harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{j=1}^N \int_{I_j}[\rho_h \partial_t \psi_h - \partial_t \rho_{h}\psi_h ]dx + \int_{\partial \Omega}\sigma \partial_t \psi_h ds =A_1(\psi_h, \partial_t \psi_h )- A_1(\partial_t \psi_h, \psi_h) =\beta_1 h \sum_{j=1}^{N-1}([\partial_x^2 \psi_h][ \partial_t \psi_h] -[\partial_x^2\partial_t \psi_h][\psi_h])_{j+1/2}=0
P^TL_5(\lambda)P = \left[\begin{array}{ccc|c}-P_4 \lambda P_4-P_3 \lambda P_3 0 \lambda P_4-P_3 \lambda P_3-P_2 \lambda P_2 -I_n \lambda P_3 \lambda P_2 \lambda P_1+P_0 \lambda I_n \hline0 -I_n \lambda I_n 0,\end{array}\right],
w^1_1+w^1w^1_2-\frac{(w^2)^2}{z_2}-w^1_{22}-\frac{w^1_2}{z_2}+\frac{w^1}{z_2^2}=0, w^2_1+w^1w^2_2+\frac{w^1w^2}{z_2}-w^2_{22}-\frac{w^2_2}{z_2}+\frac{w^2}{z_2^2}=0.
\left.\begin{array}{ll}\displaystyle 2\left(\dfrac{a_2}{2m_1}\int_{\mathbb{R}^N}|\nabla \xi|^2dx+\dfrac{a_1}{4m_2}\int_{\mathbb{R}^N}|\nabla \eta|^2dx\right)-\dfrac{N}{2}a_1a_2\int_{\mathbb{R}^N}\eta \xi^2dx=0,\end{array}\right.
Z^F_n={1\over n!}\left[\omega_{WP}^{(n)}\right]^{n-4}\cap\left[D^F\right]={\pi^2\over (n-1)n!}\sum_{k=1}^{[n/2]-1}F(n,k)k(n-k-2)\left[\omega_{WP}^{(n)}\right]^{n-4}\cap\left[D_k\right],
([\nabla_t,D]\xi)(X,X_1,\ldots,X_k) =\nabla^V_{\partial_t}(\nabla^V_X(\xi(X_1,\ldots,X_k)))-\nabla^V_X(\nabla^V_{\partial_t}(\xi(X_1,\ldots,X_k))) &\qquad-\sum_{i=1}^k\xi(X_1,\ldots,\nabla_{\partial_t}D_XX_i-D_X\nabla_{\partial_t}X_i,\ldots X_k) &\qquad-(D_{\nabla_{\partial_t}X}\xi)(X_1,\ldots,X_k).
x=\left[\begin{array}{ccccc}a_1 a_2 b_1 c_1 c_2 a_3 a_4 b_2 c_3 c_4 g_1 g_2 e f_1 f_2 h_1 h_2 l_1 d_1 d_2 h_3 h_4 l_2 d_3 d_4\end{array}\right]
f_1(t) =\sin\Big(\frac{t}{r}\Big)e_1(t)+\cos\Big(\frac{t}{r}\Big)e_3(t), f_2(t) =\sin\Big(\frac{t}{r}\Big)e_2(t)-\cos\Big(\frac{t}{r}\Big)e_4(t), f_3(t) =-\cos\Big(\frac{t}{r}\Big)e_1(t)+\sin\Big(\frac{t}{r}\Big)e_3(t)\,.
A = \left( \begin{array}{c c}{a}&{b}\\{c}&{d}\\ \end{array}\right) , R = \left( \begin{array}{c c c c}{q}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{\lambda}&{1}&{0}\\{0}&{0}&{0}&{q}\\ \end{array}\right) .
\frac{1+\sum^{n-k}_{j=1}\binom{m-i}{j}x^{j}}{\left(1+x\right)^{m-i}}=1-\frac{\sum^{m-i}_{j=n-k+1}\binom{m-i}{j}x^{j}}{\left(1+x\right)^{m-i}},
\partial_{i}Z = ( t^{I})_{, i}q_{I}= 0 \qquad \Longrightarrow \qquad g^{i j}t^{I}{}_{, i}t^{J}{}_{, j}\, q_{I}\equiv \Pi^{I J}q_{I}= -{\frac{1}{3}}( ( C^{I J}- t^{I}t^{J}) q_{I}= 0
|v_h -{\bar v}_h| \leq C \left(|\bar{u}_h (t)|+ |\bar{u}_h (t - h)|\right)^{q - 1}\left|\bar{u}_h(t)-\bar{u}_h (t-h)\right| begin{align*}2mm] \leq C \left|\bar{u}_h (t) - \bar{u}_h (t - h)\right|^q begin{align*}2mm] = C h^q \left(|\bar{u}_h (t)| + |\bar{u}_h (t - h)|\right)^{\frac{q (1- q)}{2}}\left(|\bar{u}_h (t)| + |\bar{u}_h (t - h)|\right)^{\frac{q (q - 1)}{2}}|\partial_t u_h|^q
(\tilde{\Delta}\varphi)_x=-\frac{4i q}{|\Lambda|}\sum_{k\in\mathcal{M}_\epsilon^K}\frac{e^{ik\cdot x}e^{-ik\cdot(u+u')/2}}{\sum_{i=1}^D(1-\cos k^{(i)})}\sin[k\cdot(u-u')/2].
\lim{\frac{F_{a}( \Sigma^{*}, x_{0}, \{l_{P}\}) U^{( \nu )}( C , x_{0})}{\langle F_{a}( \Sigma^{*}, x_{0}, \{l_{P}\}) \rangle \langle t r U^{( \nu )}( C , x_{0})}}\rangle ={\frac{1}{n_{\nu}}}D^{\nu}\left( a^{k ( \Sigma^{*}, C )}\right)
P_{m}^{(n-1 ,2\nu )}(\cos 2\rho)=\frac{2\Gamma (n+2\nu )\Gamma (m+2\nu+1)}{\Gamma (2\nu +\frac{1}{2})\Gamma (m+2\nu +n)}\frac{1}{\cos^{4\nu}\rho}\int_{\rho}^{\frac{\pi}{2}}\frac{\sin \psi}{\left( \cos^{2}\rho-\cos^{2}\psi \right)^{\frac{1}{2}-2\nu}}C_{2m}^{n+2\nu}(\cos \psi)d\psi .
\Phi^{s,m}_t = \begin{cases}0, \mbox{if $s < t$}; \sum_{i=0}^{r}a^{m-s}_{r,i}\Phi_{t}^{s,s-i}, \mbox{if $0 \leq s-t= r$, $0 \leq s \leq m$}; (-1)^{t}\gamma_{t-1}^{s,m}+ (-1)^s \gamma_{t-1}^{s,m-1}, \mbox{if $0 \leq s-t$, $m \leq s \leq m+n$}. \end{cases}
[a_l,a_m,a_k,a_{\infty}]= \exp(4 \pi i (\eta_{m}-\eta_{l})_1(\eta_k)_2)\left( \frac{\theta[\eta_{U_\eta\circ(V \cup \{k,l\})}](\Omega)\theta[\eta_{U_{\eta}\circ (W \cup \{k,l\})}](\Omega)}{\theta[\eta_{U_\eta \circ (V \cup \{k,m\})}](\Omega)\theta[\eta_{U_{\eta}\circ (W \cup \{k,m\})}](\Omega)}\right)^2,
\langle ap_{04}^2+ap_{01}p_{14}+bp_{04}p_{14}-dp_{14}^2-p_{01}p_{34}-au p_{04}p_{14}-bu p_{14}^2-up_{04}p_{34}+u^2p_{14}p_{34}, \;k_1,\;k_2,\;k_3, \;k_4,\;k_5,\;k_6 \rangle
\nabla^B \nabla_{\ \ \dot A}^C W_{ABC}= -2i \nabla^B_{\ \ \dot A}\nabla^{\ \ \dot B}_{\underline{A}}G_{\underline{B}\dot B}+G^B_{\ \ \dot A}\nabla^{\ \ \dot B}_{\underline{A}}G_{\underline{B}\dot B}+\frac{11}{2}i \left(\nabla^B G_{\ \ \dot A}^C \right) W_{ABC}
=\frac{1}{\bigl[(c_1+c_2)t \bigr]^{2k}}\sum_{j=1}^{k-1}\binom{2k}{j}\Bigl[ (c_1t-\beta)^{j}(c_2t+\beta)^{2k-j}- \Bigl(\frac{c_2}{c_1}\Bigr)^{2k-2j}(c_1t-\beta)^{2k-j}(c_2t+\beta)^{j}\Bigr]
\sum_{i=1}^{\left \lfloor{\frac{y^{1/2}}{a}}\right \rfloor}(y-a^{2}i^{2})^{n/2}= a^{n}\sum_{i=1}^{\left \lfloor{\frac{y^{1/2}}{a}}\right \rfloor}\bigg(\left(\frac{y^{1/2}}{a}\right)^2-i^{2}\bigg)^{n/2}.
- H_{\rho \mu \nu}H^{\rho \mu \nu}+{1 \over \Lambda}(g^{-2}_A \omega^A_3 + g^{-2}_G \omega^G_3 - g^{-2}_L \omega^L_3)_{\rho \mu \nu}(\partial_\tau \theta (x) -{c_A \Lambda \over 48 \pi^2 g_A}B_\tau) \epsilon^{\tau \rho \mu \nu}
\frac{\mathbb{P}_{\mu}((X_{T-r})^{*}_{i}= u)}{\mathbb{P}_{\mu}((X_{T-r})^{*}_{i}= (X_{T-r})^{*}_{j}= u)}= \frac{\nu(\{u\})}{\nu(\{u\})(1-e^{r-T}) + \nu(\{u\})^{2}e^{r-T}}\;.
\begin{gathered}\frac{\rm{d}\pi_f^{(D)}(\textbf{x})}{\rm{d}x_f^{(D)}}=\frac{\rm{d}{\left(\frac{p_fB_f}{N^{(F)}+N^{(D)}x_f^{(D)}}C_f^{(D)}-q_f\phi_f\right)}}{\rm{d}{x_f^{(D)}}}=-\frac{N^{(D)}p_fB_fC_f^{(D)}}{\left(N^{(F)}+N^{(D)}x_f^{(D)}\right)^2}-c_3c_4N^{(D)}\phi_f\exp\left(c_4\left(N^{(F)}+N^{(D)}x_f^{(D)}\right)\right)\end{gathered}
\bigl(\textbf{x}^{(1)}\bigr)_{k_2}\cdot \bigl(\textbf{x}^{(1)}\bigr)_{h_2}<0\,,0<\bigl|\bigl(\textbf{x}^{(1)}\bigr)_{k_2}\bigr|= \min_{k=1,\dots,N\,, \bigl(\textbf{x}^{(1)}\bigr)_{k}\ne 0}|\bigl(\textbf{x}^{(1)}\bigr)_{k}|\,.
\sum_{j'\geq j}\dot{S}_{j'+2}\dot{\Delta}_{j}\nabla\Pi\dot{\Delta}_{j'}a= \sum_{j'\geq j}(\sum_{k\le j'+1}\dot{\Delta}_{k}\dot{\Delta}_{j}\nabla\Pi)\dot{\Delta}_{j'}a = \sum_{j'\geq j}((I-\sum_{k\ge j'+2}\dot{\Delta}_{k})\dot{\Delta}_{j}\nabla\Pi)\dot{\Delta}_{j'}a=\sum_{j'\geq j}\dot{\Delta}_{j}\nabla\Pi\dot{\Delta}_{j'}a,
\xi_3^2\eta_4^2 EF \; = \; ( \xi_1\eta_4 K +\xi_3 \eta_1 K') (\xi_2\eta_4 K +\xi_3 \eta_2 K' ) \; = \; \xi_1\xi_2\eta_4^2 K^2 + \xi_3\eta_4( \xi_1\eta_2 + \xi_2\eta_1)KK' +\xi_3^2\eta_1\eta_2 K^{'2}\; = \; -\kappa^{2}\xi_3^2\eta_4^2 K^2 + \xi_3 \eta_4( \xi_1\eta_2 + \xi_2\eta_1)KK' -\kappa^{2}\xi_3^2\eta_4^2 K^{'2}
\phi\otimes \psi+ \phi^\vdash(y) = \inf_{x\in X}(\phi(x)+\psi(x)) + \phi^\vdash(y) =\inf_{x\in X}(\phi(x)+\psi(x) + \phi^\vdash(y)) \geq \inf_{x\in X}(d(x,y)+\psi(x)) \geq \psi(y),
I(f)\lesssim\int_{\mathcal{I}_{\boldsymbol{j}}}\left|\prod_{l=1}^d2^{j_l/2+j_l}\psi^{'}_{k_l}(2^{j_l}x_l)\mathrm{sgn}[\psi_{k_l}(2^{j_l}x_l)]\right|d\boldsymbol{x}\lesssim\prod_{l=1}^d2^{j_l/2},
P\left( a,v\right) :=1+\sum_{k=1}^{n}\sum_{l_{i}\in \left\{0,1,\dots,n-1\right\}}\left( \left( D^{l_{1}}p\right) \otimes \cdots \otimes \left(D^{l_{k}}p\right) \right) \left( \pi_{1}\left( a\right) \right) \sum_{\rho\in OS\left( l_{1}+1,\dots ,l_{k}+1\right)}\rho^{-1}\left( \pi_{l_{1}+\cdots +l_{k}+k}\left( v\right) \right)
B=\frac{dx_1}{dx_2}\frac{d}{dx_1}\left(\frac{b_1}{A_1}\right)=\frac{1}{A_1}\frac{d}{dx}\left(\frac{b_0}{A_0h_0}\right) =\frac{1}{A_0^3}\left(A_0\frac{d}{dx}+\gamma A_0-A'_0\right)b_0.
K ~ = ~ \frac{8 \pi^{2 \mu}( 2 \mu - 1 )^{2}( \mu - 2 )^{2}}{( 2 \mu - 3 )^{2}\Gamma ( \mu ) \Gamma ( \mu + 1 )}~ ~ , ~ ~ P ~ = ~ \frac{1 6 \pi^{2 \mu}( 2 \mu - 1 ) ( 2 - \mu )}{( 2 \mu - 3 ) \Gamma ( \mu ) \Gamma ( \mu + 1 )}
\widetilde{(\eta^{n,\pm})}_l = \begin{cases}0, n=0, p_l^{\pm}(\widetilde{(f(\psi(t_n)))}_l - \widetilde{(f(\psi^n))}_l) + q_l^{\pm}(\widetilde{(\delta_t^- f(\psi(t_n)))}_l - \widetilde{(\delta_t^- f(\psi^n))}_l), n \ge 1. \end{cases}
T r_{\{\eta , \overline{{{\eta}}}\}}( I ) = T r_{\{\eta , \overline{{{\eta}}}\}}( \theta ) = T r_{\{\eta , \overline{{{\eta}}}\}}( \overline{{{\theta}}}) = 0 \; \; , T r_{\{\eta , \overline{{{\eta}}}\}}( \overline{{{\theta}}}\theta ) = q \; .
h^1 = \big(h_1-h_2h^{-1}_4h_3\big)^{-1}= h_1^{-1}+ h_1^{-1}h_2h^4h_3 h_1^{-1}, h^2 = -h^{-1}_1h_2h^4, h^3 = - h^4 h_3h_1^{-1}, h^4 = (h_4-h_3h^{-1}_1h_2)^{-1}.
\Gamma (s_1) \Gamma (s_2) \zeta_2 (s_1,s_2 \,; a) = &\int_0^\infty \!\!\! \int_0^\infty \frac{y^{s_2-1}}{e^y-1}H(a,x+y)x^{s_1-1}dx dy +\int_0^\infty \!\!\! \int_0^\infty \frac{x^{s_1-1}}{x+y}H(1,y)y^{s_2-1}dx dy
\Psi(W_{v_m},\phi^m,\tilde f_s^i)-\Psi(W_{v_m'}, \phi^m, \tilde f_s^i) = \int_{T\times N_m}( W_{v_m}(t(a)wn)-W_{v_m'}(t(a)wn))(\omega_{\psi^{-1}}(t(a)wn)\phi^m)(1)\tilde f_s^i(t(a)wn)|a|^{-2}dn da.
\frac{c_{n}}{c_{n-2,\xi_{\boldsymbol{s}k},\xi_{\boldsymbol{t}l}}^{\ast\ast}}= \frac{2^{(n-6)/2}}{\Gamma(n-1)\pi}\bigg\{\int_{0}^{\infty}t^{\frac{n-4}{2}}\left[(2+3\sqrt{2})\left(t+\frac{\xi_{\boldsymbol{s},k}^{2}}{2}+\frac{\xi_{\boldsymbol{t},l}^{2}}{2}\right)+3\right]\exp\left(-\sqrt{2t+\xi_{\boldsymbol{s},k}^{2}+\xi_{\boldsymbol{t},l}^{2}}\right)\mathrm{d}t\bigg\}.
\nabla F(x) + \sum_{i = 1}^m \lambda_i\nabla g_i(x) + A^T v = 0, \lambda_i \geq 0, i= 1, \dots, m, s_i \geq 0, i= 1, \dots, m, \lambda_i s_i = 0 , i= 1, \dots, m, g_i(x) +s_i = 0, i= 1, \dots, m, A x = b.
\begin{array}{ll}d\omega(X_0,\cdots,X_n) = \sum_{0\leq k\leq n}(-1)^k X_k\ \omega(X_0,\stackrel{k\atop \vee}{\cdots},X_n) &+ \sum_{0\leq r<s\leq n}(-1)^{r+s}\ \omega([X_r,X_s],X_0,\stackrel{r\atop \vee}{\cdots}\stackrel{s\atop \vee}{\cdots},X_n)\end{array}
A_2 ={\rm Ch}^{NS,(2)}_{0,0},\;\;B_2 ={\rm Ch}^{NS,(2)}_{2,2},\;\;C_2 ={\rm Ch}^{NS,(2)}_{2,0},\;\;D_2 ={\rm Ch}^{NS,(2)}_{2,-2},\;\;E_2 ={\rm Ch}^{NS,(2)}_{1,1},\;\;F_2 ={\rm Ch}^{NS,(2)}_{1,-1}.
\|\epsilon_{gp}\| = L_p\|fi(fi(M_1)x_{k_{in}}) - M_1x_{k_{in}}+ fi(fi(M_2)\lambda_k) - M_2\lambda_k + (V_1 - fi(V_1))\| & \leq L_p(\|fi(fi(M_1)x_{k_{in}}) - M_1x_{k_{in}}\| + \|fi(fi(M_2)\lambda_k) - M_2\lambda_k\| + \|(V_1 - fi(V_1))\|)
\tilde{F}(P_-,x^+) ={1 \over 2 L}\langle{\cal O}(P_-,x^+){\cal O}(-P_-,0) \rangle\ =\sum_i{1 \over 2 L}\langle 0|{\cal O}(P_-) | i\rangle e^{-i P_+^i x^+}\langle i|{\cal O}(-P_-,0) |0 \rangle\ .
\kern-1em -\frac{1}{2}\int_M K_{ij}\delta \Big((d^* H)_{ij}+\delta(i_{\nabla f}H)_{ij}\Big) e^{-f}dV_g &=\frac{1}{2}\int_M h_{kl}H_{kij}(dK)_{lij}e^{-f}dV_g-\frac{1}{6}\int_M |dK|^2 e^{-f}dV_g-\frac{1}{6}\int_M \langle dK,H \rangle(\frac{_gh}{2}-\phi)e^{-f}dV_g.
F_d(s)=\frac{1}{\Gamma({\frac{d}{2}}+1)}\int_0^{{\frac{d}{s}}-\varepsilon}t^{\frac{d}{2}}\bigl(1-{\frac{s t}{d}}\bigr)^{-{\frac{d}{2}}}e^{-t}dt=1+O(s)
\alpha_{n-1}(\delta) = - \overline{S_{n}(\delta; 0)}\rho_{n}^{(2)}= \frac{S_n(\delta; 1)}{S_n^{\ast}(\delta; 1)}= \frac{\rho_{n-1}^{(2)}- \overline{\alpha}_{n-1}(\delta)}{1 - \rho_{n-1}^{(2)}\alpha_{n-1}(\delta)}, n \geq 1,
N_{C}(x_0, f_1(x_0), f_2(x_0)) = \hat T_{C}^\circ(x_0, f_1(x_0), f_2(x_0)) & =\{(x^*, s_1, s_2) \in X^*\times \mathbb{R}\times \mathbb{R}\ | \ \langle x^* v\rangle + s_1 r_1 + s_2 r_2 \le 0 & \mbox{for all}(v, r_1+r_2) \in \hat T_{epi (f_1+f_2)}(x_0, f_1(x_0)+ f_2(x_0)) \}\, .
e_0 = Z \otimes v_1 - Y \otimes v_2, e_1 = Z \otimes v_0 + a_{011}Y \otimes v_1 + (X + a_{012}Y + a_{022}Z) \otimes v_2, e_2 = (a_{011}X + a_{111}Y) \otimes v_0 + \widetilde{L}_1(X, Y, Z) \otimes v_1 + \widetilde{L}_2(X, Y, Z) \otimes v_2
g_1\cdot p = \lambda_1(-\tfrac{1}{2}e_{126}-\tfrac{1}{2}e_{349}+2e_{358}-e_{457}+e_{789}) + \lambda_2(-2e_{137}-\tfrac{1}{4}e_{249}-e_{258}-\tfrac{1}{2}e_{456}-\tfrac{1}{2}e_{689}) - \lambda_3(-e_{159}-e_{238}-\tfrac{1}{2}e_{247}-\tfrac{1}{2}e_{346}-e_{678}).
K := \frac{2L \max(M, 1)}{\min(\nu_0, 1)}e^{2 \nu_0^{-1}M}T_{m}^{\frac{s + 1}{s}}\left\| \left( (-\Delta)^{\frac{3}{4}}u_{0}, (-\Delta)^{\frac{1}{4}}u_{1}\right) \right\|_{\gamma_{\eta, L^{2}}^{s}\times \gamma_{\eta, L^{2}}^{s}}^{2}.
\underline{q}\circ D_1{\mathcal T}_1 (\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta x'}=\sum_{t=1}^{+\infty}\langle q_{t-1}-q_t \circ D_1 f(\hat{x}_t, \hat{u}_t), \delta x'_t \rangle.
\left(\mathcal{H}_{\hbar,V}+I\right)^{-1}=\left(\mathcal{H}_{\hbar,V}+I\right)^{-1}\left(-\hbar^{-2}\mathcal{L}_{\hbar}\right)\left(V+I\right)^{-1}+\left(V+I\right)^{-1}.
\Pi^{\bot}\left\{W_{\delta,q}+\delta V_{\delta,q}+\Phi-i^{*}\left(f(W_{\delta,q}+\delta V_{\delta,q}+\Phi)-\varepsilon\gamma(W_{\delta,q}+\delta V_{\delta,q}+\Phi)\right)\right\}=0; \Pi\left\{W_{\delta,q}+\delta V_{\delta,q}+\Phi-i^{*}\left(f(W_{\delta,q}+\delta V_{\delta,q}+\Phi)-\varepsilon\gamma(W_{\delta,q}+\delta V_{\delta,q}+\Phi)\right)\right\}=0.
\nabla \tilde{h}(t, x, y) = \begin{pmatrix}\partial_x \tilde h \partial_y \tilde h \end{pmatrix}= \begin{pmatrix}\partial_z h a(\partial_v-t\partial_z) h \end{pmatrix}= \begin{pmatrix}\partial^t_z h \partial^t_v h \end{pmatrix}=\nabla_t h,
\dot{\epsilon}_{\delta\sigma}=\epsilon_{\delta\sigma}\left[3{\dot{a}\over a}(1+w)+\left\{2{\dot{G}_5\over G_5}-{{6+8\epsilon}\over{1+\epsilon}}{\dot{c}\over c}\right\}(1+\epsilon_{\delta\sigma})+{{9Cc^6}\over{32\pi^2G^2_5\sigma a^4_0\varrho_m}}{\dot{c}\over c}\right].
\mathrm{E}[e_i] = \mathrm{E}[I^{d-}(y_i-x_i)^2] &= \mathrm{E}[(y_i-x_i)^2] \mathrm{E}[\alpha^-(x,y)] &= \mathrm{E}\left[ \left(-\hat{r}_i - \frac{\tilde{g}_i}{\lambda_i}\xi_i + \frac{g^{1/2}}{\tilde{\lambda}_i}\nu_i \right)^2 \right] \mathrm{E}[\alpha^-(x,y)] &= U_1 \mathrm{E}[\alpha^-(x,y)].
\mu (\phi(x_M)-\phi(x_m)) =K_r*\phi(x_M)-K_r*\phi(x_m)+\frac{1}{2}\left( \phi^2(x_M)- \phi^2 (x_m) \right) & \leq \|K_r\|_{L^1}(\phi(x_M)-\phi(x_m)) + \frac{1}{2}\left( \phi(x_M) - \phi(x_m)\right)\left( \phi(x_M) + \phi(x_m)\right),
\mathbf{v}_{m}^{\left(i+1\right)}\negmedspace=\negmedspace\left[\negmedspace\sum_{l\in\mathcal{S}^{\left(i+1\right)}}\negthickspace a_{ml}\left(\mathbf{t}_{l}-Q_{\Lambda_{s,l}}\left(\mathbf{t}_{l}-\mathbf{d}_{l}\right)\right)\right]\negthickspace\bmod\negmedspace\Lambda_{e,m}
\int \frac{x^{n-2}}{(-x^2;q^2)_\infty}\widetilde{h}_n(x;q) d_qx = \dfrac{x^n}{[n-1]_q(-x^2;q^2)_\infty}\left( \frac{\widetilde{h}_n(x;q)}{x}- \widetilde{h}_{n-1}(x;q)\right).
I_k(F) =\int_0^1\cdots\int_0^1F(t_1,\ldots,t_k)^2dt_1\ldots dt_k, J_k^{(m)}(F) =\int_0^1\cdots\int_0^1\Biggl(\int_0^1F(t_1,\ldots,t_k)dt_m\Biggr)^2dt_1\ldots dt_{m-1}dt_{m+1}dt_k.
e^{- S_{\mathrm{d u a l}}( A )}= \int D B_{\mu}D B_{\mu}^{\dagger}D \eta_{\mu}e^{- \left( S ( B , ( \partial + i \eta ) B ) + \int d^{3}x A_{\mu}\epsilon^{\mu \nu \rho}\partial_{\nu}\eta_{\rho}\right)},
L(x,\dot{x},\ddot{x})=\frac{\left\vert \ddot{x}\right\vert^{2}}{\left\vert \dot{x}\right\vert^{3}}-\frac{\left\langle \dot{x}, \ddot{x}\right\rangle^{2}}{\left\vert \dot{x}\right\vert^{5}}.
[y_1, y_2]=y_3, [y_2, y_1]=-y_3+\frac{\alpha_4+\beta_1}{\beta_3}y_4+\frac{(\alpha_5+\beta_2)\beta_3-(\alpha_4+\beta_1)\beta_4}{\beta_3}y_5, [y_2, y_2]=y_4, [y_3, y_1]=\alpha_3\gamma_2y_5, [y_2, y_3]=\alpha_3\gamma_3y_5, [y_3, y_2]=\alpha_3\gamma_4y_5, [y_1, y_4]=\beta_3\gamma_6y_5.
z_i = \frac{q^{d-i}(q^i-1)(q^{2d-i+1}-1)}{(q^{d-i}+ 1)(q^{d-i+1}+ 1)}(1 \leq i \leq d-1), z_d = \frac{(q^d-1)(q^{d+1}-1)}{q+1}.
\chi(X_{ij}) = X_{ij}+ \sum_{\substack{r\geq 1 \{u_1 < u_2 < \dots < u_r\}\subseteq U}}(-1)^r \frac{X_{a_{u_1-1},j}X_{a_{u_2-1},b_{u_1}}\cdots X_{a_{u_r-1},b_{u_{r-1}}}X_{i,b_{u_r}}}{X_{S_{u_1}}X_{S_{u_2}}\cdots X_{S_{u_r}}}.
\mathcal{F}_1:=\{Z_i: 1\leq i\leq n_1; Z_{i}=(z_{i}, 0)\in \mathbb{B}_{\rho/2}\}, \mathcal{F}_2:=\{Y^i=(y^{i}, y^i_{n+1}): 1\leq i \leq n_2, Y^{i}\in \mathbb{B}_{c_{2}\rho}^{+}\},
(\begin{matrix}1&0&0&-rcos\theta\\ 0&1&0&-rsin\theta\end{matrix})(\begin{matrix}\dot{x}\\ \dot{y}\\ \dot{\theta}\\ \dot{\phi}\end{matrix})=0=(\begin{matrix}1&0&0&-rcos\theta\\ 0&1&0&-rsin\theta\end{matrix})(\begin{matrix}dx\\dy\\d\theta\\d\phi\end{matrix})
{\rm l.h.s.}=f_{1,4}f_{2,4}f_{3,4}f_{1,3}=Q_{34}Q_{24}Q_{14}Q_{41}Q_{42}Q_{43}Q_{23}Q_{13}Q_{31}Q_{23}^{-1},{\rm r.h.s.}=f_{1,3}f_{1,4}f_{2,4}f_{3,4}=Q_{23}Q_{13}Q_{31}{\boldsymbol{Q_{23}^{-1}Q_{34}Q_{24}}}Q_{14}Q_{41}Q_{42}Q_{43}.
\sqrt{F(\rho,\sigma)}=tr(\sqrt{\rho}\sqrt{\sigma}U)=\sum_{k}tr(\sqrt{\rho}E_{k}\sqrt{\sigma}U)=\sum_{k}tr(\sqrt{\rho}\sqrt{E_{k}}\sqrt{E_{k}}\sqrt{\sigma}U)\le\sum_{k}\sqrt{tr(E_{k}\rho)tr(E_{k}\sigma)},
\sup_{v_i\in{\cal{G}}_i,\, i=1,\!...,m}\left\{-\big(f^*\Box h^*\big)\left(z-\sum_{i=1}^{m}L_i^*v_i\right)-\sum_{i=1}^{m}\big(g_i^*(v_i)+l_i^*(v_i) + \langle v_i,r_i\rangle\big) \right\}.
\nu_{>k}:=\delta_{b_{k+1}^{-1}D_{k+1}}\ast\delta_{b_{k+1}^{-1}b_{k+2}^{-1}D_{k+2}}\ast\delta_{b_{k+1}^{-1}b_{k+2}^{-1}b_{k+3}^{-1}D_{k+3}}\ast\cdots,
V(L_1, L_2, K{[n-2]})=\frac{1}{n(n-1)}V_{n-2}(\pi_{U^{\perp}}K)|det_2(u,v)|=\frac{2}{n(n-1)}V_{n-2}(\pi_{U^{\perp}}K) V_2(L_1, L_2) \, \,
\vec{\beta}= \vec{\beta}_{( n_{1}^{\prime}, n_{1}) ( n_{2}^{\prime}, n_{2}) ( n_{3}^{\prime}, n_{3})}= \sum_{a = 1}^{3}( \frac{1 - n_{a}^{\prime}}{2}\alpha_{-}+ \frac{1 - n_{a}}{2}\alpha_{+}) \vec{\omega}_{a}
\langle e_1,\ldots ,e_n\rangle \oplus Ker(T) =E,\langle e_1,\ldots ,e_n,\ldots ,e_m\rangle \oplus Ker(S)=E\{\rm with}\T(e_i)=0\{\rm for}\i>n \{\rm and}&\langle p_1,\ldots ,p_m\rangle \oplus Ker(S) =E,\langle p_1,\ldots ,p_m,\ldots ,p_l\rangle \oplus Ker(U)=E\{\rm with}\S(p_i)=0\{\rm for}\i>n.
{\sigma}_2 (|z| \to \infty) \approx \frac{2{\hbar}ce^{-\frac{\pi}{2}{\beta}}}{|{\Gamma}(1+i{\beta}) |}z^{-1/4}\cos\left(\frac{z}{2{\hbar}c}+ \beta{\rm ln}\frac{z}{{\hbar}c}+ \delta \right).
(\Delta Y_s)_+\left(f(s,\Theta_s,U_s)-f'(s,\Theta'_s, U'_s)\right) & =(\Delta Y_s)_+\left(f(s,\Theta_s,U_s)-f(s,\Theta'_s, U'_s)+f(s,\Theta'_s,U'_s)-f'(s,\Theta'_s, U'_s)\right) &\leq(\Delta Y_s)_+\left(f(s,\Theta_s,U_s)-f(s,\Theta'_s, U'_s)\right).
\psi = - 2^{-\frac{4}{3}}\frac{1}{2}\omega^2 + 2^{-\frac{4}{3}}dr^2 \wedge dr^3 \wedge \omega + 2^{\frac{2}{3}}dr^3 \wedge dr^1 \wedge \Re (\Omega) + 2^{\frac{2}{3}}dr^1 \wedge dr^2 \wedge \Im (\Omega),
T r f K ( t , x , x ) = ( 4 \pi t )^{- 3 / 2}t r ( \int_{B^{3}}d x \sqrt{g}\sum_{k = 0}^{\infty}\frac{( - t )^{k}}{k !}( f a_{k}) + \int_{S^{2}}d \theta \sqrt{\gamma}\sum_{k = 0}^{\infty}t^{\frac{k + 1 `}{2}}c_{\frac{k + 1}{2}}( f ) )
_2F_1[a,b;c;z]=\sum^{\infty}_{l=0}{{(a)_l(b)_l}\over{(c)_l}}{{z^l}\over{l!}}=1+{{ab}\over{c}}{z\over{1!}}+{{a(a+1)b(b+1)}\over{c(c+1)}}{{z^2}\over{2!}}+\cdots.
\sigma^+|1 \rangle=|0 \rangle, \ \sigma^+|0 \rangle=0, \ \langle 0|\sigma^+=\langle 1|, langle 1|\sigma^+=0, &\sigma^-|0 \rangle=|1 \rangle, sigma^-|1 \rangle=0, langle 1|\sigma^-=\langle 0|, langle 0|\sigma^-=0.
\begin{array}{rcl}V_n^\pi(\mathbf{s}) = R(\mathbf{s}_{\pi(\mathbf{s})}) +\beta \displaystyle\sum_{\mathclap{\mathcal{E}\subseteq \pi(\mathbf{s})}}q(|\mathcal{E}|,K) & \times V_{n+1}^\pi( (s_1(n+1),\ldots,s_j(n+1)=0, & \ldots,s_i(n+1)=\tau(s_i(n)), \ldots)), \end{array}
\limsup_{H \to \infty}\frac{1}{H}\sum_{h=1}^H \lim_{N \to \infty}\left| \frac{1}{N}\sum_{n=1}^N \left(f_1 \cdot f_1 \circ T^h\right)(T^nx) \, \left(f_2 \cdot f_2 \circ T^{2h}\right)(T^{2n}x) \right| \leq [N_2(f_1)]^2
L^{\frak{g}}\Bigl(\Bigl(n-\frac{1}{2}\Bigr)\omega_1\Bigr)\Bigl|_{\frak{g}^\sigma}\simeq\bigoplus_{k=0}^{\infty}A_{\frak{q}'(1)}\Bigl(\Bigl(n-2l+k-\frac{1}{2}\Bigr)\mu_1\Bigr)\boxtimes F^{\frak{so}(2n-2l+1)}(k\nu_1)
\partial_{t}\varphi + \frac{1}{2 m}({\mathrm{\boldmath \nabla}}\varphi )^{2}+ \frac{\hbar^{2}}{4 m}\left[ \frac{1}{2}\left( \frac{{\mathrm{\boldmath \nabla}}\rho}{\rho}\right)^{2}- \frac{\triangle \rho}{\rho}\right] + U = 0
\langle0|0\rangle=exp(\frac{i}{2}\int\frac{d^{4}p}{(2\pi)^{4}}[\tilde{\phi}(p)(p_{\mu}p^{\mu}-m^{2}+i\epsilon)\tilde{\phi}(-p)+J(p)\frac{1}{p_{\mu}p^{\mu}-m^{2}+i\epsilon}J(-p)])
G_{F}( x , y ) = e^{-{\frac{i e^{2}}{2}}\int d^{2}z_{1}\int d^{2}z_{2}j^{\mu}( z_{1}; x , y ) G_{\mu \nu}( z_{1}, z_{2}) j^{\nu}( z_{2}; x , y )}~ S_{F}( x , y ) ,
\overline{\gamma}:=\langle\gamma\rangle^{G}\equiv\gamma-\Xi^{Q}, \overline{\sigma}:=\langle\sigma\rangle^{G}+\frac{1}{2}\langle \{Q\wedge Q\}_{P}\rangle^{G}-\mathrm{d}_{1,0}^{\overline{\gamma}}\langle Q\rangle^{G}.
\sigma \left({t,x_1 ,x_2 ,x_3 ,m}\right) = \tilde \sigma\big(\tau_- (t,x_1 ,x_2 ,x_3 ,m),X(t,x_1,x_2 ,x_3 ,m;\tau_-(t,x_1 ,x_2 ,x_3 ,m)) \big) +
\int_{K\backslash G/H}Q(f)(\ddot{y}) d\mu_{n}(\ddot{y}) =\int_{K \backslash G / H}Q(L_nf)(\ddot{y})d \mu (y) &=\int_{G}L_n f(y).\rho(y)dy &=\int_{K \backslash G / H}Q\big(f. \lambda (n,.)\big) (\ddot{y})d \mu_\rho(\ddot{y}).
F_{\lambda}(\zeta,\Lambda):= \,k\,a_0+a_1\sum_{i=1}^k\Bigl(\Lambda_i^2\,g_{\lambda}(\zeta_i)-\sum_{j\neq i}\Lambda_i\,\Lambda_j\,G_{\lambda}(\zeta_i,\zeta_j)\Bigr)+a_2\,\lambda\,\sum_{i=1}^k \Lambda_i^4 &-a_3\,\sum_{i=1}^k\Bigl(\Lambda_i^2\,g_{\lambda}(\zeta_i)-\sum_{j\neq i}\Lambda_i \Lambda_j\,G_{\lambda}(\zeta_i,\zeta_j)\Bigr)^2.
( \kappa_{\tau}\frac{\partial}{\partial \kappa_{\tau}}+ \beta_{\tau}\frac{\partial}{\partial u_{\tau}}- \frac{1}{2}N \gamma_{\phi ( \tau )}( u_{\tau}) + L \gamma_{\phi^{2}( \tau )}( u_{\tau}) ) \Gamma_{R ( \tau )}^{( N , L )}= \delta_{N , 0}\delta_{L , 2}( \kappa_{\tau}^{- 2 \tau})^{\frac{\epsilon_{L}}{2}}B_{\tau}( u_{\tau}) ,
\gamma_k=2^{\,(2\,\cdot\,2^{C(k-1)}-2^{Ck})}\,\cdot\,10^{1-\frac{1}{p}}\cdot2^{Ck(1-\frac{1}{p})}\leq 10\cdot 2^{-\frac12\, 2^{Ck}}\cdot 2^{Ck}
\gamma =E_{n+2,1}\in \mathfrak{g}_{-2}, e_i =E_{i+1,1}\in\mathfrak{e}^1\subset \mathfrak{g}_{-1},f_i=E_{n+2,i+1}\in\mathfrak{e}^2\subset \mathfrak{g}_{-1}(1\leq i\leq n).
\hat{H}_{n}=[\begin{matrix}\hbar\omega_{c}(n-1)+\hbar\omega_{eg}&\langle e,n-1|\hat{H}_{JC}|g,n\rangle\\ \langle g,n|\hat{H}_{JC}|e,n-1\rangle&n\hbar\omega_{c}\\ \end{matrix}]
2\Re\iint_{\mathbb R^3 \times \mathbb R^3}|x-y|^{-1}f(x,y)\nabla_{x}|x-y|^{-1}\nabla_{x}\overline{f(x,y)}{\rm d}x{\rm d}y & = -\iint_{\mathbb R^3 \times \mathbb R^3}|\nabla_{x}|x-y|^{-1}|^2 |f(x,y)|^2{\rm d}x{\rm d}y.
\begin{array}{ll}\forall (X, \chi) \in \mathcal{H}, \|(X, \chi)\|_{\mathcal{H}}\!\!\!\! = |X|^2 + \frac{1}{2}\|\chi\|^2 & = |X|^2 + c^2 \| w_x \|^2 + \| w_t \|^2.\end{array}
E' = (\ 0,\I(X_1;Y_2|U_2),\ \min\{I(X_2;Y_3),\I(U_2;Y_2)+I(X_2;Y_3|U_2) \}\ ), F' = (\ 0,\ \min\{I(X_1;Y_2|U_2),\ [I(X_1,U_2;Y_2)-I(U_2;Y_3)]^+ \},\I(X_2;Y_3|U_2)+ \min\{I(U_2;Y_3) ,\I(X_1,U_2;Y_2) \}\ ).
\int^z \mathcal{C}_{\mu}(kz)\overline{\mathcal{C}}_{\nu}(kz)\frac{dz}{z}=-\frac{kz\left\{\mathcal{C}_{\mu+1}(kz)\overline{\mathcal{C}}_{\nu}(kz)-\mathcal{C}_{\mu}(kz)\overline{\mathcal{C}}_{\nu+1}(kz)\right\}}{\mu^2-\nu^2}begin{align*}-4pt] \mbox{}+\frac{\mathcal{C}_{\mu}(kz)\overline{\mathcal{C}}_{\nu}(kz)}{\mu+\nu},
\begin{aligned}\left \vert I_{f,\omega_{\pi},\psi}(g) \right \rvert = \left \vert \int_{Z_n}I_{f,\psi}(gz) \omega_{\pi}^{-1}(z) dz \right \rvert &= \left \vert \int_{P}I_{f,\psi}(gz) \omega_{\pi}^{-1}(z) dz \right \rvert & \leq C_1 C_2 \times \int_{P}\left \vert \Delta^{-\frac{1}{2}+ \delta}(gz) \right \rvert dz.\end{aligned}
S= \frac{1}{2 \lambda_{s}^2}\int d t e^{\alpha + 2 \beta-\phi}\biggl\{\frac{1}{N}\biggl[ - \dot{\phi}^2 - 2 \dot{\beta}^2 -4 \dot{\alpha}\dot{\beta}+ 2 \dot{\alpha}\dot{\phi}+ 4 \dot{\beta}\dot{\phi}\biggr] + \frac{w\lambda_{s}^2}{4 N^3}\biggl[ 8 \dot{\phi}~\dot{\alpha}~ \dot{\beta}^2 - \dot{\phi}^4\biggr]\biggr\}
p_{}^{\prime}\left( x_{1},\ldots,x_{l}\right) =\delta\left(x_{l}-f_{l}\left( x_{1},\ldots,x_{l-1}\right) \right) \delta\left(x_{l-1}-f_{l-1}\left( x_{1},\ldots,x_{l-2}\right) \right) \cdots\delta\left( x_{2}-f_{2}\left( x_{1}\right) \right) p_{1}\left(x_{1}\right)
\widetilde{\tau}_{2}=1-2t_{2}Z_{2}+\frac{3(t_{1}Z_{1})^{2}}{4}+\frac{(t_{2}Z_{2})^{2}}{4}+\frac{3}{16}\tan^{4}\left( \frac{\pi}{12}\right)(t_{1}Z_{1}t_{2}Z_{2})^{2}
T_{N}-T_{N,2}=\sum^N_{n=1}\sum^{m_0}_{j=1}\Big( \mathbb{E}\big[K(X_n+a_j\widetilde{\varepsilon}_{n-j})|\varepsilon_{n-j}\big]-\mathbb{E}\big[K(X_n+a_j\widetilde{\varepsilon}_{n-j})\big]\Big)
I_{\beta}^{n,\gamma}(f_{1},\ldots ,f_{m})=\left\{\beta\int_{0}^{z}t^{\beta -1}\prod_{i=1}^{m}\left( \frac{D_{\lambda}^{n,\gamma}f_{i}(t)}{t}\right)^{\alpha_{i}}dt\right\}^{\frac{1}{\beta}}
\P (N_{Y}= k) = \prod_{i=0}^{k-1}\frac{f(i)}{1+f(i)}- \prod_{i=0}^{k}\frac{f(i)}{1+f(i)}&= \left( 1 - \frac{f(k)}{1+f(k)}\right) \prod_{i=0}^{k-1}\frac{f(i)}{1+f(i)}=\frac{1}{1+ f(k)}\prod_{i=0}^{k-1}\frac{f(i)}{1+f(i)}= \mu_k.
\frac{dP_{Br}}{d\omega}=\frac{8\sqrt{2}}{3\sqrt{\pi}}[\frac{e^{2}}{4\pi\epsilon_{0}}]^{3}\frac{1}{(m_{e}c^{2})^{3/2}}[1-\frac{\omega_{p}^{2}}{\omega^{2}}]^{1/2}\frac{Z_{i}^{2}n_{i}n_{e}}{(k_{B}T_{e})^{1/2}}E_{1}(y),
\delta \Delta = - \Delta \biggl [ ( \epsilon^{2 n},{\cal Z})_{E}- s t r \Bigl ( \bigl ( \sum_{l = 0}^{n - 1}\kappa^{( l )}\tilde{W}D^{( 2 l + 1 )}( \epsilon^{s t}) + \sum_{l = 0}^{n}\kappa^{( l ) \prime}W D^{( 2 l )}( \epsilon^{s t}) \bigr ) W^{- 1}\Bigr ) \biggr ] ,