\begin{align*} \left \| \sum _ { j = 1 } ^ { \nu ( P ) } [ G ( \tau _ j , \alpha _ j ) - G ( \tau _ j , \alpha _ { j - 1 } ) ] - \mathcal { I } \right \| _ { X } < \varepsilon , \end{align*}


\begin{align*} \sigma _ { \mathbf { 0 } } = \sigma _ { \mathbf { 1 } } & = ( \mathbf { 0 } \mathbf { 2 } \mathbf { 4 } \mathbf { 1 } \mathbf { 3 } \mathbf { 5 } ) ; \sigma _ { \mathbf { 2 } } = \sigma _ { \mathbf { 3 } } = ( \mathbf { 0 } \mathbf { 3 } \mathbf { 5 } \mathbf { 1 } \mathbf { 2 } \mathbf { 4 } ) ; \sigma _ { \mathbf { 4 } } = \sigma _ { \mathbf { 5 } } = ( \mathbf { 0 } \mathbf { 2 } \mathbf { 5 } \mathbf { 1 } \mathbf { 3 } \mathbf { 4 } ) . \end{align*}


\begin{align*} M = \begin{pmatrix} R & S & T \\ U & W & X \\ V & Y & Z \end{pmatrix} \ ; . \end{align*}


\begin{align*} \| \omega | K \| = \| \mu | K \| = \lim _ l \mu ( f _ l ) \ \ \mbox { a n d } \ \ \| \omega | J \| = \lim _ k \mu ( \phi ( e _ k ) ) . \end{align*}


\begin{align*} \chi ( z ) = \left \{ \aligned & 1 , 0 \leq z \leq 1 , \\ & z , z \geq 2 , \endaligned \right . \end{align*}


\begin{align*} \lim _ { \stackrel { n \to \infty } { f ( a _ n ) = 1 } } \frac { | H | } { n - 1 } & \leq ( 1 - \{ \alpha \} ) \{ \beta \} / ( ( \beta - \alpha ) \beta ) \\ & < ( 1 - \{ \alpha \} ) \{ \beta \} = \lim _ { \stackrel { n \to \infty } { f ( a _ n ) = 1 } } \frac { | G | } { n - 1 } \end{align*}


\begin{align*} \Delta u _ { z _ 0 } = 0 \ \ \Omega , u _ { z _ 0 } ( z ) = { 2 \over \bar z - \bar z _ 0 } \ \ \mathbb C \setminus \Omega , \end{align*}


\begin{align*} e ( \tilde V ) & = e ( \hat V ) + 2 s \\ & = e ( V _ { t } ) + 4 s . \end{align*}


\begin{align*} \bigl \langle \partial _ t \bigl ( u _ { \delta , \lambda } ( t ) - u _ 0 \bigr ) , F ^ { - 1 } \bigl ( u _ { \delta , \lambda } ( t ) - u _ 0 \bigr ) \bigr \rangle _ { V _ 0 ^ * , V _ 0 } + \int _ \Omega \nabla \mu _ { \delta , \lambda } ( t ) \cdot \nabla F ^ { - 1 } \bigl ( u _ { \delta , \lambda } ( t ) - u _ 0 \bigr ) \ , d x = 0 , \end{align*}


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


\begin{align*} \overline { \mathbf { v } ( \overline { k } ) } = \mathbf { v } ( - k ) = \mathbf { v } ( k ) \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} . \end{align*}


\begin{align*} \partial _ \varphi | t | = 0 \end{align*}


\begin{align*} \sum _ { \pi \in \mathcal { A } _ B } e ^ { - \mu \sum _ { j = 1 } ^ { n } | x _ { b _ j } - x _ { \pi ( 2 j ) } | } \leq C _ { d , \mu } ^ n e ^ { - \frac { \mu } { 2 \sqrt { d } } D _ s ( X \setminus Y _ B , Y _ B ) } \leq C _ { d , \mu } ^ n e ^ { - \frac { \mu } { 2 \sqrt { d } } D _ s ( X ) } . \end{align*}


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


\begin{align*} \varphi ^ P _ s ( g ) = \varphi ^ P _ s ( n m k ) = | d | ^ { - r s } = | e _ r \cdot g | ^ { - r s } . \end{align*}


\begin{align*} \gamma _ { A , \ell } = \max _ { \mathcal { L } \neq \emptyset } \frac { \dim V _ { \mathcal { L } } } { \dim G _ { A , \ell } - \dim ( G _ { A , \ell } ) _ { V _ { \mathcal { L } } } } , \end{align*}


\begin{align*} \psi ( \theta ) : = \gamma \theta + \frac { \sigma ^ 2 } { 2 } \theta ^ 2 + \int _ { ( 0 , \infty ) } ( e ^ { - \theta z } - 1 + \theta z \mathbf { 1 } _ { \{ z < 1 \} } ) \upsilon ( d z ) , \theta \geq 0 , \end{align*}


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


\begin{align*} 0 & = \langle A _ 1 ( u _ 1 ) + \lambda _ 1 B _ 1 ( u _ 1 ) , ( u _ 1 - \overline { u } _ 1 ) _ + \rangle _ { \mathcal { H } _ 1 } \\ & - \langle F _ 1 ( \overline { u } _ 1 , T _ 2 u _ 2 , \nabla \overline { u } _ 1 , \nabla ( T _ 2 u _ 2 ) ) + G _ 1 ( \overline { u } _ 1 , T _ 2 u _ 2 ) , ( u _ 1 - \overline { u } _ 1 ) _ + \rangle _ { \mathcal { H } _ 1 } \end{align*}


\begin{align*} - \Delta v = s ^ { - 1 } f _ { s , \epsilon } ( x , \underline { v } _ { \epsilon } ) \equiv s ^ { - 1 } \left \{ \begin{aligned} & f _ { \epsilon } ( x , \underline { v } _ { \epsilon } ) \ , \ , & & \mbox { i n } B _ s \\ & \lambda s \hat { \phi } ( x ) \ , \ , & & \mbox { i n } \Omega \backslash B _ s \end{aligned} \right . , v = 0 \mbox { o n } \partial \Omega . \end{align*}


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


\begin{align*} \partial _ t \varphi ( t , x ) = \sigma ( \varphi ( t , x ) ) . \end{align*}


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


\begin{align*} u ( a , t ) = \alpha _ 1 , u ( b , t ) = \alpha _ 2 , u _ { x } ( a , t ) = u _ { x } ( b , t ) = 0 \end{align*}


\begin{align*} x _ i - x _ j = 0 , \ f o r \ a n y \ \{ i , j \} \in e \in E ( H ) . \end{align*}


\begin{align*} & \varLambda ( - b ^ * - ^ * a , - d ^ * - ^ * c ) \\ & = \varLambda ( b ^ * + a ^ * , d ^ * + c ^ * ) - \varLambda ( b ^ * , c ^ * ) - \varLambda ( a ^ * , d ^ * ) + \varLambda ( b ^ * , ^ * c ) + \varLambda ( ^ * a , d ^ * ) \\ & = \varLambda ( b ^ * + a ^ * , d ^ * + c ^ * ) + \langle b , c \rangle - \langle d , a \rangle . \end{align*}


\begin{align*} d ^ { \bot } = \begin{cases} 4 & { \rm i f } ~ m = 4 ~ { \rm a n d } ~ e = q + 1 , \\ 3 & { \rm o t h e r w i s e } . \end{cases} \end{align*}


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


\begin{align*} \mathrm { t r a c e } ( \mathrm { O P } _ { \hbar } ^ { T } ( \nu ) ) = \frac { 1 } { ( 2 \pi \hbar ) ^ { d } } \underset { \mathbb { R } ^ { d } \times \mathbb { R } ^ { d } } { \int } \nu ( d z ) . \end{align*}


\begin{align*} f ( - t ) = g ( i _ { \mathsf { b } } ( t ) ) \ , . \end{align*}


\begin{align*} F ( \alpha ) = F ^ { \mathrm { s e l f } } ( \alpha ) + F ^ { \mathrm { i n t } } ( \alpha ) + F ^ { \mathrm { e l a s t i c } } ( \alpha ) \end{align*}


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


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


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


\begin{align*} \rho ^ \varepsilon ( x , t ) & = \sum _ { k = 0 } ^ \infty \varepsilon ^ k \eta _ k ( \pi ( x , t ) , t , \varepsilon ^ { - 1 } d ( x , t ) ) , ( x , t ) \in \overline { Q _ { \varepsilon , T } } , \end{align*}


\begin{align*} 0 = \left [ Q ^ { q } u \right ] ( x , t ) = \left [ ( S R ) ^ { q } u \right ] ( x , t ) \mbox { f o r a l l } x \in [ 0 , 1 ] \end{align*}


\begin{gather*} d _ { r } ^ 3 \gamma + \frac { 1 } { 2 } B ( \theta \wedge ( \theta \circ \alpha ) ) = d _ { r } ^ 3 \gamma ' + \frac { 1 } { 2 } B ( \theta ' \wedge ( \theta ' \circ \alpha ) ) . \end{gather*}


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


\begin{align*} \Delta f \left ( n \right ) = f \left ( n + 1 \right ) - f \left ( n \right ) , \quad \nabla f \left ( n \right ) = f \left ( n \right ) - f \left ( n - 1 \right ) , \end{align*}


\begin{align*} S = w - \bar { w } , \end{align*}


\begin{align*} \Omega _ { n , \mathrm { k } } : = \{ ( N _ 1 , \dots , N _ n ) \in \Omega _ n : \forall 1 \le i \le \ell : N _ { k _ i } \ge 1 \} . \end{align*}


\begin{align*} \Bigl \| \sum _ { n = 1 } ^ \infty \| I _ n z \| _ Z e _ { \min I _ n } \Bigr \| _ T & = \Bigl \| \sum _ { n = 1 } ^ \infty \| I _ n z \| _ Z e _ { m _ { \min I _ n } } \Bigr \| _ U = \Bigl \| \sum _ { n = 1 } ^ \infty \| I _ n z \| _ Z e _ { \min J _ n } \Bigr \| _ U \\ & \geqslant \frac 1 2 \Bigl \| \sum _ { n = 0 } ^ \infty \| J _ n A z \| _ Y e _ { \min J _ n } \Bigr \| _ U \geqslant \frac 1 2 [ A z ] _ \wedge . \end{align*}


\begin{align*} \left \langle A ^ { 1 / 2 } \dfrac { x - x _ 0 } { | x - x _ 0 | } , A ^ { 1 / 2 } \dfrac { e } { | e | } \right \rangle & = \left \langle A ^ { 1 / 2 } \ ( \cos \delta \ , \dfrac { e } { | e | } + \sin \delta \ , \dfrac { z } { | z | } \ ) , A ^ { 1 / 2 } \dfrac { e } { | e | } \right \rangle \\ & = \cos \delta \ , \left | A ^ { 1 / 2 } \dfrac { e } { | e | } \right | ^ 2 + \sin \delta \ , \left \langle A ^ { 1 / 2 } \dfrac { z } { | z | } , A ^ { 1 / 2 } \dfrac { e } { | e | } \right \rangle . \end{align*}


\begin{align*} g _ n = \frac { d } { d t } Q _ N w _ n + \nu A Q _ N w _ n + B ( Q _ N w _ n , Q _ N w _ n ) + D B ( u ) Q _ N w _ n . \end{align*}


\begin{align*} d _ { \Gamma _ t } ( x ) \ ; = \ ; \inf \{ | x - y | \ , ; \ , y \in \Gamma _ t \} , \end{align*}


\begin{align*} h _ { R } ( x ) = \int _ { B _ { R } ( 0 ) } G _ { n - 1 , R } ( x , y ) \phi _ R ( y ) \mathrm { d } y , \end{align*}


\begin{align*} \mathsf { U p } _ { \mathsf { P S L } } ( p ) ( \lnot ^ { \mathsf { U p } _ { \mathsf { P S L } } ( \mathbb { Y } ) } ( U ) ) = \lnot ^ { \mathsf { U p } _ { \mathsf { P S L } } ( \mathbb { X } ) } \mathsf { U p } _ { \mathsf { P S L } } ( p ) ( U ) . \end{align*}


\begin{align*} J _ 2 & : = - s ^ 3 \lambda ^ 3 \int _ { \Gamma _ T } \xi ^ 3 \sigma ( \partial _ \nu ^ A \eta ^ 0 ) \psi ^ 2 \ , \d S \ , \d t + \lambda \int _ { \Gamma _ T } \left ( s \xi + \frac { \tau } { 2 } \right ) \left ( \frac { \tau } { 2 } - s \alpha \right ) ( \partial _ t \log \gamma ) ( \partial _ \nu ^ A \eta ^ 0 ) \psi ^ 2 \ , \d S \ , \d t \\ & \ge C s ^ 3 \lambda ^ 3 \int _ { \Gamma _ T } \xi ^ 3 \psi ^ 2 \ , \d S \ , \d t . \end{align*}


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


\begin{align*} f ( X + a ) + c f ( X ) = b . \end{align*}


\begin{align*} \Delta = \int _ { B _ R ( x _ 0 ) } | u ( x ) | ^ { p - 1 } \ , d x , \end{align*}


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


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


\begin{align*} f _ n ^ { \prime } ( s ) = \sum _ { \ell = 1 } ^ n \frac { \frac { d } { d s } \lambda ^ u _ { \psi ^ x _ { - \ell } ( s ) } } { \lambda ^ u _ { _ { \psi ^ x _ { - \ell } ( s ) } } } \end{align*}


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


\begin{align*} m ( R ) = R _ { 0 r } \ ; . \end{align*}


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


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


\begin{align*} \big ( \partial ^ \alpha _ j b _ i ( a ) \big ) \ , \big ( \Delta ^ { k } ( v ) \big ) = b _ i \widetilde { a } \big ( \Delta ^ { k } ( v ) \big ) = \widetilde { a } \big ( \Delta ^ { k + 1 } ( v ) \big ) = 0 \end{align*}


\begin{align*} \Delta ( v _ 1 \cdots v _ n ) = \sum _ { i = 0 } ^ n ( v _ 1 \cdots v _ i ) \otimes ( v _ { i + 1 } \cdots v _ n ) \in \mathrm { T } ( V ) \otimes \mathrm { T } ( V ) \Delta ( 1 ) = 1 \otimes 1 , \end{align*}


\begin{align*} t + \frac { 1 } { M } + \frac { M - 2 } { M } \frac { t } { 1 - t } = 1 . \end{align*}


\begin{align*} x ^ T _ j z _ j ^ * = x _ { j 0 } z _ { j 0 } ^ * + \bar x _ j ^ T \bar z _ j ^ * \end{align*}


\begin{align*} \left [ \phi \partial _ { x } - \left ( n + 1 - \frac { \tau _ { n } } { \gamma _ { n } \gamma _ { n - 1 } } \right ) x \right ] P _ { n + 1 } = \left [ \lambda _ { n } + \tau _ { n } \frac { x ^ { 2 } - \gamma _ { n } } { \gamma _ { n } \gamma _ { n - 1 } } - \left ( n + 1 \right ) \gamma _ { n + 1 } \right ] P _ { n } . \end{align*}


\begin{align*} F ( t ) = \alpha ( t , x ) \end{align*}


\begin{align*} \partial _ t p - | \nabla p | ^ 2 + p u = 0 . \end{align*}


\begin{align*} \mathcal { A } _ B = \{ \pi \in S _ { 2 n } : \pi ( 2 j - 1 ) = b _ j \ , 1 \leq j \leq n \} . \end{align*}


\begin{align*} | f _ \zeta ^ \varepsilon - \bar { f } _ \zeta | \leq c \varepsilon \sum _ { \xi = \zeta , \zeta _ 2 } \left ( | \bar { \xi } | + \Bigl | \overline { \partial ^ \circ \xi } \Bigr | + \Bigl | \overline { \nabla _ \Gamma \xi } \Bigr | + \Bigl | \overline { \nabla _ \Gamma ^ 2 \xi } \Bigr | \right ) \quad Q _ { \varepsilon , T } \end{align*}


\begin{align*} \begin{multlined} | ( A \otimes B ) ( \vec { x } ) | \leq | A ( \vec { x } \rvert _ 0 ) | \cdot | B ( \vec { x } \rvert _ 1 ) | \\ < C _ 1 \prod _ { i \in a } \phi ( \vec { x } ( i ) ) \cdot C _ 2 \prod _ { j \in b } \phi ( \vec { x } ( j ) ) \\ = C _ 1 C _ 2 \prod _ { i \in a \sqcup b } \phi ( \vec { x } ( i ) ) \ ; . \end{multlined} \end{align*}


\begin{align*} \kappa _ i ( t * x ) = \underline { \lambda } _ { i , i + 1 } \kappa _ i ( x ) , \kappa _ i ' ( t * x ) = \underline { \lambda } _ { w ( i ) , w ( i + 1 ) } \kappa _ i ' ( x ) . \end{align*}


\begin{align*} \hat { M } ^ 2 = ( M ^ 2 - b ^ 2 ) - ( a - b ) b = M ^ 2 - a b . \end{align*}


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


\begin{align*} \alpha ( q , a ) & = \begin{cases} ( q ' , b ) & ( q , a , q ' , b ) \in \Delta \\ ( q ' , a ) & ( q , \pm 1 , q ' ) \in \Delta \end{cases} \\ \beta _ { + 1 } & = \prod _ { q ' \mid \exists q , ( q , + 1 , q ' ) \in \Delta } \rho _ { q ' } \\ \beta _ { - 1 } & = \prod _ { q ' \mid \exists q , ( q , - 1 , q ' ) \in \Delta } { \rho _ { q ' } } ^ { - 1 } \end{align*}


\begin{align*} \Phi _ g ( \tau , z , \omega ) = \Phi _ { g } ( \omega , z , \tau ) . \end{align*}


\begin{align*} J _ E ( f ) ( \omega ) = \omega J _ M ( \omega ) - \int _ 0 ^ \omega J _ M ( \tilde \omega ) \ , \dd \tilde \omega . \end{align*}


\begin{align*} \mathcal { A } ^ { \mathrm { g r } } \left ( X , Y \right ) : = \bigoplus _ { s \in \mathbb { Z } } \mathcal { A } \left ( X , Y \langle s \rangle \right ) , \end{align*}


\begin{align*} F _ t ( \zeta _ { N } ^ { - 1 } ) = U _ t ( - 1 ; \zeta _ { N } ) . \end{align*}


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


\begin{align*} \Delta \log \left ( ( 1 - K ) ^ 2 \left ( 1 - 6 K + \Delta \log \left ( 1 - K \right ) \right ) \right ) = 1 2 K . \end{align*}


\begin{align*} \delta ( u - B ) _ { s , t } & = \int _ { s } ^ { t } b ( u _ r ) \dd r + \int _ { s } ^ { t } h _ 2 \big [ u _ r - B _ r \big ] \times \circ \dd W _ r \\ & = \int _ { s } ^ { t } b ( u _ r ) \dd r + \int _ { s } ^ { t } h _ 2 ^ 2 \big [ u _ r - B _ r \big ] \dd r + \int _ { s } ^ { t } h _ 2 \big [ u _ r - B _ r \big ] \times \dd W _ r \ , . \end{align*}


\begin{align*} Q _ { P _ i } . A _ { P _ j } = Q _ { P _ i } . B _ { P _ j } = - \delta _ { i j } , \end{align*}


\begin{align*} c ( \tau \sigma , \alpha ) = \frac { a p } { q b } = \frac { a } { b } \cdot \frac { p } { q } = c \big ( \tau , \sigma ( \alpha ) \big ) \cdot c ( \sigma , \alpha ) , \end{align*}


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


\begin{align*} \chi _ { s t } = \int _ s ^ t f ( s , \theta _ s - \omega _ r ) \ , d r = f ( s , \cdot ) \star \mu ^ \omega _ { s t } ( \theta _ s ) , \end{align*}


\begin{align*} h ( \lambda ) : = \frac { \lambda \wedge \lambda ^ 4 } { ( \lambda ^ 4 + 6 \lambda ^ 3 + 7 \lambda ^ 2 + \lambda ) \Bigl ( 1 + \lambda + \frac { \lambda ^ 2 } { 2 } + \frac { \lambda ^ 3 } { 6 } \Bigr ) e ^ { - \lambda } } \end{align*}


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


\begin{align*} A & = X ^ { - 1 } ( B ) \\ & = \{ \omega \in \Omega : h ( \vec { Y } ) \in B \} \end{align*}


\begin{align*} m \bigl ( u _ { \delta , \lambda } ( t ) \bigr ) = m ( u _ 0 ) = m _ 0 \end{align*}


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


\begin{align*} y = x \ , \cos ( \mu \pi / n ) - \sqrt { 1 - x ^ 2 } \ , \sin ( \mu \pi / 2 ) . \end{align*}


\begin{align*} \begin{aligned} n - ( d - 2 ) t - 2 \ & \ge \ 4 + ( 2 d - 3 ) t - ( d - 2 ) t - 2 = 2 + ( d - 1 ) t \\ & = \ \max ( w ) . \end{aligned} \end{align*}


\begin{align*} \left \{ \begin{aligned} \partial _ r ^ 2 \eta _ 2 ( r ) & = k _ d ^ { - 1 } \partial ^ \circ \eta _ 0 + k _ d ^ { - 2 } V _ \Gamma ^ 2 \eta _ 0 - \Delta _ \Gamma \eta _ 0 - k _ d ^ { - 1 } V _ \Gamma H \eta _ 0 - k _ d ^ { - 1 } f , r \in ( g _ 0 , g _ 1 ) , \\ \partial _ r \eta _ 2 ( g _ i ) & = \nabla _ \Gamma g _ i \cdot \nabla _ \Gamma \eta _ 0 - k _ d ^ { - 1 } ( \partial ^ \circ g _ i ) \eta _ 0 + k _ d ^ { - 2 } g _ i V _ \Gamma ^ 2 \eta _ 0 , i = 0 , 1 \end{aligned} \right . \end{align*}


\begin{align*} I _ { \lambda } ( \Psi _ { n } ^ { 1 } ) = I _ { \lambda } ( u _ n ) - I _ { \lambda } ( u ^ { 0 } ) + o ( 1 ) , \end{align*}


\begin{align*} \mathcal H _ i ( x , t ) = t ^ { p _ i ( x ) } + \mu ( x ) t ^ { q _ i ( x ) } . \end{align*}


\begin{align*} \int _ { Q _ T } G \partial _ t p = \sum _ { i = 1 } ^ { \ell } \int _ { Q _ T } \frac { d } { d t } \big ( c _ i \bar { G } _ i ( p , n ) \big ) - c _ i \partial _ n \bar { G } _ i ( p , n ) \partial _ t n - \bar { G } _ i ( p , n ) \partial _ t c _ i . \end{align*}


\begin{gather*} W _ 1 < W _ 2 \Longleftrightarrow \begin{cases} y _ 1 < y _ 2 \\ y _ 1 = y _ 2 , \ , x _ 1 < x _ 2 . \end{cases} \end{gather*}


\begin{align*} \{ \mu \in M _ { A , B } ( a ) : P ( f , \alpha , \mu ) \geq c , \ S _ \mu = X \} \{ \mu \in M _ { A , B } ( a ) : P ( f , \alpha , \mu ) \geq c \} \end{align*}


\begin{align*} ( \tau \sigma ) ( \alpha ) ^ { a p } = \big [ \big ( \tau ( \sigma ( \alpha ) ) \big ) ^ { a } \big ] ^ { p } = \big [ \sigma ( \alpha ) ^ { b } \big ] ^ { p } = \big [ \sigma ( \alpha ) ^ { p } \big ] ^ { b } = \big [ \alpha ^ { q } \big ] ^ { b } . \end{align*}


\begin{align*} \mathbb { M } ^ s ( \mathbb { R } ^ d ) & = \mathbb { H } ^ { s } ( \mathbb { R } ^ d ) , s \geq 0 , \\ \mathbb { M } ^ s _ { \mathrm { e } } ( \mathbb { R } ^ d ) & = \mathbb { E } ^ { \frac { 1 } { 2 } } _ q ( \mathbb { R } ^ d ) , q = \frac { s } { \sqrt { \pi } } , s \geq 0 , \\ \mathbb { M } ^ s _ { \mathrm { e } ^ 2 } ( \mathbb { R } ^ d ) & = \mathbb { E } ^ { 1 } _ q ( \mathbb { R } ^ d ) , q = \ln ( \tfrac { \pi } { \pi - s } ) , \pi > s \geq 0 . \end{align*}


\begin{align*} p ( t \ , , F ) = \P \{ X ( t ) \in F \} \qquad . \end{align*}


\begin{align*} [ A ] ( x ) = \sum _ { y \in \mathbb { N } _ 0 } A ( \bar \Phi _ \cdot ^ \infty ( x , \bar \Phi _ \cdot ^ \infty ( y , y ) ) \ ; . \end{align*}


\begin{align*} s ^ U _ i = 1 \forall i \ ; , \end{align*}


\begin{align*} \phi _ g ( \tau , z ) = - \frac { \varphi _ g \Big | _ { k _ g , 1 } T _ { - } ^ { ( N _ g ) } ( 2 ) ( \tau , z ) } { \varphi _ g ( \tau , z ) } - f _ g ( \tau , z ) , \end{align*}


\begin{align*} \sum _ { j = 1 } ^ n \nu _ j ( y , t ) \partial _ r ( \underline { D } _ j \eta ) ( y , t , r ) & = \partial _ r \Bigl ( \nu ( y , t ) \cdot \nabla _ \Gamma \eta ( y , t , r ) \Bigr ) = 0 , \\ \sum _ { j = 1 } ^ n \nu _ j ( y , t ) \underline { D } _ j ( \partial _ r \eta ) ( y , t , r ) & = \nu ( y , t ) \cdot [ \nabla _ \Gamma ( \partial _ r \eta ) ] ( y , t , r ) = 0 , \end{align*}


\begin{align*} \widetilde { \mathcal { A } _ 0 } & : = \left \{ R _ 0 \in S O ( 3 ) : R _ 0 e _ 3 = e _ 3 \right \} \times ( 0 , \infty ) \times \left ( - \frac { \ell } { 2 } , \frac { \ell } { 2 } \right ) . \end{align*}


\begin{align*} \mathcal { C } : = \frac { 1 } { \sqrt { 2 \omega ( D ) A ( D , \mathtt { m } ) } } \left ( \begin{matrix} A ( D , \mathtt { m } ) & - \tfrac { 1 } { 2 } \mathtt { m } p ' ( \mathtt { m } ) \\ - \tfrac { 1 } { 2 } \mathtt { m } p ' ( \mathtt { m } ) & A ( D , \mathtt { m } ) \end{matrix} \right ) \ , , \end{align*}


\begin{align*} \mathcal { B } ( f , \Psi ) : = - \frac { 1 } { 2 } \iint _ \Omega \partial _ r [ | t | f ] \ , \partial _ r \Psi \frac { d t d x } { | t | ^ { n - d - 1 } } . \end{align*}


\begin{align*} M ^ \chi : = \{ m \in M \colon \forall z \in Z ( \mathfrak { g } ) \ \exists n \in \mathbb { N } \ ( z - \chi ( z ) ) ^ n m = 0 \} . \end{align*}


\begin{align*} \lambda _ { n } & = 2 \left [ \left ( \gamma _ { n } + \gamma _ { n + 1 } + \gamma _ { n + 2 } \right ) - \left ( z ^ { 2 } + 1 \right ) - \frac { n } { 2 } \right ] \gamma _ { n + 1 } , \\ \tau _ { n } & = 2 \frac { h _ { n + 1 } } { h _ { n - 2 } } = 2 \gamma _ { n + 1 } \gamma _ { n } \gamma _ { n - 1 } . \end{align*}


\begin{align*} \textrm { l i } ( x ) = \textrm { L i } ( x ) + \textrm { l i } ( 2 ) < \frac { x } { \log x } + \frac { 3 x } { 2 \log ^ 2 x } + \textrm { l i } ( 2 ) \end{align*}


\begin{align*} \lim _ { k \to \infty } \Big ( m ^ k ( x ^ { k } + d ^ k ; \rho ^ \infty ) - m ^ k ( x ^ k ; \rho ^ \infty ) \Big ) = 0 . \end{align*}


\begin{align*} T _ 1 & \coloneq c _ 1 ( \chi _ 1 ) = c _ 1 ( \lambda _ 1 ^ { \alpha _ 0 } \lambda _ 2 ^ { \alpha _ 1 } ) = \alpha _ 0 l _ 1 + \alpha _ 1 l _ 2 , \\ T _ 2 & \coloneq c _ 1 ( \chi _ 2 ) = c _ 1 ( \lambda _ 1 ^ { \beta _ 0 } \lambda _ 2 ^ { \beta _ 1 } ) = \beta _ 0 l _ 1 + \beta _ 1 l _ 2 . \end{align*}


\begin{align*} A _ 1 ( r ) ^ 2 = \bigg ( \sum _ { k \ge 1 } k \sqrt { c _ k r ^ k } \cdot \sqrt { c _ k r ^ k } \bigg ) ^ 2 \le A _ 2 ( r ) A _ 0 ( r ) . \end{align*}


\begin{align*} T ( z _ 0 ) = 8 \pi \partial _ z u _ { z _ 0 } ( z _ 0 ) . \end{align*}


\begin{align*} \frac { 3 n ^ 2 t } { \binom n 2 } \approx 6 t . \end{align*}


\begin{align*} U ( x , y ) = x ^ 2 ( x ^ 2 + y ^ 2 ) . \end{align*}


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


\begin{align*} f ( u ) = \prod _ { k = 1 } ^ { \infty } ( 1 - \frac { h ^ 2 } { ( u + k h n ) ^ 2 } ) . \end{align*}


\begin{align*} \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } g _ s g _ t d \mathbb { E } ( X _ s X _ t ) = \mathbb { E } \left ( \int _ { 0 } ^ { 1 } g _ t d X _ t \right ) ^ 2 \geq \frac { \beta } { 4 } ( \sup _ { r \in [ 0 , 1 ] } \abs { g _ r } ) ^ 2 \{ \mathbb { E } ( X _ { a , b } ) ^ 2 \wedge \mathbb { E } ( X _ 1 ) ^ 2 \} . \end{align*}


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


\begin{align*} \tilde { X } ( t ) : = x _ k - ( t - k h ) \nabla f ( x _ k ) + \sigma ( k , x _ k ) ( W ( t ) - W ( k h ) ) , t \in [ k h , ( k + 1 ) h ] . \end{align*}


\begin{align*} H = \sqrt { - \Delta + m ^ 2 } - m - \frac { \kappa } { | x | } . \end{align*}


\begin{align*} \| s _ \lambda ( x _ 1 , \dots , x _ N ) s _ \lambda ( y _ 1 , \dots , y _ N ) \| = s _ \lambda ( 1 ^ N ) s _ \lambda ( 1 ^ N ) \leq \dim \mathsf { W } _ M ^ \lambda \dim \mathsf { W } _ N ^ \lambda . \end{align*}


\begin{align*} \xi _ { t } = C ( L ) \varepsilon _ { t } = \sum _ { j = - \infty } ^ { \infty } C _ { j } \varepsilon _ { t - j } , \end{align*}


\begin{align*} \mathcal { E } _ { V } ^ { \theta } ( \mu ) = \frac { 1 - \lambda } { 2 } \int _ { M } V \ , d \mu + \int _ { M \times M } g ( x - y ) d \mu _ { x } d \mu _ { y } + \lambda \int _ { M } V \ , d \mu + \frac { 1 } { \theta } { \rm e n t } [ \mu ] + \frac { 1 - \lambda } { 2 } \int _ { M } V \ , d \mu . \end{align*}


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


\begin{align*} J _ k : = \int _ { 0 = \theta _ 0 < \theta _ 1 < \ldots < \theta _ k \le 1 } \prod _ { i = 1 } ^ k ( \theta _ i - \theta _ { i - 1 } ) ^ { - 1 / 2 } d \theta _ 1 \cdots d \theta _ k \le C ^ k ( k ! ) ^ { - 1 / 2 } \end{align*}


\begin{align*} ( \varphi _ 1 ( \vec { a } ) \lor \dots \lor \varphi _ n ( \vec { a } ) ) \land b = ( \varphi _ 1 ( \vec { a } ) \land b ) \lor \dots \lor ( \varphi _ n ( \vec { a } ) \land b ) \leq c . \end{align*}


\begin{align*} \{ \mu \in M _ { A , B } ( a ) : P ( f , \alpha , \mu ) = c \} \{ \mu \in M _ { A , B } ( a ) : P ( f , \alpha , \mu ) \geq c \} . \end{align*}


\begin{align*} \int _ { \Omega \setminus \mathbb { D } } \mathrm { R e } \ , h = 0 \end{align*}


\begin{align*} \nu _ { l , \psi } : = \psi _ l \otimes \omega _ { 4 n - 2 l - 2 , \psi } , \end{align*}


\begin{align*} \pi ^ { - \frac { r s } { 2 } } \Gamma \left ( \frac { r s } { 2 } \right ) \zeta ( r s ) E ^ P _ { s , \varphi } ( g ) = \pi ^ { - \frac { r } { 2 } ( 1 - s ) } \Gamma \left ( \frac { r ( 1 - s ) } { 2 } \right ) \zeta \left ( r - r s \right ) { E ^ P _ { 1 - s , \varphi } \left ( ( g ^ \top ) ^ { - 1 } \right ) } . \end{align*}


\begin{align*} \mathcal { D } ( v ) _ { s , t } : = \int _ { s } ^ { t } \left [ \lambda _ 1 v _ r \times g ' ( v _ r ) - \lambda _ 2 v _ r \times ( v _ r \times g ' ( v _ r ) ) \right ] \dd r \ , . \end{align*}


\begin{align*} K _ { \theta _ 1 , \theta _ 2 , \Phi } : = K _ { \mathcal { G } ( \theta _ 1 , \Phi ) , \mathcal { G } ( \theta _ 2 , \Phi ) } \in X . \end{align*}


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


\begin{align*} \gamma ' ( \xi ) = ( - \sin \vartheta ( \xi ) ; \cos \vartheta ( \xi ) ) \end{align*}


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


\begin{align*} \mathbf { P } _ n ^ { \alpha , \theta , T } ( \Gamma ( x _ i : i \in \mathcal { B } \sqcup \mathcal { O } \sqcup \mathcal { U } ) ) = \lim _ { m \to \infty } m ^ { \# \mathcal { B } } P ^ { 1 , 1 - \frac { \lambda } { m } , \frac { T } { m } } _ { m , n } \left ( \sigma ( \tilde { x } ^ 1 ) = \tilde { y } ^ 1 , \ldots , \sigma ( \tilde { x } ^ p ) = \tilde { y } ^ p \right ) , \end{align*}


\begin{align*} \xi \in [ 0 , 1 ] \mapsto \theta = \omega _ k ( \eta , \xi , \omega _ j ( \xi ) ) . \end{align*}


\begin{align*} \mathring { R } _ { q + 1 } ^ B : = \mathring { R } _ { l i n } ^ B + \mathring { R } _ { o s c } ^ B + \mathring { R } _ { c o r } ^ B , \end{align*}


\begin{align*} V = C ( P ( G ( n ) , 0 ) ) \cup C ( P ( G ( n ) , 1 ) ) \cup C ( P ( G ( n ) , 2 ) ) \end{align*}


\begin{align*} \{ ( a _ n , b _ n ) \} = \{ ( [ n \alpha ] , [ n \beta ] ) \} . \end{align*}


\begin{align*} \zeta _ { i j } \left ( \frac { z } { w } \right ) = \frac { z - q _ i ^ { - c _ { i j } } w } { z - w } \forall \ , i , j \in I \ , . \end{align*}


\begin{align*} Q _ { 1 T } ( r ) & \Rightarrow A _ { m } ( r ; d ) , \\ \sup _ { 0 \leq r \leq 1 } \left \Vert Q _ { i T } ( r ) \right \Vert & \overset { P } { \rightarrow } 0 i = 2 , \ldots , 5 . \end{align*}


\begin{align*} q _ { \rho } \left ( x , t \right ) & = 2 \left [ { \displaystyle \sum \limits _ { n = 1 } ^ { N } } c _ { n } ^ { 2 } \mathrm { e } ^ { 8 \kappa _ { n } ^ { 3 } t - \kappa _ { n } x } \psi _ { \rho } \left ( x , t ; \kappa _ { n } \right ) \right ] ^ { 2 } \\ & - 4 { \displaystyle \sum \limits _ { n = 1 } ^ { N } } \kappa _ { n } c _ { n } ^ { 2 } \mathrm { e } ^ { 8 \kappa _ { n } ^ { 3 } t - \kappa _ { n } x } \psi _ { \rho } \left ( x , t ; \kappa _ { n } \right ) . \end{align*}


\begin{align*} N ( c , H ) & = { \mathcal { Q } } ( G ) + { \mathcal { C } } ( G ) \\ & = c ^ 2 \mathcal { Q } ( G _ 0 ) + \mathcal { Q } ( H ) + 2 c \mathcal { Q } ( G _ 0 , H ) + c ^ 3 \mathcal C ( G _ 0 ) + \mathcal { C } ( H ) + 3 c ^ 2 \mathcal { C } ( G _ 0 , G _ 0 , H ) + 3 c \mathcal { C } ( G _ 0 , H , H ) . \end{align*}


\begin{align*} \psi _ j = \psi _ { j , E _ 0 } + \frac { \psi _ { j , E _ 1 } } { k } + O ( k ^ { - 2 } ) , k \rightarrow \infty . \end{align*}


\begin{align*} & m _ v ( ( y , \omega ) , ( z , \eta ) ) = \max \left \{ \frac { v ( ( y , \omega ) ) } { v ( ( z , \eta ) ) } , \frac { v ( ( z , \eta ) ) } { v ( ( y , \omega ) ) } \right \} , \\ & v ( ( y , \omega ) ) : = m _ 0 ( ( y , \omega ) , ( x , \xi ) ) \cdot \max \{ w ( \Phi ( \omega ) ) , [ w ( \Phi ( \omega ) ) ] ^ { - 1 } \} , \end{align*}


\begin{align*} x \mapsto \sum a _ i ^ * x a _ i , \ \ \sum a _ i ^ * a _ i = 1 \end{align*}


\begin{align*} y = ( L \alpha ^ { - 1 } , 0 , 0 ) , \end{align*}


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


\begin{align*} \partial _ t p = \kappa \partial _ x ^ 2 \left ( x ( 1 - x ) p \right ) - \partial _ x \left ( x ( 1 - x ) \psi ( x ) p \right ) \ . \end{align*}


\begin{align*} H _ { r s } ( G , x ) = \sum _ { e u \in E ( G ) } x ^ { r s ( e ) + r s ( u ) } + \sum _ { e v \in E ( G ) } x ^ { r s ( e ) + r s ( v ) } + \sum _ { u w \in E ( G ) } x ^ { r s ( u ) + r s ( w ) } . \end{align*}


\begin{align*} \widehat { C } ^ \varepsilon ( x , t ) = C ^ \varepsilon ( x , t ) + c _ 5 + 1 \geq - | C ^ \varepsilon ( x , t ) | + c _ 5 + 1 \geq 1 \end{align*}


\begin{align*} E = 1 / ( 3 2 d ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } n \mathbb { P } ( | X | > b _ n ) = 0 . \end{align*}


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


\begin{align*} g ( t ) = 1 \ \ \forall t \in \Delta _ p \cap \Delta _ K . \end{align*}


\begin{align*} ( \pi r _ 1 ^ 2 , \pi r _ 2 ^ 2 ) = e ^ s ( \pi r ^ 2 , L ) . \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - \gamma \nabla f ( x _ k ) - \gamma \xi _ k , \end{align*}


\begin{align*} \deg ( e ( \nu ) ) = 0 , \deg ( x _ k e ( \nu ) ) = ( \alpha _ { \nu _ k } , \alpha _ { \nu _ k } ) , \deg ( \tau _ l e ( \nu ) ) = - ( \alpha _ { \nu _ { l } } , \alpha _ { \nu _ { l + 1 } } ) . \end{align*}


\begin{align*} R ^ G _ { L , \chi , \rho } : = \frac { ( - 1 ) ^ { { \rm r k } \ , G + { \rm r k } \ , L } } { | W _ L | } \sum _ { w \in W _ L } \rho ( w w _ L ^ e ) \cdot R ^ G _ { T _ w , \chi _ w } , \end{align*}


\begin{align*} \Phi ^ { \rm E } ( t , x ) = x + t \sigma ( x ) \end{align*}


\begin{align*} & G ^ { ( 4 ) } _ { \mathcal R } ( x , y ) = \\ & - \frac { i } { 2 \pi } \left [ \int _ { \theta _ c } ^ { \theta _ { \hat x } } + \int _ { \mathcal D _ { \theta _ { \hat x } , \frac \pi 2 + \theta _ { \hat x } - i \infty } } \right ] \frac { 2 i \sin \zeta \widetilde { \mathcal S } ( \cos \zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \left ( - \vert y \vert \cos ( \zeta + \theta _ { \hat y } ) + \vert x \vert \cos ( \zeta - \theta _ { \hat x } ) \right ) } } d \zeta , \end{align*}


\begin{align*} \int \sqrt { \omega _ 1 } \mathcal { C } ( f ) \ , \varphi _ 1 \ , d \omega _ 1 = \frac 1 2 \iiint \sqrt { \omega _ 1 } W f _ 1 f _ 2 ( f _ 3 + f _ 4 ) \left ( \varphi _ 3 + \varphi _ 4 - \varphi _ 1 - \varphi _ 2 \right ) \ , \dd \omega _ 1 \ , \dd \omega _ 3 \ , \dd \omega _ 4 . \end{align*}


\begin{align*} V _ C ^ { - T } = ( V _ C ^ { - 1 } ) ^ T = \begin{pmatrix} I _ { d \times d } & - C \\ 0 _ { d \times d } & I _ { d \times d } \end{pmatrix} \end{align*}


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


\begin{align*} Q ( \theta ) : = \{ k \ge 1 : \lVert \theta k \rVert \ge 1 / 4 \} = \bigcup _ { j \ge 0 } Q _ j ( \theta ) , Q _ j ( \theta ) : = \{ k \ge 1 : j + 1 / 4 \le \theta k \le j + 3 / 4 \} . \end{align*}


\begin{align*} \begin{bmatrix} 1 - 2 p & p & 0 & p \\ 0 & 1 - 2 p & p & p \\ p & 0 & 1 - 2 p & p \\ \end{bmatrix} . \end{align*}


\begin{align*} f _ { \partial } = \eta _ n + p ^ n \pi _ n . \end{align*}


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


\begin{align*} \delta = \delta ' : = \dfrac { \min \lbrace \mu , \lambda , 1 \rbrace c _ 1 } { 8 } , ~ ~ T : = \dfrac { M } { C _ { \delta ' } \mathcal H _ 2 ( C _ 0 M _ 0 , C _ 1 M _ 1 ) } , \end{align*}


\begin{align*} { [ l _ { i i } ^ { ( 0 ) } , l _ { k m } ^ { \pm } ( v ) ] = } \begin{cases} l _ { k m } ^ { \pm } ( v ) & k = i , m \neq i \\ - l _ { k m } ^ { \pm } ( v ) & k \neq i , m = i \\ 0 & \mbox { o t h e r w i s e } \end{cases} \end{align*}


\begin{align*} [ p , q ] = c [ p , c ] = 0 = [ q , c ] , \end{align*}


\begin{align*} \varphi _ { n , t } = \rho _ t \sum _ { k = 0 } ^ n \chi _ k = \sum _ { k = 0 } ^ n \chi _ k e ^ { - t k } n \in \N _ 0 . \end{align*}


\begin{align*} H _ n : = K _ n \circ \tilde { H } _ n = \tilde { H } _ n \circ K _ n \end{align*}


\begin{align*} N = N ^ { 1 } \supseteq N ^ { 2 } \supseteq N ^ { 3 } \supseteq \dots , \end{align*}


\begin{align*} \displaystyle \lim _ { \lambda \rightarrow 0 ^ + } x _ * ( \lambda ) = + \infty , \ , \displaystyle \lim _ { \lambda \rightarrow a } x _ * ( \lambda ) = 0 . \end{align*}


\begin{align*} \begin{cases} \dot { x } ( t ) = F ( x _ t ) , & t \geq 0 , \\ x _ 0 = \varphi , & \varphi \in X , \end{cases} \end{align*}


\begin{align*} 2 \mathrm { R e } \langle \kappa \Phi _ { \# } ^ y \ , | \ , H _ y R _ y ^ { \# } \rangle & = ( e _ { \alpha } + \alpha \| \lambda _ y \| ^ 2 ) \ , 2 \mathrm { R e } \langle \kappa \Phi _ { \# } ^ y \ , | \ , R ^ { \# } _ y \rangle \\ & + 2 \alpha \mathrm { R e } \langle A ^ - _ y \kappa \Phi _ { \# } ^ y \ , | \ , A ^ - _ y R ^ { \# } _ y \rangle + 2 \alpha \mathrm { R e } \langle \kappa \Phi _ { \# } ^ y \ , | \ , V _ y R ^ { \# } _ y \rangle , \end{align*}


\begin{align*} \Gamma _ \kappa : = \{ x \in \Gamma \ , : \ , u ( x ) > \kappa \} , \kappa \in \R . \end{align*}


\begin{align*} \mathfrak { D } : = ( 1 - \kappa ) \mathfrak { D } _ { i n t } + \kappa \mathfrak { D } _ { \Gamma } . \end{align*}


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


\begin{align*} \mathcal { E } \big [ k _ { n + 1 } ; H [ n ] , \Theta [ n ] \big ] & = \mathcal { E } \Big ( \{ z _ j ^ + \} , \{ e _ j \} \Big ) \\ & = - \frac { 1 } { 2 } \sum _ { j = 1 } ^ \infty { \Big \{ H [ n ] ( \overline { e _ j } , e _ j ) - \sqrt { H ^ 2 [ n ] ( \overline { e _ j } , e _ j ) - | \Theta [ n ] ( \overline { e _ j } , \overline { e _ j } ) | ^ 2 } \Big \} } \\ & : = - \frac { 1 } { 2 } \sum _ j { \mathcal { F } ( e _ j ) } . \end{align*}


\begin{align*} \sum _ { \pi \in S _ n } \prod _ { j = 1 } ^ n | m _ { j \ , \pi ( j ) } | \leq \sum _ { \ell \geq \tilde { D } } C _ d ^ n e ^ { - \frac { \mu } { 2 } \ell } \leq C _ d ^ n e ^ { - \frac { \mu } { 2 } \tilde { D } } . \end{align*}


\begin{align*} a = \sqrt { - 1 } \frac { \sqrt { 2 } } { \sqrt { | k + h ^ \vee | } } x , \end{align*}


\begin{align*} F _ \ell ^ { ( k ) } = \left \{ \begin{array} { l l } 0 , & \hbox { \rm i f \ } 0 \leq \ell < k - 1 , \\ 1 , & \hbox { \rm i f \ } \ell = k - 1 , \\ 2 ^ { \ell - k } , & \hbox { \rm i f \ } k \leq \ell \leq 2 k - 1 . \end{array} \right . \end{align*}


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


\begin{align*} \gamma _ 2 ( t ) = \begin{cases} 0 & t \le 0 , \\ \displaystyle { \exp \left ( - \exp \left ( - \left ( 1 + \frac { 1 } { ( \alpha - 1 ) t } \right ) ^ { \frac { 1 } { \alpha - 1 } } \right ) \right ) } & t > 0 , \end{cases} \end{align*}


\begin{align*} \frac { \big ( \tilde { a } ( z _ n ) - ( n - \Sigma _ { \mathrm { k } } ) \big ) ^ 2 } { 2 \tilde { b } ( z _ n ) } \sim \frac { \Sigma _ { \mathrm { k } } ^ 2 } { A _ 2 ( z _ n ) } = o ( 1 ) . \end{align*}


\begin{align*} ( \mathsf { D } _ { d } ^ { i - 1 } R _ { T , \left \lfloor T r \right \rfloor - n } ^ { ( 1 ) } ( d ) ) ^ { j } = \left ( - \psi ^ { ( i - 1 ) } ( d ) \right ) ^ { j } + o ( 1 ) + u _ { i , n } i \geq 2 . \end{align*}


\begin{align*} x y _ g = y _ g \sigma _ g ( x ) x \in D , g \in \Gamma . \end{align*}


\begin{align*} Z _ { 2 n } = \widetilde { Z } _ n \to 0 n \to \infty \end{align*}


\begin{align*} j ^ \infty _ { M , \epsilon } = j _ M ^ * ( 1 + c _ \epsilon ^ * ) ^ 3 = j _ M ^ * + 3 j _ M ^ * c _ 1 \epsilon + O ( \epsilon ^ 2 ) . \end{align*}


\begin{align*} x k _ x = ( y _ x , 0 ) \ , , \end{align*}


\begin{align*} H = C _ 0 + C _ 1 \partial _ x + C _ 2 \partial _ y \end{align*}


\begin{align*} \mathbf { \Pi } _ { N } ( | C | = N ) & \leq \mathbf { \Pi } _ { N } \left ( \frac { | C | } { N } \in ( 1 - \epsilon , 1 + \epsilon ) \right ) \\ & \leq N \mathbf { \Pi } _ { N } ( | C | = N ) . \end{align*}


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


\begin{align*} Z _ { m , \theta } ( \alpha , \beta , \gamma ) = \frac { 1 } { 2 } \sum _ { \theta \in \{ 0 , 1 \} ^ 2 } \prod _ { j \in \mathbb { T } _ { m _ 1 , m _ 2 } } \left | \alpha + \beta \exp \left \{ 2 \pi i \frac { j _ 1 + \theta _ 1 / 2 } { m _ 1 } \right \} + \gamma \exp \left \{ 2 \pi i \frac { j _ 2 + \theta _ 2 / 2 } { m _ 2 } \right \} \right | . \end{align*}


\begin{align*} k \eta \circ \nu \circ \theta ( \omega _ k ) = \tilde { \mu } ( \omega _ k ) , \end{align*}


\begin{align*} U _ { n - 1 } U _ { n } y & = \left ( A _ { n - 1 } \partial _ { x } - B _ { n - 1 } \right ) \left ( A _ { n } y ^ { \prime } - B _ { n } y \right ) \\ & = A _ { n - 1 } \left ( \partial _ { x } A _ { n } y ^ { \prime } + A _ { n } y ^ { \prime \prime } - \partial _ { x } B _ { n } y - B _ { n } y ^ { \prime } \right ) - B _ { n - 1 } \left ( A _ { n } y ^ { \prime } - B _ { n } y \right ) , \end{align*}


\begin{align*} g _ 1 ( y ) = g _ 2 ( y ) . \end{align*}


\begin{align*} b _ { i j } = h ^ { i k } g _ { j k } = h _ { i j } = D _ { i j } ^ 2 u + u \delta _ { i j } . \end{align*}


\begin{align*} \varphi _ m \left ( k _ \theta h _ a n _ r \right ) = \rho ( h _ a ) ^ { - ( \lambda + 1 ) } e ^ { i m \theta } \ , , \end{align*}


\begin{align*} q _ { \sigma _ { 1 } \sigma _ { 2 } } \left ( x , t \right ) & = q \left ( x , t \right ) - 2 \partial _ { x } ^ { 2 } \log \det \left \{ I + \mathbf { K } _ { 1 2 } \left ( x , t \right ) \right \} \\ & = q \left ( x , t \right ) - 2 \partial _ { x } ^ { 2 } \log \det \left \{ I + \mathbf { K } _ { 2 1 } \left ( x , t \right ) \right \} = q _ { \sigma _ { 2 } \sigma _ { 1 } } \left ( x , t \right ) . \end{align*}


\begin{align*} u ( t ) & = U ( t , s ) P _ 0 ( s ) u ( s ) + U ( t , s ) ( I - P _ 0 ( s ) ) u ( s ) + j ^ { - 1 } \int _ s ^ t U ^ { \odot \star } ( t , \tau ) R _ { \delta , s } ( \tau , u ( \tau ) ) d \tau \\ & = U ( t , s ) P _ 0 ( s ) u ( s ) + ( \mathcal { K } _ s ^ \eta \tilde { R } _ { \delta , s } ( u ) ) ( t ) \end{align*}


\begin{align*} \overline { \mathbf { F } } _ { N } ( C ) = \frac { 1 } { | T ^ { d } | } \int _ { [ - T , T ] ^ { d } } \delta _ { \left ( x , \theta _ { N ^ { \frac { 1 } { d } } x } \cdot C \right ) } \ , d x , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { 4 m } x _ i ^ 2 = m \ell ( \ell + 1 ) . \end{align*}


\begin{align*} \gamma _ { c , a } \circ ( F \otimes G ) = ( G \otimes F ) \circ \gamma _ { d , b } \ ; . \end{align*}


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


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


\begin{align*} \det \limits _ { x , y \in \mathbb { T } _ { m _ 1 , m _ 2 } } K ^ { \alpha , \beta , \gamma } _ { m _ 1 , m _ 2 , \theta } ( x , y ) = \prod _ { j \in \mathbb { T } _ { m _ 1 , m _ 2 } } \left ( \alpha + \beta \exp \left \{ 2 \pi i \frac { j _ 1 + \theta _ 1 / 2 } { m _ 1 } \right \} + \gamma \exp \left \{ 2 \pi i \frac { j _ 2 + \theta _ 2 / 2 } { m _ 2 } \right \} \right ) . \end{align*}


\begin{align*} ( a + g , i ) = \alpha ( ( a , i ) ) . \end{align*}


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


\begin{align*} & \int _ { \Lambda _ L ^ * } | \hat { B } _ \# ( - u _ { s } ) | \left | \nu \left ( u _ { s } + u _ 0 + \sum _ { l \in \{ 1 , . . . , s - 1 \} \cap I _ A } \sigma _ { A , s } ( l ) u _ l \right ) - E - i \eta \right | ^ { - 1 } d u _ { s } \\ & = \int _ { \Lambda _ L ^ * } | \tilde { B } ( v ) | \left | \nu \left ( v \right ) - E - i \eta \right | ^ { - 1 } d v \\ & \leq C ( E , d , L , \eta , \tilde { B } ) = { C } ( E , d , L , \eta , \hat { B } _ \# ) , \end{align*}


\begin{align*} \lim _ { T \rightarrow + \infty } \sup _ { t \geq T } | ( \langle u \rangle - B ) ^ 2 _ { t } - ( \langle u \rangle - B ) ^ 2 _ { T } | \lesssim \lim _ { T \rightarrow + \infty } 2 \int _ { T } ^ { + \infty } \| u _ r \times \partial _ x ^ 2 u _ r \| _ { L ^ 2 } \dd r = 0 \ , . \end{align*}


\begin{align*} \forall m \in \N \limsup _ { s \to \infty } \frac { \Theta ( s ) } { s ^ m } = + \infty , \end{align*}


\begin{align*} \mathcal { E } \| \tilde { x } _ l \| ^ 2 = \mathcal { E } \| \phi _ { l , k } \tilde { x } _ k \| ^ 2 , \ l \ge k , \end{align*}


\begin{align*} L \left [ \phi \partial _ { x } p \right ] = L \left [ 2 x \left ( \phi - 1 \right ) p \right ] . \end{align*}


\begin{align*} w ^ { - ( h _ j - [ h _ i + \mathbf { 1 } _ { i \in \mathcal { B } } ] ) } = ( - 1 ) ^ { \theta _ 2 \mathrm { 1 } _ { i \in \mathcal { B } , h _ i = n - 1 } } w ^ { \mathrm { 1 } _ { i \in \mathcal { B } } + h _ i - h _ j } . \end{align*}


\begin{align*} \ell _ { R } \left ( R / { I ^ { [ p ^ { e } ] } } \right ) - \ell _ { A } \left ( A / { I ^ { [ p ^ { e } ] } } \right ) = \ell _ { R } \left ( N / ( N \cap I ^ { [ p ^ { e } ] } ) \right ) \end{align*}


\begin{align*} D _ P = : \sum _ { j = 0 } ^ { s _ P } \beta _ P ^ j W _ P ^ j . \end{align*}


\begin{align*} \mu _ { \theta } ( x ) = \frac { 1 } { z } \exp \left ( - \theta { V ( x ) } \right ) , \end{align*}


\begin{align*} \omega _ { 1 2 } = - \frac { 1 } { 3 } * d \log \kappa _ 1 , \end{align*}


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


\begin{align*} \sigma _ { A , j } : \{ 1 , \ldots , 2 n \} \to \{ 0 , 1 \} \sigma _ { A , j } ( l ) = \begin{cases} 1 : \max _ { a ( l ) } > j , \\ 0 : . \end{cases} \end{align*}


\begin{align*} 1 = \{ 0 \} \ ; . \end{align*}


\begin{align*} \mu = \frac { E } { 2 ( 1 + \nu ) } \qquad \textrm { a n d } \lambda = \frac { E \nu } { ( 1 + \nu ) ( 1 - 2 \nu ) } \ , . \end{align*}


\begin{align*} M = N \prod _ { \ell _ { 1 } \in \mathcal { R } _ 1 } \ell _ { 1 } ^ { \alpha ( \ell _ 1 ) } \prod _ { \ell _ { 2 } \in \mathcal { R } _ 2 } \ell _ { 2 } ^ { \alpha ( \ell _ 2 ) } 0 \leq \alpha ( \ell _ 1 ) \leq 1 0 \leq \alpha ( \ell _ 2 ) \leq 2 , \end{align*}


\begin{align*} \langle N _ { 1 , 1 } \psi , N _ { 2 , 2 } \psi \rangle _ { L ^ 2 ( \Gamma _ T ) } & = - \frac { 1 } { 2 } \int _ { \Gamma _ T } \partial _ t \left ( \left ( \frac { \tau } { 2 } - s \alpha \right ) ( \partial _ t \log \gamma ) \right ) \psi ^ 2 \ , \d S \ , \d t \end{align*}


\begin{align*} S \cdot A = U ( S \otimes A ) \ ; . \end{align*}


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


\begin{align*} A _ k A _ k ' = \prod _ { i \in I } A _ i ^ { [ b _ { i k } ] _ + } + \prod _ { i \in I } A _ i ^ { - [ b _ { i k } ] _ - } \end{align*}


\begin{align*} \begin{bmatrix} H _ 1 ( x _ { [ 0 , N - L ] } ) \\ H _ L ( u ) \end{bmatrix} = & \begin{bmatrix} x _ 0 & x _ 1 & \dots & x _ { N - L } \\ u _ 0 & u _ 1 & \dots & u _ { N - L } \\ u _ 1 & u _ 2 & \dots & u _ { N - L + 1 } \\ \vdots & \vdots & \ddots & \vdots \\ u _ { L - 1 } & u _ L & \dots & u _ { N - 1 } \end{bmatrix} . \end{align*}


\begin{align*} \sum \limits _ { i = 0 } ^ { m _ { q + 1 } - 1 } ( 5 \theta _ { q + 1 } ) ^ \kappa \lesssim m _ { q + 1 } \theta _ { q + 1 } ^ \kappa \leq \theta _ { q + 1 } ^ { \kappa - \eta } \to 0 , \ \ a s \ q \to \infty . \end{align*}


\begin{align*} t ( w ) = a + b + \frac { a - b } { \pi } \arg { w } - \frac { a } { \pi } \arg { ( - \varepsilon - w ) } + \frac { b } { \pi } \arg { ( \varepsilon + w ) } + P _ W , \end{align*}


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


\begin{align*} \left \{ \begin{alignedat} { 2 } - { \rm d i v } \ , A ( \nabla u ) & = f ( x , u ) + g ( x , \nabla u ) & & \mbox { i n } \ ; \ ; \R ^ N , \\ u & > 0 & & \mbox { i n } \ ; \ ; \R ^ N , \end{alignedat} \right . \end{align*}


\begin{align*} | I _ 7 | & = \left | \left \langle ( 1 - A ) B h \partial _ y \tilde { h } \partial _ y u , u \right \rangle _ { H ^ { 3 , 0 } } \right | \\ & \lesssim \left \| ( 1 - A ) B h ) \right \| _ { L ^ \infty } \| \partial _ y \tilde { h } \partial _ y u \| _ { H ^ { 3 , 0 } } \| u \| _ { H ^ { 3 , 0 } } + \| \partial _ y \tilde { h } \partial _ y u \| _ { L ^ \infty } \| ( 1 - A ) B h \| _ { H ^ { 3 , 0 } } \| u \| _ { H ^ { 3 , 0 } } . \end{align*}


\begin{align*} \| M \oplus N \| _ { \mathrm { o p } } = \mathop { m a x } ( \| M \| _ { \mathrm { o p } } , \| N \| _ { \mathrm { o p } } ) < 1 \ ; . \end{align*}


\begin{align*} U ( x , t ) = \begin{pmatrix} 0 & u ( x , t ) \\ - \sigma u ( - x , - t ) & 0 \end{pmatrix} , V ( x , t ) = \begin{pmatrix} V _ { 1 1 } & V _ { 1 2 } \\ V _ { 2 1 } & - V _ { 1 1 } \end{pmatrix} \end{align*}


\begin{align*} \omega _ 0 ( h ) = - x \partial _ x - y \partial _ y - 1 \ , , \end{align*}


\begin{align*} \left [ a \right ] _ \delta = \delta ^ { - 1 } \int _ { 0 } ^ { 1 } F _ { a } ^ { - 1 } \left ( 1 - s \right ) \exp \left ( - \delta ^ { - 1 } s \right ) d s \end{align*}


\begin{align*} E _ i \ , = \ , \{ 1 , \ldots , m _ i \} \end{align*}


\begin{align*} P _ { \omega , [ a , b ] } ( E ) = \begin{cases} \det ( H _ { \omega } \upharpoonright [ a , b ] - E ) , & a \leq b \\ 1 , & a > b \end{cases} \end{align*}


\begin{align*} f ( x ) = \sum _ { \ell } \sum _ { n = 0 } ^ { 2 \ell } \langle f ( | x | \omega ) , y _ { \ell , n } ( \omega ) \rangle _ { L ^ 2 _ \omega ( \mathbb S ^ 2 ) } y _ { \ell , n } \big ( \frac { x } { | x | } \big ) . \end{align*}


\begin{align*} p ( t , x , y ) = \frac 3 2 - \frac 1 2 h ^ 2 ( t , x , y ) > 0 . \end{align*}


\begin{align*} e ^ { - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \lfloor s m \rfloor } = ( - 1 ) ^ { \theta _ 1 \mathbf { 1 } _ { s < 0 } } e ^ { - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \lfloor \tilde { s } m \rfloor } . \end{align*}


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


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


\begin{align*} \Theta _ { \pi _ { 0 , i \lambda } } = \Theta _ { \pi _ { 0 , - i \lambda } } \qquad \Theta _ { \pi _ { 1 , i \lambda } } = \Theta _ { \pi _ { 1 , - i \lambda } } \ , . \end{align*}


\begin{align*} \varphi \left ( x , k \right ) = T \left ( k \right ) \psi _ { - } ( x , k ) . \end{align*}


\begin{align*} & \prod _ { j = 1 } ^ { n + 1 } \left | \left ( e ^ { - 2 i \vartheta } \left ( u _ 0 - q + \sum _ { l = 1 } ^ { j - 1 } [ M _ A v ] _ l \right ) ^ 2 - E - i \eta \right ) ^ { - 1 } \right | \\ & \leq C _ { E , \vartheta } ^ { n - 1 } g ( q , u _ 0 , v ) . \end{align*}


\begin{align*} - \int _ { Q _ T } \rho u \eta ( p ) \log ( 1 + \frac { 1 } { p } ) ^ { r } = \int _ { Q _ T } \rho \eta ( p ) \log ( 1 + \frac { 1 } { p } ) ^ { r } \frac { \partial _ t p - | \nabla p | ^ 2 } { p } = \int _ { Q _ T } - \rho ( \frac { d } { d t } h ( p ) - \nabla h ( p ) \cdot \nabla p ) \end{align*}


\begin{align*} \alpha ( t , f ) ( x ) : = f ( x - t ) \ , . \end{align*}


\begin{align*} + \frac { \epsilon } { 2 N } \stackrel [ i = 1 ] { N } { \sum } \mathrm { t r a c e } ( v _ { i } ^ { l } \vee x _ { i } ^ { l } R _ { N } ( t ) ) . \end{align*}


\begin{align*} A _ I = \begin{pmatrix} I _ 0 \\ I _ 1 \end{pmatrix} \ ; , A _ O = \begin{pmatrix} O _ 0 & O _ 1 \end{pmatrix} \ ; . \end{align*}


\begin{align*} S ( \boldsymbol { \ell } ) _ t = S _ 0 + \sum _ { | J | \leq n } \big ( \ell ^ J _ W \langle ( \epsilon _ { J } ; \epsilon _ 0 ) ^ \thicksim , \mathbb { X } _ { t } \rangle + \ell ^ J _ \nu \langle ( \epsilon _ { J } ; \epsilon _ { 1 } ) ^ \thicksim , \mathbb { X } _ { t } \rangle \big ) . \end{align*}


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


\begin{align*} V _ k ( x ) \geq G _ k \Big ( \rho \big ( \frac { 1 } { 3 } \| \phi ( 0 , x , 0 ) \| _ X \big ) \Big ) = G _ k \circ \rho \big ( \frac { 1 } { 3 } \| x \| _ X \big ) . \end{align*}


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


\begin{align*} \Lambda _ { + } ( d ) X _ { t } = - \alpha \beta ^ { \prime } ( \Delta _ { + } ^ { - b } - 1 ) \Lambda _ { + } ( d ) X _ { t } + \varepsilon _ { t } , t = 1 , \dots , T , \end{align*}


\begin{align*} \begin{aligned} 2 \int _ { r > r _ { 0 } , 0 < \theta < \pi } \eta | \nabla \omega | ^ { 2 } d x d y = & \int _ { r > r _ { 0 } , 0 < \theta < \pi } \omega ^ { 2 } ( \Delta \eta + \boldsymbol { w } \cdot \nabla \eta ) d x d y \\ & + \int _ { r > r _ { 0 } , 0 < \theta < \pi } \nabla \eta \cdot ( 0 , b ) \omega ^ { 2 } d x d y . \end{aligned} \end{align*}


\begin{align*} \Delta \log ( 1 - K ) = 6 K - 1 . \end{align*}


\begin{align*} Q ^ 1 ( \exp ( \pi ) ) = \sum _ { n \geq 0 } \frac { 1 } { n ! } Q _ n ^ 1 ( \pi \vee \cdots \vee \pi ) = 0 . \end{align*}


\begin{align*} \big \| g _ i \big \| _ { k , L ^ 2 ( Y , L ^ p \otimes F ) } = \Big ( p ^ { \frac { n - m + k } { 2 } } \cdot \sqrt { k ! \cdot ( 2 \pi ) ^ k } + O ( p ^ { \frac { n - m + k - 2 } { 2 } } ) \Big ) \cdot \big \| B _ { i , p } ^ { \perp } f \big \| _ { L ^ 2 ( X , L ^ p \otimes F ) } . \end{align*}


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


\begin{gather*} B ( \Phi ( x ) , \Phi ( v ) ) = B ( x - \tau ( x ) - \frac { 1 } { 2 } B _ { \mathfrak a } \big ( \tau ( x ) , \tau ( \cdot ) \big ) , v + B _ { \mathfrak a } ( v , \tau ( \cdot ) ) ) \\ = - B _ { \mathfrak a } ( \tau ( x ) , v ) + B _ { \mathfrak a } ( v , \tau ( x ) ) = 0 \end{gather*}


\begin{align*} \int _ 0 ^ t T ^ { x ' } f ( d r , \cdot ) - \int _ 0 ^ t T ^ { x } f ( d r , \cdot ) = & \int _ 0 ^ t T ^ { x + \theta ' - \theta } f ( d r , \cdot ) - \int _ 0 ^ t T ^ { x } f ( d r , 0 ) \\ = & \int _ 0 ^ t T ^ { x } f ( d r , \theta ' _ r - \theta _ r ) - \int _ 0 ^ t T ^ { x } f ( d r , 0 ) \end{align*}


\begin{align*} S : = \{ Q = 0 \} \cap \{ R = 0 \} \subset \mathbb P ^ 3 \end{align*}


\begin{align*} C = \sum _ { i \in I } C _ { i } , \end{align*}


\begin{align*} R _ { k } = 2 ^ { m } . \end{align*}


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


\begin{align*} \delta _ { d a c e } = \begin{cases} 1 & d = a = c = e \\ 0 & \end{cases} \ ; . \end{align*}


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


\begin{align*} \int _ { \Omega } M ( t , \omega ) I _ { A } ( \omega ) I _ C ( \alpha ( \omega ) ) d P = \int _ { \Omega } M ( s , \omega ) I _ { A } ( \omega ) I _ C ( \alpha ( \omega ) ) d P . \end{align*}


\begin{align*} \let \veqno \eqno \forall m \in \omega _ M ^ { \vee } \cap M , \ \forall \gamma _ 1 , \gamma _ 2 \in \Gamma , \ \ \ \ \ \ \ \ h _ { \gamma _ 1 } ( m ) { \sigma _ Y } _ { \gamma _ 1 } ^ { \sharp } \left ( h _ { \gamma _ 2 } \left ( \tilde { \tau } ^ { - 1 } _ { \gamma _ 1 } ( m ) \right ) \right ) = h _ { \gamma _ 1 \gamma _ 2 } ( m ) . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{align*}


\begin{align*} \pi _ 2 ^ { * } H ^ 2 ( \tilde Z , \mathbb Z ) = \Omega _ 2 ^ { \perp _ { H ^ 2 ( X , \mathbb Z ) } } = \Omega _ 2 ^ { \perp _ { \Lambda _ { \mathrm { K 3 } } } } , \\ \widehat { \pi _ 2 } ^ { * } H ^ 2 ( \tilde Y , \mathbb Z ) = \Omega _ 2 ^ { \perp _ { H ^ 2 ( \tilde Z , \mathbb Z ) } } = \Omega _ 2 ^ { \perp _ { \Lambda _ { \mathrm { K 3 } } } } . \end{align*}


\begin{align*} & \prod _ { j = 1 } ^ d \langle v _ { 1 , j } \rangle ^ { - 1 + \epsilon } \left | \nu ( q + v _ 1 ) - E \pm i \eta \right | ^ { - 1 } \leq c _ { { \rm ( I I I ) } , d } \langle q \rangle ^ { - 2 } . \end{align*}


\begin{align*} = 2 \pi \sum \limits _ { k = 1 } ^ { \infty } \int \limits _ { \frac { 1 } { k + 1 } } ^ { \frac { 1 } { k } - 2 ^ { - 4 k - 1 } } \ , r \ , d r + 2 \pi \sum \limits _ { k = 1 } ^ { \infty } \int \limits _ { \frac { 1 } { k } - 2 ^ { - 4 k - 1 } } ^ { \frac { 1 } { k } } r \cdot 2 ^ { k - 1 } \ , d r \leqslant \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty B ( n ) \frac { x ^ n } { n ! } \frac { d } { d a } ( a ) _ n \end{align*}


\begin{align*} c _ i \mapsto \begin{cases} c _ i - 1 , & ; \\ b _ { i - 1 } , & , \end{cases} d _ j \mapsto \begin{cases} d _ j - 1 , & ; \\ a _ j , & . \end{cases} \end{align*}


\begin{align*} k _ { n } ( t , x , y ) & = \int _ 0 ^ t \int _ E k ( s , x , z ) q ( z ) k _ { n - 1 } ( t - s , x , y ) \mu ( \d z ) \d s , n \ge 1 , \end{align*}


\begin{align*} p ^ { ( \eta ) } _ j ( s ) = \sum _ { \substack { 0 \leq k \leq 3 j + 2 \eta \\ k + j } } c ^ { ( j , \eta ) } _ k s ^ k \end{align*}


\begin{align*} l _ { j i } ^ { + } ( u ) \mid 0 \rangle = 0 f o r \ a l l j > i \end{align*}


\begin{align*} = \frac { n ^ { 1 / 2 } } { m ^ 2 } \cdot \int \limits _ { 1 / R < | y | < 1 } \frac { \Vert f ^ { \ , \prime } ( y ) \Vert } { | y | ^ { 2 n } } \ , d m ( y ) \leqslant \frac { n ^ { 1 / 2 } R ^ { \ , 2 n } } { m ^ 2 } \cdot \int \limits _ { 1 / R < | y | < 1 } | \nabla f ( y ) | \ , d m ( y ) < \infty \ , . \end{align*}


\begin{align*} \sum _ { a } ( i _ a - j _ a ) + \sum _ { b } ( i _ b - j _ b ) = 0 , \end{align*}


\begin{align*} \mathrm { i d } = \mathrm { D } \Phi ( \Phi ^ { - 1 } ( \tau ) ) \cdot A ( \tau ) , A ( \tau ) = [ \mathrm { D } \Phi ( \Phi ^ { - 1 } ( \tau ) ) ] ^ { - 1 } . \end{align*}


\begin{align*} \rho = ( z _ 1 + d z ^ { 1 2 } ) \wedge e ^ { i \omega } . \end{align*}


\begin{align*} T ( i y _ 0 ) = \frac { 4 \pi ^ 3 } { \ell ^ 2 \cos ^ 2 \left ( \frac { \pi y _ 0 } { \ell } \right ) } , \end{align*}


\begin{align*} G _ \delta ( \alpha ) = \oint _ { S ^ 1 } \frac { \mathrm { d } w } { 2 \pi i w } \frac { w ^ { - \delta } } { 1 - 1 / w + \alpha } = \begin{cases} \frac { 1 } { 1 + \alpha } \mathbf { 1 } _ { | 1 + \alpha | > 0 } & , \\ - \mathbf { 1 } _ { | 1 + \alpha | < 0 } & . \end{cases} \end{align*}


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


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


\begin{align*} \sum _ { n = k _ { t o t } } ^ \infty & \frac { \Gamma ( - p - x + 2 m - n - 1 ) \Gamma ( q - x ) } { \Gamma ( - p - x + 2 m ) \Gamma ( q - x - n + k _ { t o t } ) } \frac { \Gamma ( n + 1 ) } { \Gamma ( n - k _ { t o t } + 1 ) } = - ( - 1 ) ^ { k _ { t o t } } k _ { t o t } ! \frac { \Gamma ( p + q - 2 m ) } { \Gamma ( p + q - 2 m + k _ { t o t } + 1 ) } \end{align*}


\begin{align*} \langle R ^ G _ { T , \chi } \otimes \omega _ \psi ^ \vee , R ^ G _ { S , \eta } \rangle _ { G ^ F } = \begin{cases} 0 , & \textrm { i f } \chi , \eta \textrm { i n t e r t w i n e , } \\ 1 , & \textrm { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} R _ { k + n } = c _ 1 R _ { k + n - 1 } + \cdots + c _ n R _ k ( k \geq 0 ) , \end{align*}


\begin{align*} \theta ( 0 , b ) : = \frac { \pi } { 2 } , \theta ( x , b ) : = \arctan \big ( \frac { h ' ( x , b ) } { h ( x , b ) } \big ) \end{align*}


\begin{align*} F ( x ) = \left \{ \begin{array} { r r } f ( x ) , & | x | < 1 \ , , \\ \psi ( f ( \psi ( x ) ) ) , & | x | \geqslant 1 \ , . \end{array} \right . \end{align*}


\begin{align*} \rho \ , \overline { m } _ j = \begin{cases} \lambda ^ { - 1 } \overline { m } _ { j + 1 } , & \ , j = 1 , \ldots , d - 2 ; \\ - \lambda ^ { - 1 } \sum _ { k = 1 } ^ { d - 1 } ( - q ) ^ { - k } \overline { m } _ { d - k } , & \ , j = d - 1 , \end{cases} \end{align*}


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


\begin{align*} 0 = \partial _ { \eta } ( \partial _ \xi w + \partial _ \tau w ) = v \partial _ { \eta } g _ { s u m } + g _ { s u m } \partial _ { \eta } v = v \partial _ { \eta } g _ { s u m } - \alpha g _ { s u m } o n \eta = 0 , \end{align*}


\begin{align*} \pi _ 2 ^ * f _ Y = \lambda ^ { \phi - 1 } \cdot \pi _ 1 ^ * f _ Y . \end{align*}


\begin{align*} \left | \sum _ { j = m _ 0 } ^ { m / 2 } a _ { j , m } b ^ j \right | \leq C ( s ) \left ( | a _ { m / 2 } , m | + \left | \sum _ { \ell = m _ 0 } ^ { m / 2 } c _ { \ell , m } \right | \right ) \leq C _ 1 / m _ 0 \end{align*}


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


\begin{align*} \frac { \delta _ 1 + \delta _ 2 } { 2 } + ( \delta _ 1 - \delta _ 2 ) B = \delta _ 1 \biggl ( \frac { 1 } { 2 } + B \biggr ) + \delta _ 2 \biggl ( \frac { 1 } { 2 } - B \biggr ) , \end{align*}


\begin{align*} g \left ( ( D ^ g _ X J ) Y , Z \right ) = 2 g \left ( J X , N ( Y , Z ) \right ) , \end{align*}


\begin{align*} \begin{aligned} & d ^ { 2 ^ h + 1 } = 1 & & \mbox { i f } h \mbox { i s o d d , } \\ & d ^ { 2 ^ { n + h } + 1 } = 1 & & \mbox { i f } h \mbox { i s e v e n . } \end{aligned} \end{align*}


\begin{align*} T ( k ) \psi _ { - } ( x , k ) = \overline { \psi _ { + } ( x , k ) } + R ( k ) \psi _ { + } ( x , k ) , \ ; k \in \mathbb { R } , \end{align*}


\begin{align*} g = \frac { \partial _ \tau w } { v } . \end{align*}


\begin{align*} P _ { V } ( x ) : = \{ t \in G \ , : \ , ( \alpha ( t , x ) , x ) \in V \} \end{align*}


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


\begin{align*} \hat \omega ( l ) = \exp \{ c _ { l 1 } \} = \exp \{ c _ { l 2 } \} , \ \ l \in L . \end{align*}


\begin{align*} K _ { \mathbb { D } ^ \ast , p } ( z ) \geq { K _ { \mathbb { D } , p } ( z ) } = \frac { 1 } { \pi ( 1 - | z | ^ 2 ) ^ 2 } , \ \ \ \forall \ , z \in \mathbb { D } , \end{align*}


\begin{align*} \tilde H _ 1 ( s ) : = \partial _ \nu ( h ( d _ { H } ( \eta ( s ) , O ) ) ) , \end{align*}


\begin{align*} E _ { f , p } ( A ) : = \begin{cases} \Vert D f \Vert ( A ) \quad \textrm { w h e n } p = 1 \\ \int _ { A } g _ f ^ p \ , d \mu \quad \textrm { w h e n } 1 < p < \infty . \end{cases} \end{align*}


\begin{align*} K _ { r } ^ { \perp } = 2 \kappa _ { r } \mu _ { r } . \end{align*}


\begin{align*} \frac { d } { d x } \sum _ { n = 0 } ^ \infty B ( n ) \frac { x ^ n } { n ! } = \sum _ { n = 1 } ^ \infty B ( n ) \frac { x ^ { n - 1 } } { ( n - 1 ) ! } = \sum _ { n = 0 } ^ \infty B ( n + 1 ) \frac { x ^ n } { n ! } \end{align*}


\begin{align*} \mathcal { H } ( a + \mathbf { i } b ) = \begin{pmatrix} a & b \\ - b & a \end{pmatrix} \ ; . \end{align*}


\begin{align*} F _ N ( p , q ) : = \frac { 1 } { 2 } \sum _ { - N \leq k \leq N } \log \left ( 1 + \left ( \frac { p + k } { q } \right ) ^ 2 \right ) - \frac { 1 } { 2 } \int _ { - N - 1 / 2 } ^ { N + 1 / 2 } \log \left ( 1 + \left ( \frac { p + k + \phi } { q } \right ) ^ 2 \right ) . \mathrm { d } \phi \end{align*}


\begin{align*} G _ { \mathcal R } ( x , y ) = G ^ { ( 1 ) } _ { \mathcal R } ( x , y ) + G ^ { ( 2 ) } _ { \mathcal R } ( x , y ) , \end{align*}


\begin{align*} x _ t = x _ 0 - \omega _ t + \int _ a ^ t f ( s , x _ s ) \ , d s , t \in [ a , b ] , \end{align*}


\begin{align*} \Gamma ( S , ( g _ n ) _ * F ) & = \varinjlim _ { V \supset S } \Gamma ( g _ n ^ { - 1 } ( V ) , F ) , & \Gamma ( g _ n ^ { - 1 } ( S ) , F ) & = \varinjlim _ { V ' \supset g _ n ^ { - 1 } ( S ) } \Gamma ( V ' , F ) , \end{align*}


\begin{align*} \begin{tabular} [ c ] { l } $ x ^ { 3 } P _ { n } = P _ { n + 3 } + \left ( \gamma _ { n + 2 } + \gamma _ { n + 1 } + \gamma _ { n } \right ) P _ { n + 1 } $ \\ $ + \gamma _ { n } \left ( \gamma _ { n + 1 } + \gamma _ { n } + \gamma _ { n - 1 } \right ) P _ { n - 1 } + \gamma _ { n } \gamma _ { n - 1 } \gamma _ { n - 2 } P _ { n - 3 } , $ \end{tabular} \end{align*}


\begin{align*} x ( [ 2 ] P ) = \left ( \frac { x ^ { 2 } + n ^ { 2 } } { 2 y } \right ) ^ { 2 } , \quad \forall P = ( x , y ) \in E _ { n } \end{align*}


\begin{align*} n ( x , y ) & : = \phi ( x ) \overline { \phi ( y ) } + \frac { 1 } { N } n ^ \mathrm { p a i r } ( x , y ) , ( \upsilon _ N n ) ( x , y ) : = \upsilon _ N ( x - y ) n ( x , y ) \\ m ( x , y ) & : = \phi ( x ) \phi ( y ) + \frac { 1 } { N } m ^ \mathrm { p a i r } ( x , y ) , ( \upsilon _ N m ) ( x , y ) : = \upsilon _ N ( x - y ) m ( x , y ) \\ \rho ( x ) & : = n ( x , x ) , \rho ^ \mathrm { p a i r } ( x ) : = n ^ \mathrm { p a i r } ( x , x ) . \end{align*}


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


\begin{align*} H ^ = \kappa _ \gamma ^ + ( n - 2 ) \kappa ^ ( C _ t ) . \end{align*}


\begin{align*} ( y _ 0 , z _ 0 , \dots , z _ M ) ( f ( y _ 0 ) = g ( z _ 0 ) = \dots = g ( z _ M ) ) . \end{align*}


\begin{align*} \mathfrak I ^ \theta [ F ] = \int _ 0 ^ t e ^ { - \theta i ( t - t ' ) | \nabla | } F ( t ' ) \ , d t ' . \end{align*}


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


\begin{align*} ( \tilde { \mathcal { T } } _ { \mathfrak { s } } z ) ( t ) = ( \mathcal { T } _ { \mathfrak { s } } z ) ( t ) - \int _ 0 ^ t S ( t - s ) ( i \nu z ( s ) + \epsilon ( \gamma z ( s ) - \mu \overline { z } ( s ) ) ) \d s , \end{align*}


\begin{align*} L : = \mathcal L ^ 1 ( A ) = \lim _ { i \to \infty } L _ i = 1 / 2 . \end{align*}


\begin{align*} m = 2 ^ v w + 1 . \end{align*}


\begin{align*} P \{ \sup \limits _ { C } \left | f _ k - ( f _ k ^ \nu \circ \mathcal P ^ \nu ) \right | \ge \delta \} < \delta , \ k = 1 , . . . , m , \end{align*}


\begin{align*} \bar B = \dfrac { \left \langle A ^ { 1 / 2 } \dfrac { \zeta } { | \zeta | } , A ^ { 1 / 2 } \dfrac { e } { | e | } \right \rangle } { \left | A ^ { 1 / 2 } \dfrac { e } { | e | } \right | ^ 2 } , \bar C = \dfrac { \left | A ^ { 1 / 2 } \dfrac { \zeta } { | \zeta | } \right | ^ 2 } { \left | A ^ { 1 / 2 } \dfrac { e } { | e | } \right | ^ 2 } . \end{align*}


\begin{align*} J ( v ) \lesssim \iint _ \Omega | \nabla ( e ^ { - | t | } \Delta _ i ^ h g ) | ^ 2 \frac { d t d x } { | t | ^ { n - d - 1 } } + \iint _ { \Omega } | \nabla u ( x + h e _ i , t ) | ^ 2 \frac { d t d x } { | t | ^ { n - d - 1 } } : = J _ 3 + J _ 4 . \end{align*}


\begin{align*} \Pi = \epsilon + \sum _ { \alpha = 1 } ^ { 3 N - 6 } \wedge _ { \alpha } ( x ) \dot { q } ^ { \alpha } \end{align*}


\begin{align*} r & = n + m \geq 0 , \\ s & = n - m \geq 0 , \\ n & = \frac { r + s } { 2 } , & m = \frac { r - s } { 2 } , \end{align*}


\begin{align*} \delta \Phi ( 0 , t _ 2 , x ) = \partial _ { t _ 2 } \delta \Phi ( 0 , t _ 2 , x ) = 0 \end{align*}


\begin{align*} \| x - \nu \| _ { q } = \left \lceil M + 1 + 1 + \frac { | \Lambda | \log S + \log 4 } { \min \gamma - 2 \varepsilon _ { 1 } } \right \rceil . \end{align*}


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


\begin{align*} \liminf _ { \kappa \to 0 } \frac { \mathcal { E } _ \kappa ( m _ \kappa ) - 8 \pi } { \kappa ^ 2 } = \lim _ { \kappa \to 0 } \frac { \mathcal { E } _ \kappa ( m _ \kappa ) - 8 \pi } { \kappa ^ 2 } , \end{align*}


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


\begin{align*} D _ \mathcal { G } ( T u , T w ) & = D _ { \mathcal { G } } ( ( u , F ( u ) ) , ( w , F ( w ) ) ) \\ & = \lvert u - w \rvert + H _ d ( F u , F w ) \\ & \leq \lvert u - w \rvert + l \lvert u - w \rvert \\ & \leq ( 1 + l ) \lvert u - w \rvert , \\ D _ { \mathcal { G } } ( T u , T w ) & \leq ( 1 + l ) \lvert u - w \rvert \end{align*}


\begin{align*} \partial _ { t } u + u \cdot \nabla u = - \nabla P , \ \mathrm { d i v } ( u ) = 0 . \end{align*}


\begin{align*} 0 = \partial _ { \eta } ( \partial _ \tau w ) = v \partial _ { \eta } g + g \partial _ { \eta } v = v \partial _ { \eta } g - \alpha g o n \eta = 0 , \end{align*}


\begin{align*} \bigl | K _ { \theta _ \ell , \theta _ { 3 - \ell } } \bigl ( ( y , \omega ) , ( z , \eta ) \bigr ) \bigr | = \sqrt { \ ! \frac { w ( \Phi ( \eta ) ) } { w ( \Phi ( \omega ) ) } } \cdot L ^ { ( \ell ) } _ { \Phi ( \eta ) } \bigl ( A ^ T ( \Phi ( \eta ) ) \langle z - y \rangle , \Phi ( \omega ) - \Phi ( \eta ) \bigr ) . \end{align*}


\begin{align*} = \left \{ \frac { f ( x , y , t ) } { s ( x , y ) u ( x , y , t ) } \ , \middle | \ , f ( x , y , t ) \in k [ x , y , t ] , s ( x , y ) \in S , u ( x , y , t ) \in U _ { k [ x , y ] [ t ] / k [ x , y ] } \right \} . \end{align*}


\begin{align*} \lim _ { m \rightarrow \infty } \sum _ { 0 \leq n < m } \phi ( n ) ^ 2 = S _ \phi \end{align*}


\begin{align*} \overline { ( a | b ) } = ( \phi ( a ) | \phi ( b ) ) \end{align*}


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


\begin{align*} \| \nabla \varphi _ m \| _ { L ^ { \infty } _ y } + \| \varphi _ m \| _ { \mathcal M ^ { 2 - 1 / p , p } ( \R ^ 2 ) } \leq \delta , \varphi _ m ( 0 ) = 0 , \nabla \varphi _ m ( 0 ) = 0 , \end{align*}


\begin{align*} [ e _ i ( z ) , f _ j ( w ) ] = \delta _ { i j } \delta \left ( \frac { z } { w } \right ) \left ( \psi ^ + _ i ( z ) - \psi ^ - _ i ( z ) \right ) \ , , \end{align*}


\begin{align*} ( i s ^ * \rho _ 1 + j s ^ * \overline { \omega _ 3 } ) & = | Z | ^ { - 2 } ( \overline { h _ 1 } \mathrm { d } h _ 1 + \overline { h _ 2 } \mathrm { d } h _ 2 ) + R \\ \frac { 1 } { \sqrt { 2 } a | Z | } ( s ^ * \omega _ 1 + j s ^ * \omega _ 2 ) & = - h _ 2 h _ 1 ^ { - 1 } \mathrm { d } h _ 1 + \mathrm { d } h _ 2 , \end{align*}


\begin{align*} A ^ + _ M = A _ M + ( U \oplus \mathbb { 0 } ) \end{align*}


\begin{align*} \sup _ { | z | = R _ k } \log | W ( z ) | < \frac { R _ k ^ 2 } 2 - \frac { a } 8 R _ k \log ^ { b } R _ k . \end{align*}


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


\begin{align*} \alpha = \frac { 2 - t + \sqrt { t ^ 2 + 4 } } { 2 } . \end{align*}


\begin{align*} \min _ { \emptyset \neq I \subseteq \{ 1 , \ldots , n \} } \frac { \dim H - \dim H _ { V _ I } } { \dim V _ I } = \min _ { \emptyset \neq I \subseteq \{ 1 , \ldots , n \} } \frac { \dim G _ { A _ I , \ell } } { 2 \dim A _ I } . \end{align*}


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


\begin{align*} \kappa _ \eta = \frac { 1 - \abs { \gamma ( t ) } ^ 2 } { 2 } \kappa ^ { } _ \eta + g _ { } ( \eta , \tilde \nu ) , \end{align*}


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


\begin{align*} \textit { \textsf { p r i n } } _ { \mathcal { I } _ U } \ , : \ , \tilde { M } = M _ r \xrightarrow { \pi _ r } \ldots \rightarrow M _ 2 \xrightarrow { \pi _ 2 } M _ 1 \xrightarrow { \pi _ 1 } M _ 0 = M . \end{align*}


\begin{align*} a ^ * ( k ) a ( k ) = \sum _ i a _ i ^ * ( k ) a _ i ( k ) \leq 1 , \ \ b ^ * ( k ) b ( k ) = \sum _ i b _ i ^ * ( k ) b _ i ( k ) \leq 1 . \end{align*}


\begin{align*} c _ { g , \epsilon } = \max _ { y \in \mathbb { T } ^ { d } } \left | g ^ { \epsilon } ( y ) \right | < \infty . \end{align*}


\begin{align*} \operatorname * { t r } \mathbb { G } \left ( s \right ) = \left \vert \left \vert \mathbb { G } \left ( s \right ) \right \vert \right \vert _ { 2 } = \left \vert \left \vert g \left ( \cdot , s \right ) \right \vert \right \vert ^ { 2 } = \int g \left ( x , s \right ) ^ { 2 } \mathrm { d } \mu \left ( x \right ) . \end{align*}


\begin{align*} \widetilde { B _ s } = \left ( \begin{matrix} \epsilon ^ { T } _ { { \rm { u f } } ; s } & A \\ \epsilon ^ { T } _ { { \rm f } ; s } & \ast \end{matrix} \right ) \end{align*}


\begin{align*} V _ \Gamma ( \Phi ( Y , t ) , t ) = \partial _ t \Phi ( Y , t ) \cdot \nu ( \Phi ( Y , t ) , t ) , ( Y , t ) \in \Gamma _ 0 \times [ 0 , T ] . \end{align*}


\begin{align*} \Theta ( q ^ n x ; q ) = ( - 1 ) ^ { n } q ^ { - \binom { n } { 2 } } x ^ { - n } \Theta ( x ; q ) , \end{align*}


\begin{align*} \Delta ( d ^ 2 \bar { \eta } ) = 2 | \bar { \nu } | ^ 2 \bar { \eta } + 2 d \{ ( \mathrm { d i v } \ , \bar { \nu } ) \bar { \eta } + 2 \bar { \nu } \cdot \nabla \bar { \eta } \} + d ^ 2 \Delta \bar { \eta } = 2 \bar { \eta } + 2 d ( \mathrm { d i v } \ , \bar { \nu } ) \bar { \eta } + d ^ 2 \Delta \bar { \eta } \end{align*}


\begin{align*} d = \begin{cases} 4 & { \rm i f } ~ m = 4 , ~ { \rm a n d } ~ e = q + 1 \\ 3 & { \rm o t h e r w i s e } . \end{cases} \end{align*}


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


\begin{align*} \mathcal { E } \| \tilde { x } _ l \| ^ 2 = \mathcal { E } \| \psi _ { l , k } \tilde { x } _ k \| ^ 2 , \ l \ge k , \end{align*}


\begin{align*} \partial _ { x } ^ { 2 } \log \det \left [ I + \mathbb { A } \left ( x \right ) \right ] & = - \left \langle a \left ( x \right ) , \left [ I + \mathbb { A } \left ( x \right ) \right ] ^ { - 1 } a \left ( x \right ) \right \rangle ^ { 2 } \\ & - 2 \left \langle \partial _ { x } a \left ( x \right ) , \left [ I + \mathbb { A } \left ( x \right ) \right ] ^ { - 1 } a \left ( x \right ) \right \rangle . \end{align*}


\begin{align*} Q _ { \lambda , \mu } & = - d \mu _ 1 x ^ d - \cdots - d \mu _ n x ^ d - d \lambda x _ 1 ^ { w _ 1 } \cdots x _ n ^ { w _ n } , \\ \omega _ \mu & = \frac { x _ 1 ^ { \alpha _ 1 } \cdots x _ n ^ { \alpha _ n } } { Q _ { \lambda , \mu } } \Omega . \end{align*}


\begin{align*} p ( z , t ) = \frac { 1 - 2 t ^ 2 S h ( z + t ) } { 1 + 2 t ^ 2 S h ( z + t ) } . \end{align*}


\begin{align*} \frac { 1 - \nu ^ 2 } { E } \Delta ^ 2 v = - \theta \qquad \end{align*}


\begin{align*} m ( z ^ * ) & : = \frac { 1 } { | \Omega | } \langle z ^ * , 1 \rangle _ { V ^ * , V } { \rm f o r ~ } z ^ * \in V ^ * , \\ m _ \Gamma ( z _ \Gamma ^ * ) & : = \frac { 1 } { | \Gamma | } \langle z _ \Gamma ^ * , 1 \rangle _ { V _ \Gamma ^ * , V _ \Gamma } { \rm f o r ~ } z _ \Gamma ^ * \in V _ \Gamma ^ * , \end{align*}


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


\begin{align*} E ^ 2 _ { p q } = H _ p ( K _ q ( G \ltimes P _ \bullet ) ) \Rightarrow K _ { p + q } ( G \ltimes A ) . \end{align*}


\begin{align*} \alpha : = \frac { 1 } { | D | } \int _ { D } \left [ ( u ^ 0 - B ^ 0 ) \cdot ( u ^ 0 - B ^ 0 ) + \int _ { 0 } ^ { + \infty } b ( u _ r ) \cdot ( u _ r - B _ r ) \dd r \right ] \ , . \end{align*}


\begin{align*} f ^ { M + 1 } _ i v _ { \nu , \ell _ 0 } = 0 \ l _ 0 \gg 0 , \end{align*}


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


\begin{align*} G _ s = \prod _ { [ a ] \in \Lambda _ s / \Gamma } G _ { s , [ a ] } , \end{align*}


\begin{align*} \sum _ { ( d , m ) = 1 } \mu ^ 2 ( 2 d ) \Psi ( d / D ) d ^ { - \upsilon } \sim \frac { 1 } { 2 \zeta ^ { [ 2 ] } ( 2 ) } a _ m \breve { \Psi } ( 1 - \upsilon ) D ^ { 1 - \upsilon } , \end{align*}


\begin{align*} \| f _ R \| _ \infty \le \| f _ Q \| _ \infty = | Q | ^ { - 1 } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s m _ i n _ i = 0 . \end{align*}


\begin{align*} \mathcal { K } _ { n , m } ^ { E P } [ A , D ] = \sum _ { \alpha } Z _ N ^ { \alpha } \cdot \mathcal { K } _ { m , m } [ A ^ { \alpha } , D ] . \end{align*}


\begin{align*} \lim _ { n \to \infty } \lambda _ k ( \Omega _ n , \mu _ n ) = \lambda _ k ( \Omega , \mu ) . \end{align*}


\begin{align*} \lambda : = m a x \mid \textbf { x } _ { i } - \textbf { x } _ { j } \mid \end{align*}


\begin{align*} Z _ { 2 n + 1 } = \bar { Z } _ n \to 0 n \to \infty \end{align*}


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


\begin{align*} Z _ n & = \underbrace { Z \times _ X \dots \times _ X Z } _ { ( n + 1 ) \times } , & \zeta _ n & = ( \zeta \times \dots \times \zeta ) | _ { Z _ n } , \end{align*}


\begin{align*} \omega ^ 2 - \phi _ { i j } ^ 2 = q _ i q _ j r _ { i j } s _ { i j } . \end{align*}


\begin{align*} \beta = h \sigma \dd \alpha = - \alpha \wedge \alpha + h \sigma \wedge h \sigma + \frac { 3 } { 4 } \sigma \wedge \sigma - R \sigma \wedge \sigma , 0 = ( \dd h + ( ( h \alpha + \frac { 1 } { 2 } h [ \sigma ] ) ) + S \sigma ) \wedge \sigma . \end{align*}


\begin{align*} X ( w ) = \mathrm { R e } \int ^ w { } ^ t ( 1 , - i , 2 G ) F d \zeta \end{align*}


\begin{align*} h _ { R , r } ( m , u ) \ \lesssim \ \sqrt { u + m } \mbox { a n d } h _ { R , r } ( m , u ) \ \gtrsim \ \begin{cases} m & \mbox { i f } u \ge m ^ 2 - 1 \\ \sqrt { u + m } & \mbox { i f } u \le m ^ 2 - 1 . \end{cases} \end{align*}


\begin{align*} \omega ( \pi _ Z ( a ) ) v _ \mu = ( \mu - 2 t ) v _ \mu \end{align*}


\begin{align*} \frac { 1 } { \delta } \int _ { D _ { n } ^ { c } } h ( y ) \mu ( y ) = \frac { 1 } { \delta } \left ( \mu ( h ) - \int _ { D _ { n } } h ( y ) \mu ( y ) \right ) \end{align*}


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


\begin{align*} \min _ { ( y , x , y ) \in A } J _ K ( x ) = \ell ( x ( a ) , x ( b ) ) + \int _ a ^ b f ( t , x ( t ) , y ( t ) ) d t + K \cdot d _ { S } ( x ( a ) , x ( b ) ) . \end{align*}


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


\begin{align*} L _ 1 = \langle S ( { \bf { v } } ) \partial _ t \partial _ y { \bf { v } } , \partial _ y { \bf { v } } \rangle _ { H ^ { 2 , 0 } } + \langle \partial _ y S ( { \bf { v } } ) \partial _ t { \bf { v } } , \partial _ y { \bf { v } } \rangle _ { H ^ { 2 , 0 } } = : L _ 1 ^ 1 + L _ 1 ^ 2 . \end{align*}


\begin{align*} \Delta _ k \ 1 _ { [ y , \infty ) } ( x ) = \Delta _ k ^ \ast \ 1 _ { [ 0 , x ] } ( y ) . \end{align*}


\begin{align*} W ^ { ( m ) } & : = \sum _ { i , \ell \in [ 1 , m ] } g _ { m , \alpha } ( i ) G ( S _ i , S _ { \ell + m } ) g _ { m , \alpha } ( \ell ) \circ \theta _ { m } \end{align*}


\begin{align*} ( F _ i ^ { \sigma ^ j } ) ^ { \diamondsuit } = ( \det ( f ^ { \partial } ) ) ^ { 2 s ( 1 + \phi + \cdots + \phi ^ { j - 1 } ) } \cdot ( ( F _ i ) ^ { \diamondsuit } ) ^ { \phi ^ j } . \end{align*}


\begin{align*} b _ \epsilon ( t , x ) : = \ , ( \tilde b _ \epsilon ( t , \cdot ) * \rho _ \epsilon ) ( x ) ( t , x ) \in I \times \R ^ n \ , , \end{align*}


\begin{align*} P _ { m _ 1 , m _ 2 , \theta } ^ { \alpha , \beta , \gamma } ( \sigma ) : = \frac { ( - 1 ) ^ { \theta _ 1 h _ 1 ( \sigma ) + \theta _ 2 h _ 2 ( \sigma ) } \alpha ^ { \# \{ x : \sigma ( x ) = x \} } \beta ^ { \# \{ x : \sigma ( x ) = [ x + \mathbf { e } ^ 1 ] \} } \gamma ^ { \# \{ x : \sigma ( x ) = [ x + \mathbf { e } ^ 2 ] \} } } { Y _ { m _ 1 , m _ 2 , \theta } ( \alpha , \beta , \gamma ) } . \end{align*}


\begin{align*} \mathcal C ^ + = - ( x \partial _ x + y \partial _ y + 1 ) ^ 2 \end{align*}


\begin{align*} \left | \Delta \rho _ \eta ^ \varepsilon - \overline { \Delta _ \Gamma \eta } - k _ d ^ { - 1 } \Bigl ( \overline { V _ \Gamma H \eta } \Bigr ) - \Bigl ( \bar { \zeta } _ 1 - \bar { \zeta } _ 0 \Bigr ) \right | & \leq c \varepsilon \sum _ { \xi = \eta , \zeta _ 0 , \zeta _ 1 } \left ( | \bar { \xi } | + \Bigl | \overline { \nabla _ \Gamma \xi } \Bigr | + \Bigl | \overline { \nabla _ \Gamma ^ 2 \xi } \Bigr | \right ) \end{align*}


\begin{align*} P \circ \textit { \textsf { p r i n } } _ { \mathcal { I } _ U } = \tilde { P } \qquad \mbox { o n } \tilde { M } - \tilde { E } . \end{align*}


\begin{align*} J _ { \breve { N } ^ r } ( x , t , k ) = I + O ( e ^ { - 1 6 t \xi ^ { 3 / 2 } } ) , k \in \Gamma _ 1 \cup \Gamma _ 1 ^ * , t \rightarrow \infty \end{align*}


\begin{align*} \mathfrak { D } _ 0 : = ( 1 - \kappa ) \mathfrak { D } _ { i n t , 0 } + \kappa \mathfrak { D } _ { \Gamma , 0 } \end{align*}


\begin{align*} \left \Vert \xi _ { i , \delta } \right \Vert _ { C ^ { 1 , \tau } ( \overline { \Omega } ) } , \left \Vert \xi _ { i } \right \Vert _ { C ^ { 1 , \tau } ( \overline { \Omega } ) } \leq k _ { p _ { i } } i = 1 , 2 . \end{align*}


\begin{align*} I _ { M N } = 1 + \sum _ { d = 1 } ^ \infty ( q z ) ^ d \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } s _ \lambda ( a _ 1 , \dots , a _ N ) s _ \lambda ( b _ 1 , \dots , b _ N ) \Omega _ { \frac { 1 } { M } } ^ { - 1 } ( \lambda ) \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) . \end{align*}


\begin{align*} g _ N ^ \lambda = \sum _ { \theta \in \{ 0 , 1 \} ^ 2 } g _ N ^ { \lambda , \theta } . \end{align*}


\begin{align*} \bigcap \bigl \{ \tau _ \rho ^ \dag \colon \rho \in \sigma ^ \circ [ 1 ] \bigr \} \ ; = \ ; \bigcap \bigl \{ \tau _ \rho ^ \dag \colon \rho \in ( \sigma ' ) ^ \circ [ 1 ] \bigr \} \end{align*}


\begin{align*} \int _ 0 ^ \infty t ^ { s - 1 } \partial _ t ^ k ( e ^ { - t P _ 1 } - e ^ { - t P _ 2 } ) f ( x ) \ , d t = 0 . \end{align*}


\begin{align*} F ( \textbf { a } , \textbf { b } , \textbf { x } ) = \frac { 1 } { b _ i } \left ( \sum _ { j = 1 } ^ { r } \nu _ { i j } \theta _ { x _ j } + b _ i \right ) \bullet F ( \textbf { a } , \textbf { b } + \mathbf { e _ i } , \textbf { x } ) = H ( b _ i ) \bullet F ( \textbf { a } , \textbf { b } + \mathbf { e _ i } , \textbf { x } ) \end{align*}


\begin{align*} \tilde { \tau } _ M : = \inf \{ t \in [ 0 , \tau ^ * ) , | z ( t ) | _ { H _ x ^ { \mathfrak { s } } } \geq M \} , \end{align*}


\begin{align*} \left | f \right | ^ 2 _ { L ^ 2 } = \int _ { \Xi } \left | \langle f , \Phi _ \xi \rangle \right | ^ 2 d \xi , \end{align*}


\begin{align*} \log Z _ { m , n } ^ { a , b } = \log Z _ { m , n } ^ { a - \lambda , b } + \int _ { a - \lambda } ^ a \partial _ s \log Z _ { m , n } ^ { s , b } \ , \mathrm { d } s . \end{align*}


\begin{align*} F = \sum _ { g = 0 } ^ \infty \hbar ^ { 2 g - 2 } F _ g , \end{align*}


\begin{align*} 0 = \lambda _ 0 ( \Omega , g , \mu ) < \lambda _ 1 ( \Omega , g , \mu ) \le \lambda _ 2 ( \Omega , g , \mu ) \le \dotso \nearrow \infty . \end{align*}


\begin{align*} P ^ \gamma _ { [ x ] } : = \gamma ^ { - 1 } P _ { [ x ] } \gamma \cap H _ y N _ { n - 1 } . \end{align*}


\begin{align*} \beta _ { 1 , n } ^ * ( P _ { \ell , m - \ell } ) \leq \prod _ { i = 1 } ^ { m } w _ i \leq \left ( \frac { \sum _ { i = 1 } ^ { m } w _ i } { m } \right ) ^ m \leq \frac { 1 } { m ^ m } \ ; , \end{align*}


\begin{align*} \left \{ \begin{aligned} \omega _ { t } - \mathcal { L } ( \overline { x } , y , D \omega , D ^ { 2 } \omega ) = 0 , & \ , ( 0 , T ( n ) ] \times D _ { n } , \\ \omega ( 0 , y ) = \sup \limits _ { \{ | x - \overline { x } | \leq r \} } g ( x , y ) , & D _ { n } , \\ \omega ( t , y ) = 0 , & [ 0 , T ( n ) ] \times \partial D _ { n } . \end{aligned} \right . \end{align*}


\begin{align*} \frac { d R ( g _ \tau , T , \hat y , \hat \eta _ \tau ) } { d \tau } = ( 2 H _ \tau ( y , \hat \eta _ \tau ) ) ^ { - \frac { 1 } { 2 } } \Big ( \big ( ( g _ 1 - g _ 0 ) \hat \eta _ \tau , \hat \eta _ \tau \big ) + 2 \Big ( g _ \tau \hat \eta _ \tau , \frac { \partial \hat \eta _ \tau } { \partial \tau } \Big ) \Big ) \end{align*}


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


\begin{align*} Y _ n & = \underbrace { Y \times _ X \dots \times _ X Y } _ { ( n + 1 ) \times } , & \psi _ n & = ( \psi \times \dots \times \psi ) | _ { Y _ n } . \end{align*}


\begin{align*} \| \rho \| _ { C ( \Omega ) } = \sup _ { \Omega } | \rho | , \| \rho \| _ { C ( \overline { \Omega } ) } = \sup _ { \overline { \Omega } } | \rho | . \end{align*}


\begin{align*} B : = \dfrac { \left \langle A ^ { 1 / 2 } \dfrac { z } { | z | } , A ^ { 1 / 2 } \dfrac { e } { | e | } \right \rangle } { \left | A ^ { 1 / 2 } \dfrac { e } { | e | } \right | ^ 2 } , \end{align*}


\begin{align*} \Delta ^ 2 & \geq [ \beta ] - 2 \geq 2 - 2 = 0 . \end{align*}


\begin{align*} D _ { \lambda , n } ( f ) = D _ { \sigma , n - 1 } \Big ( \frac { f ( z ) - f ^ { * } ( \lambda ) } { z - \lambda } \Big ) , \ , \ , \ , \ , f \in \mathcal H _ { \lambda , n } , \ , \ , \ , n \in \mathbb N . \end{align*}


\begin{align*} a _ { j , j ' } = \begin{cases} - 1 , & j = j ' , \\ 0 , & j \neq j ' . \end{cases} \end{align*}


\begin{align*} A + K = G \ , . \end{align*}


\begin{align*} \dd \alpha _ { i j } & = - \alpha _ { i k } \wedge \alpha _ { k j } + \beta _ { i k } \wedge \beta _ { k j } - R _ { i j p q } \sigma _ p \wedge \sigma _ q \\ \dd \beta _ { i j } & = - \beta _ { i k } \wedge \alpha _ { k j } - \alpha _ { i k } \wedge \beta _ { k j } - S _ { i j p q } \sigma _ p \wedge \sigma _ q . \end{align*}


\begin{align*} \sigma ^ { k * } f ( x ) = \int _ { \R ^ d } \sigma ^ { k * } ( x - y ) f ( y ) d y , k \in \N , \ f \in L ^ 2 ( \R ^ d ) , \end{align*}


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


\begin{align*} H ( x , y ) : = f ( x , y ) + Q ( x , y ) , \end{align*}


\begin{align*} \Phi _ { \pmb { \theta } } ( \mathcal { D } ) = \left \{ \mathrm { I d } \right \} . \end{align*}


\begin{align*} Z = \overline { Z _ 1 \oplus \ldots \oplus Z _ m } \end{align*}


\begin{align*} \frac { d ( a ) _ { 2 m } } { d a } = \frac { d } { d a } \left [ 2 ^ { 2 m } \left ( \frac { a } { 2 } + \frac { 1 } { 2 } \right ) _ m \left ( \frac { a } { 2 } \right ) _ m \right ] = 2 ^ { 2 m } \frac { d } { d a } \left [ \left ( \frac { a } { 2 } + \frac { 1 } { 2 } \right ) _ m \right ] \left ( \frac { a } { 2 } \right ) _ m + 2 ^ { 2 m } \left ( \frac { a } { 2 } + \frac { 1 } { 2 } \right ) _ m \frac { d } { d a } \left [ \left ( \frac { a } { 2 } \right ) _ m \right ] \end{align*}


\begin{align*} R ( t , 0 ) = 0 , D _ 2 R ( t , 0 ) = 0 , \forall t \in J , \end{align*}


\begin{align*} \lim _ { z \to \partial D } | h ( z ) | d ( z , \partial D ) = 0 \Longrightarrow \lim _ { z \to \partial D } | h ' ( z ) | d ( z , \partial D ) ^ 2 = 0 . \end{align*}


\begin{align*} \operatorname * { t r } \left \langle \cdot , f \right \rangle g = \left \langle g , f \right \rangle . \end{align*}


\begin{align*} \big ( b H _ { k , t } + H _ { k - 1 , t } b \big ) c & = c - \Psi _ { k , t } \ , c c \in C _ k ( A ) \ . \end{align*}


\begin{align*} \forall t \in \R , f _ i ( t ) = \int _ { \R } e ^ { \i t y } d \mu _ i ( y ) , \end{align*}


\begin{align*} \int _ 0 ^ { e ^ { - e } } | D _ x X ( t , 0 , x ) | ^ p d x = + \infty ; \end{align*}


\begin{align*} I = 1 + \sum _ { d = 1 } ^ \infty z ^ d \sum _ { \lambda \vdash d } s _ \lambda ( A ) s _ \lambda ( B ) \prod _ { \Box \in \lambda } \frac { 1 } { 1 + ( 1 + q ) \hbar c ( \Box ) + q \hbar ^ 2 c ( \Box ) ^ 2 } , \end{align*}


\begin{align*} { \displaystyle \sum \limits _ { n \geq 0 } } \frac { u _ { n + k } ( z ) } { t ^ { 2 n + 1 } } = { \displaystyle \sum \limits _ { n \geq k } } \frac { u _ { n } ( z ) } { t ^ { 2 n - 2 k + 1 } } = t ^ { 2 k } \left ( S - { \displaystyle \sum \limits _ { n = 0 } ^ { k - 1 } } \frac { u _ { n } ( z ) } { t ^ { 2 n + 1 } } \right ) . \end{align*}


\begin{align*} \displaystyle \lim _ { \eta \to 1 } \frac { \partial _ \xi w } { ( 1 - \eta ) ^ \alpha } = 0 , \displaystyle \lim _ { \eta \to 1 } \frac { \partial _ \tau w } { ( 1 - \eta ) ^ \alpha } = 0 f o r ( \tau , \xi ) \in [ 0 , T ] \times [ 0 , X ] . \end{align*}


\begin{align*} S _ \iota \subset Z _ \iota : = Z ( G _ \iota ) . \end{align*}


\begin{align*} \sup _ { \ell \le \gamma _ t } \big \{ P ( S _ { t - \ell } ^ 1 \ge \psi ( t ) - \sqrt { \ell } ) \big \} & \le C P ( A _ t ) \ , , \\ P \big ( A _ t \cap L _ t ^ c ) & \le 4 \zeta _ t P ( A _ t ) \ , , L _ t : = \bigcap _ { \ell \in ( \gamma _ t , t ] } \Big \{ S _ \ell ^ 1 \ge \frac { \psi ( t ) \ell } { ( 1 + \delta ) t } \Big \} \ , . \end{align*}


\begin{align*} \Pi _ \alpha ^ { \mathrm { p l } } \circ i _ \alpha = p _ \alpha \ , . \end{align*}


\begin{align*} X _ k = v _ k ( S _ 1 , \ldots , S _ k , R ) \forall k \in \mathbb { N } \end{align*}


\begin{align*} A \otimes B = \begin{pmatrix} A _ { 0 0 } B & \cdots & A _ { m 0 } B \\ \vdots & \ddots & \vdots \\ A _ { 0 n } B & \cdots & A _ { m n } B \end{pmatrix} \ ; . \end{align*}


\begin{align*} f = \sum _ { j \in J } c _ j \ , \psi _ j . \end{align*}


\begin{align*} C ( s ) = \{ ( p , f ( s , p ) ) \in \mathbb { R } ^ { a + b } : p \in \Omega _ P \} \forall s \in \Omega _ S \end{align*}


\begin{align*} d _ L ( C ) = \left \lfloor \frac { | C | } { | C | - 1 } n \right \rfloor = \left \lfloor n + \frac { n } { | C | - 1 } \right \rfloor = n . \end{align*}


\begin{align*} { \rm E n t } [ \mathbf { P } | \mathbf { \Pi } ^ { m } ] = m { \rm E n t } [ \sigma _ { m } \mathbf { P } | \mathbf { \Pi } ^ { 1 } ] . \end{align*}


\begin{align*} \square _ { R } = \square _ { R } ( 0 ) . \end{align*}


\begin{align*} - u _ s = - v + u _ 0 + \sum _ { l \in \{ 1 , . . . , s - 1 \} \cap I _ A } \sigma _ { A , s } ( l ) u _ l . \end{align*}


\begin{align*} \Delta \ : = \ : \coprod _ { a < b } \{ X _ a = X _ b \} \ , \coprod _ a \{ X _ a = 1 \} \ , \coprod _ a \{ X _ a = 0 \} , \end{align*}


\begin{align*} g = ( \partial _ \xi w + \partial _ \tau w ) - e ^ { - X } \frac { \delta } { 2 } w e ^ { \xi } . \end{align*}


\begin{align*} \left \vert G ^ { ( 2 ) } _ { \mathcal R , R e s } ( x , y ) \right \vert & = \left \vert \left ( I _ 1 + I _ 2 \right ) / ( 4 \pi ) \right \vert \leq C _ { R _ 0 } \left ( ( \sigma \vert x \vert ) ^ { - 3 / 2 } + e ^ { - \vert x \vert \sigma } \right ) \\ & \leq \frac { C _ { R _ 0 } } { { { \left \vert \sin \left ( { ( \theta _ c - \theta _ { \hat x } ) } / 2 \right ) \right \vert } ^ { \frac 3 2 } } \vert x \vert ^ { \frac 3 2 } } \end{align*}


\begin{align*} \mathcal { F } f = \mathcal { F } ( \Delta - \lambda ) u = ( \lambda _ \xi - \lambda ) \mathcal { F } u , \end{align*}


\begin{align*} f _ L ( r ) : = \begin{cases} \frac { 2 r } { 1 + r ^ 2 } & r < L , \\ \frac { 2 } { 1 + L ^ 2 } ( 2 L - r ) & L \leq r < 2 L , \\ 0 & 2 L \leq r , \end{cases} \end{align*}


\begin{align*} \geqslant \sum \limits _ { k = 4 } ^ { k _ 0 - 1 } \int \limits _ { \frac { 1 } { k + 1 } } ^ { \frac { 1 } { k } - 2 ^ { - 4 k - 1 } } \ , \frac { d r } { r } \ , . \end{align*}


\begin{align*} \begin{cases} i \partial _ t u + \Delta u = 0 , \\ u ( 0 , x ) = u _ 0 ( x ) , \end{cases} \end{align*}


\begin{align*} \Phi ^ u _ { \Phi ^ c _ { x _ 1 } ( \lambda ^ c _ x s ) } \left ( \lambda ^ u _ x \beta _ { x _ 1 } ( \lambda ^ c _ x s ) t \right ) = \Phi _ { x _ 1 } ( \lambda ^ u _ x t , \lambda ^ c _ s t ) , \end{align*}


\begin{align*} x = s ^ { k _ 1 } t ^ { \ell _ 1 } u ^ { m _ 1 } v ^ { n _ 1 } \alpha ^ { \epsilon _ 1 } \beta ^ { \delta _ 1 } \ \ y = s ^ { k _ 2 } t ^ { \ell _ 2 } u ^ { m _ 2 } v ^ { n _ 2 } \alpha ^ { \epsilon _ 2 } \beta ^ { \delta _ 2 } \end{align*}


\begin{align*} E [ G ( 0 , S _ \ell ) ] P ( \hat { S } ^ 1 _ { t - \ell } \ge \psi ( t ) - \sqrt { \ell } ) \le C _ 1 ( 1 + \sqrt { \ell } ) ^ { - 1 } C P ( A _ t ) \ , . \end{align*}


\begin{align*} \prod _ { k = i } ^ j D _ { \widetilde k } = L _ { \widetilde j } L _ { \widetilde { i - 1 } } ^ { - 1 } . \end{align*}


\begin{align*} Y _ t ^ { \varepsilon , \eta } = \xi + \int _ t ^ T Y _ s ^ { \varepsilon , \eta } { \dot { W } } _ { \varepsilon , \eta } ( s , B _ s ) d s - \int _ t ^ T Z _ s ^ { \varepsilon , \eta } d B _ s , t \in [ 0 , T ] \ , . \end{align*}


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


\begin{align*} \left \{ \begin{aligned} & - \Delta v = - | a _ { \epsilon , A ( y ) } \nabla \phi _ { \epsilon } | ^ 2 v ^ { - 1 } - 2 d i v \left ( a _ { \epsilon , A ( y ) } \nabla \phi _ { \epsilon } \right ) & & \mbox { i n } \Omega \\ & v > 0 & & \mbox { i n } \Omega \\ & v = 0 & & \mbox { o n } \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} \| U ( t , s _ 0 ) \| = \sup _ { \| z \| \leq 1 } \| U ( t , s _ 0 ) z \| = \sup _ { \| z \| \leq 1 } \| x ( t , s _ 0 , z ) \| \longrightarrow 0 , t \to + \infty . \end{align*}


\begin{align*} \nabla \sigma _ \varepsilon = ( d - \varepsilon \bar { g } _ 1 ) \Bigl ( \bar { \nu } - \varepsilon R \ , \overline { \nabla _ \Gamma g _ 0 } \Bigr ) + ( d - \varepsilon \bar { g } _ 0 ) \Bigl ( \bar { \nu } - \varepsilon R \ , \overline { \nabla _ \Gamma g _ 1 } \Bigr ) \quad \overline { N _ T } \end{align*}


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


\begin{align*} V _ \varphi f ( z ) = \mathrm { e } ^ { - \mathrm { i } x \xi } ( B f ) ( \bar z ) e ^ { - \frac { \pi } { 2 } | z | ^ 2 } , \end{align*}


\begin{align*} \Omega _ \varphi : = \left \{ ( z _ 1 , z _ 2 ) : \ | z _ 2 | ^ 2 < e ^ { - \varphi ( z _ 1 ) } , \ z _ 1 \in \mathbb { D } \right \} . \end{align*}


\begin{align*} Y _ 1 = 0 , \cdots , Y _ p = 0 , x _ i - x _ j = 0 , \ f o r \ a n y \ \{ i , j \} \in e \in E ( H ) \setminus E ( H ^ { \prime } ) , \end{align*}


\begin{align*} ( \tilde \Delta - \lambda _ s ) u = - ( \Delta - \lambda _ s ) h _ s = : - H _ s . \end{align*}


\begin{align*} \begin{aligned} & b _ m x _ n \leq u \leq a _ m x _ n ~ ~ ~ ~ \mbox { i n } ~ ~ ~ ~ Q ^ + _ { 2 ^ { - m } } , \\ & 0 \leq a _ m - b _ m \leq ( 1 - \mu ) ( a _ { m - 1 } - b _ { m - 1 } ) , \\ & | a _ 0 | \leq 2 C ~ ~ \mbox { a n d } ~ ~ | b _ 0 | \leq 2 C , \end{aligned} \end{align*}


\begin{align*} \Vert \alpha ^ f _ { r + 1 } \Vert ^ 2 = 2 ^ r ( \kappa _ r ^ 2 + \mu _ r ^ 2 ) . \end{align*}


\begin{align*} c ( p , t ) = c ( p _ n , \lambda ^ c _ p ( n ) t ) , \end{align*}


\begin{align*} \begin{gathered} ( A \oplus B ) _ { i / o } = A _ { i / o } \sqcup B _ { i / o } \ ; , \\ ( A \oplus B ) _ M = A _ M \oplus B _ M \ ; , \\ ( A \oplus B ) _ I = A _ I \oplus B _ I \ ; , \\ ( A \oplus B ) _ O = A _ O \oplus B _ O \ ; . \end{gathered} \end{align*}


\begin{align*} \Phi _ \nu ( Y , 0 ) = Y , \partial _ t \Phi _ \nu ( Y , t ) = ( V _ \Gamma \nu ) ( \Phi _ \nu ( Y , t ) , t ) , ( Y , t ) \in \Gamma _ 0 \times [ 0 , T ] , \end{align*}


\begin{align*} [ 3 ] = \frac { t ^ 2 x ^ 2 } { 1 - t x } . \end{align*}


\begin{align*} D _ A = D _ { i _ 1 } \cdots D _ { i _ j } \end{align*}


\begin{align*} C _ p ^ { - 1 } \| u \times \partial _ x u \| _ { L ^ 2 } \leq \| \partial _ x ( u \times \partial _ x u ) \| _ { L ^ 2 } = \| u \times \partial ^ 2 _ x u \| _ { L ^ 2 } \ , . \end{align*}


\begin{align*} \alpha _ 0 ^ t = \alpha _ 0 ^ s + \int ^ { t } _ { 0 } \alpha _ 0 ^ r \ , W ( d r , B _ r ) . \end{align*}


\begin{align*} \partial _ i \rho = \sum _ { j = 1 } ^ n R _ { i j } \overline { \underline { D } _ j \eta } ( x , t , \varepsilon ^ { - 1 } d ) + \varepsilon ^ { - 1 } \bar { \nu } _ i \partial _ r \bar { \eta } ( x , t , \varepsilon ^ { - 1 } d ) \end{align*}


\begin{align*} h ^ 1 _ { n , T ^ \ast ( 1 ) } ( \widetilde { \alpha } ( y ) ) ( \sigma ) & = \log ^ 0 _ p ( \chi ( \gamma _ 0 ^ { } ) ) \left [ \sigma \mapsto \frac { \sigma - 1 } { \gamma - 1 } \widetilde { \alpha } ( y ) - ( \sigma - 1 ) \widetilde { \alpha } ( b ) \right ] \\ & = \alpha ^ \ast \circ h ^ 1 _ { n , T } . \end{align*}


\begin{align*} \rho ( \mu ; m ) = \sum _ { x \in ( n ) _ m } \bar { \mu } ( x _ 1 ) \left ( \prod _ { i = 1 } ^ { m - 1 } \mu ( x _ i , x _ { i + 1 } ) \right ) \bar { \mu } ( x _ m ) \ ; . \end{align*}


\begin{align*} \chi ( y z ) = \chi ( y ) \chi ( z ) , \end{align*}


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


\begin{align*} \gamma _ { A , B } ( ( i , x ) ) = ( \neg i , x ) \end{align*}


\begin{align*} \mathbf { J } \left ( k , t \right ) = \left \{ \begin{array} [ c ] { c c } \mathbf { J } _ { R } \left ( k , t \right ) , & \operatorname { I m } k = 0 \\ \mathbf { J } _ { \rho } \left ( k , t \right ) , & \operatorname { R e } k = 0 \end{array} \right . , \end{align*}


\begin{align*} \pmb { p ^ * } = q ^ * _ { \tilde T } \cdot \pmb { p ^ 3 } + ( 1 - q ^ * _ { \tilde T } ) \cdot \pmb { p ^ 2 } . \end{align*}


\begin{align*} \begin{aligned} & \underset { y _ { 1 } , \hdots , y _ { m } } { } & & G ( Y ) : = \sum _ { i = 1 } ^ { m } G _ { i } ( y _ { i } ) \\ & & & y _ { 1 } = y _ { 2 } = \hdots = y _ { m } \end{aligned} \end{align*}


\begin{gather*} \epsilon _ n \colon = \begin{cases} + & \textrm { i f $ n - 2 \xrightarrow { a _ n } n $ } , \\ - & \textrm { i f $ n - 2 \xleftarrow { a _ n } n $ } . \end{cases} \end{gather*}


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


\begin{align*} { \mathcal { F } } _ { \mathcal { A } } ( J , J ' _ { \sigma , k } ) = \frac { { \mathcal { A } } _ { j ' _ { \sigma ( k ) } } ^ { J \setminus J ' _ { \sigma , k } } } { x _ { 1 } - x _ { j ' _ { \sigma ( k ) } } - 1 } - \frac { \tilde { \mathcal { D } } _ { j ' _ { \sigma ( k ) } } ^ { J \setminus J ' _ { \sigma , k } } } { x _ { 1 } + x _ { j ' _ { \sigma ( k ) } } } \ , , \end{align*}