\begin{align*} T _ 2 - T _ 1 \ge 1 \quad \mbox { a n d } \Bigl ( r ( t ) = 0 \ , \mbox { a n d } \ s ( t ) = 0 \ , \mbox { f o r } \ T _ 1 \le t \le T _ 2 \Bigr ) , \end{align*}


\begin{align*} w = \partial _ \nu u + \eta + \pi _ { \Gamma } ( v ) - g \qquad & \Sigma . \end{align*}


\begin{align*} a \circ b = a + ( 2 ^ { m - 1 } - 1 ) ^ a b = \begin{cases} a + b , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; e v e n , } \\ a + ( 2 ^ { m - 1 } - 1 ) b , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; o d d . } \end{cases} \end{align*}


\begin{align*} \begin{aligned} \Phi _ \sqcup ^ \alpha : a \sqcup ( b \sqcup c ) & \rightarrow ( a \sqcup b ) \sqcup c \\ \Phi _ \sqcup ^ \alpha ( ( 0 , x ) ) & = ( 0 , ( 0 , x ) ) \ ; , \\ \Phi _ \sqcup ^ \alpha ( 1 , ( 0 , y ) ) & = ( 0 , ( 1 , y ) ) \ ; , \\ \Phi _ \sqcup ^ \alpha ( 1 , ( 1 , z ) ) & = ( 1 , z ) \ ; , \end{aligned} \end{align*}


\begin{align*} \alpha ( t , x ) & = \frac { \mathrm { e } ^ { 2 \lambda \| \eta ^ 0 \| _ \infty } - \mathrm { e } ^ { \lambda \eta ^ 0 ( x ) } } { \gamma ( t ) } , \\ \xi ( t , x ) & = \frac { \mathrm { e } ^ { \lambda \eta ^ 0 ( x ) } } { \gamma ( t ) } , \gamma ( t ) = ( t - t _ 0 ) ( t _ 1 - t ) \end{align*}


\begin{gather*} Y _ i Y _ j = Y _ i Y _ { i + 1 } + \cdots + Y _ { j - 1 } Y _ j = s _ i + s _ { i + 1 } \cdots + s _ { j - 1 } , \end{gather*}


\begin{align*} \phi ( X ) : = \left \{ A ' \cup \{ v \} : v \in Y \right \} . \end{align*}


\begin{align*} | \Omega _ { t _ 2 } \setminus \Omega _ { t _ 1 } | \leq \sum ^ N _ { k = 1 } | B ( z _ k , t _ 2 ) \setminus { B ( z _ k , t _ 1 ) } | = N | \mathbb { B } ^ n | ( t _ 2 ^ { 2 n } - t _ 1 ^ { 2 n } ) , \end{align*}


\begin{align*} \sum _ { i = 0 } ^ d \alpha _ i \ , \Pi ^ { - 1 } S _ i , \end{align*}


\begin{align*} 0 \leq x \sqrt { b } - y \sqrt { y } = ( x - y ) \sqrt { b } + \sqrt { y } ( \sqrt { y b } - y ) \leq 2 \sqrt { b } ( x - y ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s n _ i L _ { P _ i } \simeq 0 . \end{align*}


\begin{align*} 2 \pi \int _ 0 ^ \infty e ^ { - 2 \pi t \rho } J _ k ( 2 \pi r \rho ) \ , d \rho = \int _ 0 ^ \infty e ^ { - t \rho } J _ k ( r \rho ) \ , d \rho = \frac { ( \sqrt { t ^ 2 + r ^ 2 } - t ) ^ k } { r ^ k \sqrt { t ^ 2 + r ^ 2 } } \end{align*}


\begin{gather*} W _ 1 < W _ 2 \Longleftrightarrow \begin{cases} y _ 1 < y _ 2 \\ y _ 1 = y _ 2 , \ , x _ 1 > x _ 2 , \end{cases} \end{gather*}


\begin{align*} I _ N = 1 + \sum _ { d = 1 } ^ \infty \frac { z ^ d } { d ! } \sum _ { \alpha , \beta \vdash d } p _ \alpha ( a _ 1 , \dots , a _ N ) p _ \beta ( b _ 1 , \dots , b _ N ) H _ N ( \alpha , \beta ) , \end{align*}


\begin{align*} f ( x _ 1 , y _ 1 , [ n \alpha ] ) = ( [ n \beta ] - y _ 1 ) - ( [ n \alpha ] - x _ 1 ) \end{align*}


\begin{align*} K l _ p ( \psi _ p , \psi _ p ' ; c , w _ { G _ n } ) : = \sum _ { x \in X ( w _ { G _ n } c ) } \psi _ p ( u ( x ) ) \cdot \psi _ p ' ( u ' ( x ) ) . \end{align*}


\begin{align*} V : = \sum _ { i = 0 } ^ r k \psi _ i , \sum _ { i = 0 } ^ r \sum _ { j = 1 } ^ { s _ i } k \chi ^ { m _ { i j } } _ Y = : W , \end{align*}


\begin{align*} \beta : = \partial _ 1 ^ 2 \psi ( 0 , 0 ) , \end{align*}


\begin{align*} \Delta u ( \phi ) & : = - \int _ { \mathbb { R } ^ 2 } \nabla u \cdot \nabla \phi d \mathcal { H } ^ 2 \phi \in C ^ { \infty } _ c ( \mathbb { R } ^ 2 ) \\ \Delta u ( U ) & : = \sup \{ \Delta u ( \phi ) : \phi \in C ^ { \infty } _ c ( U ) , | \phi | _ { \infty } \le 1 \} \\ & \ge \int _ { \partial _ { r e d } \{ u > 0 \} \cap U } | x _ 2 | ^ { \gamma } d \sigma ( x _ 1 , x _ 2 ) . \end{align*}


\begin{align*} \sum \limits _ { j = 1 } ^ N \langle F _ j , \bold { v } \rangle = 0 , \end{align*}


\begin{align*} \langle M _ { 1 , 2 } \psi , M _ { 2 , 3 } \psi \rangle _ { L ^ 2 ( \Omega _ T ) } & = - \frac { 1 } { 2 } \int _ { \Omega _ T } \partial _ t \left ( \left ( \frac { \tau } { 2 } - s \alpha \right ) ( \partial _ t \log \gamma ) \right ) \ , \psi ^ 2 \ , \d x \ , \d t \end{align*}


\begin{align*} P _ B = \left ( \begin{array} { c | c c c | c } & & & & \\ b & & P _ H & & 0 \\ & & & & \\ \hline c & & a & & 1 \end{array} \right ) \ , , \end{align*}


\begin{align*} \Pi _ { s ' , \ell } : = a ( 2 s ' ) Q _ \ell B ^ { A _ 0 H } ( \beta ) a ( - 2 s ' ) , \end{align*}


\begin{align*} i q _ t + q _ { x x } + 2 q ^ 2 r = 0 , - i r _ t + r _ { x x } + 2 r ^ 2 q = 0 , \end{align*}


\begin{align*} \sum _ { j = 1 } ^ n [ M _ A ( v ) ] _ j & = \sum _ { j \in I _ A } [ M _ A ( v ) ] _ j + \sum _ { j \in J _ A } [ M _ A ( v ) ] _ j \\ & = \sum _ { j \in I _ A } v _ j + \sum _ { j \in J _ A } \left ( - \sum _ { l \in a ( j ) \setminus \{ j \} } v _ l \right ) \\ & = \sum _ { j \in I _ A } v _ j - \sum _ { j \in J _ A } \sum _ { l \in a ( j ) \setminus \{ j \} } v _ l \\ & \overset { ( * ) } { = } \sum _ { j \in I _ A } v _ j - \sum _ { l \in I _ A } v _ l \\ & = 0 . \end{align*}


\begin{align*} \frac { p A \frac { \partial Q } { \partial x _ i } } { Q ^ { p + 1 } } \Omega = \frac { \frac { \partial A } { \partial x _ i } } { Q ^ { p } } \Omega \end{align*}


\begin{align*} \int _ { \R ^ d } e ^ { i \xi \cdot y } p _ t ( y ) \d y = e ^ { - t \psi ( \xi ) } , t > 0 , \ \xi \in \R ^ d , \end{align*}


\begin{align*} \left | \sum _ { l = 3 } ^ n \sigma _ l \omega _ { j _ l } + \omega _ { j _ 1 } - \omega _ { j _ 2 } \right | \ . \end{align*}


\begin{align*} \overline { E } \Big ( \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 6 , v _ j \rangle \Big ) = - \omega _ { 5 6 } ( \overline { E } ) \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 5 , v _ j \rangle - \frac { i \mu _ 2 } { \kappa _ 1 } \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 3 - i T _ { \theta } e _ 4 , v _ j \rangle , \end{align*}


\begin{align*} A ^ 2 \big ( ( 1 - | z | ^ 2 ) ^ n d A ( z ) \big ) : = \big \{ f \in \mathcal O ( \mathbb D ) : \| f \| ^ 2 = \int _ { \mathbb D } | f ( z ) | ^ 2 ( 1 - | z | ^ 2 ) ^ n d A ( z ) < \infty \big \} . \end{align*}


\begin{align*} \textbf { g } _ { \textbf { r s } } = \textbf { s o } ( 3 ) + I _ { 3 } \dot { c } \end{align*}


\begin{align*} S _ B = n \Lambda - \Delta _ B \varphi + \frac { m } { f } \Delta _ B f . \end{align*}


\begin{align*} \gamma _ { f } ( w , 8 r _ i ) \ge \gamma _ { \nu _ i } ( w , 8 r _ i ) - \gamma _ { f _ i } ( w , 8 r _ i ) - \sum _ { m = i + 1 } ^ \infty \gamma _ { \nu _ m } ( w , 8 r _ i ) . \end{align*}


\begin{align*} \begin{aligned} \tilde { E } ( z ) = \frac { \mu \left [ \frac { 1 } { \sqrt { 1 - z } } - 1 \right ] } { ( 1 - \mu + \mu \sqrt { 1 - z } ) ^ 2 } . \end{aligned} \end{align*}


\begin{align*} 0 ~ < ~ & y \sqrt { b } - x \sqrt { x } ~ = \sqrt { x } ( \sqrt { x } \sqrt { b } - x ) + \sqrt { b } ( y - x ) \\ [ . 5 e m ] \leq ~ & \sqrt { b } ( \sqrt { x } \sqrt { b } - x ) + \sqrt { b } ( y - x ) ~ \leq ~ 2 \sqrt { b } ( y - x ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} & u \in S ( \lambda , \Lambda , 0 ) & & ~ ~ \mbox { i n } ~ ~ Q _ 1 ^ + ; \\ & u = 0 & & ~ ~ \mbox { o n } ~ ~ S _ 1 . \end{aligned} \right . \end{align*}


\begin{align*} \mathcal { A } = \mathcal { A } _ { F T 2 } \mathcal { D } _ L , \end{align*}


\begin{align*} \omega + ( a ) & = s b + ( a ) \\ & = - s c + ( a ) \\ & \in I + ( a ) . \end{align*}


\begin{align*} f ( H ^ E _ b ) = \tilde { \eta } _ L f ( H _ b ) \eta _ L + \tilde { \eta } _ 0 f ( H _ b ^ E ) \eta _ 0 + \pi ^ { - 1 } \int _ { \mathcal { D } } \overline { \partial } f _ N ( z ) ( H ^ E _ b - z ) ^ { - 1 } W _ L ( z ) \d z _ 1 \d z _ 2 . \end{align*}


\begin{align*} & - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \xi w + \partial _ \tau w \leq \frac { \delta } { 2 } w i n [ 0 , T _ 1 ] \times [ 0 , X ] \times [ 0 , 1 ] , \\ & - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \tau w \leq b \delta ( 1 - \eta ) ^ { \alpha _ 0 } i n [ 0 , T _ 1 ] \times [ 0 , X ] \times [ 0 , 1 ] . \end{align*}


\begin{align*} \alpha - [ \alpha ] = \beta - [ \beta ] \text , \end{align*}


\begin{align*} w _ 1 ( f ) \deg _ f g = w _ 1 ( g ) , \end{align*}


\begin{align*} I _ { M N } = 1 + \sum _ { d = 1 } ^ \infty ( q z ) ^ d \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } s _ \lambda ( a _ 1 , \dots , a _ N ) s _ \lambda ( b _ 1 , \dots , b _ N ) \Omega _ { \frac { q } { N } } ^ { - 1 } ( \lambda ) \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) , \end{align*}


\begin{align*} \zeta _ i = \frac { 1 } { g } \{ \nabla _ \Gamma g _ i \cdot \nabla _ \Gamma \eta _ 0 - k _ d ^ { - 1 } ( \partial ^ \circ g _ i ) \eta _ 0 + k _ d ^ { - 2 } g _ i V _ \Gamma ^ 2 \eta _ 0 \} , i = 0 , 1 , \end{align*}


\begin{align*} A ( x ) \left | I \right \rangle = \alpha ( x ) \prod _ { k = 1 } ^ m \mathbf { f } ( x , x _ { k } ) | I \rangle + \sum _ { j \in I } \left [ \frac { { \mathcal { A } } ^ { I } _ j } { x - x _ { j } } - \frac { \tilde { \mathcal { D } } ^ { I } _ j } { 1 + x + x _ { j } } \right ] B ( x ) \left | I \setminus \left \{ j \right \} \right \rangle \ , , \end{align*}


\begin{align*} \pmb { p ^ * } = y ^ * _ \ell \cdot \pmb { p ^ 1 } + ( 1 - y ^ * _ \ell ) \cdot \pmb { p ^ 0 } . \end{align*}


\begin{align*} P \Big ( \frac { 1 } { b _ m } \sum _ { i = 1 } ^ { b _ m } \bar T _ i < \eta \Big ) \le e ^ { - ( 1 - \delta ) ^ 3 b _ m / ( 2 \eta ) } \ , , \end{align*}


\begin{align*} \eta _ { k , i , t } ( c ) : = \begin{cases} \Psi _ { k + 1 , i , t } \cdot ( s _ { k , i - 1 } c ) & , \\ 0 & . \end{cases} \end{align*}


\begin{align*} \mathcal { E } ^ { * } ( \mu ) = \int _ { \mathbb { R } ^ { d } \times \mathbb { R } ^ { d } } g ( x - y ) d \mu _ { x } d \mu _ { y } . \end{align*}


\begin{align*} \begin{array} { l } a _ { \epsilon } ^ 2 z ^ { - 1 } \le a _ { \epsilon } ^ 2 \underline { v } _ { \epsilon } ^ { - 1 } = | a _ { \epsilon , A ( y ) } \nabla \phi _ { \epsilon } | ^ { 2 } ( \lambda \epsilon ^ { - 1 / 2 } \psi ) ^ { - 1 } \le \epsilon ^ { 1 / 2 } \| \nabla a _ { \epsilon , A ( y ) } \| _ { \infty } | \nabla \phi _ { \epsilon } | ^ { 2 } \in L ^ { q } ( \Omega ) , \end{array} \end{align*}


\begin{align*} q _ { \delta , j , m } : = \int _ { - 1 / 2 } ^ { 1 / 2 } f _ { \delta , m } ( j + \theta _ 1 / 2 ) - f _ { \delta , m } ( j + \theta _ 1 / 2 + u ) \mathrm { d } u . \end{align*}


\begin{align*} \int _ 0 ^ T \int _ { M } \partial _ t \phi ( t , x ) + \langle \nabla _ M \phi ( t , x ) , \nabla _ M ( W \ast \mu _ t ) ( x ) \rangle _ x d \mu _ t ( x ) d t = 0 , \end{align*}


\begin{align*} X ( x + 1 , t ) - X ( x - 1 , t ) = \sum _ { z \in \Z } \sum _ { t - N ^ { \epsilon } \le s \le t } \Delta ( x - z , t - s ) \xi ( z , s ) \Gamma ( z , s ) + O ( N ^ { - 1 / 4 - \epsilon / 8 } ) \end{align*}


\begin{align*} v ^ \sharp \big ( X ( f ) \big ) = ( v ^ \sharp X ) ( f ) = ( X v ^ \sharp ) ( f ) = X \big ( v ^ \sharp ( f ) \big ) = 0 . \end{align*}


\begin{align*} \frac { \dim \mathsf { V } ^ \lambda } { \dim \mathsf { W } _ M ^ \lambda } = \frac { d ! } { M ^ d } \Omega _ { \frac { 1 } { M } } ^ { - 1 } ( \lambda ) \quad \frac { \dim \mathsf { V } ^ \lambda } { \dim \mathsf { W } _ N ^ \lambda } = \frac { d ! } { N ^ d } \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) , \end{align*}


\begin{align*} P ^ k ( \vec { y } | \vec { x } ) = \prod _ { i = 1 } ^ k P ( y _ i | x _ i ) . \end{align*}


\begin{align*} d _ H \rho : = d \rho + H \wedge \rho = ( X + \xi ) \cdot \rho , \end{align*}


\begin{align*} \begin{array} { l l } \displaystyle \partial _ { t } u _ j + a _ j ( x , t ) \partial _ { x } u _ j + \sum _ { k = 1 } ^ { n } b _ { j k } ( x , t ) u _ k + \sum _ { k = 1 } ^ { n } \int _ { 0 } ^ { x } g _ { j k } ( y , t ) u _ k ( y , t ) d y = f _ j ( x , t ) , \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ ; \ ; \ ; ( x , t ) \in ( 0 , 1 ) \times ( 0 , \infty ) , \ ; j \le n , & \end{array} \end{align*}


\begin{align*} h ^ { \pm } _ { \nu } ( a , b , x ) = a + \sqrt { b + x ^ 2 } . \end{align*}


\begin{align*} \pi _ Y ( Y ( \R ) ) = \bigsqcup _ { i = 1 } ^ { r } [ \varepsilon _ { 2 i - 1 } , \varepsilon _ { 2 i } ] . \end{align*}


\begin{align*} I _ 1 ( t \ , , x \ , ; \varepsilon ) & = \int _ t ^ { t + \varepsilon } \d s \int _ { \R ^ d } \d y \ p ( t + \varepsilon - s \ , , \d y ) \ , g ( s \ , , y \ , , u ( s \ , , y - x ) ) , \\ I _ 2 ( t \ , , x \ , ; \varepsilon ) & = \int _ 0 ^ t \d s \int _ { \R ^ d } \left [ p ( t + \varepsilon - s \ , , \d y ) - p ( t - s \ , , \d y ) \right ] g ( s \ , , y \ , , u ( s \ , , y - x ) ) . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ { \infty } \frac { A _ { n } ( \bar { r } ) } { n ! } p ^ { n } = ( 1 - p ) ^ { \bar { r } u } \end{align*}


\begin{align*} ( m ^ * ) ^ { i t } \circ \mu _ A = ( m ^ * ) ^ { i t } \circ ( \dot { + } ) ^ { i t } \circ \bigotimes _ { i = 1 } ^ l \mu _ { a _ { i } } = \left ( ( \dot { + } ) ^ { i t } \right ) ^ { \otimes k } \circ w \circ \left ( ( m ^ * ) ^ { i t } \right ) ^ { \otimes l } \circ \bigotimes _ { i = 1 } ^ l \mu _ { a _ { i } } \end{align*}


\begin{align*} h _ { \lambda } ^ { ( k ) } = \end{align*}


\begin{align*} | \hat \mu _ 1 ( y ) | = | \hat \mu _ 2 ( y ) | = 1 , \ \ y \in H . \end{align*}


\begin{align*} \lim _ { \tau \to i \infty } \Omega _ { k + 1 , D } ( \tau , w ) = 0 . \end{align*}


\begin{align*} \begin{pmatrix} f _ i ( P _ j ) \end{pmatrix} _ { \substack { 1 \leq j \leq s \\ 1 \leq i \leq t } } , \end{align*}


\begin{align*} \big | \psi _ { \omega , \varkappa } ^ { ( j ) } ( x ) \big | < \max \big \{ \big | \psi _ { \omega , \varkappa } ^ { ( j ) } \big ( x + \overline { n + 1 } _ { q } \big ) \big | , \big | \psi _ { \omega , \varkappa } ^ { ( j ) } \big ( x - \overline { n + 1 } _ { q } \big ) \big | \big \} = \big | \psi _ { \omega , \varkappa } ^ { ( j ) } \big ( x + \alpha \big ( \overline { n + 1 } _ { q } \big ) \big ) \big | \end{align*}


\begin{align*} \dd ( X _ { n + 1 } ^ { \epsilon , \Delta t } , X _ { n + 1 } ^ { 0 , \Delta t } ) & = \dd \bigl ( \Phi ( \frac { \Delta t m _ { n + 1 } ^ \epsilon } { \epsilon } , X _ n ^ { \epsilon , \Delta t } ) , \Phi ( \Delta \beta _ n , X _ n ^ { 0 , \Delta t } ) \bigr ) \\ & \le C e ^ { C | \frac { \Delta t m _ { n + 1 } ^ \epsilon } { \epsilon } | + C | \Delta \beta _ n | } \bigl ( | \frac { \Delta t m _ { n + 1 } ^ \epsilon } { \epsilon } - \Delta \beta _ n | + \dd ( X _ n ^ { \epsilon , \Delta t } , X _ n ^ { 0 , \Delta t } ) \bigr ) , \end{align*}


\begin{align*} \Delta \circ D = ( D \otimes \Phi + \Phi \otimes D ) \circ \Delta . \end{align*}


\begin{align*} \frac { 1 } { m } \sum _ { j = - m / 2 + 1 } ^ { m / 2 } \frac { e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \lfloor s m \rfloor } } { 1 - \exp \left \{ - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \right \} + \frac { z } { m } } = e ^ { \pi i \theta _ 1 \lfloor s m \rfloor / m } \sum _ { j = - m / 2 + 1 } ^ { m / 2 } a _ { j , m } b _ m ^ j \end{align*}


\begin{align*} H ( G ^ { \{ v \} } _ { - 1 / 2 } v _ { \nu , \ell _ 0 } , G ^ { \{ v \} } _ { - 1 / 2 } v _ { \nu , \ell _ 0 } ) = & - 2 ( k + h ^ \vee ) \langle \phi ( v ) , v \rangle l _ 0 + \langle \phi ( v ) , v \rangle ( \nu | \nu + 2 \rho ^ \natural ) \\ & - 2 ( k + 1 ) \langle \phi ( v ) , v \rangle ( \xi | \nu ) + 2 ( \xi | \nu ) ^ 2 \langle \phi ( v ) , v \rangle , \end{align*}


\begin{align*} \mathrm { t r a c e } ( - i \hbar \partial _ { x ^ { k } } \lor \frac { 1 } { 2 \epsilon } ( x ^ { \bot } ) ^ { k } \mathrm { O P } _ { \hbar } ^ { T } ( ( 2 \pi \hbar ) ^ { 2 } \nu ) ) = \underset { \mathbb { R } ^ { 2 } } { \int } \frac { 1 } { \epsilon } p \cdot q ^ { \bot } \nu ( d q d p ) . \end{align*}


\begin{align*} \frac { 1 } { 2 } \sum _ { j \in \mathbb { Z } } \log \left ( 1 + \left ( \frac { q } { p + j } \right ) ^ 2 \right ) = \log | \sin ( \pi ( p + q i ) | - \log | \sin ( \pi p ) | . \end{align*}


\begin{align*} \delta ( k , \xi ) = \overline { \delta ( - \bar { k } , \xi ) } , k \in \mathbb { C } \backslash \left \{ 0 \right \} . \end{align*}


\begin{align*} \rho ^ { \epsilon } = \rho \ast \delta ^ { \epsilon } . \end{align*}


\begin{align*} \lim _ { n \to \infty } \sup _ { \mu \in \mathcal { M } } \sum _ { k = n } ^ { \infty } \mu ( x \colon q _ k ( x ) > k \alpha _ k ) = 0 , \end{align*}


\begin{align*} \textbf { N } _ { s } ( v ' , u ' ) : = B _ { q } ( v , u ) \end{align*}


\begin{align*} - \Delta _ { p ( x ) } u = h ( x ) \Omega , u = 0 \partial \Omega , \end{align*}


\begin{align*} \begin{gathered} M ( A ) ( ( \mathbf { 1 } , b ) ) = \mathop { R e a l } ( A ( b ) ) \\ M ( A ) ( ( \mathbf { i } , b ) ) = \mathop { I m a g } ( A ( b ) ) \ ; . \end{gathered} \end{align*}


\begin{align*} c _ i \mapsto \begin{cases} c _ i - 1 , & ; \\ c _ i , & , \end{cases} d _ j \mapsto \begin{cases} d _ j - 1 , & ; \\ d _ j , & . \end{cases} \end{align*}


\begin{align*} \Omega \cdot \varphi ^ P _ s = r \ell ( r - \ell ) ( s ^ 2 - s ) \cdot \varphi ^ P _ { s } . \end{align*}


\begin{align*} \mathrm { W C E } ( n , \mathbb { M } ^ 2 _ { \mathrm { e } } ) & : = \sup _ { \substack { f \in \mathbb { M } ^ s _ { \mathrm { e } } , \| f \| _ { \mathbb { M } ^ s _ { \mathrm { e } } } \leq 1 } } \left | \int \limits _ { \ , - \infty } ^ \infty f ( x ) W ( x ) \mathrm { d } x - \sum _ { x \in X _ n } \omega ( x ) f ( x ) \right | ^ 2 \\ & \asymp \sum _ { x , y \in X _ n } \omega ( x ) \omega ( y ) \sum _ { k = 2 n } ^ \infty \mathrm { e } ^ { - q \sqrt { k } } h _ { k } ( x ) h _ { k } ( y ) . \end{align*}


\begin{align*} \begin{cases} & \frac { \partial u } { \partial t } = ( G ( x , X ) F ^ { \beta } ( \l _ i ) - 1 ) u , \\ & u ( \cdot , 0 ) = u _ 0 . \end{cases} \end{align*}


\begin{align*} \lim _ { n \rightarrow + \infty } ( P _ t \phi ) ( x ^ n ) = \lim _ { n \rightarrow + \infty } \mathbb { E } \left [ \phi ( u _ t ^ { x ^ n } ) \right ] = \mathbb { E } \left [ \lim _ { n \rightarrow + \infty } \phi ( u _ t ^ { x ^ n } ) \right ] = \mathbb { E } \left [ \phi ( u _ t ^ { x } ) \right ] = ( P _ t \phi ) ( x ) \ , , \end{align*}


\begin{align*} m _ i : = D . L _ { P _ i } > 0 \end{align*}


\begin{align*} P ^ { + } _ k = \left \{ \nu \in P ^ + \mid \nu ( \theta ^ \vee _ i ) \le M _ i ( k ) \right \} . \end{align*}


\begin{align*} \mu \ ; = \ ; \sum _ { j \in S _ 1 } \omega _ j \ , \pi ^ { ( 1 ) } _ j \ ; . \end{align*}


\begin{align*} d ^ \ell _ x = \frac { \lambda ^ c _ { x _ { - \ell } } ( \ell ) } { \lambda ^ u _ { x _ { - \ell } } ( \ell ) } \end{align*}


\begin{align*} C = \begin{pmatrix} C _ { 1 1 } & C _ { 1 2 } \\ C _ { 1 2 } ^ T & C _ { 2 2 } \end{pmatrix} \end{align*}


\begin{align*} | \kappa | ^ 2 - 2 \mathrm { R e } ( \overline { \Phi } ^ { ( 0 ) } _ y \kappa ) = | \kappa - \Phi ^ { ( 0 ) } _ y | ^ 2 - | \Phi ^ { ( 0 ) } _ y | ^ 2 \geq - 1 , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ k & ( 1 + y X _ i ) \times \prod _ { j = 1 } ^ { n - k } ( 1 + y \tilde { X } _ i ) \\ = & \prod _ { i = 1 } ^ n ( 1 + y T _ i ) - \frac { q } { 1 - q } y ^ { n - k } \tilde { X } _ 1 \cdot \ldots \cdot \tilde { X } _ { n - k } \bigl ( \prod _ { i = 1 } ^ k ( 1 + y X _ i ) - 1 \bigr ) \ / . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u & = f ( x , u , v ) \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; & & u > 0 \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; u \lfloor _ { \partial \Omega } = 0 , \\ - \Delta _ q v & = g ( x , u , v ) \ ; \ ; \mbox { i n $ \Omega $ , } \ , \ ; & & v > 0 \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; v \lfloor _ { \partial \Omega } = 0 , \\ \end{alignedat} \right . \end{align*}


\begin{align*} \ominus \alpha ( t , x ) = \alpha ( - t , x ) \ , . \end{align*}


\begin{align*} \eta ( z \cdot g ) & = | \det z g | \cdot h \left ( e _ r \cdot z g \right ) ^ { - r } = | \det z g | \cdot h ( t e _ r \cdot g ) ^ { - r } \\ & = | \det z | \cdot | \det g | \cdot | t | ^ { - r } \cdot h ( e _ r \cdot g ) ^ { - r } = | t | ^ r \cdot | t | ^ { - r } \cdot \eta ( g ) = \eta ( g ) \end{align*}


\begin{align*} \gamma ( x + \gamma _ x ( y ) ) = \gamma ( x ) \gamma ( y ) \end{align*}


\begin{align*} S : = \{ Q = 0 \} \cap \{ R = 0 \} \subset \mathbb P ^ 3 \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { \infty } d _ { i , j } \le \frac { 2 ^ { 1 + p } C _ d } { p } , \end{align*}


\begin{align*} \frac d { d t } \sum \limits _ { | \alpha | = 0 } ^ { 2 } ( \| \sqrt { B } \partial _ \tau ^ \alpha \partial _ y u \| _ { L ^ 2 } ^ 2 & + \| \sqrt { A } \partial _ \tau ^ \alpha \partial _ y \tilde { h } \| _ { L ^ 2 } ^ 2 ) + C _ 2 \| \partial _ y ( u , \tilde { h } ) \| _ { H ^ { 2 , 1 } } ^ 2 \\ & \le C D ( t ) ^ { \frac 1 2 } E ( t ) + C D ( t ) E ( t ) ^ { \frac 1 4 } + \frac { C } { M ^ { \frac 1 4 } } D ( t ) . \end{align*}


\begin{align*} ( F ^ * g ) _ x ( v _ i , v _ j ) = g _ { F ( x ) } ( F _ * v _ i , F _ * v _ j ) . \end{align*}


\begin{align*} m ( R ^ G _ { T , s } , \sigma ) = m ( R ^ G _ { T , s } , I ( \tau , \sigma ) ) . \end{align*}


\begin{align*} \sum _ { \pi \in S _ n } e ^ { - \mu \sum _ { j = 1 } ^ n | x _ j - y _ { \pi ( j ) } | } \leq C _ { d , \mu } ^ n e ^ { - \frac { \mu } { 2 \sqrt { d } } D _ s ( X , Y ) } . \end{align*}


\begin{align*} \begin{cases} x _ k = H _ k x _ k + M _ k x _ k w _ k , \\ y _ k = G _ k x _ k . \end{cases} \end{align*}


\begin{align*} \begin{aligned} & \delta + \varepsilon _ 0 \mathcal H _ 1 ( \norm { \xi ' } { H ^ 2 } , \norm { v ' } { H ^ 2 } ) \leq \delta + \varepsilon _ 0 \mathcal H _ 1 ( ( C _ 0 M _ 0 ) ^ { 1 / 2 } , ( C _ 1 M _ 1 ) ^ { 1 / 2 } ) \\ & ~ ~ ~ ~ \leq \dfrac { c _ 1 \min \lbrace \mu , \lambda , 1 \rbrace } { 2 } . \end{aligned} \end{align*}


\begin{align*} a ( t ) : = \frac 5 6 s ( 1 - s ) ^ 2 p ^ 5 r ^ 2 . \end{align*}


\begin{align*} \pi ( x ) = ( x ) + O ( x \exp ( - \delta _ 0 \sqrt { \log x } ) ) \end{align*}


\begin{align*} \partial _ t u - \Delta u = f \hbox { i n ~ } Q , u ( 0 ) = u _ 0 \hbox { i n ~ } \Omega , \end{align*}


\begin{align*} ( ( \phi _ { y _ 0 } ^ Y ) ^ * d v _ Y ) ( Z _ Y ) = \kappa _ { \phi , y _ 0 } ^ Y d Z _ 1 \wedge \cdots \wedge d Z _ { 2 m } . \end{align*}


\begin{align*} f _ \zeta ^ \varepsilon & = \partial _ t \rho _ \zeta ^ \varepsilon - k _ d ^ { - 1 } \Delta \rho _ \zeta ^ \varepsilon \quad Q _ { \varepsilon , T } , \psi _ \zeta ^ \varepsilon = \partial _ { \nu _ \varepsilon } \rho _ \zeta ^ \varepsilon + k _ d ^ { - 1 } V _ \varepsilon \rho _ \zeta ^ \varepsilon \quad \partial _ \ell Q _ { \varepsilon , T } . \end{align*}


\begin{align*} H ^ 2 ( \tilde Y , \mathbb Q ) & = ( \pi _ { 4 * } H ^ 2 ( X , \mathbb Z ) \oplus M ) \otimes \mathbb Q = ( \pi _ { 4 * } H ^ 2 ( X , \mathbb Z ) \oplus \widehat { \pi _ { 2 } } _ * N \oplus N _ Y ) \otimes \mathbb Q = \\ & = ( \widehat { \pi _ { 2 } } _ * ( \pi _ { 2 * } H ^ 2 ( X , \mathbb Z ) \oplus N ) \oplus N _ Y ) \otimes \mathbb Q = ( \widehat { \pi _ { 2 } } _ * H ^ 2 ( \tilde Z , \mathbb Z ) \oplus N _ Y ) \otimes \mathbb Q . \end{align*}


\begin{align*} \lambda _ m = \frac { < g , L g > } { < g , g > } = \lambda _ k . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { m } \omega _ { i } u ( \xi _ { i } ) = \frac { \Gamma ( \gamma ) } { \Gamma ( \alpha ) } \Omega \sum _ { i = 1 } ^ { m } \omega _ { i } \int ^ { \xi _ { i } } _ { 0 } s ^ { \rho - 1 } \left ( \frac { \xi _ { i } ^ { \rho } - s ^ { \rho } } { \rho } \right ) ^ { \alpha - 1 } \left ( f ( s , u ( s ) , ^ \rho D ^ { \alpha , \beta } u ( s ) ) - p ( s ) u ( s ) \right ) d s . \end{align*}


\begin{align*} \mathcal { P } _ { A , B } ( \nu ) = a \rho ( \nu , \mu _ 0 ) < \zeta . \end{align*}


\begin{align*} T _ G ( p - m ) = T _ G ( p ) \setminus \{ t \} , \end{align*}


\begin{align*} ( x , y ) _ n : = \sum _ { \gamma \in \Gamma / \Gamma _ n } \langle x , \gamma ^ { - 1 } y \rangle _ n \cdot \gamma \end{align*}


\begin{align*} \widehat { Q } = i \circ Q \circ p + i ' \circ Q ' \circ p ' \end{align*}


\begin{align*} h ( a ) = \frac { 1 } { \lambda ' } \log ( 1 + \frac { 1 } { a } ) ^ { \lambda ' } + \tilde { h } ( a ) \end{align*}


\begin{align*} L _ { S T } = \begin{pmatrix} 0 _ { d \times d } & I _ { d \times d } \\ - I _ { d \times d } & I _ { d \times d } \end{pmatrix} = J \end{align*}


\begin{align*} l _ { i } ^ { + } ( u ) = \varkappa ^ { + } ( u ) a n d l _ { i } ^ { - } ( u ) = \varkappa _ { i } ^ { - } ( u ) \end{align*}


\begin{align*} p _ H ( Z _ r , Z _ \bullet , Z _ \bullet ' ) = \left \{ \begin{array} { r c l } 1 & \mbox { f o r } & Z _ \bullet = Z _ \bullet ' , \\ 0 & \mbox { f o r } & Z _ \bullet \neq Z _ \bullet ' , \end{array} \right . \end{align*}


\begin{align*} p \{ \beta \} + q ( 1 - \{ \alpha \} ) = 1 . \end{align*}


\begin{align*} f ( \alpha , t ) = \frac { 1 } { \alpha - 1 } \sum _ { n = 0 } ^ \infty \binom { 2 \alpha + 2 n - 2 } { 2 n + 1 } t ^ { 2 n } \ . \end{align*}


\begin{align*} P \Big ( - \sum _ { j = 1 } ^ k \overline { U } _ j \ge \epsilon c _ o \bar h _ 4 ( n ) \Big ) \le \big ( \epsilon c _ o \bar h _ 4 ( n ) \big ) ^ { - 2 } \frac { C _ 1 k m ^ 2 } { ( \log m ) ^ 4 } \le \frac { 2 C _ 1 } { ( \epsilon c _ o ) ^ 2 k ( \log _ 2 n ) ^ 2 } \ , . \end{align*}


\begin{align*} 4 \ , \left ( N ( X , T ) \right ) ^ \flat \wedge \left ( J N ( X , T ) \right ) ^ \flat & = \frac { \left ( X ( \textbf { H } ) \right ) ^ 2 + \left ( J X ( \textbf { H } ) \right ) ^ 2 } { 4 } T ^ \flat \wedge ( J T ) ^ \flat , \\ & = \frac { \left ( X ( \textbf { H } ) \right ) ^ 2 + \left ( J X ( \textbf { H } ) \right ) ^ 2 } { 4 } d \theta \wedge d \varphi . \end{align*}


\begin{align*} { \displaystyle \sum \limits _ { n \geq 0 } } \left ( 2 n + 1 \right ) \frac { u _ { n } ( z ) } { t ^ { 2 n + 1 } } = - t \partial _ { t } S , \end{align*}


\begin{align*} \lim _ { T \rightarrow + \infty } \sup _ { t \geq T } \mathbb { E } \left [ \| ( u _ t - B _ t ) \cdot ( u _ t - B _ t ) - \alpha \| _ { L ^ 1 } ^ 2 \right ] = 0 \ , . \end{align*}


\begin{align*} \mathring { R } _ { q + 1 } ^ u : = \mathring { R } _ { l i n } ^ u + \mathring { R } _ { o s c } ^ u + \mathring { R } _ { c o r } ^ u , \end{align*}


\begin{align*} \Omega _ j ( \zeta ) = \sqrt { | j | _ { \bar g } ^ 4 + \zeta } , j \in \Z ^ d \ , . \end{align*}


\begin{align*} \# \left ( \{ x \in \mathbb { Z } _ q \mid s _ { h } x = 0 \} \bigcup \{ x _ 1 , \ldots , x _ { h - 1 } \} \right ) = d _ { h } + h - 1 . \end{align*}


\begin{align*} \lim _ { \delta \searrow 0 } f ( a , \delta ) = \frac { 2 \left ( 1 - \sqrt { 1 - a } \right ) } { a } . \end{align*}


\begin{align*} g ( t , s ) = \sum _ { \ell = 0 } ^ { + \infty } \left ( \log \lambda ^ u _ { f ^ { - \ell - 1 } ( \Gamma _ x ( 0 , s ) ) } - \log \lambda ^ u _ { f ^ { - \ell - 1 } ( \Gamma _ x ( t , s ) ) } \right ) . \end{align*}


\begin{align*} \frac { d } { d a } ( a ) _ { m + n } = \frac { d } { d a } \left [ ( a ) _ m ( a + m ) _ n \right ] = \frac { d } { d a } \left [ ( a ) _ m \right ] ( a + m ) _ n + ( a ) _ m \frac { d } { d a } \left [ ( a + m ) _ n \right ] \end{align*}


\begin{align*} \dot { \varphi } _ { s , t } ( z ) = p ( \varphi _ { s , t } ( z ) , t ) , 0 \leq s \leq t \leq \tau _ 0 , \varphi _ { s , s } ( z ) = z . \end{align*}


\begin{align*} X _ 1 \cdots X _ k a = \sum _ { \substack { t + s = k \\ \sigma \in W _ { t , s } } } \omega \left ( X _ { \sigma ( 1 ) } \right ) \cdots \omega \left ( X _ { \sigma ( t ) } \right ) ( a ) X _ { \sigma ( t + 1 ) } \cdots X _ { \sigma ( k ) } , \end{align*}


\begin{align*} K _ Y = \mu ^ * ( K _ X + D ) + \sum _ i a _ i E _ i . \end{align*}


\begin{align*} \mathcal { D } ( A ^ { \odot \star } ( s ) ) = \mathcal { D } ( A _ 0 ^ { \odot \star } ) , A ^ { \odot \star } ( s ) = A _ 0 ^ { \odot \star } + B ( s ) j ^ { - 1 } , \forall s \in J . \end{align*}


\begin{align*} S O _ { \alpha } ( U ( n , d ) ) = ( n - d - 1 ) \left ( ( n - d + 1 ) ^ 2 + 1 \right ) ^ { \alpha } + 2 \left ( ( n - d + 1 ) ^ 2 + 4 \right ) ^ { \alpha } + \gamma , \end{align*}


\begin{align*} \| 1 _ { \delta \Gamma _ { l , j } } \| \ = \ h _ { R , l } ( k _ j , u _ j ) . \end{align*}


\begin{align*} \begin{gathered} [ A ] _ N = A _ N / \{ A _ H ^ i ( ( 0 , ( 0 , j ) ) ) \sim A _ H ^ i ( ( 0 , ( 1 , j ) ) ) \forall i , \forall j \in b ( i ) \} \ ; , \\ [ A ] _ H ( i ) = A _ H ( ( 0 , i ) ) \ ; , \\ \end{gathered} \end{align*}


\begin{align*} g ( \theta _ 0 ) = \sqrt { a ^ 2 + b ^ 2 } . \end{align*}


\begin{align*} y = - e ^ { i u } , \end{align*}


\begin{align*} b _ i m _ j = \begin{cases} [ 2 ] m _ i , & \ ; j \equiv i \bmod d ; \\ z m _ 1 , & \ ; i - 1 \equiv 0 \equiv j \bmod d ; \\ z ^ { - 1 } m _ 0 , & \ ; i \equiv 0 \equiv j - 1 \bmod d ; \\ m _ j , & \ ; i \equiv j \pm 1 \bmod d , \ ; ; \\ 0 , & . \end{cases} \end{align*}


\begin{align*} \partial _ t \rho - \nabla \cdot ( \rho \nabla p ) = \rho G , p = \rho ^ { \gamma } , \end{align*}


\begin{align*} \# \left ( \bigcup _ { 1 \leq j \leq n } U _ j \right ) = \sum _ { 1 \leq j \leq n } \# \Big ( U _ j \Big ) - \sum _ { 1 \leq j _ 1 < j _ 2 \leq n } \# \Big ( U _ { j _ 1 } \cap U _ { j _ 2 } \Big ) + \cdots \ , . \end{align*}


\begin{align*} x \to y & : = \max \{ u \colon ( u ] \cap I ( x , y ) \cap [ y ) = \{ y \} \} \\ & ~ = \max \{ u \ge y \colon ( u ] \cap I ( x , y ) \cap [ y ) \subseteq \{ y \} \} . \end{align*}


\begin{align*} \bar R ( u ) = I + \frac { h } { u } P \in E n d ( V \otimes V ) . \end{align*}


\begin{align*} \sum _ { \substack { d \le H \\ ( d , N ) = 1 } } \mu ^ 2 ( d ) \left | \pi ( N ; d , N ) - \frac { \pi ( N ) } { \varphi ( d ) } \right | & \leq 0 . 6 5 H \left ( c _ { \pi } ( X _ 2 ) + \frac { 1 } { 1 6 \pi } \right ) \sqrt { N } \log N \\ & \leq \frac { p _ G ( X _ 2 ) N } { \log ^ A N } , \end{align*}


\begin{align*} \tilde { r } = | e ^ { - \frac { F } { 2 } } w | ^ 2 + e ^ { - v } \eta \circ \theta + \mathrm { p h } , \end{align*}


\begin{align*} F _ { 1 , \delta } ( 1 / 2 ) = F _ 1 ( 1 / 2 ) , F _ { 1 , \delta } ^ \prime ( 1 / 2 ) = F ^ \prime _ 1 ( 1 / 2 ) , \\ F _ { 1 , \delta } ^ { \prime \prime } ( s ) = \left \{ \begin{array} { l l } F _ 1 ^ { \prime \prime } ( \delta ) , & s \leq \delta \\ F _ 1 ^ { \prime \prime } ( s ) , & \delta \leq s \leq 1 - \delta \\ F _ 1 ^ { \prime \prime } ( 1 - \delta ) , & s \geq 1 - \delta . \end{array} \right . \end{align*}


\begin{align*} \frac { \partial ^ n } { \partial x ^ n } V ( y , x ) & = \delta _ n ^ 0 \Delta _ { \sigma } + O _ n \left ( \frac { y ^ { - \sigma } } { 2 \pi \sigma } \left ( x ^ { \sigma - n + 1 / 2 } \sigma ^ { n } \right ) \right ) \\ & = \delta _ n ^ 0 \Delta _ { \sigma } + O _ n \left ( x ^ { - n / 2 } ( x / y ) ^ { \sigma } \right ) \end{align*}


\begin{align*} \nu : = \frac { \sqrt { a } + i \sqrt { b } } { \sqrt { a } - i \sqrt { b } } = \frac { 1 } { d } ( a - b + 2 i \sqrt { a b } ) . \end{align*}


\begin{align*} V = \left \{ \sum _ { i = 0 } ^ { 5 } x _ i = 0 , \ \sigma _ 4 ( x _ 0 , \ldots , x _ 5 ) = 0 \right \} \subset \mathbb P ^ 5 , \end{align*}


\begin{align*} x \to y = \min \{ z * y \colon z \in [ y , x ] \cup \{ y \} \} = \min \{ z * y \colon z \in ( ( x ] \cup ( y ] ) \cap [ y ) \} . \end{align*}


\begin{align*} L ( x , v ) = Q _ { x } ( v ) - U ( x ) , ( x , v ) \in T M \end{align*}


\begin{align*} \mathrm { E } [ X _ 1 ] = - \psi ' ( 0 + ) < \infty . \end{align*}


\begin{align*} \left \{ \begin{aligned} & - \partial _ \tau w - \eta \partial _ \xi w + w ^ 2 \partial _ { \eta } ^ 2 w = 0 , ( \tau , \xi , \eta ) \in D , \\ & w \partial _ \eta w \mid _ { \eta = 0 } = 0 , w \mid _ { \eta = 1 } = 0 , \\ & w \mid _ { \tau = 0 } = w _ 0 , w \mid _ { \xi = 0 } = w _ 1 , \end{aligned} \right . \end{align*}


\begin{align*} \sigma : \ \mathbb { F } _ q & \longrightarrow \mathbb { F } _ q , \\ a & \longmapsto a ^ p , \end{align*}


\begin{align*} I ( t \ , , x ) = \int _ 0 ^ t \d s \int _ { \R ^ d } p ( t - s \ , , \d y ) \ g ( s \ , , y \ , , u ( s \ , , y - x ) ) \qquad \end{align*}


\begin{align*} ( X _ I ^ J ) ^ S = \{ w _ K \in W \mid K \subset \Delta , J \subset K \subset I \} \end{align*}


\begin{align*} p ^ 2 - 1 + 1 = p ^ 2 . \end{align*}


\begin{align*} D _ { \mathcal { G } } ( \Phi ( u , T ( u ) ) , \Phi ( w , T ( w ) ) ) & = D _ { \mathcal { G } } ( ( u , F ( u ) + T ( u ) ) , ( w , F ( w ) + T ( w ) ) ) \\ & = \lvert u - w \rvert + H _ d ( F ( u ) + T ( u ) , F ( w ) + T ( w ) ) \\ & \leq \lvert u - w \rvert + H _ d ( F ( u ) , F ( w ) ) + H _ d ( T ( u ) , T ( w ) ) \\ & \leq \lvert u - w \rvert + l \lvert u - w \rvert + H _ d ( T ( u ) , T ( w ) ) \\ & \leq ( 1 + l ) \left \{ \lvert u - w \rvert + H _ d ( T ( u ) , T ( w ) ) \right \} . \end{align*}


\begin{align*} \psi ( \xi ) \leq | \xi | ^ { \alpha } = \psi ( \xi ) + \int _ { \R ^ d \setminus \{ 0 \} } ( 1 - \cos ( \xi \cdot y ) \sigma ( y ) \d y \leq \psi ( \xi ) + | \sigma | , \xi \in \R ^ d . \end{align*}


\begin{align*} Y _ a = \{ n m k \in \mathfrak { S } : \alpha ( m ) \geq a \alpha \in \Phi \} , \end{align*}


\begin{align*} \tau = t , \ \xi = x , \ \eta = u ( t , x , y ) , \ w ( \tau , \xi , \eta ) = \partial _ y u ( t , x , y ) , \end{align*}


\begin{align*} \omega = \frac { i } { 2 } \sum _ i \zeta _ i \wedge \overline { \zeta _ i } , \psi = - i \zeta _ 1 \wedge \zeta _ 2 \wedge \zeta _ 3 . \end{align*}


\begin{align*} p \in S ^ M , M = \gamma + ( n - 1 ) / 2 . \end{align*}


\begin{align*} \tilde { x } : = x + \epsilon ( x ) , x \in X _ { n + 1 } . \end{align*}


\begin{align*} ( \Omega _ J ) ^ S = \{ w _ K \mid K \supset J \} . \end{align*}


\begin{align*} \overline { \beta _ k ( a , b ) } = \beta _ k ( \phi ( a ) , \phi ( b ) ) . \end{align*}


\begin{align*} d \mathbb { P } = Z \ , d \mathbb { Q } . \end{align*}


\begin{align*} \lambda ( x , t ) = - k _ d ^ { - 1 } d ( x , t ) \overline { V _ \Gamma } ( x , t ) , \chi ^ \varepsilon ( x , t ) = e ^ { - \lambda ( x , t ) } \rho ^ \varepsilon ( x , t ) \end{align*}


\begin{align*} b _ j : = \mathrm { R e } \int _ { - 1 / 2 } ^ { 1 / 2 } \log \left ( 1 - \frac { e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 } { m } } \left ( e ^ { 2 \pi i \phi / m } - 1 \right ) } { 1 - e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 } { m } } + \frac { x + i y } { m } } \right ) \mathrm { d } \phi . \end{align*}


\begin{align*} \Gamma = \langle a , b \mid b ^ { - 1 } a b = a ^ { - 1 } \rangle , \end{align*}


\begin{align*} a _ 1 ^ d = \cdots = a _ n ^ d = \lambda a _ 1 ^ { w _ 1 } \cdots a _ n ^ { w _ n } . \end{align*}


\begin{align*} 1 - \eta < \int _ B \psi d \mu + \int _ { X \setminus B } \psi d \mu \leq \mu ( B ) + ( 1 - \sqrt { \eta } ) ( 1 - \mu ( B ) ) = 1 - \sqrt { \eta } ( 1 - \mu ( B ) ) , \end{align*}


\begin{align*} \vert ( { A - 1 } ) / { u } \vert = \left \vert \left ( { g ( 0 ) - g ( s _ b ) } \right ) / \left ( { i \sqrt { 2 \vert x \vert } s _ b g ( s _ b ) } \right ) \right \vert \le C _ { R _ 0 } \vert x \vert ^ { - 1 / 2 } . \end{align*}


\begin{align*} T _ \theta ^ U \circ \alpha ^ { g | _ U } ( J _ \theta X , Y ) = T _ \theta ^ V \circ \alpha ^ { g | _ V } ( J _ \theta X , Y ) , \end{align*}


\begin{align*} \Delta \beta _ n = { \rm O } ( \Delta t ^ { \frac 1 2 } ) , \frac { \Delta t m _ { n + 1 } ^ \epsilon } { \epsilon } = { \rm O } ( \Delta t ^ { \frac 1 2 } ) \end{align*}


\begin{align*} \breve { a } _ 1 ( k ) = a _ 1 ( k ) \frac { k } { k - i \kappa } , \breve { a } _ 2 ( k ) = a _ 2 ( k ) \frac { k - i \kappa } { k } . \end{align*}


\begin{align*} 0 < \gamma _ 1 : = \min _ { 1 \leq i \leq m } \min \left \{ \frac { r _ i ^ - } { p _ i ^ + } , \frac { s _ i ^ - } { q _ i ^ + } \right \} - 1 \leq \gamma _ 2 : = \max _ { 1 \leq i \leq m } \max \left \{ \frac { r _ i ^ + } { p _ i ^ - } , \frac { s _ i ^ + } { p _ i ^ - } \right \} - 1 . \end{align*}


\begin{align*} & \kappa ( t ) \ ; = \ ; \frac 1 \pi \arctan ( t ) \ ; ( t \ , \in \ , \mathbb { R } ) , \ ; K ( \epsilon ) \ , = \ , 2 / \vert \log \epsilon \vert . \end{align*}


\begin{align*} | ( \varphi f ) ^ { ( n ) } ( z ) | ^ 2 \leqslant \Big ( \sum \limits _ { j = 0 } ^ n | \varphi ^ { ( n - j ) } ( z ) | ^ 2 | f ^ { ( j ) } ( z ) | ^ 2 \Big ) \Big ( \sum \limits _ { j = 0 } ^ n { \binom { n } { j } } ^ 2 \Big ) , \ , \ , \ , z \in \mathbb D . \end{align*}


\begin{align*} ( K _ { \varphi , f } u ) ( t ) : = T _ 0 ( t - s ) \varphi + j ^ { - 1 } \int _ s ^ t T _ 0 ^ { \odot \star } ( t - \tau ) [ B ( \tau ) u ( \tau ) + f ( \tau ) ] d \tau , \forall t \in I . \end{align*}


\begin{align*} : & x ' _ i = \begin{cases} ( q ' , b ) & i = i _ 0 \\ x _ i & \end{cases} \\ : & x ' _ i = \begin{cases} a & i = i _ 0 \\ ( q ' , x _ i ) & i = i _ 0 + \delta \bmod \ell \\ x _ i & \end{cases} \end{align*}


\begin{align*} \phi \Big | _ { k , m } A ( \tau ) & = ( c \tau + d ) ^ { - k } e ^ { - 2 \pi i \frac { m c z ^ 2 } { c \tau + d } } \phi \left ( \frac { a \tau + b } { c \tau + d } , \frac { z } { c \tau + d } \right ) \\ & = \sum _ { n , r } c _ A ( n , r ) q ^ n \zeta ^ r , \end{align*}


\begin{align*} \omega ( x ) = \tau ( x ) ( S _ { n } ^ { - 1 } W ) ( x ) , x \in X _ n \ , . \end{align*}


\begin{align*} \overline { \mathbf { v } ( \overline { k } ) } = \mathbf { v } ( - k ) = \mathbf { v } ( k ) \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \ \ \ \operatorname { I m } k \neq 0 . \end{align*}


\begin{align*} \Phi ( \xi ) = \bigl ( \Phi _ 1 ( \xi _ 1 ) , \ldots , \Phi _ d ( \xi _ d ) \bigr ) ^ T , \xi \in D : = D _ 1 \times \cdots \times D _ d , \end{align*}


\begin{align*} J ^ { ( T ) } _ N = \min _ { a _ j : \ , a _ 1 = 1 } \left ( \abs { F \left ( e ^ { \frac { i \pi } { T } } \right ) } \right ) , \end{align*}


\begin{align*} \int _ { u ( y ) } ^ { \bar { u } ( y ) } \frac { d \eta } { \bar { w } } - \int _ 0 ^ { u ( y ) } \frac { \bar { w } - w } { \bar { w } w } d \eta = 0 . \end{align*}


\begin{align*} J _ \alpha = \{ i = 1 , \dotsc , r - 1 \mid \alpha ^ { - 1 } ( i ) < \alpha ^ { - 1 } ( 0 ) \} . \end{align*}


\begin{align*} A ( \xi ) \xi ^ { n + 1 } = B ( \xi ) \xi ^ n + C \xi ^ { n - 1 } \end{align*}


\begin{align*} f _ i ^ + : = \max _ { x \in \overline { \Omega } _ i } f ( x ) f ^ { - } _ { i } : = \min _ { x \in \overline { \Omega } _ i } f ( x ) . \end{align*}


\begin{align*} \frac { 1 } { 2 \pi } \int _ { \mathbb { R } } \psi ^ { 2 } \left ( x \right ) \d x = 1 . \end{align*}


\begin{align*} \gamma ^ + _ { i j } \ , = \sum _ { e = \vec { i j } \in E } \log _ q ( t _ e ) \ , , \gamma ^ - _ { i j } \ , = - \sum _ { e = \vec { j i } \in E } \log _ q ( t _ e ) \ , , \gamma ^ 0 _ { i j } = \gamma ^ + _ { i j } + \gamma ^ - _ { i j } \ , . \end{align*}


\begin{align*} \gamma _ { A , \ell } : = 1 / \alpha ( G _ { A , \ell } ^ \circ ) . \end{align*}


\begin{align*} \begin{multlined} M _ \otimes M ( ( A , B ) \otimes ( C , D ) ) ( ( ( 0 , i ) , ( 1 , j ) ) \\ = 0 \neq A ( i ) D ( j ) \\ = M ( ( A , B ) ) \otimes M ( ( C , D ) ) ( ( ( 0 , i ) , ( 1 , j ) ) ) \ ; . \end{multlined} \end{align*}


\begin{align*} E ( F _ N ; G _ N ) = \int _ { [ 0 , 1 ] ^ 2 } | S ( \alpha _ 1 , \alpha _ 2 ) | ^ 4 d \boldsymbol { \alpha } . \end{align*}


\begin{align*} a \circ b + a \circ c = a \circ ( b + c ) + a . \end{align*}


\begin{align*} u ( X ) : = \int _ { \R ^ d } g ( y ) \ , d \omega ^ X ( y ) \end{align*}


\begin{align*} \partial _ t u + \Delta u = f \hbox { i n ~ } Q , u ( 0 ) = u _ 0 \hbox { i n ~ } \Omega , \end{align*}


\begin{align*} \partial _ t d ( y , t ) = - V _ \Gamma ( y , t ) , ( y , t ) \in \overline { S _ T } , \end{align*}


\begin{align*} m _ n ^ { \epsilon , \Delta t } = \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ n } m _ 0 ^ \epsilon + \frac { 1 } { \epsilon } \sum _ { \ell } ^ { n - 1 } \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ { n - \ell } } \Delta \beta _ \ell . \end{align*}


\begin{align*} \phi = k + s g ( b ) + \int ^ s _ 0 \left ( \int ^ \eta _ 0 f ( b ^ 2 - \xi ^ 2 ) d \xi \right ) d \eta + \int ^ b _ 0 \rho f ( \rho ^ 2 ) d \rho , \end{align*}


\begin{align*} [ A ] _ \psi = \sim \circ A _ \psi \rvert _ 0 \ ; , \end{align*}


\begin{align*} R ^ G _ { T , \chi } ( s u ) = \sum _ { \bar { \gamma } \in \overline { N } _ G ( s , T ) ^ F } { } ^ \gamma \chi ( s ) Q ^ { G _ s } _ { { } ^ \gamma T } ( u ) , \end{align*}


\begin{align*} S _ T = \bigcup _ { t \in ( 0 , T ] } \Gamma ( t ) \times \{ t \} , \overline { S _ T } = ( \Gamma _ 0 \times \{ 0 \} ) \cup S _ T , \end{align*}


\begin{align*} \P \left \{ H ( \cdot \ , , x ) \in C ^ { \alpha } _ { \textit { l o c } } ( \R ^ d ) \right \} = 1 \end{align*}


\begin{align*} \frac { z \exp ( t z ) } { 1 - \exp z } = - \sum _ { n = 0 } ^ { \infty } B _ n ( t ) \ , \frac { z ^ n } { n ! } \end{align*}


\begin{align*} E _ { n , L } [ \cdot ; \cdot ] & = \sum _ { A \in \mathcal { A } _ { 2 n } } D _ { n , A , L } [ \cdot ; \cdot ] . \end{align*}


\begin{align*} R ( x ) = \{ I _ n - r W ( y ) \} ^ { - 1 } . \end{align*}


\begin{align*} ( M _ \nu ) = \dim ( ( M _ \nu ) = \dim ( _ { \mathbb Z } ( a _ 1 , \dots , a _ n ) ) = \dim ( ( a _ 1 , \dots , a _ n ) ) = ( \nu ) . \end{align*}


\begin{align*} \min _ { \mathcal A _ 0 } \mathcal E _ 0 = - { 1 6 \pi ^ 2 \over \min _ { a _ 0 \in \Omega } T ( a _ 0 ) + 8 \pi \lambda } . \end{align*}


\begin{align*} \begin{gathered} ( A \otimes B ) _ V = A _ V \sqcup B _ V \ ; , \\ ( A \otimes B ) _ \psi = A _ \psi \hat \sqcup B _ \psi \ ; , \\ ( A \otimes B ) _ A ( l _ 1 \sqcup l _ 2 ) = A _ A ( l _ 1 ) \otimes B _ A ( l _ 2 ) \ ; . \end{gathered} \end{align*}


\begin{align*} \Gamma _ k & = \sum _ { ( t _ i , t _ { i + 1 } ) \subset [ a , b ] } g _ { t _ i } f _ { t _ i , t _ { i + 1 } } , \\ \mathcal { S } _ k & = \overline { S p a n } \left \{ f _ { t _ l , t _ { l + 1 } } : \ { ( t _ l , t _ { l + 1 } ) \not \subset [ a , b ] } \right \} . \end{align*}


\begin{align*} \Omega _ J = \displaystyle \coprod _ { K \supset J } ( \Omega _ J \cap X _ K ^ \circ ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \frac { 1 } { n } \int \psi _ { n } ^ i d \mu \geq 0 \mu \in M ( f , X ) , \lim _ { n \rightarrow \infty } \frac { 1 } { n } \int \varphi _ { n } ^ i d \mu = 0 \end{align*}


\begin{align*} \nabla ( \hat { \boldsymbol { w } _ \varepsilon } \phi ) = \nabla \hat { \boldsymbol { w } _ \varepsilon } \to 0 { \rm a s } r \to \infty , \end{align*}


\begin{align*} d v _ X = d v _ { g ^ { T X } } , d v _ Y = d v _ { g ^ { T Y } } . \end{align*}


\begin{align*} a \lor b = \bigwedge \{ c \in A : a , b \leq c \} , a , b \in A . \end{align*}


\begin{align*} \displaystyle \lim _ { w \to w _ 0 } \lvert t ^ * ( w ) \rvert = \infty . \end{align*}


\begin{align*} ( ^ { \rho } I ^ { \alpha } _ { a ^ + } g ) ( x ) = \dfrac { \rho ^ { 1 - \alpha } } { \Gamma ( \alpha ) } \int _ { a } ^ { x } \ ( x ^ { \rho } - \tau ^ { \rho } ) ^ { \alpha - 1 } \tau ^ { \rho - 1 } g ( \tau ) d \tau , x > a , \ \rho > 0 , \end{align*}


\begin{align*} \left ( T _ \theta ^ U - T _ \theta ^ V \right ) \circ \alpha ^ { g | _ { U \cap V } } ( X , Y ) = 0 \end{align*}


\begin{align*} \dd \phi _ { i j } = - \phi _ { i k } \wedge \phi _ { k j } + K _ { i j p q } \zeta _ { q } \wedge \overline { \zeta _ p } . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u & = f _ n ( x , u , v ) \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; u \lfloor _ \Omega = 0 , \\ - \Delta _ q v & = g _ n ( x , u , v ) \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; v \lfloor _ \Omega = 0 . \end{alignedat} \right . \end{align*}


\begin{align*} T _ R : = \sup \left \{ | t | \in \R ^ + \ : \ \left \| z ( t ) \right \| _ s < R \right \} \ , , \end{align*}


\begin{align*} \lambda _ q \coloneq \big \lceil a ^ { ( b ^ q ) } \big \rceil , \quad \tau _ q : = T \lambda ^ { - 1 5 } _ q , \delta _ q \coloneq \lambda _ 2 ^ { 3 \beta } \lambda _ q ^ { - 2 \beta } , q \ge 1 . \end{align*}


\begin{align*} H ( G , f ) = \left ( 1 + Z ^ + ( f ) ' \Gamma _ T ^ { - 1 } Z ^ + ( f ) \right ) ^ { - 1 } . \end{align*}


\begin{align*} \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) = \prod _ { \Box \in \lambda } \frac { 1 } { 1 + \frac { c ( \Box ) } { N } } \end{align*}


\begin{align*} \widetilde B ^ E _ n ( \widetilde \omega ) = B ^ E _ n ( \omega ) . \end{align*}


\begin{align*} \alpha _ { i , j } ( t _ { j , \beta } ) = \sum _ { n \in \Z } \sum _ { \alpha } r _ \alpha ( n , j , \beta ) t _ i ^ \alpha h _ { i , j } ^ n \end{align*}


\begin{align*} \begin{cases} s _ i x _ i \ne 0 , \ & \ 1 \leq i \leq h - 1 , \\ x _ i - x _ j \ne 0 \ & \ 1 \leq i < j \leq h - 1 . \end{cases} \end{align*}


\begin{align*} J = \begin{pmatrix} 0 _ { d \times d } & I _ { d \times d } \\ - I _ { d \times d } & 0 _ { d \times d } \end{pmatrix} \end{align*}


\begin{align*} H & = \langle h \rangle \times \langle h ^ g \rangle \times \langle h ^ { g ^ 2 } \rangle \times \cdots \times \langle h ^ { g ^ { s - 1 } } \rangle \\ & = \langle ( 1 , 2 ) \rangle \times \langle ( 3 , 4 ) \rangle \times \langle ( 5 , 6 ) \rangle \times \cdots \times \langle ( 2 s - 1 , 2 s ) \rangle \cong C _ 2 ^ s , \end{align*}


\begin{align*} T _ m \mu _ N ( \varphi _ j ) = \mu _ N ( \varphi _ j ) 1 \leq j \leq n \ , , \end{align*}


\begin{align*} \sigma _ 0 ( A ) ( i ) = A ( \Phi _ \cdot ^ { b , a } ( \bar { \Phi } _ { \cdot 1 } ^ { a , b } ( i ) , \bar { \Phi } _ { \cdot 0 } ^ { a , b } ( i ) ) ) \ ; , \end{align*}


\begin{align*} [ K ( P _ e ) : K ] = \frac { [ K ( A [ \ell ^ e ] ) : K ] } { [ K ( A [ \ell ^ e ] ) : K ( P _ e ) ] } \asymp _ { A , \ell } ( \ell ^ e ) ^ { \dim G _ { A , \ell } - \dim ( ( G _ { A , \ell } ) _ W ) } = ( \ell ^ e ) ^ \delta . \end{align*}


\begin{align*} S _ { k j } \cap B _ r ( x _ 0 ) = \ ( S _ { k j } \cap B _ r ( x _ 0 ) \cap \Gamma \ ) \cup \ ( S _ { k j } \cap B _ r ( x _ 0 ) \cap \Gamma ^ c \ ) = S _ { k j } \cap B _ r ( x _ 0 ) \cap \Gamma ^ c , \end{align*}


\begin{align*} 0 = b _ 0 ( X , \omega ) < b _ 1 ( X , \omega ) \le b _ 2 ( X , \omega ) \le \cdots \le + \infty . \end{align*}


\begin{align*} | \triangle _ k ^ j m ( k ) | \leq C _ N k ^ { - j } \ \ j = 0 , 1 , \dots , N , \end{align*}


\begin{align*} \gamma _ G : = \max _ { \substack { 0 \neq W \subseteq V } } \frac { \dim W } { \dim G - \dim G _ W } , \end{align*}


\begin{align*} q _ { \rho _ { N } } \left ( x , t \right ) & = q \left ( x , t \right ) \\ & + 2 \left [ \int \psi _ { \rho _ { N } } \left ( x , t ; \mathrm { i } s \right ) \psi \left ( x , t ; \mathrm { i } s \right ) \mathrm { d } \rho _ { N } \left ( s \right ) \right ] ^ { 2 } \\ & + 4 \int \psi _ { \rho _ { N } } \left ( x , t ; \mathrm { i } s \right ) \psi ^ { \prime } \left ( x , t ; \mathrm { i } s \right ) \mathrm { d } \rho _ { N } \left ( s \right ) , \end{align*}


\begin{align*} \phi ( X _ \theta ) = - ( \sqrt - 1 ) ^ 2 X _ { - \theta } = X _ { - \theta } . \end{align*}


\begin{align*} \langle { A _ \varepsilon ( \varepsilon ) } \rangle + \langle { B _ \varepsilon ( \varepsilon ) } \rangle = \langle { C _ \varepsilon ( \varepsilon ) } \rangle + \langle { D _ \varepsilon ( \varepsilon ) } \rangle . \end{align*}


\begin{align*} ( X _ I ) ^ S = \{ w _ K \mid K \subset I \} . \end{align*}


\begin{align*} \int _ { \mathbb { T } ^ { d } } { \rm i n t } \left [ \overline { \mathbf { P } } ^ { x } \right ] d x = 1 . \end{align*}


\begin{align*} R ( \pmb { \omega } ) = \begin{bmatrix} 0 & - \omega ^ { 3 } & \omega ^ { 2 } \\ \omega ^ { 3 } & 0 & - \omega ^ { 1 } \\ - \omega ^ { 2 } & \omega ^ { 1 } & 0 \end{bmatrix} \end{align*}


\begin{align*} \mathring { R } _ { q + 1 } ^ u : = \mathring { R } _ { l i n } ^ u + \mathring { R } _ { o s c } ^ u + \mathring { R } _ { c o r } ^ u , \end{align*}


\begin{align*} \widehat { \mathbf { M } } _ { d - 2 , 2 - d } : = \mathrm { a d d } \left \{ ( X _ 0 \oplus X _ 1 \oplus \cdots \oplus X _ { d - 1 } ) \langle i \rangle [ j ] \mid i , j \in \mathbb { Z } \right \} \end{align*}


\begin{align*} d ^ 1 _ { r , s } ( n , j ) = 0 ( n , j ) \ne ( - r , - s ) . \end{align*}


\begin{align*} \begin{cases} x _ { k + 1 } = & A _ 1 x _ k + A _ 2 \mathcal { E } x _ k + B u _ k ^ F \\ & + ( C _ 1 x _ k + C _ 2 \mathcal { E } x _ k + D u _ k ^ F ) w _ k , \end{cases} \end{align*}


\begin{align*} b ( x ) = \begin{cases} 0 & x \le 0 x \ge e , \\ x \log \displaystyle { \frac { e } { x } } \ , & 0 < x < e . \end{cases} \end{align*}


\begin{align*} \vert g ( 0 ) - g ( s _ b ) \vert / \vert s _ b \vert \le \left \vert \sum ^ { + \infty } _ { n = 1 } \frac { d ^ n g } { d s ^ n } ( 0 ) s ^ { n - 1 } _ b / n ! \right \vert \le C _ { R _ 0 } \sum _ { n = 1 } ^ { + \infty } \left ( \sigma ^ { ( 1 ) } _ { \theta _ { \hat { x } } } \right ) ^ { - n } \vert s _ b \vert ^ { n - 1 } \le C _ { R _ 0 } . \end{align*}


\begin{align*} \bar J : = ( j , - \sigma ) \ , . \end{align*}


\begin{align*} \mathcal { F } _ { ( w ) } ( B ; \ell ) = \prod _ { p > 2 \atop p \nmid \ell } \Big ( 1 + O ( \Big ( \frac { 1 } { p ^ { 1 + 2 \delta } } \Big ) \Big ) \prod _ { p | \ell } \frac { p ^ w } { p ^ w + 1 } \ll 1 , \end{align*}


\begin{align*} t - s \in P _ U ( x ) = \{ t \in G \ , : \ , ( \alpha ( t , x ) , x ) \in U \} \ , . \end{align*}


\begin{align*} \mu ^ { \zeta } _ x = \frac { \mu ^ s _ x | _ { \zeta ( x ) } } { \mu ^ s _ x ( \zeta ( x ) ) } = \frac { \mu ^ { \eta ^ s } _ x | _ { \zeta ( x ) } } { \mu ^ { \eta ^ s } _ x ( \zeta ( x ) ) } , \end{align*}


\begin{align*} u _ 1 ( x , 0 ) = \varphi _ 1 ( x ) , u _ 2 ( x , 0 ) = \varphi _ 2 ( x ) . \end{align*}


\begin{align*} ( x , t ) = \left ( \bar x _ n + r _ n y , \bar t _ n + \frac { r _ n ^ \gamma } { M _ n ^ { \gamma - 1 } } s \right ) , ( x , t ' ) = \left ( \bar x _ n + r _ n y , \bar t _ n + \frac { r _ n ^ \gamma } { M _ n ^ { \gamma - 1 } } s ' \right ) \end{align*}


\begin{align*} \mu \left ( A ^ { - T } ( \delta k / \sqrt { d } ) \langle \delta \cdot B _ 1 ( \ell / \sqrt { d } ) \rangle \right ) = \left | \det \left ( A ^ { - T } ( \delta \cdot k / \sqrt { d } ) \right ) \right | \cdot \mu \left ( \delta \cdot B _ 1 ( \ell / \sqrt { d } ) \right ) = \frac { \mu ( B _ 1 ( 0 ) ) \cdot \delta ^ { d } } { w ( \delta \cdot k / \sqrt { d } ) } . \end{align*}


\begin{align*} F _ { N \overline { k } } = \sum _ { g = 0 } ^ k N ^ { 2 - 2 g } F _ { N g } . \end{align*}


\begin{align*} X ^ { i } _ { \bar { g } } ( h ) = ( \bar { g } + h ) ^ { i j } ( \bar { g } + h ) ^ { p q } \left ( - \nabla ^ { \bar { g } } _ { p } h _ { q j } + \frac { 1 } { 2 } \nabla ^ { \bar { g } } _ { j } h _ { p q } \right ) , \end{align*}


\begin{align*} \left \{ \begin{array} { r l } ( \varphi _ s - \frac { 2 } { r } \phi _ r ) \phi _ { z z } & = ( \varphi _ z - 2 \phi _ { x ^ 0 } ) \phi _ { s z } , \\ ( \varphi _ z - 2 \phi _ { x ^ 0 } ) \Omega & = 2 ( r ^ 2 - s ^ 2 ) \Lambda V . \end{array} \right . \end{align*}


\begin{align*} S O _ 3 ( G ) & = \sum _ { u v \in E ( G ) } \sqrt { 2 } \frac { d _ u ^ 2 + d _ v ^ 2 } { d _ u + d _ v } \pi \\ & \leq \sqrt { 2 } \pi \left ( \sum _ { u v \in E ( G ) } \frac { | d _ u ^ 2 - d _ v ^ 2 | + 2 \Delta ^ 2 } { 2 \delta } \right ) \\ & \leq \sqrt { 2 } \pi \left ( \frac { 1 } { \delta } \left ( \frac { 1 } { 2 } \sum _ { u v \in E ( G ) } | d _ u ^ 2 - d _ v ^ 2 | \right ) + \frac { m \Delta ^ 2 } { \delta } \right ) \\ & = \sqrt { 2 } \pi \left ( \frac { S O _ 1 ( G ) + m \Delta ^ 2 } { \delta } \right ) . \end{align*}


\begin{align*} \mu : = \liminf _ { \gamma \to \infty } \ , ( \gamma ^ + - \gamma ) \ , \frac { \log \gamma } { 2 \pi } . \end{align*}


\begin{align*} [ \varphi ( x ) , [ y , z ] ] + [ \varphi ( y ) , [ z , y ] ] + [ \varphi ( z ) , [ x , y ] ] = 0 . \end{align*}


\begin{align*} \int \varphi ( x ) \ , d \mu ( x ) & = \int \varphi \circ f ^ n ( x ) \ , d \mu ( x ) = \int \left ( \int _ { \xi ( x ) } \varphi \circ f ^ n ( y ) \ , d \mu _ { x } ( y ) \right ) d \mu ( x ) \\ & = \int \left ( \int _ { \xi ( f ^ { - n } ( x ) ) } \varphi \circ f ^ n ( y ) \ , d \mu _ { f ^ { - n } ( x ) } ( y ) \right ) d \mu ( x ) \\ & = \int \left ( \int _ { \xi _ n ( x ) } \varphi ( y ) \ , d \big ( f ^ n _ { \ast } \mu _ { f ^ { - n } ( x ) } \big ) ( y ) \right ) d \mu ( x ) . \end{align*}


\begin{align*} \sum _ { 1 \leq j _ 1 < j _ 2 < \cdots < j _ i \leq n } \# \Big ( U _ { j _ 1 } \cap U _ { j _ 2 } \cap \cdots \cap U _ { j _ i } \Big ) = \binom { n - i k } { i } \ , 2 ^ { n - i ( k + 1 ) } \ , . \end{align*}


\begin{align*} \psi _ p \left ( \begin{pmatrix} 1 & u _ 1 & * & \cdots & * \\ & 1 & u _ 2 & \cdots & * \\ & & \cdots & \cdots & \cdots \\ & & & 1 & u _ { n - 1 } \\ & & & & 1 \end{pmatrix} \right ) = \xi ( u _ 1 + u _ 2 + u _ 3 + \cdots + u _ { n - 1 } ) . \end{align*}


\begin{align*} O _ \sigma \ ; : = \ ; U _ \sigma \smallsetminus \bigcup \bigl \{ U _ { \sigma ' } \colon \sigma ' \sqsubset \sigma , \ ; \sigma ' \neq \sigma \bigr \} \ ; \xhookrightarrow { \quad \textrm { c l o s e d } \quad } \ ; U _ \sigma \end{align*}


\begin{align*} f ^ * ( \lambda _ j ) = \langle f , h _ j \rangle , ~ ~ f \in \mathcal H _ { \pmb \mu } , \ , \ , \ , j = 1 , \ldots , n . \end{align*}


\begin{align*} R ( S ) : = U _ { S / R } ^ { - 1 } S . \end{align*}


\begin{align*} P _ k f ( x ) = \int _ { \mathbb R ^ d } \sum _ { | \mu | = k } \Phi _ \mu ( x ) \Phi _ \mu ( x ' ) f ( x ' ) \dd x ' . \end{align*}


\begin{align*} \dfrac { x - x _ 0 } { | x - x _ 0 | } = \cos \delta \ , \dfrac { e } { | e | } + \sin \delta \ , \dfrac { z } { | z | } . \end{align*}


\begin{align*} \begin{gathered} \Phi ^ \delta : ( a \sqcup b ) \times c \rightarrow ( a \times c ) \sqcup ( b \times c ) \ ; , \\ \Phi ^ \delta ( ( \chi , \beta ) , \gamma ) ) = ( \chi , ( \beta , \gamma ) ) \ ; . \end{gathered} \end{align*}


\begin{align*} \dot { f } _ t ( z ) = - f _ t ' ( z ) G ( z , t ) \end{align*}


\begin{align*} = \hat \mu _ 1 ( r _ 1 - r _ 2 , h _ 1 - h _ 2 ) \hat \mu _ 2 ( r _ 1 - a r _ 2 , h _ 1 - \tilde \alpha _ { G } h _ 2 ) , \ \ r _ j \in \mathbb { Q } , \ \ h _ j \in H . \end{align*}


\begin{align*} \partial _ D M _ t = \partial U _ t \subset M _ t \end{align*}


\begin{align*} ( \frac { \partial } { \partial s ^ { \alpha } } ) ^ { * } = \frac { \partial } { \partial s ^ { \alpha } } - D _ { r } ^ { - 1 } \big ( \sum _ { j = 1 } ^ { N - 1 } \pmb { r } _ { j } . \frac { d \pmb { r } _ { j } } { d s ^ { \alpha } } \big ) \frac { \partial } { \partial \lambda } \end{align*}


\begin{align*} & \frac { X ( x + 1 , t ) - X ( x - 1 , t ) } { X ( x + 1 , t ) + X ( x - 1 , t ) } \\ & = \sum _ { z \in \Z } \sum _ { t - N ^ { \epsilon } \le s \le t } \Delta ( x - z , t - s ) \xi ( z , s ) \frac { \Gamma ( z , s ) } { 2 \Gamma ( x , t + 1 ) } + O ( N ^ { - 1 / 4 - \epsilon / 1 6 } ) . \end{align*}


\begin{align*} \tilde D _ \Theta = \{ x \in V ^ 1 \cap V _ \Theta : T ^ \# _ \Theta x \in ( j _ \Theta ^ * \circ c ) V _ \Theta \} = \{ x \in V ^ 1 \cap V _ \Theta : T ^ \# _ \Theta x \in ( i ^ * _ \Theta \circ j ^ * \circ c ) V \} . \end{align*}


\begin{align*} Y ^ { q } = f ( X ) \end{align*}


\begin{align*} \forall t \in [ 0 , T ] , \ ; \ ; \ ; \ ; X _ t = \Gamma \left ( x _ 0 + \int _ 0 ^ { \cdot } f ( s , X _ s ) d s + W ^ H \right ) ( t ) , \end{align*}


\begin{align*} A = \begin{pmatrix} - m _ - & - n _ - \\ - m _ + & - n _ + \end{pmatrix} \end{align*}


\begin{align*} D ( - \Delta _ L ) = \{ f \in L ^ 2 ( \Lambda _ L ) : \nu \hat { f } \in \ell ^ 2 ( \Lambda _ L ^ * ) \} , \end{align*}


\begin{align*} x = ( t , h ) \implies \tilde { x } = ( [ [ t m ] ] , h ) , \end{align*}


\begin{align*} \overline { H } _ 5 = z ^ { m _ l } H ^ * _ 5 \overline { H } _ 6 = z ^ { m _ l } H ^ * _ 6 , \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Phi _ { t } = X _ { \bar { g } } ( g _ { t } ) \circ \Phi _ { t } , ~ ~ ~ ~ \Phi _ { 0 } = \mathrm { i d } _ { M } . \end{align*}


\begin{align*} \int _ { E } \phi \dd \left ( P ^ * _ t ( \alpha \mu + ( 1 - \alpha ) \nu ) \right ) = \alpha \int _ { E } \phi \dd \left ( P ^ * _ t \mu \right ) + ( 1 - \alpha ) \int _ { E } \phi \dd \left ( P ^ * _ t \nu \right ) = \alpha \int _ { E } \phi \dd \mu + ( 1 - \alpha ) \int _ { E } \phi \dd \nu \ , , \end{align*}


\begin{align*} \det ( g _ { A B } ) = \phi ^ { n + 2 } \Omega ^ { n - 2 } \Lambda . \end{align*}


\begin{align*} \mathsf M ( v ) ( \zeta , z , y ) = \xi _ { \Delta ^ \tau _ { g _ s } } ^ { - 1 } ( \zeta ) \mathsf M ( w ) ( \zeta , z , y ) , \end{align*}


\begin{align*} \begin{aligned} \int _ { r _ 0 } ^ r z _ { \hat \varphi } ( t ) d t \leq \beta ( C _ 2 + 2 C _ 3 ) + \sqrt { 2 C _ 1 ( n - 1 ) r } + \beta A r ^ 2 + \sqrt { ( n - 1 ) \rho A } r ^ { 2 } , \end{aligned} \end{align*}


\begin{align*} A _ M = B _ M \ ; , \end{align*}


\begin{align*} \sum ^ l _ { i = 1 } \| E _ A ( x ' _ i u _ g y ' _ i ) \| ^ 2 _ 2 \geq C ' , g \in \Gamma . \end{align*}


\begin{align*} \xi _ { \Delta ^ \tau _ { g _ s } } ( \zeta ) = \zeta ^ 2 + ( n - a ) \zeta + \frac { n ( n - 2 a ) } { 4 } + \Delta _ { g _ F } \end{align*}


\begin{align*} H ( J ^ { \{ [ [ e _ { \theta } , \phi ( v ) ] , v ] ^ { \natural } \} } _ { 0 } v _ { \nu , \ell _ 0 } , v _ { \nu , \ell _ 0 } ) = - \sum _ i ( e _ { \theta } | [ \phi ( v ) , v ] ) \xi ( u _ i ) \nu ( u ^ i ) = - \langle \phi ( v ) , v \rangle ( \xi | \nu ) . \end{align*}


\begin{align*} r ( u ) \le | r ( u ) | = \left | \int _ { \mathbb R } \widehat r ( \alpha ) \ , e ( \alpha u ) \ , d \alpha \right | \le \int _ { \mathbb R } | \widehat r ( \alpha ) | \ , d \alpha = \int _ { \mathbb R } \widehat r ( \alpha ) \ , d \alpha = r ( 0 ) = 1 \ , . \end{align*}


\begin{gather*} 4 \gamma _ { n } \left ( \phi + l _ { n - 1 } \right ) \left ( \phi + l _ { n } \right ) - \left ( n - 2 \gamma _ { n } \right ) ^ { 2 } x ^ { 2 } = 4 \gamma _ { n } \left ( \phi + l _ { n } + l _ { n - 1 } - \frac { l _ { n } l _ { n - 1 } } { z ^ { 2 } } \right ) \phi \\ = \left [ 4 \gamma _ { n } \left ( C _ { n } + l _ { n - 1 } \right ) - \left ( n - 2 \gamma _ { n } \right ) ^ { 2 } \right ] \phi . \end{gather*}


\begin{align*} P _ { n } \left ( x ; z \right ) = x ^ { n } + { \displaystyle \sum \limits _ { k = 1 } ^ { \infty } } \alpha _ { n , k } \left ( x \right ) z ^ { 2 k } , \end{align*}


\begin{align*} ( \pi _ 1 ) _ * ( \widetilde \mu ) = \mu . \end{align*}


\begin{align*} v ( y , s ) = \frac { u ( x , t ) - H _ { m _ 0 } ( x ) } { r ^ { 2 + \alpha } } . \end{align*}


\begin{align*} \| c \| _ { \ell ^ { p , q } _ { \widetilde { \kappa } } ( J ) } : = \left \| k \mapsto \| \widetilde { \kappa } ( \bullet , k ) \ , c _ { \bullet , k } \| _ { \ell ^ { p } ( J _ 1 ) } \right \| _ { \ell ^ { q } ( J _ 2 ) } < \infty . \end{align*}


\begin{align*} \aleph _ { \alpha , L } = \frac { \alpha L } { 1 2 } \| x u _ 1 \| ^ 2 + O ( \alpha ^ 2 L ) = \alpha L + O ( \alpha ^ 2 L ) , \end{align*}


\begin{align*} \beta _ { n } = \frac { \mathfrak { L } \left [ x P _ { n } ^ { 2 } \right ] } { h _ { n } } , \gamma _ { n } = \frac { \mathfrak { L } \left [ x P _ { n } P _ { n - 1 } \right ] } { h _ { n - 1 } } , n \geq 1 , \end{align*}


\begin{align*} M = \left ( \begin{matrix} A \ & B \\ C \ & D \end{matrix} \right ) \end{align*}


\begin{align*} \int _ 1 ^ \infty g ( t ) d t = \chi ^ { - \beta } \int _ 1 ^ \infty \chi ^ \beta g ( t ) d t \sim \chi ^ { - \beta } \int _ 1 ^ \infty t ^ { - ( \beta - \gamma ) } d t = \chi ^ { - \beta } \frac { 1 } { \beta - \gamma - 1 } . \end{align*}


\begin{align*} A ^ + _ M = A _ M - \mathbb { 0 } \oplus \begin{pmatrix} u _ { 0 0 } \mathbb { 1 } & u _ { 0 1 } \mathbb { 1 } \\ u _ { 1 0 } \mathbb { 1 } & u _ { 1 1 } \mathbb { 1 } \end{pmatrix} \ ; . \end{align*}


\begin{align*} \eta \cdot f = \chi _ f ( \eta ) \cdot f . \end{align*}


\begin{align*} \int _ 0 ^ \infty & \int _ { \R ^ 2 } G ( | e ^ { i t \Delta } u _ 0 | ^ 2 ) \ , d x \ , d t = \int _ 0 ^ \infty G ' ( \lambda ) \ , \bigl | \{ ( t , x ) \in ( 0 , \infty ) \times \R ^ 2 : | e ^ { i t \Delta } u _ 0 | ^ 2 > \lambda \} \bigr | \ , d \lambda . \end{align*}


\begin{align*} D _ { \lambda , n } \Big ( \frac { 1 } { 1 - z \overline { w } } \Big ) & = D _ { \sigma , n - 1 } \Big ( \frac { \frac { 1 } { 1 - z \overline { w } } - \frac { 1 } { 1 - \lambda \overline { w } } } { z - \lambda } \Big ) \\ & = D _ { \sigma , n - 1 } \Big ( \frac { \overline { w } } { ( 1 - z \overline { w } ) ( 1 - \lambda \overline { w } ) } \Big ) \\ & = \frac { | w | ^ 2 } { | 1 - \lambda \overline { w } | ^ 2 } D _ { \sigma , n - 1 } \Big ( \frac { 1 } { 1 - z \overline { w } } \Big ) . \end{align*}


\begin{align*} - \pi < \Delta ( k ) < \pi , \xi < 0 , \ | \xi | = O ( 1 ) \end{align*}


\begin{align*} m ( F ) = F _ 0 \ ; . \end{align*}


\begin{align*} | X ( n , 2 7 ) | = \frac 1 3 \left [ 1 + 2 \cdot \frac { ( \tau ( n _ q ) | 3 ) } { \tau ( n _ q ) } \right ] \cdot | X ( n , 3 ) | , \end{align*}


\begin{align*} I _ \alpha ( n ) : = [ n _ \alpha , n - n _ \alpha ] ^ 2 \cap \{ ( i , j ) : j - i \ge n _ { \alpha } \} \ , , \end{align*}


\begin{align*} J ( w - \bar { w } ) = - \partial _ \tau ( w - \bar { w } ) - \eta \partial _ \xi ( w - \bar { w } ) + \big ( ( w + \bar { w } ) \partial _ { \eta } ^ 2 \bar { w } \big ) ( w - \bar { w } ) + w ^ 2 \partial _ { \eta } ^ 2 ( w - \bar { w } ) . \end{align*}


\begin{align*} Z _ { m , n } : = \sum _ { x _ \cdot \in \Pi _ { m , n } } \prod _ { i = 1 } ^ { m + n } \omega _ { e _ i ( x _ \cdot ) } , \end{align*}


\begin{align*} E _ { \overline { k } } ^ { - 1 } I - 1 = O ( \hbar ^ { 2 k } ) , \end{align*}


\begin{align*} \begin{bmatrix} \Phi _ k \\ U _ k \end{bmatrix} d & = \begin{bmatrix} M & 0 \\ 0 & T \otimes I _ m \end{bmatrix} \begin{bmatrix} u _ { [ k - n , k ] } \\ u _ { [ k - n + 1 , k + L - 1 ] } \end{bmatrix} \\ & = Z u _ { [ k - n , k + L - 1 ] } . \end{align*}


\begin{align*} \mathcal { L } ( x ) = \left ( \begin{array} { c c } x + \frac { 1 } { 2 } + S _ 3 & S _ - \\ S _ + & x + \frac { 1 } { 2 } - S _ 3 \end{array} \right ) \ , , \end{align*}


\begin{align*} \xi _ j ( x ) = \frac { 1 } { \cosh x } C ^ { ( \frac { m + 1 } { 2 } ) } _ { j - 1 } ( \tanh x ) , \hat { \lambda } _ j = - 2 m + p + j ( j + m - 1 ) , \end{align*}


\begin{align*} \frac { \partial v _ \epsilon } { \partial t } \ , - \ , \triangle v _ \epsilon \ , + \ , \frac 2 { \epsilon ^ { 2 ( 1 - \kappa ) } } ( v _ \epsilon ^ 2 \ , - \ , 1 ) v _ \epsilon \ , - \ , \frac { \dot { \kappa } } { K ( \epsilon ) } ( v _ \epsilon ^ 2 \ , - \ , 1 ) \log \frac { 1 + v _ \epsilon } { 1 - v _ \epsilon } \ ; = \ ; 0 \end{align*}


\begin{align*} u _ { t } ( x , t ) + 6 \sigma u ( x , t ) u ( - x , - t ) u _ { x } ( x , t ) + u _ { x x x } ( x , t ) = 0 , \end{align*}


\begin{align*} D \cup E ( a _ 0 , \dots , a _ { d + e + 1 } ) = D ( a _ 0 , \dots , a _ d ) E ( a _ { d + 1 } , \dots , a _ { d + e + 1 } ) , \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes \Phi _ y ^ { ( 0 ) } \ , | \ , H _ y ( u _ { \alpha } \otimes R _ * ) \rangle = 0 , \end{align*}


\begin{align*} \sum _ i \left ( \sum _ j T _ { j i j i } \right ) = \sum _ { i , j } T _ { j i j i } = \sum _ j \left ( \sum _ i T _ { j i j i } \right ) \ ; . \end{align*}


\begin{align*} \left | z \right | \leq \sum _ { j = 1 } ^ { d } \left | z _ { j } \right | \leq d \cdot \max \bigl \{ \left | z _ { j } \right | \colon j \in \underline { d } \bigr \} < 2 d \cdot \max \bigl \{ \left | z _ { j } \right | \colon j \in \underline { d } \bigr \} . \end{align*}


\begin{align*} \left | \int _ { \Lambda } f ( x ) d x - \sum _ { j = 1 } ^ N f ( \xi _ j ) | I _ j | \right | \leq \tau \int _ { \Lambda } g ( x ) d x \end{align*}


\begin{align*} | e ^ { - i \alpha } \lambda - E - i \eta | & = | \lambda - e ^ { i \alpha } ( E + i \eta ) | \geq | \sin ( \alpha ) E + \cos ( \alpha ) \eta | . \end{align*}


\begin{align*} \partial _ { z } L \left [ \phi p \right ] = L \left [ \partial _ { z } \left ( \phi p \right ) \right ] , \end{align*}


\begin{align*} \mathbf { P r } \left ( \bigg \| \frac { d } { n \epsilon } \sum _ { i = 1 } ^ { n } J ( u _ 0 + \epsilon v _ { 0 , i } ) v _ { 0 , i } - \nabla J ( u _ 0 ) \bigg \| \geq t - \frac { L d \epsilon } { 2 } \right ) \leq 2 \exp \left ( - \frac { c n t ^ 2 } { K ^ 2 } \right ) \end{align*}


\begin{align*} \bar { \rho } : = \min ( \rho , \mu ) , \delta _ T = \Big ( \int _ { Q _ T } \mu | \nabla p + V | ^ 2 \Big ) ^ { 1 / 2 } . \end{align*}


\begin{align*} p : = \frac { 2 - 8 \varepsilon } { 2 - 9 \varepsilon } \in ( 1 , 2 ) , \end{align*}


\begin{align*} \Delta { X } = \left ( \Delta { x } , \Delta { y } , \Delta { t } \right ) \equiv \vect { 0 } , \end{align*}


\begin{align*} t _ { e ^ \ast } : = q / t _ e \ , . \end{align*}


\begin{align*} ( z - q _ 1 w ) ( z - q _ 2 w ) ( z - q _ 3 w ) \psi ^ \epsilon ( z ) e ( w ) = ( q _ 1 z - w ) ( q _ 2 z - w ) ( q _ 3 z - w ) e ( w ) \psi ^ \epsilon ( z ) \ , , \end{align*}


\begin{align*} \hat { V } _ n ( i ) : = \sum _ { \ell \in I _ n } G ( S _ i , S _ \ell ) \ , . \end{align*}


\begin{align*} V ^ o = \sum _ { \mathbf k \in 2 \pi \mathbb T ^ 2 \setminus \lbrace ( 0 , 0 ) \rbrace } \bigl ( \hat { V } _ { \mathbf k } ^ + V _ { \mathbf k } ^ + + \hat { V } _ { \mathbf k } ^ - V _ { \mathbf k } ^ - \bigr ) . \end{align*}


\begin{align*} B ( \C _ { e _ { \alpha } q ^ { i } } ) = \begin{cases} \C _ { e _ { \alpha } q ^ { i - 1 } } & i > 1 \\ 0 & i = 1 \\ \end{cases} , w ( \C _ { e _ { \alpha } q ^ { i } } ) = \begin{cases} 0 & i > 1 \\ \C _ { e _ { \alpha } } & i = 1 \\ \end{cases} , \end{align*}


\begin{align*} \operatorname { r a n k } M _ { k - d _ { K } } [ y ] \ , = \ , \operatorname { r a n k } M _ { k } [ y ] . \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes 2 \alpha ^ { 3 / 2 } \eta \tilde { \Phi } _ * ^ 1 \ , | \ , \alpha V _ y \Phi ^ y _ { \# } \rangle \geq - C | \eta | ^ 2 \alpha ^ 4 \| \tilde { \Phi } _ * ^ 1 \| ^ 2 - C \alpha | \kappa | ^ 2 \| V _ y \Phi _ { \# } ^ y \| ^ 2 = O ( \alpha ^ 4 ) . \end{align*}


\begin{align*} Z _ A ( s ) = \prod _ { \alpha \in A } \zeta ( s + \alpha ) = \sum _ { n = 1 } ^ \infty \frac { \tau _ A ( n ) } { n ^ s } . \end{align*}


\begin{align*} P ( t ) U ( t , s ) & = P ( t ) U ( t , s + j T ) U ( s + j T , s ) = U ( t , s + j T ) U ( s + T , s ) ^ j P ^ j ( s ) = U ( t , s ) P ( s ) , \end{align*}


\begin{align*} ( \det ^ { \mathrm { p l } } ) ^ * \circ \Xi = \Xi \circ \det ^ * , \end{align*}


\begin{align*} G = P _ U ( x ) + F \ , . \end{align*}


\begin{align*} e _ j \mapsto \begin{cases} \ell _ i \otimes e _ i + e _ i \otimes \ell _ i , & \mathrm { i f } \ , i = j ; \\ j \vert i \otimes i \vert j , & \mathrm { i f } \ , i \ne j . \end{cases} \end{align*}


\begin{align*} \mathcal { A } _ { F T 2 } = \begin{pmatrix} I _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } \\ 0 _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } & I _ { d \times d } \\ 0 _ { d \times d } & 0 _ { d \times d } & I _ { d \times d } & 0 _ { d \times d } \\ 0 _ { d \times d } & - I _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } \end{pmatrix} . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ m S _ { \Lambda _ i } ( C _ i ) = S _ \Lambda ( C ) , \end{align*}


\begin{align*} \sum _ { k = 1 } ^ n \sum _ { \sigma \in S h ( k , n - k ) } \epsilon ( \sigma ) Q _ { n - k + 1 } ^ 1 ( Q _ k ^ 1 ( x _ { \sigma ( 1 ) } \vee \cdots \vee x _ { \sigma ( k ) } ) \vee x _ { \sigma ( k + 1 ) } \vee \dots \vee x _ { \sigma ( n ) } ) = 0 . \end{align*}


\begin{align*} G : = \sqrt { I _ { c m } } = \sqrt { \sum _ { i = 1 } ^ { N } m _ { i } \mid \pmb { x } _ { i } - \pmb { x } _ { c m } \mid ^ { 2 } } \end{align*}


\begin{align*} = | | K \varphi | | _ { 2 } ^ { 2 } + \Re \left \langle K \varphi , | x | ^ { 2 } \varphi \right \rangle + \frac { 1 } { 4 } \left \Vert | x | ^ { 2 } \varphi \right \Vert _ { 2 } ^ { 2 } . \end{align*}


\begin{align*} R \cdot S = U ( R \otimes S ) \ ; , \end{align*}


\begin{align*} Y _ { m _ 1 , m _ 2 } ( A ) : = \frac { 1 } { 2 } \sum _ { \theta \in \{ 0 , 1 \} ^ 2 } \prod _ { j \in \mathbb { T } _ { m _ 1 , m _ 2 } } \left | \alpha + \beta \exp \left \{ 2 \pi i \frac { j _ 1 + \theta _ 1 / 2 } { m _ 1 } \right \} + \gamma \exp \left \{ 2 \pi i \frac { j _ 2 + \theta _ 2 / 2 } { m _ 2 } \right \} \right | . \end{align*}


\begin{align*} \widetilde a ( n - 1 ) u _ r ( n - 1 ) + b ( n ) u _ r ( n ) + \widetilde a ( n ) u _ r ( n + 1 ) = E u _ r ( n ) , n _ r + 1 \leq n \leq n _ { r + 1 } \end{align*}


\begin{align*} \Omega _ { x ^ 0 } - \phi _ { s z } & = 0 , \\ \Omega _ r - r \phi _ { s s } & = 0 , \end{align*}


\begin{align*} h : = \sum _ { j } f _ { B _ j } \phi _ j . \end{align*}


\begin{align*} \Phi = \varphi _ 1 \land y \leq z \ , \& \dots \& \ , \varphi _ m \land y \leq z \Longrightarrow y \leq z . \end{align*}


\begin{align*} p ^ { ( \alpha ) } _ t ( x ) = p _ t * \mu ^ \sigma _ t ( x ) = \int _ { \R ^ d } p _ t ( x - y ) \mu ^ \sigma _ t ( \d { y } ) , x \in \R ^ d , \ t > 0 , \end{align*}


\begin{align*} T _ k ( u _ k ) ( x ) = \begin{cases} \overline { u } _ k ( x ) & u _ k ( x ) > \overline { u } _ k ( x ) , \\ u _ k ( x ) & \underline { u } _ k ( x ) \leq u _ k ( x ) \leq \overline { u } _ k ( x ) , \\ \underline { u } _ k ( x ) & u _ k ( x ) < \underline { u } _ k ( x ) . \end{cases} \end{align*}


\begin{align*} \frac { d x } { d \tau } = D F ( x , t ) , \end{align*}


\begin{align*} \mathrm { W C E } ( n , \mathbb { M } ^ 2 _ { \mathrm { e } ^ 2 } ) & : = \sup _ { \substack { f \in \mathbb { M } ^ s _ { \mathrm { e } ^ 2 } , \| f \| _ { \mathbb { M } ^ s _ { \mathrm { e } ^ 2 } } \leq 1 } } \left | \int \limits _ { \ , - \infty } ^ \infty f ( x ) W ( x ) \mathrm { d } x - \sum _ { x \in X _ n } \omega ( x ) f ( x ) \right | ^ 2 \\ & = \sum _ { x , y \in X _ n } \omega ( x ) \omega ( y ) K _ { t } ( x , y ) - \frac { 1 } { \sqrt { 2 } \ , t } . \end{align*}


\begin{align*} K _ n ( x _ 1 \vee \cdots \vee x _ n ) = \frac { 1 } { n ! } \sum _ { i = 0 } ^ { n - 1 } \sum _ { \sigma \in S _ n } \frac { \epsilon ( \sigma ) } { n - i } i p X _ { \sigma ( 1 ) } \vee \cdots \vee i p X _ { \sigma ( i ) } \vee X _ { \sigma ( i + 1 ) } \vee X _ { \sigma ( n ) } . \end{align*}


\begin{align*} \Phi = \varphi _ 1 \land y \leq z \ , \& \dots \& \ , \varphi _ n \land y \leq z \Longrightarrow y \leq z , \end{align*}


\begin{align*} z = 4 C _ 2 , \tau = 1 \end{align*}


\begin{align*} S \left ( t ; z \right ) = { \displaystyle \sum \limits _ { n \geq 0 } } \frac { u _ { n } \left ( z \right ) } { t ^ { 2 n + 1 } } , \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p w & = M ( x ) \ ; \ ; & & \mbox { i n } \ ; \ ; \R ^ N , \\ w & > 0 & & \mbox { i n } \ ; \ ; \R ^ N , \\ w ( x ) & \to 0 & & \mbox { a s } \ ; | x | \to \infty \end{alignedat} \right . \end{align*}


\begin{align*} \begin{array} { l } \sum _ { k = 1 } ^ { n } a _ { \epsilon , \Omega _ k } ( x ) = a _ { \epsilon , \Omega } ( x ) . \end{array} \end{align*}


\begin{align*} \forall \varphi \in \mathcal D , \langle \sqrt { \omega } \mathcal C ( f ) , \varphi \rangle = \int _ { 0 } ^ \infty J _ M ( f ) ( \omega ) \ , \varphi ' ( \omega ) \ , \dd \omega , \\ \forall \varphi \in \mathcal D , \langle \omega ^ { 3 / 2 } \mathcal C ( f ) , \varphi \rangle = \int _ { 0 } ^ \infty J _ E ( f ) ( \omega ) \ , \varphi ' ( \omega ) \ , \dd \omega . \end{align*}


\begin{align*} \Psi _ { \theta } ( \mathcal { D } ) = \left \{ \mathrm { I d } \right \} , \end{align*}


\begin{align*} h ' _ t ( z ) : = \frac { d h _ t ( z ) } { d z } = h ' ( z + t ) \frac { 1 + 2 t ^ 2 S h ( z + t ) } { \left ( 1 + t P h ( z + t ) \right ) ^ 2 } . \end{align*}


\begin{align*} \widetilde p ( \widetilde T ^ n \widetilde \omega ) = p ( T ^ n \omega ) \widetilde q ( \widetilde T ^ n \widetilde \omega ) = q ( T ^ n \omega ) . \end{align*}


\begin{align*} 1 \leq & \| \boldsymbol { h } \| _ 2 \leq \frac { N ^ { 1 / d } ( v , w ) } { s \max ( | v | , | w | ) } : = \max \boldsymbol { h } , \\ \frac { N ^ { 1 / d } ( v , w ) } { 2 \pi s \max ( | v | , | w | ) } \leq & \| \boldsymbol { h } \| _ 2 \leq \frac { N ^ { 1 / d } ( v , w ) } { 2 \pi s \min ( | v | , | w | ) } : = \min \boldsymbol { h } , \\ & \| \boldsymbol { h } \| _ 2 \geq \frac { N ^ { 1 / d } ( v , w ) } { 2 \pi s \min ( | v | , | w | ) } . \end{align*}


\begin{align*} R _ { u \wedge v } ( x ) : = ( v \mid x ) u - ( u \mid x ) v \end{align*}


\begin{align*} \left [ Q ^ q w \right ] ( x , t ) = \left [ Q ^ { q + 1 } w \right ] ( x , t ) , \end{align*}


\begin{align*} \int _ { X \setminus A _ i } g \ , d \mathcal L ^ 1 = 0 = \int _ { X \setminus A _ i } g _ { i + 1 } \ , d \mathcal L ^ 1 . \end{align*}


\begin{align*} K = \frac { 1 } { 2 } \sum _ { \alpha = 1 } ^ { N - 1 } \mid \dot { \pmb { r } } _ { \alpha } \mid ^ { 2 } + \frac { \sum _ { i = 1 } ^ { n } m _ { i } } { 2 } \mid \dot { \mathbf { R } } _ { c m } \mid ^ { 2 } \end{align*}


\begin{align*} \eqref { i n v a r i a n t 2 } = B ( x , \left [ y , [ \alpha ( z ) , t ] \right ] ) + B ( \left [ [ y , z ] , t \right ] , \alpha ( x ) ) + B ( x , \left [ z , [ \alpha ( y ) , t ] \right ] ) \end{align*}


\begin{align*} \delta \mathbb { B } _ { s , u , t } ( x ) = B _ { u , t } ( x ) B _ { s , u } ( x ) \ , , s < u < t \in [ 0 , T ] , x \in D \ , . \end{align*}


\begin{align*} x _ 1 : = [ ( e _ 1 \otimes 1 ) \oplus 0 \oplus 0 ] , \ x _ 2 : = [ ( e _ 2 \otimes 1 ) \oplus 0 \oplus 0 ] , \ x _ 3 : = [ 0 \oplus 1 \oplus 0 ] , \ x _ 4 : = [ 0 \oplus 0 \oplus 1 ] \end{align*}


\begin{align*} \prod _ { k \in \mathbb { Z } } \left ( 1 + \left ( \frac { q } { p + k } \right ) ^ 2 \right ) = \left | \frac { \sin ( \pi ( p + q i ) } { \sin ( \pi p ) } \right | ^ 2 . \end{align*}


\begin{align*} S ( \boldsymbol { \ell } ) _ t = S _ 0 + \sum _ { | J | \leq n } & \ell ^ J _ W \langle ( \epsilon _ { J } ; \epsilon _ 0 ) ^ { \thicksim } , \mathbb { X } _ { t } \rangle + \sum _ { | J | \leq n } \ell ^ J _ \nu \langle ( \epsilon _ { J } ; \epsilon _ { 1 } ) ^ { \thicksim } - \lambda ( \epsilon _ { J } ; \epsilon _ { - 1 } ) ^ { \thicksim } , \mathbb { X } _ { t } \rangle . \end{align*}


\begin{align*} T _ \ell ( A ) : = \varprojlim _ { i } A [ \ell ^ i ] , \end{align*}


\begin{align*} \bar { \rho } = \frac { ( n + i - k - l ) - ( k - l ) } { n + i } = \frac { n + i - 2 k } { n + i } \geq 0 \end{align*}


\begin{align*} \underline { \lambda } _ { i j } \left ( \begin{pmatrix} t _ 1 & & & & \\ & t _ 2 & & & \\ & & \cdots & & \\ & & & t _ { n - 1 } & \\ & & & & t _ n \end{pmatrix} \right ) = t _ i t _ j ^ { - 1 } , 1 \leq i , j \leq n , \ i \ne j . \end{align*}


\begin{align*} p = ( 1 - \varepsilon _ p ) a _ p + \varepsilon _ p b _ p , \end{align*}


\begin{align*} \rho _ \eta ^ \varepsilon ( x , t ) = \bar { \eta } ( x , t ) + \sum _ { k = 1 , 2 } \varepsilon ^ k \eta _ k ( \pi ( x , t ) , t , \varepsilon ^ { - 1 } d ( x , t ) ) , ( x , t ) \in \overline { Q _ { \varepsilon , T } } . \end{align*}


\begin{align*} \begin{aligned} \sup _ { r \in [ 0 , t ] } \| \partial _ x u _ r \| ^ 2 _ { L ^ 2 } + 2 \lambda _ 2 \int _ { 0 } ^ { t } \| u _ r \times \partial ^ 2 _ x u _ r \| ^ 2 _ { L ^ 2 } \dd r \leq \| \partial _ x u ^ 0 \| _ { L ^ 2 } ^ 2 \ , , \end{aligned} \end{align*}


\begin{align*} 1 = 1 , a \otimes b = a b \ ; . \end{align*}


\begin{align*} \lim _ { n \to \infty } d ( \phi ^ { n } ( x ) , \phi ^ { n } ( y ) ) = 0 ; \end{align*}


\begin{align*} \begin{cases} y ' ( t ) & = - 2 \mu y ( t ) + \frac { \sigma _ * ^ 2 } { 2 } , t > 0 \\ y ( 0 ) & = \frac { \norm { X _ 0 - x ^ { \star } } ^ 2 } { 2 } . \end{cases} \end{align*}


\begin{align*} \lambda ^ u _ { f ^ { \tau ( \ell ) } ( x ^ u ) } ( k ) = \frac { \lambda ^ u _ { x ^ u } ( - k ) } { \lambda ^ u _ { x ^ u } ( \tau ( \ell ) ) } , \end{align*}


\begin{align*} \norm { \biggl ( \begin{array} { c } \xi _ \varepsilon - g ^ o \\ v _ \varepsilon - v _ p \end{array} \biggr ) - \mathcal L \bigl ( \dfrac t \varepsilon \bigr ) V ^ o } { L ^ \infty ( 0 , T ; H ^ 1 ( \mathbb T ^ 2 \times 2 \mathbb T ) ) } \rightarrow 0 \end{align*}


\begin{align*} c _ 1 ( \overline { M } _ { j , 0 , v } ) ^ { n - 1 } & = \lim _ { \epsilon \to 0 } c _ 1 ( \overline { M } _ { j , \epsilon , v } ) ^ { n - 1 } = \lim _ { \epsilon \to 0 } \frac { \deg _ { M _ { j , \epsilon } } ( H ) c _ 1 ( \overline { M } _ { k , \epsilon , v } ) ^ { n - 1 } } { \deg _ { M _ { k , \epsilon } } ( H ) } = \frac { \deg _ { M _ { j , 0 } } ( H ) } { \deg _ { M _ { k , 0 } } ( H ) } c _ 1 ( \overline { M } _ { k , 0 , v } ) ^ { n - 1 } \end{align*}


\begin{align*} \Bigl | \bar { \tau } _ \varepsilon ^ i \cdot \nabla \rho _ \eta ^ \varepsilon - \overline { \nabla _ \Gamma g _ i } \cdot \overline { \nabla _ \Gamma \eta } \Bigr | \leq c \varepsilon \sum _ { \xi = \eta , \zeta _ 0 , \zeta _ 1 } \left ( | \bar { \xi } | + \Bigl | \overline { \nabla _ \Gamma \xi } \Bigr | \right ) \end{align*}


\begin{align*} \hat { M } ^ 2 = ( M ^ 2 - b ^ 2 ) - q b = M ^ 2 - a b . \end{align*}


\begin{align*} \sum _ { - \lfloor m / 2 \rfloor \leq j \leq - m ^ { 1 / 4 } } b _ j + \sum _ { m ^ { 1 / 4 } \leq j \leq \lfloor \frac { m - 1 } { 2 } \rfloor } b _ j = O \left ( \sum _ { j \geq m ^ { 1 / 4 } } 1 / j ^ 2 \right ) = O ( m ^ { - 1 / 4 } ) . \end{align*}


\begin{align*} \int _ \R \chi ( s ) ^ p \dd s = \sum _ { j = 1 } ^ \infty \frac { x _ j ^ p } { j ^ 2 x _ j ^ 2 } = \sum _ { j = 1 } ^ \infty \frac { x _ j ^ { p - 2 } } { j ^ 2 } \end{align*}


\begin{align*} L \left [ p \right ] = { \displaystyle \int \limits _ { - z } ^ { z } } p \left ( x \right ) e ^ { - x ^ { 2 } } d x , p \in \mathbb { R } \left [ x \right ] , z > 0 , \end{align*}


\begin{align*} X _ k = v _ k ( S _ k , W _ k ) \end{align*}


\begin{align*} Q _ n ^ i ( x _ 1 \vee \cdots \vee x _ n ) = \sum _ { \sigma \in \mathrm { S h } ( n + 1 - i , i - 1 ) } \epsilon ( \sigma ) Q _ { n + 1 - i } ^ 1 ( x _ { \sigma ( 1 ) } \vee \cdots \vee x _ { \sigma ( n + 1 - i ) } ) \vee x _ { \sigma ( n + 2 - i ) } \vee \cdots \vee x _ { \sigma ( n ) } , \end{align*}


\begin{align*} E ^ * g = \lim \limits _ { n \to \infty } E ( g \circ \alpha _ n ) \equiv \lim \limits _ { n \to \infty } \int \limits _ { \Omega ^ * } g d P \alpha _ n ^ { - 1 } , \end{align*}


\begin{align*} \begin{aligned} W ^ j _ 0 ( x ) \coloneqq & \ , \frac { | b ^ j | } { 8 \pi } \frac { E } { 1 - \nu ^ 2 } \Big ( ( 1 - \log R ^ 2 ) - \frac { | x | ^ 2 } { R ^ 2 } + \log | x | ^ 2 \Big ) \Big \langle \frac { \Pi ( b ^ j ) } { | b ^ j | } , x - x ^ j \Big \rangle \ , . \end{aligned} \end{align*}


\begin{align*} \zeta _ { i j } \left ( \frac { z } { w } \right ) \psi ^ \epsilon _ i ( z ) f _ j ( w ) = \zeta _ { j i } \left ( \frac { w } { z } \right ) f _ j ( w ) \psi ^ \epsilon _ i ( z ) \ , , \end{align*}


\begin{align*} & - \sqrt { 2 / k _ { + } } e ^ { - i \frac { \pi } { 4 } } \sqrt { - s _ b } \sqrt { - s _ b ^ { * } } H _ { \theta _ c } ( 0 ) H _ { \pi - \theta _ c } ( 0 ) = \widetilde { \mathcal S } ( \cos ( \zeta ( 0 ) ) , n ) = \sqrt { \cos ^ 2 \theta _ { \hat x } - n ^ 2 } \ne 0 \end{align*}


\begin{align*} \langle p ( x ) , q ( x ) \rangle _ 0 = 0 \iff \langle \mu ( p ( x ) ) , \mu ( q ( x ) ) \rangle _ 0 = 0 . \end{align*}


\begin{align*} \liminf _ { n \to \infty , n \in N } F _ n = F = \limsup _ { n \to \infty , n \in N } F _ n . \end{align*}


\begin{align*} X : = \begin{bmatrix} D _ 1 & B \\ 0 & D _ 2 \end{bmatrix} , \end{align*}


\begin{align*} A ( x ) : = ( \mathcal { A } ( x , 0 ) ) ^ { \mathrm { W i c k } } , H _ f ^ + : = ( h _ f ) ^ { \mathrm { W i c k } } , x \in \mathbb { R } ^ 3 _ + \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes \Omega \ , | \ , H _ y u _ { \alpha } \otimes 2 \alpha ^ { 3 / 2 } \Phi _ * ^ 1 \rangle = - 4 \alpha ^ 2 \mathrm { R e } \langle P u _ { \alpha } \otimes A _ y ^ + \Omega \ , | \ , u _ { \alpha } \Phi _ * ^ 1 \rangle = 0 , \end{align*}


\begin{align*} \ell _ { R } \left ( R / { I _ { q } } \right ) = \ell _ { R } \left ( \frac { R \cap \mathbb { F } _ { \geqslant \boldsymbol { u } } } { I _ { q } \cap \mathbb { F } _ { \geqslant \boldsymbol { u } } } \right ) + \ell _ { R } \left ( \frac { R } { R \cap \mathbb { F } _ { \geqslant \boldsymbol { u } } } \right ) - \ell _ { R } \left ( \frac { I _ { q } } { I _ { q } \cap \mathbb { F } _ { \geqslant \boldsymbol { u } } } \right ) \end{align*}


\begin{align*} m : = \inf _ { \mu \in \mathcal { P } ( M ) } \int _ { M \times M } g ( x - y ) d \mu _ { x } d \mu _ { y } + \lambda \int _ { M } V \ , d \mu . \end{align*}


\begin{align*} g _ \tau = g _ 0 + \tau ( g _ 1 - g _ 0 ) , \ \ \ 0 \leq \tau \leq 1 . \end{align*}


\begin{align*} i \le i _ { 0 } = \frac 1 6 n ^ 2 - \frac 5 3 n ^ { 7 / 4 } \log ^ { 5 / 4 } n \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { N \theta } \log \left ( \int _ { \mathbb { R } ^ { d \times N } } \exp \left ( - \beta \mathcal { H } _ { N } \right ) d X _ { N } \right ) = \inf _ { \mu \in \mathcal { P } ( \mathbb { R } ^ { d } ) } \{ \mathcal { E } _ { V } ( \mu ) + \frac { 1 } { \theta } \left ( { \rm e n t } [ \mu ] \right ) \} , \end{align*}


\begin{align*} ( \mathcal E ( f _ { j ; a _ 1 , a _ 2 , \ldots , a _ { 2 s - 1 } , a _ { 2 s } } ) \Lambda _ { a _ 1 } ^ j \cdots \Lambda _ { a _ { 2 s } } ^ j ) ^ { \triangle } = z ^ { \triangle } _ { j ; a _ 1 , a _ 2 , \ldots , a _ { 2 s - 1 } , a _ { 2 s } } . \end{align*}


\begin{align*} \mathcal { E } ^ { \neq } ( { \rm e m p } _ { N } ( X _ { N } ) - \nu ) = \mathcal { E } ^ { \neq } ( { \rm e m p } _ { N } ( X _ { N } ) - \mu ) + 2 \mathcal { G } ( { \rm e m p } _ { N } ( X _ { N } ) - \mu , \mu - \nu ) + \mathcal { E } ( \mu - \nu ) . \end{align*}


\begin{align*} \dim ( \Delta ) = | F | - 1 = ( d - 1 ) t . \end{align*}


\begin{align*} \int _ \Omega | m _ p ( \cdot , z ) | ^ { p - 2 } | K _ p ( \cdot , z ) | ^ 2 = K _ p ( z ) ^ 2 \int _ \Omega | m _ p ( \cdot , z ) | ^ p = K _ p ( z ) , \end{align*}


\begin{align*} g _ H ( N ( \tilde \eta _ 1 ( t _ 1 ) ) , \tilde \nu _ 1 ( t _ 1 ) ) = \cos ( \Theta _ 1 ) , \end{align*}


\begin{align*} E _ { j n k } ^ i = \Gamma _ c ( G ^ { ( k ) } , s ^ * F _ { j n } ^ i ) \end{align*}


\begin{align*} h \cdot ( g , f ) = ( g h ^ { - 1 } , h \cdot f ) g \in G , h \in H , f \in F . \end{align*}


\begin{align*} \mathrm { \overline { d i m } _ B } \ , X \ , = \ , \limsup _ { \varepsilon \ , \to \ , 0 ^ + } \ , \frac { \log S _ X ( \varepsilon ) } { - \log \varepsilon } \end{align*}


\begin{align*} \dim _ { \kappa ( b ) } \pi _ { i } ( P \otimes _ { A } B \otimes _ { B } \kappa ( b ) ) & = \dim _ { \kappa ( b ) } \pi _ { i } ( P \otimes _ { A } \kappa ( a ) \otimes _ { \kappa ( a ) } \kappa ( b ) ) \\ & = \dim _ { \kappa ( b ) } \pi _ { i } ( P \otimes _ { A } \kappa ( a ) ) \otimes _ { \kappa ( a ) } \kappa ( b ) \\ & = \dim _ { \kappa ( a ) } \pi _ { i } ( P \otimes _ { A } \kappa ( a ) ) , \end{align*}


\begin{align*} \lambda ( P \oplus _ m P _ t ) = f ( \lambda ( P _ t ) ) \geq f ( \lambda _ 0 + \epsilon ( t ) ) . \end{align*}


\begin{align*} ( z - q _ i ^ { c _ { i j } } w ) e _ i ( z ) e _ j ( w ) = ( q _ i ^ { c _ { i j } } z - w ) e _ j ( w ) e _ i ( z ) \ , , \end{align*}


\begin{align*} d _ { r } ^ 3 \gamma ( x , y , z , t ) & = \gamma ( d ( x , y ) , \alpha ( z ) , t ) + \gamma ( [ x , z ] , \alpha ( y ) , t ) + \gamma ( [ y , z ] , \alpha ( x ) , t ) \\ + & \gamma \left ( x , y , [ \alpha ( z ) , t ] \right ) + \gamma \left ( y , z , [ \alpha ( x ) , t ] \right ) + \gamma \left ( x , z , [ \alpha ( y ) , t ] \right ) \end{align*}


\begin{align*} \rho = e ^ { B + i \omega } \wedge \Omega , \end{align*}


\begin{align*} [ a _ x , a _ y ] = 0 = [ a _ x ^ \ast , a _ y ^ \ast ] , [ a _ x , a _ y ^ \ast ] = \delta ( x - y ) . x , y \in \mathbb { R } ^ 3 . \end{align*}


\begin{align*} \mathsf h = \mathsf m / 2 , \end{align*}


\begin{align*} \mu _ G ( M ) = Z _ G ^ { - 1 } \exp \Big ( \beta \Big ( \sum _ { e \in M } w _ e + \sum _ { x \not \in M } \nu _ x \Big ) \Big ) , M \in \mathcal { M } _ G , \end{align*}


\begin{align*} \frac { d \hat \eta _ \tau } { d \tau } = ( - 1 ) \Big ( \frac { \partial \hat x _ \tau } { \partial \eta } \Big ) ^ { - 1 } \ , \frac { d x _ \tau ( T , \hat y , \hat \eta _ \tau ) } { d \tau } . \end{align*}


\begin{align*} X _ { n \delta } X _ { m \delta } & = X _ { m \delta } X _ { n \delta } . \\ X _ { n \delta } X _ { m \delta } & = X _ { ( n + m ) \delta } + X _ { ( n - m ) \delta } \ f o r \ n > m . \\ X _ { n \delta } X _ { n \delta } & = X _ { 2 n \delta } + 2 , \ \ f o r \ n \in \Z . \end{align*}


\begin{align*} ( x , t ) = \left ( \bar x _ n + r _ n y , \bar t _ n + \frac { r _ n ^ \gamma } { M _ n ^ { \gamma - 1 } } s \right ) , ( x ' , t ) = \left ( \bar x _ n + r _ n y ' , \bar t _ n + \frac { r _ n ^ \gamma } { M _ n ^ { \gamma - 1 } } s \right ) \end{align*}


\begin{align*} B ( \Phi ( u ) , \Phi ( v ) ) & = B ( u + B _ { \mathfrak a } ( u , \tau ( \cdot ) ) , v + B _ { \mathfrak a } ( v , \tau ( \cdot ) ) ) \\ & = B _ { \mathfrak a } ( u , v ) \end{align*}


\begin{align*} X _ t = X _ { \underline { n } ( t ) } + ( X _ t - X _ { \underline { n } ( t ) } ) . \end{align*}


\begin{align*} ( O p _ { w , 2 d } ( a ) f ) \otimes \bar g = O p _ { w , 4 d } ( \sigma ) ( f \otimes \bar g ) . \end{align*}


\begin{align*} h \cdot ( [ a _ g : b _ g ] ) _ { g \in G } : = ( [ a _ { h g } : b _ { h g } ] ) _ { g \in G } { \rm f o r } a _ g , b _ g \in k . \end{align*}


\begin{align*} \| \Phi _ { \# } ^ y \| ^ 2 _ { \# } & = 4 \alpha \| ( h _ { \alpha } - e _ { \alpha } + H _ f ^ + ) ^ { - 1 / 2 } P u _ { \alpha } \otimes A _ { y } ^ + \Omega \| ^ 2 \\ & = 4 \alpha \int _ { \mathbb { R } ^ 3 _ + \times \mathbb { R } ^ 3 } \mathrm { d } k \ , \mathrm { d } x \ ; \left | \frac { P u _ { \alpha } } { ( h _ { \alpha } - e _ { \alpha } + | k | ) ^ { 1 / 2 } } \lambda _ y ( k ) \right | ^ 2 + O ( \alpha ^ 3 L ^ 2 e ^ { - L } ) . \end{align*}


\begin{align*} G ( z ) = ( \tau - z ) ( 1 - \bar \tau z ) \ , p ( z ) \end{align*}


\begin{align*} \mu ( i , j ) = 0 \frac { \partial f ( \mathbf { x } ) } { \partial \mathbf { x } _ { i , j } } ( \mu ) = \lambda \ ; . \end{align*}


\begin{align*} X = \limsup _ { m \rightarrow \infty } \sum _ { i = 1 } ^ { k _ m } v _ { i , m } 1 _ { \{ X _ { i , m } \in B _ { i , m } \} } \end{align*}


\begin{align*} a _ 0 : = \sup I _ U . \end{align*}


\begin{align*} \| \rho _ t - \varphi _ { n , t } \| _ { M _ d } \leq \sum _ { k = n + 1 } ^ \infty 2 k e ^ { - t k } \rightarrow 0 \end{align*}


\begin{align*} d _ { 0 , s } ( 0 , i ) = 0 , \mbox { i f } i \ne - 2 q . \end{align*}


\begin{align*} \tilde { a } _ 1 ( k ) \tilde { a } _ 2 ( k ) = \frac { k ^ 2 } { 1 + k ^ 2 } \left ( 1 - b ^ 2 ( k ) \right ) , k \in \mathbb { R } . \end{align*}


\begin{align*} \frac { 3 n + 1 } { 2 } = 3 b 2 ^ v - 1 \equiv 1 \pmod { 2 } \end{align*}


\begin{align*} A : = \nabla ^ { T M } | _ { H } - \nabla ^ { T H } \oplus \nabla ^ { N ^ { M | H } } . \end{align*}


\begin{align*} P ^ s u : = \sum _ { k \geq 1 } \lambda _ k ^ s \pi _ k u \end{align*}


\begin{align*} \Psi = \exp _ { \bar o } ^ { \bar M } \circ ( \exp _ o ^ M ) ^ { - 1 } : M \to \bar M . \end{align*}


\begin{align*} \alpha = \frac { 2 - t + \sqrt { t ^ 2 + 4 } } { 2 } , \end{align*}


\begin{align*} \lim _ { T \rightarrow + \infty } \sup _ { t \geq T } \mathbb { E } \left [ \| ( u _ t - B _ t ) \cdot ( u _ t - B _ t ) - \alpha \| _ { L ^ 1 } \right ] = 0 \ , . \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes \Omega \ , | \ , H _ y \Phi ^ y _ { \# } \rangle & = - 8 \alpha \langle P u _ { \alpha } \otimes A ^ + _ y \Omega \ , | \ , h _ { \alpha } - e _ { \alpha } + H _ f ^ + \ , | \ , P u _ { \alpha } \otimes A ^ + _ y \Omega \rangle \\ & = - 2 \| \Phi _ { \# } ^ y \| ^ 2 _ { \# } . \end{align*}


\begin{align*} d v _ X = \kappa _ N d v _ Y \wedge d v _ N , \end{align*}


\begin{align*} J _ M ( f ) ( \omega ) = - j _ M ^ 0 + \int _ 0 ^ \omega \tilde \omega ^ { 1 / 2 } \phi ( \tilde \omega ) \ , \dd \tilde \omega J _ E ( f ) ( \omega ) = \int _ 0 ^ \omega \tilde \omega ^ { 3 / 2 } \phi ( \tilde \omega ) \ , \dd \tilde \omega . \end{align*}


\begin{align*} \begin{aligned} \left \vert I _ { f , \omega _ { \pi } , \psi } ( g ) \right \rvert & = \left \vert \int _ { Z _ n } I _ { f , \psi } ( g z ) \omega _ { \pi } ^ { - 1 } ( z ) d z \right \rvert \\ & = \left \vert \int _ { P } I _ { f , \psi } ( g z ) \omega _ { \pi } ^ { - 1 } ( z ) d z \right \rvert \\ & \leq C _ 1 C _ 2 \times \int _ { P } \left \vert \Delta ^ { - \frac { 1 } { 2 } + \delta } ( g z ) \right \rvert d z . \end{aligned} \end{align*}


\begin{align*} \partial _ t { x } \ ; = \ ; { \mathbf { H } } ( x ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { 4 } x _ i ^ 2 = 0 \end{align*}


\begin{align*} \prod _ { 1 \leq i < j \leq k } R _ { i j } ( u _ i - u _ j ) = k ! ~ A _ k , \end{align*}


\begin{align*} \| \varphi _ y \| ^ 2 = 1 + O ( \alpha ^ 3 \log ( \alpha ^ { - 1 } ) ) . \end{align*}


\begin{align*} \begin{aligned} \Lambda ( \mathbf { c } ) ( P j + k ) = & \Lambda ( \mathbf { a } ) ( j ) \Lambda ( \mathbf { b } ) ( k ) + \Lambda ( \mathbf { a } ) ( j + 1 ) \Lambda ( \mathbf { b } ) ( - P + k ) , \\ & 0 \le k < P - N < j < N . \end{aligned} \end{align*}


\begin{align*} d ( t ) , e ( t ) , f ( t ) : = \frac 5 6 s ( 1 - s ) ^ 3 p ^ 7 r ^ 3 . \end{align*}


\begin{align*} ( \overline { L } _ 0 \cdots \overline { L } _ d | Z ) = ( \overline { L } _ 0 \cdots \overline { L } _ { d - 1 } | Z \cap \mathrm { d i v } ( s _ d ) ) - \sum _ { v \in M ( K ) } \int _ { Z _ { \overline { K } _ v } ^ { \mathrm { a n } } } \log \| s _ d \| _ v \prod _ { i = 0 } ^ { d - 1 } c _ 1 ( \overline { L } _ { i , v } ) \end{align*}


\begin{align*} \partial _ { \eta } G ( z _ { m a x } ) = \partial _ { \eta } g ( z _ { m a x } ) = \frac { \alpha g } { v } ( z _ { m a x } ) > 0 . \end{align*}


\begin{align*} x = \begin{pmatrix} x _ { 1 , 1 } & x _ { 1 , 2 } & \cdots & x _ { 1 , p } \\ x _ { 2 , 1 } & x _ { 2 , 2 } & \cdots & x _ { 2 , p } \end{pmatrix} \end{align*}


\begin{align*} H [ \phi , k ; \mu ] \circ k + k \circ H ^ T [ \phi , k ; \mu ] + \Theta [ \phi , k ] + k \circ \overline { \Theta [ \phi , k ] } \circ k = 0 , k : ( \phi ) _ \perp \to ( \phi ) _ \perp . \end{align*}


\begin{align*} \partial E ( a , 1 ) = \partial X _ { \Omega _ 0 } \stackrel { \simeq } { \longrightarrow } \partial X _ { \Omega _ \tau } \end{align*}


\begin{align*} \frac { \exp \left \{ - \int \limits _ { t } ^ { \varepsilon _ 0 } \frac { d r } { r q _ { x _ 0 } ^ { \frac { 1 } { n - 1 } } ( r ) } \right \} } { { \exp \left \{ - \alpha \int \limits _ { t } ^ 1 \frac { d r } { r } \right \} } } = \exp \left \{ \int \limits _ { t } ^ { \varepsilon _ 0 } \frac { \alpha d r } { r } - \int \limits _ { t } ^ { \varepsilon _ 0 } \frac { d r } { r q _ { x _ 0 } ^ { \frac { 1 } { n - 1 } } ( r ) } \right \} = \end{align*}


\begin{align*} \hat { \partial } _ { i + 1 } = A r _ { i + 1 } \partial _ { i + 1 } = \rho ^ { - 1 } A r _ { i } ( \partial _ i - \nu _ { i } Z ) = \rho ^ { - 1 } ( \hat { \partial } _ { i } - \hat { \nu } _ i \hat { Z } _ i ) . \end{align*}


\begin{align*} L ( p ' ) = \inf \left \{ x _ n : ( p ' , x _ n ) = ( p _ 1 , \ldots , p _ { n - 1 } , x _ n ) \in U \right \} . \end{align*}


\begin{align*} { \cal Z } ( t ) = \sum _ { \alpha } { \cal Z } _ \alpha ( t ) , { \cal Z } _ \alpha ( t ) : = \Pi _ \alpha { \cal Z } ( t ) \Pi _ \alpha \ . \end{align*}


\begin{align*} \begin{aligned} & \underset { s } { } & & \sum _ { i = 1 } ^ { m } h _ { i } ( s ) \end{aligned} \end{align*}


\begin{align*} \underset { n = 1 , \ldots , N } \sup ~ \hat { E } _ { n } ^ { \epsilon , \Delta t } \le C ( T ) ( 1 + | q _ 0 ^ \epsilon | ^ 2 + | p _ 0 ^ \epsilon | ^ 2 ) \Delta t , \end{align*}


\begin{align*} h _ { k } d _ { n , k } = L \left [ \phi \partial _ { x } P _ { n + 1 } P _ { k } \right ] = 0 , n + 1 \equiv k \ \operatorname { m o d } \left ( 2 \right ) , \end{align*}


\begin{align*} \left ( c \right ) _ { n } = { \displaystyle \prod \limits _ { j = 0 } ^ { n - 1 } } \left ( c + j \right ) , n \in \mathbb { N } , \quad \left ( c \right ) _ { 0 } = 1 . \end{align*}


\begin{align*} \frac { d } { d t } p ( s - t , Y ( t , s , x ) ) = - \partial _ t p ( s - t , Y ( t , s , x ) ) + | \nabla p ( s - t , Y ( t , s , x ) ) | ^ 2 = - \gamma u ( s - t , Y ( t , s , x ) ) p ( s - t , Y ( t , s , x ) ) , \end{align*}


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


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta u = u ^ { \alpha _ 1 } + v ^ { \beta _ 1 } \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; u > 0 \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; & & u \lfloor _ { \partial \Omega } = 0 , \\ - \Delta v = v ^ { \alpha _ 2 } + u ^ { \beta _ 2 } \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; v > 0 \ ; \ ; \mbox { i n $ \Omega $ , } \ ; \ ; & & v \lfloor _ { \partial \Omega } = 0 \end{alignedat} \right . \end{align*}


\begin{align*} \int _ { \Omega } \mathbf { E } _ { \overline { \mathbf { P } } _ { N } ^ { x } } [ { \rm N u m } ( \square _ { 1 } ) ] \ , d x & = \int _ { \Omega } \left | \theta _ { N ^ { \frac { 1 } { d } } x } X _ { N } ' \big | _ { \square _ { 1 } } \right | \ , d x \\ & \geq \int _ { \Omega } { \rm e m p } _ { N } \ , d x . \end{align*}


\begin{align*} \int _ { M } ( 2 h ^ { \mu _ { \theta } } + V + \frac { 1 } { \theta } \log \mu _ { \theta } ) f \mu _ { \theta } \ , d x = 0 . \end{align*}


\begin{align*} \dim V ^ { s u } = \dim ( V ^ s \cap V ^ u ) < \dim V ^ s . \end{align*}


\begin{align*} \omega ( y ) = \frac { 1 } { 2 } | y | ^ { 2 } \log | y | \end{align*}


\begin{align*} - ( \Delta - \lambda _ s ) h _ s = ( \tilde \Delta _ a - \lambda _ s ) ( \tilde E _ { s , a } - h _ s ) = ( \Delta - \lambda _ s ) ( \tilde E _ { a , s } - h _ s ) + C _ s \cdot \eta _ a \end{align*}


\begin{align*} K _ { \mathbb { D } , \varphi } ( 0 ) = \frac { 1 } { \int _ { \mathbb { D } } e ^ { - \varphi } } . \end{align*}


\begin{align*} \Delta \log u = \frac { 2 \left \Vert \alpha _ { 2 } \right \Vert ^ 2 } { ( K _ 1 ^ \perp ) ^ 2 } ( K _ 2 ^ { \perp } - \hat { K } _ 2 ^ { \perp } ) + \frac { 2 \left \Vert \hat { \alpha } _ { 4 } \right \Vert ^ 2 } { 4 \hat { K } _ 2 ^ { \perp } } \end{align*}


\begin{align*} \eta = L C D _ { ( i , j ) \in I _ A } ( 2 ^ { j - 1 } - 2 ^ i ) \end{align*}


\begin{align*} S _ { \iota , \jmath } \subset Z _ { \iota , \jmath } : = Z ( G _ { \iota , \jmath } ' ) \subset Z _ { \jmath } \subset G _ { \iota , \jmath } ' . \end{align*}


\begin{align*} \lim _ { t \to \infty } \frac { \Psi ( x , k t ) } { \mathcal { H } _ * ( x , t ) } = 0 x \in \Omega . \end{align*}


\begin{align*} \cos F ( \delta ) = \begin{cases} \dfrac { 1 + B \ , \tan \delta } { \sqrt { 1 + C \ , \tan ^ 2 \delta + 2 \ , B \ , \tan \delta } } & \\ \\ - \dfrac { 1 + B \ , \tan \delta } { \sqrt { 1 + C \ , \tan ^ 2 \delta + 2 \ , B \ , \tan \delta } } & \end{cases} \end{align*}


\begin{align*} \Theta _ 2 - \Theta _ 1 = ( \tilde \alpha _ 1 + \alpha _ 2 ) + \int _ { s _ 2 ^ * } ^ { s _ 2 } \dot \vartheta ( s ) \ , d s + \int _ { s _ 1 } ^ { s ^ * _ 1 } \dot \vartheta ( s ) \ , d s \leq 0 , \end{align*}


\begin{align*} \partial _ t \rho _ i - \nabla \cdot ( \rho _ i \nabla p ) = \rho _ i G _ i , p ( 1 - \rho ) = 0 , \rho \leq 1 \end{align*}


\begin{align*} d ^ 0 _ { r , s } = 0 ( r , s ) . \end{align*}


\begin{align*} \mu ( H ) = \prod _ { e \in E ( H ) } \mu ( e ) \ ; . \end{align*}


\begin{align*} a ( z , \tilde { z } ) & : = \int _ \Omega \nabla z \cdot \nabla \tilde { z } \ , d x { \rm f o r ~ } z , \tilde { z } \in V , \\ a _ \Gamma ( z _ \Gamma , \tilde { z } _ \Gamma ) & : = \int _ \Gamma \nabla _ \Gamma z _ \Gamma \cdot \nabla _ \Gamma \tilde { z } _ \Gamma \ , d \Gamma { \rm f o r ~ } z _ \Gamma , \tilde { z } _ \Gamma \in V _ \Gamma , \end{align*}


\begin{align*} \begin{cases} 2 \pi \dot x = \nabla _ \xi a = B x + C \xi \\ 2 \pi \dot \xi = - \nabla _ x a = - A x - B ^ T \xi \end{cases} \end{align*}


\begin{align*} \int _ 0 ^ { T } \int _ { \R ^ n } v _ \epsilon \left ( \partial _ t \varphi + \div ( b _ \epsilon \varphi ) \right ) \dd x \dd t = - \int _ { \R ^ n } \bar u ( x ) \varphi ( 0 , x ) \dd x \ , , \end{align*}


\begin{align*} \mathcal { H } _ { N } ( X _ { N } ) = \sum _ { i \neq j } g ( x _ { i } - x _ { j } ) + N \sum _ { i = 1 } ^ { N } V ( x _ { i } ) , \end{align*}


\begin{align*} \dfrac { d x } { d \tau } = D F ( x , t ) \end{align*}


\begin{align*} f ' ( \underline { t } ) : = f ( \underline { t } ) + \alpha _ 1 t _ 1 ^ { N + 1 } + \alpha _ 2 t _ 1 ^ { N + 2 } + \cdots + \alpha _ r t _ 1 ^ { N + r } \in R [ \underline t ] . \end{align*}


\begin{align*} \langle f , g \rangle = \int _ { \mathbb { R } ^ d } f ( t ) \overline { g ( t ) } d t \ \ \ \ \ f , g \in L ^ 2 ( \mathbb { R } ^ d ) . \end{align*}


\begin{align*} \frac { d \varphi _ { s , t } ( z ) } { d t } = p ( \varphi _ { s , t } ( z ) , t ) , \varphi _ { s , s } ( z ) = z \end{align*}


\begin{align*} ( \beta _ 1 , \beta _ 2 ) \to ( \zeta _ 1 , \zeta _ 2 ) : = \Big ( \frac { \delta } { \beta _ 2 ^ 2 } , \frac { \delta } { \beta _ 1 ^ 2 } \Big ) \ , , \beta \mapsto \zeta : = \frac { \delta } { \beta ^ 2 } \end{align*}


\begin{align*} < g , B g > = \sum _ { i , j \in e \in E } d e g ( x ) ( - L _ { i j } ) f _ k ( i ) f _ k ( j ) ( g ( i ) - g ( j ) ) ^ 2 = \sum _ { i , j \in e \in E } a _ { i j } ( g ( i ) - g ( j ) ) ^ 2 \end{align*}


\begin{align*} \left \{ H _ 0 ; G _ { k + 1 } \right \} + P _ { k , e f f } = Z _ { k + 1 } \ , , \end{align*}


\begin{align*} \mathsf { e } ( \mathcal { X } \circ _ i \mathcal { Y } ) = \mathsf { e } ( \mathcal { X } ) \circ _ i \mathsf { e } ( \mathcal { Y } ) \end{align*}


\begin{align*} \Psi _ 0 = \exp ( P [ k ] ) \Psi _ { \mathrm { m f } } [ \phi ] . \end{align*}


\begin{align*} \rho ( a \cdot X ) ( n ) = a \cdot \rho ( X ) ( n ) \rho ( X ) ( a \cdot n ) = \omega ( X ) ( a ) \cdot n + a \cdot \rho ( X ) ( n ) \end{align*}


\begin{align*} \mathcal { L } g ( \omega ) = \frac { a } { \omega ^ { 1 / 3 } } g - \frac { 1 } { \omega ^ { 4 / 3 } } \int _ 0 ^ \infty k \left ( \frac { r } { \omega } \right ) f ( r ) \ , \dd r \end{align*}


\begin{align*} \tilde { \beta } _ { t , n } ^ \star = \sin \omega _ n , \tilde { \beta } _ { r , n } ^ \star = \cos \omega _ n , \end{align*}


\begin{align*} V _ { } = \{ v \in A _ V / \sim \big | \nexists x \in a _ X : A _ \psi ( ( 0 , x ) ) \in v \} \ ; . \end{align*}


\begin{align*} & d _ x - \nabla _ { x y } ^ 2 g ( x , y ) \nabla _ { y y } ^ 2 g ( x , y ) ^ { - 1 } d _ y = 0 \\ & - \nabla _ { y y } ^ 2 g ( x , y ) ^ { - 1 } \nabla _ { y x } g ( x , y ) \left ( d _ x - \nabla _ { x y } ^ 2 g ( x , y ) \nabla _ { y y } ^ 2 g ( x , y ) ^ { - 1 } d _ y \right ) + \beta \nabla _ y g ( x , y ) = 0 . \end{align*}


\begin{align*} G _ \iota = G ' _ { \iota , \jmath } \times G _ { \jmath , [ 1 ] } , \end{align*}


\begin{align*} A ^ \mathfrak { h } \ , \coloneqq \ , \{ f \in A \mid v ^ \sharp ( f ) = 0 , \ , \forall v \in \mathfrak { h } \} \end{align*}


\begin{align*} h ( \alpha ) - h ( \beta ) = \int _ 0 ^ 1 D h \ ( \beta + s ( \alpha - \beta ) \ ) \cdot ( \alpha - \beta ) \ , d s . \end{align*}


\begin{align*} \eta _ 1 = 0 , \eta _ 2 = 0 , \eta _ 3 = 0 . \end{align*}


\begin{align*} q ( t ) : = \frac 1 6 p ^ 3 \qquad y ( t ) : = p ^ 2 . \end{align*}


\begin{align*} \mathcal { F } ( \overline { \mathbf { P } } ) = \theta \mathcal { E } ( \rho - \mu _ { \theta } ) + \overline { { \rm E n t } } [ \overline { \mathbf { P } } | \overline { \mathbf { \Pi } } ^ { \mu _ { \theta } } ] , \end{align*}


\begin{align*} \begin{aligned} \Phi _ \times ^ \alpha : a \times ( b \times c ) & \rightarrow ( a \times b ) \times c \\ \Phi _ \times ^ \alpha ( ( x , ( y , z ) ) ) & = ( ( x , y ) , z ) \ ; , \end{aligned} \end{align*}


\begin{align*} \frac { 1 } { 2 \pi i } \int \limits _ { \Gamma } \tilde { w } \frac { d w } { w - z } = \sum _ { x \in w ^ { - 1 } ( \{ z \} ) } { \rm o r d } _ x ( w - z ) \tilde { w } ( x ) \end{align*}


\begin{align*} m ( a ) = | a | \ ; . \end{align*}