harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G(\bar z^{\prime\prime},z^\prime;t)=\sum_{n_1,n_2,n_3=0}^{\infty}D_{n_1n_2 n_3}\prod_{i=1}^3(\bar z^{\prime\prime}_i)^{n_i}\exp\left\{-i(\sum_{i,j=1}^3\alpha_{ij}n_in_j+\sum_{i=1}^3\beta_in_i+\gamma^\prime)t\right\}
h^{r e l}= 8 a^{2}\left[ E_{1 1}^{2}+ E_{1 1}E_{2 2}+ E_{2 2}^{2}- 3 E_{1 2}E_{3 2}- 3 E_{2 1}E_{3 1}- 9 E_{3 1}E_{3 2}+ 3 \lambda \left( E_{1 1}+ E_{2 2}\right) \right] .
\|\bar{\zeta}\|_{C(\overline{Q_{\varepsilon,T}})}= \|\zeta\|_{C(\overline{S_T})}, \|\bar{\zeta}(\cdot,0)\|_{C(\overline{\Omega_\varepsilon(0)})}= \|\zeta(\cdot,0)\|_{C(\Gamma(0))}\leq \|\zeta\|_{C(\overline{S_T})},
\zeta_{2}\left(-n_1,-n_2\right) + \zeta_{2}\left(-n_2,-n_1\right)+ \zeta_{1}\left(-n_1 - n_2\right) - \zeta_{1}\left(-n_1\right) \zeta_{1}\left(-n_2\right)= \frac{(-1)^{n_{1}+1}n_{1}!n_{2}!}{\left(n_1+n_2+2\right)!}B_{n_1+n_2+2}.
\gamma_{\pm}(k) \,=\, \sqrt{(b(k) \pm \mid \lambda \mid)^2 \,+\,\mid C_2 (a k) \,- \, \tau C_1(a k) \mid^2}\;,\;b(k) \,=\, \tau_2 \,\, B ( a k) \; , \;\lambda \,=\, \tau_2 \Lambda \;\;,
A(x_1,x_2,x_3)=\frac{1}{4}\left[X^2(x_1,x_2,x_3)-X(x_1,x_2,x_3)R_{2}R_{3}-\frac{3}{4}\right], B(x_1,x_2,x_3)=\frac{1}{4}\left[Y^2(x_1,x_2,x_3)-Y(x_1,x_2,x_3)R_{3}R_{1}-\frac{3}{4}\right], C(x_1,x_2,x_3)=\frac{1}{4}\left[Z^2(x_1,x_2,x_3)-Z(x_1,x_2,x_3)R_{1}R_{2}-\frac{3}{4}\right].
Q(x) = \sum_{x/\log x \le p \le x-\sqrt x}\pi(x-p) + O\bigg( \sum_{p\le x/\log x}\pi(x) + \sum_{x-\sqrt x\le p\le x}x \bigg) = \sum_{x/\log x \le p \le x-\sqrt x}\pi(x-p) + O\bigg( \pi(x) \pi\bigg( \frac x{\log x}\bigg) + x\sqrt x \bigg) = \sum_{x/\log x \le p \le x-\sqrt x}\pi(x-p) + O\bigg( \frac{x^2}{\log^3x}\bigg).
\begin{array}{l c r}{{P ( W_{2}) = ( y^{2}+ x^{3}+ z^{6}+ \mu x y z ) + w ( x^{{\frac{3}{2}}}z^{{\frac{3}{2}}}) + w^{2}( y + x z ) + w^{3}x^{{\frac{1}{2}}}z^{{\frac{1}{2}}}+ w^{4}.}}\\ \end{array}
\Vert g_n \Vert_{L_{\alpha +1}^2}^2 = b_{\alpha}\int_{a_n}^{a_{n-1}}\big\vert x^{\alpha / 2}\mu^{-(\alpha +2)/2}e^{-it \log_{\mu}x}\big\vert^2 x^{-(\alpha +1)}\, dx = b_{\alpha}\mu^{-(\alpha +2)}\int_{a_n}^{a_{n-1}}x^{-1}\, dx = b_{\alpha}\mu^{-(\alpha +2)}n.
\begin{aligned}h(c(y_k)+\nabla c(y_k)(x_k-y_k))+g(x_k) \leq h(c(y_k)+\nabla c(y_k)(w_k-y_k)) &+\frac{\tilde\mu}{2}(\|w_k-y_k\|^2-\|w_k-x_k\|^2-\|x_k-y_k\|^2) &+a_kg(v_k)+(1-a_k)g(x_{k-1}).\end{aligned}
\phi(A_1\otimes \cdots \otimes A_k)= M\left(\bigotimes_{i\in S_1}x_i\otimes \bigotimes_{j\in S_2}y_j\right)\left(\bigotimes_{i\in S\setminus S_1}x_i\otimes \bigotimes_{j\in S\setminus S_2}y_j\right)^TN
\|\nabla\psi_{3,\varepsilon}\|_4\leq C\|\Delta\psi_{3,\varepsilon}\|_{4/3}\leq C\left(\int_\Omega\frac{t_\varepsilon^{8/3}}{\varepsilon^{8/3}}\right)^{3/4}\leq \frac{C}{\varepsilon^2}\cdot o(\varepsilon^{9/4})=o(\varepsilon^{1/4}).
M(a_n, \ldots, a_0) :=\left[ \begin{array}{cc}a_n 1 1 0 \end{array}\right]\cdots\left[ \begin{array}{cc}a_1 1 1 0 \end{array}\right]\left[ \begin{array}{cc}a_0 1 1 0 \end{array}\right] =\left[ \begin{array}{cc}p_n q_n p_{n-1}q_{n-1}\end{array}\right] .
\delta(\phi^\rho, x,y,t):= \frac{1}{2}|\phi^\rho(x,t)-\phi^\rho(x+y,t)|^{p-2}(\phi^\rho(x,t)-\phi^\rho(x+y,t)) &+\frac{1}{2}|\phi^\rho(x,t)-\phi^\rho(x-y,t)|^{p-2}(\phi^\rho(x,t)-\phi^\rho(x-y,t)),
{\mathbf{\Omega}}= \{{{\sin}^2}{\theta_1}+{{\sin}^2}{\theta_2}\leq{\delta^2},{{\sin}^2}{\theta_2}\leq{{\sin}^2}{\theta_1}, 0 \leq{{\sin}^2}{\theta_1},{{\sin}^2}{\theta_2}\leq 1 \}.
Z_{2,1}(n,b) := \bar{F}\left( \frac{1}{a_{_{J_n}}}\left( \left( b -S_{n}^{(-J_n)}\right) \vee M_n^{(-J_n)}\right) \right) Z_{2,2}(n,b) := \bar{F}\left( \frac{1}{a_{_{J_n}}}\left( \left( b -S_{n-1}^{(-J_n)}\right) \vee M_n^{(-J_n)}\right) \right).
{{d^2x^{\rho}}\over{d\tilde{\lambda}^2}}+\tilde{\Gamma}^{\rho}_{\mu\nu}{{dx^{\mu}}\over{d\tilde{\lambda}}}{{dx^{\nu}}\over{d\tilde{\lambda}}}=-{1\over 2}{{{\cal W}^{\prime}}\over{\cal W}}{{dy}\over{d\tilde{\lambda}}}{{dx^{\rho}}\over{d\tilde{\lambda}}}.
f(x,y,z)=\dfrac{(y^2+ax^2+bxz)^2}{xz^{3}},f(x,y,z)=\dfrac{(y^2+ax^2+bxz)^3}{x^2z^{4}}, f(x,y,z)=\dfrac{(y^2+ax^2+bxz)^3}{x^{4}z^{2}},f(x,y,z)=\dfrac{(y^2+ax^2+bxz)^3}{x^{5}z}, f(x,y,z)=\dfrac{(y^2+ax^2+bxz)^4}{x^{5}z^{3}},f(x,y,z)=\dfrac{(y^2+ax^2+bxz)^4}{x^{7}z},
\sum_{i \colon \mu_{i,\ell}> k'}\langle P_{\lambda > k}f, \chi_{i,\ell}\rangle^2 \leq \sum_{i \colon \mu_{i,\ell}> 0}\langle P_{\lambda > k}f, \chi_{i,\ell}\rangle^2= \sum_{i \colon \lambda_{i,\ell}> 0}\langle P_{\lambda > k}f, \psi_{i,\ell}\rangle^2 &= \sum_{i \colon \lambda_{i,\ell}> k}\langle f, \psi_{i,\ell}\rangle^2
K_t - \tilde K_t - \int_0^t \big(Z_s - \tilde Z_s\big)\,d\hat W_s + \int_0^t\int_U \big(L_s(z) - \tilde L_s(z)\big)\,\lambda_\pi(dz)ds &= \int_0^t\int_U \big(L_s(z) - \tilde L_s(z)\big)\,\hat\pi(ds\,dz) + \int_0^t\int_\Lambda \big(R_s(b) - \tilde R_s(b)\big)\,\hat\theta(ds\,db),
\Theta_a(\rho_{j+1}) = \frac{D_a(\rho_{j+1})}{\rho_{j+1}^{m-2}}\leq \left( \frac{\rho_{j+1}}{\rho_{j}}\right)^{2N_a(\rho_{j})}\frac{D_a(\rho_{j})}{\rho_{j}^{m-2}}= \left( \frac{\rho_{j+1}}{\rho_{j}}\right)^{2N_a(\rho_{j})}\Theta_a(\rho_{j}).
J = 2\pi\Lambda \int \, d \eta d\beta\,\left[ 1 + \frac{g_1}{2}\frac{(\nabla w)^2}{1- w^2}+ \frac{g_2}{2}\left(\frac{(\nabla w)^2}{1- w^2}+ \left((\nabla v)^2+\frac{N^2\tau^2}{a^2\sinh^2\eta}\right)(1-w^2)\right)\right](1-w^2)\,.
S_{1}= S\setminus \{x,b_{1}\}=(S\setminus \{b_{1}\})\setminus \{x\}\preceq (S\setminus \{b_{1}\})\setminus \{x^\prime\}=(S\setminus \{x^\prime\})\setminus \{b_{1}\}\preceq \preceq (S\setminus \{x^\prime\})\setminus \{x^{\prime\prime}\}= S_{2}.
G ( z_{o u t}, z_{i n}) = \frac 1{1 6 \sqrt 2}e^{\gamma^{+}\frac \partial{\partial \mu}}e^{\gamma^{-}\frac \partial{\partial \nu}}\langle z_{o u t}|{({\hat{p}}^{2})}^{- 1}\left( \mu{\hat{p_{\theta}^{\prime}}}^{2}{\hat{p_{\bar{\theta}}^{\prime}}}^{2}+ \nu{\hat{p_{\bar{\theta}}^{\prime}}}^{2}{\hat{p_{\theta}^{\prime}}}^{2}\right) | z_{i n}\rangle |_{\mu = \nu = 0},
( v - v^{\prime})^{2}\rightarrow \frac{( v - v^{\prime})^{2}}{\Omega ( v ) \Omega ( v^{\prime})}\, , \quad \vec{x}_{\alpha}\rightarrow \frac{\vec{x}_{\alpha}}{\Omega ( v )}\, \quad \vec{x}_{\alpha}^{\prime}\rightarrow \frac{\vec{x}_{\alpha}^{\prime}}{\Omega ( v^{\prime})}\, .
(\mathcal{T}_{\mathfrak{s}}z)(t)= S(t)z_0-\tfrac{1}{2}\int_0^t S(t-s)(z(s)F_\Phi )\d s +i \kappa \int_0^t S(t-s)(\theta_R(|z|_{X^{\mathfrak{s}}_s})|z(s)|^2z(s))\d s -i\int_0^t S(t-s)(z(s)\d W(s)),
\alpha c= [\varphi_1, \phi_{G_i}](c)= \varphi_1[ G_i,c]+[G_i, \varphi_1(c)]=2 \sum\limits_{k\in{\mathbb Z}}\gamma_k L_{i+k}+ \frac{c}{3}\left(i^2-\frac{1}{4}\right) \gamma_{-i},
\psi^{ml}_{\lambda_1,\lambda_2}(\xi_1,\xi_2) = N \exp\left\lbrace{\xi^2_1\over{2c\theta}}-{\xi^2_2\over{2(1+c)\theta}}-{\lambda_1\xi_1\over{c\theta}}+{\lambda_2\xi_2\over{(1+c)\theta}}\right\rbrace \quad .
\sum_{j=1}^{k}\prod_{i=k}^{k+j-2}\left(1-\frac{1}{i^\beta}\right)ck^{-\beta}\le e^{ck^{1-\beta}}c^{-\beta/(1-\beta)}k^{-\beta}C\Gamma\left(\frac{1}{1-\beta},{ck^{1-\beta}}\right)\rightarrow c^{-\beta/(1-\beta)}.
\theta_{d-1}= \frac{J}{l\sqrt{2}}\left[ \sqrt{1+E}\, EllipticE\left(\frac{T}{l},\frac{1}{E+1}\right) - \frac{E}{\sqrt{1+ E}}\,EllipticF\left(\frac{T}{l},\frac{1}{E+1}\right) \right] \;,
\begin{aligned}\mathcal{A}(p) =2^n\cdot\lim\limits_{t\rightarrow+\infty}e^{-nt}Vol(\partial B_p^{g^+}(t),g^+) &=2^n\cdot\lim\limits_{t\rightarrow+\infty}Vol(\partial B(t),g^p|_{\partial B(t)}) &=2^n\cdot Vol(\partial X,\hat{g}^p)\end{aligned}
p_{n}\coloneqq\frac{41}{216}n^{4}-\frac{91}{108}n^{3}+\frac{17}{12}n^{2}+\begin{cases}-\frac{11}{3}n+9 n\equiv0\bmod{6}-\frac{107}{27}n+\frac{161}{27}n\equiv2\bmod{6}-\frac{139}{27}n+\frac{169}{27}n\equiv4\bmod{6}\end{cases}
(\pi_{j,k}) (p)_m (\pi_{i,m-n}) = b(j_k) \dots b(j_{m-1}) (p)_m (\pi_{i,m-n}) &= (p)_k b(i_{k-n}) \dots b(i_{m-n-1}) (\pi_{i,m-n}) = (p)_k (\pi_{i,k-n})
\bar{D}^{\dot{\alpha}}J_{\alpha \dot{\alpha}}^{0}= - \frac{1}{2 4}D_{\alpha}\left[ \frac{3 N_{c}- N_{f}}{2 \pi^{2}}\mathrm{T r}\, W^{2}+ \gamma Z \bar{D}^{2}\sum_{f}\left({Q}_{f}^{\dagger}e^{V}{Q}_{f}+ \bar{Q}_{f}^{\dagger}e^{- V}\bar{Q}_{f}\right) \right]
\frac{\mathcal{E}\left\{\Delta\right\}}{\log e}\approx \mathbf{C}+\ln\left(\frac{K}{\rho}\right) -2\ln \alpha^2_{1,1}- \left(\mathbf{C}+\ln\left(\frac{1}{\rho}\right)\right) + \sum^{K}_{l=1}{K \choose l}(-1)^l \ln l +2 \ln \alpha_{1,1}^2 = \ln K + \sum^{K}_{l=1}{K \choose l}(-1)^l \ln l .
\Gamma_{0}( b_{i}\to b_{f}+ \gamma ) ={\frac{4}{3}}( 4 \pi )^{- 5 / 2}\omega_{P}\left[ \ln \left({\frac{1 + N_{i}}{1 + N_{f}}}\right) \right]^{3}\left[ \ln ( 1 + N_{i}) \right]^{- 1 / 2}{\frac{N_{f}}{N_{i}}}.
\sum_{\substack{{1 \le a < b \le m}p^t_a = j(t)}}c_ad(x_{p^t_a}, x_{p^t_b}) \le \Big(\sum_{\substack{{1 \le a < b \le m}p^t_a = j(t)}}c_a\Big)d(x_{j(t)},x_{k(t)}) + \Big(\sum_{\substack{{p^t_a = j(t)}p^t_b > k(t)}}c_a\Big)d(x_{k(t)},x_{p^t_b}).
\|\tilde{D}\|_{\infty}\leq\max\{s(i\sigma M)\}=\|i\sigma M\|_{\infty}= \|M\|_{\infty}\|\tilde{D}^{-1}\|_{\infty}\leq\max\{s(i\sigma M^{-1})\}=\|i\sigma M^{-1}\|_{\infty}= \|M^{-1}\|_{\infty}
<\phi_{\mathcal{B}}{}_{\alpha_1\alpha_2\alpha_3\alpha_4}^{N_1 N_2 N_3 N_4}\ldots \phi_{\mathcal{B}}{}_{\alpha'_{10}\alpha'_8\alpha'_5\alpha'_1}^{N_{10}N_8 N_5 N_1}>\ \ = \ \ \lambda \ \ \delta^{\alpha_1\alpha'_1}\ldots \delta^{\alpha_{10}\alpha'_{10}}\ \{\mathcal{B}}_{N_1\ldots N_{10}}~~,
F_1(x,y,u,v;q)+F_1(-x,y,-u,v;q)-xy F_1(qx,qy,u,v;q)-xy F_1(-qx,qy,-u,v;q) &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad=F_2(x,y,u,v;q), &F_2(x,y,u,v;q)-uy^{-1}F_2(x,u,y,v;q)=2F_1(x,y,u,v;q).
S_{p}={\frac{1}{2 \kappa_{D}^{2}}}\int d^{D}x \sqrt{- G}\left[{\cal R}-{\frac{4}{D - 2}}( \partial \phi )^{2}-{\frac{1}{2 \cdot ( p + 2 ) !}}e^{2 a_{p}\phi}F_{p + 2}^{2}\right] ,
\begin{array}{l l}{\phi_{1}= 0}\\{\phi_{2}= \phi = \varphi ( r ) e^{i \theta}}\\{H_{\mu}= \frac{\sqrt{\beta}}{q}\frac{H ( r )}{r}\delta_{\mu}^{\theta}}\\{\tilde{D}= \frac{1}{2 \beta}\left( q ( \varphi^{2}- \eta^{2}) + \alpha e \sigma^{2}\right)}\\{G = 0 ,}\\ \end{array}
\sum n^{m}=\frac{1}{m+1}(B_{0}n^{m+1}-(\begin{matrix}m+1\\ 1\end{matrix})B_{1}n^{m}+(\begin{matrix}m+1\\ 2\end{matrix})B_{2}n^{m-1}-\cdot\cdot\cdot+(-1)^{m}(\begin{matrix}m+1\\m\end{matrix})B_{m}n)
\delta(x-[r_0]) \equiv \phi(\sum_{k=1}^\nu p^{k-1}[\phi^{-k}(r_k)]) \bmod I^\nu &\equiv \sum_{k=1}^\nu p^{k-1}[\phi^{-(k-1)}(r_k)]) \bmod I^\nu &\equiv \sum_{k=0}^{\nu-1}p^{k}[\phi^{-k}(r_{k+1})]) \bmod I^\nu
(Tw)(d):= \frac1{|d|^{\frac{2}{1-p}}}\int_0^dK_N(s,d) [ |s|^{\frac{2p}{1-p}}(c_p+w)^p-\frac{\mu}{\delta^2}|s|^{\frac2{1-p}}(c_p+w) ] \,ds-c_p.
\mathbb{E}\left[ \Vert\chi_{a}R_{z}(H_{G}^{\tau})\chi_{b}\Vert^s \right] \le \mathbb{E}\left[ \Vert\chi_{a}R_{z}(H_{G})\chi_{b}\Vert^s \right]+ \tau^{s}\mathbb{E}\left[\Vert\chi_{a}R_{z}(H_{G}) W R_{z}(H_{G}^{\tau})\chi_{b}\Vert^s\right] & =: I_{1}+ \tau^{s}I_{2}.
u_{\mu,\xi}(x, t) = \sum_{j=1}^ku_j(x, t)~\mbox{with}~ u_j(x, t):=U_{\mu_j,\xi_j}(x) + \mu_j^{\frac{n-2s}{2}}\Phi_j^*(x, t) - \mu_j^{\frac{n-2s}{2}}H(x, q_j).
D_{\mu \nu}^{p r o p}( B ) = - ( g_{\mu \rho}^{\perp}- n_{\mu}( n K n )^{- 1}( n K )_{\rho}^{\perp}) \tilde{K}_{\rho \sigma}^{- 1}( g_{\sigma \nu}^{\perp}- ( K n )_{\sigma}^{\perp}( n K n )^{- 1}n_{\nu})
G = \frac{\pi}{4 \delta ( 0 )}\biggl ( \sum_{n = 1}^{L / 2}\frac{1}{\sinh \bigl ( \kappa_{n - \frac 1 2}\pi / 2 \bigr )}- \sum_{n = 1}^{L / 2}\frac{1}{\sinh \bigl ( \kappa_{n}\pi / 2 \bigr )}\biggr ) .
\sup_{B_{R_{k}}(x_k)}u \geq C^{-1}\delta^{-\gamma}\left(\sup_{B_{\delta R_k}(x_k)}u - \inf_{B_{\delta R_k}(x_k)}u \right)-R_k^2\rho &\geq C^{-1}\delta^{-\gamma}\left(u(x_k) - \inf_{B_{\delta R_k}(x_k)}u -C \delta^{\gamma}\rho\right).
A\mathcal{X}^{A^{-1}x}_t =e^{At}x+\int^t_0e^{A(t-s)}A F(A^{-1}(A\mathcal{X}^{A^{-1}x}_s)) \, ds &\quad+\int^t_0e^{A(t-s)}A B(A^{-1}(A\mathcal{X}^{A^{-1}x}_s)) \, dW_s.
\begin{aligned}p = s_1s_2 + s_2s_3 + s_3s_1 q = s_1 + s_2 + s_3 x = 2p + q ( s_1 - s_2 - s_3 ) y = 2p + q ( s_2 - s_3 - s_1 ) z = 2p + q ( s_3 - s_1 - s_2 ) w = s_1^2 + s_2^2 + s_3^3 \end{aligned}
\begin{array}{l l l}{{d_{v}}}&{{=}}&{{2 k H + N k + N ( k + 1 ) = 2 k H + ( 2 k + 1 ) N}}\\{{d_{H}}}&{{=}}&{{[ \frac{2 k ( 2 k + 1 )}{2}2 H - 2 H k ( 2 k + 1 ) ] + [ N ] + [ 2 \frac{2 k}{2}]}}\\{{}}&{{+}}&{{[ \frac{2 k ( 2 k + 2 )}{2}2 N - N k ( 2 k + 1 ) - N ( k + 1 ) ( 2 k + 1 ) ]}}\\{{}}&{{=}}&{{2 k .}}\\ \end{array}
\left(\frac{{\rm d}}{{\rm d}t}-r\right)\int\frac{\big|\widetilde{U}\big|^2}{2h}+\int\frac{\widetilde{W}^{\rm T}Q_r(w^*)\widetilde{W}}{2h}+\int \widetilde{W}\cdot\mathrm{L}(w^*)=0.
A_{\ell+1}-B_{\ell+1}=A_\ell-B_\ell\Delta_{1,\ell}A_\ell-B_\ell\Delta_{1,\ell}B_1 &=A_\ell-B_\ell\Delta_{1,\ell}(A_\ell+B_1)=A_\ell-B_\ell=A_1-B_1.
\int d^{d}\theta \theta^{0}\theta^{1}\theta^{2}\theta^{3}\theta^{5}. . . \theta^{d}= 1 , \quad d^{d}\theta = d \theta^{d}. . . d \theta^{5}d \theta^{3}d \theta^{2}d \theta^{1}d \theta^{0}.
\overset{3}{R}(X,Y,AZ,AW) = \overset{3}{R}(AX,AY,Z,W), \overset{3}{R}(X,AY,AZ,W) = \overset{3}{R}(AX,Y,Z,AW), \overset{3}{R}(AX,AY,AZ,AW) = \overset{3}{R}(X,Y,Z,W).
\Sigma_k^{(1)}\left({\bf C}_{x_{B_k}},{\bf C}_{x_{j}}\right)=\log\left(1+\frac{{\bf h}^{H}_{kB}{\bf C}_{x_{B_k}}{\bf h}_{kB}}{\sum_{m=1,m\neq k}^{K}{\bf h}^{H}_{kB}{\bf C}_{x_{B_m}}{\bf h}_{kB}+\sum_{j=1}^{J}{\bf h}^{H}_{kj}{\bf C}_{x_{j}}{\bf h}_{kj}+\sigma^{2}_w+\sigma_n^{2}}\right).
\int d\nu_2(x) \int dt\, \widehat{g}(t)\frac{\sin{t\sqrt{x}}}{\sqrt{x}}=-i\sqrt{\frac{\pi}{2}}\int_0^{\infty}\left( g(x^{1/2})-g(-x^{1/2})\right) \frac{dx}{x}& = -i\sqrt{2\pi}\int_0^{\infty}(g(s)-g(-s))\frac{ds}{s}.
\sum_{X = 1}^\infty (Q_k(X) - \mathrm{MT}_k(X) + cX^{r - 1}q^{X/2k})u^{X - 1}= \frac{Z(u)}{Z(u^k)(1 - u)}- \frac{dq}{1 - qu}+ \frac{c}{u(1 - q^{1/2k}u)^r}\sum_{\ell = 0}^{r - 1}A(r - 1, \ell)q^{(\ell + 1)/2k}u^{\ell + 1}.
K_1=\sum_{j=1}^N \sum_{i=1}^{M-1}\left[ \sum_{k=1}^{H_0-1}(a_{ik}b_{kj}- a^0_{ik}b^0_{kj}) + c_i\right]^2, K_2= \sum_{j=1}^N \sum_{i=1}^{M-1}\left\{\sum_{k=H_0}^{H-1}a_{ik}b_{kj}\right\}^2.
E_0 = \frac{\pi}{p^+}\left( \left( \frac{p^{I_N}}{\pi}\right)^2 + \left( \frac{m}{3}\right)^2 (x^{i_N})^2 + \left( \frac{m}{6}\right)^2 (x^{i'_N})^2 + \frac{i}{\pi}\frac{m}{3}\psi_0 \gamma^4 \Omega \psi_0 - \frac{i}{\pi}\frac{m}{6}\psi'_0 \gamma^4 \Omega \psi'_0 \right) ~.
\ 0 f\Bigl(\frac{0}{0}\Bigr) = 0, f(0) = \lim_{t \rightarrow 0^+}f(t), &\ 0 f\Bigl(\frac{b}{0}\Bigr) = \lim_{t \rightarrow 0^+}t f\Bigl(\frac{b}{t}\Bigr) = b \lim_{u \rightarrow \infty}\frac{f(u)}{u}, \forall \, b > 0.
\int_{S^{2}}\mathbb{U}_{\vec{c},{\vec{c}}^{\prime}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2})\mathbb{U}_{\vec{c}^{\prime},{\vec{c}}^{\prime\prime}}^{j}(\mathbf{n}_{2},\mathbf{n}_{3})d\mathbf{n}_{2}=\mathbb{U}_{\vec{c},{\vec{c}}^{\prime\prime}}^{j}(\mathbf{n}_{1},\mathbf{n}_{3})\ .
T ( B , A ; \tau ) \equiv \int_{0}^{2 \pi}d \sigma W^{- 1}B_{\mu \nu}W \frac{\partial x^{\mu}}{\partial \sigma}\frac{\partial x^{\nu}}{\partial \tau}= \frac 1 2 \int_{0}^{2 \pi}d \sigma W^{- 1}B_{\mu \nu}W \frac{\partial ( x^{\mu}, x^{\nu})}{\partial ( \sigma , \tau )}
L(M\le k)=\sum_{i=0}^{k}L(M=i)=\sum_{i=m}^{k}{(\begin{matrix}i\\N\end{matrix})}^{-1}=(1-N^{-1})^{-1}({(\begin{matrix}m-1\\N-1\end{matrix})}^{-1}-{(\begin{matrix}k\\N-1\end{matrix})}^{-1})
\sum_{i=0}^n\sum_{j=0}^i(2(n-i) + (s+1)i)\frac{x_{i-j}}{(i-j)!}\frac{x_j}{j!}\sum_{\ell=0}^n (s+\ell)!\frac{B_{n-i, \ell}(x_1, x_2, \ldots)}{(n-i)!}= 0
(t,x,y^i,\dot{x},\dot{y}^i,\ddot{x},\ddot{y}^i)\in J^2\pi \to \left(x,y^i,\frac{\dot{y}^i}{\dot{x}},\frac{\ddot{y}^i\dot{x}-\ddot{x}\dot{y}^i}{\dot{x}^3}\right) = (x,y^i,y^i_{x},y^i_{xx})\in J^2(M,1)
\chi_{\Lambda^{\prime}}^{\mathrm{n e w}}= C \chi_{\Lambda^{\prime}}, ~ \psi_{\Lambda^{\prime}}^{\mathrm{n e w}}= C \psi_{\Lambda^{\prime}}, ~ K^{\mathrm{n e w}}= \frac{K}{C^{2}}, ~ \mu^{\mathrm{n e w}}= \frac{\mu}{C^{2}}, ~ \xi^{\mathrm{n e w}}= A ( 0 ) B ( 0 ) \xi ,
p_j(w^*)=\begin{cases}\cfrac{\overline{(i-1)b_j}+\overline{(i+1)b_j}-2\overline{ib_j}}{m}, \; \textnormal{if}1 \leq j \leq k-1, \cfrac{\overline{(i-1)a_{2k-j-1}}+\overline{(i+1)a_{2k-j-1}}-2\overline{ia_{2k-j-1}}}{m}, \textnormal{if}k \leq j \leq 2k-2, 0, \textnormal{if}j = 2k-1. \end{cases}
G_a =xy^a+\sum_{s=1}^{a-1}(a-1-s)xy^{a-s}G_b^{bt}+p\sum_{s=1}^{a-1}xy^{a-s}G_b^{t}+q\sum_{s=1}^{a-1}xy^{a-s}G_s^{b}&+xG_a+\sum_{s=2}^ax(y^{a-1}+\cdots+y^{a+1-s})G_s^t+\sum_{s=2}^ax(y^{a-1}+\cdots+y^{a+1-s})G_s^b &+\sum_{s\geq a+1}x(y^{a-1}+\cdots+y)G_s^t+\sum_{s\geq a+1}x(y^{a-1}+\cdots+y)G_s^b+(p+q)\sum_{s\geq a+1}xG_s &+\sum_{s\geq a+1}(s-1-a)xG_s.
\begin{array}{c}{{S = \frac 1{1 6 \pi G_{p + 1}}\int d^{p + 1}y \sqrt{- \tilde{g}}e^{- x^{0}}\{- \partial^{A}x^{0}\partial_{A}x^{0}+ a^{i}a^{j}\partial^{A}a_{k}\partial_{A}a_{l}\eta_{i j}^{k l}- 2 \Lambda \}}}\\{{- \frac 1{1 6 \pi G_{p + 1}}\int d^{p + 1}y \sqrt{- \tilde{g}}e^{- x^{0}}\tilde{R},}}\\ \end{array}
G_k=\Phi_0(G_k)=\langle x,y\rangle \Phi_1(G_k) G_k/\Phi_1(G_k) \cong C_p \times C_p, \Phi_1(G_k) = \langle x^p,y^p, [y,x], [y,x,x],\ldots, [y,x,\overset{p}{\ldots},x], [y,x,y] \rangle\Phi_2(G_k) \Phi_1(G_k)/\Phi_2(G_k) \cong C_p^{\, p+3},
-\frac{\partial L_{d}D_{\beta}I(p\|q)}{\partial q_{j}}=T\left\{\underbrace{\left[\frac{a-1}{a-b}(X.Y)^{a-2}-\frac{b-1}{a-b}(X.Y)^{b-2}\right]}_{Z>0}\frac{\partial (X.Y)}{\partial q_{j}}\right\}
U_{j}=\frac{1}{\alpha}\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]p^{\alpha}_{j}q^{-\alpha}_{j}&V_{j}=\frac{1}{\alpha}\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]
\Vert b-Ax_{m,k}\Vert = \Vert b-C_{m}C_{m}^{T}y_{m,k}\Vert =\left\Vert \Vert b\Vert e_{1}-H_{m}H_{m}^{T}z\right\Vert <\eta \widehat{\varepsilon}\Vert b\Vert\,,
|b(x)| = \tfrac{|l'(x)|}{|l(x)|}\le \| \phi'\|_{\infty}\tfrac{b_n}{\delta_n}= \| \phi'\|_{\infty}\tfrac{n^{2\gamma}-1}{\delta_n}= \| \phi'\|_{\infty}(n^{5 \gamma}- n^{3\gamma}) \le C n^{5 \gamma}\le C \big( n - \tfrac{1}{n^{3 \gamma}}\big)^{5 \gamma}+ 1 \le C( x^{5 \gamma}+ 1) \le C( x^{\theta}+ 1).
Q = G^{\ast}+ \bar{G}^{\ast}= \theta{\delta \over \delta X^{\ast}}-F^{\ast}{\delta \over \delta\eta}- \partial X{\delta \over \delta\psi}-\bar{\partial}X{\delta \over \delta\bar{\psi}}- (\partial\bar{\psi}+\bar{\partial}\psi){\delta \over \delta F}
I_{W_c}= \bigcap_{1 \leq i_1 < i_2 < \dots < i_c \leq s}\ \bigcap_{k = 1,\ldots,c \atop 1 \le j_k \le e_k}( A_{i_1,j_1}, A_{i_2, j_2}, \dots , A_{i_c, j_c}).
V ( \alpha^{*}, \alpha ) = \frac{g}{3 !}\left( V_{a b c}\hat{a}^{a}\hat{a}^{b}\hat{a}^{c}+ 3 \hat{a}_{a}^{\dagger}V_{\; \; b c}^{a}\hat{a}^{b}\hat{a}^{c}+ 3 \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}V_{\quad c}^{a b}\hat{a}^{c}+ \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{c}^{\dagger}V^{a b c}\right)
0 = \Lambda\circ \nabla_0(f) = \Lambda\left(\frac{df_x^0(x)}{dx}- f_\vartheta^1(x) + \frac{df_\vartheta^0(x)}{dx}\vartheta\right) = \Lambda\left(\frac{d}{dx}f_\vartheta^0(x)\vartheta\right).
B_l^{(j+1)}=\sum_{q=2}^k\sum_{(p_1,\dots,p_q)\in\mathbb{I}_{2k}^q,\, \deg(S_{p_1})+\cdots+\deg(S_{p_q})=l}\alpha_{(p_1,\dots,p_q)}[S_{p_1},\dots,S_{p_q}].
\int_{(\mathbb{R}^n)^p}\prod_{j=1}^{k}f_j\left(\sum_{m=1}^{p}a_{jm}x_m\right)dx\leq\int_{(\mathbb{R}^n)^p}\prod_{j=1}^{k}f^*_j\left(\sum_{m=1}^{p}a_{jm}x_m\right)dx,
{\Theta_{i}\over\Theta_{i_0}}={\det_{k\neq i_0,i\to i_0;j=1,\dots,N-1}\theta_{k\over N}(Nq_j| N\tau)\over\det_{k\neq i_0;j=1,\dots,N-1}\theta_{k\over N}(Nq_j| N\tau)}.
I = \int \mathrm{d}^{4}x \mathrm{d}^{4}y h_{\mu \nu}( x ) S_{s u b}^{\mu \nu \, \alpha \beta}( x - y ) h_{\alpha \beta}( y ) + \int \mathrm{d}^{4}x \mathrm{d}^{4}y J^{\mu \nu}( x ) X_{\mu \nu \, \lambda}( x - y ) \eta^{\lambda}( y ) + \cdots .
u_i = \frac{u_i^0}{\nu_{m}}, \dot{u}_i = \frac{\dot{u}_i^0}{\nu_{m}}, \tau = \frac{\tau^0}{\nu_{m}}, \dot{\tau}= \frac{\dot{\tau}^0}{\nu_{m}}, q_i = \frac{q^0_i}{\nu_{m}}\Omega \times \{0\}.
\zeta_{c y l}^{D}( - 1 ) = - \frac{1}{4 \pi a}\left[ 0 . 4 1 5 5 7 4 - 0 . 0 6 0 8 9 7 - \frac{1}{4}\ln ( 2 \pi ) + 0 . 0 4 5 6 1 1 \right] = \frac{0 . 0 0 1 2 1 3}{a^{2}}\,{.}
{\frac{1}{C_{A}}}= \frac{2^{\frac{r}{2}}\prod ( k_{j}+ 2 )}{K \kappa_{\alpha}^{A}\kappa_{\widetilde{\alpha}}^{A}}, \quad{\frac{1}{C_{B}}}= \frac{2^{\frac{r}{2}}}{\kappa_{\alpha}^{B}\kappa_{\widetilde{\alpha}}^{B}},
D_{\mu}\psi^{b}= \partial_{\mu}\psi^{b}+ \Gamma_{c d}^{b}\partial_{\mu}\varphi^{c}\psi^{d}, \qquad \Gamma_{b c}^{a}= \varphi^{a}g_{b c}= v^{a j}\partial_{( b}v_{c )}^{~ ~ j}, \qquad R_{a b c d}= g_{a c}g_{b d}- g_{a d}g_{b c}= R_{c d a b}~ .
i \frac{d b_{m}}{d t}= - \sum_{n \neq m}e^{- i \lambda \int_{0}^{t}d t^{\prime}[ E_{n}( t^{\prime}) - E_{m}( t^{\prime}) ]}e^{i [ \gamma_{n}( t ) - \gamma_{m}( t ) ]}< m ; t | i \frac{\partial}{\partial t}| n ; t > b_{n}( t ) .
J_{T^*T^*G}^{-1}(0)/G = \{(\pi, (\pi', \omega))\in \mathfrak{g}^* \times (\mathfrak{g}^*\times \mathfrak{g})\colon \pi'=0\}\cong \mathfrak{g}^*\times\mathfrak{g}, J^{-1}_{TT^*G}(0)/G = \{( \pi, (\omega, \dot \pi )) \in \mathfrak{g}^*\times (\mathfrak{g}\times \mathfrak{g}^*) \colon \dot\pi = ad^*_\omega \pi\}\cong \mathfrak{g}^*\times\mathfrak{g},
(-\Delta \tilde u, v)_{\Omega_h \setminus \Omega}+ (\partial_{n_h}\tilde u, v)_{\Gamma_h}= (-\Delta u, v)_{\Omega \setminus \Omega_h}+ (\nabla\tilde u, \nabla v)_{\Omega_h \triangle \Omega}' + (\partial_{n}u, v)_{\Gamma}&= -(u - f, v)_{\Omega \setminus \Omega_h}+ (\nabla\tilde u, \nabla v)_{\Omega_h \triangle \Omega}' + (\tau, v)_{\Gamma}.
\begin{aligned}\mathbf{u}\cdot \mathbf{n}=0,x\in \partial\Omega,0 \le t \le T, D(\mathbf{u})\cdot \mathbf{n}=0,x\in \partial\Omega,0 \le t \le T, c(x,0) =c_{0}(x),x\in \Omega,\end{aligned}
b_{1}=\Bigg(\frac{(p-1)3^{\frac{p}{p-1}}\|u_{1}\|_{2p}^{2p}}{pE_{\omega}}\Bigg)^{p-1},\ \ \b_{2}=\Bigg(\frac{(p-1)3^{\frac{p}{p-1}}\|u_{\omega}\|_{2p}^{2p}}{pE_{1}}\Bigg)^{p-1}, &b_{3}=\Bigg(\frac{(p-1)3^{\frac{3}{p-3}}\|u_{1}\|_{2p}^{2p}}{pE_{\omega}}\Bigg)^{p-1},\ \ \b_{4}=\Bigg(\frac{(p-1)3^{\frac{3}{p-3}}\|u_{\omega}\|_{2p}^{2p}}{pE_{1}}\Bigg)^{p-1}.
R_\mathsf{A}\big(p_{\tilde{U},\tilde{V},\tilde{X}|S}\big)=\min\Big\{I(U,V;Y)-I(U,V;Z),I(U,V;Y)-I(U,V;S)\Big\}\geq \tilde{R}_\mathsf{A}\left(p_{U,V,X|S}\right).
\hat{\Psi}( m , z ) = N_{m}\sqrt{\frac{g ( z )}{k}}\left[ \left\{\begin{array}{c}{{1}}\\{{B}}\\{{e^{2 i q z_{l}}}}\\{{e^{2 i q z_{l}}B}}\\ \end{array}\right\}H_{2}^{+}\left( \frac{m}{k}g ( z ) \right) + \left\{\begin{array}{c}{{A}}\\{{C}}\\{{e^{2 i q z_{l}}A}}\\{{e^{2 i q z_{l}}C}}\\ \end{array}\right\}H_{2}^{-}\left( \frac{m}{k}g ( z ) \right) \right]
\mathcal{K}^1(v,\eta_f,\theta_f) = \sum_{i,j}\int_{T_{ij}}\eta_f v\,\partial_x \theta_f\,{\rm d}x\,{\rm d}v + \sum_{i,j}\int_{J_j}\left(\widehat{v\,\eta_f}[\![\theta_f]\!]\right)_{i-1/2,v}\,{\rm d}v\, .
\cosh(x) = 1 + \frac{x^2}{2}\int_0^1\cosh(tx)(1-t) \:dt \frac{1}{\cosh(x)}= 1 + \frac{x^2}{2}\int_0^1 \Big(\frac{\tanh(tx)^2}{\cosh(tx)}- \frac{1}{\cosh(tx)^3}\Big)(1-t)\:dt,
\frac{\mu^{s + 1}}{2}\left( \frac{2 \pi}{L}\right)^{- s}\frac{1}{\Gamma \left( \frac{s}{2}\right)}\int_{0}^{\infty}d t t^{( \frac{s}{2}- 1 )}e^{- ( \frac{m L}{2 \pi})^{2}t}\Theta ( 0 , \frac{t}{\pi}\rfloor_{s = - 1},
R^{j}(\mathbf{n}_{1},\mathbf{n}_{2}) =\frac{1}{4\pi}\displaystyle{\sum_{l=0}^{2j}\sum_{m=-l}^{l}}\overline{Y_{l}^{m}}(\mathbf{n}_{1}){Y_{l}^{m}}(\mathbf{n}_{2})\ \ & =\frac{1}{4\pi}\displaystyle{\sum_{l=0}^{2j}}(2l+1)P_{l}(\mathbf{n}_{1}\cdot\mathbf{n}_{2})\ =R^{j}(\mathbf{n}_{2},\mathbf{n}_{1})
M_2=\left(\begin{aligned}-m_1 b_2 & \,\,\, b_1 -m_2\end{aligned}\right),M_3=\left(\begin{aligned}-m_1 \,\,\, 0 b_3 & b_1 -m_2 \,\, 0 & 0 \,\, b_2 -m_3\end{aligned}\right).
q_T^{(<)}= Q_T(A^{(<)}_{T,r_1(T)}) =E^{P_T}\Big[\mathbf{1}_{A^{(<)}_{T,r_1(T)}}\mathcal{E}_T\Big], q_T^{(>)}= Q_T(A^{(>)}_{T,r_2(T)}) = E^{P_T}\Big[\mathbf{1}_{A^{(>)}_{T,r_2(T)}}\mathcal{E}_T\Big].
g(x^m)-g(x^0)=\exp(-\sum\limits_{n=1}^\infty\|x_n^0\|_n)(\exp(\sum\limits_{n=1}^\infty\|x_n^0\|_n-\sum\limits_{n=1}^\infty\|x_n^m\|_n)-1)= =g(x^0)(\exp(-\ln 2)-1)=-\frac 12 g(x^0).
\lefteqn{\sum_{i=0}^n \bigl((1-T_iT_i^*)\otimes T_iT_i^*+T_iT_i^*\otimes (1-T_iT_i^*)\bigr)e_+}-\sum_{i<j}(T_iT_i^*\otimes T_jT_j^*+T_jT_j^*\otimes T_iT_i^*)e_+ =\sum_{i=0}^{n-1}(P_i-\sum_{j=i+1}^nP_iP_j)+P_n
b_{\mu}= \sum_{n = 1}^{3}\, \bigl [ \, b_{\mu}^{n}\, H_{n}+ \sum_{k = 1}^{3}\, ( \, b_{\mu}^{k n -}\, E_{n k}^{-}+ b_{\mu}^{k n +}\, E_{k n}^{+}\, ) \bigr ] .
\|W-W'\|\leq \frac{4\|\Sigma\|^{1/2}+4\sqrt{2\delta}}{\sqrt{n}}\sup_{\|u\|\leq 1}\biggl(\sum_{j=1}^n \langle X_j-X_j',u\rangle^2\biggr)^{1/2}\bigvee\frac{4}{n}\sup_{\|u\|\leq 1}\sum_{j=1}^n \langle X_j-X_j',u\rangle^2.
\sum_{n\ge0}\frac{(q^2;q^2)_nq^{2n^2}}{(-q^2;q^2)_n(q^2;q^4)_n(1-q^{2n+1})}= \frac{1}{2}\sum_{n\ge0}q^{3n^2/2}(1+q^{2n+1})(1+q^{n+1/2})\sum_{|j|\le n}q^{j^2/2}