k \partial _ { x _ i } . \end{align*}


\begin{align*} f ( A ) : = \sum _ { a \in A } \frac { 1 } { a \log a } \ < \ \infty , \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ { \frac { \alpha } { 2 } } u = \sigma u ^ q + \mu , \ , & u > 0 \ , \Omega , \\ u = 0 & , \end{cases} \end{align*}


\begin{align*} a \cdot _ { \rho } x = \rho _ a ( x ) ~ ~ \forall x \in \Bar { \Omega } . \end{align*}


\begin{align*} \boldsymbol { V } _ { A } ^ { p o s t } = \left ( \frac { \mathbf { F } ^ { H } \mathbf { F } } { \sigma _ { n } ^ { 2 } } + ( \boldsymbol { V } _ { A } ^ { p r i } ) ^ { - 1 } \right ) ^ { - 1 } . \end{align*}


\begin{align*} F _ m ( \iota ) = \begin{cases} & : 1 \leq m \leq 2 n a - 1 , \\ 0 & : . \end{cases} \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } t ^ { \frac { d } { \alpha } \left ( 1 - \frac { 1 } { q } \right ) - \frac { \delta } { \alpha } } \| u ( t , \cdot ) - A \Psi _ t ( \cdot ) \| _ { q , H _ t } = 0 , \end{align*}


\begin{align*} d _ n ^ \gamma ( { \mathcal K } ) _ X : = \inf _ { k \leq n } \inf _ { \| \cdot \| _ { Y _ k } } d ^ \gamma ( { \mathcal K } , Y _ k ) _ X , \end{align*}


\begin{align*} \sum _ { j \in J } a _ j = r ( J ) + \ell _ { 1 } ( J ) r , \sum _ { j \in J } b _ j = s ( J ) + \ell _ { 2 } ( J ) s . \end{align*}


\begin{align*} b _ i \overline b _ i = - e _ i , ~ ~ i = 0 , 1 , \cdots , r ; ~ ~ ~ ~ ~ ~ b _ { r + j } \overline b _ { r + j } = - \widehat e _ { r + j } , ~ ~ j = 1 , \cdots , s . \end{align*}


\begin{align*} | g _ m | ^ 2 \sim \frac { L \beta _ { P I } \beta _ { I S } } { 2 } \chi _ { 2 } ^ 2 \left ( \frac { 2 | h _ { P S } | ^ 2 } { L \beta _ { P I } \beta _ { I S } } \right ) , m = 1 , 2 , \ldots , M . \end{align*}


\begin{align*} C ( v ) = \ln \left ( \frac { \sinh ( z _ 0 v ) } { z _ 0 v } \right ) . \end{align*}


\begin{align*} \sum _ { n _ 1 , \ldots , n _ k = 1 } ^ { \infty } \frac { ( \mu * F ) ( n _ 1 , \ldots , n _ k ) } { n _ 1 \cdots n _ k } \end{align*}


\begin{align*} \forall \varphi \in \R ^ \Lambda , V _ 0 ( \varphi ) = \sum _ { x \in \Lambda } ( \frac 1 4 g \varphi ^ 4 _ x + \frac 1 2 \nu \varphi ^ 2 _ x ) . \end{align*}


\begin{align*} \langle \xi _ 1 \otimes \eta _ 1 , \xi _ 2 \otimes \eta _ 2 \rangle = \langle ( L _ { \xi _ 2 } ^ * L _ { \xi _ 1 } ) \eta _ 1 , \eta _ 2 \rangle \end{align*}


\begin{align*} \sum _ { k = 0 } ^ \infty \mu ( k ) ( 1 - { E _ 2 ( i , m ) } ) ^ k & \leq \sum _ { k = 0 } ^ \infty \mu ( k ) \left ( 1 - \left ( \frac { 1 } { 2 A ( m + 1 ) } \right ) ^ { i + 2 } \right ) ^ k \\ & \leq \sum _ { k = 0 } ^ \infty \mu ( k ) \left ( 1 - \left ( \frac { 1 } { 2 A ( m + 2 ) } \right ) ^ { i + 2 } \right ) ^ k . \end{align*}


\begin{align*} \theta = \log h _ 1 ^ 1 + p ( u ) + N \rho . \end{align*}


\begin{align*} B _ w ( \lambda ) : = \bigcup _ { m _ k \in \mathbb { N } } f _ { i _ 1 } ^ { m _ 1 } f _ { i _ 2 } ^ { m _ 2 } \cdots f _ { i _ \ell } ^ { m _ \ell } b _ \lambda \subset B ( \lambda ) \end{align*}


\begin{align*} \widehat { W _ t ^ \gamma u _ 0 } = e ^ { - t ( i k ^ 2 + \gamma ) } \widehat { u _ 0 } . \end{align*}


\begin{align*} \sum _ { k = N } ^ M a _ k ^ \alpha \bigg ( \sum _ { i = N } ^ k b _ i \bigg ) ^ \alpha \approx \sum _ { k = N } ^ M a _ k ^ \alpha b _ k ^ \alpha , \end{align*}


\begin{align*} \mathrm { d i s t } _ { M \setminus N _ s ( K ) } ( x , y ) & \gtrsim \mathrm { d i s t } _ { \mathbb { D } \setminus B ( 0 , s ) } ( A , B ) \gtrsim e ^ { a s } \mathrm { d i s t } ( 0 , C ) = e ^ { a s } \mathrm { d i s t } ( r ( x ) , r ( y ) ) . \end{align*}


\begin{align*} P ' ( f p ) = \sum _ { x \in N _ { s o r t e d } } \sum _ { y \in S _ x } f _ { f p } ( m _ { x } , \gamma _ { x } , k _ { x } ) f _ { f p } ( m _ { y } , \gamma _ { y } , k _ { y } ) . \end{align*}


\begin{align*} \Delta _ { r a d } = \frac { 1 } { 2 } \frac { \partial ^ 2 } { \partial r ^ 2 } + \frac { \alpha } { 2 } \frac { 1 } { \tanh ( \alpha r ) } \frac { \partial } { \partial r } \end{align*}


\begin{align*} \min \dim ( C ) = \min \{ C \} . \end{align*}


\begin{align*} H _ u ( \lambda ) = u _ 1 h _ 1 ( \lambda ) + \dots + u _ r h _ r ( \lambda ) , \lambda \in T ^ * G . \end{align*}


\begin{align*} \mathcal { J } ( \xi , s ) = 2 s \int _ { \R ^ 3 } \overline { \omega } \cdot A ( x ) e ^ { i x \cdot \xi } d x . \end{align*}


\begin{align*} \sum _ { i \in M _ C } \sum _ { j = 1 } ^ { r _ i - 1 } a _ { i r _ i } x ^ * _ { i j } + \sum _ { i \in M _ C } \sum _ { j = r _ i } ^ { n _ i } \max \left \{ a _ { i j } , b - \sum _ { k \in M _ C - i } a _ { k r _ k } \right \} x ^ * _ { i j } > b ? \end{align*}


\begin{align*} \mathrm { P f } \left [ M _ { k , l } \right ] _ { k , l = 1 } ^ { 2 N } \doteq \frac { 1 } { 2 ^ { N } N ! } \sum _ { \pi \in \mathcal { S } _ { 2 N } } \left ( - 1 \right ) ^ { \pi } \prod \limits _ { j = 1 } ^ { N } M _ { \pi \left ( 2 j - 1 \right ) , \pi \left ( 2 j \right ) } \end{align*}


\begin{align*} - \frac { 1 } { m } = m + z , \Im m \Im z > 0 . \end{align*}


\begin{align*} \frac { 1 } { 2 } | \nabla \omega | ^ 2 = W _ + ( \omega , \omega ) - \frac { s } { 3 } ~ , \end{align*}


\begin{align*} \hat { W } ( y ) : = Z ( y ) + \frac { \mu } { 2 } \| y \| ^ 2 , \end{align*}


\begin{align*} \begin{aligned} & C _ { 1 } ( \chi , a ) = \sum \limits _ { h = 1 } ^ { q } \overline { \chi } ( h ) e \left ( \frac { a h } { q } \right ) , C _ { 1 } ( q , a ) = C _ { 1 } ( \chi ^ { 0 } , a ) , \\ & C _ { 3 } ( \chi , a ) = \sum \limits _ { h = 1 } ^ { q } \overline { \chi } ( h ) e \left ( \frac { a h ^ { 3 } } { q } \right ) , C _ { 3 } ( q , a ) = C _ { 3 } ( \chi ^ { 0 } , a ) , \end{aligned} \end{align*}


\begin{align*} U _ * = \sqrt { \frac { 2 L \ \Delta } { 1 + \sum \limits _ { i = 0 } ^ { N - 1 } h _ i \big ( 2 - h _ i \frac { - \kappa } { 1 - \kappa } \big ) } } . \end{align*}


\begin{align*} g ( x , y , n , z ) & : = \frac { n ^ 2 } { \pi ^ 2 } \sin ^ 2 \Big ( \frac { \pi x } { n } \Big ) + \frac { n ^ 2 } { \pi ^ 2 } \sin ^ 2 \Big ( \frac { \pi y } { n } \Big ) \\ & - \frac { 2 n ^ 2 } { 3 \pi ^ 2 } \sin ^ 2 \Big ( \frac { \pi x } { n } \Big ) \sin ^ 2 \Big ( \frac { \pi y } { n } \Big ) + z ^ 2 . \end{align*}


\begin{align*} \delta ( Q ( f ) ) \leq \max _ t \delta _ { - t G } ( Q ^ { ( t ) } ) = \delta _ G ( Q ) \ . \end{align*}


\begin{align*} & \langle e _ { i _ 1 } + \cdots + e _ { i _ k } , e _ { j _ 1 } + \cdots + e _ { j _ d } \rangle = | \{ i _ 1 , \ldots , i _ k \} \cap \{ j _ 1 , \ldots , j _ d \} | = \\ & k + d - | \{ i _ 1 , \ldots , i _ k \} \cup \{ j _ 1 , \ldots , j _ d \} | \geq k + d - s . \end{align*}


\begin{align*} \Gamma ( h _ { \gamma _ { - } } ) \left ( \rho \right ) = f _ { \Phi + \Phi ^ { - \mathfrak { b } _ { - } , \gamma _ { - } } } \left ( \rho \right ) + \int _ { \mathbb { S } } \left \vert \rho \left ( \mathfrak { e } _ { \Psi } \right ) \right \vert ^ { 2 } \mathfrak { a } _ { \mathfrak { b } _ { + } } \left ( \mathrm { d } \Psi \right ) \ , \rho \in E _ { 1 } \ . \end{align*}


\begin{align*} \alpha _ { n + 1 } = \left ( \begin{matrix} x _ n \\ y _ n \end{matrix} \right ) - \begin{bmatrix} 2 x _ n y _ n & x _ n ^ 2 \\ 1 & 0 \end{bmatrix} ^ { - 1 } \begin{bmatrix} x _ n ^ 2 y _ n + 5 \\ x _ n + 1 \end{bmatrix} \end{align*}


\begin{align*} \Lambda ^ n _ s & = ( W _ s - W _ { \eta _ n ( s ) } ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) \\ & + ( W _ s - W _ { \eta _ n ( s ) } ) \left ( \int _ { \eta _ n ( s ) } ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) - \int _ { \eta _ n ( s ) } ^ s \psi _ { n , 1 } ( u , s ) d u . \end{align*}


\begin{align*} \frac { ( r - 2 ) k + 2 \sum _ i a _ i } { r } = 2 l + k - 2 \end{align*}


\begin{align*} \lim _ { \tau \rightarrow s _ 0 ^ - } \overline { u } ' ( \tau ) = \lim _ { \tau \rightarrow s _ 0 ^ + } \overline { u } ' ( \tau ) = 0 . \end{align*}


\begin{align*} ( u \otimes v ) ( x _ 1 , \cdots , x _ m , x _ { m + 1 } , \cdots , x _ { m + k } ) = u ( x _ 1 , \cdots , x _ m ) v ( x _ { m + 1 } , \cdots , x _ { m + k } ) . \end{align*}


\begin{align*} ( \widehat { \varPhi } ^ * _ { x , s } ) ^ { - 1 } ( t ) = \int _ { 0 } ^ { t } \dfrac { \widehat { \varPhi } _ { x } ^ { - 1 } ( \tau ) } { \tau ^ { \frac { N + s } { N } } } d \tau . \end{align*}


\begin{align*} \int x ^ { n - 2 } ( q ^ 2 x ^ 2 ; q ^ 2 ) _ \infty h _ n ( x ; q ) d _ q x = \frac { x ^ n ( x ^ 2 ; q ^ 2 ) _ \infty } { [ n - 1 ] _ q } \left ( \frac { h _ n ( \frac { x } { q } ; q ) } { x } - \frac { 1 } { q } h _ { n - 1 } ( \frac { x } { q } ; q ) \right ) . \end{align*}


\begin{align*} P _ 1 = \{ ( J \setminus A _ i , k _ 1 ( 1 ) , k _ 2 ( 1 ) ) \} \end{align*}


\begin{align*} \pi _ { \bar { Y } _ { \mathcal { T } } , \bar { X } } ( a _ { \mathcal { T } } , b ) = \pi _ { \bar { Y } _ { \mathcal { T } } | \bar { X } } ( a _ { \mathcal { T } } | b ) \pi _ { \bar { X } } ( b ) , \end{align*}


\begin{align*} 1 \le [ w ] _ { Q B _ { \beta , \psi , \infty } } : = \inf \left \{ [ w ] _ { Q B _ { \beta , \psi , p } } : w \in Q B _ { \beta , \psi , p } , ~ p > 0 \right \} . \end{align*}


\begin{align*} F ( [ a _ { j } ^ i ] ) = f ( \mu _ 1 , \cdots , \mu _ n ) , \end{align*}


\begin{align*} \lim _ { R \rightarrow \infty } \lim _ { i \rightarrow \infty } \int _ { B _ { R } ^ { c } ( 0 ) } \frac { u _ { i } ^ { p - 1 } ( x ) } { | x - y | ^ { n + \sigma p } } d y = & u ^ { p - 1 } ( x ) \lim _ { R \rightarrow \infty } \int _ { B _ { R } ^ { c } ( 0 ) } \frac { d y } { | x - y | ^ { n + \sigma p } } \\ \leq & u ^ { p - 1 } ( x ) \lim _ { R \rightarrow \infty } \int _ { B _ { R - | x | } ^ { c } ( x ) } \frac { d y } { | x - y | ^ { n + \sigma p } } = 0 . \end{align*}


\begin{align*} \{ X _ M ( 0 ) = a _ 0 , \ldots X _ M ( \ell ) = a _ { \ell } \} \end{align*}


\begin{align*} E ( \mathbf { R } _ n ) = \varepsilon n \left ( \frac { n - 1 } { 2 } \boldsymbol { a } + \boldsymbol { b } \right ) . \end{align*}


\begin{align*} \begin{aligned} 0 = \int _ { \mathbb { R } ^ N } f ( | y | ) \left ( y \cdot t _ { 1 } \right ) \bar { \phi } ( y ) d y . \end{aligned} \end{align*}


\begin{align*} v = p ^ 2 + q ^ 2 . \end{align*}


\begin{align*} ( \dot { \psi } _ \phi f ) = \frac { d } { d t } \bigg | _ { t = 0 } f ( \phi + t \psi ) , \forall f \in C ^ { \infty } ( U ) . \end{align*}


\begin{align*} - g ( T _ U V , X ) + R i c ^ \nu ( U , V ) + \mu g ( U , V ) = 0 . \end{align*}


\begin{align*} d ^ r ( \beta \circ f ) ( x , y _ 1 , \ldots , y _ r ) = \sum _ { ( I _ 1 , \ldots , I _ r ) } \beta ( d ^ { | I _ 1 | } f _ 1 ( x , y _ { I _ 1 } ) , \ldots , d ^ { | I _ k | } f _ k ( x , y _ { I _ k } ) ) \end{align*}


\begin{align*} j _ { p } ( v ) : = \begin{cases} | v | ^ { p - 2 } v & \ v \neq 0 \\ 0 & \end{cases} . \end{align*}


\begin{align*} & \eta \sqrt { - 1 } \partial \bar \partial ( \varphi _ m + \Psi _ m ) + v ' _ { \epsilon } ( \Psi _ m ) \sqrt { - 1 } \partial \bar \partial \Psi _ m \\ \ge & \frac { 1 } { \delta } \sqrt { - 1 } \partial \bar \partial ( \varphi _ m + \Psi _ m ) + v ' _ { \epsilon } ( \Psi _ m ) \sqrt { - 1 } \partial \bar \partial \Psi _ m \\ \ge & 0 \end{align*}


\begin{align*} f ( x ) = \sum _ { k = 0 } ^ n c _ k x ^ k \in K [ x ] \end{align*}


\begin{align*} - { \mathcal { A } } _ m U = F . \end{align*}


\begin{align*} \mu ^ \Lambda _ { g , \nu } ( d \varphi ) = e ^ { - \frac 1 2 ( \varphi , - \Delta \varphi ) - \sum _ { x \in \Lambda } ( \frac 1 4 g \varphi ^ 4 + \nu \varphi ^ 2 ) } , \end{align*}


\begin{align*} V = V _ 0 \supseteq V _ 1 \supseteq V _ 2 \supseteq \cdots \end{align*}


\begin{align*} \langle ( x T _ \xi ^ * y ) \widehat { a b } , \hat { c } \rangle = \langle \xi y a b , x ^ * c \xi \rangle = \langle ( x L _ \xi y ) \hat { a } , c \xi b ^ * \rangle . \end{align*}


\begin{align*} g ( S _ n ) = n \binom { n } { 3 } - ( n + 1 ) \binom { n - 1 } { 3 } , \deg ( S ) = \binom { n + 1 } { 2 } . \end{align*}


\begin{align*} \mathcal { F } = \left \{ f : \Omega \to \mathrm { D o m e } ( \Omega ) : f ( x ) = x x \in \partial \Omega f \right \} . \end{align*}


\begin{align*} F ( z ) = \sum _ { \nu > 0 } c _ \nu z ^ \nu + \sum _ { \nu \leq 0 } c _ \nu z ^ \nu = F _ { \mathrm { m } } ( z ) + F _ { \mathrm { b } } ( z ) . \end{align*}


\begin{align*} \Delta _ { { \mathcal { I } } } : = \Big \{ { \alpha } \in \mathbb { R } ^ n : \sum \limits _ { i \in \mathcal { I } } \alpha _ i = 1 , \alpha _ i { } \geq { } 0 , \forall i \in 0 , \dots , n - 1 \Big \} \end{align*}


\begin{align*} \begin{array} [ c ] { c } ( \omega _ { N } + \sqrt { - 1 } \partial _ { B } \overline { \partial } _ { B } \phi _ { T _ { 0 } , \varepsilon } ) ^ { 2 } \wedge \eta _ { N } = C _ { \varepsilon } \Omega _ { \varepsilon } \wedge \eta _ { N } ; \sup ( \phi _ { T _ { 0 } , \varepsilon } - \phi _ { T _ { 0 } } ) = \sup ( \phi _ { T _ { 0 } } - \phi _ { T _ { 0 } , \varepsilon } ) \end{array} \end{align*}


\begin{align*} g * z = \frac { 1 } { A + B z } \cdot ( C + D z ) . \end{align*}


\begin{align*} \nu _ \rho ( f ) = \nu ( a _ 0 ) = \nu ( l ( h _ \rho ) ) \end{align*}


\begin{align*} g = \prod _ { \alpha = 1 } ^ { r } \exp ( \lambda _ \alpha v _ \alpha ) , \end{align*}


\begin{align*} G = \langle g _ 1 , \ldots , g _ n | r _ 1 , \ldots , r _ m \rangle , \end{align*}


\begin{align*} - \tilde u '' + \tilde \lambda \tilde u = \tilde u ^ { p - 1 } , \tilde u \ge 0 \end{align*}


\begin{align*} & \vec { p } _ { \ell } ( z ) = { } ^ t \Biggl ( P _ { \ell } ( z ) , { P _ { \ell , 1 , r - 1 } ( z ) , \ldots , P _ { \ell , 1 , 0 } ( z ) } , \ldots , { P _ { \ell , m , r - 1 } ( z ) , \ldots , P _ { \ell , m , 0 } ( z ) } \Biggr ) \enspace , \end{align*}


\begin{align*} c _ k ( E _ n ^ * ) = e _ k ( \tau _ 1 , \tau _ 2 , \ldots , \tau _ n ) ( 1 \le k \le n ) , \end{align*}


\begin{align*} W ^ { \mathbf { s } _ 1 } = g ( W ^ { \mathbf { s } _ 0 } ) . \end{align*}


\begin{align*} J _ \gamma = \gamma - 1 + k \ . \end{align*}


\begin{align*} N _ 1 = N _ { f _ 1 } ( \Lambda _ 1 ) \end{align*}


\begin{align*} \delta ( S ) & = \lim _ { x \to \infty } \frac { 1 } { \log x } \sum _ { n \in S , n \le x } \frac { 1 } { n } , \qquad & \Delta ( S ) = \lim _ { x \to \infty } \frac { 1 } { \log \log x } \sum _ { n \in S , 1 < n \le x } \frac { 1 } { n \log n } , \end{align*}


\begin{align*} S _ { \exp _ A f } ( r ) = O ( \log ^ + T _ { \exp _ A f } ( r ) ) + O ( \log r ) + O ( 1 ) ~ ~ | | , \end{align*}


\begin{align*} \min _ { 0 { } \leq { } i { } \leq { } N } \| \nabla f ( x _ i ) \| ^ 2 { } \leq { } \frac { 2 L \ , \big ( f ( x _ 0 ) - f _ * \big ) } { 1 + \sum \limits _ { i = 0 } ^ { N - 1 } \left ( 2 h _ i - h _ i ^ 2 \frac { - \kappa } { 1 - \kappa } \right ) } \end{align*}


\begin{align*} \delta ( p _ { \vec { k } } ) & = \frac { 1 } { k ! } \sum _ { \sigma } \delta \left [ z _ { 1 \sigma ( 1 ) } \right ] \cdots \delta \left [ z _ { 1 \sigma ( k ) } \right ] , \end{align*}


\begin{align*} G ( t , x ) \frac { \left ( e ^ { t } - 1 \right ) } { t - 2 \pi i k } = - \sum _ { n = 0 } ^ { \infty } s _ n t ^ n , \end{align*}


\begin{align*} \nu _ Q ( f ) : = \min _ { 0 \leq i \leq r } \{ \nu ( a _ i Q ^ i ) \} . \end{align*}


\begin{align*} F _ { q , r } ( u _ k ) \leq F _ { q , r } ( 0 ) : = B , \end{align*}


\begin{align*} \begin{aligned} \dot { x } ( { t } ) & = f ( x ( t ) , u ( t ) ) , \end{aligned} \end{align*}


\begin{align*} \det D Y ( \cdot , u , D u ) = \psi ( \cdot , u , D u ) , \end{align*}


\begin{align*} \rho ( 0 , v ) = \rho _ { } ( v ) , v \in [ 0 , V _ F ] . \end{align*}


\begin{align*} f _ { \boldsymbol { X } ^ { \ast \ast } } ( \boldsymbol { x } ) = \frac { c _ { n } ^ { \ast \ast } } { \sqrt { | \boldsymbol { \Sigma } | } } \overline { \mathcal { G } } _ { n } \left \{ \frac { 1 } { 2 } ( \boldsymbol { x } - \boldsymbol { \mu } ) ^ { T } \mathbf { \Sigma } ^ { - 1 } ( \boldsymbol { x } - \boldsymbol { \mu } ) \right \} , ~ \boldsymbol { x } \in \mathbb { R } ^ { n } . \end{align*}


\begin{align*} C ( r ) f : = \frac 1 r \int _ 0 ^ r e ^ { s A } f \ , d s f \in E \end{align*}


\begin{align*} \Vert f \Vert _ p = \left ( \frac { 1 } { 2 \pi } \int _ 0 ^ R \int _ 0 ^ { 2 \pi } | f ( r e ^ { i \varphi } ) | ^ p d \varphi d \mu ( r ) \right ) ^ { 1 / p } \end{align*}


\begin{align*} E ( \mathbf { R } _ n ) _ 1 = \varepsilon n \left [ \frac { n - 1 } { 2 } a _ 1 + b _ 1 \right ] . \end{align*}


\begin{align*} e ^ { f _ o } v _ s = \sum _ { j = 0 } ^ { \infty } c _ j e ^ { - \lambda _ j s } \varphi _ j . \end{align*}


\begin{align*} \beta _ { n , i } ( f ^ { n } , g ^ { n } _ { i } ) \coloneqq \Pr \left ( \hat { H } _ { i } = 0 | H = 1 \right ) i \in \{ 1 , 2 \} , \end{align*}


\begin{align*} \sum _ { k = N } ^ M a _ k ^ \alpha \bigg ( \sum _ { i = k } ^ M b _ i \bigg ) ^ \alpha \approx \sum _ { k = N } ^ M a _ k ^ \alpha b _ k ^ \alpha \end{align*}


\begin{align*} C ^ n _ t = n ^ { \alpha + \frac 1 2 } \int ^ t _ 0 ( t - s ) ^ { \alpha } [ \sigma ( X ^ n _ { \eta _ n ( s ) } ) - \sigma ( X ^ n _ s ) ] \ , d W _ s \end{align*}


\begin{align*} k = k _ 2 a _ 1 a _ 2 = \textsf { 1 } _ M \end{align*}


\begin{align*} \big ( 1 + b ( u , v ) \big ) \Vert u - v \Vert ^ 2 _ { L ^ 2 ( \Omega ) } = 2 \min _ { t \in \mathbb { R } } { \Vert u - t v \Vert _ { L ^ 2 ( \Omega ) } ^ 2 } . \end{align*}


\begin{align*} \psi _ { A } = A \left [ \frac { \cosh ( k ' x ) } { \cosh ( k ' a / 2 ) } - \frac { x } { a / 2 } \frac { \sinh ( k ' x ) } { \sinh ( k ' a / 2 ) } \right ] \end{align*}


\begin{align*} \sigma ^ { ( a n , b m ) } ( y ) = y . \end{align*}


\begin{align*} | k ( t ) | \leq \frac { C _ K } { \pi } t ^ { \mu - 1 } ( \cos \varphi ) ^ { \mu - 1 } \Gamma ( 1 - \mu , 0 ) = \frac { C _ K } { \pi } t ^ { \mu - 1 } ( \cos \varphi ) ^ { \mu - 1 } \Gamma ( 1 - \mu ) . \end{align*}


\begin{align*} g ( \theta , \omega ) = \hat { g } ( \theta , \Phi \omega ) , \end{align*}


\begin{align*} { \rm R C } ( I ) = \left ( \bigcap _ { i = 1 } ^ { s + 1 } H ^ + _ { e _ i } \right ) \bigcap \left ( \bigcap _ { i = 1 } ^ m H ^ + _ { ( \gamma _ i , - d _ i ) } \right ) \bigcap \left ( \bigcap _ { i = m } ^ p H ^ + _ { ( \gamma _ i , - d _ i ) } \right ) , \end{align*}


\begin{align*} y ^ * _ i = ( i - 1 ) L + \left [ U / n - ( n - 1 ) L / 2 \right ] . \end{align*}


\begin{align*} \Lambda _ i ^ { ( s ) } : = ( x ^ { ( p ^ s ) } ) ^ { - 1 } ( \phi _ i ^ { ( s ) } ) ^ G ( x ) = 1 + \pi \Gamma _ i ^ { ( s ) t } \in \textup { G L } _ N ( \mathcal A ) . \end{align*}


\begin{align*} g ( W ^ \nu ) = W ^ { g ( \nu ) } . \end{align*}


\begin{align*} ( W f ) _ { i _ 1 \dots i _ m j _ 1 \dots j _ m } & = 2 ^ m \sigma ( i _ 1 \dots i _ m ) \sigma ( j _ 1 \dots j _ m ) ( R f ) _ { i _ 1 j _ 1 \dots i _ m j _ m } \\ ( R f ) _ { i _ 1 j _ 1 \dots i _ m j _ m } & = \frac { 1 } { ( m + 1 ) } \alpha ( i _ 1 j _ 1 ) \dots \alpha ( i _ m j _ m ) ( W f ) _ { i _ 1 \dots i _ m j _ 1 \dots j _ m } . \end{align*}


\begin{align*} k _ 1 a ^ { - 1 } = a _ 2 k _ 2 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , T ] } E \left [ | Q ^ { n , 1 } _ \tau | ^ 2 \right ] = 0 . \end{align*}


\begin{align*} \sigma _ { \pi } : = \prod _ { \substack { 1 \leq i \leq \ell \\ \mathrm { w i t h } \ a _ i \neq b _ i } } \ ( a _ i , b _ i ) \in S _ k \subseteq S _ { \infty } . \end{align*}


\begin{align*} \mathbb { E } [ \ell , k ] = \left . \left [ k \dbinom { \ell } { k } + \sum \limits _ { i = 0 } ^ { ( k ' - 1 ) / 2 } \dbinom { \ell } { k ' - 2 i - 1 } \right ] \middle / \dbinom { \ell } { k } \right . \end{align*}


\begin{align*} \sum _ { n _ 1 \cdots n _ k \le x } \tau ( [ n _ 1 , \ldots , n _ k ] ) = x \ , Q _ { \tau , 2 k - 1 } ( \log x ) + O \big ( x ^ { \theta _ { 2 k } + \varepsilon } \big ) , \end{align*}


\begin{align*} \Phi = \sum _ { k \in \mathbb { Z } } \Phi ^ { ( k ) } \end{align*}


\begin{align*} \tilde { \bf { u } } _ m ( \alpha ) = A ^ { - 1 } _ m ( \alpha ) { \bf f } _ m ( \alpha ) + A ^ { - 1 } _ m ( \alpha ) \boldsymbol { \eta } _ m = { \bf u } _ m ( \alpha ) + A ^ { - 1 } _ m ( \alpha ) \boldsymbol { \eta } _ m . \end{align*}


\begin{align*} \frac { 1 } { 2 } ( L _ \xi g ) + R i c + \mu g = 0 , \end{align*}


\begin{align*} \hat v ( t , \xi ) = e ^ { i \Theta ( t , \xi ) } \hat { h } ( t , \xi ) . \end{align*}


\begin{align*} \begin{aligned} \mathcal { E } _ { u } & : = \{ ( n , m ) \rightarrow ( n + 1 , m + 1 ) : ( n , m ) \in \mathcal { V } \} \\ \mathcal { E } _ { \ell } & : = \{ ( n , m ) \rightarrow ( n + 1 , m ) : ( n , m ) \in \mathcal { V } \} \\ \mathcal { E } _ { d } & : = \{ ( n , m ) \rightarrow ( n + 1 , m - j ) : ( n , m ) \in \mathcal { V } , \ , \ , \ , 1 \leq j \leq p \} . \end{aligned} \end{align*}


\begin{align*} C ^ * _ { \tiny \rm F i n s l e r } ( T ) = C ^ * _ { \psi , \varphi } ( \mathbf { T } ) : = \inf _ { \| \mathbf { u } \| _ { x , \varphi } = 1 } \| \mathbf { T } { \mathbf { u } } \| _ { f ( x ) , \psi } . \end{align*}


\begin{align*} a ^ \epsilon ( \lambda , m ^ 2 ) : = - 3 \lambda \big ( - \Delta ^ \epsilon + m ^ 2 ) ^ { - 1 } ( 0 , 0 ) + 6 \lambda ^ 2 \big \| \big ( - \Delta ^ \epsilon + m ^ 2 \big ) ^ { - 1 } ( 0 , \cdot ) \big \| ^ 3 _ { L ^ 3 } . \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { \# \{ i \in \N : s _ i \le N x _ { s _ i } = b \} } { \# \{ i \in \N : s _ i \le N \} } \end{align*}


\begin{align*} \tilde { c } _ s ( f ) + \tilde { c } _ s ( h ) = \tilde { c } _ s ( f + h ) , \forall \ , s \ge 0 . \end{align*}


\begin{align*} R ( e _ n , e _ { n + 2 } , e _ n , e _ { n + 2 } ) = a ^ 2 c \end{align*}


\begin{align*} \frac { d } { d t } E _ m ( t ) = c ( m - 1 ) \int _ 0 ^ L \abs { \omega _ x } ^ 2 d x + c m \int _ 0 ^ { + \infty } \int _ 0 ^ L \sigma ' ( s ) \abs { \eta _ x } ^ 2 d x d s . \end{align*}


\begin{align*} \mathbf S _ { 1 2 3 } ^ 0 : = \mathbf S _ { 1 2 3 } | _ { \mathcal H ^ 0 } \ . \end{align*}


\begin{align*} \max _ { 1 \le i \le n } \bigg | \sum _ { k = 1 } ^ i \widetilde \eta _ { n , k } \bigg | = O _ P \big ( n ^ { - 1 / 2 } ( \ln \ln n ) ^ { 1 / 2 } \big ) \quad \mbox { a s \ } n \to \infty . \end{align*}


\begin{align*} \widehat { \Theta } ^ n _ s & = \sigma ( X _ s ) \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u + X ^ n _ { \eta _ n ( s ) } - X ^ n _ s \\ & = \sigma ( X _ s ) \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u + \int ^ s _ { \eta _ n ( s ) } \left ( s - \eta _ n ( u ) \right ) ^ { \alpha } \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \\ & + \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \ , \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u , \end{align*}


\begin{align*} \widetilde { \Delta } _ r ( f ) : = \frac { 1 } { 2 } f '' + \frac { \alpha } { 2 } f ' , \end{align*}


\begin{align*} \kappa _ { \bar \epsilon _ j } ^ { - 1 } h _ L ( a _ { \epsilon _ j } ) = \kappa _ { \bar \epsilon _ j } ^ { - 1 } h _ R ( p _ { \epsilon _ j } F ^ { - 2 } a ) = \kappa _ { \bar \epsilon _ j } ^ { - 1 } h _ R ( p _ { \epsilon _ j } E ^ R _ { \epsilon _ j } ( F ^ { - 2 } ) a ) = \kappa _ { \bar \alpha } ^ { - 1 } h _ R ( p _ \alpha E ^ R _ { \epsilon _ j } ( F ^ { - 2 } ) a ) . \end{align*}


\begin{align*} \Gamma _ { P , j } : = \Gamma _ { 0 , k _ 1 ( j ) , k _ 2 ( j ) , 1 , \{ ( a _ i , b _ i ) \} _ { i \in I _ j } } ; O _ j ( P ) : = \left \langle \prod _ { i \in { I _ j } } \tau ^ { ( a _ i , b _ i ) } _ { d _ i } \sigma _ 1 ^ { k _ 1 ( j ) } \sigma _ 2 ^ { k _ 2 ( j ) } \sigma _ { 1 2 } \right \rangle ^ { \mathbf { s } ^ { \Gamma _ { P , j } } , o } \end{align*}


\begin{align*} A ^ * \phi ( x ) = \nabla \cdot ( Q \nabla \phi ) ( x ) - b \cdot \nabla \phi ( x ) + \int _ { \R ^ d } [ \phi ( x - y ) - \phi ( x ) + \nabla \phi ( x ) y 1 _ { ( 0 , 1 ] } ( | y | ) ] \nu ( d y ) . \end{align*}


\begin{align*} { \sf U } _ \infty ^ q \theta _ 0 = { a } _ 0 { \sf U } _ \infty ^ { p + 1 } \Leftrightarrow { \sf U } _ \infty ^ { - 1 } \theta _ 0 = { a } _ 0 { \sf U } _ \infty ^ { p - q } . \end{align*}


\begin{align*} a _ 0 = b _ 0 : = k + 1 , \end{align*}


\begin{align*} h = f + \lambda g . \end{align*}


\begin{align*} p _ { s u c c } = \sum _ { i = 1 } ^ { m i n ( m , ( 2 | E | - \gamma _ { R S U } ) k ) } \mbox { P r } ( C _ { i } ) \sum _ { j = 0 } ^ { k } \binom { m - i } { j } ( p _ { m a t c h , j } ) ^ { 2 } , \end{align*}


\begin{align*} E _ { F } \hat { \beta } = \beta ( F ) F \in \mathbf { F } _ { 2 } ( I _ { n } ) . \end{align*}


\begin{align*} c ( b ) & = c o a r m _ \lambda ( b ) - c o l e g _ \lambda ( b ) \\ h ( b ) & = a r m _ \lambda ( b ) + l e g _ \lambda ( b ) + 1 \end{align*}


\begin{align*} g ( x , Y , Z ) = u , g _ x ( x , Y , Z ) = p , \end{align*}


\begin{align*} \left \langle S ~ | ~ ( s t ) _ { m _ { s t } } = ( t s ) _ { m _ { t s } } \ ; \ ; s , t \in S \ ; \ ; s \neq t \right \rangle , \end{align*}


\begin{align*} \deg \mathcal { F } _ 1 ( w , x ) = \deg \mathcal { F } _ 2 ( w , x ) + 1 . \end{align*}


\begin{align*} \pi _ { [ n - 1 ] } \cdot \pi _ { i _ n } = 0 \end{align*}


\begin{align*} b = \begin{psmallmatrix} & 0 & \\ 0 & & 0 \\ & 1 & \end{psmallmatrix} \otimes \begin{psmallmatrix} & 0 & \\ 0 & & 1 \\ & 1 & \end{psmallmatrix} \otimes \begin{psmallmatrix} & 0 & \\ 1 & & 1 \\ & 1 & \end{psmallmatrix} \otimes \begin{psmallmatrix} & 1 & \\ 1 & & 1 \\ & 1 & \end{psmallmatrix} \end{align*}


\begin{align*} \left | H ( w ) \right | = \lim _ { k \rightarrow \infty } | H \left ( \omega ( z _ { n ( k ) } ) \right ) | = \lim _ { k \rightarrow \infty } | z _ { n ( k ) } | = + \infty . \end{align*}


\begin{align*} & H ^ k ( \tilde { \mathbb { P } } , \mathcal { O } ( \tilde { Y } + K _ { \tilde { \mathbb { P } } } ) ) \cong H ^ k ( \mathbb { P } , \mathcal { O } ( Y + K _ { \mathbb { P } } ) ) = 0 k > 0 \\ & H ^ k ( \tilde { \mathbb { P } } , \mathcal { O } ( \tilde { Y } + K _ { \tilde { \mathbb { P } } } - D _ j ) ) = 0 k > 0 \end{align*}


\begin{align*} \lim _ { b \to 0 ^ + } N _ \delta ( b ) = N _ \delta \end{align*}


\begin{align*} \eta _ { \nu _ { n } } ( \Phi _ { n } ( z ) ) = z \end{align*}


\begin{align*} X _ { 1 } = h X _ { 2 } \end{align*}


\begin{align*} F _ m ( \psi ) ( x , y ) = ( F _ m ( ) ( x ) , F _ m ( \widehat { \psi } ) ( y ) ) . \end{align*}


\begin{align*} \mathcal L _ { X , D } = \bigcup _ { t } z L ( t , - z ) ^ { - 1 } \mathcal H _ + . \end{align*}


\begin{align*} \Delta ( z ) = ( - 1 ) ^ { r m } \cdot \left ( \sum _ { \ell = 0 } ^ { r m } P _ { \ell } ( z ) \Delta _ { 1 , \ell + 1 } ( z ) \right ) = ( - 1 ) ^ { r m } \times \ P _ { r m } ( z ) \Delta _ { 1 , r m + 1 } ( z ) \in K \enspace . \end{align*}