harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{DTV}_{(p,q)}(X)=-\mathrm{DTE}_{(p,q)}^{2}(X)+\mu^{2}+2\mu\sigma\mathrm{DTE}_{(p,q)}(Y)+\sigma^{2}\left(L_{1}+\frac{c_{1}}{c_{(1)}^{\ast}}L_{2}\right),
U ( r ) = ( 1 + b^{2})^{2}r^{\frac{2}{1 + b^{2}}}\left( - 2 M r^{-{\frac{D - 1 - b^{2}}{1 + b^{2}}}}-{\frac{2 \Lambda}{( D - 1 - b^{2})}}\right) \, ,
(2n-4) \langle J_{ij}, F\rangle =\sum_{k\neq i,j}\langle J_{ij}, [J_{ki},\omega_{ki}] \rangle+\langle J_{ij}, [J_{kj},\omega_{kj}] \rangle &=\sum_{k\neq i,j}\langle [J_{ij}, J_{ki}] ,\omega_{ki}\rangle+\langle [J_{ij}, J_{kj}],\omega_{kj}\rangle
S_1(q) = \{j \in \mathcal{Z}_N\, : \, |q_1- q_j| < N^{-1}\}S_m(q) = \{j \in \mathcal{Z}_N \, : \, |q_j - q_k|< N^{-1}\,\, \exists \,\,k \in S_{m-1}(q)\}, \,\,\,\, m =2, \ldots, N.
g_q(W) = \frac{(q \alpha_1(q)^{-1}W, q)_\infty (q \alpha_2(q)^{-1}W, q)_\infty (q \alpha_3(q)^{-1}W, q)_\infty}{(qW, q)_\infty (iqW, q)_\infty (-qW, q)_\infty}
\| F_{u_0}(\eta) - F_{v_0}(\eta) \|_{1,\alpha}= \big\| \int_{\mathbb{R}^2}K_2\big( \eta(t, \cdot) - \eta(t,y) \big) \big( \nabla^\perp\cdot u_0(y) - \nabla^\perp\cdot v_0(y) \big) dy \big\|_{1,\alpha}\lesssim \| \nabla^\perp\cdot (u_0 - v_0) \|_\infty + \big[ \nabla^\perp\cdot (u_0 - v_0) \big]_\alpha \lesssim \| u_0 - v_0 \|_{1,\alpha}
D(\tau) = \omega_{1r}\wedge d\tau + \Big[ \left(\frac{r}{q}+t_2\right) \d\omega_{1r}+ \nu_r \Big]\wedge\tau D(\tau) = \omega_{1r}\wedge d\tau + r\left(\frac{1}{q}\d\omega_{1r}+ \omega_{2r}\right)\wedge\tau + (t_2 \d\omega_{1r}+\mu_{r})\wedge\tau.
|X| \leq210(k'+1)^2\log k + 10\ell k'+ 210(k-k')^2\log k+10\ell(k-k'-1) &\le 210\log k \left(k'^2+2k' +1 + k^2-2kk'+k'^2\right) + 10\ell(k-1) &= 210\log k \left(2k'(k'+1-k) + 1 +k^2\right) + 10\ell(k-1) &\le 210 k^2 \log k + 10\ell(k-1) = f(k,\ell)
\phi_{\xi^{\prime}}( x_{1}, x_{2}) = e^{i [ \chi_{\xi^{\prime}}( x ) - \chi_{\xi}( x ) ]}\phi_{\xi}( x_{1}, x_{2}) \ , \, p h i_{C}( x_{1}, x_{2}) = e^{i [ \chi_{C}( x ) - \chi_{\xi}( x ) ]}\phi_{\xi}( x_{1}, x_{2}) \ .
m \langle \bar{Q}T^aQ \;{\rm Tr}\; T^a A^2 \rangle =-\frac{1}{32\pi^2}\langle{\rm Tr}\; T^a W_{\alpha}W^{\alpha}\;{\rm Tr}\; T^a A^2 \rangle = \Lambda^{11/3}m^{4/3}= \Lambda_{LE}^5,
S_{p,n}=\sum_{\underset{k_{i}\ge0}{k_{1}+\dots+k_{p}=n}}\binom{n}{k_{1},\dots,k_{p}}w_{k_{1}}\dots w_{k_{p}}=\left(-1\right)^{p}n!\sum_{\underset{k_{i}\ge0}{k_{1}+\dots+k_{p}=n}}g_{k_{1}}\dots g_{k_{p}}
\Delta ={\alpha}^{2}{\gamma}^{2}-{\alpha}^{2}{\tau}^{2}-2\,\alpha\,{\delta}^{2}\gamma-4\,\alpha\,\delta\,\nu\,\tau-2\,\alpha\,\gamma\,{\nu}^{2}+{\delta}^{4}+2\,{\delta}^{2}\mu\,\tau-2\,{\delta}^{2}{\nu}^{2}+4\, \delta\,\gamma\,\mu\,\nu-{\gamma}^{2}{\mu}^{2}+{\mu}^{2}{\tau}^{2}+2\, \mu\,{\nu}^{2}\tau+{\nu}^{4}.
\lambda_{1}= \frac{\frac{\partial\overline{E}^{C}_{T}}{\partial P_{S}}\:-\frac{\partial\overline{E}^{C}_{T}}{\partial{P}_{R}}}{\left(\frac{\partial\overline{\rm BER}^{C}_{D}}{\partial{P}_{R}}-\frac{\partial\overline{\rm BER}^{C}_{D}}{\partial P_{S}}\right)}.
:\!{\cal F}[X] \! :\,\, :\!{\cal G}[X] \! :\ =\exp\biggl\{-\int d^2z'\,d^2 z'' \, \ln|z'-z''|^2\frac{\delta_F}{\delta X(z',\bar z')}\frac{\delta_G}{\delta X(z'',\bar z'')}\biggr\}:\!{\cal F}[X]\,{\cal G}[X] \! :,
\iint_{Q_T}p\Delta p (\nabla p \cdot \nabla m) = -\iint_{Q_T}|\nabla p|^2[\nabla p \cdot \nabla m] - \underbrace{\iint_{Q_T}p [\nabla p \cdot \nabla( \nabla p \cdot \nabla m)]}_{\mathcal{J}_{2,2}}.
b_i^{S}= \begin{cases}\displaystyle\theta_i^1 + \beta \, \sum_{j \in N}p_{ij}^1 \, b_j^S i\in S \cup N^{\{1\}}\displaystyle\beta \, \sum_{j \in N}p_{ij}^0 \, b^S_j, i\in N^{\{0, 1\}}\setminus S;\end{cases}
\P(Z^g_1 = Z^g_2 = \dots = Z^g_{\ell-1}=-1)>p, & \P(Z^g_1 = Z^g_2 = \dots = Z^g_{\ell-1}=0)>p, & \P(Z^g_1 = Z^g_2 = \dots = Z^g_{\ell-1}=1)>p.
j_{a_{1}}= - \frac 1{\lambda^{2}}Z_{\; a_{1}}^{a}\Delta_{a b}\left( \lambda f^{b}\left( Q \right) + Z_{\; b_{1}}^{b}y^{b_{1}}- \dot{z}^{b}\right) \equiv - \lambda Z_{\; a_{1}}^{a}j_{a},
d(v_1'\Rightarrow w_2 v_2') = d(v_1'\Rightarrow \rho^\vee_{x_1}(w_2 v_2')) + d(\rho^\vee_{x_1}(w_2 v_2')\Rightarrow w_2 v_2'), &d(v_1'\Rightarrow w_2 v_2') = d(v_1'\Rightarrow w_2 \rho_{x_2}(v_1)) + d(w_2 \rho_{x_2}(v_1)\Rightarrow w_2 v_2').
\left< \mathcal{O}_{-}^{i n}( \Omega ) \mathcal{O}_{-}^{i n}( \Omega^{\prime}) \right> = 0 \ , \left< \mathcal{O}_{+}^{i n}( \Omega ) \mathcal{O}_{-}^{i n}( \Omega^{\prime}) \right> = 0
Z_{r_i}(f,s) =\int_{F^\times \times K}\chi(a)|a|^{-s}(r_i(k)f)\left( 0_{V_i},0,a^{-1}\right) dk d^\times a &=\int_{F^\times \times K}\chi(a)|a|^{s}(r_i(k)f)\left(0_{V_i},0,a\right) dk d^\times a.
( u_{\sigma}\, p_{*}u_{0}- u_{0}\, p_{*}u_{b}) \bigg |_{- \infty}^{\infty}= ( \sigma_{0}^{2}- \sigma_{b}^{2}) \int_{- \infty}^{\infty}\frac{{\cal E}}{{\cal A}}\, u_{b}\, u_{0}\, d r^{*}
\pi_t \left(dY_{1,t}\right) = \sqrt{1 - r^2}\pi_t\left(e^{i\theta}L_t + e^{-i\theta}L_t^{\dagger}\right) dt, \pi_t \left(dY_{2,t}dY_{2,t}\right) = \pi_t \left(dY_{2,t}\right) = r^2\pi_t\left(L^{\dagger}_t L_t\right) dt, \pi_t \left(dY_{1,t}dY_{1,t}\right) = dt, \pi_t \left(dY_{2,t}dY_{1,t}\right) = \pi_t \left(dY_{1,t}dY_{2,t}\right) = 0.
\int d^{4}z_{1}d^{4}z_{2}\langle\Phi(x_{1})\Phi(x_{2})\Phi^{*}(y_{1})\Phi^{*}(y_{2}):(\Phi\Phi^{*})^{2}:(z_{1}):(\Phi\Phi^{*})^{2}:(z_{2})\rangle
[D(m,m j+k):D(m+1,n)]_q =q^{4kj}[D(m,m j -k):D(m+1,n)]_q & =\begin{cases}q^{j(m j + 2k)}n=k, q^{(j + 1)(m j+ 2k - m)}n=m-k, 0 \end{cases}
I = \frac{1}{2}\mu\int d^{3}x[-\epsilon^{mnp}A_{m}^{i}F_{np(a + v)}^{i}- \mu a_{m}^{i}a^{mi}+ \epsilon^{mnp}v_{m}^{i}(\partial_{n}v_{p}^{i}+ \frac{1}{3}\epsilon^{ijk}v_{n}^{j}v_{p}^{k}) ] ,
\begin{aligned}E^{*+}(f) =E(f)_{[x^*,x_{j+1}]}+C^+-\frac{\alpha^2}{4h}(h-\alpha)^2[f'']+O(h^4)=-\frac{1}{12}(h-\alpha)^3 f^+_{xx}(\eta^+),\textrm{with}\eta^+\in [x^*, x_{j+1}], C^+ =-\left({\frac{\left( h-{\alpha}\right)^2}{2h}}[f]-{\frac{\left( h-{\alpha}\right)^2\alpha}{2h}}[f']\right).\end{aligned}
\mu_{tor}(\Omega,\eta)=\int_{\nu^{-1}_{\Omega}(\eta)}|\overline{\nabla}U(X)|^2d\mathcal{H}^{n-1}(X)=\int_{\eta}|\overline{\nabla}U(\nu^{-1}_{\Omega}(x))|^2dS_{\Omega}(x),
|u|_0 - |v|_0 = s + (|u|_{10^{d+N}}- |v|_{10^{d+N}})(d+N) +( |u|_{0^d*}-|v|_{0^d*})\,d &= s + (|u|_{10^{d+N}}- |v|_{10^{d+N}})\,N +((|u|_{10^{d+N}}+ |u|_{0^d*}) -(|v|_{10^{d+N}}+ |v|_{0^d*}) )\,d.
\alpha_1\big(\Phi_4(\mathbf{a},\mathbf{b},c_1)\big) = a_1, \beta_1\big(\Phi_4(\mathbf{a},\mathbf{b},c_1)\big) = b_1, \gamma_1\big(\Phi_4(\mathbf{a},\mathbf{b},c_1)\big) = c_1, \alpha_2\big(\Phi_4(\mathbf{a},\mathbf{b},c_1)\big) = a_2, \beta_2\big(\Phi_4(\mathbf{a},\mathbf{b},c_1)\big) = b_2.
\Lambda_{i j , \, l m}( \hat{k}) = \delta_{i l}\delta_{j m}- 2 \hat{k}_{j}\hat{k}_{m}\delta_{i l}+ \frac{1}{D - 2}\left( - \delta_{i j}\delta_{l m}+ \hat{k}_{l}\hat{k}_{m}\delta_{i j}+ \hat{k}_{i}\hat{k}_{j}\delta_{l m}\right) + \frac{D - 3}{D - 2}\hat{k}_{i}\hat{k}_{j}\hat{k}_{l}\hat{k}_{m}\, .
y_i' = (\alpha_{i+2})^{-1}\cdot y_{i+1}^{-1}\cdot \alpha_i, y_{i^-}' = (\alpha_{i})^{-1}\cdot y_i \, y_i^- \cdot \alpha_{i+1}, y_{i^+}' = (\alpha_{i+1})^{-1}\cdot y_{i+1}\, y_i^+ \cdot \alpha_{i+2},
u_\lambda(x,y) = \mathbb{E}\int_0^\infty \exp(-\lambda t) g(X_t,Y_t) \d t, v_\mu(x,y) = \mathbb{E}\int_0^\infty \exp(-\mu t) \psi(X_t,Y_t) \d t & w_{\lambda+\mu}(x,y) = \mathbb{E}\int_0^\infty \exp(-(\lambda+\mu) t) \frac{\partial u_\lambda}{\partial y}(X_t,Y_t) \frac{\partial u_\mu}{\partial y}(X_t,Y_t) \d t
I_1 =\int_{|v_1|<\tau}\left\{\int_{0}^x\frac{1}{\tau|v_1|}e^{-\frac{a_{\ell}(x-y)}{\tau|v_1|}}dy\right\}\mathcal{M}_1(v_1)dv_1\cr =\frac{1}{a_{\ell}}\int_{|v_1|<\tau}\left\{1-e^{-\frac{a_{\ell}x}{\tau|v_1|}}\right\}\mathcal{M}_1(v_1)dv_1\cr \leq\frac{1}{a_{\ell}}\int_{|v_1|<\tau}\left\{1-e^{-\frac{a_{\ell}}{\tau|v_1|}}\right\}\mathcal{M}_1(v_1)dv_1,
I_i = [x_{i - \frac{1}{2}},x_{i+\frac{1}{2}}] \forall \,\, 1 \leq i \leq N_x; J_j = [v_{j - \frac{1}{2}}, v_{j + \frac{1}{2}}] \, \quad\forall \,\, 1 \leq j \leq N_v.
I_1\leq \frac{2}{\delta}\|u^2\|_{L^\infty(L^\infty)}\|b^1-b^2\|_{L^1(L^1)}+ 2\|g\|_{L^1(L^1)}\sum_{i=1}^2 \big\|u^i \big\|_{L^\infty(L^\infty)}.
E(\lambda n_s^{-1})E(\mu) = q_{\lambda}^{1/2}q_s^{-1}q_\mu^{1/2}\theta(\lambda)T_{n_s^{-1}}\theta(\mu) & = q_{\lambda}^{1/2}q_s^{-1}q_\mu^{1/2}\theta(\lambda n_s^{-1}(\mu))T_{n_s^{-1}}&+ \sum_{k = 1}^{-\langle \nu(\mu),\alpha\rangle}q_{\lambda}^{1/2}q_s^{-1}q_\mu^{1/2}\theta(\lambda n_s^{-1}(\mu)\mu_{n_s^{-1}}(-k))c_{s,-k}.
\boldsymbol{\psi}_\alpha(z)\boldsymbol{\psi}_\beta(w)(z-w)^{-\frac{\alpha\beta}{k}}=N\big(\boldsymbol{\psi}_\alpha(z)\boldsymbol{\psi}_\beta(w)\big) +\frac{\varepsilon_{\alpha,\beta}}{z-w}{\boldsymbol{\psi}_{\alpha+\beta}(w)}+\frac{\delta_{\alpha+\beta,0}I_{\psi_{\alpha}(z)\psi_{\beta}(w)}}{(z-w)^2}
A_1=-(n+1) a_n \int_{z_1}^{z_2}H_0 h_0^n h_1 dz+a_n \Big[\frac{h_0^n h_{0z}}{\sqrt{1+h_{0z}^2}}h_1\Big]_{z_1}^{z_2}, V_1=a_n \int_{z_1}^{z_2}h_0^n h_1 dz.
\sum_{i,j=1}^{n}c_{i}c_{j}(x_{i},x_{j})_{H}=(\sum_{i=1}^{n}c_{i}x_{i},\sum_{j=1}^{n}c_{j}x_{j})_{H}=\|\sum_{i=1}^{n}c_{i}x_{i}\|_{H}^{2}\ge0
\mathcal{A}f(x) = \frac{\lambda^3(\sigma_1^2+\sigma_{-1}^2) f''(x)}{2}+ \left[c - \lambda H_1\left(\frac{x}{\lambda}\right) 1_{\{x\ge 0\}}+ \lambda H_{-1}\left(\frac{x}{\lambda}\right) 1_{\{x<0\}}\right] f'(x).
\begin{array}{llllllccl}p s t u v x y w \lambda\ 0 1 1 n 0 2(n+2) 3(n+2) 0 \mu\ 0 0 0 1 1 4 6 0 \nu\ 0 0 0 0 0 2 3 1 \sigma\ 1 1 0 q 0 2(q+2) 3(q+2) 0 \end{array}
(w_{1}\xi)(r,s)=\xi(\overline{r-N_{1}},s) &(w_{2}\xi)(r,s)=e^{-\frac{2\pi ir}{M_{1}}}\xi(r,s) &(w_{3}\xi)(r,s)=\xi(r,\overline{s-N_{2}}) &(w_{4}\xi)(r,s)=e^{-\frac{2\pi i s}{M_{2}}}\xi(r,s),
\int \frac{d w_{2}d w_{3}d w_{4}d w}{{( 2 \pi )}^{4}}\frac{- ( w_{2}+ w_{3}+ w_{4}) X_{i}^{8}[ - ( w_{2}+ w_{3}+ w_{4}) ] w_{2}X_{j}^{8}( w_{2}) w_{3}X_{k}^{8}( w_{3}) w_{4}X_{l}^{8}( w_{4})}{[ ( w + w_{2})^{2}- R^{2}] [ ( w + w_{2}+ w_{3})^{2}- R^{2}] [ ( w + w_{2}+ w_{3}+ w_{4})^{2}- R^{2}] [ w^{2}- R^{2}]}
\omega_{h,k,n}(z) =\exp(\sum_{j\in \mathbb{Z}_k}b(j)\sum_{r=1}^{k}\frac{(ze^{2\pi \frac{i h j}{k}})^r}{k}\Phi(z^k,1, \frac{r}{k}) - 2n\pi i \frac{h}{k}) =\exp(\sum_{j\in \mathbb{Z}_k}-b(j)\ln(1-ze^{2\pi i \frac{hj}{k}}) - 2n\pi i \frac{h}{k}) &=e^{-\frac{2\pi inh}{k}}\prod_{j\in \mathbb{Z}_k}(1-ze^{2\pi i \frac{hj}{k}})^{-b(j)}
{({\mathbb D}^2)}^A{}_{B}= \,\,{\mathbb D}^{A}{}_{R}{\mathbb D}^{R}{}_{B}+{\mathbb D}^{R}{}_{B}{\mathbb D}^{A}{}_{R}=({\mathbb D}^{A}{}_{R}{\mathbb D}^{R}{}_{B}-{\mathbb D}^{R}{}_{B}{\mathbb D}^{A}{}_{R})+2\,{\mathbb D}^{R}{}_{B}{\mathbb D}^{A}{}_{R}= \,\, 3{\mathbb D}^{A}{}_{B}+2\,{\mathbb D}^{R}{}_{B}{\mathbb D}^{A}{}_{R},
\delta_0{\cal L}_0 = - \Omega [ (d+2) \chi^{\mu,\alpha\beta}(D R)_{\nu,\alpha\beta}+ 10 \chi^{\alpha,\beta\nu}D_\nu R_{\alpha\beta}+ 10 \chi^\alpha (D R)_\alpha + (d-8) \chi^\alpha D_\alpha R ]
\langle X(z)\Phi(v_{1},z_{1})\cdot\cdot\cdot\Phi(v_{n},z_{n})\rangle=\sum_{j}\langle\Phi(v_{1},z_{1})\cdot\cdot\cdot\Phi(Xv_{j},z_{j})\cdot\cdot\cdot\Phi(v_{n},z_{n})\rangle(z-z_{j})^{-1}.
F_{\mu \nu}^{4}= \frac{i}{2}\{J_{\mu}[ \Phi ] , J_{\nu}[ \Phi ] \}- \frac{i}{2}\varepsilon_{\mu \nu \alpha \beta}\{J_{\alpha}[ \Phi ] , J_{\beta}[ \Phi ] \}+ f_{\mu \nu}[ \Phi ]
G [\chi_2, \chi_1; \psi_2, \psi_1] ~ = \prod_k^\infty \biggl( \int_0^\infty dN_k N_k^{-{1\over 2}}e^{-{{(\Delta \psi_k)}^2\over 2N_k}}{\cal G}[ \chi_{2k},\chi_{1k}; N_k ] \biggr),
\dfrac{d^{k}}{dx^{k}}\phi_{\varepsilon ,\lambda}\left( x\right) =-\frac{\beta_{1}\delta^{-1}}{\sqrt{\lambda}}\dfrac{d^{k}}{dx^{k}}\left( \sin \sqrt{\lambda}\left( x-a\right) \right) +O\left( \left( \sqrt{\lambda}\right)^{k-2}e^{\left\vert t\right\vert \left( x-a\right)}\right) ,
\cos\alpha_s^0 =\frac{a_2^2+(a_1(s)/s)^2-a_3^2}{2a_2(a_1(s)/s)}\cos\alpha_t^0 =\frac{a_2^2+(a_1(t)/t)^2-a_3^2}{2a_2(a_1(t)/t)},
\left\lvert e^{-( s- s_0) A}\sum_{n=0}^{l-1}q_n\phi_n(y)\right\rvert \le\left\lvert\sum_{n=0}^{l-1}e^{-( s- s_0)\lambda_n}q_n\phi_n(y)\right\rvert \lesssim (y^{-\gamma}+y^{2\lambda_l})\sum_{n=0}^{l-1}\lvert q_n\rvert\lesssim e^{-\lambda_l s_0}e^{-\kappa s_{0}}(y^{-\gamma}+y^{2\lambda_l}).
L_{e f f}= \int d^{2}x \; [ ( \partial_{t}\mid \phi \mid )^{2}+ ( \partial_{t}\mid \chi \mid )^{2}] + \delta L_{e f f}+ \kappa \int d^{2}x \; [ \frac{B_{2}}{q_{1}}\dot{\omega}_{1}+ \frac{B_{1}}{q_{2}}\dot{\omega}_{2}] \; \; .
\sqrt{-G}W^{10}_{\mu\nu}\Big|_{y=0}=-\sqrt{-g^{0}}\frac{1}{2}V_{0}(\Phi)g^{0}_{\mu\nu}\Bigr|_{y=0}\ ,\quad\sqrt{-G}W^{10}_{\mu\nu}\Big|_{y=\infty}=\sqrt{-g^{\infty}}\frac{1}{2}V_{\infty}(\Phi)g^{\infty}_{\mu\nu}\Bigr|_{y=\infty}\ .
u^{\varepsilon}(t,x) = \frac{\sin(\delta \pi)}{\pi}\int_{0}^{t}\int_{0}^{\pi}\frac{G_{t-s}(x,y)}{(t-s)^{1-\delta}}Y_{\delta}^{\varepsilon}(s,y) \, \mathrm{d}s \, \mathrm{d}y
R_M(u/p, x, e^{i \theta + 2u}x) = \frac{1}{p}\int_0^{u}\int_X e^{-((u - s)/p) \Delta^X}( x, y) A_M(s/p ,y , e^{i \theta + 2u}x) dv (y) ds + \int_0^{u/p}\int_X e^{-((u - s)/p) \Delta^X}( x, y) B_M(s/p, y, e^{i \theta + 2u}x) dv (y) ds\,.
\Sigma_m(i, j) := \underset{}{\underbrace{\sum_{k_1=j+2}^i \sum_{k_2=j+2}^{k_1-1}\cdots \sum_{k_m=j+2}^{k_{m-1}-1}}}s_{i,k_1-1}\cdot s_{k_1-1,k_2-1}\times \cdots \times s_{k_m-1,j}.
|B'\setminus N(a_j)| \leq |B\setminus N(a)|+|T'|+|R''|+i &\le |A|+2(p-i+1)+|W'|+|R_{i-1}|+i &\le |A|+2(p-i+1)+|W'|+2i-1 &=|A'|+2p+1 &\le |A'|+2j,
{\cal F}^{\rm Bethe}_{\rm GAMP}\left(\{R_i\},\{\Sigma_i\},\{\omega_{\mu}\}, \{a_i\},\{c_i\}\right) =-\sum_{\mu}\log{\cal Z}_{\mu}&- \sum_i \frac{c_i+(a_i-R_i)^2}{2\Sigma^2_i}- \sum_{\mu}\frac{(\omega_{\mu}-\sum_i F_{\mu i}a_i)^2}{2V_{\mu}}&- \sum_i \log{{Z}(R_i,\Sigma_i)}\, V_{\mu}= \sum_iF_{\mu i}^2c_i\, ,
\quad\left(G_k(ST)^3 G^{-1}_k\right)_\mathfrak{U}&=|G_k S\,{\overline{G_k S}\,}^{-1}|_\mathfrak{U}\; |\overline{G_k ST}\,S\,{\overline{G_k STS}\,}^{-1}|_\mathfrak{U}\; |\overline{G_k STST}\,S\,{\overline{G_k STSTS}\,}^{-1}|_\mathfrak{U}&\quad(k=2,3,\ldots,p-1).
\|\tilde{f}_P\|_p = \frac{\|g\|_p}{h^d}\left(\frac{1}{S}\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}}= h^s\|g\|_p\cdot d_0Ln\ln n\left(\frac{1}{S}\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}},
\exp\left(-\frac{b^2}{4p^2}\right)\,_1\tilde{F}_1\left(\frac{\nu-\mu+2}{2};\nu+1;\frac{b^2}{4p^2}\right) =\,_1\tilde F_1\left(\frac{\nu+\mu}{2};\nu+1;-\frac{b^2}{4p^2}\right)
q_1 :=|f(i)|^2+|g(i)|^2=(a_0-a_2)^2+(a_1-a_3)^2+(b_0-b_2)^2+(b_1-b_3)^2, q_2 :=|f(i)|^2-|g(i)|^2=(a_0-a_2)^2+(a_1-a_3)^2-(b_0-b_2)^2-(b_1-b_3)^2, q_3 :=|f(i)+g(i)|^2= ((a_0-a_2)+(b_0-b_2))^2+ ((a_1-a_3)+(b_1-b_3))^2, q_4 := |f(i)-g(i)|^2= ((a_0-a_2) -(b_0-b_2))^2 +((a_1-a_3) -(b_1-b_3))^2 .
\int_\Omega |\nabla u_\varepsilon|^2 dx~ =\int_\Omega u_\varepsilon \left(\lambda_\varepsilon H(u_\varepsilon) u_\varepsilon \exp(u_\varepsilon^2)\right) dx\,, &=\int_\Omega U_{\varepsilon,x_\varepsilon}\Delta U_{\varepsilon,x_\varepsilon}dx+o\left(\check{\zeta}_\varepsilon\right)\,.
\tilde{\mathbf{H}}(q,k)=\sum_{z=1}^{\hat{Z}}\hat{\beta}_{z}\mathbf{a}_r(\hat{\mu}^{\mathrm{r}}_z)\mathbf{a}^T_t(\hat{\mu}^{\mathrm{t}}_z)\exp(jq\hat{\gamma}_{z}-jk\hat{\eta}_z)
d^{\mathcal{K}_i}-z-1 =\frac{N-T}{T}\alpha_i -\frac{N-2B-T}{T}\cdot \frac{N}{N-2B}\alpha_i &=\frac{2B}{N-2B}\alpha_i &=2B(N-2B-T)^{|\mathcal{K}_i|-1}T^{M-|\mathcal{K}_i|}>0