harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{\alpha}( z ) \, S_{A}( z ) = \left. \widetilde S_{\alpha}( \bar{z}) \, \widetilde S_{A}( \bar{z}) \right|_{z = \bar{z}}, ~ S^{\dot{\alpha}}( z ) \, S^{A}( z ) = \left. \widetilde S^{\dot{\alpha}}( \bar{z}) \, \widetilde S^{A}( \bar{z}) \right|_{z = \bar{z}}~ ~ ,
\Phi_{\alpha_{0}}^{* ( 1 )}= N_{\alpha_{1}\alpha_{0}}n_{1}^{\alpha_{1}}+ \cdots , \varphi_{\alpha_{1}}^{* ( 1 )}= \left( - \right)^{\epsilon_{\alpha_{0}}\left( \epsilon \left( f_{\alpha_{1}}\right) + 1 \right)}N_{\alpha_{1}\alpha_{0}}\nu_{1}^{\alpha_{0}}+ \cdots ,
D\log(Q) [(\xi)_{\times}Q ] = \left[ I - \frac{1}{2}(\log(Q))_\times + \frac{1}{12}(\log(Q))_\times^2 \right] \xi = & \left[ I - \frac{1}{2}(\log(Q))_\times + \frac{1}{12}\left( \log(Q) \log(Q)^T-||\log(Q)||^2 I_3 \right) \right] \xi
\frac{1}{k!}\sum_{n \leq X}a(n) \Big(1 - \frac{n}{X}\Big)^k = \frac{1}{2\pi i}\int_{(\sigma)}D(s) \frac{X^s}{s(s+1)\cdots(s+k)}ds = \frac{1}{2\pi i}\int_{(\sigma)}D(s) \frac{X^s\Gamma(s)}{\Gamma(s+k+1)}ds,
b(x',y,t)=\sum_{x\in T^{-1}(x')}\frac{1}{\vert T'(x)\vert}\left[b(x,y,t)+h(x)\frac{1}{2}(x^{2}-\langle x^{2}\rangle)\frac{\partial^{2}}{\partial y^{2}}p_{0}(y,t)\right]
\frac{1}{U}=\frac{1}{h_{o}}+\frac{1}{h_{o}d}+\frac{d_{o}ln\frac{d_{o}}{d_{i}}}{2k_{w}}+\frac{d_{o}}{d_{i}}\frac{1}{h_{i}d}+\frac{d_{o}}{d_{i}}\frac{1}{h_{i}}
d_{N}\left(\int_{c}^{t}\omega\right)+\int_{c}^{t}d\omega=\int_{c}^{t}\left(\frac{\partial\omega_{A}}{\partial t}(\tau)\right)d\tau=\omega_{A}(t)-\omega_{A}(c)=i{}_{t}^{*}\omega-i_{c}^{*}\omega.
\Sigma_{a}= \left\{\left( \begin{matrix}{{0}}&{{I}}\\{{- I}}&{{0}}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{{0}}&{{- \sigma_{1}}}\\{{- \sigma_{1}}}&{{0}}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{{\sigma_{1}}}&{{0}}\\{{0}}&{{- \sigma_{1}}}\\ \end{matrix}\right) \right\}
\sum_{h=1}^{\infty}\omega^{m^{1/B(s)}a_j\,(h/2)^{b_j}}=\omega^{m^{1/B(s)}a_j\,2^{-b_j}}\sum_{h=1}^{\infty}\omega^{m^{1/B(s)}a_j\,2^{-b_j}\,(h^{b_j}-1)}\le \omega^{m^{1/B(s)}a_j 2^{-b_j}}R.
\eta_t =\min\left(\frac{\mathfrak{y}}{d},\, d^{-\frac{2(\beta-1)}{2\beta-1}}T^{-\frac{\beta}{2\beta-1}}\right) \qquad h_t = \mathfrak{h}\, T^{-\frac{1}{2(2\beta-1)}}\enspace,
\sum_{i}\tilde{\alpha}_{i}^{\prime 2}- \tilde{\alpha}_{i}^{\prime}\sum_{j}\tilde{\alpha}_{j}^{\prime}+{\frac{1}{D - 2}}\varphi^{\prime 2}+ \sum_{a}\alpha_{a}^{\prime 2}={\frac{Q^{2}}{2}}\exp \left( - \tilde{\Phi}+ 2 \sum_{k}\tilde{\alpha}_{k}\right) \ .
\dfrac{\partial f_n^e}{\partial t}+n^2\pi^2f_n^e+\dfrac{\partial}{\partial u}\cdot\big(\mathcal{G}f_n^e\big) - \epsilon \kappa u \otimes u f_n^e+\dfrac{\epsilon}{4\pi}\int_{S_2}\kappa:v \otimes vf_n^e(v) dv - \sqrt{2}\epsilon n\pi \int_0^1 \big(f^e\kappa:\lambda(f^e)\big)(s)cos(n\pi s)ds =\dfrac{3+\epsilon}{4\pi}\kappa:u\otimes u 1_n
\begin{aligned}\tilde{S}\int_{{B_{R}}}\frac{|x|^{\frac{n\theta}{p^{*}}}|u(x)|^{{\theta}}}{\left[(q-1)\log_{q}\frac{R}{|x|}\right]^{\theta}}\frac{dx}{|x|^{{n}}}\le \int_{{B_{R}}}|x|^{\frac{n\theta}{p}}|\nabla_r u|^{\theta}\frac{dx}{|x|^{{n}}}\end{aligned}
-\int\limits_{X_{3T_1}}\frac{\partial^2}{\partial\tau\partial s}u^f\overline{v^g}dsd\tau &=\int\limits_{X_{3T_1}}\frac{\partial u^f}{\partial s}\frac{\overline{\partial v^g}}{\partial \tau}dsd\tau-\int\limits_{y_n=0}\frac{\partial u^f}{\partial s}\overline{v^g}ds+\int\limits_{\tau=0}\frac{\partial u^f}{\partial s}\overline{v^g}ds.
\hat{\Psi}^{(0)}\thicksim M_1\left(\begin{array}{cc}v+\Theta ~v+\Theta v ~v+\Theta\end{array}\right)(I+o(1))\; \Phi\left (\frac{1}{2}x\sqrt{\lambda},\Theta\right )\; u^{-\frac{1}{2}\sigma_3},
48\mathcal{F}(u^4) = 48 \mathcal{F}((1+x^2)^{-4}) = \frac{96\sqrt{\pi}}{\Gamma(4)}\left( \frac{|\xi|}{2}\right)^{\frac72}K_{\frac72}(|\xi|) = 2\sqrt{\pi}(|\xi|^3+6|\xi|^2+15|\xi|+15) \left( \frac{|\xi|}{2}\right)^{\frac12}K_{\frac12}(|\xi|) &= (|\xi|^3+6|\xi|^2+15|\xi|+15) \mathcal{F}((1+x^2)^{-1}) &= \mathcal{F}(|D|^3 u +6D^2 u+15|D|u+15u).
A_{uv}(x,y) \;=\; \int \frac{d^4q}{(2 \pi)^4}\:\hat{V}(q) \int \frac{d^4p}{(2 \pi)^4}\:A_{uv}(p+\frac{q}{2},p-\frac{q}{2}) \;e^{-ip(x-y)}\:e^{-i \:\frac{q}{2}\:(x+y)}\;,
\exp \left[\frac{n'}{(e{\cal P}')}(eQ) \sum_{q>0}\frac{1}{q}\mbox{e}^{-2iq\sigma'_{+}}\right]= \left(1-\mbox{e}^{-2i\sigma'_{+}}\right)^{-\frac{n'}{(e{\cal P}')}(eQ)}
\bar\nabla_{\dot\gamma}\bar\nabla_{\dot\gamma}\nabla_J^k J= - \bar R(\dot\gamma, \bar\nabla_J^k J) \dot\gamma +\sum (\bar\nabla^{i_r}\bar R)^{i_r'}* (\dot\gamma)^j * (\bar\nabla_J^{k_p}J)^{k_p'}* (\bar\nabla_{\dot\gamma}\bar\nabla_J^{l_q}J)^{l_q'},
F_t=\bigoplus_{1\leq j_1\leq j_2\leq\cdots\leq j_t\leq n}S(-t)^{\delta-(d_{j_1}+\cdots+d_{j_t})+n-1\choose n-1}=\bigoplus_{1\leq k_1\leq k_2\leq\cdots\leq k_{n-t}\leq n}S(-t)^{d_{k_1}+\cdots+d_{k_{n-t}}-1\choose n-1},
\phi_{Pk}=\phi_{wk}-(\frac{\delta x_{i}}{2})(\frac{\partial\phi}{\partial x})_{wk}+\frac{1}{2!}(\frac{\delta x_{i}}{2})^{2}(\frac{\partial^{2}\phi}{\partial x^{2}})_{wk}+\cdot\cdot\cdot.
\frac{1}{\kappa}\Omega_{W Z}^{\rho}( g ) = - \frac{1}{2}\int_{0}^{2 \pi}d x \, \langle ( g^{- 1}d g ) \stackrel{\wedge}{,}( g^{- 1}d g )^{\prime}\rangle - \frac{1}{2}\langle ( g^{- 1}d g ) ( 0 ) \stackrel{\wedge}{,}d M{\scriptstyle \,}M^{- 1}\rangle + \rho ( M )
\begin{aligned}\int_{\Omega}\frac{\delta\mathcal{F}}{\delta v}\wedge\partial v=\int_{\Omega}\Big(\frac{\delta\mathcal{F}}{\delta v}\wedge d(\partial\phi)+\frac{\delta_{\beta}\mathcal{F}}{\delta v}\wedge\partial\eta-\ast dw\wedge\eta'-\ast dw\wedge [\eta,u]_{1}\Big). \end{aligned}
S_{E}\rightarrow S_{E [ \partial M ]}~ = ~ \frac{1}{G}\int{\sqrt g}\left({\sqrt h}R_{h}+ \frac{1}{4}{\sqrt h}h^{i j}\partial_{i}g_{\alpha \beta}\partial_{j}g_{\gamma \delta}\epsilon^{\alpha \gamma}\epsilon^{\beta \delta}\right) ~ .
\sum_{t=1}^{T}\sum_{i=1}^n\sum_{j=1}^nG_1\|x_{i,t}-x_{j,t}\|\le n\varepsilon_1G_1+\sum_{t=1}^{T}\sum_{i=1}^n\Big(\frac{\tilde{\varepsilon}_2^2G_1^2\alpha_t}{\sigma}+\frac{\sigma\|\epsilon^z_{i,t}\|^2}{4\alpha_t}\Big).
U = \frac{r}{l_{\mathrm{s}}^{2}}\ , \quad g_{\mathrm{Y M}}^{2}= \frac{g_{\mathrm{s}}}{l_{\mathrm{s}}^{3}}\quad H - p_{1 0}= H - \frac{N}{g_{\mathrm{s}}l_{\mathrm{s}}}\ .
G(x,y) = \frac{e^{ik_{-}\vert x \vert}}{\sqrt{\vert x \vert}}G^{\infty}(\hat{x},y) + G_{Res}(x,y), \nabla_y G(x,y) = \frac{e^{ik_{-}\vert x \vert}}{\sqrt{\vert x \vert}}H^{\infty}(\hat x,y ) + H_{Res}(x,y),
\begin{array}{l}{\displaystyle | F_{\nu}\left( \frac{2\pi}{\sigma_0}|n| \tau, \frac{2\pi}{\sigma_0}|n| \tau_0 \right) |^2 \sim}\sim{\displaystyle \frac{2^{2\nu}}{\Gamma^2(1-\nu)}\left( \frac{2\pi|n|}{\sigma_0}\right)^{2-2\nu}(-\tau_0)~ J_{\nu}^2 \left( -\frac{2\pi}{\sigma_0}|n| \tau_0 \right) (-\tau)^{1-2\nu}}\end{array}
v(q) = v(o)+(X v)(o) x + (Y v)(o) y + (T v)(o) t & + \frac{1}{2}(X^2 v)(o) x^2 + \frac{1}{2}(Y^2 v)(o) y^2 + [2(Tv)(o)+(XYv)(o)]xy + O(r^3).
u^\infty_{A,q}(d ,\theta )= \underset{(\ell_1,m_1)\in \Gamma}{\sum}\; \underset{(\ell_2,m_2)\in \Gamma}{\sum}\mu_{\ell_1 m_1\ell_2m_2}Y^{m_1}_{\ell_1}(d) Y^{m_2}_{\ell_1}(\theta ),
P(\theta,t|\theta_{0})=\frac{1}{2\pi}[1+2\sum_{m=1}^{\infty}e^{-D^{r}m^{2}t}cosm(\theta-\theta_{0})]=\frac{1}{2\pi}\Theta_{3}(\frac{1}{2}(\theta-\theta_{0}),e^{-D^{r}t})
\widehat K_S(z,w) = \begin{bmatrix}C (I - zA)^{-1}B^{*}(I - zA^{*})^{-1}\end{bmatrix}\begin{bmatrix}(I - \overline{w}A^{*})^{-1}C^{*}(I - \overline{w}A)^{-1}B \end{bmatrix}.
\prod_{\substack{(a,b)=1 a,b \geq 1}}\left( 1- \frac{z^b}{2^a}\right)^{\frac{b^{2}}{a^3}}= \exp\left\{\left(\frac{7}{8}\zeta(3) -\frac{1}{12}\pi^2 \log 2 + \frac{1}{6}(\log 2)^3 \right)\left(\frac{z (1+z)}{(1-z)^3}\right) \right\},
m_1\alpha_1+m_2\alpha_2+m_3\alpha_3+m_4\alpha_4=m, &m_1\alpha_2+m_2\alpha_1+m_3\alpha_3+m_4\alpha_4=m, &m_1\alpha_1+m_2\alpha_2+m_3\alpha_4+m_4\alpha_3=m,
\begin{array}{r c l}{{}}&{{}}&{{( \epsilon \otimes \mathrm{i d}) \Delta ( H ) = \epsilon ( H ) 1 + H = H ~ ,}}\\{{}}&{{}}&{{( \epsilon \otimes \mathrm{i d}) \Delta ( X^{+}) = \epsilon ( X^{+}) q^{- H}+ \epsilon ( q^{H}) X^{+}= X^{+}~ .}}\\ \end{array}
X^\mu(\mathcal{S}_i):T^*TM\times \mathbb{R}\to M_4, & (u^{k_1},...,u^{k_N},p^{k_1},...,p^{k_N},t)\mapsto X^\mu(u^{k_1},...,u^{k_N},p^{k_1},...,p^{k_N},t)
\|\widetilde{\varphi}(T)\| = \sup\big\{\|\widetilde{\varphi}(T)(x)\| : x\in X, \|x\|<1 \big\}&= \sup\big\{\|\widetilde{\varphi}(T)(\varphi(a)(x))\| : a\in A, x\in X, \|a\|\leq 1, \|x\|<1 \big\}&= \sup\big\{\|\varphi(T(a))(x)\| : a\in A, x\in X, \|a\|\leq 1, \|x\|<1 \big\}& \leq\ \|T\|,
{C_k}={\left|{\xi_k^\star}\right|^2}\left({\frac{E}{{{N_{\rm{R}}}{\sigma^2}}}+ \sum\nolimits_{i = 1}^{{N_{\rm{R}}}}{\frac{1}{{{N_{\rm{R}}}{{\left|{\xi_i^\star}\right|}^2}}}}- \frac{1}{{{{\left|{\xi_k^\star}\right|}^2}}}}\right),
u^{(0)}_{\underline{m}}\tilde{\sigma}^{\underline{m}~\underline{\mu}\underline{\nu}}= v^{~\underline{\mu}}_{A}v_{A}^{~\underline{\nu}},\qquad u^{i}_{\underline{m}}\tilde{\sigma}^{\underline{m}~\underline{\mu}\underline{\nu}}= - v^{~\underline{\mu}}_{A}\Gamma^i_{AB}v_{A}^{~\underline{\nu}}.
0<\nu_3:=\min_{1\leq i\leq m}\min\left\{\frac{(p_*)_i^-}{p_i^+},\frac{(q_*)_i^-}{q_i^+}\right\}-1\leq\nu_4:=\max_{1\leq i\leq m}\max\left\{\frac{(p_*)_i^-}{p_i^+},\frac{(q_*)_i^-}{q_i^+}\right\}-1,
\rho^2 \asymp \begin{cases}n^{-\frac{2s}{2s+1/2}}b \geq n^{\frac{1}{2s+1/2}}, \left({b n}\right)^{-\frac{2s}{2s+3/2}}n^{\frac{1}{2s+1/2}}/m^{\frac{s+3/4}{2s+1/2}}\leq b < n^{\frac{1}{2s+1/2}}, (n/\sqrt{m})^{-\frac{2s}{2s+1/2}}b < n^{\frac{1}{2s+1/2}}/m^{\frac{s+3/4}{2s+1/2}}, \end{cases}
\frac{1}{\bar{\theta}}I(\xi,v,\theta|\bar{\xi},\bar{v},\bar{\theta}) =\bar{\theta}\tilde{H}(A(U)|A(\bar{U})) &=\bar{\theta}[\tilde{H}(A(U))-\tilde{H}(A(\bar{U}))-\tilde{H}_V(A(\bar{U}))(A(U)-A(\bar{U}))] &\geq\min_{\substack{\tilde{U}\in B_R \delta\leq\bar{\theta}\leq M}}\{\bar{\theta}\tilde{H}_{VV}(A(\tilde{U}))\}|A(U)-A(\bar{U})|^2,
\begin{cases}\phi_{A_{j}}(x)=0, |x|\leq \frac{A_{j}}{2}, \frac{1}{4_{A_{j}}}\leq |\nabla \phi_{A_{j}}|\leq |\frac{1}{2A_{j}}|, A_{j}\leq |x|\leq 2A_{j}, \phi_{A_{j}}(x)=1, |x|\geq 3A_{j},\end{cases}
\|T\|^2 = \sum_{k =0}^\infty |T\big((1+k)^{-\frac{s}{2}}h_{k}\big)|^2 = \sum_{k=0}^\infty (1+k)^{-s}\left|\int_{-\infty}^\infty h_{k}(x) W(x)\mathrm{d}x-\sum_{x\in X_n}\omega(x) h_{k}(x)\right|^2.
2 \kappa^{2}e^{- 1}L = R + 4 m^{2}e^{( 2 \lambda_{1}+ 2 \lambda_{2})}- 5 \partial_{\mu}( \lambda_{1}+ \lambda_{2})^{2}- \partial_{\mu}( \lambda_{1}- \lambda_{2})^{2}- e^{- 4 \lambda_{1}}{F_{\mu \nu}^{( 1 )}}^{2}- e^{- 4 \lambda_{2}}{F_{\mu \nu}^{( 2 )}}^{2}.
\mathcal{F}(\tau,Q) = \sum_{d,n \geq 0}\frac{1}{n!}\langle \tau, \dots, \tau \rangle^{K\textnormal{th}}_{0,n,d}Q^d \in \mathbb{Z}[ \![t_0, \dots, t_N] \! ] \otimes \mathbb{C}[\![Q]\!]
f^{2}(t)x+f^{\prime}(t)y+(g^{\prime}(t)+f(t)h(t))+p_{x}=z_{x}\Gamma_{11}(z), & f^{2}(t)y+f^{\prime}(t)x+(h^{\prime}(t)+f(t)g(t))+p_{y}=z_{y}\Gamma_{18}(z).
( \mathrm{M a s s})^{2}={\frac{1}{4 \rho_{2}^{( 0 )}}}\left[{\frac{1}{2}}( \hat{\alpha}_{L})^{2}+ N_{L}- 1 \right] ={\frac{1}{4 \rho_{2}^{( 0 )}}}\left[{\frac{1}{2}}( \hat{\alpha}_{R})^{2}+ N_{R}- \delta \right] .
\tau^{\prime}= \frac{a \tau + b}{c \tau + d}, \quad \quad \quad \left( \begin{matrix}{H_{( 3 )}^{\prime}}\\{F_{( 3 )}^{\prime}}\\ \end{matrix}\right) = \left( \begin{matrix}{d}&{c}\\{b}&{a}\\ \end{matrix}\right) \left( \begin{matrix}{H_{( 3 )}}\\{F_{( 3 )}}\\ \end{matrix}\right) \
\left(-\frac{\partial}{\partial t}+A\right)\Phi_n(t, p) = -\left(\sum_{j=1}^d \mu^{(j)}(p)e_j\right)\otimes \Phi_{n-1}(t, p) - \sum_{j=1}^de_j \otimes\frac{\partial \Phi_{n-1}(t, p)}{\partial p_j}- \left( \frac{1}{2}\sum_{j=1}^d e_{j}\otimes e_{j}\right)\otimes \Phi_{n-2}(t, p).
\tilde{M}_x-\tilde{L}_y=\tilde{\Gamma}^2_{22}\tilde{L}-2\tilde{\Gamma}^2_{12}\tilde{M}+\tilde{\Gamma}^2_{11}\tilde{N}, & \tilde{N}_x-\tilde{M}_y=-\tilde{\Gamma}^1_{22}\tilde{L}+2\tilde{\Gamma}^1_{12}\tilde{M}-\tilde{\Gamma}^1_{11}\tilde{N},
\lambda'(t) = \frac{2}{N-6}(\kappa|t|)^{-\frac{N-4}{N-6}}+ \frac{5}{2(N-6)}|t|^{-\frac{2N-7}{2(N-6)}}p_0(t) + |t|^{-\frac{5}{2(N-6)}}p_0(t).
V_{i_1...i_n}^{a_1...a_n}\left( 1;p_1,...,p_n\right)+V_{i_1...i_n}^{a_1...a_n}\left( 2a;p_1,...,p_n\right)+V_{i_1...i_n}^{a_1...a_n}\left( 2b;p_1,...,p_n\right) =0
m_{\Phi}=\sup_{n\in I}\|P_{\left\{n\right\}}\|,\kappa_{\Phi}=\sup_{k\leq\ell}\left\|P_{[k,\ell]\cap I}\right\|,ub_{\Phi}=\sup_{\sigma\subset I,|\sigma|<\infty}\|P_{\sigma}\|.
\left[ l^{\dot{I}}_{\dot{J}}, f^{\dot{K}}_{\dot{L}}\right] = \sum_{\dot{K}_1 \dot{K}_2 = \dot{K}}\delta^{\dot{K}_1}_{\dot{J}}f^{\dot{I}\dot{K}_2}_{\dot{L}}- \sum_{\dot{L}_1 \dot{L}_2 = \dot{L}}\delta^{\dot{I}}_{\dot{L}_1}f^{\dot{K}}_{\dot{J}\dot{L}_2}.
{\cal H}_{0}^{1 + 1 f l o w}= \frac{\alpha^{2}}{2 ( 1 + \alpha )^{2}}\left( \Pi^{-}\right)^{2}+ \frac{M^{2}}{\sqrt{2}}\psi_{+}^{\dag}\frac{1}{i \partial_{-}}* \psi_{+}.
h_c(\theta):=\frac{{2}^{\frac{9}{4}}e^{{5\pi i}/{8}}}{k^{\frac{3}{4}}_{+}\left[\cos\left(\frac{\theta_c-\theta}2\right)\right]^{3/2}\left(\tan\theta_c\right)^{1/2}}e^{-i k_{+}\vert y \vert\cos(\theta_c+\theta_{\hat y})},\theta\in(0,\pi/2).
G_{N}(t,\,x^{N}(t)) =\frac{1}{N}\sum_{c=1}^{m}\sum_{i\in I_{c}^{N}}g^{(c)}(t,\,z_{i}^{N}(t)) \mathbf{g}_{x}^{(c_{1},\,c_{2})}(t,\,z_{i}^{N}(t)) =g_{x}^{(c_{1})}(t,\,z_{i}^{N}(t))-g_{x}^{(c_{2})}(t,\,z_{i}^{N}(t)).
\P\left(|\hat\tau(Y_{1:n},Z_{1:n})-\tau(Y_1,Z_1)|\right. \left.\ge \gamma\sqrt{\frac{\log p}{n}}\right) &=2\left(1-\Phi\left(\frac{3}{2}\gamma\sqrt{\log p}\right)\right)+R_{n,p,\beta,\zeta,\gamma},
\mathcal B^{(k+1)}:=\left\{\omega\in\mathcal \mathcal B^{(k)}\,\Big|\,\begin{array}{c}\forall z,w\in \mathbb B_r\textrm{with}\Vert z-w\Vert<\delta_{k+1}: \sup_n\Vert f^n_\omega(z)-f^n_\omega(w)\Vert<\varepsilon_{k+1}\end{array}\right\}
\Delta_{m,p}^{k}= \frac{\beta_{p}}{L_{p}}\bigl (1 - \sqrt{1 - \tau_{m,p}^{k}}\bigr ) \leq \frac{2}{\beta_{p}}\left (\| P_{k}^{L}[F(p_{m,k}^{H})] \|_{W^{-1,p}(\Omega)}+ \| F(p_{m,k}^{H}) - P_{k}^{L}[F(p_{m,k}^{H})] \|_{W^{-1,p}(\Omega)}\right).
\varphi^{(12)}_1(\tau,z) :=3\varphi_1^{(3)}(\tau,z)\varphi_1^{(10)}(\tau,z)-8\varphi_1^{(4)}(\tau,z)\varphi_1^{(9)}(\tau,z)+\varphi_1^{(5)}(\tau,z)\varphi_1^{(8)}(\tau,z), \varphi^{(12)}_2(\tau,z) :=4\varphi_1^{(4)}(\tau,z)\varphi_1^{(9)}(\tau,z)-\varphi_1^{(5)}(\tau,z)\varphi_1^{(8)}(\tau,z)-\varphi_1^{(12)}(\tau,z).
1 = \int D \rho \delta \left( \rho ({\bf x}) - \rho_{\mathrm{g a s}}({\bf x}) \right) = \int D \rho D \lambda \exp \left\{i \left[ \sum_{a = 1}^{N}q_{a}\lambda ({\bf z}_{a}) - \int d^{3}x \lambda \rho \right] \right\}.
\begin{pmatrix}I b_2 c_A^* I \end{pmatrix}\begin{pmatrix}y_1 y_2 \end{pmatrix}= \begin{pmatrix}z_1 z_2 \end{pmatrix}, y_1 = z_1, y_2 = z_2 - b_2 c_A^* y_1.
\begin{array}{llll}{\bf G}^{(N)}= ({\bf 1}+{\bf K})[({\bf 1}+{\bf \tilde{K}})({\bf 1}+{\bf K})]^{\frac{N-1}{2}}-{\bf 1}\;\;\; \;N\in\mbox{odd}\;, \tilde{{\bf G}}^{(N)}= [({\bf 1}+{\bf K})({\bf 1}+{\bf \tilde{K}})]^{\frac{N}{2}}-{\bf 1}\;\;\; \;N\in\mbox{even}\;. \end{array}
\langle( I_{1}+ I_{2})'(u),v\rangle = \int_{M}|u(x)|^{p-2}u(x)v(x)d\mu_g(x) &+ \iint_{M\times M}\frac{\left| u(x)-u(y) \right|^{p-2}(u(x)-u(y))(v(x)-v(y))}{(d_g(x,y))^{N+ps}}d\mu_g(x)d\mu_g(y),
f(h\tilde*\pi_z(y)) = \frac{1}{\alpha_0(h)\alpha_0(\pi_z(y) )}\cdot (\delta_h*\delta_{\pi_z(y)})(\alpha_0f) &=\frac{1}{\alpha_0(h)\tilde\alpha_0(y)}(\alpha_0f)_h(\pi_z(y)) &=\frac{1}{\alpha_0(h)\tilde\alpha_0(y)}T_h(\alpha_0f)(\pi_z(y)) &= \int_X f(\pi_x(w)) \> \tilde K_h(y,dw) =(\tilde T_h(f\circ\pi_z))(y).
\begin{array}{l}\int_{\Omega}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla\varphi \mathrm{d}x+\int_{\Omega}\frac{|\nabla u_{n}|^{p}}{u_{n}+\delta +\frac{1}{n}}\varphi \mathrm{d}x=\int_{\Omega}f(x,u_{n},\nabla u_{n})\varphi \mathrm{d}x,\end{array}
C_{T}\frac{\Gamma ( c ) \Gamma ( b - a )}{\Gamma ( c - a ) \Gamma ( b )}= \frac{\Gamma ( c ) \Gamma ( a - b )}{\Gamma ( c - b ) \Gamma ( a )}\left( C_{R}+ \frac{\Gamma ( 2 - c ) \Gamma ( c - b )}{\Gamma ( c ) \Gamma ( 2 - c - b )}\right) ( \alpha - 1 )^{b - a}2^{a - b}
P_{S_\ell |\mathbf{T}}(b| \mathbf{t}) = \frac{1}{Z}\sum_{s_1,\ldots, s_N}P_{\mathbf{T}|\mathbf{S}}(\mathbf{t}| \mathbf{s}) P_{\mathbf{S}}(\mathbf{s}) \Bbb I[s_\ell = b],
\mathcal{K}_{p,q}(k) =\frac{\pi_{p,q}}{2}F\left(\frac{1}{p^*},\frac1q;\frac{1}{p^*}+\frac1q;k^q\right), \mathcal{E}_{p,q}(k) =\frac{\pi_{p,q}}{2}F\left(-\frac1p,\frac1q;\frac{1}{p^*}+\frac1q;k^q\right).
\begin{aligned}\nabla w_2=\begin{pmatrix}\frac{h^0_e}{u^0_e}\partial_x w_1 \frac{h^0_e}{u^0_e}\partial_Y w_1+\partial_Y\left(\frac{h^0_e}{u^0_e}\right)w_1\end{pmatrix}+\nabla \left(\frac{h^0_e}{u^0_e}B_v+\frac{b}{u^0_e}-B_g\right).\end{aligned}
\begin{array}{ccl}\int_{M}\gamma_{1,0}u_{\overline{1}}d\mu = -\int_{M}\gamma_{1}u_{\overline{1}0}d\mu & = -\int_{M}\gamma_{1}\left( u_{0\overline{1}}-A_{\overline{1}\overline{1}}u_{1}\right) d\mu & = \int_{M}A_{\overline{1}\overline{1}}u_{1}\gamma_{1}d\mu .\end{array}
\begin{array}{rcl}\|s_{i}\| \leq \|f'_{i}(\tilde{y}^{(k)})-f'_{i}(y_{1}^{(k)},y_{2}^{(k)},\dots,y_{i-1}^{(k)},y_{i}^{(k)},y_{i+1}^{(k-1)},\dots,y_{m}^{(k-1)})\| \leq L\underset{j=i+1}{\overset{m}{\sum}}\|y_{j}^{(k)}-y_{j}^{(k-1)}\| \leq L\underset{j=2}{\overset{m}{\sum}}\|y_{j}^{(k)}-y_{j}^{(k-1)}\|.\end{array}
I_{j j}= \beta ( x - y ) - \Gamma_{\gamma}\makebox [ . 5 i n ]{,}\Gamma_{\gamma}= - \frac{1}{2}e^{2}\int_{\gamma}\int_{\gamma}d x_{\mu}^{\prime}d x_{\mu}^{\prime \prime}\Delta_{\frac{e^{2}}{\pi}}( x^{\prime}- x^{\prime \prime}) .
S = \int d p [ A_{f}( \vec{p}) A_{f}^{\dagger}( \vec{p}) l n ( c h^{2}( g ( \vec{p}, \beta ) ) ) - A_{f}^{\dagger}( \vec{p}) A_{f}( \vec{p}) l n ( s h^{2}( g ( \vec{p}, \beta ) ) ) ] \; ,
a_{p}^{SFA}(t)=-i\int_{t_{0}}^{t}dt^{\prime}\langle p|e^{-i\int_{t^{\prime}}^{t}\hat{H}_{V}(t^{\prime\prime})dt^{\prime\prime}}\hat{H}_{I}(t^{\prime})e^{-i\int_{t_{0}}^{t^{\prime}}\hat{H}_{0}(t^{\prime\prime})dt^{\prime\prime}}|\psi(t_{0})\rangle(1.4)
{H}_{int}^{1}= \bar{\psi}e \left(\gamma^{-}\bar{B}_{-}+\gamma^{i}\bar{B}_{i}\right) \ast \left[1 + \frac{e}{2}\gamma^{+}\frac{1}{i\partial_{-}- e \bar{B}_{-}}\left( \gamma^{-}\bar{B}_{-}+ \gamma^{j}\bar{B}_{j}\right)\right]\ast \psi.
\lim_{z\rightarrow p}\frac{\mathcal{C}_{\Omega}\Big(f(z), \big(J_f(z)\tau_j\big)_N\big(f(z)\big)\Big)}{\mathcal{C}_{\Omega}\Big(f(z), \big(J_f(z)\tau_j\big)_T\big(f(z)\big)\Big)}=0.
\mu_{n}^{(a,b,c)}=\int_{cq}^{aq}x^n \rho^{(a,b,c)}(x;q)d_qx &=a q \frac{\left(a b q^{2}, a^{-1}c, a c^{-1}q ; q\right)_{\infty}}{\left(a q, b q, c q, a b c^{-1}q ; q\right)_{\infty}}\sum_{m=0}^{n}(-1)^{m}\left[\begin{array}{c}n m\end{array}\right]_{q}q^{-n m+\binom{m+1}{2}}\frac{(a q, c q ; q)_{m}}{\left(a b q^{2}; q\right)_{m}}.
\sup_{0\leq s \leq T}\|u(s)\|_{\infty}\leq 2\|u_o\|_{\infty}&{}+C^{1}_{n}\sup_{0\leq s \leq T}\|\nabla p(s)\|^{n}_{\infty}+ C^{2}_{n}\sup_{0\leq s \leq T}\|w(s)\|^{\frac{n-1}{n+1}}_{\infty}\,.
\langle \delta,\widehat{x}_0+\widehat{x}_k\rangle >0\,,1\leq m\,, &\langle \delta,\widehat{x}_0+\widehat{x}_k\rangle <0\,,1\leq m-1\,, &\langle \delta,\widehat{x}_0+\widehat{x}_m\rangle =0\,.
C^{*}=\left(\begin{array}{ccc}C_{1}^{\diamond}\cdots C_{n}^{\diamond}\end{array}\right)\cap\left(Y\oplus X_{0}\right) =\left\{\left(\left(y_{1},\ldots,y_{n}\right),x\right)\in Y\oplus X_{0}\,\left|\, x=\sum_{k=1}^{n}C_{k}^{\diamond}y_{k}\in X_{0}\right.\right\}.
X_{U}^{s,b}= \begin{cases}X_S^{s,b}\mathrm{if}\ \U(t) = e^{it\Delta}, X_{W_{\pm}}^{s,b}\mathrm{if}\ \U(t) = e^{\mp it \langle \nabla \rangle},\end{cases}Q_L^{U}= \begin{cases}Q_L^{S}\mathrm{if}\ \U(t) = e^{it\Delta}, Q_L^{W_{\pm}}\mathrm{if}\ \U(t) = e^{\mp it \langle \nabla \rangle}.\end{cases}
\begin{aligned}G(p_1,p_2) - L(p_1,p_2) = -\frac{\sqrt{g(p_1)}}{4 \pi}\log\left( 1 + \frac{\frac{1}{2}\partial_x g_{xx}\; x^3 + \frac{1}{2}\partial_y g_{xx}\; x^2 y+ \frac{1}{2}\partial_x g_{yy}\; x y^2+ \frac{1}{2}\partial_y g_{yy}\; y^3\cdots}{g_{xx}x^2 + g_{yy}y^2}\right)\end{aligned}
{\cal Z}_{\rm G-L}^{\rm 3D}=\int{\cal D}{\bf h}{\cal D}\Phi{\cal D}\Phi^{*}\exp\left\{-\int d^3x\left[\frac{1}{4\eta^2}H_{\mu\nu}^2+\frac{q^2}{2}{\bf h}^2+\left|\left(\partial_\mu+2\pi i h_\mu\right)\Phi\right|^2+m_H^2\left|\Phi\right|^2+\lambda\left|\Phi\right|^4\right]\right\}.
c_k :=\int_{\mathbb{R}}|p_{0,k}(y)|dy=\int_{\mathbb{R}}dy\left|e^{-ik\frac{2\pi}{V_F}g}\int_{0}^{V_F}p_{}(v,g)e^{-ikv\frac{2\pi}{V_F}}dv.\right| &\leq\int_{\mathbb{R}}\int_{0}^{V_F}|p_{}(v,g)|dvdg=1.
\setlength{\abovedisplayskip}{4 pt}\setlength{\belowdisplayskip}{4 pt}{\boldsymbol{v}_{t,j}^\alpha}={\mathop{f}\nolimits}\!\left({\boldsymbol{W}_j^\alpha{\boldsymbol{v}_{t - 1,j}^\alpha}+ \boldsymbol{W}_j^{\alpha,in}{\boldsymbol{x}_{t,j}}}\right),
\langle \partial_t \bar{u}_\delta , z \rangle_{V^*,V}+ \int_\Omega \nabla \bar{\mu}_\delta \cdot \nabla z \,dx=0, (\bar{\mu}_\delta, z)_H = (\nabla \bar{u}_\delta, \nabla z)_{H}- ( \partial_{\boldsymbol{\nu}}\bar{u}_\delta, z_{|_\Gamma})_{H_\Gamma}+ (\bar{\xi}_\delta,z)_{H}+ \bigl( \pi (u_\delta)-\pi(u) ,z \bigr)_{\!H}
\log\left(Z_{n}(\beta)\right) + \frac{1}{2}\log\left(1- 2\beta J\right)- (n-1)\beta^2 + \beta (J -J') - \beta C_{n,1}& ~~~~~~~~~~ - \sum_{k=2}^{m_{n}}\frac{2\mu_{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}\left| \mathbb{P}_{n}\right.\stackrel{p}{\to}0.
\left[ A_{i}( x ) , A_{j}( y ) \right] = 0 ~ ~ , ~ ~ \left[ E_{i}( x ) , E_{j}( y ) \right] = 0 ~ ~ , ~ ~ \left[ E_{i}^{A}( x ) , A_{j}^{B}( y ) \right] = - i \delta^{A B}\delta_{i j}\delta ( x - y )
{1\over 2 m_{p+1}}\,{\partial S_0\over \partial \sigma_{\mu_1\dots\mu_{p+1}}}{\partial S_0\over \partial\sigma^{\mu_1\dots\mu_{p+1}}}-\frac{m_{p+1}}{2}\, p ={\partial S_0\over\partial \Omega_{p+1}}\ .
\frac{\langle f_2,f_3\rangle}{||f_2||||f_3||}=-\frac{\overline{a}}{1+|a|^2}\sum_{j=0}^{\infty}\beta_j\overline{\gamma_j}&=-\frac{\overline{a}}{1+|a|^2}\sum_{j=0}^{\infty}e^{ij\eta_3}\sqrt{(1-\rho)\rho^j}e^{-i(\eta_1+j\eta_3)}\sqrt{(1-\rho)\rho^j}&=-e^{i\theta_1}\frac{|a|}{1+|a|^2};
{\cal H}={u \over{\cal R}}\sqrt{\left({u \over{\cal R}}\right)^2{P_u}^2 + \vec{L}^2 + \alpha^\prime\, m^2 \, L^2 g_s^{-2 \over D-3}e^{-2{D-4 \over D-3}\phi}}-qA_t \, ,
\lim_{h\rightarrow 0^+}\frac{1}{2h}\int_{x-h}^{x+h}f(t)dt=\lim_{h\rightarrow 0^+}\frac{1}{h}\int_{x}^{x+h}f(t)dt=\lim_{h\rightarrow 0^+}\frac{1}{h}\int_{x-h}^{x}f(t)dt=f(x).
\bigl\langle\omega_{d(v)e_i}^{h(v), 0}(\ldots, z_e^i)\omega_{e_i+e_j}(z_e^i,z^{(j)})\bigr\rangle_{z_e^i}=\sum_{\pm 1}\underset{z_e^i\rightarrow \pm 1}{\textrm{Res}}\omega_{d(v)e_i}^{h(v), 0}(\ldots, z_e^i)\int_{\pm 1}^{z_e^i}\omega_{e_i+e_j}(z_e^i,z^{(j)}).
\tilde{\mathcal{U}}_s^\eta =\left\{Z_s = (X_s,V_s) \in \overline{\mathcal{D}_s}\left| \inf_{1\leq i < j \leq s}\left( \frac{|v_i-v_j|}{\eta}+\frac{\iota (x_i-x_j,v_i-v_j)}{\eta \log \frac{1}{\eta}}\right) > 1 \right. \right\}
\langle 0 | T_{\mu \nu}| 0 \rangle_{\mathrm{\tiny ~ R e n .}}= \mathrm{d i a g}\left[ \langle 0 | T_{t t}| 0 \rangle , 0 , \langle 0 | T_{l l}| 0 \rangle , \langle 0 | T_{l l}| 0 \rangle \right] .
\left[ \sum_p (a_p,\alpha_p),\ \sum_p (b_p,\beta_p) \right] = \left\langle \sum_p b_p, \sum_p \alpha_p \right\rangle \left\langle \sum_p a_p, \sum_p \beta_p \right\rangle^{-1}begin{align*}5 pt] = \prod_p \langle b_p, \alpha_p \rangle \cdot \prod_p \langle a_p, \beta_p \rangle^{-1}begin{align*}5 pt] = \prod_p \left[ (a_p, \alpha_p), (b_p, \beta_p) \right]_p.
\widetilde{\mathcal{R}}_3^+=\frac{(i-1) MT}{\sqrt{2}\pi}\sum_{m\geq 1}\frac1m\sum_{c\leq C_2/m}\frac1{\sqrt c}{\mathop{\sum\nolimits^*}\limits_{d\,(\!\!\!\!\!\mod c)}}e\Big(\frac dc\Big)\sum_{n\geq1}A(m,n)e\Big(\frac{n\bar d}c\Big)\psi(n),
\langle T ( z , \bar{z}) T ( 0 , 0 ) \rangle = \frac{c / 2}{z^{4}}\qquad \langle \bar{T}( z , \bar{z}) \bar{T}( 0 , 0 ) \rangle = \frac{c / 2}{\bar{z}^{4}}\qquad \langle T \bar{T}\rangle = 0
\gamma_{\pm}( k ) \, = \, \sqrt{( b ( k ) \pm \mid \lambda \mid )^{2}\, + \, \mid C_{2}( a k ) \, - \, \tau C_{1}( a k ) \mid^{2}}\; , \; b ( k ) \, = \, \tau_{2}\, \, B ( a k ) \; , \; \lambda \, = \, \tau_{2}\Lambda \; \; ,
R \; \mathrm{V o l}= \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}\ .
\sum\limits_{2^{r}+N_0> l\geq N_0}|\widehat{\mu}(h b^l)|=\sum\limits_{2^{r}> l\geq 0}|\widehat{\mu}(hb^{N_0}b^l)|=O(2^{r(1-\eta_0)})=O(N^{(1-\eta_0)}).
\begin{array}{ccc}O3~before~splitting~~~~~ ~~~~~~~ \Delta L_{left}~~~~~~~ \Delta L_{right}O3^+ -N N \widetilde{O3^+}-N N O3^- -\frac{1}{2}-N \frac{1}{2}+N \widetilde{O3^-}\frac{1}{2}-N -\frac{1}{2}+N \end{array}
a_k(n)=2\sum_{m=1}^{\left\lfloor \sqrt{n}\right\rfloor}(-1)^{m-1}a_k(n-m^2),a_k(n)=2\sum_{m=1}^{\left\lfloor \sqrt{\frac{n}{2}}\right\rfloor}(-1)^{m-1}a_k(n-2m^2).