\begin{align*} \omega _ { t + 1 } = \omega _ t + \alpha _ t \left ( c _ t ' + \gamma \phi ( s _ t '' , a _ t '' ) ^ T \omega _ t - \phi ( s _ t ' , a _ t ' ) ^ T \omega _ t \right ) \phi ( s _ t ' , a _ t ' ) , \end{align*}


\begin{align*} X _ \omega : = [ \mathcal U _ \omega / K ] , \end{align*}


\begin{align*} k = 0 , k ' = \frac { \sqrt { 2 } } { 2 \hslash \sqrt { \beta / 3 } } . \end{align*}


\begin{align*} w ( \pi ) = \begin{cases} ( 1 + \delta s _ 1 ) ( 1 + t _ 1 ) ^ { c _ 1 / 2 } , & q \equiv 1 \pmod 4 , \\ ( 1 + v _ 1 ) ^ { c _ 1 } , & q \equiv 3 \pmod 4 , \end{cases} \end{align*}


\begin{align*} ( x y ) z = x ( y z ) + ( - 1 ) ^ { \vert y \vert \vert z \vert } x ( z y ) , \end{align*}


\begin{align*} \eta ^ 2 ( T ) = | T | ^ { 2 { m } / 3 } \Vert \lambda _ h u _ { \mathrm { n c } } \Vert ^ 2 _ { L ^ 2 ( T ) } + | T | ^ { 1 / 3 } \sum _ { F \in \mathcal { F } ( T ) } \Vert [ { D } ^ { m } _ { \mathrm { p w } } u _ { \mathrm { n c } } ] _ F \times \nu _ F \Vert ^ 2 _ { L ^ 2 ( F ) } \end{align*}


\begin{align*} \left | \left . \int u _ { 1 , t } d \nu ^ 2 _ { t } \ , \right | ^ { t = \bar s } _ { t = s } \right | \le C ( C _ 0 , D , Y ) \varepsilon ( 1 + { s ' - s } ) . \end{align*}


\begin{align*} E ^ 2 _ j = \{ Z _ { i _ j , t } ( t , B ( x _ { i _ j } , \epsilon ) ) \geq p ( t , \epsilon ) | \pi _ { i _ j } | / 2 \} , \end{align*}


\begin{align*} C _ R ( J ) R [ x ] = C _ { R [ x ] } ( J [ x ] ) = C _ { R [ x ] } ( J ' ) \end{align*}


\begin{align*} \lambda = m ^ { \frac { s / p - 1 } { 1 + N - s N / p } } . \end{align*}


\begin{align*} \bigg ( \sum _ { k = N } ^ M a _ k ^ q v _ k ^ q \bigg ) ^ { \frac { 1 } { q } } \leq C \bigg ( \sum _ { k = N } ^ M a _ k ^ p w _ k ^ p \bigg ) ^ { \frac { 1 } { p } } \end{align*}


\begin{gather*} f _ i ( v ) = \begin{cases} s _ i v & s _ i v > v s _ i v \le _ L w \\ 0 & \end{cases} e _ i ( v ) = \begin{cases} s _ i v & s _ i v < v \\ 0 & \end{cases} \end{gather*}


\begin{align*} \mathbb { E } [ \ell ] = \dfrac { \ell } { 2 ^ \ell } \left ( 2 ^ { \ell - 1 } + \dbinom { \ell - 1 } { ( \ell - 1 ) / 2 } \right ) . \end{align*}


\begin{align*} \mathrm { ( i ) } ~ ~ \mathrm { E } [ \mathbf { Y } | \boldsymbol { a } < \mathbf { Y } \leq \boldsymbol { b } ] = \boldsymbol { \mu } + \frac { \mathbf { \Sigma } ^ { \frac { 1 } { 2 } } \boldsymbol { \delta } } { F _ { \mathbf { Z } } ( \boldsymbol { \xi _ { a } } , \boldsymbol { \xi _ { b } } ) } , \end{align*}


\begin{align*} & \tilde { N } = N ^ 1 \cup N ^ 2 , \tilde { E } = E ^ 1 \cup E ^ 2 , \tilde { h } _ 0 = h _ 0 ^ 1 \cup h _ 0 ^ 2 \tilde { H } = H ^ 1 \cup H ^ 2 , \\ & \tilde { L } : \tilde { N } \to \{ 1 , . . . , d \} \mbox { s u c h t h a t } \tilde { L } | _ { N ^ i } = L ^ i . \end{align*}


\begin{align*} b ^ * = \beta ( c ^ * ) \geq g _ 0 > 0 . \end{align*}


\begin{align*} \left \vert \int _ \Omega R \xi _ E ^ 2 u ^ 2 d V _ g \right \vert = \left \vert \int _ { \Omega \cap E } R \xi _ E ^ 2 u ^ 2 d V _ g \right \vert \sim \left \vert \int _ { \tilde M } \left ( R \xi _ E ^ 2 u ^ 2 \right ) ^ { \sim } d V _ { \tilde g } \right \vert . \end{align*}


\begin{align*} u _ t = w _ t + 2 \alpha s _ t , ~ ~ ~ ~ ~ t = 1 , 2 , \ldots , n . \end{align*}


\begin{align*} r \alpha = \frac { r - 1 } { 1 + \frac 1 N - \frac 1 p } \geq p . \end{align*}


\begin{align*} \label [ I e q ] { e q : D 0 e x p r e s s i o n } D _ p ( \mu , \nu ) : = { \left ( \sum _ { k = 1 } ^ { + \infty } \frac { 1 } { 2 ^ k } | \langle \mu - \nu , \varphi ^ k \rangle | ^ p \right ) } ^ { 1 / p } . \end{align*}


\begin{align*} \begin{array} { l l } \nabla _ { e _ i } e _ j = 0 ~ . \end{array} \end{align*}


\begin{align*} S ( r ) < \sum _ { k = 2 r } ^ { \infty } \dfrac { 1 } { k \pi ^ k } \Bigl ( \dfrac { k } { 2 } \Bigr ) ^ r e ^ { \frac { r ( r - 1 ) } { k } } \dfrac { 1 } { r ^ k } \leq \dfrac { e ^ { \frac { r - 1 } { 2 } } } { 2 ^ r } \sum _ { k = 2 r } ^ { \infty } \dfrac { k ^ { r - 1 } } { ( \pi r ) ^ k } \ \ \Bigl ( , e ^ { \frac { r ( r - 1 ) } { k } } \leq e ^ { \frac { r - 1 } { 2 } } \forall \ k \geq 2 r \Bigr ) . \end{align*}


\begin{align*} A [ X _ 1 , X _ 2 ] = [ X _ 1 , X _ 2 ] \begin{pmatrix} \Lambda _ 1 & \ \\ \ & \Lambda _ 2 \end{pmatrix} , \end{align*}


\begin{align*} \left \{ \begin{aligned} \frac { d x } { d t } & = g _ 1 ( t ) + \left ( g _ 3 ( t ) + g _ 7 ( t ) \right ) x + g _ 4 ( t ) y + g _ 8 ( t ) x ^ 2 + g _ 9 ( t ) x y , \\ \frac { d y } { d t } & = g _ 2 ( t ) + g _ 5 ( t ) x + \left ( g _ 6 ( t ) + g _ 7 ( t ) \right ) y + g _ 8 ( t ) x y + g _ 9 ( t ) y ^ 2 . \end{aligned} \right . \end{align*}


\begin{align*} n ^ { \alpha + \frac 1 2 } Y ^ n _ t = R ^ n _ t + \widetilde { A } ^ n _ t + \widetilde { C } ^ n _ t + \widetilde { Z } ^ n _ t , \end{align*}


\begin{align*} \mathbf { c } _ { k } : = \begin{cases} ( 1 , 1 , \ldots , 1 ) , & 1 \leq k \leq p , \\ [ 0 . 5 e m ] ( - a _ { k - p - 1 } ^ { ( p ) } , 1 , \ldots , 1 ) , & k \geq p + 1 , \end{cases} \end{align*}


\begin{align*} \lim _ { \bar { \rho } \rightarrow ( { 1 } / { g _ 1 } ) ^ { - } } N ^ * ( \bar { \rho } ) = + \infty . \end{align*}


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


\begin{align*} \beta \frac { 1 - \varepsilon \| b \| } { 1 + \varepsilon \beta } \le \beta _ h : = \inf _ { x _ h \in X _ h \setminus \{ 0 \} } \sup _ { y _ h \in Y _ h \setminus \{ 0 \} } \frac { b ( x _ h , y _ h ) } { \| x _ h \| _ { H } \| y _ h \| _ H } . \end{align*}


\begin{align*} Q ^ { n , 4 } _ \tau = n ^ { 2 \alpha + 1 } \sum _ { j = 0 } ^ { \lfloor n \tau \rfloor } \xi _ { j , n } , \end{align*}


\begin{align*} \mathfrak { S } _ n = \mathfrak { S } _ { n - 1 } \bigsqcup ( \bigsqcup _ { r = 1 } ^ { n - 1 } s _ { r , n - 1 } \mathfrak { S } _ { n - 1 } ) \hbox { ~ ~ ~ ~ ( a d i s j o i n t u n i o n ) , } \end{align*}


\begin{align*} \prod _ { i = 1 } ^ h \# Z ( { \bf s } ^ { \Lambda _ { Q , j } } ) = \prod _ { i = 1 } ^ h \left \langle \prod _ { \ell \in I _ i } \tau ^ { ( a _ \ell , b _ \ell ) } _ { d _ \ell } \sigma _ 1 ^ { k _ 1 ( i ) } \sigma _ 2 ^ { k _ 2 ( i ) } \sigma _ { 1 2 } \right \rangle ^ { \mathbf { s } ^ { \Gamma _ { 0 , k _ 1 ( i ) , k _ 2 ( i ) , 1 , I _ i } } , o } \end{align*}


\begin{align*} \langle \mu ^ * ( J a P _ { \hat { 1 } } b J ) \hat { 1 } , \hat { 1 } \rangle & = { \rm T r } ( \mu ^ * ( J a J P _ { \hat { 1 } } J b J ) P _ { \hat { 1 } } ) = { \rm T r } ( \mu ^ * ( P _ { \hat { 1 } } ) J b J P _ { \hat { 1 } } J a J ) \\ & = \overline { { \rm T r } ( \mu ( a P _ { \hat { 1 } } b ) P _ { \hat { 1 } } ) } = \langle \hat { 1 } , \mu ( a P _ { \hat 1 } b ) \hat { 1 } \rangle . \end{align*}


\begin{align*} \psi _ n = \psi _ { n , \textrm { i n } } + \psi _ { n , \textrm { o u t } } + o ( 1 ) \end{align*}


\begin{align*} A = \left [ \begin{matrix} 1 & 0 & 0 \\ 1 & 1 - \delta & 0 \\ 0 & 0 & 1 - 2 \delta \end{matrix} \right ] , \Delta A = \left [ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & \epsilon & 0 \end{matrix} \right ] . \end{align*}


\begin{align*} \begin{aligned} a _ { n } ^ { ( k ) } & = 0 , 0 \leq k \leq p - 1 , n \in \mathbb { Z } , \\ a _ { n } ^ { ( p ) } & = a _ { n } , n \in \mathbb { Z } , \end{aligned} \end{align*}


\begin{align*} T _ h ^ { ( n ) } = \left \{ \begin{array} { l l } T _ h , & \mbox { i f $ h \neq h _ o $ , } \\ U ( \gamma _ n ) , & \mbox { i f $ h = h _ o $ . } \end{array} \right . \end{align*}


\begin{align*} D _ { B } : = B \setminus \bigcup _ { \{ \gamma \in I : B \in S _ { \gamma } \} } D _ { \gamma } . \end{align*}


\begin{align*} [ A , M ] = \{ N \in \mathcal { D } : \exists k \ , ( r _ k ( N ) = A ) \mathrm { \ a n d \ } N \le M \} , \end{align*}


\begin{align*} \begin{aligned} \dot { \delta } ( t ) = \lambda - \mathbf { u } ( t ) , \dot { w } _ { i } ( t ) = u _ { i } ( t ) - \gamma _ { i } ( t ) , \end{aligned} \end{align*}


\begin{align*} \alpha = P _ 1 \widehat { d x _ 1 } \wedge \dots \wedge d x _ n + \dots + P _ n d x _ 1 \wedge \dots \wedge \ \widehat { d x _ n } . \end{align*}


\begin{align*} \bigg \| \max _ { 1 \le k \le n } \big | V _ { k } - \frac { V _ { k } } { V _ { n } } \big | \bigg \| _ { L ^ { \gamma p } } ^ { \gamma } = \| V _ { n } - 1 \| _ { L ^ { \gamma p } } ^ { \gamma } \le C n ^ { - \frac { \gamma } { 2 } } . \end{align*}


\begin{align*} \log _ Q ( | C | ) = \sum \limits _ { j = 1 } ^ { u } \left ( \frac { \ln ( q _ j ) } { \ln ( Q ) } \right ) \dim ( C _ j ) . \end{align*}


\begin{align*} \tilde { B } S _ 1 ( t ) x = \int _ 0 ^ t \tilde { B } T ( t - s , A ) \tilde { B } T ( s , A ) x \ , \mathrm { d } s , \end{align*}


\begin{align*} \delta ^ { ( s ) } w _ k + \frac { \sum _ { i j } ( \sum _ l \Gamma ^ { k ( s ) } _ { i j , l } c _ l ) v _ i ^ { \phi ^ s } w _ j ^ { \phi ^ s } } { ( \sum _ l r _ l c _ l ) ( \sum _ l v _ l ^ { \phi ^ s } ) } = 0 , \ \ k \in \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} Y _ + = X _ { \tilde { \omega } _ + } , Y _ - = X _ { \tilde { \omega } _ - } . \end{align*}


\begin{align*} \rho ( 1 ) = \frac { 2 b d k \log ( \frac { 4 b d k } { 1 } ) } { 1 } + \frac { \log ( \frac { 3 } { \delta } ) } { 2 } \leq n . \end{align*}


\begin{align*} { \cal J } = \prod \limits _ t \left | \frac { \d A _ { i j \ , t } } { \d X _ { k m \ , t } } \right | \end{align*}


\begin{align*} \mathbb { E } _ q ( x ) \ \mathbb { E } _ q ( q x ) ^ { - 1 } = 1 - x . \end{align*}


\begin{align*} S y z ( g ) = \{ a = ( a _ 1 , \ldots , a _ n ) \in R ^ n \ : \ a _ 1 g _ 1 + \ldots + a _ n g _ n = 0 \} . \end{align*}


\begin{align*} \sum _ { 1 \leq i < j \leq n } \mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) = \sum _ { 1 \leq i \leq n } \sum _ { i + 1 < j \leq n } \mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) + O ( n ) , \end{align*}


\begin{align*} \hat U ( t _ n ) = U _ n , n = 0 , \dots , N , \hat U | _ { I _ n } \in \mathbb { P } _ 1 ( I _ n ) , \end{align*}


\begin{align*} | \mathcal { T } _ \ell | - | \mathcal { T } _ 0 | \le \Lambda _ { \mathrm { B D d V } } \sum _ { k = 0 } ^ { \ell - 1 } | \mathcal { M } _ k | \end{align*}


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


\begin{align*} A B x = x , \ x \in Y . \end{align*}


\begin{align*} \lim _ { y \rightarrow 0 ^ { + } } \int _ { \mathbb { R } } \frac { 1 + s ^ { 2 } } { | 1 - s G _ { \nu } ( i y ) | ^ { 2 } } \ , d \sigma _ { \mu } ( s ) = 0 . \end{align*}


\begin{align*} J : = \begin{pmatrix} 0 & I _ { n } \\ - I _ { n } & 0 \end{pmatrix} . \end{align*}


\begin{align*} \hat \Theta _ n : = \left \{ \theta \in \Theta : { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \nu _ { \theta } ) \leq \inf _ { \theta ' \in \Theta } { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \nu _ { \theta ' } ) + n ^ { - 1 / 2 } \lambda _ n \right \} , \end{align*}


\begin{align*} \hat \sigma ^ { - 1 } ( i ) = \begin{cases} \sigma ^ { - 1 } ( i ) , & i < j _ \sigma \\ \sigma ^ { - 1 } ( i + 1 ) , & i \in [ j _ \sigma , m - 1 ] , \end{cases} \end{align*}


\begin{align*} \langle \nabla _ i \Psi _ t , \nabla _ j \Psi _ t \rangle = \delta _ i ^ j + O ( t ) . \end{align*}


\begin{align*} f _ i ( z ) & = \left \lbrace \begin{matrix} ( f _ { i - 1 } ) _ { r , x } ( z ) & z \in B ( x , r ) x \in X _ i \\ f _ { i - 1 } ( z ) & \end{matrix} \right . \end{align*}


\begin{align*} \tilde { \mu } _ { V X ^ { ( n ) } } ( A ) & \le \mu _ { V X ^ { ( n ) } } ( A _ \varepsilon ) + \mathbb { P } \Big ( \Big \lVert \Big ( \frac { 1 } { \beta } \Big ) ^ { 1 / p } \sum _ { j = 1 } ^ n Z _ j V _ { \bullet , j } - \sum _ { j = 1 } ^ n n ^ { \kappa / p } \frac { Z _ j } { ( \lVert Z ^ { ( n ) } \rVert _ p ^ p + W _ n ) ^ { 1 / p } } V _ { \bullet , j } \Big \rVert _ 2 \ge \varepsilon \Big ) . \end{align*}


\begin{align*} [ X _ \infty ] = \bar { \tilde D } _ { m + 2 } . \end{align*}


\begin{align*} \langle \mu ^ * ( J T J ) \hat { 1 } , \hat { 1 } \rangle = \langle \hat { 1 } , \mu ( T ) \hat { 1 } \rangle , \end{align*}


\begin{align*} { \rm d } ( { \rm L } _ n ) & \ge \frac { 1 } { \sqrt { v } \ , r _ q } \sum _ { a \in A } { \rm d } ( { \rm L } _ a ) = \frac { 1 } { \sqrt { v } \ , r _ q } \ , { \rm d } ( { \rm L } _ A ) \end{align*}


\begin{align*} \frac { \mathbb { P } \Big \{ W _ j > 0 \ \Big | \ | W _ j | , \boldsymbol { W } _ { - j } \Big \} } { \mathbb { P } \Big \{ W _ j < 0 \ \Big | \ | W _ j | , \boldsymbol { W } _ { - j } \Big \} } = & \frac { \mathbb { P } \Big \{ W _ j > 0 \ \Big | \ | W _ j | \Big \} } { \mathbb { P } \Big \{ W _ j < 0 \ \Big | \ | W _ j | \Big \} } a . s . \end{align*}


\begin{align*} \mbox { H o r } D _ V W = - \frac { G ( V , W ) } { f } \mbox { g r a d } f + \mbox { H o r } P _ V W , \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \alpha ( t _ 1 , t _ 2 ) d t _ 1 d t _ 2 = \phi _ 1 * \phi _ 2 ( g ) \end{align*}


\begin{align*} \langle f , I ^ { \circ } ( w _ 1 , z ) w _ 2 \rangle = 0 , \mbox { f o r a n y } \ f \in ( M ^ 3 ) ' , \ w _ 1 \in M ^ 1 , \ w _ 2 \in M ^ 2 ( 0 ) . \end{align*}


\begin{align*} \mathbb { Z } [ \theta ] = \{ \sum _ { i = 0 } ^ { n - 1 } a _ { i } \theta ^ { i } \mid a _ { i } \in \mathbb { Z } , \ 1 \leq i \leq n \} . \end{align*}


\begin{align*} R : = \begin{pmatrix} A & { _ A M _ B } \\ 0 & B \end{pmatrix} \end{align*}


\begin{align*} \mu ( B _ 1 ( x ) ) \le \liminf _ { \ell \to \infty } \tau _ { - x _ { \sigma ( \ell ) } } \mu _ { \sigma ( \ell ) } ( B _ 1 ( x ) ) = \liminf _ { \ell \to \infty } \mu _ { \sigma ( \ell ) } ( B _ 1 ( x + x _ { \sigma ( \ell ) } ) ) = 0 , \end{align*}


\begin{align*} E ( G / A ) = E ( G / N _ G ( A ) ) E ( N _ G ( A ) / A ) = E ( N _ G ( A ) / A ) \end{align*}


\begin{align*} \sf K ( x ) = \int _ 0 ^ { r ( x ) } K ( x , s ) d s , \end{align*}


\begin{align*} I _ 5 & = \int _ 0 ^ r ( \bar \phi + e _ \delta - \delta ) ( 1 - \delta + \bar \phi + e _ \delta ) \cdot ( e '' _ \delta \phi + e '' _ \sigma \psi ) { \rm { d } } z \\ & \leq \Vert ( \phi , \psi ) \Vert _ { L ^ 2 } \cdot \Vert ( e '' _ \delta , e '' _ \sigma ) \Vert _ { L ^ 2 } \cdot \Vert ( \bar \phi + e _ \delta - \delta ) ( 1 - \delta + \bar \phi + e _ \delta ) \Vert _ { L ^ \infty } , \end{align*}


\begin{align*} \begin{aligned} - \kappa \left ( 1 + \kappa \right ) h ^ 3 + \left [ 3 \kappa + \left ( 1 + \kappa \right ) ^ 2 \right ] h ^ 2 - 4 \left ( 1 + \kappa \right ) h + 4 = 0 \end{aligned} \end{align*}


\begin{align*} \Pr \left ( \max _ { 2 \le R < r } \left | \sum _ { i = 2 } ^ R E _ i \right | \ge t n 2 ^ { ( 1 / 2 - c ( 2 - \gamma ) / 8 ) n } \right ) \ll \left ( \frac { \log H } { t H } \right ) ^ { 1 / \gamma } 2 ^ { ( 1 - 1 / ( 2 \gamma ) + c ( 2 - \gamma ) / ( 8 \gamma ) ) n } . \end{align*}


\begin{align*} P ^ { n , 2 } _ \tau = n ^ { \alpha + \frac 1 2 } \int ^ { \tau } _ 0 ( t - s ) ^ { \alpha } [ ( \sigma ' ( X _ s ) - \sigma ' ( X _ { \eta _ n ( s ) } ) ] \sigma ( X _ { \eta _ n ( s ) } ) \Xi ^ { n , 2 } _ s d s . \end{align*}


\begin{align*} { { C _ { n , u , m } = } } C _ { u , m } : = \Psi ( { \hat { P } } _ { u } ) \enspace . \end{align*}


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


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


\begin{align*} \begin{cases} i u _ t + \bigtriangleup u - | u | ^ { p - 1 } u + i \gamma u = f ( x ) \\ u ( x , 0 ) = u _ 0 \in H ^ s _ x ( \mathbb { T } ) , \end{cases} \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 + } \bigg ( \int _ t ^ { \infty } u \bigg ) ^ { \frac { 1 } { 1 - q } } V _ r ( 0 , t ) ^ { \frac { q } { 1 - q } } = 0 . \end{align*}


\begin{align*} \sum _ { n = 1 } ^ \infty a _ n ^ 2 < \infty , \end{align*}


\begin{align*} E ' ( t ) = - \int _ { \Omega } b | \nabla u _ { t } | ^ 2 d x , \end{align*}


\begin{align*} \sup _ { N \leq k \leq M } a _ k \sup _ { N \leq i \leq k } b _ i = \sup _ { N \leq k \leq M } a _ k b _ k . \end{align*}


\begin{align*} \beta ( s , r ) = ( 1 - \varphi ( r ) ) f ( s , 0 ) , \end{align*}


\begin{align*} ( L y _ i ) ( v ) & = \begin{cases} - \sum \limits _ { e \in E _ k } \frac { \delta _ E ( e ) } { c } \frac { 1 } { | e | } y _ i ( v ) & v \in W , \\ 0 & \end{cases} \end{align*}


\begin{align*} \delta ( P _ { k , l } ) = \sum _ { s - r = l - k + n } b _ { r , s } ( k , l ) P _ { r , s } \ , , \end{align*}


\begin{align*} \epsilon _ p = \frac { 2 p + 1 } { 2 } \log \left ( \frac { p + 1 } { p } \right ) - 1 = \frac { 1 } { 2 x } \log \left ( \frac { 1 + x } { 1 - x } \right ) - 1 , \end{align*}


\begin{align*} \begin{aligned} S _ { 1 } ( t ) x & = \int _ { 0 } ^ { t } T ( t - s , A ) \tilde { B } T ( s , A ) x \ , \textnormal { d } s \\ S _ { n } ( t ) x & = \int _ { 0 } ^ { t } T ( t - s , A ) \tilde { B } S _ { n - 1 } ( s ) x \ , \textnormal { d } s n > 1 . \end{aligned} \end{align*}


\begin{align*} p ( x , y , t ) \ , = \ , 2 \frac { \partial ^ 2 } { \partial x ^ 2 } \log \tau ( x , y , t ) \end{align*}


\begin{align*} \Phi _ 0 ( [ x , y ] _ \mathfrak { g } ) - [ \Phi _ 0 ( x ) , \Phi _ 0 ( y ) ] = [ Q , \Phi _ 1 ( x , y ) ] , \ ; \forall x , y \in \mathfrak { g } [ 1 ] . \end{align*}


\begin{align*} S ^ { i + 1 , j } d ^ { i , j } = - d ^ { i + 1 , j - 1 } S ^ { i , j } , \forall i , j \geq 0 . \end{align*}


\begin{align*} ( k , v , t ) \mapsto \begin{cases} ( k , v , 0 ) , & t = 0 , \\ & \\ \left ( k \exp ( t v ) , t \right ) , & t > 0 , \end{cases} \end{align*}


\begin{align*} \lim _ { R \rightarrow \infty } \sup _ { t \geq 0 } \| \chi _ { S _ { R } ^ c } e ^ { - i t H } \phi \| \geq \lim _ { R \rightarrow \infty } \| \chi _ { S _ { R } ^ c } e ^ { - i R ^ \frac { 1 } { \beta } H } \phi \| = \| \hat { f } \| \end{align*}


\begin{align*} H ^ { 0 } \left ( X , K _ X \right ) = H ^ { 0 } \left ( Y , K _ { Y } \right ) \oplus \bigoplus _ { \chi \ne \chi _ { 0 0 0 } } { H ^ { 0 } \left ( Y , K _ { Y } + L _ { \chi } \right ) } . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \frac { d } { d t } g ^ { T } ( x , t ) = - \mathrm { R i c } ^ { T } ( x , t ) + \varkappa g ^ { T } ( x , t ) \end{array} \end{align*}


\begin{align*} f ( x ) = ( x - \lambda ) ^ p g ( x ) . \end{align*}


\begin{align*} \begin{array} { c } v ( 0 , t ) = \alpha v _ x ( L , t ) - \gamma \beta p _ x ( L , t ) = 0 , \\ p ( 0 , t ) = \beta p _ x ( L , t ) - \gamma \beta v _ x ( L , t ) = 0 . \end{array} \end{align*}


\begin{align*} \mathbb S ( M ) = \{ T \in \mathbb B ( L ^ 2 M ) \mid [ T , J x J ] \in \mathbb K ^ { \infty , 1 } ( M ) , \ { \rm f o r \ a l l \ } x \in M \} . \end{align*}


\begin{align*} \left \Vert U _ { L } ^ { \Phi } - U _ { L } ^ { \Psi } \right \Vert _ { \mathcal { U } } = \left \Vert U _ { L } ^ { \Phi - \Psi } \right \Vert _ { \mathcal { U } } \leq \left \vert \Lambda _ { L } \right \vert \left \Vert \Phi - \Psi \right \Vert _ { \mathcal { W } _ { 1 } } \ , L \in \mathbb { N } _ { 0 } , \end{align*}


\begin{align*} I ' ( w ) [ \phi ] = \int _ \Omega | \nabla w ^ \frac { 1 } { p } | ^ { p - 2 } \langle \nabla w ^ \frac { 1 } { p } , \nabla ( w ^ \frac { 1 - p } { p } \phi ) \rangle , I '' ( w ) [ \phi , \phi ] = \int _ \Omega \rho ( w , \phi ) \end{align*}


\begin{align*} \pi _ \lambda ( a , b , t ) \varphi ( \xi ) = e ^ { i \lambda t } e ^ { i \lambda ( a \xi + \frac { 1 } { 2 } a b ) } \varphi ( \xi + b ) , \end{align*}


\begin{align*} H y p _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \int _ { t _ 1 } ^ { t _ 2 } \int _ { \R ^ 3 } e ^ { i X ( s ) \cdot ( \xi + \eta ) + i \mu s | \xi | + i \mu _ 1 s | \eta | } \clubsuit K _ { k _ 1 , j _ 1 ; n _ 1 } ^ { \mu _ 1 , i _ 1 } ( s , \eta , V ( s ) ) \cdot { } ^ 1 \clubsuit K ( s , \xi , X ( s ) , V ( s ) ) d \xi d \eta d s . \end{align*}


\begin{align*} \det \left ( ( \lambda I _ N - H ) _ { k | k + 1 } \right ) = - y _ { k k + 1 } \det \left ( ( \lambda I _ N - H ) _ { k k + 1 | k k + 1 } \right ) . \end{align*}


\begin{align*} Q ^ { \pi } ( s , a ) = \lim _ { T \rightarrow \infty } \mathbb { E } ^ { \pi } \left \{ \sum _ { t = 0 } ^ T \gamma ^ t c _ { a _ { \pi } } ( s _ t ) \Big | s _ 0 = s , a _ 0 = a \right \} . \end{align*}


\begin{align*} r _ { A ( U ) } ( f ) : = \sup \{ | \lambda | : \lambda \in \sigma _ { A ( U ) } ( f ) \} . \end{align*}


\begin{align*} \pi ( x , q , { \bf { a } } ) = \# \{ p \leq x : p _ n \equiv a _ n \bmod q \} . \end{align*}


\begin{align*} \boldsymbol { \delta } = \left ( \begin{array} { c c c c c c c c c c c } 0 . 4 1 7 2 \\ 0 . 4 8 7 5 \end{array} \right ) , \boldsymbol { \Omega } = \left ( \begin{array} { c c c c c c c c c c c } 0 . 7 8 8 6 & 0 . 0 9 9 3 \\ 0 . 0 9 9 3 & 0 . 6 6 8 8 \end{array} \right ) . \end{align*}


\begin{align*} \mathbb { E } ( X _ i X _ j ) = & f _ k ( j ) \bigg ( f _ k ( i ) + O _ { \alpha } \Big ( \frac { \tau _ 3 ( i ) } { \sqrt { i } } \Big ) \bigg ) + O _ { \alpha } \Big ( \frac { \tau _ 3 ( j ) } { \sqrt { j - i } } \Big ) \\ = & f _ k ( i ) f _ k ( j ) + O _ { \alpha } \Big ( \frac { \tau _ 3 ( j ) \tau _ 3 ( i ) } { \sqrt { i } } \Big ) + O _ { \alpha } \Big ( \frac { \tau _ 3 ( j ) } { \sqrt { j - i } } \Big ) , \end{align*}


\begin{align*} \Psi ( & E _ { j + 1 } ( x , d - 1 ) ) - \Psi ( E _ j ( x , d - 1 ) ) = \\ & = 2 \sqrt { d - 1 } \left ( \cot \left ( \dfrac { \pi } { 2 j + 4 } \right ) - 1 \right ) - 2 \sqrt { d - 1 } \left ( \csc \left ( \dfrac { \pi } { 2 j + 2 } \right ) - 1 \right ) \\ & = 2 \sqrt { d - 1 } \left ( \cot \left ( \dfrac { \pi } { 2 j + 4 } \right ) - \csc \left ( \dfrac { \pi } { 2 j + 2 } \right ) \right ) \\ & = f _ j \ , \sqrt { d - 1 } \ , . \end{align*}


\begin{align*} \Phi _ 2 ( \xi , \eta , \zeta ) : = \hat { \zeta } \cdot ( \xi + \eta ) + + \mu | \xi | + \mu _ 1 | \eta | . \end{align*}


\begin{align*} \mathrm { d } X = \left \{ \bar { g } ( X ) - \frac { 1 } { 2 } h ( X ) h ' ( X ) \int _ { M } v ^ { 2 } \mathrm { d } \mu \right \} \mathrm { d } t + h ( X ) \circ \mathrm { d } W , ~ X ( 0 ) = \xi . \end{align*}


\begin{align*} [ D \vec { x } ] _ { w _ i } = & \sum _ { v _ j \in V } d _ G ( v _ i , v _ j ) \vec { x } _ { v _ j } \\ = & \sum _ { \ell = 1 , \ell \neq i } ^ k \sum _ { v _ j \in W _ \ell } d _ G ( v _ i , v _ j ) \vec { x } _ { v _ j } + \sum _ { v _ j \in W _ i , v _ j \neq w _ i } d _ G ( v _ i , v _ j ) \vec { x } _ { v _ j } \\ = & \sum _ { \ell = 1 , \ell \neq i } ^ k | W _ \ell | d _ G ( v _ i , w _ \ell ) \vec { x } _ { w _ \ell } + c _ i ( | W _ i | - 1 ) \vec { x } _ { w _ i } \\ = & [ D / \mathcal { W } \vec { y } ] _ { v _ i } . \end{align*}


\begin{align*} c _ 1 \Vert f \Vert _ 1 \leq \sum _ { k = 0 } ^ { \infty } \delta _ k | \alpha _ k | \leq c _ 2 \Vert f \Vert _ 1 . \end{align*}


\begin{align*} \delta ^ { \star } _ { m , n } : = \begin{cases} 1 , & \textup { f o r } \ , \ , n \neq 1 , \\ 0 , & \textup { f o r } \ , \ , n = 1 . \end{cases} \end{align*}


\begin{align*} g _ { \dot { x } ^ { 1 } } = - \sin t + b \cos ( t + c ) , ~ ~ g _ { \dot { x } ^ { 2 } } = \cos t + b \sin ( t + c ) . \end{align*}


\begin{align*} e _ 2 ( b ) = \begin{psmallmatrix} & 0 & \\ 0 & & 0 \\ & 1 & \end{psmallmatrix} \otimes \begin{psmallmatrix} & 0 & \\ 0 & & 1 \\ & 1 & \end{psmallmatrix} \otimes \begin{psmallmatrix} & 0 & \\ 1 & & 1 \\ & 1 & \end{psmallmatrix} \otimes \begin{psmallmatrix} & 0 & \\ 1 & & 1 \\ & 1 & \end{psmallmatrix} \ , . \end{align*}


\begin{align*} u ( 0 ) = u ( L ) = y ( 0 ) = y ( L ) = 0 . \end{align*}


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


\begin{align*} \| u \| _ { F ^ 0 _ { \infty , \infty } } = \sup _ { j \in \N _ 0 , x \in \R ^ n } | u _ j ( x ) | \leq C \| u \| _ { L ^ { \infty } ( \R ^ n ) } . \end{align*}


\begin{align*} \gamma = \alpha _ 0 \bar \beta _ 1 \alpha _ 1 \cdots \bar \beta _ { n } \alpha _ n \end{align*}


\begin{align*} \abs { f ' ( z ) } = \abs { \frac { a } { 2 \lambda } } \abs { g ' ( \textbf { z } ) } \abs { \textbf { z } + \lambda i } ^ 2 . \end{align*}


\begin{align*} - \frac 1 2 \varphi - K ^ t ( \partial _ t ) \varphi + \partial _ t ^ { - 1 } W ( \partial _ t ) \psi = 0 . \end{align*}


\begin{align*} f ( t , x ) = u ( t , x ) + \Gamma ( \beta + 1 ) ( 1 + H ( t - r ) ) \sin x . \end{align*}


\begin{align*} x + y \equiv 0 \pmod 4 . \end{align*}


\begin{align*} \widetilde { H } ( \phi ) _ 0 = \left \{ \begin{array} { c c c } t ^ { n - 1 } \phi _ 2 , & \mbox { i f } & n \ \mbox { i s o d d } , \\ \det ( \widetilde { H } ( \phi ) ) t ^ { n - 1 } \phi _ 2 , & \mbox { i f } & n \ \mbox { i s e v e n } . \end{array} \right . \end{align*}


\begin{align*} [ K _ j E , e ^ { M E } ] = 0 \end{align*}


\begin{align*} \label [ I e q ] { e q : l i o u v i l l e m c k e a n } \partial _ t f ^ N _ t = - \nabla \cdot ( \mathbf { b } ^ N f ^ N _ t ) + \frac { \sigma ^ 2 } { 2 } \Delta f ^ N _ t , \end{align*}


\begin{align*} F _ { m , Q } ( z , s ) = \sum _ { \substack { \gamma \in \Gamma _ \infty \backslash \Gamma \\ \gamma \neq \gamma _ 1 , \gamma _ 2 } } f _ m ( \gamma z , s ) . \end{align*}


\begin{align*} \int _ { v ( t ) } ^ \infty \psi _ \Lambda ( u ) ^ { - 1 } d u = t . \end{align*}


\begin{align*} \left \{ \begin{array} { l r } \widehat { \varphi } _ t - A _ 1 ( i , h _ 2 ) \widehat { \varphi } _ { i i } - ( B ( i , h _ 2 ) + h _ 2 ' C ( i , h _ 2 ) + D ( i , h _ 2 ) ) \widehat { \varphi } _ i = K ( i , t ) , & t > 0 , \ 0 < i < h _ 0 , \\ \widehat { \varphi } _ i ( t , 0 ) = \widehat { \varphi } ( t , h _ 0 ) = 0 , & t > 0 , \\ \widehat { \varphi } ( 0 , i ) = 0 , & 0 \leq i \leq h _ 0 , \\ \end{array} \right . \end{align*}


\begin{align*} \sum \limits _ { 1 \leq i \leq n } f _ k ( n ) f _ k ( n + 1 ) & = N \sum _ { w \leq N } \frac { g _ k ( w ) \tau ^ * ( w ) } { w ^ 2 } + N \sum _ { w > N } \frac { g _ k ( w ) \tau ^ * ( w ; N ) } { w ^ 2 } + O _ { \varepsilon } ( N ^ { \varepsilon } ) \\ & = N \sum _ { w \leq N } \frac { g _ k ( w ) \tau ^ * ( w ) } { w ^ 2 } + O _ { \varepsilon } ( N ^ { \varepsilon } ) , \end{align*}


\begin{align*} \epsilon ^ * ( n , R ) = e ^ { - n E ( R ) + o ( n ) } . \end{align*}


\begin{align*} \delta _ X ( f \wedge g ) & = 2 [ a b , f _ 1 ] _ X \otimes g _ 1 - 2 [ a b , g _ 1 ] _ X \otimes f _ 1 - 2 a b \otimes [ f _ 1 , g _ 1 ] _ X . \end{align*}


\begin{align*} \zeta \cdot ( x , y ) = ( \zeta x , \zeta ^ { - 1 } y ) . \end{align*}


\begin{align*} E _ i = \frac { 1 } { | C | } H _ i H _ i ^ \top \end{align*}


\begin{align*} \nu ( l ( h _ { \sigma ' } ) ) \geq \beta _ b + b \gamma _ { \sigma ' } > \beta _ b + b \gamma _ { \sigma } = \nu ( f ) \end{align*}


\begin{align*} \tilde { U } = U + E , \end{align*}


\begin{align*} \mathop { \lim } \limits _ { T \to \infty } \left ( \frac 1 T \ln E _ k \right ) = \lambda _ 1 + \dots + \lambda _ k , \ \ k = 1 \dots d \end{align*}


\begin{align*} K _ r ( x , y , z ) & = \left ( \sum _ { i = 1 } ^ { n - 2 } \frac { 3 } { \varepsilon } ( z _ i - \Phi _ i ( x , y ) ) ^ 2 \right ) ^ r , \\ W _ r ( x , y , z ) & = F _ { p , q } ( x , y ) + K _ r ( x , y , z ) . \end{align*}


\begin{align*} A _ \ell ( u , v , S _ 1 \times S _ 2 ) = \int _ { S _ 1 \times S _ 2 } \frac { [ u ( x ) - u ( y ) ] ^ { \ell - 1 } ( v ( x ) - v ( y ) ) } { | x - y | ^ { N + \ell s } } ~ d x d y . \end{align*}


\begin{align*} \log \Delta ^ { \alpha \vect s + \alpha ' \vect s ' } _ \Omega = \alpha \log \Delta ^ { \vect s } _ \Omega + \alpha ' \log \Delta ^ { \vect s ' } _ \Omega \end{align*}


\begin{align*} \lim _ { | z | \to \infty } \frac { | X ^ \epsilon ( t , z ) - z | } { | z | } = 0 . \end{align*}


\begin{align*} p _ m ( \lambda , k ) = ( k \lambda ) ^ { m - 1 } + \sum _ { \substack { l = 0 \\ l \equiv m - 1 } } ^ { m - 3 } p ^ m _ l ( \lambda ^ 2 + k ^ 2 ) ( k \lambda ) ^ l , \end{align*}


\begin{align*} \Lambda ^ n _ s = ( W _ s - W _ { \eta _ n ( s ) } ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) + I _ 2 \left ( \mathbf { 1 } _ { [ \eta _ n ( s ) , s ] } \otimes \psi _ { n , 1 } ( \cdot , s ) \right ) , \end{align*}


\begin{align*} \widetilde { G } ( x , x _ 0 ) = \frac { G ( x , x _ 0 ) } { g ( x ) \ , g ( x _ 0 ) } \ge 1 , \forall x \in \Omega . \end{align*}


\begin{align*} j ( a ^ { * } ) ( \gamma ) = \overline { j ( a ) ( \gamma ^ { - 1 } ) } j ( a * b ) ( \gamma ) = \sum _ { \eta \in G ^ { r ( \gamma ) } } j ( a ) ( \eta ) \ , j ( b ) ( \eta ^ { - 1 } \gamma ) . \end{align*}


\begin{align*} v _ p ( k ! ) = \sum _ { \ell = 1 } ^ { C _ { p , k } } \left \lfloor \dfrac { k } { p ^ { \ell } } \right \rfloor \le v _ p ( N _ k ) \le \sum _ { \ell = 1 } ^ { C _ { p , k } } \left ( 1 + \left \lfloor \dfrac { k } { p ^ { \ell } } \right \rfloor \right ) = v _ p ( k ! ) + C _ { p , k } , \end{align*}


\begin{align*} \overline { e } _ { 1 } = \left ( \begin{array} { c } 1 \\ 0 \\ \vdots \\ 0 \end{array} \right ) , \overline { e } _ { 2 } = \left ( \begin{array} { l } 0 \\ 1 \\ \vdots \\ 0 \end{array} \right ) , \cdots , \overline { e } _ { n } = \left ( \begin{array} { l } 0 \\ 0 \\ \vdots \\ 1 \end{array} \right ) . \end{align*}


\begin{align*} a _ i : = \min ( V _ i ) , \ \ b _ i : = \max ( V _ i ) . \end{align*}


\begin{align*} s _ 0 - \ell _ r b = \ell e _ r + a - \ell _ r ( a h _ r - \nu e _ r ) = ( \ell + \ell ' _ r a + \ell _ r \nu ) e _ r = ( \ell + c ) e _ r , \end{align*}


\begin{align*} P _ j ( f _ j , \hat \phi ) = P _ j ( f _ j , \hat \phi _ 1 , \ldots , \hat \phi _ n ) = 0 , 1 \leq j \leq n . \end{align*}


\begin{align*} | g ( | z | \leq r ) | = | f ( \omega ( | z | \leq r ) ) | \leq \max _ { | z | = r } | f ( | z | \leq r ) | = h _ { \psi } ( r ) , \end{align*}


\begin{align*} \int _ \mathbb { T } | u | ^ { p - 1 } ( x , t ) \ , d x = 2 \pi \sum _ { 0 = k _ 1 - k _ 2 + \cdots - k _ { p - 1 } } \widehat { u } _ { k _ 1 } \overline { \widehat { u } } _ { k _ 2 } \cdots \widehat { u } _ { k _ { p - 2 } } \overline { \widehat { u } } _ { k _ { p - 1 } } = \int _ \mathbb { T } | v | ^ { p - 1 } ( x , t ) \ , d x , \end{align*}


\begin{align*} \alpha _ { x } ( a _ { y , \mathrm { s } } ) = a _ { y + x , \mathrm { s } } \ , y \in \mathfrak { L } , \ ; \mathrm { s } \in \mathrm { S } . \end{align*}


\begin{align*} \langle \Lambda _ { \gamma } f , g \rangle \vcentcolon = B _ { \gamma } ( u _ f , g ) , \langle \Lambda _ q f , g \rangle \vcentcolon = B _ q ( v _ f , g ) , \end{align*}


\begin{align*} M _ { n , k } = Y ^ { \alpha _ 1 } _ n ( t _ 1 ) \cdots Y ^ { \alpha _ k } _ n ( t _ k ) , \ M _ k = X ^ { \alpha _ 1 } _ \phi ( t _ 1 , x ) \cdots X ^ { \alpha _ k } _ \phi ( t _ k , x ) , \end{align*}


\begin{align*} w \left ( { { \eta _ 1 } , . . . , { \eta _ d } } \right ) = \mathop { \lim } \limits _ { T \to \infty } \frac 1 T \ln { \left \langle E _ { 1 } ^ { \eta _ 1 - \eta _ 2 } E _ { 2 } ^ { \eta _ 2 - \eta _ 3 } . . . E _ { d - 1 } ^ { \eta _ { d - 1 } - \eta _ d } E _ { d } ^ { \eta _ d } \right \rangle } \end{align*}


\begin{align*} E ( \rho _ R | \rho _ { \infty } ) = H '' ( \xi ) | \rho _ R - \rho _ { \infty } | ^ 2 = \frac { p ' ( \xi ) } { \xi } | \rho _ R - \rho _ { \infty } | ^ 2 , \mbox { f o r s o m e } \xi \in ( 0 , \overline { \rho } ) . \end{align*}


\begin{align*} \frac { 1 } { q ( P ) } = 1 - N + \sum _ { k = 1 } ^ N \frac { 1 } { p _ k } . \end{align*}


\begin{align*} \mathcal { E } ^ \mathrm { H } [ u _ + ] = \min \Big \{ \mathcal { E } ^ \mathrm { H } [ u ] \ ; | \ ; u \in H ^ 1 ( \mathbb { R } ^ d ) \cap L ^ 2 ( \mathbb { R } ^ d , V _ \mathrm { D W } ( x ) d x ) , \ ; \| u \| _ { L ^ 2 } = 1 \Big \} . \end{align*}


\begin{align*} a ( \hat \eta ) \Omega _ \phi x & = \check \eta x \\ a ( \check \eta ) \eta & = \hat \eta \otimes _ \phi \eta \end{align*}


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


\begin{align*} \xi _ n : = \Big ( \Big ( \frac { 1 } { \beta } \Big ) ^ { 1 / p } - \frac { n ^ { \kappa / p } } { ( \lVert Z ^ { ( n ) } \rVert _ p ^ p + W _ n ) ^ { 1 / p } } \Big ) ^ 2 \longrightarrow 0 \end{align*}


\begin{align*} \textrm { r a n k } \left ( \mathbf { V } \right ) = \sum _ { m } u \left ( \lambda _ { m } \right ) \end{align*}


\begin{align*} I ^ 0 ( \phi _ { \eta } ) = 0 \mbox { a n d } \phi _ { \eta } = 0 \mbox { i n } V _ H . \end{align*}


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


\begin{align*} \mathfrak { F } = \{ f * \phi _ \sigma : f \in \dot { H } ^ { 1 , q } ( \gamma _ { \sigma } ) , \| f \| _ { L ^ { q } ( \gamma _ { \sigma } ) } \le 1 , \| f \| _ { \dot { H } ^ { 1 , q } ( \gamma _ { \sigma } ) } \le C \} \end{align*}


\begin{align*} \left \lbrace \begin{aligned} \dot { \boldsymbol { x } } ( t ) & = A ( t ) \boldsymbol { x } ( t ) + B ( t ) \boldsymbol { u } ( t ) , \\ \boldsymbol { y } ( t ) & = C ( t ) \boldsymbol { x } ( t ) + D ( t ) \boldsymbol { u } ( t ) , \end{aligned} \right . \end{align*}


\begin{align*} \frac { \dd \Xi _ t } { \dd t } = \Bar { Q } \circ H _ t + H _ t \circ Q _ { \mathfrak { g } } . \end{align*}


\begin{align*} \sin \theta ( { \mathcal C } _ 1 , { \mathcal C } _ 2 ) = \lVert ( I - P _ { { \mathcal C } _ 2 } ) P _ { { \mathcal C } _ 1 } \rVert _ S . \end{align*}


\begin{align*} \frac { \partial \psi } { \partial t } = - { \rm i } \widehat { H } ( t ) \psi = - \sum _ { \alpha = 1 } ^ r b _ { \alpha } ( t ) { \rm i } \widehat { H } _ \alpha \psi , \psi \in \mathcal { H } , t \in \mathbb { R } . \end{align*}


\begin{align*} \frac { d x } { d t } = b _ 1 ( t ) + b _ 2 ( t ) x + b _ 3 ( t ) x ^ 2 , x \in \mathbb { R } , \end{align*}


\begin{align*} \mathcal { Q } _ { j , } ^ k \sim \mathcal { G P } ( q ^ k _ { j , } , z ^ k _ { j , } + \eta ) k = 1 , \ldots , m , \ j = 1 , \ldots , L , \end{align*}


\begin{align*} J = \dfrac { r _ + - r _ - } { 2 } \ . \end{align*}


\begin{align*} c = \liminf _ { z \rightarrow \alpha } \frac { \Im f ( z ) } { \Im z } < + \infty \end{align*}


\begin{align*} \underset { t \rightarrow 0 } { } \dfrac { \mathsf { m } ( E + t H ) - \mathsf { m } ( E ) } { t } = \underset { t \rightarrow 0 } { } \dfrac { 1 } { t } \left \{ t H - g \dfrac { E + t H } { \abs { E + t H } } + g \dfrac { E } { \abs { E } } \right \} = H - g \mathsf { F } ' ( E ; H ) . \end{align*}


\begin{align*} h ( Y ) = m n h ( X ) . \end{align*}


\begin{align*} \hat { \beta } ( y ) + \hat { \beta } ( z ) = \hat { \beta } ( y + z ) y z \mathbb { R } ^ { n } . \end{align*}


\begin{align*} \lesssim 2 ^ { 2 \epsilon M _ t } \min \big \{ 2 ^ { m - l } 2 ^ { \epsilon M _ t + 2 l + j + 2 \tilde { \alpha } _ t M _ t } , 2 ^ { - 2 m - j - 3 l } \big \} \mathbf { 1 } _ { m \in ( - 1 0 M _ t , \epsilon M _ t ] \cap \Z } + 2 ^ { m + 4 j } \mathbf { 1 } _ { m = - 1 0 M _ t } \end{align*}


\begin{align*} \sum _ { z \in \mathbb { Z } ^ { d } } \gamma ^ { d } f \left ( \gamma z + a \right ) = \sum _ { z \in \mathbb { Z } ^ { d } } \gamma ^ { d } f \left ( \gamma ( z + b ^ { ( a , \gamma ) } ) \right ) \ . \end{align*}


\begin{align*} \int _ { \{ \Psi _ 1 < - t \} \cap U } | \tilde F - F ^ { 1 + \delta } | ^ 2 e ^ { - \varphi - \Psi _ 1 } = \int _ { \{ \Psi _ 1 < - t \} \cap U } | \frac { \tilde F } { F ^ { 1 + \delta } } - 1 | ^ 2 e ^ { - \Psi _ 1 } < + \infty . \end{align*}


\begin{align*} \delta ^ { ( s ) } u = f \left ( u , u ^ { \phi ^ s } \right ) . \end{align*}


\begin{align*} C _ { M } ^ { B } = \{ [ \alpha ] _ { B } \in H _ { B } ^ { 1 , 1 } ( M , \mathbb { R } ) | \exists \omega > 0 [ \omega ] _ { B } = [ \alpha ] _ { B } \} . \end{align*}


\begin{align*} \left \| \sum _ { k = 0 } ^ \infty R ( x + i y , A ) ^ k ( x - w ) ^ k \right \| _ { \infty } \leq 2 \end{align*}


\begin{align*} g _ { i j } ( \theta ) = \sum \limits _ { k = 1 } ^ d \frac { \partial \phi ^ k } { \partial \theta _ i } ( \theta ) \frac { \partial \phi ^ k } { \partial \theta _ j } ( \theta ) , i , j = 1 , . . . , m . \end{align*}


\begin{align*} \label [ I e q ] { e q : L i n f t y a g a i n s t m o n o m i a l } \mathcal { L } _ \infty \Phi _ k ( \nu ) = \sum _ { i = 1 } ^ k \langle Q ( \nu ) , \varphi ^ i \rangle \prod _ { j \ne i } \langle \nu , \varphi ^ j \rangle , \end{align*}


\begin{align*} H / H ^ 0 = K _ 1 / K _ 1 ^ 0 = N _ K ( T ) / T = W ( K ) . \end{align*}


\begin{align*} m ^ 2 _ t : = m ^ 2 + \frac { 1 } { t } , \end{align*}


\begin{align*} \beta _ { n } ^ * ( v _ i ) = \begin{cases} v _ i ' , & , \\ 0 , & . \end{cases} \end{align*}


\begin{align*} h ( x , t ) = c ( t ) \end{align*}


\begin{align*} E ^ { T , \Upsilon , Y } \subseteq \bigg ( \bigcup _ { i = 1 } ^ n E ^ { T _ i , \Upsilon _ i , Y _ i } \bigg ) \cup \Big \{ \tilde { \phi } ^ { T , \Upsilon , Y } \big [ \tilde { h } _ j ^ { Y } \big ] : \tilde { h } _ j ^ { Y } \in Z ^ a \big [ Y _ 1 , . . . , Y _ n \big ] , \tilde { h } _ j ^ { \Upsilon } \in Z ^ a \big [ \Upsilon _ 1 , . . . , \Upsilon _ n \big ] \Big \} \end{align*}


\begin{align*} \max \{ | S | : S \in \bigcap _ { i = 1 } ^ n \mathcal { F } _ i \} & = \min \{ r _ 1 ( X ) + r _ 2 ( X _ 1 \backslash X ) + \dots + r _ { n - 1 } ( X _ { n - 2 } \backslash X _ { n - 3 } ) + r _ n ( E \backslash X _ { n - 2 } ) : \\ & X \subseteq X _ 1 \subseteq X _ 2 \subseteq \dots \subseteq X _ { n - 2 } \subseteq E \} \end{align*}


\begin{align*} \begin{array} [ c ] { c } \bigtriangledown ^ { T } H = \frac { 1 } { u + A } \bigtriangledown ^ { T } | \bigtriangledown ^ { T } u | ^ { 2 } - \frac { | \bigtriangledown ^ { T } u | ^ { 2 } } { ( u + A ) ^ { 2 } } \bigtriangledown ^ { T } u + \bigtriangledown ^ { T } t r _ { \omega } \widehat { \omega } _ { \infty } . \end{array} \end{align*}


\begin{align*} \phi ^ { - 1 } e _ { \phi ( \lambda ) } \phi = \frac { t ( \lambda , \lambda ) } { 2 } e _ \lambda ^ \vee , \end{align*}


\begin{align*} \begin{cases} ( \phi , \psi ) | _ { y = 0 } = ( \phi , \psi ) | _ { y \rightarrow \infty } = ( \phi , \psi ) | _ { x = 1 } = ( \phi , \psi ) | _ { x \rightarrow \infty } = ( 0 , 0 ) , \\ ( \phi _ x , \psi _ x ) | _ { x = 1 } = ( \phi _ x , \psi _ x ) | _ { x \rightarrow \infty } = ( 0 , 0 ) . \end{cases} \end{align*}


\begin{align*} F ^ * ( z \ , | \ , \overline \omega _ 1 , \omega _ 2 ) \cdot F ^ * ( z \ , | \ , \overline \omega _ 1 , - \omega _ 2 ) = \prod _ { k \geq 0 } \big ( 1 - x _ 2 ( q _ 2 \widetilde { q } _ 2 ) ^ { k / 2 } \big ) \cdot \prod _ { k \geq 1 } \big ( 1 - x _ 2 ^ { - 1 } ( q _ 2 \widetilde { q } _ 2 ) ^ { k / 2 } \big ) ^ { - 1 } , \end{align*}


\begin{align*} & \sum _ { t = 1 } ^ \infty \alpha _ k ^ { t } = \infty , ~ ~ \sum _ { t = 1 } ^ \infty ( \alpha _ k ^ { t } ) ^ 2 < \infty , \forall k \in { \cal K } \\ & \sum _ { t = 1 } ^ \infty \alpha ^ { t } = \infty , ~ ~ \sum _ { t = 1 } ^ \infty ( \alpha ^ { t } ) ^ 2 < \infty . \end{align*}


\begin{align*} y _ k = \frac { U } { m } - \frac { L ( m - 1 ) } { 2 } , \end{align*}


\begin{align*} \label [ I ] { d e f : m o d i f i c a t i o n } \forall t \geq 0 , P ( X _ t = Y _ t ) = 1 , \end{align*}


\begin{align*} \int _ { D _ j } ( \eta + g ^ { - 1 } ) | \bar \partial _ { \Phi } ^ * \alpha | ^ 2 e ^ { - \Phi } \ge \int _ { D _ j } v '' _ { \epsilon } ( \Psi _ m ) | \alpha \llcorner ( \bar \partial \Psi _ m ) ^ { \sharp } | ^ 2 e ^ { - \Phi } . \end{align*}


\begin{align*} \| \bar \psi ( z ) \| _ Y = \| \bar \psi ( z ) - \bar \psi ( 0 ) \| _ Y \leq \sqrt { n } \| z \| _ H \leq \sqrt { n } , \end{align*}


\begin{align*} \begin{cases} & \frac { \partial X } { \partial t } ( x , t ) = \psi u ^ \alpha \rho ^ \delta f ^ { - \beta } ( x , t ) \nu ( x , t ) , \\ & X ( \cdot , 0 ) = X _ 0 , \end{cases} \end{align*}


\begin{align*} - \frac { \hslash ^ { 2 } } { 2 m } k _ { \pm } ^ { 2 } + \frac { \beta \hslash ^ { 4 } } { 3 m } k _ { \pm } ^ { 4 } = \pm i \lambda \Leftrightarrow \frac { 2 \beta \hslash ^ { 2 } } { 3 } k _ { \pm } ^ { 4 } - k _ { \pm } ^ { 2 } \mp \frac { 2 m i \lambda } { \hslash ^ { 2 } } = 0 , \end{align*}


\begin{align*} S _ { n } : = \mathcal { T } _ { \left \{ C _ { m , r } \left ( n \right ) \left ( j \right ) \right \} _ { j = 1 } ^ { r } } \end{align*}


\begin{align*} \{ 1 , \ldots , \ell + 1 \} \setminus \{ \tau _ { \pi } ( 2 ) , \ldots , \tau _ { \pi } ( \ell + 1 ) \} = \{ 1 , \ldots , \ell + 1 \} \setminus \{ j _ 1 + 1 , \ldots , j _ { \ell } + 1 \} \end{align*}


\begin{align*} \left < \tau ( \beta ^ S _ h ) \phi , \phi \right > = \frac { 1 } { m _ { G _ S } ( B ^ S _ h ) } \int _ { G _ S / L _ S } \int _ { B ^ S _ h } \phi ( g ^ { - 1 } x L _ S ) \phi ( x L _ S ) \ , d m _ { G _ S } ( g ) d \nu ( x L _ S ) . \end{align*}


\begin{align*} \hat { H } \psi = E \psi , \end{align*}


\begin{align*} A ( T ) & = \begin{bmatrix} O & D _ { l ( T ) } & O & \cdots & O & O \\ D _ { l ( T ) } ^ T & O & D _ { l ( T ) - 1 } & \cdots & O & O \\ O & D _ { l ( T ) - 1 } ^ T & O & \cdots & O & O \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ O & O & O & \cdots & O & D _ 2 \\ O & O & O & \cdots & D _ 2 ^ T & O \end{bmatrix} , \end{align*}


\begin{align*} P _ \delta ( E ) = P ^ \parallel _ \delta ( E ^ \parallel ) \otimes P ^ \perp _ \delta ( E ^ \perp ) \end{align*}


\begin{align*} \gamma \star \delta : = \rho _ \tau ( \rho _ \tau ( \gamma ) \cup \rho _ \tau ( \delta ) ) . \end{align*}


\begin{align*} \beta ( t , d ) & = \beta ( t ) = e ^ { - t } \mbox { w h e n } \phi _ d = e ^ { \tilde \tau ( z ) } , \\ \beta ( t , d ) & = \beta ( t ) = ( 1 + t ) ^ { - \xi } , \mbox { w h e n } \phi _ d = ( 1 + \tilde \tau ( z ) ) ^ \xi , \ , \xi > 1 . \end{align*}


\begin{align*} \inf \limits _ { m \in \N } \P \{ m \ \mathrm { i s \ d r y } \} = \inf \limits _ { m \in \N } \prod _ { i = 0 } ^ { m - 1 } \left ( 1 - \P \{ \ell ^ { ( A ) } _ i > m - i \} \right ) > 0 . \end{align*}


\begin{align*} 1 = \sum \limits _ { k \in \mathbb Z } \psi _ k ( \xi ) , \end{align*}


\begin{align*} \begin{cases} - V '' + V = V ^ { p - 1 } , V \ge 0 & , \\ \sum _ { i = 1 } ^ m V _ i ' ( 0 ^ + ) = 0 ; \end{cases} \end{align*}


\begin{align*} & \mathcal { L } _ { V _ a } b - \mathcal { L } _ { V _ b } a = - 2 d ^ \Lambda a \wedge d ^ \Lambda b \\ + & \iota _ X ( d ^ \Lambda a \wedge d b - d ^ \Lambda b \wedge d a ) + d \iota _ X ( d ^ \Lambda a \wedge b - d ^ \Lambda b \wedge a ) + d d ^ \Lambda b \wedge \iota _ X a - d d ^ \Lambda a \wedge \iota _ X b \end{align*}


\begin{align*} \tau _ k = \bar \theta _ k \tau _ { k - 1 } \geq \tau _ { k - 1 } \geq \tau _ \mathrm { m i n } , \sigma _ k = \bar \theta _ k \sigma _ { k - 1 } \geq \sigma _ { k - 1 } \geq \sigma _ \mathrm { m i n } . \end{align*}


\begin{align*} \partial _ t \mathcal { U } ( s , t ) = \mathcal { L } ( t ) \mathcal { U } ( s , t ) , \mathcal { U } ( s , s ) = I . \end{align*}


\begin{align*} D P _ g ( w ) = - \left ( D L _ { ( g , P ( g ) ) } ( 0 , \cdot ) \right ) ^ { - 1 } \circ D L _ { ( g , P ( g ) ) } ( w , 0 ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : o p t i m a l c o u p l i n g j u m p T } X ^ i _ { T ^ i _ n } = \mathsf { T } { \Big ( \overline { X } { } ^ i _ { T ^ i _ n } \Big ) } , \end{align*}


\begin{align*} f ( \lambda ) = \lambda ^ N + \sum _ { k = 1 } ^ N ( - 1 ) ^ k \lambda ^ { N - k } \sum _ { 1 \le j _ 1 < \cdots < j _ k \le N } \ \prod _ { \ell = 1 } ^ k \lambda _ { j _ \ell } . \end{align*}


\begin{align*} \prod _ { k \ge 1 } \frac { \mu _ k ^ 2 } { \lambda _ { k + 1 } ^ 2 } = \infty , \end{align*}


\begin{align*} \mathrm { ( I ) } ~ \mathrm { M D T E } _ { ( \boldsymbol { a } , \boldsymbol { b } ) } ( \mathbf { Y } ) & = \boldsymbol { \mu } + \frac { \mathbf { \Sigma } ^ { \frac { 1 } { 2 } } \boldsymbol { \delta } } { F _ { \mathbf { Z } } ( \boldsymbol { \xi _ { a } } , \boldsymbol { \xi _ { b } } ) } , \end{align*}


\begin{align*} \Phi _ { \nu , \nu ' } ^ \lambda ( z ) = \lambda ^ { n / 2 } \Phi ^ 1 _ { \nu , \nu ' } ( \lambda ^ { 1 / 2 } z ) \end{align*}


\begin{align*} \varphi ( x , y ) : = \psi _ 0 ( x ) \psi _ 1 ( y ) \end{align*}


\begin{align*} \int _ D e ^ { - i \varphi } q ( x ) u _ 1 u _ 2 \ , d x = \int _ D q ( x ) e ^ { i x \cdot \xi } \ , d x + \mathcal { R } ' ( \xi , s ) , \end{align*}


\begin{align*} e _ n ( { \mathcal K } ) _ X { \geq \frac { C } { [ \log _ 2 n ] ^ \alpha } } , n = 1 , 2 , \ldots , \hbox { t h e n } d _ n ( { \mathcal K } ) _ X \geq C ' \frac { 1 } { [ \log _ 2 n ] ^ \alpha } , \end{align*}


\begin{align*} \begin{aligned} \tfrac { \sigma _ 0 L } { h _ 0 B } = 1 - \tfrac { 1 - \kappa h _ 0 } { 2 \left ( 1 - \kappa \right ) } \ , \ , , \tfrac { \alpha _ 0 } { B } = 1 \end{aligned} \end{align*}


\begin{align*} \psi ( 0 ) = \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \alpha _ { t _ 1 , t _ 2 } ( 0 ) d t _ 1 d t _ 2 \end{align*}


\begin{align*} \lfloor x \rfloor & = \max \{ m \in \mathbb { Z } \mid m \leq x \} , \lceil x \rceil = \min \{ n \in \mathbb { Z } \mid n \geq x \} . \end{align*}


\begin{align*} \sum \limits _ { 1 \leq n \leq N } f _ k ( n ) = \sum _ { w \leq N } \frac { g _ k ( w ) } { w } \Big ( \frac { N } { w } + O ( 1 ) \Big ) , \end{align*}


\begin{align*} \begin{cases} i v _ t + \bigtriangleup v + i \gamma v - \mathcal { R } ^ 2 [ v , \cdots , v ] - \mathcal { N R } [ v , \cdots , v ] = F \\ v ( x , 0 ) = u _ 0 \in H ^ 1 _ x ( \mathbb { T } ) , \end{cases} \end{align*}


\begin{align*} \Delta ( z ) = ( - 1 ) ^ { r m } \dfrac { 1 } { ( r m n + r m ) ! } \left ( \dfrac { d } { d z } \right ) ^ { r m n + r m } P _ { m , r m } ( z ) \cdot { \rm { d e t } } { \begin{pmatrix} \vec { q } _ { 0 } \ \cdots \ \vec { q } _ { r m - 1 } \end{pmatrix} } \enspace . \end{align*}


\begin{align*} \tilde { \mathbb L } ^ \vee _ \pm = \left ( H ^ 2 ( X _ \pm ; \mathbb R ) \cap \tilde { \mathbb L } ^ \vee _ \pm \right ) \oplus \bigoplus _ { j \in S _ \pm } \mathbb Z D _ j . \end{align*}


\begin{align*} \nu _ { ; i } ^ \alpha = h _ i ^ k X _ { ; k } ^ \alpha . \end{align*}


\begin{align*} e _ { \Phi } \left ( \rho \right ) = \lim \limits _ { L \rightarrow \infty } \frac { \rho \left ( U _ { L } ^ { \Phi } \right ) } { \left \vert \Lambda _ { L } \right \vert } = \rho \left ( \mathfrak { e } _ { \Phi } \right ) \ , \end{align*}


\begin{align*} I _ 2 = & \left [ \overline u _ e ^ i ( x ) \varphi ( x , 0 ) + \overline h _ e ^ i ( x ) \psi ( x , 0 ) \right ] x ^ { - \frac { 1 } { 2 } - 2 \sigma _ i } \\ \leq & | ( \varphi , \psi ) ( x , 0 ) x ^ { - \frac { 3 } { 4 } + \sigma _ { i - 1 } - \frac { 1 } { 2 } - 2 \sigma _ i } | \leq \delta _ 0 | ( \varphi , \psi ) ( x , 0 ) x ^ { - \frac { 3 } { 4 } - \sigma _ i } | ^ 2 + C | x ^ { - \frac { 1 } { 2 } - \sigma ' } | ^ 2 , \end{align*}


\begin{align*} s _ 1 ( y ) : = 1 s _ n ( y ) : = \sum _ { k \geq 1 } s ( n , k ) y ^ k ( n \geq 2 ) , \end{align*}


\begin{align*} \tau _ { - x _ n } \mu ^ i _ n = \tau _ { - x _ n } \mu _ n - \sum _ { 0 \le j \leq i } \tau _ { x ^ j _ n - x _ n } \mu ^ j , \end{align*}


\begin{align*} \int _ \Sigma \omega + \sum _ { } = 0 . \end{align*}


\begin{align*} \mathcal { K } ^ { n } _ { 2 , j } = \left \{ \pi _ { X , Y _ { [ 0 : 2 ] } , Z _ j } \in \mathcal { P } \left ( \mathcal { X } \times \mathcal { Y } _ { [ 0 : 2 ] } \times \mathcal { Z } _ j \right ) \ : \ \pi _ { Z _ j } \stackrel { \delta ^ { \prime } _ { n } } { \approx } p _ { Z _ j } \right \} . \end{align*}


\begin{align*} h _ { i j } ( \omega ) = \sum \limits _ { k = 1 } ^ d \frac { \partial \psi ^ k } { \partial \omega _ i } ( \omega ) \frac { \partial \psi ^ k } { \partial \omega _ j } ( \omega ) , i , j = 1 , . . . , m - 1 ; \end{align*}


\begin{align*} g _ k ( n ) : = \sum _ { \substack { r d = n \\ d \ { \rm { i s } } \ k - { \rm { f r e e } } } } \mu ( r ) , \end{align*}


\begin{align*} \Delta ( E ) = E \otimes 1 + 1 \otimes E \end{align*}


\begin{align*} c ( x ) = \left \{ \begin{array} { c c } 2 & x _ 1 \in ( - 2 . 5 , 1 ) x _ 2 \in ( 1 . 5 , 2 . 5 ) \\ 1 & . \end{array} \right . \end{align*}


\begin{align*} n + m = 2 p - a - 2 k \end{align*}


\begin{align*} \gamma : = \kappa ( E _ 0 ) \sqrt { t d _ { X _ 0 \times Y _ 0 } } \end{align*}


\begin{align*} P _ \delta ( E ) \psi = 0 \end{align*}


\begin{align*} W ( \lambda ^ { w _ 1 } x _ 1 , \ldots , \lambda ^ { w _ N } x _ N ) = \lambda ^ d W ( x _ 1 , \ldots , x _ N ) , \end{align*}


\begin{align*} I _ { 1 3 } ( x , t ) , I _ { 1 4 } ( x , t ) \le C \left \{ \begin{array} { l l } 1 & \mbox { i f } \ a > 0 , \\ \log ( T + 3 R ) & \mbox { i f } \ a = 0 , \\ ( T + R ) ^ { - a } & \mbox { i f } \ a < 0 . \\ \end{array} \right . \end{align*}


\begin{align*} 0 = V _ 0 \subseteq V _ 1 \subseteq \ldots \subseteq V _ n = S H _ * ( X ; \Q ) \end{align*}


\begin{align*} R ( \lambda , A ) = ( \lambda - A ) ^ { - 1 } \end{align*}


\begin{align*} [ D \vec { x } ] _ { v _ i } = \sum _ { j = 1 } ^ n d _ G ( v _ i , v _ j ) \vec { x } _ { v _ j } , \end{align*}


\begin{align*} P ( z ) = a _ n z ^ n + a _ { n - 1 } z ^ { n - 1 } + \cdots + a _ 1 z + a _ 0 , \ , \ , a _ j \in \mathbb C _ p , \ , \ , j = 0 , 1 , \dots , n . \end{align*}


\begin{align*} \tau _ x ( f \star g ) = \tau _ x ( g \star f ) , \end{align*}


\begin{align*} \P _ N \Biggl ( \prod _ { i = 1 } ^ { N } ( \Delta \tau _ i ) ^ { - \frac d 2 } > C ^ { - N } R \Biggr ) \geq C ^ N N ^ { ( 1 + \frac d 2 ) N } R ^ { - 1 - \frac 2 d } \biggl ( \log \frac { t R ^ { \frac 2 { d N } } } { C N } \biggr ) ^ { N - 1 } . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \omega ^ { \prime } = \omega _ { N } + \sqrt { - 1 } \partial _ { B } \overline { \partial } _ { B } \phi _ { T _ { 0 } } \geq 0 . \end{array} \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } E [ N ^ n _ { \eta _ n ( t ) } ( W _ b - W _ a ) ] = 0 . \end{align*}


\begin{align*} f _ i ( y , w _ i ) - f ( x , v _ i ) = f _ i ( y , t _ i p _ i ) = t _ i f _ i ( y , p _ i ) \in t _ i Q _ i \ , . \end{align*}


\begin{align*} \mathcal F [ u ] : = \det [ D ^ 2 u - g _ { x x } ( \cdot , Y ( \cdot , u , D u ) , Z ( \cdot , u , D u ) ) ] = \det E ( \cdot , Y , Z ) \psi ( \cdot , u , D u ) , \end{align*}


\begin{align*} \widehat { \lambda } ( ( n + 1 ) P ) = - \widehat { \lambda } ( ( n - 1 ) P ) + 2 \widehat { \lambda } ( n P ) + 2 \widehat { \lambda } ( P ) + v ( x ( n P ) - x ( P ) ) - v ( \Delta ) / 6 . \end{align*}


\begin{align*} \sigma ( g _ 1 , g _ 2 ) + \sigma ( g _ 1 g _ 2 , g _ 3 ) = \sigma ( g _ 1 , g _ 2 g _ 3 ) + \sigma ( g _ 2 , g _ 3 ) . \end{align*}


\begin{align*} \phi ^ { [ \lambda _ k \lambda '' v ] } : = ( \lambda _ k \lambda '' ) \cdot \phi . \end{align*}


\begin{align*} \varphi ( v ) \psi = \frac { d } { d t } \bigg | _ { t = 0 } \Phi ( { \exp ( - t v ) } , \psi ) , \forall \psi \in D _ \mathfrak { V } , \forall v \in \mathfrak { g } . \end{align*}


\begin{align*} \mathcal { G } ( 2 c ) \oplus \mathcal { G } ( 1 2 c ) = \mathcal { G } ( 4 c ) \oplus \mathcal { G } ( 1 0 c ) = \mathcal { G } ( 6 c ) \oplus \mathcal { G } ( 8 c ) = 6 \end{align*}


\begin{align*} z = H ( \omega ( z ) ) = \omega ( z ) + ( \varphi _ { \mu } \circ F _ { \mu \boxplus \nu } ) ( z ) , z \in \mathbb { C } ^ { + } . \end{align*}


\begin{align*} \Lambda _ { \Omega } ( t ) = C ( \Omega ) t ^ d + O ( t ^ { d - 1 } ) \end{align*}


\begin{align*} \sum _ { g \in \mathrm { G a l } ( K / \Q ) } g . \varphi _ k \in R _ \mathrm { r a t } ( H _ k ) = \mathcal { S } ( H _ k ) \ ; . \end{align*}


\begin{align*} B _ { j i } ( \sigma : 0 ) & = \tilde B _ { j i } ( \sigma : 0 ) + B _ { j j } ( \sigma : k _ L + 1 ) B _ { j i } ( k _ L ) B _ { i i } ( k _ L : 0 ) \cr & \geq \frac { 1 } { 2 } \eta _ i ( \eta _ j - \varepsilon ) ( \delta - \varepsilon ) + \eta _ j \varepsilon \eta _ i \cr & = \frac { 1 } { 2 } \eta _ i \varepsilon ^ 2 + \frac { 1 } { 2 } \eta _ i ( \eta _ j - \delta ) \varepsilon + \frac { 1 } { 2 } \eta _ i \eta _ j \delta \geq \frac { 1 } { 2 } \eta _ i \eta _ j \delta , \end{align*}


\begin{align*} \mathcal { S } ( L ) = \int _ { \mathcal { M } } \mathcal { I } , \mathcal { I } = \int _ { \Sigma } e ^ { - i \hat { \theta } } F \omega + \sum _ { } e ^ { - i \hat { \theta } } F f | ^ + _ - . \end{align*}


\begin{align*} ( { \rm r e s p . } ~ ~ x ( y z ) = ( x y ) z + ( - 1 ) ^ { \vert x \vert \vert y \vert } ( y x ) z ) , \end{align*}


\begin{align*} \sin \theta \big ( R ( \tilde \Xi _ 1 ) , R ( \Xi _ 1 ) \big ) \le \frac { \lVert E \rVert _ S } { \delta } = { \mathcal P } _ { B 1 } , \end{align*}


\begin{align*} \lim _ { r \to + \infty } r ^ { \frac { n - 1 } { 2 } } e ^ { r } U ( r ) = \mathfrak u > 0 \ \hbox { a n d } \ \lim _ { r \to + \infty } \frac { U ' ( r ) } { U ( r ) } = - 1 , \end{align*}


\begin{align*} 0 = H ^ { n - 1 } ( \tilde { \mathbb { P } } , \Omega _ { \tilde { \mathbb { P } } } ^ 1 \otimes \mathcal { O } ( \tilde { Y } + K _ { \tilde { \mathbb { P } } } ) ) = H ^ { n } ( \tilde { \mathbb { P } } , \Omega _ { \tilde { \mathbb { P } } } ^ 1 \otimes \mathcal { O } ( \tilde { Y } + K _ { \tilde { \mathbb { P } } } ) ) \end{align*}


\begin{align*} N _ { Y _ f / \mathcal { X } } \cong \bigoplus \limits _ { i = 1 } ^ { l ( \Delta ) - 1 } \mathcal { O } _ { Y _ f } \cong T _ { B , f } \otimes _ { \mathbb { C } } \mathcal { O } _ { Y _ f } . \end{align*}


\begin{align*} \widehat { I } _ 0 = I _ 0 \cup \mathcal { R } . \end{align*}


\begin{align*} \lim _ { | y | \to \infty } \| R ( x + i y , A _ 2 ) - R ( x + i y , A _ 1 ) \| _ q = 0 \forall x < - \omega _ 0 ( A _ j ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } } \frac { 1 + x ^ { 2 } } { | s + i t - x | ^ { 2 } } d \sigma ( x ) = 1 , t = f _ { \mu } ( s ) , \end{align*}


\begin{align*} \begin{cases} n = p - k - a \ , \ \ m = q - k \\ n = p - k - a + 1 \ , \ \ m = q - k - 1 \\ \vdots \\ n = p - k \ , \ \ m = q - a - k \\ \end{cases} \ . \end{align*}


\begin{align*} x _ { B , m } ^ { r , p o s t } = \int _ { x _ { m } ^ { r } } x _ { m } ^ { r } \hat { p } ( x _ { m } ^ { r } | \boldsymbol { y } ) , \end{align*}


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


\begin{align*} L _ { i j k } ^ { 1 } = L _ { i j k } ( 1 ) . \end{align*}


\begin{align*} C : = \left \{ y \in \Re ^ { m } | \ , - y _ { \max } e \leq y \leq y _ { \max } e \right \} \mbox { a n d } D : = \left \{ Z \in \S ^ { d } | \ , O \preceq Z \preceq z _ { \max } I \right \} , \end{align*}


\begin{align*} \lambda _ { 1 , 2 } ^ s ( t ) = \left \{ \begin{aligned} & \lambda _ 1 ( 2 s t + 1 - s ) & : 0 \leq t \leq \frac { 1 } { 2 } \\ & \lambda _ 2 ( 2 t - 1 ) & : \frac { 1 } { 2 } \leq t \leq 1 \\ \end{aligned} \right \} \lambda _ i ^ s ( t ) = \lambda _ i ( t ( 1 - s ) ) . \end{align*}


\begin{align*} V _ L = \bigoplus _ i V _ { \lambda _ i } . \end{align*}


\begin{align*} ( X _ j f ) ^ \mu = X _ j ^ \mu f ^ \mu , \end{align*}


\begin{align*} Z ^ \pm = \begin{cases} \dot { x } = - y + \sum _ { k = 2 } ^ { n } P ^ \pm _ k ( x , y ) , \\ \dot { y } = x + \sum _ { k = 2 } ^ { n } Q ^ \pm _ k ( x , y ) , \end{cases} \ \ ( x , y ) \in \Sigma ^ { \pm } , \end{align*}


\begin{align*} \Delta ^ 2 _ h ( f \ast \rho _ { \epsilon } ) = ( \Delta ^ 1 _ h f ) \ast ( \Delta ^ 1 _ h \rho _ { \epsilon } ) . \end{align*}


\begin{align*} \begin{cases} N _ 1 \ { \rm i s ~ c h o s e n ~ w i t h ~ } N _ 1 - M _ 1 = 1 0 0 , \\ N _ 2 \ { \rm i s ~ c h o s e n ~ w i t h ~ } N _ 2 - M _ 2 = 2 N _ 1 , \\ N _ k = N _ 2 , \ k = 3 , \cdots , 6 , \end{cases} \end{align*}


\begin{align*} u _ { A , q } ^ s ( x , y ) & = \frac { 1 } { 4 \pi } \frac { e ^ { i k \vert x \vert } } { \vert x \vert } \frac { e ^ { i k \vert y \vert } } { \vert y \vert } u ^ \infty _ { - A , q } \left ( \widehat { y } , - \widehat { x } \right ) + \frac { O ( 1 ) } { | x | ^ 2 } \| u _ { A , q } ( \cdot , y ) \| _ { L ^ 2 ( D ) } \cr & + \frac { O ( 1 ) } { | x | | y | ^ 2 } \| u _ { - A , q } ( \cdot , - \hat x ) \| _ { L ^ 2 ( D ) } . \end{align*}


\begin{align*} \mathbf { 1 } = ( 1 , \ldots , 1 ) . \end{align*}


\begin{align*} E ' : = \bigcup _ { x \in M } ( E _ x ) ' \ , ; \end{align*}


\begin{align*} \int \frac { f ( x ) } { g ( x / q ) } v ( x / q ) r ( x ) y ( x ) d _ q x = \frac { f ( x / q ) } { g ( x / q ) } \Big ( y ( x / q ) - v ( x / q ) D _ { q ^ { - 1 } } y ( x ) \Big ) , \end{align*}


\begin{align*} \int _ { X } f ( x ) d M _ { \mathcal { R } } ( \nu ) = \int _ { X } L _ { \mathcal { R } } ( f ) ( x ) d \nu , \end{align*}


\begin{align*} \psi _ E : = \widetilde { H } _ 0 ( [ \kappa ( E ) \sqrt { t d _ { X \times Y } } ] _ * ) . \end{align*}


\begin{align*} \delta ( P _ { k , l } ) = ( \beta _ k - \beta _ l ) P _ { k , l } \ , . \end{align*}


\begin{align*} \frac { \partial C } { \partial z ^ i _ { \epsilon } } v ^ i = 0 , \end{align*}


\begin{align*} \min _ { \mathbf { y } \in { \mathcal C } _ 1 } \lVert \mathbf { x } - \mathbf { y } \rVert _ 2 = \lVert ( I - P _ { { \mathcal C } _ 1 } \mathbf { x } ) \rVert _ 2 . \end{align*}


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


\begin{align*} \left ( \prod _ { i = 1 } ^ n r _ i ^ { s _ { i , - } } \right ) \left \langle \gamma _ 1 \bar { \psi } ^ { a _ 1 } , \ldots , \gamma _ m \bar { \psi } ^ { a _ m } \right \rangle _ { g , \{ \vec s ^ j \} _ { j = 1 } ^ m , \beta } ^ { X _ { D , \vec r } } \end{align*}


\begin{align*} ( f g , f g ) = ( g g ^ \dagger , f ^ \dagger f ) , \end{align*}


\begin{align*} \| E \| _ { L ^ 1 \cap L ^ \infty } \leq \frac { c \lambda \eta _ t } { m _ t } { \bf 1 } _ { d = 3 } + f _ { \lambda , \mu , m ^ 2 , t } \big ( \| E \| _ { L ^ 1 \cap L ^ \infty } ) . \end{align*}


\begin{align*} p ( \boldsymbol { s } , \boldsymbol { s } ^ { r } , \boldsymbol { s } ^ { c } , \boldsymbol { x } ^ { r } , \boldsymbol { x } ^ { c } ) = p ( \boldsymbol { s } ) \prod _ { m } \mathop { p ( s _ { m } ^ { r } | s _ { m } ) } \prod _ { m } \mathop { p ( s _ { m } ^ { c } | s _ { m } ) } \prod _ { m } p ( x _ { m } ^ { r } | s _ { m } ^ { r } ) p ( x _ { m } ^ { c } | s _ { m } ^ { c } ) . \end{align*}


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


\begin{align*} 0 = \left ( \Bar { Q } \circ ( \Bar \Phi \circ Q _ \mathfrak { g } - \Bar { Q } \circ \Bar \Phi ) \right ) ^ { ( n + 1 ) } & = \Bar { Q } ^ { ( 0 ) } \circ ( \Bar \Phi \circ Q _ \mathfrak { g } - \Bar { Q } \circ \Bar \Phi ) ^ { ( n + 1 ) } \\ & = [ Q , ( \Bar \Phi \circ Q _ \mathfrak { g } - \Bar { Q } \circ \Bar \Phi ) ^ { ( n + 1 ) } ] , \end{align*}


\begin{align*} F ( A x + e ) = B G ( x ) + d . \end{align*}


\begin{align*} \delta ( t _ { k + 1 } ) = \left \{ \begin{aligned} & \delta ( t _ { k } ) & & \beta ^ { \top } e ( t _ { k } ) \geq 0 \\ & \delta ( t _ { k } ) - 0 . 5 \beta ^ { \top } e ( t _ { k } ) \mathsf { T } _ { k } & & \beta ^ { \top } e ( t _ { k } ) < 0 . \end{aligned} \right . \end{align*}


\begin{align*} \Lambda ^ { ( 2 ) } _ { n , \delta } = \exp \left ( i \mu Y ^ { n , 1 } _ { t } + i \lambda n ^ { \alpha + \frac 1 2 } \int _ { t - \delta } ^ { \eta _ n ( t ) } \psi _ { n , 1 } ( s , \eta _ n ( t ) ) d W _ s \right ) . \end{align*}


\begin{align*} \Psi ^ { n , 3 } _ s = \left ( \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \ , \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) ^ 2 , \end{align*}


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


\begin{align*} & { L _ { \varepsilon , h } ( \phi ) } = \Gamma _ { \varepsilon , h } ( \phi ) \end{align*}


\begin{align*} s ^ { K e r F _ \ast } + ( m - n ) \mu - ( m - n ) ^ 2 \| H \| ^ 2 + ( m - n ) d i v ( H ) = 0 , \end{align*}


\begin{align*} W _ A \star W _ R = W _ { A R } \end{align*}


\begin{align*} \frac { 2 \ddot { x } } { C _ 1 [ f ( t ) \dot { x } - a _ o x + C _ 2 ] ^ 3 } = F ( \dot { x } , x , t ) \ , \end{align*}


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


\begin{align*} g _ { j + 1 } ( x ) = { \mu _ j } g _ j ( x ) + ( x - \lambda ) g ' _ j ( x ) \end{align*}


\begin{align*} \Psi ( y ) = x y - \Phi ( \phi ^ { - 1 } ( y ) ) = \inf _ x \left ( x y - \Phi ( x ) \right ) \ , \end{align*}


\begin{align*} I ( \nu ) = - \frac { 1 } { 2 } \log \mathrm { d e t } ( I _ k - A A ^ T ) \end{align*}


\begin{align*} & \lim _ { m \rightarrow + \infty } \int _ { D _ j } ( v '' _ { \epsilon } ( \Psi _ m ) ) | f F ^ { 1 + \delta } | ^ 2 e ^ { - \phi - \varphi _ { m } - \Psi _ m } \\ = & \int _ { D _ j } ( v '' _ { \epsilon } ( \Psi ) ) | f F ^ { 1 + \delta } | ^ 2 e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) - \varphi - \Psi } \\ < & + \infty . \end{align*}


\begin{align*} ( b \gamma + c ) a + \gamma ( b \gamma + c + 1 ) & = a ( a ^ { - 1 } \gamma + c ( \gamma + 1 ) ) + a ^ { - 1 } \gamma ^ 2 + c \gamma ( \gamma + 1 ) + \gamma \\ & = a ^ { - 1 } \left ( c ( \gamma + 1 ) a ^ 2 + c \gamma ( \gamma + 1 ) a + \gamma ^ 2 \right ) \\ ( b \gamma + 1 ) a + \gamma ( b \gamma + c + 1 ) & = a ( a ^ { - 1 } \gamma + c \gamma + 1 ) + a ^ { - 1 } \gamma ^ 2 + c \gamma ( \gamma + 1 ) + \gamma \\ & = a ^ { - 1 } \left ( ( c \gamma + 1 ) a ^ 2 + c \gamma ( \gamma + 1 ) a + \gamma ^ 2 \right ) \end{align*}


\begin{align*} \mathcal { N } _ { \tilde { A } _ j , q _ j } = \mathcal { N } _ { A _ j , q _ j } , j = 1 , 2 . \end{align*}


\begin{align*} V ( | x | ) = 1 + \frac { \mathfrak v _ \infty } { | x | ^ { \nu } } + \mathcal O \ ( \frac { 1 } { | x | ^ { \nu + \epsilon } } \ ) \hbox { $ C ^ 1 $ - u n i f o r m l y a s } \ | x | \to + \infty , \end{align*}


\begin{align*} \mathop { \lim } \limits _ { s \to - \infty } P _ { s , \tau } \varphi \left ( { \xi , j } \right ) = \left ( { \varphi , \mu _ \tau } \right ) . \end{align*}


\begin{align*} \omega _ \mu ( T _ { \bar \mu } z , T _ { \bar \mu } w ) = | \mu | \omega ( z , w ) z , w \in \R ^ { 2 n } , \end{align*}


\begin{align*} Q ^ * ( s , a ) & = \mathbb { E } \Big \{ c ( s , a ) + \gamma \min _ { a ^ \prime } Q ^ { * } ( s _ { t + 1 } , a ^ \prime ) \Big | s _ t = s , a _ t = a \Big \} , \end{align*}


\begin{align*} R _ { \ell , i , s } ( z ) & = \displaystyle P _ { \ell } ( z ) \psi _ { i , s } \left ( \dfrac { 1 } { z - t } \right ) - P _ { \ell , i , s } ( z ) = \displaystyle \psi _ { i , s } \left ( \dfrac { P _ { \ell } ( t ) } { z - t } \right ) = \sum _ { k = 0 } ^ { \infty } \dfrac { \psi _ { i , s } ( t ^ k P _ { \ell } ( t ) ) } { z ^ { k + 1 } } \enspace . \end{align*}


\begin{align*} l ( b ) = l ( a ) + \sum _ { i = 1 } ^ { \deg _ X l } \partial _ i l ( a ) ( b - a ) ^ i . \end{align*}


\begin{align*} \mathcal { G } _ 1 ( r ( T ) , x ) & = P ( T , x ) , \\ \mathcal { G } _ 2 ( r ( T ) , x ) & = \prod _ { j = 1 } ^ { d } P ( T _ j , x ) , \end{align*}


\begin{align*} ( \mathbb C ^ \times ) ^ { I } \times ( \mathbb C ) ^ { \bar { I } } : = \left \{ ( z _ 1 , \ldots , z _ m ) | z _ i \neq 0 i \in I \right \} . \end{align*}


\begin{align*} \begin{cases} & \int _ { \R ^ { d } } \langle \nabla \varphi , E \rangle d \rho = h ( \varphi ) , \forall \varphi \in C _ { 0 } ^ { \infty } , \\ & j _ p ( E ) \in \overline { \{ \nabla \varphi : \varphi \in C _ { 0 } ^ { \infty } \} } ^ { L ^ q ( \rho ; \R ^ d ) } . \end{cases} \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \log p ^ \dagger _ n ( X _ n ^ \dagger , X _ 0 ^ \dagger ) } { \log \mathbb { G } _ \dagger ( X _ n ^ \dagger , X _ 0 ^ \dagger ) } = 1 \end{align*}


\begin{align*} & \lim _ { B \rightarrow 0 + 0 } \left ( \left ( \frac { 1 } { \delta } + 1 \right ) e ^ { B } - 1 \right ) \frac { G ( 0 ) - G ( B ) } { B } \\ = & \lim _ { B \rightarrow 0 + 0 } \left ( \left ( \frac { 1 } { \delta } + 1 \right ) e ^ { B } - 1 \right ) \frac { G ( - \log 1 ) - G ( - \log e ^ { - B } ) } { 1 - e ^ { - B } } \cdot \frac { 1 - e ^ { - B } } { B } \\ \le & \frac { 1 } { \delta } G ( 0 ) . \end{align*}


\begin{align*} \mathcal { R } : = & \{ \hbox { $ q $ i s a c o n t r a c t e d b o u n d a r y n o d e o f m u l t i p l i c i t y $ 0 $ } \} \\ & \cup \{ \hbox { $ q $ i s a h a l f - n o d e o f $ \widehat { C } $ w h i c h i s o f m u l t i p l i c i t y $ 0 $ b u t n o t a n a n c h o r } \} . \end{align*}


\begin{align*} \sigma ^ { * } ( \mathcal { B } _ { d , k } ) & = \left \{ 2 \sqrt { d - 1 } \cos \left ( \frac { h } { j + 1 } \pi \right ) \colon 1 \le h \le j \le k \right \} . \end{align*}


\begin{align*} y _ i ( v ) = \begin{cases} 1 & v = v _ 0 , \\ - 1 & v = v _ i , \\ 0 & \end{cases} \end{align*}


\begin{align*} d i v ( F ) = \frac { 1 } { \sqrt { d e t \ G } } \sum \limits _ { k = 1 } ^ d \sum \limits _ { i , j = 1 } ^ m \frac { \partial } { \partial \theta _ i } ( \sqrt { d e t \ G } g ^ { i j } F ^ k ( \Phi ( \theta ) ) \frac { \partial \phi ^ k } { \partial \theta _ j } ) \end{align*}


\begin{align*} K ^ { n , 3 } _ \tau & = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ { ( \delta + \frac 1 n ) \wedge \tau } \gamma _ s [ ( \sigma ' ( X _ s ) \sigma ( X _ { \eta _ n ( s ) - \delta } ) ) ^ 2 - ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) ] \left ( \int _ { \eta _ n ( s ) - \delta } ^ { \eta _ n ( s ) } \psi _ { n , 2 } ( u , s ) d W _ u \right ) ^ 2 d s \end{align*}


\begin{align*} u = y = 0 ( c _ 1 , c _ 2 ) . \end{align*}


\begin{align*} \frac { d Y } { d t } = A ( t ) Y ( t ) , Y ( 0 ) = I , A ( t ) = \sum _ { \alpha = 1 } ^ { r } b _ \alpha ( t ) M _ \alpha , \end{align*}


\begin{align*} \P \{ N _ t \geq m + i \} \leq C _ t e ^ { - t } \frac { t ^ { m + i } } { ( m + i ) ! } < C _ t e ^ { - t } \frac { t ^ { m + i } e ^ { m + i } } { ( m + i ) ^ { m + i } } = C _ t e ^ { - t } \left ( \frac { t e } { m + i } \right ) ^ { m + i } . \end{align*}


\begin{align*} \Delta = \sum _ { i = 0 } ^ { k - 1 } \int _ { a _ i } ^ { a _ { i + 1 } } \mbox { d } s \cdot f _ S ( s ) \left [ \lambda \omega _ i s - C ( \lambda \omega _ i ) - \frac { 1 } { 2 } \lambda ^ 2 \sigma _ N ^ 2 \omega _ i ^ 2 + \frac { \rho } { 2 } ( P _ w - \omega _ i ^ 2 ) \right ] \end{align*}


\begin{align*} \mathcal { F } ( d ) = \{ C ' ( S _ { d , i } ) \mid i \in [ 1 , d ] \} \cup \{ C ( T _ { e , j } ) \mid j \in [ 1 , e ] \} \cup \{ C ( U _ { e , \ell } ) \mid \ell \in [ 1 , e ] \} \end{align*}


\begin{align*} \label [ I e q ] { e q : m o d i f i e d p h i } \left \{ \begin{array} { r c l } \partial _ s \tilde { \varphi } + a \cdot \nabla _ x \tilde { \varphi } & = & 0 \\ \tilde { \varphi } ( s = t , x ) & = & \varphi ( x ) \end{array} \right . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty u _ 0 ( n ) \ , q ^ n = \sum _ { k = 1 } ^ \infty \frac { q ^ k } { ( q ; q ) _ { k - 1 } ( q ; q ) _ k } . \end{align*}


\begin{align*} \cal L _ { \varphi + i \eta } = \lambda _ \eta Q _ \eta + N _ \eta \end{align*}


\begin{align*} \left \| f \right \| _ 1 = \left \| f \cdot \ 1 \right \| _ 1 \le \left \| f \right \| _ 2 \left \| \ 1 \right \| _ 2 = \left \| f \right \| _ 2 . \end{align*}


\begin{align*} V _ { U } = \left \lbrace v \in \mathcal { H } | \left ( \begin{array} { c c } U & - 1 \\ 0 & 0 \end{array} \right ) ( D P ) v = 0 \right \rbrace , \end{align*}


\begin{align*} \frac { 1 } { [ K : \Q ] } \sum _ { g \in \mathrm { G a l } ( K / \Q ) } g . \varphi _ k & = \frac { 1 } { p ^ { r - 1 } ( p - 1 ) } \Big ( \sum _ { k = 0 } ^ { p ^ r - 1 } \varphi _ k - \sum _ { k = 0 } ^ { p ^ { r - 1 } - 1 } \varphi _ { k p } \Big ) \\ & = \frac { 1 } { p ^ { r - 1 } ( p - 1 ) } \Big ( \mathrm { I n d } ^ G _ { \{ e \} } \mathbb { 1 } - \mathrm { I n d } ^ G _ { \langle a ^ { p ^ { r - 1 } } \rangle } \mathbb { 1 } \Big ) \in \mathcal { S } ( G ) \ ; , \end{align*}


\begin{align*} r [ B , M ] ^ * = \bigcup _ { m \le n } r _ n [ B , M ] ^ * . \end{align*}


\begin{align*} \mathrm { P } _ { \mathfrak { m } } ^ { \sharp } = - \inf f _ { \mathfrak { m } } ^ { \sharp } \left ( E _ { 1 } \right ) \in \mathbb { R } , \end{align*}


\begin{align*} k _ { \psi } ( r ^ m ) + R ^ N ( r ) + k _ { \psi } ( - 1 ) = 0 , \end{align*}


\begin{align*} \left [ ( t - \eta _ n ( s ) ) ^ { \alpha } - ( t - s ) ^ { \alpha } \right ] ^ 2 & \leq C \left [ ( t - \eta _ n ( s ) ) ^ { 2 \alpha } + ( t - s ) ^ { 2 \alpha } \right ] \\ & \leq \begin{cases} C n ^ { - 2 \alpha } & \alpha \geq 0 \\ C ( t - s ) ^ { 2 \alpha } & \alpha < 0 \end{cases} . \end{align*}


\begin{align*} y _ i & \geq 0 ( \forall i ) \\ \left | y _ { i + 1 } - y _ i \right | & \leq L ( \forall 1 \leq i \leq n - 1 ) \\ \sum _ { i = 1 } ^ n y _ i & = U . \end{align*}


\begin{align*} X _ T = \{ u \in L ^ \infty ( 0 , T ; H ^ s ( \R ) ) \mid \| u \| _ { L ^ \infty _ T H ^ s } \leq 2 \bar C , \| u \| _ { L ^ \infty _ T H ^ 3 } < 1 \} . \end{align*}


\begin{align*} \Phi ( g ) = ( b \sigma ) ^ r g = b \cdot \sigma ( b ) \cdots \sigma ^ { r - 1 } ( b ) \cdot \sigma ^ r ( g ) . \end{align*}


\begin{align*} \omega _ 0 ( A ) = \lim _ { t \to \infty } \frac { \log \phi ( t ) } { t } . \end{align*}


\begin{align*} \omega ( u ) : = \sum _ { i = 1 } ^ m \chi ( u _ i \neq y ) \cdot ( i + | \{ j \le i : u _ j = z \} | ) , \end{align*}


\begin{align*} \mathcal T _ { \kappa , 3 } ^ { 1 , 2 } = \bigcup _ { i , j \in \mathbb N \colon \atop { 2 ( i + j ) \leq \kappa } } \{ [ n _ 0 , n _ 1 , n _ 2 , n _ 3 , n _ 4 , n _ 5 , n _ 4 , n _ 7 ] \in \mathcal S _ { \kappa , 3 } : n _ 1 = n _ 2 = i , n _ 5 = n _ 6 = j \} , \end{align*}


\begin{align*} \begin{aligned} { E } _ h : = \left \{ \phi \in H _ s ~ \Big | ~ \int _ { \mathbb { R } ^ N } \sum _ { i = 1 } ^ 4 U _ { h , i } ^ { p - 1 } \frac { \partial U _ { h , i } } { \partial h } \ , \phi ~ d y = 0 \right \} , \end{aligned} \end{align*}


\begin{align*} \omega ( t ) = c _ 1 \cos t + c _ 2 \sin t + \frac { \mu a ^ 2 } { \lambda } . \end{align*}


\begin{align*} \left ( \partial _ s + L \right ) \omega = \varphi . \end{align*}


\begin{align*} \mathsf { K } _ { k + 1 } \circ \S ^ t _ k = - ( \S ^ n _ { k + 1 } ) ' \circ \mathsf { K } _ { k } , \end{align*}


\begin{align*} \mathit { \Omega } _ { \mathfrak { m } } ^ { \sharp } \doteq \left \{ \omega \in E _ { 1 } : \exists \{ \rho _ { n } \} _ { n = 1 } ^ { \infty } \subseteq E _ { 1 } \mathrm { \ } ^ { \ast } \ \omega \underset { n \rightarrow \infty } { \lim } f _ { \mathfrak { m } } ^ { \sharp } ( \rho _ { n } ) = \inf \ , f _ { \mathfrak { m } } ^ { \sharp } ( E _ { 1 } ) \right \} \end{align*}


\begin{align*} & \log F ( z + \omega _ 2 \ , | \ , \overline \omega _ 1 , \omega _ 2 ) - \log F ( z \ , | \ , \overline \omega _ 1 , \omega _ 2 ) = f _ { - 1 } ^ c ( z , \overline \omega _ 1 ) , \\ & \log G ( z + \omega _ 2 \ , | \ , \omega _ 1 , \widetilde \omega _ 1 , \omega _ 2 ) - \log G ( z + \omega _ 2 \ , | \ , \omega _ 1 , \widetilde \omega _ 1 , \omega _ 2 ) = g _ { - 1 } ^ c ( z , \omega _ 1 , \omega _ 2 ) , \end{align*}


\begin{align*} P ( Q ^ \gamma ( \infty ) \geq k ) = \rho ^ k , k \geq 0 . \end{align*}


\begin{align*} \boldsymbol { y } _ { q } ^ { c } = \mathbf { h } ^ { c } u _ { q } + \boldsymbol { n } _ { q } ^ { c } , \end{align*}


\begin{align*} \liminf _ { n \to \infty } \int _ { \Omega _ n } f ( \Phi _ n , u _ n , \nabla u _ n ) & \geq \liminf _ { n \to \infty } \int _ { \{ u _ n > 0 \} } \frac { f ( \Phi _ n , u _ n , \nabla u _ n ) } { u _ n } u _ n \\ & = \liminf _ { n \to \infty } \int _ { \R ^ N \times \R \times \R ^ N } g ( x , u , \xi ) \dd \nu _ n ( x , u , \xi ) \\ & \geq \int _ { \R ^ N } g ( x , u , \xi ) \dd \delta _ { ( x _ 0 , 0 , 0 ) } = f ' _ - ( x _ 0 , 0 ^ + , 0 ) , \end{align*}


\begin{align*} P ( h ) = \frac { 1 } { 2 L } \frac { h \big [ \kappa h ^ 2 - 2 h ( 1 + \kappa ) + 4 \big ] } { 2 - h ( 1 + \kappa ) } = \frac { 1 } { 2 L } \frac { h ( 2 - h ) ( 2 - \kappa h ) } { 2 - h ( 1 + \kappa ) } . \end{align*}


\begin{align*} \label [ I e q ] { e q : N e w t o n p o t e n t i a l } \left \{ \begin{array} { r c l } \displaystyle { \frac { \dd X ^ i _ t } { \dd t } } & = & V ^ i _ t \\ \displaystyle { \frac { \dd V ^ i _ t } { \dd t } } & = & \displaystyle { - \sum _ { j = 1 } ^ { N } \nabla V ( | X ^ j _ t - X ^ i _ t | ) } \end{array} \right . , \end{align*}


\begin{align*} \bar { r } = \sup _ { r \in \N } r _ i , \bar { a } = \left ( \sup _ { i \in \N } \sum _ { j \in \N } \lvert a _ { i j } \rvert ^ 2 \right ) ^ { \frac { 1 } { 2 } } , \alpha = 2 \bar { r } + 3 . \end{align*}


\begin{align*} u ( x , t ) \ge M _ n \left ( \frac { 1 + t - x } { 1 + l _ n R } \right ) ^ { - b \cdot a _ n } \left \{ \log \left ( \frac { 1 + t - x } { 1 + l _ n R } \right ) \right \} ^ { b _ n } \quad \mbox { i n } \ D _ n \end{align*}


\begin{align*} \frac { \partial \psi } { \partial t } = \hat { H } \psi ; \psi | _ { \ , t = 0 } = \psi _ { 0 } , \end{align*}


\begin{align*} p \left ( x _ 1 , x _ 2 , \ldots , x _ d \right ) = \sum _ { i = 1 } ^ k w _ i p _ { i , 1 } \left ( x _ 1 \right ) p _ { i , 2 } \left ( x _ 2 \right ) \cdots p _ { i , d } \left ( x _ d \right ) . \end{align*}


\begin{align*} Y ( t ) = D _ A \tilde w ( t ) - v ( t ) = : \tilde v ( t ) , \forall t \in \R , \end{align*}


\begin{align*} \bar { x } _ i : = x _ i - \frac { - \kappa } { 1 - \kappa } \frac { h _ i } { L } U \in \big [ x _ { i + 1 } , x _ { i } \big ] , i \in \big \{ 0 , \dots , N - 1 \big \} . \end{align*}


\begin{align*} \int _ { \R ^ N } u = \int _ { \R ^ N } u ^ { \alpha q } u ^ { ( 1 - \alpha ) s } \leq \left ( \int _ { \R ^ N } u ^ q \right ) ^ \alpha \left ( \int _ { \R ^ N } u ^ s \right ) ^ { 1 - \alpha } . \end{align*}


\begin{align*} h ^ { - 1 } ( s ) = \sum \nolimits _ { n = 0 } ^ { \infty } ( \Re a _ { n } ) ( s - s _ { 0 } ) ^ { n } , s \in J . \end{align*}


\begin{align*} \kappa = \begin{cases} \frac { N - 2 + p } { N - p } & \hbox { i f } 1 < p < 2 \hbox { a n d } \mathcal G = 0 \\ \frac { N ( p - 1 ) } { N ( p - 1 ) - p } & \hbox { i f e i t h e r } 2 \leq p < N \hbox { o r } p = N \geq 3 \\ 2 & \hbox { i f } p = N = 2 . \end{cases} \end{align*}


\begin{align*} ( t _ x ( \Phi ) ) ( x ) = \max _ { y \leq x } ( \Phi ( y ) ) + \min _ { z \geq x } ( \Phi ( z ) ) - \Phi ( x ) , \end{align*}


\begin{align*} A ( k ) = \begin{bmatrix} A _ S ( k ) & A _ { S \bar { S } } ( k ) \\ A _ { \bar { S } S } ( k ) & A _ { \bar { S } } ( k ) \end{bmatrix} ~ ~ ~ ~ ~ \pi ( k ) = \begin{bmatrix} \pi _ { S } ( k ) \\ \pi _ { \bar { S } } ( k ) \end{bmatrix} \end{align*}


\begin{align*} \mathcal M ^ - & = \{ \vec x : r _ j \le x _ j \le v _ j , \ r _ i \le x _ i \le s _ i \ ( i \neq j ) \} \\ \mathcal M ^ + & = \{ \vec x : v _ j \le x _ j \le s _ j , \ r _ i \le x _ i \le s _ i \ ( i \neq j ) \} \end{align*}


\begin{align*} g ( ( \nabla _ U A ) _ { X _ j } X _ j , V ) = g ( \nabla _ U A _ { X _ j } X _ j - A ( \nabla _ U X _ j , X _ j ) - A ( X _ j , \nabla _ U X _ j ) , V ) . \end{align*}


\begin{align*} \lambda _ i = r \kappa _ i + s \end{align*}


\begin{align*} Q _ { A , q } v : = i \ ; \textrm { d i v } ( A v ) + i A \cdot \nabla v - ( \abs { A } ^ 2 + q ) v , v \in H ^ 1 _ { \mathrm { l o c } } ( \R ^ 3 ) . \end{align*}


\begin{align*} \lim _ { i \to \infty } \sup _ { j } \abs { \varphi ( \mu _ j ( p _ i ^ \perp v ) ) } = 0 , \end{align*}


\begin{align*} & 0 \le s , t \le l , \\ & \{ j _ 1 , \ldots , j _ s \} \cap \{ k _ 1 , \ldots , k _ t \} = \varnothing , \\ & j _ 1 > \ldots > j _ s , k _ 1 > \ldots > k _ t , \end{align*}


\begin{align*} Q _ t : = S _ t \big ( S _ t ^ 2 \star S _ t ^ 2 \big ) . \end{align*}


\begin{align*} N ( t ) = \int _ { 0 } ^ { + \infty } g p ( t , V _ F , g ) d g = \frac { 1 } { V _ F } \int _ { 0 } ^ { + \infty } g p _ 0 ( t , g ) d g . \end{align*}


\begin{align*} \left \| \hat { p } - p \right \| ^ 2 _ 2 = \left \| \hat { p } \right \| _ 2 ^ 2 - 2 \left < p , \hat { p } \right > + \left \| p \right \| _ 2 ^ 2 . \end{align*}


\begin{align*} K ( \alpha , A , z + a f _ k ) = \operatorname { K e r } ( z + a f _ k ) \cap \bigcap _ { n = 1 } ^ \infty K ( \alpha _ n , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \end{align*}


\begin{align*} ( \delta ^ { ( s ) } ) ^ i u = f ^ { ( i ) } ( u , u ^ { \phi ^ s } , \ldots , u ^ { \phi ^ { i s } } ) , \end{align*}


\begin{align*} ( s _ i ^ { - 1 } - I d ) ( v _ i ) = \beta \circ ( I d - s _ j ) [ \beta ^ { - 1 } ( v _ 2 ) ] \end{align*}


\begin{align*} f _ { \boldsymbol { Y } } ( \boldsymbol { y } ) = \frac { 2 c _ { n } } { \sqrt { | \boldsymbol { \Sigma } | } } g _ { n } \left \{ \frac { 1 } { 2 } ( \boldsymbol { y } - \boldsymbol { \mu } ) ^ { T } \mathbf { \Sigma } ^ { - 1 } ( \boldsymbol { y } - \boldsymbol { \mu } ) \right \} H \left ( \mathbf { \Sigma } ^ { - \frac { 1 } { 2 } } ( \boldsymbol { y } - \boldsymbol { \mu } ) \right ) , ~ \boldsymbol { y } \in \mathbb { R } ^ { n } , \end{align*}


\begin{align*} S _ t u _ 0 = L _ { t } ^ { - 1 } [ S _ t u _ 0 ] W ^ \gamma _ t u _ 0 + N _ t u _ 0 . \end{align*}


\begin{align*} \Tilde { \mathcal { N } } ( f ) \circ \rho _ { \eta } ( X ) = \Pi _ { w \in W _ { \rho } ^ { m _ 0 } } f ( X + w ) , \end{align*}


\begin{align*} \Theta ( G ) & \leq \vartheta ( G ) , \\ \vartheta ( G \boxtimes H ) & = \vartheta ( G ) \vartheta ( H ) , \end{align*}


\begin{align*} \mathbf { v } _ { k } ( z ) = \frac { \mathbf { c } _ { k + 1 } } { \mathbf { d } _ { k + 1 } ( z ) + \mathbf { v } _ { k + 1 } ( z ) } . \end{align*}


\begin{align*} \int _ { X \setminus B ( \mathcal O , 1 ) } | f | d \mu = \int _ { \mathbb S } \int _ 1 ^ \infty | f ( \gamma _ \xi ( t ) ) | t ^ { Q - 1 } \ d t d \sigma ( \xi ) \end{align*}


\begin{align*} V = \{ x \in \mathbb { R } : g ( x ) > b \} . \end{align*}


\begin{align*} \deg _ 0 ( \phi ) = 2 p \phi , \phi \in H ^ { 2 p } ( D _ { I _ { \vec s } } ) ; \end{align*}


\begin{align*} \widehat { \mathbf { c } } _ { n } & = \begin{cases} \mathbf { 1 } , & 1 \leq n \leq p , \\ ( - a _ { n - p - 1 } , 1 , \ldots , 1 ) , & n \geq p + 1 , \end{cases} \\ \widehat { \mathbf { d } } _ { n } ( z ) & = ( 0 , \ldots , 0 , z ) , n \geq 1 . \end{align*}


\begin{align*} Y ^ n _ t = A ^ n _ t + Y ^ { n , 1 } _ t = : A ^ n _ t + C ^ n _ t + Z ^ n _ t , \end{align*}


\begin{align*} \mathcal { M } _ { H _ 0 } ( H ) = \mathcal { M } _ { H _ 0 ' } ( H ) + \mathcal { M } _ { H _ 0 } ( H _ 0 ' ) . \end{align*}


\begin{align*} \P ( Z \ge 1 2 \rho L _ { } \tau ) & \le e ^ { - 1 2 h \rho L _ { } \tau } 4 ^ { \tau } = \big ( 4 / e ^ { 3 } \big ) ^ { \tau } \leq e ^ { - \tau } . \end{align*}


\begin{align*} A ( t ) = ( I + \kappa U ( s , t ) ) \partial _ t { \rm L o g } ( U ( t , s ) + \kappa I ) \end{align*}


\begin{align*} a _ n = ( \lambda _ n ^ 2 - \mu _ n ^ 2 ) \prod _ { k \ne n } \left ( \frac { \lambda _ n ^ 2 - \mu _ k ^ 2 } { \lambda _ n ^ 2 - \lambda _ k ^ 2 } \right ) \end{align*}


\begin{align*} \sup _ { t > 0 } \frac { c \eta _ t } { m _ t } \leq c ( \mu ) { \bf 1 } _ { d = 2 } + c _ 0 \quad \lim _ { \mu \rightarrow \infty } c ( \mu ) = 0 , \end{align*}


\begin{align*} \overline { f } = \left ( \begin{array} { c } f _ { 0 } \\ f _ { 1 } \\ \vdots \\ f _ { n - 1 } \end{array} \right ) \in \mathbb { R } ^ { n } \stackrel { t } { \longrightarrow } f ( x ) = f _ { 0 } + f _ { 1 } x + \cdots + f _ { n - 1 } x ^ { n - 1 } \in \mathbb { R } [ x ] / \langle \phi ( x ) \rangle \end{align*}


\begin{align*} S ( \delta _ { } ) = o ( 1 ) . \end{align*}


\begin{align*} N ^ t _ s : = \# \Pi ^ t ( s ) \end{align*}


\begin{align*} \theta _ { t + 1 } = \theta _ t - \beta _ t Q _ { \theta _ t } ^ * ( s _ t , a _ t ) \frac { d } { d \theta } \log \pi _ { \theta _ t } ( a _ t | s _ t ) , \end{align*}


\begin{align*} K _ { ( i j ) ( k p ) } = - a { \delta _ { i j } } { \delta _ { k p } } + b { \delta _ { i k } } { \delta _ { j p } } + c { \delta _ { i p } } { \delta _ { j k } } \end{align*}


\begin{align*} r _ k [ A , M ] = \{ r _ k ( N ) : N \in [ A , M ] \} . \end{align*}


\begin{align*} \left | \sum _ { k = 0 } ^ { N - 1 } \varphi ( 2 \pi x ) ^ k \varphi ( 2 \pi y ) ^ { N - k - 1 } \right | \le \frac { 2 } { \left | \varphi ( 2 \pi x ) - \varphi ( 2 \pi y ) \right | } \ll \frac { 1 } { \| x - y \| } . \end{align*}


\begin{align*} ( P _ \delta ( E ) \psi ) ( y ) = [ ( | \eta _ \delta | ^ 2 * \chi _ { A } ) \psi ] ( y ) \end{align*}


\begin{align*} W _ j = H e _ j ( x ) ( \forall j = \overline { 0 , l ( T ) } ) , \end{align*}


\begin{align*} { i \choose j } a _ i h ^ j = a _ { i j 0 } + a _ { i j 1 } Q ' + \ldots + a _ { i j n } Q '^ n \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } a ( k ) - a ( k - 1 ) = + \infty \sum _ { k = 1 } ^ \infty \frac { 1 } { 1 \vee \log ( \psi ^ { - 1 } ( a ( k ) ) ) } = \infty , \end{align*}


\begin{align*} \lim _ { i \to \infty } \sup _ { j } \abs { \varphi ( \mu _ j ( ( x - x _ i ) v ) } = 0 . \end{align*}


\begin{align*} \begin{aligned} \frac { 2 - h ( 1 + \kappa ) } { 1 - \kappa h } \big ( f _ { 0 } - f _ { 1 } \big ) & { } \geq { } \frac { 1 } { 2 L ( 1 - \kappa ) } \frac { h ( 1 - \kappa ) [ \kappa h ^ 2 - 2 h ( 1 + \kappa ) + 3 ] } { 1 - \kappa h } \big \| g _ { 0 } \big \| ^ 2 \\ & + \frac { h } { 2 L \big ( 1 - \kappa \big ) } \frac { 1 - \kappa } { 1 - \kappa h } \| g _ 1 \| ^ 2 . \end{aligned} \end{align*}


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


\begin{align*} ( a , b ) = \left \{ \begin{array} { l l } ( 4 , - 2 ) , & \mbox { i f } \ell \equiv 0 \pmod { 4 } , \\ ( 2 4 , - 1 8 ) , & \mbox { i f } \ell \equiv 2 \pmod { 4 } . \end{array} \right . \end{align*}


\begin{align*} \pi _ { i _ 1 } \pi _ J = \binom { | J _ k | + 1 } { 1 } \pi _ { J ' } \end{align*}


\begin{align*} \left ( \begin{array} { c } k \\ \vec { i } \end{array} \right ) = \frac { k ! } { i _ 1 ! i _ 2 ! \cdots i _ l ! \cdots } , \end{align*}


\begin{align*} Y ^ { \prime \prime } + \frac { 1 } { t } Y ^ \prime = 0 \end{align*}


\begin{align*} D _ q F ( x ) = p ( x ) F ( x ) \end{align*}


\begin{align*} \mathtt { m } \left ( \bigwedge \limits _ { t \in J } \mathsf { c l } ' _ t \right ) = \mathtt { w } ^ + \left ( \bigwedge \limits _ { t \in J } \mathsf { c l } ' _ t \right ) - \sum \limits _ { \bigwedge \limits _ { t ' \in J ' } \mathsf { c l } ' _ { t ' } \not \dashv ~ \vdash _ { \mathsf { F D E } } \bigwedge \limits _ { t \in J } \mathsf { c l } ' _ t } \mathtt { m } \left ( \bigwedge \limits _ { t ' \in J ' } \mathsf { c l } ' _ { t ' } \right ) \end{align*}


\begin{align*} \begin{array} [ c ] { c } \psi ^ { \ast } r _ { S ^ { 1 } } ^ { 2 } = | s | _ { h } ^ { 2 } . \end{array} \end{align*}


\begin{align*} \mathrm { S p e c t r u m } ( A _ 0 ) = \{ t \in ( 0 , 1 ) \mid \ , \exists \ , 1 \leq i \leq d \mbox { s u c h t h a t } w _ i = t \} . \end{align*}


\begin{align*} f _ + ^ * K _ { X _ + } = f _ - ^ * K _ { X _ - } + \left ( \sum _ { i = 1 } ^ m D _ i \cdot e \right ) E . \end{align*}


\begin{align*} D _ q u ( x ) + u ( x ) u ( q x ) + \tilde { A } ( x ) u ( q x ) + r ( x ) = 0 , \end{align*}


\begin{align*} \mathcal { W } _ { \Lambda } \doteq \left \{ \Phi \in \mathcal { W } _ { 1 } : \Phi _ { \mathcal { Z } } = 0 \mathcal { Z } \nsubseteq \Lambda \mathcal { Z } \ni 0 \right \} \end{align*}


\begin{align*} ( \Delta - e ) ( \nabla u \cdot \nabla v ) + ( \Delta - e ) ( \nabla u \cdot \nabla v ) + ( \Delta - e ) ( \nabla v \cdot \nabla v ) = ( \Delta - e ) f , \end{align*}


\begin{align*} \Delta ' \circ \mathcal H = ( \mathcal H \otimes \Phi ) \circ \Delta + ( \Phi \otimes \mathcal H ) \circ \Delta . \end{align*}


\begin{align*} \frac { \det ( g _ { \alpha \overline { \beta } } ^ { T } + \varphi _ { \alpha \overline { \beta } } ) } { \det ( g _ { \alpha \overline { \beta } } ^ { T } ) } = e ^ { - \kappa \varphi + F } . \end{align*}


\begin{align*} ( \alpha , \beta ) _ { \rho } = ( \alpha _ n , \beta _ n ) _ { \rho , E _ { \rho } ^ n , n } = \frac { 1 } { \eta ^ n } T _ { L | K } ( \lambda _ { \rho } ( \alpha _ n ) \delta _ n ( \beta _ n ) ) \cdot _ { \rho } v _ n \end{align*}


\begin{align*} V _ k ( t ) x = \sum _ { n = k + 1 } ^ \infty S _ n ( t ) x \end{align*}


\begin{align*} \overline { \delta } ( A ) = \overline { \delta } ( A \cap { \rm L } _ { B \cap ( z , \infty ) } ) \ \le \ \delta ( { \rm L } _ { B \cap ( z , \infty ) } ) < \overline { \delta } ( A ) , \end{align*}


\begin{align*} Q B _ { \beta , \psi , \infty } : = \bigcup _ { p > 0 } Q B _ { \beta , \psi , p } \end{align*}


\begin{align*} u _ t = \Delta _ x u + { \bf b } ( x , t ) \cdot \nabla _ x u + V ( x , t ) u + f ( x , t ) \qquad Q , \end{align*}


\begin{align*} \psi ( z ) + \frac { z \psi ' ( z ) } { \psi ( z ) } = \phi ( z ) . \end{align*}


\begin{align*} f = a _ 0 + a _ 1 Q + \ldots + a _ r Q ^ r \mbox { a n d } f = b _ 0 + b _ 1 Q ' + \ldots + b _ r Q '^ r \end{align*}


\begin{align*} f _ { \mathbf { M ^ { \ast \ast } } _ { - \xi _ { \boldsymbol { s } k } , \xi _ { \boldsymbol { t } l } } } ( \boldsymbol { u } ) = c _ { n - 2 , \xi _ { \boldsymbol { s } k } , \xi _ { \boldsymbol { t } l } } ^ { \ast \ast } \exp \left \{ - \frac { 1 } { 2 } \boldsymbol { u } ^ { T } \boldsymbol { u } - \frac { 1 } { 2 } \xi _ { \boldsymbol { s } k } ^ { 2 } - \frac { 1 } { 2 } \xi _ { \boldsymbol { t } l } ^ { 2 } \right \} = \phi _ { n - 2 } ( \boldsymbol { u } ) , ~ \boldsymbol { u } \in \mathbb { R } ^ { n - 2 } , \end{align*}


\begin{align*} - \log \vert R _ { \ell , i , s } ( \beta ) \vert _ { v _ 0 } & \leq n \log \vert \beta \vert _ { v _ 0 } - ( r m + 1 ) n \log \| \boldsymbol { \alpha } \| _ { v _ 0 } - n c ( x , v _ 0 ) + ( 1 - \varepsilon _ v ) \limsup _ { n \to \infty } \dfrac { 1 } { n } \log | D _ { \boldsymbol { c } , r m n } \cdot D ' _ { \boldsymbol { c } , r m n } | ^ { - 1 } _ v + o ( n ) \\ & = \mathbb { A } _ { v _ 0 } ( \boldsymbol { \eta } , \boldsymbol { \zeta } , \boldsymbol { \alpha } , \beta ) n + o ( n ) \enspace . \end{align*}


\begin{align*} \widetilde s ( \mathcal E ) = \sum _ { 0 \leq i \leq 4 } \widetilde s _ i ( \mathcal E ) = ( 1 + H ) ^ { - n } , \end{align*}


\begin{align*} \dot x & = v , \\ \dot v & = - \varepsilon ^ 2 \nabla \Phi ( x ) , \end{align*}


\begin{align*} M : = \sup _ { N \in \mathbb { N } _ 0 } ( N + 1 ) ^ s \min _ { \mathcal { T } \in \mathbb { T } ( N ) } \eta ( \mathcal { T } ) < \infty . \end{align*}


\begin{align*} m ! \ , N _ 0 ( R f ) _ { i _ 1 j _ 1 \dots i _ m j _ m } = \sum _ { l = 0 } ^ { \lfloor \frac { m } { 2 } \rfloor } c _ { l , m } ( R ( i ^ l j ^ l N _ m f ) ) _ { i _ 1 j _ 1 \dots i _ m j _ m } , \end{align*}


\begin{align*} \mathcal U _ \omega : = \bigcup _ { I \in \mathcal A _ \omega } ( \mathbb C ^ \times ) ^ { I } \times ( \mathbb C ) ^ { \bar { I } } . \end{align*}


\begin{align*} \Psi ^ { n , 2 } _ s = \left ( \int ^ s _ { \eta _ n ( s ) } \left ( s - \eta _ n ( u ) \right ) ^ { \alpha } \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) ^ 2 , \end{align*}


\begin{align*} x _ { i + 1 } = x _ i - \tfrac { h _ i } { L } g _ i , \forall i = 0 , \dots , N - 1 \end{align*}


\begin{align*} x ( t ) = \Phi ( t , \tau ) x ( \tau ) \quad t \geq \tau \geq 0 , \end{align*}


\begin{align*} \textrm { s i n } \Theta ( Q , \widetilde Q ) = \left \{ \sin \left ( \cos ^ { - 1 } ( \zeta _ 1 ) \right ) , \sin \left ( \cos ^ { - 1 } ( \zeta _ 2 ) \right ) , \cdots , \sin \left ( \cos ^ { - 1 } ( \zeta _ r ) \right ) \right \} . \end{align*}


\begin{align*} P _ { \rm F A } ^ { } ( \lambda ) = { \rm P r } \left ( \left . { \sum _ { m = 1 } ^ { M } ( T _ { { m } } - { c } ) ^ 2 } > \frac { M } { 4 } ( \alpha - \lambda ) ^ 2 \right | \mathcal { H } _ 0 \right ) . \end{align*}


\begin{align*} \int _ { \Sigma _ { \Omega _ \Gamma } } \chi _ \alpha ( x _ n ) \big ( 1 - \chi _ \Gamma \big ) \partial _ { x _ n } ( R + V ) \ , d \mu = 0 \end{align*}


\begin{align*} \sum \limits _ { r \mid n } \mu ( r ) = \begin{cases} 1 , & n = 1 , \\ 0 , & { \rm o t h e r w i s e } , \end{cases} \end{align*}


\begin{align*} \begin{cases} & , \\ & | g ( x ) - g ( 0 ) | \le C ( 1 + | x | ^ { \frac { 2 p } { \beta } } ) | x | , \ \forall x \in \R ^ d , \\ & | \nabla g ( x ) | \le C ( 1 + | x | ^ { \frac { 2 p } { \beta } } ) \ \ x \in \R ^ d . \end{cases} \end{align*}


\begin{align*} U _ 0 = A \iota ( z _ r ) \ldots \iota ( z _ { r + s } ) B \end{align*}


\begin{align*} | W | x ^ 2 + ( t + l - l | W | - 1 ) x - l t = 0 \end{align*}


\begin{align*} & \varepsilon _ * = \varepsilon _ { \vec \gamma } , \ \ k _ * = k _ { \vec \gamma } , \ \ t _ { \mathbf { i } } = p _ { \vec \gamma } ( \mathbf { i } ) , \cr t _ { i , j } & = p _ { \vec { \gamma } } ( i , \delta _ { \vec { \gamma } } ( j ) ) \mathrm { \ f o r \ } i < \mathbf { i } , \ j < k _ * . \end{align*}


\begin{align*} \Delta = \delta ^ * \overline { \delta } , \ \ \ \ \ \zeta _ \alpha = \left ( \frac { \alpha } { \alpha + \Delta } \right ) ^ { 1 / 2 } , \ \ \ \ \ \tilde \delta _ \alpha = \frac { 1 } { \sqrt { \alpha } } \overline \delta \circ \zeta _ \alpha , \ \ \ \ \ \tilde \Delta _ \alpha = \frac { 1 } { \sqrt { \alpha } } \Delta ^ { 1 / 2 } \circ \zeta _ \alpha , \ \ \ \ \ \theta _ \alpha = 1 - \tilde \Delta _ \alpha . \end{align*}


\begin{align*} P _ { \frac { \delta } { 2 } } ( E ) P _ { \frac { \delta } { 2 } } ( F ) = 0 \end{align*}


\begin{align*} \widehat H ( t ) : = \sum _ { \alpha = 1 } ^ r b _ { \alpha } ( t ) \widehat { H } _ { \alpha } , \end{align*}


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


\begin{align*} & \int F ( x ) k ( q x ) \Big ( D _ q u ( x ) + u ( x ) u ( q x ) + \tilde { A } ( x ) u ( q x ) + r ( x ) \Big ) y ( x ) d _ q x = \\ & F ( x ) k ( x ) \Big ( y ( x ) u ( x ) - D _ { q ^ { - 1 } } y ( x ) \Big ) , \end{align*}


\begin{align*} u = U _ 0 + U _ 1 + U _ 2 , \end{align*}


\begin{align*} | g ( z ^ m ) | \leq f ( r ^ m ) \leq \hat { f } _ 0 ( r ^ m ) , f _ 0 ( z ) = \int _ { 0 } ^ { z } \frac { \psi ( t ) } { 1 - t ^ 2 } d t . \end{align*}


\begin{align*} u ^ s _ { A , q } ( x , y ) & = \underset { ( \ell _ 1 , m _ 1 ) \in \Gamma } \sum \alpha _ { \ell _ 1 m _ 1 } ( y ) h ^ { ( 1 ) } _ { \ell _ 1 } ( k \vert x \vert ) Y ^ { m _ 1 } _ { \ell _ 1 } \left ( \widehat { x } \right ) . \end{align*}


\begin{align*} \int \dfrac { ( \frac { x } { a } , \frac { x } { b } ; q ) _ \infty } { ( x ; q ) _ \infty } p _ n ( x ; a , b ; q ) d _ q x = \dfrac { a b q ^ 2 ( 1 - q ) } { ( 1 - a q ) ( 1 - b q ) } \dfrac { ( \frac { x } { a q } , \frac { x } { b q } ; q ) _ \infty } { ( x ; q ) _ \infty } p _ { n - 1 } ( x ; a q , b q ; q ) . \end{align*}


\begin{align*} d ( \mu \boxplus \nu ) _ { } ( h ( x ) ) = \pi ^ { - 1 } \left [ I _ { V } ( x ) + I _ { B \cup C } ( x ) \right ] \left ( - \Im G _ { \mu \boxplus \nu } \right ) ^ { * } ( h ( x ) ) \ , d \lambda ( h ( x ) ) . \end{align*}


\begin{align*} \mathcal { R } = - \int _ B & e ^ { - i \varphi } \big [ i ( A + \nabla \vartheta ) \cdot ( u _ 1 \nabla u _ 2 - u _ 2 \nabla u _ 1 ) \\ & + \Big ( ( A + \nabla \vartheta ) \cdot ( A _ 1 + A _ 2 ) - q + ( A + \nabla \vartheta ) \cdot \nabla \varphi \Big ) u _ 1 u _ 2 \big ] \ , d x . \end{align*}


\begin{align*} B _ 1 ( T ) & = A ( T ) + \mathrm { d i a g } ( \beta ( u _ 1 ) , \beta ( u _ 2 ) , \ldots , \beta ( u _ { n ( T ) } ) ) , \\ B _ 2 ( T ) & = - A ( T ) + \mathrm { d i a g } ( \beta ( u _ 1 ) , \beta ( u _ 2 ) , \ldots , \beta ( u _ { n ( T ) } ) ) . \end{align*}


\begin{align*} \frac { 1 } { \lambda _ N ^ 2 } \prod \limits _ { k = 1 } ^ { \infty } \Bigg ( \frac { \lambda _ N ^ 2 + \mu _ { k } ^ { 2 } } { \lambda _ N ^ 2 + \lambda _ { k } ^ { 2 } } \Bigg ) = \sum \limits _ { k = 1 } ^ { \infty } \frac { a _ k } { \lambda _ k ^ 2 ( \lambda _ N ^ 2 + \lambda _ { k } ^ { 2 } ) } . \end{align*}


\begin{align*} 0 = ( L z , z ) _ V = - \sum _ { e \in E } \frac { 1 } { 2 } \frac { \delta _ E ( e ) } { | e | ^ 2 } \sum _ { u , v \in e } ( z ( u ) - z ( v ) ) ^ 2 . \end{align*}


\begin{align*} \eta ( e ^ { 2 \pi i t } ) = [ \lambda _ 1 ( t ) , \lambda _ 2 ( t ) , \ldots , \lambda _ a ( t ) ] \end{align*}


\begin{align*} f _ i ( x ) = f \circ S ^ { ( 1 - i ) m } ( x ) , i \in \Z . \end{align*}


\begin{align*} \chi _ \alpha ( x _ n ) = \begin{cases} & 0 | x _ n | \geq 2 \alpha \\ & 1 | x _ n | \leq \alpha / 2 , \end{cases} \end{align*}


\begin{align*} m = \sum _ { j \ge 0 } m \chi _ { U _ j ( B ) } = \sum _ { j \ge 0 } m _ j . \end{align*}


\begin{align*} p _ { \kappa } ( x ) = N _ { \kappa } \ , \ , x ^ { n - 1 } \exp _ { \kappa } ( - x ) \ \ , \end{align*}


\begin{align*} \frac { d ^ k } { d z ^ k } F ( z ) = \frac { d ^ k } { d z ^ k } F _ { \mathrm { m } } ( z ) + O ( 1 ) , | z | \geq r _ 0 , ~ k \geq 0 . \end{align*}


\begin{align*} ( \alpha , \beta ) _ { \rho } = ( \alpha _ n , \beta _ n ) _ { \rho , E _ { \rho } ^ n , n } \end{align*}


\begin{align*} & \overline { \Gamma } ^ \alpha _ { 0 0 } = 0 , \ \overline { \Gamma } ^ 0 _ { 0 i } = 0 , \ \overline { \Gamma } ^ k _ { i j } = \widetilde { \Gamma } ^ k _ { i j } , \\ & \overline { \Gamma } ^ 0 _ { i j } = f ( t ) f ' ( t ) \widetilde { g } _ { i j } , \ \overline { \Gamma } ^ k _ { 0 i } = \frac { f ( t ) f ' ( t ) } { f ( t ) ^ 2 } \delta ^ k _ i . \end{align*}


\begin{align*} D _ + = \sum _ { i \in ( M _ + \cup M _ - ) } \bar { D } _ i \subset X _ + D _ - = \sum _ { i \in ( M _ + \cup M _ - ) } \bar { D } _ i \subset X _ - . \end{align*}


\begin{align*} e ^ { \max } _ { } = \max _ { j = 1 , \dots , N } E ^ 2 _ { } ( t _ j ) . \end{align*}


\begin{align*} \cos \theta : = \frac { ( v _ * ' - v ' ) \cdot ( v _ * - v ) } { | v _ * - v | ^ 2 } = \frac { v - v _ * } { | v - v _ * | } \cdot \sigma . \end{align*}


\begin{align*} \underline { \gamma } & = n \chi ( \epsilon _ n ) + O ( 1 ) \\ \overline { \gamma } & = n \chi ( \epsilon _ n ) + O ( 1 ) . \end{align*}


\begin{align*} E _ * = E _ * \otimes \mathcal { O } _ X \xhookrightarrow { } E _ * \otimes ( E _ * \otimes E ^ \vee _ * ) = ( E _ * \otimes E _ * ) \otimes E ^ \vee _ * \xrightarrow { \psi \otimes I d } L _ * \otimes E ^ \vee _ * . \end{align*}


\begin{align*} c _ d : = \frac { 1 } { Z _ N } e ^ { - d ^ 2 / 4 \sigma _ N ^ 2 } , \qquad | d | \le \sigma _ N ^ 2 . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \Phi ( \theta + t _ n h _ n ) - \Phi ( \theta ) } { t _ n } = \Phi ' _ { \theta } ( h ) \end{align*}


\begin{align*} t _ j = T ( j / N ) ^ k \end{align*}


\begin{align*} d ^ { \tt s t } = d ^ { C ^ { \tt s t } } _ { A ^ { \tt s t } } , \end{align*}


\begin{align*} C _ { \gamma _ 1 } \cap C _ { \gamma _ 2 } = \{ \gamma _ 1 \} \times C _ J = \{ \gamma _ 2 \} \times C _ J , \end{align*}


\begin{align*} ( a , x ) \ast ( b , y ) = ( a + b + h _ s ( x , y ) , \ x y ) . \end{align*}


\begin{align*} \Hat { r } ( X ) \\ & = \min _ { A \subseteq X } \bigl ( | X \backslash A | + \sum _ { i = 1 } ^ n r _ i ^ * ( A ) \bigr ) \\ & = \min _ { A \subseteq X } \bigl ( | X \backslash A | + \sum _ { i = 1 } ^ n ( | A | + r _ i ( E \backslash A ) - r _ i ( E ) \bigr ) \\ & = \min _ { A \subseteq X } \biggl ( ( n - 1 ) | A | + | X | + \sum _ { i = 1 } ^ n \bigl ( r _ i ( E \backslash A ) - r _ i ( E ) \bigr ) \biggr ) \end{align*}


\begin{align*} \varphi ( f ) ( t ) = u ( t ) \begin{pmatrix} f ( \lambda _ 1 ( t ) ) & 0 & \hdots & 0 \\ 0 & f ( \lambda _ 2 ( t ) ) & \hdots & 0 \\ \hdots & \hdots & \hdots & \hdots \\ 0 & 0 & \hdots & f ( \lambda _ a ( t ) ) \end{pmatrix} u ( t ) ^ { \ast } . \end{align*}


\begin{align*} \varphi _ { 0 } ^ { + } = A . \varphi _ { 0 } ^ { - } , \end{align*}


\begin{align*} M ^ { n , 2 } _ s = \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u , \end{align*}


\begin{align*} G ( { \bf s } _ { \alpha , \delta } ) = { \bf s } _ { \frac { \alpha } { 1 + \alpha } , \frac { \delta } { 1 + \alpha } } , \widetilde { G } ( { \bf s } _ { \alpha , \delta } ) = { \bf s } _ { \frac { \alpha } { 1 + \alpha } , \frac { \alpha + \delta } { 1 + \alpha } } E ( { \bf s } _ { \alpha , \delta } ) = { \bf s ' } _ { 1 - \alpha , 1 - \delta } . \end{align*}


\begin{align*} k ( t ) = C \mathrm { e } ^ { a t } t ^ { - d / 2 } , \end{align*}


\begin{align*} | y _ n ^ * ( x _ j ^ k ) | = a _ j | f _ { p ( j ) } ( y _ n ^ * ) | \leq C \ : a _ j . \end{align*}


\begin{align*} x \eta = x \xi ' \otimes \Omega _ \phi = \delta ( x ) \otimes \Omega _ \phi + \xi ' x \otimes \Omega _ \phi = \delta ( x ) \otimes \Omega _ \phi + \eta \sigma ^ \phi _ { i / 2 } ( x ) \end{align*}


\begin{align*} \mathbf { P } ( F _ k ^ c \cap C _ k \cap G _ k \cap B _ k \setminus \cup _ { i = k _ 1 } ^ k A _ i ) \leq \mathbf { P } ( F _ k ^ c \cap C _ k ) \leq \sqrt { e ^ { - a ( k ) + a ( k - 1 ) + 1 } } \end{align*}


\begin{align*} \phi ( \beta ( g ^ { - 1 } , \psi ^ { - 1 } ( \alpha ( g , x ) , v ) ) ) = ( x , A ( g , x , v ) ) \mbox { f o r a l l $ g \in U $ , $ x \in V $ , a n d $ v \in F $ , } \end{align*}


\begin{align*} y ( t ) & = \int _ 0 ^ 1 w ( s ) \partial _ s x ( t , s ) d s \end{align*}


\begin{align*} H ^ { * } \left ( \overline { e } _ { k } \right ) & = \left [ H \overline { e } _ { k - 1 } , H ^ { 2 } \overline { e } _ { k - 1 } , \cdots , H ^ { n } \overline { e } _ { k - 1 } \right ] \\ & = H \left [ \overline { e } _ { k - 1 } , H \overline { e } _ { k - 1 } , \cdots , H ^ { n - 1 } \overline { e } _ { k - 1 } \right ] \\ & = H H ^ { * } ( \overline { e } _ { k - 1 } ) = H H ^ { k - 2 } = H ^ { k - 1 } . \end{align*}


\begin{align*} \psi ( p , t ) = \psi _ p ( t ) = \psi _ p ( 0 ) - \int _ 0 ^ t c ( s ) \ , d s . \end{align*}


\begin{align*} B ( a , b ) = \sup _ { h \in \mathcal { M } ^ + ( a , b ) } \frac { \bigg ( \int _ a ^ b \bigg ( \int _ a ^ t h ^ r v \bigg ) ^ { \frac { q } { r } } u ( t ) d t \bigg ) ^ { \frac { 1 } { q } } } { \int _ a ^ b h } . \end{align*}


\begin{align*} I _ { D } ( y ) = \lim _ { \tilde y _ { r + 1 } \rightarrow 0 } L \circ \bar I _ { \tilde D } ( \tilde y ) , \end{align*}


\begin{align*} B ^ * _ P ( v , w ) \vcentcolon = \langle ( - \Delta ) ^ { s / 2 } v , ( - \Delta ) ^ { s / 2 } w \rangle + \sum _ { | \alpha | \leq m } \langle a _ \alpha , v ( D ^ \alpha w ) \rangle , \end{align*}


\begin{align*} { \bf U } ^ k ( \alpha _ i ) = [ \tilde u _ { k , 1 } ^ { \alpha _ i } , \tilde u _ { k , 2 } ^ { \alpha _ i } , \ldots , \tilde u _ { k , N _ h } ^ { \alpha _ i } ] ^ T \in \mathbb { R } ^ { N _ h } k = 1 , \ldots , m , \end{align*}


\begin{align*} \begin{aligned} f ( x ) & = - \tfrac { 1 } { N } \sum _ { i = 1 } ^ N y _ i \log \big ( \sigma ( A _ i x ) \big ) + ( 1 - y _ i ) \log \big ( 1 - \sigma ( A _ i x ) \big ) + \beta \sum _ { i = 1 } ^ N e _ { \lambda , \sigma } l _ 0 ( [ x ] _ i ) \\ \nabla f ( x ) & = \tfrac { 1 } { N } A ^ T \big ( \sigma ( A _ i x ) - y \big ) + \beta \nabla e _ { \lambda , \sigma } l _ 0 ( x ) \end{aligned} \end{align*}


\begin{align*} \begin{cases} u _ t = \Delta u + | u | ^ { p - 1 } u - | u | ^ { q - 1 } u & \R ^ n \times ( 0 , T ) , \\ u | _ { t = 0 } = u _ 0 ( x ) & \R ^ n . \end{cases} \end{align*}


\begin{align*} ( B ( \Lambda _ i - 1 ) ) _ { k j } - ( B ( \Lambda _ j - 1 ) ) _ { k i } = \pi L _ { i j k } ( \Lambda ) , \end{align*}


\begin{align*} f ( w , x , y , z ) = - f ( z , w , x , y ) = f ( y , z , w , x ) , \end{align*}