harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|\Box_k^{\alpha_2}f\|_{M_2}\lesssim \left(\sum_{l\in\Gamma_k^{\alpha_1,\alpha_2}}\|\Box_l^{\alpha_1}f\|^2_{L^2}\right)^{1/2}\lesssim 2^{jn\alpha_1(1/p_1-1/p_2)}\left(\sum_{l\in\Gamma_k^{\alpha_1,\alpha_2}}\|\Box_l^{\alpha_1}f\|^2_{L^{p_1}}\right)^{1/2}\lesssim 2^{jn\alpha_1(1/p_1-1/p_2)}\|f\|_{M_1}.
\partial_j(\partial_i g_{\lambda\, ij}-\partial_j g_{\lambda\,ii}) =\chi_\lambda \partial_j(\partial_i g_{ij}-\partial_j g_{ii})+\partial\chi_\lambda \partial g+\partial^2\chi_\lambda (g-\delta) &=\chi_\lambda \partial_j(\partial_i g_{ij}-\partial_j g_{ii})+O(|x|^{-q-2}).
\mathrm{\underset{x\in\pi^{-1}\left(y\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}=\underset{x\in\pi^{-1}\left(y\right)\cap T\left(\frac{\epsilon}{2},\mathbf{W}^{i}\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}
\lim_{s\to 0}\,\frac{1}{s}\,\Omega(\Phi_s(X))=\,\frac{d}{ds}\big|_{s=0}\,\int^{X(s)}_A\,F^2(\gamma,\dot{\gamma})\,dt,\quad \lim_{s\to 0}\,\frac{1}{s}\,\Omega(\Phi^c_s(X))=\,\frac{d}{ds}\big|_{s=0}\,\int^A_{X(s)}\,F^2(\tilde{\gamma},-\dot{\tilde{\gamma}})\,dt.
K(a)N^{-\frac \alpha 2}e^{-N^\alpha J(a)}N^{\alpha-1}J''\left(\frac 1\lambda\right)\left(a-\frac 1\lambda\right)= \frac{\lambda}{\sqrt{2\pi}}N^{\frac \alpha 2 - 1}e^{-N^\alpha J(a)}.
\beta_{\nu}^{( 1 )}\overline{{{\Gamma}}}_{5}= \overline{{{\Gamma}}}_{5}\beta_{\nu}^{( \widetilde{1})}, ~ ~ \beta_{\nu}^{( \widetilde{1})}\overline{{{\Gamma}}}_{5}= \overline{{{\Gamma}}}_{5}\beta_{\nu}^{( 1 )}.
W_{s,t}a\cdot a+a\cdot W_{s,t}a=0\, , \mathbb{W}_{s,t}a\cdot a+a\cdot \mathbb{W}_{s,t}a+ W_{s,t}a\cdot W_{s,t}a=-W_{s,t}a\cdot W_{s,t}a+W_{s,t}a\cdot W_{s,t}a=0\, .
\max_{\boldsymbol{j}}2^{\sum_{l=1}^dj_l/2}\max_{\boldsymbol{k}}|\theta^{*}_{\boldsymbol{j},\boldsymbol{k}}-\theta^0_{\boldsymbol{j},\boldsymbol{k}}|=R2^{-\sum_{l=1}^d\alpha_lJ_{n,l}(\boldsymbol{\alpha})\left(\frac{1}{d}+\frac{1}{2\alpha^{*}}-\frac{1}{2\alpha_l}\right)}=M\epsilon_n
\frac{\mu_k(v_{1})}{\pi(v_{1})^p}\ge C^{-1}\frac{f_{0}(w_{1})}{\pi(w_{1})^p}> C^{-1}\frac{f_{1}(w_{1})}{\pi(w_{1})^p}= C \frac{f_{1}(w_{1}')}{\pi(w_{1}')^p}> C^2 \frac{\mu_k(v_{1}')}{\pi(v_{1}')^p},
{\cal D}\left(\begin{array}{c}\delta\Psi_1 \vdots \delta\Psi_M{[\sum_Iq_1^I\delta a_1^I]+i\sigma_1[\sum_Iq_1^I\delta a_2^I]}\vdots{[\sum_Iq_M^I\delta a_1^I]+i\sigma_M[\sum_Iq_M^I\delta a_2^I]}\end{array}\right)=0\,,
\int_I a(0,t,x) \rho_3^2 |y_{xx}|^2 dx \leq - \int_I D_3 a(0,t,x) \rho_3^2 y_x y_{xx}dx + \epsilon \int_I \rho_3^2 |y_{xx}|^2 dx &+ C_\epsilon \left(\int_I \rho_3^2 |G|^2 dx + \int_\mathcal{O}\rho_3^2 |f|^2 dx + \int_I \rho_3^2 |y_t|^2dx \right. &+ \left. \int_I \rho_3^2|y|^2 dx + \int_I \rho_3^2|y_x|^2 dx + \sum_{i=1}^2 \int_I \rho_3^2 |p^i|^2 dx \right).
\frac{d}{dt}k^{(1)}_t(x) = - m k^{(1)}_t(x) - \int_{\mathbb{R}^d}a^{-}(x-y) k^{(2)}_t(x,y) dy &+ \int_{\mathbb{R}^d}a^{+}(x-y)k^{(1)}_t(y) dy .
\nu_2\left({2^k\choose 2i}\right) =\nu_2\left(\frac{2^k}{2i}{2^k-1\choose 2i-1}\right) =k-1-\nu_2(i)+s_2(2i-1)+s_2(2^k-2i)-s_2(2^k-1)=k-1-\nu_2(i), i>0
| \mathcal{T}( T ) \rangle_{\alpha_{- 1}\cdot \tilde{\alpha}_{- 1}\tilde{c}_{1}c_{1}| 0 \rangle}\sim \frac{1}{T^{2}}U_{f ( v )}^{\dagger}c ( i ) \partial X^{\mu}( i ) \tilde{c}( i ) \bar{\partial}X_{\mu}( i ) | 0 \rangle ,
\sum_{n = \frac{1}{2}}^{R / 2}~ \frac{\triangle k_{n}^{+}}{k_{n}^{+}}~ f ( k_{n}^{+}) = \sum_{n = \frac{1}{2}}^{R / 2}~ \frac{\triangle k_{n}^{+}}{k_{n}^{+}}~ e^{- \sigma^{2}/ ( \Lambda^{\prime}k_{n}^{+})}
P=Q=\sum_{k=0}^{\infty}\alpha_{2}^{\prime}(k)\frac{1}{(j_{6A}(\tau)-4)^{k+\frac{1}{2}}}=\sum_{k=0}^{\infty}\alpha_{3}^{\prime}(k)\frac{1}{(j_{6A}(\tau)+32)^{k+\frac{1}{2}}}
\xi_s = \xi_s (h_0, \tilde{h}_0) = K + \tilde{h}_0 + s (h_0 - \tilde{h}_0), \varphi = \varphi(h_0, \tilde{h}_0)= \int_0^1 p'\circ \xi_s^x \, ds + (K^y+{h}_0^y) \, \int_0^1 q'\circ \xi_s^x \, ds + \int_0^1 (D_1u+D_1g) \circ \xi_s \, ds, \psi = \psi (h_0, \tilde{h}_0) = q\circ (K^x + \tilde{h}_0^x) + \int_0^1 (D_2u+D_2g) \circ \xi_s \, ds,
\mathbf{D}_{L_I}\big(\prod_{j=1}^{2k}g^{-1}t_j\sigma(t_j) g\big)=\mathbf{D}_{L_I}\big(\prod_{j=1}^{2k}(w_js_jw_j^{-1})\sigma(w_js_jw_j^{-1})\big),
\Leftrightarrow \bar{\xi}={\frac{1}{2^{\frac{1}{4}}}}\left( \begin{matrix}{0}&{- \Psi^{T}}\\ \end{matrix}\right) , \xi \equiv{\frac{1}{2^{\frac{1}{4}}}}\left( \begin{matrix}{\Psi}\\{0}\\ \end{matrix}\right)
\sigma_0^2 = \left( \frac{R}{\xi_1 h^2}-\frac{1}{\lambda_1 h^2}+\frac{2R}{3(4\pi)^2}\right) \left( \frac{1}{6\lambda_2 h^4}-\frac{11}{12(4\pi)^2}\right)^{-1}.
{\hat{\cal T}}_{J}^{R}= \sum_{J^{1}, J^{2}}\left\langle \begin{array}{l l}{{R^{1}}}&{{R^{2}}}\\{{J^{1}}}&{{J^{2}}}\\ \end{array}\right. \left| \begin{array}{l}{{R}}\\{{J}}\\ \end{array}\right\rangle{\hat{\cal T}}_{J^{1}}^{R^{1}}( 1 ){\hat{\cal T}}_{J^{2}}^{R^{2}}( 2 ) ,
\hat{\phi}={8m^2(-k_{1,0}k_{2,0}+{\bf k_1}.{\bf k_2}+m^2)(c_{1,0}\bar{\rho}_1+c_{0,1}\bar{\rho}_2)\over (k_{1,0}k_{2,0}-{\bf k_1}.{\bf k_2}-m^2)(\lambda(c_{1,0}^2\bar{\rho}_1^2-c_{1,0}c_{0,1}\bar{\rho}_1\bar{\rho}_2+c_{0,1}^2\bar{\rho}_2^2)-8m^2) - 6m^2c_{1,0}c_{0,1}\lambda\bar{\rho}_1\bar{\rho}_2}
2\sum_{l\in\mathcal{L}}E\left[ 1_{\left\{X_{l^{+}}=I\right\}}1_{\left\{X_{l^{-}}=S\right\}}\right] =2\sum_{l\in\mathcal{L}}\Pr\left[X_{l^{+}}=I,X_{l^{-}}=S\right] =\sum_{i=1}^{N}\sum_{j=1}^{N}a_{ij}\Pr\left[X_{i}=I,X_{j}=S\right] & =\sum_{i=1}^{N}\Pr\left[ X_{i}=I\right] \sum_{j=1}^{N}a_{ij}\Pr\left[X_{j}=S|X_{i}=I\right]
G_i= \frac{1}{N^2}\frac{{\big[1 - \cos (N{\mathcal K}_1(\xi_{r_{i,o}}))\big] \big[1 - \cos ({N}{{\mathcal K}_2}(\varphi_{t_{i,o}},{\varphi_{{t_i}}}))\big]}}{{\big[1 - \cos ({\mathcal K}_1(\xi_{r_{i,o}}))\big] \big[1 - \cos ({{\mathcal K}_2}(\varphi_{t_{i,o}},{\varphi_{{t_i}}}))\big]}},
u(x)=\left\langle u, P_p(\cdot,x) \right\rangle_{\widehat{S}_p}=\frac{1}{p}\sum_{j=0}^{p-1}\int_S u(e^{\frac{j\pi i}{p}}\zeta)\overline{P_p(e^{\frac{j\pi i}{p}}\zeta ,x )}\, d\sigma(\zeta).
\tfrac{\partial}{\partial x_i}(\cos(g(r)){f_{\frak{e}_k}(x/r)_j})= -\tfrac{\sin(g(r))}{r^2}x_i{f_{\frak{e}_k}(x/r)_j}&+\sum_{\ell=1}^n\tfrac{\partial{f_{\frak{e}_k}(y)_j}}{\partial y_{\ell}}(\tfrac{1}{r}\delta_{\ell,i}-\tfrac{x_ix_{\ell}}{r^3})\cos(g(r)),
\left\|\sum_{n=1}^\infty\,n\,I_{n-1}(\phi_n(t,\ldots))\right\|_{2,q}^2\leq\,(1+\omega_{-q}^2)\sum_{n=1}^\infty\,n\,\left\|I_{n-1}(\phi_n(t,\ldots))\right\|_{2,0}^2.
2(h_{12}^3)^2-\frac{1}{\sqrt{3}}\sin{(2\alpha)}\{h_{12}^3}-\frac{1}{3}\sin^2{(2\alpha)}=0, &2(h_{12}^3)^2-\frac{1}{\sqrt{3}}\frac{\sin^2{\alpha}}{\sin (2\alpha)}\{h_{12}^3}+\frac{2}{3}\sin^2{\alpha}\cos( 2\alpha)=0,
{\bf g}( \Omega ) ={\frac{\partial^{2}\Omega}{\partial x^{i}\partial{\tilde{x}^{j}}}}d x^{i}d{\tilde{x}}^{j}\, , \quad \quad{\tilde{x}}^{i}={\tilde{x}}\, ,{\tilde{y}}\, , \quad x^{j}= x \, , y \, .
\phi_{ij}(x)=\log\frac{\lambda e^x-\mu}{e^x-1},\phi_{k\ell}(x)=\log\frac{e^x-1}{(\mu+\alpha)e^x-(\lambda+\alpha)},\phi_{i\ell}(x)=\log\frac{1+\lambda e^x}{1+(\mu+\alpha)e^x}.
\Lambda[f \Lambda f](0) = \frac 12 (Hg(0))^2 - \frac 12 (g(0))^2 - \|g\|_{\dot H^{1/2}}^2, &= \frac 12 (Hg(0))^2 - \frac 12 (g(0))^2 - \|Hg\|_{\dot H^{1/2}}^2.
G: \begin{cases}0 \to 0 1 \to 01 \end{cases}\widetilde{G}: \begin{cases}0 \to 0 1 \to 10 \end{cases}D: \begin{cases}0 \to 10 1 \to 1 \end{cases}\widetilde{D}: \begin{cases}0 \to 01 1 \to 1 \end{cases}.
\sum_{s=1}^{n-1}\frac{q^{s(n-s)+n}}{y^{s}z^{n-s}}-\frac{q^{m^2+2m}}{y^mz^m}\cdot \frac{J_2^3j(-q;q^2)j(yz;q^2)}{j(qy;q^2)j(-y;q^2)j(-z;q^2)j(qz;q^2)}
q^{-\beta}\frac{\prod_{k=1}^{2l}\phi(u+(k-l-\frac12)\eta)}{Q(u-\eta/2)Q(u+\eta/2)}=K(u)+q^{-\beta}\frac{G(u+\frac{\eta}{2})}{Q(u+\frac{\eta}{2})}-q^{\beta}\frac{G(u-\frac{\eta}{2})}{Q(u-\frac{\eta}{2})},
\text{$\langle\alpha_t(x)\eta\alpha_t(y),\eta\rangle=\lim_{n\rightarrow\mathcal U}\langle\alpha_t(x)\eta_n\alpha_t(y),\eta_n\rangle=\lim_{n\rightarrow\mathcal U}\langle x\alpha_{-t}(\eta_n)y,\alpha_{-t}(\eta_n)\rangle=\lim_{n\rightarrow\mathcal U}\langle x\zeta_ny,\zeta_n\rangle$.}
\sum \limits_{j = 1}^{k-1}\prod \limits_{i = j}^{k-1}|c(\theta_i) \cdot p'(x_i)| \leq 4^{7K_m}\cdot \sum \limits_{j = 1}^{k-1}(3/5)^{\left(1 - \frac1{M_0}\right)(k-j)/2}\leq &\leq 4^{7K_m}\cdot \sum \limits_{j = 1}^{\infty-1}(3/5)^{\left(1 - \frac1{M_0}\right)j/2}= 4^{7K_m}\cdot A
2^{\mu_*k_*t}\cdot 2^{‐m_t(s_*+1/q‐1/p_1)}= 2^{‐\alpha_*k_*t}\cdot 2^{‐m_t(1/q‐1/p_0)}\stackrel{\eqref{hat_theta_def}}{=}n^{‐\hat \theta};
2 \mathrm{I m}{\cal L}^{( 1 )}= ( 2 s + 1 ) \frac{( e \epsilon )^{2}}{( 2 \pi )^{3}}\sum_{n = 1}^{\infty}\frac{( \pm 1 )^{n + 1}}{n^{2}}e^{- \pi n / \beta}, \qquad \beta = \frac{e \epsilon}{m^{2}}.
\psi^a_{\vec{\mu}}(\theta+ 2\pi)=exp (i\vec{a}. \vec{\mu}) \psi^a_{\vec{\mu}}(\theta) \hspace{.6in}\zeta^a_{\vec{\mu}}(\theta+ 2\pi)=exp(i \vec{a}. \vec{\mu}) \zeta^a_{\vec{\mu}}(\theta)
K_{\nu}\left({\nu ze}^{-\pi i}\right) =i\left({\frac{\pi}{2\nu}}\right)^{1/2}\frac{1}{\left({1+z^{2}}\right)^{1/4}}\exp \left\{{\nu \xi +\sum\limits_{s=1}^{n-1}{\frac{\tilde{{E}}_{s}\left( p\right)}{\nu^{s}}}}\right\}\left\{{1+\eta_{n,3}\left({\nu ,\xi}\right)}\right\}.
Z(S_{n})=\frac{1}{n!}\sum_{g\in S_{n}}\prod_{k=1}^{n}a_{k}^{j_{k}(g)}=\frac{1}{n!}\sum_{l=1}^{n}(\begin{matrix}n-1\\l-1\end{matrix})(l-1)!a_{l}(n-l)!Z(S_{n-l})
A ( 1 , 2 , 3 , 4 ) : = \tilde{D}_{4 , - p_{4}}^{\alpha}\tilde{D}_{\alpha , 2 , - p_{2}}\tilde{\bar{D}}_{\dot{\alpha}, 2 , - p_{2}}\tilde{\bar{D}}_{1 , - p_{1}}^{\dot{\alpha}}\tilde{\delta}_{V}( 1 , 2 ) \tilde{\delta}_{V}( 1 , 3 ) \tilde{\delta}_{V}( 1 , 4 ) .
\ell_1+\ell_3 = 4\left((\alpha_1+\gamma_1)^2+(\alpha_2+\gamma_2)^2+(\beta_1+\delta_1)^2+(\beta_2+\delta_2)^2\right), \ell_2+\ell_4 = 4\left((\alpha_1-\gamma_1)^2+(\alpha_2-\gamma_2)^2+(\beta_1-\delta_1)^2+(\beta_2-\delta_2)^2\right).
s_L:=\min_{\footnotesize{\begin{array}{c}\{L_k\}_k; \sum_k L_k=L; \forall k, 0<L_k<L\end{array}}}\displaystyle{\sum_{i<j}}L_i.L_j=\frac12(L^2-\max_{\footnotesize{\begin{array}{c}\{L_k\}_k; \sum_k L_k=L; \forall k, 0<L_k<L\end{array}}}\sum_k L^2_k).
A_0(x) = \left[ \begin{array}{c}\frac{\pi T}g \exp\left\{\frac g{\sqrt{\pi}}(x - x_0) \right\}, \ \ \x \leq x_0 \frac{\pi T}g \left[ 2- \exp\left\{\frac g{\sqrt{\pi}}(x_0 - x) \right\}\right], \ \ \x \geq x_0\end{array}\right.
\alpha(1,3) = \alpha(2,3) = \alpha(3,5) = \alpha(4,5) = 1, \alpha(1,2) = \alpha(1,4) = \alpha(2,4) = \alpha(1,5) = \alpha(2,5) = 2, \alpha(3,4) = 3, 3 \mid r_2 \ \ \ \ 2 \mid r_3.
\textnormal{ch}(f_K(q))^{-\frac{\lambda_i}{g_H(z)}}= 1+\frac{\lambda_i}{\lambda_0 + \cdots + \lambda_N}c_1(\Lambda_0 \otimes \cdots \otimes \Lambda_N) + \cdots = 1+\lambda_i + \cdots = \textnormal{ch}(\lambda_i)
(F\bar{F})=(e^{\frac{\tau}{2}\partial^{2}}f_{\mathbb{C}})(e^{\frac{\bar{\tau}}{2}\bar{\partial}^{2}}\bar{f}_{\mathbb{C}})=e^{(\frac{\tau}{2}\partial^{2}+\frac{\bar{\tau}}{2}\bar{\partial}^{2})}f_{\mathbb{C}}\bar{f}_{\mathbb{C}}.
(\partial^\zeta_x \tilde{\phi})_t+v\cdot \nabla \partial^\zeta_x \tilde{\phi}=-(\partial^\zeta_x (v\cdot \nabla \tilde{\phi})-v\cdot \nabla \partial^\zeta_x \tilde{\phi})-\frac{\delta-1}{2}\partial^\zeta_x (\psi v).
\sum_{i=1}^{\infty}\left|\nabla_{\eta}P_{s,t}^{0}\langle f, e_{i}\rangle(x)\right|^{2}\leq\sum_{i=1}^{\infty}\frac{c}{t-s}P_{s,t}^{0}|\langle f, e_{i}\rangle|^{2}(x)|\eta|^2=\frac{c}{t-s}P_{s,t}^{0}|f|^{2}(x)|\eta|^2.
\begin{gathered}f_{n,k}(y;x)=k\binom{n}{k}F(y;x)^{k-1}f(y;x)\bigl(1-F(y;x)\bigr)^{n-k}, F_{n,k}(y;x)=\int_{x}^{y}f_{n,k}(z;x)dz.\end{gathered}
g_{n}\nu(F) = \int_{X}f(\pi(g_{n}x)) - \zeta_{g_{n}x}(f)~d\nu(x)= \int_{X}f(g_{n}\pi(x)) - g_{n}\zeta_{x}(f)~d\nu(x)= g_{n}\eta(f) - g_{n}\eta^{\prime}(f)
\mathbf{M}_{12\overline{3}}=\mathbf{F}_{12\overline{3}}-\mathbf{F}_{21\overline{3}}-\mathbf{F}_{\overline{1}2\overline{3}}+\mathbf{F}_{\overline{1}\,\overline{2}\,\overline{3}}\quad\mathbf{M}_{1\bar{3}2}=\mathbf{F}_{1\bar{3}2}-\mathbf{F}_{2\overline{3}1}-\mathbf{F}_{\overline{1}\,\overline{3}2}+\mathbf{F}_{\overline{1}\,\overline{3}\,\overline{2}}.
\begin{aligned}\int_\Omega (A \sigma : \tau + \div \tau \cdot u + \tau : p)\dx = 0, \tau \in H(\div;\mathbb{M}), \int_\Omega \div \sigma \cdot v\dx = \int_{\Omega}g \cdot v\dx, v \in L^2(\mathbb{V}), \int_\Omega \sigma : q\dx = 0, q\in L^2(\mathbb{K}). \end{aligned}
m_{ij}= (d(i)_{\lambda_i}-(-1)^\ell e(i)_{\lambda_i})( d(j)_{\lambda_j}+(-1)^\ell e(j)_{\lambda_j}) + 2\sum_{t>0}(-1)^t d(i)_{\lambda_i+t}\, d(j)_{\lambda_j-t},
S_{2^{i}}^{2}=S_{1,2^{i-1}}^{2}+S_{2,2^{i-1}}^{2}+2S_{1,2^{i-1}}S_{2,2^{i-1}}=S_{1,2^{i-2}}^{2}+S_{2,2^{i-2}}^{2}+S_{3,2^{i-2}}^{2}+S_{4,2^{i-2}}^{2}& +2\left( S_{1,2^{i-2}}S_{2,2^{i-2}}+S_{3,2^{i-2}}S_{4,2^{i-2}}+S_{1,2^{i-1}}S_{2,2^{i-1}}\right) .
\sum_{j = 0}^\infty \varrho^{2j}\left \langle \varphi_j, K(\partial_t^{\Delta t}) \varphi(t_j)\right\rangle \geq C_2(\sigma) \sum_{j = 0}^\infty \varrho^{2j}\|(\partial_t^{\Delta t})^\eta \varphi_j\|^2_X,
u_\varepsilon(z_\varepsilon)=\frac{4\pi G_{z_\varepsilon}(x_\varepsilon)}{\gamma_\varepsilon}\left(1+\frac{1}{\gamma_\varepsilon^2}+\frac{A(\gamma_\varepsilon)-2\xi_\varepsilon}{2}+o(\tilde{\zeta}_\varepsilon) \right) +o\left(\frac{1}{\gamma_\varepsilon}\right)\,.
\|p\|_K=\|p\circ \sigma\|_{I\times \Theta}\leq \alpha^{d_1+d_2}\,\beta^{d_3}\|p\circ \sigma\|_{\mathcal{B}_n}=\alpha^{d_1+d_2}\,\beta^{d_3}\|p\|_{\sigma(\mathcal{B}_n)}\;,
\tilde{\mathbf{W}}= (\mathbf{G}^H\mathbf{\Phi}^H\mathbf{H}\mathbf{H}^H\mathbf{\Phi}\mathbf{G}+ \sigma^2 \mathbf{I}_N)^{-1}\mathbf{G}^H\mathbf{\Phi}^H\mathbf{H},
D_a(u)-D_a(v) =\frac{1}{|u-a|}(u-a)\cdot \gamma-\frac{1}{|v-a|}(v-a)\cdot\gamma &=\frac{1}{|u-a|}(u-a)\cdot \gamma-\frac{1}{|u-a|}(v-a)\cdot\gamma &+\frac{1}{|u-a|}(v-a)\cdot\gamma-\frac{1}{|v-a|}(v-a)\cdot\gamma &=\frac{1}{|u-a|}(u-v)\cdot\gamma+\left(\frac{1}{|u-a|}-\frac{1}{|v-a|}\right)(v-a)\cdot\gamma.
\partial_\eta g =-c_0e^{-\beta\xi}\big(-1+(1-\eta)\frac{8}{3}\big)e^{\frac{8}{3}\eta -\frac{8}{3}}+\partial_\eta w \leq -c_0e^{-\beta\xi}e^{\frac{8}{3}\eta -\frac{8}{3}}+ \frac{b}{15}<0.
\left( \begin{matrix}{{T_{1 1}}}&{{}}&{{T_{1 2}}}\\{{T_{2 1}}}&{{}}&{{T_{2 2}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{\exp{i \alpha}\cos{\eta}}}&{{}}&{{\exp{i \beta}\sin{\eta}}}\\{{- \exp{i \gamma}\sin{\eta}}}&{{}}&{{\exp{i \delta}\cos{\eta}}}\\ \end{matrix}\right) ,
e^{i k_{1}\cdot X}e^{i k_{2}\cdot X}\sim \left( \tau - \tau^{\prime}\right)^{2 \alpha^{\prime}g^{\mu \nu}k_{1 \mu}\cdot k_{2 \nu}}\times e^{i \left( k_{1}+ k_{2}\right) \cdot X}+ \cdots .
\mathfrak{B}(\xi,t)=\begin{pmatrix}0 -i\eta\beta(\xi)e^{-t\varphi(\xi,0)}\tau^{-i\nu-\frac{1}{2}}i\eta\gamma(\xi)e^{t\varphi(\xi,0)}\tau^{i\nu-\frac{1}{2}}0 \end{pmatrix},
\lambda_{C W}\left( X ( \frac{p_{1}}{q_{1}}, \ldots , \frac{p_{n}}{q_{n}}) \right) = \frac{1}{1 2}\frac{P}{H}\left( 2 - n + \sum_{i = 1}^{n}p_{i}^{- 2}\right) - \frac{1}{1 2}\left[ 3 \mathrm{s i g n}\left( \frac{H}{P}\right) + \sum_{i = 1}^{n}\left( 1 2 s ( q_{i}, p_{i}) - \frac{q_{i}}{p_{i}}\right) \right] .
i\gamma^{\mu}\partial_{\mu}\left(\begin{array}{c}\psi_{L}\psi_{R}\psi_{\nu}\end{array}\right) =\left(\begin{array}{ccc}0 \phi^{2}0 \overline{\phi^{2}}0 \overline{\phi^{1}}0 \phi^{1}0 \end{array}\right)\left(\begin{array}{c}\psi_{L}\psi_{R}\psi_{\nu}\end{array}\right).
2N(1-\sqrt{2c}) <2\big(N-\sqrt{N[1+(N-1)c]}\big) begin{align*}2 pt] \leq\sum\limits_{i=1}^{N}\lVert x-\varphi_{i}\rVert^{2}begin{align*}2 pt] \leq2\big(N+\sqrt{N[1+(N-1)c]}\big) begin{align*}2 pt] <2N(1+\sqrt{2c}),
\displaystyle{\left| \int_{A^\ast \cap J_u}K_r(x, 0, c, d, e_N)d{\cal H}^{N-1}-\eta^{N-1}\sum_{i=1}^{h^{N-1}}K_r(a_i,0, c,d, e_N)\right|}= \displaystyle{\sum_{i=1}^{h^{N-1}}\int_{Q_i}\left|K_r(x,0, c,d, e_N)-K_r(a_i,0,c,d, e_N)\right| d{\cal H}^{N-1}=O(\varepsilon).}
A_{ut}(p_1;p_2;p_3;p_4)=-2\frac{G^2_0}{\alpha^{'}}(2\pi)^{26}\delta^{26}(\sum_i p_i)cos(\frac{p_1\theta p_4+p_2\theta p_3}{2})\frac{\Gamma(-1-\alpha^{'}u)\Gamma(-1-\alpha^{'}t)}{\Gamma(2+\alpha^{'}s)}
\int \frac{\mathrm{d}^{4}p}{( 2 \pi )^{4}}\int \frac{\mathrm{d}^{4}q}{( 2 \pi )^{4}}A_{\mu}^{a}( - p ) R_{k , \mu \nu}^{a b}( p , - q ) A_{\nu}^{b}( q ) \: .
\frac{f(v_n)}{q^{n/2}}=\frac{h(v_n)}{q^{n/2}}-\frac{a_n\delta(v_0)}{q^{n/2}(q+1)}-\sum_{k=1}^{n-1}\frac{a_{n-k}}{q^{{(n-k)}/2}}\frac{\delta(v_{k})}{q^{k/2}}.
2D(c_1 +6c_2 +42c_3)H_0^4 - \{4D- [3c_2+ 4(2D-1)c_3] \Lambda \}H_0^2-{1 \over 12}\left [ (D-2) \Lambda - (D-4) \left({2c_1 \over D(D-1)}+{c_2 \over D}+c_3 \right) \Lambda^2 \right ]=0.
\tau^1=\widehat{\bigotimes_{1\le i\le n_1}}\tau^1_i,\quad\tau^2=\widehat{\bigotimes_{1\le i\le n_2}}\tau^2_{i,1}\hat{\otimes}\tau^2_{i,2},\quad\tau^3=\widehat{\bigotimes_{1\le i\le n_3}}\tau^3_i
\mu(P^*) \approx \begin{cases}\, 1,\, \, 0<\alpha \le 1, \, R^{-\frac 12(\alpha - \ell +1)}\times R^{-\frac12(\ell-2)},\, \, 1< \alpha \le n, \, R^{-(\alpha -n)}\times R^{-\frac{n-1}{2}},\, \, n< \alpha\le n+1.\end{cases}
t(t+1) \iint_R \{v\}v^{-3/2}\{u\}u^{-3/2}\left(1+i\log\frac{u}{v}\right)^{-t-2}dudv &= it \sum_{(m,n)\in R(t)}\left(D_{m,n}(t) - E_{m,n}(t)\right) + O(1).
d:= \sup\left\{s\in \mathbb{R}: \left<\sum_{q}\l_q^{-1+s}\|u_q\|_\infty^2 \right> \leq C_{\mathrm{B}}^{3-s}\l_0^{s}\left<\sum_{q}\l_q^{2}\|u_q\|_2^2 \right> \right\},
\begin{cases}\tau_c(a)=a,\, \tau_c(b)=cbc^{-1}L=A_1\ast_{A_e}A_2,\, a\in A_1 b\in A_2 \tau_c(a)=a,\, \tau_c(t)=tc L=A\ast_{L_e}t a\in A. \end{cases}
\frac{1}{2}\frac{d}{dt}\|z(t)\|_{L^{2}}^{2}+\|\nabla z\|_{L^{2}}^{2}&=\int_{\mathbb{R}^{3}}(z\otimes u^{c,\gamma}+u^{c,\gamma}\otimes z)\cdot\nabla zdx+\int_{\mathbb{R}^{3}}(w\otimes w)\cdot\nabla zdx.
\nabla\cdot b_-= -\frac{B_0}{\gamma^2(t)+B_0^2}(-\partial_{q^1}\partial_{q^2}V(t,q)+\partial_{q^1}\tilde F_2(t,q))+\frac{B_0}{\gamma^2(t)+B_0^2}(-\partial_{q^2}\partial_{q_1}V(t,q)+\partial_{q^2}\tilde F_1(t,q)) = \frac{B_0}{\gamma^2(t)+B_0^2}\left(-\partial_{q^1}\tilde F_2(t,q)+\partial_{q^2}\tilde F_1(t,q)\right).
{\frac{d}{d t}}X = i \sum_{\rho \in \Delta_{+}}g_{| \rho |}\, ( \rho^{\vee}\! \! \cdot \hat{H}) \, y_{| \rho |}( \rho \cdot q , ( \rho^{\vee}\! \! \cdot \hat{H}) \xi ) \, ( \rho \cdot \dot{q}) \, \hat{s}_{\rho}
K=-\frac{c_{1}^{\prime\prime}(s)}{c_{1}(s)}andH=c_{1}^{\prime}(s)c_{2}^{\prime\prime}(s)-c_{2}^{\prime}(s)c_{1}^{\prime\prime}(s)+\frac{c_{2}^{\prime}(s)}{c_{1}(s)}.
\frac{\delta\bar{\mathcal{F}}}{\delta\phi_{\partial}}=\frac{\delta\tilde{\mathcal{F}}}{\delta\phi_{\partial}},\frac{\delta\bar{\mathcal{F}}}{\delta\Sigma}=\frac{\delta\tilde{\mathcal{F}}}{\delta\Sigma}=\frac{\delta\mathcal{F}}{\delta\Sigma}+(-1)^{n-1}\langle dN_{\phi}(\frac{\delta\tilde{\mathcal{F}}}{\delta\phi_{\partial}}),\eta\rangle_{\Lambda^{1}}.
\left( \ref{claim:dificil}\right) u=1\exists Z_{u=1,3}^{\ast}\mathbf{Z}_{min}\backslash \left[ \mathbf{O}_{min}\cup \left\{Z_{u=1,1}^{\ast},Z_{u=1,2}^{\ast}\right\}\right] Z_{u=1,3}^{\ast}&\text{is a descendantof}Z_{u=1,2}^{\ast}
\lefteqn{\{(r_0,r_1,r_2) \in{\cal R}_{\mathtt{GW}}^*(P_{XY}) : r_0 + r_1 = H(X) \}}&= \{(r_0,r_1,r_2) : (r_0,r_2) \in{\cal R}_{\mathtt{WAK}}^*(P_{XY}), r_0 + r_1 = H(X) \}.
\begin{aligned}\epsilon(\eta_a,t) = \sum_{n = 1}^N\int_{n-1}^{n}F(\eta_a,\tau)-F_a(\eta_a)\,d \tau & +\int_{N}^{t}F(\eta_a,\tau)-F_a(\eta_a)\,d \tau = \int_{N}^{t}F(\eta_a,\tau)-F_a(\eta_a)\,d \tau.\end{aligned}
P'_{\!r}(M,\!K,\!N)\!=\!\frac{\sum_{h=h_{\min}}^{r}\!\!\binom{\!K\!}{\!h\!}\binom{N\!-\!K}{M\!-\!h}\mathbb{P}_{r-h}(M\!-\!h,K\!-\!h)}{\binom{N}{M}}
N_{i}^{- 1}{\tilde{L}}( \lambda | x_{i}, \partial_{x_{i}}) N_{i - 1}= \left( \begin{array}{c c}{- ( y_{i - 1}- x_{i})}&{1}\\{0}&{\frac{\lambda + \beta_{i}}{y_{i - 1}- x_{i}}}\\ \end{array}\right) .
\langle U(\zeta;\eta),(\zeta;\eta)\rangle \ge\sum \limits_{k=1}^{n}\alpha_k^2+\sum \limits_{k=n+4}^{2n}\alpha_k^2 \lambda_{\lceil \frac{k-n-1}{2}\rceil}|f_k|^2 \geq \tilde c_n \sum \limits_{k=1}^{n+3}\alpha_k^2+\sum \limits_{k=n+4}^{2n}\alpha_k^2 \lambda_{\lceil \frac{k-n-1}{2}\rceil}|f_k|^2 \geq c_n \sum \limits_{k=1}^{2n}\alpha_k^2,
\sum_{\nu = 0}^{N}\omega_{\nu}^{2}\eta_{\nu \rho}\eta_{\nu \sigma}+ \sum_{k = 1}^{N}\lambda_{k}\left( \eta_{k \rho}\eta_{0 \sigma}+ \eta_{0 \rho}\eta_{k \sigma}\right) = \delta_{\rho \sigma}\Omega_{\sigma}^{2}\; ,
\tau^{e_1}(x)=\begin{cases}\tau_{a_0}^{-1}(x)\quad\;\; x\in R_{a_0}\tau_{a_1}^{-1}(x)\quad\;\; x\in R_{a_1}\tau_{c_0}^{-1}(x)\quad\;\; x\in R_{c_0}\tau_{c_1}^{-1}(x)\quad\;\; x\in R_{c_1}\end{cases}
\P (X_{n+1}=1|X_{n}=0) =\frac{f(0)}{n}\times \frac{n}{n+1}=\frac{f(0)}{n+1}, \P (X_{n+1}=0|X_{n}=0) =(1-\frac{f(0)}{n})\times \frac{n}{n+1}+\frac{1}{n+1}=1-\frac{f(0)}{n+1}.
{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b;c;z)=\frac{1}{2\pi i}\int_{-i\infty}^{+i\infty}\frac{{}^{\Psi}\Gamma^{(\alpha,\beta)}(s)B(b+s,c+s-b)}{B(b,c-b)}\Phi(b+s;c+2s;z)p^{-s}ds.
{\cal H}= \frac{1}{2}\left[ \vec{E}_{\bot}^{2}+ \vec{B}\,^{2}\right]+\frac{1}{2}{\cal E}_{3}^{2}+ \frac{1}{2 L^{2}}e_{3}^{2}\left(\vec{x}_{\perp}\right)+ \frac{1}{2}\vec{\eta}\,^{2}\left(\vec{x}_{\perp}\right) \ .
\int_{\mathcal{H}}(\mathfrak{Im}\langle e_k, T^nx \rangle)^2\,(\mathfrak{Im}\langle e_\ell, x \rangle)^2\, dm(x)=\frac{1}{2}(\mathfrak{Re}\langle RT^{*n}e_k, e_\ell \rangle)^2+\sigma_k^2\sigma_\ell^2.
\sum_{j=-\infty}^{nt}\operatorname E\Bigl\|b_n^{-1}\sum_{k=nu+1}^{nt}v_{k-j}\varepsilon_j\Bigr\|^pt=\operatorname E\|\varepsilon_0\|^p(\sqrt n\log n)^{-p}\sum_{j=-\infty}^{nt}\Bigl[\sum_{k=nu+1}^{nt}v_{k-j}\Bigr]^p.
\begin{array}{l}\widehat e_{r+j}=e_{r+j}+\overline e_{r+j},~ j=1,\cdots,s; begin{align*}4pt] \widehat E=\{e_0,\,e_1,\cdots,e_r,\, \widehat e_{r+1},\cdots,\widehat e_{r+s}\}. \end{array}
\tilde{h}_{N}f_{x}^{0}(m_{i}^{x}) = \sum_{j=1}^{d^N_x}\tilde{h}_{N}(n_j^{x}r_{j,i}^{x})=\sum_{j=1}^{d^N_y}\tilde{h}_{N}(n_j^{x})r_{j,i}^{x}= \sum_{j=1}^{d^N_x}\sum_{s=1}^{d^N_0}n_{s}^{0}\otimes u_{0,x}N^{x}_{s,j}r_{j,i}^{x};
\begin{aligned}S(U,f^{n+k}) \geq\frac13 \frac{\log M^n(\frac14 r_2,f^n)}{\log r_2}-2\frac{\log^+ M^n(r_1^2,f)}{\log r_1}-\frac{2C}{\log r_1}\geq\frac13 \frac{\log M^n(\frac14 r_2,f^n)}{\log r_2}-3\frac{\log^+ M^n(r_1^2,f)}{\log r_1},\end{aligned}
f_{k}^{\overline{{( 2 )}}}( t ) = \int_{0}^{t}\! \mathrm{d}t^{\prime}\Delta_{k , \mathrm{r e t}}( t - t^{\prime}) V ( t^{\prime}) f_{k}^{\overline{{( 1 )}}}( t^{\prime}) e^{i \omega_{k}^{0}( t - t^{\prime})}
\begin{array}{r c l}{{({\bf 1}+ \tilde{\bf K}) ({\bf 1}+{\bf K})}}&{{=}}&{{\frac{4}{( q + q^{- 1})^{2}}({\bf 1}+ ( q^{2}+ q^{- 2}) ( q + q^{- 1})^{2}{\bf P})}}\\{{({\bf 1}+{\bf K}) ({\bf 1}+ \tilde{\bf K})}}&{{=}}&{{\frac{4}{( q + q^{- 1})^{2}}({\bf 1}+ ( q^{2}+ q^{- 2}) ( q + q^{- 1})^{2}{\bf P}^{'})}}\\ \end{array}
1 = \int \frac{t}{\frac{\check{f}(\rho)}{\rho-c(1-\rho)\check{f}(\rho)}\frac{(1-\rho)\check{f}(\rho)}{\rho(1-\check{f}(\rho))}+ \frac{1-\rho}{\rho}\check{f}(\rho)t}\nu(dt).
||v_{n+2\,m}^1||^2= -\frac{\overline{d_{n+2\,m}}}{a_{n+1\,m-3}}||v_{n+1\,m-3}^1||^2= \frac{\overline{d_{n+2\,m}}}{a_{n+1\,m-3}}\cdot \frac{c_{n+1\,m-3}}{\overline{b_{nm}}}||v_{nm}^1||^2.
\nabla(\varphi.d_{/}\xi) = \nabla(d_{/}(\varphi.\xi)) = d_{/}f\wedge \frac{\partial \varphi.\xi}{\partial \lambda}- \frac{\partial f}{\partial \lambda}.\varphi.d_{/}\xi & = \frac{\partial \varphi}{\partial \lambda}.(d_{/}f\wedge \xi) + \varphi.\nabla(d_{/}\xi) = \frac{\partial \varphi}{\partial \lambda}.b[d_{/}\xi] + \varphi.\nabla(d_{/}\xi)
\sum_i \mu_i^4 = \sum_{i=1}^3 \mu_i^4 + \left(\sum_{i=1}^3 \mu_i\right)^4 &=\frac{1}{2}\left(\sum_{i=1}^3 \mu_i^2 +\left(\sum_{i=1}^3 \mu_i\right)^2 \right)^2+ 4\mu_1\mu_2\mu_3\left(\sum_{i=1}^3 \mu_i\right) &=\frac{1}{2}\vert \mathring{A}\vert^4 - 4\prod_i \mu_i:=\frac{1}{2}\vert \mathring{A}\vert^4 - 4\mathcal{K},
R^{+}_1(x) =1,R^{+}_2(x)=1+3x,R^{+}_3(x)=1+16x,R^{+}_4(x)=1+61x+41x^2; R^{-}_1(x) =x,R^{-}_2(x) =3x,R^{-}_3(x)=7x+9x^2,R^{-}_4(x)=15x+80x^2; R_1(x) =1+x,R_2(x) =1+6x,R_3(x)=1+23x+9x^2,R_4(x)=1+76x+121x^2.
{\widehat\Psi}_k{''}+\left[k^2-(S^{1/2}){''}S^{-1/2}\right]{\widehat\Psi}_k= 0 \, , ~~~~~{\widehat\Pi}_k{''}+ \left[k^2-(S^{-1/2}){''}S^{1/2}\right]{\widehat\Pi}_k = 0.
\partial_\mu J_f^\mu = \omega^\alpha_\mu \Pi^{\mu \nu}F_{\alpha \nu}\;,\quad \quad\partial_\mu J_f^\mu = a (\Pi^{\mu \nu}F_{\mu \nu}- 4{\cal L}) \;,\quad \quad\partial_\mu J_f^\mu = 2 \Pi^{\mu \nu}F_{\alpha \nu}(a^\alpha x_\mu - a_\mu x^\alpha - a \cdot x \delta^\alpha_\mu)+ 8 a \cdot x{\cal L}\;,
\mu^{k}_{i,\pm}= 1- \frac{\Delta t_k}{2}\left[ \left( \Delta t_k b^{k}_i \lambda^{k}_i + \eta \right) \pm \sqrt{\left( \Delta t_k b^{k}_i \lambda^{k}_i + \eta \right)^2 - 4 b^{k}_i\lambda^{k}_i}\right] i=1,2,\cdots, MN,
\mathrm{P}\left( \bar{E}_{m,2}\right) = \mathrm{P}\left(\log\left(1+_{2,m}\right)<R_{2,m},\right. \log\left(1+_{2,m'}\right)>R_{1,m}, \log\left(1+_{2,n'}\right)>R_{1,n}, \left.\log\left(1+_{2,n}\right)>R_{2,n}, \forall ~n\in \{1, \cdots, m-1\}\right).