\begin{align*} d i m ( V \cap V ' ) = k - i . \end{align*}


\begin{align*} L ( w _ 1 ) & = L ( x - y _ 1 ) \\ & = L ( x ) - L ( y _ 1 ) \\ & \geq f ( x ) - C - \delta r - ( f ( x ) - r + r ( \epsilon + \delta ) - C ) \\ & = - 2 \delta r - \epsilon r + r \\ & = r ( 1 - ( 2 \delta + \epsilon ) ) . \end{align*}


\begin{align*} \Delta P - \frac { n } { \alpha } \bar { g } ( \nabla P , \nabla \log u ) & = u ^ { \frac { \alpha + 1 } { \alpha } } ( | h | ^ { 2 } - n H ^ { 2 } ) - u ^ { \frac { 1 } { \alpha } } \overline { \mathrm { R i c } } ( \nu , \lambda \partial _ { r } ^ { T } ) \\ & + \frac { n - 1 } { \alpha } u ^ { \frac { \alpha + 1 } { \alpha } } | \nabla \log u | ^ { 2 } . \end{align*}


\begin{align*} K ( x , y ) = c ( x , y ) \vert x - y \vert ^ { - d - \alpha } , ~ ~ 0 < \lambda ^ { - 1 } \le c ( x , y ) = c ( y , x ) \le \lambda , ~ ~ \forall x , y \in \R ^ d , \end{align*}


\begin{align*} e _ A ( t ) = e ^ { u ( { X } _ t ) - u ( X _ 0 ) } Y _ t \exp \left ( A _ t ^ { \overline { \mu } } \right ) t \in [ \ , 0 , \ , + \infty \ , [ , \end{align*}


\begin{align*} U ^ { \mu _ T } ( \lambda ) = - \int _ \mathbb { C } \log | \lambda - z | d \mu _ T ( z ) . \end{align*}


\begin{align*} \zeta ( s ) = 1 \ ; \hbox { f o r } \ ; | s | \leq \delta ; \zeta ( s ) = 0 \ ; \hbox { f o r } \ ; | s | > 2 \delta ; 0 \leq - s \zeta ' ( s ) \leq 4 \ ; \hbox { f o r } \ ; \delta \leq | s | \leq 2 \delta . \end{align*}


\begin{align*} & \dot { \mathfrak { q } } _ 1 ( \epsilon , \partial _ y \mathcal { Z } _ 1 ) - \big ( \partial _ y ( \mathcal { L } _ 1 \epsilon ) , \mathcal { Z } _ 1 \big ) - p \big ( [ | V _ 0 | ^ { p - 1 } - R _ 1 ^ { p - 1 } ] \epsilon , \partial _ y \mathcal { Z } _ 1 \big ) \\ & = \Big ( \big [ | V _ 0 + \epsilon | ^ { p - 1 } ( V _ 0 + \epsilon ) - | V _ 0 | ^ { p - 1 } V _ 0 - p | V _ 0 | ^ { p - 1 } \epsilon \big ] , \partial _ y \mathcal { Z } _ 1 \Big ) - ( \Psi _ { V _ 0 } , \mathcal { Z } _ 1 ) , \end{align*}


\begin{align*} X \ast I = \{ ( x , i ) \in X \times I \mid d ( x , x _ 0 ) = d ( i , ( 0 , 0 ) ) \} \end{align*}


\begin{gather*} \Box \phi = \left ( | \phi _ { t } | ^ 2 - | \phi _ { x } | ^ 2 \right ) \phi + \mathbf { 1 } _ { \omega } f ^ { \phi ^ { \perp } } , \\ \phi [ 0 ] = ( a , b ) \ ; \textrm { a n d } \ ; \phi [ T ] = ( 0 , 0 ) . \end{gather*}


\begin{align*} f ' - 2 \pi i b f = - 2 \pi c ( x - a ) f . \end{align*}


\begin{align*} \mathcal { P } ^ \star = \mathcal { P } \cup \{ ( n - 1 , 1 , 2 , 3 , \ldots , n - 2 , n ) , ( 2 , n , n - 1 , \ldots , 4 , 3 , 1 ) \} . \end{align*}


\begin{align*} D _ i ( x ) \left ( \nabla \rho _ i + z _ i \rho _ i \nabla \phi \right ) \cdot \mathbf { n } = 0 , x \in \partial \Omega , i = 1 , \cdots , s . \end{align*}


\begin{align*} \Big | [ - ( - \Delta ) ^ { \gamma / 2 } G ] ( u ) \Big | = c _ { \gamma } \Big | \int _ { 0 } ^ { \infty } \frac { G ( u - w ) } { w ^ { \gamma + 1 } } d w \Big | = c _ { \gamma } \Big | \int _ { u - b _ G } ^ { u + b _ G } \frac { G ( u - w ) } { w ^ { \gamma + 1 } } d w \Big | \leq H ^ G ( u ) . \end{align*}


\begin{align*} \underset { t \to \infty } { \lim } \Delta _ { \ell , s r } ^ { a v } ( t ) = \lim _ { t \to \infty } \frac { \sum _ { i = 1 } ^ { R _ \ell ^ m ( t ) } \left ( \frac { ( T _ { \ell i } ^ m ) ^ 2 } { 2 } + T _ { \ell i } ^ m Z _ { \ell i } ^ m \right ) } { t } , \end{align*}


\begin{align*} \Lambda ( e ^ r - 1 ) = \exp \left ( - ( e ^ r - 1 ) / c \right ) \end{align*}


\begin{align*} \partial _ t H _ N + v \cdot \nabla _ x H _ N = { \rm { d i v } } _ v ^ \alpha G _ N , H _ N | _ { t = 0 } = H _ 0 \in L ^ p ( \mathbb { R } ^ 3 \times \mathbb { R } ^ 3 ) , \end{align*}


\begin{align*} | R _ 1 ( x , y ) | & \lesssim \sum \limits _ { j = - \infty } ^ \infty \int _ { r ^ { - j } } ^ { r ^ { - j + 1 } } | S _ j ( x , y ) | \frac { d t } { t } \\ & \lesssim ( r - 1 ) \sum \limits _ { j = - \infty } ^ \infty \int _ { r ^ { - j } } ^ { r ^ { - j + 1 } } \frac 1 { V ( x , y , t + d ( x , y ) ) } \Big ( \frac { t } { t + \| x - y \| } \Big ) ^ { \varepsilon } \frac { d t } { t } \\ & \lesssim ( r - 1 ) \int _ { 0 } ^ \infty \frac 1 { V ( x , y , t + d ( x , y ) ) } \Big ( \frac { t } { t + \| x - y \| } \Big ) ^ { \varepsilon } \frac { d t } { t } . \end{align*}


\begin{align*} ( u \rhd _ s v ) \rhd _ t w = ( u \rhd _ t w ) \rhd _ s ( v \rhd _ t w ) \end{align*}


\begin{align*} \hat { \phi } _ { \varepsilon } ( \mathbf { 0 } ) = 0 \end{align*}


\begin{align*} \bar \omega _ T = \frac { 1 } { T } \int _ 0 ^ T \omega \circ \tau _ \Phi ^ t \d t . \end{align*}


\begin{gather*} c ^ R _ X = c ^ L _ X \end{gather*}


\begin{align*} \Theta _ 1 ( z ) = \chi ^ - + S ( z ) \chi ^ + , \Theta _ 2 ( z ) = \chi ^ + + S ( z ) \chi ^ - , \Theta ' _ 1 ( z ) = \chi ^ - + S ^ * ( \bar z ) \chi ^ + , \Theta ' _ 2 ( z ) = \chi ^ + + S ^ * ( \bar z ) \chi ^ - , \end{align*}


\begin{align*} \forall x \in \R \colon \Phi ( x ) : = \{ 0 , x ^ 2 \} . \end{align*}


\begin{align*} \hat { \xi } ( s ) : = \left \{ \begin{aligned} & \xi ( s ) - \left ( 1 - \frac { s } { T } \right ) h , 0 \leq s \leq T , \\ & \xi ( s ) , s \geq T . \end{aligned} \right . \end{align*}


\begin{align*} ~ \begin{aligned} \tilde { \upsilon } _ { i 0 } \preceq _ i \cdots \preceq _ i \tilde { \upsilon } _ { i n } = \upsilon _ { i n } \upsilon _ { i j } \preceq _ { i } \tilde { \upsilon } _ { i j } \preceq _ i \alpha _ { i j } j \in \{ 0 , \ldots , n \} . \end{aligned} \end{align*}


\begin{align*} \forall ( t , u ) \in [ 0 , T ] \times \mathbb { R } , [ - ( - \Delta ) ^ { \gamma / 2 } G ] ( t , u ) = \sum _ { j = 0 } ^ { k } t ^ { j } [ - ( - \Delta ) ^ { \gamma / 2 } G _ j ] ( u ) . \end{align*}


\begin{align*} M _ \mu ( n , \alpha , \varepsilon , \delta , T , f ) = \inf \{ \sum _ { i } e ^ { - \alpha n _ i + f _ { n _ i } ( x _ i ) } : \mu \big ( \cup _ { i } B _ { n _ i } ( x _ i , \varepsilon ) \big ) \geq 1 - \delta \} , \end{align*}


\begin{align*} I ( \tau ) = e ^ { \left ( 1 - \frac { 2 } { \alpha } \right ) \tau } . \end{align*}


\begin{align*} x ^ \prime : y ^ \prime : z ^ \prime = \alpha x : \beta y : \gamma z \ , . \end{align*}


\begin{align*} P _ 2 ( t ) = \vartheta _ 2 ( 0 , i t ) \vartheta _ 2 ( 0 , i / t ) P _ 4 ( t ) = \vartheta _ 4 ( 0 , i t ) \vartheta _ 4 ( 0 , i / t ) , \end{align*}


\begin{align*} m ( E _ { m } ^ { [ \alpha , \beta ] , \varepsilon } ) = 1 . \end{align*}


\begin{align*} g _ \epsilon ( x ) : = ( g * \psi _ \epsilon ) ( x ) , \psi _ \epsilon ( y ) = \exp \left [ - \frac { \norm { y } ^ 2 } { 2 \epsilon } - \frac { 1 } { 2 } \log ( 2 \pi \epsilon ) \right ] . \end{align*}


\begin{align*} h ( a , b ; z ) = | ( b - z ) \vee 0 | ^ \frac { 1 } { \rho } - | ( a - z ) \vee 0 | ^ \frac { 1 } { \rho } \end{align*}


\begin{align*} \beta = ( L t _ 1 - 1 ) ^ 2 + \mu _ p t _ 1 ( 2 - L t _ 1 ) . \end{align*}


\begin{align*} \sum _ { \substack { n \geq i \geq 0 \\ m \geq j \geq 0 } } f _ { i j } X ^ i Y ^ j = 0 \end{align*}


\begin{align*} \sum _ { k = 0 } ^ m { n + k \choose k } B _ a ( \alpha + k , \beta + n + 1 ) + \sum _ { k = 0 } ^ n { m + k \choose k } B _ a ( \alpha + m + 1 , \beta + k ) = B _ a ( \alpha , \beta ) , \end{align*}


\begin{align*} R _ 1 ^ * \cup R _ 2 ^ * = \left \{ a , b \right \} \cup \left \{ c _ 1 ^ i \colon \phi _ { x y } ( c _ 1 ^ i ) \in U \cup V \right \} \cup \left \{ c _ { 2 k - 1 } ^ i \colon \phi _ { x y } ( c _ { 2 k - 1 } ^ i ) \in U \cup V \right \} . \end{align*}


\begin{align*} P ( s ) = s \int _ { \frac { \bar \rho } { 2 } } ^ s \frac { p ( z ) } { z ^ 2 } \end{align*}


\begin{align*} H = \mathcal F ( \{ p \} ) \times T \ , , T = \{ a \in H \colon p \nmid a \} \ , . \end{align*}


\begin{align*} ( - \Delta ) ^ s w ( x ) + c ( x ) w ( x ) & = ( - \Delta ) ^ s v ( x ) + c ( x ) v ( x ) + ( - \Delta ) ^ s ( w - v ) ( x ) \\ & \geqslant - C \big ( 1 + u ( a ) + ( \theta d ) ^ { - n - 2 s - 2 } \big ) \\ & \geqslant - C \big ( ( \theta d ) ^ { - n - 2 s - 2 } + u ( a ) \big ) \end{align*}


\begin{align*} | \dot { \alpha } | + | \dot { X } | + | \dot { \theta } | = O ( \delta ) , \ ; \forall t \in D _ { \delta _ 0 } . \end{align*}


\begin{align*} R ( x ) & = \{ \theta \in \Theta : f ( \theta | x ) \geq k \} . \end{align*}


\begin{align*} A + D + E = C + B + E \ , . \end{align*}


\begin{align*} w ' = ( ( \ldots ( ( ( a _ i ) c _ 1 ) c _ 2 ) \ldots ) c _ k ) ( ( \ldots ( ( ( a _ j ) d _ 1 ) d _ 2 ) \ldots ) d _ r ) \end{align*}


\begin{align*} D = D _ { G , K } \hat { \otimes } 1 + 1 \hat { \otimes } D _ { Z } , \end{align*}


\begin{align*} \begin{aligned} \omega ( B ( x _ 0 , 2 ^ { j _ k } 2 ^ { i _ 0 } \sigma ) ) & \geqslant 2 ^ k \omega ( B ( x _ 0 , 2 ^ { i _ 0 } \sigma ) ) \\ & \geqslant { C _ \omega ^ { - 1 } } 2 ^ { - \bf N } 2 ^ k \omega ( B ( x _ 0 , 2 ^ { i _ 0 + 1 } \sigma ) ) \geqslant { C _ \omega ^ { - 1 } } 2 ^ { - \bf N } 2 ^ k \sigma ^ { \bf N } . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \left \| \varphi _ { i , \delta } ^ j + \hat { f } _ i ^ { j } \right \| _ { C ^ { 2 , \alpha } \left ( B _ { R } ( \hat { x } , \hat { g } _ { i , \delta } ^ * ) \right ) } & \leq C _ { 5 } ( n , r , v , \rho , \alpha , Q ) \left \| \varphi _ { i , \delta } ^ j + \hat { f } _ i ^ { j } \right \| _ { C ^ { 0 } \left ( B _ { R } ( \hat { x } , \hat { g } _ { i , \delta } ^ * ) \right ) } \\ & \leq \Psi ( i ^ { - 1 } , \delta \ , | \ , n , r , v , \rho , \alpha , Q ) , \end{aligned} \end{align*}


\begin{align*} \varphi _ \alpha ( \xi ) = \exp \left ( - \int _ { \mathbb { S } ^ { d - 1 } } | \langle y ; \xi \rangle | ^ \alpha \lambda _ 1 ( d y ) \right ) . \end{align*}


\begin{align*} \dot { y } _ R ( t ) & = 2 R \Im \int _ { \mathbb { R } ^ N } \nabla \varphi \left ( \frac { x + X ( t ) } { R } \right ) \cdot \nabla Q \bar { v } d x + 2 R \Im \int _ { \mathbb { R } ^ N } \nabla \varphi \left ( \frac { x + X ( t ) } { R } \right ) \cdot \nabla v Q d x \\ & + 2 R \Im \int _ { \mathbb { R } ^ N } \nabla \varphi \left ( \frac { x + X ( t ) } { R } \right ) \cdot \nabla v \bar { v } d x , v = \alpha Q + h , \end{align*}


\begin{align*} v _ 0 = \frac { 1 } { 2 k + 1 } ( \rho _ 1 + k \rho _ 4 ) \end{align*}


\begin{align*} L ( r _ { \rm t h } ) = \int _ 0 ^ { \infty } \dot { r } p _ { \dot { R } R } ( \dot { r } , r _ { \rm t h } ) d \dot { r } , \end{align*}


\begin{align*} \Psi ( \nu ) = \left ( \int _ \Omega | w | ^ { 4 \chi ^ \nu } \phi ^ { 4 \alpha \chi ^ \nu } d \eta \right ) ^ { 1 / 4 \chi ^ \nu } \end{align*}


\begin{align*} C _ { \Sigma , \tau } = \{ w \in V \mid \langle w , v \rangle \geq 0 v \tau \} . \end{align*}


\begin{align*} \langle B ^ { i j } \partial _ { i } \partial _ { j } ( \partial _ { x } ^ { \alpha } u ) , \partial _ { x } ^ { \alpha } u _ { t } \rangle = - \langle B ^ { i j } \partial _ { j } \partial _ { x } ^ { \alpha } u , \partial _ { i } \partial _ { x } ^ { \alpha } u _ { t } \rangle - \langle \left ( \partial _ { i } B ^ { i j } \right ) \partial _ { j } \partial _ { x } ^ { \alpha } u , \partial _ { x } ^ { \alpha } u _ { t } \rangle , \end{align*}


\begin{align*} \Big ( 1 - \sum _ { k = 1 } ^ { L } \phi _ k q _ k \Big ) e _ j & = \Big ( 1 - \sum _ { k = 1 } ^ { L } q _ k ( a _ 1 \widetilde { \phi _ k } + p _ k ) \Big ) e _ j \\ & = \Big ( 1 - \sum _ { k = 1 } ^ { L } q _ k p _ k - a _ 1 \sum _ { k = 1 } ^ { L } q _ k \widetilde { \phi _ k } \Big ) e _ j \\ & = a _ 1 \Big ( q - \sum _ { k = 1 } ^ { L } q _ k \widetilde { \phi _ k } \Big ) e _ j . \end{align*}


\begin{align*} C _ 1 \min \Big \{ 1 , \frac { 1 } { t } \Big \} \leq \Im \omega _ 1 ( i t ) - t = s ( | \lambda | , t ) - t \leq C _ 2 \min \Big \{ 1 , \frac { 1 } { t } \Big \} . \end{align*}


\begin{align*} Z f ( x , \omega ) = \sum _ { k \in \Z ^ d } f ( x + k ) e ^ { - 2 \pi i k \cdot \omega } . \end{align*}


\begin{align*} \lim _ { y \to + \infty , y \in \R } \frac { \log | P ( i y ) | } { \log y } = - \infty , \lim _ { y \to + \infty , y \in \R } y L ( i y ) = 0 . \end{align*}


\begin{align*} \phi ( u , v ) = \int _ 0 ^ u \psi _ 0 ( - u ' ) \ , \mathrm { d } u ' + \frac { 1 } { 2 \pi } \left ( \int _ 0 ^ { 2 \pi } \psi ( x ) \ , \mathrm { d } x \right ) ( u + v ) , \end{align*}


\begin{align*} P _ i = \left \{ u _ i \in \R ^ { n + 1 } \ , \middle | \ , \begin{aligned} a _ { i 0 } \leq u _ { i j } \leq \min \{ a _ { i j } , u _ { i n } \} , \ u _ { i 0 } = a _ { i 0 } , \ ; a _ { i 0 } \leq u _ { i n } \leq a _ { i n } \end{aligned} \right \} . \end{align*}


\begin{align*} d \rho _ i ( t ) + \sum _ { j \in N ( i ) } m _ { i j } d t + \sum _ { j \in N ( i ) } m _ { i j } d W ^ { \delta } ( t ) = 0 . \end{align*}


\begin{align*} T ( k ) W ( \psi _ + ( \cdot , k ) , \psi _ - ( \cdot , k ) ) = - 2 i k . \end{align*}


\begin{align*} S ^ T S + I = S ^ T ( S + S ^ { - T } ) = S ^ T ( S + J S J ^ { - 1 } ) . \end{align*}


\begin{align*} \langle \nabla u ( x ) , \nabla v ( x ) \rangle _ x = \nabla u ( x ) \cdot \nabla v ( x ) \end{align*}


\begin{align*} 0 & = \int _ M ( - \Delta _ g r + V r + V \dot u ) r \ , d V _ g = \int _ M | \nabla r | ^ 2 \ , d V _ g + \int _ M V r ^ 2 \ , d V _ g + \int _ M V \dot u r \ , d V _ g . \end{align*}


\begin{align*} \tau _ n = ( \tau _ 0 - 1 ) \sum _ { i = 1 } ^ n k _ { i - 1 } + \tau _ 0 . \end{align*}


\begin{align*} - \Delta Q + Q = \left ( | \cdot | ^ { - ( N - \gamma ) } * | Q | ^ p \right ) | Q | ^ { p - 2 } Q , \mathbb { R } ^ N . \end{align*}


\begin{align*} \Lambda ^ { \mu _ i - 1 } \Psi ^ i \left ( { m _ { \Lambda } } _ { * } \left ( f \right ) \right ) = \mathfrak { P } _ { \ell } m _ { \Lambda } \Psi ^ i ( f ) , \end{align*}


\begin{align*} \hat q _ e = \hat q _ t ^ * \hat q = q _ e + { \epsilon \over 2 } q _ e p _ e ^ b , \end{align*}


\begin{align*} \ker L _ + = \Big \{ \partial _ { x _ 1 } Q , \partial _ { x _ 2 } Q , . . . , \partial _ { x _ N } Q \Big \} , \end{align*}


\begin{align*} M ^ P ( n , \alpha , \varepsilon , Z , T , f ) = \sup \{ \sum _ { i } e ^ { - s n _ i + f _ { n _ i } ( x _ i ) } \} , \end{align*}


\begin{align*} \begin{cases} \ \\ [ 8 p t ] \ \end{cases} . \end{align*}


\begin{align*} A = \tilde { A } + \eta \| \tilde { A } \| _ F \frac { G } { \| G \| _ F } , \ \ b = \tilde { b } + \eta \| \tilde { b } \| _ 2 \frac { \zeta } { \| \zeta \| _ 2 } , \end{align*}


\begin{align*} \sigma _ T ( { \underline T } , \mathcal X ) = \{ \lambda \in \mathbb { C } ^ n : K ( { \underline T } - \lambda , \mathcal X ) \} . \end{align*}


\begin{align*} K _ i ( \mathcal { C } ( G ) ) \cong K _ i ( C ^ * _ r ( G ) ) , \ i = 0 , 1 . \end{align*}


\begin{align*} E = \O _ C ( - e _ 1 \cdot \infty _ 1 ) \oplus \cdots \oplus \O _ C ( - e _ m \cdot \infty _ m ) \oplus \O _ C ^ { \oplus r - m } . \end{align*}


\begin{align*} \hat { p } _ \ell = \frac { ( p _ \ell / \mu _ \ell ) } { \sum _ { i = 1 } ^ { N } ( p _ i / \mu _ i ) } . \end{align*}


\begin{align*} { \bf H } = { \bf H } ^ 0 + \mu e ^ { \gamma c } V \end{align*}


\begin{align*} \det \left ( \frac { 1 } { \sqrt { 2 } } \begin{pmatrix} I & - i I \\ I & i I \end{pmatrix} \right ) = \left ( \tfrac { 1 } { \sqrt { 2 } } \right ) ^ { 2 d } \det ( 2 i I ) = 2 ^ { - d } ( 2 i ) ^ { d } = i ^ d . \end{align*}


\begin{align*} \phi _ { 0 } ( \mathbf { x } ) = \frac { \phi ^ { \mathrm { T F } } ( \mathbf { x } ) } { \| \phi ^ { \mathrm { T F } } \| } \phi ^ { \mathrm { T F } } ( \mathbf { x } ) = \begin{cases} \sqrt { \left ( \mu ^ { \mathrm { T F } } - V ( \mathbf { x } ) \right ) / \beta } , & \mbox { i f } V ( \mathbf { x } ) \leq \mu ^ { \mathrm { T F } } , \\ 0 , & \mbox { o t h e r w i s e } , \end{cases} \end{align*}


\begin{align*} q = g - \frac { \left \langle \tilde a _ i , g \right \rangle - b _ i } { \left \langle \tilde a _ i , c \right \rangle } c . \end{align*}


\begin{align*} \phi ' : T ' = K [ t _ 1 , \ldots , t _ { m + 1 } ] \rightarrow K [ H ' ] \end{align*}


\begin{align*} z = \Re ( z ) + i \ \Im ( z ) . \end{align*}


\begin{align*} \left \{ F , G \right \} _ { \ell } ^ { \Sigma } [ \psi ] = \left \langle E _ { \Sigma } , F ^ { ( 1 ) } [ \psi ] \otimes G ^ { ( 1 ) } [ \psi ] \right \rangle . \end{align*}


\begin{align*} \left \langle E ^ { i j } _ 0 , { t _ { c } } _ { * } f \otimes { t _ { c } } _ { * } g \right \rangle = \left \langle E ^ { i j } _ 0 , f \otimes g \right \rangle \end{align*}


\begin{align*} T ( a ) ( x ) & = \int _ Q K ( x , y ) a ( y ) d \omega ( y ) = \int _ Q [ K ( x , y ) - K ( x , x _ Q ) ] a ( y ) d \omega ( y ) \\ & \leqslant C \Big ( \frac { \l ( Q ) } { \| x - x _ Q \| } \Big ) ^ \varepsilon \frac { 1 } { \omega ( x , d ( x , x _ Q ) ) } \| a \| _ 1 . \end{align*}


\begin{align*} \Lambda _ { ( \infty , m ) } ( M , \Gamma ) : = \bigcup _ { n = 0 } ^ { \infty } \Lambda _ { ( n , m ) } ( M , \Gamma ) \end{align*}


\begin{align*} I _ 1 = \frac 2 3 \Re \left ( i \sum _ { \substack { m + n + k = 0 , \\ | m | , | n | , | k | \leq N } } ( m | n | | k | ^ 8 + m | k | | n | ^ 8 + k | n | | m | ^ 8 ) \xi ^ N ( m ) \xi ^ N ( n ) \xi ^ N ( k ) \right ) . \end{align*}


\begin{align*} \mathcal { I } _ { \alpha _ { 1 } \alpha _ { 2 } \dots } ^ { i _ { 1 } i _ { 2 } \dots } : = \mathcal { I } \backslash \{ i _ { 1 } , i _ { 2 } , \dots \} \cup \{ \alpha _ { 1 } , \alpha _ { 2 } , \dots \} , i _ { 1 } , i _ { 2 } , \dots \in \mathcal { I } , \ , \alpha _ { 1 } , \alpha _ { 2 } , \dots \in \mathcal { I } ^ { \mathtt { C } } \end{align*}


\begin{align*} \begin{aligned} \mathfrak { D } _ { { \rm l e f t } , a } u ( x ) & = D _ { { \rm l e f t } } u ( x ) + a ( x ) u ( x ) \\ \mathfrak { D } _ { { \rm r i g h t } , a } u ( x ) & = D _ { { \rm r i g h t } } u ( x ) + a ( x ) u ( x ) , \end{aligned} \end{align*}


\begin{align*} \beta _ { v _ i } | _ \Sigma & = \beta _ { v _ i ' } | _ { \Sigma ' } \circ \iota & & \\ \mu ^ { c } _ i | _ \Sigma & = \mu ^ { c ' } _ i | _ { \Sigma ' } \circ \iota & & i \end{align*}


\begin{align*} \Lambda ( \rho ) = \int _ 0 ^ 1 \exp ( - \rho y ^ { - 1 } ) d y \sim \exp \left ( - \rho \right ) / \rho , \textrm { a s } \rho \to \infty . \end{align*}


\begin{align*} S _ { g , \Gamma } f = \sum _ { \gamma \in \Gamma } \langle f , \pi ( \gamma ) g \rangle \ \pi ( \gamma ) g . \end{align*}


\begin{align*} x _ { n + 1 } = J ^ S _ { \mu _ n } ( x _ n + \mu _ n T _ { \lambda _ n } x _ n ) \end{align*}


\begin{align*} h ^ { j } _ { s } = \eta ^ t { \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} } _ { s } + \eta ^ t _ s \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} , \ \ \ \ \ \ \ \ h ^ { j } _ { t } = \eta ^ t { \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} } _ { t } + \eta ^ t _ t \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} , \end{align*}


\begin{align*} \partial _ t \ , c _ A + \mathbf { \tilde { v } } _ A \cdot \nabla c _ A - \Delta c _ A + \frac { 1 } { \varepsilon ^ 2 } f ' ( c _ A ) + \mathbf { w } | _ \Gamma \ , \nabla c _ { A , 0 } = \mathcal { C } + \mathbf { w } | _ \Gamma \ , \mathbf { Q } , \end{align*}


\begin{align*} \begin{aligned} \pi : X \times \R _ { \ge 0 } & \to \R _ { \ge 0 } \\ ( x , i ) & \mapsto d ( x , x _ 0 ) + i \end{aligned} & \begin{aligned} \iota : \R _ { \ge 0 } & \to X \times \R _ { \ge 0 } \\ i & \mapsto ( x _ 0 , i ) \end{aligned} \end{align*}


\begin{align*} \begin{cases} & \partial _ t \hat { V } - \mathcal L \hat { V } - g \geq 0 , \\ & \hat { V } - V ^ * \geq 0 , \\ & ( \partial _ t \hat { V } - \mathcal L \hat { V } - g ) \cdot ( \hat { V } - V ^ * ) = 0 , \end{cases} \end{align*}


\begin{align*} C ( i , j ) \ = \ \bigg [ \frac { i } { j + 1 } + \frac { 1 } { 2 } \bigg ] , \end{align*}


\begin{align*} | A \cup B | & \leq n - t ^ s n ^ { \frac { 1 } { s + 1 } } = n - \alpha ^ s n = \left ( 1 - \alpha ^ s \right ) n . \end{align*}


\begin{align*} \alpha = \inf _ { a \ge 0 } \{ a : \mathcal { L } ( \{ u ^ 2 \ge a \} ) < \mathcal { L } ( B _ 1 ( \mathbf { 0 } ) ) \} \ ; . \end{align*}


\begin{gather*} \left | \int _ { \S } \phi ( t , x ) - \phi ( 0 , x ) \ , d x \right | = \left | \int _ { \S } \int _ 0 ^ t \phi _ t ( s , x ) d s d x \right | \lesssim \int _ 0 ^ t \| \phi _ t ( s , x ) \| _ { L ^ 2 _ x } d s , \\ \left | \int _ { \S } \phi ( 0 , x ) - p \ , d x \right | \lesssim \| \phi ( 0 , x ) - p \| _ { L ^ 2 _ x } , \end{gather*}


\begin{align*} [ u , v , w ] _ T : = D ( T u , T v ) w + \theta ( T v , T w ) u - \theta ( T u , T w ) v . \end{align*}


\begin{align*} ( g _ 1 , g _ 2 ) \begin{pmatrix} P _ { { \bf b } , d } \\ P _ { { \bf b } , d } ~ _ { s + 1 } \mathcal { F } _ s ( a _ 1 , \ldots , a _ { s + 1 } ; b _ 1 , \ldots , b _ s ) ( \alpha ) ^ { q ^ d } \end{pmatrix} = 0 \end{align*}


\begin{align*} G = \{ [ ( A , B ) ] \colon A , B \in H \} \end{align*}


\begin{align*} \tau ^ i ( a , b ) = \begin{cases} ( a , b + \frac { i } { 4 k } ) & \\ ( - a , a + b + \frac { i } { 4 k } ) & . \end{cases} \end{align*}


\begin{align*} f ( o ) = \int _ D f \overline { { g } _ { ( 0 , \cdots , 0 ) } } e ^ { - \varphi } . \end{align*}


\begin{align*} \begin{aligned} \rho ^ { n + 1 } = \arg & \min _ { u \in V _ { h , \delta } ^ n } \bigg \{ \mathcal { F } _ h ( u ) \bigg \} , u : = ( \rho , m , \phi ) . \end{aligned} \end{align*}


\begin{align*} \forall n ^ 0 , x ^ X \preceq _ X L 1 _ X , y ^ X \exists z ^ X \preceq _ X L 1 _ X \exists w ^ X \left ( y \in A x \rightarrow ( w \in A z \land x = _ X z + _ X \gamma _ n w ) \right ) . \end{align*}


\begin{align*} \check H _ { c t } ^ q ( T ; A ) = \begin{cases} \bigoplus _ \N A & q = 0 \\ 0 & \mbox { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} d S _ t & = - \mu S _ t d t + F ^ 2 ( Y _ t ) d W _ t , \\ d \widetilde W _ t & = d W _ t . \end{align*}


\begin{align*} \P ( \exists B , B ' \in \mathfrak B _ { r - 1 } : \min _ { u \in B , u ' \in B ' } | u - u ' | _ { \infty } \leq 4 r \mbox { a n d } \sigma | _ B = \sigma | _ { B ' } ) \leq n ^ { - \epsilon / 4 } \ , . \end{align*}


\begin{align*} D ^ * _ R L : = \{ ( x , v ) \in T ^ * L \ | \ | v | \leq R \} \end{align*}


\begin{align*} c _ k ( \mu ) L ( \mu , \mathbf { f } , \chi ) \mathbf { f } ( g ) = \langle \mathbf { T } ( g , h ) , \mathbf { f } ( h ) \rangle . \end{align*}


\begin{align*} K _ * = 0 , C _ b = 0 , C _ W = \sigma _ 1 ^ { - 2 } . \end{align*}


\begin{align*} ( \tau + k \cdot ( v - V _ \ast ) ) \tilde { h } - \hat { \phi } ( k ) \rho [ \tilde { h } ] ( \tau , k ) k \cdot \nabla _ v \mu = \frac { - e _ 0 \hat { \Phi } ( k ) k \cdot \nabla _ v \mu } { i \tau } , \end{align*}


\begin{align*} m ' ( \mathbf { x } ) \coloneqq \begin{cases} \overline { m } & \mathbf { x } \in E _ u \ ; , \\ \underline { m } & \mathbf { x } \in \R ^ N \setminus E _ u \ ; , \end{cases} \end{align*}


\begin{align*} \mathcal { L } F ( u ) ( v ) = \int _ M ( \mathcal { L } u ) F ' ( u ) v d \mu + \int _ { \{ u > 0 \} } F '' ( u ) \Gamma ( u ) v d \mu , v \in \mathcal { D } ( \mathcal { E } ) \cap C _ c ( M ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } d X _ r ( t ) & = & \big ( A X _ r ( t ) + \alpha ( X _ r ( t ) ) \big ) d t + \sigma _ r ( X _ r ( t ) ) d W ( t ) \medskip \\ X _ r ( 0 ) & = & x , \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} & \bigl | | u + v | ^ { p _ 0 - 2 } - | u | ^ { p _ 0 - 2 } \bigr | \cdot | \mu _ n \sigma _ n ( u ) | \cdot | w | \\ & \qquad \qquad \le c \cdot \bigl ( | v | ^ { p _ 0 } \cdot ( 1 + ( n ^ { 3 / 4 } | w | ) ^ { p _ 0 } ) + 1 + | w | ^ { p _ 0 / 2 } + | u | ^ { p _ 0 } \bigr ) . \end{aligned} \end{align*}


\begin{align*} \mathcal { E } ( q , \lambda ( q , t ) ) = \frac { - 1 + e ^ { \lambda ( q , t ) \cdot ( 1 - q ) } } { 1 - q e ^ { \lambda ( q , t ) \cdot ( 1 - q ) } } = \frac { e ^ { \lambda ( q , t ) } - e ^ { \lambda ( q , t ) \cdot q } } { e ^ { \lambda ( q , t ) \cdot q } - q e ^ { \lambda ( q , t ) } } ; \end{align*}


\begin{align*} \varphi ( a ^ * ) = \overline { \varphi ( a ) } , a \in A . \end{align*}


\begin{align*} \begin{dcases} \rho _ t + ( \rho u ) _ x = 0 \ & ( 0 , + \infty ) \times ( 0 , L ) , \\ \rho \big ( u _ t + u u _ x \big ) + ( p ( \rho ) ) _ x - \nu u _ { x x } = 0 \ & ( 0 , + \infty ) \times ( 0 , L ) , \end{dcases} \end{align*}


\begin{align*} \chi ( x ) = ( - \rho \circ \pi _ { \ell } ^ { \Sigma } ( x ) , \rho \circ \pi _ r ^ { \Sigma } ( x ) ) , \end{align*}


\begin{align*} Z ( M _ l T _ k f ) ( x , \omega ) & = e ^ { 2 \pi i l \cdot x } Z f ( x - k , \omega - l ) \\ & = e ^ { 2 \pi i l \cdot x } e ^ { - 2 \pi i k \cdot \omega } Z f ( x , \omega ) . \end{align*}


\begin{align*} \omega _ i ( x , y , z ) = \varphi _ i ( [ x , y ] , z ) - \rho ( z ) \varphi _ i ( x , y ) , \ ; i = 1 , 2 \end{align*}


\begin{align*} g z = ( a z + b ) ( c z + d ) ^ { - 1 } , \lambda ( g , z ) = c z + d , g = \left [ \begin{array} { c c } a & b \\ c & d \end{array} \right ] . \end{align*}


\begin{align*} P ( \boldsymbol { \varphi } ) = \max _ { \sigma \in \mathcal { A } } \ , T _ d ^ { \sigma } . \end{align*}


\begin{align*} 1 < l : = 1 + m ( \alpha , \underline { b } _ 2 ) \sin ( \pi \delta ) \le n _ 2 ( x ) \le n _ 1 ( x ) \le \overline { b } _ 1 , x \in [ \delta , 1 - \delta ] . \end{align*}


\begin{align*} - k _ { p _ 0 } \leq \sum _ { j = 1 } ^ { p _ 0 } i _ { k _ j } \leq 0 , \end{align*}


\begin{align*} \hat { W } _ k : = \psi ( A ^ { [ k ] } _ { 1 : N } , k , F ) . \end{align*}


\begin{align*} B = \frac { 5 - 2 \sigma } { 2 } \quad C _ k = \frac { K + k } { K } + \frac { K } { K + k } - \frac { 8 } { 3 } ( 1 - \sigma ) \left ( 1 + \frac { k + 1 } { K } \right ) . \end{align*}


\begin{align*} p _ { l , m } ( f ) = \sup \{ | \partial ^ \alpha f ( x ) | \mid x \in \mathbf { K } _ l , | \alpha | \leq m \} , \end{align*}


\begin{align*} f ( z ) = \sum _ { \tau \in S _ + } c ( \tau ; v , w ) e ( \lambda ( \mathfrak { i } ( \tau ) u ) ) , \end{align*}


\begin{align*} w _ i = w ( \xi _ r ) = ( u \times v ) ( r ) = u _ r \quad w _ j = w ( \xi _ s ) = ( u \times v ) ( s ) = u _ s . \end{align*}


\begin{align*} \mathbf { f } ( g \times h ) = j ( g _ { \infty } \times h _ { \infty } , z _ 0 \times z _ 0 ) ^ { - k } \mathbf { f } ( g _ { \mathbf { h } } \times h _ { \mathbf { h } } , z \times w ) \end{align*}


\begin{align*} v ' ( x ) = \left \{ \begin{array} { l l } u ( x ) & x \not \in \Omega ' \\ \min \{ u ( x ) , v ( x ) \} & x \in \Omega ' \end{array} \right . \end{align*}


\begin{align*} \rho _ g ^ P ( \mathbf { h } ) : = \eta _ g ( D _ { X _ M } ) \end{align*}


\begin{align*} \tau ' ( b _ 0 a _ 1 b _ 1 ) = \tau ( b _ 0 \tau ' ( a _ 1 ) b _ 1 ) . \end{align*}


\begin{align*} p u b = p v c = q x c = q y \tilde { c } . \end{align*}


\begin{align*} P ( z ) y '' ( z ) = y ( z ) \end{align*}


\begin{align*} \hat { \varepsilon } _ - ( \tau ) = \varepsilon _ - ( \tau ) + \varepsilon _ + - \hat { \varepsilon } _ + \end{align*}


\begin{align*} \limsup _ { \varepsilon _ 1 \rightarrow 0 ^ { + } } \limsup _ { n \rightarrow \infty } \mathbb { E } _ { \mu _ n } \Big [ \int _ 0 ^ T n ^ { \gamma - 1 } \sum _ { x , y : | x - y | \geq \varepsilon n } & [ \overleftarrow { \eta } _ t ^ { \varepsilon _ 1 n } ( x ) \overrightarrow { \eta } _ t ^ { \varepsilon _ 1 n } ( x + 1 ) \\ - & \eta _ t ^ n ( x ) \eta _ t ^ n ( x + 1 ) ] F ( t , \tfrac { x } { n } , \tfrac { y } { n } ) ( c _ { \gamma } ) ^ { - 1 } p ( x - y ) d t \Big ] = 0 . \end{align*}


\begin{align*} P f = \pm i X f \Longleftrightarrow f ' ( x ) = \pm 2 \pi x f ( x ) \end{align*}


\begin{align*} \mathcal { L } P & = \lambda ' ( \sum _ { i } F ^ { i i } - 1 ) + u ^ { \frac { \alpha + 1 } { \alpha } } ( F ^ { i j } h _ { i l } h _ { j l } - F ^ { 2 } ) - u ^ { \frac { 1 } { \alpha } } F ^ { i j } \bar { R } _ { \nu j l i } \bar { g } ( \lambda \partial _ { r } , e _ { l } ) \\ & + \frac { 1 } { \alpha } \bar { g } ( \lambda \partial _ { r } , \nabla \log u ) - \frac { 1 } { \alpha } u ^ { \frac { \alpha + 1 } { \alpha } } F ^ { i j } \nabla _ { i } \log u \nabla _ { j } \log u . \end{align*}


\begin{align*} W ' ( \omega ) \equiv W _ { \rho ' } ( \omega ) = \bigcup _ { x \in I ( \omega ) } B ( x , \rho ' ) . \end{align*}


\begin{align*} \frac { \partial u ( \mathbf { x } , t ) } { \partial t } = - \left [ \sum _ { i = 1 } ^ { N } \frac { \partial } { \partial x _ { i } } A _ { i } ( \mathbf { x } , t ) + \sum _ { i , j = 1 } ^ { N } \frac { \partial ^ { 2 } } { \partial x _ { i } \partial x _ { j } } B ( \mathbf { x } , t ) \right ] u ( \mathbf { x } , t ) , \ ; \ , u ( \mathbf { x } , 0 ) = f ( \mathbf { x } ) \end{align*}


\begin{align*} w ^ { ( 4 ) } - ( \gamma - \eta ^ 2 ) \ , w '' + 2 i \rho ^ 2 \beta \eta w ' - \rho ^ 4 w = 0 . \end{align*}


\begin{align*} N ( T ) = \frac { T } { 2 \pi } \log \left ( \frac { T } { 2 \pi } \right ) - \frac { T } { 2 \pi } + \frac { 7 } { 8 } + \mathcal { O } ( \log T ) \end{align*}


\begin{align*} \sum _ { t = 1 } ^ { k } \bar { \sigma } ^ 2 _ { t } \geq \sum _ { t = 1 } ^ { k } \sigma ^ 2 _ { t } - 2 \sqrt { k } \theta \| S \| ^ 2 _ F . \end{align*}


\begin{align*} m _ { \eta } ( \xi , t , k ) : = P ( \xi , t , k ) \Psi ^ { A i } ( \zeta ( \xi , t , k ) ) Q ( \xi , k ) , \end{align*}


\begin{align*} ( U _ 1 ^ { q + 1 } \cup \cdots \cup U _ n ^ { q + 1 } ) ^ c \cap U ^ { q + 1 } & = \{ ( x _ 0 , \ldots , x _ q ) \in U ^ { q + 1 } : \forall i \in \{ 1 , \ldots , n \} \exists j \in \{ 0 , \ldots , q \} x _ j \not \in U _ i \} \\ & = \bigcup _ { f : \{ 1 , \ldots , n \} \to \{ 0 , \ldots , q \} } A _ f . \end{align*}


\begin{align*} q ^ { \pm \infty } : \mathbb { R } \times \mathbb { R } ^ { + } \rightarrow \mathbb { C } , q ^ { \pm \infty } ( x , t ) = C . \end{align*}


\begin{align*} { \bf L } _ n ^ 0 ( \chi _ T e ^ { i P ' c } h ) = i e ^ { i P ' c } \chi _ T ' { \bf A } _ n h + \chi _ T { \bf L } _ { n } ^ 0 ( e ^ { i P ' c } h ) . \end{align*}


\begin{align*} V _ Q D _ L U _ P = \left \{ \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in S p ( \R , 2 d ) \mid \det ( A ) \neq 0 \right \} . \end{align*}


\begin{align*} \begin{aligned} & \lim _ { n \to \infty } \sum _ { j = 1 } ^ { n - 1 } \sum _ { l = 1 } ^ k \frac { \binom { c _ n k n } l ( k ) _ l ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \frac k { k ( n - j + 1 ) - l } \frac { E _ { n , k } ( \tau ( A ^ { ( n ) } _ { c _ n k n , j , l } ) | \mathcal { W A } ^ { ( n , k ) } _ { M _ n , j , l } ) } { n k } = \\ & ( 1 - c ) \frac { p _ c ^ { ( k ) } ( 1 ) } { k + 1 } . \end{aligned} \end{align*}


\begin{align*} \mathcal R _ { N L } ( t , x ) = \frac { 1 } { R ^ 3 } \int _ 0 ^ { t } \int _ { \R ^ 3 } E ( s , \omega - ( t - R - s ) \tfrac { x - \omega } { R } + \tilde { Y } ( x , \tfrac { x - \omega } { R } ) ) \nabla _ v \mu ( \tfrac { x - \omega } { R } + \tilde { W } ( x , \tfrac { x - \omega } { R } ) ) \dd \omega \dd s . \end{align*}


\begin{align*} \sum _ { \substack { n \geq i \geq 0 \\ m \geq j \geq 0 } } f _ { i j } \bigl ( \sum _ { r _ { i j } \geq h \geq 1 } L i _ { K , \mathfrak { s } _ { i j , h } } ( { \boldsymbol \alpha _ { i j , h } } ) \bigr ) = 0 . \end{align*}


\begin{align*} \mathcal K = \overline { \textup { s p a n } } \{ f ( \underline N ) h \ , : \ ; h \in \mathcal H \textup { a n d } f \in \mathcal R ( K ) \} . \end{align*}


\begin{align*} \mu ( n - r - 1 ) = r ( r - \lambda - 1 ) . \end{align*}


\begin{align*} [ X , P ] f ( x ) = \tfrac { 1 } { 2 \pi i } ( x f ' ( x ) - ( x f ) ' ( x ) ) = \tfrac { 1 } { 2 \pi i } \left ( x f ' ( x ) - ( f ( x ) + x f ' ( x ) ) \right ) = - \tfrac { 1 } { 2 \pi i } f ( x ) . \end{align*}


\begin{align*} \tau _ \mu ( x ) = \Sigma \nabla \left ( G ^ \mu _ { 0 ^ + } ( g ) \right ) ( x ) . \end{align*}


\begin{align*} \Bigl | \exp \Bigl ( \sum \alpha _ k \Bigr ) - \prod _ { k = 1 } ^ n \exp ( \alpha _ k ) \Bigr | & = \Bigl | \exp \Bigl ( \sum _ { k = 1 } ^ n \alpha _ k + \sum _ { k = n + 1 } ^ \infty \alpha _ k \Bigr ) - \exp \Bigl ( \sum _ { k = 1 } ^ n \alpha _ k \Bigr ) \Bigr | \\ & = \Bigl | \exp \Bigl ( \sum _ { k = 1 } ^ n \alpha _ k \Bigr ) \Bigr | \Bigl | \exp \Bigl ( \sum _ { k = n + 1 } ^ \infty \alpha _ k \Bigr ) - 1 \Bigr | \xrightarrow { n \to \infty } 0 . \end{align*}


\begin{align*} \sum _ { r \in \mathbb { Z } } c ^ F _ r ( \widetilde { W _ v ^ l } , \widetilde { U _ v ^ l } ) q _ F ^ { - r / 2 } X ^ { - l r } = \omega _ { \sigma ^ { ( l ) } } ( - 1 ) ^ { n - 2 } \gamma ( X ^ l , \pi ^ { ( l ) } , \sigma ^ { ( l ) } , \psi ^ l ) \sum _ { r \in \mathbb { Z } } c ^ F _ r ( W _ v ^ l , U _ v ^ l ) q _ F ^ { r / 2 } X ^ { l r } . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { 2 | \hat { I } _ { = 2 , < \epsilon } ( k _ { n } ) | + 3 | \hat { I } _ { > 2 , < \epsilon } ( k _ { n } ) | } { k _ { n } } \leq 2 . \end{align*}


\begin{align*} \partial V = \{ y \in V : y _ 1 = 0 \} . \end{align*}


\begin{align*} \frac { 1 } { \prod \limits _ { j = i + 1 } ^ k R ( j ) } \sum \limits _ { x _ { k - 1 } < x _ k } \ldots \sum \limits _ { x _ { i + 1 } < x _ i } f ( x _ i ) & = \frac { m ( k , i ) } { \prod \limits _ { j = i + 1 } ^ k R ( j ) } \sum \limits _ { x _ { i } < x _ { k } } f ( x _ i ) \\ & = \frac { 1 } { R ( k , i ) } \sum \limits _ { x _ { i } < x _ { k } } f ( x _ i ) . \end{align*}


\begin{align*} ( K _ { H } ^ { \ast } \varphi ) ( s ) = & \ , c _ { H } \Gamma \left ( H + \frac { 1 } { 2 } \right ) \left ( D _ { T ^ { - } } ^ { \frac { 1 } { 2 } - H } \varphi ( s ) \right ) ( s ) \\ & + c _ { H } \left ( \frac { 1 } { 2 } - H \right ) \int _ { s } ^ { T } \varphi ( t ) ( t - s ) ^ { H - \frac { 3 } { 2 } } \left ( 1 - \left ( \frac { t } { s } \right ) ^ { H - \frac { 1 } { 2 } } \right ) d t . \end{align*}


\begin{align*} \begin{cases} \ X _ 1 \ \subset \ \R ^ { d _ 1 } , \ X _ 2 \ \subset \ \R ^ { d _ 1 } \\ [ 8 p t ] \ Y _ 1 \ \subset \ \R ^ { d _ 2 } , \ Y _ 2 \ \subset \ \R ^ { d _ 2 } \end{cases} \end{align*}


\begin{align*} s _ { i j } = \min \bigl \{ b _ { i j } ( \delta ) , s _ { i n } \bigr \} i \in \{ 1 , \ldots , d \} \ ; j \in \{ 0 , \ldots , n \} . \end{align*}


\begin{align*} L _ - = - \Delta + 1 - \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ { p - 2 } , \end{align*}


\begin{align*} \mathcal { S } ( T ) = \left \{ u \in \mathcal { X } _ T : \| u \| _ { \mathcal { X } _ T } \leq R , \ ; \ ; { \nu A ^ { - 1 } u } \ge \varepsilon \ ; \ ; t \in [ s , s + T ] \right \} , \end{align*}


\begin{align*} \tau ^ { Y , r } _ \varphi ( A _ 0 , \ldots , A _ k ) : = & \int _ { G ^ { \times k } } { } ^ b { \rm T r } _ Z \left ( \Phi _ { A _ 0 } ( ( g _ 1 \cdots g _ k ) ^ { - 1 } ) \circ \Phi _ { A _ 1 } ( g _ 1 ) \circ \ldots \circ \Phi _ { A _ k } ( g _ k ) \right ) \\ & \qquad \varphi ( e , g _ 1 , g _ 1 g _ 2 , \ldots , g _ 1 \cdots g _ k ) d g _ 1 \cdots d g _ k . \end{align*}


\begin{align*} \kappa _ { x , { \text r } } = \kappa _ { x , { \text t } } = \kappa _ x \end{align*}


\begin{align*} \sum _ { \mathbf { i } \in \mathcal { J } ^ 2 } | \Psi _ { \mathbf { i } } | = \sum _ { h , h ' = 2 } ^ 4 \sum _ { \mathbf { i } \in \mathcal I _ h \times \mathcal I _ { h ' } } | \Psi _ { \mathbf { i } } | = O ( n ^ 4 ) . \end{align*}


\begin{align*} S _ 1 T = \zeta _ { M T , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \sigma _ 3 ) T + O \left ( T ^ { 4 - 2 \sigma _ 1 - 2 \sigma _ 2 - 2 \sigma _ 3 } \right ) . \end{align*}


\begin{align*} B _ { a , b } = c ^ { - \frac { 1 } { \rho } } \left [ ( a \vee 0 ) ^ \frac { 1 } { \rho } , \ , ( b \vee 0 ) ^ \frac { 1 } { \rho } \right ] . \end{align*}


\begin{align*} r ^ c + r = n - 1 , \lambda ^ c - \mu = n - 2 ( r + 1 ) , \mu ^ c - \lambda = n - 2 r , \end{align*}


\begin{align*} ( 1 + X ) ^ n & = \sum _ { I \subseteq N } X ^ { | I | } = \sum _ { k = 0 } ^ n \binom { n } { k } X ^ k , \\ ( 1 - X ) ^ { - n } & = \sum _ { k _ 1 , \ldots , k _ n \ge 0 } X ^ { k _ 1 + \ldots + k _ n } = \sum _ { k = 0 } ^ \infty \binom { n + k - 1 } { k } X ^ k , \end{align*}


\begin{align*} R ^ Y ( h \overline { \mu } _ 1 ) = R ^ Y ( h \overline { \mu } _ 2 ) + h - 1 \end{align*}


\begin{align*} \lim _ { T \rightarrow \infty } \nu _ 0 \circ \alpha _ { T \Psi } ^ { 0 \to \tau } ( E _ { \Phi _ 1 - \Phi _ 0 } ) = \nu _ \tau ( E _ { \Phi _ 1 - \Phi _ 0 } ) . \end{align*}


\begin{align*} \begin{cases} t _ i = \alpha _ i + \dots + \alpha _ { n - 2 } + \frac { 1 } { 2 } ( \alpha _ { n - 1 } + \alpha _ n ) , & \ , ( 1 \leq i \leq n - 2 ) \\ t _ { n - 1 } = \frac { 1 } { 2 } ( \alpha _ { n - 1 } + \alpha _ { n } ) , & \\ t _ n = \frac { 1 } { 2 } ( - \alpha _ { n - 1 } + \alpha _ { n } ) . & \end{cases} \end{align*}


\begin{align*} S _ { 1 } = \left \{ \zeta \in \mathbb { C } \vert \textnormal { a r g } \zeta \in ( 0 , 2 \pi / 3 ) \right \} , S _ { 2 } = \left \{ \zeta \in \mathbb { C } \vert \textnormal { a r g } \zeta \in ( 2 \pi / 3 , \pi ) \right \} , \end{align*}


\begin{align*} C ^ q _ { A _ 1 , \ldots , A _ n } ( U , A ) = \{ \varphi : U ^ { q + 1 } \to A \mid \varphi | _ { A _ { i _ 0 } \times \cdots \times A _ { i _ q } } \mbox { c o n s t a n t } \forall i _ 0 , \ldots , i _ q \in \{ 1 , \ldots , q \} \} \end{align*}


\begin{align*} H _ n ( \mathbf { s } ) = H _ n ( s _ 1 , s _ 2 , \ldots , s _ r ) : = \sum \limits _ { 1 \leq k _ 1 < k _ 2 < \cdots < k _ r \leq n } \prod _ { j = 1 } ^ { r } \frac { 1 } { k _ j ^ { s _ j } } , \end{align*}


\begin{gather*} \phi [ T ] = u [ T ] . \end{gather*}


\begin{align*} \lim _ { n \to \infty } ( h _ n ^ j h _ n ^ k ) ^ { - \frac { 1 + \alpha } { 2 } } \Big | B _ n ^ j ( R ) \cap B _ n ^ k ( R ) \Big | = 0 \end{align*}


\begin{align*} F \ ! = \ ! ( f , u _ 0 ) \in \mathcal { F } ^ { k , \mathbf { s } ( s , \lambda , \lambda ' , \delta ) } _ { T , \varLambda ^ q , { \mathcal D } _ { d \oplus d ^ * } } \times C ^ { 2 s + k + 1 , \lambda , \delta } _ { \varLambda ^ q } \cap \mathcal { S } _ { d ^ \ast } \end{align*}


\begin{gather*} \| \tilde \phi _ { k + 1 } [ T ] - ( p , 0 ) \| _ { H ^ 1 _ x \times L ^ 2 _ x } = \| \left ( \phi _ k [ T ] - ( p , 0 ) \right ) ^ p \| _ { H ^ 1 _ x \times L ^ 2 _ x } \lesssim \varepsilon \| \phi _ k [ T ] - ( p , 0 ) \| _ { H ^ 1 _ x \times L ^ 2 _ x } , \end{gather*}


\begin{align*} s ( c , d ) = { 1 \over 4 d } \sum _ { n = 1 } ^ { \vert d \vert - 1 } \cot \left ( { \pi n \over d } \right ) \cot \left ( { \pi n c \over d } \right ) \ \ \ \ \ ( c \in { \mathbb Z } , \ d \in { \mathbb Z } \setminus \{ 0 \} , \ \gcd ( c , d ) = 1 ) , \end{align*}


\begin{align*} A = R _ k ^ { \alpha } \left ( \partial _ k ( p _ \alpha ) , f \right ) ( x ) + R _ k ^ { \alpha } \left ( \partial _ k ( f ) , p _ \alpha \right ) ( x ) , \end{align*}


\begin{align*} \sigma ^ { A } _ q > \sigma _ { q + 1 } = \sigma _ { q + 2 } = \cdots = \sigma _ { n + 1 } , \ \ q \leq n , \end{align*}


\begin{align*} I \bigg ( X , \frac { P } { Q } \bigg ) \ = \ \sum \limits _ { i = 1 } ^ m \ I \bigg ( K _ i , \frac { P _ i } { Q _ i } \bigg ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { j - 1 } w ( v _ i v _ j ) \geq ( j - 1 ) \max \biggl \{ ( h - 1 ) , \frac { r - 2 } { r - 1 } k \biggr \} . \end{align*}


\begin{align*} Z f ( x , \omega ) = | Z f ( x , \omega ) | e ^ { 2 \pi i \varphi ( x , \omega ) } . \end{align*}


\begin{align*} \sum _ { \l \in S \L } | \langle f , \rho ( \l ) \widehat { S } g \rangle | ^ 2 & = \sum _ { \l \in \L } | \langle f , \rho ( S \l ) \widehat { S } g \rangle | ^ 2 \\ & = \sum _ { \l \in \L } | \langle f , \widehat { S } \rho ( \l ) \widehat { S } ^ { - 1 } \widehat { S } g \rangle | ^ 2 \\ & = \sum _ { \l \in \L } | \langle \widehat { S } ^ { - 1 } f , \rho ( \l ) g \rangle | ^ 2 . \end{align*}


\begin{align*} X ( \theta ) \log X ( \theta ) = \theta X ^ 0 \log ( X ( \theta ) ) + ( 1 - \theta ) X ^ 1 \log ( X ( \theta ) ) . \end{align*}


\begin{align*} & \langle K ( x , y ) , \lambda _ \delta ( x , y ) \chi _ 0 ( y ) [ \phi ( y ) - \phi ( x ) ] \psi ( x ) \rangle \\ & = \sum \limits _ { j \in \Gamma } \langle K ( x , y ) , \lambda _ \delta ( x , y ) \bar \eta _ j ( y ) \chi _ 0 ( y ) [ \phi ( y ) - \phi ( x ) ] \psi ( x ) \rangle . \end{align*}


\begin{align*} C _ { S _ 0 } : = \left ( \frac { 2 4 C _ H } { S _ 0 } \right ) ^ { \frac { 1 } { 2 } } . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ { n } k ^ { \ell - 1 } S ( k ) & = \sum _ { k = 1 } ^ { n } k ^ { \ell - 1 } \sum _ { \substack { a \in S \\ 1 \leqslant a \leqslant k } } 1 = \sum _ { a \in S } \sum _ { a \leqslant k \leqslant n } k ^ { \ell - 1 } . \end{align*}


\begin{align*} [ 2 \cdot 4 ^ k q + 1 / 2 ] \ = \ [ 2 \cdot 4 ^ k \alpha ] + d ( d = 0 \ \ d = 1 ) \end{align*}


\begin{align*} R ^ { 2 , \operatorname { l i n } } & = \lbrack \beta ( \rho _ 0 + \nu + \nu _ c ) - \beta ( \rho _ 0 ) \rbrack u _ 0 + [ \beta ( \rho _ 0 + \nu + \nu _ c ) - \beta ( \nu ) ] w , \\ R ^ { 2 , \operatorname { c o r r } } & = \beta ( \rho _ 0 + \nu + \nu _ c ) w _ c , \\ R ^ h & = h _ 0 - h ^ \ast . \end{align*}


\begin{align*} \psi ( x , y ) = L ( | x - y | ^ 2 + 1 ) ^ { 1 \over 2 } . \end{align*}


\begin{align*} & \frac { \tilde { \phi } ^ { n + 1 } - 2 \phi ^ n + \phi ^ { n - 1 } } { \tau ^ 2 } + \eta ^ n \frac { \tilde { \phi } ^ { n + 1 } - \phi ^ { n - 1 } } { 2 \tau } \\ & = \left ( \frac 1 2 \Delta - \vartheta ^ n \right ) \frac { \tilde { \phi } ^ { n + 1 } + \phi ^ { n - 1 } } { 2 } + \vartheta ^ n \phi ^ n - V \phi ^ n - \beta | \phi ^ n | ^ 2 \phi ^ n + \Omega L _ z \phi ^ n + \lambda ^ n \phi ^ n , \\ & \phi ^ { n + 1 } = \tilde { \phi } ^ { n + 1 } / \| \tilde { \phi } ^ { n + 1 } \| , n \geq 1 , \end{align*}


\begin{align*} ( - \Delta ) ^ s _ \delta u ( x ) : = c _ { n , s } \int _ { \R ^ n \setminus B _ \delta ( x ) } \frac { u ( x ) - u ( y ) } { \vert x - y \vert ^ { n + 2 s } } \dd y . \end{align*}


\begin{align*} { C } _ { n } ( \kappa , { \bf P } _ { Y _ 1 } ) = \sup _ { ( \Lambda _ t , K _ { \overline { Z } _ t } ) , t = 1 , \ldots , n , \frac { 1 } { n } { \bf E } \big \{ \sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \mathbb R } ^ { n _ x } } ^ 2 \big \} \leq \kappa } \sum _ { t = 1 } ^ n \Big \{ H ( I _ t ) - H ( \hat { I } _ t ) \Big \} \end{align*}


\begin{align*} \flat _ { \omega _ Q } ( \pmb { X } ) = d H , \end{align*}


\begin{align*} \Sigma _ { \sigma _ 2 } ^ { 1 } = 2 \sum _ { \substack { 0 < \gamma < T \\ \sigma _ 2 \le \beta < 1 } } \frac { x ^ { \beta - 1 } } { \gamma } \le 2 N ( \sigma _ 2 , T ) \frac { x ^ { - \frac { 1 } { R \log t _ 0 } } } { t _ 0 } , \end{align*}


\begin{align*} \lim _ { T \to \infty } \frac { N _ 0 ( T ) } { N ( T ) } = 1 \end{align*}


\begin{align*} & \mathcal H _ 0 ( \rho , S ) = \mathcal K ( S , \rho ) + \mathcal F ( \rho ) - \alpha L ( \rho ) , \ ; \\ & \mathcal H _ 1 ( \rho , S ) = \eta _ 1 \mathcal K ( S , \rho ) + \eta _ 2 I ( \rho ) + \eta _ 3 \mathcal V ( \rho ) + \eta _ 4 \mathcal W ( \rho ) - \eta _ 5 L ( \rho ) \end{align*}


\begin{align*} r = \sqrt { P _ t } \Biggl ( \sum _ { n = 1 } ^ N g _ n c _ n \mathbf { h } _ n ^ T + \mathbf { h } _ d ^ T \Biggr ) \mathbf { w } s + n , \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 ! } , ( k \ge 0 ) . \end{align*}


\begin{align*} \int _ 0 ^ 1 \left ( \frac { 1 } { n } - \frac { 1 } { n ^ 3 } \right ) n _ x \varphi _ x d x + \alpha \int _ 0 ^ 1 \frac { \varphi _ x } { n } d x + \int _ 0 ^ 1 \left ( n - b \right ) \varphi d x = 0 , \end{align*}


\begin{align*} [ W ] = \frac { \langle W \rangle } { \langle W ' \rangle } . \end{align*}


\begin{align*} \abs { x } = \prod _ { s \in V } \abs { x _ s } _ s \end{align*}


\begin{align*} A ( \Gamma \bigodot m ) = \left ( \begin{array} { c c c c } A _ { 1 1 } & A _ { 1 2 } & \cdots & A _ { 1 n } \\ A _ { 2 1 } & A _ { 2 2 } & \cdots & A _ { 2 n } \\ \vdots & \vdots & \ddots & \vdots \\ A _ { n 1 } & A _ { n 2 } & \cdots & A _ { n n } \end{array} \right ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \ddot { u } ( t ) + \eta ( t ) \dot { u } ( t ) = - \nabla f ( u ( t ) ) , t > 0 , \\ & u ( 0 ) = u _ 0 , \dot { u } ( 0 ) = v _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} \widehat { V _ g f } ( \xi , \eta ) = e ^ { 2 \pi i \xi \cdot \eta } f ( - \eta ) \overline { \widehat { g } ( \xi ) } . \end{align*}


\begin{align*} \langle \Phi , \phi _ i \rangle _ { L ^ 2 _ \rho } = \left \{ \begin{array} { r c l } & & \mathcal { I } _ 0 \left [ \frac { b _ \tau } { b } - 1 \right ] b ^ \alpha + O \left ( b ^ { \alpha + 1 - \frac { \epsilon } { 2 } } \right ) , \\ & & O ( b ^ { \alpha + 1 - \frac { \epsilon } { 2 } } ) . \end{array} \right . \end{align*}


\begin{align*} N ^ { t - T _ 3 ^ { \sigma } } \big ( N a ^ t + c ^ t \big ) = 1 \ ; \ ; N ^ { 1 - T _ 3 ^ { \omega } } \big ( N b + d \big ) = 1 , \end{align*}


\begin{align*} L _ z = - i ( x \partial _ y - y \partial _ x ) \end{align*}


\begin{align*} E ^ { i j } _ 0 ( x , x ' ) = a \left ( \frac { \partial } { \partial x } \right ) ^ { \mu _ i + \mu _ j - 1 } \delta ( x - x ' ) . \end{align*}


\begin{align*} \dot { x } _ i \sim \mu _ i , \quad \dot { \mu } _ i + \sum ^ n _ { \substack { j = 1 , \\ j \not = i } } \frac { a _ { i j } } { ( x _ i - x _ j ) ^ 3 } + \sum _ { \substack { j , k = 1 , \\ j \not = i } } ^ n \frac { b _ { i j k } \mu _ k } { ( x _ i - x _ j ) ^ 3 } \sim 0 , \end{align*}


\begin{align*} \begin{aligned} & u _ { i n } ( x ) = f _ n ( x ) , u _ { i j } ( x ) \leq f _ { i } ( x ) , & & i = 1 , \ldots , d , \ j = 0 , \ldots , n , \\ & u ( x ) \leq a ( x ) , f ^ L _ i = u _ { i 0 } ( x ) = a _ { i 0 } ( x ) \leq \cdots \leq a _ { i n } ( x ) = f ^ U _ { i } & & i = 1 , \ldots , d , \\ & \deg ( u _ { i j } ) \leq \tau , \deg ( a _ { i j } ) \leq \tau & & i = 1 , \ldots , d , \ j = 0 , \ldots , n . \end{aligned} \end{align*}


\begin{align*} L A ( \phi ) = L \phi \quad \mbox { f o r a l l } \quad \phi \in C ^ 1 . \end{align*}


\begin{align*} ( ( y + x ) - x ) \in A x & \leftrightarrow x = _ X J ^ A _ \gamma ( y + x ) \\ & \leftrightarrow x ' = _ X J ^ A _ \gamma ( y ' + x ' ) \\ & \leftrightarrow ( ( y ' + x ' ) - x ' ) \in A x ' \end{align*}


\begin{align*} f _ { \alpha } ^ { ( \gamma ) } ( o ) = \sum _ { \beta < \gamma } a _ { \beta } g _ { \beta } ^ { ( \gamma ) } ( o ) + a _ { \gamma } g _ { \gamma } ^ { ( \gamma ) } ( o ) . \end{align*}


\begin{align*} S ( H , f ) = \frac { f - 2 d + 1 } { 4 } \hbox { a n d } N ' ( f , H ) = - 2 d + 1 , \end{align*}


\begin{align*} H = - ( h _ { i j } \dot { M } ^ i _ k \dot { ) } M ^ j _ l x ^ k x ^ l \end{align*}


\begin{align*} \arg \left ( \frac { \sigma + i T } { 2 } \right ) + \arg \left ( \sigma - 1 + i T \right ) = \pi + \mathcal { O } \left ( \frac { 1 } { T } \right ) \end{align*}


\begin{align*} f ( t ) = \sum _ { k \in \Z } f ( k ) g ( t - k ) , \forall t \in \R \forall f \in B _ I ^ 1 ( \R ) , \end{align*}


\begin{align*} \Phi _ { p , q } ( \sigma , \tau ) = \gamma ( \sigma ) + \tau \eta ( \sigma ) \ , . \end{align*}


\begin{align*} \tau _ i f ( t , y ) = f ( t , y - x _ i ( t ) ) ; \end{align*}


\begin{align*} \det Q _ S = \frac { 1 } { 2 } \Big [ 2 + ( - 1 ) ^ { k } ( \det V - \det U ) \Big ] , \end{align*}


\begin{align*} \hat { \overline { u } } _ n ( \xi ) = \frac { ( - i \xi ) ^ n } { n ! } \left [ \alpha - i \pi \ , \mathrm { s g n } ( \xi ) \right ] . \end{align*}


\begin{align*} \L ^ \circ _ { a , b } = \{ \l ^ \circ \in \R ^ 2 \mid \pi ( \l ^ \circ ) \pi ( \l ) = \pi ( \l ) \pi ( \l ^ \circ ) \forall \l \in \L _ { a , b } \} . \end{align*}


\begin{align*} \sup _ { | y | = 1 } \sum \limits _ { j = 1 } ^ { \infty } \int _ { \mathfrak { D } ^ * } ~ | \Omega ( x + \mathfrak { p } ^ j y ) - \Omega ( x ) | ~ d x < \infty . \end{align*}


\begin{align*} C _ { v } ( u , w , z ) = \frac { 1 } { 4 } \left . \frac { \partial ^ 3 } { \partial r \partial s \partial t } L ( v + t u + s w + r z ) \right | _ { t = s = r = 0 } , \end{align*}


\begin{align*} & T _ 1 ( u ) - T ^ { ' } _ 1 ( u ) = T D ( \mathfrak { X } ) u - [ \mathfrak { X } , T u ] = ( d _ T \mathfrak { X } ) ( u ) . \end{align*}


\begin{align*} \varphi ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \cdots a _ n b _ n ) & = 0 + \varphi ( b _ 0 a _ 1 a _ 2 \cdots a _ n b _ n ) \tau ( b _ 1 ) \cdots \tau ( b _ { n - 1 } ) \\ & \varphi ( b _ 0 a _ 1 a _ 2 \cdots a _ n b _ n ) \varphi ( b _ 1 ) \cdots \varphi ( b _ { n - 1 } ) . \end{align*}


\begin{align*} ( u \otimes \overline { v } ) ( h ) = \langle h , v \rangle _ \mathcal { H } u , u , v , h \in \mathcal { H } . \end{align*}


\begin{align*} _ { s + 1 } F _ { s } ( a _ 1 , \ldots , a _ { s + 1 } ; b _ 1 , \ldots , b _ s ; x ) : = \sum \limits _ { i = 0 } ^ \infty \frac { ( a _ 1 ) _ i \cdots ( a _ { s + 1 } ) _ i } { ( b _ 1 ) _ i \cdots ( b _ s ) _ i } \frac { x ^ i } { i ! } , \end{align*}


\begin{align*} ( M ) \in \{ 1 \} \cup \bigcup _ { i = 1 } ^ k \left [ \lambda _ i ( M ) - e , \lambda _ i ( M ) + e \right ] \end{align*}


\begin{align*} S _ { g , \widetilde { g } , \Gamma } f = \sum _ { \gamma \in \Gamma } \langle f , \pi ( \gamma ) g \rangle \pi ( \gamma ) \widetilde { g } . \end{align*}


\begin{align*} m _ { 0 , \lambda } ( \Z ( \{ P _ 1 , \ldots , P _ q \} , B ) ) = 0 . \end{align*}


\begin{align*} \langle W \rangle \langle ( W ' ) '' \rangle - \langle W ' \rangle \langle W '' \rangle = ( - 1 ) ^ { | W | } \mbox { f o r } | W | \geq 2 . \end{align*}


\begin{align*} & \sum _ { i = \lfloor N - \eta ^ { - 1 } \rfloor + 1 } ^ { N - 1 } \sup _ { h \in \mathrm { L i p } ( 1 ) } \left | \tilde { Q } _ { i - 1 } \big ( P _ { \eta } - \tilde { Q } _ { 1 } \big ) P _ { ( N - i ) \eta } h ( x ) \right | \\ & \qquad \leq C ( 1 + | x | ) \ , \eta ^ { 2 / \alpha } \sum _ { i = \lfloor N - \eta ^ { - 1 } \rfloor + 1 } ^ { N - 1 } [ ( N - i ) \eta ] ^ { - 1 / \alpha } \leq C \frac { \alpha } { \alpha - 1 } \ , ( 1 + | x | ) \ , \eta ^ { 2 / \alpha - 1 } . \end{align*}


\begin{align*} A ( N ( x ) , x ) = A ( x , N ( x ) ) = 1 , \ \forall x \in [ 0 , 1 ] . \end{align*}


\begin{align*} \chi _ { | | ; \alpha } ^ { ( \ell ) } : = \chi _ { | | } ( K _ { \alpha \alpha } ^ { ( \ell ) } ) = 1 = \chi _ { \perp ; } ( K _ { \alpha \alpha } ^ { ( \ell ) } ) = : \chi _ { \perp ; \alpha } ^ { ( \ell ) } , \forall \ell \geq 1 , \ , x _ \alpha \in \R ^ { n _ 0 } . \end{align*}


\begin{align*} 1 + R _ - ( \kappa _ x ) = T _ + ( \kappa _ x ) . \end{align*}


\begin{align*} 0 \le s ( \nu _ + | \nu _ 1 ) \le \liminf _ { T \to \infty } s ( \nu _ 0 \circ \alpha _ { T \Psi } ^ { 0 \to 1 } | \nu _ 1 ) = 0 . \end{align*}


\begin{align*} P _ n = \sum _ { i = 1 } ^ n X _ i ^ 2 ( X _ i - 1 ) ^ 2 \end{align*}


\begin{align*} \sum _ { \rho \in R _ \epsilon } ( \rho ) & = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { T } \left [ \log \left | \frac { \frac { 1 } { 2 } - \epsilon + i t } { 2 } \right | - \log \left | \frac { \frac { 1 } { 2 } + \epsilon + i t } { 2 } \right | \right ] \ d t + \\ & \frac { 1 } { 2 \pi } \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \left [ \arg \left ( \frac { \sigma + i T } { 2 } \right ) - \arg \frac { \sigma } { 2 } \right ] \ d \sigma \end{align*}


\begin{align*} \prod _ { i = 1 } ^ n \prod _ { j \ne i } | \alpha _ i - \alpha _ j | _ v \leq \delta . \end{align*}


\begin{align*} \check { \varphi } ( \omega ) = \varepsilon ( a ) , \omega = \varphi ( - a ) , a \in A , \end{align*}


\begin{align*} \Lambda _ n = \mathrm { R } [ X _ 1 , \ldots , X _ n ] ^ { \mathfrak { S } _ n } \end{align*}


\begin{align*} s B s _ i & = s X _ i U _ i C s _ i \\ & = { } ^ s X _ { i } s s _ i U _ i C \\ & \subseteq B s s _ i B , \end{align*}


\begin{align*} \Phi ( x , \omega ) = e ^ { - \tfrac { \pi } { 2 } ( x ^ 2 + \omega ^ 2 ) } \Phi ^ { \xi , \eta } ( x , \omega ) = e ^ { \pi i ( x \cdot \eta - \xi \cdot \omega ) } \Phi ( x - \xi , \omega - \eta ) \end{align*}


\begin{align*} \partial _ { \Sigma , \epsilon } ^ { * } \circ \Psi _ { ( \Sigma , U ) } \circ \eta _ { ( \Sigma , U ) } = \partial _ { \widetilde { \Sigma } , \widetilde { \epsilon } } ^ { * } \circ \Psi _ { ( \widetilde { \Sigma } , U ) } \circ \eta _ { ( \widetilde { \Sigma } , U ) } , \end{align*}


\begin{align*} \varPhi _ q \ , d _ q \ , \varPsi _ { \mu , q } { \mathcal N } _ q u = \varPhi _ q \ , d _ q \ , \varPsi _ \mu M _ 1 ^ { ( q ) } ( ( d _ q \oplus d _ { q - 1 } ^ * u , u ) , \end{align*}


\begin{align*} x \in \mathrm { d o m } ( J ^ A _ \gamma ) : = \exists y ^ X \left ( \gamma ^ { - 1 } ( x - _ X y ) \in A y \right ) . \end{align*}


\begin{align*} \phi _ c ^ 1 ( S , t ) = \theta ^ 1 _ c ( \| S \| _ { \mathcal C ( [ 0 , t ] ; \mathbb R ^ N ) } ) , \ ; \ ; \phi _ c ^ 2 ( \rho , t ) = \theta ^ 2 _ c \Big ( \min _ { i = 1 } ^ N \min _ { s \in [ 0 , t ] } \rho _ i ( s ) \Big ) , \end{align*}


\begin{align*} \omega ^ { ( j ) } = ( \delta _ { 1 , j } , \ldots , \delta _ { q , j } ) \in \Omega . \end{align*}


\begin{align*} \gamma _ F ( P \cap \tau _ v P ) = ( 2 \pi ) ^ { - m k } ( \det \tilde { \Lambda } ) ^ { - \frac { 1 } { 2 } } \int _ { B \times B } e ^ { - \frac { 1 } { 2 } \tilde { \Lambda } ^ { - 1 } t \cdot t } \ d t , \end{align*}


\begin{align*} \partial _ t u = \partial _ r ^ 2 u + \frac { d + 1 } { r } \partial _ r u - ( d - 1 ) \frac { \sin ( 2 u ) } { 2 r ^ 2 } , ( r , t ) \in \R _ + \times \R _ + . \end{align*}


\begin{align*} \check { \varphi } ( \varphi ( - a ) \varphi ( - b ) ) = \varphi ( S ^ { - 1 } ( b ) a ) . \end{align*}


\begin{align*} | E _ 1 \cup E _ 2 | \le \varepsilon n ^ 2 . \end{align*}


\begin{align*} \begin{aligned} Y & \le c \cdot \ , \int _ s ^ { s + \delta } | X _ u - X _ s | ^ { p - 2 } \cdot \bigl ( 1 + X _ u ^ 2 + | X _ s | \cdot ( 1 + | X _ u | ^ { \ell _ \mu + 1 } ) \bigr ) \ , d u \\ & + \sup _ { s \le t \le s + \delta } \Bigl | \int _ s ^ t p \cdot ( X _ u - X _ s ) \cdot | X _ u - X _ s | ^ { p - 2 } \cdot \sigma ( X _ u ) \ , d W _ u \Bigr | \end{aligned} \end{align*}


\begin{align*} \mathrm { V o l } ( N ) & \leq \mathrm { V o l } \left ( B ^ { M ^ k ( - \lambda ) } _ { \mathrm { D i a m } ( N ) } ( x ' ) \right ) = \frac { 2 \pi ^ { \frac { k } { 2 } } } { \Gamma ( \frac { k } { 2 } ) } \int _ 0 ^ { \mathrm { D i a m } ( N ) } \left ( \frac { \sinh ( t \sqrt { \lambda } ) } { \sqrt { \lambda } } \right ) ^ { k - 1 } d t . \end{align*}


\begin{align*} P \pi ( \mathbf { h } ) f = P \pi ( \mathbf { h } ) P f = \pi ( \mathbf { h } ) P f , \forall \mathbf { h } \in \mathbf { H } . \end{align*}


\begin{align*} J _ 3 : = \sum _ { \substack { \alpha \in A \\ \beta \in B } } \sum _ { \substack { \alpha ' \neq \alpha \\ \beta ' \neq \beta } } \frac { 1 } { 4 \pi i } \int _ { ( \epsilon ) } \underset { s _ 2 = - s _ 1 - \alpha ' - \beta ' } { } \ \mathcal { J } \ , d s _ 1 . \end{align*}


\begin{align*} & \widetilde m _ { 1 2 } = ( \Sigma _ 1 - \Sigma _ 2 ) \theta _ { 1 2 } ( \rho ) \dot W ^ { \delta } , \ ; \widetilde m _ { 2 3 } - ( \Sigma _ 2 - \Sigma _ 3 ) \dot W ^ { \delta } = m _ { 2 3 } - ( \Sigma _ 2 - \Sigma _ 3 ) \theta _ { 2 3 } ( \rho ) \dot W ^ { \delta } . \end{align*}


\begin{align*} \int _ 0 ^ \tau \int _ \Omega p ( \rho ) b _ \alpha ( \rho ) - I ^ \alpha _ 1 = - \int _ 0 ^ \tau \langle \nabla p ( \rho ) , \mathcal { B } ( b _ \alpha ( \rho ) - \langle b _ \alpha ( \rho ) \rangle \rangle = \sum _ { j = 2 } ^ 6 I ^ { \alpha } _ { j } + J ^ { \alpha } ( \tau ) , \end{align*}


\begin{align*} \eta _ { { \rm r h s } , 2 } ( E ) = h _ E \Vert f - \Pi _ { E , q - 1 } f \Vert _ { 0 , E , \omega } + \Vert f - \Pi _ { E , q } f \Vert _ { 0 , E , \omega } \ , . \end{align*}


\begin{align*} X _ 1 ^ T ( \tau ) = x _ 1 - ( t - \tau ) v _ 1 . \end{align*}


\begin{align*} \zeta _ { s , t , x } ( v ) : = v - \frac { \tilde Y _ { s , t } ( x , v ) } { t - s } . \end{align*}


\begin{align*} \mathcal { L } _ \beta = \Delta - \beta ( \tau ) z \cdot \nabla - \beta ( \tau ) I d \end{align*}


\begin{align*} \mathcal { F } _ N ( \mathbf { h } , \mathbf { g } ) = \mathbb { E } \left [ \int _ 0 ^ T \mathcal { L } \left ( t , \mathbf { Y } ( t ) , \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \delta _ { X ^ n ( t ) } \right ) d t + \int _ 0 ^ T \Psi \left ( \mathbf { h } ( t ) , \mathbf { g } \left ( \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \delta _ { X ^ n ( t ) } \right ) \right ) d t \right ] , \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - A ^ \intercal S _ { i _ k } ( S _ { i _ k } ^ \intercal A B ^ { - 1 } A ^ \intercal S _ { i _ k } ) ^ \dagger S _ { i _ k } ^ \intercal ( A x _ k - b ) , \end{align*}


\begin{align*} L ' \otimes \Z _ 2 & = \bigoplus _ { j = 0 } ^ 1 L ' _ { 2 , j } ( \pi ^ j ) , \\ K _ { \ell } \otimes \Z _ 2 & = K _ { \ell , 2 , 0 } , \end{align*}


\begin{align*} \langle S f , f \rangle = \sum _ { \gamma \in \Gamma } | \langle f , e _ \gamma \rangle | ^ 2 , \end{align*}


\begin{align*} \bar { \Delta } _ { \rm R e a c t i v e } = \frac { \mathbb { E } G ^ { \rm R e a c } _ j } { \mathbb { E } T ^ { \rm R e a c } _ j } = \frac { \mathbb { E } T ^ { \rm P r o a c } _ j } { 2 \mathbb { E } \left ( T ^ { \rm R e a c } _ j \right ) ^ 2 } + \mathbb { E } \tau ^ { \rm R e a c } _ { V _ j } - \tau _ { \rm f } - \frac { 1 } { 2 } . \end{align*}


\begin{align*} x _ t = x _ 0 + \int _ 0 ^ t \big ( \alpha _ { - 1 } x ^ { - 1 } _ s - \alpha _ 0 + \alpha _ 1 x _ s - \alpha _ 2 s ^ { 2 H - 1 } x _ s ^ { \rho } \big ) d s + \int _ 0 ^ t \sigma x _ s ^ { \theta } d B _ s ^ H , \end{align*}


\begin{align*} \mathrm { E C H } ( Y , \lambda , \Gamma ) : = \bigoplus _ { * : \ , \ , \mathbb { Z } \mathrm { - g r a d i n g } } \mathrm { E C H } _ { * } ( Y , \lambda , \Gamma ) . \end{align*}


\begin{align*} \pi : M \to \mathbf { H } , \pi ( C ) = \left ( C \cdot D ^ x \right ) \ell + \left ( C \cdot \mathbb { D } \right ) \gamma . \end{align*}


\begin{align*} L ' _ { \tilde { b } } \wedge L ' _ { c } = L ' _ { s y \tilde { b } } . \end{align*}


\begin{align*} E [ ( S _ n ) ^ 2 ] & = \dfrac { \Gamma ( n + 2 \alpha ) } { \Gamma ( n ) } \sum _ { \ell = 1 } ^ n \dfrac { \Gamma ( \ell ) } { \Gamma ( \ell + 2 \alpha ) } \\ & = \begin{cases} \dfrac { n } { 1 - 2 \alpha } + \dfrac { 1 } { 2 \alpha - 1 } \cdot \dfrac { \Gamma ( n + 2 \alpha ) } { \Gamma ( n ) \Gamma ( 2 \alpha ) } & ( \alpha \neq 1 / 2 ) , \\ \displaystyle n \sum _ { \ell = 1 } ^ n \dfrac { 1 } { \ell } & ( \alpha = 1 / 2 ) . \end{cases} \end{align*}


\begin{align*} \left \{ \begin{array} { l } i \partial _ t u - \Delta u = \rho ^ 2 | u | ^ 2 + \langle \nabla \rangle ^ { - \alpha } \dot { W } \ , , t \in [ 0 , T ] \ , , \ , x \in \R ^ d \ , , \\ u _ 0 = \phi \ , , \end{array} \right . \end{align*}


\begin{align*} \lambda h _ q + ( 1 - \lambda ) k _ q = \psi _ t ( x _ q , t _ q ) , \eta _ q = \zeta _ q = \nabla \psi ( x _ q , t _ q ) , \end{align*}


\begin{align*} \kappa ^ N _ { \sigma } ( \mathbf { M } ) = \sum _ { g \ge 0 } N ^ { - 2 g } \kappa _ { \sigma } ^ { ( 2 g ) } ( \mathbf { M } ) . \end{align*}


\begin{align*} a [ p ] : = a + { \rm s g n } ( a ) \cdot p = \begin{cases} a + p , & a > 0 , \\ a - p , & a < 0 . \end{cases} \end{align*}


\begin{align*} ( f \otimes g ) ( x , \omega ) = f ( x ) g ( \omega ) \end{align*}


\begin{align*} l \lambda _ p - m s _ \sigma + t _ \sigma + 1 = \lambda _ q \ , , \end{align*}


\begin{align*} \alpha _ { \rm H J M } ( h ) = \sum _ { j = 1 } ^ { \infty } \sigma ^ j ( h ) \int _ 0 ^ { \bullet } \sigma ^ j ( h ) ( \eta ) d \eta , h \in H . \end{align*}


\begin{align*} \left \{ \begin{aligned} & u _ { k } = w _ { k - 1 } - s \nabla f \left ( w _ { k - 1 } \right ) , \\ & w _ { k } = u _ { k } + \frac { k - 1 } { k + 2 } \left ( u _ { k } - u _ { k - 1 } \right ) . \end{aligned} \right . \end{align*}


\begin{align*} \int _ E \Pi _ { E , k } \varphi = \int _ E \varphi \forall \varphi \in L ^ 2 ( E ) \ , . \end{align*}


\begin{align*} \inf _ { B _ 1 ( 2 e _ 1 ) } \tilde u _ { \bar M } = 0 . \end{align*}


\begin{align*} u ( a ) & = \tau \zeta ( a ) ( \rho - \vert a \vert ) ^ { - n - 2 } . \end{align*}


\begin{align*} x ^ r + \gamma _ 1 \omega ( x ) x ^ { r - 1 } + \cdots + \gamma _ { r - 1 } \omega ( x ) ^ { r - 1 } x = 0 , \end{align*}


\begin{align*} & \Gamma : = \bigcup _ { i \in I } P _ i ^ * ( \Gamma _ i ) , \\ & \rho ( P _ i ^ * ( x ^ * ) ) : = \rho _ i ( x ^ * ) \ , \ , i \in I x ^ * \in \Gamma _ i . \end{align*}


\begin{align*} \mathcal { E } ( h , k ) = \mathcal { L } ^ r ( h , k ) + \mathcal { U } ^ r ( h , k ) + O \bigg ( \bigg ( Q + \frac { Q ^ 2 } { C } \bigg ) \frac { ( X C Q h k ) ^ { \varepsilon } ( h , k ) } { \sqrt { h k } } \bigg ) \\ + O \Big ( ( X C Q h k ) ^ { \varepsilon } \big ( X C + X ^ { - \frac { 1 } { 2 } } Q ^ { \frac { 5 } { 2 } } + Q ^ { \frac { 3 } { 2 } } + X ^ 2 h k Q ^ { - 9 6 } \big ) \Big ) . \end{align*}


\begin{align*} { \bar { \Delta } _ { \rm T A R Q } } = - \frac { 1 } { 2 } + { n _ 1 } \left ( { \frac { 2 } { { 1 - { \epsilon _ 1 } } } - \frac { 1 } { 2 } - \frac { { m \epsilon _ 1 ^ m } } { { 1 - \epsilon _ 1 ^ m } } } \right ) . \end{align*}


\begin{align*} \int _ 2 ^ T \abs { E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 & = O \left ( \int _ 2 ^ T t _ 3 ^ { 3 - 2 \sigma _ 1 - 2 \sigma _ 2 - 2 \sigma _ 3 } d t _ 3 \right ) \\ & \ll \begin{cases} T ^ { 4 - 2 \sigma _ 1 - 2 \sigma _ 2 - 2 \sigma _ 3 } & ( \frac { 3 } { 2 } < \sigma _ 1 + \sigma _ 2 + \sigma _ 3 < 2 ) \\ \log T & ( \sigma _ 1 + \sigma _ 2 + \sigma _ 3 = 2 ) . \\ \end{cases} \end{align*}


\begin{align*} F _ { n , \lambda } ( 1 ) = \sum _ { k = 0 } ^ { \infty } ( k ) _ { n , \lambda } \bigg ( \frac { 1 } { 2 } \bigg ) ^ { k + 1 } , ( n \ge 0 ) . \end{align*}


\begin{align*} S _ { g , \L _ { a , b } } f = \sum _ { \l \in \L _ { a , b } } \langle f , \pi ( \l ) g \rangle \ , \pi ( \l ) g . \end{align*}


\begin{align*} e ^ { - t \mathfrak { D } _ { { \rm l e f t } , a } } u ( x ) & = \exp \bigg [ - \int _ { x - t } ^ { x } a ( y ) \ , d y \bigg ] u ( x - t ) \\ e ^ { - t \mathfrak { D } _ { { \rm r i g h t } , a } } v ( x ) & = \exp \bigg [ - \int _ { x } ^ { x + t } a ( y ) \ , d y \bigg ] v ( x + t ) , \end{align*}


\begin{align*} \frac { B } { A } = e ^ { 2 a \gamma } . \end{align*}


\begin{align*} { \bf H } [ g _ b ] ( x ) ~ = ~ { 1 \over \pi } \cdot \int _ { 0 } ^ { 2 } g _ b ' ( y ) \cdot \ln | x - y | ~ d y . \end{align*}


\begin{align*} x _ { k + 1 } = x _ k + w _ k \frac { w _ k ^ \intercal A ^ \intercal ( b - A x _ k ) } { \norm { A w _ k } _ 2 ^ 2 } , \end{align*}


\begin{align*} \phi = s \chi _ A + t \chi _ B + \chi _ C \end{align*}


\begin{align*} & \int ^ { T } _ 0 \int _ { \Omega \times \mathbb { R } ^ 3 } f _ N \big ( \partial _ t \varphi + v \cdot \nabla _ x \varphi + ( u _ { N } \chi _ { \{ | u _ N | \leq N \} } - v ) \cdot \nabla _ v \varphi + L [ f _ N ] \cdot \nabla _ v \varphi + \Delta _ v \varphi \big ) \ , d x d v d t \\ & = - \int _ { \Omega \times \mathbb { R } ^ 3 } f _ { 0 , N } \varphi ( 0 , x , v ) \ , d x d v + \int _ 0 ^ T \int _ { \Sigma ^ - } ( v \cdot \nu ( x ) ) g _ N \varphi \ , d \sigma ( x ) d v d t ; \end{align*}


\begin{align*} w ( t ) = \ & U ( t ) \sigma - \int _ 0 ^ t U ( t - \tau ) z ( \tau ) d \tau . \end{align*}


\begin{align*} U ( w ) \Phi ( z ) U ( w ) ^ { - 1 } = \left ( \frac { d w } { d z } \right ) ^ { \mu } \Phi ( w ( z ) ) \ , , \end{align*}


\begin{align*} \nabla P = \lambda \partial _ { r } ^ { T } - u ^ { \frac { 1 } { \alpha } } \nabla u . \end{align*}


\begin{align*} \rho ^ { n + 1 } = _ { \rho \in K } \left \{ \frac { 1 } { 2 \tau } W ^ 2 _ 2 ( \rho ^ n , \rho ) + E ( \rho ) \right \} , \rho ^ 0 = \rho ^ { i n } ( x ) , \end{align*}


\begin{align*} D _ { 1 , q } : = | \nabla \varphi ( x _ q , t _ q ) | \left ( m ( \{ u ( \cdot , t _ q ) < u ( y _ q , t _ q ) \} ) - m ( \{ u _ { \star , \lambda } ( \cdot , t _ 0 ) < u _ { \star , \lambda } ( x _ 0 , t _ 0 ) \} ) \right ) , \end{align*}


\begin{align*} R _ { n + 1 } ( x , y ) = \sum _ { k = 0 } ^ n { n \choose k } L _ { k } ( x , y ) R _ { n - k } ( x , y ) . \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } t } \mathcal { F } ( t ) & = 2 \ , \mathrm { R e } \langle \dot { \phi } , \ddot { \phi } \rangle + 2 \ , \mathrm { R e } \left \langle - \frac 1 2 \Delta \phi + V \phi + \beta | \phi | ^ 2 \phi - \Omega L _ z \phi , \ , \dot { \phi } \right \rangle \\ & = 2 \ , \mathrm { R e } \left \langle \dot { \phi } , - \eta ( t ) \dot { \phi } + \lambda _ { \phi } ( t ) \phi \right \rangle \\ & = - 2 \ , \eta ( t ) \| \dot { \phi } \| ^ 2 . \end{align*}


\begin{align*} f ( x ) = \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } \omega ( Q ) \psi _ { Q } ( x , x _ { Q } ) q _ { Q } h ( x _ { Q } ) , \end{align*}


\begin{align*} \varphi ( a _ 1 \cdots a _ n ) = \frac { 1 } { \tau ' ( p ) } \tau ' ( ( p a _ 1 ) a _ 2 \cdots a _ n ) = \frac { 1 } { \tau ' ( p ) } \tau ' ( p a _ 1 ) \tau ( a _ 2 \cdots a _ n ) = 0 . \end{align*}


\begin{align*} \sideset { } { ^ { \flat } } \sum _ { \chi \bmod q } \chi ( m ) \overline { \chi ( n ) } = \frac { 1 } { 2 } \Bigg ( \sum _ { \substack { d | q \\ d | ( m \pm n ) } } \phi ( d ) \mu \left ( \frac { q } { d } \right ) \Bigg ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \mathrm { d } X _ { t } = f ( X _ { t } , \theta ) \mathrm { d } t + \sigma \mathrm { d } W _ { t } + \mathrm { d } L _ { t } - \mathrm { d } R _ { t } , \\ & X _ { 0 } = x \in [ a , b ] , \end{aligned} \right . \end{align*}


\begin{align*} \mathcal { N } ( \mathfrak { c } _ { c } ) = \mathcal { I } \setminus \mathfrak { c } _ { r } , \mathcal { N } ( \mathfrak { c } _ { r } ) = \mathcal { I } ^ { \mathtt { C } } \setminus \mathfrak { c } _ { c } . \end{align*}


\begin{align*} \norm { u } _ { L ^ \alpha ( E ) } : = \left ( \int _ E | u ( x ) | ^ \alpha d x \right ) ^ \frac { 1 } { \alpha } , \ ; \norm { u } _ { E , \alpha } : = \left ( \frac { 1 } { | E | } \int _ E | u ( x ) | ^ \alpha d x \right ) ^ \frac { 1 } { \alpha } . \end{align*}


\begin{align*} Y _ { \alpha \beta } ^ { i j } Y _ { \beta \gamma } ^ { i j } = - Y _ { \alpha \gamma } ^ { i j } , Y _ { \alpha \beta } ^ { i m } \cdot Y _ { \alpha \beta } ^ { m j } = - Y _ { \alpha \beta } ^ { i j } . \end{align*}


\begin{align*} w _ t + a _ n ( t , x ) \cdot w _ x ~ = ~ F ^ { ( k ) } ( t , x ) , w ( 0 , \cdot ) ~ = ~ \overline { w } ( \cdot ) \end{align*}


\begin{align*} \sum _ { \gamma } c _ \gamma z _ \gamma = \sum _ { \gamma } c _ \gamma x ^ \gamma = p ( x ) \geq 0 z \in M _ { ( n , d ) } , \end{align*}


\begin{align*} E ^ { [ \sigma ] } ( f , G \rtimes S ) ( t , \overline { t } ) = ( - 1 ) ^ n E ^ { [ \sigma ] } ( \widetilde { f } , \widetilde { G } \rtimes S ) ( t ^ { - 1 } , \overline { t } ) \ , . \end{align*}


\begin{align*} \mathcal { E } _ { m } ^ { 2 } ( u , v , w ) : = \sum _ { | \alpha | = 0 } ^ { m } \langle A _ { 1 } ^ { 0 } \partial _ { x } ^ { \alpha } u , \partial _ { x } ^ { \alpha } u \rangle + \sum _ { | \alpha | = 0 } ^ { m } \langle A _ { 2 } ^ { 0 } \partial _ { x } ^ { \alpha } u , \partial _ { x } ^ { \alpha } u \rangle + \sum _ { | \alpha | = 0 } ^ { m } \langle A _ { 3 } ^ { 0 } \partial _ { x } ^ { \alpha } u , \partial _ { x } ^ { \alpha } u \rangle . \end{align*}


\begin{align*} \mathcal { I } _ \ell : = \mathcal { I } _ { \ell , n } : = \left \{ \mathbf { i } = ( i _ 1 , i _ 2 , i _ 3 , i _ 4 ) \in \mathcal J : | \mathbf i | = \ell \right \} , \ell \in \{ 2 , 3 , 4 \} , \end{align*}


\begin{align*} m ^ { m o d } ( k ) = m ^ { P C } _ { \eta } ( \zeta ( k ) , \xi ) + m ^ { P C } _ { - \eta } ( \zeta ( k ) , \xi ) + \mathcal { O } ( t ^ { - 1 / 2 } ) . \end{align*}


\begin{align*} \overline { \mathbb { K } } _ { \psi _ E } ^ { \pi _ E } ( w ) \xi \{ \chi , 0 \} = r _ l \big ( \epsilon ( \widetilde { \chi } _ 0 ^ { - 1 } \pi _ E , \psi _ E ) \big ) \xi \bigg \{ \chi , \cfrac { - n ( \widetilde { \chi } _ 0 ^ { - 1 } \pi _ E , \psi _ E ) } { e } \bigg \} \end{align*}


\begin{align*} F _ { j k } = \int _ { \Gamma ^ { b o d y } } p _ k n _ { j } \ d \Gamma , \mathcal { T } , \end{align*}


\begin{align*} ( K _ \alpha * f ) '' ( x ) = \int _ { \R } \big [ f ( x + z ) - f ( x ) - z f ' ( x ) \big ] \ , K _ \alpha '' ( z ) d z . \end{align*}


\begin{align*} \int _ { \Omega _ R } | b ' ( \rho ) | ^ { \beta _ 0 - 1 } \leq & c \int _ { \{ \rho > \overline \rho - \alpha _ 1 \} } P ( \rho ) \leq c \int _ { \{ \rho > \overline \rho - \alpha _ 1 \} } \left ( P ( \rho ) - P ( r ) - P ' ( r ) ( \rho - r ) \right ) , \\ \int _ { \Omega _ R } | b ' ( \rho ) | ^ { \beta _ 0 } \leq & c \int _ { \Omega _ R } p ( \rho ) . \end{align*}


\begin{align*} \alpha \phi + \beta \epsilon ( x ) \frac { \partial \phi } { \partial \mathbf { n } } = \phi ^ b , x \in \partial \Omega . \end{align*}


\begin{align*} J _ 4 = \int _ 0 ^ t \ ! \int _ { \Gamma _ { t } ( 2 \delta ) } ( \zeta \circ d _ \Gamma ) \mathbf { w } | _ \Gamma \theta _ { 0 } ^ { \prime } ( \rho ) ( \mathbf { n } - \varepsilon \nabla _ \tau h _ { \varepsilon } ) ( \mu _ { \varepsilon } - \mu _ A ) \ , \mathrm { d } x \ , \mathrm { d } \varsigma \triangleq \ ; J _ { 4 1 } + J _ { 4 2 } , \end{align*}


\begin{align*} \int _ { \R ^ n } \nabla \bar { u } \cdot \nabla v = \tilde { \lambda } _ 0 \int _ { \R ^ n } \tilde { m } _ 0 \bar { u } v \forall v \in H _ 0 ^ 1 ( T ) \ ; . \end{align*}


\begin{align*} Q _ \alpha ( - 1 ) = 2 \int _ { - 1 } ^ 0 Q _ \alpha ( - 1 ) x \ , d x \geq 2 \int _ { - 1 } ^ 0 Q _ \alpha ( x ) \ , d x = 2 \frac { - P ( - \alpha ) } { P ( \alpha ) } \geq 2 , \end{align*}


\begin{align*} \Lambda ( \alpha , w ) p ( x , w ) = p ( x \alpha , w ) , \ , \ , \ , \ , \ , \ , \alpha \in h ( Y ^ 1 ) , \ , \ , \ , x \in \C ^ { 2 n } \end{align*}


\begin{align*} \Big | \frac { 1 } { 2 m \ell n } \sum _ { x } \Phi _ { n } ( s , \tfrac { x } { n } ) \sum _ { j = 1 } ^ { m - 1 } \sum _ { z = x - \ell } ^ { x - 1 } \int _ { \Omega - \Omega _ 2 ^ j ( x ) } [ \eta ( z ) - \eta ( z - j \ell ) ] \overrightarrow { \eta } ^ { \varepsilon n } ( x + 1 ) [ f ( \eta ) - f ( \eta ^ { z , z - j \ell } ) ] d \nu _ { b } \Big | \end{align*}


\begin{align*} R _ \eta ( M ) = | \{ W _ k ^ i : \exists f \in V ^ i , M f = \lambda f , \lambda > \eta \} | . \end{align*}


\begin{align*} \kappa _ { 2 k ; \alpha } ^ { ( \ell ) } : = \kappa _ { ( \alpha \alpha ) \cdots ( \alpha \alpha ) } ^ { ( 0 0 ) \cdots ( 0 0 ) , ( \ell ) } = \frac { 1 } { ( 2 k - 1 ) ! ! } \kappa \bigg ( \underbrace { z _ { i ; \alpha } ^ { ( \ell + 1 ) } , \ldots , z _ { i ; \alpha } ^ { ( \ell + 1 ) } } _ { 2 k } \bigg ) \end{align*}


\begin{align*} s ( 0 , \cdot ) = \rho ^ { ( 1 ) } _ 0 , \ \nabla \Psi ( 0 , \cdot ) = \nabla \Psi _ 0 = u _ 0 - H ( u _ 0 ) . \end{align*}


\begin{gather*} \Box \tilde \phi = \left ( | \tilde \phi _ { t } | ^ 2 - | \tilde \phi _ { x } | ^ 2 \right ) \tilde \phi + \mathbf { 1 } _ { \omega } \tilde f ^ { \tilde \phi ^ { \perp } } , \\ \tilde \phi [ 0 ] = ( a , b ) \textrm { a n d } \tilde \phi [ T ] = ( p , 0 ) . \end{gather*}


\begin{align*} R _ 0 = R _ 0 ( \alpha , \alpha _ 0 ) : = \sup \big \{ r ( { \bf y } ) : { \bf y } \in \mathcal A \big \} . \end{align*}


\begin{align*} \hat { \varepsilon } _ j ( \tau ) = \| \phi _ { j , \infty } \| ^ { - 2 } _ \rho \langle \varepsilon , \phi _ { j , \infty } \rangle _ { L ^ 2 _ \rho } , \end{align*}


\begin{align*} x \in ( 0 , \pi / 2 ) \mapsto \partial \varphi ( x ) : = \left \{ \frac { 1 } { \cos ^ 2 x } \right \} . \end{align*}


\begin{align*} I ( \hat { \alpha } \cup { ( \gamma , N ) } , \hat { \alpha } \cup { ( \gamma , N - p _ { i } ) } \cup { ( \delta _ { 1 } , 1 ) } \cup { ( \delta _ { 2 } , 1 ) } ) = 2 . \end{align*}


\begin{align*} G _ \lambda f ( x ) = \int _ M g _ \lambda ( x , y ) f ( y ) \mu ( d y ) , x \in M , \end{align*}


\begin{align*} S = & \sum _ { \ell = - \lfloor 1 / 2 \delta _ K \rfloor } ^ { 1 / \delta _ K - 1 - \lfloor 1 / 2 \delta _ K \rfloor } h _ K G \big ( h _ K \ell \big ) e ^ { h _ K \ell \partial _ x \varphi ( t _ K , x _ K ) + C \frac { ( h _ K \ell ) ^ 2 } { \log K } } . \end{align*}


\begin{align*} \widetilde { h } ^ { j } = D ( \zeta ) ^ q g ^ j ( \zeta ) \quad \textrm { w i t h } D ( \zeta ) = { \prod _ { i = 1 } ^ { n } ( \zeta - \zeta _ { i } ) } . \end{align*}


\begin{align*} T _ t ^ { \alpha } ( f ) \geq 0 , T _ t ^ { \alpha } ( 1 ) = e ^ { t d } , T _ t ^ { \alpha } ( p _ \alpha ) = p _ \alpha . \end{align*}


\begin{align*} \nu _ { \alpha , d } ( d u ) = \sum _ { k = 1 } ^ d \delta _ 0 ( d u _ 1 ) \otimes \dots \otimes \delta _ 0 ( d u _ { k - 1 } ) \otimes \nu _ { \alpha , 1 } ( d u _ k ) \otimes \delta _ 0 ( d u _ { k + 1 } ) \otimes \dots \otimes \delta _ 0 ( d u _ d ) , \end{align*}


\begin{align*} & - \infty < \alpha = \min _ { M } \theta _ { V } ^ { ( \omega ) } \leqq \max _ { M } \theta _ { V } ^ { ( \omega ) } < \beta \leqq + \infty \\ & \dot { \sigma } \leqq { 0 } \leqq \ddot { \sigma } , \\ & \lim _ { t \to \alpha + 0 } \sigma ( t ) = + \infty . \end{align*}


\begin{align*} \tilde { f } ( s , y ) & = \alpha _ { - 1 } ( - \tilde { \theta } y ^ { 2 \tilde { \theta } + 1 } ) + \alpha _ 0 y ^ { \tilde { \theta } + 1 } - \alpha _ 1 \frac { y } { \tilde { \theta } } + \alpha _ 2 \frac { 1 } { \tilde { \theta } ^ { \rho } } y ^ { - \tilde { \theta } \rho + \tilde { \theta } + 1 } \end{align*}


\begin{align*} \dfrac { S _ n } { I _ n } = 1 + O ( ( I _ n ) ^ { - 1 } ) \mbox { a s $ n \to \infty $ . } \end{align*}


\begin{align*} U _ { x } ( y ) : = \{ z \in H : P _ { [ x , y ] } z \neq y \} , \end{align*}


\begin{align*} \nu _ * = ‐ \mu _ * ‐ \left ( \frac { \gamma _ * } { q } ‐ \frac { \gamma _ * } { p _ 1 } \right ) _ + . \end{align*}


\begin{align*} p = P _ C x \left ( p \in C \forall q \in C \left ( \langle q - p , x - p \rangle \leq 0 \right ) \right ) . \end{align*}


\begin{align*} x = ( x _ 1 , x ^ \perp ) . \end{align*}


\begin{align*} R ( k , i ) = \frac { \prod _ { j = i + 1 } ^ k R ( j ) } { m ( k , i ) } . \end{align*}


\begin{align*} 1 - \frac { \sigma _ { k - 1 } ( \kappa | i ) \kappa _ { i } } { \sigma _ { k } ( \kappa ) } = \frac { \sigma _ { k } ( \kappa | i ) } { \sigma _ { k } ( \kappa ) } \geq 0 . \end{align*}


\begin{align*} V ^ * _ g F = \iint _ { \R ^ { 2 d } } F ( x , \omega ) M _ \omega T _ x g \ , d ( x , \omega ) . \end{align*}


\begin{align*} u \ ! + \ ! \varPsi _ { \mu , q } B _ q ( w , u ) = v ^ { ( 0 ) } . \end{align*}


\begin{align*} \Psi ^ { P } ( f ) [ \psi ] = \int _ { \mathbb { R } } f ( x ) P ( \phi ( x ) , \partial _ x \phi ( x ) , \ldots , \partial _ x ^ n \phi ( x ) ) \ , \mathrm { d } x , \end{align*}


\begin{align*} \sum _ { \substack { m \mid n \\ m \neq n / d _ p } } \nu _ p ( \Phi _ m ( \beta ) ) \le \sum _ { m \mid n } \nu _ p ( m ) + \sum ^ 7 _ { m = 1 } \nu _ p ( \Phi _ m ( \beta ) ) . \end{align*}


\begin{align*} w ( t , k ) : = \mathcal { F } f ( t , k ) \exp \Big ( i \int _ 0 ^ t \big | \mathcal { F } f ( s , k ) \big | ^ 2 \ , \frac { d s } { 2 ( s + 1 ) } \Big ) , | \mathcal { F } f ( t , k ) | = | w ( t , k ) | . \end{align*}


\begin{align*} \begin{aligned} u _ { \star , \lambda } ( x , t ) = \inf \bigg \{ \max \{ u ( y , t ) , u ( z , t ) \} : x = \lambda y + & ( 1 - \lambda ) z \bigg \} , \\ & ( x , t ) \in \R ^ n \times ( 0 , \infty ) , \end{aligned} \end{align*}


\begin{align*} ( \lambda I - A ^ * ) X _ n = Y _ n . \end{align*}


\begin{align*} \pi ( p ' ) = \sum _ { i = 1 } ^ d \alpha _ i \pi ( p _ i ) , h ( p ' ) < \sum _ { i = 1 } ^ d \alpha _ i h ( p _ i ) . \end{align*}


\begin{align*} f = \sum _ { k = 1 } ^ n c _ k \pi ( \mathbf { h _ k } ) g \in \mathcal { H } _ g , \end{align*}


\begin{align*} \mathcal H f = \frac { 1 } { \pi } P . V . \int _ { - \infty } ^ { \infty } \frac { f ( y ) } { x - y } \ , d y . \end{align*}


\begin{align*} s ( 1 , d ) = \frac { d ^ 2 - 3 \vert d \vert + 2 } { 1 2 d } s ( 2 , d ) = \frac { d ^ 2 - 6 \vert d \vert + 5 } { 2 4 d } \ \ \ \ \ ( d \in { \mathbb Z } \setminus \{ 0 \} ) . \end{align*}


\begin{align*} \ell _ { \beta } ( x ) : = \left \{ \begin{array} { l l } x \log | x | - x , & \beta = - 1 \\ \frac { 1 } { \beta + 1 } \int _ 0 ^ x | y | ^ { \beta + 1 } { \rm d } y , & \beta \in \ , ] \ , - 3 / 2 , \ , + \infty \ , [ \ , \setminus \ , \{ \pm 1 \} \end{array} \right . , \end{align*}


\begin{align*} \overline { \zeta } \left ( A , T ^ d _ N , u \right ) = \det \Big ( I _ { d _ c N ^ d } - u M _ A \Big ) ^ { - 1 / N ^ d } . \end{align*}


\begin{align*} \partial _ t P ' ( \rho ^ { \epsilon } ) = - u ^ { \epsilon } \cdot \nabla P ' ( \rho ^ { \epsilon } ) - P '' ( \rho ^ { \epsilon } ) \rho ^ { \epsilon } \mathrm { d i v } u ^ { \epsilon } . \end{align*}


\begin{align*} \big < \Gamma _ 1 , \Gamma _ 2 \big > _ { \mathcal { L } _ 2 ( H ) } = \sum _ { j = 1 } ^ \infty \big < \Gamma _ 1 \phi _ j , \Gamma _ 2 \phi _ j \big > , \| \Gamma \| ^ 2 _ { \mathcal { L } _ 2 ( H ) } = \sum _ { j = 1 } ^ \infty \big \| \Gamma \phi _ j \| ^ 2 , \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } d X ( t ) & = & \mu X ( t ) d t + \sigma X ( t ) d W ( t ) \\ X ( 0 ) & = & x \end{array} \right . \end{align*}


\begin{align*} ( \rho ^ { * } _ { ( 1 ) } \otimes \rho ^ { * } _ { ( 1 ) } ) E _ { \widetilde { \Sigma } } = ( \partial _ { \Sigma } \otimes \partial _ { \Sigma } ) E _ { \mathcal { M } } = E _ { \Sigma } , \end{align*}


\begin{align*} \P ( | U _ { \mathsf k } | \geq N \mbox { f o r a l l } \mathsf k = \mathsf 1 , \ldots , \mathsf q ) \geq 1 - n ^ { - 4 } \ , , \mbox { w h e r e } N \stackrel { \Delta } { = } \frac { n ^ d } { q ( 1 6 r ) ^ d } \ , . \end{align*}


\begin{align*} \sum _ { \substack { 0 \leq a < p \\ \gcd ( n , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi a ( \tau ^ { n } - u ) } { p } } = 0 \end{align*}


\begin{align*} d _ i z _ { i ; \alpha } ^ { ( \ell ) } : = \partial _ { x _ { i ; \alpha } } z _ { i ; \alpha } ^ { ( \ell ) } , i = 1 , \ldots , n _ 0 . \end{align*}


\begin{align*} C = ( a _ 1 , a _ 2 , \ldots , a _ k , \overline { a _ 1 } , \overline { a _ 2 } , \ldots , \overline { a _ k } ) \end{align*}


\begin{align*} \varphi ( k , l ) & = \varphi ( 0 , l ) + k \cdot l + \kappa ( k , 0 ) \\ & = \left ( \varphi ( 0 , 0 ) + \kappa ( 0 , l ) \right ) + k \cdot l + \kappa ( k , 0 ) \end{align*}


\begin{align*} N _ 3 ( f , H ) = - f - 3 S ( H , f ) + \frac { 9 } { 2 } S ( H _ 3 , 3 f ) = \frac { c _ a '' + 1 } { 4 } = \begin{cases} - 2 a - 1 & \hbox { i f $ a \equiv 0 \pmod 3 $ } , \\ 2 a + 1 & \hbox { i f $ a \equiv 2 \pmod 3 $ } , \end{cases} \end{align*}


\begin{align*} & \det ( I _ { n + 2 M } - u { \cal A } ) = ( 1 + 2 ( \beta + 2 \delta ) u + 4 \delta ( \delta + 1 ) u ^ 2 ) ^ { M - N } \\ & \times \det ( ( 1 + \delta u ) ( 1 + 2 ( \beta + 2 \delta ) u + 4 \delta ( \delta + 1 ) u ^ 2 ) I _ N \\ & - ( 1 + 2 \delta u ) \beta u ( 1 + ( \beta + 2 \delta ) u ) { \bf A } ( G ) + \beta { } ^ 2 ( 1 + 2 \delta u ) u ^ 2 { \bf D } _ G ) . \end{align*}


\begin{align*} l _ { \pi _ { \lambda , \beta } } ( f ( z ) ) & = E _ { \beta } ( \lambda z ) = \sum _ { m = 0 } ^ { \infty } \frac { ( \lambda z ) ^ { m } } { \Gamma ( m \beta + 1 ) } \\ & = \sum _ { m = 0 } ^ { \infty } \frac { z ^ { m } } { m ! } \underbrace { \frac { \lambda ^ { m } m ! } { \Gamma ( m \beta + 1 ) } } _ { = : \tilde { M } _ { \lambda , \beta } ( m ) } = \sum _ { m = 0 } ^ { \infty } \frac { z ^ { m } } { m ! } \tilde { M } _ { \lambda , \beta } ( m ) . \end{align*}


\begin{align*} y _ { i } = \left \langle \mathcal { X } _ { i } , \mathcal { A } ^ { * } \right \rangle + \varepsilon _ { i } \quad \quad \mathbb { E } \left ( \varepsilon _ { i } \mid \mathcal { X } _ { i } \right ) = 0 \quad M _ { \delta } = \sqrt [ k ] { \mathbb { E } \left ( \mathbb { E } \left ( | \varepsilon _ { i } | ^ { 1 + \delta } | \mathcal { X } _ { i } \right ) ^ k \right ) } < \infty \end{align*}


\begin{align*} S = \{ ( r x , ( r a - r ^ { - 1 } ) v ) : r > 0 , \ , x \in \Sigma \} . \end{align*}


\begin{align*} x ^ i x ^ j = 0 \mbox { f o r a l l } x \in N \mbox { a n d } i , j \geq 2 . \end{align*}


\begin{align*} S ( H _ 6 , 6 f ) = \frac { 1 0 f + c _ a ''' } { 1 8 } \hbox { w h e r e } c _ a ''' : = \begin{cases} - 1 9 a - 1 0 & \hbox { i f $ a \equiv 0 \pmod 6 $ } , \\ a - 1 8 & \hbox { i f $ a \equiv 2 \pmod 6 $ } , \\ - a - 1 9 & \hbox { i f $ a \equiv 3 \pmod 6 $ } , \\ 1 9 a + 9 & \hbox { i f $ a \equiv 5 \pmod 6 $ } , \end{cases} \end{align*}


\begin{align*} \mu _ \beta ( A ) : = \int _ A e ^ { - \beta d ( x , v _ 0 ) } \nu ( B _ Z ( \zeta _ x , \alpha ^ { - n _ x } ) ) \ , d \mathcal { H } ^ 1 ( x ) , \end{align*}


\begin{align*} Q \left ( x \right ) = \int _ { x } ^ { \infty } \frac { 1 } { \sqrt { 2 \pi } } \exp \left ( - \frac { t ^ 2 } { 2 } \right ) d t . \end{align*}


\begin{align*} T _ n ^ { \sigma } + \sum _ { k = n + 1 } ^ { d } C _ { k } ^ { ( d ) , \sigma } ( \mathbf { P } _ { \ ! \sigma } ) \cdot \bigg ( H ( \mathbf { p } _ { \sigma _ { k } } ) + \int \ ! \varphi _ k ^ { \sigma } \ , \mathrm { d } \mathbf { p } _ { \sigma _ k } - \sum _ { \ell = 1 } ^ { n } \chi _ { \ell } ^ { \sigma } ( \mathbf { p } _ { \sigma _ k } ) \big ( T _ { \ell } ^ { \sigma } - T _ { \ell - 1 } ^ { \sigma } \big ) \bigg ) , \end{align*}


\begin{align*} J ' ( \alpha , \beta ) = \sum _ { t \neq 0 , 1 } \alpha ( t ) \beta ( 1 - t ) \end{align*}


\begin{align*} C ' = \min _ { k \in \{ 0 , \ldots , K - 1 \} } C _ k \end{align*}


\begin{align*} \begin{array} { l l @ { } l l } & \displaystyle n \cdot ( t _ { 1 } + \dots + t _ { m } ) & \\ & \displaystyle v _ { i j } - x _ j \ \leq \ t _ i \ , , & i \in [ m ] j \in [ n ] \\ & \displaystyle x _ { 1 } + \dots + x _ { n } \ = \ 0 & \end{array} \end{align*}


\begin{align*} U = \bigg \{ x \in B _ \rho ^ + u ( x ) > \frac 1 2 \tau d ^ { - n - 2 } \zeta ( x ) \bigg \} \end{align*}


\begin{align*} \begin{aligned} & \lim _ { n \to \infty } P _ { n , k } ( J _ { n , k } ^ { \mathcal { S } } = l | W _ { \mathcal { S } ( n , k ; M _ n ) } ) = \frac 1 { \lim _ { n \to \infty } P _ { n , k } ( W _ { \mathcal { S } ( n , k ; M _ n ) } ) } \binom k { l - 1 } ( \frac c { 1 - c } ) ^ { l - 1 } ( 1 - c ) ^ { k } = \\ & \frac 1 { \lim _ { n \to \infty } P _ { n , k } ( W _ { \mathcal { S } ( n , k ; M _ n ) } ) } \binom k { l - 1 } c ^ { l - 1 } ( 1 - c ) ^ { k - l + 1 } , \ l \in \{ 2 , \cdots , k \} , \end{aligned} \end{align*}


\begin{align*} M _ \omega T _ x = e ^ { 2 \pi i \omega \cdot x } \ , T _ x M _ \omega , \end{align*}


\begin{align*} \| f ( x ) \| _ p ^ p \leq \sum \limits _ { l = - \infty } ^ \infty \mu ( \widetilde { \Omega } _ l ) ^ { 1 - \frac { p } { 2 } } \| \sum \limits _ { Q \in B _ l } \omega ( Q ) D _ k ( x , x _ Q ) { \widetilde D } _ k ( f ) ( x _ Q ) \| _ 2 ^ p . \end{align*}


\begin{align*} & ( \widetilde { x } _ 1 ( t ) , \ldots , \widetilde { x } _ n ( t ) ) ^ \intercal = T ( x _ 1 ( t ) , \ldots , x _ n ( t ) ) ^ \intercal , \\ & ( \widetilde { \mu } _ 1 ( t ) , \ldots , \widetilde { \mu } _ n ( t ) ) ^ \intercal = T ( \mu _ 1 ( t ) , \ldots , \mu _ n ( t ) ) ^ \intercal , \\ & ( \widetilde { \alpha } _ 1 , \ldots , \widetilde { \alpha } _ n ) ^ \intercal = T ( \alpha _ 1 , \ldots , \alpha _ n ) ^ \intercal . \end{align*}


\begin{align*} \mathcal { D } ( \mathbf { x } _ k , { \mathbf { u } } _ k ) _ { \{ i , j \} } = { \partial h _ i ( \boldsymbol { f } _ O ^ { d _ i - 1 } \left ( \boldsymbol { f } ( \mathbf { x } _ k , { \mathbf { u } } _ k ) \right ) ) } \big / { \partial { u } _ { j , k } } . \end{align*}


\begin{align*} \lambda w _ j + ( 1 - \lambda ) h _ j = \frac { 1 } { 2 } \mp b i \end{align*}


\begin{align*} u \times v : = ( u _ 1 , \ldots , u _ p , v _ 1 [ p ] , \ldots , v _ { q } [ p ] ) . \end{align*}


\begin{align*} \begin{aligned} & e ^ { - t } C _ p \left | \dot { \xi } ( t ) \right | ^ { q } \leq e ^ { - t } \phi \left ( \xi ( t ) \right ) - e ^ { - t } D \phi \left ( \xi ( t ) \right ) \cdot \dot { \xi } ( t ) \\ \Longrightarrow & C _ p \left | \dot { \xi } ( t ) \right | ^ { q } \leq \phi \left ( \xi ( t ) \right ) - D \phi \left ( \xi ( t ) \right ) \cdot \dot { \xi } ( t ) . \end{aligned} \end{align*}


\begin{align*} \frac { F \rightarrow s = _ X s ' F \rightarrow t = _ X t ' } { F \rightarrow ( s \in A t \leftrightarrow s ' \in A t ' ) } \end{align*}


\begin{align*} \int _ { \R ^ N } \nabla \tilde { u } _ n \cdot \nabla v = \tilde { \lambda } _ n \int _ { \R ^ N } \tilde { m } _ 0 \tilde { u } _ n v \ ; . \end{align*}


\begin{align*} G ( y ) - \frac { C _ 0 } { y ^ 2 } = \Bigg [ \int _ 0 ^ { \infty } t e ^ { - t } \bigg ( \frac { C _ 0 } { t ^ 2 + y ^ 2 } - \frac { C _ 0 } { y ^ 2 } \bigg ) \ , \dd t \Bigg ] \lesssim \frac { 1 } { y ^ 4 } \int _ 0 ^ \infty t ^ 3 e ^ { - t } \ , \dd t = O \bigg ( \frac { 1 } { y ^ 4 } \bigg ) , \end{align*}


\begin{align*} F \star G [ j ] : = \sum _ { n = 0 } ^ { \infty } \frac { \hbar ^ n } { n ! } \left \langle ( \tfrac { i } { 2 } E _ { \mathcal { I } } + H _ c ) ^ { \otimes n } , F ^ { ( n ) } [ j ] \otimes G ^ { ( n ) } [ j ] \right \rangle . \end{align*}


\begin{align*} \overline { H } _ n ( \overline { s } ) = \sum \limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \binom { n } { k } { _ { s + 1 } F _ s } \left ( \left \{ \frac { 1 } { 2 } \right \} ^ { s } , 1 - k ; \left \{ \frac { 3 } { 2 } \right \} ^ { s } ; - 1 \right ) . \end{align*}


\begin{align*} \tilde { \lambda } _ { n } \le \lambda ^ 1 ( B ^ 1 ( \mathbf { y } _ n ) , B ^ { \delta / k _ n } ( \mathbf { y } _ n ) ) = \lambda ^ 1 ( B ^ 1 ( \mathbf { 0 } ) , B ^ { \delta / k _ n } ( \mathbf { 0 } ) ) . \end{align*}


\begin{align*} u _ { k } = u _ { k - 1 } - s \nabla f \left ( u _ { k - 1 } \right ) , k = 1 , 2 , \ldots , \end{align*}


\begin{align*} A \otimes B = \begin{pmatrix} a _ { 1 1 } B & \cdots & a _ { 1 m } B \\ \vdots & \ddots & \vdots \\ a _ { m 1 } B & \cdots & a _ { m n } B \end{pmatrix} . \end{align*}


\begin{align*} \iint _ { \R ^ { 2 d } } | V _ g f ( x , \omega ) | ^ p \ , d ( x , \omega ) \begin{cases} \leq \left ( \tfrac { 2 } { p } \right ) ^ d ( \norm { f } _ 2 \norm { g _ 2 } ) ^ p & 2 \leq p < \infty \\ \geq \left ( \tfrac { 2 } { p } \right ) ^ d ( \norm { f } _ 2 \norm { g _ 2 } ) ^ p & 1 \leq p \leq 2 \\ \end{cases} \end{align*}


\begin{align*} \lim _ { s \to 0 } \int _ { \left \lbrace u > \varepsilon \right \rbrace } \frac { 1 } { u ^ p } ( P _ s u ) ^ p \Big ( \sup _ { t > 0 } P _ t f \Big ) ^ p \ : d \mu = \int _ { \left \lbrace u > \varepsilon \right \rbrace } \Big ( \sup _ { t > 0 } P _ t f \Big ) ^ p \ : d \mu . \end{align*}


\begin{align*} \langle T ( 1 - \eta ) b , f \rangle & = \int _ { \R ^ N } \int _ { \R ^ N } K ( x , y ) ( 1 - \eta ( y ) ) b ( y ) f ( x ) d \omega ( y ) d \omega ( x ) \\ & = \int _ { \R ^ N } \int _ { \R ^ N } [ K ( x , y ) - K ( x _ 0 , y ) ] ( 1 - \eta ( y ) ) b ( y ) f ( x ) d \omega ( y ) d \omega ( x ) . \end{align*}


\begin{align*} ( ( \phi ^ { - 1 } ) ^ * ( \phi ^ * ( r _ { g } ) \circ \sigma ) ) ^ { - 1 } \circ ( r _ g \circ \psi ) = ( r _ g \circ ( \phi ^ { - 1 } ) ^ * \sigma ) ^ { - 1 } \circ r _ g \circ \psi = ( ( \phi ^ { - 1 } ) ^ * \sigma ) ^ { - 1 } \circ \psi . \end{align*}


\begin{align*} \psi ( \xi , t ) : = e ^ { - i t \xi | \xi | ^ { 1 + a } } . \end{align*}


\begin{align*} D ( X ) : = \sup _ { x \in \mathbb { R } } | P ( X \leq x ) - P ( Z \leq x ) | . \end{align*}


\begin{align*} \varphi ^ { \operatorname { r o t } } _ \alpha ( \xi ) = \hat { \mu } _ { \alpha } ^ { \operatorname { r o t } } \left ( \xi \right ) = \exp \left ( - \frac { \| \xi \| ^ \alpha } { 2 } \right ) . \end{align*}


\begin{align*} I ( \hat { \alpha } \cup { ( \gamma , p _ { i + 1 } ) } , \hat { \alpha } \cup { ( \delta , 1 ) \cup { ( \gamma , p _ { i + 1 } - p _ { i } ) } } ) = 4 . \end{align*}


\begin{align*} \int _ { \R ^ { d - 1 } } \tilde \varphi \ , d x = 0 . \end{align*}


\begin{align*} d _ H ( N , N ' ) : = \inf \left \{ | | \phi | | _ H \ \middle | \ \phi \in \mathrm { H a m } ( M ) , \ \phi ( N ) = N ' \right \} , \end{align*}


\begin{align*} \mathbf { M } _ { 1 } \mathbf { M } _ { \bar { 2 } \ , \bar 1 } = { } & \mathbf { M } _ { 1 \bar { 3 } \ , \bar 2 } + \mathbf { M } _ { \bar 3 1 \bar 2 } + \mathbf { M } _ { \bar { 3 } \ , \bar 2 1 } . \end{align*}


\begin{align*} S _ { \tau ' } ( Y _ 0 , d W _ \cdot ) = e ^ { - \mu ( \tau ' - \tau ) } S _ { \tau } ( Y _ 0 , d W _ \cdot ) + S _ { \tau ' - \tau } ( Y _ { \tau } , d ( \Theta ^ { \tau } W ) _ \cdot ) . \end{align*}


\begin{align*} \frac { 1 } { L ^ n } \int _ U \varphi ( y ) \ d n _ L ( y ) - \int _ U \varphi ( x ) \bar { \nu } ( x ) d x & \geq - ( \sup _ U \varphi ) \cdot \int _ { Q _ 1 } \big | \nu _ { R , L } ( x , w ) - \bar { \nu } ( x ) \big | \ d x \\ & - \omega _ \varphi ( \delta ) \cdot | | \bar { \nu } | | _ { L ^ \infty ( Q _ 1 ) } \cdot | Q _ 1 | . \end{align*}


\begin{align*} \lambda _ 1 F _ 1 + \lambda _ 2 F _ 2 + \lambda _ 3 F _ 3 = 0 \end{align*}


\begin{align*} \sqrt { \sum _ { i = 1 } ^ { n } a _ i } \leq \sum _ { i = 1 } ^ { n } \frac { a _ i } { \sqrt { \sum _ { k = 1 } ^ { i } a _ k } } \leq 2 \sqrt { \sum _ { i = 1 } ^ { n } a _ i } \end{align*}


\begin{align*} \omega ( z \cdot q ) = \kappa _ G ( z ) + \omega ( q ) \end{align*}


\begin{align*} \mu ( k , \ell , m , n ) : = \int \overline { \mathcal { K } ( x , k ) } \mathcal { K } ( x , \ell ) \overline { \mathcal { K } ( x , m ) } \mathcal { K } ( x , n ) \ , d x . \end{align*}


\begin{align*} E ( x , t , k ) : = \left \{ \begin{aligned} & m ^ { ( 3 ) } ( x , t , k ) \left ( m ^ { L C } \left ( x , t , k \right ) \right ) ^ { - 1 } , k \in U _ { \xi } \\ & m ^ { ( 3 ) } ( x , t , k ) \left ( m ^ { G P } \left ( x , t , k \right ) \right ) ^ { - 1 } , e l s e w h e r e , \end{aligned} \right . \end{align*}


\begin{align*} ( \partial _ t + A ) ^ \gamma X ( t ) = \dot { W } ^ { Q } ( t ) , t \in [ 0 , T ] , X ( 0 ) = 0 . \end{align*}


\begin{align*} W _ { 1 } = \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle , W _ { 2 } = \left \langle \left [ \vartheta _ { 1 } \right ] , \left [ \vartheta _ { 2 } \right ] , \dots , \left [ \vartheta _ { s } \right ] \right \rangle \in G _ { s } \left ( { \rm H ^ { 2 } } \left ( { \bf A } , \mathbb C \right ) \right ) , \end{align*}


\begin{align*} \int _ 0 ^ { \infty } \ ! \ ! \dot { x } p _ { | \dot { h _ i } | } ( \dot { x } ) d \dot { x } = \frac { \sigma _ { \dot { X } } } { \sqrt { 2 \pi } } = \sqrt { \frac { \pi } { 2 } } \sigma f _ D , \ : \ : \forall i = 1 , \cdots , N . \end{align*}


\begin{align*} \left \Vert \theta \right \Vert _ { g _ t } = \left \Vert \theta _ { t } ^ { c } \right \Vert _ { g _ t } = 1 . \end{align*}


\begin{align*} \tilde \theta = \sum _ { j \in \tilde J } \left ( \frac { \bar \alpha _ j } { \rho _ j } - 1 \right ) = \sum _ { j \in J \cap \{ i + 1 , \dots , n \} } \left ( \frac { \bar \alpha _ j } { \rho _ j } - 1 \right ) . \end{align*}


\begin{align*} h _ \lambda ( v ) = ( h \circ W ) ( v ) = \frac { W ( v ) - v } { ( W ( v ) - v ) ^ 2 + | \lambda | ^ 2 } . \end{align*}


\begin{align*} \begin{array} { l l } \displaystyle \sum _ { j = 1 } ^ { n } a _ { i j } w _ j = \theta , \ ; \ ; \ ; i = 1 , 2 , . . . , m \\ \displaystyle \sum _ { j = 1 } ^ { n } w _ j = 1 \end{array} \end{align*}


\begin{align*} \rho ( 0 , \cdot ) = \rho _ { 0 , \varepsilon } = \varrho + \varepsilon \rho ^ { ( 1 ) } _ { 0 , \varepsilon } , \ u ( 0 , \cdot ) = u _ { 0 , \varepsilon } \end{align*}


\begin{align*} P _ { n , k } ( W _ { \mathcal { S } ( n , k ; M ) } | A ^ { ( n ) } _ { M , n , l } ) = \begin{cases} 1 , \ \ l \in [ k - 1 ] ; \\ 0 , \ \ l = k . \end{cases} \end{align*}


\begin{align*} \underset { s = 0 } { } \ \widetilde { V } ( s ) = \lim _ { s \rightarrow 0 } s \widetilde { V } ( s ) = 1 \end{align*}


\begin{align*} \bar { \Delta } \triangleq \mathop { \lim } \limits _ { N \to \infty } \frac { 1 } { N } \sum \nolimits _ { t = 1 } ^ N { \Delta \left ( t \right ) } . \end{align*}


\begin{align*} S _ \varepsilon = \left \{ R _ { \varepsilon } ( s , t ) = R ( s , t ) + 2 \varepsilon U ^ { 1 } ( s , t ) + \ldots + 2 \varepsilon ^ { m } U ^ { m } ( s , t ) \right \} , \end{align*}


\begin{align*} \lim _ { t \to \infty } t ^ { \frac { 1 } { 2 } } \left \| u ( \cdot , t ) - t ^ { - \frac { 1 } { 2 } } f _ { M } \left ( ( \cdot ) t ^ { - \frac { 1 } { 2 } } \right ) \right \| _ { L ^ { \infty } } = 0 , \end{align*}


\begin{align*} W _ { \Sigma } ( s , s ' ) = \frac { - 1 } { 4 \pi } \frac { 1 } { \sqrt { \gamma ' ( s ) \gamma ' ( s ' ) } } \left ( \frac { 1 } { ( s - s ' ) ^ 2 } + i \delta ' ( s - s ' ) \right ) . \end{align*}


\begin{align*} G = \Big ( \int _ { \mathbb { R } ^ 3 } | v | ^ { \kappa _ 0 } f \ , d v \Big ) ^ \frac { 1 } { \kappa _ 0 + 3 } , \end{align*}


\begin{align*} u _ 1 ( 0 ) = \phi , u _ 2 ( 0 ) = \varphi , \phi \not = \varphi , \end{align*}


\begin{align*} \vartheta _ 3 ( i s ; \omega ) = Z \varphi _ s ( 0 , - \omega ) . \end{align*}


\begin{align*} p _ 1 : = p + 2 \alpha \tilde { y } \neq 0 , p _ 2 : = p - 2 \alpha \tilde { x } \neq 0 \quad . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ j { r \choose { i - 1 } } { { r - i } \choose { j - i } } ( - 1 ) ^ { j - i } = 1 \end{align*}


\begin{align*} \sum _ { \mathbf { v } \in P _ { \mathbf { c } } } \exp { \Big ( 2 \pi i \frac { \langle \pmb { \xi } , \mathbf { v } - \mathbf { c } \rangle } { p ^ { 2 k } } \Big ) } = \exp { \Big ( 2 \pi i \frac { \langle \pmb { \xi } , \mathbf { u } - \mathbf { c } \rangle } { p ^ { 2 k } } \Big ) } \sum _ { \mathbf { v } \in P _ { \mathbf { c } } } \exp { \Big ( 2 \pi i \frac { \langle \pmb { \xi } , \mathbf { v } - \mathbf { c } \rangle } { p ^ { 2 k } } \Big ) } \end{align*}


\begin{align*} R _ \pi = \sum _ { i = 1 } ^ { n } w _ { i \pi ( i ) } . \end{align*}


\begin{align*} d _ j = \partial _ { x _ { j ; \alpha } } , j = 1 , 2 . \end{align*}


\begin{align*} f _ { s + t } ( u , u ' ) = \begin{array} { l } f _ t ( u , u ' ) + s A \cdot f _ t ( u , u ' ) \\ - s f _ t ( u , A \cdot u ' ) - s f _ t ( A \cdot u , u ' ) \\ + s ^ 2 f _ t ( A \cdot u , A \cdot u ' ) - s ^ 2 A \cdot f _ t ( u , A \cdot u ' ) \\ - s ^ 2 A \cdot f _ t ( A \cdot u , u ' ) + s ^ 3 A \cdot f _ t ( A \cdot u , A \cdot u ' ) . \end{array} \end{align*}


\begin{align*} \Big ( x \frac { d } { d x } \Big ) _ { n , \lambda } f ( x ) = \sum _ { k = 1 } ^ { n } S _ { 2 , \lambda } ( n , k ) x ^ { k } \Big ( \frac { d } { d x } \Big ) ^ { k } f ( x ) , \end{align*}


\begin{align*} \mathbf { F } = \left ( \R [ X _ 1 , \ldots , X _ n ] _ { \leq 2 } \right ) _ { n > 0 } \end{align*}


\begin{align*} R ^ 2 _ 1 = - \left ( R ^ { 2 , \operatorname { q u a d } } + R ^ { 2 , \operatorname { m e a n } } + R ^ { \chi } + R ^ { 2 , \operatorname { l i n } } + R ^ { 2 , \operatorname { c o r r } } + R ^ h \right ) , \end{align*}


\begin{align*} \Sigma _ 2 \ll p ^ { ( \frac { 3 } { 2 } - \epsilon ) h _ p } \sum _ { m = \max \{ 0 , k _ p - h _ p \} } ^ { \infty } \frac { p ^ { \frac { 1 } { 2 } - \epsilon } } { p ^ { m ( \frac { 1 } { 2 } - \varepsilon + \epsilon ) } p ^ { ( h _ p - k _ p + 1 ) ( 1 - \varepsilon ) } } . \end{align*}


\begin{align*} \hat q _ { j i } = ( \hat q _ j ) ^ { - 1 } \hat q _ i = ( \hat q _ { i j } ) ^ { - 1 } . \end{align*}


\begin{align*} E _ 0 ( f , p ) = \sum _ { z \leq p } \frac { 1 } { p } \sum _ { \substack { 0 \leq a < p \\ \gcd ( n , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi a ( \tau ^ { n } - u ) } { p } } \cdot \frac { 1 } { p } \sum _ { \substack { 0 < \leq b < p \\ g c d ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { m } - v ) } { p } } = 0 . \end{align*}


\begin{align*} \forall \delta > 0 \forall n , m \geq \theta ( \delta ) : = \tau \left ( \phi \left ( \alpha _ G \left ( \frac { \beta _ H ( \delta / 2 ) } { 2 } \right ) \right ) , \xi \left ( \frac { \beta _ H ( \delta / 2 ) } { 2 } \right ) \right ) ( d ( x _ n , x _ m ) < \delta ) . \end{align*}


\begin{align*} \Re \left ( \frac { 1 } { s - \rho } \right ) = \frac { 2 - \beta } { ( 2 - \beta ) ^ 2 + ( T - \gamma ) ^ 2 } \geq \frac { 1 } { 4 + ( T - \gamma ) ^ 2 } \geq \frac { 1 } { 4 ( 1 + ( T - \gamma ) ^ 2 ) } \end{align*}


\begin{align*} \sigma ( \l ' , \l ) = \l ' \cdot J \l , \end{align*}


\begin{align*} \mathcal { A } ^ c _ G ( X ) : = \big ( C ^ \infty _ c ( G ) \hat { \otimes } \Psi ^ { - \infty } ( S ) \big ) ^ { K \times K } \subset \mathcal { A } _ G ( X ) . \end{align*}


\begin{align*} x ^ X = _ X y ^ X : = \norm { x - _ X y } _ X = _ \mathbb { R } 0 \end{align*}


\begin{align*} \begin{array} { c c c c c } B ^ i ( X , Y ) & : = & - g ( A _ i ( X ) , Y ) , & & \\ B ^ i ( X , e _ j ) & : = & B ^ i ( e _ j , X ) & = & B ^ i ( e _ i , e _ l ) = 0 , \end{array} \end{align*}


\begin{align*} \sum _ { \gamma \in \Gamma _ { 5 } ( P G ) } c ( g ( \gamma ) ) = 5 \equiv 1 \pmod 2 . \end{align*}


\begin{align*} \partial _ t u = \partial _ r ^ 2 u + \frac { d + 1 } { r } \partial _ r u - 3 ( d - 2 ) u ^ 2 - ( d - 2 ) r ^ 2 u ^ 3 , ( r , t ) \in \R _ + \times \R _ + . \end{align*}


\begin{align*} \pi ( F \natural G ) = \pi ( ( F \natural G ) _ r ) = \pi ( F _ r * _ { \mathbf { H } _ r } G _ r ) = \pi ( F _ r ) \pi ( G _ r ) = \pi ( F ) \pi ( G ) . \end{align*}


\begin{align*} V ^ * _ g F = \iint _ { \R ^ { 2 d } } F ( x , \omega ) M _ \omega T _ x g \ , d ( x , \omega ) \end{align*}


\begin{align*} F _ w ( t ) = \sum _ { n = 0 } ^ \infty \# \{ \} t ^ n . \end{align*}


\begin{align*} P ( x ) = \sum _ { k = 0 } ^ n { n + m + 1 \choose k } x ^ { n - k } ( 1 - x ) ^ k \quad \mbox { a n d } Q ( x ) = \sum _ { k = 0 } ^ m { n + m + 1 \choose k } x ^ k ( 1 - x ) ^ { m - k } . \end{align*}


\begin{align*} \mu ^ { \# } _ { 0 } ( k , \ell , m , n ) = \sqrt { 2 \pi } \ , \delta _ 0 ( k - \ell + m - n ) . \end{align*}


\begin{align*} b = \frac { \lambda } { T - t } . \end{align*}


\begin{align*} \Lambda _ 1 ( f , g ) = \sigma ( t , 1 ) , \end{align*}


\begin{align*} F / x ^ \ell F \simeq L / x ^ \ell L = L / y ^ \ell L \simeq G / y ^ \ell G \end{align*}


\begin{align*} [ J ^ { \chi _ A } ] _ \mathcal { M } : = \lambda \alpha \in \mathbb { N } ^ \mathbb { N } , x \in X . \begin{cases} J ^ A _ { r _ \alpha } x & r _ \alpha > 0 , \\ 0 & , \end{cases} \end{align*}


\begin{align*} \psi ( \alpha ) : = \psi ( \overline { i _ { c } } ) + \bar { \psi } _ { 2 } ( \overline { i _ { c } } ; \alpha ) . \end{align*}


\begin{align*} \mathcal { L } _ { \sigma } ( \phi , \chi ) : = E ( \phi ) - \chi ( \| \phi \| ^ 2 - 1 ) + \sigma R ( \| \phi \| ^ 2 - 1 ) , \end{align*}


\begin{align*} \lim _ { y \to + \infty , y \in \R } \frac { \log | f ( i y ) / f '' ( i y ) | } { \log y } = + \infty . \end{align*}


\begin{align*} X _ 1 ^ T ( \tau _ { x _ 1 } ) = x _ 1 , \end{align*}


\begin{align*} U ^ k _ i f ( y ) \coloneqq U _ { k - 1 } \circ \ldots \circ U _ i f ( y ) = \frac { 1 } { R ( k , i ) } \sum \limits _ { x \in X ( i ) : x < y } f ( x ) \end{align*}


\begin{align*} \P ( \sigma | _ A = \sigma | _ { A ' } \mbox { a n d } \sigma | _ { B } = \sigma | _ { B ' } ) = q ^ { - 2 ( r - 1 ) ^ d } \ , . \end{align*}


\begin{align*} Q ( \xi ) = - \frac { ( \epsilon ( x ) + 1 ) } { e ^ { 2 x } } , \xi = e ^ x . \end{align*}


\begin{align*} A \ = \ \{ n : f ( n ) = 1 \} . \end{align*}


\begin{align*} ( 1 \otimes b ^ * ) \Delta ( p _ i ) = \sum _ { j = 1 } ^ { m _ i } a _ { i j } \otimes b _ { i j } , \end{align*}


\begin{align*} \langle a \rangle & : = \sqrt { 1 + | a | ^ 2 } . \end{align*}


\begin{align*} \| ( \mathcal { M } ( \tilde u _ 0 , a _ n , b ) u ) ( t , \cdot ) - ( \mathcal { M } ( \tilde u _ 0 , a _ 0 , b ) u ) ( t , \cdot ) \| _ { C ^ 0 ( \bar \Omega ) } & = \| \int _ s ^ t e ^ { - A ( t - \tau ) } u ( \tau , \cdot ) \big [ a _ n ( \tau , \cdot ) - a _ 0 ( \tau , \cdot ) \big ] d \tau \| _ { C ^ 0 ( \bar \Omega ) } \\ & \le R \| a _ n - a _ 0 \| _ { \mathcal { X } _ T } T . \end{align*}


\begin{align*} \partial _ s v = \Delta v + \frac { 1 } { 2 } \left ( \frac { b _ s } { b } - b \right ) \Lambda _ \xi v - 3 ( n - 4 ) v ^ 2 - ( n - 4 ) | \xi | ^ 3 v ^ 3 \end{align*}