\begin{align*} \frac { [ x ^ { n - \Sigma _ { \mathrm { k } } } ] S ( x ) } { [ x ^ n ] S ( x ) } = w _ n ^ { \Sigma _ { \mathrm { k } } } \bigg ( \exp \bigg \{ - \frac { \big ( A _ { 1 , 1 } ( w _ n ) - ( n - \Sigma _ { \mathrm { k } } ) \big ) ^ 2 } { 2 A _ { 2 , 2 } ( w _ n ) } \bigg \} + o ( 1 ) \bigg ) \end{align*}


\begin{align*} \mathbb { E } _ { v _ 0 \in \mathbb { S } ^ { d - 1 } } [ v _ 0 ^ T v _ 0 ] = \frac { 1 } { d } \cdot \mathbf { I } . \end{align*}


\begin{align*} \widetilde { b _ n } = a _ n + b _ { n - 1 } - a _ { n - 1 } + f ( a _ n ) . \end{align*}


\begin{align*} \nabla \lambda = - k _ d ^ { - 1 } \overline { V _ \Gamma \nu } - k _ d ^ { - 1 } d \ , \nabla \overline { V _ \Gamma } \quad \overline { Q _ { \varepsilon , T } } \end{align*}


\begin{align*} c = \sum _ { i = 1 } ^ m a _ i \ , \chi _ { C _ i } \ , , \end{align*}


\begin{align*} I ( f ) = \lim _ { n } I ( f _ n ) . \end{align*}


\begin{align*} & \sigma ^ { h - e } ( H ) \sigma ^ { h - e } ( ( \sigma ^ { h - e } ( H ) ) ^ { T } ) = \sigma ^ { h - e } ( H ) \sigma ^ { 2 h - 2 e } ( H ^ T ) , \\ & \sigma ^ { h - e } ( H ) \sigma ^ { e } ( ( \sigma ^ { h - e } ( H ) ) ^ { T } ) = \sigma ^ { h - e } ( H ) H ^ T . \end{align*}


\begin{align*} a _ { i j } ( q ) = \sum _ { k = 1 } ^ { d } \sigma _ { i k } ( q ) \sigma _ { j k } ( q ) . \end{align*}


\begin{align*} \displaystyle \lambda _ o = \frac { r ( r - 1 ) \pi ^ { r / 2 } } { 2 \Gamma ( r / 2 ) \zeta ( r ) } . \end{align*}


\begin{align*} \chi _ t ^ { - 1 } = \begin{pmatrix} X _ t & Y _ t \\ W _ t & Z _ t \end{pmatrix} . \end{align*}


\begin{align*} g ( y ) = g ( \kappa y ) , \ \ y \in Y , \end{align*}


\begin{align*} \pi _ g ( w ) _ j = \begin{cases} g ( w _ j ) & w _ j \in ( Q \times \Gamma ) \\ w _ j & w _ j \in \Gamma \end{cases} \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u & = a ( x ) u ^ { - \gamma } + \lambda f ( x , u ) \ ; \ ; & & \mbox { i n } \ ; \ ; \Omega , \\ u & > 0 & & \mbox { i n } \ ; \ ; \Omega , \\ u & = 0 & & \mbox { o n } \ ; \ ; \partial \Omega , \end{alignedat} \right . \end{align*}


\begin{align*} \mathcal { E } _ \mathrm { t o t } [ \phi , k ; N ] : = \Big \{ E _ \mathrm { H } [ \phi ] + \frac { 1 } { N } \mathcal { E } \big [ k ; \ , H _ 0 [ \phi , k ] , \Theta [ \phi , k ] \big ] \Big \} , \end{align*}


\begin{align*} \left [ a \right ] _ \delta = \mathbb { E } \max _ { 0 \leq j \leq N } \sum _ { i = 1 } ^ n \left \vert a _ i \right \vert X _ i ^ { ( j ) } \end{align*}


\begin{align*} \partial _ t n - \alpha \Delta n = - n \sum _ { i = 1 } ^ { \ell } \beta _ i \rho _ i . \end{align*}


\begin{align*} V = \bigoplus _ { i = 1 } ^ n V _ i \end{align*}


\begin{align*} \operatorname { I m } \varphi \left ( x , \mathrm { i } k + 0 \right ) \mathrm { d } k = \pi \psi \left ( x , \mathrm { i } k \right ) \mathrm { d } \rho \left ( k \right ) , \ \ \ k \geq 0 , \end{align*}


\begin{align*} \nu _ M ( \pi ( f ) ) = \max _ { \prec } \{ M b : c _ b \not = 0 \} . \end{align*}


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


\begin{align*} H _ { \left \lfloor T r \right \rfloor } ( d ) = T ^ { 1 / 2 - d } \sum _ { n = 0 } ^ { \left \lfloor T r \right \rfloor - 1 } \pi _ { n } ( d ) ( - \sum _ { k = n } ^ { T } ( k + d ) ^ { - 1 } ) \xi _ { \left \lfloor T r \right \rfloor - n } \Rightarrow A ( r ; d ) , \end{align*}


\begin{align*} f ( \partial _ y { \bf { v } } ) = \left ( \frac { \theta ( P - q ) } { R } f _ 1 , \frac { P - q } { a } f _ 2 + \theta f _ 3 , \theta f _ 2 + \frac { \theta ^ 2 } { 2 q } \frac { P + q } { P - q } f _ 3 \right ) ^ T , \end{align*}


\begin{align*} \langle s _ 1 , s _ 2 \rangle = \int _ X \langle s _ 1 ( x ) , s _ 2 ( x ) \rangle _ h d v _ X ( x ) \end{align*}


\begin{align*} f ( x ) \ ; = \ ; \bigl ( x ^ 2 \ , - \ , 1 \bigr ) x , \ ; g ( x ) \ ; = \ ; ( x ^ 2 \ , - \ , 1 ) \log \Bigl \vert \dfrac { 1 + x } { 1 - x } \Bigr \vert , \ ; \phi _ \epsilon ( x ) \ ; = \ ; \dfrac 2 { \epsilon ^ { 2 ( 1 - \kappa ) } } f ( x ) \ , - \ , \dfrac { \dot { \kappa } } { K ( \epsilon ) } g ( x ) . \end{align*}


\begin{align*} = | | \Pi ^ { k } \Pi ^ { l } \varphi | | _ { 2 } ^ { 2 } + \left \langle \Pi ^ { l } \varphi , \frac { i \hbar } { \epsilon } \mathbf { J } _ { k l } \Pi ^ { k } \varphi \right \rangle . \end{align*}


\begin{align*} \varphi ^ * \theta = \mu \theta , \end{align*}


\begin{align*} c _ { h , k } = \sum _ { h _ 0 + \dots + h _ k = h } \prod _ { j = 1 } ^ k 2 ^ { j h _ j } \leq 2 ^ k \cdot 2 ^ { h k } . \end{align*}


\begin{align*} \beta \wedge \sigma & = 0 \\ \dd \sigma & = - \alpha \wedge \sigma - \frac { 1 } { 2 } [ \sigma ] \wedge \sigma \\ \dd \alpha & = - \alpha \wedge \alpha + \beta \wedge \beta - R \sigma \wedge \sigma \\ \dd \beta & = - \beta \wedge \alpha - \alpha \wedge \beta - S \sigma \wedge \sigma \end{align*}


\begin{align*} S _ { k j } = S _ k \cap B _ j ^ * . \end{align*}


\begin{align*} ( h _ f z ) ( \omega ) = f ( \omega , z ( \omega ) ) \in K _ \sigma \ \ \ \mbox { i f } \ \ \ \omega \in \Omega _ \varepsilon \equiv \Omega ^ 1 _ \varepsilon \cap \Omega ^ 2 _ \varepsilon , \end{align*}


\begin{align*} s _ n \sim \frac { S ( z _ n ) } { \sqrt { 2 \pi z _ n ^ 2 C '' ( z _ n ) } } \cdot z _ n ^ { - n } \cdot n ! , z _ n z _ n C ' ( z _ n ) = n \end{align*}


\begin{align*} \Omega _ j = | j | ^ 2 , j \in \Z ^ 2 , \end{align*}


\begin{align*} \left ( p + \frac 1 2 h ^ 2 \right ) ( t , x , y ) \equiv \left ( ( \rho ^ 0 ) ^ \gamma + \frac 1 2 ( h ^ 0 ) ^ 2 \right ) ( t , x , 0 ) = \frac 3 2 , \end{align*}


\begin{align*} \tfrac { 1 + m + x _ N } { k + \ell m + \ell x _ N } \geq \min \Bigl \{ \tfrac { 1 + m + x } { k + \ell m + \ell x } : x \in [ 0 , 1 ] \Bigr \} = \left \{ \begin{array} { c l } \frac 1 k > \beta & \ k = \ell \ , ; \\ & \\ \frac { 1 + m } { k + \ell m } \geq \beta & \ k > \ell \ , ; \\ & \\ \frac { 2 + m } { k + \ell ( m + 1 ) } \geq \beta & \ k \ < \ell \ , . \end{array} \right . \end{align*}


\begin{align*} \widehat { A ^ { k / 2 } u } ( n ) : = | n | ^ k \hat { u } ( n ) . \end{align*}


\begin{align*} B ( t ) \varphi : = [ L ( t ) \varphi ] r ^ { \odot \star } , \forall t \in \mathbb { R } , \ \varphi \in X , \end{align*}


\begin{align*} X _ t ^ i = x _ 0 ^ i + \int _ 0 ^ t b ^ i ( s , X _ s ) d s + W _ t ^ i + \alpha _ i \max _ { 0 \leq s \leq t } X ^ i _ s + \beta _ i \min _ { 0 \leq s \leq t } X ^ i _ s . \end{align*}


\begin{align*} z _ i = 1 - X _ i \ / , ( 1 \le i \le k ) \ / ; \zeta _ j = 1 - T _ j \ / , ( 1 \le j \le n ) \ / . \end{align*}


\begin{align*} \mathbb { E } \big [ | \zeta _ X ( \alpha ) \zeta _ Y ( \alpha ) D ( X , Y ) | ^ 2 \big ] = & \sum _ { h _ 1 , h _ 2 , a , b , c , d } f ( a , b ) \overline { f ( c , d ) } \frac { ( a , c ) ^ { 2 \alpha } ( b , d ) ^ { 2 \alpha } } { ( a c b d ) ^ \alpha } \frac { 1 } { h _ 1 ^ { 2 \alpha } h _ 2 ^ { 2 \alpha } } \\ = & \displaystyle \zeta ( 2 \alpha ) ^ 2 S _ f ( 2 ; \alpha ) . \end{align*}


\begin{align*} \beta ^ * ( \mu ; P _ { ( s , t ) } ) = \sum _ { P \in \mathbf { C } ^ * ( P _ { ( s , t ) } , n ) } \mu ( P ) \ ; , \end{align*}


\begin{align*} \int _ 0 ^ \infty \int _ { \R ^ 2 } G ( | e ^ { i t \Delta } u _ 0 ^ \sigma | ^ 2 ) \ , d x \ , d t = \sigma ^ 4 \int _ 0 ^ \infty \int _ { \R ^ 2 } G ( | e ^ { i t \Delta } u _ 0 ^ 1 | ^ 2 ) \ , d x \ , d t , \end{align*}


\begin{align*} Y _ { m , n } \left ( 1 , 1 - \frac { \lambda } { m } , \frac { T } { m } \right ) = \frac { 1 } { 2 } \sum _ { \theta \in \{ 0 , 1 \} ^ 2 } \prod _ { w ^ n = ( - 1 ) ^ { \theta _ 2 } } \prod _ { j = 0 } ^ { m - 1 } \left | 1 - \frac { \lambda } { m } + \exp \left \{ 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \right \} + \frac { T w } { m } \right | . \end{align*}


\begin{align*} \partial _ { \varphi _ { i j } } : = - \frac { t _ i } { | t | } \partial _ j + \frac { t _ j } { | t | } \partial _ i . \end{align*}


\begin{align*} S = \{ x \in ( T ) : T ( x ) \} , \end{align*}


\begin{align*} d ( T _ * \mu _ h ) = ( \L _ T h ) d x . \end{align*}


\begin{align*} \mathcal { C } ( g _ 0 ) = - \mathcal { C } ( g _ 1 ) - 3 \mathcal { C } ( g _ 0 , g _ 1 , g _ 1 ) - 3 \mathcal { C } ( g _ 0 , g _ 0 , g _ 1 ) . \end{align*}


\begin{align*} g ( S , W ) & = f ( S , W ) \\ & \overset { ( a ) } { = } X \end{align*}


\begin{align*} & \Phi ( \mbox { s i g n } ( f _ 1 ( \mathbf { p } ) ) , \dots , \mbox { s i g n } ( f _ m ( \mathbf { p } ) ) ) \\ & = \Phi ' \big ( f _ 1 ( \mathbf { p } ) < 0 , - f _ 1 ( \mathbf { p } ) < 0 , \cdots , f _ m ( \mathbf { p } ) < 0 , - f _ m ( \mathbf { p } ) < 0 \big ) . \end{align*}


\begin{align*} L \bullet ~ _ 2 F _ 1 ( a , b ; c + 1 ; x ) = \left [ - a b + \left ( - x ( a + b + 1 ) + c + 1 \right ) \partial _ x - ( x - 1 ) x \partial _ x ^ 2 \right ] \bullet ~ _ 2 F _ 1 ( a , b ; c + 1 ; x ) = 0 \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*} E _ w ( Z ) = \sum _ T z ^ T \end{align*}


\begin{align*} \partial _ t u - \Delta \mu = 0 & { \rm i n ~ } Q , \\ \mu = - \delta \Delta u + \beta ( u ) + \delta \pi ( u ) - f & { \rm i n ~ } Q , \end{align*}


\begin{align*} \tilde { g } _ { \ell } ( z , \zeta , q ) \ : = \ : e _ { \ell } ( \zeta ) \ : - \ : \frac { q } { 1 - q } \sum _ { s = n - k + 1 } ^ { \ell } ( - 1 ) ^ s \binom { n - s } { \ell - s } \Delta _ { s + k - n } , \end{align*}


\begin{align*} \frac { d } { d s } \big | _ { s = 0 } h _ s ( x ) = \xi ( x ) . \end{align*}


\begin{align*} y + \mathbb { K } y = \psi , \end{align*}


\begin{align*} \psi ( t ) = t \log \left ( x + \sigma \right ) + O \left ( | t | ^ 3 x ^ { - 2 } \right ) . \end{align*}


\begin{align*} N ^ 2 = \int _ { [ 0 , 1 ] ^ 2 } S ^ 2 ( \alpha _ 1 , \alpha _ 2 ) \sum _ { m = 0 } ^ { 2 f ( N ) } e ( - m \alpha _ 1 ) \sum _ { n = 0 } ^ { 2 g ( N ) } e ( - n \alpha _ 2 ) d \boldsymbol { \alpha } \end{align*}


\begin{align*} K _ { \theta _ 1 , \theta _ 2 } \bigl ( ( y , \omega ) , ( z , \eta ) \bigr ) & = K _ { \theta _ 1 , \theta _ 2 } \bigl ( ( - z , \omega ) , ( - y , \eta ) \bigr ) \\ & = \overline { K _ { \theta _ 2 , \theta _ 1 } \bigl ( ( - y , \eta ) , ( - z , \omega ) \bigr ) } = \overline { K _ { \theta _ 2 , \theta _ 1 } \bigl ( ( z , \eta ) , ( y , \omega ) \bigr ) } . \end{align*}


\begin{align*} M _ \otimes ^ { \mathcal { M } \times \mathcal { N } } ( a ) = ( M _ \otimes ^ { \mathcal { M } } ( a ) , M _ \otimes ^ { \mathcal { N } } ( a ) ) \ ; . \end{align*}


\begin{align*} \begin{aligned} - \operatorname { d i v } \mathcal { A } ( x , u , \nabla u ) & = \mathcal { B } ( x , u , \nabla u ) & & \Omega , \\ \mathcal { A } ( x , u , \nabla u ) \cdot \nu & = \mathcal { C } ( x , u ) & & \partial \Omega , \end{aligned} \end{align*}


\begin{align*} - \partial _ s w _ n - \Delta w _ n + \theta _ n h ( \bar x _ n + r _ n y , \bar t _ n + r _ n ^ 2 s ) | D w _ n | ^ \gamma = g _ n , \end{align*}


\begin{align*} P \Big ( \sum _ { u = 1 } ^ p ( Z _ u - \underline { Z } _ u ) \ge \epsilon \bar h _ 4 ( n ) \Big ) & \le \sum _ { u = 1 } ^ p P ( Z _ u - \underline { Z } _ u \ge \epsilon p ^ { - 1 } \bar h _ 4 ( n ) ) \\ & \le C p ^ 3 ( \epsilon \bar h _ 4 ( n ) ) ^ { - 2 } n ^ 2 ( \log n ) ^ { - \alpha / 2 } \le C ' \epsilon ^ { - 2 } ( \log n ) ^ { - c _ 1 \lambda } . \end{align*}


\begin{align*} \mathcal { P } _ { A , B } ( \mu ) = \lim _ { n \rightarrow \infty } \left ( \frac { \int \varphi _ { n } ^ { 1 } d \mu } { \int \psi _ { n } ^ { 1 } d \mu } , \ldots , \frac { \int \varphi _ { n } ^ { d } d \mu } { \int \psi _ { n } ^ { d } d \mu } \right ) . \end{align*}


\begin{align*} \inf _ { K \Subset \Omega } \liminf _ { \epsilon \to 0 } \int _ { A ^ \epsilon \setminus K } \int _ I \ell _ { A ^ \epsilon } ^ { \alpha } ( v , y ) f ( v , y ) \dd v \dd y = 0 . \end{align*}


\begin{align*} \Theta _ \alpha ( \nu ) = \sum _ { \jmath \in J ( S _ \iota ) } \Theta _ { \alpha , \jmath } ( \nu ) , \end{align*}


\begin{align*} T _ p ( y , x ) = ( \nabla ^ { 1 , 0 } ) ^ k B _ p ^ X ( y , x ) . \end{align*}


\begin{align*} \langle \varphi , \psi \rangle = \int _ G \psi \varphi d x \varphi \in M _ d ( G ) , \psi \in L ^ 1 ( G ) . \end{align*}


\begin{align*} \| \phi \| _ { L ^ { p } ( \mathbb C ^ n ) } \le \frac { C _ n | \Omega | ^ { \frac 1 { 2 n } } } { p - 1 } \sum _ { j = 1 } ^ n \| \partial \phi / \partial \bar { z } _ j \| _ { L ^ p ( \mathbb C ^ n ) } , \ \ \ \forall \ , \phi \in C ^ 1 _ 0 ( \mathbb C ^ n ) , \end{align*}


\begin{align*} \norm { A - C M R } & = \norm { A - C C ^ + A R ^ + R } = \norm { ( I - C C ^ + ) A + C C ^ + A ( I - R ^ + R ) } \\ & \le \norm { ( I - C C ^ + ) A } + \norm { C C ^ + } \ , \norm { A ( I - R ^ + R ) } . \end{align*}


\begin{align*} Z _ { \left \lfloor T r \right \rfloor } ^ { \ast } ( d ) & = T ^ { 1 / 2 - d } ( \log T ) ^ { - 1 } \mathsf { D } _ { d } \Delta _ { + } ^ { - d } \xi _ { \left \lfloor T r \right \rfloor } \\ & = T ^ { 1 / 2 - d } ( \log T ) ^ { - 1 } \sum _ { n = 1 } ^ { \left \lfloor T r \right \rfloor - 1 } \pi _ { n } ( d ) \sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - 1 } \xi _ { \left \lfloor T r \right \rfloor - n } \Rightarrow W ( r ; d ) . \end{align*}


\begin{align*} c _ { d , \alpha } = \frac { \alpha 2 ^ { \alpha - 1 } \Gamma \big ( ( d + \alpha ) / 2 \big ) } { \pi ^ { d / 2 } \Gamma ( 1 - \alpha / 2 ) } , \end{align*}


\begin{align*} B _ \alpha : & = B \cup \bigcup \left \{ A ( \beta , \delta ) : \beta \in I \ \wedge \ \beta < \alpha \ \wedge \ \delta \in J _ \beta \right \} \ \\ J _ \alpha : & = \left \{ \gamma \in J _ \alpha ' : \left ( A ( \alpha , \gamma ) \setminus A _ \alpha \right ) \cap B _ \alpha = \emptyset \right \} . \end{align*}


\begin{align*} x ( d ) - x ( c ) = \int _ { c } ^ { d } D F ( x ( \tau ) , t ) , \end{align*}


\begin{align*} M f ^ { \mathcal { A } } = f ^ { \mathcal { B } } M \ ; . \end{align*}


\begin{align*} X \circ \Pi ( 0 , \theta ) = \left ( h ( 0 ) , t \circ \Pi ( 0 , \theta ) \right ) = \left ( z _ 0 , a \cfrac { \theta } { \pi } + b \left ( 1 - \cfrac { \theta } { \pi } \right ) \right ) . \end{align*}


\begin{align*} \omega = \Xi \biggl ( { } \prod _ { j = 1 } ^ { g } S _ { j , 1 , 1 } \ , S _ { j + g , 1 , 1 } \biggr ) . \end{align*}


\begin{align*} \left | \omega ^ { ( \nu ) } _ \psi ( g ) \right | = q ^ { \frac { \nu } { 2 } \dim V ^ g } , \end{align*}


\begin{align*} \partial _ t f = \frac { 1 } { 2 } \partial _ x ^ 2 f + \frac { \beta } { 2 } ( \partial _ x f ) ^ 2 + \sqrt { 2 } \xi . \end{align*}


\begin{align*} \dd \begin{pmatrix} \zeta _ 1 \\ \zeta _ 2 \\ \zeta _ 3 \end{pmatrix} = - \phi \wedge \begin{pmatrix} \zeta _ 1 \\ \zeta _ 2 \\ \zeta _ 3 \end{pmatrix} + \begin{pmatrix} \overline { \zeta } _ 2 \wedge \overline { \zeta } _ 3 \\ \overline { \zeta } _ 3 \wedge \overline { \zeta } _ 1 \\ \overline { \zeta } _ 1 \wedge \overline { \zeta } _ 2 \end{pmatrix} \end{align*}


\begin{align*} \int _ 0 ^ { \infty } e ^ { - s x } W _ { q } ( x ) d x = \frac { 1 } { \psi ( s ) - q } , s > \Phi ( q ) , \end{align*}


\begin{align*} N ^ { \ell , r - \ell } = \left \{ \begin{pmatrix} 1 _ \ell & * \\ 0 & 1 _ { r - \ell } \end{pmatrix} \right \} . \end{align*}


\begin{align*} \sum _ { n \le x } a _ n \log \frac x n = \frac 1 { 2 \pi i } \int _ { a - i \infty } ^ { a + i \infty } \alpha ( s ) \frac { \big ( x ^ { s / 2 } - x ^ { - s / 2 } \big ) ^ 2 } { s ^ 2 } \ , d s \end{align*}


\begin{align*} \begin{gathered} \mathcal { H } ( a + \mathbf { i } b + \mathbf { j } c + \mathbf { k } d ) = \begin{pmatrix} x & y \\ - y ^ * & x ^ * \end{pmatrix} \ ; , \\ x = a + \mathbf { i } b y = c + \mathbf { i } d \ ; . \end{gathered} \end{align*}


\begin{align*} u ^ h ( t ) - v ^ h ( t ) = & \int _ { 0 } ^ { t } \big ( \Delta u ^ h ( s ) - \Delta v ^ h ( s ) \big ) d s + \int _ { 0 } ^ { t } \big ( u ^ h ( s ) \log | u ^ h ( s ) | - v ^ h ( s ) \\ & \log | v ^ h ( s ) | \big ) d s + \int _ { 0 } ^ { t } \big ( \sigma ( u ^ h ( s ) ) - \sigma ( v ^ h ( s ) ) \big ) h ( s ) d s . \end{align*}


\begin{align*} & \delta \left ( \alpha \right ) \left [ \psi \left ( \mathrm { i } \alpha \right ) - \int K \left ( \alpha , s \right ) \mathrm { d } \mu \left ( s \right ) \right ] + \mu ^ { \prime } \left ( \alpha \right ) \\ & = \widetilde { \delta } \left ( \alpha \right ) \left \{ \psi \left ( \mathrm { i } \alpha \right ) - \int K \left ( \alpha , s \right ) \mathrm { d } \mu \left ( s \right ) \right \} , \end{align*}


\begin{align*} \kappa ^ \pi = \exp ( - \pi \vee ) \kappa \exp ( \pi \vee ) \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \gamma _ s = \sum \limits _ { j = s } ^ { N } { ( - 1 ) ^ { s + j } a _ j } , \\ s = 1 , \ldots , N , \end{array} \right . \end{align*}


\begin{align*} \begin{cases} ( \partial _ t + P ) u ( t , x ) = 0 \ , ( 0 , \infty ) \times M , \\ u | _ { t = 0 } = f . \end{cases} \end{align*}


\begin{align*} \mathbf { l } = ( l _ 0 , l _ 1 , \ldots ) \ ; . \end{align*}


\begin{align*} \bar { \mu } ( x ) = \sum _ { y \in [ n ] \setminus \{ x \} } \mu ( x , y ) \ ; . \end{align*}


\begin{align*} \mathfrak { B } ^ r ( \xi , t ) = \begin{pmatrix} 0 & i \eta \beta ^ r ( \xi ) e ^ { - t \varphi ( \xi , 0 ) } \tau ^ { - i \nu - \frac { 1 } { 2 } } \\ - i \eta \gamma ^ r ( \xi ) e ^ { t \varphi ( \xi , 0 ) } \tau ^ { i \nu - \frac { 1 } { 2 } } & 0 \end{pmatrix} , \end{align*}


\begin{align*} 2 e ^ { x / 2 } \left | \sin \left ( \frac { \pi \theta _ 1 } { 2 } + \frac { y } { 2 } + \frac { x i } { 2 } \right ) \right | = \left | e ^ z - ( - 1 ) ^ { \theta _ 1 } \right | \end{align*}


\begin{align*} h ( \sigma ) _ i : = \# \{ x \in \mathbb { T } _ { m } : \sigma ( x ) = [ x + \mathbf { e } ^ i ] \sigma ( x ) _ i = 0 \} . \end{align*}


\begin{align*} \Delta \omega - \boldsymbol { w } \cdot \nabla \omega - ( a y \partial _ 1 + b \partial _ 2 ) \omega = 0 , { \rm i n } ~ ~ \Omega _ 0 , \end{align*}


\begin{align*} \mu _ A \colon R \otimes _ A R \to R , \mu _ A ( x \otimes _ A y ) = x y . \end{align*}


\begin{align*} \int _ { \beta ^ { - 1 } } \omega _ 1 \cdots \omega _ m = ( - 1 ) ^ m \int _ \beta \omega _ m \cdots \omega _ 1 \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { h ( \lambda x ) } { h ( x ) } = 1 , \lambda > 0 . \end{align*}


\begin{align*} & \varepsilon ^ { - 1 } \partial _ r \eta _ 0 ( r ) + \partial _ r \eta _ 1 ( r ) + k _ d ^ { - 1 } V _ \Gamma \eta _ 0 ( r ) \\ & + \varepsilon \{ \partial _ r \eta _ 2 ( r ) - \nabla _ \Gamma g _ i \cdot \nabla _ \Gamma \eta _ 0 ( r ) + k _ d ^ { - 1 } ( \partial ^ \circ g _ i ) \eta _ 0 ( r ) + k _ d ^ { - 1 } V _ \Gamma \eta _ 1 ( r ) \} = O ( \varepsilon ^ 2 ) . \end{align*}


\begin{align*} c _ k ( Y , \lambda ) = \sup _ { R > R _ 0 } b _ k ( [ - R , 0 ] \times Y ) . \end{align*}


\begin{align*} \Lambda _ { i j } \cdot T _ { i ' j ' } ^ { ( l ) } = T _ { i ' j ' } ^ { ( l ) } . \end{align*}


\begin{align*} V _ g f ( x , \xi ) = \int _ { \mathbb { R } ^ d } f ( t ) \overline { g ( t - x ) } e ^ { - 2 \pi i \xi \cdot t } d t , x , \xi \in \mathbb { R } ^ d . \end{align*}


\begin{align*} A ( t ) x ^ 2 + B ( t ) x y + C ( t ) y ^ 2 = H ( t ) z ^ 2 \end{align*}


\begin{align*} \Delta \log ( 1 - K ) = 2 K + 1 . \end{align*}


\begin{align*} \mathcal U _ \varepsilon ( t = 0 ) = \mathcal U _ 0 = \biggl ( \begin{array} { c } \xi _ 0 \\ \mathcal P _ \tau v _ 0 \end{array} \biggr ) . \end{align*}


\begin{align*} \begin{cases} B \partial _ { t } u + B u \partial _ { x } u - A B h \partial _ { x } \tilde { h } + ( 1 - A ) B h \partial _ { y } \tilde { h } \partial _ { y } u - A B h ^ 2 \partial _ { y } ^ 2 u = 0 , \\ A \partial _ { t } \tilde { h } + A u \partial _ { x } \tilde { h } - A B h \partial _ { x } u + ( 1 - B ) A h ( \partial _ { y } \tilde { h } ) ^ 2 - A B h ^ 2 \partial _ { y } ^ 2 \tilde { h } = 0 , \\ ( u , \tilde { h } ) | _ { t = 0 } = ( u _ 0 , \tilde { h } _ 0 ) ( x , y ) , \end{cases} \end{align*}


\begin{align*} O p _ { \mathcal { A } } ( a ) = O p _ \mathcal { B } ( b ) \Longleftrightarrow \ , \ , b = \mu ( \mathcal { B } \mathcal { A } ^ { - 1 } ) ( a ) . \end{align*}


\begin{align*} \beta _ 2 ( \tilde V ) & = 1 + d + s \\ \beta _ 4 ( \tilde V ) & = 1 + d + s \\ \beta _ 3 ( \tilde V ) & = \beta _ 3 ( \hat V ) . \end{align*}


\begin{align*} X _ n = h ^ { ( n ) } ( ( \phi _ n ( W _ j ) ) _ { j \in J } ) \end{align*}


\begin{align*} E ( x , t , k ) = I - \frac { 1 } { 2 \pi i } \oint _ { | s + k _ 0 | = \epsilon } \frac { \mathfrak { B } ^ r ( \xi , t ) } { ( s + k _ 0 ) ( s - k ) } d s + \frac { 1 } { 2 \pi i } \oint _ { | s - k _ 0 | = \epsilon } \frac { \overline { \mathfrak { B } ^ r ( \xi , t ) } } { ( s - k _ 0 ) ( s - k ) } d s + R ( \xi , t ) , | k \pm k _ 0 | > \epsilon , \end{align*}


\begin{align*} N _ { \sigma } ( D ) : = \sum _ { \ ; \Gamma } \sigma _ { \Gamma } ( D ) \cdot \Gamma , \end{align*}


\begin{align*} \dd \bigl ( \varphi ( t , x _ 1 ) , \varphi ( t , x _ 2 ) \bigr ) = \eta ( t ; x _ 1 , x _ 2 ) \le e ^ { C t } \dd ( x _ 1 , x _ 2 ) . \end{align*}


\begin{align*} | \int _ \Lambda f d x - \sum _ { i = 1 } ^ N f ( \xi _ j ) | I _ j | | & \leq \sum _ { i = 1 } ^ N \left | \int _ { I _ j } f ( x ) d x - f ( \xi _ j ) | I _ j | \right | \\ [ \sup _ { \xi \in I _ j } f ( \xi ) - \inf _ { \xi \in I _ j } f ( \xi ) ] \Delta x _ j . \end{align*}


\begin{align*} \lambda _ { \infty } = ( \lambda _ { \infty , \gamma } ) _ { \gamma = 1 , 2 } , \lambda _ { \infty , \gamma } : = \lambda _ { \infty , \gamma } ( 0 ) = \frac { \chi _ { \Lambda } ( k ) } { 2 \pi | k | ^ { 1 / 2 } } \mathbf { e } _ { \gamma } ( k ) , \end{align*}


\begin{align*} 0 = \partial _ \eta G = \partial _ \eta g a t z _ { m a x } . \end{align*}


\begin{align*} B _ \mathcal { A } : = \begin{pmatrix} A _ { 1 3 } & \frac { 1 } { 2 } I _ { d \times d } - A _ { 1 1 } \\ \frac { 1 } { 2 } I _ { d \times d } & - A _ { 2 1 } \end{pmatrix} , \end{align*}


\begin{align*} \overline { \omega ( u ) } = \overline { \omega ( u ) | _ \mathsf { D } } \ , . \end{align*}


\begin{align*} e ^ { - t P _ 1 } ( x , y ) = e ^ { - t P _ 2 } ( x , y ) \ , ( t , x , y ) \in ( 0 , \infty ) \times U \times U . \end{align*}


\begin{align*} W ( T ) = \cap \{ W ( U ) : U \hbox { i s a u n i t a r y d i l a t i o n o f } T \} . \end{align*}


\begin{align*} u _ { n + 1 } ^ { \prime } = z ^ { 2 } u _ { n } ^ { \prime } , n \geq 0 . \end{align*}


\begin{align*} \mathbf { v } ( k + \mathrm { i } 0 ) = \mathbf { v } ( k - \mathrm { i } 0 ) \mathbf { J } ( k , t ) , \ \ \ \operatorname { I m } k = 0 . \end{align*}


\begin{align*} \sum _ { i , j = 1 } ^ n ( \nu _ i W _ { i j } ) ( y , t ) \underline { D } _ j \eta ( y , t , r ) & = ( W ^ T \nu ) ( y , t ) \cdot \nabla _ \Gamma \eta ( y , t , r ) = 0 , \\ \sum _ { i , j = 1 } ^ n ( W _ { i j } W _ { j i } ) ( y , t ) & = \sum _ { i , j = 1 } ^ n W _ { i j } ( y , t ) ^ 2 = | W ( y , t ) | ^ 2 \end{align*}


\begin{align*} | \overline { N } _ G ( s , T ) ^ F | = \sum _ { \upsilon \in j _ { G _ s } ^ { - 1 } ( \omega ) } \frac { | W _ G ( T ) ^ F | } { | W _ { G _ s } ( T _ \upsilon ) ^ F | } , \end{align*}


\begin{align*} \limsup _ { R \to \infty } \left ( \max _ { | z | = R } | W ( z ) | \right ) \exp \left ( - \frac { R ^ 2 } { 2 } \right ) ( R + 1 ) ^ { 3 / p } \le 1 , \end{align*}


\begin{align*} \delta \left ( \alpha \right ) \left [ \psi \left ( \mathrm { i } \alpha \right ) - \int K \left ( \alpha , s \right ) \mathrm { d } \mu \left ( s \right ) \right ] + \mu ^ { \prime } \left ( \alpha \right ) = \widetilde { \delta } \left ( \alpha \right ) \widetilde { \psi } \left ( \mathrm { i } \alpha \right ) . \end{align*}


\begin{align*} \eta : = e ^ { - \psi } - \frac { 1 } { \pi } \int _ { \mathbb { D } } e ^ { - \psi } \end{align*}


\begin{align*} | \nabla _ s ( t ^ { \frac { d } { 2 } } u ( t x ) ) | _ 2 ^ 2 & = \int _ { \mathbb { R } ^ d \times \mathbb { R } ^ d } \frac { t ^ d | u ( t x ) - u ( t y ) | ^ 2 } { | x - y | ^ { d + 2 s } } d x d y \\ & = t ^ { 2 s } \int _ { \mathbb { R } ^ d \times \mathbb { R } ^ d } \frac { | u ( t x ) - u ( t y ) | ^ 2 } { | t x - t y | ^ { d + 2 s } } d ( t x ) d ( t y ) \\ & = t ^ { 2 s } | \nabla _ s u | _ 2 ^ 2 . \end{align*}


\begin{align*} \mathcal { K } _ { n , m } [ 1 , z _ i ^ a \overline { z } _ i ^ b ] = \sum _ { l + k = b } \frac { 1 } { \pi ^ { k } } \frac { a ! b ! } { ( a - k ) ! l ! k ! } z _ i ^ { a - k } \overline { z } ' _ i { } ^ { l } . \end{align*}


\begin{align*} \langle \widehat { \varphi } _ i , f \rangle = \langle \varphi _ i , \Phi ^ * ( f ) \rangle \rightarrow \langle \varphi _ \infty , \Phi ^ * ( f ) \rangle = \langle \widehat { \varphi } _ \infty , f \rangle , \end{align*}


\begin{align*} \int _ X \mu = \int _ Y f _ * \mu , \end{align*}


\begin{align*} { \rm e m p } _ { N } ^ { \delta } = { \rm e m p } _ { N } ( X _ { N } ^ { \delta } ) . \end{align*}


\begin{align*} w ^ f ( t , x ) = \int _ 0 ^ t G ( t - s , P ) f ( s , x ) \ , d s , \end{align*}


\begin{align*} h _ d ( n ) = \hat { h } _ d ( n ) : = \sigma _ d \sqrt { 2 n ( 1 + 1 _ { \{ d = 5 \} } \log n ) \log _ 2 n } \ , , d \ge 5 , \end{align*}


\begin{align*} \partial _ t \psi + ( u \partial _ x + v \partial _ y ) \psi - \partial _ y ^ 2 \psi = 0 . \end{align*}


\begin{align*} \phi ( [ x , y , z ] ) = [ r x , r y , r z ] . \end{align*}


\begin{align*} f ( x ) = \sum _ { i < j } \mid \mid \pmb { x } _ { i } - \pmb { x } _ { j } \mid \mid ^ { - 2 } \end{align*}


\begin{align*} \mathbf { P } _ n ^ { \lambda , T } = \sum _ { \theta \in \{ 0 , 1 \} ^ 2 } \frac { Z _ { n , \theta } ( \lambda , T ) } { Z _ n ( \lambda , T ) } \mathbf { P } _ n ^ { \lambda , \theta , T } . \end{align*}


\begin{align*} R _ r = \varprojlim _ { L } R _ { r , L } \ , . \end{align*}


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


\begin{align*} \int _ \Omega \mathrm { R e } \ , h = \mathrm { R e } \ , \int _ \Omega { h } = | \Omega | \cdot \mathrm { R e } \ , h ( 0 ) = 0 . \end{align*}


\begin{align*} & I _ j = \{ k \in [ 1 , \mu _ j ] : \eta _ { j k } \in \Gamma _ G \cdot \chi _ { j l } \vartheta _ j \textrm { f o r s o m e } l \in [ 1 , \lambda _ j ] \} , \\ & I _ j ' = \{ k \in [ 1 , \mu ' _ j ] : \eta ' _ { j k } \in \Gamma _ G \cdot \chi ' _ { j l } \vartheta _ j ' \textrm { f o r s o m e } l \in [ 1 , \lambda ' _ j ] \} . \end{align*}


\begin{align*} L _ \tau = \begin{pmatrix} I _ { d \times d } & \tau I _ { d \times d } \\ I _ { d \times d } & - ( 1 - \tau ) I _ { d \times d } \end{pmatrix} \end{align*}


\begin{align*} \mathbb { X } ^ { N } _ t = 1 + \int _ 0 ^ t \mathbb { X } ^ { N } _ { s ^ - } \otimes d \mathbf { X } _ s + \sum _ { 0 < s \leq t } \mathbb { X } ^ { N } _ { s ^ - } \otimes \left ( - \frac 1 2 ( \Delta X _ s ) ^ { \otimes 2 } + \sum _ { k = 2 } ^ N \frac { 1 } { k ! } ( \Delta X _ s , A n t i ( \Delta \mathbb { X } _ s ) ) ^ { \otimes k } \right ) . \end{align*}


\begin{gather*} \circlearrowleft _ { x , y , z } B \left ( f ( x , y ) , g ( [ \alpha ( z ) , t ] ) \right ) = \circlearrowleft _ { x , y , z } B \left ( f ( \alpha ( x ) ) , \alpha ( y ) , g ( [ z , a ] ) ) \right ) . \end{gather*}


\begin{align*} \dfrac { \lambda } { \Lambda } = \dfrac { \lambda \ , \Phi } { \Lambda \ , \Phi } \leq C : = \dfrac { \left | A ^ { 1 / 2 } \dfrac { z } { | z | } \right | ^ 2 } { \left | A ^ { 1 / 2 } \dfrac { e } { | e | } \right | ^ 2 } \leq \dfrac { \Lambda \ , \Phi } { \lambda \ , \Phi } = \dfrac { \Lambda } { \lambda } . \end{align*}


\begin{align*} \Tilde { \omega } = d \log r \wedge d y ^ 2 + d \theta \wedge d x ^ 2 + \omega _ 0 \end{align*}


\begin{align*} 1 - K = | z | ^ { 2 k _ j } u _ 0 , \end{align*}


\begin{align*} I _ n - d ( x ) \overline { W } ( x ) = I _ n - r W ( y ) \end{align*}


\begin{align*} \widehat Z = \left [ \begin{array} { c } Z \\ Z ' \end{array} \right ] \in \Omega ' , Z \in \C ^ { m , n - k } , \ Z ' \in \C ^ { n - m , n - k } \end{align*}


\begin{align*} u ( x , 0 ) = f ( x ) , a \leq x \leq b \end{align*}


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


\begin{align*} \displaystyle \lim _ { \eta \to 1 } \partial _ \xi w = \displaystyle \lim _ { \eta \to 1 } \partial _ \tau w = 0 , \end{align*}


\begin{align*} \nabla \bar { \eta } ( y ) = \nabla _ \Gamma \eta ( y ) , \partial _ i \bar { \eta } ( y ) = \underline { D } _ i \eta ( y ) , y \in \Gamma , \ , i = 1 , \dots , n \end{align*}


\begin{align*} \det \limits _ { x , y \in \mathbb { T } _ { m _ 1 , m _ 2 } } K _ m ^ { A , \theta } ( x , y ) = \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*} & \psi _ { l , k } : = \left [ \begin{array} { c c c c } \psi _ { l , k + 1 } ( \sqrt { \bar { \alpha } } \mathcal { A } _ { 1 , k } + \sqrt { \bar { \alpha } } \mathcal { A } _ { 2 , k } ) \\ \psi _ { l , k + 1 } \sqrt { 1 - \bar { \alpha } } \mathcal { A } _ { 1 , k } \\ \psi _ { l , k + 1 } ( \sqrt { \bar { \alpha } } \mathcal { C } _ { 1 } + \sqrt { \bar { \alpha } } \mathcal { C } _ { 2 , k } ) \\ \psi _ { l , k + 1 } \sqrt { 1 - \bar { \alpha } } \mathcal { C } _ { 1 } \\ \end{array} \right ] , \end{align*}


\begin{align*} s = s ^ { ( 0 ) } \times s ^ { ( - ) } \times s ^ { ( + ) } \end{align*}


\begin{align*} \partial _ t u = \lambda _ 1 u \times \mathcal { H } ( u ) - \lambda _ 2 u \times ( u \times \mathcal { H } ( u ) ) \ , , \end{align*}


\begin{align*} | \widehat { f } _ \# ( k ) | = \left | \int _ { - L / 2 } ^ { L / 2 } e ^ { - i 2 \pi k x } f ( x ) d x \right | \leq \int _ { - L / 2 } ^ { L / 2 } \underbrace { \left | e ^ { - i 2 \pi k x } \right | } _ { \leq 1 } | f ( x ) | d x \leq \pi \sup _ { x \in \R } | \langle x \rangle ^ 2 f ( x ) | . \end{align*}


\begin{align*} \eta ( r ) = \left \{ \begin{array} { l l } 1 , r _ { 1 } < r < \rho , \\ 0 , r \leq r _ { 0 } , r \geq R , \end{array} \right . \end{align*}


\begin{align*} \underset { n \rightarrow + \infty } { \lim } \int _ { G } ( R _ { \alpha } \ast | u _ { \lambda _ n } | ^ p ) | u _ { \lambda _ n } | ^ p \ , d \mu = \int _ { G } ( R _ { \alpha } \ast | u | ^ p ) | u | ^ p \ , d \mu . \end{align*}


\begin{align*} d _ { Q } \Sigma \big ( P , Q \big ) : = d x _ Q \frac { \partial } { \partial x _ Q } \Sigma \big ( P , Q \big ) \end{align*}


\begin{align*} E _ W ( \rho ) = \frac { 1 } { 2 } \int _ M \int _ M W ( x , y ) d \rho ( x ) d \rho ( y ) . \end{align*}


\begin{align*} \zeta = \frac { a / s } { t } = \frac { a / s } { t ' / s ' } . \end{align*}


\begin{align*} \begin{aligned} u _ \ell \ & = \ x _ 1 \big ( \textstyle \prod _ { s = \ell } ^ { d - 2 } x _ { n - s t - 1 } \big ) \big ( \prod _ { s = 0 } ^ { \ell - 1 } x _ { n - s t } \big ) \\ & = \ x _ 1 x _ { n - ( d - 2 ) t - 1 } \cdots x _ { n - \ell t - 1 } x _ { n - ( \ell - 1 ) t } \cdots x _ { n - t } x _ n . \end{aligned} \end{align*}


\begin{align*} J _ { X , t } = \ , J _ { X ( 0 , t , \cdot ) } \R ^ n t \in [ 0 , T ] \ , \end{align*}


\begin{align*} \R ^ { j } G ( F M ) [ x _ j ] = \R ^ j G ( F M ) , \end{align*}


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


\begin{align*} \beta : = \ , \frac { \ell \ , p ^ 2 } { p - 1 } < 1 . \end{align*}


\begin{align*} \mathcal { E } = \mathcal { L } _ 0 \oplus \cdots \oplus \mathcal { L } _ N / S \end{align*}


\begin{align*} \left ( \alpha ( t , x ) \oplus \alpha ( s , x ) \right ) \oplus \alpha ( r , x ) = \alpha ( ( t + s ) + r , x ) = \alpha ( t + ( s + r ) , x ) \\ = \alpha ( t , x ) \oplus \left ( \alpha ( s , x ) \oplus \alpha ( r , x ) \right ) \ , . \end{align*}


\begin{align*} ( a , b ) \otimes ( c , d ) = ( a \sqcup c , b \sqcup d ) \ ; , \end{align*}


\begin{align*} U _ 0 ( t , s ) & = U _ 0 ( t , s + m _ t k _ \varepsilon T + m _ t ^ \star T ) U _ 0 ( s + m _ t k _ \varepsilon T + m _ t ^ \star T , s + m _ t k _ \varepsilon T ) U _ 0 ( s + m _ t k _ \varepsilon T , s ) \\ & = U _ 0 ( t - m _ t k _ \varepsilon T - m _ t ^ \star T , s ) U _ 0 ( s + m _ t ^ \star T , s ) U _ 0 ( s + m _ t k _ \varepsilon T , s ) \\ & = U _ 0 ( t - m _ t k _ \varepsilon T - m _ t ^ \star T , s ) U _ 0 ( s + T , s ) ^ { m _ t ^ \star } U _ 0 ( s + T , s ) ^ { m _ t k _ \varepsilon } . \end{align*}


\begin{align*} P \ , ( A - \lambda B ) \ , Q = \left [ \begin{array} { c c } R ( \lambda ) & 0 \\ 0 & S ( \lambda ) \end{array} \right ] , R ( \lambda ) = \left [ \begin{array} { c c } J - \lambda I _ r & 0 \\ 0 & I _ s - \lambda N \end{array} \right ] , \end{align*}


\begin{align*} I ( w a , f ) : = \int f ( n _ 1 ^ t w a n _ 2 ) \psi ^ { - 1 } ( n _ 1 n _ 2 ) d n _ 1 d n _ 2 , \end{align*}


\begin{align*} \int _ { G } ( R _ \alpha \ast u ) ( x ) v ( x ) \ , d \mu : = \underset { x , y \in G , x \neq y } { \sum } R _ \alpha ( x , y ) u ( y ) v ( x ) \leq C _ { r , s , \alpha , N } \| u \| _ r \| v \| _ s , \end{align*}


\begin{align*} & W \left \{ \operatorname { R e } \varphi \left ( \mathrm { i } \alpha - 0 \right ) , \psi \left ( \mathrm { i } \alpha \right ) \right \} \\ & = \operatorname { R e } W \left \{ \varphi \left ( \mathrm { i } \alpha - 0 \right ) , \psi \left ( \mathrm { i } \alpha \right ) \right \} = - 2 \alpha . \end{align*}


\begin{align*} \mathbb P ( f ^ * ) : \mathbb P ( V ) ^ * = { \rm P r o j } ( S ( V ^ * ) ) \dashrightarrow \mathbb P ( W ) ^ * = { \rm P r o j } ( S ( W ^ * ) ) . \end{align*}


\begin{align*} \eqref { 1 c o h o m o l o g y O c t 1 4 } & = B \Big ( d ^ 2 f ( x , y , z ) , g ( t ) \Big ) - \circlearrowleft _ { x , y , z } B \Big ( \rho \left ( \alpha ( x ) \right ) f ( y , z ) , g ( t ) \Big ) \\ + & \circlearrowleft _ { x , y , z } B \left ( \beta ( f ( [ x , y ] , t ) ) , g ( z ) \right ) + \circlearrowleft _ { x , y , z } B ( f ( \alpha ( a ) , \alpha ( z ) ) , g ( [ x , y ] ) ) . \end{align*}


\begin{align*} \sum _ 2 : = \sum _ { \ell > B _ { d , \mu } n } C _ d ^ n \left ( \frac { 2 \ell + n } { n } \right ) ^ { d n } e ^ { - \mu \ell } , \end{align*}


\begin{align*} \Delta \rho ^ \varepsilon ( x , t ) & = \varepsilon ^ { - 2 } \partial _ r ^ 2 \eta _ 0 ( r ) + \varepsilon ^ { - 1 } \{ - H \partial _ r \eta _ 0 ( r ) + \partial _ r ^ 2 \eta _ 1 ( r ) \} \\ & - r | W | ^ 2 \partial _ r \eta _ 0 ( r ) + \Delta _ \Gamma \eta _ 0 ( r ) - H \partial _ r \eta _ 1 ( r ) + \partial _ r ^ 2 \eta _ 2 ( r ) + O ( \varepsilon ) . \end{align*}


\begin{align*} c _ k \left ( \partial B ^ 4 ( a ) \right ) = d a , \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { N \to \infty } \frac { 1 } { N } \log \left ( \mathbf { \Pi } _ { N } \left ( \frac { | C | } { N } \in ( 1 - \epsilon , 1 + \epsilon ) \right ) \right ) = 0 . \end{align*}


\begin{align*} \lim _ { T _ 0 \to \infty } \frac { 1 } { T _ 0 } \ , \mathbb { E } \big [ \widetilde { v } _ { , , m } ( f ) \ , \widetilde { v } ^ * _ { , n } ( f ) \big ] & = 0 \quad \forall n , m , \\ \lim _ { T _ 0 \to \infty } \frac { 1 } { T _ 0 } \ , \mathbb { E } \big [ \widetilde { v } _ { , , m } ( f ) \ , \widetilde { v } _ { , m } ^ * ( f ) \big ] & = 0 \quad \forall m . \end{align*}


\begin{align*} \phi _ \alpha ( x ) = e ^ { - \alpha x } \end{align*}


\begin{align*} \sum _ { k = n } ^ \infty \mathrm { e } ^ { - q k ^ p } | h _ { k } ( x ) | ^ 2 & \lesssim \sum _ { k = n } ^ \infty \mathrm { e } ^ { - q k ^ p } k ^ { \frac { 1 } { 3 } - \frac { 1 } { \alpha } } . \end{align*}


\begin{align*} G = \left ( \begin{array} { c c c c } 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ \end{array} \right ) \end{align*}


\begin{align*} \partial _ t v - \Delta _ \Gamma w = 0 & { \rm o n ~ } \Sigma , \\ w = - \delta \Delta _ \Gamma v + \beta _ \Gamma ( v ) + \pi _ \Gamma ( v ) - g + \partial _ { \boldsymbol { \nu } } u & { \rm o n ~ } \Sigma , \\ v ( 0 ) = v _ 0 & { \rm o n ~ } \Gamma , \end{align*}


\begin{align*} L _ 1 \ge \frac 1 2 \frac d { d t } \sum \limits _ { | \alpha | = 0 } ^ { 2 } \left \| \sqrt { S ( { \bf { v } } ) } \partial _ \tau ^ \alpha \partial _ y { \bf { v } } \right \| _ { L ^ 2 } ^ 2 - C D ( t ) ^ { \frac 1 2 } E ( t ) - C f ( t ) E ( t ) . \end{align*}


\begin{align*} \begin{cases} \dot X _ t = b ( t , X _ t ) + \dot \omega _ t , \\ X _ 0 = x , \end{cases} \end{align*}


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


\begin{align*} \overline { \partial } f = \mu \ , \partial f , \end{align*}


\begin{align*} \left \{ \int _ { 0 } ^ { 1 } g _ t d D X _ t = 0 , \ g _ t \neq 0 \right \} \Rightarrow \left \{ D X _ t = 0 , \ \forall t \in [ l ( \omega ) , u ( \omega ) ] \right \} . \end{align*}


\begin{align*} \partial _ t \nabla d ( x , t ) = \nabla \partial _ t d ( x , t ) = - \nabla \Bigl ( \overline { V } _ \Gamma ( x , t ) \Bigr ) = - R ( x , t ) \overline { \nabla _ \Gamma V _ \Gamma } ( x , t ) . \end{align*}


\begin{align*} \dot { f } _ t ( z ) = - f _ t ' ( z ) ( \tau ( t ) - z ) ( 1 - \overline { \tau ( t ) } z ) p ( z , t ) \end{align*}


\begin{align*} u ( t + \varepsilon \ , , x ) - u ( t \ , , x ) = J ( t \ , , x \ , ; \varepsilon ) + [ I ( t + \varepsilon , x ) - I ( t , x ) ] + [ H ( t + \varepsilon \ , , x ) - H ( t \ , , x ) ] , \end{align*}


\begin{align*} f ( O ( \gamma _ { \sigma _ 1 } ) ) = f ( T _ 1 \cdot \gamma _ { \sigma _ 1 } ) = T _ 1 \cdot f ( \gamma _ { \sigma _ 1 } ) \subset T _ 2 \cdot f ( \gamma _ { \sigma _ 1 } ) = O ( \gamma _ { \sigma _ 2 } ) \end{align*}


\begin{align*} v _ 0 = \left [ \frac { v - 1 } { 2 } \right ] . \end{align*}


\begin{align*} \sum _ { i , j = d + 1 } ^ n ( \partial _ { \varphi _ { i j } } u ) ( \partial _ { \varphi _ { i j } } v ) = \sum _ { i , j = d + 1 } ^ n \Big \{ \frac { t _ i ^ 2 } { | t | ^ 2 } ( \partial _ { t _ j } u ) ( \partial _ { t _ j } v ) - 2 \frac { t _ i t _ j } { | t | ^ 2 } ( \partial _ { t _ i } u ) ( \partial _ { t _ j } v ) + \frac { t ^ 2 _ j } { | t | ^ 2 } ( \partial _ { t _ i } u ) ( \partial _ { t _ i } v ) \Big \} . \end{align*}


\begin{align*} \gamma ^ \varepsilon \geq c _ 4 \cdot c _ 3 \varepsilon - c _ 2 \varepsilon = \varepsilon > 0 \quad \partial _ \ell Q _ { \varepsilon , T } . \end{align*}


\begin{align*} \widetilde { \psi } \left ( \mathrm { i } \alpha \right ) = \psi \left ( \mathrm { i } \alpha \right ) - \int K \left ( \alpha , s \right ) \mathrm { d } \mu \left ( s \right ) . \end{align*}


\begin{align*} d g g ^ { - 1 } = : \sum _ { a = 1 } ^ { 3 } \psi ^ { a } R ( \textbf { e } _ { a } ) \end{align*}


\begin{align*} P _ \ell ( X ) = \ell - a _ \ell ( g ) X + X ^ 2 . \end{align*}


\begin{align*} \mathbb { E } [ ( X ( x + 1 , t ) - X ( x - 1 , t ) ) ^ { 2 k } ] = \sum _ { ( x _ 1 , s _ 1 ) , \ldots , ( x _ { 2 k } , s _ { 2 k } ) } \mathbb { E } \left [ \prod _ { i } \Delta ( x _ i , s _ i ) \xi ( x _ i , s _ i ) \prod _ i \Gamma ( x _ i , s _ i ) \right ] . \end{align*}


\begin{align*} R _ { s , t } : = Y _ { s , t } - Y _ s ' X _ { s , t } , \ ( s , t ) \in \Delta _ 1 , \end{align*}


\begin{align*} \tilde E _ { a , s } = h _ s - ( \tilde \Delta _ a - \lambda _ s ) ^ { - 1 } ( \Delta - \lambda _ s ) h _ s . \end{align*}


\begin{align*} - \int _ 0 ^ T \int _ { \R ^ d } \partial _ j \varphi ( x ) u ( s , x ) \d x \ ; \psi ( s ) \d s & = - \lim _ n \int _ 0 ^ T \int _ { \R ^ d } D ^ j _ { - h _ n } \varphi ( x ) u ( s , x ) \d x \ ; \psi ( s ) \d s \\ & = \lim _ n \int _ 0 ^ T \int _ { \R ^ d } \varphi ( x ) D ^ j _ { h _ n } u ( s , x ) \d x \ ; \psi ( s ) \d s . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ 4 \ell _ i = q _ 1 ^ 2 - q _ 2 ^ 2 ; \end{align*}


\begin{align*} \int _ { \R ^ d } G ( \Psi _ { n } ^ { 1 } ) d x = \int _ { \R ^ d } G ( u _ n ) d x - \int _ { \R ^ d } G ( u ^ 0 ) d x + o ( 1 ) . \end{align*}


\begin{align*} F ( s ) & = \frac { 2 e ^ { \gamma } } { s } \left ( 1 + \int _ 2 ^ { s - 1 } \frac { \log ( t - 1 ) } { t } \mathrm { d } t + \int _ 2 ^ { s - 3 } \frac { \log ( t - 1 ) } { t } \left ( \int _ { t + 2 } ^ { s - 1 } \frac { 1 } { u } \log \frac { u - 1 } { t + 1 } \mathrm { d } u \right ) \mathrm { d } t \right ) \end{align*}


\begin{align*} f ( \zeta ) \longmapsto ( N _ { - k _ 0 } f ) ( \zeta ) = f \left ( \frac { \zeta } { \sqrt { - 4 8 t k _ 0 } } - k _ 0 \right ) . \end{align*}


\begin{align*} \mathcal { T } ^ * ( x , t ) : = t ^ { p _ * ( x ) } + \mu ( x ) ^ { \frac { q _ * ( x ) } { q ( x ) } } t ^ { q _ * ( x ) } , ( x , t ) \in \overline { \Omega } \times [ 0 , \infty ) , \end{align*}


\begin{align*} \begin{gathered} ( A \otimes B ) _ S = A _ S \oplus B _ S \ ; , \\ ( A \otimes B ) _ \sigma ( ( \beta , g ) ) = \begin{cases} A _ \sigma ( g ) & \ \beta = 0 \\ B _ \sigma ( g ) & \ \beta = 1 \\ \end{cases} \ ; . \end{gathered} \end{align*}


\begin{align*} X \smallsetminus { \downarrow } ^ { \mathbb { X } } ( p ^ { - 1 } [ { \downarrow } ^ { \mathbb { Y } } U ] ) = X \smallsetminus { \downarrow } ^ { \mathbb { X } } ( X \smallsetminus { \downarrow } ^ { \mathbb { X } } ( p ^ { - 1 } [ Y \smallsetminus U ] ) ) . \end{align*}


\begin{align*} \varphi ( x _ 1 \cdots x _ d ) = \xi _ 1 ( x _ 1 ) \cdots \xi _ d ( x _ d ) ( 1 ) x _ 1 , . . . , x _ d \in G . \end{align*}


\begin{align*} \langle \Lambda _ { M _ 1 } , \Lambda _ { M _ 2 } \rangle = | M _ 1 | | M _ 2 | + | M _ 1 \cap M _ 2 | \pmod 2 . \end{align*}


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


\begin{align*} w ( S ) \ge \sum _ { i = 1 } ^ t w ( S _ i ) . \end{align*}


\begin{align*} F ^ 1 \circ \widetilde { Q } ( v _ 1 \vee \cdots \vee v _ n ) & = F ^ 1 ( Q _ 0 \vee v _ 1 \vee \cdots \vee v _ n + Q ( v _ 1 \vee \cdots \vee v _ n ) ) \\ & = F _ { n + 1 } ^ 1 ( Q _ 0 \vee v _ 1 \vee \cdots \vee v _ n ) + ( Q ' ) ^ 1 \circ F ( v _ 1 \vee \cdots \vee v _ n ) \\ & \stackrel { ! } { = } \widetilde { Q ' } ^ 1 \circ F ( v _ 1 \vee \cdots \vee v _ n ) . \end{align*}


\begin{align*} f : X \to X / G = : Y , \end{align*}


\begin{align*} \frac { d H ( x ( t ) , \xi _ 0 ( t ) , \xi ( t ) ) } { d t } = \frac { \partial H } { \partial x } \frac { \partial H } { \partial \xi } + \frac { \partial H } { \partial \xi } \Big ( - \frac { \partial H } { \partial x } \Big ) + \frac { \partial H } { \partial \xi _ 0 } \frac { d \xi _ 0 } { d t } = 0 . \end{align*}


\begin{align*} \dim G _ B \geq \dim G _ { B _ 1 \times B _ 2 } = 3 \end{align*}


\begin{align*} \rho _ { N } ( x ) = \frac { 1 } { \omega _ { N } } \exp \left ( - 2 N \beta \zeta ( x ) \right ) . \end{align*}


\begin{align*} q _ T : = \begin{cases} 0 & \textbf { \textit { y } } ( T ) + \textbf { \textit { y } } ( \delta ( T ) ) = 0 \\ \frac { \textbf { \textit { y } } ( T ) } { \textbf { \textit { y } } ( T ) + \textbf { \textit { y } } ( \delta ( T ) ) } & \end{cases} \end{align*}


\begin{align*} \langle 0 \mid \sum _ { \sigma } s g n ( \sigma ) l _ { \sigma ( 1 ) , 1 } ^ { - } ( u ) \cdots l _ { \sigma ( j ) , i } ^ { - } ( u + ( i - 1 ) h ) = 0 \end{align*}


\begin{align*} \langle x \rangle = ( 1 + x ^ 2 ) ^ { 1 / 2 } , \end{align*}


\begin{align*} { \mathcal { K } } ( x ) = \left ( \begin{array} { c c } p + x + 1 & 0 \\ 0 & p - x - 1 \end{array} \right ) \ , , \end{align*}


\begin{align*} n - k - \frac { n } { r + 1 } = d - 2 - \lfloor \frac { d - 2 } { r + 1 } \rfloor \end{align*}


\begin{align*} R i c _ { B } + \nabla _ { B } ^ { 2 } \varphi - \frac { m } { f } \nabla _ { B } ^ { 2 } f = \Lambda g _ { B } , \end{align*}


\begin{align*} \kappa = \frac { A } { 2 } { \rm e x p } \left \{ - \frac { 1 } { 2 \pi i } { \rm p . v . } \int _ { - \infty } ^ { \infty } \frac { { \rm l o g } \frac { s ^ 2 } { 1 + s ^ 2 } \left ( 1 - b ^ 2 ( s ) \right ) } { s } d s \right \} . \end{align*}


\begin{align*} \mathbf { E } _ { \overline { \mathbf { P } } } [ F ] = \frac { 1 } { | \Omega | } \int _ { \Omega } \mathbf { E } _ { \overline { \mathbf { P } } ^ { x } } [ F ( x , \cdot ) ] \ , d x . \end{align*}


\begin{align*} c _ k ( \partial E ( a , b ) ) = N _ k ( a , b ) , \end{align*}


\begin{align*} N _ { s } ( v ' , u ' ) : = \textbf { M } ' _ { x } ( v , u ) = f ( x ) \textbf { M } _ { x } ( v , u ) \end{align*}


\begin{align*} T ^ * = \sup \{ s \in [ 0 , T ] | w > 0 f o r ( \tau , \xi , \eta ) \in [ 0 , s ] \times [ 0 , X ] \times \{ \eta = 0 \} \} \end{align*}


\begin{align*} L _ z ( f ) : = f ( z ) , \ \ \ L _ { z , j } ( f ) : = \frac { \partial { f } } { \partial { x _ j } } ( z ) , \ \ \ f \in { A ^ p ( \Omega ) } \end{align*}


\begin{align*} \mathbf { \Pi } ^ { \mu } \left \{ | C | ( B ) = n \right \} = \frac { ( m ( B ) ) ^ { n } } { n ! } \exp \left ( - \lambda | B | \right ) , \end{align*}


\begin{align*} V = k v _ 0 \oplus \cdots \oplus k v _ n . \end{align*}


\begin{align*} c _ A ( Z ) : = \min \{ \dim ( Z ) - \dim \mathcal { O } _ M \ , | \ , M \in Z \} \in \{ 0 , 1 \} . \end{align*}


\begin{align*} f ^ { h ; w } : = \sum c _ a x ^ a t ^ { \max \{ w \cdot b : c _ b \not = 0 \} - w \cdot a } . \end{align*}


\begin{align*} \beta \pi _ 1 ^ * f = \pi _ 2 ^ * f . \end{align*}


\begin{align*} \int _ { \mathbb { D } } \frac { | h ( w ) | ^ p } { | w | ^ { p k _ p } } \leq \liminf _ { k \rightarrow \infty } \int _ { \mathbb { D } } \frac { | g _ { z _ { j _ k } } ( w ) | ^ p } { | w | ^ { p k _ p } } = \frac { 1 } { K _ { p , \varphi _ p } ( 0 ) } . \end{align*}


\begin{align*} \frac { d _ n } { r _ n } \to + \infty , r _ n \to 0 , \frac { M _ n ^ { \gamma - 1 } } { r _ n ^ { \gamma - 2 } } = \left ( \frac { M _ n } { r _ n ^ { \alpha _ 0 } } \right ) ^ { \gamma - 1 } \to + \infty , \end{align*}


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


\begin{align*} \Omega = & k + g _ 1 ( z ) - z g _ 1 ^ \prime ( z ) + \dfrac { 1 } { 2 } \displaystyle \int ^ { r ^ 2 - s ^ 2 } _ 0 g _ 6 ( \xi ) d \xi > 0 , \ ; \forall z \in \mathbb { R } , \ ; r \geq | s | ; \\ \Lambda = & \Omega \phi _ { z z } + ( r ^ 2 - s ^ 2 ) g _ 6 ( r ^ 2 - s ^ 2 ) \phi _ { z z } > 0 , \ ; \forall z \in \mathbb { R } , \ ; r \geq | s | . \end{align*}


\begin{align*} ^ { \rho } D _ { a ^ + } ^ { \alpha , \beta } \ ^ { \rho } I _ { a ^ + } ^ { \alpha } g = g , x \in [ a , b ] . \end{align*}


\begin{align*} \mathcal { I } ( g ) = \int _ 0 ^ \infty t ^ { 2 s } ( \Theta ( t g ) - 1 ) \frac { d t } { t } = \int _ 0 ^ \infty t ^ { 2 s } \left ( \sum _ { 0 \neq v \in \Z ^ r } e ^ { - \pi | v \cdot t g | ^ 2 } \right ) \frac { d t } { t } \end{align*}


\begin{align*} z _ 0 : = \mathcal { T } _ r ( u _ 0 ) \ , , \end{align*}


\begin{align*} ( L _ { y _ 1 } \pi ) ^ { \varepsilon _ 1 } ( L _ { y _ 2 } \pi ) ^ { \varepsilon _ 2 } \cdots ( L _ { y _ k } \pi ) ^ { \varepsilon _ k } ( \widetilde 0 ) = \alpha \pi ^ k ( \widetilde 0 ) . \end{align*}


\begin{align*} f \circ H ^ s _ { x , y } = H ^ s _ { f ( x ) , f ( y ) } \circ f , \end{align*}


\begin{align*} K _ n \circ Q _ { B , n } ^ n = Q _ { B , n } ^ n \circ K _ n K _ n \circ \tilde { H } _ n = \tilde { H } _ n \circ K _ n , \end{align*}


\begin{align*} [ L _ { m , i } , L _ { n , j } ] = ( n ( i + q ) - m ( j + q ) ) L _ { m + n , i + j } \end{align*}


\begin{align*} A \subseteq \bigcup _ { j = 1 } ^ n U _ j \mbox { a n d } \bigcup _ { j = 1 } ^ n ( U _ j \times U _ j ) \subseteq U \ , . \end{align*}


\begin{align*} \vec { W } ^ r ( \alpha , \beta ; 0 ) = \vec { W } ^ r ( \alpha , \beta ; r ) = \vec { W } ^ r ( \alpha , \beta ) . \end{align*}


\begin{align*} d _ M ( x , z ) & \leqslant d _ M ( x , f ( x ) ) + d _ M ( f ( x ) , y ) + d _ M ( y , f ( z ) ) + d _ M ( f ( z ) , z ) \\ & = d _ M ( x , f ( x ) ) + d _ M ( f ( x ) , \eta ( x ) ) + d _ M ( \eta ( z ) , f ( z ) ) + d _ M ( f ( z ) , z ) \\ & \leqslant 4 \delta . \end{align*}


\begin{align*} \int _ 0 ^ 1 \frac { 1 } { \omega ( t ) } \dd t = + \infty . \end{align*}


\begin{align*} \beta = 2 - \frac 2 { \pi } \arcsin \frac { 2 \sqrt { p - 1 } } { p } \ , . \end{align*}


\begin{align*} \overline { g } ( { x } ) = \lim \limits _ { t \to + \infty } \omega ( t , y ; { x } ) \end{align*}


\begin{align*} \left \{ \begin{array} { l } - i \partial _ t u + \sqrt { m ^ 2 - \Delta } u = ( V _ b * | u | ^ 2 ) u , \\ u | _ { t = 0 } = u _ 0 \end{array} \right . \end{align*}


\begin{align*} f ( 2 y ) g ( ( I + \tilde \alpha ) y ) = g ( ( I - \tilde \alpha ) y ) , \ \ y \in Y . \end{align*}


\begin{align*} \mu _ { S } ( M ) = \mu _ { S } ( A n n _ { S } M ) \end{align*}


\begin{align*} \lambda ( ( P \oplus _ m P _ t ) [ S ] ) & = \lambda _ { \widehat { \mathcal { E } } _ { t } [ S ] } ( x _ { s _ 1 } , \dots , x _ { s _ k } ) \\ & = \lambda _ { \widehat { \mathcal { E } } _ t } ( x ' _ 1 , \dots , x ' _ { m + m _ t - 1 } ) \\ & = \lambda _ { \mathcal { E } } ( x ' _ 1 , \dots , x ' _ { m - 1 } , y ) + \lambda _ { \mathcal { E } _ { t } } \left ( x ' _ { m } , \dots , x ' _ { m + m _ t - 1 } \right ) . \end{align*}


\begin{align*} A ^ * P + P A - P B B ^ * P + C ^ * C = 0 . \end{align*}


\begin{align*} z & = x _ 1 x _ { n - d + 1 } \cdots x _ { n - \ell - 2 } \cdot x _ { n - \ell } \cdots x _ { n - 1 } x _ n \\ & = x _ 1 \big ( \textstyle \prod _ { s = \ell + 1 } ^ { d - 2 } x _ { n - s - 1 } \big ) \big ( \prod _ { s = 0 } ^ { \ell } x _ { n - s } \big ) . \end{align*}


\begin{align*} \begin{aligned} & A ( U ) v = A ( \overline { u } ) v , & & A ( U ) \overline { v } = A ( u ) \overline { v } , & & & V \in N ^ { M | H } , \\ & A ( U ) v = A ( u ) v , & & A ( U ) \overline { v } = A ( \overline { u } ) \overline { v } , & & & V \in T H . \end{aligned} \end{align*}


\begin{align*} \widetilde { V } _ { 0 , n } \le 2 \sum _ { x \in \R _ n } \sum _ { y \in \R ( n , \infty ) } G ( x , y ) \stackrel { d } { = } 2 \sum _ { x \in \R _ n } \sum _ { y \in \hat { \R } _ \infty } G ( x , y ) , \ , \end{align*}


\begin{align*} \mathcal { A } = \bigcup _ { b = 1 } ^ { t } A _ b = \{ a _ 1 , a _ 2 , \cdots , a _ n \} . \end{align*}


\begin{align*} \phi \left ( t ; z \right ) \partial _ { t } S + \psi \left ( t ; z \right ) S = \left [ 2 \phi \left ( t ; z \right ) - 1 \right ] u _ { 0 } \left ( z \right ) + 2 u _ { 1 } \left ( z \right ) , \end{align*}


\begin{align*} E _ { \mathfrak { s } } ( z ) = | z | ^ 2 _ { H _ x ^ { \mathfrak { s } } } - \mathfrak { s } \tfrac { \kappa } { 2 } | z | ^ 4 _ { L _ x ^ 4 } , \end{align*}


\begin{align*} 0 = L ( \partial _ \xi w + \partial _ \tau w ) = v L _ 0 g _ { s u m } + g _ { s u m } L v + 2 w ^ 2 \partial _ { \eta } v \partial _ { \eta } g _ { s u m } i n D _ { T ^ * } , \end{align*}


\begin{align*} D . L _ P > 0 \quad i = 1 , \ldots , s . \end{align*}


\begin{align*} ( X _ - , \omega _ - ) \circ ( X _ + , \lambda _ + ) = X _ - \cup _ { Y _ 0 } Y _ + \end{align*}


\begin{align*} \partial \operatorname * { t r } \log \left ( I + \mathbb { A } \right ) = \operatorname * { t r } \left ( I + \mathbb { A } \right ) ^ { - 1 } \partial \mathbb { A } , \end{align*}


\begin{align*} \ell _ { i , k } ^ { - 2 } & = \big ( \ell _ { i , j } ^ { 2 } \underbrace { ( s _ { k } \hdots s ^ { 2 } _ { i } \hdots s _ { k } ) \hdots ( s _ { j + 1 } \hdots s ^ { 2 } _ { i } \hdots s _ { j + 1 } ) } _ { k - j \ , \mbox { \scriptsize t e r m s } } \big ) ^ { - 1 } \\ \ell _ { a , p } ^ { - 2 } & = \big ( \underbrace { ( s _ { j + 1 } \hdots s ^ { 2 } _ { p } \hdots s _ { j + 1 } ) \hdots ( s _ { a } \hdots s ^ { 2 } _ { p } \hdots s _ { a } } _ { j + 2 - a \ , \mbox { \scriptsize t e r m s } } ) \ell _ { j + 2 , p } ^ { 2 } \big ) ^ { - 1 } \end{align*}


\begin{align*} \left \{ \begin{aligned} \omega _ { y } + ( u + a y ) v _ { y } - ( v + b ) u _ { y } - a v & = p _ { x } , \\ - \omega _ { x } - ( u + a y ) v _ { x } + ( v + b ) u _ { x } & = p _ { y } , \\ u _ { x } + v _ { y } & = 0 . \end{aligned} \right . \end{align*}


\begin{align*} X _ k = h _ k ( Y _ 1 , \ldots , Y _ k , R ) \forall k \in \{ 1 , 2 , 3 , \ldots \} \end{align*}


\begin{align*} \limsup _ { k \rightarrow \infty } n ! k ^ { - n } \dim H _ { ( 2 ) } ^ { j } ( X _ c , L ^ k ) \leq \int _ { K ' ( j , h ^ L _ \chi ) } ( - 1 ) ^ j c _ 1 ( L , h _ \chi ^ L ) ^ n \leq \int _ { X _ c ( j , h ^ L _ \chi ) } ( - 1 ) ^ j c _ 1 ( L , h _ \chi ^ L ) ^ n . \end{align*}


\begin{align*} E [ | S _ i - x | _ + ^ { - 2 } ] \le C E [ G ( S _ i , x ) ] = C \sum _ { \ell = i } ^ \infty P ( S _ \ell = x ) . \end{align*}


\begin{align*} \exp \Bigl ( a \ , \frac { \partial } { \partial x } \Bigr ) \ , f ( x ) \Big | _ { x = 0 } = f ( a ) \end{align*}


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


\begin{align*} \Z ^ d = \bigcup _ { \alpha } \Omega _ \alpha \ , \end{align*}


\begin{align*} u ( t ) = U ( t , s ) u ( s ) + j ^ { - 1 } \int _ s ^ t U ^ { \odot \star } ( t , \tau ) f ( \tau ) d \tau , \forall t \in I , \end{align*}


\begin{align*} D ^ + V ( \phi ( t , x , u ) ) = \dot { V } _ { u ( \cdot + t ) } ( \phi ( t , x , u ) ) \leq a V ( \phi ( t , x , u ) ) + M . \end{align*}


\begin{align*} \underset { k = i \kappa } { \rm R e s } \breve { N } ^ { ( 1 ) } ( x , t , k ) = c _ 2 ( x , t ) \breve { N } ^ { ( 2 ) } ( x , t , i \kappa ) , \end{align*}


\begin{align*} \Psi ^ h _ 1 ( s ) & = i \Delta I _ h z _ R ( s ) , \\ \Psi ^ h _ 2 ( s ) & = i \kappa \theta _ R ( | z _ R | _ { X _ s } ) I _ h ( | z _ R ( s ) | ^ 2 z _ R ( s ) ) , \\ \Psi ^ h _ 3 ( s ) & = - i \nu I _ h z _ R ( s ) - \epsilon I _ h ( \gamma z _ R ( s ) - \mu \overline { z _ R } ( s ) ) , \\ \Psi ^ h _ 4 ( s ) & = - \tfrac { 1 } { 2 } I _ h ( z _ R ( s ) F _ \Phi ) . \end{align*}


\begin{align*} \tilde { D } _ s ( X , Y ) : = \min _ { \pi \in S _ n } \sum _ { j = 1 } ^ n | x _ j - y _ { \pi ( j ) } | _ { \infty } . \end{align*}


\begin{align*} G ( s , \lambda ) : = \sum _ { k = 0 } ^ \infty \frac { s ^ { 2 k + 1 } \lambda ^ k } { ( 2 k + 1 ) ! } = \begin{cases} \sin { ( s \sqrt { \lambda } ) } / \sqrt { \lambda } \lambda > 0 , \\ \sinh { ( s \sqrt { - \lambda } ) } / \sqrt { - \lambda } \lambda < 0 . \end{cases} \end{align*}


\begin{align*} \Delta ( e ^ { - v } \eta \circ \theta ) = \Delta ( e ^ { - v } ) \eta \circ \theta + 2 \nabla ( e ^ { - v } ) \cdot \nabla ( \eta \circ \theta ) + e ^ { - v } \Delta ( \eta \circ \theta ) , \end{align*}


\begin{align*} T _ m f ( x ) = \sum _ { \mu \in \N ^ d } m ( 2 | \mu | + d ) ( f ( x ) , \Phi _ { \mu } ) \Phi _ { \mu } ( x ) , \end{align*}


\begin{align*} e ^ { t z ^ n a _ n } g ( b ) & = \sum _ r \frac { 1 } { r ! } t ^ r z ^ { n r } a ^ r _ n g ( b ) = \sum _ r \frac { 1 } { r ! } ( - 1 ) ^ { - n r } z ^ { n r } g ( t ^ r a ^ r _ n b ) = e ^ { t ( - z ) ^ { n } a _ n } b . \end{align*}


\begin{align*} \frac { \partial { K _ p ^ { q / p } } } { \partial { x _ j } } ( z ) = q \ , \mathrm { R e } \ , \int _ \Omega | g _ z | ^ { q - 2 } \overline { g } _ z g _ { z , j } . \end{align*}


\begin{align*} \lambda _ { \infty , \gamma } ( x ) = \frac { \chi _ { \Lambda } ( k ) } { 2 \pi | k | ^ { 1 / 2 } } \mathbf { e } _ { \gamma } ( k ) e ^ { i k x } , \gamma = 1 , 2 , \end{align*}


\begin{align*} P \bigg ( \liminf _ { n \to \infty } ( r ( s _ n ) ^ { - 1 } \sup _ { t < s _ n } | B _ t | ) \le 1 \bigg ) = 1 . \end{align*}


\begin{align*} ( L N R ) _ { i j } & = \sum _ { k , \ell } L _ { i k } N _ { k \ell } R _ { \ell j } , & ( \Upsilon L ' N ' R ' ) _ { i j } & = \sum _ { r , k , \ell } \Upsilon _ { i r } L ' _ { r k } N ' _ { k \ell } R ' _ { \ell j } . \end{align*}


\begin{align*} ( Y Z ) _ i = \begin{cases} Z _ i & , \\ Y _ i & . \end{cases} \end{align*}


\begin{align*} \sigma : = \norm { \nu } \ , , \qquad \nu : = U _ 0 q B \varphi + \overline { V _ 0 } \overline { q B \varphi } \ , , \end{align*}


\begin{align*} S _ { \ell } ( m ) = m \end{align*}


\begin{align*} u ( x ) = \int \limits _ { S ( a , R ) } P ( x , \eta ) u ( \eta ) d \sigma ( \eta ) \end{align*}


\begin{align*} E ( M ) = \bigl \{ \sup _ { a \le s \le b } | \omega _ { s } | \le M \bigr \} , F ( M _ 0 ) = \bigl \{ \sup _ { a \le s , t \le b } | \omega _ { s t } | \le M _ 0 \bigr \} , \end{align*}


\begin{align*} E _ p ( \phi ) = \frac { 1 } { p } \int _ { 0 } ^ { \pi } \big ( \dot r ^ 2 ( t ) + ( m - 1 ) \frac { \sin ^ 2 r ( t ) } { \sin ^ 2 t } \big ) ^ \frac { p } { 2 } \sin ^ { m - 1 } t d t . \end{align*}


\begin{align*} ( \mathcal { A } \otimes \mathcal { F } ) \vee ( \mathcal { A } \otimes \mathcal { G } ) = \mathcal { A } \otimes \left ( \mathcal { F } \vee \mathcal { G } \right ) , \end{align*}


\begin{align*} P = \left \{ \begin{pmatrix} ( r - 1 ) \times ( r - 1 ) & * \\ 0 & 1 \times 1 \end{pmatrix} \right \} \end{align*}


\begin{align*} \partial \left ( I + \mathbb { A } \right ) ^ { - 1 } = - \left ( I + \mathbb { A } \right ) ^ { - 1 } \partial \mathbb { A } \left ( I + \mathbb { A } \right ) ^ { - 1 } . \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { N \to \infty } \frac { | C _ { N , \delta } | } { N } = 1 . \end{align*}


\begin{align*} r _ { 1 } ( k ) : = \frac { b ( k ) } { a _ 1 ( k ) } , r _ { 2 } ( k ) : = \frac { \overline { b ( - k ) } } { a _ 2 ( k ) } = \frac { b ( k ) } { a _ 2 ( k ) } \ \left ( \textnormal { f o l l o w s \ f r o m } \overline { b ( - k ) } = b ( k ) \right ) . \end{align*}


\begin{align*} j ( v ) ^ 2 = - q ( v ) 1 _ { A } , \end{align*}


\begin{align*} K _ p ( z ) = \frac { K _ { p , \varphi _ p } ( z ) } { | z | ^ { p k _ p } } , \ \ \ z \in \mathbb { D } ^ \ast . \end{align*}


\begin{align*} 1 _ S \otimes f \circ \rho _ M ( S ) ( g ) = \rho _ { P } ( S ) ( g ) \circ 1 _ S \otimes f : S \otimes _ R M \rightarrow S \otimes _ R P . \end{align*}


\begin{align*} \bar { d } ( x , y ) : = \sup \{ d ( \alpha ( t , x ) , \alpha ( t , y ) ) \ , : \ , t \in G \} \ , . \end{align*}


\begin{align*} X \smallsetminus { \downarrow } ^ { \mathbb { X } } p ^ { - 1 } [ Y \smallsetminus ( U \cap V ) ] = ( X \smallsetminus { \downarrow } ^ { \mathbb { X } } p ^ { - 1 } [ Y \smallsetminus U ] ) \cap ( X \smallsetminus { \downarrow } ^ { \mathbb { X } } p ^ { - 1 } [ Y \smallsetminus V ] ) , \end{align*}


\begin{align*} x _ { k + 1 } & = A x _ k + B u _ k , \\ \phi _ k & = \tilde { C } x _ k + \tilde { D } u _ k , \end{align*}


\begin{align*} g \cdot \mu _ { s , t } : = \int _ s ^ t g ( \tau ) \delta _ { \omega _ { \tau } } d \tau . \end{align*}


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


\begin{align*} \max _ { x \in \Delta } f ( x ) \ ; , \Delta = \left \{ x : \sum _ { i = 1 } ^ { n } x _ i = 1 x _ 1 , \ldots , x _ n \geq 0 \right \} \ ; . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ { N - 1 } \dd _ { 2 p } ( R _ { n , 1 } ^ { \epsilon , \Delta t } , 0 ) \le C _ p ( T ) \Delta t ^ { \frac 1 2 } . \end{align*}


\begin{align*} c h \ , H _ 0 ( \overline M ( \widehat \nu _ h ) ) = \sum _ { w \in \widehat W ^ \natural } d e t ( w ) c h \ , M ^ W ( w . \widehat \nu _ h ) . \end{align*}


\begin{align*} F _ { k , m } : = \big \{ \max _ { j \le 2 k m } | S _ j | \le \sqrt { m } \big \} \Longrightarrow \inf _ { j , \ell \in [ 1 , 2 k m ] } \{ G ( S _ j , S _ \ell ) \} \ge ( 4 C m ) ^ { - 1 } \ , . \end{align*}


\begin{align*} \left | q ( x , y ) \right | \leq C \prod ^ d _ { i = 1 } ( 1 + | x _ i | ) ^ { 2 H _ i - \beta _ i } ( 1 + | y _ i | ) ^ { 2 H _ i - \beta _ i } , \end{align*}


\begin{align*} \lim \limits _ { t \rightarrow 0 + } P _ \infty ( t * u ) = 0 \ \ \mbox { a n d } \ \ P _ \infty ( t * u ) > 0 , \ \ \mbox { f o r } \ \ t \ \ \mbox { s m a l l e n o u g h } . \end{align*}


\begin{align*} \mathcal P _ \sigma v _ \varepsilon \rightarrow v _ { p , \sigma } ~ ~ ~ L ^ 2 ( 0 , T ; H ^ 2 _ { \sigma , l o c } ) \cap C ( [ 0 , T ] ; H ^ 1 _ { \sigma , l o c } ) . \end{align*}


\begin{align*} K ( \tilde { M } ( p ) ) = M ( K ( p ) ) \end{align*}


\begin{align*} \log \left ( \prod _ { \substack { p \leq x \\ p \equiv a \pmod m } } \left | n ! \right | _ p \right ) = \sum _ { \substack { p \leq x \\ p \equiv a \pmod m } } \log | n ! | _ p . \end{align*}


\begin{align*} B _ { \mathfrak a } ( f \wedge g ) ( x _ 1 , \cdots , x _ { p + q } ) = \sum _ { \sigma \in S h ( p , q ) } B _ { \mathfrak a } \left ( f ( x _ { \sigma ( 1 ) } , \cdots , x _ { \sigma ( p ) } ) , g ( x _ { \sigma ( p + 1 ) } , \cdots , x _ { \sigma ( p + q ) } ) \right ) \end{align*}


\begin{gather*} i ' \left ( \theta ( x , y ) \right ) - \theta ' \left ( s ( x ) , s ( y ) \right ) = - i \left ( [ x , y ] \right ) + \rho \left ( s ( x ) \right ) i ( y ) + \rho \left ( s ( y ) \right ) i ( x ) . \end{gather*}


\begin{align*} \Delta h ( z + y + r \omega - A y - b ) & = \Delta h \ ( | z + y + r \omega - A y - b | \ , \dfrac { z + y + r \omega - A y - b } { | z + y + r \omega - A y - b | } \ ) \\ & = | z + y + r \omega - A y - b | ^ { p - 2 } \ , \Delta h \ ( \dfrac { z + y + r \omega - A y - b } { | z + y + r \omega - A y - b | } \ ) . \end{align*}


\begin{align*} \varphi _ { i j } \left ( \frac { z } { w } \right ) = \left ( \frac { z - w } { z q ^ { - 1 } - w } \right ) ^ { \delta _ { i j } } \prod _ { e = \vec { i j } \in E } \left ( \frac { 1 } { t _ e } - \frac { z } { w } \right ) ^ { - 1 } \ , . \end{align*}


\begin{align*} \log \det \left ( I + \mathbb { A } \right ) = \operatorname * { t r } \log \left ( I + \mathbb { A } \right ) , \end{align*}


\begin{align*} L ^ { \pm } ( u ) _ { b _ { 1 } \cdots b _ { k } } ^ { a _ { 1 } \cdots a _ { k } } = \sum _ { \sigma \in \mathfrak { S } _ { k } } s g n ( \sigma ) l _ { a _ { k } b _ { \sigma ( k ) } } ^ { \pm } ( u + ( k - 1 ) h ) \cdots l _ { a _ { 1 } b _ { \sigma ( 1 ) } } ^ { \pm } ( u ) \end{align*}


\begin{align*} \begin{cases} - \Delta u ( x ) = v ^ p ( x ) , & x \in \Omega , \\ - \Delta v ( x ) = u ^ q ( x ) , & x \in \Omega , \\ u ( x ) = v ( x ) = 0 , & x \in \partial \Omega . \\ \end{cases} \end{align*}


\begin{align*} \mathcal { F } ( \Delta ) \subseteq \mathcal { G } \cup \mathcal { H } \cup \big \{ [ n ] \setminus F _ p : p = 1 , \dots , d \big \} \cup \mathcal { F } . \end{align*}


\begin{align*} f ' \colon \{ \ , M ' , M ' \cup \Psi ' \mid M ' \subset Z ' \smallsetminus \Psi ' , \ | M ' | \ \} & \rightarrow \{ \ , M '' \mid M '' \subset Z '^ { ( 1 ) } \ , \} \\ M ' , M ' \cup \Psi ' & \mapsto M ' \smallsetminus \textstyle \binom { - } { 2 m } . \end{align*}


\begin{align*} x = x _ f + x _ M \in \left ( \frac { V } { 2 } + \frac { U } { 2 } \right ) \subseteq \left ( \frac { U } { 2 } + \frac { U } { 2 } \right ) = U . \end{align*}


\begin{align*} J _ { \breve { N } ^ r } ( x , t , k ) = I + O \left ( e ^ { - 8 t \kappa _ { \delta } ( 3 \xi - \kappa _ { \delta } ^ 2 ) } \right ) , k \in \Gamma _ 1 \cup \Gamma _ 1 ^ * , t \rightarrow \infty , \end{align*}


\begin{align*} - 2 n q \cdot d _ { 0 , q } ( n , j - 2 q ) = - n q \cdot d _ { 0 , q } ( n , j ) . \end{align*}


\begin{align*} \Delta ( x ) = x \otimes 1 + 1 \otimes x , \end{align*}


\begin{align*} T N = N \oplus T ^ H N \end{align*}


\begin{align*} & b ( 1 - \eta ) \leq w \leq c _ 0 ^ { - 1 } ( 1 - \eta ) \sqrt { - \ln ( \mu ( 1 - \eta ) ) } , - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \xi w + \partial _ \tau w \leq \frac { \delta } { 2 } w , \\ & w \partial _ \eta ^ 2 w \leq 2 \delta , - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \tau w \leq b \delta ( 1 - \eta ) ^ { \alpha _ 0 } i n D . \end{align*}


\begin{align*} \int _ { D _ 1 \times \cdots \times D _ k } e ^ { 1 , 2 , \cdots , k } _ k ( w , z ) \frac { \partial ^ { k - 1 } f _ k ( z ) } { \partial \bar z _ 1 \cdots \partial \bar z _ { k - 1 } } d V ( z ) = \int _ { D _ 1 \times \cdots \times D _ k } ( - 1 ) ^ k f _ k ( z ) \frac { \partial ^ { k - 1 } e ^ { 1 , 2 , \cdots , k } _ k ( w , z ) } { \partial \bar z _ 1 \cdots \partial \bar z _ { k - 1 } } d V ( z ) . \end{align*}


\begin{align*} & b ( x , \xi , u , v ) = ( \sigma \circ \mathcal { A } ^ { - 1 } ) ( x , \xi , u , v ) , \\ & \tilde b ( x , \xi , u , v ) = ( \tilde \sigma \circ \mathcal { A } ^ { - 1 } ) ( x , \xi , u , v ) , \\ & c ( x , \xi , u , v ) = b ( x , \xi , u , v ) \tilde b ( x , \xi , u , v ) . \end{align*}


\begin{align*} Z _ t : = \int _ 0 ^ t Y _ { s ^ - } \otimes d \mathbf { X } _ s , t \in [ 0 , 1 ] , \end{align*}


\begin{align*} c _ { i + 1 , j } = c _ { i , j } + c _ { 1 , j } . \end{align*}


\begin{align*} R ^ G _ { T , \chi } ( s u ) = \sum _ { \upsilon \in j _ { G _ s } ^ { - 1 } ( \omega ) } Q ^ { G _ s } _ { T _ \upsilon } ( u ) \chi _ \upsilon ( s ) , \end{align*}


\begin{align*} c _ k ( Y ) \le c _ k ( B ^ 4 ( a ) ) = d a \le L . \end{align*}


\begin{align*} \gamma _ { n } \left ( n + \frac { 1 } { 2 } - \gamma _ { n } - \gamma _ { n + 1 } \right ) \left ( n - \frac { 1 } { 2 } - \gamma _ { n } - \gamma _ { n - 1 } \right ) = z ^ { 2 } \left ( \frac { n } { 2 } - \gamma _ { n } \right ) ^ { 2 } . \end{align*}


\begin{align*} S _ \ell ^ { ( k ) } = \sum _ { i = 0 } ^ \ell F _ { i } ^ { ( k ) } . \end{align*}


\begin{align*} - \partial _ s w _ n - \sigma _ n \Delta w _ n + h \left ( \bar x _ n + r _ n y , \bar t _ n + \frac { r _ n ^ \gamma } { M _ n ^ { \gamma - 1 } } s \right ) | D w _ n | ^ \gamma = g _ n , \end{align*}


\begin{align*} \varphi _ t + f \varphi _ x - \varphi f _ x + \frac { 1 } { 2 } \sigma ^ 2 \varphi _ { x x } = & 0 \\ \sigma \varphi _ x - \varphi \sigma _ x = & 0 \end{align*}


\begin{align*} \alpha + \beta = 2 \nu , \ , \beta / \gamma ^ 2 = \nu + 1 , \ , \beta ^ 3 / \gamma ^ 4 = ( \nu + 1 ) ^ 2 ( \nu + 2 ) \end{align*}


\begin{align*} A _ { \infty } ( x ) = A _ { \infty } ^ + ( x ) + A _ { \infty } ^ - ( x ) , \end{align*}


\begin{align*} \kappa _ { k , k ' + 3 } = \begin{cases} & 1 ( k ' = k ) , \\ & 0 ( k ' \not = k ) . \end{cases} \end{align*}


\begin{align*} J _ \nu ( t ) = \sqrt { \frac { 2 } { \pi t } } \cos { \left ( t - \frac { \pi \nu } { 2 } - \frac { \pi } { 4 } \right ) } + \O _ \nu ( t ^ { - 3 / 2 } ) . \end{align*}


\begin{align*} \begin{aligned} & | u ( x , t ) - H ( x , t ) | \leq C | ( x , t ) | ^ { k + 1 + \alpha } , ~ ~ \forall ~ ( x , t ) \in \Omega _ { 1 } , \\ & | D ^ { k + 1 } u ( 0 ) | \leq C \end{aligned} \end{align*}


\begin{align*} P ( x , t ) = P _ g ( x , t ) + \mathbf { \Pi } _ { k + l } \left ( \sum _ { \mathop { k + 1 \leq | \sigma | + 2 \gamma \leq k + l , } \limits _ { \sigma _ n \geq 1 } } \frac { a _ { \sigma \gamma } } { \sigma ! \gamma ! } x ^ { \sigma - e _ n } t ^ { \gamma } \left ( x _ n - P _ { \Omega } ( x ' , t ) \right ) \right ) , \end{align*}


\begin{align*} P = - g ^ { i j } \operatorname { I d } _ E \partial _ i \partial _ j + b ^ j \partial _ j + c , \ , b ^ j , c \in C ^ \infty ( U ; \operatorname { E n d } ( E ) ) , \end{align*}


\begin{align*} \varphi ( t _ 2 , x ) - \varphi ( t _ 1 , x ) = \int _ { t _ 1 } ^ { t _ 2 } \sigma ( \varphi ( t , x ) ) d t \end{align*}


\begin{align*} \phi \left ( x ; z \right ) \partial _ { x } P _ { n + 1 } = { \displaystyle \sum \limits _ { k = 0 } ^ { n + 2 } } d _ { n , k } \left ( z \right ) P _ { k } , \end{align*}


\begin{align*} \widehat { T f _ k } ( \xi ) & = \int [ m ( \xi ) - m ( \xi - \eta ) ] \hat { \chi } _ { < k - 3 } ( \eta ) \hat { f _ k } ( \xi - \eta ) \dd \eta \\ & = \int _ 0 ^ 1 \int _ { \eta \ll 2 ^ k } \nabla m ( \xi - t \eta ) \eta \hat { \chi } _ { < k - 3 } ( \eta ) \hat { f _ k } ( \xi - \eta ) \dd \eta \dd t \end{align*}


\begin{align*} \left \lbrace \begin{aligned} q ^ \epsilon ( t ) & = q _ 0 ^ \epsilon + \frac { 1 } { \epsilon } \int _ { 0 } ^ { t } p ^ \epsilon ( s ) d s , \\ p ^ \epsilon ( t ) & = e ^ { - \frac { t } { \epsilon ^ 2 } } p _ 0 ^ \epsilon + \frac { 1 } { \epsilon } \int _ { 0 } ^ { t } e ^ { - \frac { t - s } { \epsilon ^ 2 } } f ( q ^ \epsilon ( s ) ) d s + \frac { 1 } { \epsilon } \int _ { 0 } ^ { t } e ^ { - \frac { t - s } { \epsilon ^ 2 } } d \beta ( s ) . \end{aligned} \right . \end{align*}


\begin{align*} M ( \nu ) : = \langle R ^ G _ { T , \chi ^ { ( \nu ) } } \otimes \omega _ \psi ^ { ( \nu ) , \vee } , R ^ G _ { S , \eta ^ { ( \nu ) } } \rangle _ { G ^ { F ^ \nu } } \end{align*}


\begin{align*} \det ^ * S _ { j , k , l } = \frac { 1 } { l ! } \biggl ( \sum _ { ( j ' , k ' ) \in J } \epsilon _ { j ' , k ' } \boxtimes S _ { j ' , k ' , 1 } \biggr ) ^ l \setminus \epsilon _ { j , k } \ , , \end{align*}


\begin{align*} \begin{gathered} R ( \mathbf { 1 } ) = \sqrt { A ( \mathbf { 1 } ) } \ ; , \\ R ( \mathbf { i } ) = 0 \end{gathered} \end{align*}


\begin{align*} S ( n ) = \frac { 3 n + 1 } { 8 } \equiv 1 \pmod { 4 } . \end{align*}


\begin{align*} \rho _ t : = ( \bar \rho J _ { X , t } ) \circ X ( 0 , t , \cdot ) \ , , \end{align*}


\begin{align*} K ^ { \alpha , \beta , \gamma } _ { m _ 1 , m _ 2 , \theta } ( x , y ) : = ( - 1 ) ^ { \theta _ 1 \mathbf { 1 } \{ x _ 1 = m _ 1 - 1 , y _ 1 = 0 \} + \theta _ 2 \mathbf { 1 } \{ x _ 2 = m _ 2 - 1 , y _ 2 = 0 \} } \left ( \alpha \mathbf { 1 } _ { y = x } + \beta \mathbf { 1 } _ { y = x + \mathbf { e } ^ 1 } + \gamma \mathbf { 1 } _ { y = x + \mathbf { e } ^ 2 } \right ) . \end{align*}


\begin{align*} \| f \| _ { H ^ { k , l } } ^ 2 = \sum \limits _ { | \alpha | = 0 } ^ { k } \sum \limits _ { \beta = 0 } ^ { l } \| \partial _ \tau ^ \alpha \partial _ y ^ \beta f \| _ { L ^ 2 ( \Omega ) } ^ 2 < + \infty . \end{align*}


\begin{align*} | \pi _ { \xi } ^ { \perp } ( \eta _ \circ ) | ^ 2 = | \eta _ \circ | ^ 2 - | \pi _ { \xi } ( \eta _ \circ ) | ^ 2 = 1 - | \langle \xi _ \circ , \eta _ \circ \rangle \cdot \xi _ \circ | ^ 2 = 1 - | \langle \xi _ \circ , \eta _ \circ \rangle | ^ 2 \ , . \end{align*}


\begin{align*} \sum _ { z \in \Z } \Delta ( x - z , t - s ) = 0 , \end{align*}


\begin{align*} x \to y : = \min \{ z * y \colon z \in I ( x , y ) \cap [ y ) \} ; \end{align*}


\begin{align*} \varphi ( \lambda ) : = \sup _ { u , n \in \N } E [ \exp ( \lambda \ , 2 ^ u \ , \overline { \alpha } ^ { ( n ' _ u ) } _ { 1 } ) ] \le 1 + c \lambda ^ 2 < \infty \ , , \end{align*}


\begin{align*} \mathbf { J } _ 1 & = \lambda ^ { - 4 + \epsilon } \left [ \frac { 1 } { 2 } \int _ { \Gamma _ { t _ 0 , \theta } } \partial _ t ( \psi _ \Gamma ^ 2 ) \ , \d S \ , \d t - \lambda \int _ { \Gamma _ { t _ 0 , \theta } } \left ( s \xi - \frac { 3 } { 2 } \right ) \partial _ \nu ^ { A _ 1 } \eta ^ 0 \psi _ \Gamma ^ 2 \ , \d S \ , \d t \right ] . \end{align*}


\begin{align*} \acute { \chi } \bigl ( ( ( r , d ) , e ) , ( ( r ' , d ' ) , e ' ) \bigr ) = ( 1 - g ) r r ' + ( r - e ) d ' - ( d + \nu e ) r ' + e e ' . \end{align*}


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


\begin{align*} \omega ( x ) = \Lambda _ { n } ( x ) W ( x ) = \big ( \sum _ { k = 0 } ^ n \left | h _ { k } ( x ) \right | ^ 2 \big ) ^ { - 1 } W ( x ) , x \in X _ n \ , , \end{align*}


\begin{align*} \tau f ' + \overline { \tau g ' } = - { 2 \bar \tau \over ( \bar z - \bar z _ 0 ) ^ 2 } \end{align*}


\begin{align*} G _ 1 \sigma ^ e ( G _ 1 ^ T ) = G \sigma ^ e ( G ^ T ) + \mathbf { g } \sigma ^ e ( \mathbf { g } ^ T ) = { \rm d i a g } ( \underbrace { 0 , \ 0 , \ \cdots , \ 0 } _ { k - 1 } , \gamma ^ { p ^ e + 1 } ) \end{align*}


\begin{align*} \begin{aligned} \eta _ 2 ^ \varepsilon ( x , t ) & = \varepsilon d ( x , t ) \Bigl ( \bar { g } _ 1 \bar { \zeta } _ 0 - \bar { g } _ 0 \bar { \zeta } _ 1 \Bigr ) ( x , t ) + \frac { 1 } { 2 } d ( x , t ) ^ 2 \Bigl ( \bar { \zeta } _ 0 - \bar { \zeta } _ 1 \Bigr ) ( x , t ) , \\ \rho _ \eta ^ \varepsilon ( x , t ) & = \bar { \eta } ( x , t ) - k _ d ^ { - 1 } d ( x , t ) \Bigl ( \overline { V _ \Gamma \eta } \Bigr ) ( x , t ) + \eta _ 2 ^ \varepsilon ( x , t ) . \end{aligned} \end{align*}


\begin{align*} u _ { z _ 0 } ( z ) = \frac { 2 } { \bar z - \bar z _ 0 } . \end{align*}


\begin{align*} \partial _ 3 \tilde { \omega } _ { k } ( \tau , x , t ) = a _ k ( x , t ) \exp { \int _ { \tau } ^ { t } \partial _ 1 a _ k ( \tilde { \omega } _ k ( \rho , x , t ) , \rho ) \ , d \rho } . \end{align*}


\begin{align*} \left ( \phi \partial _ { x } ^ { \ast } + \psi \right ) L = 0 . \end{align*}


\begin{align*} | M ^ * \cap M _ 1 | + | M ^ * \cap M _ 2 | = \begin{cases} | M \cap M _ 1 | + | M \cap M _ 2 | , & ; \\ | M \cap M _ 1 | + | M \cap M _ 2 | + 2 , & \end{cases} \end{align*}


\begin{align*} \begin{aligned} & S O ( z ) = \frac { 1 } { 1 6 } z ^ 2 O ( z ) + \frac { z } { 2 } , \\ & O ( z ) = S O ( z ) O ( z ) + 1 . \end{aligned} \end{align*}


\begin{align*} \mathfrak { D } _ { \Gamma } & : \ \mathcal { E } ( M ) \times \mathcal { E } ' ( M ' ) \mapsto [ 0 , + \infty ) , \\ \mathfrak { D } _ \Gamma & ( \xi , \xi ' ) : = ( { \rm d i s t } _ { \Gamma } ( l ' ( \xi ' ) , l ( \xi ) ) ^ 2 + ( r ' ( \xi ' ) - r ( \xi ) ) ^ 2 ; \end{align*}


\begin{align*} \tilde E _ s = h _ s - ( \tilde \Delta - \lambda _ s ) ^ { - 1 } ( \Delta - \lambda _ s ) h _ s . \end{align*}


\begin{align*} \mu _ { 2 n } \left ( z \right ) = 2 { \displaystyle \int \limits _ { 0 } ^ { z } } x ^ { 2 n } e ^ { - x ^ { 2 } } d x = { \displaystyle \int \limits _ { 0 } ^ { z ^ { 2 } } } s ^ { n - \frac { 1 } { 2 } } e ^ { - s } d s = \widehat { \gamma } \left ( n + \frac { 1 } { 2 } , z ^ { 2 } \right ) , \end{align*}


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


\begin{align*} \limsup _ { \varepsilon \ , \to \ , 0 ^ + } \ , \frac { \log \log \mathcal E _ { \mathrm { t o p } } ( f ) ( \varepsilon ) } { - \log \varepsilon } \ , = \ , \overline { \mathrm { m o } } \ , ( \mathcal M _ f ^ { \mathrm { e r g } } ( X ) , \mathrm W _ p ) \ , \leqslant \ , \overline { \mathrm { m o } } \ , ( \mathcal M _ 1 ( X ) , \mathrm W _ p ) \ , \leqslant \ , \mathrm { \overline { d i m } _ B } \ , X . \end{align*}


\begin{align*} \| u _ s ^ \star ( \varphi ) - u _ s ^ \star ( \psi ) \| _ { \eta , s } & = \| \mathcal { G } _ s ( u _ s ^ \star ( \varphi ) , \varphi ) - \mathcal { G } _ s ( u _ s ^ \star ( \psi ) , \psi ) \| _ { \eta , s } \\ & \leq K _ \varepsilon \| \varphi - \psi \| + \frac { 1 } { 2 } \| u _ s ^ \star ( \varphi ) - u _ s ^ \star ( \psi ) \| _ { \eta , s } . \end{align*}


\begin{align*} r = \left ( \rho ^ \nabla \right ) ^ { J , + } - \frac { 1 } { 4 } \| N \| ^ 2 \omega + 4 \ , \left ( N ( X , T ) \right ) ^ \flat \wedge \left ( J N ( X , T ) \right ) ^ \flat , \end{align*}


\begin{align*} t ^ j _ i = \left \{ \begin{array} { l l } x _ i & \mbox { i f } i \in I \setminus ( A \cup \{ i _ 0 , \dots , i _ j \} ) , \\ y _ i & \mbox { i f } i \in A \cup \{ i _ 0 , \dots , i _ j \} . \end{array} \right . \end{align*}


\begin{align*} X ^ \epsilon ( t _ { n + 1 } ) = \varphi \bigl ( X ^ \epsilon ( t _ n ) , \zeta ^ \epsilon ( t _ { n + 1 } ) - \zeta ^ \epsilon ( t _ n ) \bigr ) = \varphi \bigl ( X ^ \epsilon ( t _ n ) , \frac { 1 } { \epsilon } \int _ { t _ n } ^ { t _ { n + 1 } } m ^ \epsilon ( s ) d s \bigr ) \end{align*}


\begin{align*} \nabla \hat { J } ( u _ { 0 } ) = \mathbb { E } _ { v _ 0 \in \mathbb { B } ^ d } \left [ \nabla J ( u _ { 0 } + \epsilon v _ 0 ) \right ] = \frac { d } { \epsilon } \cdot \mathbb { E } _ { v _ 0 \in \mathbb { S } ^ { d - 1 } } \left [ J ( u _ { 0 } + \epsilon v _ 0 ) v _ 0 \right ] , \end{align*}


\begin{align*} V _ \varphi \widetilde { g } ( x , \xi ) & = \int _ { \mathbb { R } ^ d } g ( C t ) e ^ { - 2 \pi i \xi \cdot t } \overline { \varphi ( t - x ) } d t \asymp \int _ { \mathbb { R } ^ d } g ( s ) e ^ { - 2 \pi i ( ( C ^ { - 1 } ) ^ T \xi ) \cdot s } \overline { \varphi ( C ^ { - 1 } ( s - C x ) ) } d s \\ & = V _ { \widetilde { \varphi } } g ( B ( x , \xi ) ) , \end{align*}


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


\begin{align*} \int _ \R \frac { 1 } { \lambda ^ 2 - z ^ 2 } f _ 1 ( \lambda ) \lambda \coth ( \frac { \pi \lambda } { 2 } ) \ , d \lambda = \int _ { \R + i } \frac { 1 } { \lambda ^ 2 - z ^ 2 } f _ 1 ( \lambda ) \lambda \coth ( \frac { \pi \lambda } { 2 } ) \ , d \lambda + \pi i f _ 1 ( z ) \coth ( z ) \ , . \end{align*}


\begin{align*} S _ { \mathbf { t } } = \left \{ \Psi _ { 2 ^ { m - 2 } - 1 } \left ( \mathcal { F } _ d ^ { \mathbf { t } } ( \mathbf { x } , \mathbf { y } ) \right ) : d \in \{ 1 , 2 \} , \mathbf { y } \in \mathbb { Z } _ { 2 } ^ { n } \right \} , \end{align*}


\begin{align*} [ F ( n ) ] + [ F ( m + l ) ] = [ F ( n + l ) ] + [ F ( m ) ] , \mbox { w i t h } 1 \leq n < n + l \leq m < m + l \leq N . \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*} B = \begin{bmatrix} B _ 1 & 0 \\ 0 & B _ 2 , \end{bmatrix} \end{align*}


\begin{align*} H ^ k ( \mathbb { S } ^ 2 ) : = H ^ k ( D ; \R ^ 3 ) \cap \{ g : D \rightarrow \R ^ 3 \ , \textrm { s . t . } \ , | g ( x ) | = 1 \ , \ , x \in D \} \ , , \end{align*}


\begin{align*} \mathbb { E } ( \overline { \Vert X ( t ) \Vert } ^ 2 ) & = \mathbb { E } \left ( \left [ \frac { 1 } { t } \int _ 0 ^ t \Vert X ( s ) \Vert \right ] ^ 2 \right ) \\ & = \frac { 1 } { t ^ 2 } \mathbb { E } \left ( \left [ \int _ 0 ^ t \Vert X ( s ) \Vert \right ] ^ 2 \right ) \\ & \leq \frac { 1 } { t } \mathbb { E } \left ( \int _ 0 ^ t \Vert X ( s ) \Vert ^ 2 \right ) = \mathbb { E } ( \overline { \Vert X ( t ) \Vert ^ 2 } ) . \end{align*}


\begin{align*} \beta ' ( Z ) = Z \left ( \alpha ( \cdot ) \right ) \end{align*}


\begin{align*} ( \widetilde f _ 2 \otimes \widetilde f ' _ 2 ) ( \omega _ { Z , Z ' } ) = 2 \ , \rho _ { Z ^ { ( 2 ) } } \otimes \left [ \rho _ { \binom { 3 } { 2 } } + \rho _ { \binom { 2 } { 3 } } \right ] = 2 \sqrt 2 \ , \left [ \rho ^ { ( 3 ) } _ { Z ^ { ( 2 ) } } \otimes \rho ^ { ( 3 ) } _ { Z '^ { ( 2 ) } } \right ] . \end{align*}


\begin{align*} g = S + \varepsilon + \mu _ 1 \tau + \mu _ 1 ( 1 - \eta ) . \end{align*}


\begin{align*} ( \mathbf { a } ) \le \frac { 1 } { N } \sum _ { \tau = - ( N - 1 ) } ^ { N - 1 } | \Lambda ( \mathbf { a } ) ( \tau ) | . \end{align*}


\begin{align*} \mathbb { F } _ { \geqslant \boldsymbol { u } } : = \left \{ \ x \in { \mathbb { F } } \ | \ \vartheta ( x ) \geqslant _ { \boldsymbol { a } } \boldsymbol { u } \ \right \} \cup \left \{ 0 \right \} \mathbb { F } _ { > \boldsymbol { u } } : = \left \{ \ x \in { \mathbb { F } } \ | \ \vartheta ( x ) > _ { \boldsymbol { a } } \boldsymbol { u } \ \right \} \cup \left \{ 0 \right \} \end{align*}


\begin{align*} k _ { 0 } : = C \left \Vert h \right \Vert _ { L ^ { N } ( \Omega ) } ^ { \frac { 1 } { p ^ { \pm } - 1 } } , \end{align*}


\begin{align*} \begin{multlined} [ [ A ] ] ( i ) = \sum _ { j \in b } \sum _ { k \in c } A ( ( ( i , ( j , j ) ) , ( k , k ) ) ) \\ = \sum _ { ( j , k ) \in b \times c } X ( A ) ( ( i , ( ( j , k ) , ( j , k ) ) ) ) = [ X ( A ) ] ( i ) \ ; , \end{multlined} \end{align*}


\begin{align*} X _ n : = \frac { 1 } { n } \sum _ { i , \ell \in [ 1 , n ] } G ( S _ i , \tilde { S } _ \ell ) , Y : = \int _ 0 ^ 1 \int _ 0 ^ 1 G _ \beta ( \beta _ s , \tilde { \beta } _ t ) d s d t \ , . \end{align*}


\begin{align*} \begin{cases} g ( x ) = \frac { 1 } { | x | ^ { d - 2 } } d \geq 3 \\ g ( x ) = - \log ( | x | ) d = 1 , 2 \end{cases} \end{align*}


\begin{align*} \frac { 1 } { \pi \varepsilon ^ 2 } \int \limits _ { B ( \zeta _ 0 , \varepsilon ) \cap { \Bbb B } ^ 2 } Q ( z ) \ , d m ( z ) \leqslant \frac { 3 } { \pi \varepsilon } \int \limits _ { 1 - \varepsilon } ^ 1 \beta ( r ) \ , d r \leqslant \frac { 6 } { \pi } : = c \ , , 0 < \varepsilon < \varepsilon _ 0 \end{align*}


\begin{align*} [ \partial _ x ^ \alpha , b ] f ( x ) & = c \Bigl ( \int _ { \R ^ d } \frac { b ( y ) f ( y ) - b ( x ) f ( x ) } { | y - x | ^ { d + \alpha } } \d y - b ( x ) \int _ { \R ^ d } \frac { f ( y ) - f ( x ) } { | y - x | ^ { d + \alpha } } \d y \Bigr ) \\ & = c \int _ { \R ^ d } \frac { ( b ( y ) - b ( x ) ) f ( y ) } { | y - x | ^ { d + \alpha } } \d y . \end{align*}


\begin{align*} [ F , G ] = - ( - 1 ) ^ { \abs { F } } ( Q ' ) ^ 1 _ 2 \circ ( F \star G ) . \end{align*}


\begin{align*} { \mathcal { A } } _ { k } ^ { J } = \frac { 2 x _ k \alpha ( x _ k ) } { 1 + 2 x _ k } \prod _ { j \in J \setminus k } \mathbf { f } ( x _ k , x _ { j } ) \ , , \tilde { \mathcal { D } } _ { k } ^ { J } ( x ) = \tilde { \delta } ( x ) \prod _ { j \in J \setminus k } \mathbf { h } ( x _ k , x _ { j } ) \ , . \end{align*}


\begin{align*} Q _ i = \bigcup _ { \substack { \beta \in P _ p \\ \beta ^ { - 1 } ( 0 ) = j _ 0 } } Q _ { i , \beta } \ , . \end{align*}


\begin{align*} M ( ( \chi , \alpha ) , ( \xi , \beta ) ) = M \rvert _ { \chi \xi } ( \alpha , \beta ) \ ; , \end{align*}


\begin{align*} G ^ { ( n ) } = \{ ( g _ 1 , \dots , g _ n ) \in G ^ n \colon s ( g _ i ) = r ( g _ { i + 1 } ) \} . \end{align*}


\begin{align*} \Sigma _ { L , M } & = Y _ L \times _ X Z _ M , & \sigma _ { L , M } & = ( \psi _ L \times \zeta _ M ) | _ { \Sigma _ { L , M } } , \end{align*}


\begin{align*} g _ { s u m } = \frac { \partial _ \xi w + \partial _ \tau w } { v } \end{align*}


\begin{align*} - d u ( t , x ) = \frac 1 2 \Delta u ( t , x ) d t + u ( t , x ) W ( d t , x ) , \ u ( T , x ) = \phi ( x ) \ , . \end{align*}


\begin{align*} s \mapsto \ln ( f \circ \tanh ( s / 2 ) ) = ( d - 1 ) \ln ( \cosh ( s ) ) , \end{align*}


\begin{align*} R _ g = e ^ { - \beta ( t ) g ( r ) } \left ( \left ( 1 + r ^ 2 \beta ( t ) g ' ( r ^ 2 ) \right ) \frac { \partial } { \partial t } - \frac { r \beta ' ( t ) g ( r ^ 2 ) } { 2 } \frac { \partial } { \partial r } - 2 \beta ( t ) g ' ( r ^ 2 ) \frac { \partial } { \partial \theta } \right ) . \end{align*}


\begin{align*} h ^ \nu x = h ( x \circ \alpha _ \nu ) = h \sum \limits _ { i = 1 } ^ s h ( c _ i ) I _ { A _ i } \ \ \ P \alpha _ \nu ^ { - 1 } - \mbox { a . s . } \end{align*}


\begin{align*} \sup _ { s \geq 0 } \ , ( w B ( s ) - u s ) ~ \overset { { \rm d } } { = } ~ \theta _ { 1 , \ , 2 u / w ^ 2 } . \end{align*}


\begin{align*} \mathcal { A } ^ { \mathrm { s h } } \left ( X \langle r \rangle , Y \langle s \rangle \right ) : = \mathcal { A } \left ( X , Y \right ) _ { s - r } \end{align*}


\begin{align*} \begin{cases} d X _ t = b ( t , X _ t ) \ , d t + \sigma ( t , X _ t ) \ , d B _ t , \\ X _ 0 = x _ 0 , \end{cases} \end{align*}


\begin{align*} \alpha _ 0 ( A ) ( x ) = A ( \hat \Phi _ \cdot ^ \infty ( \hat \Phi _ \cdot ^ \infty ( \Phi _ { \cdot 0 } ^ \infty ( x ) , \Phi _ { \cdot 0 } ^ \infty ( \Phi _ { \cdot 1 } ^ \infty ( x ) ) ) , \Phi _ { \cdot 1 } ^ \infty ( \Phi _ { \cdot 1 } ^ \infty ( x ) ) ) ) \ ; . \end{align*}


\begin{align*} \sigma _ { \sigma ^ { - 1 } _ y ( x ) } = \sigma _ x . \end{align*}


\begin{align*} a _ i x _ i - b _ { i + 1 } x _ { i + 1 } = h _ i . \end{align*}


\begin{align*} \sigma _ 2 = \sigma _ 1 \circ \chi . \end{align*}


\begin{align*} \frac { 1 } { 3 } \left ( \frac { \delta } { a } \right ) ^ 3 = 2 \ , \zeta ( 3 ) , \end{align*}


\begin{align*} ( T ^ \# x ) ( y ) = \langle x , \bar y \rangle _ 1 = ( ( j ^ * \circ c ) x ' ) ( y ) = ( ( j ^ * \circ c \circ \tilde T ) x ) ( y ) , \end{align*}


\begin{align*} \begin{cases} \dot { y } ( t ) = L ( t ) y _ t , & t \geq s , \\ y _ s = \varphi , & \varphi \in X , \end{cases} \end{align*}


\begin{align*} \dfrac { d x } { d \tau } = D [ A ( t ) x ] . \end{align*}


\begin{align*} X _ t = - c t + S _ t , t \geq 0 , \end{align*}


\begin{align*} \begin{aligned} p _ { r } = \frac { 1 } { r } [ \omega _ { \theta } + u v _ { \theta } - ( v + b ) u _ { \theta } ] . \end{aligned} \end{align*}


\begin{align*} \tilde H '' _ 1 ( 0 ) = \underbrace { \bigl ( h '' g _ H ( T _ * N , N ) + h ' \kappa _ \eta - h ' g _ H ( T _ * N , N ) \kappa _ 1 \bigr ) } _ { > 0 } \frac { d } { d s } \Big \lvert _ { s = 0 } g _ H ( T _ * N , N ^ \perp ) . \end{align*}


\begin{align*} \lim _ { i \to \infty } u _ i = - 1 \{ \theta _ 2 \} \times M . \end{align*}


\begin{align*} \lim _ { j \rightarrow \infty } \frac { \eta _ j } { k _ j } \ = \ \infty \mbox { a n d } \frac { h _ r ( k _ j ) } { h _ l ( \eta _ j ) } \ \ge \ \frac { \omega ( \eta _ j ) } { \omega ( k _ j ) } . \end{align*}


\begin{align*} u ( t ) = U ( t , s ) \varphi + j ^ { - 1 } \int _ s ^ t U ^ { \odot \star } ( t , \tau ) f ( \tau ) d \tau , \varphi \in X . \end{align*}


\begin{align*} d ( f ( x ) , x ^ * ) = d ( f ( x ) , f ( x ^ * ) ) \leq c \cdot d ( x , x ^ * ) . \end{align*}


\begin{align*} L _ { \infty , k + 1 } \circ Q _ { B , k + 1 } ^ { k + 1 } = - Q _ { A , 1 } ^ 1 \circ L _ { \infty , k + 1 } . \end{align*}


\begin{align*} S ( 3 ) = S ( 1 3 ) = 5 \end{align*}


\begin{align*} & \mathcal { H } _ \mathrm { a p p r o x } : = \mathrm { c o n s t } + \int { d x \big \{ \overline { B ( x ) } b _ x + B ( x ) b _ x ^ \ast \big \} } \\ & + \int { d x d y \Big \{ ( b _ x , b _ x ^ \ast ) \begin{pmatrix} - H ^ T ( x , y ) & ( \upsilon _ N \overline { m } ) ( x , y ) \\ - ( \upsilon _ N m ) ( x , y ) & H ( x , y ) \end{pmatrix} \begin{pmatrix} - b _ y ^ \ast \\ b _ y \end{pmatrix} \Big \} } , \end{align*}


\begin{align*} & J ( x , t , k ) = \begin{pmatrix} 1 & 0 \\ \frac { r _ 1 ( k ) } { 1 + r _ 1 ( k ) r _ 2 ( k ) } e ^ { 2 i t \theta } & 1 \end{pmatrix} \begin{pmatrix} 1 + r _ 1 ( k ) r _ 2 ( k ) & 0 \\ 0 & \frac { 1 } { 1 + r _ 1 ( k ) r _ 2 ( k ) } \end{pmatrix} \begin{pmatrix} 1 & \frac { r _ 2 ( k ) } { 1 + r _ 1 ( k ) r _ 2 ( k ) } e ^ { - 2 i t \theta } \\ 0 & 1 \end{pmatrix} . \end{align*}


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


\begin{align*} c _ { i , m } = \frac { \rho _ { i , m } } { \rho _ m } , G ^ m = \sum _ { i = 1 } ^ { \ell } c _ { i , m } G _ i ( p _ m , n _ m ) , \end{align*}


\begin{align*} \widetilde { \alpha } : ( a \otimes b ) \otimes c = a \otimes ( b \otimes c ) \ ; . \end{align*}


\begin{align*} g ( x ) = \frac { 1 } { | x | ^ { d - 2 s } } , \end{align*}


\begin{align*} \forall \omega > 0 , J _ M ( \mathfrak f ) ( \omega ) = - j _ M ^ * < 0 . \end{align*}


\begin{align*} \lim _ { \| u \| _ X \to \infty } \frac { \langle A u , u \rangle } { \| u \| _ X } = + \infty . \end{align*}


\begin{align*} P _ k = \begin{pmatrix} p _ { 1 1 } & \cdots & p _ { 1 \ell _ k } \\ p _ { 2 1 } & \cdots & p _ { 2 \ell _ k } \\ p _ { 3 1 } & \cdots & p _ { 3 \ell _ k } \end{pmatrix} , \end{align*}


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


\begin{align*} y _ 2 - y _ 1 = e . \end{align*}


\begin{align*} ( u _ l ) _ t - a _ { i j } ( u _ l ) _ { i j } = G ( x , t ) , \end{align*}


\begin{align*} z ( t ) = & S ( t ) z _ 0 - \int _ 0 ^ t S ( t - s ) ( i \nu z ( s ) + \epsilon ( \gamma z ( s ) - \mu \overline { z } ( s ) ) ) \d s - \tfrac { 1 } { 2 } \int _ 0 ^ t S ( t - s ) ( z ( s ) F _ \Phi ) \d s \\ & + i \kappa \int _ 0 ^ t S ( t - s ) ( | z ( s ) | ^ 2 z ( s ) ) \d s - i \int _ 0 ^ t S ( t - s ) ( z ( s ) \d W ( s ) ) \end{align*}


\begin{align*} 1 = { K _ 1 } \leq { K _ 2 } \leq . . . \leq { K _ n } = K \end{align*}


\begin{align*} ( R / \mathfrak { p } \otimes _ R A , R / \mathfrak { p } \otimes _ R P ) = ( R / \mathfrak { p } _ { n - 1 } / \mathfrak { p } / \mathfrak { p } _ { n - 1 } \otimes _ { R / \mathfrak { p } _ { n - 1 } } R / \mathfrak { p } _ { n - 1 } \otimes _ R A , R / \mathfrak { p } _ { n - 1 } / \mathfrak { p } / \mathfrak { p } _ { n - 1 } \otimes _ { R / \mathfrak { p } _ { n - 1 } } R / \mathfrak { p } _ { n - 1 } \otimes _ R P ) \end{align*}


\begin{align*} Y \colon \begin{cases} & x ^ 2 + y ^ 2 + \prod _ { i = 1 } ^ { n } ( t - \epsilon _ i ) = 0 , \\ & w ^ 2 + ( t - \lambda _ 1 ) ( t - \lambda _ 2 ) = 0 , \end{cases} \quad Y ' \colon \begin{cases} & x ^ 2 + y ^ 2 + \prod _ { i = 1 } ^ { n } ( t - \epsilon _ i ' ) = 0 , \\ & w ^ 2 + ( t - \lambda _ 1 ' ) ( t - \lambda _ 2 ' ) = 0 , \end{cases} \end{align*}


\begin{align*} a _ i = 3 ^ { v _ i } \end{align*}


\begin{align*} ( u ( \theta ) \psi ) ^ { \wedge } = u ( - \theta ) \hat { \psi } . \end{align*}


\begin{align*} \sum _ { j \in J _ i } | Q _ { i , j } | = 1 - | S _ i | \ge \frac { 1 } { 2 } . \end{align*}


\begin{align*} \bar M _ p ( f ) \ge \frac { 1 } { p ^ { n - d } } M _ p ( f ) = \infty . \end{align*}


\begin{align*} \mathcal { M } ( A , \textbf { d } ) ^ { s s } _ { \theta } : = ( \bigoplus _ { n \geq 0 } ( A , \textbf { d } ) _ { n \theta } ) . \end{align*}


\begin{align*} C ( \theta _ 1 , x + i y ) : = \frac { 1 } { 2 } \sum _ { k \in \mathbb { Z } } \int _ { - 1 / 2 } ^ { 1 / 2 } \left \{ \log \left ( 1 + \left ( \frac { y / 2 \pi + \theta _ 1 / 2 + k } { x / 2 \pi } \right ) ^ 2 \right ) - \log \left ( 1 + \left ( \frac { y / 2 \pi + \theta _ 1 / 2 + k + \phi } { x / 2 \pi } \right ) ^ 2 \mathrm { d } \phi \right ) \right \} . \end{align*}


\begin{align*} d _ { r , s } ( n , i ) = 0 , \mbox { i f } r \ne 0 \mbox { a n d } s \ne 0 . \end{align*}


\begin{align*} G ( | u | ^ 2 ) : = F ( u ) \bar u , \end{align*}


\begin{align*} \Delta = \{ x _ i + \cdots + x _ { r - 1 } \mid i = 1 , \dotsc , r - 1 \} . \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { N } \log \left ( \mathfrak { R } _ { N } ( B ( \overline { \mathbf { P } } , \delta ) ) \right ) = \lim _ { N \to \infty } \frac { 1 } { N } \log \left ( \mathfrak { Q } _ { N } ( B ( \overline { \mathbf { P } } , \delta ) ) \right ) . \end{align*}


\begin{align*} \chi ( \rho ) ^ { 1 / 2 } \ , \chi \Big ( 1 - \rho - \frac { 2 \pi i \kappa } { \log T } \Big ) ^ { 1 / 2 } & = \Big ( \frac { \gamma } { 2 \pi } \Big ) ^ { \pi i \kappa / \log T } \bigg ( 1 + O \Big ( \frac { 1 } { T } \Big ) \bigg ) \\ & = e ^ { \pi i \kappa } \ , \Big ( 1 + O \big ( ( \log T ) ^ { - 1 } \big ) \Big ) \end{align*}


\begin{align*} \psi ^ * ( u ) : = \sup _ { | \xi | \leq u } \psi ( \xi ) , u \geq 0 , \end{align*}


\begin{align*} n & = y _ n n + y _ n z _ n ^ 2 C '' ( z _ n ) \delta _ n + o ( C '' ( z _ n ) \delta _ n ) + m c _ m S _ n \\ & = n + n \frac { L - c _ m S } { C ( z _ n ) } + \delta _ n \bigg ( z _ n ^ 2 C '' ( z _ n ) - n \frac { z _ n C ' ( z _ n ) } { C ( z _ n ) } \bigg ) + o ( C '' ( z _ n ) \delta _ n ) + m c _ m S _ n . \end{align*}


\begin{align*} \lim _ { \tau \to 0 } \int _ { Q _ { \infty } } ( m ^ { \tau } - \nu \nabla p ) \cdot \nabla \psi = 0 \end{align*}


\begin{align*} \frac { 2 } { r } = \frac { 1 } { 2 } - \frac { 1 } { p } . \end{align*}


\begin{align*} H [ n + 1 ] ( e , \overline { e } ) - | \Theta [ n + 1 ] ( \overline { e } , \overline { e } ) | & \ge h [ n + 1 ] ( e , \overline { e } ) - \mu _ { n + 1 } \\ & - \frac { 1 } { N } | ( \upsilon _ N \Big ( \frac { k _ { n + 1 } } { \delta - k _ { n + 1 } \circ \overline { k _ { n + 1 } } } \Big ) ) ( \overline { e } , \overline { e } ) | , \end{align*}


\begin{align*} M ( t ) = \int ^ { 2 \pi } _ 0 | g ( t e ^ { i \theta } ) | ^ p d \theta , \end{align*}


\begin{align*} \begin{aligned} M ( ( A , B ) ) ( ( 0 , i ) ) & = A ( i ) \ ; , \\ M ( ( A , B ) ) ( ( 1 , i ) ) & = B ( i ) \ ; . \end{aligned} \end{align*}


\begin{align*} G = 2 \left ( \bar { x } _ 4 x _ 3 g + x _ 4 \bar { x } _ 3 \left ( g + x _ 1 + 3 \right ) \right ) , \end{align*}


\begin{align*} H ^ \ell \circ ( \lambda ^ 1 ( t ) \star \exp _ \star ( F ^ 1 ( t ) ) ) & = \left ( ( H ^ 1 \circ ( \lambda ^ 1 ( t ) \star \exp _ \star F ^ 1 ) ) \star \frac { 1 } { ( \ell - 1 ) ! } ( H ^ 1 F ) ^ { \star ( \ell - 1 ) } \right ) . \end{align*}


\begin{align*} & \mathring { R } ^ u _ { q + 1 } = \mathcal { R } ^ u \mathbb { P } _ H \div \mathring { R } ^ u _ { q + 1 } , \\ & \mathring { R } _ { q + 1 } ^ B = \mathcal { R } ^ B \mathbb { P } _ H \div \mathring { R } _ { q + 1 } ^ B . \end{align*}


\begin{align*} \chi _ { \lambda _ 1 D _ 1 + Q _ 2 } ( x ) = \chi _ { \lambda _ 1 D _ 1 + D _ 2 } ( x ) \ ; \ ; \textup { f o r a l l } \ ; \ ; \lambda _ 1 \in \mathbb K , \end{align*}


\begin{align*} \dd \sigma _ i & = - \alpha _ { i j } \wedge \sigma _ j + \beta _ { i j } \wedge \eta _ j + \sigma _ k \wedge \sigma _ l - \eta _ k \wedge \eta _ l \\ \dd \eta _ i & = - \beta _ { i j } \wedge \sigma _ j - \alpha _ { i j } \wedge \eta _ j - \sigma _ k \wedge \eta _ l - \eta _ k \wedge \sigma _ l \end{align*}


\begin{align*} A \left ( \Delta \right ) \subset ( - 1 , 1 ) ^ d , \Delta = [ 0 , \delta ) ^ n , \end{align*}


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


\begin{align*} L \left [ x ^ { 2 } P _ { n } ^ { 2 } \right ] = h _ { n + 1 } \left ( z \right ) + \gamma _ { n } ^ { 2 } \left ( z \right ) h _ { n - 1 } \left ( z \right ) = h _ { n + 1 } \left ( z \right ) + \gamma _ { n } \left ( z \right ) h _ { n } \left ( z \right ) . \end{align*}


\begin{align*} I ( \rho , r , y ) = \int _ { | z | \leq \rho } \ ( \Gamma ( z ) - \Gamma ( \rho ) \ ) \ , \ ( \Delta h ( z + y + r \omega - A y - b ) - \Delta h ( z + y + r \omega - A y - b - u ( y ) ) \ ) \ , d z . \end{align*}


\begin{align*} a _ i z _ i = b _ { i + 1 } z _ { i + 1 } . \end{align*}


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


\begin{align*} \psi ( x ) w = w x x \in a _ 0 A a _ 0 . \end{align*}


\begin{align*} B _ t = B _ s + \int _ { s } ^ { t } B _ r \times \circ \dd W _ r \ , , \end{align*}


\begin{align*} \alpha _ 3 = \frac { \sqrt { 2 - \sqrt { 2 } } } { 2 } = - \alpha _ 5 . \end{align*}


\begin{align*} \tilde \sigma ( r , y , \rho , \eta ) = 1 _ { ( r , \rho ) } \otimes \bar a ( y , - \eta ) . \end{align*}


\begin{align*} \underline { u } _ { i } = C ^ { - 1 } \xi _ { i , \delta } \overline { u } _ { i } = C \xi _ { i } . \end{align*}


\begin{align*} H _ N ( \alpha , \beta ) = \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } \frac { ( \dim \mathsf { V } ^ \lambda ) ^ 2 } { d ! } \omega _ \alpha ( \lambda ) \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) \omega _ \beta ( \lambda ) , \end{align*}


\begin{align*} \exists \alpha _ n \in \mathcal P a ( X ^ n \cap Q ^ n ) \ \ \ \mbox { s u c h t h a t } \ \ \ h _ n \alpha _ n = \alpha _ n \ P - \mbox { a . s . } \end{align*}


\begin{align*} \tilde { R } = \begin{pmatrix} \tilde { g } _ { n - k + 1 } \\ \tilde { g } _ { n - k + 2 } \\ \vdots \\ \tilde { g } _ { n } \end{pmatrix} \ / ; \hat { R } = \begin{pmatrix} \hat { g } _ { n - k + 1 } \\ \hat { g } _ { n - k + 2 } \\ \vdots \\ \hat { g } _ { n } \end{pmatrix} \ / ; E _ { n - k + 1 } ^ \zeta = \begin{pmatrix} e _ { n - k + 1 } ( \zeta ) \\ \vdots \\ \vdots \\ e _ n ( \zeta ) \end{pmatrix} \end{align*}


\begin{align*} \begin{aligned} \Phi _ \sqcup ^ U : a \sqcup \{ \} & \rightarrow a \\ \Phi _ \sqcup ^ U ( ( 0 , x ) ) & = x \ ; . \end{aligned} \end{align*}


\begin{align*} z = \sum _ { i = 1 } ^ { n } z _ i \cdot \frac { \partial } { \partial z _ i } , \overline { z } = \sum _ { i = 1 } ^ { n } \overline { z } _ i \cdot \frac { \partial } { \partial \overline { z } _ i } . \end{align*}


\begin{align*} \partial _ t U + U \partial _ x U + \partial _ x p = 0 . \end{align*}


\begin{align*} x _ i ( y _ j ) = \frac { \partial } { \partial ( y _ i ) } ( y _ j ) = \delta _ { i j } . \end{align*}


\begin{align*} f _ { 2 r } ( t , s ) = ( t - \varepsilon _ 1 s ) \cdot \ldots \cdot ( t - \varepsilon _ { 2 r } s ) . \end{align*}


\begin{align*} \rho = { \rm i n t } [ \overline { \mathbf { P } } ^ { x } ] , \end{align*}


\begin{align*} V _ * | _ C = \sum _ { i = 1 } ^ { k } \ , g _ i \ , \partial _ { x _ i } . \end{align*}


\begin{align*} E ^ P _ { s , \varphi } ( g ) = \sum _ { { \gamma \in P \cap \Gamma \backslash \Gamma } } \varphi ^ P _ { s } ( \gamma \cdot g ) . \end{align*}


\begin{align*} e _ i ^ \star = \ell _ i ; \ell _ i ^ \star = e _ i ; ( 0 \vert ( d - 1 ) ) ^ { \star } = ( - 1 ) ^ d ( ( d - 1 ) \vert 0 ) ; ( i \vert ( i \pm 1 ) ) ^ \star = ( i \pm 1 ) \vert i , \ ; i = 1 , \ldots , d - 1 . \end{align*}


\begin{align*} A _ { n } \left ( x ; z \right ) = \frac { \phi \left ( x ; z \right ) } { 2 \gamma _ { n } \left ( z \right ) C _ { n } \left ( x ; z \right ) } , B _ { n } \left ( x ; z \right ) = \frac { n - 2 \gamma _ { n } \left ( z \right ) } { 2 \gamma _ { n } \left ( z \right ) C _ { n } \left ( x ; z \right ) } x , \end{align*}


\begin{align*} \frac { d \varphi _ { s , t } ( z ) } { d t } = ( \tau ( t ) - \varphi _ { s , t } ( z ) ) ( 1 - \overline { \tau ( t ) } \varphi _ { s , t } ( z ) ) p ( \varphi _ { s , t } ( z ) , t ) , \varphi _ { s , s } ( z ) = z \end{align*}


\begin{align*} \sigma _ { ( a , i ) } ( ( b , j ) ) = ( \alpha ^ { - h } ( b ) - \alpha ^ { i - h } ( g ) , j - h ) , \end{align*}


\begin{align*} H ( x , t ) = \sum _ { | \sigma | + 2 \gamma = k , \sigma _ n \geq 1 } \frac { a _ { \sigma \gamma } } { \sigma ! \gamma ! } x ^ { \sigma } t ^ { \gamma } . \end{align*}


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


\begin{align*} G ( x , u , \rho ) F ^ \beta = c \end{align*}


\begin{align*} { \bf T } f = \sum _ { s = 1 } ^ n ( - 1 ) ^ { s - 1 } \sum _ { 1 \leq i _ 1 \leq \cdots \leq i _ s \leq n } { \bf T } _ { i _ 1 } \cdots { \bf T } _ { i _ s } \left ( \frac { \partial ^ { s - 1 } f _ { i _ s } } { \partial \bar z _ { i _ 1 } \cdots \partial \bar z _ { i _ { s - 1 } } } \right ) , \end{align*}


\begin{align*} M \Z ^ { n + 1 } = \Z ^ n ? \end{align*}


\begin{align*} \omega \cdot \ell + \Omega \cdot L = \omega \cdot \ell \pm \Omega _ { j _ 1 } \pm \Omega _ { j _ 2 } \pm \dots \pm \Omega _ { j _ k } \end{align*}


\begin{align*} \hat \mu _ j ( s , l ) = \exp \{ a _ { l j } s ^ 2 + b _ { l j } s + c _ { l j } \} , \ \ s \in \mathbb { R } , \ \ l \in L , \ \ j = 1 , 2 , \end{align*}


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


\begin{align*} J = \frac { 1 } { \delta } \ , \iint _ R | v _ 1 - \tau \ , v _ 2 | ^ { p - 2 } \ , d \sigma d \tau = \frac { 1 } { \delta } \ , \int _ 0 ^ 1 \ ( \int _ { \sigma - \delta } ^ \sigma | v _ 1 - \tau \ , v _ 2 | ^ { p - 2 } \ , d \tau \ ) d \sigma . \end{align*}


\begin{align*} I _ m : = \frac { 1 } { m } \sum _ { j = - m / 2 + 1 } ^ { m / 2 } \frac { e ^ { - 2 \pi i \delta \frac { j + \theta _ 1 / 2 } { m } } } { 1 - \exp \left \{ - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \right \} + \frac { z } { m } } = G _ \delta ( z / m ) + B _ { \delta , m } , \end{align*}