\begin{align*} \mathsf { D } _ { d } ^ { m } Z _ { \left \lfloor T r \right \rfloor } ( d ) = \sum _ { n = 0 } ^ { \left \lfloor T r \right \rfloor - 1 } \mathsf { D } _ { d } ^ { m } ( T ^ { 1 / 2 - d } \pi _ { n } ( d ) ) \xi _ { \left \lfloor T r \right \rfloor - n } = \sum _ { n = 0 } ^ { \left \lfloor T r \right \rfloor - 1 } T ^ { 1 / 2 - d } \pi _ { n } ( d ) R _ { T n } ^ { ( m ) } ( d ) \xi _ { \left \lfloor T r \right \rfloor - n } , \end{align*}


\begin{align*} \overline { u } _ { \mathbf k } \cdot \mathbf k = 0 . \end{align*}


\begin{align*} a V ( h ) = \rho V ( S _ F ) . \end{align*}


\begin{align*} ( \mathrm { I } ) ~ = ~ \sum _ { x = 4 } ^ \infty \sum _ { y = 4 } ^ \infty \underbrace { \Bigl ( x \sqrt { ( x \wedge a ) ( x \wedge b ) } - y \sqrt { ( y \wedge a ) ( y \wedge b ) } \Bigr ) ^ 2 p ( x ) p ( y ) } _ { : = u _ { x , y , a , b } } . \end{align*}


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


\begin{align*} K = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { f } p _ { i } \dot { q } _ { i } \end{align*}


\begin{align*} \mathbb { X } _ t = & 1 + \int _ { 0 } ^ { t } \mathbb { X } _ { s ^ - } \otimes d X _ s + \frac { 1 } { 2 } \int _ { 0 } ^ { t } \mathbb { X } _ { s ^ - } \otimes d [ X , X ] ^ c _ s \\ & + \sum _ { 0 < s \leq t } \mathbb { X } _ { s ^ - } \otimes \{ \exp ( \Delta X _ s ) - \Delta X _ s - 1 \} , \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 ^ + } N _ { \sigma } ( D + \epsilon A ) = N _ { \sigma } ( D ) . \end{align*}


\begin{align*} \frac { d } { d x } \big ( \frac { \cosh x } { m } ( \xi ' ( x ) - ( m - 1 ) \tanh x \xi ( x ) ) \big ) = \frac { f '' ( x ) } { m } - \frac { f ( x ) } { \cosh ^ 2 ( x ) } - \tanh x f ' ( x ) \end{align*}


\begin{align*} B _ { \mathfrak a } \left ( \beta ( v ) , w \right ) = B _ { \mathfrak a } \left ( v , \beta ( w ) \right ) . \end{align*}


\begin{align*} l _ { i i } ^ { + } ( u ) \mid 0 \rangle = \varkappa ^ { + } ( u ) \mid 0 \rangle \end{align*}


\begin{align*} \begin{aligned} & x ^ { k + 1 } = \mathrm { p r o x } _ g ^ { T _ k } ( x ^ k - T _ k ( A ^ T \Bar { y } ^ k + \nabla h ( x ^ k ) ) ) \\ & \hat { y } _ i ^ { k + 1 } = \mathrm { p r o x } _ { f _ i ^ * } ^ { S _ k ^ i } ( y _ i ^ k + S _ k ^ i A _ i x ^ { k + 1 } ) , \ \ i = 1 , . . . , n , \end{aligned} \end{align*}


\begin{align*} C _ 1 - C _ 2 \varepsilon _ 1 - C \varepsilon _ 8 - C \alpha > 0 , C _ 3 - \alpha - \sum _ { j = 2 } ^ 7 \varepsilon _ j > 0 , 1 - C \alpha - C \alpha \varepsilon _ 7 > 0 . \end{align*}


\begin{align*} u _ t + u u _ x = \lambda u _ { x x } \end{align*}


\begin{align*} \mathsf { D } _ { d } Z _ { \left \lfloor T r \right \rfloor } ( d ) & \Rightarrow - \psi ( d ) W ( r ; d ) + A ( r ; d ) = \Gamma ( d ) ^ { - 1 } \int _ { 0 } ^ { r } ( - \psi ( d ) + \log ( r - s ) ) ( r - s ) ^ { d - 1 } \mathsf { d } W ( s ) \\ & = \int _ { 0 } ^ { r } \mathsf { D } _ { d } ( \Gamma ( d ) ^ { - 1 } ( r - s ) ^ { d - 1 } ) \mathsf { d } W ( s ) = \mathsf { D } _ { d } W ( r ; d ) . \end{align*}


\begin{align*} f \left ( U _ 1 ^ { k _ 1 } \ldots U _ n ^ { k _ n } x \right ) = \bar f \left ( U _ 1 ^ { k _ 1 } \ldots U _ d ^ { k _ d } \bar U _ { d + 1 } ^ { \bar k _ { d + 1 } } \ldots \bar U _ { n } ^ { \bar k _ { n } } x \right ) \end{align*}


\begin{align*} \delta ^ e = ( \delta _ { m - 2 } , \delta _ { m - 1 } , \delta _ m , \delta _ { m + 1 } , \delta _ { m + 2 } , \delta _ { m + 3 } ) \end{align*}


\begin{align*} \int _ \Sigma \omega - \int _ { \partial _ + \Sigma } \lambda _ + + \int _ { \partial _ - \Sigma } \lambda _ - = \rho ( A ) . \end{align*}


\begin{align*} P = Z ( Z ^ { T } Z ) ^ { - 1 } Z ^ { T } \mbox { f o r } Z \in \mathcal { M } ( L ) \end{align*}


\begin{align*} \left ( ^ { \rho } D _ { b ^ - } ^ { \alpha } g \right ) ( x ) = \frac { \rho ^ { \alpha - n - 1 } } { \Gamma ( n - \alpha ) } \left ( - x ^ { 1 - \rho } \frac { d } { d x } \right ) ^ { n } \int _ { x } ^ { b } \ ( x ^ { \rho } - \tau ^ { \rho } ) ^ { n - \alpha + 1 } \tau ^ { \rho - 1 } g ( \tau ) d \tau . \end{align*}


\begin{align*} \delta u _ { s , t } + \int _ s ^ t \left ( - \lambda _ 2 [ \partial _ x ^ 2 u _ r + u _ r | \partial _ x u _ r | ^ 2 ] + \lambda _ 1 u _ r \times \partial _ x ^ 2 u _ r \right ) \dd r = h W _ { s , t } u _ s + h ^ 2 \mathbb W _ { s , t } u _ s + u ^ \natural _ { s , t } \ , , \end{align*}


\begin{align*} \sum _ { \Omega _ { n , \mathrm { k } } } \prod _ { i \in [ n ] _ { \neq \mathrm { k } } } \frac { c _ i ^ { N _ i } } { N _ i ! } \cdot \prod _ { 1 \le i \le \ell } \frac { c _ { k _ i } ^ { N _ { k _ i } - 1 } } { N _ i ! } & \le \sum _ { \Omega _ { n - \Sigma _ { \mathrm { k } } } } \prod _ { i = 1 } ^ { n - \Sigma _ { \mathrm { k } } } \frac { c _ i ^ { N _ i } } { N _ i ! } = [ x ^ { n - \Sigma _ { \mathrm { k } } } ] S ( x ) . \end{align*}


\begin{align*} T ^ * = \sup \{ s \in [ 0 , T ] | w \geq 2 b f o r ( \tau , \xi , \eta ) \in [ 0 , s ] \times [ 0 , X ] \times \{ \eta = 0 \} \} \end{align*}


\begin{align*} P _ k = P _ { k , e f f } + R _ { k , \perp } \ , \end{align*}


\begin{align*} F _ n ^ \pi ( x _ 1 , \dots , x _ n ) = \sum _ { k = 0 } ^ \infty \frac { 1 } { k ! } F _ { n + k } ( \pi , \dots , \pi , x _ 1 , \dots , x _ n ) \end{align*}


\begin{align*} X = ( x , \xi ) , Y = ( y , \eta ) , Z = ( z , \zeta ) , \end{align*}


\begin{align*} \gamma _ 0 = b c e ^ { - C y _ 0 ^ 2 } \end{align*}


\begin{align*} \hat C = \begin{pmatrix} g & f \cr 1 & g \cr \end{pmatrix} \quad \hbox { a n d } \hat B = \begin{pmatrix} 0 & 1 \cr x _ 2 \sin \theta & 0 \end{pmatrix} . \end{align*}


\begin{align*} L _ { \lambda } ( f ) ( z ) : = \frac { f ( z ) - f ^ * ( \lambda ) } { z - \lambda } , \ , \ , \ , z \in \mathbb D . \end{align*}


\begin{align*} C ^ S = \sum _ { | I | \leq m } h ^ I \langle \epsilon _ { I } , \hat { \mathbb { S } } ( \boldsymbol { \ell } ) _ T \rangle , \end{align*}


\begin{align*} u & = x _ 1 \big ( \textstyle \prod _ { s = \ell + 1 } ^ { d - 2 } x _ { n - s - 1 } \big ) \big ( \prod _ { s = 0 } ^ { \ell } x _ { n - s } \big ) , \ \ \ \ & v = \big ( \textstyle \prod _ { s = \ell } ^ { d - 1 } x _ { n - s - 1 } \big ) \big ( \prod _ { s = 0 } ^ { \ell - 1 } x _ { n - s } \big ) ; \end{align*}


\begin{align*} H _ { L } : = H _ { \lambda , L } : = - \frac { \hbar ^ 2 } { 2 m } \Delta _ L + \lambda V _ { L } \end{align*}


\begin{align*} R : = k [ x , y ] { \rm a n d } S : = R \setminus \{ 0 \} . \end{align*}


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


\begin{align*} \mathbf { P } _ n ^ { \lambda , \theta , T } ( \Gamma ( x _ i : i \in \mathcal { B } \cup \mathcal { O } \cup \mathcal { U } ) ) = \left ( \det \limits _ { i , j \in \mathcal { O } \sqcup \mathcal { U } \sqcup \mathcal { B } } K ^ * _ { i , j } \right ) \prod _ { i \in \mathcal { B } } \mathrm { d } t _ i \end{align*}


\begin{align*} S _ i \cap \{ \Psi \neq 0 \} \cap \{ \Psi \neq 1 \} = \emptyset , \ \ S _ i \subset \{ \Psi \equiv 1 \} S _ i \subset \{ \Psi \equiv 0 \} . \end{align*}


\begin{align*} y _ n = 1 + \frac { L - c _ m S } { C ( z _ n ) } - \frac { z _ n C ' ( z _ n ) } { C ( z _ n ) } \delta _ n + o \bigg ( \frac { C ' ( z _ n ) } { C ( z _ n ) } \delta _ n \bigg ) . \end{align*}


\begin{align*} C ( v , v ' ) = - \frac { 1 } { 2 } \omega _ { s } ( [ v , v ' ] ) \end{align*}


\begin{align*} \lambda _ i = \prod _ { h = 0 } ^ { \ell _ k } c _ h ^ { \kappa _ { i + 3 , h } } . \end{align*}


\begin{align*} \begin{cases} \tilde { x } _ { k + 1 } = \tilde { A } _ k \tilde { x } _ k + \tilde { C } _ k \tilde { x } _ k w _ k , \\ \tilde { x } _ { 0 } = \left [ \begin{array} { c c c } \xi \\ 0 \end{array} \right ] , \ k \in { \mathcal N } _ { T - 1 } , \end{cases} \end{align*}


\begin{align*} F | _ U = f _ 2 + f _ 3 \end{align*}


\begin{align*} \sum _ { d \mid k ' } \sum _ { e \mid d } \left ( \frac { 1 } { e } - \frac { 1 } { d } \right ) \leq \sum _ { d \mid k ' } \sum _ { e \mid d } \frac { 1 } { e } = \left ( \frac { 7 } { 3 } \right ) ^ L , \end{align*}


\begin{align*} [ \widetilde J _ { ( \omega , x , y ) } \psi ] ( n ) & = \overline { \widetilde p ( \widetilde T ^ { n - 1 } ( \omega , x , y ) ) } \psi ( n - 1 ) + \widetilde q ( \widetilde T ^ n ( \omega , x , y ) ) \psi ( n ) \\ & \qquad \qquad + \widetilde p ( \widetilde T ^ n ( \omega , x , y ) ) \psi ( n + 1 ) . \end{align*}


\begin{align*} h ( \tau ) = \det ( \tfrac { \tau } { i } ) ^ { 1 / 2 } \end{align*}


\begin{align*} \begin{aligned} & \int _ { B _ { R } ^ { + } \backslash B _ { r _ { 0 } } ^ { + } } \eta ^ 6 | \nabla \omega | ^ { 2 } d x d y \\ & = \frac { 1 } { 2 } \int _ { B _ { R } ^ { + } \backslash B _ { r _ { 0 } } ^ { + } } \omega ^ { 2 } ( \Delta \eta ^ 6 + \boldsymbol { w } \cdot \nabla \eta ^ 6 ) d x d y + \frac 1 2 \int _ { B _ { R } ^ { + } \backslash B _ { r _ { 0 } } ^ { + } } \nabla \eta ^ { 6 } \cdot ( a y , b ) \omega ^ { 2 } d x d y . \end{aligned} \end{align*}


\begin{align*} \dd P = \sum _ { i = 1 } ^ n \frac { \partial P } { \partial X _ i } ( 0 , \ldots , 0 ) \ , \dd X _ i . \end{align*}


\begin{align*} \alpha \left ( [ x , y ] \right ) = \left [ \alpha ( x ) , \alpha ( y ) \right ] \end{align*}


\begin{align*} u = \sum _ { F } \frac { \langle f , F \rangle \cdot F } { \lambda _ { s _ F } - \lambda _ w } + \frac { \langle f , 1 \rangle \cdot 1 } { ( \lambda _ 1 - \lambda _ w ) \cdot \langle 1 , 1 \rangle } + \frac { 1 } { 4 \pi i } \int _ { ( \frac { 1 } { 2 } ) } \frac { \langle f , E _ s \rangle \cdot E _ s } { \lambda _ s - \lambda _ w } d s , \end{align*}


\begin{align*} ( x \cdot f ) ( g ) = \frac { d } { d t } \bigg { | } _ { t = 0 } f ( g \cdot e ^ { t x } ) \end{align*}


\begin{align*} \mathcal { P } _ { A , B } \left ( \nu _ \xi \right ) \in a ^ \xi , \ h _ { \nu _ \xi } ( f ) = 0 \rho ( \nu _ \xi , \mu _ \xi ) < \frac { \zeta } { 2 } \xi \in \{ + , - \} ^ d . \end{align*}


\begin{align*} \int _ { \mathbb { P } ^ { n - 1 } } C ( \xi ) d V _ { F S } ( \xi ) = | \Omega | , \end{align*}


\begin{align*} & 2 ( \nu + \rho ^ { \natural } | \gamma ) + 2 m ( k + h ^ \vee ) = \\ & 2 ( \nu + \rho ^ { \natural } | \gamma ) + ( k + h ^ \vee ) + ( 2 m - 1 ) ( k + h ^ \vee ) \leq 2 ( \nu + \rho ^ { \natural } | \gamma ) + ( k + h ^ \vee ) \leq ( k + 1 ) - 2 ( \xi | \nu ) \leq 0 , \end{align*}


\begin{align*} H _ \mu ( m , b ^ { \mu , t } _ n m ' ) = H _ \mu ( g ( b ) ^ { \mu , s } _ { - n } m , m ' ) . \end{align*}


\begin{align*} { \mathcal { L } } G - e ^ { x / 3 } { \mathcal { Q } } ( G ) - e ^ { x / 3 } { \mathcal { C } } ( G ) = e ^ { x / 3 } \Phi , \end{align*}


\begin{align*} \nabla \bar { \eta } ( x ) = R ( x ) \overline { \nabla _ \Gamma \eta } ( x ) , \bar { \nu } ( x ) \cdot \nabla \bar { \eta } ( x ) = 0 \end{align*}


\begin{align*} \omega _ { j _ 1 } \pm \omega _ { j _ 2 } \pm . . . \pm \omega _ { j _ l } \not = 0 \ , . \end{align*}


\begin{align*} \left \{ \begin{array} { l } \Box u = | \nabla | ^ { - 1 } Q ( \overline u , u ) , \\ ( u , \partial _ t u ) | _ { t = 0 } = ( u _ 0 , u _ 1 ) , \end{array} \right . \end{align*}


\begin{align*} { \Theta ( t g ) = { t ^ { - r } } ( g ) ^ { - 1 } \Theta \left ( \frac { ( g ^ \top ) ^ { - 1 } } { t } \right ) } . \end{align*}


\begin{align*} \lVert { L _ { z + t e _ j } - L _ z - t L _ { z , j } } \rVert = O ( | t | ^ 2 ) . \end{align*}


\begin{align*} \Phi _ y = \Phi _ y ^ { ( 0 ) } + 2 \eta \alpha ^ { 3 / 2 } \tilde { \Phi } _ * ^ 1 + R _ * , \end{align*}


\begin{align*} \frac { \partial } { \partial Z _ N } \kappa _ N = 0 , \end{align*}


\begin{align*} \underset { k = i \kappa } { \rm R e s } M ^ { ( 1 ) } ( x , t , k ) = \frac { \gamma _ 0 } { a _ { 1 } ' ( i \kappa ) } e ^ { - 2 \kappa x + 8 \kappa ^ 3 t } M ^ { ( 2 ) } ( x , t , i \kappa ) , \gamma _ 0 ^ 2 = 1 , \end{align*}


\begin{align*} g ( y ) = f ( 2 ( I - \tilde \alpha ) ^ { - 1 } y ) g ( ( I + \tilde \alpha ) ( I - \tilde \alpha ) ^ { - 1 } y ) , \ \ y \in Y . \end{align*}


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


\begin{align*} \mathsf { b t w } _ n \coloneqq \bigvee _ { i = 1 } ^ { n + 1 } \lnot ( \lnot x _ i \land \bigwedge _ { 0 < j < i } x _ j ) \end{align*}


\begin{align*} t = s + f + u \ , . \end{align*}


\begin{gather*} \omega _ i = \frac { u _ i } { u } , \\ \omega _ { i j } = \frac { u _ { i j } } { u } - \frac { u _ i u _ j } { u ^ 2 } , \\ h _ { i j } = u _ { i j } + u \delta _ { i j } = u ( \omega _ { i j } + \omega _ i \omega _ j + \delta _ { i j } ) . \end{gather*}


\begin{align*} G ( x , y ) & = \frac { e ^ { i k _ { + } \vert x \vert } } { \sqrt { \vert x \vert } } G ^ { \infty } ( \hat { x } , y ) + G _ { R e s } ( x , y ) , \\ \nabla _ y G ( x , y ) & = \frac { e ^ { i k _ { + } \vert x \vert } } { \sqrt { \vert x \vert } } H ^ { \infty } ( \hat x , y ) + H _ { R e s } ( x , y ) , \end{align*}


\begin{align*} S ^ i _ { \ell } \left ( 2 \cdot 2 ^ { \ell v } q + 1 \right ) = 2 \cdot \left ( 2 ^ { \ell } - 1 \right ) ^ i 2 ^ { \ell ( v - i ) } q + 1 \end{align*}


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


\begin{align*} c _ 1 ^ S ( L _ { \alpha _ j } ^ * ) | _ { w _ K } = t \ \ \ \textrm { f o r a l l } \ K \subset \Delta \ \textrm { w i t h } \ J \subset K \subset I \end{align*}


\begin{align*} ( D ^ 2 _ \epsilon \times T ^ 2 \times \Sigma , \ , \Tilde { \omega } : = r \ , d r \ , d \theta ^ 0 + d \theta ^ { 1 2 } + \omega _ \Sigma ) . \end{align*}


\begin{align*} \min _ { \| \phi \| = 1 } \Big \{ E _ \mathrm { H } [ \phi ] + \frac { 1 } { N } \mathcal { E } \big [ k _ { n + 1 } ; \ , H [ \phi , k _ { n + 1 } ] , \ , \Theta [ \phi , k _ { n + 1 } ] \big ] , \phi \in \mathfrak { h } ^ 1 _ V \Big \} , \end{align*}


\begin{align*} | \xi _ 1 | , | \xi _ 2 | \leq \min \{ \frac { \sqrt { \tau } } { 2 } , \tau \frac { 1 } { 8 \sqrt { s E } } \} = : C _ 2 ( s , \tau ) \end{align*}


\begin{align*} ( m _ L ) _ { i _ r , j _ r } = \left \{ \begin{matrix} 1 & ( i _ r , j _ r ) = ( p _ t , l _ t ) s + 1 \le t \le k \\ 0 & . \end{matrix} \right . \end{align*}


\begin{align*} f = \mathfrak f + g , g = c ^ * G _ 0 + H ^ * , \end{align*}


\begin{align*} \mathrm { r a d } ( \langle \cdot , \cdot \rangle ) = \left \{ m \in \widehat { M } _ { z , \lambda } \mid \langle m , m ' \rangle = 0 , \ ; \forall m ' \in \widehat { M } _ { z ^ { - 1 } , \lambda ^ { - 1 } } \right \} \end{align*}


\begin{align*} \lim _ { \omega \downarrow 0 } J ( f ) ( \omega ) = - c _ 0 ^ { 3 } j _ M ^ * \mbox { a n d } \lim _ { \omega \rightarrow \infty } J ( f ) ( \omega ) = - c _ \infty ^ { 3 } j _ M ^ * . \end{align*}


\begin{align*} & v _ 1 ( s ) : = \frac { \cos ( 2 \zeta ( s ) ) - n ^ 2 } { n ^ 2 - 1 } , ~ v _ 2 ( s ) : = \frac { 2 i \sin \zeta ( s ) \mathcal S ( \cos ( \zeta ( s ) ) , n ) } { n ^ 2 - 1 } , \\ & F ( s ) : = e ^ { - i k _ { + } \vert y \vert \cos ( \zeta ( s ) + \theta _ { \hat y } ) } . \end{align*}


\begin{align*} \begin{aligned} | H ( y ) | & \leq ( n - 1 ) \max _ { \alpha = 1 , \dots , n - 1 } \max _ { z \in \Gamma } | \kappa _ \alpha ( z ) | , \\ | W ( y ) | & \leq ( n - 1 ) ^ { 1 / 2 } \max _ { \alpha = 1 , \dots , n - 1 } \max _ { z \in \Gamma } | \kappa _ \alpha ( z ) | \end{aligned} \end{align*}


\begin{align*} \begin{cases} J _ h ^ 1 = \{ I = [ t _ k ^ 1 , t _ { k + 1 } ^ 1 ] : I \subseteq [ s _ h , s _ { h + 1 } ] \} , \\ J _ h ^ N = \{ I = [ t _ k ^ N , t _ { k + 1 } ^ N ] : I \subseteq [ s _ h , s _ { h + 1 } ] , \ I \not \subseteq J \ \forall J \in \cup _ { M < N } J _ h ^ M \} , \end{cases} \end{align*}


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


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


\begin{align*} 2 n q \cdot d _ { r , s } ( 0 , 0 ) & = ( 2 n q + n s + r i - r q ) d _ { r , s } ( - n , i ) + ( 2 n q + n s + r i + r q ) d _ { r , s } ( n , - i ) . \end{align*}


\begin{align*} G _ { \neq \mathrm { k } } ( x ) = G ( x ) \cdot T _ 2 ( x ) , \quad T _ 2 ( x ) = \exp \bigg \{ - \sum _ { j \ge 1 } \sum _ { 1 \le i \le \ell } c _ { k _ i } x ^ { j k _ i } \bigg \} . \end{align*}


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


\begin{align*} x = \Gamma \left ( x _ 0 + \int _ 0 ^ { \cdot } f ( s , x _ s ) d s + w \right ) \end{align*}


\begin{align*} L ( t ) = \sum _ { s \in [ 0 , t ] } ( \Delta Y ) ( s ) \ 1 _ { \{ ( \Delta Y ) ( s ) > 1 \} } \qquad , \end{align*}


\begin{align*} \begin{aligned} \partial _ t \Phi ( t = 0 , x ) & = \partial _ t \varphi ( t = 0 , x ) = \sigma ( x ) \\ \partial _ t ^ 2 \Phi ( t = 0 , x ) & = \partial _ t ^ 2 \varphi ( t = 0 , x ) = \sigma ' ( x ) \sigma ( x ) \\ \partial _ t \partial _ x \Phi ( t = 0 , x ) & = \partial _ t \partial _ x \varphi ( t = 0 , x ) = \sigma ' ( x ) . \end{aligned} \end{align*}


\begin{align*} 0 = \phi \partial _ { x } ^ { \ast } L _ { H } + 2 x \phi L _ { H } = \left ( \phi \partial _ { x } ^ { \ast } + \psi \right ) L _ { H } , \end{align*}


\begin{align*} \partial _ x ^ 2 w _ t ( x ) = w _ t ( x ) | \partial _ x w _ t ( x ) | ^ 2 + w _ t ( x ) \times \partial _ x ^ 2 w _ t ( x ) \ , , \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u = a _ 1 ( x ) f ( u + \epsilon , v ) \ ; \ ; & \mbox { i n $ \R ^ N $ , } \ ; \ ; u > 0 \ ; \ ; & & \mbox { i n $ \R ^ N $ , } \\ - \Delta _ q v = a _ 2 ( x ) g ( u , v + \epsilon ) \ ; \ ; & \mbox { i n $ \R ^ N $ , } \ ; \ ; v > 0 \ ; \ ; & & \mbox { i n $ \R ^ N $ , } \end{alignedat} \right . \end{align*}


\begin{align*} O _ { a \sqcup b } = O _ a O _ b \ ; . \end{align*}


\begin{align*} \int _ \Omega \varphi = \int _ { \mathbb { P } ^ { n - 1 } } \int _ { F ^ { - 1 } ( \xi ) } \varphi ( z ) | z | ^ { 2 n - 2 } d \sigma ( z ) d V _ { F S } ( \xi ) , \end{align*}


\begin{align*} \left . \partial _ \nu u _ { z _ 0 } \right | _ \Omega = \nu f ' + \overline { \nu g ' } , \left . \partial _ \nu u _ { z _ 0 } \right | _ { \Omega ^ c } = - { 2 \bar \nu \over ( \bar z - \bar z _ 0 ) ^ 2 } \end{align*}


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


\begin{align*} \Gamma _ 1 ^ h ( u ) & = \langle ( I _ h - I ) z , I _ h ( z u ) \rangle _ { H _ x ^ \mathfrak { s } } , \\ \Gamma _ 2 ^ h ( u ) & = \langle z , ( I _ h - I ) ( z u ) \rangle _ { H _ x ^ \mathfrak { s } } , \\ \Gamma _ 3 ^ h ( u ) & = \mathfrak { s } \langle | I _ h z | ^ 2 I _ h z - | z | ^ 2 z , I _ h ( z u ) \rangle _ { L _ x ^ 2 } , \\ \Gamma _ 4 ^ h ( u ) & = \mathfrak { s } \langle | z | ^ 2 z , ( I _ h - I ) ( z u ) \rangle _ { L _ x ^ 2 } . \end{align*}


\begin{align*} \lim _ { r \to \infty } \frac { | \boldsymbol { w } ( r , \theta ) | } { \sqrt { \log r } } = 0 . \end{align*}


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


\begin{align*} r ( z ) : = \sup \{ r > 0 \colon \ B _ X ( z , r ) \} \end{align*}


\begin{align*} \phi ( g ( P ( x ) ) + g ( Q ( x ) ) ) = \psi ( \varphi ( \phi ( P ( x ) ) ) ) + \psi ( \varphi ( \phi ( Q ( x ) ) ) ) \end{align*}


\begin{align*} | \mathcal { S } _ { \ell } | = { \ell + n - 1 \choose n - 1 } \leq \frac { C } { \sqrt { n } } e ^ n \left ( \frac { \ell + n } { n } \right ) ^ n . \end{align*}


\begin{align*} 1 = \{ \} \ ; . \end{align*}


\begin{align*} \varphi : X \to \mathbb P ^ 2 _ { x _ 0 , x _ 1 , x _ 2 } = \mathbb P ^ 2 _ { R } , \end{align*}


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


\begin{align*} & \int _ E k ( s , x , z ) k ( t , z , y ) \mu ( \d z ) = k ( t + s , x , y ) , & & x , y \in E , \\ & k ( t , x , y ) = k ( t , y , x ) \ge 0 , & & t > 0 , \ ; x , y \in E , \end{align*}


\begin{align*} \int _ \Omega | m _ p ( \cdot , z ) | ^ { p - 2 } \overline { ( K _ p ( \cdot , z ) - K _ { 2 , p , z } ( \cdot , z ) ) } f = 0 , \ \ \ \forall \ , f \in A ^ 2 _ { p , z } ( \Omega ) . \end{align*}


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


\begin{align*} \begin{aligned} \Lambda ( Z _ { \alpha } , Z _ { \beta } ) ( P j + k ) & = \Lambda ( { C _ { k _ 1 } , C _ { k _ 2 } } ) ( j ) \Lambda ( \mathbf { b _ p } , \mathbf { b _ p ' } ) ( k ) \\ & + \Lambda ( C _ { k _ 1 } , C _ { k _ 2 } ) ( j + 1 ) \Lambda ( \mathbf { b _ p } , \mathbf { b _ p ' } ) ( - P + k ) , \end{aligned} \end{align*}


\begin{align*} \begin{cases} \displaystyle \frac { 1 - \nu ^ 2 } E \Delta ^ 2 v ^ e = 0 & \\ [ 2 m m ] \displaystyle \nabla ^ 2 v ^ e \ , t = - \nabla ^ 2 v ^ p \ , t & \end{cases} \end{align*}


\begin{align*} \underset { \Delta t \to 0 } \lim ~ \underset { \epsilon \to 0 } \lim ~ \dd _ p ( X _ N ^ { \epsilon , \Delta t } , X ^ \epsilon ( T ) ) = \underset { \Delta t \to 0 } \lim ~ \dd _ p ( X _ N ^ { 0 , \Delta t } , X ^ 0 ( T ) ) = 0 . \end{align*}


\begin{align*} \mathcal { G } ^ * ( 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*} \mathcal S ( \cos ( \zeta ( s ) ) , n ) = \left \{ \begin{array} { l l } \sqrt { 2 / k _ { + } } e ^ { - i \frac { \pi } { 4 } } H _ { \theta _ c } ( s ) H _ { \pi - \theta _ c } ( s ) \sqrt { s - s _ b ^ { * } } \sqrt { s } , & s \geq 0 , \\ \sqrt { 2 / k _ { + } } e ^ { - i \frac { \pi } { 4 } } H _ { \theta _ c } ( s ) H _ { \pi - \theta _ c } ( s ) \sqrt { s - s _ b ^ { * } } \sqrt { - s } i , & s < 0 . \end{array} \right . \end{align*}


\begin{align*} h _ t ( z ) = h ( z + t ) - \frac { 2 t h ' ( z + t ) } { 1 + t P h ( z + t ) } \end{align*}


\begin{align*} \Delta = \bigcap _ { i \geq 1 } \big ( ( A _ i \times A _ i ) \cup ( A _ i ^ c \times A _ i ^ c ) \big ) . \end{align*}


\begin{align*} [ X , T ] = [ X , J T ] & = \frac { X ( \textbf { H } ) } { 2 } \left ( T - J T \right ) , \\ [ J X , T ] = [ J X , J T ] & = \frac { J X ( \textbf { H } ) } { 2 } \left ( T - J T \right ) . \end{align*}


\begin{align*} L ^ { + } ( u ) _ { 1 \cdots i - 1 , i } ^ { 1 \cdots i - 1 , j } \mid 0 \rangle = 0 f o r \ a l l i < j . \end{align*}


\begin{align*} \omega _ 0 ( s ) \omega _ 0 ( h _ a ) \omega _ 0 ( s ) ^ { - 1 } = \omega _ 0 ( h _ { a ^ { - 1 } } ) ( a \in \R ^ \times ) \ , . \end{align*}


\begin{align*} \omega _ k ( \tilde x ( \theta , \eta , x , t ) , \eta , \theta ) = \omega _ j ( \tilde x ( \theta , \eta , x , t ) , x , t ) . \end{align*}


\begin{align*} & 1 \times 3 + 1 \times \frac { q ( q - \xi ) } { 2 } + 2 \times ( q - 2 ) + 3 \times k _ { 3 , 3 } = ( q + 1 ) ^ 2 . \end{align*}


\begin{align*} p ^ * : k [ N ] \to k [ M ^ \circ ] , X _ { i ; s } \mapsto \prod _ { j = 1 } ^ { n + m } A _ { j ; s } ^ { b _ { j i } } . \end{align*}


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


\begin{align*} ( A + B ) ( x ) \ni 0 \iff x - J _ { \mu A } ( x - \mu B ( x ) ) = 0 \iff M _ { A , B , \mu } ( x ) = 0 , \end{align*}


\begin{align*} \bar \Phi _ \cdot ^ \infty ( x , y ) \coloneqq ( \Phi _ \cdot ^ \infty ) ^ { - 1 } ( x , y ) = \frac { ( x + y ) ( x + y + 1 ) } { 2 } + y \ ; . \end{align*}


\begin{align*} E ( a , b ) = \left \{ z \in \C ^ 2 \ ; \bigg | \ ; \frac { \pi | z _ 1 | ^ 2 } { a } + \pi \frac { | z _ 2 | ^ 2 } { b } \le 1 \right \} . \end{align*}


\begin{align*} \forall J _ 1 , . . . , J _ r \ \ ; \ ; \ ; \ \ ; \ ; \ ; \left | J _ l \right | \leq N \ , \ \ ; \ ; \forall l = 1 , . . . , r \\ \sum _ { l = 1 } ^ r \sigma _ { j _ l } \omega _ { j _ l } \not = 0 \Longrightarrow \left | \sum _ { l = 1 } ^ r \sigma _ { j _ l } \omega _ { j _ l } \right | \geq \frac { \gamma _ r } { N ^ { \tau _ r } } \ , . \end{align*}


\begin{align*} K ( \mathbb P ( V ) ) = k ( z _ 1 , . . . , z _ n ) \hookrightarrow \kappa ( \lambda ) \xrightarrow { \simeq } k ' , z _ j \mapsto t _ { i _ j } + t _ { i _ 0 } ^ { d _ j } . \end{align*}


\begin{align*} g _ { m + 1 } ( d ) = \mathsf { D } _ { d } g _ { m } ( d ) + g ( d ) g _ { m } ( d ) . \end{align*}


\begin{align*} n _ k ^ 0 = \eta _ { \frac { 1 } { k } } * n ^ 0 , \end{align*}


\begin{gather*} \theta ( x , y ) = \theta ' ( x , y ) - \tau ( \delta ( x , y ) ) + \rho ( x ) \tau ( y ) + \rho ( y ) \tau ( x ) = \theta ' ( x , y ) - d ^ 1 \tau ( x , y ) \end{gather*}


\begin{align*} \omega _ { s } = D _ { x } ^ { - 1 } \bigg ( \sum _ { j = 1 } ^ { N - 1 } \pmb { r } _ { j } . d \pmb { r } _ { j } \bigg ) \end{align*}


\begin{align*} \Delta _ { l _ { 2 1 } } \Delta _ { l _ { 1 1 } } P ( u + v ) + \Delta _ { l _ { 2 2 } } \Delta _ { l _ { 1 2 } } Q ( u + \tilde \alpha v ) = 0 , \ \ u , v \in Y , \end{align*}


\begin{align*} \pi ( x ) = \frac { 1 } { 2 } \sum _ { i , j = 1 } ^ n \pi ^ { i j } ( x ) \frac { \partial } { \partial x ^ i } \wedge \frac { \partial } { \partial x ^ j } , \pi ^ { i j } ( x ) = \{ x ^ i , x ^ j \} . \end{align*}


\begin{align*} X _ { \mathrm { i n } } : = \theta ^ { - 1 } ( I ^ \circ ) \subset \subset Y . \end{align*}


\begin{align*} \Phi = x \land y \leq z \ , \& \ , \lnot x \land y \leq z \Longrightarrow y \leq z . \end{align*}


\begin{align*} \nabla \pi ( x ) \Bigl \{ I _ n - d ( x ) \overline { W } ( x ) \Bigr \} = \overline { P } ( x ) , \nabla \pi ( x ) = \overline { P } ( x ) R ( x ) = R ( x ) \overline { P } ( x ) , \end{align*}


\begin{align*} \delta : = \frac { 1 } { \mathtt { m } } \| \rho \| _ { s } + \frac { 1 } { \sqrt { \kappa } } \| \Pi _ 0 ^ { \bot } \phi \| _ { s } \ , , \theta : = \Pi _ 0 \phi \ , . \end{align*}


\begin{align*} \omega _ \psi ^ \flat : = \omega _ \psi \cdot \chi _ G , \end{align*}


\begin{align*} & \zeta _ { i , i + 1 } \left ( \frac { z } { w } \right ) = \frac { d ^ { - 1 } z - q w } { z - w } \ , , \zeta _ { i , i - 1 } \left ( \frac { z } { w } \right ) = \frac { z - q d ^ { - 1 } w } { z - w } \ , , \\ & \zeta _ { i i } \left ( \frac { z } { w } \right ) = \frac { z - q ^ { - 2 } w } { z - w } \ , , \zeta _ { i j } \left ( \frac { z } { w } \right ) = 1 \mathrm { i f } j \ne i , i \pm 1 \ , . \end{align*}


\begin{align*} \begin{array} { l } \Vert \nabla u _ { i } \Vert _ { p _ { i } ( x ) } ^ { p _ { i } \left ( x _ { 4 } ^ { i } \right ) } = \int _ { \Omega } | \nabla u _ { i } | ^ { p _ { i } ( x ) } d x . \end{array} \end{align*}


\begin{align*} A ( \tau ) : = \mathrm { D } \Phi ^ { - 1 } ( \tau ) \forall \ , \tau \in \mathbb { R } ^ d \end{align*}


\begin{align*} D = \sum _ { i = 0 } ^ 3 \alpha _ i D _ i , \end{align*}


\begin{align*} \mathbb { E } \left ( \int _ { 0 } ^ { 1 } g _ t d X _ t \right ) ^ 2 = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } g _ t g _ s d \mathbb { E } ( X _ s X _ t ) = \frac { 1 } { n } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } g _ t g _ s d \mathbb { E } \langle D X _ s , D X _ t \rangle _ { \mathcal { H } } = \frac { 1 } { n } \mathbb { E } \norm { \int _ { 0 } ^ { 1 } g _ t d D X _ t } ^ 2 _ { \mathcal { H } } . \end{align*}


\begin{align*} \operatornamewithlimits { R e s } _ { w = - \alpha } U _ c ( s , w ) = - \frac { 1 } { 2 } \cdot \frac { 1 - 2 ^ { 1 - 2 \alpha - 2 s } } { 1 - 2 ^ { - 2 \alpha - 2 s } } Z ^ { [ 2 ] } _ { A ' } ( 1 - \alpha ) \zeta ^ { [ 2 ] } ( 2 s + 2 \alpha ) V _ c ( s , - \alpha ; \ell ) , \end{align*}


\begin{align*} \lambda _ { \widehat { \mathcal { E } } } ( x ) = \lambda _ { \mathcal { E } _ 1 } \left ( x _ 1 , \dots , x _ { m _ 1 } \right ) + \lambda _ { \mathcal { E } _ 2 } ( x _ { i _ 1 } , \dots , x _ { i _ { m _ 2 } } ) , \end{align*}


\begin{align*} \alpha _ 5 ' = \frac 3 2 - 3 \delta _ 1 , \beta _ 5 ' = \frac 3 2 + 3 \delta _ 2 , \| \mathcal C ( H ) \| _ { \alpha _ 5 ' , \beta _ 5 ' } \lesssim \| H \| _ { \alpha , \beta } ^ 3 . \end{align*}


\begin{align*} \lim _ { r \to \infty } r ^ { \rho - \rho ( r ) } \left | \sum _ { r < | z _ k | } \frac 1 { z _ k ^ \rho } \right | & \le \lim _ { r \to \infty } r ^ { \rho - \rho ( r ) } \left ( \left | \sum _ { R _ n < | z _ k | } \frac 1 { z _ k ^ \rho } \right | + \left | \sum _ { R _ n < | z _ k | \le r } \frac 1 { z _ k ^ \rho } \right | \right ) = 0 . \end{align*}


\begin{align*} \lim _ { k \to \infty } \frac { n ^ k } { h _ l ( n , k ) } = \lim _ { k \to \infty } \frac { n ^ k } { \frac { n ^ k - 1 } { n - 1 } } = n - 1 . \end{align*}


\begin{align*} \Psi ( Z ) = q ^ A \zeta ^ B s ^ C \prod _ { ( n , r , m ) > 0 } \exp \left ( - \sum _ { a = 1 } ^ \infty c _ { a } ( n m , r ) \frac { ( q ^ n \zeta ^ r s ^ { t m } ) ^ a } { a } \right ) . \end{align*}


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


\begin{align*} \psi ( \theta ) = c \theta + \int _ { ( 0 , \infty ) } ( e ^ { - \theta z } - 1 ) \upsilon ( d z ) , \theta \geq 0 . \end{align*}


\begin{align*} \Omega ( P _ 1 , P _ 2 ) = \frac { F _ \Omega ( P , Q ) d x _ P \otimes d x _ Q } { ( x _ P - x _ Q ) ^ 2 h _ X ( x _ { P } , y _ { \bullet P } ) h _ X ( x _ { Q } , y _ { \bullet Q } ) } , \end{align*}


\begin{align*} m ( \pi , \sigma ) : = \langle \pi \otimes \omega _ { 4 n - 2 , \psi } ^ \vee , \sigma \rangle _ { G ^ F } . \end{align*}


\begin{align*} ( \phi \cdot M ) ( a ) : = \phi ( h a ) M ( a ) \phi ^ { - 1 } ( t a ) , a \in Q _ 1 \phi \in ( \textbf { d } ) . \end{align*}


\begin{align*} \log _ \star ( 1 \epsilon + \phi ) = \sum _ { n = 1 } ^ \infty \frac { ( - 1 ) ^ { n - 1 } } { n } \phi ^ { \star n } \exp _ \star ( \phi ) = \sum _ { n = 0 } ^ \infty \frac { 1 } { n ! } \phi ^ { \star n } \end{align*}


\begin{align*} C _ { 1 - \gamma , \rho } ^ { \alpha , \beta } [ a , b ] = \left \{ g \in C _ { 1 - \gamma , \rho } [ a , b ] , \ ^ { \rho } D _ { a ^ + } ^ { \alpha , \beta } g \in C _ { \mu , \rho } [ a , b ] \right \} , \end{align*}


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


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


\begin{align*} \bigl ( q _ n ^ { \epsilon , \Delta t } , p _ n ^ { \epsilon , \Delta t } \bigr ) = \bigl ( \tilde { q } ^ { \epsilon , \Delta t } ( t _ n ) , \tilde { p } ^ { \epsilon , \Delta t } ( t _ n ) \bigr ) . \end{align*}


\begin{align*} ( D \vee ( D ' \cap q P q ) ) t = t P t . \end{align*}


\begin{align*} \langle j u ( t ) - U ^ { \odot \star } ( t , s ) j u ( s ) , x ^ \odot \rangle & = \langle w ( t ) , x ^ \odot \rangle - \langle w ( s ) , x ^ \odot \rangle \\ & = \int _ s ^ t \langle d ^ \star w ( \tau ) , x ^ \odot \rangle d \tau = \langle \int _ s ^ t U ^ { \odot \star } ( t , \tau ) f ( \tau ) d \tau , x ^ \odot \rangle . \end{align*}


\begin{align*} Z = \int _ 0 ^ \cdot \mathbb { X } ^ { N } _ { s ^ - } \otimes d \mathbf { X } _ s , \end{align*}


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


\begin{align*} g _ s : = x ^ { 2 s - 2 } { \left ( \overline g + o ( 1 ) \right ) } , s \in \mathbb R , \end{align*}


\begin{align*} \dim W _ n = \frac { n - 1 } { 2 } . \end{align*}


\begin{align*} & \frac d { d t } \sum \limits _ { \beta = 0 } ^ { 1 } \sum \limits _ { | \alpha | = 0 } ^ { 3 - \beta } \left ( \| \sqrt { G _ 1 } \partial _ \tau ^ \alpha \partial _ y ^ \beta \tilde { u } \| _ { L ^ 2 } ^ 2 + \| \sqrt { G _ 2 } \partial _ \tau ^ \alpha \partial _ y ^ \beta \tilde { \theta } \| _ { L ^ 2 } ^ 2 + \| \sqrt { G _ 3 } \partial _ \tau ^ \alpha \partial _ y ^ \beta \tilde { q } \| _ { L ^ 2 } ^ 2 \right ) + \frac { C _ 0 } { 2 } D ( t ) \\ \le & C E ( t ) ^ { \frac 5 3 } + C f ( t ) E ( t ) ^ { \frac 1 2 } . \end{align*}


\begin{align*} f ( \gamma ) : = [ \phi _ 2 ( \gamma _ 1 \cdot \gamma _ 2 \cdot \gamma _ 3 ) ] = [ \phi _ 2 ( \gamma _ 1 ) + \phi _ 2 ( \gamma _ 2 ) + \phi _ 2 ( \gamma _ 3 ) ] \in \Z / 3 \Z , \end{align*}


\begin{align*} ( - 1 ) ^ { \mathrm { 1 } _ { i \in \mathcal { O } } } \frac { T ^ { \mathrm { 1 } _ { i \in \mathcal { B } } } } { n } \sum _ { w ^ n = ( - 1 ) ^ { \theta _ 2 } } \frac { w ^ { \mathrm { 1 } _ { i \in \mathcal { B } } + h _ i - h _ j } e ^ { - ( T w + \theta _ 1 \pi i + \lambda ) [ t _ j - t _ i ] _ i } } { 1 - ( - 1 ) ^ { \theta _ 1 } e ^ { - ( T w + \lambda ) } } = H _ n ^ { \lambda , \theta , T } ( x _ i , x _ j ) . \end{align*}


\begin{align*} W _ L ^ { } = E _ y - E _ { \infty } = & - \aleph _ { \alpha , L } \frac { \alpha } { L ^ 4 } + O ( \alpha ^ 4 \log ( \alpha ^ { - 1 } ) ) \\ & + O \big ( \alpha ^ 2 L e ^ { - L / 2 } \big ) + O \Big ( \frac { \alpha ^ 2 } { L ^ 5 } \Big ) + O \Big ( \frac { \alpha ^ 3 } { L ^ 2 } \log ( \alpha ^ { - 1 } ) \Big ) . \end{align*}


\begin{align*} \Phi _ 1 ^ * = H _ f ^ { - 1 } P _ f A _ { \infty } ^ - H _ f ^ { - 1 } A _ { \infty } ^ + A _ { \infty } ^ + \Omega , \end{align*}


\begin{align*} \dot { z } ( t ) = ( A ^ * - P ( t ) B B ^ * ) z ( t ) , \ ; z ( \tau ) = x _ 0 , \ ; \tau \leq t \leq T . \end{align*}


\begin{align*} ( x y z ) ^ U : = \prod _ i x _ i ^ { n ' _ i } \prod _ i y _ i ^ { n '' _ i } \prod _ j z _ j ^ { n _ j } \end{align*}


\begin{align*} \theta _ n ^ { \frac 1 { \gamma - 1 } } = \frac { | u _ n ( x _ n , t _ n ) - | u _ n ( \bar x _ n , \bar t _ n ) | } { ( | x _ n - \bar x _ n | + | t _ n - \bar t _ n | ^ { \frac 1 2 } ) ^ { \alpha _ 0 } } \le K . \end{align*}


\begin{align*} \int _ 0 ^ 1 \frac 1 { \omega ( \delta ) } \dd \delta \ge \int _ 0 ^ { \bar \delta } \frac { 1 } { \delta \Theta ^ { - 1 } ( ( 1 / \delta ) ^ { \frac { \alpha } { \alpha - 1 } } ) } \dd \delta = \frac { \alpha - 1 } { \alpha } \int _ { \bar s } ^ \infty \frac { \Theta ' ( s ) } { s \Theta ( s ) } \dd s = + \infty . \end{align*}


\begin{align*} E _ { \mathcal { A } } : = \begin{pmatrix} A _ { 1 1 } & 0 _ { d \times d } \\ 0 _ { d \times d } & A _ { 2 3 } \end{pmatrix} \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { \log \left ( K _ { N , \beta } \right ) } { N } = 0 . \end{align*}


\begin{align*} \dim _ { \kappa ( a ' ) } \pi _ i ( M \otimes _ { A } \widetilde { A } \otimes _ { \widetilde { A } } \kappa ( a ' ) ) & = \dim _ { \kappa ( a ' ) } \pi _ i \left ( M \otimes _ A \kappa ( a ) \otimes _ { \kappa ( a ) } \kappa ( a ' ) \right ) \\ & = \dim _ { \kappa ( a ' ) } \pi _ i \left ( M \otimes _ A \kappa ( a ) \right ) \otimes _ { \kappa ( a ) } \kappa ( a ' ) \\ & = \dim _ { \kappa ( a ) } \pi _ i \left ( M \otimes _ { A } \kappa ( a ) \right ) , \end{align*}


\begin{align*} \xi _ d ( x ) ( 1 ) & = \oplus _ { k = 0 } ^ n \delta _ { \gamma _ { x v _ 0 } ( k ) } \\ \xi _ i ( x ) \left ( \oplus _ { k = 0 } ^ n \delta _ { \gamma _ { y v _ 0 } ( k ) } \right ) & = \oplus _ { k = 0 } ^ n \delta _ { \gamma _ { x y v _ 0 } ( k ) } \\ \xi _ 1 ( x ) \left ( \oplus _ { k = 0 } ^ n \delta _ { \gamma _ { y v _ 0 } ( k ) } \right ) & = \langle \oplus _ { k = 0 } ^ n \delta _ { \gamma _ { x y v _ 0 } ( k ) } , \oplus _ { k = 0 } ^ n \delta _ { \gamma _ { v _ 0 } ( n - k ) } \rangle \end{align*}


\begin{align*} \exp \biggl [ - w \biggl ( s _ { { + } , 0 , 1 } + \sum _ { l = 1 } ^ { \infty } s _ { { + } , 0 , l + 1 } \ , \frac { \partial } { \partial s _ { { + } , 0 , l } } \biggr ) \biggr ] \ f = \rho ( w ) \ , f \end{align*}


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


\begin{align*} L ^ { \pm } ( u ) = \begin{pmatrix} k _ { 1 } ^ { \pm } ( u ) & k _ { 1 } ^ { \pm } ( u ) e _ { 1 } ^ { \pm } ( u ) & \ldots \\ f _ { 1 } ^ { \pm } ( u ) k _ { 1 } ^ { \pm } ( u ) & \vdots & \ldots \\ \vdots & \vdots & \ldots \end{pmatrix} \end{align*}


\begin{align*} \begin{aligned} \| C ^ { \theta } _ d u \| _ { L ^ q _ t L ^ 2 _ x } \lesssim d ^ { - \frac 1 q } \| u \| _ { V ^ 2 _ \theta } , \end{aligned} \end{align*}


\begin{align*} h = \begin{pmatrix} 1 & 0 \\ 0 & - 1 \end{pmatrix} \ , . \end{align*}


\begin{align*} \mathbb { E } \bigg ( \Big | \sum _ { k = 1 } ^ N a _ k \omega _ k \Big | ^ p \bigg ) \sim \left ( \sum _ { k = 1 } ^ N | a _ k | ^ 2 \right ) ^ { \frac { p } { 2 } } , \end{align*}


\begin{align*} X ^ S _ i = \left \{ \begin{array} { c c } X _ i ' & i \in S , \\ X _ i & i \not \in S . \end{array} \right . \end{align*}


\begin{align*} E _ y : = \inf \sigma ( H _ y ) , \end{align*}


\begin{align*} \rho ( \alpha ( x ) ) \circ \beta & = \beta \circ \rho ( x ) \\ \rho \left ( [ x , y ] \right ) \circ \beta & = - \rho \left ( \alpha ( x ) \right ) \rho ( y ) - \rho \left ( \alpha ( y ) \right ) \circ \rho ( x ) . \end{align*}


\begin{align*} \mathcal { K } _ { n , m } [ 1 , P _ i ( Z ) z _ i ^ a \overline { z } _ i ^ b ] = \mathcal { K } _ { n , m } [ 1 , P _ i ( Z ) ] \cdot \mathcal { K } _ { n , m } [ 1 , z _ i ^ a \overline { z } _ i ^ b ] , \end{align*}


\begin{align*} F ( z , s ) : = \sum _ { j \in \mathbb { Z } } \frac { e ^ { 2 \pi i j s } } { 2 \pi i j + z } s \in [ 0 , 1 ) , z \in \mathbb { C } - 2 \pi i \mathbb { Z } , \end{align*}


\begin{align*} | f | _ { V ^ n } = \langle S ^ n f , f \rangle _ { L ^ 2 } . \end{align*}


\begin{align*} \begin{cases} \begin{aligned} d X ( t ) & = - M _ { A , B , \mu } ( X ( t ) ) d t + \sigma ( t , X ( t ) ) d W ( t ) , t \geq 0 \\ X ( 0 ) & = X _ 0 , \end{aligned} \end{cases} \end{align*}


\begin{align*} u ( a , t ) = \alpha _ 1 , u ( b , t ) = \alpha _ 2 \end{align*}


\begin{align*} \begin{aligned} & \norm { ( \xi , v ) } { \mathfrak Y _ { T _ \varepsilon } } ^ 2 : = \norm { \xi } { L ^ \infty ( 0 , T _ \varepsilon ; L ^ 2 ( \Omega _ h \times 2 \mathbb T ) ) } ^ 2 + \norm { v } { L ^ \infty ( 0 , T _ \varepsilon ; L ^ 2 ( \Omega _ h \times 2 \mathbb T ) ) } ^ 2 \\ & ~ ~ ~ ~ + \norm { \nabla v } { L ^ 2 ( 0 , T _ \varepsilon ; L ^ 2 ( \Omega _ h \times 2 \mathbb T ) ) } ^ 2 . \end{aligned} \end{align*}


\begin{align*} \mathcal { A } ( x , t ) = \left ( { \begin{array} { c c } \mathcal { A } _ 1 ( x , t ) & \mathcal { A } _ 2 ( x , t ) \frac { t } { | t | } \\ \frac { t ^ T } { | t | } \mathcal { A } _ 3 ( x , t ) & b ( x , t ) I d _ { ( n - d ) \times ( n - d ) } \\ \end{array} } \right ) , \end{align*}


\begin{align*} ( - 1 ) ^ { \sum _ { k = 1 } ^ p ( x ^ k + y ^ k ) } \det \limits _ { i \notin X , j \notin Y } M _ { i , j } . \end{align*}


\begin{align*} x ( t , s _ 0 , x _ 0 ) = U ( t , s _ 0 ) x _ 0 , \end{align*}


\begin{align*} N ( t ) : = u ( t ) - e ^ { i t \Delta } u _ 0 ^ \sigma = - i \int _ 0 ^ \infty e ^ { i ( t - s ) \Delta } F ( u ( s ) ) \ , d s \end{align*}


\begin{align*} H ( \xi ) E ( \ominus \xi ) \ : = \ : \left ( \prod _ { i = 1 } ^ k \ominus \xi \right ) \xi ^ { n - k } ( 1 - H _ 1 ) + q \ / . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n b _ i W _ i \leq C _ q \left \vert b \right \vert _ 1 + C _ q \left ( t ^ 2 + \ln \left \vert b \right \vert _ 0 \right ) e ^ { t ^ 2 / q } \left \vert b \right \vert _ { q / 2 } \end{align*}


\begin{align*} \phi _ R ( x ) = \begin{cases} ( R ^ 2 - | x | ^ 2 ) ^ { \frac { 1 } { 2 } } , & x \in B _ { R } ( 0 ) , \\ 0 , & x \in \mathbb { R } ^ n \backslash B _ { R } ( 0 ) . \end{cases} \end{align*}


\begin{align*} D _ { T _ * } = ( 0 , T _ * ] \times ( 0 , X ] \times ( 0 , 1 ) . \end{align*}


\begin{align*} W ( P f ) = K W ( f ) \end{align*}


\begin{align*} \int _ { Q _ T } \eta ' ( p ) | \nabla p | ^ m = - \int _ { Q _ T } \eta ( p ) ( \Delta p | \nabla p | ^ { m - 2 } + ( m - 2 ) | \nabla p | ^ { m - 4 } D ^ 2 p : \nabla p \otimes \nabla p ) \end{align*}


\begin{align*} H ( k ) : = G ' ( e ^ { - k } ) e ^ { - k } . \end{align*}


\begin{align*} \Big ( \frac { k _ { n + 1 } } { \delta - k _ { n + 1 } \circ \overline { k _ { n + 1 } } } \Big ) ( x , y ) = \sum _ j { \Big ( \frac { z _ j } { 1 - | z _ j | ^ 2 } \Big ) e _ j ( x ) e _ j ( y ) } . \end{align*}


\begin{align*} S _ { q , r } ( m ) = \frac { \left ( q - 1 \right ) m + r } { p ^ e } \end{align*}


\begin{align*} ( \hat { u } , \hat { \theta } , \hat { h } ) ( \bar { t } , \bar { x } , \bar { y } ) : = ( u , \theta , h ) ( t , x , y ) . \end{align*}


\begin{align*} ( I + \ell ^ j B ) ^ \ell = I + \ell ^ { j + 1 } B + \sum _ { k = 2 } ^ \ell \binom { \ell } { k } \ell ^ { j k } B ^ k . \end{align*}


\begin{align*} \mathbb { P } \{ \det ( C _ t ) = 0 \} & = \mathbb { P } \left \{ v ^ T \cdot C _ t \cdot v = 0 , \ \ v \neq 0 \right \} . \end{align*}


\begin{align*} \alpha \nabla _ B \Lambda = \nabla _ B \left ( \Delta _ B \varphi + \frac { \nabla _ B \varphi ( f ) } { f } + ( m - 1 ) \left ( \frac { \Delta _ B f } { f } + ( m - 1 ) \frac { | \nabla _ B f | ^ 2 } { f ^ 2 } \right ) \right ) , \end{align*}


\begin{align*} \begin{cases} ( \partial _ t ^ 2 - \partial _ s ^ 2 ) W = F \ , ( 0 , 2 T ) \times ( 0 , \infty ) , \\ W | _ { \{ t = 0 \} } = \partial _ t W | _ { \{ t = 0 \} } = 0 , \\ \end{cases} \end{align*}


\begin{align*} U _ k ^ { l _ k } = U _ 1 ^ { a _ { 1 , k } } \circ \cdots \circ U _ d ^ { a _ { d , k } } , d < k \le n , \end{align*}


\begin{align*} \frac { 1 } { 2 \pi i } \int _ { ( a ) } \breve { W } ( s ) N ^ s \sum _ { U \subset A _ s \atop | U | = j } \prod _ { u \in U } X _ d ( 1 / 2 + u ) \mathcal { B } ^ { ( d ) } ( A _ s - U + U ^ { - } ; \ell ) \ , d s \ll D ^ { - \epsilon } \end{align*}


\begin{align*} \ell _ { R } \left ( M / { I ^ { [ q ] } M } \right ) = \ell _ { R } \left ( N / { I ^ { [ q ] } N } \right ) + \ell _ { R } \left ( K / { I ^ { [ q ] } K } \right ) + O ( q ^ { d - 1 } ) \end{align*}


\begin{align*} \binom { 3 n / 2 - 1 } { 3 } - s + d . \end{align*}


\begin{align*} m \otimes n = m n \ ; . \end{align*}


\begin{align*} \frac { [ x ^ { n - \Sigma _ { \mathrm { k } } } ] S _ { \neq \mathrm { k } } ( x ) } { [ x ^ n ] S ( x ) } = \frac { [ x ^ { n - \Sigma _ { \mathrm { k } } } ] S ( x ) } { [ x ^ n ] S ( x ) } + \sum _ { 1 \le u \le n } \frac { [ x ^ { n - \Sigma _ { \mathrm { k } } - u } ] S ( x ) } { [ x ^ n ] S ( x ) } [ x ^ u ] T _ 1 ( x ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \tilde { v } _ 1 ( x , t ) = \int _ { - \infty } ^ { x } u ( y , t ) \tilde { v } _ 2 ( y , t ) d y , \\ & \tilde { v } _ 2 ( x , t ) = 1 - \sigma \int _ { - \infty } ^ { x } \left ( u ( - y , - t ) - A \right ) \tilde { v } _ 1 \left ( y , t \right ) d y , \\ \end{aligned} \right . \end{align*}


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


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


\begin{align*} T _ i ^ { - 1 } y _ i T _ i ^ { - 1 } & = y _ { i + 1 } \intertext { a n d } T _ i y _ i ^ * T _ i & = y _ { i + 1 } ^ * , \end{align*}


\begin{align*} \left \{ \begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = f ~ ~ & & \mbox { i n } ~ ~ \Omega \cap Q _ 1 ; \\ & u = g ~ ~ & & \mbox { o n } ~ ~ \partial \Omega \cap Q _ 1 . \end{aligned} \right . \end{align*}


\begin{align*} & I _ 2 ( \langle s , t \rangle ) = \{ s ^ 2 , t , s t , s ^ 2 t , s ^ 3 t \} , I _ 2 ( N \alpha ) = \{ s ^ j u ^ k \alpha \mid j , k \in \{ 0 , 1 , 2 , 3 \} \} , \\ & I _ 2 ( N \beta ) = \{ s ^ j t u ^ j v \beta , s ^ j u ^ k \beta \mid j , k \in \{ 0 , 1 , 2 , 3 \} , \ , j + k \equiv 0 \pmod { 4 } \} , \\ & I _ 2 ( N \alpha \beta ) = \{ s ^ j u ^ j \alpha \beta \mid j \in \{ 0 , 1 , 2 , 3 \} \} \cup \{ ( s t v \alpha \beta ) ^ { \beta ^ k } , ( s ^ 2 t u ^ 3 v \alpha \beta ) ^ { \beta ^ k } \mid k \in \{ 0 , 1 \} \} . \end{align*}


\begin{align*} ( { \rm R e } z ) | P f ( z ) | = d ( z , \partial D ) | P f ( z ) | \leq \rho _ D ^ { - 1 } ( z ) | P f ( z ) | \leq \hat \beta _ D ( t _ 1 ) . \end{align*}


\begin{align*} \varepsilon _ 3 = k - f _ 2 ( r ) - f _ 6 ( s ) + s f _ { 1 0 } ( r ) + \int ^ s _ 0 \left ( \int ^ \eta _ 0 f _ 9 ( r ^ 2 - \xi ^ 2 ) d \xi \right ) \ , d \eta + \int ^ r _ 0 \rho f _ 9 ( \rho ^ 2 ) \ , d \rho . \end{align*}


\begin{align*} \Lambda \overline { \psi _ { 1 } ( - x , - t , - \bar { k } ) } \Lambda ^ { - 1 } = \psi _ 2 ( x , t , k ) , k \in \mathbb { C } \backslash \{ 0 \} \end{align*}


\begin{align*} \tilde { H } _ 6 = - \sin \varphi H _ 5 + \cos \varphi H _ 6 , \end{align*}


\begin{align*} S ( n ) = \frac { 3 n + 1 } { 2 } \equiv - 1 \pmod { 4 } . \end{align*}


\begin{align*} V \circ V : = \{ ( x , y ) \in X \times X \ , : \ , \mbox { t h e r e e x i s t s } z \in X \mbox { w i t h } ( x , y ) , ( y , z ) \in V \} \end{align*}


\begin{align*} [ A ] ( i ) = \sum _ { 0 \leq j < b } A ( \Phi _ \cdot ^ { a , b b } ( i , \Phi _ \cdot ^ { b , b } ( j , j ) ) ) \ ; . \end{align*}


\begin{align*} g _ i ^ { ( m ) } ( y _ i ) = g _ i ( y _ i ) P _ m \ ! \left ( h _ i ( y _ i ) , \ldots , h _ i ^ { ( m - 1 ) } ( y _ i ) \right ) \qquad ( 1 \leq i \leq s ) , \end{align*}


\begin{align*} ( x y ) ^ U : = \prod _ i x _ i ^ { n ' _ i } \prod _ i y _ i ^ { n _ i } \end{align*}


\begin{align*} \mathcal { E } _ \eta = \mathcal { L } _ 0 ^ { \otimes \eta _ 0 } \oplus \cdots \oplus \mathcal { L } _ N ^ { \otimes \eta _ N } / S \ , . \end{align*}


\begin{align*} u ( x , t ) = o ( 1 ) , x < 0 , \ t > 0 \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u & = \lambda u ^ { - \gamma } + u ^ { q - 1 } \ ; \ ; & & \mbox { i n } \ ; \ ; \Omega , \\ u & > 0 & & \mbox { i n } \ ; \ ; \Omega , \\ u & = 0 & & \mbox { o n } \ ; \ ; \partial \Omega , \end{alignedat} \right . \end{align*}


\begin{align*} \gamma _ A = \max _ { \emptyset \neq I \subseteq \{ 1 , \ldots , n \} } \frac { 2 \dim A _ I } { \dim G _ { A _ I } } , \end{align*}


\begin{align*} \langle A ( x , y ; \xi , \zeta ) ( x - y ) , \xi - \zeta \rangle \geq 0 , \forall x , y \in ( T ) , \xi \in T ( x ) , \zeta \in T ( y ) \footnote { W h e n $ h ( x ) = | x | ^ 2 $ , t h i s i s t h e w e l l - k n o w n n o t i o n o f m o n o t o n e m a p t h a t h a s b e e n t h e s u b j e c t o f l a r g e a m o u n t o f r e s e a r c h a n d a p p l i c a t i o n s , s e e f o r e x a m p l e t h e c l a s s i c a l a n d p o l i s h e d b o o k b y H . B r \ ' e z i s \cite { b r e z i s - b o o k - m o n o t o n e - m a p s } . } \end{align*}


\begin{align*} d _ E ^ * ( x , y ) = d _ E ^ \nu ( \hat x , \hat y ) < \varepsilon , \ \ \ y | _ { B } = 0 , \end{align*}


\begin{align*} d _ p ( \mu , \nu ) = \left ( \inf _ { \pi \in \Pi ( \mu , \nu ) } \int _ { M \times M } d ( x , y ) ^ p d \pi ( x , y ) \right ) ^ { 1 / p } , \end{align*}


\begin{align*} ( I u ) _ { J } : = \overline { u _ { \bar J } } \ . \end{align*}


\begin{align*} A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{pmatrix} . \end{align*}


\begin{align*} \omega ( r ) : = \sqrt { \abs { g _ { x } } } = \frac { \sinh ^ { n - 1 } ( r ) \cosh ^ { d - 1 } ( r ) } { r ^ { n - 1 } } . \end{align*}


\begin{align*} \exp \left \{ - \int \limits _ { a } ^ { \varepsilon _ 0 } \frac { d r } { r q _ 0 ^ { 1 / 2 } ( r ) } \right \} \leqslant e ^ { \ , c } \cdot ( 4 / k _ 0 ) = 4 e ^ { \ , c } \cdot \frac { k _ 0 + 1 } { k _ 0 ( k _ 0 + 1 ) } \leqslant \frac { 8 e ^ { \ , c } } { k _ 0 + 1 } \leqslant 8 e ^ { \ , c } \cdot a \ , , \end{align*}


\begin{align*} \delta _ 0 y _ \varepsilon \geq | \int _ 0 ^ { y _ 1 } \partial _ y \bar { u } _ 1 - \partial _ y u _ 1 d y ' | = | \int _ { y _ 1 } ^ { y _ 2 } \partial _ y u _ 1 d y ' | \geq | y _ 2 - y _ 1 | \gamma _ \varepsilon . \end{align*}


\begin{align*} \begin{gathered} \vec \gamma \rvert _ 0 ( i ) = \vec \gamma ( ( 0 , i ) ) \ ; , \\ \vec \gamma \rvert _ 1 ( i ) = \vec \gamma ( ( 1 , i ) ) \ ; . \end{gathered} \end{align*}


\begin{align*} \Phi _ \varepsilon ^ i ( X , t ) = \Phi _ \nu ( Y , t ) + \varepsilon g _ i ( \Phi _ \nu ( Y , t ) , t ) \nu ( \Phi _ \nu ( Y , t ) , t ) , \end{align*}


\begin{align*} | \alpha \langle V _ y \rangle _ { \Phi _ { \# } ^ { y } } | & \leq C \frac { \alpha } { y } \| \Phi ^ y _ { \# } \| ^ 2 + \langle \alpha V ^ > _ y \rangle _ { \Phi _ { \# } ^ y } \\ & \leq C \frac { \alpha ^ 5 } { L } \log ( \alpha ^ { - 1 } ) + \alpha \| \Phi _ { \# } ^ y \| \| P \Phi _ { \# } ^ y \| = O ( \alpha ^ 5 \log ( \alpha ^ { - 1 } ) ) . \end{align*}


\begin{align*} \lim _ { L \to \infty } T _ { n , L } [ z ; \cdot , \cdot ] & = \sum _ { A \in \mathcal { A } _ n } \lim _ { L \to \infty } C _ { n , A , L } [ z ; \cdot , \cdot ] \end{align*}


\begin{align*} \eta _ 0 : = \min _ { F \in \mathcal { F } _ h ^ 0 \cup \mathcal { F } _ { h , D } ^ \partial } \eta _ F , \tau _ 0 : = \min _ { F \in \mathcal { F } _ h ^ 0 } \tau _ F . \end{align*}


\begin{align*} \chi _ { s , N } ( \boldsymbol { \alpha } ) \sim \displaystyle \sum _ { \substack { \boldsymbol { r } \in \mathbb { Z } ^ d \\ \boldsymbol { r } = ( r _ 1 , \cdots , r _ d ) } } c _ { \boldsymbol { r } } e ( \boldsymbol { r } . \boldsymbol { \alpha } ) , \end{align*}


\begin{align*} \tilde { g } _ { \ell } ( z , \zeta , q ) \ : = \ : e _ { \ell } ( \zeta ) + ( - 1 ) ^ { n - k } \frac { q } { 1 - q } e _ { n - k } ( 1 - \tilde { z } ) \Bigl ( { k \choose \ell + k - n } - e _ { \ell + k - n } ( z ) \Bigr ) \ / . \end{align*}


\begin{align*} \phi \left ( r e ^ { i \theta ^ 0 } , \ , \theta ^ 1 , \ , \theta ^ 2 , \sigma , h , \theta ^ S \right ) = \left ( \sqrt { 2 \log ( r \epsilon ) } \ , e ^ { i \left ( p \theta ^ 0 + a \theta ^ 2 \right ) } , \ , \theta ^ 1 , \ , q \theta ^ 0 + b \theta ^ 2 , \varphi ( \sigma ) , h , \theta ^ S + \theta ^ 0 \right ) , \end{align*}


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


\begin{align*} & \mathcal { E } ( k _ n ) = \\ & \mathrm { t r } \Big \{ \big ( \delta - k _ n \circ \overline { k _ n } ) ^ { - 1 } \circ \Big ( k _ n \circ H [ n - 1 ] \circ \overline { k _ n } + \frac { 1 } { 2 } k _ n \circ \overline { \Theta [ n - 1 ] } + \frac { 1 } { 2 } \Theta [ n - 1 ] \circ \overline { k _ n } \Big ) \Big \} \le 0 . \end{align*}


\begin{align*} \frac { \partial \ } { \partial \phi _ n } = \frac { - \sum \limits _ { k = 1 } ^ K \frac { e ^ { - \rho _ k } } { \rho _ k ^ 2 } \frac { \partial \rho _ k } { \partial \phi _ n } } { \sum \limits _ { k = 1 } ^ K \exp ( - \rho _ k ) } \end{align*}


\begin{align*} L ( n ) \leq \lim _ { k \to \infty } \bigg ( n - 1 + \frac { 1 } { h _ l ( n , k ) } \bigg ) - \frac { 2 \cdot n ^ 2 } { 3 \cdot n + 2 } = \frac { n ^ 2 - n - 2 } { 3 \cdot n + 2 } . \end{align*}


\begin{align*} \begin{array} { c c c } & \forall P , Q \in \mathcal { A } _ { d , q } , \partial _ i ( P Q ) = \big ( \partial _ i P \big ) \big ( 1 \otimes Q \big ) + \big ( P \otimes 1 \big ) \big ( \partial _ i Q \big ) , \\ & \\ & \forall i , j , \partial _ i X _ j = \delta _ { i , j } 1 \otimes 1 , \partial _ i Z _ j = 0 . \end{array} \end{align*}


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


\begin{align*} X ^ { \lambda } _ { F _ { X / k } } = \{ a _ 0 x ^ p _ 0 + \cdots + a _ n x ^ p _ n = 0 \} \subset \mathbb P ^ n _ k . \end{align*}


\begin{align*} \| { { \bf x } } \| _ \gamma = \gamma \max _ { i \in I } \{ | x _ i | \} , \end{align*}


\begin{align*} \partial _ t \nu - \nabla \cdot ( \nu \nabla p ) = 0 , \end{align*}


\begin{align*} \Theta _ { \alpha , \jmath } ( \nu ) : = \sum _ { s \in S ^ { F ^ \nu } _ { \iota , \jmath } } \overline { \chi _ { \upsilon } ^ { ( \nu ) } ( s ) } \cdot \eta _ { \varsigma } ^ { ( \nu ) } ( s ) \cdot \omega _ { \psi } ^ { ( \nu ) } ( s u ) . \end{align*}


\begin{align*} g ( x ) = g ( - x ) . \end{align*}


\begin{align*} c _ { n } = \left ( \int \psi _ { + } ( x , \mathrm { i } \kappa _ { n } ) ^ { 2 } \mathrm { d } x \right ) ^ { - 1 / 2 } \end{align*}


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


\begin{align*} e _ { i j } : = \begin{cases} 1 , & j = i , \\ 0 , & j \neq i . \end{cases} \end{align*}


\begin{align*} Y _ { n , m } : = \sum _ { \substack { i \in [ 1 , n ] , \\ \ell \in [ 1 , m ] } } g _ { n , \alpha } ( i ) G ( S _ i , \tilde { S } _ \ell ) \tilde { g } _ { m , \alpha } ( \ell ) , \underline { Y } _ { n , m } = \overline { g } _ { n , \alpha } \overline { g } _ { m , \alpha } \sum _ { \substack { i \in [ 1 , n ] , \\ \ell \in [ 1 , m ] } } G ( S _ i , \tilde { S } _ \ell ) . \end{align*}


\begin{align*} \int _ 0 ^ t f ( s , \theta _ s ^ 1 - \omega _ s ) \ , d s - \int _ 0 ^ t f ( s , \theta _ s ^ 2 - \omega _ s ) \ , d s = \int _ 0 ^ t \bigl ( f ( s , \theta _ s ^ 1 - \omega _ s ) - f ( s , \theta _ s ^ 2 - \omega _ s ) \bigr ) \ , d s , \end{align*}


\begin{align*} S ^ j ( m ) = 2 ^ { v - j } 3 ^ j w - 1 \end{align*}


\begin{align*} \begin{gathered} A _ M ( i , j ) = A _ M ( j , i ) \ ; , \\ A _ i = A _ o \ ; , \ A _ I ( i , l ) = A _ O ( l , i ) \ ; , \end{gathered} \end{align*}


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


\begin{align*} J _ N = 1 + \sum _ { d = 1 } ^ \infty \frac { z ^ { 2 d } } { d ! d ! } N ^ { 2 d } \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } s _ \lambda ( P Q ) \frac { ( \dim \mathsf { V } ^ \lambda ) ^ 2 } { \dim \mathsf { W } _ N ^ \lambda } , \end{align*}


\begin{align*} K _ { p , \varphi _ p } ( 0 ) = \frac { 2 - p k _ p } { 2 \pi } . \end{align*}


\begin{align*} \iint _ { Q _ { 1 , \bar s } } \tilde m ^ { q ' _ 0 } = | \zeta ( 0 , 0 ) | - \iint _ { Q _ { 1 , \bar s } } \tilde b \tilde m \cdot D \zeta . \end{align*}


\begin{align*} b ( x ) = \begin{cases} 0 & x \le 0 x \ge e ^ { - e } , \\ \beta x \log \frac { 1 } { x } ( \log \log \frac { 1 } { x } - 1 ) ^ \alpha & 0 < x < e ^ { - e } . \end{cases} \end{align*}


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


\begin{align*} \mathbb { E } ( \det C ( Y ) ) = \mathbb { E } ( \langle D Y , D Y \rangle _ \mathcal { H } ) > 0 . \end{align*}


\begin{align*} \lim _ { z \rightarrow z _ 0 } \frac { K _ { \Omega , p } ( z ) } { K _ { \Omega ' , p } ( z ) } = 1 \end{align*}


\begin{align*} \begin{array} { c c } \displaystyle c _ j ( \xi , x , t ) = \exp \int _ x ^ \xi \left ( \frac { b _ { j j } } { a _ { j } } \right ) ( \eta , \omega _ j ( \eta ) ) \ , d \eta , d _ j ( \xi , x , t ) = \frac { c _ j ( \xi , x , t ) } { a _ j ( \xi , \omega _ j ( \xi ) ) } . \end{array} \end{align*}


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


\begin{align*} { \bf e } _ i ^ 2 = \left \{ \begin{array} { l l l } \sum \limits _ { j = i + 1 } ^ n a _ { i j } { \bf e } _ j , & i \leq n - 1 ; \\ { \bf 0 } , & i = n . \end{array} \right . \end{align*}


\begin{align*} \beta ^ 2 + b \beta - \mu = 0 , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s m _ i \alpha ( P _ i , t ) \simeq 0 \end{align*}


\begin{align*} ( 2 A + \nabla ( t ) B - \nu \nabla ( t ) D ) \delta ^ { n + 1 } = ( 2 A - \nabla ( t ) B \\ + \nu \nabla ( t ) D ) \delta ^ { n } \end{align*}


\begin{align*} U ( t , s ) = \left \{ \begin{array} { c c l } \Phi ( t , s ) & , & \tau _ { j } < t , s \leq \tau _ { j + 1 } , \\ \Phi ( t , s ) \prod \limits _ { k = j + 1 } ^ { i } ( 1 + b _ { k } ) & , & \tau _ { j } < s \leq \tau _ { j + 1 } \leq \tau _ { i } < t \leq \tau _ { i + 1 } , \\ \Phi ( t , s ) \prod \limits _ { k = j + 1 } ^ { i } \left ( \frac { 1 } { 1 + b _ { k } } \right ) & , & \tau _ { j } < t \leq \tau _ { j + 1 } \leq \tau _ { i } < s \leq \tau _ { i + 1 } , \end{array} \right . \end{align*}


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


\begin{align*} \small \mathbf { v } _ { i , j } = \mathbf { y } _ { i , j } - \mathbf { H } _ { i , j } \mathbf { x } _ { i , j } = \begin{cases} \mathbf { n } _ { i , j } , ( i , j ) \in \mathbb { S } _ b , b \notin \mathbb { I } , \\ \mathbf { H _ I } _ { i , j } \mathbf { x _ I } _ { i , j } + \mathbf { n } _ { i , j } , ( i , j ) \in \mathbb { S } _ b , b \in \mathbb { I } , \end{cases} \end{align*}


\begin{align*} g * \rho _ \delta ( t ) = \int _ \R g ( t - s ) \rho _ \delta ( s ) \ , d s , \end{align*}


\begin{align*} \breve { M } ( x , t , k ) = \begin{pmatrix} 1 & 0 \\ 0 & \frac { k - i \kappa } { k } \end{pmatrix} + \frac { \breve { M } ^ r _ { 1 } ( x , t ) } { k } + \frac { i \kappa } { k - i \kappa } P ( x , t ) + O ( k ^ { - 2 } ) , { \rm a s } \ k \rightarrow \infty , \end{align*}


\begin{align*} \mathcal { I } ( a + b i ) : = \int _ 0 ^ 1 \mathrm { d } \theta \log \left | a + i b + e ^ { 2 \pi i \theta } \right | . \end{align*}


\begin{align*} E ( z ) = \mu E ( z ) P ( z ) + R ( z ) + 1 , \end{align*}


\begin{align*} \sup _ { n \in \N } E [ e ^ { c _ 2 \overline { \Theta } _ n } ] < \infty , \Theta _ n : = \sum _ { u = 1 } ^ p \sum _ { j = 1 } ^ { 2 ^ { u - 1 } } \alpha ^ { ( n ' _ u ) } _ { j } \ , . \end{align*}


\begin{align*} 0 = \R ^ l G ( F P ) \rightarrow \R ^ l G ( F X ) \rightarrow \R ^ { l + 1 } G ( F Q ) = 0 , 1 \leq l \leq i - 1 . \end{align*}


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


\begin{align*} \lim _ { k \to \infty } \frac { 3 ^ k } { h ( 3 , k ) } = \lim _ { k \to \infty } \frac { 3 ^ k } { \frac { 3 ^ k - 1 } { 2 } } = 2 , \end{align*}


\begin{align*} D _ x X ( t , s , x ) & = \exp \left ( \int _ s ^ t D _ x b ( v , X ( v , s , x ) ) \dd v \right ) , \\ J _ X ( t , s , x ) & = \exp \left ( \int _ s ^ t \div _ x b ( v , X ( v , s , x ) ) \dd v \right ) . \end{align*}


\begin{align*} S ( f \star g ) = S f \star ' S g . \end{align*}


\begin{align*} \mathrm { s c l } _ { G ( K ) } ( \lambda ) = g ( K ) - \frac { 1 } { 2 } \ge \frac { 1 } { 2 } \end{align*}


\begin{align*} B ' \cap Q ^ \omega = ( B ' \cap Q ) ^ \omega . \end{align*}


\begin{align*} \begin{aligned} d Y ( t ) & = - \nabla f ( X ( t ) ) d t + \sigma ( t , X ) d W ( t ) , \\ X ( t ) & = Q ( \eta Y ( t ) ) , \end{aligned} \end{align*}


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


\begin{align*} M ( A ) _ M = \begin{pmatrix} 0 & A _ M \\ - A _ M ^ T & 0 \end{pmatrix} \ ; . \end{align*}


\begin{align*} \mathbf { P r } \left ( \bigg \| \frac { d } { n \epsilon } \sum _ { i = 1 } ^ { n } J ( u _ 0 + \epsilon v _ { 0 , i } ) v _ { 0 , i } - \nabla J ( u _ 0 ) \bigg \| \geq t - \Omega ( d \epsilon ) \right ) \rightarrow 0 , n \rightarrow \infty . \end{align*}


\begin{align*} \lim _ { | \eta | \to \infty , \ , \eta \in J } ~ \eta ^ { - 2 k } \sum \nolimits _ { i = 0 } ^ { 2 k } \psi ( a _ i ) \eta ^ i = \psi ( a _ { 2 k } ) \geq 0 . \end{align*}


\begin{align*} \lambda \partial _ i Q ( \alpha ) + \partial _ i R ( \alpha ) = 0 \end{align*}


\begin{align*} Y _ { i j } = & \frac { 1 } { \phi \Omega } ( u ^ i , x ^ i ) \left ( \begin{array} { c c } y _ { 1 1 } & y _ { 1 2 } \\ y _ { 1 2 } & y _ { 2 2 } \end{array} \right ) \left ( \begin{array} { c } u ^ j \\ x ^ j \end{array} \right ) \end{align*}


\begin{align*} \lambda ( g ) + \sum _ { \ell \mid M } \delta ( g , \ell ) = \lambda ( f ) + \sum _ { \ell \mid M } \delta ( f , \ell ) , \end{align*}


\begin{align*} \delta _ { * , L } ( u ) = | \Lambda _ L | \delta _ * ( u ) , \end{align*}


\begin{align*} P ^ * \left ( ( \Omega _ { i j } \times B _ { i j } ) \triangle ( \Omega _ { i j } \times C _ { i j } ' ) \right ) = P ^ * \left ( \bar B _ { i j } \triangle \bar C _ { i j } ' \right ) < \frac { \varepsilon } { s m } \ \ \ ( 1 \le i \le m , \ 1 \le j \le s ) . \end{align*}


\begin{align*} k ( \gamma ) = \sum _ { i = 1 } ^ s m _ i P _ i , \end{align*}


\begin{align*} G : A ^ * ( S , \mathbb { Q } ) [ \zeta ] / ( Q ) \to \bigoplus _ { i = 0 } ^ N A ^ * ( S , \mathbb { Q } ) , \alpha \mapsto \left ( \pi _ * ( \Psi ( \alpha ) \cdot \zeta ^ i ) \right ) _ { i = 0 , \ldots , N } \ , . \end{align*}


\begin{align*} H _ 3 ^ \theta = e ^ { - i \theta } H _ 3 , \ , \ , H _ 4 ^ \theta = e ^ { - i \theta } H _ 4 , \ , \ , H _ 5 ^ \theta = e ^ { - i \theta } H _ 5 \ , H _ 6 ^ \theta = e ^ { - i \theta } H _ 6 . \end{align*}


\begin{align*} \begin{cases} & \frac { \partial X } { \partial t } ( x , t ) = \psi u ^ \alpha \rho ^ \delta F ^ { \beta } ( x , t ) \nu , \\ & X ( \cdot , 0 ) = X _ 0 . \end{cases} \end{align*}


\begin{align*} \Sigma ( x ) = \begin{cases} \{ 0 \} \cup \{ x \} & \mbox { i f $ x \leq 0 $ , } \\ \emptyset & \mbox { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} ( 1 - \tanh ^ 2 x ) u '' ( \tanh x ) - ( 2 + m ) \tanh x u ' ( \tanh x ) + \frac { m } { m + p - 2 } ( \hat { \lambda } + m - p ) u ( \tanh x ) = 0 . \end{align*}


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


\begin{align*} \mathcal Q ( \Delta _ { g _ s } ) : = \frac { D _ { \rm m a x } ( \Delta _ { g _ s } ) } { D _ { \rm m i n } ( \Delta _ { g _ s } ) } . \end{align*}


\begin{align*} \ell _ k ( u ) \ell _ m ( v ) \textbf { 1 } = \ell _ k ( u ) \bar { \ell } _ m ( v ) \textbf { 1 } = \bar { \ell } _ m ( v ) \ell _ k ( u ) \textbf { 1 } = \bar { \ell } _ m ( v ) \bar { \ell } _ k ( u ) \textbf { 1 } . \end{align*}


\begin{align*} \Phi ( P ) \in A ( R ) _ { \textup { t o r s } } \ \ \Longleftrightarrow \ \Phi ^ + ( P ) = 0 . \end{align*}


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


\begin{align*} | W _ \mathcal { A } ( f , g ) ( x , \xi ) | = | W _ { \mathcal { A } _ { F T 2 } \mathcal { D } _ L } ( f , g ) ( x , \xi ) | . \end{align*}


\begin{align*} w _ l = ( s _ { \alpha _ 1 } \ldots s _ { \alpha _ n } ) ( s _ { \alpha _ 1 } \ldots s _ { \alpha _ ( n - 1 ) } ) \ldots ( s _ { \alpha _ 1 } s _ { \alpha _ 2 } ) s _ { \alpha _ 1 } . \end{align*}


\begin{align*} [ x , y ] _ { \mathfrak n } & = [ x , y ] + \theta ( x , y ) + \gamma ' ( x , y ) ; \\ [ x , v ] _ { \mathfrak n } & = \rho ( x ) v + \gamma ' ( x , v ) ; \\ [ v , w ] _ { \mathfrak n } & = \gamma ' ( v , w ) ; \\ [ Z , x ] _ { \mathfrak n } & = \rho ^ { * } ( x ) Z \\ [ Z , v ] _ { \mathfrak n } & = \rho ^ { * } ( v ) Z \\ [ Z _ 1 , Z _ 2 ] _ { \mathfrak n } & = 0 \end{align*}


\begin{align*} \Vert D f \Vert ( X ) \le \lim _ { i \to \infty } \int _ 0 ^ 1 g _ i \ , d \mu = \lim _ { i \to \infty } \int _ 0 ^ 1 g _ i \ , d \mathcal L ^ 1 = 1 . \end{align*}


\begin{align*} - \int _ { \R ^ d } \partial _ j \varphi ( x ) u ( s , x ) \d x = \int _ { \R ^ d } \varphi ( x ) v ( s , x ) \d x s \in ( 0 , T ) , \end{align*}


\begin{align*} A ( t ) = \int _ { 0 } ^ { t } a ( r ) \ , d r + \sum \limits _ { k = 1 } ^ { \infty } b _ { k } H _ { \tau _ { k } } ( t ) , t \geq 0 , \end{align*}


\begin{align*} z . x & = \big ( [ z _ { p ^ k - 1 } , \ldots , z _ 0 ] \cdot ( { A ^ { - 1 } } ) ^ { \textrm { t r } } \big ) \cdot \big ( A ^ { \textrm { t r } } \cdot [ x _ { p ^ k - 1 } , \ldots , x _ 0 ] ^ { \textrm { t r } } \big ) \\ & = [ z _ { p ^ k - 1 } , \ldots , z _ 0 ] \cdot [ x _ { p ^ k - 1 } , \ldots , x _ 0 ] ^ { \textrm { t r } } = 0 . \end{align*}


\begin{align*} m = 2 \cdot 2 ^ { \ell v } q + 1 . \end{align*}


\begin{align*} J _ { \breve { N } ^ r } ( x , t , k ) = \begin{pmatrix} 1 & 0 \\ 0 & \frac { k - i \kappa } { k } \end{pmatrix} J _ { \breve { N } } ( x , t , k ) \begin{pmatrix} 1 & 0 \\ 0 & \frac { k } { k - i \kappa } \end{pmatrix} . \end{align*}


\begin{align*} \mathcal { A } _ t = \begin{pmatrix} A _ { t , 1 1 } & I _ { d \times d } - A _ { t , 1 1 } & A _ { t , 1 3 } & A _ { t , 1 3 } \\ A _ { t , 2 1 } & - A _ { t , 2 1 } & I _ { d \times d } - A _ { t , 1 1 } ^ T & - A _ { t , 1 1 } ^ T \\ 0 _ { d \times d } & 0 _ { d \times d } & I _ { d \times d } & I _ { d \times d } \\ - I _ { d \times d } & I _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } \end{pmatrix} , \end{align*}


\begin{align*} k _ { i } ^ { + } ( u ) \mid 0 \rangle = \varkappa _ { i } ^ { + } ( u ) \mid 0 \rangle a n d k _ { i } ^ { - } ( u ) \mid 0 \rangle = \varkappa _ { i } ^ { - } ( u ) \mid 0 \rangle \end{align*}


\begin{align*} \d z = ( i \Delta z - i \nu z - \epsilon ( \gamma z - \mu \overline { z } ) ) \d t + i \kappa | z | ^ 2 z \d t - i ( z \circ \d W ) ( x , t ) \in \R \times \R ^ + , \end{align*}


\begin{align*} u \times ( \beta \partial _ x u ^ { 1 } , 0 , 0 ) \cdot \partial _ x u = \beta ( \partial _ x u ^ { 1 } , 0 , 0 ) \cdot ( u \times \partial _ x u ) = \beta \partial _ x u ^ 1 ( u ^ 2 \partial _ x u ^ 3 - u ^ 3 \partial _ x u ^ 2 ) \ , . \end{align*}


\begin{align*} \mathcal L : t \mapsto \mathcal L ( t ) = \exp ( - L t ) . \end{align*}


\begin{align*} H \ = \ H _ 0 + H _ 1 Y _ k + H _ 2 Y _ k ^ 2 + \cdots . \end{align*}


\begin{align*} \chi _ i = - \xi ( \theta _ i ^ \vee ) , \end{align*}


\begin{align*} R ( x ) \Bigl \{ I _ n - d ( x ) \overline { W } ( x ) \Bigr \} = I _ n , x \in \overline { N } \end{align*}


\begin{align*} \rho _ { s , t } ^ \lambda ( x , y ) = \frac { \lambda ^ { - d } } { ( 2 \pi ) ^ \frac d 2 } \int _ { \R ^ d } \rho _ { s , t } ( x - z , y - z ) \exp \big ( - \frac { | z | ^ 2 } { 2 \lambda ^ 2 } \big ) d z . \end{align*}


\begin{align*} ( x , y ) = ( \alpha ( t + s , \alpha ( - t - s , x ) ) , y ) \in V [ O , U ] \subseteq V \ , . \end{align*}


\begin{align*} q _ i ^ + : = \max _ { x \in \overline { \Omega } _ i } ~ q ( x ) < { ( p ^ * ) } ^ { - } _ { i } : = \min _ { x \in \overline { \Omega } _ i } p ^ * ( x ) i \in \{ 1 , \cdots , m \} . \end{align*}


\begin{align*} \Delta \log u _ 1 = 4 K _ 1 ^ * , \end{align*}


\begin{align*} \mathbb { E } [ \log X ] & = \psi _ 0 ^ f ( a ) , \\ \mathrm { V a r } ( \log X ) & = \psi _ 1 ^ f ( a ) . \end{align*}


\begin{align*} D _ + ( \sigma ) \ ; = \ ; \bigsqcup \bigl \{ O ( \sigma ) \colon \sigma ' \sqsubset \sigma \bigr \} . \end{align*}


\begin{align*} - \omega ^ { 1 / 2 } \mathcal C ( f ) & = \omega ^ { 1 / 2 } \phi + j _ M ^ \infty \delta _ \infty - j _ M ^ 0 \delta _ 0 \\ - \omega ^ { 3 / 2 } \mathcal C ( f ) & = \omega ^ { 3 / 2 } \phi - \left ( \int _ 0 ^ \infty \tilde \omega ^ { 3 / 2 } \phi \ , \dd \tilde \omega \right ) \delta _ \infty , \end{align*}


\begin{align*} d ^ 2 _ { Q } ( \theta , \gamma ) & = ( d ^ 2 \theta , d _ { r } ^ 3 \circ d _ { r } ^ 2 \sigma - \frac { 1 } { 2 } B _ { \mathfrak a } \left ( d ^ 1 \tau \wedge ( d ^ 1 \tau \circ \alpha ) \right ) + \frac { 1 } { 2 } B \left ( d ^ 1 \tau \wedge ( d ^ 1 \tau \circ \alpha ) \right ) ) \\ & = ( 0 , 0 ) . \end{align*}


\begin{align*} \widetilde { \mathcal { E } } ^ { ( j ) } ( x , t ) : = \begin{cases} \mathcal { E } ^ { ( j ) } ( x , t ) , & x _ 1 \geq 0 , \\ - \mathcal { E } ^ { ( j ) } ( - x _ 1 , x _ 2 , x _ 3 , t ) , & x _ 1 < 0 . \end{cases} \end{align*}


\begin{align*} \lambda \| S ( V | \Psi ^ 3 ) \| ^ 2 _ { 2 } = \lambda \iint _ \Omega | \nabla V | ^ 2 \Psi ^ 3 \frac { d t d x } { | t | ^ { n - d - 2 } } \leq \iint _ \Omega \mathcal { A } \nabla V \cdot \nabla V \Psi ^ 3 \frac { d t d x } { | t | ^ { n - d - 2 } } \end{align*}


\begin{align*} S _ 1 : = \{ b \mid \textrm { $ b $ - o r b i t e x i t s } \Gamma \textrm { v i a } h = \frac { \pi } { 2 } \textrm { w i t h } \Omega ( b ) \leq \tfrac { 1 } { 2 } \} . \end{align*}


\begin{align*} V ( h ) \nabla ^ 2 _ B f ( X , Y ) = \rho V \left ( \frac { S _ F } { f ^ 2 } \right ) g _ B ( X , Y ) , \end{align*}


\begin{align*} c _ 4 = \frac { c _ 2 + 1 } { c _ 3 } , \zeta ^ \varepsilon ( x , t ) = e ^ { - c _ 4 \sigma _ \varepsilon ( x , t ) } \chi ^ \varepsilon ( x , t ) , ( x , t ) \in \overline { Q _ { \varepsilon , T } } . \end{align*}


\begin{align*} \sum _ { \substack { 0 < \gamma , \gamma ' \le T \\ | \gamma - \gamma ' | \le \frac { 2 \pi \lambda } { \log T } } } \ ! \ ! \ ! \ ! \ ! \ ! 1 \ \ & \ge \ N ( T ) \int _ { - 1 } ^ 1 \widehat r ( \alpha ) F ( \alpha ) \ , d \alpha \\ & = N ( T ) \bigg ( \widehat r ( 0 ) + 2 \int _ 0 ^ 1 \alpha \ , \widehat r ( \alpha ) \ , d \alpha + o ( 1 ) \bigg ) \\ & = N ( T ) \bigg ( 1 + c ( \lambda ; r ) + o ( 1 ) \bigg ) . \end{align*}


\begin{align*} b ^ + _ i = \sum _ { j \in I } a _ j \cdot \# \left \{ e = \vec { i j } \in E \right \} - a _ i \ , , b ^ - _ i = - N _ i - \sum _ { j \in I } a _ j \cdot \# \left \{ e = \vec { j i } \in E \right \} + a _ i \ , . \end{align*}


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


\begin{align*} D = \sum _ { j + k \leq l } g _ { j , k } ( D ) \hat { Z } _ i ^ k \hat { \partial } _ i ^ j , \end{align*}


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


\begin{align*} \theta _ n = \frac { M _ n ^ { \gamma - 1 } } { r _ n ^ { \gamma - 2 } } , g _ n ( y , s ) = \frac { r _ n ^ 2 } { M _ n } f _ n \left ( \bar x _ n + r _ n y , \bar t _ n + r _ n ^ 2 s \right ) \end{align*}


\begin{align*} \varphi ( \xi ; \zeta ) : = 2 i \theta \left ( \xi , - k _ 0 + \frac { \eta } { \rho } \right ) = 1 6 i k _ 0 ^ 3 - \frac { i } { 2 } \zeta ^ 2 + \frac { i \zeta ^ 3 } { 1 2 \rho } . \end{align*}


\begin{align*} Z _ \pm ^ \varepsilon ( x , t ) = \pm e ^ { - ( c _ 5 + 1 ) t } \zeta ^ \varepsilon ( x , t ) - \left ( \sup _ { \overline { \Omega _ \varepsilon ( 0 ) } } | e ^ { - c _ 4 \sigma _ \varepsilon ( \cdot , 0 ) } \chi _ 0 ^ \varepsilon | + \sup _ { Q _ { \varepsilon , T } } | e ^ { - c _ 4 \sigma _ \varepsilon } f ^ \varepsilon | + \frac { 1 } { \varepsilon } \sup _ { \partial _ \ell Q _ { \varepsilon , T } } | \psi ^ \varepsilon | \right ) \end{align*}


\begin{align*} \mu ( - \bar z ) = - \frac { 1 } { 2 } ( 2 { \rm R e } z ) ^ 2 S f ( z ) . \end{align*}


\begin{align*} [ z ] _ \wedge & \leqslant \Bigl \| \sum _ { n = 1 } ^ \infty \| I _ n z \| _ Z e _ { \min I _ n } \Bigr \| _ T = \Bigl \| \sum _ { n = 1 } ^ \infty \| I _ n z \| _ Z e _ { m _ { \min I _ n } } \Bigr \| _ U = \Bigl \| \sum _ { n = 1 } ^ \infty \| I _ n z \| _ Z e _ { \min K _ n } \Bigr \| _ U \\ & \leqslant 2 \Bigl \| \sum _ { n = 1 } ^ \infty \| K _ n A z \| _ W e _ { \min K _ n } \Bigr \| _ U = 2 \Bigl \| \sum _ { n = 1 } ^ \infty \| J _ n A z \| _ W e _ { \min K _ n } \Bigr \| _ U . \end{align*}


\begin{align*} { \sf P } ^ \gamma : = \gamma ^ { - 1 } { \sf P } \gamma \cap { \sf H N } . \end{align*}


\begin{align*} d \gamma _ { N } = \exp \left ( - \beta \mathcal { H } ^ { * } _ { N } \right ) d \pi ^ { \otimes N } , \end{align*}


\begin{align*} \int _ { M } d \mu = 0 , \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } X ( \omega , t _ k ) = \bar { x } ( \omega ) . \end{align*}


\begin{align*} \forall \varphi \in \mathcal D , \langle \delta _ 0 , \varphi \rangle = \varphi ( 0 ) \mbox { a n d } \langle \delta _ \infty , \varphi \rangle = \lim _ { \omega \to \infty } \varphi ( \omega ) . \end{align*}


\begin{align*} Z _ { n + 1 } \leq K b ^ n \left ( Z _ n ^ { 1 + \mu _ 1 } + Z _ n ^ { 1 + \mu _ 2 } \right ) , n = 0 , 1 , 2 , \ldots , \end{align*}


\begin{align*} \rho _ i ( x , y ) = d ( x , y ) ^ p \frac { \nu _ i ( ( d ( x , y ) , \infty ) ) } { \mu ( B ( y , d ( x , y ) ) ) } \textrm { w i t h } d \nu _ i ( t ) : = p s _ i ( 1 - s _ i ) t ^ { - p s _ i - 1 } \ , d t , \end{align*}


\begin{align*} H _ b ^ E \tilde { \eta } _ L \psi = H _ b \tilde { \eta } _ L \psi , \end{align*}


\begin{align*} f ( \mathrm { k } ) : = \prod _ { 1 \le i \le k } c _ { k _ i } w _ n ^ { k _ i } , \quad \mathrm { k } \in { B } . \end{align*}


\begin{align*} w \cdot e _ j = e _ { w ( j ) } , \end{align*}


\begin{align*} \overline { \dim } _ B ( \mathcal { G } ( T ) ) & = \varlimsup _ { \delta \rightarrow 0 } \frac { \log N _ { ( 1 + l ) \eta } ( \mathcal { G } ( T ) ) } { - \log ( ( 1 + l ) \eta ) } \\ & \le \varlimsup _ { \eta \rightarrow 0 } \frac { \log N _ { \eta } ( \mathcal { G } ( F + T ) ) } { - \log ( ( 1 + l ) \eta ) } = \varlimsup _ { \eta \rightarrow 0 } \frac { \log N _ { \eta } ( \mathcal { G } ( F + T ) ) } { - \log ( \eta ) } = \overline { \dim } _ B ( \mathcal { G } ( F + T ) ) , \end{align*}


\begin{align*} Z _ r \left ( g g ^ \top , \frac { r s } { 2 } \right ) = 2 \zeta \left ( { r s } \right ) E ^ P _ { s , \varphi } ( g ) , \end{align*}


\begin{align*} \Lambda \overline { \psi _ { 1 } ( - x , - t , - k ) } \Lambda ^ { - 1 } = \psi _ 2 ( x , t , k ) , k \in \mathbb { R } \backslash \{ 0 \} \end{align*}


\begin{align*} I \langle x \rangle = \{ g \in G \mid g ^ n \in \langle x \rangle \ \} \end{align*}


\begin{align*} \operatorname { p r o x } _ g ^ \tau ( x ) : = \arg \min _ { z \in C } \frac { 1 } { 2 } \| z - x \| _ { \tau ^ { - 1 } } ^ 2 + g ( z ) \end{align*}


\begin{align*} \mathcal { L } _ { \Box } \ ! \ ! \ ! : : = & x \mid \varphi \land \psi \mid \varphi \lor \psi \mid \varphi \to \psi \mid \lnot \varphi \mid \Box \varphi \mid \Diamond \varphi \mid 0 \mid 1 . \end{align*}


\begin{align*} u = ( \nabla \psi ) ^ { \bot } , \ \omega = \Delta \psi , \ \partial _ { t } \omega + \mathrm { d i v } ( u \omega ) = 0 , \end{align*}


\begin{align*} P _ 2 \left ( n + 1 \right ) = S ^ { ( k ) } ( n ) \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { h _ n ^ { \alpha _ 1 } \hat { k } _ n ^ { \beta _ 1 } } { C _ n ^ { p - 1 } } = 0 , \lim _ { n \to \infty } \frac { \hat { h } _ n ^ { \alpha _ 2 } k _ n ^ { \beta _ 2 } } { C _ n ^ { q - 1 } } = 0 . \end{align*}


\begin{align*} & ( g ( \xi _ 0 , \xi ) , ( \xi _ 0 , \xi ) ) = \Big ( g \ , g ^ { - 1 } \Big ( \frac { d x _ 0 } { d t } , \frac { d x } { d t } \Big ) , g ^ { - 1 } \Big ( \frac { d x _ 0 } { d t } , \frac { d x } { d t } \Big ) \Big ) \\ = & \Big ( \Big ( \frac { d x _ 0 } { d t } , \frac { d x } { d t } \Big ) , g ^ { - 1 } \Big ( \frac { d x _ 0 } { d t } , \frac { d x } { d t } \Big ) \Big ) \end{align*}


\begin{align*} \left \Vert u \right \Vert _ { p ( x ) } = a \rho _ { p ( x ) } ( \frac { u } { a } ) = 1 . \end{align*}


\begin{align*} \lim _ { r \to \infty } \bar { p } ( r ) = p _ { \infty } < \infty . \end{align*}


\begin{align*} ( d _ { p + 1 } , w _ { q + 1 } ) ( t , x ) = 0 , \quad \forall t \in [ 0 , 2 T + \tfrac { 7 \tau _ q } { 6 } ] \cup [ 3 T - \tfrac { \tau _ q } { 6 } , 1 \big ] . \end{align*}


\begin{align*} A _ k = ( a ^ k _ { i j } ) _ { i , j } \in C ^ 1 \left ( \overline { \Omega } ; \mathbb { R } ^ { N \times N } \right ) , a ^ k _ { i j } = a ^ k _ { j i } , 1 \leq i , j \leq N , \\ D _ k = ( d ^ k _ { i j } ) _ { i , j } \in C ^ 1 \left ( \Gamma ; \mathbb { R } ^ { N \times N } \right ) , d ^ k _ { i j } = d ^ k _ { j i } , 1 \leq i , j \leq N , \end{align*}


\begin{align*} P _ { i , j } = \{ I \subset [ i , j ] \mid \min I = i , \ , \max I = j \} , \end{align*}


\begin{align*} J \phi & = X ^ \flat \wedge ( J T ) ^ \flat + ( J X ) ^ \flat \wedge T ^ \flat , \\ & = { e ^ u } \left ( d y \wedge \left ( \frac { 1 } { 2 } ( \textbf { H } - 1 ) d \theta + d \varphi \right ) - d x \wedge \left ( \frac { 1 } { 2 } ( \textbf { H } + 1 ) d \theta + d \varphi \right ) \right ) . \end{align*}


\begin{align*} \mathbb { L } _ { q } u = - u ^ { \prime \prime } + q \left ( x \right ) u = k ^ { 2 } u , \end{align*}


\begin{align*} \mathcal { E } _ { V } ^ { \theta } ( { \mu } _ { \theta } ^ { \epsilon } ) = \mathcal { E } _ { V } ^ { \theta } ( \mu _ { \theta } ) - \epsilon | X | \left ( \mathcal { E } _ { V } ^ { \theta } ( \mu _ { \theta } ) \right ) + \epsilon \int _ { X } h ^ { \mu _ { \theta } } ( x ) + V ( x ) \ , d x + \frac { 1 } { \theta } | X | \epsilon \log \epsilon + O ( \epsilon ^ { 2 } ) . \end{align*}


\begin{align*} \begin{aligned} & { C 1 } : A ( { \mathbf { c } } ) ( \tau ) + A ( { \mathbf { d } } ) ( \tau ) = 0 | \tau | \in \mathcal { U } _ { 1 } \cup \mathcal { U } _ { 2 } ; \\ & { C 2 } : C ( { \mathbf { c } , \mathbf { d } } ) ( \tau ) + C ( { \mathbf { d } , \mathbf { c } } ) ( \tau ) = 0 , | \tau | \in \mathcal { U } _ { 2 } . \end{aligned} \end{align*}


\begin{align*} \phi = k + \sqrt { z ^ 2 + \varepsilon } + s g _ 5 ( r ) + \frac { ( r ^ 2 - s ^ 2 ) ^ { m + 1 } } { m + 1 } + \frac { 2 s } { 1 + 2 m } I _ m ( r , s ) , \end{align*}


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


\begin{align*} \begin{cases} \partial _ s \mu - \sigma \Delta \mu = 0 & \\ \mu ( y , s ) = 0 & \\ \mu ( 0 ) = \delta _ 0 \end{cases} \end{align*}


\begin{align*} \omega _ j = \sqrt { | j | _ g ^ 4 - f ' ( a ^ 2 ) | j | _ g ^ 2 } \ . \end{align*}


\begin{align*} \varinjlim _ { k \to \infty } C _ c ( \Q _ q , \Z ) _ { \Gamma _ k } = C _ c ( \Q _ q , \Z ) _ { \Gamma } . \end{align*}


\begin{align*} & \prod _ { j = 1 } ^ d \langle v _ { 1 , j } \rangle ^ { - 1 + \epsilon } \left | \nu ( q + v _ 1 ) - E \pm i \eta \right | ^ { - 1 } \leq \eta ^ { - 1 } c _ { { \rm ( I ) } , d } \prod _ { j = 1 } ^ d \langle q _ j \rangle ^ { - 1 + \epsilon } . \end{align*}


\begin{align*} D _ { T ^ * } = ( 0 , T ^ * ] \times ( 0 , X ] \times ( 0 , 1 ) \end{align*}


\begin{align*} \pm k _ { 0 } = \pm \sqrt { - \frac { x } { 1 2 t } } , \end{align*}


\begin{align*} \Delta _ { N k } = F _ N - F _ { N \overline { k } } , k \in \N _ 0 , \end{align*}


\begin{align*} x ^ * : \ \Omega ^ * \to X \ \ \mbox { d e f i n e d a s } \ \ x ^ * ( \omega , x ) = x , \ \ \mbox { w h e r e } \ \ x \in X = Z , \end{align*}


\begin{align*} R _ { n , 1 } ^ { \epsilon , \Delta t } & = - \delta \Phi \Bigl ( \epsilon ( m _ { n + 1 } ^ { \epsilon , \Delta t } - m ^ \epsilon ( t _ { n + 1 } ) ) , \frac { \Delta t m _ { n + 1 } ^ { \epsilon , \Delta t } } { \epsilon } , X _ n ^ \epsilon \Bigr ) \\ R _ { n , 2 } ^ { \epsilon , \Delta t } & = \delta \Phi \Bigl ( \zeta ^ \epsilon ( t _ { n + 1 } ) - \zeta ^ \epsilon ( t _ n ) , \epsilon ( m _ n ^ { \epsilon , \Delta t } - m ^ \epsilon ( t _ n ) ) , X _ n ^ \epsilon \Bigr ) . \end{align*}


\begin{align*} \| \zeta \| _ { C ( \overline { S _ T } ) } \leq c \| \zeta ( \cdot , 0 ) \| _ { C ( \Gamma ( 0 ) ) } + c \varepsilon \sum _ { \xi = \zeta , \zeta _ 2 } \Bigl ( \| \xi \| _ { C ( \overline { S _ T } ) } + \| \xi \| _ { C ^ { 2 , 1 } ( S _ T ) } \Bigr ) \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u & = h ( x , u , \nabla 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*} I _ N = 1 + \sum _ { d = 1 } ^ \infty z ^ d \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } s _ \lambda ( a _ 1 , \dots , a _ N ) s _ \lambda ( b _ 1 , \dots , b _ N ) \Omega _ { \frac { q } { N } } ^ { - 1 } ( \lambda ) \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) , \end{align*}


\begin{align*} x ^ 1 = \Re \frac { z - z _ 0 } { \partial _ { \gamma } \eta _ i ( l _ 0 ) } , x ^ 2 = \Im \frac { z - \eta _ i \circ \tilde { l } ^ { - 1 } ( x ^ 1 ) } { \partial _ { \gamma } \eta _ i ( l _ 0 ) } \end{align*}


\begin{align*} ( u , v , \partial _ y \tilde { h } , g ) | _ { y = 0 } = \bf { 0 } , \end{align*}


\begin{align*} \frac { f - p } { \prod _ { j = 1 } ^ k ( z - \lambda _ j ) } = \sum _ { j = 1 } ^ k \alpha _ j \frac { f - p } { z - \lambda _ j } \end{align*}


\begin{align*} d { \mathbf { P } } _ { N , \beta } = \mu _ { \theta } ^ { \otimes N } \end{align*}


\begin{align*} \breve { M } ^ { r } ( x , t , k ) = \overline { \breve { M } ^ { r } ( x , t , - \bar { k } ) } . \end{align*}


\begin{align*} \alpha ( t , x ) = \alpha ( s , y ) \mbox { a n d } ( y , x ) \in U \end{align*}


\begin{align*} R _ { a } b = a \times b \\ R _ { g a } = g R _ { a } g ^ { - 1 } \\ R _ { a } . R _ { b } = < a \mid b > \\ A _ { g x } ( a ) = g A _ { x } ( g ^ { - 1 } a ) : = A d _ { g } A _ { x } ( a ) \\ ( x \mid R _ { \xi } y ) = ( x \wedge y \mid \xi ) \\ ( R _ { \xi } x \mid R _ { \eta } y ) = ( R _ { \xi } x \wedge y \mid \eta ) \end{align*}


\begin{align*} \left | \sum _ { \ell = 0 } ^ { n - 1 } \varphi ( f ^ \ell ( x ) ) - \varphi ( f ^ \ell ( y ) ) \right | \leq C . \end{align*}


\begin{align*} u ( x , t ) = \frac { A } { 1 - \gamma _ 0 e ^ { - A x + A ^ 3 t } } . \end{align*}


\begin{align*} f ( 1 ) = [ \beta ] - [ \alpha ] = [ \beta ] - 1 \geq 2 - 1 = 1 , \end{align*}


\begin{align*} I u _ 0 = u _ 0 \ , \epsilon : = \left \| u _ 0 \right \| _ { \ell ^ 2 _ s } < \epsilon _ 0 \ , \end{align*}


\begin{align*} W _ L ^ { } \simeq \begin{dcases} - \frac { C _ 1 ( \alpha ) } { L ^ 6 } , & \\ - \frac { C _ 2 ( \alpha ) } { L ^ 3 } , & , \end{dcases} \end{align*}


\begin{align*} \exp ( \hbar [ T , \cdot ] _ G ) \star = \star ' , \end{align*}


\begin{align*} T ^ \flat & = \frac { 1 } { 2 } ( \textbf { H } - 1 ) d \theta + d \varphi , \\ ( J T ) ^ \flat & = - \frac { 1 } { 2 } ( \textbf { H } + 1 ) d \theta - d \varphi . \end{align*}


\begin{align*} R _ { n _ k } : = \sum _ { j = 1 } ^ k U _ j - \Delta _ { n _ k , k } \ , , \end{align*}


\begin{align*} M _ { x y } & = \{ 1 \leq r \leq k \mid \sigma ^ r ( y ) = x \} , \\ N _ { x y } & = \{ 1 \leq s \leq k \mid \tau ^ s ( y ) = x \} , \end{align*}


\begin{align*} I _ { s , N } ( \boldsymbol { \alpha } ) \sim \displaystyle \sum _ { \substack { \boldsymbol { r } \in \mathbb { Z } ^ d \\ \boldsymbol { r } = ( r _ 1 , \cdots , r _ d ) } } c _ { \boldsymbol { r } } e ( \boldsymbol { r } . \boldsymbol { \alpha } ) , \end{align*}


\begin{align*} \rho _ t = \ , \rho ( t , \cdot ) = \ , X ( t , 0 , \cdot ) _ \# \bar \rho t \in [ 0 , T ] \end{align*}


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


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


\begin{align*} x ^ \star = \min _ { x \in X } f ( A x , b ) + g ( x ) + h ( x ) . \end{align*}


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


\begin{align*} \mathfrak l _ 3 = ( v t - w , u ) \subset \C [ u , v , w ] [ t ] = R [ t ] . \end{align*}


\begin{align*} | \Omega _ \varphi | = \pi \int _ { \mathbb { D } } e ^ { - \varphi } , \end{align*}


\begin{align*} H = \bigoplus _ { i = 1 } ^ r \langle \ell ^ { m - m _ i } \pi _ m ( e _ i ) \rangle \subseteq ( \Z / \ell ^ m \Z ) ^ n . \end{align*}


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


\begin{align*} \tilde { J } ^ { \pm } ( u ) = \begin{pmatrix} 1 & 0 \\ f _ { 2 } ^ { \pm } ( u ) & 1 \end{pmatrix} \begin{pmatrix} k _ { 2 } ^ { \pm } ( u ) & 0 \\ 0 & k _ { 3 } ^ { \pm } ( u ) \end{pmatrix} \begin{pmatrix} 1 & e _ { 2 } ^ { \pm } ( u ) \\ 0 & 1 \end{pmatrix} \end{align*}


\begin{align*} \{ g , h \} \mu = \pi ( d g , d h ) \mu = k \ , d g \wedge d h \wedge d F _ 1 \wedge \dots \wedge d F _ { n - 2 } \end{align*}


\begin{align*} x _ { i } ^ { * } = \displaystyle \arg \min _ { x _ { i } } ( f _ { i } ( x _ { i } ) - x _ { i } ^ { T } y ^ { * } ) \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u & = a ( x ) u ^ { - \gamma } + \lambda b ( x ) u ^ { q - 1 } \ ; \ ; & & \mbox { i n } \ ; \ ; \R ^ N , \\ u & > 0 & & \mbox { i n } \ ; \ ; \R ^ N , \end{alignedat} \right . \end{align*}


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


\begin{align*} \ell = \begin{pmatrix} \ell _ 0 \\ \ell _ 1 \end{pmatrix} \end{align*}


\begin{align*} \sum _ { j , k = 0 } ^ n g _ { j k } ( x ) d x _ j d x _ k \end{align*}


\begin{align*} \d z = ( i \Delta z - i \nu z - \epsilon ( \gamma z - \mu \overline { z } ) ) \d t + i \kappa | z | ^ 2 z \d t ( x , t ) \in \R \times \R ^ + . \end{align*}


\begin{align*} \theta : V = \bigoplus _ { i = 0 } ^ n k e _ i \to H ^ 0 ( X , L ) , e _ i \mapsto s _ i . \end{align*}


\begin{align*} \ddot { x } _ { \varepsilon _ n } ^ { a } + \Gamma _ { i j } ^ { a } \ , \dot { x } ^ { i } _ { \varepsilon _ n } \ , \dot { x } ^ { j } _ { \varepsilon _ n } + \varepsilon _ n ^ { - 2 } \ , g ^ { a b } \ , \partial _ b \tilde { U } = 0 \end{align*}


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


\begin{align*} C \left ( { \mathbf { c } , \mathbf { d } } \right ) ( \tau ) = \begin{cases} \sum _ { i = 0 } ^ { L - 1 - \tau } \omega ^ { c _ { 1 i } - c _ { 2 i + \tau } } , & 0 \leq \tau \leq L - 1 , \\ \sum _ { i = 0 } ^ { L - 1 + \tau } \omega ^ { c _ { 1 i - \tau } - c _ { 2 i } } , & - L + 1 \leq \tau \leq - 1 , \\ 0 , & | \tau | \geq L , \end{cases} \end{align*}


\begin{align*} X = X _ { - } ( s ) \oplus X _ 0 ( s ) \oplus X _ { + } ( s ) , \forall s \in \mathbb { R } , \end{align*}


\begin{align*} \delta : = 1 0 ^ { - 7 } s ( 1 - s ) ^ 4 \end{align*}


\begin{align*} \theta _ { \tau } ( x ) = x + \tau . \end{align*}


\begin{align*} P _ { \ ! \mu } ( z ) : = \int _ { \mathbb T } \frac { 1 - | z | ^ 2 } { | z - \zeta | ^ 2 } d \mu ( \zeta ) , \ , \ , \ , z \in \mathbb D . \end{align*}


\begin{align*} s _ \lambda ( x _ 1 , \dots , x _ N ) = \frac { \dim \mathsf { V } ^ \lambda } { d ! } \sum _ { \alpha \vdash d } \omega _ \alpha ( \lambda ) p _ \alpha ( x _ 1 , \dots , x _ N ) , \end{align*}


\begin{align*} N _ G ( s , T ) ^ F : = \{ \gamma \in G ^ F : s ^ \gamma \in T \} , \end{align*}


\begin{align*} \epsilon : = \left \| ( \psi _ 0 , \dot \psi _ 0 ) \right \| _ s : = \left \| \psi _ 0 \right \| _ { H ^ { s + 2 } } + \left \| \dot \psi _ 0 \right \| _ { H ^ { s } } < \epsilon _ { s r } \ , \end{align*}


\begin{align*} \sum _ { 0 < \gamma \le T } n ( \gamma , k ) ^ { 2 k } \le \frac { \log T } { 2 \pi } \sum _ { \gamma \le T } \int _ { J ( \gamma ) } n ( t , k + 1 ) ^ { 2 k } \ , d t = \frac { \log T } { 2 \pi } \int _ 0 ^ T n ( t , k + 1 ) ^ { 2 k } \Big ( \sum _ { \gamma \in I ( t ) } 1 \Big ) \ , d t \ , . \end{align*}


\begin{align*} \liminf _ { i \to \infty } \int _ { 0 } ^ { \delta } t ^ p \ , d \nu _ i = p \liminf _ { i \to \infty } s _ i ( 1 - s _ i ) \int _ { 0 } ^ { \delta } t ^ { - p ( s _ i - 1 ) - 1 } \ , d t = 1 \quad \textrm { f o r a l l } \delta > 0 , \end{align*}


\begin{align*} E _ p ( h ) = \frac { 1 } { p } \int _ { - \infty } ^ { \infty } \big ( h '^ 2 ( x ) + ( m - 1 ) \cos ^ 2 h ( x ) \big ) ^ \frac { p } { 2 } \cosh ^ { p - m } x d x . \end{align*}


\begin{gather*} \gamma _ { \mathfrak n } ( v , w , u ) = B \left ( d ' ( v , w ) , u \right ) = B \left ( B _ { \mathfrak a } \left ( \rho ( \cdot ) v , w \right ) , u \right ) = 0 . \end{gather*}


\begin{align*} \gamma _ { n } \left ( z \right ) = { \displaystyle \sum \limits _ { k = 1 } ^ { \infty } } \eta _ { n , k } z ^ { 2 k } , \end{align*}


\begin{align*} \delta _ n = o ( \eta _ n ) . \end{align*}


\begin{align*} \sum _ { l = 1 } ^ { p } \sigma _ l \omega _ { j _ l } \not = 0 \ \Longrightarrow \ \left | \sum _ { l = 1 } ^ { p } \sigma _ l \omega _ { j _ l } \right | \geq \frac { \gamma ' _ r } { N ^ { \tau ' _ r } } \ . \end{align*}


\begin{align*} R i c _ B ( X , Y ) - \frac { m } { f } \nabla _ B ^ 2 f ( X , Y ) + \nabla _ B ^ 2 \varphi ( X , Y ) + h \nabla _ B ^ 2 f ( X , Y ) = ( \lambda + \rho S _ g ) g _ B ( X , Y ) \end{align*}


\begin{align*} u \otimes v : = ( u _ i v _ j ) _ { 1 \leq i , j \leq 3 } . \end{align*}


\begin{align*} \sigma _ \varepsilon ( x , t ) = \{ d ( x , t ) - \varepsilon \bar { g } _ 0 ( x , t ) \} \{ d ( x , t ) - \varepsilon \bar { g } _ 1 ( x , t ) \} . \end{align*}


\begin{align*} Y _ { i j } = \begin{cases} 0 & \ ; \\ 0 & \ ; \\ 1 & \ . \end{cases} \end{align*}


\begin{align*} \sigma _ 0 ( A ) ( a k + j ) = A ( b j + k ) \ ; , \end{align*}


\begin{align*} \ell ( h ) = \frac { ( \widehat \nu _ h | \widehat \nu _ h + 2 \widehat \rho ) } { 2 ( k + h ^ \vee ) } - h . \end{align*}


\begin{align*} f ( \lambda ) = e v _ { \lambda } ( f ) = \big ( \tau ^ { - 1 } ( \lambda ) \big ) ( f ) . \end{align*}


\begin{align*} \lim _ { m \to \infty } \left | \prod _ { 0 \leq j \leq m - 1 } \left ( 1 + e ^ { 2 \pi i \frac { j + \theta / 2 } { m } } + \frac { x + i y } { m } \right ) \right | = 2 e ^ { x / 2 } \left | \sin \left ( \frac { \pi \theta _ 1 } { 2 } + \frac { y } { 2 } + \frac { x i } { 2 } \right ) \right | . \end{align*}


\begin{align*} R _ j ^ h ( \ell ) : = \sum _ { \ell _ 1 + \ell _ 2 = \ell \atop \pi ( \ell _ 2 ) \nparallel v ( j ) } \frac { - \left ( 2 \pi ( \ell _ 1 ) \cdot \pi ( \ell _ 2 ) \right ) V ( \ell _ 1 ) V ( \ell _ 2 ) } { ( \omega \cdot \ell _ 2 + 2 j \cdot \pi ( \ell _ 2 ) - | \pi ( \ell _ 2 ) | ^ 2 ) ( \omega \cdot \ell _ 2 + 2 ( j - \pi ( \ell _ 1 ) ) \cdot \pi ( \ell _ 2 ) - | \pi ( \ell _ 2 ) | ^ 2 ) } \ , , \end{align*}


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


\begin{align*} \{ X \in I _ { i , m } \} = \{ \omega \in \Omega : X _ { i , m } ( \omega ) \in B _ { i , m } \} \end{align*}


\begin{align*} ( - 1 ) ^ { \mathrm { 1 } _ { i \in \mathcal { O } } } \frac { T ^ { \mathrm { 1 } _ { i \in \mathcal { B } } } } { n } \sum _ { w ^ n = ( - 1 ) ^ { \theta _ 2 } } \frac { w ^ { \mathrm { 1 } _ { i \in \mathcal { B } } + h _ i - h _ j } e ^ { - ( T w + \theta _ 1 \pi i + \lambda ) [ t _ j - t _ i ] _ i } } { 1 - ( - 1 ) ^ { \theta _ 1 } e ^ { - ( T w + \lambda ) } } = K _ n ^ { \lambda , \theta , T } ( x _ i , x _ j ) , \end{align*}


\begin{align*} \bar { u } _ 1 ( t , y _ 1 ) = u _ 1 ( t , y _ 2 ) , \end{align*}


\begin{align*} \epsilon ( \sigma ) x _ { \sigma ( 1 ) } \vee \cdots \vee x _ { \sigma ( n ) } = x _ 1 \vee \cdots \vee x _ n . \end{align*}


\begin{align*} \hat { \kappa } _ 2 ^ 2 + \hat { \mu } _ 2 ^ 2 = \kappa _ 2 ^ 2 + \mu _ 2 ^ 2 . \end{align*}


\begin{align*} P : = ( m _ J : J \subset [ n ] ) . \end{align*}


\begin{align*} R ( a ) = \sqrt { A ( a ) } \end{align*}


\begin{align*} \Sigma ( \kappa ; \eta , P ) & = - i e ^ { \pi i \kappa } \sum _ { T < \gamma \le 2 T } \zeta ' ( 1 - \rho ) \ , \zeta \Big ( \rho + \frac { 2 \pi i \kappa } { \log T } \Big ) \ , M \Big ( 1 - \rho - \frac { 2 \pi i \eta } { \log T } , P \Big ) \ , M \Big ( \rho + \frac { 2 \pi i \eta } { \log T } , P \Big ) \\ & + O \big ( T \log T \big ) \\ & = - i e ^ { \pi i \kappa } \ , ( \log T ) \ , I ( 2 \pi i \kappa , 0 , g _ 1 , g _ 2 , 1 , - x ) + O \big ( T \log T ) \end{align*}


\begin{align*} K _ I ( x , f ) = \quad \left \{ \begin{array} { r r } \frac { | J ( x , f ) | } { { l \left ( f ^ { \ , \prime } ( x ) \right ) } ^ n } , & J ( x , f ) \ne 0 , \\ 1 , & f ^ { \ , \prime } ( x ) = 0 , \\ \infty , & \end{array} \right . \ , . \end{align*}


\begin{align*} g _ { a , b , c } ( x , y ; q ) : = \left ( \sum _ { r , s \geq 0 } + \sum _ { r , s < 0 } \right ) ( - 1 ) ^ { r + s } x ^ r y ^ s q ^ { a \binom { r } { 2 } + b r s + c \binom { s } { 2 } } . \end{align*}


\begin{align*} \omega _ 0 = \chi _ + ^ { - 1 } \omega \end{align*}


\begin{align*} \left \{ \begin{array} { l } ( - i \gamma ^ \mu \partial _ \mu + M ) \psi = g \phi \Gamma \psi , \\ ( \Box + m ^ 2 ) \phi = - g \psi ^ \dagger \gamma ^ 0 \Gamma \psi . \end{array} \right . \end{align*}


\begin{align*} P _ \ell ( X ) = \ell - a _ \ell ( f ) X . \end{align*}


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


\begin{align*} \mathbb { P } \left \{ \sum _ { i = 1 } ^ n a _ i X _ i > C t ^ { 2 / q } e ^ { t ^ 2 / ( 2 q ) } \left \vert a \right \vert _ q ) \right \} \leq C e ^ { - t ^ 2 / 2 } \end{align*}


\begin{align*} S ^ { v _ 0 } ( m ) = 2 \cdot 3 ^ { v _ 0 } w + 1 . \end{align*}


\begin{align*} \Theta ( g ) = \sum _ { v \in \mathbb { Z } ^ r } \varphi ( v \cdot g ) = \sum _ { v \in \mathbb { Z } ^ r } e ^ { - \pi | v \cdot g | ^ 2 } \end{align*}


\begin{align*} { \mu \choose \nu } { \mu ' \choose \nu ' } : = \prod _ j { \mu _ j \choose \nu _ j } { \mu _ j ' \choose \nu _ j ' } \end{align*}


\begin{align*} m ^ { p c } _ { - k _ 0 } \left ( \xi , \zeta ( k ) \right ) = I + \frac { i } { \zeta } \begin{pmatrix} 0 & - \beta ^ r ( \xi ) \\ \gamma ^ r ( \xi ) & 0 \end{pmatrix} + O ( \frac { 1 } { \zeta ^ 2 } ) , \zeta \rightarrow \infty . \end{align*}


\begin{align*} \Delta _ x G ( x - A y - b , y - A y - b , u ( y ) ) = \Delta h ( x - A y - b ) - \Delta h ( x - A y - b - u ( y ) ) . \end{align*}


\begin{align*} ( - 1 ) ^ { k - 1 } q \left ( X _ a \right ) ^ k \ : = \ : \left ( \prod _ { b = 1 } ^ k X _ b \right ) \left ( 1 - X _ a \right ) ^ n . \end{align*}


\begin{align*} N _ 1 ( a , R ) & : = a \left ( \log c _ 2 ( \overline { \alpha } ; R ) + k \left ( \log k + a _ 2 ( m ) \varphi ( m ) + 1 \right ) \right ) + \frac { a _ 6 ( m ) k \varphi ( m ) a } { \log a } \\ & \quad + 2 . 5 \log a + \log ( k + 1 ) + 2 . 5 \log k + \log H + ( k - 1 ) \log \left ( \max _ { 1 \leq j \leq k } \{ | \alpha _ j | \} \right ) \\ & \quad + \log \sqrt { 2 \pi } + 1 + k \left ( a _ 2 ( m ) \varphi ( m ) + 2 \right ) + \frac { a _ 6 ( m ) k \varphi ( m ) } { \log a } + \frac { 1 9 } { 1 2 a } . \end{align*}


\begin{align*} \int _ E | f _ E | ^ p = \lim _ { j \rightarrow \infty } \int _ E | f _ j | ^ p = s _ p ( E , \Omega ) > 0 . \end{align*}


\begin{align*} \delta \langle \partial _ x z ^ { \otimes 2 } , \mathbf { 1 } \rangle _ { s , t } = \langle ( \mathcal D + I + \mathbb { I } + \tilde { \mathbb { I } } ) _ { s , t } ( 1 , 1 ) , \mathbf { 1 } \rangle + \langle ( \partial _ x z ) _ { s , t } ^ { \natural , 2 } , \mathbf { 1 } \rangle \ , . \end{align*}


\begin{align*} u \mapsto \Phi ( u ) : = e ^ { i t \Delta } u _ 0 - i \int _ 0 ^ t e ^ { i ( t - s ) \Delta } F ( u ( s ) ) \ , d s \end{align*}


\begin{align*} \partial _ t d ( x , t ) & = - \overline { V _ \Gamma } ( x , t ) , \\ \partial _ t \pi ( x , t ) & = \overline { V _ \Gamma } ( x , t ) \bar { \nu } ( x , t ) + d ( x , t ) R ( x , t ) \overline { \nabla _ \Gamma V _ \Gamma } ( x , t ) , \end{align*}


\begin{align*} \frac { 1 } { 2 } \sum _ { - N \leq k \leq N } \log \left ( 1 + \left ( \frac { p + k } { q } \right ) ^ 2 \right ) = \sum _ { - N \leq j \leq N } \log | p + j | - ( 2 N + 1 ) \log q + \frac { 1 } { 2 } \sum _ { - N \leq j \leq N } \log \left ( 1 + \left ( \frac { q } { p + j } \right ) ^ 2 \right ) . \end{align*}


\begin{align*} u ( x ) = C _ 1 \int _ \Omega \ln \frac { 1 } { | x - y | } v ^ p ( y ) \mathrm { d } y - \int _ \Omega h ( x , y ) v ^ p ( y ) \mathrm { d } y . \end{align*}


\begin{align*} \hat \mu _ j ( s , 0 ) = \exp \{ - \sigma _ j s ^ 2 \} , \ \ s \in \mathbb { R } , \end{align*}


\begin{align*} N _ i ( k , \nu ) = ( \widehat \nu _ h + \widehat \rho | \eta _ i ^ \vee ) . \end{align*}


\begin{align*} C : = \{ ( z , w ) \in \partial W \colon \theta ( z ) \in J \setminus I , \ \partial _ z \theta ( z ) \neq 0 \} . \end{align*}


\begin{align*} X _ { N } ^ { \delta } = { \rm a r g m i n } _ { \overline { \mathbf { P } } _ { N } \in B ( \overline { \mathbf { P } } , \delta ) } { \rm F } _ { N } ( X _ { N } , \mu _ { \theta } ) . \end{align*}


\begin{align*} S ( { \bf { v } } ) : = \left ( \begin{array} { c c c } \frac { \theta ( P - q ) } { R } & 0 & 0 \\ 0 & \frac { P - q } { a } & \theta \\ 0 & \theta & \frac { \theta ^ 2 } { 2 q } \frac { P + q } { P - q } \end{array} \right ) \end{align*}


\begin{align*} d ( \alpha ( t , x ) , \alpha ( t , y ) ) = d ( x , y ) \ , . \end{align*}


\begin{align*} \Tilde { H } : = \begin{cases} \pi ^ * H & \Tilde { M } \setminus \pi ^ { - 1 } ( V ) , \\ & \\ \pi ^ * H - d ( \pi ^ * \Tilde { B } _ 0 ) & \pi ^ { - 1 } ( N ) . \end{cases} \end{align*}


\begin{align*} \langle f \rangle _ { \Lambda , \beta } = \mathrm { t r } _ { \Lambda } ( f ( \sigma ) e ^ { - \beta H ( \sigma ) } ) / Z _ { \Lambda } ( \beta ) , \end{align*}


\begin{align*} Y _ n ^ { \epsilon , \Delta t } = \Phi \bigl ( \epsilon ( m _ n ^ \epsilon - m ^ \epsilon ( t _ n ) ) , X _ n ^ { \epsilon , \Delta t } \bigr ) , \end{align*}


\begin{align*} ( r + 1 ) \chi ( M ) - \chi ( N _ r ^ f M ) = - N ( a _ r ^ + ) . \end{align*}


\begin{align*} \lim _ { N \to \infty } \log \omega _ { N } = \log | \omega | \end{align*}


\begin{align*} & \varpi _ k = 1 + \gamma _ k , \\ & \upsilon _ k = \frac { \boldsymbol { \theta } _ i ^ T \mathrm { d i a g } ( \mathbf { h } _ k ^ H ) \mathbf { G } \mathbf { w } _ k } { \sum _ { \ell \in \mathcal { K } } | \boldsymbol { \theta } _ i ^ T \mathrm { d i a g } ( \mathbf { h } _ k ^ H ) \mathbf { G } \mathbf { w } _ \ell | ^ 2 + \sigma _ k ^ 2 } . \end{align*}


\begin{align*} \int _ { 0 } ^ { + \infty } \| \partial _ x h ( r ) \| _ { L ^ \infty } \dd r = M \ , , M \in [ 0 , + \infty ) \ , . \end{align*}


\begin{align*} m _ 1 \prec _ { s } m _ 2 \ \Leftrightarrow \ \lambda m _ 2 = \lambda m _ 1 + { p } ^ { \ast } _ 1 ( n ) n \in { N } ^ + _ { s } \lambda \in \mathbb Z _ { > 0 } , \end{align*}