\begin{align*} [ t ^ k ] \circ \mathcal { T } _ { \bold { c } } ( t ^ m ) & = \dfrac { t ^ { k + m } } { c _ m } = \dfrac { 1 } { c _ { m + k } } \dfrac { A ( m + k - 1 ) \cdots A ( m ) } { B ( m + k ) \cdots B ( m + 1 ) } t ^ { k + m } \\ & = \mathcal { T } _ { \bold { c } } \circ A ( \theta _ t - 1 ) \circ \cdots \circ A ( \theta _ t - k ) \circ B ( \theta _ t ) ^ { - 1 } \circ \cdots \circ B ( \theta _ t - k + 1 ) ^ { - 1 } \circ [ t ^ k ] ( t ^ m ) \enspace . \end{align*}


\begin{align*} \mathbb { E } ^ 2 \Big ( \sum _ { 1 \leq i \leq n } { X _ i } \Big ) = \Big ( \sum _ { 1 \leq i \leq n } f _ k ( i ) \Big ) ^ 2 + O _ { \varepsilon } ( n ^ { 3 / 2 + \varepsilon } ) , \end{align*}


\begin{align*} F _ { \nu _ { n } } ( H _ { \nu _ { n } } ( s + i t ) ) = s + i t , s + i t \in Q , \ , t > f _ { n } ( s ) , \ , n \ge N . \end{align*}


\begin{align*} f = b _ 0 + b _ 1 \phi + \cdots + b _ s \phi ^ s + \cdots , b _ s \in \Z _ p [ x ] . \end{align*}


\begin{align*} 1 / g ( t ) = \begin{cases} \frac { C } { t ^ { 3 / 2 } } & 1 \leq t \leq f ( n ) ^ 2 \\ \frac { C } { t f ( n ) } & f ( n ) ^ 2 \leq t \leq D ^ 2 . \end{cases} \end{align*}


\begin{align*} 0 \leq \gamma _ t \leq \begin{cases} \frac { 1 } { t m ^ 2 m _ t ^ 2 } & d = 2 , \\ c \log \Big ( 1 + \frac { 1 } { m ^ 2 t } \Big ) & d = 3 , \end{cases} \end{align*}


\begin{align*} \Sigma _ \omega : = \{ \sigma _ I | \overline { I } \in \mathcal A _ \omega \} . \end{align*}


\begin{align*} f ^ { \tilde { \phi } ^ { - n } } \circ \rho _ { \eta } ( X ) = \Pi _ { w \in W _ { \rho } ^ { m _ 0 } } f ^ { \tilde { \phi } ^ { - ( n + 1 ) } } ( X + w ) . \end{align*}


\begin{align*} \mu \int _ { \Omega } \varepsilon z : \varepsilon v + \nu \int _ { \Omega } \mathsf { m } ' \left ( \varepsilon y ; \varepsilon z \right ) : \varepsilon v = \int _ { \Omega } h \cdot v , \forall v \in Y , \end{align*}


\begin{align*} K ^ 2 ( z , w ) B _ m ^ { - 2 } ( z , w ) \Big ( \ ! \ ! \Big ( \ , \partial _ i \bar { \partial } _ j \log B _ m ( z , w ) \Big ) \ ! \ ! \Big ) _ { i , j = 1 } ^ m \preceq B _ m ^ { - 2 } ( z , w ) \mathcal G _ { K ^ { - 1 } } ( z , w ) . \end{align*}


\begin{align*} E ^ { T , \Upsilon , Y } \cup Z ^ a \big [ T _ 1 , . . . , T _ n \big ] = \bigg ( \bigcup _ { i = 1 } ^ n E ^ { T _ i , \Upsilon _ i , Y _ i } \bigg ) \cup \Big \{ \tilde { \phi } ^ { T , \Upsilon , Y } \big [ \tilde { h } _ j ^ { Y } \big ] : \tilde { h } _ j ^ { Y } \in Z ^ a \big [ Y _ 1 , . . . , Y _ n \big ] \Big \} . \end{align*}


\begin{align*} \overline { G } _ { ( k ) } ( u ) = \int _ { u } ^ { \infty } \overline { G } _ { ( k - 1 ) } ( s ) \mathrm { d } s , ~ k \geq 2 . \end{align*}


\begin{align*} w _ { n , i } ^ * & \in \left ( Q _ { a _ { n , i } } ( \alpha - 1 , A _ i , u _ i + a _ i f _ { q ( i ) } ) \right ) ^ { ( \alpha - 1 ) } \\ & \subseteq \left ( \bigcap _ { j = 1 } ^ \infty K ( \alpha - 1 , A _ j , u _ j + a _ j f _ { q ( j ) } ) \right ) ^ { ( \alpha - 1 ) } = K ^ { ( \alpha - 1 ) } . \end{align*}


\begin{align*} p ( t , v , g ) = \rho ( t , v ) \frac { 1 } { \sqrt { 2 \pi a ( t ) } } \exp \left ( - \frac { ( g - g _ { } ( t ) ) ^ 2 } { 2 a ( t ) } \right ) . \end{align*}


\begin{align*} H f ( x ) : = \frac { 1 } { x } \int _ 0 ^ x f ( t ) d t \end{align*}


\begin{align*} ( - \Delta ) ^ s u + \sum _ { | \alpha | \leq m } a _ \alpha D ^ \alpha u & = 0 \mbox { i n } \ ; \ ; \Omega , u - f \in \widetilde H ^ s ( \Omega ) \end{align*}


\begin{align*} \lim _ { r \to 0 } \frac { 1 } { | B _ r ( 0 ) | } \int _ { B _ r ( x ) } f ( y ) \ , d y = f ( x ) . \end{align*}


\begin{align*} \widehat { u } ( x , t ) = u ( \widehat { x } , t ) , \widehat { x } = x - 2 < x , \vec { a } > \vec { a } . \end{align*}


\begin{align*} \Psi ^ { n , 4 } _ s = 2 \sigma ( X _ s ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) \left ( \int ^ s _ { \eta _ n ( s ) } \left ( s - \eta _ n ( u ) \right ) ^ { \alpha } \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) , \end{align*}


\begin{align*} A _ 1 ( t ) = \psi ^ { - 1 } \partial _ t \psi = ( I - \kappa e ^ { - a ( t , s ) } ) ^ { - 1 } \partial _ t a ( t , s ) \end{align*}


\begin{align*} P _ \delta ( E ^ \parallel \times E ^ \perp ) = P ^ \parallel _ \delta ( E ^ \parallel ) \otimes P ^ \perp _ \delta ( E ^ \perp ) \end{align*}


\begin{align*} f _ i \circ S ^ j = f _ { i - \ell } i \in \Z . \end{align*}


\begin{align*} d \Theta _ n ( e _ \lambda ) = e _ { \theta ( \lambda ) } . \end{align*}


\begin{align*} | y ^ * ( u _ n ) | = a _ n | f _ { q ( n ) } ( y ^ * ) | \leq C a _ n \end{align*}


\begin{align*} \begin{array} { c c c c c } \mathcal { M } _ { A , q } & : & H ^ 1 ( D ) & \rightarrow & H ^ 2 _ { \mathrm { l o c } } ( \R ^ 3 ) , \\ & & v & \mapsto & \mathcal { M } _ { A , q } v : = u _ { A , q } ^ s \end{array} \end{align*}


\begin{align*} d _ { n - 1 } ( { \mathcal K } , N ) _ H \geq C '' \frac { 1 } { [ \log _ 2 ( n + \lceil \log _ 2 N \rceil ) ] ^ \alpha } , n = 1 , 2 , \ldots . \end{align*}


\begin{align*} \label [ I e q ] { e q : b o l t z m a n n g e n e r a t o r } \mathcal { L } _ N \varphi _ N = \sum _ { i = 1 } ^ N L ^ { ( 1 ) } \diamond _ i \varphi _ N + \frac { 1 } { N } \sum _ { i < j } L ^ { ( 2 ) } \diamond _ { i j } \varphi _ N , \end{align*}


\begin{align*} A _ 1 ( t ) = \psi ^ { - 1 } \partial _ t \psi = ( I + \kappa U ( s , t ) ) \partial _ t { \rm L o g } ( U ( t , s ) + \kappa I ) \end{align*}


\begin{align*} g _ i = t \tilde { g } _ i . \end{align*}


\begin{align*} c _ k : = \int _ { \mathbb { R } } \int _ { 0 } ^ { V _ F } p _ { } ( v , g ) e ^ { - i k v \frac { 2 \pi } { V _ F } } d v , k \in \mathbb { Z } . \end{align*}


\begin{align*} E _ { \mbox { \tiny M D } } ( \theta ) = \sup _ { \lambda \ge 0 } \sup _ { P \le P _ w } \left \{ \lambda ( \sqrt { P _ s P } - \theta ) - \tilde { C } _ \lambda ( P ) - \frac { \lambda ^ 2 \sigma _ N ^ 2 P } { 2 } \right \} . \end{align*}


\begin{align*} \tau _ j : = \eta _ \varepsilon \rho _ j \eta _ \varepsilon : = ( 1 - \varepsilon ) \frac { \alpha _ { \min } - \Lambda _ 0 } { 2 \Lambda _ 1 ( 1 + \delta _ { \min } ^ { - 1 } ) } \end{align*}


\begin{align*} m = n _ { j _ 1 } + \ldots + n _ { j _ s } , 1 \le s \le l , \end{align*}


\begin{align*} \dim _ { \rm H } B \cap W ( \mathbf { t } ) \geq \min _ { A \in \mathcal { A } } \left \{ \# \mathcal { K } _ 1 + \# \mathcal { K } _ 2 + \frac { \sum _ { k \in \mathcal { K } _ 3 } a _ k - \sum _ { k \in \mathcal { K } _ 2 } t _ k } { A } \right \} : = s , \end{align*}


\begin{align*} d V _ { H T } = d V _ { \alpha } . \end{align*}


\begin{align*} I ^ { ( n ) } = ( \{ t ^ a \mid a / n \in \mathcal { Q } ( I ^ \vee ) \} ) = ( \{ t ^ a \mid \langle a , u _ i \rangle \geq n \mbox { f o r } i = 1 , \ldots , m \} ) , \end{align*}


\begin{align*} \widetilde { K } ^ { m ; p , q , i ; a } _ { k , j ; n , l , r } ( t , x , \zeta ) = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ 0 ^ { 2 \pi } \int _ 0 ^ { \pi } \big [ ( t - s ) K _ { k ; n } ^ { \mu } ( \mathfrak { m } ) ( y , \zeta , \omega ) + \widetilde { K } _ { k ; n } ^ { \mu } ( \mathfrak { m } ) ( y , \zeta ) \big ] f ( s , x - y + ( t - s ) \omega , v ) \end{align*}


\begin{align*} \phi _ { j } ( z ) : = \langle ( z I - H ) ^ { - 1 } e _ { j } , e _ { 0 } \rangle = \sum _ { n = 0 } ^ { \infty } \frac { \langle H ^ { n } e _ { j } , e _ { 0 } \rangle } { z ^ { n + 1 } } , 0 \leq j \leq p - 1 , \end{align*}


\begin{align*} M _ { n + 1 } = C \eta ^ { - n } \mu ^ { - n } M _ n ^ p . \end{align*}


\begin{align*} d i m _ x ^ { l o g } f ^ { - 1 } ( f ( x ) ) : = \end{align*}


\begin{align*} \sum _ { j = 1 } ^ m s _ i ^ j = \int _ \beta [ D _ i ] , i \in \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} \tau \log | \phi _ \tau | + \Re \mathcal { Q } _ \tau = \breve { Q } _ \tau , \end{align*}


\begin{align*} ( ( \exp , 0 ) \cdot X _ 0 - x _ 0 ) ( ( f , a ) ) = \exp ( a ) . \end{align*}


\begin{align*} \psi _ { \mathrm { d i s c } } ( k ) : = \max _ { 0 \le a < q } \left | \Pr ( \{ S _ k p / q \} \le a / q ) - ( a + 1 ) / q \right | , \end{align*}


\begin{align*} & s _ { ( k ) } = \frac 1 { k ! } \prod _ { b = 2 } ^ k \frac { ( y _ b + 1 ) ( y _ b + \omega + b - 3 ) } { 2 b + \omega - 4 } , \end{align*}


\begin{align*} \left \lbrace \begin{array} { c } \phi ( + a / 2 ) = \phi ( - a / 2 ) \\ \partial _ { x } \phi ( + a / 2 ) = \partial _ { x } \phi ( - a / 2 ) \\ \partial _ { x } ^ { 2 } \phi ( + a / 2 ) = \partial _ { x } ^ { 2 } \phi ( - a / 2 ) \\ \partial _ { x } ^ { 3 } \phi ( + a / 2 ) = \partial _ { x } ^ { 3 } \phi ( - a / 2 ) \end{array} \right . . \end{align*}


\begin{align*} M _ 1 = \sum _ { \substack { 0 \leq l \leq i \\ \gcd ( l , i ) \ { \rm { i s } } \ k - { \rm { f r e e } } \\ \gcd ( l , i + 1 ) \ { \rm { i s } } \ k - { \rm { f r e e } } } } C _ { \alpha } ( i , l ) \end{align*}


\begin{align*} C _ { M } ^ { B } = \{ [ \alpha ] _ { B } \in H _ { B } ^ { 1 , 1 } ( M , \mathbf { R } ) | \exists \omega > 0 [ \omega ] _ { B } = [ \alpha ] _ { B } \} . \end{align*}


\begin{align*} \deg a _ i \leq \frac { 1 } { \mu } ( \delta _ A ( a ) + \deg A ) i = 0 , \dots , \mu - 1 \ , \end{align*}


\begin{align*} m _ i = p + i , R _ i = \frac { 7 \bar R _ 0 } { 8 } - 4 ( 2 i + 1 ) h _ 0 \quad \mbox { f o r a l l } i = 0 , \dots , i _ \infty . \end{align*}


\begin{align*} \rho _ { i c } ( I ) = \max \left \{ ( d _ i f _ \ell ) / \langle \gamma _ i , \delta _ \ell \rangle \right \} _ { i , \ell } . \end{align*}


\begin{align*} E ^ \mathrm { B o g } = \ ; & - \frac { 1 } { 2 } \mathrm { T r } _ { \perp , r } \left [ D _ r + \lambda Q _ r K _ { 1 1 } Q _ r - \sqrt { D _ r ^ 2 + 2 \lambda D _ r ^ { 1 / 2 } Q _ r K _ { 1 1 } Q _ r D _ r ^ { 1 / 2 } } \right ] \\ & - \frac { 1 } { 2 } \mathrm { T r } _ { \perp , \ell } \left [ D _ \ell + \lambda Q _ \ell K _ { 2 2 } Q _ \ell - \sqrt { D _ \ell ^ 2 + 2 \lambda D _ \ell ^ { 1 / 2 } Q _ \ell K _ { 2 2 } Q _ \ell D _ \ell ^ { 1 / 2 } } \right ] . \end{align*}


\begin{align*} \partial _ { i _ 1 } ^ { k _ 1 } \cdots \partial _ { i _ r } ^ { k _ r } \left ( I _ Q \right ) ( f ) = 0 \end{align*}


\begin{align*} l ^ \gamma ( t \circ \varphi ( \lambda _ 1 ) ) = l ^ \tau ( t ) \circ \varphi ( \lambda _ 2 ) . \end{align*}


\begin{align*} K ' = \{ r e ^ { i \theta } : | r - r _ { 0 } | \le \varepsilon , \theta \in [ h _ { \nu } ( r ) + \varepsilon , \pi ] \} . \end{align*}


\begin{align*} \mathrm { M D T E } _ { ( \boldsymbol { a } , \boldsymbol { b } ) } ( \mathbf { X } ) & = \mathrm { E } \left [ \mathbf { X } | \boldsymbol { a } < \mathbf { X } \leq \boldsymbol { b } \right ] \end{align*}


\begin{align*} \psi _ { B _ { k } } = B _ { k } \left [ \frac { \cos ( k x ) . \sinh ( k ' x ) } { \cos ( k a / 2 ) . \sinh ( k ' a / 2 ) } - \frac { \sin ( k x ) . \cosh ( k ' x ) } { \sin ( k a / 2 ) . \cosh ( k ' a / 2 ) } \right ] , \end{align*}


\begin{align*} R _ { \widetilde { G } } = \left ( \begin{array} { l l l } 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{array} \right ) , \ R _ G = \left ( \begin{array} { l l l } 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right ) , \ R _ { \widetilde { D } } = \left ( \begin{array} { l l l } 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right ) , \ \ R _ D = \left ( \begin{array} { l l l } 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array} \right ) . \end{align*}


\begin{align*} \phi ^ { [ \lambda \lambda _ k \lambda '' v ] } & = ( \lambda \lambda _ k \lambda '' ) \cdot \phi \\ & = \lambda \cdot ( ( \lambda _ k \lambda '' ) \cdot \phi ) \\ & = \lambda \cdot \phi ^ { [ \lambda _ k \lambda '' v ] } . \end{align*}


\begin{align*} 0 = h ( \boldsymbol { R } _ 1 ( \tau ; t , x ) , p ( \boldsymbol { R } _ 1 ( \tau ; t , x ) ; t , x ) ) \ , , \end{align*}


\begin{align*} G ( t , x ) \frac { \left ( e ^ { t } - 1 \right ) } { t - 2 \pi i k } = \sum _ { n = 0 } ^ { \infty } \left ( \int _ { 0 } ^ { 1 } e ^ { - 2 \pi i k z } \bar { P } _ { n } ( x , z ) d z \right ) \frac { t ^ n } { n ! } , \end{align*}


\begin{align*} { { \mathcal { L } } } _ m ( { { \mathfrak { A } } } ( \boldsymbol { t } ) ) = { \rm { d e t } } ( \psi _ s ( t ^ { u + \ell } ( t - 1 ) ^ { r n } ) ) _ { \substack { 0 \le s \le r - 1 \\ 0 \le \ell \le r - 1 } } \enspace . \end{align*}


\begin{align*} \Im H _ { \mu } ( s + i t ) = t \left [ 1 - \int _ { \mathbb { R } } \frac { 1 + x ^ { 2 } } { | s + i t - x | ^ { 2 } } d \sigma ( x ) \right ] . \end{align*}


\begin{gather*} \left ( Q \left | \left ( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right ) \right . \right ) ( x , y ) : = Q ( a x + b y , c x + d y ) , \end{gather*}


\begin{align*} | f g | ^ 2 = ( f g , f g ) \leq ( f ^ \dagger f , f ^ \dagger f ) ( g g ^ \dagger , g g ^ \dagger ) = | f | ^ 2 | g | ^ 2 . \end{align*}


\begin{align*} \Psi _ { n + 1 } ( d _ n f \otimes P ) = \Psi _ n ( f \otimes \partial _ { n + 1 } P ) , \end{align*}


\begin{align*} \begin{array} { l l } \partial _ t ^ 2 \psi = A _ 2 ( t ) \psi \end{array} \end{align*}


\begin{align*} c ^ { m , E } _ { j , l , r ; e r r } ( t - s , v , \omega ) : = \big ( \frac { \hat { v } + \omega } { ( 1 + \hat { v } \cdot \omega ) ^ 2 } \big ( 1 - | \hat { v } ^ 2 | \big ) + \omega - \frac { ( \omega + \hat { v } ) \hat { v } \cdot \omega } { 1 + \hat { v } \cdot \omega } \big ) \varphi _ { l ; r } ( \tilde { v } + \omega ) \varphi _ j ( v ) \varphi _ { m ; - 1 0 M _ t } ( t - s ) \end{align*}


\begin{align*} \nabla _ i u = & g ^ { k l } h _ { i k } \nabla _ l \Phi , \\ \nabla _ i \nabla _ j u = & g ^ { k l } \nabla _ k h _ { i j } \nabla _ l \Phi + \phi ' h _ { i j } - ( h ^ 2 ) _ { i j } u , \end{align*}


\begin{align*} \sum \limits _ { i = 2 } ^ { \infty } \frac { 1 } { \alpha _ i } \sum \limits _ { k : k \geq 0 , k \leq \alpha _ i } \mu ( k ) k \leq c _ { \alpha } \frac { \alpha _ 1 } { \alpha _ 2 } + c _ { \alpha } ( 1 - \beta _ 1 ) . \end{align*}


\begin{align*} \| u \| _ { \alpha } : = \| u \| _ { \infty } + [ u ] _ { \alpha } . \end{align*}


\begin{align*} \Psi ( C _ 1 \cap C _ 2 ) = \left \langle \left \{ \gamma ^ { j } \texttt { l c m } ( \widehat { \ , F _ { 1 , j } \ ; } , \widehat { \ , F _ { 2 , j } \ ; } ) \ ; : \ ; 0 \leq j < t \right \} \right \rangle . \end{align*}


\begin{align*} \begin{cases} - \Delta _ p G - \lambda G ^ { p - 1 } = \delta _ 0 & \Omega \\ G \geq 0 & \Omega \\ G = g & \partial \Omega \end{cases} \end{align*}


\begin{align*} c = I _ 1 ( u ) \leqslant \liminf _ { n \rightarrow \infty } I _ 1 \left ( \dfrac { u _ n + u } { 2 } \right ) . \end{align*}


\begin{align*} \tilde { \Sigma } : = \{ ( a , b ) \in \Sigma ( \beta , \mathcal { L } , a _ { } , a _ 0 , a _ 1 , b _ 0 , b _ 1 , \tilde { C } , \tilde { \rho } ) : \ , a , b \mbox { a r e } \mathcal { C } ^ 3 \} . \end{align*}


\begin{align*} \begin{aligned} 0 & = \int _ { \mathbb { R } ^ N } \sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \frac { \partial U _ { h _ n , i } } { \partial h _ n } \ , \phi _ n ~ d y \\ & = \sum _ { k = 1 } ^ 4 \int _ { { \C } _ k } \left ( \sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \frac { \partial U _ { h _ n , i } } { \partial h _ n } \ , \phi _ n \right ) d y \\ & = \sum _ { k = 1 } ^ 4 \int _ { { \C } _ k } \left ( \sum _ { i = 1 } ^ 4 f ( | y - h _ n t _ i | ) ( y - h _ n t _ i ) \cdot t _ i \phi _ n ( y ) \right ) d y , \end{aligned} \end{align*}


\begin{align*} c _ 1 \cos t _ 0 + c _ 2 \sin t _ 0 + \frac { \mu a ^ 2 } { \lambda } = 0 , \end{align*}


\begin{align*} \tilde { r } _ * e _ \alpha = \sum _ i \tilde { r } ^ i _ \alpha e _ i . \end{align*}


\begin{align*} \sigma ( i _ 1 \cdots i _ { r - 1 } ) \{ \delta _ { i _ 1 i _ 2 } \cdots \delta _ { i _ { r - 1 } i _ r } \} = \sigma ( i _ 1 \cdots i _ r ) \{ \delta _ { i _ 1 i _ 2 } \cdots \delta _ { i _ { r - 1 } i _ r } \} . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ 3 \partial _ k ^ 1 { \partial _ k ^ 1 } ^ \ast \psi = \sum _ { k = 1 } ^ 3 \partial _ k ^ 2 { \partial _ k ^ 2 } ^ \ast \psi = 0 \ , \mbox { a n d } \end{align*}


\begin{align*} I _ { D _ { Z , + } , d } = \frac { \prod _ { 0 < a < v _ 2 \cdot d } ( v _ 2 + a z ) } { \prod _ { i \in I _ + , D _ i \cdot d > 0 } ( \bar D _ i + ( D _ i \cdot d ) z ) } [ \textbf { 1 } ] _ { ( - D _ i \cdot d ) _ { i \in I _ + } , v _ 2 \cdot d } , \end{align*}


\begin{align*} \begin{aligned} A _ { j k } ^ \dagger = A _ { k j } \ & , \ \ A _ { 1 1 } + A _ { 2 2 } + A _ { 3 3 } = 0 \ , \\ [ A _ { j k } , A _ { l m } ] & = \delta _ { l , k } A _ { j m } - \delta _ { j , m } A _ { l k } \ . \end{aligned} \end{align*}


\begin{align*} f ( x ) = x ^ 4 + a x ^ 3 + b x \end{align*}


\begin{align*} U _ k & : = \ell ( \gamma _ 2 \gamma _ 1 \phi _ k \gamma _ 1 \gamma _ 2 ) \end{align*}


\begin{align*} L _ 1 : = \sup _ { n \ge 1 } \sup _ { s \in [ 0 , T ] } E ( \sigma ^ 2 ( X ^ n _ { \eta _ { n } ( s ) } ) ) < \infty . \end{align*}


\begin{align*} \varepsilon & = \sqrt { \frac { 2 b d k \log ( \frac { 4 b d k } { \varepsilon } ) } { n } + \frac { \log ( \frac { 3 } { \delta } ) } { 2 n } } \\ & \le \sqrt { \frac { 2 b d k \log ( 4 b d k n ) } { n } + \frac { \log ( \frac { 3 } { \delta } ) } { 2 n } } \\ & \leq \sqrt { \frac { 2 b d k \log ( 4 b d k n ) } { n } } + \sqrt { \frac { \log ( \frac { 3 } { \delta } ) } { 2 n } } . \end{align*}


\begin{align*} F ( n ( z , x ) ) = \begin{pmatrix} z & \frac { x } { 2 } & 0 & 0 \end{pmatrix} , F ( n ( z , x ) ^ t ) = \begin{pmatrix} 0 & 0 & z & \frac { x } { 2 } \end{pmatrix} . \end{align*}


\begin{align*} u = u _ R ~ \textrm { o n } ~ \Omega _ R a . s . , \textrm { a n d } ~ \lim _ { R \rightarrow \infty } \mathbb P ( \Omega _ R ) = 1 . \end{align*}


\begin{align*} \tilde { \mathcal { H } } ^ { j ; a } _ { k , \tilde { k } ; m , l } ( t _ 1 , t _ 2 ) = \int _ 0 ^ { t _ a } \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } E ( \tau , X ( t _ a ) - y + ( t _ a - \tau ) \omega ) \cdot \nabla _ v \big [ \big ( \tilde { \mathcal { K } } ^ { 0 } _ { k , \tilde { k } ; l } ( y , v , V ( t _ a ) ) + ( t _ a - \tau ) \tilde { \mathcal { K } } ^ { 1 } _ { k , \tilde { k } ; l } ( y , v , V ( t _ a ) ) \big ) \varphi _ { j } ( v ) \big ] \end{align*}


\begin{align*} N ( t , u , z ) = 1 + z ( N ( t , t , z ) - 1 + u ) N ( t , u , z ) , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , t ] } E [ | S ^ { n , M , 1 } _ \tau | ^ 2 ] = 0 . \end{align*}


\begin{align*} \mathrm { P } _ { \Phi _ { \mathfrak { m } } ( c ) } = - \inf f _ { \Phi _ { \mathfrak { m } } ( c ) } \left ( E _ { 1 } \right ) , c \in L ^ { 2 } ( \mathbb { S } ; \mathbb { C } ; | \mathfrak { \mathfrak { a } } | ) , \end{align*}


\begin{align*} \langle \Psi _ \varphi ( \xi ) \hat y , \hat x \rangle = \varphi ( x ^ * \xi y ) . \end{align*}


\begin{align*} b _ { i j } h _ \theta ^ j = a _ { i 0 j } + a _ { i 1 j } Q _ \theta + \ldots + a _ { i s j } Q _ \theta ^ s . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } n ^ { \alpha + \frac 1 2 } \int _ 0 ^ { \tau } ( t - s ) ^ \alpha \sigma ' ( X _ s ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) d s = 0 , \end{align*}


\begin{align*} & \partial _ t \mathcal { S } = - \int _ { L _ t } ( \theta _ t - \hat { \theta } ) ( e ^ { - i \hat { \theta } } \Omega ) = - \int _ { L _ t } ( \theta _ t - \hat { \theta } ) ( e ^ { i ( \theta - \hat { \theta } ) } ) d v o l _ { L _ t } \\ = & - \int _ { L _ t } ( \theta _ t - \hat { \theta } ) \sin ( \theta _ t - \hat { \theta } ) d v o l _ { L _ t } . \end{align*}


\begin{align*} \mathcal { D } _ 1 ^ { n s } \cong \langle T _ { a } , T _ { b } ~ \mid ~ T _ { a } * T _ { b } * T _ { a } = T _ { b } , T _ { b } * T _ { a } * T _ { b } = T _ { a } \rangle . \end{align*}


\begin{align*} y '' ( \delta ) = y '' ( 1 ) - \int _ \delta ^ 1 y ''' ( x ) d x , \end{align*}


\begin{align*} \int _ { \partial \omega _ c } ( m \cdot \nu ) \abs { \nabla u } ^ 2 d x = \int _ { \Gamma _ 0 } ( m \cdot \nu ) \left | \frac { \partial u } { \partial \nu } \right | ^ 2 d \Gamma \ \ \ \ \Re \left ( \int _ { \Gamma _ 0 } \frac { \partial u } { \partial \nu } \left ( m \cdot \nabla \bar { u } \right ) d \Gamma \right ) = \int _ { \Gamma _ 0 } ( m \cdot \nu ) \left | \frac { \partial u } { \partial \nu } \right | ^ 2 d \Gamma . \end{align*}


\begin{align*} a _ { \mathrm { p w } } ( \phi , v _ { \mathrm { n c } } ) = \mu ( \phi , v _ { \mathrm { n c } } ) _ { 1 + \delta } \quad v _ { \mathrm { n c } } \in V ( { \mathcal { T } } ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : a b s t r a c t p d e } \partial _ t f _ t = Q ( f _ t ) , \end{align*}


\begin{align*} \frac { \partial u } { \partial t } + \frac { \partial ^ 2 } { \partial x ^ 2 } \Biggl ( a \frac { \partial ^ 2 u } { \partial x ^ 2 } \Biggr ) = f . \end{align*}


\begin{align*} L _ { \xi } v = \Delta _ \xi v - q { \sf U } ^ { q - 1 } ( 1 - \chi _ 1 ) v - \eta \dot \eta ( \Lambda _ \xi v ) , \Lambda _ \xi = \tfrac { 2 } { 1 - q } - \xi \cdot \nabla _ \xi . \end{align*}


\begin{align*} v _ { i , t } & = \rho u _ { i , t } + \lambda > \lambda \ ; , \ , \end{align*}


\begin{align*} N ^ n _ \tau : = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s \left ( \sigma ' ( X _ s ) \right ) ^ 2 \left ( \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) ^ 2 \ , d s . \end{align*}


\begin{align*} ( L ( x ) , x ) _ V = - \sum _ { e \in E } \frac { \delta _ E ( e ) } { | e | ^ 2 } \sum _ { \{ u , v \} \subset e } ( x ( u ) - x ( v ) ) ^ 2 . \end{align*}


\begin{align*} \mathcal { F } ( w , x ) & = \dfrac { \mathcal { F } _ 1 ( w , x ) } { \mathcal { F } _ 2 ( w , x ) } , \end{align*}


\begin{align*} \epsilon _ h = s i g n ( b _ h \tilde c _ 1 ( \alpha , h ) , \tilde c _ 2 ( \alpha , h ) ) \end{align*}


\begin{align*} k _ 1 ( x , t ) - k _ 2 ( x , t ) = U ( \phi _ t x ) - U ( x ) . \end{align*}


\begin{align*} m = n _ { j _ 1 } + \ldots + n _ { j _ s } , \end{align*}


\begin{align*} n = x ^ 2 + 1 6 y ^ 2 + 4 8 z ^ 2 . \end{align*}


\begin{align*} \psi _ { A _ { 0 } } = A _ { 0 } \left [ 1 - \frac { \cosh ( k ' x ) } { \cosh ( k ' a / 2 ) } \right ] \end{align*}


\begin{align*} \ell ' _ 2 ( \ell _ 2 ' ( x , y ) , z ) _ { | _ m } + \circlearrowleft ( x , y , z ) = 0 \ , & \Longrightarrow \ , \ell ' _ 2 ( [ x , y ] _ \mathfrak { g } , z ) _ { | _ m } + \ell ' _ 2 ( \eta ( x , y ) , z ) _ { | _ m } + \circlearrowleft ( x , y , z ) = 0 , \\ & \Longrightarrow \ , \nu ( z ) \left ( \eta ( x , y ) _ { | _ m } \right ) - \eta ( [ x , y ] _ \mathfrak { g } , z ) _ { | _ m } + \circlearrowleft ( x , y , z ) = 0 . \end{align*}


\begin{align*} \mathcal U _ n : = \Big \{ ( X , u ) \in \mathfrak X _ 1 \times \mathcal C \ \Big | \ \big ( u ( \cdot ) , \ , ( F ) \setminus \{ x , y \} \big ) > \frac 1 n \Big \} , \forall n \in \N . \end{align*}


\begin{align*} \sum _ { i \le k } c _ i d _ i = \sum _ { i \le k } c _ i \Big ( \sum _ { j \le i } d _ j - \sum _ { j \le i - 1 } d _ j \Big ) = \sum _ { i \le k - 1 } ( c _ i - c _ { i + 1 } ) \sum _ { j \le i } d _ j \ + \ c _ k \sum _ { i \le k } d _ i . \end{align*}


\begin{align*} G / Z ( G ) = \dfrac { G / Z ( G ) ^ 0 } { Z ( G ) / Z ( G ) ^ 0 } . \end{align*}


\begin{align*} \dim ( C ^ \perp \cap D ^ \perp ) = n \omega - \dim ( C ) - \dim ( D ) + \dim ( C \cap D ) . \end{align*}


\begin{align*} & \int \frac { x ^ { n - 2 } } { ( - x ^ 2 ; q ^ 2 ) _ \infty } \widetilde { h } _ n ( x ; q ) d _ q x = \frac { x ^ n } { [ n - 1 ] _ q ( - x ^ 2 ; q ^ 2 ) _ \infty } \left ( \frac { \widetilde { h } _ n ( x ; q ) } { x } - \widetilde { h } _ { n - 1 } ( x ; q ) \right ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } u _ t - \Delta u = v ^ { p } , \ & t > 0 , x \in \R ^ N \\ v _ t - \Delta v = u ^ { q } , & t > 0 , x \in \R ^ N \end{array} \right . \end{align*}


\begin{align*} \sum _ { m = 1 } ^ { \infty } \frac { c _ m ( n ) } { m } = 0 . \end{align*}


\begin{align*} \| h \partial _ { x _ 2 } \phi _ h ( x _ 1 , x _ 2 ) \| _ { L ^ 2 ( \Gamma ) } = 0 \end{align*}


\begin{align*} \Phi _ 3 ( \xi , \eta , \sigma , \zeta ) : = \hat { \zeta } \cdot ( \xi + \eta + \sigma ) + \mu _ 2 | \sigma | + \mu | \xi | + \mu _ 1 | \eta | . \end{align*}


\begin{align*} \dot { x } ( t , s ) & = - \partial _ s x ( t , s ) + f ( s ) x ( t , s ) , s \in [ 0 , 1 ] , \\ x ( t , 0 ) & = \int _ 0 ^ 1 h ( s ) x ( t , s ) d s , \end{align*}


\begin{align*} b ( \rho ) = \begin{cases} \log ( \overline { \rho } - \rho ) & \mbox { i f } \rho \in [ 0 , \overline { \rho } - \delta ) , \\ \log \delta & \mbox { i f } \rho \geq \overline { \rho } - \delta , \end{cases} \end{align*}


\begin{align*} m = \varepsilon _ 1 n _ { k _ 1 } + \varepsilon _ 2 n _ { k _ 2 } + \ldots + \varepsilon _ { l } n _ { k _ l } , \end{align*}


\begin{align*} \frac { \partial } { \partial \tau } p ( \tau ; \cdot ) = f ( \tau , p ( \tau ; \cdot ) ) + g ( \tau , p ( \tau ; \cdot ) ) \mu ( \tau , p ( \tau ; \cdot ) ) \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } E [ \Lambda ^ { ( 1 ) } _ { n , \delta } ] = 0 . \end{align*}


\begin{align*} H _ { k , j ; n , l , r ; p , q ; \star } ^ { \mu , m , i ; l i n } ( t , x , \zeta ) : = \int _ { 0 } ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \big [ ( t - s ) \mathcal { K } _ { k , n , l , r } ^ { \star ; \mu , m } ( y , v , \omega , \zeta ) + \mathcal { K } _ { k , n , l , r } ^ { \star ; e r r , \mu , m } ( y , v , \omega , \zeta ) \big ] f ( s , x - y + ( t - s ) \omega , v ) \end{align*}


\begin{align*} K _ n ( z , w ) = \sum \ , \frac { z ^ k } { \norm { z ^ k } } \cdot \frac { \overline { w } ^ k } { \norm { \overline { w } ^ k } } = \frac { n ! } { \pi ^ n } \cdot \frac { 1 } { ( 1 - z \cdot \overline { w } ) ^ { n + 1 } } \end{align*}


\begin{align*} F _ { \ , \kappa } \ , ( x ) = \frac { 1 } { 2 } + \frac { 1 } { 2 } \ , \ , { \rm e r f } _ { \kappa } \ ! \left ( \sqrt { \beta } \ , x \right ) \ \ , \end{align*}


\begin{align*} \nabla _ z g _ { r , x } & = \nabla _ { \chi _ r ( d ( z ) ) \Phi _ x ( g ( z ) ) } \Phi _ x ^ { - 1 } \left ( \chi _ r ' ( d ( z ) ) \left ( \nabla _ z d \right ) \Phi _ x ( g ( z ) ) + \chi _ r ( d ( z ) ) \nabla _ { g ( z ) } \Phi _ x \nabla _ z g \right ) , \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty p ( N , M , n ) q ^ n = \frac { ( q ; q ) _ { N + M } } { ( q ; q ) _ N ( q ; q ) _ M } . \end{align*}


\begin{align*} \mathsf { F r } _ x ( a ) = \mu ( [ a ] ) . \end{align*}


\begin{align*} f _ 0 ( z ) = \left \{ \begin{array} { l r } z ( 1 + E z ) ^ { \frac { D - E } { E } } , & E \neq 0 ; \\ z e ^ { D z } , & E = 0 . \end{array} \right . \end{align*}


\begin{align*} u _ t = \Delta u + | u | ^ { p - 1 } u \end{align*}


\begin{align*} \frac { d } { d t } \mathcal { S } ( L _ t ) = \int _ { L _ t } h _ t ( e ^ { - i \hat { \theta } } \Omega ) \end{align*}


\begin{align*} f = r _ s + q _ s g ^ s , r _ s = \sum \nolimits _ { 0 \le t < s } a _ s g ^ t . \end{align*}


\begin{align*} \Omega \wedge \eta _ { 0 } = ( \sqrt { - 1 } ) ^ { n } F ( z _ { 1 } , . . . , z _ { n } ) d z _ { 1 } \wedge d \overline { z } _ { 1 } \wedge . . . \wedge d z _ { n } \wedge d \overline { z } _ { n } \wedge d x \end{align*}


\begin{align*} \Delta ( z ) = ( - 1 ) ^ { r m } \left ( \sum _ { \ell = 0 } ^ { r m } P _ { \ell } ( z ) \Delta _ { 1 , \ell + 1 } ( z ) \right ) \enspace . \end{align*}


\begin{align*} { \mathfrak m } { \mathfrak c } _ { Z , Y } : = - | [ Z , Y ] | ^ { 2 } - \langle Z , [ [ Z , Y ] , Y ] \rangle \end{align*}


\begin{align*} Q ( T , x ) & = \prod _ { v \in V ( T ) } \mathcal { H } ( v , x ) . \end{align*}


\begin{align*} f _ 1 ( x ) = x ^ 4 + x ^ 3 \end{align*}


\begin{align*} V ( x ) = O ( | x | ^ { - ( 1 + \epsilon ) } ) x \rightarrow \infty \end{align*}


\begin{align*} \tilde { \omega } _ + = ( \omega _ + , \epsilon ) , \tilde { \omega } _ - = ( \omega _ - , \epsilon ) , \end{align*}


\begin{align*} q _ { \gamma _ 1 , \gamma _ 2 } = \Omega ( e , v _ 1 + v _ 2 ) \end{align*}


\begin{align*} Q = \sum _ { \substack { a , b \geq 0 \\ a + b \geq s } } c _ { a , b } \phi ^ a ( z - r ) ^ b \end{align*}


\begin{align*} \frac { \partial C } { \partial z ^ j _ { \eta } } \frac { \partial ^ 2 \varphi ^ j } { \partial x ^ { \epsilon } \partial x ^ { \eta } } = \frac { f ' ( x ^ 1 ) } { \sqrt { 1 + f '^ 2 ( x ^ 1 ) } } \delta _ { \epsilon 1 } \left [ f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \right ] , \end{align*}


\begin{align*} F ( A x + e ) = B G ( x ) + d . \end{align*}


\begin{align*} \sum _ { i = 0 , 1 , 2 } | \nabla _ v \mathcal { K } ^ { \mu , i } _ { k , j ; n , l } ( y , v , V ( s ) ) | + \sum _ { a = 3 , 4 } 2 ^ { - \max \{ n , ( \gamma _ 1 - \gamma _ 2 ) M _ { t ^ \star } \} - \epsilon M _ t } | \nabla _ v \mathcal { K } ^ { \mu , a } _ { k , j ; n , l } ( y , v , V ( s ) ) | \end{align*}


\begin{align*} \begin{array} { c } \cdots \stackrel { \dd = \ell _ 1 } { \longrightarrow } E _ { - 3 } \stackrel { \dd = \ell _ 1 } { \longrightarrow } E _ { - 2 } \stackrel { \dd = \ell _ 1 } { \longrightarrow } \mathfrak g [ 1 ] \oplus E _ { - 1 } \stackrel { { \rho ' } } { \longrightarrow } T M , \end{array} \end{align*}


\begin{align*} I ( \nu ) = - \frac { 1 } { 2 } \log \mathrm { d e t } ( I _ k - A A ^ T ) \end{align*}


\begin{align*} \ddot { \vec { r } } = - \frac { \mu } { r ^ 3 } \vec { r } + \vec { a } _ O , \end{align*}


\begin{align*} \mathrm { E } [ X ^ { n } | x _ { p } < X < x _ { q } ] = \frac { \int _ { x _ { p } } ^ { x _ { q } } x ^ { n } \frac { c _ { 1 } } { \sigma } g _ { 1 } \left ( \frac { 1 } { 2 } \left ( \frac { x - \mu } { \sigma } \right ) ^ { 2 } \right ) \mathrm { d } x } { F _ { X } ( x _ { p } , x _ { q } ) } . \end{align*}


\begin{align*} \begin{array} { l l } \sum \limits _ { i = n + 1 } ^ { m } g ( \nabla _ { U _ i } U _ i , U _ i ) + \sum \limits _ { j = 1 } ^ { n } g ( \nabla _ { X _ j } X _ j , X _ j ) \\ + s ^ { K e r F _ \ast } + \frac { 1 } { \lambda ^ 2 } s ^ { N } + \mu ( m - n + n ) = 0 , \end{array} \end{align*}


\begin{align*} \| u \| ^ 2 _ { L ^ 2 _ \delta ( \Omega ) } = \int _ \Omega u ^ 2 \rho ^ { - 2 \delta - n } d V _ { g } \leq \left [ \int _ { \Omega } | u | ^ q d V _ { g } \right ] ^ { \frac { 2 } { q } } \left [ \int _ { \Omega } \rho ^ { - \frac { 2 \delta q } { q - 2 } - \frac { n q } { q - 2 } } d V _ { g } \right ] ^ { \frac { q - 2 } { q } } \end{align*}


\begin{align*} \sup \left \{ \int _ 0 ^ \infty f g \ , d r : \ g \geq 0 \ , , \ \overline \nu _ \alpha ( g ) \leq 1 \right \} = \alpha \ , \underline \mu _ \alpha ( f ) \end{align*}


\begin{align*} L _ 0 ^ \lambda \Phi _ { \nu , \nu ' } ^ \lambda = ( 2 | \nu | _ 1 + n ) \lambda \Phi _ { \nu , \nu ' } ^ \lambda , \end{align*}


\begin{align*} \lim \limits _ { r \rightarrow \infty } \left . \sum \limits _ { i = 0 } ^ { b r } \dbinom { a r } { i } \middle / r \dbinom { a r } { b r } \right . \leq \lim \limits _ { r \rightarrow \infty } \dfrac { a r - b r + 1 } { r ( a r - 2 b r + 1 ) } = 0 . \end{align*}


\begin{align*} \theta _ { t + 1 } = \theta _ t - \beta _ t \nabla V ( \theta _ t ) - \beta _ t d _ t - \beta _ t w _ { \rm a } ( t ) , \end{align*}


\begin{align*} \Psi ( \lambda _ 1 \cdots \lambda _ k ) = e _ { \lambda _ 1 } \cdots e _ { \lambda _ k } ( \alpha ^ n / n ! ) . \end{align*}


\begin{align*} x ^ { l - 1 } = x ^ { l - 2 } + \ldots + 1 , x > 1 , \end{align*}


\begin{align*} \mbox { F o r } x _ 0 \notin N , \tilde { N } = N \cup \{ x _ 0 \} , \quad & \tilde { h _ 0 } = h _ 0 \cup \{ x _ 0 \} , \\ \mbox { F o r } \{ x _ 1 , . . . , x _ n \} \subseteq N \mbox { s u c h t h a t } \quad & \forall j = 1 , . . . , n \mbox { a n d } \forall y \in N , x _ j \leq y . \quad \mbox { T h e n } \\ \tilde { E } = E \cup & \{ ( x _ 1 , x _ 0 ) , . . . , ( x _ n , x _ 0 ) \} . \end{align*}


\begin{align*} ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \ldots , y _ i ) = 0 ( 1 \le k \le \min \{ i , n - i \} ) . \end{align*}


\begin{align*} \Gamma ^ { k ( s ) } _ { i j } - \Gamma ^ { k ( s ) } _ { j i } = L ^ { k ( s ) } _ { i j } ( \Lambda ^ { ( s ) } ) . \end{align*}


\begin{align*} D \cdot h _ s ^ + = 0 , D \cdot h _ s ^ - = 2 , D \cdot l _ s = 1 . \end{align*}


\begin{align*} F ( z + \omega _ 2 \ , | \ , \overline \omega _ 1 , \omega _ 2 ) \cdot F ( z \ , | \ , \overline \omega _ 1 , - \omega _ 2 ) = \prod _ { k \geq 0 } \big ( 1 - x _ 2 ( q _ 2 \widetilde { q } _ 2 ) ^ { k / 2 } \big ) \cdot \prod _ { k \geq 1 } \big ( 1 - x _ 2 ^ { - 1 } ( q _ 2 \widetilde { q } _ 2 ) ^ { k / 2 } \big ) ^ { - 1 } , \end{align*}


\begin{align*} I _ { 3 } = \int _ { \mathbb { R } } \frac { 1 + s ^ { 2 } } { s ^ { 2 } } \ , d \sigma _ { \mu } ( s ) < + \infty , \end{align*}


\begin{align*} u ( t ) = \frac 1 { 2 \pi i } \int _ { \sigma \pm i \infty } K ( s ) \int _ 0 ^ t e ^ { s ( t - \tau ) } g ( t ) d s = \frac 1 { 2 \pi i } \int _ { \sigma \pm i \infty } K ( s ) y ( t ; s ) d s , \end{align*}


\begin{align*} \zeta \cdot ( X , T ) = ( e ^ { \frac { 2 \pi i } { d } } X , e ^ { m \frac { 2 \pi i } { d } } T ) . \end{align*}


\begin{align*} f ( t , i , v ) = \tilde R _ i M _ i ( v ) , M _ i = \frac { 1 } { T _ i } e ^ { - \frac { v ^ 2 } { 2 T _ i } } \end{align*}


\begin{align*} f _ { \sigma } ^ { ( j ) } ( \kappa _ i ) & = f _ { \sigma } ^ { ( j ) } ( \sigma ( \lambda _ i ) ) = \sum _ { k = j } ^ n ( k ) _ j \sigma ( c _ k ) \sigma ( \lambda _ i ) ^ { k - j } \\ & = \sigma \left ( \sum _ { k = j } ^ n ( k ) _ j c _ k \lambda _ i ^ { k - j } \right ) = \sigma \left ( f ^ { ( j ) } ( \lambda _ i ) \right ) . \end{align*}


\begin{align*} \int f \ d \xi = \int ( f ( x ) - f ( y ) ) \ a ( d x , d y ) , \end{align*}


\begin{align*} \eta _ { \mu } ( z \Sigma _ { \mu } ( z ) ) = z , z \in V . \end{align*}


\begin{align*} \int _ 0 ^ \infty \ 1 ( c _ d ' r | \nabla f ( a ) | - C A r ^ 2 > \kappa ) \ , \frac { d r } { r ^ { p + 1 } } & = p ^ { - 1 } \left ( R _ - ^ { - p } - R _ + ^ { - p } \right ) . \end{align*}


\begin{align*} Y \cdot ( f , a ) = a + Y . \end{align*}


\begin{align*} \Psi ( E _ j ( x , a ) ) & = 4 \sqrt { a } \sum _ { h = 1 } ^ { \lfloor \frac { j } { 2 } \rfloor } \cos \left ( \frac { h } { j + 1 } \pi \right ) . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ n b _ j ( g \circ S ^ { 1 - j } ) ( x ) + d ' = f ( A x + e ) . \end{align*}


\begin{align*} \xi _ Q ( s ) : = \pi ^ { - s } \Gamma ( s ) \zeta _ Q ( s ) . \end{align*}


\begin{align*} \rho = \sum _ { k = 1 } ^ m \mathrm { I n d } _ { H _ k } ^ G \varphi _ k \ ; . \end{align*}


\begin{align*} \varphi _ k ^ \lambda ( z ) = \lambda ^ n L _ k ^ { n - 1 } ( \tfrac 1 2 \lambda | z | ^ 2 ) e ^ { - \frac 1 4 \lambda | z | ^ 2 } , z \in \R ^ { 2 n } , \end{align*}


\begin{align*} I _ { k } ( c ) : = h ^ 4 \int _ { \mathbb { R } ^ 4 } w ( x ) H \big ( h ^ { - 1 } k , N ( x ) - 1 \big ) e _ { k } ( - h c \cdot x ) \ , d x . \end{align*}


\begin{align*} \int _ L \lambda \wedge \chi = - \int _ L f _ L d \chi . \end{align*}


\begin{align*} f _ m = g _ 0 + g _ 1 + \cdots + g _ m . \end{align*}


\begin{align*} \frac { 1 } { \sqrt { n } } \ , \sum _ { i = 1 } ^ n ( U ( \gamma _ i ) - A _ 0 ) \end{align*}


\begin{align*} \psi _ n + c _ n \left ( \sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \frac { \partial U _ { h _ n , i } } { \partial h _ n } \right ) \in E _ n . \end{align*}


\begin{align*} \dot { x } ( t , s ) & = - \partial _ s x ( t , s ) + f ( s ) x ( t , s ) , s \in [ 0 , 1 ] , \\ x ( t , 0 ) & = \int _ 0 ^ 1 h ( s ) x ( t , s ) d s , \end{align*}


\begin{align*} \phi _ { x _ { 0 } } ( N \times \{ 0 \} ) = S ^ { 1 } \cdot x _ { 0 } \subset U ^ { + } \end{align*}


\begin{align*} & \int _ { \left [ 0 , 1 \right ] ^ d } \left ( p ( x ) - \hat { p } \left ( x \right ) \right ) ^ 2 d x \\ & = \int _ { \left [ 0 , 1 \right ] ^ d } \hat { p } \left ( x \right ) ^ 2 d x - 2 \int _ { \left [ 0 , 1 \right ] ^ d } p ( y ) \hat { p } ( y ) d y + \int _ { \left [ 0 , 1 \right ] ^ d } p ( z ) ^ 2 d z . \end{align*}


\begin{align*} x ' = \exp \left ( \operatorname { a d } _ h \right ) \left ( x \right ) . \end{align*}


\begin{align*} I _ 3 & = \int _ 0 ^ r \left [ ( \bar \phi + e _ \delta - \delta ) ( 2 - \delta + \bar \phi + e _ \delta ) ( \bar \phi '' , \bar \psi '' ) \cdot ( \phi , \psi ) \right ] { \rm { d } } z \\ & \leq \Vert ( \phi , \psi ) \Vert _ { L ^ 2 } \cdot \Vert ( \bar \phi '' , \bar \psi '' ) \Vert _ { L ^ 2 } \cdot \Vert ( \bar \phi + e _ \delta - \delta ) ( 2 - \delta + \bar \phi + e _ \delta ) \Vert _ { L ^ \infty } , \end{align*}


\begin{align*} \int _ { \Psi _ j } x ^ k e ^ { x ^ r / \hbar } d x & = \int _ 0 ^ \infty t ^ k e ^ { ( \pi i + \arg \hbar i + 2 \pi i j ) ( k + 1 ) / r } e ^ { t ^ r \hbar ^ { - 1 } e ^ { \pi i + \arg \hbar i } } d t \\ & = \exp \left ( ( \pi i + \arg \hbar i + 2 \pi i j ) \frac { k + 1 } { r } \right ) \int _ 0 ^ \infty t ^ k e ^ { - t ^ r / | \hbar | } d t \end{align*}


\begin{align*} \min _ { u \in B ( \overline { u } , r ) } j _ \delta ( u ) \end{align*}


\begin{align*} \psi _ { A _ { k } } = A _ { k } \left [ \frac { \cos ( k x ) . \cosh ( k ' x ) } { \cos ( k a / 2 ) . \cosh ( k ' a / 2 ) } - \frac { \sin ( k x ) . \sinh ( k ' x ) } { \sin ( k a / 2 ) . \sinh ( k ' a / 2 ) } \right ] \end{align*}


\begin{align*} \label [ I e q ] { e q : s p a t i a l l y h o m o g e n e o u s b o l t z m a n n } \partial _ t f ( t , v ) = \int _ { \R ^ d } \int _ { \mathbb { S } ^ { d - 1 } } B ( v - v _ * , \sigma ) \Big ( f _ t ( v _ * ' ) f _ t ( v ' ) - f _ t ( v _ * ) f _ t ( v ) \Big ) \dd v _ * \dd \sigma , \end{align*}


\begin{align*} e ^ { - i t H } \varphi = e ^ { - i t H _ x } \psi _ 0 \otimes e ^ { - i t H _ y } \psi _ 1 = e ^ { - i t E _ 0 } \psi _ 0 \otimes e ^ { - i t H _ y } \psi _ 1 \end{align*}


\begin{gather*} h _ { r } ( \tau ) = ( - 1 ) ^ k h _ { - r } ( \tau ) \end{gather*}


\begin{align*} \Sigma _ t ( \varphi ) = \Big ( \big < \zeta _ x ; \zeta _ y \big > ^ { C _ t ^ { - 1 } \varphi } _ { A , g , \nu + 1 / t } \Big ) _ { x , y \in \Lambda } , \end{align*}


\begin{align*} 2 V _ k ( n ) & = \Big ( \sum _ { 1 \leq i \leq n } f _ k ( i ) f _ k ( i + 1 ) \Big ) ^ 2 - \sum _ { 1 \leq i \leq n } \big ( f _ k ( i ) f _ k ( i + 1 ) \big ) ^ 2 + O _ \varepsilon ( n ^ { 1 + \varepsilon } ) \\ & = \Big ( \sum _ { 1 \leq i \leq n } f _ k ( i ) f _ k ( i + 1 ) \Big ) ^ 2 + O _ \varepsilon ( n ^ { 1 + \varepsilon } ) . \end{align*}


\begin{align*} \bigg \| \max _ { 0 \le k \le n - 1 } \big | V _ { k + 1 } - V _ { k } \big | \bigg \| _ { L ^ { \gamma p } } ^ { \gamma } = \bigg \| \max _ { 1 \le k \le n } \big | \mathbf { E } \big ( \frac { m ^ 2 } { n \sigma ^ 2 } | f ^ { - 1 } \mathcal { M } \big ) \circ f ^ { n - k } \big | \bigg \| _ { L ^ { \gamma p } } ^ { \gamma } \le C n ^ { - \left ( \gamma - \frac { 2 \gamma } { p } \right ) } , \end{align*}


\begin{align*} a _ 0 ( s ) = \int _ 0 ^ 1 z ^ { 2 \alpha - 2 s - 1 } \int _ 0 ^ 1 \int _ 0 ^ 1 \Big ( \frac { \sin ^ 2 ( \pi x ) } { \pi ^ 2 } + \frac { \sin ^ 2 ( \pi y ) } { \pi ^ 2 } + z ^ 2 \Big ) ^ { - \alpha } d x d y d z \end{align*}


\begin{align*} Y ( t ) = \exp ( \Omega ( t ) ) . \end{align*}


\begin{align*} \frac { \partial f _ 0 } { \partial x ^ 1 } & = \frac { \partial f _ 2 } { \partial x ^ 2 } & \frac { \partial f _ 0 } { \partial x ^ 2 } & = - \frac { \partial f _ 2 } { \partial x ^ 1 } \end{align*}


\begin{align*} N _ d ( \mathrm { C H } ( S ) ) \cap \mathrm { A n } ( R ) & \subseteq \mathrm { A n } ( R ) \cap \bigcup _ { i = 1 } ^ N N _ { C + d } ( \mathrm { C o n e } ( x , B ( y _ i , \varepsilon ) ) ) \setminus B ( x , R ) \\ & \subseteq \mathrm { A n } ( R ) \cap \bigcup _ { i = 1 } ^ N \mathrm { C o n e } ( x , B ( y _ i , \varepsilon ( 1 + A C ' ) ) ) , \end{align*}


\begin{align*} I _ { ( Z _ + , D _ { Z , + } ) } ( y , z ) = z e ^ { t _ + / z } \sum _ { d \in \mathbb K _ { + } } y ^ { d } \left ( \prod _ { i \in M _ 0 } \frac { \prod _ { a \leq 0 , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } { \prod _ { a \leq D _ i \cdot d , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } \right ) \\ \left ( \prod _ { 0 < a \leq v _ 1 \cdot d } ( v _ 1 + a z ) \right ) \textbf { 1 } _ { [ - d ] } I _ { D _ { Z , + } , d } , \end{align*}


\begin{align*} \phi _ 1 ( s ) = s ^ { - a } \log ( 2 + s ) , \end{align*}


\begin{align*} P ( h ) = \frac { 1 } { 2 L } \frac { h ( \kappa h - 2 \kappa + 2 ) } { 1 - \kappa } = \frac { 1 } { 2 L } \big [ 2 h - h ^ 2 \frac { - \kappa } { 1 - \kappa } \big ] . \end{align*}


\begin{align*} \Big | E _ { 2 - \mathrm { m o d e } } - E _ 0 - E ^ w _ N - N \frac { \mu _ + - \mu _ - } { 2 } \Big | \le C _ \varepsilon \max \{ T ^ { 1 / 2 - \varepsilon } , N ^ { - 1 + \varepsilon \delta } \} . \end{align*}


\begin{align*} \begin{aligned} p ( t , v , g ) & = \frac { 1 } { V _ F } \sum _ { k = - \infty } ^ { + \infty } p _ k ( t , g ) e ^ { i k v \frac { 2 \pi } { V _ F } } , \\ p _ k ( t , g ) : & = \int _ { 0 } ^ { V _ F } p ( t , v , g ) e ^ { - i k v \frac { 2 \pi } { V _ F } } d v , k \in \mathbb { Z } . \end{aligned} \end{align*}


\begin{gather*} e ( \partial B _ { R _ \ell } ( x _ 0 ) ) = u ( \partial B _ { r _ \ell } ( x _ 0 ) ) = 0 , \\ \shortintertext { a n d } \lim _ { \ell \to \infty } \frac { e ( B _ { R _ \ell } ( x _ 0 ) ) } { u ( B _ { r _ \ell } ( x _ 0 ) ) } = \limsup _ { R \to 0 ^ + } \frac { e ( B _ R ( x _ 0 ) ) } { u ( B _ R ( x _ 0 ) ) } . \end{gather*}


\begin{align*} X ^ 3 + Y ^ 3 + Z ^ 3 = n X Y Z , \end{align*}


\begin{align*} \beta | f ' ( z ^ m ) | + ( 1 - \beta ) | f ( z ^ m ) | + \sum _ { n = 1 } ^ { \infty } | a _ n | \phi _ { n } ( r ) \leq d ( 0 , \partial { \Omega } ) \end{align*}


\begin{align*} ( a \# b ) ( n \otimes y ) = \sum _ { ( b ) } ( S ^ { - 1 } ( b _ { - 1 } y _ { - 1 } ) \rightharpoonup a ) n \otimes b _ 0 y _ 0 . \end{align*}


\begin{align*} \left \lbrace \begin{aligned} \boldsymbol { x } ( t ) - \boldsymbol { x } _ c & \to \hat { \boldsymbol { x } } ( t ) , \\ \boldsymbol { u } ( t ) - \boldsymbol { u } _ c & \to \hat { \boldsymbol { u } } ( t ) , \end{aligned} \right . \end{align*}


\begin{align*} H = H _ { \phi } = \left ( \begin{array} { c c c | c } 0 & \cdots & 0 & \phi _ 0 \\ \hline & & & \phi _ 1 \\ & I _ { n - 1 } & & \vdots \\ & & & \phi _ { n - 1 } \\ \end{array} \right ) \in \mathbb { Z } ^ { n \times n } , \end{align*}


\begin{align*} F _ { \mu } ( z + \varphi _ { \mu } ( z ) ) = z \end{align*}


\begin{align*} \nu _ Q \left ( f \right ) = \min _ { 0 \leq i \leq r } \{ \nu _ Q ( f _ i Q '^ i ) \} . \end{align*}


\begin{align*} U ( t ) : = \exp \left ( { \rm i } \alpha ( t ) \widehat { H } _ 2 \right ) \exp \left ( { \rm i } \beta ( t ) \widehat { H } _ 3 \right ) . \end{align*}


\begin{align*} \Omega ( f , g ) = _ { z = 0 } ( f ( - z ) , g ( z ) ) d z , f , g \in \mathcal H , \end{align*}


\begin{align*} w \left ( \eta \right ) = \mathop \sum \limits _ n \frac { w ^ { ( n ) } } { { n ! } } \eta ^ n \end{align*}


\begin{align*} \int _ { U } | f | ^ 2 e ^ { - k \varphi - r _ 1 \psi } \le \left ( \int _ { U } | f | ^ { 2 q _ 1 } e ^ { - q _ 1 k \varphi } \right ) ^ { \frac { 1 } { q _ 1 } } \cdot \left ( \int _ U e ^ { - q _ 2 r _ 1 \psi } \right ) ^ { \frac { 1 } { q _ 2 } } < + \infty , \end{align*}


\begin{align*} w t ( b _ 1 \otimes b _ 2 ) & = w t ( b _ 1 ) + w t ( b _ 2 ) , \\ \varepsilon _ i ( b _ 1 \otimes b _ 2 ) & = \max \left ( \varepsilon _ i ( b _ 1 ) , \varepsilon _ i ( b _ 2 ) - \langle \alpha _ i ^ \vee , w t ( b _ 1 ) \rangle \right ) , \\ \varphi _ i ( b _ 1 \otimes b _ 2 ) & = \max \left ( \varphi _ i ( b _ 2 ) , \varphi _ i ( b _ 1 ) + \langle \alpha _ i ^ \vee , w t ( b _ 2 ) \rangle \right ) , \end{align*}


\begin{align*} U = U _ { + } \oplus U _ { - } , \end{align*}


\begin{align*} - g ( A _ { \mathcal { H } \nabla _ X W } X , W ) = 0 . \end{align*}


\begin{align*} \mathcal { M } ( p _ 1 , \ldots p _ k , q ) = \deg q - \sum _ 1 ^ k \deg p _ i + k - 2 . \end{align*}


\begin{align*} ( f \star g ) ( x ) = \epsilon ^ d \sum _ { y \in \Lambda _ { \epsilon , L } } f ( x - y ) g ( y ) , f , g \in \R ^ { \Lambda _ { \epsilon , L } } . \end{align*}


\begin{align*} \label [ I e q ] { e q : l i n e a r P m u } \langle Q ( \mu ) , \tilde { \varphi } \rangle = \langle \mu , K \star \tilde { \varphi } \rangle - \langle \mu , \tilde { \varphi } \rangle . \end{align*}


\begin{align*} \frac { 1 } { 2 } \{ g ( \nabla _ X \xi , Y ) + g ( \nabla _ Y \xi , X ) \} + R i c ( X , Y ) + \mu g ( X , Y ) = 0 , \end{align*}


\begin{align*} \varkappa ( B \cap E , \sigma ) = \varkappa ( B , \sigma _ E ) . \end{align*}


\begin{align*} \sum \limits _ { i = 1 } ^ { \infty } \mu \left ( \left [ C \alpha _ i , \infty \right ) \right ) \precsim \sum \limits _ { i = 1 } ^ { \infty } \mu \left ( \left [ \alpha _ i , \infty \right ) \right ) \end{align*}


\begin{align*} F ( D ^ 2 u ) = g \quad \ \mathbb { R } ^ n \setminus \overline { B } _ { r _ 0 } \end{align*}


\begin{align*} \Pr \big ( \Delta ( C _ A ) \le \delta \big ) = \Pr ( X \ge 1 ) . \end{align*}


\begin{align*} C ^ 2 _ { T , \beta } = 2 \frac { T ^ { 1 - \beta } } { \Gamma ( 2 - \beta ) } \max ( 2 ^ { 1 - \beta } \Gamma ( 1 - \beta ) T ^ { \beta } , 1 ) . \end{align*}


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


\begin{align*} \liminf _ { n \to \infty } | T _ { \beta } ^ n x - x | = 0 . \end{align*}


\begin{align*} & x \cdot m = x m - m x , \end{align*}


\begin{align*} \nu \left ( \partial _ i L ( h _ \sigma ) - \partial _ i L ( h _ \rho ) \right ) = \min _ { 1 \leq k \leq n - i } \left \{ \nu \left ( { i + k \choose i } \partial _ { i + k } L ( h _ \rho ) ( h _ \sigma - h _ \rho ) ^ k \right ) \right \} . \end{align*}


\begin{align*} \delta ( a ) = \left [ \sum _ { n \ge 0 } \sum _ { j = 0 } ^ \infty \beta _ { n , j } P _ { j , j + n } + \sum _ { n < 0 } \sum _ { j = 0 } ^ \infty \beta _ { n , j } P _ { j - n , j } , a \right ] \end{align*}


\begin{align*} [ \sigma , e ] [ \tau , f ] = [ \sigma \tau , e \alpha _ { \sigma } ( f ) ] \quad [ \sigma , e ] ^ { * } = [ \sigma ^ { - 1 } , \alpha _ { \sigma } ^ { - 1 } ( e ^ { * } ) ] . \end{align*}


\begin{align*} \Phi ( x , y ) : = \frac { 1 } { 4 \pi } \frac { e ^ { i k \abs { x - y } } } { \abs { x - y } } , x \neq y \end{align*}


\begin{align*} \operatorname { s g n } ( Z ) : = \begin{cases} \operatorname { s g n } ( F ) & U _ { 0 } : = \{ x \in M \mid \operatorname { g r a d } ( \| X \| ^ { 2 } ) \neq 0 \} \\ 0 & M \backslash U _ { 0 } , \end{cases} \end{align*}


\begin{align*} \beta _ i : = \nu ( \partial _ i L ( Q ) ) = \nu ( \partial _ i L ( h _ { \rho } ) ) \end{align*}


\begin{align*} \mathcal { V } _ { \left \{ C _ { i } \right \} _ { i = 1 } ^ { r } } : = \mathcal { R } _ { C _ { r } } \dots \mathcal { R } _ { C _ { 1 } } , \end{align*}


\begin{align*} D _ { q ^ { - 1 } } u ( x ) + u ( x ) u ( x / q ) = 0 , \end{align*}


\begin{align*} R ^ { N } _ { \alpha \beta \gamma } { } ^ \delta = \overline { \Gamma } _ { \beta \gamma , \alpha } ^ \delta - \overline { \Gamma } _ { \alpha \gamma , \beta } ^ \delta + \sum _ \eta \overline { \Gamma } _ { \beta \gamma } ^ \eta \overline { \Gamma } _ { \alpha \eta } ^ \delta - \overline { \Gamma } _ { \alpha \gamma } ^ \eta \overline { \Gamma } _ { \beta \eta } ^ \delta . \end{align*}


\begin{align*} A . a = \lambda a \Rightarrow \left ( \begin{array} { l r } G & O \\ O & - G \end{array} \right ) \left ( \begin{array} { l r } a _ { + } \\ a _ { - } \end{array} \right ) = \lambda \left ( \begin{array} { l r } a _ { + } \\ a _ { - } \end{array} \right ) , \end{align*}


\begin{align*} \rho = \min _ { 0 \leq n \leq n _ 0 } \left ( \frac { m _ n } { 2 } \prod _ { j = 0 } ^ { n - 1 } \sqrt { \frac { m _ n } { 2 M _ 0 + m _ n } } \right ) , \end{align*}


\begin{align*} M _ \Omega ( H ^ s \to H ^ { - s } ) \vcentcolon = \{ \ , a \in M ( H ^ s \to H ^ { - s } ) \ , ; \ , \inf _ { a ' \in M _ 0 ( H ^ s \to H ^ { - s } ) } \norm { a - a ' } _ { s , - s } < \delta ( \Omega ) \ , \} . \end{align*}


\begin{align*} P _ { n , s , p } ( \Omega ) \vcentcolon = \inf _ { u \in W _ \Omega ^ { s , p } ( \R ^ n ) , u \neq 0 } \frac { [ u ] _ { s , p , \R ^ n } ^ p } { \norm { u } _ { L ^ p ( \Omega ) } ^ p } , \end{align*}


\begin{align*} u _ t - \Delta u ^ m = \dfrac { u ^ p } { ( 1 + | x | ) ^ \gamma } \quad \R ^ N \times ( 0 , T ) , \textrm { f o r s o m e $ T > 0 $ , } \end{align*}


\begin{align*} I ( n ) : = \int _ { - 1 } ^ { 1 } e ^ { n \cdot f ( z ) } g ( z ) \ , d z \end{align*}


\begin{align*} \alpha _ { k , j } & = \begin{cases} \sum \limits _ { i = 1 } ^ { n } a _ i \cdot \mathtt { b } ^ + ( \phi _ i ) \geqslant 1 - c & t _ { k , j } = f ^ \sharp _ { k , j } \\ \sum \limits _ { i = 1 } ^ { n } a _ i \cdot \mathtt { b } ^ + ( \phi _ i ) < - c & \end{cases} \end{align*}


\begin{align*} G ( y ) : = \sum _ { j = 1 } ^ m \max \{ F ( x ) : Z ( y ) = h ^ 1 , h ^ 2 , \ldots , h ^ m \land x \in P ( h ^ j ) \} . \end{align*}


\begin{align*} \Lambda = \Lambda _ { \epsilon , L } , A = \epsilon ^ d \big ( - \Delta ^ \epsilon + m ^ 2 \big ) , g = \epsilon ^ d \lambda , \nu = \epsilon ^ d ( \mu - m ^ 2 + a ^ \epsilon ( \lambda , m ^ 2 ) ) . \end{align*}


\begin{align*} H / H ^ 0 = W ( K ) , \end{align*}


\begin{align*} f _ { \lambda } ( \lambda ) ^ 2 - \dfrac { 1 } { 2 } \nabla f \cdot \nabla f ( \lambda ) = - f ( \lambda ) \Delta f ( \lambda ) + 2 \sum _ { \l - k > 1 } \det \left ( ( \lambda I _ N - H ) _ { k | k } \right ) \det \left ( ( \lambda I _ N - H ) _ { \l | \l } \right ) . \end{align*}


\begin{align*} 2 \nabla _ { X } X = - \operatorname { g r a d } ( \| X \| ^ { 2 } ) . \end{align*}


\begin{align*} V a r ( \hat { \pi } _ { h , n } ( x ) ) & \le \frac { c } { T _ n } \big [ \frac { 1 } { \prod _ { l = 1 } ^ d h _ l } ( \delta _ 1 + \Delta _ n ) + \frac { 1 } { \prod _ { l \ge 3 } h _ l } \log ( \frac { \delta _ 2 } { \delta _ 1 } ) + \delta _ 2 ^ { 1 - \frac { d } { 2 } } + D + \frac { 1 } { ( \prod _ { l = 1 } ^ d h _ l ) ^ 2 } e ^ { - \rho D } \big ] . \end{align*}


\begin{align*} \Delta = \langle \begin{pmatrix} - 1 \\ - 1 \\ - 1 \end{pmatrix} , \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix} , \begin{pmatrix} - 1 \\ 1 0 \\ 0 \end{pmatrix} , \begin{pmatrix} - 1 \\ - 1 \\ 0 \end{pmatrix} \rangle \end{align*}


\begin{align*} \psi _ { B _ { k } } = B _ { k } \left [ \frac { \sin ( k x ) } { \sin ( k a / 2 ) } - \frac { \sinh ( k ' x ) } { \sinh ( k ' a / 2 ) } \right ] , \end{align*}


\begin{align*} \mathfrak { M } _ { i } = \bigcup \limits _ { q _ { i } \leq P _ { i } } \bigcup \limits _ { \substack { 1 \leq a _ { i } \leq q _ { i } \\ ( a _ { i } , q _ { i } ) = 1 } } \mathfrak { M } _ { i } ( a _ { i } , q _ { i } ) , \mathfrak { m } _ { i } = \left [ \frac { 1 } { Q _ { i } } , 1 + \frac { 1 } { Q _ { i } } \right ] \backslash \mathfrak { M } _ { i } , \end{align*}


\begin{align*} \sqrt { 1 - 1 6 i m \beta \lambda / 3 } = \pm \left ( \sqrt { \frac { \sqrt { 1 ^ { 2 } + ( 1 6 m \beta \lambda / 3 ) ^ { 2 } } + 1 } { 2 } } - i \sqrt { \frac { \sqrt { 1 ^ { 2 } + ( 1 6 m \beta \lambda / 3 ) ^ { 2 } } - 1 } { 2 } } \right ) , \end{align*}


\begin{align*} \frac { \partial \psi } { \partial t } = - { \rm i } \widehat { H } ( t ) \psi , \psi \in \mathcal { H } , t \in \mathbb { R } , \end{align*}


\begin{align*} \nabla u = \frac { \widetilde { \nabla } u } { f ( u ) ^ 2 - | \widetilde { \nabla } u | _ { \widetilde { g } } ^ 2 } , | \nabla u | _ g ^ 2 = \frac { | \widetilde { \nabla } u | _ { \widetilde { g } } ^ 2 } { f ( u ) ^ 2 - | \widetilde { \nabla } u | _ { \widetilde { g } } ^ 2 } , \end{align*}


\begin{align*} 2 \Re \left ( \int _ { \omega _ c } \Delta u ( m \cdot \nabla \bar { u } ) d x \right ) = ( d - 2 ) \int _ { \omega _ c } \abs { \nabla u } ^ 2 d x + \int _ { \Gamma _ 0 } ( m \cdot \nu ) \left | \frac { \partial u } { \partial \nu } \right | ^ 2 d \Gamma . \end{align*}


\begin{align*} \sum _ { \substack { i \in M _ P , \\ i \neq i ^ * } } \sum _ { j \in N _ i } a _ { i j } \tilde x _ { i j } + \sum _ { \substack { i j \in P , \\ i \notin M _ 0 , i \neq i ^ * } } ( b - s ) \tilde x _ { i j } + a _ { i ^ * j ^ * } \tilde x _ { i ^ * j ^ * } & \leq s + ( b - s ) | \bar P | \\ & = s + ( | M _ P - M _ 0 | - 1 ) ( b - s ) \\ & = b + ( | M _ P - M _ 0 | - 2 ) ( b - s ) . \end{align*}


\begin{align*} y _ k = y _ 1 = \frac { U } { n } - \frac { L ( n - 1 ) } { 2 } , \end{align*}


\begin{align*} \frac { d P } { d t } = P B R ( t ) ^ { - 1 } B ^ \textsf { T } P - P A - A ^ \textsf { T } P - Q ( t ) , P ( t _ f ) = S . \end{align*}


\begin{align*} \Big ( ( \gamma _ j , \gamma ' _ j ) \big | ( \boldsymbol { \gamma } _ { - j } , \boldsymbol { \gamma } ' _ { - j } ) = ( \boldsymbol { a } , \boldsymbol { b } ) \Big ) \sim \mathcal { N } ( \boldsymbol { \eta } ^ { ( j ) } , \sigma ^ 2 \ , \mathbf { R } ^ { ( j ) } ) \ , \end{align*}


\begin{align*} f _ { ( s , r ) } = \tilde { f } _ { \tilde { \nu } ( s , r ) } \circ f _ r : \Big ( \dot r U ^ - P ^ + _ K \Big ) \bigcap f _ r ^ { - 1 } \Big ( \dot { \tilde { \nu } } ( s , r ) \tilde { U } ^ { - } \tilde { B } ^ + \Big ) \rightarrow \tilde { G } . \end{align*}


\begin{align*} \Upsilon _ { \mathrm { i n d e x } \ 3 } ( a , b , c , d ) = \{ g \in \Upsilon : a \cdot g _ { 1 , 2 } + b \cdot g _ { 1 , 3 } + c \cdot g _ { 2 , 1 } + d \cdot g _ { 3 , 1 } \equiv 0 \bmod 3 \} . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , T ] } E [ | N ^ { n , 3 } _ \tau | ^ 2 ] = 0 . \end{align*}


\begin{align*} \textup { d c o v } ( P ) = \sum _ { p \ , \in \ , P } x ^ { \textup { d c o v } ( p ) } = 1 + 7 x + 7 x ^ 2 + x ^ 3 \ , , \end{align*}


\begin{align*} G / A = ( G / A ) ^ 0 \rtimes F ' \end{align*}


\begin{gather*} \phi ( \tau , z ) = h _ 0 ( \tau ) \theta _ { 1 , 0 } ( \tau , z ) + h _ 1 ( \tau ) \theta _ { 1 , 1 } ( \tau , z ) \end{gather*}


\begin{align*} ( \widehat { \Theta } ^ n _ s ) ^ 2 = \sum _ { j = 1 } ^ 6 \Psi ^ { n , j } _ s , \end{align*}


\begin{align*} f _ { \mathbf { M ^ { \ast } } _ { - \xi _ { \boldsymbol { s } k } } } ( \boldsymbol { w } ) = c _ { n - 1 , \xi _ { \boldsymbol { s } k } } ^ { \ast } \overline { G } _ { n } \left \{ \frac { 1 } { 2 } \boldsymbol { w } ^ { T } \boldsymbol { w } + \frac { 1 } { 2 } \xi _ { \boldsymbol { s } k } ^ { 2 } \right \} , ~ \boldsymbol { w } \in \mathbb { R } ^ { n - 1 } , \end{align*}


\begin{align*} \frac { d } { d \theta } V _ { \theta } = \sum _ { s , t = 0 , 1 , \ldots } \gamma ^ t p _ { t , \theta } ( s ) \sum _ { a } \pi _ { \theta } ( a | s ) Q _ { \theta } ^ * ( s , a ) \frac { d } { d \theta } \log \pi _ { \theta } ( a | s ) . \end{align*}


\begin{align*} \partial _ t ^ n u ( t ) = A _ n ( t ) u ( t ) \end{align*}


\begin{align*} A = - \Delta _ z + \tfrac 1 4 | z | ^ 2 - i N , \end{align*}


\begin{align*} \nabla _ W h ( Y , Z ) - \nabla _ Z h ( Y , W ) = - \bar { R } ( \nu , Y , Z , W ) = 0 \end{align*}


\begin{align*} \overline { V } = \bigcup _ { k \in K } [ a _ { k } , b _ { k } ] \cup ( - \infty , a ] \cup [ b , \infty ) , \end{align*}


\begin{align*} & \left < \frac { i } h [ - h ^ 2 \Delta + V ( x ) - E ( h ) , \chi _ \alpha ( x _ n ) h D _ n ] u _ h , \ , u _ h \right > _ { L ^ 2 ( \Omega _ \Gamma ) } \\ = & \int _ { \Sigma _ { \Omega _ \Gamma } } \left ( 2 \chi _ \alpha ' ( x _ n ) \xi _ n ^ 2 - \chi _ \alpha ( x _ n ) \partial _ { x _ n } ( R + V ) \right ) \ , d \mu + o ( 1 ) \\ = & I _ 1 - I _ 2 + o ( 1 ) \end{align*}


\begin{align*} F ( x ) = \sum _ { k \ge 1 } \frac { a _ k } { \lambda _ k ^ 2 - x } . \end{align*}


\begin{align*} \log _ { \lambda } ( 1 + t ) = \sum _ { n = 1 } ^ { \infty } \frac { \lambda ^ { n - 1 } ( 1 ) _ { n , 1 / \lambda } } { n ! } t ^ { n } , ( \mathrm { s e e } \ [ 8 ] ) . \end{align*}


\begin{align*} \bar { \mathcal { S } } = \sum _ 1 ^ { N - 1 } ( \sup _ { L _ i } f _ { L _ i } - \sup _ { L _ { i + 1 } } f _ { L _ { i + 1 } } ) \left ( e ^ { - i \hat { \theta } } \int _ { \mathcal { E } _ i } \Omega \right ) . \end{align*}


\begin{align*} \partial _ t u ( t ) = A _ i ( t ) u ( t ) \end{align*}


\begin{align*} \P ( D _ n ) \geq c , \ \ \ n \in \N \end{align*}


\begin{align*} \bar \nabla _ Y \nu = A ( Y ) , \end{align*}


\begin{align*} m _ { \beta , \gamma , 1 , \eta } ( \theta ) = \frac { 1 } { K ^ * _ { \beta , \gamma , 1 , \eta } } \sum _ { k = 0 } ^ { \gamma - 1 } { \gamma - 1 \choose k } ( - 1 ) ^ { k } \theta ^ { - 1 - k + \beta } , 1 < y < \eta . \end{align*}


\begin{align*} \mathfrak { H i g h } ^ { j } _ { k , \tilde { k } } ( t _ 1 , t _ 2 ) = \int _ { t _ 1 } ^ { t _ 2 } \langle V ( s ) \rangle ^ { - 1 } E ( s , X ( s ) ) \cdot \widetilde { \mathfrak { H i g h } } ^ { j } _ { k , \tilde { k } } ( s ) d s , \end{align*}


\begin{align*} \texttt { t } ' _ { g f } = t _ { g \cdot w _ 0 ^ \zeta , f \cdot w _ 0 ^ \zeta } , f , g \in \Z ^ { m | n } _ { \zeta - } . \end{align*}


\begin{align*} \boldsymbol { y } _ { t , p } ^ { r } = \mathbf { H } ^ { r } \boldsymbol { v } _ { t , p } + \boldsymbol { n } _ { t , p } ^ { r } , \end{align*}


\begin{align*} \lim _ { | y | \to \infty } \operatorname { d i s t } \left ( x + i y , \operatorname { S p e c } A \right ) = \infty . \end{align*}


\begin{align*} \xi _ 2 ( s ) : = \pi ^ { - s } \ , \Gamma ( s ) \ , \zeta ( \Delta , s ) , \Omega ( s ) : = \frac { 1 } { 3 } \ , s \ , \pi ^ { 2 - s } \ , \Gamma ( s ) \ , \zeta ( \Delta , s - 1 ) . \end{align*}


\begin{align*} \lambda \cdot \bar { \psi } = \bar { \psi } \end{align*}


\begin{align*} \R _ { \geq 0 } ^ n = \{ x \in \mathbb { R } ^ n : \} \end{align*}


\begin{align*} \Delta _ { \lambda } ( \dot { t } ^ { \sharp } \dot { u } ^ { \flat , - 1 } g f _ { ( s , r ) } ( h ) ) & = \Delta _ { \lambda } ( \dot { t } ^ \sharp h _ 5 ^ { \sharp , - 1 } z _ 5 ^ \sharp \dot { r } ^ { - 1 , \sharp } ) = \Delta _ { \lambda } ( \iota ( \dot { t } ^ \sharp h _ 5 ^ { \sharp , - 1 } z _ 5 ^ \sharp \dot { r } ^ { - 1 , \sharp } ) ) \\ & = \Delta _ { \lambda } ( \iota ( \dot { t } ^ \sharp ) \iota ( h _ 5 ^ { \sharp , - 1 } ) z _ 5 ^ \sharp \iota ( \dot { r } ^ { - 1 , \sharp } ) ) . \end{align*}


\begin{align*} G ( 0 ; \Psi _ 1 , I _ + ( \Psi _ 1 ) _ o , f ) \le \int _ { \{ \Psi < 0 \} } | \tilde f _ 0 | ^ 2 \le \liminf _ { j \rightarrow + \infty } \int _ { \{ \Psi _ 1 < 0 \} \cap D _ j } | \tilde f _ j | ^ 2 = \lim _ { j \rightarrow + \infty } C _ j . \end{align*}


\begin{align*} - \frac { \hslash ^ { 2 } } { 2 m } \partial _ { x } ^ { 2 } \psi + \frac { \beta \hslash ^ { 4 } } { 3 m } \partial _ { x } ^ { 4 } \psi = E \psi , \end{align*}


\begin{align*} \iota _ j \circ \pi _ j \circ \varphi _ { n _ j , n _ { j - 1 } } ^ { m } = \varphi _ { n _ j , n _ { j - 1 } } ^ { m } \end{align*}


\begin{align*} \int _ { \Xi _ { \mu } } x ^ { \mu ' } e ^ { W / \hbar } d x _ 1 \wedge \cdots \wedge d x _ a = \delta _ { \mu \mu ' } , \end{align*}


\begin{align*} G \supset \bigcap _ { N = 1 } ^ { \infty } \bigcup _ { n = N } ^ { \infty } \bigcup _ { w \in \Sigma _ { \beta _ 1 } ^ n , v \in \Sigma _ { \beta _ 2 } ^ n } \tilde { J } _ { n , 1 } ( w , v ) \times \tilde { J } _ { n , 2 } ( w , v ) . \end{align*}


\begin{align*} P ( Q ( \infty ) \geq k ) = \rho ^ k , k \geq 0 . \end{align*}


\begin{align*} \delta ( a b ) - \delta ( a ) \delta ( b ) & = ( \delta ( a b ) - a b ) - ( \delta ( a ) - a ) \delta ( b ) - \delta ( a ) ( \delta ( b ) - b ) \\ & + ( \delta ( a ) - a ) ( \delta ( b ) - b ) \in \mathbb K ( H ) , \end{align*}


\begin{align*} ( P ( \Psi _ t ) P ^ T ( \Psi _ t ) ( x ) ) ^ { - 1 } = \begin{bmatrix} I _ n + O ( t ) & O ( t ^ 2 ) \\ O ( t ^ 2 ) & 2 t \cdot \begin{pmatrix} \begin{bmatrix} I _ { \frac { n ( n - 1 ) } { 2 } } & 0 \\ 0 & ( 3 \cdot \Xi ( \frac { 1 } { 3 } ) ) ^ { - 1 } \end{bmatrix} + O ( t ) \end{pmatrix} \end{bmatrix} _ . \end{align*}


\begin{align*} \tau ^ M : = \tau + M ; M = \mathrm { d i a g } ( M _ 1 \rho _ 1 , \dots , M _ n \rho _ n ) . \end{align*}


\begin{align*} \lim \limits _ { | \zeta | \to \infty } f ( \zeta ) = \mathfrak c , \end{align*}


\begin{align*} \log P ( r , k ) = \sum _ { i = 0 } ^ { r - 1 } \log \Bigl ( 1 + \dfrac { 2 i } { k } \Bigr ) < \sum _ { i = 0 } ^ { r - 1 } \dfrac { 2 i } { k } = \dfrac { r ( r - 1 ) } { k } \ \implies P ( r , k ) < e ^ { \frac { r ( r - 1 ) } { k } } . \end{align*}


\begin{align*} \psi _ { \mu } ( z ) = \int _ { \mathbb { T } } \frac { z \zeta } { 1 - z \zeta } \ , d \mu ( \zeta ) , \quad \eta _ { \mu } ( z ) = \frac { \psi _ { \mu } ( z ) } { 1 + \psi _ { \mu } ( z ) } , \end{align*}


\begin{align*} & \eta _ { x , p ; \delta } ( y ) = e ^ { i p ( y - x ) } \eta _ \delta ( y - x ) \end{align*}


\begin{align*} F ( \dot { x } , x , t ) = - \frac { \partial } { \partial x } \left [ \frac { \partial \Phi _ { g n } ( x , t ) } { \partial t } \right ] = \frac { \partial E _ { g n } ( \dot { x } , x , t ) } { \partial x } \ . \end{align*}


\begin{align*} \iota _ \Gamma ^ * \tilde { \mathfrak { o } } ^ \pi = o _ N \otimes ( \tilde { \mathfrak { o } } ^ \pi _ { v ^ o } \boxtimes \tilde { \mathfrak { o } } _ { v ^ c } ) , \end{align*}


\begin{align*} \tilde { \alpha } = \frac { \sqrt { ( d \tilde { x } ^ 1 ) ^ 2 + ( d \tilde { x } ^ 2 ) ^ 2 + ( d \tilde { x } ^ 3 ) ^ 2 } } { \tilde { x } ^ 3 } . \end{align*}


\begin{align*} \rho \left ( A \right ) = \int _ { E _ { 1 } } \hat { \rho } \left ( A \right ) \mu _ { \rho } \left ( \mathrm { d } \hat { \rho } \right ) , A \in \mathcal { U } . \end{align*}


\begin{align*} ( A x , y ) _ V = \sum \limits _ { \begin{subarray} { 1 } { u , v \in V ; } \\ { u \neq v } \end{subarray} } x ( u ) y ( v ) \sum \limits _ { e \in E _ u \cap E _ v } \frac { \delta _ E ( e ) } { | e | ^ 2 } . \end{align*}


\begin{align*} S _ { \alpha , \beta } = \left \{ x = \left ( \begin{matrix} x _ { 1 } \\ x _ { 2 } \end{matrix} \right ) \in \mathbb { Z } ^ { 2 } \mid 0 \leq x _ { 1 } , x _ { 2 } < p q \right \} . \end{align*}


\begin{align*} f \mapsto \sup _ k \left ( ( 2 k ) ! ^ { - 1 } \int f ^ { 2 k } \ d \mu \right ) ^ { 1 / 2 k } = \left \bracevert f \right \bracevert _ { \cosh _ 2 } \ . \end{align*}


\begin{align*} G ( s ) = \sum _ { n = 1 } ^ { N } \frac { 1 } { s ^ { 2 } + 2 \zeta _ n \omega _ { n } s + \omega _ { n } ^ { 2 } } , \end{align*}


\begin{align*} f _ { \ , \kappa } \ , ( x ) = \sqrt { \frac { 2 \beta \kappa } { \pi } } \left ( 1 + \frac { 1 } { 2 } \kappa \right ) \frac { \Gamma \ ! \left ( \frac { 1 } { 2 \kappa } + \frac { 1 } { 4 } \right ) } { \Gamma \ ! \left ( \frac { 1 } { 2 \kappa } - \frac { 1 } { 4 } \right ) } \ , \ , \ , \exp _ { \kappa } ( - \beta \ , x ^ 2 ) \ \ . \end{align*}


\begin{align*} \psi _ n ( y ) = \psi _ n ( T _ i y ) \quad \textrm { f o r a l l } \ \ i \in \{ 1 , \cdots , 1 2 \} \ \ \textrm { a n d } \ y \in \mathbb { R } ^ N . \end{align*}


\begin{align*} { \lambda _ k } = \frac { \partial } { { \partial { \eta _ k } } } w \left ( 0 \right ) \end{align*}


\begin{align*} \frac { \partial E } { \partial z ^ j _ { \eta } } \frac { \partial ^ 2 \varphi ^ j } { \partial x ^ { \epsilon } \partial x ^ { \eta } } = 2 b ^ 2 f ( x ^ 1 ) \Big [ \delta _ { \epsilon 1 } f ' \left ( x ^ 1 \right ) \left \lbrace f ( x ^ 1 ) f '' \left ( x ^ 1 \right ) + f '^ 2 ( x ^ 1 ) + \sin ^ 2 \left ( x ^ 2 \right ) \right \rbrace + \delta _ { \epsilon 2 } f ( x ^ 1 ) \sin \left ( x ^ 2 \right ) \cos \left ( x ^ 2 \right ) \Big ] . \end{align*}


\begin{align*} f ( w , x , y , z ) = f ( w , x , z , y ) . \end{align*}


\begin{align*} i \widehat { u } _ t = \left \{ \mp \frac { p + 1 } { 4 \pi } \int _ \mathbb { T } | u | ^ { p - 1 } ( x , t ) \ , d x \right \} \frac { \widehat { v } } { L _ t [ u ] } + i \frac { \widehat { v } _ t } { L _ t [ u ] } = \mp \widehat { \mathcal { R } ^ 1 } [ v , \cdots , v ] \frac { 1 } { L _ t [ u ] } + i \frac { \widehat { v } _ t } { L _ t [ u ] } , \end{align*}


\begin{align*} \hat { A } : = J ^ { - 1 } A J , \end{align*}


\begin{align*} C ( u , v ) = u ^ { \dagger } \cdot A \cdot v , \end{align*}


\begin{align*} y = \frac { t ( t ' - y ) + t ' f _ t ( y ) } { t ' - t } \leq \frac { t \max ( - f ( y ) , 0 ) } { - f ( t ) } + \frac { t ' f _ t ( y ) } { t ' - t } . \end{align*}


\begin{align*} E ( s ) : = \ker ( | \Gamma | - s I ) = \ker ( H _ u ^ 2 - s ^ 2 I ) , E _ 1 ( s ) : = \ker ( | \Gamma _ 1 | - s I ) = \ker ( K _ u ^ 2 - s ^ 2 I ) ; \end{align*}


\begin{align*} \left \{ \begin{array} { l } a _ { n + 1 } = p a _ n + 1 , \ a _ 1 = 0 , \\ b _ { n + 1 } = p b _ n + 1 , \ b _ 1 = 1 \end{array} \right . \end{align*}


\begin{align*} \Theta \equiv \Theta ( p , s _ 1 , q , s _ 2 ) = \begin{cases} \min \{ 1 , p s _ 1 / ( p - 1 ) \} \mbox { i f } q s _ 2 < p s _ 1 + 2 ( 1 - s _ 1 ) , \\ \min \{ 1 , q s _ 2 / ( q - 1 ) \} \mbox { i f } p s _ 1 < q s _ 2 . \end{cases} \end{align*}


\begin{align*} u ^ s _ { A , q } ( x , y ) = \sum _ { \substack { ( \ell _ 1 , m _ 1 ) \in \Gamma \\ ( \ell _ 2 , m _ 2 ) \in \Gamma } } \gamma _ { \ell _ 1 m _ 1 \ell _ 2 m _ 2 } h ^ { ( 1 ) } _ { \ell _ 1 } ( k \vert x \vert ) Y ^ { m _ 1 } _ { \ell _ 1 } \left ( \widehat { x } \right ) h ^ { ( 1 ) } _ { \ell _ 2 } ( k \vert y \vert ) Y ^ { m _ 2 } _ { \ell _ 2 } \left ( \widehat { y } \right ) . \end{align*}


\begin{align*} k = \frac { 2 } { \sqrt { 4 b - 1 } } , l = \frac { 1 } { \sqrt { 4 b - 1 } } . \end{align*}


\begin{align*} G _ p = p \delta _ 1 + ( 1 - p ) \delta _ \infty , p > p _ c \end{align*}


\begin{align*} u ( z , l ) = { \underset { \bar { z } \in \{ 1 , . . . , Z \} } { \textrm { m a x } } } \biggl ( \textrm { m i n } \biggl \{ g [ \bar { z } , z , l ] , u ( \bar { z } , l - 1 ) \biggr \} \biggr ) , \end{align*}


\begin{align*} P ( E _ { x \rightarrow y } | S _ { i } ) = { \left ( \frac { i } { m _ x } \right ) } ^ { k _ x } , \end{align*}


\begin{align*} \beta = \frac { h - 1 } { 1 - \kappa h } . \end{align*}


\begin{align*} { \rm d } ( { \rm L } _ n ) & \ge { \rm d } \bigg ( \bigcup _ { a \in A } \bigcup _ { c \in C _ a ^ v } { \rm L } _ { a c } \bigg ) = \sum _ { a \in A } \sum _ { c \in C _ a ^ v } { \rm d } ( { \rm L } _ { a c } ) = \sum _ { a \in A } { \rm d } ( { \rm L } _ a ) \ , \sum _ { c \in C _ a } \frac { 1 } { c } , \end{align*}


\begin{align*} \label [ I e q ] { e q : T N h a t } \widehat { T } _ { N , t } \Phi ( \mu _ { \mathbf { x } ^ N } ) = T _ { N , t } [ \Phi \circ \boldsymbol { \mu } _ N ] ( \mathbf { x } ^ N ) , \end{align*}


\begin{align*} \sigma = \sigma ( \Lambda ) = \Lambda ( [ 0 , 1 ] ) . \end{align*}


\begin{align*} \mathsf { T } _ { n } ( R ) = \begin{pmatrix} R & 0 & \cdots & 0 \\ R & R & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ R & R & \cdots & R \\ \end{pmatrix} \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta _ p u _ 1 + \Delta _ p u _ 2 = \hat f & \hbox { i n } \Omega \\ u _ 1 - u _ 2 = g _ 1 - g _ 2 & \hbox { o n } \partial \Omega . \end{array} \right . \end{align*}


\begin{align*} \frac { 1 } { \gamma ^ { 1 / 2 } } = 1 - \frac { m } { m + 1 } \end{align*}


\begin{align*} \mathbf { H } ^ { r } = \sum _ { k = 1 } ^ { K } x _ { k } ^ { r } \boldsymbol { a } \left ( \theta _ { k } ^ { r } \right ) \boldsymbol { a } ^ { H } \left ( \theta _ { k } ^ { r } \right ) , \end{align*}


\begin{align*} \varphi ( t ) = \ ( ( \alpha + \delta + \beta - 1 ) \eta ( T ^ * - t ) \ ) ^ { \frac { 1 } { 1 - \delta - \beta - \alpha } } , \end{align*}


\begin{align*} { n + 1 \brack k } = n { n \brack k } + { n \brack k - 1 } , { n \brack 0 } = \begin{cases} 1 , & n = 0 , \\ 0 , & n \neq 0 . \end{cases} \end{align*}


\begin{align*} H \Phi _ { \nu , \nu ' } = ( | \nu | + | \nu ' | + n ) \Phi _ { \nu , \nu ' } . \end{align*}


\begin{align*} \omega ( a ) = \omega ( c ) = 0 \end{align*}


\begin{align*} \left \lbrace \begin{array} { l l } G . a _ { + } = \lambda a _ { + } ; \\ G . a _ { - } = - \lambda a _ { - } . \end{array} \right . \end{align*}


\begin{align*} \Pi _ 1 u ( t _ n ) = u ( t _ n ) , n = 0 , \dots , N , \Pi _ 1 u | _ { I _ n } \in \mathbb { P } _ 1 ( I _ n ) . \end{align*}


\begin{align*} \frac { b + n ( b _ 0 + b _ 1 ) + \mu } { 4 n + 1 } = \theta < \frac { 4 n + 3 } { 4 ( 4 n + 1 ) } + \epsilon . \end{align*}


\begin{align*} \tau \left ( \theta ^ { k } \beta \right ) = H ^ { k } \tau ( \beta ) = H ^ { k } \overline { \beta } , 0 \leq k \leq n - 1 . \end{align*}


\begin{align*} \alpha _ { k , j } & = \begin{cases} \sum \limits _ { i = 1 } ^ { n } a _ i \cdot \mathtt { w } ^ + ( \phi _ i ) \geqslant 1 - c & t _ { k , j } = f ^ \sharp _ { k , j } \\ \sum \limits _ { i = 1 } ^ { n } a _ i \cdot \mathtt { w } ^ + ( \phi _ i ) < - c & \end{cases} \end{align*}


\begin{align*} \dot C _ t ( V _ t ) ( \varphi ) \dot C _ t - \frac { 1 } { 2 } \ddot { C } _ t & \geq \Big [ \frac { C _ t } { t } \big ( C _ t ^ { - 1 } - \chi _ t C _ t ^ { - 2 } \big ) \frac { C _ t } { t } + \frac { 1 } { t } A C _ t \Big ] \dot C _ t \\ & = \Big [ \frac { 1 } { t } \Big ( \frac { 1 } { t } + A \Big ) C _ t - \frac { \chi _ t } { t ^ 2 } \Big ] \dot C _ t \\ & = \Big [ \frac { 1 } { t } - \frac { \chi _ t } { t ^ 2 } \Big ] \dot C _ t = \dot \kappa _ t \dot C _ t . \end{align*}


\begin{align*} x ( t ) = \frac { 2 t ^ 3 - 2 t ^ 2 } { 2 t - 1 } \end{align*}


\begin{align*} p ( t , v , g ) = \frac { 1 } { V _ F } p _ 0 ( t , g ) . \end{align*}


\begin{align*} _ { \Gamma } \circ \gamma = ( \gamma _ t \circ \nabla ) ' & & & & \mathbf { c u r l } _ { \Gamma } \circ \gamma _ t = ( \gamma _ n \circ \mathbf { c u r l } ) ' . \end{align*}


\begin{align*} A ^ { k l } _ { i j } \xi _ i \xi _ j \eta _ k \eta _ l : = ( D _ { p _ k p _ l } A _ { i j } ) \xi _ i \xi _ j \eta _ k \eta _ l \ge 0 \end{align*}


\begin{align*} \prescript { } { \gamma } { \mathcal { F } } _ \tau f ( \lambda ) : = \int _ G e ^ \tau _ { \lambda , 1 } ( g ) f ( g ) \ ; d g , \ ; \ ; \ ; \ ; \lambda \in \mathfrak { a } ^ * _ \mathbb { C } . \end{align*}


\begin{align*} \kappa _ 4 = \sum _ { k = 0 } ^ { \infty } \int _ 0 ^ 1 \int _ { 0 } ^ { 1 \wedge ( y + k ) } [ ( y + k ) ^ \alpha - ( y + k - z ) ^ \alpha ] ^ 2 d z d y . \end{align*}


\begin{align*} S ( x ) = S _ \infty ( x ) ( 1 + \eta N * \rho ) , \end{align*}


\begin{align*} \iint _ { E \times E } \varphi _ 2 ( z _ 1 ' , z _ 2 ' ) \Gamma ^ { ( 2 ) } ( z _ 1 , z _ 2 , \dd z _ 1 ' , \dd z _ 2 ' ) & = \int _ { \Theta } \varphi _ 2 ( \psi _ 1 ( z _ 1 , z _ 2 , \theta ) , \psi _ 2 ( z _ 1 , z _ 2 , \theta ) ) \nu ( \dd \theta ) \\ & = \int _ { \Theta } \varphi _ 2 ( \psi _ 2 ( z _ 2 , z _ 1 , \theta ) , \psi _ 1 ( z _ 2 , z _ 1 , \theta ) ) \nu ( \dd \theta ) , \end{align*}


\begin{align*} \varphi ( u _ j ) = ( w _ { 1 , j } , w _ { 2 , j } , \hdots , w _ { L , j } ) \in B \end{align*}


\begin{align*} \mathrm { T V } _ { q } ( X ) = \mathrm { E } [ ( X - \mathrm { T C E } _ { X } ( x _ { q } ) ) ^ { 2 } | X > x _ { q } ] \end{align*}


\begin{align*} \{ x y = 0 \} / \mu _ d , \end{align*}


\begin{align*} k = \frac { 1 } { 2 \hslash \sqrt { \beta / 3 } } \sqrt { 1 - \sqrt { 1 - \frac { 1 6 } { 3 } m \beta | E | } } \end{align*}


\begin{gather*} H ( \Q ^ E | \R ^ E ) = \log 2 > 0 , \\ H ( \Q ^ { D \cup E \cup F } | \R ^ { D \cup E \cup F } ) = H ( \Q ^ { D \cup E } | \R ^ { D \cup E } ) = H ( \Q ^ { E \cup F } | \R ^ { E \cup F } ) = 0 . \end{gather*}


\begin{align*} \begin{cases} i u _ t + \bigtriangleup u \pm | u | ^ { p - 1 } u = 0 \\ u ( x , 0 ) = u _ 0 \in H ^ s _ x ( \mathbb { T } ) , \end{cases} \end{align*}


\begin{align*} H _ { w } e _ { ( k ) } H _ { \omega _ { ( d ) } } H _ { y } = \sum \limits _ { \substack { ( \omega ' , \chi , \varpi ' ) \in B _ { k , n } ^ { * } \times \mathfrak { S } _ { 2 k + 1 , n } \times B _ { k , n } \\ \ell ( \omega ' ) + \ell ( \chi ) + \ell ( \varpi ' ) \leq \ell ( w ) + \ell ( \omega _ { ( d ) } ) + \ell ( y ) } } r _ { \omega ' } s _ { \varpi ' } t ^ { \chi } H _ { \omega ' } e _ { ( k ) } H _ { \chi } H _ { \varpi ' } \end{align*}


\begin{align*} \frac { 1 } { k ! } \big ( \log _ { \lambda } ( 1 + t ) \big ) ^ { k } = \sum _ { n = k } ^ { \infty } S _ { 1 , \lambda } ( n , k ) \frac { t ^ { n } } { n ! } , \end{align*}


\begin{align*} K ^ { G ( K / E ) } = F _ \infty ^ { G ( F _ \infty / E ) } = E . \end{align*}


\begin{align*} \partial _ s \left ( e ^ { f _ o } v _ s \right ) = \Delta _ { g _ o } ^ { f _ o } \left ( e ^ { f _ o } v _ s \right ) . \end{align*}


\begin{align*} a _ n = \frac { p ^ { n - 1 } - 1 } { p - 1 } . \end{align*}


\begin{align*} \mathcal A _ k : = \big \{ f \in ( \mathcal H , K _ 1 ) \otimes ( \mathcal H , K _ 2 ) : \big ( \big ( \tfrac { \partial } { \partial \zeta } \big ) ^ { \boldsymbol i } f ( z , \zeta ) \big ) _ { | \Delta } = 0 , \ ; | \boldsymbol i | \leq k \big \} , \end{align*}


\begin{align*} & c _ 2 ( F ( \mathbf { x } _ 0 ) , F ( \mathbf { y } _ 0 ) , \delta ) = c _ 1 ( g _ \delta ( F ( \mathbf { x } _ 0 ) ) , g _ \delta ( F ( \mathbf { y } _ 0 ) ) ) \\ \neq \ , & c _ 2 ( F ( \mathbf { x } _ 1 ) , F ( \mathbf { y } _ 1 ) , \delta ) = c _ 1 ( g _ \delta ( F ( \mathbf { x } _ 1 ) ) , g _ \delta ( F ( \mathbf { y } _ 1 ) ) ) \end{align*}


\begin{align*} u ^ \infty _ { A , q } ( d , \theta ) = \underset { ( \ell _ 1 , m _ 1 ) \in \Gamma } { \sum } \ ; \underset { ( \ell _ 2 , m _ 2 ) \in \Gamma } { \sum } \mu _ { \ell _ 1 m _ 1 \ell _ 2 m _ 2 } Y ^ { m _ 1 } _ { \ell _ 1 } ( d ) Y ^ { m _ 2 } _ { \ell _ 1 } ( \theta ) , \end{align*}


\begin{align*} \tilde { w } ( \kappa , h ) : = \ \frac { 2 ( 1 - h ) + 2 ( 1 - \kappa h ) } { h ( 2 - h ) ( 2 - \kappa h ) } \end{align*}


\begin{align*} { R } ^ { i } _ { k } = 2 \frac { \partial { G } ^ i } { \partial x ^ k } - y ^ j \frac { \partial ^ 2 { G } ^ i } { \partial x ^ j \partial y ^ k } + 2 { G } ^ j \frac { \partial ^ 2 { G } ^ i } { \partial y ^ j \partial y ^ k } - \frac { \partial { G } ^ i } { \partial y ^ j } \frac { \partial { G } ^ j } { \partial y ^ k } . \end{align*}


\begin{align*} \Omega _ { \mu } = \{ r \zeta : \zeta \in \mathbb { T } , 0 \le r < R _ { \mu } ( \zeta ) \} . \end{align*}


\begin{align*} C _ 1 - \alpha \langle x \rangle \leq M _ \infty ( x ) : = \log ( S _ \infty ( x ) ) \leq C _ 2 - \alpha \langle x \rangle . \end{align*}


\begin{gather*} \Box _ { g _ { i , t } } u _ { i , t } = 0 M \times [ s , s ' ] , \\ u _ { i , s } = u ^ 0 _ i , \end{gather*}


\begin{align*} \widetilde U _ \tau = \bigsqcup _ { \sigma } H _ \sigma , \end{align*}


\begin{align*} & \sup _ { x \geq 1 } \Vert ( w , \Omega ) x ^ { - \sigma _ 0 } \Vert _ { L _ \eta ^ 2 } ^ 2 + \Vert ( w , \Omega ) x ^ { - \sigma _ 0 - \frac { 1 } { 2 } } \Vert _ { L _ x ^ 2 L _ \eta ^ 2 } ^ 2 + \Vert ( w _ \eta , \Omega _ \eta ) x ^ { - \sigma _ 0 } \Vert _ { L _ x ^ 2 L _ \eta ^ 2 } ^ 2 \\ & \lesssim R ( \delta , \sigma ) ^ 3 + R ( \delta , \sigma ) ^ 4 + R ( \delta , \sigma ) ^ 6 . \end{align*}


\begin{align*} k _ { v , n } = \min \left \{ v \left ( \psi _ n ^ 2 ( x ( P ) ) \right ) , v \left ( \phi _ n ( x ( P ) ) \right ) \right \} . \end{align*}


\begin{align*} \begin{aligned} \frac { h \big [ \kappa h ^ 2 - 2 h ( 1 + \kappa ) + 3 \big ] } { 2 L \big [ 2 - h ( 1 + \kappa ) \big ] } \big \| g _ { 0 } \big \| ^ 2 + \frac { h } { 2 L \big [ 2 - h ( 1 + \kappa ) \big ] } \| g _ 1 \| ^ 2 { } \leq { } f _ { 0 } - f _ { 1 } \end{aligned} \end{align*}


\begin{align*} \sigma _ 0 = \frac 1 2 ( \sigma + \sigma ' ) , \end{align*}


\begin{align*} U = \Xi \Lambda K ^ T , \end{align*}


\begin{align*} \left \langle u , v \right \rangle _ { H ^ 2 _ { a } ( 0 , 1 ) } = \int _ { 0 } ^ { 1 } f v \ , d x \Longleftrightarrow \int _ { 0 } ^ { 1 } \biggl ( u v + a u '' v '' \biggr ) d x = \int _ { 0 } ^ { 1 } f v \ , d x , \end{align*}


\begin{align*} R ( \delta , \sigma ) : = { \rm { m a x } } \{ \Vert ( e _ \sigma , z ^ l ( e _ \delta - \delta ) ) \Vert _ { L ^ \infty } , \Vert z ^ l ( e ' _ \delta , e ' _ \sigma ) \Vert _ { L ^ \infty } , \Vert z ^ l ( e ' _ \delta , e ' _ \sigma , e '' _ \delta , e '' _ \sigma ) \Vert _ { L ^ 2 } \} . \end{align*}


\begin{align*} \int _ M ^ x e ^ { \lambda y } d F _ Z ( y ) & \leq \ , e ^ { \lambda M } ( 1 - F _ Z ( M ) ) + \lambda \int _ M ^ x e ^ { \lambda y } ( 1 - F _ Z ( y ) ) d y \\ [ 3 p t ] & \leq \ , V e ^ { \lambda M } M ^ { - \gamma } + \lambda V \int _ M ^ x e ^ { \lambda y } y ^ { - \gamma } d y \\ [ 3 p t ] & = \ , V e ^ { 2 \gamma } \left ( \frac { \lambda } { 2 \gamma } \right ) ^ { \gamma } + V e ^ { \lambda x } x ^ { - \gamma } \int _ 0 ^ { \lambda ( x - M ) } e ^ { - w } \left ( 1 - \frac { w } { \lambda x } \right ) ^ { - \gamma } d w , \end{align*}


\begin{gather*} t _ 1 ( \tau , z ) = \overline { \theta _ { 1 , 0 } ^ 0 ( \tau ) } \theta _ { 1 , 0 } ( \tau , z ) + \overline { \theta _ { 1 , 1 } ^ 0 ( \tau ) } \theta _ { 1 , 1 } ( \tau , z ) \end{gather*}


\begin{align*} p ( t , g ) : = p _ 0 ( t , g ) . \end{align*}


\begin{align*} \norm { \rho _ t ( \beta ) f _ t } ^ 2 & = \Lambda _ \Omega ( t ) \int _ { \chi \in \mathcal { X } _ p ^ { t e m p } / W } \chi ( \phi _ { \beta ^ \ast \ast \beta } ) \ , d \nu _ p ( \chi ) + O _ \Omega \Big ( \| \beta ^ \ast \ast \beta \| _ { L ^ 1 } t ^ { d - \delta } \Big ) \\ & = \Lambda _ \Omega ( t ) \int _ { \chi \in \mathcal { X } _ p ^ { t e m p } / W } | \beta ( \omega _ { \chi } ) | ^ 2 \ , d \nu _ p ( \chi ) + O _ \Omega \Big ( \| \beta \| ^ 2 _ { L ^ 1 } t ^ { d - \delta } \Big ) . \end{align*}


\begin{align*} h ^ * _ P = ( h _ 1 ^ * , \ldots , h _ s ^ * , 0 , \ldots , 0 ) \in \Z ^ d \ , , \end{align*}


\begin{align*} \lim \limits _ { r \rightarrow \infty } \left . \sum \limits _ { i = 0 } ^ { b r } \dbinom { a r } { i } \middle / r \dbinom { a r } { b r } \right . = 0 . \end{align*}


\begin{align*} \left ( \bigcup _ { j = 1 } ^ e T _ { e , j } \right ) \cup \left ( \bigcup _ { \ell = 1 } ^ e U _ { e , \ell } \right ) = [ 4 k d + 1 , 4 k d + 8 k e ] \cup [ 4 k N + 1 , 4 k N + 4 e ] = [ 4 k d + 1 , 4 k N + 4 e ] , \end{align*}


\begin{align*} P _ { o u t } & = 1 - \textrm { P r } \bigg [ \bar { \gamma _ 1 } \ge 1 , \bar { \gamma _ 2 } \ge 1 , . . . , \bar { \gamma _ i } \ge 1 , . . . , \bar { \gamma _ N } \ge 1 \bigg ] \\ & = 1 - \textrm { P r } \bigg [ { \underset { i \in \{ 1 , . . . , N \} } { \textrm { m i n } } } \{ \bar { \gamma _ i } \} \ge 1 \bigg ] . \end{align*}


\begin{align*} \Phi ( t , \tau ) = I + \int _ \tau ^ { t } A ( \tau ' ) \Phi ( \tau ' , \tau ) d \tau ' \quad t \geq \tau \geq 0 . \end{align*}


\begin{align*} \Bigl ( { \rm t w o \ s u m m a t i o n s \ i n \ \eqref { b 1 } } \Bigr ) = \sum _ { k = 1 } ^ N & \det \left ( ( \lambda _ i ( t ) I _ N - H _ { \alpha } ( t ) ( \lambda _ j ( t ) I _ N - H _ { \alpha } ( t ) ) _ { k | k } \right ) \\ - 2 \sum _ { \l - k > 1 } & \det \left ( ( \lambda _ i ( t ) I _ N - H _ { \alpha } ( t ) ) _ { k | \l } \right ) \det \left ( ( \lambda _ j ( t ) I _ N - H _ { \alpha } ( t ) ) _ { \l | k } \right ) . \end{align*}


\begin{align*} \begin{aligned} c ' ( h ) = - \kappa \left ( 1 + \kappa \right ) h ^ 3 + \left [ 3 \kappa + \left ( 1 + \kappa \right ) ^ 2 \right ] h ^ 2 - 4 \left ( 1 + \kappa \right ) h + 4 \end{aligned} \end{align*}


\begin{align*} u ( t ) = 1 + t ^ \beta + H ( t - r ) t ^ \beta , \end{align*}


\begin{align*} - \frac { \mathrm { d } } { \mathrm { d } x } \log \widetilde { \Phi } ( x ) \geq - \frac { \mathrm { d } } { \mathrm { d } x } \log f ( x ) = \frac { 1 } { 2 ( x + 2 ) \sqrt { \log ( x + 2 ) } } . \end{align*}


\begin{align*} \Lambda _ k ^ \lambda g = g \times _ \lambda \varphi _ k ^ \lambda . \end{align*}


\begin{align*} b _ 1 \sqrt { n _ 1 } + b _ 2 \sqrt { n _ 2 } + \cdots + b _ k \sqrt { n _ k } = 0 \end{align*}


\begin{align*} R ( r ) = \begin{cases} \frac { 1 } { 2 } ( 1 + \cos \pi r ) , & 0 \leq r \leq 1 , \\ 0 , & r > 1 . \end{cases} \end{align*}


\begin{align*} \begin{bmatrix} - A ^ T \hat z \\ A \hat x \end{bmatrix} \in \begin{bmatrix} \partial f ( \hat x ) + \nabla h ( \hat x ) \\ \partial g ^ \ast ( \hat z ) \end{bmatrix} \end{align*}


\begin{align*} \theta _ { N + 2 \ , B } = \sum _ { C = 1 } ^ { N + 1 } h _ { B C } \theta _ C \end{align*}


\begin{align*} = s ^ 2 \sin \theta \big [ 1 - \cos ( \theta - \tilde { \theta } ) + \sin \theta \sin \tilde { \theta } ( 1 - \cos ( \phi - \tilde { \phi } ) ) \big ] \gtrsim s ^ 2 \sin \theta | 1 - \cos ( \theta - \tilde { \theta } ) | . \end{align*}


\begin{align*} \langle \Phi _ 0 ( x ) ^ { ( 0 ) } ( \alpha ) , e \rangle & = \Phi _ 0 ( x ) ^ { ( 0 ) } [ \langle \alpha , e \rangle ] - [ \Phi _ 0 ( x ) ^ { ( 0 ) } , \iota _ e ] ^ { ( - 1 ) } ( \alpha ) \\ & = \varrho ( x ) [ \langle \alpha , e \rangle ] - \langle \alpha , \nabla _ x ( e ) \rangle . \end{align*}


\begin{align*} \begin{array} { l l } \partial _ t ^ n \psi ( t ) = A _ { n } ( t ) \psi ( t ) \end{array} \end{align*}


\begin{align*} \lim \limits _ { n \rightarrow \infty } \left \langle J ' _ \lambda ( v _ n ) , v _ n - v \right \rangle = 0 . \end{align*}


\begin{align*} \mathcal { G } ( r ( T ) , x ) & = x - \sum _ { j = 1 } ^ { k } \dfrac { \alpha _ j } { \mathcal { G } ( r ( T _ j ) , x ) } . \end{align*}


\begin{align*} V _ { \theta } ^ { \eta } = \sum _ { s } \eta ( s ) J _ { \theta } ^ * ( s ) , \end{align*}


\begin{gather*} \lim _ { | y | \to \infty } \left \| ( 1 + ( w - x ) R ( \lambda , A _ 2 ) ) ^ { - 1 } \right \| _ { \infty } = 1 \\ \lim _ { | y | \to \infty } \left \| ( 1 + ( x - w ) R ( \mu , A _ 1 ) ) \right \| _ { \infty } = 1 . \end{gather*}


\begin{align*} \varphi ( T ) = \lim _ { N \to \infty } \frac { 1 } { N } \sum _ { n = 1 } ^ N \varphi ( u _ n ^ * T u _ n ) = 0 . \end{align*}


\begin{align*} [ \alpha , v _ n ] _ { L , n } = [ \alpha _ m , v _ m ] _ { E _ { \rho } ^ m , m } & = \dfrac { 1 } { \eta ^ m } \operatorname { T } _ m ( \lambda _ { \rho } ( \alpha _ m ) v _ m ^ { - 1 } ) \cdot _ { \rho } v _ m \\ & = \dfrac { 1 } { \eta ^ m } \operatorname { T } _ m ( \lambda _ { \rho } ( \alpha _ m ) v _ m ^ { - 1 } ) \cdot _ { \rho } ( \eta ^ m \cdot _ { \rho } v _ { 2 m } ) \\ & = \operatorname { T } _ m ( \lambda _ { \rho } ( \alpha _ m ) v _ m ^ { - 1 } ) \cdot _ { \rho } v _ { 2 m } . \end{align*}


\begin{align*} u \wedge u ^ t ( \cdot ) = \min \{ u ( \cdot ) , u ^ t ( \cdot ) \} , u \vee u ^ t ( \cdot ) = \max \{ u ( \cdot ) , u ^ t ( \cdot ) \} . \end{align*}


\begin{align*} \lambda _ { n , k } = \int _ 0 ^ 1 x ^ { k - 2 } ( 1 - x ) ^ { n - k } \Lambda ( d x ) . \end{align*}


\begin{align*} \widetilde { H } ( S _ { [ n ] } ( f _ 2 ) \circ \phi \circ S _ { [ n ] } ( f _ 1 ) ) = \left \{ \begin{array} { c c c } f _ 2 \circ \widetilde { H } ( \phi ) \circ f _ 1 & \mbox { i f } & n \ \mbox { i s o d d } , \\ \det ( f _ 1 ) \det ( f _ 2 ) f _ 2 \circ \widetilde { H } ( \phi ) \circ f _ 1 & \mbox { i f } & n \ \mbox { i s e v e n } . \end{array} \right . \end{align*}


\begin{align*} \norm { I _ t - \Tilde { \Psi } _ t } _ { C ^ { s , \alpha } ( M ) } = O ( t ^ { l + \frac { 1 } { 2 } - \frac { s + \alpha } { 2 } } ) , \end{align*}


\begin{align*} \sum _ { m = 1 } ^ \infty \exp \Big \{ - \sum _ { i = 1 } ^ m \mu ( ( A ( m ) ^ { { \rho } i } , \infty ) ) \Big \} < \infty . \end{align*}


\begin{align*} L f = \mathcal G f \end{align*}


\begin{align*} \frac { \partial C } { \partial z ^ j _ { \eta } } \frac { \partial ^ 2 \varphi ^ j } { \partial x ^ { \epsilon } \partial x ^ { \eta } } = \frac { f ' ( x ^ 1 ) } { \sqrt { 1 + f '^ 2 ( x ^ 1 ) } } \delta _ { \epsilon 1 } \left [ f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \right ] , \end{align*}


\begin{align*} \delta ^ { ( s ) } v _ k + \frac { \sum _ { i j } ( \sum _ l \Gamma ^ { k ( s ) } _ { i j , l } c _ l ) v _ i ^ { \phi ^ s } v _ j ^ { \phi ^ s } } { ( \sum _ l r _ l c _ l ) ( \sum _ l v _ l ^ { \phi ^ s } ) } = 0 , \ \ k \in \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} 2 g ( \nabla _ X Y , Z ) = X g ( Y , Z ) + Y g ( X , Z ) - Z g ( X , Y ) + g ( [ X , Y ] , Z ) - g ( Y , [ X , Z ] - 2 P _ X Z ) - g ( X , [ Y , Z ] ) . \end{align*}


\begin{align*} K _ { r s } = - \alpha ( \beta _ { r r } + ( \beta _ r ) ^ 2 ) \end{align*}


\begin{align*} \widetilde { B _ 0 } S _ { n } ( t ) x = \int _ 0 ^ t \widetilde { B _ 0 } T ( t - s , A ) \tilde { B } S _ { n - 1 } ( t ) x \ , \mathrm { d } s , \end{align*}


\begin{align*} \tilde B ( | v - v _ * | , | \tilde r | ) = | v - v _ * | ^ \gamma \tilde b ( | \tilde r | ) , \tilde r = u / | u | \cdot \omega . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \log \sup _ { u \in V } p _ n ( u , v ) } { \log n } = 0 \end{align*}


\begin{align*} \Re \left \langle \begin{pmatrix} \varphi \\ \psi \end{pmatrix} , B ( s ) \begin{pmatrix} \varphi \\ \psi \end{pmatrix} \right \rangle _ \Gamma \geq \beta \ , \min ( 1 , | s | ^ 2 ) \frac { \Re s } { | s | ^ 2 } \left ( \| \varphi \| ^ 2 _ { - 1 / 2 , \Gamma } + \| \psi \| ^ 2 _ { 1 / 2 , \Gamma } \right ) \end{align*}


\begin{align*} \left ( \int _ { \R ^ N } u ^ q \right ) ^ { \frac { 1 } { q } } = \norm { u } _ { L ^ { q } } \leq C \norm { \nabla u } _ { L ^ p } ^ { \beta } \norm { u } _ { L ^ 1 } ^ { 1 - \beta } , \end{align*}


\begin{align*} \phi ( N \times { \mathbb R } ^ { \dim M - 1 } ) \subset U ^ { + } : = \{ x \in M \mid \operatorname { s g n } ( Z ) > 0 \} . \end{align*}


\begin{align*} \mathbb { E } ( \overline { T } _ n ) = \frac { 1 } { n } \sum _ { 1 \leq i \leq n } \mathbb { E } ( X _ i X _ { i + 1 } ) . \end{align*}


\begin{align*} P _ { \kappa } ( x ) = N _ { \kappa } \int _ 0 ^ { x } t ^ { n - 1 } \exp _ { \kappa } ( - x ) \ \ , \end{align*}


\begin{align*} E ( G ^ 0 / A ) = E ( G ^ 0 / R ) E ( R / A ) \neq 0 \end{align*}


\begin{align*} ( L ( t , - z ) \alpha , L ( t , z ) \beta ) = ( \alpha , \beta ) , \end{align*}


\begin{align*} \begin{cases} f ^ k \to f F ^ s _ { p _ 1 , q } , \ , & \| f ^ k \| _ { L ^ { \infty } ( \R ^ n ) } \leq C \\ g ^ k \to g F ^ s _ { p , q } , \ , & \end{cases} \end{align*}


\begin{align*} \sum _ { i \in I } a _ i = \sum _ { j = 1 } ^ h \left [ r ( I _ j ) + ( | I _ j | - 1 ) r \right ] = r ( I ) + ( | I | - 1 ) r \end{align*}


\begin{align*} \boldsymbol { F } ( n ) = \beta \mathbf { a } ( n ) . \end{align*}


\begin{align*} \begin{aligned} u \mapsto \int _ { \mathbb { R } ^ { N } } \frac { 1 } { 2 } \left ( | \nabla u | ^ { 2 } + { V _ { \varepsilon } } ( y ) u ^ { 2 } \right ) - \frac { 1 } { p + 1 } | u | ^ { p + 1 } d y , \end{aligned} \end{align*}


\begin{align*} \langle \mathcal { N } _ \perp \rangle _ { \psi _ \mathrm { g s } } \le \ ; & C \\ \langle \mathrm { d } \Gamma _ \perp ( h _ \mathrm { M F } - \mu _ + ) \rangle _ { \psi _ \mathrm { g s } } \le \ ; & C \\ \langle \mathcal { N } _ - \rangle _ { \psi _ \mathrm { g s } } \le \ ; & C _ \varepsilon \min \{ N , T ^ { - 1 - \varepsilon } \} . \end{align*}


\begin{align*} \alpha _ { i , v } = \zeta ^ { a _ i v ^ T } . \end{align*}


\begin{align*} \frac { \nu ( \left \{ \alpha \right \} ) } { \omega ^ { \prime } ( h ( \alpha ) ) } = \nu ( \left \{ \alpha \right \} ) + \mu ( \{ s _ { \mu } \} ) = \mu \boxplus \nu ( \{ \alpha + s _ { \mu } \} ) . \end{align*}


\begin{align*} \begin{vmatrix} \det \left ( ( \lambda _ i ( t ) I _ N - H _ { \alpha } ( t ) ) _ { k | k } \right ) & \det \left ( ( \lambda _ i ( t ) I _ N - H _ { \alpha } ( t ) ) _ { k | \l } \right ) \\ \det \left ( ( \lambda _ i ( t ) I _ N - H _ { \alpha } ( t ) ) _ { \l | k } \right ) & \det \left ( ( \lambda _ i ( t ) I _ N - H _ { \alpha } ( t ) ) _ { \l | \l } \right ) \end{vmatrix} & = f ( \lambda _ i ( t ) ) \det \left ( ( \lambda _ i ( t ) I _ N - H _ { \alpha } ( t ) ) _ { k \l | k \l } \right ) \\ & = 0 , \end{align*}


\begin{align*} ( w x , y , z ) - ( w , x y , z ) + ( w , x , y z ) = w ( x , y , z ) + ( w , x , y ) z . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { g } e _ { i } f _ { i } = n . \end{align*}


\begin{align*} \frac { d } { d x } \Big ( e ^ { - x } y _ { p , \lambda } ( x ) \Big ) = \phi _ { p , \lambda } ( x ) . \end{align*}


\begin{align*} \# Z ( H ^ { \Lambda _ Q } ) = - \# Z ( H ^ { \Lambda _ { Q ^ - } } ) . \end{align*}


\begin{align*} F ( z \ , | \ , \overline \omega _ 1 , \omega _ 2 ) = \exp \Bigg ( \int _ C \frac { e ^ { z s } } { ( e ^ { \overline \omega _ 1 s } - 1 ) ( e ^ { \omega _ 2 s } - 1 ) } \frac { d s } { s } \Bigg ) , \end{align*}


\begin{align*} U _ t = \mathcal { A } U , U ( 0 ) = U _ 0 , \end{align*}


\begin{align*} X _ { w _ J } \cong \prod _ { k = 1 } ^ { m } F l _ { n _ k } . \end{align*}


\begin{align*} x ^ 1 = c _ 1 t + c _ 2 , ~ x ^ 2 = k _ 1 \textnormal { a n d } x ^ 1 = k _ 2 , ~ x ^ 2 = c _ 3 t + c _ 4 , \end{align*}


\begin{align*} f _ { \mathbf { M ^ { \ast } } _ { - \xi _ { \boldsymbol { s } k } } } ( \boldsymbol { w } ) = c _ { n - 1 , \xi _ { \boldsymbol { s } k } } ^ { \ast } \left ( 1 + \sqrt { \boldsymbol { w } ^ { T } \boldsymbol { w } + \xi _ { \boldsymbol { s } , k } ^ { 2 } } \right ) \exp \left \{ - \sqrt { \boldsymbol { w } ^ { T } \boldsymbol { w } + \xi _ { \boldsymbol { s } , k } ^ { 2 } } \right \} , ~ \boldsymbol { w } \in \mathbb { R } ^ { n - 1 } , \end{align*}


\begin{align*} \lim _ { N \to \infty } \sum \limits _ { k \ge 1 } \frac { a _ k } { \lambda _ k ^ 2 ( \lambda _ N ^ 2 + \lambda _ { k } ^ { 2 } ) } = \sum \limits _ { k \ge 1 } \frac { a _ k } { \lambda _ k ^ 4 } , \end{align*}


\begin{align*} h _ { i j ; k } - h _ { i k ; j } = - \bar R _ { \alpha \beta \gamma \delta } \nu ^ \alpha X _ { ; i } ^ \beta X _ { ; j } ^ \gamma X _ { ; k } ^ \delta = 0 , \end{align*}


\begin{align*} \det ( x I - B _ 1 ( T ) ) & = \prod _ { j = 1 } ^ { l ( T ) } \det ( R _ j ) . \end{align*}


\begin{align*} B _ { i i } ( t _ 1 : t _ 0 ) & \geq \eta _ i 0 \leq t _ 0 \leq t _ 1 \leq t _ L , \cr B _ { j j } ( t _ 1 : t _ 0 ) & \geq \eta _ j 0 \leq t _ 0 \leq t _ 1 \leq \sigma , \cr \sum _ { t = 0 } ^ { \sigma - 1 } B _ { j i } ( t ) & \geq \delta \delta \in ( 0 , \eta _ j ) . \end{align*}


\begin{align*} \big ( \frac { v } { w - t \tau / 2 } \big ) > 0 , \big ( \frac { v } { w + t \tau / 2 } \big ) > 0 , \big ( \frac { v } { - t } \big ) > 0 , \\ \big ( \frac { - t \tau / 2 } { w - t \tau / 2 } \big ) > 0 , \big ( \frac { - t \tau / 2 } { w + t \tau / 2 } \big ) > 0 , \big ( \frac { - t \tau / 2 } { - t } \big ) = ( \tau / 2 ) > 0 . \end{align*}


\begin{align*} R ( w + i y , A ) \left ( 1 - ( x - w ) R ( x + i y , A ) \right ) = R ( x + i y , A ) \end{align*}


\begin{align*} \P ( Y ( t , x ) > R ) & \geq \P ( A _ N ) \P _ N \Biggl ( Y _ < ( \tau _ N , \eta _ N ) \prod _ { i = 1 } ^ { N } u _ < ( \tau _ { i } , \eta _ { i } ; \tau _ { i - 1 } , \eta _ { i - 1 } ) \zeta _ i > R \Biggr ) \\ & \geq C ^ N N ^ { - N } R ^ { - 1 - \frac 2 d } \log ^ { N - 1 } ( R ) \\ & = \frac { R ^ { - 1 - \frac 2 d } ( C \log R ) ^ { K \exp ( W ( \frac 1 { d K } \log R ) ) } } { \log R ( K \exp ( W ( \frac 1 { d K } \log R ) ) ) ^ { K \exp ( W ( \frac 1 { d K } \log R ) ) } } . \end{align*}


\begin{align*} \bigcup _ { i = 1 } ^ N B ( x _ i , \frac { 1 } { 2 } c ' ) \subseteq B ( 0 , c + \frac { 1 } { 2 } c ' ) . \end{align*}


\begin{align*} \kappa ^ p \nu _ p ( \{ ( a , r ) \in \omega \times \R _ + : \ c _ d ' r | \nabla f ( a ) | - C A r ^ 2 > \kappa \} ) = I _ 1 ( \kappa ) - I _ 2 ( \kappa ) \ , , \end{align*}


\begin{align*} N _ 2 = N _ { f _ 2 } ( \Lambda _ 2 ) . \end{align*}


\begin{align*} \mathbb { A } _ { v _ 0 } ( \boldsymbol { \eta } , \boldsymbol { \zeta } , \boldsymbol { \alpha } , \beta ) - \lim _ { n \to \infty } \dfrac { 1 } { n } \sum _ { v \neq v _ 0 } { { F _ v } } ( \boldsymbol { \alpha } , \beta ) ( n ) = V ( \boldsymbol { \eta } , \boldsymbol { \zeta } , \boldsymbol { \alpha } , \beta ) \enspace . \end{align*}


\begin{align*} & { \rm ( a ) } \ \rho ( I ) = 1 ; \quad { \rm ( b ) } \ \rho _ { i c } ( I ) = 1 ; { \rm ( c ) } \ G \mbox { i s b i p a r t i t e } ; \quad { \rm ( d ) } \ \rho ( I ^ \vee ) = 1 . \end{align*}


\begin{align*} T _ { f w ^ \zeta _ 0 } = M _ { f w ^ \zeta _ 0 } = M _ { f w _ 0 ^ \zeta } + \sum _ { g ' \prec f w _ 0 ^ \zeta } t _ { g ' f w _ 0 ^ \zeta } ( q ) M _ { g ' } , \end{align*}


\begin{align*} \tau ( \alpha \beta ) = \tau ( \alpha ) \otimes \tau ( \beta ) = \overline { \alpha } \otimes \overline { \beta } . \end{align*}


\begin{align*} \begin{cases} e '' _ \sigma + \frac { z } { 2 } e ' _ \sigma = 0 , \\ e _ \sigma ( 0 ) = 0 , e _ \sigma ( \infty ) = \sigma . \end{cases} \end{align*}


\begin{align*} ( \Delta - f _ 2 ' ( { \sf U } _ \infty ) ) \theta & = - f ( { \sf U } _ \infty ) - \sum _ { i = 0 } ^ N \tfrac { f ^ { ( i ) } ( { \sf U } _ \infty ) } { i ! } ( \theta _ 0 + \cdots + \theta _ { L - 1 } ) ^ i + \sum _ { i = 2 } ^ N \tfrac { f _ 2 ^ { ( i ) } ( { \sf U } _ \infty ) } { i ! } ( \theta _ 0 + \cdots + \theta _ { L - 1 } ) ^ i . \end{align*}


\begin{align*} R _ { g ^ { - 1 } * g } \frac { d g } { d t } = \mathcal { P } ( { \rm A d } _ g a ( t ) ) , \end{align*}


\begin{align*} \left \langle V _ { \psi _ 1 } \xi _ 1 , V _ { \psi _ 2 } \xi _ 2 \right \rangle _ { _ { L ^ 2 ( G _ 2 ) } } = \langle \xi _ 1 , \xi _ 2 \rangle _ { _ { L ^ 2 ( \widehat { \R ^ 2 } \times \widehat { \R } ) } } \langle T \psi _ 2 , T \psi _ 1 \rangle _ { _ { L ^ 2 ( \widehat { \R ^ 2 } \times \widehat { \R } ) } } . \end{align*}


\begin{align*} W _ j = E _ j ( x , d - 1 ) ( \forall j = \overline { 0 , l ( T ) } ) , \end{align*}


\begin{align*} E \ni f \mapsto P f : = \lim _ { r \to \infty } C ( r ) f \end{align*}


\begin{align*} \alpha = s + y , \ \beta = s - y \end{align*}


\begin{align*} G ( x , y ) = \int _ 0 ^ \infty H ( x , y , t ) d t . \end{align*}


\begin{align*} S ^ { ( q ) } _ { [ n , j ] } & : = \sum _ { \gamma \in \mathcal { S } _ { [ n , j ] } } w ( \gamma + q ) , \\ T ^ { ( q ) } _ { [ n , j ] } & : = \sum _ { \gamma \in \widehat { \mathcal { S } } _ { [ n , j ] } } w ( \gamma - q ) . \end{align*}


\begin{align*} \langle \mathcal { H } _ { q } ^ { n + 1 } e _ { r } , e _ { 0 } \rangle = \langle \mathcal { H } _ { q } ^ { n } ( \mathcal { H } _ { q } \ , e _ { r } ) , e _ { 0 } \rangle = \begin{cases} \sum _ { k = 0 } ^ { p } a _ { q } ^ { ( k ) } \langle \mathcal { H } _ { q } ^ { n } e _ { k } , e _ { 0 } \rangle , & \mbox { i f } \ , \ , r = 0 , \\ [ 0 . 5 e m ] \langle \mathcal { H } _ { q } ^ { n } e _ { r - 1 } , e _ { 0 } \rangle + \sum _ { k = 0 } ^ { p } a _ { r + q } ^ { ( k ) } \langle \mathcal { H } _ { q } ^ { n } e _ { r + k } , e _ { 0 } \rangle , & \mbox { i f } \ , \ , r \geq 1 . \end{cases} \end{align*}


\begin{align*} \frac { \log _ 2 ( \widetilde { N _ H } ( q \alpha ) ) } { ( R + q ) \alpha } = \frac { q } { q + R } \frac { \log _ 2 ( \widetilde { N _ H } ( q \alpha ) ) } { q \alpha } > \frac { t } { t + 1 } ( 1 + \frac { 1 } { t } ) h = h . \end{align*}


\begin{align*} H ^ D = \left \{ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} : a b \neq 0 \right \} , H ^ S = \left \{ \begin{pmatrix} a & b \\ - b & a \end{pmatrix} : a ^ 2 + b ^ 2 > 0 \right \} , H ^ \alpha = \left \{ \begin{pmatrix} a & b \\ 0 & | a | ^ \alpha \end{pmatrix} : a \neq 0 \right \} , \end{align*}


\begin{align*} ( B ( \Lambda _ i ^ { \nu } - 1 ) ) _ { k j } - ( B ( \Lambda _ j ^ { \nu } - 1 ) ) _ { k i } = \pi L _ { i j k } ( \Lambda ^ { \nu } ) - \pi ^ { \nu } L _ { i j k } ^ { \nu } , \end{align*}


\begin{align*} \lim \limits _ { t \rightarrow \infty } \| \chi _ { S _ n } e ^ { - i t H _ 0 } \varphi \| = 0 \end{align*}


\begin{align*} w _ 2 ( \pi ) & = \sum \limits _ { j = 1 } ^ n ( c _ j / 2 ) \left ( \sum \limits _ { 1 \leq i _ 1 < i _ 2 < \cdots < i _ j \leq n } ( t _ { i _ 1 } + t _ { i _ 2 } + \cdots + t _ { i _ j } ) \right ) \\ & = \frac { 1 } { 2 } \sum \limits _ { j = 1 } ^ n c _ j \binom { n - 1 } { j - 1 } \left ( \sum \limits _ { i = 1 } ^ n t _ i \right ) \\ & = \frac { 1 } { 2 } \sum \limits _ { j = 1 } ^ n c _ j \binom { n - 1 } { j - 1 } a _ 2 . \end{align*}


\begin{align*} Z ^ k _ { j , , r s } = \exp \Big [ - \frac { 1 } { 2 } \big ( Q _ j ^ k ( \alpha _ r ) - Q _ j ^ k ( \alpha _ s ) \big ) ^ 2 \Big ] r , s = 1 , \ldots , p . \end{align*}


\begin{align*} S = \begin{bmatrix} - v _ { i } \sin ( \psi _ i ) \sec ^ 2 ( \beta _ i ) & \cos ( \psi _ i ) - \sin ( \psi _ i ) \tan ( \beta _ i ) \\ v _ { i } \cos ( \psi _ i ) \sec ^ 2 ( \beta _ i ) & \sin ( \psi _ i ) + \cos ( \psi _ i ) \tan ( \beta _ i ) \end{bmatrix} , \end{align*}


\begin{align*} E ( G / H ) = E ( ( N H ) / H ) = E ( N / ( N \cap H ) ) = \pm 1 \end{align*}


\begin{align*} J ' ( v ; w ) : = \lim _ { t \to 0 } \frac { J ( v + t w ) - J ( v ) } { t } = \int _ { \R ^ { d } } \langle w , j _ { q } ( v ) \rangle d \rho - L ( w ) . \end{align*}


\begin{align*} a ( u , J I u - J z _ { \mathrm { n c } } ) + a _ { \mathrm { p w } } ( z _ { \mathrm { n c } } , z _ { \mathrm { n c } } - I u ) & = b ( \lambda u , J I u - J z _ { \mathrm { n c } } + ( 1 + \delta ) ( z _ { \mathrm { n c } } - I u ) ) \\ & = \lambda b ( u , ( J - 1 ) ( I u - z _ { \mathrm { n c } } ) ) + \lambda b ( \delta u , z _ { \mathrm { n c } } - I u ) . \end{align*}


\begin{align*} w \left ( { { \eta _ 1 } , . . . , { \eta _ d } } \right ) = \mathop { \lim } \limits _ { T \to \infty } \frac 1 T \ln { \left \langle D _ { 1 } ^ { \eta _ 1 } . . . D _ { d } ^ { \eta _ d } \right \rangle } \end{align*}


\begin{align*} \begin{cases} r _ + + r _ - = 2 a + 4 k \\ 0 \le r _ - \le a + 3 k \le r _ + \le 2 a + 3 k \end{cases} \ . \end{align*}


\begin{align*} & \int _ 0 ^ r \left [ ( \phi '' + \frac { z } { 2 } \phi ' - \frac { z } { 2 } e _ \sigma \psi ' ) \cdot z \phi ' + ( \psi '' + \frac { z } { 2 } \psi ' - \frac { z } { 2 } e _ \sigma \phi ' ) \cdot z \psi ' \right ] { \rm { d } } z \\ & = \frac { 1 } { 2 } \int _ 0 ^ r | z ( \phi ' , \psi ' ) | ^ 2 + \int _ 0 ^ r z ( \phi ' \phi '' + \psi ' \psi '' ) - \int _ 0 ^ r z ^ 2 e _ \sigma \phi ' \psi ' . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \abs { w _ i - \lambda \frac { z _ i - a } { 1 - \bar { a } z _ i } } ^ 2 \end{align*}


\begin{align*} d ^ n ( \Phi ) ( \vec { p } \ , ) : = \sum _ { i = 0 } ^ { n + 1 } ( - 1 ) ^ i \pi _ { \bigwedge \vec { p } , \bigwedge \vec { p } ^ { \ , i } } ( \Phi ( \vec { p } ^ { \ : i } ) ) \end{align*}


\begin{align*} y ^ d \mapsto \prod _ { i = 1 } ^ { l _ + } y _ i ^ { p _ i ^ + \cdot d } \prod _ { j \in S _ + } x _ j ^ { D _ j \cdot d } . \end{align*}


\begin{align*} \sum _ { m = m _ 0 } ^ \infty \prod _ { i = i _ 0 } ^ m \sum _ { k = 0 } ^ \infty \mu ( k ) ( 1 - E _ 2 ( i , m ) ) ^ i \precsim \sum _ { m = m _ 0 } ^ \infty \prod _ { i = i _ 0 } ^ m \mu ( [ 0 , 2 A ( m ) ^ { { { \rho } } i } ] ) . \end{align*}


\begin{align*} H _ { ( X , D ) } ( y ) = \sum _ { d \in \mathbb K } \sum _ { \vec d = ( D _ i \cdot d ) _ { i \in I } \in ( \mathbb Z ) ^ { | I | } , f = [ d ] \in \mathbb K / \mathbb L } H _ { ( X , D ) , f , \vec d } \textbf { 1 } _ { f } [ \textbf { 1 } ] _ { \vec d } . \end{align*}


\begin{align*} p ( t , V _ F , g ) - p ( t , 0 , g ) = 0 , \ \forall t > 0 , g \in \mathbb { R } . \end{align*}


\begin{align*} z ^ k _ { j , } ( \alpha ^ * ) = z ^ k _ { j , } ( \alpha ^ * , \alpha ^ * ) - [ Z ( \boldsymbol { \alpha } , \alpha ^ * ) ] ^ T [ Z _ { j , } ^ k ] ^ { - 1 } \ , Z ( \boldsymbol { \alpha } , \alpha ^ * ) , \end{align*}


\begin{align*} \begin{aligned} \frac { 1 } { 2 } \frac { d ( b ^ 2 ( t ) ) } { d t } & \leq C | b | - ( 1 - g _ 1 ) b ^ 2 + g _ 1 \delta _ 1 | b c | , \\ \frac { 1 } { 2 } \frac { d ( c ^ 2 ( t ) ) } { d t } & \leq C | c | + 2 a _ 1 | b c | - c ^ 2 . \end{aligned} \end{align*}


\begin{align*} r ( g ) = \min \{ j \in \N \ : \ S y z ( g ) _ j \ne 0 \} \leq d - 1 . \end{align*}


\begin{align*} \nu _ p \left ( \binom { m } { t } \right ) = \frac { \sigma _ p ( t ) + \sigma _ p ( m - t ) - \sigma _ p ( m ) } { p - 1 } . \end{align*}


\begin{align*} S ( x _ 1 , \dotsc , x _ { n } ) = ( x _ { n } , x _ 1 , \dotsc , x _ { n - 1 } ) . \end{align*}


\begin{align*} [ K _ j J , e ^ { M J } ] = 0 \end{align*}


\begin{align*} a = H _ { \mu } ( A + i f _ { \mu } ( A ) ) , b = H _ { \mu } ( B + i f _ { \mu } ( B ) ) , \end{align*}


\begin{align*} R ( z , A _ 2 ) - R ( z , A _ 1 ) = R ( z , A _ 2 ) B R ( z , A _ 1 ) , \end{align*}