harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[ \bar{\gamma}^{2}\bar{\gamma}^{0}\bar{\gamma}^{1}\partial_{\vartheta}+ \frac{\bar{\gamma}^{3}\bar{\gamma}^{0}\bar{\gamma}^{1}}{\sin \vartheta}\partial_{\varphi}+ \bar{\gamma}^{2}\bar{\gamma}^{0}\bar{\gamma}^{1}S \partial_{\vartheta}S^{- 1}+ \frac{\bar{\gamma}^{3}\bar{\gamma}^{0}\bar{\gamma}^{1}}{\sin \vartheta}S \partial_{\varphi}S^{- 1}\right] \tilde{\Phi}= - i \kappa \tilde{\Phi}
\|\Re_A(\mathbb{T})\|_\mathbb{A}^2 =\max\left\{\left\|P_A+\frac{(T^{\sharp_A})^{\sharp_A}T^{\sharp_A}}{4}\right\|_A ,\left\|P_A+\frac{T^{\sharp_A}(T^{\sharp_A})^{\sharp_A}}{4}\right\|_A\right\}=\frac{\|T^{\sharp_A}\|_A^2}{4}+1=\frac{\|T\|_A^2+4}{4}.
\begin{array}{l}b^-(i) = w(2i) r^-(i) = w(2i+1)\end{array},\begin{array}{l}b^+(i) = w(2i+1) r^+(i) = w(2i)\end{array},\quad \begin{array}{l}b^0(i) = w(2i) r^0(i) = w(2i)\end{array}, i\geq 0.
J_{1-r^2(1-l_1^2)/4}\left(\frac{d}{2},\frac{1}{2}\right)=1-\frac{1}{D_{d}}\int_{1-r^2(1-l_1^2)/4}^1 t^{\frac{d}{2}-1}(1-t)^{-\frac{1}{2}}dt.
\hat{\mathbf{x}}_{blue}=\left(\mathbf{H}^*\mathbf{\Sigma}_z^{-1}\mathbf{H}\right)^{-1}\mathbf{H}^*\mathbf{\Sigma}_z^{-1}\mathbf{y}&=\mathbf{x}+\left(\mathbf{H}^*\mathbf{\Sigma}_z^{-1}\mathbf{H}\right)^{-1}\mathbf{H}^*\mathbf{\Sigma}_z^{-1}\mathbf{z}&=\mathbf{x}+\mathbf{e}_{blue},
\begin{array}{ll}\widehat D_{\Omega,\varepsilon}^* S(\bar x|\bar y)(y^*):=\big\{x^*\in X^*~\big| (x^*,-y^*)\in\widehat N_\varepsilon\big((\bar x,\bar y);\operatorname{gph}S|_\Omega\big) &\cap\big[J\big(T(\bar x;\Omega)\big)\times Y^*\big]\big\}.\end{array}
R_{1,2}^1 =R_{N_1}(u^1,u^2)=\frac{1}{N_1}\sum_{i=1}^{N_1}u_i^1 u_i^{2}\,\,\mbox{and}\,\,R_{1,2}^2=R_{N_2}(v^{1},v^{2})=\frac{1}{N_2}\sum_{i=1}^{N_2}v_i^1 v_i^{2}.
\left( \begin{array}{c}U^{+}_{k,0}U^{-}_{k,0}\end{array}\right)=\left( \begin{array}{c}u^{+}_{k,0}r^{k-\frac{1}{2}}u_{k,0}^{-}r^{-k-\frac{1}{2}}\end{array}\right).
c^{\left(0,J\right)}_{ij}W^{J}= \left<d^{\left(I',J'\right)}_{i,j}Z_{i},\Xi_{j}\right>W^{J'}+\dots,\quad\mbox{for suitable multi-indexes $J,J'\in\mathbb{N}^{q^{2}}$ and $I'\in\mathbb{N}^{q\left(p-q\right)}$,}
(2 \pi)^3 N_0 = (2 \pi)^3 \tilde{N}_0 (t) - \int \frac{d^3 k}{2 k^0}\tilde{a}_0 (t,\vec{k})^\dagger \tilde{q}_0 (t, \vec{k})- \int \frac{d^3 k}{2 k^0}\tilde{a}_0 (t,\vec{k}) \tilde{q}_0 (t, \vec{k})^*+ \int \frac{d^3 k}{2 k^0}|\tilde{q}_0 (t, \vec{k})|^2 \, .
\frac{\partial \varepsilon_{+}}{\partial \theta}= \mathrm{e}^{\theta}\left( 1 - \frac{C}{2 \pi}\int_{- \infty}^{\infty}d \theta^{\prime}\, \mathrm{e}^{- \theta^{\prime}}\frac{\partial L_{+}}{\partial \theta^{\prime}}\right)
\left( \begin{array}{c c c}{{\gamma^{0 0}}}&{{\gamma^{0 1}}}&{{\gamma^{0 2}}}\\{{\gamma^{1 0}}}&{{\gamma^{1 1}}}&{{\gamma^{1 2}}}\\{{\gamma^{2 0}}}&{{\gamma^{2 1}}}&{{\gamma^{2 2}}}\\ \end{array}\right) = \left( \begin{array}{c c c}{{\eta}}&{{\eta^{\prime}}}&{{\eta^{\prime}}}\\{{\eta^{\prime}}}&{{\eta}}&{{\eta^{\prime}}}\\{{\eta^{\prime}}}&{{\eta^{\prime}}}&{{\eta}}\\ \end{array}\right)
T = -\frac{1}{2}(\partial X)^2-\frac{1}{2}(\partial \phi)^2 -\frac{1}{\sqrt{2k}}\partial^2\phi-\frac{1}{2}(\partial u)^2 -\frac{1}{2}(\partial v)^2-\frac{1}{2}(\partial H)^2 -\eta\partial \xi,
\delta_{ij}^{(2s)}a = \frac{1}{\pi}((a^{p^s}+\pi\delta^{(s)}_{i}a)^{p^s}-a^{p^{2s}})+(\pi^{p^s-1}+ \delta^{(s)}_{i}\pi)((\delta^{(s)}_{j}a)^{p^s}+\pi (\delta^{(s)})_{ij}a).
\displaystyle \sum_{\gamma \in \Lambda}\left( \prod_{l=1}^{m}p_{j_l}\right) = \sum_{\gamma \in \Lambda}\mathbb{P}\left( \Tilde{Y}_1 = a_{j_1}, \ldots, \Tilde{Y}_m= a_{j_m}\right) \leq \mathbb{P}\left( \prod_{l=1}^m \Tilde{Y_l}\geq 1 \right),
\mathbf{B}^\emptyset = (1-\beta) \mathbf{I}, \mathbf{N}^\emptyset = \mathbf{I}- \beta \mathbf{P}, \mathbf{H}^\emptyset = \frac{1}{1-\beta}\mathbf{I}, \mathbf{A}^\emptyset = \frac{1}{1-\beta}(\mathbf{I}- \beta \mathbf{P}).
D_{1}={\pi^{-1}(1-w\bar \eta)^{-(2+\alpha\cdot\mathbf 1)}}\Big(I+\sum_{j=1}^{n}\alpha_j\Big(I+z_j\frac{\partial}{\partial z_j}\Big)\Big), &A_1(z,w,\eta)=\Big(z_1\Big(\frac{\sqrt{1-|\eta|^2}}{1-w\bar \eta}\Big)^{\alpha_1},\dots,z_n\Big(\frac{\sqrt{1-|\eta|^2}}{1-w\bar \eta}\Big)^{\alpha_n}\Big).
\rho_Y(x) = \begin{cases}\frac{\sqrt[3]{2 A^2}- 2 \big( q \gamma (x^2 - 3) - q \sqrt{\gamma}x + q + 3 \gamma \big)}{\pi 2^{\frac{5}{3}}\sqrt{3 q \gamma}\sqrt[3]{A}}x \in{\rm Supp}(\rho_Y) 0 \end{cases}
\frac 12\triangle|\mathring{Ric}|^2 \geq\frac{n+2}{n}|\nabla |\mathring{Ric}||^2-\sqrt{\frac{n-2}{2(n-1)}}\left(|W|^2+\frac{2n}{n-2}|\mathring{Ric}|^2\right)^{\frac 12}|\mathring{Ric}|^2+\frac{R}{n-1}|\mathring{Ric}|^2.
\bigl( \lambda_{t}(aw_2,w_2a,w_1,w_2,w_1,w_2,\ldots), \rho_{t}(aw_2,w_2a,w_1,w_2,w_1,w_2,\ldots)\bigr) = \bigl( \lambda_{r}(aw_2,w_2a,w_1,w_2,w_1,w_2,\ldots), \rho_{r}(aw_2,w_2a,w_1,w_2,w_1,w_2,\ldots)\bigr).
(\tilde{y}_{a_1}^M \cdots \tilde{y}_{a_k}^M) \underbrace{\tilde{x}_1 \cdots \tilde{x}_{\nu_1-kM}}_{{\rm all}\;{\rm in}\;C_1^*}\cdots \underbrace{\tilde{x}_{n-kM-\nu_r+1}\cdots \tilde{x}_{n-kM}}_{{\rm all}\;{\rm in}\; C_r^*}= z,
\mathcal{C}^{\pm}(n^2_1n_2,m;q) = \sideset{}{^\star}{\displaystyle\sum}_{a \, \mathrm{mod}\, q}\,\,\,\sum_{b \, \mathrm{mod}\, p_1}\,S\left( r\overline{(a+bq)}, \pm n_{2}; p_1 qr/n_{1}\right)\,e\left(-m \frac{\overline{(a+bq) p_2}}{p_1q}\right),
N = 6 : \quad A ={\frac{1}{3}}\left( \begin{matrix}{- 1}&{1}&{2}\\{2}&{1}&{0}\\{3}&{{\frac{3}{2}}}&{0}\\ \end{matrix}\right) ; \quad B ={\frac{1}{3}}\left( \begin{matrix}{1}&{2}&{0}\\{1}&{- 1}&{2}\\{{\frac{3}{2}}}&{3}&{0}\\ \end{matrix}\right)
\mu^{3/2}\iint_{Q_T}|\mathbf{w}_{x}|^4dxdt \leq C \mu \int_0^T \|\mathbf{w}_{x}\|_{L^\infty(\Omega)}^2 \left(\sqrt{\mu}\int_{Q_T}|\mathbf{w}_x|^2dx\right)dt\leq C.
\left( \begin{array}{c}{{M_{1}}}\\{{M_{2}}}\\{{M_{3}}}\\{{M_{4}}}\\ \end{array}\right) = \frac{1}{\sqrt{2}}\left( \begin{array}{c r r r}{{1}}&{{1}}&{{- 1}}&{{- 1}}\\{{1}}&{{- 1}}&{{1}}&{{- 1}}\\{{1}}&{{1}}&{{1}}&{{1}}\\{{1}}&{{- 1}}&{{- 1}}&{{1}}\\ \end{array}\right) \left( \begin{array}{c}{{L_{1}}}\\{{L_{2}}}\\{{L_{3}}}\\{{L_{4}}}\\ \end{array}\right)
I_5 =\int_0^r (\bar\phi+e_\delta-\delta)(1-\delta+\bar\phi+e_\delta)\cdot(e''_\delta \phi+e''_\sigma\psi){\rm{d}}z \leq\Vert (\phi,\psi)\Vert_{L^2}\cdot \Vert (e''_\delta,e''_\sigma)\Vert_{L^2}\cdot\Vert (\bar\phi+e_\delta-\delta)(1-\delta+\bar\phi+e_\delta)\Vert_{L^\infty},
S_{\mathrm{B I - Y M}}^{( \mathrm{q})}= \sqrt{\,{\frac{N \, V}{4 g_{0}^{2}}}\, \mathbf{F}_{c d \, \mu \nu}^{( 0 )}\, \mathbf{F}^{( 0 )}{}^{c d \, \mu \nu}\,}\equiv \sqrt{\,{\frac{N \, V}{4 g_{0}^{2}}}\mathrm{T r}\, \mathbf{F}_{\mu \nu}^{( 0 )}\, \mathbf{F}^{( 0 )}{}^{\mu \nu}\,}
U=\left(\begin{array}{cccc}-\sqrt{\frac{r_0}{2M}}\sqrt{\frac{2M}{r_0}}0 0 \sqrt{\frac{r_0}{2M}}0 0 0 0 0 0 \sqrt{\frac{r_0}{2M}}0 0 \sqrt{\frac{2M}{r_0}}\sqrt{\frac{r_0}{2M}}\end{array}\right)\,.
{\cal L}=-2f_\phi^2 \ln(S+S^+)|_{\theta^2\bar{\theta^2}}+\frac{1}{2}SW^\alpha_j W_\alpha^j|_{\theta^2}+\frac{1}{2}S^+\overline{W}{}^{\dot{\alpha}}_j \overline{W}{}_{\dot{\alpha}}^j|_{\bar{\theta}^2}
\psi ( \alpha , \beta , \beta^{\prime}, s , s^{\prime}) = \frac{\varphi_{( \alpha + \beta + \beta^{\prime}, s + s^{\prime})}\varphi_{( \alpha + \beta - \beta^{\prime}, s - s^{\prime})}}{\varphi_{( - \alpha + \beta + \beta^{\prime}, - s + s^{\prime})}\varphi_{( \alpha - \beta + \beta^{\prime}, - s - s^{\prime})}}.
\Phi^{1,2,3}\ \Phi^{3,2,1}=1 e^{\frac{t_{13}+t_{23}}2}=\Phi^{3,1,2}\e^{\frac{t_{13}}2}\ \left(\Phi^{1,3,2}\right)^{-1}\e^{\frac{t_{23}}2}\ \Phi \Phi^{1,2,34}\ \Phi^{12,3,4}=\Phi^{2,3,4}\ \Phi^{1,23,4}\ \Phi^{1,2,3}.
T_{f}{\Gamma^{*}_{\bar{g}}}+{\Gamma^{*}_{\bar{f}}}{S_{g}}= T_{g}{\Gamma^{*}_{\bar{f}}}+{\Gamma^{*}_{\bar{g}}}{S_{f}}, \Gamma_{f}T_{g}+S_{f}{\Gamma_{g}}=\Gamma_{g}T_{f}+S_{g}{\Gamma_{f}}.
\begin{matrix}I\equiv|E_{x}|^{2}+|E_{y}|^{2}=|E_{a}|^{2}+|E_{b}|^{2}=|E_{r}|^{2}+|E_{l}|^{2}\\Q\equiv|E_{x}|^{2}-|E_{y}|^{2},\\U\equiv|E_{a}|^{2}-|E_{b}|^{2},\\V\equiv|E_{r}|^{2}-|E_{l}|^{2}.\end{matrix}
\cosh(s_{k+1}) = \cosh(s_k) / \cosh(\eta_k \|g_k\|) = \cosh(s_k) \sqrt{1 - \tanh(\eta_k \|g_k\|)^2}&= \cosh(s_k) \sqrt{1 - \frac{1}{\zeta_{s_k}^2}\cdot \frac{(f(x_k) - f^*)^2}{\|g_k\|^2}},
\begin{cases}\mathcal{M}(\Omega^q_+) ={}\{(i,j) \in \Omega^q_+ \cap{\cal A}(t_q)_> \mid x^q_{i}(t_q + \Delta t_q ) = x^*_j(t_q) \}, \mathcal{M}(\Omega^q_-) ={}\{(i,j) \in \Omega^q_- \cap{\cal A}(t_q)_< \mid x^q_{j}(t_q + \Delta t_q ) = x^*_i(t_q) \}. \end{cases}
\frac{V_{A}}{V_{B}}= \frac{\frac{m_{A}}{c_{A}}}{\frac{m_{B}}{c_{B}}}= \frac{\frac{0 . 1}{2 0 0}}{\frac{1 . 7}{2 0 0 0}}= \frac{1}{2 0 0 0}\times \frac{2 0 0 0}{1 . 7}= \frac{1 0}{1 7}.
\begin{cases}\frac{d}{dt}y(t)=\frac{2A\sqrt{6}\,y(t)z(t)^6+2z(t)^2+y(t)^2}{12\,y(t)^3z(t)^8}, \frac{d}{dt}z(t)=-\frac{2A\sqrt{6}\,y(t)z(t)^4+1}{12\,y(t)^2z(t)^7}, y(0)=1, z(0)=1.\end{cases}
| \Phi \rangle = \sum_{r , s}\, b_{\overline{{\jmath}}_{1}. . . \overline{{\jmath}}_{s}}^{i_{1}. . . i_{r}}\, \lambda_{i_{1}}\dots \lambda_{i_{r}}\, \psi^{\overline{{\jmath}}_{1}}\dots \psi^{\overline{{\jmath}}_{s}}| 0 \rangle ~ ,
\frac1N\sum_{k=0}^N (-1)^k\binom{N}{k}g^kD^N(g^{N+1-k}f) &-\frac{N+1}N g\sum_{k=0}^N(-1)^k\binom{N}{k}g^kD^N(g^{N-k}f) &+\sum_{k=0}^N(-1)^k\binom{N}{k}g^{k+1}D^N(g^{N-k}f)=0.
D_{\xi}\,\phi_\xi(x)=\sum_{k\in \mathbb{Z}^3}\left[\sum_{j=1}^3\Big(\frac{2\pi}{\ell}k_j+\xi_j\Big)\cdot\alpha_j+\beta\right]\widehat{\phi}_{\xi}(k)\,e^{i\big(\frac{2\pi k}{\ell}+\xi\big)\cdot x}.
\sum_{i=0,1,2}|\nabla_v \mathcal{K}^{\mu,i}_{k,j; n,l}(y, v, V(s))| + \sum_{a=3,4}2^{-\max\{n, (\gamma_1-\gamma_2) M_{t^\star}\}-\epsilon M_t}|\nabla_v \mathcal{K}^{\mu,a}_{k,j ; n,l}(y, v, V(s))|
\left|\partial^\beta f(y)-\partial^\beta f(z)\right| \leq\sum_{k=1}^7\left|\partial^\beta f\left(re^{i\phi\frac{k}{7}}\right)-\partial^\beta f\left( r e^{i\phi\frac{k-1}{7}}\right)\right| &\lesssim\sum_{k=1}^7\left(\frac{r}{2}\right)^{\varepsilon}r^{-n-N-\varepsilon}\sim|y|^{-n-N},
\lim_{N \to \infty}\frac{1}{N \theta}\log \left( \int_{\mathbb{R}^{d \times N}}\exp \left( - \beta \mathcal{H}_{N}\right) d X_{N}\right) = \inf_{\mu \in \mathcal{P}(\mathbb{R}^{d})}\{\mathcal{E}_{V}(\mu) +\frac{1}{\theta}\left({\rm ent}[\mu] \right) \},
\times\left[\vert\beta\vert^2\log (1+\vert\beta\vert^2\vert z\vert^2 + \vert\lambda\vert^2 \vert a\vert^{3/2}\vert z\vert^2) - \frac{\vert\lambda\vert^2 \vert a\vert^{3/2}}{1+\vert\beta\vert^2\vert z\vert^2 + \vert\lambda\vert^2\vert a\vert^{3/2}\vert z\vert^2}\right]_{\vert a\vert^{3/2}}^{\vert a\vert^4}.
A_m(x)=S_{m}(x)+S_{k_{1}m}(x)+\cdot\cdot\cdot+S_{k_{t-1-j_0}m}(x) &=\Bigr(S_m(x_2)+S_{k_{1}m}(x_2)+\cdot\cdot\cdot+S_{k_{t-1-j_0}m}(x_2)\Bigr) &\quad\quad+\Bigr(R_m(x,x_2)+R_{k_{1}m}(x,x_2)+\cdot\cdot\cdot+R_{k_{t-1-j_0}m}(x,x_2)\Bigr) &=A_{m}(x_{2})(1+v_{m}(x,x_2)),
\sigma^{1}= \left( \begin{array}{}{{c ]{c c}0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) , \quad \sigma^{2}= \left( \begin{array}{}{{c ]{c c}0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) , \quad \sigma^{3}= \left( \begin{array}{}{{c ]{c c}1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right)
\int_{\Gamma_0}\widehat{\zeta}_N\widehat{\psi}_N\gamma\,\, d \widehat{s}=\int_{\Gamma(t)}\zeta_N\psi_N\,\, d s=\int_{\Gamma(t)}\zeta \psi_N\,\, d s=\int_{\Gamma_0}\widehat{\zeta}\widehat{\psi}_N\gamma\,\,d \widehat{s}\quad\forall~\widehat{\psi}_N\in X_N(0),
L_{\mathrm{f l u c}}^{\mathrm{s t r i n g}}= \frac{\pi}{1 2 R_{p}}- \sum_{i = 1}^{2}\frac{\omega v_{i}\bar{\gamma}_{i}}{\pi}\left[ \ln \left( \frac{M m_{i}}{\sigma}\right) + 1 \right] + \frac{1}{2}\omega + \omega f ( v_{1}, v_{2}) .
( \partial_{\bar{l}}A_{0}^{z}) ( \partial_{l}A_{0}^{\bar{z}}) - ( \partial_{\bar{l}}A_{0}^{\bar{z}}) ( \partial_{l}A_{0}^{z}) = \frac{{\cal L}_{0}}{2}= \frac{4}{\cosh ( l )^{2}}- 1 \, .
\|B^k\|_{L^{\infty}(0,T; H^{\frac52-\alpha})}\leq \ \|B(0)\|_{H^{\frac{5}{2}-\alpha}}+C \|B^{k-1}\|_{L^{2}(0,T; H^{\frac52-\frac{\alpha}{2}})}\|B^{k}\|_{L^{2}(0,T; H^{\frac52-\frac{\alpha}{2}})}.
\big | \big | \tilde{\delta}_{~ v}^{u}\big | \big | = \left( \begin{matrix}{{0_{[ N - 2 ] \times [ N - 2 ]}}}&{{0_{[ N - 2 ] \times [ 2 ]}}}\\{{0_{[ 2 ] \times [ N - 2 ]}}}&{{1_{[ 2 ] \times [ 2 ]}}}\\ \end{matrix}\right) ~ ~ ,
\begin{array}[c]{rl}A(t)= \alpha_{3}(t)+\beta_{3}(t)p(t)+\gamma_{3}(t)K_{1}(t)+\alpha_{1}(t)p(t)+\beta_{1}(t)p^{2}(t) & +\gamma_{1}(t)K_{1}(t)p(t)+\alpha_{2}(t)q(t)+\beta_{2}(t)p(t)q(t)+\gamma_{2}(t)K_{1}(t)q(t),\end{array}
[e^{iz_1s}\tilde{E}_i(-iz_1s) + e^{-iz_1s}\tilde{E}_i(iz_1s)] - [e^{iz_0s}\tilde{E}_i(iz_0s) + e^{-iz_0s}\tilde{E}_i(-iz_0s)] = 2 \pi i [ e^{iz_0s}+ e^{-iz_0s}]
\hat{E}_n(W_\ell|Z=z^r) =\frac{\sum_{i=1}^n W_{\ell,i}\mathbf{1}(Z_i=z^r)}{\sum_{i=1}^n \mathbf{1}(Z_i=z^r)}, \hat{E}_n(Y_1=s,Y_2=t|Z=z^r) =\frac{\sum_{i=1}^n \mathbf{1}(Y_{1,i}=s,Y_{2,i}=t,Z_i=z^r)}{\sum_{i=1}^n \mathbf{1}(Z_i=z^r)},
Z_{1}X_{2}Z_{3}=[\begin{matrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}\\c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}\\s_{2}s_{3}&c_{3}s_{2}&c_{2}\end{matrix}]
\left[\mathfrak{S}^{r,t}_{g,d}(C)\right] = g! \cdot n \frac{\prod\limits_{i=2}^t \dfrac{i! \cdot (n-i)}{(s-1+i)!}\cdot \prod\limits_{j=2}^{r+1-t}\dfrac{j! (n+j)!}{(s+n-1+j)! (n-t-1+j)!(n-1+j)}}{g(s-1)!(s+n-1)!(t-1)!(r-t)!}\left(\theta -\frac{g}{n}x \right).
E_{1}( m , \lambda ) = \frac{3}{2}m + \frac{1 5}{1 6}\frac{\lambda}{m^{2}}- \frac{1 6 5}{1 2 8}\frac{\lambda^{2}}{m^{5}}+ \frac{3 9 1 5}{1 0 2 4}\frac{\lambda^{3}}{m^{8}}
\delta \kappa = - \kappa_{r}- \frac{\lambda_{r}^{2}}{6 0 \left[ \frac{4 5 \lambda_{r}^{2}}{4 \pi}\left( 2 + \ln \left[ \frac{m_{r}^{2}+ \frac Q C}{4 \Lambda^{2}}\right] \right) + \frac{5 4 \kappa_{r}^{2}}{\pi ( m_{r}^{2}+ \frac Q C )}\right]}
\mathbf{E}\left(e^{-\lambda b_n^{1/\alpha}\nu_n(B)}\right)=\mathbf{E}\left(e^{-\lambda b_n^{1/\alpha}\sum_{x\in B}\xi^n_x}\right)=\left(1-\lambda^\alpha b_n \Gamma(1-\alpha)+o(b_n)\right)^{\mu_n(B)}\rightarrow e^{-\lambda^\alpha \Gamma(1-\alpha)\mu(B)}.
\mathcal{L}_{\rm CP-even}=i\,\bar\psi_i\gamma^\mu\partial_\mu \psi^i+\bar\psi_2\partial_5\psi^1-\bar\psi_1\partial_5\psi^2-\omega(\bar\psi_1\psi^1+\bar\psi_2\psi^2)-2\Lambda\delta(y)\left[\bar\psi_1\psi^1+\rho\,\bar\psi_2\psi^2\right]
|n({\vec p}_1^+)n({\vec p}_1^-) \cdots n({\vec p}_N^+)n({\vec p}_N^+)\rangle \propto [d^{\dag}({\vec p}_1^+)]^{n({\vec p}_1^+)}[c^{\dag}({\vec p}_1^-)]^{n({\vec p}_1^-)}\cdots [d^{\dag}({\vec p}_N^+)]^{n({\vec p}_N^+)}[c^{\dag}({\vec p}_N^+)]^{n({\vec p}_N^+)}|0 \rangle
\mathrm{}\begin{array}{lll}[H(z),X(w)]=2X(w)\delta(\frac{w}{z}) &[H(z),Y(w)]=-2X(w)\delta(\frac{w}{z}) &[X(z),Y(w)]=H(w)\delta(\frac{w}{z})-kw\partial_w\delta(\frac{w}{z})\end{array}
\beta(m)\beta(\theta(m)+n)+\beta(n)\beta(m+\theta(n)) =\frac{b_ic_i}{c_i+l_m}\frac{b_jc_j}{c_i+c_j+l_m+l_n}+\frac{b_jc_j}{c_j+l_n}\frac{b_ic_i}{c_i+c_j+l_m+l_n}&=\frac{b_ib_jc_ic_j}{(c_i+l_m)(c_j+l_n)}.
a^{\mp}= a^{\mp}(\vec \sigma \cdot \vec L +{\bf 1}) =\frac{1}{\sqrt{2}}\left\{\mp \Sigma_1 \left(\frac{\partial}{\partial r}+\frac{1}{r}\right) \pm \frac{1}{r}(\vec \sigma \cdot \vec L +{\bf 1})\Sigma_1 \Sigma_3 - \Sigma_1 r \right\}
V(P, I) = \sum_{\{n_{i, k}\}}\sum_{j \geq 0}\left(\frac{1}{j!2^j}\prod_{i, k \geq 0}\frac{1}{n_{i, k}!}\right) \hbar^{p(\{n_{i, k}\}, j)}\left(\sum_l \partial_{P_l^{(1)}}\partial_{P_l^{(2)}}\right)^j\prod_{i, k \geq 0}I_{i, k}^{n_{i, k}}.
{\cal L}^{q , 1}\! ( u , v ) ={\cal R}^{q , 1}\! ( u , v ) \quad \mathrm{a n d}\quad{\cal L}^{q , r \! - \! 1}\! ( u , v ) ={\cal R}^{q , r \! - \! 1}\! ( u , v ) \quad \mathrm{i m p l y ~ t h a t}\quad{\cal L}^{q , r}\! ( u , v ) ={\cal R}^{q , r}\! ( u , v )
\alpha_p^{\sharp}(\mathbf h, \breve{\mathbf g}, \pmb{\phi}) = \Omega_p(1)\mathrm{vol}(\Gamma_{00}) + \Omega_p(r_1)\mathrm{vol}(\Gamma_{00}r_1\Gamma_{00}) + \Omega_p(r_u)\mathrm{vol}(\Gamma_{00}r_u\Gamma_{00}),
\phi ( \psi \otimes \bar{v}; z , \bar{z}) \; \phi ( \chi \otimes \bar{w}; \zeta , \bar{\zeta}) \; = \; \phi \left( \phi \left( \psi \otimes \bar{v}; z - \zeta , \bar{z}- \bar{\zeta}\right) ( \chi \otimes \bar{w}) ; \zeta , \bar{\zeta}\right) ,
\alpha_{( \cdot ) , ( \cdot )}^{- 1 , 1}= A_{( \cdot ) , ( \cdot )}^{- 1 , 1}+ \frac{1}{N^{2}}A_{( \cdot ) , ( \cdot )}^{0 , 0}- \frac{1}{N}B_{( \cdot ) , ( \cdot )}^{0 , 0}- \frac{1}{N}B_{( \cdot ) , ( \cdot )}^{- 1 , 1}.
{\cal Z}=\frac{1}{2\pi i}\int_{{\cal C}_0}\frac{dz}{z+e^{-\frac{\pi}{g^2_{nc}\theta}}}\exp\left[z\Phi(-z;\frac{3}{2};1)\frac{A_{nc}}{\sqrt{2\pi g_{nc}^2\theta^3}}\,\,\right],
\lim_{l \to \infty}\limsup_{n \to \infty}\| r_n^l \|_{L^s (\mathbb{R}^N)}=0, \forall s \in \left( 2, 2^* \right), 2^* = \begin{cases}\frac{2N}{N-2}, N\ge 3, +\infty, N=1,2.\end{cases}
\arg\min_{\vec{X}\in \mathbb{R}^{m \times n}}\Biggl\lbrace F(\vec{X}):= \dfrac{1}{2}\|\vec{Y - X}\|_F^2 + \lambda_0\sum_{i=1}^{k}\phi(\sigma_i(\vec{X});a_0) &+ \lambda_1\sum_{i=1}^{m}\sum_{j=1}^{n}\phi(\vec{X}_{i,j};a_1) \Biggr\rbrace,
\begin{aligned}\frac{d}{dt}\mathcal{F}^N_{l+1}(t)+\frac{C_7}{2}\left(\|\nabla^{l+2}\varrho\|_{L^2}^2+\|\nabla^{l+2}\varrho\|_{H^{N-l-2}}^2+\|\nabla^{l+2}u\|_{H^{N-l-1}}^2+\|\nabla^{l+2}n\|_{H^{N-l}}^2\right)\le 0.\end{aligned}
S_{2m+1,\sigma}(t,\pi)\leq\dfrac{1}{2\pi(2m+2)!}\,\bigg(\dfrac{3}{2}-\sigma\bigg)^{2m+2}\log C(t,\pi) -\dfrac{1}{\pi(2m)!}\,\displaystyle\sum_{\gamma}g^{-}_{2m+1,\sigma,\Delta}(t-\gamma)+ O_m(d).
p =(1,1) = H_1 \cap H_2 \cap H_3 q =(1, \zeta_3^2)=H_1 \cap H_3 \cap H_4 r =(1, \zeta_3) \subset H_1 \cap H_3 s =(\zeta_3,1) = H_2 \cap H_4 t =(1, -\zeta_3) \subset H_1 \cap H_4
\mathbf{E}( \mathbf{x}) = \frac{1}{4 \pi}\int \rho_{e}( \mathbf{x}^{\prime}) \frac{\mathbf{r}}{r^{3}}\, d^{3}x^{\prime}\, - \frac{1}{4 \pi}\int \mathbf{J}_{m}( \mathbf{x}^{\prime}) \,{\bf \times}\, \frac{\mathbf{r}}{r^{3}}\, d^{3}x^{\prime}
q^{2dk-d}\left| \sum_{{\bf m}\in S_0}\left(\prod_{j=1}^k \widehat{E_j}({\bf m})\right)\right|^2 -\nu^2_k(0) \le -\mbox{A}\overline{\mbox{B}}- \overline{\mbox{A}}\mbox{B}\le 2 |\mbox{A}| |\mbox{B}|.
{}{}\textrm{maximize :}\;\;\;E \left\{\sum_{i=1}^{M_R}\ln \left( 1 + \frac{P}{N_0}l_i \right) \right\}&\textrm{s.t.}\;\;\;{\mathcal V}= \bar{\eta}E\left\{P \right \}\leq Q, \;\; 0 \leq P \leq P_{max}.
M_{t_i^k, t_{i+1}^k}= \int_{t_i^k}^{u^k_i}\int_{u^k_i}^{t^k_{i+1}}(P_{zr}\nabla \bar{f}_r)(W^H_r + \phi_s) (z-r)^{H-1/2}dr dW_z = \int_{t_i^k}^{u^k_i}h^{(x,y),k}_z dW_z
\|\triangle^T\triangle e_\lambda\|_{L^2(\partial M)}=\|\triangle^T\Xi\partial_\nu e_\lambda\|_{L^2(\partial M)}\lesssim\lambda^3\|\partial_\nu e_\lambda\|_{L^2(\partial M)}\lesssim\lambda^2\|\triangle e_\lambda\|_{L^2(\partial M)}.
x_\gamma = \left[ \begin{array}{ccccccc}1 V_{d1}\ldots V_{qn}V_{d1}^2 V_{d1}V_{d2}\ldots \end{array}\right. \left.\begin{array}{cccccc}\ldots V_{qn}^2 V_{d1}^3 V_{d1}^2 V_{d2}\ldots V_{qn}^\gamma \end{array}\right]^\intercal
\Phi_{2 ,\frac{N}{2}}(t) := e^t - \displaystyle{\sum_{j = 0}^{j_{\frac{N}{2}}- 2}\frac{t^j}{j !}}\ \ \ \mbox{and}\ \j_{\frac{N}{2}}= \min \left\{j : j \geq \frac{N}{2}\right\}.
\small Tr \left ( e^{t\Delta_{\Omega, K}}- e^{t\Delta_{\Omega, 0}}\right ) = \frac{1}{2 \pi}\int_{\partial \Omega}K(\sigma) d\sigma + \frac{\sqrt{t}}{8 \sqrt{\pi}}\int_{\partial \Omega}\left ( K(\sigma) \kappa(\sigma)+2 K^2(\sigma) \right )d\sigma+ O(t).
P(y) =c_0+\sum_{i=1}^{n+1}a_iy_i+\sum_{k\in \{1,\dots,n-1\},\ell\in \{n,n+1\}}a_{k\ell}y_ky_\ell &+ \left(c_1y_n^3+c_2y_n^2y_{n+1}+c_3y_ny_{n+1}^2+c_4y_{n+1}^3\right).
\|u\|_{C^{0,\alpha}(B_{d/8}(x))}\le \frac{C (N,\mu,\alpha)}{d^{\alpha}}\left ( \inf_{B_{d/8}(x)}u+d\|F\|_{L^N(B_{d/4}(x))}\right ) &\le \frac{C (N,\mu,\alpha)}{d^{\alpha}}\left ( u(x)+d\|F\|_{L^N(B_{d/4}(x))}\right ).
W(\xi_0,n,x,\bar{\theta})= E_{\xi_0}\biggl[\varphi\biggl(x+\sum_{i=1}^n Z^{\bar{\theta}}\biggr)\biggr] = E_{\xi_0}\biggl[V\biggl(\frac{\xi_0f_1(X_1)}{\xi_0 f_1(X_1)+(1-\xi_0)f_2(X_1)}, n-1, x+X_1\biggr)\biggr].
\left. K_{f}e^{{\frac{i}{\hbar}}S ( \Phi , \Phi_{a}^{*}, \bar{\Phi})}\right|_{\Phi_{a}^{*}= \bar{\Phi}= 0}= e^{{\frac{i}{\hbar}}[ S ( F ( \Phi ) , 0 , 0 ) + O ( \hbar ) ]}.
\left[ 2 \left( 3 - q \right) - p \right] H_{3}+ \left[ 2 \left( 3 - q \right) - 3 p \right] H_{4}= m_{0}\sum_{j = 1}^{N}\left( \frac{m_{0}}{\vert E_{j}\vert}- \frac{\vert E_{j}\vert}{m_{0}}\right) .
g(x+\alpha_{0},y+\beta_{0}) = f(x+\alpha_{0},y+\beta_{0}) - (x+\alpha_{0}+c(y+\beta_{0}))^{k}\cdot f_{k}(y) &= f(x,y) - \delta' + \delta - (x+cy)^k \cdot f_{k}(y) = g(x,y) - \delta' + \delta.
[ \hat{\cal H}_{0}^{\pm}( n ) , \hat{\cal H}_{0}^{\pm}( m ) ]_{-}= \pm \frac{2 \pi}{2 \pi}{L}( n - m ) \hat{\cal H}_{0}^{\pm}( n + m ) \pm \frac{1}{3}( \frac{2 \pi}{2 \pi}{L})^{2}n ( n^{2}- 1 ) \delta_{n , - m}, \,
\displaystyle \left( 1-a \right) \mbox{$_5$F$_4$}(1,1,1,1,a;\,2,2,2,2;\,1)=1/6\, \left( \Psi \left( 2-a \right) +\gamma \right)^{3}+ \left( 1/12\,{\pi}^{2}-1/2\,\Psi^{\prime}\left(2-a \right) \right) \left( \Psi \left( 2-a \right) +\gamma \right) \mbox{}+1/3\,\zeta \left( 3 \right) +1/6\,\Psi^{(2)}\left(2-a \right)
\left| - 2\int \Delta \xi \cdot G_{h/2}\ast u \,\nabla G_{h/2}\ast u\,dx \right|\lesssim \left\|\nabla^3 \xi \right\|_\infty \int \left(1- \left|G_{h/2}\ast u\right|^2\right) dx = \left\|\nabla^3 \xi \right\|_\infty h E_h(u) =o(1),
\sum_{n = 2}^N \frac{\psi(n)}{\log n}= N \frac{\psi(N)}{\log N}- 2 \frac{\psi(2)}{\log 2}+ \sum_{n = 2}^{N-1}n \bigg(\frac{\psi(n)}{\log n}- \frac{\psi(n+1)}{\log(n+1)}\bigg).
\begin{cases}\begin{array}{ccllll}x_{1}' = 0 x_{2}' = \frac{1-\eta}{2-\eta}x_{1}y_{1}+\frac{1-\eta}{2-\eta}x_{2}y_{1}y_{1}' = 0 y_{2}' = \frac{1}{2-\eta}x_{1}y_{1}+x_{1}y_{2}+\frac{1}{2-\eta}x_{2}y_{1}+x_{2}y_{2}\end{array}\end{cases}
p_{2}= \int_{0}^{\infty}\, \frac{d \omega}{2 \pi}\int_{0}^{\omega}\, \frac{d \omega^{\prime \prime}}{2 \pi}\vert \langle o u t \, \omega \omega^{\prime \prime}\vert i n \rangle \vert^{2}= \frac 1 2 \, \int_{0}^{\infty}\, \frac{d \omega}{2 \pi}p_{2 \omega}.
D^l(\varphi_1^{s_1}\varphi_2^{r_2}\dotsm \varphi_k^{r_k})=\sum_{\substack{j_1+j_2+\cdots+j_k=l j_1,j_2,\ldots,j_k\geq0}}\,\frac{l!}{j_1!j_2!\cdots j_k!}\;D^{j_1}(\varphi_1^{s_1})D^{j_2}(\varphi_2^{r_2})\cdots D^{j_k}(\varphi_k^{r_k}).
\hat{\delta}_{V}(z-z')=\int dk\Theta (\Lambda -|k|)\xi^{*}_{V}(k,z')\xi_{V}(k,z)=\frac{1}{2\pi}\int_{-\Lambda}^{\Lambda}dke^{ik(z-z')}=\frac{1}{\pi}\frac{\sin [\Lambda (z-z')]}{(z-z')}
(\begin{matrix}|\varphi_{\alpha}\rangle\\ |\varphi_{\beta}\rangle\\ |\varphi_{\gamma}\rangle\\ \end{matrix})=(\begin{matrix}\Omega_{\alpha1}&\Omega_{\alpha2}&\Omega_{\alpha3}\\ \Omega_{\beta1}&\Omega_{\beta2}&\Omega_{\beta3}\\ \Omega_{\gamma1}&\Omega_{\gamma2}&\Omega_{\gamma3}\\ \end{matrix})(\begin{matrix}|\psi_{1}\rangle\\ |\psi_{2}\rangle\\ |\psi_{3}\rangle\\ \end{matrix})
\left\|\frac{\partial^d}{\partial x_1\cdots\partial x_d}K_{\boldsymbol{J}_n}(f)\right\|_2\lesssim\sqrt{\sum_{j_1=N_1-1}^{J_{n,1}-1}\cdots\sum_{j_d=N_d-1}^{J_{n,d}-1}\sum_{\boldsymbol{k}}2^{2\sum_{l=1}^dj_l}\theta_{\boldsymbol{j},\boldsymbol{k}}^2}
U_{\phi}= \frac{1}{V_{p + 1}}\,{\underbrace{J_{\phi}~ J_{\phi}}}= \frac{T_{p}^{2}}{8}\, V_{p + 1}\, \frac{\left[ n^{2}\, ( 3 - p ) - 2 m^{2}\right]^{2}}{\Delta_{m , n}}\,{\underbrace{\phi ~ \phi}}
\left|(Kv)(t)-u_1\right| = \left| \frac{\lambda}{\Gamma (2\alpha )}\int_{\beta}^{t}(t-s)^{2\alpha -1}\frac{f(s, v(s))}{\left( \int_{0}^{t}f(x, v)\, dx\right)^{2}}ds\right| &\leq \frac{\lambda M}{\Gamma (2\alpha +1) c_{1}^{2}}h^{2\alpha}\leq b.
h_{0}\propto \frac{1}{\sqrt{2 n}}\quad , \quad \dot{h}_{0}\propto \sqrt{\frac{n}{2}}\quad , \quad \ddot{h}_{0}\propto \frac{n^{3 / 2}}{\sqrt{2}}\quad , \quad{\stackrel{. . .}h}_{0}\propto \frac{n^{5 / 2}}{\sqrt{2}}\quad .
s_{0,i}\cdot_\delta\widehat{\lambda^{(i-1)}}=(\lambda_i^{(i-1)}-i,\lambda^{(i-1)}_1-1,\dots,-|\lambda^{(i-1)}|+\delta,\dots,\lambda^{(i-1)}_n-n)-\rho(\delta) &=(\lambda_i^{(i-1)}-i-\delta,\lambda^{(i-1)}_1,\lambda^{(i-1)}_2,\dots,-|\lambda^{(i-1)}|+\delta+i,\dots,\lambda^{(i-1)}_n).
\sum_{j=1}^2 \| D^s \partial_j (gf ) - g D^s \partial_j f \|_{L^2(\mathbb R^2)}\lesssim_s \| D^s f \|_2 \| \widehat{D g}(\eta) \|_{L^1_{\eta}}+ \| f \|_2 \| \widehat{D^{1+s}g}(\eta) \|_{L^1_{\eta}}& \lesssim_{s,\epsilon}\| D^s f \|_2 \| g \|_{H^{2+\epsilon}}+ \| f \|_2 \| g \|_{H^{2+s+\epsilon}},
(1-z)^p\cdot\Phi(p)\equiv (1-z)^p\cdot p\Phi'(0)+(1-z)^p\cdot\Phi(0) \equiv p\big(\Psi_1'(0)-z^p\cdot\Psi_2'(0)\big)+\big(1-z^p-(1-z)^p\big)\cdot\Psi_1(0)+(1-z)^p\cdot\Phi(0) \equiv \big(\Psi_1(p)-\Psi_1(0)\big)-z^p\cdot\big(\Psi_2(p)-\Psi_2(0)\big)+(1-z^p)\cdot\Psi_1(0) = \Psi_1(p)-z^p\cdot\Psi_2(p)\pmod{p^2}.
S = \frac{1}{4 g_{Y M}^{2}}\mathrm{T r}\left( V \sum_{\alpha \beta}( F_{\alpha \beta}+ \phi_{\alpha \beta}\cdot{\bf 1}) g^{\alpha \mu}g^{\beta \nu}( F_{\mu \nu}+ \phi_{\mu \nu}\cdot{\bf 1}) \right) .
\underbrace{\phi^{i_{2 ( k + r )}^{\prime}}\phi^{i_{2 ( k + r ) - 1}^{\prime}}}\ldots \underbrace{\phi^{i_{2}^{\prime}}\phi^{i_{1}^{\prime}}}F_{a_{p - 2 r + 1}a_{p - 2 r + 2}}\ldots F_{a_{p - 1}a_{p}}) .