harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\eta_i \ = \ \int_{-T}^0 \eta(x)e_i(x) dx \ = \ \eta(0)\tilde e_i(0) - \int_{[-T,0]}\tilde e_i(x)d^-\eta(x) &= \ \int_{[-T,0]}(\tilde e_i(0) - \tilde e_i(x)) d^- \eta(x),
R_3 =\frac{1}{n}\sum_{j=1}^m\sum_{k=1}^m\sum_{l=1}^{m}\sum_{t=1}^{m}\frac{1}{\sqrt{p_jp_kp_lp_t}}\mathbb{E}\tilde{I}_j(1)I_k(1)\mathbb{E}I_l(1)I_t(1)\frac{\partial^4g}{\partial s_j\partial s_k\partial s_l\partial s_t}(\boldsymbol{\xi}).
\tilde \nabla_c \alpha_{ab}= \tilde \nabla_c \bar W_{aLbL}= \bar W(\tilde{\sigma}_{ca}\underline{L}, L , e_b, L) + \bar W(e_a, e_c, e_b, L)+\bar W(e_a, L, \tilde{\sigma}_{cb}\underline L, L)+ \bar W(e_a, L, e_b, e_c) = \tilde{\sigma}_{ca}\beta_b+\tilde{\sigma}_{cb}\beta_a+\bar W_{acbL}+\bar W_{bcaL}.
(m+\eta_1-1)\gamma(\eta_1-1) =\frac{(m+\eta_1-1)\theta^{\eta_1-1}\Gamma(m+\eta_1-1)}{z_\theta (\eta_1-1)!\Gamma(m)}&=\frac{\theta^{\eta_1-1}}{z_\theta\Gamma(m)(\eta_1-1)!}\Gamma(m+\eta_1) &=\frac{1}{\theta}\eta_1\gamma(\eta_1).
i\,\Gamma_{0,0}^{\,\infty,\,\alpha_{0}}(0):=-c_{1}^{\,R}\quad\mbox{and}\quad\Gamma_{l,2l}^{\,\infty,\,\alpha_{0}}(0)=0\,,\,\Gamma_{l,2l-1}^{\,\infty,\,\alpha_{0}}(0)=0\,,\,\partial_{\mu}\,\partial_{\nu}\,\Gamma_{l,2l-1}^{\,\infty,\,\alpha_{0}}(0)=0\quad\mbox{for}\quad l>0\ ,
A := \{(1-\epsilon)\|u'+v'\|^2 \le \|f(u'+v')\|^2 \le (1+\epsilon)\|u'+v'\|^2\}B := \{(1-\epsilon)\|u'-v'\|^2 \le \|f(u'-v')\|^2 \le (1+\epsilon)\|u'-v'\|^2\}
f_{1,3,1}(q^2,q^2;q^2) + q^3f_{1,3,1}(q^6,q^6;q^2) &=\frac{J_{4,8}J_{16,32}j(q^2;q^{16})j(q^{10};q^{16})J_{32}J_{1,4}}{j(-q^6;q^{16})j(-q^{14};q^{16})J_{16}^2}.
H = t_s \sum_{i=0}^{s}\frac{a^{-i}q^{i(2s-d)}(A-\theta_s I )( A-\theta_{s-1}I ) \cdots (A-\theta_{s-i+1}I)}{(q^2;q^2)_i}, H^{-1}= t^{-1}_s \sum_{i=0}^{s}\frac{a^{i}q^{i(d-2s)}(A-\theta_s I )( A-\theta_{s-1}I ) \cdots (A-\theta_{s-i+1}I)}{(q^{-2};q^{-2})_i}.
[\begin{matrix}T_{1}^{(e)}&T_{2}^{(e)}&T_{3}^{(e)}\end{matrix}]=[\begin{matrix}e_{1}&e_{2}&e_{3}\end{matrix}]\cdot[\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\\ \end{matrix}].
(\lambda-\frac{1}{2})\varphi=\partial_{n}D(\psi)|_{\partial\Omega}-(\lambda-\frac{1}{2})(T^{*}\varphi-\frac{1}{2}\varphi),(\lambda+\frac{1}{2})\varphi=-\partial_{n}D(\psi)|_{\partial\Omega}+(\lambda+\frac{1}{2})(T^{*}\varphi+\frac{1}{2}\varphi).
\mathbb{P}(\{T_1 = d_0 + 1\}\cap \mathcal{A}) \geq \mathbb{P}(\{T_1 = \check{d}_{\xi}+ 1\}\cap \{\check{d}_{\xi}= d_0\}\cap \mathcal{A}) = \mathbb{P}( \{\check{d}_{\xi}= d_0\}\cap \mathcal{A}) \geq \mathbb{P}( \mathcal{A}\cap \mathcal{B}) \geq 1 - 4\delta,
D_{\mathrm{H}}^{2}[ \Phi , \Phi^{\prime}] = \sum_{x}\sum_{\alpha = 0}^{3}\left[ ( ( \phi_{\alpha})_{x}- ( \phi_{\alpha}^{\prime})_{x})^{2}+ ( ( \dot{\phi}_{\alpha})_{x}- ( \dot{\phi}_{\alpha}^{\prime})_{x})^{2}\right] \, ,
\left(\partial_{0,\nu}\left(N(\mathrm{m}_{0})\begin{pmatrix}\frac{1}{2}0 0 \frac{1}{2}\end{pmatrix}+\partial_{0,\nu}^{-1}\begin{pmatrix}\frac{1}{2}0 0 \frac{1}{2}\end{pmatrix}\right)+\begin{pmatrix}0 \partial_{1}\partial_{1,c}0\end{pmatrix}\right)\begin{pmatrix}u v\end{pmatrix}=\begin{pmatrix}f g\end{pmatrix}.
\begin{cases}\partial_t\rho+\nabla\cdot(\rho u)=\alpha\Delta\rho \{\rm{in}}\Q_T, \rho(x,0)=\rho_{0,\delta}(x) \{\rm{in}}\ \Omega, \qquad\frac{\partial\rho}{\partial\nu}= 0 \{\rm{on}}\ \partial\Omega\times (0,T).\end{cases}
g_{1+\tau+\tau^3}(x)=\, (x-1)\sum_{0\le i_0,i_1,i_3\le 1}(-1)^{i_0+i_1+i_3}\frac{x^{r(i_0+i_1+i_3)}-1}{x^r-1}\cr \cdot\Bigl[\sum_{l=0}^{r-1}\bigl(i_0+1+(l-2)k\bigr)\bigl(i_1+1-(l-1)k\bigr)(i_3+1-lk)x^l\cr -(i_0+1-2k)(i_1+i_3+k+1)-(i_0+1-k)(i_3+1-k)x\Bigr]-1.
{\mathcal D}^{( 1 / 2 , 1 / 2 )}\otimes{\mathcal D}^{( 1 / 2 , 1 / 2 )}={\mathcal D}^{( 1 , 1 )}\oplus{\mathcal D}^{( 1 , 0 )}\oplus{\mathcal D}^{( 0 , 1 )}\oplus{\mathcal D}^{( 0 , 0 )}\, .
N^{(1)}_{(f,H_i),\leqslant k_i}(r) \geqslant\sum_{s\ne i,t}N^{(1)}_{(f,H_s),\leqslant k_s}(r)-N^{(1)}_{(f,H_t),\leqslant k_t}(r)-2(N(r,S_i)+N(r,S_t)) &\geqslant \sum_{s\ne i,t}N^{(1)}_{(f,H_s),\leqslant k_s}(r)-N^{(1)}_{(f,H_t),\leqslant k_t}(r)-2\sum_{u=1}^3\sum_{s=i,t}N_{(f^u,H_s),\leqslant k_s}(r).
\left( \beta x^{1-\alpha/\beta}D\right)^{n}\left[ x^{r/\beta}\psi\left(x+1\right) \right] =x^{\left( r-n\alpha\right) /\beta}\sum_{k=0}^{n}S\left( n,k;\alpha,\beta,r\right) \beta^{k}x^{k}\psi^{\left( k\right)}\left( x+1\right) ,
\int_{G / H}d x \sqrt g D_{A \zeta , q}^{( n ) *}( g_{x}^{- 1}) D_{A \xi , p}^{( n^{\prime})}( g_{x}^{- 1}) = V d_{n}^{- 1}d_{D}\delta_{\zeta \xi}\delta_{p q}\delta_{n n^{\prime}}.
c_n=2^{\frac{n}{2}}c_{\frac{n}{2}}=2^{\frac{n}{2}+\frac{n}{4}}c_{\frac{n}{4}}=\ldots=2^{\frac{n}{2}+\frac{n}{4}+\cdots+\frac{n}{2^t}}c_{\frac{n}{2^t}}.
{\cal M}= e^{2 \phi}\left( \begin{array}{c c}{{| \lambda |^{2}}}&{{a}}\\{{}}&{{}}\\{{a}}&{{1}}\\ \end{array}\right) \, , \hspace{1 c m}{\cal M}^{- 1}= e^{2 \phi}\left( \begin{array}{c c}{{1}}&{{- a}}\\{{}}&{{}}\\{{- a}}&{{| \lambda |^{2}}}\\ \end{array}\right) \, ,
\sum_{i=1}^N\widehat\Gamma_i\widehat\Sigma_{ii}^{-1}\bigl(\widehat BM_{z z}^{i j}\widehat B{}'-\widehat\Sigma_{z z}^{i j}\bigr)\widehat\Sigma_{j j}^{-1}I_{K+1}^{-}=0,
\phi_{t}\left( x,y\right) :=\min_{\omega\in\left\{\mathcal{H}_{x}^{t}\longleftrightarrow\mathcal{H}_{y}^{t}\right\}}\left\vert \left\{e^{\ast}\in\mathbf{S}_{\left\{x,y\right\}}\left( \omega\right) :e^{\ast}\subset\mathcal{S}_{\left\{x,y\right\}}^{t}\right\}\right\vert \ ,
\dim\mathcal{M}(L) = \dim \mathcal{M}(L/I)+( \dim L/\gamma_{2}(L)-1) \dim I - \dim\ker(\lambda_{c}) \cr \leq \frac{1}{2}(n-m-1) (n+m-2)-(\sum\limits_{i=2}^{c-1}n-m-i) +(n-m-1)-(n-m-c) \cr = \dim \mathcal{M}(L),
b_n (\hat{\varDelta}-a_n ) = \max_{u \in\mathcal{D}^{*}}b_n \bigl(L_n^{*}(u) -a_n\bigr) = \max_{u \in\mathcal{D}^{*}}g_n (u) , g_n (u) = \sum_{l=1}^k\bigl[u_l b_n (Z_{nl}-a_n ) +u'_l b_n (-W_{nl}-a_n ) \bigr] .
\partial_{t}^{j}\partial_{r}^{k}v_{2}(t,r) = \frac{\partial_{t}^{j}\partial_{r}^{k}\left(\frac{-b r}{t^{2}}\right)}{\log^{b}(t)}+ E_{\partial_{t}^{j}\partial_{r}^{k}v_{2}}(t,r)
C_{T,K}:= C(\|K\|_{\operatorname{CZ}_{\alpha}}+ \sup_{\varepsilon>0}\|T_{\varepsilon}(1,1)\|_{\operatorname{BMO}}+ \sup_{\varepsilon>0}\|T^{1*}_{\varepsilon}(1,1)\|_{\operatorname{BMO}}&+ \sup_{\varepsilon>0}\|T^{2*}_{\varepsilon}(1,1)\|_{\operatorname{BMO}}+ \sup_{\varepsilon>0}\|T_{\varepsilon}\|_{\operatorname{WBP}})
\int_{-\infty}^{\infty}e^{-\phi(x-b)}\left(V_q(x)-\mathbb P_x\left(X_{e(q)}> y\right)\right)dx &=\mathbb E\left[e^{\phi \underline{X}_{e(q)}}\right]\mathbb E\left[e^{\phi\overline{X}_{e(p+q)}}\right]\int_{0}^{\infty}F_0(x+y-b)e^{\phi x}dx, \ \Re(\phi)=0,
W(u_+ (z, \cdot \,), u_- (z, \cdot \,)) = [m_-(z,x_0) - m_+(z,x_0)] u_+ (z, x_0) u_- (z, x_0), m_{\pm}(z,x_0) = [p(x_0) u_{\pm}^{\prime}(z, x_0)] / u_{\pm}(z, x_0).
Z^i_t = Z^i_0 + W^{i, 1}_t -W^{i, 2}+ \mu^i \Hat{Z}^i - \sum_{j: j\neq i}\mu^{ij}\, \int_0^t v_{ij}(s) \, d{\Hat{Z}^j_s}, \int_0^t Z^i_s \, d{\Hat{Z}^i_s}=0,\forall \; t\geq 0,
\begin{array}{r c l}{{\hat{X}}}&{{=}}&{{X Y ( Y^{2}- 9 Z^{2})}}\\{{\hat{Y}}}&{{=}}&{{4 8^{1 / 3}( 3 Z^{2}+ Y^{2})}}\\{{\hat{Z}}}&{{=}}&{{3 X^{2}}}\\ \end{array}
\sum\limits_{j=1}^n1_{\left\{X_i\leq X_j\right\}}=\sum\limits_{j=1}^{k^*}1_{\left\{\mu+G(\xi_i)+\Delta\leq \mu+G(\xi_j)\right\}}+\sum\limits_{j={k^*}+1}^n1_{\left\{\mu+G(\xi_i)+\Delta\leq \mu+G(\xi_j)+\Delta\right\}}&=\sum\limits_{j=1}^n1_{\left\{G(\xi_i)\leq G(\xi_j)\right\}}-\sum\limits_{j=1}^{k^*}1_{\{G(\xi_i)\leq G(\xi_j)<G(\xi_i)+\Delta\}}.
\mathrm{Term}_k=\sum_{\alpha'+\alpha''=\alpha-1}(\mathrm{ad}x_i)^{\alpha'}(-\mathrm{ad}x_j)^{\alpha''}([t_{ki},t_{kj}])=\sum_{\gamma+\delta=\alpha-1}[(\mathrm{ad}x_i)^\gamma(t_{ki}),(-\mathrm{ad}x_j)^\delta(t_{kj})] &=\sum_{\gamma+\delta=\alpha-1}(-1)^\gamma[(\mathrm{ad}x_k)^\gamma(t_{ki}),(\mathrm{ad}x_k)^\delta(t_{kj})],
\vartheta_X(\xi) -\sum_{i=1}^{s-r}\left( \kappa(p_i) b_y + \kappa(p_i) \right) \langle w_i , (\nu,\ell) \rangle = \sum_{j=1}^a (-1-c_j) \langle u_j,\nu \rangle - \sum_{j=a+1}^r \langle u_j,\nu \rangle - \sum_{i=1}^{s-r}(1+d_i) \langle w_i , (\nu,\ell) \rangle.
f_n = B_{n-2,1}( 2! a_1, 3! a_2, \ldots) + B_{n-2,2}(2! a_1, 3! a_2, \ldots ), c_{n-2}=B_{n,2}\left( 1, 2! a_1, \ldots \right) , g_n := n f_n - c_{n-2}= \frac{n(n-2)!}{2}\sum_{k=0}^{n-2}a_{n-k-2}a_k (n-k-2) k .
\kappa_m \left( \sum_{i=1}^n c_i Y_\rho(t_i) \right)= \frac{\left[(\rho+1)^{-1}+(2\sqrt{\rho})^{-1}\right]^m+\left[(\rho+1)^{-1}-(2\sqrt{\rho})^{-1}\right]^m}{\left[(\rho+1)^{-2}+(4\rho)^{-1}\right]^{m/2}}&\times\left(\sum_{i=1}^n c_it_i\right)^m \frac{(m-1)!}{2}.
h_v (z-b,D(b,z)) = h_{E_1 ,v}(f_1 (z)-\infty )-h_{E_1 ,v}(f_1 (b)-\infty )-h_{E_2 ,v}(f_2 (z)-\infty ) &+h_{E_2 ,v}(f_2 (b)-\infty )+ 2\chi (x(b))-2\chi (x(z)).
\left( \partial_\mu 1\otimes 1 +\frac{1}{4}\epsilon_{\alpha \beta \delta}\omega^{\alpha \beta}_{\mu}1\otimes \gamma^{\delta}+\frac{i}{2\ell}A^{I}_{\mu}(-\frac{1}{\ell}N^c_I \mp K^c_I) \sigma_c \otimes 1 \pm \frac{1}{2\ell}e_{\mu\alpha}1\otimes \gamma^{\alpha}\right) \varepsilon_r = 0 \ ,
d\mu_{N, \beta}^{V;\boldsymbol{B}}(\lambda)=\frac{1}{Z_{N, \beta}^{V;\boldsymbol{B}}}\prod_{i=1}^{N}d\lambda_{i}e^{-\frac{N\beta}{2}V(\lambda_{i})}\mathbf{1}_{\boldsymbol{B}}(\lambda_{i})\prod_{1\leq i<j\leq N}\left|\lambda_{i}-\lambda_{j}\right|^{\beta},
\chi_{p_0,q_0}= \Big[\prod_{j\in p_0}\chi_{1,j}^{0}\Big] \cdot1\cdot \Big[\prod_{j\in q_0}\chi_{2,j}^{0}\Big] &=\sum_{q_1\cup p_1=p_0, q_1\cap p_1=\emptyset}\Big[\prod_{j\in p_0}\chi_{1,j}^{0}\Big] \Big[\prod_{j\in p_1}\chi_{1,j}^{1}\Big] \Big[\prod_{j\in q_1}\chi_{2,j}^{1}\Big] \Big[\prod_{j\in q_0}\chi_{2,j}^{0}\Big]\,.
W_L^{}= E_y -E_{\infty}= -\aleph_{\alpha,L}\frac{\alpha}{L^4}+ O(\alpha^4 \log(\alpha^{-1})) &+ O\big( \alpha^2 L e^{-L/2}\big) + O \Big( \frac{\alpha^2}{L^5}\Big) + O\Big( \frac{\alpha^3}{L^2}\log(\alpha^{-1})\Big).
\mathbb P (v_{n+1}v_i) = \lambda \frac{(v_i)}{2n-2}+ (1-\lambda)\frac{1}{n}&= \frac{n\lambda (v_i) + (2n-2)(1-\lambda)}{(2n-2)n}&\propto \deg(v_i) + \frac{2(1-\lambda)}{\lambda}\cdot \frac{n-1}{n}.
\mathbf g_p(h\nu_{\gamma}\beta_1\nu_{\delta}\varpi_p) = \begin{cases}\xi_1(p)^{-2}\xi_2(\gamma t)|p^{-1}|_p^{1/2}= p^{1/2}\xi_1(p)^{-2}\underline{\chi}_p(\gamma)\underline{\chi}_p(h) \gamma\delta \equiv 1 \pmod p, 0 . \end{cases}
\tau_{Y}(u_{1},u_{2}) = L_{\operatorname{hor}^{\gamma}(u_{1})}\beta_{Y}(u_{2}) -L_{\operatorname{hor}^{\gamma}(u_{2})}\beta_{Y}(u_{1}) - \beta_{Y}([u_{1},u_{2}]) + L_{Y}\sigma(u_{1},u_{2}).
L_{u}(\beta_2)= u \beta_2^{q^2}+ (u+1)\beta_2^q+ (u^2+u) \beta_2 = \beta_2\left [ u \beta_2^{q^2-1}+ (u+1)\beta_2^{q-1}+ (u^2+u) \right ] = \beta_2\left [ u \left ( u^2 \right )^{q+1}+ (u+1)u^2+ (u^2+u) \right ] = \beta_2\left [ u (u+1)^2+ (u+1)u^2+ (u^2+u) \right ] = 0.
\int_{v\in \S_{a}^{m-2}}\|\nabla F_{v}\|^2 (x) \, dV_{a}=\frac{\mathrm{Vol}(\S^{m-2})}{m-1}\Big[ m\| \nabla F_{a}\|^2 (x)+ \mathrm{trace}(A_{Q_{1}^{x}}) \Big]
{\bf{H}}= \sum\limits_{k = 1}^K{\sum\limits_{{l}= 1}^{{L_{{\rm R}}}}{\sum\limits_{{j}= 0}^{{L_{{\rm T}}}}{{\xi_{k,{l},{j}}}{{\bf{a}}_{\rm{R}}}\left({\Phi_{{k},{l}}^{\rm{A}}}\right){\bf{a}}_{\rm{T}}^H\left({\Theta_{k,{j}}^{\rm{D}}}\right)}}}={{\bf{R}}}{\bf{\Xi T}}^H,
\Phi(\omega,-x,z)\tilde{\Phi}(-\omega,-x,-y)=\frac{1}{\sqrt{2\pi(z-y)^3}}\mathbb{E}[ \left(-\int_{y}^{z}\phi''(u)\textbf{e}(u)du \right)|\textbf{e}(\omega)=x]\times \frac{\mathbb{P}[\textbf{e}(\omega)\in dx]}{dx}
F^{\{i\}}= \sum_{k=1}^{d}\frac{\partial e_{k}}{\partial x_{i}}S_{k}(x_{1},\dots,x_{n})=\sum_{k=1}^{d}\sum_{r=0}^{k-1}(-1)^{r}x_{i}^{r}e_{k-1-r}S_{k}=\sum_{r=0}^{d-1}x_{i}^{r}\left( \sum_{k=r+1}^{d}(-1)^{r}e_{k-1-r}S_{k}\right).
\sum_{\mathbf{v}\in[n]^{I_s}}\Bigg(\sum_j b_{v_{i_s}j}^{(e_s)}\Bigg)^{q_s}\prod_{e\in E(G_{I_s})}b_{v(e)}^{(e)}= \qquad \sum_{\mathbf{v}\in[n]^{I_s'}}\Bigg(\sum_j b_{v_{i_s}j}^{(e_s)}\Bigg)^{q_s}\Bigg(\sum_j b_{v_{i'}j}^{(e')}\Bigg) \prod_{e\in E(G_{I_s'})}b_{v(e)}^{(e)},
q_{k}= q_{0 \left( k \right)}- \frac \nu{2 W \left( q_{0 \left( k \right)}\right)}, \; \psi_{+ \left( k \right)}= \lambda_{+}W^{k}\left( q_{0 \left( k \right)}\right) , \; \psi_{- \left( k \right)}= \lambda_{-}W^{- k}\left( q_{0 \left( k \right)}\right) ,
\partial_t V^{\sigma}(m_t) =\int_{\mathbb{R}^d}a(m_t,x) \partial_t m_t(x) dx = \int_{\mathbb{R}^d}a(m_t,x) \div \left( \nabla a(m_t,x)m_t(x)\right) dx &= - \int_{\mathbb{R}^d}\left| \nabla a(m_t,x)\right|^2 m_t(dx) \,.
\mathcal{L}(\mathcal{X}) :=\lim_{t\to \infty}\frac{1}{t}\mathbb{E}\bigl(\log \Vert \mathbf{X}_{i_1}\cdots \mathbf{X}_{i_t}\Vert \bigr) =\lim_{t\to \infty}\frac{1}{t}\int_{\underline{i}\in \Sigma}\log\Vert \mathbf{X}_{i_1}\cdots \mathbf{X}_{i_t}\Vert d\mu(\underline{i}) \,.
= \int d^{n}x \left\{\frac{1}{2}F_{\alpha \beta}F^{\alpha \beta}+ \frac{1}{2}G_{\alpha \beta}G^{\alpha \beta}+ 2 D_{\alpha}\phi^{\dagger}D^{\alpha}\phi + 2 ( m^{2}- \phi^{\dagger}\phi )^{2}\right\}
(\mathfrak{F}, \mathfrak{F}_1,\dots, \mathfrak{F}_m)=\bigg(\Big(\sum_j (F^j)^{t}\Big)^\frac1{t},\Big(\sum_j (F_1^j)^{t_1}\Big)^\frac1{t_1},\dots,\Big(\sum_j (F_m^j)^{t_m}\Big)^\frac1{t_m}\bigg)
d s^{2}= \left({\frac{r_{0}^{2}}{r^{2}}}- 1 + \Delta \right)^{- 1}d u \left( d v + \left({\frac{p}{r^{2}}}+ k \right) d u + \left( 2 r_{0}^{2}{\frac{\dot{f}_{i}}{r^{2}}}+ 2 a_{i}\right) d y^{i}\right) + d y^{2}+ H_{5}d x^{2}.
\Theta(z,\tau) = \prod_{k \geq 1}\left(1 - e^{2 k \pi i \tau}\right) \left(1 + e^{(2k-1) \pi i \tau}e^{2\pi i z}\right) \left(1 + e^{(2k-1) \pi i \tau}e^{-2\pi i z}\right).
\begin{aligned}\alpha \left( \tau+t \right)^{-\alpha-1}\left[ \log\left( t+\tau \right) \right]^{\frac{\beta}{m-1}}\ge C^{p-1}\left( t+\tau \right)^{-\alpha p}\left[ \log(\tau+t) \right]^{\frac{\beta p}{m-1}}t>0 \, ,\end{aligned}
\Pi_{\mu}^{\lambda}\Theta_{\lambda}^{\nu}= 0 \; , \qquad \Pi_{\mu}^{\nu}+ \Theta_{\mu}^{\nu}= \delta_{\mu}^{\nu}\left( T_{1}+ T_{2}+ T_{3}+ T_{4}\right) + T_{5}+ T_{6}+ T_{7}+ T_{8}
I= E_\lambda(x) R Q(x),Q(x):= \int_0^x \varphi_1(t)\varphi_2(t)\,dt, II:= -\int_0^xE_{\lambda}(x-x_1)R \Phi(x_1) E_\lambda(x_1)\Phi(0)\,dx_1 = -M_1(x,\lambda)\Phi(0)
\mathrm{\mathrm{}^* \!}F_{\mu \nu}( x ) \stackrel{?}{=}\partial_{\nu}\tilde{A}_{\mu}( x ) - \partial_{\mu}\tilde{A}_{\nu}( x ) + i \tilde{g}[ \tilde{A}_{\mu}( x ) , \tilde{A}_{\nu}( x ) ] ,
\stackrel{\wedge}{h}= \frac{1}{2}\{A^{-},\,A^{+}\}_{q}=\frac{1}{2}\left( q\,A^{-}A^{+}+ q^{-1}A^{+}A^{-}\right) =2\left( \stackrel{\wedge}{T}^{2}- \cosh\frac{\omega}{4}\right)
\int_{\mathfrak m}g_k(\alpha;N)^h\mathcal G(\alpha)^2e(-n\alpha)\,{\rm d}\alpha =\sum_{N<p_1,p_2\le 2N}\int_{\mathfrak m}\mathcal G(\alpha)^2e((p_1^k+p_2^k-n)\alpha)\,{\rm d}\alpha ,
|\mathbb{E}\Delta_{3}(s)|^{2}\ll\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{k_{i}}\sum_{\mathbf{u}\in I_{ij}}\Vert{\mathbb{E}}({\widetilde{X}}_{\mathbf{u}}|\mathcal{F}_{i,j-1}^{X})\Vert_{2}^{2}\leq2\left(L(A)+\max_{1\leq j\leq i\leq n}\Vert{\mathbb{E}}({X}_{ij}|\mathcal{G}_{i,j-q})\Vert_{2}^{2}\right) & \ll\left( L(A)+\eta_{q}^{2}\right) \,.
\Vert Df(y)[h]-Df(x)[h]\Vert = \left\Vert \int_{0}^1 \frac{d}{dt}Df(x+t(y-x))[h]dt\right\Vert &= \left\Vert \int_{0}^1 Hf(x+t(y-x))[h,y-x]dt\right\Vert &\leq \int_0^1 \Vert Hf(x+t(y-x))[h,y-x]\Vert dt &\leq M\Vert h\Vert \Vert y-x\Vert.
U_{i,m,n}=\left(1-\frac{1}{q_i}\right)^{-1}\left(U_i^{m}(U_i^*)^{n}-\frac{1}{q_i}U_i^*U_i^{m}(U_i^*)^{n}U_i\right) &= \begin{cases}\left(1-\frac{1}{q_i}\right)^{-1}\left(U_i^{m}(U_i^*)^{n}-\frac{1}{q_i}U_i^{m-1}(U_i^*)^{n-1}\right) m,n\geq 1 U_i^{m}(U_i^*)^{n}\min\{m,n\}=0,\end{cases}
K \equiv - 6 M^{3}\int_{- y_{c}}^{y_{c}}W^{4}R d y = \frac{2 4}k M^{3}\widetilde{\lambda}\left( 1 - e^{- 2 k y_{c}}\right) - 3 6 M^{3}k \left( 1 - e^{- 4 k y_{c}}\right) \; .
\| \partial_d(I_{h,T}u-u)\|_{0,T_{non}^s}\leqslant \frac{C}{h}\left(\sum_{i\in\mathcal{I}^{s'}\cup\mathcal{I}^{int}}\left(\| \mathcal{E}_i\|_{0,T_{non}^s}+\|\mathcal{F}_i\|_{0,T_{non}^s}\right)+\sum_{i\in \mathcal{I}}\| \mathcal{R}_i\|_{0,T_{non}^s}\right),~d = 1, 2.
\vec{\Gamma}_{\mu}^{\prime}= \vec{e}_{3}( \vec{e}_{3}\vec{\Omega}_{\mu}) = ( \cos \theta - 1 ) \vec{e}_{3}[ \vec{e}_{3}\cdot \vec{e}_{\phi}\wedge \partial_{\mu}\vec{e}_{\phi}] = ( \cos \theta - 1 ) \partial_{\mu}\phi \vec{e}_{3}.
\|Dh\|_{L^2}^2 =\frac{1}{3}\int_M (\nabla_ih_{jk}+\nabla_jh_{ki}+\nabla_kh_{ij})^2 dV_g=\|\nabla h\|_{L^2}^2+2\int_M \nabla_ih_{jk}\nabla_jh_{ki}dV_g.
\frac{d}{d \tau}\left( \frac{\dot{x}_{i}^{\mu}( \tau )}{\sqrt{\dot{x}_{i}^{2}( \tau )}}\right) = ( - 1 )^{i + 1}K_{i}( \tau ) \acute{x}_{i}^{\mu}( \tau ) ( i = 1 , 2 ) ,
H^{(i)}(t) \le C_1 \left( \lVert a^*_0\rVert_{L^\infty_H L^p_z}+ K^{(i)}(t) H^{(i)}(t)+ R^* t^{1/2}H^{(i)}(t)+ R^* t^{1/2}K^{(i)}(t)\right),
\left( \begin{array}{l}f^{1/2}_\pm (f^{-1/2}_\pm)^* \end{array}\right)(\theta,\phi) =v\left( \begin{array}{l}D^{1/2}_{1/2,1/2}D^{1/2}_{-1/2,1/2}\end{array}\right)(\theta,\phi,\mp \phi) =v e^{\pm i \phi/2}\left( \begin{array}{l}e^{- i \phi/2}\cos (\theta/2) e^{i \phi/2}\sin (\theta/2) \end{array}\right).
\eta ( x , t ) = \mathrm{R e}\left( \frac{- 3 A_{0}^{2}}{2 i q ( 2 - q^{2}- 3 i q )}\exp i ( \omega t - q x ) \int_{- \infty}^{\infty}\phi_{K}( \xi ) \eta_{D}^{2}( \xi ) \eta_{q}( \xi ) \mathrm{d}\xi \right)
I_{1}( \mu ) - I_{1}( \mu_{0}) ={\frac{1}{2}}( \mu^{2}- \mu_{0}^{2}) I_{0}( \mu_{0}) - \frac{\mu_{0}^{3}}{8 \pi}\left[{\frac{1}{3}}\left( \sqrt{\frac{\mu^{2}}{\mu_{0}^{2}}}- 1 \right)^{2}\left( 2 \sqrt{\frac{\mu^{2}}{\mu_{0}^{2}}- 1}\right) \right] ,
\|D_{\hbar^{-1/2}}f\|_{L^p}=\hbar^{d(1/2-1/p)/2}\|f\|_{L^p}\quad{\rm and}\|D_{\hbar^{-1/2}}f\|_{L^p_k}=\hbar^{d(1/2-1/p)/2}\|f\|_{\tilde{L}^p_k}.
\Big( \sum_j u_j(t) \Big)^2 = \big( u_{j_1}(t) + \cdots + u_{j_m}(t)\big)^2\lesssim \big( \max_\ell u_{j_\ell}(t) \big)^2 \leq \sum_{\ell}u_{j_\ell}(t)^2 \leq \sum_j u_j(t)^2.
\big\| E_{\mathbb{P}^d}g \big\|_{L^p( Q_{N_1^2})}= \big\| e^{i \frac{t}{N_1^2}\Delta}\phi(\frac{x}{N_1}) \big\|_{L^p( Q_{N_1^2})}= N_1^{\frac{2}{p}}N_1^{\frac{d}{p}}\big\| e^{i t \Delta}\phi \big\|_{L^p( [0,1]\times [0, N_1]^d )}
\begin{cases}\rho_t=R^I_0\wedge b+2\int_0^t\sqrt{\rho_s}dW_s+(|I|-1)\Big(2-\frac{\chi|I|}{4\pi N}\Big)t-\Lambda_t \forall t\geq 0,\;\rho_t\leq b (\Lambda_s)_{s\geq 0}\mbox{is an adapted increasing process s.t.}\Lambda_0=0\mbox{and}\int_0^t(b-\rho_s)d\Lambda_s=0.\end{cases}.
\begin{array}{l l}\min z begin{align*}1ex] \textnormal{s.t.}\sum_{v\in V^k}\tilde{c}_j^v \alpha_v \leq z,j=1,\ldots,m begin{align*}1.2ex] \sum_{v\in V^k}\alpha_v=1 begin{align*}1.2ex] \alpha_v\geq 0, v\in V^k,\end{array}
\sigma_{\partial}(A \cdot B) = \sigma_{\partial}((A_x \cdot B_x)_{x \in X}) = (\sigma_{\partial}(A_x \cdot B_x))_{x \in X}&= (\sigma_{\partial}(A_x) \cdot \sigma_{\partial}(B_x))_{x \in X}= \sigma_{\partial}(A) \cdot \sigma_{\partial}(B).
\Delta_2\le2 (X\cdot Y) (|X|^{\gamma/2}-|Y|^{\gamma/2})^2 \leq 2 |X\|Y| (|X|^{\gamma/2}-|Y|^{\gamma/2})^2=2 (|X\|Y|)^{1+\gamma}\bigl(|X|^{|\gamma|/2}- |Y|^{|\gamma|/2}\bigr)^2.
d_{A_\infty}\dot \eta + [ \dot A_\infty \wedge \eta] = d_{A_\infty}[ \eta \wedge \gamma_\infty] + [\dot A_\infty \wedge \eta ] &= [d_{A_\infty}\eta \wedge \gamma_\infty ] - [ \eta \wedge d_{A_\infty}\gamma_\infty] + [\dot A_\infty \wedge \eta] = 0
\Lambda^{-m_i}\cdot \tilde{\pi}\left( \Theta_2\left((g_1^2,g_2^2)^{m_i}w_i \right) \right) = \Lambda^{-m_i}\cdot \Lambda^{m_i}\cdot \tilde{\pi}(\Theta_2(w_i)) = \tilde{\pi}\left( \Theta_2(w_i) \right) \in \mathcal{L}(N(T)).
T:CF(H^{(k)},J)\longrightarrow CF(H^{(k)},J^{+\frac{1}{k}}) T(z_-)=\sum_{z_+\in \widetilde{P}(H^{(k)})}\sharp \mathcal{N}(z_-,z_+,[0,1],\rho,H^{(k)},J)\cdot z_+
\big( 1 \boxtimes c_1(N_{A/B_2})\big) \cdot (\mathrm{id}\boxtimes \pi^{\ast}\pi_{\ast})\Delta_D = \big( 1 \boxtimes c_1(N_{A/B_2})\big) \cdot \Delta_A & = \big( c_1(N_{A/B_2}) \boxtimes 1 \big) \cdot \Delta_A & = - \big( c_1(N_{A/B_1}) \boxtimes 1 \big) \cdot \Delta_A.
\tilde{y}=\frac{1}{\tilde{\tau}_{w}}\int_{0}^{\tilde{u}}\tilde{\mu}d\tilde{u},\tilde{\tau}_{w}=\int_{0}^{1}\tilde{\mu}d\tilde{u},q_{w}=-\frac{1}{Pr}\tau_{w}(\frac{dh}{du})_{w},
\begin{array}{cccccc}0 x_1 0 0 \cdots 0 0 0 x_2 0 \cdots 0 0 0 0 x_3 \cdots 0 \vdots \vdots \vdots \vdots \vdots \vdots 0 0 0 0 \cdots x_{p-1}x_0 0 0 0 \cdots 0\end{array}
a_k=\begin{cases}t(2r+1)!! (k=0), (t(2r+1) +s)(2r-1)!! (k=1), \left(t \binom{2(r+1)-k}{2(r+1-k)}+s \binom{2r+1-k}{2(r+1-k)}\right) (2(r+1-k)-1)!! (2 \leq k \leq r+1). \end{cases}
\dim \Pi'= \prod_{\alpha >0}\frac{\langle \mu', \alpha\rangle}{\langle\rho,\alpha\rangle}=\frac{\prod_{k=2}^{l'}(\mu'_1-\mu'_k) \dots \prod_{k=l+1}^{l'}(\mu'_{l}-\mu'_k)}{\prod_{k=2}^{l'}(k-1) \dots \prod_{k=l+1}^{l'}(k-l)}\,,
{\cal G}=-\frac{(2N-1)V\beta^2}{16\pi G N}\left [\frac{\Lambda e^{2b\phi_0}}{4N-1}r_+^{4N-1}+\frac{16\pi^2 Q^2 e^{2a\phi_0}}{V^2 \beta^4}\frac{1}{r_+}\right],
P^{y^{\prime}z^{\prime}}T^{A , b}( P^{y^{\prime}z^{\prime}})^{- 1}= T^{A , b}, P^{y^{\prime}z^{\prime}}T^{A , \hat{b}}( P^{y^{\prime}z^{\prime}})^{- 1}= - T^{A , \hat{b}}. \,
\left| \, \begin{matrix}{{a_{0}\hfill}}&{{a_{- 1}\hfill}}&{{a_{- 5}}}\\{{a_{1}\hfill}}&{{a_{0}\hfill}}&{{a_{- 4}}}\\{{a_{5}\hfill}}&{{a_{4}\hfill}}&{{a_{0}}}\\ \end{matrix}\right| .
\lim_{t\to 0}t^{1-\frac1p}\|\omega_\theta(t)\|_{L^p(\Omega)}\, =\, 0~, \hbox{for}1 < p \le \infty~, \lim_{t\to \infty}t^{1-\frac1p}\|\omega_\theta(t)\|_{L^p(\Omega)}\, =\, 0~, \hbox{for}1 \le p \le \infty~.
a_{( N )}^{[ 2 ] e f f}( P_{N}) = z_{S}^{2}[ a_{( S )}^{[ 2 ] e f f}( \tilde{P}_{N}) +{\frac{e Q}{2 \pi \mu c}}i \partial_{( S )}\ln ({\frac{z_{S}}{\bar{z}_{S}}}) ]
(\int_{\mathbb{R}}\phi(s)dB(s))\diamond(\int_{\mathbb{R}}\psi(s)dB(s))=(\int_{\mathbb{R}}\phi(s)dB(s)).(\int_{\mathbb{R}}\psi(s)dB(s))-\int_{\mathbb{R}}\phi(s)\psi(s)ds
\sum_{x,x'=0}^n \Big(\delta_{x,x'}-M_{x,x'}(m) \Big)\tilde V_{x'}(m) \tilde V^{\star}_{x}(m) &={- \tilde V_{0}(m) \sum_{x=0}^n M_{x,0}(m) \tilde V^{\star}_{x}(m)}+ \sum_{x=0}^n\tilde v_x(m) \tilde V^{\star}_{x}(m).
|\frac{d}{dt}f_{i,\lambda}(u(t)) | = |\left\langle \nabla f_{i,\lambda}(u(t)), \dot{u}(t) \right\rangle | & \leq \|\left(\partial f_{i}(u(t))\right)^{0}\| \|\dot{u}(t) \| \leq C \|\dot{u}(t) \|,
\begin{array}{l}g_0(x)=(x-1)^{t_0}+u(x-1)^{k_1}p_1(x)+u^2(x-1)^{k_2}p_2(x)+u^3(x-1)^{k_3}p_3(x), g_1(x)=u(x-1)^{t_1}+u^2(x-1)^{k_4}p_4(x)+u^3(x-1)^{k_5}p_5(x), g_2(x)=u^2(x-1)^{t_2}+u^3(x-1)^{k_6}p_6(x), g_3(x)=u^3(x-1)^{t_3},\end{array}
=\frac{1}{1-p^2x}\sum_{r=1}^{\infty}x^r\bigg(\frac{(px)^{r+1}(1-(px)^r)}{1-px}- (p^2x)^{2r+1}\bigg(\frac{p^{-r-1}(1-p^{-r})}{1-p^{-1}}\bigg)\bigg)
\begin{aligned}R_k(t)-R(t)= \int_0^t(u_k-u)|_{x=0}d\tau = \frac{1}{N}\int_0^t\int_0^N(u_k-u)dxd\tau+\frac{1}{N}\int_0^t\int_0^N(x-N)(\partial_xu_k-\partial_xu)dxd\tau \rightarrow 0,\;k\rightarrow+\infty\end{aligned}
d_i = \lim_{\Omega_i \ni y \to x}v_i \circ H(y) - a \leq \lim_{\Omega_i \ni y \to x}v^- (y) \leq \lim_{\Omega_i \ni y \to x}v^+ (y) \leq \lim_{\Omega_i \ni y \to x}W_i \circ H (y) = d_i.
0\, =_{q/2}\Bigl(\frac{E_2^2D_1^2+E_1E_2^2D_1+F_3^2+E_1E_2F_3+E_1^2F_4}{E_1^2E_2^2}\Bigr)\cr =_{q/2}\Bigl(\frac{D_1^2}{E_1^2}+\frac{D_1}{E_1}+\frac{F_3^2}{E_1^2E_2^2}+\frac{F_3}{E_1E_2}+\frac{F_4}{E_2^2}\Bigr)\cr =_{q/2}\Bigl(\frac{F_4}{E_2^2}\Bigr).
\left|\sum_{\substack{|\Im(\rho)| \leq T \:\Re(\rho) > \sigma}}\frac{x^{\rho - 1}}{\rho}\right| \leq\frac{2N(\sigma, T)}{T}&\le 2\bigg[C_1(\sigma)\exp\left(\frac{B_2(5-8\sigma)}{3}r\right)\left(B_2r\right)^{5 - 2\sigma}&\qquad\qquad\qquad+ C_2(\sigma)\exp\left(-B_2r\right)(B_2)^2r^2\bigg] &= s_2''(x, \sigma),\ .
\lambda_\alpha\lambda_\beta-\lambda_\beta\lambda_\alpha=0, \lambda_\alpha\theta_\beta^i-\theta_\beta^i\lambda_\alpha=0, \theta_\alpha^i\theta_\beta^j+\theta_\beta^j\theta_\alpha^i=0 (\alpha, \beta\in A, \i, j\in[N]).
\{\omega^{i_{0}, \alpha_{0}}; \omega^{i_{1}, \alpha_{1}}\}= \{\omega^{i_{0}, 0}, \omega^{i_{0}, 1}, \ldots , \omega^{i_{0}, N_{0}}; \omega^{i_{1}, 0}, \omega^{i_{1}, 1}, \ldots , \omega^{i_{1}, N_{1}}\}
\div^{P}\left\{\frac{\nabla^{P}u}{\sqrt{e^{2\psi}+\left\vert\nabla^{P}u\right\vert^{2}}}\right\}=nH+\frac{\left\langle\nabla^{P}u,\nabla^{P}\psi\right\rangle}{\sqrt{e^{2\psi}+\left\vert \nabla^{P}u\right\vert^{2}}}
C_5^{\frac{pq}{p-q}}\lesssim \sum_{k=-\infty}^{M-1}2^{-k\frac{q}{p-q}}\bigg( \int_0^{{x_{k+1}}}\bigg( \int_t^{{x_{k+1}}}u \bigg)^{\frac{1}{1-q}}d\big[ V_r(0,t)^{\frac{q}{1-q}}\big] \bigg)^{\frac{p(1-q)}{p-q}}= D_1.
[\begin{matrix}p_{1}&p_{2}\\p_{2}+1&p_{1}\end{matrix}][\begin{matrix}u_{1}\\u_{2}\end{matrix}]=[\begin{matrix}\frac{p_{1}+6p_{2}}{5.0}\\ 2p_{1}-6\end{matrix}],forp_{1}\in[2,4],p_{2}\in[-2,1].
\Big( \frac{1+h_q^2}{2h_p^2}- \frac 1{2H_p^2}\Big) H_p = -\frac{h_p-H_p}{H_p^2}+ \frac{H_p^3 h^2_q + (2h_p + H_p)(h_p-H_p)^2}{2h_p^2 H_p^2},
0\leq I_2(s)\leq \frac{Cf(x+s)}{s}\int_0^s\frac{tdt}{t^2+v(s)^2}=\frac{C\xi v(s)}{\pi s}\int_{0}^s\frac{t dt}{t^2+v(s)^2}=\frac{C\xi v(s)}{2\pi s}\log\bigg(1+\frac{s^2}{v(s)^2}\bigg)\to 0