\begin{align*} S _ { k } ( c ; \xi ) = & { \sum _ { a \ , \hbox { \tiny m o d } \ , k } } ^ { \ ! \ ! \ ! * } \ ; \sum _ { z '' \ , \hbox { \tiny m o d } \ , k } e _ { k \ell } \big ( a \ell F _ h ( z '' ) + c \cdot ( \xi + \ell \bar \ell ( z '' - \xi ) ) \big ) , \end{align*}


\begin{align*} \lambda _ * H ^ { ( 1 ) } ( \mu _ N ) - \lambda _ * \Psi ( - 1 ) & \cong \Psi _ * \binom { N } { 2 } , \\ \Psi ^ * \Psi _ * \binom { N } { 2 } & \cong * . \end{align*}


\begin{align*} v = v ( \cdot , \xi ) = \log ( w _ { i j } \xi _ i \xi _ j ) + \tau | D u | ^ 2 + e ^ { \kappa \phi } + \beta \log ( u _ 0 - u ) \end{align*}


\begin{align*} ( D / \mathcal { W } ) _ { i , j } = \begin{cases} | W _ j | d _ G ( w _ i , w _ j ) & \\ ( | W _ i | - 1 ) & \\ 2 ( | W _ i | - 1 ) & \\ \end{cases} \end{align*}


\begin{align*} \widetilde { Q } _ { ( X , \eta , \omega ) } : = \{ ( X _ t , \eta _ t ) \ : \ t \in Q _ { ( X , \eta , \omega ) } \} \end{align*}


\begin{align*} R _ k : = A [ x _ 1 , \ldots , x _ n ] / ( x _ 1 , \ldots , x _ n ) ^ { k + 1 } , \end{align*}


\begin{gather*} \# \Gamma _ 0 ( N ) _ Q = \begin{cases} 6 & \\ 4 & \\ 2 & \end{cases} \end{gather*}


\begin{align*} \tilde { \pi } _ t = \tilde { F } ( \tilde { \pi } _ { t - 1 } , e _ t , y _ t ) \end{align*}


\begin{align*} \inf _ { 2 \leq l \leq s } \inf _ { ( x , y ) \in \bar B } \sum _ { i = 1 } ^ { n - 2 } ( \varphi _ i ^ 1 ( x , y ) - \varphi _ i ^ l ( x , y ) ) ^ 2 > \varepsilon . \end{align*}


\begin{align*} | a - b | + | b - c | - | a - c | = 2 \left ( \max ( a , b , c ) - \max ( a , c ) + \min ( a , c ) - \min ( a , b , c ) \right ) . \end{align*}


\begin{align*} \alpha ( k ) = \beta ( k ) \frac { \mu ( k + 1 ) } { \mu ( k ) } \textrm { w h e r e } \mu ( 0 ) = 1 . \end{align*}


\begin{align*} x \begin{pmatrix} P _ { 0 } ( x ) \\ P _ { 1 } ( x ) \\ P _ { 2 } ( x ) \\ \vdots \end{pmatrix} = \begin{pmatrix} b _ { 0 } & 1 & & & \\ a _ { 0 } & b _ { 1 } & 1 & \\ & a _ { 1 } & b _ { 2 } & 1 & \\ & & \ddots & \ddots & \ddots \end{pmatrix} \begin{pmatrix} P _ { 0 } ( x ) \\ P _ { 1 } ( x ) \\ P _ { 2 } ( x ) \\ \vdots \end{pmatrix} \end{align*}


\begin{align*} B Z ^ { \nu } _ i = D ^ { \nu t } _ i + F _ i ^ { \nu t } , \end{align*}


\begin{align*} & \lim _ { n \rightarrow \infty } n ^ { 2 \alpha + 1 } \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' ( X _ s ) \Theta ^ n _ s ) ^ 2 d s \\ & = \kappa _ 2 ^ 2 \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' \sigma ) ^ 2 ( X _ s ) d s , \end{align*}


\begin{align*} { \partial _ t } x \left ( t \right ) = \xi \left ( t \right ) x \left ( t \right ) \ , \ \ x ( 0 ) = 1 \end{align*}


\begin{align*} \widetilde { N } _ f : = M _ f S _ \zeta ; \widetilde { M } _ f : = \frac { 1 } { [ | W _ f | ] } M _ f S _ \zeta , f \in \Z ^ { m | n } _ { \zeta - } . \end{align*}


\begin{align*} \Delta \left ( a \right ) & = 2 \ln \left ( \frac { \sqrt { 2 } \Gamma \left ( \frac { 2 \left | a \right | + 1 } { 4 } + \frac { 1 } { 2 } \right ) } { \sqrt { \left | a \right | } \Gamma \left ( \frac { 2 \left | a \right | + 1 } { 4 } \right ) } \right ) + \ln { \left | a \right | } = 2 \ln \left ( \frac { \sqrt { 2 } \Gamma \left ( \frac { \left | a \right | } { 2 } + \frac { 3 } { 4 } \right ) } { \Gamma \left ( \frac { \left | a \right | } { 2 } + \frac { 1 } { 4 } \right ) } \right ) , \end{align*}


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


\begin{align*} b ( x ) \geq b _ 0 > 0 \ \ \ \ \omega _ b \subset \Omega , c ( x ) \geq c _ 0 \neq 0 \ \ \ \ \omega _ c \subset \Omega c ( x ) = 0 \ \ \ \ \Omega \backslash \omega _ c \end{align*}


\begin{align*} \Re \left ( \left < { \mathcal { A } } _ m U , U \right > _ { \mathcal { H } } \right ) = c ( m - 1 ) \int _ 0 ^ L \abs { \omega _ x } ^ 2 d x + \frac { c m } { 2 } \int _ 0 ^ { L } \int _ 0 ^ \infty \sigma ' ( s ) \abs { \eta _ x } ^ 2 d s d x \leq 0 , \end{align*}


\begin{align*} h _ t ( x , y ) = p _ t ( x , y ) - \frac { 1 } { n } . \end{align*}


\begin{align*} E _ { n u l l } ( \dot x , x , t ) = - \frac { \partial \Phi _ { n u l l } ( x , t ) } { \partial t } \ . \end{align*}


\begin{align*} \lim _ { z _ { k } \rightarrow + \infty } H ( \boldsymbol { z } ) \overline { \mathcal { G } } _ { n } \left ( \frac { 1 } { 2 } \boldsymbol { z } ^ { T } \boldsymbol { z } \right ) = 0 , ~ k \in \{ 1 , 2 , \cdots , n \} . \end{align*}


\begin{align*} A = \frac { 1 } { n } \sum _ { t = 1 } ^ n ( w _ t s _ t + \alpha s _ t ^ 2 ) \end{align*}


\begin{align*} \begin{cases} f ( k , v , 0 ) = F ( k , v , 0 ) , \\ & \\ f ( k \exp ( t v ) , t ) = F ( k , v , t ) , & t > 0 , \end{cases} \end{align*}


\begin{align*} \| u \| = \sqrt { M / N } , \| u ' \| = \sqrt { 1 - M / N } . \end{align*}


\begin{align*} X _ M : = \left \{ x \in X \mid M _ { x } \not = 0 \right \} . \end{align*}


\begin{align*} K = \bigcap _ { n = 1 } ^ \infty K ( \alpha - 1 , A _ n , u _ n + a _ n f _ { q ( n ) } ) . \end{align*}


\begin{align*} D = \{ \mu = ( \mu _ 1 , \ldots , \mu _ n ) \ , | \ , \hbox { $ 0 \le \mu _ i \le r _ i - 2 $ f o r $ 1 \le i \le n $ } \} . \end{align*}


\begin{align*} x _ j ^ { d + 1 } = x _ j ^ 2 \cdot x _ j ^ { d - 1 } = \varpi _ j x _ j ^ { d - 1 } x _ { j + 1 } - \varpi _ { j - 1 } x _ j ^ d ( 1 \le j \le n - 1 ) \end{align*}


\begin{align*} \operatorname { d i v } ( Y ) \mu = L _ { Y } ( \mu ) = d \circ i _ { Y } ( \mu ) , \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } ( a u '' v ' ) ( x _ 0 - \delta ) = \lim _ { \delta \rightarrow 0 } ( a u '' v ' ) ( x _ 0 + \delta ) . \end{align*}


\begin{align*} \alpha : = \psi _ { x , y } \circ \iota \circ \psi _ { y , x } ^ { - 1 } : \mathcal E \to \mathcal E \end{align*}


\begin{align*} \partial ( \overline { \mathcal { M } ( \alpha , \beta ) / \R } ) = \bigcup _ r \overline { \mathcal { M } ( \alpha , r ) / \R } \times \overline { \mathcal { M } ( r , \beta ) / \R } . \end{align*}


\begin{align*} R _ { \widetilde { G } } R _ { \widetilde { D } } ^ k R _ G = R _ { G } R _ D ^ k R _ { \widetilde { G } } R _ { \widetilde { D } } { R _ { \widetilde { G } } ^ k } R _ D = R _ { D } R _ G ^ k { R _ { \widetilde { D } } } . \end{align*}


\begin{align*} \overline { \omega } ^ { n + 1 } = r ^ { 2 n + 1 } ( \Pi ^ { \ast } \omega _ { h } ) ^ { n } \wedge d r \wedge \overline { \eta } , \end{align*}


\begin{align*} \begin{aligned} P _ c & ( \Psi _ t ) P _ c ^ T ( \Psi _ t ) ( x ) \\ & = \begin{bmatrix} I _ n + O ( t ) & \Upsilon _ c ^ T + O ( t ) \\ \Upsilon _ c + O ( t ) & \frac { 1 } { 2 t } \cdot \begin{pmatrix} \begin{bmatrix} I _ { \frac { n ( n - 1 ) } { 2 } } & 0 \\ 0 & 3 \cdot \Xi ( \frac { 1 } { 3 } ) - \frac { n + 2 } { n } \cdot J _ n \end{bmatrix} + O ( t ) \end{pmatrix} + \Gamma _ c ( x ) \end{bmatrix} , \end{aligned} \end{align*}


\begin{align*} \mathbf { H } ^ { r } & = \mathbf { A } ( \boldsymbol { x } ^ { r } ) \mathbf { A } ^ { H } = \sum _ { m = 1 } ^ { \widetilde { M } } x _ { m } ^ { r } \boldsymbol { a } \left ( \overline { \theta } _ { m } \right ) \boldsymbol { a } ^ { H } \left ( \overline { \theta } _ { m } \right ) , \\ \mathbf { h } ^ { c } & = \mathbf { A } \boldsymbol { x } ^ { c } = \sum _ { m = 1 } ^ { \widetilde { M } } x _ { m } ^ { c } \boldsymbol { a } \left ( \overline { \theta } _ { m } \right ) , \end{align*}


\begin{align*} k ' = \frac { 1 } { 2 \hslash \sqrt { \beta / 3 } } \sqrt { 1 + \sqrt { 1 - \frac { 1 6 } { 3 } m \beta | E | } } . \end{align*}


\begin{align*} \sum _ j k _ 1 ( j ) = r ( J ) + n _ 1 r , \sum _ j k _ 2 ( j ) = s ( J ) + n _ 2 s . \end{align*}


\begin{align*} \varphi ( \lambda ) = h ( \lambda ) \cdot q _ { n , m } ( \lambda ) , \ ; \ ; \ ; \lambda \in \mathfrak { a } ^ * _ \mathbb { C } , \end{align*}


\begin{align*} & \Delta ( E _ i ) = E _ i \otimes 1 + K _ i \otimes E _ i , \\ & \Delta ( F _ i ) = F _ i \otimes K _ i ^ { - 1 } + 1 \otimes F _ i , \\ & \Delta ( K _ i ^ { \pm 1 } ) = K _ i ^ { \pm 1 } \otimes K _ i ^ { \pm 1 } . \end{align*}


\begin{align*} a _ j ( x ) = ( A _ 2 - A _ 1 ) ( x ) \cdot e _ j = A ( x ) \cdot e _ j , j = 1 , 2 , 3 , x \in \R ^ 3 , \end{align*}


\begin{align*} M ^ { n , 1 } _ s = \int ^ s _ { \eta _ n ( s ) } ( s - \eta _ n ( u ) ) ^ { \alpha } \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \end{align*}


\begin{align*} \sum _ { h = 1 } ^ { + \infty } \frac { 1 } { h ^ \alpha ( k + h ) ^ \beta } & \leq \sum _ { h = 1 } ^ { + \infty } \frac { 1 } { h ^ { \alpha + \beta - \gamma } ( k + h ) ^ \gamma } \\ & = \frac { 1 } { k ^ \gamma } \sum _ { h = 1 } ^ { + \infty } \frac { 1 } { h ^ { \alpha + \beta - \gamma } ( 1 + h / k ) ^ \gamma } \\ & \leq \frac { 1 } { k ^ \gamma } \sum _ { h = 1 } ^ { + \infty } \frac { 1 } { h ^ { \alpha + \beta - \gamma } } \end{align*}


\begin{align*} E : = s x \partial _ x + r y \partial _ y - \sum _ { \stackrel { 0 \le \alpha \le r - 2 , \ 0 \le \beta \le s - 2 , } { d \ge 0 } } ( s \beta + r \alpha + r s ( d - 1 ) ) t _ { \alpha , \beta , d } \partial _ { t _ { \alpha , \beta , d } } . \end{align*}


\begin{align*} A _ 3 & = \{ x \in X _ n \mid ( x , y ) \in B y \sim x \} , \\ A _ 1 & = \{ x \in X _ n \setminus A _ 3 \mid v ( x , y ) \neq 0 y \sim x \} , \\ A _ 2 & = X _ n \setminus ( A _ 1 \cup A _ 3 ) . \end{align*}


\begin{align*} \mathcal { F } f ( p ; \nu ) = \int d u \ , f ( u ) \ , ( p \cdot u ) ^ { i \nu - 1 } , \end{align*}


\begin{align*} ( \alpha , \beta ) _ { L , n } = ( \alpha v _ n \delta _ n ( \beta ) , v _ n ) _ { L , n } . \end{align*}


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


\begin{align*} E [ | Q ^ { n , 4 } _ \tau | ^ 2 ] \le C \sum _ { j = 0 } ^ { \lfloor n \tau \rfloor } \left ( \int _ { \frac j n } ^ { \frac { j + 1 } n \wedge \tau } \gamma _ s d s \right ) ^ 2 \le C \sup _ { 0 \le j \le \lfloor n \tau \rfloor } \left ( \int _ { \frac j n } ^ { \frac { j + 1 } n \wedge \tau } \gamma _ s d s \right ) \int _ 0 ^ { \tau } \gamma _ s d s , \end{align*}


\begin{align*} \P \{ \ell ^ { ( A ) } _ i \leq m - i \} = \sum \limits _ { k = 0 } ^ \infty \mu ( k ) \left ( \P \left \{ \forall t > 0 : t > \sum \limits ^ { i + S _ { t } } _ { z = i + 1 } \frac { 1 } { A ( z ) } S _ { t } < m - i \right \} \right ) ^ k , \end{align*}


\begin{align*} \begin{aligned} \mathcal { N } ^ s _ p u ( x ) = \int _ { \Omega } \dfrac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) } { | x - y | ^ { N + p s } } d y , ~ ~ \forall x \in \R ^ N \setminus \Omega , \end{aligned} \end{align*}


\begin{align*} \Upsilon _ c : = \begin{bmatrix} \cdots & \cdots & \cdots & \cdots \\ \Gamma _ { i j } ^ 1 & \Gamma _ { i j } ^ 2 & \cdots & \Gamma _ { i j } ^ n \\ \cdots & \cdots & \cdots & \cdots \\ \Gamma _ { i i } ^ 1 & \Gamma _ { i i } ^ 2 & \cdots & \Gamma _ { i i } ^ n \\ \cdots & \cdots & \cdots & \cdots \end{bmatrix} , i \neq j , \end{align*}


\begin{align*} K _ { + } = \{ s + i t \in K : t \ge f _ { \nu } ( s _ { 0 } ) + \varepsilon \} \end{align*}


\begin{align*} A ^ { \tt s t } = ( A ^ { \circ } ) ^ { \circ ' } . \end{align*}


\begin{align*} Y _ { ; i j } ^ k - Y _ { ; j i } ^ k = - { R _ { i j m } } ^ k Y ^ m \end{align*}


\begin{align*} g _ { 0 } = d r ^ 2 + \sin ^ 2 r \cdot h _ { \mathbb { S } ^ { n - 2 } } + \cos ^ 2 r \cdot d s ^ 2 \end{align*}


\begin{align*} b = - 3 = - 1 \end{align*}


\begin{align*} \log \ , D _ { 2 , n } = \sum _ { p : } \max _ { 0 \le k \le n } \log \ , \left | \dfrac { k ! } { ( \beta ) _ k } \right | _ p & \le \log ( | c | + ( n - 1 ) d ) \pi _ { | c | , d } ( | c | + ( n - 1 ) d ) \enspace , \end{align*}


\begin{align*} \sum _ { u = i } ^ s \epsilon ^ { \bullet } ( i ) \epsilon ^ { \bullet } ( s ( \alpha _ i ) ) ^ { - 1 } q _ { \alpha _ u } \alpha _ u \alpha ^ * _ u - \sum _ { v = i } ^ t \epsilon ^ { \bullet } ( t ( \beta _ i ) ) \epsilon ^ { \bullet } ( i ) ^ { - 1 } \beta _ v \beta _ v ^ * \end{align*}


\begin{align*} \begin{aligned} x & \longmapsto x ( 1 + ( b - a ) g x ^ a y ^ b ) ^ { ( b + 1 ) / ( b - a ) } \\ y & \longmapsto y ( 1 + ( b - a ) g x ^ a y ^ b ) ^ { ( a + 1 ) / ( a - b ) } . \end{aligned} \end{align*}


\begin{align*} \Delta _ n = \frac { n - 2 } { n - 1 } \Delta _ { n - 2 } - \frac { 1 } { n - 1 } \lambda _ { n - 2 } , n \geq 3 , \end{align*}


\begin{align*} \lim \limits _ { \ell \rightarrow \infty } \dfrac { \mathbb { E } [ \ell , k ] } { \ell } = \dfrac { 1 } { 2 } \end{align*}


\begin{align*} \sum a _ m x ^ m . v + a _ 0 v + \sum b _ n y ^ n . v = 0 . \end{align*}


\begin{align*} \sigma _ p ( N , \xi , h ) = h ^ { - 2 } \tau _ p ( B _ h ) . \end{align*}


\begin{align*} \sqrt { n } { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \nu _ { \hat \theta _ n } ) = \mathbb { M } _ n ( \hat t _ n ) + o _ { \mathbb { P } } ( 1 ) . \end{align*}


\begin{align*} D _ V X = D _ X V = \frac { X f } { f } V + P _ X V . \end{align*}


\begin{align*} \int _ 0 ^ 1 \| \tilde { B } _ 0 S _ n ( s ) \| _ { r _ n } \ , \mathrm { d } s < \infty , n = 0 , 1 , \ldots , \ell , \end{align*}


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


\begin{align*} \phi ( x ) = A e ^ { i k x } + B e ^ { - i k x } + C e ^ { k ' x } + D e ^ { - k ' x } , \end{align*}


\begin{align*} H ^ { * } ( \overline { \alpha } \otimes \overline { \beta } ) = H ^ { * } ( \overline { \alpha } ) H ^ { * } ( \overline { \beta } ) = \left ( \begin{matrix} p q & 0 \\ 0 & p q \end{matrix} \right ) , \ \ L _ { \alpha , \beta } = L \left ( H ^ { * } ( \overline { \alpha } \otimes \overline { \beta } ) \right ) \end{align*}


\begin{align*} \lim _ { n \to \infty } \mu ( A \cap T ^ { - n } B ) = \mu ( A ) \mu ( B ) . \end{align*}


\begin{align*} \dim ( \mathcal { F } _ 1 ) = | E ( G ) | - \# \{ v \in V ( G ) : v \} - 1 \ , . \end{align*}


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


\begin{align*} \widetilde { \mathcal { D } } _ { [ n , j , k ] } = \{ \gamma \in \mathcal { D } _ { [ n , j ] } : \kappa _ { i } ( \gamma ) = k \} , i \leq k \leq n - j + i . \end{align*}


\begin{align*} _ r ( z \ , | \ , \omega _ 1 , \cdots , \omega _ r ) = \Gamma _ r ( z \ , | \ , \omega _ 1 , \cdots , \omega _ r ) ^ { - 1 } \ \Gamma _ r \big ( \sum _ { i = 1 } ^ { r } \omega _ i - z \ , | \ , \omega _ 1 , \cdots , \omega _ r \big ) ^ { ( - 1 ) ^ r } \ . \end{align*}


\begin{align*} \left [ \frac { \AA ( n + 1 ) } { \AA ( n ) } \right ] ^ n = \left [ 1 + \frac { A ^ { - 1 } ( n + 1 ) } { \sum \limits _ { j = 1 } ^ n A ^ { - 1 } ( j ) } \right ] ^ n \leq \left [ 1 + \frac { \varepsilon } { n } \right ] ^ n \leq e ^ \varepsilon . \end{align*}


\begin{align*} { \mathcal L } _ v ( d x _ 1 \wedge \cdots \wedge d x _ n ) = d ( \iota ( v ) ( d x _ 1 \wedge \cdots \wedge d x _ n ) ) . \end{align*}


\begin{align*} \mathcal { G } ( n ) \oplus \mathcal { G } ( n ) \oplus \mathcal { G } ( 2 a c ) = \mathcal { G } ( n + 1 ) \oplus \mathcal { G } ( n + 1 ) \oplus \mathcal { G } ( 2 a c - 2 ) \end{align*}


\begin{align*} \lim _ { z \rightarrow _ { \sphericalangle } s } ( z - s ) G _ { \mu } ( z ) = \mu ( \{ s \} ) , s \in \mathbb { R } . \end{align*}


\begin{align*} f _ { \mathbf { M ^ { \ast \ast } } _ { - \xi _ { \boldsymbol { s } k } } } ( \boldsymbol { v } ) = c _ { n - 1 , \xi _ { \boldsymbol { s } k } } ^ { \ast \ast } \overline { \mathcal { G } } _ { n } \left \{ \frac { 1 } { 2 } \boldsymbol { v } ^ { T } \boldsymbol { v } + \frac { 1 } { 2 } \xi _ { \boldsymbol { s } k } ^ { 2 } \right \} , ~ \boldsymbol { v } \in \mathbb { R } ^ { n - 1 } , \end{align*}


\begin{align*} - v '' + v = v ^ { p - 1 } v \ge 0 ; \end{align*}


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


\begin{align*} \delta ( t _ { k + 1 } ) & = \delta ( t _ { k } ) + \frac { ( 2 - \phi ) \lambda - \beta ^ { \top } u ( t _ { k } ) } { 2 } \mathsf { T } _ { k } , \end{align*}


\begin{align*} \delta ( \textbf { a } ) = \lim _ { x \to \infty } \frac { \# \{ p \leq x : p _ n \equiv a _ n \bmod q \} } { x } = \frac { 1 } { \varphi ( q ) ^ k } . \end{align*}


\begin{align*} J ( v , u ) = \Delta v ( 1 ) ^ 2 + \int _ 0 ^ 1 \Delta v ( s ) ^ 2 d s + \int _ 0 ^ 1 \Delta u ( s ) ^ 2 d s . \end{align*}


\begin{align*} X _ t = X _ 0 + \int _ 0 ^ t b ( X _ s ) d s + \int _ 0 ^ t a ( X _ s ) d W _ s , t \in [ 0 , T ] , \end{align*}


\begin{align*} \Phi ( x ) = \frac { \langle x \rangle ^ \gamma } { \gamma } , \mbox { a n d } \Psi ( v ) = \frac { \langle v \rangle ^ \beta } { \beta } , \ , \beta \geq 2 . \end{align*}


\begin{align*} \int _ { I } \int _ { \R ^ { d } } ( \partial _ { t } \varphi ( x , t ) + \langle v _ { t } ( x ) , \nabla _ { x } \varphi ( x , t ) \rangle ) d \mu _ { t } ( x ) d t = 0 , \forall \varphi \in C _ { 0 } ^ { \infty } ( \R ^ { d } \times I ) . \end{align*}


\begin{align*} { \mathfrak X } _ { \omega , c } ( G / H \times { \mathbb R } ^ { k } ) ^ { X } : = \{ Y \in { \mathfrak X } _ { \omega , c } ( G / H \times { \mathbb R } ^ { k } ) \mid [ X , Y ] = 0 \} . \end{align*}


\begin{align*} \lim _ { \zeta _ i \to z _ i } \left ( \frac { f ( z , \zeta ) } { ( z _ i - \zeta _ i ) } \bigg | _ { z _ l = \zeta _ l , l \neq i } \right ) = 0 , \ ; \ ; i = 1 , \ldots , m . \end{align*}


\begin{align*} \sigma ( T _ 0 ) \supseteq \bigcup _ { j = 1 } ^ { k } \sigma ( T _ j ) . \end{align*}


\begin{align*} \abs { \gamma ' ( 0 ) } = 1 - \abs { a } ^ 2 = 1 - \abs { \gamma ( 0 ) } ^ 2 . \end{align*}


\begin{align*} \widetilde { M } _ { n } ( t ) = \sup _ { s \in [ 0 , t ] } M _ { n } ( s ) + ( 1 + t ) ^ { n ^ 3 } \lesssim \big ( \widetilde { M } _ { n } ( t ) \big ) ^ { 1 - \epsilon } + ( 1 + t ) ^ { n ^ 3 } , \Longrightarrow \widetilde { M } _ { n } ( t ) \lesssim ( 1 + t ) ^ { n ^ 3 } . \end{align*}


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


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


\begin{align*} F _ \rho = \widetilde { M } \times _ \rho \C ^ n \ ; , \end{align*}


\begin{align*} \mathbf { N } _ { \Lambda } \doteq \left \{ \begin{array} { l l } \sum _ { \mathrm { s } \in \mathrm { S } } a _ { x , \mathrm { s } } ^ { \ast } a _ { x , \mathrm { s } } & \Lambda = \left \{ x \right \} x \in \mathfrak { L } \\ 0 & \end{array} \right . , \Lambda \in \mathcal { P } _ { \mathrm { f } } \ , \end{align*}


\begin{align*} \left | \sum _ { k = 1 } ^ \infty \frac { \beta ( \sigma , t ) ^ { k } } { k ! } \right | \leq \left | \frac { \beta ( \sigma , t ) } { 1 - \beta ( \sigma , t ) } \right | : = \beta _ 1 ( \sigma , t ) . \end{align*}


\begin{align*} c _ k & : = \int _ { \mathbb { R } } | p _ { 0 , k } ( y ) | d y = \int _ { \mathbb { R } } d y \left | e ^ { - i k \frac { 2 \pi } { V _ F } g } \int _ { 0 } ^ { V _ F } p _ { } ( v , g ) e ^ { - i k v \frac { 2 \pi } { V _ F } } d v . \right | \\ & \leq \int _ { \mathbb { R } } \int _ { 0 } ^ { V _ F } | p _ { } ( v , g ) | d v d g = 1 . \end{align*}


\begin{align*} H { ( a , c ) } : = \{ x \in { \mathbb R } ^ s \vert \ , \langle x , a \rangle = c \} \ \mbox { a n d } \ H ^ + { ( a , c ) } : = \{ x \in { \mathbb R } ^ s \vert \ , \langle x , a \rangle \geq c \} . \end{align*}


\begin{align*} f ( t , x , v ) = f ( t , x , R _ x v ) , R _ x v = v - 2 ( v \cdot n _ x ) n _ x , \end{align*}


\begin{align*} T _ U V = g ( U , V ) H ~ ~ T _ U X = - g ( H , X ) U , \end{align*}


\begin{align*} M _ { \mathbb { R } } ^ { * } = \left \{ H ^ { * } ( \overline { f } ) \mid \overline { f } \in \mathbb { R } ^ { n } \right \} \ \ \ M _ { Q } ^ { * } = \left \{ H ^ { * } ( \overline { f } ) \mid \overline { f } \in Q ^ { n } \right \} . \end{align*}


\begin{align*} \lambda _ 0 = \frac { \beta } { \delta } , \lambda _ 1 = - 2 \log \delta \ , , \lambda _ 2 = - \frac { \gamma } { \delta } . \end{align*}


\begin{align*} \begin{aligned} & \max \limits _ { \alpha \in \mathfrak { m } _ { i } } | f _ { i } ( \alpha ) | \ll N _ { i } ^ { 1 - 1 / 1 8 + \varepsilon } , \\ & \max \limits _ { \alpha \in \mathfrak { m } _ { i } } | S _ { i } ( \alpha ) | \ll N _ { i } ^ { 1 / 3 - 1 / 4 2 + \varepsilon } . \end{aligned} \end{align*}


\begin{align*} | I _ 3 | \le \frac { c } { T _ n } ( \frac { 1 } { \prod _ { l \ge 4 } h _ l } \frac { 1 } { \delta _ 2 ^ { \frac { 1 } { 2 } } } + D ) . \end{align*}


\begin{align*} ( m \otimes x ) ( a \# b ) = \sum _ { ( b ) } m ( a \# b _ { - 1 } ) \otimes x b _ 0 . \end{align*}


\begin{align*} \mathfrak { a = } \underset { } { \underbrace { \mathfrak { a } _ { + } } } - \underset { } { \underbrace { \mathfrak { a } _ { - } } } \end{align*}


\begin{align*} h _ i = w _ i ( g _ 1 , \ldots , g _ r ) . \end{align*}


\begin{align*} u \left ( s \right ) = \xi . \end{align*}


\begin{align*} \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } \ln ( \vert \varphi _ { a , b } ' ( y _ k ) \vert ) & = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } \ln \left ( \left \vert \varphi _ { 0 , 1 } ' \left ( \frac { y _ k - a } { b } \right ) \right \vert \right ) = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } \ln \left ( \left \vert \varphi _ { 0 , 1 } ' \left ( \varphi _ { 0 , 1 } ^ { ( k ) } \left ( \frac { y _ 0 - a } { b } \right ) \right ) \right \vert \right ) \\ & \to \ln ( 2 ) \end{align*}


\begin{align*} \int f ( x ) r ( x ) y ( x ) d _ q x = - f ( x / q ) D _ { q ^ { - 1 } } y ( x ) . \end{align*}


\begin{align*} \lim \limits _ { T \to \infty } \frac 1 T \ln \langle { x ^ \eta } ( T ) \rangle = w ( \eta ) \ , \end{align*}


\begin{align*} \dot { C } ( \lambda W ) + \frac { \rho } { \lambda } \cdot W = \zeta \cdot W , \end{align*}


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


\begin{align*} \int ^ { \eta _ n ( s ) - \delta } _ 0 \psi _ { n , 2 } ^ 2 ( u , s ) d u & = \int ^ { \eta _ n ( s ) - \delta } _ 0 [ ( s - \eta _ n ( u ) ) ^ { \alpha } - ( \eta _ n ( s ) - \eta _ n ( u ) ) ^ \alpha ] ^ 2 d u \\ & \le C n ^ { - 2 } \delta ^ { 2 \alpha - 2 } . \end{align*}


\begin{align*} ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \ldots , y _ i ) + \sum _ { \substack { 0 \le p \le i , \ 1 \le q \le n - i \\ p + q = k } } ( y _ i - y _ { i + 1 } ) e _ { p } ( y _ 1 , \ldots , y _ i ) e _ { q } ( y _ { i + 1 } , \ldots , y _ n ) = 0 . \end{align*}


\begin{align*} \boldsymbol { a _ h } ( \boldsymbol { u _ h } , \boldsymbol { v _ h } ) = \lambda _ h \boldsymbol { b _ h } ( \boldsymbol { u _ h } , \boldsymbol { v _ h } ) \quad \boldsymbol { v _ h } \in \boldsymbol { V _ h } . \end{align*}


\begin{align*} \tilde { K } _ { k , l _ 2 } ( y , v , V ( s ) ) = \int _ { \R ^ 3 } e ^ { i y \cdot \xi } i \xi | \xi | ^ { - 2 } \varphi _ k ( \xi ) \psi _ { \leq l _ 2 } ( \theta _ { V ( s ) } ( v ) \cdot \tilde { \xi } ) d \xi . \end{align*}


\begin{align*} u \left ( \lambda _ { m } \right ) = \frac { \log \left ( 1 + \frac { \lambda _ { m } } { \varepsilon } \right ) } { \log \left ( 1 + \frac { 1 } { \varepsilon } \right ) } + o \left ( \varepsilon \right ) , \end{align*}


\begin{align*} \begin{aligned} \mathcal { T } ( \Lambda ) & \to \ , \widehat { \Lambda } \\ M ( w ) & \mapsto M ( \widehat { w } ) \end{aligned} \end{align*}


\begin{align*} \Delta _ 1 = - \frac { \pi \gamma } { 2 } + 2 \beta ^ { \prime } ( 1 ) . \end{align*}


\begin{align*} t _ n ^ a = \inf \{ m > t _ { n - 1 } ^ a : Z _ m \leq a Z ^ * _ { m - 1 } \} \end{align*}


\begin{align*} M _ { \operatorname { A d } _ { g } \left ( x \right ) } = g M _ { x } , \end{align*}


\begin{align*} S _ { n , j } = \sum _ { i = 1 } ^ n A _ { i , j } , j \in [ d ] , \end{align*}


\begin{align*} S _ { k } ( a , k , \xi \ , ( \ell ) ) : = \sum _ { { \tiny \begin{tabular} { c } $ z \ , \hbox { \tiny m o d } \ , k \ell $ \\ $ z = \xi ( \ell ) $ \end{tabular} } } e _ { k \ell } ( a \ell N ( z ) + c \cdot z ) , \end{align*}


\begin{align*} F _ { \nu _ { n } } ( z ) = H _ { \nu _ { n } } ^ { - 1 } ( z ) = \frac { 1 } { 2 \pi i } \int _ { \partial Q } \frac { \zeta H _ { \nu _ { n } } ( \zeta ) } { H _ { \nu _ { n } } ( \zeta ) - z } \ , d \zeta \end{align*}


\begin{align*} \psi _ \mathrm { g a u s s } : = \sum _ { \substack { - \sigma _ N ^ 2 \le d \le \sigma _ N ^ 2 \\ N + d } } c _ d \ , u _ 1 ^ { \otimes ( N + d ) / 2 } \otimes _ \mathrm { s y m } u _ 2 ^ { \otimes ( N - d ) / 2 } , \end{align*}


\begin{align*} & \left < \frac { i } h [ - h ^ 2 \Delta + V ( x ) - E ( h ) , \chi _ \alpha ( x _ n ) h D _ n ] u _ h , \ , u _ h \right > _ { L ^ 2 ( \Omega _ \Gamma ) } \\ & = \left < O p _ h \left ( \{ \sigma ( - h ^ 2 \Delta + V ( x ) - E ( h ) ) , \sigma ( \chi _ \alpha ( x _ n ) h D _ n ) \} \right ) u _ h , u _ h \right > _ { L ^ 2 ( \Omega _ \Gamma ) } + O ( h ) , \end{align*}


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


\begin{align*} \P ( Z _ s = k ) \to \exp ( - e ^ { - s } ) \frac { e ^ { - k s } } { k ! } . \end{align*}


\begin{align*} \frac { d U } { d t } ( t ) \psi = \sum _ { j = 1 } ^ r \frac { d f _ j } { d t } ( t ) { \rm A d } _ { \prod _ { k = 1 } ^ { j - 1 } \exp ( { \rm i } f _ k ( t ) \widehat H _ k ) } ( { \rm i } \widehat H _ j ) U ( t ) \psi , \forall \psi \in \mathcal { H } , \end{align*}


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


\begin{align*} E = b ^ 2 f ^ 2 ( x ^ 1 ) \left [ f '^ 2 ( x ^ 1 ) + \sin ^ { 2 } \left ( x ^ 2 - \theta \right ) \right ] . \end{align*}


\begin{align*} \frac { \partial E } { \partial z ^ i _ { \epsilon } } v ^ i = 2 b ^ 2 \delta _ { \epsilon 1 } f ^ 2 ( x ^ 1 ) f ' ( x ^ 1 ) . \end{align*}


\begin{align*} \left ( \bigcap _ { e = 1 } ^ \infty K _ e \right ) S _ 2 \subseteq \left ( \bigcap _ { e = 1 } ^ n K _ e \right ) S _ 2 = \bigcap _ { e = 1 } ^ n ( K _ e S _ 2 ) \end{align*}


\begin{align*} \phi ( N \times \{ 0 \} ) = S ^ { 1 } \cdot x _ { 0 } \end{align*}


\begin{align*} \varrho ( t ) : = ( t + 1 ) ^ { - 0 . 4 5 } . \end{align*}


\begin{align*} \prescript { } { l } { \mathcal { F } } _ { n } ( f ) ( \lambda ) = h _ { \lambda _ 2 } ( \lambda _ 1 ) \cdot q _ { l _ 1 , n _ 1 } = h _ { \lambda _ 1 } ( \lambda _ 2 ) \cdot q _ { l _ 2 , n _ 2 } ( \lambda _ 2 ) , \end{align*}


\begin{align*} h _ { d , b , A } \triangleq \prod _ { i = 1 } ^ d h _ { 1 , b , A _ i } , \end{align*}


\begin{align*} H _ { ( X , D ) } ( y ) = e ^ { \frac { t } { 2 \pi i } } \sum _ { \substack { d \in \mathbb K \\ ( k _ 1 , \ldots , k _ m ) \in ( \mathbb Z _ { \geq 0 } ) ^ m } } y ^ d \left ( \frac { \Gamma ( 1 + \frac { D } { 2 \pi i } + D \cdot d ) } { \prod _ { i = 1 } ^ m \Gamma ( 1 + \frac { \bar D _ i } { 2 \pi i } + D _ i \cdot d ) } \right ) \textbf { 1 } _ { [ d ] } [ \textbf { 1 } ] _ { D \cdot d - \sum _ { i = 1 } ^ m k _ i a _ i } \frac { \prod _ { i = 1 } ^ m x _ i ^ { k _ i } } { \prod _ { i = 1 } ^ m ( k _ i ! ) } . \end{align*}


\begin{align*} \lambda \leq \sum _ { t = 1 } ^ k e ^ T A _ { \bar { S } S } ( k + 1 ) e , \sum _ { t = 1 } ^ k e ^ T A _ { S \bar { S } } ( k + 1 ) e \leq \Lambda , \end{align*}


\begin{align*} \tilde { f } _ \mu ( p ) = \sum \limits _ { i } ^ { N } \ , \alpha _ i \frac { u _ { i \mu } } { u _ i \cdot p } , \ , \ , \ , \ , \ , \ , \ , \sum \limits _ { i } ^ { N } \alpha _ i = 0 , \ , \ , u _ i \cdot u _ i = 1 , \ , i = 1 , \ldots , N . \end{align*}


\begin{align*} \varphi _ { a , b } ( x ) : = \begin{cases} \frac { b } { 2 } \left ( \frac { x - a } { b } - \frac { 1 } { \frac { x - a } { b } } \right ) + a = \frac { x } { 2 } + \frac { a } { 2 } - \frac { \frac { b ^ 2 } { 2 } } { x - a } & \\ a & \end{cases} \end{align*}


\begin{align*} \widetilde { \Phi } ( x ) = \Phi ( ( x + M ) ^ 4 ) f ( x ) , \end{align*}


\begin{align*} p ( x , y ) = \sum _ { i = 1 } ^ \infty w _ i f _ i ( x ) g _ i ( y ) \end{align*}


\begin{align*} \beta _ r = \varphi ' ( r ) \big [ \log \cos r - f ( s , 0 ) \big ] - \varphi ( r ) \tan r \end{align*}


\begin{align*} s ^ { - 1 } \mathcal F = t ^ { - 1 } \mathcal F = \Gamma ( \ker ( \dd s ) ) + \Gamma ( \ker ( \dd t ) ) . \end{align*}


\begin{align*} G _ y f ( x ) = \frac { 1 } { x } ( f ( x e ^ { - y } ) - f ( x ) + y x f ' ( x ) ) \end{align*}


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


\begin{align*} \gamma ^ { 1 / 2 } h = ( m + 1 ) h = m h + h = h \end{align*}


\begin{align*} ( A ^ { M \sharp M } \cap A ^ { \tilde M \sharp \tilde M } ) ^ * = ( A ^ { M \sharp M } ) ^ * \cap ( A ^ { \tilde M \sharp \tilde M } ) ^ * \subset M ( A ) ^ { * * } . \end{align*}


\begin{align*} ( x ) _ { 0 , \lambda } = 1 , ( x ) _ { n , \lambda } = x ( x - \lambda ) ( x - 2 \lambda ) \cdots \big ( x - ( n - 1 ) \lambda \big ) , ( n \ge 1 ) , \end{align*}


\begin{align*} \mathcal { S } = \int _ 0 ^ 1 d t \int _ { L _ t } h _ t ( e ^ { - i \hat { \theta } } \Omega ) . \end{align*}


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


\begin{align*} \psi _ i ^ k ( \beta _ k ( g , ( \psi _ j ^ k ) ^ { - 1 } ( x , v ) ) ) = ( \alpha ( g , x ) , a _ k ( g , x , v ) ) \end{align*}


\begin{align*} p _ x [ T ] = p _ 1 [ T ] \wedge p _ 3 [ T ] , \quad & p _ y [ T ] = p _ 2 [ T ] \wedge p _ 4 [ T ] , \\ p [ T ] = p _ 5 [ T ] \wedge p _ 6 [ T ] , \quad & q [ T ] = q _ 1 [ T ] \wedge q _ 2 [ T ] . \end{align*}


\begin{align*} \int _ 0 ^ 1 d t \int _ { L _ t } h _ t ( e ^ { - i \hat { \theta } } \Omega ) = \int _ { L _ t } f _ { L _ t } ( e ^ { - i \hat { \theta } } \Omega ) | ^ { t = 1 } _ { t = 0 } - \int _ { \cup _ t L _ t } \lambda \wedge ( e ^ { - i \hat { \theta } } \Omega ) . \end{align*}


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


\begin{align*} i \int _ D A ( x ) \cdot \left ( u _ 2 \nabla u _ 1 - u _ 1 \nabla u _ 2 \right ) d x = 2 s \int _ D \overline { \omega } \cdot A ( x ) e ^ { i x \cdot \xi } d x + \mathcal { R } ( \xi , s ) , \end{align*}


\begin{align*} & \gamma ^ { \mathcal O \rtimes E \mathcal M } \big ( ( o , u _ \bullet ) ; ( f ^ { ( 1 ) } , v ^ { ( 1 ) } _ \bullet ) , \dots , ( f ^ { ( r ) } , v ^ { ( r ) } _ \bullet ) \big ) \\ & \quad { } = \big ( \gamma ^ { \mathcal O } ( o , u _ 1 . f ^ { ( 1 ) } , \dots , u _ r . f ^ { ( r ) } ) ; \gamma ^ { E \mathcal M ^ \bullet } ( u _ \bullet ; v _ \bullet ^ { ( 1 ) } , \dots , v _ \bullet ^ { ( r ) } ) \big ) \end{align*}


\begin{align*} { \Delta \left ( \frac { \partial W _ { h } } { \partial h } \right ) - \frac { \partial W _ { h } } { \partial h } + p \left ( \sum _ { i = 1 } ^ 4 ( U _ { h , i } ) ^ { p - 1 } \frac { \partial U _ { h , i } } { \partial h } \right ) = 0 . } \end{align*}


\begin{align*} \textstyle \sum _ { j = 1 } ^ n - p _ j ( 0 ) \log _ q p _ j ( 0 ) \le - n ( 1 - \omega ) \log _ q ( 1 - \omega ) . \end{align*}


\begin{align*} E _ { F } \hat { \beta } = \beta ( F ) F \in \mathbf { F } _ { 2 } ( \Sigma ) , \end{align*}


\begin{align*} \kappa _ 2 = \sqrt { \kappa _ 3 + \frac { 2 \alpha + 1 } { 2 ( \alpha + 1 ) ^ 2 } + \kappa _ 4 + \kappa _ 5 } , \end{align*}


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


\begin{align*} S _ { 1 } ( N , \alpha ) = \sum \limits _ { p \sim N } ( \log p ) e ( p \alpha ) \end{align*}


\begin{align*} Z _ { i , s } ( t , \phi ) = \liminf _ { n \to \infty } Z _ { i , s } ^ { ( n ) } ( t , \phi ) . \end{align*}


\begin{align*} \dot C _ t = ( 1 - A ) C _ t ^ 2 , \ddot C _ t = 2 ( 1 - A ) C _ t \dot C _ t . \end{align*}


\begin{align*} X _ { \mathcal { F } } = \{ x \in \mathcal { A } ^ { \Z ^ d } \mid \forall p \in \mathcal { F } , p \not \sqsubset x \} \end{align*}


\begin{align*} k \left [ \tan ( k a / 2 ) + \cot ( k a / 2 ) \right ] = - k ' \left [ \tanh ( k ' a / 2 ) - \coth ( k ' a / 2 ) \right ] \end{align*}


\begin{align*} b ^ { * } j ( a ) ( \gamma \beta ^ { - 1 } ) c = \lim _ { n } b ^ { * } j ( f _ { n } ) ( \gamma \beta ^ { - 1 } ) c . \end{align*}


\begin{align*} s ( x , y ) = x \displaystyle { t ( x , y ) = \exp _ x \left ( \sum _ { i = 1 } ^ n y _ i X _ i \right ) } = \varphi _ 1 ^ { \sum _ { i = 1 } ^ n y _ i X _ i } ( x ) \end{align*}


\begin{align*} \int _ a ^ b f ( x ) d x = \left [ \frac { 1 } { 2 } L x ^ 2 + c x \right ] _ { x = a } ^ { x = b } = \frac { 1 } { 2 } L \left ( b ^ 2 - a ^ 2 \right ) + c ( b - a ) = \left ( L \frac { a + b } { 2 } + c \right ) ( b - a ) . \end{align*}


\begin{align*} K _ r ( x , y , \varphi ^ l ( x , y ) ) = \left ( \sum _ { i = 1 } ^ { n - 2 } \frac { 3 } { \varepsilon } ( \varphi _ i ^ l ( x , y ) - \Phi _ i ( x , y ) ) ^ 2 \right ) ^ r > \left ( \frac 4 3 \right ) ^ r . \end{align*}


\begin{align*} J _ 1 = 2 \int _ 0 ^ \infty ( \sigma + h _ p ^ 0 ) h _ { p x } ^ 1 u _ { p x } ^ 1 x ^ { 1 - 2 \sigma _ 1 } { \rm d } y \leq \Vert \sigma + h _ p ^ 0 \Vert _ { L ^ \infty } \Vert ( u _ { p x } ^ 1 , h _ { p x } ^ 1 ) x ^ { \frac { 1 } { 2 } - \sigma _ 1 } \Vert _ { L _ y ^ 2 } ^ 2 , \end{align*}


\begin{align*} L _ { f \cdot \tau } = M _ { f \cdot \tau } + \sum _ { g \prec f \cdot x } \ell _ { g , f \cdot \tau } M _ g . \end{align*}


\begin{align*} A _ i - A _ i ^ * = \sum _ { 0 < | h | < H } \left ( 1 - \frac { | h | } { H } \right ) \hat { f } ( h ) \left ( e ( h Y _ i \alpha ) - e ( h ( Y _ i \alpha - \delta _ i ) ) \right ) \sum _ { k \in H _ i } e ( h ( S _ k - Y _ i ) \alpha ) , \end{align*}


\begin{align*} { \rm L } _ a & : = \big \{ b a \in \N \ ; : \ ; p \mid b \implies p \ge P ( a ) \big \} , \end{align*}


\begin{align*} M ^ { ( a ) } = \sum _ { i , \alpha } M _ \alpha ^ i ( 1 \otimes \cdots \otimes E _ i ^ \alpha \otimes \cdots \otimes 1 ) , \end{align*}


\begin{align*} p _ p ^ { 1 , a } = \int _ y ^ \infty ( R ^ { v , 0 } + R _ E ^ { v , 1 } ) \end{align*}


\begin{align*} { R ^ N _ { \alpha \beta \gamma } } ^ \delta \partial _ \delta = R ^ N ( \partial _ \alpha , \partial _ \beta ) \partial _ \gamma . \end{align*}


\begin{align*} \frac { f ( r ) } { f ' ( r ) } = \frac { k _ { \phi } ( r ) } { k ' _ { \phi } ( r ) } = \frac { r } { \psi ( - r ) } . \end{align*}


\begin{align*} \dot { C } ( \lambda w ) = \frac { \delta z _ 0 \sinh ( z _ 0 \lambda w ) + ( 1 - \delta ) \cdot 2 \lambda q ^ 2 / ( q ^ 2 - \lambda ^ 2 w ^ 2 ) ^ 2 } { \delta \cosh ( z _ 0 \lambda w ) + ( 1 - \delta ) q ^ 2 / ( q ^ 2 - \lambda ^ 2 w ^ 2 ) } . \end{align*}


\begin{align*} \sigma ( \mathcal L _ n ) = \bigg \{ \left . \frac { n ^ 2 } { \pi ^ 2 } \sin ^ 2 \Big ( \frac { \pi k } { n } \Big ) \right | k = 0 \ , , \cdots , n - 1 \bigg \} . \end{align*}


\begin{align*} \langle \Phi _ 0 ( x ) ^ { ( 0 ) } ( \alpha ) , e \rangle = \varrho ( x ) [ \langle \alpha , e \rangle ] - \langle \alpha , \nabla _ x ( e ) \rangle , \ ; \ ; . \end{align*}


\begin{align*} F _ 0 ( z ) = F ( z ) , \ \ \ F _ { s } ( z ) = \sum _ { k = 0 } ^ { \infty } ( k + \gamma _ 1 ) \cdots ( k + \gamma _ s ) c _ k z ^ { k + 1 } \ \ 1 \le s \le r - 1 \enspace . \end{align*}


\begin{align*} \int _ { U } ( - \Delta ) ^ s m _ 0 g \ , d x = 0 \end{align*}


\begin{align*} H = H _ 0 + z H _ 1 \in \N _ { A , E } ( a ) \ , \end{align*}


\begin{align*} \rho _ 1 = s \Big ( i \omega _ 2 + \big ( \frac { \xi } { 2 s } - \sqrt { 1 - \frac { \vert \xi \vert ^ 2 } { 4 s ^ 2 } } \omega _ 1 \big ) \Big ) : = s \omega ^ { * } _ 1 ( s ) , \end{align*}


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


\begin{align*} \phi ( \phi ^ { - 1 } ( \gamma _ 1 ) \cup \phi ^ { - 1 } ( \gamma _ 2 ) ) = \gamma _ 1 \star \gamma _ 2 . \end{align*}


\begin{align*} \left ( \Phi _ 0 ( x ) - \Psi _ 0 ( x ) - [ Q , \iota _ { \varphi ( x ) } ] \right ) ( f ) & = \rho ( \varphi ( x ) ) [ f ] - \langle Q ( f ) , \varphi ( x ) \rangle \\ & = 0 . \end{align*}


\begin{align*} E _ \beta ( z ) = \sum _ { m = 0 } ^ \infty \frac { z ^ m } { \Gamma ( \beta m + 1 ) } \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty \int _ { - \infty } ^ \infty | \langle f , \rho [ x , a ] g \rangle | ^ 2 \ , \frac { d x \ , d a } { a ^ 2 } = \| f \| _ 2 ^ { \ , 2 } \int _ { - \infty } ^ \infty \frac { | \widehat { g } ( \omega ) | ^ 2 } { | \omega | } \ , d \omega , f , g \in L ^ 2 ( \R ) . \end{align*}


\begin{align*} \sigma _ \infty ( N , w ) & = \lim _ { \epsilon \to 0 ^ + } ( 2 \epsilon ) ^ { - 1 } \int _ { | N ( x ) - 1 | \le \epsilon } w ( x ) \ , d x \\ & = \lim _ { \epsilon \to 0 ^ + } ( 2 \epsilon ) ^ { - 1 } \int _ { 1 - \epsilon } ^ { 1 + \epsilon } \left ( \int _ { N ^ { - 1 } ( y ) } w \ , d \tau ^ { ( y ) } _ \infty \right ) \ , d y = \int _ { N ^ { - 1 } ( 1 ) } w \ , d \tau ^ { ( 1 ) } _ \infty , \end{align*}


\begin{align*} s = - \frac { H } { z } - \frac { m } { 3 } , m \geq 1 \end{align*}


\begin{align*} \int _ B \left ( Q _ { A , q } v _ 1 u _ { - A , q } - Q _ { - A , q } v _ 2 \right ) d x = \int _ B \left ( Q _ { A , q } v _ 1 v _ 2 - Q _ { - A , q } v _ 2 v _ 1 \right ) d x . \end{align*}


\begin{align*} \widehat { H } _ { N H } ( t ) : = \frac 1 2 \sum _ { i = 1 } ^ n \widehat { p } _ i ^ 2 + \omega ( t ) \frac 1 2 \sum _ { i = 1 } ^ n \widehat { x } ^ 2 _ i + \tilde { \omega } ( t ) \sum _ { i = 1 } ^ n \widehat { x } _ i ^ \alpha , \end{align*}


\begin{align*} & \varphi ( x ) - \varphi ( y ) = - \varphi ( y ) \\ & \le \langle \nabla \varphi ( y ) , x - y \rangle + \left ( ( \| \varphi \| _ { \infty } + \| \nabla \varphi \| _ { \infty } ) \bigvee C _ 1 C _ 4 ^ { 2 - p } \right ) | x - y | ^ p , \ \forall x \in S ^ c , y \in S \end{align*}


\begin{align*} | A | = q ^ { n \left ( 1 - \frac { 1 } { k ^ { 2 } } \right ) } . \end{align*}


\begin{align*} \min \limits _ { i = 2 , \dots , d \atop h , \ell = 1 , \dots , k } \left | \xi _ h - \eta _ { i \ell } \right | = \left | \xi _ 1 - \eta _ { 2 1 } \right | = 2 \sin { \pi \over d k } . \end{align*}


\begin{align*} ( i \partial _ t + \partial _ x ^ 2 + i \gamma ) G = \frac { p + 1 } { 2 } \| v \| _ { L _ x ^ { p - 1 } } ^ { p - 1 } G + F , \end{align*}


\begin{align*} y ' = s y + g y ( 0 ) = 0 . \end{align*}


\begin{align*} ( X , \alpha ) = ( - 1 ) ^ p \int _ M ( \iota _ X \alpha ) \mu . \end{align*}


\begin{align*} \sigma ^ { * } ( T ) & = \bigcup _ { j \in \Phi } \{ x \in \mathbb { R } \colon W _ j ( x ) = 0 \} . \end{align*}


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


\begin{align*} \int _ { \R ^ { 2 N } \setminus ( C \Omega ) ^ 2 } \varPhi _ { x , y } \left ( \dfrac { | u ( x ) - u ( y ) | } { | x - y | ^ s } \right ) \dfrac { d x d y } { | x - y | ^ N } = 0 . \end{align*}


\begin{align*} \frac { 1 } { t ^ 2 } \left \{ G ( V , V ) G ( U , U ) - G ( U , V ) ^ 2 \right \} - G ( \mbox { H o r } ( P _ V V ) , \mbox { H o r } ( P _ U U ) ) + G ( \mbox { H o r } ( P _ V U ) , \mbox { H o r } ( P _ V U ) ) = 0 . \end{align*}


\begin{align*} \mathbf { u } _ { k + 1 } & = R \mathbf { u } _ k + a _ { k + 1 } \mathbf { v } _ k + a _ { k + 1 } ^ 2 \mathbf { u } _ { k - 1 } \\ \mathbf { v } _ { k + 1 } & = ( 2 R + 1 ) \mathbf { u } _ k + a _ { k + 1 } \mathbf { v } _ k + a _ { k + 1 } \mathbf { w } _ k \\ \mathbf { w } _ { k + 1 } & = - \mathbf { u } _ k - a _ { k + 1 } \mathbf { w } _ k , \end{align*}


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


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


\begin{align*} \mathcal { G } ( 1 0 c + 1 ) \oplus \mathcal { G } ( 1 ) & = 6 = \mathcal { G } ( 8 c + 1 ) \oplus \mathcal { G } ( 2 c + 1 ) \\ \mathcal { G } ( 1 0 c + 1 ) \oplus \mathcal { G } ( 4 c + 1 ) & = 7 = \mathcal { G } ( 8 c + 1 ) \oplus \mathcal { G } ( 6 c + 1 ) \end{align*}


\begin{align*} r _ n = \min \{ x : x \in \{ x _ j \} _ { j \in \Z _ + } , x \geq n \} . \end{align*}


\begin{align*} R _ \mathrm { r a t } ( G ) = R _ \mathrm { r a t } ( G ' ) \otimes R _ \mathrm { r a t } ( G '' ) = \mathcal { S } ( G ' ) \otimes \mathcal { S } ( G '' ) \subseteq \mathcal { S } ( G ) \ ; . \end{align*}


\begin{align*} \begin{cases} - \Delta _ p \hat { u } _ n - a _ n | \hat { u } _ n | ^ { p - 2 } \hat { u } _ n = \hat { f } _ n \qquad & \Omega \\ \hat { u } _ n = \hat { g } _ n & \partial \Omega \end{cases} \end{align*}


\begin{align*} \Omega _ { \bullet } ( q ) : = & \sum _ { \lambda \in \mathcal { P } } q ^ { | \lambda | ( 1 - g ) } ( q ^ { - n ( \lambda ) } H _ { \lambda } ( q ) ) ^ { 2 g + k - 2 } \prod ^ k _ { j = 1 } s _ { \lambda } [ \frac { \Z _ j } { 1 - q } ] \\ \Omega _ { \ast } ( q ) : = & \sum _ { \lambda \in \mathcal { P } } q ^ { | \lambda | ( 2 - 2 g ) } ( q ^ { - n ( \lambda ) } H _ { \lambda } ( q ) ) ^ { 4 g + 2 k - 4 } \prod ^ k _ { j = 1 } s _ { \lambda } [ \frac { \Z _ j } { 1 - q } ] ^ 2 , \end{align*}


\begin{align*} K _ { M } ^ { T } = \pi ^ { \ast } ( K _ { Z } ^ { o r b } ) = \pi ^ { \ast } ( \varphi ^ { \ast } ( K _ { Z } + [ \Delta ] ) ) . \end{align*}


\begin{align*} \tilde { T } _ { \vec s ^ 1 , k _ 1 } \star \tilde { T } _ { \vec s ^ 2 , k _ 2 } = \sum _ { \vec s ^ 3 , k _ 3 } \frac { \partial ^ 3 \Phi _ 0 } { \partial t _ { \vec s ^ 1 , k _ 1 } \partial t _ { \vec s ^ 2 , k _ 2 } \partial t _ { \vec s ^ 3 , k _ 3 } } \tilde { T } _ { - \vec s ^ 3 } ^ { k _ 3 } . \end{align*}


\begin{align*} y \cdot \overline \xi \cdot x = \overline { x ^ * \xi y ^ * } , \end{align*}


\begin{align*} \delta _ { J } \geq D n ^ { - \sqrt { b ( n ) } / \log n } = D e ^ { - \sqrt { b ( n ) } } . \end{align*}


\begin{align*} \int _ { \Gamma } a _ j ( \gamma ) d { \gamma _ j } _ * \mu ( \gamma ) = 0 . \end{align*}


\begin{align*} \begin{aligned} ( \rho _ \alpha - \alpha ( k + d - 2 ) ) x _ { v } & \leq ( \rho _ \alpha - \alpha d ( v ) ) x _ { v } = ( 1 - \alpha ) ( k - 1 ) x _ { u _ 1 } + ( 1 - \alpha ) \sum _ { u v \in E ( H ) } x _ u \\ & \leq ( 1 - \alpha ) ( k - 1 ) x _ { u _ 1 } + ( 1 - \alpha ) ( d - 1 ) x _ v , \end{aligned} \end{align*}


\begin{align*} \nu \left ( \sum _ { j \in J } \partial _ j L ( h _ \sigma ) ( h _ \rho - h _ \sigma ) ^ j \right ) \geq \min _ { j \in J } \{ \beta _ j + j \gamma _ \sigma \} > \min _ { i \in I } \{ \beta _ i + i \gamma _ \rho \} = \beta _ b + i \gamma _ b . \end{align*}


\begin{align*} g ( x ) & = \prod _ { j = 1 } ^ n ( 1 + r _ j ( t _ 0 ) \cos 2 \pi n _ j x ) \\ & = 1 + \sum _ { A \subset \{ 1 , 2 , \ldots , n \} } \left ( \prod _ { j \in A } r _ j ( t _ 0 ) \cos 2 \pi n _ j x \right ) \\ & = 1 + \sum _ { A \subset \{ 1 , 2 , \ldots , n \} } \left ( w _ A ( t _ 0 ) \prod _ { j \in A } \cos 2 \pi n _ j x \right ) , \end{align*}


\begin{align*} \sum \limits _ { l \ , | \ , i _ l = i } { t _ i } = 1 , p _ { i j } = \sum \limits _ { l < m \ , | \ , i _ l = i , i _ m = j } { t _ l t _ m } . \end{align*}


\begin{align*} \lim _ { \alpha \to 0 } \frac { i } h \int _ { \Omega _ \Gamma } [ - h ^ 2 \Delta + V ( x ) - E ( h ) , \chi _ \alpha ( x _ n ) h D _ n ] u _ h \overline { u _ h } d x = o ( 1 ) . \end{align*}


\begin{align*} S _ { R , C } ( x , y ) = \mathrm { C o n e } \left ( x , \{ z \in \partial _ \infty M : \mathrm { d i s t } ( x , [ y , z ] ) \geq R \} \right ) \cap B ( x , R + C ) \setminus B ( x , R ) . \end{align*}


\begin{align*} \nabla _ { Z } Z = f ^ { 2 } \nabla _ { \partial _ { \theta } } \partial _ { \theta } = - f ^ { 2 } c _ { 1 } \partial _ r c _ { 1 } \partial _ { 2 } \end{align*}


\begin{align*} P : = \left [ \begin{array} { l l l l l } g _ 0 & g _ 1 & \dots & g _ N & x _ 0 \end{array} \right ] \end{align*}


\begin{align*} \vec { v } \left ( { \bf s ' } _ { \alpha , 0 } \right ) = \left ( 1 - \alpha , \alpha , 1 \right ) \vec { v } \left ( { \bf s } _ { \alpha , 0 } \right ) = \left ( 1 - \alpha , \alpha , 0 \right ) . \end{align*}


\begin{align*} \phi \tilde c _ 1 ( \alpha , g ) = \tilde c _ 1 ( \alpha , h ) \ ; \ ; \ ; \ ; \ ; \mbox { a n d } \ ; \ ; \ ; \ ; \ ; \phi \tilde c _ 2 ( \alpha , g ) = \tilde c _ 2 ( \alpha , h ) . \end{align*}


\begin{align*} f _ j & = \begin{cases} 2 \csc \left ( \dfrac { \pi } { 2 j + 4 } \right ) - 2 \cot \left ( \dfrac { \pi } { 2 j + 2 } \right ) , & 2 \nmid j , \\ 2 \cot \left ( \dfrac { \pi } { 2 j + 4 } \right ) - 2 \csc \left ( \dfrac { \pi } { 2 j + 2 } \right ) , & 2 \mid j . \end{cases} \end{align*}


\begin{align*} m = l ( m ) r ( m ) = l ( m ' ) r ( m ' ) = m ' . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } n ^ { 2 \alpha + 1 } \int ^ { \tau _ 2 } _ { \tau _ 1 } \gamma _ s M ^ { n , 1 } _ s M ^ { n , 2 } _ s d s = 0 , \end{align*}


\begin{align*} \psi _ \Lambda ( u ) = \Lambda ( \{ 0 \} ) u ^ 2 + \int _ { ( 0 , 1 ] } \left ( e ^ { - u x } - 1 + u x \right ) x ^ { - 2 } \Lambda ( d x ) . \end{align*}


\begin{align*} \partial _ { x _ i } \partial _ { x _ j } \bigl ( a ^ { i j } \varrho \bigr ) - \partial _ { x _ i } \bigl ( b ^ i \varrho \bigr ) + c \varrho = 0 \end{align*}


\begin{align*} \sum _ { j = 0 } ^ p a _ j ( j + 1 ) = 1 \ . \end{align*}


\begin{align*} [ x y , z ] = - [ y z , x ] - [ z x , y ] \mbox { i f } \mathsf { c h a r } ( F ) = 3 ; \end{align*}


\begin{align*} R = \begin{pmatrix} A & B & 0 \\ C & D & 0 \\ E & F & 1 \end{pmatrix} \end{align*}


\begin{align*} E = \{ Z ( t , B _ \epsilon ) \geq b \cdot p ( t , \epsilon ) \} . \end{align*}


\begin{align*} ( a v g ( x ) , \beta ) _ E & = \sum \limits _ { e \in E } \delta _ E ( e ) ( a v g ( x ) ) ( e ) \beta ( e ) \\ & = \sum \limits _ { e \in E } \delta _ E ( e ) \beta ( e ) \frac { \sum \limits _ { v \in e } x ( v ) } { | e | } \\ & = \sum \limits _ { v \in V } \delta _ V ( v ) x ( v ) \sum \limits _ { e \in E _ v } \frac { \beta ( e ) } { | e | } \frac { \delta _ E ( e ) } { \delta _ V ( v ) } \\ & = ( x , a v g ^ * ( \beta ) ) _ V . \end{align*}


\begin{align*} e ^ { i \tau \Phi ( \xi , \eta _ 1 , \eta _ 2 ) } = \frac { 1 } { i \Phi ( \xi , \eta _ 1 , \eta _ 2 ) } \left [ \partial _ \tau e ^ { i \tau \Phi ( \xi , \eta _ 1 , \eta _ 2 ) } \right ] , \end{align*}


\begin{align*} C = C ^ { ( < j ) } \oplus C ^ { ( j ) } \oplus C ^ { ( > j ) } . \end{align*}


\begin{align*} & \# ( z ) \le l - 1 , \\ & u \wedge v = u \wedge z = v \wedge z = 0 , \\ & 0 \le \# ( u ) , \# ( v ) \le l - \# ( z ) , u + v > 0 . \end{align*}


\begin{align*} \langle u ( n ) f , a \rangle = \langle f , \theta ( u ( n ) ) a \rangle . \end{align*}


\begin{align*} \lambda = - \frac { a ( 1 - b ^ 2 ) ^ { \frac { 3 } { 2 } } } { 1 + b \dot { \theta } \sin ( \theta - t ) } . \end{align*}


\begin{align*} B _ q ( v , v ) & \geq \norm { ( - \Delta ) ^ { s / 2 } v } _ { L ^ 2 ( \R ^ n ) } ^ 2 + \langle q _ { s , 1 } , v ^ 2 \rangle + \langle q _ { s , 2 } , v ^ 2 \rangle \\ & \geq \norm { ( - \Delta ) ^ { s / 2 } v } _ { L ^ 2 ( \R ^ n ) } ^ 2 + \langle q _ { s , 1 } , v ^ 2 \rangle - \abs { \langle q _ { s , 2 } , v ^ 2 \rangle } \\ & \geq \left ( \delta ( \Omega ) - \norm { q _ { s , 2 } } _ { s , - s } \right ) \norm { v } _ { H ^ s ( \R ^ n ) } ^ 2 + \langle q _ { s , 1 } , v ^ 2 \rangle \end{align*}


\begin{align*} \mathrm { A r e a } ( \mathbb { S } ^ { n - 1 } ) = \frac { 2 \pi ^ { \frac { n } { 2 } } } { \Gamma ( \frac { n } { 2 } ) } , \end{align*}


\begin{align*} m A = l ( m ) r ( m ) A & \subseteq l ( m ) A r ( m ) \\ & = A r ( m ) = ( A l ( m ) ^ { - 1 } ) l ( m ) r ( m ) = A m . \end{align*}


\begin{align*} \left | \sum _ { 1 \leq i _ 1 < i _ 2 < \cdots < i _ { n - m } \leq n - 1 } \omega _ { i _ 1 } \omega _ { i _ 2 } \cdots \omega _ { i _ { n - m } } \right | = \left | \omega _ m \omega _ { m + 1 } \cdots \omega _ { n - 1 } \right | \geq \left | \omega _ m \right | ^ { n - m } . \end{align*}


\begin{align*} u ( t ) = \frac { d ^ m } { d t ^ m } \int _ 0 ^ t k _ m ( t - \tau ) g ( \tau ) d \tau , \end{align*}


\begin{align*} k _ { \pm } ^ { ' 2 } = \frac { 1 - \sqrt { 1 \pm 1 6 i m \beta \lambda / 3 } } { 4 \beta \hslash ^ { 2 } / 3 } . \end{align*}


\begin{align*} q _ c + q _ { c + 1 } + \cdots + q _ { d - 1 } = \sum _ { \{ i , c \} \in E ( \mathsf { P } _ { n + 2 } ) \atop b \leq i < c } x _ { i c } - \sum _ { \{ j , d \} \in E ( \mathsf { P } _ { n + 2 } ) \atop c \leq j < d } x _ { j d } \end{align*}


\begin{align*} \mathcal { H } _ { A + \nabla \varphi , q } \tilde { u } : = - ( \nabla + i ( A + \nabla \varphi ) ) ^ 2 \tilde { u } + q ( x ) \tilde { u } = e ^ { - i \varphi ( x ) } \mathcal { H } _ { A , q } u . \end{align*}


\begin{align*} { v ' } ^ + = \phi ^ + \mbox { o n $ \partial D $ } \end{align*}


\begin{align*} u ^ * ( \xi , t ) = \frac { 1 } { \rho ( \xi , t ) } . \end{align*}


\begin{align*} \delta W _ + : = - \nabla \cdot W _ + = 0 . \end{align*}


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


\begin{align*} G = \left \{ \begin{pmatrix} a & x \\ 0 & 1 \end{pmatrix} : x \in \R , a = \pm 1 \right \} \end{align*}


\begin{align*} \delta x = \frac { \partial f } { \partial x } \bigg | _ { ( x _ 0 , u _ 0 ) } \delta x + \frac { \partial f } { \partial u } \bigg | _ { ( x _ 0 , u _ 0 ) } \delta u . \end{align*}


\begin{align*} ( e ^ { - i \hat { \theta } } ( \omega _ { X ^ \vee } + \sqrt { - 1 } \alpha _ \phi ) ^ n ) = 0 , [ \alpha _ \phi = \alpha + \sqrt { - 1 } \partial \bar { \partial } \phi ] \in - c _ 1 ( E ) . \end{align*}


\begin{align*} \tau ( A ) = L \left ( H ^ { * } ( \overline { \alpha } ) \right ) + L \left ( H ^ { * } ( \overline { \beta } ) \right ) . \end{align*}


\begin{align*} \tilde L = C _ 0 \oplus L , ~ ~ ~ ~ ~ \tilde L ^ * = \{ c \in \tilde L \ , | \ , \ell _ c = \ell \} . \end{align*}


\begin{align*} \mathbb { E } ( \overline { S } _ n ) = \frac { 1 } { n } \sum _ { 1 \leq i \leq n } \mathbb { E } ( X _ i ) . \end{align*}


\begin{align*} \lesssim \| \mathfrak { m } ( \cdot , \zeta ) \| _ { \mathcal { S } ^ \infty } 2 ^ { m / 4 + k / 2 + n + \max \{ - j , p , n + \epsilon M _ t \} / 2 - p / 2 + \max \{ l , p \} + 8 \epsilon M _ t } \big ( 2 ^ { \min \{ n , p \} + j } \mathbf { 1 } _ { l = - j , l > n + \epsilon M _ t } \end{align*}


\begin{align*} c _ p : = ( \hat { v } _ 2 \delta _ { 3 p } - \hat { v } _ 3 \delta _ { 2 p } ) - \frac { \hat { v } _ 2 \omega _ 3 - \hat { v } _ 3 \omega _ 2 } { 1 + \hat { v } \cdot \omega } \hat { v } _ p . \end{align*}


\begin{align*} a _ { \mathrm { p w } } ( z _ { \mathrm { n c } } , v _ { \mathrm { n c } } ) = ( \lambda u , v _ { \mathrm { n c } } ) _ { 1 + \delta } \quad v _ { \mathrm { n c } } \in V ( { \mathcal { T } } ) . \end{align*}


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


\begin{align*} \Delta _ n ^ { ( \alpha ) } = \displaystyle \prod _ { j = 1 } ^ { n } j ^ j ( \alpha + j ) ^ { j - 1 } \end{align*}


\begin{align*} \frac { 1 - C _ a } { 1 + C _ b } = \frac { 1 - \big ( ( 1 + \alpha ) \varkappa ^ 2 + ( 1 + 1 / \alpha ) \Lambda _ 1 ^ 2 ( \epsilon _ 3 + \widehat \Lambda _ 3 \varkappa ^ 2 ) \big ) } { 1 + ( 1 + 1 / \alpha ) \Lambda _ 1 ^ 2 \Lambda _ 3 } = \theta _ 0 ( \varkappa , \alpha ) < 1 . \end{align*}


\begin{align*} \mu \int _ { \Omega } \varepsilon S ' ( \overline { u } , h ) : \varepsilon p + \nu \int _ { \Omega } \mathsf { m } ' \left ( \varepsilon \overline { y } ; \varepsilon S ' ( \overline { u } , h ) \right ) : \varepsilon p = \int _ { \Omega } h \cdot p . \end{align*}


\begin{align*} m _ { j , , r } ^ k = q ^ k _ { j , } ( \alpha _ r ) = \frac { 1 } { 4 } \alpha _ r ^ 2 r = 1 , \ldots , p , \end{align*}


\begin{align*} \bar s ^ 0 = \min _ { k \in \mathcal K } \ \max _ { \Vert \vec p _ c \Vert ^ 2 \le P } \ | { \vec { h } } _ { k } ^ { H } \vec { p } _ { c } | ^ 2 = \min _ { k \in \mathcal K } P \Vert \vec h _ k \Vert ^ 2 . \end{align*}


\begin{align*} p ( 0 , v , g ) = p _ { } ( v , g ) , v \in ( 0 , V _ F ) , g \in \mathbb { R } . \end{align*}


\begin{align*} \mathop { \lim } \limits _ { \varepsilon \to \varepsilon _ 0 } \mathop { \sup } \limits _ { ( x , j ) \in K \times S } P \left ( { \| { y ^ \varepsilon \left ( { t , s , \xi , j } \right ) - y ^ { \varepsilon _ 0 } \left ( { t , s , \xi , j } \right ) } \| _ H \ge \eta } \right ) = 0 . \end{align*}


\begin{align*} l ( h _ \rho ) = a _ { \rho 0 } ( f ) + b _ 1 Q _ \rho + \ldots + b _ l Q _ \rho ^ l \end{align*}


\begin{align*} x + i f ( x ) = \omega \left ( H ( x + i f ( x ) ) \right ) = \omega \left ( h ( x ) \right ) , \end{align*}


\begin{align*} \psi _ \mathrm { t r i a l } : = \sum _ { s = 0 } ^ N \sum _ { | d | \le \sigma _ N ^ 2 } c _ { d , s } u _ 1 ^ { \otimes ( N - s + d ) / 2 } \otimes _ { \mathrm { s y m } } u _ 2 ^ { \otimes ( N - s - d ) / 2 } \otimes _ { \mathrm { s y m } } \Phi _ { \mathrm { t r i a l } , s } . \end{align*}


\begin{align*} \int _ { 0 } ^ { a } D _ q f ( t ) d _ q t = f ( a ) - \lim _ { n \rightarrow \infty } f ( a q ^ n ) . \end{align*}


\begin{align*} \sum _ { g \in \mathrm { G a l } ( K / \Q ) } \tau ^ \mathrm { a n } ( M / S , g . \rho ) = \sum _ { g \in \mathrm { G a l } ( K / \Q ) } \tau ^ \mathrm { t o p } ( M / S , g . \rho ) \ ; . \end{align*}


\begin{align*} n _ i = i + \sum _ { j = 1 } ^ { i - 1 } p _ j \end{align*}


\begin{align*} \nabla g : = \left ( \frac { \partial g } { \partial x _ { 1 } } , \frac { \partial g } { \partial x _ { 2 } } , \dots , \frac { \partial g } { \partial x _ { N } } , \frac { \partial g } { \partial y _ { 1 2 } } , \dots , \frac { \partial g } { \partial y _ { N - 1 N } } \right ) , \ \ \Delta g : = \sum _ { k = 1 } ^ N \frac { \partial ^ 2 g } { \partial x _ { k } ^ 2 } + \sum _ { k = 1 } ^ { N - 1 } + \frac { \partial ^ 2 g } { \partial y _ { k k + 1 } ^ 2 } . \end{align*}


\begin{align*} \int _ { \Omega _ \epsilon } \eta ^ p \langle | \nabla G _ \epsilon | ^ { p - 2 } \nabla G _ \epsilon , \nabla ( G _ \epsilon - g ) \rangle = - p \int _ { \Omega _ \epsilon } \eta ^ { p - 1 } ( G _ \epsilon - g ) \langle | \nabla G _ \epsilon | ^ { p - 2 } \nabla G _ \epsilon , \nabla \eta \rangle \leq C \end{align*}


\begin{align*} b = l ^ C _ A ( b ) r ^ C _ A ( b ) . \end{align*}


\begin{align*} \int _ 0 ^ { T } \big [ \left ( u , \partial _ t \phi \right ) - \nu \left ( \nabla u , \nabla \phi \right ) - \left ( ( u \cdot \nabla ) \ , u , \phi \right ) \big ] \ , d t + \left ( u _ 0 , \phi ( 0 ) \right ) = 0 , \end{align*}


\begin{align*} u _ m \big ( n \big ) = \sum _ { j = 0 } ^ { \infty } C _ m \big ( n + k ( 3 k + 1 ) / 2 - j \big ) \ , \big ( M _ k ( j ) + \widetilde { P } _ k ( j ) \big ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , T ] } E \left [ | Q ^ { n , 4 } _ \tau | ^ 2 \right ] = 0 . \end{align*}


\begin{align*} \rho _ { k } ( x ) = \left \{ \begin{array} { l l } 0 & 1 \leq x \leq 5 0 + 5 0 ( k - 2 ) , \\ 1 & x \geq 6 0 + 5 0 ( k - 2 ) , \end{array} \right . \end{align*}


\begin{align*} ( \alpha , \beta ) _ { \rho , L , n } = T _ { L | K } ( \lambda _ { \rho } ( \alpha ) \psi _ { L , n } ( \beta ) ) \cdot _ { \rho } v _ n \end{align*}


\begin{align*} \mathcal { G } ( O _ n ) & = \mathcal { G } ( h _ 0 ) \oplus \mathcal { G } ( h _ 1 ) \oplus \dots \oplus \mathcal { G } ( h _ d ) \\ & = ( k _ 0 \oplus k _ 1 \oplus \dots \oplus k _ d ) s + [ \mathcal { G } ( r _ 0 ) \oplus \mathcal { G } ( r _ 1 ) \oplus \dots \oplus \mathcal { G } ( r _ d ) ] & \mathcal { G } ( r _ i ) < s = 2 ^ t \\ & = ( k _ 0 \oplus k _ 1 \oplus \dots \oplus k _ d ) s + \mathcal { G } ( ( r _ 0 , r _ 1 , \dots , r _ d ) ) \end{align*}


\begin{align*} \theta ^ { \ell _ i - i - 1 } _ t & = \bigcirc _ { \ell = 0 } ^ { i } ( \theta _ t + \gamma _ { r - i + \ell } ) ^ { - 1 } \circ ( \theta _ t + \xi _ { i , 1 } ) \circ \cdots \circ ( \theta _ t + \xi _ { i , \ell _ i } ) \\ & = \bigcirc _ { \ell = 0 } ^ { j } ( \theta _ t + \gamma _ { r - j + \ell } ) ^ { - 1 } \circ ( \theta _ t + \xi _ { j , 1 } ) \circ \cdots \circ ( \theta _ t + \xi _ { j , \ell _ j } ) = \theta ^ { \ell _ j - j - 1 } _ t \enspace . \end{align*}


\begin{align*} - 2 \ln \left ( \frac { 2 \Gamma \left ( \frac { 3 } { 4 } \right ) } { \Gamma \left ( \frac { 1 } { 4 } \right ) } \right ) = \frac { \gamma } { 2 } + \frac { 1 } { 2 } \ln \left ( \frac { \pi } { 2 } \right ) - \frac { 1 } { \pi } \left ( \Delta _ 1 + \frac { \pi \gamma } { 2 } \right ) . \end{align*}


\begin{align*} d \lambda _ i ( t ) = \sqrt { \frac { 2 } { \beta } } d B _ i ( t ) + \sum _ { j ( \neq i ) } \frac { 1 } { \lambda _ i ( t ) - \lambda _ j ( t ) } d t - \frac { \lambda _ i ( t ) } { 2 } d t , \ \ i = 1 , \dots , N , \end{align*}


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


\begin{align*} \| H _ { \lambda } \| _ \infty < + \infty \Rightarrow | \nabla H _ { \lambda } ( x ) | = o ( | \nabla \Gamma ( x ) | ) \hbox { a s } x \to 0 . \end{align*}


\begin{align*} R ^ i _ j = K ( x , y ) \left \lbrace F ^ 2 \delta ^ i _ j - F F _ { y ^ j } y ^ i \right \rbrace . \end{align*}


\begin{align*} M & \le \left ( \frac { 4 b d } { \varepsilon } \right ) ^ { b d k } \left ( \frac { 4 k } { \varepsilon } \right ) ^ k \\ & \le \left ( \frac { 4 b d k } { \varepsilon } \right ) ^ { b d k } \left ( \frac { 4 b d k } { \varepsilon } \right ) ^ { b d k } \\ & = \left ( \frac { 4 b d k } { \varepsilon } \right ) ^ { 2 b d k } . \end{align*}


\begin{align*} | \Gamma | ^ 2 - | \Gamma _ 1 | ^ 2 = \Gamma ^ * \Gamma - \Gamma ^ * S S ^ * \Gamma = \Gamma ^ * ( I - S S ^ * ) \Gamma = \Gamma ^ * ( e _ 0 e _ 0 ^ * ) \Gamma , \end{align*}


\begin{align*} F _ { p } ( s , f ) ( z ) = f ( e ^ { 2 \pi i s d _ { p } } z ^ { b _ { p } } ) \end{align*}


\begin{align*} \begin{array} { l l } \nabla _ { e _ 1 } e _ 1 = \nabla _ { X } X = e _ 1 , \\ \nabla _ { e _ 2 } e _ 2 = \nabla _ { U } U = 0 , \\ \nabla _ { e _ 1 } e _ 2 = \nabla _ { X } U = 0 , \\ \nabla _ { e _ 2 } e _ 1 = \nabla _ { U } X = 0 . \end{array} \end{align*}


\begin{align*} T r ( \boldsymbol { X } ) = E ( \boldsymbol { X } , \boldsymbol { X } ) - ( \boldsymbol { M } , \boldsymbol { M } ) . \end{align*}


\begin{align*} L ^ { \prime } _ { n s } [ \dot { x } ^ { \prime } ( t ^ { \prime } ) , x ^ { \prime } ( t ^ { \prime } ) , t ^ { \prime } ] = \frac { 1 } { C ^ { \prime } _ 1 f ^ { \prime \ 2 } ( t ^ { \prime } ) [ f ^ { \prime } ( t ^ { \prime } ) \dot { x } ^ { \prime } ( t ) - a ^ { \prime } _ o x ^ { \prime } ( t ^ { \prime } ) + C ^ { \prime } _ 2 + v ^ { \prime } _ o V _ 0 ] } \ . \end{align*}


\begin{align*} \nu _ { s _ { m } \rightarrow \eta _ { m } ^ { r } } ( s _ { m } ) \propto & \nu _ { h _ { m } \rightarrow s _ { m } } ( s _ { m } ) \times \nu _ { h _ { m + 1 } \rightarrow s _ { m } } ( s _ { m } ) \times \nu _ { \eta _ { m } ^ { c } \rightarrow s _ { m } } ( s _ { m } ) \\ = & \pi _ { s , m } ^ { r , o u t } \delta ( s _ { m } - 1 ) + ( 1 - \pi _ { s , m } ^ { r , o u t } ) \delta ( s _ { m } ) , \end{align*}


\begin{align*} \left \| \frac { f } { y } x ^ { \alpha } \right \| _ { L ^ { 2 } } = \left \| \left \| \frac { f } { y } \right \| _ { L _ { y } ^ { 2 } } x ^ { \alpha } \right \| _ { L _ { x } ^ { 2 } } \leq \left \| f _ { y } x ^ { \alpha } \right \| _ { L ^ { 2 } } \end{align*}


\begin{align*} \| \mu \| _ { C o n } = \max \{ \| \mu \| _ { M \otimes _ { e h } N } , \| \mu ^ * \| _ { M \otimes _ { e h } N } \} . \end{align*}


\begin{align*} { \sf U } ( \xi ) = { \sf U } _ \infty ( \xi ) + { \sf B } _ 1 | \xi | ^ \gamma + O ( | \xi | ^ { \gamma - { \sf k } _ 1 } ) | \xi | \to \infty , \end{align*}


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


\begin{align*} \mathbb { V } _ { n , k } = \{ V \in \mathbb { R } ^ { k \times n } \colon V V ^ T = I _ k \} , \end{align*}


\begin{align*} 0 \leq A : = \frac { \varrho _ 1 - \varrho _ 2 } { ( \varrho _ 1 - \varrho _ 2 ) + ( p _ 1 - p _ 2 ) } \leq 1 0 \leq B : = \frac { p _ 1 - p _ 2 } { ( \varrho _ 1 - \varrho _ 2 ) + ( p _ 1 - p _ 2 ) } \leq 1 . \end{align*}


\begin{align*} u ^ s _ { - A , q } ( y , x ) & = \underset { ( \ell _ 2 , m _ 2 ) \in \Gamma } \sum \beta _ { \ell _ 2 m _ 2 } ( x ) h ^ { ( 1 ) } _ { \ell _ 2 } ( k \vert y \vert ) Y ^ { m _ 2 } _ { \ell _ 2 } \left ( \widehat { y } \right ) . \end{align*}


\begin{align*} \textstyle \big ( \sum _ { x \in G } a _ x x \big ) \cdot \big ( \sum _ { y \in G } b _ y y \big ) = \sum _ { z \in G } \big ( \sum _ { x y = z } a _ x b _ y \big ) z . \end{align*}


\begin{align*} \omega _ i = \frac { \lambda ( a _ i + a _ { i + 1 } ) } { 2 [ ( \sigma _ N ^ 2 + \sigma _ Z ^ 2 ) \lambda ^ 2 + \rho ) ] } . \end{align*}


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


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


\begin{align*} \int _ \Delta \omega = \int _ { S ^ { n - 1 } } f ^ * \alpha , \end{align*}


\begin{align*} \Lambda _ { r , s } = ( \kappa _ 1 , \kappa _ 2 , \ldots , \kappa _ { m } ) \end{align*}


\begin{align*} H _ { s , c } = \{ n \in \mathbb { N } , n > 1 2 7 \ p | n , p ^ { \nu _ p ( n ) } \leq s \ \ p > s c ^ { - 1 } \ \ d + \left [ \frac { s } { p } \right ] \geq p \} \end{align*}


\begin{align*} f = f _ 0 \oplus f _ 1 \oplus f _ 2 \oplus \cdots \end{align*}


\begin{align*} \Delta v = \lambda ^ 2 \Big ( \hat { V } ( | y | ) v - | v | ^ { p - 1 } v \Big ) \mathbb { R } ^ N , \end{align*}


\begin{align*} \begin{cases} u _ p ^ 0 ( x , \eta ) + \delta = \phi _ * ( z ) + w ( x , \eta ) , \\ h _ p ^ 0 ( x , \eta ) + \sigma = \psi _ * ( z ) + \Omega ( x , \eta ) . \end{cases} \end{align*}


\begin{align*} \nabla ^ 2 u = & - k | \nabla u | , \\ \mu | \nabla u | = & - \langle J , \nabla u \rangle . \end{align*}


\begin{align*} c _ { N , i } = \frac { N ^ i \Gamma ( N - i + \beta ) } { i ! ( N - i - 1 ) ! ( \alpha + 1 ) ^ { N - i + \beta } } . \end{align*}


\begin{align*} a = a _ 1 \mathbf { e } _ 1 + a _ 2 \mathbf { e } _ 2 , c = \cos \theta \cos \phi \mathbf { e } _ 1 + \cos \theta \sin \phi \mathbf { e } _ 2 + \sin \theta \mathbf { e } _ 3 , ( a - c ) \times b = - ( a _ 2 - \cos \theta \sin \phi ) \mathbf { e } _ 3 + \sin \theta \mathbf { e } _ 2 \end{align*}


\begin{align*} H = A + \nabla \vartheta . \end{align*}


\begin{align*} \mathrm { c a r d } ( \mathcal { R } _ { [ n , j ] } ) = \binom { m ( p + 1 ) + j } { m } , \mathrm { c a r d } ( \mathcal { S } _ { [ n , j ] } ) = \frac { j + 1 } { p m + j + 1 } \binom { m ( p + 1 ) + j } { m } . \end{align*}


\begin{align*} t _ j = T ( j / N ) ^ k k \geq \frac { 2 - \beta } { \beta } , j = 0 , \dots , N . \end{align*}


\begin{align*} \begin{aligned} \sum _ { v \in V ( G ) } x _ v & \leq \frac { 2 e ( L ) + 2 \epsilon e ( S ) + ( 1 + \epsilon ) e ( L , S ) } { \rho _ \alpha } \\ & \leq \frac { \epsilon n + \epsilon C _ 1 n + ( 1 + \epsilon ) ( r - 2 + \epsilon ) n } { \rho _ \alpha } \\ & = \frac { ( ( 1 + C _ 1 ) \epsilon + ( 1 + \epsilon ) ( r - 2 + \epsilon ) ) n } { \rho _ \alpha } . \end{aligned} \end{align*}


\begin{align*} S : = \{ i \in \{ 1 , \ldots , m \} | \overline { \{ i \} } \not \in \mathcal A _ \omega \} . \end{align*}


\begin{align*} \int _ { B _ R ( h _ n t _ 1 ) } \phi _ n ^ 2 d y = o _ n ( 1 ) . \end{align*}


\begin{align*} \mathcal { H } ( v , x ) & = x - d ( v ) - \sum _ { w \in c ( v ) } \dfrac { 1 } { \mathcal { H } ( w , x ) } ( \forall v \in V ( T ) ) , \end{align*}


\begin{align*} \begin{aligned} x _ i { } & = { } \frac { U } { L } + \frac { U } { L } \sum \limits _ { j = i } ^ { N - 1 } h _ j \\ f _ i { } & = { } \Delta - \frac { U ^ 2 } { L } \sum \limits _ { j = 0 } ^ { i - 1 } h _ j \ , \left ( 1 - \frac { h _ j } { 2 } \frac { - \kappa } { 1 - \kappa } \right ) \\ g _ i { } & = { } U \end{aligned} \end{align*}


\begin{align*} \delta ^ { k + 1 } N _ m ^ k f = 0 . \end{align*}


\begin{align*} S ^ { j \ell } F ( A x + e ) = F ( A S x + e ) = B G ( S x ) + d = S ^ { k m } B G ( x ) + d = S ^ { k m } F ( A x + e ) , \end{align*}


\begin{align*} \langle f , \theta ( u ( n ) ) I ^ { \circ } ( w _ 1 , z ) w _ 2 \rangle = 0 . \end{align*}


\begin{align*} \partial _ { y _ k } \partial _ { y _ m } \bigl ( q ^ { k m } \sigma \bigr ) - \partial _ { y _ k } \bigl ( h ^ k \sigma \bigr ) = 0 , \end{align*}


\begin{align*} \sin ( \alpha _ k ^ * ) \Re \{ e _ k \} = \cos ( \alpha _ k ^ * ) \Im \{ e _ k \} \end{align*}


\begin{align*} ( \zeta - 1 ) \left ( \frac { 1 } { \zeta } - 1 \right ) = 2 - \left ( \zeta + \frac { 1 } { \zeta } \right ) = 2 - 2 \cos \left ( \frac { 1 } { j + 1 } \pi \right ) , \end{align*}


\begin{align*} & \Delta ^ { \sigma ( 2 + k ) } _ { h } \cdots \Delta ^ { \sigma ( m - 1 + k ) } _ { h } H ( x _ 0 , \ldots , x _ { m - 1 } ) \\ & = \Delta ^ { \sigma ( 2 + k ) } _ { h } \cdots \Delta ^ { \sigma ( m - 1 + k ) } _ { h } \Bigg [ \sum \limits _ { s = 0 } ^ { m - 1 } \prod \limits _ { i = 0 } ^ { m - 3 } x _ { i + s } - \sum \limits _ { s = 0 } ^ { m - 1 } \prod \limits _ { i = 0 } ^ { m - 3 } x _ { \sigma ( i + s ) } \Bigg ] . \end{align*}


\begin{align*} S ( A ) = \{ g : \mathbb { D } \rightarrow \mathbb { C } \mbox { h o l o m o r p h i c : t h e r e i s } f \in A \mbox { w i t h } | \hat { g } ( k ) | \leq | \hat { f } ( k ) | \mbox { f o r a l l } k \} . \end{align*}


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


\begin{align*} \tilde \Lambda _ l \coloneqq \sum _ { j = 0 } ^ { \min ( l , k ) } \frac { \gamma ^ { 2 k + l - 2 j } \kappa _ { k , j , l } } { ( l - j ) ! j ! ( ( k - j ) ! ) ^ 2 } . \end{align*}


\begin{align*} S ( x _ 0 , x _ 1 , x _ 2 , \ldots ) & = ( 0 , x _ 0 , x _ 1 , x _ 2 , \ldots ) \ , , \intertext { a n d b y $ S ^ * $ i t s a d j o i n t ( t h e \emph { b a c k w a r d s h i f t } ) , } S ^ * ( x _ 0 , x _ 1 , x _ 2 , \ldots ) & = ( x _ 1 , x _ 2 , x _ 3 , \ldots ) , \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \Psi ( t _ n h _ n ) - \Psi ( 0 ) } { t _ n } = h ( g ) . \end{align*}


\begin{align*} f ( x ) = \delta _ H ( x ) , g ( y ) = \lambda \| y \| _ 1 , g ^ \ast ( z ) = \begin{cases} 0 & \| z \| _ \infty \leq \lambda \\ + \infty & \mbox { o t h e r w i s e , } \end{cases} h ( x ) = \frac { 1 } { 2 } \| C x - b \| ^ 2 , \end{align*}


\begin{align*} - \int _ C \frac { e ^ { \omega s } \cdot s ^ { 1 - d } } { ( e ^ { \omega s } - 1 ) ^ 2 } \ , d s = \frac { ( d - 1 ) \zeta ( d ) } { 2 \pi i } \cdot \big ( \frac { \omega } { 2 \pi i } \big ) ^ { d - 2 } , \end{align*}


\begin{align*} \frac { \| \mbox { g r a d } b \| ^ 2 } { b ^ 2 } \left \{ G ( V , V ) G ( U , U ) - G ( U , V ) ^ 2 \right \} - G ( \mbox { H o r } ( P _ V V ) , \mbox { H o r } ( P _ U U ) ) + G ( \mbox { H o r } ( P _ V U ) , \mbox { H o r } ( P _ V U ) ) = 0 , \end{align*}


\begin{align*} & \zeta ( \Delta _ n , s ) = V _ \alpha ( s ) \Bigg ( a ( s ) \ , n ^ { \alpha - 2 s } + \sum _ { m = 0 } ^ { M - 1 } b _ m ( s ) \ , n ^ { - 2 m } \Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) , \\ & \zeta ( \widetilde { \Delta } _ n , s ) = V _ \alpha ( s ) \Bigg ( \widetilde { a } ( s ) \ , n ^ { \alpha - 2 s } + \sum _ { m = 0 } ^ { M - 1 } \widetilde { b } _ m ( s ) \ , n ^ { - 2 m } \Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) . \end{align*}


\begin{align*} \Sigma _ \pm = \{ \sigma _ I : \bar { I } \in \mathcal A _ \pm \} . \end{align*}


\begin{align*} & | p | _ v = p ^ { - \tfrac { [ K _ v : \Q _ p ] } { [ K : \Q ] } } \ \ v \in { { \mathfrak { M } } } ^ { f } _ K \ \ v \mid p \enspace , \\ & | x | _ v = | \iota _ v x | ^ { \tfrac { [ K _ v : \R ] } { [ K : \Q ] } } \ \ v \in { { \mathfrak { M } } } ^ { \infty } _ K \enspace , \end{align*}


\begin{align*} \lambda ^ { - \frac { n - 2 } { 2 } } \sigma { \sf A } _ 1 & = - \eta ^ \frac { 2 } { 1 - q } | y | \to \infty , \ | \xi | \to 0 , \\ - \eta ^ \frac { 2 } { 1 - q } { \sf B } _ 1 & = K { \sf D } _ J ( T - t ) ^ J \eta ^ \gamma | \xi | \to \infty , \ | z | \to 0 . \end{align*}


\begin{align*} \P ( T _ { d e t } \geq \ss ) = \exp \Big ( - \int _ { B _ O ( R ) } \P _ { x _ 0 } ( T _ { d e t } \leq \ss ) d \mu ( x _ 0 ) \Big ) \end{align*}


\begin{align*} g ( \nabla _ { V } X , W ) = - g ( \nabla _ { W } X , V ) \end{align*}


\begin{align*} \omega | _ L = 0 , ( e ^ { - i \hat { \theta } } \Omega | _ L ) = 0 . \end{align*}


\begin{align*} \mathcal Q ( x ) ( v ) = ( a v g ^ * \circ a v g ) ( x ) ( v ) = \sum \limits _ { e \in E _ v } \frac { ( a v g ( x ) ) ( e ) } { | e | } \frac { \delta _ E ( e ) } { \delta _ V ( v ) } , \end{align*}


\begin{align*} g _ l = \sum _ 0 ^ l h _ k l = 0 , 1 , 2 , \ldots \end{align*}


\begin{align*} ( x , y , z ) = - [ x , y ] z + x [ y , z ] + [ x z , y ] ; \end{align*}


\begin{align*} x y z = a , x + y + z = a n . \end{align*}


\begin{align*} E ( c _ p u _ p ) = c _ p ^ 2 E ( u _ p ) \geq L + \epsilon , \end{align*}


\begin{align*} \prod \limits _ { k = 1 } ^ { \infty } \frac { \mu _ { k } ^ { 2 } } { \lambda _ { k } ^ { 2 } } = 1 - \sum \limits _ { k = 1 } ^ { \infty } \frac { a _ { k } } { \lambda _ k ^ 2 } ; \end{align*}


\begin{align*} f _ { r , s } ( x ) = f ( r x + s ) \in K [ x ] . \end{align*}


\begin{align*} P ( S _ i ) = \frac { { m _ x \choose i } \sum _ { j = 0 } ^ { i } ( - 1 ) ^ { j } { { i } \choose { j } } ( i - j ) ^ { \gamma _ x k _ x } } { m _ x ^ { \gamma _ x k _ x } } . \end{align*}


\begin{align*} u _ { A , q } ( \cdot , d ) = u ^ i ( \cdot , d ) + u _ { A , q } ^ s ( \cdot , d ) \mathrm { i n } \ ; \R ^ 3 , \end{align*}


\begin{align*} & \partial _ s u + \Delta u = \frac { f ' ( u ) } { f ( u ) } \left ( m + \frac { | \widetilde { \nabla } u | _ { \widetilde { g } } ^ 2 } { f ( u ) ^ 2 - | \widetilde { \nabla } u | _ { \widetilde { g } } ^ 2 } \right ) + \mathcal { H } \ , \frac { f ( u ) } { \sqrt { f ( u ) ^ 2 - | \widetilde { \nabla } u | _ { \widetilde { g } } ^ 2 } } , \\ & u ( 0 , \cdot ) = u _ 0 , \end{align*}


\begin{align*} \hat w ( s ) = ( s ^ \beta { + } A ) ^ { - 1 } \left ( \hat f ( s ) - \frac 1 { s } A u _ 0 \right ) , \end{align*}


\begin{align*} f ^ h = t ^ { \deg f } f \left ( \frac { x _ 1 } { t } , \ldots , \frac { x _ n } { t } \right ) . \end{align*}


\begin{align*} \nu _ { \eta _ { m } ^ { c } \rightarrow s _ { m } } ( s _ { m } ) = \pi _ { s , m } ^ { c , i n } \delta ( s _ { m } - 1 ) + ( 1 - \pi _ { s , m } ^ { c , i n } ) \delta ( s _ { m } ) , \end{align*}


\begin{align*} \norm { \nabla _ x \tilde { f } } \lesssim L , \\ \norm { H ( \tilde { f } ) _ x } \leq \sum _ { i = 0 } ^ { N _ 0 } P ^ { \circ i } ( L ) , \end{align*}


\begin{align*} V = \sum _ { d _ 1 , \ldots , d _ k \le x } h _ { f , k } ( d _ 1 , \ldots , d _ k ) \Big ( \Big ( \frac { x } { d _ 1 \cdots d _ k } \Big ) ^ { r + 1 } T _ { k - 1 } \Big ( \log \frac { x } { d _ 1 \cdots d _ k } \Big ) + O \Big ( \Big ( \frac { x } { d _ 1 \cdots d _ k } \Big ) ^ { r + \theta _ { k } + \varepsilon } \Big ) \Big ) , \end{align*}


\begin{align*} u _ k = p u _ { k - 1 } + a _ k u _ { k - 2 } , k \geq 1 , \end{align*}


\begin{align*} P _ a = \begin{cases} P ^ { - 1 } , & a \in P ^ { - 1 } \\ P \cup \{ b \in A \mid b < f ( a ) ^ { - 1 } \} & a \in P . \end{cases} \end{align*}


\begin{align*} \mathrm { D T M } _ { ( p , q ) } ( X ^ { n } ) = \mathrm { E } \left [ ( X - \mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { n } | x _ { p } < X < x _ { q } \right ] , \end{align*}


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


\begin{align*} J _ { w , \tau , \lambda } \psi ( \lambda , \cdot ) = \mathbf { c } _ { w ^ { - 1 } , \tau } ( - w \lambda ) \psi ( w \lambda , \cdot ) , \ ; \ ; \ ; \lambda \in \mathfrak { a } ^ * _ \mathbb { C } , w \in W _ A . \end{align*}


\begin{align*} \operatorname { t r } ( \alpha ) = \operatorname { T r } \left ( H ^ { * } ( \overline { \alpha } ) \right ) , \ \ \ N ( \alpha ) = \operatorname { d e t } \left ( H ^ { * } ( \overline { \alpha } ) \right ) . \end{align*}


\begin{align*} 0 = \frac { \partial \tau ( t , x ) } { \partial t } + \frac { \partial \tau ( t , x ) } { \partial x } \frac { \partial p ( t ; t , x ) } { \partial \tau } \ , . \end{align*}


\begin{align*} P _ { \rm F A } ^ { } ( \lambda ) = Q \left ( \frac { { ( 1 - \alpha ) \left ( { \lambda } - 1 \right ) \bar { N } { M } - { M } } } { \sqrt { 2 { M } + \left ( 2 - \alpha - { \lambda } \right ) ^ 2 \bar { N } { M } } } \right ) \end{align*}


\begin{align*} \psi _ s : K [ t ] \longrightarrow K ; \ t ^ k \mapsto \dfrac { 1 } { ( k + \gamma _ { r - s } ) \cdots ( k + \gamma _ r ) } = \dfrac { 1 } { ( k + \zeta _ 1 ) \cdots ( k + \zeta _ { s + 1 } ) } \enspace , \end{align*}


\begin{align*} A = \left [ \begin{matrix} B & { \bf 0 } \\ { \bf 0 } & C \end{matrix} \right ] , B = \left [ \begin{matrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ \epsilon & 0 & 1 \end{matrix} \right ] , \end{align*}


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


\begin{align*} \sum _ { { z = 1 } } ^ \infty \frac { 1 } { A ( z ) } = \infty \end{align*}


\begin{align*} & \left . \frac { d } { d t } ( X , \delta _ X D f \wedge \delta _ X D g ) \right | _ { t = 0 } = ( \dot X , \delta _ X D f \wedge \delta _ X D g ) + ( X , \dot { \delta _ X } D f \wedge \delta _ X D g ) \\ & + ( X , \delta _ X \dot { D f } \wedge \delta _ X D g ) + ( X , \delta _ X { D f } \wedge \dot { \delta _ X } D g ) + ( X , \delta _ X { D f } \wedge \delta _ X \dot { D g } ) . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ g [ \alpha _ i , \beta _ i ] \prod _ { j = 1 } ^ k \gamma _ j = 1 , \end{align*}


\begin{align*} d \nu ( s ) = \frac { 1 8 } { \pi ^ { 4 } } \sum \nolimits _ { m \in \mathbb { Z } } \frac { 1 } { m ^ { 2 } } I _ { [ m , m + 1 ] } ( s ) \left \{ \sum \nolimits _ { n \in \mathbb { N } } \frac { 1 } { 2 ^ { n } n ^ { 2 } } \sum \nolimits _ { j = 1 } ^ { 2 ^ { n } } d \tau \left ( s - m - j 2 ^ { - n } \right ) \right \} , \end{align*}


\begin{align*} M _ { n , k } ( k ) = \dfrac { k - n + 1 } { k + 1 } \dbinom { k + n } { n } = \dbinom { k + n } { n } - \dbinom { k + n } { n - 1 } . \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { N } \ , \left \langle \left ( \mathcal { N } _ 1 - \frac { N } { 2 } \right ) ^ 2 \right \rangle _ { \psi _ \mathrm { g s } } = 0 , \end{align*}


\begin{align*} H _ { k , j ; n , l , r ; \star } ^ { \mu , m , i ; l i n } ( t , x , \zeta ) = \int _ { 0 } ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \big [ ( t - s ) \mathcal { K } _ { k , n , l , r } ^ { \star ; \mu , m } ( y , v , \omega , \zeta ) + \mathcal { K } _ { k , n , l , r } ^ { \star ; e r r , \mu , m } ( y , v , \omega , \zeta ) \big ] f ( s , x - y + ( t - s ) \omega , v ) \end{align*}


\begin{align*} \varphi _ n = \varphi _ { n ; \textrm { o u t } } + \varphi _ { n ; \textrm { i n } } + \varphi _ { n ; \textrm { s u r } } \end{align*}


\begin{align*} S _ 2 = \sum _ { u , v , z } b _ { u \vee z } b _ { v \vee z } \int _ { E ^ c } e ^ { 2 \pi i ( u - v ) x } d x , \end{align*}


\begin{align*} a ^ \epsilon ( \lambda , m ^ 2 ) = \begin{cases} - c _ 1 \lambda \log ( \epsilon ^ { - 2 } ) + O _ { m ^ 2 , \lambda } ( 1 ) & ( d = 2 ) , \\ - c _ 1 \lambda \epsilon ^ { - 1 } + c _ 2 \lambda ^ 2 \log ( \epsilon ^ { - 2 } ) + O _ { m ^ 2 , \lambda } ( 1 ) & ( d = 3 ) . \end{cases} \end{align*}


\begin{align*} b _ { i - l + k - n , i } ( k , l ) P _ { i - l + k - n , j } + b _ { l , l + j - i + n } ( i , j ) P _ { k , l + j - i + n } = \chi _ { l , i } \delta ( P _ { k , j } ) \ , . \end{align*}


\begin{align*} \theta _ 1 ( x ) = \left \{ \begin{array} { c l c } 1 & & x \in ( b _ 1 + \varepsilon , b _ 2 - \varepsilon ) , \\ 0 & & x \in ( 0 , b _ 1 ) \cup ( b _ 2 , L ) , \\ 0 \leq \theta _ 1 \leq 1 & & . \end{array} \right . \end{align*}


\begin{align*} & c _ { n - 2 , \xi _ { \boldsymbol { s } k } , \xi _ { \boldsymbol { t } l } } ^ { \ast \ast } = \frac { \Gamma ( ( n - 2 ) / 2 ) } { ( 2 \pi ) ^ { ( n - 2 ) / 2 } } \left \{ \int _ { 0 } ^ { \infty } t ^ { ( n - 4 ) / 2 } \ln \left [ 1 + \exp \left ( - \frac { 1 } { 2 } \xi _ { \boldsymbol { s } , k } ^ { 2 } - \frac { 1 } { 2 } \xi _ { \boldsymbol { t } , l } ^ { 2 } \right ) \exp ( - t ) \right ] \mathrm { d } t \right \} ^ { - 1 } . \end{align*}


\begin{align*} Q ( a ) ^ k = b ( a \otimes a ) ^ { \binom { k } { 2 } } Q ( a ^ k ) . \end{align*}


\begin{align*} \Phi _ { \nu , \nu ' } ^ \lambda ( z ) : = ( 2 \pi ) ^ { - n / 2 } \lambda ^ { n / 2 } ( \pi _ \lambda ( z , 0 ) \Phi _ \nu ^ \lambda , \Phi _ { \nu ' } ^ \lambda ) , z \in \R ^ { 2 n } \end{align*}


\begin{align*} W _ A ^ u = T _ { u ( 0 ) } W ^ u ( x ) , W ^ s _ A = T _ { u ( 0 ) } W ^ s ( y ) , \end{align*}


\begin{align*} \lambda _ * = \dfrac { \rho ^ { \varphi ^ + - { q ^ \pm } } } { 2 c _ 2 c ^ { q ^ \pm } } . \end{align*}


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


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


\begin{align*} L _ { f \cdot \tau } S _ \zeta = q ^ { - \ell ( \tau ) } \widetilde { N } _ { f \cdot \tau } + \sum _ { g \prec f \cdot \tau } \ell _ { g , f \cdot \tau } M _ g S _ \zeta , \end{align*}


\begin{align*} p _ { \mu \boxplus \nu } ( h ( x ) ) = \frac { f ( x ) } { \pi } \int _ { \mathbb { R } } \frac { d \nu ( s ) } { ( x - s ) ^ { 2 } + f ( x ) ^ { 2 } } , x \in V , \end{align*}


\begin{align*} \Lambda _ 0 = ( 0 , 1 ) \end{align*}


\begin{align*} Z ( \overline \delta , \overline \mu , \beta ) = \exp ( F ^ { P } \big ( g _ s , Q , \beta ) + F ^ { N P } ( g _ s , Q , \beta ) \big ) . \end{align*}


\begin{align*} \langle Y _ M ' ( u , z ) f , a \rangle = \langle f , Y _ { M } ( e ^ { z L ( 1 ) } ( - z ^ { - 2 } ) ^ { L ( 0 ) } u , z ^ { - 1 } ) a \rangle \end{align*}


\begin{align*} B ^ { i } \phi : = P _ { \Upsilon ^ { i } } ( I - d _ { V } ^ { i - 1 } G ^ { i } ) \phi , \end{align*}


\begin{align*} \varphi ( \lambda ) \gamma ( m _ w ^ { - 1 } ) \mathbf { c } _ { w ^ { - 1 } , \gamma } ( - w \lambda ) = \tau ( m _ w ^ { - 1 } ) \mathbf { c } _ { w ^ { - 1 } , \tau } ( - w \lambda ) \circ \varphi ( w \lambda ) , \ ; \ ; \ ; \lambda \in \mathfrak { a } ^ * _ \mathbb { C } , w \in W _ A . \end{align*}


\begin{align*} \| Y ( t ) - u ( t ) \| & = \| Y _ i - \tilde v ( t ) \| \leq \| Y _ i - \tilde v ( \sigma _ i ) \| + \| \tilde v ( \sigma _ i ) - \tilde v ( t ) \| < \frac { \epsilon } { 2 } , \end{align*}


\begin{align*} \left | 1 - M ( x + i f ( x ) ) \right | = \frac { 2 } { \sqrt { x ^ { 2 } + [ f ( x ) + 1 ] ^ { 2 } } } \leq \frac { 2 } { | x | } \rightarrow 0 \quad ( | x | \rightarrow \infty ) . \end{align*}


\begin{align*} \sum \limits _ { i , j = 1 } ^ { 2 } h _ { \dot { x } ^ { i } \dot { x } ^ { j } } y ^ { i } y ^ { j } \leq 0 , \end{align*}


\begin{align*} N ^ n _ \tau = N ^ { n , 1 } _ \tau + N ^ { n , 2 } _ \tau , \end{align*}


\begin{align*} \hat { h } _ i : = ( h _ 1 , \ldots , h _ { i - 1 } , h _ { i + 1 } , \ldots , h _ n ) \end{align*}


\begin{align*} g ^ { - 1 } ( s | \rho , \lambda ) = \frac { \lambda s } { ( \sigma _ N ^ 2 + \sigma _ Z ^ 2 ) \lambda ^ 2 + \rho } . \end{align*}


\begin{align*} \hat f ( \xi ) = \int _ { \R ^ n } f ( x ) e ^ { - i \xi x } \ , d x , \xi \in \R ^ n , \end{align*}


\begin{align*} \langle f , \sum _ { i \geq 0 } \binom { \frac { j _ 1 } { T } } { i } ( - 1 ) ^ i z _ 2 ^ { \frac { j _ 1 } { T } - i } I ^ { \circ } ( w _ 1 , z _ 2 ) u _ { p + \frac { j _ 2 } { T } + i } w _ 2 \rangle = 0 , \end{align*}


\begin{align*} \Psi ( \omega ) = h _ { 2 } \omega - \frac { d } { d t } ( h _ { 1 } \omega ' ) . \end{align*}


\begin{align*} A _ \ell ( y ) : = \{ ( x , u ) \in B _ { 3 R } ^ { d _ { \mathrm { C C } } } ( 0 ) : | x - y | \ge 2 ^ { \gamma \iota } C R _ \ell \} \end{align*}


\begin{align*} F _ p ( x ) = L ( h _ \theta ) + \sum _ { i \in I } \partial _ { i } L ( h _ \theta ) ( Q - h _ \theta ) ^ { i } = : L _ p ( Q ) . \end{align*}


\begin{align*} C ( v ) = \ln \cosh ( z _ 0 v ) . \end{align*}


\begin{align*} W \left [ { \eta \left ( t \right ) } \right ] = \ln \left \langle { e ^ { \mathop \int d t \eta ( t ) \xi ( t ) } } \right \rangle \end{align*}


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


\begin{align*} x _ { n + 1 } = f ( x _ { n } ) + \epsilon ( L _ G ( g ( x _ n ) ) ) , \end{align*}


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


\begin{align*} \| T x \| = \| x \| \end{align*}


\begin{align*} ( T _ p \varphi ) ( z ) = 2 \sqrt y \sum _ { n \neq 0 } \left ( a _ \varphi ( p n ) + p ^ { - 1 } a _ \varphi ( n / p ) \right ) K _ { i r } ( 2 \pi | n | y ) e ( n x ) . \end{align*}


\begin{align*} | \mathcal { S } _ { \kappa , 3 } ^ B | = 3 \left ( \sum _ { i , j \in \mathbb N \colon \atop { 2 ( i + j ) \leq \kappa } } \binom { \kappa - 2 ( i + j ) + 3 } { 3 } - A _ \kappa \right ) = 3 b _ \kappa . \end{align*}


\begin{align*} W ( \sigma ) = R _ { \varphi ( \sigma ) } ( u ) \wedge v + u \wedge R _ { \varphi ( \sigma ) } ( v ) . \end{align*}


\begin{align*} \int \limits _ { V } \left ( \overline { \rho _ R \log \rho _ R } - \rho _ R \log \rho _ R \right ) ( \tau , \cdot ) \ d x = \frac { 1 } { 2 \mu + \lambda } \int \limits _ 0 ^ { \tau } \int \limits _ { V } \left ( \overline { p ( \rho _ R ) } \rho _ R - \overline { p ( \rho _ R ) \rho _ R } \right ) \ d x \ d t . \end{align*}


\begin{align*} \epsilon _ g = s i g n ( b _ g \tilde c _ 1 ( \alpha , g ) , \tilde c _ 2 ( \alpha , g ) ) \end{align*}


\begin{align*} k _ 1 ( \Gamma ) = r ( J ) + p r k _ 2 ( \Gamma ) = s ( J ) + ( \ell _ 1 + \ell _ 2 - | J | + 1 + \sum _ { j \in J } d _ j - p ) s , \end{align*}


\begin{align*} \alpha _ j \varpi _ j = 0 ( 1 \le j \le n - 1 ) , \end{align*}


\begin{align*} E _ \mathcal { N } = | | x _ \mathcal { N } - x ( b ) | | , \end{align*}


\begin{align*} \widehat { \left ( | u | ^ { p - 1 } u \right ) } _ k = 2 \pi \sum _ { k = k _ 1 - k _ 2 + \cdots + k _ { p } } \widehat { u } _ { k _ 1 } \overline { \widehat { u } } _ { k _ 2 } \cdots \widehat { u } _ { k _ { p - 2 } } \overline { \widehat { u } } _ { k _ { p - 1 } } \widehat { u } _ { k _ p } = \frac { 1 } { L _ t [ u ] } \widehat { \left ( | v | ^ { p - 1 } v \right ) } _ k . \end{align*}


\begin{align*} N ^ { - 1 } _ \omega ( g ) ( x ) = \frac { 1 } { ( 2 \pi ) ^ 3 } \int _ { \R ^ 3 } e ^ { - i x \cdot \xi } \Big ( \frac { \hat { g } ( \xi ) } { \omega \cdot \xi } \Big ) d \xi , \end{align*}


\begin{align*} \lambda _ n ( x ) \triangleq \sum _ { i = 0 } ^ \infty a _ i x ^ i \end{align*}


\begin{align*} t ( v ) = t ' ( h _ k v ) \end{align*}


\begin{align*} \mathcal { E } _ u ^ { ( n ) } = & ( \chi - 1 ) f _ u ^ { ( n ) } + \mathcal { \tilde E } _ u ^ { ( n ) } \end{align*}


\begin{align*} \mathfrak H : = \bigoplus _ { \vec s \in \mathbb Z ^ n } \mathfrak H _ { \vec s } , \end{align*}


\begin{align*} \Phi _ b - \Phi _ a & = \int _ a ^ b \frac { \dd J _ t } { \dd t } \ , d t \\ \Psi - \Phi & = \int _ a ^ b \left ( \Bar Q \circ H _ t + H _ t \circ Q _ { \mathfrak { g } } \right ) d t \\ & = \Bar Q \circ \left ( \int _ a ^ b H _ t \ , d t \right ) + \left ( \int _ a ^ b H _ t \ , d t \right ) \circ Q _ { \mathfrak { g } } . \end{align*}


\begin{align*} \begin{aligned} & \langle \nabla _ i \nabla _ j \Psi _ t - \nabla _ l \Psi _ t \Gamma _ { i j } ^ l , \nabla _ m \nabla _ m \Psi _ t - \nabla _ l \Psi _ t \Gamma _ { m m } ^ l - \frac { \sum _ k ( \nabla _ k \nabla _ k \Psi _ t - \nabla _ l \Psi _ t \Gamma _ { k k } ^ l ) } { n } \rangle \\ & = \frac { 1 } { 2 t } O ( t ) + \sum _ l \Gamma _ { i j } ^ l ( \Gamma _ { m m } ^ l - \frac { \sum _ k \Gamma ^ l _ { k k } } { n } ) . \end{aligned} \end{align*}


\begin{align*} i u _ t + \bigtriangleup u + i \gamma u = 0 , \end{align*}


\begin{align*} | \pi _ i ( t ) | : = \lim _ { n \to \infty } n ^ { - 1 } \# ( \pi _ i ( t ) \cap [ n ] ) . \end{align*}


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


\begin{align*} a _ n ( p ^ e ) = \binom { n - 1 + e } { e } _ p , \end{align*}


\begin{align*} T _ n ( f ) : = n \wedge S _ n ( f ) , f \in C . \end{align*}


\begin{align*} \mathcal { W } _ { \frac { p } { 2 } } ( g \circ W _ { n } , g \circ W ) & = \mathcal { W } _ { \frac { p } { 2 } } ( g \circ W _ { n } \circ \pi _ \Delta , W ) \\ & \le \mathcal { W } _ { \frac { p } { 2 } } ( g \circ W _ { n } \circ \pi _ \Delta , \sigma X _ n ) + \mathcal { W } _ { \frac { p } { 2 } } ( \sigma X _ n , \sigma B ) \\ & \le C n ^ { - \frac { 1 } { 4 } + \frac { 1 } { 4 ( p - 1 ) } } + C n ^ { - \frac { 1 } { 4 } + \delta } \le C n ^ { - \frac { 1 } { 4 } + \frac { 1 } { 4 ( p - 1 ) } } , \end{align*}


\begin{align*} \dim \left [ \gg , x \right ] = | \Delta _ { 1 } \backslash { A } ^ { \perp } | + 2 k . \end{align*}


\begin{align*} \Psi ( x , y ) = \begin{cases} \Psi ^ + ( x , y ) , & y > 0 , \\ \Psi ^ - ( x , y ) , & y < 0 , \end{cases} \end{align*}


\begin{align*} \label [ I e q ] { e q : N p a r t i c l e m e a n f i e l d g e n e r a t o r } \mathcal { L } _ N \varphi _ N ( \mathbf { x } ^ N ) = \sum _ { i = 1 } ^ N L _ { \mu _ { \mathbf { x } ^ N } } \diamond _ i \varphi _ N ( \mathbf { x } ^ N ) , \end{align*}


\begin{align*} \Lambda _ n = \Lambda ^ { ( 1 ) } _ { n , \delta } + \Lambda ^ { ( 2 ) } _ { n , \delta } , \end{align*}


\begin{align*} \lim _ { i \to \infty } \| e _ i ^ { n , g } J e _ i ^ { n , g } J ( K _ { g , n } ^ i - [ T _ g ^ i , J x _ n J ] ) J e _ i ^ { n , g } J e _ i ^ { n , g } \| = 0 . \end{align*}


\begin{align*} \label [ I e q ] { e q : T i n f t y t p h i k M c K e a n } T _ { \infty , t } \Phi _ k ( \nu ) = \prod _ { i = 1 } ^ k \big \langle \nu , T ^ { \nu } _ t \varphi ^ i \big \rangle , \end{align*}


\begin{align*} \Lambda _ { \mu } : = D _ { \mu } \cap \big \{ h = ( \rho - \mu ) * \gamma _ { \sigma } : \big \} . \end{align*}


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


\begin{align*} c _ { 1 } = \frac { \Gamma \left ( t \right ) } { \Gamma ( t - 1 / 2 ) \pi ^ { \frac { 1 } { 2 } } } , ~ t > \frac { 1 } { 2 } , \end{align*}


\begin{align*} J ( \overline { y } , \overline { u } ) \leq \liminf _ { n } J ( y _ n , u _ n ) = \hat { J } . \end{align*}


\begin{align*} \omega = \sum _ { i = 1 } ^ g [ \alpha _ i , \alpha ^ \# _ i ] \in \pi _ { 2 n } ( Z _ { g , 1 } ) , \end{align*}


\begin{align*} C _ \omega : = \bigcap _ { I \in \mathcal A _ \omega } \angle _ I \subset \mathbb L ^ \vee \otimes \mathbb R . \end{align*}


\begin{align*} H _ { ( X _ + , D _ + ) } ( y ) = e ^ { \frac { t _ + } { 2 \pi i } } \sum _ { d \in \mathbb K _ { + } } y ^ d \left ( \frac { 1 } { \prod _ { i \in M _ 0 } \Gamma ( 1 + \frac { \bar D _ i } { 2 \pi i } + D _ i \cdot d ) } \right ) \textbf { 1 } _ { [ d ] } [ \textbf { 1 } ] _ { ( D _ i \cdot d ) _ { i \in I _ + } } . \end{align*}


\begin{align*} \inf _ { \theta \in N _ 1 \cap \Theta _ n ^ c } { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \nu _ { \theta } ) & > { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \mu ) + o _ { \mathbb P } ( n ^ { - 1 / 2 } ) \\ & \geq \inf _ { \theta \in N _ 1 \cap \Theta _ n } { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \nu _ \theta ) + o _ { \mathbb P } ( n ^ { - 1 / 2 } ) . \end{align*}


\begin{align*} \begin{aligned} \bar { y } _ j & = r \ , \left ( \sqrt { 1 - h ^ 2 } \cos \left ( \frac { 2 ( j - 1 ) \pi } { k } \right ) , \sqrt { 1 - h ^ 2 } \sin \left ( \frac { 2 ( j - 1 ) \pi } { k } \right ) , h \right ) \\ \underline { y } _ j & = r \ , \left ( \sqrt { 1 - h ^ 2 } \cos \left ( \frac { 2 ( j - 1 ) \pi } { k } \right ) , \sqrt { 1 - h ^ 2 } \sin \left ( \frac { 2 ( j - 1 ) \pi } { k } \right ) , - h \right ) \end{aligned} \end{align*}


\begin{align*} \delta ^ { ( s ) } c _ k = v _ k , \ \ k \in \{ 1 , \ldots , n \} \end{align*}


\begin{align*} f ( a ) = \frac { 1 } { a \log a } \ \le \ \frac { 1 } { a \log 2 P ( a ) } \ < \ \frac { e ^ \gamma } { a } \prod _ { p < P ( a ) } \Big ( 1 - \frac { 1 } { p } \Big ) \ = \ e ^ \gamma \ , { \rm d } ( { \rm L } _ a ) . \end{align*}


\begin{align*} \frac { d x } { d t } = \sum _ { \alpha = 1 } ^ r b _ { \alpha } ( t ) X _ { \alpha } ( x ) , x ( a ) = x _ 0 , \end{align*}


\begin{align*} C _ m ( n ) = \sum _ { j = - \infty } ^ \infty ( - 1 ) ^ j u _ m \big ( n - j ( 3 j - 1 ) / 2 \big ) . \end{align*}


\begin{align*} \tilde \lambda ( h \iota ) = \lambda ( h ) , h \in H \end{align*}


\begin{align*} S ( \partial _ t ) \varphi ( x , t ) : = \int _ 0 ^ t \int _ \Gamma k ( | x - y | , t - \tau ) \varphi ( y , \tau ) d \Gamma _ y d \tau x \in \R ^ d \setminus \Gamma \end{align*}


\begin{align*} D w ( x _ 0 ) = 0 , { \rm a n d } \ \ L w ( x _ 0 ) \le 0 , \end{align*}


\begin{align*} [ E _ { j k } , E _ { l m } ] = \delta _ { l , k } E _ { j m } - \delta _ { j , m } E _ { l k } . \end{align*}


\begin{align*} D L _ { ( g , f ) } ( 0 , h ) = & \ 2 \Delta ^ f _ g \left ( 2 \Delta _ g ^ f h + h \right ) - ( 4 \pi ) ^ { - \frac { n } { 2 } } \int _ { M } h e ^ { - f } d v _ g \\ & - 2 \left \langle \nabla _ g h , \nabla _ g \left ( 2 \Delta _ g f + f - | \nabla f | ^ 2 + R _ g \right ) \right \rangle . \end{align*}


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


\begin{align*} P ( X _ t , \eta _ t ) = t , \end{align*}


\begin{align*} \norm { h } ^ 2 = u ( a ) . \end{align*}


\begin{align*} \dim _ { \rm H } D _ { \beta } ( \varphi , x _ 0 ) = \frac { 1 } { 1 + \alpha } \alpha = \liminf _ { n \to \infty } \frac { \log _ { \beta } \varphi ( n ) ^ { - 1 } } { n } , \end{align*}


\begin{align*} { \partial _ t } { x _ k } \left ( t \right ) = { A _ { k p } } \left ( t \right ) { x _ p } \left ( t \right ) , \ \ k = 1 \dots d \end{align*}


\begin{align*} q ( P ) = \left ( 1 - \frac { p ' } { 2 } + \frac { p ' } { 2 } \cdot \frac { 1 } { p } \right ) ^ { - 1 } = 2 \end{align*}


\begin{align*} \lim _ { n \to \infty } \min _ { 1 \le i \le k _ { n } } \mu _ { n , i } \left ( ( - \varepsilon , \varepsilon ) \right ) = 1 \end{align*}


\begin{align*} B _ { \gamma } ( u , \phi ) = F ( \phi ) \quad u - f \in \Tilde { H } ^ s ( \Omega ) \end{align*}


\begin{align*} A _ { \mu } ( \zeta ) = \sum \limits _ { k = 1 } ^ \infty \frac 1 { k ! } K _ 0 ^ { ( k ) } ( \zeta ) | \zeta | ^ { k - 2 \mu } \left ( \sum \limits _ { | \vec { i } | = k , \sum i _ l l = \mu } ^ \infty \left ( \begin{array} { c } k \\ \vec { i } \end{array} \right ) \left [ \prod \limits _ { l = 1 } ^ \infty \left ( \begin{array} { c } \frac 1 2 \\ l \end{array} \right ) ^ { i _ l } \right ] \right ) . \end{align*}


\begin{align*} W _ i = \mathsf { s g n } ( | \hat \theta _ i | - | \hat \theta ' _ i | ) \ , \max ( | \hat \theta _ i | , | \hat \theta ' _ i | ) \ . \end{align*}


\begin{align*} \pi _ { \emptyset } = 1 \in M . \end{align*}


\begin{align*} \tau ( t , x ) = \tau ( \sigma , p ( \sigma ; t , x ) ) \ , . \end{align*}


\begin{align*} \mathbb P \big ( \sup _ { t , x } | u ^ n ( t , x ) - u ( t , x ) | ^ { 2 p } \ge 1 \big ) \le \sum _ { i = 1 } ^ 3 \mathbb P ( R _ i ^ { ( n ) } \ge 3 ^ { - 2 p } ) \end{align*}


\begin{align*} ( x ) _ { n } = \sum _ { k = 0 } ^ { n } S _ { 1 , \lambda } ( n , k ) ( x ) _ { k , \lambda } , ( n \ge 0 ) , \end{align*}


\begin{align*} \begin{array} { l l } A _ n ( t ) = \Pi _ { k = 1 } ^ n \left [ ( \kappa { \mathcal U } _ k ( s , t ) + I ) \partial _ t { \rm L o g } ( { \mathcal U _ k } ( t , s ) + \kappa I ) \right ] , \end{array} \end{align*}


\begin{align*} f ^ * g _ N - \frac { \mathrm { t r } _ { g _ M } f ^ * g _ N } { m } g _ M = 0 , \end{align*}


\begin{align*} g _ 4 ( K ( 3 , n \tilde { q } , n \tilde { p } ) ) = d - 1 \end{align*}


\begin{align*} g ^ c ( y ) = \inf _ { x \in \R ^ d } \big [ c ( x , y ) - g ( x ) \big ] , y \in \R ^ d . \end{align*}


\begin{align*} \Tilde { \mathcal { N } } ( f ^ { \tilde { \phi } ^ { - ( n + 1 ) } } ) = f ^ { \tilde { \phi } ^ { - n } } . \end{align*}


\begin{align*} \label [ I e q ] { e q : g r a d i e n t m c k e a n } b ( x , \mu ) = - \nabla V ( x ) - \int _ { \R ^ d } \nabla W ( x - y ) \mu ( \dd y ) , \end{align*}


\begin{align*} \mu _ { V X ^ { ( n ) } } ( A ) = \mathbb { P } ( V X ^ { ( n ) } \in A ) = \mathbb { P } \Big ( \sum _ { j = 1 } ^ n n ^ { 1 / p } \frac { Z _ j } { ( \lVert Z ^ { ( n ) } \rVert _ p ^ p + W _ n ) ^ { 1 / p } } V _ { \bullet , j } \in A \Big ) , \end{align*}


\begin{align*} \ker ( \kappa _ f ) \cong & \langle \ , x _ i \frac { \partial f } { \partial x _ i } , i = 1 , . . . , n \\ & \ , \ , \ , w _ { - \alpha } ( f ) , \alpha \in R ( N , \Sigma _ { \Delta } ) \ , \rangle . \end{align*}


\begin{align*} \frac { \tilde p ^ \# _ m ( x , y ) } { \tilde \pi ( y ) } = 1 + \sum _ { j = 2 } ^ { n } f _ j ( x ) f _ j ( y ) \lambda _ j ^ m , \end{align*}


\begin{align*} A = ( D P ) ^ { \dagger } \left ( \begin{array} { c c } I & 0 \\ 0 & - I \end{array} \right ) ( D P ) , \end{align*}


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


\begin{align*} \nu _ p \left ( \binom { \alpha \beta } { p ^ y } \right ) = x , \end{align*}


\begin{align*} k ! ( x _ i - x _ { i + 1 } ) e _ k ( x _ 1 , \ldots , x _ i ) = \alpha _ i \varpi _ { i - k + 1 } \varpi _ { i - k + 2 } \cdots \varpi _ { i } . \end{align*}


\begin{align*} { \hat { A } } ^ n x & = J ^ { - 1 } A ^ n J = \left ( y _ k + y _ 0 \right ) _ { k \in \N } \\ & = \left ( \left [ \prod _ { j = k } ^ { k + n - 1 } w ^ j \right ] \left ( x _ { k + n } - l ( x ) \right ) + \left [ \prod _ { j = 0 } ^ { n - 1 } w ^ j \right ] \left ( x _ { n } - l ( x ) \right ) \right ) _ { k \in \N } \in c ( \N ) . \end{align*}


\begin{align*} ( \theta _ t - k ) ^ n \circ [ t ^ k ] ( t ^ m ) = ( k + m - k ) ^ n t ^ { m + k } = m ^ n t ^ { m + k } \enspace . \end{align*}


\begin{align*} \mathcal { F } ( f ) ( \xi ) = \int e ^ { - i x \cdot \xi } f ( x ) d x . \end{align*}


\begin{align*} u \times v \mapsto \langle u | v \rangle _ t = e ^ { t ( ( \tilde { f } ^ { | u \rangle } , \tilde { f } ^ { | v \rangle } ) _ { { } _ { \mathfrak { J } } } + 4 \pi ) } = e ^ { - t 4 \pi ( \lambda \textrm { c o t h } \lambda - 1 ) } \end{align*}


\begin{align*} \boldsymbol { M } = \sum _ { j = 0 } ^ 6 \mathbf { s } ( j ) P ( \boldsymbol { X } = \mathbf { s } ( j ) ) . \end{align*}


\begin{align*} \frac { F ( z + \overline \omega _ 1 \ , | \ , \overline \omega _ 1 , \omega _ 2 ) } { F ( z \ , | \ , \overline \omega _ 1 , \omega _ 2 ) } = \frac { 1 } { 1 - x _ 2 } , \frac { F ( z + \omega _ 2 \ , | \ , \overline \omega _ 1 , \omega _ 2 ) } { F ( z \ , | \ , \overline \omega _ 1 , \omega _ 2 ) } = \frac { 1 } { 1 - x _ 1 } . \end{align*}


\begin{align*} \widetilde { f } _ \mu ( p ) = \sum \limits _ { i } ^ { N } \ , \alpha _ i { \textstyle \frac { u _ { i \mu } } { u _ i \cdot p } } , \ , \ , \ , \sum \limits _ { i } ^ { N } \alpha _ i = 0 , \end{align*}


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


\begin{align*} \widetilde A \widetilde D - \widetilde B \widetilde C = 1 , \widetilde { E } < \widetilde { A } + \widetilde { C } , \widetilde { F } < \widetilde { B } + \widetilde { D } , - \widetilde C \leq \widetilde C \widetilde F - \widetilde D \widetilde E < \widetilde D . \end{align*}


\begin{align*} ( a b + c d ) ^ 2 = \Big ( ( a + d ) ( b + c ) \Big ) \cdot ( a b + c d ) - \Big ( ( a + d ) ^ 2 b c + a d ( b + c ) ^ 2 - 4 ( a d ) ( b c ) \Big ) \cdot 1 . \end{align*}


\begin{align*} u _ { { \rm H i P O D } } ^ { L , M _ L } ( \alpha ^ * ) = \displaystyle \sum _ { k = 1 } ^ { m } \Big [ \sum _ { j = 1 } ^ { N _ h } { u } _ { { \rm P O D } , k , j } ^ { \alpha ^ * } \ , \vartheta _ j ( x ) \Big ] \varphi _ k ( \psi _ x ( { \bf y } ) ) , \end{align*}


\begin{align*} \alpha _ i '^ { - 1 } \alpha _ i c _ i = c _ i \beta _ i ' \beta _ i ^ { - 1 } , ~ ~ ~ ~ ~ ~ i = 0 , 1 , \cdots , r . \end{align*}


\begin{gather*} t ^ s _ { d } ( \tau , z ) : = \overline { \theta _ { d ^ 2 , s d ^ 2 } ^ 0 ( \tau ) } \theta _ { 1 , s d } ( \tau , z ) \end{gather*}


\begin{align*} \mu ( y , x ) & = \begin{cases} 1 & y = x , \\ - \sum \limits _ { y \leq t < x } \mu ( y , t ) & y < x , \\ 0 & y > x . \end{cases} \end{align*}