harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Phi(z,n,a) = \frac{(-1)^{n-1}}{(n-1)!}\left\{P\int_0^{\infty{e}^{{i}\varphi}}\frac{t^{n-1}\,{e}^{at}}{z\,{e}^t-1}\,{d}t+ \pi\,\frac{\partial^{n-1}}{\partial a^{n-1}}\left(z^{-a}\,\cot(\pi a)\right)\right\},
\widetilde \theta_\Gamma^{\frac 1 2}= \log(r_B)\sum_{e'\in E_3}(-1)^{e'-1}\prod_{e\in E_1}d\arg(z_{s(e)}-z_{t(e)}) \prod_{e\in E_2}d\arg(z_{s(e)}-\zeta_B) \prod_{e\in E_3\smallsetminus\{e'\}}d\arg(z_{s(e)}-z_{t(e)}) + ().
\mathrm{Var}(T_\tau) :={n \choose 4}^{-1}\sum_{c = 1}^4{4 \choose c}{n - 4 \choose 4 - c}\zeta_c^{\tau}\leq \frac{16\zeta^{h^T_\tau}_1}{n}+ \frac{\widetilde{C}}{n^2}\left(\zeta^{h^T_\tau}_2 + \zeta^{h^T_\tau}_3+ \zeta^{h^T_\tau}_4\right)
\lim_{n\to\infty}\int_\Omega\Phi(f_n(t))\, \mathrm{d}x = \int_\Omega\Phi(f(t))\, \mathrm{d}x \;\;\;\; \lim_{n\to\infty}\int_\Omega\Phi(f_n(t_0))\, \mathrm{d}x = \int_\Omega \Phi(f(t_0))\, \mathrm{d}x\,.
[\widetilde{\nabla}_{j},\widetilde{\nabla}_{k}]=\frac{2\pi i}{(\det\mathcal{T})^{2}}(\mathcal{C}_{3j}\mathcal{C}_{1k}+\mathcal{C}_{4j}\mathcal{C}_{2k}-\mathcal{C}_{3k}\mathcal{C}_{1j}-\mathcal{C}_{4k}\mathcal{C}_{2j})\widetilde{\mathcal{I}},
i{\cal{M}}_{1}= 1 6 i \alpha g^{4}\sin ({\frac{q \tilde{l}}{2}}) \epsilon_{\nu}q_{\mu}\left[ g^{\mu \nu}( q - p )^{\rho}+ g^{\nu \rho}( p - l )^{\mu}+ g^{\rho \mu}( l - q )^{\nu}\right]{\frac{\tilde{l}_{\rho}( \tilde{l}j )}{l^{4}\tilde{l}^{4}}}.
\ldots , \; \xi^{6}q^{- 2}w_{1}, \; \xi^{4}q^{- 2}w_{1}, \; \xi^{2}q^{- 2}w_{1}; \; q^{2}w_{1}, \; \xi^{- 2}q^{2}w_{1}, \; \xi^{- 4}q^{2}w_{1}, \; \ldots .
W(k_1,n_2)=\frac{1}{4\pi^2 R^2_{nc}}\int_0^{2\pi R_{nc}}d^2x\,\Omega(x_1;x_2\to x_2+2\pi R_{nc}n_2-2\pi R_{nc}\frac{c}{N}k_1)\star \exp(i\frac{k_1 x}{R_{nc}}).
\mathcal{D}(e^{i}\wedge e^{j})=2e^{i}\wedge e^{j}+O(1-\coth^{2}(r)) &\mathcal{D}(e^{i}\wedge e^{0})=(m-2)(e^{i}\wedge e^{0})+O(1-\coth^{2}(r)),
H_{3}\! = \! \! ( q - q^{- 1})^{- 1}\! x_{+}( k^{2}A^{2}+ k^{- 2}D^{2}) - ( q - q^{- 1}) x_{-}C^{2}+ 2 x_{0}( s k^{- 1}\! D C \! - s^{- 1}\! k A C ) ,
\langle \overline{{{\phi}}}_{1}, \overline{{{\phi}}}_{2}\rangle = \overline{{{\phi}}}_{2}( \phi_{1}^{*}) = \int \d v \ \overline{{{\phi}}}_{2}( x ) \phi_{1}^{*}( x ) = \int \d v \ \phi_{2}( x ) \overline{{{\phi_{1}^{*}}}}( x ) .
\sup_{x\in X}\int_Yw(x,y)^\frac1p\big|K(x,y)\big|d\nu(y)=C_0<\infty, \sup_{y\in Y}\int_Xw(x,y)^{-\frac1{p'}}\big|K(x,y)\big|d\mu(x)=C_1<\infty,
\inf_{x,p,\theta}\f(x), \mathrm{s.t.}\ \sum_{i \in [m]}g_i(x,\bar{u}^i)+p_i + \theta \Gamma \le 0, &p_i +\theta \ge \sup_{u^i \in \mathcal{U}_i}g_i(x,u^i)-g_i(x,\bar{u}^i) \ \forall i \in [m], &p,\theta \ge 0, &x \in \mathcal{X}.
I(a,b) = \int_{\o}\psi \left( x^2 \left( y+\frac{1}{2}\right) + 2 x \left( y+\frac{1}{2}\right) \left( z+\frac{1}{2}\right) \right) dx = \int_{\o}\psi \left( \frac{1}{2}x^2 + \frac{1}{2}x \right) dx = 1
\begin{aligned}u(t, E_{t}, W_{E_t}) = u(0, 0, 0) + \int_{0}^{t}u_{t_{1}}(s, E_{s}, W_{E_{s}})\mathrm{d}s + \int_{0}^{t}u_{t_{2}}(s, E_{s}, W_{E_{s}})\mathrm{d}E_{s}+ \frac{1}{2}\int_0^t tr[u_{xx}(s, E_{s}, W_{E_s})Q]\mathrm{d}E_s + \int_0^t \langle u_x(s, E_{s}, W_{E_s}), dW_{E_s}\rangle_K, \end{aligned}
e^{(x+y)/2}\cdot D^t_{\partial}\left(e^{(-x-y)/2}x^n y^m \right) = \sum\limits_{i=0}^{\min(t,m)}\binom{t}{i}\cdot\frac{m!}{(m-i)!}\cdot \biggl(e^x D^{t-i}\left( e^{-x}x^n\right)\biggr)\cdot y^{m - i},
g_{f\circ A}(\infty)\biggl(=\sum_{j=1}^\infty\Bigl(\frac{\log\|(F_A)^j(0,1)\|}{p^j}-\frac{\log\|(F_A)^{j-1}(0,1)\|}{p^{j-1}}\Bigr)\biggr)=\frac{\log p}{2(p-1)}.
\begin{array}{r c l}{{\bf{Q}}_{-{\frac{1}{k}}}^{+}\vert \Omega_{- s}^{-}\rangle}&{=}&{0}\\{\{{\frac{({\bf{Q}}_{-{\frac{1}{k}}}^{-})^{r}}{[ r ] !}}\vert \Omega_{- s}^{-}\rangle}&{;}&{o \le r \le k - 1 \}}\\ \end{array}
R = \; (n_5^{-1}n_3^{-1})^3 n_2^{-1}n_3^2 n_4 n_2 n_1 n_3 n_5^{-1}n_1^{-1}n_4^{-1}n_5 n_2 n_4^{-1}n_5(n_1^{-1}n_4^{-1})^2 & n_5 n_1^{-1}n_2^{-1}n_5 n_3 n_5 n_1 n_4 n_5^{-1}n_2^{-1}n_4 n_3^{-1}n_1 n_4 n_2 n_1 .
Z\left({\cal A},G,p,t^{(A)}_{n,l}\right) = \sum_{\{r^{(A)}_{n,l}\}}Z\left({\cal A},G,p,r^{(A)}_{n,l}\right) \prod_{A=1}^p \prod_{l=1}^{\infty}\prod_{n=-\infty}^{+\infty}\left[t_{n,l}^{(A)}\right]^{r^{(A)}_{n,l}}~.
A ( x_{1}, x_{2}, x_{3}, x_{4}) = \langle{\cal C}^{1 1}( x_{1}){\cal C}_{1 1}^{\dagger}( x_{2}) : t r ( \phi_{2}^{\dagger}( x_{3}) \phi^{2}( x_{3}) ) : : t r ( \phi_{3}^{\dagger}( x_{4}) \phi^{3}( x_{4}) ) : \rangle
\sum_{j=1}^{d_k}k\langle (k+\Delta')^{-1}i_{\mu}u_j,i_{\mu}u_j\rangle= \sum_{j=1}^{d_k}k\langle \sum_{n=0}^{N}a_n\square_{n}i_{\mu}u_j,i_{\mu}u_j\rangle &+\sum_{j=1}^{d_k}k\langle (k+\Delta')^{-1}a_N\square_{N+1}i_{\mu}u_j,i_{\mu}u_j\rangle.
[(0,1-a,0,0)\otimes(0,1,0,0)+(0,1,0,0)\otimes(0,1-a,0,0)] = [(1-b,0,0,0)\otimes(0,1,0,0)+(0,1,0,0)\otimes(1-b,0,0,0)] = [(1,0,0,0)\otimes(0,1-b^{-1},0,0)+(0,1-b^{-1},0,0)\otimes(1,0,0,0)].
dY=\sum_i \omega_i Y_i, &dN=\sum_{ij}A_{ij}\omega_i Y_j+\sum_{i,r}C^r_i\omega_i \xi_{r}, &d Y_i=-\sum_j A_{ij}\omega_j Y-\omega_i N+\sum_j\omega_{ij}Y_j+\sum_{j,r}B^{r}_{ij}\omega_j \xi_{r}, &d \xi_{r}=-\sum_i C^{r}_i\omega_i Y-\sum_{i,j}\omega_iB^{r}_{ij}Y_j +\sum_{s}\theta_{rs}\xi_{s},
\left\langle \vartheta_{n}\right| V_{m}\left| \vartheta_{r}\right\rangle = A_{m}( r ) \delta_{n , m + r}\, , \quad \left\langle n \right| V_{m}\left| r \right\rangle = B_{m}( r - n ) e^{\frac{2 \pi i}{k}m n}\, ,
\overline{\widehat{w}}_k := \langle k \rangle^{-s}\langle \tau-k^2\rangle^{b'}\overline{\widehat{v}}_k, \qquad\widehat{z^\ell}_{k_\ell}:= \langle k_\ell\rangle^{s}\langle \tau_\ell +(-1)^{\ell+1}k_\ell^2\rangle^{b}\widehat{u^\ell}_{k_\ell},
{\gamma_{D,1}^M}= \frac{{{P_s}{{\left|{{{\bf{w}}^T}{{\bf{h}}_{SD}}}\right|}^{\rm{2}}}}}{{\mathbb{E}\left\{{{{\left|{{{\bf{w}}^T}{{\bf{n}}_{SD}}}\right|}^2}}\right\}}}= \frac{{{P_s}{{\left|{{{\bf{w}}^T}{{\bf{h}}_{SD}}}\right|}^{\rm{2}}}}}{{{\sigma^2}{{\left\|{\bf{w}}\right\|}^2}}}
\textbf{S}(t,z)(\alpha) = \alpha -\sum_{\substack{d \in H_2(X;\mathbb{Z})-\{0\}l \geq 0}}\, \sum_{k=0}^N \frac{1}{l!}\left\langle T_k, \tau ', \dots , \tau ', \frac{\alpha}{z+\psi}\right\rangle^\textnormal{coh}_{0,l+2,d}T^k e^{\tau_2(d)}
\hat{Q}=:\left[\begin{array}{cc}\hat{Q}_{1}\hat{Q}_{2}\hat{Q}_{2}^{\top}\hat{Q}_{3}\end{array}\right],\quad\tilde{A}_{12}:=\left[\begin{array}{cc}\hat{A}_{1}0 0 \hat{A}_{2}\end{array}\right],\quad\hat{A}_{p}:=\left[\begin{array}{c}\hat{A}_{3,p}\hat{A}_{4,p}\end{array}\right].
\nabla^{\mathrm{restr}}_{\partial z}(v_{1})=( \Omega^{\mathrm{restr}}_{\partial z})_{11}v_{1}+ ( \Omega^{\mathrm{restr}}_{\partial z})_{21}v_{2}&\nabla^{\mathrm{restr}}_{\partial z}(v_{2})=( \Omega^{\mathrm{restr}}_{\partial z})_{12}v_{1}+ ( \Omega^{\mathrm{restr}}_{\partial z})_{22}v_{2}.
\aligned \frac d{dt}\mathcal{YM}_e^0(\nabla^t)_{\big{|}_{s =0}}= \frac d{dt}\mathcal{YM}_e^0(\nabla^t)_{\big{|}_{s =0}}=\int_M\langle\delta^\nabla\big(\exp (\frac 12||R^\nabla ||^2)R^\nabla \big), A \rangle \, dv\, .\endaligned
\begin{aligned}b_mx_n\leq u\leq a_mx_n~~~~\mbox{in}~~~~Q^+_{2^{-m}}, 0\leq a_m-b_m\leq (1-\mu)(a_{m-1}-b_{m-1}), |a_0|\leq 2C~~\mbox{and}~~|b_0|\leq 2C,\end{aligned}
Pr(\Theta,\overline{\theta})=f^{\prime}(\Theta)\prod_{i=1}^{n}g_{i}^{\prime}(\Phi_{i}\cap\Theta,\overline{\theta}[\Phi_{i}])=\prod_{j=1}^{m}h_{j}(\Psi_{j}\cap\Theta,\overline{\theta}[\Psi_{j}])
\mathcal{L}\left(x,\psi\right) =f\left(x\right)+\psi\left(Lx\right)-g_{\Psi}^{*}\left(\psi\right) =\inf_{y\in Y}f\left(x\right)+g\left(y\right)+\psi\left(Lx\right)-\psi\left(y\right) \leq f\left(x\right)+g\left(Lx+y\right)+\psi\left(Lx\right)-\psi\left(Lx+y\right),
M_Q(n,j,0;a) =\sum_{k=j}^{n-j}\binom{n-j}{k-j}\binom{n-k}{n-k-j}\binom{a+k}{a}\binom{a+n-k}{a}G(n,k,a) &=\sum_{k=j}^{n-j}\binom{n-j}{k-j}\bigl{(}\binom{a+n-k}{a}\binom{n-k}{n-k-j}\bigr{)}\binom{a+k}{a}G(n,k,a)
\langle \varphi_i^{(0)}(X_0) \varphi_j^{(0)}(X_0) \rangle_{\zeta c}\simeq - \frac{\lambda}{1 9 2 \pi}(N + 2) \frac{T}{\omega^2}\frac{\delta_{i j}}{V}e^{2 \omega X_0}.
\frac{\partial^2 P}{\partial \tilde{r}^2}\left( \tilde{r}^*;0 \right) =\frac{\tilde{\gamma}}{\tilde{r}^* \left( \tilde{\gamma}- \tilde{r}^* \right)}-\frac{\tilde{\gamma}\,\frac{\partial P_R}{\partial \tilde{r}}}{P_R \left( \tilde{\gamma}- P_R \right)}+2 \frac{d h}{d \tilde{r}}.
\int_{\Sigma}|A|^2u^2e^{\frac{|x|^2}{4}}d\sigma \leq\int_{\Sigma}(\dfrac12-\mu_1)u^2e^{\frac{|x|^2}{4}}d\sigma+\int_{\Sigma}|\nabla u|^2e^{\frac{|x|^2}{4}}d\sigma &=\int_{\Sigma}(\dfrac12-\mu_1+\lambda_1)u^2e^{\frac{|x|^2}{4}}d\sigma.
\Pi_1^{\delta^2}=\delta^2\frac{\lambda T\eta^2}{8\pi\Omega}\frac{1}{\sqrt{1-r^2}}-\delta^2\frac{\lambda\eta^2}{16\pi^2}\left[-\frac{1}{\epsilon}+\ln\left(\frac{4\pi T^2}{M^2}\right)-\gamma\right].
\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in I_{\ell}}[g(W_{i},\hat{\gamma}_{\ell},\theta_{0})+\phi(W_{i},\hat{\gamma}_{\ell},\hat{\alpha}_{\ell},\tilde{\theta}_{\ell})]=\hat{\psi}(\theta_{0})=\frac{1}{n}\sum_{i=1}^{n}\psi(W_{i},\gamma_{0},\alpha_{0},\theta_{0})+o_{p}(n^{-1/2}),
\{L_x,\L_y\}=\alpha\Big(\frac{y}{xJ_x^2J_y}\{J_x,\ \vert Z_y\vert^2\}-\frac{x}{yJ_y^2J_x}\{J_y,\ \vert Z_x\vert^2\}\Big)+\alpha^2\{E_x,\E_y\}\ .
s_0 = \sum_{i=0}^n a_i \begin{bmatrix}x^iy^{n-i}z^iw^{n-i}\end{bmatrix}~ ~ s_1 = \sum_{i=0}^n b_i \begin{bmatrix}x^ny^{n-i}z^iw^{n-i}\end{bmatrix}.
W_{p,\xi}(t(a))= \chi_{\psi}(a^{-1})\chi_{\delta}(a^{-1})|a|_p^{1/2}W_{p,a^2\xi}(1) = W_{p,a^2\xi}(1) = \Psi_p(a^2\xi;\alpha_p) = \Psi_p(\xi;\alpha_p).
0=(\frac{1}{2}v_{1}^{2}+\Psi_{1}+\epsilon_{1}+\frac{p_{1}}{\rho_{1}})\rho_{1}A_{1}v_{1}\Delta t-(\frac{1}{2}v_{2}^{2}+\Psi_{2}+\epsilon_{2}+\frac{p_{2}}{\rho_{2}})\rho_{2}A_{2}v_{2}\Delta t
0 = K(Z,Z)(I) = \Gamma_1 (Z,Z)(Q(Z,Z)(I) \succeq \epsilon_0 \Gamma_1(Z,Z)(I) 0 = K(Z,Z)(I) = - \Gamma_3 (Z,Z)(Q(Z,Z)(I) \preceq -\epsilon_0 \Gamma_3(Z,Z)(I)
(\sigma^{ab}n_{ab})^{(1)}= \rho + \kappa^2 (\sigma^{ab}n_{ab})^{(2)}= \frac{2}{3}D\rho (\sigma^{ab}n_{ab})^{(3)}= \frac{3}{8}D^2\rho+\frac{1}{30}|\alpha|^2- \frac{11}{45}|\beta|^2 .
p^{0}( b_{r}^{0 , 1}+ \cdot \cdot \cdot + b_{r}^{0 , 6}) \mid \Phi \rangle + \mathrm{( t e r m s ~ c o n t a i n i n g ~ b o t h ~ \a l p h a ~ a n d ~ b ~ o s c i l l a t o r s )}\mid \Phi \rangle = 0
\int_{S^2}(h_0^{(3)}- h^{(3)}) dS^2 = \frac{1}{2}\int_{S^2}|\AA^{'(3)}|^2_{\tilde{\sigma}}dS^2-\frac{1}{4}\int_{S^2}(\tilde{\Delta}Y_0^{(3)})^2 dS^2-\int_{S^2}\tilde X^i \tilde{\Delta}(Y_i^{(5)}-Y_i^{'(5)}) dS^2+\int_{S^2}(k^{(3)}- \frac{1}{4}- h^{(3)}) dS^2.
\exp\frac{S(C|M)_{(\Phi\otimes\mathbb{I}_M)(\hat{\rho}_{AM})}}{n}\ge \eta\exp\frac{S(A|M)_{\hat{\rho}_{AM}}}{n}+ \left|1-\eta\right| \exp S_0 &= \eta\exp\frac{-S(\hat{\rho}_A)}{n}+ \left|1-\eta\right| \exp S_0 &\ge \eta\exp(-g(E))+ \left|1-\eta\right| \exp S_0\;,
A_3 =\{1,3\}, q(3,1) =d_1^{-1}d_{8-1}=d_1^{-1}d_7q(3,3)=d_3^{-1}d_{8-3}=d_3^{-1}d_5, F_3 =\langle q(3,1)^8\rangle\times\langle q(3,3)^8\rangle.
\left(\frac{1}{3 \alpha}(e x^2+\beta x+\alpha r)\right)^3+y_1^3+y_2^3+\frac{1}{3 \alpha}(e x^2+\beta x+\alpha r) * y_1 * y_2= r y_0^2+r y_1 * y_2+t y_0 +\delta.
F=\prod^m_{i=1}T^{\varepsilon_i}_{\alpha_i},F^{*}=\prod^{m-2}_{i=1}T^{\eta_i}_{\beta_i},W_j=\prod^j_{l=1}T^{\varepsilon_l}_{\alpha_l},W^{*}_j=\prod^{j}_{l=1}T^{\eta_l}_{\beta_l},
i\partial_t \Psi[\xi ,t)=\int d^2 x \biggl[i\biggl(A_0+{1\over\theta}J_i\epsilon_{ij}{\partial_j\over\nabla^2}\biggr){\delta\over\delta\xi}+ A_0\rho-\xi{\dot \rho}\biggr]\Psi[\xi,t)\,,
\left( 1 -{\frac{r_{0}}{r}}\right){\frac{r}{r_{0}}}{\frac{d}{d r}}\left( r^{2}{\frac{d f_{\ell}}{d r}}\right) + \left( 2 \ell + ( 2 \ell + 1 + 2 i k r_{0}\cosh \gamma ){\frac{r_{0}}{r}}\right){\frac{r^{2}}{r_{0}}}\;{\frac{d f_{\ell}}{d r}}+ \ell ( - \ell + 2 i k r_{0}\cosh \gamma ) \; f_{\ell}= 0
q(x,A):= \sum_{1\leq i_1< i_2\leq|x|}1_A\bigl(C_{i_1,i_2}(x) \bigr)+\frac{\rho}{2}\sum_{i=1}^{|x|}\int_{b_i(x)}^{e_i(x)}p(s)I_A\bigl(R_{is}(x)\bigr)\,ds,
x+y\ge2\lfloor\frac{b+\lceil\frac{b}{5}\rceil}{2}+1\rfloor=2\lfloor\frac{10n+7+\lceil\frac{10n+7}{5}\rceil}{2}+1\rfloor=2\lfloor\frac{10n+7+\lceil2n+\frac{7}{5}\rceil}{2}+1\rfloor
|\frac{-4(1-\alpha) \lambda(t)^{-\alpha}\lambda'(t)}{\log(\lambda_{0,0}(t))}\int_{t}^{\infty}\frac{e''(s) ds}{(\lambda(t)^{1-\alpha}+s-t)^{2}(1+s-t)^{3}}|\leq \frac{C}{t^{3}\log^{b+2}(t) (\log(\log(t)))^{3/2}}
\bar{S}_{f r e e}^{( 0 )}[ \phi ] = -{\frac{1}{2}}\mathrm{T r}\left( [ \hat{x}_{\mu}^{\prime}, \phi ( \hat{x}) ] [ \hat{x}_{\mu}^{\prime}, \phi ( \hat{x}) ] \right) ={\frac{1}{2}}\mathrm{T r}\, ( \hat{\partial}_{\mu}\phi ( \hat{x}) )^{2}.
\frac{dP}{d\bar P}\Bigg|_{\mathcal{F}_{t}}=\exp\Bigg\{\int_{0}^{t}[\lambda(\sigma_s)-\bar\lambda(\sigma_s)]ds-\sum_{s\leq t}\ln\frac{\lambda(\sigma_{s^-})p(\sigma_{s^-},\sigma_s)}{\bar \lambda(\sigma_{s^-})\bar p(\sigma_{s^-},\sigma_s)}\Bigg\},
\Theta\, \cot \left(\Theta\left[\ln \left( \frac{z}{2}\right) + \gamma\right]\right)\stackrel{(a \rightarrow 0)}{=}\tilde{k}a\,\frac{J'_{l}( \tilde{k}a )}{J_{l}( \tilde{k}a )}\,\left[ 1 + O \left( [\kappa a]^{2}\right) \right] \; .
W_{H}=\left[ \begin{array}{cccccccc}1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1\end{array}\right]
\frac{\pi}{2}=\sum_{n=1}^{\infty}\frac{(-1)^{\epsilon(n)}}{n}=1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\cdot\cdot\cdot
a_{\chi, \mathfrak{p}}= \frac{\varOmega^{j-k}}{\varOmega_p^{j-k}}\left( \frac{\sqrt{d_E}}{2\pi}\right)^j G(\chi) \left(1 - \frac{\chi(\mathfrak{p})}{p}\right) L_{\infty, \mathfrak{g}\overline{\mathfrak{p}}}(\chi^{-1}, 0).
\mathcal K(x_1-y_1, x_2-y_2, x_3-y_3) \geq \frac{1}{|7\ell(I)|^{2}|7\ell(J)|^{2}+(49\ell(S))^2}={1\over 2\cdot 49^2 \ell(I)^2\ell(J)^2}={1\over 2\cdot 49^2 |R|},
(1-x)B(x,u) =\frac{1}{u}\left(B(x,u)-B_1(x)u\right)+\frac{x^5u^2C^2(x)}{(1-xu)(1-xuC(x))}+\frac{x^3(1-x)u}{(1-2x)(1-xu)}, A(x,u) =\frac{1}{u(1-xu)}A(x,u)+\frac{x^2u}{1-xu}(F_T(x)-1)-((1-2x)F_T(x)-1+x)+B(x,u).
g_{\mu \nu}= \left( \begin{array}{c c}{{h_{i j}- B a_{i}a_{j}}}&{{- B a_{i}}}\\{{- B a_{j}}}&{{- B}}\\ \end{array}\right) ~ ~ , ~ ~ g^{\mu \nu}= \left( \begin{array}{c c}{{h^{i j}}}&{{- a^{i}}}\\{{- a^{j}}}&{{a^{2}- B^{- 1}}}\\ \end{array}\right) ,
\mathbf{E}\le 2^{4k+2}\left( C_{3}k \right)^{C_{4}k}\sum_{t=1}^{2k-1}\left( C_{1}(4k+2) \right)^{C_{2}}\left( \frac{\left( 4k+2 \right)^{6}}{n}\right)^{2k-t}\to 0
C(\underbrace{1,\cdots,1}_s) =\frac{1}{4^s3^k}\bigl(3^k+(-1)^{k-s}(3^{s-1}+w(k))\bigr) &= \frac{1}{4^s3^k}\bigl( 3^k +(-1)^{k-s}(3^{s-1}+w(0))+(-1)^{k-s}(w(k)-w(0))\bigr) =\frac{1}{4^s3^k}\bigl(3^k-(-1)^k+(-1)^{k-s}(w(k)-w(0))\bigr).
\partial_a \ln V_N \Gamma_{\hat{a}}\eta-{1\over V_N}\partial_a E_N\Gamma_{\hat{a}}\Gamma_N \eta-{1\over 2}\sum_{N'\neq N}\bar{q}_{NN'}{1\over V_{N'}}\partial_a E_{N'}\Gamma_{\hat{a}}\Gamma_{N'}\eta \qquad\qquad
\begin{tabular}{c|cccc}$l$ 0 1 2 3 \hline $M_l^2$ 11.59 19.43 29.26 41.10 ${M_{l^{*}}}^2$ 34.53 48.07 63.60 81.11 ${M_{l^{**}}}^2$ 68.98 88.24 109.5 132.7 \end{tabular},
{\mathsf{C}}^{\rm{FD}}_{\mathcal{N}_{\mathcal{M}_1}}+{\mathsf{C}}^{\rm{FD}}_{\mathcal{N}_{\mathcal{M}_2}}={\mathsf{C}}^{\rm{FD}}_{\mathcal{N}_{\mathcal{K}}}+{\mathsf{C}}^{\rm{FD}}_{\mathcal{N}_{[1:N]\backslash \mathcal{K}}}\geq\mathsf{C}_{\mathcal{N}_{[1:N]}}^{\rm{FD}}.
\deg\left(\frac{\textbf{Q}^{\lambda_j}}{Q_i^{\lambda_j(Q_i)}}\right)=\deg\left(\frac{\tilde{\textbf{Q}}^{\lambda_j}}{\tilde Q_i^{\lambda_j(Q_i)}}\right)<\deg(Q_i)\mbox{for every}j, 1\leq j\leq r.
\psi(c_1c_2\otimes d_1d_1) =\theta(c_1c_2)d_1d_1=\theta(c_1)\theta(c_2)d_1d_1=\theta(c_1)d_1\theta(c_2)d_2 =\psi(c_1\otimes d_1)\psi(c_2\otimes d_2),
\|\widetilde{K}_{j,\textbf{r}}(\widehat{f_1}, \dots , \widehat{f_j})\|_{L^2_\alpha}\leq \sum_{k=1}^j \|\widetilde{K}^k_{j,\textbf{r}}(\widehat{f_1}, \dots , \widehat{f_j})\|_{L^2_\alpha},
\psi_U \star (D_1 \star D_2) \,=\, (\psi_U \star D_1) \star D_2 &\,=\, (\Theta_1.\psi_U) \star D_2 &\,=\, \Theta_1.(\psi_U \star D_2) &\,=\, \Theta_1\Theta_2.\psi_U \ .
\sum_{x \in E_N/\pm 1}\int_\rho^\infty F_x \cdot (\eta(u,v)|g_x) = \sum_{x \in E_N/\pm 1}\int_{\rho^2}^\infty F_x | T \cdot (\eta(u,v)|g_x T) & = \sum_{x \in E_N/\pm 1}\int_{\rho^2}^\infty F_{xT}\cdot (\eta(u,v)|g_{xT}) & = \sum_{x \in E_N/\pm 1}\int_{\rho^2}^\infty F_x \cdot (\eta(u,v)|g_x).
\delta= 16777216(d_{1, 0}^2 d_{-1, 0}^2-2 d_{1, 0}d_{-1, 0}d_{1, -1}d_{-1, 1}+d_{1, -1}^2 d_{-1, 1}^2) d_{1, 0}^2 d_{-1, 0}^2 d_{1, -1}^2 d_{-1, 1}^2.
L_{n-m}^{(\alpha-1)}(x)\big(xL_{m-1}^{(\alpha+1)}(-x)+\alpha L_m^{(\alpha)}(-x)\big)+(n-m)L_m^{(\alpha-1)}(-x)L_{n-m}^{(\alpha-1)}(x){}=L_{n-m}^{(\alpha-1)}(x)\big(xL_{m-1}^{(\alpha+1)}(-x)+\alpha L_m^{(\alpha)}(-x)+(n-m)L_m^{(\alpha-1)}(-x)\big).
\Big\Vert \nabla_{\mathrm{x}}\mathbb{I}_{(\Omega)}\Big[\psi\Big] \Big\Vert_{\mathrm{L}^2(\Omega\times\mathbb{R_+})}^2 = \alpha\delta^2 \Big\Vert\nabla_{\xi}\mathbb{I}_{(B)}\Big[\hat{\psi}\Big] \Big\Vert_{\mathrm{L}^2(B\times\mathbb{R_+})}^2,
\ell(\lambda n_s^{-1}) = \sum_{\beta\in\Sigma^+\setminus\{\alpha\}}\lvert\langle\nu(\lambda),\beta\rangle\rvert + \lvert\langle\nu(\lambda),\alpha\rangle - 1\rvert & = \sum_{\beta\in\Sigma^+\setminus\{\alpha\}}\lvert\langle\nu(\lambda),\beta\rangle\rvert + \lvert\langle\nu(\lambda),\alpha\rangle\rvert - 1 & = \ell(\lambda) - 1.
\nu_n = -\frac{1}{n}\sum \limits_{i = 1}^n \Big \{\sum \limits_{j = 1}^i \log(4q_{2n, 2j - 1}p_{2n, 2j - 1}) + \sum \limits_{j = 1}^{i - 1}\log(4q_{2n, 2j}p_{2n, 2j}) + \log(2p_{2n, 2i})\Big\}\delta_{\frac{i}{n}}~,
\min_{\tilde{\boldsymbol{\theta}}_t, \tilde{\boldsymbol{\theta}}_r}\sum_{i \in \{t,r\}}\|\tilde{\boldsymbol{\theta}}_i + \boldsymbol{\vartheta}_i \|^2 \mathrm{s.t.}\tilde{\beta}_{t,n}^2 + \tilde{\beta}_{r,n}^2 = 1, \forall n \in \mathcal{N}, \cos(\tilde{\phi}_{t,n}- \tilde{\phi}_{r,n}) = 0, \forall n \in \mathcal{N},
(a;q)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k}),|q|<1, (a_{1},a_{2},\cdots,a_{n};q)_{\infty}=(a_{1};q)_{\infty}(a_{2};q)_{\infty}\cdots(a_{n};q)_{\infty}.
w_{0}(x)\triangleq\{\begin{matrix}\frac{1}{L}(\frac{1}{2}+\frac{1}{2}cos(\frac{2\pi x}{L}))=\frac{1}{L}cos^{2}(\frac{\pi x}{L}),&|x|\le L/2\\ 0,&|x|>L/2\end{matrix}\}.
\tilde{g}_{ij}(\xi) := \int\partial_i\tilde{l}^{(\alpha)}\partial_j\tilde{l}^{(-\alpha)}\, dx,\quad\tilde{\Gamma}^{(\alpha)}_{ij,k}= \int\partial_i\partial_j \tilde{l}^{(\alpha)}\partial_k\tilde{l}^{(-\alpha)}\, dx.
\begin{array}{rcl}1(x,y) := \delta_e(y) (f_1\bullet f_2)(x,y) := \int_H f_1(x,z)\,f_2(z^{-1}xz,z^{-1}y)\,dz \epsilon(f) := \int_H f(e,y)\,dy (\Delta f)(x_1,y_1;x_2,y_2) := f(x_1x_2,y_1)\,\delta_{y_1}(y_2) (S f)(x,y) := f(y^{-1}x^{-1}y,y^{-1}).\end{array}
{}_{p+}F^2_{q+}(a_+,b_+;X,Y) =\frac{\Gamma(m+1)}{x^m \rho_{m+}'}\frac{1}{2\pi i}\int_{\Gamma}{}_{p+}F_{q+}\left(a_+ -m, b_+ -m;xs\right) \Delta_y(s){\rm d}s
l\Biggl( \sqrt{\frac{\alpha^2M_1+(1-\alpha)^2 2m-\frac{4\alpha^2m^2}{n}}{n}}+\frac{2\alpha m}{n}\Biggr)+(n-l)\Biggl(-\sqrt{\frac{\alpha^2M_1+(1-\alpha)^2 2m-\frac{4\alpha^2m^2}{n}}{n}}+\frac{2\alpha m}{n}\Biggr) & =\sqrt{\frac{\alpha^2M_1+(1-\alpha)^2 2m-\frac{4\alpha^2m^2}{n}}{n}}(2l-n)+2\alpha m,
p(d_{i})=\frac{d_{i}}{\sigma_{i}^{2}}\,\exp\Bigl(-\frac{\Vert\mathbf{z}_{i}\Vert{}^{2}+d_{i}^{2}}{2\,\sigma_{i}^{2}}\Bigr)\, I_{0}\Bigl(\frac{\Vert\mathbf{z}_{i}\Vert\, d_{i}}{\sigma_{i}^{2}}\Bigr)\: d_{i}>0,
[i\gamma_{\mu}\partial^{\mu}, M^{kl}]=i\gamma_{\rho}g^{\rho l}\partial^{k}-i\gamma_{\rho}g^{\rho k}\partial^{l}+i\gamma_{\rho}Z^{l \rho}_{\nu}\partial^{\nu}\partial^{k}-i\gamma_{\rho}Z^{k \rho}_{\nu}\partial^{\nu}\partial^{l}
\frac{1}{2}\frac{d}{dt}\|\Delta v\|_{L^{2}}^{2}+\|\nabla\Delta v\|_{L^{2}}^{2}=-\int_{\mathbb{R}^{3}}\Delta((u\cdot\nabla) v)\Delta v dx-\int_{\mathbb{R}^{3}}\Delta\nabla\cdot(v\nabla\Psi)\Delta v dx :=L_{1}+L_{2}.
e^{-|O(at)x-y|^2/4s}&=e^{-|x|^2/4s}+e^{-|x|^2/4s}\,\frac{\left(O(at)x\right)\cdot y}{2s}&+\frac{1}{2}\,e^{-|O(at)x-\theta y|^2/4s}\; y^T\,\frac{\left(O(at)x-\theta y\right)\otimes \left(O(at)x-\theta y\right)-2s\,\mathbb I}{4s^2}\, y
|\alpha^{-1}(\beta-\varepsilon)|_{\mathfrak{p}}\leq|\beta-\varepsilon|_{\mathfrak{p}}=|\beta^{(\mathfrak{p})}-\varepsilon^{(\mathfrak{p})}|<\max\left\{|\beta|_{\mathfrak{p}},c_{\varepsilon,\mathfrak{p}}\right\}.
\begin{aligned}\Psi_{n_1,\ldots,n_r}(\tau)= \sum_{q=1}^r\sum_{\substack{k_1+\cdots+k_r=N k_1,\ldots,k_r\ge1}}\bigg( (-1)^{N+n_q+k_{q+1}+\cdots+k_r}\prod_{\substack{j=1 j\neq q}}^{r}\binom{k_j-1}{n_j-1}&\times\zeta(\underbrace{k_{q-1},k_{q-2},\ldots,k_1}_{q-1})\Psi_{k_q}(\tau) \zeta(\underbrace{k_{q+1},k_{q+2},\ldots,k_r}_{r-q})\bigg),\end{aligned}
\Delta_{\gamma_2}\bigcap (\mathbb{N}^2\setminus \{0\})=\bigcup\limits_{m=0}^{j_0-1}\Big\{(m+bj_0, \frac{mi_0+n_m}{j_0}+bi_0+a)|a, b\in \mathbb{Z}^{+}\Big\}
(\cosh(t) x_0 + \sinh(t) x_1 - y_0)^2 - (\cosh(t) x_1 + \sinh(t) x_0 - y_1)^2 \sim \Big(e^{-t}\frac{x_0 - x_1}{2}-y_0\Big)^2 - \Big(e^{-t}\frac{x_1 -x_0}{2}-y_1\Big)^2 \sim e^{-t}(x_0 - x_1) (-y_0) - e^{-t}(x_1 - x_0) (-y_1) & = e^{-t}(x_1 - x_0) (y_0 + y_1) \to\infty.
\|p(v)-p(x)\|_{\infty}= |v_1-x_1| = |v_1|, \|p(v)-p(y)\|_{\infty}= |v_2-y_2|= |v_2-3|, \|p(v)-p(z)\|_{\infty}= |v_3-z_3|=|v_3+3|.
q\left(\xi+\sum_{i=1}^r s_id\log f_i+\sum_{i=1}^{r}s_i (-d\log t_i)\right)(v)= \left(\xi+\sum_{i=1}^r s_id\log f_i\right)(v)-\sum_{i=1}^{r}s_i \frac{df_i}{t_i}(v) = \xi(v)+\sum_{i=1}^r s_i\left(\frac{df_i}{f_i}(v)-\frac{df_i}{t_i}(v)\right) = \xi(v) = 0
T(\lambda) = 8N(\lambda) + 4T_{x_{1}}^{+}(\lambda) + 4T_{x_{2}}^{+}(\lambda) + 4T_{x_{3}}^{+}(\lambda) &\ \ \ +2\left \lfloor{\frac{a_{1}\lambda^{1/2}}{\pi}}\right\rfloor +2\left \lfloor{\frac{a_{2}\lambda^{1/2}}{\pi}}\right\rfloor+2\left \lfloor{\frac{a_{3}\lambda^{1/2}}{\pi}}\right\rfloor + 1,
d^2_{Z_i}(x_{kp}) \leq (m_k+1-kp)\big( d^2_{Z_i}(x_{m_k}) +\sum_{n=kp}^{m_k-1}\|x_n-x_{n+1}\|^2\big) &\leq \underbrace{p\big(\omega_+^{-1}\sigma_+^{-1}\beta_+^{-1}+\sigma_+^{-1}\big)}_{=\lambda}\Big(\|x_{kp}-z\|^2 - \|x_{(k+1)p}-z\|^2\Big).
T^* \hat{H}_n(k_x, k_y) T =\left(\begin{array}{cc}\sin n k_x u + \cos nk_x + e^{ik_y}u + \cos n k_x + e^{-ik_y}- \sin n k_x\end{array}\right) &=\hat{H}_{\mathrm{loc}}(a_n(k_x), b_n(k_x), k_y),
f(A_1,B_1)-f(A_2,B_2) =\!\iiint\frac{f(x_1,y)-f(x_2,y)}{x_1-x_2}dE_{A_1}(x_1)(A_1-A_2)dE_{A_2}(x_2)dE_{B_1}(y) begin{align*}.2cm] +\iiint\frac{f(x,y_1)-f(x,y_2)}{y_1-y_2}dE_{A_2}(x)dE_{B_1}(y_1)(B_1-B_2)dE_{B_2}(y_2).
\int_{[Q^{\epsilon_{-}}_{g,n}([E^{ss}/G],\beta)]^{}}1-\int_{[Q^{\epsilon_{+}}_{g,n}([E^{ss}/G],\beta)]^{}}1 &=\sum_{k\geq 1}\sum_{\overline{\beta}}\dfrac{1}{k!}\int_{[Q^{\epsilon_{+}}_{g,n+k}([E^{ss}/G],\overline{\beta})]^{}}\prod_{i=1}^{k}\mathrm{ev}^*_{n+i}[(z-\psi_{n+i})I_{\tilde{\beta}_i}(z-\psi_{n+i})]_0
[Q_{i},\;a^{\dagger}_{m}] = \psi^{\dagger}_{\dot{\alpha}}(\gamma^{\dagger}_{i})_{\dot{\alpha}m}, \;\;\;\;\;\;[Q_{i},\; \psi^{\dagger}_{\dot{\alpha}}]_{+}= a^{\dagger}_{m}(\gamma_{i})_{m \dot{\alpha}}
\Pr\Bigl\{\Big|\frac{\dim(V_1 \cap V_2)}{m}-{\xi_1}\boxdot{\xi_2}\Big|< \frac{k}{m}\Bigr\}&\ge\Pr\{{d_1}\boxdot{d_2}\le{\dim(V_1 \cap V_2)}<{d_1}\boxdot{d_2}+k \}&\ge 1-q^{-k-m\max(0,1-\xi_1-\xi_2)}.
\left|\frac{2^k\|\lambda^{(k-1)}\|+2^{k+1}q}{\lambda[0,T_{k+1}]+2^{k+1}(q+1)}-1\right| = \frac{2^{k+1}}{\lambda[0,T_{k+1}]+2^{k+1}(q+1)}< \frac{2^{k+1}}{2^{\frac{k(k+1)}{2}}+2^{k+1}(q+1)}.
\begin{aligned}\beta_0 =h(x_0),\beta_1=x_1+\ln|h'(x_0)|, \beta_n =n!\sum_{k=1}^{n-1}\frac{1}{k!}\frac{h^{(k)}(x_0)}{(h'(x_0))^n}\sum_{i_1+\dots+i_k=n}\frac{x_{i_1}}{i_1!}\dots\frac{x_{i_k}}{i_k!}+\frac{h^{(n)}(x_0)}{(h'(x_0))^n},n\geq 2,\end{aligned}
f_m(n)=m^{n-1}+\sum_{k=1}^{n-1}\sum_{i=k}^{n-1}\sum_{j=0}^{i-1}(m-1)^{i-k}a^{i-j}(a-1)^{n-2i+j}{i-1\choose k-1}{i\choose j}{n-i-1\choose i-j-1}.
\Lambda_{\alpha}\bar{\Lambda}_{\dot{\beta}}\bar{\mathrm{\raisebox{0 e x}[ 1 . 7 5 e x ] [ 0 e x ]{~ D ~}}}\bar{\Theta}^{\dot{\beta}}= 0 , \ \, L a m b d a_{\alpha}\bar{\Lambda}_{\dot{\beta}}D \Theta^{\alpha}= 0
\Lambda^{+}_{0}(x) N_{2,\varphi}^{+}(x) + N_{2,\varphi}^{-}(x) ={{M}_{1,\varphi}(x)\equiv - N_{1,\varphi}^{-}(x) N_{1,\varphi}^{+}(x),}\; x\in{\mathbb R},
\Phi_{\mathbf h_p}(\nu_{\gamma}\beta_1\nu_{\delta}) = \frac{\chi_{\psi}(p)\underline{\chi}_p(\gamma) p^{-1/2}}{p-1}(p - G(-\gamma,p)), \Phi_{\pmb{\phi}_p}(\nu_{\gamma}\beta_1\nu_{\delta}) = \chi_{\overline{\psi}_p}(p)p^{-1/2}, \Phi_{\breve{\mathbf g}_p}(\nu_{\gamma}\beta_1\nu_{\delta}) = \underline{\chi}_p(\gamma).
w(z)={\hbar\over 2 \pi^2}\int_0^{\infty}dk_z\int_0^{\infty}dk_{\|}{k_{\|}\over k}\left[\left(k^2+k_{\|}^2\right)\sin^2(k_z z) +k_z^2 \cos^2(k_z z)\right]\left[{1\over 2}+{\overline n}(k)\right].
\Vert (\bar{R}_j\,\cdot +\bar{c}_j) - (\bar{R}^\iota\,\cdot +\bar{c}^\iota)\Vert^2_{L^2(\tilde{Q})}\le Cm^{-2}k^2 G, \Vert \bar{R}_j - \bar{R}^\iota \Vert^p_{L^p(\tilde{Q})}\le Cm^{-1}\hat{H}_p