harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\hat{H}_{back}=\frac{e^{2}}{2}\int_{\Omega}dR\int_{\Omega}dR^{\prime}\frac{n(R)n(R^{\prime})}{|R-R^{\prime}|}=\frac{e^{2}}{2}(\frac{N}{\Omega})^{2}\int_{\Omega}dR\int_{\Omega}dR^{\prime}\frac{1}{|R-R^{\prime}|}
z^{L_{0}}e^{xL_{\pm 1}}=e^{xz^{\mp 1}L_{\pm 1}}z^{L_0}, e^{xL_1}e^{zL_{-1}}=e^{(1-xz)^{-1}zL_{-1}}(1-xz)^{-2L_0}e^{(1-xz)^{-1}xL_1}.
\left( \begin{array}{c}{{F}}\\{{* F}}\\ \end{array}\right) \rightarrow \left( \begin{array}{l l}{{\cos \alpha}}&{{\sin \alpha}}\\{{- \sin \alpha}}&{{\cos \alpha}}\\ \end{array}\right) \left( \begin{array}{c}{{F}}\\{{* F}}\\ \end{array}\right) \, .
f\circ (\overline{u_1}\otimes\dots\otimes \overline{u_n})= \overline{f\circ(u_1\otimes\dots\otimes u_n)}\circ\varphi =\overline{g\circ(u_1\otimes\dots\otimes u_n)}\circ\varphi =g\circ (\overline{u_1}\otimes\dots\otimes \overline{u_n}).
C_{x_1}C_{x_l}=C_{\tau s_3}C_{s_0}C_{s_2}C_{s_1}C_{s_3}C_{x_l}+\Box=\xi^2C_{\tau s_3}C_{s_0}C_{s_2}C_{x_l}+\Box\in \xi^2C_{\tau s_3}C_{s_0}C_{s_2}C_{x_l}+H^{<13}.
n_1:=\max\{n_0,\varphi_1(\delta/2),\varphi_2(\tilde\varepsilon/4M^2),\varphi_3(\tilde\varepsilon^2/8M^2)\}, k_0=\max\{\varphi_2(\varepsilon^2/6M^2),\varphi_3(\varepsilon^2/12M^2)\}, d:=\max\{f(k_0),\lceil\log_c(\varepsilon/8M)\rceil\}, \tilde\varepsilon=\frac{1-c}{16}\cdot\varepsilon, c:=\exp(-\delta/2).
\|\tilde X(\phi^{(1)},\ldots,\phi^{(r)})\|_s \leq K_s \sum_{l=1}^r \|\phi^{(1)}\|_1 \ldots \|\phi^{(l-1)}\|_1 \|\phi^{(l)}\|_s \|\phi^{(l+1)}\|_1 \ldots \|\phi^{(r)}\|_1, &\forall \; \phi^{(1)},\ldots,\phi^{(r)}\in H^s \oplus H^s.
\bigl(\mathcal{S}_N L_M\bigr)(q\,|\,a, b) - \bigl(\mathcal{S}_N L_M\bigr)(0\,|\,a, b) = \sum\limits_{k=0}^\infty \Bigl[\bigl(\mathcal{S}_N L_{M-1}\bigr)(q+ka_M\,|\,a, b) - & -\bigl(\mathcal{S}_N L_{M-1}\bigr)(ka_M\,|\,a, b)\Bigr],
\phi^{i}( x , z ) \phi^{j}( x , z^{\prime}) |_{c t}= \frac{b^{i j}}{| z - z^{\prime}|^{2}}\delta ( z - z^{\prime}) + c^{i j}\delta^{2}( z - z^{\prime}) + \delta^{( 2 )}( z - z^{\prime}) \ \Gamma_{k}^{i j}\phi^{k}( x , z )
\begin{aligned}q^{a(r)}f_{m,k}e_{m,j-k-1}= q^{N \frac{k(k+1)}{2}+(k+1) a(r)}{m-1 \brack k}_{q^N}& \times \left( d^{m-1}\sum_{\substack{\alpha < a(r) w(\alpha)=j+m-k-2}}q^{\alpha}+d^m \sum_{\substack{\alpha < a(r) w(\alpha)=j+m-k-1}}q^{\alpha}\right){j+m-k-2 \brack m-1}_{q^N}.\end{aligned}
\|f\|^2_{H^{m, m-1}_{k+\ell}(\mathbb{R}^2_+)}=\sum_{|\alpha_1+\alpha_2|\le m, \alpha_1\le m-1}\|\langle y\rangle^{k+\ell+\alpha_2}\,\partial^{\alpha_1}_x \partial^{\alpha_2}_y f\|_{L^{2}(\mathbb{R}^2_+)}^2,
S_2[h]=\left( 2\chi\partial_t\chi - 2\chi\Delta_{g_0(t)}\chi - 2|\nabla^{g_0(t)}\chi|^2 \right)h - 2\chi \nabla^{g_0(t)}\chi * \nabla^{g_0(t)}h & - 2\chi\nabla^{g_0(t)}\chi*R_1[h].
P(u,x)=\lim_{N\rightarrow\infty}\left.\,P_{2n}(\frac{x}{N})\right|_{n=Nt/2}=\cos u x =\Gamma(\frac12) \frac{J_{-\frac12}\left( u x \right)}{\left( u x/2 \right)^{-\frac12}}.
V ( \overline{{\rho}}) = \frac{3}{2}\int \frac{d^{4}p}{( 2 \pi )^{4}}\ln ( p^{2}+ e^{2}\overline{{\rho}}^{2}) + \frac{1}{2}m^{2}\overline{{\rho}}^{2}+ \frac{\lambda}{2 4}\overline{{\rho}}^{4}.
\langle \hat V, \varphi\rangle = \Big(\frac{1}{2}+\gamma+\log\pi \Big) \varphi(0)+ \frac1\pi \int_{|\xi|\leq 1}(\varphi(\xi)-\varphi(0))\frac{\xi_2^2}{|\xi|^4}\, d\xi+ \frac1\pi \int_{|\xi|>1}\varphi(\xi)\frac{\xi_2^2}{|\xi|^4}\,d\xi
\frac{y^{n}}{n^{r}( n + p )^{k}}= \int_{0}^{y}\Omega^{r - 1}\int_{0}^{x}\Omega^{k - 1}\Omega_{n}x^{p}\frac{d x}{x}= \int_{0}^{y}\Omega^{r - 1}\Omega_{p}\Omega^{k - 1}\Omega_{n}.
Q_R(sp\otimes b) =(s\otimes1)(\operatorname{id}\otimes\operatorname{id}\otimes\varphi)\bigl(\Delta_{13}(a^*)(1\otimes E)(1\otimes1\otimes x)\bigr)(1\otimes b) &=(\operatorname{id}\otimes\operatorname{id}\otimes\varphi)\bigl((s\otimes1\otimes1)\Delta_{13}(a^*)(1\otimes E)(1\otimes b\otimes x)\bigr).
E_{n}( \alpha ,\beta ) =(\alpha_{n}+\beta_{n}+1)^{2}-(\alpha +\beta +1)^{2}=4n(\alpha +\beta +1+n), \psi_{n}( x;\alpha ,\beta ) =\psi_{0}( x;\alpha ,\beta ) P_{n}^{( \alpha ,\beta )}(z), n\in \mathbb{N},
a_{0}=\int_{\mathbb{R}_{+}}\frac{\alpha t}{1-\alpha t}\,d\mu_{1}(t),a_{1}=\begin{cases}\int_{\mathbb{R}_{+}}\frac{t}{(1-\alpha t)^{2}}\,d\mu_{1}(t) a_{0}\neq-1, \frac{1}{\alpha}\int_{\mathbb{R}_{+}}\frac{1}{(1-\alpha t)^{2}}\,d\mu_{1}(t) a_{0}=-1,\end{cases}
\lim_{\varepsilon \searrow 0}\epsilon ( q^{2}) \: \int_{- \infty}^{\infty}d \tau \: \frac{1}{( ( \tau^{2}- \frac{1}{4}) \: q^{2}\pm i \varepsilon )^{\alpha + 1}}\: H_{z}^{( \beta )}\:{\mathcal{P}}( \tau ) \: e^{- i q \tau ( y - x )}\; .
A =-\frac{\varphi(2)}{\Gamma_p(\frac{1}{2})}\sum_{j=0}^{p-2}\omega^j(x)~\pi^{(p-1)\ell_j}~\Gamma_p\left(\left\langle\frac{1}{4}-\frac{j}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac{3}{4}-\frac{j}{p-1}\right\rangle\right) &\times\Gamma_p\left(\left\langle\frac{1}{2}+\frac{j}{p-1}\right\rangle\right)\Gamma_p\left(\frac{j}{p-1}\right),
f_X^{\rm G}\left( \lambda;\nu ;y\right)=\frac{\lambda^\lambda}{\Gamma( \lambda) \nu^\lambda}y^{\lambda-1}\exp\left( - \frac{\lambda y}{\nu}\right), F_X^{\rm G}\left( \lambda;\nu ;y\right)=\frac{1}{\Gamma(\lambda)}\gamma\left ( \lambda,\frac{\lambda y}{\nu}\right ).
\begin{aligned}\mathcal{M}_j = \iint\left|\int_{O(d)}\int_{|\xi|\approx 2^j}\widehat{f\,d\mu}(\xi)\,e^{-2\pi i \xi\cdot u'}\, \widehat{f\,d\mu}(-\theta^t\xi) \,e^{2\pi i \xi\cdot (\theta u)}\,d\xi\,d\theta\right|^2\,d\mu(u)\,d\mu(u') \lesssim \ 2^{-j\gamma}||f||^2_{L^2(\mu)}\end{aligned}
\mathrm{Hom}_B(M, \tau \mathcal{GF}(M)) \simeq \mathrm{Hom}_B(M, \mathcal{G}(\tau \mathcal{F}( M))) & \simeq \mathrm{Hom}_A(\mathcal{F}(M), \tau \mathcal{F}( M)) =0.
\|\zeta^l_l\|=\sqrt{1-\sum_{m=1}^{l-1}\|\zeta^m_l\|^2}>\sqrt{1-\theta(\varepsilon)^2\sum_{m=1}^{l-1}c_m^2(\theta(\varepsilon))}=\sqrt{1-s_{l-1}(\theta(\varepsilon))}>0.
e^{ih}=e^{i[z_1,z_1^*]+\cdots +i[z_m,z_m^*]}&=\prod_{k=1}^{M}(e^{ix_k},e^{iy_k})\cdot \prod_{k=1}^{m}e^{i[z_k^*,z_k]}\cdot e^{ic},
\displaystyle\prod_{i = 0}^{\ell-1}P(q_i^-, q_i^+) = \prod P(q_i^-, v_i)P(v_i, q_i^+) &= \prod P(q_{i + 1}^-, v_{i+1}) P(v_i, q_i^+) &= \prod P(v_i, q_i^+)^2 &= 1,
r\approx(\frac{9F}{4})^{1/3}(\tau_{0}-\tau)^{2/3},e^{\lambda/2}\approx(\frac{2F}{3})^{1/3}\frac{\tau_{0}^{\prime}}{\sqrt{1+f}}(\tau_{0}-\tau)^{-1/3},4\pi\rho\approx\frac{F^{\prime}}{3F\tau_{0}^{\prime}(\tau_{0}-\tau)}
\overline{{C}}( \alpha_{l}) = \frac{\Gamma ( \frac{D}{2}- 1 )}{2^{D}\pi^{\frac{3}{2}D - 2}}\sum_{m = 0}^{M}( L_{m}+ 2 \delta_{m , l}) \sum_{n = 1}^{\infty}\frac{\cos ( 2 n \pi \alpha_{m})}{n^{D - 2}}.
C_{IJK}\,h^Ih^Jh^K = 3\left\{h^1\,\big(h^2\big)^2 -h^1\,\big(h^\mu\big)^2 -h^2\,\big(h^i\big)^2 +\gamma_{\mu ij}\,h^\mu\, h^i\,h^j\right\}.
_{a^{+}}\mathbf{I}_{x}^{q}[f(x)] =\frac{1}{\Gamma(q)}\int_{a}^{x}(x-\tau)^{q-1}f(\tau)d\tau,_{b^{-}}\mathbf{I}_{x}^{q}[f(x)] =\frac{1}{\Gamma(q)}\int_{x}^{b}(\tau-x)^{q-1}f(\tau)d\tau,
h_n(x, y,t) := h_n(r,t) = (4 \pi t)^{-\frac{n}{2}}e^{- \frac{(n-1)^2 t}{4}- \frac{(n-1) r}{2}- \frac{r^2}{4t}}( 1 + r + t)^{\frac{(n-3)}{2}}(1 + r),
l_{ab}= - r \tilde \sigma_{ab}+\frac{2}{3}r^3 \bar R_{La b L}+ O(r^4) \sigma_{ab}= r^2\tilde\sigma_{ab}- \frac{1}{3}r^4\bar R_{La b L}+ O(r^5) l_a^c= -r^{-1}\delta_a^c+\frac{1}{3}r \bar R_{La b L}\tilde{\sigma}^{bc}+ O(r^2) \eta_a = \frac{1}{3}r^2 \bar R_{La L\underline L}+O(r^3).
\overline{Q}(t) = (t-a)(b-t)\det\begin{bmatrix}s_0' s_1' \ldots s_{m-2}' 1 s_1' s_2' \ldots s_{m-1}' t \vdots \vdots \vdots \vdots \vdots s_{m-1}' s_{m}' \ldots s_{2m-3}' t^{m-1}\end{bmatrix},
dY_t = 2\kappa\|X_t\|^2\,dt - 2\nu\sqrt{Y_t}X_t^\top dW_t - d\,\nu^2Y_t\,dt - 2X_t^\top A_0 X_t\,dt - 2\sum_{p=1}^m X_t^\top A_p X_t\, d\widehat W^p_t.
\hspace{. 7 5 i n}+ \sum_{j = 1}^{l - 1}\gamma^{( l - j ) / 2}\varphi_{i}( - l + j ) \otimes \gamma^{- l + j}x_{i}^{+}( - j ) \quad{\it m o d}U N_{-}\otimes U N_{+}^{2}
\sup_{\substack{\alpha\neq \beta 0\leq t<T_0^*}}\Big |\frac{\alpha-\beta}{z(\alpha,t)-z(\beta,t)}\Big|+ \sup_{\substack{\alpha\neq \beta 0\leq t<T_0^*}}\Big |\frac{z(\alpha,t)-z(\beta,t)}{\alpha-\beta}\Big|=\infty.
S=\int d^{2}x\left[{\frac 1 2}{\partial}_{\mu}{\phi}{\partial}^{\mu}{\phi}+{\frac 1 2}{\lambda}_{{\mu}{\nu}}\left({\partial}^{\mu}{\phi}-{\epsilon}^{{\mu}{\sigma}}{\partial}_{\sigma}{\phi}\right)\left({\partial}^{\nu}{\phi}-{\epsilon}^{{\nu}{\rho}}{\partial}_{\rho}{\phi}\right)\right],
\|1_{|k|\neq 1}(Z, Q)(t)\| \leq C_{\alpha}\exp(10) \sqrt{\frac{t}{\xi}}\|(Z, Q)(2\xi)\|, \|1_{|k|=1}(Z, Q)(t)\| \leq C_{\alpha, \gamma}(\frac{t}{\xi})^{\gamma}\exp(10) \sqrt{\frac{t}{\xi}}\|(Z, Q)(2\xi)\|,
\phi\left (\bigotimes\limits_{i=1}^{m-1}X_iE_{j_ij_i}X_i^*\otimes B\right )= \left(Z_{X}\left (\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\right )Z_{X}^*\right) \otimes \varphi_{j_1,\ldots,j_{m-1},X}(B)
t(t+1) \iint_R \{v\}v^{-3/2}\{u\}u^{-3/2}\left(1+i\log\frac{u}{v}\right)^{-t-2}dudv = \sum_{(m,n)\in R(t)}\left(\widetilde{F}_{m,n}(t) - \widetilde{G}_{m,n}(t)\right) + O(1),
diag(\mathbf{V}_{\overline{\mathbf{x}}_k}^{post}) = [diag( \mathbf{V}_{\mathbf{x}_{k-J}}^{post}) \; diag( \mathbf{V}_{\mathbf{x}_{k-J+1}}^{post}) \ldots diag( \mathbf{V}_{\mathbf{x}_{k}}^{post})], &diag( \mathbf{V}_{\mathbf{x}_{k}}^{post}) = [v_{x_{k,1}}^{post}\; v_{x_{k,2}}^{post}\: \ldots \: v_{x_{k,N_t}}^{post}].
B_j^-=\mathbf{p}B_j^- + \sum_{i=j+1}^{n}\sum_{s=2}^{i-j+1}\sum_{I\in \mathcal{I}_{ji}(s)}E^{e_I}\mathbf{p}B_i^-\frac{1}{\prod_{\ell\in I,\ell\neq i}(h_i-h_\ell)}.
\begin{array}{llll}\eta_z^+\rightarrow\eta_z,\quad \quad\lambda_+\rightarrow \lambda,\quad \quad \beta_{+z}\rightarrow \beta_z,\quad \quad\gamma^+\rightarrow \gamma, \eta^-_{z}\rightarrow\eta,\quad \quad\lambda_-\rightarrow \lambda_z,\quad \quad \beta_{-z}\rightarrow \beta_{zz},\quad \quad\gamma^-\rightarrow \gamma^z.\end{array}
E(T) + \int_0^T \int_\Omega |\partial_t m|^2 \d x\d t = E(0) + \int_0^T \int_\Omega f\cdot\partial_t m \d x\d t \leq E(0) + \frac12 \int_0^T \int_\Omega |f|^2 \d x\d t + \frac12 \int_0^T \int_\Omega |\partial_t m|^2 \d x\d t,
\sigma(1)\tau^{+}=\frac{1}{2}\tau^{+}(0)\overline{\tau}^{-}(1)\tau^{+}-\frac{1}{2}\tau^{-}(0)\overline{\tau}^{+}(1)\tau^{+}&=-\frac{1}{2}\tau^{+}(0)h-\frac{1}{2}\tau^{-}(0)(2x^{+}) &=\frac{1}{2}\tau^{+}+\tau^{+}=\frac{3}{2}\tau^{+},
\big|[(f_i+g_j,A_i\cap B_j)_{i,j}]\big|_* \leq \big|[((f_i,A_i)_i]\big|_*+\big|[(g_j,B_j)_j]\big|_*, \big|\lambda[(f_i,A_i)_i]\big|_* =|\lambda|\,\big|[(f_i,A_i)_i]\big|_*, \big|h[(f_i,A_i)_i]\big|_* =|h|\,\big|[(f_i,A_i)_i]\big|_*,
BA_t(g_1,g_2)(\omega) = \frac{g_1\bigl(t\omega + \sqrt{1-t^2}\omega^\perp\bigr)g_2\bigl(t\omega - \sqrt{1-t^2}\omega^\perp\bigr)}{\sqrt{1-t^2}}& \;\;\;\;\; + \frac{g_1\bigl(t\omega - \sqrt{1-t^2}\omega^\perp\bigr)g_2\bigl(t\omega + \sqrt{1-t^2}\omega^\perp\bigr)}{\sqrt{1-t^2}}.
[D\widetilde{\psi}:D\tau_M]_t = s(\psi)[D\widetilde{\chi}:D\tau_M]_t = s(\psi)[D\widetilde{\chi}:D\widetilde{\varphi}]_t\,[D\widetilde{\varphi}:D\tau_M]_t &= [D\widetilde{\psi}:D\widetilde{\varphi}]_t\,\lambda^{\varphi}(t)
R_{H}= \prod_{i = 1}^{N_{f}}\left[ \frac{\sqrt{| m_{i}|^{2}+ ({\tilde{m}}_{i}+ \frac{1}{2}M_{W})^{2}}+{\tilde{m}}_{i}+ \frac{1}{2}M_{W}}{\sqrt{| m_{i}|^{2}+ ({\tilde{m}}_{i}- \frac{1}{2}M_{W})^{2}}+{\tilde{m}}_{i}- \frac{1}{2}M_{W}}\right]^{- k / 2}\ .
(\phi/\psi_{\gamma_n}^2, Q^n_t(\phi/\psi_{\gamma_n}^2))_{H_n}=(Q^n_{t/2}(\phi/\psi_{\gamma_n}^2), Q^n_{t/2}(\phi/\psi_{\gamma_n}^2))_{H_n}\leq (\phi/\psi_{\gamma_n}^2, \phi/\psi_{\gamma_n}^2)_{H_n}.
T^{(2)}(A,\phi \circ j_A, \gamma \circ j_A)(t) = T^{(2)}\left (M_{L''},(n_1, \ldots, n_{e+2}) \circ \alpha_L \circ j_A, \gamma \circ j_A\right )(t) = T^{(2)}\left (M_{L''},(n_1+ \ldots+ n_e, n_{e+1}, n_{e+2}) \circ \alpha_{L''}, \gamma \circ j_A\right )(t) & \ \dot{=}\ \max(1,t)^{|p n_{e+1}+ q n_{e+2}+ pq(n_1 + \ldots + n_e)|},
f_{i+1/2}^{}(q_{i+1/2}^{L},q_{i+1/2}^{R}) =\frac{1}{2}\left(f(q_{i+1/2}^{L})+f(q_{i+1/2}^{R})\right)-\frac{|a_{i+1/2}|}{2}\left(q_{i+1/2}^{R}-q_{i+1/2}^{L})\right),
(\Delta_{1,1})=\frac{8}{p^{2}}\bigg[\frac{1}{n(n-1)}\{f^{(1)}(\tau_{1})\}^{2}(\Sigma_{1}^{2})+\frac{1}{m(m-1)}\{f^{(1)}(\tau_{2})\}^{2}(\Sigma_{2}^{2})+\frac{2}{nm}\{f^{(1)}(\tau_{3})\}^{2}(\Sigma_{1}\Sigma_{2})\bigg]
\int \varphi^s(A)d\mu_n(A) =\int \sigma_1(A)\sigma_2(A)^{s-1}d\mu_n(A) &= \int \|A\|^{2-s}|\det A|^{s-1}d\mu_n(A) = \int \|A\|^{2-s}d\hat\mu_n(A)
e^{\tau L_{0}}= \left( \begin{matrix}{{\strut 1}}&{{\tau}}&{{\tau}}\\{{\strut \tau}}&{{1 +{\frac{\tau^{2}}{2}}}}&{{{\frac{\tau^{2}}{2}}}}\\{{\strut - \tau}}&{{-{\frac{\tau^{2}}{2}}}}&{{1 -{\frac{\tau^{2}}{2}}}}\\ \end{matrix}\right) \ .
\hspace{10mm}\times\prod_{i = 1}^{k}\frac{\Gamma( i\rho )}{(\rho)}\prod_{i =0}^{k - 1}\frac{\Gamma(1 + a + i\rho)\Gamma(1 + c + i\rho)}{\Gamma(2 + a + c +(k - 2 + i)\rho)}
<\overline{T}(k_{1})T(k_{2})\phi_{a}(k_{3})\phi_{b}(k_{4})>_{trunc}=-i(2\pi )^{D}\delta \left( \sum k_{i}\right) \left( 2\nu_{ab}+\frac{\nu_{a}\nu_{b}}{u-m^{2}}+\frac{\nu_{a}\nu_{b}}{t-m^{2}}\right)
w(t,x)=\frac{\partial}{\partial t}\left(\frac{1}{2\pi}\int\limits_{|y-x|< t}\frac{w^e_0(y)dy}{\sqrt{t^2-|y-x|^2}}\right)+\frac{1}{2\pi}\int\limits_{|y-x|< t}\frac{w^e_1(y)dy}{\sqrt{t^2-|y-x|^2}}.
h^*_d \le h^*_{d-1}\le \dots \le h^*_{\lfloor \frac{d+1}{2}\rfloor}h^*_{j+1}\ge h^*_{d-j}\ \ 0 \le j \le \tfrac d 2 - 1 h^*_j \le \binom{h^*_1 + j - 1}{j}\ \ 0 \le j \le d \, .
\frac{g_t}{g}=\;P^3_1(\varphi)+P_2^2(\varphi)+P_3^1(\varphi)+P_1^1(\varphi), \varphi_t =\;-3\varphi_{\xi^5}+P_2^3(\varphi)+P_3^2(\varphi)+P_1^2(\varphi)+P_4^1(\varphi)+P_2^1(\varphi).
\left\{A,B\right\}=\Sigma_{\lambda}C^{\lambda}\!_{\mu\nu}\frac{\partial A}{\partial\Sigma_{\mu}}\frac{\partial B}{\partial\Sigma_{\nu}}-\frac{\partial A}{\partial\Sigma_{\mu}}\mathbf{L}_{\mu}B+\frac{\partial B}{\partial\Sigma_{\mu}}\mathbf{L}_{\mu}A,
\varphi^{2d-2}g_{e_j}=b\varrho^{-1}t^{n_{e_j}+1-p^{2d-2}}g_{e_j}\varphi^{2d-2}g_{f_j}=b\varrho^{-1}t^{n_{f_j}+1-p^{2d-2}}g_{f_j}\gamma(x)(g_{e_j})-x^{s_{e_j}}g_{e_j}\in t\cdot k_{\mathcal E}^+\cdot g_{e_j}\gamma(x)(g_{f_j})-x^{s_{f_j}}g_{f_j}\in t\cdot k_{\mathcal E}^+\cdot g_{f_j}
\int_{0}^{y}\nu^{+}(E,\mathrm{i}\tilde{y}) \mathrm{d}\tilde{y}+L(E,\mathrm{i}0)= \int_{\mathbb{C}}\log |\tilde{E}-E|\mathrm{d}\mathcal{N}^{\mathrm{i}y}(\tilde{E}).
B_j^+\Omega_\lambda = \sum_{i=1}^{j}\sum_{s=1}^{j-i+1}\sum_{I\in\mathcal{I}_{ij}(s)}d_{i}^+(\lambda)\frac{\prod_{\ell\in I^\complement}(\lambda_i-\lambda_\ell-i+\ell+1)}{\prod_{\ell=i+1}^{j}(\lambda_i-\lambda_\ell-i+\ell)}E^{e_I}\Omega_{\lambda+\epsilon_i}
\langle C^{a}( x ) \bar{C}^{b}( y ) \rangle = i \int{\frac{d^{4}k}{i ( 2 \pi )^{4}}}{\frac{- k^{2}\delta^{a b}- v \epsilon^{a b}}{( - k^{2})^{2}+ v^{2}}}e^{i k ( x - y )}.
h_k = \frac{m(H \cap I_{k})}{m(I_{k})}= \frac{m((A \cup (H \setminus A)) \cap I_k)}{m(I_k)}= \frac{m(A \cap I_{k})}{m(I_{k})}+\frac{m((H\setminus A) \cap I_{k})}{m(I_{k})}= \frac{m(A \cap I_{k})}{m(I_{k})}= a_k
\lim_{n \to \infty}\Big\| \pi_{n}\Big(\Big(\big(\Gamma_{n}^{2n}\circ P_n + \Gamma_{\lceil \frac{3n}{2}\rceil}^{\lceil \frac{5n}{2}\rceil}\circ Q_n\big) - \iota_{n}\Big) (t_{\mu}t_{\nu}^{*})\Big)\Big\| = 0,
(a_1 \cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)}) = \frac{(10t + 1)t}{2^4}a_1 - \frac{(2t - 1)t}{2^4}a_{-1}+ \frac{t}{2^4}v_{(2,3)}+ \frac{t}{2^2}a_1 \cdot v_{(2,3)}(a_{-1}\cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)}) = -\frac{(6t - 1)t}{2^4}a_1 - \frac{(2t - 1)t}{2^4}a_{-1}+ \frac{t}{2^4}v_{(2,3)}+ \frac{t}{2^2}a_1 \cdot v_{(2,3)}
49(j_{n+r}^{(3)}J_{n+s}^{(3)}) =2^{2n+4+r+s}-2^{n+r+3}(a\omega_{1}^{n+s}+b\omega_{2}^{n+s})+3\cdot 2^{n+s+1}(a\omega_{1}^{n+r}+b\omega_{2}^{n+r}) &\ \ -3(a^{2}\omega_{1}^{2n+r+s}+b^{2}\omega_{2}^{2n+r+s})-7(\omega_{1}^{r}\omega_{2}^{s}+\omega_{1}^{s}\omega_{2}^{r}),
\sum_{k=0}^{\infty}\frac{\binom{2k}{k}^2}{32^k(1+k)}\bigg\{H_{2k}^{(2)}-\frac{1}{4}H_{k}^{(2)}\bigg\}=\frac{\Gamma(\frac{3}{2})}{16\Gamma(\frac{5}{4})^2}\psi\,'\bigg(\frac{5}{4}\bigg).
\int_{\Xi}e^{W'/\hbar}dx_1\wedge\cdots\wedge dx_n=\int_{\Xi}\psi^*\left(e^{W'/\hbar}dx_1\wedge\cdots\wedge dx_n\right)=\int_{\Xi}e^{\psi(W')/\hbar}dx_1\wedge\cdots\wedge dx_n.
0\le\Phi_{I,\lambda,P}(f)-\Phi_{I,\lambda,Q}(f) =f(t_{j})-f(t_{j-1})+f(t_{j-2})-f(t_{j-1})-\left[|f(t_{j})-f(x)|+|f(x)-f(t_{j-2})|\right] \le f(t_{j})-2f(t_{j-1})-f(t_{j-2})-[f(t_{j})-2f(x)+f(t_{j-2})] =2[f(x)-f(t_{j-1})],
e(M_{1},M_{1}) =1+(p^{3}-p^{2}-1)(p+1)+(p^{4}-p^{3}-2p^{2}+2p+1)p+(p^{2}-1)p^{3}&=(2p^{3}-3p+1)p^{2}, e(C_{p}^{3},M_{1}) =(p^{3}-1)(p+1)p^{2}+(p^{3}-1)(p^{2}-2)p^{2}=(p^{3}-1)(p^{2}+p-1)p^{2}.
|I_3| \le C_3 |\hat v|_\theta\mu_\Delta(\hat Y\cap D_n)\,d_\theta((z_a,\varphi(a)-n),(z_a',\varphi(a)-n)) = C_3\theta |\hat v|_\theta\mu_\Delta(\hat Y\cap D_n)\,d_\theta(z,z').
A= \frac{i}{2}\{\frac{1}{2}d\ln (\frac{d\overline{g}_{2}}{d\overline{z}_{2}}\frac{dz_{2}}{dg_{2}}) + \frac{(g_{2}d\overline{g}_{2}- \overline{g}_{2}dg_{2})}{(1- g_{2}\overline{g}_{2})}\},
P_2(t/3) = (1 - t)^2 \left( 1 - \frac{t}3 + t^2 - \frac{5}3 t^3 + \frac{4}{3}t^4 - \frac43 t^5 + \frac43 t^6 - \frac53 t^7 + t^8 - \frac13 t^9 + t^{10}\right),
\mathbf{A}=A_{i}\mathbf{E}^{i}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathbf{A}=A^{i}\mathbf{E}_{i}
n_1(\xi,\eta) := \frac{|\xi|^{1/2}(\xi-\eta) \big[\xi |\xi|^{-1}\varphi_t(\xi) \chi(\xi-\eta,\eta) - \eta |\eta|^{-1}\varphi_t(\eta) \chi(\eta-\xi,\xi)\big]}{2|\xi-\eta|^{1/2}(|\xi|^{3/2}-|\xi-\eta|^{3/2}- |\eta|^{3/2})}.
E[S^{env,m}_{s,t}]=E\left[S^{env,0}_{s,t}\right]+\frac{n}{2}E\left[\ln(\beta(t,q_t)/\beta(s,q_s))\right] &+\int_s^tE\left[\left( \beta^{-3}\nabla_q\beta\cdot \left( \frac{3n+2}{6}- \int_0^\infty Tr[\gamma e^{-2y \gamma}] e^{-y \gamma}dy\right)\gamma^{-1}\nabla_q\beta\right)(r,q_r) \right]dr &+O(m^\delta)
Z_{2 \mathrm{i n s t}}\simeq - \frac{2 N}{\pi}[ \quad \frac{\pi}{2}- \arcsin ( \frac{2 \pi^{2}m^{2}}{A}- 1 ) - \frac{2 \pi^{2}m^{2}}{A}\sqrt{\frac{A}{\pi^{2}m^{2}}- 1}\quad ] .
q^{n^{2}/2}S_{n}\left(xq^{-n-1/2};q\right)=\frac{1}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\left(xe^{iy};q\right)_{n}}{\left(q;q\right)_{n}}\exp\left(\frac{y^{2}}{\log q^{2}}-iny\right)dy,
U(y ,\mathbb{I}) \tilde{a}( \mathbf{x}) U( y ,\mathbb{I})^{-1}=\frac{iy^{0}}{2\pi^2}\left(\frac{m^2K_{2}(m\sqrt{\vert \mathbf{x+y}\vert^{2}-(y^{0})^2})}{\vert \mathbf{x+y}\vert^{2}-(y^{0})^2}\right)\ast\tilde{a}(\mathbf{x+y}) ,
\sum_{i\in\Gamma_{P^{\psi},Q^{\psi}}}(-1)^{\sum_{j=1}^{i-1}(q_j^{\psi}-p_j^{\psi})}\lambda_{P^{\psi}(i),Q^{\psi}(i)}=\sum_{i\in\Gamma_{P,Q}}(-1)^{\sum_{j=1}^{i-1}(q_j^{\psi}-p_j^{\psi})}\lambda_{P^{\psi}(i),Q^{\psi}(i)}.
\int_G \int_X \int_X h(g^{-1}.(x-y)) u(x)v(y) \, dx \, dy \, dg = \int_G \int_X \int_X h(x'-y') u(g.x')v(g.y') \, dx' \, dy' \, dg = \int_X \int_X h(x'-y') \left( \int_G u(g.x')v(g.y') \, dg \right) dx' \, dy'.
E\left(X_{N_1 + 1}^{t_0}X_{2N_1 + 1}^{t_0}\cdots X_{(k-1)N_1 + 1}^{t_0}\right)= E\left(X_1^{t_0}X_{N_1 + 1}^{t_0}\cdots X_{(k-2)N_1 + 1}^{t_0}\right),
|T_{k,S^\ast}(S)| \equiv \begin{cases}(-1)^{z(S)}|F'(\pi(S))| \mod{Deg(F^\ast)_{|S|}}\mbox{if}S\mbox{is non-degenerate}\mbox{and}|I(S)|=k; 0 \mod{Deg(F^\ast)_{|S|}}\mbox{otherwise.}\end{cases}
\big(\sum_{j=1}^d|x_j+y_j|^{2R_j}\big)^{1/(2p)}\le \big(\sum_{j=1}^d|x_j|^{2R_j}\big)^{1/(2p)}+\big(\sum_{j=1}^d|y_j|^{2R_j}\big)^{1/(2p)}
D_t v = \bar\nabla_{F_t}\dot\gamma \big|_{s=0}= \bar\nabla_{\dot\gamma}J_t \big|_{s=0}, \bar\nabla_i v = \bar\nabla_{F_i}\dot\gamma \big|_{s=0}= \bar\nabla_{\dot\gamma}J_i \big|_{s=0}
V_{E}(\beta,L,\phi)=\mu^{4}\biggl(2\pi^{2}c^{2}\phi^{2}+\frac{1}{3}\pi^{2}g\phi^{4}-\frac{1}{2\mu^{2}}\delta m^{2}\phi^{2}-\frac{1}{4!}\delta\lambda\phi^{4}\biggr)+G_{E}(\beta,L,\phi)
(h^{2}+ 2h-\mu)(h+4)^{n}=h^{n+2}+2(2n+1)h^{n+1}+ &\sum_{k=0}^{n-2}\bigg(-\mu{n\choose k}(4)^{k}+2{n\choose k+1}(4)^{k+1}+{n\choose k+2}(4)^{k+2}\bigg)h^{n-k}&+\big(-\mu n(4)^{n-1}+2(4)^{n}\big)h-\mu(4)^{n}.
\Lambda(p,q):= \left(p^{\mu}q_{\mu}\right)^2\big[\left(p^{\mu}q_{\mu}\right)^2-1\big]^{-\frac{3}{2}}, \mathcal{S}(p,q):= \big[\left(p^{\mu}q_{\mu}\right)^2-1\big]^2I_3-\big(p-q\big)\otimes\big(p-q\big)-\big[\left(p^{\mu}q_{\mu}\right)-1\big]\big(p\otimes q+q\otimes p\big).
\int_{0}^{t}ds\int_{0}^{\lambda(t-s)}\bigl[z^{d-\frac{1}{2}}K_{\frac{1}{2}-d}(z)\bigr]^{k}\,dz &\leq \biggl[2^{-(\frac{1}{2}+d)}\varGamma\biggl(\frac{1}{2}-d\biggr) \biggr]^{k}\int_{0}^{t}ds\int_{0}^{\lambda(t-s)}z^{k(2d-1)}\,dz &=\frac{[\lambda^{2d-1}2^{-(\frac{1}{2}+d)}\varGamma(\frac{1}{2}-d) ]^{k}\lambda}{(k(2d-1)+1)(k(2d-1)+2)}t^{2kd-k+2},
\mathcal{L}=\frac{_1}{^2}i\hbar (\Psi^{*}\partial_t\Psi -\partial_t\Psi^{*}\Psi )-\frac{\hbar^2}{2m}\Psi^{*}\overleftarrow{\nabla}\cdot \overrightarrow{\nabla}\Psi -V\left( \mathbf{x}\right) \Psi^{*}\Psi .
\varphi_{h}=A_{0}+B_{0}\theta+C_{0}lnr+\sum_{n=1}^{\infty}(A_{n}r^{n}+B_{n}r^{-n})cosn\theta+\sum_{n=1}^{\infty}(C_{n}r^{n}+D_{n}r^{-n})sinn\theta,
{\frac{d \sigma_{\pi}}{d E_{q}d \varphi_{q}d \varphi_{p}d q_{3}}}\approx{\frac{e^{2}\; \sin^{2}\pi \delta}{2 5 6 \pi^{4}}}( 1 - s_{3}r_{3}){\frac{c^{- \delta}( 1 + s_{3}) + c^{\delta}( 1 - s_{3})}{M^{3}}}.
(\widetilde{Sq}^0_*)_{(5,9)}= (QP_5^0)_{23}\bigoplus \big((\widetilde{Sq}^0_*)_{(5,9)}\cap (QP_5^+)_{23}\big), &(\widetilde{Sq}^0_*)_{(5,9)}\cap (QP_5^+)_{23}= \bigoplus_{j=1}^4 QP_5^+(\omega_{(j)}).
\int_{-\infty}^{0}\frac{1+s^{2}}{(x-s)^{2}}\,d\sigma_{\mu_{n}}(s) =nb_n^2\int_{-\infty}^0\frac{1+s^2}{(x-b_ns)^2}\;d\sigma_\mu^-(s) &\leq\int_{-\infty}^{-1}\frac{2ns^{2}}{(x/b_n-s)^{2}}\,d\sigma_{\mu}^-(s).
\Big\|\sup_{N\ge 1}\big|\sum_{n=1}^N \alpha_n D(\lambda_n t)\big|\, \Big\|_{\S^2}\le C \Big(\sum_{n\ge 1}|\beta_n|\Big) \Big(\sum_{n\ge 1}\big(\sum_{k\,:\, n\le \lambda_k< n+1}|\alpha_k|\big)^2\Big)^{1/2}\, .
\begin{aligned}-\partial_x^3w_1+\frac{h^0_e}{u^0_e}\partial_x^3w_2 =\partial_{xYY}w_1-\frac{h^0_e}{u^0_e}\partial_{xYY}w_2+(-\Delta \partial_xw_1+\frac{h^0_e}{u^0_e}\Delta \partial_xw_2) &=\partial_{xYY}w_1-\frac{h^0_e}{u^0_e}\partial_{xYY}w_2+\partial_xG_e.\end{aligned}
g_{1}(\nabla^{^{M_1}}_{X}Y,W) =\cos^{2}\theta g_{1}(\nabla^{^{M_1}}_{X}Y,W)-g_{1}(\pi_*(\mathcal{H}\nabla^{^{M_1}}_{X}Y),\pi_*(\omega\phi W))+g_{1}(\mathcal{A}_{X}\mathcal{B}Y,\omega W) &+g_{1}(\pi_*(\mathcal{H}\nabla^{^{M_1}}_{X}\mathcal{C}Y),\pi_*(\omega W))-\eta (Y)g_{1}(X,\omega W)
\alpha^*\tilde \Theta \cdot v_q = \tilde \Theta(\alpha_q) \cdot T\alpha(v_q) = \alpha_q \cdot T\pi_{T^*Q}\circ T\alpha(v_q) = \alpha_q \cdot T(\pi_{T^*Q}\circ \alpha)(v_q) = \alpha_q \cdot v_q,
\begin{array}{l}{{\{\Omega_{n}^{a}, \phi_{n^{\prime}}^{b}\}\approx 0 \a \ne b}}\\{{\{\Omega_{n}^{a}, \phi_{n^{\prime}}^{a}\}\approx 0 \n^{\prime}< n}}\\{{\{\Omega_{n}^{a}, \phi_{n}^{a}\}\approx ( - 1 )^{N_{a}- n}\eta^{a}( q , p )}}\\ \end{array}
S^{2n+1}_{\varepsilon_1, \varepsilon_2, a}=\big\{(x, y) \in{\mathbf{R}}_r^{n+2}\times{\mathbf{R}}_r^{n+2}\,|\, x\cdot x = \varepsilon_1, \, y\cdot y = \varepsilon_2, \,x\cdot y = a , \, x \not= 0, \, y \not= 0 \big\},
||\gamma^V(t)||^2\leq -\lambda_1||\gamma^V(t)||^2_{L^2(\mathbb{R}^N)}\leq -\lambda_1||\gamma(t)||^2_{L^2(\mathbb{R}^N)}= -\lambda_1(tL)^N||\tilde{u}||^2_{L^2(\mathbb{R}^N)}.
|\tilde{r}| \le \sum_{j=1}^k p^j \sum_{i=0}^{j-1}\binom{j}{i}C(k)^{i}\kappa(k)^{j-i}\|S_k-I_k\|^{(j-i)(k+3)+i}& \quad\le p^{k+1}\|S_k-I_k\|^{k+3}\sum_{j=1}^k\sum_{i=0}^{j-1}\binom{j}{i}C(k)^{i}\kappa(k)^{j-i}.
{\bf P}={\bf p},\quad{\bf M}={\bf p}\times{\bf x}+\sum_{i=1}^N{\bf p}_i\times{\bf e}_i={\bf p}\times{\bf x}+\sum_{i=1}^N{{\bf p}^\bot}_i\times{\bf e}_i -\frac 12\sum_{i,j=1}^N{\phi}_{ij}{\bf e}_i\times{\bf e}_j.
\begin{cases}\Delta_g u_\alpha+m_0^2 u_\alpha=u_\alpha^{2^\star-1}+\omega_\alpha^2(1-qv_\alpha)^2 u_\alpha, &\Delta_g v_\alpha+(m_1^2+q^2u_\alpha^2) v_\alpha= q u_\alpha^2,\end{cases}
H_{R_t(X_0^{N}(t))}(x)-H_X(x)= H_{X_0^{N}(t)}(x-t)-H_{R_{-t}(X)}(x-t) \Rightarrow d_H\bigl(R_t\bigl(X_0^{N}(t)\bigr),X\bigr)=d_H\bigl(X_0^{N}(t),R_{-t}(X)\bigr) \Rightarrow \min_{t\in\mathbb{R}}d_H\bigl(R_t\bigl(X_0^{N}(t)\bigr),X\bigr)=\min_{t\in\mathbb{R}}d_H\bigl(X_0^{N}(t),R_{-t}(X)\bigr).
K_{p}( z ) \stackrel{( z \rightarrow 0 )}{\sim}\frac{1}{2}\left[ \Gamma ( p ) \left( \frac{z}{2}\right)^{- p}+ \Gamma ( - p ) \left( \frac{z}{2}\right)^{p}\right] \left[ 1 + O ( z^{2}) \right]
\inf_{\substack{\varphi(t) \in C_{0T}([0,T], \mathbb{R}^d) \varphi(0)=x,\,\varphi(T)=y}}\biggl \{S_{0T}^{\mathcal{K}}(\varphi(t))\, \Bigl \vert \,\varphi(t) \in D_{\pm\delta}\cup \partial D_{\pm\delta}, \, \, \forall t \in [0, T] \biggr\},
a^+(\varphi)f^{(n)}:=\varphi\otimes f^{(n)}, a^-(\varphi) f_1\otimes\dots\otimes f_n :=\sum_{i=1}^n q^{i-1}(\varphi,f_i)_\mathcal{H}\, f_1\otimes\dots\otimes \check f_{i}\otimes \dots\otimes f_n
f(\lambda(A_{g_\varphi})) = \varphi^{-\frac{4}{n-2}}\chi_1 f(\chi_1^{-1}(\chi_1 - \chi_2),1, \ldots, 1) + O( \varphi^{-\frac{4}{n-2}}) &= \varphi^{-\frac{4}{n-2}}\chi_1 f(-1,1, \ldots, 1) + O( \varphi^{-\frac{4}{n-2}}) &\geq \frac{1}{Cr^2}\varphi^{-\frac{4}{n-2}}\{0 < r < r_1\},