\begin{align*} \delta ( a ) = \sum _ { n \in \Z } \delta _ n ( a ) \end{align*}


\begin{align*} { { \tilde { \psi } } } _ { \alpha , s } \bigcirc _ { w ' = 0 } ^ s ( \theta _ { X _ s } + \gamma _ { r - s + w ' } ) \bigcirc _ { w = 1 } ^ { \ell } ( \theta _ { X _ s } + \xi _ w ) ( P ) = { \rm { E v a l } } _ { { { X _ s \rightarrow \alpha } } } \bigcirc _ { w = 1 } ^ { \ell } ( \theta _ { X _ s } + \xi _ w ) ( P ) \enspace . \end{align*}


\begin{align*} \begin{dcases} & d X _ k ( t ) = d B _ { k k + 1 } ( t ) + \dfrac { \alpha _ k - 1 } { 2 } \frac { 1 } { X _ k ( t ) } d t \\ & X _ k ( 0 ) = x _ k \end{dcases} , \end{align*}


\begin{align*} f ^ { * } \left ( K _ Y + L _ { 1 0 0 } \right ) = f ^ { * } \left ( E + \sum \limits _ { i = 1 } ^ { n } { F _ { i i } } \right ) . \end{align*}


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


\begin{align*} \gamma _ { \mu _ { t } } = t / 2 , \quad \sigma _ { \mu _ { t } } = ( t / 2 ) \ , \delta _ { 1 } , \end{align*}


\begin{align*} \widehat { T ( u ^ 1 , \cdots , u ^ p ) } ( k ) = \sum _ { k = k _ 1 - k _ 2 + \cdots + k _ p } m ( k _ 1 , \cdots , k _ n ) \prod _ { \substack { \ell = 1 \\ o d d } } ^ p \widehat { u ^ \ell } _ { k _ \ell } \prod _ { \substack { \ell = 2 \\ e v e n } } ^ p \overline { \widehat { u ^ \ell } _ { k _ \ell } } \end{align*}


\begin{align*} \mathcal M _ 0 = [ \vec r ^ 0 , \vec s ^ 0 ] = \{ \vec x : r ^ 0 _ i \le x _ i \le s ^ 0 _ i \} \end{align*}


\begin{align*} \Omega ( t ) = \sum _ { k = 0 } ^ { \infty } H _ k ( t ) , \end{align*}


\begin{align*} u _ { i _ 1 \cdots i _ m } = u _ { i _ { \pi ( 1 ) } \cdots i _ { \pi ( m ) } } \end{align*}


\begin{align*} \begin{cases} & , \\ & | g _ t ( x ) | \le C ( 1 + | x | ^ { K } ) | x | , \ \forall x \in \R ^ d , \\ & | \nabla g _ t ( x ) | \le C ( 1 + | x | ^ { K } ) \ \ x \in \R ^ d . \end{cases} \end{align*}


\begin{align*} \left | \frac 1 { 2 \pi i } \int _ { \Gamma _ - \cup \Gamma _ + } e ^ { s t } K ( s ) d s \right | & \leq \frac { C _ K } { \pi } \int _ { r _ 0 } ^ \infty r ^ { - \mu } e ^ { - r t \cos \varphi } d r \\ & = \frac { C _ K } { \pi } t ^ { \mu - 1 } ( \cos \varphi ) ^ { \mu - 1 } \int _ { r _ 0 \cos \varphi } ^ \infty r ^ { - \mu } e ^ { - r } d r \\ & = \frac { C _ K } { \pi } t ^ { \mu - 1 } ( \cos \varphi ) ^ { \mu - 1 } \Gamma ( 1 - \mu , r _ 0 \cos \varphi ) , \end{align*}


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


\begin{align*} | \pi | _ 1 + | \pi | _ 2 = | \pi | , \mbox { t h e t o t a l n u m b e r o f b l o c k s o f $ \pi $ , w h i l e } \end{align*}


\begin{align*} \rho ' _ \tau = \rho _ { f ' } R ' = \rho _ { f \widetilde { H } ( R R ' ) } R ' = \rho _ f \rho _ { \widetilde { H } ( R R ' ) } R ' = \rho _ f R R \rho _ { \widetilde { H } ( R R ' ) } R ' = \rho _ \tau ( R \rho _ { \widetilde { H } ( R R ' ) } R ' ) . \end{align*}


\begin{align*} B _ q ( u , v ) - \lambda \langle u , v \rangle = \tilde { F } ( v ) \ \ \ v \in \widetilde { H } ^ s ( \Omega ) \end{align*}


\begin{align*} \partial _ { i _ 1 } ^ { k _ 1 } \ldots \partial _ { i _ s } ^ { k _ s } S ( h ) = n ! \cdot \partial _ { i _ 1 } ^ { k _ 1 } \ldots \partial _ { i _ s } ^ { k _ s } V o l ( h ) \end{align*}


\begin{align*} ( ( A _ G ) x ) ( v ) : = ( L _ G ( x ) ) ( v ) + r ( v ) x ( v ) , \end{align*}


\begin{align*} P _ { j } = g _ { x ^ { j } } - \frac { d } { d t } g _ { \dot { x } ^ { j } } , \textnormal { f o r } ~ ~ j = 1 , 2 , \end{align*}


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


\begin{align*} & \langle y , ( - u _ i , 1 , 0 , 0 ) \rangle \leq 0 , \ i = 1 , \ldots , m , \\ & \langle y , - e _ i \rangle \leq 0 , \ , i = 1 , \ldots , s , \end{align*}


\begin{align*} u _ s = v _ { r - 1 } \left ( a _ s g ^ s \right ) = v _ { r - 1 } ( a _ s ) + s v _ { r - 1 } ( g ) = v _ { r - 1 } ( a _ s ) + s V _ r . \end{align*}


\begin{align*} \kappa _ i = \sigma ( \lambda _ i ) \end{align*}


\begin{align*} A = \begin{pmatrix} a & b & c & d \\ e & f & g & h \\ c & d & a & b \\ g & h & e & f \end{pmatrix} , \end{align*}


\begin{align*} \delta _ n \otimes \delta _ m = \bigoplus _ { 0 \leq j \leq \min ( n , m ) } \delta _ { n + m - 2 j } , \ ; \ ; \ ; \ , n , m \in \mathbb { N } _ 0 . \end{align*}


\begin{align*} f ( p , v ) = \big ( \sigma ( p ) , v - \sf K ( p ) \big ) . \end{align*}


\begin{align*} \sum _ { i = 3 , 4 } \big | T _ { k , j ; n } ^ { \mu , i } ( \mathfrak { m } , E ) ( t , x , \zeta ) + \hat { \zeta } \times T _ { k , j ; n } ^ { \mu , i } ( \mathfrak { m } , B ) ( t , x , \zeta ) \big | \end{align*}


\begin{align*} \tilde \lambda _ { j } = \frac { \gamma ^ { - j } \kappa _ { k , j } } { j ! \big ( ( k - j ) ! \big ) ^ 2 } , \end{align*}


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


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


\begin{align*} g ( y ) = \int \limits _ { | x | > | y | } | \varphi _ 0 ( x ) | ^ 2 \ , d x \end{align*}


\begin{align*} \dot { C } ( \lambda w ) = \left ( \zeta - \frac { \rho } { \lambda } \right ) w . \end{align*}


\begin{align*} \xi _ h : = \Theta _ { { 2 \pi \over k } ( h - 1 ) } \xi _ 1 , \ h = 1 , \ldots , k , \ \hbox { w i t h } \ \xi _ 1 : = ( 1 , 0 , 0 ) \in \mathbb R ^ 2 \times \mathbb R ^ { n - 2 } , \ \end{align*}


\begin{align*} \left ( \sum _ { 1 \le i , j \le 3 } C _ { i j } x _ i x _ j \right ) \cdot L ( x ) = 0 \end{align*}


\begin{align*} \Psi + \mu U = 0 , \end{align*}


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


\begin{align*} D _ - = D _ { - , 1 } + \cdots + D _ { - , n } , \end{align*}


\begin{align*} ( y _ i - y _ { i + 1 } ) \cdot \pi _ { [ a , b ] } = 0 ( a \le i \le b ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \abs { w _ i - \lambda \frac { z _ i - a } { 1 - \bar { a } z _ i } } ^ 2 . \end{align*}


\begin{align*} \mu _ 1 \star \mu _ 2 ( g ) = \sum _ { h \in G } \mu _ 1 ( g ^ { - 1 } h ) \mu _ 2 ( h ) \end{align*}


\begin{align*} J _ 4 & = \int _ 0 ^ r - \frac { z ^ 3 } { 2 } ( e _ \sigma + \bar \psi ) \cdot ( e ' _ \sigma \phi + e ' _ \delta \psi ) { \rm { d } } z \\ & \leq \Vert z ( \phi , \psi ) \Vert _ { L ^ 2 } \cdot \Vert z ^ { 2 } ( e ' _ \delta , e ' _ \sigma ) \Vert _ { L ^ 2 } \cdot \frac { 1 } { 2 } ( \Vert e _ \sigma \Vert _ { L ^ \infty } + \Vert \bar \psi \Vert _ { L ^ \infty } ) , \end{align*}


\begin{align*} v _ r ( \pi _ r ) = e _ r \ , v _ { r - 1 } ( \pi _ r ) = e _ r . \end{align*}


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


\begin{align*} \angle _ I : = \left \{ \sum _ { i \in I } a _ i D _ i | a _ i \in \mathbb R _ { > 0 } \right \} \subset \mathbb L ^ \vee \otimes \mathbb R . \end{align*}


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


\begin{align*} \begin{aligned} w ( ( n , m ) \rightarrow ( n + 1 , m + 1 ) ) & = 1 , \\ w ( ( n , m ) \rightarrow ( n + 1 , m - j ) ) & = a _ { m - j } ^ { ( j ) } , 0 \leq j \leq p . \end{aligned} \end{align*}


\begin{align*} ( - 2 ) ^ m m ! \ , I _ 0 ( ( R f ) _ { i _ 1 j _ 1 \cdots i _ m j _ m } ) = J _ { i _ 1 j _ 1 } \cdots J _ { i _ m j _ m } ( J _ m f ) , \end{align*}


\begin{align*} Q ( f _ 1 , \ldots , f _ n ) = 0 . \end{align*}


\begin{align*} \sigma _ { p , \beta } ^ 2 : = \Big ( \frac { p } { \beta } \Big ) ^ { 2 / p } \frac { \Gamma ( 3 / p ) } { \Gamma ( 1 / p ) } , \end{align*}


\begin{align*} C = \sqrt { \det A } = \sqrt { \sum \limits _ { k \neq l } ^ { } \left ( z ^ k _ 1 \right ) ^ 2 \left ( z ^ l _ 2 \right ) ^ 2 - \sum \limits _ { k \neq l } ^ { } z ^ k _ 1 z ^ k _ 2 z ^ l _ 1 z ^ l _ 2 } . \end{align*}


\begin{align*} \mathcal { T } _ { i j } ( x , p ) \underset { } { = } \phi _ j ( [ \tilde { p } _ { i j } ] _ + ) , \end{align*}


\begin{align*} U ( t ) = \exp \left ( { \rm i } \alpha ( t ) \widehat { H } _ 2 \right ) \exp \left ( { \rm i } \beta ( t ) \widehat { H } _ 3 \right ) , \end{align*}


\begin{align*} \rho = \sum _ { k \ge 1 } w _ k \delta _ { \lambda _ k } , \rho ^ { [ 1 ] } = \sum _ { k \ge 1 } w _ k ^ { [ 1 ] } \delta _ { \mu _ k } \end{align*}


\begin{align*} f _ \lambda ( \lambda ) & = N \lambda ^ { N - 1 } + \sum _ { k = 1 } ^ { N - 1 } ( - 1 ) ^ k ( N - k ) \lambda ^ { N - k - 1 } \sum _ { 1 \le j _ 1 < \cdots < j _ k \le N } \underset { 1 \le l , m \le k } { { \rm d e t } } ( H _ { j _ l j _ m } ) , \\ f _ { \lambda \lambda } ( \lambda ) & = N ( N - 1 ) \lambda ^ { N - 2 } + \sum _ { k = 1 } ^ { N - 2 } ( - 1 ) ^ k ( N - k ) ( N - k - 1 ) \lambda ^ { N - k - 2 } \sum _ { 1 \le j _ 1 < \cdots < j _ k \le N } \underset { 1 \le l , m \le k } { { \rm d e t } } ( H _ { j _ l j _ m } ) . \end{align*}


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


\begin{align*} 1 = \sum \limits _ { k \in \mathbb N } \psi _ k ( \xi ) + \psi _ { \leq 0 } ( \xi ) . \end{align*}


\begin{align*} \sum _ { i \in M _ P } \sum _ { j \in N _ i } a _ { i j } x _ { i j } + \sum _ { \substack { i j \in P , \\ i \in M _ P - M _ 0 } } ( b - s ) x _ { i j } \leq b + ( | M _ P - M _ 0 | - 1 ) ( b - s ) \end{align*}


\begin{align*} \mathbb { P } ( ) \leq \frac { \mathbb { G } ( n ) } { \mathbb { G } ( m ) } = \frac { \widetilde { \Phi } ( n ) } { \widetilde { \Phi } ( m ) } \leq \frac { f ( n ) } { f ( m ) } \end{align*}


\begin{align*} E ( \mathbf { S } _ n ) = \tau \mathbf { v } _ n = \varepsilon \sum \limits _ { i = 1 } ^ 3 \big [ p _ n ( i ) - p _ n ( i + 3 ) \big ] \mathbf { e } _ i = \varepsilon \sum \limits _ { i = 1 } ^ 3 ( a _ i n + b _ i ) \mathbf { e } _ i . \end{align*}


\begin{align*} A _ 2 & = \{ ( P _ f ) _ { f \in G } \in \wp ( G ) ^ G \ ; | \ ; \mbox { t h e r e e x i s t s $ f \in G $ s u c h t h a t $ i d \in P _ f \cup P _ f ^ { - 1 } $ } \} \\ & = \bigcup _ { f \in G } W _ { ( f , i d ) } . \end{align*}


\begin{align*} Y = \left ( \rho X + \lambda \right ) \ ; , \ , \end{align*}


\begin{align*} \operatorname { t r } X \mathcal { L } ( \rho ) & = - \operatorname { t r } ( X \frac { 1 } { 2 } \sum \limits _ { j } [ C _ { j } , [ C _ { j } , \rho ] ] ) = - \frac { 1 } { 2 } \sum \limits _ { j } \operatorname { t r } ( X ( C _ { j } ^ { 2 } \rho - 2 C _ { j } \rho C _ { j } + \rho C _ { j } ^ { 2 } ) \\ & = - \operatorname { t r } ( \rho \frac { 1 } { 2 } \sum \limits _ { j } [ C _ { j } , [ C _ { j } , X ] ] ) = \operatorname { t r } \mathcal { L } ( X ) \rho . \end{align*}


\begin{align*} \lim \limits _ { Q \to \infty } \frac 1 Q \sum \limits _ { k : k \geq 0 , k < Q } \mu ( k ) k \leq \lim \limits _ { Q \to \infty } \sum \limits _ { k : k \geq 0 } \mu ( k ) \left [ \frac k Q \wedge 1 \right ] = 0 . \end{align*}


\begin{align*} \psi ( t ) = \exp \left ( \int _ s ^ t A _ 1 ( \tau ) d \tau \right ) . \end{align*}


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


\begin{align*} \left \{ d _ { + } \in L _ { + } ^ { 2 } ( \mathbb { S } ; \mathbb { C } ) : \max _ { c _ { + } \in L _ { + } ^ { 2 } ( \mathbb { S } ; \mathbb { C } ) } \mathfrak { f } _ { \mathfrak { m } } \left ( c _ { - } , c _ { + } \right ) = \mathfrak { f } _ { \mathfrak { m } } \left ( c _ { - } , d _ { + } \right ) \right \} \end{align*}


\begin{align*} S ( t + s ) \rho _ 0 = S ( t ) S ( s ) \rho _ 0 ; \ t , s > 0 , \end{align*}


\begin{align*} \sum _ { x \in \lambda } h ( x ) = | \lambda | + n ( \lambda ) + n ( \lambda ^ { \ast } ) . \end{align*}


\begin{align*} \Gamma = \langle \gamma _ 1 , \ldots , \gamma _ r | \delta _ 1 = \cdots = \delta _ s = 1 \rangle . \end{align*}


\begin{align*} \hat { \beta } ( y ) + \hat { \beta } ( z ) = \sum _ { i = 1 } ^ { n } \hat { \beta } ( ( y _ { i } + z _ { i } ) e _ { i } ( n ) ) . \end{align*}


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


\begin{align*} \forall \varphi \in \R ^ { \Lambda _ { \epsilon , L } } , ( \Delta ^ \epsilon \varphi ) _ x = \epsilon ^ { - 2 } \sum _ { y \sim x } \big [ \varphi _ y - \varphi _ x \big ] , \end{align*}


\begin{align*} f ^ { \prime } ( \alpha ) = c . \end{align*}


\begin{align*} \begin{aligned} { Q } ^ { + r } & = Q \cup \{ ( \emptyset , r , 0 ) \} \\ { Q } ^ { + s } & = Q \cup \{ ( \emptyset , 0 , s ) \} , \end{aligned} \end{align*}


\begin{align*} g _ { k - 2 } ( z , \omega _ 1 , \widetilde \omega _ 1 ) = \int _ C \frac { - e ^ { ( z + \overline \omega _ 1 ) s } \ , s ^ { k - 2 } } { ( e ^ { \omega _ 1 s } - 1 ) ( e ^ { \widetilde \omega _ 1 s } - 1 ) } \ d s , \end{align*}


\begin{align*} f _ t \circ \left ( \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \alpha ( t _ 1 , t _ 2 ) d t _ 1 d t _ 2 \right ) = \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } f _ { T ( t _ 1 , t _ 2 ) } \circ ( \alpha ( t _ 1 , t _ 2 ) ) ( x ) d t _ 1 d t _ 2 \end{align*}


\begin{align*} ( L y _ i ) ( v ) & = - \frac { \delta _ E ( e _ 0 ) } { \delta _ V ( v ) } \frac { 1 } { | e _ 0 | } y _ i ( v ) . \end{align*}


\begin{align*} b _ j = \frac { c _ j ( b _ i - 1 ) } { ( c _ i - 1 ) } . \end{align*}


\begin{align*} \partial _ { i _ 1 } ^ { k _ 1 } \cdots \partial _ { i _ r } ^ { k _ r } \left ( I _ Q \right ) ( f ) = 0 \end{align*}


\begin{align*} s = \log ( \lambda | t | ) , t = - \lambda ^ { - 1 } e ^ s = - e ^ { s - s _ 0 } , \end{align*}


\begin{align*} v _ j : = \mathcal { S } f _ j \quad \mbox { i n } D \mbox { a n d } u _ j : = v _ j + u ^ s _ j \mbox { i n } D . \end{align*}


\begin{align*} H _ Z ^ k ( X , \mathcal { F } ) = 0 k = 0 , . . . , r - 1 . \end{align*}


\begin{align*} \phi ( x ) = x ^ { n } - \phi _ { n - 1 } x ^ { n - 1 } - \cdots - \phi _ { 1 } x - \phi _ { 0 } \in \mathbb { Z } [ x ] , \end{align*}


\begin{align*} \beta ^ \vee ( x ) : = \beta ( x , \cdot ) \colon E _ j \times \cdots \times E _ k \to F \end{align*}


\begin{align*} \begin{cases} & \Bar { \Phi } ^ { ( r ) } \colon S _ \mathbb { K } ^ { r + 1 } ( \mathfrak { g } [ 1 ] ) \xrightarrow { \Phi _ { r } } \mathfrak { X } _ { - r } ( E ) [ 1 ] \\ & \Bar { \Psi } ^ { ( r ) } \colon S _ \mathbb { K } ^ { r + 1 } ( \mathfrak { g } [ 1 ] ) \xrightarrow { \Psi _ { r } } \mathfrak { X } _ { - r } ( E ) [ 1 ] \end{cases} , \ ; \end{align*}


\begin{align*} \sum _ { m = 1 } ^ \infty \prod _ { i = 0 } ^ m ( 1 - r _ { m - i } ^ i ) < \infty , \end{align*}


\begin{align*} F _ { \alpha \beta } = \nabla ^ \Sigma _ { \alpha \beta } F = | \nabla u | ^ { - 2 } ( \nabla _ 3 k _ { \alpha \beta } - \nabla _ \alpha k _ { \beta 3 } ) = O ( | x | ^ { - \tau - 2 } ) . \end{align*}


\begin{align*} \mathbb { E } ( X _ i X _ { i + 1 } ) = \mathbb { P } ( P _ i , P _ { i + 1 } \ { \rm { a r e \ b o t h } } \ k - { \rm { f r e e } } ) , \end{align*}


\begin{align*} \int _ \mathbb { R } K ( x ) d x = 1 , \left \| K \right \| _ \infty < \infty , \mbox { s u p p } ( K ) \subset [ - 1 , 1 ] , \int _ \mathbb { R } K ( x ) x ^ l d x = 0 , \end{align*}


\begin{align*} S ^ m _ t ( f _ 1 , \dots , f _ m ) ( x ) = \int _ { \S ^ { m - 1 } } \prod _ { i = 1 } ^ m f ( x - t y _ i ) d \sigma ( y ) , \end{align*}


\begin{align*} a _ \ell c _ j ( c _ i - b _ i ) ( c _ j + c _ i - 1 ) = 0 . \end{align*}


\begin{align*} T _ { A , q } w ( x ) : = \int _ { D } \Phi ( x , y ) Q _ { A , q } w ( y ) d y , x \in D . \end{align*}


\begin{align*} d ( c _ k ) + k \le d ( c _ { k - 1 } ) + k - 1 \le \dots \le d ( c ) = d _ 0 ( c _ k ) + k . \end{align*}


\begin{align*} C ^ { v } : = \{ f : \lVert f ( x + h ) - f ( x ) \rVert = O ( v ( h ) ) \} . \end{align*}


\begin{align*} s = \theta s _ 1 + ( 1 - \theta ) s _ 2 \quad \frac { 1 } { p } = \frac { \theta } { p _ 1 } + \frac { 1 - \theta } { p _ 2 } , \end{align*}


\begin{align*} \Gamma _ { i j } ^ { k ( s ) } : = ( \Gamma _ i ^ { ( s ) } ) _ { j k } = ( \Gamma _ i ^ { ( s ) t } ) _ { k j } \in \mathcal A . \end{align*}


\begin{align*} \| \chi _ { S _ r ^ c } V \| = r ^ { - 2 } \in L ^ 1 ( r ) \end{align*}


\begin{align*} \sum _ { g \in \mathrm { G a l } ( K / \Q ) } g . \varphi _ k = f ^ * \Big ( \sum _ { g \in \mathrm { G a l } ( K / \Q ) } g . \varphi _ k ' \Big ) \in f ^ * R _ \mathrm { r a t } ( G ' ) \ ; . \end{align*}


\begin{align*} u _ { x y } = l _ 1 ( x _ 0 , x _ 1 ) \ , \begin{vmatrix} x _ 0 & x _ 1 \\ a _ 0 & a _ 1 \end{vmatrix} \ , \ u _ { y z } = l _ 2 ( y _ 0 , y _ 1 ) \ , \begin{vmatrix} y _ 0 & y _ 1 \\ b _ 0 & b _ 1 \end{vmatrix} \ , \end{align*}


\begin{align*} \Psi ( y ) = c _ 1 ^ \pm e ^ y + c _ 2 ^ \pm e ^ { - y } + q _ \pm , y \gtrless 0 , \end{align*}


\begin{align*} \delta ( p , \pi _ 1 , \ldots , \pi _ d ) S ( p ) ^ d \lesssim _ { n } \prod _ { j = 1 } ^ d s _ { k _ j } ( p , e ( \pi _ j ) ) , \end{align*}


\begin{align*} \alpha _ i = \begin{cases} n _ 1 - 2 & \\ n _ i & \\ - n _ i / 2 & \\ ( 2 - n _ { 2 k } ) / 2 & \end{cases} \end{align*}


\begin{align*} ( V \tilde v _ 1 ( \eta ^ { - 1 } \lambda _ i y , t ) , \psi _ 2 ) _ { L _ y ^ 2 ( B _ { 4 { \sf R } _ { \sf i n } } ) } & = \int _ { | y | < 4 { \sf R } _ { \sf i n } } V ( y ) \tilde v _ 1 ( \eta ^ { - 1 } \lambda _ i y , t ) \psi _ 2 ( y ) d y \\ & = \eta ^ n \lambda _ i ^ { - n } \int _ { | \xi | < 4 { \sf R } _ { \sf i n } \eta \lambda _ i ^ { - 1 } } V ( \eta \lambda _ i ^ { - 1 } \xi ) \tilde v _ 1 ( \xi , t ) \psi _ 2 ( \eta \lambda _ i ^ { - 1 } \xi ) d \xi . \end{align*}


\begin{align*} \beta = \prod _ { i = 1 } ^ k \gamma _ { j _ i } \sigma _ { j _ i } \gamma _ { j _ i } ^ { - 1 } \end{align*}


\begin{align*} m ^ \# = \frac { f \cdot m } { \int f d m } . \end{align*}


\begin{align*} \begin{aligned} T _ 3 ( h _ { N - 1 } { } \geq { } 1 ) = 0 \end{aligned} \end{align*}


\begin{align*} \nu ( L _ p ( h _ \rho ) ) = \beta _ b + b \gamma _ \rho . \end{align*}


\begin{align*} B ( \cdot , u , p ) = \det E ( \cdot , Y ( \cdot , u , p ) , Z ( \cdot , u , p ) ) \psi ( \cdot , u , p ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } n ^ { - j } \alpha _ { k , j } = \lim _ { n \to \infty } ( n r ^ d ) ^ { - j } r ^ { d ( 2 k - 1 ) } \kappa _ { k , j } , \end{align*}


\begin{align*} \mathbb { P } _ x ( X _ n = o ) \leq \sqrt { \frac { 2 c ( o ) } { c _ { \min } } } \left ( \exp \left [ - \frac { r ^ 2 } { 2 n } \right ] + \exp \left [ - \frac { c _ 1 ( \log r ) ^ \gamma n } { r ^ 2 } \right ] \right ) , \end{align*}


\begin{align*} \big | E r r ^ { \mu , a ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) \big | \lesssim \sum _ { i = 1 , 2 } \int _ { \R ^ 3 } \int _ { \R ^ 3 } f ( t _ i , X ( t _ i ) - y , v ) \big | K ^ a _ { k , n } ( y , v , V ( t _ i ) ) \big | d y d v \end{align*}


\begin{align*} x ( t ) = \Phi ( t , t _ 0 ) x _ 0 + \int ^ { t } _ { t _ 0 } \Phi ( t , s ) B ( s ) u ( s ) d s . \end{align*}


\begin{align*} \lim _ { n \to \infty } \int _ 0 ^ L \abs { y ^ n } ^ 2 d x = 0 . \end{align*}


\begin{align*} D ^ C & = \{ ( P _ f ) _ { f \in G } \in \wp ( G ) ^ G \ ; | \ ; \mbox { $ \exists g , h \in G $ w i t h $ g \neq h $ s u c h t h a t $ g h ^ { - 1 } \notin P _ h $ a n d $ h g ^ { - 1 } \notin P _ g $ } \} \\ & = \bigcup _ { g , h \in G , g \neq h } ( W _ { ( h , g h ^ { - 1 } ) } ^ C \cap W _ { ( g , h g ^ { - 1 } ) } ^ C ) , \end{align*}


\begin{align*} \lim _ { t \to 0 } \frac { N _ t } { v ( t ) } = 1 \end{align*}


\begin{align*} \begin{array} { l l } A _ n ( t ) = \Pi _ { k = 1 } ^ n \left [ ( I - \kappa e ^ { - \alpha _ k ( t , s ) } ) ^ { - 1 } \partial _ t \alpha _ k ( t , s ) \right ] , \end{array} \end{align*}


\begin{align*} d _ { n - 1 } ( { \mathcal K } , \lambda ^ n ) _ X \geq C '' \frac { 1 } { [ \log _ 2 n ] ^ \alpha } , n = 2 , \ldots . \end{align*}


\begin{align*} f ( x ) = x ^ { p } ( x - 1 ) ^ { q } . \end{align*}


\begin{align*} \rho _ k = \dfrac { 1 } { \varepsilon } \left ( \dfrac { 1 } { \widetilde { v } - \varepsilon } \sum _ { l = 1 } ^ n v ^ l \dfrac { f ^ h _ { j i } } { f _ { j i } } ( w _ l ) - \dfrac { f ^ h _ { j i } } { f _ { j i } } ( w _ k ) \right ) , \end{align*}


\begin{align*} X _ c \cdot ( f , a ) = f ( c ) + X _ { c + a } . \end{align*}


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


\begin{align*} \mathrm { W O T } - \lim _ { n \to \infty } T _ { h _ o } ^ { ( n ) } = \mathrm { W O T } - \lim _ { n \to \infty } U ( \gamma _ n ) = A _ 0 = T _ { h _ o } . \end{align*}


\begin{align*} L ( h _ \rho ) - L _ p ( h _ \rho ) = \sum _ { j \in J } \partial _ { j } L ( h _ \theta ) ( h _ \rho - h _ \theta ) ^ { j } . \end{align*}


\begin{align*} \mathbf { S } _ { d , 0 } ( s ) \mathcal { K } _ { t } u & = \mathbf { S } _ { d , 0 } ( s ) \mathbf { S } _ { d , 0 } ( t ) | \cdot | ^ { \alpha } | u | ^ p + \mathbf { S } _ { d , 0 } ( s ) \mathbf { S } _ { d , 0 } ( t ) \zeta ( \cdot ) \mathbf { w } \\ & = \mathbf { S } _ { d , 0 } ( t + s ) | \cdot | ^ { \alpha } | u | ^ p + \mathbf { S } _ { d , 0 } ( t + s ) \zeta ( \cdot ) \mathbf { w } \\ & = \mathbf { S } _ { d , - \alpha } ( t + s ) | u | ^ p + \mathbf { S } _ { d , 0 } ( t + s ) \zeta ( \cdot ) \mathbf { w } \\ & = \mathcal { K } _ { t + s } u . \end{align*}


\begin{align*} F \left ( f \right ) \left ( k \right ) \equiv \hat { f } \left ( k \right ) \doteq \int _ { \mathbb { R } ^ { d } } f \left ( x \right ) \mathrm { e } ^ { - i k \cdot x } \mathrm { d } x = \frac { ( - 1 ) ^ { n } } { k _ { j _ { 1 } } ^ { 2 } \cdots k _ { j _ { n } } ^ { 2 } } F \left ( \frac { \partial ^ { 2 n } f } { \partial x _ { j _ { 1 } } ^ { 2 } \cdots \partial x _ { j _ { n } } ^ { 2 } } \right ) \left ( k \right ) \ , \end{align*}


\begin{align*} { \left ( { { \eta _ 0 } } \right ) _ { k p } } = \left ( { \frac { { d + 1 } } { 2 } - k } \right ) { \delta _ { k p } } \end{align*}


\begin{align*} r ( x , y , Z ) : = r _ { \mathrm { V } } ( x ) + r _ { \mathrm { O } } ( x , y , Z ) \end{align*}


\begin{align*} \sum _ { \pi : \lambda _ { \pi _ { \infty } } \in t \Omega } \chi _ { \pi _ p } ( \phi _ \beta ) = \Lambda _ \Omega ( t ) \int _ { \chi \in \mathcal { X } _ p ^ { t e m p } / W } \chi ( \phi _ { \beta } ) \ , d \nu _ p ( \chi ) + O _ \Omega \Big ( \| \beta \| _ { L ^ 1 } t ^ { d - \delta } \Big ) , \end{align*}


\begin{align*} Q ^ \beta \hat U ( t _ { n + 1 } ) + A U _ { n + 1 } = Q ^ \beta \hat U ( t _ { n + 1 } ) + A \hat U ( t _ { n + 1 } ) = f ( t _ { n + 1 } ) , \end{align*}


\begin{align*} W : = \Sigma \circ h : X ^ \vee \rightarrow \mathbb C . \end{align*}


\begin{align*} C _ t = ( t A + ( \alpha - t ) ) ^ { - 1 } . \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*} ( \nu , s ) \cdot ( \mu , w ) = ( s \mu + F \nu - s w F ( s ) ^ { - 1 } ( \nu ) , s w F ( s ) ^ { - 1 } ) . \end{align*}


\begin{align*} \lambda _ { 1 } = + \lambda _ { + } : c \lambda _ { 2 } = + \lambda _ { - } : d \lambda _ { 3 } = - \lambda _ { + } : a \lambda _ { 4 } = - \lambda _ { - } : b . \end{align*}


\begin{align*} - \Delta _ p v ^ 0 - a | v ^ 0 | ^ { p - 2 } v ^ 0 = 0 \qquad \Omega \end{align*}


\begin{align*} \lim _ { h \to 0 } \left ( h ^ { - 1 } ( \mu ( t + h ) - \mu ( t ) ) - \dot \mu ( t ) \right ) = 0 \end{align*}


\begin{align*} e _ k ( y _ 1 , y _ 2 , \ldots , y _ n ) = \sum _ { \substack { 0 \le p \le i , \ 0 \le q \le n - i \\ p + q = k } } e _ { p } ( y _ 1 , y _ 2 , \ldots , y _ i ) e _ { q } ( y _ { i + 1 } , y _ { i + 2 } , \ldots , y _ n ) \end{align*}


\begin{align*} \Gamma ( B G , \mathbb Z ( n ) ) : = \lim _ { [ k ] \in \Delta ^ { \mathrm { o p } } } \Gamma ( G ^ { \times k } , \mathbb Z ( n ) ) , \end{align*}


\begin{align*} & a _ \ell ( c _ i ^ 2 - c _ i ) = 0 i > \ell \\ & a _ \ell c _ i c _ j = 0 i > \ell , j > i . \end{align*}


\begin{align*} \left \| \sum _ { k = 1 } ^ n a _ k \phi _ k \right \| _ p \le c \left \| \sum _ { k = 1 } ^ n a _ k \phi _ k \right \| _ 2 \end{align*}


\begin{align*} G ^ n _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s \left ( \sigma ' ( X _ s ) \right ) ^ 2 M ^ { n , 1 } _ s M ^ { n , 2 } _ s d s . \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { \partial ^ 2 C ^ 2 } { \partial z ^ i _ { \epsilon } \partial z ^ j _ { \eta } } \frac { \partial ^ 2 \varphi ^ j } { \partial x ^ { \epsilon } \partial x ^ { \eta } } v ^ i = f ( x ^ 1 ) \left [ - f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \right ] , \end{align*}


\begin{align*} \mathfrak g ( S ) \times \mathfrak g ( S ) \rightarrow \mathfrak g ( S ) , \ \ ( a , b ) \mapsto a + _ { { \pi } } b : = a + b + \pi a b . \end{align*}


\begin{align*} \int _ 0 ^ { c _ 1 } \left ( \abs { z } ^ 2 + \abs { y _ x } ^ 2 \right ) d x = o ( 1 ) \int _ { c _ 2 } ^ L \left ( \abs { v } ^ 2 + a \abs { u _ x } ^ 2 \right ) d x = o ( 1 ) . \end{align*}


\begin{align*} \| \sin \Theta ( Q , \widetilde Q ) \| = \| Q _ \perp ^ * \widetilde Q \| = \| \widetilde Q _ \perp ^ * Q \| , \end{align*}


\begin{align*} \widehat { ( r _ j f ) } ( \xi ) = i ( 1 - \phi ( \xi ) ) \xi _ i / | \xi | \widehat { f } ( \xi ) . \end{align*}


\begin{align*} \psi ^ { - 1 } ( x ) = 8 \phi ^ { - 1 } \Bigl ( a \left ( 8 a ^ { - 1 } ( x ) \right ) \Bigr ) , \end{align*}


\begin{align*} h ( X ) = \frac { \log _ 2 ( \widetilde { N _ H } ( q \alpha ) ) } { ( q + r ) \alpha } + \frac { h ( W ) } { ( q + r ) \alpha } = h . \end{align*}


\begin{align*} T x = ( c ( x ) E x , s ( x ) F x ) . \end{align*}


\begin{align*} A ( \rho ) = t _ \rho \zeta _ \phi ^ { p , \infty } ( \rho ) \end{align*}


\begin{align*} \sum _ { r \leq s } \Phi ( x _ i ^ r ) & = \sum _ { k = 0 } ^ n \dim ( M _ i ^ k \cap \ker A _ i ^ s ) \end{align*}


\begin{align*} \sum _ { \mu \not = 0 } \dim \left [ \gg ^ { \mu } , x \right ] = \sum _ { \mu \not = 0 } \dim \gg _ { 1 } ^ { \mu } = | \Delta _ { 1 } \backslash { A } ^ { \perp } | . \end{align*}


\begin{align*} \varphi _ 0 ( x ) = C e ^ { - c | x | } | x | \geq 1 \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } \| \chi _ { S _ { v t } } e ^ { - i t H } \psi \| = \| \psi \| \end{align*}


\begin{align*} V ^ * A _ 1 & = x \tilde \Delta _ \alpha ( a ) & & + \alpha ^ { - 1 / 2 } \Delta ^ { 1 / 2 } ( x ) \zeta _ \alpha ( a ) & & - \alpha ^ { 1 / 2 } \Gamma ( x , \zeta _ \alpha ( a ) ) \\ & = : B _ 1 & & + B _ 2 & & - B _ 3 . \end{align*}


\begin{align*} s _ { n } : = \int x ^ { n } d \mu ( x ) = \langle J ^ { n } e _ { 0 } , e _ { 0 } \rangle , n \geq 0 , \end{align*}


\begin{align*} & x _ 1 \cdot \frac { \partial f } { \partial x _ 1 } , . . . , x _ n \cdot \frac { \partial f } { \partial x _ n } , \\ & w _ { - \alpha } ( f ) : = \sum \limits _ { m \in \Delta \cap M } h t _ { - \alpha } ( m ) \cdot a _ m \cdot x ^ { m - \alpha } , \alpha \in R ( N , \Sigma _ { C ( \Delta ) } ) . \end{align*}


\begin{align*} \psi _ { B _ { 0 } } = B _ { 0 } \left [ \frac { x } { a / 2 } - \frac { \sinh ( k ' x ) } { \sinh ( k ' a / 2 ) } \right ] , \end{align*}


\begin{align*} \Delta _ n \underset { \sim } { < } ( \frac { 1 } { n } ) ^ { \frac { \bar { \beta } } { 2 \bar { \beta } + d } ( \frac { 1 } { \beta _ 1 } + \frac { 1 } { \beta _ 2 } ) } \mbox { f o r } \beta _ 2 = \beta _ 3 . \end{align*}


\begin{align*} t _ { i j } = \begin{cases} v _ k , ~ j = i + k ~ ~ k \in \mathcal { I } , \\ 0 , \quad , \end{cases} \end{align*}


\begin{align*} \Omega _ t ^ \pm = \{ ( x , y ) \in \R ^ 2 : y \gtrless \varphi ( t , x ) \} \end{align*}


\begin{align*} \Omega _ { \mu } \cap \mathbb { H } = \{ r e ^ { i \theta } : r > 0 , h _ { \mu } ( r ) < \theta < \pi \} . \end{align*}


\begin{align*} \begin{cases} ( \lambda \dot \lambda - \sigma ) ( \Lambda _ y { \sf Q } , \psi _ 2 ) _ { L _ y ^ 2 ( B _ { 4 { \sf R } _ { \sf i n } } ) } + \lambda ^ \frac { n - 2 } { 2 } ( V ( \eta ^ \frac { 2 } { 1 - q } \tilde v _ 1 + \tilde w _ 1 ) , \psi _ 2 ) _ { L _ y ^ 2 ( B _ { 4 { \sf R } _ { \sf i n } } ) } = 0 t \in ( 0 , T ) , \\ \lambda ( t ) = 0 t = T . \end{cases} \end{align*}


\begin{align*} q \equiv 3 \pmod 8 , \left ( \frac { p } { q } \right ) = - 1 . \end{align*}


\begin{align*} M _ 0 ( H ^ s \to H ^ { - s } ) \subset M _ \Omega ( H ^ s \to H ^ { - s } ) = M _ 0 ( H ^ s \to H ^ { - s } ) + M _ { \delta ( \Omega ) } ( H ^ s \to H ^ { - s } ) . \end{align*}


\begin{align*} { \mathcal L } _ H = \left \{ ( q , p , P _ q , P _ p ) \ ; \left | \exists \ ; u \in U \ ; \mbox { s . t . } \begin{array} { l } P _ q = \frac { \partial H } { \partial q } ( q , p , u ) , P _ p = \frac { \partial H } { \partial p } ( q , p , u ) , \\ \frac { \partial H } { \partial u } ( q , p , u ) = 0 \ , \end{array} \right . \right \} \ , . \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*} \textrm { r a n k } \left ( \mathbf { V } \right ) & = \sum _ { m } \frac { \log \left ( 1 + \frac { \lambda _ { m } } { \varepsilon } \right ) } { \log \left ( 1 + \frac { 1 } { \varepsilon } \right ) } + o \left ( \varepsilon \right ) \\ & = \frac { M \log \left ( \frac { 1 } { \varepsilon } \right ) + \log \left ( \left | \mathbf { V } + \varepsilon \mathbf { I } \right | \right ) } { \log \left ( 1 + \frac { 1 } { \varepsilon } \right ) } + o \left ( \varepsilon \right ) \end{align*}


\begin{align*} \left \{ \begin{array} { c l } i \partial _ t u + \Delta u & = \pm | u | ^ 2 u , ( t , x ) \in \R \times \R , \\ u ( 0 ) & = u _ 0 \in M ^ s _ { p , 2 } ( \R ) . \end{array} \right . \end{align*}


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


\begin{align*} \abs { U _ 0 } & \leq 2 \mu + \mu ( s + 1 ) \\ & = 2 \mu + \mu \abs { z _ r \ldots z _ { r + s } } _ { Y \cup \mathcal { O } } \\ & \leq 2 \mu + \mu ( \lambda \abs { z _ r \ldots z _ { r + s } } _ { X \cup \mathcal { P } } + \lambda c ) \\ & \leq 2 \mu + \mu ( \lambda ( \abs { U _ 0 } _ { X \cup \mathcal { P } } + 2 \mu ) + \lambda c ) \end{align*}


\begin{align*} \left \{ \begin{array} { l l l } l ( X ) = ( 0 \to 0 \to \cdots \to X ) & \hbox { } \\ & \hbox { } \\ e ( X _ 1 \to X _ 2 \to \cdots \to X _ n ) = X _ n & \hbox { } \\ & \hbox { } \\ r ( X ) = ( X \xrightarrow { 1 _ { X } } X \xrightarrow { 1 _ { X } } \cdots \xrightarrow { 1 _ { X } } X ) & \hbox { } \end{array} \right . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ \infty t _ { n , k } a _ k w _ { m _ k } ( x ) \to 0 \end{align*}


\begin{align*} N \leq \left ( \frac { c + \frac { 1 } { 2 } c ' } { \frac { 1 } { 2 } c ' } \right ) ^ \rho = ( 1 + 2 c / c ' ) ^ \rho . \end{align*}


\begin{align*} l ^ B _ A ( m ) \cdot r ^ B _ A ( m ) = l ^ B _ A ( m ) \cdot q ( r ^ B _ A ( m ) ) \cdot q ( r ^ B _ A ( m ) ) ^ { - 1 } \cdot r ^ B _ A ( m ) \end{align*}


\begin{align*} \sum _ { q ' = 0 } ^ q \binom { q } { q ' } \cdot 2 ^ { q ' } \cdot ( - 1 ) ^ { q ' } , \end{align*}


\begin{align*} g = \begin{pmatrix} a & * & * \\ b & * & * \\ c & * & * \end{pmatrix} , a \equiv 1 \bmod 3 , b \equiv c \equiv 0 \bmod \sqrt { - 3 } . \end{align*}


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


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


\begin{align*} \mathbb { L } f _ ! B ( G , G , * ) _ { \mathrm { t r i v } } \simeq f _ ! B ( G , G , * ) _ { \mathrm { t r i v } } = B ( H , G , * ) _ { \mathrm { t r i v } } \end{align*}


\begin{align*} \sum _ { w \in W _ { \rho } ^ { m _ 0 } } \dfrac { f ' ( v _ { n + 1 } + w ) } { f ( v _ { n + 1 } + w ) } = \eta \delta _ n ( \beta ) . \end{align*}


\begin{align*} \rho _ f ( x ) : = \int _ \R f \d v P _ f ( x ) : = \rho _ f ( x ) T _ f ( x ) : = \int _ \R v ^ 2 f \d v , \end{align*}


\begin{align*} B R ( z , A ) = \int _ 0 ^ \infty \mathrm { e } ^ { z s } \tilde { B } T ( s , A ) \ , \mathrm { d } s , \end{align*}


\begin{align*} \nabla \phi _ \mathrm { p d } ( x ^ { ( k _ i ) } , z ^ { ( k _ i ) } ) - \nabla \phi _ \mathrm { p d } ( x ^ { ( k _ i + 1 ) } , z ^ { ( k _ i + 1 ) } ) + \begin{bmatrix} - A ^ T z ^ { ( k _ i + 1 ) } \\ A x ^ { ( k _ i + 1 ) } \end{bmatrix} \in \begin{bmatrix} \partial f ( x ^ { ( k _ i + 1 ) } ) + \nabla h ( x ^ { ( k _ i + 1 ) } ) \\ \partial g ^ \ast ( z ^ { ( k _ i + 1 ) } ) \end{bmatrix} , \end{align*}


\begin{align*} F _ { \mu \boxplus \nu } ^ { \prime } ( s _ { 0 } ) = F _ { \nu } ^ { \prime } ( \alpha ) \omega ^ { \prime } ( s _ { 0 } ) \end{align*}


\begin{align*} T [ p q ] = T [ p ] T [ q ] \end{align*}


\begin{align*} \frac { d _ { i + 1 , k } } { d _ { i , k } } \leq \Psi ( k ) : = \exp \left [ - \frac { k } { 2 \psi ^ { - 1 } ( k ) } \right ] \end{align*}


\begin{align*} \mathrm { M T C E } _ { \boldsymbol { q } } ( \mathbf { X } ) & = \mathrm { E } \left [ \mathbf { X } | \mathbf { X } > V a R _ { \boldsymbol { q } } ( \mathbf { X } ) \right ] \\ & = \mathrm { E } [ \mathbf { X } | X _ { 1 } > V a R _ { q _ { 1 } } ( X _ { 1 } ) , \cdots , X _ { n } > V a R _ { q _ { n } } ( X _ { n } ) ] , \end{align*}


\begin{align*} Q ^ \phi _ t p _ k ( x ) = \sum _ { l = 0 } ^ k \dbinom { k } { l } \frac { W _ \phi ( k + 1 ) } { W _ { \phi } ( l + 1 ) } x ^ l t ^ { k - l } . \end{align*}


\begin{align*} j _ n ^ * ( t _ l ) = \begin{cases} t _ l ' , & \\ 0 , & \end{cases} \end{align*}


\begin{align*} f ( A x + e ) = g ( x ) + d \end{align*}


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


\begin{align*} \mathbf r ^ 1 & = d \mathbf e ^ 1 \times \mathbf v = ( 0 , - d v _ 3 , d v _ 2 ) , \\ \mathbf s ^ 1 & = ( \mathbf r ^ 1 \times \mathbf v ) / d = ( \mathbf e ^ 1 \times \mathbf v ) \times \mathbf v = \left ( - v _ 2 ^ 2 - v _ 3 ^ 2 , v _ 1 v _ 2 , v _ 1 v _ 3 \right ) . \end{align*}


\begin{align*} \begin{aligned} a ( [ r ] \boxplus [ { r ' } ] ) ) & = \{ a [ { x + x ' } ] : x \in r G , \ x ' \in r ' G \} \\ & = \{ [ { a x + a x ' } ] : x \in r G , \ x ' \in r ' G \} = [ a [ r ] \boxplus a [ { r ' } ] ) ; \end{aligned} \end{align*}


\begin{align*} I _ { ( Z _ - , D _ { Z , - } ) } ( y , z ) = z e ^ { t _ - / z } \sum _ { d \in \mathbb K _ { - } } \tilde y ^ { d } \left ( \prod _ { i \in M _ 0 } \frac { \prod _ { a \leq 0 , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } { \prod _ { a \leq D _ i \cdot d , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } \right ) \\ \left ( \prod _ { 0 < a \leq v _ 2 \cdot d } ( v _ 2 + a z ) \right ) \textbf { 1 } _ { [ - d ] } I _ { D _ { Z , - } , d } , \end{align*}


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


\begin{align*} I _ { D _ + , d } = \frac { 1 } { \prod _ { i \in I _ + , D _ i \cdot d > 0 } ( \bar D _ i + ( D _ i \cdot d ) z ) } [ \textbf { 1 } ] _ { ( - D _ i \cdot d ) _ { i \in I _ + } } , \end{align*}


\begin{align*} \P \{ m \} = \prod \limits _ { i = 0 } ^ { m - 1 } \left ( 1 - \P \{ \ell ^ { ( A ) } _ i > m - i \} \right ) \geq \exp \left ( - g _ m \sum \limits _ { i = 0 } ^ { m - 1 } \P \{ \ell ^ { ( A ) } _ i > m - i \} \right ) , \end{align*}


\begin{align*} \varphi _ k ( D ) = 2 ^ { - 2 \ell k } ( - \Delta ) ^ \ell \widetilde \varphi _ j ( D ) \end{align*}


\begin{align*} & { \rm ( a ) } \ \rho ( I ( G ) ) = 1 ; \quad { \rm ( b ) } \ \rho _ { i c } ( I ( G ) ) = 1 ; { \rm ( c ) } \ G \mbox { i s b i p a r t i t e } ; \quad { \rm ( d ) } \ \rho ( I ( G ) ^ \vee ) = 1 ; \end{align*}


\begin{align*} I ( f ) = \left \{ b , 1 \leq b \leq \deg ( f ) \mid \epsilon ( f ) = \frac { \nu ( f ) - \nu ( \partial _ b f ) } { b } \right \} . \end{align*}


\begin{align*} B \wedge \langle S \rangle & = B \wedge \left ( \bigvee _ { s \in S } \langle s \rangle \right ) \\ & = \bigvee _ { s \in S } ( B \wedge \langle s \rangle ) \end{align*}


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


\begin{align*} \Pi _ \xi ^ { \mu , \nu } = \Pi ^ { \mu , \nu } \circ \Big ( u + \xi ( v - u ) \Big ) ^ { - 1 } . \end{align*}


\begin{align*} \mu _ n ( \alpha ) = - \frac { \log \left | \alpha - p _ n / q _ n \right | } { \log q _ n } , \end{align*}


\begin{align*} \begin{aligned} & \frac { \Delta t \| u _ 0 \| _ 2 ^ 3 } { \nu \ , h ^ { 1 / 2 } } = o ( 1 ) , \end{aligned} \end{align*}


\begin{align*} \Theta _ \omega ( \alpha ) = \sum _ i \arctan \lambda _ i = \hat { \theta } \mod \pi \Z . \end{align*}


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


\begin{align*} J ( S _ \delta ( u _ \delta ) , u _ \delta ) \leq j _ \delta ( u _ \delta ) \leq j _ \delta ( \overline { u } ) = J ( S _ \delta ( \overline { u } ) , \overline { u } ) . \end{align*}


\begin{align*} \sum _ { i \in M _ P } \sum _ { j \in N _ i } a _ { i j } \tilde x _ { i j } + ( b - s ) \sum _ { i j \in \bar P } \tilde x _ { i j } & \leq \sum _ { i \in M _ P } \sum _ { j \in N _ i } a _ { i j } \tilde x _ { i j } + ( b - s ) | \bar P | \\ & \leq s + ( b - s ) | \bar P | \\ & = b + ( | \bar P | - 1 ) ( b - s ) \\ & = b + ( | M _ P - M _ 0 | - 1 ) ( b - s ) . \end{align*}


\begin{align*} V ( s ) \ge \begin{cases} c ( m ) s ^ { m + 1 } & s \le R \\ c ( m ) R s ^ { m } & s > R \\ \end{cases} , \end{align*}


\begin{align*} \hat { \beta } = A ^ { 0 } Y + ( Y ^ { \prime } H _ { 1 } ^ { 0 } Y , \ldots , Y ^ { \prime } H _ { k } ^ { 0 } Y ) ^ { \prime } , \end{align*}


\begin{align*} \Phi ( x ) + \Psi ( \phi ( x ) ) = x \phi ( x ) \ , x \geq 0 \ , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { n - 1 } \left ( - \frac { 1 } { 2 } \right ) ^ k P _ k \left ( n - 1 , \frac { n } { 2 } \right ) \left ( \zeta \left ( k - n + 2 , \frac { n } { 4 } \right ) - \zeta \left ( k - n + 2 , \frac { n + 2 } { 4 } \right ) \right ) = 2 ^ { 1 - n } \Gamma ^ 2 \left ( \frac { n } { 2 } \right ) . \end{align*}


\begin{align*} \nabla _ { \vec s , k } \left ( L ( t , z ) z ^ { - \mu } z ^ { \rho } \alpha \right ) = 0 , \nabla _ { z \partial z } \left ( L ( t , z ) z ^ { - \mu } z ^ { \rho } \alpha \right ) = 0 \end{align*}


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


\begin{align*} \psi _ { n , 1 } ( s , t ) : = ( t - \eta _ n ( s ) ) ^ { \alpha } - ( t - s ) ^ { \alpha } , 0 \le s < t \le T , \end{align*}


\begin{align*} H _ { \gamma } ( Y | X ) = \textstyle \sum _ { i = 1 } ^ s \sum _ { j = 1 } ^ t - p _ { i , j } \log _ { \gamma } \frac { p _ { i , j } } { p _ i } ; \\ I _ { \gamma } ( X ; Y ) = \textstyle \sum _ { i = 1 } ^ s \sum _ { j = 1 } ^ t - p _ { i , j } \log _ { \gamma } \frac { p _ i p ' _ j } { p _ { i , j } } . \end{align*}


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


\begin{align*} \mathcal { G } ( n ) \oplus \mathcal { G } ( n ) \oplus \mathcal { G } ( m + 1 ) \oplus \mathcal { G } ( m + 1 ) = 0 = \mathcal { G } ( n + 1 ) \oplus \mathcal { G } ( n + 1 ) \oplus \mathcal { G } ( m ) \oplus \mathcal { G } ( m ) \end{align*}


\begin{align*} \begin{array} { l l } \dot { x } _ i = u _ i , & i = 1 , \dots , r , \\ \dot { y } _ { i j } = x _ i u _ j - x _ j u _ i , & i < j \\ \end{array} \end{align*}


\begin{align*} a = \frac { \frac { t } { \sqrt { n - 1 } } } { \sqrt { 1 + \frac { t ^ 2 } { n - 1 } } } . \end{align*}


\begin{align*} \phi ( t ) = \| F ( t ) \| _ q \psi ( t ) = \| G ( t ) \| _ r . \end{align*}


\begin{align*} U _ \Gamma = \{ ( x ' , x _ n ) : \ , x ' \in \Gamma \ , \ , \ , \ , x _ n \in ( - c , c ) \} \end{align*}


\begin{align*} \widetilde { C } ( m , p ) : = C ( m , p ) ^ { - 1 } = C \left ( \frac { 1 } { 4 m } , \frac { p - m } { m ( 3 p - 4 m ) } \right ) . \end{align*}


\begin{align*} q ( y ) = \max \{ q _ j ( y _ j ) \colon j \in \Phi \} \quad \mbox { f o r a l l $ \ , y = ( y _ j ) _ { j \in J } \in F $ , } \end{align*}


\begin{align*} ( \partial _ s + \Delta ) w & = v ( \partial _ s + \Delta ) \varphi + \varphi ( \partial _ s + \Delta ) v - 2 g ( \nabla \varphi , \nabla v ) \\ & = v ( \partial _ s + \Delta ) \varphi + \varphi ( \partial _ s + \Delta ) v - 2 \rho \lambda e ^ { \lambda u } \varphi g ( \nabla u , \nabla v ) . \end{align*}


\begin{align*} \beta ^ * - \sigma & = \beta ^ * ( I ^ * + R ^ * ) - \delta I ^ * \\ 0 & = \gamma I ^ * - \omega R ^ * + \delta R ^ * I ^ * \end{align*}


\begin{align*} \xi \colon H _ * ( \textstyle { \prod _ { k = 1 } ^ { m } } F l _ { n _ k } ; \Z ) \cong H _ * ( F l _ { J } ; \Z ) . \end{align*}


\begin{align*} P ( h ) u _ h = E ( h ) u _ h \ , \ , M , \\ \end{align*}


\begin{align*} \mathcal { G } ( r ( T ) , x ) = \frac { \mathcal { G } _ 1 ( r ( T ) , x ) } { \mathcal { G } _ 2 ( r ( T ) , x ) } , \end{align*}


\begin{align*} \begin{cases} r ^ * _ i < \tilde { c } _ i & \\ r ^ * _ i = \tilde { c } _ i & \end{cases} , 1 \leq i \leq n \end{align*}


\begin{align*} F ^ { ( \iota ) } ( L ) = \sum _ { \ell = - 1 } ^ \infty F ^ { ( \iota ) } ( L ) \chi _ \ell ( L / U ) , \end{align*}


\begin{align*} \sigma _ { p , \alpha } ^ 2 : = \Big ( \frac { p } { 1 + \alpha } \Big ) ^ { 2 / p } \frac { \Gamma ( 3 / p ) } { \Gamma ( 1 / p ) } , \end{align*}


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


\begin{align*} \Sigma _ i = ( X _ i , U _ i , F _ i ) i \in \{ 1 , 2 \} . \end{align*}


\begin{align*} \partial _ t ^ k u ( t ) = A _ k ( t ) u ( t ) ~ \Leftrightarrow ~ \partial _ t ^ k U ( t , s ) u _ s = A _ k ( t ) U ( t , s ) u _ s \end{align*}


\begin{align*} \cosh d _ H : = \cosh r \cosh r ' - \sinh r \sinh r ' \cos ( \theta { - } \theta ' ) . \end{align*}


\begin{align*} \| u \| ^ q _ { L ^ q ( \Omega ) } + b \| \gamma u \| ^ r _ { L ^ r ( \Sigma ) } = 1 < 2 , \end{align*}


\begin{align*} \sum _ { n = 1 } ^ \infty \frac { 1 } { 1 \vee \log \left ( \phi ^ { - 1 } ( n ) \right ) } = \infty . \end{align*}


\begin{align*} \hat { H } \phi ( x ) = E \phi ( x ) \end{align*}


\begin{align*} \mu ( E ) = \int _ E f ( \mathrm { d } z ) , \end{align*}


\begin{align*} \frac { d g } { d t } = - e ^ { - 2 \gamma t } X _ 2 ^ R ( g ) - \omega ^ 2 e ^ { 2 \gamma t } X _ 1 ^ R ( g ) , g \in \widetilde { S L } ( 2 , \mathbb { R } ) , t \in \mathbb { R } , \end{align*}


\begin{align*} \kappa = \frac { \mu } { \delta ^ { - 1 } \ , \| A ^ T A \| _ 2 + \mu } \Leftrightarrow \frac { 1 } { \kappa } = \frac { 1 } { \mu ^ { - 1 } \ , \delta ^ { - 1 } \ , \| A ^ T A \| _ 2 + 1 } . \end{align*}


\begin{align*} m = ( l ^ B _ A ( m ) q ( r ^ B _ A ( m ) ) ( q ( r ^ B _ A ( m ) ) ^ { - 1 } r ^ B _ A ( m ) ) \end{align*}


\begin{align*} \frac { 1 } { q } D _ { q ^ { - 1 } } f ( x ) = p ( x ) f ( x ) \end{align*}


\begin{align*} \langle \mathcal { W } ^ { n } \hat { e } _ { r } , \hat { e } _ { 0 } \rangle = \sum _ { \gamma \in \mathcal { S } _ { 3 } ( n , r ) } w ( \gamma ) \end{align*}


\begin{align*} y ^ d \mapsto \prod _ { i = 1 } ^ { l _ - } \tilde y _ i ^ { p _ i ^ - \cdot d } \prod _ { j \in S _ - } \tilde x _ j ^ { D _ j \cdot d } . \end{align*}


\begin{align*} \underbrace { \varpi _ { a } \varpi _ { a + 1 } \cdots \varpi _ { b } } _ { k } = k ! e _ k ( x _ 1 , x _ 2 , \ldots , x _ b ) , \end{align*}


\begin{align*} E = Q [ \theta ] = \left \{ \sum _ { i = 0 } ^ { n - 1 } a _ { i } \theta ^ { i } \mid a _ { i } \in Q \right \} . \end{align*}


\begin{align*} \mu \int _ { \Omega } \varepsilon w : \varepsilon v + \nu \int _ { \Omega } \mathsf { m } _ \delta ' ( \varepsilon y _ \delta ) \varepsilon w : \varepsilon v = \int _ { \Omega } h \cdot v , \forall v \in Y , \end{align*}


\begin{align*} J = J _ 1 \sqcup J _ 2 \sqcup \cdots \sqcup J _ { m } , \end{align*}


\begin{align*} { \mathcal C } _ X ( A , z ) : = \sum _ { n = 1 } ^ \infty \frac { c _ n ( A ) } { n ! } z ^ n , z \in \mathbb C . \end{align*}


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


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


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


\begin{align*} \sum _ { j = 1 } ^ { k } \alpha _ j \ \mu ( T _ j , \lambda ) - \sum _ { j = 1 } ^ { k } \mu ( T _ j , \lambda ) & = \sum _ { j = 1 } ^ { k } ( \alpha _ j - 1 ) \mu ( T _ j , \lambda ) . \end{align*}


\begin{align*} \int _ { B _ 1 } \Big ( A ( x ) \nabla u \cdot \nabla ( \eta u ) \Big ) \ , d x = - \int _ { B _ 1 } f ( x ) ( \eta u ) \ , d x . \end{align*}


\begin{align*} f ( u ) = \frac { 1 } { ( 2 \pi ) ^ 3 } \int \limits _ { 0 } ^ { \infty } \ , d \nu \ , \nu ^ 2 \ , \int \limits _ { \mathbb { S } ^ 2 } \ , d ^ 2 p \ , \mathcal { F } f ( p ; \nu ) \ , ( p \cdot u ) ^ { - i \nu - 1 } \end{align*}


\begin{align*} B u ' ( t ) + A u ( t ) + B Q u ( t ) = f ( t ) , t \in \ , ] 0 , T [ , u ( 0 ) = u _ 0 , \end{align*}


\begin{align*} \mathbf { g r a d } ' \gamma _ { t } ' \circ \gamma _ t = \gamma ' \circ \gamma _ n \mathbf { c u r l } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \left ( g ^ n _ i \right ) ^ 2 \frac { 1 } { n } \rightarrow \int _ 0 ^ 1 g ( x ) ^ 2 d x . \end{align*}


\begin{align*} \mathrm { P } _ { ( \Phi , \mathfrak { a } _ { \mathfrak { b } } ) } = - \inf _ { \rho \in E _ { 1 } } \left \{ f _ { \Phi } \left ( \rho \right ) + \Delta _ { \mathfrak { a } _ { \mathfrak { b } } } \left ( \rho \right ) \right \} \ . \end{align*}


\begin{align*} e ^ { i \Delta t } F _ c ( \frac { | x | } { ( t + 1 ) ^ { \alpha } } \leq 1 ) e ^ { - i \Delta t } = F _ c ( \frac { | x - 2 P t | } { ( t + 1 ) ^ { \alpha } } \leq 1 ) , \end{align*}


\begin{align*} \lambda ( u , s ) = \mu ( s ) \cdot \lim _ { n \to + \infty } \left ( \frac { u _ 0 ^ { ( n ) } } { s _ 0 ^ { ( n ) } } \right ) ^ { 2 ^ n } = \mu ( s ) \cdot \lim _ { n \to + \infty } \left ( \frac { u _ 0 ^ { ( n ) } } { \mu ( s ) } \right ) ^ { 2 ^ n } . \end{align*}


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


\begin{align*} \phi [ \alpha ] g [ \alpha ] ^ { - 1 } g ^ { - 1 } \phi ^ { - 1 } = [ \alpha ] h [ \alpha ] ^ { - 1 } h ^ { - 1 } \end{align*}


\begin{align*} \sum _ { m \ge 0 } m ^ n t ^ m = \frac { t A _ n ( t ) } { ( 1 - t ) ^ { n + 1 } } , \end{align*}


\begin{align*} \hat { \beta } ( - z ) = - \hat { \beta } ( z ) z \in \mathbb { R } ^ { n } , \end{align*}


\begin{align*} w _ \ell & : = \left ( \sum _ { q \in \left [ b \right ] } p _ { n } \left ( q , \ell \right ) \right ) \\ f _ { \ell , 1 } & : = \left ( \sum _ { j \in \left [ b \right ] } \frac { p _ { n } \left ( j , \ell \right ) } { \sum _ { q \in \left [ b \right ] } p _ { n } \left ( q , \ell \right ) } h _ { 1 , b , j } \right ) \\ f _ { \ell , 2 } & : = h _ { 1 , b , \ell } \\ f _ { i , j } & : = h _ { 1 , b , 1 } , \forall j > 2 , \forall i . \end{align*}


\begin{align*} ( i \partial _ t + \partial _ x ^ 2 + i \gamma ) T [ W _ t ^ \gamma & u _ 0 , v , \cdots , v ] = - \widehat { \mathcal { H L } _ B } [ W _ t ^ \gamma u _ 0 , v \cdots , v ] \\ & \qquad - \frac { p - 1 } { 2 } \widehat { T } [ W _ t ^ \gamma u _ 0 , ( i \partial _ t + \bigtriangleup ) v , v , \cdots , v ] \\ & \qquad + \frac { p - 1 } { 2 } \widehat { T } [ W _ t ^ \gamma u _ 0 , v , ( i \partial _ t + \bigtriangleup ) v , v , \cdots , v ] , \end{align*}


\begin{align*} \Phi _ { g n } ( x , t ) = { \frac { h _ 1 ( t ) } { h _ 2 ( t ) } } \ln \vert h _ 2 ( t ) x + h _ 4 ( t ) \vert \ , \end{align*}


\begin{align*} C ^ * _ \lambda \mathbb F _ 2 \otimes C ^ * _ \rho \mathbb F _ 2 \ni \sum _ { i = 1 } ^ n a _ i \otimes x _ i \mapsto \sum _ { i = 1 } ^ n a _ i x _ i \in \mathbb B ( \ell ^ 2 \mathbb F _ 2 ) , \end{align*}


\begin{align*} D ( m ) = 1 + \sum _ { j = 1 } ^ { \infty } \prod _ { i = 1 } ^ { j } \bigg ( \frac { 1 } { E _ { m + i } } - 1 \bigg ) . \end{align*}


\begin{align*} \exp \left ( \sum _ { r = 1 } ^ { \infty } x _ r \frac { t ^ r } { r ! } \right ) = \sum _ { n = 0 } ^ { \infty } B _ { n } ( x _ 1 , x _ 2 , \ldots , x _ n ) \frac { t ^ n } { n ! } , \end{align*}


\begin{align*} E = b ^ 2 f ^ 2 ( x ^ 1 ) . \end{align*}


\begin{align*} H _ V = \int _ 1 ^ t \frac { \d s } { V ( s ) } . \end{align*}


\begin{align*} w ^ { ( n ) } = \mathop \int { W ^ { \left ( n \right ) } } \left ( { { t _ 1 } - { t _ 2 } , \ldots , { t _ 1 } - { t _ n } } \right ) d { t _ 2 } . . . d { t _ n } \end{align*}


\begin{align*} \| j _ { p } ( v ) \| _ { L ^ { q } ( \rho ; \R ^ d ) } ^ { q } = \| v \| _ { L ^ p ( \rho ; \R ^ d ) } ^ p = \int _ { \R ^ d } \langle j _ p ( v ) , v \rangle d \rho . \end{align*}


\begin{align*} E _ { \gamma _ 1 , \gamma _ 2 } = I m ( \gamma _ 1 ^ { - 1 } - I d ) \cap I m ( \gamma _ 2 - I d ) \end{align*}


\begin{align*} \begin{cases} e '' _ \delta + \frac { z } { 2 } e ' _ \delta = 0 , \\ e _ \delta ( 0 ) = 0 , e _ \delta ( \infty ) = \delta , \end{cases} \end{align*}


\begin{align*} \begin{aligned} \delta ( x _ 1 ^ { a _ 1 } \cdots x _ n ^ { a _ n } d x _ 1 \wedge \cdots \wedge \widehat { d x _ i } \wedge \cdots \wedge d x _ n ) = & { } ( - 1 ) ^ { i - 1 } a _ i x _ 1 ^ { a _ i } \cdots x _ i ^ { a _ i - 1 } \cdots x _ n ^ { a _ n } \Omega \\ & { } + ( - 1 ) ^ { i - 1 } \hbar ^ { - 1 } r _ i x _ 1 ^ { a _ 1 } \cdots x _ i ^ { r _ i + a _ i - 1 } \cdots x _ n ^ { a _ n } \Omega \end{aligned} \end{align*}


\begin{align*} [ u ] _ { W ^ { s , p } ( \Omega ) } \vcentcolon = \left ( \int _ { \Omega } \int _ { \Omega } \frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { n + s p } } \ , d x d y \right ) ^ { 1 / p } \end{align*}


\begin{align*} \Big \langle f , T _ 2 \circledast T _ 2 \Big \rangle = \Big \langle f , T _ 1 \Big \rangle \otimes \Big \langle f , T _ 2 \Big \rangle \end{align*}


\begin{align*} v _ { i j } = \left \{ \begin{array} { l l } \displaystyle \sqrt { \frac { | ( \lambda _ i ( A ) E _ n - A ) _ { m m } | ^ { r o w } } { \prod _ { k = 1 ; k \ne i } ^ n ( \lambda _ i ( A ) - \lambda _ k ( A ) ) } } , & j = m , \\ & \\ \displaystyle \frac { - v _ { i m } ^ { - 1 } \overline { | ( \lambda _ i ( A ) E _ n - A ) _ { j m } | ^ { r o w } } } { \prod _ { k = 1 ; k \ne i } ^ n ( \lambda _ i ( A ) - \lambda _ k ( A ) ) } , & j \ne m . \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} S _ { k } ( N , \alpha ) & \ll N ^ { 1 - \varrho + \varepsilon } + \frac { q ^ { \varepsilon } N L ^ { c } } { \sqrt { q ( 1 + | \lambda | N ^ { k } ) } } , \\ & \varrho = \left \{ \begin{array} { l l l } { 1 / 8 } , & i f \ k = 2 , \\ 1 / 1 4 , & i f \ k = 3 . \end{array} \right . \end{aligned} \end{align*}


\begin{align*} \Tilde { \Phi } _ L ( \beta ) \Tilde { \sigma } _ j ^ { - 1 } ( \xi ) & = \Tilde { \sigma } _ i ^ { - 1 } h _ i ( \xi ) \\ & = \Tilde { \sigma } _ i ^ { - 1 } ( ( \alpha , h _ i ) _ { M , n } + \xi ) \\ & = ( \alpha , h _ i ) _ { M , n } + \Tilde { \sigma } _ i ^ { - 1 } ( \xi ) \end{align*}


\begin{align*} V [ g ] = \mathcal { L } ( V , g ) / D \mathcal { L } ( V , g ) . \end{align*}


\begin{align*} E _ { 1 } = \overline { \mathrm { c o } } \left ( \mathcal { E } \left ( E _ { 1 } \right ) \right ) = \overline { \mathcal { E } \left ( E _ { 1 } \right ) } \ , \end{align*}


\begin{align*} \xi _ { j , n } = \int _ { \frac j n } ^ { \frac { j + 1 } n \wedge \tau } \gamma _ s ( s - \frac j n ) ^ { 2 \alpha } ( \sigma ' \sigma ) ^ 2 ( X _ { \frac j n } ) \left [ ( W _ s - W _ { \frac j n } ) ^ 2 - ( s - \frac j n ) \right ] d s . \end{align*}


\begin{align*} \iint _ { Q _ T } p \Delta p ( \nabla p \cdot \nabla m ) = - \iint _ { Q _ T } | \nabla p | ^ 2 [ \nabla p \cdot \nabla m ] - \underbrace { \iint _ { Q _ T } p [ \nabla p \cdot \nabla ( \nabla p \cdot \nabla m ) ] } _ { \mathcal { J } _ { 2 , 2 } } . \end{align*}


\begin{align*} { \downarrow _ 1 } ( v \Uparrow u ) = \{ u ' \in { \downarrow _ 1 } ( v ) \mid q ( u ) \prec q ( u ' ) \} \cup \{ u \} . \end{align*}


\begin{align*} \Phi \circ Q = Q ' \circ \Phi . \end{align*}


\begin{align*} \Phi _ 5 ( \xi , \eta , \sigma , \kappa , \chi , \zeta ) : = \hat { \zeta } \cdot ( \xi + \eta + \sigma + \kappa + \chi ) + \mu _ 4 | \chi | + \mu _ 3 | \kappa | + \mu _ 2 | \sigma | + \mu _ 1 | \eta | + \mu | \xi | . \end{align*}


\begin{align*} D ^ { \nu } _ i + D _ i ^ { \nu t } = C ^ { \nu } _ i . \end{align*}


\begin{align*} B ( 3 , 5 , 7 ) = \sigma _ 2 \sigma _ 1 ^ { - 1 } \sigma _ 2 \sigma _ 1 ^ { - 1 } \sigma _ 2 \sigma _ 1 \sigma _ 2 ^ { - 1 } \sigma _ 1 \sigma _ 2 ^ { - 1 } \sigma _ 1 \end{align*}


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


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


\begin{align*} \mathrm { v o l } ( N _ d ( \mathrm { C H } ( S ) ) \cap B ( x , \rho ) ) & = \mathrm { v o l } ( N _ d ( \mathrm { C H } ( \gamma S ) ) \cap B ( \gamma x , \rho ) ) \\ & \lesssim e ^ { a \rho \beta } V \left ( 1 + \frac { 1 } { a } \log A ( 1 + A C ' ) \right ) \lesssim e ^ { a \rho \beta } , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \frac { H _ n ( 1 - s ) } { H _ n ( s ) } = 1 . \end{align*}


\begin{align*} \frac { 1 } { \kappa _ n } \left ( 1 - \frac { 1 } { x \sqrt { n \cdot \kappa _ n } } \right ) = \sum _ { j = 0 } ^ { J - 1 } \alpha _ j ( x ) \cdot n ^ { - j / 2 } + O ( n ^ { - J / 2 } ) . \end{align*}


\begin{align*} \mathcal { F } ( x , z ) = \frac { 2 C ^ 3 } { E } . \end{align*}


\begin{align*} e ^ { h , u } _ n : = u ^ h _ n - R _ h u ( t _ n ) , e ^ { h , \varphi } : = \varphi ^ h _ n - P ^ 2 _ h \varphi ( t _ n ) , e ^ { h , \psi } : = \psi ^ h _ n - P ^ 3 _ h \psi ( t _ n ) , \end{align*}


\begin{align*} \bar { I } _ { K _ S } = - H \frac { \Gamma ( 1 + \frac H z ) ^ 4 \Gamma ( 1 - \frac H z ) } { \Gamma ( 1 + \frac { 3 H } { z } ) } \sum _ { m \geq 0 } ( - 1 ) ^ m \frac { ( 2 \pi i ) e ^ { \frac { 2 \pi i m } { 3 } } } { e ^ { - 2 \pi i \frac { H } { z } } - e ^ { \frac { 2 \pi i m } { 3 } } } \frac { q ^ { - \frac m 3 } } { \Gamma ( m ) \Gamma ( 1 - \frac m 3 ) ^ 4 \Gamma ( 1 + \frac m 3 ) } . \end{align*}


\begin{align*} \zeta ( a , b ) - \zeta \left ( a , b + \frac { 1 } { 2 } \right ) = 2 ^ a \Phi ( - 1 , a , 2 b ) , \end{align*}


\begin{align*} u _ 0 ( x ) = \exp ( - 2 | x | ^ 2 ) , v _ 0 \equiv 0 . \end{align*}


\begin{align*} \mathcal T _ { \kappa , 2 } = \bigcup _ { i = 0 } ^ { \lfloor \kappa / 2 \rfloor } \left \{ [ n _ 0 , n _ 1 , n _ 2 , n _ 3 ] \in \mathcal S _ { \kappa , 2 } \colon n _ 1 = n _ 2 = i \right \} \triangleq \bigcup _ { i = 0 } ^ { \lfloor \kappa / 2 \rfloor } \mathcal { Q } _ { \kappa , i } . \end{align*}


\begin{align*} F ^ * ( k , 2 , 3 ) = 8 \left \lceil \frac { k } { 4 } \right \rceil . \end{align*}


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


\begin{align*} R & \leq \frac { 1 } { n \log n } \left [ n \log P _ { \ , } - n \log r _ 0 - n \log \sqrt { \pi } + \frac { 1 } { 2 } n \log \frac { n } { 2 } - \frac { n } { 2 } \log e + o ( n ) - 0 . 5 9 9 n \right ] \\ & = \frac { 1 } { n \log n } \left [ \left ( \frac { 1 } { 2 } + \left ( 1 + b \right ) \right ) \ , n \log n - n \left ( \log \frac { \lambda \sqrt { \pi e } } { 2 \rho } + 1 . 0 5 9 9 \right ) + o ( n ) \right ] \ ; , \ , \end{align*}


\begin{align*} \lim \limits _ { L \rightarrow \infty } \hat { \rho } ( \left \vert A _ { L } \right \vert ^ { 2 } ) = | \hat { \rho } ( A ) | ^ { 2 } \ , \mathrm { } A \in \mathcal { U } . \end{align*}


\begin{align*} h & = \sum _ { A \in \left [ B \right ] ^ { d } } w _ A h _ { d , B , A } . \end{align*}


\begin{align*} \beta _ h & = \inf _ { ( \tau _ h , v _ h ) \in S ( V _ h ) } \sup _ { ( \sigma _ h , u _ h ) \in S ( V _ h ) } b _ 1 ( ( \tau _ h , - v _ h ) , ( \sigma _ h , - u _ h ) ) . \end{align*}


\begin{align*} Y \cdot ( f , a ) = a + Y . \end{align*}


\begin{align*} \nabla \phi _ \mathrm { p d } ( x ^ { ( k ) } , z ^ { ( k ) } ) - \nabla \phi _ \mathrm { p d } ( x ^ { ( k + 1 ) } , z ^ { ( k + 1 ) } ) \in \begin{bmatrix} A ^ T z ^ { ( k + 1 ) } + \partial f ( x ^ { ( k + 1 ) } ) + \nabla h ( x ^ { ( k + 1 ) } ) \\ - A x ^ { ( k + 1 ) } + \partial g ^ \ast ( z ^ { ( k + 1 ) } ) \end{bmatrix} . \end{align*}


\begin{align*} \phi = [ \gamma _ h r \gamma _ g ^ { - 1 } ] \in \mathcal L _ x ( M ) \end{align*}


\begin{align*} D ( A ^ { 2 } ) & = \left \{ x : = \left ( x _ k \right ) _ { k \in \N } \in D ( A ) \ , \middle | \ , A x \in D ( A ) \right \} \\ & = \left \{ x : = \left ( x _ k \right ) _ { k \in \N } \in c _ 0 \ , \middle | \ , \left ( w ^ { k } w ^ { k + 1 } x _ { k + 2 } \right ) _ { k \in \N } \in c _ 0 \right \} \end{align*}


\begin{align*} \Phi ( z ) = 1 - \sum _ { n \ge 1 } \frac { a _ n } { \lambda _ n ^ 2 - z } , \end{align*}


\begin{align*} \varphi ( Y , x _ 0 ) = \frac { \alpha x _ 0 + \beta } { \gamma x _ 0 + \delta } . \end{align*}


\begin{align*} & J _ m ( x , r \xi ) = \frac { r ^ { m } } { | r | } ( J _ m ^ k f ) ( x , \xi ) \mbox { f o r } r \neq 0 , \\ & J _ m ( x + s \xi , \xi ) = J _ m ( x , \xi ) . \end{align*}


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


\begin{align*} \psi ( f ) = f \circ \lambda , \end{align*}


\begin{align*} r _ k ( x \oplus 2 ^ { - s } ) = \left \{ \begin{array} { r l l } r _ k ( x ) & & k \neq s , \\ - r _ k ( x ) & & k = s , \end{array} \right . \end{align*}


\begin{align*} N ( t ) = \rho ( t , V _ F ) \int _ 0 ^ { + \infty } g \frac { 1 } { \sqrt { 2 \pi a ( t ) } } \exp \left ( - \frac { ( g - g _ { } ( t ) ) ^ 2 } { 2 a ( t ) } \right ) d g , \end{align*}


\begin{align*} B ( z ) = \frac { a - z } { 1 - \bar a z } \end{align*}


\begin{align*} f ( b ) : = \begin{cases} \frac { \int \limits _ { 0 } ^ { \pi } \sin ^ { n - 2 } ( t ) d t } { \int \limits _ { 0 } ^ { \pi } \frac { \sin ^ { n - 2 } ( t ) } { [ \phi ( b \cos ( t ) ) ] ^ n } d t } , & \mbox { i f } d V = d V _ { B H } , \\ \frac { \int \limits _ { 0 } ^ { \pi } \sin ^ { n - 2 } ( t ) T ( b \cos ( t ) ) d t } { \int \limits _ { 0 } ^ { \pi } \sin ^ { n - 2 } ( t ) d t } , & \mbox { i f } d V = d V _ { H T } , \end{cases} \end{align*}


\begin{align*} \textrm { t h e r e e x i s t s N s u c h t h a t } \sum _ { k = 0 } ^ { \infty } \frac { ( 1 + k ) ^ { 2 n } } { | \mu ( k ) | ^ { 2 } } w ( k ) < \infty \textrm { f o r } n < N \textrm { a n d i n f i n i t e f o r } n \geq N . \end{align*}


\begin{align*} x _ { k + 1 } : = x _ k , \overline { y } _ { k + 1 } : = y _ k - \frac { 1 } { \sigma _ k } g ( x _ { k + 1 } ) , \overline { Z } _ { k + 1 } : = \left [ Z _ k - \frac { 1 } { \sigma _ k } X ( x _ { k + 1 } ) \right ] _ { + } \end{align*}


\begin{align*} | D \widehat \psi _ s | _ { g _ o } ^ 2 = ( g _ { 0 } ) _ { a b } | _ { \widehat \psi _ s } \frac { \partial \widehat \psi _ s ^ a } { \partial x ^ i } \frac { \partial \widehat \psi _ s ^ b } { \partial x ^ j } g _ o ^ { i j } = { \rm t r } _ { g _ o } ( \widehat \psi _ s ^ * g _ o ) . \end{align*}


\begin{align*} \dfrac { ( \Tilde { \mathcal { N } } ( f ) \circ \rho _ { \eta } ( X ) ) ' } { \Tilde { \mathcal { N } } ( f ) \circ \rho _ { \eta } ( X ) } = \eta \dfrac { ( \Tilde { \mathcal { N } } ( f ) ) ' } { \Tilde { \mathcal { N } } ( f ) } \circ \rho _ { \eta } ( X ) . \end{align*}


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


\begin{gather*} W _ D : = \sum _ { i , \chi } n _ { i , \chi } C _ { \varphi _ i } ( D ) \chi \end{gather*}


\begin{align*} \frac { k ' a } { 2 } \left [ \tanh ( k ' a / 2 ) - \coth ( k ' a / 2 ) \right ] = 1 , \end{align*}


\begin{align*} A = \left ( A _ { \epsilon \delta } \right ) = \left ( \sum \limits _ { i = 1 } ^ { 3 } z ^ i _ { \epsilon } z ^ i _ { \delta } \right ) . \end{align*}


\begin{align*} H ( \lambda ) : & = \Phi ( \lambda ) ( \phi ( \lambda ) + \lambda ( 1 - \Phi ( \lambda ) ) ) - \frac { 1 } { 2 } { \phi ( \lambda ) } . \end{align*}


\begin{align*} \omega = \sum _ i [ \alpha _ i ^ \# , \alpha _ i ] \end{align*}


\begin{align*} \overline { \widehat { w } } _ k : = \langle k \rangle ^ { - s } \langle \tau - k ^ 2 \rangle ^ { b ' } \overline { \widehat { v } } _ k , \qquad \widehat { z ^ \ell } _ { k _ \ell } : = \langle k _ \ell \rangle ^ { s } \langle \tau _ \ell + ( - 1 ) ^ { \ell + 1 } k _ \ell ^ 2 \rangle ^ { b } \widehat { u ^ \ell } _ { k _ \ell } , \end{align*}


\begin{align*} \det ( x I - B _ 1 ( T ) ) & = \prod _ { j = 1 } ^ { l ( T ) } \prod _ { v \in V ( T , j ) } \mathcal { F } ( v , x ) \\ & = \prod _ { v \in V ( T ) } \mathcal { F } ( v , x ) . \end{align*}


\begin{align*} F _ { q , r } ( u ) = E ( u + 1 ) - \frac { n - 2 } { 2 q ( n - 1 ) } \int _ M R ' | u + 1 | ^ q d V _ g - \frac { n - 2 } { r } \int _ { \partial M } H ' | \gamma ( u + 1 ) | ^ r d \sigma _ g \end{align*}


\begin{align*} a \mapsto \begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix} . \end{align*}


\begin{align*} \partial _ s \left ( \widehat \psi _ { s * } X \right ) = \widehat \psi _ { s * } [ X _ s , X ] = \widehat \psi _ { s * } [ X _ s - X , X ] \end{align*}


\begin{align*} \tilde { N } _ i ( E _ * ) = H ^ 0 ( \textnormal { P E n d } _ { \mathrm { S p } } ( E _ { * _ i } ) \otimes K ( D ) ) . \end{align*}


\begin{align*} \{ \mathbf { S } _ n = \mathbf { s } ( j ) \} = C ( n , j ) . \end{align*}


\begin{align*} { \rm K B S M } \left ( L ( p , q ) \right ) = \frac { { \rm K B S M } ( { \rm S T } ) } { < a - b b m ( a ) > } , { \rm w h e r e } \ a \ { \rm b a s i s \ e l e m e n t \ o f \ K B S M ( S T ) } . \end{align*}


\begin{align*} H ( p , G ) = \tilde { Y } _ O ( ( \underbrace { p , p , \cdots , p } _ { p ' / 2 } ) , G ) ^ { 2 / p ' } \end{align*}


\begin{align*} e ^ { f _ o ( \cdot ) } v ( \cdot , s _ 0 - s _ 0 ^ { \theta / 4 } ) = e ^ { f _ o ( \cdot ) } w ( \cdot , s _ 0 - s _ 0 ^ { \theta / 4 } ) . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { \infty } \sum _ { j = 0 } ^ { \infty } \frac { 1 } { ( \textrm { m a x } \ , ( j , k ) + 1 ) ^ 2 } \left | \frac { \mu ( j ) } { \mu ( k ) } \right | ^ 2 \frac { w ( k ) } { w ' ( j ) } < \infty , \end{align*}


\begin{align*} \mathcal { I } _ x ( f ( x ) ) = \int _ \mathbb { R } \frac { 1 + s ^ 2 } { ( x - s ) ^ 2 + f ( x ) ^ 2 } \ ; d \rho ( s ) \leq b , \ ; \ ; \ ; \ ; \ ; x \in \mathbb { R } . \end{align*}


\begin{align*} x ^ 3 + y ^ 3 + n ^ 2 z ^ 3 = n x y z \end{align*}


\begin{align*} \lambda \triangleq p \left ( s _ { m } = 1 \right ) = \frac { \rho _ { 0 , 1 } } { \rho _ { 0 , 1 } + \rho _ { 1 , 0 } } . \end{align*}


\begin{align*} G _ { \mu } ( z ) = \int _ { \mathbb { R } } \frac { d \mu ( t ) } { z - t } , F _ { \mu } ( z ) = \frac { 1 } { G _ { \mu } ( z ) } , z \in \mathbb { H } . \end{align*}


\begin{align*} P ^ f _ H = \{ ( x , 0 , p _ x , p _ y ) \in { \mathbb R } ^ 4 \} \equiv { \mathbb R } ^ 3 \end{align*}


\begin{align*} \begin{array} { l l } \left . \partial _ s ^ 2 { \rm L o g } \left ( e ^ { \int a _ 1 ( t , s ) d s } e ^ { \int a _ 2 ( t , s ) d s } \right ) \right | _ { s = 0 } - \left . \partial _ s ^ 2 { \rm L o g } \left ( e ^ { \int a _ 2 ( t , s ) d s } e ^ { - \int a _ 1 ( t , s ) d s } \right ) \right | _ { s = 0 } = 0 \end{array} \end{align*}


\begin{align*} P ( T _ 0 , x ) & = \dfrac { \mathcal { G } _ 1 ( r ( T ) , x ) \displaystyle \prod _ { j = 1 } ^ { k } P ( T _ j , x ) ^ { \alpha _ j } } { \displaystyle \prod _ { j = 1 } ^ { k } \mathcal { G } _ 1 ( r ( T _ j ) , x ) } . \end{align*}


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


\begin{align*} \varphi _ { \mu } ( z ) = c + \int _ { \mathbb { R } } \frac { 1 + z x } { z - x } \ , d \sigma ( x ) , \end{align*}


\begin{align*} q _ s = a _ s + a _ { s + 1 } g _ r + \cdots + a _ m g _ r ^ { m - s } . \end{align*}


\begin{align*} ( P _ h u , v ) _ { L ^ 2 ( \Omega ) } = ( u , v ) _ { L ^ 2 ( \Omega ) } v \in V _ h . \end{align*}


\begin{align*} P _ { \vect { s ' } } ( f ( \Delta _ \Omega ^ { \vect { s '' } } \circ \rho ) ) * I ^ { - \vect { s '' } } _ \Omega = c P _ { \vect { s ' } - \vect { s '' } } f \end{align*}


\begin{align*} d \Theta _ n \circ A d _ \phi = A d _ { \Theta _ n ( \phi ) } \circ d \Theta _ n \end{align*}


\begin{align*} \frac { \sum _ { i = 1 } ^ n a _ i } { \sum _ { i = 1 } ^ n b _ i } \le \max _ { j = 1 , \ldots , n } \frac { a _ j } { b _ j } . \end{align*}


\begin{align*} \mathbb { P } ( H ^ 0 ( \mathcal { M } , \mathcal { L } ) ^ \vee ) = \mathbb { P } ( H ^ 0 ( \mathcal { U } , \mathcal { L } ) ^ \vee ) \cong \mathbb { P } ( H ^ 0 ( \mathcal { U } ' , \mathcal { L } ' ) ^ \vee ) = \mathbb { P } ( H ^ 0 ( \mathcal { M } ' , \mathcal { L } ' ) ^ \vee ) \end{align*}


\begin{align*} \norm { u \wedge u ( \cdot + \delta e _ 2 ) } _ { L ^ 1 ( \R ^ N ) } > m \quad \delta \in ( 0 , \delta _ 0 ) u \in \mathcal { M } ^ f _ { m + \frac { \eta } { 2 } } . \end{align*}


\begin{align*} W ( y | x ) = \frac { e ^ { - ( \rho x + \lambda ) } ( \rho x + \lambda ) ^ y } { y ! } \ , . \ , \end{align*}


\begin{align*} \sum \limits _ { i } \alpha _ i = 0 \end{align*}


\begin{align*} \gamma _ 1 = ( 1 , 2 ) , \ , \gamma _ 2 = ( 1 , 3 ) , \ldots , \gamma _ n = ( 1 , n + 1 ) , \ldots , \end{align*}


\begin{align*} \mathcal { P } _ { m , s } ( \Omega ) \vcentcolon = \{ \ , P = \sum _ { | \alpha | \leq m } a _ \alpha D ^ \alpha \ , ; \ , \norm { P } _ { m , s } < \delta ( \Omega ) , a _ { \alpha } \in M ( H ^ { s - | \alpha | } \rightarrow H ^ { - s } ) \ , \} \end{align*}


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


\begin{align*} \frac { f ^ \vee ( x + t y ) - f ^ \vee ( x ) } { t } \ ; = \ ; g ^ \vee ( t ) \ ; \to \ ; g ^ \vee ( 0 ) \end{align*}


\begin{align*} \| \nabla ^ 2 \mathcal { H } \| ^ 2 & = g ^ { i k } g ^ { j l } \nabla ^ 2 \mathcal { H } ( \partial _ i , \partial _ j ) \nabla ^ 2 \mathcal { H } ( \partial _ k , \partial _ l ) , \\ \nabla ^ 2 \mathcal { H } ( \partial _ i , \partial _ j ) & = \partial _ i \partial _ j \mathcal { H } - \nabla _ { \nabla _ { \partial _ i } \partial _ j } \mathcal { H } = \mathcal { H } _ { i j } - \Gamma _ { i j } ^ k \mathcal { H } _ k . \end{align*}


\begin{align*} \frac { 2 \ddot { x } } { C _ 3 \dot { x } ^ 2 } = F ( \dot { x } , x , t ) \ , \end{align*}


\begin{align*} \langle \mu _ i ( x \cdot \phi _ 0 ( T ) - \phi _ 0 ( T ) \cdot x ) \hat { a } , \hat { b } \rangle = \mu _ i ( [ x , \varphi _ 0 ] ) ( b ^ * T a ) \to \langle \mu ( x \cdot \phi _ 0 ( T ) - \phi _ 0 ( T ) \cdot x ) \hat { a } , \hat { b } \rangle . \end{align*}


\begin{align*} u = u \cdot \omega \omega + \sum _ { i = 1 , 2 , 3 } ( u \cdot ( \omega \times \mathbf { e } _ i ) ) ( \omega \times \mathbf { e } _ i ) \Longrightarrow B _ 3 = B \cdot \omega ( \omega \cdot \mathbf { e } _ 3 ) - \sum _ { i = 1 , 2 , 3 } ( B \cdot ( \omega \times \mathbf { e } _ i ) ) ( \omega \times \mathbf { e } _ 3 ) \cdot \mathbf { e } _ i . \end{align*}


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


\begin{align*} { } E ( - t ) = \sum _ { n \geq 0 } ( - 1 ) ^ n e _ n t ^ n = \ , \ , ( - \sum _ { n \geq 1 } \frac { p _ n t ^ n } { n } ) = \sum _ { \lambda } ( - 1 ) ^ { ( 2 | \lambda | - l ( \lambda ) ) } \frac { p _ { \lambda } } { z _ { \lambda } } t ^ { | \lambda | } . \end{align*}


\begin{align*} \eta ( \cdot , s ) = s w ( x ) + \int _ 0 ^ s f ^ 6 ( x , \tau ) d \tau . \end{align*}


\begin{align*} F ( t , x , v ) = \begin{cases} x / v , & \mbox { f o r } x / v \leq t , \\ t , & \mbox { f o r } x / v > 0 , \ , v > 0 \\ t , & \mbox { f o r } v = 0 \\ t , & \mbox { f o r } ( 1 - x ) / | v | > t , \ , v < 0 \\ ( 1 - x ) / | v | & \mbox { f o r } - ( 1 - x ) / v \leq t . \\ \end{cases} \end{align*}


\begin{align*} \sum _ { j = 1 } ^ n ( { \widehat { x } } _ j { \widehat { p } } _ j + { \widehat { p } } _ j { \widehat { x } } _ j ) = 2 \sum _ { j = 1 } ^ n { \widehat { x } } _ j { \widehat { p } } _ j - { \rm i } n \widehat { I } , \end{align*}


\begin{align*} \begin{aligned} \hat { W } ( x _ i ) = f _ i \nabla \hat { W } ( x _ i ) = g _ i . \end{aligned} \end{align*}


\begin{align*} f = q Q ' + r \mbox { w i t h } \deg ( r ) < \deg ( Q ' ) = \deg ( Q _ 0 ) . \end{align*}


\begin{align*} \Omega ( \alpha , A , z + a f _ k ) = \{ z + a f _ k \} \cup \bigcup _ { n = 1 } ^ \infty \Omega ( \alpha _ n , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \end{align*}


\begin{align*} \begin{array} { l c l } \sigma _ { i , i + 1 } & : = & 1 + u \ ( \sigma _ i + \sigma _ { i + 1 } ) + u ^ 2 \ ( \sigma _ i \sigma _ { i + 1 } + \sigma _ { i + 1 } \sigma _ i ) + u ^ 3 \ \sigma _ i \sigma _ { i + 1 } \sigma _ i . \\ \end{array} \end{align*}


\begin{align*} T _ { q ^ { \prime } } \cdots T _ { p } : = \begin{cases} I d & \mathrm { i f } \ , q ^ { \prime } < p \\ T _ { q ^ { \prime } } \circ \left ( T _ { q ^ { \prime } - 1 } \cdots T _ { p } \right ) & \mathrm { i f } \ , q ^ { \prime } \ge p \end{cases} . \end{align*}


\begin{align*} g ( w ^ * ) = g ( w _ j ^ * ) = 0 . \end{align*}


\begin{align*} \tilde { I } _ 1 & \le \frac { c } { T _ n } \frac { \sum _ { j = 1 } ^ d | \log h _ j ^ * | } { \prod _ { l = 3 } ^ d h _ l ^ * } . \end{align*}


\begin{align*} \varphi ( z ) = 2 \sqrt y \sum _ { n \neq 0 } a ( n ) K _ { i r } ( 2 \pi | n | y ) e ( n x ) . \end{align*}


\begin{align*} E ^ 1 _ j = \{ x _ { i _ j } \in B \} \end{align*}


\begin{align*} \cap _ { i = 1 } ^ { m } C _ { i } = \cap _ { n = 0 } ^ { \infty } \mathrm { F i x } ( S _ { n } ) = \cap _ { n = 0 } ^ { j _ { f } } \mathrm { F i x } ( S _ { n } ) . \end{align*}


\begin{align*} f ( x ) - f ( y ) & = \biggl ( \frac { x } { \lvert x \rvert } - \frac { y } { \lvert y \rvert } \biggr ) h \biggl ( \frac { \lvert x \rvert } { \lVert x \rVert } , \lVert x \rVert \biggr ) + \frac { y } { \lvert y \rvert } \Biggl ( h \biggl ( \frac { \lvert x \rvert } { \lVert x \rVert } , \lVert x \rVert \biggr ) - h \biggl ( \frac { \lvert y \rvert } { \lVert y \rVert } , \lVert y \rVert \biggr ) \Biggr ) , \end{align*}


\begin{align*} \partial _ t u ( t ) = A _ 1 ( t ) u ( t ) \end{align*}


\begin{align*} B _ { k } ^ { * } = \{ t _ { n - 1 } t _ { n - 2 } \cdots t _ { 2 k } t _ { 2 k - 2 } t _ { 2 k - 4 } \cdots t _ 2 \ : | \ : \forall ~ j , ~ t _ j = 1 \mbox { o r } t _ j = s _ { i _ j , j } \mbox { f o r s o m e } 1 \leq i _ j \leq j \} , \end{align*}


\begin{align*} P ( E _ { x \rightarrow y } | S _ { i } ) = { \left ( \frac { i } { m _ x } \right ) } ^ { k _ x } , \end{align*}


\begin{align*} \sum _ { \substack { S \subseteq \{ 1 , \ldots , k \} \ \mathrm { w i t h } \\ \rho \wedge \pi _ { { } _ S } = \pi } } \ ( - 1 ) ^ { | S | } \ = \ ( - 1 ) ^ { | \pi | _ 1 / 2 } . \end{align*}


\begin{align*} E _ \mathcal { N } = \max _ { k = 1 , \ldots , \mathcal { N } } | | x ( t _ k ) - x _ k | | . \end{align*}


\begin{align*} b _ { k , n } ^ j = ( - 1 ) ^ { \lfloor ( j + 1 ) / 2 \rfloor } 2 ^ j ( \kappa _ k + 1 / 2 ) _ { \lfloor ( j + 1 ) / 2 \rfloor } ( \gamma _ { k } + n + k / 2 ) _ { \lfloor j / 2 \rfloor } . \end{align*}


\begin{align*} R _ j : = \{ v \in F _ j : v F _ j F _ j \setminus v \subseteq F _ i 1 \leq i < j \} \ , . \end{align*}


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


\begin{align*} W : = \{ w \in X ^ * \mid \abs { w } \leq n , \ , w \in H \} , \end{align*}


\begin{align*} A C ( r ) = \frac { A } r \int _ 0 ^ r e ^ { s A } \ , d s = \frac { e ^ { r A } - I } { r } \to 0 = A P \end{align*}


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


\begin{align*} \overline { u } ( t , r ) = C _ 1 e ^ { \gamma _ 0 t } [ h ( r ) - h ( \delta _ 1 ) ] \ , \ 0 < r < \delta _ 2 ( t ) , \ \ \overline { v } ( t , r ) = C _ 2 e ^ { \gamma _ 0 t } [ h ( r ) - h ( \delta _ 2 ) ] , \ 0 < r < \delta _ 2 ( t ) , \end{align*}


\begin{align*} L _ f = M _ f + \sum _ { g \prec f } \ell _ { g f } ( q ) M _ g , \ell _ { g f } ( q ) \in q ^ { - 1 } \Z [ q ^ { - 1 } ] . \end{align*}


\begin{align*} \mathcal { F } ( v , x ) & = x - \beta ( v ) - \sum _ { w \in c ( v ) } \dfrac { 1 } { \mathcal { F } ( w , x ) } \\ & = x - \beta ( v _ 1 ) - \dfrac { c _ j } { \mathcal { F } ( v _ 2 , x ) } . \end{align*}


\begin{align*} \left | \left . \int u _ { 1 , t } d \nu ^ 2 _ { t } \ , \right | ^ { t = \bar s } _ { t = s } - \left . \int u _ { 1 , t } v _ { 2 , t } \ , d g _ { 1 , t } \ , \right | ^ { t = \bar s } _ { t = s } \right | & \le C \varepsilon \end{align*}


\begin{align*} g \equiv \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & \sqrt { - 3 } x \\ 0 & \sqrt { - 3 } y & 1 \end{pmatrix} \mod 3 . \end{align*}


\begin{align*} \mathcal { F } ( x , z ) = \frac { 2 C ^ 3 } { E } . \end{align*}


\begin{align*} & \langle x , A _ n F _ n ( x ) \rangle = \langle x , A _ n ( x _ 1 ^ 3 , \ldots , x _ { n - 1 } ^ 3 ) ^ \top \rangle - \langle x , A _ n x \rangle \le - \langle x , A _ n x \rangle , \end{align*}


\begin{align*} B _ { - \lambda _ Y / r } \left ( r \alpha _ Y + \lambda _ Y + \frac { ( \lambda _ Y , \lambda _ Y ) } { 2 r } \beta _ Y \right ) = r \alpha _ Y . \end{align*}


\begin{align*} A = H _ { i _ 1 } \cap \ldots \cap H _ { i _ n } \end{align*}


\begin{align*} N ^ { n , 5 } _ \tau : = \frac 1 n \sum _ { j = 0 } ^ { \lfloor n \tau \rfloor } ( \sigma ' \sigma ) ^ 2 ( X _ { \frac j n } ) \int _ { 0 } ^ { 1 \wedge ( n \tau - j ) } \gamma _ { \frac { x + j } n } \sum _ { k = j + 1 } ^ { \infty } [ ( x + k ) ^ \alpha - k ^ { \alpha } ] ^ 2 d x = 0 . \end{align*}


\begin{align*} \Psi ^ * _ t g _ { \mathrm { c a n } } = g + \frac { t } { 3 } ( \frac { 1 } { 2 } \mathrm { S c a l } _ g \cdot g - \mathrm { R i c } _ g ) + O ( t ^ 2 ) , \end{align*}


\begin{align*} W = x _ 1 ^ { d / w _ 1 } + \cdots + x _ n ^ { d / w _ N } , \end{align*}


\begin{align*} \Lambda ^ * : = \Big \{ k \in \frac { 2 \pi } { L } \Z ^ d : \forall i \in \{ 1 , \dots , d \} , \ - \frac { \pi } { \epsilon } < k _ i \leq \frac { \pi } { \epsilon } \Big \} , \end{align*}


\begin{align*} \label [ I e q ] { e q : m a r t i n g a l e p a r t i c l e s } M ^ { \varphi _ N } _ t : = \varphi _ N { \left ( \mathbf { X } ^ N _ t \right ) } - \varphi _ N { \left ( \mathbf { X } ^ N _ 0 \right ) } - \int _ 0 ^ t \mathcal { L } _ N \varphi _ N { \left ( \mathbf { X } ^ N _ s \right ) } \dd s , \end{align*}


\begin{align*} \alpha _ { j } = ( z _ { \mathrm { n c } } , \phi _ { j } ) _ { 1 + \delta } = \frac { 1 } { \mu _ { j } } a _ { \mathrm { p w } } ( z _ { \mathrm { n c } } , \phi _ { { j } } ) = \frac { \lambda } { \mu _ { j } } ( u , \phi _ { { j } } ) _ { 1 + \delta } . \end{align*}


\begin{align*} X \begin{pmatrix} * & * & * \\ * & * & * \\ a & b & c \end{pmatrix} = \begin{cases} - a & \\ c & \end{cases} \end{align*}


\begin{align*} \Delta _ { A } \left ( \rho \right ) = \left \Vert P _ { \rho } \pi _ { \rho } \left ( A \right ) \Omega _ { \rho } \right \Vert _ { \mathcal { H } _ { \rho } } ^ { 2 } \ , \rho \in E _ { 1 } \ , \end{align*}


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


\begin{align*} \langle { x ^ \eta } \left ( T \right ) \rangle = \left \langle { e ^ { \eta \mathop \int \limits _ 0 ^ T \xi \left ( t \right ) d t } } \right \rangle = e ^ { W \left [ { \eta { { \rm { I } } _ { \left [ { 0 , 1 } \right ] } } \left ( t / T \right ) } \right ] } \end{align*}


\begin{align*} D _ { q ^ { - 1 } } \sin _ q z = \cos _ q ( \frac { z } { q } ) , D _ { q ^ { - 1 } } \cos _ q z = - \sin _ q ( \frac { z } { q } ) . \end{align*}


\begin{align*} \partial _ t ^ 2 \tilde u + L \tilde u = f \end{align*}


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


\begin{align*} N ( t , t ) = t ^ { n + 1 } N ( 1 / t , 1 / t ) . \end{align*}


\begin{align*} \frac { \partial C } { \partial z ^ i _ { \epsilon } } = \frac { 1 } { C } \sum \limits _ { i \neq l } ^ { } \left ( z ^ i _ 1 z ^ l _ 2 - z ^ i _ 2 z ^ l _ 1 \right ) \left ( \delta _ { \epsilon 1 } z ^ l _ 2 - \delta _ { \epsilon 2 } z ^ l _ 1 \right ) . \end{align*}


\begin{align*} \theta ( t ) = \int _ 0 ^ t \phi ( t - s ) \theta _ 0 ( s ) \textnormal { d } s \leq M _ \alpha \int _ 0 ^ t \theta _ 0 ( s ) \textnormal { d } s < \infty , \end{align*}


\begin{align*} \label [ I e q ] { e q : e m p i r i c a l m e a s u r e i n f i n i t e } \mu _ { \overline { \mathcal { X } } { } ^ N } = \frac { 1 } { N } \sum _ { i = 1 } ^ N \delta _ { \overline { X } { } ^ i } . \end{align*}


\begin{align*} \mathcal { U } _ { [ n , j , k ] } : = \{ \gamma \in \mathcal { U } _ { [ n , j ] } : \lambda ( \gamma ) = k \} , j \leq k \leq n - 1 . \end{align*}


\begin{align*} \tau = \begin{cases} t & \alpha + \delta + \beta = 1 , \\ \frac { \log ( ( 1 - \alpha - \delta - \beta ) \eta t + 1 ) } { ( 1 - \alpha - \delta - \beta ) \eta } & \alpha + \delta + \beta < 1 , \\ \frac { \log ( \frac { T ^ * - t } { T ^ * } ) } { ( 1 - \alpha - \delta - \beta ) \eta } & \alpha + \delta + \beta > 1 . \end{cases} \end{align*}


\begin{align*} 0 < \beta : = \inf _ { x \in H \setminus \{ 0 \} } \sup _ { y \in H \setminus \{ 0 \} } \frac { b ( x , y ) } { \| x \| _ { H } \| y \| _ H } = \inf _ { y \in H \setminus \{ 0 \} } \sup _ { x \in H \setminus \{ 0 \} } \frac { b ( x , y ) } { \| x \| _ { H } \| y \| _ H } . \end{align*}


\begin{align*} & & h _ i ^ 0 & = 0 , & \forall i \in [ m ] \\ & & h _ i ^ { j } & \leq h _ i ^ { j + 1 } , & \forall j \in [ m - 1 ] , i \in [ m ] \\ & & \sum _ { i \in [ m ] } h _ i ^ { j } & = j , & \forall j \in [ m ] \\ & & \sum _ { k \in [ j ] } h _ i ^ { k } & \geq j - y _ i + 1 , & \forall j \in [ m ] , \forall i \in [ m ] \end{align*}


\begin{align*} \beta : = c _ { n , \sigma p } \left ( \int _ { B _ { 4 } \setminus B _ { 3 } } \frac { \phi ^ { p - 1 } ( y ) } { | y | ^ { n + \sigma p } } d y + \int _ { B _ { 6 } \setminus B _ { 4 } } \frac { d y } { | y | ^ { n + \sigma p } } + \int _ { B _ { 6 } ^ { c } } \frac { ( 1 - \psi ( y ) ) ^ { p - 1 } } { | y | ^ { n + \sigma p } } d y \right ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : p d m p f l o w } \forall n \in \N , \quad \forall t \in [ T ^ i _ { n } , T ^ i _ { n + 1 } ) , \dd X ^ i _ t = a ( X ^ i _ t ) \dd t , \end{align*}


\begin{align*} \lim _ { \lambda \rightarrow \infty } \mathbb { \hat { E } } [ ( | X _ 1 | ^ 2 - \lambda ) ^ + ] = 0 \end{align*}


\begin{align*} \prescript { } { n } { \mathcal { A } } _ { m } \ni \varphi ( \lambda ) = \begin{cases} h ( \lambda ) q _ { n , m } ( \lambda ) , & m < n , \\ q _ { n , m } ( \lambda ) h ( \lambda ) , & m > n , \\ \end{cases} \ ; \ ; \ ; \ ; \lambda \in \mathfrak { a } ^ * _ \mathbb { C } . \end{align*}


\begin{align*} \mathcal { H } ^ s _ { \infty } ( E _ n ^ { ( 1 ) } ) \ll \begin{cases} \left ( \frac { \delta } { 3 ^ n } \right ) ^ s & \tilde { \delta } \leq \frac { 1 } { 3 ^ n } , \\ 3 ^ n \cdot \delta \cdot \left ( \frac { \delta } { 3 ^ n } \right ) ^ s & \tilde { \delta } > \frac { 1 } { 3 ^ n } . \end{cases} \end{align*}


\begin{align*} M _ \ell \lesssim ( R / R _ \ell ) ^ { d _ 1 } = 2 ^ { d _ 1 ( \iota - \ell ) } . \end{align*}


\begin{align*} \frac { \partial ^ 2 E } { \partial z ^ i _ { \epsilon } \partial z ^ j _ { \eta } } \frac { \partial ^ 2 \varphi ^ j } { \partial x ^ { \epsilon } \partial x ^ { \eta } } v ^ i = 2 b ^ 2 f ( x ^ 1 ) \left [ 1 + 2 f '^ 2 ( x ^ 1 ) \right ] . \end{align*}


\begin{align*} F ( L , \mathbf U ) f = f * \mathcal K _ { F ( L , \mathbf U ) } f \in \S ( G ) , \end{align*}


\begin{align*} I K = \left \{ \sigma \in \Lambda ^ 2 X \mid A ( \sigma ) \le 1 \right \} . \end{align*}


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


\begin{align*} \left \| g ( j ) \right \| ^ 2 = \sum \limits _ { i \in \mathbb Z } { \left | { g _ i ( j ) } \right | ^ 2 } < \infty \left \| h ( j ) \right \| ^ 2 = \sum \limits _ { i \in \mathbb Z } { \sum \limits _ { k \in \mathbb N } { \left | { h _ { i , k } ( j ) } \right | ^ 2 } } < \infty . \end{align*}


\begin{align*} \mathrm { g } _ { \theta } ( a _ { x , \mathrm { s } } ) = \mathrm { e } ^ { - i \theta } a _ { x , \mathrm { s } } \ , x \in \mathbb { Z } ^ { d } , \ \mathrm { s } \in \mathrm { S } , \end{align*}


\begin{align*} \int _ { \R ^ N } u \wedge u ^ t + \int _ { \R ^ N } u \vee u ^ t = 2 \int _ { \R ^ N } u = 2 m . \end{align*}


\begin{align*} v _ 0 * ^ { \delta _ 1 } v _ 1 * ^ { \delta _ 2 } \cdots * ^ { \delta _ m } v _ m = w _ 0 * ^ { \delta _ 1 } w _ 1 * ^ { \delta _ 2 } \cdots * ^ { \delta _ m } w _ m , \end{align*}


\begin{align*} \epsilon ' _ n = \frac { P _ { \ , } } { n ^ { 1 + b } } \ , , \ , \end{align*}


\begin{align*} { \cal G } _ t = \big \{ \mbox { \rm p r i m e $ p $ } \ , \big | \ , q < p \le t , ~ \mu _ q ( p ) \ge ( \log _ q t ) ^ 2 \big \} ; \\ \overline { \cal G } _ t = \big \{ \mbox { \rm p r i m e $ p $ } \ , \big | \ , q < p \le t , ~ \mu _ q ( p ) < ( \log _ q t ) ^ 2 \big \} . \end{align*}


\begin{align*} V ( s ) \colon & H ^ { - 1 / 2 } ( \Gamma ) \rightarrow H ^ { 1 / 2 } ( \Gamma ) \\ K ( s ) \colon & H ^ { 1 / 2 } ( \Gamma ) \rightarrow H ^ { 1 / 2 } ( \Gamma ) \\ K ^ t ( s ) \colon & H ^ { - 1 / 2 } ( \Gamma ) \rightarrow H ^ { - 1 / 2 } ( \Gamma ) \\ W ( s ) \colon & H ^ { 1 / 2 } ( \Gamma ) \rightarrow H ^ { - 1 / 2 } ( \Gamma ) \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*} \mathcal K _ { \kappa , 2 } = \left \{ \left [ n _ 0 , n _ 1 , n _ 2 , n _ 3 \right ] \colon n _ i \geq 0 , \ ; \sum _ { i = 0 } ^ { 3 } n _ i = \kappa , \ ; n _ 1 \leq n _ 2 \right \} , \end{align*}


\begin{align*} 2 K _ X & \equiv f ^ * \left ( 2 K _ Y + \sum \limits _ { \sigma \ne 0 } { D _ { \sigma } } \right ) ; \\ f _ { * } \mathcal { O } _ X & = \mathcal { O } _ Y \oplus \bigoplus \limits _ { \chi \ne \chi _ { 0 0 0 } } L _ { \chi } ^ { - 1 } . \end{align*}


\begin{align*} \partial \varphi \left ( x \right ) = x \varphi \left ( x \right ) \end{align*}


\begin{align*} N \leq \left ( \frac { 1 + \frac { 1 } { 2 } \sin \theta } { \frac { 1 } { 2 } \sin \theta } \right ) ^ \rho = ( 1 + 2 / \sin \theta ) ^ \rho \leq ( 1 + \sqrt { 8 c } ) ^ \rho . \end{align*}


\begin{align*} \Delta ^ m ( ^ s f ) = 2 ^ m \delta _ e ^ m R f , \end{align*}


\begin{align*} 1 2 \pi ^ { 1 / 2 } | d | ^ { \frac 3 2 } | b _ { \psi } ( d ) | ^ 2 = \frac { 1 } { \langle \varphi , \varphi \rangle } \sum _ { Q \in \Gamma \backslash \mathcal Q _ { d ^ 2 } } \chi _ d ( Q ) \begin{dcases} \int _ { C _ Q } \varphi ( z ) y ^ { - 1 } \ , | d z | & d > 0 , \\ \int _ { C _ Q } i \partial _ z \varphi ( z ) \ , d z & d < 0 . \end{dcases} \end{align*}


\begin{align*} [ \Phi ( T ) , J x _ 1 J ] = \lim _ i [ A _ i , J x _ 1 J ] \otimes T = 0 , \end{align*}


\begin{align*} \sum _ { Q \in \Gamma \backslash \mathcal Q _ D } \chi _ d ( Q ) \int _ { C _ Q } i \partial _ z \varphi ( z ) y ^ { s } \ , d z = \pi ^ { - s - \frac 1 2 } \sqrt { | d | } \ , \Gamma ( \tfrac s 2 + \tfrac { i r } 2 + \tfrac 3 4 ) \Gamma ( \tfrac s 2 - \tfrac { i r } 2 + \tfrac 3 4 ) L ( s + \tfrac 1 2 , \varphi \times \chi _ d ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : m a x w e l l m o l e c u l e s c u t o f f } \Phi ( | u | ) = 1 , \int _ 0 ^ \pi \Sigma ( \theta ) \dd \theta < + \infty . \end{align*}


\begin{align*} g _ { \beta , T } ( t ) = \frac 1 { \Gamma ( 1 - \beta ) } \left ( ( T - t ) ^ { - \beta } + t ^ { - \beta } \right ) . \end{align*}


\begin{align*} u ( x , t ) & = \pm \eta ( t ) ^ \frac { 2 } { 1 - q } { \sf U } ( \xi ) + o ( \eta ( t ) ^ \frac { 2 } { 1 - q } ) | x | \sim \eta ( t ) , \\ u ( x , t ) & = \pm { \sf U } _ \infty ( x ) + \Theta _ J ( x , t ) + o ( \Theta _ J ( x , t ) ) | x | \sim \sqrt { T - t } ( J \in \N ) . \end{align*}


\begin{align*} \forall t > 0 , \| E \| _ { L ^ 1 \cap L ^ \infty } \leq c ( \mu ) \lambda , \lim _ { \mu \rightarrow \infty } c ( \mu ) = 0 . \end{align*}


\begin{align*} f \left ( \left \{ 1 , \dots , j _ { f } \right \} \right ) = \left \{ 1 , \dots , m \right \} . \end{align*}


\begin{align*} \label [ I e q ] { e q : O m e g a N } \Omega _ N \left ( f ^ N , f \right ) : = W _ p { \left ( f ^ { N } , f ^ { \otimes N } \right ) } \underset { N \to + \infty } { \longrightarrow } 0 . \end{align*}


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


\begin{align*} f ( A ) = \sum _ p f ( A _ p ) & = \sum _ { \substack { p < x \\ p \notin A } } f ( A _ p ) + \sum _ { p \in A , p \ge x } f ( p ) \\ & \le e ^ \gamma \sum _ { \substack { p < x \\ p \notin A } } { \rm d } ( { \rm L } _ p ) + \sum _ { p \in A , p \ge x } f ( p ) \\ & \le e ^ \gamma \sum _ { p } { \rm d } ( { \rm L } _ p ) + \big ( e ^ \gamma + o _ x ( 1 ) \big ) \sum _ { p \ge x } { \rm d } ( { \rm L } _ p ) \ \le \ e ^ \gamma + o _ x ( 1 ) \end{align*}


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


\begin{align*} S _ \tau f _ 0 ( z ) : = f _ \tau ( z ) \geq \eta ( z ) \int _ { \{ \tilde \tau ( z ' ) \leq R \} } f _ 0 ( z ' ) \d z ' . \end{align*}


\begin{align*} \mathfrak { M } ( \xi , v , V ( s ) ) = \frac { \mathbf { e } _ 3 \langle V ( s ) \rangle } { | \xi | ( | \xi | + \hat { V } ( s ) \cdot \xi ) } - \frac { V _ 3 ( s ) } { | \xi | ( | \xi | + \hat { V } ( s ) \cdot \xi ) ^ 2 } \big ( \xi - \hat { V } ( s ) \hat { V } ( s ) \cdot \xi \big ) . \end{align*}


\begin{align*} \bar { \psi } _ \theta ( z ) = \bar { f } _ \theta ( \gamma ) ( \gamma ^ { - 1 } z ) , \end{align*}


\begin{align*} \mu ( ( b , \infty ) ) \geq ( \ln b ) ^ { - a } . \end{align*}


\begin{align*} \mu \left ( \bigcup \limits _ { i = 1 } ^ { n } R _ i \right ) = \sum \limits _ { J \subseteq \{ 1 , \ldots , n \} , J \ne \varnothing } ( - 1 ) ^ { | J | + 1 } \mu \left ( \bigcap \limits _ { j \in J } R _ i \right ) . \end{align*}


\begin{align*} \Delta f = d i v ( \nabla f ) . \end{align*}


\begin{align*} ( { \bf 1 } ^ \epsilon _ 0 f ) ( x ) : = \epsilon ^ { - d } { \bf 1 } _ { x = 0 } f ( 0 ) . \end{align*}


\begin{align*} \varphi _ n ( f ) ( t ) = ( u ( t ) \otimes I _ { 2 ^ n } ) \begin{pmatrix} f ( e ^ { \pi i t } ) & 0 \\ 0 & f ( e ^ { \pi i ( 1 + t ) } ) \end{pmatrix} ( u ( t ) \otimes I _ { 2 ^ n } ) ^ { \ast } \end{align*}


\begin{align*} b _ { i , j } = b _ { j , i } \geq 0 \ \hbox { f o r a n y } \ i , j = 1 , \cdots , n ; b _ { i , i } = - \sum _ { j = 1 , j \neq i } ^ { } b _ { i , j } . \end{align*}


\begin{align*} p ( n ) = \frac { 1 } { 4 \sqrt { 3 } n \cdot \kappa _ n } \left ( 1 - \frac { 1 } { x \sqrt { n \cdot \kappa _ n } } \right ) e ^ { x \sqrt { n \cdot \kappa _ n } } \left ( 1 + O ( e ^ { - x \sqrt { n } / 2 } ) \right ) \end{align*}


\begin{align*} \big ( E ( L ) \big ) ( \nu ) = \frac { \overline { \eta } ( ( 1 , 0 ) L ^ { - 1 } ) } { | \det ( L ) | } \ , g ( \nu ) , L \in K _ 0 \nu \in \R ^ * . \end{align*}


\begin{align*} \nu _ { x _ { m } ^ { r } \rightarrow f _ { m } ^ { r } } ( x _ { m } ^ { r } ) = \mathcal { C N } ( x _ { m } ^ { r } ; x _ { B , m } ^ { r , p r i } , v _ { B , m } ^ { r , p r i } ) . \end{align*}


\begin{align*} z _ { n _ k } ^ * \overset { w ^ * } { \underset { k \rightarrow \infty } { \longrightarrow } } y ^ * - \sum _ { m = 1 } ^ \infty c _ m \widetilde { u } _ m = : z ^ * . \end{align*}


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


\begin{align*} g ( A _ { X _ j } U , A _ { X _ j } V ) = \sum \limits _ { j = 1 } ^ { n } g ( A _ { X _ j } X _ j , U ) g ( A _ { X _ j } X _ j , V ) . \end{align*}


\begin{align*} C ( \lambda , 0 ) = \int F ( \lambda f ) \ p ( x ) \ \mu ( d x ) \leq 2 \ , - 1 \leq \lambda \leq 1 \ . \end{align*}


\begin{align*} \{ e ^ { \pm i n _ k x } : \ , k = 1 , 2 , \ldots \} \end{align*}


\begin{align*} \int _ \Omega \abs { F ' } ^ p d x d y = \int _ \mathbb { D } \abs { f ' } ^ { 2 - p } d x d y . \end{align*}


\begin{align*} f ^ { ( j ) } ( \lambda ) & = 0 \quad \qquad \qquad \\ f ^ { ( p ) } ( \lambda ) & \neq 0 \\ f ^ { ( j ) } ( \lambda ) & = ( j ) _ p g ^ { ( j - p ) } ( \lambda ) . \end{align*}


\begin{align*} g \in L ^ \infty ( \Omega ) \cap W ^ { 1 , p } ( \Omega ) \hbox { $ p - $ h a r m o n i c i n } \Omega , \ g \hbox { n o n - c o n s t a n t u n l e s s } g = 0 . \end{align*}


\begin{align*} \hat { B } ( z ) = \lambda ^ B \frac { z - a ^ B } { 1 - \overline { a ^ B } z } . \end{align*}


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


\begin{align*} - \omega _ 0 ( A _ j ) = \inf \Re \left ( \operatorname { S p e c } ( A _ j ) \right ) . \end{align*}


\begin{align*} L _ { n } ( \dot { x } , x ) = \frac { a _ 1 \dot x } { a _ 2 x + a _ 4 } , \end{align*}


\begin{align*} S _ 1 = \frac { 1 } { d _ 1 \cdots d _ l } \sum \limits _ { r _ j \mid ( u _ j / d _ j ) , 1 \leq j \leq l } \frac { \mu ( r _ 1 ) \cdots \mu ( r _ l ) } { r _ 1 \cdots r _ l } + O _ { \alpha , l } \Big ( \frac { 1 } { \sqrt { n } } \prod \limits _ { 1 \leq j \leq l } \tau _ 2 ( u _ j / d _ j ) \Big ) . \end{align*}


\begin{align*} \chi _ { B _ { 3 R } ^ { d _ { \mathrm { C C } } } ( 0 ) } ( F ^ { ( \iota ) } \psi ) ( \sqrt L ) f & = \chi _ { B _ { 3 R } ^ { d _ { \mathrm { C C } } } ( 0 ) } \bigg ( \sum _ { \ell = - 1 } ^ \iota + \sum _ { \ell = \iota + 1 } ^ \infty \bigg ) F _ \ell ^ { ( \iota ) } ( L , U ) f \\ & = : g _ { \le \iota } + g _ { > \iota } . \end{align*}


\begin{align*} \Sigma ^ * \Sigma - \Sigma ^ * \Sigma \Sigma ^ * \Sigma = \Sigma ^ * \big ( q q ^ * \big ) \Sigma = \Sigma ^ * q ( \Sigma ^ * q ) ^ * = 0 , \end{align*}


\begin{align*} F ( x ) & = \left ( f ( x ) , f \circ S ^ { - 1 } ( x ) , f \circ S ^ { - 2 } ( x ) , \dotsc , f \circ S ^ { - n + 1 } ( x ) \right ) \\ & = \left ( f ( x _ 1 , x _ 2 \ldots , x _ n ) , f ( x _ 2 , \ldots , x _ { n } , x _ 1 ) , \ldots , f ( x _ n , x _ 1 , \ldots , x _ { n - 1 } ) \right ) . \end{align*}


\begin{align*} f ( A ) & = \sum _ { 0 \le i \le k } f ( A _ { ( i ) } ) \ \le \ \frac { e ^ \gamma } { m _ q } \sum _ { 0 \le i \le k } \frac { { \rm d } ( { \rm L } _ { A _ { ( i ) } } ) } { 1 + v _ i } . \end{align*}


\begin{align*} ( \omega - \omega _ { \theta } ) ^ T A _ { \theta } ( \omega - \omega _ { \theta } ) = - ( 1 - \gamma ) | | \Phi ( \omega - \omega _ { \theta } ) | | _ { \rm D } ^ 2 - \gamma | | \Phi ( \omega - \omega _ { \theta } ) | | _ { \rm D i r } ^ 2 , \end{align*}


\begin{align*} f _ i ( b _ 1 \otimes b _ 2 ) & = \begin{cases} b _ 1 \otimes f _ i ( b _ 2 ) & \varepsilon _ i ( b _ 1 ) < \varphi _ i ( b _ 2 ) \\ f _ i ( b _ 1 ) \otimes b _ 2 & \varepsilon _ i ( b _ 1 ) \geq \varphi _ i ( b _ 2 ) \end{cases} \end{align*}


\begin{align*} \rho \left | u _ { i _ 1 , t } - u _ { i _ 2 , t } \right | & = \left | v _ { i _ 1 , t } - v _ { i _ 2 , t } \right | \\ & \stackrel { ( a ) } { > } \epsilon ' _ n v _ { i _ 1 , t } \\ & \stackrel { ( b ) } { > } \lambda \epsilon ' _ n \ ; , \ , \end{align*}


\begin{align*} \sum _ { \varepsilon _ j = 0 , 1 } ( - 1 ) ^ { \varepsilon _ 1 + \ldots + \varepsilon _ l } w _ n \left ( \alpha \oplus \frac { \varepsilon _ 1 } { 2 ^ { k _ 1 } } \oplus \ldots \oplus \frac { \varepsilon _ l } { 2 ^ { k _ l } } \right ) = \left \{ \begin{array} { l l l } 0 & & n \neq m , \\ \pm 2 ^ l & & n = m . \end{array} \right . \end{align*}


\begin{align*} 0 \le \zeta _ j \le 1 , \ \zeta _ j ( x ) = 1 \ , \ \sup _ { j , x } | \nabla \zeta _ j ( x ) | < \infty . \end{align*}


\begin{align*} \{ \lambda \in P _ + \mid \mathcal { C } _ { X ^ + } ^ \lambda \neq 0 \} = \{ \lambda \in P _ + \mid \mathcal { C } _ { X ^ - } ^ \lambda \neq 0 \} = R , \end{align*}


\begin{align*} \varphi ( \lambda _ 1 , \lambda _ 2 ) : = h ( \lambda _ 1 , \lambda _ 2 ) \cdot q _ { l , n } ( \lambda _ 1 , \lambda _ 2 ) . \end{align*}


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


\begin{align*} W _ A = \left \{ w \in W \mid w \left ( { A } \right ) \subset { A } \cup - { A } \right \} . \end{align*}


\begin{align*} c _ { i j } ( t ) : = \frac { 1 } { N ( \lambda _ i ( t ) - \lambda _ j ( t ) ) ^ 2 } . \end{align*}


\begin{align*} \beta _ \mu ^ t = t ^ 2 \cdot \beta _ \mu . \end{align*}


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


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


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


\begin{align*} H = H _ 0 + V \end{align*}


\begin{align*} u _ i \left ( s \right ) = \xi _ i r ( s ) = j \in S , \end{align*}


\begin{align*} { \rm { \Delta } } _ { i j . . . k p } ^ { \left ( n \right ) } = w _ { i j . . . k p } ^ { \left ( n \right ) } \delta \left ( { { t _ 1 } - { t _ 2 } } \right ) . . . \delta \left ( { { t _ 1 } - { t _ n } } \right ) \end{align*}


\begin{align*} \phi ( N \times { \mathbb R } ^ { \dim M - 1 } ) \subset U ^ { + } : = \{ x \in M \mid \operatorname { s g n } ( Z ) > 0 \} . \end{align*}


\begin{align*} Q ( g ) & = Q ( g _ o ) + Q ( g ) - Q ( g _ o ) \\ & = Q ( g _ o ) \circ \left [ I + Q ( g _ o ) ^ { - 1 } \circ \left ( Q ( g ) - Q ( g _ o ) \right ) \right ] , \end{align*}


\begin{align*} t _ { - 1 } = \begin{pmatrix} - 1 \\ & 1 \\ & & - 1 \end{pmatrix} . \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*} & ( - 1 ) ^ { k - 1 } \left ( C _ m ( n ) - \sum _ { j = - k } ^ k ( - 1 ) ^ j u _ m \big ( n - j ( 3 j - 1 ) / 2 \big ) \right ) \geqslant 0 . \end{align*}


\begin{align*} \partial \overline { \mathcal { M } } ( p _ 1 , \ldots p _ k , q ) = & \bigcup \overline { \mathcal { M } ( p _ l , r ) / \R } \times \overline { \mathcal { M } } ( p _ 1 , \ldots p _ { l - 1 } , r , p _ { l + 1 } , \ldots p _ k , q ) \\ & \cup \bigcup \overline { \mathcal { M } } ( p _ 1 , \ldots , p _ j , r , p _ { l + j + 1 } , \ldots p _ k , q ) \times \overline { \mathcal { M } } ( p _ { j + 1 } , \ldots p _ { l + j } , r ) . \end{align*}


\begin{align*} \rho \ , L = \sqrt { \frac { 2 A } { A + G } } L = \sqrt { \frac { 2 L ^ 2 A } { A + G } } \leq \sqrt { L A } \leq A \end{align*}


\begin{align*} \| Q _ n \pi ( a ) - Q _ n \pi ( a ) Q _ n \| & = \| V _ n ^ * P _ n V _ n \pi ( a ) - V _ n ^ * P _ n V _ n \pi ( a ) V _ n ^ * P _ n V _ n \| \\ & = \| V _ n ^ * P _ n \sigma ( a ) V _ n - V _ n ^ * P _ n \sigma ( a ) P _ n V _ n \| \\ & \leq \| P _ n \sigma ( a ) - P _ n \sigma ( a ) P _ n \| \approx 0 , \end{align*}


\begin{align*} \Phi _ R ( x ) : = \eta \left ( \frac { x _ 1 } { R } \right ) \zeta \left ( \frac { x ' } { R ^ { \alpha } } \right ) , x ' = ( x _ 2 , x _ 3 ) , R > 1 , \alpha > 0 , \end{align*}