harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\hat{J}_{rstw}(\gamma) = \sum_{j,k,l,m}\hat{v}^k_r \hat{v}^k_s \hat{v}^l_t \hat{v}^m_w \hat{\omega}^{jk}\hat{\omega}^{jl}\hat{\omega}^{kl}\hat{\omega}^{km}\left[ h_1 (\gamma)\sum_{o,q}\hat{\alpha}_{jko}\hat{\omega}^{oq}\hat{\alpha}_{lmq}- h_2(\gamma) \hat{\alpha}_{jklm}\right] \,.
M_{N N}= N m_{0}\, \frac{\sqrt{2}}{3}\left( \, \sqrt{1 - \sqrt{1 - \left( \frac{g^{2}N}{4 \pi m_{0}^{2}\alpha_{0}}\right)^{2}}}+ \, \frac{4 \pi m_{0}^{2}\alpha_{0}}{g^{2}N}\, \sqrt{1 + \sqrt{1 - \left( \frac{g^{2}N}{4 \pi m_{0}^{2}\alpha_{0}}\right)^{2}}}\; \right)
\frac{\partial \mathcal{E}_t}{\partial t}= \dot \beta_t e^{\beta_t}\left( f(X_t) - f(x^\ast) + \frac{\mu}{2}\|x^\ast - Z_t\|^2\right), \frac{\partial \mathcal{E}_t}{\partial X_t}=e^{\beta_t}\nabla f(X_t), \frac{\partial \mathcal{E}_t}{\partial Z_t}= \mu(x^\ast - Z_t).
\left( \hat{V}^{( B = 0 )}\right)^{- 1}\partial_{\pm}\hat{V}^{( B = 0 )}={\frac{1 \mp t}{1 \pm t}}\left( \begin{array}{c c}{- E^{- 1}\partial_{\pm}E}&{0}\\{0}&{E^{- 1}\partial_{\pm}E}\\ \end{array}\right) ={\frac{1 \mp t}{1 \pm t}}P_{\pm}
(1-\gamma) (\partial_y + 2\ell^{-1})\,h|_{0_+}-\frac{2}{3}\,\gamma\,\ell\,\big(\partial_\lambda\partial_\rho h^{\lambda\rho}-\partial_\lambda^2 h\big)|_{0_+}=\frac{1}{6M_*^3}\,S_\mu^\mu\,.
\gamma\left\{\Vert\partial_x^m\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\partial_x^m\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\le\frac{C_0}{\gamma}\left\{\sum\limits_{j=1}^5\Vert G_j\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert \partial_x^m g\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,,
(\nu\otimes\operatorname{id})\bigl((\sigma^{\nu}_t\otimes\sigma^{\mu}_{-t})(E)(b\otimes1)\bigr) =\sigma^{\mu}_{-t}\bigl(\gamma_B(\sigma^{\nu}_{-t}(b))\bigr) &=(R\circ\sigma^{\nu}_{i/2})(b)=(\nu\otimes\operatorname{id})\bigl(E(b\otimes1)\bigr).
\textnormal{tr}\big ( r_{n}(x, \xi, \lambda) \big ) = \sum_{\substack{n=2j - |\alpha|-2 |\alpha| \leq 3n}}r_{n, j , \alpha}(x) \, \xi^\alpha \, \textnormal{tr}\left (r_0(x, \xi, \lambda)^j \right ).
\Vert u_{i+1}-u_i\Vert <\tau\Vert u_i-u_{i-1}\Vert+\tilde\varepsilon &<\tau^2\Vert u_{i-1}-u_{i-2}\Vert+\tau\tilde\varepsilon+\tilde\varepsilon &<\ldots &<\tau^{i+1}\cdot\Vert u_0-p\Vert+\tilde\varepsilon\cdot\sum_{k=0}^{i-1}\tau^k &<\tau^{i+1}d+\frac{\tilde\varepsilon}{1-\tau}
X_{j i}= \frac{1}{q - q^{- 1}}( X_{j - 1 \, i + 1}\Phi ( F_{i}) X_{j - 1 \, i + 1}^{- 1}- X_{j \, i + 1}\Phi ( F_{i}) X_{j \, i + 1}^{- 1}) , \; 2 \leq j \leq n - i .
H_{x,x'}^{(n)}=\frac{1}{(n+1)^2}\sum_{j,j'=0}^n\Phi\left(\frac{\theta j}{n+1}, \frac{\theta' j'}{n+1}\right) \exp\left\{\frac{2i\pi j x}{n+1}\right\}\exp\left\{\frac{2i\pi j'x'}{n+1}\right\}
\sum_{\sigma \in \mathfrak{S}_{k}}\prod_{1\le i<j \le k}\frac{qz_{\sigma(i)}-z_{\sigma(j)}}{z_{\sigma(i)}-z_{\sigma(j)}}\prod_{i=1}^{k}\left(\frac{1-\nu z_{\sigma(i)}}{1-z_{\sigma(j)}}\right)^{\!m_{i}}((m_{1}, \ldots , m_{k}) \in \mathbb{W}^{k}),
f(X)\mapsto[\begin{matrix}f(1)\\f(\gamma)\\ \vdots\\f(\gamma^{m-1})\end{matrix}],[\begin{matrix}f(\gamma^{m})\\f(\gamma^{m+1})\\ \vdots\\f(\gamma^{2m-1})\end{matrix}],...,[\begin{matrix}f(\gamma^{n-m})\\f(\gamma^{n-m+1})\\ \vdots\\f(\gamma^{n-1})\end{matrix}]
\mathbb{E}_0\left[\left(\bar{U}^{(pl)}_h\right)^2 \middle| \mathcal{F}_{n, l-1}\right] = \mathbb{E}_0\left[\left(\bar{U}^{(pl)}_h\right)^2 \middle| \mathbf{X}^{(p)}\right] = \mathbb{E}_0 \left[ \left(\bar{U}^{(pl)}_h\right)^2\right].
\frac{\ell(\ell+1)}{\sin^4 \phi}[ \cos^2 \phi \, p_{1,\ell}(\phi) - 2 \cos \phi \, p_{1,\ell+1}(\phi)+ p_{1,\ell+2}(\phi)] &=\frac{\ell^2}{\sin^4 \phi}[- \sin^2 \phi \cos \psi_{1,\ell}h_1(0) s_{1,0}(\ell,\phi)]+ \phi^{-4-1/2}O(\ell^{-1/2})+O(\phi^{-2}).
w^{(p)}_{q+1}:=\sum_{k\in \Lambda_v}a_{(v,k)}\phi_{{(\gamma,\tfrac{1}{2},k)}}g_{{(2,\sigma)}}\psi_k\bar{k}+\sum_{k\in \Lambda_b}a_{(b,k)}\phi_{{(\gamma,\tfrac{1}{2},k)}}g_{{(2, \sigma)}}\psi_k\bar{k}, &d^{(p)}_{q+1}:=\sum_{k\in \Lambda_b}a_{(b,k)}\phi_{{(\gamma,\tfrac{1}{2},k)}}g_{{(2, \sigma)}}\psi_k\bar{\bar{k}},
S_{0}= \left\{x_{s}, x_{s}^{-1}\right\}_{s\in S}\cup \left\{w_{r}, w_{r}^{-1}\right\}_{r\in R}\cup\left\{e\right\}, R_{0}= \left\{x_{s}x_{s}^{-1}e=e\right\}_{s\in S}\cup\left\{w_{s}w_{s}^{-1}e=e\right\}_{r\in R},
\frac{1}{\Gamma(k+j+2)}y^{j+1}{}_2F_1\left(j+\frac{3}{2}, j+1, j+k+2; y\right) = \frac{1}{\Gamma(k+1/2)\Gamma(j+3/2)}\int_0^1\frac{(1-w)^{k-1/2}(wy)^{j+1}}{w^{1/2}(1-wy)^{j+1}}dw.
\overline{J}_1^N(m_0^{\otimes N}, \gamma^N,\widetilde{\gamma}^N):= \mathbb{E}_N\left[ \sum_{t=0}^T f(t,\overline X_t^{1,N},\mu_t, \widetilde u^{1,N}_t) + F(\overline X_T^{1,N},\mu_T)\right],
(i) \ (-\Delta_p)^s U = U^{p_s^* -1}\ \mbox{in}\ \mathbb{R}^{N}&(ii) \ \vert \vert U \vert \vert_{s,p}^p = \left\Vert U \right\Vert_{p_s^*}^{p_s^*}= S^{\frac{N}{sp}}.
0= -4\langle \tau_{H}, \alpha_{1}\rangle\{m_{1}(\cot\langle 2\alpha_{1}, H \rangle )^{2}+2m_{1}(\cot\langle 4\alpha_{1}, H \rangle )^{2}&+9m_{2}(\cot\langle 6\alpha_{1}, H \rangle )^{2}\}(2\alpha_{1}+\alpha_{2}).
- \epsilon_{b k}^{q}\hat{R}_{i b , a m}\epsilon_{c m}^{q}K_{a c}= ( \epsilon^{q})_{k b}^{t}\hat{R}_{i b , a m}( \epsilon^{q})_{m c}^{t - 1}K_{a c}= \hat{R}^{\epsilon}_{i k , a c}K_{a c}= K_{i k}^{\epsilon}
f_1 = 0; \sum_{i=1}^{n+m}f_i = d-g+m; a_i + f_i = d_i, 2 \leq i \leq n; a_{n+j}+ f_{n+j}= e_j +1, 1 \leq j \leq m; a_1 = 2g-1 - \sum_{j=2}^m a_j.
L_{1}=\frac{\Gamma(t-1)\left[\xi_{p}\left(1+\xi_{p}^{2}\right)^{-(t-1)}-\xi_{q}\left(1+\xi_{q}^{2}\right)^{-(t-1)}\right]}{2\Gamma(t-\frac{1}{2})\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},~ L_{2}=\frac{F_{Y_{(1)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},
( - 1 )^{l_{0}+ . . . + l_{N}+ . . .}( - 1 )^{n_{1}+ . . . + n_{L}+ . . .}( - 1 )^{k_{0}+ . . . + k_{M}+ . . .}A_{{\bf l},{\bf n},{\bf k}}( q ) > 0 \, ~ ~ \mathrm{f o r}~ ~ 0 < q < 1
\begin{array}{c c c}{{{\cal E}= \int_{M_{3}}< ( H_{[ \mu \nu}\otimes I_{E}- I_{E}\otimes K_{[ \mu \nu})}}&{{\wedge}}&{{( D_{\rho ]}^{A}\Phi_{A}\otimes I_{E}- I_{E}\otimes D_{\rho ]}^{B}\Phi_{B}) >}}\\{{- \int_{M_{3}}< ( D_{[ \mu}^{A}\Phi_{A}\otimes I_{E}) \wedge ( H_{\nu \rho ]}\otimes I_{E}) >}}&{{+}}&{{( A \leftrightarrow B , \Phi_{A}\leftrightarrow \Phi_{B}) .}}\\ \end{array}
\frac{(7+2+1)^{2}}{3}+\frac{(7+6)^{2}}{2}+\frac{(11+6)^{2}}{2}+\frac{(10+7+3)^{2}}{3}+\frac{(5+3+4)^{2}}{3}+\frac{(11+4)^{2}}{2}
N^{A B}= \bar{\epsilon}_{1}\Gamma^{A B C}D_{C}\epsilon_{2}+ \textstyle{\frac{1}{8}}\bar{\epsilon_{1}}\Gamma^{C_{1}C_{2}}\epsilon_{2}F_{C_{1}C_{2}}^{A B}+ \textstyle{\frac{1}{9 6}}\bar{\epsilon}_{1}\Gamma^{A B C_{1}\cdots C_{4}}\epsilon_{2}F_{C_{1}\cdots C_{4}}.
f(\gamma):=a_1-\frac{\gamma(a_2)_-}{\gamma+1}-\frac{(b_1)_-}{\gamma+1}-\frac{\chi_1k(\gamma-1)}{d_3\gamma}-\frac{\chi_1l(\gamma-1)}{d_3(\gamma+1)}-\frac{\chi_2k(\gamma-1)}{d_3\gamma(\gamma+1)},
\left( \int_{B(0,1)}\rho^{q}\right)^{1/q}= \left( \frac{p-1}{p-k}\right)^{1/q}\left( \frac{\Gamma \left( \frac{k+1}{2}\right)}{\pi^{k-1}}\right)^{1/p}\leq \left( \frac{1+\alpha}{\alpha}\right)^{1/q}\left( \frac{k^{k/2}}{\pi^{k-1}}\right)^{1/p}\leq \frac{1+\alpha}{\alpha}\cdot k^{\frac{1}{2(1 + \alpha)}}.
\left\|\frac{\partial}{\partial t}\psi_{t/2}\right\|_{L^p(\mathbb{R}^N)}\le C\,[\psi]_{\mathcal{B}^{\alpha,p}_\infty(\mathbb{R}^N)}\,\left(\int_{\mathbb{R}^N}\mathcal{K}_1(z)\, \Big||z|^{2}-N\Big|\,|z|^\alpha\,dz\right)\,t^{\frac{\alpha}{2}-1}.
S_\tau(x,y) = S(x,y) -S(x-\tau,y+\tau), &F_\tau(x) = f(x)\bar f(x-\tau), &G_\tau(y) =g(y)\bar g(y+\tau), &\phi_\tau(x,y) = \phi(x,y)\phi(x-\tau,y+\tau).
\langle \bar{\chi}_{\sigma_{1}u_{1}}( x_{1}) \chi_{\sigma_{2}u_{2}}( x_{2}) \rangle = \int \!{\cal D}z{\cal D}z^{*}\exp ( - S_{C P_{1}}) \bar{\chi}_{\sigma_{1}u_{1}}( x_{1}) \chi_{\sigma_{2}u_{2}}( x_{2}) ~ .
\frac{g^{1-m}(r)}{mr^{n-1-\delta_1}}\int_0^r\rho^{n-2+\frac{n-2-nm}{m}-\delta_1}\rho^{\delta_1}|h(\rho)|d\rho\leq \frac{(3\tilde\eta)^{1-m}(\tilde\eta+|\zeta|) r^{\frac{n-2-nm}{m}}}{m\left(n-1+\frac{n-2-nm}{m}-\delta_1\right)}= \frac{\left({3\tilde\eta}\right)^{1-m}\left(\tilde\eta+|\zeta|\right)}{m \left(n-2\delta_1\right)}{r^{\frac{n-2-nm}{m}}}
(eS)e =\underset{a\in S}{\cup}(ea)e=\underset{a\in S}{\cup}(ee\cdot a)e=\underset{a\in S}{\cup}(ea)(ee)=\underset{a\in S}{\cup}(ee)(ae) =\underset{a\in S}{\cup}e(ae)=\underset{a\in S}{\cup}e(ea)=\underset{a\in S}{\cup}(ee)(ea)=\underset{a\in S}{\cup}(ae)(ee) =\underset{a\in S}{\cup}(ae)e=\underset{a\in S}{\cup}(ee)a=\underset{a\in S}{\cup}ea=eS.
L ={1 \over 2}(\partial_{\mu}\Phi) (\partial^{\mu}\Phi) +{1 \over 2}(\partial_{\mu}\Psi) (\partial^{\mu}\Psi) +{1 \over 2}m^2 \Phi^2+{1 \over 2}\omega^2 \Psi^2+{\lambda_\Phi \over 24}\Phi^4+{\lambda_\Psi \over 24}\Psi^4+{\lambda_{\Phi \Psi}\over 4}\Phi^2 \Psi^2~,
2\int_{Z_\lambda}|\langle\nabla\triangle e_\lambda,N\rangle|ds\geq -2\int_{Z_\lambda}\langle\nabla\triangle e_\lambda, N\rangle ds=\int_{\partial M}\sigma(\partial_\nu e_\lambda)\partial_\nu\triangle e_\lambda ds \gtrsim \lambda^2 \|\partial_\nu e_\lambda\|_{L^1(\partial M)}.
\int_{\mathbb{R}}J(x+\Delta)dF(x) =-\int_{\mathbb{R}}\int_{\mathbb{R}}1_{\left\{x+\Delta\leq G(y)\right\}}H_m(y)\varphi(y)dy dF(x) &=-\int_{\mathbb{R}}\int_{\mathbb{R}}1_{\left\{x+\Delta\leq G(y)\right\}}dF(x)H_m(y)\varphi(y)dy &=-\int_{\mathbb{R}}F(G(y)-\Delta)H_m(y)\varphi(y)dy.
\left\|(\bar{u}_h)_+-(u_h)_+ \right\|_{L^1(K_T)}begin{align*}3mm] \leq \left\|(\bar{u}_h)_+-(\bar{u}_h)_+^{(\ell)}\right\|_{L^1(K_T)}+\left\|(\bar{u}_h)_+^{(\ell)}-(u_h)_+^{(\ell)}\right\|_{L^1(K_T)}+\left\|(u_h)_+^{(\ell)}-(u_h)_+ \right\|_{L^1(K_T)}begin{align*}3mm] =:\mathbf{I}_{h,\ell}+\mathbf{II}_{h,\ell}+\mathbf{III}_{h,\ell}.
Z_{\alpha_{L}}^{\alpha_{L - 1}}Z_{\alpha_{L + 1}}^{\alpha_{L}}= C_{\alpha_{L + 1}}^{\alpha_{L - 1}\beta_{0}}\frac{\delta S_{0}}{\delta \Phi^{\beta_{0}}}, \alpha_{L + 1}= 1 , \ldots , M_{L + 1},
(u_{\neq}\cdot \nabla u^1_{\neq})_0 = (\bar{u}_{\neq}^2 \cdot (\partial_y - t\partial_y u_0^1\partial_x)\bar{u}^1_{\neq})_0 + (\bar{u}_{\neq}^3 \cdot \partial_z\bar{u}^1_{\neq})_0,
(A+\frac{k^2}{2}I_n)^T P^{-1}+P^{-1}(A+\frac{k^2}{2}I_n)+\sum_{i=1}^m N_i^T P^{-1}N_i \leq -P^{-1}BB^T P^{-1}, (A+\frac{k^2}{2}I_n)^T Q+Q (A+\frac{k^2}{2}I_n)+\sum_{i=1}^m N_i^T Q N_i = -C^T C.
|\nu(S_1 \cap S_3) - \nu(S_2 \cap S_3)| = |\nu(S_1 \cap S_2 \cap S_3) + \nu((S_1 \setminus S_2) \cap S_3) - \nu(S_1 \cap S_2 \cap S_3) - \nu((S_2 \setminus S_1) \cap S_3)| & \leq \nu(S_1 \triangle S_2),
\limsup_{|z| \to 1}(1-|z|)v(z)|(g \circ \varphi)^{\prime}(z)| = \lim_{k \to \infty}(1-|z_{n_k}|)v(z_{n_k})|(g \circ \varphi)^{\prime}(z_{n_k})| &= \lim_{k \to \infty}(1-|z_{n_k}|)v(z_{n_k})|\varphi^{\prime}(z_{n_k})||g^{\prime}(\varphi(z_{n_k}))| &= 0 \cdot |g^{\prime}(y)| = 0,
\overline{L}v = w^{ij}F_{p_lp_m}D_{li}uD_{mj}u+F_{p_l}w^{ij}(D_{lij}u-D_{p_k}A_{ij}D_{lk}u) + w^{ij}(F_{up_l}u_jD_{li}u+F_{up_l}u_iD_{lj}u+F_zu_{ij}+F_{jp_l}D_{li}u+F_{ip_l}D_{lj}u) + w^{ij}\big(F_{uu}u_{i}u_j+F_{iu}u_j+F_{ju}u_i+F_{ij}-D_{p_k}A_{ij}(F_k+F_zu_k)\big).
\frac12\int_{\Sigma}\int_{\Sigma}\frac{(\phi(x) - \phi(y))^2}{|x-y|^{N+s}}d x d y =\int_{\Sigma}\frac{H}{|x|^{1+s}}\phi(x)^2 dx & +\frac12\int_{\Sigma}\int_{\Sigma}\frac{(\psi(x) - \psi(y) )^2 w(x) w(y)}{|x-y|^{N+s}}d x d y .
M^j_{(0,1)\times (0,1)}\left [\frac{D_m(\lambda)}{W_m(\lambda)}\u_m(\lambda, \cdot) \v_m(\lambda, \cdot) - \frac{D_m(0)}{W_m(0)}\, u_m(0, \cdot) \v_m(0, \cdot) \right] \lesssim \, \frac{\lambda}{1+|m|}, j=1,2
S=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int\cdot\cdot\cdot\int d^{4}x_{1}d^{4}x_{2}...d^{4}x_{n}T[H_{int}(x_{1})H_{int}(x_{2})\cdot\cdot\cdot H_{int}(x_{n})]
\left(-q^{\ell+1};q\right)_{\infty}=(-1)^{\ell}q^{-\binom{\ell}{2}}\sum_{n=0}^{\infty}\left[\frac{a_{2n+\ell}(q)q^{2n}}{(q,q^{4};q^{5})_{\infty}(q^{2};q^{2})_{n}}-\frac{b_{2n+\ell}(q)q^{2n}}{(q^{2},q^{3};q^{5})_{\infty}(q^{2};q^{2})_{n}}\right].
f^{(1)}= \epsilon p_\Gamma -\sum_{k=m+1}^t\sum_{i=1}^n M_{2k}x^{2ke_i}, f^{(2)}= \epsilon p_\Gamma + \sum_{k=m+1}^t\sum_{i=1}^n M_{2k}x^{2ke_i}+ \sum_{|\alpha|\geq 2m+1}f_\alpha x^\alpha.
\int_{|y| \le t^{\beta/\alpha}}|I_{B}|dy \le C\int_{|r|\le t^{\beta/\alpha}}t^{-\beta - 1}|r|^{-d+2\alpha + d - 1}dr \le C t^{-\beta - 1}(t^{\beta/\alpha})^{2\alpha}= C t^{\beta - 1}.
\sum_{(k_1,\ldots,k_s)}(-1)^i \frac{1}{2^i k_1! \cdots k_s!}\prod_{r=1}^s \alpha^{m_r}(n)^{k_r}= (-1)^i \frac{1}{2^i i!}\sum_{(k_1,\ldots,k_s)}\frac{i!}{k_1! \cdots k_s!}\prod_{r=1}^s \alpha^{m_r}(n)^{k_r}&= (-1)^i \frac{1}{2^i i!}A_1(n)^i,
G = \prod^{2d-1}_{j=0}(\xi^{j}x+\xi^{-j}y+z) = \prod^{d-1}_{j=0}(\xi^{j}x+\xi^{-j}y+z)(\xi^{j+d}x+\xi^{-j+d}y+z) = \prod^{d-1}_{j=0}(\xi^{j}x+\xi^{-j}y+z)(-\xi^{j}x-\xi^{-j}y+z) = (-1)^d\prod^{d-1}_{j=0}(\xi^{j}x+\xi^{-j}y+z)(\xi^{j}x+\xi^{-j}y-z)=F
2R \leq C_1+C_2+I(X_1X_2;Y|Q)-I(X_1X_2;U|Q)+I(X_1;U|X_2Q)+I(X_2;U|X_1Q) &= C_1+C_2+I(X_1X_2;Y|UQ)+I(X_1;U|X_2Q)+I(X_2;U|X_1Q) &\leq C_1+C_2+I(X_1X_2;Y|U)+I(X_1;U|X_2)+I(X_2;U|X_1).
\frac{2n(2n-1)(2n-2)(2n-3)}{16n^4}- 4 \frac{2n(2n-1)(2n-2)}{8n^3}+ 6 \Big(1 - \frac{1}{2n}\Big) - 3 = \frac{3}{4n^2}- \frac{3}{4n^3}\in \mathcal{O}(n^{-2}).
D_{m,\lambda}^{(k)}= m! \sum_{n=0}^m \sum_{i_1+i_2+ \cdots+i_n=m}\frac{D_n^{(k)}}{n!}\left(^{\lambda}_{i_1}\right)\left(^{\lambda}_{i_2}\right) \cdots \left(^{\lambda}_{i_n}\right).
\begin{cases}u_{t}(t,x) - u_{xx}(t,x) + b(u(t,x)) = f(t,x), (t,x) \in (0,1)^2, u(t,0) = u(t,1) = 0, t \in (0,1), u(0,x) = u_0(x), x\in (0,1), \end{cases}
: \!{\cal F}[ X ] \! : \, \, : \!{\cal G}[ X ] \! : \ = \exp \biggl \{- \int d^{2}z^{\prime}\, d^{2}z^{\prime \prime}\, \ln | z^{\prime}- z^{\prime \prime}|^{2}\frac{\delta_{F}}{\delta X ( z^{\prime}, \bar{z}^{\prime})}\frac{\delta_{G}}{\delta X ( z^{\prime \prime}, \bar{z}^{\prime \prime})}\biggr \}: \!{\cal F}[ X ] \,{\cal G}[ X ] \! : ,
-\frac{1}{4}\det \left( e_{M}^{A}\right) e^{-\phi}Tr\left(G_{MN}G_{PQ}g^{MN}g^{PQ}\right) =-\frac{e}{4}e_{\stackrel{.}{0}}^{0}e^{-\phi}Tr\left( G_{ab}G_{ab}-2G_{a0}G_{a0}\right)
\frac{1}{|{\rm Aut}(\Gamma)|}\sum_{j=0}^{n}{\xi_{\Gamma}}_*\left(\prod_{i=1}^{n}\left(1+3\omega_i\right) \prod_{(h,h')\in E_2(\Gamma)}\frac{1}{\psi_h-(1+3\omega_{h'})}\right)\cdot (-\lambda - \delta_1)^{j}.
m_{A \widetilde{\otimes}B}= (m_A \otimes m_B) \circ (\mathrm{id}_A \otimes c_{B,A}\otimes \mathrm{id}_B) \: : \: A \otimes B \to A \otimes B, 1_{A \widetilde{\otimes}B}= 1_A \otimes 1_B \: : \: \mathbb{C}\to A \otimes B.
\|g_n\|_{L^{q_0}(Q_{R,\tau})}= \sigma_n^{\gamma'\frac{N+1}{N+2}}\|f_n\|_{L^{q_0}( \widetilde{Q}_{n,R,\tau})}, \widetilde{Q}_{n,R,\tau}= B_{Rr_n}(\bar x_n) \times (\bar t_n, \bar t_n + r_n^\gamma M_n^{1-\gamma}\tau)
\begin{minipage}[c]{35pc}For{\em disjoint}subsets $S$ and $T$ of $\omega,$we have $a_S + a_T=a_{S\cup T},$ $b_S + b_T=b_{S\cup T},$and $0=a_S\,a_T=a_S\,b_T=b_S\,a_T=b_S\,b_T.$\end{minipage}
\mathfrak{L}[\log F(z)]-2|z|^{2}(\log F(z) )_{z\overline{z}}= 2\sum_{k=3}^{p}(3k-k^{2}-2)|z|^{2(k-1)}\log G_{k}(z)+\sum_{k=1}^{p}(3-2k)|z|^{2(k-1)}\mathfrak{L}[\log G_{k}(z)].
-\nabla \circ \left( \left\vert \nabla u\right\vert^{p-2}\nabla u\right)-\lambda \left\vert u\right\vert^{p_{0}-2}u\left\vert \nabla u\right\vert^{p_{1}}=0,u\left\vert_{\ \partial \Omega}\right. =0,\ \lambda \in \mathbb{C},
I^{\pm}_{H}=\sum^{N^{\pm}_{H}}_{j=1}I_{h_{j}}^{\pm}\, ,\, I^{\pm}_{C}=\sum^{M^{\pm}_{C}}_{j=1}I_{c_{j}}^{\pm}\, ,\, I^{\pm}_{W}=\sum^{M^{\pm}_{W}}_{j=1}I_{w_{j}}^{\pm}\, \mathrm{and}\, I^{\pm}_{S}=\sum^{N^{\pm}_{S}}_{j=1}I_{y_{j}}^{\pm}
\partial_{\Sigma^mG}(\Sigma^m1)=0, \quad\partial_{\Sigma^mG}(\Sigma^{m+1}\lambda_{i_1})=(-1)^m\Sigma^m1(1\otimes y_{i_1}-y_{i_1}\otimes 1), \forall \lambda_{i_1}\in \Lambda_1 &\partial_{\Sigma^m G}(\Sigma^{m+k}\lambda_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{m+k-j}\Sigma^{m+k-1}\lambda_{i_1\cdots \hat{i_j}\cdots i_k}(1\otimes y_{i_j}-y_{i_j}\otimes 1),
{\cal A}_{fi}=\int d{\bf x\,}\left[ -\frac e{w_{{\bf p}}}{\bf A}({\bf x})\,{\bf \cdot \,p}+V({\bf x})\right] \,{\rm e}^{-\,i\,({\bf p}^{\prime}-{\bf p})\,{\bf \cdot \,x}}=-\frac e{w_{{\bf p}}}\,{\bf \tilde A(q})\,{\bf \cdot \,p}+\tilde V({\bf q})\;\;\;
\begin{array}{rcl}\displaystyle{\frac{dw}{d\tau}}= \displaystyle{w''' + \tilde\beta z'' + \frac32{w'}^2 + 2\tilde\beta zw' -\frac12\tilde\alpha z^2 ,}\displaystyle{\frac{dz}{d\tau}}= \displaystyle{w''w+K\tilde\beta w''+w'z'+2\tilde\beta zz'-K \tilde\alpha z'\,,}\end{array}
\frac{p_i}{p_{N+1}}= \frac{1}{\Lambda_0 \cdots \Lambda_N}\sum_{0 \leq j_1 < \cdots < j_i \leq N}\Lambda_{j_1}\cdots \Lambda_{j_i}\prod_{k \in \{0,\dots,N\}-\{j_1, \dots, j_i\}}\frac{1-\Lambda_k}{1-q}
p\nmid N(\delta), \p\equiv 1\hskip-.07in\pmod 4 \Longrightarrow \left| I_p\right|=2; p| N(\delta), \p\equiv 1\hskip-.07in\pmod 4 \Longrightarrow\left| I_p\right|=1; p\nmid \Delta, \p\equiv 3\hskip-.07in\pmod 4 \Longrightarrow \left| I_p\right|=0; p| \Delta, \p\equiv 3\hskip-.07in\pmod 4 \Longrightarrow\left| I_p\right|=1.
\binom{r}{i/2}\binom{n-r}{i/2}\leq \binom{r}{r(n-r)/n}\binom{n-r}{r(n-r)/n}\cdot \left( \frac{r(n-r)}{n}\right)^4 &\leq \binom{r}{r(n-r)/n}\binom{n-r}{r(n-r)/n}\cdot n^4.
\begin{array}{l}(x-1)^{\epsilon_1}h_1(x) =\begin{cases}(x-1)^{n-r+k_2}p_2(x)-(x-1)^{n-2r+2k_1}p_1^2(x) \mbox{if~}n-r+k_1>r, (x-1)^{r-k_1+k_2}p_2(x) -(x-1)^{k_1}p_1^2(x) \mbox{if~}n-r+k_1\le r,\end{cases}\end{array}
e^{i\tilde Hs}e^{-i H_\Lambda s}\psi_0 - \psi_0 = -i\int_0^s e^{i\tilde H\tau}\delta H_\Lambda e^{-i H_\Lambda\tau}\psi_0\, d\tau e^{i(\tilde H-E_p)s}Q_0 = -i\frac{d}{ds}e^{i(\tilde H-E_p)s}\tilde{R}Q_0,
\phi_{n}(1) = \chi \Big(\int_{b \in supp(f_{n})}f_{n}(b)E_{b}db \Big) = \int_{b \in supp(f_{n})}f_{n}(b)1_{A}(b)db = \int_{b \in supp(f_{n})}f_{n}(b)db ~ (\textrm{Since $supp(f_{n}) \subset Int(A)$}) =1.
\begin{array}{lcll}\mbox{floor}\mbox{SU(2/1)}\mbox{field}\mbox{helicity}\mid \mbox{gnd}> (0~+\frac{1}{2}) \phi ~0 \mid \mbox{1st}> (0~+\frac{1}{2}) 2\lambda -\frac{1}{2}\mid \mbox{2nd}> (0~+\frac{3}{2}) A_\mu -1. \end{array}
\frac{1}{2}\frac{d}{dt}\|\Lambda^{\frac{\alpha}2}B\|^2_{L^2}+\mu \|\Lambda^{\alpha}B\|^2_{L^2}=-\int_{\mathbb S^1}BJ_x\Lambda^{\alpha}B\, dx+\int_{\mathbb S^1}B_xJ\Lambda^{\alpha}B\, dx.
I(n_b,n_b;a^{(b)}) = \int d\mu_{\tilde{Q}}[\varphi]d\mu_{\tilde{C}}[\theta] \;\exp \left( - 2 \sum_{b=1}^N \sum_{j=1}^{n_b}\Big( \chi^{(b)}, \delta_n(x_j^{(b)})\!-\!\delta_n(y_j^{(b)}) \Big) \right)\times
M(q^2x,y,z;q) =\frac{xq}{yz}M(x,y,z;q) +\frac{J_4^3j(y^2z^2;q^4)}{j(y^2;q^4)j(z^2;q^4)}&\ \ \ \ \ - \frac{qx}{yz}\frac{J_4^3j(x^2y^2;q^4)}{j(x^2;q^4)j(y^2;q^4)}-\frac{qx}{yz}\frac{J_4^3j(x^2z^2;q^4)}{j(x^2;q^4)j(z^2;q^4)}.
\prod_{u = 1}^{N}d v_{u}^{2}d v_{u}^{2 *}= d \xi_{1}^{2}d \xi_{1}^{2 *}\cdot 2^{N - 1}( \xi_{2}^{2})^{2 N - 3}\, d \xi_{2}^{2}\, d^{2 N - 3}\hat{\Omega}_{2}\ .
T(B_1^+\setminus B'_1) \subset \{y_n>0, y_{n+1}<0\}, T(\Lambda_w) \subset\{y_n=0, y_{n+1}\leq 0\}, \T(B'_1\setminus \Lambda_w)\subset \{y_n>0, y_{n+1}=0\}, T(\Gamma_w) \subset\{y_n=y_{n+1}=0\}.
=(-1)^{k'+1+\alpha}\left[2(m_{k'+1}-f(b))\mathbf{1}_{|m_{k'+1}-f(b)|\ge\lambda/2}+f(b)-(f(\tau_{k'})+(-1)^{k'+\alpha}\lambda)\right]=\lambda+(-1)^{k'+1+\alpha}\left[2(m_{k'+1}-f(b))\mathbf{1}_{|m_{k'+1}-f(b)|\ge\lambda/2}+f(b)-f(\tau_{k'})\right]
{\rm I}_{2}\le c \| \sum_{j=0}^{n-1}\chi_{[t_{j}, t_{j+1})}(s) B_{n-j}P_{h}\|_{L^2(0, T;\mathcal{L}_2^0)}= c\big(\tau \sum_{j=0}^{n-1}\| B_{n-j}P_{h}\|_{\mathcal{L}_2^0}^2 \big )^{\frac{1}{2}}< \infty.
0 ={\cal M}^4 - ( \tilde{\gamma}\tilde g^2 k_q + \frac1{k_q k_T}) ( q_0^2 + \bar{q}_0^2 ){\cal M}^2 + \tilde{\gamma}\frac1{k_T}\tilde g^2 (q_0^2 + \bar{q}_0^2)^2.
a\times b=|\begin{matrix}e_{1}&e_{2}&e_{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{matrix}|,a\wedge b=|\begin{matrix}e_{23}&e_{31}&e_{12}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{matrix}|,
\begin{aligned}P^U = -D_r(r^{1-n}E_\rho(r^{n-1}\Phi)) +r^{1-n}(S_r/\rho) E_S(r^{n-1}\Phi), P^\rho = -r^{1-n}D_r E_U(r^{n-1}\Phi), P^S = -r^{1-n}(S_r/\rho) E_U(r^{n-1}\Phi). \end{aligned}
n \ge |k *_C A| + |\ell *_C A| = (k + \ell)|A| + (k + \ell - 2)|C| - (2k + 2\ell - 4)|K| \Longleftrightarrow \frac{|A|}{|K|}\le \frac{n - (k + \ell - 2)|C| + (2k + 2\ell - 4)|K|}{(k + \ell)|K|}.
\lambda(K_{2}^{\{1,2\}}) = \max_{\vec{x}\in S_{2}}\lambda(K_{2}^{\{1,2\}}, \vec{x}) = \max_{\vec{x}\in S_{2}}x_1+x_2+2x_1 x_2 = 1 + 2\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) = \frac{3}{2}.
(\nabla_v \log f)\cdot \Delta_v(\nabla_v f) = |\nabla_v \log f|^2 \Delta_v f + 2(\nabla_v \log f) \cdot (\nabla_v f \cdot \nabla_v ) (\nabla_v \log f) & + f \nabla_v \log f \cdot \Delta_v (\nabla_v \log f).
V = \left( \begin{matrix}{{\frac{1}{\sqrt{1 - Y \bar{Y}}}}}&{{\frac{Y}{\sqrt{1 - Y \bar{Y}}}}}\\{{\frac{\bar{Y}}{\sqrt{1 - Y \bar{Y}}}}}&{{\frac{1}{\sqrt{1 - Y \bar{Y}}}}}\\ \end{matrix}\right) \ .
\langle \widehat{p}_1 f ,f \rangle_{L^2 (\mathbb{R}^2)}= \int_{\mathbb{R}^2}\left[\left(-i \mu \frac{\partial}{\partial x_1}+ \frac{\eta}{2 \mu}x_2 \right)f (x_1, x_2) \right] \overline{f (x_1, x_2)}dx_1 dx_2 = \frac{\eta}{2 \mu}x_2^{(0)}
(e_{j,k+n}- \delta_{j,k}) \cdot f = [e_{j,k+n},f] = \sum_{i}P_{i}[e_{j,k+n},e_{ij}^{l_{ij}}] Q_{i}= P_{j}((e_{jj}-1)^{l_{jj}}-e_{jj}^{l_{jj}}) e_{j,k+n}\cdot Q_{j}+ \sum_{i \neq j}-l_{ij}\: P_{i}e_{ij}^{l_{ij}-1}e_{i,k+n}\cdot Q_{i}
R_{r \theta}^{r \theta}= R_{r \phi}^{r \phi}= \frac{B^{\prime}}{2 A B r}= - M / r^{3}, \hspace{0 . 3 c m}R_{\theta \phi}^{\theta \phi}= \frac{1 - 1 / B}{r^{2}}= 2 M / r^{3}
\frac{(MQ^2T)^{\frac{3\delta}{2-3\delta}}}{QT}\ll\frac{(MQ^2)^\frac{3\delta}{2-3\delta}}{Q}&\ll M^{\frac{12}{5}\delta}\left(\frac{X^{1/2}}{M^\frac{6}{5}(\log X)^B}\right)^{\frac{4}{5}\delta}&\ll M^{\frac{36}{25}\delta}X^{\frac{2}{5}\delta}(\log X)^{-\frac{4}{5}\delta B},
\begin{pmatrix}\partial_r+\dfrac{ak_{1/2}}{m r}-(m+a) -(m-a) \partial_r+\dfrac{ak_{1/2}}{m r}\end{pmatrix}\cdot\begin{pmatrix}f^+_{m_{1/2},k_{1/2}}f^-_{m_{1/2},k_{1/2}}\end{pmatrix}=0.
D_{\alpha}=\frac{\partial}{\partial \theta^{\alpha}}+i\sigma^{\mu}_{\alpha{\dot \alpha}}{\overline \theta}^{\dot \alpha}\partial_{\mu}\, ,\hspace{1cm}{\overline D}_{\dot \alpha}=-\frac{\partial}{\partial{\overline \theta}^{\dot \alpha}}-i \theta^{\alpha}\sigma^{\mu}_{\alpha{\dot \alpha}}\partial_{\mu}
e^{t_\nu}=(1+o_\nu(1))\Bigl(\frac{\nu(\gamma_p+\gamma_q)\int_{\mathbb{R}^N}(I_\mu*|u_0|^{p})|v_0|^{q}}{|\nabla u_0|^2_2+|\nabla v_0|^2_2}\Bigr)^{\frac{1}{2-\gamma_p-\gamma_q}}=(1+o_\nu(1))\nu^{\frac{1}{2-\gamma_p-\gamma_q}}.
B_2 - A_2 = \frac 12 \int_0^x \int_0^{t_1}(1 - e^{-2k(x-t_1)}) V(t_1) (1 - e^{-2k(t_1 -t_2)}) V(t_2) e^{-2kt_2}\,dt_2 \,dt_1 & = \frac 12 \int_0^x \int_0^{t_1}(e^{-2kt_2}- e^{-2kt_1}- e^{-2k(x-t_1+t_2)}) V(t_1) V(t_2) \,dt_2 \,dt_1 + O(e^{-2\Re k x})
=\mathbb{E}\left[\sum_{k=1}^{K}z(a^k)- Z(a^*)\right] &=\mathbb{E}\left[\sum_{n\in \mathcal{A}(L^m)}\theta_{n,K}\frac{Z(n)}{K}- \theta_{a^*,K}\frac{Z(a^*)}{K}\right] &=\sum_{n\neq a^*}\beta\delta_n\mathbb{E}[\theta_{n,K}].
\frac{i}{\sqrt{2 p^+}}\left[ \gamma_0 (\tau) -{\gamma_{0}^\dagger}(\tau) \right] = \frac{i}{2 \sqrt{\mu p^+}}\left( e^{-2 i \mu \tau}a_0 + e^{2 i \mu \tau}\bar{a}_0 \right) - \frac{m R_1}{2 \mu l_s^2 p^+}.
K(q) = \frac 12 \Im \sum_j w_j\bar z_j = \frac i4 \sum_j (z_j\bar w_j- \bar z_j w_j) &= \frac i4 \sum_j \int_{\alpha_j}\tau_q \int_{\beta_j}\bar \tau_q - \int_{\alpha_j}\bar\tau_q \int_{\beta_j}\tau_q &= \frac i4 \int_{S_q}\tau_q \wedge \bar \tau_q = \frac 14 \int_{S_q}|\tau_q|^2\, dA = \frac 12 \int_X |q|\, dA.
\sum_{k=0}^n \binom{a}{k}\binom{b}{n-k}\binom{k}{c}\binom{n-k}{d}k^3 = \binom{a+b-c-d}{n-c-d}\binom{a}{c}\binom{b}{d}\cdot \frac{f_1(n,a,b,c,d)}{(a+b-c-d)(a+b-c-d-1)(a+b-c-d-2)}
\lim_{n\uparrow\infty}\sup_{0<s<\varepsilon_{n},\x\in B_{n},\r\geq h_{\beta}(s)}\mathbf{E}_{r\delta_{x}}\exp\left\{-\int_{B}\mathrm{d}y\X_{s}(y)p_{s}(y-x)\right\}=0.
x_{1}+ x_{2}+ \dots + x_{k}= n x_{i}\leq a_{i}x_{1}+ x_{2}+ \dots + x_{k}= n a_{i}- x_{i}\geq 0 a_{1}- x_{1}+ a_{2}-x_{2}+ \dots + a_{k}-x_{k}= \sum_{j=1}^{k}a_{j}- n a_{i}- x_{i}\geq 0
\sin(\theta)+\sin(\theta+\alpha) =2\sin\biggl(\theta+\frac{\alpha}{2}\biggr)\cos \biggl(\frac{\alpha}{2}\biggr), \cos(\theta)+\cos(\theta+\alpha) =2\cos\biggl(\theta+\frac{\alpha}{2}\biggr)\cos \biggl(\frac{\alpha}{2}\biggr)
\begin{aligned}\log (f^n)^\#(z) \leq \log\!\left( C^n \prod_{j=0}^{n-1}|z_{j}|^{\rho}\right) = n\log C + \rho \sum_{j=0}^{n-1}t_{j}\leq n\log C + \rho c_0(1+\rho)^{n-1}.\end{aligned}
E_{C}^{( 1 ) r e n}( R ) = \frac{1}{2 \pi}\int_{0}^{\infty}\ln \left( 1 - e^{- 2 R \omega}\right) d \omega = - \frac{R}{\pi}\int_{0}^{\infty}\frac{\omega d \omega}{e^{2 R \omega}- 1}.
\tilde{Y}_{w_{j}}\stackrel{\sigma_{j}}{\rightarrow}\tilde{Y}_{w_{j-1}}\stackrel{\sigma_{j-1}}{\rightarrow}\cdots\stackrel{\sigma_{2}}{\rightarrow}\tilde{Y}_{w_{1}}\stackrel{\sigma_{1}^{-1}}{\rightarrow}Y_{i_{1}}\rightarrow F .