\begin{align*} | W ( g , f ) ( x , \omega ) | & = 2 ^ d | V _ { g ^ \vee } f ( 2 x , 2 \omega ) | = 2 ^ d | \langle f , M _ { 2 \omega } T _ { 2 x } g ^ \vee \rangle | \\ & \leq 2 ^ d \norm { f } _ 2 \norm { M _ { 2 \omega } T _ { 2 x } g ^ \vee } _ 2 = 2 ^ d \norm { f } _ 2 \norm { g } _ 2 . \end{align*}


\begin{align*} \frac { \partial g ^ { i } } { \partial y _ { j k } ^ { \sigma } } + \frac { \partial g ^ { j } } { \partial y _ { k i } ^ { \sigma } } + \frac { \partial g ^ { k } } { \partial y _ { i j } ^ { \sigma } } = 0 . \end{align*}


\begin{align*} A - \lambda = 0 \end{align*}


\begin{align*} y [ n ] & = \sum _ { i \in \mathcal { K } } \nolimits \sqrt { p _ i } s [ n ] h _ i + w [ n ] = \sqrt { K \bar { p } } h ' s [ n ] + w [ n ] , \ n = 1 , 2 , \cdots , N , \end{align*}


\begin{align*} \Lambda ( \rho ) = \int _ 0 ^ \infty \exp \left ( - \rho y ^ { - 1 } - y \right ) d y \sim \rho ^ { 1 / 4 } \exp ( - 2 \sqrt { \rho } ) \sqrt { \pi } , \textrm { a s } \rho \to \infty . \end{align*}


\begin{align*} | D _ k f ( x ) - D _ k f ( y ) | & = \bigg | \int _ { \Bbb R ^ N } ( D _ k ( x , z ) - D _ k ( y , z ) ) f ( z ) d \omega ( z ) \bigg | \\ & \leqslant \| f \| _ { L ^ \infty } \bigg ( \int _ { \| x - z \| \leqslant r ^ { 4 - k } } + \int _ { \| y - z \| \leqslant r ^ { 4 - k } } \bigg ) \frac { ( r ^ k \| x - y \| ) ^ s } { V _ k ( x ) + V _ k ( z ) } d \omega ( z ) , \end{align*}


\begin{align*} \sigma _ r ( h ) : = S _ r \sigma ( h ) , h \in H . \end{align*}


\begin{align*} \partial _ t u + \partial _ x ( - | D | u + u ^ 2 ) = 0 , \end{align*}


\begin{align*} \left | \sum _ { \substack { | \Im ( \rho ) | \leq T \\ \ : \Re ( \rho ) > \sigma } } \frac { x ^ { \rho - 1 } } { \rho } \right | & \leq \frac { 2 N ( \sigma , T ) } { T } \\ & \le 2 \bigg [ C _ 1 ( \sigma ) \exp \left ( \frac { B _ 2 ( 5 - 8 \sigma ) } { 3 } r \right ) \left ( B _ 2 r \right ) ^ { 5 - 2 \sigma } \\ & \qquad \qquad \qquad + C _ 2 ( \sigma ) \exp \left ( - B _ 2 r \right ) ( B _ 2 ) ^ 2 r ^ 2 \bigg ] \\ & = s _ 2 '' ( x , \sigma ) , \ . \end{align*}


\begin{align*} B _ t + a B J _ x + b J B _ x + \mu \Lambda ^ \alpha B = & \ 0 , \ \ \ \ a , b \in \mathbb R \\ B _ x = & \ \mathcal H J \end{align*}


\begin{align*} \frac { 1 } { y ^ { \sigma _ 1 } } \sum _ { n \leq x } \frac { 1 } { n ^ { \sigma _ 2 } ( y + n ) ^ { \sigma _ 3 } } & \ll \frac { 1 } { y ^ { \sigma _ 1 + \sigma _ 3 } } \sum _ { n \leq x } \frac { 1 } { n ^ { \sigma _ 2 } } \\ & \ll \begin{cases} x ^ { - \sigma _ 1 - \sigma _ 3 } & ( \sigma _ 2 > 1 ) \\ x ^ { - \sigma _ 1 - \sigma _ 3 } ( \log x ) & ( \sigma _ 2 = 1 ) \\ x ^ { 1 - \sigma _ 1 - \sigma _ 2 - \sigma _ 3 } & ( \sigma _ 2 < 1 ) , \\ \end{cases} \end{align*}


\begin{align*} f ( x ) & = - \ln r \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q } w ( Q ) \psi _ { j } ( x , x _ { Q } ) q _ { j } f ( x _ { Q } ) + R _ 1 ( f ) ( x ) + R _ M ( x ) \\ & = T _ M ( f ) ( x ) + R _ 1 ( f ) ( x ) + R _ M ( x ) . \end{align*}


\begin{align*} \nabla ( f _ h ) ( x ) = - \int _ 0 ^ { + \infty } e ^ { - t } P _ t ^ { \Sigma } ( \nabla ( h ) ) ( x ) d t . \end{align*}


\begin{align*} \left \{ \mathfrak { P } _ { \ell } \chi F , \mathfrak { P } _ { \ell } \chi G \right \} _ { \ell } ^ { \widetilde { \Sigma } } [ \widetilde { \psi } ] = \left \langle E _ { \widetilde { \Sigma } } , \left ( \mathfrak { P } _ { \ell } \chi F \right ) ^ { ( 1 ) } [ \widetilde { \psi } ] \otimes \left ( \mathfrak { P } _ { \ell } \chi G \right ) ^ { ( 1 ) } [ \widetilde { \psi } ] \right \rangle . \end{align*}


\begin{align*} \sup _ { f \in M } \| f \| _ { Y _ q ( \tilde \Omega _ { [ \hat t ( n ) ] } ) } \stackrel { ( \ref { n u 2 } ) , ( \ref { e m b _ n u } ) } { \underset { \mathfrak { Z } _ 0 } { \lesssim } } 2 ^ { ( ( 1 ‐ \lambda ) \mu _ * ‐ \lambda \alpha _ * ) k _ * \hat t ( n ) } \stackrel { ( \ref { m t p r 0 e q } ) } { = } n ^ { ‐ \hat \nu } . \end{align*}


\begin{align*} ( A \otimes B ) ( A \ , { } ^ I \ ! \otimes ^ I B ) = ( A \otimes B ) E , ( A \ , { } ^ I \ ! \otimes ^ I B ) ( A \otimes B ) = E ( A \otimes B ) . \end{align*}


\begin{align*} H \psi _ { \pm } ( x , k ) = \left ( - \partial _ { x x } + V \right ) \psi _ { \pm } ( x , k ) = k ^ { 2 } \psi _ { \pm } ( x , k ) \end{align*}


\begin{align*} 0 = \lim _ { n \to \infty } \frac { C \sqrt { k _ { n } } } { \epsilon k _ { n } } \geq \lim _ { n \to \infty } \frac { | \hat { I } _ { \geq \epsilon } ( k _ { n } ) | } { k _ { n } } . \end{align*}


\begin{align*} \mathrm { C h } _ \lambda ( u , v ) : = \mathrm { C h } ( u , v ) + \lambda \left \langle u , v \right \rangle _ { L ^ 2 ( M ) } , u , v \in W ^ { 1 } ( M ) , \end{align*}


\begin{align*} s = \frac { u + 2 + 2 \sqrt { u + 1 } } { u } , \end{align*}


\begin{align*} \mathrm { v a r } [ Z _ i ] & \stackrel { ( a ) } { = } \mathbb { E } [ X ^ 2 ] \mathbb { E } [ Y ^ { - 2 } ] = \int _ { \sqrt { \mu _ i / \vartheta _ i } } ^ { \infty } y ^ { - 2 } f _ Y ( y ) \mathrm { d } y \stackrel { ( b ) } { = } 2 \int _ { \sqrt { \mu _ i / \vartheta _ i } } ^ { \infty } \frac { 1 } { y } e ^ { \frac { \mu _ i } { \vartheta _ i } - y ^ 2 } \mathrm { d } y \stackrel { ( c ) } { = } e ^ { \frac { \mu _ i } { \vartheta _ i } } \mathrm { E i } \Big ( \frac { \mu _ i } { \vartheta _ i } \Big ) , \end{align*}


\begin{align*} \frac { n - d } { n - 1 } R _ 1 + \frac { 1 } { n - 1 } \left ( R _ 2 + \cdots + R _ n \right ) = ( 1 , \cdots , 1 ) . \end{align*}


\begin{align*} ( q _ 1 , r _ 1 ) = \left ( \frac { 2 p } { 1 + s _ c ( p - 1 ) } , \frac { 2 N p } { N + \gamma } \right ) ( q _ 2 , r _ 2 ) = \left ( \frac { 2 p } { 1 - s _ c } , \frac { 2 N p } { N + \gamma + 2 s _ c p } \right ) ; \end{align*}


\begin{align*} h ( x , m ) \ = \ x m + x + m , \end{align*}


\begin{align*} \tilde M _ 1 ( p , \tau , s , u _ 0 ) = p b _ { \sup } | 1 - p | \Big ( m ^ * ( \tau , s , u _ 0 ) - \frac { a _ { \sup } } { b _ { \inf } } | \Omega | \Big ) . \end{align*}


\begin{align*} & \frac { d \rho } { d t } = \frac { \partial } { \partial S } \mathcal H _ 0 ( \rho , S ) + \frac { \partial } { \partial S } \mathcal H _ 1 ( \rho , S ) \circ d W ( t ) , \\ & \frac { d S } { d t } = - \frac { \partial } { \partial \rho } \mathcal H _ 0 ( \rho , S ) - \frac { \partial } { \partial \rho } \mathcal H _ 1 ( \rho , S ) \circ d W ( t ) . \end{align*}


\begin{align*} \widehat { \mathbf { 1 } _ { P _ { \textbf { c } } } } ( \pmb { \xi } ) = \frac { \exp { ( 2 \pi i \langle \pmb { \xi } , \mathbf { c } \rangle ) } } { p ^ { 2 k n } } \sum _ { \mathbf { v } \in P _ { \mathbf { c } } } \exp { \Big ( 2 \pi i \frac { \langle \pmb { \xi } , \mathbf { v } - \mathbf { c } \rangle } { p ^ { 2 k } } \Big ) } . \end{align*}


\begin{align*} p = \frac { 4 \beta } { ( \chi - \beta ) ^ 2 } . \end{align*}


\begin{align*} m : = \frac { y _ 2 + y _ 1 } 2 \ell : = y _ 2 - y _ 1 \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d } } \int _ { \mathbb { R } ^ { d } } g \left ( \frac { u ( x ) - u ( y ) } { \vert x - y \vert ^ { \alpha } } \right ) \frac { v ( x ) - v ( y ) } { \vert x - y \vert ^ { \alpha + d } } d x d y + \int _ { \mathbb { R } ^ { d } } g ( u ) v \ d x = \int _ { \mathbb { R } ^ { d } } K ( x ) f ( x , u ) v \ d x , \ \ W ^ { \alpha , G } ( \mathbb { R } ^ { d } ) . \end{align*}


\begin{align*} \lim _ { t \rightarrow + \infty } \frac { x _ 1 ( t ) - x _ 2 ( t ) } { \sqrt { t } } = \alpha _ 1 - \alpha _ 2 , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 3 \Phi ^ { i j } ( p , q ) \left ( \frac { q _ i } { q ^ 0 } - \frac { p _ i } { p ^ 0 } \right ) = \sum _ { j = 1 } ^ 3 \Phi ^ { i j } ( p , q ) \left ( \frac { q _ j } { q ^ 0 } - \frac { p _ j } { p ^ 0 } \right ) = 0 , \end{align*}


\begin{align*} ( 1 + \varepsilon ) \vert B \vert ^ d & \geq ( 1 + \varepsilon ) \mathcal { H } ^ { d } _ { \infty } ( B ) \geq ( 1 + \varepsilon ) \mathcal { H } ^ { d } _ { \infty } ( E ) \geq \sum _ { n \geq 0 } \vert L _ n \vert ^ d \\ & \geq \sum _ { n \geq 0 } \mathcal { L } ^ d ( L _ n ) \geq \mathcal { L } ^ d ( E ) = \mathcal { L } ^ d ( B ) = \vert B \vert ^ d . \end{align*}


\begin{align*} \frac { \partial u } { \partial x _ 1 } ( 0 ) & = 2 n \gamma _ { n , s } \int _ { \R ^ n _ + \setminus B _ r ^ + } \frac { r ^ { 2 s } y _ 1 u ( y ) } { ( \vert y \vert ^ 2 - r ^ 2 ) ^ s \vert y \vert ^ { n + 2 } } \dd y , \end{align*}


\begin{align*} V _ g f ( x , \omega ) = e ^ { - 2 \pi i x \cdot \omega } V _ { \widehat { g } } \widehat { f } ( \omega , - x ) . \end{align*}


\begin{align*} \mathbf { B } = ( [ - 1 , 1 ] ^ n ) _ { n > 0 } . \end{align*}


\begin{align*} \norm { f } _ { S _ 0 } = \norm { V _ g f } _ { L ^ { 1 , 1 } } = \iint _ { \R ^ { 2 d } } | V _ g f ( x , \omega ) | \ , d ( x , \omega ) < \infty . \end{align*}


\begin{align*} \langle f , \rho ( x , \omega , \tau ) g \rangle = e ^ { - 2 \pi i \tau } \langle f , \rho ( x , \omega ) g \rangle = e ^ { - 2 \pi i \tau } A ( f , g ) ( x , \omega ) = e ^ { - 2 \pi i \tau } e ^ { \pi i x \cdot \omega } V _ g f ( x , \omega ) . \end{align*}


\begin{align*} A _ { n } ^ { \lambda , \beta } ( x ) = \sum _ { m = 0 } ^ { n } \frac { B _ { n } ^ { m } } { m ! } C _ { m } ^ { \lambda , \beta } ( x ) . \end{align*}


\begin{align*} \lambda ^ { \Q _ { \alpha } } ( \nu ) : = \inf \left \{ { \Q } _ { \alpha } ( f , f ) \ ; \Biggl | \ ; f \in \C , ~ \int _ { E } f ^ { 2 } { \rm d } \nu = 1 \right \} . \end{align*}


\begin{align*} G \cap \partial \pi _ v ( Q ) = \pi _ v ( G ^ + ) \cap \pi _ v ( G ^ - ) \cap \partial \pi _ v ( Q ) = \pi _ v ( G ^ + \cap Q _ v ^ - ) \cap \pi _ v ( G ^ - \cap Q _ v ^ + ) = \pi _ v ( G ^ + \cap G ^ - ) . \end{align*}


\begin{align*} 2 R < \int _ { y _ 0 } ^ \infty \rho d y \leq \int _ { y _ 0 } ^ \infty y ^ { - \beta } d y = \frac { 1 } { \beta - 1 } y _ 0 ^ { 1 - \beta } . \end{align*}


\begin{align*} ( x , \omega , e ^ { 2 \pi i \tau } ) \circledcirc ( x ' , \omega ' , e ^ { 2 \pi i \tau ' } ) = ( x + x ' , \omega + \omega ' , e ^ { 2 \pi i ( \tau + \tau ' + x ' \cdot \omega ) } ) . \end{align*}


\begin{align*} \partial _ t \zeta ( x , t , y , s ; v ) = v _ 1 \frac { \sqrt { s } } { 2 \sqrt { t } } \ , \Psi - v _ 2 \frac { \sqrt { 1 - s } } { 2 \sqrt { 1 - t } } , \end{align*}


\begin{align*} \tilde \mu _ { | S + T | } = \tilde \mu _ { | S | } \boxplus \tilde \mu _ { | T | } . \end{align*}


\begin{align*} L _ + Q = - ( 2 p - 2 ) \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ { p } \right ) Q ^ { p - 1 } , \end{align*}


\begin{align*} p _ 1 : = ( - \Delta ) ^ { - 1 } \partial _ i \partial _ j ( \rho u _ i u _ j ) \mbox { a n d } p _ 2 : = ( \lambda + 2 \nu ) \operatorname { d i v } \mathbf { u } . \end{align*}


\begin{align*} A ^ K _ i ( t ) = \widetilde A ^ K _ i ( t ) + 2 C ( \bar b + \bar d ) t + 2 \bar p t \leq & \max _ { 0 \leq j \leq 1 / \delta _ K } \widetilde A ^ K _ j ( 0 ) + 2 C ( \bar b + \bar d ) t + 2 \bar p t \\ = & \max _ { 0 \leq j \leq 1 / \delta _ K } \beta ^ K _ j ( 0 ) + 2 C ( \bar b + \bar d ) t + 2 \bar p t . \end{align*}


\begin{align*} \partial ^ 2 _ { x _ l x _ k } \Phi \in T _ { y _ 0 } ^ { \perp } X , \forall \ l , k = 1 , \dots n , \end{align*}


\begin{align*} I ( \alpha \cup { \gamma _ { 0 } } , \alpha ) = 2 + 2 w ( \alpha ) . \end{align*}


\begin{align*} F = \int _ { l } ^ { u } f ^ { 2 } ( x ) \pi _ { l , u } ( x ) \mathrm { d } x . \end{align*}


\begin{align*} 2 ^ { ‐ m _ 0 ' ( ( 1 ‐ \lambda ) ( s _ * + 1 / q ‐ 1 / p _ 1 ) + \lambda ( 1 / q ‐ 1 / p _ 0 ) ) } = 2 ^ { ‐ m _ 0 ' ( 1 ‐ \lambda ) s _ * } \stackrel { ( \ref { h a t _ s i g m a _ d e f } ) } { = } n ^ { ‐ \hat \sigma } . \end{align*}


\begin{align*} \langle f , h \rangle = \iint _ { \R ^ { 2 d } } V _ { g _ 0 } f ( z ) \overline { V _ { g _ 0 } h ( z ) } \ , d z , \end{align*}


\begin{align*} T ^ { \epsilon , L } : = \lim _ { t \uparrow \infty } T _ t , \end{align*}


\begin{align*} c _ { i , j } = \dfrac { 4 } { \pi ^ 2 } \int _ D f ( x , y ) T _ i ( x ) T _ j ( y ) \omega ( x , y ) d x d y = \dfrac { 4 } { \pi ^ 2 } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { \pi } f ( \cos \theta _ x , \cos \theta _ y ) \cos i \theta _ x \cos j \theta _ y d \theta _ x d \theta _ y , \end{align*}


\begin{align*} ( D \setminus \{ \bar { b } _ n \} ) \cup ( E \setminus \{ \bar { b } _ n \} ) \cup ( F \setminus \{ \bar { b } _ n \} ) & = ( D \cup E \cup F ) \setminus \{ \bar { b } _ n \} \\ & = \mathbb { N } \setminus \{ \bar { b } _ n \} \end{align*}


\begin{align*} \sum _ { \rho \in R } ( \rho ) = \frac { 1 } { 2 \pi i } \int _ { \partial { R } } \log F ( s ) \ d s \end{align*}


\begin{align*} \widetilde { K } = \{ Q ^ - ( \cdot + x ( t ) , t ) , \ ; t \in \mathbb { R } \} \end{align*}


\begin{align*} D ^ * f = ( \langle f , e _ \gamma \rangle ) _ { \gamma \in \Gamma } . \end{align*}


\begin{align*} f ( x _ 1 , \ldots , x _ n ) \ = \ g ( h _ 1 ( x _ 1 , \ldots , x _ n ) , \ldots , h _ m ( x _ 1 , \ldots , x _ n ) ) . \end{align*}


\begin{align*} t ^ c _ j = Q ^ c \left ( \prod _ { i \in \mathcal { N } _ c \setminus \{ j \} } \operatorname { s g n } ( \phi ^ v _ i ( t ^ v _ i ) ) \sum _ { i \in \mathcal { N } _ c \setminus \{ j \} } | \phi ^ v _ i ( t ^ v _ i ) | \right ) . \end{align*}


\begin{align*} \lambda = \frac { 1 } { \| \tilde a _ i \| ^ 2 } \in \mathbb { R } _ { > 0 } . \end{align*}


\begin{align*} \widetilde { L } _ n ' - \widetilde { L } _ { n - 1 } ' - \widetilde { L } _ { n - 1 } = 0 . \end{align*}


\begin{align*} - \begin{bmatrix} H & - A ^ T \\ A & 0 \end{bmatrix} \begin{bmatrix} \mathbf { x } \\ \mathbf { y } \end{bmatrix} + \begin{bmatrix} - \mathbf { g } \\ \mathbf { b } \end{bmatrix} \in N _ D ( \mathbf { x } , \mathbf { y } ) , \end{align*}


\begin{align*} | \partial _ t \widetilde { N } _ 3 | \leq \frac { | \dot { \mathfrak { q } } _ 1 | + | \dot { \mathfrak { q } } _ 2 | } { ( \mathfrak { q } _ 1 - \mathfrak { q } _ 2 ) ^ 4 } \ll \frac { 1 } { ( \mathfrak { q } _ 1 - \mathfrak { q } _ 2 ) ^ 4 } = N _ 3 . \end{align*}


\begin{align*} f ( \underline T ) = V ^ * f ( \underline N ) V , \textup { f o r e v e r y } f \in \mathcal R ( X ) , \end{align*}


\begin{align*} H ( v ) : = \beta ( v ) \omega _ 0 ( v , u ) , \end{align*}


\begin{align*} \int _ { \Omega _ R } | b ( \rho ) | ^ \gamma = \int _ { \{ \rho > \overline \rho - \alpha _ 1 \} } | b ( \rho ) | ^ \gamma \leq c \int _ { \{ \rho > \overline \rho - \alpha _ 1 \} } P ( \rho ) \leq c \int _ { \{ \rho > \overline \rho - \alpha _ 1 \} } \left ( P ( \rho ) - P ( r ) - P ' ( r ) ( \rho - r ) \right ) \end{align*}


\begin{align*} B ( y , u ) : = B _ + ( y , u ) \cup B _ - ( y , u ) , \end{align*}


\begin{align*} \mu ^ { \# } ( k , \ell , m , n ) : = \int \overline { \mathcal { K } ^ { \# } ( x , k ) } \mathcal { K } ^ { \# } ( x , \ell ) \overline { \mathcal { K } ^ { \# } ( x , m ) } \mathcal { K } ^ { \# } ( x , n ) \ , d x \end{align*}


\begin{align*} \left \{ \begin{aligned} \Ddot { \xi } ( s ) & = \dot { \xi } ( s ) + f ^ \prime ( \xi ( s ) ) , s \in ( 0 , b ) , \\ \xi ( 0 ) & = x _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} \bar { \Delta } ^ { \rm P } _ { \rm P r o a c t i v e } = - 1 - \tau _ { \rm f } + \mathbb { E } T ^ { \rm P r o a c } _ j + \mathbb { E } \tau ^ { \rm P r o a c } _ { V _ { j - 1 } } . \end{align*}


\begin{align*} \mathcal { T } ( * ^ V , L ) & = - \tfrac 1 4 \sum _ { i < j } \left ( \lambda _ i \lambda _ j \ , w _ i w _ j \otimes 1 + 1 \otimes v _ i v _ j \right ) \circ \left ( \mu _ { i j } \ , w _ i w _ j \otimes 1 + 1 \otimes v _ i v _ j \right ) \\ & = \tfrac 1 4 \sum _ { i < j } \left ( - \lambda _ i \lambda _ j + w _ i w _ j \otimes v _ i v _ j \right ) \circ \left ( - \mu _ { i j } + w _ i w _ j \otimes v _ i v _ j \right ) , \end{align*}


\begin{align*} \tau ' _ { K } = \inf \Big \{ t \ge 0 , \exists i \in \{ 0 , 1 , \cdots , { 1 \over \delta _ K } - 1 \} ; N ^ K _ i ( t \log K ) < K ^ a \Big \} . \end{align*}


\begin{align*} \mathcal { C } _ R = \{ u \in C _ c ^ \infty ( \overline { W _ R } ) \mid u = 0 \} . \end{align*}


\begin{align*} \begin{cases} - \Delta P _ { \tilde { \Omega } _ { \varepsilon } } w + \tilde { \lambda } _ 0 \underline { m } P _ { \tilde { \Omega } _ { \varepsilon } } w = f _ 0 ( w ) & \tilde { \Omega } _ { \varepsilon } \ ; , \\ P _ { \tilde { \Omega } _ { \varepsilon } } w = 0 & \partial \tilde { \Omega } _ { \varepsilon } \ ; . \end{cases} \end{align*}


\begin{align*} \mathcal A = \big \{ { \bf y } \in \R ^ { 4 d } : 4 \ , C _ 0 \ge c _ 0 \mathcal W ( x , v ) + c _ 0 \mathcal W ( x ' , v ' ) \big \} , \Gamma = \big \{ { \bf y } \in \R ^ { 4 d } : r ( { \bf y } ) \ge R _ 0 \big \} , \end{align*}


\begin{align*} & r _ { \eta } = r ( \eta ) e ^ { 2 i t g ( \eta ) } D ^ { - 2 } _ { 0 } ( \eta , \xi ) \exp \left [ i \nu ( \eta ) \log ( 2 t g '' \left ( \eta \right ) ) \right ] , \\ & r _ { - \eta } = r ( - \eta ) e ^ { 2 i t g ( - \eta ) } D ^ { - 2 } _ { 0 } ( - \eta , \xi ) \exp \left [ i \nu ( - \eta ) \log ( - 2 t g '' \left ( - \eta \right ) ) \right ] . \end{align*}


\begin{align*} L = \omega ^ 2 - F ^ 2 - d x ^ 2 - d y ^ 2 \end{align*}


\begin{align*} \mu \left ( \bigcup _ { i \in \mathbb { N } } \bigcup _ { L \in \mathcal { F } _ { 2 , i } } L \right ) = 1 . \end{align*}


\begin{align*} P _ { i n } = P _ t | h _ 1 | ^ 2 . \end{align*}


\begin{align*} \tau ( y ) & = \sup \{ 0 < \delta < c \ ; | \ ; ( \mathcal B _ d ( x , \delta ) \times \mathcal B _ { n - d } ( y , \varepsilon ) ) \cap X \\ & \} . \end{align*}


\begin{align*} E _ n = { A _ n ( i ) \over ( 1 + i ) ^ { n - 1 } } , \end{align*}


\begin{align*} \sum _ { \gamma \in \Gamma _ { 9 } ( P G ) } c ( g ( \gamma ) ) = 3 5 \equiv 1 \pmod 2 . \end{align*}


\begin{align*} \omega _ Q ^ a = d q ^ i \wedge d p _ i ^ a = - d \theta ^ a _ Q , \end{align*}


\begin{align*} \Gamma : = \mathrm { S L } _ n ( { \Z [ 1 / p ] } ) < \mathrm { S L } _ n ( \R ) \times \mathrm { S L } _ n ( \Q _ p ) , \end{align*}


\begin{align*} \mu _ { j k } = \frac { a _ { j k } } { 2 \rho a d } , \nu _ { j k } = \frac { b _ { j k } } { 2 \rho \omega a d } , ( j , k ) = 1 , 3 . \end{align*}


\begin{align*} \begin{aligned} \Delta \left ( t \right ) & = t - U \left ( t \right ) \\ & = t - t ^ S _ { j - 1 } - \tau _ { \rm f } + \tau ^ { \rm R e a c } _ { V _ { j - 1 } } , t \in \mathcal { I } _ j . \end{aligned} \end{align*}


\begin{align*} J ^ m ( x \cdot y ) \cdot J ^ { m + 1 } x = J ^ m y \end{align*}


\begin{align*} X = \begin{pmatrix} ( g ^ 1 ) ^ T \\ ( g ^ 2 ) ^ T \end{pmatrix} \begin{pmatrix} g ^ 1 & g ^ 2 \end{pmatrix} = \begin{pmatrix} \| g ^ 1 \| ^ 1 & \langle g ^ 1 , g ^ 2 \rangle \\ \langle g ^ 1 , g ^ 2 \rangle & \| g ^ 2 \| ^ 2 \end{pmatrix} , \end{align*}


\begin{align*} \alpha U ( z ) = \left [ \begin{array} { c } a z + b u _ 0 \\ c z + d u _ 0 \end{array} \right ] = \left [ \begin{array} { c } ( a z + b u _ 0 ) ( c z + d u _ 0 ) ^ { - 1 } u _ 0 \\ u _ 0 \end{array} \right ] u _ 0 ^ { - 1 } ( c z + d u _ 0 ) . \end{align*}


\begin{align*} \nu _ F = \sum _ { H \in H ( n - m ) } \nu _ { F , H } . \end{align*}


\begin{align*} N _ i ^ { { \bf m } * } = \sum _ { j \in \bf m } a _ { i j } ^ { \bf m } N _ j , \end{align*}


\begin{align*} \begin{aligned} | N _ { G } ( h ) \backslash V ( T ' \cup B ) | & \geq d _ { G } ( h ) - | N _ { G } ( h ) \cap V ( B ) | \\ & - | N _ { G } ( h ) \cap ( V ( T ' ) \backslash \{ u _ { l + 1 } , \cdots , u _ { j - 1 } \} | \\ & \geq ( t + 3 ) - 2 - ( t - \lfloor \frac { j - l - 1 } { 2 } \rfloor ) \\ & \geq 1 + \lfloor \frac { j - l - 1 } { 2 } \rfloor . \end{aligned} \end{align*}


\begin{align*} i _ a : ( U , \tau _ 0 \restriction U ) & \to ( G , \tau ) \\ i _ a ( x ) & = a \cdot x \end{align*}


\begin{align*} \frac { 1 } { \phi ( d ) } \sum _ { \psi \bmod d } \psi ( m h ) \overline { \psi } ( - n k ) = \left \{ \begin{array} { c l } 1 & d | m h + n k \\ \\ 0 & . \end{array} \right . \end{align*}


\begin{align*} \Gamma ( 3 , x ) = \Big ( x ^ 2 + 2 ( x + 1 ) \Big ) e ^ { - x } , \ \ \Gamma ( s , x ) \sim x ^ { s - 1 } e ^ { - x } \ \ x \to \infty , \ \ s > 1 . \end{align*}


\begin{align*} h _ A ( \tfrac { 1 } { x } ) = h _ A ( x ) = h _ A ( y ) . \end{align*}


\begin{align*} F ( x , \omega ) \langle \pi ( x , \omega ) f , g \rangle = 0 , ( x , \omega ) \end{align*}


\begin{align*} \langle \widetilde { f } , \varphi \rangle = \frac { 1 } { \overline { \langle g , \widetilde { g } \rangle } } \iint _ { \R ^ { 2 d } } V _ g f ( x , \omega ) \langle M _ \omega T _ x \widetilde { g } , \varphi \rangle \ , d ( x , \omega ) . \end{align*}


\begin{align*} \int _ { t _ 1 } ^ { T _ { x , \xi } } p \left ( f \left ( \xi ( s ) \right ) - u \left ( \xi ( s ) \right ) \right ) d s = u \left ( \xi ( t _ 1 ) \right ) , \end{align*}


\begin{align*} \mathcal { Z } _ K = \bigcup _ { \sigma \in K } ( D ^ 2 , S ^ 1 ) ^ \sigma \subseteq ( D ^ 2 ) ^ m \end{align*}


\begin{align*} \L ^ \circ = r ^ { - 2 } \L . \end{align*}


\begin{align*} E _ { 1 } = - \frac { 1 } { 2 \pi i } \int _ { \Gamma _ E } \left ( I + \mu ( s ) \right ) ( J ^ E ( s ) - I ) d s : = I _ 1 + I _ 2 + I _ 3 . \end{align*}


\begin{align*} M _ { p , \alpha } = \{ a \in S ( B ) : p ( a ) = \alpha \} \end{align*}


\begin{align*} \langle V ^ * _ g F , f \rangle & = \iint _ { \R ^ { 2 d } } F ( x , \omega ) \langle M _ \omega T _ x g , f \rangle \ , d ( x , \omega ) \\ & = \iint _ { \R ^ { 2 d } } F ( x , \omega ) \overline { V _ g f ( x , \omega ) } \ , d ( x , \omega ) \\ & = \langle F , V _ g f \rangle . \end{align*}


\begin{align*} \sup _ { B _ R ( y ) } \frac { u ( x ) } { x _ 1 } \leqslant C \sup _ { B _ R ( y ) } u = C \sup _ { B _ { R / y _ 1 } ( e _ 1 ) } \tilde u & \leqslant C \inf _ { B _ { R / y _ 1 } ( e _ 1 ) } \tilde u = C \inf _ { B _ R ( y ) } u \leqslant C \inf _ { B _ R ( y ) } \frac { u ( x ) } { x _ 1 } \end{align*}


\begin{align*} \hat { K } = \arg \max _ { k } f ( \mathbf { y } ; \mathcal { H } _ k ) . \end{align*}


\begin{align*} A = n \ , \frac { 2 \gamma a } { 1 - e ^ { - 2 \gamma a } } \ , e ^ { - 2 \gamma a } B = n \ , \frac { 2 \gamma a } { 1 - e ^ { - 2 \gamma a } } . \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - \alpha _ k \frac { \alpha _ k ^ \intercal x _ k - \beta _ k } { \norm { \alpha _ k } _ 2 ^ 2 } . \end{align*}


\begin{align*} \sum _ { \substack { F \in \mathcal { F } \\ { G \in \mathcal { G } } } } \varphi ( F , G ) \geq \rho ( d , d - k - 1 ) | \mathcal { F } | = \rho ( d , d - k - 1 ) f _ { d - 1 } ( P ) . \end{align*}


\begin{align*} = \int _ { G ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \phi ( \tilde { \tau } _ m ( g \times h ) , s ) \mathbf { f } ( h ) \mathbf { d } h . \end{align*}


\begin{align*} 0 & = - \int _ M | \nabla \dot u | ^ 2 \ , d V _ g + 2 \int _ M V \dot u \ , r \ , d V _ g + \int _ M V \dot u ^ 2 \ , d V _ g \\ & \ , \ , + \int _ M V r ^ 2 \ , d V _ g + \int _ M | \nabla r | ^ 2 \ , d V _ g . \end{align*}


\begin{align*} \begin{aligned} & \begin{bmatrix} \mathcal { B G } & \mathcal { A B G } & \dots & \mathcal { A } ^ { n - 1 } \mathcal { B G } \end{bmatrix} \\ & = \begin{bmatrix} B _ 1 g _ 1 ^ \top & A _ 1 B _ 1 g _ 1 ^ \top & \cdots & A _ 1 ^ { n - 1 } B _ 1 g _ 1 ^ \top \\ \vdots & \vdots & \ddots & \vdots \\ B _ m g _ m ^ \top & A _ m B _ m g _ m ^ \top & \cdots & A _ m ^ { n - 1 } B _ m g _ m ^ \top \end{bmatrix} . \end{aligned} \end{align*}


\begin{align*} L \sigma _ { \widetilde { X } } = \left ( \mathcal { O } _ { \widetilde { X } } , \alpha ^ \ast \mathcal { T } ( - 1 ) , M , \alpha ^ \ast \mathcal { O } ( 1 ) , \mathcal { O } _ { L _ 1 } , \dots , \mathcal { O } _ { L _ 7 } , \mathcal { B } _ 1 , \mathcal { B } _ 2 , \mathcal { B } _ 3 , \mathcal { B } _ 4 \right ) \end{align*}


\begin{align*} f ^ { - 1 } ( Z ) \ = \ \Pi ( \Gamma _ f \ \cap \ ( \R ^ n \times Z ) ) \end{align*}


\begin{align*} \theta \Big ( \sum _ { i = 0 } ^ { + \infty } x _ i t ^ { i } , \sum _ { j = 0 } ^ { + \infty } y _ j t ^ { j } \Big ) \Big ( \sum _ { k = 0 } ^ { + \infty } V _ k t ^ { k } \Big ) = \sum _ { s = 0 } ^ { + \infty } \sum _ { i + j + k = s } \theta ( x _ i , y _ j ) v _ k t ^ { s } , \ ; \forall x _ i , y _ j \in L , \ ; v _ k \in V . \end{align*}


\begin{align*} \norm { f } _ 2 = \norm { \widehat { f } } _ 2 . \end{align*}


\begin{align*} X _ { t + s } - X _ t = \begin{cases} 1 & G ^ + _ { t , s } ( x ) \cap \{ X _ t = x \} , \\ - 1 & G ^ - _ { t , s } ( x ) \cap \{ X _ t = x \} , \end{cases} \end{align*}


\begin{align*} S _ q = \begin{cases} U _ 0 \bot p U _ 0 & q = p , \\ H _ 2 & . \end{cases} \end{align*}


\begin{align*} & \left \| \lambda _ q ^ s e ^ { - \mu \lambda _ q ^ \alpha t } \| B _ q ( 0 ) \| _ { L ^ 2 } \right \| _ { L ^ { \frac { \alpha } { s + \alpha - \frac 5 2 } } ( 0 , T ; l ^ 2 ) } \\ \leq & \ \left \| e ^ { - \mu \lambda _ q ^ \alpha t } B _ q ( 0 ) \right \| _ { L ^ { \frac { \alpha } { s + \alpha - \frac 5 2 } } ( 0 , T ; H ^ s ) } \\ \leq & \ C ( T ) \| B ( 0 ) \| _ { H ^ { \frac { 5 } { 2 } - \alpha } } \end{align*}


\begin{align*} t _ i = t _ j \textrm { f o r } 0 \leq i < j \leq n - 1 . \end{align*}


\begin{align*} \mathcal { A } _ \delta ( f ) ( x ) = - f ^ \prime ( x ) + \delta | x | ^ { \delta - 1 } \operatorname { s i g n } ( x ) f ( x ) . \end{align*}


\begin{align*} P ^ { K B } _ \mu ( T , f ) & = P ^ { B } _ \mu ( T , f ) , ~ \underline { C P } ^ K _ \mu ( T , f ) = \underline { C P } _ \mu ( T , f ) , \\ \overline { C P } ^ K _ \mu ( T , f ) & \leq \overline { C P } _ \mu ( T , f ) , ~ P ^ { K P } _ \mu ( T , f ) = P ^ { P } _ \mu ( T , f ) . \end{align*}


\begin{align*} D ^ n ( x ) = x \sum _ { k = 0 } ^ { \lfloor n / 2 \rfloor } L ( n , k ) x ^ { 2 k } y ^ { n - 2 k } . \end{align*}


\begin{align*} \varphi \circ S ^ 2 = \varphi ( - \nu ) , \psi \circ S ^ 2 = \psi ( - \nu ) . \end{align*}


\begin{align*} \P ( B \mbox { i s n o t u n i q u e } ) \leq \sum _ { B ' \in \mathfrak B _ { r - 1 } , B ' \neq B } \P ( \sigma | _ B = \sigma | _ { B ' } ) \leq n ^ d q ^ { - ( r - 1 ) ^ { d } } \ , , \end{align*}


\begin{align*} ( \underbrace { 1 , 1 , \cdots , 1 } _ { n } ) = \frac { 1 } { ( n - 1 ) } \sum \limits _ { i = 1 } ^ n R _ { i } . \end{align*}


\begin{align*} H ( d ( x _ n , y ) ) & \leq G ( d ( x _ N , y ) ) + \sum _ { i = N } ^ \infty \varepsilon _ i \\ & \leq \beta _ H ( \delta / 2 ) / 2 + \beta _ H ( \delta / 2 ) / 2 \\ & \leq \beta _ H ( \delta / 2 ) \end{align*}


\begin{align*} \left ( x ^ 3 - \omega ( x ) x ^ 2 \right ) \left ( x - \frac 1 2 \omega ( x ) \right ) ^ k = L _ v ^ k ( x ^ 3 - \omega ( x ) x ^ 2 ) , \mbox { f o r a l l } k \geq 1 . \end{align*}


\begin{align*} \{ \lambda ^ h _ k \} _ { | k | \geq k _ 0 } \cap \{ \lambda ^ p _ k \} _ { k \geq k _ 0 } = \emptyset , \ \ \{ \lambda ^ h _ k \} _ { | k | \geq k _ 0 } \cap \big ( \{ \lambda _ 0 \} \cup \{ \widehat \lambda _ n \} _ { n = 1 } ^ { n _ 0 } \big ) = \emptyset , \ \ \{ \lambda ^ p _ k \} _ { k \geq k _ 0 } \cap \big ( \{ \lambda _ 0 \} \cup \{ \widehat \lambda _ n \} _ { n = 1 } ^ { n _ 0 } \big ) = \emptyset . \end{align*}


\begin{align*} \dfrac { d } { d \omega } M _ \omega f \Big | _ { \omega = 0 } = 2 \pi i X f \dfrac { d } { d x } T _ x f \Big | _ { x = 0 } = - 2 \pi i P f . \end{align*}


\begin{align*} E _ 1 ( f , p ) = \sum _ { z \leq p } \frac { 1 } { p } \sum _ { \substack { 0 < b < p \\ g c d ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { m } - v ) } { p } } \cdot \frac { 1 } { p } \sum _ { \substack { 0 \leq b < p \\ g c d ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { e m } - v ) } { p } } = 0 . \end{align*}


\begin{align*} w ( \alpha ) - ( w ( \beta ) + w i n d ( \eta _ { \kappa _ { + } } ) - 1 ) = \# ( \gamma _ { 0 } \cap { \pi ( v \cap { [ - s _ { * } , s _ { * } ] \times S ^ { 3 } } ) } ) = \# ( \mathbb { R } \times \gamma _ { 0 } \cap v ) . \end{align*}


\begin{align*} \widehat { D } ^ * \Phi ( x , y ) ( y ^ * ) \subset \begin{cases} \widehat { D } ^ * g ( x ) ( y ^ * ) & y ^ * \in \widehat { \mathcal N } _ D ( g ( x ) - y ) , \\ \varnothing & \end{cases} \end{align*}


\begin{align*} 1 \geq \mu _ { \mathrm { d i s c } } ( \gamma _ { 3 } ) - \mu _ { \mathrm { d i s c } } ( \gamma _ { 0 } \cup { \gamma _ { 2 } } ) = \mu _ { \mathrm { d i s c } } ( \gamma _ { 3 } ) - 1 0 . \end{align*}


\begin{align*} \left . \begin{aligned} & \nabla g ( \bar x ) ^ * y ^ * = 0 , \ , \nabla g ( \bar x ) ^ * \hat z ^ * = 0 , \\ & y ^ * \in \mathcal N _ D ( g ( \bar x ) ; \nabla g ( \bar x ) u ) \end{aligned} \right \} \quad \Longrightarrow \hat z ^ * \notin D _ \textup { s u b } \mathcal N _ D ( g ( \bar x ) , y ^ * ) \left ( \frac { \nabla g ( \bar x ) u } { \norm { \nabla g ( \bar x ) u } } \right ) \end{align*}


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


\begin{align*} \ < P s i > _ n ( t , x ) = - i \int _ 0 ^ t \int _ { \mathbb { R } ^ d } a _ { n } ( t - t ' , x - x ' ) \dot { W } ( t ' , x ' ) d t ' d x ' = - i \int _ 0 ^ t \int _ { \mathbb { R } ^ d } a _ { n } ( t - t ' , x - x ' ) W ( d t ' , d x ' ) , \end{align*}


\begin{align*} P _ n : = P ^ 0 _ n = \left \{ \left [ \begin{array} { c c } a & b \\ c & d \end{array} \right ] \in G _ { n } : c = 0 \right \} , \end{align*}


\begin{align*} \int _ \sigma ^ \tau \delta ( t ) d t \lesssim R \left ( \delta ( \sigma ) + \delta ( \tau ) \right ) = R _ 2 \left ( 1 + \sup _ { \sigma \le t \le \tau } | x ( t ) | \right ) \left ( \delta ( \sigma ) + \delta ( \tau ) \right ) . \end{align*}


\begin{align*} \int _ { \Omega } g _ 1 ( w ) d \mu = \sum \limits _ { k \in T } \int _ { [ k ] } g _ 1 ( w ) d \mu = \sum \limits _ { k \in T } \mu ( [ k ] ) \log p _ { k } = \sum \limits _ { k \in T } p _ k \log p _ k . \end{align*}


\begin{align*} \langle \ell _ { } ' , f \rangle = - \langle \ell _ { } , f ' \rangle = \int _ { - a } ^ 0 \overline { f ' ( x ) } \ , d x - \int _ 0 ^ a \overline { f ' ( x ) } \ , d x = 2 \overline { f ( 0 ) } . \end{align*}


\begin{align*} \begin{cases} B _ x ( x , t ) \geq 0 \ \ \mbox { f o r } \ x \in ( - X ( x _ 0 , t ) , X ( x _ 0 , t ) ) ; \\ B _ x ( x , t ) \leq 0 \ \ \mbox { f o r } \ x \in ( - \infty , - X ( x _ 0 , t ) ) \cap ( X ( x _ 0 , t ) , \infty ) . \end{cases} \end{align*}


\begin{align*} U T = V S \in R [ X _ 1 , \dots , X _ { n - 2 } ] . \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { M ^ { 2 } } { | \Lambda ( M , \Gamma ) | } = 2 \mathrm { V o l } ( Y , \lambda ) . \end{align*}


\begin{align*} n a + ( n - 1 ) b = c \end{align*}


\begin{align*} \min _ { x \in \mathbb R ^ d } f ( x ) : = \mathbb E _ { z \sim \mathcal D } \left [ f ( x ; z ) \right ] , \end{align*}


\begin{align*} \theta = \sum _ { i \in J } \left ( \frac { \bar \alpha _ { i } } { \rho _ { i } } - 1 \right ) > 0 . \end{align*}


\begin{align*} A _ \alpha ( p _ \alpha f ) ( x ) & = \sum _ { k = 1 } ^ d \left ( R _ k ^ { \alpha } ( \partial _ k ( p _ \alpha ) , f ) ( x ) + R _ k ^ { \alpha } ( \partial _ k ( f ) , p _ \alpha ) ( x ) \right ) \\ & \quad + \sum _ { k = 1 } ^ d \left ( \partial _ k ( p _ \alpha ) ( x ) D _ k ^ { \alpha - 1 } ( f ) ( x ) + p _ \alpha ( x ) \partial _ k D ^ { \alpha - 1 } _ k ( f ) ( x ) \right ) . \end{align*}


\begin{align*} f ( n , k , \ell ) = \left ( \sum _ { t \leq j \leq \ell } \binom { \ell } { j } \right ) \binom { \ell } { t } \ell k ^ { \ell - t - 1 } \sum _ { 0 \leq i \leq k - \ell } \binom { n - t } { i } \end{align*}


\begin{align*} \mathcal { Q } ^ \bullet = \bigoplus _ { p = 0 } ^ \infty \biguplus _ { q = 0 } ^ \infty \biguplus _ { r = 0 } ^ \infty \mathcal { Q } ^ { p q r } . \end{align*}


\begin{align*} \Psi _ { \mathbf i } = \varphi _ 2 \left ( \mathbf { i } _ { 1 : 4 } \right ) \cdot \varphi _ 2 \left ( \mathbf { i } _ { 5 : 8 } \right ) . \end{align*}


\begin{align*} \begin{gathered} ( 1 + X ^ 1 + X ^ { 1 + 1 } + \ldots ) ( 1 + X ^ 2 + X ^ { 2 + 2 } + \ldots ) \ldots ( 1 + X ^ k + X ^ { k + k } + \ldots ) \\ = \frac { 1 } { 1 - X } \frac { 1 } { 1 - X ^ 2 } \ldots \frac { 1 } { 1 - X ^ k } = \frac { 1 } { X ^ k ! } . \end{gathered} \end{align*}


\begin{align*} \mbox { f o r e v e r y $ U \in \mathcal { W } _ 1 $ , } \ \ \ \ \eta ( U ) = \frac { \mu ( B ^ { [ U ] } ) } { \sum _ { \widetilde { U } \in \mathcal { W } _ 1 } \mu ( B ^ { [ \widetilde { U } ] } ) } . \end{align*}


\begin{align*} \psi \colon \thinspace \begin{cases} \widehat { \delta } \longmapsto \alpha \\ a _ i \longmapsto g ^ { b _ i } , \ ; \end{cases} \end{align*}


\begin{align*} \begin{aligned} & \phi ( a _ { 1 p ^ t _ 1 } , \ldots , a _ { \omega ^ t - 1 p ^ t _ { \omega ^ t - 1 } } , u _ { \omega ^ t p ^ t _ { \omega ^ t } } , a _ { \omega ^ t + 1 p ^ t _ { \omega ^ t + 1 } } , \ldots , a _ { d p ^ t _ d } ) \\ & - \phi ( a _ { 1 p ^ t _ 1 } , \ldots , a _ { \omega ^ t - 1 p ^ t _ { \omega ^ t - 1 } } , u _ { \omega ^ t p ^ t _ { \omega ^ { t } } - 1 } , a _ { \omega ^ t + 1 p ^ t _ { \omega ^ t + 1 } } , \ldots , a _ { d p ^ t _ d } ) , \end{aligned} \end{align*}


\begin{align*} m _ 1 = \inf \limits _ { \mathcal { M } } J = J ( t _ { \widehat { w } ^ + } \widehat { w } ^ + + s _ { \widehat { w } ^ - } \widehat { w } ^ - ) . \end{align*}


\begin{align*} C _ h ( X ) | _ A = C _ h = C _ h ( A ) \end{align*}


\begin{align*} \gamma ^ \ast ( t ) = \frac { R ^ 2 } { \pi R ^ 2 t - r + 2 R } . \end{align*}


\begin{align*} f = \sum \limits _ { k = - \infty } ^ { \infty } \phi _ { k } * \psi _ { k } * f , \end{align*}


\begin{align*} \mathcal { \psi } = \zeta ^ { i \nu \sigma _ 3 } e ^ { - \frac { i } { 4 } \zeta ^ 2 \sigma _ 3 } \left ( I + m ^ { P C } _ { \eta , 1 } \zeta ^ { - 1 } + \mathcal { O } ( \zeta ^ { - 2 } ) \right ) , { \rm a s } \zeta \rightarrow \infty . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ { s + m + 1 } a _ i e _ i \mapsto h _ 1 | _ { \bar { \sigma } } \left ( \sum _ { i = 0 } ^ m a _ i e _ i + a g \left ( \frac { 1 } { a } \sum _ { i = 0 } ^ { s } a _ { i + m + 1 } e _ i \right ) \right ) . \end{align*}


\begin{align*} d _ { i } ^ { \prime } = \frac { \partial } { \partial x ^ { i } } + \sum _ { k = 0 } ^ { r - 1 } \sum _ { j _ { 1 } \leq \ldots \leq j _ { k } } \frac { \partial } { \partial y _ { j _ { 1 } \ldots j _ { k } } ^ { \sigma } } y _ { j _ { 1 } \ldots j _ { k } i } ^ { \sigma } . \end{align*}


\begin{align*} { { d \hat z _ i } \over d t } = \hat u _ i , \end{align*}


\begin{align*} \forall i = 1 , \ldots , n - 1 : \left \langle c , c ^ { ( i ) } \right \rangle = 0 . \end{align*}


\begin{align*} a _ { \inf } > \begin{cases} \frac { \mu \chi ^ 2 } { 4 } , & \\ \mu ( \chi - 1 ) , & \\ \end{cases} \end{align*}


\begin{align*} C ^ { o } ( \kappa , { \bf P } _ { Y _ 1 } ) = C ^ { \infty } ( \kappa , { \bf P } _ { Y _ 1 } ) = C ^ { \infty } ( \kappa ) = \mbox { ( \ref { l l _ 3 } ) } , \ : \forall { \bf P } _ { Y _ 1 } \end{align*}


\begin{align*} U = [ U _ 1 , u _ { n + 1 } ] , \ \ V = \left [ \begin{array} { c c } V _ { 1 1 } & v _ { 1 2 } \\ v _ { 2 1 } & v _ { 2 2 } \end{array} \right ] , \ \ \Sigma = \left [ \begin{array} { c c } \Sigma _ { 1 } & 0 \\ 0 & \sigma _ { n + 1 } \end{array} \right ] , \end{align*}


\begin{align*} \chi _ { \ell / r } = \widetilde { \pi } _ { \ell / r } \circ \chi \circ \pi _ { \ell / r } | _ { \Sigma } ^ { - 1 } . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \binom { c _ n k n } l } { \prod _ { i = 1 } ^ l ( k n - c _ n k n + i ) } = \frac 1 { l ! } ( \frac c { 1 - c } ) ^ l . \end{align*}


\begin{align*} \frac { \partial } { \partial t } ( e ^ { \lambda ( q , t ) } - 1 ) - 1 = \mathcal { E } ( q , \lambda ( q , t ) ) . \end{align*}


\begin{align*} \mathcal { L } ^ \omega \chi ^ k ( x , \omega ) = \mathcal { L } ^ \omega \Pi ^ k ( x ) , \ ; ( k = 1 , \dots , d ) \end{align*}


\begin{align*} \arg \zeta ( \sigma ) = \pi \end{align*}


\begin{align*} X _ 1 ( f ) = X _ 2 ( f ) = X _ 3 ( f ) \quad . \end{align*}


\begin{align*} G : = \left \{ \begin{aligned} & \begin{pmatrix} 1 & 0 \\ - r e ^ { 2 i t g } & 1 \end{pmatrix} , & k \in U _ 1 \cup U _ 2 , \\ & \begin{pmatrix} 1 & - r ^ { * } e ^ { - 2 i t g } \\ 0 & 1 \end{pmatrix} , & k \in U _ 1 ^ { * } \cup U _ 2 ^ { * } , \\ & I , & \textnormal { e l s e w h e r e } \end{aligned} \right . \end{align*}


\begin{align*} \lim _ { \lambda \rightarrow + \infty } \| f \| ^ 2 _ { p , \lambda , t } = \int _ { \{ p \Psi _ 1 < - t \} } | f | ^ 2 , \ \forall t \in [ 0 , + \infty ) . \end{align*}


\begin{align*} \begin{aligned} \max _ { \boldsymbol { \Theta } , \ : \mathbf { w } } & \biggl | \Bigl ( \mathbf { g } ^ T \boldsymbol { \Theta } \mathbf { H } + \mathbf { h } _ d ^ T \Bigr ) \mathbf { w } \biggr | ^ 2 \\ \textrm { s . t . } & \| \mathbf { w } \| ^ 2 \leqslant 1 \\ & \phi _ n \in [ 0 , 2 \pi ) , \ : \forall n = 1 , 2 , \ldots , N . \end{aligned} \end{align*}


\begin{align*} A \otimes ^ I B = E ( A \otimes B ) E \end{align*}


\begin{align*} \Phi _ m [ C ] = \Theta ( C ) . \end{align*}


\begin{align*} f _ { R | C } = \int f _ { R | C B } f _ { B | C } d b , \end{align*}


\begin{gather*} \gamma _ X \alpha _ X = \alpha _ X ( S ^ 2 \otimes \gamma _ X ) , \\ \beta _ X \gamma _ X = ( S ^ 2 \otimes \gamma _ X ) \beta _ X , \\ c _ X ^ R = c _ X ^ L , \end{gather*}


\begin{align*} t ( \mathbf { P } _ { \ ! \sigma } ) = t ( \mathbf { p } _ { \sigma _ d } ; \ldots ; \mathbf { p } _ { \sigma _ 1 } ) = \sum _ { n = 1 } ^ { d } C _ { n } ^ { ( d ) , \sigma } ( \mathbf { P } _ { \ ! \sigma } ) \cdot \bigg ( H ( \mathbf { p } _ { \sigma _ { n } } ) + \int \ ! \varphi _ n ^ { \sigma } \ , \mathrm { d } \mathbf { p } _ { \sigma _ n } \bigg ) , \end{align*}


\begin{align*} \Gamma _ 1 ^ 0 = \Gamma _ 1 ( \mathcal { N } ) = \left \{ \gamma = \left [ \begin{array} { c c c } a & b & c \\ g & e & f \\ h & l & d \end{array} \right ] \in G ( \mathcal { O } ) : \gamma \equiv \left [ \begin{array} { c c c } 1 _ m & \ast & \ast \\ 0 & 1 _ r & \ast \\ 0 & 0 & 1 _ m \end{array} \right ] \mathcal { N } \right \} . \end{align*}


\begin{align*} 1 + u ^ m \alpha _ m t + \cdots + u ^ { n - 1 } \alpha _ { n - 1 } t ^ { n - 1 } = 1 + u ^ { m - 1 } \beta _ m t ^ m + \cdots + u ^ { n - 2 } \beta _ { n - 1 } t ^ { n - 1 } \end{align*}


\begin{align*} \| Y \| ^ p _ { W ^ { k , p } \delta } = \sum _ { \ell = 0 } ^ { k } \int _ { \Sigma _ \pm } \vert \nabla ^ \ell Y \vert ^ p _ { \Sigma _ i } . \end{align*}


\begin{align*} \gamma = \min \{ \gamma _ \circ , \gamma _ \star \} , \kappa = \tfrac 1 9 \min \{ \kappa _ \circ , \kappa _ \star \} . \end{align*}


\begin{align*} \Gamma ^ { k } = \{ \chi \in \Gamma : \forall x \in \mathfrak { P } ^ { k } \implies \chi ( x ) = 1 \} . \end{align*}


\begin{align*} - 2 ( 1 - \alpha ) + m ( 1 - 2 \alpha ) \geq - 2 ( 1 - \alpha ) + 2 ( 1 - 2 \alpha ) = - 2 \alpha > - 1 . \end{align*}


\begin{align*} \mathbf { 1 } _ r = ( 0 , 0 , 1 ) \mathbf { h } _ r ^ { - 1 } = ( - x , - \omega , e ^ { - 2 \pi i \tau } ) , \end{align*}


\begin{align*} \omega _ { p } \left ( \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix} \right ) \geq \frac { 1 } { 2 ^ { 1 - \frac { 1 } { p } } } \omega _ { p } ( A + B ) , \end{align*}


\begin{align*} \lim _ { r \to \infty } \| \sigma _ r ( h ) - \sigma ( h ) \| _ { L _ 2 ^ 0 ( H ) } = 0 \end{align*}


\begin{align*} m _ 0 = \inf \limits _ { \mathcal { N } } J \geq \inf \limits _ { \mathcal { M } } J = m _ 1 . \end{align*}


\begin{align*} \left ( \lambda E - \mathcal { A } \right ) ^ { - \frac { r } { 2 } } f = \frac { 1 } { \Gamma ( \frac { r } { 2 } ) } \int _ 0 ^ { + \infty } \dfrac { e ^ { - \lambda t } } { t ^ { 1 - \frac { r } { 2 } } } P _ t ( f ) d t , \end{align*}


\begin{align*} \sum _ { n \geq 0 , \lambda \in \mathcal { P } ( n ) } t ^ n \ = \prod _ { n \geq 1 } \frac { 1 } { ( 1 - t ^ n ) } \end{align*}


\begin{align*} \check { S } ( \varphi ( - a ) ) = \varphi ( - \sigma ^ { \varphi } ( \delta _ { \varphi } S ( a ) ) ) . \end{align*}


\begin{align*} \| 1 _ { \varepsilon A } + x \| \ = \ \| y - P _ { B \cup D } ( y ) \| & \ \leqslant \ \mathbf C _ \ell \| y - P _ { \Lambda } ( y ) \| \\ & \ \leqslant \ \mathbf C _ \ell \mathbf C _ { \lambda , p } \widehat { \sigma } _ m ( y ) \ \leqslant \ \mathbf C _ \ell \mathbf C _ { \lambda , p } \| x + 1 _ { \delta B } \| . \end{align*}


\begin{align*} \mathbf A \tilde { \mathbf V } ^ { n + 4 } = 4 \tilde { \mathbf V } ^ { n + 3 } - 3 \tilde { \mathbf V } ^ { n + 2 } + \frac { 4 } { 3 } \tilde { \mathbf V } ^ { n + 1 } - \frac { 1 } { 4 } \tilde { \mathbf V } ^ { n } + k \mathbf b ^ { n + 4 } + k \mathbf q ( \tilde { \mathbf V } ^ { n + 4 } ) , \end{align*}


\begin{align*} \gamma _ { k } ^ { \langle n \rangle } = \frac { \left ( \sum _ { m \in \mathbb { M } _ k } \sqrt { \eta _ { m k } } \left | \hat { g } _ { m k , n } \right | ^ 2 \right ) ^ 2 } { \sum _ { m \in \mathbb { M } _ k } \eta _ { m k } ( \beta _ { m k } - \alpha _ { m k } ) \alpha _ { m k } + \frac { 1 } { \gamma _ t } } . \end{align*}


\begin{align*} \frac { c _ k ( \mu ) L ( \mu , \mathbf { f } , \chi ) } { \pi ^ { \beta } } \mathbf { f } ( g _ { \mathbf { h } } \cdot g _ { \infty } ) = \mathfrak { P } _ k ( w ) \langle \mathbf { h } '' , \mathbf { f } \rangle . \end{align*}


\begin{align*} 0 = \int _ { \mathbb { R } } \rho _ 3 ( t , u ) G ( t , u ) d u - \int _ 0 ^ t \int _ { \mathbb { R } } \rho _ { 3 } ( s , u ) \partial _ s G ( s , u ) d u d s - \int _ 0 ^ t \int _ { \mathbb { R } } \rho _ { 5 } ( s , u ) [ - ( - \Delta ) ^ { \gamma / 2 } G ] ( s , u ) d u d s . \end{align*}


\begin{align*} 0 & = \widetilde { H } ' _ { n + 1 } + 2 \widetilde { H } _ n + 2 x \widetilde { H } ' _ n + \widetilde { H } '' _ n \\ & = - 2 ( n + 1 ) \widetilde { H } _ n + 2 \widetilde { H } _ n + 2 x \widetilde { H } _ n ' + \widetilde { H } '' _ n \\ & = \widetilde { H } '' _ n + 2 x \widetilde { H } ' _ n - 2 n \widetilde { H } _ n . \end{align*}


\begin{align*} \log ( \kappa ( \tfrac { \eta } { n } , \tfrac { 1 } { \eta } ) ) = - \log ( A ( \tfrac { \eta } { n } ) ) + \log ( B ( \tfrac { \eta } { n } ) ) = - \log ( A ( \tfrac { \eta } { n } , \tfrac { 1 } { \eta } ) ) - 2 \log ( \tanh ( \tfrac { \eta } { 2 } ) ) + \log ( h ( \eta ) ) , \end{align*}


\begin{align*} \Phi _ m [ C ] = \Theta ( C ) . \end{align*}


\begin{align*} f _ \varepsilon ( x ) & = \int _ { \R ^ n } \bar f ( y ) \eta _ \varepsilon ( x - y ) \dd y = \int _ { \R ^ n _ + } \bar f ( y ) \big ( \eta _ \varepsilon ( x - y ) - \eta _ \varepsilon ( x _ \ast - y ) \big ) \dd y . \end{align*}


\begin{align*} a | z \rangle = z | z \rangle = z \sum _ { n = 0 } ^ { \infty } A _ { n } | n \rangle = \sum _ { n = 1 } ^ { \infty } z A _ { n - 1 } | n - 1 \rangle . \end{align*}


\begin{align*} 2 ( p - 1 ) = 2 n ( p ^ \prime - 1 ) + \sum \frac { n } { n _ i } ( n _ i - 1 ) \ , . \end{align*}


\begin{align*} \partial _ \tau w = \partial _ y ^ 2 w + \frac { d + 1 } { y } \partial _ y w - \beta ( \tau ) \Lambda _ y w - 3 ( d - 2 ) w ^ 2 - ( d - 2 ) y ^ 2 w ^ 3 , \end{align*}


\begin{align*} K _ { \mathfrak { G } } = ( 2 \eta + 1 ) ^ { n - 1 } \end{align*}


\begin{align*} \Vert \eta \Vert _ \gamma = \max _ { t \in [ 0 , T ] } e ^ { - \gamma t } \ , \vert \eta ( t ) \vert , \eta \in C ^ 0 ( [ 0 , T ] ; \R ^ d ) , \end{align*}


\begin{align*} \mathbb { E } G ^ { \rm R e a c } _ j = \frac { \mathbb { E } \left ( T ^ { \rm R e a c } _ j \right ) ^ 2 } { 2 } + \mathbb { E } T ^ { \rm R e a c } _ j \left ( \mathbb { E } \tau ^ { \rm R e a c } _ { V _ { j - 1 } } - \tau _ { \rm f } - \frac { 1 } { 2 } \right ) . \end{align*}


\begin{align*} & \frac { 1 } { 2 \pi } \int _ { 0 } ^ { T } \left [ \log \left | - \frac { 1 } { 2 } - \epsilon + i t \right | - \log \left | - \frac { 1 } { 2 } + \epsilon + i t \right | \right ] \ d t + \\ & \frac { 1 } { 2 \pi } \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \left [ \arg \left ( \sigma - 1 + i T \right ) - \arg ( \sigma - 1 ) \right ] \ d \sigma = 0 \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty d ^ 2 ( \rho ^ n , \rho ^ { n + 1 } ) \leq 2 \tau ( E ( \rho ^ 0 , \phi ^ { 0 } ) - \inf _ { ( \rho , \phi ) \in \mathcal { A } } E ( \rho , \phi ) ) . \end{align*}


\begin{align*} \mathcal { W } _ m : = \{ { \bf c } \in \Z ^ n : | c _ i | \leq H ^ i , m ^ 2 \mid \mathrm { d i s c } ( f _ { { \bf c } } ) \} . \end{align*}


\begin{align*} \mathcal { Q } ^ \bullet _ \star = \bigoplus _ { p = 0 } ^ \infty \biguplus _ { q = 0 } ^ \infty \biguplus _ { r = 0 } ^ \infty \mathcal { Q } ^ { p q r } _ \star , \end{align*}


\begin{align*} \lim _ { \eta \to 0 } v _ \eta ( \xi _ 0 ) \stackrel { a . s . } { = } 0 . \end{align*}


\begin{align*} D _ { \underline T } = \sum _ { i = 1 } ^ n T _ i \otimes E _ i . \end{align*}


\begin{align*} \begin{aligned} & \Phi \ \ ( x _ \varepsilon , y _ \varepsilon , t _ \varepsilon ) \in \R ^ n \times \R ^ n \times ( 0 , \infty ) \ \\ & ( x _ \varepsilon , y _ \varepsilon , t _ \varepsilon ) \to ( x _ 0 , x _ 0 , t _ 0 ) , u ( x _ \varepsilon , t _ \varepsilon ) \to u ( x _ 0 , t _ 0 ) \quad \ \varepsilon \to 0 . \end{aligned} \end{align*}


\begin{align*} L \sigma ^ \prime _ \mathcal { X } = \left ( \mathcal { E } ^ \prime _ p , \mathcal { E } _ { p _ 1 } , \mathcal { E } _ { p _ 2 } , \mathcal { E } _ { p _ 3 } , \mathcal { E } _ { p _ 4 } , \Phi ( L \sigma _ { \widetilde { X } } ) \right ) \end{align*}


\begin{align*} p _ { \widehat { K } } ( \hat { k } ) & = \mathbb { P } \big ( \mathrm { r o u n d } ( \widehat { K } ^ \mathbb { R } ) = \hat { k } \big ) , \end{align*}


\begin{align*} \begin{dcases} \rho _ { t } + \rho _ { x } = 0 \quad \mbox { i n } ( 0 , T ) \times ( 0 , 1 ) , \\ \rho ( t , 0 ) = \rho ( t , 1 ) , t \in ( 0 , T ) , \\ \rho ( 0 , x ) = \rho _ { 0 } ( x ) , x \in ( 0 , 1 ) , \end{dcases} \end{align*}


\begin{align*} c ( \alpha ) = - \frac { 2 ( 2 \alpha ^ 2 + 1 ) } { 3 ( 1 - 4 \alpha ) } . \end{align*}


\begin{align*} \left \langle h , \int _ 0 ^ t \left [ \int _ \R e ^ { - i k ^ { 2 } s } \overline { \phi } ( k ) \overline { \mathcal { Q } } ( y , k ) F ( s , y ) \ , d y \right ] d s \right \rangle = & \int _ 0 ^ t \int _ \R \Big ( \int _ \R e ^ { i k ^ { 2 } s } \phi ( k ) \mathcal { Q } ( y , k ) h ( k ) \ , d k \Big ) \overline { F } ( s , y ) \ , d y d s . \end{align*}


\begin{align*} \boldsymbol { A } \Psi _ { r n j } = \mathfrak { a } ( p ^ { 1 - r } ) \Psi _ { r n j } r n j \end{align*}


\begin{align*} \begin{aligned} \sup _ { Q _ s \in \mathfrak { Q } _ s , s \in \lbrace 1 , \ldots , k \rbrace } \min _ { G \in \mathcal { G } ( Q _ 0 , \ldots , Q _ k ) } \det ( G ^ \intercal G ) \geq \sup _ { Q _ s \in \mathfrak { Q } _ s ' , s \in \lbrace 1 , \ldots , k \rbrace } \min _ { G \in \mathcal { G } ( Q _ 0 , \ldots , Q _ k ) } \det ( G ^ \intercal G ) , \end{aligned} \end{align*}


\begin{align*} - S J S ^ T = - J \ ; \Leftrightarrow S J S ^ T = J \end{align*}


\begin{align*} \left | \int _ { \mathbb S ^ 1 } B _ x J _ { x } B _ x \ , d x \right | \leq & \ \| J _ x \| _ { L ^ \infty } \| B _ x \| ^ 2 _ { L ^ 2 } , \\ \left | \int _ { \mathbb S ^ 1 } B J _ x B _ { x x } \ , d x \right | \lesssim & \ \| J _ x \| _ { L ^ \infty } \| B _ { x x } \| _ { L ^ \infty } \| B _ x \| _ { L ^ 2 } . \end{align*}


\begin{align*} G = \{ a \rightarrow a x , \ ; \ ; x \rightarrow 1 + x ^ 2 \} . \end{align*}


\begin{align*} m ^ * ( \tau , s , u _ 0 ) = \frac { a _ { \sup } } { b _ { \inf } } { \rm a n d } \tilde M _ 1 ( p , \tau , s , u _ 0 ) = 0 \forall \ , \ , \tau \ge s . \end{align*}


\begin{align*} \frac { d ^ 2 } { d \lambda ^ 2 } \Big | _ { \lambda = 0 } I [ Q + \lambda h ] \ge 0 . \end{align*}


\begin{align*} h _ t ( x ) = \sum _ { p = 1 } ^ { q } \sum _ { s = 1 } ^ m \bigg ( e ^ { - i \langle x , \xi _ p \rangle } g _ s ( \xi _ p ) \rho _ { s t } ( V _ p ) + e ^ { i \langle x , \xi _ p \rangle } g _ s ( - \xi _ p ) \rho _ { s t } ( - V _ p ) \bigg ) , \end{align*}


\begin{align*} \tilde { f } ^ { \prime } ( s , y ) = f _ 1 ( s , y ) + f _ 2 ( s , y ) , \end{align*}


\begin{align*} { \rm S h t } _ \infty ( G , b , \mu ) = \varprojlim _ K { \rm S h t } _ K ( G , b , \mu ) , \end{align*}


\begin{align*} \int _ { \mathbb S ^ 1 } B _ { x x } J B _ x \ , d x = - \int _ { \mathbb S ^ 1 } B _ x J _ { x } B _ x \ , d x - \int _ { \mathbb S ^ 1 } B _ x J B _ { x x } \ , d x \end{align*}


\begin{gather*} e _ { P _ j } ( K / k ) = l ^ { \beta _ j } , 1 \leq \beta _ j \leq n , 1 \leq j \leq r , \quad e _ { \infty } ( K / k ) = l ^ { t } , 0 \leq t \leq n \end{gather*}


\begin{align*} | \check x ^ \perp | = | x ^ \perp - \check { \mathcal T } _ { t , x _ 1 , v _ 1 } v ^ \perp | = \check { \mathcal T } _ { t , x _ 1 , v _ 1 } | v _ * ^ \perp ( t , x , v _ 1 ) - v ^ \perp | \geq \frac 1 2 \check \tau _ { t , x } | v _ * ^ \perp ( t , x , v _ 1 ) - v ^ \perp | . \end{align*}


\begin{align*} \inf _ { ( \rho , m ) } \sup _ { S \in H ^ 1 _ R } \mathcal L ( \rho , m , S ) = \sup _ { S \in H ^ 1 _ R } \inf _ { ( \rho , m ) } \mathcal L ( \rho , m , S ) \end{align*}


\begin{align*} & S \Big ( \sum \limits _ { | j | > n } \sum \limits _ { Q \in Q ^ j } \omega ( Q ) \psi _ { Q } ( \cdot , x _ { Q } ) q _ { Q } h ( x _ { Q } ) \Big ) ( x ) \\ & \qquad = \Big ( \sum \limits _ { Q ' } \Big | q _ { Q ' } \Big ( \sum \limits _ { | j | > n } \sum \limits _ { Q \in Q ^ j } \omega ( Q ) \psi _ { Q } ( \cdot , x _ { Q } ) q _ { Q } h ( x _ { Q } ) \Big ) ( x _ { Q ' } ) \Big | ^ 2 \chi _ { Q ' } ( x ) \Big ) ^ { \frac { 1 } { 2 } } . \end{align*}


\begin{align*} I ( \alpha , \beta , Z ) - I ( \alpha , \beta , Z ' ) = < c _ { 1 } ( \xi ) + 2 \mathrm { P D } ( \Gamma ) , Z - Z ' > . \end{align*}


\begin{align*} \begin{aligned} & { 1 \over q } \varphi ( x _ q , t _ q ) ^ { 1 - q } ( \lambda h _ q + ( 1 - \lambda ) k _ q ) + \mu ( w _ q ( x _ q , t _ q ) ) \\ & \geq - w _ q ( x _ q , t _ q ) ^ { 1 - q } \left ( \lambda u ( y _ q , t _ q ) ^ { q - 1 } + ( 1 - \lambda ) u ( z _ q , t _ q ) ^ { q - 1 } \right ) \omega ( | X _ q | ) . \end{aligned} \end{align*}


\begin{align*} R ( z ) = \frac { z - \beta _ { k } } { \beta _ 1 - \beta _ { k } } r ( z ) - \frac { z - \beta _ { 1 } } { \beta _ 1 - \beta _ { k } } s ( z ) , \end{align*}


\begin{align*} s '' ( t + k \tau '' ) & = s ( t + k \tau '' ) + s ' ( t + k \tau '' ) = s ( t + k \alpha \tau ) + s ' ( t + k \alpha ' \tau ' ) \\ & = k \alpha \nu + s ( t ) + k \alpha ' \nu ' + s ' ( t ) = k ( \alpha \nu + \alpha ' \nu ' ) + s '' ( t ) . \end{align*}


\begin{align*} \hat { V } ( t , S _ f ( t ) ) = V ^ * ( S _ f ( t ) ) , \frac { \partial \hat { V } } { \partial S } ( t , S _ f ( t ) ) = \frac { \partial V ^ * } { \partial S } ( S _ f ( t ) ) , \end{align*}


\begin{align*} 2 - J _ { 0 } ( u ) = I ( \delta _ { k + 1 } , \delta _ { k } ) - J _ { 0 } ( \delta _ { k + 1 } , \delta _ { k } ) = \mu _ { \mathrm { d i s c } } ( \delta _ { k + 1 } ) - \mu _ { \mathrm { d i s c } } ( \delta _ { k } ) . \end{align*}


\begin{align*} \gamma _ M ( g ) = g - \sigma _ M ( g ) c , \end{align*}


\begin{align*} \varphi ( \theta ) = \sum _ { n \not = 0 } \varphi _ n e ^ { i n \theta } , \varphi _ n = \frac { x _ n + i y _ n } { 2 \sqrt { n } } \textrm { f o r } n > 0 \end{align*}


\begin{align*} \alpha _ n ( x y ) = x \alpha _ n ( y ) + \alpha _ 1 ( x ) \alpha _ { n - 1 } ( y ) + \dots + \alpha _ n ( x ) y . \end{align*}


\begin{align*} \lambda _ \ell ( N ^ j _ k ) & = \prod \limits _ { s = 1 } ^ j \left ( 1 - \prod ^ { k - s + j } _ { i = \ell } \delta _ i \right ) \\ & = \prod \limits _ { s = 1 } ^ j \left ( 1 - \prod ^ { k - s + j } _ { i = \ell } \frac { ( q ^ i - 1 ) ( q ^ { n - i + 1 } - 1 ) } { ( q ^ { i + 1 } - 1 ) ( q ^ { n - i } - 1 ) } \right ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \partial _ t f + v \cdot \nabla _ x f + { \rm { d i v } } _ v \big ( ( u - v ) f \big ) - \Delta _ v f + { \rm { d i v } } _ v ( f L [ f ] ) = 0 , \\ & \partial _ t \rho + { \rm { d i v } } _ x ( \rho u ) = 0 , \\ & \partial _ t ( \rho u ) + { \rm { d i v } } _ x ( \rho u \otimes u ) + \nabla _ x \rho ^ \gamma - { \rm { d i v } } _ x \mathbb { S } ( \nabla u ) = - \int _ { \mathbb { R } ^ 3 } ( u - v ) f \ , d v , \end{aligned} \right . \end{align*}


\begin{align*} u \leqq v & \ , : \Longleftrightarrow \ , u _ i \leq v _ i , \ ; i = 1 , \ldots , K , \\ u < v & \ , : \Longleftrightarrow \ , u _ i < v _ i , \ ; i = 1 , \ldots , K . \end{align*}


\begin{align*} S _ n : = \sum _ { j = 1 } ^ n f ( j ) \mbox { a n d } I _ n : = \int _ 1 ^ n f ( t ) \ , d t . \end{align*}


\begin{align*} \mathbb { K } _ \psi ^ \pi ( w ) \xi \{ \chi , k \} = \epsilon ( \chi ^ { - 1 } \pi , \psi ) \xi \{ \chi ^ { - 1 } \varpi _ \pi , - n ( \chi ^ { - 1 } \pi , \psi ) - k \} , \end{align*}


\begin{align*} P ( \xi , k ) : = \left \{ \begin{aligned} & \Delta _ { \eta } Q _ 1 ^ { - 1 } \left ( \Psi ^ { A i } _ 0 \right ) ^ { - 1 } \zeta ^ \frac { \sigma _ 3 } { 4 } , & \mathbb { C } _ { + } \cap U _ \delta ( \eta ) , \\ & \Delta _ { \eta } Q _ 2 ^ { - 1 } \left ( \Psi ^ { A i } _ 0 \right ) ^ { - 1 } \zeta ^ \frac { \sigma _ 3 } { 4 } , & \mathbb { C } _ { - } \cap U _ \delta ( \eta ) . \end{aligned} \right . \end{align*}


\begin{align*} \mathbf { R } = \big ( 1 _ { \{ q _ \ell \in \mathbf { p } _ { k , j } \} } \big ) _ { \ell = 1 , \dots , 2 k , j = 1 , \dots , 4 } . \end{align*}


\begin{align*} G ( z , \chi ) : & = \prod _ { p } \left ( 1 - \frac { \chi ( p ) } { p ^ { z } } \right ) \left ( 1 - \frac { \chi ( p ) ^ { 2 } } { p ^ { 2 z } } \right ) \left ( \sum _ { \alpha = 0 } ^ { \infty } \frac { a ( p ^ { \alpha } ) \chi ( p ^ { \alpha } ) } { p ^ { \alpha z } } \right ) , \end{align*}


\begin{align*} \sum _ { j = 1 } ^ n \sigma _ j ( e _ i v ) = 0 \sum _ { j = 1 } ^ n \sigma _ j ( e _ i ) \sigma _ j ( v ) = 0 \end{align*}


\begin{align*} \varphi \in C ^ \infty ( \R ) , \varphi ( s ) = 0 ~ \forall s \in ( - \infty , - 1 ] , \varphi ( s ) = 1 ~ \forall s \in [ 0 , \infty ) , \varphi ' ( s ) \geq 0 ~ \forall s \in \R . \end{align*}


\begin{align*} 2 \pi ( 2 - 2 g ) = \sum _ { p \in \sigma } 2 \pi - k _ p \pi \end{align*}


\begin{align*} \int _ 0 ^ \infty e ^ { ( c _ 2 - \lambda ) t } t ^ { - 1 / 2 } P _ { s + \alpha t } f ( x ) d t = \int _ 0 ^ \delta e ^ { ( c _ 2 - \lambda ) t } t ^ { - 1 / 2 } P _ { s + \alpha t } f ( x ) d t + \int _ \delta ^ \infty e ^ { ( c _ 2 - \lambda ) t } t ^ { - 1 / 2 } P _ { s + \alpha t } f ( x ) d t \end{align*}


\begin{align*} T _ x f ( t ) = f ( t - x ) \textnormal { a n d } M _ \omega f ( t ) = f ( t ) \ , e ^ { 2 \pi i \omega \cdot t } . \end{align*}


\begin{align*} Y ( \mathcal { I } ) _ { \beta \omega } ^ { m j } = - Y ( \mathcal { I } ) _ { \beta \alpha } ^ { m i } \cdot Y ( \mathcal { I } ) _ { \alpha \omega } ^ { m i } \cdot Y ( \mathcal { I } ) _ { \beta \alpha } ^ { i j } \cdot Y ( \mathcal { I } ) _ { \alpha \omega } ^ { i j } \in \mathbb { C } , ( j , \beta ) \in \mathfrak { c } . \end{align*}


\begin{align*} \varphi ( S ^ { - 1 } ( b ) a ) = \varphi ( S ^ { - 1 } ( a ' ) b ) , \forall b \in A . \end{align*}


\begin{align*} C ^ k _ { \rm d i f f } ( G ) : = \left \{ \begin{array} { l } c : G ^ { \times ( k + 1 ) } \to \mathbb { C } ~ { \rm s m o o t h } , \\ ~ c ( g g _ 0 , \ldots , g g _ k ) = c ( g _ 0 , \ldots , g _ k ) , \ \forall g , g _ 0 , \ldots , g _ k \in G \end{array} \right \} . \end{align*}


\begin{align*} \Phi ( p , q ) : = \frac { \Lambda ( p , q ) } { p ^ 0 q ^ 0 } \mathcal { S } ( p , q ) \end{align*}


\begin{align*} \dfrac { d l } { d \theta } = & \dfrac { \partial l } { \partial \theta } + \dfrac { \partial l } { \partial p ( \theta ) } \dfrac { \partial p ( \theta ) } { \partial \theta } . \end{align*}


\begin{align*} \exists y ^ \sigma A _ 0 ( y ) \rightarrow s = _ \rho t \equiv \forall y ^ \sigma ( A _ 0 ( y ) \rightarrow s = _ \rho t ) \end{align*}


\begin{align*} \lambda _ i \ : x _ i \ : y _ { i + 1 } = \mu _ i \ : x _ { i + 1 } \ : y _ i \end{align*}


\begin{align*} \mathcal { V } _ { i j } = \psi _ j ( r _ i , s _ i ) , ( i , j ) = 1 , . . . , N _ { e p } . \end{align*}


\begin{align*} \mathcal { L } ^ { \Sigma } ( f ) ( x ) = - \langle x ; \nabla ( f ) ( x ) \rangle + \Delta ^ { \Sigma } ( f ) ( x ) , \end{align*}


\begin{align*} \begin{cases} | \alpha _ + ( t ) | \lesssim e ^ { - c _ 1 ^ - t } , & c _ 0 < c _ 1 \le e _ 0 e _ 0 < c _ 0 < c _ 1 , \\ | \alpha _ + ( t ) - A e ^ { - e _ 0 t } | \lesssim e ^ { - c _ 1 t } , & c _ 0 \le e _ 0 < c _ 1 . \end{cases} \end{align*}


\begin{align*} \begin{aligned} \Xi _ k & = \begin{bmatrix} y _ { 1 , [ k , k + d _ 1 - 1 ] } ^ \top & \dots & y _ { m , [ k , k + d _ m - 1 ] } ^ \top \end{bmatrix} ^ \top . \end{aligned} \end{align*}


\begin{align*} \mu \Big ( \bigcup _ { i = 1 } ^ { N _ \Omega } B _ { n _ i } \Big ) \geq C \mu ( \Omega ) . \end{align*}


\begin{align*} \frac { \mathbf { P } ( S _ n / ( \sigma \sqrt { n } ) > x ) } { 1 - \Phi ( x ) } = 1 + o \big ( 1 \big ) , \ \ \ \ \ n \rightarrow \infty . \end{align*}


\begin{align*} \underset { n \rightarrow + \infty } { \lim } \left ( \psi \left ( \frac { t } { n } \right ) \right ) ^ n = \exp \left ( \psi ' ( 0 ) t \right ) . \end{align*}


\begin{align*} \left \langle a , f \right \rangle : = \sum _ { n \in \mathbb { Z } } a _ n f _ { - n } . \end{align*}


\begin{align*} \int _ { \Omega } | \nabla u | ^ { p - 2 } \nabla u \cdot \nabla \phi \ , d \mu = \int _ { \partial \Omega } \phi f \ , d \nu \end{align*}


\begin{align*} T ( u , p ) = \sum _ { \substack { 0 < a < p \\ g c d ( n , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi a ( \tau ^ { n } - u ) } { p } } . \end{align*}


\begin{align*} E [ X _ { n + 1 } \mid \mathcal { F } _ n ] & = \alpha \cdot \dfrac { S _ n } { n } , \\ \intertext { a n d } E [ S _ { n + 1 } \mid \mathcal { F } _ n ] & = \left ( 1 + \dfrac { \alpha } { n } \right ) S _ n . \end{align*}


\begin{align*} y _ R ( 0 ) = \int R ^ 2 \varphi \left ( \frac { x } { R } \right ) | u _ 0 | ^ 2 d x \le y _ R ( t ) , \forall t \ge 0 , \ ; \forall R \ge R _ 0 , \end{align*}


\begin{align*} \overline { H } _ n ^ { \star } ( \mathbf { s } ) = \overline { H } _ n ^ { \star } ( s _ 1 , s _ 2 , \ldots , s _ r ) : = \sum \limits _ { 0 \leq k _ 1 \leq k _ 2 \leq \cdots \leq k _ r \leq n - 1 } \prod _ { j = 1 } ^ { r } \frac { 1 } { ( 2 k _ j + 1 ) ^ { s _ j } } . \end{align*}


\begin{align*} L ' \otimes \Z _ v & = \begin{cases} \displaystyle { \bigoplus _ { j = 0 } ^ 1 } L ' _ { v , j } ( \pi ^ j ) & ( v = 2 , p _ 1 , \cdots , p _ k ) , \\ L ' _ { v , 0 } & ( \mathrm { o t h e r w i s e } ) , \\ \end{cases} \\ K _ { \ell } \otimes \Z _ v & = \begin{cases} \displaystyle { \bigoplus _ { j = 0 } ^ 1 } K _ { \ell , v , j } ( \pi ^ j ) & ( v = 2 , p _ 1 , \cdots , p _ k ) , \\ K _ { \ell , v , 0 } & ( \mathrm { o t h e r w i s e } ) , \end{cases} \end{align*}


\begin{align*} \theta _ k = \sum _ { j : \ j \le k , \ j \in J } \left ( \frac { \bar \alpha _ { j } } { \rho _ { j } } - 1 \right ) . \end{align*}


\begin{align*} \limsup _ { t \to T _ \ast } \| \rho ( t , \cdot ) \| _ { W ^ { 1 , \infty } } = \infty . \end{align*}


\begin{align*} \forall x ^ * \in \Gamma _ k \forall l \leq m \left ( \norm { x _ { n + l } - x ^ * } < e ^ A \norm { x _ n - x ^ * } + ( 2 M + 1 ) e ^ A \sum _ { i = n } ^ { n + l - 1 } \mu _ i + \frac { 1 } { r + 1 } \right ) \end{align*}


\begin{align*} D _ { U } Y _ { t } ^ { x ; l ^ { \epsilon } } ( W ) = \lim _ { \delta \to 0 } \frac { Y _ { t } ^ { x ; l ^ { \epsilon } } ( W + \delta U ) - Y _ { t } ^ { x ; l ^ { \epsilon } } ( W ) } { \delta } . \end{align*}


\begin{align*} L _ + f _ * = \lambda _ 0 \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ { p - 1 } . \end{align*}


\begin{align*} \det \left ( \frac { 1 } { \sqrt { 2 } } \begin{pmatrix} I & I \\ i I & - i I \end{pmatrix} \right ) = \left ( \tfrac { 1 } { \sqrt { 2 } } \right ) ^ { 2 d } \det ( - 2 i I ) = 2 ^ { - d } ( - 2 i ) ^ { d } = ( - i ) ^ d \end{align*}


\begin{align*} a | m \rangle = \sqrt { m } | m - 1 \rangle , a ^ { \dagger } | m \rangle = \sqrt { m + 1 } | m + 1 \rangle . \end{align*}


\begin{align*} \mathcal { C } \coloneqq \{ \mathbf { x } \in \mathcal { A } : \nabla f = 0 \} \ ; . \end{align*}


\begin{align*} C = - x _ k p _ \alpha ( x ) \partial _ k ( f ) ( x ) + f ( x ) \left ( - p _ \alpha ( x ) - x _ k \partial _ k ( p _ \alpha ) ( x ) \right ) , \end{align*}


\begin{align*} \liminf _ { T \to \infty } \frac { N _ 0 ( T ) } { N ( T ) } = 1 \end{align*}


\begin{align*} x \widetilde { m } : = \sum _ i z _ i ( y _ i m ) , x \in A \otimes B , x E = \sum _ i z _ i y _ i \textrm { f o r } y _ i \in A \ , { } ^ I \ ! \otimes ^ I B , z _ i \in A \otimes B . \end{align*}


\begin{align*} \underline { k } = \max \{ j \in J : j \leq k \} . \end{align*}


\begin{align*} H ( Z _ t ) & = H ( Z _ 0 ) + \int _ 0 ^ t ( H ' ( Z _ s ) \cdot \widetilde \mu ( Z _ s ) + \tfrac { 1 } { 2 } H '' ( Z _ s ) \cdot \widetilde \sigma ^ 2 ( Z _ s ) ) \ , d s + \int _ 0 ^ t H ' ( Z _ s ) \cdot \widetilde \sigma ( Z _ s ) \ , d W _ s \\ & = x _ 0 + \int _ 0 ^ t \mu ( H ( Z _ s ) ) \ , d s + \int _ 0 ^ t \sigma ( H ( Z _ s ) ) \ , d W _ s . \end{align*}


\begin{align*} \begin{aligned} & \frac { e ^ { - t } u \left ( \xi ( t ) \right ) - e ^ { - ( t + h ) } u ( \xi ( t + h ) ) } { h } \leq \frac { e ^ { - t } \phi \left ( \xi ( t ) \right ) - e ^ { - ( t + h ) } \phi ( \xi ( t + h ) ) } { h } \\ \Longrightarrow & \frac { 1 } { h } \int _ t ^ { t + h } e ^ { - s } C _ p \left | \dot { \xi } ( s ) \right | ^ { q } d s \leq \frac { e ^ { - t } \phi \left ( \xi ( t ) \right ) - e ^ { - ( t + h ) } \phi ( \xi ( t + h ) ) } { h } \end{aligned} \end{align*}


\begin{align*} H '' _ f ( t ) & = - 2 \int _ M ( | \nabla v | ^ 2 + V _ 1 v ^ 2 ) \ , d V _ g + 6 t \int _ M q v ^ 2 \ , d V _ g \\ & + 1 2 t ^ 2 \int _ M q v r _ t \ , d V _ g + 8 t ^ 3 \int _ M q v \dot { r } _ t \ , d V _ g + t ^ 4 \int _ M q v \ddot { r } _ t \ , d V _ g . \end{align*}


\begin{align*} a _ 0 + a _ 1 \phi + \cdots + a _ d \phi ^ d \ = \ 0 , \end{align*}


\begin{align*} \frac { T } { 4 } \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \log \left ( \frac { \sigma ^ 2 + T ^ 2 } { 4 } \right ) \ d \sigma & = \epsilon T \log \left ( \frac { T } { 2 } \right ) + \epsilon \ \mathcal { O } \left ( \frac { 1 } { T } \right ) \end{align*}


\begin{align*} \pi _ G ( G \times I K ^ { - 1 } \cap \Gamma ) & \subseteq \pi _ G ( G \times ( W \tau ( F ) ) \cap \Gamma ) \\ & = \pi _ G ( G \times W \cap \Gamma ) F \\ & = \Lambda F \end{align*}


\begin{align*} G = \{ x \rightarrow x y , \ ; \ ; y \rightarrow z , \ ; \ ; z \rightarrow 2 y z \} . \end{align*}


\begin{align*} \widehat { F _ { ( \xi , \eta ) } } ( x , \omega ) = F _ { ( \xi , \eta ) } ( - \omega , x ) , \forall ( \xi , \eta ) \in \R ^ { 2 d } . \end{align*}


\begin{align*} S + J S J ^ { - 1 } = \begin{pmatrix} E & F \\ - F & E \end{pmatrix} = \frac { 1 } { \sqrt { 2 } } \begin{pmatrix} I & I \\ i I & - i I \end{pmatrix} \begin{pmatrix} E + i F & 0 \\ 0 & E - i F \end{pmatrix} \frac { 1 } { \sqrt { 2 } } \begin{pmatrix} I & - i I \\ I & i I \end{pmatrix} \end{align*}


\begin{align*} \begin{cases} \ ( W _ 1 \times I _ 1 ) \cup ( Z _ 1 \times J _ 1 ) \\ [ 8 p t ] \ ( W _ 2 \times I _ 2 ) \cup ( Z _ 2 \times J _ 2 ) \end{cases} \end{align*}


\begin{align*} \left ( \int _ { \overline \Omega } g _ k ^ p \ , d \mu \right ) ^ { 1 / p } \le \sum _ { j = k } ^ { N ( k ) } \lambda _ { j , k } \left ( \int _ { \overline \Omega } g _ { u _ j } ^ p \ , d \mu \right ) ^ { 1 / p } = \sum _ { j = k } ^ { N ( k ) } \lambda _ { j , k } \left ( I _ { f _ j } ( u _ j ) + p \int _ { \partial \Omega } u _ j f _ j \ , d \nu \right ) ^ { 1 / p } . \end{align*}


\begin{align*} E ( k ) = I + \frac { 1 } { 2 \pi i } \int _ { \Gamma _ E } \frac { ( I + \mu ( s ) ) ( J ^ E ( s ) - I ) } { s - k } d s , \end{align*}


\begin{align*} \omega _ \ell ( u , v ) = \omega _ { \ell } ( u ) = \rho ' ( - u ) , \omega _ { r } ( u , v ) = \omega _ r ( v ) = \frac { \widetilde { \gamma } ' ( \rho \gamma ^ { - 1 } ( v ) ) } { \gamma ' ( \gamma ^ { - 1 } ( v ) ) } \rho ' ( \gamma ^ { - 1 } ( v ) ) . \end{align*}


\begin{align*} \varepsilon _ 0 = \varepsilon _ 0 ( \delta ) = C _ p \delta ^ { p } . \end{align*}


\begin{align*} { k + j \choose k } _ q \left ( N _ k ^ j ( V , V ' ) - \frac { 1 } { { k + j \choose k } _ q } I ( V , V ' ) \right ) = \sum \limits _ { i = 1 } ^ j ( - 1 ) ^ { i } { j \choose i } _ q a _ i . \end{align*}


\begin{align*} \mathfrak { i } ( x ) ^ { \ast } = I ^ { - 1 } \mathfrak { i } ( x ^ { \ast } ) I , \ , \ , \ , \ , I : = \left [ \begin{array} { c c } - \alpha & 0 \\ 0 & 1 \end{array} \right ] , \end{align*}


\begin{align*} \langle T _ x f , T _ x g \rangle = \langle f , g \rangle \textnormal { a n d } \langle M _ \omega f , M _ \omega g \rangle = \langle f , g \rangle . \end{align*}


\begin{align*} \mathbf { M } _ { u } \mathbf { M } _ { v } = \sum _ { w \in \mathfrak { B } _ { p + q } } c _ { u , v } ^ w \mathbf { M } _ { w } . \end{align*}


\begin{align*} \partial _ t ^ 2 u + H _ 1 u + u = a ( x ) u ^ 2 + u ^ 3 , \end{align*}


\begin{align*} \prod _ { \iota \in 2 ^ { V ' } } h _ { x y } ^ \iota ( c ) = \sum _ { d \in D ( \Gamma _ { x y } ) ( c ) } \prod _ { \iota \in 2 ^ { V ' } } g ^ { \iota } ( d _ \iota ) . \end{align*}


\begin{align*} \widetilde { e } _ m ^ { M , N } - \widetilde { e } _ { m - 1 } ^ { M , N } + k A ^ 2 \widetilde { e } _ m ^ { M , N } = - k P _ N A P F ( X _ m ^ { M , N } ) + k P _ N A P F ( X ( t _ m ) ) , \widetilde { e } _ 0 ^ { M , N } = 0 , \end{align*}


\begin{align*} n ( 0 ) = n ( 1 ) = 1 . \end{align*}


\begin{align*} q = L \cap H _ i . \end{align*}


\begin{align*} \Big ( r _ l \big ( \epsilon ( X , \Pi _ { \chi _ 0 } \otimes \Pi _ { \pi _ F } , \psi _ F ) \big ) \Big ) ^ l = \epsilon \big ( X , \chi _ 0 r _ l ( \pi _ F ) , \overline { \psi } _ F \big ) ^ l . \end{align*}


\begin{align*} S _ t ( ( Y , S , \widetilde W ) _ 0 , d W _ \cdot ) = e ^ { - \mu t } S _ 0 + e ^ { - \mu t } \int _ 0 ^ t e ^ { \mu s } F ^ 2 \left ( Y _ s \left ( ( Y , S , \widetilde W ) _ 0 , d W _ \cdot \right ) \right ) d W _ s , t \ge 0 . \end{align*}


\begin{align*} { \widetilde { E } _ { m } ^ { [ \alpha , \beta ] , \rho , \varepsilon } = \left \{ x \in \mathbb { R } ^ d : \ \underline \dim _ { { \rm l o c } } ( m , x ) \in [ \alpha , \beta ] \ a n d \ \ \forall r \leq \rho , \ m ( B ( x , r ) ) \leq r ^ { \underline { \dim } _ { \mathrm { l o c } } ( m , x ) - \varepsilon } \right \} } \end{align*}


\begin{align*} U ^ i V ^ j = 0 \mbox { f o r a l l } i \geq 1 \mbox { a n d } j \geq 2 \end{align*}


\begin{align*} \mathrm { E C C } ( Y , \lambda , \Gamma ) = \bigoplus _ { * : \ , \ , \mathbb { Z } / d \ , \ , \mathrm { g r a d i n g } } \mathrm { E C C } _ { * } ( Y , \lambda , \Gamma ) . \end{align*}


\begin{align*} K _ { 1 } = \frac { 2 - \sigma } { \sigma } K n _ { x } { R e _ { x } } ^ { 1 / 2 } \end{align*}


\begin{align*} C ( x , t ) = \frac { C _ { 0 } } { 2 ( \pi D t ) ^ { 1 / 2 } } \exp \left ( - \frac { x ^ { 2 } } { 4 D t } \right ) \end{align*}


\begin{align*} K _ { s , t , x } : = \{ v : s < { \mathcal T } _ { t , x , v } - 1 , | v | < \delta ^ { - \beta } , \check \tau _ { t , x } \langle v ^ \perp \rangle < \langle x ^ \perp \rangle / 4 \} , \end{align*}


\begin{align*} { \bar { \Delta } ^ { \rm P } _ { \rm P r o a c t i v e } } = - 1 - { \tau _ { \rm f } } - \frac { { \left ( { { \tau _ { \rm c } } + { n _ m } + \mathcal { T } } \right ) { \epsilon _ m } } } { { 1 - { \epsilon _ m } } } + \frac { 2 } { { 1 - { \epsilon _ m } } } \left ( { { \tau _ { \rm c } } + { n _ 1 } + \mathcal { T } + \sum \limits _ { i = 1 } ^ { m - 1 } { \left ( { { n _ { i + 1 } } - { n _ i } } \right ) { \epsilon _ i } } } \right ) . \end{align*}


\begin{align*} \chi _ { \mathbf L } = \P ( \mathcal E _ { \mathsf k } ( U , \mathbf L ) \mid \mathcal L ( U ) = \mathbf L ) \ , , \end{align*}


\begin{align*} \beta ( \tau ) = - \frac { 1 } { 2 } \frac { \mu _ \tau } { \mu ( \tau ) } , \end{align*}


\begin{gather*} \int _ { 0 } ^ 1 \Lambda ( x ) d x = \sum _ { n = 1 } ^ \infty | I _ n | c _ n = \alpha ^ { - 1 } \sum _ { n = 1 } ^ \infty \lambda _ n - \lambda _ { n + 1 } \\ = \lim _ { M \to \infty } \alpha ^ { - 1 } \sum _ { n = 1 } ^ M \lambda _ n - \lambda _ { n + 1 } = \lim _ { M \to \infty } \alpha ^ { - 1 } ( \lambda _ 1 - \lambda _ { M + 1 } ) \\ = \alpha ^ { - 1 } \lambda _ 1 \end{gather*}


\begin{align*} F _ { \alpha , x , t } ( u ) = \frac { 1 } { \left ( 1 - e ^ { - \alpha t } \right ) ^ { \frac { d } { \alpha } } } \frac { p _ \alpha ( u ) } { p _ \alpha ( x ) } p _ \alpha \left ( \dfrac { x - u e ^ { - t } } { ( 1 - e ^ { - \alpha t } ) ^ { \frac { 1 } { \alpha } } } \right ) . \end{align*}


\begin{align*} b ( \tau ) \partial _ \tau z = \partial ^ 2 _ \xi z + \frac { d + 1 } { \xi } \partial _ \xi z - 3 ( d - 2 ) ( 2 Q + \xi ^ 2 Q ^ 2 ) z + \bar { B } ( z ) + \mu ( \tau ) \Lambda _ \xi Q + \mu ( \tau ) \Lambda _ \xi z , \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \dot { \gamma } ^ \ast ( s ) = - \pi \gamma ^ \ast ( s ) ^ 2 & \ s > 0 , \\ \gamma ^ \ast ( 0 ) = r , & \end{array} \right . \end{align*}


\begin{align*} & \theta _ { i j } ( \rho ) [ v _ { i j } - ( \lambda _ i - \lambda _ j ) ] = 0 , \ ; \forall ( i , j ) \in E , \\ & \ < \dot \lambda , \rho \ > - \frac 1 4 \sum _ { i j } v _ { i j } ^ 2 \theta _ { i j } ( \rho ) + \frac 1 2 \sum _ { i j } ( \Sigma _ i - \Sigma _ j ) ( \lambda _ { i } - \lambda _ { j } ) \theta _ { i j } ( \rho ) d W ^ { \delta } ( t ) = 0 , \ ; \mathcal L ^ 1 \ ; \end{align*}


\begin{align*} \boxed { \alpha = \sum _ { n = 0 } ^ \infty \frac { \alpha ^ { ( n ) } ( 0 ) } { n ! } X ^ n . } \end{align*}


\begin{align*} z ^ 3 y ^ 2 + z y ^ 4 + x ^ 5 = 0 \ , . \end{align*}


\begin{align*} u + \left | u ' \right | ^ 2 - \left ( 1 - | x | \right ) = 0 \end{align*}


\begin{align*} \delta ^ * ( \alpha \delta ) ^ * = ( \alpha \delta ) ^ * . \end{align*}


\begin{align*} 2 ^ n { { \rm d } ^ n \over { \rm d } x ^ n } E ( x ) = P _ n ( E ( x ) ) . \end{align*}


\begin{align*} \frac { - V ( w ) } { U ( w ) } S ( w ) + T ( w ) = 0 , \end{align*}


\begin{align*} \nabla \varphi ( y _ \varepsilon , t _ \varepsilon ) = { 4 \over \varepsilon } | x _ \varepsilon - y _ \varepsilon | ^ 2 ( y _ \varepsilon - x _ \varepsilon ) , \nabla ^ 2 \varphi ( y _ \varepsilon , t _ \varepsilon ) \leq { 4 \over \varepsilon } | x _ \varepsilon - y _ \varepsilon | ^ 2 I + { 8 \over \varepsilon } ( x _ \varepsilon - y _ \varepsilon ) \otimes ( x _ \varepsilon - y _ \varepsilon ) . \end{align*}


\begin{align*} T B ( f ) = \sum _ { \gamma \in \Gamma ( G ) } t b ( f ( \gamma ) ) . \end{align*}


\begin{align*} Y ( \mathcal { I } ) _ { \alpha \beta } ^ { i j } : = c _ { 1 } c _ { 2 } \cdot \frac { \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } _ { \alpha } ^ { i } ) } { \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } _ { \beta } ^ { i } ) } \cdot \frac { \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } _ { \beta } ^ { j } ) } { \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } _ { \alpha } ^ { j } ) } , i , j \in \mathcal { I } , \ , \alpha , \beta \in \mathcal { I } ^ { \mathtt { C } } \end{align*}


\begin{align*} ( - \Delta ) ^ s u + c u & = 0 B _ \rho ^ + . \end{align*}


\begin{align*} F _ * J = \tilde J F _ * . \end{align*}


\begin{align*} x ^ m w _ { y y } \ , + \varepsilon y ^ n w _ { x x } = 0 , \varepsilon = 1 \ \textrm { o r } \ - 1 \ , . \end{align*}


\begin{align*} \varphi ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \cdots a _ n b _ n ) & = \frac { 1 } { \omega ( p ) } \omega ( p b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \cdots a _ n b _ n ) \\ & = \frac { 1 } { \omega ( p ) } \omega ( p a _ 1 \cdots a _ n ) \tau ( b _ 0 ) \cdots \tau ( b _ n ) \\ & = \varphi ( a _ 1 \cdots a _ n ) \varphi ( b _ 0 ) \cdots \varphi ( b _ n ) . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n \binom { n } { k } a _ k x ^ k = \sum _ { m = 0 } ^ n \binom { n } { m } \nabla ^ m b _ n ( x - 1 ) ^ { m } . \end{align*}


\begin{align*} \nu ^ + : k & \longrightarrow [ 0 , \infty ] , \\ \alpha & \mapsto \max \{ \nu ( \alpha ) , 0 \} , \end{align*}


\begin{align*} \hbox { i f } f = a ^ 2 + a b + b ^ 2 , \hbox { t h e n } \vert a \vert + \vert b \vert \leq \sqrt { 4 f } \hbox { a n d } \vert a b \vert \geq \sqrt { f / 3 } . \end{align*}


\begin{align*} & e _ { r } ^ { n } : = \left \| G ( \phi ^ { n } ) - \mu ^ n \phi ^ { n } \right \| _ { \infty } < \varepsilon _ { r } \end{align*}


\begin{align*} & I _ { 1 } = - \frac { 1 } { 2 \pi i } \oint _ { \partial U _ { \delta } ( \pm \eta ) } \left ( J ^ E ( s ) - I \right ) d s , \\ & I _ { 2 } = - \frac { 1 } { 2 \pi i } \int _ { \Gamma ^ { ( 3 ) } \backslash U _ { \delta ( \pm \eta ) } } \left ( J ^ E ( s ) - I \right ) d s , \\ & I _ { 3 } = - \frac { 1 } { 2 \pi i } \int _ { \Gamma _ E } \mu ( s ) ( J ^ E ( s ) - I ) d s . \end{align*}


\begin{align*} \left \{ \aligned & \partial _ t u - \Delta u + ( u \cdot \nabla ) u + \nabla p = 0 , \\ & \mbox { d i v } u = 0 , \\ & u ( 0 , x ) = f ^ { \omega } ( x ) . \endaligned \right . \end{align*}


\begin{align*} \Delta ( \mathbf { F } _ { 1 \bar { 4 } \ , \bar { 2 } 3 } ) = \iota \otimes \mathbf { F } _ { 1 \bar { 4 } \ , \bar { 2 } 3 } + \mathbf { F } _ { 1 } \otimes \mathbf { F } _ { \bar { 3 } \ , \bar { 1 } 2 } + \mathbf { F } _ { 1 \bar { 2 } } \otimes \mathbf { F } _ { \bar { 1 } 2 } + \mathbf { F } _ { 1 \bar { 3 } \ , \bar { 2 } } \otimes \mathbf { F } _ { 1 } + \mathbf { F } _ { 1 \bar { 4 } \ , \bar { 2 } 3 } \otimes \iota . \end{align*}


\begin{align*} \left \{ ~ ~ \begin{aligned} & \int _ 0 ^ T ( v - y ) \dd ( y - u ) \geq 0 & & \forall v \in C ( [ 0 , T ] ; Z ) , \\ & y ( t ) \in Z \forall t \in [ 0 , T ] , & & y ( 0 ) = y _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} \mu = \sum _ { i = 1 } ^ m p _ i \mu \circ f _ i ^ { - 1 } . \end{align*}


\begin{align*} \bigg | \ln \frac { \mathbf { P } ( S _ n / ( \sigma \sqrt { n } ) > x ) } { 1 - \Phi ( x ) } \bigg | = O \bigg ( \frac { 1 + x ^ 3 } { \sqrt { n } } \bigg ) , \ \ \ \ n \rightarrow \infty . \end{align*}


\begin{align*} \frac { 1 } { n } \Delta P & = u ^ { \frac { \alpha + 1 } { \alpha } } ( \frac { 1 } { n } | h | ^ { 2 } - H ^ { 2 } ) - \frac { 1 } { n } u ^ { \frac { 1 } { \alpha } } \overline { \mathrm { R i c } } ( \nu , \lambda \partial _ { r } ^ { T } ) \\ & + \frac { 1 } { \alpha } \bar { g } ( \lambda \partial _ { r } , \nabla \log u ) - \frac { 1 } { n \alpha } u ^ { \frac { \alpha + 1 } { \alpha } } | \nabla \log u | ^ { 2 } . \end{align*}


\begin{align*} \mathcal { L } ( g _ j ) ( x ) = - \langle x ; \nabla ( g _ j ) ( x ) \rangle + \Delta ( g _ j ) ( x ) . \end{align*}


\begin{align*} \begin{aligned} K : & = \{ \rho = ( \rho _ 1 , \cdots , \rho _ s ) , u = ( u _ 1 , \cdots , u _ s ) : \\ & \partial _ t \rho _ i + \nabla \cdot ( \rho _ i u _ i ) = 0 , ( \rho _ i u _ i ) \cdot \mathbf { n } = 0 \partial \Omega \times [ 0 , 1 ] , \\ & \rho _ i \in \mathcal { P } ( \Omega ) , \rho _ i ( x , 0 ) = \rho _ i ^ 0 ( x ) , \rho _ i ( x , 1 ) = \rho _ i ^ 1 ( x ) \} . \end{aligned} \end{align*}


\begin{align*} f _ n = V ^ * _ g F _ n = \iint _ { \R ^ { 2 d } } F _ n ( x , \omega ) M _ \omega T _ x g \ , d ( x , \omega ) = \iint _ { K _ n } V _ g f ( x , \omega ) M _ \omega T _ x g \ , d ( x , \omega ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \Lambda _ { n , 1 } = 0 , \lim _ { n \to \infty } \Lambda _ { n , 2 } = 4 . \end{align*}


\begin{align*} P _ n ( x ) = 2 ^ n x ^ { n + 1 } E _ n \left ( { 1 + x ^ 2 \over 2 x ^ 2 } \right ) . \end{align*}


\begin{align*} f ( x ^ i ) = f ^ i , \nabla f ( x ^ i ) = g ^ i \ \ i \in I , \end{align*}


\begin{align*} \mathrm { e } _ { \pi _ { \lambda , \beta } } ( z ; x ) = \frac { \mathrm { e } ^ { z x } } { l _ { \pi _ { \lambda , \beta } } ( z ) } = \sum _ { n = 0 } ^ { \infty } \frac { z ^ { n } } { n ! } A _ { n } ^ { \lambda , \beta } ( x ) , \end{align*}


\begin{align*} \log \Gamma ( s ) = \left ( s - \frac { 1 } { 2 } \right ) \log s - s - \frac { 1 } { 2 } \log ( 2 \pi ) + \mathcal { O } \left ( \frac { 1 } { s } \right ) \end{align*}


\begin{align*} j ( ( \eta , \xi ) ) = \left ( \left ( { \eta ( x ) \atop \eta ( x ) } \right ) , \left ( { \xi ( x ) \atop \xi ( x ) } \right ) \right ) . \end{align*}


\begin{align*} \binom { j + m } { j } & \sim \dfrac { j ^ m } { m ! } ( j \to \infty ) , \\ t _ n ^ { ( 2 m ) } & \sim m ! \cdot ( 2 m - 1 ) ! ! \cdot ( \log n ) ^ { m } ( n \to \infty ) . \end{align*}


\begin{align*} D ^ * \Phi ( ( x , y ) ; ( u , v ) ) ( y ^ * ) \subset \begin{cases} \bigcup \limits _ { w \in D g ( x ) ( u ) } D ^ * g ( x ; ( u , w ) ) ( y ^ * ) & y ^ * \in \mathcal N _ D ( g ( x ) - y ; w - v ) , \\ \varnothing & \end{cases} \end{align*}


\begin{align*} b _ j : = q _ j + \Big ( 1 - \sum _ { k = 1 } ^ { L } \phi _ k q _ k \Big ) e _ j . \end{align*}


\begin{align*} \begin{aligned} V _ { r r } ( r , p ) & = 2 \beta r ^ { - 3 } | p | ^ 2 + a ( 1 - \beta ) ^ { \alpha - 1 } \beta ( 1 - \alpha ) ( \beta - \alpha \beta - 1 ) r ^ { \beta ( 1 - \alpha ) - 2 } | p | ^ \alpha \\ & = r ^ { - 3 } | p | ^ \alpha \left ( 2 \beta | p | ^ { 2 - \alpha } + a ( 1 - \beta ) ^ { \alpha - 1 } ( \beta - \alpha \beta ) ( \beta - \alpha \beta - 1 ) r ^ { \beta ( 1 - \alpha ) + 1 } \right ) . \end{aligned} \end{align*}


\begin{align*} \mathcal L ( \rho , m , S ) & : = \ < S ( 1 ) , \rho ^ b \ > - \ < S ( 0 ) , \rho ^ a \ > + \mathcal A ( \rho , m ) \\ & \quad - \int _ 0 ^ 1 ( \ < \dot S , \rho \ > + \ < m , \nabla _ G S \ > ( 1 + \dot W ^ { \delta } ( t ) ) ) d t , \end{align*}


\begin{align*} ( d \theta _ { t } ^ { c } ) ^ { n } \wedge \theta \wedge \theta _ { t } ^ { c } = \sum ^ n _ { k = 1 } \binom n k ( d \theta _ { t } ^ { c } ) ^ { n - k } \wedge d \zeta ^ k _ t \wedge \theta \wedge \theta ^ c + \sum ^ n _ { k = 1 } \binom n k ( d \theta _ { t } ^ { c } ) ^ { n - k } \wedge d \zeta ^ k _ t \wedge \theta \wedge \zeta _ t \end{align*}


\begin{align*} \partial _ y ( B \circ A ^ { r _ 0 } \circ H ) ( x , y ) = \partial _ y B ( x , \tilde { y } ) = \partial _ 2 B ( x , \tilde { y } ) \cdot \partial _ y \tilde { y } = O ( \varepsilon ) \cdot O ( \varepsilon ) = O ( \varepsilon ^ { 2 } ) \end{align*}


\begin{align*} \int _ A \langle \nabla u ( x , y ) , \nabla v ( x , y ) \rangle _ { \rho , ( x , y ) } \mu _ \omega ( x , y ) = \int _ A \langle \nabla u ( x , y ) , \nabla v ( x , y ) \rangle _ { ( x , y ) } \ , d ( \mu \times \mathcal { L } ^ 1 ) ( x , y ) . \end{align*}


\begin{align*} \begin{aligned} & \varphi _ t ( x _ q , t _ q ) + F ( w _ q ( x _ q , t _ q ) , \nabla \varphi ( x _ q , t _ q ) , \nabla ^ 2 \varphi ( x _ q , t _ q ) - C _ q \varepsilon I , W _ \star [ x _ 0 , t _ 0 ] ) \\ & \geq \lambda \frac { u ( y _ q , t _ q ) ^ { q - 1 } } { \varphi ( x _ q , t _ q ) ^ { q - 1 } } D _ { 1 , q } + ( 1 - \lambda ) \frac { u ( z _ q , t _ q ) ^ { q - 1 } } { \varphi ( x _ q , t _ q ) ^ { q - 1 } } D _ { 2 , q } , \end{aligned} \end{align*}


\begin{align*} \mathbf { a } _ M ( z , \tau ) \mathbf { a } _ M ( z ' , \tau ' ) = ( M z , \tau ) ( M z ' , \tau ' ) = ( M ( z + z ' ) , \tau + \tau ' + \tfrac { 1 } { 2 } \sigma ( M z , M z ' ) ) \end{align*}


\begin{align*} R ^ { 1 , \operatorname { l i n } } & = \nu u _ 0 + \rho _ 0 w , \\ R ^ { 1 , \operatorname { c o r r } } & = \rho _ 0 w _ c + \nu w _ c . \end{align*}


\begin{align*} L _ { i j } \rhd x ^ h = ( a _ { i k } x ^ k + a _ { 0 i } ) \delta ^ h _ j - ( a _ { j k } x ^ k + a _ { 0 j } ) \delta ^ h _ i , \end{align*}


\begin{align*} G _ \lambda ^ { B , T } f ( x ) : = \int _ B g _ \lambda ^ { B , T } ( x , y ) f ( y ) \ : \mu ( d y ) \end{align*}


\begin{align*} ( a ^ { \dagger } a ) _ { k , \lambda } | m \rangle = \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) ( a ^ { \dagger } ) ^ { l } a ^ { l } | m \rangle = \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) ( m ) _ { l } | m \rangle . \end{align*}


\begin{align*} c l ( G _ i ) = \sum _ p n _ { i , p } c l ( G _ { i , p } ) \end{align*}


\begin{align*} I _ { 2 1 } \leq & \sum _ { q \geq 0 } \int _ 0 ^ { \frac { t } 2 } c _ q \lambda _ q ^ { \alpha + \beta - 2 \gamma } e ^ { - \mu \lambda _ q ^ \alpha ( t - \tau ) } \| B ( \tau ) \| ^ 2 _ { H ^ { \frac 5 2 - \alpha + \gamma } } \ , d \tau \\ \lesssim & \int _ 0 ^ { \frac { t } 2 } ( t - \tau ) ^ { - \frac { \alpha + \beta - 2 \gamma } { \alpha } } \| B ( \tau ) \| ^ 2 _ { H ^ { \frac 5 2 - \alpha + \gamma } } \ , d \tau . \end{align*}


\begin{align*} \mathbf { L } _ { \mathrm { e x } } : = \begin{array} { c } \begin{pmatrix} 1 & 0 & 0 & 0 & 1 & 1 & 3 \\ 0 & 1 & 0 & 0 & 2 & 2 & 4 \\ 0 & 0 & 1 & 1 & 4 & 5 & 0 \end{pmatrix} \\ \begin{array} { c c c c c c c } { \scriptstyle i } & { \scriptstyle j } & { \scriptstyle m } & { \scriptstyle \alpha } & { \scriptstyle \beta } & { \scriptstyle \gamma } & { \scriptstyle \delta } \end{array} \end{array} \end{align*}


\begin{align*} \Big [ \sum _ { i = 0 } ^ { + \infty } x _ i t ^ { i } , \sum _ { j = 0 } ^ { + \infty } y _ j t ^ { j } , \sum _ { k = 0 } ^ { + \infty } z _ k t ^ { k } \Big ] = \sum _ { s = 0 } ^ { + \infty } \sum _ { i + j + k = s } [ x _ i , y _ j , z _ k ] t ^ { s } , \ ; \forall x _ i , y _ j , z _ k \in L . \end{align*}


\begin{align*} \check H _ { c t } ^ q ( X ; A _ X ) = H Y ^ q _ b ( X , A ) . \end{align*}


\begin{align*} R e ( ( z ^ 0 ) ^ * A w ) = c _ 1 \ ; \ ; \forall w \in S ^ { n } _ \beta \Rightarrow R e ( ( z ^ 0 ) ^ * A w ^ 0 ) = c _ 1 \end{align*}


\begin{align*} \begin{array} { l l } z _ 1 2 + z _ 2 ( 3 + i ) = \eta \\ z _ 1 ( 1 + i ) + z _ 2 3 = \eta \\ z _ 1 + z _ 2 = 1 \end{array} \end{align*}


\begin{align*} T ( t ) f ( x ) = \begin{cases} f ( x - t ) & \mbox { f o r } x > t \\ 0 & \mbox { f o r } x \leq t . \end{cases} \end{align*}


\begin{align*} \hat { \theta } _ n = \frac { \sum _ { k = 1 } ^ n X _ { k - 1 } X _ k } { \sum _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } . \end{align*}


\begin{align*} M ^ { ( 2 \ell - 1 ) } _ n & \sim C ' _ \ell \cdot n ^ { \ell - 1 / 2 } ( \log n ) ^ { \ell - 1 } ( n \to \infty ) \end{align*}


\begin{align*} t _ I = \prod _ { i \in I } \Biggl ( \sum _ { j = 0 } ^ n a _ { i j } \lambda _ { i j } \Biggr ) \cdot \prod _ { i ' \notin I } \Biggl ( \sum _ { j = 0 } ^ n \lambda _ { i ' j } \Biggr ) = \sum _ { e \in E } \Biggl ( \prod _ { i \in I } a _ { i e ( i ) } \Biggr ) y _ e = : L _ I ( y ) I \subseteq \{ 1 , \ldots , d \} . \end{align*}


\begin{align*} \lambda ^ p _ k : = - \mu _ k : = - k ^ 2 \pi ^ 2 - 2 c _ k k \pi - 2 i d _ k k \pi - ( c _ k ^ 2 - d _ k ^ 2 ) - 2 i c _ k d _ k , \forall k \geq k _ 0 . \end{align*}


\begin{align*} & \eta ^ { \prime \prime \prime } ( x ) - \lambda \eta ^ { \prime \prime } ( x ) - 2 \lambda \eta ^ { \prime } ( x ) + \lambda ^ 2 \eta ( x ) = 0 , \ \ \forall x \in ( 0 , 1 ) , \\ & \eta ( 0 ) = 0 , \ \ \eta ( 1 ) = 0 , \ \ \eta ^ { \prime \prime } ( 0 ) = \eta ^ { \prime \prime } ( 1 ) . \end{align*}


\begin{align*} \begin{aligned} | \sigma _ n ( x ) | ^ 2 & \le c \cdot \min \bigl ( \sqrt { n } \ , ( 1 + x ^ 2 ) , \sigma ^ 2 ( x ) \bigr ) , \\ | \mu _ n ( x ) | & \le c \cdot \min \bigl ( \sqrt { n } \ , ( 1 + | x | ) , | \mu ( x ) | \bigr ) . \end{aligned} \end{align*}


\begin{align*} \mathcal S ( \rho _ t , \Phi _ t ) & = \ < \rho ( 0 ) , \Phi ( 0 ) \ > - \ < \rho ( T ) , \Phi ( T ) \ > + \int _ 0 ^ T \ < \partial _ t \Phi ( t ) , \rho _ t \ > + \mathcal H _ 0 ( \rho _ t , \Phi _ t ) d t \\ & + \int _ 0 ^ T \mathcal H _ 1 ( \rho _ t , \Phi _ t ) \dot W _ { \delta } d t . \end{align*}


\begin{align*} S ( H _ 5 , 5 f ) = \frac { 7 f - 1 0 d + 2 + \varepsilon _ d } { 5 } \hbox { a n d } N _ 5 ' ( f , H ) = - \frac { 4 } { 3 } d + \frac { 1 + \varepsilon _ d } { 6 } , \end{align*}


\begin{align*} \sum _ i \varphi ( A b _ i ) a _ i = 0 , \end{align*}


\begin{align*} \theta _ 1 ( a _ 1 ) & = \\ \theta _ 2 ( c ) & = . \end{align*}


\begin{align*} ( [ a , b ] [ c , d ] ) [ e , f ] = [ u a , v d ] [ e , f ] = [ i ( u a ) , ( j q ) f ] , \end{align*}


\begin{align*} \Bigl \{ ( f , \phi , \lambda , \delta ) \Bigm | \check { b } \bigl ( V ( \lambda ) \bigr ) \leq \phi \leq \hat { b } \bigl ( V ( \lambda ) \bigr ) , \ f = A ( \lambda ) , \ ( \lambda _ i , \delta _ i ) \in ( \ref { e q : S O S 2 - l o g } ) , \ ; i = 1 , 2 \Bigr \} . \end{align*}


\begin{align*} \Xi _ { \perp \star } : = \{ X \in \Xi _ \star ~ | ~ \mathbf { g } _ \star ( X , \Xi _ { t \star } ) = 0 \} \Omega _ { t , \star } : = \{ \omega \in \Omega _ \star ~ | ~ \mathbf { g } _ \star ^ { - 1 } ( \omega , \Omega _ { \perp \star } ) = 0 \} . \end{align*}


\begin{align*} \int _ { P _ R } F ( z ) \overline { z ^ \beta } e ^ { - \pi | z | ^ 2 } \ , d z = \sum _ { \alpha \geq 0 } c _ \alpha \int _ { P _ R } z ^ \alpha \overline { z ^ \beta } e ^ { - \pi | z | ^ 2 } \ , d z = c _ \beta \ , \mu _ { \beta , R } . \end{align*}


\begin{align*} \int _ { \mathbf { H } _ r } F ( \mathbf { h } _ r ) \ , d \mathbf { h } _ r = \int _ { \R ^ { 2 d } } \int _ 0 ^ 1 F ( x , \omega , e ^ { 2 \pi i \tau } ) \ , d \tau \ , d ( x , \omega ) . \end{align*}


\begin{align*} e ^ { - \lambda ^ 2 / 2 } \mathbf { E } \left ( e ^ { - \lambda \mathcal { N } } \mathbf { 1 } _ { \{ \mathcal { N } > 0 \} } \right ) = \frac { 1 } { \sqrt { 2 \pi } } \int _ 0 ^ { \infty } e ^ { - ( y + \lambda ) ^ 2 / 2 } d y = 1 - \Phi \left ( \lambda \right ) \end{align*}


\begin{align*} M : = M _ 0 \max \lbrace ( 3 d ) ^ \frac { 1 } { p } , ( 3 d ) ^ \frac { 1 } { p ' } \rbrace , \end{align*}


\begin{align*} I _ 3 & \leq | \partial _ { \phi } \mathcal { F } _ h ( u ^ * ) | \cdot | \tilde \phi | \\ & \leq \frac { 1 } { h } ( | \epsilon | + \beta _ a ^ { - 1 } + \beta _ b ^ { - 1 } ) C _ \phi ^ * \cdot \gamma | z | ( | c | + k | d | ) h ^ 2 = C _ 0 C ^ * _ \phi h . \end{align*}


\begin{align*} \div ( \beta ( \rho ) u ) = \beta ' ( \rho ) \div ( \rho u ) + \left [ \beta ( \rho ) - \rho \beta ' ( \rho ) \right ] \div u , \end{align*}


\begin{align*} a ' + a '' & = 1 , \\ b ' + b '' & = \tfrac { \sin \alpha } { \sin \gamma } , \\ c ' + c '' & = \tfrac { \sin \beta } { \sin \gamma } . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ \infty \lambda _ j ^ { 2 \beta \left ( \frac { \sigma } { 2 } + n + \tau + \frac { 1 } { 2 } - \gamma \right ) } \widetilde { \lambda } _ j ^ { - \alpha } \eqsim _ { \text ( \alpha , \beta , \gamma , \sigma , n , \tau ) } \sum _ { j = 1 } ^ \infty j ^ { \frac { 4 } { d } \left [ \beta \left ( n + \tau + \frac { 1 + \sigma } { 2 } \right ) - \beta \gamma - \frac { \alpha } { 2 } \right ] } , \end{align*}


\begin{align*} { ( \mathcal { M } ( \tilde u _ 0 , a , b ) u ) ( t , \cdot ) } & = e ^ { - A ( t - s ) } \tilde u _ 0 - \chi \int _ s ^ t e ^ { - A ( t - \tau ) } \nabla \cdot \left ( \frac { u ( \tau , \cdot ) } { v ( \tau , \cdot ) } \nabla v ( \tau , \cdot ) \right ) d \tau \\ & + \int _ s ^ t e ^ { - A ( t - \tau ) } u ( \tau , \cdot ) \big [ { \mu } + a ( \tau , \cdot ) - b ( \tau , \cdot ) u ( \tau , \cdot ) \big ] d \tau , \end{align*}


\begin{align*} \sigma ( X , [ \alpha ] ) : = \int _ X c _ X L ( X ) \wedge \psi _ X ^ * ( \alpha ) , \end{align*}


\begin{align*} S _ { \varepsilon } = \left \{ R _ { \varepsilon } ( s , t ) = R ( s , t ) + 2 \varepsilon U ^ { 1 } ( s , t ) \right \} \end{align*}


\begin{align*} S ^ { A , B } _ { \varepsilon } = \left \{ R ^ { A , B } _ { \varepsilon } ( s , t ) = R ( s , t ) + \varepsilon \left ( A \times R ( s , t ) + B \right ) \right \} \end{align*}


\begin{align*} \Gamma ^ + : = \{ \xi \in \Gamma ( S ^ + ( T N ) \otimes E ) : P _ { > 0 } \left ( \xi | _ { \partial N } \right ) = 0 \} . \end{align*}


\begin{align*} | \Lambda ( M , \Gamma ) | = \sum _ { n = 0 } ^ { \infty } | \Lambda _ { ( n , \infty ) } ( M , \Gamma ) | = \sum _ { n = 0 } ^ { \infty } | \Lambda _ { ( 0 , \infty ) } ( M - n R , \Gamma - n [ \gamma ] ) | . \end{align*}


\begin{align*} [ \pi ( \l ' ) , \pi ( \l ) ] = \left ( 1 - e ^ { 2 \pi i \sigma ( \l ' , \l ) } \right ) \pi ( \l ' ) \pi ( \l ) . \end{align*}


\begin{align*} \mu ( x _ 0 ) = u _ 0 ^ \star ( x _ 0 ) , \end{align*}


\begin{align*} D _ { k n } \le { \log 2 \over \pi K } + { 1 \over \pi | \bar { u } _ { k n } | } \sum _ { k = 1 } ^ K { 1 \over c } = { \log 2 \over \pi K } + { K \over \pi | \bar { u } _ { k n } | c } . \end{align*}


\begin{align*} \forall x ^ X \left ( \exists y ^ X , z ^ X ( z \in A y \land x = _ X y + _ X z ) \rightarrow ( x - _ X J ^ A _ { 1 } x ) \in A ( J ^ A _ { 1 } x ) \right ) , \end{align*}


\begin{align*} \widetilde { f } = \frac { 1 } { \langle g , \widetilde { g } \rangle } V _ g ^ * V _ { \widetilde { g } } f \in M ^ { p , q } \end{align*}


\begin{align*} Q ( x ) = \sum _ { k = 0 } ^ m q _ k x ^ k \end{align*}


\begin{align*} \begin{cases} \ E ^ \prime ( n ) \ = \ \min \{ E ( i ) : 0 \leq i \leq n \} \\ [ 8 p t ] \ k ( n ) \ = \ \min \{ i : 0 \leq i \leq n , \ E ( i ) = E ^ \prime ( n ) \} \\ [ 8 p t ] \ A ^ \prime ( n ) \ = \ A ( k ( n ) ) \end{cases} . \end{align*}


\begin{align*} u = x + t , \ ; v = t - x , \end{align*}


\begin{align*} ( x , b ) + ( x ' , b ' ) = ( \rho ( x + \varphi ( b , x ' ) + \gamma ( b , b ' ) , b + b ' ) , b + b ' ) \end{align*}


\begin{align*} 0 \leq | \xi | \leq \epsilon t + L _ \epsilon t ^ { 1 / 2 } , \ \forall t \geq 0 , \ \xi \in \partial F ^ + ( t ) = \partial F ( t ) . \end{align*}


\begin{align*} G = & \{ ( x , r ) : x \in \mathbb { R } ^ k , 0 < r < d ( x , K ) , z _ 1 , z _ 2 \in K \\ & d ( x , z _ 1 ) , d ( x , z _ 2 ) \leq d ( x , K ) + \epsilon r \\ & | \theta | > \cos ^ { - 1 } \left ( 2 \left ( \frac { 1 - ( 2 \delta + \epsilon ) } { 1 + 2 \delta } \right ) ^ 2 - 1 \right ) \} \end{align*}


\begin{align*} C ( k ) \ = \ \frac { 1 } { A ( k + c ) } ( k + c \geq c \implies A ( k + c ) \neq 0 ) . \end{align*}


\begin{align*} X = X ( 0 ) \cup \ldots \cup X ( d ) , \end{align*}


\begin{align*} T _ h ( \lambda ) ( u ) ( { x } ) & = \frac { \eta _ 0 ^ 1 } { 4 \pi } \int _ { B _ 1 } \frac { \exp ( { i \sqrt { \lambda } h | x - y | ) } } { | x - y | } u ( y ) d y \\ & + \frac { \eta _ 0 ^ 2 } { 4 \pi } \int _ { B _ 2 } \frac { \exp ( { i \sqrt { \lambda } h | x - y | ) } } { | x - y | } u ( y ) d y \end{align*}


\begin{align*} R ( Q ) = R _ 0 ( Q ) \cap \{ \C ^ \vee ( Q ) Q \} \cap \{ \C ^ \vee ( Q ^ + ) Q ^ + \} . \end{align*}


\begin{align*} \partial _ s v = \partial _ { \xi } ^ 2 v + \frac { d + 1 } { \xi } \partial _ \xi v + \frac { 1 } { 2 } \frac { \lambda _ s } { \lambda } \Lambda _ \xi v - 3 ( d - 2 ) v ^ 2 - ( d - 2 ) \xi ^ 2 v ^ 3 , \end{align*}


\begin{align*} \bar { \Delta } _ { \rm P r o a c t i v e } = \frac { \mathbb { E } \sum _ { t \in \mathcal { I } _ j } \Delta { \left ( t \right ) } } { \mathbb { E } T ^ { \rm P r o a c } _ j } = \frac { \mathbb { E } T ^ { \rm P r o a c } _ j } { 2 \mathbb { E } \left ( T ^ { \rm P r o a c } _ j \right ) ^ 2 } + \mathbb { E } \tau ^ { \rm P r o a c } _ { V _ j } - \tau _ { \rm f } - \frac { 1 } { 2 } . \end{align*}


\begin{align*} ( \lambda x ^ \rho . t ) ( s ^ \rho ) = _ \tau t [ s / x ] . \end{align*}


\begin{align*} \widehat { \kappa } _ { 2 k ; \alpha } ^ { ( L ) } : = \frac { \kappa _ { 2 k ; \alpha } ^ { ( \ell ) } } { ( K _ { 2 ; \alpha } ^ { ( \ell ) } ) ^ k } . \end{align*}


\begin{align*} \alpha _ { n - j } ( F ) = \alpha _ { n - j } ( F _ 1 ) + \alpha _ { n - j } ( F _ 2 ) - \alpha _ { n - j } ( F | _ { D _ 1 \cap D _ 2 } ) . \end{align*}


\begin{align*} g ^ K _ { i } ( t ) = \Delta _ K A ^ K _ { i } ( t \wedge \theta _ K ) + \frac { \| p \| _ { } } { \underline p } A ^ K _ { i + 1 } ( t \wedge \theta _ K ) , \end{align*}


\begin{align*} E _ { \beta } ( z ) : = \sum _ { n = 0 } ^ { \infty } \frac { z ^ { n } } { \Gamma ( \beta n + 1 ) } , z \in \mathbb { C } . \end{align*}


\begin{align*} H o m _ { S h } ( \Z _ { U , X } ^ \# , \mathcal I ) = H o m _ { P S h } ( \Z _ { U , X } , \mathcal I ) = \mathcal I ( U ) \end{align*}


\begin{align*} E ( \phi ( \cdot , t ) ) + \| \dot { \phi } ( \cdot , t ) \| ^ 2 + 2 \int _ 0 ^ t \eta ( s ) \| \dot { \phi } ( \cdot , s ) \| ^ 2 \mathrm { d } s = \mathcal { F } ( 0 ) = E ( \phi _ 0 ) + \| v _ 0 \| ^ 2 , \forall t > 0 . \end{align*}


\begin{align*} \frac { 1 } { g ( t ) } \exp ( x t ) = \sum _ { n = 0 } ^ { \infty } A _ { n } ( x ) \frac { t ^ { n } } { n ! } , \end{align*}


\begin{align*} \mathcal { P } _ i ( t , y ) = \frac { \mathcal { R } _ i ( y - \mathfrak { q } _ i ( t ) ) } { [ \mathfrak { q } _ 1 ( t ) - \mathfrak { q } _ 2 ( t ) ] ^ 2 } \end{align*}


\begin{align*} E ( \phi ) = \int _ { \mathbb { R } ^ d } \left ( \frac 1 2 | \nabla \phi ( \mathbf { x } ) | ^ 2 + V ( \mathbf { x } ) | \phi ( \mathbf { x } ) | ^ 2 + \frac { \beta } { 2 } | \phi ( \mathbf { x } ) | ^ 4 - \Omega \overline { \phi } ( \mathbf { x } ) L _ z \phi ( \mathbf { x } ) \right ) \mathrm { d } \mathbf { x } \end{align*}


\begin{align*} a | m \rangle = \sqrt { m } | m - 1 \rangle , a ^ { + } | m \rangle = \sqrt { m + 1 } | m + 1 \rangle . \end{align*}


\begin{align*} S ^ { - 1 } f = S ^ { - 1 } S ( S ^ { - 1 } f ) = \sum _ { \gamma \in \Gamma } \langle f , S ^ { - 1 } e _ \gamma \rangle S ^ { - 1 } e _ \gamma . \end{align*}


\begin{align*} a ( \widetilde E _ 2 ^ { ( 3 ) } , T ) = 0 . \end{align*}


\begin{align*} \omega _ { p } ^ { p } ( A ) \leq \frac { 1 } { 2 ^ { p - 2 } } \underset { i , j = 1 } { \overset { 2 } { \sum } } \omega _ { p } ^ { p } ( a _ { i j } ) \end{align*}


\begin{align*} K _ { g _ { \textnormal { p o i n } } } = \frac { 2 \pi \ , \chi ( \Sigma ) } { \mu _ { g _ { \textnormal { p o i n } } } ( \Sigma ) } = \frac { 4 \pi \ , ( 1 - p ) } { \mu _ { g _ { \textnormal { p o i n } } } ( \Sigma ) } , \end{align*}


\begin{align*} W ' & = w _ 2 , w _ 3 , \ldots , w _ n \mbox { o r } W ' = \emptyset \mbox { i f } | W | = 1 . \\ W '' & = w _ 1 , w _ 2 , \ldots , w _ { n - 1 } \mbox { o r } W '' = \emptyset \mbox { i f } | W | = 1 . \\ W ^ * & = w _ n , w _ { n - 1 } , \ldots , w _ 1 . \end{align*}


\begin{align*} \bar { g } _ n ^ { ( 2 m ) } \sim ( j _ 0 + 1 ) ^ m A _ { 2 m \alpha } n ^ { - m ( 1 - 2 \alpha ) } \end{align*}


\begin{align*} \kappa _ 2 ( y ) & = \lim _ { N \to \infty } \left ( 2 H _ { 2 N + 1 } - 2 \ln \left ( 2 N + 1 \right ) - H _ { N } + \ln { \frac { N } { \epsilon _ 2 } } + 2 \ln \left ( \frac { 2 N + 1 } { N } \right ) \right ) + \ln { \lvert y \rvert } \\ & = \gamma + \ln { \left ( \frac { 4 \lvert y \rvert } { \epsilon _ 2 } \right ) } . \end{align*}


\begin{align*} \dot { y } _ R ( t ) = 2 R \Im \int e ^ { - i t } Q \nabla \varphi \left ( \frac { x } { R } \right ) \cdot e ^ { i t } \nabla Q d x = 0 \delta ( u ( t ) ) = 0 u ( t ) = e ^ { i t } Q . \end{align*}


\begin{align*} \mathcal S ( \rho _ t , \Phi _ t ) & = \ < \rho ( 0 ) , \Phi ( 0 ) \ > - \ < \rho ( 1 ) , \Phi ( 1 ) \ > + \int _ 0 ^ 1 \ < \partial _ t \Phi ( t ) , \rho _ t \ > + \mathcal H _ 0 ( \rho _ t , \Phi _ t ) d t \\ & + \int _ 0 ^ 1 \mathcal H _ 1 ( \rho _ t , \Phi _ t ) \circ d W ( t ) . \end{align*}


\begin{align*} \tilde { f } ' ( s , y ) \le f _ 2 ( s , y ) \le s ^ { 2 H - 1 } K = K _ s \end{align*}


\begin{align*} P _ { \mathcal { M } } = \sum _ { I } { X ^ I } \in \R [ X _ 1 , \ldots , X _ n ] , \end{align*}


\begin{align*} s ( r , 0 ) = \begin{cases} 0 , 0 < r \leq \lambda _ 1 ( \mu ) ; \\ s _ 0 , \quad r \in ( \lambda _ 1 ( \mu ) , \lambda _ 2 ( \mu ) ) ; \\ + \infty , , r \geq \lambda _ 2 ( \mu ) . \end{cases} \end{align*}


\begin{align*} m _ { 0 , \lambda } ( V ) = 0 , \end{align*}


\begin{align*} u \ ! + \ ! \varPhi _ q \ , d _ q \ , \varPsi _ { \mu , q } W _ q u = v ^ { ( 0 ) } . \end{align*}


\begin{align*} \begin{aligned} \mathbf e & \approx \mathcal O ( h ) \begin{bmatrix} \mathbf 0 \\ G ( \mathbf S _ 2 , S _ { m + 1 } ) \end{bmatrix} + \mathcal O ( h ) \begin{bmatrix} \mathbf 0 \\ G ( \mathbf S _ 2 , S _ { m + 2 } ) \end{bmatrix} + \sum _ { j = m + 3 } ^ M \mathcal O ( h ^ 5 ) \begin{bmatrix} \mathbf 0 \\ G ( \mathbf S _ 2 , S _ { j } ) \end{bmatrix} . \end{aligned} \end{align*}


\begin{align*} C _ { \mathbf { F } _ N } ^ T C _ { \mathbf { F } _ N } = \begin{pmatrix} \sum _ { k = 0 } ^ { N - 1 } \cos ( 2 \pi k / N ) ^ 2 & \sum _ { k = 0 } ^ { N - 1 } \cos ( 2 \pi k / N ) \sin ( 2 \pi k / N ) \\ \sum _ { k = 0 } ^ { N - 1 } \cos ( 2 \pi k / N ) \sin ( 2 \pi k / N ) & \sum _ { k = 0 } ^ { N - 1 } \sin ( 2 \pi k / N ) ^ 2 \end{pmatrix} . \end{align*}


\begin{align*} A _ f ( X , Y ) : = D _ X ( D _ Y ( f ) ) - P ^ { \textnormal { T a n } ( f ) } ( D _ X ( D _ Y ( f ) ) ) \equiv ( D _ X ( D _ Y ( f ) ) ) ^ { \perp _ { f } } , \end{align*}


\begin{align*} \left | p ( ( \ell + i + 1 ) \delta _ K ) - p ( ( \ell + i ) \delta _ K ) \right | = & p ( ( \ell + i ) \delta _ K ) \frac { \left | p ( ( \ell + i + 1 ) \delta _ K ) - p ( ( \ell + i ) \delta _ K ) \right | } { p ( ( \ell + i ) \delta _ K ) } \\ \leq & \frac { \| p \| _ { } \delta _ K } { \underline { p } } p ( ( \ell + i ) \delta _ K ) \end{align*}


\begin{align*} J _ { \ast } : = \chi _ { \ast } ( x ) \int _ { k \geq 0 } \mathcal { K } _ { \ast } ^ { \# } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \ , d k , \end{align*}


\begin{align*} s ( \nu ) = s ( \nu \circ \alpha _ { \Phi } ^ { \sigma \to \tau } ) . \end{align*}


\begin{align*} \dot { y } = \Re \int _ { \mathbb { R } ^ N } | x | ^ 2 \dot { u } \bar { u } d x \\ = - 2 \Im \int _ { \mathbb { R } ^ N } | x | ^ 2 ( \Delta u ) \bar { u } d x \\ = 4 \Im \int _ { \mathbb { R } ^ N } x \cdot \nabla u \bar { u } d x , \end{align*}


\begin{align*} L ' _ { b } \wedge L ' _ { c } = ( L ' _ { b } \wedge L ' _ { b } ) \wedge L ' _ { c } = L ' _ { s x b } . \end{align*}


\begin{align*} \sup _ { S \in H _ R ^ 1 } \mathcal L ( \rho , m , S ) = \mathcal A ( \rho , m ) + R \mathcal E ( \rho , m ) , \end{align*}


\begin{align*} \langle \Delta _ { \emph { C K } } z , x \otimes y \rangle = \langle z , x \star y \rangle , \langle x \otimes y , \Delta _ { \emph { G L } } z \rangle = \langle x y , z \rangle \end{align*}


\begin{align*} M _ \alpha ( g ) ( x ) = g ( x ) p _ \alpha ( x ) . \end{align*}


\begin{align*} h _ { s t } = g _ { s t } , s , t = 1 , \dots , m . \end{align*}


\begin{align*} \pi _ { \lambda , \beta } = \int _ { 0 } ^ { \infty } \pi _ { \lambda \tau } \ , \mathrm { d \nu _ { \beta } ( \tau ) , } \quad \forall \lambda > 0 . \end{align*}


\begin{align*} D ( u ^ { \epsilon } , \eta ^ { \epsilon } , \tau ^ { \epsilon } ) = \mu _ 1 \| \nabla u ^ { \epsilon } \| _ { H ^ 3 } ^ 2 + \mu _ 2 \| \mathrm { d i v } u ^ { \epsilon } \| _ { H ^ 3 } ^ 2 + \nu [ \beta ( L - 1 ) + 2 \bar { \mathfrak { z } } ] \| \nabla \eta ^ { \epsilon } \| _ { H ^ 3 } ^ 2 + \frac { \beta A _ 0 } { 4 k ^ 2 } \| \tau ^ { \epsilon } \| _ { H ^ 3 } ^ 2 + \frac { \beta \nu } { 2 k ^ 2 } \| \nabla \tau ^ { \epsilon } \| _ { H ^ 3 } ^ 2 . \end{align*}


\begin{align*} h ( \mathcal { I } ) \cdot h ( \mathcal { I } _ { \alpha \beta } ^ { i j } ) = h ( \mathcal { I } _ { \alpha } ^ { i } ) \cdot h ( \mathcal { I } _ { \beta } ^ { j } ) + Y _ { \alpha \beta } ^ { i j } \cdot h ( \mathcal { I } _ { \beta } ^ { i } ) \cdot h ( \mathcal { I } _ { \alpha } ^ { j } ) + \frac { h ( \mathcal { I } _ { \alpha } ^ { i } ) \cdot h ( \mathcal { I } _ { \beta } ^ { j } ) } { Y _ { \alpha \beta } ^ { i j } } + h ( \mathcal { I } _ { \beta } ^ { i } ) \cdot h ( \mathcal { I } _ { \alpha } ^ { j } ) . \end{align*}


\begin{gather*} \phi [ T ] = u [ T ] . \end{gather*}


\begin{align*} & \epsilon \phi ^ { \epsilon } \partial _ t u ^ { \epsilon } \rightarrow 0 \ ; \ ; \ ; \ ; L ^ \infty ( 0 , T ; H ^ 1 ) , \\ & \epsilon \phi ^ { \epsilon } u ^ { \epsilon } \cdot \nabla u ^ { \epsilon } \rightarrow 0 \ ; \ ; \ ; \ ; L ^ \infty ( \mathbb { R } ^ + ; H ^ 2 ) , \ ; \ ; \ ; \ ; T > 0 , \end{align*}


\begin{align*} \int _ { \R ^ d } \psi _ z ( u ) \ , \d u \ge & \int _ { \R ^ d } \big ( \varphi ( | u | ) \wedge \varphi ( | u | + | z | ) \big ) \ , \d u \ge \int _ { \R ^ d } \varphi ( | u | + | z | ) \ , \d u = \int _ { \{ | u | \ge | z | \} } \varphi ( u ) \ , \d u , \end{align*}