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


\begin{align*} \Omega & = \sum _ { k } x _ k x ^ * _ k = \sum _ { 1 \leq i \leq r } h _ i ^ 2 + \sum _ { i < j } x _ { i j } y _ { j i } + \sum _ { i > j } y _ { i j } x _ { j i } = \sum _ { 1 \leq i \leq r } h _ i ^ 2 + \sum _ { i < j } x _ { i j } y _ { j i } + y _ { j i } x _ { i j } \\ & = \sum _ { 1 \leq i \leq r } h _ i ^ 2 + \sum _ { i < j } 2 x _ { i j } y _ { j i } + [ x _ { i j } , y _ { j i } ] = \sum _ { 1 \leq i \leq r } h _ i ^ 2 + \sum _ { i < j } 2 x _ { i j } y _ { j i } - h _ i + h _ j \end{align*}


\begin{align*} w \to u _ w ^ { } = \sum _ { F } \frac { \langle f , F \rangle \cdot F } { \lambda _ { s _ F } - \lambda _ w } \end{align*}


\begin{align*} \partial _ \xi \omega _ j ( \xi , x , t ) = \frac { 1 } { a _ j ( \xi , \omega _ j ( \xi , x , t ) ) } , \ ; \ ; \omega _ j ( x , x , t ) = t . \end{align*}


\begin{align*} \frac { 1 } { s _ 1 \ldots s _ { n } } \sum _ { j _ 1 = 0 } ^ { s _ 1 - 1 } \ldots \sum _ { j _ n = 0 } ^ { s _ n - 1 } f \left ( U _ 1 ^ { j _ 1 } \ldots U _ n ^ { j _ n } x \right ) . \end{align*}


\begin{align*} \omega ^ 2 - F ^ 2 = x _ 0 x _ 1 x _ 2 x _ 3 . \end{align*}


\begin{align*} \left \vert x \right \vert _ { r , q } \leq 4 q ^ { - 1 } \left ( \left \vert x \right \vert _ { 1 } + r \sum _ { i = 1 } ^ { n } i ^ { - 1 + 1 / q } x _ { [ i ] } \right ) \leq 1 6 \left \vert x \right \vert _ { r , q } \end{align*}


\begin{align*} X ( t ) - X ( \xi ) = \int _ { \xi } ^ { t } { \rm d } [ A ( s ) ] X ( s ) \end{align*}


\begin{align*} R _ { 2 , \infty } ^ { ( d ) } ( s , \boldsymbol { \alpha } , N ) = \frac { 1 } { N } \displaystyle \sum _ { 1 \leq m \neq n \leq N } \chi _ { s , N } ( \boldsymbol { \alpha } ( \boldsymbol { a } _ m - \boldsymbol { a } _ n ) ) , \end{align*}


\begin{align*} \Phi ( x , y ; \xi , \zeta ) = \int _ 0 ^ 1 \int _ 0 ^ 1 | y - \zeta + s ( \zeta - \xi ) + t ( x - y ) | ^ { p - 2 } d t \ , d s . \end{align*}


\begin{align*} \beta _ { x _ 1 } ( \lambda ^ c _ x s ) = \prod _ { \ell = 1 } ^ { \infty } \frac { \lambda ^ u _ { \psi ^ { x _ 1 } _ { - \ell } ( \lambda ^ c _ x s ) } } { \lambda ^ u _ { x _ { - \ell + 1 } } } = \prod _ { \ell = 1 } ^ { \infty } \frac { \lambda ^ u _ { \psi ^ x _ { - \ell + 1 } ( s ) } } { \lambda ^ u _ { x _ { - \ell + 1 } } } = \frac { \lambda ^ u _ { \Phi ^ c _ x ( s ) } } { \lambda ^ u _ x } \beta _ x ( s ) , \end{align*}


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


\begin{gather*} i \vert i + 1 \vert i + 2 = 0 = i \vert i - 1 \vert i - 2 , i = 0 , \ldots , d - 1 ; \\ i \vert i + 1 \vert i = i \vert i - 1 \vert i , i = 1 , \ldots , d - 1 ; \\ 0 \vert 1 \vert 0 = ( - 1 ) ^ d ( 0 \vert d - 1 \vert 0 ) . \end{gather*}


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


\begin{align*} \| A \| _ { p - v a r } : = \sup _ { \mathcal { D } \subset [ 0 , 1 ] } \left ( \sum _ { t _ i \in \mathcal { D } } \| A _ { t _ i , t _ { i + 1 } } \| ^ p \right ) ^ { \frac { 1 } { p } } . \end{align*}


\begin{align*} \tilde { \mu } [ \dd y ] = \frac { \exp ( - \mathcal { E } ( y ) ) \dd y } { \int _ { D } \exp ( - \mathcal { E } ( z ) ) \dd z } \ , . \end{align*}


\begin{align*} \begin{aligned} \frac 1 2 \int _ { B _ { R } ^ { + } \backslash B _ { \rho } ^ { + } } \nabla \eta ^ { 6 } \cdot ( a y , b ) \omega ^ { 2 } d x d y \leq \frac { C R } { R - \rho } \int _ { B _ { R } ^ { + } \backslash B _ { \rho } ^ { + } } | \nabla \boldsymbol { w } | ^ { 2 } d x d y \leq \frac { C R } { R - \rho } . \end{aligned} \end{align*}


\begin{align*} \frac { [ x ^ { n - \Sigma _ { \mathrm { k } } } ] S _ { \neq \mathrm { k } } ( x ) } { [ x ^ n ] S ( x ) } = ( 1 + o ( 1 ) ) \cdot z _ n ^ { \Sigma _ { \mathrm { k } } } , s _ n < k _ 1 < \cdots < k _ \ell < s _ n + o ( s _ n ) n \to \infty , \end{align*}


\begin{align*} G = \bigcup _ { j = 1 } ^ n ( t _ j + P _ V ( x ) ) \ , . \end{align*}


\begin{align*} G ^ A = P y ^ A + Q ^ A , \end{align*}


\begin{align*} \sigma _ a \sigma _ 0 ^ { - 1 } ( b ) = \begin{cases} b , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; e v e n , } \\ ( 2 ^ m - 1 ) ( 1 - b ) = b - 2 , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; o d d . } \end{cases} \end{align*}


\begin{align*} \rho _ s ( x ) & = \frac { ( x + s + \tfrac 1 2 ) ^ { x + s } } { ( x + \tfrac 1 2 ) ^ x } e ^ { - s } \left ( 1 + O ( x ^ { - 1 } ) \right ) \\ & = ( x / e ) ^ { s } \left ( 1 + s / x \right ) ^ { x + s } \left ( 1 + O \left ( x ^ { - 1 } \right ) \right ) \end{align*}


\begin{align*} \varphi ( 0 ) = 0 , \nabla \varphi ( 0 ) = 0 . \end{align*}


\begin{align*} y ( t ) : = V ( \phi ( t , x , u ) ) , t \in [ t _ - , t _ + ) , \end{align*}


\begin{align*} \vartheta ( x ) = \sum _ { p \leq x } \log p , \end{align*}


\begin{align*} \inf \Bigl \{ P _ f ( F ) : V _ f ( F ) = v , F \subset H _ { \R } ^ n \Bigr \} . \end{align*}


\begin{align*} C _ 1 ( b ^ * ) = \frac { 1 } { \psi ' ( b ^ * ) } \end{align*}


\begin{align*} F ( f ^ { \langle 1 \rangle } , \ldots , f ^ { \langle r \rangle } ) ^ { \phi } = \det ( f ^ { \partial } ) ^ { - 2 s } \cdot F ^ { \tau } ( f ^ { \langle 2 \rangle } , \ldots , f ^ { \langle r + 1 \rangle } ) ; \end{align*}


\begin{align*} \left \{ \begin{array} { l l } a _ s = \gamma _ s + \gamma _ { s + 1 } , \\ s = 1 , \ldots , N . \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} [ x , y ] _ { \mathfrak n } & = [ x , y ] + \theta ( x , y ) + \gamma ( x , y , \cdot ) ; \\ [ x , v ] _ { \mathfrak n } & = \rho ( x ) v + B _ { \mathfrak a } \left ( \theta ( \cdot , x ) , v \right ) ; \\ [ v , w ] _ { \mathfrak n } & = B _ { \mathfrak a } \left ( \rho ( \cdot ) v , w \right ) ; \\ [ Z , x ] _ { \mathfrak n } & = Z \left ( [ x , \cdot ] \right ) ; \\ [ Z _ 1 , v + Z _ 2 ] _ { \mathfrak n } & = 0 . \end{aligned} \end{align*}


\begin{align*} \left \lbrace \begin{aligned} X _ { n + 1 } ^ { \epsilon , \Delta t } & = \Phi ( \frac { \Delta t m _ { n + 1 } ^ { \epsilon , \Delta t } } { \epsilon } , X _ n ^ { \epsilon , \Delta t } ) \\ m _ { n + 1 } ^ { \epsilon , \Delta t } & = m _ n ^ { \epsilon , \Delta t } - \frac { \Delta t } { \epsilon ^ 2 } m _ { n + 1 } ^ { \epsilon , \Delta t } + \frac { \Delta \beta _ n } { \epsilon } , \end{aligned} \right . \end{align*}


\begin{align*} f ( \omega ) = \begin{cases} \omega ^ { - 7 / 6 } + O ( \epsilon \omega ^ { - 7 / 6 + \delta } ) & \mbox { f o r } \omega \leq 1 , \\ ( 1 + c ^ * ) \omega ^ { - 7 / 6 } \left ( 1 + O ( \epsilon \omega ^ { - \delta } ) \right ) & \mbox { f o r } \omega > 1 . \end{cases} \end{align*}


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


\begin{align*} Y ( t ) = \phi ( t , X ( t ) ) = \phi _ 1 ( t ) \phi _ 2 ( X ( t ) ) , \end{align*}


\begin{align*} \langle N _ { 1 , 1 } \psi , N _ { 2 , 1 } \psi \rangle _ { L ^ 2 ( \Gamma _ T ) } & = 0 , \end{align*}


\begin{align*} \lambda ( e ) = h [ n + 1 ] ( e , \overline { e } ) + \Theta ^ \mathrm { p a i r } [ n + 1 ] ( \overline { e } , \overline { e } ) . \end{align*}


\begin{align*} C ( \mu ) : = \prod ^ { s } _ { t = 1 } C _ { \eta _ t } \wr S _ { m _ t } , \end{align*}


\begin{align*} \begin{gathered} ( \vec { \alpha } \sqcup \vec { \beta } ) ( ( 0 , i ) ) = \vec \alpha ( i ) \ ; , \\ ( \vec { \alpha } \sqcup \vec { \beta } ) ( ( 1 , i ) ) = \vec \beta ( i ) \\ \end{gathered} \end{align*}


\begin{align*} \phi ( a , b ) ^ { \ast } ( \omega _ { j k } ) = \zeta _ 7 ^ { a ( j + 1 ) + b ( k - 6 ) } \omega _ { j k } . \end{align*}


\begin{align*} \mathcal { E } ( \phi _ v ) = \int _ { D } g ( \phi _ v ) \cdot \phi _ v \dd x = | D | g ( \phi _ v ) \cdot \phi _ v = | D | g ( v ) \cdot v = \bar { \mathcal { E } } ( v ) \ , , \quad \forall v \in \mathbb { S } ^ 2 \ , . \end{align*}


\begin{align*} i \partial _ t u + \Delta u = F ( u ) , ( t , x ) \in \R \times \R ^ 2 , \end{align*}


\begin{align*} \big ( w ^ * , ( u ^ * , 0 ) \big ) \in L ^ * \ \Rightarrow \ w ^ * = 0 , \ u ^ * = 0 \end{align*}


\begin{align*} \tilde b ( \tilde y , s ) = b ( \tilde y - \xi _ s , s ) - \xi ' _ s \ , = h _ 1 \gamma | D w ( \tilde y - \xi _ s , s ) | ^ { \gamma - 2 } D w ( \tilde y - \xi _ s , s ) + \frac { y _ 0 } \tau . \end{align*}


\begin{align*} \psi : = \Big ( \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 6 , v _ j \rangle \Big ) ^ 2 , \end{align*}


\begin{align*} u _ i = \prod _ { 1 \leq j \leq n , i \neq j } ( a _ i - a _ j ) ^ { - 1 } , \end{align*}


\begin{align*} \mathcal { E } _ { \Lambda _ { k + 1 , D } } ( \tau ) = \sum _ { n \geq 1 } \frac { c ( n ) } { n ^ { 2 k + 1 } } e ^ { 2 \pi i n \tau } , \Lambda _ { k + 1 , D } ^ * ( \tau ) = \sum _ { n \geq 1 } \frac { c ( n ) } { ( 4 \pi n ) ^ { 2 k + 1 } } \Gamma ( 2 k + 1 , 4 \pi n v ) e ^ { - 2 \pi i n \tau } . \end{align*}


\begin{align*} N ( T ) = \sum _ { 0 < \gamma _ d \le T } m ( \gamma _ d ) \end{align*}


\begin{align*} G _ { s u m } = g _ { s u m } - \frac { ( \delta b ) ^ { \frac { 1 } { \alpha _ 0 } } } { K } . \end{align*}


\begin{align*} \begin{multlined} [ \widetilde { C } \alpha _ 0 ^ { - 1 } \widetilde { D _ I } \alpha _ 0 \sigma ( A ) ] ( i ) \\ = \sum _ j ( - 1 ) ^ { | j | | j | } A ( ( i , ( j , j ) ) ) = \sum _ j ( - 1 ) ^ { | j | } A ( ( i , ( j , j ) ) ) \\ \neq \sum _ j A ( ( i , ( j , j ) ) ) = [ A ] ( i ) \ ; . \end{multlined} \end{align*}


\begin{align*} \varphi _ \omega ( M ) \ : = \ \inf _ { k \ge 1 } \frac { \omega ( M k ) } { \omega ( k ) } \mbox { a n d } \Phi _ \omega ( M ) \ : = \ \sup _ { k \ge 1 } \frac { \omega ( M k ) } { \omega ( k ) } , M = 1 , 2 , 3 , \ldots . \end{align*}


\begin{align*} \left ( \begin{smallmatrix} c & 0 \\ 0 & c ' \end{smallmatrix} \right ) A \left ( \begin{smallmatrix} \lambda \\ \kappa \end{smallmatrix} \right ) = B \left ( \begin{smallmatrix} \lambda \\ \kappa \end{smallmatrix} \right ) \ \ { \rm m o d } \left ( \begin{smallmatrix} Y \\ Y ' \end{smallmatrix} \right ) \ , . \end{align*}


\begin{align*} | \widetilde { h } ( x ) - \widetilde { h } ( y ) | \le \limsup _ { i \to \infty } \left | \sum _ { l = i } ^ { N _ i } a _ { i , l } h _ l ( x ) - \sum _ { l = i } ^ { N _ i } a _ { i , l } h _ l ( y ) \right | \le \limsup _ { i \to \infty } \int _ { \gamma } \sum _ { l = i } ^ { N _ i } a _ { i , l } g _ { h _ l } \ , d s = \int _ { \gamma } g \ , d s \end{align*}


\begin{align*} \sum _ { 0 < \gamma \le T } m ( \gamma ) ^ { 2 k - 1 } = \sum _ { 0 < \gamma _ d \le T } m ( \gamma _ d ) ^ { 2 k } < ( C k ) ^ { 2 k - 1 } \ , T \log T \ , . \end{align*}


\begin{align*} T _ i ^ { \pm 1 } m _ j = \begin{cases} q ^ { \mp 1 } m _ i , & \ ; j = i ; \\ m _ i - q ^ { \pm 1 } m _ j , & \ ; j = i \pm 1 ; \\ - q ^ { \pm 1 } m _ j , & . \end{cases} \end{align*}


\begin{align*} x \ , \theta _ 2 - y \ , \theta _ 1 = 0 , x \ , \theta _ 3 - z \ , \theta _ 1 = 0 , y \ , \theta _ 3 - z \ , \theta _ 2 = 0 \end{align*}


\begin{align*} V = \left \{ y _ 4 ^ 2 \sigma _ 2 + \left ( \sigma _ 1 \sigma _ 3 - \sigma _ 4 + \frac { \sigma _ 1 ^ 2 \sigma _ 2 } { 4 } = 0 \right ) \right \} . \end{align*}


\begin{align*} M _ i ( k ) = \frac { 2 } { u _ i } \left ( k + \frac { h ^ \vee - \bar h ^ \vee _ i } { 2 } \right ) , i \ge 0 , \end{align*}


\begin{align*} \partial _ { z } L \left [ p \left ( x ; z \right ) \right ] = e ^ { - z ^ { 2 } } \left [ p \left ( z ; z \right ) + p \left ( - z ; z \right ) \right ] + L \left [ \partial _ { z } p \left ( x ; z \right ) \right ] , \end{align*}


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


\begin{align*} \theta ( \Psi _ \phi ( v ) ) = \theta ( v ) Y ^ { - \phi ( v ) + y ( v ) } = \Psi _ \phi ( \theta ( v ) ) Y ^ { - \phi ( v ) - \phi ( \theta ( v ) ) } , \end{align*}


\begin{align*} f _ r = \# [ \sigma ( J _ r ) \cap ( E , \infty ) ] . \end{align*}


\begin{align*} \mathcal { R } ( x , \xi ) = \frac { 1 } { p ( x ) } | \xi | ^ { p ( x ) } + \frac { \mu ( x ) } { q ( x ) } | \xi | ^ { q ( x ) } ( x , \xi ) \in \Omega \times \R ^ N \end{align*}


\begin{align*} \phi _ i \lambda \ = \ \left \{ \begin{array} { l l } \phi _ i ( \lambda ) \phi _ i & \mbox { i f $ \phi _ i $ i s a n e n d o m o r p h i s m o f $ K $ } \\ \phi _ i ( \lambda ) + \lambda \phi _ i & \mbox { i f $ \phi _ i $ i s a d e r i v a t i o n o n $ K $ } , \end{array} \right . \end{align*}


\begin{align*} \frac { 1 } { n } \sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \end{align*}


\begin{align*} F _ 2 \left ( a , b _ 1 , b _ 2 ; c _ 1 , c _ 2 ; x , y \right ) = \sum _ { m = 0 } ^ { \infty } \sum _ { n = 0 } ^ { \infty } \frac { ( a ) _ { m + n } \left ( b _ 1 \right ) _ m \left ( b _ 2 \right ) _ n } { \left ( c _ 1 \right ) _ m \left ( c _ 2 \right ) _ n } \frac { x ^ m y ^ n } { m ! n ! } \end{align*}


\begin{align*} \widetilde { \mathbf { v } } \left ( k \right ) = \mathbf { v } \left ( k \right ) + \int \frac { W \left \{ \mathbf { v } \left ( k , \cdot \right ) , \psi \left ( \cdot , \mathrm { i } s \right ) \right \} } { k ^ { 2 } + s ^ { 2 } } \mathrm { d } \mu \left ( s \right ) , \ k \notin \Sigma , \end{align*}


\begin{align*} \dfrac { \max _ { 1 \le j < 2 ^ n } \left | \sum _ { i = 1 } ^ { j } \left ( X _ i - X _ { i , 2 ^ n } \right ) \right | } { b _ { 2 ^ n } } \overset { \mathbb { P } } { \to } 0 \ n \to \infty . \end{align*}


\begin{align*} \langle f , g \rangle _ { n } = \sum _ { x \in X _ n } \tau ( x ) f ( x ) g ( x ) , \| f \| _ { n } ^ 2 = \langle f , f \rangle _ { n } . \end{align*}


\begin{align*} S ^ j _ { q , r } \left ( m \right ) = p \cdot \left ( q - 1 \right ) ^ j q ^ { v - j } w + r \in \Omega _ p \end{align*}


\begin{align*} \Gamma _ 0 = \mathcal { V } ^ 0 \cup \mathcal { V } ^ 1 \cup \cdots \cup \mathcal { V } ^ { m - 1 } , \end{align*}


\begin{align*} \widetilde I ^ k = I ^ k . \end{align*}


\begin{align*} ( R / I ) ( \underline t ) = U _ { ( R / I ) [ \underline { t } ] / ( R / I ) } ^ { - 1 } ( ( R / I ) [ \underline { t } ] ) . \end{align*}


\begin{align*} m _ { 0 , + } ( \zeta ) = m _ { 0 , - } ( \zeta ) J _ 0 , \zeta \in \mathbb { R } , \end{align*}


\begin{align*} ( u \wedge v \mid x \wedge y ) = \begin{vmatrix} ( u \mid x ) & ( u \mid y ) \\ ( v \mid x ) & ( v \mid y ) \end{vmatrix} \end{align*}


\begin{align*} \varphi ( x ^ { k + 1 } ; \rho ^ \infty ) - \varphi ( x ^ { K _ \rho } ; \rho ^ \infty ) & = \sum _ { t = K _ \rho } ^ k \Big ( \varphi ( x ^ { t + 1 } ; \rho ^ \infty ) - \varphi ( x ^ t ; \rho ^ \infty ) \Big ) \\ & \leq c _ { } \sum _ { t = K _ \rho } ^ k \Big ( m ^ t ( x ^ { t } + d ^ t ; \rho ^ \infty ) - m ^ t ( x ^ t ; \rho ^ \infty ) \Big ) \end{align*}


\begin{align*} \Gamma = & \begin{bmatrix} \Gamma _ 0 & \Gamma _ 1 & \dots & \Gamma _ n \end{bmatrix} \\ = & \begin{bmatrix} D & C B & C A B & \dots & C A ^ { n - 1 } B \end{bmatrix} . \end{align*}


\begin{align*} K _ { \Omega _ \varphi , 2 } ( 0 ) = \frac { 1 } { \pi } K _ { \mathbb { D } , \varphi } ( 0 ) , \end{align*}


\begin{align*} \begin{bmatrix} p & 0 & q \\ 0 & 1 & 0 \\ a & 0 & b \end{bmatrix} \in S L ( 3 , \Z ) . \end{align*}


\begin{align*} \pi L _ x ^ { - 1 } = L _ { \pi ( x ) } ^ { - 1 } L _ { \widetilde { 1 } } \pi . \end{align*}


\begin{align*} \sum _ { d \mid k / 2 } \sum _ { e \mid d } \frac { 1 } { \varphi \left ( d / e \right ) } \sum _ { \substack { a > B \sqrt { e / d } \\ ( a , d ) = e } } \frac { \mu ^ 2 ( a ) } { \varphi ( a ^ 2 ) } & \leq \sum _ { a > B / \sqrt { k ' } } \frac { \mu ^ 2 ( a ) } { \varphi ( a ^ 2 ) } \sum _ { e \mid ( a , k ' ) } \sum _ { \substack { ( a , d ) = e \\ d \mid k ' } } 1 \\ & \leq 2 ^ L \sum _ { a > B / \sqrt { k ' } } \frac { \mu ^ 2 ( a ) } { \varphi ( a ^ 2 ) } , \end{align*}


\begin{align*} | F ( z ) - w _ 0 | + | F ( 1 / z ) - w _ \infty | = O ( | z | ^ \varepsilon ) \quad z \to 0 \end{align*}


\begin{align*} \begin{cases} \partial _ t \rho + \div _ x ( b \rho ) = 0 & \\ \rho ( 0 , \cdot ) = \bar \rho & \end{cases} \end{align*}


\begin{align*} A _ { y } ^ + ( x ) = a ^ { \dagger } ( \lambda _ { y } ( x ) ) , A _ { y } ^ - ( x ) = a ( \lambda _ { y } ( x ) ) , \end{align*}


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


\begin{align*} L V = L ( \partial _ v u ) = \partial _ v ( L u ) + [ L , \partial _ v ] = [ L , \partial _ v ] \ \end{align*}


\begin{align*} v _ { i j } v _ { i k } = - v _ { i k } v _ { i j } & v _ { j i } v _ { k i } = - v _ { k i } v _ { j i } j \neq k \\ v _ { i j } v _ { k l } = v _ { k l } v _ { i j } & i \neq k j \neq l \\ \sum _ { \sigma \in S _ 3 } v _ { 1 \sigma ( 1 ) } & v _ { 2 \sigma ( 2 ) } v _ { 3 \sigma ( 3 ) } = 1 . \end{align*}


\begin{align*} a _ n & = \operatorname { m e x } \{ a _ k , b _ k \ ; | \ ; k < n \} \\ b _ n & = f ( a _ n ) + b _ { n - 1 } + a _ n - a _ { n - 1 } . \end{align*}


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


\begin{align*} | \Lambda _ t ( f ) | = \left | \frac { f ( z + t e _ j ) - f ( z ) } { t } - \frac { \partial { f } } { \partial { x _ j } } ( z ) \right | = O ( | t | ) \ \ \ ( t \rightarrow 0 ) , \end{align*}


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


\begin{align*} \langle O p _ \mathcal { A } ( a ) f , g \rangle & = \langle a , W _ \mathcal { A } ( g , f ) \rangle = \langle a , \mu ( V _ C ^ T \mathcal { A } _ { F T 2 } \mathcal { D } _ L ) ( g \otimes \bar f ) \rangle = \langle \mathfrak { T } _ { L ^ { - 1 } } \mathcal { F } _ 2 ^ { - 1 } \mu ( V _ C ^ { - T } ) a , g \otimes \bar f \rangle , \end{align*}


\begin{align*} \mathcal { D } : = \sigma ( f ) = \left \{ f ^ { - 1 } ( B ) : B \in \mathcal { B } \right \} . \end{align*}


\begin{align*} F ( A ) _ M = A _ M + a \mathbb { 1 } \ ; , \end{align*}


\begin{align*} S _ { \mathbf { t } } = \left \{ \Psi _ 7 \left ( G _ d + \frac { q } { 2 } \mathbf { t } \cdot \mathbf { y } ) \right ) : d \in \{ 1 , 2 \} , \mathbf { y } \in \mathbb { Z } _ { 2 } ^ { 2 } \right \} , \end{align*}


\begin{align*} M = \| u \| _ { L ^ { \infty } ( \overline { \Omega } ) } \leq C . \end{align*}


\begin{align*} & u _ { t } ( x , t ) + 6 u ( x , t ) u ( - x , - t ) u _ { x } ( x , t ) + u _ { x x x } ( x , t ) = 0 , \ \ x \in \mathbb { R } , \ t \in \mathbb { R } , \\ & u ( x , t = 0 ) = u _ { 0 } ( x ) \rightarrow \left \{ \begin{aligned} & 0 , \ \ x \rightarrow - \infty , \\ & A , \ \ x \rightarrow + \infty , \end{aligned} \right . , \end{align*}


\begin{align*} t ^ { 2 } \equiv \begin{cases} 0 \mod 8 , & n \equiv 1 \mod 8 , \\ 4 \mod 8 , & n \equiv 5 \mod 8 . \end{cases} \end{align*}


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


\begin{align*} A = \{ a _ { k j } : \ , 1 \le j \le n , \ , 1 \le k \le d \} \end{align*}


\begin{align*} \limsup \limits _ { \substack { \varepsilon \to 0 \\ t ' \to t , x ' \to x , y ' \to y } } V ^ { \varepsilon } ( t ' , x ' , y ' ) - \psi ^ { \varepsilon } ( t ' , x ' , y ' ) = \overline { V } ( t , x ) - \psi ( t , x ) \end{align*}


\begin{align*} \rho ^ { ( 1 ) } _ { \Lambda _ M } : = \frac { 1 } { \sqrt 2 } ( \rho _ { \Lambda _ M } + \rho _ { \Lambda _ { M \cup \Psi } } ) , R ^ { ( 1 ) } _ { \Lambda _ M } : = \frac { 1 } { \sqrt 2 } ( R _ { \Lambda _ M } + R _ { \Lambda _ { M \cup \Psi } } ) , \end{align*}


\begin{align*} D _ { \mu , 0 } ( f ) : = \displaystyle \lim _ { R \to 1 ^ { - } } \displaystyle \int _ { \mathbb T } \big | f ( R \zeta ) \big | ^ 2 P _ { \ ! \mu } ( R \zeta ) d \sigma ( \zeta ) , \end{align*}


\begin{align*} \lambda _ o = \sup \{ \lambda > 0 : M _ G ( \lambda ) < \infty \} \ , \end{align*}


\begin{align*} { \rm N E } ( X / Y ) = \sum _ { i = 1 } ^ m \R _ { \geq 0 } [ \Gamma _ i ] . \end{align*}


\begin{align*} M ( \nu ) = \sum _ { \alpha } \sum _ { \jmath \in J ( S _ \iota ) ^ F } \Psi _ \alpha ( q ^ \nu ) \Theta _ { \alpha , \jmath } ( \nu ) \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*} \C _ 1 : = \bigg \{ \sum _ { j } \underline { W } _ j > & \varepsilon \frac { n \log _ 3 n } { ( \log m ) ^ 2 } \bigg \} , \C _ 2 : = \bigg \{ \sum _ { j } \underline { W } _ j > \varepsilon \frac { n \log _ 3 n } { ( \log m ) ^ 2 } \bigg \} , \\ \C _ 3 & : = \bigg \{ \max _ { j < k } \{ W _ j - \underline { W } _ j - \underline { \hat { W } } _ j \} > \frac { m } { ( \log m ) ^ 2 } \bigg \} \ , . \end{align*}


\begin{align*} \chi _ i = \frac { h ^ \vee - \bar h ^ \vee _ i } { u _ i } , \ i \geq 0 . \end{align*}


\begin{align*} \psi _ j ( \Phi ( P ) ) = \Phi _ j ^ + ( P ) . \end{align*}


\begin{align*} \Delta ( g ) = g \otimes g \epsilon ( g ) = 1 . \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { s _ 1 ( x ) } { \pi ( x ) } = \frac { 2 ( p - 3 ) } { ( p - 1 ) ^ 3 } . \end{align*}


\begin{align*} \Gamma ( x , t ) & = 1 + \sum _ { z \in \Z } \sum _ { s = 1 } ^ { t - 1 } p ( x - z , t - s ) \xi ( z , s ) \Gamma ( z , s ) . \end{align*}


\begin{align*} \left | \int \hat { B } ( q ) \left ( q ^ 2 - E - i \eta \right ) ^ { - 1 } d q \right | & \leq \int \left | \hat { B } ( q ) \right | \left | \left ( q ^ 2 - E - i \eta \right ) ^ { - 1 } \right | d q \\ & \leq \int \left | \hat { B } ( q ) \right | \left | \eta ^ { - 1 } \right | d q \\ & = \eta ^ { - 1 } \int \left | \hat { B } ( q ) \right | d q \\ & = \| \hat { B } \| _ 1 \eta ^ { - 1 } \end{align*}


\begin{align*} \epsilon _ { 1 , 0 } & = 1 , \\ \textstyle \int _ X \epsilon _ { 1 , 2 } & = 1 , \\ \textstyle \int _ X \epsilon _ { j , 1 } \ , \epsilon _ { j ' , 1 } & = \begin{cases} 1 , & j ' = j + g , \\ - 1 , & j = j ' + g , \\ 0 , & . \end{cases} \end{align*}


\begin{align*} H _ y = h _ { \alpha } + H _ f ^ + + \alpha V _ { y } - 2 \alpha ^ { 1 / 2 } \mathrm { R e } P A _ { y } ( x ) + \alpha \| \lambda _ { y } \| ^ 2 + 2 \alpha A _ { y } ^ { + } A _ { y } ^ - . \end{align*}


\begin{align*} \mu ( \dd y ) = \frac { \exp ( - \frac { \lambda _ 2 } { h _ 2 } \bar { \mathcal { E } } ( y ) ) \dd y } { \int _ { \mathbb { S } ^ 2 } \exp ( - \frac { \lambda _ 2 } { h _ 2 } \bar { \mathcal { E } } ( v ) ) \dd v } \ , , \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*} g _ z : = \frac { | m _ p ( \cdot , z ) | ^ { p - 2 } m _ p ( \cdot , z ) } { m _ p ( z ) ^ p } . \end{align*}


\begin{align*} \operatorname * { t r } \left ( I + \mathbb { A } \right ) ^ { - 1 } \partial \mathbb { A } & = - \operatorname * { t r } \left \langle \cdot , a \right \rangle \left ( I + \mathbb { A } \right ) ^ { - 1 } a \\ & = - \left \langle \left ( I + \mathbb { A } \right ) ^ { - 1 } a , a \right \rangle . \end{align*}


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


\begin{align*} \alpha ( t , U ' ) : = \{ ( \alpha ( t , x ) , \alpha ( t , y ) ) : ( x , y ) \in U ' \} \subseteq U \ , \end{align*}


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


\begin{gather*} d ^ 2 _ { Q } \left ( \theta , \gamma \right ) ( x , y , z ) ( t ) \\ = \left ( d ^ 2 \theta ( x , y , z ) , d ^ 3 _ { r } \gamma ( x , y , z , t ) + \frac { 1 } { 2 } B _ { \mathfrak a } \left ( \theta \wedge ( \theta \circ \alpha ) \right ) ( x , y , z , a ) \right ) \end{gather*}


\begin{align*} K _ 1 ^ { \ast } = \frac { 1 } { 2 } - K . \end{align*}


\begin{align*} \alpha ( t , x ) = F ( t ) = \Psi ( i _ { \mathsf { b } } ( t ) ) \mbox { f o r a l l } t \in G \ , . \end{align*}


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


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


\begin{align*} \hat { S } ( \boldsymbol { \ell } ) _ t : = ( t , S ( \boldsymbol { \ell } ) _ t ) , \end{align*}


\begin{align*} H _ b ^ E ( \tilde { \eta } _ L \psi ) = \lim _ { n \to \infty } \tilde { H } _ b ( \tilde { \eta } _ L \psi _ n ) = \tilde { H } _ b ( \tilde { \eta } _ L \psi ) . \end{align*}


\begin{align*} \sigma ^ 0 _ \Gamma ( \alpha ) ( \mathcal { F } _ q ( \delta _ \gamma \otimes x ) ) & = \sigma _ { \alpha } ( \sigma _ \gamma ( q ) x ) = \sigma _ { \alpha \gamma } ( q ) x = \mathcal { F } _ q ( \delta _ { \alpha \gamma } \otimes x ) \\ & = \mathcal { F } _ q ( ( \lambda _ \Gamma \otimes i d ) ( \alpha ) ( \delta _ \gamma \otimes x ) ) \end{align*}


\begin{align*} S _ j ^ { j - \pi ( \ell ) } ( \ell ) = \dfrac { P _ j ^ { j - \pi ( \ell ) } ( \ell ) } { \omega \cdot \ell + \Omega _ j - \Omega _ { j - \pi ( \ell ) } } \mbox { i f } \ ; 0 < | \ell | \le K , S _ j ^ { j - \pi ( \ell ) } ( \ell ) = 0 \mbox { o t h e r w i s e } . \end{align*}


\begin{align*} \langle f , g \rangle _ { \mathcal { H } _ \lambda } : = \sum _ { k \in \N } \hat { f } _ k \ , \overline { \hat { g } _ k } \ , \lambda _ k , f , g \in \mathcal { H } _ \lambda . \end{align*}


\begin{align*} e ^ { - t P _ i } P ^ k f = P ^ k e ^ { - t P _ i } f = ( - \partial _ t ) ^ k e ^ { - t P _ i } f , \end{align*}


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


\begin{align*} s _ * ( \lambda ) = \sqrt { ( \nu + \lambda ) ^ 2 + c _ * ( \lambda ) x _ * ( \lambda ) ^ 2 } \end{align*}


\begin{align*} \gamma _ { \mathfrak n } ( x , y , v ) & = B \left ( d ' ( x , y ) , v \right ) = B _ { \mathfrak a } ( \theta ( x , y ) , v ) ; \\ \gamma _ { \mathfrak n } ( x , v , y ) & = B \left ( d ' ( x , v ) , y \right ) = B _ { \mathfrak a } ( \theta ( x , y ) , v ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ k \| x b _ i - b _ i x \| _ 2 \geq C \| x - E _ A ( x ) \| _ 2 . \end{align*}


\begin{align*} = \exp \{ - \sigma _ 1 ( s _ 1 - s _ 2 ) ^ 2 \} \hat \mu _ 2 ( s _ 1 - a s _ 2 , l ) , \ \ s _ j \in \mathbb { C } , \ \ l \in L . \end{align*}


\begin{align*} \mathrm { t r } ( \gamma ( B ) ^ T \nabla ^ 2 f ( B ) \gamma ( B ) ) = \Delta f - \sum _ { i = 1 } ^ { 3 } B ^ i ( B \cdot [ \partial _ { i , 1 } f , \partial _ { i , 2 } f , \partial _ { i , 3 } f ] ) \ , . \end{align*}


\begin{align*} [ [ A ] ] _ P = [ \widetilde { A } ] _ P \ ; . \end{align*}


\begin{align*} | ( \vec \alpha _ i , \vec \alpha _ o ) | = | \vec { \alpha } _ i | + | \vec { \alpha } _ o | = 0 \ ; . \end{align*}


\begin{align*} \begin{cases} b _ k = a _ k + b _ { k - 1 } - a _ { k - 1 } + f ( a _ k ) \quad \\ b _ k - a _ k > b _ { k - 1 } - a _ { k - 1 } \end{cases} \end{align*}


\begin{align*} \tilde { a } _ \ell ( x ) : = x \tilde { f } _ \ell ' ( x ) , \tilde { b } _ \ell ( x ) : = x ^ 2 \tilde { f } _ \ell '' ( x ) + x \tilde { f } _ \ell ' ( x ) \quad \tilde { c } _ \ell ( x ) : = x ^ 3 \tilde { f } _ \ell ''' ( x ) + 3 x ^ 2 \tilde { f } _ \ell '' ( x ) + x \tilde { f } _ \ell ' ( x ) . \end{align*}


\begin{align*} H = \left ( 1 - \frac { a b x } { ( c - 1 ) c ( x - 1 ) } \right ) - \frac { x ( a + b + 1 ) - 2 c x + c - 1 } { ( c - 1 ) c ( x - 1 ) } \theta \end{align*}


\begin{align*} h _ { k } d _ { n , k } & = L \left [ \phi \partial _ { x } \left ( P _ { n + 1 } P _ { k } \right ) \right ] - L \left [ \phi P _ { n + 1 } \partial _ { x } P _ { k } \right ] \\ & = L \left [ 2 x \left ( \phi - 1 \right ) P _ { n + 1 } P _ { k } \right ] - L \left [ \phi P _ { n + 1 } \partial _ { x } P _ { k } \right ] . \end{align*}


\begin{align*} \phi ( u ) : = \psi \biggl ( \frac { u } { 2 } , - \frac { u } { 2 } \biggr ) = \psi ( u , 0 ) - \frac { u } { 2 } , \end{align*}


\begin{align*} \| P _ { B \setminus A } ( x - \mathcal { T } ( x - z , \Delta _ \alpha ) ) \| & \ \le \ 2 \mathbf { A _ p } \alpha \sup _ { | \eta | = 1 } \| 1 _ { \eta ( B \setminus A ) } \| \\ & \ \le \ 2 \mathbf { A _ p } C _ { s d } \min _ { n \in \Lambda } | e _ n ^ * ( x - z ) | \| 1 _ { \varepsilon \Lambda } \| \\ & \ \le \ 2 \mathbf { A _ p } C _ { s d } C _ { q } ^ 2 \eta _ p ( C _ { q } ) \| x - z \| . \end{align*}


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


\begin{align*} \{ g , h \} = \omega ( X _ g , X _ h ) . \end{align*}


\begin{align*} \lim _ { T \to \infty } \ , \sup _ { s \geq T } \ , ( \sigma B ( s ) - u s ) = - \infty , \end{align*}


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


\begin{align*} P _ { A , \theta } \left ( \sigma ( x ^ 1 ) = y ^ 1 , \ldots , \sigma ( x ^ p ) = y ^ p \right ) = ( - 1 ) ^ { \theta _ 1 g _ 1 + \theta _ 2 g _ 2 } \prod _ { i = 1 } ^ k w _ { A } ( x ^ i , y ^ i ) \det \limits _ { i , j = 1 } ^ k \left ( K ^ { - 1 } _ { A , \theta } ( y ^ i , x ^ j ) \right ) . \end{align*}


\begin{align*} \bar { \nu } \cdot \nabla \rho _ \eta ^ \varepsilon = - k _ d ^ { - 1 } \Bigl ( \overline { V _ \Gamma \eta } \Bigr ) + \varepsilon \Bigl ( \bar { g } _ 1 \bar { \zeta } _ 0 - \bar { g } _ 0 \bar { \zeta } _ 1 \Bigr ) + d \Bigl ( \bar { \zeta } _ 1 - \bar { \zeta } _ 0 \Bigr ) \quad \overline { Q _ { \varepsilon , T } } . \end{align*}


\begin{align*} d : = \deg ( \Phi _ { K _ S } ) \leq 9 + \frac { 2 7 - 9 q } { p _ g - 2 } \leq 3 6 . \end{align*}


\begin{align*} \begin{gathered} G ^ A : A _ o \times B _ o \rightarrow K \ ; , \\ G ^ B : B _ o \times A _ o \rightarrow K \ ; , \\ H ^ A : B _ i \times A _ i \rightarrow K \ ; , \\ H ^ B : A _ i \times B _ i \rightarrow K \ ; , \end{gathered} \end{align*}


\begin{align*} K ( \omega ) : = & 1 + | z | ^ 3 _ { L ^ \infty ( 0 , \tilde { \tau } _ M ; H _ x ^ { \mathfrak { s } } ) } + \left | \int _ 0 ^ \cdot S ( \cdot - s ) ( z ( s ) \d W ( s ) ) \right | _ { L ^ r ( 0 , \tilde { \tau } _ M ; W _ x ^ { \mathfrak { s } , p } ) } \\ & + \sup _ { 0 \leq t _ 0 \leq \tilde { \tau } _ M } \left | \int _ 0 ^ { t _ 0 } S ( \cdot - s ) ( z ( s ) \d W ( s ) ) \right | _ { L ^ r ( 0 , \tilde { \tau } _ M ; W _ x ^ { \mathfrak { s } , p } ) } , \end{align*}


\begin{align*} \overline { U } _ { R [ \underline { t } ] / R } : = \pi ' ( U _ { R [ \underline { t } ] / R } ) . \end{align*}


\begin{align*} E ^ 2 _ { p q } = H _ p ( G , L _ q \phi _ ! F ) \Rightarrow H _ { p + q } ( G \ltimes E , F ) \end{align*}


\begin{align*} | \alpha | = \sum _ { i = 1 } ^ { k } \alpha _ i , \alpha ! = \prod _ { i = 1 } ^ { k } \alpha _ i ! , B ^ { \alpha } = \prod _ { i = 1 } ^ { k } B _ i ^ { \alpha _ i } . \end{align*}


\begin{align*} a \lor b = a + b + a \cdot b , a \land b = a \cdot b \lnot a = a + 1 , \end{align*}


\begin{align*} \phi _ { \tau } \left ( \upsilon \right ) : = \left ( A ^ { - 1 } ( \tau ) A ( \upsilon + \tau ) \right ) ^ T = A ^ T ( \upsilon + \tau ) \cdot A ^ { - T } ( \tau ) , \end{align*}


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


\begin{align*} \langle \lambda _ i ^ * e _ N \xi , \lambda _ i ^ * e _ N \eta \rangle & = \langle e _ N \lambda _ i \lambda _ i ^ * e _ N \xi , \eta \rangle \\ & = \langle E _ N ( \lambda _ i \lambda _ i ^ * ) e _ N \xi , \eta \rangle \\ & = \langle e _ N \xi , \eta \rangle \\ & = \langle e _ N \xi , e _ N \eta \rangle . \end{align*}


\begin{align*} \phi \left ( r e ^ { i \theta ^ 0 } , \ , \theta ^ 1 , \ , \theta ^ 2 , \sigma \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 , \sigma \right ) , \end{align*}


\begin{align*} N = \bigoplus _ { i = 1 } ^ t N _ i \end{align*}


\begin{align*} S O _ { \alpha } ( G ^ * ) - S O _ { \alpha } ( G ^ { * ' } ) = & \left ( d _ { G ^ * } ^ 2 ( v _ { i - 1 } ) + 9 \right ) ^ \alpha - \left ( d _ { G ^ * } ^ 2 ( v _ { i - 1 } ) + 4 \right ) ^ \alpha + 2 \cdot 1 3 ^ \alpha + 2 \cdot 8 ^ \alpha - 3 \cdot 2 0 ^ \alpha - 1 7 ^ \alpha \\ < & 1 0 ^ \alpha - 5 ^ \alpha + 2 \cdot 1 3 ^ \alpha + 2 \cdot 8 ^ \alpha - 3 \cdot 2 0 ^ \alpha - 1 7 ^ \alpha < 0 . \end{align*}


\begin{align*} Z _ { t } : = R \vee \| y _ { o } \| + \sqrt { 2 } M _ { t } - \eta \xi _ { t } + L _ { t } , \end{align*}


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


\begin{align*} [ 2 b ] = \frac { q ^ 3 t x ^ 4 } { ( 1 - q x ) ( 1 - q t x ) } . \end{align*}


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


\begin{align*} \xi _ m ^ n = \hat { \epsilon } ^ n e ^ { i m k h } \end{align*}


\begin{align*} \omega ( \epsilon ) : = \iint _ \Omega | \nabla u | ^ 2 _ \kappa \partial _ r \Psi ^ { 3 } _ \epsilon \frac { d t d x } { | s | ^ { n - d - 2 } } , \end{align*}


\begin{align*} H _ { M _ 1 } ( x ) = \begin{cases} x & | x | \le M _ 1 \\ \frac { M _ 1 x } { | x | } & x \ge M _ 1 . \end{cases} \end{align*}


\begin{align*} \theta ( u _ w ) \cdot \eta _ a ( v _ w ) - \eta _ a ( u _ w ) \cdot \theta ( v _ w ) = 0 . \end{align*}


\begin{align*} p ^ i ( \tilde x ) | _ A = p ^ i ( \tilde y ) | _ A , \ ( \tilde x , \tilde y \in \mathcal P a ( E ) , \ A \in \mathcal F _ { i } , \ A \subset \Omega _ i ) \ \ \mbox { i m p l i e s } \ \ p ^ i \circ h ( \tilde x ) | _ { A } = p ^ i \circ h ( \tilde y ) | _ { A } \ \mbox { a . s . } \end{align*}


\begin{align*} \sigma ^ 1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , \ \sigma ^ 2 = \begin{bmatrix} 0 & - i \\ i & 0 \end{bmatrix} , \ \sigma ^ 3 = \begin{bmatrix} 1 & 0 \\ 0 & - 1 \end{bmatrix} . \end{align*}


\begin{align*} \lim _ { n \to \infty } \alpha _ n ^ { - 1 } \sup _ { \mu \in \mathcal { M } } \int _ { \{ x \colon q _ n ( x ) > n \alpha _ n \} } q _ n ( x ) \ , \mu ( d x ) = 0 . \end{align*}


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


\begin{align*} \{ ( x , g ) \mid x \in H ^ { ( 0 ) } , g \in G , f ( x ) = r ( g ) \} \to G ^ { ( 0 ) } , ( x , g ) \mapsto s ( g ) \end{align*}


\begin{align*} f _ j ( x ) = \frac { 1 } { \sqrt { m _ 1 m _ 2 } } \exp \left \{ 2 \pi i \left ( \frac { j _ 1 + \theta _ 1 / 2 } { m _ 1 } x _ 1 + \frac { j _ 2 + \theta _ 2 / 2 } { m _ 2 } x _ 2 \right ) \right \} . \end{align*}


\begin{align*} B _ f ( b ) = \frac { b } { 2 \pi } \sum _ { k \in \Z } f \big ( { \rm s g n } ( k ) \sqrt { 2 | k | b } \big ) . \end{align*}


\begin{align*} ( f \otimes g ) ( x , y ) = f ( x ) g ( y ) , x , y \in \mathbb { R } ^ d . \end{align*}


\begin{align*} w : = \begin{cases} 2 \quad \textrm { i n } A , \\ 1 \quad \textrm { i n } X \setminus A , \end{cases} \end{align*}


\begin{align*} \Delta ( \tau ) & = \tau \otimes 1 + 1 \otimes \tau , & S ( \tau ) & = - \tau , \\ \Delta ( \theta ) & = \theta \otimes 1 + 1 \otimes \theta , & S ( \theta ) & = - \theta , \\ \Delta ( \sigma _ i ) & = \textstyle \sum _ { u + v = i } \sigma _ u \otimes \sigma _ v + \sum _ { u + v + p = i } \sigma _ u \tau \otimes \sigma _ v \tau , & S ( \sigma _ i ) & = ( - 1 ) ^ i \sigma _ i . \end{align*}


\begin{align*} \lim _ { k \to \infty } H ^ k d ^ k = 0 . \end{align*}


\begin{align*} \partial _ { x } P _ { n } \left ( x ; z \right ) = n x ^ { n - 1 } + O \left ( x ^ { n - 2 } \right ) = n P _ { n - 1 } \left ( x ; z \right ) + O \left ( x ^ { n - 2 } \right ) , \end{align*}


\begin{align*} \mathbf { P } _ { N , \beta } \left ( \overline { \mathbf { P } } _ { N } ( X _ { N } ) \in B ( \overline { \mathbf { P } } , \delta ) \right ) = \mathbf { \Pi } _ { N } \left ( \overline { \mathbf { F } } _ { N } ( C ) \in B ( \overline { \mathbf { P } } , \delta ) \Big | | C | = N \right ) . \end{align*}


\begin{align*} ( L _ n \phi _ ! F _ \bullet ) _ x = H _ n ( x / \phi , \pi _ x ^ * F _ \bullet ) . \end{align*}


\begin{align*} P = B V ^ { - 1 } A \quad Q = C V D . \end{align*}


\begin{align*} \displaystyle \Phi ( t , s ) = e ^ { \int _ { s } ^ { t } a ( \tau ) \ , d \tau } \leq e ^ { L _ { 1 } } \textnormal { a n d } \prod \limits _ { k = j + 1 } ^ { i } | 1 + b _ { k } | \leq e ^ { \sum \limits _ { k = 1 } ^ { \infty } | \ln ( | 1 + b _ { k } | ) | } \leq e ^ { L _ { 2 } } . \end{align*}


\begin{align*} \Phi _ { \# } ^ y : = 2 \alpha ^ { 1 / 2 } ( h _ { \alpha } - e _ { \alpha } + H _ f ^ + ) ^ { - 1 } P u _ { \alpha } \otimes A _ { y } ^ + \Omega . \end{align*}


\begin{align*} ( ( i ^ * _ \Theta \circ T ^ \# \circ i _ \Theta ) x ) ( y ) = ( T ^ \# \circ i _ \Theta x ) ( i _ \Theta y ) = \langle i _ \Theta x , i _ \Theta \bar y \rangle _ 1 = \langle x , \bar y \rangle _ 1 = ( T ^ \# _ \Theta x ) ( y ) \end{align*}


\begin{align*} ( \omega - Q ) ( \omega + Q ) = x _ 0 x _ 1 x _ 2 x _ 3 , \end{align*}


\begin{align*} \partial _ t \rho _ { \gamma _ k } + \frac { 1 } { \gamma _ k } \rho _ { \gamma _ k } u _ { \gamma _ k } = \nabla \rho _ { \gamma _ k } \cdot \nabla p _ { \gamma _ k } . \end{align*}


\begin{align*} K _ { \sigma , j } ( z , w ) = \sum _ { k = j } ^ \infty \binom { k } { j } ^ { - 1 } z ^ k \overline { w } ^ k , z , w \in \mathbb D . \end{align*}


\begin{align*} \det ( x I _ M - Z _ 1 Z _ 2 ) = x ^ { M - N } \det ( x I _ N - Z _ 2 Z _ 1 ) , \end{align*}


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


\begin{align*} [ \psi _ i ^ \epsilon ( z ) , \psi _ j ^ { \epsilon ' } ( w ) ] = 0 \ , , \psi ^ \pm _ { i , \mp b ^ \pm _ i } \cdot ( \psi ^ \pm _ { i , \mp b ^ \pm _ i } ) ^ { - 1 } = ( \psi ^ \pm _ { i , \mp b ^ \pm _ i } ) ^ { - 1 } \cdot \psi ^ \pm _ { i , \mp b ^ \pm _ i } = 1 \ , , \end{align*}


\begin{align*} ( A ' - A ) ^ \gamma \Phi _ k ( x , x ' ) & = \sum _ { | \mu | = k } \sum _ { \tau + \sigma = \gamma } A ^ \tau \Phi _ \mu ( x ) ( A ' ) ^ \sigma \Phi _ \mu ( x ' ) \\ & = \big ( 2 ( k + 1 ) \big ) ^ \frac { | \gamma | } 2 \sum _ { | \mu | = k } \sum _ { \tau + \sigma = \gamma } \Phi _ { \mu + \tau } ( x ) \Phi _ { \mu + \sigma } ( x ' ) . \end{align*}


\begin{align*} { \displaystyle \sum \limits _ { m = - \infty } ^ { \infty } } \left ( \int _ { m } ^ { m + 1 } \left \vert q \left ( x \right ) \right \vert d x \right ) ^ { 2 } < \infty , \end{align*}


\begin{align*} \widehat { K \cup Q } = \widehat K \cup Q , \mbox { a n d } ( K \cup Q ) ^ \circ = K ^ \circ . \end{align*}


\begin{align*} \phi \left ( x ; z \right ) = x ^ { 2 } - z ^ { 2 } , \quad \psi \left ( x ; z \right ) = 2 x \phi \left ( x ; z \right ) . \end{align*}


\begin{align*} H = \left [ \prod _ { i = 1 } H ( b _ i ) \right ] / \langle L _ 1 , \dots , L _ r \rangle \end{align*}


\begin{align*} t \circ \Pi ( - r , \pi - \theta ) + t \circ \Pi ( r , \theta ) = a + b , ( r , \theta ) \in D ^ + . \end{align*}


\begin{align*} \overline { C } _ { 2 } ( z ; \tau ) & = \tilde { \theta } _ { 2 } ( z ; \tau ) \prod _ { n \geq 1 } \frac { 1 } { \left [ 1 - q ^ { 3 n } + \Phi _ { 3 } ( \zeta ) g _ { n } ( z ; \tau ) \right ] } \end{align*}


\begin{align*} - q n \cdot d ^ 1 _ { 0 , \frac q 2 } \left ( n , j - \frac { 3 q } 2 \right ) & = - \frac { q n } 2 \cdot d ^ 1 _ { 0 , \frac q 2 } ( n , j ) , \\ 0 & = q n \cdot d ^ 1 _ { 0 , \frac q 2 } ( n , j ) . \end{align*}


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


\begin{align*} { \bf E } _ v ^ { \otimes M } \ { \bf E } _ y ^ { \otimes M } \sum _ { \gamma _ 1 , . . . , \gamma _ n = 1 } ^ M \prod _ { j = 1 } ^ n v _ { \gamma _ j } \exp ( 2 \pi i q _ j \cdot y _ { L , \gamma _ j } ) = \sum _ { A \in \mathcal { A } _ n } 1 _ { \{ | A | \leq M \} } \frac { M ! } { ( M - | A | ) ! } \prod _ { a \in A } \left \{ m _ { | a | } \delta _ * \left ( \sum _ { l \in a } q _ l \right ) \right \} \end{align*}


\begin{align*} \begin{cases} \Delta _ 0 F ( r ) = F ( r ) - F ( r / 2 ) \\ \Delta _ { k + 1 } F ( r ) = \Delta _ k F ( r ) - 2 ^ { k + 1 } \Delta _ k F ( r / 2 ) . \end{cases} \end{align*}


\begin{align*} E = \prod _ { i = 1 } ^ { s } \left [ \dfrac { a _ { i } } { b ^ { d _ { i } } } , \dfrac { a _ { i + 1 } } { b ^ { d _ { i } } } \right ] \end{align*}


\begin{align*} a _ 1 ( k ) = \frac { A ^ 2 } { 4 k ^ 2 } a _ 2 ( 0 ) ( 1 + o ( k ) ) , k \rightarrow 0 . \end{align*}


\begin{align*} \partial _ x ( u \times ( u \times g ( u ) ) ) \cdot \partial _ x u = 0 \ , , \end{align*}


\begin{align*} \overline { \mathbf { P } } _ { N } ( X _ { N } ) = \frac { 1 } { | \Omega | } \int _ { \Omega } \delta _ { \left ( x , \theta _ { N ^ { \frac { 1 } { d } } x } \cdot X _ { N } ' \right ) } \ , d x , \end{align*}


\begin{align*} \langle \theta , v \rangle = \sum _ { k = 1 } ^ { K } s ^ k v ( x ^ k ) \ge - C _ 2 \| v \| _ { H ^ 2 ( \Omega ) } \ , . \end{align*}


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


\begin{align*} \alpha ^ 2 - ( m - p ) \alpha + ( m - 1 ) = 0 \end{align*}


\begin{align*} \sum _ l \left ( t _ { l i l } M _ { j k } \right ) = \left ( \sum _ l t _ { l i l } \right ) M _ { j k } \ ; . \end{align*}


\begin{align*} & d \mu _ t ^ { \epsilon , K } \ , = \ , \left ( \frac { \epsilon ^ { 1 - \kappa } } 2 | D v _ { \epsilon , K } | ^ 2 \ , + \ , \frac 1 { 2 \epsilon ^ { 1 - \kappa } } ( v _ { \epsilon , K } ^ 2 - 1 ) ^ 2 \right ) \ , d x , \\ & d \xi _ t ^ { \epsilon , K } \ , = \ , \left ( \frac 1 { 2 \epsilon ^ { 1 - \kappa } } ( v _ { \epsilon , K } ^ 2 - 1 ) ^ 2 \ , - \ , \frac { \epsilon ^ { 1 - \kappa } } 2 | D v _ { \epsilon , K } | ^ 2 \right ) \ , d x . \end{align*}


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


\begin{align*} m ( a ) = 1 ^ { \mathcal { H } } \ ; . \end{align*}


\begin{align*} \delta ( g , \ell ) = 0 a _ \ell ( g ) \not \equiv \ell \pmod { p } \end{align*}


\begin{align*} ( T _ t \mu ) \oplus ( T _ s \mu ) : = T _ { t + s } \mu \ , . \end{align*}


\begin{align*} ( a , i ) \circ ( b , j ) = ( a + \alpha ^ i ( b ) , i + _ n j ) , ( a , i ) ^ { - } = ( - \alpha ^ { - i } ( a ) , - i ) , \quad { \rm a n d } \lambda _ { ( a , i ) } ( ( b , j ) ) = ( \alpha ^ i ( b ) , j ) . \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ { \frac { n } { 2 } } h ( x , y ) = 0 , & y \in \Omega , \\ ( - \Delta ) ^ i h ( x , y ) = ( - \Delta ) ^ i \left ( C _ n \ln \left ( \frac { 1 } { | x - y | } \right ) \right ) , \ , \ , i = 0 , 1 , \cdots , \frac { n } { 2 } - 1 , & y \in \partial \Omega . \end{cases} \end{align*}


\begin{align*} [ 1 a ] = \frac { 1 - t x } { 1 - t x - t x ^ 2 } . \end{align*}


\begin{align*} p = R \rho \theta , \end{align*}


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


\begin{align*} J ' _ { \sigma , k } = \left ( { j ' } _ { \sigma ( 1 ) } , { j ' } _ { \sigma ( 2 ) } , \dots , { j ' } _ { \sigma ( k ) } \right ) \ , . \end{align*}


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


\begin{align*} D _ { \mathfrak p } ( g , r , s ) : = \dim _ B ( \mathbb H ^ r _ g ( s ) _ B ) , \end{align*}


\begin{align*} d _ A ( \mu ) = \sup _ { \lambda \in \Lambda } | a _ { \lambda , \lambda - \mu } | . \end{align*}


\begin{align*} \alpha = ( \phi ( \jmath ) , [ 1 ] , \upsilon ( \jmath ) , \varsigma ( \jmath ) ) , \end{align*}


\begin{align*} \langle \Lambda _ 1 , \Lambda _ 2 \rangle = \langle \theta ^ + ( \Lambda _ 1 ) , \theta ^ + ( \Lambda _ 2 ) \rangle = \langle \theta ^ - ( \Lambda _ 1 ) , \theta ^ - ( \Lambda _ 2 ) \rangle \end{align*}


\begin{align*} \frac { 1 } { \left ( 1 - 2 x t + t ^ 2 \right ) ^ \alpha } = \sum _ { n = 0 } ^ \infty C _ n ^ \alpha ( x ) t ^ n \ . \end{align*}


\begin{align*} X ( x , t ) = ( 1 + \xi ( x , t ) ) \Gamma ( x , t ) = \Gamma ( x , t ) + \sum _ { z \in \Z } p ( x - z , t - t ) \xi ( z , t ) \Gamma ( z , t ) . \end{align*}


\begin{align*} & \Upsilon _ { \mu , s } Y ^ { \mu , t } ( b , z ) = Y ^ { \overline \mu + 2 s , t + s - \sqrt { - 1 } \Im ( \mu ) } ( b , z ) \Upsilon _ { \mu , s } \end{align*}


\begin{align*} \varepsilon ( X a ) \stackrel { \scriptscriptstyle \eqref { e q : c o m p L R a l g } } { = } \varepsilon ( a X ) + X ( a ) \stackrel { \scriptscriptstyle \eqref { m a i n s o m m a 1 } } { = } X ( a ) \end{align*}


\begin{align*} M ^ { \ell , r - \ell } = \left \{ \begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix} \right \} . \end{align*}


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


\begin{align*} b _ k ( x , s ) = \begin{cases} ( s - \overline { u } _ k ( x ) ) ^ { q _ k ( x ) - 1 } & s > \overline { u } _ k ( x ) , \\ 0 & \underline { u } _ k ( x ) \leq s \leq \overline { u } _ k ( x ) , \\ - ( \underline { u } _ k ( x ) - s ) ^ { q _ k ( x ) - 1 } & s < \underline { u } _ k ( x ) . \end{cases} \end{align*}


\begin{align*} - \partial _ t u - \sigma \Delta u + | D u | ^ \gamma , \sigma = o ( 1 ) , \end{align*}


\begin{align*} g \circ k _ 0 ( x ) = g ( x , 0 ) & = k \circ p _ 0 ' ( x ) \\ \pi \circ g ( x , y ) = \pi ( k \circ p _ 0 ' ( x ) + f ( y ) ) = \pi \circ f ( y ) = p _ 0 '' ( y ) & = p _ 0 '' \circ \pi _ 0 ( x , y ) . \end{align*}


\begin{align*} \begin{multlined} A _ 1 \sim A _ 2 B _ 1 \sim B _ 2 \\ \Rightarrow A _ 2 \otimes B _ 2 = U ( S _ A \otimes A _ 1 ) \otimes U ( S _ B \otimes B _ 1 ) \\ = U ( U ( S _ A \otimes S _ B ) \otimes ( A _ 1 \otimes B _ 1 ) ) \\ \Rightarrow A _ 1 \otimes B _ 1 \sim A _ 2 \otimes B _ 2 \ ; . \end{multlined} \end{align*}


\begin{align*} H _ \tau ( y , \hat \eta _ \tau ) = \frac { 1 } { 2 } \sum _ { j , k = 0 } ^ n g _ \tau ^ { j k } ( y ) \eta _ { j \tau } \eta _ { k \tau } . \end{align*}


\begin{align*} \begin{aligned} v _ 1 & = \textstyle \phantom { \big ( } \prod _ { s = 0 } ^ { d - 1 } x _ { 2 + s t } , \\ v _ 2 & = \textstyle \big ( \prod _ { s = 0 } ^ { d - 2 } x _ { 2 + s t } \big ) x _ { 3 + ( d - 1 ) t } . \end{aligned} \end{align*}


\begin{align*} \sum _ { { B } _ { = } } f ( \mathrm { k } ) \le \binom { \ell } { 2 } \sum _ { s _ n < k _ 1 , \dots , k _ { \ell - 1 } \le n } f ( k _ 1 , k _ 1 , k _ 2 , \dots , k _ \ell ) \le \binom { \ell } { 2 } \sum _ { s _ n < k \le n } c _ k ^ 2 w _ n ^ { 2 k } \cdot I _ { 1 } ^ { \ell - 2 } . \end{align*}


\begin{align*} d _ n = d ( \bar x _ n , \partial \Omega ) , M _ n = \frac { | u ( \bar x _ n , t _ n ) | } z , r _ n = ( t _ n - \bar t _ n ) ^ { \frac 1 \gamma } M _ n ^ { \frac { \gamma - 1 } { \gamma } } , \end{align*}


\begin{align*} \delta ( \partial _ x z ) ^ { \natural , 2 , i , i } _ { s , u , t } & = - 2 \sum _ { l = 1 } ^ 3 \partial _ x W _ { u , t } ^ { i , l } \delta ( \partial _ x z ^ l z ^ i ) _ { s , u } + 2 \sum _ { l = 1 } ^ 3 \delta \partial _ x \mathbb { W } ^ { i , l } _ { s , u , t } z _ s ^ l z _ s ^ i - 2 \sum _ { l = 1 } ^ 3 \partial _ x \mathbb { W } ^ { i , l } _ { u , t } \delta ( z ^ l z ^ i ) _ { s , u } \\ & \quad + \end{align*}


\begin{align*} [ L _ { m , i } , G _ { n , j } ] & = \left ( n ( i + q ) - m \left ( j + \frac q 2 \right ) \right ) G _ { m + n , i + j } , \\ [ G _ { m , i } , G _ { n , j } ] & = 2 q L _ { m + n , i + j } . \end{align*}


\begin{align*} V _ { L , \omega } ( x ) : = \int _ { \Lambda _ L } B _ \# ( x - y ) d \mu _ { L , \omega } ( y ) , \end{align*}


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


\begin{align*} \overline { V ^ \lambda } ( y _ 0 ) = \min _ { \Sigma _ \lambda \cap \Omega } \overline { V ^ \lambda } ( x ) < 0 . \end{align*}


\begin{align*} ( X , \omega ) = \left ( [ a , b ] \times Y , d \left ( e ^ s \lambda \right ) \right ) \end{align*}


\begin{align*} \left \Vert \alpha _ { 3 } \right \Vert ^ 2 = 1 - K . \end{align*}


\begin{align*} \sum _ { m = 1 } ^ n \lambda _ { m , n } \le \dfrac { 2 ^ b \varepsilon _ 1 b _ { 2 ^ { n } } } { 2 ^ b - 1 } : = C _ 1 ( b ) \varepsilon _ 1 b _ { 2 ^ { n } } . \end{align*}


\begin{align*} w _ A ( x , y ) : = \alpha _ x \mathbf { 1 } _ { y = x } + \beta _ x \mathbf { 1 } _ { y = x + \mathbf { e } ^ 1 } + \gamma _ x \mathbf { 1 } _ { y = x + \mathbf { e } ^ 2 } \end{align*}


\begin{align*} c ( \tau \sigma , \alpha ) = c \big ( \tau , \sigma ( \alpha ) \big ) \cdot c ( \sigma , \alpha ) \end{align*}


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


\begin{align*} H ( t ) = H \colon V \oplus W \ni ( v , w ) \longmapsto ( - h _ V ( v ) , 0 ) \in V \oplus W . \end{align*}


\begin{align*} v ( w ) = \frac { 1 } { 2 \pi } \int _ \R v _ \lambda ( w ) \ , d \lambda \end{align*}


\begin{align*} ( \L _ T h ) \varphi = h ( \varphi \circ T ) \varphi \in \C ^ { q + i } , i \in \{ 0 , 1 , 2 \} . \end{align*}


\begin{align*} \mathcal { N } = \bigcup _ { i = 1 } ^ { n _ 1 } N _ i = \{ a _ 1 , a _ 2 , \cdots , a _ n \} , \end{align*}


\begin{align*} \left | a \alpha ^ { k } - \frac { 2 ^ { n } } { 3 } \right | = \left | \frac { 2 ^ { m } } { 3 } - \frac { \left ( ( - 1 ) ^ { n } + ( - 1 ) ^ { m } \right ) } { 3 } - b \beta ^ { k } - c \gamma ^ { k } \right | . \end{align*}


\begin{align*} H = \mathrm { t r } [ W ] = \sum _ { \alpha = 1 } ^ { n - 1 } \kappa _ \alpha , | W | ^ 2 = \mathrm { t r } [ W ^ 2 ] = \sum _ { \alpha = 1 } ^ { n - 1 } \kappa _ \alpha ^ 2 \quad \Gamma . \end{align*}


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


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


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


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


\begin{align*} \widehat { \varphi } ( \gamma _ 1 \cdots \gamma _ d ) = \widehat { \xi } _ 1 ( \gamma _ 1 ) \cdots \widehat { \xi } _ d ( \gamma _ d ) \gamma _ 1 , . . . , \gamma _ d \in \Gamma . \end{align*}


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


\begin{align*} Y ^ 2 Z = X ^ 3 + a _ 2 X ^ 2 Z + a _ 4 X Z ^ 2 \ , . \end{align*}


\begin{align*} \phi _ i ( v ) \ = \ \sum _ { j = 1 } ^ n \phi _ i ( \lambda _ j \cdot v _ j ) \ = \ \sum _ { j = 1 } ^ n \phi _ i ( \lambda _ j ) \cdot v _ j + \lambda _ j \cdot \phi _ i ( v _ j ) , \end{align*}


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


\begin{align*} 0 < \delta \leq \frac { d - \alpha } { 2 } \kappa = \frac { 2 ^ \alpha \Gamma \left ( \frac { \alpha + \delta } { 2 } \right ) \Gamma \left ( \frac { d - \delta } { 2 } \right ) } { \Gamma \left ( \frac { \delta } { 2 } \right ) \Gamma \left ( \frac { d - \alpha - \delta } { 2 } \right ) } . \end{align*}


\begin{align*} \Lambda : = \{ \lambda \in ( \mathbb P ^ n _ { k ' } ) ^ * ( k ' ) \ , | \ , ( I ) , ( I I ) \} \end{align*}


\begin{align*} \mathcal { L } = x \mid \varphi \land \psi \mid \varphi \lor \psi \mid \varphi \to \psi \mid \lnot \varphi \mid 0 \mid 1 . \end{align*}


\begin{align*} r ( T , J T ) & = \sum _ { l , k = 1 } ^ { 6 } g \left ( J T , N ( e _ l , e _ k ) \right ) g \left ( J T , N ( e _ l , e _ k ) \right ) + \sum _ { l , k = 1 } ^ { 6 } g \left ( N ( T , e _ k ) , e _ l \right ) g \left ( N ( J T , e _ k ) , J e _ l \right ) , \\ & = \sum _ { l , k = 1 } ^ { 6 } g \left ( J T , N ( e _ l , e _ k ) \right ) ^ 2 - \sum _ { l , k = 1 } ^ { 6 } g \left ( N ( T , e _ k ) , e _ l \right ) ^ 2 , \\ & = \frac { 1 } { 8 } { \textbf { H } ^ \prime } ^ 2 . \end{align*}


\begin{align*} [ M : N ] ^ { - 2 } \Psi _ i \circ \Psi \circ T _ i ( x ) = \sum _ { j = 1 } ^ J [ M : N ] ^ { - 2 } \Psi _ i \circ \Psi \circ T _ i ( z _ j x ) = \sum _ { j = 1 } ^ J \mu _ i ^ j z _ j x = \sigma _ i x . \end{align*}


\begin{align*} \left \lbrace \begin{aligned} Q _ n ^ { \epsilon , \Delta t } & = q _ n ^ { \epsilon , \Delta t } + \epsilon p _ n ^ { \epsilon , \Delta t } \\ P _ n ^ { \epsilon , \Delta t } & = \epsilon p _ n ^ { \epsilon , \Delta t } . \end{aligned} \right . \end{align*}


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


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


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


\begin{align*} \mathcal { F } ^ { - 1 } ( \omega ) = \mathcal { E } ^ * ( \omega ) \wedge \Phi . \end{align*}


\begin{align*} \langle \nabla ^ 2 v ( \gamma ( \xi ) ) \ , \gamma ' ( \xi ) , \gamma ' ( \xi ) \rangle = \langle \nabla ^ 2 v ( \gamma ( \xi ) ) \ , \gamma ' ( \xi ) , - ( \gamma ' ( \xi ) ) ^ { \perp } \rangle = 0 \textrm { f o r e v e r y } \xi \in [ 0 , \ell ] \ , . \end{align*}


\begin{gather*} \gamma _ { \mathfrak n } ( x , y , z ) = B \left ( d ' ( x , y ) , z \right ) = B \left ( \delta ( x , y ) + \theta ( x , y ) + \gamma ( x , y , \cdot ) , z \right ) = \gamma ( x , y , z ) , \end{gather*}


\begin{align*} \langle f , g \rangle = \langle \hat f , \hat g \rangle , \ \ \ \ \ f , g \in L ^ 2 ( \mathbb { R } ^ d ) \end{align*}


\begin{align*} \Tilde { M } : = M \setminus \left ( D ^ 2 _ { \epsilon _ 0 } \times T ^ 2 \times \Sigma \right ) \cup _ \phi \left ( D ^ 2 _ 1 \times T ^ 2 \times \Sigma \right ) , \end{align*}


\begin{align*} d _ { r , s } ( r , i ) = 0 , \mbox { i f } i \ne 3 q + s . \end{align*}


\begin{align*} { \rm g c d } ( v _ 1 , v _ 2 ) = 1 \mbox { a n d } v _ 1 > 0 \mbox { o r } v = ( 0 , 1 ) \ . \end{align*}


\begin{align*} \psi ( u , v ) = \frac { u + v } { 2 } + \phi ( u - v ) \end{align*}


\begin{align*} \begin{aligned} p ( P ) & = \frac { 2 } { \pi R ^ { 2 } } \int _ { 0 } ^ { R } \int _ { 0 } ^ { \pi } p ( r ' , \theta ' ) r ' d r ' d \theta ' \\ & + \frac { 4 } { \pi R ^ { 2 } } \int _ { 0 } ^ { R } \int _ { 0 } ^ { r ' } \int _ { 0 } ^ { \pi } \frac { ( \tilde { u } - u ) v _ { \theta ' } } { \rho } r ' d \theta ' d \rho d r ' \doteq I _ { 1 } + I _ { 2 } . \end{aligned} \end{align*}


\begin{align*} L \circ H = H \circ f , \quad f = H ^ { - 1 } \circ L \circ { H } . \end{align*}


\begin{align*} \begin{aligned} \theta ( a b ) & = \varphi ( a b ) - \varepsilon _ m ( a b ) \\ & = \varphi ( a ) \varphi ( b ) - \varepsilon _ m ( a ) \varepsilon _ m ( b ) \\ & = ( \theta ( a ) + \varepsilon _ m ( a ) ) ( \theta ( b ) + \varepsilon _ m ( b ) ) - \varepsilon _ m ( a ) \varepsilon _ m ( b ) \\ & = \varepsilon _ m ( a ) \theta ( b ) + \varepsilon _ m ( b ) \theta ( a ) , \end{aligned} \end{align*}


\begin{align*} D ( f ) = \int _ { \mathbb T } \int _ { \mathbb T } \frac { | f ^ { * } ( \zeta ) - f ^ { * } ( \lambda ) | ^ 2 } { | \zeta - \lambda | ^ 2 } d \sigma ( \zeta ) d \sigma ( \lambda ) , \ , \ , f \in \mathcal D , \end{align*}


\begin{align*} r _ i = \frac { h _ i w _ i } { \gcd ( v _ i , w _ i ) } , t _ i = \frac { h _ i v _ i } { \gcd ( v _ i , w _ i ) } , \end{align*}


\begin{align*} h _ { t + j } = \binom { a _ t + j } { t + j } + \binom { a _ { t - 1 } + j - 1 } { t + j - 1 } + \cdots \end{align*}


\begin{align*} \kappa = - 1 / 2 \cdot \kappa _ b - \kappa _ c . \end{align*}


\begin{align*} { } _ 2 F _ 1 ( 1 , 1 ; 2 + \epsilon ; z ) = - \frac { 1 + \epsilon } { z } \ln ( 1 - z ) + O ( \epsilon ^ 2 ) \end{align*}


\begin{align*} \int _ { a } ^ { \infty } F _ { 0 } \left ( x \right ) \mathrm { d } x & = \int \left [ \int _ { a } ^ { \infty } \mathrm { e } ^ { - 2 s x } \mathrm { d } x \right ] \mathrm { d } \sigma \left ( s \right ) \\ & = \int \mathrm { e } ^ { - 2 a s } \frac { \mathrm { d } \sigma \left ( s \right ) } { 2 s } \leq \max _ { s \in \operatorname * { S u p p } \sigma } \mathrm { e } ^ { - 2 a s } \cdot \int \frac { \mathrm { d } \sigma \left ( s \right ) } { 2 s } < \infty . \end{align*}


\begin{align*} h ( t , 0 ) = 0 \mbox { f o r a l l } t \ge 0 , \end{align*}


\begin{align*} X ^ n = \Gamma \left ( x _ 0 + \int _ 0 ^ { \cdot } T ^ W f ^ n ( d s , X ^ n _ s - W _ s ) + W \right ) . \end{align*}


\begin{align*} \frac { \partial { K _ p } } { \partial { x _ j } } ( z ) = \frac { p } { q } K _ p ( z ) ^ { 1 - q / p } \frac { \partial { K _ p ^ { q / p } } } { \partial { x _ j } } ( z ) = \frac { p } { q } m _ p ( z ) ^ { q - p } \frac { \partial { K _ p ^ { q / p } } } { \partial { x _ j } } ( z ) . \end{align*}


\begin{align*} \mathcal { F } _ 2 \Phi ( x , \xi ) = \int _ { \mathbb { R } ^ d } \Phi ( x , y ) e ^ { - 2 \pi i \xi \cdot y } d y . \end{align*}


\begin{align*} [ D , E ] _ G = ( - 1 ) ^ { \abs { E } \abs { D } } \left ( D \circ E - ( - 1 ) ^ { \abs { D } \abs { E } } E \circ D \right ) . \end{align*}


\begin{gather*} \left \{ \begin{array} { l l } \alpha = \alpha ' + d ^ 1 \left ( - \tau \right ) \\ \gamma = \gamma ' ( x , y , \cdot ) - B _ { \mathfrak a } \left ( ( \alpha ' + \frac { 1 } { 2 } d ( - \tau ) ) \wedge ( - \tau ) \right ) ( x , y , \cdot ) \end{array} \right . \end{gather*}


\begin{align*} L _ 0 g = - ( g \frac { L v } { v } + 2 \frac { w ^ 2 } { v } \partial _ { \eta } v \partial _ { \eta } g ) = - g \frac { L v } { v } < 0 a t z _ { m i n } , \end{align*}


\begin{align*} \langle \epsilon _ { I } , \hat { \mathbb { S } } ( \boldsymbol { \ell } ) _ t \rangle = \langle \epsilon _ { U _ I ( \boldsymbol { \ell } ) } , \mathbb { X } _ t \rangle \end{align*}


\begin{align*} f \ , = \ , F ^ \prime , g \ , = \ , G ^ \prime . \end{align*}


\begin{align*} \ell _ i : = ( 0 , \dots , 0 , 1 , \dots , 1 , 0 \dots , 0 ) \in \mathcal L _ J \end{align*}


\begin{align*} ( H _ { b _ 0 + \epsilon } ^ E - z ) ^ { - 1 } = S _ \epsilon ( z ) - S _ \epsilon ( z ) T _ \epsilon ( z ) + ( H _ { b _ 0 + \epsilon } ^ E - z ) ^ { - 1 } T ^ 2 _ \epsilon . \end{align*}


\begin{align*} L ^ 2 ( M , \tau ) = \bigoplus _ i u _ i ^ * L ^ 2 ( N , \tau ) \end{align*}


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


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u = f ( 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 ( 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*} ( A \oplus B ) _ P = A _ P \cdot B _ P \ ; . \end{align*}


\begin{align*} \frac { 2 \lambda } { \lambda ^ 2 - z ^ 2 } = \frac { 1 } { \lambda - z } + \frac { 1 } { \lambda + z } \ , . \end{align*}


\begin{align*} \mathrm { d } \rho \left ( s \right ) : = \sum _ { n } c _ { n } ^ { 2 } \left ( t \right ) \delta _ { \kappa _ { n } } \left ( s \right ) \mathrm { d } s , \end{align*}


\begin{align*} v _ s - \tilde { F } ( D ^ 2 v , y , s ) = 0 ~ ~ ~ ~ \mbox { i n } ~ Q _ { x _ n } ( p ) , \end{align*}


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


\begin{align*} \| ( H _ { q } [ \omega , 0 ] \upharpoonright \Lambda - E ) \widetilde \psi \| & = ( | \widetilde \psi ( { \nu - M - 1 } ) | ^ { 2 } + | \widetilde \psi ( { \nu + M + 1 } ) | ^ { 2 } ) ^ { \frac { 1 } { 2 } } \\ & < 2 e ^ { - ( \min \gamma - 2 \varepsilon _ { 1 } ) ( M + 1 ) } \| \psi \upharpoonright \Lambda \| ^ { - 1 } < 4 e ^ { - ( \min \gamma - 2 \varepsilon _ { 1 } ) M } < 4 \delta _ { 0 } \end{align*}


\begin{align*} \frac { \partial ^ n } { \partial \hat { t } ^ n } = \left ( \frac { \partial \hat { z } _ i } { \partial \hat { t } } \right ) ^ n \frac { \partial ^ n } { \partial \hat { z } _ i ^ n } + \sum _ { j = 1 } ^ { n - 1 } O _ { d _ 0 } ( \rho ^ { - 1 } ) \frac { \partial ^ j } { \partial \hat { z } _ i ^ j } . \end{align*}


\begin{align*} \psi _ { - \sigma } \left ( x , t , k \right ) & = \psi \left ( x , t , k \right ) \\ & + { \displaystyle \sum \limits _ { n = 1 } ^ { N } } c _ { n } ^ { 2 } e ^ { 8 \kappa _ { n } ^ { 3 } t } y _ { n } \left ( x , t \right ) \int _ { x } ^ { \infty } \psi \left ( s , t , k \right ) \psi \left ( s , t , \mathrm { i } \kappa _ { n } \right ) \mathrm { d } s \end{align*}


\begin{align*} \operatorname * { S u p } \limits _ { \left \vert I \right \vert = 1 } \int _ { I } \max \left ( - q \left ( x \right ) , 0 \right ) \mathrm { d } x < \infty . \end{align*}


\begin{align*} Y _ { t } = b ( x , Y _ { t } ) \ , t + \sqrt { 2 } \varrho ( x , Y _ { t } ) \ , W _ { t } \end{align*}


\begin{align*} \bar { t } = t , \bar { x } = x , \bar { y } = \psi ( t , x , y ) , \end{align*}


\begin{align*} 2 \left ( n ( i + q ) - m \left ( j + \frac q 2 \right ) \right ) d ^ 1 _ { r , s } ( m + n , i + j ) & = \left ( ( n + r ) ( i + q ) - m \left ( j + s + \frac q 2 \right ) \right ) d ^ 1 _ { r , s } ( n , j ) , \\ q ( d ^ 1 _ { r , s } ( m , i ) + d ^ 1 _ { r , s } ( n , j ) ) & = 0 . \end{align*}


\begin{align*} \begin{gathered} \Phi _ { + 1 } ^ { a , b } : \{ 0 , \ldots , b - 1 \} \rightarrow \{ 0 , \ldots , a + b - 1 \} \ ; , \\ \Phi _ { + 1 } ^ { a , b } ( i ) = a + i \ ; , \end{gathered} \end{align*}


\begin{align*} \underset { k = 0 } { \rm R e s } \breve { M } ^ { ( 2 ) } ( x , t , k ) = c _ 0 ( \xi ) \breve { M } ^ { ( 1 ) } ( x , t , 0 ) , \end{align*}


\begin{align*} _ p ( \psi , \psi ' , n ) = \sum _ { x = u ( x ) n u ' ( x ) \in X ( n ) } \psi ( u ( x ) ) \psi ' ( u ' ( x ) ) . \end{align*}


\begin{align*} d _ R ^ * ( \alpha , \alpha ' ) = E ^ * ( \min \{ | \alpha - \alpha ' | ; 1 \} . \end{align*}


\begin{align*} \begin{aligned} & \partial ^ \circ \eta _ 0 ( r ) - V _ \Gamma \partial _ r \eta _ 1 ( r ) + k _ d r | W | ^ 2 \partial _ r \eta _ 0 ( r ) \\ & - k _ d \Delta _ \Gamma \eta _ 0 ( r ) + k _ d H \partial _ r \eta _ 1 ( r ) - k _ d \partial _ r ^ 2 \eta _ 2 ( r ) = f \end{aligned} \end{align*}


\begin{align*} s \circ t = \tau ( s , t ) s t s , t \in \mathcal { S } , \end{align*}


\begin{align*} a _ n & = \frac { 1 } { 2 \pi } \int _ { - 1 } ^ { 0 } \int _ { \mathbb R } e ^ { - i \lambda x } \bigg ( \frac { \sin ( \lambda / 2 ) } { \lambda / 2 } \bigg ) ^ n \ , d \lambda \ , d x \\ & = \frac { 1 } { 2 \pi } \int _ 0 ^ 1 \int _ { \mathbb R } \cos ( \lambda x ) \bigg ( \frac { \sin ( \lambda / 2 ) } { \lambda / 2 } \bigg ) ^ n \ , d \lambda \ , d x . \end{align*}


\begin{align*} \Xi _ { [ s ] } ( R _ x ) = R _ { x ^ { ( 0 ) } } \otimes R _ { \Lambda _ x ^ { ( - ) } } \otimes R _ { \Lambda _ x ^ { ( + ) } } \end{align*}


\begin{align*} \| \bar { \zeta } \| _ { C ( \overline { Q _ { \varepsilon , T } } ) } = \| \zeta \| _ { C ( \overline { S _ T } ) } , \| \bar { \zeta } ( \cdot , 0 ) \| _ { C ( \overline { \Omega _ \varepsilon ( 0 ) } ) } = \| \zeta ( \cdot , 0 ) \| _ { C ( \Gamma ( 0 ) ) } \leq \| \zeta \| _ { C ( \overline { S _ T } ) } , \end{align*}


\begin{align*} 2 \int _ { D } \phi u \cdot \partial _ x u d x = \int _ { D } \phi \partial _ x | u | ^ 2 d x = - \int _ { D } \partial _ x \phi | u | ^ 2 d x = 0 \ , , \end{align*}


\begin{align*} \big ( M ^ * \setminus \pi ^ { - 1 } ( F ' ) \big ) \setminus K _ k = \bigcup _ { x ' \not \in F ' } \big [ \{ * \} \cup \big ( \pi ^ { - 1 } ( x ' ) \setminus K _ k \big ) \big ] . \end{align*}


\begin{align*} \| \Gamma _ j ^ h \circ \Phi \| _ { \mathcal { L } _ 2 ( L _ x ^ 2 ; \mathbb { C } ) } \leq \| \Phi \| _ { \gamma ( L _ x ^ 2 ; W _ x ^ { \mathfrak { s } , r / 2 } ) } \| \Gamma _ j ^ h \| _ { \mathcal { L } ( W _ x ^ { \mathfrak { s } , r / 2 } ; \mathbb { C } ) } , j = 1 , 2 , 3 , 4 . \end{align*}


\begin{align*} Z _ 1 = \bigcup _ { i \in ( j , J - j ] } \hat { Z } _ i , Z _ 2 = \bigcup _ { i \in [ 1 , j ] \cup ( J - j , J ] } \hat { Z } _ i \ , , \end{align*}


\begin{align*} \mathrm { r t } _ R ( A ) = \min \{ n \geq 1 \ \colon \ ( 0 : _ R f ^ { n + 1 } ) = ( 0 : _ R f ^ n ) \} . \end{align*}


\begin{align*} F _ { k , t } ( r ) = 2 \pi i ^ k \int _ 0 ^ \infty e ^ { - 2 \pi t \rho } J _ { - k } ( 2 \pi r \rho ) \ , d \rho = 2 \pi ( - 1 ) ^ k i ^ k \int _ 0 ^ \infty e ^ { - 2 \pi t \rho } J _ k ( 2 \pi r \rho ) \ , d \rho \ , . \end{align*}


\begin{align*} \Pi _ \alpha : \Pi ^ { \perp } \ell ^ 2 \to \Pi ^ { \perp } \ell ^ 2 , \Pi _ \alpha u : = \left \{ \begin{matrix} z _ { ( j , \sigma ) } & & j \in \Omega _ \alpha \\ 0 & & j \not \in \Omega _ \alpha \end{matrix} \right . \ . \end{align*}


\begin{align*} \partial _ i R ( x ) = \bar { \nu } _ i ( x ) R ( x ) \overline { W } ( x ) R ( x ) + d ( x ) R ( x ) \partial _ i \overline { W } ( x ) R ( x ) . \end{align*}


\begin{align*} d ( 0 , \partial \Omega _ n ) = d \left ( 0 , \frac { \partial \Omega - \bar x _ n } { r _ n } \right ) = \frac 1 { r _ n } d ( \bar x _ n , \partial \Omega ) \to + \infty \end{align*}


\begin{align*} f _ i ( x ) = \int _ 0 ^ x g _ i ( s ) \ , d s , x \in [ 0 , 1 ] , i \in \N . \end{align*}


\begin{align*} \lim \limits _ { y \to + \infty } ( u , \tilde { h } ) = ( 0 , 0 ) . \end{align*}


\begin{align*} \delta ( k , \xi ) = \left ( \frac { k + k _ 0 } { k - k _ 0 } \right ) ^ { ^ { i \nu ( - k _ 0 ) } } e ^ { \chi ( \xi , k ) } , \end{align*}


\begin{align*} | \det \tilde { M } | \leq \prod _ { j = 1 } ^ p \| v _ j \| \leq \| v _ 1 \| , \end{align*}


\begin{align*} & r _ { q + 1 , d } = \begin{cases} 1 & \mbox { i f } d = q + 1 , \\ 0 & \mbox { i f } d = 0 , \\ \frac { q ( q - 1 ) } { 2 } & \mbox { i f } d \in \{ 1 , 3 \} , \\ 2 q & \mbox { i f } d = 2 . \end{cases} \end{align*}


\begin{align*} a _ n & : = \lim _ { d \to \infty } \frac { A _ n ( d ) } { d ^ n } , b _ n ^ 1 : = \varlimsup _ { d \to \infty } \frac { \mu _ n ( d ) } { d ^ n } \\ [ 1 e x ] b _ n ^ 2 & : = \varliminf _ { d \to \infty } \frac { \mu _ n ( d ) } { d ^ n } , c _ n : = \binom { n } { \lfloor n / 2 \rfloor } 2 ^ { - n } . \end{align*}


\begin{align*} \alpha ( t , x ) \oplus \alpha ( s , x ) : = \alpha ( s + t , x ) \ , \end{align*}


\begin{align*} p ^ { ( \alpha ) } _ t ( x ) = e ^ { - | \sigma | t } p _ t ( x ) + e ^ { - | \sigma | t } \sum _ { k = 1 } ^ \infty \frac { t ^ k ( p _ t * \sigma ^ { k * } ) ( x ) } { k ! } \end{align*}


\begin{align*} p _ Y \ , Q [ H ] = [ H ] \ , p _ X . \end{align*}


\begin{align*} \frac { \zeta \dim V / P + \sum _ { k = 1 } ^ { n } \eta _ { k } \dim ( F _ { k } / P \cap F _ { k } ) } { \dim V / P } \ge \frac { \sum _ { k = 1 } ^ { n } \eta _ { k } \dim F _ { k } } { n + 1 } . \end{align*}


\begin{align*} \tilde p _ { s _ i } | _ { w _ K } = 0 \ \ \ \textrm { f o r a l l } \ K \subset \Delta \ \textrm { w i t h } \ J \subset K \subset I \end{align*}


\begin{align*} \frac { 1 } { \epsilon ^ 2 } \int _ 0 ^ \infty e ^ { - \frac { 2 r } { \epsilon ^ 2 } } d r = \frac 1 2 . \end{align*}


\begin{align*} X ^ 0 ( t ) = \varphi ( \beta ( t ) , X ( 0 ) ) , \end{align*}


\begin{align*} c _ \delta = M _ 0 = E [ M _ { \bar { T } _ 1 } ] \ge e ^ { \lambda ( 1 - \delta ) } E [ e ^ { - \lambda ^ 2 \bar { T } _ 1 / 2 } ] \ , . \end{align*}


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


\begin{align*} j ( x ; q ) = j ( x ; q ^ 2 ) j ( q x ; q ^ 2 ) \frac { J _ 1 } { J _ 2 ^ 2 } . \end{align*}


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


\begin{align*} { \rm D i s c } ( \eta , \delta , N ) = \max _ { X _ { N } \in \Lambda _ { N } } \max _ { i } \left | { \rm e m p } _ { N } ( K _ { i } ) - \rho ( K _ { i } ) \right | . \end{align*}


\begin{align*} \mathfrak { B } ^ r _ { 1 2 } ( \xi , t ) = \frac { k _ 0 } { k _ 0 + i \kappa } \mathfrak { B } _ { 1 2 } ( \xi , t ) , \mathfrak { B } ^ r _ { 2 1 } ( \xi , t ) = \frac { k _ 0 + i \kappa } { k _ 0 } \mathfrak { B } _ { 2 1 } ( \xi , t ) . \end{align*}


\begin{align*} \sup _ { 0 \leq r \leq 1 } \sup _ { 1 \leq n \leq \left \lfloor T r \right \rfloor - 1 } ( \sum _ { k = n } ^ { T } ( k + d ) ^ { - 1 } ) ^ { m } \leq ( \sum _ { k = 1 } ^ { T } ( k + d ) ^ { - 1 } ) ^ { m } \sim ( \log T ) ^ { m } , \end{align*}


\begin{align*} \frac { s } { 1 } = \frac { f ^ b t s } { f ^ b t } = 0 { \rm i n } R ( S ) = U _ { S / R } ^ { - 1 } S . \end{align*}


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


\begin{align*} W _ j - \underline { W } _ j - \underline { \hat { W } } _ j \stackrel { d } { = } Y _ { j m , m } - \underline { Y } _ { j m , m } \ , . \end{align*}


\begin{align*} { \mu } _ { \theta } ^ { \epsilon } = \frac { \mu _ { \theta } + \epsilon \mathbf { 1 } _ { X } } { 1 + \epsilon | X | } . \end{align*}


\begin{align*} m _ p ( z ) : = \inf \left \{ \| f \| _ p : f \in A ^ p ( \Omega ) , f ( z ) = 1 \right \} \end{align*}


\begin{align*} i \in U ( i ) & = U ( 0 , i ) = U \big ( L ( a , a ' ) , i \big ) = U L \big ( U ( a , i ) , U ( a ' , i ) \big ) = U L \big ( 1 , U ( a ' , i ) \big ) = \\ & = U L U ( a ' , i ) = U ( a ' , i ) \subseteq U ( a ' ) , \end{align*}


\begin{align*} \mu _ { t _ n } ( \ , \cdot \ , ) = \frac { 1 } { t _ n } \int _ { 0 } ^ { t _ n } \mathbb { P } ( u ^ { x } _ s \in \cdot \ , ) \dd s \ , . \end{align*}


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


\begin{align*} \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 3 - i T _ { \theta _ j } e _ 4 , v _ j \rangle = 0 . \end{align*}


\begin{align*} \begin{array} { l } d _ X ( y \circ \alpha _ n , \alpha _ n ) = E \min \{ \| y \circ \alpha _ n - \alpha _ n \| _ X ; 1 \} \\ = E \min \{ \| y \circ \alpha _ n - x ^ * \circ \alpha _ n \| _ X ; 1 \} = E ^ n \min \{ \| y - x ^ * \| _ X ; 1 \} < \delta \ \ \ \mbox { f o r a l l } \ \ \ n \ge n _ 2 \end{array} \end{align*}


\begin{align*} \begin{bmatrix} \alpha _ { 1 1 } & \cdots & \alpha _ { 1 n } & 0 & 0 & \cdots & 1 \\ \vdots & & \vdots & \vdots & \vdots & & \vdots \\ \alpha _ { ( m - 1 ) 1 } & \cdots & \alpha _ { ( m - 1 ) n } & 0 & 1 & \cdots & 0 \\ \alpha _ { m 1 } & \cdots & \alpha _ { m n } & 1 & 0 & \cdots & 1 \\ \end{bmatrix} , \end{align*}


\begin{align*} \begin{gathered} \operatorname { E v a l } ( N , M _ 0 + \epsilon M _ 1 ) \\ = \operatorname { E v a l } ( N , M _ 0 ) + \epsilon \sum _ { a \in N } \operatorname { E v a l } ( N _ a , M _ 0 \sqcup M _ 1 ) + \mathcal { O } ( \epsilon ^ 2 ) \ ; . \end{gathered} \end{align*}


\begin{align*} S ( n ) = \frac { 3 n + 1 } { 2 ^ v } \end{align*}


\begin{align*} ( \mathcal C ^ + - z ^ 2 ) ^ { - 1 } \left ( \frac { 1 } { 2 \pi } \int _ \R v _ \lambda \ , d \lambda \right ) = \frac { 1 } { 2 \pi } \int _ \R ( \lambda ^ 2 - z ^ 2 ) ^ { - 1 } v _ \lambda \ , d \lambda \ , . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ n \mu _ i ^ { - w _ i } \prod _ { j = 0 } ^ { w _ i - 1 } ( - \frac { w _ i } { d } D _ \lambda - \frac { 1 + \alpha _ i } { d } - j ) \omega _ \mu = \prod _ { i = 1 } ^ n \left ( - \frac { d } { w _ i } \mu _ i \right ) ^ { - w _ i } \prod _ { j = 0 } ^ { w _ i - 1 } ( D _ \lambda + \frac { 1 + \alpha _ i + d j } { w _ i } ) \omega _ \mu , \end{align*}


\begin{align*} \begin{gathered} ( A \otimes B ) _ G = A _ G \sqcup B _ G \ ; , \\ ( A \otimes B ) _ H = A _ H \hat \sqcup B _ H \ ; . \\ \end{gathered} \end{align*}


\begin{align*} n : = \max \left \{ a \in \mathbb { Z } : N _ 2 ( a ) \geq 0 \right \} . \end{align*}


\begin{align*} \lim _ { m \to \infty } A _ m ( x + i y ) = \max \{ x , 0 \} . \end{align*}


\begin{align*} \omega _ H ( \mathbf { c } ) & = \omega _ H ( ( \mathbf { u } G ' \ | \ \mathbf { u } A \ | \ \mathbf { u } \mathbf { a _ 1 } \ | \ \mathbf { u } \mathbf { a _ 2 } \ | \ \cdots \ | \ \mathbf { u } \mathbf { a _ { i } } ) ) \\ & \geq \omega _ H ( ( \mathbf { u } G ' \ | \ \mathbf { u } A ) ) \\ & \geq d , \end{align*}


\begin{align*} \begin{array} { l l } \mbox { t h e r e i s } T > 0 \mbox { a n d } q \in \N \mbox { s u c h t h a t , f o r a l l } k \in \N , \ w \in C _ h ( \Pi ) ^ n , \mbox { a n d } x \in [ 0 , 1 ] , \\ \left [ Q ^ { k q } w \right ] \left ( x , k T \right ) = 0 \mbox { w h e r e } Q = Q _ \varphi \mbox { f o r } \varphi ( x ) = w ( x , 0 ) . \end{array} \end{align*}


\begin{align*} Z = m _ 1 Z _ 1 \dot { + } \cdots \dot { + } m _ r Z _ r . \end{align*}


\begin{align*} \phi ( x ^ 0 , z , r , s ) = \varepsilon _ 1 ( x ^ 0 , r ) + \varepsilon _ 2 ( x ^ 0 , z ) + \varepsilon _ 3 ( r , s ) + \varepsilon _ 4 ( s , z ) , \end{align*}


\begin{align*} M ( A ) _ M ( i , j ) = \mathcal { H } ( A _ M ( i , j ) ) \ ; , \end{align*}


\begin{align*} \mathcal { V } ^ { 1 - } _ { 2 , \mathrm { l o c } } ( J ; E ) : = \bigcup _ { 0 < p < 1 } \mathcal { V } _ { 2 , \mathrm { l o c } } ^ { p } ( J ; E ) \ , . \end{align*}


\begin{align*} f ( \Phi ^ t _ G ( u ) ) = \sum _ { k = 0 } ^ n \frac { t ^ k } { k ! } ( A d _ G ^ k f ) ( u ) + \frac { 1 } { n ! } \int _ 0 ^ t ( t - \tau ) ^ n ( A d _ G ^ { n + 1 } f ) \left ( \Phi ^ \tau _ G ( u ) \right ) d \tau \ , \end{align*}


\begin{align*} \aleph _ { \alpha , L } = \frac { 1 } { 6 \pi } \left \langle \alpha L \arctan \left ( \frac { 1 } { \alpha L ( h _ 1 - e _ 1 ) } \right ) \right \rangle _ { x u _ 1 } , \end{align*}


\begin{align*} N _ m : = ( T _ m ( K \cdot m ) ) ^ { \perp } / ( T _ m ( K \cdot m ) \cap ( T _ m ( K \cdot m ) ^ { \perp } ) , \end{align*}


\begin{align*} d \xi _ t ^ \epsilon \ , = \ , \frac { F ( v _ \epsilon ) } { 2 \epsilon ^ { 1 - \kappa } } \ , ( 1 \ , - \ , \vert D r _ \epsilon \vert ^ 2 ) \ , d x . \end{align*}


\begin{align*} A ( { \mathbf { c } } ) ( \tau ) + A ( { \mathbf { d } } ) ( \tau ) = 0 , 0 < \tau < Z . \end{align*}


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


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


\begin{align*} d _ i + a _ { i - 1 } y _ { i - 1 } = c _ i + b _ { i + 1 } y _ i \end{align*}


\begin{align*} \left \{ D X _ t = 0 , \ \forall t \in [ l ( \omega ) , u ( \omega ) ] \right \} & \Rightarrow \left \{ \exists t \in \mathbb { Q } \cap [ 0 , 1 ] , \ D X _ t = 0 \right \} \\ & \Rightarrow \bigcup _ { t \in \mathbb { Q } \cap [ 0 , 1 ] } \left \{ D X _ t = 0 \right \} . \end{align*}


\begin{align*} \Delta _ k ( r ^ d F ( r ) ) = \sum _ { h = 0 } ^ { d } c _ { h , d } r ^ d \Delta _ { k - d } F \big ( \frac { r } { 2 ^ h } \big ) . \end{align*}


\begin{align*} \mathrm { t r a c e } ( - \hbar ^ { 2 } \Delta _ { x _ { k } } \mathrm { O P } _ { \hbar } ^ { T } ( ( 2 \pi \hbar ) ^ { 2 } \nu ) ) = \underset { \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 } } { \int } | p | ^ { 2 } \nu ( d q d p ) + \hbar . \end{align*}


\begin{align*} \chi ^ 1 _ { A _ m } ( q ) = \prod _ { i = 1 } ^ { m } ( q - i + 1 ) \end{align*}


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


\begin{align*} a _ i ^ * x a _ i & = [ M : N ] E _ M ( a _ i ^ * e _ N ) x a _ i = [ M : N ] E _ M ( a _ i ^ * e _ N x a _ i ) = [ M : N ] E _ M ( a _ i ^ * e _ N \gamma _ 0 ( x ) a _ i ) \ \ \ ( \textnormal { L e m m a \ref { l : m e } } ) \\ & = [ M : N ] E _ M ( a _ i ^ * e _ N a _ i \gamma _ 0 ( x ) ) \ \ \ ( \gamma _ 0 ( x ) \in M ' \cap M _ 1 ) , \end{align*}


\begin{align*} \| U ( t , s _ 0 ) \| \leq C = C e ^ { \alpha ( t - s _ 0 ) } e ^ { - \alpha ( t - s _ 0 ) } \leq C e ^ { \alpha T } e ^ { - \alpha ( t - s _ 0 ) } = K e ^ { - \alpha ( t - s _ 0 ) } , \end{align*}


\begin{align*} d X ^ 0 ( t ) = \sigma ( X ^ 0 ( t ) ) \circ d \beta ( t ) \end{align*}


\begin{align*} & - 2 \int _ 0 ^ t \operatorname { I m } \langle ( 1 - \mathfrak { s } \kappa | z | ^ 2 ) z , z \d W ( s ) \rangle _ { L _ x ^ 2 } - 2 \mathfrak { s } \int _ 0 ^ t \operatorname { I m } \langle \partial _ x z , \partial _ x ( z \d W ( s ) ) \rangle _ { L _ x ^ 2 } \\ = & - 2 \mathfrak { s } \int _ 0 ^ t \operatorname { I m } \langle \partial _ x z , z \partial _ x \d W ( s ) \rangle _ { L _ x ^ 2 } , \end{align*}


\begin{align*} \det ( y _ 0 1 _ g + y _ 1 \lambda ^ { - 1 } D _ 1 ) = \lambda ^ { - g } \Big ( \sum _ { i = 0 } ^ g y _ 0 ^ { g - i } y _ 1 ^ i \Theta _ i ( \lambda 1 _ g , D _ 1 ) \Big ) , \end{align*}


\begin{align*} \{ g \} ( y , Z _ N ) : = \rho \Big ( \frac { | Z _ N | } { r _ { \perp } } \Big ) \cdot g ( y ) \cdot Z _ N ^ { \otimes k } , \end{align*}


\begin{align*} \inf \{ 2 t \in \mathbb { N } \colon 1 + q _ x + \cdots + q _ x ^ t = 0 , \ t < d \} \leq 2 s \end{align*}


\begin{align*} e _ 1 ( Z , \overline { L } ) \ge h _ { \overline { L } } ( Z ) \ge \frac { \sum _ { i = 1 } ^ { \dim Z + 1 } e _ i ( Z , \overline { L } ) } { \dim Z + 1 } . \end{align*}


\begin{align*} \delta ' _ k : = \min _ { i \in I } \delta ' _ { k , i } \in \Q _ { > 0 } , \Delta ' : = \sum _ k \delta ' _ k \Delta _ k . \end{align*}


\begin{align*} | \nabla f | ^ 2 + \frac { 2 a } { f } = b , \end{align*}


\begin{align*} \int _ { t - a ^ { - 1 } } ^ { \infty } \left ( x - t \right ) d \mu ( x ) = 0 \end{align*}


\begin{align*} \frac { d x _ i } { d t } = \underbrace { \left ( x _ { i + 1 } - x _ { i - 2 } \right ) x _ { i - 1 } } _ { } - \underbrace { \mathrel { \phantom { \left ( x _ i \right ) } } x _ { i } \mathrel { \phantom { \left ( x _ i \right ) } } } _ { d a m p i n g } + \underbrace { \mathrel { \phantom { \left ( x _ i \right ) } } F \mathrel { \phantom { \left ( x _ i \right ) } } } _ { } \end{align*}


\begin{align*} \Q \pi ( U ; x , y ) ^ \wedge : = \varprojlim _ n \Q \pi ( U ; x , y ) / I ^ n \pi ( U ; x , y ) . \end{align*}


\begin{align*} \norm { ( I - C C ^ + ) A } & \le \norm { A ( I - \mathbb { P } ) } = \norm { A ( I - V V ^ T ) ( I - \mathbb { P } ) } \\ & \le \norm { ( V ^ T \ ! P ) ^ { - 1 } } \ , \norm { A ( I - V V ^ T ) } , \\ \norm { A ( I - R ^ + \ ! R ) } & \le \norm { ( I - \mathbb { S } ) A } = \norm { ( I - \mathbb { S } ) ( I - U U ^ T ) A } \\ & \le \norm { ( S ^ T U ) ^ { - 1 } } \ , \norm { ( I - U U ^ T ) A } , \end{align*}


\begin{align*} T _ { 2 n } = \underbrace { - 0 + T _ 0 } _ { \Delta T _ 0 } + \underbrace { - T _ 0 + T _ 1 } _ { \Delta T _ 1 } + \underbrace { - T _ 1 + T _ 2 } _ { \Delta T _ 2 } + \dots + \underbrace { - T _ { 2 n - 1 } + T _ { 2 n } } _ { \Delta T _ { 2 n } } = T _ 0 + \sum _ { i = 0 } ^ { 2 n } \Delta T _ i , \end{align*}


\begin{align*} ( \gamma . L _ P ) = ( k ( \gamma ) , P ) , \end{align*}


\begin{align*} \mp \partial _ \mu p ^ { e _ \pm } ( b , T , \mp \mu ) = n ^ { e _ \pm } ( b , T , \mp \mu ) \end{align*}


\begin{align*} x \to y : = \left \{ \begin{array} { l } x * y \mbox { i f } y \le x , \\ 1 _ x \mbox { i f } x < y , \\ y \mbox { o t h e r w i s e . } \\ \end{array} \right . \end{align*}


\begin{align*} \mathrm { p r o j } _ { m / m - 1 } ( \mathcal { L } _ { T , F _ m } ^ { G _ m ^ \prime } ( z _ m ) ) = \mathcal { L } _ { T , F _ { m - 1 } } ^ { G _ { m - 1 } ^ \prime } ( z _ { m - 1 } ) . \end{align*}


\begin{align*} ( y ^ 2 \frac { \partial ^ 2 } { \partial y ^ 2 } - \lambda _ s ) u = 0 \end{align*}


\begin{align*} \langle \epsilon _ { J } , \mathbb { X } _ { t } \rangle = \langle \epsilon _ { J } ^ \P , \mathbb { Y } _ { t } \rangle \end{align*}


\begin{align*} & Z = \left [ \begin{array} { c c } M & 0 _ { ( m + n ) \times m ( L - 1 ) } \\ \hline 0 _ { m ( L - 1 ) \times m } & T \otimes I _ m \end{array} \right ] \\ & = \begin{bmatrix} M _ n & M _ { n - 1 } & \dots & M _ 1 & M _ 0 & 0 & \dots & 0 \\ 0 & d _ n I & d _ { n - 1 } I & \dots & d _ 1 I & d _ 0 I & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & 0 \\ 0 & \dots & 0 & d _ n I & d _ { n - 1 } I & \dots & d _ 1 I & d _ 0 I \end{bmatrix} , \end{align*}


\begin{align*} b _ \epsilon ( t , \cdot ) \in C ^ \infty ( \R ^ n ; \R ^ n ) D _ x b _ \epsilon ( t , x ) = \ , \int _ { \R ^ n } D _ x \rho _ \epsilon ( x - y ) \ , \tilde b _ \epsilon ( t , y ) \ , d y \quad \forall ( t , x ) \in I \times \R ^ n \ , , \end{align*}


\begin{align*} e ^ B - e ^ A = \int _ { 0 } ^ 1 e ^ { \alpha B } ( B - A ) e ^ { ( 1 - \alpha ) A } \ d \alpha . \end{align*}


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


\begin{align*} - K _ { X ' } - D _ \epsilon = \frac { 1 } { 1 + \epsilon } ( \mu ^ * ( - K _ X - D ) + \epsilon H ) = : L _ \epsilon . \end{align*}


\begin{align*} \tilde { M } : = R M = \left ( \begin{matrix} U ^ T A \ & U ^ T B \\ C & D \end{matrix} \right ) , \end{align*}


\begin{align*} x = x '' = ( \alpha _ 1 '' , . . . , \alpha _ m '' ) , \ \ \ \mbox { a n d } \ \ \ A = \bigcap \limits _ { i = 1 } ^ m \Omega _ i ^ { * , 3 } , \ \ \ \mbox { r e s p e c t i v e l y } . \end{align*}


\begin{align*} \theta _ x ( \underline { \lambda } \times \underline { \lambda } ' ) = \prod _ { i = 1 } ^ { n - 1 } \xi ( \lambda _ i \kappa _ i ( x ) ) \cdot \prod _ { i = 1 \atop w ( i + 1 ) < w ( i ) } ^ { n - 1 } \xi ( \lambda _ i ' \kappa _ i ' ( x ) ) . \end{align*}


\begin{align*} a _ { i j } ^ \varepsilon & = k _ d \delta _ { i j } , b _ i ^ \varepsilon = - 2 k _ d \partial _ i \lambda , c ^ \varepsilon = \partial _ t \lambda - k _ d ( \Delta \lambda + | \nabla \lambda | ^ 2 ) \quad Q _ { \varepsilon , T } , \\ \beta ^ \varepsilon & = \nu _ \varepsilon \cdot \nabla \lambda + k _ d ^ { - 1 } V _ \varepsilon \quad \partial _ \ell Q _ { \varepsilon , T } , \end{align*}


\begin{align*} Q _ { i , j } : = R _ { i , j } ( [ 0 , A r _ i ] \times [ 0 , r _ i ^ 2 ] ) . \end{align*}


\begin{align*} f - f _ n = \sum _ { a _ 1 , b _ 1 , \ldots a _ k , b _ k = - n } ^ n \sum _ { i = 1 } ^ k \Big ( g _ i \otimes h _ i - \frac { a _ i b _ i } { n ^ 2 } g \otimes h \Big ) 1 _ { A _ { a _ 1 \ldots a _ k n } \times B _ { b _ 1 \ldots b _ k n } } , \end{align*}


\begin{align*} \lambda \mathcal { B } ( u ) = ( \lambda _ 1 B _ 1 ( u _ 1 ) , \lambda _ 2 B _ 2 ( u _ 2 ) ) . \end{align*}


\begin{align*} \mathbb { W } _ { s , t } = \frac { 1 } { 2 } W _ { s , t } W _ { s , t } + \mathbb { L } _ { s , t } \ , , \end{align*}


\begin{align*} \sum _ { \substack { 0 < \gamma , \gamma ' \le T \\ 0 < | \gamma - \gamma ' | \le \frac { 2 \pi \lambda } { \log T } } } \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! 1 \ = \ 2 \sum _ { 0 < \gamma \le T } n ( \gamma , \lambda ) \ = \ 2 \sum _ { 0 < \gamma _ d \le T } m ( \gamma _ d ) \ , n ( \gamma _ d , \lambda ) . \end{align*}


\begin{align*} ( M _ V ) _ j ^ { j ' } ( \ell ) = V _ { \ell , j - j ' } = \begin{cases} V ( \ell ) & \mbox { i f } j - j ' = \pi ( \ell ) \\ 0 & \mbox { o t h e r w i s e } \end{cases} \ , \end{align*}


\begin{align*} S = \bigsqcup _ { \jmath \in J ( S ) } S _ \jmath . \end{align*}


\begin{align*} \frac { d } { d t } w _ n + \nu A w _ n + B ( w _ n , w _ n ) + D B ( u ) w _ n = g _ { n - 1 } - \mu P _ N w _ n , w _ n ( t _ { n - 1 } ) = v _ n ^ 0 - u ( t _ { n - 1 } ) , \end{align*}


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


\begin{align*} \lim _ { m \to \infty } G _ \delta ( z / m ) = \mathbf { 1 } _ { \mathrm { R e } ( z ) > 0 , \delta = 0 } - \mathbf { 1 } _ { \mathrm { R e } ( z ) < 0 , \delta = 1 } \end{align*}


\begin{align*} \begin{cases} 0 = \theta ( x u _ w + y v _ w ) = \theta ( u _ w ) x + \theta ( v _ w ) y \\ 0 = \eta _ a ( x u _ w + y v _ w ) = \eta _ a ( u _ w ) x + \eta _ a ( v _ w ) y \end{cases} \end{align*}


\begin{align*} e _ k = & | \upsilon _ k | ^ 2 \big ( \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 \big ) \\ [ - 0 . 5 e m ] & - 2 \mathrm { R e } \{ \upsilon _ k ^ * \boldsymbol { \theta } _ i ^ T \mathrm { d i a g } ( \mathbf { h } _ k ^ H ) \mathbf { G } \mathbf { w } _ k \} + 1 , \end{align*}


\begin{align*} \overline { R } _ { r , r + 2 n } & = 2 \varphi _ { n } - \varphi _ { 2 n } + \overline { R } _ { r , r + n } + \overline { R } _ { r + n , r + 2 n } - V _ { r , r + n , r + 2 n } \ , . \end{align*}


\begin{align*} \begin{dcases} d ^ \star ( j \circ u ) ( t ) = A ^ { \odot \star } ( t ) j u ( t ) + f ( t ) , & t \geq s , \\ u ( s ) = \varphi , & \varphi \in X , \end{dcases} \end{align*}


\begin{align*} _ 2 F _ 1 ( a , b ; c ; x ) = H ( c ) \bullet ~ _ 2 F _ 1 ( a , b ; c + 1 ; x ) \end{align*}


\begin{align*} - \Delta _ { p _ { i } ( x ) } \xi _ { i } ( x ) = 1 \Omega , \xi _ { i } ( x ) = 0 \partial \Omega \end{align*}


\begin{align*} d J _ { t \leftarrow 0 } = \sum _ { i = 1 } ^ { d } D V _ i ( Y _ t ) J _ { t \leftarrow 0 } d X _ t + D V _ 0 ( Y _ t ) J _ { t \leftarrow 0 } d t , \ J _ { 0 \leftarrow 0 } = I _ { d \times d } . \end{align*}


\begin{align*} x \to y : = \max \{ z * y \colon x , y \le z \} . \end{align*}


\begin{align*} u _ { N } ( x , t ) = \sum _ { m = - 1 } ^ { N } \delta _ { m } ( t ) Q _ { m } ( x ) , \end{align*}


\begin{align*} \frac { \partial X } { \partial r } & = A ( X ) , X ( 0 , \mu ) = X _ 0 ( \mu ) , \\ \frac { \partial Y } { \partial r } & = B ( Y ) , Y ( 0 , \mu ) = Y _ 0 ( \mu ) , \end{align*}


\begin{align*} \chi ( + i 0 ) + \chi ( - i 0 ) = \frac { 1 } { i \pi } { \rm p . v . } \int _ { - \infty } ^ { \infty } \frac { { \rm l o g } \frac { s ^ 2 } { 1 + s ^ 2 } \left ( 1 - b ^ 2 ( s ) \right ) } { s } d s . \end{align*}


\begin{align*} \begin{cases} ( \partial _ t ^ 2 + P ) u = 0 \\ u | _ { t = T } = 0 , \partial _ t u | _ { t = T } = \phi . \end{cases} \end{align*}


\begin{align*} \alpha ( A ) ( ( x , y ) ) = A ( ( x , \bar { \Phi } _ \times ^ \alpha ( y ) ) ) \ ; , \end{align*}


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


\begin{align*} \kappa _ { 2 ; 0 } = \sigma ^ 2 \ , , \kappa _ { 3 ; 0 } = \alpha _ 3 \end{align*}


\begin{align*} ( \alpha ( t - r , x ) , x ) = ( \alpha ( - r , \alpha ( t , x ) ) , \alpha ( - r , \alpha ( r , x ) ) ) \in U \ , , \end{align*}


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


\begin{align*} f = c _ P f = \int _ { ( N \cap \Gamma ) \backslash N } f ( n x ) d n . \end{align*}


\begin{align*} D _ r ^ B Y _ t = \tilde Y _ t = & D _ r ^ B \xi + \int _ t ^ T \tilde Y _ s { { W } } ( d s , B _ s ) \\ & + \int _ r ^ T Y _ s \nabla _ x { { W } } ( d s , B _ s ) - \int _ t ^ T \tilde Z _ s d B _ s , \ 0 \leq t \leq r \leq T . \end{align*}


\begin{align*} N _ x \circ \mathcal { H } _ x | _ { \mathcal { W } ^ { c u } ( x ) } = \mathcal { H } _ { f ( x ) } \circ f | _ { \mathcal { W } ^ { c u } ( x ) } , N _ x \colon ( t , s ) \mapsto ( \lambda _ x ^ u t , \lambda _ x ^ c s ) . \end{align*}


\begin{align*} w > 0 o n [ 0 , T ] \times [ 0 , X ] \times \{ \eta = 0 \} . \end{align*}


\begin{align*} \Gamma _ F ( f ) = - F , x > 0 . \end{align*}


\begin{align*} \lim _ { m \to \infty , m } ( - 1 ) ^ { \lfloor s m \rfloor } \frac { 1 } { m } \sum _ { j = 0 } ^ { m - 1 } \frac { e ^ { - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \lfloor s m \rfloor } } { 1 + \exp \left \{ 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \right \} + \frac { z } { m } } = ( - 1 ) ^ { \theta _ 1 \mathbf { 1 } _ { s < 0 } } \frac { e ^ { - z ( s + \mathbf { 1 } _ { s < 0 } ) } } { 1 - ( - 1 ) ^ { \theta _ 1 } e ^ { - z } } . \end{align*}


\begin{align*} _ n ( k ) \cdot A _ r = \{ M \in _ { n \times n } ( k ) \ , | \ , M ^ 2 = M , \ ( M ) = r \} . \end{align*}


\begin{align*} \bigl \langle \partial _ t \bigl ( u _ { \delta , \lambda } ( t ) - m _ 0 \bigr ) , 1 \bigr \rangle _ { V ^ * , V } & = \frac { d } { d t } \int _ \Omega u _ { \delta , \lambda } ( t ) \ , d x = 0 , \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 4 } & A x _ e + B ( K ^ * K ) ^ { - 1 } B ^ * w - B ( K ^ * K ) ^ { - 1 } v = 0 , \\ & C ^ * C x _ e + z - A ^ * w = 0 . \end{alignedat} \right . \end{align*}


\begin{align*} F _ 0 ( D ^ 2 H _ m ) = 0 , \end{align*}


\begin{align*} G \sim G \circ ( F \circ G ' ) = ( G \circ F ) \circ G ' \sim G ' \end{align*}


\begin{align*} \partial ^ \circ \eta ( y , t ) = \partial _ t \tilde { \eta } ( y , t ) + ( V _ \Gamma \nu ) ( y , t ) \cdot \nabla \tilde { \eta } ( y , t ) , ( y , t ) \in \overline { S _ T } \end{align*}


\begin{align*} \overline g | _ { U ' } = d x ^ 2 + x ^ 2 g _ F ( z ) + g _ Y ( y ) , U ' = U \backslash Y , \end{align*}


\begin{align*} x \to y : = \max \{ u \colon ( u ] \cap L ( ( [ x ) \cap [ y ) ) \cap [ y ) = \{ y \} \} , \end{align*}


\begin{align*} \| d z _ N ^ { \odot \beta } \| ^ 2 = 2 ^ k \cdot \frac { \beta ! } { k ! } . \end{align*}


\begin{align*} P ( x , t ) = \sum _ { i = 1 } ^ { k } H _ i ( x , t ) , ~ H _ i \mbox { i s a n } i \mbox { - f o r m } \end{align*}


\begin{align*} \mathfrak { W } ( f _ k ) = m \leq k . \end{align*}


\begin{align*} \lim \limits _ { y \to \infty } ( \rho , u , \theta , h ) = ( \rho ^ 0 , u ^ 0 , \theta ^ 0 , h ^ 0 ) ( t , x ) . \end{align*}


\begin{align*} \overline { N _ T } = \bigcup _ { t \in [ 0 , T ] } \overline { N ( t ) } \times \{ t \} . \end{align*}


\begin{align*} \nu : \R ^ d \to \R _ + , p \mapsto \nu ( p ) : = \frac { 1 } { 2 } p ^ 2 , \end{align*}


\begin{align*} \partial _ t p _ { m + 1 } - \gamma ( p _ { m + 1 } + \frac { 1 } { m } ) \Delta p _ { m + 1 } - | \nabla p _ { m + 1 } | ^ 2 = \gamma p _ { m + 1 } G ^ m , \end{align*}


\begin{align*} q \left ( x , t \right ) = - 2 \partial _ { x } \int y \left ( \kappa , x , t \right ) \mathrm { d } \sigma \left ( \kappa \right ) \end{align*}


\begin{align*} S ^ { F ^ \nu } = \bigsqcup _ { \jmath \in J ( S ) ^ F } S _ \jmath ^ { F ^ \nu } . \end{align*}


\begin{align*} \deg \Psi _ \alpha ( t ) \leq - \overline { \rm r k } \ , G _ \iota = - \overline { \rm r k } \ , G = - n . \end{align*}


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


\begin{align*} Y \left ( \alpha \right ) + \int _ { 0 } ^ { 1 } 2 s \sqrt { 1 - s ^ { 2 } } \frac { \mathrm { e } ^ { 8 s ^ { 3 } t - 2 s x } } { s + \alpha } Y \left ( s \right ) \mathrm { d } s = 1 , \ \ \ \alpha \in \left [ 0 , 1 \right ] , \end{align*}


\begin{align*} \alpha ( G _ A ) = \min _ { \emptyset \neq I \subseteq \{ 1 , \ldots , n \} } \frac { \dim G _ { A _ I } } { 2 \dim A _ I } = \frac { 1 } { \gamma _ { A } } . \end{align*}


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


\begin{align*} \Phi _ t \circ Q ^ { \pi _ 0 } = Q ^ { \pi ( t ) } \circ \Phi _ t . \end{align*}


\begin{align*} 4 = \int _ { B _ { \omega , q } ^ { ( j ) } } | \Delta ' _ { \omega , q } ( E ) | \ , d E \leq m ( B _ { \omega , q } ^ { ( j ) } ) \max _ { E \in B _ { \omega , q } ^ { ( j ) } } | \Delta ' _ { \omega , q } ( E ) | \end{align*}


\begin{align*} H ^ 0 ( X _ P , \mathcal { O } _ { X _ P } ( - K _ { X _ P } ) ) = \bigoplus _ { m \in P \cap M } \mathbb { C } \chi ^ m . \end{align*}


\begin{align*} H = - \frac { 1 } { 2 } \sum _ { \{ x , y \} \subset \Lambda } J _ { x , y } \sigma _ x \sigma _ y + h \sum _ { x \in \Lambda } \sigma _ x , \end{align*}


\begin{align*} \frac { n ! } { \prod _ { 1 \le i \le n } i ! ^ { N _ i } } \cdot \prod _ { 1 \le i \le n } \frac { C _ i ^ { N _ i } } { N _ i ! } = n ! \prod _ { 1 \le i \le n } \frac { ( C _ i / i ! ) ^ { N _ i } } { N _ i ! } ; \end{align*}


\begin{align*} \mathcal { I } ( \Pi _ b ) \coloneqq \lim _ { L \to \infty } \frac { 1 } { | S _ L | } { \rm T r } ( \chi _ L \Pi _ b ) = \int _ \Omega { \rm t r } _ { \C ^ 2 } \Pi _ b ( { x } , { x } ) \d { x } , \end{align*}


\begin{align*} \begin{array} { r r } u _ j ( x , t ) = [ Q u ] _ j ( x , t ) \end{array} \end{align*}


\begin{align*} g \geq 0 o n ( \{ \tau = 0 \} \cup \{ \xi = 0 \} \cup \{ \eta = 1 \} ) \cap \overline { D _ { T ^ * } } \end{align*}


\begin{align*} \widetilde { P } _ k = \begin{pmatrix} 1 & 1 & \cdots & 1 \\ p _ { 1 0 } & p _ { 1 1 } & \cdots & p _ { 1 \ell _ k } \\ p _ { 2 0 } & p _ { 2 1 } & \cdots & p _ { 2 \ell _ k } \\ p _ { 3 0 } & p _ { 3 1 } & \cdots & p _ { 3 \ell _ k } \end{pmatrix} \end{align*}


\begin{align*} \left \{ \begin{aligned} \frac { \partial } { \partial \ , t } u ^ { T , n } + \mathcal { L } ( x , y , D u ^ { T , n } , D ^ { 2 } u ^ { T , n } ) = 0 , & [ 0 , T ) \times D _ { n } , \\ u ^ { T , n } ( T , y ) = g ( x , y ) , & D _ { n } , \\ u ^ { t , n } ( t , y ) = 0 , & [ 0 , T ] \times \partial D _ { n } , \end{aligned} \right . \end{align*}


\begin{align*} c _ P \tilde E _ { a , s } = \begin{cases} A _ s y ^ s + B _ s y ^ { 1 - s } & ( 0 < y < a ) \\ y ^ s & ( y > a ) . \end{cases} \end{align*}


\begin{align*} I _ { M N } = 1 + \sum _ { d = 1 } ^ \infty \frac { z ^ d } { d ! d ! } N ^ { 2 d } \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } s _ \lambda ( a _ 1 , \dots , a _ N ) s _ \lambda ( b _ 1 , \dots , b _ N ) \frac { ( \dim \mathsf { V } ^ \lambda ) ^ 2 } { ( \dim \mathsf { W } _ M ^ \lambda ) ( \dim \mathsf { W } _ N ^ \lambda ) } . \end{align*}


\begin{align*} t ^ * ( w ) = - \frac { a - b } { \pi } \log { \lvert w \rvert } + \frac { a } { \pi } \log { \lvert - \varepsilon - w \rvert } - \frac { b } { \pi } \log { \lvert \varepsilon + w \rvert } + P ^ * _ W + c , \end{align*}


\begin{align*} [ u \times ( u \times A \partial _ x u ) ] \cdot \partial _ x u = [ u ( A \partial _ x u \cdot u ) - A \partial _ x u ] \cdot \partial _ x u = - A \partial _ x u \cdot \partial _ x u \ , . \end{align*}


\begin{align*} ( i d \otimes r ) ( r \otimes i d ) ( i d \otimes r ) = ( r \otimes i d ) ( i d \otimes r ) ( r \otimes i d ) . \end{align*}


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


\begin{align*} \log ( N ! ) = N \log ( N ) - N + O ( \log ( N ) ) . \end{align*}


\begin{align*} F _ \xi ( x , 0 ) = x . \end{align*}


\begin{align*} T ^ \omega _ { s , t } b ( x ) = \int _ s ^ t b ( x + \omega _ r ) \ , d r = b \star \mu _ { s , t } ^ { - \omega } . \end{align*}


\begin{align*} x _ 4 ^ { n - 2 } = \lambda . \end{align*}


\begin{align*} \mathcal E _ 0 ( \mathrm { i d } , r _ 0 , y _ 0 ) : = \inf _ { m _ { \kappa _ n } } \liminf _ { n \to \infty } { E _ { \kappa _ n } ( m _ { \kappa _ n } ) - 8 \pi \over \kappa _ n ^ 2 } \leq \lim _ { n \to \infty } { E _ { \kappa _ n } ( \phi _ n ) - 8 \pi \over \kappa _ n ^ 2 } = - { 2 \pi r _ 0 ^ 2 \over y _ 0 ^ 2 } - 8 \pi r _ 0 , \end{align*}


\begin{align*} & H ( q , u _ 0 , v ) : = \left | \prod _ { j = 1 } ^ n \widehat { B } ( - e ^ { - i \vartheta } [ M _ A ( v ) ] _ j ) \widehat { \overline { \psi } } _ 1 ( e ^ { i \vartheta } u _ 0 ) \widehat { \psi } _ 2 \left ( e ^ { - i \vartheta } u _ 0 \right ) \right | g ( q , u _ 0 , v ) , \end{align*}


\begin{gather*} \epsilon _ i \colon = \begin{cases} + & \textrm { i f $ i - 1 \xrightarrow { a _ i } i $ } , \\ - & \textrm { i f $ i - 1 \xleftarrow { a _ i } i $ } . \end{cases} \end{gather*}


\begin{align*} | \nu | ^ 2 = 1 , \underline { D } _ j \nu _ i = - W _ { j i } , \sum _ { i = 1 } ^ n \underline { D } _ i \nu _ i = \mathrm { d i v } _ \Gamma \nu = - H \quad \overline { S _ T } , \end{align*}


\begin{align*} \lambda _ { k + 1 , D } \left ( \tau + 1 , w + 1 \right ) = \lambda _ { k + 1 , D } \left ( \tau , w \right ) , \lambda _ { k + 1 , D } \left ( - \frac { 1 } { \tau } , - \frac { 1 } { w } \right ) = \tau ^ { 2 k + 2 } \lambda _ { k + 1 , D } \left ( \tau , w \right ) . \end{align*}


\begin{align*} \begin{dcases} d ^ \star ( j \circ u ) ( t ) = A _ 0 ^ { \odot \star } j u ( t ) + B ( t ) u ( t ) + f ( t ) , & t \geq s , \\ u ( s ) = \varphi , & \varphi \in X , \end{dcases} \end{align*}


\begin{align*} X _ n : = Z _ n + Y _ n , \end{align*}


\begin{align*} M ( \alpha , \Phi , \mathcal { U } ) = \lim _ { n \rightarrow \infty } \inf _ { \Gamma } \sum _ { U \in \Gamma } \exp ( - \alpha m ( U ) + \varphi ( U ) ) \end{align*}


\begin{align*} \mathrm { t r } ( \ell _ i ) = 1 \ ; \ ; i = 0 , \ldots , d - 1 ; \mathrm { t r } ( a ) = 0 \ ; \deg ( a ) \ne 2 . \end{align*}