harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
z'_i=z_i+\sqrt{2}a_iz_{m+2},\ \ 1\leq i\leq n &z'_l=\sqrt{2}a_lz_{m+2},\ \n+1\leq l\leq m &z'_{m+1}=\sqrt{2}\sum\limits_{i=1}^{n}{a_iz_i}+1+(a^2_1+\cdots+a^2_m)z_{m+2}&z'_{m+2}=z_{m+2}
\dot{\beta}(t, u^{1}, \dotsc, u^{m-d-1}) + u^{m-d}\dot{X}_{m-d}(t) + \dotsb + u^{m-1}\dot{X}_{m-1}(t) &= a_{1}X_{1}(t) + \dotsb + a_{m-1}X_{m-1}(t)
\left\langle \prod_{b = 1}^{N}\left[ \psi_{1}^{( b )}( - L , 0 ) \; U \Big ({\cal C}( L ) \Big ) \; \overline{{{\psi}}}_{2}^{( b )}( + L , 0 ) \right] \prod_{c = 1}^{l}\left[ \psi_{1}^{( c )}( - L , 0 ) \; U \Big ({\cal C}( L ) \Big ) \; \overline{{{\psi}}}_{2}^{( c )}( + L , 0 ) \right] \right\rangle_{0}^{\theta}
\mathbb{P}\left\{\int_{0}^{1}g_tdDX_t=0 \right\}=\mathbb{P}\left\{\int_{0}^{1}g_tdDX_t=0,\g_t\neq 0 \right\}+\mathbb{P}\left\{\int_{0}^{1}g_tdDX_t=0,\g_t=0 \right\}=\mathbb{P}\left\{\int_{0}^{1}g_tdDX_t=0,\g_t=0 \right\}=\mathbb{P}\left\{\g_t=0 \right\}.
\rho^{*}\equiv\frac{\rho}{\rho_{0}},u^{*}\equiv\frac{u}{u_{0}},r^{*}\equiv\frac{r}{r_{0}},t^{*}\equiv\frac{u_{0}}{r_{0}}t,\nabla^{*}\equiv r_{0}\nabla,g^{*}\equiv\frac{g}{g_{0}},\sigma^{*}\equiv\frac{\sigma}{p_{0}},
D_{r}= ( \gamma^{\mu}\otimes \mathbf{1}) ( \mathbf{f}_{\mu}- \frac{1}{4}\Gamma_{\mu \nu \rho}( x ) \gamma^{\nu}\gamma^{\rho}) - \frac{1}{8}F_{\mu \nu k}( x ) \omega \gamma^{\mu}\gamma^{\nu}\otimes \gamma^{k}+ \frac{1}{4}A_{\mu}^{\; m}( x ) t_{m l k}\gamma^{\mu}\otimes \gamma^{k}\gamma^{l}+ \frac{1}{4}t_{m l k}\omega \otimes \gamma^{k}\gamma^{l}\gamma^{m}\; ,
S_6:= \sum_{\substack{w_1 R< |w_1|\leq N^{2b}}}\frac{\alpha(|w_1|)}{|w_1|}S_7:= \sum_{\substack{w_1 Y\leq |w_1|\leq R}}\frac{\alpha(|w_1|)}{|w_1|}\left(V^{(m)}\left(\frac{\log |w_1|}{\log R}\right)\right)^2
\frac{4(n-1)}{n-2}\Delta_{\hat{g}}\hat{\varphi}+R_{\hat{g}}\hat{\varphi}+\frac{n-1}{n}\hat{\tau}^{2}\hat{\varphi}^{N-1}=\frac{\hat{w}^{2}}{\hat{\varphi}^{N+1}}.
\sum^{\infty}_{m,n=-\infty}q^{n^2+mn+m^2}=1+6\sum^{\infty}_{n=0}\frac{q^{3n+1}}{1-q^{3n+1}}-6\sum^{\infty}_{n=0}\frac{q^{3n+2}}{1-q^{3n+2}}.
\zeta ( \{2 p \}_{n}) = \frac{2 p \cdot ( 2 \pi )^{2 p n}}{( 2 p n + p ) !}\left( \frac{1}{2 \sin \frac{\pi}{2 p}}\right)^{2 p n + p}\bigg \{1 + \sum_{k = 2}^{N_{p}}R_{p , k}^{2 p n + p}\bigg \},
\Lambda^{-}(X) = \sum_{i\in\mathcal{I}^-}(A_i-\overline{X})^T \otimes \Phi^-_i(X) \overline{M}^{+}+ \sum_{i\in\mathcal{I}^+}(A_i-\overline{X})^T\otimes \Phi^-_i(X) - \left( (X-\overline{X})^T\otimes I_2 \right) \overline{M}^+.
\|\langle x \rangle^{s}(P(h)-E \pm i \varepsilon)v\|_{L^2}\le \|(P(h)-E \pm i \varepsilon)\langle x \rangle^{s}v \|_{L^2}+ \|[P(h),\langle x \rangle^{s}]\langle x \rangle^{-s}\langle x \rangle^{s}v \|_{L^2}& \le C_{E_{\max}, \varepsilon, h}\| \langle x \rangle^{s}v \|_{H^2},
\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \operatorname*{l i m}_{\mu_{f}, \mu_{g}\to 0}\int_{0}^{\infty}d |{\bf p}| \:{\tilde{f}}_{{\mu}_{f}}( |{\bf p}| ) \cdot \frac{\partial}{\partial |{\bf p}|}{\tilde{g}}_{{\mu}_{g}}( |{\bf p}| ) .
u_{M}^{N}(x):=\frac{1}{N}\sum_{i=1}^N\left[ g(X^{x,i}_{M})+\frac{1}{d}\sum\limits_{k= 1}^M \widetilde{r}^2(X^{x,i}_{k-1})f\left(X^{x,i}_{k-1}+\widetilde{r}(X^{x,i}_{k-1})Y^i\right)\right], x\in D,
\exp\bigr(\mathcal{D}_\mathbb{P}(s)\bigr) =\exp\Biggr(\sum_{p\in\mathbb{P}}p^{-s}\Biggr)=\prod_{p\in\mathbb{P}}\exp\bigr({p^{-s}}\bigr)=\zeta(s)\prod_{p\in\mathbb{P}}\exp(p^{-s})(1-p^{-s})
i \frac{d}{dt}\widehat{\psi}_l(t) - \alpha \frac{d^2}{dt^2}\widehat{\psi}_l(t) - |\mu_l|^2 \widehat{\psi}_l(t) - \varepsilon^{2}\widehat{(f(\psi_M))}_l(t) =0, l\in \mathcal{T}_M,\t > 0.
\left | \int_{S_R}A_n(\gamma_v)\cdot (Id-T)(\mu_i-\mu_0) \right| =\left |\int_{S_R}(Id-R^*)\left( A_n(\gamma_v)\right)\cdot(\mu_i-\mu_0)\right| &\leq \frac{2\|\gamma_v\|}{n}\| \mu_i-\mu_0\|.
{F(x) \over x^{\ell}}\int_{0}^{x}dy y^{\ell-1}F(y) ={F(x) \over x^{\ell}}\int_{\Lambda^2}^{x}dy y^{\ell-1}F(y) +{F(x) \over x^{\ell}}\int_{0}^{\Lambda^2}dy y^{\ell-1}F(y) ,
(T_1,T_2,T_3) = K_1 \begin{pmatrix}0 \omega \omega' -\frac{\imath \pi}{\alpha}\eta \eta' \end{pmatrix}, (H_1,H_2,H_3) = K_2 (T_1,T_2,T_3) + K_3 \begin{pmatrix}0 \omega \omega' 0 \eta \eta' \end{pmatrix},
R= a^6 \left( D^6+ \tilde{R}^{(5)}D^5 + \tilde{R}^{(4)}D^4 + \tilde{R}^{(3)}D^3 + \tilde{R}^{(2)}D^2 + \tilde{R}^{(1)}D + \tilde{R}^{(0)}\right) &+\sigma^{(1)}D^{-1}\gamma^{(1)}+\sigma^{(2)}D^{-1}\gamma^{(2)}.
U_{k}= \left( \left| \Psi \right\rangle \left\langle \Psi \right| \right)_{k , k + 1}= \left( \begin{array}{l l l}{{0}}&{{0}}&{{0}}\\{{0}}&{{V}}&{{0}}\\{{0}}&{{0}}&{{0}}\\ \end{array}\right)_{k , k + 1}
\begin{matrix}p^\dagger_0 = 2a_0^{\dagger}, \displaystyle p^\dagger_k = \frac{1}{\prod_{i=1}^{k}\sqrt{(s+i)(s-i+1)}}\left( a_{-k}^{\dagger}J_+^k + a_k^\dagger J_-^k \right), \displaystyle m^\dagger_k = \frac{1}{\prod_{i=1}^{k}\sqrt{(s+i)(s-i+1)}}\left( a_{-k}^{\dagger}J_+^k - a_k^\dagger J_-^k \right).\end{matrix}
\varphi_n(\mu)\left(\begin{array}{cc}\ast \ast r s\end{array}\right):=\Delta^{\frac{\underline{k}}{2}}\frac{\mu\left(P_n\right)}{(|r|^2+|s|^2)^{M+n+2}},P_n(r,s):=\left|\begin{array}{cc}\bar s -\bar r x y\end{array}\right|^{n+\lambda}\left|\begin{array}{cc}r s x y\end{array}\right|^{n-\lambda}.
\int_{0}^{\infty}dr~r^{\frac{n}{2}}~(c^2 + r^2)^{m}~J_{\frac{n-2}{2}}(r|p|) ~=~ (2/|p|)^{m+1}~\frac{c^{\frac{n}{2}+m}}{\Gamma(-m)}~K_{\frac{n}{2}+m}(c|p|) ~~~,
c_{n-2,\xi_{\boldsymbol{s}k},\xi_{\boldsymbol{t}l}}^{\ast\ast}=\frac{\Gamma((n-2)/2)}{(2\pi)^{(n-2)/2}}\left\{\int_{0}^{\infty}t^{(n-4)/2}\ln\left[1+\exp\left(-\frac{1}{2}\xi_{\boldsymbol{s},k}^{2}-\frac{1}{2}\xi_{\boldsymbol{t},l}^{2}\right)\exp(-t)\right]\mathrm{d}t\right\}^{-1}.
\mathcal K_N:=\{\beta=(\beta_t)_{t=0}^{T-1}: \beta_t (\widehat{\mathcal{R}}_N,\mathcal B(\widehat{\mathcal{R}}_N)) (\widehat{\mathcal{E}}_{t,N},\mathcal B(\widehat{\mathcal{E}}_{t,N})), & t \in [\![0,T]\!]\}.
\zeta_R'(r)=\frac1r\int_{\partial D_r}\nabla z_R\cdot\nu \, \overline{\psi^k_2(\theta_0(x))}\,ds=-\frac{i}{\sqrt\pi}r^{-\frac k2-1}\int_{\partial D_r}e^{-\frac{i}{2}\theta_0}(i\nabla+A_{0})z_R\cdot\nu\, \psi_k\, ds.
L^X f = \sum_{i=1}^n b_X^i \partial_{x_i}f + \sum_{i,j=1}^n a_X^{ij}\partial_{x_i x_j}^2 f, L^Y f = \sum_{i=1}^m b_Y^i \partial_{x_i}f + \sum_{i,j=1}^m a_Y^{ij}\partial_{x_i x_j}^2 f, L f = (L^X + L^Y)f.
f=\sum\limits_{k=-\infty}^{\infty}\sum\limits_{Q}\mu(Q){\widetilde{\widetilde{D}}}_{k}(x, x_Q) D_{k}(f)(x_Q)=\sum\limits_{k=-\infty}^{\infty}\sum\limits_{Q}\mu(Q) D_{k}(x,x_Q){\bar{\bar{D}}}_{k}(f)(x_Q),
\frac{\Im w_{n+1}}{10}=\Re \frac{4(w_1+1)\sum\limits_{j=3}^{n}|w_j|^2 +|w_1|^2(2w_2+w_1\overline{w_2})+4w_1\overline{w_2}+2(w_1)^2\overline{w_2}}{(2+w_1)(2+\overline{w_1})(20-d_nw_1\overline{w_1})}.
I_{\alpha \rho}= - \int_{0}^{1}d h \int_{0}^{h}d t\frac{\partial c_{t}^{\mu}}{\partial x^{\rho}}\frac{\partial c_{t}^{\nu}}{\partial t}\frac{\partial c_{h}^{\delta}}{\partial x^{\alpha}}\frac{\partial c_{h}^{\beta}}{\partial h}[G_{\beta \delta}(c_{h}), G_{\mu \nu}(c_{t})]
\varphi(t,s)-\varphi(t,0)= \sum_{i=1}^5\Big(x^i(t,s)-x^i(t,0)\Big)\frac{\partial}{\partial x^i}= \sum_{i=1}^4\Big(x^i(t,s)-x^i(t,0)\Big)e_i(t)+\Big(x^5(t,s)-x^5(t,0) &+\frac 1 2\Big((x^1(t,s)-x^1(t,0))x^3(t,0)+(x^2(t,s)-x^2(t,0))x^4(t,0) &-(x^3(t,s)-x^3(t,0))x^1(t,0)-(x^4(t,s)-x^4(t,0))x^2(t,0)\Big)\Big)e_5(t)\,.
\int_{V}\frac{e^{\frac{q_1}{\alpha_0^2}(u \circ \tau_{b_i})^2}- 1}{(1 + |u \circ \tau_{b_i}|)^2}\dx \leq C_1 \int_{U}|u \circ \tau_{b_i}|^2 \dv_g , = C_1 \int_{\tau_{b_i}(U)}u^2 \dv_g .
\mathbf{p}E_{jm}=E_{jm}+\sum_{i=1}^{j-1}\sum_{s=2}^{j-i+1}\sum_{I\in\mathcal{I}_{ij}(s)}E_{i_si_{s-1}}\cdots E_{i_2i_1}E_{i_1 m}\frac{1}{\prod_{\ell\in I,\ell\neq j}(h_j-h_\ell)}
U_n:= \left(\frac{2^{n+2}}{T}+C(\varepsilon, p)\right) \int_{T_n}^{T}\int \eta_n^2u_n^p\;dxdt \;+(C(p)+1)2^{2n+2}\int_{T_n}^{T}\int_{B_n}a \eta_n^2 u_n^p \;dxdt +2 p k_n^2 \int_{T_n}^{T}\int \eta_n^2 u_n^{p-1}\;dxdt.
( X_{\cdot, i}^{(u)}, X_{\cdot, i+1}^{(u)}, \ldots , X_{\cdot, i+k-1}^{(u)}) \, =\, ( X_{\cdot, 1}^{(u)}, X_{\cdot, 2}^{(u)}, \ldots , X_{\cdot, k}^{(u)}) \, ; i=1, \ldots , n-k +1\, .
\begin{aligned}\lim_{\theta\to\infty}-\frac{2b_n+2a(\theta)^2}{\theta}= \frac{N-1+2\Delta}{2}+\frac{(2\Delta-1)(N+1)}{4}\left(\frac{1}{2n-N-1}-\frac{1}{2n-N+1}\right), \lim_{\theta\to\infty}\frac{4 u_n}{\theta^2}= \frac{n(N+1-n)(2n-N-2\Delta)(2n-N-2+2\Delta)}{4(2n-N-1)^2}. \end{aligned}
\frac{d^{n-3}}{dv^{n-3}}\left(F_t\frac{df}{dv}\right)=\sum_{i=0}^{n-3}\binom{n-3}{i}\frac{d^iF_t}{dv^i}\frac{d^{n-2-i}f}{dv^{n-2-i}}=P_2.
\tilde D_{k+1}^{\ell}(\Gamma_{k+1}^{\ell}) = \sum_{j_{\ell}=1}^{N_{k+1}^{\ell}}\sum_{i \in I_k}\Psi_{\ell,j_{\ell}}(x_k^{i}) \, p_k^i = \sum_{j_{\ell}=1}^{N_{k+1}^{\ell}}\mathbb E \Psi_{\ell,j_{\ell}}(\hat X_k),
0 < p \big(\phi(p-1) - \phi(p+1)\big) + \phi(p-1) + \phi(p+1) &\leq p \big(\phi(p-1) - \phi(p+1)\big) + \tfrac{1}{2}(p-1) + \tfrac{1}{2}(p+1) &= p \big(\phi(p-1) - \phi(p+1) + 1\big)
E \left [ |\Delta Y_t|^2+\int_t^T |\Delta Z_s|^2 ds + \int_t^T \int_U |\Delta U_s(e)|^2 \lambda(de)ds \right ] = 2 E \left [ \int_t^T \langle \Delta Y_s, f(s,Y_s^1, Z_s^1, U_s^1)-f(s, Y_s^2,Z_s^2,U_s^2) \rangle ds \right ] &+ 2 E \left [\int_t^T \langle \Delta Y_s, d\Delta\Lambda_s \rangle \right].
\begin{array}{l l}{{d s_{6}^{2}=}}&{{e^{2 \lambda}\left( - d t^{2}+ d z^{2}\right) + e^{2 \varphi}d x^{\mu}d x^{\mu},}}\\{{H_{e}=}}&{{d \left( e^{2 \lambda}d t \wedge d z \right) ,}}\\ \end{array}
\begin{bmatrix}\pi(P_{s(e)}) 0 0 *\end{bmatrix}=\rho(S_e^*S_e)=\rho(S_e)^*\rho(S_e)=\begin{bmatrix}\pi(S_e)^*\pi(S_e)+Y_e^*Y_e * * *\end{bmatrix},
\Delta E=\int_{1}^{2}IVdt=\int_{1}^{2}Id\Phi=\int_{\varphi_{1}}^{\varphi_{2}}I_{c}sin\varphi d(\Phi_{0}\frac{\varphi}{2\pi})=-\frac{\Phi_{0}I_{c}}{2\pi}\Delta cos\varphi.
\left.\begin{array}{ll}p_{n-1}\left(\lambda_{n+1}-\lambda_{n-1}\right)+a_{n+1}p_n\left(\lambda_{n+1}-\lambda_n\right) =0 q_{n-1}\left(\lambda_{n+1}-\lambda_{n-1}\right)+a_{n+1}q_n\left(\lambda_{n+1}-\lambda_n\right) =0\end{array}\right.
\phi^k_{n}=\psi_n\circ\varphi_{k+1}^{-1}\circ\phi^k\circ \varphi_{k+1}\circ\psi_n^{-1}\mbox{and}\phi_{n}=\varphi_n\circ\varphi_0^{-1}\circ\phi\circ\varphi_0\circ \varphi_n^{-1}\ .
\sum_{n = 1 / 2 , \cdots}{\frac{\Delta p^{+}}{p_{n}^{+}}}= \sum_{n = n_{\mathrm{I R}}+ 1}^{n_{\mathrm{U V}}- 1}\frac{1}{n}\simeq \mathrm{l n}\frac{n_{\mathrm{U V}}}{n_{\mathrm{I R}}}\simeq \mathrm{l n}\frac{2 \Lambda^{2}}{M^{2}+{\bf p}_{\bot}^{2}},
E_1^D * q_{k+1}^D(\boldsymbol{\mathcal{T}}_n) = E_{k+1}^D * \boldsymbol{\mathcal{T}}_n - \boldsymbol{\alpha}_{k}\times E_{k+1}^D - \boldsymbol{\beta}_{k}\times E_{k}^D, p_{k+1}(\boldsymbol{\mathcal{T}}_n)\times\boldsymbol{\beta}_{k+1}= \boldsymbol{\mathcal{T}}_n * E_{k+1}- E_{k+1}\times\boldsymbol{\alpha}_{k}- E_k.
R_{M N}-{\frac{1}{2}}G_{M N}R +{\frac{1}{2}}{\frac{\sqrt{-{\widehat G}}}{\sqrt{- G}}}{\delta_{M}}^{\alpha}{\delta_{N}}^{\beta}{\widehat G}_{\alpha \beta}{\widetilde f}_{D - 2}\delta^{( 2 )}( x^{i}) = 0 ,
c_p(n) = \frac{1}{p2^p}\Bigl[\bigl(\zeta(p,\,1+2\lambda)-2\zeta(p, 1+\lambda)\bigr)\frac{B_{p+1}(n)-B_{p+1}}{p+1}+\zeta(p)\frac{B_{p+1}(n+1)-B_{p+1}}{p+1}-n\zeta(p)\Bigr].
\mathbf{X}_{\mathrm{t}}=\begin{bmatrix}\mathbf{B}_1 \mathbf{B}_2 \cdots \mathbf{B}_{T}\mathbf{B}_2 \mathbf{B}_3 \cdots \mathbf{B}_{T+1}\vdots \vdots \ddots \vdots \mathbf{B}_{U}\mathbf{B}_{U+1}\cdots \mathbf{B}_{K}\end{bmatrix}
|(b_{\sigma_2})_{2B,w}-(b_{\sigma_2})_{2B}|\leq \frac{1}{w(2B)}\int_{2B}|b_{\sigma_2}(z)-(b_{\sigma_2})_{2B}|w(z)dz &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq \frac{C}{w(2B)}w(2B)\|b_{\sigma_2}\|_*=C\|b_{\sigma_2}\|_*
|x|^{\frac{n-p}{p}}\lambda^{-\frac{\theta-1}{\theta}}=|z|^{\frac{n-p}{p}}\left[(q-1)\log_{q}\frac{R}{|z|}+\lambda\left(\frac{R}{|z|}\right)^{1-q}\right]^{-\frac{\theta-1}{\theta}}.
\nabla ( V , V_{\bar{I}}, \bar{V}, \bar{V}_{I}) = ( V , V_{\bar{J}}, \bar{V}, \bar{V}_{J}) \left( \begin{matrix}{{0}}&{{0}}&{{0}}&{{\bar{P}_{I}}}\\{{0}}&{{0}}&{{\bar{P}^{\bar{J}}}}&{{\bar{P}_{\I}^{\bar{J}}}}\\{{0}}&{{P_{\bar{I}}}}&{{0}}&{{0}}\\{{P^{J}}}&{{P_{\ \bar{I}}^{J}}}&{{0}}&{{0}}\\ \end{matrix}\right)
|\xi^\pm(s)-x| =|\xi^\pm(0)-\int_s^0\dot{\xi}^\pm(\sigma)d\sigma-x| &\leqslant |z|+|\int_s^0(\dot{\xi}^*(\sigma)\pm\frac z \tau)d\sigma| &\leqslant 2|z|+\tau K &\leqslant \frac \rho 2.
\left\langle \left(W_{(0)}(x_{\mu}) - W_{(0)}(x_{\mu}^{'})\right) \prod_{i=1}^{k}W_{(0)}(x_{k}) \right\rangle = \left\langle \left\{Q_{W}, \int_{x_{\mu}}^{x_{\mu}^{'}}\partial_{\mu}W_{(0)}dx^{\mu}\cdot \prod_{i=1}^{k}W_{(0)}(x_{k})\right\}\right\rangle = 0,
| \Psi_{s,t}(x,v) - v| = | \Psi_{s,t}(x,v) - \zeta( \Psi_{s,t}(x,v))| \leq \frac{|\tilde Y_{s,t}(x,\Psi_{s,t}(x,v))|}{t-s}\leq \frac{|\tilde Y_{s,t}(x,v)|}{t-s}+ \frac 1 2 | \Psi_{s,t}(x,v) - v|,
\rho(\mu)=(\begin{matrix}cos(\frac{2\pi}{3})&0&-sin(\frac{2\pi}{3})\\ 0&1&0\\sin(\frac{2\pi}{3})&0&cos(\frac{2\pi}{3})\end{matrix}),\rho(\nu)=(\begin{matrix}-1&0&0\\ 0&-1&0\\ 0&0&1\end{matrix}).
P_{\ \nu}^{\mu}= -{\frac{1}{( d - 2 )^{2}}}\left[ R_{\ \lambda}^{\mu}R_{\ \nu}^{\lambda}-{\frac{d}{2 ( d - 1 )}}R R_{\ \nu}^{\mu}-{\frac{1}{2}}\delta_{\nu}^{\mu}R_{\ \beta}^{\alpha}R_{\ \alpha}^{\beta}+{\frac{d + 2}{8 ( d - 1 )}}\delta_{\ \nu}^{\mu}R^{2}\right] \ .
F_{\rm KD}(z;{\boldsymbol{t}}):= F_{\rm KD}(z;t_1,\ldots,t_n)=\sum_{1 \le i \not= j \le n}{c_{ij}\over (z-t_i)(t_i-t_j)}= \sum_{1 \le i < j \le n}{c_{ij}\over(z-t_i)(z-t_j)}.
Q(x)=x^{T}Lx+\gamma((1-x)^{T}F(1-x)+x^{T}Bx)=\sum_{e_{ij}}w_{ij}(x_{i}-x_{j})^{2}+\gamma(\sum_{v_{i}}f_{i}(1-x_{i})^{2}+\sum_{v_{i}}b_{i}x_{i}^{2}),
\left\| R_{22}\right\|_2 \leq \sum\limits_{j=1}^{m}\left\| \int_0^t AE(t-s)\omega_{\alpha_j}(s) (B_jA^{-1}) Au_0 ds \right\| \leq C \sum\limits_{j=1}^{m}t^{\alpha_j}\left\| Au_0\right\| \leq C t^{\alpha-1}\left\| u_0\right\|_2,
{\cal B}\Bigl ( \frac{v^{2}}{N_{F}}, \tilde{\alpha}, \tilde{\lambda}, \{\zeta_{i}\}\Bigr ) = \frac{v^{2}}{N_{F}}{\cal E}_{\mathrm{c l}}\bigl ( \tilde{\alpha}, \tilde{\lambda}, \{\zeta_{i}\}\bigr ) +{\cal E}^{F}( \{\zeta_{i}\}) - 1 .
\biggl [ \, G_{i j k l}( x ) \frac{\delta}{\delta h_{k l}( x )}\frac{\delta}{\delta h_{i j}( x )}- \frac{1}{( 1 6 \pi G )^{2}}\, \sqrt{h ( x )}\Bigl ( \! ~ R ( x ) + 2 \Lambda \Bigr ) \biggr ] \, \Psi [ h ] = 0 \ ,
\partial_{T}^{5}f = - \frac{1}{1 8 \pi i}\left\{\frac{j_{T}( T )}{j ( T ) - j ( i )}\right\}^{3}\left\{\frac{j ( i )}{j ( T )}\right\}^{2}\left\{5 + 1 3 \frac{j ( T )}{j ( i )}\right\}\, ,
D(u^{\epsilon},\eta^{\epsilon},\tau^{\epsilon})=\mu_1\|\nabla u^{\epsilon}\|_{H^3}^2+\mu_2\|\mathrm{div}u^{\epsilon}\|_{H^3}^2+\nu[\beta(L-1)+2\bar{\mathfrak{z}}]\|\nabla\eta^{\epsilon}\|_{H^3}^2+\frac{\beta A_0}{4k^2}\|\tau^{\epsilon}\|_{H^3}^2+\frac{\beta\nu}{2k^2}\|\nabla\tau^{\epsilon}\|_{H^3}^2.
\prod_{p|a}(1 - p^{-1})^{-1}=\prod_{p|a}(1 + p^{-1})(1 - p^{-2})^{-1}&\leq \exp\bigg( \sum_{p|a}\frac{2}{p^2}\bigg) \prod_{p|a}(1 + p^{-1}) =\Big(1+ O\Big(\frac{1}{w(x)}\Big)\Big) \prod_{p|a}(1 + p^{-1}).
|I_3(\xi)| \leq p_{\L,k+1}^{(2)}\cdot p_{\L,k}^{(2)}\cdot \ldots \cdot p_{\L,m+2}^{(2)}|I_3| &\leq \frac{k}{k+2}\cdot \frac{k-1}{k+1}\cdot \ldots \cdot \frac{m+1}{m+3}|I_3| &= \frac{(m+2)(m+1)}{(k+2)(k+1)}|I_3|.
\mathrm{Var}_0\left[\bar{U}^{(12)}_h\right] = \mathbb{E}_0\left[\left( U^{(12)}_h \right)^4 \right]- \mu_h^2 &= \begin{cases}\frac{2k^{4}(\zeta^h_1)^2}{n^2}+ O\left(n^{-3}\right) d = 1, \frac{8}{n^{4}}{k \choose 2}^4\{(\zeta_2^h)^2+ 6\eta^h \}+ O\left(n^{-5}\right) d = 2. \end{cases}
d s^{2}={\frac{l_{1}l_{2}}{\sqrt{\alpha}}}\, e^{Q ( \alpha , \beta ) / 2}\, ( - d \alpha^{2}+ d \beta^{2}) + \alpha{\left( e^{V ( \alpha , \beta )}d x^{2}+ e^{- V ( \alpha , \beta )}d y^{2}\right)},
D=\left( \begin{array}{ccccc}0 0 \textsf{a}\textsf{b}\textsf{c}0 0 \textsf{a}\textsf{b}\textsf{c}-\textsf{a}-\textsf{a}0 0 0 -\textsf{b}-\textsf{b}0 0 0 -\textsf{c}-\textsf{c}0 0 0 \end{array}\right)
d(y,\alpha) \geq d(y,z)-6C_\eta &=d(o,y)-d(o,z)-6C_\eta &\geq d(o,x)-d(o,z)-6C_\eta-2C_\zeta &\geq (\xi\cdot\zeta)_o-(\zeta\cdot\eta)_o-K_\eta-K_\zeta-6C_\eta-2C_\zeta &\geq R_\zeta-(\zeta\cdot\eta)_o-K_\eta-K_\zeta-6C_\eta-2C_\zeta \text{since}\xi\in U(\zeta,R_\zeta) &=2C_\zeta=d(x,y)
\frac{h_{T,S}R_{T,S}}{h_TR_T}= \prod_{v\in S-S_{\infty}}\frac{[H^0(\hat{\mathbb{Z}},H^1(I_v,\hat{T}))]}{[H^1(K_v,\hat{T})]}\frac{(f_w\log N(v))^{r_v(\hat{T})}}{h(D_w/I_w,\hat{T}^{I_w})}.
\lim_{z\rightarrow z_j}\frac{g_1p_*\left(f_{u_1}\left(\prod_{1\le j\le m}f_{z_j}\right)\left(\sum_{1\le j\le m}p_j\frac{df_{z_j}}{f_{z_j}}\right)\right)}{\sum_{0\le l\le k_j}a_{j,l}(z-z_j)^{l}dz}=c_0
\theta(\mathsf{P}_{j}^{i}) = \sum_{m, n}c^m_n q^{-(2\rho, \lambda_m - \lambda_n)}\pi(K_{2\rho}^{-1})_{i}^{i}(\mathsf{M}_{m}^{n})_{j}^{i}\pi(K_{2\rho})_{j}^{j}.
|<m,m^{\prime}>|^{2}=|h_{n+1}(\tilde{Z},\tilde{Z}^{\prime})|^{2}=|(h_{2}(\mathbf{z},\mathbf{z}^{\prime}))^{n}|^{2}=(|h_{2}(\mathbf{z},\mathbf{z}^{\prime})|^{2})^{n},
\left(e^{s\bar{D}^2}\right)_{Ax,By}=\int d^4z_1\dots\int d^4z_n\left(e^{\frac{s\bar{D}^2}{n+1}}\right)_{Ax,C_1z_1}\left(e^{\frac{s\bar{D}^2}{n+1}}\right)_{C_1z_1,C_2z_2}\dots\left(e^{\frac{s\bar{D}^2}{n+1}}\right)_{C_nz_n,By}
S_{e f f}= N S_{0}( \sigma^{0}) + S_{1}( \sigma^{0}) +{\frac{i}{2}}B_{2}+{\frac{1}{2}}A_{2}+ \sum_{n = 1}^{\infty}{\frac{1}{\sqrt{N^{n}}}}\big [ - i{\frac{( - 1 )^{n + 1}}{( n + 2 )}}B_{n + 2}+ i L^{d}\delta^{d}( 0 ){\frac{( - 1 )^{n + 1}}{n}}B_{n}+{\frac{1}{( n + 2 ) !}}A_{n + 2}\big ]
\Phi(t)= \sum_{r\ge 1}\int_0^t dt_1\cdots \int_0^{t_{r-1}}dt_r\sum_{\sigma\in \S_r}{(-1)^{d(\sigma)}\over r}{r-1\choose d(\sigma)}^{-1}\psi(t_{\sigma(r)})\cdots \psi(t_{\sigma(1)}) \ .
|B_{\delta^{-1}}|^{-(k-1)}\prod_{j=1}^k \int_{B_{\delta^{-1}}}\Big(\sum_{K_j \in P_{\delta}(I_j)}|g_{K_j}|^2\Big) = \int_{B_{\delta^{-1}}}\prod_{j=1}^k \Big(\sum_{K_j \in P_{\delta}(I_j)}|g_{K_j}|^2\Big).
\left(\begin{array}{cc}0 - t \sum_{j}\mbox{\Large $e^{i{\bf k \cdot u}_{j}}$}- t \sum_{j}\mbox{\Large $e^{i{\bf k \cdot v}_{j}}$}0\end{array}\right) \left(\begin{array}{c}c_{\bullet}c_{\circ}\end{array}\right) = E({\bf k})\left(\begin{array}{c}c_{\bullet}c_{\circ}\end{array}\right)
Q/Q(2) \downarrow L' = (00,\lambda_3) + (10,\lambda_4), Q(2)/Q(3) \downarrow L' = (10,\lambda_1) + (01,0), Q(3)/Q(4) \downarrow L' = (01,\lambda_3), Q(4)/Q(5) \downarrow L' = (00,\lambda_1) + (01,0), Q(5)/Q(6) \downarrow L' = (00,\lambda_4), Q(6)/Q(7) \downarrow L' = (10,0).
\left[\begin{array}{cc}\frac{\partial\mathcal{H}}{\partial \tilde{x}}\frac{\partial\mathcal{H}}{\partial \tilde{\lambda}}\end{array}\right]\mu=0, \mu^t\mu=1, \frac{d\nu}{ds}=\mu, \nu(0)=\left[\begin{array}{c}\tilde{x}^{(0)}e \end{array}\right]
F_{x_{1}x_{2}x_{3}}^{\Delta_{1}\Delta_{2}\Delta_{3}}= \frac{1}{| x_{1 2}|^{\Delta_{1}+ \Delta_{2}- \Delta_{3}}| x_{2 3}|^{\Delta_{2}+ \Delta_{3}- \Delta_{1}}| x_{3 1}|^{\Delta_{3}+ \Delta_{1}- \Delta_{2}}}.
i_R \pi_2^*\Omega_2 = i_R d(x\cdot y)\wedge\sum_{i=1}^{n+1}\big((-1)^{i+1}y_idy_1\wedge \cdots \wedge \breve{dy_i}\wedge\cdots \wedge dy_{n+1}\big) \hphantom{i_R \pi_2^*\Omega_2}{}=\sum_{i=1}^{n+1}((-1)^{i+1}y_idy_1\wedge \cdots \wedge \breve{dy_i}\wedge\cdots \wedge dy_{n+1}),
A_{q,2}= \biggl( \frac{(p+2) (p-2)}{16 \pi}\biggr)^{\frac{1}{4}- \frac{1}{2p}}\biggl( \frac{4}{p+2}\biggr)^\frac{1}{p}\biggl( \frac{4}{p-2}\biggr)^{\frac{1}{4}- \frac{1}{2p}}\leq C \, p^\frac{1}{4}.
R ( \theta ) \rightarrow R^{'}= \left( \begin{array}{c c}{{f ( \theta )}}&{{0}}\\{{0}}&{{g ( \theta )}}\\ \end{array}\right) R ( \theta ) \left( \begin{array}{c c}{{f^{\dagger}( - \theta )}}&{{0}}\\{{0}}&{{g^{\dagger}( - \theta )}}\\ \end{array}\right)
S = S^{\prime}+ \int d \tau \left( \sum_{a}q_{s}^{*}\delta_{a}q^{s}\eta_{a}- \sum_{a b}\alpha_{b}^{*}\left( \zeta^{- 1}\right)_{a b}\eta_{a}+ \sum_{a b c}\frac{1}{2}C_{a b}{}^{c}\eta_{c}^{*}\eta_{a}\eta_{b}+ \sum_{a}\bar{\eta}_{a}^{*}b_{a}\right) .
\sum_{i,j}\beta_{ij}k_{ij}^m+K_m=0,\sum_{i,j}\beta_{ij}l_{ij}^{n}+L_n=0,\qquad\forall m\in\left\{1,2,\ldots,\frac{N(N-1)}{2}\right\},\, n\in\{1\ldots,N\}.
\vec{V}_i =I_{n_d}\otimes \cdots \otimes I_{n_{i+1}}\otimes V_i \otimes I_{n_{i-1}}\otimes \cdots \otimes I_{n_1}, \overline{\vec{V}}_i ={V}_d\otimes \cdots \otimes V_{i+1}\otimes I_{n_i}\otimes V_{i-1}\otimes \cdots \otimes V_1.
\begin{array}{l c r}{{\partial_{\sigma}^{2}X^{9}+ \sum_{\nu = 1}^{8}{\frac{1}{2}}( \{X^{\nu + ( - )^{\nu + 1}}, \{X^{\nu + ( - )^{\nu + 1}}, X^{9}\}\}+ \{X^{\nu}, \{X^{\nu}, X^{9}\}\}) = 0 ,}}\\ \end{array}
\frac{\partial}{\partial\theta}\left( \frac{{\dot\kappa}^{-S}(\theta)}{{\dot\kappa}^{A}(\theta)}\right)= \frac{{\ddot\kappa}^{-S}(\theta) \cdot{\dot\kappa}^{A}(\theta) -{\ddot\kappa}^{A}(\theta) \cdot{\dot\kappa}^{-S}(\theta)}{\left[{\dot\kappa}^{A}(\theta) \right]^{2}}\ge 0,
\left\{{\frac{\partial^{2}}{\partial r^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partial r}}-{\frac{l ( l + 4 )}{r^{2}}}+ \tilde{\omega}^{2}\left( 1 +{\frac{\tilde{R}_{B}^{4}}{r^{4}}}\right) \right\}\varphi_{B}^{l}= 0 ,
\partial_t{w}^{(i)}_{\alpha-1}(x_i,t) \, + \, \big( v_i^{(i)}(x_i)\, w^{(i)}_{\alpha-1}(x_i,t) \big)^\prime = - \widehat{\varphi}^{(i)}(x_i, t), (x_i, t) \in I_\varepsilon^{(i)}\times (0, T),
\frac{1}{i \partial_{-}- e \bar{B}_{-}}- \frac{1}{i\partial_{-}}= \left(\frac{1}{i \partial_{-}- e \bar{B}_{-}}e\bar{B}_{-}\right) * \frac{1}{i \partial}= \frac{1}{i\partial}\ast \left( e \bar{B}_{-}\frac{1}{i \partial_{-}- e\bar{B}_{-}}\right)\,
\lim_{M\to\infty}D\left(T\right) =\lim_{M\to\infty}\frac{\sigma_{\theta}^4\mathbf{p}^T\mathbf{d}\mathbf{d}^T\mathbf{p}}{\mathbf{p}^T\mathbf{B}\mathbf{p}+\frac{2\sigma_n^2}{\sqrt{M}}\mathbf{b}^T\mathbf{p}+\sigma_n^4}&=\frac{\sigma_{\theta}^4\mathbf{p}^T\mathbf{d}\mathbf{d}^T\mathbf{p}}{\mathbf{p}^T\mathbf{B}\mathbf{p}+\sigma_n^4}\;,
{\hat{F}[ L , ~ f ]}~ = ~ \alpha^{- \sigma [ L , ~ f ]}~ \sum_{\{j_{i}\}}~ ~ \mu_{j_{1}}~ \mu_{j_{2}}~ . . . . ~ \mu_{j_{r}}~ V [ L ; ~ n_{1}, ~ n_{2}, ~ . . . ~ n_{r}; ~ j_{1}, ~ j_{2}, ~ . . . . ~ j_{r}]
Z_{0}(s,\chi,f)=\sum\limits_{k\in\mathbb{N}^n}\int\limits_{\substack{x\in (R_{K}\setminus\{0\})^n ord (x)=k}}\omega(f(x))\ |dx| =\int\limits_{(R_{K}^\times)^{n}}\omega(f(x))\ \vert dx\vert+\sum\limits_{\Delta\in\mathcal{C}}\ \sum\limits_{k\in\Delta\cap\mathbb{N}^n}\ \int\limits_{\substack{x\in (R_{K})^n ord (x)=k}}\omega(f(x))\ \vert dx\vert.
e^{-2\hat \phi}= 1+{2\tilde m\over r}\ , \qquad K = 1+\sum_1^s{2\hat m\over r}\ , \qquad A_n ={2 q_n\over r}\ ,\qquad n=4, \dots 8, \qquad r^2 \equiv \vec x^2.
\prod_{i \neq k}\mu_i^{2n_i}\cdot \iota(\mu_k)^{2n_k}= \sigma\left( \prod_{i=1}^g \mu_i^{2n_i}\right) = \prod_{i=1}^g \mu_i^{2n_i}\Rightarrow (\mu_k/\iota(\mu_k))^{2n_k}= 1.
\left(\prod^{n-1}_{j=0}z_j\right) A_n^{(E-\l v)}(x) =\prod^{0}_{j=n-1}\left[z_j\cdot\begin{pmatrix}E-\l(z_{j-1}-z_{j-1}^{-1}) -1 1 0\end{pmatrix}\right] &=\begin{pmatrix}E-\l(z^2_{n-1}-1) -1 1 0\end{pmatrix}\cdots \begin{pmatrix}E-\l(z^2_0-1) -1 1 0\end{pmatrix}&=\begin{pmatrix}\l^n+M^{(n)}_1 M^{(n)}_2 M^{(n)}_3 M^{(n)}_4\end{pmatrix},
(a u_x)_x + b u_x + c u - u_{t}= f,~~\Omega u(x,0) = \phi (x),~0 \leq x \leq s(0)=s_0 a(0,t) u_x (0,t) = g(t),~0 \leq t \leq T a(s(t),t) u_x (s(t),t) + \gamma (s(t),t) s'(t) = \chi (s(t),t),~0 \leq t \leq T u(s(t),t) = \mu (t),~0 \leq t \leq T, \intertext{where}\Omega = \{(x,t): 0 < x < s(t),~0 < t \leq T\}
\int_{Z_{\overline{K}_v}^{\mathrm{an}}}\log\|s_d\|_v \prod_{i=0}^{d-1}c_1(\overline{L}_{i,v})=\frac{1}{e^d}c_1(\widetilde{L}_0)\dots c_1(\widetilde{L}_{n-1})[V_v]\log N(v).
q^j(\xi^j):=\frac{[n^j]''(r)}{4[n^j(r)]^2}-\frac{5}{16}\frac{\{[n^j]'(r)\}^2}{[n^j(r)]^3}+\frac{l(l+1)}{r^2n^j(r)}-\frac{l(l+1)}{[\xi^j]^2}.
\frac{1}{(j-1)x^{j-1}}=\int_x^{\infty}\frac{dt}{t^j}<\sum_{m=0}^{\infty}\frac{1}{(x+m)^j}&<\sum_{m=0}^{\infty}\int_{x+m-\frac 12}^{x+m-\frac 12}\frac{dt}{t^j}=\int_{x-\frac 12}^{\infty}\frac{dt}{t^j}=\frac{1}{(j-1)(x-\frac 12)^{j-1}}.
(2n-1-q) a \|u\|^{2n}+(n-1-q) b \|u\|^n +q \int_{\Omega}(|x|^{-\mu}*F(u)) f(u).u &- \left(\int_{\Omega}(|x|^{-\mu}*f(u)u)f(u).u + \int_{\Omega}(|x|^{-\mu}*F(u)) f'(u) u^2\right) <0.
\ln \left( x^{[ n + 1 ]}( \theta ) \right) = - r m \cosh \theta - \frac{2 \sqrt{3}}{\pi}\int_{- \infty}^{\infty}\mathrm{d}\theta^{\prime}\frac{\cosh ( \theta - \theta^{\prime})}{1 + 2 \cosh 2 ( \theta - \theta^{\prime})}\ln \left( f_{\scriptscriptstyle G}\bigl ( x^{[ n ]}( \theta^{\prime}) \bigr ) \right) .
\left\| \nabla' w \right\|_{L^2(B_{\mu})}\lesssim \ \mu^s \left( \left\| y_{n+1}^{\frac{1-2s}{2}}\nabla \nabla' w \right\|_{L^2(B_{2\mu}^+)}+\left\| y_{n+1}^{\frac{1-2s}{2}}w \right\|_{L^2(B_{2\mu}^+)}\right) & + \mu^{-1}\left\| w \right\|_{L^2(B_{2\mu})}.
y^j y^i \frac{\partial \Gamma^k_{\,\, ij}}{\partial y^l}=y^j\frac{\partial N^k_{\,\,j}}{\partial y^l}-y^j\Gamma^k_{\,\, lj}=y^j\frac{\partial N^k_{\,\,j}}{\partial y^l}-N^k_{\,\,l}.