harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\alpha\left( \exp\left( z_{1}Z_{1}\right) \exp\left( z_{2}Z_{2}\right)\exp\left( x_{1}X_{1}\right) \exp\left( x_{2}X_{2}\right) \exp\left(yY\right) \right) & =\exp\left( 4z_{1}Z_{1}\right) \exp\left( 4z_{2}Z_{2}\right) \exp\left(2x_{1}X_{1}\right) \exp\left( 2x_{2}X_{2}\right) \exp\left( 2yY\right) .
\delta R_{AB}= \partial_{C}\delta \Gamma_{A B}^{C}- \partial_{B}\delta \Gamma_{A C}^{C}+ \overline{\Gamma}_{A B}^{C}\delta \Gamma_{C D}^{D}+ \delta\Gamma_{A B}^{C}\overline{\Gamma}_{C D}^{D}- \delta\Gamma_{B C}^{D}\overline{\Gamma}_{A D}^{C}- \overline{\Gamma}_{B C}^{D}\delta\Gamma_{A D}^{C}.
\xi(r)=\frac{1}{N-2}+o(r), \xi'(r)=\xi''(0)r+o(r), \quad r\to0 \zeta(r)=\frac{1}{r^{N-2}}+o\left(\frac{1}{r^{N-2}}\right), \zeta'(r)=-\frac{N-2}{r^{N-1}}+o\left(\frac{1}{r^{N-1}}\right) \quad r\to0.
\{g ( \Lambda^{0}) , q ( \Lambda ) \}= \frac k{2 \pi}\int d^{2}x d^{2}y \Lambda^{( 0 )}( x ) \epsilon^{i j}[ \partial_{j}\Lambda ( y ) ] \left[ \frac \partial{\partial x^{i}}\delta^{2}( x - y ) \right]
\partial_t \bar{v}-\partial_{x}^2 \bar{v}=(1-v_0^2)\bar{v}- v_0 \bar{v}^2-\frac{1}{3}\bar{v}^3-\bar{w}-\bar{v}X_1(x,t) -Z(\bar{v},x,t)-v_0 J_0^2=\psi_1(\bar{v},\bar{w},x,t), \partial_t\bar{w}-\rho\partial_{x}^{2}\bar{w}=\varepsilon(\bar{v}-\gamma \bar{w}) +\varepsilon J_0= \psi_2(\bar{v},\bar{w},x,t),
|Z_0|_{\delta_1}, |Z_0^{-1}|_{\delta_1}<\varepsilon^{-\frac{1}{2}}=\epsilon_1^{-1}, \ \ \ |Z_0|_{\delta_1}^{\frac{1}{2}}|F_1|_{h_1,\delta_1}, |Z_0^{-1}|_{\delta_1}^{\frac{1}{2}}|F_1|_{h_1,\delta_1}<1.
\langle Ux_{n-1}+\theta_i(x_{n-1}-z),x_{n-1}-y_i\rangle =\langle Ux_{n-1}+\theta_i(y_i-z),x_{n-1}-y_i\rangle+\theta_i\Vert x_{n-1}-y_i\Vert^2 =\langle Ux_{n-1}-Uy_i,x_{n-1}-y_i\rangle+\theta_i\Vert x_{n-1}-y_i\Vert^2 \ge\theta_i\Vert x_{n-1}-y_i\Vert^2.
U_{l,k}^{\delta}:= I_{F,l}^{\delta}\times \left[\frac{\delta^2 k}{|I_{F,l}^{\delta}|},\frac{\delta^2 (k+1)}{|I_{F,l}^{\delta}|}\right], I_{F,l}^{\delta}:= \left[F^{-1}(\delta l),F^{-1}(\delta (l+1))\right].
\left\| AXB - XBX^TAX \right\|_F^2 = \left\| AX_1\bar{B}- X_1\bar{B}X_1^TAX_1 \right\|_F^2 + \left\|X_1\bar{B}X_1^TAX_2\right\|_F^2 &= \left\| AX_1\bar{B}- X_1\bar{B}X_1^TAX_1 \right\|_F^2 + \left\| X_2^T\left( AX_1\bar{B}- X_1\bar{B}X_1^TAX_1 \right)X_1^T \right\|_F^2 &= \Theta\left( \left\| AX_1\bar{B}- X_1\bar{B}X_1^TAX_1 \right\|_F^2 \right).
\phi_1(c,r,p) = \left(\frac{c^3}{2}+c+3\right)^2+\left(4-c^2\right)^2 r^2 \left(-3 c^2 p r+\frac{9}{4}c^2 r^2+c^2-3 c p r+2 c+1\right) +2 \left(\frac{c^3}{2}+c+3\right) \left(4-c^2\right) r \left(\frac{3}{2}c r-3 c p^2r-1+c p+p\right).
[\nabla^{k,g_P}(\hat\eta_i')^t ]_{C^{\alpha}(B^{g_P}(\tilde{x}_i, \frac{i}{4}))}\leq C, \sum_{j=0}^k \lambda_i^{-k+j-\alpha}|(\nabla^{j,g_P}(\hat\eta_i')^t)(\tilde{x}_i)|_{g_P(\tilde{x}_i)}\leq C,
\mathbf{P_*}(||\tilde{Z}_n^*|| >C_1.n^{-1/2}(logn)^{-1}) &\leq \mathbf{P_*}(||L_n^* - \mathbf{E_*}L_n^*||>C_1.n^{-1/4}(log n)^{-1/2}) &= o_p(n^{-1/2})
|f_L(z)|\geq |a_0|-\left|\sum_{\alpha\in J_1}a_\alpha\prod_{j=1}^n \left(\frac{\Gamma(L_j+\alpha_j)}{\alpha_j! \Gamma(L_j)}\right)^{1/2}z_j^{\alpha_j}\right|- \left|\sum_{\alpha\in J_2}a_\alpha\prod_{j=1}^n \left(\frac{\Gamma(L_j+\alpha_j)}{\alpha_j! \Gamma(L_j)}\right)^{1/2}z_j^{\alpha_j}\right|\ ,
G_{tt}=-\kappa e^{4\kappa \Psi(r)}\left[2\Psi^{''}+4\frac{\Psi^{'}}{r}-\kappa (\Psi^{'})^2\right] \ , \ \G_{rr}=-\frac{G_{\theta\theta}}{r^2}=\frac{G_{\varphi\varphi}}{r^2\sin^2\theta}=\kappa^2 (\Psi^{'})^2
A_H(x_+)-A_H(x_-) \geq \int_{\Sigma}u^*(d\lambda-dH\wedge dt) = \int_{\Sigma}u^*d\lambda\cr = \int_{\Sigma}(a,f)^*d(e^{\phi_R}\alpha) = \int_{\Sigma}(a,f)^*d(\psi\alpha)\cr = \int_{\Sigma}\Bigl(\psi'(a)da\wedge f^*\alpha+\psi(a)f^*d\alpha\Bigr).
h(\chi(u^x_r)^*\chi(u_s^y)) =h(\alpha(\chi(\overline{x}))u_{r^{-1}s}\alpha(\chi(y)))=h(u_{r^{-1}s}\alpha(\alpha_{r^{-1}s}(\chi(\overline{x}))\chi(y)))=\delta_{r,s}h_G(\chi(\overline{x})\chi(y)) &=\delta_{r,s}\delta_{x,y}.
\begin{array}{rl}\left( h(t) - \mu(t,\theta^H_t) \right)\frac{\partial}{\partial x}\mu(t,\theta^H_t) \equiv 0 \Leftrightarrow \left( \sum_{j=1}^n\theta_j(t)\mathbb{E}[\frac{\partial}{\partial y_j}q_{\gamma}(\Theta)|\mathcal{F}_t] - \sum_{i=0}^k\tilde{a}_i (\theta^H_t)^i \right)\sum_{i=1}^k i\tilde{a}_i (\theta^H_t)^{i-1}\equiv 0 \end{array}
\begin{aligned}S_K(u,v)= \;h_K^{-1}\;\sum_{F\subset \partial K}\Bigl[\big(Q_K u-Q_F u,Q_K v-Q_F v\big)_{F}+ & \epsilon_F h_F\sum_{e\subset \partial F}\big( u-Q_F u, v-Q_F v\big)_e\Bigr],\end{aligned}
g^{(q)}(b,a) =-\tilde{r}^{(q)}(b)+\frac{\tilde{r}^{(q)}(a)}{(c-\delta) \mathbb{W}^{(q)}(a-b)}+\frac{\tilde{r}^{(q)}(b)+g^{(q)}(b,a)}{Z^{(q)}(b)}\left(Z^{(q)}(b)-\frac{r^{(q)}(a)}{(c-\delta)\mathbb{W}^{(q)}(a-b)}\right).
\left( (j+1 \ldots n-1)(j \ldots n) \right) \odot \hat{\mu}_{r_{n-2}+j}\odot \mu = \left( (j+1 \ldots n)(j \ldots n-1) \right) \odot \mu > \left( (j+1 \ldots n-1)(j \ldots n) \right) \odot \mu.
{\frac{1}{4 \pi G_{4}}}\int_{\partial \Sigma}d S_{\mu \nu}e \hat{E}^{\mu \nu}= \bar{\varepsilon}_{\infty}\left[ P_{\mu}^{A D M}\gamma^{\mu}+{\frac{1}{2 \sqrt{2}G_{4}}}e^{- \varphi_{\infty}/ 2}\{V_{R \, \infty}L ( \vec{Q}- i \gamma^{5}\vec{P}) \}^{a}\Gamma^{a}\right] \varepsilon_{\infty},
\begin{array}{c}{{\left({\frac{D}{D x^{\mu}}}- i^{\frac{p}{2}}{\frac{\partial f}{\partial R}}\Gamma^{\mu}\prod_{I = 1}^{p - 1}\Gamma^{I}\right) \Psi_{+ +}= 0 ( \mu = 1 , \ldots , p - 1 )}}\\{{\left({\frac{D}{D y^{\mu}}}- i^{\frac{q}{2}}{\frac{\partial f}{\partial R}}\Gamma^{p + \mu}\prod_{I = 1}^{q - 1}\Gamma^{p + I}\right) \Psi_{+ +}= 0 ( \mu = 1 , \ldots , q - 1 )}}\\ \end{array}
\gamma_{i}^u = \frac{P_i^u G_{ib}}{\sigma^2 + \sum_{j=1}^J x_{ij}P_j^d\beta}, \; \gamma_{j}^d = \frac{P_j^d G_{bj}}{\sigma^2 + \sum_{i=1}^I x_{ij}P_i^u G_{ij}},
\big\|b_{k,\ell}(\rho)\chi(\frac{\rho}M)\rho^{\frac{n-1}{2}}\big\|_{L_\rho^2}= \big\|\big[\mathcal{H}_{\nu}\chi(\frac{\rho}M)\mathcal{H}_{\nu}\big]\big(Y_{k,l}(\theta)a_{k,\ell}(r)\big)\big\|_{L_x^2}= \big\|\tilde{P}_j\big(Y_{k,l}(\theta)a_{k,\ell}(r)\big)\big\|_{L_x^2}.
x\left( \phi-1\right) P_{k}=\left[ \allowbreak x^{3}-\left(z^{2}+1\right) x\right] P_{k}=P_{k+3}+\gamma_{k}\gamma_{k-1}\gamma_{k-2}P_{k-3}& +\left[ \gamma_{k}+\gamma_{k+1}+\gamma_{k+2}-\left( z^{2}+1\right)\right] P_{k+1}& +\gamma_{k}\left[ \gamma_{k}+\gamma_{k+1}+\gamma_{k-1}-\left(z^{2}+1\right) \right] P_{k-1}.
S\circ T((a_{\xi})_{\xi<\omega_1})=\left(T((a_{\xi})_{\xi<\omega_1})(z_\xi)\right)_{\xi<\omega_1}=\left(a_{\xi}\cdot\chi_{A_\xi}(z_\xi)\right)_{\xi<\omega_1}=(a_{\xi})_{\xi<\omega_1}
r(t)=\frac{\sum_{n=1}^{\infty}\varphi_{n}e^{-\lambda_{n}t}\phi_{n}(q)}{w(t)}+\frac{1}{w(t)}\int_{0}^{t}\left( \sum_{n=1}^{\infty}\phi_{n}(q)f_{n}(\tau )e^{-\lambda_{n}(t-\tau )}\right) r(\tau )d\tau
2^{2n}A^{2,2}a'_2 = -2^{n}A^{2,1}a'_1 - A^{2,0}a_0 &= A^{2,1}\Big(A^{1,1}\Big)^{-1}\Big(A^{1,0}a_0\Big) - A^{2,0}a_0.
\frac{3}{2}X_h^{j}- 2 X_h^{j-1}+ \frac{1}{2}X_h^{j-2}= h f(X_h^j) + \frac{3}{2}g( X_h^{j-1}) \Delta_h W^j - \frac{1}{2}g(X_h^{j-2}) \Delta_h W^{j-1},
\frac{d}{dt}\mathbb{E}_{z\sim\mu}f(g(\theta_t,z))= \frac{d}{dt}\int_{Z}f(g(\theta_t,z))\mu(z)dz = \frac{d}{dt}\int_\Omega f(x)\rho(\theta_t,x)dx = \int_\Omega f(x) \frac{\partial}{\partial t}\rho(\theta_t,x)dx,
\frac{\partial \rho(\tau_{k}, \alpha)}{\partial A_{k}}= \frac{2 \log(2)}{\alpha}\left[ \frac{(1 + \tau_{k})\rho(\tau_{k}, \alpha) + \tau_{k}}{\tau_{k}}\left(\frac{R_{k}\lambda_{u}}{\lambda_{k}Ww_{k}}\right) \right].
V_n=(2 \cdot 4^{\ell(n)}-n) \frac 1{36^{\ell(n)}}\frac 14+(n-4^{\ell(n)}) \frac 1{36^{\ell(n)}}\frac 5{36}=\frac 1{36^{\ell(n)+1}}(13 \cdot 4^{\ell(n)}-4n).
\operatorname{tr}(T)=-R^2=O\left(r^{-2}\right), T(\nabla f, \cdot)=(n-1) \operatorname{Ric}(\nabla f, \cdot)-R^2 \nabla f=O\left(r^{-\frac{3}{2}}\right), T(\nabla f, \nabla f)=(n-1) \operatorname{Ric}(\nabla f, \nabla f)-R^2|\nabla f|^2=O\left(r^{-2}\right).
\varphi\left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{\ell_1}c_1 \, a^{\ell_1}\, b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{- \ell_2}0 0 0 d^{- \ell_2}\end{array}\right) .
J_2(\chi^+_2,\chi^-_2)= \,\frac{1}{2}\int_0^\infty\Big[\,\big|{\chi^+_2}'(r)\big|^2{U^+}^2+\big|{\chi^-_2}'(r)\big|^2{U^-}^2+2A_+{U^+}^4{\chi^+_2}^2+2A_-{U^-}^4{\chi^-_2}^2\,\Big]r\,{\mathrm d}r begin{align*}1mm] +\int_0^\infty\Big[\,4B{U^+}^2{U^-}^2\chi^+_2\chi^-_2+{\mathfrak h}_2^+\chi^+_2{U^+}^2+{\mathfrak h}_2^-\chi^-_2{U^-}^2\,\Big]r\,{\mathrm d}r.
I_1\leq \frac2\delta \|u^2\|_{L^\infty(L^{p'_2})}\|b^1-b^2\|_{L^1(L^{p_2})}+ C_{d,p_2}\bigg(\sum_{i=1}^2\|u^i\|_{L^\infty(L^{p'_2})}\bigg) \big\| \nabla b^1\big\|_{L^1(L^{p_2})}.
f_N \stackrel{q}{\smallfrown}f_N (t_1,\dots, t_{d-q},s_1,\dots,s_{d-q}) = &= \sum\limits_{i_{1},\dots,i_q=1}^{N}f_N(t_1,\dots,t_{d-q},i_1,\dots,i_q)f_N(i_q,\dots,i_1,s_1,\dots,s_{d-q}).
\Sigma_{\bar{X}^j}= \mbox{cov}(Y_0^j + P^{'}(P P^{'})^{-1}\bar{V}^j + \bar{U}^j , Y_0^j + P^{'}(P P^{'})^{-1}\bar{V}^j + \bar{U}^j) & = \Sigma_{\bar{V}^j}P^{'}(P P^{'})^{-1}(P P^{'})^{-1}P^{'}+\Sigma_{\bar{U}^j},
s\sum^{N-1}_{n=0}\int_\Omega |\hat{A}^{\epsilon,s}_n|^2 d\hat x \leq 2\sum_{n=0}^{N-1}\int_{ns}^{(n+1)s}\int_{\Omega}\left(|\hat A^{\epsilon,s}-\hat A^{\epsilon,s}_n|^2+|\hat A^{\epsilon,s}|^2\right) d\hat x dx_3.
f_3(\rho^{(1)}) =\frac{1}{2}\log\left(1+2P(1+\rho^{(1)})\right) &\geq \frac{1}{2}\log\left(1+2P(1+\rho^{(1)})\right)-\frac{1}{2}\log\left(\frac{1}{1-{\left(\rho^{(1)}\right)}^2}\right) &= 2f^\prime_4(C,\rho^{(1)})-2C &{\geq}f^\prime_4(C,\rho^{(1)}).
\partial_t g_{\mu \nu}= k_{\mu \nu}, g^{ij}\partial_t g_{\mu \nu, i}= g^{ij}\partial_{i}k_{\mu \nu}, - g^{00}\partial_t k_{\mu \nu}= 2 g^{0 j}\partial_{j}k_{\mu \nu}+ g^{ij}\partial_{j}g_{\mu \nu, i}- 2 H_{\mu \nu}[g,k] - 2 \nabla_{(\mu}F_{\nu)}[g,k] + 2f \cdot V_{\mu}V_{\nu}[g,\widetilde{\phi}],
\begin{cases}a \cdot ( u \cdot b ) - (u \cdot b ) \cdot a + H (a, T (u \cdot b)) - H ( T (u \cdot b), a) = u \cdot ( ab -ba) + ( a \cdot u - u \cdot a + H (a, Tu) - H (Tu, a)) \cdot b, (a \cdot u -u \cdot a + H (a, Tu) - H (Tu, a)) \cdot (ab - ba) = 0, ~ b \in A, u \in M.\end{cases}
\int_0^{\frac{t}{2}}t^{\frac{\beta}{\alpha}}(t-\tau)^{-1+\frac{2\gamma-\beta}{\alpha}}\tau^{-\frac{2\gamma}{\alpha}}\, d\tau = \int_0^{\frac{1}{2}}(1-\tau')^{-1+\frac{2\gamma-\beta}{\alpha}}(\tau')^{-\frac{2\gamma}{\alpha}}\, d\tau' \lesssim \int_0^{\frac{1}{2}}(\tau')^{-\frac{2\gamma}{\alpha}}\, d\tau'\lesssim 1
\ddot{w}= 2 \frac{w ( k - w^{2})}{a^{2}}\left( \frac{1 - K^{2}}{1 + V^{2}}\right) + \frac{2 \dot{a}\dot{w}}{a}\left( \frac 1 2 - \frac{1 - K^{2}}{1 + V^{2}}\right) .
s_{\psi(G)}(\varphi) =s_G(\varphi)+\varphi \cdot s_G(\psi)-s_G(\psi) &= \varphi \cdot \xi_G-\xi_G +\varphi \cdot (\psi \cdot \xi_G -\xi_G)-(\psi \cdot \xi_G -\xi_G) &=\varphi \cdot (\psi \cdot \xi_G)-\psi \cdot \xi_G &=\delta (\psi \cdot \xi_G) (\varphi).
\mathfrak{N}_{s}= -\psi T T_{s}= \big[\psi\mathfrak{N}\big] = \frac{1}{2}\Big(\bar{\psi}\mathfrak{N}+\psi\bar{\mathfrak{N}}\Big) T_{t}= \big[i\psi'\bar{\mathfrak{N}}\big] = \frac{1}{2}i\big(\psi'\bar{\mathfrak{N}}-\bar{\psi}'\mathfrak{N}\big).
\left( \begin{matrix}{{X_{1}}}\\{{X_{2}}}\\{{X_{3}}}\\{{X_{4}}}\\ \end{matrix}\right) = \frac{1}{2 r}\left( \begin{matrix}{{3}}&{{1}}&{{1}}\\{{1}}&{{- 1}}&{{- 3}}\\{{- 1}}&{{- 1}}&{{3}}\\{{- 3}}&{{1}}&{{- 1}}\\ \end{matrix}\right) \left( \begin{matrix}{{\varphi^{( 1 )}}}\\{{\varphi^{( 2 )}}}\\{{\varphi^{( 3 )}}}\\ \end{matrix}\right) ,
y=\frac{\frac{a}{K_{1}}+\frac{3a^{2}}{K_{1}K_{2}}+\frac{3a^{3}}{K_{1}K_{2}K_{3}}+\frac{a^{4}}{K_{1}K_{2}K_{3}K_{4}}}{1+\frac{4a}{K_{1}}+\frac{6a^{2}}{K_{1}K_{2}}+\frac{4a^{3}}{K_{1}K_{2}K_{3}}+\frac{a^{4}}{K_{1}K_{2}K_{3}K_{4}}}
\mathbb{P}\Big[ \|T(E)\|_1 \leq \frac{c}{c+1}\Big] = 1-\mathbb{P}\Big[ \|T(E)\|_1 > \frac{c}{c+1}\Big]\geq 1- \frac{c+1}{c}\,\mathbb{E}\big[ \|T(E)\|_1 \big]
{_pF_q}\left(\left.\begin{array}{c}a_{1},\dots,a_p begin{align*}4pt]b_{1},\dots,b_q \end{array}\right| x\right):=\sum_{n=0}^\infty\frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n}\frac{x^n}{n!},
\Gamma^{2}= - i \lim_{\theta \rightarrow 2 i \pi / 3}( \theta - \frac{2 \pi i}{3}) \, S ( \theta ) = 2 \, \sqrt{3}\, \frac{( c + 1 ) ( 1 + 2 c )}{( c - 1 ) ( 1 - 2 c )}\: ,
u_+(x):=\frac{1}{\sqrt 2}\left(\left(\Delta_{g}^2+m\right)^{1/4}\varphi+i\left(\Delta_{g}^2+m\right)^{-1/4}\psi\right)\ , u_-(x):=\frac{1}{\sqrt 2}\left(\left(\Delta_{g}^2+m\right)^{1/4}\varphi-i \left(\Delta_{g}^2+m\right)^{-1/4}\psi\right)\ ,
r_n^{\mathrm{mu}}=\delta_n\!\cdot\!\left(\sum_{\iota=1}^{M}h_{\iota n}x_\iota+z_n\right)+\bar{\delta}_n\!\cdot\!\mathrm{sgn}\left(\sum_{\iota=1}^{M}h_{\iota n}x_\iota+z_n\right),
H \; = \; \Delta_{q}\sum_{i = 1}^{L}\sigma_{i}^{z}\; + \; \frac{\Delta_{\eta}}{2}\sum_{i = 1}^{L - 1}\: [ ( 1 + u ) \sigma_{i}^{x}\sigma_{i + 1}^{x}\: + \: ( 1 - u ) \sigma_{i}^{y}\sigma_{i + 1}^{y}] \; + \; B \; + \; S ,
C_{A^* + B^*}\ =\ \bigcup_{1 \leq i \leq k, 1 \leq j \leq \ell}C_{I_i^* + J_j^*}&\ =\ \bigcup_{1 \leq i \leq k, 1 \leq j \leq \ell}I_i^* + J_j^* &\ =\A^* + B^*.
\sum_{n\geq t_j^{k}}\frac{d_{2k}(n^2)}{n^2}\leq t_j^{-k\nu}\sum_{n=1}^{\infty}\frac{d_{2k}(n^2)}{n^{2-\nu}}\ll \exp\left(-\frac{j k\log_3 k}{4\log k}+O(k\log_2 k)\right).
\|f\|_*=\max\big(\sup_{t\in (-\infty,0)}\max\big(1,\big(1-r+{|x_-|\over \sqrt{2}}+({|v|\over 2^{3\over 2}}-r)|t|\big)\big)|\dot f(t)|, \sup_{(0,+\infty)}|\dot f|,\sup_{(-\infty,0)}|f|\Big).
\rho_{\pm}e_{7}= \mp i \rho_{\pm}, \; \rho_{\pm}e_{5}= \mp i \rho_{\pm}e_{1}, \; \rho_{\pm}e_{3}= \mp i \rho_{\pm}e_{2}, \; \rho_{\pm}e_{6}= \mp i \rho_{\pm}e_{4},
h\left(0\right) =\int_{0}^{\frac{1}{2}}\left(1-y\right)\varphi\left(y\right)dy h'\left(0\right) =\int_{0}^{\frac{1}{2}}\varphi\left(y\right)dy h\left(\tfrac{1}{2}\right) =\int_{0}^{\frac{1}{2}}\left(\tfrac{1}{2}+y\right)\varphi\left(y\right)dy h'\left(\tfrac{1}{2}\right) =-\int_{0}^{\frac{1}{2}}\varphi\left(y\right)dy
1 - \rho\sigma\varphi(a_1(t_1)b_1(s_1))\varphi(a_2(t_2)b_2(s_2)) = &1 - \frac{1}{z(z+1)w(w+1)}(H_{a_1,b_1}(t_1,s_1) &\ \ - (z+1)(w+1))(H_{a_2,b_2}(t_2,s_2) - (z+1)(w+1))
n_i^{\rm imp}= \int_{\lambda_{iL}}^{\lambda_{iR}}d\lambda\, L(\lambda|\lambda_p) = -F(\lambda_{iR}|\lambda_p) + F(\lambda_{iL}|\lambda_p), d_i^{\rm imp}= \left\{\int_{-\infty}^{\lambda_{iL}}-\int_{\lambda_{iR}}^{\infty}\right\}d\lambda\, L(\lambda|\lambda_p) = - F(\lambda_{iR}|\lambda_p) - F(\lambda_{iR}|\lambda_p).
S(\lambda_{za}) = S(\lambda_z \lambda_a + [\lambda_z,\rho_a]) = S(\lambda_a)S(\lambda_z) + [S(\rho_a),S(\lambda_z)]\cr = \lambda_{S(a)}\lambda_{S(z)}+ [\rho_{S(a)},\lambda_{S(z)}] = \lambda_{S(a)S(z)}= \lambda_{S(za)},
0=<<\left\{J_{i}D_{i}^{\alpha}+igK_{i}^{*}T^{\alpha}_{ij}\psi_{j}-ig\psi_{i}^{*}T^{\alpha}_{ij}K_{j}\right\}-i\varepsilon \frac{\delta}{\delta c^{\alpha}}\delta_{BRS}O-\delta_{BRS}Y^{\alpha}>>
\left(1-\sqrt{b/d}\right)\left( \frac{y+\sqrt{bd}}{2}\right)^{2m}<\frac{-b+\sqrt{bd}}{\sqrt{bd}}\left( \frac{y+\sqrt{bd}}{2}\right)^{2m}<Y_{2m}^{(b,d)}=Y_{2h}^{(a,b)}< &<\frac{b+\sqrt{ab}}{\sqrt{ab}}\left(\frac{r+\sqrt{ab}}{2}\right)^{2h}\leq \left(1+\sqrt{b/a}\right)\left(\frac{r+\sqrt{ab}}{2}\right)^{2h}.
P( \psi(X_{t_{n+1}})\in A|\>\sigma(\psi(X_{t_n}))) =K_{\psi(X_{t_n})}\circ K^X_{\mu_{t_{n+1}-t_n}}(x_0,\psi^{-1}(A)) &= P( \psi(X_{t_{n+1}})\in A|\>\sigma(\psi(X_{t_0}),\ldots\psi(X_{t_n}) )) \quad\quad
\left\{\prod_{X_{i}\in ({\it G}_{1})_{X'_{i'}}}\;e^{\eta'_1\sum_{\stackrel{i_{\alpha_{1}}< j_{\alpha_{1}}< k_{\alpha_{1}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}},k_{\alpha_{1}}\right\}\in J_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}t_{k_{\alpha_{1}}}{\bf 1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}})=w(t_{k_{\alpha_{1}}}))}}\right\}.
I = \sum_{n=0}^\infty \frac{(\beta)_n}{(\gamma)_n}\, x^n \, I_n & = \frac{\Gamma(\beta+2)}{\Gamma(\beta+\gamma+1)}\, \int_0^1 v^{\beta+\gamma}\, (1-xv)^{-\gamma}\, dv \, \sum_{n=0}^\infty \frac{(\beta)_n\, \Gamma(n+\gamma)}{\Gamma(n+1)\, (\gamma)_n}\, \bigg( x \, \frac{1-v}{1-xv}\bigg)^n
\int_{M}\bigg( | D \mathrm{u}( z ) |^{p( z ) - 2}D \mathrm{u}( z ) + \mu ( z ) \, | D \mathrm{u}( z ) |^{q( z ) - 2}D \mathrm{u}( z ) \, \bigg) \, D \varphi ( z ) \,\, dv_{g}( z ) & = \int_{M}g( z ) \, | \mathrm{u}( z ) |^{- \gamma ( z )}\, \varphi ( z ) \,\, dv_{g}( z ) + \lambda \int_{M}| \mathrm{u}( z ) |^{r( z ) -1}\varphi ( z ) \,\, dv_{g}( z ),
\kappa ( u ) = \exp \int_{- \infty}^{+ \infty}\sinh u t \; \frac{\sinh ( \pi - \lambda ) t \; \sinh ( \pi - 3 \lambda + u ) t - \sinh \lambda t \; \sinh ( u + \lambda ) t}{t \sinh \pi t \; \sinh ( \pi - 2 \lambda ) t}\; d t .
|\partial_{r}\left(\frac{\phi(r,\xi)}{\sqrt{r}}\right)| \leq \begin{cases}C\left(|\phi_{0}'(r)| + \xi \log(1+r^{2})\right), r^{2}\xi \leq 4 \frac{C |a(\xi)| \xi^{1/4}}{r^{1/2}}, r^{2}\xi > 4\end{cases}
m_\varphi:=\inf_{\textup{Amp}(\{\theta\})}\frac{\varphi_a-\varphi_b}{a-b}= \inf_{\textup{Amp}(\{\theta\})}\frac{\varphi_c-\varphi_d}{c-d}, \ \M_\varphi:=\sup_{\textup{Amp}(\{\theta\})}\frac{\varphi_a-\varphi_b}{a-b}= \sup_{\textup{Amp}(\{\theta\})}\frac{\varphi_c-\varphi_d}{c-d}.
\sum_{n=m}^{\lfloor m^2/q \rfloor}\Phi_{n,m}^2 \leq \sum_{n=m}^{\lfloor m^2/q \rfloor}\left( 2 e^{-m^2/(2n)}\right)^2 begin{align*}1.5ex] \leq 4 \sum_{n=m}^{\lfloor m^2/q \rfloor}e^{-m^2/n}
\frac12\int_\Omega|x|^{-s}(u(x,\widetilde{T}))^2dx-\frac12\int_\Omega|x|^{-s}(u_0(x))^2dx+\int_0^{\widetilde{T}}(\xi,\nabla u)dt=\int_0^{\widetilde{T}}(|u|^{q-2}u\ln|u|,u)dt.
\mu\left(\{\underline{x}:\sigma^{i+k}\underline{x}\in C_{r_n}(\sigma^i\underline{x})\}\right)=\mu(S_k(r_n))\leq\mu(S_{\ell_k}(r_n))\leq B^{6}\sum_{C_{\ell}\in\mathcal{C}_{\ell}}\mu(C_{\ell})^{\omega_{\ell}+1}\rho^{\gamma_{\ell}}\leq B^6Z_{\ell}(\omega_{\ell}),
L_{T}:={\sigma_{\tau}^{*}}(t)\left[P_{a}\dot{Q}^{a}+\Theta_{\alpha}\dot{\Theta}^{\alpha}-H_{T}(\Theta^{\alpha},\Theta_{\alpha},Q^{a},P_{a})\right] &=P_{a}(t)\dot{Q}^{a}(t)-H_{T}(\Theta^{\alpha}(t)=\epsilon^{\alpha},\Theta_{\alpha}(t)=\epsilon_{\alpha},Q^{a}(t),P_{a}(t)).
[ \chi_{\alpha}( p ) , \chi^{\dagger \beta}( q ) ]_{+}= \delta_{\alpha}^{\beta}2 \pi \delta ( p - q ) \, , \qquad [ a_{\alpha}( p ) , a^{\dagger \beta}( q ) ] = \mathrm{s g n}( p ) \delta_{\alpha}^{\beta}2 \pi \delta ( p - q ) \, .
\int_\Omega \Big(\big\langle\frac{\partial d}{\partial x_i}, \frac{\partial d}{\partial x_j}\big\rangle \frac{\partial Y^i}{\partial x_j}-\frac{|\nabla d|^2}2{\rm{div}}Y+\big\langle\tau, Y\cdot\nabla d\big\rangle\Big)=0,\ \forall \Y\in C_0^\infty(\Omega,\mathbb R^3).
\begin{aligned}\frac{1}{2}\frac{d}{dt}\int ( |\nabla \varrho|^2+\!|\nabla u|^2+|\nabla B|^2 ) dx +\int ( \mu |\nabla^2 u|^2+(\mu+\nu)|\nabla{\rm div}u|^2+|\nabla^2 B|^2)dx &=\int \nabla S_1 \cdot \nabla \varrho \dx+\int \nabla S_2 \cdot \nabla u \dx+\int \nabla S_3 \cdot \nabla B \dx=I\!I_1+I\!I_2+I\!I_3. \end{aligned}
e^{tA}\left(\begin{array}[c]{c}y z\end{array}\right) =\left(\begin{array}[c]{cc}\cos\sqrt{{\Lambda}}t \frac{1}{\sqrt{{\Lambda}}}\sin\sqrt{{\Lambda}}t -\sqrt{{\Lambda}}\sin\sqrt{{\Lambda}}t \cos\sqrt{{S}}t\end{array}\right) \left(\begin{array}[c]{c}y z\end{array}\right) ,t\in\mathbb{R}.
A_J =\{P\in \hat{X}_K|\,\,\, |y_j(P)|<1 \,\,\,(j\in J)\,\,\, |y_j(P)|\geq \lambda\,\,\, (j \notin J) \}= \{P\in \hat{X}_K|\,\,\, y_j(P)=0 \,\,\,(j\in J),\,\,\,|y_j(P)|\geq \lambda\,\,\, (j \notin J)\}\times A^{|J|}[0,1).
\begin{array}{lll}\beta_1: X_1=0 \beta_5 : a_{11}X_1 + a_{12}X_2 + a_{13}X_3 = 0 \beta_2: X_2=0 \beta_6 : a_{21}X_1 + a_{22}X_2 + a_{23}X_3 = 0 \beta_3: X_3=0 \beta_7 : a_{31}X_1 + a_{32}X_2 + a_{33}X_3 = 0 \beta_4: X_1+X_2+X_3=0\end{array}
\phi_1 = \frac{m}{\sqrt\lambda}\left\{- \frac{3}{4}m^2 t^2{\eta}_0 - \frac{1}{4}{\eta}_{k{_0}}+ a_1 e^{i \omega_1 t}\frac{~{\rm{sh}}~z}{{\rm{ch}}^2~z}+ \sum_{k=0}^{\infty}a_k{\widetilde C}_k(t) \eta_k (x)\right\}
U^{*}=U-\frac{\partial U}{\partial Q}|_{x}\cdot Q=U-\frac{1}{2C(x)}\frac{\partial Q^{2}}{\partial Q}|_{x}\cdot Q=U-QV=\frac{1}{2}QV-QV=-\frac{1}{2}QV=-\frac{1}{2}V^{2}C(x).
\begin{aligned}\mathcal{F}_{1}:=P^{\ast}\Lambda^{1}(\Omega)\times H^{-\frac{1}{2}}\Lambda^{n-1}(\Sigma)\times H^{-\frac{1}{2}}\Lambda^{n-1}(\Gamma), \mathcal{E}_{1}:=P\Lambda^{n-1}(\Omega)\times H^{\frac{1}{2}}\Lambda^{0}(\Sigma)\times H^{\frac{1}{2}}\Lambda^{0}(\Gamma),\end{aligned}
((u-c) \varrho )_x + (v \varrho)_y = 0, \textrm{in}\Omega_i, -c(\varrho u)_x + (\varrho u^2)_x + (\varrho uv)_y = -P_x , \textrm{in}\Omega_i, -c(\varrho v)_x + (\varrho uv)_x + (\varrho v^2)_y = -P_y - g \varrho, \textrm{in}\Omega_i.
{\cal F}_{i j}= \left( \begin{array}{c c c c}{{0}}&{{{\cal F}_{0 1}}}&{{0}}&{{0}}\\{{-{\cal F}_{0 1}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{{\cal F}_{2 3}}}\\{{0}}&{{0}}&{{-{\cal F}_{2 3}}}&{{0}}\\ \end{array}\right) .
Z = \Sigma_{\mathrm{i n s t a n t o n}}( \mathrm{d e t}\Delta_{1})^{-{\frac{1}{2}}}( \mathrm{d e t}\Delta_{0})^{- 1}( \mathrm{d e t}\Delta_{1})^{\frac{1}{2}}( \mathrm{d e t}\Delta_{0}) = \Sigma_{\mathrm{i n s t a n t o n}}\pm 1 = \pm 1
\frac{M\circ\sigma_\lambda(Z(\nu))}{M(Z(\nu))}=\delta_{s(\lambda), r(\nu)}\frac{\rho(\Lambda)^{-d(\lambda\nu)}x^\Lambda_{s(\nu)}}{\rho(\Lambda)^{-d(\nu)}x^\Lambda_{s(\nu)}}=\delta_{s(\lambda), r(\nu)}\rho(\Lambda)^{-d(\lambda)}.
\begin{aligned}x_1=\exp(2\pi i z/\overline\omega_1), x_2=\exp(2\pi i z/\omega_2), q_1=\exp(2\pi i{\omega_2}/{\overline\omega_1}), q_2=\exp(2\pi i{\omega_1}/{\omega_2}) , \widetilde{q}_2=\exp(2\pi i{\widetilde\omega_1}/{\omega_2}) .\end{aligned}
\mathcal{I}^\star (\lambda) = \sup_{\gamma \in \bar{\mathcal{P}}}I_{\lambda}[\gamma] = \sup_{\pi \in \Gamma(\Pi, \phi_\lambda )}\int_{\mathcal{V}\times \mathcal{V}}\varphi_\lambda d\pi = \sup_{\pi \in \Pi}\int_{\mathcal{V}}f_\lambda d\pi.
(\partial^h\Psi)_{A_{p+1}\overline{B}_q}= \sum_{i=1}^{p+1}(-1)^{i-1}\delta_{\alpha_i}(\psi_{\alpha_1\dots\hat{\alpha}_i\dots\alpha_{p+1}\overline{B}_q}), (\bar{\partial}^h\Psi)_{A_p\overline{B}_{q+1}}= (-1)^p\sum_{i=1}^{q+1}(-1)^{i-1}\delta_{\bar{\beta}_i}(\psi_{A_p\bar{\beta}_1\dots\hat{\bar{\beta}}_i\dots\bar{\beta}_{q+1}}).
Y_{l_{1}}^{m_{1}}\star_{1}^{n}Y_{l_{2}}^{m_{2}}=Y_{l_{1}}^{m_{1}}Y_{l_{2}}^{m_{2}}|_{n}+\frac{i}{n}\{Y_{l_{1}}^{m_{1}},Y_{l_{2}}^{m_{2}}\}|_{n}+ o(1/n) \ ,
[D_{x^\prime}D^{m-1}u]_{\frac{a}{2m},a,Q_R^+}&\le N(\sum_{|\alpha|\le m}\|D^\alpha u\|_{L_\infty(Q_{4R}^+)}+\sum_{|\alpha|=m}[f_\alpha]_{a,Q_{4R}^+}^\ast+\sum_{|\alpha|<m}\|f_\alpha\|_{L_\infty(Q_{4R}^+)}).
{\cal L}\;=\;{1\over 2}( \tilde \Psi\sp\ast (i\tilde \partial_t - \tilde A_0)\tilde \Psi \;+\; c.c.)\;+\;\tilde A_0 -\;{1\over 2}\;|\tilde D_i\tilde \Psi|^2 +{1\over 2}({1\over \beta}{\bf \tilde E}\sp2 - \tilde{\bf B}\sp2 )\;-\;{1\over 8}\kappa\sp2 (\tilde \Psi \tilde \Psi\sp\ast-1)\sp2
\phi\big(\Delta(s^{\Lambda^i})^{E\cap \Lambda^i}\big)=\Delta(s^\Lambda)^{E\cap \Lambda^i}=\sum_{\substack{\emptyset\neq G\subseteq E G\cap \Lambda^{e_i}\neq \emptyset \lambda\in \mathrm{MCE}(G)}}(-1)^{(|G|+1)}s_\lambda^\Lambda{s_\lambda^\Lambda}^*,
\acute{x}^{A A^{\prime}}= ( p + \bar{p}) o^{A}\bar{o}^{A^{\prime}}+ | q | ( \textrm{e}^{- \i \varphi}o^{A}\bar{\iota}^{A^{\prime}}+ \textrm{e}^{\i \varphi}\iota^{A}\bar{o}^{A^{\prime}}) .
\begin{cases}\mathbf{S}= \{(x_1, x_2) \in \mathbb{R}^2 \mid x_1^2 + x_2^2 = 2 \}, \mathbf{n}= (1, 1), \mathbf{P}= \{(x_1, x_2) \in \mathbb{R}^2 \mid x_1 + x_2 = 0 \}.\end{cases}
\|Kh\|^2=\int_{B}\nabla (Kh)\cdot \nabla (Kh)=\int_{B}(Kh) h \leq |Kh|_\frac{2n}{n-2}|h|_\frac{2n}{n+2}\leq c\|Kh\|\, |h|_\frac{2n}{n-2},
\Lambda= \Lambda(a,b)=\int_{(2a-b)^+}^a (h(x,a)-h(a,b)){\rm d}x+\int_a^{\frac{a+b}2}(h(a,x)-h(a,b)){\rm d}x & +\int_{\frac{a+b}2}^b (h(x,b)-h(a,b)){\rm d}x+\int_{b}^{2b-a}(h(b,x)-h(a,b)){\rm d}x.
d\rho_i=\sum_{j\in N(i)}\omega_{ij}(S_i-S_j)\theta_{ij}(\rho) dt; d S_i+(\sum_{j\in N(i)}\frac{\omega_{ij}}{2}(S_i-S_j)^2 \frac{\partial \theta_{ij}}{\partial \rho_i}+\frac 18 \frac{\partial}{\partial \rho_i}I(\rho)+\mathbb V_i +\sum_{j\in N(i)}\mathbb W_{ij}\rho_j-\log(\rho_i))dt+\sigma_idW_t=0.
0 =\langle N_i, DF(x)E_1\rangle + \langle D_vN_i, F(x)E_1\rangle - \langle F(x)N_i, D_vE_1\rangle &= \langle N_i, DF(x)E_1\rangle - \langle N_i, D_vE_2\rangle - \sum_{j=1,2}\langle F(x)N_i,N_j\rangle \langle N_j, D_vE_1\rangle &= \langle N_i, DF(x)E_1\rangle -\left( w_2^{i+2}(v) + \sum_{j=1,2}\langle F(x)N_i,N_j\rangle w_1^{j+2}(v)\right)
(2n+1)N^{(1)}_{(f,H_1),\leqslant k_1}(r)\ge (2n-1)\sum_{i=2}^{2n+2}N^{(1)}_{(f,H_i),\leqslant k_i}(r)-2\sum_{u=1}^3\sum_{i=2}^{2n+2}N^{(1)}_{(f^u,H_i),>k_s}(r) &-2(2n+1)\sum_{u=1}^3N^{(1)}_{(f^u,H_1),>k_1}(r)+o(T(r)).
\tilde\varphi\circ\bar\partial_E\circ\tilde\varphi^{-1}= d\bar z\left[\frac{\partial}{\partial\bar z}+\begin{pmatrix}-\bar Z\log (u_1/|z|^{q}) -\bar Qw^{p+q}/u_1u_2 0 \bar Z\log(u_2/|z|^{p}) \end{pmatrix}\right].
\Lambda'(u)v=C_{\alpha,\beta,\eta}Q(u) \left(\frac{\beta-\alpha}{\beta-\eta}A(u)B(u)N'(u)v-N(u)B(u)A'(u)v-\frac{\eta-\alpha}{\beta-\eta}N(u)A(u)B'(u)v\right),