harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L=\int\limits_{t_{0}}^{t_{1}}\left( \sqrt{({\dot{x}^{1}})^2+{(\dot{x}^{2}})^2}+b\left( \cos \theta(x^1,x^2) \dot{x}^{1}+ \sin \theta(x^1,x^2) \dot{x}^{2}\right) \right) dt.
\phi_{(\alpha,\beta,\gamma)}^j(x) = \alpha \psi_1(e_1),\phi_{(\alpha,\beta,\gamma)}^j(y) = \beta \psi_1(e_2),\phi_{(\alpha,\beta,\gamma)}^j(z) = \gamma \rho^j \psi_1(e_2^{-1}e_1^{-1})
\sum_{2r_0+r_1=d+1}{d+1 \choose r_0}{d+1-r_0 \choose r_1}\frac{(d+2)!}{2^{r_2}}\left[{n_1 \choose d+1}{n_2 \choose d+2}+{n_1 \choose d+2}{n_2 \choose d+1}\right],
\gamma_k = \frac{|\mathbf{h}_k^H \mathbf{\Theta}_i \mathbf{G}\mathbf{w}_k|^2}{\sum_{\ell \in \mathcal{K}\setminus k}|\mathbf{h}_k^H \mathbf{\Theta}_i \mathbf{G}\mathbf{w}_\ell|^2 + \sigma_k^2}.
\hat h_{\rm red}= \sum_{\vec{b}}\left\{-\frac{g^2 LM}{4\sqrt2}\frac{1}{J}\frac{\delta}{\delta\varphi_{\vec b}}J\frac{\delta}{\delta\varphi_{\vec b}}+\left(\frac{4\sqrt2}{g^2LM}\right) \sum_{\vec{\varepsilon}}(\varphi_{\vec b}-\varphi_{\vec b+\vec\varepsilon})^2\right\}\, .
(a(x)-a(y))\langle x,x-y\rangle+a(y)|x-y|^{2}=a(y)(|x-y|^{2}+\langle x,x-y\rangle) &=a(y)(\langle x-y,x-y\rangle-\langle x,x-y\rangle & a(y)\langle x-y,-y\rangle &=a(y)(|y|^2-\langle x,y\rangle) &\geq a(y)(|y|^2- |x||y|)\geq0
\alpha\odot x=[\frac{x_{1}^{\alpha}}{\sum_{i=1}^{D}x_{i}^{\alpha}},\frac{x_{2}^{\alpha}}{\sum_{i=1}^{D}x_{i}^{\alpha}},...,\frac{x_{D}^{\alpha}}{\sum_{i=1}^{D}x_{i}^{\alpha}}]=C[x_{1}^{\alpha},...,x_{D}^{\alpha}]\forall x\in S^{D},\alpha\in\mathbb{R}
\displaystyle{\mathcal{G}_{K , L}\left( A , B ; \omega^{m}\right) \equiv \frac{1}{\sqrt{( K + L ) ! K ! L !}}\partial_{x}^{K}\partial_{y}^{L}\prod_{s = 0}^{K + L - 1}\left( \omega^{- s / 2}x \mathbf{A}+ \omega^{s / 2}y \mathbf{B}\right) \vert_{x = y = 0}.}
t ( 0 ) = 0 , \; \; t ( \tau_{\mathrm{h o r}}) = \infty , \; \; t ( \frac{\omega_{2}}{2}) = \frac{E l r_{m}}{2 \sqrt{M H_{2}}\sqrt{r_{m}^{2}- M l^{2}}}\frac{Z [ \epsilon , k ]}{\sqrt{r_{m}^{2}( 1 - k^{2}) + k^{2}M l^{2}}},
{\cal I}_{n}( \kappa ) ={\frac{n}{( n + 1 ) \kappa}}+ \sum_{k = 0}^{K}{\frac{\Gamma ( k - 1 / n )}{\Gamma ( - 1 / n ) \Gamma ( k + 1 )}}I_{k}\kappa^{k}+{\cal O}( \kappa^{K + 1})
\Delta_{P}(f_{X}(X|\alpha,p,\nu))=\frac{\Gamma(\frac{1}{2}(\nu+p))}{(\nu\pi)^{p/2}\alpha^{p/2}\Gamma(\frac{1}{2}\nu)}(1+\frac{X^{T}X}{\alpha\nu})^{-(\nu+p)/2}dx_{1}...dx_{p}
\frac{d^{2}x^{i}}{ds^{2}}+\Gamma_{jk}^{i}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds}+\frac{1}{\left( \frac{dS\left( t\right)}{dt}\right)^{2}}\left(\left( \frac{d^{2}S\left( t\right)}{dt^{2}}\right) +\phi\left( t\right)\frac{dS\left( t\right)}{dt}\right) \frac{dx^{i}}{ds}+\frac{1}{\left(\frac{dS\left( t\right)}{dt}\right)^{2}}V^{,i}=0.
a(\rho) =\frac{1}{2\sqrt{\pi}}\Gamma(1+2ic(\rho)){e}^{\frac{i\pi}{4}}{e}^{-6ic(\rho)\ln2}{e}^{\pi c(\rho)}\left [\cos\pi\Theta-\frac{\rho}{\Theta}\sin\pi\Theta\right ], b(\rho) =\frac{1}{2\sqrt{\pi}}\Gamma(1-2ic(\rho)){e}^{-\frac{i\pi}{4}}{e}^{6ic(\rho)\ln2}{e}^{\pi c(\rho)}\left [\cos\pi\Theta+\frac{\rho}{\Theta}\sin\pi\Theta\right ],
G_{B}^{( 1 )}( x_{n}, x_{m}) = G_{B}( x_{n}, x_{m}) -{\frac{\vert \tau_{a}- \tau_{b}\vert^{2}}{T_{3}+ G_{B}( \tau_{a}, \tau_{b})}}{\frac{( x_{n}- x_{m})^{2}}{T^{2}}}
L_{\eta}g_{ij}=2\left( \frac{1}{2}\xi_{,t}\right) g_{ij}V_{,k}\eta^{k}+\left( \xi_{,t}+\left( \ln\omega\right)_{,t}\xi\right) V =-\frac{f_{,t}}{\omega}\eta_{i,t}=f_{,i}.
pa_{r}=\frac{n^{2}a^{2}b^{2}}{r^{2}}\epsilon cos(\theta)-p\frac{n^{2}a^{2}b^{2}}{r^{3}}=\frac{n^{2}a^{2}b^{2}}{r^{2}}(\epsilon cos(\theta)-\frac{p}{r}).
\int_{-1}^{0}e^{-t(1-x)}(1-x)^{-1/2}\,dx =\int_{0}^{1}e^{-t(1+x)}(1+x)^{-1/2}\,dx &\le \int_{0}^{1}e^{-t(1-x)}(1-x)^{-1/2}\,dx,
\begin{aligned}a_{1,3}=\gamma \;a_{1,2}a_{2,3}=\delta \;a_{1,2}\end{aligned}\begin{aligned}b_{1,3}=\gamma \;b_{1,2}b_{2,3}=\delta \;b_{1,2}\end{aligned}\begin{aligned}c_{1,3}=\gamma \;c_{1,2}c_{2,3}=\delta \;c_{1,2}\end{aligned}
\mathbb{P}_0\left(\mu^{(2)}=m_+\right)= \mathbb{P}_0\left(\mu^{(2)}=m_-\right)= \frac{\beta_2}{2(\beta_2+\beta_4)}= \frac{1}{4}, & \mathbb{P}_0\left(\mu^{(2)}=\widehat{m}_-\right)= \mathbb{P}_0\left(\mu^{(2)}=\widehat{m}_+\right)= \frac{\beta_4}{2(\beta_2+\beta_4)}= \frac{1}{4},
A_+(\omega,k) = \{(\gamma , m_ke) \mid \omega(k) \in h_{\gamma,m_{k}e}\}\setminus \{(\gamma , m_{k-1}e) \mid \omega(k-1) \in h_{\gamma,m_{k -1}e}\}, A_{-}(\omega,k) = \{(\gamma , m_{k-1}e) \mid \omega(k-1) \in h_{\gamma,m'_{k-1}e}\}\setminus \{( \gamma, m_{k}e) \mid \omega(k) \in h_{\gamma,m_{k}'e}\}.
w_t(0,0)=\lim_{k\to\infty}u_t(\xi_{b_{j-1}}(t_k), t_k)=\lim_{k\to\infty}-u_r(\xi_{b_{j-1}}(t_k), t_k)\xi'_{b_{j-1}}(t_k)\geq -w_r(0,0)\sigma_0>0.
\int\limits_{SM}( \delta X)\eta = - \int\limits_{SM}{({g^{ij}}}{D_i}{X_j})\eta = 0,\ \textrm{and}\int\limits_{SM}( \dot \delta X)\eta = - \int\limits_{SM}{F{g^{ij}}}\frac{{\partial{X_i}}}{{\partial{y^j}}}\eta = 0,
{\mathbb E}_{\mathbb Q}\left[{\rm e}^{\vartheta S_N}\right]={\displaystyle \prod_{i=1}^N M_X\left(\omega_i(N) ({\rm e}^{\vartheta+\vartheta^\star}-1)\right)}\left/{\displaystyle\prod_{i=1}^N M_X\left(\omega_i(N) ({\rm e}^{\vartheta^\star}-1)\right)}\right..
E_{1,l}= \lambda[2a(2l+d+4) + 2b_1(4a-2l-d+7-4z_1) - 16b_2z_1(z_1-1){}- 24b_3z_1^2(z_1-1)- l(l+d-1) + 6], \psi_{1,l}(r) \propto r^l (1+\lambda r^2)^{-a-1}[1 - z_1(1+\lambda r^2)]{}\times \exp\left(- \frac{b_1}{1+\lambda r^2}- \frac{b_2}{(1+\lambda r^2)^2}- \frac{b_3}{(1+\lambda r^2)^3}\right),
R_{k}^{b}\propto - i \Bigg ( \frac{g}{3 2 \pi^{2}}\Bigg )^{k}\int d t \: e^{- 2 t}\Bigg [ ( 2 t )^{k - 1}+ ( k - 1 ) ( 2 t )^{k - 2}e^{- 2 t}f_{2}( \beta ) + \cdot \cdot \cdot \Bigg ] ,
\lambda_{\phi}(t)=\frac{E(\phi)+\frac{\beta}{2}\int_{\mathbb{R}^d}|\phi(\mathbf{x},t)|^4\mathrm{d}\mathbf{x}-\|\dot{\phi}(\cdot,t)\|^2}{\|\phi(\cdot,t)\|^2}=\mu_{\phi}(t)-\frac{\|\dot{\phi}(\cdot,t)\|^2}{\|\phi(\cdot,t)\|^2},
T_{ab}{}^{c}\;=\; -\left[ i_{k_{(m)}}\hat{C}\right]_{ab}Q^{mn}k_{(n)}^{c}\hspace{.2cm},\hspace{.2cm}2 K_{abc}\;=\; -T_{abc}+T_{bca}-T_{cab}\; ,\; \check{\omega}\;=\; \omega + K \; .
\langle E_k ,\nabla_{E_j}^T x^T \rangle = \langle E_k , \nabla_{E_j}x^T \rangle = \langle E_k , \nabla_{E_j}(x-x^{\perp}) \rangle = \delta_{jk}- \langle E_k , \nabla_{E_j}x^{\perp}\rangle &= \delta_{jk}+\langle \nabla_{E_j}E_k , x^{\perp}\rangle = \delta_{jk}+ A^{x^{\perp}}(E_j , E_k) \, .
\psi(t, \xi) =|\xi|^{2k+1}\psi\Big(t, \frac{\xi}{|\xi|}\Big) + |\xi|^{2k}\ln |\xi| \int_{\partial B_1}(w,\xi) m(t,w)~S_1(dw) &=|\xi|^{2k+1}\psi\Big(t, \frac{\xi}{|\xi|}\Big).
\dot{\xi}^{i}= \partial_{\mu}\xi^{i}\dot{x}^{\mu}= \partial_{\mu}\xi^{i}\frac{p^{\mu}}{N \cdot p}= \frac{p^{\mu}e_{\mu}^{\; \; i}}{N \cdot p}\equiv \frac{\pi^{i}}{N \cdot p}\; ,
S_{\mathrm{g f}}+ S_{\mathrm{F P}}= \int_{M}g^{- 2}( \varphi_{\mathrm{o}}) \left( \frac{1}{2}\nabla^{\mu}A_{\mu}\nabla^{\nu}A_{\nu}+ \partial_{\mu}\bar{\theta}\partial^{\mu}\theta \right) \sqrt{h}d^{3}x ,
{f}_{ik}(a)=[{f}_{ik}(a),{u}_{kk}(1)]\in[\mathfrak{osp}_{m|2n}(R,{}^-),\mathfrak{osp}_{m|2n}(R,{}^-)],{g}_{ki}(a)=[{u}_{kk}(1),{g}_{ki}(a)]\in[\mathfrak{osp}_{m|2n}(R,{}^-),\mathfrak{osp}_{m|2n}(R,{}^-)],
\left[ F , \pi_{a_{2 k + 1}}\right] = \left[ F , \nu_{a_{2 k + 1}}^{a_{2 j}}\right] \gamma_{a_{2 j}}+ \left[ F , \gamma_{a_{2 j}}\right] \nu_{a_{2 k + 1}}^{a_{2 j}}\approx 0 , \; k = 0 , \cdots , \Lambda ,
P^\top =[e_{i_1},\ldots,e_{i_k},e_{i_{k+1}},\ldots,e_{i_\ell}]\intertext{an consider}P@@A_{I(u)}P^\top = \begin{bmatrix}A_{I(v)}A_{I(v)I(w)}A_{I(w)I(v)}A_{I(w)}\end{bmatrix}.
F(g) = b_1^-F(g_1),E(f) = (-1)^{s-j}b_1 E(f_1), & \varphi(E(f),F(g)) = (-1)^{s-j+ |F(g_1)|}\varphi(b_1,b_1^-) \varphi(E(f_1),F(g_1))
\frac{1}{2}\frac{d}{dt}\int_{\Omega}\rho |u_t|^2 x+\int_{\Omega}|\nabla u_t|^2x\leq Cc^{8}_1|\sqrt{\rho}u_t|^2_{2}+Cc^4_1|\nabla v_t|^2_{2}+Cc^{8}_1.
\|S*g(t)\|^2_{\mathbb H^\mu}\le \sum_{n=1}^\infty \lambda_n^{\mu-1-\delta}\int_0^t (t-\tau)^{-\delta}|g_n(\tau)|^2 d\tau & = \int_0^t (t-\tau)^{-\delta}\|g(\tau)\|^2_{\mathbb H^{\mu-1-\delta}}d\tau.
\Gamma_{M}\bigl(w\,|\,a,a_{M}\bigr) = e^{\phi_{M}(w,x\,|\,a, a_{M})}\, \Gamma_{M-1}(w\,|\,a)\prod\limits_{k=1}^\infty \frac{\Gamma_{M-1}(w+ka_{M}\,|\,a)}{\Gamma_{M-1}(x+ka_{M}\,|\,a)}\times \times \exp\Bigl(\Psi_{M}(x,ka_{M}\,|\,a)-\Psi_{M}(w,ka_{M}\,|\,a)\Bigr).
S_{ij,k}= - \frac{2h_{2+p\,\, k i}\lambda_p \delta_{pj}}{(1+\lambda_p^2)}- \frac{2h_{2+p\,\, k j}\lambda_p \delta_{pi}}{(1+\lambda_p^2)}= - h_{2+p\,\, k i}T_{pp}\delta_{pj}- h_{2+p\,\, k j}T_{pp}\delta_{pi}= - h_{2+j \,\,k i}T_{jj}- h_{2+i\,\, k j}T_{ii}.
P(\xi^{c}) =1-\exp\left(-\frac{\tau}{2\sigma^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\,\left(\frac{\tau}{2\sigma^{2}}\right)^{k}, P(\xi) =P(\chi_{1}>\tau)=\exp\left(-\frac{\tau}{2\sigma^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\,\left(\frac{\tau}{2\sigma^{2}}\right)^{k},
\int_0^{1-t_n}\left(\left(\frac{1}{r_{jn}}\partial_rQ_j(\frac{r}{r_{jn}})\right)^2+\frac{g^2(Q_j(\frac{r}{r_{jn}}))}{r^2}\right)\,rdr=o_n(1),\,\,{\rm as}\,\,n\to\infty.
\Phi = \left( \begin{array}{c}{\Theta}\\{\chi}\\ \end{array}\right) = \left( \begin{array}{c}{\Theta}\\{\frac{i E \sigma^{3}}{k_{z}- \lambda}\Theta}\\ \end{array}\right) , \Theta = \left( \begin{array}{c}{\Theta_{1}}\\{\Theta_{2}}\\ \end{array}\right)
q\left( \bar{\lambda}\left( D^{2}\bar{w}_{\infty}\right) \right)=1/\left[ \left( n-1\right) K\right] , D^{2}\bar{w}_{\infty}\left( y\right) \geq0,\text{also one and only oneeigenvalue of}D^{2}\bar{w}_{\infty},\D_{11}\bar{w}_{\infty}\ 0\ 0.
0\rightarrow K_{0}(\mathsf{\acute{E}t}_{c}(X,\mathbf{F}_{q}))^{\operatorname*{Tr}=0}\rightarrow K_{0}(\mathsf{\acute{E}t}_{c}(X,\mathbf{F}_{q}))\rightarrow\bigoplus_{x\in X(\mathbf{F}_{q})}\mathbf{F}_{q}\rightarrow0
\frac 1 3 \varepsilon^{i j}\left( \begin{array}{l}{2 \bar{\varepsilon}\gamma_{\mu \rho}\partial_{i}\theta \bar{\theta}\gamma^{\mu \nu}\partial_{j}\theta}{- \bar{\varepsilon}\gamma_{\mu \rho}\theta \partial_{i}\bar{\theta}\gamma^{\mu \nu}\partial_{j}\theta}\end{array}\right) \lambda_{\nu}\lambda^{\rho},
y_1=\frac{1}{\sqrt{\mu}}x_2, \, y_2=\frac{1}{\sqrt{2}}(x_1+x_3), \, y_3=\frac{1}{\sqrt{2}}(x_1-x_3); [y_1,y_2]=[y_3,y_1]=-\frac{1}{\sqrt{2}}y_1-\frac{1}{2\sqrt{\mu}}(y_2+y_3), \, [y_2,y_3]=\frac{1}{\sqrt{2}}(y_2+y_3).
\bar{\partial}P_{s}^{j , m}= \lambda [ \phi_{s - 1}^{l , m + 1 / 2}\otimes \overline{{{\phi}}}^{1 / 2 , - 1 / 2}- \phi_{s - 1}^{l , m - 1 / 2}\otimes \overline{{{\phi}}}^{1 / 2 , 1 / 2}]
\,\, c_i\,\,\,\, \gamma, c_i\gamma\in \Gamma c_j\,\,\,\, j; p^{-1}(p(\Gamma g))\cap \Omega=\{\Gamma c_1gu_{p_1},\ldots, \Gamma c_ngu_{p_n}\}\,\, \,\,\tilde{m}\, g.
\frac{\partial}{\partial \xi}F = S (\frac{\partial}{\partial \xi}f) S^{-1}+\frac{\bar \xi-\xi}{2}\; \frac{\partial S}{\partial \xi}S^{-1}, \quad\frac{\partial}{\partial\bar \xi}F = S(\frac{\partial}{\partial\bar \xi}f) S^{-1}-\frac{\bar \xi- \xi}{2}\;\frac{\partial S}{\partial \bar \xi}S^{-1}.
D_i^{-1}= \sum_{\gamma\in Q^+}\dim (U^-_{z,-\gamma}\cap T_i(U_z^-))e(-\gamma) = \sum_{\gamma\in Q^+}\dim (U^-_{z,-\gamma}\cap T_i^{-1}(U_z^-))e(-\gamma), \overline{D}_i^{-1}= D_i^{-1}-\sum_{\gamma\in Q^+}\dim (J^-_{z,-\gamma}\cap T_i(U_z^-))e(-\gamma) = D_i^{-1}-\sum_{\gamma\in Q^+}\dim (J^-_{z,-\gamma}\cap T_i^{-1}(U_z^-))e(-\gamma).
A'_{i}(\vec{x},t) = e^{-it\tilde{A}_{0}}\left[A'_{i}(\vec{x},0) +\int_{0}^{t}dt'\, e^{it'\tilde{A}_{0}}(\partial_{i}\tilde{A}_{0})e^{-it'\tilde{A}_{0}}\right] e^{it\tilde{A}_{0}}
B_{\mathcal{H}^{\infty}}=\left(\prod_{\phi\in H_{1},\psi\in H_{2}}B_{\mathcal{H}(E)}\left(\|\phi\|\|\psi\|\right)\right)\cap\left\{M\colon E\to L(H_{1},H_{2})\,|\, M(z)(z\in E)\right\},
\big |(f_0\psi_j)^{(r)}(x) \big| \leq [(R+\delta/2)^{\frac{r}{\beta}}-R^{r/\beta}] |f_0(x)|^{\frac{\beta-r}{\beta}}, \ \x\in [x_{j-1},x_j], \j=1,\ldots,m,
R \; \mbox{Vol}= \frac{1}{2}d\Phi \wedge{*}d\Phi + \frac{g_s e^{-\Phi}}{4}H_3 \wedge{*}H_3 +\frac{3 g_s^{1/2}e^{3\Phi/2}}{8}F_2 \wedge{*}F_2 +\frac{g_s^{3/2}e^{\Phi/2}}{8}\tilde F_4 \wedge{*}\tilde F_4 \ .
T_{l}= \frac{1}{8}\left( \begin{array}{c c c c c}{{0}}&{{- 1}}&{{1}}&{{- 2}}&{{2}}\\{{1}}&{{- 1}}&{{0}}&{{1}}&{{- 1}}\\{{2}}&{{- 3}}&{{1}}&{{0}}&{{0}}\\ \end{array}\! \right)^{\! \! \! \sim},
\mathcal A_n\left(\sum_{1\leq i\leq a_1}\Psi_{\{[\{1\},i]\}}, \sum_{1\leq i\leq a_2}\Psi_{\{[\{1,2\},i]\}},\dots,\sum_{1\leq i\leq a_m}\Psi_{\{[\{1,\dots,m\},i]\}},\dots\right)=\sum_{\Pi\vDash n}\Psi_\Pi
{\cal W}( M_{\alpha}) = \int_{M_{\alpha}}H_{\alpha}^{2}d A_{\alpha}+ \int_{\partial M_{\alpha}}\kappa^{\alpha}d s = \bar{\cal W}( \bar{M}_{\alpha}) = \int_{\bar{M}_{\alpha}}\left( \bar{H}_{\alpha}^{2}+ \bar{R}_{\alpha}\right) d \bar{A}_{\alpha}+ \int_{\partial \bar{M}_{\alpha}}\bar{\kappa}^{\alpha}d \bar{s},
\begin{aligned}\sum_{j \notin L_{ik}}\lambda_{ij}\leq \delta_{ik},\ \sum_{j \notin R_{ik}}\lambda_{ij}\leq 1- \delta_{ik}k \in K_i, &\lambda_i \in \Lambda_i,\ \delta_i \in \{0,1\}^{\lceil \log_2l_i \rceil},\end{aligned}
\frac{1}{1+H^2r^2}\left(\frac{dr}{d\xi^1}\right)^2-(1+H^2r^2)\left(\left(\frac{dt}{d\xi^1}\right)^2+c_1^2\right)+r^2\left(\left(\frac{d\phi}{d\xi^1}\right)^2+c_2^2\right)=0.
\Vert u_{k}\Vert_{\varepsilon_{k}}^{2}=\frac{1}{\varepsilon_{k}^{n}}\int_{M}b(x)\left( f^{\prime}(W_{\varepsilon_{k},\xi_{k}})u_{k}^{2}+f^{\prime}(W_{\varepsilon_{k},\xi_{k}})(\psi_{k}+\zeta_{k})u_{k}\right) .
\lim_{n \uparrow \infty}\mathbb E\left[e^{-s \overline{X}^n_{e(q)}}\right]=q\lim_{n \uparrow \infty}\int_{0}^{\infty}e^{-qt}\mathbb E\left[e^{-s\overline{X}^n_{t}}\right]dt=\mathbb E\left[e^{-s \overline{X}_{e(q)}}\right],
\begin{array}{c}s_i^2=e \ \ \ \ (i\geq 0) ,\ \ \ \ \ \ \s_is_j=s_js_i\ \ \ \ (|i-j|\geq 2), s_0s_1s_0s_1=s_1s_0s_1s_0, \ \s_is_{i+1}s_i=s_{i+1}s_is_{i+1}\ \ \ \ (i\geq 1). \end{array}
\frac{\partial^{2}\tilde{\phi}}{\partial \tilde{t}^{2}}- \frac{\partial^{2}\tilde{\phi}}{\partial \tilde{x}^{2}}+ \tilde{\eta}_{m}\tilde{\phi}^{2}\frac{\partial \tilde{\phi}}{\partial \tilde{t}}= - \tilde{\phi}+ \tilde{\alpha}\tilde{\phi}^{2}- \tilde{\lambda}\tilde{\phi}^{3}+ \tilde{\xi}_{m}\tilde{\phi}.
F\Big(\frac{t}{c\tau+d},\frac{a\tau+b}{c\tau+d}\Big) =(c\tau+d)^le^{2\pi\sqrt{-1}m(ct^2/(c\tau+d))}F(t,\tau), F(t+\lambda\tau+\mu,\tau) =e^{-2\pi\sqrt{-1}m(\lambda^2\tau+2\lambda t)}F(t,\tau),
\begin{cases}|\dot{u}| + |\nabla u|<A, \frac{1}{A}\leq \left. \int_M u d\mu \right|_{t}\leq A, \forall \; t \in [-1, 1], \frac{1}{A}\leq \frac{|B_{g(t)}(x, r)|}{r^m}, \forall \; x \in M, \; t \in [-1, 1], r \in (0, 1). \end{cases}
iA^{+}=\left [ \begin{array}{ll}<\psi^{(0)}_1|d\psi^{(0)}_1> <\psi^{(0)}_2|d\psi^{(0)}_1> <\psi^{(0)}_1|d\psi^{(0)}_2> <\psi^{(0)}_2|d\psi^{(0)}_2>\end{array}\right ]=iA_a dx^a,
\nabla = \beta_g g \frac{\partial}{\partial g}+\beta_\xi \xi \frac{\partial}{\partial \xi}+\beta_\sigma \sigma \frac{\partial}{\partial \sigma}+\beta_\rho \rho \frac{\partial}{\partial \rho}+\gamma_Q \mathbf{N}_Q + \gamma_{\bar Q}\mathbf{N}_{\bar Q}+ \gamma_B \mathbf{N}_B.
D_L\equiv D_L(\Omega_B)= \frac{V_0 '(1+z)}{2H_0(\omega+1)V_0^{\frac{1}{2}+\frac{1}{3(\omega+1)}}}\int{\phi(0)}^{\phi(z)}\frac{d\phi}{V^{\frac{3\omega+1}{6(\omega+1)}}},
T_{a, \vec{v}}^{j, l}f:=\sum_{\substack{Q\in\mathcal{D}_{\vec{v}}l(Q)=2^{\lfloor{-j\rho+j\epsilon +10}\rfloor}}}T_{a}^{j, l}(f\chi_{\frac{1}{3}Q}) \:\: l \leq j \epsilon,
e^{\sum V \left( x_{i}^{( a )}- y_{j}^{( b )}\right) - \frac{1}{2}\sum V \left( x_{i}^{( a )}- x_{j}^{( b )}\right) - \frac{1}{2}\sum V \left( y_{i}^{( a )}- y_{j}^{( b )}\right)}\left( \frac{1}{2 \pi}\right)^{N ( n + m )}\left[ \frac{e^{2}N}{4 \pi}e^{2 \gamma}\right]^{\frac{N ( n - m )^{2}}{2}}.
\begin{array}[c]{cl}d{y}(t)= \big{[}b_x(\bar{X}{(t)},\bar{u}(t))y(t)+b(\bar{X}{(t)},u^{\varepsilon}(t))-b(\bar{X}{(t)},\bar{u}(t))\big{]}dt, y(0)= 0,t\in (0,T].\end{array}
A^{'^{\prime}}+ \frac{7}{4}( A^{\prime})^{2}= - \frac{1}{1 2 M^{3}}\left( \frac{1}{2}( \pi^{\prime})^{2}+ V ( \pi ) e^{A / 2}\right) - \frac{\vartheta}{1 2 M^{3}}\delta ( y )
\sum_{u < n \leq v}\tau(n) = v \log v - u \log u + (2 \gamma - 1)(v-u) + O(v^{1/3}) = (v-u) \log u + v \log (v/u) + (2 \gamma - 1)(v-u) + O(v^{1/3}) &\ll (v-u) \log u + O(v^{1/3}) .
S_{\omega}(u)=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx +\frac{\omega+1}{2}\int_{\mathbb{R}^N}|u|^2dx -\frac{1}{2}\int_{\mathbb{R}^N}|u|^2\log |u|^2dx, I_{\omega}(u)=\int_{\mathbb{R}^N}|\nabla u|^2dx +\omega\int_{\mathbb{R}^N}|u|^2dx-\int_{\mathbb{R}^N}|u|^2\log |u|^2dx.
p^{n_{s}^{\prime}-n_{1}^{\prime}}q^{n_{2}^{\prime}-n_{1}^{\prime}}a_{n_{2}}N_{n_{1}}-p^{n_{2}^{\prime}+n_{s}^{\prime}-2n_{1}^{\prime}}a_{n_{1}}{N_{n_{2}}}=p^{n_{2}^{\prime}-n_{1}^{\prime}}q^{n_{s}^{\prime}-n_{1}^{\prime}}a_{n_{s}}\frac{N_{n_{2}}N_{n_{1}}}{N_{n_{s}}}.
{\cal T}_{\mu \nu}^{\alpha \beta}\sim \epsilon_{\mu}^{\alpha \rho \sigma}\epsilon_{\nu}^{\beta \lambda \gamma}\big ( A F_{\rho \sigma}F_{\lambda \gamma}+ B F_{\rho \sigma}\tilde{F}_{\lambda \gamma}+ C \tilde{F}_{\rho \sigma}\tilde{F}_{\lambda \gamma}\big ) ,
\psi_{x_1}=(x_2x_5),\psi_{x_2}=(x_1x_4),\psi_{x_3}=(x_1x_2)(x_4x_5), \psi_{x_4}=(x_2x_5)(x_3x_6),\psi_{x_5}=(x_1x_4)(x_3x_6),\psi_{x_6}=(x_1x_2)(x_4x_5).
[f_1 ,\ldots ,f_n] = \hbox{det}\left( \begin{array}{ccc}f_1 \ldots f_n D_1 (f_1) \ldots D_1 (f_n) \hdotsfor{3}D_{n-1}(f_1) \ldots D_{n-1}(f_n)\end{array}\right) \, .
\mathbf{D}_0:=\nu(1-\nu)p'\begin{pmatrix}0 0 0 -1 1 0 0 0 0 \end{pmatrix}\quad\textrm{and}\mathbf{D}_1:= \begin{pmatrix}0 0 0 0 \nu^2 0 -(1-\nu)u 0 1-\nu \end{pmatrix}\,.
T_{z}^{\prime}C_{\alpha}\otimes T_{z}^{\prime}C_{\beta}\mapsto e^{( \frac{\lambda_{i}- \lambda_{j}}{d_{\alpha}}+ \frac{\lambda_{i}- \lambda_{l}}{d_{\beta}}) t}T_{z}^{\prime}C_{\alpha}\otimes T_{z}^{\prime}C_{\beta}
\mathbb{E}\{tr(\tilde{\boldsymbol{H}}^H\tilde{\boldsymbol{H}})^{-1}\}\approx \mathbb{E}\{tr(\hat{\boldsymbol{H}}^H\hat{\boldsymbol{H}})^{-1}\}&=\frac{K}{F-K}\frac{1}{2\sigma^2(1+\sqrt{\frac{M-F}{FK}})}.
R_{\mu \nu \rho\sigma}R^{\mu \nu \rho\sigma}= \lambda_s^2 \,(\gamma_2 +\frac{\gamma_1^2}{2})^2 + \frac{2\gamma_1^2 \lambda_s^2}{r_s^2}+\frac{4\,(1-\lambda_s)^2}{r_s^4}+{\cal{O}}(r-r_s)
\begin{aligned}\nabla(\|(x_{q,\sigma}^*)_{\mathcal S}\|_{q}^{q})-\lambda^*{A_{\mathcal S}^T}(y_{\mathcal S}-{A_{\mathcal S}}(x_{q,\sigma}^*)_{\mathcal S}) =0, \lambda^*(\|y-{A}x_{q,\sigma}^*\|_2^2-\sigma^2) =0, \|y-{A}x_{q,\sigma}^*\|_2^2 \leq\sigma^2, \lambda^* \geq0.\end{aligned}
d s^{2}~ = ~ - f ( r ) ~ \left( 1 - \left( \frac{r_{0}}{r}\right)^{2}\right) ~ d t^{2}+ f ( r )^{- 1}\left( 1 - \left( \frac{r_{0}}{r}\right)^{2}\right)^{- 1}~ d r^{2}+ r^{2}d \Omega_{3}^{2}~ ,
X = \bigg\{\psi(t)\bigg|\ \psi(0) = \psi_0,\ \|\psi(t)\|_2 \leq 2C_1(1 + t)^{-\frac{d}{4}}\|\psi_0\|_1\ \{\rm and}\ &\sqrt{Q(\psi(t))}\leq 2C_1C_Q(1 + t)^{-\frac{d}{4}-\frac{\eta}{2}}\|\psi_0\|_1,\ \forall \t\geq 0 \bigg\}
W_{\mathrm{e f f}}= \int d^{4}x \left[ - \frac{1}{4}F_{\mu \nu}^{2}- \frac{1}{2}m^{2}{\left( A_{\mu}-{\partial_{\mu}\phi^{2}}\right)}^{2}+ i \bar{\psi}\gamma^{\mu}D_{\mu}^{+}\psi \right] + M_{1}+ S_{\mathrm{g f}}
\sum_{1 \leq l \leq p-1, 1 \leq t \leq l+1,0 \leq q \leq t-1}{(\theta \sum_{1 \leq i \leq k}{v^2_i})^{2p-2l}\binom{t}{q+1}\binom{2p-2l-1}{q}b_{l,t}}:=s_2(\theta\sum_{1 \leq i \leq k}{v^2_i},p).
\Phi\left(\bigvee_{l\geq j+1}p_l\right)^{\perp}\check{y}h_n = \Phi\left(\bigvee_{l\geq j+1}p_l\right)^{\perp}\check{y}(h_{n, 1}+\cdots+h_{n, j}) &= \Phi\left(\bigvee_{l\geq j+1}p_l\right)^{\perp}\check{y}(p_1+\cdots+p_j)h_n.
\mathbb{E}\left[y_i^* y_i\right]=a_c^2\lvert\mathbf{h}_{i,*}\mathbf{p}_c\rvert^2+\sum_{l=1}^M a_l^2\lvert\mathbf{h}_{i,*}\mathbf{p}_l\rvert^2+\sigma_n^2.
|\delta_hu(x)-\delta_h u(y)|^p\leq c_p^{\frac{p}{2}}\left[\left( (u_h(x)-u_h(y))^{p-1}-(u(x)-u(y))^{p-1}\right)(\delta_hu(x)-\delta_hu(y))\right]^{\frac{p}{2}}\times &\times \left[ |u_h(x)-u_h(y)|^{2}+|u(x)-u(y)|^{2}\right]^{\frac{2-p}{2}\frac{p}{2}}.
+ \sqrt{| g_{E}|}\, \textstyle{\frac{1}{2}}e^{\varphi}a^{2}\left\{\left[ \mathrm{T r}\, \left( Q \mathcal{M}\right) \right]^{2}- 2 \mathrm{T r}\, \left( Q \mathcal{M}Q \mathcal{M}\right) \right\}\; ,
A(i,j,X_{m,l}) = \sum_{Y\subseteq X_{m,l}}(-1)^{|Y|}a_{iy_1}a_{y_1y_2}\cdots a_{y_kj}; A'(i,j,X_{m,l}) = \sum_{Y\subseteq X_{m,l}}(-1)^{|Y|}a_{iy_k}a_{y_ky_{k-1}}\cdots a_{y_1j}.
f_{\eta,\lambda}(n)=\frac{\sum_{i=1}^{\lfloor\frac{n-\ell(\eta)}{2}\rfloor}\sum_{\mu\rhd_{2i+1}\eta}\kappa_{\mu,\eta}f_{\mu,\lambda}(n)+\sum_{i=1}^{\lfloor\frac{n-\ell(\lambda)}{2}\rfloor}\sum_{\mu\rhd_{2i+1}\lambda}\kappa_{\mu,\lambda}f_{\mu,\eta}(n)}{n+1-\ell(\eta)-\ell(\lambda)}.
x\ast_{n, n}x = (a_{1}\ast a_{2}\ast \cdots \ast a_{n}) \ast (a_{1}\ast a_{2}\ast \cdots \ast a_{n}) = (a_{1}\ast a_{1}) \ast (a_2 \ast a_2) \ast \cdots (a_n \ast a_{n})=(\top_1 \ast a_1) \ast (\top_1 \ast a_2) \ast \cdots \ast (\top_1 \ast a_n) = (\top_n) \ast (a_{1}\ast a_{2}\ast \cdots \ast a_{n});
\begin{pmatrix}\Re P_{11}+ \Re P_{12}\Im P_{12}- \Im P_{11}\Re P_{21}+ \Re P_{22}\Im P_{22}- \Im P_{21}\Im P_{11}+ \Im P_{12}\Re P_{11}- \Re P_{12}\Im P_{21}+ \Im P_{22}\Re P_{21}- \Re P_{22}\end{pmatrix}= \{0\}
\sigma_n^2\equiv \langle E^2\rangle_n-\langle E\rangle_n^2= 4\frac{n(N-n)}{N-1}\left ( \langle e^2\rangle-\langle e\rangle^2\right ),\langle e^2\rangle =\frac{1}{4N}\sum_j e_j^2= 1+\lambda^2\ .
0 =-\lim_{x\rightarrow-1^{+}}[f,\widetilde{g}](x) \Longleftrightarrow0 =\lim_{x\rightarrow-1^{+}}(1-x)^{-1}(1+x)^{\beta+1}(f^{\prime}(x)\widetilde{g}(x)-f(x)\widetilde{g}^{\prime}(x)) \Longleftrightarrow0 =\lim_{x\rightarrow-1^{+}}(1+x)^{\beta+1}f^{\prime}(x).
\sum_{n=0}^{\infty}\frac{\left(a,b;q\right)_{n}z^{n}}{\left(q,c;q\right)_{n}}=\frac{\left(b;q\right)_{\infty}}{\left(c,z;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b,z;q\right)_{n}b^{n}}{\left(q;q\right)_{n}}\left(azq^{n};q\right)_{\infty}
\varphi_{il}^{\star \star}\left(Z\right)=\displaystyle\sum_{i_{1},i_{2}=1}^{q}\varphi_{i l}^{i_{1}i_{2}}\left(Z_{i_{1}},Z_{i_{2}}\right),\quad\mbox{for all $i=1,\dots,q$ and $l=1,\dots,p-q$},
\sum_{\sigma=0}^{p^k-1}\mathcal{A}(\mathbf{a}_\sigma^t)(\tau)=\sum_{r=0}^{p^{m}-\tau-1}\sum_{\sigma=0}^{p^{k}-1}\omega_q^{\left(a^t_{\sigma,r}-a^t_{\sigma,r+\tau}\right)}=0,
|(E_1-E_2)_x|(x)= |n_1-n_2-(b_1-b_2)|(x)\le|n_1-n_2|(x)+|b_1-b_2|(x) \le \frac{|n_1-n_2|(x)}{(1-x)^{\frac{1}{2}}}+\|b_1-b_2\|_{C[0,1]}\le C\|b_1-b_2\|_{C[0,1]},x\in(1-\delta_1,1].
G_{(1,2)}(\mathbf{x}) = F_{(1,2)}(\mathbf{x}) + 2 \beta F_{(1,1,2)}(\mathbf{x}) + \beta F_{(1,2,1)}(\mathbf{x}) + 3 \beta^2 F_{(1,1,1,2)}(\mathbf{x}) + 2 \beta^2 F_{(1,1,2,1)}(\mathbf{x}) + \beta^2 F_{(1,2,1,1)}(\mathbf{x}),
{\hat \Psi}\ =\ \sum_{j=|k|-1/2}^{\infty}\sum_{|m|\le j}[a^{j+}_{km}{\hat D}^j_{k-1/2,m}a\ +\a^{j-}_{km}{\hat D}^j_{k+1/2,m}a^* ]\ \in \{\hat{\cal S}}_k \ ,
\begin{cases}\dot{u}\leq M(\sum\log_{-}|z-a_j|+1)\;\; \mbox{on}\;\bar{\Omega}\times [\epsilon,T), \dot{u}+Au-g\leq M(\sum\log_{-}|z-a_j|+1)\;\;\mbox{on}\; \bar{\Omega}\times [\epsilon,T), \end{cases}
\|\nabla (u\,\eta)\|^p_{L^p(\mathbb{R}^N)}\le C\,\left(\frac{R}{r}\right)^N\,\left(\frac{R}{h_0}\right)^{1+p}\,\left[\frac{(1-s)\,\mathcal{M}_\gamma}{(\Gamma-1)^p}+\left(1+\frac{h_0^{-\Gamma\,p}}{(\Gamma-1)^p}\right)\,\|u\|^p_{L^p(B_{R+h_0})}\right].
\sum_{k=0}^{(p^r+1)/2}(4k-1)\frac{(-\frac{1}{2})_k^6}{k!^6}\equiv\frac{15p^r(p^{2r}-p^{4r}-1)}{64(p^{2r}-1)}\sum_{k=0}^{(p^r-3)/2}\frac{(\frac{3}{2})_k^3(\frac{7}{2})_k}{k!(k+2)!^3}\pmod{p^{r+3}}.
0 = \left[u_i(a) - u_i(a^j)\right] - \left[u_i(a^i) - u_i(a^{ij})\right] = \left[u_i(a) - u_i(a^i)\right] - \left[u_i(a^j) - u_i(a^{ij})\right] = \Phi_i(a) - \Phi_i(a^j) = \Phi_{ij}(a).
\sum_{j=1}^{n}\theta\left(x - \frac{j}{n}+ \varepsilon_j \right) = \sum_{k \in \mathbb{Z}}e^{-\pi \alpha k^2}\left( \sum_{j=1}^{n}e^{2 \pi i k \varepsilon_j}e^{-2 \pi i k \frac{j}{n}}\right) e^{2 \pi i k x}.
2^3 6^2 V^2 = \left|\begin{array}{ccc}2l_1^2 -l_{12}^2+l_1^2+l_2^2 -l_{13}^2+l_1^2+l_3^2 -l_{12}^2+l_1^2+l_2^2 2l_2^2 -l_{23}^2+l_2^2+l_3^2 -l_{13}^2+l_1^2+l_3^2 -l_{23}^2+l_2^2+l_3^2 2l_3^2\end{array}\right|
d^2_C(x_m)\leq\|x_m -p\|^2 \leq (\gamma_2 \cdots \gamma_m) \|x_1-p\|^2-\beta\nu^2\max_{i\in I}d_{C_i}^2(x_1) &\leq (\gamma_2 \cdots \gamma_m) d_C^2(x_1) -\frac{\beta\nu^2}{\kappa^2}d_C^2(x_1) &= \left[\gamma_2 \cdots \gamma_m -\frac{\beta\nu^2}{\kappa^2}\right]_+d_C^2(x_1),
a_1=\|Q_1\left(\cdot;\mu_{K(\gamma)}\right)\|_{L^{2}\left(\mu_{K(\gamma)}\right)}, a_2=\|Q_2\left(\cdot;\mu_{K(\gamma)}\right)\|_{L^{2}\left(\mu_{K(\gamma)}\right)}/\|Q_1\left(\cdot;\mu_{K(\gamma)}\right)\|_{L^{2}\left(\mu_{K(\gamma)}\right)}.
I_n^{(1)}\leq c_4 R^{-\delta}(A_{2+\delta}+ 4A_\delta) \int_0^1 s^{-3\delta}ds + c_4 R^2 e^{-R/2}\int_1^\infty e^{-sR/2}\bigl[1+ 4s^{-2}R^{-2}\bigr] ds = \hbox{o}(R)
\mathcal{F}(g_{\nu}^{+})=(2\pi)^{\frac{m}{2}}\mathcal{F}(T_{\nu}^{+})\cdot \mathcal{F}((f_{\nu})_N) &\mathcal{F}(g_{\nu}^{-})=(2\pi)^{\frac{m}{2}}\mathcal{F}(T_{\nu}^{-})\cdot \mathcal{F}((f_{\nu})_N).