\begin{align*} h _ { K \oplus _ 2 L } ( x ) = \sqrt { h _ K ( x ) ^ 2 + h _ L ( x ) ^ 2 } \ge \sqrt { \rho _ K ( u ) ^ 2 + \rho _ L ( u ) ^ 2 } | \langle x , u \rangle | = \sqrt { \rho _ K ( u ) ^ 2 + \rho _ L ( u ) ^ 2 } h _ { [ - u , u ] } ( x ) . \end{align*}


\begin{align*} \lambda _ { 1 2 } + \lambda _ { 1 2 3 ' } & = \log \left ( \frac { 8 1 } { \zeta } \frac { \zeta } { 3 6 } \right ) = \log \frac { 9 } { 4 } \end{align*}


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


\begin{align*} Z _ n = M _ n + r _ n N _ n , \end{align*}


\begin{align*} h _ t ( n , k ) \le \frac { n } { n - k } \binom { n } { k } ^ { - 1 } \binom { n - 1 } { k } h _ t ( n - 1 , k ) = h _ t ( n - 1 , k ) \enspace . \end{align*}


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


\begin{align*} S _ 2 : 2 + 3 X ^ { p - 1 } Y = 0 \end{align*}


\begin{align*} a _ { ( n ) } \cdot v = 0 , n \geq N , \end{align*}


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


\begin{align*} \sum _ { l \neq j } \ln | \sin \pi ( \theta _ j - \theta _ l ) | = \sum _ { l \in I _ 1 , l \neq j } \ln { | \sin \pi ( j - l ) \alpha | } + \sum _ { l \in I _ 2 } \ln { | \sin \pi ( j - l ) \alpha | } \triangleq \sum _ { 1 } + \sum _ { 2 } . \end{align*}


\begin{align*} Z ^ i ( v ^ \mathrm { a p p } ( s ) , s ) - Z ^ i ( v ( s ) , s ) = \mathcal O ( s ^ { \min ( \frac 3 2 , \frac { i + 1 } 2 ) } ) . \end{align*}


\begin{align*} [ X + \xi , Y + \eta ] _ H = [ X , Y ] - \mathrm { a d } _ X ^ * \eta - \iota _ Y d \xi + H ( X , Y , \cdot ) \in \mathfrak { g } \oplus \mathfrak { g } ^ * , \end{align*}


\begin{align*} \tilde { \lambda } ^ { k } = \lambda ^ k - \beta ( \tilde { x } ^ { k } - \tilde { y } ^ { k } ) . \end{align*}


\begin{align*} \mu _ { ( \boldsymbol { x } , \boldsymbol { \phi } ( t , \boldsymbol { x } ) ) } ^ { N , i } + \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } h \Big ( \mu _ { ( \boldsymbol { x } , \boldsymbol { \phi } ( t , \boldsymbol { x } ) ) } ^ { N , j } \Big ) = - R ^ { - 1 } B \Big [ P _ t \nu ^ { N , i } _ { \boldsymbol { x } } + \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } \phi ^ { N , j } ( t , \boldsymbol { x } ) \Big ] . \end{align*}


\begin{align*} & ( i ) \ \ \ \nabla ( J + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ) ( x ^ * ) = 0 \\ & ( i i ) \ \ \gamma _ k f _ k ( x ^ * ) = 0 \ \ k \in \{ 1 , . . . , m \} \\ & ( i i i ) \ A _ J + \sum \limits _ { k = 1 } ^ m \gamma _ k A _ k \succeq 0 \end{align*}


\begin{align*} A ( \theta , E ) = \left ( \begin{matrix} E - \lambda \tan { \pi \theta } \ & - 1 \\ 1 & 0 \end{matrix} \right ) . \end{align*}


\begin{align*} \Phi _ { \rho } ( w ) : = \int _ { 1 } ^ { w } \frac { 1 } { \rho ( v ) } \ , d v \end{align*}


\begin{align*} \bar q _ t ^ n ( x , y ) : = \int _ 0 ^ t \langle \Sigma _ s \delta _ { x _ { j _ 1 } } , \delta _ { x _ { j _ 2 } } \rangle d s . \end{align*}


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


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


\begin{align*} g ( N ( X , Y ) , Z ) + g ( N ( Y , Z ) , X ) + g ( N ( Z , X ) , Y ) = 0 , \end{align*}


\begin{align*} S _ \sigma D = \sigma ( D ) , \end{align*}


\begin{align*} \log \det ( u _ { , i j } ) = - v _ j x ^ j + u _ { , i } \xi ^ i + c , \xi \in \R ^ n , v \in ( \R ^ n ) ^ * . \end{align*}


\begin{align*} \int _ { \C ^ n } | g ( z ) | ^ 2 d z = ( 2 \pi ) ^ { - 2 n } \ , | \lambda | ^ { 2 n } \ , \sum _ { k = 0 } ^ \infty \int _ { \C ^ n } | g \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ( z ) | ^ 2 d z . \end{align*}


\begin{align*} \alpha _ x = \mu _ x , \end{align*}


\begin{align*} Y _ t = \mathcal S ( t ) Y _ 0 + \int _ 0 ^ t \mathcal S ( t - s ) \alpha _ s d s + \int _ 0 ^ t \mathcal S ( t - s ) \sigma _ s d W _ s , t \in [ 0 , T ] . \end{align*}


\begin{align*} \sum _ { u = 1 } ^ t d _ u = 2 j , \end{align*}


\begin{align*} \Sigma _ 0 = \sum \limits _ { n \leq X } \sum \limits _ { D \leq d < 2 D \atop { n ^ 2 + n + 1 \equiv 0 \ , ( d ^ 2 ) } } 1 \ , , \end{align*}


\begin{align*} h ( x ) & = \frac { \left [ \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \alpha - 1 } \ , \left [ 1 - \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \beta - 1 } \ , \phi \left ( \frac { x - \mu } { \sigma } \right ) } { \sigma \ , \mathrm { B } ( \alpha , \beta ) \ , \left [ 1 - I _ { \Phi ( \frac { x - \mu } { \sigma } ) } ( \alpha , \beta ) \right ] } , \end{align*}


\begin{align*} g \circ L _ f = f \bullet _ \delta g \ \mbox { a n d } \ g \circ R _ f = g \bullet _ \delta f . \end{align*}


\begin{align*} T _ { \xi } ( t ) = \exp \left ( \ - 0 . 5 \ \gamma ^ { - 1 } \ ( \ln ^ 2 t ) \ \right ) , \ t \ge e , \end{align*}


\begin{align*} \tilde { \sigma } ( u _ 1 , \dots , u _ n ) : = ( \tilde { \sigma } _ 1 ( u _ 1 , u _ 2 ) , \dots , \tilde { \sigma } _ { n - 1 } ( u _ { n - 1 } , u _ n ) ) . \end{align*}


\begin{align*} \| P \| _ K = \max _ { x \in K } \sum _ { j = 1 } ^ { n + 1 } | \lambda _ j ( x ) | = \max _ { x \in K } \sum _ { j = 1 } ^ { n + 1 } \frac { | \Delta _ j ( x ) | } { | \Delta | } . \end{align*}


\begin{align*} m = \left \lfloor { \frac { { i + { N _ { { \rm { C P } } } } } } { { K + { N _ { { \rm { C P } } } } } } } \right \rfloor , \end{align*}


\begin{align*} [ v _ a , v _ b ] = \alpha _ c \varepsilon _ c v _ c , \forall \quad \mbox { c y c l i c } \quad ( a , b , c ) \in \mathfrak { S } _ 3 , H = h \mathrm { v o l } _ g , \end{align*}


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


\begin{align*} f ( x ) & = \sum _ { i = 1 } ^ n V ( x _ i ) + \sum _ { 1 \le i < j \le n } J _ { i j } K ( x _ i - x _ j ) , \end{align*}


\begin{align*} \prescript J { } \ell ( w ) : = \# \{ \beta \in \Phi ^ + _ J \mid w ^ { - 1 } \beta \in \Phi ^ - \} . \end{align*}


\begin{align*} \theta _ n ( f ) ( \gamma _ 1 , \dots , \gamma _ n ) = f ( \rho ( \gamma _ 1 ) , \dots , \rho ( \gamma _ n ) ) . \end{align*}


\begin{align*} 2 H = \eta z ^ m + \lambda . \end{align*}


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


\begin{align*} \kappa = \frac { \alpha } { f \sqrt { 1 + { f ' } ^ 2 } } + \varpi \end{align*}


\begin{align*} A '' _ n ( j ) = A _ n ( j ) A ' _ n ( j ) \end{align*}


\begin{align*} B _ { G ' , \infty } = ( B _ { G ' , n } \widehat { \otimes } _ { A _ n } A ) ^ { \wedge - u } , \end{align*}


\begin{align*} b _ n - b _ { 1 , n } - b _ { 2 , n } = \nabla \times ( \phi _ n ( \chi _ { R _ n / 2 } - \chi _ { R _ 0 } ) \nabla \theta ) + G _ n ( \chi _ { R _ n / 2 } - \chi _ { R _ 0 } ) \nabla \theta , \end{align*}


\begin{align*} \left \Vert \varphi \right \Vert _ { p , \beta } = \max _ { 0 \leq q \leq p } | \sup _ { x \in \mathbb { R } } \left ( e ^ { p \left \vert x \right \vert ^ { \beta } } \varphi ^ { ( q ) } \left ( x \right ) \right ) | , \ ; \ ; \ ; \beta > 1 \end{align*}


\begin{align*} \begin{aligned} \theta _ { i , m } ^ \star = & \begin{cases} \angle \chi _ { i , m } + \pi , & \angle \chi _ { i , m } \in [ 0 , \pi ) , \\ \angle \chi _ { i , m } - \pi , & \angle \chi _ { i , m } \in [ \pi , 2 \pi ) , \end{cases} \\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \forall i \in \{ \mathrm { t , r } \} , \forall m \in \mathcal { M } . \end{aligned} \end{align*}


\begin{align*} \left ( d L _ { \left ( x , p \right ) } \right ) _ { \left ( 0 , 1 \right ) } \left ( \xi , \eta \right ) = \left . \frac { d } { d t } \right \vert _ { 0 } \left ( x + p t \xi \bar { p } , p e ^ { t \eta } \right ) = \left ( p \xi \bar { p } , p \eta \right ) \end{align*}


\begin{align*} d _ I ( \C ) = \min _ { \mathbf { a } , \mathbf { b } \in \C , \mathbf { a } \neq \mathbf { b } } \left \{ \ , \ , d _ I ( \mathbf { a } , \mathbf { b } ) \ , \ , \right \} . \end{align*}


\begin{align*} B _ { r } = \{ g : \rho ( g ) < r \} . \end{align*}


\begin{align*} \mathcal { M _ { \mathbb { N } } } & = \{ \omega = \frac { \sum _ { h = 1 } ^ { H } c _ h \delta _ { a _ h } } { \sum _ { h = 1 } ^ { H } c _ h } : \ ; c _ h \in \mathbb { N } ^ + \} , \\ \mathcal { M _ { \mathbb { Q } } } & = \{ \omega = \frac { \sum _ { h = 1 } ^ { H } c _ h \delta _ { a _ h } } { \sum _ { h = 1 } ^ { H } c _ h } : \ ; c _ h \in \mathbb { Q } ^ + \} , \\ \mathcal { M _ { \mathbb { R } } } & = \{ \omega = \frac { \sum _ { h = 1 } ^ { H } c _ h \delta _ { a _ h } } { \sum _ { h = 1 } ^ { H } c _ h } : \ ; c _ h \in \mathbb { R } ^ + \} , \end{align*}


\begin{align*} v : = - d + \frac { 1 } { 8 b _ { n - 1 } } d ^ 2 . \end{align*}


\begin{align*} F _ { q } ( s ) = \int _ { 0 } ^ { \infty } \exp _ { q } ( - s t ) f ( t ) d t , \end{align*}


\begin{align*} \mathbf { g } = \pm u ^ 2 d t ^ 2 + g . \end{align*}


\begin{align*} | \Gamma _ { \ell , m , n } | = ( q - 1 ) ^ 3 q ^ { 3 \ell + m - n + 6 } \end{align*}


\begin{align*} \mathcal W ( E ) = \bigcap _ { p \ge 1 } \mathcal L _ 1 ^ p ( E ) \cap \{ \nabla : \mathcal { Y M } _ e ^ 0 ( \nabla ) < \infty \} \ , . \end{align*}


\begin{align*} \beta ^ m _ k \stackrel { \rm d e f } { = } \max \left \{ \| g _ k \| , - \lambda _ { \rm m i n } ( H _ k ) \right \} . \end{align*}


\begin{align*} \theta \mapsto P _ \theta ( A ) = \sum _ { i = 1 } ^ k w _ i F _ { v _ i } ( A ) \end{align*}


\begin{align*} \kappa ^ { n } _ m ( t ) : = \sum _ { i = m } ^ { 2 m } i \psi ^ { n } _ i + 2 m \sum _ { i = 2 m + 1 } ^ { n } \psi ^ { n } _ i , \end{align*}


\begin{align*} G = \{ ( \omega , x ) \in \Omega \times R ^ n \mid x \in C ( \omega ) , ~ \langle \mathcal { F } ( x ^ * ) ( \omega ) , x - x ^ * ( \omega ) \rangle < 0 \} . \end{align*}


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


\begin{align*} \bar { w } _ 0 ( 0 ) = \int _ Y w ^ 0 d y . \end{align*}


\begin{align*} B ( x , y ) = \begin{pmatrix} x _ 1 y _ 3 \\ - x _ 2 y _ 3 \\ ( x _ 2 - x _ 1 ) ( y _ 2 + y _ 1 ) \end{pmatrix} . \end{align*}


\begin{align*} u _ { n n } \sigma _ 1 ( b ) - \sum _ { \alpha \leq n - 1 } u _ { n \alpha } ^ 2 + \sigma _ 2 ( b ) = f . \end{align*}


\begin{align*} C \log ( 2 N + 1 ) \leq \Vert M _ { H _ 0 } : S _ { 1 } ^ { 2 N + 1 } \rightarrow S _ { 1 } ^ { 2 N + 1 } \Vert . \end{align*}


\begin{align*} { \bf g } | _ { \partial \bf N } & = \pm u ^ 2 d t ^ 2 + g ^ \intercal \\ A _ { \bf g } & = \pm u \nu ( u ) d t ^ 2 + A _ g . \end{align*}


\begin{align*} \omega _ { 1 1 } = \partial _ { z _ 1 } \bar \partial _ { z _ 1 } \left ( \frac { i } { z _ 1 - \bar z _ 1 } + \vert z _ 2 \vert ^ 2 \right ) = \frac { 1 } { 4 } \left ( \frac { \partial } { \partial x _ 1 } - i \frac { \partial } { \partial y _ 1 } \right ) \left ( \frac { \partial } { \partial x _ 1 } + i \frac { \partial } { \partial y _ 1 } \right ) \frac { 1 } { 2 y _ 1 } = \frac { 1 } { 4 } \frac { 1 } { y _ 1 ^ 3 } \end{align*}


\begin{align*} \langle T ^ { ( \lambda ) } _ a f , g \rangle _ \lambda = \langle a f , g \rangle _ \lambda , \end{align*}


\begin{align*} ( \mathcal { F } _ N ) = \begin{pmatrix} N + 2 \\ 2 \end{pmatrix} \ , . \end{align*}


\begin{align*} g ( u _ 1 ) = \begin{cases} e ^ { - \frac { 1 } { u _ 1 } } & \mbox { i f } u _ 1 > 0 ; \\ 0 & \mbox { i f } u _ 1 \leq 0 , \end{cases} \end{align*}


\begin{gather*} \inf _ { Q \in \Gamma } Q ( c ) = \sup _ { f _ 1 , \ldots , f _ n } \ , \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \end{gather*}


\begin{align*} \deg T _ { \varphi _ A ^ * ( f ) } = \deg T _ f . \end{align*}


\begin{align*} \abs { \hat { c } _ \Delta ( \eta , \xi _ \Delta ) } = \abs { c _ \Delta ( \xi _ \Delta \eta _ { \Delta ^ c } , \eta _ \Delta ) \frac { \gamma _ \Delta ( \xi _ { \Delta } | \eta _ { \Delta ^ c } ) } { \gamma _ \Delta ( \eta _ \Delta | \eta _ { \Delta ^ c } ) } } \leq \frac { 1 } { \delta } e ^ { \abs { \Delta } } \norm { c _ \Delta ( \cdot , \eta _ { \Delta } ) } _ \infty \leq \frac { 1 } { \delta } e ^ R \norm { c _ \Delta ( \cdot , \eta _ \Delta ) } _ \infty . \end{align*}


\begin{align*} \begin{aligned} \dot { \zeta } = & \ , A \zeta + B z \\ x = & \ , C \zeta \end{aligned} \end{align*}


\begin{align*} \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi _ j ( \partial _ t \varphi + ( v _ j + u _ { \infty } ) \cdot \nabla \varphi ) \dd x \dd s + \int _ { \mathbb { R } ^ { 3 } } \Phi _ 0 \varphi _ 0 \dd x = - \mu _ j \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \left ( \Phi _ j \Delta \varphi - \frac { 2 } { r } \partial _ r \Phi _ j \varphi \right ) \dd x \dd s . \end{align*}


\begin{align*} v ^ { - 1 } \alpha = \beta _ 1 > \cdots > \beta _ \ell \in \Phi ^ + \end{align*}


\begin{align*} \dot { s } \varepsilon ( z ) \sum _ { x \in \bar { \mathbb { F } } _ q } a _ x \varepsilon ( x ) \xi _ 1 = \sum _ { x \in \bar { \mathbb { F } } _ q } a _ x ( \varepsilon ( x ( z ^ 2 - z ) ^ { - 1 } - z ^ { - 1 } ) + \varepsilon ( x z ^ { - 1 } ) - \varepsilon ( x ( z - 1 ) ^ { - 1 } ) ) \xi _ 1 \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 4 D ^ g \theta ( J e _ i , \left ( D ^ g _ { e _ i } J \right ) X ) & = \frac { 1 } { 2 } g ( D ^ g _ { J X } \theta , J \theta ) + \frac { 1 } { 2 } g ( D ^ g _ X \theta , \theta ) + g ( d \theta , N _ X ) . \end{align*}


\begin{align*} x _ 2 v _ 1 - x _ 1 v _ 2 + k v _ 3 = 0 . \end{align*}


\begin{align*} N \mapsto \mu _ N = \frac { 1 } { | \Phi _ N | } \sum _ { n \in \Phi _ N } \delta _ { T ^ n a } \end{align*}


\begin{align*} & a ( k ) = e ^ { - i c } + \mathcal { O } \left ( k ^ { - 1 } \right ) , \ \ b ( k ) = \mathcal { O } \left ( k ^ { - 1 } \right ) , k \rightarrow \infty , \\ & a ( k ) = e ^ { - i c } \left ( 1 + \mathcal { O } \left ( k ^ 2 \right ) \right ) , \ \ b ( k ) = \mathcal { O } \left ( k ^ 3 \right ) , k \rightarrow 0 , \end{align*}


\begin{align*} \Delta v + \big ( \Delta ' ( h ) \big ) \bar u = 0 \mbox { i n } M \setminus \Omega . \end{align*}


\begin{align*} S t _ { D N D C } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( 1 , z ) - 1 | | _ { \infty } = 0 , \end{align*}


\begin{align*} x l _ 1 + y k _ 1 = 1 \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } \mu \xi ( \mu , t ) \ d \mu = \int _ { 0 } ^ { \infty } \mu \xi ^ { \mathrm { i n } } ( \mu ) \ d \mu = \varrho _ { 0 } , \forall ~ t \geq 0 . \end{align*}


\begin{align*} \gamma _ { w , k , m } \le \epsilon = 1 / 2 ^ b ~ { \rm f o r } ~ k \ge ( b + 2 ) \log _ { \Delta _ { v , m } } ( 2 ) ~ { \rm i f } ~ 2 \Delta _ { v , m } ^ k \le 1 . \end{align*}


\begin{align*} I : = \{ \beta \in \Phi ^ + \setminus \{ \alpha \} \mid s _ \alpha ( \beta ) \in \Phi ^ - \} . \end{align*}


\begin{align*} \phi _ 4 ' = - \frac { 1 } { \epsilon } [ c \phi _ 4 - \kappa \phi _ 2 g ( \phi _ 1 ) ] . \end{align*}


\begin{align*} 0 = \varphi ' ( s ) = \beta \left [ ( M s - A ) ^ { 2 } - ( B - M s ) ^ { 2 } \right ] \end{align*}


\begin{align*} \prod _ { j = 1 } ^ m \ell ( D F _ j ( h _ { j - 1 } ( u ) ) ) \leq \ell ( D h _ m ( u ) ) \leq | D h _ m ( u ) | \leq \prod _ { j = 1 } ^ m | D F _ j ( h _ { j - 1 } ( u ) ) | , \end{align*}


\begin{align*} c _ { b } = \mathop { \inf } \limits _ { ( u , v ) \in E \setminus \{ ( 0 , 0 ) \} } \mathop { \max } \limits _ { t > 0 } I ( \gamma _ { u , v } ( t ) ) . \end{align*}


\begin{align*} - \frac { 1 } { \sqrt { 1 + { f ' } ^ 2 } } = \frac { \eta } { m + 1 } f ^ { m + 1 } + \lambda f + c , \end{align*}


\begin{align*} 2 c ( \gamma , x ) = h ( \gamma x , \gamma y , \gamma z ) - h ( x , y , z ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \sup _ { y \in \R ^ N } \int _ { B _ r ( y ) } u _ { \lambda _ n } ^ 2 d x = 0 , \end{align*}


\begin{align*} \overline { W ^ { k , i } } = W ^ { - k , i } , \quad \big [ W ^ { k , i } , W ^ { l , j } \big ] _ { t } = 2 t \delta _ { k , - l } \delta _ { i , j } . \end{align*}


\begin{align*} \lim _ { \alpha \rightarrow 0 ^ + } \left ( \lim _ { n \rightarrow \infty } \sqrt [ n ] { \binom { n + \lceil \alpha n \rceil } { \lceil \alpha n \rceil } } \right ) = \lim _ { \alpha \rightarrow 0 ^ + } \left ( \lim _ { n \rightarrow \infty } \sqrt [ n ] { \binom { n } { \lceil \alpha n \rceil } } \right ) = 1 . \end{align*}


\begin{align*} 0 & = \nabla _ { X _ { k + 1 } } ( \Omega _ { k - 1 } ( X _ 0 , \ldots , X _ k ) \varphi ) = \nabla _ { X _ { k + 1 } } \Omega _ { k - 1 } ( X _ 0 , \ldots , X _ k ) \varphi - \Omega _ { k - 1 } ( X _ 0 , \ldots , X _ k ) \circ \omega ( X _ { k + 1 } ) \varphi \\ & = \Omega _ k ( X _ 0 , \ldots , X _ { k + 1 } ) \varphi , \quad \forall X _ i \in \Gamma ( T L ) , \end{align*}


\begin{align*} C _ 0 = \frac { n ( n - 1 ) } { 2 ( n + 2 ) \omega _ n ^ { \frac 2 n } } \ , . \end{align*}


\begin{align*} g & = t ^ { - \sigma ^ - } \cdot \prod _ { i = \sigma ^ - } ^ { \sigma ^ + - 1 } \big ( w _ i ( a _ 1 , \dots , a _ d ) \ , t ^ { - 1 } \big ) \cdot w _ { \sigma ^ + } ( a _ 1 , \ldots , a _ d ) \cdot t ^ { \sigma ^ + } , \\ g & = t ^ { - \sigma ^ + } \cdot \prod _ { j = \sigma ^ - } ^ { \sigma ^ + - 1 } \big ( w _ { \sigma ^ + + \sigma ^ - - j } ( a _ 1 , \dots , a _ d ) \ , t \big ) \cdot w _ { \sigma ^ - } ( a _ 1 , \ldots , a _ d ) \cdot t ^ { \sigma ^ - } , \end{align*}


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


\begin{align*} d X _ t ^ i & = b ( t , X _ t ^ i , \overline \mu _ t ^ N , u _ t ^ i ) d t + \sigma d W _ t , \end{align*}


\begin{align*} \widetilde { Q } = \frac { 1 } { 2 i } \begin{pmatrix} | u _ x | ^ 2 & u _ x \\ - 2 i \bar { u } _ { x x } - \bar { u } _ x | u _ x | ^ 2 & - | u _ x | ^ 2 \end{pmatrix} . \end{align*}


\begin{align*} \begin{cases} - \delta ^ 2 ( K ( x ) \nabla w ) = ( w - q ) ^ { p } _ + , \ \ & x \in \Omega , \\ w = 0 , \ \ & x \in \partial \Omega . \end{cases} \end{align*}


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


\begin{align*} [ M , v \wedge \partial _ v ] = 2 \sum _ { 1 \leq i \leq 2 } \big ( [ \Lambda _ i , v ] \wedge \partial _ v \big ) \Lambda _ i = 2 \sum _ { 1 \leq i \leq 2 } \Lambda _ i \big ( [ \Lambda _ i , v ] \wedge \partial _ v \big ) , \textrm { w i t h } [ \Lambda _ i , v ] = c _ i \big ( ( t - t _ 0 ) ^ { 1 \over 2 } , 0 , 0 \big ) , \end{align*}


\begin{align*} S \subset \bigcup _ { i = 1 } ^ { C _ { B C } } \bigcup _ { B \in \mathcal { B } ^ { ( i ) } } B . \end{align*}


\begin{align*} a _ 1 x ^ q + a _ 1 x = c _ 1 y ^ q + b _ 1 y . \end{align*}


\begin{align*} \rho ( x ) = ( \rho _ k ( x ) : k \in \N ) \end{align*}


\begin{align*} g ( t ) = \frac { | A \oplus _ 2 \sqrt { t } B | ^ 2 } { | \partial ( A \oplus _ 2 \sqrt { t } B ) | ^ 2 _ { n - 1 } } \end{align*}


\begin{align*} f ( x ) = \left ( \log \left ( 1 + \frac { 1 } { 2 x } \right ) + \frac { 1 } { 4 x } \right ) - \left ( \psi \left ( x + \frac 1 2 \right ) - \psi ( x ) \right ) . \end{align*}


\begin{align*} z : = \zeta ^ { 2 } | \tilde w ^ \varepsilon | ^ 2 - \lambda \tilde w ^ \varepsilon , \end{align*}


\begin{align*} \begin{aligned} \Delta u _ 1 ^ t \wedge \dots \wedge \Delta u _ k ^ t & \wedge \beta _ n ^ { n - m } \mapsto \Delta u _ 1 \wedge \dots \wedge \Delta u _ k \wedge \beta _ n ^ { n - m } , k = 1 \dots , m . \end{aligned} \end{align*}


\begin{align*} d u = \Delta _ p u \ , d t + \sum _ { n = 0 } ^ { + \infty } \xi _ { n } \cdot \nabla u \circ d B _ t ^ { n } , \end{align*}


\begin{align*} K ( c _ 1 , \dots , c _ n , 0 ) = & p _ i \circ H ( c _ 1 , \dots , c _ n , m ) \\ = & p _ i \circ \iota _ { U _ k } ( c 1 , \dots , c _ n ) = p _ i ( c _ 1 , \dots , c _ n ) , \end{align*}


\begin{align*} \gamma ( X _ i , X _ j ) = 0 \quad \forall i \neq j , \end{align*}


\begin{align*} F ^ \eta ( x , y ) = \big \{ l \in \mathbb R ^ n \colon \| l \| \leq c _ H ( 1 + \| x \| + \| y \| ) + \eta \big \} , \ \ x , y \in \mathbb R ^ n , \ \ \eta \in [ 0 , 1 ] . \end{align*}


\begin{align*} \beta _ i : = \beta + i w ^ { - 1 } \gamma \in \Phi , i = 0 , \dotsc , k . \end{align*}


\begin{align*} \operatorname { \mathbb { J } } _ { 1 / 2 } ( \Omega , u ) : = \sum _ { s \in S _ { u } \cap \Omega } | J ( s ) | ^ { 1 / 2 } , \end{align*}


\begin{align*} J _ { l , 0 } \left ( \rho \right ) = \int _ { 0 } ^ { \infty } \left ( q ( s ) I _ { l } ^ { \left ( 1 \right ) } ( s , \rho ) + Q ^ { \prime } ( s ) I _ { l } ^ { \left ( 2 \right ) } ( s , \rho ) \right ) d s , \end{align*}


\begin{align*} \mathcal S ( t + \Delta _ n ) e _ i - \mathcal S ( t ) e _ i = \int _ t ^ { t + \Delta _ n } \mathcal S ( u ) \mathcal A e _ i d u \end{align*}


\begin{align*} \begin{aligned} \phi _ { \mathrm { t } , m } ^ \star = \sqrt { \alpha _ { \mathrm { t } , m } ^ \star } e ^ { \jmath \theta _ { \mathrm { t } , m } ^ \star } , \phi _ { \mathrm { r } , m } ^ \star = \sqrt { 1 - \alpha _ { \mathrm { t } , m } ^ \star } e ^ { \jmath \theta _ { \mathrm { r } , m } ^ \star } , \forall m \in \mathcal { M } . \end{aligned} \end{align*}


\begin{align*} \frac { d } { d x } f ( x ) = & \frac { \theta ^ { 2 } } { B ( a , b ) \ , x ^ 6 } \exp \left ( - \frac { a \theta } { x ^ 2 } \right ) \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } \\ & \times \left [ 4 a - \frac { 6 x ^ 2 } { \theta } + \frac { 4 ( b - 1 ) } { 1 - \exp \left ( \frac { \theta } { x ^ 2 } \right ) } \right ] , x > 0 . \end{align*}


\begin{align*} i ( x ) = d ( x ) \end{align*}


\begin{align*} | x _ t | ^ 2 - | x _ 0 | ^ 2 = 2 \int _ 0 ^ t x _ s \cdot \sigma d W _ s - 2 \int _ 0 ^ t A x _ s \cdot x _ s d s + t \sum _ { i , j = 1 } ^ n | \sigma _ { i j } | ^ 2 . \end{align*}


\begin{align*} \langle \langle Z \cdot \varphi , \varphi ' \rangle \rangle = \langle \langle \varphi , Z \cdot \varphi ' \rangle \rangle . \end{align*}


\begin{align*} \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) = \int \eta ( \lambda | \bar { U } ; x , t ) d \boldsymbol { \nu } ( \lambda ) \end{align*}


\begin{align*} \langle i _ p ( T _ \phi ( x _ 1 , \ldots , x _ n ) ) , i _ { p ' } ( y ^ * ) \rangle _ { p , p ' } = \left \langle \left ( M _ { \widetilde { \phi } } ( i _ { p _ 1 , \alpha } ( x _ 1 ) , \ldots , i _ { p _ n , \alpha } ( x _ n ) ) \right ) _ { \alpha \in I } , i _ { p ' } ( y ^ * ) \right \rangle _ { p , p ' } , \end{align*}


\begin{align*} F _ { q } ( s - s _ { 0 } ) = L _ { q } \bigg [ f ( t ) \exp _ { q } \bigg ( \frac { s _ { 0 } t } { 1 - ( 1 - q ) s t } \bigg ) \bigg ] . \end{align*}


\begin{align*} f ( t ) = \displaystyle \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { \left ( \alpha t \right ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! } = \sin \alpha t . \end{align*}


\begin{align*} \mathcal { J } ( \mu , \nu ; \alpha ) : = \frac { 1 } { 2 } \mathbb { E } \Big \{ \int _ 0 ^ T \Big [ Q \left ( x _ t ^ { \xi , \alpha } + l ( \nu _ t ) \right ) ^ 2 + R \left ( \alpha _ t + h ( \mu _ t ) \right ) ^ 2 \Big ] d t + G \left ( x _ T ^ { \xi , \alpha } + g ( \nu _ T ) \right ) ^ 2 \Big \} , \end{align*}


\begin{align*} 1 - \mathbb { P } _ { Y _ n } [ K _ { \epsilon } ] \leq \sum _ { k = 1 } ^ { \infty } \mathbb P _ { Y _ n } [ ( A _ k ^ { \epsilon } ) ^ c ] \leq \epsilon , \end{align*}


\begin{align*} x _ n : = \prescript { \sigma ^ { 1 - n } } { } { } \left ( \left ( x ^ { \ast , \sigma , n - 1 } \right ) ^ { - 1 } x ^ { \ast , \sigma , n } \right ) , \end{align*}


\begin{align*} \partial _ { i i } f ( x ) & = V '' ( x _ i ) + \sum _ { j \neq i } J _ { i j } K '' ( x _ i - x _ j ) , \partial _ { i j } f ( x ) = - J _ { i j } K '' ( x _ i - x _ j ) . \end{align*}


\begin{align*} F _ j \subseteq \bigcup _ { n = 0 } ^ \infty [ x _ { n , j } , y _ { n , j } ] \end{align*}


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


\begin{align*} \ell _ \eta : = \sup _ { \substack { g , q \in \phi ( Y ) \\ \pi ( g ) = \pi ( q ) } } \eta \left ( \frac { d ( g , \pi ^ { - 1 } ( \bar y ) ) } { d ( q , \pi ^ { - 1 } ( \bar y ) ) } \right ) < \infty , \end{align*}


\begin{align*} \int _ X | | \xi ( x ) | | G \ \mu ( d x ) = \int _ X \beta ( x ) \ \mu ( d x ) , \end{align*}


\begin{align*} \left [ A ( y _ 1 x ) - A ( y _ 1 x ' ) \right ] + \sum _ { n = 2 } ^ \infty \left [ A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) \right ] \end{align*}


\begin{align*} X _ \beta = \{ ( x , y , w , z ) \in X : w - x = \beta \} \end{align*}


\begin{align*} x \in X = p _ X [ \mu \cdot ( \nu \cdot \gamma ) ] \end{align*}


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


\begin{align*} \null _ { j } \eta = \eta _ { j , 1 } \ , \null _ { j , 2 } \eta \quad \quad \null _ { k } \hat { \eta } = \hat { \eta } _ { k , 1 } \ , \null _ { k , 2 } \hat { \eta } \end{align*}


\begin{align*} \Gamma _ { i k } ^ k = \left ( \log \det g ^ { 1 / 2 } \right ) _ { , i } = \frac { 1 } { 2 } ( - v _ i + u _ { , i q } \xi ^ q ) . \end{align*}


\begin{align*} \Lambda _ { g } = A Q - D \end{align*}


\begin{align*} \widetilde { a } _ k ( x ) h _ k '' ( x ) + \widetilde { b } _ k ( x ) h _ k ' ( x ) + ( 2 k - 1 ) \widetilde { c } _ k ( x ) h _ k ( x ) = 0 , \end{align*}


\begin{align*} \partial _ t T ( t , u ) & = \frac { D } { 4 \gamma } \partial _ u ^ 2 T ( t , u ) u \in ( 0 , 1 ) \\ T ( t , 0 ) & = T _ - , \partial _ u T ( t , 1 ) = - \frac { 4 \gamma J } { D } , T ( 0 , u ) = T _ 0 ( u ) , \end{align*}


\begin{align*} \rho _ - \left \| Q _ { j , - } \right \| _ { L ^ 2 } ^ 2 \leq \operatorname { R e } \int _ { \mathbb { R } } Q _ { j , - } ( I + J ) Q _ { j , - } ^ H d z = \operatorname { R e } \int _ { \mathbb { R } } \mathcal { P } ^ - ( D _ j ) Q _ { j , - } ^ H d z \leq \| D _ j \| _ { L ^ 2 } \left \| Q _ { j , + } \right \| _ { L ^ 2 } . \end{align*}


\begin{align*} \varphi ^ { \pm } ( x , t ; k ) = I + k \int _ { \pm \infty } ^ { x } e ^ { - 2 i k ^ 2 ( x - y ) \widehat { \sigma } _ 3 } P _ y ( y ) \varphi ^ { \pm } ( y , t ; k ) d y . \end{align*}


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


\begin{align*} d = 2 n = 4 l + 2 \equiv 2 \mod 3 \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 5 } } \nabla _ y \varphi \cdot \nabla _ y \tilde { \varphi } \dd y = \mu ^ { 2 } \int _ { \mathbb { R } ^ { 5 } } \left ( \varphi - 1 - \frac { \gamma } { r ^ { 2 } } \right ) _ { + } \tilde { \varphi } \dd y , \end{align*}


\begin{align*} a \wedge b : = ( a \cdot b ) a = ( b \cdot a ) b . \end{align*}


\begin{align*} M _ n = n \sup _ { Q \in \P ( \R ) } \left ( \int _ { \R } V \ , d Q + \frac 1 2 \int _ { \R } \int _ { \R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \right ) , \end{align*}


\begin{align*} \nabla _ H \psi ( q _ 0 ) = \nabla _ H \varphi ( q _ 0 ) . \end{align*}


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


\begin{align*} \frac { 1 } { \varepsilon ^ 2 } \int _ { A _ \varepsilon } \left ( u _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } q d x = \frac { 2 c _ \varepsilon } { \ln \frac { 1 } { \varepsilon } } + O \left ( \frac { 1 } { \ln \frac { 1 } { \varepsilon } } \right ) \leq C . \end{align*}


\begin{align*} \begin{array} { l l l } f _ L ( L \circ s ) & = & f ^ M _ L ( \mathcal { M } _ { \mathbb { C } , \mathsf { S } } ( L \circ s ) ) \\ & = & f ^ M _ L ( ( L , R ) \star \mathcal { M } _ { \mathbb { C } , \mathsf { S } } ( s ) ) \\ & = & f ^ M _ L ( L , R ) \circ f ^ M _ L ( \mathcal { M } _ { \mathbb { C } , \mathsf { S } } ( s ) ) \\ & = & f ^ M _ { L } ( L , R ) \circ f _ L ( s ) . \end{array} \end{align*}


\begin{align*} \left ( d J \theta \right ) ^ { ( 2 , 0 ) + ( 0 , 2 ) } = \left ( \mathcal { L } _ { \theta ^ \sharp } J \right ) ^ { a n t i - s y m } . \end{align*}


\begin{align*} \alpha _ w = \sum _ { i = 0 } ^ { n } w _ i \frac { d x _ i } { x _ i } . \end{align*}


\begin{align*} \det ( A + u \otimes v ) = ( 1 + v ^ T A ^ { - 1 } u ) \det ( A ) , \end{align*}


\begin{align*} m : = \inf _ { \partial \Omega } \frac { G ( ( D ^ 2 u ) ' ) } { \widetilde { f } } , \end{align*}


\begin{align*} a ( U Z ) = a ( Z ) , \end{align*}


\begin{align*} X = \{ \sum _ { n = 0 } ^ { \infty } a _ n z ^ n \mid \left ( a _ n / w _ n \right ) _ { n \in \mathbb { N } } \in \ell ^ 2 \} \end{align*}


\begin{align*} \left | \frac { \partial ^ 2 \varphi } { \partial x _ \beta \partial x _ \mu } ( x ' , \rho ( x ' ) ) \right | \leq C \delta \sqrt { \tilde { b } _ \beta \tilde { b } _ \mu } , \ \beta , \mu = 1 , \ldots , n - 1 ; \beta \neq \mu \end{align*}


\begin{align*} \alpha ( g , Z ) = ( \xi + z _ 1 \psi _ 1 ) \alpha ( g , Z ) - \alpha ( g , Z ) \cdot \left ( \sum _ { \begin{smallmatrix} 1 \in I \subseteq [ 1 , n ] \\ | I | > 1 \end{smallmatrix} } ( z _ { I ' } + 1 ) \cdot \delta _ { I ' } \right ) \end{align*}


\begin{align*} \frac { 1 } { m } \sum _ { i = 1 } ^ m U _ i < \left ( \frac { \pi } { 4 } + \frac { 1 9 9 \pi } { 3 2 0 0 k } \right ) \| \mathbf { x } \| . \end{align*}


\begin{align*} d _ { 4 } ( n ) = d _ { e } ( n ) . \end{align*}


\begin{align*} H ( Q _ k \ , | \ , P ) = \frac { 1 } { Q ( B _ k ) } \int _ { B _ k } \log \frac { d Q } { d P } \ , d Q - \log Q ( B _ k ) \end{align*}


\begin{align*} R _ { \max } ^ { \rm { N } } = W { \log _ 2 } \left ( { 1 + \frac { { \left ( { P - { p _ K } \left ( { \overline \varepsilon , W } \right ) } \right ) { { \left | { { { \widetilde h } _ b } } \right | } ^ 2 } } } { { { p _ K } \left ( { \overline \varepsilon , W } \right ) { { \left | { { { \widetilde h } _ b } } \right | } ^ 2 } + W { N _ 0 } } } } \right ) . \end{align*}


\begin{align*} A _ { \mathcal N } ( a ) : = \bigoplus _ { \lambda \in { \mathcal N } } A _ \lambda ( a ) . \end{align*}


\begin{align*} F ( P , \mathcal { U } ) : = \{ ( \pi , \tau ) \in X ( \Gamma _ d ( q ) ) \mid P \in \pi \tau \in \mathcal { U } \} . \end{align*}


\begin{align*} ( \mathbf { u } G _ 2 , \mathbf { u } G _ 3 , \cdots , \mathbf { u } G _ { 2 k } ) = ( \mathbf { v } G _ 2 , \mathbf { v } G _ 3 , \cdots , \mathbf { v } G _ { 2 k } ) = ( 1 , 1 , \cdots , 1 ) , \end{align*}


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


\begin{align*} \bigl ( T \xRightarrow { Q _ 1 ^ 0 } Q _ 1 \circ A \xRightarrow { \beta * 1 _ A } Q _ 2 \circ A \bigr ) = \bigl ( T \xRightarrow { Q _ 2 ^ 0 } Q _ 2 \circ A \bigr ) \end{align*}


\begin{align*} S t _ { D N P } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( z ^ { 2 } , z ) - z ^ { 2 } | | _ { \infty } = 0 . \\ \end{align*}


\begin{gather*} \mathcal { U } = \bigl \{ Q \in \mathbb { P } : Q \ll P ^ * ( \pi _ 1 , \ldots , \pi _ n ) Q \bigr \} . \end{gather*}


\begin{align*} d ( \bar x , y ) & \le d ( \bar x , x ) + d ( x , y ) \le \bar K r _ { m ' } + ( 6 \lambda r _ i + 3 \lambda r _ m ) \le \left ( 6 \lambda + \frac 1 { \bar K } + \frac { 3 \lambda } { \bar K ^ 2 } \right ) r _ i \le 7 \lambda r _ i . \end{align*}


\begin{align*} \int _ \Omega \varphi _ n ( \eta ) \mu ( d \eta ) = 0 . \end{align*}


\begin{align*} \omega _ { i j } V _ i V _ j = \tfrac { 1 } { 2 } h _ { i j } V _ i V _ j . \end{align*}


\begin{align*} r a + 1 = k q \ , , \end{align*}


\begin{align*} U \backslash \partial A = ( U \cap A ) \cup ( U \backslash \bar A ) , \end{align*}


\begin{align*} g ( z ) = ( 2 \pi ) ^ { - n } \ , | \lambda | ^ n \ , \sum _ { k = 0 } ^ \infty g \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ( z ) . \end{align*}


\begin{align*} \begin{aligned} & S _ h \subset T _ h \subset \left \{ b \in L ^ { 2 } _ { \sigma , \textrm { a x i } } ( \mathbb { R } ^ { 3 } ) \ \middle | \ H [ b ] = h \right \} , \\ & I _ h = \inf \left \{ E [ b ] \ \middle | \ b \in T _ h \right \} . \end{aligned} \end{align*}


\begin{align*} \hat { q } _ i = q ( z _ i ) + \frac { \hat { q } _ i } { \ln \frac { R } { \varepsilon } } g _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ i ) - \Sigma _ { j \neq i } \frac { \hat { q } _ j } { \ln \frac { R } { \varepsilon } } \bar { G } _ { z _ j } ( T _ { z _ j } z _ i , T _ { z _ j } z _ j ) , \end{align*}


\begin{align*} \mathbb { B } _ { k , S ; ( n , g ) } ( \tau , z ) = \mathbb { B } _ { k , S [ u ] ; ( n , g ) } ( \tau , ( u ^ t ) ^ { - 1 } z ) , \end{align*}


\begin{align*} \gamma _ 0 J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \ge \gamma _ 0 J ( x ^ * ) \end{align*}


\begin{align*} \begin{aligned} & \omega _ { S ^ 1 } = ( x - i y ) d x + ( y + i x ) d y \\ & = ( \cos \theta - i \sin \theta ) d ( \cos \theta ) + ( \sin \theta + i \cos \theta ) d ( \sin \theta ) \\ & = - \cos \theta \sin \theta d \theta + \sin \theta \cos \theta d \theta + i ( \cos ^ 2 \theta d \theta + \sin ^ 2 \theta d \theta ) = i d \theta . \end{aligned} \end{align*}


\begin{align*} \sum _ { x , y = 0 } ^ n ( \delta _ { x , y } - M _ { x , y } ( m ) ) f _ y ^ \star f _ x = \sum _ { j , j ' = 0 } ^ n \left ( 1 - \Theta _ m ( \mu _ j , \mu _ { j ' } ) \right ) \left | \sum _ { x = 0 } ^ n \psi _ j ( x ) f _ x \psi _ { j ' } ( x ) \right | ^ 2 . \end{align*}


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


\begin{align*} \omega = \eta _ 1 ^ { * } \widetilde \omega ^ 0 \wedge \eta _ 1 ^ { * } \widetilde \omega ^ 1 \wedge \cdots \wedge \eta _ k ^ { * } \widetilde \omega ^ 0 \wedge \eta _ k ^ { * } \widetilde \omega ^ 1 , \end{align*}


\begin{align*} \sigma ( \sqrt { a } ) = - \sqrt { a } , \ & \sigma ( \sqrt { d } ) = \sqrt { d } \\ \tau ( \sqrt { a } ) = \sqrt { a } , \ & \tau ( \sqrt { d } ) = - \sqrt { d } . \end{align*}


\begin{align*} \frac { a ( x _ a ; x _ c ; \beta ) } { a ( x _ a ; x _ b ; \alpha ) a ( x _ a ; x _ d ; { \beta - \alpha } ) } = 1 , \\ \frac { c ( x _ a ; x _ c ; \beta ) } { c ( x _ a ; x _ b ; \alpha ) a ( x _ a ; x _ d ; { \beta - \alpha } ) } = 1 , \end{align*}


\begin{align*} h ( T ) & = x y \log ( x y ) - \sum _ { k = 1 } ^ { r } \left ( \Delta ( i _ k ) + \Delta ( j _ k ) \right ) \\ & \le x y \log ( x y ) - \sum _ { k = 1 } ^ { r } \left ( \Delta ( x ) + \Delta ( y + 1 - k ) \right ) \\ & = h ( T ' ) , \end{align*}


\begin{align*} \left \langle l _ 1 , m _ 1 \right | e ^ { i k \phi } \sin ( n \theta ) \left | l _ 2 , m _ 2 \right \rangle = \end{align*}


\begin{align*} M _ { \pm } ( x ; z ) = I + \mathcal { P } ^ { \pm } \left ( M _ { - } ( x ; \cdot ) R ( x ; \cdot ) \right ) ( z ) , z \in \mathbb { R } , \end{align*}


\begin{align*} \int _ { \Omega } n \xi ( \cdot , t _ { 2 } ) - \int _ { t _ { 1 } } ^ { t _ { 2 } } \int _ { \Omega } n \xi _ { t } + \int _ { t _ { 1 } } ^ { t _ { 2 } } \int _ { \Omega } ( \nabla n - n S ( x , n , c ) \cdot \nabla c ) \cdot \nabla \xi = \int _ { \Omega } n \xi ( \cdot , t _ { 1 } ) , \end{align*}


\begin{align*} \ell \binom { k } { 2 } m ^ { k - 2 } < s \binom { k } { 2 } m ^ { k - 2 } \leq \frac { d m ^ { k - 1 } } { 4 } \leq \frac { m ^ k } { 4 } \end{align*}


\begin{align*} T ( z , t ) = \left ( \begin{matrix} 1 & - \bar z & \frac { - | z | ^ 2 + i t } { 2 } \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix} \right ) \end{align*}


\begin{align*} \nu _ z = - \frac { 1 } { 2 } h _ z - ( \alpha - I \beta ) h _ { \overline { z } } . \end{align*}


\begin{align*} \lim _ { n \to \infty } \lim _ { N \to \infty } \sum _ { m = 1 } ^ { n } \sum _ { i = N } ^ { \infty } \mathbb { E } _ { n , m } [ \langle \chi _ { n , m } , e _ i \rangle ^ 2 ] = 0 \end{align*}


\begin{align*} \hat { k } _ d ( \Lambda ) = \prod _ { \sigma \in \Lambda _ d } \prod _ { v _ { q , i } \in \sigma } x _ { q , i } = \prod _ { q , i } x _ { q , i } ^ { f ( q , i ) } = \prod _ { q , i } x _ { q , i } ^ { e ( q , i ) } \prod _ { \rho \in \Gamma _ { d - 1 } } x _ { m ( \rho ) , 1 } . \end{align*}


\begin{align*} ( x _ 0 f _ { x _ 0 } ( x ) : \cdots : x _ n f _ { x _ n } ( x ) ) = ( \lambda _ 0 : \cdots : \lambda _ n ) . \end{align*}


\begin{align*} \delta _ { r j } = \sum _ { h = 1 } ^ { n } p _ { r j h } \theta ^ { h - 1 } , \end{align*}


\begin{align*} x \Longleftrightarrow y = \frac { x - c } { \rho } , \end{align*}


\begin{align*} | | y ( t ) - y ( s ) | | _ { L ^ 2 } = & \left | \left | \int _ s ^ t ( \partial _ t y ) ( \tau ) d \tau \right | \right | _ { L ^ 2 } \le \left | \int _ s ^ t | | ( \partial _ t y ) ( \tau ) | | _ { L ^ 2 } d \tau \right | \\ \le & | t - s | | | \partial _ t y | | _ { L ^ { \infty } ( 0 , \tau _ 1 ; L ^ 2 ) } \le M _ 1 | t - s | , \end{align*}


\begin{align*} \partial _ v \Gamma _ { u v } ^ v - \Gamma _ { v u } ^ u \Gamma _ { u v } ^ v + g _ { u v } = 0 . \end{align*}


\begin{align*} m = \int _ { \Gamma ( t ) } c ( t ) \ , \mathrm { d } \mathcal { H } ^ d = c ( t ) \mathcal { H } ^ d \big ( \Gamma ( t ) \big ) = \alpha _ d c ( t ) R ( t ) ^ d \phantom { x x } \Leftrightarrow \phantom { x x } c ( t ) = \frac { m } { \alpha _ d } R ( t ) ^ { - d } \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { c _ \varepsilon } { \ln \frac { 1 } { \varepsilon } } = \pi \min \limits _ { \overline { \Omega } } q ^ 2 \sqrt { d e t ( K _ H ) } . \end{align*}


\begin{align*} \begin{aligned} & \| x ( t ) - z ( t ) \| \le \| x ( 0 ) - z ( 0 ) \| + \gamma ^ 2 3 L _ \mathcal { D } M _ \mathcal { D } T t \\ & + \gamma L _ \mathcal { D } \int _ 0 ^ t ( \| x ( 0 ) - z ( 0 ) \| + \gamma ^ 2 3 L _ \mathcal { D } M _ \mathcal { D } T \tau ) \\ & \cdot e ^ { \gamma L _ \mathcal { D } ( t - \tau ) } \ , d \tau \\ & = \| x ( 0 ) - z ( 0 ) \| e ^ { \gamma L _ \mathcal { D } t } + \gamma 3 M _ \mathcal { D } T \left ( e ^ { \gamma L _ \mathcal { D } t } - 1 \right ) \end{aligned} \end{align*}


\begin{align*} \Lambda _ C : = \{ ( x , y ) \in \Lambda : x \in C \} \end{align*}


\begin{align*} 3 \left ( { k \choose 2 } + { l + 1 \choose 2 } + { k + 1 \choose 2 } + { l \choose 2 } \right ) = 3 ( k ^ 2 + l ^ 2 ) \geq ( k + l ) ^ 2 . \end{align*}


\begin{align*} \chi ( s ) : = \beta ^ { - 1 } ( s , 0 ) . \end{align*}


\begin{align*} [ F , G ] ( x ) = \sum _ { { \bf k , n } \in \Gamma _ { r } } [ \widehat { F } _ { \bf k } , \widehat { G } _ { \bf n } ] { \rm e } ^ { { \rm i } \langle { \bf k + n } , x \rangle } . \end{align*}


\begin{align*} \tilde { z } = z + \varepsilon ( x \wedge _ g y ) z + O ( \varepsilon ^ 2 ) . \end{align*}


\begin{align*} T _ 1 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , v , J v ) = 0 \textrm { f o r e v e r y } x _ 1 , x _ 2 , x _ 3 , x _ 4 , v \in V . \end{align*}


\begin{align*} \begin{aligned} \partial _ x ^ 2 \left ( e ^ { i c _ + ( x ) } u _ x ( x ) \right ) & = I _ 3 ' ( x ) + I _ 4 ' ( x ) , \end{aligned} \end{align*}


\begin{align*} \Gamma _ 0 ^ \varepsilon = \big \{ \theta _ { \rho ^ \varepsilon } ( 0 , z ) \ , \big | \ , z \in M \big \} \end{align*}


\begin{align*} { \bf e } = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} , \ \ \ \ { \bf f } = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} , \ \ \ \ { \bf h } = \begin{bmatrix} 1 & 0 \\ 0 & - 1 \end{bmatrix} . \end{align*}


\begin{align*} \int _ \Omega u \Delta v _ 1 \wedge \dots \wedge \Delta v _ { m - 1 } \wedge \beta _ n ^ { n - m } \wedge \Delta \omega = \int _ \Omega \omega \Delta u \wedge \Delta v _ 1 \wedge \dots \wedge \Delta v _ { m - 1 } \wedge \beta _ n ^ { n - m } \end{align*}


\begin{align*} \chi _ j = \chi - \sum _ { \substack { i = 1 , \\ i \neq j } } ^ { n } \chi _ i + ( n - 1 ) r . \end{align*}


\begin{align*} | \{ j \le 2 : j \not = i , m _ j > 0 \} | = \begin{cases} 0 \\ 1 \end{cases} \end{align*}


\begin{align*} \lim \nolimits _ { k \to \infty } C ( l ' , r ' ; P _ { k , S } ^ { l , r } ) = \delta _ S ( l , r ; \ , l ' , r ' ) , \end{align*}


\begin{align*} \int _ U ^ T \Sigma _ s ^ T d s : = \int _ U ^ T \mathcal S ( T - s ) \Sigma _ s \mathcal S ( T - s ) ^ * d s . \end{align*}


\begin{align*} W ( r ) : = M _ { 0 } r ^ { d } . \end{align*}


\begin{align*} T ( F r o b _ v ) = T _ v \end{align*}


\begin{align*} \beta _ n : = \frac { \ln q _ { n + 1 } } { q _ n } , \end{align*}


\begin{align*} f \cdot g : = \int _ \S f ( x ) g ( x ) \ \d x + \int _ \S D f ( x ) D g ( x ) \ \d x \end{align*}


\begin{align*} \mathbf { \Phi } _ { \mathrm { r } } ^ { H } \mathbf { \Phi } _ { \mathrm { r } } + \mathbf { \Phi } _ { \mathrm { t } } ^ { H } \mathbf { \Phi } _ { \mathrm { t } } = \mathbf { I } _ { M } . \end{align*}


\begin{align*} \textstyle \Pi _ \lambda f = \sum _ { \alpha : \ , d + 2 | \alpha | = \lambda } \ , \langle f , \Phi _ \alpha \rangle \Phi _ \alpha . \end{align*}


\begin{align*} \chi _ K ( \alpha ) + \chi _ K ( - \alpha ) = 1 - \Phi _ K ^ + ( \alpha ) - \Phi _ K ^ + ( - \alpha ) = \begin{cases} 1 , & \alpha \in \Phi \setminus \Phi _ K , \\ 0 , & \alpha \in \Phi _ K . \end{cases} \end{align*}


\begin{align*} \left ( d b \right ) _ { p } \left ( y \right ) = \left ( d \mathcal { L } _ { g } \right ) _ { \iota } \left ( d \bar { b } _ { o } \left ( x \right ) \right ) y ^ { b } = \left ( d g \right ) _ { o } \left ( x ^ { \bar { b } } \right ) \end{align*}


\begin{align*} r ( k ) = \begin{cases} - 2 i k r _ 1 ( z ) & | k | \leq 1 \\ ( 2 i k ) ^ { - 1 } r _ 2 ( z ) & | k | \geq 1 . \end{cases} \end{align*}


\begin{align*} u _ x ( x , t ) = 2 i m ( x , t ) e ^ { 4 i \int _ { - \infty } ^ x | m ( s , t ) | ^ 2 d s } , \end{align*}


\begin{align*} P ( \lambda ) = \left \{ ( x _ \alpha ) \in \mathbb { R } ^ { \sharp R ^ + } _ { \geq 0 } \mid \forall \ ; \mathbf { p } , \sum _ { \alpha _ \in \mathbf { p } } x _ \alpha \leq m _ { e ( p ) } - m _ { s ( p ) } \right \} \end{align*}


\begin{align*} \begin{aligned} | x ( t ) | _ { \mathcal { A } _ \delta } & \le | x _ 1 ( t ) | _ { \mathcal { A } _ \delta } + | x ( t ) - x _ 1 ( t ) | \\ & \le \kappa \bar c _ 0 e ^ { - \gamma \lambda ( t - \bar { t } / \gamma ) } + \bar c \\ & \le \kappa \bar c _ 0 + \bar c \leq d & & \forall \ , t \in I _ 1 \end{aligned} \end{align*}


\begin{align*} \phi ( u _ 0 e _ 1 u _ 1 ) = - \tau ( u _ 0 , e _ 1 ) \phi ( e _ 1 ) . \end{align*}


\begin{align*} N _ + ( x ; k ) = N _ - ( x ; k ) V ( x ; k ) , \end{align*}


\begin{align*} \mu ( A _ { h , j } ) \geq c h ^ d , j = 1 , \ldots , N _ n , \end{align*}


\begin{align*} \varDelta : = \sqrt { \sum _ { i = 1 } ^ n \| F _ i ( x ) - \tilde { F } _ i ( x ) \| ^ 2 } \end{align*}


\begin{align*} \lim _ { n \to \infty } ( T _ i ( x ^ \pm _ n ) - x ^ \pm _ n ) Q _ i ( x ^ \pm _ n ) = 0 . \end{align*}


\begin{align*} 1 + \gamma _ { w , k , m } = \frac { 1 + \Delta _ { w , k + 1 , m } } { 1 + \Delta _ { w , k , m } } , ~ | \Delta _ { w , i , m } | \le \frac { w - m } { m } \Big | \frac { z _ { w } } { z _ { w - m } } \Big | ^ i , ~ i = k , k + 1 , \end{align*}


\begin{align*} \frac { \partial ^ 2 \varphi } { \partial x _ \beta ^ 2 } ( x ' , \rho ( x ' ) ) \geq b _ \beta - C \delta \tilde { b } _ \beta , \ \beta = 1 , \ldots , n - 1 \end{align*}


\begin{align*} C _ { * } ( n ) : = ( C _ { * } ( R ^ { 2 n } ) , d ) = \cdots \rightarrow C _ { 2 } ( R ^ { 2 n } ) \rightarrow C _ { 1 } ( R ^ { 2 n } ) \xrightarrow { \varepsilon } \mathbb { Z } \rightarrow 0 \end{align*}


\begin{align*} A _ k \ = \ \sum _ { j = 0 } ^ { k - 1 } \binom { k } { j } B _ j . \end{align*}


\begin{align*} y \left [ i \right ] = \sum \limits _ { t = 0 } ^ { { n ' _ { { \rm { s p a n } } } } } { \left ( { \sum \limits _ { l ' = 1 } ^ { L ' } { { \bf { g } } _ { l ' } ^ H \left [ t \right ] { { \bf { \bar H } } _ { l ' } ^ \bot { \bf { \bar X } } _ { l ' } } } } \right ) { { \bf { d } } \left [ { i - t } \right ] } } + z \left [ { i } \right ] . \end{align*}


\begin{align*} \left | \dfrac { - \gamma \omega } { - \gamma \alpha \omega } \right | & = \left | 1 + \dfrac { \gamma \alpha \omega - \gamma \omega } { - \gamma \alpha \omega } \right | \leq \max \left ( 1 , \left | \dfrac { \gamma \alpha \omega - \gamma \omega } { - \gamma \alpha \omega } \right | \right ) , \end{align*}


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


\begin{align*} x _ n = \rho ( x ' ) = \frac { 1 } { 2 } \sum _ { \alpha , \beta < n } B _ { \alpha \beta } x _ \alpha x _ \beta + O ( | x ' | ^ 3 ) , \end{align*}


\begin{align*} \varphi ( [ m , n ] ) = \int _ { \widehat { N } } \chi ( [ m , n ] ) \ , \mathrm { d } \mu _ \varphi ( \chi ) = \int _ { \widehat { N } } \chi ^ m ( n ) \chi ^ { - 1 } ( n ) \ , \mathrm { d } \mu _ \varphi ( \chi ) = 1 . \end{align*}


\begin{align*} f ( t ) = \exp \left ( \pm \sigma _ { \epsilon } \ ; t \right ) = { } _ { 0 } F _ { 0 } \left ( \ ; ; \ ; ; \pm \sigma _ { \epsilon } \ ; t \right ) . \end{align*}


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


\begin{align*} C _ p ( C _ { \ast } ) = B _ { p } ( C _ { \ast } ) \oplus \ell _ p ( H _ p ( C _ { \ast } ) ) \oplus s _ p ( B _ { p - 1 } ( C _ { \ast } ) ) . \end{align*}


\begin{align*} [ L _ \lambda L ] = ( \partial + 2 \lambda ) L , [ L _ \lambda H ] = ( \partial + \lambda ) H , [ H _ \lambda L ] = \lambda H , [ H _ \lambda H ] = 0 . \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } s } \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) & = \frac { 1 } { 2 \Gamma ( 1 / s ) } \begin{cases} s ^ 2 \tilde \psi \left ( s , ( \frac { \mu - x } { \sigma } ) ^ s \right ) + \psi ( 1 / s ) \Gamma ( 1 / s , ( \frac { \mu - x } { \sigma } ) ^ s ) , & x \leq \mu , \\ - [ s ^ 2 \tilde \psi \left ( s , ( \frac { x - \mu } { \sigma } ) ^ s \right ) + \psi ( 1 / s ) \Gamma ( 1 / s , ( \frac { x - \mu } { \sigma } ) ^ s ) ] , & x > \mu . \end{cases} \end{align*}


\begin{align*} \left \{ \begin{aligned} d \bar { x } ^ i _ t = & ~ \Big [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \bar { x } ^ i _ t - B ^ 2 R ^ { - 1 } \Phi ( t , \bar { \nu } _ t ^ i ) - B h \Big ( k \big ( t , \bar { \nu } _ t ^ i , \Phi ( t , \bar { \nu } _ t ^ i ) \big ) \Big ) \Big ] d t \\ & + \sigma d W _ t ^ i + \sigma _ 0 d W _ t ^ 0 , \\ \bar { x } ^ i _ { t _ 0 } = & ~ \xi ^ i , \end{aligned} \right . \end{align*}


\begin{align*} - L \eta = G \textrm { i n } \ \mathbb { R } ^ { 2 } _ { + } , \eta = 0 \textrm { o n } \ \partial \mathbb { R } ^ { 2 } _ { + } , \end{align*}


\begin{align*} F ^ { i j } v _ { i j } \geq \delta _ 1 \left ( \frac { 1 } { b _ { n - 1 } ^ { 1 / k } } + \sum _ { i = 1 } ^ n F ^ { i i } \right ) \mbox { i n } \omega . \end{align*}


\begin{align*} u _ 2 ( t ) = U ( t - t _ 0 ) u _ 2 ( t _ 0 ) - i \lambda _ 6 \int ^ t _ { t _ 0 } U ( t - \tau ) \mathcal { N } _ 2 ( u _ 1 ( \tau ) , u _ 2 ( \tau ) ) \ d \tau \end{align*}


\begin{align*} \min _ { r \in [ 0 , R ] } c ( r , t ) = c ( 0 , t ) . \end{align*}


\begin{align*} \begin{aligned} \Rightarrow & B _ 0 r ^ q \big ( g - d r \otimes d r \big ) \leq H e s s ( r ) \leq ( \sqrt { A } \coth \sqrt { A } ) r ^ q \big ( g - d r \otimes d r \big ) \ , , \operatorname { f o r } \ , r \geq 1 \ , , \end{aligned} \end{align*}


\begin{align*} A = \{ n \in \N : T ^ n a \in E \} . \end{align*}


\begin{align*} | q _ { t , \gamma } ( x , y ) | \leq | q _ t ( x , y ) | \leq \frac { C _ N } { t ^ { n / 2 + 1 } } e ^ { - \frac { c | x - y | ^ 2 } { t } } \Big ( 1 + \frac { \sqrt { t } } { \rho ( x ) } + \frac { \sqrt { t } } { \rho ( y ) } \Big ) ^ { - N } \end{align*}


\begin{align*} \ker ( D _ { \varphi } ) = \bigg \{ \sum _ j \varphi _ j e _ j \bigg \} . \end{align*}


\begin{align*} \min _ { \overline { \mathbf { \Phi } } } ~ ~ & \mathsf { T r } ( \overline { \mathbf { \Phi } } \mathbf { Y } \overline { \mathbf { \Phi } } ^ H \mathbf { Z } ) - 2 \Re \{ \mathsf { T r } ( \overline { \mathbf { \Phi } } \mathbf { X } ) \} \\ \mathrm { s . t . } ~ ~ & \overline { \mathbf { \Phi } } ^ H \overline { \mathbf { \Phi } } = \mathbf { I } _ M , \end{align*}


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


\begin{align*} G _ { \tilde u } = T G _ u { } ^ t T \end{align*}


\begin{align*} a \circ \widehat { \varphi } ( [ U , V ] , x ) = a ( U D ( x ) V ) = a ( D ( x ) V ) = a \circ \widehat { \varphi } ( [ I _ n , V ] , x ) , \end{align*}


\begin{align*} b ^ \Lambda _ \lambda = \eta ( \tau ) ^ \ell c ^ \Lambda _ \lambda \end{align*}


\begin{align*} ( \omega \kappa ) ^ { - 1 } \alpha \omega \kappa \begin{pmatrix} 1 \\ \vdots \\ \theta ^ { n - 1 } \end{pmatrix} = \theta \begin{pmatrix} 1 \\ \vdots \\ \theta ^ { n - 1 } \end{pmatrix} . \end{align*}


\begin{align*} \sum _ { k = 2 } ^ n \frac { \mu ( k ) } { k } ( I - S ) h _ k ( z ) = \sum _ { k = 1 } ^ n \frac { \mu ( k ) } { k } \left [ \log ( 1 - z ^ k ) - \log ( 1 - z ) - \log k \right ] \end{align*}


\begin{align*} ( \nabla ^ { \ell + 2 } _ \nu h ) ( e _ a , e _ b ) = 0 , \mbox { o r i n t h e l o c a l f r a m e } h _ { a b ; \tiny \underbrace { 0 \cdots 0 } _ { \ell + 2 - \mbox { t i m e s } } } = 0 . \end{align*}


\begin{align*} M = X ^ { \frac { 1 } { 2 } } \end{align*}


\begin{align*} \dfrac { \hat { k } _ d ( \Delta ) } { \hat { k } _ d ( \Psi ) } = x _ \sigma . \end{align*}


\begin{align*} a ( z ) = 1 + \frac { 1 } { 2 i } \int _ { \mathbb { R } } \left ( | u _ y | ^ 2 \Psi ^ - _ { 1 1 } ( y ; z ) + u _ y \Psi ^ - _ { 2 1 } ( y ; z ) \right ) \mathrm { d } y , \end{align*}


\begin{align*} \frac { \partial \textbf { \textit { U } } } { \partial x _ n } + Q \textbf { \textit { U } } = \textbf { \textit { W } } \end{align*}


\begin{align*} \bar { R } _ { c , k } & = \mathbb { E } \left [ R _ { c , k } | \mathbf { \hat { G } } \right ] & \bar { R } _ { k } = \mathbb { E } \left [ R _ { k } | \mathbf { \hat { G } } \right ] , \end{align*}


\begin{align*} \pi \circ \gamma ( a , y ) = \pi ( a , y ) = \pi ( \gamma ' ( a , y ) ) = \pi ( a ' , y ' ) . \end{align*}


\begin{align*} \mathcal D _ { \psi } : = \psi _ 0 \frac { \partial } { \partial x } + \psi _ 1 \frac { \partial } { \partial y } . \end{align*}


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


\begin{align*} w ( t , y ) & \geq w _ 0 ( y ) + \left ( - g \big ( c _ 0 ( y ) \big ) \frac { 1 } { w _ 0 ( y ) } - \varepsilon \right ) \cdot t = 1 + ( - g _ O - \varepsilon ) \cdot t \\ & > 1 + ( - g _ I + \varepsilon ) \cdot t = w _ 0 ( x ) + \left ( - g \big ( c _ 0 ( x ) \big ) \frac { 1 } { w _ 0 ( x ) } + \varepsilon \right ) \cdot t \geq w ( t , x ) \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } \beta } \ell = & \psi ( \alpha + \beta ) - \psi ( \beta ) + \log \left [ 1 - \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \right ] . \end{align*}


\begin{align*} f '' ( x ) = a + b ' x + c ' x ^ q , \end{align*}


\begin{align*} G _ { u } : = c \left ( R ^ { \frac { 6 } { r } } \left ( \frac { 1 } { R ^ 3 } \int _ { \mathcal { C } ( R / 2 , R ) } \vert \nabla u \vert ^ { \frac { p } { 2 } } d x \right ) ^ { \frac { 2 } { p } } , S _ { u } : = R ^ { \frac { 6 } { r } } \left ( \frac { 1 } { R ^ 3 } \int _ { \mathcal { C } ( R / 2 , R ) } \vert u \vert ^ { p } d x \right ) ^ { \frac { 2 } { p } } \right ) , \end{align*}


\begin{align*} \alpha ( \log x ) : = \begin{cases} \ 0 & \ x = x _ 0 \\ \frac { - 1 } { \log | x - x _ 0 | } & \ | x - x _ 0 | < 2 \delta . \end{cases} \end{align*}


\begin{align*} \| ( \lambda x _ v + ( 1 - \lambda ) x _ w ) \| ^ 2 = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 - \lambda ( 1 - \lambda ) \| x _ v - x _ w \| ^ 2 . \end{align*}


\begin{align*} \tilde { V } _ n ( \varphi ) = V ( \varphi ) - V _ n ( \varphi ) , \forall \varphi \in \mathcal { C } _ { p w } ( - h , 0 ; \mathbb { R } ^ m ) . \end{align*}


\begin{align*} c _ { b _ { k } } & = \frac { 1 } { 2 } \| ( \widetilde { u } _ { k } , \widetilde { v } _ { k } ) \| _ { E } ^ { 2 } + \frac { 1 } { 2 } \Big ( B ( \widetilde { u } _ { k } ) + B ( \widetilde { v } _ { k } ) \Big ) - \frac { 1 } { 2 p } F ( \widetilde { u } _ { k } , \widetilde { v } _ { k } ) \\ & \geq \Big ( \frac { 1 } { 2 } - \frac { \alpha } { 2 ( p \alpha - 1 ) } \Big ) \| ( \widetilde { u } _ { k } , \widetilde { v } _ { k } ) \| _ { E } ^ { 2 } , \end{align*}


\begin{align*} v _ i = \deg \left ( \overline { \gamma _ Z ^ { - 1 } ( \P ^ { n - i } ) } \right ) \end{align*}


\begin{align*} \frac { 1 } { \varepsilon ^ 2 \ln \frac { 1 } { \varepsilon } } \int _ \Omega \left ( v ^ \tau _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + q \ln \frac { 1 } { \varepsilon } d x \rightrightarrows q ^ { p + 1 } ( \bar { x } ) \int _ { \hat { B } _ \tau ( 0 ) } ( \hat { U } ( y ) + \ln \tau ) ^ { p } _ + d y . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ { 3 m } \alpha _ j + \sum _ { k = 1 } ^ l \beta _ k \leq \sum _ { i = 1 } ^ { n + 1 } \lambda _ i + \varepsilon . \end{align*}


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


\begin{align*} a _ i = - \frac { 1 } { \norm { C _ j ^ { \top } b _ i } ^ 2 } S _ j ^ { \top } R _ j C _ j ^ { \top } b _ i i = 1 , \dots , r . \end{align*}


\begin{align*} \begin{array} { l l l } L _ f \circ g & = & \mu \circ ( f \otimes i d _ S ) \circ \lambda _ S ^ { - 1 } \circ g \\ & = & \mu \circ ( f \otimes i d _ S ) \circ ( i d _ I \otimes g ) \circ \lambda _ I ^ { - 1 } \\ & = & f \bullet _ \mu g \end{array} \end{align*}


\begin{align*} { \bf { u } } _ k ^ { \star } = { \left [ { \bf { \bar U } } ^ { \star } \right ] _ { : , k + 1 } } = { \left [ \begin{array} { l } { { \bf { I } } _ K } \\ { { \bf { 0 } } _ { \left ( { { M _ t } - K } \right ) \times K } } \end{array} \right ] _ { : , k + 1 } } , \ \forall k , \end{align*}


\begin{align*} P _ { \min } ^ { \rm { O } } \left ( { \overline S , \overline \varepsilon , \overline R } \right ) = \mathop { \arg \min } \limits _ { { W _ s } \in \left [ { { W ^ { l o w } } , { W ^ { u p } } } \right ] } P _ { \min } ^ { \rm { O } } \left ( { \overline S , \overline \varepsilon , \overline R | { W _ s } } \right ) . \end{align*}


\begin{align*} A ( t ) y ( t , \xi ) & = - i e ^ { - W } \Delta ( e ^ W y ) - ( \mu + \tilde { \mu } ) y \\ & = - i ( \Delta + b ( t , \xi ) \cdot \nabla + c ( t , \xi ) ) y ( t , \xi ) , \end{align*}


\begin{align*} \delta ^ g D ^ g _ { \theta ^ \sharp } \theta = \frac { 1 } { 2 } \delta ^ g \left ( d \theta ( \theta ^ \sharp , \cdot ) \right ) = - \frac { 1 } { 2 } \delta ^ g d ( \| \theta \| ^ 2 ) = - \frac { 1 } { 2 } \Delta ^ g ( \| \theta \| ^ 2 ) . \end{align*}


\begin{align*} \begin{aligned} F _ a ( \eta _ a ) & : = \left ( \begin{array} { c } - \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \\ - \bar { y } _ a + \dfrac { a _ { 0 , \delta } ( x _ a ) } { 2 } \end{array} \right ) . \end{aligned} \end{align*}


\begin{align*} { y _ { \rm f } } \left [ { m , k } \right ] = { { { \bf { \tilde h } } } ^ H } \left [ k \right ] { { \bf { u } } _ k } s \left [ { m , k } \right ] + z \left [ m , k \right ] , \ \forall k \in \left [ 0 , { K } - 1 \right ] . \end{align*}


\begin{align*} \begin{aligned} & \rho _ - \left \| Q _ { j , + } \right \| _ { L ^ 2 } ^ 2 \leq \operatorname { R e } \int _ { \mathbb { R } } Q _ { j , + } ( I + J ) ^ H Q _ { j , + } ^ { H } d z \\ & = \operatorname { R e } \int _ { \mathbb { R } } \mathcal { P } ^ + ( D _ j ) ( I + J ) ^ { H } Q _ { j , + } ^ { H } d z \leq \rho _ + \| D _ j \| _ { L ^ 2 } \left \| Q _ { j , + } \right \| _ { L ^ 2 } . \end{aligned} \end{align*}


\begin{align*} W '' = e ^ { - \lambda ^ * T } \big ( W ' + W ^ D \big ) . \end{align*}


\begin{align*} \partial _ t c = \Delta _ \Gamma \big ( G ' ( c ) \big ) \end{align*}


\begin{align*} N K B L ( m , n ) & = \sum _ { A \in M N A P P ( m , n ) } N ( A ) \cdot ( m - 1 ) ! \\ & \ge \sum _ { i = 1 } ^ { 6 } N ( A _ i ) \cdot ( m - 1 ) ! \ge ( 6 \cdot 2 ^ m ) \cdot ( m - 1 ) ! . \end{align*}


\begin{align*} { \mathcal { E } } = \frac { 1 } { 2 } \int _ { \Omega } \left ( | u | ^ { 2 } + | B | ^ { 2 } \right ) \dd x , { \mathcal { H } } = \int _ { \Omega } A \cdot B \dd x , \end{align*}


\begin{align*} \left \{ \begin{aligned} F _ 2 ^ { X , D } ( \alpha ) _ x & = H ^ 0 \left ( P ^ D _ X ( \alpha ) _ x , T _ { P ^ D _ X ( \alpha ) _ x } ( - [ s ( x ) ] ) \right ) \\ F _ 1 ^ { X , D } ( \alpha ) _ x & = H ^ 0 \left ( P ^ D _ X ( \alpha ) _ x , T _ { P ^ D _ X ( \alpha ) _ x } ( - 2 [ s ( x ) ] ) \right ) \end{aligned} \right . \end{align*}


\begin{align*} v _ n ^ { - 1 } \lambda = \sigma ^ n v ^ { - 1 } ( \sigma \circ w ) ^ { - n } ( \lambda ) . \end{align*}


\begin{align*} - \varepsilon ^ 2 ( K ( x ) \nabla u ) + V ( x ) u = u ^ { p } , \ \ \ \ x \in \mathbb { R } ^ n , \end{align*}


\begin{align*} W ^ { \mu * } \otimes W ^ \mu = \bigoplus _ { j = 1 } ^ { \dim W ^ \mu } W ^ { \mu * } \otimes w _ j \simeq \bigoplus _ { j = 1 } ^ { \dim W ^ \mu } W ^ { \mu * } \end{align*}


\begin{align*} \abs { t _ n } _ p & = \abs { t _ \ell } _ p \abs { k } _ p , \intertext { w h i l e f o r $ p = 2 $ , } \abs { t _ n } _ 2 & = \begin{cases} \abs { t _ \ell } _ 2 & k \\ 2 \abs { t _ { 2 \ell } } _ 2 \abs { k } _ 2 & k \end{cases} \end{align*}


\begin{align*} \widehat { f _ \delta } ( a , z ) = ( 2 \pi ) ^ { - p - q } | \lambda | ^ { p + q } \ , \Big ( \sum _ { j = 1 } ^ { d ( p , q ) } P _ { p , q } ^ j ( z ) \ , \mathcal { G } _ { n + p + q } ( g _ { \delta , j } ) ( a ( p , q ) ) \Big ) \ , \varphi _ { k - q , \lambda } ^ { n + p + q - 1 } ( z ) . \end{align*}


\begin{align*} \sum \mu _ p ( C ) = 2 , \mathfrak { i } = 2 , \quad \mathfrak { t } = 3 \end{align*}


\begin{align*} n = \dim ( X _ \ast ( T ) _ { \Gamma _ 0 } \otimes \mathbb Q ) ^ { \sigma } - ( X _ \ast ( T ) _ { \Gamma _ 0 } \otimes \mathbb Q ) ^ { \sigma w } . \end{align*}


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


\begin{align*} N + E _ N ^ + - E _ N ^ - = f ^ * ( f _ * N ) . \end{align*}


\begin{align*} \begin{aligned} \norm { M ^ { i } ( \phi f ) } _ { ( 2 , 0 ) } \leq \norm { \phi M ^ { i } f } _ { ( 2 , 0 ) } + \sum _ { 1 \leq j \leq 2 i } \frac { 1 } { j ! } \norm { ( \partial _ { v _ 1 } ^ j \phi ) P _ j f } _ { ( 2 , 0 ) } , \end{aligned} \end{align*}


\begin{align*} K _ { \lambda , \beta } = \sum _ { J \subset \varrho ^ { - 1 } ( \beta ) } \tilde \kappa _ { \lambda , J } , \end{align*}


\begin{align*} \hat \Phi \left ( A _ \phi \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) = A _ \phi \hat \Phi \left ( \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) , \end{align*}


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


\begin{align*} & \frac { 1 } { \varepsilon ^ 2 } \int _ \Omega \left ( v ^ \tau _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p + 1 } _ + d x = \int _ { \hat { B } _ \tau ( 0 ) } q ^ { p + 1 } ( \bar { x } + \varepsilon y ) ( \hat { U } ( y ) + \ln \tau ) ^ { p + 1 } _ + d y \\ \rightrightarrows & q ^ { p + 1 } ( \bar { x } ) \int _ { \hat { B } _ \tau ( 0 ) } ( \hat { U } ( y ) + \ln \tau ) ^ { p + 1 } _ + d y , \end{align*}


\begin{align*} N e w t o n ( F ) = C o n v ( \bigcup \limits _ { \mu } S u p p ( s _ \mu ) ) . \end{align*}


\begin{align*} \mathbf { u } _ { t } ( t , \mathbf { x } ) = D \Delta _ { \mathbf { x } } \mathbf { u } ( t , \mathbf { x } ) + f ( \mathbf { u } ( t , \mathbf { x } ) ) , \mathbf { u } \in \mathbb { R } ^ { n } , d \geq 2 , t \geq 0 , n \geq 2 , \end{align*}


\begin{align*} X = r \nabla r , X ^ { \flat } = r d r \nabla X ^ { \flat } = ( \frac 1 2 r ^ 2 ) = d r \otimes d r + r ( r ) \ , . \end{align*}


\begin{align*} \delta ( \rho , \varrho \gamma ^ + ) = \delta ( \rho \varrho , \gamma ^ + ) - \delta ( \varrho , \gamma ^ + ) \end{align*}


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


\begin{gather*} \mathcal { A } ( x , \xi ) = D _ { \xi } F ( x , \xi ) . \end{gather*}


\begin{align*} e ^ { \gamma x } f _ r ( x ) = \int _ { - \infty } ^ \infty e ^ { \gamma ( x - y ) } f _ { r + } ( x - y ) e ^ { \gamma y } \mu _ - ( d y ) \le c \int _ { - \infty } ^ \infty e ^ { \gamma y } \mu _ - ( d x ) < \infty . \end{align*}


\begin{align*} \begin{aligned} u ( x ) \geq u ( x _ 0 ) - C b _ \alpha | x - x _ 0 | = \ , & u ( x _ 0 ) - C b _ \alpha ( x _ n - \rho ( 0 , \tilde { x } ) ) \\ \geq \ , & u ( x _ 0 ) - C \epsilon _ 0 \delta ^ 2 b _ \alpha ^ 2 . \end{aligned} \end{align*}


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


\begin{align*} \tilde H ( p , s ) = \tilde G ( p , s ) - F ( p , s ) . \end{align*}


\begin{align*} d k ( \gamma , x ) = k ( \gamma x ) - k ( x ) \qquad ( \gamma \in \Gamma , \ x \in X ) . \end{align*}


\begin{align*} d _ { i j } : = \langle \eta _ i , \eta _ j \rangle = 1 + \frac { \delta _ { i j } } { \varphi _ i } , \end{align*}


\begin{align*} E : = \begin{bmatrix} 1 & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ \\ \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \dots & 0 \end{bmatrix} . \end{align*}


\begin{align*} \begin{aligned} v _ t + ( v + u _ { \infty } ) \cdot \nabla v + \nabla p & = ( b + B _ { \infty } ) \cdot \nabla b + \nu \Delta v , \\ b _ t + ( v + u _ { \infty } ) \cdot \nabla b & = ( b + B _ { \infty } ) \cdot \nabla v + \mu \Delta b , \\ \nabla \cdot v = \nabla \cdot b & = 0 . \end{aligned} \end{align*}


\begin{align*} & ( I - Q ^ \dagger ) F ^ { q , \dagger } _ { v v } ( v + \psi ^ \dagger ( v , p ) , p ) [ v _ 1 + \psi ^ \dagger _ v ( v , p ) v _ 1 , v _ 2 + \psi ^ \dagger _ v ( v , p ) v _ 2 ] \\ & \quad + ( I - Q ^ \dagger ) F ^ { q , \dagger } _ v ( v + \psi ^ \dagger ( v , p ) , p ) \psi ^ \dagger _ { v v } ( v , p ) [ v _ 1 , v _ 2 ] = 0 . \end{align*}


\begin{align*} E X ^ { m } Y ^ { n - m } = \sum _ { j = 0 } ^ { \min ( m , n - m ) } \rho ^ { j } H _ { m , j } H _ { n - m , j } . \end{align*}


\begin{align*} 1 \leq \sum _ { j = 1 } ^ { N L } x _ j \leq \sqrt { N L } , 1 \leq \sum _ { j = 1 } ^ { N L } y _ j \leq \sqrt { N L } . \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \bar { \theta } ) = & E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ n Z ^ { \bar { \theta } } \biggr ) \biggr ] \\ = & E _ { \xi _ 0 } \biggl [ V \biggl ( \frac { \xi _ 0 f _ 1 ( X _ 1 ) } { \xi _ 0 f _ 1 ( X _ 1 ) + ( 1 - \xi _ 0 ) f _ 2 ( X _ 1 ) } , n - 1 , x + X _ 1 \biggr ) \biggr ] . \end{align*}


\begin{align*} 4 \sqrt { 3 } \pi ^ { 3 } \int _ { 0 } ^ { \infty } \left ( \sum _ { n , s = 1 } ^ { \infty } \chi _ { - 3 } ( n ) s ^ { 2 } e ^ { - 2 \pi u n s } \right ) \left ( \sum _ { k , r = 1 } ^ { \infty } \frac { \chi _ { - 3 } ( k r ) } { k } \left ( e ^ { - \frac { 2 \pi k r } { 9 u } } - e ^ { - \frac { 2 \pi k r } { 3 u } } \right ) \right ) u d u . \end{align*}


\begin{align*} \sigma ( F ^ q _ v ( 0 , p ) ) = \operatorname { c l } ( \{ \xi _ k , k \in \N \} ) . \end{align*}


\begin{align*} \langle \overline { \alpha } ' _ 1 , \overline { \alpha } ' _ 1 \rangle \langle \overline { \alpha } ' _ 2 , \overline { \alpha } ' _ 2 \rangle < \langle \overline { \alpha } ' _ 1 , \overline { \alpha } ' _ 2 \rangle ^ 2 = \langle \alpha ' _ 1 , \alpha ' _ 2 \rangle ^ 2 = \cosh ( \theta ) ^ 2 = 1 - s . \end{align*}


\begin{align*} h ( x ) = h _ 0 + \left \{ \begin{array} { c c } ( x - \pi ) ^ 2 - 1 & x < \pi \\ \cos ( x - \pi ) - 2 & x \in [ \pi , \ , 2 \pi ) \\ ( x - 2 \pi ) ^ 2 - 3 & x \ge 2 \pi \end{array} \right . \end{align*}


\begin{align*} \aligned 0 & = \delta ^ \nabla \big ( \exp \ , ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) R ^ \nabla \big ) \\ & = - \sum _ { i = 1 } ^ m \big ( \nabla _ { e _ i } \exp \ , ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) R ^ \nabla \big ) ( e _ i , \cdot ) \\ & = \exp \ , ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) \delta ^ \nabla R ^ \nabla - i _ { \big ( \exp \ , ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) \big ) } R ^ \nabla \ , . \endaligned \end{align*}


\begin{align*} \dot \theta _ k = \frac 1 M \sum _ { j = 1 } ^ M a _ { k - j } \sin ( \theta _ j - \theta _ k ) - \frac 1 M \sum _ { j = 1 } ^ M a _ { 1 - j } \sin ( \theta _ j ) . \end{align*}


\begin{align*} \prod ^ { n e } _ { \substack { j = 1 \\ j \equiv i \bmod e } } ( x - \zeta ^ j _ { n e } y ) = x ^ n - \zeta ^ i _ e y ^ n \end{align*}


\begin{align*} \begin{array} { c } \frac { v _ 0 } { v _ 2 } = \frac { - | z | ^ 2 + i \tilde { t } \sqrt { d } } { 2 } \\ \frac { v _ 1 } { v _ 2 } = z \end{array} . \end{align*}


\begin{align*} \partial _ { t } P = \Theta \partial _ { t } \sigma _ { k } + \sigma _ { k } \partial _ { t } \Theta , \end{align*}


\begin{align*} \sum _ { k = 1 } ^ n \frac { \mu ( k ) } { k } \log ( 1 - z ^ k ) \to - z \ \ell ^ q \end{align*}


\begin{align*} u ( t ) = U _ \alpha ( t ) u _ 0 + \int _ 0 ^ t U _ \alpha ( t - s ) F ( u ( s ) ) d s . \end{align*}


\begin{align*} \sigma ^ { - 1 } \left ( r \right ) - \sigma ^ { - 1 } \left ( s \right ) = \int _ { s } ^ { r } \tfrac { 1 } { \sqrt { 2 } } \left ( \ell ^ { \prime } \left ( t \right ) \right ) ^ { 1 / 2 } ~ d t \end{align*}


\begin{align*} \int _ a ^ b e ^ { i \lambda \phi ( x ) } \psi ( x ) d x = \int _ a ^ b e ^ { i \lambda \phi ( x ) } ( D ^ t ) ^ N ( \psi ) ( x ) d x . \end{align*}


\begin{align*} \| ( \ell _ n ^ { ( 1 ) } - \ell ) q _ n \alpha \| = | \ell _ n ^ { ( 1 ) } - \ell | \cdot \| q _ n \alpha \| . \end{align*}


\begin{align*} \left \Vert \mathcal { L } _ { 1 } ^ { + } \left ( \rho \right ) \right \Vert _ { L _ { 2 } \left ( \mathbb { R } \right ) } ^ { 2 } = \int _ { - \infty } ^ { \infty } \left ( \int _ { M ^ { + } } \left \vert Q ^ { \prime } ( s ) \right \vert \beta ^ { k - 2 } ( s ) d s \right ) \left ( \int _ { M ^ { + } } \left \vert Q ^ { \prime } ( s ^ { \prime } ) \right \vert \overline { \beta } ^ { k - 2 } ( s ^ { \prime } ) d s ^ { \prime } \right ) d \rho . \end{align*}


\begin{align*} X = X _ 1 \boxtimes X _ 2 \boxtimes \cdots \boxtimes X _ k , \end{align*}


\begin{align*} \begin{aligned} \Big | \langle \boldsymbol { \nu } , \eta _ t \rangle - \bar \eta _ t + \langle \boldsymbol { \nu } , q _ { \alpha , x _ \alpha } \rangle - \bar q _ { \alpha , x _ \alpha } \Big | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; , \end{aligned} \end{align*}


\begin{align*} \delta + x = \lambda . \end{align*}


\begin{align*} | | \mathbf { v } ( T + t , \mathbf { v } ^ 0 ) | | _ { \mathcal { E } } = | | \mathbf { v } ( t , \mathbf { v } ^ 1 ) | | _ { \mathcal { E } } \leq \delta . \end{align*}


\begin{align*} \hat W _ r ( 0 ) = k \hat W _ \frac { r } { k } ( 0 ) . \end{align*}


\begin{align*} g ^ * f _ ! = f ' _ ! g '^ * \end{align*}


\begin{align*} R _ c = \pi \sqrt { 5 0 / 2 } \approx 1 5 . 7 \ , , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \int _ { \mathbb { T } ^ d } \langle a ( u _ n ( x ) ) \nabla u ( x ) , \nabla u _ n ( x ) - \nabla u ( x ) \rangle d x = 0 . \end{align*}


\begin{align*} \| P _ 0 \Psi _ { R ( \lambda ) } \| _ { H ^ 1 } = O ( b - \lambda ) ~ ~ \lambda \to b . \end{align*}


\begin{align*} f ^ \lambda \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ( z ) = ( 2 \pi ) ^ { - p - q } \lambda ^ { p + q } \ , P ( z ) \ , g ^ \lambda \ast _ \lambda \varphi _ { k - p , \lambda } ^ { n + p + q - 1 } ( z ) \end{align*}


\begin{align*} \Omega _ { N } : = \{ \omega \in \Omega : | \omega ( t ) | \leq N e ^ { | t | } , t \in \R \} , \ N \in \N . \end{align*}


\begin{align*} ( B ^ { \circ } S _ { T - ( \cdot ) } ^ { \circ } x ^ { \circ } \odot \lambda ) ( \boldsymbol { \cdot } ) u = \int _ { ( \boldsymbol { \cdot } ) } \langle B ^ { \circ } S _ { T - t } ^ { \circ } x ^ { \circ } , u \rangle \d t \qquad ( u \in U ) \end{align*}


\begin{align*} t = r _ 0 + r _ 1 q + \cdots + r _ { n - 1 } q ^ { n - 1 } , \end{align*}


\begin{align*} X ( x , \alpha ) = \begin{cases} + , & \ell ( x , \alpha ) > 0 , \\ \bigcirc , & \ell ( x , \alpha ) = 0 , \\ - , & \ell ( x , \alpha ) < 0 . \end{cases} \end{align*}


\begin{align*} z ^ \top \nabla ^ 2 f ( x ) z & \le \sum _ { i = 1 } ^ n z _ i ^ 2 V '' ( x _ i ) \le - \kappa | z | ^ 2 , \end{align*}


\begin{align*} \Delta _ { 1 } ( x , \xi _ 0 ) = E _ { \xi _ 0 } \left [ \varphi \left ( x + X _ { 1 } \right ) \right ] - E _ { \xi _ { 0 } } \left [ \varphi \left ( x + Y _ { 1 } \right ) \right ] \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ { T } \Vert P _ n B ( t , X _ n ( t ) ) Q _ n - B ( t , X ( t ) ) \Vert _ { L _ 2 } ^ 2 d t = 0 . \end{align*}


\begin{align*} L ( p \mid u ) = \prod _ { i = 1 } ^ m p _ j ^ { u _ j } . \end{align*}


\begin{align*} h ( k , k ) = 1 \ \mbox { a n d } \ h _ t ( n , k ) \le h _ t ( n - 1 , k ) \le h _ t ( k + 1 , k ) = \frac { t } { k + 1 } \enspace . \end{align*}


\begin{align*} L _ { q } \left [ t ^ { m - 1 } \right ] = \displaystyle \int _ { 0 } ^ { \infty } d t \exp _ { q } ( - s t ) t ^ { m - 1 } . \end{align*}


\begin{align*} \partial ^ 2 _ { j k } v _ i - \Gamma _ { k j } ^ j \partial _ j v _ i - \Gamma _ { j k } ^ k \partial _ k v _ i = 0 . \end{align*}


\begin{align*} \frac { 1 } { N ( f ( x ) ) } : = - \frac { f ' ( x ) } { f ( x ) } \end{align*}


\begin{align*} \Phi = y _ i \xi ^ i + c , \end{align*}


\begin{align*} T ( t ) x = \tau \lim _ { n \to \infty } \left ( \frac { n } { t } R \left ( \frac { n } { t } , A \right ) \right ) ^ { n } x \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } \Psi ^ - _ 2 ( x ; z ) = e ^ { i c _ - ( x ) } e _ 2 . \end{align*}


\begin{align*} \omega ( E , \epsilon ) = 0 , L ( E , \epsilon ) = \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | . \end{align*}


\begin{align*} d e t ^ { S ^ 2 } ( \otimes _ { 1 \leq i < j \leq 2 d } ( v _ { i , j } ) ) = \sum _ { ( \Gamma _ 1 , . . . , \Gamma _ d ) \in \mathcal { P } ^ { h , c f } _ d ( K _ { 2 d } ) } \varepsilon _ d ^ { S ^ 2 } ( ( \Gamma _ 1 , . . . , \Gamma _ d ) ) M _ { ( \Gamma _ 1 , . . . , \Gamma _ d ) } ( ( v _ { i , j } ) _ { 1 \leq i < j \leq 2 d } ) , \end{align*}


\begin{align*} F _ p C ^ \ast ( \C { O } _ h ( G ) , M ) : = h ^ p C ^ \ast ( \C { O } _ h ( G ) , M ) , p \geq 0 , \end{align*}


\begin{align*} \left ( \Delta K _ m \right ) ^ m \wedge \beta _ n ^ { n - m } = 0 \mathbb { H } ^ n \setminus \{ 0 \} . \end{align*}


\begin{align*} \int _ 0 ^ { 2 - i \Delta _ n + s } | x - y - i \Delta _ n + s | ^ { 2 \mathfrak H - 1 } d y = \int _ { i \Delta _ n - s } ^ 2 | x - y | ^ { 2 \mathfrak H - 1 } d y \leq \int _ 0 ^ 2 | x - y | ^ { 2 \mathfrak H - 1 } d y \leq \frac { 2 ^ { 2 \mathfrak H } } { \mathfrak H } , \end{align*}


\begin{align*} \mathbf { g } = & [ \log 9 , \log 9 , \log 6 , \log \zeta , \log 5 4 , \log 5 4 , \log 2 1 6 ] ^ \intercal \end{align*}


\begin{align*} x \cdot B ( x , x ) = 0 . \end{align*}


\begin{align*} & \int _ { B _ r } d i v _ { \mathbb { G } } F | \nabla _ { H } v | ^ 2 - 2 \sum _ { i = 1 } ^ { m } \int _ { B _ r } X _ i v [ X _ i , F ] v - 2 \int _ { B _ r } F v \Delta _ { H } v \\ & = \int _ { \partial B _ r } | \nabla _ { H } v | ^ 2 < F , \nu > - 2 \sum _ { i = 1 } ^ { m } \int _ { \partial B _ r } F v X _ i v < X _ i , \nu > . \end{align*}


\begin{align*} \overbrace { [ x + a , y + b ] } & = ( x + a ) \cdot _ { \ltimes } ( y + b ) - ( y + b ) \cdot _ { \ltimes } ( x + a ) \\ & = x \cdot y + \ell ( x ) b + r ( y ) a - y \cdot x - \ell ( y ) a - r ( x ) b \\ & = [ x , y ] + ( \ell - r ) ( x ) b - ( \ell - r ) ( y ) a . \end{align*}


\begin{align*} g _ 2 = - \frac { 1 } { \nu } I \widehat { g _ 1 } I + \frac { 2 i } { \sqrt { \tau _ 0 } } h _ z J g _ 1 . \end{align*}


\begin{align*} u _ k = \frac { - 1 } { 8 \cos ^ { 2 } ( \frac { \pi k } { n - 1 } ) } S v _ k , \end{align*}


\begin{align*} D _ n ( x , t _ X , t _ Y ) = \begin{cases} ( t _ X + t _ Y ) \Delta _ n ( x , \frac { t _ X } { t _ X + t _ Y } ) & \\ 0 & \end{cases} \end{align*}


\begin{align*} \Big \| \sum \limits _ { k = 2 } ^ { n } \mu ( k ) ( I - S ) h _ k - ( 1 - z ) \Big \| _ { H ^ 2 ( \mathbb { D } ) } \longrightarrow 0 n \to \infty \end{align*}


\begin{align*} \mathcal { A } _ q ( p , x ) = \mathcal { A } _ q ^ p ( x ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { \overline { p } ^ n } { \sqrt { \Gamma ( q n + 1 ) } } \psi _ n ( x ) , p \in \mathbb { H } , x \in \mathbb { R } \end{align*}


\begin{align*} f ( x ) = \sum _ { n = 0 } ^ \infty c _ n ( a , b ) \ , g ( x ; ( a + n ) \theta ) , \end{align*}


\begin{align*} \lim _ { n \to \infty } \mu \left ( \left \{ \tau \in \partial _ e ( T ( A ) ) \mid \frac { 1 } { \sqrt { n } } \sum _ { i = 0 } ^ { n - 1 } \tau ( \alpha ^ i ( a ) ) \le z \right \} \right ) = \frac { 1 } { \sigma \sqrt { 2 \pi } } \int _ { - \infty } ^ z \exp \left ( - \frac { t ^ 2 } { 2 \sigma ^ 2 } \right ) d t \end{align*}


\begin{align*} \gamma ^ i ( x ) = \frac 1 { p - 1 } \sum _ { j \neq i , j \in \C } K ( x ^ i - x ^ j ) , ~ ~ ~ ~ \forall x \in \mathbb R ^ { N d } , \end{align*}


\begin{align*} \overline { \nu _ { \alpha } ^ { \prime } } \left ( x \right ) = \nu _ { \alpha } ^ { \prime } \left ( x , \infty \right ) = \frac { 1 } { \alpha \log ^ { \alpha } \left ( 1 + x \right ) } \end{align*}


\begin{align*} \Phi _ N = \{ L _ N , L _ N + 1 , \dots , M _ N - 1 \} \end{align*}


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


\begin{align*} \overline { \chi } = ( T \cap \overline { \psi } ) \cup ( T \setminus \overline { \psi } ) = T . \end{align*}


\begin{align*} \alpha _ { 0 } : = \frac { 1 6 } { \sqrt { 3 } } . \end{align*}


\begin{align*} \rho ^ * \omega & = \rho ^ * \left ( \frac { 4 d x \wedge d z } { x } \right ) = \frac { 4 t ( 1 + \cos \theta ) } { t ( 1 + \cos \theta ) } d t \wedge d \theta \\ & = 4 \ d t \wedge d \theta . \end{align*}


\begin{align*} \displaystyle T ( f ) ( x ) = \int _ { \mathbb { C } } { A _ q ( z , x ) } f ( z ) | z | ^ { \frac { 2 } { q } - 2 } e ^ { - \frac { | z | ^ 2 } { q } } d A ( z ) , \end{align*}


\begin{align*} \frac { 1 } { 2 } \underset { j = 1 } { \overset { N } { \sum } } \partial _ { x ^ j x ^ j } \phi ^ { N , i } ( t , \boldsymbol { x } ) \sigma ^ 2 + \frac { 1 } { 2 } \underset { j , \tau = 1 } { \overset { N } { \sum } } \partial _ { x ^ j } \partial _ { x ^ \tau } \phi ^ { N , i } ( t , \boldsymbol { x } ) \sigma _ 0 ^ 2 = \frac { 1 } { 2 } \partial _ { \nu \nu } \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) \sigma _ 0 ^ 2 + O ( \frac { 1 } { N } ) . \end{align*}


\begin{align*} \left | \mathbb E \left [ \langle B _ 1 , B _ 2 \rangle \sum _ { i = 1 } ^ n \int _ { ( i - 1 ) \Delta _ n } ^ { i \Delta _ n } \Delta _ n ^ { - ( \mathfrak H + \epsilon ) } \langle ( \mathcal S ( i \Delta _ n - s ) - I ) B _ 1 , B _ 2 \rangle d s \right ] \right | > c , \end{align*}


\begin{align*} \langle \vartheta _ l , \varphi _ l \rangle _ { \C } = 0 \ \ \textrm { o n } \ \C \ \textrm { f o r } \ l = 1 , 2 . \end{align*}


\begin{align*} { S } ^ { \rm P R } ( { \cal A } ) = \kappa ( \rho ) \| W | _ { \mathcal D } \| _ F \frac { W | _ { \mathcal D } } { \| W | _ { \mathcal D } \| _ F } . \end{align*}


\begin{align*} \dim K = | \sigma | - 1 > | \tau | - 1 = \dim L . \end{align*}


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


\begin{align*} ( V , \partial ^ \square c ) = - \mathrm { E } ( \Sigma , c ) = - ( V _ g , w _ g ) \end{align*}


\begin{align*} \tilde { d } _ H ( \xi , \eta ) = \tilde { d } _ H ( p _ 0 , q _ { \delta , \beta } ) . \end{align*}


\begin{align*} \abs { \log \left ( \frac { \rho ( \xi _ { \Delta } \eta _ { \Lambda \setminus \Delta } ) } { \rho ( \eta _ { \Lambda } ) } \right ) } = \abs { \log \left ( \frac { \rho ( \xi _ { \Delta } \lvert \eta _ { \Lambda \setminus \Delta } ) } { \rho ( \eta _ { \Delta } \lvert \eta _ { \Lambda \setminus \Delta } ) } \right ) } \leq \abs { \Delta } \log \left ( \frac { 1 } { \delta ( \rho ) } \right ) . \end{align*}


\begin{align*} g ( x ) = G \bigg ( \frac { 1 } { n } \sum _ { i = 1 } ^ n \delta _ { x _ i } \bigg ) , \ \ G : \P ( \R ) \to \R , \end{align*}


\begin{align*} M _ { I J } ^ \prime = \begin{pmatrix} G _ { I _ 1 } ^ \prime & G _ { I _ 2 } ^ \prime & \cdots & G _ { I _ { 2 k } } ^ \prime \\ G _ { J _ 1 } ^ \prime & G _ { J _ 2 } ^ \prime & \cdots & G _ { J _ { 2 k } } ^ \prime \end{pmatrix} \end{align*}


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


\begin{align*} ( \langle z _ 1 , \ldots , z _ n \rangle \in \mathfrak { C } ) \implies ( z _ 1 = z _ n ) z _ 1 \ldots , z _ n \in X . \end{align*}


\begin{align*} R _ { c , k } = \log _ 2 \left ( 1 + \gamma _ { c , k } \right ) . \end{align*}


\begin{align*} P = \sum _ { k = 0 } ^ d p _ { k } x _ 1 ^ { d - k } x _ 2 ^ k , \end{align*}


\begin{align*} x = \sqrt { { p _ s } } { x _ s } + \sqrt { { p _ b } } { x _ b } , \end{align*}


\begin{align*} Q ( h , v ) = \Big ( v A + \bar u A ' ( h ) - ( \nu ( u ) ) ' \bar g ^ \intercal , \ , 2 v \Big ) \mbox { o n } \hat \Sigma . \end{align*}


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


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


\begin{align*} \begin{aligned} & ( i ) & \left | \tilde w ^ \varepsilon ( x , t ) \right | & \leq K , \medskip \\ & ( i i ) & \left | \partial _ x \tilde w ^ \varepsilon ( x , t ) \right | & \leq K , \medskip \\ & ( i i i ) & \left | \tilde w ^ \varepsilon ( x , t ) - \tilde w ^ \varepsilon ( x , s ) \right | & \leq K \left ( \left | t - s \right | ^ { 1 / 2 } + | t - s | \right ) . \end{aligned} \end{align*}


\begin{align*} \sum _ { j = 1 } ^ d y _ { n , j } < \min _ { k = 1 , 2 , \ldots , d } \left ( x _ { n + 1 , k } + \sum _ { j \neq k } x _ { 0 , j } \right ) \forall n \geq N _ 0 . \end{align*}


\begin{align*} \psi _ n ( k h ) = \psi ( h ) \forall k \in K _ n , \forall h \in H . \end{align*}


\begin{align*} \mathcal { P } ^ - ( D _ j ) = \mathcal { P } ^ - \left ( R ( x ; z ) V _ { j } ( k ) \right ) ( k ) , j = 1 , 2 , \end{align*}


\begin{align*} R _ { k , j } ^ m = B _ { \rm R F } \log _ 2 \left ( 1 + \frac { p _ { k , j } ^ m \left | G _ { k , j } ^ m \right | ^ 2 } { { \sum \limits _ { j ' \ne j } \sum \limits _ { k ' \ne { { k } } } { p _ { k ' , j ' } ^ { { m } } { { { \left | G _ { k ' , j } ^ m \right | ^ 2 } } } } } + N _ { \rm R F } B _ { \rm R F } } \right ) , k \ne 0 , \end{align*}


\begin{align*} | C _ G ( x ) | = 1 + \deg ( x ) = { \frac { | G | } { | x ^ G | } } , \textrm { f o r a l l } x \in G . \end{align*}


\begin{align*} | y - s | \geq | s | - | y | \geq | s | - \frac 3 2 | x | > | s | - \frac 3 4 | s | = \frac 1 4 | s | . \end{align*}


\begin{align*} v ( x ) \left \{ \begin{array} { c l } \ge 0 & \bar u ( x ) = u _ a , \\ \le 0 & \bar u ( x ) = u _ b . \end{array} \right . \end{align*}


\begin{align*} R _ { \bar A } ( c ( x _ n ) ) = \Phi ( t _ n , c ( x _ n ) ) \in \partial A . \end{align*}


\begin{align*} B _ q ( \psi _ m ) ( z ) = \displaystyle \frac { z ^ m } { \sqrt { \Gamma ( q m + 1 ) } } , \end{align*}


\begin{align*} \begin{aligned} \mathcal { I } ( x ) = & \inf \left \{ \int ^ { 1 } _ { 0 } \Gamma ( \phi ( r ) ) d r ; ~ \phi \in L ^ 2 ( [ 0 , 1 ] ) \ { \rm i s \ s u c h \ t h a t \ } \right . \\ & \left . x ( t ) = x _ { 0 } + \int ^ { t } _ { 0 } b ( x ( s ) ) d s + \int ^ { t } _ { 0 } \sigma ( x ( s ) ) \phi ( s ) d s \right \} . \end{aligned} \end{align*}


\begin{align*} { \bf { h } } \left [ n \right ] = \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l \delta \left [ { n - { n _ l } } \right ] } , \end{align*}


\begin{align*} R & = ( a _ 1 , b _ 1 ) \times \cdots \times ( a _ d , b _ d ) , \\ R ' & = ( a ' _ 1 , b ' _ 1 ) \times \cdots \times ( a ' _ d , b ' _ d ) . \end{align*}


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


\begin{align*} \frac { \partial ^ 2 f ( z ) } { \partial z ^ 2 } = & \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \ , \phi ( z ) \ , [ \Phi ( z ) ] ^ { \alpha - 2 } \ , [ 1 - \Phi ( z ) ] ^ { \beta - 2 } \ , \frac { \partial s ( z ) } { \partial z } . \end{align*}


\begin{gather*} T ( A ) = 1 \quad A \in \mathcal { R } A \supset \{ \pi _ 1 > \pi _ 2 \} . \end{gather*}


\begin{align*} \sum \limits _ { n = 0 } ^ { \infty } d _ { o } ( n ) q ^ { n } \sum _ { n = 0 } ^ { \infty } q ^ { n ( n + 1 ) / 2 } \equiv \sum \limits _ { n = 1 } ^ { \infty } \frac { q ^ { 2 n } } { 1 - q ^ { 2 n } } - \sum \limits _ { n = 1 } ^ { \infty } \frac { q ^ { 4 n } } { 1 - q ^ { 4 n } } \pmod { 2 } \end{align*}


\begin{align*} & \Sigma _ 1 = \sum \limits _ { D \leq d < 2 D } \sum \limits _ { n \in \mathcal { N } ' ( d ) } \psi \left ( \frac { - n } { d ^ 2 } \right ) \ , , \\ & \Sigma _ 2 = \sum \limits _ { D \leq d < 2 D } \sum \limits _ { n \in \mathcal { N } ' ( d ) } \psi \left ( \frac { X - n } { d ^ 2 } \right ) \ , . \end{align*}


\begin{align*} \phi = \frac { 1 } { 2 } \left ( u _ { , p } \xi ^ p + v _ q x ^ q \right ) . \end{align*}


\begin{align*} \Theta f ( x ' , t ) : = \int _ { \R ^ { n - 1 } } \theta ( x ' , t ; y ' ) f ( y ' ) \ , d y ' , ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} \kappa _ 1 = a \kappa _ 2 + b , \end{align*}


\begin{align*} W ( t , x ) = \sqrt { C _ { d } \kappa } \ , \sum _ { k \in \Z ^ d _ 0 } \sum _ { i = 1 } ^ { d - 1 } \theta _ { k } \sigma _ { k , i } ( x ) W ^ { k , i } _ { t } , \end{align*}


\begin{align*} \left . \frac { \partial s ( z ) } { \partial z } \right | _ { z = 0 } = - \frac { 1 } { 4 } + \frac { 1 - \alpha } { \pi } . \end{align*}


\begin{align*} ( F ^ { \bullet , R l a } ) ^ { R l a } & = \varinjlim _ { h , h ' \to \infty } ( F ^ { R G _ { ( h ' ) } - a n } ) ^ { R G _ { ( h ) } - a n } \\ & = \varinjlim _ { h \to \infty } ( F ^ { R G _ { ( h ) } - a n } ) ^ { R G _ { ( h ) } - a n } \\ & = \varinjlim _ { h \to \infty } F ^ { R G _ { ( h ) } - a n } \\ & = F ^ { R l a } , \end{align*}


\begin{align*} \hat { Y } ^ { \epsilon } ( t ) & = - \big ( 3 \delta ^ { 1 / 2 } + \sup _ { t \in [ 0 , 1 ] } | \varphi ( t ) - \varphi _ m ( t ) | ^ { 1 / 2 } \big ) + \int ^ { t } _ { 0 } \big ( b ( \hat { X } ^ { \epsilon } ( s ) ) - b ( X ^ { \epsilon } ( s ) ) \big ) d s \\ & \quad - \lambda ( \epsilon ) \epsilon \int ^ { t } _ { 0 } \big ( \sigma ( \hat { X } ^ { \epsilon } ( s - ) ) - \sigma ( X ^ { \epsilon } ( s - ) ) \big ) \epsilon \theta ^ { \epsilon } ( s - ) \tilde { N } ^ { \epsilon ^ { - 2 } } ( d s ) . \end{align*}


\begin{align*} \Sigma _ 0 = \Sigma \setminus ( S _ p ( F ) \cup S _ { \infty } ( F ) ) \supset S . \end{align*}


\begin{align*} d ( z ) & = \sum _ { \alpha \in S _ + } : X ^ \alpha \varphi ^ \alpha : - \frac { 1 } { 2 } \sum _ { \alpha , \beta , \gamma \in S _ + } { f ^ { \alpha \beta } } _ \gamma : \varphi _ \gamma \varphi ^ \alpha \varphi ^ \beta : + \sum _ { \alpha \in S _ + } ( f | q ^ \alpha ) \varphi ^ \alpha + \sum _ { \alpha \in S _ { \frac { 1 } { 2 } } } : \varphi ^ \alpha \Phi _ \alpha : . \end{align*}


\begin{align*} \widetilde { \tau _ 2 } { \widetilde { \sigma } } ^ { - 1 } { \widetilde { \tau _ 2 } } ^ { - 1 } { \widetilde { \tau _ 1 } } ^ { - 1 } = { \widetilde \sigma } ^ { - 1 } { \widetilde { \tau _ 1 } } ^ { - 1 } [ \widetilde \sigma , \widetilde { \tau _ 2 } ] . \end{align*}


\begin{align*} H _ r ( E ^ { \vee } ) = H _ r ( E ) ^ { - 1 } \end{align*}


\begin{align*} \big [ M ^ k , v \wedge \partial _ v \big ] = k M ^ { k - 1 } \big [ M , v \wedge \partial _ v \big ] = 2 k M ^ { k - 1 } \sum _ { 1 \leq i \leq 2 } \Lambda _ i \big ( \big [ \Lambda _ i , v \big ] \wedge \partial _ v \big ) , \end{align*}


\begin{align*} \delta = \frac { \gamma } { 1 + 2 \gamma } , \end{align*}


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


\begin{align*} \mathcal { E } : = u _ 1 ^ t \Delta u _ 2 ^ t \wedge \dots \wedge \Delta u _ { k } ^ t \wedge \beta _ n ^ { n - m } - u _ 1 \Delta u _ 2 \wedge \dots \wedge \Delta u _ { k } \wedge \beta _ n ^ { n - m } \longrightarrow 0 , t \to \infty , \end{align*}


\begin{align*} T _ { \delta } - \mathcal { P } ^ - \left ( T _ { \delta } R _ { \delta } \right ) = F _ { \delta } , \end{align*}


\begin{align*} \| ( j - \ell _ w ) \alpha \| \geq \| q _ n \alpha \| \geq \frac { 1 } { 2 q _ { n + 1 } } = \frac { 1 } { 2 } e ^ { - \beta _ n q _ n } \geq \frac { 1 } { 2 } e ^ { - 3 0 0 \varepsilon q _ n } . \end{align*}


\begin{align*} D _ x ( A ^ q \eta ) ( x ) & = D _ x \left ( \int _ { x - r } ^ { x + r } ( \eta ( y ) - \eta ( x ) ) \cos ( 2 \pi q ( y - x ) ) \ \d y \right ) \\ & = ( \eta ( x + r ) - \eta ( x ) ) \cos ( 2 \pi q r ) - ( \eta ( x - r ) - \eta ( x ) ) \cos ( 2 \pi q r ) \\ & + \int _ { x - r } ^ { x + r } - \partial \eta ( x ) \ \cos ( 2 \pi q ( y - x ) ) + 2 \pi q ( \eta ( y ) - \eta ( x ) ) \sin ( 2 \pi q ( y - x ) ) \ \d y . \end{align*}


\begin{align*} 0 = u c \lim _ i \| \varpi \otimes \varpi ( \cdot ) \xi _ i - \xi _ i \| \end{align*}


\begin{align*} \begin{cases} a \mapsto a b , \\ c \mapsto c , & c \neq a , \end{cases} \begin{cases} a \mapsto b a , \\ c \mapsto c , & c \neq a . \end{cases} \end{align*}


\begin{align*} \ell ( x ) = a _ n x ^ n + a _ { n - 1 } x ^ { n - 1 } + \cdots + a _ 0 \end{align*}


\begin{align*} 1 \le \frac { \sup _ { t \ge x } f ( t ) } { f ( x ) } = \sup _ { t \ge x } \frac { f ( t ) } { \alpha ( x ) } \frac { \alpha ( x ) } { f ( x ) } \le \sup _ { t \ge x } \frac { f ( t ) } { \alpha ( t ) } \cdot \frac { \alpha ( x ) } { f ( x ) } \to 1 \end{align*}


\begin{align*} | p | _ { A ( x ) } : = \left ( a _ { i j } ( x ) p _ i p _ j \right ) ^ { 1 / 2 } \end{align*}


\begin{align*} E ( G ) = \{ V _ i \cup V _ j : \ : \{ i , j \} \in F \} . \end{align*}


\begin{align*} s _ \alpha ( - w ^ { - 1 } \gamma ) = \langle \alpha ^ \vee , w ^ { - 1 } \gamma \rangle \alpha - w ^ { - 1 } \gamma \in \Phi ^ - . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } i . ~ ~ \phi = 2 \pi N , ~ N ~ i s ~ i n t e g e r \\ i i . ~ ~ \phi = \frac { 2 \pi Z } { N } , \frac { Z } { N } ~ i s ~ n o t ~ i n t e g e r . \\ \end{array} \right . \end{align*}


\begin{align*} \hat z = \Phi ( z ) = \frac { 1 } { \Phi _ 1 ( z ) } \end{align*}


\begin{align*} G r _ w ( E ) = G r _ w ( F ) . \end{align*}


\begin{align*} f ( z , t ) = \int _ \Omega \int _ { \C ^ n } \widehat { f } ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \ , d \nu ( a ) . \end{align*}


\begin{align*} y \left [ i \right ] = \left ( { \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { \bf { H } } _ l ^ \bot { \bf { X } } _ l } } \right ) { \bf { d } } \left [ { i - { n _ { \max } } } \right ] + z \left [ i \right ] . \end{align*}


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


\begin{align*} \dot g ( J ) \in A t ^ { \rho ( \dot g ( J ) ) } , \end{align*}


\begin{align*} | A | ^ m | A ' + B _ 1 + \ldots + B _ m | \le ( 1 + \varepsilon ) | A ' | \prod _ { i = 1 } ^ m | A + B _ i | . \end{align*}


\begin{align*} { \rho ^ { \circ } ( K ) } = { \sup _ { x \in K } \rho ( \{ x \} ) } < { \infty } , \end{align*}


\begin{align*} \chi ^ g = \chi \Rightarrow \chi ( n ^ g ) = \chi ( n ) \forall n \in N \Rightarrow n ^ g = n \forall n \in N . \end{align*}


\begin{align*} F ( z ) ( 0 ) = \lambda _ { \Theta } [ ( \Theta \circ h ) ( z ) , \alpha _ z ] ( 0 ) = ( \Theta \circ h ) ( z ) \end{align*}


\begin{align*} \# \{ \beta \in \Phi ^ + \setminus I \mid \varphi ( v ' \beta ) < 0 \} = & \# \{ \beta \in \Phi ^ + \setminus ( I \cup \{ \alpha \} ) \mid \varphi ( v s _ \alpha \beta ) < 0 \} \\ = & \# \{ \beta \in \Phi ^ + \setminus ( I \cup \{ \alpha \} ) \mid \varphi ( v \beta ) < 0 \} \\ = & \# \{ \beta \in \Phi ^ + \setminus I \mid \varphi ( v \beta ) < 0 \} - 1 . \end{align*}


\begin{align*} \varphi ^ L _ { \sigma , s ' } ( D ) = \varphi ^ L _ { \sigma , s ' } ( \{ c \} ) \varphi ^ L _ { \sigma , s ' } ( E ) = \varphi ^ L _ { \sigma , s ' } ( C _ i ) . \end{align*}


\begin{align*} U ^ { \dagger } = A + \sum _ { k = 1 } ^ { n - 2 } \Delta _ k . \end{align*}


\begin{align*} \psi _ { \tau } ( p ) \stackrel { d e f } { = } \left [ \ p \int _ 0 ^ { \infty } \ t ^ { p - 1 } \ T _ { \tau } ( t ) \ d t \ \right ] ^ { 1 / p } = | | \tau | | L _ p ( \Omega ) , \end{align*}


\begin{align*} u = z \cdot ( n _ 1 , \dotsc , n _ l ) \in \C ^ l \end{align*}


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


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


\begin{align*} H ( E ) = \left ( \N \left ( \prod _ { i = 1 } ^ { \ell } \mathfrak a _ i \right ) \prod _ { v \in V _ \infty } \det ( h _ { v } ( b _ i , b _ j ) ) ^ { e _ { v } / 2 } \right ) ^ { 1 / [ K : \mathbb { Q } ] } \end{align*}


\begin{align*} t ( y ) = \frac { 6 } { y ^ 2 } + 4 ( b - 1 ) \frac { \exp ( y ) } { \left [ 1 - \exp ( y ) \right ] ^ 2 } . \end{align*}


\begin{align*} \lambda ^ * = - 3 \sqrt 2 \frac { \sqrt { 1 + a ^ 2 } } { 2 + \frac { 1 9 7 } { 1 9 9 } a ^ 2 } \eta . \end{align*}


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


\begin{align*} h _ s ( i ) : = & \exp \Big ( - \max \Big \{ \frac { 1 2 8 \alpha \ , \log n } { \tilde \varepsilon ^ 2 ( 1 + \tilde \varepsilon ) ^ { - i } m _ s } , \ , C _ h \Big \} \Big ) , \end{align*}


\begin{align*} p x E \big [ X Y ^ { p - \gamma } ] E \big [ Y ^ { p } \big ] ^ { \frac { \beta } { p } } & \leq p x E \big [ X \big ] _ { \frac { p } { \gamma } } E \big [ Y ^ { p } \big ] ^ { 1 - \frac { \gamma - \beta } { p } } \\ & \leq \big [ X \big ] _ { \frac { p } { \gamma } } \big ( ( \gamma - \beta ) x ^ { \frac { p } { \gamma - \beta } } + ( p + \beta - \gamma ) E \big [ Y ^ { p } \big ] \big ) \end{align*}


\begin{align*} ( x _ 1 , x _ 2 , \cdots , x _ k ) \bar { G } = { \bf 0 } , \end{align*}


\begin{align*} ( a / b ) ^ { k _ 1 } = ( \alpha / \beta ) ^ { l _ 1 } \zeta \end{align*}


\begin{align*} \prescript { L _ i } { } { } x { } ^ { R _ j } = \prescript { L _ i } { } { } \left ( \prescript L { } { } x { } ^ R \right ) { } ^ { R _ j } \underset { } \leq \prescript { L _ i } { } { } \left ( \prescript L { } { } y { } ^ R \right ) { } ^ { R _ j } = \prescript { L _ i } { } { } y { } ^ { R _ j } . \end{align*}


\begin{align*} \hat { f } _ i ( x _ i ) = \int _ { \R ^ { n - 1 } } f ( x _ 1 , \ldots , x _ n ) \ , \prod _ { j \neq i } Q _ j ( d x _ j ) \ge - c _ 1 e ^ { c _ 2 x _ i ^ 2 } \prod _ { j \neq i } \int _ { \R } e ^ { c _ 2 x _ j ^ 2 } Q _ j ( x _ j ) \ , d x _ j \end{align*}


\begin{align*} Z _ { i } ^ { \theta } : = \theta _ { i } X _ { i } + ( 1 - \theta _ { i } ) Y _ { i } , ~ 1 \leq i \leq n . \end{align*}


\begin{align*} X _ n = ( \mathfrak X , \mathcal O _ { \mathfrak X } \otimes _ R R _ n ) , \end{align*}


\begin{align*} \begin{aligned} B & = \nabla \times ( \Phi \nabla \theta ) + G \nabla \theta , \\ A & = \nabla \times ( \eta \nabla \theta ) + \Phi \nabla \theta , \\ \Phi & = \phi - \phi _ { \infty } . \end{aligned} \end{align*}


\begin{align*} [ x , e ^ f ( x ) ] = s ( \pi _ \mathcal Z ( x ) ) = s ( \pi _ \mathcal Z ( \gamma x ) ) = [ \gamma x , e ^ f ( \gamma x ) ] = [ x , \rho ( \gamma ) ^ { - 1 } e ^ f ( \gamma x ) ] , \end{align*}


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


\begin{align*} \begin{aligned} & \dfrac { b _ { 1 , \delta } ( x ) } { 2 } = \ , \int _ 0 ^ 1 h ( x + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \\ = & \ , \int _ 0 ^ 1 [ \tilde { h } ( x + \delta \sin ( 2 \pi t ) ) + m ( x + \delta \sin ( 2 \pi t ) ) ] \sin ( 2 \pi t ) \ , d t \\ = & \ , \int _ 0 ^ 1 \tilde { h } ( x + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \\ & \ , + \int _ 0 ^ 1 [ m ( x + \delta \sin ( 2 \pi t ) ) - m ( x ) ] \sin ( 2 \pi t ) \ , d t . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} S _ { e , \mathcal { Y M } ^ 0 } ( X , Y ) & = \big ( \exp ( \frac { | | R ^ { \nabla } | | ^ 2 } 2 ) - 1 \big ) g ( X , Y ) - \exp ( \frac { | | R ^ { \nabla } | | ^ 2 } 2 ) \langle i _ X R ^ { \nabla } , i _ Y R ^ { \nabla } \rangle \ , , \\ \end{aligned} \end{align*}


\begin{align*} \ddot { \kappa } + \frac { 1 } { \rho } \dot { \kappa } + \left ( 1 - \frac { ( 3 / 2 ) ^ { 2 } } { \rho ^ { 2 } } \right ) \kappa = 0 , \kappa & > 0 , 0 < \rho < R _ 0 , \\ \kappa ( R _ 0 ) & = 0 . \end{align*}


\begin{align*} \dot { x } = \gamma f ( x , t ) x ( 0 ) = x _ 0 . \end{align*}


\begin{align*} \widetilde \mu _ { ( x , t ) } = ( \phi _ t ) _ * \widetilde \mu _ { ( x , 0 ) } = ( \phi _ t ) _ * ( \widehat \mu _ { x } \times \delta _ 0 ) = \widehat \mu _ { x } \times \delta _ t \end{align*}


\begin{align*} E _ 1 = i \sum _ \alpha \frac { \partial ( q _ 1 - b _ 1 ) } { \partial \xi _ \alpha } \frac { \partial q _ 1 } { \partial x _ \alpha } + b _ 0 q _ 1 + \frac { \partial q _ 1 } { \partial x _ n } - c _ 1 . \end{align*}


\begin{align*} \begin{cases} \partial _ t \mathbf { w } + ( \mathbf { v } \cdot \nabla ) \mathbf { w } = ( \mathbf { w } \cdot \nabla ) \mathbf { v } , \ \ & D \times ( 0 , T ) , \\ \nabla \cdot \mathbf { v } = 0 , \ \ & D \times ( 0 , T ) , \\ \mathbf { v } \cdot \mathbf { n } = v _ n , \ \ & \partial D \times ( 0 , T ) , \\ \mathbf { w } ( x , 0 ) = \nabla \times \mathbf { v } _ 0 ( x ) , \ \ & D . \end{cases} \end{align*}


\begin{align*} i \frac { \partial u } { \partial t } & = \Delta u + c u + b \cdot \nabla u + f \ ( s , T ) \times \mathbb { R } ^ d , \\ u ( s ) & = x , \end{align*}


\begin{align*} \begin{aligned} & \int _ { \Omega } \Delta u _ { \epsilon } ( x ) \wedge \Delta v _ 1 ( x ) \wedge \dots \wedge \Delta v _ { m - 1 } ( x ) \wedge \beta _ n ^ { n - m } \wedge \omega ( x ) \\ = & \int _ { B ( 0 , \epsilon ) } \chi _ { \epsilon } ( y ) d V ( y ) \int _ { \Omega } u ( z ) \Delta v _ 1 ( z + y ) \wedge \dots \wedge \Delta v _ { m - 1 } ( z + y ) \wedge \beta _ n ^ { n - m } \wedge \Delta \omega ( z + y ) \geq 0 , \end{aligned} \end{align*}


\begin{align*} \forall x \in E _ v \ , , \ \Vert x \Vert _ v = \begin{cases} \sqrt { h _ v ( x , x ) } v \in V _ { \infty } \\ \inf \left \lbrace \vert \alpha \vert _ v \ , , \ \alpha \in K _ v \ , , \ x \in \alpha L \right \rbrace v \in V _ f . \end{cases} \end{align*}


\begin{align*} \Sigma ( x , \mu ) = \hat { \kappa } + \mathrm { d i a g } ( x ) \hat { \eta } + \sum _ { n = 1 } ^ { N } \null _ { n } \eta \int _ { \R ^ { m } } g _ { n } ( x , y ) \ , \mu ( d y ) \end{align*}


\begin{align*} & x ^ { t + 1 } ( \gamma ) = ( I + \gamma B ) ^ { - 1 } \left ( x ^ t + \alpha ( \gamma ) ( x ^ t - x ^ { t - 1 } ) - \gamma \left ( F ( x ^ t ) + \beta ( \gamma ) ( F ( x ^ t ) - F ( x ^ { t - 1 } ) ) \right ) \right ) \forall \gamma > 0 , \\ & L _ { t + 1 } ( \gamma ) = \| F ( x ^ { t + 1 } ( \gamma ) ) - F ( x ^ { t } ) - \kappa _ t \gamma ^ { - 1 } ( x ^ { t + 1 } ( \gamma ) - x ^ t ) \| / \| x ^ { t + 1 } ( \gamma ) - x ^ t \| \forall \gamma > 0 , \end{align*}


\begin{align*} \delta = \limsup \max ( 0 , \delta _ n ) . \end{align*}


\begin{align*} \langle \langle \varphi , \varphi ' \rangle \rangle = \tau \langle \langle \varphi ' , \varphi \rangle \rangle \end{align*}


\begin{align*} R _ j = \sum \limits _ { \forall v } \sum \limits _ { \forall q } R _ { v , j } ^ q + \sum \limits _ { \forall k , k \ne 0 } \sum \limits _ { \forall m } R _ { k , j } ^ m + \sum \limits _ { \forall n } R _ { 0 , j } ^ n , \end{align*}


\begin{align*} \sum _ { x \in V _ i } \deg _ { F _ \ell } ( x ) = \ell < s = \frac { d m } { 2 k ^ 2 } . \end{align*}


\begin{align*} \eta ^ { D } ( A \times [ b , \infty ) ) = e ^ { - \pi \lambda b } \langle \eta ^ { D } , f _ { b } \rangle & = e ^ { - \pi \lambda b } \langle \eta ^ { D } , f \rangle \\ & = ( \pi \lambda ) ^ { - 1 } e ^ { - \pi \lambda b } Z _ { \lambda } ^ { D } ( A ) = \int _ { A \times [ b , \infty ) } Z _ { \lambda } ^ { D } ( \mathrm { d } x ) e ^ { - \pi \lambda h } \mathrm { d } h \end{align*}


\begin{align*} \sum _ { \lambda , \mu } \exp \Big ( \frac { - \pi ( u ^ 2 + v ^ 2 ) } { v } S [ \lambda ] - \frac { \pi } { v } S [ \mu ] - \frac { 2 \pi } { v } u \lambda ^ t S \mu \Big ) = \sum _ { \lambda , \mu } \exp \Big ( - \pi v S [ \lambda ] - \frac { \pi } { v } S [ u \lambda + \mu ] \Big ) . \end{align*}


\begin{align*} \sum _ { y = 0 } ^ n M _ { x , y } ( m ) g _ y = S ^ { p } _ { x , x } ( m ) . \end{align*}