harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{u_{-}(\theta)}^{u_{+}(\theta)}\phi(u) d(\pi_\theta\widetilde{\mu})(u) =\int_{(u,v)\in D}\phi(u) d\widetilde{\mu}(u,v) = \int_{u_{-}(\theta)}^{u_{+}(\theta)}\phi(u)Y_{(\theta,u)}du.
\begin{cases}\tfrac{2}{3}d^\ast d \beta_0 = (\lambda-1)(\lambda+5)\beta_0 -\tfrac{1}{3}(\lambda-5)d^\ast \beta_1, (dd^\ast + \tfrac{2}{3}d^\ast d) \beta_1= \tfrac{2}{3}(\lambda+1)(\lambda+3)\beta_1 + \tfrac{1}{3}(\lambda+9)d\beta_0.\end{cases}
\begin{aligned}\mathbf{Ax}-\mathbf{b}\tau -{{\mathbf{r}}_{p}}\nu =0, -{{\mathbf{A}}^{T}}\mathbf{y}+\mathbf{c}\tau -\mathbf{s}-{{\mathbf{r}}_{d}}\nu =0,{{\mathbf{b}}^{T}}\mathbf{y}-{{\mathbf{c}}^{T}}\mathbf{x}-\kappa -{{r}_{g}}\nu =0, \mathbf{x}\circ\mathbf{s}=\mu\mathbf{e}, \kappa \tau =\mu\end{aligned}
\frac{2Li_{(k_1,...,k_r)}(1-(ab)^{-t})}{(a^{-t}+b^t)^r}=\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_1,...,k_r)}}\left(\frac{\ln a}{\ln a +\ln b}\right)(\ln a +\ln b)^n{t^{n}\over n!}
\mathbb{E}\left[\mathbf{y'}^H\mathbf{s}^{\left(\right)}\right. \left.|\hat{\mathbf{H}}\right]=a_c\sum_{i=1}^M\hat{\phi}^{\left(i,c\right)^*}+\sum_{j=1}^{M}a_j\hat{\phi}^{\left(j,j\right)^*}.
w_0^\pm\circ X^{-1}_\gamma (s,t) =w_0^\pm \circ X^{-1}_\gamma (s, 0)+t\partial_t w_0^\pm \circ X^{-1}_\gamma(s,t)\mid_{t=0}+\dots &=w_0^\pm (s, 0)+t \partial_n w_0^\pm \circ X^{-1}_\gamma(s,0)+\dots
< \exp \{i \sqrt{\pi}[ \phi ( z_{1}) + \phi ( z_{2}) - \phi ( z_{3}^{*}) - \phi ( z_{4}^{*}) ] \}> \propto \left[{\frac{( z_{1}- z_{2}) ( z_{3}^{*}- z_{4}^{*})}{( z_{1}- z_{3}^{*}) ( z_{1}- z_{4}^{*}) ( z_{2}- z_{3}^{*}) ( z_{2}- z_{4}^{*})}}\right]^{1 / 4}
h(x_1,\ldots,x_r):=\sum_{m=1}^r\sum_{(i_1,i_2,\ldots,i_m)}\frac{1}{i_1!i_2!\cdots i_m!}H\tbinom{i_1,i_2,\ldots,i_m}{x_{i_1}',x_{i_2}',\ldots,x_{i_m}'},
|00\rangle:=(\begin{matrix}1\\ 0\\ 0\\ 0\end{matrix});|01\rangle:=(\begin{matrix}0\\ 1\\ 0\\ 0\end{matrix});|10\rangle:=(\begin{matrix}0\\ 0\\ 1\\ 0\end{matrix});|11\rangle:=(\begin{matrix}0\\ 0\\ 0\\ 1\end{matrix}).
[ \, A_{{\sf G I}\, i}^{{\gamma}}({\bf{r}}) \,{\textstyle \frac{{\lambda}^{\gamma}}{2}}\, ] = V_{\cal{C}}({\bf{r}}) \, [ \, A_{i}^{\gamma}({\bf{r}}) \,{\textstyle \frac{\lambda^{\gamma}}{2}}\, ] \, V_{\cal{C}}^{- 1}({\bf{r}}) +{\textstyle \frac{i}{g}}\, V_{\cal{C}}({\bf{r}}) \, \partial_{i}V_{\cal{C}}^{- 1}({\bf{r}}) \, ,
E_{n,l}= \lambda \Bigl[2a(4n+2l+d) + 2n(2n+1) + 2b_1\Bigl(4a+4n-2l-d+3-4\sum_i z_i\Bigr){}- 16b_2\Bigl(\sum_i z_i^2 - \sum_i z_i\Bigr) -24b_3\Bigl(\sum_i z_i^3 - \sum_i z_i^2\Bigr) - l(l+d-1)\Bigr]
E\left(X_1^\lambda X_2^\lambda \cdots X_n^\lambda\right) \leq \left(\prod_{i=1}^{N_1}\left(E\left(X_1^{t_0}\right)\right)^k(1 + \varepsilon)^{2(k-1)}\right)^{\frac{1}{N_1}}& = \left(\left(E(X_1^{t_0})\right)^n (1 + \varepsilon)^{2(n - N_1)}\right)^{\frac{1}{N_1}}\leq \left(\left(E(X_1^{t_0})\right)^n (1 + \varepsilon)^{2n}\right)^{\frac{1}{N_1}}.
W = W^{\mathrm{c l}}\frac{H}{H_{0}}\frac{3 n \left( 1 - \beta^{2}\right)}{1 0}\left( \frac{1 + \zeta}{2}\frac{n + \mu - 1}{n + \mu}+ \frac{1 - \zeta}{2}\right) \left( \frac{5}{6}l_{2}^{2}+ \frac{1}{6}l_{3}^{2}\right) \; .
=\int_{D(0,R)}\frac{|\det D\Phi_{jk}(s,t)| dsdt}{|H_{jk}(Z_j(s,t))-H_{jk}(Z_j(s_1,t_1))|^q|H_{jk}(Z_j(s,t))-H_{jk}(Z_j(s_0,t_0))|^q}
\begin{pmatrix}-a_{0}(0) -a_{0}(0)^{2}-a_{0}(0)^{3}-a_{0}(0)^{4}a_{0}^{(1)}(b_{0}(0)) - a_{0}^{(1)}(0) -2a_{0}(0)a_{0}^{(1)}(0) -3a_{0}(0)^{2}a_{0}^{(1)}(0) -4a_{0}(0)^{3}a_{0}^{1}(0) A B C D E F G H \end{pmatrix}
0 =8\left( N\kappa+n-j+1\right) \left( N\kappa+n-j\right) c_{n+1-2j}& -4\left( N\kappa-\kappa+2n-1-3j\right) \left( N\kappa+n-j\right)c_{n-1-2j}& +\left( n-2j-2\right) \left( n-2j-1\right) c_{n-3-2j}
\Gamma_1= (\zeta^5+\zeta)x^2-(\zeta^4+\zeta^3-1)xy+(\zeta^5+\zeta^3)y^2+ (-\zeta^3+\zeta^2-\zeta)xz+(\zeta^6-\zeta^4-\zeta)yz+(\zeta^5+\zeta^4)z^2.
\varrho_1 E^0_{\tilde{\chi}, \theta_2, \phi_2}(\psi_0)= \int_X(\phi_2-\psi_0)(\theta_1+dd^c\varphi)^n\leq\int_X(-\psi_0)(\theta_1+dd^c\phi_1)^n+nA^n\int_X\omega^n\leq B,
\|I_2\|_{L^2}\leq C\int_{t-h}^{t+h}(t+h-\tau)^{-1}\cdot \frac{(t+h-\tau)^{\beta}}{\tau^{-(\beta+1/2)}}R_1R_2 \leq CR_1R_2h^{\beta}t^{\beta+1/2},
\begin{array}{r l}X_i(x)= \int_{B_0(R)}\mathcal{G}_i(x-y)_j\overrightarrow{\Delta}_\xi X(y)^j\,dx &+\int_{\partial B_0(R)}\mathcal{L}_\xi X(y)^{kl}\nu_k(y)\mathcal{G}_i(x-y)_l\,d\sigma -\int_{\partial B_0(R)}\mathcal{L}_\xi\left(\mathcal{G}_i(x-\cdot)\right)_{kl}(y)\nu(y)^k X(y)^l\,d\sigma.\end{array}
\overline{PR}^2 + \overline{QS}^2 = \frac 12 \left( |A+B|^2 + |C+D|^2 +|B+C|^2+|D+A|^2\right) &= |A|^2+|B|^2+ |C|^2+|D|^2+\left( A \cdot B + C \cdot B + D \cdot A +D\cdot C\right) & = |A|^2+|B|^2+ |C|^2+|D|^2 - |B+D|^2 A+B+C+D=0 & \leq |A|^2+|B|^2+ |C|^2+|D|^2 - \left||B| - |D| \right|^2.
K_{\alpha}(t) =\log M_{\alpha}(t) &=\log(\frac{\sqrt{\pi}}{\Gamma(1/\alpha)}) &\qquad+\log{\rm{H}}_{1,2}^{1,1}\left[\frac{-\sigma^2\Gamma(1/\alpha)}{4\Gamma(3/\alpha)}t^2\left|\genfrac{}{}{0pt}{}{(1-\frac{1}{\alpha},\frac{2}{\alpha})}{(0,1),(\frac{1}{2},1)}\right.\right].
E \left\{e^{i2\pi\frac{d_0}{N}mu_k}\right\}=\frac{1}{2}\int_{-1}^{1}e^{i2\pi\frac{d_0}{N}mx}dx=\frac{\sin(2\pi\frac{d_0}{N}m)}{2\pi\frac{d_0}{N}m}.
1 + \| T\| \geqslant \| T + \lambda I\| \geqslant \| Tx_n + \lambda x_n\| \geqslant \| \lambda x_n + \lambda \| T\| x_n - (-Tx_n + \lambda \| T\| x_n)\| &\geqslant \| \lambda x_n + \lambda \| T\| x_n\| - \| Tx_n - \lambda \| T\| x_n\| = 1+ \| T\| - \| Tx_n - \lambda \| T\| x_n\|.
V = \left( \frac{b}{6}\right)^{2}\frac{| z |^{4}}{( 9 / \gamma )^{2}( 1 - | z |^{2})^{2}}\left\{3 \left| 1 + \ln ( c \omega ( S ) h ( T ) e^{- \tilde{K}/ 2}z^{3}) \right|^{2}+ C_{2}| z |^{2}\right\}
|\sum_{i=1}^{n}\sum_{j=1}^{n}\lambda_{i}\lambda_{j}log\frac{g(z_{i})-g(z_{j})}{z_{i}-z_{j}}|\le\sum_{i=1}^{n}\sum_{j=1}^{n}\lambda_{i}\overline{\lambda_{j}}log\frac{z_{i}\overline{z_{j}}}{z_{i}\overline{z_{j}}-1},
h(u_{ij}^{(\pi)}(u_{lm}^{(\pi')})^{*})=\delta_{\pi\pi'}\delta_{il}\dfrac{(Q_{\pi})_{mj}}{d_{\pi}},h((u_{ij}^{(\pi)})^{*}u_{lm}^{(\pi')})=\delta_{\pi\pi'}\delta_{jm}\dfrac{(Q_{\pi}^{-1})_{li}}{d_{\pi}},
\gamma_{i}^{k}= \begin{cases}\|f_{i}^{k}\|_{L^{\infty}}|B(y_{i},2^{k}r)|^{1/p}k=1,\cdots,J_{0}\|f_{i}^{k}\|_{L^{\infty}}|B(\frac{y_{1}+y_{2}}{2}, 2^{J_{0}+1}r)|^{1/p}k=J_{0}+1.\end{cases}
\hat{\sigma}^{2}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}=\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}-\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}x_{i}x_{j}.
x\cdot \nabla u(x) ={}(\alpha - \sigma)\int_{\mathbb{R}^n}\frac{u(z)^p}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz + \int_{\mathbb{R}^n}\frac{p u(z)^{p-1}(z\cdot \nabla u)}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz ={}(\alpha - \sigma)u(x) + \int_{\mathbb{R}^n}\frac{z\cdot \nabla u(z)^{p}}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz.
J_2^n(\theta',\theta) := \int_{\Delta_{\theta,t}^m}\frac{\partial}{\partial x_{l_1}}b_n^{(i)}(s_1,x+B_{s_1}^H) \cdots \frac{\partial}{\partial x_j}b_n^{(l_{m-1})}(s_m,x+B_{s_m}^H) \left( K_H(s_m,\theta') - K_H (s_m,\theta) \right) ds.
x^{*}\left(\int\limits_{a}^{b}f^{\prime}(t)dt\right) =\int\limits_{a}^{b}x^{*}\circ f^{\prime}(t)dt &=\int_{a}^{b}(x^{*}\circ f)^{\prime}(t)dt &=(x^{*}\circ f)(b)-(x^{*}\circ f)(a) &=x^{*}( f(b)- f(a)).
\vert I_1^{(n)}\vert \leq \int_{0}^{dv_n}t\bigg\vert \frac{d}{dt}\bigg(\frac{t}{t^2+v_n^2}\bigg)\bigg\vert dt=\int_{0}^{dv_n}t\bigg\vert \frac{v_n^2-t^2}{(t^2+v_n^2)^2}\bigg\vert dt &\leq \int_{0}^{dv_n}\frac{d}{v_n}dt=d^2.
z_{\alpha \beta}= m \left( \begin{matrix}{\gamma_{a b}^{0}\delta_{q p}+{\cal P}_{a q b p}^{( - )}}&{0}\\{0}&{{\cal P}^{( - )}{}_{\dot{q}\dot{p}}^{a b}}\\ \end{matrix}\right) = \Sigma_{i = 1}^{2 4}\lambda_{\alpha}^{i}\lambda_{\beta}^{i},
\begin{array}{rcl}\mathcal{F}_{1,N}(\mathbb{Q}) \stackrel{\sim}{\rightarrow}\mathcal{F}^1_N(\mathbb{Q}) h(\tau)=\sum_{n\gg-\infty}c_nq^n \mapsto (h\circ\omega_N)(\tau)=h(\tau/N)=\sum_{n\gg-\infty}c_nq^{n/N}\end{array}
j^{-\frac{3r+(e-1)n}{3n}}=q^{\frac{r}{n}+\frac{e}{3}-1}\left(1+\sum_{x\geq 1}\frac{\prod_{i = 0}^{x-1}(3r+(2i+e-1)n)}{(3n)^xx!}(qg(q))^x\right)
V^{g^4}\left( L \right) = \frac{1}{2}g^4 C_A C_F \left({-\frac{1}{L}}\right) \int\limits_{\bf 0}^{\bf y}{dz^i}\int\limits_{\bf 0}^{\bf y}{dz^{\prime j}}D_{ij}(z,z^\prime) .
Z_{r_\ell}(f,2-\ell) =\int_{F^\times}f(0,0,a)|a|^s d^\times a \bigg |_{s=2-\ell}=\int_{|a|\le q^{-A}}f(0)|a|^s d^\times a \bigg |_{s=2-\ell}+\sum_{j=-B}^{A-1}q^{j(\ell-2)}f(0,0,\varpi^j) =\frac{q^{A(\ell-2)}}{1-q^{\ell-2}}f(0)+\sum_{j=-B}^{A-1}q^{j(\ell-2)}f(0,0,\varpi^j),
W(n) := \sum_{y_0 <k \leqslant n/2}\frac{1}{2k^2}\bigg| \sum_{\substack{|\lambda|=n-2k \lambda_1 < k}}a(\lambda) \bigg|^2 \leqslant \frac{1}{2y_0}\sum_{y_0 <k \leqslant n}\frac{1}{k}\bigg| \sum_{\substack{|\lambda|=n-k \lambda_1 < k/2}}a(\lambda) \bigg|^2.
\mathbf{P}^{-\frac{1}{2}}\mathbf{A}\mathbf{P}^{-\frac{1}{2}}\mathbf{y}= \mathbf{P}^{-\frac{1}{2}}\mathbf{b}, \mathbf{x}= \mathbf{P}^{-\frac{1}{2}}\mathbf{y}
\left. \begin{array}{c}\max_{\overline{Q}_{T}}\varphi_{\lambda}\left( x_{1},t\right) =\varphi_{\lambda}\left( b,T/2\right) =e^{2\lambda b^{2}}, \min_{\overline{Q}_{T,\varepsilon}}\varphi_{\lambda}\left( x_{1},t\right)=\exp \left[ 2\lambda \left( a^{2}-\alpha \left( T/2-\varepsilon \right)^{2}\right) \right] .\end{array}\right.
\displaystyle Z^{\scriptscriptstyle BFV}_{\scriptscriptstyle CPI}=\int{\cal D}\Lambda_{\scriptscriptstyle A}{\cal D}\xi^{\scriptscriptstyle A}{\cal D}\bar{\Gamma}_{\scriptscriptstyle A}{\cal D}\Gamma^{\scriptscriptstyle A}e^{i \int d^4x \widetilde{\cal L}^{\scriptscriptstyle BFV}}
k = \max\{8\delta + 1, \log_2(2\delta_\mathcal{H}+ 1), T + R\}, \rho = (2R + \epsilon)\lambda^2 + \epsilon + R \eta = \max\{(8\delta + 1)/2, \lambda(T + K) + \lambda\epsilon, \lambda(R + r) + \lambda\epsilon, \lambda(R + \rho) + \lambda\epsilon\}.
N^{-1}\psi(R(2^{2n_1+1}))N = \left(\begin{array}{cc}1 2^{n_1}0 1 \end{array}\right), N^{-1}\psi(L(2^{2n_1+1}))N = \left(\begin{array}{cc}1 0 2^{3 n_1+2}1 \end{array}\right),
r \sum_{\tau=t+h}^{n}Z_{N,J}^{\tau}+n Z_{I,J}^{t+h-1}+\sum_{\tau=t}^{t+h-2}n^{t+h-\tau}Z_{i_{\tau},J}^{\tau}\leq r+\sum_{\tau=1}^{h-1}n^{\tau}, & \forall \J\subseteq V, \lvert J \rvert = h\in[n-1], \I\subseteq N, \lvert I \rvert = r\in [n-1], t\in [n-h], \i_t,\dotsc,i_{t+h-2}\in N.
\overline{g}(q,p,z) = \overline{g}(q,p,0)+\overline{g}_z(q,p,0)z+\frac{1}{2}\overline{g}_{zz}(q,p,\tau z)z^2 = \overline{g}(q,p,0) + \overline{g}_z(0,0,0)z+\overline{g}_{q_i,z}(0,0,0)q_iz+\overline{g}_{p_i,z}(0,0,0)p_iz +z[b_{ij}(q,p)q_ip_j+c_{ij}(q,p)q_jq_i+d_{ij}(q,p)p_jp_i]+f(q,p,z)z^2.
\begin{array}{r}{\overline{{M}}^{n}: 1 + \displaystyle \sum_{i , j = 0}^{n}\eta_{j k}x^{j}x^{k}= 0 ,}\\{1 + \displaystyle \sum_{i , j = 0}^{n}\eta_{j k}y^{j}y^{k}= 0 .}\\ \end{array}
\frac{|h(z_1+t) - h(z_2 + t)|}{|h(z_1) - h(z_1 + t)|}= \frac{|F(z_1+t) - F(z_2 + t)|}{|F(z_1) - F(z_1 + t)|}& \geq \lambda\left( \frac{|(z_1 + t) - (z_2 + t)|}{|z_1 - (z_1 + t)|}\right) = \lambda\left( \frac{|z_1 - z_2|}{t}\right) \geq \lambda\left( \frac{r}{t}\right).
H_{c}= \int ( \frac{g^{2}}{6 4 \pi^{2}}G_{i j}^{a}G_{i j}^{a}+ \frac{8 \pi^{2}}{g^{2}}( \pi_{i}^{a}+ \frac{1}{8 \pi}\epsilon_{i j}A^{j , a})^{2}- A_{0}^{a}\wedge^{a}) d^{2}x
M_j = \left[\begin{array}{ccccc}M_{11}0 M_{13}M_{14}M_{15}0 M_{22}M_{23}0 M_{25}M_{31}M_{32}0 0 M_{35}M_{41}0 0 0 M_{45}M_{51}M_{52}M_{53}M_{45}M_{55}\end{array}\right]
\begin{array}{l}L(U^\pm, \Psi^\pm)U^\pm=A_1(U^{\pm})\partial_xU^{\pm}+\frac{1}{\Psi_y^{\pm}}(A_2(U^{\pm})-\Psi_x^{\pm}A_1(U^{\pm})-\Psi_z^{\pm}A_3(U^{\pm}))\partial_y U^{\pm}+A_3(U^{\pm})\partial_zU^{\pm}\end{array},
\lbrace A , B \rbrace{}^{*}= \lbrace A , B \rbrace - i [ \lbrace A , \pi_{\theta \, i}+{\frac{i}{2}}\theta_{i}^{*}\rbrace \lbrace \pi_{\theta^{*}\, i}+{\frac{i}{2}}\theta_{i}, B \rbrace + \lbrace A , \pi_{\theta^{*}\, i}+{\frac{i}{2}}\theta_{i}\rbrace \lbrace \pi_{\theta \, i}+{\frac{i}{2}}\theta_{i}^{*}, B \rbrace ] ,
\langle\nabla_{X_{1}}\nabla_{X_{2}}X_{2},X_{1}\rangle=\displaystyle\frac{c\lambda_{1}(\nu^{2}-b_{1}^{2})}{\lambda_{2}-\lambda_{1}}-\displaystyle\frac{\nu^{2}b_{2}^{2}}{(\lambda_{2}-\lambda_{1})^{2}}.
\mbox{ind}_{C(\lambda_0; \varepsilon)}(\mathbf{T}-W_\lambda)=\mbox{tr}_{\mathbb{C}^m}\oint_{C(\lambda_0; \varepsilon)}{W_\xi'}(W_\xi-\mathbf{T})^{-1}d\xi, 0<\varepsilon<\varepsilon_{0}.
\quad\left( \partial_t u^*, A^{2s}u^* \right)_H + \nu \left( Au^*, A^{2s}u^* \right)_H &= - \left( e^{\varphi A^{\alpha}}B(u) , A^{2s}u^* \right)_H - \left( h^*, A^{2s}u^* \right)_H + \frac{d \varphi}{dt}\left( A^{\alpha}u^* , A^{2s}u^* \right)_H .
\frac{\partial}{\partial t}\varepsilon^{klm}=\frac{\partial}{\partial t}\left(\frac{\eta^{klm}}{\sqrt{\det (g_{ij})}}\right)=-\frac{\eta^{klm}}{\det (g_{ij})}\cdot\left(-R\sqrt{\det (g_{ij})}\right)=R\varepsilon^{klm}.
T_{+ 2 - 2}^{- q}= \frac 1 2 \gamma_{q \dot{q}}^{i}\Omega_{- 2}^{- 2 i}\psi_{+ 2 \dot{q}}^{2 +}- \frac 1 2 \gamma_{q \dot{q}}^{i}\Omega_{+ 2}^{- 2 i}\psi_{- 2 \dot{q}}^{1 -},
RT_e(\Phi) =\sum_{\gamma_0}\frac{\tau(U^{\sigma}\times U^{\sigma})}{d(U_{\gamma_0},U^{\sigma}\times U^{\sigma})}SRO_{\gamma_0}(\Phi) TRT_e(f) =\sum_{\delta_0}\frac{\tau(G^{\sigma}\times G^{\theta})}{d(G_{\delta_0},G^{\sigma}\times G^{\theta})}STRO_{\delta_0}(f).
\begin{array}{lr}(P_jQ_{jk}P_{k}w)(x,t) = c_j(x_j,x,t) [2mm]\displaystyle\quad\times\int_{\omega_k(x_k,0,t)}^{\omega_k(x_k,1,t)}\hskip-5mm r_{jk}(\tilde{\omega}_k(t,x_k,z),z)c_k(x_k,\tilde{\omega}_k(t,x_k,z),z) [2mm]\times \displaystyle \int_{0}^{1}\partial_{3}\tilde{\omega}_k(t,x_k,z)w(\eta,z)\, d\eta dz, \end{array}
\lim_{N\to\infty}\Bigl( A^{(+)}q^Ncq^{-2N-1}(1-q)[\mathcal{P},\mathcal{Q}](0)q^N +A^{(-)}cq^Nc^{-1}q^{-2N-1}(1-q) [\mathcal{P},\mathcal{Q}](0)(-c)q^N\Bigr) =\bigl(A^{(+)}-A^{(-)}\bigr)cq^{-1}(1-q) [\mathcal{P},\mathcal{Q}](0),
\left( \begin{array}{c}a_{D}(u) a(u) \end{array}\right)\rightarrow M_{i}^{Higgs}\left( \begin{array}{c}a_{D}(u) a(u) \end{array}\right) = \left( \begin{array}{c}a_{D}(u e^{2 \pi i}) a(ue^{2 \pi i})=-a \end{array}\right),
R^{*}(n, D, \epsilon, \delta) := \inf \{R :\mbox{$\exists$ an $(n, D, R, \epsilon, \delta)${\rm code}}\}. \tilde{R}(n, D, \epsilon, \delta) := \inf \{R : \mbox{$\exists${\rm a deterministic}$(n, D, R, \epsilon, \delta)${\rm code}}\}.
\|\nabla u\|^{p}_{L^p(\Omega)}\lesssim \sum_{Q_i}\sum_{Q_j\cap Q_i\neq \emptyset}\ell(Q_i)^{1-p}P(A,Q_i\cup Q_j) \lesssim \sum_{Q_i}\ell(Q_i)^{1-p}P(A,Q_i).
{\rm Tr}\;S\Delta\vartheta_a^{(1)}P_-\vartheta_a^{(3)}\cdots P_-\vartheta_a^{(2n+1)}P_-\vartheta_a^{(2)}&={\rm Tr}\;\vartheta_a^{(2)}S\Delta\vartheta_a^{(1)}P_-\vartheta_a^{(3)}\cdots P_-\vartheta_a^{(2n+1)}P_- &={\rm Tr}\;S\Delta\vartheta_a^{(1)}\vartheta_a^{(2)}P_-\vartheta_a^{(3)}\cdots P_-\vartheta_a^{(2n+1)}P_-.
\left\langle{\bf A},{\bf B}\right\rangle:=\sum\limits_{i=1}^{R_a}\sum\limits_{j=1}^{R_b}c_i c_j \prod\limits_{\ell=1}^d \left\langle{\bf a}_i^{(\ell)},{\bf b}_j^{(\ell)}\right\rangle,
[e_r, e_r]=\theta_re_n, [e_i, e_j]=\sum^{n-1}_{t=1}\alpha^t_{ij}e_t+\beta_{ij}e_n, [e_j, e_i]=-\sum^{n-1}_{t=1}\alpha^t_{ij}e_t+\gamma_{ji}e_n.
\mathcal{Q}'(g)\phi = \frac{n-4}{2}\int \phi P_g \big( P_g^{-1}\big( Q_g - \overline{Q}_g \big)\big)\dv_g &= \frac{n-4}{2}\int (P_g \phi) \big( P_g^{-1}\big( Q_g - \overline{Q}_g \big)\big)\dv_g &= \frac{n-4}{2}\big\langle \phi, P_g^{-1}\big( Q_g - \overline{Q}_g \big) \big\rangle_{W^{2,2}}.
\Big\| \sum_{n = N_1}^{N_2-1}\big| H_{n+1}f - H_n f \big| \Big\|_{\ell^p}\leq\Big\| \sum_{n = N_1}^{N_2-1}\big| h_{n+1}- h_n \big| \Big\|_{\ell^1}\|f\|_{\ell^p}&\lesssim N_1^{-k}\big(\vartheta_B(N_2) - \vartheta_B(N_1)\big) \|f\|_{\ell^p},
\mathbb{P}\left( \bigcap_{k =1}^j A_{n,i_k}\right) =\mathbb{P}\left( \min_{1 \leq k \leq j}\left(\frac{\varepsilon_{i_k}}{P_{i_k}}\right) > \frac{\varepsilon}{T_n}\right) & = \mathbb{E}\exp\left(-\frac{\varepsilon( P_{i_1}+ \cdots + P_{i_j})}{T_n}\right) & = \mathbb{E}\left( \frac{T_n}{T_n + P_{i_1}+ \cdots + P_{i_j}}\right),
\begin{aligned}f_t-\frac{1}{2\sqrt{t}}f'' + \frac{1}{2\sqrt{t}}\geq 0, f - f^* \geq 0, \left(f_t-\frac{1}{2\sqrt{t}}f'' + \frac{1}{2\sqrt{t}}= 0\right) \vee (f-f^* =0),\end{aligned}
{\frac{1}{A}}{\frac{\partial{\cal E}}{\partial m^{2}}}={\frac{1}{2}}\sum_{n = 1}^{\infty}\int \int{\frac{d k_{1}d k_{2}}{( 2 \pi )^{2}}}{\frac{1}{\sqrt{k_{1}^{2}+ k_{2}^{2}+{\frac{n^{2}\pi^{2}}{a^{2}}}+ m^{2}}}}.
\widehat K_{S}(z,w) = \begin{bmatrix}K_{S}(z,w) \frac{S(z) - S(\overline{w})}{z - \overline{w}}\frac{\widetilde S(z) - \widetilde S(\overline{w})}{z - \overline{w}}K_{\widetilde S}(z,w) \end{bmatrix}\quad\mbox{where}\widetilde S(z):= S(\overline{z})^{*}.
\int_{X^n}{dz_2\ldots dz_{n+1}\over \Pi_{i=1}^{n+1}|z_i-z_{i+1}|^{n-1}}\le C\int_{X^{n-l+2}}{dz_l\ldots dz_{n+1}\over |z_1-z_l|^{n-l+1}\Pi_{i=l}^{n+1}|z_i-z_{i+1}|^{n-1}}
T r K_{H_{2}^{\alpha}}= \alpha T r K_{H_{2}}+ \alpha{\frac{e^{-{\bar{s}/ 4}}}{( 4 \pi \bar{s})^{1 / 2}}}\int_{0}^{\infty}d y \cosh y f ( y , \alpha ) e^{-{y^{2}/ \bar{s}}}~ ~ ,
g_{D+d}= ||g_{MN}|| = \left( \begin{array}{ccc}g_1 g_1 K g_1{\cal A}K^T g_1 g_2 + K^T g_1 K g_2{\cal B}+K^T g_1{\cal A}{\cal A}^T g_1{\cal B}^T g_2 +{\cal A}^T g_1 K e^{-\psi/2}g_D +{\cal A}^T g_1{\cal A}+{\cal B}^T g_2{\cal B}\end{array}\right),
L_i(\lambda)=\left(\begin{array}{cccccc}0 0 0 \dots x_i x_i+y_i -\lambda 0 0 \dots 0 0 0 1 0 \dots 0 0 0 0 1 \dots 0 0 \dots \dots \dots \dots \dots \dots 0 0 0 \dots 1 1 \end{array}\right)
\|\rho^N\|_{H^m}\leq \|\rho_0\|_{H^m}\exp\left(C\|\rho_0\|_{L^\infty}t+\frac{\|\nabla\rho_0\|_{L^\infty}}{\|\rho_0\|_{L^\infty}}\left(\exp(C\|\rho_0\|_{L^\infty}t)-1\right)\right) =:B_{\rho_0}(t).
\tilde{\psi}_n\left( \mathbf{p}\right) =\tilde{\psi}_n^{\left( 0\right)}\left( \mathbf{p}\right) -T\sum_{m=-\infty}^\infty \tilde{S}_F^{n-m}\left( \mathbf{p}\right) \tilde{\eta}_m\left( \mathbf{p}\right) ,
R_{\mu\nu}- \frac{1}{2}\, G_{\mu\nu}\,{\cal R}- \left(\partial_\mu\varphi\,\partial_\nu\varphi -\frac{1}{2}\,G_{\mu\nu}\,\partial_\rho\varphi\,\partial^\rho\varphi \right) -\left( \partial_\mu\eta_a\,\partial_\nu\eta_a -\frac{1}{2}\,G_{\mu\nu}\,\partial_\rho\eta_a\,\partial^\rho\eta_a\right)
(i\hbar)^{2}\frac{\partial^{2}\psi(x,t)}{\partial t^{2}}=(i\hbar)\frac{\partial}{\partial t}(i\hbar)\frac{\partial\psi(x,t)}{\partial t}=(i\hbar)\frac{\partial}{\partial t}(\beta mc^{2}-i\hbar\sum_{k=1}^{3}\alpha_{k}\nabla_{k}c)\psi(x,t)
\frac{\partial}{\partial t}\int_{-(q_{0}+l)}^{(q_{0}+l)}\Psi^{*}\Psi dx=2\cdot\frac{\hbar}{2mi}(\Psi_{1}^{*}\frac{\partial\Psi_{1}}{\partial x}-\Psi_{1}\frac{\partial\Psi_{1}^{*}}{\partial x}),
G(z,t)= e^{-t (c_{1}\{(\lambda_{1}+\mu(1-z))^{\alpha_{1}}-{\lambda_{1}}^{\alpha_{1}}\}+c_{2}\{(\lambda_{2}+\mu(1-z))^{\alpha_{2}}-{\lambda_{2}}^{\alpha_{2}}\})}, \; |z|\leq 1, \;\; \mu \leq \frac{\lambda_{i}}{2},\; i=1,2.
\begin{array}{lllllll}\omega(e_r,f_r) = 1, \omega(e_r,\phi_2) = 1, \omega(e_r,\phi_3) = \varepsilon a_2, \omega(f_r,\phi_2) = \varepsilon, \omega(f_r,\phi_3) = \varepsilon a_1, \omega(\phi_2,\phi_3) = \varepsilon. \end{array}
T_R\bigl((\operatorname{id}\otimes\omega)(V)\Lambda_{\psi}(y)\bigr) =T_R\bigl(\Lambda_{\psi}((\operatorname{id}\otimes\omega)(\Delta y))\bigr) =\Lambda_{\psi}\bigl((\operatorname{id}\otimes\bar{\omega})(\Delta(y^*))\bigr) &=(\operatorname{id}\otimes\bar{\omega})(V)\Lambda_{\psi}(y^*)=(\operatorname{id}\otimes\bar{\omega})(V)T_R\Lambda_{\psi}(y).
e_h(t) =F_h(t)u_0+\int_0^t F_h(t-s)\tilde \xi(s)ds+\int_0^t F_h(t-s)b( \tilde u (s))ds &+\int_0^t E_h(t-s)P_h \Big( b( \tilde u (s))-b( \tilde u_h(s))\Big)ds.
Z^+(f, L; s, \phi) = \int_{G_+/\Gamma, \chi(g)\geq 1}\chi(g)^s\phi(g)\sum_{x \in L \setminus L_0}f(g\cdot x)dg, Z^+(f, \hat{L}; s, \phi) = \int_{G_+/\Gamma, \chi(g)\geq 1}\chi(g)^s\phi(g)\sum_{x \in \hat{L}\setminus \hat{L}_0}f(g\cdot x)dg.
\frac{j-1}{n-j}\sum_{i=j+1}^{n}\frac{1}{i-2}\;\;\mbox{is increasing in}\;\;2\le j<n; \mbox{and}\;\; \frac{1}{n-j}\sum_{i=j+1}^{n}\frac{1}{i-2}\;\;\mbox{is decreasing in}\;\;2\le j<n.
\int_{\Omega_{\varepsilon_{n_k}}}|2w - e^{-\beta_{\varepsilon_{n_k}}\Psi_{\varepsilon_{n_k}}(\mathbf{q})}V_{\varepsilon_{n_k}}| |V_{\varepsilon_{n_k}}| = o\left(e^{D \sigma_1 \beta_{\varepsilon_{n_k}}\Psi_{\varepsilon_{n_k}}(\mathbf{q})}\right)
\digamma_{t}=\sqrt{D_{\left\{0\right\}}}B_{t}+\int\nolimits_{0}^{t}\int\nolimits_{\left\vert \nu \right\vert \geq 1}\nu N\left( \mathrm{d}s\mathrm{d}\nu \right) +\int\nolimits_{0}^{t}\int\nolimits_{\left\vert \nu \right\vert<1}\nu M\left( \mathrm{d}s\mathrm{d}\nu \right) \ ,t\in \mathbb{R}_{0}^{+}\ .
|C_m(\gamma_1,\gamma_2;\mathbf{t},\mathbf{c})| \le C \sum_{\sigma \in \{1,2\}^m}\int_{[0,T]^m}d\mathbf{s}\prod_{j=1}^{m}|s_j-s_{j-1}|^{\gamma_{\sigma_j}+\gamma_{\sigma_{j-1}'}+1}\le C \sum_{\sigma \in \{1,2\}^m}\int_{[0,1]^m}d\mathbf{s}\prod_{j=1}^{m}|s_j-s_{j-1}|^{\gamma_{\sigma_j}+\gamma_{\sigma_{j-1}'}+1},
\langle J'(u,v), (u,v)\rangle= 2\big[b+\kappa 2^*(s_2)c\big]-2^*(s_1)b-k\big(2^*(s_2)\big)^2c = \big[2-2^*(s_1)\big]b+\big(2-2^*(s_2)\big) 2^*(s_2)\kappa c < \max\{2-2^*(s_1),2-2^*(s_2)\}\big[b+ 2^*(s_2)\kappa c\big] = \max\{2-2^*(s_1),2-2^*(s_2)\}a.
\begin{tabular}[c]{l}$x^{n}=P_{n}(x;z)+c_{n}\left( z\right) P_{n-2}(x;z)$ $-\left[ d_{n}\left( z\right) -c_{n}\left( z\right) c_{n-2}\left(z\right) \right] P_{n-4}(x;z)+O\left( x^{n-6}\right) .$\end{tabular}
\begin{gathered}s_\alpha^j \colon C^n(\mathcal{C}) \to C^{n+1}(\mathcal{C})\,; j=1,\dots,n\,,\; \alpha = -1,1\,, s_\alpha^j (S) (a_1,\dots,a_{n+1}) = \begin{cases}S(a_1,\dots,a_{j-1},a_{j+1}\dots,a_{n+1}) a_j \neq \alpha 0 a_j = \alpha \,. \end{cases}\end{gathered}
\kappa_m\left(\sum_{i=1}^n c_i Z_{\gamma_1,\gamma_2}(t_i)\right)\rightarrow \frac{(a+b)^m - (a-b)^m}{(a^2+b^2)^{m/2}}\left(\sum_{i=1}^n c_i t_i\right)^m \frac{(m-1)!}{2},
a(\Lambda_1,\dotsc,\Lambda_n) \cdot T = a(\Lambda_1,\dotsc,\Lambda_n)\Bigl(-\sum_{k=1}^n \Lambda_k\Bigr), a(\Lambda_1,\dotsc,\Lambda_n) \cdot S^i = a(\Lambda_1,\dotsc,\Lambda_n)\Bigl(-\sum_{k=1}^n \theta_k^i\Bigr)
\begin{array}{l}q_0 = \sqrt{\frac{25-D}{6}- \vec{q}^{\;2}}\ , p_{0_{\pm}}= q_0 \pm \sqrt{q_0^{2}- \vec{p}^{\;2}+ 2\vec{q}\vec{p}- 4}= \frac{1}{\sqrt{6}}\left( \sqrt{25-D - 6\vec{q}^{\;2}}\pm \sqrt{1-D - 6(\vec{q}- \vec{p})^2}\right) \ . \end{array}
t_2:= \big(\lambda_hu_{\mathrm{nc}}, (I-\widehat{I})(u-\widehat{u}_{\mathrm{nc}})\big)_{1+\delta}\le2\Vert h_{\mathcal{T}}^{m}\lambda_h u_{\mathrm{nc}}\Vert_{L^2(\mathcal{T}\setminus\widehat{\mathcal{T}})}\Vert h_{\mathcal{T}}^{-{m}}(I-\widehat{I})(u-\widehat{u}_{\mathrm{nc}}) \Vert_{L^2(\Omega)}.
w_+ = \frac{1-\rho^n}{1-\rho^{n+1}}= \frac{\sum_{m=0}^{n-1}\rho^m}{\sum_{m=0}^n \rho^m}, w_- = \frac{\rho(1-\rho^n)}{1-\rho^{n+1}}= \frac{\sum_{m=0}^{n-1}\rho^{m+1}}{\sum_{m=0}^n \rho^m}, K = \frac{(1-\rho)\rho^n (1-\rho^n)^n}{(1-\rho^{n+1})^{n+1}}= \frac{( \sum_{m=0}^{n-1}\rho^{m+1})^n}{( \sum_{m=0}^n \rho^m )^{n+1}}.
\mathbf{A}\cdot\left(\mathbf{B}\pm\mathbf{C}\right)=\mathbf{A}\cdot\mathbf{B}\pm\mathbf{A}\cdot\mathbf{C}\,\,\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,\,\,\,\left(\mathbf{B}\pm\mathbf{C}\right)\cdot\mathbf{A}=\mathbf{B}\cdot\mathbf{A}\pm\mathbf{C}\cdot\mathbf{A}
= \frac{1}{( 2 \pi )^{l}}\left( \frac{- 1}{2 L}\right)^{l}\exp \left( \frac{1}{2}l^{2}e^{2}\frac{1}{\pi^{2}}\int d^{2}p \frac{1}{p^{2}+ m_{d}^{2}}\frac{1}{p^{2}}\sin^{2}( p_{1}L ) \right)
\begin{aligned}\int_0^{\tau}U(t_n+\tau,t_n+r)\left [| U(t_n+r,t_n)u(t_n) |^2 U(t_n+r,t_n)u(t_n)\right] d r &\approx\tau U(t_n+\tau,t_n+\tau\xi_n)\left [| U(t_n+\tau\xi_n,t_n)u(t_n) |^2 U(t_n+\tau\xi_n,t_n)u(t_n)\right].\end{aligned}
\begin{array}{ll}i_{\Delta f}(\Delta g) (X_0, X_1, \cdots, X_{p+q}) &= \sum_{k=0}^p (-1)^{kq}\sum_{\sigma \in Sh(q, p-k)}\epsilon(\sigma) \Delta f \circ_k^\sigma \Delta g(X_0, X_1, \cdots, X_{p+q}).\end{array}
\begin{array}{c}{{L = g_{\alpha \beta}( d X^{\alpha}\cdot d X^{\beta}) + 2 g_{\alpha a}( d X^{\alpha}\cdot d X^{a}) + g_{a b}( d \Phi^{a}\cdot d \Phi^{b}) +}}\\{{+ b_{\alpha \beta}d X^{\alpha}\wedge d X^{\beta}+ 2 b_{\alpha a}d X^{\alpha}\wedge d \Phi^{a}+ b_{a b}d \Phi^{a}\wedge d \Phi^{b}}}\\ \end{array}
G_{\alpha}^{( n )}\equiv \sum_{m = 0}^{n}\{\tau_{\alpha}^{( n - m )}, \tilde{H}^{( m )}\}+ \sum_{m = 0}^{n - 2}\{\tau_{\alpha}^{( n - m )}, \tilde{H}^{( m + 2 )}\}_{( \Phi )}+ \{\tau_{\alpha}^{( n + 1 )}, \tilde{H}^{( 1 )}\}_{( \Phi )}, ~ ~ n \geq 2
\mbox{\mathversion{bold}$w$}^{1}=\mbox{\mathversion{bold}$v$}_{1}\times \mbox{\mathversion{bold}$e$}^{1}\mbox{\mathversion{bold}$w$}^{2}=\mbox{\mathversion{bold}$v$}_{2}\times \mbox{\mathversion{bold}$e$}^{2}.
K=\frac{3}{5}E_{F}=\frac{3}{5}\frac{\hbar^{2}k_{F}^{2}}{2m_{e}}=\frac{3}{5}(\frac{9\pi}{4})^{\frac{2}{3}}\frac{1}{(r_{s}/a_{0})^{2}}Ry\approx\frac{2.21}{(r_{s}/a_{0})^{2}}Ry
\left\langle A_{\alpha}\mathbf{x},\mathbf{x}\right\rangle =\alpha\left(2m+\chi\left( \chi-2\right) e_{k}\right) +2\left( 1-\alpha\right) \left(m-\chi e_{k}\right) & =2m+\left( \alpha\chi-2\right) \chi e_{k}.