harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{1}= 2 \, \frac{\sqrt{( | y |^{2}+ | x_{0}|^{2})^{3}+ ( | y |^{2}+ | x_{0}|^{2})^{2}I_{2}+ ( | y |^{2}+ | x_{0}|^{2}) ( I_{2}^{2}- I_{4}) / 2 + | I_{3}|^{2}}}{| y |^{2}+ | x_{0}|^{2}}\, ,
X(u;A,B)=\sum_{n>0}\frac{X_{-n}}{n}(u+Ah)^{n}-\sum_{n>0}\frac{X_{n}}{n}(u+Bh)^{-n}+log(u+Bh)p_{X}+q_{X}, &X_{+}(u;B)=-\sum_{n>0}\frac{X_{n}}{n}(u+Bh)^{-n}+log(u+Bh)p_{X}, &X_{-}(u;A)=\sum_{n>0}\frac{X_{-n}}{n}(u+Ah)^{n}+q_{X}, &X(u;A)=X(u;A,A),X(u)=X(u;0).
\begin{aligned}\phi_p(z, y) = (zy)^k + \sum_{i=0}^{k-1}(zy)^{i}(z^{2(k-i)}+ y^{2(k-i)}) = p(zy)^k + \sum_{i=0}^{k-1}(zy)^{i}(z^{k-i}- y^{k-i})^2.\end{aligned}
M_{N}\leq D^p(1-r^{-p})\sum_{n\geq 0}a^{pn}\mu(G^a_n)\sum_{\ell\geq 0}r^{-p\ell}\Big(1\land\Big[\frac{\sqrt A r^{d\ell/2+d}}{2(r^{d/2}-1)}\frac1{\sqrt{N\mu(G_n^a)}}\Big]\Big) = D^p(1-r^{-p})\sum_{n\geq 0}a^{pn}\mu(G^a_n)\Psi_{r,p,d/2}\Big(\frac{\sqrt A r^{d}}{2(r^{d/2}-1)\sqrt{N\mu(G_n^a)}}\Big).
-\frac{1}{c^{2}}(i\frac{2mc^{2}}{\hbar}\frac{\partial\phi}{\partial t}+(\frac{mc^{2}}{\hbar})^{2}\phi)e^{-\frac{i}{\hbar}mc^{2}t}\approx(\nabla^{2}-(\frac{mc}{\hbar})^{2})\phi e^{-\frac{i}{\hbar}mc^{2}t}
\qquad \qquad \qquad \qquad +{\frac{1}{2}}\sum_{M^{\prime}\neq M}( \bar{q}_{M M^{\prime}}- 3 ){\frac{1}{V_{M^{\prime}}}}\partial_{a}\tilde{E}_{M^{\prime}}\Gamma_{\hat{a}}\Gamma_{M^{\prime}}^{\perp}\eta = 0 .
f'_m(r_e)=\dfrac{m h_m}{r_e}\big( \alpha_m r_e^m-\beta_m r_e^{-m}\big)=-\dfrac{1}{r_e}\,\dfrac{\displaystyle\int_a^b \bigg(\Big( \dfrac{R}{s}\Big)^{|m|}-\Big( \dfrac{s}{R}\Big)^{|m|}\bigg) s\,ds}{\Big( \dfrac{R}{r_e}\Big)^{|m|}-\Big( \dfrac{r_e}{R}\Big)^{|m|}}.
\frac{\partial u(\mathbf{x},t)}{\partial t}=-\left[ \sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}A_{i}(\mathbf{x},t)+\sum_{i,j=1}^{N}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}B(\mathbf{x},t)\right] u(\mathbf{x},t),\;\,u(\mathbf{x},0)=f(\mathbf{x})
C_{p}\left( \frac{d}{\phi d \phi}\right) f_{p , m}^{( 2 p )}= f_{p + 1 , m}^{( 2 p + 2 )}~ ~ \rightarrow ~ ~ k \left( \frac{d}{\phi d \phi}\right)^{4}f_{m}^{( 0 )}= f_{4 , m}^{( 8 )}\ ,
\frac 1 3 ( \lambda^{0^{\prime}})^{2}\varepsilon^{i j}\, \, \sum_{\pm}\, \left( \begin{array}{l}{2 \bar{\varepsilon}_{\pm}\gamma^{m}\partial_{i}\theta_{\pm}\, \, \, \, \bar{\theta}_{\pm}\gamma_{m}\partial_{j}\theta_{\pm}}{- \bar{\varepsilon}_{\pm}\gamma^{m}\theta_{\pm}\, \, \, \, \partial_{i}\bar{\theta_{\pm}}\gamma_{m}\partial_{j}\theta_{\pm}}\end{array}\right) ,
\partial L/\partial{\rm{\bf R}}=({\rm{\bf I}}+{\rm{\bf W}}_2{\rm{\bf R}})^{-1}{\rm{\bf W}}_2 -({\rm{\bf I}}+{\rm{\bf W}}_1{\rm{\bf R}})^{-1}{\rm{\bf W}}_1 +\lambda{\rm{\bf I}}-{\rm{\bf M}}={\rm{\bf 0}}
\sum_{n=0}^\infty \Big(\frac{1}{\lambda_n+\lambda +1}-\frac{1}{\lambda_n+\lambda}\Big) =-\ \frac{\displaystyle{\sum_{n=0}^\infty \frac{|\langle 1| f_n \rangle|^2}{(\lambda_n+\lambda)^2}}}{\displaystyle{\sum_{n=0}^\infty \frac{|\langle 1 | f_n \rangle|^2}{\lambda_n+\lambda}}}=\frac{d}{d\lambda}\log \mathcal H_\lambda (u)\ .
\frac{2\pi^{(d-2)/2}}{\Gamma((d-2)/2)}\int_\alpha^\pi f(\eta) \sin^{d-2}\eta \, d\eta \, \int_0^\pi \frac{\sin^{d-3}\xi \,d\xi}{(2 - 2\gamma)^{(d-2)/2}}= F_Q - Q(\theta), \alpha \leq \theta \leq \pi,
\liminf_{j\to\infty}\frac{|x_j|^{n-2}}{\log \frac1{a_b(x_j)}}u(x_j) = \infty, ( >0) i_0=0 \liminf_{j\to\infty}|x_j|^{n-2+i_0}a_b(x_j)^{i_0}|D^{i_0}u(x_j)| = \infty,( >0) i_0\in \{1,2,\ldots, n-1\}.
G(x,y) =\frac12\sum_{\substack{j\ge0 k\ge2}}\frac{(2k)!(k-1)(2k+j)}{k!(k+2)!}{2k-1+j\choose j}x^k(x+1)^jy^{2k+j}&=\sum_{\substack{j\ge0 k\ge2}}{2k+j\choose k+2,k-2,j}x^k(x+1)^jy^{2k+j}&=\sum_{\substack{n\ge 4 k\ge2}}{n\choose k+2,k-2,n-2k}x^k(x+1)^{n-2k}y^n,
\tilde{e}_{i_2}^{}\tilde{e}_{i_1}^{}\left( b \otimes b^\prime \right) = \tilde{f}_{i_3}^{a_3}\cdots \tilde{f}_{i_l}^{a_l}(b_\lambda) \otimes \tilde{e}_{i_2}^{r_2}\tilde{e}_{i_1}^{r_1}(b^\prime).
\prod_{j=1}^d\prod_{\mathit{v}\in S_j}|\rho_\mathit{v}(\alpha_1)q \sigma_j(u_1)+\cdots+\rho_\mathit{v}(\alpha_d)q \sigma_j(u_d)-p|_\mathit{v}=|\alpha_1 q u_1+\cdots +\alpha_d q u_d-p|.
\frac{\overline{p(z,\eta)}p(z,w)-\overline{\tilde{p}(z,\eta)}\tilde{p}(z,w)}{1-\bar{\eta}w}= (1,\bar{\eta},\dots,\bar{\eta}^{m-1})T_p(z) (1,w,\dots, w^{m-1})^t.
\varkappa(m,p_1,p_2):=\sum_{s=0}^n\sum_{\nu\in I_s}{k}^{\frac{sr}{2}(\frac{1}{p_2}-\frac{3}{4})}(\ln k)^{sr}\cdot \prod_{\nu_j>k}[{\nu_j}^{\frac{r}{2}(\frac{1}{p_2}-\frac{3}{4})}(\ln(\nu_j))^r]|m(\nu)|^r<\infty,
\,\,\, F^{\bf a}_k=a_k\,\,\,\,\,-n+1\le k\le0\,\,\,\,\,\, F^{\bf a}_k = b_1F^{\bf a}_{k-1}+\cdots + b_nF^{\bf a}_{k-n}\,\,\,\,\,k>0.
||\dot\gamma||_2 \leq \frac{1}{\sqrt{c_1}}\Big[T\max_{\overline{\Omega}}L(x,0)+2T\max_{\overline{\Omega}\times \mathcal{P}(\overline{\Omega})}|F|+2\max_{\overline{\Omega}\times \mathcal{P}(\overline{\Omega})}|G|+Tc_0\Big]^{\frac{1}{2}}=K,
\underline{\lim}_{t\to\infty}\frac{h(t)}{t}=\underline{\lim}_{t\to\infty}\frac{\tilde{h}(t-T)+l^*}{t}\geq\underline{\lim}_{t\to\infty}\frac{\eta(t)}{t}&=\underline{\lim}_{t\to\infty}\frac{(1-\epsilon)^{2}\int^t_{\tilde{T}+T}\mu(z_\epsilon)_x(s,0)ds+\tilde{h}(\tilde{T})}{t}&=(1-\epsilon)^{2}\lim_{t\to\infty}\frac{\int^t_0\mu(z_\epsilon)_x(s,0)ds}{t}.
\lim_{x\to a}\frac{f_y(x)-f_y(a)-T(a)(x-a)}{\left\|x-a\right\|}=\lim_{x\to a}\frac{\varphi\left((\delta(x)-\delta(a)-Dg_U(a)(x-a))\otimes y\right)}{\left\|x-a\right\|}&=\lim_{x\to a}\varphi\left(\frac{\delta(x)-\delta(a)-Dg_U(a)(x-a)}{\left\|x-a\right\|}\otimes y\right) &=\varphi(0\otimes y)=\varphi(0)=0.
\textbf{H}(x_0)\otimes\mathbf{\Psi}(t)\textbf{J}^{1,T}_{N_t\times N_t}\cdot\textbf{C}+\varphi(x_0)=g_1(t), &\textbf{H}(x_{N_h})\otimes\mathbf{\Psi}(t)\textbf{J}^{1,T}_{N_t\times N_t}\cdot\textbf{C}+\varphi(x_{N_h})=g_2(t).
\Gamma^{(0)}{}_{ij}^{k}=g^{k\bar{m}}\Gamma^{(0)}_{ij,\bar{m}}=g^{k\bar{m}}\partial_i\partial_j\partial_{\bar{m}}\mathcal{K}=g^{k\bar{m}}\partial_i g_{j\bar{m}}=\partial_i (\log{g_{m\bar{n}}})^{k}_{\phantom{1}j}=\Gamma_{ij}^{k}
\Gamma_{s}( u^{- 1})^{T}\mathcal{C}\gamma^{a}= \Gamma_{s}( u^{- 1})^{T}\tilde{\mathcal{C}}\Gamma_{s}( P^{a}) = \tilde{\mathcal{C}}\Gamma_{s}( u ) \Gamma_{s}( P^{a}) \, .
\sum_{{u,w}\in V(G)\backslash\{x_1,x_2\}}d_G(u,w) =\sum_{{u,w}\in V(G)\backslash\{x_1,x_2,v\}}d_G(u,w)+\sum_{u\in V(G)\backslash\{x_1,x_2\}}d_G(u,v) &=\sum_{{u,w}\in V(G')\backslash\{x\}}d_{G'}(u,w)+\sum_{u\in V(G')\backslash\{x\}}(d_{G'}(u,x)+1) &=\sum_{{u,w}\in V(G')}d_{G'}(u,w)+(n-3) &=W(G')+(n-3)
\lVert \tilde{f}(u+h)-\sum_{i=0}^k \frac{\tilde{f}^{(i)}(u)}{i!}[h^i] \rVert_r\le C\,\lVert f \rVert_{C^{r+2}}\lVert h \rVert^k_{L^{\infty}}(\lVert h \rVert_{r+p}+\lVert h \rVert_{L^{\infty}}\lVert u \rVert_{r+p}).
\frac{\gamma\left(n,\frac{n(\rho+\sigma_w^2)}{\rho}\ln(1+\frac{\rho}{\sigma_w^2})\right)}{\Gamma(n)}-\frac{\gamma\left(n,\frac{n\sigma_w^2}{\rho}\ln(1+\frac{\rho}{\sigma_w^2})\right)}{\Gamma(n)}=\epsilon,
U_{j}=-\frac{1}{\alpha}\left[\left(\frac{a-1}{a-b}\right)(X.Y)^{a-2}-\left(\frac{b-1}{a-b}\right)(X.Y)^{b-2}\right]\times M &V_{j}=-\frac{1}{\alpha}\left[\left(\frac{a-1}{a-b}\right)(X.Y)^{a-2}-\left(\frac{b-1}{a-b}\right)(X.Y)^{b-2}\right]\times N
J_{3}\le\frac{1}{2N^{2}}\sum_{i=1}^{N}\int_{G_{N}}\left|\nabla_{i}f(x)\right|M_{\epsilon,i}(x)dx J_{4}\le\frac{1}{2N^{2}}\sum_{i\neq j}\int_{F_{jk}}\left|f_{ij}(x)-f_{ij}(x)\right|M_{\epsilon,i}(x)dS_{ij}(x).
g_{a}(\lambda )=\frac{4\lambda^{2}-\eta^{2}}{4{\zeta_{a}^{(0)}}^{2}-\eta^{2}}\,\prod_{\substack{b=1 b\neq a}}^{\mathsf{N}}\frac{\lambda^{2}-{\zeta_{b}^{(0)}}^{2}}{{\zeta_{a}^{(0)}}^{2}-{\zeta_{b}^{(0)}}^{2}}\text{\for}a\in \{1,...,\mathsf{N}\},
2 \frac{\ddot{b}}{b}+ \frac{{\dot{b}}^{2}- 1}{b^{2}}+ \frac{f^{\prime \prime \prime}( R )}{f^{\prime}( R )}\dot{R}^{2}+ \frac{f^{\prime \prime}( R )}{f^{\prime}( R )}\ddot{R}+ 2 \frac{f^{\prime \prime}( R )}{f^{\prime}( R )}\frac{\dot{b}}{b}\dot{R}- \frac{1}{2}\frac{f ( R )}{f^{\prime}( R )}+ \frac{1}{2}R = 0 .
\chi ( r ) = \frac{\sqrt{2}}{2}\left( \begin{array}{c}{1}\\{e^{i \phi}}\\ \end{array}\right) \delta_{r ( + )}+ \frac{\sqrt{2}}{2}\left( \begin{array}{c}{- e^{- i \phi}}\\{1}\\ \end{array}\right) \delta_{r ( - )},
c_n^{gr}(A)= c_{m+p}^{gr}(A) > \left( \frac{1}{(8k(m+p))}\right)^p (1-\delta)^n(a-\varepsilon)^m>\left(\frac{1}{8kn}\right)^d \frac{1}{(a-\varepsilon)^d}(1-\delta)^n(a-\varepsilon)^n.
\frac{d^l}{dx^l}v^{n,k}(x)\Big|_{x=a}=\frac{d^l}{dx^l}\theta_{v}^{n,k}(x)\Big|_{x=a}&=\frac{1}{n}\frac{d^l}{dx^l}\left(\exp(x-a)\left(1-F_{n-1,k}(a;x)\right)\right)\Big|_{x=a}&+\left(1-\frac{1}{n}\right)\frac{d^l}{dx^l}\left(\exp(2(x-a))\left(1-F_{n-2,k}(a;x)\right)\right)\Big|_{x=a}&=\frac{1}{n}+\left(1-\frac{1}{n}\right)2^{l},
\begin{aligned}\frac{1}{\log(s_n)}\int_{f_n}^{s_nf_n}\frac{\rho_B(\lambda(A) y)}{y}\mathrm{d}y \sim \frac{1}{\log(s_n)}\int_{f_n}^{s_nf_n}\frac{\rho_B(u)}{u}\mathrm{d}u \sim \alpha^{s,f}(B).\end{aligned}
-B_i C_j+B_j C_i-AD_k \epsilon_{kij}= 0 (Ay^4-B_l y^l) D_k\epsilon_{kij}-(C_i y^4+D_k \epsilon_{kli}y^l)B_j + (C_j y^4+D_k \epsilon_{klj}y^l)B_i = 0 (Ay^0+C_l y^l) D_k\epsilon_{kij}+(B_i y^0+D_k \epsilon_{kli}y^l)C_j -(B_j y^0+D_k \epsilon_{klj}y^l)C_i = 0 (B_i y^0+C_i y^4)(D_p\epsilon_{pqr}) \epsilon_{iqr}= 0.
\psi ( p ) \Bigr |_{U_{+}}= \left( \begin{matrix}{{\omega ( p )}}\\{{1}}\\ \end{matrix}\right) \varphi_{+}( p ) \delta ( p^{2}) , \qquad \qquad \psi ( p ) \Bigr |_{U_{-}}= \left( \begin{matrix}{{1}}\\{{\tilde{\omega}( p )}}\\ \end{matrix}\right) \varphi_{-}( p ) \delta ( p^{2}) ,
Pr_{\rho_m}\{|v_1|\leq k\}\leq Pr_{\rho_m}\{v_1\leq k\}&=Pr_{\rho_m}\{E_{\rho_m}[v_1]-v_1\geq E_{\rho_m}[v_1]-k\}&\leq Pr_{\rho_m}\{|E_{\rho_m}[v_1]-v_1|\geq |E_{\rho_m}[v_1]-k|\}&=Pr_{\rho_m}\{(E_{\rho_m}[v_1]-v_1)^2\geq (E_{\rho_m}[v_1]-k)^2\}&\leq \frac{Var_{\rho_m}(v_1)}{(E_{\rho_m}[v_1]-k)^2}&\leq \frac{B}{(\sqrt{N}\lambda_1^m-k)^2}
K = \sigma_{\mu}x^{\mu}= \left[ \begin{array}{c c}{{q ( x^{0}- x^{3})}}&{{- \left[ 2 \right]^{1 / 2}q^{- 1 / 2}x^{+}}}\\{{- \left[ 2 \right]^{1 / 2}q^{1 / 2}x^{-}}}&{{q x^{0}+ q^{- 1}x^{3}}}\\ \end{array}\right] \equiv \left[ \begin{array}{c c}{{q D}}&{{B}}\\{{A}}&{{C / q}}\\ \end{array}\right] \; ,
<\dot{x}_\mu(t_1)\dot{x}_\nu(t_2)>_{S_2}=\hat{{\cal N}}(\alpha)\left[\dot{x}_\mu^{cl}(t_1)\dot{x}_\nu^{cl}(t_2)-2\delta_{\mu\nu}\frac{\partial^2}{\partial t_1\partial t_2}\Delta^{(N)}(t_1,t_2)\right]e^{-S_2[x^{cl}]},
|b_{sc}+y_+(0)|=|A(y_-)(0)|\le{2n^{3\over 2}\left({1-{|v_-|^2\over c^2}}\right)^{1\over 2}\big({r\beta_1^l\over c}+2\beta_2(n^{1\over 2}(|x_-|+r)+1)\big)\over \alpha(\alpha+1)\big({|v_-|\over 2\sqrt{2}}-r\big)^2(1-r)^\alpha}.
R =\frac{1}{NB}\left[|\mathcal{M}_2|\times|\mathcal{A}| +|\mathcal{M}_1|\times|\mathcal{A}^{*c}|\right] =[H(p_1^*)-H(p_1)]\times q_1 +[H(p_2^*)-H(p_2)]\times q_2^*.
\begin{aligned}0 = W_i = - (1 + \beta\rho) \left(\sum_k u_{ki}\rho_k + \sum_k u_{k}\rho_{ki}- \phi_{i}- \phi_{u}u_{i}- q_{i}\right) &- \beta \rho_{i}(\sum_{k}u_{k}\rho_{k}- \phi - q) + (A + \frac{M}{2}) \rho_{i},\end{aligned}
\begin{array}{l}\lim\limits_{t\rightarrow\infty}\mathbb{E}^{\hat{\psi}_0}[f(\psi_\tau^{t,e})]= \mathbb{E}^{\hat{\psi}_0}[f\circ F(e^*_0, e^*_{\varsigma_k- \varsigma_{k-1}}...,e^*_{\varsigma_k- \varsigma_1},\tau,J_\tau,\kappa,T_\tau^{sel})]\end{array}
-q_*^{-1}t^{\frac{N+2A}{2}+1}(\nabla^2 u_0)(x,t) =-t[M_{0,1}+o(1)]\nabla^2 U(|x|)+\left[\frac{N+2A}{2}M_{0,1}+o(1)\right]\nabla^2[UF_0](|x|)+O(t^{-1}) \ge\left[\frac{N+2A}{2}M_{0,1}+o(1)\right]\nabla^2[UF_0](|x|)+O(t^{-1})
\underline{Q}_{i}(t) = \det\begin{bmatrix}1 1 \lambda_{i,2}^{2}t\end{bmatrix}, i\in\mathbb{N}\cap[1,\eta], \underline{P}_{i}(0) = \sigma\left(\frac{\underline{Q}_{i}(t)-\underline{Q}_{i}(0)}{t}\right),i\in\mathbb{N}\cap[1,\eta].
-t^{1-2\alpha}u'(t) = \frac{1}{\Gamma(\alpha) 4^{\alpha}}\Bigl( -2\alpha \int\limits_0^{\infty}r^{-\alpha-1}\exp^{-\frac{t^2}{4r}}T(r) x \, dr + \frac{t^2}{2}\int\limits_0^{\infty}r^{-\alpha-2}\exp^{-\frac{t^2}{4r}}T(r) x \, dr \Bigr).
{\cal L}_{{B I}}^{{S U S Y}}= \frac{1}{4 e^{2}}\left[ \int \! d^{2}\theta ~ W^{2}+ \int \! d^{2}\bar{\theta}~ \bar{W}^{2}\right] + \sum_{s , t = 0}^{\infty}a_{s t}^{( \beta )}\int \! d^{4}\theta ~ W^{2}\bar{W}^{2}~ X^{s}Y^{t}
\sum_{0\leq n<m <\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n})}{p^{m(1-\alpha)}p^{n\alpha}}= \sum_{m=0}^{\infty}\frac{\tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^m)}{p^{m}}- \sum_{m=0}^{\infty}\frac{\tau_{A}(p^m) \tau_B(p^m)}{p^{m}}.
W_0(\psi^\rho\tilde h) \le\left(1+\frac{C_0\,\alpha_1\,\rho^{2(\alpha_1-1)+d}}{\alpha_1-1}\right)W_0(\tilde h) &\le\left(1+\frac{C_0\,\rho^{d}}{\alpha_1-1}\right)\left(1-\frac{\alpha_1-1}{d+\alpha_1-1}\right)W_0(z)=\left(1+\frac{C_0\rho^{d}}{\gamma_0}\right)\left(1-\frac{\gamma_0}{d+\gamma_0}\right)W_0(z)\,.
\sum_{A\subset P(\sigma,\tau), |A|=|\sigma \cap P|}(-1)^{{\rm sign}(\sigma,\tau,P(\sigma,\tau),A)}\frac{\partial^{k'}X_{\eta_1(\sigma,\tau,P(\sigma,\tau),A)}(s)}{\partial s^{k'}}X_{\eta_2(\sigma,\tau,P(\sigma,\tau),A)}(s)=0.
T_iT_j\big(x_{i,1}^+\big) = T_i\big(\big[x_{j}^+,x_{i,1}^+\big]\big) = \big[\big[x_i^+,x_j^+\big],-x_{i,1}^-+\tfrac{\hbar}{2}\big\{h_i,x_i^{-}\big\}\big],
\big(\mathbf{H}(\mathbf{u})\big)(t, \cdot) = \exp(-t/\tau) \mathbf{H}_{0}+ \int_{0}^{t}\exp\big(-(t - s)/\tau\big) \mathbf{F}\big(\nabla \mathbf{u}(s, \cdot)\big) \mathrm{d}s t \in [0, T].
\sum_{t = 1}^{N}\sum_{n = 1}^{\infty}V_{m n}^{r t}V_{n 0}^{t s}= V_{m 0}^{r s}, \ \ \ \sum_{t = 1}^{N}\sum_{m = 1}^{\infty}V_{n m}^{r t}V_{m l}^{t s}= \delta_{n , l}\delta_{r , s}.
M_z (s,x_0) :={1 \over (1 + s)^2}\chi_{|\tau| \le 5}\Vert \phi \Vert_{H^{11}(\Sigma_0)}+ \sup_{|\alpha| \le 10}\sup_{\psi_1}\int_0^s \int_{\Sigma_t}|\partial^\alpha (\Box \phi)| |\partial_z \psi| d x d t.
b_{\mathcal{T}}(\widehat{\boldsymbol{\sigma}}_{\mathcal{T}}-\boldsymbol{\sigma}_{\mathcal{T}}, v)=\sum_{K\in\mathcal{T}}\int_{\partial K}M_n(\boldsymbol{\sigma}_{\mathcal{T}}-\widehat{\boldsymbol{\sigma}}_{\mathcal{T}})\partial_{\boldsymbol{n}}vds \forall~v\in V_{\mathcal{T}}.
{\widehat{\mathcal{Q}}_n}(z) = \frac{(z-j^{(0)}+j^{(4)})(z+j^{(0)}+j^{(4)}+1)(z-j^{(3)}+j_n^{(12)})(z-j^{(3)}-j_n^{(12)}-1)}{2z\,(2z-1)}\,.
\left(\sum_{l=0}^{k}\underline{M_{l}}\right) \underline{m_{k,r+2r_i}^{(i)}}=t^{\alpha/2}\underline{m_{k+1,r}^{(i)}}\left(\sum_{l=0}^{k-1}\underline{M'_{l}}\right)+t^{\beta'/2}\underline{m_{k, r}^{(i)}m_{k,r+2r_i}^{(i)}\prod_{s=0}^{k-1}A_{i, r+(2k-1-2s)r_i}^{-1}},
\textbf{w}= \left [ \sum_{n=1}^{N}G^{C}_{\sigma}(e_{n})\textbf{X}_{n}\textbf{X}_{n}^{H}\right ]^{-1}\left [ \sum_{n=1}^{N}G^{C}_{\sigma}(e_{n})\textit{D}_{n}^{*}\,\textbf{X}_{n}\right ]
\begin{aligned}\boldsymbol{v}^{\perp}(\boldsymbol{x},t) = \sum_{k=1}^{\infty}\left(- \boldsymbol{v}_k^c(\boldsymbol{x}) \sin(k \omega t) + \boldsymbol{v}_k^s(\boldsymbol{x}) \cos(k \omega t)\right).\end{aligned}
\zeta^{rot}=\frac{k_{B}T}{B}+\frac{1}{3}+\frac{1}{15}(\frac{B}{k_{B}T})+\frac{4}{315}(\frac{B}{k_{B}T})^{2}+\frac{1}{315}(\frac{B}{k_{B}T})^{3}+\cdot\cdot\cdot.
\Xi_2(z) =\mathbb{E}\left[e^{-z\mathcal{I}}\right]=\mathbb{E}\left[{e^{-z(\mathcal{I}_{\mathrm{LoS}}+\mathcal{I}_{\mathrm{NLoS}})}}\right] &=\mathbb{E}\left[{e^{-z\mathcal{I}_{\mathrm{LoS}}}}\right] \mathbb{E}\left[{e^{-z\mathcal{I}_{\mathrm{NLoS}}}}\right]
\Theta_h = - H d^mx + p_\alpha^id u^\alpha \wedge d^{m-1}x_i + \frac{1}{n(ij)}\, p_\alpha^{1_i+1_j}d u_i^\alpha \wedge d^{m-1}x_j \, , \Omega_h = d H \wedge d^mx - d p_\alpha^i \wedge d u^\alpha \wedge d^{m-1}x_i- \frac{1}{n(ij)}\, d p_\alpha^{1_i+1_j}\wedge d u_i^\alpha \wedge d^{m-1}x_j \, .
\begin{array}{ccl}\lambda_m(\tilde{\mathbf A}) = a_0+\sum_{k=1}^r a_k\left(e^{-\frac{2\pi ik(m-1)}{n}}+ e^{-\frac{2\pi i(n-k)(m-1)}{n}}\right) & = a_0 + 2\sum_{k=1}^r a_k\cos\left(\frac{2\pi k(m-1)}{n}\right), m=1\ldots n\end{array}
\|T_{\lambda^0}\|_{L^1\cap L^\infty}\leq |\mu-m^2|\Big[\frac{\|E\|_{L^1\cap L^\infty}}{m^2_t}+ \frac{1}{m^4_t}+\frac{c}{m^2_t}{\bf 1}_{d=2}+ \frac{c}{m_t}{\bf 1}_{d=3}\Big].
A_2=\widetilde{m}^{q+1}K^i_{y_j}K^j_{y_i}=\widetilde{m}^{q+1}\widetilde{w}_{y_iy_j}\widetilde{w}_{y_jy_i}=\widetilde{m}^{q+1}|\nabla_y^2\widetilde{w}|^2,
\frac{d(\tilde\alpha(h),y),(x,\tilde\beta(h)))}{h}=\left\|\left(\frac{d_X(\tilde\alpha(h),x)}{h},\frac{d_Y(\tilde\beta(h),y)}{h}\right)\right\| \stackrel{h\to 0^+}{\longrightarrow}\|(a,b)\|.
\beta(p,k)=-\frac{(T'_0-T_0)^2\omega_k q}{4\pi}\sqrt{\frac{1}{\omega_k\omega_p}}\int_{-\infty}^{\infty}\!\!\!dy \frac{\sin(\frac{2p}{q}y)}{{\rm cosh}^2(y)}\tanh^2(y)\sin(\frac{2k}{q}y).
\tau_{4}(n) =\sum_{\substack{n_1 \geq 0 n_{1}^2 + 3 n_{2}^2 =n}}\left(\frac{n_1}{3}\right) \left(\frac{4}{n_{1}^2 + n_{2}^2}\right) n_1 &= -\frac{1}{6}\sum_{\substack{n_1 \geq 0 n_{1}^2 +3 n_{2}^2 =4n}}\left( \frac{-12}{n_1}\right) \left( \frac{4}{n_2}\right)n_1 - 3 \left( \frac{12}{n_1}\right) \left( \frac{-4}{n_2}\right)n_2.
(\overline{\rho}(ar_1,r_1a,r_1^2,r_1) r_1^2 \overline{\lambda}(ar_1,r_1a,r_1^2,r_1))^{\omega+1}\quad\quad=\overline{\rho}(ar_1,r_1a,r_1^2,r_1) r_1^2 \overline{\lambda}(ar_1,r_1a,r_1^2,r_1).
\widehat{x}_p =\mathbf{q}_p^{\mathrm{H}}\mathbf{y}&=\mathbf{q}_p^{\mathrm{H}}\mathbf{H}_{\mathrm{e}}^{\mathrm{H}}\mathbf{V}\mathbf{x}+\mathbf{q}_p^{\mathrm{H}}\mathbf{z},
\Delta_n(\tau_1,\tau_2) \!=\! \begin{bmatrix}\frac{2}{h}U^\top(h+\tau_2)A_d 2A_d^\top U(\tau_1-\tau_2)A_d A_d^{\top}U(\tau_1-\tau_2)A_d \frac{1}{h}I_m \end{bmatrix}.
\theta(S)\theta(T) =L(\zeta_1,\cdots \zeta_l)^*F(S)L(\eta_1,\cdots, \eta_m)L(\eta_1,\cdots, \eta_m)^*F(T)L(\xi_1,\cdots, \xi_n) &=L(\zeta_1,\cdots \zeta_l)^*F(S)F(T)L(\xi_1,\cdots, \xi_n) &=\theta(ST).
\left( \tilde{M}_{\xi}^{2}( X ) \right)_{\mathrm{\scriptsize{I}}}+ \left( \tilde{M}_{\xi}^{2}( X ) \right)_{\mathrm{\scriptsize{I I}}}= M_{\xi}^{2}( X ) - \mu^{2}( \xi = \pi , \sigma ) .
\Upsilon^{g_1 g_2}_{\alpha \beta \gamma}\: = \:\left( \Upsilon^{g_2}_{\alpha \beta \gamma}\right) \,\left( g_2^* \Upsilon^{g_1}_{\alpha \beta \gamma}\right) \,\left( \Omega^{g_1, g_2}_{\alpha \beta}\right) \,\left( \Omega^{g_1, g_2}_{\beta \gamma}\right) \,\left( \Omega^{g_1, g_2}_{\gamma \alpha}\right)
a=\frac{j-i+1}{2},b= -n+\frac{i+j}{2},c=-\frac{j}{2},d=n+\frac{3-j}{2}, e=1,f=\frac{1}{2}, g=\frac{3}{2}, h=\frac{\sqrt{2(n-i)+5}(-2n+i+j-1)}{\sqrt{2(n-i)+1}(i-j-2)}, &l=\frac{\sqrt{2(n-i)+5}(2n-i-j-1)(2n-i-j+1)}{\sqrt{2(n-i)-3}(i-j)(i-j-2)}.
\prod_{j=1}^{n}(\lambda-X_{j})=\lambda^{n}-e_{1}(X_{1},...,X_{n})\lambda^{n-1}+e_{2}(X_{1},...,X_{n})\lambda^{n-2}+\cdot\cdot\cdot+(-1)^{n}e_{n}(X_{1},...,X_{n}).
\begin{aligned}\hat{f}_{k,h}=\mathbb{E}[f_{k,h}] &+\frac{\sqrt{\beta_h^kp_d}\mathrm{Var}[f_{k,h}]}{\beta_h^kp_d\mathrm{Var}[f_{k,h}]+1}\left(y_{dpk,h}-\sqrt{\beta_h^kp_d}\mathbb{E}[f_{k,h}]\right),\end{aligned}
\chi_{(a_1,\ldots,a_n)}^{-1}(1)=\left[0,\frac{\pi}{2 a_n}\right) \cup \left(\pi-\frac{\pi}{2 a_n},\pi +\frac{\pi}{2 a_n}\right) \cup \left(2\pi-\frac{\pi}{2a_n}, 2 \pi \right].
\frac\pi2\le W^V(u_k,s,0) =W^V(u,r_ks,x_k) &=\underbrace{W^V(u,r_ks,x_k)-W^V(u,\rho,x_k)}_{\le0}+W^V(u,\rho,x_k)-W^V(u,\rho,x_0)+W^V(u,\rho,x_0) &\le W^V(u,\rho,x_k)-W^V(u,\rho,x_0)+W^V(u,\rho,x_0),
\begin{aligned}\Delta\dot{y}=A_{yy}\Delta y+A_{yz}\Delta z-\sigma_y \Delta y +\sigma_x\Delta x+B_{yh}\Delta u_h+d_y-d_b \Delta\dot{z}=A_{zy}\Delta y+A_{zz}\Delta z+B_{zh}\Delta u_h+d_z \Delta\dot{x}=\sigma_y \Delta y -(\sigma_x+a_{xx})\Delta x+d_x+d_b\end{aligned}
d |X^{i,N}_t-\bar X^I_t|^2 = 2\sqrt 2 (X^{i,N}_t-\bar X^I_t)\cdot (dB^i_t-d\bar B^I_t) +4\frac{|I|-1}{|I|}dt &+\frac{2\chi}N (X^{i,N}_t-\bar X^I_t)\cdot\Big[\sum_{j\ne i}K(X^{i,N}_t-X^{j,N}_t) - |I|^{-1}Z^I_t\Big]dt
\begin{array}{c}{\Omega = d \left( p_{\mu}d x^{\mu}+ \theta_{i}^{\mu}g_{\mu \nu}D \theta_{i}^{\nu}\right) =}\\{= d p_{\mu}\wedge d x^{\mu}+ \frac 1 2 R_{\mu \nu \lambda \rho}\theta_{i}^{\mu}\theta_{i}^{\nu}d x^{\lambda}\wedge d x^{\rho}}\\{+ g_{\mu \nu}D \theta_{i}^{\mu}\wedge D \theta_{i}^{\nu},}\\ \end{array}
ln(\frac{T_{\infty}-T_{d}+\frac{Q_{g}}{\dot{m}_{F}C_{p,F}}}{T_{g}-T_{d}+\frac{Q_{g}}{\dot{m}_{F}C_{p,F}}})=\frac{1}{r}\frac{\dot{m}_{F}C_{p,F}}{4\pi\lambda_{g}}
\sqrt{1 - v^{2}}\sum_{n}\exp \Big [ - \frac{\alpha^{\prime}( 1 - v^{2})}{2 R^{2}t}( n - \tilde{q}R \hat{J}_{0})^{2}\Big ] = \sqrt{\frac{2 R^{2}t}{\alpha^{\prime}}}\sum_{w}\exp \Big [ - \frac{2 \pi R^{2}t}{\alpha^{\prime}( 1 - v^{2})}w^{2}+ 2 \pi i \tilde{q}R w \hat{J}_{0}\Big ] .
\frac{\frac{n_{1}}{r_{1}}}{\frac{n}{r}}\geq\frac{1-\frac{k}{ln^{2}}}{1+\frac{1}{ln_{1}^{2}}}=1-\frac{\frac{k}{ln^{2}}+\frac{1}{ln_{1}^{2}}}{1+\frac{1}{ln_{1}^{2}}}.
\frac{\partial \gamma_{}}{\partial n_p}= &\frac{\lambda_{ab}^2{\rho^\ast}^2\left[\!-\!\lambda_{ab}\rho^\ast n_p^2-(2\sigma_b^2+2\lambda_{ab}\rho^\ast)n_p+n\sigma_b^2+\lambda_{ab}\rho^\ast n\right]}{n\ln2\left[\sigma_b^3+\lambda_{ab}\rho^\ast\sigma_b(n_p+1)\right]^2}.
\sum_{i=0}^{p-1}\frac{p(-1)^{i}}{2i-3}=-\frac{p}{3}+\frac{p}{2p-3}-\sum_{i=0}^{p-1}\frac{p(-1)^{i}}{2i-1}&\equiv \frac{5p}{3}+\frac{p}{2p-3}+(-1)^{\frac{p-1}{2}}-\frac{1}{2}\sum_{k=1}^{\frac{p-3}{2}}(-1)^{\frac{p-1}{2}+k}\frac{p^2}{k^2}\pmod{p^3},
\mathcal{F}^{i}= H_{-i}\left(\check{C}(\underline{a})\otimes_{R}-\right) = H_{-i}\left(\left(\varinjlim \Sigma^{-n}K^{R}(\underline{a}^{k})\right)\otimes_{R}-\right) \cong \varinjlim H_{n-i}\left(K^{R}(\underline{a}^{k})\otimes_{R}-\right) \cong \varinjlim H_{n-i}\left(\underline{a}^{k};-\right)
P_{n}\left( I+\mathbf{H}(\varphi)\right) P_{n}\cdot P_{n}A^{-1}P_{n}=P_{n}\left( I+\mathbf{H}(\varphi)\right) A^{-1}P_{n}-P_{n}\left(I+\mathbf{H}(\varphi)\right) Q_{n}A^{-1}P_{n}& =P_{n}-P_{n}\mathbf{H}(\varphi)Q_{n}A^{-1}P_{n}.
E_{in}^{-1}(y_n) R^{-1}(y_n) R(x_n) E_{in}(x_n) = E_{in}^{-1}(y_n) \left(\mathbb{I}+\mathcal{O}((x-y) n^{-5/2})\right) E_{in}(x_n) & = \mathbb{I}+ \mathcal{O}\left((x-y)n^{-\frac{1}{2}}\right) + E_{in}^{-1}(y_n) \mathcal{O}\left((x-y) n^{- \frac{5}{2}}\right) E_{in}(x_n).
N ( E ) \simeq{\frac{2 \sqrt{2}}{3 \pi}}{\frac{E^{3 / 2}}{g \hbar^{2}}}\left( \operatorname{l o g}{\frac{g^{2}E}{v^{4}}}+ 5 \operatorname{l o g}2 -{\frac{8}{3}}\right) .
a_{14}^{4123}=a_{14}^{4132}=a_{14}^{4213}=a_{14}^{4231}=a_{14}^{4312}=a_{14}^{4321}= a_{14}^{2413}=a_{14}^{2431}=a_{14}^{2341}=a_{14}^{3412}=a_{14}^{3421}=a_{14}^{3241}
III_1 \leq \int_{\big( B(g, 4 C_2 r) \big)^c}\int_{B(g,\eta r)}\Big|K_j(\tilde g,\tilde{\tilde g})-K_j(\tilde g,g)\Big| \, d\tilde{\tilde g}\,d\tilde g &\quad+ \int_{B(g, 4 C_2 r) \backslash B(g, c_* \eta r)}\int_{B(g,\eta r)}\Big|K_j(\tilde g,\tilde{\tilde g})-K_j(\tilde g,g)\Big| \, d\tilde{\tilde g}\,d\tilde g &=: III_{11}+III_{12},
i_Z^! p_{13*}(-) = q_{T*}(e(N_{Y^{4T}}Y^4)^{-1}f_T^*(e(N_{Y^{2T}}Y^2))\cdot (i_{Z^{(2)}})^!_{i_{Y^4}}(-)) & = q_{T*}\left(e(Y)^{-2}\cdot_2 (i_{Z^{(2)}})^!_{i_{Y^4}}(-)\right)
\|f_1f_2\|_{L^2}^2= \sum\limits_{k\in\mathbb{Z}}\|f_1f_{2,k}\|_{L^2}^2\leq \sum\limits_{k\in\mathbb{Z}}C\|f_1\|_{L^2}^2(\|f_{2,k}\|_{H^{1}}^2+\|\partial_zf_{2,k}\|_{H^{1}}^2) \leq C\|f_1\|_{L^2}^2\big(\|f_{2}\|_{H^{1}}^2+\|\partial_zf_{2}\|_{H^{1}}^2\big),
\begin{cases}^{C}_{0}D_{t}^{0.6}y(x,t)= \displaystyle \frac{\partial^{2}}{\partial x^{2}}y(x,t) \hbox{in}[0,1]\times[0,T] y(0,t) \;= \;y(1,t) = 0 \hbox{in}[0,T] y(x,0) = y_{0}(x) \hbox{in}[0,1],\end{cases}
\Omega_p(\nu_{\gamma}\beta_1\nu_{\delta}) = \overline{\Phi_{\mathbf h_p}(\nu_{\gamma}\beta_1\nu_{\delta})}\Phi_{\breve{\mathbf g}_p}(\nu_{\gamma}\beta_1\nu_{\delta})\Phi_{\pmb{\phi}_p}(\nu_{\gamma}\beta_1\nu_{\delta}) = \frac{p - \overline{G(-\gamma,p)}}{p(p-1)}= \frac{p - G(\gamma,p)}{p(p-1)}.
D_{\mu}^{A C}J^{\mu \, C}= \partial_{\mu}J^{\mu \, A}-{\frac{i g}{2}}d^{A B C}[ A_{\mu}^{B}, J^{\mu \, C}]_{\mathrm{M}}+{\frac{g}{2}}\, f^{A B C}\{A_{\mu}^{B}, J^{\mu \, C}\}_{\mathrm{M}}= 0
G = \int_{x_{i n}}^{x_{o u t}}D^{\prime}x \int D v \exp \left\{i \int_{t_{i n}}^{t_{o u t}}\left[ L ( x , v ) + \frac{\partial L ( x , v )}{\partial v}( \dot{x}- v ) \right] d t \right\}J ( x , v ) \; .
\Lambda^{b}= ( M_{\mathrm{P V}})^{b}\left( \frac{1 6 \pi^{2}}{b g^{2}}\right)^{T_{G}}\! \exp \left( - \frac{8 \pi^{2}}{g^{2}}\right) \, \prod_{i}Z_{i}^{- T ( R_{i})}\, ,
U(\phi)=\frac{1}{2}\sum_a \frac{\partial W^1}{\partial\phi^a}\frac{\partial W^1}{\partial \phi^a}=\frac{1}{2}\left(1- 2(\phi_1^2+ \phi_2^2)^{\frac{n-1}{2}}\cos \left[ (n-1) \arctan\frac{\phi^2}{\phi^1}\right] + (\phi_1^2 +\phi_2^2)^{n-1}\right)
\left\lbrace\begin{array}{ll}y_{ii}= \sum\limits_{k = 1,k \neq i}^K \left| z_{ik}\right|^2, i = 1, \ldots, K, y_{ij}= \sum\limits_{k = 1,k \neq i,j}^K z_{ik}z_{jk}^*, j = 1, \ldots, K, \,\, j \neq i.\end{array}\right.
O ({\vec{x}}) = \sum_{n_{1}, n_{2}, n_{3}= -{N}}^{+{N}}\frac{1}{X^{3 / 2}}O_{n_{1}, n_{2}, n_{3}}e x p \left( - 2 \pi i ( \sum_{j = 1}^{3}n_{j}x_{j}) / X \right) ,
I_{non-planar}^{T=0}= \frac{g}{2}\int \frac{d^2 p}{(2\pi)^2}\frac{d^2k}{(2\pi)^2}\frac{e^{i\tilde{p}k}}{e^{\beta(\frac{k^2}{2}-\mu)}-1}= \frac{g}{8\pi^2 \theta^2}\frac{1}{e^{-\frac{\mu}{T}}-1}\,.