harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\max_{\mathbf{W}, \boldsymbol{\theta}_t, \boldsymbol{\theta}_r}\sum_{k \in \mathcal{K}}R_k \mathrm{s.t.}\mathrm{tr}(\mathbf{W}\mathbf{W}^H) \le P_t, \beta_{t,n}^2 + \beta_{r,n}^2 = 1, \forall n \in \mathcal{N}, \cos(\phi_{t,n}- \phi_{r,n}) = 0, \forall n \in \mathcal{N},
\left| \frac{1}{\|G_n v_1\|^{-s}}- \frac{1}{\|G_n v_2\|^{-s}}\right| = \frac{\left| \|G_n v_1\|^{-s}- \|G_n v_2\|^{-s}\right|}{\|G_n v_1\|^{-s}\|G_n v_2\|^{-s}}\leq \frac{c_s}{\|G_n\|^{-s}}\mathbf{d}(v_1, v_2)^{\bar{s}}.
f_1\leq -\mu_0\int(\varepsilon_y^2+\varepsilon^2)\varphi_B'+Cb_c^{\frac{7}{2}}, &f_k\leq \frac{\mu_0}{10}\int(\varepsilon_y^2+\varepsilon^2)\varphi_B'+Cb_c^{\frac{7}{2}},k=2,3.
B_{N1}^R(1,\xi_n)= 2i\xi_n M_1(1,\xi_n)\begin{pmatrix}0 \int_0^1 e^{-2i\xi_nt}r_{N+1}(t)dt\end{pmatrix}=\begin{pmatrix}\ell^2_n 0\end{pmatrix}.
dh=\sum_{n=0}^{\left[\frac{p+2}{2}\right]}\frac{1}{n!}dh^{(p+2-2n)}{\cal F}^n +\sum_{n=1}^{\left[\frac{p+2}{2}\right]}\frac{1}{(n-1)!}(-1)^p h^{(p+2-2n)}(C\Gamma_\mu\Gamma_{11})_{\alpha\beta}\Pi^\mu \Pi^\alpha\Pi^\beta{\cal F}^{n-1}\quad ,
I_{\textup{nested}, 2}= \Big\| 1_{Q_0}\Big( \mathop{\sum_{R \in \mathcal{D}_{\textup{good}}:\, R \subset Q^*}}_{2^{-\kappa}< \ell(R) < 2^{s-r}}1_{R}\int_{\ell(R)/2}^{\ell(R)}\Big| \mathop{\sum_{\ell=r+1}^{s-\textup{gen(R)}}}\theta_t (1_{R^{(\ell-1)}}\Delta_{R^{(\ell)}}f) \Big|^2\, \frac{dt}{t}\Big)^{1/2}\Big\|_{L^q(\mu)}.
Z_{n}(T,x)= 2\pi\lambda\sum_{k=1}^{4}b_k \int_{x}^{\infty}F\left(N_\mathrm{N},\frac{n\eta_\mathrm{N}\bar{a}_kTx^{\alpha_\mathrm{N}}}{N_\mathrm{N}t^{\alpha_\mathrm{N}}}\right) &(1-p(t))t\mathrm{d}t,
a_{h,1}(\ell,k)=\frac 1{(2\pi i)^4}\int_{\Gamma_{D_1}}dz\int_{\Gamma_{D_2}}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega\frac{G_{n_1,\mu_1,\xi_1}(z)G_{\Delta n,\Delta\mu,\Delta\xi}(w)\left(\frac 1{z-w}-h\right)}{G_{k,\mu_1,\xi_1}(\zeta)G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)(z-\zeta)(w-\omega)}
\tilde G(s) =\lim_{b\to0}\mu_U\left(r_U \leq \frac s{\mu(U)}\right)=\lim_{b\to0}\frac1{\mu(U)}\mu\left(\left\{r_U \leq \frac s{\mu(U)}\right\}\cap U\right) &\geq \lim_{b\to0}\frac{\mu(U\setminus A)}{\mu(U)}=\lim_{b\to0}\frac{\mu([0,a))}{\mu([0,b))}=1
\mathfrak{g}_{p}[{\Sigma}^o ]^\Gamma:= \left(\mathfrak{g}\otimes \mathbb{C}_{p}[{\Sigma}^o]\right)^\Gamma,\,\,\,\,\, \mathfrak{g}_{p''}[{\Sigma}'^o]^\Gamma:= \left(\mathfrak{g}\otimes \mathbb{C}_{p''}[{\Sigma}'^o]\right)^\Gamma.
d s^{2}= \left( 1 + \frac{u^{4}+ 2 u^{2}v^{2}}{\beta^{2}}\right) d t^{2}- \frac{d u^{2}+ 2 d v^{2}}{\beta^{2}}- \frac{u^{2}( 2 v d u + u d v )^{2}}{\beta^{4}}.
\begin{aligned}\left(1-\lambda \right) \max \{e^{D_{\varpi}(P_0 \parallel Q_0)}, e^{D_{\varpi}(P_1 \parallel Q_1)}\}+ \lambda \max \{e^{D_{\varpi}(P_0 \parallel Q_0)}, e^{D_{\varpi}(P_1 \parallel Q_1)}\}\ge e^{D_{\varpi}(P_{\lambda}\parallel Q_{\lambda})}. \end{aligned}
\begin{aligned}\mathbb F^g_r(a-2\epsilon, b+2\epsilon) = \mathbb F^f_r(c_{i-1}, c_{j+1}) \mathbb F^g_r(a+2\epsilon, b-2\epsilon)= \mathbb F^f_r(c_i,c_j) \mathbb F^g_r(a+2\epsilon, b+2\epsilon) = \mathbb F^f_r(c_{i},c_{j+1}) \mathbb F^g_r(a-2\epsilon, b-2\epsilon)= \mathbb F^f_r(c_{i-1}, c_{j}).\end{aligned}
\frac{1}{|{\rm Aut}(\Gamma)|}\sum_{j=0}^{n-1}{\xi_{\Gamma}}_*\left(\prod_{i=1}^{n-1}\left(1+3\omega_i\right) \prod_{(h,h')\in E_2(\Gamma)}\frac{1}{\psi_h-(1+3\omega_{h'})}\right)\cdot (-\lambda - \delta_1)^{j}\cdot (3\omega_n -\lambda - \delta_1).
d^2\!f(\overline{x}|\overline{v})(w) =\min_{z\in\mathbb{R}^n}\big\{d^2\vartheta(F(\overline{x})) (dF(\overline{x})(w)\,|\,d^2F(\overline{x})(w|z))-\langle\overline{v},z\rangle\}=d^2\!f(F(\overline{x}))(dF(\overline{x})(w))+\langle \overline{\xi},d^2 F(\overline{x})(w)\rangle,
\Gamma_{\underline{\alpha}\underline{\beta}}^{a}= \left( \begin{array}{c c}{\gamma_{\alpha \beta}^{a}\delta_{q}^{r}}&{0}\\{0}&{\gamma^{a \alpha \beta}\delta_{q^{\prime}}^{r^{\prime}}}\\ \end{array}\right) , \quad a = 0 , 1 , . . . , p ,
w(g_1\times g_2)=\left[\begin{array}{cccccccccc}0 0 0 -1_t 0 -1 0 0 0 0 0 0 0 0 0 0 u 0 0 u' 0 0 0 0 0 0 0 -\zeta 0 0 1_t 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 v 0 0 v'\end{array}\right].
P_{2N}(x) = x \sum_{j=0}^{2N-1}{2N-1\choose j}x^j \delta_2^{2N-1-j}x^{\frac12(2N-1-j)}. &= \sum_{j=0}^{2N-1}{2N-1\choose j}\delta_2^{j-1}x^{N+1 + \frac12(j-1)}\intertext{Now put $j = 2\ell-1$ where $\ell=1,2,\dots, N$ to get}P_{2N}(x) = \sum_{\ell=1}^{N}{2N-1\choose 2\ell-1}x^{N+\ell}.
\cos ( \frac{k_{1}^{\prime}\wedge k_{2}^{\prime}}{2 s^{2}}) \cos ( \frac{k_{3}^{\prime}\wedge k_{4}^{\prime}}{2 s^{2}}) = \cos ( \frac{k_{1}^{\prime}\wedge k_{2}^{\prime}}{2}- t k_{1}^{\prime}\wedge k_{2}^{\prime}) \cos ( \frac{k_{3}^{\prime}\wedge k_{4}^{\prime}}{2}- t k_{3}^{\prime}\wedge k_{4}^{\prime})
Z=\sum_{J_+,J_-}\frac{1}{J_+!J_-!}(\frac{V}{V_c})^J \exp-[JRe\;I_c]e^{ik\pi (J_+-J_-)}=\exp \{e^{i\pi k}2(\frac{V}{V_c})\exp -[Re\;I_c]\};\;\;(J=J_++J_-).
Z(s,f,\chi)=\frac{(1-q^{-1})^2q^{-6-14s}}{(1-q^{-1-2s})(1-q^{-5-12s})}-\frac{(1-q^{-1})^2q^{-9-20s}}{(1-q^{-1-2s})(1-q^{-9-20s})}.
1+\sum_{\lambda_{\underline{j}}\neq \emptyset}(1-\xi_m)^{k}(q^{\lambda_{j_k}}-1)(-1)^{k-1}= 1+\sum_{k=1}^{\l(\lambda)}(1-\xi_m)^{k}\sum_{l(\lambda_{\underline{j}})=k}(q^{\lambda_{j_k}}-1)(-1)^{k-1}
L:= -\frac{2H}{H^2-4K}F^{ij, rs}\nabla_m h_{ij}\nabla_m h_{rs}+\frac{2H}{H^2-4K}(1-\frac1q)\frac{|\nabla K|^2}{K}&+\left(\frac{\alpha}{K}-\frac{4}{H^2-4K}\right) (1-\frac1q)\frac{F^{ij}\nabla_i K\nabla_j K}{K}+\frac{2F^{ij}\nabla_iH\nabla_j H}{H^2-4K}-(\alpha^2+\alpha)\frac{F^{ij}\nabla_iK\nabla_j K}{K^2}.
\tilde{\rho}(y_{1}(\cdot,\star,\ast),y_{2}(\cdot,\star,\ast)):=\sup\left\{\mathrm{e}^{at}\left\Vert y_{1}(t,\xi,\eta)-y_{2}(t,\xi,\eta)\right\Vert :(t,\xi,\eta)\in\mathbb{R}_{+}\times\mathcal{N}_{r}(\mathcal{M})\right\}.
\partial_{\alpha}^k \tilde{\theta}= \partial_{\alpha}^k (I-\mathcal{H})(\zeta-\bar{\zeta})=\partial_{\alpha}^k (I-\mathcal{H})(\zeta-\alpha) = \partial_{\alpha}^k(I+\bar{\mathcal{H}}-(\bar{\mathcal{H}}+\mathcal{H}))(\zeta-\alpha) = 2\partial_{\alpha}^{k-1}(\zeta_{\alpha}-1)-\partial_{\alpha}^k(\bar{\mathcal{H}}+\mathcal{H}))(\zeta-\alpha).
(A_{\infty},u_{\infty}) = \lim_{t\rightarrow \infty}e^{\textbf{i}t\xi_0}h(t_1)^{-1}(A(t_1), u(t_1)) &= k^{-1}\lim_{t\rightarrow \infty}e^{k\xi_0 k^{-1}}k h(t_1)^{-1}(A(t_1),u(t_1)) \in (\mathcal{G}^c)^{2,2}(A_0,u_0)
R(z)=\begin{cases}T(z)P^{(\infty)}(z)^{-1}, \mbox{for $z \in \mathbb{C}\setminus \overline{U}$,}T(z)P^{(0)}(z)^{-1}, \mbox{for $z\in D_{0}$,}T(z)P^{(4)}(z)^{-1}, \mbox{for $z\in D_{4}$.}\end{cases}
{\cal H}_{\mathrm{6 - v e r t e x}}= \sum_{j = 0}^{l_{1}}\left[ \left( \begin{array}{c}{2 l_{1}}\\{l_{1}- j}\\ \end{array}\right) - \left( \begin{array}{c}{2 l_{1}}\\{l_{1}- j - 1}\\ \end{array}\right) \right] \rho_{j},
1=R(\lambda)^{1/2}\times R(\lambda)^{-1/2}\mathop{=}\limits_{\lambda\rightarrow\infty}\left(\sum_{l=0}^{\infty}{c}_{l-1}(\underline{\Lambda})\lambda^{n+1-l}\right)\left(\sum_{l=0}^{\infty}\hat{c}_{l-1}(\underline{\Lambda})\lambda^{-n-1-l}\right)
\Delta \hat{G}^{\xi = 0}[ a , b ; E ] = \lim_{\epsilon \rightarrow 0^{+}}\frac{\hat{G}_{F}^{\xi = 0}[ a , \epsilon ; E ] \hat{G}_{F}^{\xi = 0}[ \epsilon , b ; E ]}{\frac{1}{v}- \hat{G}_{F}^{\xi = 0}[ \epsilon , \epsilon ; E ]}.
\frac{\partial A}{\partial q_{j}}=\frac{\partial A(p_{j},q_{j})}{\partial q_{j}}\equiv \frac{\partial Aj}{\partial q_{j}}&\frac{\partial B}{\partial q_{j}}=\frac{\partial B(p_{j},q_{j})}{\partial q_{j}}\equiv \frac{\partial Bj}{\partial q_{j}}
\Delta_{\lambda}(\dot{t}^{\sharp}\dot{u}^{\flat, -1}g f_{(s,r)}(h)) = \Delta_{\lambda}(\dot{t}^\sharp h_5^{\sharp, -1}z_5^\sharp \dot{r}^{-1, \sharp})= \Delta_{\lambda}( \iota(\dot{t}^\sharp h_5^{\sharp, -1}z_5^\sharp \dot{r}^{-1, \sharp})) = \Delta_{\lambda}( \iota(\dot{t}^\sharp ) \iota(h_5^{\sharp, -1}) z_5^\sharp \iota(\dot{r}^{-1, \sharp})).
|\nabla_\xi (I)| = \left|\nabla_\xi \ln\left(\frac{(8 + 8\alpha)|\ln |x + \xi\sqrt{\theta(x)}||}{|\frac{K_0}{4}\sqrt{|\ln \theta(x)|}+ \xi|^2}\right) \right| & \leq \frac{C\sqrt{\theta(x)}}{|x + \xi\sqrt{\theta(x)}||\ln|x + \xi \sqrt{\theta(x)}||}+ \frac{C}{|\frac{K_0}{4}\sqrt{|\ln \theta(x)|}+ \xi|}\leq \frac{C}{\sqrt{|\ln \theta(x)|}},
\mathfrak{S}^{r,t}_{g,d}(C) := \left\{D\in C_n : \exists \, \ell \in G^r_{d}(C) \,\,\mbox{with}\,\, h^0\left( C, \ell\otimes \mathcal{O}_C (-D) \right) \geq r+1-t \right\}.
\Delta^{\!{\scriptscriptstyle 0}}_n=\sum_{j=1}^{n-1}\frac{1}{\lambda_j \pi_j u_j^{{\scriptscriptstyle 0}}u_{j+1}^{{\scriptscriptstyle 0}}}+ \sum_{j=1}^{n-1}\sum_{p=2}^{j}\frac{(u_p^{{\scriptscriptstyle 0}})^2 \pi_p}{\lambda_j\pi_j u_j^{{\scriptscriptstyle 0}}u_{j+1}^{{\scriptscriptstyle 0}}}\textup{for all}\;n\geq 2
T_{0}(x,\lambda)\prod(x-z)^{\kappa}=\left\{-T_{0}(z,\frac{\kappa^{2}}{\lambda})-\frac{1}{2}\left[\kappa MN+\epsilon_{0}(N,\lambda)+\epsilon_{0}(M,\frac{\kappa^{2}}{\lambda})\right]\right\}\prod(x-z)^{\kappa},
\frac{\Gamma_q(\lambda_N - z)}{\Gamma_q(\lambda_1 + \theta N - z)}= \frac{[-z + \lambda_N - 1]_q\cdots [-z+1]_q[-z]_q}{[-z + \lambda_1 + \theta N - 1]_q\cdots [-z + \theta N + 1]_q[-z + \theta N]_q}\frac{\Gamma_q(-z)}{\Gamma_q(\theta N - z)},
M=\mu(x,\sigma_{I},\sigma_{D})=\sigma_{D}^{2}g(\sigma_{I})\otimes[\begin{matrix}L_{x}^{2}(x,\sigma_{D})&L_{x}L_{y}(x,\sigma_{D})\\L_{x}L_{y}(x,\sigma_{D})&L_{y}^{2}(x,\sigma_{D})\end{matrix}]
(1)\prod\limits_{i=2p}^{mp+2p-1}([\omega]- h_{i,1}) \prod\limits_{1\leq i\leq m^2p, m\nmid i}\bigg([\omega]-h_{p+\tfrac{i}{m},1}\bigg)* [H]=0, (2)\prod\limits_{i=1}^{p}([\omega]- h_{i,1})\prod\limits_{i=1}^{p-1}([\omega]- h_{mp+p+i,1}) \prod\limits_{i=1}^{m^2p}\bigg([\omega]-h_{p+\tfrac{i}{m},1}\bigg)=0.
\Vert u_0(t)\Vert_{\rho_t}= \Vert u_0^\#(t)\Vert_\rho \leq e^{-\kappa t}\Vert \eta\Vert_\rho + \Big\Vert \int_0^t P_{t-s}^\# \Pi_0\Phi_s dW(s)\Big\Vert_\rho\, .
\left. \hat{F}_{\alpha \beta}^{I}\right|_{\mathrm{r e g}}= \frac{( \partial \Pi )^{2}}{2 \Pi^{2}}e_{\alpha}^{a \,}\, \hat{\xi}_{a b}^{+}\, e_{\beta}^{b \,}- \frac{D_{\alpha}\, \partial_{\gamma}\Pi}{2 \Pi}e^{a \, \gamma}\, \hat{\xi}_{a b}^{+}\, e_{\beta}^{b}- e_{\alpha}^{a \,}\, \hat{\xi}_{a b}^{+}\, e^{b \, \delta}\frac{D_{\beta}\, \partial_{\delta}\Pi}{2 \Pi}
\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d) =\Big\{\nabla g(\overline{x})\xi\ |\ \langle\xi,g'(\overline{x})d\rangle=0,\, \xi\in\mathcal{T}_{\mathcal{N}_{K}(g(\overline{x}))}(\lambda)\Big\}.
\Psi_{\kappa}^{( P )}= \frac{i}{2^{3 / 2}}\, \exp \left( - \frac{\pi \kappa}2 - i \frac{\kappa}2 \ln \left( \frac{- x_{+}}{- x_{-}}\right) \right) H_{i \kappa}^{( 1 )}\left( m \sqrt{( - x_{-}) ( - x_{+})}\right) .
RE_{\hat{g}}=\frac{\hat{\sigma}_{g}}{\hat{g}}\approx\sqrt{(\frac{s_{L}}{n_{L}\overline{L}})^{2}+4(\frac{s_{T}}{n_{T}\overline{T}})^{2}+(\frac{\overline{\theta}}{2})^{4}(\frac{s_{\theta}}{n_{\theta}\overline{\theta}})^{2}}
t_{b}t_{v}(a) = t_{b}t_{a}t_{c}t_{b}t_{c}^{-1}t_{a}^{-1}(a) = t_{b}t_{a}t_{c}t_{b}(a) = c, t_{b}t_{v}(c) = t_{b}t_{a}t_{c}t_{b}t_{c}^{-1}t_{a}^{-1}(c) = t_{b}t_{a}t_{c}t_{b}(c) = a.
\begin{aligned}I_1(u) :=\int_{-\infty}^{\infty}u(t,x)dx=I_1(u_0),\;\;\;I_2(u):=\int_{-\infty}^{\infty}u^2(t,x)dx=I_2(u_0), I_3(u) :=\int_{-\infty}^{\infty}\left(\frac12 (\partial_xu)^2 -G(u)\right)(t,x)dx=I_3(u_0),\end{aligned}
E = \gamma_2([0,1] \times \mathbb{S}_2^{r-1}\times \mathbb{S}_2^{r-1},d_2) + \gamma_1([0,1] \times \mathbb{S}_2^{r-1}\times \mathbb{S}_2^{r-1}, d_\infty), V = \Delta_2([0,1] \times \mathbb{S}_2^{r-1}\times \mathbb{S}_2^{r-1},d_2), U= \Delta_\infty([0,1] \times \mathbb{S}_2^{r-1}\times \mathbb{S}_2^{r-1},d_\infty).
J_{1}^{n}(\theta):=\int_{0}^{(nc_{J})^{-1}}\,\mathrm{d}x_{1}\,x_{1}^{-\psi_{p}(\theta)_{+}-1}\int_{0}^{c_{J}x_{1}}\,\mathrm{d}x_{2J}\,x_{2J}L_{n}(x_{2J})^{2}, &J_{2}^{n}(\theta):=\int_{(nc_{J})^{-1}}^{\pi}\,\mathrm{d}x_{1}\,x_{1}^{-\psi_{p}(\theta)_{+}-1}\int_{0}^{c_{J}x_{1}}\,\mathrm{d}x_{2J}\,x_{2J}L_{n}(x_{2J})^{2}.
P_{e|\gamma}(x) =w_1 Q_z\left(a_0\sqrt{x},\pi/2\right)+w_2 Q_z\left(a_1\sqrt{x},\pi/2\right)-w_3 Q_z\left(a_0\sqrt{x},\pi/4\right) &-2 w_2 \sum_{l=1}^{L-1}Q_z\left(a_0\sqrt{x},\alpha_l\right)-w_2\sum_{l=1}^{L-1}Q_z\left(a_l\sqrt{x},\beta_l^+\right)+w_2\sum_{l=2}^{L}Q_z(a_l\sqrt{x},\beta_l^-),
\begin{cases}[1]\dot{q}^1\frac{\partial \gamma_1(q)}{\partial q^1}+\dot{q}^2\frac{\partial \gamma_2(q)}{\partial q^1}-2q^2q^1=0 [2]\dot{q}^1\frac{\partial \gamma_1(q)}{\partial q^2}+\dot{q}^2\frac{\partial \gamma_2(q)}{\partial q^2}-(q^1)^2=0 [3]\gamma_1(q)-\dot{q}^1=0 [4]\gamma_2(q)=0. \end{cases}
F_{\mu \nu \lambda \rho}=\partial_\mu A_{\nu \lambda \rho}-\partial_\nu A_{\mu \lambda \rho}+\partial_\lambda A_{\rho \mu \nu}-\partial_\rho A_{\lambda \mu \nu}\equiv \partial_{\left[ \mu \right.}A_{\left. \nu \lambda \rho \right]}.
f(v) =\sum_{i=-d}^{d}(\left\vert v-i\right\vert +1)+\sum_{i=-(k-1)}^{k-1}(\left\vert v-i\right\vert +2)+ & (\left\vert x^{\prime}-v\right\vert +2+\left\vert -x^{\prime}-v\right\vert+2)
\delta_{B}^{r}L_{B}^{r}= \eta \; \frac{d}{d \tau}\; \Bigl [ q^{1 / 2}\lambda \; p_{\mu}\dot{x}^{\mu}- \; \frac{\lambda e}{1 + q^{2}}\; ( p^{2}- m^{2}) +{\cal B}\frac{d}{d \tau}( \lambda \; e ) \Bigr ] .
\sum_{x^{\varepsilon}<m\le x^{1-\varepsilon}}\frac{(\log\log x-\log\log m)^2}{\varphi(m)}= \int_{x^{\varepsilon}}^{x^{1-\varepsilon}}\frac{(\log\log x-\log\log t)^2}{t}d\sum_{x^{\varepsilon}<m\le t}\frac{m}{\varphi(m)}= Af(\varepsilon)\log x+O((\log\log x)^2)
(g)^{\mu\nu}=\left(\begin{array}{ccc}0 e^{-\psi/2}0 e^{-\psi/2}(f^{2}x^{2}_{I}-h_{++})e^{-\psi}+g_{+I}\gamma^{IJ}g_{J+}-e^{-\frac{3}{2}\psi}g_{+I}\gamma^{IJ}0 -e^{-\frac{3}{2}\psi}g_{+J}\gamma^{JI}e^{-\psi}\gamma^{IJ}\end{array}\right)
\mathcal{D}_{k,\ell}(0,0) = C_{k,\ell}\prod_{p^\alpha \mid \mid h}\frac{\sum_{j=0}^{\alpha}\frac{\sigma_{k-1}(p^j,1)\sigma_{\ell-1}(p^j,1)}{p^{j}}- \frac{\sigma_{k-1}(p^{j+1},1)\sigma_{\ell-1}(p^{j+1},1)}{p^{j+2}}}{(1-p^{-1})^{k-1}+(1-p^{-1})^{\ell-1}- (1-p^{-1})^{k+\ell-2}}.
S^{TD}(\beta)=\bar{S}^{BH}(\beta)+\left(-\mbox{Tr}(\hat{\rho}^H_\epsilon(\beta)\log\hat{\rho}^H_\epsilon(\beta)\right)-\left(-\mbox{Tr}(\hat{\rho}^R_\epsilon(2\pi\mu)\log\hat{\rho}^H_\epsilon(2\pi\mu))\right)~~~
\Omega_{n}(z_1,\bar{z_2},s)=\sum \limits_{\gamma \in \Gamma}\frac{\mu_{\gamma}(\bar{z_1},-\bar{z_2})^{n-1}\mu_{\gamma}(\bar{z_1},-z_2)^{n+1}}{|\mu_{\gamma}(z_1, -z_2)|^{2s-2}|\mu_{\gamma}(z_1, -\bar{z_2})|^{2s+2}}.
V^{\ast}\Psi A^{\dagger}=V^{\ast}A^{\dagger}+(I+F_{S_{A}}K^{\ast}K)^{-1}F_{S_{A}}K^{\ast}A^{\dagger}, V^{\ast}\Psi KS_{A}^{\dagger}H =(S_{A}-I)S_{A}^{\dagger}H+(I+F_{S_{A}}K^{\ast}K)^{-1}F_{S_{A}}K^{\ast}KS_{A}^{\dagger}H.
E_{1}^{rs}=F(]Q_{\Gamma}^{r+1}[_{{\mathfrak Q}_{\Gamma}})\otimes_{K}H^s_{dR}(]Q_{\Gamma}^{r+1}[_{{\mathfrak Q}_{\Gamma}})\Longrightarrow H^{s+r}_{dR}(X_{\Gamma},F).\tag*{$(C)_{\pi}$}
3(k-1){\rm e}^{9A}[ 2 k r A'' + 2 k^2 A' + 6 kr A'^2 - 9 r^2 A'^3 ] + 4 b^2 r [ 3 r^2 A'' - 3 (2k-1) r A' + 9r^2 A'^2 - 2 k(k-1) ] = 0,
\lambda_j = \exp(4 \pi i (\eta_j-\eta_2)_1(\eta_1)_2)\left( \frac{\theta[\eta_{U_\eta\circ(V \cup \{1,2\})}](\Omega)\theta[\eta_{U_{\eta}\circ (W \cup \{1,2\})}](\Omega)}{\theta[\eta_{U_\eta \circ (V \cup \{1,j\})}](\Omega)\theta[\eta_{U_{\eta}\circ (W \cup \{1,j\})}](\Omega)}\right)^2.
B_2(S^2(\mathcal{H}) \times S^2(\mathcal{H}), S^2(\mathcal{H})) =B(S^2(\mathcal{H}) \overset{\wedge}{\otimes}S^2(\mathcal{H}),S^2(\mathcal{H})) & = \bigl(S^2(\mathcal{H}) \overset{\wedge}{\otimes}S^2(\mathcal{H}) \overset{\wedge}{\otimes}S^2(\mathcal{H})\bigr)^*.
\hat\gamma (E+i\epsilon)-\hat\gamma (E) =\frac{1}{2}\int\ln(1+\frac{\epsilon^{2}}{(E-E')^{2}})d\hat{\mathcal N}(E') &>\frac{\ln 2}{2}(\hat{\mathcal N}(E+\epsilon)-\hat{\mathcal N}(E-\epsilon)).
{}\hat{\pi}_{h,{T_n}}^a(x) = \frac{1}{T_n \prod_{l = 1}^d h_l}\int_0^{T_n}\prod_{l = 1}^d K(\frac{x_l - X_{\varphi_{n, l}(u)}^l}{h_l}) du = : \frac{1}{T_n}\int_0^{T_n}\prod_{l = 1}^d K_{h_l}(x_l - X_{\varphi_{n, l}(u)}^l) du.
\begin{array}{l}\Phi_{2}\bar{\Phi}_{2}-q^{2}\bar{\Phi}_{2}\Phi_{2}=1, \Phi_{1}\bar{\Phi}_{1}-q^{2}\bar{\Phi}_{1}\Phi_{1}=1+(q^{2}-1)\bar{\Phi}_{2}\Phi_{2}, \Phi_{2}\Phi_{1}=q\Phi_{1}\Phi_{2}, \Phi_{2}\bar{\Phi}_{1}=q\bar{\Phi}_{1}\Phi_{2},\end{array}
E_{s}(\alpha_{N})=\sum_{k=1}^{t-1}\left(\sum_{j=k+1}^{t}2^{n_{j}-n_{k}}\right)\mathcal{L}_{s}(2^{n_{k}+1})+\sum_{k=1}^{t}\left(1-\sum_{j=k+1}^{t}2^{n_{j}-n_{k}+1}\right)\mathcal{L}_{s}(2^{n_{k}}),
e^n_s \bar b^n_s = (x_s-x_s^n)(b_s(x_s)-b_s(x^n_s)) + (x_s-x^n_s)(b_s(x^n_s)-b_s(x^n_{\kappa(n,s)})) & +(x_s-x^n_s)(b_s(x^n_{\kappa(n,s)})-b_s^n(x^n_{\kappa(n,s)}))
\begin{cases}\tilde{\gamma}_i=\gamma_i+O(1)Z_{a}h+O(1)|\gamma_4|h,\,\,i=1,5, &\tilde{\sigma}_j=\sigma_j+O(1)Z_{a}h+O(1)|\gamma_4|h,\,\,j=2,3, &\tilde{\gamma}_4=(1-{\phi(T_b)h}/{u_b})\gamma_4+O(1)Z_{a}h.\end{cases}
\tilde{S}( x ) = \exp \left( - i \frac{x_{1}}{\sqrt{x_{1}^{2}+ \lambda^{2}}}\left[ \mathrm{a r c t a n}\left( \frac{x_{2}}{\sqrt{x_{1}^{2}+ \lambda^{2}}}\right) - \pi \Big ( \frac{1}{2}+ m \Big ) \right] \right) ,
(\alpha_{i}\wedge a_{i})\cdot\psi=\frac{1}{4}(2^{\frac{1}{2}})^{2}(\iota(\alpha_{i})(a_{i}\wedge\psi)-a_{i}\wedge(\iota(\alpha_{i})\psi))=\frac{1}{2}\psi-a_{i}\wedge(\iota(\alpha_{i})\psi).
B_n^{\overbrace{\scriptstyle{(0,\ldots,0,-1,0,\ldots,0)}}^r}=\displaystyle \sum_{l=1}^{1}{\alpha}_l^{\overbrace{\scriptstyle{(0,\ldots,0,-1,0,\ldots,0)}}^r}(l+r)^n &={\alpha}_1^{\overbrace{\scriptstyle{(0,\ldots,0,-1,0,\ldots,0)}}^r}(r+1)^n &={\alpha}_1^{\overbrace{\scriptstyle{(0,\ldots,0}}^{i-1},1)}(r+1)^n.
F_1(x,y,t) = x e^{-t}- \frac{e^{-\alpha t}y}{(1-e^{- \alpha t})^{1 - \frac{1}{\alpha}}}, & F_2(x,y,t) = \nabla(f) \left(xe^{-t}+ (1- e^{-\alpha t})^{\frac{1}{\alpha}}y\right).
\rho_N(\lambda_1,\cdots,\lambda_p)\equiv\left\langle \prod_{a=1}^p \frac{1}{N}{\rm tr}\, \delta (\lambda_a-M)\right\rangle\stackrel{\mbox{\small Wick's theorem}\atop{\downarrow}}{=}\det_{1\leq a,b \leq p}K_N(\lambda_a,\lambda_b)
P_1 = \max \left\{\exp\left(\left(\frac{\gamma^2}{2}- \varepsilon^2\right) L^2\right), 1\right\}, P_2 = \max \left\{\exp\left(\varepsilon^2 L^2\left(\frac{\varepsilon^2}{\gamma^2}- 1\right)\right),1 \right\}.
\exp\left(-\bigtriangleup S_1[\overline X + \xi]\right) = ig\pi\alpha'\frac{1}{2}N_{div}\int_0^{2\pi}d\theta ~{\rm Tr}~{\cal P}\left( U[A] ~\left[(D_a D^c F_{cb}- D_b D^c F_{ca}) \xi^a \partial_{\theta}\xi^b\right.\right.
\mathfrak{a}_n(x) = \sum_{j=1}^n \binom{n}{j}(x+x^2+\cdots+x^j) d_{n-j}(x) = \sum_{i=0}^{n-1}\binom{n}{i}d_i(x) (x+x^2+\cdots+x^{n-i}) = xA_n(x),
h'_{p}\left(x\right) =-2\int_{0}^{x}\sinh\left(y-x\right)f\left(y\right)dy h''_{p}\left(x\right) =2\int_{0}^{x}\sinh\left(y-x\right)f\left(y\right)-2f\left(x\right)=h\left(x\right)-2f\left(x\right)
\tilde{{\cal L}}\rightarrow - \frac{1}{4}{\cal F}^{2}+{\cal A}J + a J - \frac{1}{4 g}\epsilon^{\mu \nu \rho}a_{\mu}\partial_{\nu}a_{\rho}+{\cal L}_{M}+ \frac{g^{2}}{2}J^{2}
\frac{-1}{4(4\pi^2)^2}\int_{R\leq\left|u\right|\leq R_2}\mathrm{d}^4u\left(\frac{\delta_{\nu\sigma}\delta_{\tau\mu}}{4\left|u\right|^4}-\frac{\delta_{\nu\sigma}u_\tau u_\mu+\delta_{\tau\mu}u_\nu u_\sigma}{2\left|u\right|^6}+\frac{u_\nu u_\sigma u_\tau u_\mu}{\left|u\right|^8}\right)F_{1\mu\nu ax}F_{1\sigma\tau ax}
\Delta_{\nu}=4(1+\langle z,z\rangle )\left( \sum_{i,j=1}^{n}(\delta_{ij}+z_{i}\bar{z_{j}})\frac{\partial^{2}}{\partial z_{i}\partial \bar{z_{j}}}-\nu \sum_{j=1}^{n}\left( z_{j}\frac{\partial}{\partial z_{j}}-\bar{z_{j}}\frac{\partial}{\partial \bar{z_{j}}}\right) -\nu^{2}\right) +4\nu^{2}
\phi_{\nu_{1}}^{a n}T_{0}\left( M_{\nu_{1}, \nu_{2}}, Z \right) = T_{0}\left( M_{1 , \nu_{2}}, Z_{1}\cup N_{1}\right) \otimes T_{0}\left( M_{2}, Z_{2}\cup N_{1}\right) \otimes T_{0}( N_{1}) .
x_{N}^{n-j}D_{x_{N}}^{m-j}u(x,t)=-x_{N}^{n-j}{\displaystyle\int\limits_{x_{N}}^{\infty}}\xi_{N}^{-n+j-1}\left[\xi_{N}^{n-j+1}D_{\xi_{N}}^{m-j+1}u(x^{\prime},\xi_{N},t)\right] d\xi_{N}.
\exp \left[ \int_{0}^{1}\rho_{n}( \tau ) \gamma^{n}d \tau \right] = \left. \exp \left( i \gamma^{n}\frac{\partial_{l}}{\partial \theta^{n}}\right) \exp \left[ - i \int_{0}^{1}\rho_{n}( \tau ) \theta^{n}d \tau \right|_{\theta = 0}\right] ,
{\cal S}= \int d^{5}x \sqrt{- g}\left[{\frac{\cal R}{1 6 \pi G_{N}}}-{\frac{1}{2}}\partial_{\mu}\Phi^{a}\partial^{\mu}\Phi^{a}-{\frac{\lambda}{4}}( \Phi^{a}\Phi^{a}- \eta^{2})^{2}\right] - \int_{i}d^{4}x \sqrt{- h}\sigma_{i}( \Phi^{a}) \, ,
\mu \left( \left\{\omega \in \Omega : \frac{1}{n}\sum_{i=1}^n h(Z_i(\omega)) \geq \sqrt{\frac{8 (\log n)^{\frac{2}{\gamma}}\tau \sigma^2}{n}}+ \frac{8 (\log n)^{\frac{2}{\gamma}}B \tau}{3 n}\right\}\right) \leq 2 e^{- \tau}
Z[{\overline{A}}]=\int\!\!{\mathcal{D}}\psi{\mathcal{D}}{\overline\psi}\exp\left[\int\!\!d^{4}x_{_{E}}\left\{-{\overline\psi}\gamma_{\mu}(\partial_{\mu}-i{\overline{A}}_{\mu})\psi+{g^{2}\over2}\left[({\overline\psi}\psi)^{2}+({\overline\psi}i\gamma_{5}\psi)^{2}\right]\right\}\right],
\Theta_{2\kappa,2\lambda}^\circlearrowleft = \sum_{r\geq 0}\frac{2r^3}{(N+1)(N+2)^2}\binom{N+2}{\kappa+1}\binom{N+2}{\kappa-r+1}\binom{N+2}{\kappa+r+1}=\frac{\binom{N+1}{\kappa}\binom{N+1}{\kappa +1}\binom{N}{\kappa}}{(N+ 1)}
F^{-1}(Mt-a(t)) =y\left(t-\frac{a(t)}{M}\right)=y(t)+y'\left(t-\frac{\theta_t a(t)}{M}\right)\cdot \frac{-a(t)}{M}&=F^{-1}(Mt)-f\left(y\left(t-\frac{\theta_t a(t)}{M}\right)\right)\cdot a(t).
\left((\Phi_j^*f)^\Omega_a(z)\right)^r\leq c\sum_{k=j}^\infty2^{(j-k)L_\Psi r}2^{kn}\int_\Omega\frac{|(\Phi_k\ast f)(y)|^r}{(1+2^j|y-z|)^{ar}}dy
Shew[X]=\frac{2(1-\kappa^{2})(144\kappa^{8}+23\kappa^{6}+27\kappa^{4}-6\kappa^{2}+1)}{\beta^{3}\sigma_{\kappa}^{3}(4\kappa^{2}-1)^{3}(144\kappa^{4}-25\kappa^{2}+1)}for0\le\kappa<1/4
\lim_{( p_{1}+ p_{2})^{2}\rightarrow m_{b_{n}}^{2}}Z_{n}( \theta_{1}, \theta_{2}) \equiv \lim_{\theta_{1 2}\rightarrow \theta + \theta_{1 2}^{b_{n}}}Z_{n}( \theta_{1}, \theta_{2}) = Z_{n}( \theta ) ~ .
- \chi^{A}\epsilon_{B}\{Q_{A}, Q^{B}\}\, \equiv \, [ \delta_{\chi}, \delta_{\epsilon}] \, = \, \frac{i}{1 6}( \chi^{\top}C \Gamma_{M N}\epsilon ) J^{M N}+ \frac{i}{1 6}( \chi^{\top}C \Gamma_{M_{1}\ldots M_{6}}\epsilon ) J^{M_{1}\ldots M_{6}}\; ,
\langle M',a^{m}\rangle = \langle a^{m}, b^{m+1}\rangle=\langle b^{m+1}, M'\rangle =\langle a^{m+1}\rangle^{\perp}\langle M,b^{m}\rangle = \langle b^{m}, a^{m}\rangle=\langle a^{m}, M\rangle =\langle b^{m+1}\rangle^{\perp}.
x_1^{p^2}=x_2^{p^2}=x_3^{p^2}=x_4^{p^2}=x_5^p=1, & [x_1,x_2]=x_1^p, \, [x_1,x_3]=x_3^p, \, [x_1,x_4]=1,\, [x_1,x_5]=x_1^p,\,[x_2,x_3]=x_2^p, &[x_2,x_4]=1, \, [x_2,x_5]=x_4^p, \, [x_3,x_4]=1, \, [x_3,x_5]=x_4^p,\,[x_4,x_5]=1.
\begin{array}{l}{T_{0}| \Omega \rangle = \frac{1}{2 ( k + \check{g})}S_{a 0}^{a}| \Omega \rangle = \frac{p}{2 ( k + \check{g})}| \Omega \rangle}\\{\}\\{J_{0}| \Omega \rangle = \frac{1}{( k + \check{g})}S_{a 0}^{a}| \Omega \rangle = \frac{p}{( k + \check{g})}| \Omega \rangle}\\ \end{array}
\chi^1 + \mathrm{span}_{C^\infty(M)}\{\chi^j- \chi^1, \, 2\leq j\leq n\}= \left\{\sum_{k=1}^n f^k \chi^k \, : \, f^k \in C^\infty(M), \, \sum_{k=1}^n f^k \equiv 1 \right\}.
A=A1=A1(t)=(A1_{ij})_{i,j=0}^{\infty}=\begin{pmatrix}0 0 0 0 0 0 \dots \vdots \vdots \vdots \vdots \vdots \vdots \dots 0_{r 0}0 0 0 0 0 \dots a_0\cdots a_r 0 0 0 0 0 \dots 0 a_1\cdots a_{r+1}0 0 0 0 \dots \vdots \vdots \ddots \vdots \vdots \vdots \dots \end{pmatrix}.
a(y) = \psi(y) y^{-\frac{1}{3}}= g \Bigl( \frac{m^2 y}{N}\Bigr) \widehat{k^*}\Bigl( \frac{M T c}{2 \pi^2 \sqrt{y p}}\Bigr) e \Bigl( - \frac{T^2 c}{4 \pi^2 \sqrt{y p}}\Bigr) y^{-\frac{13}{12}}.
\phi_{1}'(\kappa;u)=\kappa \phi_{1}(\kappa;u) -\frac{1}{2}\int_{\mathcal{C}_{<}}\frac{dw}{2\pi i}\sqrt{2w}e^{\frac{1}{3}(w-\kappa)^3-u(w-\kappa)}, \phi_{1}'''(\kappa;u)=\kappa \phi_{1}''(\kappa;u) -\frac{1}{2}\int_{\mathcal{C}_{<}}\frac{dw}{2\pi i}\sqrt{2w}(w-\kappa)^2 e^{\frac{1}{3}(w-\kappa)^3-u(w-\kappa)}.
\left\Vert x-x_{k}\right\Vert_{A^{T}A}^{2}-\left\Vert x-x_{k+1}\right\Vert_{A^{T}A}^{2}= \Vert x_{k+1}-x_{k}\Vert_{A^{T}A}^2 + 2\left(x_{k+1}-x_{k}\right)^{T}A^{T}A\left(x-x_{k+1}\right).
\sum_{v \in A_r}\deg^-(v) = O(n^{pA_1}\log^2 n) + \sum_{k=2}^{k_0}O((n/(k \omega \log^2 n) )^{pA_1}\log^2 n) \cdot O(\log^2 n) &= O(n^{pA_1}\log^2 n) + O((n/(\omega \log^2 n) )^{pA_1}\log^4 n) \cdot O( k_0^{1-pA_1}) = O(n \log^2 n / \omega),
\begin{array}{ccc}X_{1}(b_{1}) = \nu\lambda_{1}-\displaystyle\frac{c\nu (b_{2})^{2}}{\lambda_{2}-\lambda_{1}}, X_{2}(b_{2}) = \nu\lambda_{2}+\displaystyle\frac{c\nu (b_{1})^{2}}{\lambda_{2}-\lambda_{1}}. \end{array}
\ell(x',s_\alpha u\beta) = \ell(x,u\beta) + \ell(s_\alpha, -u\beta) = \begin{cases}\ell(x,u\beta), u\beta \neq \pm \alpha, -\ell(x,\alpha)+1>0, u\beta = -\alpha, \ell(x,\alpha)-1<0, u\beta = \alpha.\end{cases}