harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
y_0 x-\nu_2(y_0)= \left( \frac{b-a}{V(P)}\right)b - \left[ \left(\frac{b-a}{V(P)}\right)a+ \left(\frac{b-a}{V(P)}\right)^2 \frac{V(P)}{2}\right]=\frac{(b-a)^2}{2V(P)}.
y^{2}t^{2}+ t ( y^{2}- a_{1}^{2}) ( y^{2}- a_{2}^{2}) \ldots ( y^{2}- a_{n}^{2}) + \Lambda_{S U}^{2 b_{0}^{S U}}\Lambda_{S O}^{b_{0}^{S O}}y^{2}= 0 .
\Lambda_{I R}^{\prime +}( p , p ) = -{\frac{\lambda e^{3}}{2 \pi^{3}}}\int{\frac{d k^{+}}{k^{+}}}\int d^{2}k_{\perp}{\frac{\Theta_{\Delta}( k )}{k_{\perp}^{2}+{\frac{m^{2}k^{+ 2}}{p^{+ 2}}}}}\biggl [ 1 -{\frac{\frac{m^{2}k^{+ 2}}{2 p^{+}}}{k_{\perp}^{2}+{\frac{m^{2}k^{+ 2}}{p^{+ 2}}}}}\biggr ] \; .
M_{11}=\left(\begin{array}{ccc}1 * 0 * 0 0 * * 0\end{array}\right),\,\,M_{12}=\left(\begin{array}{ccc}0 * 1 * 1 * * * 0\end{array}\right), M_{21}=\left(\begin{array}{ccc}0 0 0 * 0 * * * 0\end{array}\right),\,\,M_{22}=\left(\begin{array}{ccc}0 0 0 * 0 * * * 1\end{array}\right).
\alpha \cdot e_i= \alpha\cdot (\overline{\omega_i-\omega_1})=\overline{\alpha\cdot (\omega_i-\omega_1)}=\overline{\omega_{\alpha(i)}-\omega_{\alpha(1)}}=\overline{\omega_{i+1}-\omega_2}=\overline{(\omega_{i+1}-\omega_1)-(\omega_2-\omega_1)}= e_{i+1}-e_2.
\Phi_{pq}(X) = \sum_{i=0}^{r}\sum_{j=0}^{s}X^{ip+jq}- \sum_{i=r+1}^{q-1}\sum_{j=s+1}^{p-1}X^{ip+jq-pq}= \sum_{i=0}^{[p^{q-2}]_q- 1}\sum_{j=0}^{[q^{p-2}]_p- 1}X^{ip+jq}- \sum_{i= [p^{q-2}]_q}^{q-1}\sum_{j=[q^{p-2}]_p}^{p-1}X^{ip+jq-pq}
I(\lambda)\equiv\int_{I_{x}}f(x)e^{\lambda S(x)}dx=(\frac{2\pi}{\lambda})^{\frac{n}{2}}e^{\lambda S(x^{0})}(f(x^{0})+O(\lambda^{-1}))\prod_{j=1}^{n}(-\mu_{j})^{-\frac{1}{2}},\lambda\rightarrow\infty,
\lim_{t\rightarrow \infty}t^{\frac{d-2\delta}{\alpha}(1-\frac{1}{q})}\| u(t, \cdot)-A\Psi_t\|_{q,h}= \lim_{t\to \infty}t^{\frac{d-2\delta}{\alpha}(1-\frac{1}{q})}\|u(t+1,\cdot) -A\Psi_{t+1}\|_{q,h}&= \lim_{t\to \infty}t^{\frac{d-2\delta}{\alpha}(1-\frac{1}{q})}\|\tilde P_t\left( \tilde P_1f-A\Psi_1\right)\|_{q,h}=0.
t_1 =-\frac{(x_1x_2+1)(x_1x_3-1)(x_2x_3-1)}{(x_1-x_3)(x_1x_2^2x_3-1)}, t_2 =\frac{(x_2x_3-1)(1-2x_1+x_1x_2+x_1x_3-2x_1x_2x_3+x_1^2x_2x_3)}{(x_1x_2-1)(1-2x_3+x_1x_3+x_2x_3-2x_1x_2x_3+x_1x_2x_3^2)}, t_3 =-\frac{(x_1x_2-1)(x_1x_3-1)(x_2x_3+1)}{(x_1-x_3)(x_1x_2^2x_3-1)}.
\binom{|A^{\ast}|}{w_1}\binom{\ell}{w_2}^{\rho}\!\!\leq \exp \left[ |A^{\ast}| H_2 \Big( \frac{w_1}{|A^{\ast}|}\Big) + \rho \ell H_2 \left( \frac{w_2}{\ell}\right) \right].
\lim_{t\rightarrow\infty}u_{2k-1}^{(t)}=\frac{\check{c}_k^{(1)}\check{c}_{k-1}^{(0)}}{\check{c}_k^{(0)}\check{c}_{k-1}^{(1)}}, &\lim_{t\rightarrow\infty}u_{2k}^{(t)}=0.
\alpha_{- 1}^{\mu}\bar{\alpha}_{- 1}^{2 5}\, | \Omega ; p > \ \ , \ \ \alpha_{- 1}^{2 5}\bar{\alpha}_{- 1}^{\mu}\, | \Omega ; p > \ \ , \ \ \alpha_{- 1}^{2 5}\bar{\alpha}_{- 1}^{2 5}\, | \Omega ; p > \ ,
Y_n(r) = \frac{2}{\pi}\left[ \log\left(\frac{r}{2}\right) + \gamma \right] - \frac{1}{\pi}\sum_{k=0}^{n-1}\frac{(n-1-k)!}{k!}\left( \frac{2}{r}\right)^{n-2k}-\frac{1}{\pi}\sum_{k=0}^{+\infty}\frac{(-1)^k}{k!(n+k)!}\left( \frac{r}{2}\right)^{2k+n}\left[ \psi(k+n)+\psi(k) \right],
\langle f,g\rangle=\int_{[0,2\pi]^{n}}f(e^{i\theta_{1}},...,e^{i\theta_{n}})\overline{g(e^{i\theta_{1}},...,e^{i\theta_{n}})}\prod_{1\le j<k\le n}|e^{i\theta_{j}}-e^{i\theta_{k}}|^{\frac{2}{\alpha}}d\theta_{1}\cdot\cdot\cdot d\theta_{n}
V_{\mathcal{F}_{k l,I}}^{en}:=\mathrm{span}\{\xi^i_{\mathcal{F}_{k l,I}}\}_{i=1}^{n_{k l}}\mbox{and}V_{\mathcal E_k}^{en}:=\mathrm{span}\{\xi^i_{\mathcal E_k}\}_{i=1}^{m_k},
\ \|(\langle x_1, x \rangle+\langle \xi_1,D_x \rangle)\ ...\ (\langle x_m, x \rangle+\langle \xi_m,D_x \rangle)e^{-tq^w}u\|_{L^2}\leq \C_5\tilde{c}^m(m!)^{\frac{2k_{0}+1}{2}}\Big(\prod_{j=1}^{m}|X_j|\Big)e^{-\omega_0 (t-1)}\|u\|_{L^2}\leq \ \tilde{C}^m(m!)^{\frac{2k_{0}+1}{2}}\Big(\prod_{j=1}^{m}|X_j|\Big)e^{-\omega_0 t}\|u\|_{L^2}.
H_{B}^{\mathrm{b a r e}}= \frac{1}{2}\sum_{k = 1}^{\infty}\Bigl ( - \sigma_{k + 1}^{x}\sigma_{k}^{x}- \sigma_{k + 1}^{y}\sigma_{k}^{y}+ \sigma_{k + 1}^{z}\sigma_{k}^{z}\Bigr ) + h \sigma_{1}^{z},
Q= \left(\begin{array}[c]{cc}\Lambda^{-1}0 0 \Lambda^{-1}\end{array}\right),\;\;\;\; Q h = \Big(\begin{array}[c]{c}\Lambda^{-1}h_1 \Lambda^{-1}h_2\end{array}\Big),\;\;\; h = (h_1, h_2 ) \in H = U \times V'.
\lim_{k\to \infty}\left( \left[ \prod_{j=k}^{k+n-1}w^j \right]\left(x_{k+n}-l(x)\right)+\left[ \prod_{j=0}^{n-1}w^j \right]\left(x_{n}-l(x)\right) \right) &=\left[ \prod_{j=0}^{n-1}w^j \right]\left(x_{n}-l(x)\right).
A_\infty = D_1^2 + \sum_{1\le j\le p}\bigl( D_{2j}^2 + (D_{2j+1}+f_j x_{2j})^2\bigr). \shortintertext{Then}A_\infty-f_\infty = D_1^2 +\sum_{1\le j\le p}Z_j^* Z_j \shortintertext{with}Z_j= iD_{2j}+ (D_{2j+1}+f_j x_{2j}).
T_{j}( s > 0 ) = \left( \begin{array}{c c}{- i \rho_{j}/ 2 s + O ( 1 )}&{s^{( k - 3 ) / 2}p_{j}+ O ( s^{( k - 1 ) / 2})}\\{- s^{( k - 3 ) / 2}\bar{p}_{j}^{T}+ O ( s^{( k - 1 ) / 2})}&{T_{j}( 0 - ) + O ( s )}\\ \end{array}\right) ,
s_2(a + b) = \sum_{i<j}\alpha_i(a+b)\alpha_j(a+b) = &= \sum_{i<j}\alpha_i(a)\alpha_j(a) + \sum_{i<j}\alpha_i(b)\alpha_j(b) +\sum_{i\neq j}\alpha_i(a)\alpha_j(b)= &= s_2(a) + s_2(b) + \sum_{i\neq j}\alpha_i(a)\alpha_j(b)
\theta_s^1( a_pa_0\otimes a_1\otimes a_2\otimes\cdots\otimes a_{p-1}) ={\rm Tr}\left(\Gamma a_0\left(\prod_{l=1}^{m-2}b_l(a_l)\right)\delta^2(a_{m-1})\left(\prod_{l=m}^{p-1}b_{l+1}(a_l)\right)[T,a_p]\right)+Y^1_p.
\Gamma_{jk|v}(z_j,s)=\sum_{\beta=1}^n \frac{\partial z_j}{\partial z_k^{\beta}}g_{k|v}^{\beta}(z_k,s)-g_{j|v}(z_j,s)+\sum_{u=1}^m \left(\frac{\partial g_{jk}(z_k,t)}{\partial t_u}\right)_{t=0}h_{u|v}(s)
\begin{array}{r l}\hat{g}_{ab}=u^{q-2}g_{ab}\hat{K}_{ab}=u^{-2}[\frac{\tilde{N}}{2}(\mathcal{L}_gW)_{ab}+U_{ab}]+\frac{1}{n}u^{q-2}g_{ab}\left(\tau^*+\tilde{N}div_g(V+Q)\right).\end{array}
\int_{G^2}{P^l(x_1,x_2){d^2}_{{\mu}_{G^2}}\left(x_1,x_2\right)}&=\int_{G^2}{P^l_i\left(x_1\right)P^l_j\left(x_2\right){d^2}_{{\mu}_{G^2}}\left(x_1,x_2\right)}&={\Phi}_1\left(0\right){\Phi}_2\left(0\right)=1,
S_{\mathrm{q Y M}}^{\mathrm{r e d .}}= -{\frac{1}{4}}\left({\frac{2 \pi}{a}}\right)^{4}{\frac{N}{g_{\mathrm{Y M}}^{2}}}\mathrm{T r}{\mathrm{\boldmath{~ F ~}}}_{\mu \nu}{\mathrm{\boldmath{~ F ~}}}^{\mu \nu},
\overline{s}(x)=\left.\arraycolsep=0.6pt\def\arraystretch{1.4}\begin{array}{|c|c|c|c| c|c|c|c| c|c|c|c| c|c|c|c|}\hline16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 \hline 3 3 1 2 2 3 1 2 3 2 2 3 1 3 3 3 \hline\end{array}\right..
H_{\rm free}=\int\frac{1}{2}\left\{|\pi(\stackrel{\rightharpoonup}{x})|^2+|\bigtriangledown\varphi(\stackrel{\rightharpoonup}{x})|^2+m|\varphi(\stackrel{\rightharpoonup}{x})|^2\right\}d^3\stackrel{\rightharpoonup}{x}\ \ \ .
\mu_1 =\max_{f>0}\frac{\big|H\big|\cdot\biggl(\sum_{v\in V}f(v)\biggr)^2}{\big|H\big|\cdot\sum_{v\in V}\cdot f(v)^2}&=\max_{f>0}\frac{\biggl(\sum_{v\in V}f(v)\biggr)^2}{\sum_{v\in V}f(v)^2}&=\mu_1',
\int_{R}y^\gamma|\nabla(u-\tilde u)|^2 =-\int_{R}\left[(u-\tilde u)(y^\gamma(u-\tilde u)_{x})_{x}+(u-\tilde u)(y^\gamma(u-\tilde u)_{y})_{y}\right] =-\int_{R}(u-\tilde u)[y^\gamma\Delta(u-\tilde u)+\gamma y^{\gamma-1}(u-\tilde u)_{y}] =I_1+I_2,
\delta \int_M \mathcal{E}(H,K) \, dS &= \int_M \bigg(\frac{1}{2}\mathcal{E}_H+ 2H\mathcal{E}_K\bigg) \Delta u + \bigg((2H^2-K+2k_0)\mathcal{E}_H+2HK\mathcal{E}_K-2H\mathcal{E}\bigg)u &\phantom{\int_M h}- \mathcal{E}_K \langle h, \,u \rangle \,\, dS,
(\frac 12[\xi^{*},\xi ])^n=n!\;\sum_{i_1<\cdots <i_n}\xi_{i_1}^{*}\cdots\xi_{i_n}^{*}\xi_{i_1}\cdots \xi_{i_n},\;\cdots ,(\frac 12[\xi^{*},\xi])^p=p!\;\xi_1^{*}\cdots \xi_p^{*}\xi_1\cdots \xi_p,
f_{\gamma_z}(\gamma)=A_xA_y\sum^\infty_{k=0}\sum^k_{n=0}\tilde{B}_x(n)\tilde{B}_y(k-n)\Big(a^{n+1}_x\gamma^ne^{-a_x\gamma}&\times\Gamma\left(k-n+1,a_y\gamma\right)+a^{k-n+1}_y\gamma^{k-n}e^{-a_y\gamma}\Gamma\left(n+1,a_x\gamma\right)\Big).
\mathop{\lim}\limits_{k \to +\infty}\frac{k}{MN}\mathbb{E}\left[{\left\| \hat{\textbf{h}}_k - \textbf{h}\right\|}_2^2 \bigg| \hat{\boldsymbol{\psi}}_k \to \boldsymbol{\psi}\right] ={C}_S^{\min}(\boldsymbol{\psi}).
\delta P_{\mu ( 1 )}^{A}= \frac{1}{2 \pi \alpha^{\prime}}\sqrt{- \bar{G}}\left( ( \bar{D}_{i j}^{A}\Phi_{( 1 )}^{j}) \bar{n}_{\mu}^{i}+ ( \bar{\Omega}_{i}^{A B}- \bar{G}^{A B}\bar{\Omega}_{i C}^{C}) \Phi_{( 1 )}^{i}\bar{x}_{\mu , B}\right)
\mathbf{a}_1= (\mathbf{A}_1)^T = [\mathbf{a}_{11}^T,\mathbf{a}_{12}^T, \cdots, \mathbf{a}_{1n_2}^T], &\vdots \mathbf{a}_M= (\mathbf{A}_M)^T = [\mathbf{a}_{M1}^T,\mathbf{a}_{M2}^T, \cdots, \mathbf{a}_{Mn_2}^T ],
\begin{aligned}\dot{y}= p_y(z,u_h)-r_y(y,u_h)+d_y(t)\mathbf{-g_y(y,x)+g_x(y,x)-d_b(t)}\dot{z}= p_z(y,z,u_h)-r_z(y,z,u_h)+d_z(t) \mathbf{\dot{x}}= \mathbf{g_y(y,x)-g_x(y,x)-r_x(x)+d_x(t)+d_b(t)}\end{aligned}
u_{ij}^{(\alpha)}u_{jk}^{(\beta)}+u_{ij}^{(\beta)}u_{jk}^{(\alpha)}+u_{k,i}^{(\alpha)}u_{ij}^{(\beta)}u_{ki}^{(\alpha)}+ u_{jk}^{(\alpha)}u_{ki}^{(\beta)}+ u_{jk}^{(\beta)}u_{ki}^{(\alpha)}=0,
\begin{array}{r c l}{\mathrm{H t r}[ T_{a}, \cdots ] = 0 ,}&{{\qquad}}&{\mathrm{H t r}[ \Sigma_{\alpha}, \cdots ] = 0 ,}\\{\mathrm{S t r}[ T_{a}, \cdots ] = 0 ,}&{{\qquad}}&{\mathrm{S t r}\{\Sigma_{\alpha}, \cdots \}= 0 ,}\\ \end{array}
\Phi_{pq}(X) = \sum_{\substack{k=0 \exists i\in[0,r], j\in[0,s] : k = ip + jq}}^{(p-1)(q-1)}X^k - \sum_{\substack{k=0 \exists i\in[r+1,q-1], j\in[s+1,p-1] : k+pq = ip + jq}}^{(p-1)(q-1)}X^k
\frac{i}{2}Tr D^{-1}G= \frac{1}{12N}\int d^Dx \left( \lambda_0+\frac{\eta_0\phi^2}{10N}\right) [G_{aa}(x,x)]^2 +\frac{3\eta_0}{6!N^2}\int d^{D}x [G_{aa}(x,x)]^3+ cte .
\int_{\mathcal{O}(B(\mathbf{y},r))^{c}}|\tau_{-\mathbf y}(f*\phi)(\mathbf x)| \, dw(\mathbf x) &\leq Cr^{-\delta'}\|f(\cdot)(1+\|\cdot\|)^{\mathbf{N}/2+\delta}\|_{L^1(dw)}\|\phi(\cdot)(1+\|\cdot\|)^{\mathbf{N}+\delta}\|_{L^{\infty}}.
\begin{aligned}c(\xi) =\sum_{k\in S^+}(- 4 c_6\,k^3-24 c_7 k+\frac{16}{3}c_2^2\,k^3+16 c_2 c_3 k )\,\xi_k begin{align*}2mm] =(\frac{16}{3}c_2^2-4 c_6) v_3\cdot \xi+(16 c_2 c_3-24 c_7) v_1\cdot \xi,\end{aligned}
( p^{+}+ \gamma^{+}- 1) \int_{M}| D \mathrm{u}( z )|^{p( z )}\,\, dv_{g}( z ) + ( q^{+}+ \gamma^{+}- 1) \int_{M}\mu ( z ) \, | D \mathrm{u}( z ) |^{q( z )}\,\, dv_{g}( z ) = \lambda ( r^{-}+ \gamma^{-}- 1 ) \int_{M}| \mathrm{u}( z ) |^{r( z )}\,\, dv_{g}( z ),
\theta^{\mu \nu}= \frac{2 \pi \alpha^{\prime}}{g_{2}g ( 1 - e^{2}) + g^{2}b^{2}}\left( \begin{array}{c c c}{{0}}&{{g_{2}e}}&{{0}}\\{{- g_{2}e}}&{{0}}&{{- g b}}\\{{0}}&{{g b}}&{{0}}\\ \end{array}\right) .
J ( x ) = \tau ( x ) + \theta_{i}^{\alpha}\zeta_{\alpha}^{i}- 4 \theta_{i j}m^{i j}( x ) + \theta^{\alpha \beta}w_{\alpha \beta}+ \vartheta_{i}^{\alpha}\kappa_{\alpha}^{i}+ 4 \theta^{4}M^{2}( x )
\|\tilde\omega - \omega\|_\pi^2 = \|\tilde\omega\|_\pi^2 - (2c-1)\|\omega\|_\pi^2 = \|\tilde\omega\|_\pi^2 - (2c-1) = \|\tilde\omega\|_\pi^2 - 1 + O\left(\frac{d}{n}\right).
\begin{array}{l l l}{E_{\pm}^{\dagger}= E_{\mp},}&{\quad \quad H^{\dagger}= H ,}&{\quad \quad \quad \mathrm{f o r}\quad s u ( 2 ) ,}\\{E_{\pm}^{\dagger}= - E_{\mp},}&{\quad \quad H^{\dagger}= H ,}&{\quad \quad \quad \mathrm{f o r}\quad s u ( 1 , 1 ) .}\\ \end{array}
R_1(v)= \sum_{j=1}^k M_{11}(e_j) = \sum_{j=1}^k M_{11}(e_j^{-1})= \sum_{j=1}^k m_j M_{12}(e_j^{-1})=m_1 \sum_{j=1}^k M_{12}(e_j^{-1}) - C,
S_{G F}^{( 0 )}= - \frac{1}{4}\int d x^{4}\sqrt{- \bar{g}}\Bigl [ \bar{g}^{\mu \nu}\bar{g}^{\lambda \sigma}f_{\mu \lambda}^{( 0 )}f_{\nu \sigma}^{( 0 )}+ \frac{I_{2}}{I_{1}}\bar{g}^{\mu \nu}a_{\mu}^{( 0 )}a_{\nu}^{( 0 )}\Bigr ] .
\frac{d^2 x^a}{d\hat s^2}+\{^{\;\;a}_{mn}\}_{hat}\frac{dx^m}{d\hat s}\frac{dx^n}{d\hat s}-\frac{\psi_{,n}}{2}(\frac{dx^n}{d\hat s}\frac{dx^a}{d\hat s}-\hat g^{na})=0,
\lambda^{p}\mu^{a_{2}p-Q}\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}u(x)|^{p}dx=\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}\phi(x)|^{p}dx
P_{\leq M}(Vu) = P_{\leq M}( P_{\leq M_0}V u) \sim \sum_{i+j \leq M, i \leq M_0}P_i V P_j u \sim \sum_{i \leq M_0}P_i V \sum_{j =0}^{M-i}P_j u \sim \sum_{i \leq M_0}P_i V P_{\leq M}u \sim P_{\leq M_0}V P_{\leq M}u = V P_{\leq M}u,
\psi_{u}(t)=\left(t^{\frac{\beta_0}{e_j}},\displaystyle\sum_{\beta_1 \leq l <\beta_{j}}c_lt^{\frac{l}{e_j}}+ ut^{\frac{\beta_{j}}{e_j}}+\displaystyle\sum_{l > \frac{\beta_{j}}{e_j}}s_l(u)t^{l}\right),
\frac54 2\frac{5^4}{4^4}a^4N^{4/4}\frac{750}{4^4}a^3N^{-1/4}(x-N)^2e^{2aN^{5/4}}\le c_0 \frac{5^3}{4^3}a^3 \frac{N^{3/4}}{2}(x-N)^2e^{2aN^{5/4}}&\le c_0\, \partial_xP_N e^{aN^{5/4}}\le c_0\, P_N \partial_xP_N ,
\mathcal{C}^{\perp_{\mathrm{s},t}}\ = \ \{(v',w') \in \mathbb{Z}_{p^a}^{2n}~:~ \langle(v',w') \mid (v,w) \rangle_{\mathrm{s}}\equiv 0 \!\!\!\! \pmod{p^{a-t}}(v,w) \in \mathcal{C}\}.
\begin{array}{c}{{8 t_{2}< O_{2}O_{k_{1}}. . . O_{k_{p - 1}}P >_{g}}}\\{{+ 4 t_{0}< O_{k_{1}}. . . O_{k_{p - 1}}P P >_{g}+ ( 4 ( p - 1 ) + 4 \Sigma ) < O_{k_{1}}. . . O_{k_{p - 1}}P >_{g}= 0 ,}}\\ \end{array}
(t-x)\int_{x-\delta}^{x}\frac{\varphi(y)dy}{(t-y)^2}=\int_{x-\delta}^{x}\frac{(t-y)\varphi(y)dy}{(t-y)^2}-\int_{x-\delta}^{x}\frac{(x-y)\varphi(y)dy}{(t-y)^2}&=\int_{x-\delta}^{x}\frac{\varphi(y)dy}{t-y}-\int_{x-\delta}^{x}\frac{(x-y)\varphi(y)dy}{(t-y)^2}.
\Delta_{0}^{( 1 )}= \frac 1{2 m^{2}}i p^{( 1 )}\left( i p^{( 1 )}- \varepsilon m \right) \left( 1 - \sigma_{p}^{( 1 ) 2}\right) = \Psi_{0}^{( 1 )}\cdot \overline{{\Psi}}_{0}^{( 1 )}
\lambda_\star:=\lambda_{\theta_\star}\quad\mbox{where}\quad\theta_\star:=\frac{(d-2)\,(n-1)\,\big(3\,n+1-d\,(3\,n+5)\big)}{(d+1)\,\big(d\,(n^2-n-4)-n^2+3\,n+2\big)}\,.
\Pi^{\rm{\rm Gvac}}_{\mu\nu\rho\sigma}(p)\sim\kappa^2\Lambda_{{\rm Gvac}}^4N_{\mu\nu\rho\sigma}(p^2)\sim\frac{\Lambda_{{\rm Gvac}}^4}{M^2_{\rm PL}}N_{\mu\nu\rho\sigma}(p^2),
\begin{array}{rl}g_{w(l)}(y)= \prod\limits_{i=1}^n h_{d_i}(ny+i) = (\prod\limits_{p=1}^r(\prod\limits_{i \in L(p)}h_{b_i}(ny+i)) \ast h_l(ny+s(p))) \ast \prod\limits_{i \in{L(r+1)}}h_{b_i}(ny+i) = (\prod\limits_{p=1}^r c_y(p) \ast h_l(z_y(p))) \ast c_y(r+1)\end{array}
|c_j| \geq (j_1 - i_0) + (j_2 - i_1) + \cdots (j_0 - i_m) & \geq (j_1 - i_0) + (j_2 - j_1) + \cdots + (j_0 - j_m) & = j_0 - i_0 = j_0 -i_0 +1 - 1 & = l(c_j) - 1.
\tilde A_{h,\tilde h}= e^{-m\hat{\tilde h}s_{\tilde h}}D_{s_{\tilde h}}^j \hat{\tilde h}^{m-j}P_{m-j}(h,h e^{\hat{\tilde h}s_{\tilde h}}\hat x',y,\hat{\tilde h}{}^{-1}D_{Y_{\tilde h}})\bigl(\delta(s_{\tilde h})\delta(Y_{\tilde h})\bigr)\cdot\hat{\tilde h}{}^{-n}.
\int_{X_E}r_l(V)\begin{pmatrix}g 0 0 1\end{pmatrix}r_l(\mathcal{S})(g) \,dg = \int_{X_F}r_l(V)\begin{pmatrix}g 0 0 1\end{pmatrix}r_l(\mathcal{S})(g) \,dg
\mathcal{E}=C+a_1(\mathcal{T}_1-\mathcal{T}_\delta)^2+a_2(\mathcal{T}_2-\mathcal{T}_\delta)^2+2b_1\mathcal{T}_1+2b_2\mathcal{T}_2+2c_{12}(\mathcal{T}_1-\mathcal{T}_\delta)(\mathcal{T}_2-\mathcal{T}_\delta),
\varphi_{\xi^3}=\;\frac{1}{9}(3\varphi\varphi_{\xi^2}+6\varphi_{\xi}^2+2\varphi^2\varphi_{\xi}), \varphi_{\xi^4}=\;\frac{\varphi^2+5\varphi_{\xi}}{3}\varphi_{\xi^2}+\frac{2\varphi(\varphi^2+9\varphi_{\xi})}{27}\varphi_{\xi}.
M_p(T) \,=\, \sup_{0 < t \le T}t^{1-\frac1p}\|S(t)\omega_0\|_{L^p (\Omega)}~, \hbox{and}N_p(T) \,=\, \sup_{0 < t \le T}t^{1-\frac1p}\|\omega_\theta(t)\|_{L^p (\Omega)}~.
\ln \mbox{det}= \frac{1}{2}\int_0^\infty \frac{dt}{t}\left\{\mbox{Tr}\left(\!\e^{-P^2t}- \exp[-(D^2 + \frac{1}{2}\sigma F) t\!\ ]\!\ \right)+ \frac{\|F\|^2}{24\pi^2}\right\}e^{-tm^2}.
{\Phi}^\ell={\mathcal{M}}^{\ell+1}{\Phi}^{\ell+1},{\mathcal{M}}^{\ell+1}=\left(\begin{array}{cccc}\mathbb{I}_p \mathcal{B}_0^{\ell+1}-\mathcal{A}_1^\ell \mathbb{I}_p &\mathcal{B}_1^{\ell+1}-\mathcal{A}_2^\ell \mathbb{I}_p & \ddots \ddots\end{array}\right).
[f]_{\gamma,\alpha}=\sup_{x\not=y}\, \min(1+|x|,1+|y|)^{\gamma+\alpha}\frac{|f(x) - f (y)|}{|x-y|^\alpha}, \|f\|_{\gamma,\alpha}= \| (1+|x|)^\gamma f\|_{L^\infty}+ [f]_{\gamma,\alpha},
\| \nabla (u-M^{-1}) \|_{L^2(S^2)}^2 = \| \nabla u \|_{L^2(S^2)}^2 + 2 \langle \nabla u, \nabla M^{-1}\rangle_{L^2(S^2)}+ \| \nabla M^{-1}\|_{L^2(S^2)}^2
X_{t}=e^{-bt}\left(X_{0}+a\int_{0}^{t}e^{bs}\mathrm{d}s+\sigma\int_{0}^{t}e^{bs}\sqrt{X_{s}}\mathrm{d}B_{s}+\int_{0}^{t}e^{bs}\mathrm{d}J_{s}\right),t\geqslant0.
v = \frac{1}{5}\left[ \frac{1}{1 2}v_{0}^{\prime \prime \prime}+ \frac{1}{1 2}u_{0}^{\prime \prime}v_{0}^{\prime}+ \frac{1}{4}u_{0}^{\prime}v_{0}^{\prime \prime}+ \frac{1}{6}\left( u_{0}^{\prime}\right)^{2}v_{0}^{\prime}- \frac{8}{2 7}\left( v_{0}^{\prime}\right)^{3}\right] \quad e t c \quad ,
\lim_{\Omega\to 0}\Phi^{\Delta,J}(t,\bar t,\psi; \tau, \bar\tau, \phi) =e^{i(\mu\hat\psi + \lambda \hat\phi)}\, \sum_{m,n=0}^\infty c_{mn}t^m \tau^n e^{-\frac{1}{2}|t|^2-\frac{1}{2}|\tau|^2}\,.
g(q, p, t) = \sum_{n=1}^N\left[i \lambda \epsilon_n t - \mu_n{\cal F}^{(n)}\cdot q^{(n)}+ \mu_n \left( \frac{p^{(n)}}{\sqrt{m_n}}- \nu_n{\cal F}^{(n)}\right)^2 \right],
\xi\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u},\,\,\,\,\,\xi^2 \frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u},\,\,\,\,\,\wp^{(k)}(u-p)\xi^2\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u}\in H^0(S,\wedge^2 \Theta_S),\,\,\,\,\,k=0,...,n-2,
U(x) = \overline\psi^{a}(x) \psi^{a}(x) \overline\psi^{b}(x) \psi^{b}(x) - \overline\psi^{a}(x) \gamma_{s}\psi^{a}(x) \overline\psi^{b}(x) \gamma_{s}\psi^{b}(x) .
E_{\alpha}= (-1)^{\alpha_1}\sigma_{\alpha}(0)\sum_j\prod_{k \neq j}\frac{\sigma_{\alpha}(z_{jk})}{\sigma_0(z_{jk})}\left [ \frac{l}{n}\sum_{i \neq j}\frac{\sigma_{\alpha}^\prime(z_{ji})}{\sigma_{\alpha}(z_{ji})}- \partial_j \right ],\alpha \equiv (\alpha_1, \alpha_2) \neq (0, 0) \equiv (n, n),
K_2 (z_2 )R_{21}(z_1 +z_2 )K_1 (z_1 )R_{12}(z_1 -z_2 )=R_{21}(z_1 -z_2 )K_1 (z_1 )R_{12}(z_1 +z_2 )K_2 (z_2 ).
W_{f i}[ K ] = \frac{1}{\sqrt{2 \pi \hbar}}\exp \left\{- \frac{i}{\hbar}\left( p_{f}q_{i}+ \sum_{j = 1}^{N}\epsilon H ( p_{f}- \epsilon K_{j - 1}- \sum_{l = 1}^{j}\epsilon J_{l}) \right) \right\}.
\begin{array}[c]{cl}\dfrac{\mathbf{H}\times\mathbf{C}}{\mathbf{Z}[\overline{T}]\rtimes U}=X \downarrow \dfrac{\mathbf{H}\times\mathbf{C}}{\mathcal{O}_{K}\rtimes\mathcal{O}_{K}^{\times,+}}=X(K,\mathcal{O}_{K}^{\times,+})\end{array}
{\frac{R R^{\prime 2}}{N^{2}}}- R -{\frac{1}{1 2}}{\frac{R^{3}}{N^{2}}}D^{\prime 2}+{\frac{Q^{2}}{4 8 \pi^{4}}}{\frac{1}{R^{3}}}e^{{\frac{2}{\sqrt 3}}D}= 0 ,
\langle T^k({\bf 1}_{\Delta}),{\bf 1}_{\Delta}\rangle=\sum_{i_1,...,i_k\in\{1,...,m\}}\mu(\{(x\in X|\psi_{i_k}...\psi_{i_1}(x)=x\})\geq\sum_{i_1,...,i_k\in\{1,...,m\}}\mu(A_{i_1,...,i_k}).
{\rm Hilb}\big(4T_{3}^{0},t\big)=[2]^{2}[3],{\rm Hilb}\big(4T_{4}^{0},t\big)=(1,6,19,42,70,90,87,57,23,6,1),{\rm Hilb}\big(\big(4T_{3}^{0}\big)^{!},t\big)(1-t)=(1,2,2,1),{\rm Hilb}\big(\big(4T_{4}^{0}\big)^{!},t\big)(1-t)^2=(1,4,6,2,-4,-3),{\rm Hilb}\big(\big(4T_{5}^{0}\big)^{!},t\big)(1-t)^2=(1,8,26,40,24,-3,-6).
n_1(T)+n_2(T)+n_3(T)+n_4(T) = n, n_1(T)+2n_2(T)+3n_3(T)+4n_4(T) = 2(n-1), m_{12}(T)+m_{13}(T)+m_{14}(T) = n_1(T), m_{12}(T)+2m_{22}(T)+m_{23}(T)+m_{24}(T) = 2n_2(T), m_{13}(T)+m_{23}(T)+2m_{33}(T)+m_{34}(T) = 3n_3(T), m_{14}(T)+m_{24}(T)+m_{34}(T)+2m_{44}(T) = 4n_4(T).
\Psi_{bv}(\boldsymbol{y}) = \begin{pmatrix}\sqrt{l_{e_{2}}\dots l_{e_{N}}}\psi_{e_{1}\dots e_{N}}(0,l_{e_{2}}y_{1},\dots,l_{e_{N}}y_{N-1}) \sqrt{l_{e_{2}}\dots l_{e_{N}}}\psi_{e_{1}\dots e_{N}}(l_{e_{1}},l_{e_{2}}y_{1},\dots,l_{e_{N}}y_{N-1}) \end{pmatrix},
(\psi_- + \phi_-+\tau\zeta_-)(\psi_+ + \phi_+ + \tau\zeta_+)-(\phi_-+\tau\zeta_-)(\phi_+ +\tau\zeta_+) = (\psi_-+\phi_-)(\psi_+ + \phi_+) -\phi_-\phi_+ + \tau(\zeta_-\psi_+ + \zeta_+\psi_-) +\mbox{h.o.}
T_{+}\mathrm{d}S_{+}+ T_{-}\mathrm{d}S_{-}= \mathrm{d}E_{+}+ \mathrm{d}E_{-}- \Phi_{+}\mathrm{d}Q_{+}- \Phi_{-}\mathrm{d}Q_{-}+ ( p_{+}+ p_{-}) \mathrm{d}V .
M_{1,k}=\int_{-\infty}^{+\infty}x^{k}\varphi(x)dx=-\int_{-\infty}^{+\infty}x^{k-1}\varphi^{\prime}(x)dx=(k-1)\int_{-\infty}^{+\infty}x^{k-2}\varphi(x)dx=(k-1)M_{1,k-2}
Q'(s) = \frac{Q'(0)e^{f(s)}}{b(s)^{2n+1}}+ (2n+2)\frac{e^{f(s)}}{b(s)^{2n+1}}\int_{0}^s b(t)^{2n-1}e^{-f(t)}\left[Q(t)^3-Q(t)\right]\,\mathrm{d}t.
x_{n-1}-x_{n}=\frac{A_{n-1}}{B_{n-1}}-\frac{A_{n}}{B_{n}}=(-1)^{n}\frac{a_{1}a_{2}\cdot\cdot\cdot a_{n}}{B_{n}B_{n-1}}=\frac{\prod_{i=1}^{n}(-a_{i})}{B_{n}B_{n-1}}.
|\frac{1}{n}\log Pr((X^n,Y^n) = (x^n,y^n)) + H(X,Y)| < \delta |\frac{1}{n}\log Pr(X^n=x^n) + H(X)| < \delta |\frac{1}{n}\log Pr(Y^n=y^n)+ H(Y)| < \delta
h^{1,2}(Z_\Delta) = h^{2,1}(Z_{\Delta}) = \sum_{F \in \Delta[2]}\ell^*(F) h^{1,1}(Z_\Delta) = h^{2,2}(Z_\Delta) = 2\ell(\Delta^\circ) - 5 - \sum_{F \in \Delta^\circ[2]}\ell^*(F) + \sum_{F \in \Delta^\circ[1]}\ell^*(F) \ell^*(F^\circ)
u_{l,m,n}(x)= \frac1{\sqrt{2\pi R^2L\omega_{l,m,n}}J_{l+1}\left(\alpha_m^{(l)}\right)}J_l\left(\frac{\alpha_m^{(l)}}Rr\right) e^{-i\omega_{l,m,n}t+il\theta +i\frac{n\pi}{2L}z}
\left| \sum_{k = k_\ell}^n \log \frac{|T'(x_k)|}{|T'(y_k)|}\right| \le \left| \log \frac{| T'(x_{k_\ell}) |}{| T'(y_{k_\ell}) |}\right| + \left| \log \frac{(T^p)' (x_{k_\ell +1})}{(T^p)' (y_{k_\ell +1})}\right|,
\P_{x+h(s)}(\tau_{h(s)}> u_0) = \P_{x}(\tau^{(h(s))}_{0}> u_0) \geq \P_x(\tau^{(h(0))}_0 > u_0) &\geq \frac{\P_x(\tau^{(h(0))}_0 > u_0)}{\P_x(\tau^{(0)}_{u \to -Lu}> u_0)}\P_{x+h(s)}(\tau_{u \to h(s)-Lu}> u_0).
|h(z) - h_{\beta}(z)| = |g \circ f(z) - g \circ \Gamma(f, \beta)(z)| \leq 3C_1(g) \|f(z) - \Gamma(f, \beta)(z)\|_{\infty}&\leq 3C_1(g) \cdot \frac{1}{\beta}\log s,
A = \frac{\pi}{4}\left( L^{2}- \frac{1}{\mu^{2}}\right) + \frac{\pi \left( y^{2}+ z^{2}\right)}{\ln \left( \mu L \left( 1 - \frac{4 \left( w^{2}+ x^{2}\right)}{L^{2}}\right) \right)}
\sum_{r,s=1}^n [\frac{\partial h_{rs}}{\partial \bar{z}_v}+\sum_{c=1}^n (h_{cs}\frac{\partial \varphi^{r}_v}{\partial z_c}-\varphi^c_v\frac{\partial h_{rs}}{\partial z_c}+h_{r c}\frac{\partial \varphi^{s}_v}{\partial z_c})]\frac{\partial \xi_j^\alpha}{\partial z_{r}}\frac{\partial \xi_j^\beta}{\partial z_{s}}=0 \,\,\,\,\,
\begin{cases}\bar h^{1^*}_{11k}+\bar h^{1^*}_{22k}=0, &\bar h^{2^*}_{11k}+\bar h^{2^*}_{22k}=0, &(\bar h^{1^*}_{11}-3\bar h^{1^*}_{22})\bar h^{1^*}_{11k}+4\bar h^{2^{\ast}}_{11}\bar h^{2^*}_{11k}=0.\end{cases}
\Omega=-H\left(t,q,p=\psi(t,q)\right){\mbox d}t+\psi(t,q)\,\mbox{d}q\stackrel{\textstyle !}{=}\mbox{d}S(t,q)=\partial_t S(t,q)\,\mbox{d}t+\partial_q S(t,q)\,\mbox{d}q\,.
\det L(x) =c\prod\limits_{1\leq i<j\leq N}\left( -\frac{(x_{i}-x_{j})^{2}}{x_{i}x_{j}}\right)^{\kappa\lambda/2},\lambda:=\frac{\gamma n_{\tau}}{2(N-1)}=\operatorname{tr}(\tau(( 1,2))) ,
F_{c}(\tilde{J}_k)-F_{c}(\tilde{J}_{k-1}) =F(\tilde{J}_k)-F(\tilde{J}_{k-1})-K_z e^{-lkh}h\|Z_{0}\|_{\infty}&\leq (M\big(F(J_k)+2\big)^2-K_z) e^{-lkh}h\|Z_{0}\|_{\infty}.
\lefteqn{\langle\xi(k^1)\xi(k^2)\xi(k^3)\xi(k^4)\rangle}&=\left\{\begin{array}{cl}\langle|\xi(k^1)|^2|\xi(k^2)|^2\rangle \mbox{for}\;\{-k^1,-k^2\}=\{k^3,k^4\}0 \mbox{if $\{k^1,k^2,k^3,k^4\}$ is not composed}begin{align*}-0.5ex] \mbox{of two pairs that sum to zero}\end{array}\right\}.
\begin{array}{r@{\}l}\left( 1 - \alpha^{3n-3j}\right) \kappa^n_{3n-2j, j, j}= \displaystyle \alpha^{3n-5j-3}\left( \sum_{\nu=0}^{j}\alpha^{3\nu}\right) \kappa^n_{3n-2j-2, j+1, j+1}& = \displaystyle \alpha^{3n-5j-3}(1-\alpha^3)^{-1}(1-\alpha^{3j+3})\kappa^n_{3n-2j-2, j+1, j+1},\quad\cdots (\ast)\end{array}
\mathfrak{M}\bigl(\frac{q}{\tau}\,\Big|\,\frac{1}{\tau},\tau\lambda_1,\tau\lambda_2\bigr) \,\Gamma^{\frac{q}{\tau}}(1-\tau) \Gamma(1-\frac{q}{\tau}) = \mathfrak{M}(q\,|\,\tau,\lambda_1,\lambda_2) \Gamma^{q}(1-\frac{1}{\tau}) \Gamma(1-q).
V^{[n]}_{z_1z_1z_1}- V^{[n]}_{z_1}-U_{z_2}=0, & V^{[j]}_{z_1z_1}-V^{[j]}+U\,\Omega^{(j+1)}=0, j=1\dots n-1, & \left(\Omega^{[1]}\,U\right)_{z_1}=0, & \Omega^{[j]}_{z_1}=UV^{[j]}_{z_1}, j=1\dots n.
\begin{gathered}\hat{R}(v_{1}\otimes v_{1})=q^{2}v_{1}\otimes v_{1},\quad\hat{R}(v_{0}\otimes v_{1})=v_{1}\otimes v_{0},\quad\hat{R}(v_{-1}\otimes v_{1})=q^{-2}v_{1}\otimes v_{-1}, \hat{R}(v_{-1}\otimes v_{0})=v_{0}\otimes v_{-1},\quad\hat{R}(v_{-1}\otimes v_{-1})=q^{2}v_{-1}\otimes v_{-1}.\end{gathered}
\frac{d}{d\epsilon}K(\epsilon)K(\sqrt{1-\epsilon^{2}})=\frac{1}{\epsilon(1-\epsilon^{2})}[E(\epsilon)K(\sqrt{1-\epsilon^{2}})-K(\epsilon)E(\sqrt{1-\epsilon^{2}})-(1-2\epsilon^{2})K(\epsilon)K(\sqrt{1-\epsilon^{2}})]
\quad(-1)^{n+1}\left(\sum\limits_{r+s=n-2-m}S_{r,s}x_rx_s\right)=m!\left.\frac{\partial^m}{\partial z_n^m}\right|_{\{z_n=0\}}\left(\sum_{j=2}^{n-2}(-1)^r\left(H_{\mathcal{L}}\right)_{n-j,n}\left(H_{\mathcal{L}}\right)_{n,j}\right).
x^{\tilde{\alpha}}D^{\tilde{\beta}}\left[\L_0\phi\right](x) = A\sum_{i=1}^{d}x^{\tilde{\alpha}}D^{\tilde{\beta}}D_i^2\phi(x) +\sum_{i=1}^{d}\sum_{j=1}^{d}S_{ij}x_jx^{\tilde{\alpha}}D^{\tilde{\beta}}D_i\phi(x) -Bx^{\tilde{\alpha}}D^{\tilde{\beta}}\phi(x)
L_\pi(Z)= \sum_{k\ge 0}\, (-1)^k s_{(1^k)}[s_\pi(Z)] = \sum_\nu\, \ell_{\pi\nu}\, s_\nu(Z)\quad\mbox{and}M_\pi(Z)= \sum_{k\ge 0}\, s_{(k)}[s_\pi(Z)] = \sum_\nu\, m_{\pi\nu}\, s_\nu(Z)\,.