\begin{align*} ( d / d t ) v + A v = 0 , v \in \R ^ D , \end{align*}


\begin{align*} \delta _ 1 ( X ) = 2 \mu F ( \mu ) - 2 J ( \mu ) \delta _ 2 ( X ) = \mu - 2 J ( m ) , \end{align*}


\begin{align*} ( B \Psi - \Phi ) _ n = - B + 2 B K x _ n - \Phi _ n = 0 , \end{align*}


\begin{align*} | a ( z ) | \geq 1 / 2 , \operatorname { I m } z = 0 . \end{align*}


\begin{align*} \begin{aligned} \int _ { \Gamma _ h ^ { \pm } } e ^ { \rho \cdot \mathbf x } \mathrm d \sigma & = \frac { \Gamma ( 1 ) } { \tau } \frac { 1 } { ( \mathbf d + \mathrm i \mathbf d ^ { \perp } ) \cdot \mathbf { \hat x _ 1 } } - I _ { R _ 1 } + \frac { \Gamma ( 1 ) } { \tau } \frac { 1 } { ( \mathbf d + \mathrm i \mathbf d ^ { \perp } ) \cdot \mathbf { \hat x _ 2 } } - I _ { R _ 2 } , \end{aligned} \end{align*}


\begin{align*} \mathbb { E } \left [ \lvert y _ k \rvert ^ 2 \right ] = a _ c ^ 2 \lvert \mathbf { h } _ k ^ { \textrm { T } } \mathbf { p } _ c \rvert ^ 2 + \sum _ { i = 1 } ^ K a _ i ^ 2 \lvert \mathbf { h } _ k ^ { \textrm { T } } \mathbf { p } _ i \rvert ^ 2 + \sigma _ w ^ 2 . \end{align*}


\begin{align*} \mathcal { W } ^ 2 ( \mu _ 1 , \mu _ 2 ) = \inf \mathbb { E } [ | X - Y | ^ 2 ] . \end{align*}


\begin{align*} P _ n u : = \sum _ { i = 1 } ^ { n } \langle u , e _ i \rangle e _ i . \end{align*}


\begin{align*} A = \left ( \begin{array} { c c } B & C \\ C ^ T & D \end{array} \right ) \end{align*}


\begin{align*} N ( f ) & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert ( 1 - m ) \det ( I - A ^ i M ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert ( 1 - m ) \det ( I - B ^ i C ^ i M ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert ( 1 - m ) \det ( I - C ^ i M ) \rvert . \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Theta ( t , x ) = \int _ \S W _ r ( x - y ) \sin ( \Theta ( t , y ) - \Theta ( t , x ) ) \ \d y . \end{align*}


\begin{align*} \begin{cases} \delta _ { + } ( z ) = ( 1 + \bar { r } _ 1 ( z ) r _ 2 ( z ) ) \delta _ { - } ( z ) , z \in \mathbb { R } \\ \delta _ { \pm } ( z ) \rightarrow 1 \quad | z | \rightarrow \infty . \end{cases} \end{align*}


\begin{align*} \alpha _ t ( x ) : = - R ^ { - 1 } B ( P _ t x + \varphi _ t ^ \xi ) - h ( \mu _ t ) . \end{align*}


\begin{align*} _ { \rm m i n } \left ( a _ j ^ { \left ( o \right ) } \right ) = 2 a _ j ^ { \left ( o \right ) } \left ( a _ j ^ { \left ( o \right ) } \sum _ { i = 1 } ^ { M } \lvert \phi ^ { \left ( i , j \right ) } \rvert ^ 2 - \Re \left \{ \phi ^ { \left ( j , j \right ) } \right \} \right . \left . + \sum _ { q = 1 } ^ { M - 1 } \sum _ { r = q + 1 } ^ { M } f _ { q , r } ^ { \left ( j \right ) } \right ) . \end{align*}


\begin{align*} \nabla _ i A _ { j k } = \nabla _ j A _ { i k } . \end{align*}


\begin{align*} h _ { i j k } = h _ { i k j } . \end{align*}


\begin{align*} u _ j ( x ) & = \int _ { x - r } ^ { x + r } \prod _ { \stackrel { i = 1 } { i \neq j } } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) ( - \partial \eta _ j ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( y ) - \tilde \Psi ( x ) ) \ \d y , \\ q ( x ) & = \int _ { x - r } ^ { x + r } \prod _ { i = 1 } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) \sin ^ { [ n + 1 ] } ( \tilde \Psi ( y ) - \tilde \Psi ( x ) ) ( - \partial \tilde \Psi ( x ) ) \ \d y \end{align*}


\begin{align*} \delta _ \rho G ( y , z ) = - \delta _ \rho h ( y , z ) = \int _ { \partial \Omega } \frac { \partial G ( y , x ) } { \partial \mathbf { n } _ x } \frac { \partial G ( z , x ) } { \partial \mathbf { n } _ x } \rho ( x ) d \sigma _ x , \end{align*}


\begin{align*} \lim _ { \ell \to \infty } \alpha _ \ell = + \infty . \end{align*}


\begin{align*} x = \bigwedge \{ p \in P ( X ) \ : | \ : x \le p \} \end{align*}


\begin{align*} H ( \overline { Q } ) \le \frac 1 n \sum _ { i = 1 } ^ n H ( Q _ i ) . \end{align*}


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


\begin{align*} X _ n : = \{ v \in L ^ 2 ( D ) : v | _ { K } \in P ^ \ell ( K ) , K \in { \mathcal T } _ n \} , \end{align*}


\begin{align*} & \left \vert \int _ { \epsilon } ^ { \infty } r ^ { \alpha } e ^ { - \mu r } \mathrm d r \right \vert \leq \frac { 2 } { \Re \mu } e ^ { \frac { \epsilon } { 2 } \Re { \mu } } , \\ & \int _ { 0 } ^ { \epsilon } r ^ { \alpha } e ^ { - \mu r } \mathrm d r = \frac { \Gamma ( \alpha + 1 ) } { \mu ^ { \alpha + 1 } } + \mathcal O \left ( \frac { 2 } { \Re { \mu } } e ^ { - \frac { \epsilon } { 2 } \Re { \mu } } \right ) , \end{align*}


\begin{align*} \Psi ( r , \xi ) = \Phi ( r ) + \Gamma ( \xi ) \end{align*}


\begin{align*} | ( \mathcal { R } _ \lambda ( \sigma ) v _ T , v _ T ) | = \prod _ { j } \frac { 1 } { | a _ { i _ j + 1 } - a _ { i _ j } | } , \end{align*}


\begin{align*} \begin{aligned} | x ( t ) - x _ a ( t ) | & = | x ( t ) - z ( t ) - x _ a ( t ) + z ( t ) | \\ & \leq | x ( t ) - z ( t ) | + \gamma \bar \epsilon ( L _ r , M _ r , \delta ) . \end{aligned} \end{align*}


\begin{align*} G ( U , x , t ) = ( \nabla _ V \tilde { \eta } ) ( A ( U , x , t ) , x , t ) \end{align*}


\begin{align*} \delta z = - \frac { 1 } { 8 } { F _ z } . \end{align*}


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


\begin{align*} ( - R _ { k \ell j i } + R _ { \ell i k j } ) V _ j V _ \ell = ( R _ { k \ell i j } + R _ { \ell i k j } ) V _ j V _ \ell = - R _ { i k \ell j } V _ j V _ \ell = 0 \end{align*}


\begin{align*} \psi _ \epsilon = \sup _ { \| x - x ' \| < \delta , \| y - y ' \| \le \delta } u ( x ' , y ' ) \end{align*}


\begin{align*} f ( x ) = \log \Gamma ( x ) - \log \Gamma \left ( x + \frac 1 2 \right ) - \log \Gamma \left ( x + \frac 3 2 \right ) \end{align*}


\begin{align*} \mathcal { G } ^ { \mathrm { e n d } } = \left ( \begin{array} { c c } A & g ^ { - 1 } \\ h & A ^ * \end{array} \right ) \end{align*}


\begin{align*} \begin{cases} - \cdot ( K _ H ( x ) \nabla \varPhi ) = \frac { 1 } { \varepsilon ^ 2 } \left ( \varPhi - \left ( \frac { \alpha } { 2 } | x | ^ 2 + \beta \right ) | \ln \varepsilon | \right ) ^ p _ + , \ & x \in B _ { R ^ * } ( 0 ) , \\ \varPhi ( x ) = 0 , \ & x \in \partial B _ { R ^ * } ( 0 ) , \end{cases} \end{align*}


\begin{align*} \begin{aligned} 0 < 2 \sigma _ 2 ( \lambda ( D ^ 2 u ) ) = \ , & 2 \sum _ { 1 \leq i < j \leq n } ( u _ { i i } u _ { j j } - u _ { i j } ^ 2 ) \\ = \ , & ( \Delta u ) ^ 2 - \sum _ i u _ { i i } ^ 2 - \sum _ { i \neq j } u _ { i j } ^ 2 . \end{aligned} \end{align*}


\begin{align*} K ( c _ 1 , \dots , c _ n , 2 m ) = & p _ j \circ H ( c _ 1 , \dots , c _ n , m ) \\ = & p _ j \circ i _ { U _ k } ( c 1 , \dots , c _ n ) = p _ j ( c _ 1 , \dots , c _ n ) . \end{align*}


\begin{align*} G _ 1 ( Q _ 1 , \dots , Q _ n ) \coloneqq \int _ { \R ^ n } f ( x _ 1 , \dots , x _ n ) \prod _ { i = 1 } ^ { n } Q _ i ( d x _ i ) , G _ 2 ( Q _ 1 , \dots , Q _ n ) \coloneqq H ( Q _ 1 \times \cdots \times Q _ n ) . \end{align*}


\begin{align*} \sigma _ \ell = \sigma _ { \tilde X _ \ell } = \int _ { \mathbb S ^ d } | D _ y \tilde X _ \ell | ^ 2 \ , d y . \end{align*}


\begin{align*} C _ 2 = \sum _ { \beta \in F ^ { \times } } \sum _ { X \in S ( \beta ) } \Theta _ { \theta } \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = q ^ 8 ( 1 - q ) ^ 2 ( 1 - q ^ 2 ) . \end{align*}


\begin{align*} a _ { j k } : = \int _ { T } \mathrm { d i v } ( { \boldsymbol \psi } ^ { \mathrm { R T } _ 2 } _ { j } ) \chi _ k \ , d { \boldsymbol x } = \frac { 4 \delta _ { j k } - 3 } { 1 8 0 } j , k = 1 , 2 , 3 , \end{align*}


\begin{align*} ( B \Psi - \Phi ) _ n = - B + 2 B K x _ n - \Phi _ n = 0 , \end{align*}


\begin{align*} \tilde { Q } = \ln \tilde { \kappa } _ 1 - N \ln \nu ^ { n + 1 } . \end{align*}


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


\begin{align*} Z _ { i , j } = X _ { i , j } \ , { \rm c h } _ { \rm r e f } [ B ( m _ i , m _ j , s _ j - s _ i ) ] . \end{align*}


\begin{align*} \partial _ s \rho ( s , u ) - m \cdot \nabla \big ( \rho ^ 2 ( s , u ) \big ) = 0 , \rho ( 0 , u ) = \rho ^ { \rm i n i } ( u ) , \end{align*}


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


\begin{align*} \operatorname { r a n k } E ( M _ { k , \mathrm { i n } } ( N _ { ( k , - ) } ) ) = r . \end{align*}


\begin{align*} \log ( h _ { t } + h ) = \log \sigma _ { n - 1 } + \chi ( x , t ) , \end{align*}


\begin{align*} I _ 1 = y \cdot f ( y - 1 ) + ( y - 1 ) \left ( f ( ( a + 1 ) y + b ) - f ( a y + b ) \right ) \end{align*}


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


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


\begin{align*} C _ m \left ( \bigcup _ { N = 1 } ^ \infty U _ { p , N } ( \sigma ) \right ) = C _ m ( U _ p ( \sigma ) ) = \lim \limits _ { N \to \infty } C _ m ( U _ { p , N } ( \sigma ) ) \end{align*}


\begin{align*} c = c _ - ( x ) - c _ + ( x ) = \frac { 1 } { 2 } \int _ { - \infty } ^ \infty | u _ x | ^ 2 d x , \end{align*}


\begin{align*} v [ j ] \mathrel { : = } v [ j ] + v [ t ] w _ { t } [ j ] \end{align*}


\begin{align*} c _ { k + 1 } - c _ { k } = ( c _ { k + 1 } - c _ { k } ) \left ( \sum _ { i \in \Omega _ t ^ o } a _ i - \sum _ { i \in \Omega _ t ^ e } a _ i \right ) . \end{align*}


\begin{align*} \mathcal K ^ h = \mathcal K _ { \mathbf 0 } ^ h , \ \Gamma _ h = \partial \mathcal K \cap B _ h , \ \Lambda _ h = \mathcal K \cap \partial B _ h . \end{align*}


\begin{align*} \bar \mu _ t ^ { * , t _ 0 , \xi } : = \rho \big ( - R ^ { - 1 } B \mathbb E \big [ \bar U ( t , \bar x _ t ^ { * , t _ 0 , \xi } , \nu _ t ^ { * , t _ 0 , \xi } ) | \mathcal { F } _ t ^ { W ^ 0 } \big ] \big ) \quad \bar \nu _ t ^ { * , t _ 0 , \xi } : = \mathbb E \big [ \bar x _ t ^ { * , t _ 0 , \xi } | \mathcal { F } _ t ^ { W ^ 0 } \big ] . \end{align*}


\begin{align*} F ^ * = \min _ { x } \ \{ F ( x ) \coloneqq f ( x ) + P ( x ) \} , \end{align*}


\begin{align*} \frac { \| \hat x - x \| _ 2 } { \| \hat x \| _ 2 } = O \big ( n ^ 4 \ , { \bf u } \ , \kappa ( A ) \ , { \bf g _ { \rm G E P P } } ( A ) \big ) , \end{align*}


\begin{align*} \mathbb { E } \left [ Z _ { i , j } \right ] = \frac { \pi } { 4 } \| \mathbf { x } \| . \end{align*}


\begin{align*} \begin{cases} \dim H ^ 1 ( S , T _ S ) = 4 0 ; \\ \dim H ^ 2 ( S , T _ S ) = 0 , \end{cases} \end{align*}


\begin{align*} \mathcal { L } _ H \varPhi = W = f _ \varepsilon \left ( \varPhi - \frac { \alpha } { 2 } | x ' | ^ 2 | \ln \varepsilon | \right ) \ \ \ B _ { R ^ * } ( 0 ) , \end{align*}


\begin{align*} \gamma _ t \beta _ t = \frac { \gamma _ { t - 1 } ( 1 - \kappa _ t ) } { ( 1 - \kappa _ { t - 1 } ) } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 3 } { 2 } \gamma _ 0 , \quad \alpha _ t = \frac { \kappa _ { t - 1 } \gamma _ t \beta _ t } { \gamma _ { t - 1 } } = \frac { \kappa _ { t - 1 } ( 1 - \kappa _ { t } ) } { 1 - \kappa _ { t - 1 } } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 1 } { 2 } . \end{align*}


\begin{align*} \begin{aligned} & w ( j ) < w ( i _ 2 ) = w ( l ) w ( j ) > w ( i _ 1 ) i _ 1 < j < i _ 2 , \\ & w ( i _ 1 ) > w ( l + 1 ) > w ( i _ 2 ) = w ( l ) . \end{aligned} \end{align*}


\begin{align*} \int _ { \Omega } \tilde { A } \cdot B \dd x = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x , \end{align*}


\begin{align*} \forall ( x , y ) \in E \times E \ , \ \ b _ { \sigma } ( x , y ) = \sigma ( x ) ( y ) . \end{align*}


\begin{align*} \int _ { \mathcal K _ 0 } e ^ { \eta \cdot \mathbf x } \mathrm d \mathbf x = \frac { \vert \omega _ 1 \land \omega _ 2 \land \omega _ 3 } { ( - \eta \cdot \omega _ 1 ) ( - \eta \cdot \omega _ 2 ) ( - \eta \cdot \omega _ 3 ) } \end{align*}


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


\begin{align*} \begin{cases} \partial _ t u = \Delta u - ( u \cdot \nabla ) u - \nabla p - \nabla \cdot ( \nabla n \otimes \nabla n ) , \\ \nabla \cdot u = 0 , \\ \partial _ t n = \Delta n - ( u \cdot \nabla ) n - \Phi ( n ) , \\ u = 0 \quad \frac { \partial n } { \partial \nu } = 0 \quad \partial \mathcal { O } , \\ u ( 0 ) = u _ 0 , n ( 0 ) = n _ 0 , \end{cases} \end{align*}


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


\begin{align*} \begin{aligned} \left . \dfrac { \partial h ( s ) } { \partial s } \right | _ { s = x + \delta \sin ( 2 \pi t ) } = & \ , \left . \dfrac { \partial h ( s ) } { \partial s } \right | _ x \\ & + \left . \dfrac { \partial ^ 2 h ( s ) } { \partial s ^ 2 } \right | _ { x } \delta \sin ( 2 \pi t ) + O ( \delta ^ 2 ) \\ \end{aligned} \end{align*}


\begin{align*} & \frac { ( n - 2 ) ( 1 0 - n ) } { 4 } \int _ { B _ { \rho } } r ^ { 2 - n } u _ r ^ 2 \d x \\ & \leq C \int _ { B _ { 2 \rho } \setminus B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon \int _ { B _ { 2 \rho } } r ^ { 4 - n } | D ^ 2 u | | \nabla u | \d x \\ & + C \varepsilon \int _ { B _ { 2 \rho } } \big ( r ^ { 3 - n } + \varepsilon r ^ { 4 - n } \big ) | \nabla u | ^ 2 \d x . \end{align*}


\begin{align*} - \Delta u = \lambda u & D , \\ [ 1 m m ] u = 0 & \partial D . \end{align*}


\begin{align*} a ( u , v ) : = ( \triangle u , \Delta v ) . \end{align*}


\begin{align*} V _ { \alpha , \beta , \gamma } = \mathbb { C } [ \partial ] v , L _ \lambda v = ( \partial + \alpha \lambda + \beta ) v , H _ \lambda v = \gamma v . \end{align*}


\begin{align*} T ( g , \rho ) = \sum _ { l \geq 1 } ( U ( l ) - U ( l - 1 ) ) ^ 2 \widehat { g } \left ( \frac { l } { \pi _ N ( x ) } \right ) , \end{align*}


\begin{align*} \begin{aligned} \int u _ k ^ s \Delta u _ 1 ^ t \wedge \dots \wedge \Delta u _ { k - 1 } ^ t \wedge \beta _ n ^ { n - m } \wedge \Delta \omega = \int \Delta u _ 1 ^ t \wedge \dots \wedge \Delta u _ { k - 1 } ^ t \wedge \Delta u _ k ^ s \wedge \beta _ n ^ { n - m } \wedge \omega \geq 0 , \end{aligned} \end{align*}


\begin{align*} \dot { s } \varepsilon ( x ) \dot { s } = \displaystyle \varepsilon ( - x ^ { - 1 } ) \dot { s } h ( - x ) \varepsilon ( - x ^ { - 1 } ) . \end{align*}


\begin{align*} \Im m ( a _ 0 \overline { a _ 1 } ) = \Im m ( a _ 2 \overline { a _ 3 } ) ; \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n \binom n k \frac { F _ { j k - 1 } } { L _ j ^ k } B _ { n - k } = B _ n \Big ( \frac { \alpha ^ j } { L _ j } \Big ) , \qquad \mbox { $ n $ e v e n } , \end{align*}


\begin{align*} ( \widehat { h } ^ i , \widehat { h } ^ i { } _ j , \widehat { h } ^ i { } _ { j k } ) & = ( \phi ( h ^ a ) ^ i , D \phi ( h ^ a ) ^ i { } _ \alpha h ^ \alpha { } _ j , H \phi ( h ^ a ) ^ i { } _ { \alpha \beta } h ^ \alpha { } _ j h ^ \beta { } _ k + D \phi ( h ^ a ) ^ i { } _ \alpha h ^ \alpha { } _ { j k } ) \\ * & = ( \phi ( h ^ a ) ^ i , ( D \phi ( h ^ a ) ^ i { } _ j , H \phi ( h ^ a ) ^ i { } _ { j k } ) ( h ^ i { } _ j , h ^ i { } _ { j k } ) ) , \end{align*}


\begin{align*} \left ( \mathfrak { M } ^ \prime ( f ) \eta \right ) ( x , m ) = f \left ( O ^ m x \right ) \eta ( x , m ) , \ ; \left ( \mathcal { F } ' ( O ^ k ) \eta \right ) ( x , m ) = \eta ( x , m - k ) . \end{align*}


\begin{align*} ( \phi | _ { k , S } \gamma ) ( \tau , z ) : = J _ { k , S } ( \gamma , ( \tau , z ) ) \phi ( \gamma ( \tau , z ) ) , \end{align*}


\begin{align*} \sum _ { K \ \ L } \ \sum _ { \widetilde { K } \ \ \pi ^ { - 1 } ( K ) } ( e _ { \widetilde { K } } - 1 ) \langle [ \widetilde { K } ] , \alpha \rangle = 0 . \end{align*}


\begin{align*} & \sup _ { x \in \left ( - \infty , x _ { 0 } \right ) } \left \| \langle x \rangle \mathcal { P } ^ { + } \left ( z ^ { - i } \bar { r } _ 1 ( z ) e ^ { 2 i z x } \right ) \right \| _ { L _ z ^ 2 } \leq \left \| z ^ { - i } \bar r _ 1 ( z ) \right \| _ { H ^ 1 } , i = 0 , 1 , \\ & \sup _ { x \in \left ( - \infty , x _ { 0 } \right ) } \left \| \langle x \rangle \mathcal { P } ^ - \left ( r _ 2 ( z ) e ^ { - 2 i z x } \right ) \right \| _ { L _ { z } ^ 2 } \leq \left \| r _ 2 ( z ) \right \| _ { H ^ 1 } . \end{align*}


\begin{align*} \lim _ { k \to \infty } \deg _ p ( N _ k ' ) = \lim _ { k \to \infty } \deg _ p ( N _ k '' ) ; \end{align*}


\begin{align*} f ( \lambda _ i ( A ) ) = f ( n + 1 - 2 i ) = \frac { 1 } { 2 n } \prod _ { k = 1 ; k \neq i } ^ { n } \left ( k - i \right ) , \ 1 \leq i \leq n . \end{align*}


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


\begin{align*} D = 3 \left ( \left ( a - b \right ) ^ { 4 } - \left ( a ^ { 4 } + b ^ { 4 } + \left ( a - 1 \right ) ^ { 4 } + \left ( b - 1 \right ) ^ { 4 } \right ) + 1 \right ) \end{align*}


\begin{align*} \begin{aligned} F _ \kappa = \sum _ { n = 0 } ^ \infty \kappa ^ n F _ n \ , , \end{aligned} \end{align*}


\begin{align*} \theta _ 0 ( x ) - x = - \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } \frac { x ^ { s } } { s } \bigg \{ \frac { \zeta ' ( s ) } { \zeta ( s ) } + g ( s ) \bigg \} d t . \end{align*}


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


\begin{align*} \langle u , v \rangle : = \ , \ , ^ { t } u \psi _ { 2 n } v \end{align*}


\begin{align*} G : = { \rm G a l } ( M ' / K ' ) = { \rm G a l } ( M ' / M _ 1 ' ) \times { \rm G a l } ( M ' / M _ 2 ' ) \cong C _ 2 \times C _ 2 \end{align*}


\begin{align*} \begin{aligned} & F ( s ) \lesssim \frac { 1 } { s ^ { \tau } } , 0 < \tau \leq \frac { 3 } { 2 } , \\ & F ^ { ( k ) } ( s ) \lesssim \frac { 1 } { s ^ { k + \tau } } , 0 \leq \tau \leq \frac { 3 } { 2 } , \ k \in \mathbb { N } . \end{aligned} \end{align*}


\begin{align*} \rho ( g ) & = g & g \in G \\ \rho ^ { \vee } ( g ) & = ( g ^ { - 1 } ) ^ { \vee } ( \varphi ) & g \in G . \end{align*}


\begin{align*} \left | \left . \dfrac { \partial \varepsilon _ \delta ( x , \bar { y } _ a , t ) - b _ { 1 , \delta } ( x ) / 2 } { \partial x } \right | _ { x = x _ a } \right | \le 2 L _ r . \end{align*}


\begin{align*} \tilde { \kappa } _ i = \begin{cases} \kappa _ 1 , & i = 1 , \\ \kappa _ i - 1 , & i > 1 . \end{cases} \end{align*}


\begin{align*} \det D ^ { 2 } u = \mu , \end{align*}


\begin{align*} A ^ { 1 , \dots , k } : = A ^ { 1 } A ^ { 2 } \dots A ^ { k } . \end{align*}


\begin{align*} \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi ( \partial _ t \varphi + ( v + u _ { \infty } ) \cdot \nabla \varphi ) \dd x \dd s = - \int _ { \mathbb { R } ^ { 3 } } \Phi _ 0 \varphi _ 0 \dd x , \end{align*}


\begin{align*} \mathbf { T } = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} . \end{align*}


\begin{align*} \alpha = { { \mathbb { E } } \left \{ x \mu _ { \rm a } ( x ) \right \} } / { { \mathbb { E } } \left \{ x ^ 2 \right \} } = { { \mathbb { E } } \left \{ x \mu _ { \rm a } ( x ) \right \} } , \end{align*}


\begin{align*} & \lim _ { m ' \to + \infty } \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ { h _ { m ' } } e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } \\ = & \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } < + \infty . \end{align*}


\begin{align*} \Sigma _ { \mu , \nu , \lambda } : = \left \{ \omega \in \mathcal { A } _ { \mu , \nu } \mid \mathcal { E } _ { \lambda } \left ( \omega \right ) = S _ { \mu , \nu , \lambda } \right \} . \end{align*}


\begin{align*} \mathcal { P } ' \left ( \partial _ Y \Phi ( X ) \right ) = \partial _ Y \left ( \Phi \left ( \mathcal { P } ( X ) \right ) \right ) . \end{align*}


\begin{align*} q _ { n , m } \cdot q _ { l , n } = r ^ { l } _ { n , m } \cdot q _ { l , m } \end{align*}


\begin{align*} C F ( F ) = \widehat { t e l } ( C F ( F ) , i d ) \rightarrow C ^ { L , k } = \varinjlim _ { p } \varprojlim _ { q } \varinjlim _ { n _ { i } } C F ^ { L , k } _ { [ p , q ] } ( H _ { n _ { i } , t } ) . \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) = \lambda _ n ( u _ n , v _ n ) \quad \ , v _ n \in X _ n . \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ g \textbf { \textit { U } } = 0 & \ \Omega , \\ \Lambda _ { g } ( \textbf { \textit { U } } ) = \tau \textbf { \textit { U } } & \ \partial \Omega , \end{cases} \end{align*}


\begin{align*} \langle f \rangle ( a ) = \lim _ { { T ' } \to \infty } \frac { 1 } { { T ' } } \int _ 0 ^ { T ' } f ( \Phi _ { - t \Lambda } a ) d t , a \in \C ^ n . \end{align*}


\begin{align*} v = 0 , \nu ' ( h ) ( \bar u ) + \nu ( v ) = 0 , A ' ( h ) = 0 \mbox { o n } \hat \Sigma , \end{align*}


\begin{align*} \omega _ \epsilon : = \{ x \in \Omega : | x _ \beta | < \epsilon \delta \frac { b _ \alpha } { \sqrt { b _ \beta } } , \beta = \alpha + 1 , \ldots , n - 1 , x _ n < \epsilon ^ 2 \delta ^ 2 b _ \alpha \} \end{align*}


\begin{align*} \sum _ { r _ { Q } ( n ) = 0 } n \ll \max \left \{ \frac { p ^ { 3 + \epsilon } } { ( \min Q ^ { * } ) ^ { 2 } } , p ^ { 5 / 2 + \epsilon } \right \} . \end{align*}


\begin{align*} \Phi ^ g ( x , y ) = ( g _ 1 ( x ) e ^ { y _ 1 } , \dots , g _ m ( x ) e ^ { y _ m } ) \ , . \end{align*}


\begin{align*} m \left | \sum _ { k = 1 } ^ m \frac { \mu ( k ) } { k } \right | \leq \frac { 1 } { 2 } . \end{align*}


\begin{align*} F ^ L = \C [ z _ { s , k } | 1 \leq s \leq m , 1 \leq k \leq L ] \otimes C ^ { \C } ( r _ { t , k } , c _ k | 1 \leq t \leq n , 1 \leq k \leq L ) . \end{align*}


\begin{align*} \omega = \sum _ { i = 1 } ^ n d x ^ i \wedge d \theta _ i . \end{align*}


\begin{align*} \begin{cases} d Y _ t = L _ { z } ^ \perp Y _ t d t + \Pi _ { \mathrm { k e r } A ^ \perp } \sigma d W _ t \\ Y _ t | _ { t = 0 } = Y _ 0 \in \mathrm { k e r } A ^ \perp . \end{cases} \end{align*}


\begin{align*} c _ \pm ( x ) = \frac { 1 } { 2 } \int _ { \pm \infty } ^ x | u _ y ( y , t ) | ^ 2 d y . \end{align*}


\begin{align*} { \rm I m } ( M ( w ) ) = \frac { { \rm I m } ( w ) } { | c w + d | ^ { 2 } } . \end{align*}


\begin{align*} \frac { 1 } { 3 } \frac { d } { d t } \int _ { \Omega } n ^ { 3 } + \frac { 8 } { 9 } \int _ { \Omega } | \nabla n ^ { \frac { 3 } { 2 } } | ^ { 2 } = \frac { 4 } { 3 } \int _ { \Omega } n ^ { \frac { 3 } { 2 } } \nabla n ^ { \frac { 3 } { 2 } } \cdot ( S ( x , n , c ) \cdot \nabla c ) . \end{align*}


\begin{align*} N _ f ( z ) & = \dfrac { 1 - z } { 1 - m z } . \end{align*}


\begin{align*} \begin{aligned} | \zeta _ \shortparallel | ^ 2 \geq \tfrac { 1 } { 2 } | \zeta _ \shortparallel - \tfrac { 1 } { 2 } V _ \shortparallel | ^ 2 - \tfrac { 1 } { 4 } | V _ \shortparallel | ^ 2 \ , . \end{aligned} \end{align*}


\begin{align*} \psi _ { K , H } ( t ) = ( n - 1 ) \sqrt K \cdot \frac { ( n - 1 ) \sqrt K \sinh \sqrt K t + H \cosh \sqrt K t } { ( n - 1 ) \sqrt K \cosh \sqrt K t + H \sinh \sqrt K t } . \end{align*}


\begin{align*} R _ c = \pi / 0 . 0 6 7 \approx 4 6 . 9 . \end{align*}


\begin{align*} \norm { \tfrac { 1 } { p _ n } - \tfrac { 1 } { f } } _ \infty = \norm { \tfrac { p _ n - f } { p _ n f } } _ \infty \leq \tfrac { 2 } { \delta ^ 2 } \| p _ n - f \| _ \infty \to 0 . \end{align*}


\begin{align*} L _ X g ( \nu , \cdot ) = h ( \nu , \cdot ) \mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \Sigma . \end{align*}


\begin{align*} \lambda _ i \neq 0 , i = 1 , \dots , n . \end{align*}


\begin{align*} & \| \alpha _ * u _ * + \lambda x _ v + ( 1 - \lambda ) x _ w \| ^ 2 = \\ & = \alpha _ * ^ 2 \| u _ * \| ^ 2 + \| \lambda x _ v + ( 1 - \lambda ) x _ w \| ^ 2 + 2 \alpha _ * \langle u _ * , \lambda x _ v + ( 1 - \lambda ) x _ w \rangle = \\ & = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 \end{align*}


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


\begin{align*} \sum _ { j = 1 } ^ m \sum _ { h = 1 } ^ 2 C _ { j , h } \int _ { \Omega } \frac { \partial } { \partial z _ { j , h } } ( - \delta ^ 2 ( K ( z _ j ) \nabla V _ { \delta , Z , j } ) ) \frac { \partial V _ { \delta , Z , i } } { \partial z _ { i , \hbar } } = \int _ { \Omega } u \frac { \partial V _ { \delta , Z , i } } { \partial z _ { i , \hbar } } , \ \ \forall i = 1 , \cdots , m , \ \hbar = 1 , 2 . \end{align*}


\begin{align*} \mathcal { F } _ { \mathcal { R } } & = \{ f = \frac { \sum _ { h = 1 } ^ { H } c _ h \chi _ { Q _ h } } { \sum _ { i = 1 } ^ { H } c _ h \nu ( Q _ h ) } : \ ; c _ h \in \mathbb { R } ^ + , \ ; Q _ i \cap Q _ j = \emptyset \} , \end{align*}


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


\begin{align*} \Delta _ { \Psi } X ( u ) = X ( \Delta _ { \Psi } u ) . \end{align*}


\begin{align*} g ( x ^ { \mu } ) : = \sum _ { i _ 1 = 0 } ^ { \infty } \cdots \sum _ { i _ n = 0 } ^ { \infty } \frac { 1 } { i _ 1 ! \cdots i _ n ! } \partial _ 1 ^ { i _ 1 } \cdots \partial _ n ^ { i _ n } g ( \epsilon ( x ^ { \mu } ) ) ( x ^ 1 - \epsilon ( x ^ 1 ) ) ^ { i _ 1 } \cdots ( x ^ n - \epsilon ( x ^ n ) ) ^ { i _ n } . \end{align*}


\begin{align*} h ( s ) = h _ { a , \Delta } ( s ) = \left ( \dfrac { a } { a ^ 2 + s ^ 2 } \right ) \left ( \dfrac { e ^ { 2 \pi a \Delta } + e ^ { - 2 \pi a \Delta } - 2 \cos ( 2 \pi \Delta s ) } { \left ( e ^ { \pi a \Delta } - e ^ { - \pi a \Delta } \right ) ^ 2 } \right ) \end{align*}


\begin{align*} \begin{aligned} \frac { \partial u } { \partial \overline { q _ l } } = \partial _ { x _ { 4 l } } + \mathbf { i } \partial _ { x _ { 4 l + 1 } } + \mathbf { j } \partial _ { x _ { 4 l + 2 } } + \mathbf { k } \partial _ { x _ { 4 l + 3 } } , \end{aligned} \end{align*}


\begin{align*} { f _ i } \left ( { { \bf { x } } + { { \bf { c } } _ i } \Delta t , t + \Delta t } \right ) - { f _ i } \left ( { { \bf { x } } , t } \right ) = { \left ( { { { { \bf { \hat M } } } ^ { - 1 } } { \bf { \hat \Lambda \hat M } } } \right ) _ { i j } } \left [ { { f _ i } \left ( { { \bf { x } } , t } \right ) - f _ i ^ { \left ( { e q } \right ) } \left ( { { \bf { x } } , t } \right ) } \right ] + { { \bf { F } } _ i } \left ( { { \bf { x } } , t } \right ) , \end{align*}


\begin{align*} A _ i f ( x ) : = \sum _ { \substack { y \sim x \\ \tau ( y ) = \tau ( x ) + i } } f ( y ) , \end{align*}


\begin{align*} t = \sum _ { j = 1 } ^ \infty \frac { b _ j } { 2 ^ j } \end{align*}


\begin{align*} \mathcal { H } ^ 2 ( t ; c , f ) : = \Bigg \{ \tilde { f } : \int _ { \{ \Psi < - t \} } | \tilde { f } | ^ 2 _ h c ( - \Psi ) < + \infty , \ \tilde { f } \in H ^ 0 ( \{ \Psi < - t \} , \mathcal { O } ( K _ M \otimes E ) ) \\ \& ( \tilde { f } - f ) _ { z _ 0 } \in \mathcal { O } ( K _ M ) _ { z _ 0 } \otimes J _ { z _ 0 } , z _ 0 \in Z _ 0 \Bigg \} , \end{align*}


\begin{align*} \tilde G ( p , s ) = s ^ { - p / 2 } 2 ^ { 3 p / 2 - 1 } \Gamma ( p / 2 ) D ( p ) \end{align*}


\begin{align*} \Vert M _ { \widetilde { \phi } } ( a _ 1 , \ldots , a _ n ) \Vert _ { S _ p ^ N } \leq \Vert T _ { \phi } ^ { ( N ) } \Vert \Vert a _ 1 \Vert _ { S _ { p _ 1 } ^ N } \ldots \Vert a _ n \Vert _ { S _ { p _ n } ^ N } . \end{align*}


\begin{align*} Z ( v , s ) = \Psi ^ q + \mathcal O ( \norm { ( v , s ) } ) . \end{align*}


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


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


\begin{align*} & \phi _ x + i k ^ 2 \sigma _ 3 \phi = k P _ x \phi , \\ & \phi _ t + i \eta ^ 2 \sigma _ 3 \phi = H \phi , \end{align*}


\begin{align*} | \rho ( x ) | = ( x , \Sigma ) : = \inf \{ d ( x , y ) \mid y \in \Sigma \ , \} . \end{align*}


\begin{align*} F ( u , v ) & = \frac { p } { p \alpha - 1 } \Big ( \alpha a ( u , v ) + ( \alpha - 1 ) b ( u , v ) + ( 3 \alpha - 2 ) c ( u , v ) \Big ) \\ & \geq \frac { p } { p \alpha - 1 } ( \alpha - 1 ) \| ( u , v ) \| _ { E } ^ { 2 } . \end{align*}


\begin{align*} I ' ( u , v ) = \mu J ' ( u , v ) . \end{align*}


\begin{align*} \int _ { \{ \Psi < - t \} } | F _ t + \alpha q | ^ 2 _ h c ( - \Psi ) - \int _ { \{ \psi < - t \} } | F _ t | ^ 2 _ h c ( - \Psi ) \geq 0 \end{align*}


\begin{align*} \begin{pmatrix} \theta ^ 2 & \theta \cdot \sigma \\ \theta \cdot \sigma & \sigma ^ 2 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & - 2 \end{pmatrix} . \end{align*}


\begin{align*} E ^ { 1 } _ { p , q } = H _ { q } ( G , C _ { p } ) \Rightarrow H _ { p + q } ( G , C _ { * } ) \end{align*}


\begin{align*} 0 = \frac { ( \tilde { \kappa } _ 1 ) _ i } { \tilde { \kappa } _ 1 } - N \frac { ( \nu ^ { n + 1 } ) _ i } { \nu ^ { n + 1 } } , \end{align*}


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


\begin{align*} \rho ^ { \lambda } _ { k } ( f ) e ^ { n - 1 } _ { k , \lambda } ( z , t ) = e ^ { i \lambda t } f ^ { - \lambda } \ast _ { - \lambda } \varphi _ { k , \lambda } ^ { n - 1 } ( z ) . \end{align*}


\begin{align*} d = d _ \alpha + b - a _ { \alpha + 1 } + \sum _ { j = 2 } ^ \infty ( a _ j - a _ { \alpha + j } ) \ . \end{align*}


\begin{align*} \displaystyle \int _ { \Omega } n ( \cdot , t ) = \int _ { \Omega } n _ { 0 } , n ( x , t ) \ge 0 , 0 < c ( x , t ) < \gamma \mbox { f o r } x \in \Omega . \end{align*}


\begin{align*} A ( p , s ) = 2 ^ { - p / 2 } \left ( \frac { 3 s } { 2 } - p \right ) \frac { 1 2 s - \left ( \frac { p } { 2 } + 2 \right ) \left ( \frac { p } { 2 } + 3 \right ) } { 1 4 4 } \end{align*}


\begin{align*} \begin{aligned} v ^ { ( 1 ) } = \textup { R e } \ , u ^ { ( 1 ) } & = \big ( 1 , \cos ( \pi / N ) , \cos ( 2 \pi / N ) , \ldots , \cos ( ( N - 1 ) \pi / N ) \big ) \\ v ^ { ( 2 ) } = \textup { I m } \ , u ^ { ( 1 ) } & = \big ( 0 , \sin ( \pi / N ) , \sin ( 2 \pi / N ) , \ldots , \sin ( ( N - 1 ) \pi / N ) \big ) \end{aligned} \end{align*}


\begin{align*} \mathbb B _ { k , S , g } ( \tau , z ) = v ^ { k } e ^ { - 4 \pi S [ y ^ t ] / v } B _ { k , S , g } ( \tau , z ; \tau , z ) . \end{align*}


\begin{align*} \tilde p : = p _ \beta ( \beta ) , \end{align*}


\begin{align*} ( x ) _ k = \frac { \Gamma ( x + k ) } { \Gamma ( x ) } = x ( x + 1 ) \dots ( x + k - 1 ) \end{align*}


\begin{align*} \dot { { x } } _ a = - \gamma \int _ 0 ^ 1 y _ \delta ( { x } _ a , t ) u ( t ) d t \ , . \end{align*}


\begin{align*} \sum _ { r _ { Q } ( n ) = 0 } n \ll p ^ { 1 + \epsilon } m p ^ { - 1 / 4 + \epsilon } \ll p ^ { 5 / 2 + \epsilon } . \end{align*}


\begin{align*} \Delta \rho _ 1 ( x ) \leq \psi _ { K , - H } ( \rho _ 1 ( x ) ) = ( n - 1 ) \sqrt K \cdot \frac { ( n - 1 ) \sqrt K \sinh \sqrt K \rho _ 1 - H \cosh \sqrt K \rho _ 1 } { ( n - 1 ) \sqrt K \cosh \sqrt K \rho _ 1 - H \sinh \sqrt K \rho _ 1 } . \end{align*}


\begin{align*} \begin{aligned} \int _ { \R ^ 3 } | v | ^ k e ^ { - a | v | ^ 2 } \d v = \Big \{ \int _ { | v | \leq 1 } + \int _ { | v | > 1 } \Big \} | v | ^ k e ^ { - a | v | ^ 2 } \d v \ , . \end{aligned} \end{align*}


\begin{align*} \nabla _ V X & = \omega ( V ) \\ \nabla _ V \omega & = - R ( X , V ) + T _ h ( V , \cdot ) . \end{align*}


\begin{align*} \deg _ H ( x ) = \deg _ { H _ i } ( x ) + \deg _ F ( x ) \leq \binom { p - 1 } { k - 1 } + \frac { d } { 2 } \leq \frac { d } { 2 } + \frac { d } { 2 } = d . \end{align*}


\begin{align*} Y _ t - y _ t = \int _ 0 ^ t e ^ { L _ { z _ 0 } ^ \perp ( t - s ) } \Pi _ { \mathrm { k e r } A ^ \perp } ( B ( y _ s , z _ 0 - z _ s ) + B ( z _ 0 - z _ s , y _ s ) - B ( y _ s , y _ s ) + A y _ s ) d s . \end{align*}


\begin{align*} \sum _ { k } \varphi _ k ( d \phi _ { i k } + \Omega _ { i j } ) & = \sum _ k \varphi _ k d \phi _ { i k } + \sum _ l \varphi _ l \phi _ { i l } \wedge \Big ( \sum _ k \varphi _ k \phi _ { k l } \Big ) = \sum _ k \varphi _ k d \phi _ { i k } + \sum _ l \varphi _ l \phi _ { i l } \wedge ( 2 \varphi _ l \phi _ { l l } ) \\ & = \sum _ k \varphi _ k d \phi _ { i k } - \sum _ { l } \phi _ { i l } \wedge d \varphi _ l = d \Big ( \sum _ k \varphi _ k \phi _ { i k } \Big ) = 0 . \end{align*}


\begin{align*} a _ { \ell } = R _ 1 ( \ell ) \beta _ 1 ^ { \ell } + \cdots + R _ s ( \ell ) \beta _ s ^ { \ell } \end{align*}


\begin{gather*} f \in L f \le c P ( c > f + \epsilon ) = 0 \epsilon > 0 . \end{gather*}


\begin{align*} \sum _ { p \leq x } \frac { \chi _ { - 4 } ( p ) ^ { 2 } } { 2 p } = \sum _ { p \leq x } \frac { 1 } { 2 p } = \frac { 1 } { 2 } \log \log x + c + o ( 1 ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 3 \arctan \lambda _ i = \hat \theta _ 0 \in \left ( \pi , \frac { 3 \pi } { 2 } \right ) . \end{align*}


\begin{align*} V ( A , - C ) = V ( A , C ) + \sqrt { | A | \ , | C | } . \end{align*}


\begin{align*} \hat U ( \xi , t ) = \exp ( E ( \xi ) t ) \hat w _ 0 ( \xi ) . \end{align*}


\begin{align*} v ( r , \xi ) = \Delta ^ { N } _ \Gamma u ( r , \xi ) . \end{align*}


\begin{align*} H _ { m i n } \left ( ( E _ 1 [ t ] \times E _ 1 [ t ] ^ { \vee } ) \otimes F \right ) & = H _ { m i n } ( E _ 1 [ t ] \times E _ 1 [ t ] ^ { \vee } ) H _ { m i n } ( F ) \\ & = H _ { m i n } ( E _ 1 [ t ] ) H _ { m i n } ( F ) \\ & = t H _ { m i n } ( E _ 1 ) H _ { m i n } ( F ) . \end{align*}


\begin{align*} ( a b ) ^ i { } _ { j k l } & = a ^ i { } _ { \alpha \beta \gamma } b ^ \alpha { } _ j b ^ \beta { } _ k b ^ \gamma { } _ l + a ^ i { } _ { \alpha \beta } b ^ \alpha { } _ { j l } b ^ \gamma { } _ k + a ^ i { } _ { \alpha \beta } b ^ \alpha { } _ j b ^ \gamma { } _ { k l } + a ^ i { } _ { \alpha \beta } b ^ \beta { } _ l b ^ \gamma { } _ { j k } + a ^ i { } _ \alpha b ^ \alpha { } _ { j k l } \end{align*}


\begin{align*} x _ n ( t ) : = x _ a ( t - n { \bar { t } } / { \gamma } , x ( n { \bar { t } } / { \gamma } ) ) \forall \ , t \in I _ n . \end{align*}


\begin{align*} \partial _ j \varphi _ i = 2 \Gamma ^ i _ { j i } \varphi _ i \quad \forall i \neq j . \end{align*}


\begin{align*} \pi = \pi _ 1 \oplus \pi _ 2 \oplus \pi _ 3 \tau = t s _ 1 \tau _ 1 \oplus ( 1 - t ) s _ 2 \tau _ 2 \oplus ( t ( 1 - s _ 1 ) + ( 1 - t ) ( 1 - s _ 2 ) ) \tau _ 3 . \end{align*}


\begin{align*} \hat { p } ^ i _ { j k } = \frac { \hat { p } ^ i _ j ( B ) \hat { p } ^ i _ k ( C ) } { \hat { p } ^ i ( D ) } . \end{align*}


\begin{align*} F ( X ) = \sum _ { i = 1 } ^ n f _ i ( x _ i ) , ~ G ( Y ) = \sum _ { i = 1 } ^ n g _ i ( y _ i ) . \end{align*}


\begin{align*} u _ N = z + v _ N \end{align*}


\begin{align*} \left | \sum _ { i = 1 } ^ n [ 0 , z _ i ] \right | _ n ^ 2 = \sum _ { | I | = n } ( \det \ U _ I ) ^ 2 . \end{align*}


\begin{align*} { \mathcal { S } } _ 1 = \left \{ U \in L ^ { 2 } _ { \sigma } ( \Omega ) \ \middle | \ \nabla \times U = f _ 1 ^ { + } U , \ \ \int _ { \Omega } \textrm { c u r l } ^ { - 1 } U \cdot U \dd x = 1 \right \} . \end{align*}


\begin{align*} [ q ^ \alpha , q ^ \beta ] = \sum _ { \gamma \in S } { f ^ { \alpha \beta } } _ \gamma q ^ \gamma . \end{align*}


\begin{align*} \forall x \in E , \forall \lambda \in K _ v , \ \sigma _ v ( x \otimes \lambda ) = \sigma ( x ) \otimes \lambda \end{align*}


\begin{align*} \sum _ { k = 1 } ^ \infty \frac { \mu ( k ) } { k } = 0 \ \ \mathrm { a n d } \ \ \sum _ { k = 1 } ^ \infty \frac { \mu ( k ) \log k } { k } = - 1 \end{align*}


\begin{align*} j = j _ k = \Big \lfloor \frac { \omega _ k ( k + 1 ) } { 1 + \omega _ k } \Big \rfloor . \end{align*}


\begin{align*} \delta ^ 2 ( 1 - \epsilon ^ 2 ) b _ { \alpha } u _ n ( x ) \leq u ( y ) - u ( x ) \leq C b _ { \alpha } ^ 2 , \mbox { w i t h } y = ( x ' , \delta ^ 2 b _ \alpha ) \in \partial _ 2 \omega \end{align*}


\begin{align*} d ^ D _ t = \sum _ { u , v : t ( u , v ) = t } \sum _ { s = 0 } ^ { \omega _ { u v } - 1 } d _ { ( u , v , s ) } . \end{align*}


\begin{align*} \langle \lambda , \lambda ' \rangle : = \sum _ { \alpha = 1 } ^ N \lambda _ { \alpha } \lambda ' _ { \alpha } , \quad \ , \lambda = ( \lambda _ { \alpha } ) _ { 1 \le \alpha \le N } , \ , \ , \ , \lambda ' = ( \lambda ' _ { \alpha } ) _ { 1 \le \alpha \le N } \in \mathbb { C } ^ N . \end{align*}


\begin{align*} L u : = \big ( a _ { j k } ^ { \alpha \beta } \partial _ j \partial _ k u _ \beta \big ) _ { 1 \leq \alpha \leq N } , \end{align*}


\begin{align*} [ X _ 1 ] = ( \frac { \mu } { 2 } d \overline { z } - \frac { 2 } { \mu } h _ z ^ 2 d z + \frac { \tau _ 0 } { 2 \mu } d \overline { z } ) J - \frac { 1 } { \mu } d h \sqrt { \tau _ 0 } I . \end{align*}


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


\begin{align*} u = X \cdot \mathbf { e } , \nu ^ { n + 1 } = \mathbf { e } \cdot \nu , \end{align*}


\begin{align*} N = n + \frac { n ( n - 1 ) } { 2 } , \end{align*}


\begin{align*} \xi _ 2 = \kappa _ 0 \xi _ 1 ^ 2 + O ( | \xi _ 1 | ^ 3 ) \mbox { a s } \xi _ 1 \rightarrow 0 \end{align*}


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


\begin{align*} J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \ge J ( x ^ * ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ^ * ) = J ( x ^ * ) \ \ \forall \ , x \in \mathbb { R } ^ n . \end{align*}


\begin{align*} V ( r ) = \frac { L ( L + 1 ) } { r ^ 2 } - \frac { Q } { r } f + \kappa B _ 1 \frac { r } { f } + \kappa B _ 2 \frac { 1 } { f ^ 2 } , B _ 2 > 0 , \end{align*}


\begin{align*} \mathcal { J } ' ( \overline { u } ; h ) & = \partial _ { y } J ( \overline { y } , \overline { u } ) S ' ( \overline { u } ; h ) + \partial _ { u } J ( \overline { y } , \overline { u } ) h \\ & = \langle \overline { y } - g , S ' ( \overline { u } ; h ) \rangle + \alpha \langle \overline { u } , h \rangle , \end{align*}


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


\begin{align*} \phi \rightarrow \phi e ^ { i \chi } , \ \ A _ { \mu } = A _ { \mu } - \partial _ { \mu } \chi , \end{align*}


\begin{align*} M _ { I } = \bigoplus _ { J \subset I } R _ { J } , \end{align*}


\begin{align*} \begin{aligned} v ^ { ( 1 ) } = \textup { R e } \ , u ^ { ( 1 ) } & = \big ( 1 , \cos ( \pi / N ) , \cos ( 2 \pi / N ) , \ldots , \cos ( ( N - 1 ) \pi / N ) \big ) \\ v ^ { ( 2 ) } = \textup { I m } \ , u ^ { ( 1 ) } & = \big ( 0 , \sin ( \pi / N ) , \sin ( 2 \pi / N ) , \ldots , \sin ( ( N - 1 ) \pi / N ) \big ) \end{aligned} \end{align*}


\begin{align*} I _ { h , W , \gamma } = \left ( \frac { W } { 2 } \right ) ^ { - 2 } I _ { \tilde { h } , 2 , \tilde { \gamma } } . \end{align*}


\begin{align*} W = \bigoplus _ { \alpha = 1 } ^ r \Omega ^ { a ( \alpha ) } \end{align*}


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


\begin{align*} \sum \limits _ { i = j + 1 } ^ { n - 1 } - k _ i \geq \sum \limits _ { i = j + 1 } ^ { n - 1 } - r = - ( n - j - 1 ) r . \end{align*}


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


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


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


\begin{align*} \hat { A } ( y | x ) = A ( x ) \ , \ \forall \ ( y | x ) \in \hat { \Omega } \ . \end{align*}


\begin{align*} \left \| x \right \| _ { - 1 } : = \left \| A ^ { - 1 } x \right \| , x \in X \end{align*}


\begin{align*} \gamma \alpha \omega - \gamma \omega = \dfrac { ( \alpha \omega - \omega ) \det ( \gamma ) } { ( c \omega + d ) ( c \alpha \omega + d ) } . \end{align*}


\begin{align*} d u = \Delta _ p u \ , d t + \nabla u \circ d W , u ( 0 , \cdot ) = u _ 0 , \end{align*}


\begin{align*} \xi ' = - L - 2 , \eta ' = \frac { Q } { 2 ( L + 2 ) } , \zeta ' = \frac { \kappa ( L + 2 ) B _ 1 } { Q } , \end{align*}


\begin{align*} F _ n ( \eta ) = T _ n - \frac { 1 } { \eta } I _ n . \end{align*}


\begin{align*} | | B _ q ( \psi ) | | _ { M L _ q ( \mathbb { C } ) } = | | \psi | | _ { L ^ 2 ( \mathbb { R } ) } . \end{align*}


\begin{align*} I _ x = \bar { x } x = x \bar { x } . \end{align*}


\begin{align*} x _ { 2 i , n - j } + x _ { 2 i , n - j + 1 } = a _ j , \mbox { a n d } \end{align*}


\begin{align*} L ( x , \gamma ) : = J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) = x ^ T A ( \gamma ) x + 2 b ( \gamma ) ^ T x + c ( \gamma ) \end{align*}


\begin{align*} X _ { i ; j k } - X _ { i ; k j } & = R _ { k j \ell i } X ^ \ell \\ X _ { j ; k i } - X _ { j ; i k } & = R _ { i k \ell j } X ^ \ell \\ X _ { k ; i j } - X _ { k ; j i } & = R _ { j i \ell k } X ^ \ell . \end{align*}


\begin{align*} j ( \tau , s ) = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } ( U + \tau \tilde { U } + s U _ 0 ) \cdot ( U + \tau \tilde { U } + s U _ 0 ) \dd x . \end{align*}


\begin{align*} \varphi ( t ) : = \mathrm { K L } \left ( \rho _ { t } \mid \pi \right ) \end{align*}


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


\begin{align*} x _ { \rm c } [ n ] = \begin{cases} \kappa _ { \rm t } , & \mathrm { i f } x [ n ] \geq \kappa _ { \rm t } , \\ x [ n ] , & \mathrm { i f } \kappa _ { \rm b } < x [ n ] < \kappa _ { \rm t } , \\ \kappa _ { \rm b } , & \mathrm { i f } x [ n ] \leq \kappa _ { \rm b } , \end{cases} \end{align*}


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


\begin{align*} A = \frac { N ^ { \frac 1 d } } { ( \log N ) ^ { \frac 1 { 1 6 d } } } , R = 1 , T = \frac 1 { N ( \log N ) ^ { \frac 1 { 1 6 } } } . \end{align*}


\begin{align*} ( S _ { \omega } ) ^ { \delta } : = \{ u \in H _ { r } ^ { 1 } ( \R ^ { 2 } ) | u = \widetilde { u } + \overline { u } , \widetilde { u } \in S _ { \omega } , \| \overline { u } \| _ { H _ { r } ^ { 1 } ( \R ^ { 2 } ) } \leq \delta \} , \end{align*}


\begin{align*} \nabla _ i u : = \nabla _ { e _ i } u = e _ i ^ j \partial _ j u = e _ i ^ j u _ j \end{align*}


\begin{align*} \hat M = ( M \setminus \Omega ) \cup \overline U \end{align*}


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


\begin{align*} i \leq N _ v / 2 , \ \beta _ i ^ v = \beta _ i ^ { v _ l } \oplus \beta _ i ^ { v _ r } , i > N _ v / 2 , \ \beta _ i ^ v = \beta ^ { v _ r } _ { i - N _ v / 2 } . \end{align*}


\begin{align*} ( L _ { X } g ) ( Y , Z ) = g ( \nabla _ { Y } X , Z ) + g ( \nabla _ { Z } X , Y ) = 0 . \end{align*}


\begin{align*} \mathcal { C } _ n ^ { ( k ) } \ = \ A _ k F _ n + B _ k F _ { n + 1 } - \sum _ { j = 0 } ^ k \binom { k } { j } B _ j n ^ { k - j } . \end{align*}


\begin{align*} f \vee g = \frac { 1 } { 2 } \big ( f + g + | f - g | \big ) , \end{align*}


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


\begin{align*} \vartheta : = \ , \mathbf { A } \ , , \qquad \textbf { A n t i } \ , \vartheta = \mathbf { A } \ , , \end{align*}


\begin{align*} \begin{aligned} & A _ { b e n } y '' _ { b e n } + M _ { b e n } y _ { b e n } = E y _ { b e n } \\ & A _ { n b } y '' _ { n b } + M _ { n b } y _ { n b } = E y _ { n b } . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \geq \underline { \delta } ^ \star ( \delta ) & \forall \ , x \geq x ^ \star + \delta \\ \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \leq - \underline { \delta } ^ \star ( \delta ) & \forall \ , x \leq x ^ \star - \delta \ , . \end{aligned} \end{align*}


\begin{align*} K _ { V , I } = \sum _ { J \subset I } \kappa _ { V , J } . \end{align*}


\begin{align*} \rho ^ { \circ } ( A ) = \inf \{ t > 0 : A \mbox { i s c o v e r e d b y f i n i t e l y m a n y s u b s e t s o f s i z e } \leqslant t \} , \end{align*}


\begin{align*} [ a _ { ( m ) } , b _ { ( n ) } ] = \sum _ { j \in \mathbb { Z _ { + } } } \left ( \begin{array} { c c c } m \\ j \end{array} \right ) ( a _ { ( j ) } b ) _ { ( m + n - j ) } . \end{align*}


\begin{align*} & \psi \equiv 1 \ ; \ ; o n Q _ 0 a n d s u p p ( \psi ) \subset Q , \\ [ 2 m m ] & \phi \equiv 1 \ ; \ ; o n \ ; \ ; Q _ 1 a n d s u p p ( \phi ) \subset Q _ 0 \subset Q . \end{align*}


\begin{align*} \tilde { \xi } ( g \gamma ) = \tilde { \xi } ( g ) , \qquad g \in G ^ \circ , \gamma \in \Gamma ^ \circ , \end{align*}


\begin{align*} V & = \Big ( Z ( T ' , M _ n ) + e _ 1 M _ n ( F ) e _ 3 + e _ 3 M _ n ( F ) e _ 2 \Big ) \cap \Lambda ^ \perp \\ & = \left ( Z ( T ' , M _ n ) \cap \Lambda ^ \perp \right ) + e _ 1 M _ n ( F ) e _ 3 + e _ 3 M _ n ( F ) e _ 2 \end{align*}


\begin{align*} & \sum _ { j = 0 } ^ { \infty } r _ { j } ^ { 2 - n } \int _ { B _ { r _ { j + 1 } } \setminus B _ { r _ { j + 2 } } } | \nabla u | ^ 2 \d x \\ & \leq C \int _ { B _ { 1 } } r ^ { 2 - n } u _ r ^ 2 \d x + C \varepsilon ^ 2 \int _ { B _ { 1 } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \varepsilon \leq \varepsilon _ 0 . \end{align*}


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


\begin{align*} \partial _ Y \left ( \Phi ( \mathcal { P } ( X ) ) \right ) = \Phi ( \nabla _ Y \mathcal { P } ( X ) ) - \sum _ { i = 1 , 2 } \sqrt { c _ i } \ \langle Y _ i , \mathcal { P } ( X ) \rangle N _ i \\ + \Phi ( B ( Y , \mathcal { P } ( X ) _ T ) - B ^ * ( Y , \mathcal { P } ( X ) _ N ) ) \end{align*}


\begin{align*} \tau _ { n , H } ^ M : = \inf \{ t \geq 0 : \Vert Y _ n ( t ) \Vert _ H > M \} \wedge T . \end{align*}


\begin{align*} d g _ 1 '' = ( \log \sqrt { \tau _ 0 } ) _ z d z \ g '' _ 1 + e ^ { 2 i \theta } \frac { 1 } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) \ J \widehat { g _ 1 '' } I ; \end{align*}


\begin{align*} y ^ q ( x ^ { q + 1 } + d ) = ( x + d ) . \end{align*}


\begin{align*} \begin{aligned} f ( t ) \le & \ , f ( 0 ) + t \gamma ^ 2 \left ( \bar { k } _ 3 ( L _ r , \delta ) + \bar { k } _ 2 ( L _ r , \delta ) \right ) \\ & + \gamma A \int _ 0 ^ t f ( \tau ) \ , d \tau , \end{aligned} \end{align*}


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


\begin{align*} 2 \widetilde y _ n & = 2 e ^ { r n ( A + a ) + r A } \\ & > ( e ^ { - r A } + e ^ { r a } ) e ^ { r n ( A + a ) + r A } \\ & = e ^ { r n ( A + a ) } + e ^ { r ( n + 1 ) ( A + a ) } \\ & = \widetilde x _ n + \widetilde x _ { n + 1 } . \end{align*}


\begin{align*} \xi _ p ( V ) = \left ( \sum _ { l \ge n } ( l + 2 ) ( l + n + 3 ) ( l - n + 1 ) a _ { l + 3 } z ^ l \right ) d z ^ { \otimes 2 } \end{align*}


\begin{align*} h _ d ( q ) = \log ( c _ { d , 2 } ( q ) ^ q ) - \log ( c _ { d , \infty } ( q ) ^ q ) . \end{align*}


\begin{align*} a ( u , v ) = \lambda ( u , v ) ~ v \in H ^ 1 _ 0 ( D ) , \end{align*}


\begin{align*} p \Delta \left ( H - \frac { \mu } { 2 } \right ) ^ { p - 1 } + 2 p \left ( H - \frac { \mu } { 2 } \right ) ^ { p - 1 } \left ( 2 H ^ 2 - K \right ) - 4 H \left ( \left ( H - \frac { \mu } { 2 } \right ) ^ { p } + \varsigma \right ) = 0 , \end{align*}


\begin{align*} p ( 1 v , 0 w ) = p ( v 1 , w \bullet ) = \psi \left ( d _ { - } ^ { \ell + 1 } L ( v 1 , w \bullet ) \right ) . \end{align*}


\begin{align*} \sigma _ \ell = \sigma _ { \tilde X _ \ell } = \int _ { \mathbb S ^ d } | D _ y \tilde X _ \ell | ^ 2 \ , d y . \end{align*}


\begin{align*} \begin{aligned} Z _ { I } ( 1 , \{ 2 \} ^ { n } ; ( \alpha , \beta ) ) = Z _ { I I } ( \{ 2 \} ^ { n - 1 } , 3 ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \Gamma ^ k _ { i j } = 0 \quad \forall i \neq j \neq k \neq i , \end{align*}


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


\begin{align*} \widehat { \iota } ( D ) = \frac { \beta m + \gamma } { m + b _ 1 + 1 } . \end{align*}


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


\begin{align*} \begin{aligned} \abs { c _ 2 ( S ^ 2 \setminus S _ i ) } \le \abs { c _ 1 ( S ^ 1 \setminus S _ i ) } + \abs { c _ 2 ( P ^ 2 ) } & \le \abs { S } - \abs { S \cap S _ i } + q - q _ i + s _ 2 \\ & = ( 2 k - t ' - p ) - \abs { S \cap S _ i } + q - q _ i + ( p - q - s _ 1 ) \\ & = 2 k - t ' - s _ 1 - \abs { S \cap S _ i } - q _ i . \end{aligned} \end{align*}


\begin{align*} \nu _ { y _ 2 } = \mu * \delta _ { s ( y _ 2 ) } = \mu * \delta _ { g k _ 0 s ( y _ 1 ) } = g \left ( \mu * k _ 0 \delta _ { s ( y _ 1 ) } \right ) = g \nu _ { y _ 1 } . \end{align*}


\begin{align*} L K ^ L _ { \cdot \beta } = 0 \ , \ , \ , \ , \R ^ n _ { + } \beta \in \{ 1 , \dots , N \} , \end{align*}


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


\begin{align*} R _ { \beta } : = L _ { T ' } , \end{align*}


\begin{align*} \epsilon _ r P ( x _ 1 , \ldots , x _ n ) = P ( v _ 1 , \ldots , v _ n ) . \end{align*}


\begin{align*} \mathcal { P } ' \left ( \partial _ Y \Phi ( X ) \right ) = \partial _ Y \left ( \Phi \left ( \mathcal { P } ( X ) \right ) \right ) . \end{align*}


\begin{align*} A ^ { 1 } A ^ { 2 } : = \begin{bmatrix} A ^ { 1 } \\ \hline A ^ { 2 } \end{bmatrix} . \end{align*}


\begin{align*} \log L ( s , \rho ) & = \log \left ( \prod _ { v } \det ( 1 - \rho ( \mathrm { F r o b } _ { v } ) \mathrm { N } ( v ) ^ { - s } ) ^ { - 1 } \right ) \\ & = \sum _ { \ell = 1 } ^ { \infty } \frac { 1 } { \ell } \left ( \sum _ { \deg ( v ) \mid \ell } \deg ( v ) \mathrm { t r } \left ( \rho ( \mathrm { F r o b } _ { v } ) ^ { \frac { \ell } { \deg ( v ) } } \right ) \right ) q ^ { - \ell s } . \end{align*}


\begin{align*} T ( \delta _ \xi ) = \frac { \delta _ \xi } { \sqrt { 1 + \gamma ( \xi ) ^ 2 } } . \end{align*}


\begin{align*} \tilde { A } _ n ( \ell ) a _ { \ell } = \tilde { Q } _ { n , 1 } ( \ell ) \tilde { \beta } _ { n , 1 } ^ { \ell } + \cdots + \tilde { Q } _ { n , \tilde { s } ( n ) } ( \ell ) \tilde { \beta } _ { n , \tilde { s } ( n ) } ^ { \ell } \end{align*}


\begin{align*} S H ^ { * } _ { M } ( K ; \Lambda _ { 0 } ) : = H ^ { * } ( \widehat { t e l } ( \mathcal { C } ; \Lambda _ { 0 } ) ) . \end{align*}


\begin{align*} ( L _ { X } g ) ( Y , Z ) = g ( \nabla _ { Y } X , Z ) + g ( \nabla _ { Z } X , Y ) = 0 . \end{align*}


\begin{align*} S _ { h } = \mathcal { K } \cap B _ { h } , \ \Gamma _ { h } ^ { \pm } = \Gamma ^ { \pm } \cap B _ { h } , \ \Lambda _ { h } = \mathcal { K } \cap \partial B _ { h } . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } ( \sigma _ { n } ^ M < T ) = 0 . \end{align*}


\begin{align*} \frac { 1 } { ( 1 + \varepsilon ) ^ { j - 1 } } \frac { 1 } { E ( t _ { i _ { j } } ) } ( i _ { j + 1 } - i _ j ) = \Omega \left ( \frac { 1 } { j ^ 2 } \right ) . \end{align*}


\begin{align*} [ v ] _ { B ^ { 1 + \gamma } _ { p , \infty } ( B _ \rho ) } = \displaystyle \sup _ { h \in \mathbb { R } ^ { n } } \biggl ( \displaystyle \int _ { B _ \rho } \dfrac { | \tau ^ 2 _ h v ( x ) | ^ { p } } { | h | ^ { ( 1 + \gamma ) p } } d x \biggr ) ^ { \frac { 1 } { p } } . \end{align*}


\begin{align*} \sum \limits _ { i = j + 1 } ^ n \chi _ i \leq ( \sum \limits _ { i = j + 1 } ^ n w _ i ) \chi + \sum \limits _ { i = j + 1 } ^ { n - 1 } k _ i + ( n - j ) r . \end{align*}


\begin{align*} X _ i = 0 , \omega _ { i j } V _ i V _ j = \tfrac { 1 } { 2 } h _ { i j } V _ i V _ j , \omega _ { i a } V _ i = h _ { i a } V _ i , \omega _ { a b } = 0 \end{align*}


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


\begin{align*} \sum _ { i = 1 } ^ M \mathbb { E } \left [ y _ i ^ * y _ i | \right . \left . \hat { \mathbf { H } } \right ] = & \sum _ { j = 1 } ^ { M } a _ j ^ 2 \left ( \sum _ { l = 1 } ^ M \lvert \hat { \phi } ^ { \left ( l , j \right ) } \rvert ^ 2 + M \sigma _ { e _ i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 \right ) + a _ c ^ 2 \left ( \sum _ { i = 1 } ^ M \lvert \hat { \phi } ^ { \left ( i , c \right ) } \rvert ^ 2 + M \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ c \rVert ^ 2 \right ) + M \sigma _ n ^ 2 . \end{align*}


\begin{align*} - K _ X = ( n - 1 ) A _ X , \end{align*}


\begin{align*} \frac { \partial u } { \partial t } = \mathcal { L } ( u ) + f ( t , x ) , \mathcal { B } ( u ) = g ( t , x ) , \end{align*}


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


\begin{align*} a _ { n + 1 } = a _ { \alpha + n + 1 } + d _ { \alpha + n + 1 } - d _ { \alpha + n } \iff a _ { \alpha + n + 1 } - a _ { n + 1 } = d _ { \alpha + n } - d _ { \alpha + n + 1 } \ , \ \forall \ n \in \N \ . \end{align*}


\begin{align*} B = \begin{bmatrix} F & M \\ W & Q \end{bmatrix} , \end{align*}


\begin{align*} - \Delta u = F \quad \Omega , u = g \quad \partial \Omega \end{align*}


\begin{align*} \int _ { 0 } ^ { T } \sum _ { j = 1 } ^ { \infty } V _ { i , j } \psi _ j ( t ) d t < \infty , \end{align*}


\begin{align*} H _ 2 = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 3 } } \Theta \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = \theta ( a ) q ^ 8 ( 1 - q ) ( 1 - q ^ 2 ) . \end{align*}


\begin{align*} ( u _ { h } = \mathrm { p r } _ { \underline { S } } \colon \underline { S } \times _ { h , \mathfrak { Z } , u } \mathfrak { U } _ { X } \to \underline { S } , \Theta _ { u , h } = \mathrm { i d } _ { \underline { S } _ { \mathbb { C } } } \times _ { \mathfrak { Z } _ { \mathbb { C } } } \Theta _ { u } ) \end{align*}


\begin{align*} & \partial _ { t } Q - \Theta \sigma ^ { i j } _ { k } \nabla _ { i j } Q \\ & = \frac { 1 } { P } ( \partial _ { t } P - \Theta \sigma ^ { i j } _ { k } \nabla _ { i j } P ) - A \left [ \partial _ { t } \left ( \frac { \rho ^ { 2 } } { 2 } \right ) - \Theta \sigma ^ { i j } _ { k } \nabla _ { i j } \left ( \frac { \rho ^ { 2 } } { 2 } \right ) \right ] + \frac { \Theta } { P ^ { 2 } } \sigma ^ { i j } _ { k } \nabla _ { i } P \nabla _ { j } P . \end{align*}


\begin{align*} \begin{aligned} f _ n = K i n e t i c ( n ) + F l u i d ( n ) \ , , \end{aligned} \end{align*}


\begin{align*} \begin{cases} c _ { \alpha + n + 1 } - c _ { n + 1 } = b _ { \alpha + n } - b _ { \alpha + n + 1 } \\ d _ n = c _ { \alpha + 1 } + ( b _ { \alpha + 1 } - b _ \alpha ) = d \\ b _ { \alpha + n } = b _ \alpha + ( d _ 1 - c _ { \alpha + 1 } ) + \sum _ { j = 2 } ^ n ( c _ j - c _ { \alpha + j } ) \\ b = b _ \alpha + d - c _ { \alpha + 1 } + \sum _ { j = 2 } ^ \infty ( c _ j - c _ { \alpha + j } ) \end{cases} \end{align*}


\begin{align*} - \epsilon ^ 2 \delta ^ 2 b _ { \alpha } u _ n ( x ) \leq u ( 0 ) - u ( x ) + \sum _ { \beta = 1 } ^ { n - 1 } x _ \beta u _ \beta ( x ) \leq C b _ \alpha ^ 2 \end{align*}


\begin{align*} [ \sqrt { - 1 } \theta \wedge \bar { \theta } , \Lambda _ \omega ] \alpha = \bar { \theta } \wedge ( \alpha \llcorner ( \bar { \theta } ) ^ \sharp ) , \end{align*}


\begin{align*} \hat \Phi ^ \dagger ( 0 , 0 ) & = 0 \end{align*}


\begin{align*} \mathcal { D } + \mathcal { E } = 0 . \end{align*}


\begin{align*} ( f _ { I _ 1 } ( \mathbf { x } ) , f _ { I _ 2 } ( \mathbf { x } ) , \cdots , f _ { I _ s } ( \mathbf { x } ) ) = ( f _ { J _ 1 } ( \mathbf { y } ) , f _ { J _ 2 } ( \mathbf { y } ) , \cdots , f _ { J _ s } ( \mathbf { y } ) ) . \end{align*}


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


\begin{align*} \phi ( z , r ) \longmapsto \varphi ( y ) = \frac { \phi ( z , r ) } { r ^ { 2 } } , y = { } ^ { t } ( y _ 1 , y ' ) , \ y _ 1 = z , \ | y ' | = r , \end{align*}


\begin{align*} \varrho \ , { u } _ { i , t t } & = \sigma _ { j i , j } , \ \ , i = 1 , 2 , 3 , \\ \varrho \ , j \ , { \vartheta } _ { i , t t } & = m _ { j i , j } ^ * + \epsilon _ { i j k } \ , \sigma _ { j k } , i = 1 , 2 , 3 , \end{align*}


\begin{align*} q - h \ge \log _ { \theta } \Big ( 1 + \frac { d } { \epsilon _ 0 } \Big ) = \log _ { \theta } ( 1 + b _ 0 d ) ~ { \rm f o r } ~ h = 0 , 1 , \dots , q - 1 . \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Theta ( t , x ) = \int _ \S \int _ \S W _ r ( z - x ) W _ r ( y - x ) \sin ( \Theta ( t , z ) + \Theta ( t , y ) - 2 \Theta ( t , x ) ) \ \d y \d z \end{align*}


\begin{align*} \langle \nabla f _ { \delta } ( x ^ g _ k ) - \nabla f ( x ^ * ) , x _ f ^ k - x _ g ^ k \rangle & \leq f ( x _ f ^ k ) - f _ { \delta } ( x _ g ^ k ) - \langle \nabla f ( x ^ * ) , x _ f ^ k - x _ g ^ k \rangle \\ & \leq f ( x _ f ^ k ) - f ( x _ g ^ k ) - \langle \nabla f ( x ^ * ) , x _ f ^ k - x _ g ^ k \rangle + \delta _ x \\ & = B _ f ( x _ f ^ k , x ^ * ) - B _ f ( x _ g ^ k ) + \delta _ x . \end{align*}


\begin{align*} ( d _ { i } y ) ( [ \mathbf { a } ] \otimes [ \mathbf { b } ] ) = y ( d ^ i [ \mathbf { a } ] \otimes [ \mathbf { b } ] ) = 0 , \end{align*}


\begin{align*} \dot { s } ( e - \varepsilon ( y ) ) \lambda _ i = \sum _ { j , \mu } g ^ i _ { j , \mu } ( e - \varepsilon ( y ^ { - 1 } x ^ i _ { j , \mu } ) ) \lambda _ j . \end{align*}


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


\begin{align*} ( \partial _ v \Gamma _ { u v } ^ v - 2 \Gamma _ { u v } ^ v \Gamma _ { v u } ^ v ) \tau = \partial _ u \Gamma _ { v u } ^ u - 2 \Gamma _ { u v } ^ u \Gamma _ { v u } ^ v . \end{align*}


\begin{align*} I _ f ( p ' , q ' ) \geq \sum _ { j = 0 } ^ { D - 1 } q '' _ j f \left ( \frac { p '' _ j } { q '' _ j } \right ) . \end{align*}


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


\begin{align*} \overline { T } _ { n } [ { \bf \Lambda } ] = \lambda S _ n [ 1 ] , \end{align*}


\begin{align*} \tilde X _ { n + 1 } ^ i = \tilde X _ n ^ i + b ( \tilde X _ n ^ i ) \tau + \gamma ^ i ( \tilde X _ n ) \tau + \sigma W _ \tau ^ i , \end{align*}


\begin{align*} \nabla _ Y \mathcal { P } ( X ) = \mathcal { P } ( B ( Y , X ) ) - B ( Y , f ( X ) ) + B ^ * ( Y , h ( X ) ) . \end{align*}


\begin{align*} \ddot { x } + \frac { 3 } { t } \dot { x } + \nabla f ( x ) = 0 \end{align*}


\begin{align*} \lambda _ 0 = 1 , \lambda _ t = \prod _ { i = 1 } ^ { t } ( 1 - \alpha _ i ) . \end{align*}


\begin{align*} c _ 1 = & \frac { | z _ 1 | ^ 4 } { | | z _ 1 | ^ 2 - \sin \theta _ \xi \cos \theta _ \xi + \mathrm i \cos \theta _ \xi ( \cos \theta _ \xi + \sin \theta _ \xi ) | ^ 4 } , \\ z _ 4 = & \left ( | z _ 1 | ^ 2 - 1 / 2 \sin 2 \theta _ \xi - \mathbf i \cos \theta _ \xi ( \cos \theta _ \xi + \sin \theta _ \xi ) \right ) ^ 2 . \end{align*}


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


\begin{align*} \left ( Q _ - J + D _ j \right ) _ { i 1 } = \left ( M _ - V _ j J \right ) _ { i 1 } = \left ( M _ - R V _ j \right ) _ { i 1 } = \left ( M _ - R \right ) _ { i 1 } , i = 1 , 2 , \end{align*}


\begin{align*} \partial _ t U + \sum _ { j = 1 } ^ d A ^ j ( U ) \frac { \partial U } { \partial x _ j } + B ( x ) V = 0 . \end{align*}


\begin{align*} \overline { \mathrm { S } ( V ) } = \prod _ { n \in \mathbb { N } } \mathrm { S } ^ n ( V ) \end{align*}


\begin{align*} C ( z ) = \sum _ { i } c _ { i } g _ { i } ( z ) \end{align*}


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


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \xi _ 2 = \varepsilon ( - z ^ { - 1 } ) \dot { s } h ( - z ) \varepsilon ( - z ^ { - 1 } ) \varepsilon ( 1 ) \xi _ 2 = \varepsilon ( - z ^ { - 1 } ) \dot { s } \varepsilon ( z ^ 2 - z ) \xi _ 2 . \end{align*}


\begin{align*} & S ' ( h , v ) = 0 \mbox { i n a c o l l a r n e i g h b o r h o o d $ U $ o f $ \hat \Sigma $ i n } M \setminus \Omega \\ & \left \{ \begin{array} { l } h ^ \intercal = 0 \\ H ' ( h ) = 0 \end{array} \right . \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} y = \exp { \left ( \frac { \lambda } { \sigma - \frac { 1 } { 2 } } \right ) } , x = y ^ { - 1 } \log ^ { 2 } { q } , \end{align*}


\begin{align*} \sum _ { j = 1 } ^ { D - 1 } \left ( \nu _ { j } - \nu _ { j - 1 } \right ) \P \left \{ Y \geq \nu _ j \right \} , \end{align*}


\begin{align*} g = D ^ 2 u . \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } z ( \Psi ^ - _ { 1 1 } ( x ; z ) - e ^ { - i c _ - ( x ) } ) = \widehat { \Psi } ^ - _ { 1 1 } ( x ) . \end{align*}


\begin{align*} u _ n = T _ n f _ n . \end{align*}


\begin{align*} \left \{ \begin{array} { l r } \partial _ u \tau = 2 \Gamma _ { u v } ^ v \tau ( 1 - \tau ) \\ \partial _ v \tau = 2 \Gamma _ { v u } ^ u ( 1 - \tau ) , \end{array} \right . \end{align*}


\begin{align*} \frac { 1 } { 1 - y } = \sum _ { n = 0 } ^ \infty y ^ n . \end{align*}


\begin{align*} Q _ { j , \pm } ( x ; k ) = \mathcal { C } \left ( Q _ { j , - } ( x ; k ) J + D _ j \right ) ( z ) , z \in \mathbb { C } ^ { \pm } . \end{align*}


\begin{align*} z ^ j \cdot \left ( \prod _ { i = 1 } ^ n x _ i ^ { \alpha _ i } . y _ i ^ { \beta _ i } \right ) ^ k , 0 \le j \le p - 1 , 1 \le k \le p - 1 . \end{align*}


\begin{align*} w _ \Gamma ( x ) = j p ( x ) / q ( x ) \in [ \gamma _ { j } , \gamma _ { j + 1 } ) . \end{align*}


\begin{align*} \int _ { B _ { 1 / 2 } } r ^ { 2 - n } | \nabla u | ^ 2 \d x \leq C \int _ { B _ { 1 } } r ^ { 2 - n } u _ r ^ 2 \d x + C \varepsilon ^ 2 \int _ { B _ { 1 } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \varepsilon \leq \varepsilon _ 0 , \end{align*}


\begin{align*} W ( 0 ^ \infty | [ 0 ^ k 1 ] ) = \sum _ { n \in \N } ( a _ { n + k } - a ) \ . \end{align*}


\begin{align*} m ( x ; z ) - \widetilde { m } ( x ; z ) = \int _ { - \infty } ^ x e ^ { 2 i z ( x - y ) } [ w ( y ) - \widetilde { w } ( y ) ] d y , \end{align*}


\begin{align*} \Phi _ { n , e } ^ { ( i ) } ( x , y ) \Psi _ { n , e } ^ { ( i ) } ( x , y ) = x ^ n - \zeta ^ i _ { e } y ^ n . \end{align*}


\begin{align*} \bar { y } _ t = - \int _ 0 ^ t \exp \left ( - \int _ s ^ t Z _ { t ' , 3 } d t ' \right ) \bar { z } _ { s , 3 } y _ s d s . \end{align*}


\begin{align*} \sum _ { n < x ^ 2 } \frac { \Lambda _ x ( n ) ^ 2 } { n ^ { 2 \sigma } } & < \sum _ { n = 1 } ^ \infty \frac { \Lambda ( n ) \log n } { n ^ { 2 \sigma } } = \frac { d } { d s } \left [ \frac { \zeta ' } { \zeta } ( s ) \right ] _ { s = 2 \sigma } . \end{align*}


\begin{align*} \Delta u \wedge \Delta v _ 1 \wedge \dots \wedge \Delta v _ { m - 1 } \wedge \beta _ n ^ { n - m } ( \omega ) = \int u \Delta v _ 1 \wedge \dots \wedge \Delta v _ { m - 1 } \wedge \beta _ n ^ { n - m } \wedge \Delta \omega , \omega \in C _ 0 ^ { \infty } ( \Omega ) , \end{align*}


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


\begin{align*} x _ n ( t ) : = x _ a ( t - n { \bar { t } } / { \gamma } , x ( n { \bar { t } } / { \gamma } ) ) \forall \ , t \in I _ n . \end{align*}


\begin{align*} D '' u _ { \epsilon , j } = D '' \left ( ( 1 - v ' _ { \epsilon } ( \Psi ) ) f F ^ { 1 + \delta } \right ) . \end{align*}


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


\begin{align*} \frac { d \psi _ i } { d t } = \Psi ^ { n } _ i ( \psi ) , \end{align*}


\begin{align*} a ( u , v ) = ( f , v ) v \in H _ 0 ^ 2 ( D ) . \end{align*}


\begin{align*} \underline { u } = \frac { 1 } { 4 } \left ( \sum _ { j = 1 } ^ p e _ j \cdot u ( e _ j ) + \sum _ { j = p + 1 } ^ n e _ j \cdot ( - u ^ * ( e _ j ) ) \right ) . \end{align*}


\begin{align*} Z _ { V } ( [ \alpha ] ) = - \int _ { V } \alpha + \sqrt { - 1 } \int _ { V } \omega , \end{align*}


\begin{align*} \int _ { 0 < x _ { 2 } < \frac { 2 \mu } { \nu } } \omega \mathrm { d } x = o ( 1 ) \nu \end{align*}


\begin{align*} \sup _ { Q \in \P ( \R ^ n ) } \left ( \int _ { \R ^ n } f \ , d Q - H ( Q ) \right ) = \log Z \in ( - \infty , \infty ) , \end{align*}


\begin{align*} \begin{aligned} \nabla \times U = f U , \nabla \cdot U & = 0 \textrm { i n } \ \mathbb { R } ^ { 3 } , \\ U & \to B _ { \infty } \textrm { a s } \ | x | \to \infty . \end{aligned} \end{align*}


\begin{align*} V \left ( \mathcal { K } _ { m , n } \right ) = \left \{ \left ( i , j \right ) : 1 \leq i \leq m , 1 \leq j \leq n \right \} , \end{align*}


\begin{align*} e _ c \left [ t + 1 \right ] = & \left \{ 1 - 4 \mu \sum _ { j = 1 } ^ { M } \lvert \phi ^ { \left ( j , c \right ) } \rvert ^ 2 - 2 \mu f ^ { \left ( c \right ) } \vphantom { \sum _ { j = 1 } ^ { M } } \right \} e _ c \left [ t \right ] - 2 \mu \left ( 2 \sum _ { j = 1 } ^ { M } \lvert \phi ^ { \left ( j , c \right ) } \rvert ^ 2 + \mu f ^ { \left ( c \right ) } \right ) a _ c ^ { \left ( o \right ) } \\ & - 2 \mu \sum _ { l } ^ { M } \Re \left \{ \phi ^ { \left ( l , c \right ) } \right \} . \end{align*}


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


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


\begin{align*} v = 0 , \nu ' ( h ) ( \bar u ) + \nu ( v ) = 0 , A ' ( h ) = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} F _ m ( n ) = 1 + H ( n , m ) - r _ m ( n ) \sum _ { k = 1 } ^ m \frac { \mu ( k ) } { k } , \end{align*}


\begin{align*} \begin{aligned} J _ { 1 } & = \int _ { \Lambda _ { h } } ( w - v ) \partial _ { \nu } u _ { 0 } \mathrm d \sigma , J _ { 2 } = - \int _ { \Lambda _ { h } } u _ { 0 } \partial _ { \nu } ( w - v ) \mathrm d \sigma . \end{aligned} \end{align*}


\begin{align*} \kappa _ { V , I } : = \sum _ { J \subset I } ( - 1 ) ^ { | J | - | I | } K _ { V , J } . \end{align*}


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


\begin{align*} X '' \coloneqq \{ B '' \mid B '' \subset ( C / E ) / ( A / E ) , \ \dim B '' = b - a , \ B '' \cap ( D / E ) = 0 \} , \end{align*}


\begin{align*} \sum _ { \alpha , \beta = 1 } ^ d \xi _ \alpha \cdot \left ( \nabla G ( U , x , t ) ^ T B _ { \alpha \beta } ( U , x , t ) \ , \xi _ \beta \right ) = \sum _ { \alpha , \beta = 1 } ^ d \sum _ { i , j = 1 } ^ n \xi _ \alpha ^ i D _ { \alpha \beta } ^ { i j } \xi _ \beta ^ j > 0 \end{align*}


\begin{align*} A _ 1 \circ F \circ A _ 2 ( x ) + A _ 3 ( x ) = F ' ( x ) , \end{align*}


\begin{align*} \nabla \times ( a _ { t } + ( B \times u ) + \mu \nabla \times b ) = 0 \textrm { o n } \ ( L ^ { 2 } _ { \sigma } \cap H ^ { 1 } ) ( \mathbb { R } ^ { 3 } ) ^ { * } . \end{align*}


\begin{align*} h ( H _ { \rho } ( x ) ) = h ( x ) \end{align*}


\begin{align*} m ( x , t ) = \lim _ { k \rightarrow \infty } ( k \psi ( x , t ; k ) ) _ { 1 2 } . \end{align*}


\begin{align*} \P \left ( \frac { 1 } { m } \sum _ { i = 1 } ^ m U _ i \geq \left ( \frac { \pi } { 4 } + \frac { 1 9 9 \pi } { 3 2 0 0 k } \right ) \| \mathbf { x } \| \right ) \leq 2 \exp \left ( - \frac { c _ 8 m } { k } \right ) . \end{align*}


\begin{align*} g ^ { q p } u _ { , p q i } = - v _ i + u _ { , i q } \xi ^ q \end{align*}


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


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


\begin{align*} \partial _ \tau \gamma = \frac { c \bar { K } } { 4 \pi } \mathbf { b } _ { \gamma ( \tau ) } . \end{align*}


\begin{align*} \sum _ { w \in E } w t ( w ) = \sum _ { w \in O } w t ( w ) . \end{align*}


\begin{align*} \delta _ i = \begin{cases} \tilde { \delta } _ i , & i \in [ m ] \\ - \tilde { \delta } _ { i - m } , & i \in [ k ] \setminus [ m ] . \end{cases} \end{align*}


\begin{align*} \mathcal { V } \subset \mathrm { S O } _ { g \mathrm { T } ( X ) } ( \widehat { \Z } ) = g \mathrm { S O } _ X ( \widehat { \Z } ) g ^ { - 1 } . \end{align*}


\begin{align*} \bar { v } _ { i } : = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ v _ { i } \end{pmatrix} \in R ^ { 2 n } . \end{align*}


\begin{align*} G ^ { \alpha \beta } _ 0 = \frac { \partial G } { \partial r _ { \alpha \beta } } ( \nabla _ { \alpha \beta } u ( x _ 0 ) ) , 1 \leq \alpha , \beta \leq n - 1 . \end{align*}


\begin{align*} \lim \limits _ { \varepsilon \to 0 } \frac { | A \oplus _ p ( \varepsilon ^ { \frac { 1 } { p } } \ , B ) | - | A | } { \varepsilon } = \frac { 1 } { p } \int \limits _ { S ^ { n - 1 } } h ^ p _ B ( u ) h _ A ^ { 1 - p } d S _ A ( u ) , \end{align*}


\begin{align*} d ( X ( \bar u ) ) - L _ X g ( \nabla \bar u , \cdot ) & = \nabla ^ 2 \bar u ( X , \cdot ) - \nabla X ( \nabla \bar u , \cdot ) . \end{align*}


\begin{align*} \Gamma ( x , y ) & = - \frac { 1 } { e ^ { - { \psi } } } \left ( d \phi ( x ) \int _ { - \infty } ^ y e ^ { - { \psi ( x , v ) } } d v - \int _ { - \infty } ^ y e ^ { - { \psi ( x , v ) } } d _ x { \psi } ( x , v ) d v \right ) \\ & = \frac { 1 } { e ^ { - \psi } } \left ( d \phi ( x ) \int _ { y } ^ \infty e ^ { - { \psi ( x , v ) } } d v - \int _ { y } ^ \infty e ^ { - { \psi ( x , v ) } } d _ x { \psi } ( x , v ) d v \right ) . \end{align*}


\begin{align*} \left \| f \right \| _ X : = \sum _ { j = 1 } ^ m \int _ 0 ^ 1 { \left | f _ j ( t ) \right | \ \dd { t } } . \end{align*}


\begin{align*} \begin{cases} x _ 1 \mapsto \frac { 4 a ^ 2 } { s ^ 2 } ( x _ 1 - \tilde { \nu _ 1 } ' ( l s ) x _ 2 - \tilde { \nu _ 1 } ( l s ) x _ 3 + \tilde { \nu _ 1 } ' ( l s ) l s x _ 3 ) \\ x _ 2 \mapsto \frac { 2 a } { s } ( x _ 2 - l s x _ 3 ) \\ x _ 3 \mapsto x _ 3 \end{cases} \end{align*}


\begin{align*} m _ { \mathrm A D M } ( g ) = \frac { 1 } { 2 ( n - 1 ) \omega _ { n - 1 } } \lim _ { r \to \infty } \int _ { | x | = r } \sum _ { i , j = 1 } ^ n \left ( \frac { \partial g _ { i j } } { \partial x _ i } - \frac { \partial g _ { i j } } { \partial x _ j } \right ) \frac { x _ j } { | x | } \ , d \sigma . \end{align*}


\begin{align*} a \in F \textup { a n d } \pi ( b ) = \pi ( a ) \implies b \in F \end{align*}


\begin{align*} W ( t ) = \sum _ { l = 1 } ^ { n _ 1 } \Psi _ { k l } \beta ^ c _ l ( t ) , k = 1 , \dots , n . \end{align*}


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


\begin{align*} u ( \mathbf x ) = \int _ { \mathbb S ^ { 1 } } e ^ { \mathrm { i } k \xi \cdot \mathbf { x } } g ( \xi ) \mathrm d \xi , \ \xi \in \mathbb S ^ { n - 1 } , \ \mathbf { x } \in \mathbb { R } ^ n , \ g \in L ^ { 2 } ( \mathbb S ^ { n - 1 } ) , n = 2 \mbox { o r } 3 , \end{align*}


\begin{align*} \omega = \sum _ { i = 1 } ^ n d x ^ i \wedge d \theta _ i . \end{align*}


\begin{align*} D _ { \xi _ 0 } ( k ) : = \left \lbrace z \in \mathbb { C } : \ | z - \xi _ 0 | < k \mu ( \xi _ 0 ) \right \rbrace \ . \end{align*}


\begin{align*} \kappa ( s ) : = \sup _ { \stackrel { k \in \N } { k \neq \ell } } c _ 1 ( q , k , p ( s ) ) \end{align*}


\begin{align*} { { \bf { h } } _ l } = { \alpha _ l } { \bf { a } } \left ( { { \theta _ l } } \right ) = { \alpha _ l } { \left [ { 1 , { e ^ { - j \pi \cos { \theta _ l } } } , \cdots , { e ^ { - j \pi \left ( { { M _ t } - 1 } \right ) \cos { \theta _ l } } } } \right ] ^ T } . \end{align*}


\begin{align*} \widetilde { \Phi } ( x ' ) : = \Phi ^ { \top } ( - x ' ) , \widetilde { \Psi } _ { \varepsilon , \varepsilon ^ { - 1 } } ( x ' ) : = \Psi ^ { \top } _ { \varepsilon , \varepsilon ^ { - 1 } } ( - x ' ) , \widetilde { P } ^ L ( x ' ) : = ( P ^ L ) ^ { \top } ( - x ' ) , x ' \in \R ^ { n - 1 } , \end{align*}


\begin{align*} y ( q ) = \psi ( q ^ 3 ) \left ( \frac { \varphi _ 6 \varphi _ 9 ^ 2 } { \varphi _ 3 \varphi _ { 1 8 } } - 2 q \psi ( q ^ 9 ) \right ) = \frac { \varphi _ 6 ^ 2 } { \varphi _ 3 } \frac { \varphi _ 6 \varphi _ 9 ^ 2 } { \varphi _ 3 \varphi _ { 1 8 } } - 2 q \psi ( q ^ 3 ) \psi ( q ^ 9 ) = z ( q ^ 3 ) - 2 q \psi ( q ^ 3 ) \psi ( q ^ 9 ) \end{align*}


\begin{align*} \Sigma ^ { \pm } ( s - w , \phi , F _ { \pm } ) = & \lim _ { z \downarrow 0 } \sum _ { \varepsilon = \pm } \int _ 0 ^ \infty w _ z ( \lambda ) \lambda ^ { 4 ( 1 - s + w ) - 1 } \int _ 0 ^ \infty \\ & \times \sum _ { \ell , m = 1 } ^ \infty \rho _ \phi ( 3 \ell m ) K _ \nu ( 6 \pi \ell m t ^ 2 ) \hat { F } _ 4 \left ( 0 , 0 , \frac { 3 m \lambda } { t } , \frac { \varepsilon 3 \ell m t ^ 3 } { \lambda } \right ) \frac { d \lambda } { \lambda } t d t . \end{align*}


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


\begin{align*} f _ k ( x ) : = x ^ T A _ k x + 2 b _ k ^ T x + c _ k \end{align*}


\begin{align*} \nabla \mathsf { f } = \left [ \frac { \partial f _ i } { \partial y _ j } = 2 m _ i \pi \frac { \dot { \mathsf { H } } ( z ) } { \mathsf { H } ( b ) } \frac { y _ j } { z } : 1 \le i \le d , 1 \le j \le N \right ] = \frac { 2 \pi } { \mathsf { H } ( b ) } \frac { \mathsf { m } \otimes y } { z ^ { n + 2 } H ( z , z ^ 2 ) } . \end{align*}


\begin{align*} \begin{aligned} \det M = \sum \limits _ { \sigma \in S _ n } \sigma { M } _ { n _ { 1 1 } n _ { 1 2 } } \cdots { M } _ { n _ { 1 l _ 1 } n _ { 1 1 } } M _ { n _ { 2 1 } M _ { 2 2 } } \cdots { M } _ { n _ { r l _ r } n _ { r 1 } } . \end{aligned} \end{align*}


\begin{align*} \frac { \partial s _ { \delta , i } } { \partial z _ { i , h } } = O ( \delta | \ln \delta | ^ { \frac { p - 1 } { 2 } } ) . \end{align*}


\begin{align*} - \Phi ^ * ( x , u ) = { \sup } \{ f \in U S C ( \overline { \Omega } ) : f \in F ( \Omega ) f ( \tau ) \le - { \phi ^ * _ \tau } ( u ) \tau \in \partial \Omega \} \end{align*}


\begin{align*} f ' ( x ) = f ( x + r ) = a x ^ { q + 1 } + b ' x ^ q + c ' x . \end{align*}


\begin{align*} ( 1 + \rho ) ^ { k } = \frac { 1 } { 2 ^ { k } } \sum _ { j = 0 } ^ { 2 k } \sum _ { m = 0 } ^ { \left \lfloor j / 2 \right \rfloor } ( 2 \rho ) ^ { j - 2 m } \binom { k } { j - 2 m } \binom { k - j + 2 m } { m } . \end{align*}


\begin{align*} q _ { \pi ( a ) } = p _ { \pi ( a ) } . \end{align*}


\begin{align*} \| f | \sigma _ 1 \| _ { B V ( \sigma _ 1 ) } = \lim _ { n \to \infty } \| p _ n | \sigma _ 1 \| _ { B V ( \sigma _ 1 ) } \leq \lim _ { n \to \infty } \| p _ n \| _ { B V ( \sigma ) } = \| f \| _ { B V ( \sigma ) } . \end{align*}


\begin{align*} T h _ k = \Psi ( r _ k ) . \end{align*}


\begin{align*} \gamma ( x ) : = p ( x , e ) \equiv p ( x , 1 ) . \end{align*}


\begin{align*} A _ j = \textrm { c u r l } ^ { - 1 } B _ j \to A = \textrm { c u r l } ^ { - 1 } B \textrm { i n } \ L ^ { 2 } _ { \textrm { l o c } } ( 0 , T ; L ^ { 2 } ( \Omega ) ) , \end{align*}


\begin{align*} C _ 1 = \sum _ { X \in S ( 0 ) } \Theta _ { \theta } \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = - q ^ { 8 } ( 1 - q ) ^ 3 ( 1 - q ^ 2 ) . \end{align*}


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


\begin{align*} \hat { f } ( \pi ) = \int _ { G } f ( g ) \pi ( g ) d g \end{align*}


\begin{align*} \alpha _ n ( 0 ) = h [ P _ { \leq N } \psi _ 0 ] ( h n ) + ( - 1 ) ^ n h [ P _ { \leq N } \phi _ 0 ] ( h n ) . \end{align*}


\begin{align*} c _ { k + 1 } = \sum _ { i = 1 } ^ { t - 1 } c _ i a _ i + c _ { k } \sum _ { i \in \Omega _ t ^ e } a _ i + c _ { k + 1 } \sum _ { i \in \Omega _ t ^ o } a _ i , \end{align*}


\begin{align*} & w _ 0 = e _ 1 , \ \ w _ { n + 1 } ( x ; z ) = F w _ n = \int _ { - \infty } ^ x F ( x , y ; z ) w _ n ( y ) d y . \end{align*}


\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \in \R ^ n _ + , \end{align*}


\begin{align*} \begin{aligned} \| \eta ( t ) - \eta _ a ( t ) \| _ 2 \le & \ , \| \eta ( t ) - z ( t ) \| _ 1 + \gamma \bar { k } _ 3 ( L _ r , \delta ) \\ = & \ , f ( t ) + \gamma \bar { k } _ 3 ( L _ r , \delta ) . \end{aligned} \end{align*}


\begin{align*} \sigma ( P ^ { - 1 } ) ( x , \xi ) = \mathfrak b _ { 2 } + \mathfrak b _ { 3 } + \mathfrak b _ { 4 } + . . . , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { \pi ( a ) } ( x - p _ i ) = \prod _ { i = 1 } ^ { \pi ( a ) } p _ i ^ { \alpha _ i } . \end{align*}


\begin{align*} C ^ { \alpha } _ { i } G _ { \alpha \beta } ^ T H ^ { \beta } _ 1 \overline { G _ { \alpha \beta } } = G _ { \alpha \beta } ^ T H ^ { \beta } _ 1 \overline { G _ { \alpha \beta } } ( \overline { C ^ { \alpha } _ { i } } ^ { T } ) \end{align*}


\begin{align*} p _ { _ { l , 2 } } \varphi _ l = - { \bf i } q _ { _ { l , 1 } } \vartheta _ l \ \textrm { o n } \ \C \ \textrm { f o r } \ l = 1 , 2 . \end{align*}


\begin{align*} A \cdot ( v _ { 1 } , \dots , v _ { k } ) : = ( A v _ { 1 } , \dots , A v _ { k } ) . \end{align*}


\begin{align*} \P ( X \geq \delta _ i ) & = \P ( Y \geq \delta _ i ' ) = \P ( Y \geq 2 ^ { - j } ) = \sum _ { i \in [ j ] } \P \left \{ Y = 2 ^ { - i } \right \} = \sum _ { i \in [ j ] } r 2 ^ i = 2 r \left ( 2 ^ { j } - 1 \right ) \leq 2 r 2 ^ j . \end{align*}


\begin{align*} \overline { M } = \overline { C } \cap ( N K \cap H ) / K . \end{align*}


\begin{align*} \operatorname { g r a d } \operatorname { d i v } \textbf { \textit { u } } = g ^ { i j } \Bigl ( \frac { \partial ^ 2 u ^ k } { \partial x _ j \partial x _ k } + \Gamma ^ l _ { k l } \frac { \partial u ^ k } { \partial x _ j } + \frac { \partial \Gamma ^ l _ { k l } } { \partial x _ j } u ^ k \Bigr ) \frac { \partial } { \partial x _ i } . \end{align*}


\begin{align*} \frac { p ' _ { h + 1 } ( x ) } { p _ { h + 1 } ( x ) } = \frac { 1 } { 2 \sqrt x } \Big ( \frac { p _ h ( \sqrt x ) } { p _ h ( \sqrt { - x } ) } - \frac { p _ h ( \sqrt { - x } ) } { p ' _ h ( \sqrt x ) } \Big ) , ~ h = 0 , 1 , \dots , \end{align*}


\begin{align*} \mu _ { ( \boldsymbol { x } , \boldsymbol { \phi } ( t , \boldsymbol { x } ) ) } ^ { N , i } : = k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } } , \lambda ^ { N , 1 } _ { \boldsymbol { \phi } ( t , \boldsymbol { x } ) } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \phi } ( t , \boldsymbol { x } ) } ) , \quad \mu ^ i : = k \big ( t , \nu ^ { N , i } _ { \boldsymbol { x } } , \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) \big ) , \end{align*}


\begin{align*} \epsilon ( x , t ) : = \int _ 0 ^ t f ( x , \tau ) - f _ a ( x ) \ , d \tau , \end{align*}


\begin{align*} | \mathfrak { I } _ { q _ 1 , \ldots , q _ 4 , \kappa } ( \ell ) | \leq \frac { C } { \ell ^ 4 } \log \ell , \end{align*}


\begin{align*} \Sigma _ t = \{ ( x ' , x _ n ) \in \mathbb { R } ^ n \ | \ x ' \in \mathbb { S } _ t ^ { n - 1 } , \ x _ n \in \mathbb { R } \} \end{align*}


\begin{align*} B = \sum _ { j } c _ j \bold { B } _ j , c _ j = \int _ { \Omega } B \cdot \bold { B } _ { j } \dd x , A = \sum _ { j } c _ j \bold { A } _ { j } . \end{align*}


\begin{align*} \int _ 0 ^ \infty \left ( 1 - \frac { u ^ 2 } { 6 s } + \frac { u ^ 4 } { 7 2 s ^ 2 } \right ) e ^ { - u } u ^ { p / 2 - 1 } \dd u = \Gamma \left ( p / 2 \right ) - \frac { \Gamma ( p / 2 + 2 ) } { 6 s } + \frac { \Gamma ( p / 2 + 4 ) } { 7 2 s ^ 2 } \end{align*}


\begin{align*} \int _ { 1 } ^ \infty \frac { d s } { \Big ( \displaystyle \int _ 0 ^ s F ( t ) d t \Big ) ^ { p / ( 2 p - q + 1 ) } } < \infty \quad \mbox { w h e r e } \ ; F ( t ) = \int _ 0 ^ t f ( k ) d k . \end{align*}


\begin{align*} u _ { \varepsilon } ( x ' , t ) : = u ( x ' , t + \varepsilon ) \quad f _ { \varepsilon } ( x ' ) : = u ( x ' , \varepsilon ) , x ' \in \R ^ n , \ , t > 0 . \end{align*}


\begin{align*} D _ { \eta } = A ^ { - 1 } A _ { \eta } : \Gamma _ { \mathbb { C } } ^ \perp \rightarrow \Gamma _ { \mathbb { C } } ^ \perp , \end{align*}


\begin{align*} V _ { \lambda / \mu } : = L _ T , \end{align*}


\begin{align*} H ( d G , d G ) = Q _ 0 d z ^ 2 + \left ( \frac { \tau _ 0 } { 4 } + \frac { 4 | Q _ 0 | ^ 2 } { \tau _ 0 } \right ) d z d \overline { z } + \overline { Q _ 0 } d \overline { z } ^ 2 . \end{align*}


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


\begin{align*} v = 0 \partial \O . \end{align*}


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


\begin{align*} \begin{cases} x _ { 2 n + 1 } & = ( 1 - \alpha _ n ) T x _ { 2 n } + \alpha _ n u , \\ x _ { 2 n + 2 } & = ( 1 - \beta _ n ) U x _ { 2 n + 1 } + \beta _ n x _ { 2 n + 1 } . \end{cases} \end{align*}


\begin{align*} Z ^ { ( i ) } = \partial _ i \mbox { a n d } Z ^ { ( i , j ) } = x _ i \partial _ j - x _ j \partial _ i \mbox { f o r } i , j = 1 , \dots , n . \end{align*}


\begin{align*} \begin{aligned} | | S ( t ) \varphi _ 0 | | _ { T } & \leq C | | \varphi _ 0 | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } , \\ \left \| \int _ { 0 } ^ { t } S ( t - s ) \mathbb { P } \nabla \cdot F _ 0 ( s ) \dd s \right \| _ T & \leq C ' | | F _ 0 | | _ { L ^ { 2 } ( 0 , T ; L ^ { 2 } ( \mathbb { R } ^ { 3 } ) ) } . \end{aligned} \end{align*}


\begin{align*} \frac { \tilde { u } ^ { [ n ] } ( x ) - \tilde { u } ^ { [ n - 1 ] } ( x ) } { \Delta t } = \theta \left [ \mathcal { L } ( \tilde { u } ^ { [ n ] } ) + f ( t _ n , x ) \right ] + ( 1 - \theta ) \left [ \mathcal { L } ( \tilde { u } ^ { [ n - 1 ] } ) + f ( t _ { n - 1 } , x ) \right ] , \end{align*}


\begin{align*} [ \omega ] = 2 [ H ] - [ E ] , [ \alpha ] = 6 [ H ] + [ E ] . \end{align*}


\begin{align*} \partial _ x U ( t , x , \nu ) = P _ t , \quad \partial _ { x x } U ( t , x , \nu ) = \partial _ { x \nu } U ( t , x , \nu ) = 0 , \quad \partial _ { \nu } U ( t , x , \nu ) = \partial _ { \nu } \Phi ( t , \nu ) , \quad \partial _ { \nu \nu } U ( t , x , \nu ) = \partial _ { \nu \nu } \Phi ( t , \nu ) \end{align*}


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


\begin{align*} z m ( x ; z ) e _ 2 - ( I - F ) \left ( \widehat { \Psi } ^ - _ { 1 1 } e _ 1 + \widehat { \Psi } ^ - _ { 2 1 } e _ 2 \right ) = \widetilde { m } ( x ; z ) e _ 2 , \end{align*}


\begin{align*} \lim _ { x \to 0 } \frac { h ( x ) } { \exp ( - a \theta / x ^ 2 ) / x ^ 3 } = & \frac { 2 \theta } { \operatorname { B } ( a , b ) } \lim _ { x \to 0 } \frac { \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } } { \operatorname { I } _ { 1 - \exp ( - \theta / x ^ 2 ) } ( b , a ) } \\ = & \frac { 2 \theta } { \operatorname { B } ( a , b ) } . \end{align*}


\begin{align*} & C _ r = ( i _ r , j _ r ) \\ & i _ r < j _ r < i _ s < j _ s i _ s < i _ r < j _ r < j _ s \ , . \end{align*}


\begin{align*} \begin{cases} - \cdot ( K _ H ( x ) \nabla \varPhi ) = \frac { 1 } { \varepsilon ^ 2 } \left ( \varPhi - \left ( \frac { \alpha } { 2 } | x | ^ 2 + \beta \right ) | \ln \varepsilon | \right ) ^ p _ + , \ & x \in B _ { R ^ * } ( 0 ) , \\ \varPhi ( x ) = 0 , \ & x \in \partial B _ { R ^ * } ( 0 ) , \end{cases} \end{align*}


\begin{align*} \sum _ { 1 \le i < j < k \le M } \left | \det \ ! \ ! \begin{pmatrix} x _ i & x _ j & x _ k \\ y _ i & y _ j & y _ k \\ z _ i & z _ j & z _ k \end{pmatrix} \right | \sum _ { i = 1 } ^ M | z _ i | \le \ ! \ ! \ ! \sum _ { 1 \le i < j \le M } \left | \det \ ! \ ! \begin{pmatrix} y _ i & y _ j \\ z _ i & z _ j \end{pmatrix} \right | \sum _ { 1 \le i < j \le M } \left | \det \ ! \ ! \begin{pmatrix} x _ i & x _ j \\ z _ i & z _ j \end{pmatrix} \right | \end{align*}


\begin{align*} \epsilon _ { f \otimes \chi _ D } = \begin{cases} - \lambda _ f ( p ) p ^ { 1 / 2 } & ( p , D ) = 1 D \equiv \square \mod 4 p ; \\ 1 & p | D . \end{cases} \end{align*}


\begin{align*} \prod _ { j \in J _ i } \left ( n \sum _ { p \mid j } \frac 1 p \right ) & \le \left ( \frac n { e ^ { \alpha ^ { i - 1 } } } \right ) ^ { \# J _ i } = \exp ( \# J _ i \log n - \# J _ i \alpha ^ { i - 2 } ) \\ & = \exp ( \# J _ i \log n - { ( 1 + o ( 1 ) ) n ( \alpha - 1 ) / e ^ \gamma } ) \end{align*}


\begin{align*} & \Delta _ j \Delta _ k f = 0 , ~ ~ ~ | j - k | \geq 2 , \\ & \Delta _ j ( S _ { k - 1 } f \Delta _ k f ) = 0 , ~ ~ ~ | j - k | \geq 5 . \end{align*}


\begin{align*} \int _ { 0 } ^ { 1 } m ( s ) \sin ( 2 \pi t ) \ , d t = 0 \end{align*}


\begin{align*} \begin{aligned} \left | e ^ { - i c _ - ( x ) } - e ^ { - i \tilde { c } _ - ( x ) } \right | & = \left | e ^ { \frac { 1 } { 2 i } \int _ { - \infty } ^ x \left ( | u _ y ( y ) | ^ 2 - | \tilde { u } _ y ( y ) | ^ 2 \right ) d y } - 1 \right | \leq 2 \delta c _ 1 \| u _ y - \tilde { u } _ y \| _ { L ^ 2 } . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \nabla ^ 2 _ U \eta ( U , x , t ) - ( \nabla _ V \tilde { \eta } ) & ( A ( U , x , t ) , x , t ) \cdot \nabla ^ 2 _ U A ( U , x , t ) = \\ & = ( \nabla _ V ^ 2 \tilde { \eta } ) ( A ( U , x , t ) , x , t ) : ( \nabla _ U A ( U , x , t ) , \nabla _ U A ( U , x , t ) ) \ ; . \end{aligned} \end{align*}


\begin{align*} x = a _ 2 ^ 2 + \dots + a _ n ^ 2 + a _ { n + 1 } ^ 2 \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \mathsf { U } ^ n ) = W ( \xi _ 0 , n , x , \mathsf { V } ^ n ) . \end{align*}


\begin{align*} & \mathcal { P } ^ + \left ( f ( z ) e ^ { - 2 i z x } \right ) = \int _ { 2 x } ^ { + \infty } \widehat { f } ( \xi ) e ^ { i z ( \xi - 2 x ) } \mathrm { d } \xi , \\ & \mathcal { P } ^ - \left ( f ( z ) e ^ { 2 i z x } \right ) = - \int ^ { \infty } _ { 2 x } \widehat { f } ( \xi ) e ^ { - i z ( \xi - 2 x ) } \mathrm { d } \xi , \end{align*}


\begin{align*} \lambda _ 1 \alpha ^ { m - 1 } + \cdots + \lambda _ m \beta ^ { m - 1 } = 0 , \end{align*}


\begin{align*} \widehat { \mathbf { Q } } = \begin{footnotesize} \begin{pmatrix} \widehat { \mathbf { Q } } _ 1 & 0 \\ 0 & \widehat { \mathbf { Q } } _ 2 \end{pmatrix} \end{footnotesize} \end{align*}


\begin{align*} \theta ^ 0 { } _ u ( X ) & = v ^ i { } _ \alpha d u ^ \alpha , \\ * \theta ^ 1 { } _ u ( X ) & = v ^ i { } _ \alpha d u ^ \alpha { } _ j - v ^ i { } _ \alpha u ^ \alpha { } _ { j \beta } v ^ \beta { } _ \gamma d u ^ \gamma . \end{align*}


\begin{align*} x = \bigwedge \{ p \in P ( X ) \ : | \ : x \le p \} \end{align*}


\begin{align*} d z _ t = \Pi _ { \mathrm { k e r } A } ( B ( y _ t , y _ t ) - A y _ t ) d t + \Pi _ { \mathrm { k e r } A } \sigma d W _ t . \end{align*}


\begin{align*} \partial _ { L _ y } f ( x ) & = \{ l \in L _ y : \forall z \in X : f ( z ) \geq f ( x ) + l ( z ) - l ( x ) \} \\ & = \{ \begin{multlined} [ t ] l - l ( y ) : l \in L , \forall z \in X : \\ f ( z ) \geq f ( x ) + l ( z ) - l ( y ) - l ( x ) + l ( y ) \} \end{multlined} \\ & = \{ l - l ( y ) : l \in \partial _ L f ( x ) \} \end{align*}


\begin{align*} ( \nabla _ V ( \nabla X ) ) ^ i _ j = V _ k \ , g ( \nabla _ { e _ k } \nabla _ { e _ j } X - \nabla _ { \nabla _ { e _ k } e _ j } X , e _ i ) : = V _ k X _ { i ; j k } . \end{align*}


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


\begin{align*} ( 1 + \hat y _ k ) ^ { - m _ k } F _ M ( \hat y _ 1 , \dots , \hat y _ n ) = F _ { M ' } ( \hat y _ 1 ' , \dots , \hat y _ n ' ) , \end{align*}


\begin{align*} \mu | _ { E _ { 2 , m } } = \lambda _ y = \lambda _ x | _ { E _ { 2 , m } } . \end{align*}


\begin{align*} ( \mathbf { x } , - \mathbf { y } ) \underbrace { \begin{pmatrix} G _ { 4 } & G _ { 5 } & \cdots & G _ { 2 k + 2 } \\ G _ { 5 } & G _ { 6 } & \cdots & G _ { 2 k + 3 } \end{pmatrix} } _ M = 0 . \end{align*}


\begin{align*} \mathrm { Q u a d } ^ i : = \{ z ^ i _ { j k } z ^ i _ { j ' k ' } - z ^ i _ { j k ' } z ^ i _ { j ' k } \mid j , j ' \in [ s _ i ] , k , k ' \in [ t _ i ] \} . \end{align*}


\begin{align*} r ^ \vee ( \kappa + h ^ \vee ) ( \check { \kappa } + \check { h } ^ \vee ) = 1 , r ^ \vee ( \check { \kappa } + n + \check { h } ^ \vee ) ( \varkappa + h ^ \vee ) = 1 . \end{align*}


\begin{align*} h _ { i j k } = h _ { i k j } . \end{align*}


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


\begin{align*} l . \psi \varphi = \psi \underbrace { c _ { g ' } ( l ) } _ { \in K } \varphi = \psi \varphi c _ { g } ( c _ { g ' } ( l ) ) = \psi \varphi c _ { g ' g } ( l ) \end{align*}


\begin{align*} Z _ { V , [ \alpha ] } ( t ) = \frac { 1 } { 2 } \int _ { V } ( t ^ { 2 } \omega ^ { 2 } - \alpha ^ { 2 } ) + \sqrt { - 1 } t \int _ { V } \alpha \wedge \omega . \end{align*}


\begin{align*} \sup _ { t < T _ { \rm m a x } } \int _ { \Omega } ( n \log n ( \cdot , t ) + e ^ { - 1 } ) \le \sum _ { i = 1 } ^ { N } \sup _ { t < T _ { \rm m a x } } \int _ { \Omega \cap B _ { \frac { \delta } { 2 } } ( q _ { i } ) } ( n \log n ( \cdot , t ) + e ^ { - 1 } ) \le C . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { m } \gamma _ { k } f _ { k } ( x ) \ge 0 \ \ \forall \ x \in \mathbb { R } ^ { n } . \end{align*}


\begin{align*} r s [ R , G ] = s r [ R , G ] r s r ' s ' [ R , G ] = r r ' s s ' [ R , G ] . \end{align*}


\begin{align*} \left ( \frac { d } { d \lambda } \right ) ^ r \left [ I ( \lambda ) - \lambda ^ { - \frac { n } { 2 } } \sum _ { j = 0 } ^ N a _ j \lambda ^ { - j } \right ] = O \left ( \lambda ^ { - \frac { n } { 2 } - r - N - 1 } \right ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta u + V ( x ) u + \gamma \phi ( x ) u = b \ , | u | ^ { p - 2 } u & \mathbb { R } ^ 2 \\ \Delta \phi = u ^ 2 & \mathbb { R } ^ 2 , \end{array} \right . \end{align*}


\begin{align*} X _ { n + 1 } = A X _ { n } + B u _ n + \xi _ { n + 1 } \end{align*}


\begin{align*} r = \frac { h } { h + h ^ q } . \end{align*}


\begin{align*} & d _ { ( \psi , a ) } \mathcal { F } ( \phi , b ) = \lim _ { t \to 0 } \frac { \mathcal { F } ( \gamma _ t ) - \mathcal { F } ( \psi , a ) } { t } \\ & = \left ( Q ^ + \phi + \lim _ { t \to 0 } \frac { \pi ^ { - } ( a _ t \cdot \psi _ t - a \cdot \psi ) } { t } , \ , \ , d ^ * b , \ , \ , d ^ + b - \lim _ { t \to 0 } \frac { \rho ^ { - 1 } ( \mu ( \psi _ t ) - \mu ( \psi ) ) } { t } \right ) . \end{align*}


\begin{align*} v = 0 \Omega _ { 1 } \setminus D \end{align*}


\begin{align*} F ( r , t ) : = ( Q ( r , t ) - W ( r , t ) ) \exp ( - t ) \end{align*}


\begin{align*} D _ \mu & = \partial _ \mu - i g \frac { \lambda ^ a } { 2 } A ^ a _ \mu \\ F ^ a _ { \mu \nu } & = \partial _ \mu A ^ a _ \nu - \partial _ \nu A ^ a _ \mu + g f ^ { a b c } A ^ b _ \mu A ^ a _ \nu \end{align*}


\begin{align*} D '' u _ { m ' , \epsilon , j } + P _ { m ' } \big ( \sqrt { N _ 1 \tilde { \lambda } _ { m ' } } h _ { m ' , \epsilon , j } \big ) = D '' \left ( ( 1 - v ' _ { \epsilon } ( \Psi ) ) f F ^ { 1 + \delta } \right ) . \end{align*}


\begin{align*} \varphi _ { c , d } ( u ) = \varphi _ { c , d } ( v ) , \end{align*}


\begin{align*} d \geq 2 : 0 < s < s _ s \leq - \frac d 2 , d = 1 : s < \frac 1 6 s \ne - \frac 1 2 , \end{align*}


\begin{align*} \Pi _ S ( X ) \rightarrow \bigoplus _ { i \in I } \Pi _ S ( Z _ i , N _ i ) \rightrightarrows \cdots \rightrightarrows \bigoplus _ { J \subset I , \sharp J = n } \Pi _ S ( Z _ J , N _ J ) \rightrightarrows \cdots \end{align*}


\begin{align*} s ( G ^ { \sigma , \tau } ) & = \sum _ { \gamma \in \Gamma } s ( G ^ { ( \gamma ) } ) + \sum _ { v \in V } \sigma ( v ) - 2 ( \abs { \Gamma } - 1 ) . \end{align*}


\begin{align*} A \circ \widetilde \Phi \circ \widetilde { \mathcal { P } } \circ \widetilde \Phi ^ { - 1 } \circ A ^ { - 1 } = \mathcal { P } ' \end{align*}


\begin{align*} \| \psi ( \mathbf x ) \| _ { H ^ { 1 , 8 } } = \mathcal O ( \tau ^ { - \frac { 2 } { 5 } } ) , \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) = ( f _ n , v _ n ) v _ n \in X _ n , \end{align*}


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


\begin{align*} S _ r ( a , b ) = \Gamma ( b ) \Gamma \left ( 1 - \frac { r } { 2 } \right ) \sum _ { n = 0 } ^ \infty \frac { ( - 1 ) ^ n } { ( a + n ) ^ { 1 - r / 2 } \Gamma ( b - n ) n ! } , r < 2 . \end{align*}


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


\begin{align*} \mathcal { S } _ n ^ { ( p ) } \ & = \ \sum _ { i = 0 } ^ { n } \ , i ^ p \cdot F _ { i } , \\ \intertext { a n d r e p l a c i n g $ i $ w i t h $ n - i $ w e h a v e } \mathcal { S } _ n ^ { ( p ) } \ & = \ \sum _ { i = 0 } ^ { n } \ , ( n - i ) ^ p \cdot F _ { n - i } \ = \ \sum _ { i = 0 } ^ { n } \left ( \sum _ { k = 0 } ^ p \binom { p } { k } n ^ { p - k } ( - 1 ) ^ k i ^ k \right ) \cdot F _ { n - i } . \end{align*}


\begin{align*} \tilde { \rho } ( X _ { \alpha } ) = \sum _ { k = 1 } ^ L \frac { \partial } { \partial z _ { s , k } } r _ { t , k } ( 1 \pm \alpha _ { t , k } ) . \end{align*}


\begin{align*} f _ \xi ( z ) = \begin{cases} \frac { f ( z ) } { \xi - z } & , \\ 0 & . \end{cases} \end{align*}


\begin{align*} - \left ( \partial _ { \rho } ^ { 2 } + \frac { 4 } { \rho } \partial _ { \rho } \right ) \varphi = \mu ^ { 2 } \left ( \varphi - \frac { W } { 2 } \right ) _ { + } , \rho > 0 . \end{align*}


\begin{align*} X _ { n + 1 } = L ( X _ n ) + F ( X _ n ) + G ( X _ n ) \xi _ { n + 1 } , \end{align*}


\begin{align*} H _ { V } = \Delta + V ( \cdot ) , \end{align*}


\begin{align*} \begin{aligned} \int _ { \Gamma _ h } e ^ { \rho \cdot \mathbf x } \mathrm d \sigma & = \sin \theta _ 0 \frac { 2 \pi \Gamma ( 2 ) } { \tau ^ 2 } \frac { 1 } { ( ( \mathbf d + \mathrm i \mathbf d ^ { \perp } ) \cdot \mathbf { \hat { x } } ( \theta _ 0 , \varphi _ \xi ) ) ^ 2 } - \sin \theta _ 0 \int _ { 0 } ^ { 2 \pi } I _ { R } \mathrm d \varphi , \end{aligned} \end{align*}


\begin{align*} y _ { n b } = A \exp \{ \sqrt { \frac { \mu + E } { b } } x \} + B \exp \{ - \sqrt { \frac { \mu + E } { b } } x \} , x \in ( R , R + r ) . \end{align*}


\begin{align*} B ( t ) : = \mathrm { d i a g } ( M _ { q _ j ( t ) } ) , \end{align*}


\begin{align*} \sigma ( t ) = \max ( t , 0 ) , t \in \mathbb { R } . \end{align*}


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


\begin{align*} \sup _ { x \in \mathbb { R } , t > 0 } | \mathcal { I } ( x , t ) | \leq \bar { \tau } = \max \left ( \sum _ { i = 1 } ^ p \frac { 2 R } { \lambda _ i } , \sum _ { i = p + 1 } ^ n \frac { 2 R } { \lambda _ i } \right ) . \end{align*}


\begin{align*} v _ 0 = \sum Y _ { \ 1 } ^ { - } v _ k . \end{align*}


\begin{align*} ( d _ i y ) ( [ \mathbf { a ' } , \mathbf { a } ] \otimes [ 0 , \mathbf { b } ] ) = y ( d ^ i [ \mathbf { a ' } , \mathbf { a } ] \otimes [ 0 , \mathbf { b } ] ) = 0 , \end{align*}


\begin{align*} { \rm V o l } ( \tilde A \setminus { \tilde \Sigma } ^ { r } ) & \geq { \rm V o l } ( A \cap ( \tilde A \setminus { \tilde \Sigma } ^ { r } ) \ , ) = { \rm V o l } ( A ) - { \rm V o l } ( A \cap ( \tilde B \cup { \tilde \Sigma } ^ { r } ) ) \\ & \geq { \rm V o l } ( A ) - { \rm V o l } ( { \tilde \Sigma } ^ { r } ) - \sum _ { i = s + 1 } ^ m { \rm V o l } ( A \cap B _ { p _ i } ( r ) ) . \\ \end{align*}


\begin{align*} 0 < \Psi \leq \frac { 3 } { 2 } ( a _ 3 = 0 ) ( \beta - \gamma = 0 , \ ) . \end{align*}


\begin{align*} \partial _ t \Phi ^ { 2 } _ { + } + u \cdot \nabla \Phi ^ { 2 } _ { + } & = 0 \textrm { i n } \ \mathbb { R } ^ { 3 } \times ( 0 , T ) , \end{align*}


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