\begin{align*} G : = G _ n : = \{ g \in \mathrm { G L } ( V , \mathbb { D } ) : \langle g x , g y \rangle = \langle x , y \rangle \} . \end{align*}


\begin{align*} \left ( x ^ 3 - \omega ( x ) x ^ 2 \right ) \left ( x - \frac 1 2 \omega ( x ) \right ) ^ k = L _ v ^ k ( x ^ 3 - \omega ( x ) x ^ 2 ) , \mbox { f o r a l l } k \geq 1 . \end{align*}


\begin{align*} \partial \varphi ( x , y ) & = \varphi ( \overline { [ x , y ] } ) - \overline { [ \varphi ( x ) , y ] } - \overline { [ x , \varphi ( y ) ] } \\ \theta _ \varphi ( x , y ) & = \theta ( \varphi ( x ) , y ) + \theta ( x , \varphi ( y ) ) . \end{align*}


\begin{align*} & \eta ^ { \prime \prime \prime } ( x ) - \lambda \eta ^ { \prime \prime } ( x ) - 2 \lambda \eta ^ { \prime } ( x ) + \lambda ^ 2 \eta ( x ) = 0 , \ \ \forall x \in ( 0 , 1 ) , \\ & \eta ( 0 ) = 0 , \ \ \eta ( 1 ) = 0 , \ \ \eta ^ { \prime \prime } ( 0 ) = \eta ^ { \prime \prime } ( 1 ) . \end{align*}


\begin{align*} \delta ( u ( t ) ) & = \Big | \| \nabla u ( t ) \| _ 2 ^ 2 - \| \nabla Q \| _ 2 ^ 2 \Big | \\ & = \Big | 2 \alpha \| \nabla Q \| _ 2 ^ 2 + 2 \int Q h _ 1 d x \Big | + O ( \widetilde { \delta } ^ 2 ) \\ & = 2 \left ( \| \nabla Q \| _ 2 ^ 2 + \| Q \| _ 2 ^ 2 \right ) | \alpha ( t ) | + O ( \widetilde { \delta } ^ 2 ) . \end{align*}


\begin{align*} \begin{aligned} ( \rho ^ { n + 1 } , \phi ^ { n + 1 } ) = & \arg \min _ { ( \rho , \phi ) \in \mathcal { A } , m } \left \{ \frac { 1 } { 2 \tau } \sum _ { i = 1 } ^ s \int _ 0 ^ 1 \int _ { \Omega } F ( \rho _ i , m _ i ) D ^ { - 1 } _ i d x d t + E ( \rho ( \cdot , 1 ) , \phi ( \cdot , 1 ) ) \right \} , \\ & \partial _ t \rho _ i + \nabla \cdot ( m _ i ) = 0 , m _ i \cdot \mathbf { n } = 0 , x \in \partial \Omega , \ ; \rho ( x , 0 ) = \rho ^ n . \end{aligned} \end{align*}


\begin{align*} \norm { K } _ { o p } ^ 2 = \sup _ { \stackrel { f \in \mathcal { H } } { \norm { f } _ \mathcal { H } = 1 } } \norm { K f } _ \mathcal { H } ^ 2 \leq \sum _ { n \in \N } | c _ n | ^ 2 \norm { K e _ n } _ \mathcal { H } ^ 2 \leq \sum _ { n \in \N } | c _ n | ^ 2 \sum _ { n \in \N } \norm { K e _ n } _ \mathcal { H } ^ 2 = \norm { K } _ { H . S . } ^ 2 . \end{align*}


\begin{align*} \limsup _ { \varepsilon \rightarrow 0 ^ { + } } \limsup _ { n \rightarrow \infty } \mathbb { E } _ { \mu _ n } \Big [ \Big | \int _ { 0 } ^ { t } \frac { 1 } { n } \sum _ { x } \Phi _ { n } ( s , \tfrac { x } { n } ) [ \eta _ { s } ( x ) - \overleftarrow { \eta } _ { s } ^ { \ell } ( x ) ] \eta _ { s } ( x + 1 ) d s \Big | \Big ] = 0 . \end{align*}


\begin{align*} \inf _ { 0 \not = f \in H ^ 1 ( \mathbb { R } ^ N ) } \frac { \| \nabla f \| _ 2 ^ { N p - ( N + \gamma ) } \| f \| _ 2 ^ { N + \gamma - ( N - 2 ) p } } { \int \left ( | \cdot | ^ { - ( N - \gamma ) } * | f | ^ p \right ) | f | ^ p d x } , \end{align*}


\begin{align*} u ( x ) \leqslant \tau \bigg ( 1 - \frac \theta 2 \bigg ) ^ { - n - 2 } d ^ { - n - 2 } = \bigg ( 1 - \frac \theta 2 \bigg ) ^ { - n - 2 } u ( a ) x \in B _ { \theta d / 2 } ( a ) . \end{align*}


\begin{align*} \lim _ { k \rightarrow 0 ^ + } \mathcal { K } ( x , k ) = - \lim _ { k \rightarrow 0 ^ - } \mathcal { K } ( x , k ) . \end{align*}


\begin{align*} f ( s ) = e ^ { A + B s } \prod _ { n = 1 } ^ \infty ( 1 - s / \rho _ n ) e ^ { s / \rho _ n } \end{align*}


\begin{align*} \tilde c _ { n , s } : = c _ { n , s } \int _ { \R ^ n _ + } \frac { \dd z } { \vert e _ 1 + z \vert ^ { n + 2 s } } . \end{align*}


\begin{align*} - \Phi _ r ( \{ X _ I \} ) = \frac { \sum _ { I \in \mathcal { P } [ r ] } ( - 1 ) ^ { | I | + 1 } X _ I \prod _ { J \in \mathcal { P } [ r ] \atop J \neq I } ( 1 - X _ J ) } { \prod _ { J \in \mathcal { P } [ r ] } ( 1 - X _ J ) } . \end{align*}


\begin{align*} 1 _ { S _ a } S 1 _ { S _ b } = S _ { G ( a , b ) } . \end{align*}


\begin{align*} \lim _ { h \to 0 } \frac { ( - \Delta ) ^ s v ( h e _ 1 ) } h = - 2 c _ { n , s } ( n + 2 s ) \int _ { \R ^ n _ + } \frac { y _ 1 v ( y ) } { \vert y \vert ^ { n + 2 s + 2 } } \dd y . \end{align*}


\begin{align*} i k \mathcal { K } ^ { \# } _ { R } ( x , k ) = \partial _ x \mathcal { K } _ { \# , R } ' ( x , k ) - \mathcal { K } _ { \# , R } '' ( x , k ) , \end{align*}


\begin{align*} [ J ^ { \chi _ A } ] _ \mathcal { M } : = \lambda \alpha \in \mathbb { N } ^ \mathbb { N } , x \in X . \begin{cases} J ^ A _ { r _ \alpha } x & r _ \alpha > 0 , \rho > - r _ \alpha / 2 x \in \mathrm { d o m } ( J ^ A _ { r _ \alpha } ) , \\ 0 & , \end{cases} \end{align*}


\begin{align*} \langle f , g \rangle _ { X ( k ) } = \sum \limits _ { \tau \in X ( k ) } \pi _ k ( \tau ) f ( \tau ) g ( \tau ) , \end{align*}


\begin{align*} x ^ * \in \Gamma : = \{ x ^ * \in X \mid T ( x ^ * ) \cap S ( x ^ * ) \neq \emptyset \} . \end{align*}


\begin{align*} \begin{aligned} u ( x + h ) + u ( x - h ) - 2 u ( x ) & \leq \int _ 0 ^ T e ^ { - s } L \left ( \xi ( s ) + \left ( 1 - \frac { s } { T } \right ) h , - \left ( \dot { \xi } ( s ) - \frac { h } { T } \right ) \right ) \\ & + e ^ { - s } L \left ( \xi ( s ) - \left ( 1 - \frac { s } { T } \right ) h , - \left ( \dot { \xi } ( s ) + \frac { h } { T } \right ) \right ) - 2 e ^ { - s } L \left ( \xi ( s ) , - \dot { \xi } ( s ) \right ) d s . \end{aligned} \end{align*}


\begin{align*} N _ { t , : } = \frac { M _ { i _ t , : } } { \sqrt { p P _ { i _ t } } } , \ \ t \in [ p ] . \end{align*}


\begin{align*} \eta ( A ) & = \eta ( B ) \cdot \frac { \sum _ { V \in \mathcal { W } _ { p + 1 } : V \cap A \neq \emptyset } \mu ( B ^ { [ V ] } ) } { \sum _ { V ' \in \mathcal { F } ^ { B } } \mu ( B ^ { [ V ^ { \prime } ] } ) } \leq 2 Q _ { d , 4 } \frac { \eta ( B ) } { \mu ( B ) } \mu ( 5 A ) . \end{align*}


\begin{align*} \begin{aligned} E = E _ 0 - \frac { 1 } { 2 } \left [ \int _ { \Gamma _ D } \epsilon ( x ) \phi ^ b _ D \partial _ n \phi d s - \int _ { \Gamma _ N } \phi ^ b _ N \phi d s - \frac { 1 } { \beta _ R } \int _ { \Gamma _ R } \phi ^ b _ R \phi d s \right ] . \end{aligned} \end{align*}


\begin{align*} y ^ { [ j ] } = y ^ { ( j ) } , j = \overline { 0 , m - 1 } . \end{align*}


\begin{align*} M ( k ) = \left ( \begin{array} { c c c } - 1 & 0 & 0 \\ 0 & \cos ( \frac { 2 \pi k } { n } ) & \sin ( \frac { 2 \pi k } { n } ) \\ 0 & \sin ( \frac { 2 \pi k } { n } ) & - \cos ( \frac { 2 \pi k } { n } ) \end{array} \right ) . \end{align*}


\begin{align*} \omega ^ 2 a _ { j k } - i \omega b _ { j k } = \frac { \mathcal { F } \{ F _ { j k } \} } { \mathcal { F } \{ x _ k \} } , \end{align*}


\begin{align*} Y _ { \alpha \beta } ^ { i j } = - \frac { h ( \mathcal { I } _ { \alpha } ^ { i } ) \cdot h ( \mathcal { I } _ { \beta } ^ { j } ) } { h ( \mathcal { I } _ { \beta } ^ { i } ) \cdot h ( \mathcal { I } _ { \alpha } ^ { j } ) } \end{align*}


\begin{align*} & \sum _ { i , j = 1 } ^ 3 \sigma ^ { i j } \big ( u _ 0 \hat { p } _ i - u _ i \big ) \big ( u _ 0 \hat { p } _ j - u _ j \big ) \geq \sum _ { i , j = 1 } ^ 3 \frac 1 2 u _ 0 ^ 2 \sigma ^ { i j } \hat { p } _ i \hat { p } _ j - \sum _ { i , j = 1 } ^ 3 \sigma ^ { i j } u _ i u _ j . \end{align*}


\begin{align*} | \det { \mathfrak { S } } | = | B _ K | ^ N | A ( \beta _ 1 ) \cdots A ( \beta _ K ) | = | A _ N | ^ K | B ( \alpha _ 1 ) \cdots B ( \alpha _ N ) | . \end{align*}


\begin{align*} a = \prod _ { p \in P } p ^ { \mathsf v _ p ( a ) } \ , , \quad \end{align*}


\begin{align*} \int _ { P _ R } | z ^ \alpha | ^ 2 e ^ { - \pi | z | ^ 2 } \ , d z = \prod _ { k = 1 } ^ d \left ( 2 \pi \int _ 0 ^ R r ^ { 2 \alpha _ k + 1 } e ^ { - \pi r _ k ^ 2 } \ , d r _ k \right ) = \mu _ { \alpha , R } \end{align*}


\begin{align*} \Sigma _ { c , a , e , d , g _ 1 , g _ 2 , g _ 3 , g _ 4 } \ll ( X Q L a e d g _ 1 g _ 3 g _ 4 ) ^ { \varepsilon } \left ( \frac { L Q } { c } \right ) \mathop { \sum _ { S _ 5 > 0 } \sum _ { S _ 6 > 0 } } _ { S _ 6 \leq S _ 5 } \frac { S _ 6 ^ { \varepsilon } } { S _ 5 ^ { j _ 1 - \varepsilon } } \bigg ( 1 + \frac { X c Q ^ { \vartheta - 1 } } { L } \bigg ) ^ { j _ 1 - 1 } \\ \times \Big ( Q ^ { \vartheta } L S _ 5 + a e L ^ 2 S _ 5 ^ 2 \Big ) . \end{align*}


\begin{align*} \mathcal { E } ( f , g ) = - \left \langle \mathcal { L } f , g \right \rangle _ { L ^ 2 ( M ) } , f \in \mathcal { D } ( \mathcal { L } ) , \ g \in \mathcal { D } ( \mathcal { E } ) . \end{align*}


\begin{align*} \sum _ { \gamma < \beta } b _ { \gamma } ^ { \beta } g _ { \alpha } ^ { ( \gamma ) } ( o ) + g _ { \alpha } ^ { ( \beta ) } ( o ) = \int _ D g _ { \alpha } \overline { { g } _ { \beta } } e ^ { - \varphi } , \ \forall \beta < \alpha . \end{align*}


\begin{align*} \begin{cases} u _ j / d _ j \to \hat { u } _ \infty & \ C ^ 2 _ { l o c } ( \bar { \Sigma } ' ) ; \\ ( f _ j - f ) / ( ( 1 + f ) d _ j ) \to \hat { f } _ \infty & \ C ^ { k , \alpha } ( M ) . \end{cases} \end{align*}


\begin{align*} \mu _ { \varepsilon } = - \varepsilon \Delta c _ { \varepsilon } + \frac { 1 } { \varepsilon } f ^ { \prime } ( c _ { \varepsilon } ) , \end{align*}


\begin{align*} \Phi ( h ) = B ( h , h ) = \lambda _ { N + 3 } ^ 2 B ( f , f ) = \lambda _ { N + 3 } ^ 2 \Phi ( f ) \le 0 . \end{align*}


\begin{align*} \frac { d } { d t } \sum _ { | \alpha | = 0 } ^ { m } \langle A _ { 3 } ^ { 0 } \partial _ { x } ^ { \alpha } u , \partial _ { x } ^ { \alpha } u \rangle \leq \frac { C \epsilon } { 2 } \| \nabla v \| _ { m } ^ { 2 } + C \left \lbrace \| f _ { 3 } \| _ { m } ^ { 2 } + ( \mu _ { 0 } ( t ) + \mu _ { 1 } ( t ) ) \left ( \| w \| _ { m } ^ { 2 } + \| v \| _ { m } ^ { 2 } \right ) \right \rbrace . \end{align*}


\begin{align*} \omega _ { p } \left ( \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix} \right ) & = \omega _ { p } \left ( U \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix} U ^ { \ast } \right ) \\ & = \omega _ { p } \left ( \begin{bmatrix} 0 & B \\ A & 0 \end{bmatrix} \right ) \end{align*}


\begin{align*} m = \begin{cases} 3 , & \rho < 1 , \\ 4 , & \rho \ge 1 . \end{cases} \end{align*}


\begin{align*} \mathbf { y } _ { n _ k } \cap B _ 1 ( \mathbf { 0 } ) = \emptyset , \varepsilon < \frac { \underline { m } } { \underline { m } + \overline { m } } \ ; . \end{align*}


\begin{align*} \inf _ { ( \rho , m ) } \sup _ { S \in H ^ 1 } \mathcal L ( \rho , m , S ) = \sup _ { S \in H ^ 1 } \inf _ { ( \rho , m ) } \mathcal L ( \rho , m , S ) . \end{align*}


\begin{align*} I _ { 2 , \lambda } ( f ) ( x , y ) = \int _ { 0 } ^ { + \infty } \dfrac { e ^ { - ( 1 + \lambda ) t } } { \sqrt { 1 - e ^ { - 2 t } } } f \left ( x e ^ { - t } + y \sqrt { 1 - e ^ { - 2 t } } \right ) d t . \end{align*}


\begin{align*} \alpha _ n & = \sqrt { 2 \log n } , & \beta _ n & = \alpha _ n - ( 2 \alpha _ n ) ^ { - 1 } \bigl ( \log \log n + \log ( 4 \pi ) \bigr ) . \end{align*}


\begin{align*} B _ i f = \sum \limits _ { j = k } ^ { \infty } q ^ { - ( j + 1 ) } ~ \int \limits _ { | y | = q ^ { j + 1 } } f ( x - y ) a _ i ( y ) d y . \end{align*}


\begin{align*} f ( t ) = \int _ { - 1 / 2 } ^ { 1 / 2 } \widehat { f } ( \omega ) e ^ { 2 \pi i t \cdot \omega } \ , d \omega , \end{align*}


\begin{align*} f ( \xi _ k ) = \lim _ { r \to 1 ^ { - } } f ( r \xi _ k ) = p ( \xi _ k ) , \end{align*}


\begin{align*} y _ 1 = 0 , \dotsc , \ y _ n = 0 , \ y _ i - y _ j = 0 \textrm { f o r } 1 \leq i < j \leq n \end{align*}


\begin{align*} & U _ { \xi } = U _ { \eta } \cup U _ { - \eta } , \\ & U _ { \eta } = \left \{ k : | k - \eta | < \varrho \right \} , U _ { - \eta } = \left \{ k : | k + \eta | < \varrho \right \} , \end{align*}


\begin{align*} \frac { A _ { \sigma _ n } ( i ) } { B _ { \sigma _ n } } \leq p _ { \sigma _ n } ( i ) < \frac { A _ { \sigma _ n } ( i ) + 1 } { B _ { \sigma _ n } } \ ; \ ; A _ { \sigma _ n } ( 1 ) = B _ { \sigma _ n } - \sum _ { i \in \mathcal { I } _ n ^ { \sigma } \setminus \{ 1 \} } A _ { \sigma _ n } ( i ) , \end{align*}


\begin{align*} | V _ g f ( x , \omega ) | = | \langle f , M _ \omega T _ x g \rangle | \leq \norm { f } _ 2 \norm { g } _ 2 = 1 . \end{align*}


\begin{align*} w _ i = w ( \xi _ r ) = ( u \times v ) ( r ) = v _ { r - p } [ p ] \quad w _ j = w ( \xi _ s ) = ( u \times v ) ( s ) = v _ { s - p } [ p ] . \end{align*}


\begin{align*} \mu _ m ( d x ) = c _ { m , d } \left ( 1 + \| x \| ^ 2 \right ) ^ { - m - d / 2 } d x , \end{align*}


\begin{align*} \frac { d } { d t } \int x w ( x , t ) d x = \frac 1 2 ( \| \phi \| ^ 2 - \| \varphi \| ^ 2 ) . \end{align*}


\begin{align*} b _ z ( x ) : = b ( x - z ) . \end{align*}


\begin{align*} \Omega _ m = \{ \theta \ : | \ : \theta = x _ { 1 : m } , x \in \Omega _ n \} . \end{align*}


\begin{align*} \forall k \in \N \colon \eta _ k \in \widehat { \partial } \varphi ( x _ k ' ) + \widehat { D } ^ * \Phi ( x _ k , y _ k ) \left ( \frac { \norm { x _ k - \bar x } } { \norm { y _ k - \bar y } } y _ k ^ * \right ) . \end{align*}


\begin{align*} A ( y + z ) = A ( y ) y \in \R ^ d z \in \Gamma ; \end{align*}


\begin{align*} \rho ^ k _ \ell = \prod \limits ^ { k - \ell } _ { i = 1 } \left ( 1 - \delta _ { k - \ell - i } ^ { k - i } \right ) , \ \ \rho _ { } = \min _ { 0 \leq \ell \leq k } \{ \rho ^ k _ \ell \} . \end{align*}


\begin{align*} C \sqrt { \frac { n ^ 3 \log n } { P ^ { - 1 } ( n ) } } \bigg / \bigg ( c \frac { n ^ 2 } { P ^ { - 1 } ( n ) } \bigg ) = \frac C c \sqrt { P ^ { - 1 } ( n ) \frac { \log n } n } . \end{align*}


\begin{align*} F ( t ^ { \ast } ) = 0 . \end{align*}


\begin{align*} a = a ^ \sigma , \epsilon ( a ) = 0 , a \odot a = a , a ^ 2 = ( \lambda - \mu ) a + \mu w + ( r - \mu ) e , \end{align*}


\begin{align*} \int _ M | \nabla _ g u ( x ) | ^ 2 { \rm d } v _ g - \mu \int _ M \frac { u ^ 2 ( x ) } { d ^ 2 _ g ( x _ 0 , x ) } { \rm d } v _ g + \int _ M u ^ 2 ( x ) { \rm d } v _ g = \lambda \int _ M \alpha ( x ) \xi _ x u ( x ) { \rm d } v _ g , \end{align*}


\begin{align*} ( - \Delta ) ^ s v ( x ) + c ( x ) v ( x ) & = - ( - \Delta ) ^ s u ( x ) - c ( x ) u ( x ) + c ( x ) \tau d ^ { - n - 2 } \bigg ( 1 - \frac \theta 2 \bigg ) ^ { - n - 2 } \zeta ( x ) \\ & \geqslant - x _ 1 - C \tau d ^ { - n - 2 } \| c ^ - \| _ { L ^ \infty ( B _ \rho ^ + ) } \bigg ( 1 - \frac \theta 2 \bigg ) ^ { - n - 2 } \zeta ( x ) . \end{align*}


\begin{align*} & T _ 1 ( u ) = T ^ { ' } _ 1 ( u ) + T D ( \mathfrak { X } ) u - [ \mathfrak { X } , T u ] _ \mathfrak { g } \\ & \ ; \ ; \ ; \ ; \ ; = T ^ { ' } _ 1 ( u ) + ( d _ T \mathfrak { X } ) ( u ) , \ ; \forall u \in V , \end{align*}


\begin{align*} \ ; [ T u + u , T v + v , T w + w ] _ { L \oplus V } & = T \Big ( \theta ( T v , T w ) u - \theta ( T u , T w ) v + D ( T u , T v ) w \Big ) \\ & + \theta ( T v , T w ) u - \theta ( T u , T w ) v + D ( T u , T v ) w \in G r ( T ) . \end{align*}


\begin{align*} \mathcal { X } = f ( W ( h _ 1 ) , \dots , W ( h _ m ) ) \end{align*}


\begin{align*} D ^ { x } = \sum _ { \substack { \mathbf { m } \\ } } \nu _ x ( \mathbf { m } ) \ , D _ { \mathbf { m } } , \end{align*}


\begin{align*} b _ \alpha ( s ) = \begin{cases} b ( s ) & s \in [ 0 , \bar \rho - \alpha ] \\ b ( \bar \rho - \alpha ) & s \in ( \bar \rho - \alpha , \bar \rho ) . \end{cases} \end{align*}


\begin{align*} \boxed { \frac { 1 } { \det ( 1 _ n - D A ) } = \sum _ { k _ 1 , \ldots , k _ n \ge 0 } c _ { k _ 1 , \ldots , k _ n } X _ 1 ^ { k _ 1 } \ldots X _ n ^ { k _ n } , } \end{align*}


\begin{align*} \widetilde \Gamma ^ l _ { i j k } \stackrel { ( i j k ) } { = } \partial _ \lambda ^ l \widetilde \Gamma ^ \lambda _ { \alpha \beta \gamma } \partial ^ \alpha _ i \partial ^ \beta _ j \partial ^ \gamma _ k + 2 \partial ^ l _ \lambda \Gamma ^ \lambda _ { ( \alpha \beta ) } \partial ^ \alpha _ i \partial ^ \beta _ { j k } + \partial ^ l _ \lambda \partial ^ \lambda _ { i j k } \end{align*}


\begin{align*} K _ { v } ^ N ( \mathfrak { n } ) = \left \{ \gamma = \left [ \begin{array} { c c } a & b \\ c & d \end{array} \right ] \in G _ N ( \mathbb { A } _ { v } ) \cap M _ N ( \mathcal { O } _ v ) : \gamma \equiv 1 _ N \mathfrak { n } _ v \right \} . \end{align*}


\begin{align*} k [ K ] = S / \big ( v _ { i _ 1 } \ ! \cdots v _ { i _ t } \ , : \ , \{ i _ 1 , \ldots , i _ t \} \in { \rm M F } ( K ) \big ) . \end{align*}


\begin{align*} \underbrace { \frac { 1 } { n ! } \sum _ { g \in \mathrm { S y m } ( n ) } 2 ^ { c ( g _ E ) } } _ { } = \underbrace { \frac { 1 } { n ! } \sum _ { \substack { g \in \mathrm { S y m } ( n ) \\ | g | \ \mathrm { o d d } } } 2 ^ { c ( g _ E ) } } _ { } + \underbrace { \frac { 1 } { n ! } \sum _ { \substack { g \in \mathrm { S y m } ( n ) \\ | g | \ \mathrm { e v e n } } } 2 ^ { c ( g _ E ) } } _ { } . \end{align*}


\begin{align*} z ^ 2 = z \cdot z = z ^ T z , z \in \C ^ d . \end{align*}


\begin{align*} ( ( W _ 1 \times I _ 1 ) \cup ( Z _ 1 \times J _ 1 ) ) \cap ( ( W _ 2 \times I _ 2 ) \cup ( Z _ 2 \times J _ 2 ) ) \ = \ \emptyset . \end{align*}


\begin{align*} \rho _ { \beta * } ( E _ { \theta } ) \cdot ( \rho _ { \beta } ( g ) \cdot v _ { \beta } ) & = { \frac { d } { d t } } _ { | t = 0 } \left ( \rho _ { \beta } ( g g ^ { - 1 } \exp ( t \ , E _ { \theta } ) g ) \cdot v _ { \beta } \right ) \\ & = \rho _ { \beta } ( g ) \cdot \rho _ { \beta * } ( \mathrm { A d } ( g ^ { - 1 } ) \cdot E _ { \theta } ) \cdot v _ { \beta } \\ & = \rho _ { \beta } ( g ) \cdot 0 = 0 \ , . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { E _ { n , k } ( \tau ( A ^ { ( n ) } _ { c _ n k n , n , l } ) | \mathcal { W A } ^ { ( n , k ) } _ { M _ n , n , l } ) } { n k } = \frac { 1 - c } { k - l + 1 } . \end{align*}


\begin{align*} \sum _ { G ( \mathcal { O } _ { \mathbf { h } } ) \backslash \mathfrak { X } / G ( \mathcal { O } _ { \mathbf { h } } ) } \lambda _ { \mathbf { f } } ( \xi ) \cdot \chi _ { \mathbf { h } } ( \det ( r ) ) | \det ( r ) | _ { \mathbf { h } } ^ { - s } \mathbf { f } ( g ) = D ( s , \mathbf { f } , \chi ) \mathbf { f } ( g ) . \end{align*}


\begin{align*} \overline { S } _ n ^ { \boldsymbol { \mu } , \sigma } - \sum _ { k = n + 1 } ^ { d } C _ { k } ^ { ( d ) , \sigma } ( \mathbf { P } _ { \ ! \sigma } ) \cdot \bigg ( \int \ ! \log \boldsymbol { \mu } _ k ^ { \sigma } \ , \mathrm { d } \mathbf { p } _ { \sigma _ k } + \sum _ { \ell = 1 } ^ { n } \chi _ { \ell } ^ { \sigma } ( \mathbf { p } _ { \sigma _ k } ) \big ( \overline { S } _ { \ell } ^ { \boldsymbol { \mu } , \sigma } - \overline { S } _ { \ell - 1 } ^ { \boldsymbol { \mu } , \sigma } \big ) \bigg ) , \end{align*}


\begin{align*} h = - \tilde \sigma _ 0 ( 0 ) + \tilde h . \end{align*}


\begin{align*} g ( \chi ) = \inf \limits _ { \phi } \mathcal { L } _ { \sigma } ( \phi , \chi ) , \end{align*}


\begin{align*} \begin{aligned} & u _ { \star , \lambda } ( x , t ) - u _ { q , \lambda } ( x , t ) \\ & \leq { 1 \over q } \max \left \{ u ( y _ j , t ) , u ( z _ j , t ) \right \} \max \{ { \lambda } ^ { { 1 \over q } - 1 } ( 1 - \lambda ) , ( 1 - \lambda ) ^ { { 1 \over q } - 1 } \lambda \} ( 1 - r ^ q ) + { 1 \over j } , \end{aligned} \end{align*}


\begin{align*} | z | ^ 2 = \overline { z } \cdot z = z ^ * z , z ^ * = \overline { z } ^ T . \end{align*}


\begin{align*} a \ , \gamma = \frac { \log \left ( n + \sqrt { n ^ 2 + 4 } \right ) - \log ( 2 ) } { n } . \end{align*}


\begin{align*} \lambda _ 1 ( T ) = \lambda _ 2 ( \mu _ { | T | } ) , \qquad \lambda _ 2 ( T ) = \lambda _ 1 ( \mu _ { | T | } ) . \end{align*}


\begin{align*} \frac { 1 } { 2 } \sum _ { \ell = 1 } ^ n d _ \ell - \sum _ { d _ \ell \geq \delta } d _ \ell = \frac { 1 } { 2 } \left ( \sum _ { d _ \ell < \delta } d _ \ell - \sum _ { d _ \ell \geq \delta } d _ \ell \right ) \geq \frac { \delta d } { 2 } \end{align*}


\begin{align*} g ( k , \xi ) : = 4 X ^ 3 _ \eta ( k ) , X _ { \eta } ( k ) = \sqrt { k ^ 2 - \eta ^ 2 } , \eta = \sqrt { - 2 \xi } \in ( C _ R , C _ L ) . \end{align*}


\begin{align*} M = M _ 1 \oplus \big ( \bigoplus _ { \alpha \in \Phi } M _ { \overline { \chi } ^ { \langle \alpha , \lambda \rangle } } \big ) \end{align*}


\begin{align*} x ^ \prime = j x \ , , y ^ \prime = y \ , , z ^ \prime = j ^ 2 z \ , , w ^ \prime = w \ , . \end{align*}


\begin{align*} ( D , D ^ + ) = ( ( \prod _ { i \in I } C ^ + _ i ) [ 1 / ( \varpi _ i ) ] , \prod _ { i \in I } C ^ + _ i ) . \end{align*}


\begin{align*} \lim \limits _ { x \to 0 ^ + } \sinh ( x ) - x ^ 2 \coth ( x ) = 0 - \lim \limits _ { x \to 0 ^ + } x \cdot x \ , \coth ( x ) = - 0 \cdot 1 = 0 . \end{align*}


\begin{align*} f _ 1 \geq 1 = I , f _ 4 , f _ 5 \geq h - 1 \geq h - e _ 6 = e _ 1 + \dots + e _ 5 , f _ 6 \geq h . \end{align*}


\begin{align*} d _ { q - 1 } \tilde \varphi ( x _ 0 , \ldots , x _ q ) & = \sum _ { i = 0 } ^ q ( - 1 ) ^ i \varphi ( 0 , x _ 0 , \ldots , \hat x _ i , \ldots , x _ q ) \\ & = \sum _ { i = 1 } ^ { q + 1 } ( - 1 ) ^ { i + 1 } \varphi ( 0 , x _ 0 , \ldots , \hat x _ { i - 1 } , \ldots , x _ q ) + d _ q \varphi ( 0 , x _ 0 , \ldots , x _ q ) \\ & = \varphi ( x _ 0 , \ldots , x _ q ) \end{align*}


\begin{align*} B f ( x ) = \sum \limits _ { j = k } ^ { \infty } q ^ { - ( j + 1 ) } g _ j \ast f ( x ) . \end{align*}


\begin{align*} ( \varphi ( - a ) \check { \delta } _ { \check { \varphi } } ) ( b ) = \varphi ( \kappa ^ { - 1 } ( b ) a ) = ( \varphi \circ S ^ 2 ) ( b \kappa ( a ) ) ) = \varphi ( b \kappa ( a ) \nu ^ { - 1 } ) , a , b \in A . \end{align*}


\begin{align*} \sigma _ T ( { \underline T } , \mathcal X ) = \{ \lambda = ( \lambda _ 1 , . . . , \lambda _ n ) \in \mathbb { C } ^ n : { \underline T } - \lambda \} . \end{align*}


\begin{align*} B = W _ 1 ^ c \cap \cdots \cap W _ n ^ c = \left ( \bigcup _ i W _ i \right ) ^ c = \emptyset . \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \langle K ( x , y ) , \lambda _ \delta ( x , y ) \chi _ 0 ( y ) [ \phi ( y ) - \phi ( x ) ] \psi ( x ) \rangle = 0 . \end{align*}


\begin{align*} M ^ p _ p ( \mu , x _ 0 ) = \int _ { B } ( d ( x , x _ 0 ) ) ^ p \ , d \mu < \infty , \end{align*}


\begin{align*} M _ 2 ( p , H ) = { \pi ^ 2 \over 8 } \left ( 1 - ( - 1 ) ^ { a } \frac { 2 a + 1 } { p } \right ) , \end{align*}


\begin{align*} \Omega = \sum _ { i = 1 } ^ { \infty } \lambda _ i a _ i , \end{align*}


\begin{align*} ( \partial _ t - \sigma \Delta ) \ \widehat { \mu } _ t = - \div \left ( \widehat { v } ( t , \cdot ) \ , \widehat { \mu } _ t \right ) . \end{align*}


\begin{align*} \Phi ( x , \omega , \tau ) = \sum _ { k \in \Z } \Phi _ k ( x , \omega ) e ^ { 2 \pi i k \tau } , \end{align*}


\begin{align*} u ( t ) = \boldsymbol { V } ( t ) f _ { 0 } + \int \nolimits _ { 0 } ^ { t } \boldsymbol { V } ( t - \tau ) F ( u ( \tau ) ) d \tau , \end{align*}


\begin{align*} \langle X g , P g \rangle = \sum _ { k , l \in \Z } \langle P g , T _ { - k } M _ { - l } g \rangle \langle T _ { - k } M _ { - l } g , X g \rangle = \langle P g , X g \rangle . \end{align*}


\begin{align*} M _ { d _ 0 } ( p , H ) = { 2 \pi ^ 2 \mu ( d _ 0 ) \over d _ 0 ^ 2 p } \sum _ { \delta \mid d _ 0 } \delta \mu ( \delta ) \sum _ { h \in H _ { d _ 0 } } s ( h , \delta p ) \end{align*}


\begin{align*} ( a ^ { \dagger } a ) _ { k , \lambda } = \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) ( a ^ { \dagger } ) ^ { l } a ^ { l } , ( k \in \mathbb { N } ) . \end{align*}


\begin{align*} \mu _ T : = \int _ 0 ^ T \delta _ { B _ t ^ H } \ , \d t , \end{align*}


\begin{align*} \Phi _ { \delta } ^ { x _ 0 } ( d , e , t ) = e ^ { \beta t } d + \varphi _ { \beta } ( t ) ( e + 2 B ) \end{align*}


\begin{align*} x = v + \sqrt { v ^ 2 - u } , \ ; \ ; y = v - \sqrt { v ^ 2 - u } . \end{align*}


\begin{align*} u _ x + \rho _ x + \lambda \rho = f , \end{align*}


\begin{align*} \dfrac { 1 } { \Gamma ( s ) } = 0 \mbox { f o r $ s = 0 , - 1 , - 2 , \ldots $ } . \end{align*}


\begin{align*} S _ n = \frac { 1 } { \sqrt { n } } \sum _ { k = 1 } ^ n X _ k , \end{align*}


\begin{align*} \limsup _ { M \rightarrow \infty } \limsup _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { | x | \geq M n } \sup _ { s \in [ 0 , T ] } | [ - ( - \Delta ) ^ { \gamma / 2 } G _ s ] ( \tfrac { x - 1 } { n } ) - [ - ( - \Delta ) ^ { \gamma / 2 } G _ s ] ( \tfrac { x } { n } ) | = 0 . \end{align*}


\begin{align*} Y ( \mathcal { A } ) _ { \omega \beta } ^ { i v } = \left ( Y ( \mathcal { A } ) _ { \alpha \kappa _ { 2 } } ^ { g v } \cdot Y ( \mathcal { A } ) _ { \kappa _ { 2 } \beta } ^ { g v } \right ) \cdot \left ( Y ( \mathcal { A } ) _ { \beta \kappa _ { 2 } } ^ { g i } \cdot Y ( \mathcal { A } ) _ { \kappa _ { 2 } \alpha } ^ { g i } \right ) \cdot Y ( \mathcal { A } ) _ { \omega \alpha } ^ { i v } \end{align*}


\begin{align*} u _ 0 ( x ) = | x | + 1 , \end{align*}


\begin{align*} D = \sum c _ i D _ i , \end{align*}


\begin{align*} N _ { f } ( p ) = c ( f , p ) \left ( \frac { \varphi ( p - 1 ) } { p - 1 } \right ) ^ 2 p + O \left ( p ^ { 1 - 2 \varepsilon } \right ) , \end{align*}


\begin{align*} \varphi ( a ^ 2 \cdot v _ 0 ) = [ \varphi ( a ) , a \cdot v _ 0 ] + [ a , \varphi ( a \cdot v _ 0 ) ] = 2 \lambda ( a \cdot v _ 0 ) a \cdot v _ 0 + [ a , [ \zeta , a ] ] , \end{align*}


\begin{align*} \int _ { \Omega \cap B _ { \gamma / 2 } } e ^ { \delta | w | } d x \leq \sum _ { j = 0 } ^ \infty \int _ { \Omega \cap B _ { \gamma / 2 } } \delta ^ j | w | ^ j / j ! \leq c \sum _ { j = 0 } ^ \infty ( \delta j ) ^ j / j ! \leq c ( d , \chi ) . \end{align*}


\begin{align*} g ( x _ k ) \leq g ( x ) = f ( x ) . \end{align*}


\begin{align*} V = c ^ { - \frac { n - 2 } { 4 } } \Delta _ g ( c ^ { \frac { n - 2 } { 4 } } ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \iint \frac { | u _ n ( x ) | ^ p | u _ n ( y ) | ^ p } { | x - y | ^ { N - \gamma } } d x d y = \lim _ { l \to \infty } \sum _ { j = 1 } ^ l \iint \frac { | U ^ j ( x ) | ^ p | U ^ j ( y ) | ^ p } { | x - y | ^ { N - \gamma } } d x d y . \end{align*}


\begin{align*} \Psi ( P ) : = \left \{ \mathbf { e } \in \mathbb { Z } ^ { d } : \ , \exists c _ { \mathbf { e } } \in \mathbb { C } ^ { \times } : c _ { \mathbf { e } } \mathbf { t } ^ { \mathbf { e } } \in \mathrm { S u p p } ( P ) \right \} . \end{align*}


\begin{align*} \nabla _ { j } \Phi & = \bar { g } ( \lambda \partial _ { r } , e _ { j } ) , \\ \nabla _ { i } \nabla _ { j } \Phi & = \lambda ' \delta _ { i j } - h _ { i j } \bar { g } ( \lambda \partial _ { r } , \nu ) = \lambda ' \delta _ { i j } - u h _ { i j } , \end{align*}


\begin{align*} P = P ^ T , Q = Q ^ T , D = Q L P + L ^ { - T } = C A ^ { - 1 } B + A ^ { - T } . \end{align*}


\begin{align*} \frac r a = \alpha _ 1 - \frac { 1 } { \alpha _ 2 - \frac { 1 } { \alpha _ 3 - \frac { 1 } { ( . . . ) } } } \eqqcolon [ \alpha _ 1 , . . . , \alpha _ n ] \end{align*}


\begin{align*} \R _ i ( t , y ) = Q _ { 1 + \mu _ i ( t ) } ( y - x _ i ( t ) ) . \end{align*}


\begin{align*} 0 < \eta \le \inf \Big \{ | A ( z ) | : z \notin \bigcup _ { j = 1 } ^ { n } \mathfrak B _ j \Big \} . \end{align*}


\begin{align*} K _ { \alpha \alpha } ^ { ( 0 ) } = \frac { 1 } { n _ 0 } \norm { x _ \alpha } ^ 2 = K = \frac { 1 } { n _ 0 } \norm { x _ \beta } ^ 2 = K _ { \beta \beta } ^ { ( 0 ) } , K > 0 . \end{align*}


\begin{align*} I ( A ( x , y ) , z ) = I ( x , I ( y , z ) ) . \end{align*}


\begin{align*} \max \big \{ \max \mathsf L ( a ) / \min \mathsf L ( a ) \colon a \in S \big \} = n _ t / n _ 1 \min \Delta ( S ) = \gcd ( n _ 2 - n _ 1 , \ldots , n _ t - n _ { t - 1 } ) \ , . \end{align*}


\begin{align*} \int _ \gamma g _ 0 \ , d s _ \rho = \int _ \gamma g _ 0 \ , \rho \ , d s \end{align*}


\begin{align*} \max ( A ) - \max ( B ) & = \max ( C ) - \max ( D ) \ , \\ \min ( A ) - \min ( B ) & = \min ( C ) - \min ( D ) \ , . \end{align*}


\begin{align*} E _ 0 = \int _ { \Omega } \bigg ( \sum _ { i = 1 } ^ s \rho _ i \log \rho _ i + \frac { 1 } { 2 } ( f + \sum _ { i = 1 } ^ s z _ i \rho _ i ) \phi \bigg ) d x , \end{align*}


\begin{align*} \rho ( F _ r ) & = \iint _ { \R ^ { 2 d } } \int _ 0 ^ 1 F ( x , \omega ) e ^ { - 2 \pi i \tau } \rho ( x , \omega ) e ^ { 2 \pi i \tau } \ , d \tau \ , d ( x , \omega ) \\ & = \iint _ { \R ^ { 2 d } } F ( x , \omega ) \rho ( x , \omega ) \ , d ( x , \omega ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta u & = \varepsilon ^ { - 2 } \left ( e ^ { u } - \left ( x ^ { 2 } + y ^ { 2 } \right ) e ^ { - u } \right ) & \Omega , \\ u & = g & \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} V ^ \omega : = \{ w \in T _ y M \ | \ \omega _ y ( w , v ) = 0 , \ \forall v \in V \} \end{align*}


\begin{align*} g ( x - \alpha \nabla g ( x ) ) & \leq g ( x ) + \langle - \alpha \nabla g ( x ) , \nabla g ( x ) \rangle + \dfrac { L _ g } { 2 } \norm { \alpha \nabla g ( x ) } ^ 2 \\ & = g ( x ) + \ ! \left ( \dfrac { \alpha L _ g } { 2 } - 1 \right ) \ ! \alpha \norm { \nabla g ( x ) } ^ 2 . \end{align*}


\begin{align*} x _ 1 ^ \prime = a ^ 2 \varepsilon ^ 4 x _ 2 \ , , x _ 2 ^ \prime = \frac { \varepsilon } { a } x _ 1 \ , , y _ 1 ^ \prime = \varepsilon ^ 3 y _ 2 \ , , y _ 2 ^ \prime = \frac { \varepsilon ^ 2 } { a } y _ 1 \ , , \end{align*}


\begin{align*} | S | = K \frac { n ! } { 2 } = \sum _ { \substack { g \in \mathrm { S y m } ( n ) \\ | g | \ \mathrm { e v e n } } } 2 ^ { c ( g _ E ) - 1 } \end{align*}


\begin{align*} \Phi ( S ) = \underset { v \sim \pi _ k | _ S } { \mathbb { E } } \left [ M ( v , X ( k ) \setminus S ) \right ] , \end{align*}


\begin{align*} D _ { m } = \hat { D } _ { m } \cap \bar { D } _ { m } \end{align*}


\begin{align*} \frac { 1 } { 1 - x ( e _ { \lambda } ( t ) - 1 ) } = \sum _ { n = 0 } ^ { \infty } F _ { n , \lambda } ( x ) \frac { t ^ { n } } { n ! } , ( \mathrm { s e e } \ [ 1 2 ] ) . \end{align*}


\begin{align*} \left ( \forall y \in A \otimes ^ I B : x y = 0 \right ) \Rightarrow x E = 0 , \left ( \forall y \in A \otimes ^ I B : y x = 0 \right ) \Rightarrow E x = 0 . \end{align*}


\begin{align*} \mathcal { U } ( h , k ) : = \frac { 1 } { 2 } \sum _ { \substack { 1 \leq q < \infty \\ ( q , h k ) = 1 } } W \left ( \frac { q } { Q } \right ) \sum _ { \substack { 1 \leq m , n < \infty \\ ( m n , q ) = 1 \\ m h \neq n k } } \frac { \tau _ A ( m ) \tau _ B ( n ) } { \sqrt { m n } } V \left ( \frac { m } { X } \right ) V \left ( \frac { n } { X } \right ) \sum _ { \substack { 1 \leq c \leq C , d \geq 1 \\ c d = q \\ d | m h \pm n k } } \phi ( d ) \mu ( c ) , \end{align*}


\begin{align*} p ( x ) = \phi ( g _ { j ( 1 ) } ( x ) , \ldots , g _ { j ( d ) } ( x ) ) , \deg ( p ) \leq \delta , \end{align*}


\begin{align*} [ x , y , z ] = [ [ x , y ] , z ] , \forall x , y , z \in L . \end{align*}


\begin{align*} \Delta _ { \ell } ^ { a v } ( t ) = \frac { 1 } { t } \int _ { 0 } ^ { t } \Delta _ { \ell } ( i ) d i . \end{align*}


\begin{align*} \mathcal { M } \left ( \lambda \right ) = \lambda ^ 2 G + \lambda D + C , \lambda \in \mathbf { C } , \end{align*}


\begin{align*} F ( r , t ) : = F ( f , r , t ) = \sup _ { s > 0 } \left [ \frac { r } { s } - \frac { t } { f ( s ) } \right ] . \end{align*}


\begin{align*} \lim _ { M \to \infty } \mu ( \mathbb { E } ^ M ) = \lim _ { n \to \infty } \mu ( \mathbb { P } ^ M ) = 0 , \end{align*}


\begin{align*} M = \left \{ f : \Omega \rightarrow \R , \ ; \ ; \left \| \frac { \nabla ^ r f } { g } \right \| _ { L _ { p _ 1 } ( \Omega ) } \le 1 , \ ; \ ; \| w f \| _ { L _ { p _ 0 } ( \Omega ) } \le 1 \right \} , \end{align*}


\begin{align*} \nabla _ { \mathbf { x } } \mathcal { L } _ k ( \mathbf { x } , \mathbf { y } ) = ( H + \rho I ) \mathbf { x } - A ^ T \mathbf { y } + \mathbf { g } - \rho { \mathbf { x } _ k } - \begin{bmatrix} 0 \\ \frac { \mu } { x _ { \bar { d } + 1 } } \\ \vdots \\ \frac { \mu } { x _ { d } } \end{bmatrix} = 0 ; \\ - \nabla _ { \mathbf { y } } \mathcal { L } _ k ( \mathbf { x } , \mathbf { y } ) = ( A \mathbf { x } + \delta ( \mathbf { y } - \mathbf { y } _ k ) - \mathbf { b } ) = 0 . \end{align*}


\begin{gather*} e _ { \infty } ( K / k ) = \frac { l ^ n } { \gcd \big ( \frac { d _ P } w \frac { { \eta } ^ w - 1 } { { \eta } - 1 } , l ^ n \big ) } . \end{gather*}


\begin{align*} \left ( \begin{array} { c c c c } A _ { 1 1 } & A _ { 1 2 } & \cdots & A _ { 1 n } \\ A _ { 2 1 } & A _ { 2 2 } & \cdots & A _ { 2 n } \\ \vdots & \vdots & \ddots & \vdots \\ A _ { n 1 } & A _ { n 2 } & \cdots & A _ { n n } \end{array} \right ) \left ( \begin{array} { c c c c } C _ 1 \\ C _ 2 \\ \vdots \\ C _ n \end{array} \right ) = \left ( \begin{array} { c c c c } X _ 1 \\ X _ 2 \\ \vdots \\ X _ n \end{array} \right ) . \end{align*}


\begin{align*} \mathfrak L ( U , \iota ) = \mathfrak L ( | U | , \iota ) = \{ \mathbf L \in \mathfrak L ( U ) : \mu _ U ( \mathbf L ) \geq \iota \} \ , . \end{align*}


\begin{align*} \widehat B _ t ^ N ( k , t ) = i \sum _ { m + n = k , | m | , | n | , | k | \leq N } m | n | \widehat B ( m ) \widehat B ( n ) - \mu | k | ^ \alpha \widehat B ( k ) . \end{align*}


\begin{align*} \sum _ { \substack { m \geq 1 \\ ( m , 2 ) = 1 } } \frac { \lambda _ f ( l _ 1 m ^ 2 ) } { m ^ { 1 + 2 s } } = \lambda _ f ( l _ 1 ) \zeta ( 2 + 4 s ) ^ { - 1 } L ( 1 + 2 s , \operatorname { s y m } ^ 2 f ) \prod _ { p | 2 l } \left ( 1 + O \left ( \frac { 1 } { p ^ { 1 + 2 s } } \right ) \right ) ^ { - 1 } \end{align*}


\begin{align*} | S | & \leqslant \int _ { \| u - y \| \leqslant t } | f _ t ( x , u ) - f _ t ( x , y ) | \cdot | g _ s ( u , y ) | d \omega ( u ) \\ & \quad + \int _ { \| u - y \| > t } \big ( | f _ t ( x , u ) | + | f _ t ( x , y ) | \big ) \cdot | g _ s ( u , y ) | d \omega ( u ) \\ & = : I + I \ ! I , \end{align*}


\begin{align*} c _ j = \inf _ { z \in Z _ j ( V _ j ) } \abs { H _ j ' ( z ) } \quad M _ j = \sup _ { ( s , t ) \in V _ j } \abs { D \Phi _ j ( s , t ) } \end{align*}


\begin{align*} - \Delta Q + Q - \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ { p - 1 } = 0 . \end{align*}


\begin{align*} \left \{ \aligned & \partial _ { t } u + ( u \cdot \nabla ) u + \nabla p = ( b \cdot \nabla ) b , x \in \mathbb { R } ^ { 2 } , \ , t > 0 , \\ & \partial _ { t } b + ( u \cdot \nabla ) b + \mathcal { L } b = ( b \cdot \nabla ) u , \\ & \nabla \cdot u = 0 , \ \ \ \nabla \cdot b = 0 , \\ & u ( x , 0 ) = u _ { 0 } ( x ) , \ , \ , b ( x , 0 ) = b _ { 0 } ( x ) , \endaligned \right . \end{align*}


\begin{align*} \alpha \left [ \begin{array} { c c } z & 1 \\ 1 & - \overline { z } \end{array} \right ] = \left [ \begin{array} { c c } \alpha z & 1 \\ 1 & - \overline { \alpha z } \end{array} \right ] \left [ \begin{array} { c c } \lambda ( \alpha , z ) & 0 \\ 0 & \overline { \lambda ( \alpha , z ) } \end{array} \right ] , \end{align*}


\begin{align*} f _ k ( Q ) - f _ k ( \pi _ v ( Q ) ) \geq { \lceil \frac { d } { 2 } \rceil \choose d - k - 1 } + { \lfloor \frac { d } { 2 } \rfloor \choose d - k - 1 } = 2 \rho ( d , d - k - 1 ) . \end{align*}


\begin{align*} \max _ { j \in \lbrace 1 , \ldots , \epsilon \rbrace } \frac { \norm { E _ j ^ \intercal ( A x _ 0 - b ) } _ 2 ^ 2 } { \norm { A x _ 0 - b } _ 2 ^ 2 \norm { A ^ \intercal E _ j } _ F ^ 2 } \geq \sum _ { j = 1 } ^ \epsilon \frac { \norm { A ^ \intercal E _ j } _ F ^ 2 } { \norm { A } _ F ^ 2 } \frac { \norm { E _ j ^ \intercal ( A x _ k - b ) } _ 2 ^ 2 } { \norm { A x _ 0 - b } _ 2 ^ 2 \norm { A ^ \intercal E _ j } _ F ^ 2 } = \frac { 1 } { \norm { A } _ F ^ 2 } . \end{align*}


\begin{align*} \norm { f - \sum _ { k = 1 } ^ N e _ { \pi ( k ) } } _ \mathcal { B } < \varepsilon , N \geq N _ 0 . \end{align*}


\begin{align*} \Delta | \nabla u | ^ 2 & = < \nabla \Delta u , \nabla u > + | \nabla ^ 2 u | ^ 2 \\ & = \varepsilon ^ { - 2 } ( e ^ u + r ^ 2 e ^ { - u } ) | \nabla u | ^ 2 - \varepsilon ^ { - 2 } e ^ { - u } < \nabla r ^ 2 , \nabla u > + | \nabla ^ 2 u | ^ 2 . \end{align*}


\begin{align*} \begin{array} { l c l } \sum _ { i , j = 1 } ^ { 2 } \langle [ R ( X _ i , X _ j ) , A ] X _ i , A X _ j \rangle & = & \sum _ { i , j = 1 } ^ { 2 } \langle [ R ( \widetilde X _ i , \widetilde X _ j ) , A ] \widetilde X _ i , A \widetilde X _ j \rangle \\ \\ & = & - \langle R ( \widetilde X _ 1 , \widetilde X _ 2 ) \widetilde X _ 1 , \widetilde X _ 2 \rangle ( \lambda _ 1 - \lambda _ 2 ) ^ 2 \\ \\ & = & 2 K ( | A | ^ 2 - 2 f ^ 2 ) , \end{array} \end{align*}


\begin{align*} \Omega _ { \perp \star } : = \{ \omega \in \Omega _ \star ~ | ~ \langle \Xi _ { t \star } , \omega \rangle _ \star = 0 \} , \end{align*}


\begin{align*} u _ t + | \nabla u | m ( K \cap \{ u ( \cdot , t ) < u ( x , t ) \} ) = 0 , \end{align*}


\begin{align*} \Psi ^ { - 1 } | _ { t = \theta } = \begin{pmatrix} \tilde { \pi } ^ s & 0 \\ - \tilde { \pi } ^ s L i _ { K , s } ( \alpha ) & \tilde { \pi } ^ s \end{pmatrix} . \end{align*}


\begin{align*} \mathcal A ( \rho , m ) = \frac 1 2 \int _ 0 ^ 1 \Big ( \frac { m _ { 1 2 } ^ 2 } { \theta _ { 1 2 } ( \rho ) } + \frac { m _ { 2 3 } ^ 2 } { \theta _ { 2 3 } ( \rho ) } \Big ) d t , \end{align*}


\begin{align*} I ( \alpha _ { k _ { n } } ) - J _ { 0 } ( \alpha _ { k _ { n } } ) = 2 n c _ { s \gamma } + 2 \lfloor n \theta \rfloor + 1 \end{align*}


\begin{align*} 0 \ne g \cdot v = g \cdot ( u + a e _ t ) = g \cdot u + g \cdot a e _ t = a g _ t . \end{align*}


\begin{align*} ( \lambda + 2 \nu ) \partial _ { 1 1 } \phi ^ 1 + ( \lambda + \nu ) \partial _ { 1 2 } \phi ^ 2 + \nu \partial _ { 2 2 } \phi ^ 1 & = 0 , \\ \nu \partial _ { 1 1 } \phi ^ 2 + ( \lambda + \nu ) \partial _ { 1 2 } \phi ^ 1 + ( \lambda + 2 \nu ) \partial _ { 2 2 } \phi ^ 2 & = 0 . \end{align*}


\begin{align*} \Big ( \lambda + c _ \alpha \ , \lambda ^ 2 \epsilon _ n \Big ) \Big ( 1 + \delta _ n ( \lambda ) \Big ) = x \sqrt { 1 - \delta _ n ( \lambda ) } . \end{align*}


\begin{align*} \mathcal { E } _ m ( g , \tilde { f } _ m ) = \langle f ; g \rangle _ { L ^ 2 ( \mu _ m ) } , \tilde { f } _ m = \int _ 0 ^ { + \infty } \mathcal { P } ^ m _ t ( f ) d t . \end{align*}


\begin{align*} \left \langle E ^ { i j } _ 0 , ( { m _ { \Lambda } } _ * f ) \otimes ( { m _ { \Lambda } } _ * g ) \right \rangle = \Lambda ^ { 2 - \mu _ i + \mu _ j } \left ( \mathfrak { P } _ { \ell } m _ { \Lambda } \left \{ \Psi ^ i ( f ) , \Psi ^ j ( g ) \right \} _ { \ell } ^ { \Sigma _ 0 } \right ) [ 0 ] . \end{align*}


\begin{align*} P ( x ) = A x ^ 4 + B x ^ 3 + C x ^ 2 + 1 \end{align*}


\begin{align*} \frac { f ' ( \gamma ) } { f ( \gamma ) } = - a + \frac { 1 } { \gamma } - a \coth ( \gamma a ) = - a + \frac { 1 } { \gamma } \underbrace { ( 1 - \gamma a \coth ( \gamma a ) ) } _ { < 0 , \ , \searrow } , \end{align*}


\begin{align*} \mathcal { K } ( s _ 1 , - s _ 1 - \alpha - \beta , 2 ) = \left ( \frac { h } { k } \right ) ^ { s _ 1 } \mathcal { K } ( 0 , - \alpha - \beta , 2 ) . \end{align*}


\begin{align*} & \bullet \limsup _ { B \in \mathcal { Q } } B = [ 0 , 1 ] . \\ & \bullet \delta \geq 1 , \dim _ H ( \limsup _ { B \in \mathcal { Q } } B ^ { \delta } ) = \frac { 1 } { \delta } . \end{align*}


\begin{align*} k n & < e ( S _ 1 ( x ) , L _ 1 ( x ) \setminus \{ z \} ) + | N _ 1 ( x ) \cap N _ 1 ( z ) | \mathbf { \mathrm { v } } _ z + \frac { \epsilon ^ 2 n } { 2 } \\ & \leq ( k + \epsilon ) n - | N _ 1 ( x ) \cap N _ 1 ( z ) | + \left ( 1 - \frac { 1 } { 1 6 k ^ 3 } \right ) | N _ 1 ( x ) \cap N _ 1 ( z ) | + \frac { \epsilon ^ 2 n } { 2 } \\ & = k n + \epsilon n + \frac { \epsilon ^ 2 n } { 2 } - \frac { | N _ 1 ( x ) \cap N _ 1 ( z ) | } { 1 6 k ^ 3 } , \end{align*}


\begin{align*} u ( x ) = \inf \left \lbrace \int _ 0 ^ \infty e ^ { - s } \left ( C _ p | \dot { \gamma } ( s ) | ^ { q } + f ( \gamma ( s ) ) \right ) d s : \gamma \in \mathrm { A C } ( [ 0 , \infty ) ; \overline { \Omega } ) , \gamma ( 0 ) = x \right \rbrace , \end{align*}


\begin{align*} \bar \varphi ( z ) = \frac { \varphi ( z _ - ) ( z _ + - z ) + \varphi ( z _ + ) ( z - z _ - ) } { z _ + - z _ - } . \end{align*}


\begin{align*} \lim _ { t \to \infty } \sum _ { \ell = 1 } ^ { N } \frac { \gamma _ \ell R _ \ell ^ \pi ( t ) } { t } \le 1 , \ \ \end{align*}


\begin{align*} ( - \Delta ) ^ s w + c w & = ( - \Delta ) ^ s v + c v + ( - \Delta ) ^ s ( w - v ) \\ & \geqslant - C \bigg ( 1 + \tau d ^ { - n - 2 } + ( \theta d ) ^ { - n - 2 s - 2 } \bigg ) x _ 1 \\ & \geqslant - C \bigg ( ( \theta d ) ^ { - n - 2 s - 2 } + \tau d ^ { - n - 2 } \bigg ) x _ 1 \end{align*}


\begin{align*} \int _ { H _ \rho } \frac { ( y _ 1 + 1 / \rho ) \tilde u ( y ) } { \vert Q _ \rho ( a ) - y \vert ^ { n + 2 s + 2 } } \dd y & = \rho ^ { - n - 1 } \int _ { \R ^ n _ + } \frac { z _ 1 u ( z ) } { \vert Q _ \rho ( a ) - z / \rho + e _ 1 / \rho \vert ^ { n + 2 s + 2 } } \dd z . \end{align*}


\begin{align*} I ( y ) : = \{ t \in [ 0 , T ] \colon | y ( t ) | < r \} , \end{align*}


\begin{align*} \phi = \phi ^ \sigma , \epsilon ( \phi ) = 0 , \langle \phi , 1 \rangle = 0 , \phi ^ 2 + \phi * \phi = \theta \phi + | | \phi | | _ 2 ^ 2 \delta _ e , \end{align*}


\begin{align*} \mathrm { v a r } \big [ \widehat { K } _ { \mathrm { a - c p t - f } } ^ \mathbb { R } \big ] = \upsilon ^ 2 \bigg ( \ ! \Big ( 1 \ ! - \ ! \frac { \pi } { 4 } \Big ) \ ! \sum _ { i \in \mathcal { K } } \ ! \Big ( 1 \ ! + \ ! \frac { 1 } { N _ 1 \bar { \gamma } _ i } \Big ) \ ! + \ ! \sum _ { i \in \mathcal { K } } \ ! \frac { 1 } { 2 N _ 1 \bar { \gamma } _ i } \ ! + \ ! \frac { 1 } { 2 N _ 2 \bar { \gamma } _ c } \ ! \bigg ) , \end{align*}


\begin{align*} \liminf _ { n \to \infty } \norm { x _ n - x _ { n + 1 } } / \mu _ n = 0 . \end{align*}


\begin{align*} m _ 1 = \lim _ { n \rightarrow + \infty } J ( w _ n ) & \geq \liminf _ { n \rightarrow + \infty } J ( t _ { \widehat { w } ^ + } w ^ + _ n + s _ { \widehat { w } ^ - } w ^ - _ n ) \\ & \geq J ( t _ { \widehat { w } ^ + } \widehat { w } ^ + + s _ { \widehat { w } ^ - } \widehat { w } ^ - ) \\ & \geq m _ 1 . \end{align*}


\begin{align*} \ker L _ + = \{ \partial _ { x _ 1 } Q , . . . , \partial _ { x _ N } Q \} L _ + ( \ref { l i n e a r i z e d o p e r a t o r L + } ) , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { t \in [ 0 , T ] } \| Z ^ m ( t ) - Z ^ { n } ( t ) \| _ { L ^ p ( \Omega ; H ^ \beta ) } = & \lim _ { n \rightarrow \infty } \sup _ { t \in [ 0 , T ] } \| P ^ m ( I - P _ n ) Z ( t ) \| _ { L ^ p ( \Omega ; H ^ \beta ) } \\ \leq & C \lim _ { n \rightarrow \infty } \lambda _ { n + 1 } ^ { - \frac { \gamma - \beta } 2 } \sup _ { t \in [ 0 , T ] } \| Z ( t ) \| _ { L ^ p ( \Omega ; H ^ \gamma ) } . \end{align*}


\begin{align*} h _ \lambda ( t ) = h ( - i \omega _ 1 ( i t ) ) = h ( W ( t ) ) . \end{align*}


\begin{align*} \phi ( x , t ) : = \sigma \left ( \max \{ \gamma ( x ) - C t , 0 \} \right ) , \end{align*}


\begin{align*} | A f ( x , \omega ) | < A f ( 0 , 0 ) = \norm { f } _ 2 ^ 2 , \end{align*}


\begin{align*} \begin{aligned} | X _ t - X _ s | ^ { p } & = \int _ s ^ t p \cdot | X _ u - X _ s | ^ { p - 2 } \cdot \bigl ( ( X _ u - X _ s ) \cdot \mu ( X _ u ) + \frac { 1 } { 2 } ( p - 1 ) \cdot \sigma ^ 2 ( X _ u ) \bigr ) \ , d u \\ & + \int _ s ^ t p \cdot ( X _ u - X _ s ) | X _ u - X _ s | ^ { p - 2 } \sigma ( X _ u ) \ , d W _ u . \end{aligned} \end{align*}


\begin{align*} D ^ j \left ( X ^ { q ( m + 2 ) } \right ) = \frac { A ^ m _ q } { A ^ m _ { q - j } } X ^ { ( q - j ) ( m + 2 ) } \ , . \end{align*}


\begin{align*} I _ 2 \leq & \sum _ { q \geq 0 } \int _ 0 ^ t c _ q \lambda _ q ^ { \alpha - \beta } e ^ { - \mu \lambda _ q ^ \alpha ( t - \tau ) } \| B ( \tau ) \| ^ 2 _ { H ^ { \frac 5 2 - \alpha + \beta } } \ , d \tau \\ \lesssim & \int _ 0 ^ t ( t - \tau ) ^ { - 1 + \frac { \beta } { \alpha } } \| B ( \tau ) \| ^ 2 _ { H ^ { \frac 5 2 - \alpha + \beta } } \ , d \tau . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty p ( n ) X ^ n = \alpha ^ { - 1 } = \frac { ( \alpha ^ 3 ) ^ 3 } { ( \alpha ^ 5 ) ^ 2 } \equiv \frac { ( \alpha _ 0 + \alpha _ 1 ) ^ 3 } { \alpha ( X ^ 5 ) ^ 2 } \pmod { 5 } . \end{align*}


\begin{align*} w ( x , y ) = \sum _ { j = 0 } ^ \infty \alpha _ j ( x ) y ^ j \ , , \end{align*}


\begin{align*} \dim _ K ( E \cap I ) _ d & = \dim _ K I _ d - \dim _ K ( x _ 1 ^ 2 , \ldots , x _ n ^ 2 ) _ d \\ & = ( \dim _ K R _ d - 1 ) - \dim _ K ( x _ 1 ^ 2 , \ldots , x _ n ^ 2 ) _ d \\ & = \dim _ K E _ d - 1 . \end{align*}


\begin{align*} \widehat { K '' } _ \mathrm { a - c p t - d } ^ \mathbb { R } \stackrel { K _ l \rightarrow \infty } { = } \xi \big ( 1 + \widehat { K ' } _ \mathrm { a - c p t - d } ^ \mathbb { R } \big ) / ( 1 - \xi ) ^ 2 \end{align*}


\begin{align*} \langle ( P _ i ^ { \perp } + z P _ i ) \xi , U _ i ^ * \xi \rangle = z _ i . \end{align*}


\begin{align*} V = \{ ( z , w ) \in \mathbb D ^ 2 \ , : \ , \det ( A ^ * + A z w - ( z + w ) I ) = 0 \} , \end{align*}


\begin{align*} \Delta \circ \sigma ^ { \varphi } = ( S ^ 2 \otimes \sigma ^ { \varphi } ) \circ \Delta , \Delta \circ \sigma ^ { \psi } = ( \sigma ^ { \psi } \otimes S ^ { - 2 } ) \circ \Delta \end{align*}


\begin{align*} \begin{cases} & - \mathcal L \hat { V } - g \geq 0 , \\ & \hat { V } - V ^ * \geq 0 , \\ & ( - \mathcal L \hat { V } - g ) \cdot ( \hat { V } - V ^ * ) = 0 , \end{cases} \end{align*}


\begin{align*} \left | \phi _ { i , o u t , \beta } ( y ) - \sum _ { j = 0 } ^ i c _ { i , j } ( \sqrt { b } ) ^ { 2 j - \gamma } T _ { j } \left ( \frac { y } { \sqrt { b } } \right ) \right | \le C y ^ { 2 i + 2 - \gamma } | \ln y | b ^ { 1 - \frac { \epsilon } { 2 } } . \end{align*}


\begin{align*} \norm { f - \sum _ { k = 1 } ^ N e _ { \pi ( k ) } } _ \mathcal { B } < \frac { \varepsilon } { 2 } , \forall N \geq N _ 0 . \end{align*}


\begin{align*} z ^ { n } = \sum _ { k = 0 } ^ { n } \binom { n } { k } A _ { k } ^ { \lambda , \beta } ( x ) M _ { \lambda , \beta } ( n - k ) . \end{align*}


\begin{align*} \bar { g } ( \nabla P , \nabla \log u ) = \bar { g } ( \lambda \partial _ { r } , \nabla \log u ) - u ^ { \frac { \alpha + 1 } { \alpha } } | \nabla \log u | ^ { 2 } . \end{align*}


\begin{align*} I [ g _ 1 , g _ 2 , g _ 3 ] ( t , k ) = \frac { \pi } { | t | } e ^ { - i t k ^ 2 } \int e ^ { i t ( - p + \epsilon _ 0 k ) ^ 2 } g _ 1 ( \epsilon _ 1 ( - p + \epsilon _ 0 k ) ) \overline { g _ 2 ( \epsilon _ 2 ( - p + \epsilon _ 0 k ) ) } \\ \times g _ 3 ( \epsilon _ 3 ( - p + \epsilon _ 0 k ) ) \mathrm { p . v . } \frac { \widehat { \phi } ( p ) } { p } \ , d p + \mathcal { O } ( | t | ^ { - 1 - \rho } ) \end{align*}


\begin{align*} f ( t ) = e ^ { - \pi t \cdot A t + 2 \pi b \cdot t + c } , \end{align*}


\begin{align*} - \dot { \theta } \| Q \| _ 2 ^ 2 = O ( \delta + \delta \delta ^ * ) \ ; \Rightarrow \ ; | \dot { \theta } | = O ( \delta + \delta \delta ^ * ) . \end{align*}


\begin{align*} \gamma ^ 2 - d \gamma + 3 ( d - 2 ) = 0 . \end{align*}


\begin{align*} B _ { I _ j } ( v ) \cap B ' _ { I _ { j ' } } = \emptyset \mbox { f o r a l l } 1 \leq j , j ' \leq \ell \mbox { a n d } B ' _ { I _ j } \cap B ' _ { I _ { j ' } } = \emptyset \mbox { f o r } 1 \leq j \neq j ' \leq \ell \ , . \end{align*}


\begin{align*} \langle z | ( a ^ { \dagger } a ) _ { k , \lambda } | z \rangle & = \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) \langle z | ( a ^ { \dagger } ) ^ { l } a ^ { l } | z \rangle = \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) ( \overline { z } ) ^ { l } z ^ { l } \langle z | z \rangle \\ & = \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) | z | ^ { 2 l } = \phi _ { k , \lambda } ( | z | ^ { 2 } ) . \end{align*}


\begin{align*} x - ( t - s ) v - X ( s ) = \check x _ { t , x , v } - ( \mathcal T _ { t , x , v } - s ) v + X ( \mathcal T _ { t , x , v } ) - X ( s ) . \end{align*}


\begin{align*} \Sigma _ 0 ^ { \sigma _ 1 } = 2 \sum _ { \substack { 0 < \gamma < T \\ 0 \le \beta < \sigma _ 1 } } \frac { x ^ { \beta - 1 } } { \gamma } \le \varepsilon _ 2 ( x , \sigma _ 1 , T _ 0 , T ) , \end{align*}


\begin{align*} \begin{cases} - \Delta h _ { \varepsilon } + \tilde { \lambda } _ 0 \underline { m } \ , h _ { \varepsilon } = 0 & \Omega _ { \varepsilon } \ ; , \\ h _ { \varepsilon } = w & \partial \Omega _ { \varepsilon } \ ; , \end{cases} \end{align*}


\begin{align*} s ( | \lambda | , 0 ) ^ 2 + | \lambda | ^ 2 = \frac { s ( | \lambda | , 0 ) } { h ( s ( | \lambda | , 0 ) ) } , \end{align*}


\begin{align*} \eta _ s ( t ; x ) : = \xi ( s + t ; s , x ) , t \in [ 0 , u - s ] . \end{align*}


\begin{align*} U _ P = \left \{ \begin{pmatrix} I & P \\ 0 & I \end{pmatrix} \mid P = P ^ T \right \} , V _ Q = \left \{ \begin{pmatrix} I & 0 \\ Q & I \end{pmatrix} \mid Q = Q ^ T \right \} \end{align*}


\begin{align*} d _ k ( u , v ) = \| u - v \| _ { X ^ k } . \end{align*}


\begin{align*} { \rm s p } _ 0 ( \tau ) = p ^ e \big ( \mathbb { N } + \frac { n } { 2 } \big ) + \frac { 1 } { 2 } { \bigcup } _ { z = 0 } ^ { e } \Lambda _ z = p ^ e \mathbb { N } + \big \{ \overline { \lambda } ( x ) \colon 0 \le x < p ^ e \big \} . \end{align*}


\begin{align*} \int _ 0 ^ 1 \overline { \eta ( x ) } \eta ^ \prime ( x ) d x = i \int _ 0 ^ 1 \Im ( \overline { \eta ( x ) } \eta ^ \prime ( x ) ) d x . \end{align*}


\begin{align*} \begin{cases} c ( \mu , d , s ) \mathcal { H } ^ s _ { \infty } ( \Omega ) \leq \mathcal { H } ^ { \tilde { \mu } , s } _ { \infty } ( \Omega ) \leq \mathcal { H } ^ { s } _ { \infty } ( \Omega ) s < \alpha \\ \mathcal { H } ^ { \tilde { \mu } , s } _ { \infty } ( \Omega ) = 0 s > \alpha . \end{cases} \end{align*}


\begin{align*} s _ { \cdot 0 } = a _ { i 0 } , s _ { \cdot j } = \sum _ { k = 0 } ^ n \min \{ a _ { \cdot k } , a _ { \cdot j } \} \lambda _ { \cdot k } j = 1 , \ldots , n . \end{align*}


\begin{align*} a _ j = \sum \limits _ { i = 1 } ^ { j } ( - 1 ) ^ i { j \choose i } _ q b _ i , \end{align*}


\begin{align*} & M \left ( a ( \mu + \alpha ) ^ 2 + b ( \mu + \alpha ) \right ) - M \left ( a ( \mu + M M ^ * \alpha ) ^ 2 + b ( \mu + M M ^ * \alpha ) \right ) & & \\ = \ , & M \left ( ( 1 - ( M M ^ * ) ^ 2 ) a \alpha ^ 2 + ( 1 - M M ^ * ) ( b + 2 a \mu ) \alpha \right ) & & \\ \equiv \ , & ( 1 - M M ^ * ) \left ( ( 1 + M M ^ * ) a \alpha ^ 2 + b \alpha \right ) & & \bmod k \Z . \end{align*}


\begin{align*} u _ { k + 1 } = u _ { k } - s \nabla f \left ( u _ { k } \right ) + \gamma \left ( u _ { k } - u _ { k - 1 } \right ) , k = 1 , 2 , \ldots , \end{align*}


\begin{align*} \Phi \natural \Phi ^ { \xi , \eta } = e ^ { - \tfrac { \pi } { 2 } ( \xi ^ 2 + \eta ^ 2 ) } \Phi . \end{align*}


\begin{align*} J _ { 0 } ( \alpha , \beta , Z ) + J _ { 0 } ( \beta , \gamma , Z ' ) = J _ { 0 } ( \alpha , \gamma , Z + Z ' ) . \end{align*}


\begin{align*} & \dot { \widetilde \rho _ 2 } + \widetilde m _ { 2 3 } ( 1 + \dot W ^ { \delta } ) = 0 , \\ & \dot { \widetilde \rho _ 3 } + \widetilde m _ { 3 2 } ( 1 + \dot W ^ { \delta } ) = 0 . \end{align*}


\begin{align*} ( x , 0 , 1 ) \circ ( 0 , \omega , 1 ) = ( x , \omega , e ^ { - \pi i x \cdot \omega } ) ( 0 , \omega , 1 ) \circ ( x , 0 , 1 ) = ( x , \omega , e ^ { \pi i x \cdot \omega } ) . \end{align*}


\begin{align*} \begin{aligned} c _ 0 f \left ( \xi ( t _ 1 ) \right ) & \leq \left ( T _ { x , \xi } - t _ 1 \right ) p ( 1 - c _ 0 ) f ( \xi ( t _ 1 ) ) \\ \frac { c _ 0 } { p ( 1 - c _ 0 ) } & \leq T _ { x , \xi } - t _ 1 \\ \frac { c _ 0 } { p ( 1 - c _ 0 ) } & \leq T _ { x , \xi } . \end{aligned} \end{align*}


\begin{align*} \Omega _ 1 & = \{ x \in \Omega : \rho ( t , x ) \in [ 0 , \alpha _ 1 ] \} , \\ \Omega _ 2 & = \{ x \in \Omega : \rho ( t , x ) \in ( \alpha _ 1 , \bar \rho - \alpha _ 1 ] \} , \\ \Omega _ 3 & = \{ x \in \Omega : \rho ( t , x ) \in ( \bar \rho - \alpha _ 1 , \bar \rho ) \} , \\ \end{align*}


\begin{align*} d _ { q - 1 } \tilde p = ( I - \varPhi _ q \ , d _ q ) \Big ( \varPsi _ { \mu , q } f + \varPsi _ { \mu , q , 0 } u _ 0 - \varPsi _ \mu B _ q ( w , u ) \Big ) . \end{align*}


\begin{align*} \nabla ( \tilde { h } _ R ) ( x ) = \frac { 1 } { R } \nabla ( \psi ) ( \frac { x } { R } ) h ( x ) + \psi \left ( \frac { x } { R } \right ) \nabla ( h ) ( x ) . \end{align*}


\begin{align*} \mu _ { j k } = \frac { a _ { j k } } { \frac { 1 } { 2 } \pi \rho R ^ 2 } , \nu _ { j k } = \frac { b _ { j k } } { \frac { 1 } { 2 } \pi \rho \omega R ^ 2 } , ( j , k ) = 1 , 3 . \end{align*}


\begin{align*} \mathcal { E } = \left \{ u \in C ^ 0 ( \bar \Omega ) \ , | \ , u \ge 0 , \ , \ , { \int _ \Omega u ( x ) d x \le M _ 0 ^ * } , \int _ \Omega u ^ { - p } ( x ) d x \le M _ 1 ^ * , \ , \ , \int _ \Omega u ^ { q } ( x ) d x \le M _ 2 ^ * \right \} \end{align*}


\begin{align*} \det S = ( - 1 ) ^ { n - 2 k } \det Q _ S . \end{align*}


\begin{align*} \mathbf { P } \Big ( \frac { 1 } { \sigma } ( \hat { \theta } _ n - \theta ) \sqrt { \Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } > \Phi ^ { - 1 } ( 1 - \kappa _ n / 2 ) \Big ) \sim \kappa _ n / 2 \end{align*}


\begin{align*} \dim ( X \cup Y ) = \max \{ \dim ( X ) , \dim ( Y ) \} \end{align*}


\begin{align*} \norm { X g } _ 2 \norm { P g } _ 2 = \infty . \end{align*}


\begin{align*} \left \{ \begin{aligned} u + H ( x , D u ) & \leq 0 \Omega , \\ u + H ( x , D u ) & \geq 0 \overline { \Omega } , \end{aligned} \right . \end{align*}


\begin{align*} u < c , u = c \quad u > c , \quad . \end{align*}


\begin{align*} \int _ \R M \leq \int _ { - \sigma } ^ \sigma K = 2 \int _ 0 ^ \sigma K \leq C \int _ 0 ^ \sigma - \log x \ , d x = C \sigma ( | \log \sigma | + 1 ) . \end{align*}


\begin{align*} e ^ { \gamma ( t - s ) } \varphi _ { \gamma } ( s - r ) + \varphi _ { \gamma } ( t - s ) = \varphi _ { \gamma } ( t - r ) . \end{align*}


\begin{align*} R _ 1 = \begin{bmatrix} \frac { 4 b ^ 2 - 3 \varphi ^ 2 } { 4 b ^ 4 } & - \frac { \varphi ' } { b ^ 2 } \\ - \frac { \varphi ' } { b ^ 2 } & - \frac { \varphi '' } { \varphi } \end{bmatrix} , R _ 2 = R _ 3 = \begin{bmatrix} \frac { \varphi ^ 2 } { 4 b ^ 4 } & \frac { \varphi ' } { 2 b ^ 2 } \\ \frac { \varphi ' } { 2 b ^ 2 } & 0 \end{bmatrix} , \end{align*}


\begin{align*} p _ 1 : U _ 2 & \to X \times \R _ { \ge 0 } \\ ( x , t ) & \mapsto \begin{cases} ( x , t ) & t \ge 0 \\ ( x , 0 ) & t < 0 \end{cases} \end{align*}


\begin{align*} \eta _ k \coloneqq \begin{cases} 1 & \overline { B } _ { R _ k } ( \mathbf { 0 } ) \ ; , \\ 1 - \bigl ( | \mathbf { x } | - R _ k ) ^ 2 & B _ { R _ k + 1 } ( \mathbf { 0 } ) \setminus B _ { R _ k } ( \mathbf { 0 } ) , \\ 0 & . \end{cases} \end{align*}


\begin{align*} Q ( x ) = C x + O ( x ^ { 3 / 4 + \varepsilon } ) , \end{align*}


\begin{align*} \phi \leq \sqrt { ( 1 - \delta ) ( z _ 1 - \delta ) } + z _ 2 , \ 1 = z _ 0 \geq z _ 1 \geq \delta \geq z _ 2 \geq 0 , \ \delta \in \{ 0 , 1 \} , \ f = z _ 1 + z _ 2 . \end{align*}


\begin{align*} \langle \varepsilon _ - , \phi _ { j , b , \beta } \rangle _ { L ^ 2 _ { \rho _ \beta } } = 0 , j = 0 j = 1 , \end{align*}


\begin{align*} M _ { n , k } = \sum _ { i = 1 } ^ { k } \Big ( a _ n \alpha _ { i } X _ { \beta _ { i } } ( Z _ { i } - 1 ) + a _ i \varepsilon _ i \Big ) , \end{align*}


\begin{align*} \lambda a _ q + ( 1 - \lambda ) b _ q = w _ q ( x _ q , t _ q ) ^ q , \end{align*}


\begin{align*} \partial _ x ^ k \epsilon ( x ) = \partial _ x ^ k ( h _ + e ^ { \lambda _ 1 x } ) + O ( e ^ { 3 \lambda _ 1 x } ) , x \to + \infty . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } d X ( t ) & = & \big ( A X ( t ) + \alpha ( X ( t ) ) \big ) d t + \sigma ( X ( t ) ) d W ( t ) \medskip \\ X ( 0 ) & = & x \end{array} \right . \end{align*}


\begin{align*} \widetilde { \mathcal { O } } f & = - \widetilde { U } ^ { - 1 } \mathcal { H } ( \widetilde { U } f ) - f \\ & = - \widetilde { U } ^ { - 1 } U \mathcal { O } U ^ { - 1 } \widetilde { U } f - 2 f \\ & = - e ^ { - x ^ 2 } \mathcal { O } ( e ^ { ( \cdot ) ^ 2 } f ( \cdot ) ) - 2 f \end{align*}


\begin{align*} \tilde { p } ( A , B ) & = \left ( H ^ { - 1 } \circ ( B \circ A ^ { r _ 0 } ) ^ { r _ 1 } \circ A \circ H , H ^ { - 1 } \circ B \circ A ^ { r _ 0 } \circ H \right ) , \end{align*}


\begin{align*} \min _ { \sigma \in \mathbb { R } } f ( \sigma ; k ) \coloneqq \frac { 1 } { m } \sum _ { j = 1 } ^ m F ( \sigma , \xi _ j ; k ) + \delta _ { \left [ \sigma _ { \min } , \sigma _ { \max } \right ] } \left ( \sigma \right ) . \end{align*}


\begin{align*} \Psi _ t ( x , y ) : = f ( t , x ) g ( t , d ( x , y ) ) . \end{align*}


\begin{align*} \min \left \{ I _ b [ \gamma ] ; \gamma \in \mathrm { A C } \left ( [ 0 , b ] ; \Omega \right ) , \gamma ( 0 ) = x , \gamma ( b ) = \xi ( b ) \right \} . \end{align*}


\begin{align*} \sum \limits ^ { 2 k } _ { i = 1 } \ , \lambda ^ 2 _ i = t r a c e \ Q _ S ^ 2 = n ^ 2 - 2 n + 2 k . \end{align*}


\begin{align*} u ^ { T } ( \sum _ { i } p _ i y _ i ) ( \sum _ { i } p _ i y _ i ) ^ { T } u & = \left ( \sum _ { i } p _ i ( u ^ { T } y _ i ) \right ) ^ { 2 } \\ & \leq \left ( \sum _ { i } p _ { i } \right ) \cdot \left ( \sum _ { i } p _ { i } ( u ^ { T } y _ i ) ^ { 2 } \right ) & \textrm { ( C a u c h y - - S c h w a r z ) } \\ & = u ^ { T } \left ( \sum _ { i } p _ i y _ { i } y _ { i } ^ { T } \right ) u . \end{align*}


\begin{align*} \beta ( \tau ) = - \frac { 1 } { 2 } \frac { \mu _ \tau } { \mu ( \tau ) } , \end{align*}


\begin{align*} { \displaystyle \sum _ { n = 0 } ^ { \infty } \left ( \frac { x ^ { n } } { n ! } \right ) ^ { \gamma } } & \leq \left ( \sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n ! } \right ) ^ { 2 - \gamma } \left ( \sum _ { n = 0 } ^ { \infty } \left ( \frac { x ^ { n } } { n ! } \right ) ^ { 2 } \right ) ^ { \gamma - 1 } \\ & \leq ( \mathrm { e } ^ { x } ) ^ { 2 - \gamma } ( \mathrm { e } ^ { 2 x } ) ^ { \gamma - 1 } = \mathrm { e } ^ { \gamma x } \end{align*}


\begin{align*} \check { \varphi } ( \omega ^ * \omega ) = \omega ^ * ( S ^ { - 1 } ( a ) ) = \varphi ( a ^ * a ) , \omega = \varphi ( - a ) , a \in A , \end{align*}


\begin{align*} f = \sum _ { k = 1 } ^ n c _ k \pi ( x _ k , \omega _ k , 0 ) g \in \mathcal { H } _ g . \end{align*}


\begin{align*} ( - \Delta ) ^ s \tilde u ( x ) + \tilde c \tilde u = 0 , B _ { 2 R / y _ 1 } ( e _ 1 ) . \end{align*}


\begin{align*} \varepsilon _ { k + 1 } & = - i \left ( R ( \mathcal { V } _ { k + 1 } ^ A ) - R ( \mathcal { V } _ k ^ A ) \right ) + \varepsilon _ k + e ^ { - ( k + 1 ) e _ 0 t } \left ( \mathcal { L } - ( k + 1 ) e _ 0 \right ) \mathcal { Z } _ { k + 1 } ^ A \\ & = - i \left ( R ( \mathcal { V } _ { k + 1 } ^ A ) - R ( \mathcal { V } _ k ^ A ) \right ) + O \left ( e ^ { - ( k + 2 ) e _ 0 t } \right ) . \end{align*}


\begin{align*} ( u + \Delta y ) ^ \perp : = \left \{ z \in H _ 0 ^ 1 ( \Omega ) \colon \int _ \Omega u z - \nabla y \cdot \nabla z \dd x = 0 \right \} \end{align*}


\begin{align*} A b _ 0 ( x , 0 ) = \lim _ { \omega \to 0 } \frac { \sin ( \pi \omega ) ( 1 - | x | ) } { \pi \omega } = 1 - | x | . \end{align*}


\begin{align*} \Pr [ T ' _ i = T ~ | ~ W ] = \Pr [ V ' = T ~ | ~ W ~ ~ \mathcal { E } _ i ( W ) ] \end{align*}


\begin{align*} m \left ( \bigcap _ { \mu > 0 } \bigcup _ { \delta \leq \mu } K \cap \left ( \{ f < a \} \setminus \{ f _ \delta < a _ \delta \} \right ) \right ) = 0 . \end{align*}


\begin{align*} \begin{array} { l l @ { } l l } & \displaystyle x _ k - x _ \ell & \\ & \displaystyle v _ { i j } - x _ j \leq t _ i \ , , & i \in [ m ] j \in [ n ] \\ & \displaystyle x _ { 1 } + \dots + x _ { n } = 0 & \\ & \displaystyle t _ { 1 } + \dots + t _ { m } = p ^ * / n & \end{array} \end{align*}


\begin{align*} \phi _ { i , b , \beta } ( y ) = \sum _ { j = 0 } ^ i c _ { i , j } ( 2 \beta ) ^ j ( \sqrt { b } ) ^ { 2 j - \gamma } T _ { j } \left ( \frac { y } { \sqrt { b } } \right ) + \tilde \phi _ { i , b } ( y ) , \end{align*}


\begin{align*} M _ { p } : = \sum _ { p \leq p ' } \mu _ { P } ( p , p ' ) F _ { p ' } , F _ p : = \sum _ { p \leq p ' } M _ { p ' } . \end{align*}


\begin{align*} \| \Psi \| _ { Y _ T } = \sup _ { s \in [ 0 , T ] , x \in \R ^ 3 } | \Psi ( s , x ) | \langle \check \tau _ { s , x } ^ 2 + d _ { s , x } ^ 2 + | x ^ \perp | ^ 2 \rangle + | \nabla \Psi ( s , x ) | \langle \check \tau _ { s , x } ^ 3 + d _ { s , x } ^ 3 + | x ^ \perp | ^ 3 \rangle . \end{align*}


\begin{align*} \left | L \left ( \frac { w _ 1 + w _ 2 } { 2 } \right ) \right | \geq L \left ( \frac { w _ 1 + w _ 2 } { 2 } \right ) = \frac { L ( w _ 1 ) + L ( w _ 2 ) } { 2 } \geq r ( 1 - ( 2 \delta + \epsilon ) ) . \end{align*}


\begin{align*} h _ K ^ - = w _ K \left ( \frac { p } { 4 \pi ^ 2 } \right ) ^ { m / 4 } \prod _ { \chi \in X _ K ^ - } L ( 1 , \chi ) \leq w _ K \left ( { p M ( p , H ) \over 4 \pi ^ 2 } \right ) ^ { m / 4 } , \end{align*}


\begin{align*} F _ \nu ( t ) : = \begin{cases} t ^ { 2 - 2 \nu } & \nu < 1 \\ \log t & \nu = 1 \\ - t ^ { 2 - 2 \nu } & \nu > 1 . \end{cases} \end{align*}


\begin{align*} \begin{aligned} A ^ { 0 } ( U ) V _ { t } + A ^ { i } ( U ) \partial _ { i } V - B ^ { i j } ( U ) \partial _ { i } \partial _ { j } V + D ( U ) V & = F ( U ; D _ { x } U ) , \\ \left . V \right \rvert _ { t = 0 } & = U _ { 0 } \end{aligned} \end{align*}


\begin{align*} \partial _ t u - L ^ { ( n ) } u = f , - L ^ { ( n ) } u = f \end{align*}


\begin{align*} \partial _ \beta \| \phi _ { 1 , b , \beta } \| ^ { 2 } _ { L ^ 2 _ { \rho _ \beta } } = \left ( 2 \frac { 1 } { \beta } + \frac { d + 2 } { 2 \beta } - \left ( \frac { 2 } { \beta } \left ( \frac { d } { 2 } - \gamma + 2 \right ) - \frac { 1 } { \beta } \left ( \frac { d } { 2 } - \gamma + 1 \right ) \right ) \right ) \| \phi _ { 1 , b , \beta } \| _ { \rho _ \beta } ^ 2 + O ( b ^ { 1 - \frac { \epsilon } { 2 } } ) \end{align*}


\begin{align*} Z ( r , \theta ) = Z ( 0 , 0 ) + r ^ { \mu _ { j } } \cdot e ^ { i \theta } + O ( r ^ { 2 \mu _ { j } } ) \end{align*}


\begin{align*} J _ { t } ( U ) = V , \ , J _ { t } ( V ) = - U . \end{align*}


\begin{align*} | U | \geq \sup _ { p > 2 } \left ( \tfrac { p } { 2 } \right ) ^ { \frac { 2 d } { p - 2 } } ( 1 - \varepsilon ) ^ { \frac { p } { p - 2 } } \geq 2 ^ d ( 1 - \varepsilon ) ^ 2 , ( p = 4 ) . \end{align*}


\begin{align*} \sigma = d x \wedge d \omega = \sum _ { k = 1 } ^ d d x _ k \wedge d \omega _ k . \end{align*}


\begin{align*} x = \sum _ { i = 0 } ^ { 2 j + t } \alpha _ { i } e _ { i } , \quad \alpha _ { 2 j + t } \neq 0 . \end{align*}


\begin{align*} 2 ( x y ) ( x z ) + x ^ 2 ( y z ) = 0 . \end{align*}


\begin{align*} \underbrace { \begin{bmatrix} H + \rho I + \Theta ^ \dagger & - A ^ T \\ A & \delta I \end{bmatrix} } _ { \mathcal { N } _ { \rho , \delta , \Theta } } \begin{bmatrix} \Delta \mathbf { x } \\ \Delta \mathbf { y } \end{bmatrix} = \begin{bmatrix} \xi _ d ^ 1 \\ \xi ^ 2 _ d + X ^ { - 1 } _ { \mathcal { C } } \xi _ { \mu , \sigma } \\ \xi _ p \\ \end{bmatrix} , \end{align*}


\begin{align*} ( x , y ) \ = \ x ^ y \end{align*}


\begin{align*} \mathcal { L } _ { m a p } = \left [ 1 , \ldots , m \right ] . \end{align*}


\begin{align*} \widehat { ( \partial _ { \Sigma _ 0 , \epsilon } ^ { * } ) ^ { \otimes 2 } \overline { u } _ n } ( \xi , \eta ) & = - 4 \xi _ u \eta _ u \delta ( \xi _ u + \eta _ u ) \widehat { \overline { u } } ( \xi _ u ) \\ & = \frac { ( - i ) ^ { n } \xi _ u ^ { n + 2 } } { n ! } [ \alpha - i \pi \mathrm { s g n } ( \xi _ u ) ] . \end{align*}


\begin{align*} ( \phi - 1 ) \kappa ( g ) = \kappa ( k ) - ( \phi - 1 ) c _ { b , \mu } . \end{align*}


\begin{align*} C _ { \mathbf { F } } = \begin{pmatrix} v _ { 1 , 1 } & \ldots & v _ { 1 , d } \\ & \ddots & \\ v _ { K , 1 } & \ldots & v _ { K , d } \end{pmatrix} D _ \mathbf { F } = C _ \mathbf { F } ^ T . \end{align*}


\begin{align*} \widetilde { s } _ { N } = \sum _ { \substack { k , l \in \Z ^ d \\ | k | ^ 2 + | l | ^ 2 \leq N } } c _ { k , l } M _ { \beta l } T _ { \alpha k } g \end{align*}


\begin{align*} \frac { \kappa } { 2 } u '' + u ' - \lambda u = 0 , u ( 0 ) = u ( 1 ) = 0 \end{align*}


\begin{align*} T _ k \bigg ( \frac 1 { T _ k ( 1 ) } \bigg ) ( x ) & = \int _ { \Bbb R ^ N } \theta ( r ^ k \| x - y \| ) \frac 1 { T _ k ( 1 ) ( y ) } d \omega ( y ) \\ & \sim \int _ { \Bbb R ^ N } \theta ( r ^ k \| x - y \| ) \frac 1 { V _ k ( y ) } d \omega ( y ) \\ & \sim \frac 1 { V _ k ( x ) } \int _ { \Bbb R ^ N } \theta ( r ^ k \| x - y \| ) d \omega ( y ) \\ & = \frac 1 { V _ k ( x ) } T _ k ( 1 ) ( x ) \sim 1 . \end{align*}


\begin{align*} \mathcal { V } ^ \omega \vdash \forall x ^ X , \gamma ^ 1 , { \gamma ' } ^ 1 \left ( \gamma > _ \mathbb { R } 0 \land \gamma ' > _ \mathbb { R } 0 \land \gamma = _ \mathbb { R } \gamma ' \rightarrow J ^ A _ \gamma x = _ X J ^ A _ { \gamma ' } x \right ) . \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { \tilde { \Psi } _ { \varepsilon } ( \mathbf { x } ) } { | \mathbf { x } - \mathbf { x } _ { \varepsilon } | } = \sqrt { \tilde { \lambda } _ 0 \underline { m } } , \qquad \mathbf { x } \in \partial \Omega . \end{align*}


\begin{align*} \chi ( \xi ) = \begin{cases} 1 , \ \ \ \mbox { f o r } \ \ | \xi | \leq \frac 3 4 \\ 0 , \ \ \ \mbox { f o r } \ \ | \xi | \geq 1 . \end{cases} \end{align*}


\begin{align*} \begin{aligned} & 2 \beta | p | ^ { 2 - \alpha } + a ( 1 - \beta ) ^ { \alpha - 1 } ( \beta - \alpha \beta ) ( \beta - \alpha \beta - 1 ) r ^ { \beta ( 1 - \alpha ) + 1 } \\ & \leq 2 L ^ { 2 - \alpha } + a ( 1 - \beta ) ^ { \alpha - 1 } ( \alpha - \alpha ^ 2 ) ( \alpha - \alpha ^ 2 - 1 ) \min \{ c _ 0 ^ { \alpha ( 1 - \alpha ) + 1 } , c _ 0 ^ { 2 - \alpha } \} \end{aligned} \end{align*}


\begin{align*} \gamma _ { k } ^ { \langle n \rangle } = \frac { \left ( \sum _ { m = 1 } ^ M \sqrt { \eta _ { m k } } \alpha _ { m k } \right ) ^ 2 } { \sum _ { m = 1 } ^ M \beta _ { m k } \sum _ { k ' = 1 } ^ { K } \eta _ { m k ' } \alpha _ { m k ' } + \frac { 1 } { \gamma _ t } } , \end{align*}


\begin{align*} \langle \phi _ 0 , \partial _ \tau \phi _ 0 \rangle _ { L ^ 2 _ \rho } = - \frac { \gamma b _ \tau } { b } \| \phi _ 0 \| ^ 2 _ { L ^ 2 _ \rho } ( 1 + O ( b ^ { 1 - \frac { \epsilon } { 2 } } ) ) . \end{align*}


\begin{align*} { \mathcal N } _ 1 u = ( u \cdot \nabla ) u = \star ( \star d _ 1 u \wedge u ) + d _ 0 | u | ^ 2 / 2 , \end{align*}


\begin{align*} \begin{aligned} T ^ { \rm R e a c } _ j & = t _ j ^ S - t _ { j - 1 } ^ S = \tau ^ { \rm R e a c } _ { m } R _ j + \tau ^ { \rm R e a c } _ { V _ j } + \tau _ { \rm f } . \end{aligned} \end{align*}


\begin{align*} & f ( 0 , x , v ) = f _ 0 ( x , v ) , ( x , v ) \in \Omega \times \mathbb { R } ^ 3 , \\ & \rho ( 0 , x ) = \rho _ 0 ( x ) , u ( 0 , x ) = u _ 0 ( x ) , x \in \Omega , \end{align*}


\begin{align*} \Vert v - I _ h ^ C v \Vert _ { k , E } \lesssim h _ E ^ k \vert v \vert _ { 1 , D _ E } , k = 0 , 1 \ , , \end{align*}


\begin{align*} \begin{cases} \forall x ^ X , y ^ X \left ( \langle x - _ X y , J ^ A _ { 1 } x - _ X J ^ A _ { 1 } y \rangle _ X \geq _ \mathbb { R } \norm { J ^ A _ { 1 } x - _ X J ^ A _ { 1 } y } ^ 2 _ X \right ) , \\ \forall p ^ X , x ^ X \left ( ( x - _ X p ) \in A p \rightarrow p = _ X J ^ A _ { 1 } x \right ) . \end{cases} \end{align*}


\begin{align*} \mu ( E \cap B \cap A ) & = \mu ( E \cap B ) + \mu ( A \cap B ) - \mu \left ( ( E \cap B ) \cup ( A \cap B ) \right ) \\ & \geq ( c + 1 - \frac { c } { 2 } - 1 ) \mu ( B ) = \frac { c } { 2 } \mu ( B ) > 0 , \end{align*}


\begin{align*} \lim \limits _ { \vert ( t , s ) \vert \rightarrow + \infty } \frac { F ( x , t w ^ + + s w ^ - ) } { \max \lbrace \vert t \vert ^ { g ^ + } , \vert s \vert ^ { g ^ + } \rbrace } = + \infty , \ \ \ x \in \mathbb { R } ^ d . \end{align*}


\begin{align*} \chi = \sum _ { i = 1 } ^ n \frac { \alpha _ { i - 1 } } { \lambda _ i } . \end{align*}


\begin{align*} z _ i [ n ] = \sqrt { p } h _ i v [ n ] + w _ i ' [ n ] , \ n = 1 , 2 , \cdots , N _ 1 , \end{align*}


\begin{align*} \binom { c _ n k n } l \frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \le C \frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n - l } } \le C \frac { ( k j ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n - l } } . \end{align*}


\begin{align*} \tilde \gamma = \frac { 1 } { 2 } ( d + \sqrt { d ^ 2 - 1 2 d + 2 4 } ) . \end{align*}


\begin{align*} A _ n = \{ ( u , v ) : u , v \in [ n ] , u \ne v \} , \end{align*}


\begin{align*} \mathrm { v a r } \big [ \widehat { K } _ { \ ! \mathrm { a - c p t - f } } ^ \mathbb { R } \big ] \ ! = \ ! \upsilon ^ 2 \bigg ( \ ! \Big ( 1 \ ! - \ ! \frac { \pi } { 4 } \Big ) \Big ( 1 \ ! + \ ! \frac { 1 } { N _ 1 \bar { \gamma } ' } \Big ) \ ! K \ ! + \ ! \frac { 1 } { 2 N _ 1 \bar { \gamma } ' } \ ! + \ ! \frac { 1 } { 2 N _ 2 \bar { \gamma } _ c } \bigg ) = \frac { 4 - \pi } { \pi } K + \frac { 1 + \frac { N _ 1 \bar { \gamma } ' } { N _ 2 \bar { \gamma } _ c } } { \pi ( 1 + N _ 1 \bar { \gamma } ' ) } \end{align*}


\begin{align*} S \lesssim \sum \limits _ { j = 1 } ^ \infty 2 ^ { - j ( \varepsilon ) } + 1 \lesssim 1 . \end{align*}


\begin{align*} \frac { d } { d t } \int u ^ 2 ( x , t ) d x = 0 . \end{align*}


\begin{align*} S _ k ^ j = \frac { 1 } { q ^ { j ^ 2 } { k \choose k - j } _ q } \sum \limits _ { i = 0 } ^ { j } ( - 1 ) ^ { j - i } q ^ { { j - i \choose 2 } } { j \choose i } _ q { k + i \choose i } _ q N _ k ^ { i } , \end{align*}


\begin{align*} x = \sum _ { i = 0 } ^ { 3 j + 1 } \alpha _ { i } e _ { i } , ~ \alpha _ { 3 j + 1 } \neq 0 \quad ~ \langle x \rangle _ { T ^ { * 3 } } = j + 1 . \end{align*}


\begin{align*} \widehat { D } _ { L , m } f ( x ) = \mu ( D _ L ) f ( x ) = i ^ m \sqrt { | \det ( L ^ { - 1 } ) | } \ , f ( L ^ { - 1 } x ) = i ^ m \mathcal { D } _ L f ( x ) , \det ( L ) \neq 0 . \end{align*}


\begin{align*} \norm { V _ g f } _ 2 = \norm { f } _ 2 \norm { g } _ 2 \end{align*}


\begin{align*} v _ k : = \sum _ { j = k } ^ { N ( k ) } \lambda _ { j , k } u _ j , g _ k : = \sum _ { j = k } ^ { N ( k ) } \lambda _ { j , k } g _ { u _ j } \end{align*}


\begin{align*} \rho : H _ 2 ( X _ \psi , Q ) \to \mathbf { N } : = W / \left ( n \left ( \ker r \right ) \right ) . \end{align*}


\begin{align*} \mathcal { T } ^ { p , r } : = \underbrace { \Omega \otimes _ \mathcal { X } \ldots \otimes _ \mathcal { X } \Omega } _ { p } \otimes _ \mathcal { X } \underbrace { \Xi \otimes _ \mathcal { X } \ldots \otimes _ \mathcal { X } \Xi } _ { r } \end{align*}


\begin{align*} \| \phi ( \cdot , t ) \| ^ 2 \equiv \| \phi _ 0 \| ^ 2 = 1 , \frac { \mathrm { d } } { \mathrm { d } t } E ( \phi ( \cdot , t ) ) = - 2 \| \dot { \phi } ( \cdot , t ) \| ^ 2 , \forall t \geq 0 . \end{align*}


\begin{align*} \rho ( \mathbf { h } ) = \rho ( x , \omega , \tau ) = e ^ { 2 \pi i \tau } M _ { \omega / 2 } T _ x M _ { \omega / 2 } = e ^ { 2 \pi i \tau } e ^ { \pi i x \cdot \omega } T _ x M _ \omega . \end{align*}


\begin{align*} D ^ k ( \bar { x } \bar { y } ^ { - 1 } ) = \bar { x } \bar { y } ^ { - 1 } ( \bar { y } - \bar { x } ) ^ k , \end{align*}


\begin{align*} \hat u \circ \gamma = C \hat u \end{align*}


\begin{align*} E ^ { [ 1 ] } ( f , G \rtimes S ) ( t , \overline { t } ) = \sum \widehat { m } _ { \underline { \lambda } , \widetilde { \underline { \lambda } } } ( - 1 ) ^ { n _ { \underline { \lambda } } } ( t \overline { t } ) ^ { { \rm a g e } ( \underline { \lambda } ) - \frac { n - n _ { \underline { \lambda } } } { 2 } } \left ( \frac { \overline { t } } { t } \right ) ^ { { \rm a g e } ( \widetilde { \underline { \lambda } } ) - \frac { n - n _ { \widetilde { \underline { \lambda } } } } { 2 } } , \end{align*}


\begin{align*} \Delta _ G & = \left \{ \widehat { x } _ i - \widehat { x } _ { i + 1 } \ , , 1 \leq i \leq n - 1 \ , , \ ; 2 \ , \widehat { x } _ { n } \right \} \ , , \\ \intertext { a n d } \Delta _ K & = \left \{ \widehat { x } _ i - \widehat { x } _ { i + 1 } \ , , 1 \leq i \leq n - 1 \right \} \ , , \end{align*}


\begin{align*} m _ { 0 , \lambda } ( V ) = 0 \ \mbox { i f $ \lambda = 1 ^ n $ } . \end{align*}


\begin{align*} p _ { t } + \Delta y _ { t + 1 } & \in \partial _ { x } H _ t ( x _ { t } , y _ t ) , \\ \Delta x _ t & \in \partial _ { y } [ - H _ t ] ( x _ { t } , y _ t ) , \end{align*}


\begin{align*} f \cdot g ( \xi ) = f ( \xi ) g ( \xi ) , \ \mathcal { F } ( f \ast g ) = \mathcal { F } ( f ) \cdot \mathcal { F } ( g ) . \end{align*}


\begin{align*} J ( z ) = | 1 + z | ^ p - \left ( 1 + \frac { p } { 2 } z + \frac { p } { 2 } \bar { z } \right ) , z \in \mathbb { C } \end{align*}


\begin{align*} \mu ( A _ m ) \leq \sum _ { \ell = 1 } ^ \infty \mu ( C _ { m , \ell } ) & \leq \sum _ { \ell = 1 } ^ \infty 4 ^ { \ell } \mu ( B _ { m , \ell } ) \leq C _ 5 \sum _ { \ell = 1 } ^ \infty 4 ^ { \ell } e ^ { - c _ 1 m \ell } \leq C _ 5 \sum _ { \ell = 1 } ^ \infty e ^ { ( 2 - c _ 1 m ) \ell } . \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 ^ + } \frac { i t } { \omega _ 2 ( i t ) } = \frac { | \lambda | ^ 2 - \lambda _ 2 ( \mu ) } { | \lambda | ^ 2 } . \end{align*}


\begin{align*} \frac { 1 } { k + 1 } & \geq \norm { T _ { \lambda _ N } x _ N - z } \\ & \geq \vert \norm { T _ { \lambda _ N } x _ N } - \norm { z } \vert \\ & = \norm { z } - \norm { T _ { \lambda _ N } x _ N } \\ & \geq \norm { T ^ \circ x _ N } - \norm { T _ { \lambda _ N } x _ N } \\ & = \vert \norm { T ^ \circ x _ N } - \norm { T _ { \lambda _ N } x _ N } \vert \end{align*}


\begin{align*} & \partial _ { x } \int \mathbf { 1 } _ { \{ k \geq 0 \} } e ^ { i k x } \mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \ , d k \\ & = \int \mathbf { 1 } _ { \{ k \geq 0 \} } i k e ^ { i k x } \mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \ , d k + \int \mathbf { 1 } _ { \{ k \geq 0 \} } e ^ { i k x } \partial _ { x } \mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \ , d k . \end{align*}


\begin{align*} A ( e _ 1 ) = \begin{bmatrix} 1 & e _ 1 \cdot w _ 2 & \cdots & e _ 1 \cdot w _ n \\ w _ 2 \cdot e _ 1 & 1 & \cdots & w _ 2 \cdot w _ n \\ \vdots & \vdots & \cdots & \vdots \\ w _ n \cdot e _ 1 & w _ n \cdot w _ 2 & \cdots & 1 \end{bmatrix} . \end{align*}


\begin{align*} \zeta ^ \wedge _ { \mathcal { L } _ f , p } ( s ) = \int _ { \Z _ p \setminus \{ 0 \} } | a | _ p ^ { ( n + 2 ) s - 4 n } \mu _ 2 ( G _ 2 ^ + ( a ) ) d \mu _ 1 ( a ) . \end{align*}


\begin{align*} \| g _ n \| ^ 2 = \int _ 0 ^ 1 g _ n ' ( x ) ^ 2 e ^ { \gamma x } d x = \int _ 0 ^ 1 f _ n ( x ) ^ 2 e ^ { \gamma x } d x \leq e ^ { \gamma } \int _ 0 ^ 1 f _ n ( x ) ^ 2 d x = \frac { e ^ { \gamma } } { 2 } . \end{align*}


\begin{align*} \sum _ { \alpha , n } c _ { \alpha , n } x _ { \alpha , n } = 0 \ \ ( c _ { \alpha , n } \in K ) . \end{align*}


\begin{align*} T _ { s + t } ^ { \alpha } ( f ) = ( T _ t ^ { \alpha } \circ T _ s ^ \alpha ) ( f ) = ( T _ s ^ \alpha \circ T _ t ^ { \alpha } ) ( f ) . \end{align*}


\begin{align*} g ( t ) = \sum _ { n = 0 } ^ { \infty } g _ { n } \frac { t ^ { n } } { n ! } \end{align*}


\begin{align*} T _ k ( f ) ( x ) & = \sum \limits _ { i = 1 } ^ { \infty } \sum \limits _ { j = k } ^ { \infty } q ^ { - ( j + 1 ) } \lambda _ i \int \limits _ { | y | = q ^ { j + 1 } } f ( x - y ) a _ i ( y ) d y , \\ & = \sum \limits _ { i = 1 } ^ { \infty } \lambda _ i B _ i f , \end{align*}


\begin{align*} \left \{ \aligned & \partial _ t u - \Delta u + ( u \cdot \nabla ) u + \nabla p = 0 , \\ & \mbox { d i v } u = 0 , \\ & u ( 0 , x ) = u _ 0 ( x ) , \endaligned \right . \end{align*}


\begin{align*} 2 R \leq \int _ { y _ 0 } ^ \infty \rho d y = 1 - y _ 0 + \int _ 1 ^ \infty y ^ { - \beta } d y = 1 - y _ 0 + \frac { 1 } { \beta - 1 } . \end{align*}


\begin{align*} \Theta ( g ) : = \langle v _ 0 , \pi ( g ) v _ 0 \rangle . \end{align*}


\begin{align*} \left \langle \left ( | \cdot | ^ { - ( N - \gamma ) } * | Q | ^ p \right ) Q ^ { p - 1 } , h _ 1 \right \rangle = 0 , \end{align*}


\begin{align*} \Gamma = \{ ( \gamma , \gamma ^ * ) \mid \gamma \in \Gamma _ G \} \end{align*}


\begin{align*} \bar \delta _ n ( k ) = \sqrt { \sigma _ n ^ 2 ( k ) \cdot \binom { d } k } , \delta _ n ( k ) = \sqrt { \frac { 2 } { 9 0 ^ k } \cdot \binom { d } k } . \end{align*}


\begin{align*} - \Delta u _ { \ast } = \tilde { \lambda } _ 0 \tilde { m } _ 0 u _ { \ast } \R ^ N \ , . \end{align*}


\begin{align*} \xi ( y , h , s + l , s - l ) = \frac { \Gamma _ t ( 2 s - 2 m + 1 ) } { \Gamma _ { m - q } ( s + l ) \Gamma _ { m - p } ( s - l ) } \omega ( y , h , s + l , s - l ) , \end{align*}


\begin{align*} m ^ 3 + \mu m ^ 2 + 2 \mu m + \mu ^ 2 = 0 . \end{align*}


\begin{align*} \widetilde \mu = ( G ' \cdot \mu + \tfrac { 1 } { 2 } G '' \cdot \sigma ^ 2 ) \circ G ^ { - 1 } \ , \ , \widetilde \sigma = ( G ' \cdot \sigma ) \circ G ^ { - 1 } \end{align*}


\begin{align*} E ( k ) = I + \frac { 1 } { 2 \pi i } \int _ { \Gamma _ E } \frac { ( I + \mu ( s ) ) ( J ^ E ( s ) - I ) } { s - k } d s , \end{align*}


\begin{align*} W _ k , \zeta _ k = \varphi _ C ( A , b , \lbrace x _ j : j \leq k \rbrace , \lbrace W _ j : j < k \rbrace , \lbrace \zeta _ j : j < k \rbrace ) \in \mathbb { R } ^ { d \times n _ k } \times \mathfrak { Z } . \end{align*}


\begin{align*} \mathcal { A } _ G ( X ) : = \big ( C _ c ( G ) \hat { \otimes } C ( S \times S ) \big ) ^ { K \times K } . \end{align*}


\begin{align*} \begin{aligned} \left | \dot { \xi } ( t ) \right | ^ { q - 1 } & = \frac { e ^ t } { C _ p q } \left | \int _ 0 ^ t e ^ { - s } D f \left ( \xi ( s ) \right ) d s + \tilde { c } \right | \\ \left | \dot { \xi } ( t ) \right | & = \left ( \frac { e ^ t } { C _ p q } \right ) ^ \frac { 1 } { q - 1 } \left | \int _ 0 ^ t e ^ { - s } D f \left ( \xi ( s ) \right ) d s + \tilde { c } \right | ^ \frac { 1 } { q - 1 } , \end{aligned} \end{align*}


\begin{align*} S _ { \mu _ { T ^ * T } } \left ( \psi _ { \mu _ { T ^ * T } } \left ( - \frac { 1 } { s ( | \lambda | , 0 ) ^ 2 } \right ) \right ) = - \frac { 1 } { s ( | \lambda | , t ) ^ 2 } \cdot - \frac { s ( | \lambda | , t ) ^ 2 } { | \lambda | ^ 2 } = \frac { 1 } { | \lambda | ^ 2 } \end{align*}


\begin{align*} \mathcal { F } _ N ( \mathbf { h } _ * , \mathbf { g } _ * ) = \min _ { ( \mathbf { h } , \mathbf { g } ) \in E ^ m } \mathcal { F } _ N ( \mathbf { h } , \mathbf { g } ) . \end{align*}


\begin{align*} ( f \smile g ) ( x _ 1 \otimes \cdots x _ { n + m } ) = f ( x _ 1 \otimes \cdots x _ n ) g ( x _ { n + 1 } \otimes \cdots x _ { n + m } ) . \end{align*}


\begin{align*} \sum _ { \chi \in X _ p } \chi ( q _ 1 ) \overline { \chi } ( q _ 2 ) \sum _ { k = 1 } ^ { p - 1 } \vert S ( k , \chi ) \vert ^ 2 = \sum _ { \chi \in X _ p } \chi ( q _ 1 ) \overline { \chi } ( q _ 2 ) \sum _ { k = 1 } ^ { p - 1 } \left \vert \sum _ { l = 1 } ^ k \chi ( l ) \right \vert ^ 2 \\ = \sum _ { \chi \in X _ p } \sum _ { k = 1 } ^ { p - 1 } \sum _ { 1 \leq l _ 1 , l _ 2 \leq k } \chi ( q _ 1 l _ 1 ) \overline { \chi ( q _ 2 l _ 2 ) } = ( p - 1 ) ^ 2 { \mathcal { A } } ( q _ 1 , q _ 2 , p ) , \end{align*}


\begin{align*} E _ i ( 1 ) = \alpha + \int _ 0 ^ 1 ( n _ i - b _ i ) ( y ) d y , i = 1 , 2 . \end{align*}


\begin{align*} \Bigg | \ln \frac { \mathbf { P } \big ( ( \hat { \theta } _ n - \theta ) \sqrt { \Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } > x \sigma \big ) } { 1 - \Phi ( x ) } \Bigg | \leq c \ , \Bigg ( \frac { x ^ 3 } { \sqrt { n } } + ( 1 + x ) \frac { \ln n } { \sqrt { n } } \Bigg ) . \end{align*}


\begin{align*} - \gamma ^ { - 1 } \norm { p - J ^ A _ \gamma x } ^ 2 & = \gamma ^ { - 1 } \langle J ^ A _ \gamma x - p , p - J ^ A _ \gamma x \rangle \\ & = \langle J ^ A _ \gamma x - p , \gamma ^ { - 1 } ( x - J ^ A _ \gamma x ) - \gamma ^ { - 1 } ( x - p ) \rangle \\ & \geq \tilde { \rho } \norm { \gamma ^ { - 1 } ( x - J ^ A _ \gamma x ) - \gamma ^ { - 1 } ( x - p ) } ^ 2 \\ & = \tilde { \rho } / \gamma ^ 2 \norm { p - J ^ A _ \gamma x } ^ 2 . \end{align*}


\begin{align*} \arg \left ( \frac { \sigma + i T } { 2 } \right ) = \frac { \pi } { 2 } + \mathcal { O } \left ( \frac { 1 } { T } \right ) \end{align*}


\begin{align*} \mathcal L V + g + \rho \max \{ V ^ * - V , 0 \} = 0 , \end{align*}


\begin{align*} \frac { \partial V } { \partial S } ( t , S _ f ( t ) ) = \frac { \partial V ^ * } { \partial S } ( S _ f ( t ) ) . \end{align*}


\begin{align*} P _ { n + 1 } ( x ) = ( 1 + x ^ 2 ) \sum _ { k = 0 } ^ n { n \choose k } Q _ k ( x ) Q _ { n - k } ( x ) . \end{align*}


\begin{align*} \begin{aligned} U ^ { T } C V & = \Sigma = { \rm d i a g } \{ \sigma _ 1 , \cdots , \sigma _ t \} , \\ { U } ^ { T } _ { A } A V _ { A } & = \Sigma _ { A } = { \rm d i a g } \{ \sigma ^ { A } _ 1 , \cdots , \sigma ^ { A } _ n \} , \end{aligned} \end{align*}


\begin{align*} \psi ( x , t ) = U [ \psi _ { 0 } ] ( x , t , 0 ) + D [ u , \psi ] ( x , t ) , \end{align*}


\begin{align*} f ( n ) \ = \ 2 ^ n . ] \end{align*}


\begin{align*} \frac { 1 } { n h } \sum _ { k = 0 } ^ { n - 1 } \partial _ { \theta ^ { 2 } } f ( X _ { t _ { k } } , \theta _ { 0 } ) \delta B _ { t _ { k } } \rightarrow 0 , \end{align*}


\begin{align*} & \partial _ t f + v \cdot \nabla _ x f + { \rm { d i v } } _ v ( ( u - v ) f ) + { \rm { d i v } } _ v ( f L [ f ] ) - \Delta _ v f = 0 , \\ & \partial _ t \rho + { \rm { d i v } } _ x ( \rho u ) = 0 , \\ & \partial _ t ( \rho u ) + { \rm { d i v } } ( \rho u \otimes u ) + \nabla \rho ^ { \gamma } + \delta \nabla \rho ^ { \beta } - { \rm { d i v } } \mathbb { S } ( \nabla u ) = - \int _ { \mathbb { R } ^ 3 } ( u - v ) f \ , d v , \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ d } D _ i v ( \tau _ x \omega ) \partial _ j \eta ( x ) d x & = - \int _ { \mathbb { R } ^ d } v ( \tau _ x \omega ) \partial _ i \partial _ j \eta ( x ) d x \\ & = - \int _ { \mathbb { R } ^ d } v ( \tau _ x \omega ) \partial _ j \partial _ i \eta ( x ) d x \\ & = \int _ { \mathbb { R } ^ d } D _ j v ( \tau _ x \omega ) \partial _ i \eta ( x ) d x . \end{align*}


\begin{align*} \lim _ { | x | , | \omega | \to 0 } \langle M _ \omega T _ x f , g \rangle = \lim _ { | x | , | \omega | \to 0 } \langle f , T _ { - x } M _ { - \omega } g \rangle = \langle f , g \rangle , \end{align*}


\begin{align*} d ^ 2 f = - ( \partial _ y F ) ^ { - 1 } \cdot \left ( \partial _ x ^ 2 F + \partial _ y ( \partial _ x F ) \cdot d f \right ) - ( \partial _ y F ) ^ { - 1 } \cdot \left ( \partial _ x ( \partial _ y F ) \cdot d f + \partial _ y ^ 2 F \cdot ( d f ) ^ 2 \right ) , \end{align*}


\begin{align*} \lim _ { t \to \infty } \| P _ { t } ( x , \cdot ) - \mu \| _ { \mathrm { T V } } = 0 , \end{align*}


\begin{align*} \begin{cases} - \Delta u = f & \Omega \ , , \\ \ \ \ , u = g & \Gamma \ , , \end{cases} \end{align*}


\begin{align*} \Phi ( R , r ) \equiv \frac { 1 } { | C _ r | } \int _ { C _ { R - r } } N ( x , r ; F ) \ d x = \frac { 1 } { | C _ r | } \int _ { C _ { R - r } } N ( r ; \tau _ x ( F ) ) \ d x , \end{align*}


\begin{align*} \forall x ^ X , p ^ X \left ( \gamma > _ \mathbb { R } 0 \land \tilde { \rho } > _ \mathbb { R } - \gamma \land p = _ X J ^ A _ { \gamma } x \rightarrow \gamma ^ { - 1 } ( x - _ X p ) \in A p \right ) , \end{align*}


\begin{align*} d _ j ^ { ( 2 \ell ) } \sim m ! \cdot ( 2 \ell - 1 ) ! ! \cdot j ^ { - ( m - \ell ) } ( \log j ) ^ { \ell } ( j \to \infty ) . \end{align*}


\begin{align*} c _ { \ell , 0 } \| \phi _ { \ell , \phi , \beta } \| ^ { - 2 } _ { L ^ 2 _ { \rho _ \beta } } \langle \varepsilon , \phi _ { \ell , b , \beta } \rangle _ { L ^ 2 _ { \rho _ \beta } } = - \| \phi _ { 0 , \phi , \beta } \| ^ { - 2 } _ { L ^ 2 _ { \rho _ \beta } } \langle \varepsilon , \phi _ { 0 , b , \beta } \rangle _ { L ^ 2 _ { \rho _ \beta } } \end{align*}


\begin{align*} \int _ { 0 } ^ { 1 } \overline { \xi ( x ) } \xi ^ { \prime } ( x ) d x + \int _ { 0 } ^ { 1 } \overline { \xi ( x ) } \eta ^ { \prime } ( x ) d x & = \lambda \int _ { 0 } ^ { 1 } | \xi ( x ) | ^ 2 d x \\ \int _ { 0 } ^ { 1 } \overline { \eta ( x ) } \eta ^ { \prime \prime } ( x ) d x + \int _ { 0 } ^ { 1 } \overline { \eta ( x ) } \eta ^ { \prime } ( x ) d x + \int _ { 0 } ^ { 1 } \overline { \eta ( x ) } \xi ^ { \prime } ( x ) d x & = \lambda \int _ { 0 } ^ { 1 } | \eta ( x ) | ^ 2 d x . \end{align*}


\begin{align*} \omega _ 2 ( t ) & \triangleq \mathbb { E } [ e ^ { \imath t T } ] = \mathbb { E } \big [ e ^ { \imath t \sum _ { i \in \mathcal { K } ' } Z _ i } \big ] = \mathbb { E } [ e ^ { \imath t Z } ] ^ { K ' } \stackrel { ( a ) } { = } \bigg [ \int _ { - \infty } ^ { \infty } \big ( \cos ( t z ) + \imath \sin ( t z ) \big ) f _ Z ( z ) \mathrm { d } z \bigg ] ^ { K ' } \\ & \stackrel { ( b ) } { = } \bigg [ 2 \int _ { 0 } ^ { \infty } \cos ( t z ) f _ Z ( z ) \mathrm { d } z \bigg ] ^ { K ' } , \ \forall t \ge 0 , \end{align*}


\begin{align*} \lim \limits _ { t \to + \infty } \frac { x ( t ) } { t } = 0 . \end{align*}


\begin{align*} \omega ^ * ( a ) = \overline { \omega ( S ( a ) ^ * ) } , \omega \in \check { A } , a \in A . \end{align*}


\begin{align*} T ^ \circ x ^ * = \mu _ 0 ^ { - 1 } ( x ^ * + \mu _ 0 T ^ \circ x ^ * - x ^ * ) \in S x ^ * . \end{align*}


\begin{align*} A _ 1 ( X ) = A _ \infty \cap e ^ { B L O ( X ) } , R H _ \infty ( X ) = e ^ { B U O ( X ) } . \end{align*}


\begin{align*} C _ { n } ^ { \lambda , \beta } ( x ) = \frac { \mathrm { d } ^ { n } } { \mathrm { d } z ^ { n } } \mathrm { e } _ { \pi _ { \lambda , \beta } } ( f ( z ) ; x ) \Big | _ { z = 0 } . \end{align*}


\begin{align*} \begin{array} { l l } \displaystyle \sum _ { j = 1 } ^ { n } a _ { i j } w _ j = \theta , \ ; \ ; \ ; i = 1 , 2 , . . . , m \\ \displaystyle \sum _ { j = 1 } ^ { n } w _ j = 1 \\ \lvert a r g w _ j \rvert \leq \beta , \ ; \ ; \ ; j = 1 , 2 , . . . , n \end{array} \end{align*}


\begin{align*} \mathcal { N } _ S = \mathcal { N } _ + ^ \delta + \mathcal { N } _ - ^ \delta + \mathcal { N } _ + ^ { \mathrm { p . v . } } + \mathcal { N } _ - ^ { \mathrm { p . v . } } \end{align*}


\begin{align*} n = m ^ 2 , \theta = m - 2 \ell + 1 , \hat \theta = \frac { | m - 2 \ell + 1 | } { m } \sqrt { \frac { m + 1 } { \ell ( m - 1 ) ( m - \ell + 1 ) } } . \end{align*}


\begin{align*} { { x } } \left ( t \right ) = \mathcal { U } \left ( t \right ) x _ 0 , t \ge 0 , \end{align*}


\begin{align*} \sum _ { n = 1 } ^ \infty A _ n ( x , y ) { t ^ n \over n ! } = x y \ , { e ^ { x t } - e ^ { y t } \over x e ^ { y t } - y e ^ { x t } } . \end{align*}


\begin{align*} U ^ 2 \subseteq V , \ ; U V \subseteq U , \ ; V ^ 2 \subseteq U , \ ; U V ^ 2 = 0 . \end{align*}


\begin{align*} \log \zeta ( s ) = \sum _ { | \gamma - t | \leq 1 } \log ( s - \rho ) + \mathcal { O } ( \log t ) \ \ \ \ \ \ \ \ \ ( - 1 \leq \sigma \leq 2 ) \end{align*}


\begin{align*} \mathfrak { G } _ p = \Sigma _ 1 \times \left ( 1 - \frac { 1 } { p } \right ) \prod _ { \substack { \hat { \alpha } \neq \alpha \\ \hat { \beta } \neq \beta } } \left ( 1 - \frac { 1 } { p ^ { 1 + \hat { \alpha } + \hat { \beta } } } \right ) \prod _ { \hat { \alpha } \neq \alpha } \left ( 1 - \frac { 1 } { p ^ { 1 + \hat { \alpha } - \alpha } } \right ) \prod _ { \hat { \beta } \neq \beta } \left ( 1 - \frac { 1 } { p ^ { 1 + \hat { \beta } - \beta } } \right ) , \end{align*}


\begin{align*} C _ { G N } ^ { - 1 } \triangleq \inf _ { 0 \not = u \in H ^ 1 ( \mathbb { R } ^ N ) } Z [ u ] \end{align*}


\begin{align*} \rho ( x , \omega ) = \rho ( x , \omega , e ^ { 2 \pi i 0 } ) \end{align*}


\begin{align*} Z ( t ) = e ^ { B t } \bigg [ z + e ^ { - B t } c W ( t ) + \int _ 0 ^ t e ^ { - B s } ( b + B c W ( s ) ) d s \bigg ] , t \in [ 0 , T ] . \end{align*}


\begin{align*} x _ { k + 1 } = x _ k + W _ k ( W _ k ^ \intercal A ^ \intercal A W _ k ) ^ \dagger W _ k ^ \intercal A ^ \intercal ( b - A x _ k ) . \end{align*}


\begin{align*} w ^ l = \frac { \varphi ^ l _ s + \psi ^ l _ t - F ^ l - H ^ l } { z _ { s s } + z _ { t t } } \ , , \end{align*}


\begin{align*} \int _ 0 ^ 1 \left ( \int _ 0 ^ { \frac { w } { t } } z e ^ { - z } e ^ { z t } d z \right ) d t & = \int _ 0 ^ 1 \left ( \int _ 0 ^ w \frac { y } { t } e ^ { - \frac { y } { t } } e ^ { y } \frac { d y } { t } \right ) d t , \\ & = \int _ 0 ^ w y e ^ { y } \left ( \int _ 0 ^ 1 e ^ { - \frac { y } { t } } \frac { d t } { t ^ 2 } \right ) d y = \int _ 0 ^ w y e ^ { y } \frac { e ^ { - y } } { y } d y = w . \end{align*}


\begin{align*} & \alpha + \alpha ' = \pi & & & & \alpha + \alpha ' < \pi & & & & \alpha + \alpha ' < \pi \\ & \beta + \beta ' < \pi & & o r & & \beta + \beta ' = \pi & & o r & & \beta + \beta ' < \pi \\ & \gamma + \gamma ' < \pi & & & & \gamma + \gamma ' < \pi & & & & \gamma + \gamma ' = \pi \end{align*}


\begin{align*} \Gamma ( \pi ) & \ge \frac { 1 } { N } \sum _ { \ell = 1 } ^ { N } \Bigg ( \lim _ { t \to \infty } \frac { \sum _ { i = 1 } ^ { R _ \ell ^ \pi ( t ) } \rho _ \ell ( { T _ { \ell i } ^ \pi } ) ^ 2 } { 2 t } + \lim _ { t \to \infty } \frac { c _ \ell R _ \ell ^ \pi ( t ) } { t } \Bigg ) . \end{align*}


\begin{align*} \ddot { y } _ R ( t ) = - 8 s _ c ( p - 1 ) \delta ( t ) + A _ R ( u ( t ) ) , \end{align*}


\begin{align*} \mu _ x ^ s = \abs { x } _ s \cdot \mu ^ s \end{align*}


\begin{align*} E f ( x , \bar u ) + E f ^ * ( p , y ) = E [ x \cdot p ] + E [ \bar u \cdot y ] . \end{align*}


\begin{align*} \left \| u ( \cdot , t ) - t ^ { - \frac { 1 } { 2 } } f _ { M } \left ( ( \cdot ) t ^ { - \frac { 1 } { 2 } } \right ) \right \| _ { L ^ { \infty } } = \left ( \left \| \widetilde { f } _ { M } ( \cdot ) \right \| _ { L ^ { \infty } } + o ( 1 ) \right ) t ^ { - 1 } \log t a s \ \ t \to \infty . \end{align*}


\begin{align*} \left \{ \begin{array} { l } i \partial _ t u - \Delta u = \rho ^ 2 | u | ^ 2 + \langle \nabla \rangle ^ { - \alpha } \dot { W } \ , , t \in [ 0 , T ] \ , , \ , x \in \R ^ d \ , , \\ u _ 0 = \phi \ , , \end{array} \right . \end{align*}


\begin{align*} \underline { \dim } _ B \big ( G ( h _ i ) ) \big ) = \varliminf _ { \delta \rightarrow 0 } \frac { \log \Big ( N _ { \delta } ( G ( h _ i ) ) \Big ) } { - \log ( \delta ) } \geq 2 - \sigma ~ ~ ~ ~ ~ ~ \forall ~ ~ i \in \{ 1 , 2 , \cdots , M \} . \end{align*}


\begin{align*} \int _ { \mathbb S ^ 1 } B _ x J B \ , d x = - \frac 1 2 \int _ { \mathbb S ^ 1 } B J _ x B \ , d x . \end{align*}


\begin{align*} \left \{ \begin{aligned} u + \left | u ' \right | ^ 2 - \left ( 1 - | x | \right ) & \leq 0 ( - 1 , 1 ) , \\ u + \left | u ' \right | ^ 2 - \left ( 1 - | x | \right ) & \geq 0 [ - 1 , 1 ] . \end{aligned} \right . \end{align*}


\begin{align*} L _ t ^ h = \frac { h ( X _ t ) } { h ( X _ 0 ) } \exp \left ( A _ t ^ { \overline { \mu } } \right ) . \end{align*}


\begin{align*} 0 < \alpha _ { 0 } : = C T _ { 0 } < \tfrac { 1 } { 2 } , \end{align*}


\begin{align*} B ^ H _ t = \int ^ t _ 0 K _ H ( t , s ) d W _ s , t \ge 0 . \end{align*}


\begin{align*} \lambda _ { ( t - 1 ) k + i } - \lambda _ { ( t - 1 ) k + ( i + 1 ) } = a _ { i , t } . \end{align*}


\begin{align*} \begin{aligned} & h _ q + G _ \beta ( a _ q , \eta _ q , Y _ q , K \cap \{ v ( \cdot , t _ q ) \leq v ( y _ q , t _ q ) \} ) \geq 0 , \\ & k _ q + G _ \beta ( b _ q , \zeta _ q , Z _ q , K \cap \{ v ( \cdot , t _ q ) \leq v ( z _ q , t _ q ) \} ) \geq 0 , \end{aligned} \end{align*}


\begin{align*} \lim _ { y \to \infty } \kappa _ 1 ( y ) & = \lim _ { y \to \infty } \left ( \psi \left ( \frac { 1 + \lvert y \rvert } { 2 } \right ) - \ln { \lvert y \rvert } + \ln { \epsilon _ 1 } \right ) , \end{align*}


\begin{align*} \left [ R _ m ( x , y ) , R _ m ( z , t ) \right ] = R _ m ( R _ m ( x , y ) z , t ) + R _ m ( z , R _ m ( x , y ) t ) , \ ; \forall ( x , y , z , t ) \in ( T _ m M ) ^ 4 , \end{align*}


\begin{align*} \overline { A f ( - x , - \omega ) } = A f ( x , \omega ) . \end{align*}


\begin{align*} S _ 1 = \{ j \in S _ 0 ; | I _ j \cap Z _ { \psi _ \lambda } | \geq \kappa p _ 0 \} , \end{align*}


\begin{align*} \sum _ { \gamma \in \Gamma } | \langle f , S ^ { - 1 } e _ \gamma \rangle | ^ 2 = \sum _ { \gamma \in \Gamma } | \langle S ^ { - 1 } f , e _ \gamma \rangle | ^ 2 = \langle S ( S ^ { - 1 } f ) , S ^ { - 1 } f \rangle = \langle S ^ { - 1 } f , f \rangle . \end{align*}


\begin{align*} \lim _ { s \rightarrow \infty } \nabla \phi * _ x \rho [ h _ { V _ \ast } ( s , \cdot ) ] ( 0 ) = \lim _ { R \rightarrow \infty } \nabla \phi * _ x \rho [ g _ { R , X _ \ast , V _ \ast } ( R , \cdot ) ] ( X _ \ast ) . \end{align*}


\begin{align*} V _ { g _ 0 } ( M _ \eta T _ \xi g _ 0 ) ( x , \omega ) = e ^ { - \pi i ( x + \xi ) \cdot ( \omega - \eta ) } e ^ { - \frac { \pi } { 2 } ( ( x - \xi ) ^ 2 + ( \omega - \eta ) ^ 2 ) } . \end{align*}


\begin{align*} E [ R ] \begin{cases} < + \infty & ( - 1 < \alpha < - 1 / 2 ) , \\ = + \infty & ( - 1 / 2 \leq \alpha \leq 1 / 2 ) . \end{cases} \end{align*}


\begin{align*} \sigma | _ { B _ 1 } = \sigma | _ { B _ 4 } \mbox { a n d } \sigma | _ { B _ 3 } = \sigma | _ { B _ 6 } \end{align*}


\begin{align*} h _ i ( s ) : = c \varphi \left ( \frac { s - t } { 1 / i } \right ) \forall s \in [ 0 , T ] \forall i \in \mathbb { N } . \end{align*}


\begin{align*} & \lim _ { \mu \to 0 } \lambda _ { m i n } ( \delta I + ( \Theta ^ { - 1 } + \rho I ) ^ { - 1 } ) ^ { - 1 } = \frac { \rho } { \delta \rho + 1 } \\ & \lim _ { \mu \to 0 } \lambda _ { m a x } ( \delta I + ( \Theta ^ { - 1 } + \rho I ) ^ { - 1 } ) ^ { - 1 } = \frac { 1 } { \delta } . \end{align*}


\begin{align*} p _ k = ( - 1 ) ^ k \frac { n + m + 1 } { k + m + 1 } { n + m \choose m } { n \choose k } \quad ( 0 \leq k \leq n ) \end{align*}


\begin{align*} | B f ( z ) | e ^ { - \frac { \pi } { 2 } | z | ^ 2 } \leq \norm { V _ { g _ 0 } f } _ \infty \leq \norm { f } _ 2 \norm { g _ 0 } _ 2 = \norm { f } _ 2 , \end{align*}


\begin{align*} x _ t - y _ t = 0 , 0 \le t \le 1 . \end{align*}


\begin{align*} \widehat { K } = \arg \max _ { k \in \mathbb { N } } f _ { \widehat { K } ^ \mathbb { R } | K = k } ( \hat { k } ) . \end{align*}


\begin{align*} \partial _ k f ^ \# ( k ) & = \int _ \R \partial _ k \overline { \mathcal { K } ^ \# _ 0 ( x , k ) } f ( x ) \ , d x + \int _ \R \chi _ + ( x ) \partial _ k \overline { \mathcal { K } ^ \# _ + ( x , k ) } f ( x ) \ , d x \\ & + \int _ \R \chi _ + ( x ) \partial _ k \overline { \mathcal { K } ^ \# _ - ( x , k ) } f ( x ) \ , d x + \int _ \R \partial _ k \overline { \mathcal { K } ^ \# _ R ( x , k ) } f ( x ) \ , d x . \end{align*}


\begin{align*} N \rho ( [ \mathfrak { A } _ c i _ 1 ( F ) , \mathfrak { A } _ c i _ 2 ( G ) ] _ { \mathfrak { A } _ c ( \mathcal { I } ) } ) & = [ \mathfrak { A } \chi _ 1 \circ N \rho _ 1 ( F ) , \mathfrak { A } \chi _ 2 \circ N \rho _ 2 ( G ) ] _ { \mathfrak { A } ( \mathcal { M } ) } \\ & = 0 . \end{align*}


\begin{align*} g _ b ( x ) ~ = ~ \chi _ { \strut [ 0 , \infty [ } ( x ) \cdot \phi ( x , b ) , x \in \R , b \geq 0 , \end{align*}


\begin{align*} r ^ 2 & = \lim _ { t \rightarrow 0 ^ + } \left ( - ( s ( r , t ) - t ) ^ 2 + \frac { s ( r , t ) - t } { h ( s ( r , t ) ) } \right ) \\ & = \lim _ { s \rightarrow 0 ^ + } \frac { s } { h ( s ) } \lim _ { t \rightarrow 0 ^ + } \left ( 1 - \frac { t } { s ( r , t ) } \right ) . \end{align*}


\begin{align*} \begin{aligned} p & = \langle K ( x , y ) , ( 1 - \lambda _ \delta ( x , y ) ) \chi _ 0 ( y ) [ \phi ( y ) - \phi ( x ) ] \psi ( x ) \rangle \\ & + \langle K ( x , y ) , \lambda _ \delta ( x , y ) \chi _ 0 ( y ) [ \phi ( y ) - \phi ( x ) ] \psi ( x ) \rangle \\ & = : p _ { 1 , \delta } + p _ { 2 , \delta } . \end{aligned} \end{align*}


\begin{align*} a _ k = ( - 1 ) ^ k \sum _ { \nu = k } ^ n { m + \nu \choose \nu } { \nu \choose k } . \end{align*}


\begin{align*} s ( c ^ * , d ) = s ( c , d ) c c ^ * \equiv 1 \pmod d \end{align*}


\begin{align*} W _ L = ( \nabla ^ X F _ L , ( \nabla ^ X ) ^ 2 F _ L ) \end{align*}


\begin{align*} & | D _ k T _ { M } f ( x ) | ^ 2 \chi _ { Q ' } ( x ) \\ & \lesssim \sum \limits _ { \sigma \in G } \sum \limits _ { j = - \infty } ^ \infty r ^ { - | j - k | \varepsilon } r ^ { [ - k - ( - k \vee - j ) ] { \bf N } ( 1 - \frac 1 \theta ) } \bigg \{ M \Big ( \sum \limits _ { Q _ j } | q _ { j } f ( x _ { Q _ j } ) | ^ \theta \chi _ { Q } \Big ) ( x ) \bigg \} ^ { 2 / \theta } \chi _ { Q ' } ( x ) , \end{align*}


\begin{align*} U \Big ( \sum _ { k = 1 } ^ n c _ k \pi ( x _ k , \omega _ k , 0 ) g \Big ) = \sum _ { k = 1 } ^ n c _ k \rho ( x _ k , \omega _ k ) \varphi . \end{align*}


\begin{align*} A _ l \coloneqq \left [ 0 \to M _ { l - a - 1 } \to M _ { l - 1 } \oplus M _ { l - a } \to M _ l \to 0 \right ] , \end{align*}


\begin{align*} H = - \tilde \sigma _ 0 ( 1 ) + \tilde H \end{align*}


\begin{align*} \norm { u _ \beta ( t , \cdot ) } { L ^ 2 } \leq M ( t ) \Bigg \{ a _ { \beta } ( t ) + \sum \limits _ { l = 1 } ^ { | \beta | } \tilde { M } ( t ) ^ l \bigg ( \sum _ { \substack { 0 \leq | \alpha | \leq | \beta | - l \\ \alpha < \beta } } a _ \alpha ( t ) \sum _ { \substack { \theta _ 1 + \dots + \theta _ l = \beta - \alpha \\ \theta _ i \not = \mathbf { 0 } , \ , i = 1 , \dots , l } } \prod _ { i = 1 } ^ { l } \| q _ { \theta _ i } \| _ { L ^ \infty } \bigg ) \Bigg \} , \end{align*}


\begin{align*} \begin{aligned} | N _ { G } ( v ) \cap ( V ( T ' ) \backslash \{ u _ { 1 } , \cdots , u _ { s } \} ) | & = | N _ { G } ( v ) \cap ( X \backslash \{ u _ { 1 } , \cdots , u _ { s } \} ) | \\ & \leq | X | - \lfloor \frac { s } { 2 } \rfloor \\ & \leq t - \lfloor \frac { s } { 2 } \rfloor \end{aligned} \end{align*}


\begin{align*} \bigl [ A ^ { b + \frac { \sigma } { 2 } } S ( \ , \cdot \ , ) \bigr ] ^ { ( j ) } ( t ) & = \bigl [ S ( \ , \cdot \ , - \varepsilon ) A ^ { b + \frac { \sigma } { 2 } } S ( \varepsilon ) \bigr ] ^ { ( j ) } ( t ) \\ & = ( - A ) ^ j S ( t - \varepsilon ) A ^ { b + \frac { \sigma } { 2 } } S ( \varepsilon ) = ( - 1 ) ^ j A ^ { j + b + \frac { \sigma } { 2 } } S ( t ) . \end{align*}


\begin{align*} \partial _ t z _ k = \frac { ( 1 + \omega _ k ^ 1 ) z _ k } { t } + O \bigg ( \frac { 1 } { t ^ { 5 / 4 - \delta _ 0 } } \bigg ) . \end{align*}


\begin{align*} \langle \widehat { g _ 1 } , T _ x M _ \omega \widehat { g _ 2 } \rangle = \langle g _ 1 , M _ x T _ { - \omega } g _ 2 \rangle = V _ { g _ 2 } f _ 2 ( - \omega , x ) . \end{align*}


\begin{align*} w = P _ { \tilde { \Omega } _ { \varepsilon } } w + e ^ { - \beta _ { \varepsilon } \tilde { \Psi } _ { \varepsilon } ( \mathbf { x } _ { \varepsilon } ) } \tilde { V } _ { \varepsilon } \tilde { \Omega } _ { \varepsilon } \ ; . \end{align*}


\begin{align*} & ( - \Delta ) ^ s ( \tilde u - \tau \varphi ^ { ( 1 ) } ) ( a ) + c _ \rho ( a ) ( \tilde u - \tau \varphi ^ { ( 1 ) } ) ( a ) \\ & \qquad = - C \int _ { H _ \rho } \left ( \frac 1 { \vert a - y \vert ^ { n + 2 s } } - \frac 1 { \vert Q _ \rho ( a ) - y \vert ^ { n + 2 s } } \right ) \big ( \tilde u ( y ) - \tau \varphi ^ { ( 1 ) } ( y ) \big ) \dd y . \end{align*}


\begin{align*} U _ { i J ( i ) } : = \bigl \{ ( x , u _ i ) \bigm | u _ { i j } \geq a _ { i j } j \leq \min ( J _ i ) , \ u _ { i j } = u _ { i n } j \geq \max ( J _ i ) , \ u _ { i j } ( x ; J ) = u _ { i j } ( x ) , \ ( x , u _ { \cdot n } ) \in W _ J \bigr \} \end{align*}


\begin{align*} \sum _ l ( \partial _ { x _ l } f _ \alpha ) ( \partial _ { t _ j } \varphi ^ l ) = 0 , \end{align*}


\begin{align*} P _ T ( X ) = P ( X ) - T Q ( X ) \end{align*}


\begin{align*} \mathcal { L } ^ 0 ( h , k ) : = \sum _ { \substack { 1 \leq q < \infty \\ ( q , h k ) = 1 } } W \left ( \frac { q } { Q } \right ) \sum _ { \substack { 1 \leq m , n < \infty \\ ( m n , q ) = 1 } } \frac { \tau _ A ( m ) \tau _ B ( n ) } { \sqrt { m n } } V \left ( \frac { m } { X } \right ) V \left ( \frac { n } { X } \right ) \sum _ { \substack { c > C , d \geq 1 \\ c d = q } } \mu ( c ) \end{align*}


\begin{align*} \mathbb { E } \Big [ \sup _ { 1 \leq m \leq M } \| Y _ m ^ { M , N } \| ^ p _ { L ^ 6 } \Big ] \leq C ( p , T , \gamma , X _ 0 ) . \end{align*}


\begin{align*} \Psi ( \nu ) \leq \sum _ { \mu = 1 } ^ \nu c \chi ^ \mu \leq c ( d , \chi ) \chi ^ \nu . \end{align*}


\begin{align*} \delta ( t ) = \Big | \| \nabla u ( t ) \| _ 2 ^ 2 - \| \nabla Q \| _ 2 ^ 2 \Big | , \end{align*}


\begin{align*} I m ( \displaystyle \sum _ { i = 1 } ^ { m } \overline { z _ i } a _ { i k } ) = I m ( \displaystyle \sum _ { i = 1 } ^ { m } \overline { z _ i } a _ { i l } ) : = p _ 1 \ ; \ ; \forall k \not = l \end{align*}


\begin{align*} \phi ^ \sigma ( g ) = \phi ( g ^ { - 1 } ) \quad \quad \epsilon ( \phi ) = \phi ( e ) . \end{align*}


\begin{align*} \int \int \nabla \phi _ k \cdot \nabla v ~ d x d z = \int _ { \Gamma ^ { N B C } } q ~ v ~ d \Gamma , \end{align*}


\begin{align*} S _ U : = \bigcup _ { 0 \leq k \leq \lfloor \frac { \beta - \alpha } { \varepsilon _ 2 } \rfloor + 1 } E ^ { [ \theta _ k , \theta _ { k + 1 } ] , \rho _ { U } , \varepsilon _ 2 } _ { \mu } \cap E _ U \cap \left \{ x \in \R ^ d : \ \overline { \dim } _ { \mathrm { l o c } } ( m _ { E _ U } ^ { s _ 2 } , x ) \leq d \right \} . \end{align*}