\begin{align*} \varrho ( [ x , y ] ) \varrho ( x ) = 0 , \quad \forall \ y \in A . \end{align*}


\begin{align*} ( I - Q ^ \dagger ) F ^ { q , \dagger } _ v ( v + \psi ^ \dagger ( v , p ) , p ) [ v _ 1 + \psi ^ \dagger _ v ( v , p ) v _ 1 ] = 0 \end{align*}


\begin{align*} \rho _ \phi ( m ) \rho _ \phi ( n ) = \sum _ { d | ( m , n ) } \rho _ \phi \left ( \frac { m n } { d ^ 2 } \right ) . \end{align*}


\begin{align*} \tilde { M } _ \tau ^ \dagger \ , \tilde { M } _ \tau = \begin{pmatrix} X ^ 2 & 0 \\ 0 & 0 \end{pmatrix} + \tau \begin{pmatrix} 0 & e ^ { i \varphi } X D \\ e ^ { - i \varphi } D ^ \dagger X & 0 \end{pmatrix} + \tau ^ 2 \ : \begin{pmatrix} D D ^ \dagger + E & 0 \\ 0 & D ^ \dagger D \end{pmatrix} \ : . \end{align*}


\begin{align*} s _ { r D } ( r x ) = s _ { D } ( x ) + \gamma \log r \end{align*}


\begin{align*} R _ { n + 1 } ( s ) = f ' \bigl ( { \xi _ n ( s ) } \bigr ) R _ n ( s ) , \end{align*}


\begin{align*} \sum _ { j \neq k } \ln | \sin \pi ( \theta _ k - \theta _ j ) | = \sum _ { j \in I _ 0 , j \neq k } \ln { | \sin \pi ( k - j ) \alpha | } + \sum _ { j \in I _ { \ell } } \ln { | \sin \pi ( k - j ) \alpha | } = : \sum _ { 1 } + \sum _ { 2 } . \end{align*}


\begin{align*} \Delta _ n ( \tau _ 1 , \tau _ 2 ) \ ! = \ ! \begin{bmatrix} \frac { 2 } { h } U ^ \top ( h + \tau _ 2 ) A _ d \\ 2 A _ d ^ \top U ( \tau _ 1 - \tau _ 2 ) A _ d \\ A _ d ^ { \top } U ( \tau _ 1 - \tau _ 2 ) A _ d \\ \frac { 1 } { h } I _ m \end{bmatrix} . \end{align*}


\begin{align*} y _ { m } = \mathbf { h } _ { m } ^ H \mathbf { w } _ { m } x _ { m } + n _ { m } , \end{align*}


\begin{align*} B _ n = ( \mathbb Z / 2 \mathbb Z ) ^ n \rtimes S _ n \end{align*}


\begin{align*} h ( T ) & = y q \log ( y q ) + \sum _ { k = 1 } ^ { r } \left ( \Delta ( i _ k ) + \Delta ( j _ k ) \right ) \\ & \le y q \log ( y q ) + \sum _ { k = 1 } ^ { r } \left ( \Delta ( q + 1 ) + \Delta ( k ) \right ) \\ & = h ( T ' ) , \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty b _ k ( n ) q ^ n = \prod _ { n = 1 } ^ \infty \frac { 1 - q ^ { k n } } { 1 - q ^ n } . \end{align*}


\begin{align*} \widetilde { E } _ { \omega } \leq J _ { \omega } ( u ) \leq \liminf _ { n \rightarrow \infty } J _ { \omega } ( u _ { n } ) = \widetilde { E } _ { \omega } . \end{align*}


\begin{align*} ( \lambda _ { S } ) _ { i j } : = \delta _ { i , 0 } L _ { ( P _ S ) _ i } \mbox { a n d } ( \rho _ { S } ) _ { i j } : = \delta _ { j , 0 } R _ { ( P _ S ) _ j } , \end{align*}


\begin{align*} k _ { t } ( x , y ) = ( \sinh 2 t ) ^ { - d / 2 } e ^ { \Psi _ t ( x , y ) } , \end{align*}


\begin{align*} a c - b d = \frac { 1 } { 2 } [ ( a - b ) ( c + d ) + ( a + b ) ( c - d ) ] , \end{align*}


\begin{align*} q = \frac { a _ { 0 } ^ { \prime \prime } - a _ { 0 } ^ { \prime } } { a _ { 0 } + 1 } \end{align*}


\begin{align*} \Psi _ { n , e } ^ { ( i ) } ( x , y ) = \prod ^ { n e } _ { \substack { j = 1 \\ ( j , n e ) > 1 \\ j \equiv i \bmod e } } ( x - \zeta ^ j _ { n e } y ) . \end{align*}


\begin{align*} \pi _ 1 ^ * ( d \log \varphi _ { \lambda , z } ) = \left ( \sum _ { Z \in \mathcal I } m _ Z \rho _ { Z } \right ) \frac { d g } { g } + \eta \end{align*}


\begin{align*} \hat { \mathbf { D } } = \begin{pmatrix} \begin{aligned} & \mathbf { D } & 0 \\ & 0 & 1 \end{aligned} \end{pmatrix} . \end{align*}


\begin{align*} & M i n i m i z e _ { x \in \mathbb { R } ^ n } \ , T r ( A _ J , X ) + 2 b _ J ^ T x \\ & s . t . \ \ T r ( A _ k , X ) + 2 b _ k ^ T x + c _ k \le 0 \ , k = 1 , . . . , m \\ & \begin{pmatrix} 1 & x ^ T \\ x & X \end{pmatrix} \succeq 0 \end{align*}


\begin{align*} \mu _ m ^ { ( 1 2 ) } \big ( \mathfrak { s } \boldsymbol { J } \big ) = \mu _ m ^ { ( 1 2 3 ) } \big ( \boldsymbol { J } \big ) \mu _ m ^ { ( 1 2 3 ) } \big ( \mathfrak { s } \boldsymbol { J } \big ) = \mu _ m ^ { ( 1 2 ) } \big ( \boldsymbol { J } \big ) . \end{align*}


\begin{align*} g _ \xi ( z ) = \begin{cases} 0 & z \in U ^ c , \\ \frac { g ( z ) } { \xi - z } & z \in U \end{cases} \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { T - 1 } - \Theta _ k \Lambda _ k ' + ( 1 - \Theta _ k ) \Lambda _ k \le - \frac { \zeta _ 0 } { \log \gamma } = \max \left \{ \log _ \gamma \frac { \bar \Delta ' } { \delta _ 0 } , 0 \right \} . \end{align*}


\begin{align*} N e w t o n ( F _ t ) & = C o n v ( \bigcup \limits _ { i = 0 } ^ m N e w t o n ( s _ { \Lambda ^ { ( i ) } } ) ) \\ & = t C o n v ( \bigcup \limits _ { i = 0 } ^ m N e w t o n ( s _ { \lambda ^ { ( i ) } } ) ) \\ & = t N e w t o n ( F ) . \end{align*}


\begin{align*} H = \left \{ S \in V ^ { ( k - 1 ) } : \ : \deg _ { G } ( S ) \geq \sqrt { d } \right \} , \end{align*}


\begin{align*} j ( \tau , s ) = H [ b + \tau \tilde { b } + s b _ 0 ] , \tau , s \in \mathbb { R } . \end{align*}


\begin{align*} a _ { i } = Q _ { i } \kappa . \end{align*}


\begin{align*} 1 6 \ , \mathcal A = 1 6 \ , a \cdot b \le ( 2 a + 2 b ) ^ 2 \ = { \mathcal L } ^ 2 \end{align*}


\begin{align*} c = \prod _ { i = 1 } ^ { n - 1 } \norm { \eta _ i } _ { H ^ 1 } ^ 2 \left ( 1 + ( n + 3 ) \left ( \frac { n + 6 } { 4 } + \frac 1 4 \norm { \tilde \Psi } _ { H ^ 1 } \right ) \right ) \end{align*}


\begin{align*} g ( x - k \gamma ) & = g ( x ) - \gamma \sum _ { j = 1 } ^ k \frac { g ( x - ( j - 1 ) \gamma ) - g ( x - j \gamma ) } { \gamma } \\ & < g ( x ) - \gamma \sum _ { j = 1 } ^ k ( \Lambda + \varepsilon ) ^ { - j } \frac { \Lambda - 1 + \delta } { \gamma } g ( x ) \\ & = g ( x ) \left ( 1 - ( \Lambda - 1 + \delta ) \sum _ { j = 1 } ^ k ( \Lambda + \varepsilon ) ^ { - j } \right ) \\ & < 0 , \end{align*}


\begin{align*} \varphi _ n ( p ) = \frac { 1 } { n } \langle p , \xi \rangle + n \left ( ( x _ p y _ \xi - y _ p x _ \xi ) ^ 2 + z _ p ^ 2 \right ) , p \in \Omega . \end{align*}


\begin{align*} ( \sigma _ 1 , \dots , \sigma _ { n - 1 } ) \subset \bigoplus _ { i = 1 } ^ { n - 1 } ( E _ { i , q _ i } , E _ { i + 1 , q _ i ' } ) . \end{align*}


\begin{align*} \int _ { \Omega } ( K _ H ( x ) \nabla v ^ \tau _ \varepsilon | \nabla v ^ \tau _ \varepsilon ) d x = & 2 \pi q ^ 2 \sqrt { ( K _ { H } ) _ { 1 1 } ( K _ { H } ) _ { 2 2 } } ( \bar { x } ) \ln \frac { 1 } { \varepsilon } + O ( 1 ) + C | \ln \tau | \\ = & 2 \pi q ^ 2 \sqrt { d e t ( K _ { H } ) } ( \bar { x } ) \ln \frac { 1 } { \varepsilon } + O ( 1 ) + C | \ln \tau | , \end{align*}


\begin{align*} \mathcal D _ { \psi , z ' } ( g ) = \left ( \frac { d } { d \bar z } g \circ T ^ { - 1 } \right ) \circ T . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { \infty } | J _ { n } ( i ) | \leq \left ( \sum _ { i = 1 } ^ { \infty } | J _ { n } ( i ) | ^ { 1 / 2 } \right ) ^ { 2 } \leq M ^ { 2 } \forall n \geq 1 . \end{align*}


\begin{align*} M _ I : = M _ { \varrho ( I ) } \textrm { a n d } R _ I : = R _ { \varrho ( I ) } . \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty \left | \frac { \theta _ 0 ( e ^ { \kappa + \tau } ) - \theta _ 0 ( e ^ \tau ) - \delta e ^ \tau } { e ^ { \sigma \tau } } \right | ^ 2 d \tau = \frac { 1 } { 2 \pi } \int _ { - \infty } ^ \infty \left | \frac { e ^ { \kappa s } - 1 } { s } \right | ^ 2 \left | \frac { \zeta ' } { \zeta } ( s ) + g ( s ) \right | ^ 2 d t . \end{align*}


\begin{align*} A ^ { - 1 } \mathcal { L } _ g = \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + B - Q \Bigr ) \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + Q \Bigr ) \end{align*}


\begin{align*} \sum _ { j = 0 } ^ n a _ { i j } = k , \forall \ , i = 0 , \dotsc , n . \end{align*}


\begin{align*} \xi = - L - 1 , \eta = \frac { Q } { 2 ( L + 1 ) } - \sqrt { \kappa B _ 4 } , \zeta = \kappa \left ( \frac { B _ 3 } { 2 \sqrt { \kappa B _ 4 } } + 1 \right ) , \sigma = \sqrt { \kappa B _ 4 } , \end{align*}


\begin{align*} \bar a _ { i , j } \big ( \sqrt { \mu } \partial _ { v _ j } g \big ) = a _ { i , j } * \Big ( \partial _ { v _ j } ( \sqrt { \mu } g ) - g \partial _ { v _ j } \sqrt { \mu } \Big ) = \big ( \partial _ { v _ j } a _ { i , j } \big ) * \big ( \sqrt { \mu } g \big ) + \frac 1 2 a _ { i , j } * \big ( v _ j \sqrt { \mu } g \big ) , \end{align*}


\begin{align*} \nabla _ { H } u = \sum _ { i = 1 } ^ m X _ i u X _ i \end{align*}


\begin{align*} \begin{aligned} \frac { \partial u } { \partial \overline { q _ l } } = \partial _ { x _ { 4 l } } + \mathbf { i } \partial _ { x _ { 4 l + 1 } } + \mathbf { j } \partial _ { x _ { 4 l + 2 } } + \mathbf { k } \partial _ { x _ { 4 l + 3 } } , \end{aligned} \end{align*}


\begin{align*} \mathbf { \Phi } _ { \mathrm { r } , g } ^ { H } \mathbf { \Phi } _ { \mathrm { r } , g } + \mathbf { \Phi } _ { \mathrm { t } , g } ^ { H } \mathbf { \Phi } _ { \mathrm { t } , g } = \mathbf { I } _ { \bar { M } } , \forall g \in \mathcal { G } . \end{align*}


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


\begin{align*} \begin{cases} \dim H ^ 1 ( S , T _ S ) = 6 8 ; \\ \dim H ^ 2 ( S , T _ S ) = 6 . \end{cases} \end{align*}


\begin{align*} \varphi ( f \circ g ) = ( ( f \circ g ) _ { 1 1 } , \ldots , ( f \circ g ) _ { n n } ) = ( f _ { 1 1 } \circ g _ { 1 1 } , \ldots , f _ { n n } \circ g _ { n n } ) \\ = ( f _ { 1 1 } , \ldots , f _ { n n } ) \circ ( g _ { 1 1 } , \ldots , g _ { n n } ) = \varphi ( f ) \circ \varphi ( g ) . \end{align*}


\begin{align*} \pi _ { \lambda , c _ 1 } ' ( \prod _ { i = 1 } ^ g [ \alpha _ i , \beta _ i ] \prod _ { j = 1 } ^ n \gamma _ j ) = \pm \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} . \end{align*}


\begin{align*} u _ { n n } \sigma _ { k - 1 } ( b ) - \sum _ { \alpha = 1 } ^ { n - 1 } u _ { \alpha n } ^ 2 \sigma _ { k - 2 ; \alpha } ( b ) + \sigma _ k ( b ) = f . \end{align*}


\begin{align*} \ell ( x ) + 1 = \ell ( x ' ) \geq \ell ( x ) + d ( v ' \Rightarrow v ) + d ( w v \Rightarrow w ' v ' ) \geq \ell ( x ) + 1 . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { \pi ( a ) - 1 } q _ i \neq \prod _ { i = 1 } ^ { \pi ( a ) - 1 } p _ i ^ { \alpha _ i } . \end{align*}


\begin{align*} D ( x ) : = x \cdot A x . \end{align*}


\begin{align*} \{ u _ { \alpha \beta } ( 0 ) \} _ { 1 \leq \alpha , \beta \leq n - 1 } = \mathrm { d i a g } \{ b _ 1 , \ldots , b _ { n - 1 } \} . \end{align*}


\begin{align*} u ( x ) \geq \left ( \frac { 1 } { 2 } - C \delta \right ) \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta x _ \beta ^ 2 - C b _ \alpha | \hat { x } | ^ 2 \geq \frac { 1 } { 4 } \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta x _ \beta ^ 2 - C b _ \alpha | \hat { x } | ^ 2 \end{align*}


\begin{align*} v ( x ) & = \underbrace { v ( x _ 0 ) } _ { = 0 } + \nabla v ( x _ 0 ) \cdot ( x - x _ 0 ) + o \left ( \abs { x - x _ 0 } \right ) \\ [ 7 p t ] & = - \frac { \partial v } { \partial \nu } ( x _ 0 ) \abs { x - x _ 0 } + o \left ( \abs { x - x _ 0 } \right ) \\ [ 5 p t ] & \ge \tau \abs { x - x _ 0 } + o \left ( \abs { x - x _ 0 } \right ) \\ [ 5 p t ] & > \frac { \tau } { 2 } \abs { x - x _ 0 } = \frac { \tau } { 2 } d ( x , \partial \Omega ) \end{align*}


\begin{align*} \kappa _ { p , d } = 2 ^ { 1 - p } \frac { \Gamma \left ( \frac { d - p } { 2 } \right ) } { \Gamma \left ( \frac { d } { 2 } \right ) \Gamma \left ( \frac { p } { 2 } \right ) } . \end{align*}


\begin{align*} V & = \big ( G ( c ) - G ' ( c ) c \big ) H , \\ \partial ^ \square c & = \Delta _ \Gamma \big ( G ' ( c ) \big ) + c H V . \end{align*}


\begin{align*} x _ { N ( \ell - 1 ) + j } = \max _ { 1 \leq q \leq N L } x _ q , y _ { N ( k - 1 ) + i } = \max _ { 1 \leq q \leq N L } y _ q . \end{align*}


\begin{align*} \mu _ 1 \prec \mu _ 2 \Longleftrightarrow \mu _ 2 - \mu _ 1 = \alpha _ 1 + \dots + \alpha _ m \alpha _ 1 , \dots , \alpha _ m \in \Phi _ + . \end{align*}


\begin{align*} & b _ k ^ T \Tilde { x } \le b _ k ^ T ( \lambda x _ v + ( 1 - \lambda ) x _ w ) , \\ & b _ k ^ T ( \Tilde { x } - ( \lambda x _ v + ( 1 - \lambda ) x _ w ) ) \le 0 . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac 1 n \sum _ { x = 0 } ^ n \int _ { 0 } ^ { + \infty } \varphi \left ( t , \frac x n \right ) \mathcal E _ x ( n ^ 2 t ) \dd t = \int _ { 0 } ^ { + \infty } \int _ 0 ^ 1 \varphi ( t , u ) T ( t , u ) \dd t \dd u , \quad \mbox { i n p r o b a b i l i t y , } \end{align*}


\begin{align*} f ( u , y ) = \mp \sum _ { u < v < y } f ( u , v ) f ( v , y ) . \end{align*}


\begin{align*} D = \{ c \} \cup \{ d \in C _ i \mid \varphi _ { \sigma , s } ^ L ( d ) = a \} \end{align*}


\begin{align*} f ( x + y ) = \frac { f ( x ) f ( y ) } { f ( x ) + f ( y ) } \ \ \ ( x , y \in \mathbb { R } - \{ 0 \} ) . \end{align*}


\begin{align*} \triangledown \tilde { f } _ g ( \mathbf { \Phi } _ { g } ) = 2 \mathbf { Z } _ { g } \mathbf { \Phi } _ { g } \mathbf { Y } _ { g } - 2 \mathbf { X } _ { g } ^ H , \forall g \in \mathcal { G } . \end{align*}


\begin{align*} F ( X ) = \sum _ { i = 0 } ^ { 2 ^ n - 1 } A _ i X ^ i . \end{align*}


\begin{align*} \sum \limits _ { n = 0 } ^ { \infty } g _ { o } ( n , \alpha , k , p ) - g _ { e } ( n , \alpha , k , p ) q ^ { n } = \frac { ( q ^ { k } ; q ^ { k } ) _ { \infty } } { ( q ; q ) _ { \infty } } \sum _ { n = 0 } ^ { \infty } \frac { q ^ { p n + \alpha } } { 1 + q ^ { p n + \alpha } } . \end{align*}


\begin{align*} \phi ( m _ n + 1 ) + \phi ( m _ n - 1 ) + \lambda \tan ( \pi \theta _ { m _ n } ) \phi ( m _ n ) = E \phi ( m _ n ) \end{align*}


\begin{align*} \operatorname { T r } \textbf { \textit { K } } ( t , x ^ { \prime } , x ^ { \prime } ) = \frac { 1 } { ( 2 \pi ) ^ { n - 1 } } \int _ { \mathbb { R } ^ { n - 1 } } \bigg ( \frac { i } { 2 \pi } \int _ { \mathcal { C } } e ^ { - t \tau } \sum _ { j \leqslant - 1 } \operatorname { T r } \phi _ { j } \ , d \tau \bigg ) \ , d \xi . \end{align*}


\begin{align*} \mathsf { T } _ { \mathbf { \Phi } _ { g } } \mathcal { M } _ g = \{ \mathbf { T } _ { g } \in \mathbb { C } ^ { 2 \bar { M } \times \bar { M } } : \Re \{ \mathbf { \Phi } _ { g } ^ H \mathbf { T } _ { g } \} = \mathbf { 0 } _ { \bar { M } } \} , \forall g \in \mathcal { G } . \end{align*}


\begin{align*} & \prescript L { } { } \ell { } ^ R ( x , \cdot ) : \Phi \rightarrow \mathbb Z , \alpha \mapsto \prescript L { } { } \ell { } ^ R ( x , \alpha ) , \\ & \prescript L { } { } \ell { } ^ R ( x , \alpha ) : = \langle \mu , \alpha \rangle + \chi _ R ( \alpha ) - \chi _ L ( w \alpha ) . \end{align*}


\begin{align*} a ( - k ) = a ( k ) , k \in \overline { D } _ + , b ( - k ) = - b ( k ) , \operatorname { I m } k ^ { 2 } = 0 . \end{align*}


\begin{align*} d = \left ( \Lambda ^ { - \frac { 1 } { 2 } } ( s ) \beta ( s ) \right ) \beta ^ { \frac { 1 } { 2 } } ( s ) = \left ( \frac { \beta ^ { 3 } ( s ) } { \Lambda ( s ) } \right ) ^ { \frac { 1 } { 2 } } = \frac { 1 } { 4 \sqrt [ 4 ] { t / 2 } } . \end{align*}


\begin{align*} \varphi _ { | N } = \frac { 1 } { n } ( \psi _ 1 + \cdots + \psi _ n ) \end{align*}


\begin{align*} { { \bf { \bar X } } ^ { \star } } = \left [ { { { \bf { W } } ^ { \star } } , { { \bf { 0 } } _ { \left ( { \sum { { { \bar r } _ { l ' } } } } \right ) \times \left ( { { M _ t } - K } \right ) } } } \right ] , \end{align*}


\begin{align*} \int | h _ X | ^ 2 w _ U & = \int h _ X \overline { h _ X w _ U } = \int \widehat { h _ X } \overline { \widehat { h _ X } * \widehat { w _ U } } \end{align*}


\begin{align*} ( a \wedge b ) \cdot c & = ( a \cdot b ) \ : \cdot \ : ( a \cdot c ) \\ a \cdot ( b \wedge c ) & = ( a \cdot b ) \wedge ( a \cdot c ) . \end{align*}


\begin{align*} V ^ { H ' } = \varinjlim _ { n } X _ { H ' , n } \widehat { \otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) \oplus \varinjlim _ { n } A _ { H ' , n } \widehat { \otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) . \end{align*}


\begin{align*} R ^ \nabla _ { X , Y } Z = R ^ W _ { X , Y } Z & - \frac { 1 } { 2 } ( d J \theta ) ( X , Y ) J Z - \frac { 1 } { 2 } ( d \theta ) ( X , Y ) Z \\ & - \frac { 1 } { 2 } \left ( D ^ W _ X \left ( D ^ W _ Y J \right ) - D ^ W _ Y \left ( D ^ W _ X J \right ) - D ^ W _ { [ X , Y ] } J \right ) J Z \\ & + \frac { 1 } { 4 } \left ( \left ( D ^ W _ X J \right ) \left ( D ^ W _ Y J \right ) - \left ( D ^ W _ Y J \right ) \left ( D ^ W _ X J \right ) \right ) Z . \end{align*}


\begin{align*} V ' _ i ( u ) [ v ] = 4 B _ i ( u ^ 2 , u v ) , \ \forall u , v \in X , \ i = 0 , 1 , 2 . \end{align*}


\begin{align*} 2 \frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \kappa _ 1 ( \kappa _ 1 - \tilde { \kappa } _ i ) } - \frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \kappa _ 1 ^ 2 } = \frac { F ^ { i i } ( \kappa _ 1 + \tilde { \kappa } _ i ) } { \kappa _ 1 ^ 2 ( \kappa _ 1 - \tilde { \kappa } _ i ) } h _ { 1 1 i } ^ 2 \geq c ( n ) \frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \kappa _ 1 ^ 2 } , \end{align*}


\begin{align*} e ( \rho , x ) = e ^ { i \rho x } \left ( 1 + \left ( z + 1 \right ) \sum _ { n = 0 } ^ { \infty } \left ( - 1 \right ) ^ { n } z ^ { n } a _ { n } ( x ) \right ) \end{align*}


\begin{align*} Q ^ 2 - B Q - \Bigl [ I _ { n + 1 } \frac { \partial } { \partial x _ n } , Q \Bigr ] + C = 0 , \end{align*}


\begin{align*} C o n v ( \bigcup \limits _ { \mu } S u p p ( s _ \mu ) ) = C o n v ( \bigcup \limits _ { i = 0 } ^ l N e w t o n ( s _ { \lambda ^ i } ) ) . \end{align*}


\begin{align*} n = \frac { \beta \left ( 3 \beta ^ { 2 } + 4 \right ) + 1 } { 2 } \end{align*}


\begin{align*} \nabla _ \Sigma f ( p ) = 0 \phantom { x x } \phantom { x x } D _ \Sigma ^ 2 f ( p ) \leq 0 . \end{align*}


\begin{align*} g _ 0 = \begin{bmatrix} \sqrt { x _ 0 } & 0 \\ \sqrt { - y _ 0 } & \frac { 1 } { \sqrt { x _ 0 } } \end{bmatrix} = \begin{bmatrix} \sqrt { x _ 0 } & 0 \\ 0 & \frac { 1 } { \sqrt { x _ 0 } } \\ \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ \sqrt { - { x _ 0 } { y _ 0 } } & 1 \\ \end{bmatrix} . \end{align*}


\begin{align*} \forall \psi \in \Omega _ { \Tilde { \Lambda } _ m \setminus \Lambda _ n } \ \forall \Delta \subset \Lambda _ n \ \forall \xi _ { \Delta } : \nu ( \xi _ { \Delta } \eta _ { \Lambda _ n \setminus \Delta } \psi _ { \Tilde { \Lambda } _ m \setminus \Lambda _ n } ) = 0 . \end{align*}


\begin{align*} \lim _ { \lambda _ 1 , \ldots , \lambda _ d \rightarrow 1 } \frac { h ^ { ( \lambda _ 1 , \ldots , \lambda _ d ) } ( x ) } { \Delta ( \lambda ) } = \frac { h ( x ) } { \prod _ { j = 1 } ^ { d - 1 } j ! } , \lim _ { \lambda _ 1 , \ldots , \lambda _ d \rightarrow 1 } \frac { \mathfrak { h } ^ { ( \lambda _ 1 , \ldots , \lambda _ d ) } ( x ) } { \Delta ( \lambda ) } = \frac { \mathfrak { h } ( x ) } { \prod _ { j = 1 } ^ { d - 1 } j ! } . \end{align*}


\begin{align*} \left | \left . \dfrac { \partial \varepsilon _ \delta ( x , \bar { y } _ a , t ) - b _ { 1 , \delta } ( x ) / 2 } { \partial x } \right | _ { x = x _ a } \right | \le 2 L _ r . \end{align*}


\begin{align*} \rho ( g ) = \tilde { \Gamma } ( g ) ^ { \frac { - 1 } { Q - 2 } } \end{align*}


\begin{align*} e ^ { ( \delta _ n - L + 5 0 \varepsilon ) t q _ n } r _ { \ell _ t } \leq e ^ { ( \delta _ n - L + 5 0 \varepsilon ) t _ 0 q _ n } \max _ { 1 \leq j \leq y _ n - 1 } r _ j \leq & C _ 0 y _ n q _ n e ^ { ( \delta _ n - L + 5 0 \varepsilon ) t _ 0 q _ n } \\ \leq & e ^ { \varepsilon y _ n q _ n } e ^ { ( \delta _ n - L + 5 0 \varepsilon ) \frac { y _ n } { 2 } q _ n } \\ = & e ^ { ( \delta _ n - L + 5 2 \varepsilon ) \frac { y _ n } { 2 } q _ n } \\ \leq & e ^ { ( \delta _ n - L + 5 2 \varepsilon ) \ell _ 0 q _ n } . \end{align*}


\begin{align*} \sigma ( ( u ^ i , u ^ i { } _ j ) . a ) & = \sigma ( u ^ i , u ^ i { } _ \alpha a ^ \alpha { } _ j ) \\ * & = ( u ^ i , u ^ i { } _ \alpha a ^ \alpha { } _ j , - \Gamma ^ i { } _ { \alpha \beta } u ^ \alpha { } _ \gamma u ^ \beta { } _ \delta a ^ \gamma { } _ j a ^ \delta { } _ k ) \\ * & = \sigma ( u ^ i , u ^ i { } _ j ) . a \end{align*}


\begin{align*} & \langle \mu , \alpha \rangle + \chi _ R ( v ' \alpha ) + \chi _ L ( - w ' v ' \alpha ) \\ \leq & \langle \mu , \alpha \rangle + \chi _ R ( v ' \alpha ) + 1 - \chi _ L ( w ' v ' \alpha ) \\ = & \prescript L { } { } \ell { } ^ R ( x ' , \alpha ) + 1 \leq 0 . \end{align*}


\begin{align*} \int _ { 0 } ^ { t _ { 0 } } \bigg ( \int _ { R _ { 0 } } ^ { \infty } \nu \xi ( \nu , s ) d \nu \bigg ) ^ { 2 } \ d s = 0 \end{align*} % \end{align*}


\begin{align*} \mathcal { Q } ( u ) : = u _ t - \frac { \partial } { \partial x } \left ( a ( x , u , u _ x ) \right ) + b ( x , u , u _ x ) , \end{align*}


\begin{align*} & \int _ { B _ { 1 / 2 } } | \nabla v | ^ { 3 - 3 \theta } \d x \leq C \end{align*}


\begin{align*} \begin{aligned} & \tilde { \Gamma } \triangleq \frac { 4 \left ( \gamma ^ 2 + \delta ^ 2 \right ) } { \left ( 1 - 3 \gamma \right ) ^ 2 } \left ( 1 - \frac { \delta } { \sqrt { 2 k } } \right ) ^ { - 1 } < \gamma , \\ & \nu = 1 - \delta / \sqrt { 2 k } . \end{aligned} \end{align*}


\begin{align*} \Sigma _ t : = \sigma _ t \sigma _ t ^ * \forall t \in [ 0 , T ] , \end{align*}


\begin{align*} \bar \beta _ n ( G ) = O ( n ^ { e ( G ) } ) , \end{align*}


\begin{align*} a _ g = \abs v ^ \gamma * ( \sqrt \mu g ) , \ \boldsymbol { A } _ g = ( a _ { 1 , g } , a _ { 2 , g } , a _ { 3 , g } ) = \abs v ^ \gamma * ( \sqrt \mu \partial _ v g ) \textrm { a n d } \boldsymbol { B } _ { g } = ( b _ { 1 , g } , b _ { 2 , g } , b _ { 3 , g } ) = \abs v ^ \gamma * ( v \sqrt \mu g ) , \end{align*}


\begin{align*} & \mathbb { E } \widetilde { g } _ { \delta , L , \mu } ( x ) = g _ { \delta , L , \mu } ( x ) , \\ & \mathbb { E } \| \widetilde { g } _ { \delta , L , \mu } ( x ) - g _ { \delta , L , \mu } ( x ) \| ^ 2 \leq \frac { \sum _ { i = 1 } ^ n \sigma _ { f , i } ^ 2 } { n ^ 2 r } = \frac { \sigma _ f ^ 2 } { n r } . \end{align*}


\begin{align*} \frac { \alpha } { 2 \beta } \min \{ \kappa - r , \lambda \} \leq \frac { \alpha } { 2 \beta } \min \{ \kappa - r , r \} \leq \frac { 1 } { 2 } \frac { \alpha \kappa } { 2 \beta } = \frac { 1 } { 2 } \min \{ \alpha + \gamma - \frac { 1 } { 2 } , \alpha \} \leq \frac { \alpha } { 2 } \leq \min \{ \alpha - \varepsilon , \frac { 1 } { 2 } \} . \end{align*}


\begin{align*} \mathbb P \left [ \int _ 0 ^ T \| \alpha _ s \| ^ { \frac { 2 m } { 2 + m } } d s + \int _ 0 ^ T \| \sigma _ s \| _ { L _ { } ( U , H ) } ^ m d s < \infty \right ] = 1 . \end{align*}


\begin{align*} A ^ { - 1 } \mathcal { L } _ g = I _ { n + 1 } \frac { \partial ^ 2 } { \partial x _ n ^ 2 } + B \frac { \partial } { \partial x _ n } + C = \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + B - Q \Bigr ) \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + Q \Bigr ) , \end{align*}


\begin{align*} \mathcal { P } _ { \widetilde { \mu } } = \mathcal { D } \times \mathcal { C } . \end{align*}


\begin{align*} \hat y _ t ^ \varepsilon = \frac { y ^ \varepsilon _ t - y _ t } { \varepsilon } - K _ t ~ ~ \hat z _ t ^ \varepsilon = \frac { z ^ \varepsilon _ t - z _ t } { \varepsilon } - L _ t . \end{align*}


\begin{align*} A ( G ) : = L _ 2 ( G ) \ast L _ 2 ( G ) = \{ s \mapsto \langle \l _ s \xi , \eta \rangle \mid \xi , \eta \in L _ 2 ( G ) \} . \end{align*}


\begin{align*} \sigma _ { e _ c } ^ 2 \left [ t + 1 \right ] = & \left \{ 1 - 2 \mu \left ( 2 \sum _ { j = 1 } ^ { M } \lvert \phi ^ { \left ( j , c \right ) } \rvert ^ 2 - f ^ { \left ( c \right ) } \right ) \right \} \sigma _ { e _ c } ^ 2 \left [ t \right ] \\ & - 4 \left ( \frac { _ { } \left ( a _ c ^ { \left ( o \right ) } \right ) } { a _ c ^ { \left ( o \right ) } } + a _ c ^ { \left ( o \right ) } \sum _ { j = 1 } ^ { M } \Re \left \{ \phi ^ { \left ( j , c \right ) } \right \} \right ) \end{align*}


\begin{gather*} f \in L f \le c P ( c > f + \epsilon ) = 0 \epsilon > 0 . \end{gather*}


\begin{align*} \deg _ G ( v ) = \frac { \sum _ { v \in S \in V ^ { ( k - 1 ) } } \deg _ G ( S ) } { k - 1 } \geq \frac { \deg _ H ( v ) \sqrt { d } } { k - 1 } > d , \end{align*}


\begin{align*} | Z _ t | ^ 2 = | Z _ 0 | ^ 2 \end{align*}


\begin{align*} C ^ { \alpha } _ { i } G _ { \alpha \beta } ^ T H ^ { \beta } _ 1 \overline { G _ { \alpha \beta } } = G _ { \alpha \beta } ^ T H ^ { \beta } _ 1 \overline { G _ { \alpha \beta } } ( \overline { C ^ { \alpha } _ { i } } ^ { T } ) \end{align*}


\begin{align*} \kappa = \sum _ { j = 1 } ^ { n } P _ { j } a _ { j } , \end{align*}


\begin{align*} \left \langle \overset { \cdot } { K } _ { \alpha } \left ( \omega \right ) , \varphi \right \rangle : = - \left \langle K _ { \alpha } \left ( \omega \right ) , \varphi ^ { \prime } \right \rangle : = \int _ { \mathbb { R } _ { + } } K _ { \alpha } \left ( t , \omega \right ) \varphi ^ { \prime } \left ( t \right ) d t \end{align*}


\begin{align*} - \infty < \inf _ { \xi : \xi _ { \Lambda } = \eta _ { \Lambda } } f ( \xi ) \leq f ( \eta ) \leq \sup _ { \xi : \xi _ { \Lambda } = \eta _ { \Lambda } } f ( \xi ) < \infty . \end{align*}


\begin{align*} [ X \times _ k \mathbb A _ k ^ n \to S , \sigma ] = [ X \times _ k \mathbb A _ k ^ n \to S , \sigma ' ] \end{align*}


\begin{align*} \left | \int _ M f ( x ) \right | = \left | \int _ M ( f ( x ) - \tilde f ( x ) ) \right | \leq { \rm V o l } ( \tilde \Sigma ^ r ) \cdot { \rm V o l } ( M ) . \end{align*}


\begin{align*} g ( u , c ) = \frac { \vartheta } { 1 - \frac { 1 } { \psi } } ( 1 - r ) u \bigg [ \Big ( \frac { c } { \left ( ( 1 - r ) u \right ) ^ { \frac { 1 } { 1 - r } } } \Big ) ^ { 1 - \frac { 1 } { \psi } } - 1 \bigg ] , \end{align*}


\begin{align*} f _ { 1 } ^ { + } = 2 \mathcal { I } _ 1 = \int _ { \Omega } | U | ^ { 2 } \dd x = \int _ { \Omega } U \cdot \nabla \times ( \textrm { c u r l } ^ { - 1 } U ) \dd x = f \int _ { \Omega } U \cdot \textrm { c u r l } ^ { - 1 } U \dd x = f . \end{align*}


\begin{align*} \partial \varrho ( \xi ) = 1 , \bar { \partial } \partial \varrho \rfloor \xi = 0 . \end{align*}


\begin{align*} \varphi ( x ' , \rho ( x ' ) ) = \frac { 1 } { 2 } \sum _ { \alpha \leq n - 1 } b _ \alpha x _ \alpha ^ 2 + \frac { 1 } { 6 } \sum _ { \alpha , \beta , \gamma \leq n - 1 } \varphi _ { \alpha \beta \gamma } ( 0 ) x _ \alpha x _ \beta x _ \gamma + O ( | x ' | ^ 4 ) , \end{align*}


\begin{align*} L _ { e v e n } : = A + \sum _ { k = 1 } ^ { \frac { n - 2 } { 2 } } B _ k . \end{align*}


\begin{align*} h ( G ) \le ( m - y ) \log ( m - y ) + y \log ( y ) + m \left ( \log ( m ) - \log ( y ) \right ) - ( m - y ) \left ( \log ( m - y ) - \log ( y ) \right ) = m \log m . \end{align*}


\begin{align*} L ( E ) = \max \bigg \{ \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | , \log | \lambda | \bigg \} , \forall \ E \in \mathbb { C } . \end{align*}


\begin{align*} \mathbb { E } _ { n , m } ' [ \beta \gamma ] \\ = \mathbb { E } _ { \mathcal { M } } [ \mathbb { 1 } _ { X _ 0 ( M ) } ( x _ 1 ) \mathbb { 1 } _ { X _ 1 ( M ) } ( x _ 2 ) \mathbb { E } _ { n , m } [ \chi _ { n , m } ( w , x _ 1 ) \chi _ { n , m } ( w , x _ 2 ) | M ] \end{align*}


\begin{align*} ( D _ { a ^ + } ^ { \alpha } f ) ( x ) : = \frac { d } { d x } \left [ ( { \bf I } _ { a ^ + } ^ { 1 - \alpha } f ) ( x ) \right ] \end{align*}


\begin{align*} \partial _ t c _ { \min \ , | t } & = \partial ^ \square c _ { | ( t , p ) } = \Delta _ \Gamma G ' ( c ) + c V H _ { \ , | ( t , p ) } \\ & = G '' ( c ) \Delta _ \Gamma c + G ''' ( c ) \big | \nabla _ \Gamma c \big | ^ 2 + g ( c ) H ^ 2 c _ { \ , | ( t , p ) } \geq g ( c ) H ^ 2 c _ { \ , | ( t , p ) } . \end{align*}


\begin{align*} \frac { \sum \limits _ { i = 1 } ^ j \chi _ i - r j } { \chi } < \sum \limits _ { i = 1 } ^ { j } w _ i < \frac { \sum \limits _ { i = 1 } ^ j \chi _ i - r ( j - 1 ) } { \chi } . \end{align*}


\begin{align*} \begin{aligned} \Big [ | \langle \boldsymbol { \nu } , A \rangle - \bar A | + | \langle \boldsymbol { \nu } , f _ \alpha \rangle - \bar f _ \alpha | \Big ] & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} | x ( 2 \bar { t } / \gamma ) | _ { \mathcal { A } _ \delta } & \le \kappa \bar c _ 0 e ^ { - \gamma \lambda \bar { t } / \gamma } + \bar c \\ & \le \kappa \bar c _ 0 e ^ { - \lambda \bar { t } _ 2 } + \bar c < \bar c _ 0 . \end{aligned} \end{align*}


\begin{align*} \eta ^ \lambda = ( 1 - \lambda ) p _ 0 + \lambda \eta _ 0 \in \Omega \end{align*}


\begin{align*} G = \begin{pmatrix} \mathbf { 1 } _ { n - 1 } & 1 \\ G _ { n - 1 } & \mathbf { 0 } _ { n - 1 } ^ T \end{pmatrix} , \end{align*}


\begin{align*} F ( x + F ( x ) ) = - F ( x ) , \end{align*}


\begin{align*} S _ { R } ^ { \alpha } ( x , y ) = \sum _ { n = 0 } ^ { \infty } \Big ( 1 - \frac { 2 n + d } { R } \Big ) _ { + } ^ { \alpha } \Phi _ { n } ( x , y ) , \end{align*}


\begin{align*} \| g \| _ { 1 } \geq \int _ { \triangle \setminus \triangle _ { \delta } } g d U ( x ) = \int _ { \triangle \setminus \triangle _ { \delta } } g ^ { + } d U ( x ) + \int _ { \triangle \setminus \triangle _ { \delta } } g ^ { - } d U ( x ) \geq ( 3 / 2 0 ) \cdot \| g \| _ { 1 } . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { n - 1 } } e ^ { - C | \xi | } \ , d \xi = \frac { \Gamma ( n - 1 ) \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) } { C ^ { n - 1 } } , C > 0 , \ n \geqslant 2 , \end{align*}


\begin{align*} L _ { k } ^ { \left ( 2 \right ) } \left ( s , \rho \right ) = \int _ { - \infty } ^ { \infty } \frac { e ^ { i S ( \tau , x - s ) } \tau ^ { k - 2 } } { \tau - \rho } d \tau . \end{align*}


\begin{align*} v _ \epsilon ( t ) : = & \int _ { - \infty } ^ { t } ( \int _ { - \infty } ^ { t _ 1 } ( \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ) ( s ) d s ) d t _ 1 \\ & - \int _ { - \infty } ^ { - t _ 0 } ( \int _ { - \infty } ^ { t _ 1 } ( \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ) ( s ) d s ) d t _ 1 - t _ 0 , \end{align*}


\begin{align*} \nu ( x ) = \pi _ J ( v ^ { - 1 } \mu ) . \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } \Psi ^ \pm _ 2 ( x ; z ) = e ^ { - i c _ \pm } e _ 2 , \end{align*}


\begin{align*} \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi _ { + } ^ { 2 } ( \partial _ t \varphi + ( v + u _ { \infty } ) \cdot \nabla \varphi ) \dd x \dd s = - \int _ { \mathbb { R } ^ { 3 } } \Phi _ { 0 , + } ^ { 2 } \varphi _ 0 \dd x . \end{align*}


\begin{gather*} f = \bigl ( \pi _ { i + 1 } - \pi _ i \bigr ) \ , 1 ( \pi _ 1 \in A _ 1 ) \ , \ldots \ , 1 ( \pi _ i \in A _ i ) , \end{gather*}


\begin{gather*} \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \ge \inf _ { Q \in \mathcal { U } } Q \Bigl ( \ , \sum _ { i = 1 } ^ n f _ i \circ \pi _ i \Bigr ) \end{gather*}


\begin{align*} \begin{cases} \mathcal { L } _ H q = 0 , \\ \nabla ^ { \perp } q \cdot \nu | _ { \partial \Omega } = - v _ n | _ { \partial \Omega } . \\ \end{cases} \end{align*}


\begin{align*} M _ { I J } = \begin{pmatrix} G _ { I _ 1 } & G _ { I _ 2 } & \cdots & G _ { I _ { 2 k } } \\ G _ { J _ 1 } & G _ { J _ 2 } & \cdots & G _ { J _ { 2 k } } \end{pmatrix} , \end{align*}


\begin{align*} \begin{aligned} Y ^ { \epsilon } ( t ) & = y ^ { \epsilon } _ { 0 } + \int ^ { t } _ { 0 } h ^ { \epsilon } ( s ) d s + \lambda ( \epsilon ) \epsilon \int ^ { t } _ { 0 } \gamma ^ { \epsilon } ( s ) \tilde { N } ^ { \epsilon ^ { - 2 } } ( d s ) . \end{aligned} \end{align*}


\begin{align*} \log \det ( u _ { , i j } ) = - v _ j x ^ j + u _ { , i } \xi ^ i + c , \end{align*}


\begin{align*} J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \ge J ( x ^ * ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ^ * ) = J ( x ^ * ) \ \ \forall \ , x \in \mathbb { R } ^ n . \end{align*}


\begin{align*} A _ t & = a _ t , \\ B \times u & = b \times v + b \times u _ { \infty } + B _ { \infty } \times v , \\ \nabla \times B & = \nabla \times b . \end{align*}


\begin{align*} \beta ( z , w ) = \dfrac { 1 } { 2 } \log \dfrac { 1 + | \varphi _ z ( w ) | } { 1 - | \varphi _ z ( w ) | } , z , w \in \mathbb { B } _ n . \end{align*}


\begin{align*} f _ e ( x ) = ( e ^ { \lambda \Lambda _ e } - 1 ) ^ { - 1 } \sum _ { n = 1 } ^ \infty ( ( \lambda \Lambda _ { e } ) ^ n / n ! ) \ , g _ { e } ^ { \ast n } ( x ) \end{align*}


\begin{align*} \pi ^ G _ k ( X ) = \begin{cases} \displaystyle \operatorname * { c o l i m } _ { V \in s ( \mathcal { U } _ G ) } \ ; \ ; [ S ^ { k + V } , X ( V ) ] _ * ^ G & \mathrm { f o r } \ ; \ ; k \geq 0 \\ \displaystyle \operatorname * { c o l i m } _ { { V \in s ( \mathcal { U } _ G ) } } \ ; \ ; [ S ^ { k + V } , X ( \mathbb { R } ^ { - k } \oplus V ) ] _ * ^ G & \mathrm { f o r } \ ; \ ; k \leq 0 \end{cases} \end{align*}


\begin{align*} \pi \circ \phi = \mbox { i d } _ Y , \end{align*}


\begin{align*} a _ { n , k } = \frac 1 n \sum _ { j \in [ n ] ^ k } \rho _ H ( j _ 1 - j _ 2 ) \rho _ H ( j _ 2 - j _ 3 ) \cdots \rho _ H ( j _ { k - 1 } - j _ k ) \rho _ H ( j _ k - j _ 1 ) . \end{align*}


\begin{align*} & \partial _ { t } \left ( \frac { \rho ( u , t ) ^ { 2 } } { 2 } \right ) - \Theta \sigma ^ { i j } _ { k } \nabla _ { i j } \left ( \frac { \rho ( u , t ) ^ { 2 } } { 2 } \right ) \\ & = ( k + 1 ) h \Theta \sigma _ { k } - \rho ^ { 2 } + \sigma _ { k } \nabla _ { i } h \nabla _ { j } \Theta - \Theta \sigma ^ { i j } _ { k } w _ { m i } w _ { m j } . \end{align*}


\begin{align*} Z _ t ^ n = 1 + \sum _ { i = 1 } ^ n \int _ 0 ^ t Z _ s ^ n \triangle K _ s ^ { i , n } \cdot d W _ s ^ i . \end{align*}


\begin{align*} V _ n ( \varphi ) = \begin{bsmallmatrix} \varphi ( 0 ) \\ \Phi _ n \end{bsmallmatrix} ^ { \top } \mathbf { P } _ n \begin{bsmallmatrix} \varphi ( 0 ) \\ \Phi _ n \end{bsmallmatrix} , \end{align*}


\begin{align*} S t _ { D N M } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( f , z ) - f ( z ) | | _ { \infty } = 0 \end{align*}


\begin{align*} \Phi ( x ' ) : = \partial _ n K ^ L ( x ' , 1 ) , x ' \in \R ^ { n - 1 } . \end{align*}


\begin{align*} \tilde { \kappa } _ i = \begin{cases} \kappa _ 1 , & i = 1 , \\ \kappa _ i - 1 , & i > 1 . \end{cases} \end{align*}


\begin{align*} | \frac { 1 } { \lambda ( t ) } \mathcal { G } ( t ) - \mathcal { E } ( t ) | & \leq ( \mu c _ 1 + \nu ) \frac { \alpha ( t ) \delta ^ { \frac { 1 } { \ell } } ( t ) } { \lambda ( t ) } \mathcal { E } ( t ) \mathcal { E } ^ { ( r + \frac { 1 } { \ell ^ { \prime } } + \frac { 1 } { q } - 1 ) } ( t ) \\ & \leq ( \mu c _ 1 + \nu ) c ^ * _ { _ E } \mathcal { E } ( t ) , \end{align*}


\begin{align*} ( A , B ) \cdot Z = A Z B ^ { - 1 } \end{align*}


\begin{align*} \langle z _ 1 ^ * , v \rangle = \hat z _ 1 \cdot v , \langle z _ 2 ^ * , v \rangle = \hat z _ 2 \cdot v , \end{align*}


\begin{align*} e _ { \rm t h m } ( u ) = T ( u ) , u \in [ 0 , 1 ] , \end{align*}


\begin{align*} \int _ { \mathfrak Y ( n ) } | \mathfrak h ( n ) _ { \eta } ^ * \omega ( n ) | = ( \phi _ 0 ) _ ! \int _ { S m ( \widetilde { \mathfrak Y ( n ) } ) } | \phi _ { \eta } ^ * \mathfrak h ( n ) _ { \eta } ^ * \omega ( n ) | . \end{align*}


\begin{align*} \varphi ^ L _ { \sigma , s } ( x ) = a < b = \varphi ^ L _ { \sigma , s } ( y ) . \end{align*}


\begin{align*} \mathcal { S } ^ { k - 1 } : = \{ \underline { x } \in [ 0 , 1 ] ^ { k } : | | \underline { x } | | = 1 \} , ~ \mathcal { D } ^ { k } ( A ) : = \{ \rho ( \underline { a } ) : \underline { a } \in A ^ { k } \} ~ \mbox { a n d } ~ \mathcal { D } ^ { \underline { k } } ( A ) : = \{ \rho ( \underline { a } ) : \underline { a } \in A ^ { \underline { k } } \} , \end{align*}


\begin{align*} \underset { 0 \leq t \leq T } { \sup } \Bigg [ \frac { 1 } { N } \underset { i = 1 } { \overset { N } { \sum } } \mathbb { E } ^ { \boldsymbol { \xi } } [ | \phi ^ { N , i } ( t , \boldsymbol { x } ^ * _ t ) - \Psi ^ { N , i } ( t , \boldsymbol { x } ^ * _ t ) | ^ 2 ] \Bigg ] \leq \frac { C } { N ^ 2 } \Big ( 1 + \frac { 1 } { N ^ 2 } \underset { i , j = 1 } { \overset { N } { \sum } } | \xi ^ { i } - \xi ^ { j } | ^ 2 \Big ) \end{align*}


\begin{align*} \mathrm { L i f t } ( f ) = \left \lbrace \prod _ { \alpha = 1 } ^ d z ^ { i _ { \alpha } } _ { j _ { \alpha } k _ { \alpha } } - \prod _ { \alpha = 1 } ^ d z ^ { i _ { \alpha } } _ { j ' _ { \alpha } k _ { \alpha } } \mid k _ { \alpha } \in [ t _ { i _ { \alpha } } ] \alpha \in [ d ] \right \rbrace . \end{align*}


\begin{align*} \frac { 2 + h \chi _ 4 ( x ) + \overline { h } \overline { \chi _ 4 } ( x ) } { 4 } = \left \{ \begin{array} { l l l } 1 , & \hbox { i f $ \chi _ 4 ( x ) \in \{ 1 , \chi _ 4 ( g ) \} $ ; } \\ 0 , & \hbox { } \end{array} \right . \end{align*}


\begin{align*} \int _ { 1 } ^ \infty \frac { d s } { \Big ( \displaystyle \int _ 0 ^ s F ( t ) d t \Big ) ^ { p / ( 2 p - q + 1 ) } } = \infty . \end{align*}


\begin{align*} \begin{aligned} f ( t , s ) = F _ { \boldsymbol { \omega } \boldsymbol { \eta } } \big ( L ^ { - 1 } _ { \boldsymbol { \omega } } ( t ) , K ^ { - 1 } _ { \boldsymbol { \eta } } ( s ) , f \big ( L ^ { - 1 } _ { \boldsymbol { \omega } } ( t ) , K ^ { - 1 } _ { \boldsymbol { \eta } } ( s ) \big ) \big ) , ~ ~ ~ ~ ~ \forall ~ ( t , s ) \in L _ { \boldsymbol { \omega } } ( S G ' ) \times K _ { \boldsymbol { \eta } } ( S G '' ) , \end{aligned} \end{align*}


\begin{align*} N : = \left \{ n \in \N _ 0 \colon f ( \Phi ^ n ( \alpha ) ) \in G \right \} \end{align*}


\begin{align*} H = \Theta : L ^ 2 ( \R ^ { n - 1 } , \mathbb { C } ^ N ) \to L ^ 2 ( \R ^ n _ + , d x ' d t / t ) ^ N . \end{align*}


\begin{align*} \chi _ i = \left \{ \begin{aligned} & \phi ^ s _ i & & ( i \neq j ) \\ & \mu & & ( i = j ) \end{aligned} \right . \end{align*}


\begin{align*} - \Delta u = \lambda u & D , \\ [ 1 m m ] u = 0 & \partial D . \end{align*}


\begin{align*} C ( \alpha ) : = \sum _ { j = 1 } ^ \infty \frac { 1 } { 1 + 0 . 1 j } \frac { 1 } { j ^ \alpha } < \infty . \end{align*}


\begin{align*} X ^ v \circ X ^ w = t ^ { - \langle v , w \rangle } X ^ { v + w } \ , v , w \in \mathbb { N } ^ n . \end{align*}


\begin{align*} k ( \theta ) = I _ f ( \theta ) \end{align*}


\begin{align*} R ( \tau ) \Psi ^ \prime ( \tau ) + \ell \sin \Psi ( \tau ) - \kappa \ell R ( \tau ) U ( \tau ) = 0 , \end{align*}


\begin{align*} ( \theta _ 1 , \theta _ 2 ) = ( b ^ { - x } \alpha ^ y , a ^ { - x } \beta ^ y ) . \end{align*}


\begin{align*} g ( x ) = \left \{ \begin{array} { r c l } \frac { 1 } { \inf \{ \beta : x \in Q _ { \beta } \} } & \mbox { i f } & x \in H \\ 0 & \mbox { i f } & x \notin H \end{array} \right . \end{align*}


\begin{align*} \mathbf { F } _ d ^ H \mathbf { H } _ { s f } \mathbf { F } _ a = \mathbf { H } _ { a d } , \end{align*}


\begin{align*} \mu _ { ( \boldsymbol { x } , \boldsymbol { \Psi } ( t , \boldsymbol { x } ) ) } ^ { N , i } : = k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } } , \lambda ^ { N , 1 } _ { \boldsymbol { \Psi } ( t , \boldsymbol { x } ) } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \Psi } ( t , \boldsymbol { x } ) } ) , \end{align*}


\begin{align*} \int _ { t _ 1 } ^ { t _ 2 } \sum _ { i = 1 } ^ { \infty } \sum _ { j = 1 } ^ { i } g _ j V _ { i , j } \psi _ i ( s ) \psi _ j ( s ) d s < \infty \end{align*}


\begin{align*} \begin{aligned} \frac { d } { d s } \mathcal { Y M } _ { e , g _ s } ( { \nabla } ) _ { \big { | } _ { s = 0 } } & = \frac 1 2 \int _ M \langle S _ { e , \mathcal { Y M } } , \delta g \rangle d v _ g \end{aligned} \end{align*}


\begin{align*} ( \mathbf { I } - \mathbf { P } ) ( v _ i v _ j \sqrt { \mu } ) = \Big \{ v _ i v _ j - \frac { | v | ^ 2 } { 3 } \delta _ { i j } \Big \} \sqrt { \mu } . \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } s } \Gamma ( 1 / s , x ^ s ) & = - \frac { 1 } { s ^ 2 } \Big [ \log ( x ^ s ) \Gamma ( 1 / s , x ^ s ) + x ^ s T ( 3 , 1 / s , x ^ s ) \Big ] - x \exp ( - x ^ s ) \log ( x ) \\ & \triangleq \tilde \psi ( s , x ) , \end{align*}


\begin{align*} X _ j ( X _ 1 ^ r b _ 1 ) = \left \{ \begin{array} { l l } \sum _ { i = 2 } ^ { k } X _ i ^ { p - 1 } b _ i & \\ 0 & \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} & \mathbf { A x } - \mathbf { b } \tau - { { \mathbf { r } } _ { p } } \nu = 0 , \\ & - { { \mathbf { A } } ^ { T } } \mathbf { y } + \mathbf { c } \tau - \mathbf { s } - { { \mathbf { r } } _ { d } } \nu = 0 , \\ & { { \mathbf { b } } ^ { T } } \mathbf { y } - { { \mathbf { c } } ^ { T } } \mathbf { x } - \kappa - { { r } _ { g } } \nu = 0 , \\ & \mathbf { x } \circ \mathbf { s } = \mu \mathbf { e } , \\ & \kappa \tau = \mu \end{aligned} \end{align*}


\begin{align*} \gamma _ { p , c } : = \sum _ { k = 1 } ^ { l } ( p - 2 + \alpha _ { k } + \beta _ { k } ) c _ { k } ^ { \beta _ { k } } \big [ \null _ { k } \eta \big ] _ { \frac { p } { 2 - \alpha _ { k } } } \end{align*}


\begin{align*} \frac { 1 } { \rho _ K } \| ( x ^ { K + 1 } , \lambda ^ { K + 1 } ) - ( x ^ K , \lambda ^ K ) \| \leq \frac { r _ 0 + \theta } { \rho _ 0 \zeta ^ K } \overset { \eqref { K } } { \leq } \frac { \varepsilon } { 2 } , \eta _ K = \eta _ 0 \sigma ^ K \overset { \eqref { K } } { \leq } \frac { \varepsilon } { 2 } , \end{align*}


\begin{align*} \mathcal { A } _ { H _ { t } } ( [ \gamma , u ] ) = \int _ { \gamma } H _ { t } + \int u ^ { * } \omega . \end{align*}


\begin{align*} \mu ( \gamma _ \Lambda ( \eta _ \Lambda | \cdot ) ) = \mu ( \eta _ \Lambda ) . \end{align*}


\begin{align*} \begin{aligned} \frac { v ' ( r ) ^ { 2 } } { 2 } & \leq \int _ { 0 } ^ { r } t ^ { b } v ^ { q } ( t ) v ' ( t ) f ( w ( t ) ) d t \\ & \leq r ^ { b } f ( w ( r ) ) \int _ { 0 } ^ { r } \left ( \frac { v ^ { q + 1 } ( t ) } { q + 1 } \right ) ' d t \\ & \leq C f ( w ( r ) ) v ^ { q + 1 } ( r ) \quad \mbox { f o r a l l } \ 0 < r < R , \end{aligned} \end{align*}


\begin{align*} \int _ \Omega u ( t , x ) \cdot \nabla h ( x ) \ , d x = 0 \forall h \in H ^ 1 _ { \rm l o c } ( \Omega ) \end{align*}


\begin{align*} \Delta \rho _ 1 ( x ) \leq \psi _ { K , - H } ( \rho _ 1 ( x ) ) = - \psi _ { K , H } ( \rho ( x ) ) . \end{align*}


\begin{align*} F _ i : = Y _ { \frac i 4 } = & \mathcal S \left ( \frac 1 4 \right ) Y _ { \frac { i - 1 } 4 } + \int _ { \frac { i - 1 } 4 } ^ { \frac i 4 } \mathcal S \left ( \frac i 4 - s \right ) \alpha _ s d s + \int _ { \frac { i - 1 } 4 } ^ { \frac i 4 } \mathcal S \left ( \frac i 4 - s \right ) \sigma _ s d W _ s \\ = & \mathcal S \left ( \frac 1 4 \right ) Y _ { \frac { i - 1 } 4 } + \mu _ i + \epsilon _ i , \end{align*}


\begin{align*} C ^ r _ { N , H } ( \alpha ) : = \{ p \in G = H \ltimes N \ , : \ , d ( 1 , \tilde \pi _ N ( p ) ) \leq \alpha d ( 1 , \tilde \pi _ H ( p ) ) \} , \end{align*}


\begin{align*} I _ 1 = y \cdot f ( y - 1 ) + ( y - 1 ) \left ( f ( ( a + 1 ) y + b ) - f ( a y + b ) \right ) \end{align*}


\begin{align*} \phi = \left \{ \begin{array} { l l } u , & { \rm o n } \Omega \setminus D , \\ \max \{ u , v \} , & { \rm o n } D , \end{array} \right . \end{align*}


\begin{align*} S _ { e , \mathcal { Y M } ^ 0 } = S _ { e , \mathcal { Y M } } = 0 \ , . \end{align*}


\begin{align*} \nu = [ h , \eta ] = h \eta h ^ { - 1 } \eta ^ { - 1 } = \rho \eta ^ { g } \rho ^ { - 1 } \eta ^ { - 1 } \in \Gamma ^ G . \end{align*}


\begin{align*} \sum _ { s = 1 } ^ { n } k _ { r s } \theta ^ { s - 1 } & \overset { \eqref { e q : a _ i - k _ i j - t h e t a } } { { = } } a _ { r } = \sum _ { j = 1 } ^ { n } \Big ( \sum _ { h = 1 } ^ { n } p _ { r j h } \theta ^ { h - 1 } \Big ) a _ { j } = \sum _ { j = 1 } ^ { n } \Big ( \sum _ { h = 1 } ^ { n } p _ { r j h } \theta ^ { h - 1 } \Big ) \Big ( \sum _ { l = 1 } ^ { n } k _ { j l } \theta ^ { l - 1 } \Big ) \\ & \overset { \eqref { e q : t h e t a - i - t h e t a - j - c _ i j k } } { = } \sum _ { j , h , l = 1 } ^ { n } \sum _ { s = 1 } ^ { n } p _ { r j h } k _ { j l } c _ { h l s } \theta ^ { s - 1 } . \end{align*}


\begin{align*} \begin{aligned} \mathcal { P } ^ { + } \left ( z ^ { - 1 } \bar { r } _ 1 ( z ) e ^ { - 2 i z x } \right ) & = \int _ { 2 x } ^ { \infty } \widehat { ( z ^ { - 1 } \bar { r } _ 1 ( z ) ) } ( \xi ) e ^ { i z ( y - 2 x ) } d \xi . \end{aligned} \end{align*}


\begin{align*} 2 a = ( a + b ) + ( a - b ) \end{align*}


\begin{align*} v [ j ] \mathrel { : = } v [ j ] + v [ t ] w _ { t } [ j ] \end{align*}


\begin{align*} \mathcal { A } = \frac { 1 } { 2 } \sum _ { j = 1 } ^ p e _ j \cdot \big ( ( \widetilde { \nabla } _ Y B ) ( X , e _ j ) - ( \widetilde { \nabla } _ X B ) ( Y , e _ j ) \big ) , \end{align*}


\begin{align*} S _ p = S _ { i n } \cup S _ { s p } , \end{align*}


\begin{align*} a ( Z ) = a \big ( ( Z Z ^ * ) ^ \frac { 1 } { 2 } \big ) \end{align*}


\begin{align*} X - W = c _ i W ^ { ( i ) } + O ^ { 3 , \alpha } ( | x | ^ { - q } ) . \end{align*}


\begin{align*} \mathcal { B } _ q ( \psi _ m ) ( p ) = \displaystyle \frac { p ^ m } { \sqrt { \Gamma ( q m + 1 ) } } , \end{align*}


\begin{align*} h ( x , y ) : = \begin{cases} ( x , y ) & ( x < 0 ) \\ ( x , x + y ) & ( x \geq 0 ) . \end{cases} \end{align*}


\begin{align*} h ( x ) & = \prod _ { j = 1 } ^ d \lambda _ j \int _ 0 ^ \infty d a _ d \int _ 0 ^ { x _ d - x _ { d - 1 } + a _ d } d a _ { d - 1 } \ldots \int _ 0 ^ { x _ { 2 } - x _ 1 + a _ { 2 } } d a _ 1 \\ & \times \left ( e ^ { - \sum _ { i = 1 } ^ d \lambda _ i a _ i } h ( x _ 1 + a _ 1 , \ldots , x _ d + a _ d ) \right ) . \end{align*}


\begin{align*} f = \prod _ { g \in \mathcal G } g . \end{align*}


\begin{align*} \textbf { I } = \sum _ { k = 1 } ^ T \langle \lambda _ k F ( x _ k ) , x _ 0 - x \rangle = \sum _ { k = 1 } ^ T \langle s _ { k - 1 } - s _ k , x _ 0 - x \rangle = \langle s _ 0 - s _ T , x _ 0 - x \rangle = \langle s _ T , x - x _ 0 \rangle . \end{align*}


\begin{align*} \forall z \in \Z ^ d : \sum _ { \Delta \Subset \Z ^ d } \sum _ { \xi _ \Delta } \sum _ { i = 1 } ^ q \norm { \nabla ^ i _ z c _ \Delta ( \cdot , \xi _ \Delta ) } _ \infty < \infty , \end{align*}


\begin{align*} & \langle H \rangle \\ = & \frac { 1 } { 2 \times 4 ^ 3 } \sum \limits _ { x \neq 0 } [ ( 2 + h \chi _ 4 ( x ) + \overline { h } \overline { \chi _ 4 } ( x ) ) ( 4 ( q - 3 ) - 4 h \chi _ 4 ( x ) - 4 \overline { h } \overline { \chi _ 4 } ( x ) ) ] \\ = & \frac { 1 } { 2 \times 4 ^ 3 } \sum \limits _ { x \neq 0 } [ 8 ( q - 5 ) + ( 4 h ( q - 3 ) - 8 h ) \chi _ 4 ( x ) + ( 4 \overline { h } ( q - 3 ) - 8 \overline { h } ) \overline { \chi _ 4 } ( x ) ] \\ = & \frac { ( q - 1 ) ( q - 5 ) } { 1 6 } . \end{align*}


\begin{align*} \mathbb B _ { k , S , g } ( \tau , z ) = v ^ { k } e ^ { - 4 \pi S [ y ^ t ] / v } B _ { k , S , g } ( \tau , z ; \tau , z ) . \end{align*}


\begin{align*} x y ' - x ' y = - n \ , . \end{align*}


\begin{align*} \delta ( \eta \gamma ^ n , \gamma ^ + ) + \delta ( \eta \gamma ^ n , ( \eta \gamma ^ n ) ^ - ) = f ( \eta \gamma ^ n \gamma ^ + , \eta \gamma ^ n ( \eta \gamma ^ n ) ^ - ) ) - f ( \gamma ^ + , ( \eta \gamma ^ n ) ^ - ) ) . \end{align*}


\begin{align*} D ( M _ { q } ) = \{ f \in \ell ^ { \infty } \ ; | \ ; q f \in \ell ^ { \infty } \} \end{align*}


\begin{align*} H _ { V } = \Delta + V ( \cdot ) , \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { f _ s ( x ) } { f ( x ) } = \lim _ { x \to \infty } \frac { f _ { + a } ( x ) } { f ( x ) } \cdot \frac { f _ { a r } ( x ) } { f _ { + a } ( x ) } \cdot \frac { f _ s ( x ) } { f _ { a r } ( x ) } = ( 1 - e ^ { - \lambda _ s } ) ^ { - 1 } . \end{align*}


\begin{align*} L b _ { 0 } + b _ { 0 } ^ { \prime } = q , \end{align*}


\begin{align*} \left ( \mathcal { L } _ { J \theta ^ \sharp } J \right ) ^ { a n t i - s y m } = J \left ( \mathcal { L } _ { \theta ^ \sharp } J \right ) ^ { a n t i - s y m } . \end{align*}


\begin{align*} T _ u v = \sum _ j S _ { j - 1 } u \Delta _ j v , R ( u , v ) = \sum _ { | j - j ' | \leq 1 } \Delta _ j u \Delta _ j ' v . \end{align*}


\begin{align*} 2 & = \limsup _ { x \to \infty } \frac { g _ u ^ { \ast 2 } \ast f _ r ( x ) } { g _ u ( x ) } \\ & \ge \limsup _ { x \to \infty } \frac { g _ u ^ { \ast 2 } ( x ) } { g _ u ( x ) } \liminf _ { x \to \infty } \frac { g _ u ^ { \ast 2 } \ast f _ r ( x ) } { g _ u ^ { \ast 2 } ( x ) } \\ & \ge \limsup _ { x \to \infty } \frac { g _ u ^ { \ast 2 } ( x ) } { g _ u ( x ) } , \end{align*}


\begin{align*} ( g _ { j k } ) = 2 \left ( \begin{array} { c c c } \omega ^ 2 & 0 & 0 \\ 0 & 0 & 2 \omega ^ { - 2 p + 2 } \\ 0 & 2 \omega ^ { - 2 p + 2 } & 0 \end{array} \right ) . \end{align*}


\begin{align*} ( x _ j ^ 2 - x _ k ^ 2 ) a _ { j k } b _ { j k } = 0 . \end{align*}


\begin{align*} \mathcal { E } _ { j } : = - { { \mathcal L } } \widetilde { u } _ { \mathrm { a p } , j } + \tilde { { \mathcal N } } _ { j } ( u _ { \mathrm { a p } , 1 } , u _ { \mathrm { a p } , 2 } ) . \end{align*}


\begin{align*} \partial _ t R _ 0 = g ( u _ 0 , R _ 0 ) . \end{align*}


\begin{align*} - x \cdot \nabla ( L u ) = f ' ( u ) \left ( x \cdot \nabla u \right ) , \end{align*}


\begin{align*} I ( x ) = \int _ { - \pi } ^ { \pi } s \left ( \frac { i ( 1 - e ^ { i \theta } ) } { 2 ( 1 + e ^ { i \theta } ) } \right ) \exp \left ( \frac { e ^ { i \theta } - 1 } { e ^ { i \theta } + 1 } x \right ) e ^ { i \theta ( m + 1 ) } \left ( e ^ { i \theta } + 1 \right ) ^ { n - m - 2 } d \theta . \end{align*}


\begin{align*} [ x \cdot y , \alpha ( z ) ] = \varepsilon ( y , z ) [ x , z ] \cdot \alpha ( y ) + \alpha ( x ) \cdot [ y , z ] . \end{align*}


\begin{align*} ( e ^ { \lambda \Lambda _ 1 } - 1 ) f _ 1 ( x ) / \breve g _ 1 ( x ) - \lambda \Lambda _ 1 g _ 1 ( x ) / \breve g _ 1 ( x ) & = \sum _ { n = 2 } ^ \infty ( ( \lambda \Lambda _ 1 ) ^ n / n ! ) \ , g ^ { \ast n } _ 1 ( x ) / \breve g _ 1 ( x ) \\ & = \sum _ { n = 2 } ^ \infty ( ( \lambda \Lambda _ 1 ) ^ n / n ! ) \ , \breve g ^ { \ast n } _ 1 ( x ) / \breve g _ 1 ( x ) \end{align*}


\begin{align*} f ( t ) = \exp _ { q ^ { \prime } } \left ( \pm \sigma _ { \epsilon } \ ; t \right ) = { } _ { 1 } F _ { 0 } \left ( - \frac { 1 } { 1 - q ^ { \prime } } ; \ ; ; \mp ( 1 - q ^ { \prime } ) \sigma _ { \epsilon } \ ; t \right ) . \end{align*}


\begin{align*} \textbf { \textit { K } } ( t , x ^ { \prime } , y ^ { \prime } ) = e ^ { - t \Lambda _ g } \delta ( x ^ { \prime } - y ^ { \prime } ) = \sum _ { k = 1 } ^ { \infty } e ^ { - t \tau _ k } \textbf { \textit { U } } _ k ( x ^ { \prime } ) \otimes \textbf { \textit { U } } _ k ( y ^ { \prime } ) . \end{align*}


\begin{align*} | a _ { C } ( n ) | \leq \left ( \sum _ { i = 1 } ^ s | c _ { i } | \right ) \tau ( n ) \sqrt { n } \end{align*}


\begin{align*} \mathcal { Z } \left ( \hat { \mu } \right ) \subset \bigcup _ { i = 1 } ^ { r } \rho ^ { - i } \bigcup _ { j = 0 } ^ { \infty } \frac { b _ { j } } { p ^ { j } N _ { j } } , \ 1 \leq i \leq r , \ b _ { j } \in \mathbb { Z } \setminus N _ { j } \mathbb { Z } . \end{align*}


\begin{align*} C : \mathbb { R } ^ { 3 } \rightarrow \mathfrak { s o } _ { 3 } \xi \mapsto C _ { \xi } C _ { \xi } \left ( y \right ) = \xi \times y \end{align*}


\begin{align*} A ( X ) : = \left ( \begin{array} { c } P _ H [ \Delta u - ( u \cdot \nabla ) u - \nabla n \cdot \Delta n ] \\ \Delta n - ( u \cdot \nabla ) n - \varphi ( n ) \end{array} \right ) . \end{align*}


\begin{align*} \gamma _ k = \frac { a _ k ^ 2 \lvert \mathbf { \hat { h } } _ k ^ { \textrm { T } } \mathbf { p } _ k \rvert ^ 2 } { \sum \limits _ { \substack { i = 1 \\ i \neq k } } ^ K a _ i ^ 2 \lvert \mathbf { h } _ k \mathbf { p } _ i \rvert ^ 2 + \sigma _ w ^ 2 } . \end{align*}


\begin{align*} \begin{aligned} P ( s I + x ) = \det ( s I + \operatorname { d i a g } ( \lambda _ 1 , \dots , \lambda _ n ) ) = \prod \limits _ 1 ^ n ( s + \lambda _ k ) . \end{aligned} \end{align*}


\begin{align*} \mbox { $ ( ( X _ { 2 , 6 } \setminus N ( c _ 2 ) ) \cup \{ c _ 6 \} ) \cap { \cal N } = \emptyset $ . } \end{align*}


\begin{align*} \sup _ { x } | G _ x ^ { ( n ) } | \le \frac { C } { n } , n = 1 , 2 , \ldots . \end{align*}


\begin{align*} \aligned \Phi _ \lambda ( u ) = \frac { 1 } { 2 } Q _ \lambda ( u ) - \frac { 1 } { 2 p } J ( u ) , \endaligned \end{align*}


\begin{align*} & \prod _ { j = k } ^ { x _ 2 } | \cos ( \pi \theta _ j ) ) | \leq C ( \varepsilon ) c _ { n , \ell } c _ { n , \ell + 1 } e ^ { ( - \ln { 2 } + \varepsilon ) | x _ 2 - k | } , \ \ \\ & \prod _ { j = x _ 1 } ^ k | \cos ( \pi \theta _ j ) ) | \leq C ( \varepsilon ) e ^ { ( - \ln 2 + \varepsilon ) | k - x _ 1 | } . \end{align*}


\begin{align*} g _ I = \big ( g _ { \{ i \} } \cup g _ { I \setminus \{ i \} } \big ) \setminus \big ( g _ { \{ i \} } \cap g _ { I \setminus \{ i \} } \big ) \ , , \forall i \in I . \end{align*}


\begin{align*} \lim _ { n \rightarrow + \infty } \int _ { \{ \Psi < - l \} } | f | _ h ^ 2 c _ t ^ n ( - \Psi ) = \int _ { \{ - t \leq \Psi < - l \} } | f | _ h ^ 2 . \end{align*}


\begin{align*} \epsilon _ 1 b _ { n - 1 } u _ n ( x ) \leq u ( y ) - u ( x ) + \sum _ { \alpha = 1 } ^ { n - 1 } ( y _ \alpha - x _ \alpha ) u _ \alpha ( x ) \leq C b _ { n - 1 } ^ 2 \end{align*}


\begin{align*} \begin{aligned} h ( x ) = & \ , h _ 0 + A \sin ( 1 0 x ) \\ & \ , + \left \{ \begin{array} { c c } ( x - \pi ) ^ 2 - 1 & x < \pi \\ \cos ( x - \pi ) - 2 & x \in [ \pi , \ , 2 \pi ) \\ ( x - 2 \pi ) ^ 2 - 3 & x \ge 2 \pi \end{array} \right . \end{aligned} \end{align*}


\begin{align*} \eta _ i : = \frac { \alpha ^ g ( X _ i , X _ i ) } { \langle A X _ i , X _ i \rangle } \in \Gamma ( T ^ \perp _ g M \otimes \mathbb { C } ) . \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \nabla ^ k [ P _ t ^ L ( x ' - y ' ) ] d y ' = 0 , \quad \forall t > 0 , \ , k \in \N . \end{align*}


\begin{align*} \widetilde { X } = \varprojlim _ { G ' \subset G } X _ { G ' } . \end{align*}


\begin{align*} \varphi _ 2 ( x ) = u ( x ) + t v _ 2 ( x ) \end{align*}


\begin{align*} T x = \frac { 1 } { \lambda } x . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } \mu ( h ^ { - k } ( A ) \cap B ) = \mu ( A ) \mu ( B ) ; \end{align*}


\begin{align*} \delta _ n : = \frac { \ln \| q _ n ( \theta - \frac { 1 } { 2 } ) \| - \ln \| q _ n \alpha \| } { q _ n } . \end{align*}


\begin{align*} a _ { i , i } ( z ) = \abs { z } ^ \gamma \sum _ { \stackrel { 1 \leq j \leq 3 } { j \neq i } } z _ j ^ 2 , a _ { i , j } ( z ) = - \abs { z } ^ \gamma z _ i z _ j \ \ { \rm f o r } \ \ i \neq j , \end{align*}


\begin{align*} A _ t + \mathbb { P } ( B \times u ) = - \mu \nabla \times B \textrm { o n } \ L ^ { 2 } _ { \sigma } ( \mathbb { R } ^ { 3 } ) . \end{align*}


\begin{align*} h = \nu & = 0 \\ \lambda \varepsilon _ 1 \delta _ 1 = \lambda \varepsilon _ 1 \delta _ 4 & = 0 \\ \varepsilon _ 2 \delta _ 2 & = \varepsilon _ 3 \delta _ 3 \\ \varepsilon _ 2 \delta _ 5 & = \varepsilon _ 3 \delta _ 6 \\ \lambda ^ 2 + \rho ^ 2 - \varepsilon _ 1 \mu \delta _ 1 + \varepsilon _ 2 \rho \delta _ 2 & = 0 \\ \lambda ^ 2 + \rho ^ 2 - \varepsilon _ 1 \mu \delta _ 4 + \varepsilon _ 2 \rho \delta _ 5 & = 0 . \end{align*}


\begin{align*} f ( x ) = ( e ^ \lambda - 1 ) ^ { - 1 } \sum _ { n = 1 } ^ \infty ( \lambda ^ n / n ! ) \ , g ^ { \ast n } ( x ) . \end{align*}


\begin{align*} I _ \delta ( u ) = \frac { \delta ^ 2 } { 2 } \int _ { \Omega } \left ( K ( x ) \nabla u | \nabla u \right ) - \frac { 1 } { p + 1 } \int _ { \Omega } ( u - q ) ^ { p + 1 } _ + . \end{align*}


\begin{align*} \lambda _ { m - l + 1 } + \mu _ { 1 } = l - 1 . \end{align*}


\begin{align*} \langle V , z _ \tau \rangle = v _ \tau + \sum _ { q = 1 } ^ { s + 1 } ( - 1 ) ^ q v _ { \tau _ q } , \end{align*}


\begin{align*} K _ i = \langle \{ a _ i , b _ i , c _ i \} , \{ a _ i , c _ i , d _ i \} , \{ b _ i , d _ i \} \rangle , \ L _ i = \langle \{ a _ i , b _ i , c _ i \} , \{ b _ i , d _ i \} \rangle . \end{align*}


\begin{align*} 8 ^ { n - m } ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } = n ! \det ( A , \dots , A , 8 I , \dots , 8 I ) \Omega _ { 2 n } , \end{align*}


\begin{align*} c = 4 + ( n + 3 ) \left ( n + 2 + \norm { \tilde \Psi } _ { H ^ 1 } ^ 2 \right ) . \end{align*}


\begin{align*} \hat { f } ( \pi ) = \int _ { G } f ( g ) \pi ( g ) d g \end{align*}


\begin{align*} f ' ( x ) = f ( x + r ) = a x ^ { q + 1 } + b ' x ^ q + c ' x . \end{align*}


\begin{align*} - H ( \rho ^ \varepsilon _ 0 ) ( z _ l ) \geq \frac { 1 } { 2 R } \phantom { x x } \phantom { x x } H ( \rho ^ \varepsilon _ 0 ) ( z _ l ) + H ( \rho ^ \varepsilon _ 0 ) ( z _ r ) = \frac { 2 \varepsilon \rho _ 0 } { R ^ 2 - \varepsilon ^ 2 \rho _ 0 ^ 2 } \leq \frac { 4 \varepsilon \rho _ 0 } { R ^ 2 } \end{align*}


\begin{align*} \operatorname { E } ( X ^ r ) = \frac { \theta ^ { r / 2 } } { \operatorname { B } ( a , b ) } \ , S _ r ( a , b ) , r < 2 . \end{align*}


\begin{align*} \lim _ { n \to \infty } | g _ { n + 1 } | \sum _ { i = n + 1 } ^ { \infty } \psi _ i ( t _ p ) = 0 . \end{align*}


\begin{align*} d \mu ( \xi , \eta ) = d _ \epsilon ( \xi , \eta ) ^ { - 2 h ( g ) / \epsilon } \cdot d v _ p ( \xi ) d v _ p ( \eta ) \end{align*}


\begin{align*} & P \otimes _ { G } Q \rightarrow P \otimes _ { G } Q \\ & p \otimes q \mapsto p \kappa ^ { - 1 } \otimes \kappa q = p \otimes q \end{align*}


\begin{align*} \begin{aligned} \mathsf { M } ^ n = & \{ \mathsf { m } _ 1 , \cdots , \mathsf { m } _ n \} \\ = & \left \{ g \bigl ( \xi _ 0 \bigr ) , g \bigl ( \xi _ 1 ^ { \mathsf { M } } \bigr ) , \cdots , g \bigl ( \xi _ { n - 1 } ^ { \mathsf { M } } \bigr ) \right \} . \end{aligned} \end{align*}


\begin{align*} p _ { S } ( x ) \le \sum _ { i = 1 } ^ { d + 1 } \frac { d y } { \Omega _ d \cdot d ( y , H _ i ) ^ d } \le \frac { d + 1 } { \Omega _ d } \cdot \sup _ { x \in \mathbb R ^ n } \int _ { P _ { 1 , x } } \frac { d y } { d ( y , H _ 1 ) ^ d } . \end{align*}


\begin{align*} A = \left ( \begin{array} { c c } B & C \\ C ^ T & D \end{array} \right ) \end{align*}


\begin{align*} I ( u ) = m | | u | | ^ 2 _ { L ^ 2 ( \mathbb { H } ^ n ) } + | | \nabla _ { H } u | | ^ 2 _ { L ^ 2 ( \mathbb { H } ^ n ) } - { \rm R e } \int _ { \mathbb { H } ^ n } f ( u ) \overline { u } d x . \end{align*}


\begin{align*} n _ t = \nabla \cdot ( \nabla n - \chi ( c ) n \nabla c ) , \quad c _ { t } = \Delta c - n c , \qquad x \in \Omega , \ , t > 0 , \end{align*}


\begin{align*} C \geq h ( x , t ) - h ( x , 0 ) = \int _ { 0 } ^ { t } ( P - h ) ( x , t ) d t . \end{align*}


\begin{align*} \sum _ { k } \varphi _ k ( d \phi _ { i k } + \Omega _ { i j } ) & = \sum _ k \varphi _ k d \phi _ { i k } + \sum _ l \varphi _ l \phi _ { i l } \wedge \Big ( \sum _ k \varphi _ k \phi _ { k l } \Big ) = \sum _ k \varphi _ k d \phi _ { i k } + \sum _ l \varphi _ l \phi _ { i l } \wedge ( 2 \varphi _ l \phi _ { l l } ) \\ & = \sum _ k \varphi _ k d \phi _ { i k } - \sum _ { l } \phi _ { i l } \wedge d \varphi _ l = d \Big ( \sum _ k \varphi _ k \phi _ { i k } \Big ) = 0 . \end{align*}


\begin{align*} & \overline { \mathcal { P } ^ - ( f ( z ) e ^ { 2 i z x } ) } = - \lim _ { \varepsilon \rightarrow 0 } \frac { 1 } { 2 \pi i } \int _ { R } \frac { \bar { f } ( s ) e ^ { - 2 i s x } } { s - ( z + i \varepsilon ) } d s = - \mathcal { P } ^ + ( \bar f ( z ) e ^ { - 2 i z x } ) \\ & = - \int _ { 2 x } ^ { + \infty } \widehat { \bar { f } \ } ( \xi ) e ^ { i z ( \xi - 2 x ) } \mathrm { d } \xi . \end{align*}


\begin{align*} Z ^ 1 ( v , s ) = \Psi ^ q + v \end{align*}


\begin{align*} u _ { t t } - \Delta u + b ( t , x ) | u _ t | ^ { m - 1 } u _ t + | u | ^ { p - 1 } u = 0 , \end{align*}


\begin{align*} G ^ { ( 1 ) } \vee . . . \vee G ^ { ( k ) } = \vee _ { \gamma \in \Gamma } G ^ { ( \gamma ) } . \end{align*}


\begin{align*} q = \frac { b _ { 0 } ^ { \prime \prime } + b _ { 0 } ^ { \prime } } { b _ { 0 } + 1 } . \end{align*}


\begin{align*} \sum _ { q = r + 1 - \sum _ { d = j } ^ { i + 1 } \ell _ { d } } ^ { r - \sum _ { d = j + 1 } ^ { i + 1 } \ell _ { d } } a _ q ^ 2 \leq \frac { \beta ^ 2 } { g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor \big ) ^ 2 } \leq \frac { h ^ 2 } { 1 6 ^ 2 } \cdot 1 6 ^ { j - i } . \end{align*}


\begin{align*} { \bf c } _ { 2 , \mu } ( t ) = \left ( \begin{array} { c } C ^ { ( p ) } _ { 0 , 0 , \mu } ( t ) \\ \vdots \\ C ^ { ( p ) } _ { n , n , \mu } ( t ) \end{array} \right ) \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { r _ n ^ 2 } { h _ n } = 0 . \end{align*}


\begin{align*} \rho ^ { \lambda } _ { k } ( f ) e ^ { n - 1 } _ { k , \lambda } ( z , t ) = e ^ { i \lambda t } f ^ { - \lambda } \ast _ { - \lambda } \varphi _ { k , \lambda } ^ { n - 1 } ( z ) . \end{align*}


\begin{align*} S _ { R } ^ { \alpha } f ( x ) = \sum _ { k = - \infty } ^ { \infty } S _ { R } ^ { \alpha } f _ { k } ( x ) . \end{align*}


\begin{align*} \mathbb { E } { \bf g } _ { \rm G E } ( G ) = + \infty . \end{align*}


\begin{align*} g _ d ( \ell ) = \begin{cases} \ell , & d = 1 , \\ \log \ell , & d = 2 , \\ 1 , & d \geq 3 , \end{cases} \end{align*}


\begin{align*} \Gamma = \bigoplus _ { n = 1 } ^ \infty \Gamma _ n , F = \prod _ { n = 1 } ^ \infty F _ n \quad G = \Gamma \rtimes F \end{align*}


\begin{align*} _ { \rm r e f } [ V _ \lambda ] = q ^ { - \frac { c } { 2 4 } } q ^ { \frac { ( \lambda | \lambda + 2 \rho ) } { 2 ( k + h ^ \vee ) } - \lambda ( x + y ) } \ \prod _ { i = 1 } ^ L X _ i \ \prod _ { 1 \leq i < j \leq L } X _ { i , j } \end{align*}


\begin{align*} K f = \int _ { - \infty } ^ x u _ y ( y ) e ^ { 2 i z y } d y \int _ { - \infty } ^ { y } \bar { u } _ s ( s ) e ^ { - 2 i z s } f d s , \end{align*}


\begin{align*} f ( z ) - f ( 0 ) & = \int _ { 0 } ^ 1 \left ( \frac { \partial f } { \partial x } ( t z ) x - \frac { \psi _ 0 } { \psi _ 1 } y \frac { \partial f } { \partial x } ( t z ) \right ) d t \\ & = \int _ { 0 } ^ 1 \left ( x - \frac { \psi _ 0 } { \psi _ 1 } y \right ) \frac { \partial f } { \partial x } ( t z ) d t = \left ( x - \frac { \psi _ 0 } { \psi _ 1 } y \right ) h _ 1 ( z ) , \\ \end{align*}


\begin{align*} \mathcal { H } ^ { \prime } ( t ) = & \lambda ( t ) \mathcal { E } ^ { \prime } ( t ) + \lambda ^ { \prime } ( t ) \mathcal { E } ( t ) + \mu \alpha ^ { \prime } ( t ) \mathcal { E } ^ r ( t ) \langle P ( u _ t ) , u \rangle + \mu \alpha ( t ) \mathcal { E } ^ r ( t ) \frac { d } { d t } \langle P ( u _ t ) , u \rangle \\ & + \mu r \alpha ( t ) \mathcal { E } ^ { r - 1 } ( t ) \mathcal { E } ^ { \prime } ( t ) \langle P ( u _ t ) , u \rangle . \end{align*}


\begin{align*} \chi _ i \cap \psi _ i & = \left \{ \begin{aligned} & \psi _ i & & ( i \neq j ) \\ & \sigma \cap \psi _ j & & ( i = j ) \end{aligned} \right . \\ & = \left \{ \begin{aligned} & \psi _ i & & ( i \neq j ) \\ & \psi _ j \setminus \{ x \} & & ( i = j ) , \end{aligned} \right . \end{align*}


\begin{align*} P _ { \min } ^ { \rm { S - N } } \left ( { \overline S , \overline \varepsilon , \overline R } \right ) = \mathop { \arg \min } \limits _ { { W _ m } \in \left [ { 0 , W } \right ] } \ ; \left ( { { p _ K } \left ( { \bar S , \bar \varepsilon , { W _ m } } \right ) + p _ { b , m } ^ * + p _ { b , o } ^ * } \right ) . \end{align*}


\begin{align*} \sum _ { m = 0 } ^ { n } \left ( - 1 \right ) ^ { m } \sum _ { k = 0 } ^ { m } \binom { m } { k } \binom { n - m } { k } \rho ^ { k } = \left \{ \begin{array} { c c c } 0 & & n \\ ( 1 - \rho ) ^ { n / 2 } & & n \end{array} \right . . \end{align*}


\begin{align*} \begin{aligned} & A _ { b e n } y '' _ { b e n } + M _ { b e n } y _ { b e n } = E y _ { b e n } \\ & A _ { n b } y '' _ { n b } + M _ { n b } y _ { n b } = E y _ { n b } . \end{aligned} \end{align*}


\begin{align*} \gamma ( u ^ q + u ) + ( \gamma + 1 ) ( v ^ q + v ) & = c , \\ \gamma u ^ { q + 1 } + ( \gamma + 1 ) v ^ { q + 1 } & = d , \end{align*}


\begin{align*} F _ i ( x ) = \tilde { F } _ i ( x ) H ( x ) + \frac { e } { \| \hat { F } _ i ( x ) \| } \hat { F } _ i ( x ) . \end{align*}


\begin{align*} \norm { h _ 1 } _ { L ^ 2 } ^ 2 & = \int _ \S \left [ \prod _ { i = 1 } ^ n ( \eta _ i ( x + r ) - \eta _ i ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( x + r ) - \tilde \Psi ( x ) ) \right ] ^ 2 \ \d x \\ & \le \int _ \S \prod _ { i = 1 } ^ n ( \eta _ i ( x + r ) - \eta _ i ( x ) ) ^ 2 \ \d x \\ & \le \prod _ { i = 1 } ^ n \norm { \partial \eta _ i } _ { L ^ 2 } ^ 2 . \end{align*}


\begin{align*} \lim _ { n \to \infty } \Pr ( K = k ( \theta _ 0 ) \mid X _ 1 , \ldots , X _ n ) = 1 ~ ~ ~ \mathrm { a . s . } [ P _ { \theta _ 0 } ] . \end{align*}


\begin{align*} \phi = \frac { ( 2 n + 1 ) \pi } { N } , \end{align*}


\begin{align*} ( \sqrt { - 1 } & \Theta _ { T M } + \varpi \otimes I d _ { T M } ) ( \kappa _ 1 \otimes \kappa _ 2 , \kappa _ 1 \otimes \kappa _ 2 ) \ge 0 \\ & \forall \kappa _ 1 , \kappa _ 2 \in T M w i t h \langle \kappa _ 1 , \kappa _ 2 \rangle = 0 \end{align*}


\begin{align*} \tilde { \xi } ( g \gamma ) = \tilde { \xi } ( g ) , \qquad g \in G ^ \circ , \gamma \in \Gamma ^ \circ , \end{align*}


\begin{align*} \kappa _ { V , I } : = \sum _ { J \subset I } ( - 1 ) ^ { | J | - | I | } K _ { V , J } . \end{align*}


\begin{align*} L _ 2 ( x ) = C \displaystyle | b ( x + h ) - b ( x ) | g _ \gamma ( f ) ( x ) \leq C \displaystyle | h | g ( f ) ( x ) . \end{align*}


\begin{align*} \Omega : = \bigcup _ { \theta \colon e ^ { i \theta } [ 0 , + \infty ) \subset \bar { \Delta } } \Omega _ { \theta } , \end{align*}


\begin{align*} H _ { x , u } ^ + & = \{ y \in \mathbb R ^ n : \langle y , u \rangle \ge \langle x , u \rangle \} \\ H _ { x , u } & = \partial H _ { x , u } ^ + = \{ y \in \mathbb R ^ n : \langle y , u \rangle = \langle x , u \rangle \} , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { m } \alpha _ i f _ i ( x , t , u ) \le k _ 0 + k _ 1 \sum _ { i = 1 } ^ m u _ i , \forall ( x , t ) \in \Omega \times \R _ + , \end{align*}


\begin{align*} d ( x ) & : = \max \{ 1 \le l \le n \ , : \ , \ , \\ & \qquad \} . \end{align*}


\begin{align*} \mathcal { Q } ( u ) : = u _ t - A ( x , u , u _ x ) u _ { x x } + B ( x , u , u _ x ) , \end{align*}


\begin{align*} \begin{aligned} \abs { c _ 2 ( S ^ 2 \setminus S _ i ) } \le \abs { c _ 1 ( S ^ 1 \setminus S _ i ) } + \abs { c _ 2 ( P ^ 2 ) } & \le \abs { S } - \abs { S \cap S _ i } + q - q _ i + s _ 2 \\ & = ( 2 k - t ' - p ) - \abs { S \cap S _ i } + q - q _ i + ( p - q - s _ 1 ) \\ & = 2 k - t ' - s _ 1 - \abs { S \cap S _ i } - q _ i . \end{aligned} \end{align*}


\begin{align*} | F _ y ( x , y ) - p _ y ( x , y ) | = | F _ y ( x , y ) - h ^ y ( x , y ) | < 2 \varepsilon . \end{align*}


\begin{align*} ( I - Q ) F ^ q ( B _ \phi u + \psi ( B _ \phi u , p ) , p ) = 0 . \end{align*}


\begin{align*} p _ k : = \sum _ { i = 1 } ^ { k } \left ( x _ 1 \cdots \widehat { x _ i } \cdots x _ k \right ) ^ { p - 1 } \end{align*}


\begin{align*} o _ { k } ( 1 ) = \int _ { \R ^ { 2 } } | \nabla \widetilde { u } _ { k } | ^ { 2 } + \widetilde { u } _ { k } ^ { 2 } + 3 B ( \widetilde { u } _ { k } ) - \int _ { \R ^ { 2 } } \widetilde { u } _ { k } ^ { 2 p } , \end{align*}


\begin{align*} F ( \eta ) : = T - \frac { 1 } { \eta } I , \eta \in \Omega . \end{align*}


\begin{align*} \mathbf { y } ( t ) = \mathbf { a } ( \theta _ { 0 } ) s ( t ) + \mathbf { v } ( t ) , \end{align*}


\begin{align*} \sum _ { \sigma \subset \mu _ \ell } c _ { \ell } x _ { \ell , \omega ( \mu _ \ell ) - \omega ( \kappa ) } + c _ { \kappa } & = 0 \sigma \subset \kappa \\ \sum _ { \sigma \subset \mu _ \ell } c _ { \ell } x _ { \ell , \omega ( \mu _ \ell ) - \omega ( \kappa ) } & = 0 \sigma \not \subset \kappa . \end{align*}


\begin{align*} ( \Lambda - 1 + \delta ) \sum _ { j = 1 } ^ k ( \Lambda + \varepsilon ) ^ { - j } > 1 , \end{align*}


\begin{align*} \begin{aligned} \Delta u _ 1 \wedge \dots \wedge \Delta u _ k & \wedge \beta _ n ^ { n - m } ( \omega ) = \int u _ k \Delta u _ 1 \wedge \dots \wedge \Delta u _ { k - 1 } \wedge \beta _ n ^ { n - m } \wedge \Delta \omega \end{aligned} \end{align*}


\begin{align*} \beta _ 1 \oplus \beta _ 2 = \begin{cases} \emptyset & i _ 2 < j _ 1 , \\ \{ ( \alpha _ { i _ 1 , \overline { j } _ 1 } , \alpha _ { j _ 2 , \overline { i } _ 2 } ) , ( \alpha _ { i _ 1 , \overline { j } _ 2 } , \alpha _ { j _ 1 , \overline { i } _ 2 } ) \} & j _ 1 \leq j _ 2 < i _ 2 , \\ \{ ( \alpha _ { i _ 1 , \overline { j } _ 1 } , \alpha _ { i _ 2 , \overline { j } _ 2 } ) \} & j _ 1 \leq i _ 2 \leq j _ 2 . \\ \end{cases} \end{align*}


\begin{align*} \nabla _ { V } \hat X & = \omega ( V ) \\ \nabla _ V \omega & = - R ( \hat X , V ) + T _ h ( V ) . \end{align*}


\begin{align*} \mathbf { a } ( \theta _ { 0 } ) & = [ 1 , e ^ { - j 2 \pi \frac { d } { \lambda } { s i n } \theta _ 0 } , \cdots , e ^ { - j 2 \pi ( N - 1 ) \frac { d } { \lambda } { s i n } \theta _ 0 } ] ^ { T } \\ & = [ 1 , e ^ { - j \phi } , e ^ { - j 2 \phi } , \cdots , e ^ { - j ( N - 1 ) \phi } ] ^ { T } , \end{align*}


\begin{align*} \delta _ { o p t } ( \Gamma ) = \frac { \frac { 1 } { 1 6 \sqrt { 3 } } + \frac { 1 } { 2 4 } \left ( 2 \sqrt { 3 } + 3 \right ) } { v o l ( \widehat { A V } _ 3 ) } \approx 0 . 8 3 8 8 2 5 . \end{align*}


\begin{align*} \widetilde { H } ( x _ 0 + \Delta x , Y _ 0 + \Delta Y ) - \widetilde { H } ( x _ 0 , Y _ 0 ) = [ D \widetilde { H } ] ( x _ 0 , Y _ 0 ) ( \Delta x , \Delta Y ) + o ( | \Delta x | + \| \Delta Y \| _ { L ^ 2 } ) . \end{align*}


\begin{align*} \min _ { \xi , \eta } p ( \xi , \eta ) = \min \left \{ \min _ { \xi } p \left ( \xi , \frac 1 2 \right ) , \min _ { \xi } p \left ( \xi , \frac { 9 9 } { 1 0 0 } \right ) \right \} . \end{align*}


\begin{align*} \omega : = \R \setminus B _ R ( 0 ) = \{ x \in \R : \Vert x \Vert \ge R \} \end{align*}


\begin{align*} \mathbb { E } \left [ y _ l ^ * y _ j | \hat { \mathbf { H } } \right ] = & a _ c ^ 2 \hat { \phi } ^ { \left ( l , c \right ) ^ * } \hat { \phi } ^ { \left ( j , c \right ) } + \sum _ { q = 1 } ^ M a _ q ^ 2 \hat { \phi } ^ { \left ( l , q \right ) ^ * } \hat { \phi } ^ { \left ( j , q \right ) } . \end{align*}


\begin{align*} \vert \{ g \in G \ \vert \ ( g K ) \cap K \neq \emptyset \} \vert = \sum \limits _ { j \in J } \vert I _ j \vert < + \infty , \end{align*}


\begin{align*} \kappa : = D _ { a , n } = \begin{pmatrix} a \\ & a ^ { - 1 } \\ & & \ddots \\ & & & a \\ & & & & a ^ { - 1 } \end{pmatrix} . \end{align*}


\begin{align*} c _ k : = \frac { N _ k ( A ) } { r ( A ) ^ k } \ \end{align*}


\begin{align*} S = \{ s _ \alpha \mid \alpha \in \Delta \} \subseteq W . \end{align*}


\begin{align*} V _ p ( A [ n - 1 ] , B ) = \frac { 1 } { n } \int \limits _ { S ^ { n - 1 } } h ^ p _ B ( u ) h _ A ^ { 1 - p } d S _ A ( u ) . \end{align*}


\begin{align*} \phi ( y ) = \sum _ { z \in \partial I ( y ) } G _ { I ( y ) } ( z , y ) \phi ( z ' ) . \end{align*}


\begin{align*} 1 = \sum _ { y = 0 } ^ n \int _ 0 ^ { \theta } { \frak G } _ { x , y } ( s ) \dd s , x = 0 , 1 , \ldots , n \end{align*}


\begin{align*} \int _ { \partial \Omega } \nabla c \cdot \nu c \psi ^ { 2 } = \int _ { \partial \Omega \cap B _ { \eta } } \nabla c \cdot \nu c \psi ^ { 2 } . \end{align*}


\begin{align*} H \bigg ( \mathrm { L a w } \bigg ( \sum _ { i = 1 } ^ n t _ i X _ i \bigg ) \bigg ) \le \sum _ { i = 1 } ^ n t _ i H ( Q _ i ) . \end{align*}


\begin{align*} \chi \cdot ( [ K \delta K ] ( \phi ) ) ( \sigma , g ) & = \chi ( \det g ) ( [ K \delta K ] ( \phi ) ) ( \sigma , g ) \\ & = \chi ( \det g ) \sum _ i \delta _ i \cdot \phi ( \sigma , g \delta _ i ) . \end{align*}


\begin{align*} \mathbf { q } _ c = ( \mathbf { x } _ c , \mathbf { y } _ c , \mathbf { s } _ c , \kappa _ c , \tau _ c ) = ( \mathbf { e } , 0 , \mathbf { e } , 1 , 1 ) . \end{align*}


\begin{align*} L _ x v = B ( x , v ) + B ( v , x ) \end{align*}


\begin{align*} \tilde h _ d ( \alpha _ d ) & \leq O ( 1 ) + \log \frac { d \cdot \Gamma ( \frac { d } { 2 } ) } { \alpha _ d d ^ { \frac { d - 1 } { 2 } } } + d \log 2 \\ & = O ( 1 ) + \log \frac { \Gamma ( \frac { d } { 2 } ) } { C e ^ { - c d } d ^ { \frac { d - 1 } { 2 } } } + d \log 2 . \end{align*}


\begin{align*} \left | \frac { e ^ { \kappa s } - 1 } { s } \right | & = \left | \sum _ { n = 0 } ^ \infty \frac { \kappa ^ { n + 1 } s ^ n } { ( n + 1 ) ! } \right | \leq e ^ { \frac { 3 } { 4 } \kappa } \kappa , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { k _ 2 } \left ( 1 - \frac { i } { n - X ( t ) } \right ) = 1 + O ( 1 / n ) . \end{align*}


\begin{align*} \mathbf { w } _ { x s } ^ T \mathbf { e } = \mathbf { s } ^ T \mathbf { T } _ x \mathbf { e } = \mathbf { s } ^ T \mathbf { x } = k \mu , \end{align*}


\begin{align*} H ^ { \lambda } _ { G } ( \sigma ) = \left \{ \begin{array} { l l } J \sum \limits _ { x \in { V } } { \ln \lambda _ { \sigma ( x ) } , } \ \ \ $ i f $ \sigma \in \Omega ^ G $ , $ \\ + \infty , \ \ \ \ \ \ \ \ \ \ \ \ \ $ \ i f $ \sigma \ \notin \Omega ^ G $ , $ \\ \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} \alpha & : = \sup \{ t \in [ 0 , 1 ) \mid F _ E ( t ) > t \} , \\ \beta & : = \inf \{ t \in [ 0 , 1 ) \mid F _ E ( t ) < t \} , \end{aligned} \end{align*}


\begin{align*} \mu ( T ^ { 2 r _ n } A \cap A ) & \geq \mu ( A \cap T ^ { - r _ n } A \cap T ^ { r _ n } A ) \\ & > 2 \big ( \beta - \frac { 1 } { 2 } \big ) \mu ( A ) \\ & = \big ( 2 \beta - 1 \big ) \mu ( A ) > \mu ( A ) ^ 2 . \end{align*}


\begin{align*} & \limsup _ { N \to \infty } \frac { 1 } { \# ( \Lambda _ N ^ { \vec { v } } ( 1 ) ) } \sum _ { ( m , n ) \in \Lambda _ N ^ { \vec { v } } ( 1 ) } \min _ { 1 \leq i < j \leq k } d ( T ^ { ( m , n ) } x _ i , T ^ { ( m , n ) } x _ j ) \\ \geq & \limsup _ { N \to \infty } \frac { 1 } { 2 N } \sum _ { n = 0 } ^ { N - 1 } \min _ { 1 \leq i < j \leq k } { d } ( T ^ { ( n , [ n \beta + t ] ) } x _ i , T ^ { ( n , [ n \beta + t ] ) } x _ j ) \geq \eta / 2 > 0 . \end{align*}


\begin{align*} d I _ 3 \cap ( I _ 1 \cup I _ 2 \cup I _ 3 ) = \emptyset , \end{align*}


\begin{align*} \frac { \partial s _ { \delta , i } } { \partial z _ { i , h } } = O ( \delta | \ln \delta | ^ { \frac { p - 1 } { 2 } } ) . \end{align*}


\begin{align*} \alpha v = \theta v , \end{align*}


\begin{align*} \begin{aligned} h \left ( E ^ \Phi _ A ( x ) \right ) & = E ^ \Psi _ B \left ( h ( x ) \right ) \\ h \left ( P ^ \Phi _ { \bar A } ( x ) \right ) & = P ^ \Psi _ { \bar B } \left ( h ( x ) \right ) . \\ \end{aligned} \end{align*}


\begin{align*} X _ n : = \{ v \in L ^ 2 ( D ) : v | _ { K } \in P ^ \ell ( K ) , K \in { \mathcal T } _ n \} , \end{align*}


\begin{align*} F ^ q ( v , p ) : = F ( \Psi ^ q + v , p ) , F ^ q \colon X \times \mathcal P \to X . \end{align*}


\begin{align*} \ell \left ( \bar \phi \left ( X ^ { ( N _ { r _ 0 } ( v ) ) } \right ) \right ) \leq \ell \left ( X ^ { ( N _ { r _ 0 } ( v ) ) } \right ) - 1 = | N _ { r _ 0 } \left ( B \right ) | \ell ( X ) - 1 . \end{align*}


\begin{gather*} \gamma ^ + ( u ) ( x ) \ne \gamma ^ - ( u ) ( x ) , \\ [ 5 p t ] \gamma ^ + ( u ) + s \gamma ^ + ( v ) ( x ) = \gamma ^ - ( u ) + s \gamma ^ - ( v ) ( x ) , \end{gather*}


\begin{align*} \sum _ i F ^ { i i } h _ { i i 1 1 } = - \sum _ { p , q , r , s } F ^ { p q , r s } h _ { p q 1 } h _ { r s 1 } = - \sum _ { p \neq q } F ^ { p p , q q } h _ { p p 1 } h _ { q q 1 } + \sum _ { p \neq q } F ^ { p p , q q } h _ { p q 1 } ^ 2 . \end{align*}


\begin{align*} Z _ { q } ( \beta ) = \frac { 1 } { h ^ { D N } } \displaystyle \int _ { 0 } ^ { \infty } \ ; d \bar { x } \ ; d \bar { p } \ ; \exp _ { q } \left ( - \beta \displaystyle \sum _ { i = 1 } ^ { D N } \left ( \frac { p _ { i } ^ { 2 } } { 2 m } + \frac { 1 } { 2 } m \omega ^ { 2 } x _ { i } ^ { 2 } \right ) \right ) , \end{align*}


\begin{align*} d _ \square ( W _ 1 , W _ 2 ) : = \sup _ { S , T \subset [ 0 , 1 ] } \bigg | \int _ { S \times T } \big ( W _ 1 ( u , v ) - W _ 2 ( u , v ) \big ) d u d v \bigg | , \end{align*}


\begin{align*} \sup _ { n } \mathbb { E } _ { \mathbb { P } } \left [ \exp \left \{ \kappa \sum _ { i = 1 } ^ { n } \int _ { T _ { 0 } } ^ { T _ { 0 } + \delta } \triangle K _ { t } ^ { i , n } \cdot d W _ { t } ^ { i } - \frac { \kappa } { 2 } \sum _ { i = 1 } ^ n \int _ { T _ { 0 } } ^ { T _ { 0 } + \delta } \left | \triangle K _ { t } ^ { i , n } \right | ^ { 2 } d t \right \} \right ] \end{align*}


\begin{align*} \begin{aligned} b _ { 1 , \delta } ( x ^ \star ) = & \ , 2 \int _ { 0 } ^ { 1 / 2 } h ( x ^ \star + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \\ & - 2 \int _ { 0 } ^ { 1 / 2 } h ( x ^ \star + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t = 0 . \end{aligned} \end{align*}


\begin{align*} - K _ F = ( n - 1 ) A _ X \vert _ F , \end{align*}


\begin{align*} \Big ( ( \bar { \bf q } ( 0 ) , \bar { \bf p } ( 0 ) ) , C ( 0 ) \Big ) = \Big ( ( \bar { \bf q } ( \theta ) , \bar { \bf p } ( \theta ) ) , C ( \theta ) \Big ) . \end{align*}


\begin{align*} \phi _ { \lambda } ( [ a _ { \mu } b ] ) = [ \phi _ { \lambda } ( a ) _ { \lambda + \mu } b ] + [ a _ { \mu } \phi _ { \lambda } ( b ) ] , \forall \ a , b \in \mathcal { A } . \end{align*}


\begin{align*} \left ( q , \left ( x , p \right ) \right ) \mapsto L _ { \left ( 0 , q \right ) } R _ { \left ( 0 , \bar { q } \right ) } \left ( x , p \right ) = \left ( q x \bar { q } , q p \bar { q } \right ) = \left ( I _ { q } \left ( x \right ) , q p \bar { q } \right ) \end{align*}


\begin{align*} \begin{aligned} & z '' _ { b e n } + ( M _ { b e n } - E ) A _ { b e n } ^ { - 1 } z _ { b e n } = 0 \\ & z '' _ { n b } + ( M _ { n b } - E ) A _ { n b } ^ { - 1 } z _ { n b } = 0 , \end{aligned} \end{align*}


\begin{align*} \varphi _ \alpha ( z ) = ( ( z + \alpha ) \vee 0 ) + ( ( z - \alpha ) \wedge 0 ) . \end{align*}


\begin{align*} \prod _ { j = x _ 1 } ^ k | \cos ( \pi \theta _ j ) | \leq e ^ { - ( \ln 2 - \varepsilon ) | x _ 1 - k | } c _ { n , \ell - 1 } , \prod _ { j = k } ^ { x _ 2 } | \cos ( \pi \theta _ j ) | \leq e ^ { - ( \ln 2 - \varepsilon ) | x _ 2 - k | } c _ { n , \ell } . \end{align*}


\begin{align*} \mbox { \rm g r a p h } \left . \tau \right | _ { { \cal A } _ \delta } = \{ ( x _ a , y _ a ) \in { \cal A } _ \delta \times \mathbb { R } \ ; : \ ; y _ a = \tau ( x _ a ) \} \end{align*}


\begin{align*} \nabla \partial _ t R _ 0 = \partial _ 1 g ( u _ 0 , R _ 0 ) \nabla u _ 0 + \partial _ 2 g ( u _ 0 , R _ 0 ) \nabla R _ 0 \in L ^ 2 ( ( 0 , T ) \times \Omega ) \end{align*}


\begin{align*} q - h \ge \log _ { \theta } \Big ( 1 + \frac { d } { \epsilon _ 0 } \Big ) = \log _ { \theta } ( 1 + b _ 0 d ) ~ { \rm f o r } ~ h = 0 , 1 , \dots , q - 1 . \end{align*}


\begin{align*} g ' ( s ) & = \frac { ( \alpha - 1 ) ( s - a ) ^ { \alpha - 2 } ( b - s ) ^ { \alpha - 1 - \beta } - ( \alpha - 1 - \beta ) ( s - a ) ^ { \alpha - 1 } ( b - s ) ^ { \alpha - 2 } } { ( b - a ) ^ { \alpha - 1 - \beta } \Gamma ( \alpha ) } \\ & = \frac { ( s - a ) ^ { \alpha - 2 } ( b - s ) ^ { \alpha - 2 - \beta } \left [ ( \alpha - 1 ) ( b - s ) - ( \alpha - 1 - \beta ) ( s - a ) \right ] } { ( b - a ) ^ { \alpha - 1 - \beta } \Gamma ( \alpha ) } . \end{align*}


\begin{align*} D _ { x , 2 } & = \frac { 1 } { 2 L _ { x y } } \left ( \sqrt { n \sigma _ { G , r } ^ 2 } + L _ { y } \sqrt { n } M _ y + \| \nabla G ( Y ^ * ) \| \right ) \\ & + \frac { 1 } { 2 \mu _ { x } } \left ( \sqrt { n \sigma _ { F , r } ^ 2 } + L _ { x } \sqrt { n } M _ x + \| \nabla F ( X ^ * ) \| \right ) , \end{align*}


\begin{align*} L ^ { * } : = \left \lbrace y \in E \mid h \left ( L , y \right ) \subset { \mathcal O } _ K \right \rbrace . \end{align*}


\begin{align*} \frac { d y } { d t } & = A ( t ) y , \\ y ( s ) & = x , s \le t < \infty , \end{align*}


\begin{align*} f ( z , t ) = \int _ \Omega \int _ { \C ^ n } F ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \ , d \nu ( a ) . \end{align*}


\begin{align*} E = ( \pi ( 0 , r ) \times ( - \delta , 0 ) ) \cup ( \pi ( 0 , r ) \times ( t , t + \delta ) ) \end{align*}


\begin{align*} \Phi ^ \dagger _ { v p } ( v , p ) [ v _ 1 , p _ 1 ] & = Q ^ \dagger F ^ { q , \dagger } _ { v v } ( v + \psi ^ \dagger ( v , p ) , p ) [ v _ 1 + \psi _ v ^ \dagger ( v , p ) v _ 1 ] \psi ^ \dagger _ p ( v , p ) p _ 1 \\ & + Q ^ \dagger F ^ { q , \dagger } _ { v p } ( v + \psi ^ \dagger ( v , p ) , p ) [ v _ 1 + \psi _ v ^ \dagger ( v , p ) v _ 1 , p _ 1 ] \\ & + Q ^ \dagger F ^ { q , \dagger } _ { v } ( v + \psi ^ \dagger ( v , p ) , p ) \psi ^ \dagger _ { v p } ( v , p ) [ v _ 1 , p _ 1 ] \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } t } \int _ { \Gamma } 1 \ , \mathrm { d } \mathcal { H } ^ d = - \int _ { \Gamma } H V \ , \mathrm { d } \mathcal { H } ^ d = - \int _ { \Gamma } g ( c ) H ^ 2 \ , \mathrm { d } \mathcal { H } ^ d . \end{align*}


\begin{align*} \beta _ { \infty } ( n ) = \frac { 4 \pi ^ { 2 } n } { \sqrt { \det ( A ) } } . \end{align*}


\begin{align*} P ( x , y ) & : = \pi _ x y \ : : \ : S _ y \rightarrow S _ x \\ & \ , \ , = \pi _ x y \ , U _ { \xi } \ : : \ : S _ x \rightarrow S _ x \ : . \end{align*}


\begin{align*} & = \frac { 1 } { ( n + 1 ) ^ 2 [ \exp \left \{ \frac { i \pi x } { n + 1 } \right \} - 1 ] } \sum _ { j , j ' = 0 } ^ n \Big [ \Phi \left ( \frac { j - 1 } { n + 1 } , \frac { j ' } { n + 1 } \right ) - \Phi \left ( \frac { j } { n + 1 } , \frac { j ' } { n + 1 } \right ) \Big ] \\ & \times \exp \left \{ \frac { 2 i \pi j x } { n + 1 } \right \} \exp \left \{ \frac { 2 i \pi j ' x } { n + 1 } \right \} \end{align*}


\begin{align*} f ( r ) \frac { d W _ + ( r ) } { d r } = W _ + ( r ) W _ - ( r ) + E _ 1 - E _ 0 . \end{align*}


\begin{align*} \alpha \left ( R ( r + s ) \right ) & = \int ^ { T _ 2 } _ { T _ 1 } G \left ( \frac { 3 T _ 1 + T _ 2 } { 4 } , \tau \right ) f _ \uparrow \left ( r ( \tau ) + s ( \tau ) \right ) d \tau \\ & \ge \int ^ { T _ 2 } _ { \frac { 3 T _ 1 + T _ 2 } { 4 } } G \left ( \frac { 3 T _ 1 + T _ 2 } { 4 } , \tau \right ) f _ \uparrow \left ( r ( \tau ) + s ( \tau ) \right ) d \tau \\ & \ge f _ \uparrow \left ( a + 4 ^ { - k } d \right ) \int ^ { T _ 2 } _ { \frac { 3 T _ 1 + T _ 2 } { 4 } } G \left ( \frac { 3 T _ 1 + T _ 2 } { 4 } , \tau \right ) d \tau \\ & > a . \end{align*}


\begin{align*} \ell _ { \lambda ' } ( \rho ) = \begin{cases} \ \ \ ( \lambda / \lambda ' ) ^ { \frac 2 3 } \rho , & \lambda \ge \lambda ' , \\ [ 2 p t ] \ , ( \lambda ' - \lambda ) / \lambda ' + \rho , & \lambda < \lambda ' . \end{cases} \end{align*}


\begin{align*} W : = \max _ { \overline { \omega } _ \delta } ( A \Psi \pm T _ \alpha ( u - \varphi ) ) \end{align*}


\begin{align*} L ( x ) : = \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( y ' ) ( x _ \beta - y _ \beta ) . \end{align*}


\begin{align*} \mu _ t = j _ t - \frac { L } { 2 } \medspace ( 1 \leq t \leq n ) , \end{align*}


\begin{align*} E _ a ( x ) \ll \begin{cases} \pi _ N ( x ) ^ { j - a } L ^ { 2 r } & a \leq j \\ L ^ { 2 r - ( a - j ) } & a > j . \\ \end{cases} \end{align*}


\begin{gather*} y \left ( \rho \right ) = e ^ { - i \rho x } e \left ( \rho , x \right ) - 1 , \\ \left ( \mathbf { H } ( \varphi ) y \right ) \left ( \rho \right ) = \frac { 1 } { 2 \pi i } \int _ { - \infty } ^ { \infty } \frac { \varphi \left ( \tau \right ) y \left ( \tau \right ) } { \tau + \rho } d \tau , \quad \rho \in \mathbb { R } \end{gather*}


\begin{align*} \lim _ { n \to \infty } \int _ X ( f \circ h ^ n ) g \ ; d \mu = \int _ X f \ , d \mu \cdot \int _ X g \ , d \mu \end{align*}


\begin{align*} A S ( m ) + S ( m ) A ^ T + \frac { 2 \pi i m } { \theta } S ( m ) = \Sigma _ 2 ( g ) . \end{align*}


\begin{align*} V ( A , - C ) - \sqrt { | A | \ , | C | } = | A | . \end{align*}


\begin{align*} \Im m ( a _ 0 \overline { a _ 1 } ) = \Im m ( a _ 2 \overline { a _ 3 } ) ; \end{align*}


\begin{align*} a _ j \left [ t + 1 \right ] = & a _ j \left [ t \right ] - 4 \mu a _ j \left [ t \right ] \sum _ { l = 1 } ^ { M } \lvert \phi ^ { \left ( l , j \right ) } \rvert ^ 2 + 2 \mu \Re \left \{ \phi ^ { \left ( j , j \right ) } \right \} - 4 \mu a _ j \left [ t \right ] \sum _ { q = 1 } ^ { M - 1 } \sum _ { r = q + 1 } ^ { M } \Re \left \{ \phi ^ { \left ( q , j \right ) ^ * } \phi ^ { \left ( r , j \right ) } \right \} . \end{align*}


\begin{align*} \Delta \mathbf { A } = \mathbf { A } - \mathbf { A } _ o , \ \Delta \mathbf { b } = \mathbf { b } - \mathbf { b } _ o , \ \Delta \mathbf { c } = \mathbf { c } - \mathbf { c } _ o . \end{align*}


\begin{align*} \begin{array} { c } \forall k \in \mathbb { N } , \ ; l _ k ( \tau ) = ( - 1 ) ^ k \overset { k } { \underset { j = 0 } { \sum } } ( - 1 ) ^ j ( \begin{smallmatrix} k \\ j \end{smallmatrix} ) ( \begin{smallmatrix} k + j \\ j \end{smallmatrix} ) \left ( \frac { \tau + h } { h } \right ) ^ j , \end{array} \end{align*}


\begin{align*} \lim _ { p \to \infty } \frac { \psi ( p ) } { p } = 0 . \end{align*}


\begin{align*} { \rm O } ( \epsilon ^ 0 ) : g _ i ^ { ( 0 ) } = g _ i ^ { ( e q ) } , \end{align*}


\begin{align*} U ^ \ast = ( U ^ \ast ) _ \ast = K \end{align*}


\begin{align*} \mathbb { E } _ { \mathbb { P } } \left [ \left | \frac { 1 } { t ^ { \frac { 1 } { 2 } - H } } \sum _ { j = 2 } ^ n \triangle k _ t ^ { 1 , j , n } \right | ^ { 2 p } \right ] \leq p ! \left ( ( n - 1 ) \beta \right ) ^ { p } . \end{align*}


\begin{align*} \Delta _ { \lambda } \xi = \lambda { \xi } _ 1 + . . . . . \lambda ^ { h } { \xi } _ h , \end{align*}


\begin{align*} g _ { x _ j } ( x ) = 0 , x _ j = x _ { j - 1 } \end{align*}


\begin{align*} \begin{array} { r l } \min \limits _ { x \in X } \{ f ( x ) : g _ i ( x ) \leq 0 , \ i = 1 , \ldots , m _ 1 ; h _ j ( x ) = 0 , \ j = 1 , \ldots , m _ 2 \} . \end{array} \end{align*}


\begin{align*} e ^ { \gamma x } f _ r ( x ) = \int _ { - \infty } ^ \infty e ^ { \gamma ( x - y ) } f _ { r + } ( x - y ) e ^ { \gamma y } \mu _ - ( d y ) \le c \int _ { - \infty } ^ \infty e ^ { \gamma y } \mu _ - ( d x ) < \infty . \end{align*}


\begin{align*} \Phi ( t , x , y ) : = \begin{cases} ( x + t , y - t ) & ( x \geq 0 x + t \geq 0 ) \\ ( x + t , y - x - t ) & ( x < 0 x + t \geq 0 ) \\ ( x + t , x + y ) & ( x \geq 0 x + t < 0 ) \\ ( x + t , y ) & ( x < 0 x + t < 0 ) \ \end{cases} \end{align*}


\begin{align*} \aligned \exp \ , ( \frac 1 2 | | R ^ { \nabla ^ t } | | ^ 2 ) = \exp \ , ( \frac 1 2 | | R ^ \nabla | | ^ 2 + t \langle R ^ \nabla , d ^ \nabla A \rangle + \varepsilon ( t ^ 2 ) ) \ , , \endaligned \end{align*}


\begin{align*} Z g \left ( X , Y \right ) = g \left ( \nabla _ { Z } X , Y \right ) + g \left ( X , \nabla _ { Z } ^ { \ast } Y \right ) \end{align*}


\begin{align*} \begin{cases} 2 + \gamma + \frac { 2 \theta } { \alpha } < p , & \alpha \leq 2 , \\ 4 + \gamma + \theta - \alpha < p , & \alpha > 2 . \end{cases} \end{align*}


\begin{align*} \begin{array} { c c c } C _ { \ast } : = C _ { \ast } ( K ; \mathrm { A d } _ { \rho } ) = ( 0 \stackrel { \partial _ 1 } { \longrightarrow } C _ { 0 } ( K ; { \mathrm { A d } _ \rho } ) \stackrel { \partial _ 0 } { \longrightarrow } 0 ) . \end{array} \end{align*}


\begin{align*} - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + ( j - 1 ) r < \chi _ j < - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + j r , \end{align*}


\begin{align*} \mu = \frac { 2 } { \partial _ { s } j ( 0 , 0 ) } \left ( \int _ { \mathbb { R } ^ { 3 } } b \cdot b _ 0 \dd x \right ) . \end{align*}


\begin{align*} \mu \Big ( \bigcap _ { j = 0 } ^ { t } T ^ { j h ( C ) } J \Big ) \geq ( 1 - \epsilon ) \mu ( J ) . \end{align*}


\begin{align*} ( H f ) ( z _ 1 , \ldots , z _ N ) & = \frac { d } { d s } ( e ^ { s H } f ) ( z _ 1 , \ldots , z _ N ) | _ { s = 0 } \\ & = \frac { d } { d s } \big [ ( w \chi ) ( e ^ { s H } ) \cdot f \big ( ( w \alpha _ 1 ) ( e ^ { - s H } ) z _ 1 , \ldots , ( w \alpha _ N ) ( e ^ { - s H } ) z _ N \big ) \big ] | _ { s = 0 } \\ & = d ( w \chi ) ( H ) f ( z _ 1 , \ldots , z _ N ) - \sum _ { r = 1 } ^ N \sum _ { s = 1 } ^ { \ell } d ( w \alpha _ r ) ( H ) z _ { r s } \frac { \partial f } { \partial z _ { r s } } ( z _ 1 , \ldots , z _ N ) , \end{align*}


\begin{align*} f ( x ) = \frac { 1 } { \sigma \mathrm { B } ( \alpha , \beta ) } \ , \left [ \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \alpha - 1 } \ , \left [ 1 - \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \beta - 1 } \ , \phi \left ( \frac { x - \mu } { \sigma } \right ) , x \in \mathbb { R } , \end{align*}


\begin{align*} - \left ( \partial _ { \rho } ^ { 2 } + \frac { 4 } { \rho } \partial _ { \rho } \right ) \varphi = \mu ^ { 2 } \left ( \varphi - \frac { W } { 2 } \right ) _ { + } , \rho > 0 . \end{align*}


\begin{align*} \begin{aligned} ( \Delta u ) ^ { k } \wedge \beta _ n ^ { n - k } \geq 0 , k = 1 , 2 , \dots , m . \end{aligned} \end{align*}


\begin{align*} \deg _ H ( S ) = \deg _ F ( S ) \leq 1 , \end{align*}


\begin{align*} \begin{bmatrix} D \Psi - \kappa \ell U & - \kappa \ell R & R D + \ell \cos \Psi & \sin \Psi - \kappa U R \end{bmatrix} . \end{align*}


\begin{align*} \psi _ o = \frac { { { \mathbf { e } } ^ { T } } } { k } \left ( { { \mathbf { x } } _ { o } } + { { \mathbf { s } } _ { o } } \right ) . \end{align*}


\begin{align*} \begin{aligned} \dot { x } ( t ) & - \dot { z } ( t ) = \\ & - \gamma ( h ( x ( t ) + \delta u ( t ) ) - h ( z ( t ) + \delta u ( t ) ) ) u ( t ) \\ & - \gamma ( h ( z ( t ) + \delta u ( t ) ) - h ( x _ a ( t ) + \delta u ( t ) ) ) u ( t ) \\ & - \left . \gamma ^ 2 \dfrac { \partial \epsilon ( x , t ) } { \partial x } \right | _ { x = x _ a ( t ) } \dfrac { b _ { 1 , \delta } ( x _ a ( t ) ) } { 2 } . \end{aligned} \end{align*}


\begin{align*} \chi _ j \leq \chi ( \sum \limits _ { i = 1 } ^ { j } w _ i ) - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + \sum \limits _ { i = 1 } ^ { j } k _ i + ( j - 1 ) r . \end{align*}


\begin{align*} X _ { \alpha } v _ k = \sum _ { i = 1 } ^ { k } X _ { m - k + 1 } \cdots X _ { m - k + i - 1 } [ X _ { \alpha } , X _ { m - k + i } ] X _ { m - k + i + 1 } \cdots X _ m v _ 0 \end{align*}


\begin{align*} & \psi ^ { \pm } ( x , t ; k ) = \sigma _ 2 \overline { \psi ^ { \pm } ( x , t ; \bar { k } ) } \sigma _ 2 , \ \ \ \psi ^ { \pm } ( x , t ; k ) = \sigma _ 3 \psi ^ { \pm } ( x , t ; - k ) \sigma _ 3 , \end{align*}


\begin{align*} \partial _ j \varphi _ i = 2 \Gamma ^ i _ { j i } \varphi _ i \quad \forall i \neq j . \end{align*}


\begin{align*} \lim \limits _ { R \to \infty } R ^ { N p - N - \alpha } J ( \Psi _ { R } ) = [ M ( | \Psi | ^ p ) ] ^ 2 J ( \eta ) . \end{align*}


\begin{align*} \frac { g ' ( x + \gamma ) } { g ' ( x ) } = e ^ { a \gamma } > 1 , \end{align*}


\begin{align*} F ( r , t ) : = ( Q ( r , t ) - W ( r , t ) ) \exp ( - t ) \end{align*}


\begin{align*} a ( u , v ) = \int _ D \nabla u \cdot \nabla \bar { v } ~ d x , ( f , v ) = \int _ D f \bar { v } ~ d x . \end{align*}


\begin{align*} g = \dot x ( J ) \ , \dot y ( J ) \ , \dot z ( J ) . \end{align*}


\begin{align*} \langle \omega _ a , \beta \rangle = \begin{cases} 1 , & \alpha = \beta , \\ 0 , & \alpha \neq \beta . \end{cases} \end{align*}


\begin{align*} \lim _ { z \to - \infty } \lim _ { x \to \infty } I _ k ^ x ( z ) = \lim _ { z \to - \infty } \lim _ { x \to \infty } J _ k ^ x ( z ) = 1 , \end{align*}


\begin{align*} s _ \alpha ( \mu ) = \mu - \langle \mu , \alpha \rangle \alpha ^ \vee , \alpha \in \Phi , ~ \mu \in X _ \ast . \end{align*}


\begin{align*} \hat W _ r ( m _ 2 ) = \frac { m _ 1 } { m _ 2 } \hat W _ { \frac { r m _ 2 } { m _ 1 } } ( m _ 1 ) . \end{align*}


\begin{align*} c _ \varepsilon = I _ \varepsilon ( u _ \varepsilon ) \geq \inf \limits _ { w \in \mathcal { N } _ \varepsilon } I _ \varepsilon ( w ) . \end{align*}


\begin{align*} s ( q _ 1 ) s ( q _ 2 ) = c ( q _ 1 , q _ 2 ) s ( q _ 1 q _ 2 ) \forall q _ 1 , q _ 2 \in Q . \end{align*}


\begin{align*} \frac { d } { d b } \left ( R L ( b , c ) - L L ( b , c ) \right ) = J _ 1 ( b , c ) + J _ 2 ( b , c ) + J _ 3 ( b , c ) , \end{align*}


\begin{align*} \frac { d \mu _ g ^ n } { d t } = \sum _ { i = 1 } ^ { n - 1 } \sum _ { j = i } ^ { n - 1 } ( i g _ { j + 1 } - i g _ j - g _ i ) V _ { i , j } \psi _ i \psi _ j \end{align*}


\begin{align*} g ^ { \ast 2 } ( x ) = ( 2 / \lambda ^ 2 e ^ { \lambda } ) \big \{ ( 1 - e ^ { - \lambda } ) f ( x ) - e ^ { - \lambda } \sum _ { n \neq 2 } ^ \infty ( \lambda ^ n / n ! ) \ , g ^ { \ast n } ( x ) \big \} . \end{align*}


\begin{align*} \begin{aligned} \binom { n } { m } \det ( A _ 1 , \dots , A _ m , I , \dots , I ) \geq \mathcal { H } _ m ( A _ 1 ) ^ { \frac { 1 } { m } } \dots \mathcal { H } _ m ( A _ m ) ^ { \frac { 1 } { m } } . \end{aligned} \end{align*}


\begin{align*} \lambda _ { \phi _ { \psi } } ( w ) = \bigoplus _ { i = 0 } ^ { a } \bigoplus _ { j = 0 } ^ { d } | w | ^ { \frac { a + d } { 2 } - ( i + j ) } \end{align*}


\begin{align*} & \ \dot { s } \epsilon ( z ) \dot { s } ( e - \varepsilon ( 1 ) ) \lambda _ i = \varepsilon ( - z ^ { - 1 } ) \dot { s } ( \varepsilon ( - z ) - \varepsilon ( z ^ 2 - z ) ) \lambda _ i \\ & \ = \sum _ { k , \nu } g ^ i _ { k , \nu } ( \varepsilon ( - z ^ { - 1 } ( 1 + x ^ i _ { k , \nu } ) ) - \varepsilon ( z ^ { - 1 } ( z - 1 ) ^ { - 1 } ( x ^ i _ { k , \nu } - z + 1 ) ) ) \lambda _ k . \end{align*}


\begin{align*} \mu = ( \mathbf { x } ^ T \mathbf { s } + \kappa \tau ) / ( k + 1 ) = \hat { \mathbf { x } } ^ T \hat { \mathbf { s } } / ( k + 1 ) \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta y + \mathcal { N } ( \cdot , y ) & = u , \ ; \ ; \Omega , \\ y & = 0 , \ ; \ ; \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} r _ { i , j } ( w ) . : = | \{ k ; k \leq i , w ( k ) \leq j \} | . \end{align*}


\begin{align*} Q ^ c ( g , R ) ( \pi ; \bar { \pi } ^ h ) = 2 Q ( g , R ) ( \pi ; \bar { \pi } ^ h ) , \end{align*}


\begin{align*} \lambda _ { 1 2 3 ' } = \log m , m \in \mathbb { N } . \end{align*}


\begin{align*} \# R ^ i _ { n _ 0 ; l _ 1 , \cdots , l _ { n _ 0 } } = \underset { \tilde { l } = ( \tilde { l } _ { i } , \tilde { l } _ { i + 1 } , \cdots , \tilde { l } _ { n _ 0 } ) \in L } { \sum } c _ { \tilde { l } } \# P _ { l _ 1 , l _ 2 , \cdots , l _ { i - 1 } , \tilde { l } _ i , \tilde { l } _ { i + 1 } , \cdots , \tilde { l } _ { n _ 0 } } \end{align*}


\begin{align*} A x ^ 2 + B y ^ 2 + C u ^ 2 + D v ^ 2 = 1 , A , B , C , D \in \mathbb { R } _ { > 0 } . \end{align*}


\begin{align*} a _ { n - i } ( \gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \end{align*}


\begin{align*} \left | \Pi _ { \mathrm { k e r } A ^ \perp } \frac { d ^ \ell } { d t ^ \ell } X _ t | _ { t = 0 } \right | \ge \frac { \gamma } { C _ J } K ^ { \ell + 1 } , \end{align*}


\begin{align*} x = \begin{pmatrix} - \sigma & - \tau & - 1 \\ - \tau & - 1 & - \sigma \\ 1 & \sigma & \tau \end{pmatrix} , \ y = \begin{pmatrix} 1 & \sigma & \tau \\ \sigma & \tau & 1 \\ \tau & 1 & \sigma \end{pmatrix} , \end{align*}


\begin{align*} \mathrm { e } ^ { \lambda \sum _ { i = 1 } ^ d ( y _ i - x _ i ) - d n \ln ( 1 + \lambda ) } \le \mathrm { e } ^ { \lambda d ( y _ 1 - x _ 1 ) + d ^ 2 - n d \lambda + n d \lambda ^ 2 } \le C _ d \mathrm { e } ^ { - d \frac { ( y _ 1 - x _ 1 - n ) } { \sqrt n } } . \end{align*}


\begin{align*} \begin{aligned} \Psi ^ - _ { 2 1 } ( x ; z ) & = \int _ { - \infty } ^ { x - \delta } e ^ { 2 i z ( x - y ) } \nu ( y ; z ) d y + \nu ( x ; z ) \int _ { x - \delta } ^ { x } e ^ { 2 i z ( x - y ) } \mathrm { d } y \\ & + \int _ { x - \delta } ^ { x } e ^ { 2 i z ( x - y ) } ( \nu ( y ; z ) - \nu ( x ; z ) ) \mathrm { d } y \equiv I _ 1 + I _ 2 + I _ 3 , \end{aligned} \end{align*}


\begin{align*} 0 = \varrho _ { Z Z } ( L , L ) , \end{align*}


\begin{align*} | A | \ , | B _ 1 + \dots + B _ m | \le \prod \limits _ { i = 1 } ^ m | A + B _ i | , \end{align*}


\begin{align*} \mathbf { D } = \mathbf { I } . \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } P ^ L ( x ' ) \ , d x ' = I _ { N \times N } , \end{align*}


\begin{align*} - \epsilon ^ 2 \delta ^ 2 b _ { \alpha } u _ n ( x ) \leq u ( 0 ) - u ( x ) + \sum _ { \beta = 1 } ^ { n - 1 } x _ \beta u _ \beta ( x ) \leq C b _ \alpha ^ 2 \end{align*}


\begin{align*} A _ { p , N } & \rightarrow \sum _ { A = 0 } ^ { 2 n - 1 } \int _ S \left [ v _ p ( d _ 1 v _ { p } \wedge \alpha ) _ A \tau ( \mathbf { { n } } ) _ { A 0 } + v _ { p } ( d _ 0 v _ p \wedge \alpha ) _ A \tau ( \mathbf { { n } } ) _ { A 1 } \right ] d S \\ & = \int _ B ( d _ 0 v _ p \wedge d _ 1 v _ { p } + v _ p d _ 0 d _ 1 v _ { p } ) \wedge \alpha + \int _ B ( d _ 1 v _ { p } \wedge d _ 0 v _ p + v _ p d _ 1 d _ 0 v _ { p } ) \wedge \alpha = 0 , \end{align*}


\begin{align*} h ( \gamma \bar \Delta ) \sum _ { k = 0 } ^ { T - 1 } \Theta _ k \Lambda _ k ' < \phi ( x _ 0 ) - \hat \phi + \sum _ { k = 0 } ^ { T - 1 } \Theta _ k ' \left ( \mathcal { E } _ k + \mathcal { E } _ k ^ + + r \right ) . \end{align*}


\begin{align*} \langle e _ { i } , A e _ { j } \rangle = \langle A e _ { i } , A e _ { j } \rangle = \langle e _ { i } , e _ { j } \rangle = 0 \end{align*}


\begin{align*} \begin{aligned} G ^ { \alpha \beta } _ 0 \sigma _ { \alpha \beta } \geq \ , & G ( \sigma _ { \alpha \beta } - \theta \delta _ { \alpha \beta } ) + \theta \sum _ { \alpha = 1 } ^ { n - 1 } G ^ { \alpha \alpha } _ 0 \\ \geq \ , & \gamma + \theta \sum _ { \alpha = 1 } ^ { n - 1 } G ^ { \alpha \alpha } _ 0 \end{aligned} \end{align*}


\begin{align*} D _ { n + 1 } ( x ) = m _ \tau ( x ) D _ n ( R ^ { t } x ) . \end{align*}


\begin{align*} \mu _ k ^ { \mathrm { o p t } } = \frac { \sum _ { l \in \mathcal { M } _ { k } } \overline { \mathbf { h } } _ { k l } ^ { \mathbf { H } } \mathbf { z } _ { k l } } { \sum _ { i \in \mathcal { K } _ { l } } \sum _ { l \in \mathcal { M } _ { i } } \overline { \mathbf { h } } _ { k l } ^ { \mathbf { H } } \mathbf { z } _ { i l } \mathbf { z } _ { i l } ^ { \mathbf { H } } \overline { \mathbf { h } } _ { k l } + \delta _ { } ^ 2 } . \end{align*}


\begin{align*} { \rm E E } _ { v , j } = \frac { { \overline { \rho } _ { v , j } B _ { \rm V L C } \log _ 2 \left ( { 1 + { \frac { \exp ( 1 ) } { 2 \pi } } \frac { { \overline P } _ v { \left ( { { R _ { \rm { P D } } } { \overline { G } _ { v , j } } } \right ) } ^ 2 } { { \sum \limits _ { v ' \ne v } { \overline P } _ { v ' } { \left ( { { R _ { \rm { P D } } } { \overline { G } _ { v ' , j } } } \right ) } ^ 2 } + N _ { \rm V L C } B _ { \rm V L C } } } \right ) } } { P _ { \rm V L C } + P _ v } . \end{align*}


\begin{align*} \frac { 1 } { t } \int _ 0 ^ t \vert \langle \psi _ { \beta , f _ \delta } , e ^ { - i s H _ N } \psi _ { \beta , f _ \delta } \rangle \vert ^ 2 \ , d s = O ( t ^ { - 2 ( 1 - \beta ) } ) \end{align*}


\begin{align*} \Psi ^ q + v + \psi ( v , p ( s ) ) = \Psi ^ q + v + \frac 1 2 ( v , p ( s ) - p ( 0 ) ) H \begin{pmatrix} v \\ p ( s ) - p ( 0 ) \end{pmatrix} + O ( \norm { ( v , s ) } ^ 3 ) , \end{align*}


\begin{align*} \int _ { 0 } ^ { \lambda } \mu [ \xi ( \mu , t ) - \xi ( \mu , \tau ) ] \ d \mu = - \int _ { \tau } ^ { t } \int _ { \lambda } ^ { \infty } \int _ { 0 } ^ { \lambda } \nu \Lambda ( \mu , \nu ) \xi ( \mu , s ) \xi ( \nu , s ) \ d \nu d \mu d s . \end{align*} % \end{align*}


\begin{align*} f _ k ( x ) : = x ^ T A _ k x + 2 b _ k ^ T x + c _ k \end{align*}


\begin{align*} C _ H ( C , M ^ \vee ) : = \bigoplus _ { n \geq 0 } \ , C _ { n , H } ( C , M ^ \vee ) \end{align*}


\begin{align*} \frac { 1 } { z } = - \int _ 0 ^ { + \infty } e ^ { \omega z } d \omega , \quad R e ( z ) < 0 . \end{align*}


\begin{align*} \Gamma ( x _ 0 , y ) = d _ x T ( x _ 0 , y ) . \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) - x _ a ( t ) \| \le & \ , \| x ( 0 ) - z ( 0 ) \| e ^ { \gamma L _ \mathcal { D } t } \\ & \ , + \gamma 3 M _ \mathcal { D } T \left ( e ^ { \gamma L _ \mathcal { D } t } - 1 \right ) \\ & \ , + \gamma 2 M _ \mathcal { D } T . \end{aligned} \end{align*}


\begin{align*} { \bf X } ' ( t ) = e ^ { - A t } { \bf X } ' ( 0 ) + \int _ { 0 } ^ t e ^ { - A ( t - s ) } \Sigma \Big ( { \bf p } ( s - ) \Big ) \dd M ( s ) , t \ge 0 . \end{align*}


\begin{align*} \mathcal S ( t ) f = \sum _ { j = 1 } ^ { \infty } e ^ { t \lambda _ j } \langle e _ j , f \rangle e _ j , \end{align*}


\begin{align*} a _ { j k } x _ k = - x _ j b _ { j k } , x _ j a _ { j k } = - b _ { j k } x _ k , \end{align*}


\begin{align*} | G _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | + | \nabla G _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | \le C , \ \ 1 \leq i \neq j \leq m , \end{align*}