\begin{align*} \mathbb { Z } ^ n \cap \Omega = \mathrm { v o l } ( \Omega ) + O ( \max _ { \pi } \mathrm { v o l } \big ( \pi ( \Omega ) ) \big ) , \end{align*}


\begin{align*} \Lambda ( p , q ) : = & \left ( p ^ { \mu } q _ { \mu } \right ) ^ 2 \big [ \left ( p ^ { \mu } q _ { \mu } \right ) ^ 2 - 1 \big ] ^ { - \frac { 3 } { 2 } } , \\ \mathcal { S } ( p , q ) : = & \big [ \left ( p ^ { \mu } q _ { \mu } \right ) ^ 2 - 1 \big ] ^ 2 I _ 3 - \big ( p - q \big ) \otimes \big ( p - q \big ) - \big [ \left ( p ^ { \mu } q _ { \mu } \right ) - 1 \big ] \big ( p \otimes q + q \otimes p \big ) . \end{align*}


\begin{align*} ( u , v ) _ { H ^ 4 _ * } = \int _ U u _ { x x x x } v _ { x x x x } d x . \end{align*}


\begin{align*} \tilde \gamma \subset \bigcup _ { k = M + 1 } ^ N B ^ { ( k ) } \Subset \Omega ^ + . \end{align*}


\begin{align*} \mathbb { L } \leq & \Big \| \int _ 0 ^ t E ( t - s ) A ( P _ n - I ) P F ( X ^ { n } ( s ) ) \ , \dd s \Big \| _ { L ^ p ( \Omega ; H ) } \\ & + \Big \| \int _ 0 ^ t E ( t - s ) A P \big ( F ( X ^ { n } ( s ) ) - F ( X ( s ) ) \big ) \ , \dd s \Big \| _ { L ^ p ( \Omega ; H ) } = \mathbb { L } _ 1 + \mathbb { L } _ 2 . \end{align*}


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


\begin{align*} 2 \operatorname { R e } \left < T x , x \right > = \left < T x , x \right > + \left < x , T x \right > = - 2 \beta \eta \left | v \left ( 1 \right ) \right | ^ 2 - 2 \kappa \ , | v ' \left ( 1 \right ) \ ! | ^ 2 , \end{align*}


\begin{align*} \sum _ { k = 1 } ^ \infty \alpha _ k = \sum _ { n = 0 } ^ \infty \Bigl ( \sum _ { k = 1 } ^ \infty a _ { k , n } \Bigr ) X ^ n \end{align*}


\begin{align*} \Phi _ { R , L + } = \Phi _ { R , L - } \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix} . \end{align*}


\begin{align*} { d \widetilde g ^ K _ { i } ( t ) \over d t } < \bar p \log K \sum _ { \ell = - \lfloor 1 / 2 \delta _ K \rfloor } ^ { 1 / \delta _ K - 1 - \lfloor 1 / 2 \delta _ K \rfloor } h _ { K } G ( h _ { K } \ell ) \left [ \widetilde g ^ K _ { \ell + i } ( t ) - \widetilde g ^ K _ { i } ( t ) \right ] ^ + \ , e ^ { h _ { K } L | \ell | } . \end{align*}


\begin{align*} \widetilde { V } ( s ) = - \frac { 1 } { s } \int _ 0 ^ { \infty } V ' ( x ) x ^ s \ , d x . \end{align*}


\begin{align*} \begin{aligned} & ( 1 - \beta ) h _ q + v ( y _ q , t _ q ) ^ { \beta } \mu ( v ( y _ q , t _ q ) ^ { 1 - \beta } ) \geq - v ( y _ q , t _ q ) ^ { \beta } \omega ( | X _ q | ) , \\ & ( 1 - \beta ) k _ q + v ( z _ q , t _ q ) ^ { \beta } \mu ( v ( z _ q , t _ q ) ^ { 1 - \beta } ) \geq - v ( z _ q , t _ q ) ^ { \beta } \omega ( | X _ q | ) . \end{aligned} \end{align*}


\begin{align*} a ^ n - \gamma _ { n - 1 } a ^ { n - 1 } - \dots - \gamma _ { 1 } a = 0 , \end{align*}


\begin{align*} d \Phi _ t = \nabla \log \left ( \frac { d \gamma ^ { \epsilon , L } } { d \varphi } \right ) ( \Phi _ t ) d t + \sqrt { 2 } d B _ t , \Phi _ 0 \sim \nu ^ { \epsilon , L } , \end{align*}


\begin{align*} d _ T = \begin{cases} \delta _ T , n \geq 1 , \\ \partial _ T , \ ; \ ; \ ; n = 0 . \end{cases} \end{align*}


\begin{align*} k [ P ] = k [ K ] \quad \textrm { a n d } \mathcal { Z } _ P = \mathcal { Z } _ K . \end{align*}


\begin{align*} \sum _ { \rho \in R _ \epsilon } ( \rho ) = - \frac { \epsilon T } { 2 \pi } \log \pi + \frac { \epsilon T } { 2 \pi } \log \left ( \frac { T } { 2 } \right ) - \frac { \epsilon T } { 2 \pi } + \frac { \epsilon } { \pi } \ \mathcal { O } ( \log T ) \end{align*}


\begin{align*} U ( x _ 0 , \delta ) : = \{ x \in H : \| x - x _ 0 \| < \eta \} . \end{align*}


\begin{align*} ( \mathbf { v } \mathrel { \trianglelefteq } ( A , B , C ) \ , \& \ , \mid B \cap \mathbf { v } _ 2 \mid = \omega ) \Rightarrow \mid E \cap \mathbf { v } _ 2 \mid = \omega \end{align*}


\begin{align*} & d \rho ( t ) + d i v _ G ^ { \theta } ( \rho ( t ) \nabla _ G S ( t ) ) ( d t + d W _ t ^ { \delta } ) = 0 , \\ & \ < \dot S ^ { \textrm { a b s } } , \rho \ > + \frac 1 4 \sum _ { i j } ( S _ i - S _ j ) ^ 2 \theta _ { i j } ( \rho ) ( 1 + \dot { W _ t ^ { \delta } } ) ^ 2 = 0 , \ ; \mathcal L ^ 1 \ ; \\ & \ < \frac { d \dot S ^ { \textrm { \textrm { s i n g } } } } { d \mu } , \rho \ > = 0 , \ ; \forall \mu \ ; , \ ; \mu \ ; \bot \ ; \mathcal L _ 1 . \end{align*}


\begin{align*} \operatorname { K o s } ^ { { k [ K ] } } ( v _ 1 , \ldots , v _ m ) = \big ( k [ K ] \otimes \Lambda ( u _ 1 , \ldots , u _ m ) , \ d \big ) , d ( u _ i ) = v _ i , \end{align*}


\begin{align*} u ( \cdot + x ( t _ n ' ) , t _ n ' ) = u ( \cdot + x ( t _ n ) + x ( t _ n ' ) - x ( t _ n ) , t _ n ' ) \to e ^ { i t _ 0 } e ^ { i \theta _ 0 } Q ( \cdot - x _ 0 - x _ 1 ) H ^ 1 . \end{align*}


\begin{align*} g ( t ) = \sqrt { \gamma } \ , e ^ { - \gamma | t | } , \ ; \gamma > 0 . \end{align*}


\begin{align*} \begin{cases} e ^ { - Q } \delta _ { k l } \leq \hat { h } _ { \delta , k l } = \hat { h } _ { \delta } ( \frac { \partial } { \partial \hat { f } ^ { k } } , \frac { \partial } { \partial \hat { f } ^ { l } } ) \leq e ^ { Q } \delta _ { k l } ; \\ R ^ { 1 + \alpha } \left \| \frac { \partial \hat { h } _ { \delta , k l } } { \partial \hat { f } ^ { j } } \right \| _ { C ^ { 0 , \alpha } ( B _ { R } ( \hat x , \hat h _ \delta ) ) } \leq \Psi ( \delta \mid n , r , v , \rho , \alpha , Q ) \end{cases} \end{align*}


\begin{align*} R _ H ( s _ 1 , s _ 2 ) = \int _ 0 ^ { s _ 1 \wedge s _ 2 } K _ H ( s _ 1 , u ) K _ H ( s _ 2 , u ) d u . \end{align*}


\begin{align*} | \langle V _ g ^ * F , \varphi \rangle | & = | \langle F , V _ g \varphi \rangle | \\ & \leq \norm { F } _ { L ^ { p , q } } \norm { V _ g \varphi } _ { L ^ { p ' , q ' } } \\ & \leq \norm { F } _ { L ^ { p , q } } \norm { ( 1 + | z | ) ^ n V _ g \varphi } _ \infty \norm { ( 1 + | z | ) ^ { - n } } _ { L ^ { p ' , q ' } } . \end{align*}


\begin{align*} { \rm G e n } ( a , t ) = { 1 \over { \rm G e n } ( a ^ { - 1 } , t ) } \end{align*}


\begin{align*} \bullet ' = f ( \bullet ) , \ \ \bullet = f ^ { - 1 } ( \bullet ' ) ; \widetilde { \bullet } = \tau ( \bullet ' ) , \ \ \bullet ' = \tau ^ { - 1 } ( \widetilde { \bullet } ) . \end{align*}


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


\begin{align*} \int _ { \R ^ N } | \nabla u _ { \ast } | ^ 2 \le \int _ { \R ^ N } | \nabla u | ^ 2 = \tilde { \lambda } _ 0 \ ; , \end{align*}


\begin{align*} S ( k ) = \Phi _ { R } ^ { - 1 } ( k ) \Phi _ { L } ( k ) \end{align*}


\begin{align*} \rho ( \alpha ; u ) : = \tilde { \rho } ( u ) + \overline { \rho } ( \alpha ; u ) \ \ \Vert u \Vert = \inf \left \lbrace \lambda > 0 : \rho ( \alpha ; \frac { u } { \lambda } ) \leq 1 \right \rbrace . \end{align*}


\begin{align*} \pi ( \gamma ) = M _ \omega T _ x . \end{align*}


\begin{align*} L ' _ b \wedge L ' _ { \tilde { c } } = L ' _ { p u b } . \end{align*}


\begin{align*} \Theta _ n & = \frac { 1 6 } { n ^ 4 \delta ^ 4 _ n ( k ) } \sum _ { \substack { \mathbf { P } _ { k } \in \mathcal P ( d , k ) ^ 4 : \\ p _ { k , 1 } = p _ { k , 2 } = p _ { k , 3 } = p _ { k , 4 } } } \sum _ { \mathbf { i } \in \mathcal { J } ^ 2 } \varphi _ { k - 1 } \left ( \mathbf { P } _ { k } , \mathbf { i } \right ) \cdot \varphi _ { 4 } \left ( \mathbf { i } \right ) . \end{align*}


\begin{align*} \widehat { f } ( \xi ) = \sum _ { \substack { I , J \\ \abs { I , J } \leq o ( f ) } } \xi ^ I D ^ J u _ { I , J } ( \xi ) , \end{align*}


\begin{align*} \| f \| _ { S ( \dot { H } ^ { s _ c } ) } = \| f \| _ { L _ t ^ { q _ 2 } L _ x ^ { r _ 1 } } \lesssim \big \| | \nabla | ^ { s _ c } f \big \| _ { L _ t ^ { q _ 2 } L _ x ^ { r _ 2 } } \lesssim \big \| \left \langle \nabla \right \rangle f \| _ { L _ t ^ { q _ 2 } L _ x ^ { r _ 2 } } \lesssim \big \| \left \langle \nabla \right \rangle f \big \| _ { S ( L ^ 2 ) } . \end{align*}


\begin{align*} \frac { z _ k - g ( \bar x ) } { \norm { x _ k - \bar x } } = \frac { g ( x _ k ) - g ( \bar x ) } { \norm { x _ k - \bar x } } - \frac { y _ k } { \norm { x _ k - \bar x } } \to v - 0 = v \end{align*}


\begin{align*} \varphi _ C ( A , b , \lbrace x _ j : j \leq k \rbrace , \lbrace W _ j : j < k \rbrace , \lbrace \zeta _ j : j < k \rbrace ) = ( W _ k , \emptyset ) . \end{align*}


\begin{align*} \left | \int _ \R x f ( x ) \overline { f ' ( x ) } \ , d x \right | = | \langle x f , f ' \rangle | \leq \norm { x f } _ 2 \norm { f ' } _ 2 = \left ( \int _ \R x ^ 2 | f ( x ) | ^ 2 \ , d x \right ) ^ { 1 / 2 } \left ( \int _ \R | f ' ( x ) | ^ 2 \ , d x \right ) ^ { 1 / 2 } . \end{align*}


\begin{align*} u b = v c , \ ; u ^ + = v ^ + = ( v c ) ^ + \mbox { a n d } b c ^ * = u ^ * b . \end{align*}


\begin{gather*} \left | \left ( u _ x ( x ) \right ) ^ { p } \right | = | \langle u _ x , p \rangle p | = | \langle u _ x , p - u \rangle | \leq 3 \varepsilon | u _ x ( x ) | , \\ \left | \left ( u _ t ( x ) \right ) ^ { p } \right | \leq 3 \varepsilon | u _ t ( x ) | \ ; \textrm { a n d } \ ; \left | \left ( u ( x ) - p \right ) ^ { p } \right | \leq \frac { 3 \varepsilon } { 2 } \left | u ( x ) - p \right | . \end{gather*}


\begin{align*} \mathsf P _ k ( x ) = \# \{ a \in H \colon \mathsf N ( a ) \le x , a \ \ \mathsf P _ k \} \ , . \end{align*}


\begin{align*} \pi _ { k } ( X _ x ) = \sum \limits _ { y \in X ( k ) : y > x } \pi _ { k } ( y ) = R ( k , i ) \pi _ i ( x ) \end{align*}


\begin{align*} f ( X _ { t } , \theta ) = - \theta X ^ { \gamma } _ { t } , \end{align*}


\begin{align*} \rho ( X ) = \rho ( Y ) - n = \rho ( \mathcal { Y } _ { \overline K } ) - n = \rho ( \mathcal { X } _ { \overline K } ) , \end{align*}


\begin{align*} f = \frac { 1 } { \langle \widetilde { g } , g \rangle } \iint _ { R ^ { 2 d } } V _ g f ( x , \omega ) M _ \omega T _ x \widetilde { g } \ , d ( x , \omega ) . \end{align*}


\begin{align*} P ' ( s ) s - P ( s ) = p ( s ) , \ P '' ( s ) = \frac { p ' ( s ) } { s } s \in [ 0 , \bar \rho ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \dot { \xi } ( t ) & = & \beta \xi ( t ) \\ \xi ( 0 ) & = & x \end{array} \right . \end{align*}


\begin{align*} \sum _ { n \geq 0 } \vert B _ n \vert ^ { \varepsilon } \mu ( B _ n ) \leq \sum _ { k \geq 0 } \sum _ { 1 \leq j \leq J _ k } 2 ^ { - k \varepsilon } = \sum _ { k \geq 0 } J _ k 2 ^ { - k \varepsilon } < + \infty . \end{align*}


\begin{align*} \int R _ 1 R _ 2 ^ p = \frac { \kappa _ 0 } { ( x _ 2 - x _ 1 ) ^ 2 } \int Q ^ p + O \bigg ( \frac { 1 } { | x _ 1 - x _ 2 | ^ 3 } \bigg ) . \end{align*}


\begin{align*} t ^ c _ j = \prod _ { i \in \mathcal { N } _ c \setminus \{ j \} } \operatorname { s g n } ( t ^ v _ i ) \min _ { i \in \mathcal { N } _ c \setminus \{ j \} } | t ^ v _ i | , \end{align*}


\begin{align*} H _ n ( w ) = \frac { d ^ n } { d w ^ n } \left ( ( w - 1 ) ^ { n - 1 } ( w + 1 ) ^ { n + 1 } \right ) , \end{align*}


\begin{align*} { \bf D } & = ( \delta { } _ { v _ 1 } ) \otimes \cdots \otimes ( \delta { } _ { v _ N } ) , \\ D ^ { \prime } _ 1 & = ( \delta { } _ { o ( e _ 1 ) } ) \otimes \cdots \otimes ( \delta { } _ { o ( e _ M ) } ) , \\ D ^ { \prime } _ 2 & = ( \delta { } _ { t ( e _ 1 ) } ) \otimes \cdots \otimes ( \delta { } _ { t ( e _ m ) } ) , \\ B ^ { \prime } & = ( \beta { } _ { e _ 1 } ) \otimes \cdots \otimes ( \beta { } _ { e _ m } ) . \end{align*}


\begin{align*} \mathcal { C } _ L = \mathcal { C } \smallsetminus \{ 1 , - 1 \} . \end{align*}


\begin{align*} n = \binom { m } { 2 } , \theta = 5 - m , \hat \theta = | 5 - m | \sqrt { \frac { m + 1 } { m ( m - 1 ) ( m - 2 ) ( m - 3 ) } } . \end{align*}


\begin{align*} \frac { | m h \pm n k | } { g \ell } W \left ( \frac { c | m h \pm n k | } { g \ell Q } \right ) = \frac { Q } { 2 \pi i c } \int _ { ( \varepsilon ) } \ell ^ { - w } \Upsilon _ { \pm } ( w ; m h , n k ) \ , d w , \end{align*}


\begin{align*} & \partial _ t \eta + \mathrm { d i v } ( \eta u ) = \nu \Delta \eta , \\ & \partial _ t \tau + \mathrm { d i v } ( u \tau ) - ( \nabla u \tau + \tau \nabla ^ T u ) - k \eta ( \nabla u + \nabla ^ T u ) = \nu \Delta \tau - \frac { A _ 0 } { 2 } \tau , \end{align*}


\begin{align*} & \Theta \in \mathcal C ^ { \infty } ( ( 0 , \infty ) \times ( 0 , \infty ) ) ; \\ & \Theta \ ; \ ; [ 0 , \infty ) \times [ 0 , \infty ) ; \\ & \Theta ( s , t ) = \Theta ( t , s ) ; \ ; \Theta ( s , t ) > 0 , \ ; \ ; s , t > 0 , \\ & \min ( s , t ) \le \Theta ( s , t ) \le \max ( s , t ) , s , t \ge 0 . \\ & \Theta \ ; \ ; ( 0 , \infty ) \times ( 0 , \infty ) . \end{align*}


\begin{align*} ( \xi , \eta ) \mapsto ( u , v ) : = ( \xi + \eta , - | \xi | ^ { \alpha } - | \eta | ^ { \alpha } ) . \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \frac { \dd \tilde z ' } { \big ( 1 + \vert \tilde z ' \vert ^ 2 \big ) ^ { \frac { n + 2 s } 2 } } & = \frac { \displaystyle \pi ^ { \frac { n - 1 } 2 } \Gamma \big ( \frac { 1 + 2 s } 2 \big ) } { \displaystyle \Gamma \big ( \frac { n + 2 s } 2 \big ) } . \end{align*}


\begin{align*} \int _ I \hat { V } ( 0 , \gamma ( s ) ) \ , d s = 0 , \end{align*}


\begin{align*} R ( f , g ) ( x , \omega ) = f ( x ) \overline { \widehat { g } ( \omega ) } e ^ { - 2 \pi i x \cdot \omega } = \widehat { V _ g f } ( \omega , - x ) . \end{align*}


\begin{align*} \Psi _ { ( \Sigma , U ) } ( f ) [ \psi ] : = \int _ { \Sigma } f \psi \mathrm { d } V _ { \Sigma } . \end{align*}


\begin{align*} p _ { \widehat { K } } ( \hat { k } ) & = \mathbb { P } \big ( \hat { k } - 1 / 2 \le \widehat { K } ^ \mathbb { R } \le \hat { k } + 1 / 2 \big ) = F _ { \widehat { K } ^ \mathbb { R } } ( \hat { k } + 1 / 2 ) - F _ { \widehat { K } ^ \mathbb { R } } ( \hat { k } - 1 / 2 ) . \end{align*}


\begin{align*} Y _ { \delta _ { t } \delta _ { s } } ^ { a _ { s } a _ { t } } + 1 = c _ { 1 } ^ { ( s , t ) } \frac { \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } ) \cdot \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } _ { \delta _ { s } \delta _ { t } } ^ { a _ { s } a _ { t } } ) } { \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } _ { \delta _ { s } } ^ { a _ { s } } ) \cdot \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } _ { \delta _ { t } } ^ { a _ { t } } ) } . \end{align*}


\begin{align*} f = \frac { 1 } { \norm { g } _ 2 ^ 2 } \iint _ { \R ^ { 2 d } } V _ g f ( x , \omega ) M _ \omega T _ x g \ , d ( x , \omega ) = \frac { 1 } { \norm { g } _ 2 ^ 2 } \iint _ { \R ^ { 2 d } } V _ g f ( \l ) \ , \pi ( \l ) g \ , d \l , \l = ( x , \omega ) \in \R ^ { 2 d } . \end{align*}


\begin{align*} \theta ( a _ x ) = \theta \left ( \frac { f ( \pi _ n ) } { [ \omega _ x ] } \right ) = \frac { f ( \theta ( \pi _ n ) ) } { x } = \frac { f ( \zeta _ { 2 ^ n } - 1 ) } { x } = 1 . \end{align*}


\begin{align*} \Psi _ { r n j } \left ( x \right ) = p ^ { \frac { - r } { 2 } } \chi _ { p } \left ( p ^ { - 1 } j \left ( p ^ { r } x - n \right ) \right ) \Omega \left ( \left \vert p ^ { r } x - n \right \vert _ { p } \right ) , \end{align*}


\begin{align*} T ( f _ j ) ( x ) & = T ( g ) ( x ) + T ( h _ j ) ( x ) = T ( g ) ( x ) + \int _ { 2 R \leqslant d ( 0 , y ) \le j } K ( x , y ) f ( y ) d \omega ( y ) \\ & = T ( g ) ( x ) + \int _ { 2 R \leqslant d ( 0 , y ) \le j } [ K ( x , y ) - K ( 0 , y ) ] f ( y ) d \omega ( y ) + c _ j - C ( R ) , \end{align*}


\begin{align*} \int _ 0 ^ T h \dd ( y - u ) = 0 \int _ 0 ^ T ( v - y ) \dd ( y - u ) \geq 0 ~ ~ \forall v \in C ( [ 0 , T ] ; Z ) . \end{align*}


\begin{align*} | I _ { 3 , c } | \leq & \varrho \overline P \int _ 0 ^ \tau \int _ { \Omega _ R } \left | \frac { \varrho - \rho } { \varepsilon } \right | | s | | \Delta \Psi | \\ \leq & c \left \| \frac { \varrho - \rho } { \varepsilon } \right \| _ { L ^ 2 ( 0 , T ; L ^ 2 ( \Omega ) ) } \| s \| _ { L ^ { q _ 1 } ( 0 , T ; L ^ { q _ 1 } ( \Omega ) ) } \| \Delta \Psi \| _ { L ^ { q _ 2 } ( 0 , T ; L ^ { q _ 2 } ( \Omega ) ) } \leq c ( D , T ) \varepsilon ^ { 2 ( 1 - \frac { 1 } { q _ 1 } - \frac { 1 } { q _ 2 } ) } \end{align*}


\begin{align*} \lim _ { t \to 0 } t ^ { \frac { \beta } { \alpha } } \| B ( t ) \| _ { H ^ { \beta } ( \mathbb S ^ 1 ) } = 0 \ \ \forall \ \ \beta > 0 . \end{align*}


\begin{align*} \tau ^ 1 = & \inf \{ t \ge 0 : \ Y \notin \Pi _ { 1 } \} , & \\ \tilde \tau ^ 2 = & \inf \{ t \ge \tau ^ 1 : \ Y \notin \Pi _ 2 \} , & \tau ^ 2 = & \inf \{ t \ge 0 : \ Y \notin \Pi _ 2 \} , \\ \tilde \tau ^ 3 = & \inf \{ t \ge \tau ^ 2 : \ Y \notin \Pi _ 3 \} , & \tau ^ 3 = & \inf \{ t \ge 0 : \ Y \notin \Pi _ 3 \} , \end{align*}


\begin{align*} \norm { f - \sum _ { \gamma \in F _ { 2 n + 1 } } e _ \gamma } _ \mathcal { B } = \norm { f - \sum _ { k = 1 } ^ { N _ { 2 n + 1 } } e _ { \pi ( k ) } } _ \mathcal { B } < \frac { \varepsilon } { 2 } . \end{align*}


\begin{align*} c _ { i , j } ^ { ( r , s ) } = \dfrac { 4 } { \pi ^ 2 } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { \pi } f _ { x ^ { r } y ^ s } ( \cos \theta _ x , \cos \theta _ y ) \cos i \theta _ x \cos j \theta _ y d \theta _ x d \theta _ y , \end{align*}


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


\begin{align*} \sum _ { n \geq 0 } \mu ( L _ { B , n } ) = + \infty \end{align*}


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


\begin{align*} _ r F _ s ( z ) \begin{cases} z = 0 & , \\ z \in \mathbb { C } _ { \infty } & , \\ z \in \mathbb { C } _ { \infty } | z | _ { \infty } < q ^ { \sum _ { j = 1 } ^ s ( b _ j - 1 ) - \sum _ { j = 1 } ^ r ( a _ j - 1 ) } & . \end{cases} \end{align*}


\begin{align*} \mathcal { H } ^ { \frac { 1 } { \delta } } _ { \infty } ( U _ { p , q , \delta } ) \geq ( \mathcal { H } ^ { \frac { \log 2 } { \log 3 } } _ { \infty } ( U _ { p , q , \delta } ) ) ^ { \frac { \log 3 } { \delta \log 2 } } \geq \widetilde { C } ( q ^ { - 2 \delta \frac { \log 2 } { \log 3 } } ) ^ { \frac { \log 3 } { \delta \log 2 } } = \widetilde { C } \mathcal { L } \Big ( B \Big ( \frac { p } { q } , q ^ { - 2 } \Big ) \Big ) . \end{align*}


\begin{align*} \mathcal { L } _ { d , \delta } ( f ) ( x ) = \Delta ( f ) ( x ) - \delta \| x \| ^ { \delta - 2 } \langle x ; \nabla ( f ) ( x ) \rangle . \end{align*}


\begin{align*} | \mathcal { D D } _ { ( s , t ) } | = \binom { \lfloor ( s - 1 ) / 2 \rfloor + \lfloor ( t - 1 ) / 2 \rfloor } { \lfloor ( s - 1 ) / 2 \rfloor } , \end{align*}


\begin{align*} \mathcal { H } ^ { \mu , s } _ { t } ( A ) = \inf \left \{ \mathcal { H } ^ { s } _ { t } ( E ) : \ E \subset A , \ \mu ( E ) = \mu ( A ) \right \} . \end{align*}


\begin{align*} \lambda ( t ) = C ( u _ 0 ) ( 1 + o ( 1 ) ) ( T - t ) ^ { \frac { 2 } { \alpha } } . \end{align*}


\begin{align*} G = \{ a \rightarrow a v , \ ; \ ; v \rightarrow u , \ ; \ ; u \rightarrow 2 u v \} . \end{align*}


\begin{align*} \int _ 0 ^ 1 ( J ^ n _ 1 ( 1 , u ) ) ^ 2 d u & \le \Big ( \int _ 0 ^ 1 \Big ( \exp \Big ( \int _ s ^ t \tilde { f } ' ( r , x _ r ) d r \Big ) \Big ( \tilde { f } ' _ n ( s , x _ s ^ n ) - K _ s \Big ) d s \Big ) ^ 2 \\ & = \Big \{ - \exp \Big ( \int _ s ^ 1 \tilde { f } ' ( r , x _ r ) d r \Big ) \Big \vert _ { s = 0 } ^ 1 - \int _ 0 ^ 1 K _ s \exp \Big ( \int _ s ^ 1 \tilde { f } ' ( r , x _ r ) d r \Big ) d s \Big \} ^ 2 \\ & \le \Big ( \exp ( \int _ 0 ^ 1 K _ r d r ) + \int _ 0 ^ 1 K _ s d s \cdot \exp ( \int _ 0 ^ 1 K _ r d r ) \Big ) ^ 2 . \end{align*}


\begin{align*} R _ 0 : = \sup _ { { \bf y } \in \mathcal A } r ( { \bf y } ) & \le \big ( 1 + \alpha _ 0 + \alpha ^ { - 1 } \big ) \sup _ { { \bf y } \in \mathcal A } \big ( | x | + | v | + | x ' | + | v ' | \big ) \\ & \le 2 \big ( 1 + \alpha _ 0 + \alpha ^ { - 1 } \big ) \sup _ { ( x , v ) \in \mathcal A _ 0 } \big ( | x | + | v | \big ) \\ & \le 2 R ^ * \big ( 1 + \alpha _ 0 + \alpha ^ { - 1 } \big ) . \end{align*}


\begin{align*} \beta ^ { m a t } : = \left ( \begin{array} { c c } 0 & \beta ^ { ( \eta ) } _ { 1 2 } \\ \beta ^ { ( \eta ) } _ { 2 1 } & 0 \end{array} \right ) = \frac { i } { 2 } \left [ \sigma _ 3 , m ^ { P C } _ { \eta , 1 } \right ] = \left ( \begin{array} { c c } 0 & i [ m ^ { P C } _ { \eta , 1 } ] _ { 1 2 } \\ - i [ m ^ { P C } _ { \eta , 1 } ] _ { 2 1 } & 0 \end{array} \right ) . \end{align*}


\begin{align*} \int _ \Omega & \left ( \frac { 1 } { 2 } \rho | u | ^ 2 + P ( \rho ) \right ) ( \tau ) + \int _ 0 ^ \tau \int _ \Omega \mathbb S ( \nabla u ) \cdot \nabla u \\ & \leq \int _ \Omega \left ( \frac { 1 } { 2 } \rho _ 0 | u _ 0 | ^ 2 + P ( \rho _ 0 ) \right ) + \int _ 0 ^ \tau \int _ \Omega \rho f \cdot u . \end{align*}


\begin{align*} \mathbb G _ n = \left \{ \left ( \sum _ { 1 \leq i \leq n } z _ i , \sum _ { 1 \leq i < j \leq n } z _ i z _ j , \dots , \prod _ { i = 1 } ^ n z _ i \right ) : \ , | z _ i | < 1 , i = 1 , \dots , n \right \} . \end{align*}


\begin{align*} | C \cap L | = O _ d ( \log | L | ) = O _ d ( \log \ell _ 1 ) , \end{align*}


\begin{align*} \prod _ \ast a _ j ( k _ j ) : = \overline { a _ 1 ( k _ 1 ) } a _ 2 ( k _ 2 ) \overline { a _ 3 ( k _ 3 ) } a _ 4 ( k _ 4 ) . \end{align*}


\begin{align*} \frac { 1 } { k } \big ( e _ { \lambda } ( t ) - 1 \big ) ^ { k } = \sum _ { n = k } ^ { \infty } S _ { 2 , \lambda } ( n , k ) \frac { t ^ { n } } { n ! } , ( k \ge 0 ) . \end{align*}


\begin{align*} E \left [ \left ( \dfrac { S _ n } { \sqrt { n / ( 1 - 2 \alpha ) } } \right ) ^ 2 \right ] - 1 \sim - \dfrac { 1 } { \Gamma ( 2 \alpha ) } \cdot n ^ { - ( 1 - 2 \alpha ) } \qquad ( n \to \infty ) . \end{align*}


\begin{align*} K = \langle R ( X _ 1 , X _ 2 ) X _ 2 , X _ 1 \rangle = 2 \langle \xi , E _ 3 ^ { \perp } \rangle ^ 2 - 1 + 2 f ^ 2 - \frac { 1 } { 2 } | A | ^ 2 , \end{align*}


\begin{align*} \check H _ { c t } ^ q ( F _ n ; A ) = \begin{cases} \bigoplus _ \N A & q = 0 \\ 0 & \mbox { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} \alpha \alpha ' = 1 + ( \alpha _ m + \alpha ' _ m ) t ^ m + \cdots \quad \alpha ^ { - 1 } = 1 - \alpha _ m t ^ m + \cdots \end{align*}


\begin{align*} \tilde { \Omega } _ { \varepsilon } & \coloneqq \{ \mathbf { x } \in \R ^ N : k _ { \varepsilon } \mathbf { x } + \mathbf { x } _ { \varepsilon } \in \Omega \} \ ; , \\ \Omega _ { \varepsilon } & \coloneqq \{ \mathbf { x } \in \R ^ N : k _ { \varepsilon } \mathbf { x } + \mathbf { q } \in \Omega \} \ ; . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { | X | } | \mathcal { X } ( L ^ { ( i ) } ) | \end{align*}


\begin{align*} \nu _ F = \sum _ { H \in H ( n - m ) } \nu _ { F , H } . \end{align*}


\begin{align*} \begin{cases} E _ { \text i } ^ + ( \kappa _ x ) + E _ { \text r } ^ - ( \kappa _ x ) = E _ { \text t } ^ + ( \kappa _ x ) \\ E _ { \text i } ^ + ( \kappa _ x ) - E _ { \text r } ^ - ( \kappa _ x ) = \alpha ( \kappa _ x ) E _ { \text t } ^ + ( \kappa _ x ) \end{cases} \end{align*}


\begin{align*} \lambda ( \alpha _ 1 \alpha _ 2 , z ) = \lambda ( \alpha _ 1 , \alpha _ 2 z ) \lambda ( \alpha _ 2 , z ) \alpha _ 1 , \alpha _ 2 \in \mathcal { G } , z \in \mathcal { H } . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n \binom { n } { k } e _ k x ^ k = \sum _ { n = 0 } ^ n \binom { n } { j } f _ j x ^ j ( 1 - x ) ^ { n - j } = ( 1 - x ) ^ n \sum _ { m = 0 } ^ n \binom { n } { j } f _ j \left ( \frac { x } { 1 - x } \right ) ^ j . \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 ^ + } - i t \cdot \omega _ 2 ( i t ) & = \lambda _ 1 ( \mu ) ^ 2 - | \lambda | ^ 2 . \end{align*}


\begin{align*} \eta ^ 2 _ { \alpha ^ { - 1 } A _ Y } ( U ) : \alpha ^ { - 1 } A _ Y ( U ) = A _ Y ( \alpha ( U ) ) & \to \alpha ^ * A _ Y ( U ) = A _ X ( U ) \\ ( \varphi : \alpha ( U ) \to A ) & \mapsto ( \varphi \circ \alpha : U \to A ) \end{align*}


\begin{align*} \dim _ H ( \limsup _ { n \rightarrow + \infty } R _ n ) = \min _ { 1 \leq i \leq d } \left \{ \frac { \dim ( \mu ) + \sum _ { 1 \leq j \leq i } \tau _ i - \tau _ j } { \tau _ i } \right \} . \end{align*}


\begin{align*} \tilde \varphi : \Z _ { \ge 0 } & \to A \\ x & \mapsto \sum _ { i = 0 } ^ { x - 1 } \varphi ( i , i + 1 ) . \end{align*}


\begin{align*} \left [ ( 1 - \Delta ) ^ 2 + e _ 0 ^ 2 \right ] \mathcal { Y } _ 1 = F ( x ) , \end{align*}


\begin{align*} D _ n ( u , v ) = A _ n ( x , y ) , \end{align*}


\begin{align*} \vartheta ^ n = \frac 1 2 \Big ( ( V + \beta | \phi ^ n | ^ 2 ) _ { \max } + ( V + \beta | \phi ^ n | ^ 2 ) _ { \min } \Big ) , \end{align*}


\begin{align*} \begin{aligned} \left | \| W W ^ { T } \bar { U } \bar { \Sigma } ^ { - 1 } \| _ F - \| ( S S ^ { T } - W W ^ { T } ) \bar { U } \bar { \Sigma } ^ { - 1 } \| _ F \right | & \leq \| W W ^ { T } \bar { U } \bar { \Sigma } ^ { - 1 } + ( S S ^ { T } - W W ^ { T } ) \bar { U } \bar { \Sigma } ^ { - 1 } \| _ F = \| S \hat { V } \| _ F , \end{aligned} \end{align*}


\begin{align*} u \leq v & \Leftrightarrow \ell ( v ) = \ell ( u ) + \ell ( v u ^ { - 1 } ) \\ & \Leftrightarrow \ell ( v \xi ^ { - 1 } ) = \ell ( u \xi ^ { - 1 } ) + \ell ( v u ^ { - 1 } ) \\ & \Leftrightarrow \ell ( v \xi ^ { - 1 } ) = \ell ( u \xi ^ { - 1 } ) + \ell ( v \xi ^ { - 1 } ( u \xi ^ { - 1 } ) ^ { - 1 } ) \\ & \Leftrightarrow u \xi ^ { - 1 } \leq v \xi ^ { - 1 } \\ & \Leftrightarrow \rho _ \xi ( u ) \leq \rho _ \xi ( v ) . \end{align*}


\begin{align*} \mathrm { s g n } _ X ( g h ) & = \prod _ { \{ u , v \} \in E ( X ) } \frac { u ^ { g h } - v ^ { g h } } { u - v } \\ & = \prod _ { \{ u , v \} \in E ( X ) } \frac { u ^ g - v ^ g } { u - v } \cdot \frac { ( u ^ g ) ^ h - ( v ^ g ) ^ h } { u ^ g - v ^ g } \\ & = \prod _ { \{ u , v \} \in E ( X ) } \frac { u ^ g - v ^ g } { u - v } \cdot \prod _ { \{ u , v \} \in E ( X ) } \frac { ( u ^ g ) ^ h - ( v ^ g ) ^ h } { u ^ g - v ^ g } \\ & = \mathrm { s g n } _ X ( g ) \ \mathrm { s g n } _ X ( h ) , \end{align*}


\begin{align*} \begin{aligned} & \psi _ t ( x _ q , t _ q ) + \lambda G _ \beta ( a _ q , \xi _ q , Y _ q , W _ \star [ x _ 0 , t _ 0 ] \} ) + ( 1 - \lambda ) G _ \beta ( b _ q , \xi _ q , Z _ q , W _ \star [ x _ 0 , t _ 0 ] ) ) \\ & \geq \lambda \left ( G _ \beta ( a _ q , \xi _ q , Y _ q , W _ \star [ x _ 0 , t _ 0 ] ) - G _ \beta ( a _ q , \xi _ q , Y _ q , U [ y _ q , t _ q ] ) \right ) \\ & \qquad + ( 1 - \lambda ) \left ( G _ \beta ( b _ q , \xi _ q , Z _ q , W _ \star [ x _ 0 , t _ 0 ] ) - G _ \beta ( b _ q , \xi _ q , Z _ q , U [ z _ q , t _ q ] ) \right ) , \end{aligned} \end{align*}


\begin{align*} c _ { x , y } ( \eta ) : = \tilde { c } _ { x , y } ( \eta ) \xi _ { x , y } ( \eta ) , \end{align*}


\begin{align*} f ( t z ) & = t ^ { - k } f ( z ) , \\ f ( \gamma z ) & = \chi ( \gamma ) f ( z ) , \end{align*}


\begin{align*} \left \{ \begin{array} { l l l l } H _ \mu u \ ! + \ ! B _ q ( w , u ) & = & f & ( x , t ) \in { \mathbb R } ^ n \times ( 0 , T ) , \\ u & = & u _ 0 , & ( x , t ) \in \mathbb { R } ^ n \times \{ 0 \} . \end{array} \right . \end{align*}


\begin{align*} \psi \left ( \begin{pmatrix} x _ 0 \\ y _ 0 \end{pmatrix} , \ldots , \begin{pmatrix} x _ { q - 1 } \\ y _ { q - 1 } \end{pmatrix} \right ) : = \psi _ { \max ( y _ i ) } \left ( \begin{pmatrix} x _ 0 \\ y _ 0 \end{pmatrix} , \ldots , \begin{pmatrix} x _ { q - 1 } \\ y _ { q - 1 } \end{pmatrix} \right ) \end{align*}


\begin{align*} \tilde M _ { n , A } = \frac 2 n \sum _ { i < j } \prod _ { p \in A } \tilde I _ { i , j } ^ { ( p ) } , \tilde N _ { n , A } = \frac 1 n \sum _ { i = 1 } ^ n \prod _ { p \in A } \tilde I _ { i , i } ^ { ( p ) } . \end{align*}


\begin{align*} \nabla _ U U = \nabla _ V U = 0 , \nabla _ U V = \nabla _ V V = 0 . \end{align*}


\begin{align*} \mathcal { H } ^ { \zeta } ( E ) = \lim _ { t \to 0 ^ + } \mathcal { H } ^ { \zeta } _ t ( E ) . \end{align*}


\begin{align*} 4 B ^ { [ \widetilde { U } ] } \cap 4 B ^ { [ \widehat { U } ] } = \emptyset ; \end{align*}


\begin{align*} V \cdot W = \left \{ \sum _ i v _ i w _ i \mid v _ i \in V , w _ i \in W \right \} . \end{align*}


\begin{align*} \begin{aligned} & ~ ~ ~ ~ \epsilon + 2 ( 2 + \xi ) ( 2 \xi \sqrt { 1 + \xi } + \xi ^ 2 ) + ( 2 \xi \sqrt { 1 + \xi } + \xi ^ 2 ) ^ 2 \\ & = \epsilon + \xi ^ 4 + 6 \xi ^ 3 + 8 \xi ^ 2 + 4 \xi ^ 3 \sqrt { 1 + \xi } + 4 \xi ^ 2 \sqrt { 1 + \xi } + 8 \xi \sqrt { 1 + \xi } \\ & < \epsilon + 1 5 \xi + 1 6 \xi \sqrt { 1 + \xi } < \epsilon + 3 6 \xi = \frac { \epsilon ^ 2 + 2 0 \epsilon } { \epsilon + 2 } < 1 0 \epsilon . \end{aligned} \end{align*}


\begin{align*} \widehat { ( \partial _ { \Sigma _ 0 , \epsilon } ^ { * } ) ^ { \otimes 2 } \overline { u } _ n } ( \xi , \eta ) = - 4 \xi _ u \eta _ u \hat { \delta } _ { \epsilon } ( 2 \xi _ v ) \hat { \delta } _ { \epsilon } ( 2 \eta _ v ) \int _ { \mathbb { R } ^ { 2 } } \overline { u } _ n ( s - s ' ) e ^ { i ( \xi _ u - \xi _ v ) s } e ^ { i ( \eta _ u - \eta _ v ) s } \ , \mathrm { d } s \ , \mathrm { d } s ' . \end{align*}


\begin{align*} \psi | q _ i | \rightarrow 0 , \ ; \ ; \nabla _ { q _ i } \psi \cdot \frac { q _ i } { | q _ i | } \rightarrow 0 , \ ; \ ; \ ; \ ; | q _ i | \rightarrow \infty , \ ; \ ; \ ; \ ; ( x , t ) \in \mathbb { R } ^ 3 \times ( 0 , T ] , \ ; \ ; i = 1 , \ldots , K . \end{align*}


\begin{align*} f = \iint _ { \R ^ { 2 d } } V _ g f ( \l ) \ , \pi ( \l ) g \ , d \l . \end{align*}


\begin{align*} \nabla \varphi ( y _ \varepsilon , t _ \varepsilon ) = 0 , \nabla ^ 2 \varphi ( y _ \varepsilon , t _ \varepsilon ) \leq 0 . \end{align*}


\begin{align*} 0 = { \rm t r . d e g } _ { \overline { k } ( t ) } \overline { k } ( t ) \bigl \{ 1 , \ _ { s + 1 } \mathcal { F } _ s ( { \bf a } ; { \bf b } ) ( \alpha ) ^ { q ^ { d } } \bigr \} = { \rm t r . d e g } _ { \overline { k } } \overline { k } \bigl \{ 1 , \ _ { s + 1 } F _ s ( { \bf a } ; { \bf b } ) ( \alpha ) ^ { q ^ { d } } \bigr \} . \end{align*}


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


\begin{align*} \theta _ K ( L ) = \tau _ K ( L ) \wedge \tau ' _ { K } . \end{align*}


\begin{align*} F _ r ( x , \omega , e ^ { 2 \pi i \tau } ) = e ^ { - 2 \pi i \tau } F ( x , \omega ) . \end{align*}


\begin{align*} u _ \varepsilon = \varepsilon + \varepsilon ^ m v _ m ( x ) + r _ \varepsilon ( x ) \forall \ , x \in M , \end{align*}


\begin{align*} \begin{cases} \ A ( x ) - f ( n ) > 2 ^ { - x } \implies x > f ( n ) \\ [ 8 p t ] \ A ( x ) - f ( n ) < 2 ^ { - x } \implies x < f ( n ) \end{cases} . \end{align*}


\begin{align*} E ^ { i j } _ { 0 } ( x , x ' ) = \sum _ { k = 0 } ^ n a _ k \left ( \frac { \partial } { \partial x } \right ) ^ k \delta ( x - x ' ) , \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } d Z ( t ) & = & ( B Z _ t + b ) d t + c \ , d W ( t ) \medskip \\ Z ( 0 ) & = & z . \end{array} \right . \end{align*}


\begin{align*} G : = \left \{ \begin{aligned} & \begin{pmatrix} 1 & 0 \\ - r e ^ { 2 i t g } & 1 \end{pmatrix} , & k \in U _ 1 \cup U _ 2 , \\ & \begin{pmatrix} 1 & - r ^ { * } e ^ { - 2 i t g } \\ 0 & 1 \end{pmatrix} , & k \in U _ 1 ^ { * } \cup U _ 2 ^ { * } , \\ & I , & \textnormal { e l s e w h e r e } \end{aligned} \right . \end{align*}


\begin{align*} ( A \setminus \{ \bar { a } _ n \} ) \cup C _ 1 \cup C _ 2 & = ( A \setminus \{ \bar { a } _ n \} ) \cup ( C \setminus \{ \bar { a } _ n \} ) \\ & = ( A \cup C ) \setminus \{ \bar { a } _ n \} = \mathbb { N } \setminus \{ \bar { a } _ n \} \end{align*}


\begin{align*} - k _ { \varepsilon } \Delta v _ { \varepsilon } ( \mathbf { x } ) + | \nabla v _ { \varepsilon } ( \mathbf { x } ) | ^ 2 - \tilde { \lambda } _ 0 \underline { m } = - k _ { \varepsilon } \sigma _ 3 \sqrt { \tilde { \lambda } _ 0 \underline { m } } \frac { ( N - 1 ) } { | \mathbf { x } - \mathbf { y } _ { \varepsilon } | } + \sigma _ 3 ^ 2 \tilde { \lambda } _ 0 \underline { m } - \tilde { \lambda } _ 0 \underline { m } < 0 \end{align*}


\begin{align*} e '' _ { f _ { \lambda + 1 } } = ( n - 1 ) f _ { \lambda + 1 } + g _ { \lambda + 1 } = ( n - 1 ) f _ { \lambda + 1 } + ( f _ { \lambda + 1 } - f _ \lambda ) = f _ { \lambda + 1 } + f _ { \lambda + 2 } \end{align*}


\begin{align*} \varphi ( x ) = \begin{cases} | x | ^ 2 , & | x | \le 1 , \\ 0 , & | x | \ge 2 , \end{cases} \varphi ( r ) \ge 0 , \varphi '' ( r ) \le 2 , \end{align*}


\begin{align*} m _ { i , \lambda } ( S ( \underline { P } , B ) ) = 0 . \end{align*}


\begin{align*} ( a \omega ) ^ * = \omega ^ * a ^ * , ( \omega a ) ^ * = a ^ * \omega ^ * . \end{align*}


\begin{align*} \Lambda _ \xi = 2 + \xi \partial _ \xi \end{align*}


\begin{align*} F _ { \mu } ( \omega _ 1 ( z ) ) = F _ { \nu } ( \omega _ 2 ( z ) ) = F _ { \mu \boxplus \nu } ( z ) , z \in \mathbb { C } ^ + . \end{align*}


\begin{align*} \sum _ { v _ i , v _ j \notin V ( C ) } \beta _ i \beta _ j - \sum _ { v _ i \in V ( C ) , v _ j \notin V ( C ) } \alpha _ i \beta _ j = \sum _ { v _ j \notin V ( C ) } \beta _ j ( \sum _ { v _ r \notin V ( C ) } \beta _ r - \sum _ { v _ l \in V ( C ) } \alpha _ l ) \end{align*}


\begin{align*} \sum _ { [ K _ p : \mathbb { Q } _ p ] = n } \frac { | \mathrm { D i s c } ( K _ p ) | _ { p } ^ { 1 / 2 } } { | \mathrm { A u t } ( K _ p ) | } \int _ { \mathcal { O } _ { K _ p } } | \mathrm { D i s c } ( f _ \alpha ) | _ p ^ { 1 / 2 } \psi _ { p ^ { 2 k } } ( f _ \alpha ) e _ { p ^ { 2 k } } ( - u _ 1 \sigma _ 1 ( \pmb { \lambda } ) + u _ 2 \sigma _ 2 ( \pmb { \lambda } ) ) d \mu ( \alpha ) , \end{align*}


\begin{align*} - \int _ { H _ { \sigma _ 2 } } ^ T N ( \sigma _ 2 , t ) { d } \Big ( \frac { x ^ { - \frac { 1 } { R \log t } } } { t } \Big ) \le - N ( \sigma _ 2 , T ) \int _ { t _ 0 } ^ T { d } \Big ( \frac { x ^ { - \frac { 1 } { R \log t } } } { t } \Big ) = N ( \sigma _ 2 , T ) \Big ( \frac { x ^ { - \frac { 1 } { R \log t _ 0 } } } { t _ 0 } - \frac { x ^ { - \frac { 1 } { R \log T } } } { T } \Big ) . \end{align*}


\begin{align*} & \inf _ { \rho , \widehat v } \ ; [ \int _ 0 ^ 1 \frac 1 2 \frac 1 { ( 1 + \epsilon \dot W ^ { \delta } ) ^ 2 } \ < \widehat v _ t , \widehat v _ t \ > _ { \theta ( \rho _ t ) } d t ] \\ & \ ; \ ; d \rho ( t ) + d i v _ G ^ { \theta } ( \rho ( t ) \widehat v ( t ) ) = 0 \\ & \ ; \ ; \rho ( 0 ) = \rho _ a , \ ; \rho ( 1 ) = \rho _ b . \end{align*}


\begin{align*} & \forall \gamma ^ 1 , { \gamma ' } ^ 1 , x ^ X , { x ' } ^ X \big ( \gamma > _ \mathbb { R } 0 \land { \gamma ' } > _ \mathbb { R } 0 \land \tilde { \rho } > _ \mathbb { R } - \gamma \\ & \qquad \qquad \qquad \qquad \land x = _ X x ' \land \gamma = _ \mathbb { R } \gamma ' \rightarrow J ^ A _ \gamma x = _ X J ^ A _ { \gamma ' } x ' \big ) . \end{align*}


\begin{align*} I _ 1 \leq C \int _ 0 ^ { 4 \epsilon } \int _ { \Gamma _ t } t ^ { - p s } d S d t \leq C \int _ { 0 } ^ { 4 \epsilon } t ^ { - p s } d t = C \epsilon ^ { 1 - p s } . \end{align*}


\begin{align*} \dot { W } _ { s , p } ^ \nu ( x ) : = \sum _ { j = - \infty } ^ \infty 2 ^ { j s q } \int _ { B ( x , 2 ^ { - j } ) } \left ( \nu ( B ( y , 2 ^ { - j } ) ) \right ) ^ { q - 1 } \mu ( d y ) , x \in M , \end{align*}


\begin{align*} \gamma ^ { \epsilon , L } ( d \varphi ) \propto \exp \left [ - \frac { 1 } { 2 } \sum _ { x \in \Lambda _ { \epsilon , L } } \left ( \varphi _ x ( - \Delta ^ 1 + \epsilon ^ 2 m ) \varphi _ x \right ) \right ] d \varphi = \exp \left [ - \frac { 1 } { 2 } ( \varphi , A _ { \epsilon } \varphi ) \right ] d \varphi , \end{align*}


\begin{align*} ( \Psi _ { \Sigma } ( f ) \star _ { H , \ell } \Psi _ { \Sigma } ( g ) ) [ \psi ] = ( \Psi _ { \Sigma } ( f ) \cdot \Psi _ { \Sigma } ( g ) ) [ \psi ] - \frac { \hbar } { 4 \pi } \int _ { \mathbb { R } ^ { 2 } } \frac { f ( s ) g ( s ' ) } { ( s - s ' ) ^ 2 } \ , \mathrm { d } s \ , \mathrm { d } s ' , \end{align*}


\begin{align*} S ( k ) : = \left ( \begin{array} { c c } T ( k ) & R _ { + } ( k ) \\ R _ { - } ( k ) & T ( k ) \end{array} \right ) , \ \ S ^ { - 1 } ( k ) : = \left ( \begin{array} { c c } \overline { T ( k ) } & \overline { R _ { - } ( k ) } \\ \overline { R _ { + } ( k ) } & \overline { T ( k ) } \end{array} \right ) . \end{align*}


\begin{align*} \omega _ 2 ( i t ) = i \left [ t - s + \frac { ( s - t ) ^ 2 + | \lambda | ^ 2 } { s - t } \right ] = i \frac { | \lambda | ^ 2 } { s ( | \lambda | , t ) - t } = \frac { | \lambda | ^ 2 } { i t - \omega _ 1 ( i t ) } . \end{align*}


\begin{align*} G _ { k , l } ( x ) = \sum _ { s = 1 } ^ \infty \frac { x ^ { k ( s + 1 ) } } { k ( s + 1 ) - l } . \end{align*}


\begin{align*} & \frac { 1 } { 2 \pi } \int _ { 0 } ^ { T } \left [ \log \left | \pi ^ { - \left ( \frac { \frac { 1 } { 2 } - \epsilon + i t } { 2 } \right ) } \right | - \log \left | \pi ^ { - \left ( \frac { \frac { 1 } { 2 } + \epsilon + i t } { 2 } \right ) } \right | \right ] \ d t + \\ & \frac { 1 } { 2 \pi } \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \left [ \arg \pi ^ { - \left ( \frac { \sigma + i T } { 2 } \right ) } - \arg \pi ^ { - \frac { \sigma } { 2 } } \right ] \ d \sigma = 0 \end{align*}


\begin{align*} X _ 1 \ \cap \ X _ n \ = \ \emptyset , Y _ 1 \ \cap \ Y _ 2 = \emptyset . \end{align*}


\begin{align*} \frac { 1 } { t } d _ K ( x + t \beta x ) = 0 . \end{align*}


\begin{align*} W _ { i } \upharpoonright t ' = \Phi _ i ^ { \Gamma _ A } \upharpoonright t ' , 0 = \Gamma _ A ( \langle x _ m , y _ m \rangle ) = \Psi _ i ^ { W _ i \upharpoonright t ' } ( \langle x _ m , y _ m \rangle ) . \end{align*}


\begin{align*} u ( t , x ) \sim \sum _ { i = 1 } ^ n Q ( x - t - x _ i ( t ) ) , \end{align*}


\begin{align*} \begin{cases} \ f ( x ) = q - 2 ^ { - x } \\ [ 8 p t ] \ g ( x ) = q + 2 ^ x \end{cases} . ] \end{align*}


\begin{align*} g _ 1 ( \theta ^ { q ^ N } ) \bigr ( ( \theta ^ { q ^ N } - \theta ^ q ) ^ n \cdots ( \theta ^ { q ^ N } - \theta ^ { q ^ { N - 1 } } ) ^ n \bigr ) \alpha ^ { q ^ N } + g _ 2 ( \theta ^ { q ^ N } ) L i _ { C , n } ( \alpha ) ^ { q ^ N } - g _ 2 ( \theta ^ { q ^ N } ) \beta ^ { q ^ N } = 0 . \end{align*}


\begin{align*} u ( x ) = \int _ 0 ^ { T } e ^ { - s } L \left ( \xi ( s ) , - \dot { \xi } ( s ) \right ) d s + e ^ { - T } u ( \xi ( T ) ) . \end{align*}


\begin{align*} L ( \rho ) = \sum _ { i = 1 } ^ N ( \log ( \rho _ i ) \rho _ i - \rho _ i ) , \end{align*}


\begin{align*} \sup _ { f \in M } \| f \| _ { Y _ q ( \tilde \Omega _ { [ \hat t ( n ) ] } ) } \stackrel { ( \ref { n u 2 } ) , ( \ref { e m b _ n u } ) } { \underset { \mathfrak { Z } _ 0 } { \lesssim } } 2 ^ { ( ( 1 ‐ \lambda ) \mu _ * ‐ \lambda \alpha _ * ) k _ * \hat t ( n ) } \stackrel { ( \ref { h a t _ m t } ) } { = } n ^ { ( ( 1 ‐ \lambda ) \mu _ * ‐ \lambda \alpha _ * ) / \gamma _ * } = n ^ { ‐ \theta _ 3 } \end{align*}


\begin{align*} \chi _ { f , G } ^ I ( \sigma , \sigma ' ) = \chi _ { f , G } ^ I ( s ^ { m ^ * } , 1 ) \end{align*}


\begin{align*} [ x + u , y + v , z + w ] _ { L \oplus V } = [ x , y , z ] + \theta ( y , z ) u - \theta ( x , z ) v + D ( x , y ) w , \end{align*}


\begin{align*} g ( \lambda - \overline { \lambda } ) L h = P h + Q \overline { h } + g ( \lambda - \overline { \lambda } ) \left ( g ^ { 2 } F ^ { j } + g \lambda G ^ { j } + \lambda ^ { 2 } H ^ { j } \right ) , \end{align*}


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


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


\begin{align*} r ( t , y ) = & \sum _ { \substack { i , j = 1 \\ j \not = i } } ^ n \frac { A _ { i j } ( t , y - x _ { i } ( t ) ) } { x ^ 2 _ { i j } ( t ) } + \sum _ { \substack { i , j = 1 \\ j \not = i } } ^ n \frac { B _ { i j } ( t , y - x _ { i } ( t ) ) } { x _ { i j } ^ 3 ( t ) } \varphi _ { i j } ( t , y ) , \end{align*}


\begin{align*} \mathcal A ( \rho ^ * , m ^ * ) & = \sup _ { S \in \mathbb H ^ 1 _ R } \mathcal L ( \rho ^ * , m ^ * , S ) \ge \inf _ { \rho , m } \sup _ { S \in \mathbb H ^ 1 _ R } \mathcal L ( \rho , m , S ) \\ & = \mathcal A ( \rho ^ { * , R } , m ^ { * , R } ) + R \mathcal E ( \rho ^ { * , R } , m ^ { * , R } ) \\ & \ge \mathcal A ( \rho ^ { * , R } , m ^ { * , R } ) . \end{align*}


\begin{align*} D u \left ( \xi ( t ) \right ) = - C _ p q \left | \dot { \xi } ( t ) \right | ^ { q - 2 } \dot { \xi } ( t ) . \end{align*}


\begin{align*} M [ u ] = M [ Q ] , \ ; E [ u ] = E [ Q ] , \end{align*}


\begin{align*} e _ k & = \phi _ k \left ( \langle 2 \phi _ { k , x } + \varphi _ { k , x } , \varphi _ { k , x } \rangle - \langle 2 \phi _ { k , t } + \varphi _ { k , t } , \varphi _ { k , t } \rangle \right ) \\ & \ ; \ ; \ ; \ ; \ ; \ ; - \varphi _ k \left ( | \tilde \phi _ { k + 1 , t } | ^ 2 - | \tilde \phi _ { k + 1 , x } | ^ 2 \right ) + \mathbf { 1 } _ { \omega } \left ( h _ k ^ { p ^ { \perp } } + f _ { k - 1 } ^ { \phi _ k ^ { \perp } } - f _ { k } ^ { \tilde \phi ^ { \perp } _ { k + 1 } } \right ) . \end{align*}


\begin{align*} P = \frac { | X | ^ { 2 } } { 2 } - \frac { \alpha } { \alpha + 1 } \langle X , \nu \rangle ^ { \frac { \alpha + 1 } { \alpha } } , \end{align*}


\begin{align*} F \natural G ( x , \omega ) = \int _ { \R ^ { 2 d } } F ( x ' , \omega ' ) G ( x - x ' , \omega - \omega ' ) e ^ { \pi i ( x ' \cdot \omega - x \cdot \omega ' ) } \ , d ( x ' , \omega ' ) . \end{align*}


\begin{align*} \Phi = \varepsilon _ { \mathbb C } + \kappa \prec \Phi \end{align*}


\begin{align*} \gamma _ { ( M , g ) } : = \inf _ { u \in H ^ 1 ( M ) \setminus \{ 0 \} } \frac { \displaystyle \int _ M | \nabla _ g u | ^ 2 { \rm d } v _ g } { \displaystyle \int _ M u ^ 2 { \rm d } v _ g } \geq \frac { ( n - 1 ) ^ 2 } { 4 } \kappa . \end{align*}


\begin{align*} \mu _ { 0 } : = \left ( c _ { \alpha } ^ { + } - c _ { \alpha } ^ { - } \right ) \Gamma \biggl ( \frac { 3 - \alpha } { 2 } \biggl ) + \frac { \left ( c _ { \alpha } ^ { + } + c _ { \alpha } ^ { - } \right ) \beta \chi _ { * } ( 0 ) } { 2 - \alpha } \Gamma \biggl ( 2 - \frac { \alpha } { 2 } \biggl ) , \mu _ { 1 } : = \frac { c _ { \alpha } ^ { + } + c _ { \alpha } ^ { - } } { 2 } - \kappa d . \end{align*}


\begin{align*} X ^ k _ \sigma = \{ y \in X ( k ) : y > \sigma \} . \end{align*}


\begin{align*} m ^ { G P } = \Delta _ L ( k ) = \frac { 1 } { 2 } \left ( \begin{array} { c c } \chi _ L ( k ) + \chi _ L ^ { - 1 } ( k ) & i \left ( \chi _ L ( k ) - \chi _ L ^ { - 1 } ( k ) \right ) \\ - i \left ( \chi _ L ( k ) - \chi _ L ^ { - 1 } ( k ) \right ) & \chi _ L ( k ) + \chi _ L ^ { - 1 } ( k ) \end{array} \right ) , \end{align*}


\begin{align*} \sup _ { \mathbf i \in \mathcal J : | \mathbf i | = \mu } | \varphi _ 2 \left ( \mathbf { i } \right ) | & = O \left ( n ^ { 2 - \mu } \right ) . \end{align*}


\begin{align*} \mathbf { v } _ A ^ { \pm } ( x , t ) = \mathbf { v } _ 0 ^ { \pm } ( x , t ) + \varepsilon \mathbf { v } _ 1 ^ { \pm } ( x , t ) + \varepsilon ^ 2 \mathbf { v } _ 2 ^ { \pm } ( x , t ) , p _ A ^ { \pm } ( x , t ) = p _ 0 ^ { \pm } ( x , t ) + \varepsilon p _ 1 ^ { \pm } ( x , t ) , c _ { \pm } ^ { o u t } = \pm 1 . \end{align*}


\begin{align*} ( m ) _ { k , \lambda } = \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) ( m ) _ { l } , ( k \ge 1 ) . \end{align*}


\begin{align*} \frac { d } { d t } \left ( e ^ { - t } C _ p q \left | \dot { \xi } ( t ) \right | ^ { q - 2 } \dot { \xi } ( t ) \right ) = e ^ { - t } D f \left ( \xi ( t ) \right ) . \end{align*}


\begin{align*} \alpha ' = \alpha \rho \wedge 1 , \end{align*}


\begin{align*} J _ L ( g , g \cdot \xi _ 1 , \dots , g \cdot \xi _ k ) = \mu \iff \pmb { F } \ell ( \xi _ 1 , \dots , \xi _ k ) = \big ( { \rm A d } ^ * _ { g } \big ) ^ k \mu = ( { \rm A d } ^ * _ { g } \mu _ 1 , \dots , { \rm A d } ^ * _ { g } \mu _ k ) . \end{align*}


\begin{align*} \begin{aligned} \partial _ s g _ { R , X _ \ast , V _ \ast } + v \cdot \nabla _ x g _ { R , X _ \ast , V _ \ast } - \nabla ( \phi * _ x \rho [ g _ { R , X _ \ast , V _ \ast } ] ) \cdot \nabla _ v \mu & = - e _ 0 \nabla \Phi ( ( x - X _ \ast + ( R - s ) V _ \ast ) \cdot \nabla _ v \mu , \\ g _ { R , X _ \ast , V _ \ast } ( 0 , \cdot ) & = 0 . \end{aligned} \end{align*}


\begin{align*} \ln q = { \theta \over 2 } x . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ \infty \| b _ j \| _ b ^ 2 \le \frac { C } { \delta ^ { 2 \max m _ j } } \Big ( 1 + \frac { 1 } { \delta ^ 2 } \log \frac { 1 } { \delta } \Big ) ^ 2 . \end{align*}


\begin{align*} \varepsilon _ \ell ( \tau ) = - \frac { 2 } { \alpha } m _ 0 b ^ { \frac { \alpha } { 2 } } ( \tau ) . \end{align*}


\begin{align*} \sum _ { j \in N ( i ) } m _ { i j } ( t ) = \sum _ { j \in N ( i ) } ( \Sigma _ i - \Sigma _ j ) \theta ( \rho _ i , \rho _ j ) \dot W ^ { \delta } ( t ) . \end{align*}


\begin{align*} \sum _ { p \le x } \frac { 1 } { p } = & \log \log x + b _ 1 + O \Big ( \frac { 1 } { \log x } \Big ) , \mbox { a n d } \\ \sum _ { p \le x } \frac { \lambda ^ 2 _ f ( p ) } { p } = & \log \log x + b _ 2 + O \Big ( \frac { 1 } { \log x } \Big ) . \end{align*}


\begin{align*} \vec { B _ 0 } ( \mathbf { 0 } ) = ( \tilde { g } _ u ^ { i j } - g ^ { i j } ) \partial _ j \psi \cdot \partial _ i | _ { \mathbf { 0 } } ; & \ & B _ 1 ( \mathbf { 0 } ) = \frac 1 2 \tilde { g } _ u ^ { i j } \partial _ i ( \log \det [ \tilde { g } _ u ] ) \partial _ j \psi ( \mathbf { 0 } ) \ , , \end{align*}


\begin{align*} c _ { \alpha } = \left ( \dfrac { - \alpha ( \alpha - 1 ) } { 4 \Gamma ( 2 - \alpha ) \cos \left ( \frac { \alpha \pi } { 2 } \right ) } \right ) . \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d t } E ( t ) = - \int _ { \mathbb { S } ^ 1 } a ( x ) | \phi _ t | ^ 2 ( t , x ) d x \leq 0 , \end{align*}


\begin{align*} v = \frac { 1 } { 4 k + 1 } \left ( 4 \rho _ 1 + \rho _ 2 + \rho _ 3 \right ) \end{align*}


\begin{align*} \mathcal { L } ^ 0 ( h , k ) = - \sum _ { \substack { 1 \leq c \le C \\ ( c , h k ) = 1 } } \mu ( c ) \sum _ { \substack { 1 \leq m , n < \infty \\ ( m n , c ) = 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 { 1 \le d < \infty \\ ( d , m h n k ) = 1 } } W \left ( \frac { c d } { Q } \right ) . \end{align*}


\begin{align*} a _ j ( X _ j + \overline { X _ j } ) = 0 . \end{align*}


\begin{align*} p _ t ^ A ( x , y ) = h ( x ) h ( y ) p _ t ^ { Y , h } ( x , y ) . \end{align*}


\begin{align*} G \cup H \cup I & = ( \mathbb { N } \setminus \{ \bar { b } _ n \} ) \cup \{ \bar { b } _ n \} = \mathbb { N } , \end{align*}


\begin{gather*} \begin{matrix} \Xi _ { k + 1 } = \mathcal { A } \Xi _ k + \mathcal { B } \mathbf { v } _ k , & \mathbf { y } _ k = \mathcal { C } \Xi _ k , \end{matrix} \\ \mathbf { v } _ k \coloneqq \begin{bmatrix} v _ { 1 , k } \\ v _ { 2 , k } \end{bmatrix} = \begin{bmatrix} 2 \xi _ { 2 , k } - \xi _ { 1 , k } + T _ s ^ 2 z _ { 1 , k } \\ 2 \xi _ { 4 , k } - \xi _ { 3 , k } + T _ s ^ 2 z _ { 2 , k } \end{bmatrix} . \end{gather*}


\begin{align*} U _ \Sigma ( \gamma _ \Sigma ) = 1 . \end{align*}


\begin{align*} u _ n = \sum _ { \beta = 1 } ^ { N } f _ n ^ { \beta } + q _ n ^ N \end{align*}


\begin{align*} \sum _ { \substack { 2 < p \leq X } } \log p \sim X . \end{align*}


\begin{align*} & q _ t ( x , t ) - 6 q ^ 2 ( x , t ) q _ { x } ( x , t ) + q _ { x x x } ( x , t ) = 0 , ( x , t ) \in \mathbb { R } \times \mathbb { R } ^ { + } , \\ & q ( x , 0 ) = q _ { 0 } ( x ) = \left \{ \begin{aligned} & C _ { L } , x < 0 , \\ & C _ { R } , x > 0 , \end{aligned} \right . , \end{align*}


\begin{align*} i _ { v } \omega = \sqrt { - 1 } \overline { \partial } \theta _ { v } ^ { ( \omega ) } \quad \int _ { M } \theta _ { v } ^ { ( \omega ) } \omega ^ { n } = 0 . \end{align*}


\begin{align*} f _ 1 ( s , y ) : = - \alpha _ { - 1 } \tilde { \theta } ( 2 \tilde { \theta } + 1 ) y ^ { 2 \tilde { \theta } } + \alpha _ 0 ( \tilde { \theta } + 1 ) y ^ { \tilde { \theta } } - \frac { \alpha _ 1 } { \tilde { \theta } } \end{align*}


\begin{align*} ( \partial _ t - \sigma \Delta ) \ \eta _ t = - \div \left ( V ( t , \cdot ) \ , \eta _ t \right ) , \end{align*}


\begin{align*} \beta ( z , \tau ) F ( w ) = e ^ { 2 \pi i \tau } e ^ { \pi w \cdot z } e ^ { - \frac { \pi } { 2 } | z | ^ 2 } F ( w - \overline { z } ) , \end{align*}


\begin{align*} ( d _ { \alpha } T ) ( \phi ) : = ( - 1 ) ^ { k - 1 } T ( d _ { \alpha } \phi ) , \ \forall \phi \in \mathcal { E } ^ { k - 1 } ( M , \R ) , \end{align*}


\begin{align*} \det ( \mathbf { L } \cdot \mathbf { R } ) = \sum _ { \mathcal { I } \in \wp _ { k } [ n ] } \Delta _ { \mathbf { L } } ( \mathcal { I } ) \cdot \Delta _ { \mathbf { R } } ( \mathcal { I } ) \end{align*}


\begin{align*} \eta _ t + \mathrm { d i v } _ x ( u \eta ) = \nu \Delta _ x \eta . \end{align*}


\begin{align*} { \bf P } _ { e r r o r } ^ { ( n ) } \triangleq & \frac { 1 } { M ^ { ( n ) } } \sum _ { m \in { \cal M } ^ { ( n ) } } { \bf P } ^ g \Big \{ d _ { n } ( Y ^ { n } ) \neq m \big | M = m \Big \} \leq \epsilon _ n . \end{align*}


\begin{align*} s = _ \rho t : = \forall y _ 1 ^ { \rho _ 1 } , \dots , y _ k ^ { \rho _ k } ( s y _ 1 \dots y _ k = _ 0 t y _ 1 \dots y _ k ) \end{align*}


\begin{align*} X M _ l T _ k g ( x ) = ( k + x - k ) e ^ { 2 \pi i l \cdot x } g ( x - k ) = k M _ l T _ k g ( x ) + M _ l T _ k X g ( x ) , \end{align*}


\begin{align*} \L ^ \perp = M ^ { - T } \Z ^ d . \end{align*}


\begin{align*} \langle a ( x ) x - a ( y ) y , x - y \rangle = & \langle ( a ( x ) - a ( y ) ) x , x - y \rangle + a ( y ) \langle x , x - y \rangle \\ = & ( a ( x ) - a ( y ) ) \langle x , x - y \rangle + a ( y ) | x - y | ^ { 2 } \end{align*}


\begin{align*} F _ { \widehat { K } ^ \mathbb { R } _ { \mathrm { u - c p t } } | K } ( \hat { k } ) = 1 - e ^ { - \frac { 1 + N \bar { \gamma } \hat { k } } { 1 + N \bar { \gamma } K } } , \ \ f _ { \widehat { K } ^ \mathbb { R } _ { \mathrm { u - c p t } } | K } ( \hat { k } ) = \frac { N \bar { \gamma } } { 1 + N \bar { \gamma } K } e ^ { - \frac { 1 + N \bar { \gamma } \hat { k } } { 1 + N \bar { \gamma } K } } , \ \ \hat { k } \ge - 1 / ( N \bar { \gamma } ) . \end{align*}


\begin{align*} \int _ { X _ E } r _ l ( \widetilde { V } ) \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} r _ l ( \widetilde { \mathcal { S } } ) ( g ) \ , d g = \int _ { X _ F } r _ l ( \widetilde { V } ) \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} r _ l ( \widetilde { \mathcal { S } } ) ( g ) \ , d g , \end{align*}


\begin{align*} S : = { \sqrt { \Delta ( { \mathbb K } ) } } \sum _ { d \vert m } \mu ( d ) \lambda ^ { ( n / d ) ^ s } . \end{align*}


\begin{align*} & \gamma ^ - f ( t , x , v ) = g ( t , x , v ) , ( t , x , v ) \in ( 0 , T ) \times \Sigma ^ - , \\ & \rho ( t , x ) = \rho _ B ( x ) , ( t , x ) \in ( 0 , T ) \times \Gamma _ { \rm { i n } } , \\ & u ( t , x ) = u _ B ( x ) , ( t , x ) \in ( 0 , T ) \times \partial \Omega , \end{align*}


\begin{align*} L ( A ) = \sum \limits _ { i = 1 } ^ r ( L _ { v _ 1 } , \ldots , L _ { v _ n } ) . u _ i , \end{align*}


\begin{align*} G : = G ( T ) : = \{ g \in \mathrm { S L } _ n ( \mathbb { B } ) : g ^ { \ast } T g = T \} . \end{align*}


\begin{align*} P _ { h + 1 } = \frac { x _ { h + 1 } z } { P _ { k + 1 - h } + \frac { x _ h z } { P _ h } } . \end{align*}


\begin{align*} \overline Y ^ 1 _ t & = e ^ { \lambda t } Y ^ 1 _ 0 , \\ \overline Y ^ 2 _ t & = e ^ { - \mu t } Y ^ 2 _ 0 = e ^ { - \mu t } L , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { t } i a _ i ^ { \ell } = \sum _ { a \in S } a ^ { \ell } \sum _ { \substack { b \in S \\ b \leqslant a } } 1 = \sum _ { a \in S } a ^ \ell \left ( \frac { a } { n } t + E _ a \right ) = \frac { t } { n } \sum _ { a \in S } a ^ { \ell + 1 } + \sum _ { a \in S } a ^ \ell E _ a \end{align*}


\begin{align*} \lim _ { x \to \infty } \tanh ( x ) = 1 \end{align*}


\begin{align*} x _ { i _ 1 } \cdots x _ { i _ n } \triangleleft x _ { j _ 1 } \cdots x _ { j _ m } = \sum \limits _ { k = 0 } ^ n x _ { i _ 1 } \cdots x _ { i _ k } x _ { j _ 1 } \cdots x _ { j _ m } x _ { i _ { k + 1 } } \cdots x _ { i _ n } . \end{align*}


\begin{align*} 0 = & f _ c ( y ) \equiv \frac { 1 } { 2 } y ^ - \star y ^ + + \frac { a } { 2 } y ^ 0 \star y ^ 0 - c , \\ [ 6 p t ] 0 = & \mathrm { d } f _ c = \frac { 1 } { 2 } ( y ^ - \star \eta ^ + + \eta ^ - \star y ^ + ) + a y ^ 0 \star \eta ^ 0 , \\ [ 6 p t ] 0 = & y ^ - \star E ^ + - y ^ + \star E ^ - - \sqrt { a } \ , y ^ 0 \star H + \mathrm { i } \nu y ^ + \star H - 2 \mathrm { i } \nu ( 1 + \mathrm { i } \nu ) y ^ + \star E ^ + . \end{align*}


\begin{align*} & [ T u , T v , z ] = - [ T v , z , T u ] - [ z , T u , T v ] = [ z , T v , T u ] - [ z , T u , T v ] . \end{align*}


\begin{align*} [ u , v ] _ T = \rho ( T u ) v - \rho ( T v ) u \ ; \forall u , v \in V . \end{align*}


\begin{align*} 0 \leq \frac { d } { d r } \Big | _ { r = 1 } r ^ { - 1 } v ^ \gamma ( r x ) = - v ^ \gamma ( x ) + x \cdot D v ^ \gamma ( x ) . \end{align*}


\begin{align*} C _ 2 & \ll \sum _ { n = N + 1 } ^ \infty \frac { 1 } { n ^ { \sigma _ 2 } } \int _ n ^ \infty \frac { 1 } { u ^ { \sigma _ 1 + 1 } ( u + n ) ^ { \sigma _ 3 } } d u \\ & \ll \sum _ { n = N + 1 } ^ \infty \frac { 1 } { n ^ { \sigma _ 2 } } \int _ n ^ \infty \frac { 1 } { u ^ { \sigma _ 1 + \sigma _ 3 + 1 } } d u . \end{align*}


\begin{align*} \hat { \theta } _ { n } = \arg \min _ { \theta \in \Theta } \Phi _ { n } ( \theta ) . \end{align*}


\begin{align*} \mathrm { d i v } u = 0 \ ; \ ; \ ; \ ; ( x , t ) \in \mathbb { R } ^ 3 \times [ 0 , T ] . \end{align*}


\begin{align*} \nabla _ { k l } ^ F ( V ) \equiv \nabla _ k ^ { F } \nabla _ l ^ { F } ( V ) : = \nabla ^ F _ k ( \nabla ^ F _ l ( V ) ) - ( \Gamma _ F ) ^ m _ { k l } \ , \nabla ^ F _ m ( V ) . \end{align*}


\begin{align*} Z f ( x , \omega ) & = \sum _ { k \in \Z ^ d } g ( k ) = \sum _ { k \in \Z ^ d } \widehat { g } ( k ) = \sum _ { k \in \Z ^ d } T _ { - \omega } M _ x \widehat { f } ( k ) = \sum _ { k \in \Z ^ d } \widehat { f } ( k + \omega ) e ^ { 2 \pi i x \cdot ( k + \omega ) } \\ & = e ^ { 2 \pi i x \cdot \omega } Z \widehat { f } ( \omega , - x ) . \end{align*}


\begin{align*} \check H _ { c t } ^ 1 ( T ; A ) = 0 . \end{align*}


\begin{align*} P ^ { \gamma } _ t ( f ) ( x ) = \int _ { \mathbb { R } ^ d } f ( x e ^ { - t } + \sqrt { 1 - e ^ { - 2 t } } y ) \gamma ( d y ) , \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d t } \| \Lambda ^ { \frac { \alpha } 2 } B \| ^ 2 _ { L ^ 2 } + \mu \| \Lambda ^ { \alpha } B \| ^ 2 _ { L ^ 2 } = - \int _ { \mathbb S ^ 1 } B J _ x \Lambda ^ { \alpha } B \ , d x + \int _ { \mathbb S ^ 1 } B _ x J \Lambda ^ { \alpha } B \ , d x . \end{align*}


\begin{align*} E _ 1 = t ^ { - 1 } \left ( \textnormal { R e s } _ { k = \eta } T _ { \eta , 1 } + \textnormal { R e s } _ { k = - \eta } T _ { - \eta , 1 } \right ) + \mathcal { O } ( t ^ { - 2 } ) , \end{align*}


\begin{align*} \inf \{ S ( \rho _ t , \Phi _ t ) : ( - \Delta _ { \rho _ t } ) ^ \dagger \Phi _ t \in \mathcal T _ { \rho _ t } \mathcal P _ o ( G ) , \rho ( 0 ) = \rho ^ a , \rho ( 1 ) = \rho ^ b \} . \end{align*}


\begin{align*} u ( x , t ) = e ^ { i t } \left ( Q ( x ) + h ( t , x ) \right ) . \end{align*}


\begin{align*} \kappa _ 1 ( y ) & = \psi \left ( \frac { 1 + \lvert y \rvert } { 2 } \right ) + \ln { \frac { 2 } { \lvert y \rvert } } . \end{align*}


\begin{align*} U _ i f ( x ) & = \underset { y \lessdot x } { \mathbb { E } } [ f ( y ) ] , \\ D _ i f ( y ) & = \underset { x \gtrdot y } { \mathbb { E } } [ f ( x ) ] . \end{align*}


\begin{align*} \Omega _ { k } = \bigcup _ { I \in \mathcal { K } _ k } \overset { \circ } { I } . \end{align*}


\begin{align*} \langle \varphi , \rho ( x , \omega ) \varphi \rangle = A \varphi = \Phi ( x , \omega ) . \end{align*}


\begin{align*} \partial _ s h + ( v - V _ \ast ) \cdot \nabla _ x h - \nabla ( \phi * _ x \rho [ h ] ) \cdot \nabla _ v \mu & = - e _ 0 \nabla \Phi ( x ) \cdot \nabla _ v \mu , h ( 0 , \cdot ) = 0 . \end{align*}


\begin{align*} V = V _ 0 + V _ 1 + \ldots + V _ s + V _ 0 ' \end{align*}


\begin{align*} \frac { c _ k ( \mu ) L ( \mu , \mathbf { f } , \chi ) } { \pi ^ { \beta } } \frac { \mathbf { f } ( g _ { \mathbf { h } } \cdot g _ { \infty } ) } { \mathfrak { P } _ k ( w ) } = \langle \mathbf { h } _ w , \mathbf { f } \rangle . \end{align*}


\begin{align*} \big ( g \curvearrowleft f \big ) ( x ) : = ( g - 1 ) f _ { x g } ( x ) = ( g ( x ) - 1 ) f ( x g ( x ) ) = \sum _ { \substack { w \in \N ^ * \\ u \in \N ^ * } } f _ w g _ u x _ u ( x g ( x ) ) _ w \in G ^ 0 . \end{align*}


\begin{align*} \mu : \prod _ { i = 1 } ^ { m } G _ { b _ i } \to G . \end{align*}


\begin{align*} f ( z ) & : = \mathrm { e } _ { \pi _ { \lambda , \beta } } ( z ; 0 ) = \frac { 1 } { l _ { \pi _ { \lambda , \beta } } ( z ) } = \frac { 1 } { E _ { \beta } ( \lambda ( \mathrm { e } ^ { z } - 1 ) ) } , \\ g ( z ) & : = \mathrm { \mathrm { e } } ^ { z x } . \end{align*}


\begin{align*} \rho _ 1 ( t ) - \rho _ 1 ( 0 ) = \int _ 0 ^ t m _ { 2 1 } ( s ) d s + \int _ 0 ^ t \frac 1 2 ( \Sigma _ 1 - \Sigma _ 2 ) d W ^ { \delta } ( s ) . \end{align*}


\begin{align*} \Im \left \{ \log \zeta ( \sigma + i T ) \right \} = \mathcal { O } ( \log T ) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( - 1 \leq \sigma \leq 2 ) \end{align*}


\begin{align*} \int _ 0 ^ s \delta _ + \dd ( \delta - h ) = 0 \forall s \in ( 0 , T ] . \end{align*}


\begin{align*} \left | \frac { \beta _ k } { \alpha _ i - \beta _ k } \right | \geqslant \frac { | \beta _ k | } { | \beta _ k | + M } \geqslant \frac { | \beta _ k | } { \frac { 3 } { 2 } | \beta _ k | } = \frac { 2 } { 3 } , \end{align*}


\begin{align*} | \Omega | = \int _ \Omega u ^ { \frac { p } { p + 1 } } u ^ { - \frac { p } { p + 1 } } \le \Big ( \int _ \Omega u \Big ) ^ { \frac { p } { p + 1 } } \Big ( \int _ \Omega u ^ { - p } \Big ) ^ { \frac { 1 } { p + 1 } } \forall \ , t > s . \end{align*}


\begin{align*} n ^ g _ { m , n } = \sum _ { \pi ( \beta ) = m \ell + n \gamma } n ^ g _ \beta , \end{align*}


\begin{gather*} F ( \psi _ { X , Y } ) = \psi _ { F ( X ) , F ( Y ) } \end{gather*}


\begin{align*} \mathcal { U } ^ 0 ( h , k ) = \mathcal { U } ^ 1 ( h , k ) + \mathcal { U } ^ 2 ( h , k ) , \end{align*}


\begin{align*} f = \iint _ { \R ^ { 2 d } } F ( x , \omega ) M _ \omega T _ x g \ , d ( x , \omega ) , \end{align*}


\begin{align*} \tau _ E ( p ^ m ) = \tau _ { E \smallsetminus \{ \gamma \} } ( p ^ m ) + p ^ { - \gamma } \tau _ E ( p ^ { m - 1 } ) . \end{align*}


\begin{align*} \begin{aligned} \Phi ^ { P } _ { g } ( f _ 0 , f _ 1 , & \dots , f _ m ) \\ : = & \int _ { h \in M / Z _ M ( x ) } \int _ { K N } \int _ { G ^ { \times m } } C \big ( H ( g _ 1 . . . g _ m k ) , H ( g _ 2 . . . g _ m k ) , \cdots , H ( g _ m k ) \big ) \\ & f _ 0 \big ( k h g h ^ { - 1 } n k ^ { - 1 } ( g _ 1 \dots g _ m ) ^ { - 1 } \big ) f _ 1 ( g _ 1 ) \dots f _ m ( g _ m ) d g _ 1 \cdots d g _ m d k d n d h , \end{aligned} \end{align*}


\begin{align*} f ( z ) - 1 \sim c z ^ { - s } , L ( z ) = \frac { f ' ( z ) } { f ( z ) } \sim f ' ( z ) \sim - c s z ^ { - 1 - s } , F ( z ) \sim \frac { z ^ { 1 + s } } { c s } , \end{align*}


\begin{align*} \min \{ v _ i ( \mathbf { c } ) , k \} = \min \{ v _ 1 ( \mathbf { c } ) + ( i - 1 ) b , k \} . \end{align*}


\begin{align*} p = \Lambda ( e ^ r - 1 ) \end{align*}


\begin{align*} n ( s ) = \int _ \mathbb { R } \frac { u ^ 2 } { s ^ 2 + u ^ 2 } d \mu ( u ) \qquad d ( s ) = \int _ \mathbb { R } \frac { 1 } { s ^ 2 + u ^ 2 } d \mu ( u ) . \end{align*}


\begin{align*} N ^ j _ t = \int _ 0 ^ t R ^ j ( s ) u ^ j _ l ( s ) d W ^ l _ s , j = 1 , 2 , \dots , d , \end{align*}


\begin{align*} \int _ { \Omega } \left ( f ( x ) + \sum _ { i = 1 } ^ { s } z _ i \rho ^ { i n } _ i \right ) d x + \int _ { \partial \Omega } \epsilon ( x ) \phi ^ b d s = 0 , \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \dot { \xi } ( t ) & = & A \xi ( t ) + \alpha ( \xi ( t ) ) \\ \xi ( 0 ) & = & x \end{array} \right . \end{align*}


\begin{align*} \mathcal W _ 0 ( x , v ) = 1 + 2 U ( x ) + \theta _ 0 | x | ^ 2 + | v | ^ 2 + \theta ^ * \ < x , v \ > , x , v \in \R ^ d , \end{align*}


\begin{align*} | \partial _ y ^ k \tilde \phi _ { i , b , \beta } ( y ) | \le C \left ( y ^ { - \gamma + 2 - k } I _ { y \in [ 0 , y _ 0 ] } + y ^ { - \gamma + 2 i + 2 - k } I _ { y \in [ y _ 0 , + \infty ) } \right ) b ^ { 1 - \frac { \epsilon } { 2 } } , y \in \R k = 0 , 1 . \end{align*}


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


\begin{align*} { ( z - w ) } ^ N [ a ( z ) , b ( w ) ] = 0 . \end{align*}


\begin{align*} \begin{aligned} d Z _ t & = \widetilde \mu ( Z _ t ) \ , d t + \widetilde \sigma ( Z _ t ) \ , d W _ t , t \in [ 0 , \infty ) , \\ Z _ 0 & = G ( x _ 0 ) , \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \int _ { t _ 1 } ^ { T _ { x , \xi } } p \left ( f \left ( \xi ( s ) \right ) - u \left ( \xi ( s ) \right ) \right ) d s & \leq \int _ { t _ 1 } ^ { T _ { x , \xi } } p ( 1 - c _ 0 ) f \left ( \xi ( s ) \right ) d s \\ & \leq \left ( T _ { x , \xi } - t _ 1 \right ) p ( 1 - c _ 0 ) f ( \xi ( t _ 1 ) ) \end{aligned} \end{align*}


\begin{align*} \lim _ { x \to 0 ^ + } \left ( \sinh ( x ) - x ^ 2 \coth ( x ) \right ) = 0 . \end{align*}


\begin{align*} \nu _ 2 ( t ) = \frac { 1 } { 3 . 3 5 9 \log | t | } \left ( 1 - \frac { 8 . 0 2 \log \log | t | } { \log | t | } \right ) . \end{align*}


\begin{align*} C : = C _ { 0 } ^ { \prime } \left ( T _ { 0 } ^ { \prime } + C _ { 0 } \left ( \sqrt { \frac { 2 T _ { 0 } ^ { \prime } } { \gamma } } \right ) \right ) < 1 , \end{align*}


\begin{align*} ( \alpha ^ * \beta ^ * \delta ) ^ * = ( \delta ( \alpha \delta ) ^ * ( \beta \delta ) ^ * ) ^ * = ( \delta ^ * ( \alpha \delta ) ^ * ( \beta \delta ) ^ * ) ^ * = \delta ^ * ( \alpha \delta ) ^ * ( \beta \delta ) ^ * . \end{align*}


\begin{align*} \{ z ' = z _ i ' \} \subseteq \{ \log | z ' | \leq \log | z _ i ' | \} \subseteq \{ \psi _ V ( z ) \leq \log | z _ i ' | + C \} , \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty \frac { \mu _ n } { \lambda _ n } \leq A < \infty \end{align*}


\begin{align*} T f ( x ) = \lim \limits _ { k \to - \infty } \int _ { | y | > q ^ { k } } f ( x - y ) \dfrac { \Omega ( y ) } { | y | } d y , ~ ~ ~ ~ ~ k \in \mathbb { Z } , \end{align*}


\begin{align*} \alpha ^ p = 1 + \alpha _ m ^ p t ^ { p m } + \cdots \end{align*}


\begin{align*} u _ { p , \tau } = ( u ^ 1 , \dots , u ^ m ) : B _ { 2 0 0 } ( p , g _ \tau ) \to \mathbb R ^ m . \end{align*}


\begin{align*} \frac 1 2 \frac { d } { d t } \| \Lambda ^ { \frac { \alpha } { 2 } } B \| _ { L ^ 2 } ^ 2 + \mu \| \Lambda ^ { \alpha } B \| _ { L ^ 2 } ^ 2 = \int _ { \mathbb S ^ 1 } B _ x J \Lambda ^ \alpha B \ , d x . \end{align*}


\begin{align*} B _ 2 , B _ 3 , B _ 4 \ll \left ( 1 + ( \abs { s _ 3 } / \sigma _ 3 ) \right ) \begin{cases} N ^ { - \sigma _ 1 - \sigma _ 3 } & ( \sigma _ 2 > 1 ) \\ N ^ { - \sigma _ 1 - \sigma _ 3 } \log N & ( \sigma _ 2 = 1 ) \\ N ^ { 1 - \sigma _ 1 - \sigma _ 2 - \sigma _ 3 } & ( \sigma _ 2 < 1 ) , \\ \end{cases} \end{align*}


\begin{align*} \Delta t = \frac { \operatorname { C r } \Delta x _ { \min } } { u _ { \max } } , \end{align*}


\begin{align*} g ( x ) & + \sum _ { \substack { u , u _ 1 , \dotsc , u _ k \in \{ \ 1 \} \cup \N ^ * \\ v = i _ 1 \dotsm i _ k \in \N ^ * } } f _ v g _ u g _ { u _ 1 } \dotsm g _ { u _ k } x _ { u } x _ { i _ 1 } x _ { u _ 1 } \dotsm x _ { i _ k } x _ { u _ k } \\ & = 1 + \sum _ { w \in \N ^ * } ( \phi * \psi ) ( w ) x _ w = \Lambda _ { \mathrm { g r } } ( \phi * \psi ) ( x ) . \end{align*}


\begin{align*} | \hat q | : = \left \{ \begin{aligned} | q | + { ( q q _ d ^ * + q _ d q ^ * ) \over 2 | q | } \epsilon , & \ { \rm i f } \ q \not = 0 , \\ | q _ d | \epsilon , & \ { \rm o t h e r w i s e } , \end{aligned} \right . \end{align*}


\begin{align*} W _ { n + 2 } & = \sigma ( \sigma ^ { n + 1 } ( a ) ) '' \\ & = \sigma ( W _ { n + 1 } ) a \\ & = \sigma ( W _ n ) , \sigma ( \varepsilon _ n ) , \sigma ( W _ n ) a \\ & = W _ { n + 1 } , \varepsilon _ { n + 1 } , W _ { n + 1 } . \end{align*}


\begin{align*} m _ { i , \lambda } ( \Z ( P , B ) ) = 0 , \end{align*}


\begin{align*} E [ B ] : = \{ x \in X \mid ( x , y ) \in E , y \in B \} \end{align*}


\begin{align*} f ( k ) : = - \left ( \frac { 3 } { 2 } g ( k ) \right ) ^ { \frac { 2 } { 3 } } , k \in \mathbb { C } \backslash ( - \infty , \eta ] . \end{align*}


\begin{align*} E _ i = \left \lbrace ( [ x _ 0 : \cdots : x _ n ] , v ) : \ \left | \frac { x _ 0 } { x _ i } \right | _ { v } \leq 1 , \cdots , \left | \frac { x _ n } { x _ i } \right | _ { v } \leq 1 \right \rbrace , \end{align*}


\begin{align*} \Theta ^ { Q , j } : = \Theta _ { a , k , \zeta , \mu , \sigma } , \\ W ^ { Q , j } : = W _ { a , k , \zeta , \mu , \sigma } . \end{align*}


\begin{align*} A ' ( \gamma ) = - 2 a n \ , \frac { e ^ { 2 a \gamma } ( 2 a \gamma - 1 ) + 1 } { \left ( e ^ { 2 a \gamma } - 1 \right ) ^ 2 } < 0 . \end{align*}


\begin{align*} \mathfrak { k } : = \{ Z \in \Xi ~ | ~ \mathcal { L } _ Z \mathbf { g } ( X , Y ) = \mathbf { g } ( [ Z , X ] , Y ) + \mathbf { g } ( X , [ Z , Y ] ) X , Y \in \Xi \} . \end{align*}


\begin{align*} ( h \varphi ) ( x _ 0 , \ldots , x _ { q - 1 } ) = \sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \varphi ( \alpha ( x _ 0 ) , \ldots , \alpha ( x _ i ) , \beta ( x _ i ) , \ldots , \beta ( x _ { q - 1 } ) ) . \end{align*}


\begin{align*} \prod _ { i ' = 1 } ^ d a _ { i ' 0 } + \sum _ { i = 1 } ^ d \sum _ { t : \omega _ t = i } \biggl ( \prod _ { i ' \neq i } a _ { i ' p ^ t _ { i ' } } \bigl ( s _ { i p ^ t _ i } - s _ { i p ^ { t - 1 } _ { i } } \bigr ) \biggr ) . \end{align*}


\begin{align*} C : \mathcal { H } \to \ell ^ 2 ( \Gamma ) , C f = ( \langle f , e _ \gamma \rangle ) _ { \gamma \in \Gamma } . \end{align*}


\begin{align*} \mu \otimes k ( A \times B ) = \int _ A k ( x , B ) \ , \mu ( d x ) A \in \mathcal B ( X _ { 1 : t } ) , B \in \mathcal B ( X _ { t + 1 } ) . \end{align*}


\begin{align*} \begin{aligned} \begin{bmatrix} i \delta _ 1 & \lambda \\ \overline { \lambda } & \delta _ 2 \end{bmatrix} ^ { - 1 } = \begin{bmatrix} - i \delta _ 2 ( | \lambda | ^ 2 + \delta _ 1 \delta _ 2 ) ^ { - 1 } & \lambda ( | \lambda | ^ 2 + \delta _ 1 \delta _ 2 ) ^ { - 1 } \\ \overline { \lambda } ( | \lambda | ^ 2 + \delta _ 1 \delta _ 2 ) ^ { - 1 } & - i \delta _ 1 ( | \lambda | ^ 2 + \delta _ 1 \delta _ 2 ) ^ { - 1 } \end{bmatrix} . \end{aligned} \end{align*}


\begin{align*} f ( x ) = f _ 0 + \sum _ { k = 1 } ^ \infty \sum _ { ( i _ 1 , \dotsc , i _ k ) \in \N ^ k } f _ { i _ 1 \dotsm i _ k } x _ { i _ 1 } \dotsm x _ { i _ k } \end{align*}


\begin{align*} 2 i t \int \mathbf { 1 } _ { \{ k \geq 0 \} } \phi ( k ) \mathcal { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \ , d k = \int \mathbf { 1 } _ { \{ k \geq 0 \} } \phi ( k ) k ^ { - 1 } \mathcal { Q } ( x , k ) h ( k ) \ , \partial _ k e ^ { i k ^ { 2 } t } \ , d k \\ = \phi ' ( 0 ) h ( 0 ) \mathcal { Q } ( x , 0 ) - \int \mathbf { 1 } _ { \{ k \geq 0 \} } e ^ { i k ^ { 2 } t } \partial _ { k } \big ( \phi ( k ) k ^ { - 1 } \mathcal { Q } ( x , k ) h ( k ) \big ) \ , d k . \end{align*}


\begin{align*} \frac { T } { 4 } \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \log \left ( \frac { \sigma ^ 2 + T ^ 2 } { 4 } \right ) \ d \sigma = \frac { T } { 4 } \left [ \sigma \log \left ( \frac { \sigma ^ 2 + T ^ 2 } { 4 } \right ) - 2 \sigma + 2 T \arctan \left ( \frac { \sigma } { T } \right ) \right ] \Biggr | _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \end{align*}


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


\begin{align*} - \lim _ { \varepsilon \to 0 } \int _ { | x | > \varepsilon } \log | x | \overline { f ' ( x ) } \ , d x = - \int _ \R \log | x | \overline { f ' ( x ) } \ , d x , \end{align*}


\begin{align*} G = \{ x \rightarrow x y , \ ; \ ; y \rightarrow x y \} . \end{align*}


\begin{align*} \| \left \langle \nabla \right \rangle ( R ( h ^ A ) - R ( h ) ) \| _ { S ' ( ( t , + \infty ) , L ^ 2 ) } & \lesssim \sum _ { n = 0 } ^ \infty \| \left \langle \nabla \right \rangle ( R ( h ^ A ) - R ( h ) ) \| _ { S ' ( ( t + n , t + n + 1 ) , L ^ 2 ) } \\ & \lesssim \sum _ { n = 0 } ^ \infty e ^ { - \lambda e _ 0 ^ - ( t + n ) } \lesssim e ^ { - \lambda e _ 0 ^ - t } , \forall t \ge t _ 0 , \end{align*}


\begin{align*} \Sigma _ { \sigma _ 2 } ^ { 1 } \le 2 \int _ { H _ { \sigma _ 2 } } ^ T \frac { x ^ { - \frac { 1 } { R \log t } } } { t } d N ( \sigma _ 2 , t ) = 2 \Big ( \frac { x ^ { - \frac { 1 } { R \log T } } } { T } N ( \sigma _ 2 , T ) - \int _ { H _ { \sigma _ 2 } } ^ T N ( \sigma _ 2 , t ) { d } \Big ( \frac { x ^ { - \frac { 1 } { R \log t } } } { t } \Big ) \Big ) . \end{align*}


\begin{align*} \begin{aligned} & u _ { i n } ( x ) = f _ n ( x ) , u _ { i j } ( x ) \leq f _ { i } ( x ) , & & i = 1 , \ldots , d , \ j = 0 , \ldots , n , \\ & u ( x ) \leq a ( x ) , f ^ L _ i = u _ { i 0 } ( x ) = a _ { i 0 } ( x ) \leq \cdots \leq a _ { i n } ( x ) = f ^ U _ { i } & & i = 1 , \ldots , d , \\ & \deg ( u _ { i j } ) \leq \tau , \deg ( a _ { i j } ) \leq \tau & & i = 1 , \ldots , d , \ j = 0 , \ldots , n . \end{aligned} \end{align*}


\begin{align*} | \mathrm { D i s c } ( \alpha ) | _ v = \prod _ { i = 1 } ^ n \prod _ { j \ne i } | \lambda _ i - \lambda _ j | _ v . \end{align*}


\begin{align*} \beta _ j ( t ) = \left \langle h ( t ) , Q _ j \right \rangle - \alpha _ + ( t ) \left \langle \mathcal { Y } _ + , Q _ j \right \rangle - \alpha _ - ( t ) \left \langle \mathcal { Y } _ - , Q _ j \right \rangle . \end{align*}


\begin{align*} \widehat { \Phi } ( k , n ) : = \Phi \left ( \left \lceil 2 C ( k + 1 ) \right \rceil - 1 , \max \{ \theta ( M \varpi ( k ) + M - 1 ) , n \} \right ) . \end{align*}


\begin{align*} \langle u , y \rangle : = E [ u \cdot y ] . \end{align*}


\begin{align*} D _ i ( c ) = \begin{cases} \dfrac { \alpha ^ { i + 1 } - \beta ^ { i + 1 } } { \alpha - \beta } & c \neq \pm 2 , \\ [ 3 m m ] i + 1 & c = 2 , \\ [ 2 m m ] ( - 1 ) ^ i ( i + 1 ) & c = - 2 , \end{cases} \end{align*}


\begin{align*} v ( r , t ) = \frac { 1 } { 2 \varkappa ( T - t ) } w \Big ( \frac { r } { \sqrt { 2 \varkappa ( T - t ) } } \Big ) \end{align*}


\begin{align*} \int _ G c _ X ( g ^ { - 1 } x ) d g = 1 , \quad \mbox { f o r a l l } ~ x \in X \end{align*}


\begin{align*} L ( A ) ( U \oplus U ^ 2 ) = 0 , \ ; V ^ 2 \subseteq L ( A ) , \ ; v ( v u ) \in L ( A ) \mbox { f o r a l l } u \in U \mbox { a n d } v \in V \end{align*}


\begin{align*} x M _ n ( x ) = W _ n ( x ) , \end{align*}


\begin{align*} m ^ { L C } ( x , t , k ) : = m ^ { G P } ( x , t , k ) \cdot m ^ { m o d } ( k , \xi ) . \end{align*}


\begin{align*} & L ( x _ { \rm t h } ) = \sqrt { \frac { \pi } { 2 } } \sigma f _ D \\ [ - 0 . 2 e m ] & \times \sum _ { i = 1 } ^ N \underbrace { \int _ 0 ^ { x _ { \rm t h } } \ ! \ ! \cdots \int _ 0 ^ { x _ { \rm t h } } } _ { ( N - 1 ) - { \rm f o l d } } p _ { | h _ { 1 } | , \cdots , | h _ { N } | } ( x _ 1 , \dots , x _ i = x _ { \rm t h } , \cdots , x _ N ) \\ [ - 0 . 2 e m ] & \times \underbrace { d x _ 1 \cdots d x _ k \cdots d x _ N } _ { \substack { ( N - 1 ) - { \rm f o l d } \\ k \neq i } } . \end{align*}


\begin{align*} \Phi _ H = \{ \pm t _ i \pm t _ j \mid 1 \leq i \neq j \leq n \} \subset X ^ * ( T ) . \end{align*}


\begin{align*} \check { A } = \{ \varphi ( - a ) \mid a \in A \} = \{ \psi ( a - ) \mid a \in A \} = \{ \psi ( - a ) \mid a \in A \} . \end{align*}


\begin{align*} \widetilde { U } ^ p ( t ) = U ^ p ( t ) - U ^ p ( t - 1 ) = \sum _ { k \geq N } a _ k \left ( 1 - e ^ { \tilde { \lambda } _ k } \right ) e ^ { \tilde { \lambda } _ k ( T - t ) } , \\ \widetilde { U } ^ h ( t ) = U ^ h ( t ) - U ^ h ( t - 1 ) = \sum _ { \mod { k } \geq N } b _ k \left ( 1 - e ^ { \tilde { \gamma } _ k } \right ) e ^ { \tilde { \gamma } _ k ( T - t ) } , \end{align*}


\begin{align*} \varepsilon = \varepsilon _ l ( \phi _ l - \phi _ 0 ) + \sum _ { j = 1 , j \ne l } ^ M \varepsilon _ j \phi _ j + \varepsilon _ - , \end{align*}


\begin{align*} T f _ 1 ( x ) & = \int _ { \R ^ N } K ( x , y ) u ( y ) [ f _ 1 ( y ) - f _ 1 ( x ) ] d \omega ( y ) \\ & \qquad + \int _ { \R ^ N } K ( x , y ) v ( y ) f _ 1 ( y ) d \omega ( y ) + \int _ { \R ^ N } K ( x , y ) u ( y ) f _ 1 ( x ) d \omega ( y ) , \end{align*}


\begin{align*} [ - ( - \Delta ) ^ { \gamma / 2 } G ] ( u ) = & c _ { \gamma } \int _ { 0 } ^ { \infty } \frac { G ( u + w ) + G ( u - w ) - 2 G ( u ) } { w ^ { \gamma + 1 } } d w . \end{align*}


\begin{align*} \operatorname { C o v } ( f ( X _ { \alpha , \beta } ) , g ( X _ { \alpha , \beta } ) ) = \int _ { [ - 1 , 1 ] } ( 1 - x ^ 2 ) g ' ( x ) \tilde { f } ' ( x ) \mu _ { \alpha , \beta } ( d x ) , \tilde { f } = \int _ { 0 } ^ { + \infty } Q _ t ^ { \alpha , \beta } ( f ) d t , \end{align*}


\begin{align*} \nabla \Psi ( t , x ) = \nabla \Psi _ 0 ( x ) , \ \Delta \Psi ( t , x ) = s ( t , x ) = 0 t \geq 0 , \ | x | > D + \frac { \sqrt { p ' ( \varrho ) } } { \varepsilon } t . \end{align*}


\begin{align*} P _ \gamma ^ { - 1 } ( [ x , y ] ) = \left ( H \setminus U _ a ( x ) \right ) \cap \left ( H \setminus U _ b ( y ) \right ) . \end{align*}


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


\begin{align*} \sum _ { \substack { 1 \leq w \leq h \\ 1 \leq z \leq s } } ( - 1 ) ^ { w + z } \sum _ { \substack { y _ 1 + . . . + y _ w = h \\ x _ 1 + . . . + x _ z = s } } \sum _ { \substack { 0 \leq p _ w \leq . . . \leq p _ 1 \leq i - h d \\ 0 \leq v _ z \leq . . . \leq v _ 1 \leq r - i - s d } } \prod _ { a = 1 } ^ w i _ a \alpha ^ { i _ a } _ { r - ( h - y _ 1 - . . . - y _ a ) d - p _ a } \prod _ { b = 1 } ^ z x _ b \alpha ^ { x _ b } _ { r - ( s - x _ 1 - . . . - x _ b ) d - v _ b } . \end{align*}


\begin{align*} \overline { \lambda } = x - c _ { 2 } \theta ( x ) \big ( x ^ 2 ( \epsilon _ n + \delta _ n ) + x \delta _ n \sqrt { | \ln \delta _ n | } \big ) \in [ 0 , \ , o ( \min \{ \epsilon _ n ^ { - 1 } , \delta _ n ^ { - 1 } \} ) ) , \end{align*}


\begin{align*} \begin{aligned} | | y | ^ { \ell _ \mu } - | x | ^ { \ell _ \mu } | & \leq \ell _ \mu \cdot | x - y | \cdot ( | x | ^ { \ell _ \mu - 1 } + | y | ^ { \ell _ \mu - 1 } ) . \end{aligned} \end{align*}


\begin{align*} I : = \{ \tau \geqslant 0 u ( x ) \geqslant \tau \varphi ( x ) x \in U \} . \end{align*}


\begin{align*} y _ n = g _ n y _ { n - 1 } + h _ n , \end{align*}


\begin{align*} c ^ { - \alpha } T _ c ( \nu _ \alpha ) ( d u ) = \nu _ \alpha ( d u ) , \end{align*}


\begin{align*} s '' ( t + k ' \bar { \tau } ) = \bar { \nu } ^ { \kappa + k ' - 1 } \left ( \bigwedge _ { i = \bar { t } _ \textup { p } } ^ { \bar { t } _ { \textup { p } } + \bar { \tau } - 1 } \bar { s } ( i ) \right ) \wedge \bar { \nu } ^ { \kappa + k ' } \left ( \bigwedge _ { i = \bar { t } _ \textup { p } } ^ { \bar { t } _ \textup { p } + h } s ( i ) \right ) = \bar { \nu } ^ { k ' } s '' ( t ) , \end{align*}


\begin{align*} \sum _ { 0 \leq m < n < \infty } \frac { \tau _ A ( p ^ { m } ) \tau _ B ( p ^ { n } ) \left ( 1 - \frac { p ^ w } { p ^ { 2 } } \right ) } { p ^ { m ( 2 - \alpha - w ) } p ^ { n ( 1 - \beta ) } } = \frac { \tau _ B ( p ) } { p ^ { 1 - \beta } } - \frac { \tau _ B ( p ) } { p ^ { 3 - \beta - w } } + O \left ( \frac { 1 } { p ^ { 1 + \varepsilon } } \right ) . \end{align*}


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


\begin{align*} L _ + \mathcal { Y } _ 1 = e _ 0 \mathcal { Y } _ 2 L _ - \mathcal { Y } _ 2 = - e _ 0 \mathcal { Y } _ 1 . \end{align*}


\begin{align*} \mathcal { Q } ^ \bullet _ { M _ c \star } = \bigoplus _ { p = 0 } ^ { n - 1 } \biguplus _ { q = 0 } ^ \infty \biguplus _ { r = 0 } ^ \infty \mathcal { Q } ^ { p q r } _ { M _ c \star } . \end{align*}


\begin{align*} I ( \hat { \alpha } \cup { ( \gamma , N ) } , \hat { \alpha } \cup { ( \delta , 1 ) \cup { ( \gamma , N - p _ { i } ) } } ) = 2 . \end{align*}


\begin{align*} Q _ { b ( \tau ) } ( y ) = \frac { 1 } { b ( \tau ) } Q \left ( \frac { | y | } { \sqrt { b ( \tau ) } } \right ) . \end{align*}


\begin{align*} \langle f , U _ { k - 1 } g \rangle _ { X ( k ) } = \langle D _ k f , g \rangle _ { X ( k - 1 ) } . \end{align*}


\begin{align*} \frac { d } { d t } E ( \rho , \phi ) ( t ) = - \int _ { \Omega } \sum _ { i = 1 } ^ s D _ i ( x ) \rho _ i | \nabla ( \log \rho _ i + z _ i \phi ) | ^ 2 d x \leq 0 , t > 0 , \end{align*}


\begin{align*} \min \{ v _ i ( \mathbf { c } ) , k \} = \min \{ v _ n ( \mathbf { c } ) + ( n - i ) a , k \} \end{align*}


\begin{align*} & ( \phi ^ \epsilon , u ^ \epsilon , \eta ^ \epsilon - 1 , \tau ^ \epsilon ) \rightarrow ( \phi , u , \eta - 1 , \tau ) \ ; \ ; \ ; \ ; L ^ { \infty } ( \mathbb { R } ^ + ; H ^ 3 ) , \\ & ( \nabla u ^ \epsilon , \nabla \eta ^ \epsilon , \tau ^ \epsilon , \nabla \tau ^ \epsilon ) \rightarrow ( \nabla u , \nabla \eta , \tau , \nabla \tau ) \ ; \ ; \ ; \ ; L ^ 2 ( \mathbb { R } ^ + ; H ^ 3 ) , \end{align*}


\begin{align*} \chi _ { \ell } = \begin{cases} 1 & A x _ { \ell + 1 } - b \neq A x _ { \ell } - b , \\ 0 & A x _ { \ell + 1 } - b = A x _ { \ell } - b . \end{cases} \end{align*}


\begin{align*} \frac { d } { d t } \int _ { \Omega } u & = \int _ { \Omega } \Delta u - \chi \int _ { \Omega } \nabla \cdot \Big ( \frac { u } { v } \nabla v \Big ) + \int _ { \Omega } a ( x , t ) u - b ( x , t ) u ^ 2 \\ & = \int _ { \Omega } a ( x , t ) u ( x , t ) d x - \int _ { \Omega } b ( x , t ) u ^ 2 ( x , t ) d x \\ & \leq a _ { \sup } \int _ { \Omega } u - \frac { b _ { \inf } } { | \Omega | } \Big ( \int _ { \Omega } u \Big ) ^ 2 . \end{align*}


\begin{align*} B _ t + ( B J ) _ x + \mu \Lambda ^ \alpha B = & \ 0 , \\ B _ x = & \ \mathcal H J . \end{align*}


\begin{align*} \begin{aligned} & \mathbb { P } \left ( R _ j = a \right ) = \left ( 1 - \epsilon _ m \right ) \epsilon ^ a _ m , a \in \mathbb { N } , \\ & \mathbb { P } \left ( V _ j = i \right ) = \frac { \epsilon _ { i - 1 } - \epsilon _ i } { 1 - \epsilon _ m } , i \in [ m ] . \end{aligned} \end{align*}


\begin{align*} p \in C ^ 1 ( [ 0 , \bar \rho ) ) , \ p ( 0 ) = 0 , \ p ' > 0 ( 0 , \bar \rho ) , \ \lim _ { s \to \bar \rho _ - } p ( s ) = + \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*} \lim _ { n \rightarrow \infty } X ^ { n } ( \cdot ) = X ( \cdot ) \ ; \ ; C _ W ( [ 0 , T ] ; H ^ \frac d 3 ) . \end{align*}


\begin{align*} d \Omega = - \frac 1 2 [ \Omega , \Omega ] . \end{align*}


\begin{align*} { A ( r ) = \left \{ x \in A \ : \ \forall \tilde { r } \leq r , \ m ( B ( x , \tilde { r } ) \cap A ) \geq c . m ( B ( x , \tilde { r } ) ) \right \} } \end{align*}


\begin{align*} \partial _ t p _ t ( \varphi ) = \nabla \cdot \left ( p _ t ( \varphi ) \nabla u _ t ( \varphi ) \right ) \forall \varphi \in \R ^ { \Lambda _ { \epsilon , L } } , t \ge 0 , \end{align*}


\begin{align*} \begin{cases} - \Delta g _ k + \tilde { \lambda } _ 0 \underline { m } g _ k = f _ 0 ( g _ k ) + \frac { 1 } { M _ k } F _ k \tilde { \Omega } _ { \varepsilon _ k } \\ g _ k = 0 \partial \tilde { \Omega } _ { \varepsilon _ k } \end{cases} \end{align*}


\begin{align*} \varrho ( x , y ) = 2 ^ { 1 - \nu } \sigma ^ 2 [ \Gamma ( \nu ) ] ^ { - 1 } ( \kappa \norm { x - y } { \R ^ d } ) ^ \nu K _ \nu ( \kappa \norm { x - y } { \R ^ d } ) , x , y \in \mathcal D , \end{align*}


\begin{align*} \sum _ { \gamma \in \Gamma _ { 6 } ( P G ) } c ( g ( \gamma ) ) = 5 \equiv 1 \pmod 2 \end{align*}


\begin{align*} g h _ { s } ^ { j } + \lambda h _ { t } ^ { j } = g \eta ^ t _ s \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} + \lambda \eta ^ t _ t \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} - \eta ^ { t } \Lambda \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} + g ^ { 2 } F ^ { j } + g \lambda G ^ { j } + \lambda ^ { 2 } H ^ { j } . \end{align*}


\begin{align*} \forall t \in [ 0 , T ] , \forall x \in \mathbb { Z } , \Phi _ n ( t , \tfrac { x } { n } ) : = n ^ { \gamma } \sum _ { y : | x - y | \geq \varepsilon n } F ( t , \tfrac { x } { n } , \tfrac { y } { n } ) p ( y - x ) . \end{align*}


\begin{align*} g ( p ) = r \sim \log | \log p | , \qquad \textrm { a s } p \to 0 . \end{align*}


\begin{align*} \left \| x - \sum _ { n = 1 } ^ { \lceil \lambda m \rceil } e ^ * _ { \rho _ x ( n ) } ( x ) e _ { \rho _ x ( n ) } \right \| & \ \leqslant \ \mathbf C _ \ell \left \| x - \sum _ { n = 1 } ^ { m } e ^ * _ { \rho _ x ( n ) } ( x ) e _ { \rho _ x ( n ) } \right \| \\ & \ \leqslant \ \mathbf C _ \ell \mathbf C _ { 1 , r p } \sigma _ m ^ { \mathcal { U } _ { X } , R } ( x ) . \end{align*}


\begin{align*} \sum _ { \ell = 1 } ^ k d _ \ell \leq k ( k - 1 ) + ( n - k ) , \end{align*}


\begin{align*} \frac { \partial u } { \partial x _ 1 } ( 0 ) & = \int _ { \R ^ n } y _ 1 \psi _ s ( y ) u ( y ) \dd y . \end{align*}


\begin{align*} r ( t ) = \Re \left [ \left ( \sum _ { m = 1 } ^ M h _ m ( t ) s _ m [ t ] \textbf { a } _ m ^ T ( \theta ) \textbf { w } _ m \right ) e ^ { j 2 \pi f _ 0 t } \right ] + n ( t ) , \end{align*}


\begin{align*} E = I + k ^ { - 1 } E _ 1 + \mathcal { O } ( k ^ { - 2 } ) . \end{align*}


\begin{align*} \mathcal { I } _ 1 ^ * ( h , k ) = \sum _ { \substack { \alpha \in A \\ \beta \in B } } \frac { 1 } { 2 ( 2 \pi i ) ^ 2 } \int _ { ( \epsilon ) } \int _ { ( \epsilon ) } \mathcal { J } \ , d s _ 2 \ , d s _ 1 + O \big ( X ^ { \varepsilon } Q ^ { \frac { 3 } { 2 } + \varepsilon } ( h k ) ^ { \varepsilon } \big ) , \end{align*}


\begin{align*} \mathcal { S } _ k & \coloneqq \left \{ \mathbf { P } _ k \in \mathcal { W } _ { k } ( 0 ) : \mathbf { p } _ { k - 1 , 1 } = \mathbf { p } _ { k - 1 , 2 } , ~ \mathbf { p } _ { k - 1 , 3 } = \mathbf { p } _ { k - 1 , 4 } \right \} , \\ & = \big \{ \mathbf { P } _ k \in \mathcal { W } _ { k } ( 0 ) : ( \mathbf p _ { k - 1 , 1 } \cup \mathbf p _ { k - 1 , 2 } ) \cap ( \mathbf p _ { k - 1 , 3 } \cup \mathbf p _ { k - 1 , 4 } ) = \varnothing \big \} . \end{align*}


\begin{align*} \lim _ { t \to 0 } \varphi P _ t ^ \mathcal { R } u = u \end{align*}


\begin{align*} l ( x , y , \omega ) & = \sum _ { t = 0 } ^ T \left [ \Delta x _ t \cdot y _ t + H _ t ( x _ { t } , y _ t , \omega ) ) \right ] \\ & = \sum _ { t = 0 } ^ T \left [ - x _ { t } \cdot \Delta y _ { t + 1 } + H _ t ( x _ { t } , y _ t , \omega ) \right ] , \end{align*}


\begin{align*} K _ Z + g _ { * } ^ { - 1 } \varepsilon D _ 1 + \sum f ^ { - 1 } _ { * } a _ i E _ i + g _ { * } ^ { - 1 } \varepsilon D _ 2 + ( 2 \varepsilon - 1 ) F = g ^ { * } ( K _ X + \varepsilon D ) . \end{align*}


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


\begin{align*} \alpha = P ^ { - 1 } \Big ( \frac n \beta \Big ) , \gamma = 2 \frac { n ^ 2 } \alpha . \end{align*}


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


\begin{align*} \left \langle \varepsilon _ - , \phi _ { i , b , \beta } \right \rangle _ { L ^ 2 _ { \rho _ \beta } } = 0 , \forall j = 0 , . . . , \ell . \end{align*}


\begin{align*} [ A , B ] = A B - B A . \end{align*}


\begin{align*} e ^ { - Q } \delta _ { s t } \leq h _ { s t } = h ( \frac { \partial } { \partial y ^ { s } } , \frac { \partial } { \partial y ^ { t } } ) \leq e ^ { Q } \delta _ { s t } \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 ^ + } s ( r , t ) t = r ^ 2 - \lambda _ 2 ( \mu ) ^ 2 . \end{align*}


\begin{align*} M _ t = 1 + \sum _ { n = 1 } ^ \infty m _ n ( t ) x ^ n . \end{align*}


\begin{align*} x ' ( t ) = g ( x ( t - d ( L x _ t ) ) ) \end{align*}


\begin{gather*} d _ { k [ S ] } : k [ S ] ^ * \otimes k [ S ] \to k , d _ { k [ S ] } ( f \otimes x ) = f ( x ) , \\ b _ { k [ S ] } : k \to k [ S ] \otimes k [ S ] ^ * , b _ V ( 1 ) = \sum _ { g \in S } g \otimes g ^ * , \\ d _ { k [ S ] ^ * } : k [ S ] \otimes k [ S ] ^ * \to k , d _ { k [ S ] ^ * } ( x \otimes f ) = f ( x ) , \\ b _ { k [ S ] ^ * } : k \to k [ S ] ^ * \otimes k [ S ] , b _ { k [ S ] ^ * } ( 1 ) = \sum _ { g \in S } g ^ * \otimes g , \end{gather*}


\begin{align*} E \left [ \left ( \dfrac { S _ n } { \sqrt { n / ( 1 - 2 \alpha ) } } \right ) ^ 2 \right ] - 1 \sim - \dfrac { 1 } { \Gamma ( 2 \alpha ) } \cdot n ^ { - ( 1 - 2 \alpha ) } \qquad ( n \to \infty ) . \end{align*}


\begin{align*} d _ q : C ^ q ( U , A ) & \to \mathcal C ^ { q + 1 } ( U , A ) \\ \varphi & \mapsto ( ( x _ 0 , \ldots , x _ { q + 1 } ) \mapsto \sum _ { i = 0 } ^ { q + 1 } ( - 1 ) ^ i \varphi ( x _ 0 , \ldots , \hat x _ i , \ldots , x _ { q + 1 } ) . \end{align*}


\begin{align*} & \int ^ { T } _ 0 \int _ { \Omega \times \mathbb { R } ^ 3 } f \big ( \partial _ t \varphi + v \cdot \nabla _ x \varphi + ( u - v ) \cdot \nabla _ v \varphi + L [ f ] \cdot \nabla _ v \varphi + \Delta _ v \varphi \big ) \ , d x d v d t \\ & + \int _ { \Omega \times \mathbb { R } ^ 3 } f _ 0 \varphi ( 0 , x , v ) \ ; d x d v = \int ^ { T } _ { 0 } \int _ { \Sigma ^ - } \big ( v \cdot \nu ( x ) \big ) g \varphi \ , d \sigma ( x ) \ , d v d t , \end{align*}


\begin{align*} \norm { f - f _ n } _ { M ^ { p , q } } = \norm { V _ g ^ * ( V _ g f - F _ n ) } _ { M ^ { p , q } } \leq C \norm { V _ g f - F _ n } _ { L ^ { p , q } } . \end{align*}


\begin{align*} \norm { ( X - a ) f } _ 2 ^ 2 = \int _ \R ( x - a ) ^ 2 | f ( x ) | ^ 2 \ , d x \end{align*}


\begin{align*} A g _ 0 ( x , 0 ) = e ^ { - \frac { \pi } { 2 } x ^ 2 } . \end{align*}


\begin{align*} \mathbf { E } ( \mathfrak { g } , s ) = \sum _ { t = 0 } ^ m \sum _ { \gamma \in P _ N \backslash P _ N \tilde { \tau } _ t ( G \times G ) } \phi ( \gamma \mathfrak { g } , s ) = : \sum _ { t = 0 } ^ m \mathbf { E } _ t ( \mathfrak { g } , s ) . \end{align*}


\begin{align*} d ( \phi \wedge \psi ) = ( d _ { \alpha } \phi ) \wedge \psi + ( - 1 ) ^ { k - 1 } \phi \wedge d _ { - \alpha } \psi . \end{align*}


\begin{align*} \mu ( \Omega ) \geq \mu \left ( \bigcup _ { L \in \mathcal { F } } L \right ) \geq \mu ( \Omega ) \sum _ { k \geq 1 } C ( 1 - C ) ^ { k - 1 } = \mu ( \Omega ) , \end{align*}


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


\begin{align*} \mathcal { A } _ \delta ( f _ \delta ) = g , \end{align*}


\begin{align*} \phi ( u - \delta f _ J ) \ = \ \phi ( u ) + \delta ( n - | J | ) \sigma _ J - \delta | J | ( m - \sigma _ J ) \ = \ \phi ( u ) + \delta \left ( n \sigma _ J - m | J | \right ) \end{align*}


\begin{align*} p ( x ) = \phi \bigl ( g _ { j ( 1 ) } ( x ) , \ldots , g _ { j ( d ) } ( x ) \bigr ) j ( 1 ) , \ldots , j ( d ) \in \{ 1 , \ldots , m \} , \deg ( p ) = \delta . \end{align*}


\begin{align*} \kappa ( f ) = \sum _ { ( d , e ) \in D _ { 2 } ( H G ) } | f ( d ) \cap f ( e ) | \end{align*}


\begin{align*} R f ( x , \omega ) = f ( x ) \overline { \widehat { f } ( \omega ) } e ^ { - 2 \pi i x \cdot \omega } . \end{align*}


\begin{align*} u ( t , r ) = \frac { 1 } { \lambda ( t ) } v ( \xi , s ) \end{align*}


\begin{align*} \lim \limits _ { r \to 0 } \frac { \log \prod \limits _ { i = 1 } ^ { q } \frac { p _ { w _ i } } { C _ { w _ i } } } { \log r } = \frac { \sum \limits _ { k \in T } p _ k \log p _ k } { \sum \limits _ { k \in T } p _ k \log C _ k } - 1 . \end{align*}


\begin{align*} \lim _ { | \tau _ n | \to 0 } \norm { e ^ { 2 \pi i \tau _ n } f - f } _ 2 = 0 , \ , \lim _ { | x _ n | \to 0 } \norm { T _ x f - f } _ 2 = 0 \ , \ , \lim _ { | \omega _ n | \to 0 } \norm { M _ \omega f - f } _ 2 = 0 . \end{align*}


\begin{align*} \coth ( x ) < 1 + \frac { 2 } { 2 1 - 1 } = 1 . 1 , \forall x > \log ( \sqrt { 2 1 } ) , \end{align*}


\begin{align*} \frac { \epsilon \big ( X , { \rm i n d } _ { \mathcal { W } _ { K ' } } ^ { \mathcal { W } _ { K } } ( \rho ) , \psi _ K \big ) } { \epsilon ( X , \rho , \psi _ { K ' } ) } = \frac { \epsilon \big ( X , { \rm i n d } _ { \mathcal { W } _ { K ' } } ^ { \mathcal { W } _ { K } } ( 1 _ { K ' } ) , \psi _ K \big ) } { \epsilon ( X , 1 _ { K ' } , \psi _ { K ' } ) } , \end{align*}


\begin{align*} \delta _ m & : = \frac { T } { m } , m \in \mathbb { N } , \\ [ t ] _ m ^ - & : = k \delta _ m , k \delta _ m \leq t < ( k + 1 ) \delta _ m , k = 0 , \ldots , m - 1 , \\ [ t ] _ m ^ + & : = ( k + 1 ) \delta _ m , k \delta _ m \leq t < ( k + 1 ) \delta _ m , k = 0 , \ldots , m - 1 , \end{align*}


\begin{align*} \mathcal { E } _ p ( u ) ^ p : = \inf _ g \int _ X g ^ p \ , d \mu , \end{align*}


\begin{align*} \Phi ^ { + } _ G & = \left \{ \widehat { x } _ i - \widehat { x } _ j \ , , \ ; 1 \leq i < j \leq n \ , , \ ; \widehat { x } _ i + \widehat { x } _ j \ , , \ ; 1 \leq i \leq j \leq n \right \} \\ \intertext { a n d } \Phi ^ { + } _ K & = \left \{ \widehat { x } _ i - \widehat { x } _ j ) \ , , \ ; 1 \leq i < j \leq n \right \} \ , , \end{align*}


\begin{align*} b \mathfrak { h } _ \omega ( a _ 0 , a _ 1 , \ldots , a _ { p + 1 } , a _ { p + 2 } ) = p \mathrm { S T r } _ \omega \left ( [ F , a _ { p + 2 } ] a _ 0 [ F , a _ 1 ] \cdots [ F , a _ { p + 1 } ] \right ) = 0 . \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 ( x ) = \sum _ { j } e _ j ( x ) h _ j . \end{align*}


\begin{align*} \int _ 0 ^ t ( t - \tau ) ^ { \frac { m - \frac 5 2 } { \alpha } } \tau ^ { \frac { \frac 5 2 - \alpha - m } { \alpha } } \ , d \tau = \int _ 0 ^ 1 ( 1 - \tau ' ) ^ { \frac { m - \frac 5 2 } { \alpha } } ( \tau ' ) ^ { \frac { \frac 5 2 - \alpha - m } { \alpha } } \ , d \tau ' \leq C \end{align*}


\begin{align*} \frac { 1 } { \displaystyle { \frac { 2 k ( 2 k - 1 ) } { 2 } } } \ , \sum \limits _ { i < j } \ , | \lambda _ i | \ , | \lambda _ j | \geq \left ( \prod ^ { 2 k } _ { i = 1 } \ , | \lambda _ i | \right ) ^ { \frac { 1 } { k } } , \end{align*}


\begin{align*} & \| \mathcal { P } ( \mathbf { x } _ k ) - \mathbf { x } _ k \| \leq \| \mathcal { P } ( \mathbf { x } _ k ) - \mathcal { P } ( \mathbf { x } _ { k - 1 } ) \| + \| \mathcal { P } ( \mathbf { x } _ { k - 1 } ) - \mathbf { x } _ { k } \| \\ & \leq \eta \| \mathbf { x } _ k - \mathbf { x } _ { k - 1 } \| + \| \mathcal { P } ( \mathbf { x } _ { k - 1 } ) - \mathbf { x } _ { k } \| , \end{align*}


\begin{align*} \varphi _ C ( A , b , \lbrace x _ j : j \leq k \rbrace , \lbrace W _ j : j < k \rbrace , \lbrace \zeta _ j : j < k \rbrace ) = ( e _ { \mathrm { r e m } ( \zeta _ { k - 1 } , d ) + 1 } , \zeta _ { k - 1 } + 1 ) . \end{align*}


\begin{align*} S _ { 2 , \lambda } ( n + 1 , k ) = S _ { 2 , \lambda } ( n , k - 1 ) + ( k - n \lambda ) S _ { 2 , \lambda } ( n , k ) , \end{align*}


\begin{align*} 0 \leq | \xi | \leq \epsilon t + l _ \epsilon t ^ { q - 1 } , \ \forall t \geq 0 , \ \xi \in \partial F ^ + ( t ) = \partial F ( t ) . \end{align*}


\begin{align*} \mathcal { D } \mathcal { X } _ t = \left ( \mathcal { D } ^ { j } _ { r } \mathcal { X } _ t \right ) _ { j \in \{ 1 , 2 , \dots , d \} , \ , r \in [ 0 , T ] } . \end{align*}


\begin{align*} \Bigl | \Bigl ( \prod ( 1 + \alpha _ k ) \Bigr ) \circ \beta - \prod _ { k = 1 } ^ n ( 1 + \alpha _ k ( \beta ) ) \Bigr | & = \Bigl | \Bigl ( \prod _ { k = 1 } ^ \infty ( 1 + \alpha _ k ) - \prod _ { k = 1 } ^ n ( 1 + \alpha _ k ) \Bigr ) \circ \beta \Bigr | \\ & \le \Bigl | \prod _ { k = 1 } ^ \infty ( 1 + \alpha _ k ) - \prod _ { k = 1 } ^ n ( 1 + \alpha _ k ) \Bigr | \xrightarrow { n \to \infty } 0 . \end{align*}


\begin{align*} P _ { n } ( \lambda ) = & ( \lambda ^ 2 - 4 ) ^ { n - 1 } \left [ \lambda ^ 3 - 6 \lambda - 4 \right ] - 2 ( n - 1 ) ( \lambda + 2 ) ( \lambda ^ 2 - 4 ) ^ { n - 1 } \end{align*}


\begin{align*} \mu ( x , t ) : = \dfrac { 1 } { 2 } \norm { r ( x , t ) } ^ 2 : = \dfrac { 1 } { 2 } \norm { \begin{pmatrix} h ( x ) \\ f ( x ) - t \end{pmatrix} } ^ 2 , \end{align*}


\begin{align*} | P _ 1 \rho | \leq ( | p _ 1 | + | p _ 2 | ) . \end{align*}


\begin{align*} \Phi _ { \lambda ^ h _ k } : = ( \xi _ { \lambda ^ h _ k } , \eta _ { \lambda ^ h _ k } ) \ \ \lambda ^ h _ k , \ \ \forall | k | \geq k _ 0 . \end{align*}


\begin{align*} \mathcal { F } ( f ) ( \xi ) = \int _ { \mathbb { R } ^ n } f ( x ) \exp ^ { - 2 \pi i \xi \cdot x } d x , \end{align*}


\begin{align*} P = 1 1 - \delta \ , , \end{align*}


\begin{align*} ( \rho ^ { * } _ { ( 1 ) } \otimes \rho ^ { * } _ { ( 1 ) } ) E _ { \Sigma _ 0 } & = ( \rho ^ { * } _ { ( 1 ) } \otimes \rho ^ { * } _ { ( 1 ) } ) ( \partial _ { \Sigma _ 0 } \otimes \partial _ { \Sigma _ 0 } ) E _ { \mathcal { M } _ 0 } \\ & = ( \partial _ { \Sigma } \otimes \partial _ { \Sigma } ) ( \chi ^ { * } \otimes \chi ^ { * } ) E _ { \mathcal { M } _ 0 } \\ & = : E _ { \Sigma } . \end{align*}


\begin{align*} \frac { ( q ^ 2 ; q ^ 2 ) _ \infty ^ k } { ( q ; q ) _ \infty ^ { 3 k + 1 } } = \frac { ( - q ; q ) _ \infty ^ k } { ( q ; q ) _ \infty ^ k } \frac { 1 } { ( q ; q ) _ \infty ^ { k + 1 } } , \end{align*}


\begin{align*} \Delta _ K \beta _ i ^ K ( t \wedge \theta _ K ) & = g _ i ^ K ( t ) - \frac { \| p \| _ { } } { \underline { p } } A _ { i + 1 } ^ K ( t \wedge \theta _ K ) + \Delta _ K M _ i ^ K ( t \wedge \theta _ K ) \\ & \leq \max _ { 0 \leq j \leq 1 / \delta _ K } g ^ K _ j ( 0 ) + 2 C ( K , L ) t + \frac { \| p \| _ { } } { \underline { p } } \left ( \max _ { 0 \leq j \leq 1 / \delta _ K - 1 } \beta ^ K _ i ( 0 ) + C _ 1 t \right ) + \varepsilon _ K . \end{align*}


\begin{align*} \int _ { G ( \Q ) \backslash G ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \mathbf { E } _ t ( g \times h , s ) \mathbf { f } ( h ) \mathbf { d } h = 0 . \end{align*}


\begin{align*} \lambda _ { \phi } ( t ) & = \frac { \int _ { \mathbb { R } ^ d } \left ( \frac 1 2 | \nabla \phi | ^ 2 + V | \phi | ^ 2 + \beta | \phi | ^ 4 - \Omega \overline { \phi } L _ z \phi \right ) \mathrm { d } \mathbf { x } - \| \dot { \phi } ( \cdot , t ) \| ^ 2 } { \| \phi ( \cdot , t ) \| ^ 2 } . \end{align*}


\begin{align*} K _ I = \{ \sigma \in K \ , : \ , \sigma \subseteq I \} . \end{align*}


\begin{align*} \lambda _ h T _ h ( \lambda _ h ) u _ h = u _ h \end{align*}


\begin{align*} \lim _ { k \to \infty } \int _ { \partial \Omega } \phi _ k \ , f \ , d \nu = \int _ { \partial \Omega } v \ , f \ , d \nu . \end{align*}


\begin{align*} H ^ \infty _ L ( G ) = \left \{ f \in L ^ 2 ( G ) | \int _ G \big ( 1 + L ( g ) \big ) ^ { 2 k } | f ( g ) | ^ 2 d g < \infty , \ \forall k \in \mathbb { N } \right \} . \end{align*}


\begin{align*} \int _ { \mathfrak { Z } } \delta ( w , z ) ^ { - k } | \delta ( w , z ) | ^ { - 2 s } \delta ( w ) ^ { s + k } f ( w ) d w = \widetilde { c } _ k ( s ) f ( z ) \delta ( z ) ^ { - s } , \end{align*}


\begin{align*} P _ n ^ { ( \alpha , \beta ) , * } ( x ) = ( 1 - x ) ^ \alpha ( 1 + x ) ^ \beta P _ n ^ { ( \alpha , \beta ) } ( x ) , \qquad \hbox { f o r } ~ x \in ( - 1 , 1 ) , ~ n \geq 0 , \end{align*}


\begin{align*} Q _ v ^ + & = \langle F \in \mathcal { F } \mid v \cdot u _ F > 0 \rangle , \\ Q _ v ^ - & = \langle F \in \mathcal { F } \mid v \cdot u _ F < 0 \rangle . \end{align*}


\begin{align*} G _ \mu ( z ) = \int _ \R \frac { 1 } { z - x } d \mu ( x ) , z \in \mathbb { C } ^ + ; \end{align*}


\begin{align*} 0 = | \langle f , \rho ( \mathbf { h } ) g \rangle | = | A ( f , g ) ( x , \omega ) | = | V _ g f ( x , \omega ) | \forall ( x , \omega ) \in \R ^ { 2 d } . \end{align*}


\begin{align*} \sum _ i \omega ( b _ i d ) a _ i c = 0 , \forall c \in A _ u ^ s , \end{align*}


\begin{align*} \begin{cases} \ i = ( x + 1 ) ( g _ 0 ( x , n ) - f _ 0 ( x , n ) ) \\ [ 8 p t ] \ j = h _ 0 ( x , n ) \end{cases} . \end{align*}


\begin{align*} u ( t , y + t ) = \Theta ( \vec { \mathfrak { q } } ( t ) , 0 , y ) + \epsilon ( t , y ) , \end{align*}


\begin{align*} \omega ( B ( x _ 0 , 2 ^ { j _ { k + 1 } } 2 ^ { i _ 0 } \sigma ) ) = \omega ( B ( x _ 0 , 2 ^ { j _ k + 1 } 2 ^ { i _ 0 } \sigma ) ) = \omega \Big ( \bigcup \limits _ { \ell = j _ { k } + 1 } ^ { j _ { k + 1 } } \chi _ \ell \Big ) \geqslant 2 \omega ( B ( x _ 0 , 2 ^ { j _ k } 2 ^ { i _ 0 } \sigma ) ) \end{align*}


\begin{align*} \{ f _ { W _ { V , i } } : X _ { W _ { V , i } } \to W _ V \} _ { 1 \leq i \leq k } . \end{align*}


\begin{align*} \lambda _ h = \lambda _ 0 - i \frac { { \lambda _ 0 } ^ \frac { 5 } { 2 } } { 4 \pi } \left [ { U _ 0 \sum _ { k = 1 } ^ { k = n } \eta _ 0 ^ k U _ k } \right ] h + \mathcal { O } ( h ^ 2 ) \end{align*}


\begin{align*} s _ { i j } = m _ { i j t _ i ( H ) } = a _ { i j } j \leq \tau ( i , t _ i ( H ) - 1 ) s _ { i j } = m _ { i j t _ i ( H ) } \leq a _ { i j } \tau ( i , t _ i ( H ) - 1 ) \leq j < j _ i ' s _ { i j } = s _ { i n } \end{align*}


\begin{align*} \frac { \sqrt { \Delta ( { \mathbb K } ) } } { n } \sum _ { d \vert n } \mu ( n / d ) \lambda ^ { d ^ s } = \frac { { \sqrt { \Delta ( { \mathbb K } ) } } } { n } \sum _ { d \vert n } \mu ( d ) \lambda ^ { ( n / d ) ^ s } \end{align*}


\begin{align*} { \bar { \Delta } ^ { \rm P } _ { \rm R a t e l e s s } } = - 1 - { \tau _ { \rm f } } + { 2 } \left ( { { \tau _ { \rm c } } + { n _ 1 } + \mathcal { T } + \sum \limits _ { i = 1 } ^ { \infty } { \left ( { { n _ { i + 1 } } - { n _ i } } \right ) { \epsilon _ i } } } \right ) . \end{align*}


\begin{align*} \frac { 1 } { ( k + 1 ) ^ 2 } \geq \langle T ^ \circ x - z , - z \rangle + \langle z - T ^ \circ x , - T ^ \circ x \rangle = \norm { T ^ \circ x - z } ^ 2 \end{align*}


\begin{align*} u _ t = u _ 0 + t \ , v + t ^ 2 \ , r _ t , \forall \ , t \in [ 0 , 1 ] \end{align*}


\begin{align*} \int ( w ( \tau _ 0 ) - Q _ b ) \phi _ { 1 , b , \beta } \frac { y ^ 2 } { 2 } \rho _ \beta d y = \varepsilon _ 0 ( \tau _ 0 ) \int \phi _ { 0 , b , \beta } \phi _ { 1 , b , \beta } \frac { y ^ 2 } { 2 } \rho _ \beta d y + \varepsilon _ 1 ( \tau _ 0 ) \int \phi _ { 1 , b , \beta } ^ 2 \frac { y ^ 2 } { 2 } \rho _ \beta d y + o ( b ^ { \frac { \alpha } { 2 } } ) . \end{align*}


\begin{align*} L _ { \Sigma , g } \hat { u } _ \infty = \frac { n } { 2 } \nu _ { \Sigma , g } ( \hat { f } _ \infty ) . \end{align*}


\begin{align*} \int _ { G _ { \infty } / K _ { \infty } } \phi _ { \infty } ( \tilde { \tau } _ m ( g _ { \infty } \times h _ { \infty } ) , s ) \mathbf { f } ( h _ { \mathbf { h } } \cdot h _ { \infty } ) \mathbf { d } h _ { \infty } = c _ k ( s ) \mathbf { f } ( h _ { \mathbf { h } } \cdot g _ { \infty } ) , \end{align*}