harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\beta \big( (v_1 \, \cdots\, v_m) \otimes v_{m+1}\big) \ =\ (v_1\, \cdots\, v_m\,v_{m+1}) - (-1)^{|v_{m+1}|(|v_1|+\cdots+|v_m|)}(v_{m+1}\, v_1\, \cdots\, v_m) \, .
\lim_{t\to\infty}\int_{t-A}^{\infty}\left(|\partial_r(\psi-\phi^L)|^2+|\partial_t(\psi-\phi^L)|^2+\frac{|\psi-\ell-\phi^L|^2}{r^2}\right)\,rdr=0,\,\,\,{\rm for\,\,any\,\,}A>0.
{\bar{{\cal G}}}_{( \mathrm{I})}^{\left( \rho \right) a}\Gamma^{\left( \rho \right)}= \Delta_{\mathrm{c l}( \mathrm{I})}^{\left( \rho \right) a}+ O \left( \rho^{2}\right) \, \, ,{\bar{{\cal G}}}_{( \mathrm{I I})}^{\left( \rho \right) a}\Gamma^{\left( \rho \right)}= \Delta_{\mathrm{c l}( \mathrm{I I})}^{\left( \rho \right) a}+ O \left( \rho^{2}\right) \, \, ,
\partial_{\xi}\left(\frac{\lambda^{\frac{3}{2}}(\lambda\xi)}{\sqrt{2m+2}((\lambda\xi)^2+m+1)^{\frac{1}{4}}}\right)=\frac{\lambda^{\frac{9}{2}}\frac{\xi^2}{2}+2\lambda^{\frac{5}{2}}(m+1)}{\sqrt{2m+2}((\lambda\xi)^{2}+2m+2)^{\frac{5}{4}}}.
\kappa T_{00}= 0, \, \kappa T_{11}= -\frac{t}{r^3}, \, \kappa T_{22}= \frac{t (2 r+t)}{4 r^2}, & \kappa T_{33}= \kappa T_{22}\sin^2 \theta, \kappa T = -\frac{t^2}{2 r^4}.
\sigma^i\underline{x}= \left(x_i^{r_n-r},x_j^{r_n},\dots,x_i^{r_n},\dots,x_j^{r_n},\dots\right)=\left(x_i^{r},x_{j}^{r_n-r},x_{j+r_n}^r,\dots,x_i^{r},x_{j}^{r_n-r},\dots,x_{j}^{r_n-r},x_{j+r_n}^r,\dots\right).
D_aB =D_a\gamma^{(2n+1)}(1-P_{\rm F}-D_aP_{\rm F}D_a) &=-\gamma^{(2n+1)}D_a(1-P_{\rm F}-D_aP_{\rm F}D_a) &=-\gamma^{(2n+1)}(1-P_{\rm F}-D_aP_{\rm F}D_a)D_a=-BD_a,
\big|[\Phi^{-1}u^{N,k, r}] (v)\big| =|u^{N,k, r}_{m-N}\varphi_{N_i,k,m-N}(v)|\leq\frac{C(\varepsilon)e^{(L(E)-\varepsilon)N}e^{-(L(E)-\varepsilon)m}}{2^{(m-N)/2}}&\lesssim_N C(\varepsilon)e^{-\big(L(E)+\frac{\log2}{2}-\varepsilon\big)m}.
R_{ij}= -\frac{1}{3}\alpha^{ab}\tilde{X}_a^i \tilde{X}_b^j =\sigma^{(0) ab}\tilde{X}^i_a \tilde{X}^j_b S_j= -\frac{4}{3}\beta^c \tilde X_c^j=\tilde{\sigma}^{ab}\gamma_{ab}^{(2)c}\tilde{X}^j_c.
\sum_{n=0}^\infty B(n) \frac{x^n}{n!}\frac{d}{da}(a)_n =\sum_{n=0}^\infty B(n+1) \frac{x^{n+1}}{(n+1)!}(a)_{n+1}\frac{1}{a}\sum_{k=0}^{n}\frac{(a)_{k}}{(a+1)_{k}}
X_2 \ = \X_1 \times \{Y_x \}\ = \ \ \bigcup_{x \in X}\ \{x \}\times Y_x, R_2 \ = \R_1 \ltimes \{S_x \}\ \ (x_1,y_1), (x_2,y_2) \in X_2, ((x_1,y_1), (x_2,y_2)) \in R_2 \Longleftrightarrow \begin{cases}(x_1,x_2) \in R_1^{\circ}x_1 = x_2 \ \ (y_1,y_2) \in S_{x_1}. \end{cases}
\lim_{S \rightarrow 0}\sup_n \sup_{\tau \in \mathcal{T}_n^{M,T}}\mathbb{P}\bigg( \sup_{t \in [0,\tau \wedge S]}\lVert u_{\phi_n}\rVert_V^2 + \int_0^{\tau \wedge S}\lVert A(u_{\phi_n}\rVert_H^2 dt > \tilde{M}^2 + (M-1)^2 \bigg) = 0.
\displaystyle h(a) ={_3F_2}\left( a,a,a;2\,a,a+1;1 \right) \mbox{}-\frac{{a}^{2}}{2}\,{\frac{{_4F_3}\left( a+1,a+1,a+1,a+1\,;a+2,a+2,2\,a+1;1 \right)}{\left( a+1 \right)^{2}}}\,.
\tau_1=\inf\Big\{t\in [0,t_0]\; : \;|X^{1,N}_t-X^{2,N}_t|\geq \frac{2a+1}2d \Big\}, \tau_2=\inf\Big\{t\in [0,t_0] : \;|X^{1,N}_t+X^{2,N}_t - 2 \bar x| \geq \frac{2a-1}2d \Big\}, \tau_3=\inf\Big\{t\in [0,t_0] : \;\min_{j=3,\dots,N}|X^{j,N}_t-\bar x| \leq bd\Big\},
\begin{aligned}|s_\lambda(y',y_d,z_d)| \leq |s_{\lambda,low}(y',y_d,z_d)| + |s_{\lambda,high}(y',y_d,z_d)| &\leq \frac{C y_d \, e^{-c z_d}}{(y_d + z_d +|y'|)^{d-1}(1+y_d + z_d+|y'|) (1+y_d+ z_d)}.\end{aligned}
J^{(1,1)}_2 \leq 32\max\{1,C^2_{\textsf{KMT}}\}L^{-2}_*\sum\limits_{n=2}^{\infty}p_n(T_*)\ln^2(n)+84\sqrt{C^{(2)}_{\textsf{KMT}}}L^{-2}_*\sum\limits_{n=2}^{\infty}{p_n(T_*)}n^{-1}&\leq 32\max\{1,C^2_{\textsf{KMT}}\}L^{-2}_*\sum\limits_{n=2}^{\infty}p_n(T_*)\ln^2(n)+252\sqrt{C^{(2)}_{\textsf{KMT}}}L^{-2}_*T^{-1}_*\,.
\frac{1}{2}\frac{d}{dt}\int_{\Omega}\rho |u_t|^2 x+\int_{\Omega}|\nabla u_t|^2x \leq C(\sigma)(\|\nabla u\|^2_1+|D(u)|_\infty+1)(|\sqrt{\rho}u_t|^2_{2}+1)+C|H_t|^2_2+C|\sqrt{\rho}\theta_t|^2_2.
|\check{f}_{k,l}^{m+2,M}- \check{f}_{k,l}^{m,M}| = \left |\frac{k(k-1)}{m(2k+m-1)}\check{f}_{k,l}^{m,M}- \frac{(k+1)(k+2)}{m(2k+m+3)}\check{f}_{k+2,l}^{m,M}\right | & \leq \left[ \frac{k(k-1)}{m(2k+m-1)}+ \frac{(k+1)(k+2)}{m(2k+m+3)}\right]f(1,1),m\geq 1.
\Gamma = \frac{8}{m}\, \frac{k_{0}^{2}\, ( V_{1}- k_{0}^{2})^{3 / 2}\, ( V_{2}+ k_{0}^{2})^{1 / 2}}{V_{1}\, ( V_{1}+ V_{2}) \, ( 1 + \sqrt{V_{1}- k_{0}^{2}}\, r_{1})}\, \exp \left\{- \, 2 \, \sqrt{V_{1}- k_{0}^{2}}\, \Delta r \right\}.
\begin{array}{rl}vxv^{-1}\displaystyle=(v\cdot x \ + \v \wedge x)v^{-1}=(v\cdot x \ - \x \wedge v)v^{-1}= &\displaystyle=\big[v\cdot x \ - \ (xv- \x\cdot v)\big] v^{-1}=(2 v\cdot x \ - \xv)v^{-1}=2\frac{x\cdot v}{|v|^2}v - x\end{array}
\Big|\int_{E_m+y}g(x)dx-\int_{E_m}g(x)dx\Big| &= \int_{E_m}g(x+y)-g(x)dx &=y\cdot \int_{E_m}\nabla g(x)dx+\int_{E_m}o_x(|y|)dx &=\int_{E_m}o_x(|y|)dx.
z_{0}^{-1}\left(\frac{z_1-z_2}{z_0}\right)^{r/2}\delta\left(\frac{z_{1}-z_{2}}{z_{0}}\right)Y(u,z_{1}-z_2)=z_1^{-1}\delta\left(\frac{z_{0}+z_{2}}{z_1}\right)Y(u,z_0).
\mathcal L_{N,n}=\sum\limits_{k=1}^d \sum_{x,x+e_k\in\Lambda_{N,n}}\mathcal L^{x,x+e_k}\, ,\mathbb L_{N,n}=\sum_{x\in \Lambda_{N,n}}\mathbb L^{x}_{\Lambda_{N,n}}\, ,L_{\widehat b,N,n}= \sum_{x\in \Lambda_{N,n}\cap\Gamma_N}L^{x}_{\widehat b,N}\, .
\kappa(x,\omega) = \bar\kappa(x) + \sum_{\ell=1}^{\infty}\sqrt{\mu_\ell}v_\ell(x) \eta_\ell(\omega)\approx \bar\kappa(x) + \sum_{\ell=1}^{L}\sqrt{\mu_\ell}v_\ell(x) \eta_\ell(\omega),
\mathcal{H}_k(t,\mathfrak{X})=\sum_{i}\left[X_i(t)D_i \log U\big(T-t,\mathfrak{X}(t)\big)+\frac{X_i(t)}{X(t)}\right]\sigma_{ik}(t,\mathfrak{X}) -\vartheta_k(t,\mathfrak{X})\,,{\mathrm{by}}\ \eqref{eq:pi^U}\,.
\int_{-\infty}^0 |D_\lambda(x,y)|\,\mathrm dx = \frac{1}{2\sqrt{|\lambda|}}\int_{-\infty}^0 e^{-\operatorname{Re}\sqrt{\lambda}|x-y|}\,\mathrm dx=\frac{2-e^{\operatorname{Re}\sqrt{\lambda}y}}{2\sqrt{|\lambda|}\operatorname{Re}\sqrt{\lambda}}\leq \frac{1}{\sqrt{|\lambda|}\operatorname{Re}\sqrt{\lambda}}.
H_x(x,y,t) = 0 H_y(x,y,t) = -\frac{(x-\gamma-t)}{\sigma^2}\exp\left(-\left( \frac{x-\gamma-t}{\sigma}\right)^2\right) E_z(x,y,t) = \frac{(x-\gamma-t)}{\sigma^2}\exp\left(-\left( \frac{x-\gamma-t}{\sigma}\right)^2\right)
\begin{array}{c}{\phi^{\kappa}( \lambda ) = q^{\kappa}( \lambda ) + \pi^{\nu}( \lambda )}\\{\phi^{\nu}( \lambda ) = q^{\nu}( \lambda ) + \pi^{\kappa}( \lambda )}\\{\left[ \pi^{\kappa}( \lambda ) , q^{\kappa}( \mu ) \right] = \ln h ( \mu , \lambda )}\\{\left[ \pi^{\nu}( \lambda ) , q^{\nu}( \mu ) \right] = \ln h ( \lambda , \mu )}\\ \end{array}
\begin{array}{r c l}{R_{\mu \nu}+ \frac{1}{2}\partial_{\mu}\varphi \partial_{\nu}\varphi + \frac{1}{1 2}{\cal V}_{0}e^{- \varphi}g_{\mu \nu}}&{=}&{0 ,}\\{}&{}&{}\\{\nabla^{2}\varphi + \frac{1}{2}{\cal V}_{0}e^{- \varphi}}&{=}&{0 .}\\ \end{array}
\mathcal{H}{\tilde \nabla}_{{\tilde U}_i}{\tilde U}_i = \mathcal{H}\nabla_{\frac{1}{\phi}U_i}( \frac{1}{\phi}U_i) - \frac{1}{\phi^2}g(U_i , U_i) \mathcal{H}\nabla \log \phi &= \frac{1}{\phi^2}( - \nabla f - \nabla \log \phi).
\begin{array}{l}{{Y^{( 1 / 2 )}= - 2 D_{2}\Phi ;}}\\{{Y^{( n + 1 / 2 )}= \partial_{-}Y^{( n - 1 / 2 )}- 2 i \partial_{-}\Phi \cdot \sum_{l = 1}^{n - 1}D_{2}Y^{( l - 1 / 2 )}\cdot Y^{( n - l - 1 / 2 )};}}\\{{n = 1 , 2 , 3 , \cdots}}\\ \end{array}
a^{n+1}[a, b] = \sum_{i+j=n}\binom{n}{i}a [a,b]^{(i+1)}a^j = \sum_{i+j=n}\binom{n}{i}\Big[ [a,b]^{(i+2)}a^j + [a,b]^{(i+1)}a^{j+1}\Big]\,.
\Big( \displaystyle\sum_{\sigma \subset U \atop |\sigma|=k/2}\prod_{u_i \in \sigma}(d_i)(d_i-1) \Big)\Big( \displaystyle\sum_{\sigma \subset W \atop |\sigma|=k/2}\prod_{w_i\in \sigma}(d_i')(d_i'-1) \Big)
f^{p-1}= \sum_{i_0+i_1+\cdots+i_s = p-1}{p-1 \choose i_0, \ldots, i_s}\lambda_0^{i_0}\lambda_1^{i_1}\cdots \lambda_s^{i_s}(x_1\cdots x_n)^{i_0}m_1^{i_1}\cdots m_s^{i_s}.
{\cal L}( J ) = \frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi + \frac{J}{2}\phi^{2}+ \frac{\lambda \mu^{2 \varepsilon}}{4 !}\phi^{4}+{\cal L}_{\mathrm{C T}}( J ) - \mu^{- 2 \varepsilon}\frac{\zeta}{2}J^{2}.
{\rm minimize}\hat{m}_{((A_r)_k,(B_r)_k,(C_r)_k)}(\xi,\eta,\zeta) &{\rm subject\, to}||(\xi,\eta,\zeta)||_{((A_r)_k,(B_r)_k,(C_r)_k)}\leq \Delta_k, &{\rm where}\hat{m}_k(\xi,\eta,\zeta):=\hat{m}_{((A_r)_k,(B_r)_k,(C_r)_k)}(\xi,\eta,\zeta), &\quad\quad (\xi,\eta,\zeta)\in T_{((A_r)_k,(B_r)_k,(C_r)_k)}M.
\lim_{t \rightarrow t^{\prime}}{\frac{d}{d t^{\prime}}}t a n^{- 1}\biggl ({\frac{x_{2}( t^{\prime}) - x_{2}( t )}{x_{1}( t^{\prime}) - x_{1}( t )}}\biggr ) ={\frac{1}{2}}{\frac{d}{d t}}t a n^{- 1}\biggl ({\frac{{\dot{x}}_{2}( t )}{{\dot{x}}_{1}( t )}}\biggr ) \, .
\left\langle \partial_t \left(|u|^{m-2}u\right)(t), \phi \right\rangle_{H^1_0}+ \int_\Omega \nabla u(x,t) \cdot \nabla \phi(x) \,\d x = 0 \mbox{for a.e.~}\t \in (0, \infty) \mbox{and}\ \phi \in H^1_0(\Omega),
\Psi_{\kappa}^{( R )}= \frac 1{\pi \sqrt{2}}\, \exp \left( \frac{\pi \kappa}2 - i \frac{\kappa}2 \ln \left( \frac{x_{+}}{- x_{-}}\right) \right) K_{i \kappa}\left( m \sqrt{- x_{-}x_{+}}\right) .
\lim_{\delta \to 0}~~\Big|~~y_0+\int_{0}^{\cdot}b_{j^{\delta}(s)}(s,y^{\delta}(s))ds+\int_{0}^{\cdot}\sigma_{j^{\delta}(s)}(s,y^{\delta}(s))dB^{\delta}(s) ~~ -~~\Big(y_0+\int_{0}^{\cdot}b_{j(s)}(s,y(s))ds+\int_{0}^{\cdot}\sigma_{i(s)}(s,y(s))dB(s)~~\Big)~~\Big|_{(0,T)}~~=~~0.
\frac{\sum\limits_{j=0}^{N-1}\ln w_j}{\ln|z|}\leq \frac{\sum\limits_{j=0}^{N-1}\ln w_j}{\ln\delta_2}&=\big(\sum\limits_{j=0}^{N-1}\ln w_j\big)\bigg(\frac{\varepsilon(\sum\limits_{j=0}^{N-1}\ln w_j)}{\varepsilon{N}-1}\bigg)^{-1}=N-\frac{1}{\varepsilon}.
\sum_{x\in\Bbb F_{q^2}}f(x)^{2+(q-3)q}\cr=\, \frac{3b^{8-7q}(b^{1-q}-a)(b^2-a^2b^{q-1}-3a)(9a-2b^2+a^3b^{2q-2}-6a^2b^{q-1}+ab^{q+1})}{a^4(b^2-4a)^2}.
D_n^{(n)}(x_1^{(n)}) D_{n-1}^{(n)}(x_2^{(n)}) - D_n^{(n)}(x_2^{(n)}) D_{n-1}^{(n)}(x_1^{(n)}) = D_n^{(n)}(x_2^{(n)}) \Big[ D_n^{(n)}(x_1^{(n)}) - D_{n-1}^{(n)}(x_1^{(n)})\Big] - D_n^{(n)}(x_1^{(n)}) \Big[ D_n^{(n)}(x_2^{(n)}) - D_{n-1}^{(n)}(x_2^{(n)})\Big].
\mathcal{I}\begin{pmatrix}w^T \bar{f}-f - w\end{pmatrix}= -\frac{1}{2}\int dxdy\left\{w(y, x)a_xa^\ast_y + w(x, y)a^\ast_x a_y -f(x, y) a^\ast_x a^\ast_y -\bar{f}(x, y)a_x a_y \right\}
\big| |a(T(s),s)| - |a(s)| \big| \lesssim \min \Big( \max \big( \delta_1 , \delta_2 \big), \max \big( \epsilon^{\frac{1}{4}}, \delta_3 \big) \Big)^{\frac{1}{2}}+ \min \Big( \max \big( \tilde{\delta}_1 , \tilde{\delta}_2 \big), \max \big( \tilde{\epsilon}^{\frac{1}{4}}, \tilde{\delta}_3 \big) \Big)^{\frac{1}{2}}=: \Delta + \tilde{\Delta}.
\begin{aligned}\phi^j_0(x) = \begin{cases}1, \x_j \le x < x_{j+1}, 0, , \end{cases}\phi^j_p(x) = \frac{x - x_j}{x_{j+p}- x_j}\phi^j_{p-1}(x) + \frac{x_{j+p+1}- x}{x_{j+p+1}- x_{j+1}}\phi^{j+1}_{p-1}(x), \end{aligned}
\left\{- \frac{\gamma}{8 N^{2}}\int d \theta \rho ( \theta ) \left\{\left( \frac{\partial}{\partial \theta}\frac{\delta}{\delta \rho ( \theta )}\right)^{2}-{\cal V}^{2}( \theta ) \right\}+ V [ \rho ] \right\}~ \tilde{\psi}[ \rho ] ~ = ~ E ~ \tilde{\psi}[ \rho ] \; .
\frac{C}{w(B(\mathbf{x},t))^{1/2}}\geq \|h_{t^2}(\mathbf{x},\cdot)\|_{L^2(dw)}=\|E(i\cdot,\mathbf{x})e^{-t^2\|\cdot\|^2}\|_{L^2(dw)}\geq e^{-1}\|E(i\cdot,\mathbf{x})\|_{L^2(B(0,t^{-1}),dw)}.
\int_{\Pi}\Big(\zeta\varrho\,\partial_t\psi+ \zeta \varrho\mathbf u\cdot\nabla\psi +\psi \varrho \nabla\zeta\mathbf u\Big)\,dxdt+\int_{\mathbb R^2}\psi(x,0)\zeta(x)\varrho_0(x)\, dx= 0.
\begin{array}{l l l}{L_{N S}}&{=}&{\frac{1}{2}( R_{D 5}- L_{D 5}) + ( L_{D 3}- R_{D 3})}\\{L_{D 5}}&{=}&{\frac{1}{2}( R_{N S}- L_{N S}) + ( L_{D 3}- R_{D 3})}\\ \end{array}
\int \! \! \int d^{d}k_{1}d^{d}k_{2}\left( \frac{\partial}{\partial k_{1 \mu}}+ \frac{\partial}{\partial k_{2 \mu}}\right) \left[ k_{1 \mu}P_{k_{1}, m_{1}}^{\nu_{1}}P_{k_{1}- k_{2}, m_{2}}^{\nu_{2}}P_{k_{2}, m_{3}}^{\nu_{3}}\right] \equiv 0 ,
\sum_{n \geq 1}\frac{a_n \sigma_{1,\textbf{1},\alpha}(d'n)}{n^{s + 1}}=\sum_{\gcd(n,d')=1}\frac{a_n \sigma_{1,\textbf{1},\alpha}(n)}{n^{s + 1}}\prod_{p\mid d'}\left(\sum_{b=0}^{\infty}\frac{a_{p^b}\sigma_{1,\textbf{1},\alpha}(p^{b+e_p})}{(p^{b})^{s + 1}}\right).
-\Delta u \leq \max_{t=0}\left( -\Delta u -\frac{R}{4}-|\nabla u|^2 \right) + \frac{R}{4}+|\nabla u|^2 \leq \max_{t=0}\{-\Delta u-|\nabla u|^2\}+ \max_{t=0}|\nabla u|^2+\frac{R+1}{4}\leq \max_{t=0}\left( -\Delta u \right) + \max_{t=0}|\nabla u|^2 + \frac{R+1}{4}.
P^{(-)}_{I_{r;j}}\Big[P^{(+)}_{I_{l;i}}V^{(k,q)}_{I_{l;i}}P^{(+)}_{I_{l;i}}+P^{(-)}_{I_{l;i}}V^{(k,q)}_{I_{l;i}}P^{(-)}_{I_{l;i}}\Big]P^{(-)}_{I_{r;j}}=P^{(-)}_{I_{r;j}}\,V^{(k,q)}_{I_{l;i}}\,P^{(-)}_{I_{r;j}}
Y\underline{\mathbf{d}}w =\sup_{y\in Y,z\in X}(z\mathbf{d}w-z\mathbf{d}y)_+ &=\sup_{z\in X}(z\mathbf{d}w-\inf_{y\in Y}z\mathbf{d}y)_+ &=\sup_{z\in X}(z\mathbf{d}w-z\mathbf{d}Y)_+ &\geq\sup_{z\in X}(z\mathbf{d}w-z\mathbf{d}x)_+ &=x\underline{\mathbf{d}}w.
{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z)= \sum_{n=0}^{\infty}(a)_n \frac{{}^{M\!L\!\!}B_{p}(b+n,\;c-b)}{B(b,\;c-b)}\frac{z^n}{n!}(p\geq0; \ |z|<1;\Re(c)>Re(b)>0,\Re(\alpha)>0)
\frac{qx^2 \nu_z\nu_x - q^{-1}x^2 \nu_x \nu_z}{q-q^{-1}}= q^{-5}\nu_z x^2 + q^{-3}\nu_z - q^{-4}\nu_z^2 - \frac{q\nu_y \nu_z - q^{-1}\nu_z\nu_y}{q-q^{-1}}+ \frac{\nu_y\nu_z^2-q^{-2}\nu_z\nu_y\nu_z}{q-q^{-1}}.
\{0 \}\; \subseteq \; \underbrace{[\widetilde{\mathcal{W}}, \widetilde{\mathcal{L}}]}_{= z_1 \mathcal{F}}\; \subseteq \; \widetilde{\mathcal{Z}}\; \subseteq \; \widetilde{\mathcal{W}}\; \subseteq \; \widetilde{\mathcal{L}}
= \frac{-\xi\cdot(\xi-2\eta)}{8 |\xi|^2}\underbrace{(1-\cos(\angle(\xi-\eta, \eta)))\Big((\xi-\eta)\times\eta\Big)}_{}- \frac{\xi}{8|\xi|}\cdot \underbrace{\Big(\frac{\xi-\eta}{|\xi-\eta|}-\frac{\eta}{|\eta|}\Big) \Big( (\xi - \eta) \times \eta \Big)}_{}.
\mathcal{I}:=\int_{\mathbb{R}}f(te_{j_1}+\omega)H_{\ell_1}\Big(\frac{t}{\sigma_{j_1}}\Big)\,d\gamma_{j_1}(t) =\int_{\mathbb{R}}f(te_{j_1}+\omega)H_{\ell_1}\Big(\frac{t}{\sigma_{j_1}}\Big)e^{-t^2/2\sigma_{j_1}^2}\,\frac{dt}{\sigma_{j_1}\sqrt{2\pi}}&=\int_{\mathbb{R}}f(\sigma_{j_1}se_{j_1}+\omega)H_{\ell_1}(s)e^{-s^2/2}\frac{ds}{\sqrt{2\pi}}\cdot
F(n,k) =(-1)^{n+k}\frac{[6n-2k+1](q;q^2)_{n-k}(-q;q^2)_{n-k}(-q;q^2)_{n+k}}{(q^4;q^4)_{n}(-q^4;q^4)_{n}(-q^4;q^4)_{n-k}}, G(n,k) =\frac{(-1)^{n+k}(q;q^2)_{n-k}(-q;q^2)_{n-k}(-q;q^2)_{n+k-1}}{(1-q)(q^4;q^4)_{n-1}(-q^4;q^4)_{n-1}(-q^4;q^4)_{n-k}},
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|^m \geq \sqrt{\frac{\frac{1}{{d+m-1 \choose m}}\left(\sum_{j=1}^n f_j(\tau_j)^m\right)^2-\sum_{j=1}^n |f_j(\tau_j)|^{2m}}{n^2-n}}=\sqrt{\frac{\frac{1}{(^m(\mathcal{X}))}\left(\sum_{j=1}^n f_j(\tau_j)^m\right)^2-\sum_{j=1}^n |f_j(\tau_j)|^{2m}}{n^2-n}}.
{\rm{E}}[ \left. G_{\alpha}(t, V(t)) \right| \mathcal{F}_s ] = \frac{{\rm{E}}[\left. e^{\alpha V(t)}\right| \mathcal{F}_s ]}{{\rm{E}}[ e^{\alpha V(t)}]}&= \frac{e^{\alpha V(s)}{\rm{E}}[ e^{\alpha (V(t)-V(s))}]}{{\rm{E}}[ e^{\alpha V(s)}]{\rm{E}}[ e^{\alpha (V(t)-V(s))}]}&= \frac{e^{\alpha V(s)}}{{\rm{E}}[ e^{\alpha V(s)}]}= G_{\alpha}(s, V(s)).
\Phi^+(B_\ell) = \{\epsilon_i=\sum_{i\le k \le \ell}\alpha_k \,(1 \le i \le \ell), \epsilon_i-\epsilon_j=\sum_{i\le k < j}\alpha_k \,(1 \le i < j \le \ell), & \epsilon_i+\epsilon_j=\sum_{i\le k<j}\alpha_k+2 \sum_{j\le k \le \ell}\alpha_k\,(1 \le i < j \le \ell) \}.
\widehat{{\boldsymbol{\omega}}}(j):= \,(\omega_{1},\cdots, \omega_{j-1}, \omega_{j+1},\cdots, \omega_{r})\in \mathbb{C}^{r-1}= \,(\omega_{1},\cdots,\widehat{\omega}_{j},\cdots, \omega_{r}),{\boldsymbol{\omega}}^{-}[j]:= \,(\omega_{1},\cdots,-\omega_{j},\cdots,\omega_{r})\in \mathbb{C}^{r}.
\int_{L_k\cup M_k}H_{Q_k}(x, t)\, dx \geq (4\pi t)^{-1}\int_{L_k\cup M_k}\bigl(1-e^{-d(x_1)^2/t}-e^{-d(x_2)^2/t}-4e^{-1/(2t)}\bigr)\, dx = (4\pi t)^{-1}\biggl[|L_k\cup M_k|(1-4e^{-1/(2t)}) -\int_{L_k\cup M_k}\bigl(e^{-d(x_1)^2/t}+e^{-d(x_2)^2/t}\bigr)\, dx\biggr]\, .
U_{t}=U_{xx}+(c_{\mu}+\partial_{x}(\chi_2 V_2-\chi_1 V_1)(\cdot;u))U_{x}+(a+(\chi_2\lambda_2V_2-\chi_1\lambda_1V_1)(\cdot;u)-(b+\chi_2\mu_2-\chi_1\mu_1)U)U,
\Delta E=-\frac{\beta_0^4}8\left(\frac 1{m^3}+\frac 1{M^3}\right)+\frac{\beta_0^2}{2mM}\left<\phi_0\right|I\left|\phi_0\right>+\frac 1{2\mu}\left(\frac{\mu^2}{m^2}+\frac{\mu^2}{M^2}\right)\left<\phi_0\right|I^2\left|\phi_0\right>
F(X)= X^{3^m}+(\delta^{2\cdot{3^{m-1}}}+\delta^{3^{m-1}+3^{2m-1}})(X^{3^{m-1}}-X^{3^{2m-1}}) + (\delta^{3^{m-1}}+\delta^{3^{2m-1}})(X^{2\cdot3^{m-1}}+X^{2\cdot3^{2m-1}}+X^{3^{m-1}+3^{2m-1}})+\delta^{3^{2m-1}+2\cdot3^{m-1}}.
u(x)=\frac{4^{d+2}}{\Sigma_n^2\cdot\Gamma(2d+4)}\cdot\int_{\mathcal S^n}f(z)\int_0^\infty \int_{\mathcal S^n}\rho\cdot\theta_\rho^d(x\cdot y)\cdot\overline{\psi^d(y\cdot z)}\,d\sigma(y)\,d\rho\,d\sigma(z).
Z_{\gamma_1,\gamma_2}(t)= \frac{A}{2}~ \int_{\mathbb{R}^2}' \left[\int_0^t [(s-x_1)_+^{\gamma_1}(s-x_2)_+^{\gamma_2}+(s-x_1)_+^{\gamma_2}(s-x_2)_+^{\gamma_1}] ds \right]B(dx_1) B(dx_2),
H_{d}(F^{\alpha}_{n}(x),F^{\alpha}(x)) = H_{d}\bigg(F_{n}(x)+\alpha\left[F^{\alpha}_{n}(L^{-1}_{j}(x))-S(L^{-1}_{j}(x))\right], F(x)+\alpha\left[F^{\alpha}(L^{-1}_{j}(x))-S(L^{-1}_{j}(x))\right]\bigg) \leq H_{d}(F_{n}(x),F(x))+\lvert \alpha \rvert H_d(F^{\alpha}_{n}(L^{-1}_{j}(x)),F^{\alpha}(L^{-1}_{j}(x))).
T^{\Delta}_{\alpha}\Big(k(t)v(t)K^p(t)\Big) \leq\beta k^\sigma(t)v^\sigma(t)r(t)f(t)K^{p-\alpha}(t)+(p-\alpha+1)k^\sigma(t)v^\sigma(t)r(t)f(t)\big(K^\sigma(t)\big)^{p-\alpha}&\leq(p+\beta-\alpha+1)k^\sigma(t)v^\sigma(t)r(t)f(t)\big(K^\sigma\big(t))^{p-\alpha}.
{\rm either}\;\; \tilde{{\cal S}}^{ab}= -\frac{i}{4}[dx^a \; \wedge + \;e^a, \; dx^b\; \wedge + \; e^b ] = -\frac{i}{4}[\tilde{\gamma}^a, \tilde{\gamma}^b], \quad{\rm or}\quad \tilde{\tilde{\cal S}}{}^{ab}= \frac{i}{4}[dx^a\; \wedge -\; e^a ,\; dx^b \; \wedge - \;e^b ].
(h^{2}+ 2h-\mu)(h+4)^{n}=h^{n+2}-2(2n+1)h^{n+1}&+\sum_{k=0}^{n-2}\bigg(-\mu{n\choose k}(4)^{k}+2{n\choose k+1}(4)^{k+1}&+{n\choose k+2}(4)^{k+2}\bigg)h^{n-k}&+(-\mu n(4)^{n-1}+2(4)^{n})h-\mu(4)^{n}.
\left[ e^{+ 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) , C_{m_{d}}\alpha \big ) + i \theta}+ e^{- 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) , C_{m_{d}}\alpha \big ) - i \theta}\right] .
\left((1+\sqrt{x})^{4k+4}+(1-\sqrt{x})^{4k+4}\right)=2\sum_{j=0}^{2k+2}{4k+4\choose 2j}x^j = 2(1+x)^{2k+2}+2\sum_{j=1}^{2k+1}\left({4k+4\choose 2j}-{2k+2\choose j}\right)x^j=2(1+x)^{2k+2}+8x\sum_{j=0}^{2k}\alpha_jx^j = 2(1+x)^{2k+2}+8x\sum_{j=0}^{k}a'_jx^j(1+x)^{2k-j},
\Delta E_{\Delta R}^{( 6 )}= \frac 1{( 2 \mu )^{2}}\int \frac{d \,{\bf p}}{( 2 \pi )^{3}}\phi_{0}^{*}({\bf p}) \left[{\bf p}^{2}+ \beta_{0}^{2}\right]^{2}\Delta R ({\bf p}^{2}) \phi_{0}({\bf p})
W^{\Omega_{z_{1}}^{*}, \Omega_{z_{2}}}\left( \begin{array}{c c}{{c}}&{{d}}\\{{b}}&{{a}}\\ \end{array}\right) = ( - 1 )^{n - 1 + \mu + \kappa}W^{( n - 1 , 1 )}\left( \left. \begin{array}{c c}{{c}}&{{d}}\\{{b}}&{{a}}\\ \end{array}\right| z_{1}- z_{2}\right) ,
\{g_{\mu,\lambda_1}\star g_{\mu,\lambda_2}\}(x) = \frac{\sqrt{\pi}}{\Gamma(\mu)}\, e^{-(\lambda_1+\lambda_2)x/2}\, \left(\frac{x}{\lambda_2-\lambda_1}\right)^{\mu-\tfrac{1}{2}}\, I_{\mu-\tfrac{1}{2}}\left(\frac{\lambda_2-\lambda_1}{2}\, x\right)
\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|}.
\int_I \frac{|w'|^2 V_{\beta_0}}{|\beta_0'|}+ \frac{k^2 |w|^2 | \beta_0'|}{V_{\beta_0}}\,dt = \int_I \frac{k^2 |w|^2 | \beta_0'|}{V_{\beta_0}}\,dt & \le \lambda_{k,n}(\omega) \int_I |w|^2 V_{\beta_0}| \beta_0'| \,dt
\left\|\vec{t}\right\|_{\dot f^{s+n(\tau-\frac{1}{p})}_{\infty,\infty}(\mathbb A)}=\left\|\left\{2^{j[s+n(\tau-\frac{1}{p})]}\left|A_j\vec{t}_j\right|\right\}_{j\in\mathbb Z}\right\|_{L\dot A_{\infty,\infty}}=\sup_{j\in\mathbb Z}2^{j[s+n(\tau-\frac{1}{p})]}\left\|\,\left|A_j\vec{t}_j \right|\,\right\|_{L^\infty}.
N^*(1-\delta_\lambda,M,Q_\mu,T_\nu)=\sum_{\substack{Q_\mu/2<q\leq Q_\mu (q,Mp_0)=1}}\:\sideset{}{^*}\sum_{\psi\pmod{q}}\sum_{\xi\pmod{M}}N(1-\delta_\lambda,T_\nu,\psi\xi).
\mathcal W(\epsilon_4)=\bigcup_{{\bf x}\in \mathcal V_{(1)}}\mathcal W(\epsilon_4;\psi({\bf x}),(\tilde{\psi}\vert_{\Sigma_1({\bf x})})^{-1};((\Sigma_1({\bf x}),\vec z_1({\bf x})),u_{{\bf x}}))\times \{{\bf x}\}.
<\phi_\omega,\phi_\sigma>=\int_{\Sigma}\sqrt{h}d^{D-1}x\left[(\omega+\sigma)\phi^{*}_\omega \phi_{\sigma}+i\phi^{*}_\omega a^i(\nabla_i+i\sigma a_i)\phi_\sigma-i(\nabla_i-i\omega a_i)\phi^{*}_\omega a^i\phi_\sigma \right]~~~,
2 \ \Re \left\langle N x , G \left( t \right) x \right\rangle + \sum_{\ell = 1}^{3}\left\langle N^{1/2}L_{\ell}\, x , N^{1/2}L_{\ell}\, x \right\rangle= \left\langle x , \left( \alpha \left( t \right) a^\dagger + \overline{\alpha \left( t \right)}a -2 \kappa N \right) x \right\rangle
l(n)=\psi^{-1/2}+ \xi_c p\psi^{\frac{q}{q+1}}M^{\frac{q(-\tau+1)}{q+1}}+\xi_c M_x^{-1}\psi^2 p^4+\frac{M^2}{\sqrt{n}}\psi^3 p^6 & +p \psi \Big(\frac{\xi_c^{1/2}}{M_x^{5/6}}+\frac{\sqrt{M}}{M_x^3}\Big) \sqrt{\log \frac{p}{\gamma}}+\gamma,
|E(H_i)|\le s\cdot \frac{q\ell}{8640Cp\varepsilon\ln\left(\frac{1}{\varepsilon}\right)}\le\frac{\varepsilon n^2}{12C\ln \left(\frac{1}{\varepsilon}\right)(1+\varepsilon)}\le \frac{\varepsilon}{6C\ln \left(\frac{1}{\varepsilon}\right)}\cdot \binom{|V_{i-1}|}{2}.
\left(z_{4}sz_{1}s\right)^{-1}z_{4}\left(sz_{1}s\right) =e\mbox{or}z_{4}\left(z_{4}sz_{1}s\right)^{-1}\left(sz_{1}z\right) =\left[z_{4},\left(sz_{1}s\right)^{-1}\right].
\mathbf{w}_{k,j}= \left( \mathbf{V}_{\xi_{k,j}}+ \mathbf{h}_{k,j}\mathbf{h}_{k,j}^H\right)^{-1}\mathbf{h}_{k,j}, \mathbf{V}_{\xi_{k,j}}\triangleq N_0 \mathbf{I}_{N_r(N\!+\!J)}+ \sum_{\substack{i=1, i \neq k}}^N v_{x_{i,j}}^{prio}\mathbf{h}_{i,j}\mathbf{h}_{i,j}^H.
\frac{d^2\xi}{dr^2}+\left( \frac{\omega^2}{\lambda^4}-\frac{\frac{l(l+1)}{r^2}+\frac{2M}{r^3}-\frac{2Q^2}{r^4}+m^2}{\lambda^2}-\frac{1}{\lambda}\frac{d^2\lambda}{dr^2}\right)\xi=0.
\mathcal{M}^+_{\lambda,\Lambda}(M)\,:=\,\Lambda\sum_{e_i>0}e_i\,+\,\lambda\sum_{e_i<0}e_i\;\;\;\;\mbox{and}\;\;\;\;\mathcal{M}^-_{\lambda,\Lambda}(M)\,:=\,\Lambda\sum_{e_i<0}e_i\,+\,\lambda\sum_{e_i>0}e_i,
\forall \varphi\in \mathcal{W}^{3,\alpha}_{0}, \left< \tilde{\eta}_{t},\varphi \right>-\int_{0}^{t}\left< \tilde{\eta}_{z},L_{z}\varphi \right> dz - \int_{0}^{t}\left< u_{z},\frac{\partial \Psi}{\partial y}(\cdot,\overline{\gamma}(z))R\varphi \right> \tilde{\Gamma}_{z}dz =0,
\begin{cases}b_i^+=\sum_{k=1}^q \beta_k b_{i-k}^+ + \alpha_i (1-\gamma_i)^\delta \ \ \ \mbox{with $\ \alpha_i (1-\gamma_i)=0 $ for \ $i>p$}b_i^-=\sum_{k=1}^q \beta_k b_{i-k}^- + \alpha_i (1+\gamma_i)^\delta \ \ \ \mbox{with $\ \alpha_i (1+\gamma_i)=0$ for \ $i>p$}\end{cases}
\int_{\Omega}\left| \frac{\nabla u}{K}\right|^{p(x)-2}\langle \frac{\nabla u}{K}, \nabla \eta \rangle = \lambda S \int_{\Omega}\left| \frac{u}{k}\right|^{p(x)-2}\frac{u}{k}\eta, \quad\,\, \forall\eta \in C^{\infty}_0 (\Omega).
e_{_{\mu}}^{\; a}= \left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{\rho \frac{1}{\beta}\cos \phi}}&{{\rho \frac{1}{\beta}\sin \phi}}\\{{0}}&{{- \rho \frac{\sin \alpha \theta}{\alpha}\sin \phi}}&{{\rho \frac{\sin \alpha \theta}{\alpha}\cos \phi}}\\ \end{array}\right) .
O(\ell k(\delta^2 m\ell + k) \log (\ell) ) =O(k\delta^2 m\ell^2 \log(\ell)) +O(\ell k^2 \log (\ell) ) \cr = O(k^3 m \frac{\log \ell}{\log^2 m}) + O(\delta mn)= O(k mn \delta) .
\check{R}_{2,3}\left(y/x\right)\check{R}_{1,2}\left(y/z\right)\check{R}_{2,3}\left(x/z\right)=\check{R}_{1,2}\left(x/z\right)\check{R}_{2,3}\left(y/z\right) \check{R}_{1,2}\left(y/x\right),
\widetilde{\boldsymbol{e}}_{u,h}^n:=j_u\widetilde{\boldsymbol{w}}_t^n-\widetilde{\boldsymbol{u}}_h^n, \boldsymbol{e}_{u,h}^n:=j_u\boldsymbol{w}_t^n - \boldsymbol{u}_h^n, e_{p,h}^n:=j_p r_t^n - p_h^n
\,f_{n+s}=x_{s}\quad s=1,\cdots,j-1,\, s\neq n+l-\delta_{l,j}-\delta_{l,k}\,f_{n+l-\delta_{l,j}-\delta_{l,k}}=x_lx_j, \,f_{n+s}=x_{s+1}s=j,\cdots k-2, \,f_{n+s}=x_{s+2}s=k-1,\cdots n-3, f_{2n-2}=1.
\psi_{\mu}(z)=\int_{\mathbb{R}_{+}}\frac{zt}{1-zt}\,d\mu(t),\quad\eta_{\mu}(z)=\frac{\psi_{\mu}(z)}{1+\psi_{\mu}(z)},z\in\mathbb{C}\backslash\mathbb{R}_{+},
\times \prod_{b=1}^N e^{-2\pi\big(1-\frac{g}{\pi+gN}\big)\sum_{j=1}^{n_b}\sum_{l=1}^{n_b}\Big[\big(1-\delta_{jl}\big)\big(C^{0,S}(x_j^{(b)},x_l^{(b)})+C^{0,S}(y_j^{(b)},y_l^{(b)})\big)-2C^{0,S}(x_j^{(b)},y_l^{(b)}) \Big]}
\langle g^{\mu\nu}({\bf x}_{F})g^{\sigma\rho}({\bf y}_{F})\rangle=- D^{\mu\nu;\sigma\rho}({\bf x}_{F}-{\bf y}_{F})=- C^{\mu\nu;\sigma\rho}\vert{\bf y}_{F}-{\bf x}_{F}\vert^{-4\gamma}
\frac{1}{r}\int_{r-t}^{r+t}\widetilde{\Omega}_s[u_1+\mu u_0](x)K_1(t,r;s;\mu,\nu^2)\, \mathrm{d}s \xrightarrow [r\to 0]{}2 \int_{0}^{t}\frac{\partial}{\partial s}\, \Omega_{s}[u_1+\mu u_0](x) K_1(t,0;s;\mu,\nu^2)\, \mathrm{d}s.
\dot{\overline{S}}(t,x) = \bigg[ \frac{-\dot{s}(t) (1 - |\dot{s}(t)|^2) + \dot{s}(t) \ddot{s}(t)(x_1 - (s(t)-s(0)))}{(1 - |\dot{s}(t)|^2)^{3/2}}\bigg]{\partial_1}S( \Phi_1(t,x), x_2 ) &= \dot{\Phi}_1(t,x)\sqrt{1-|\dot s(t)|^2}{\partial_1}\bar S(t,x)\,.