harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_{1}=\begin{bmatrix}0 -1 -2 \varepsilon \varepsilon -3 \varepsilon \varepsilon \varepsilon -3 \varepsilon \varepsilon \varepsilon -4 \varepsilon \varepsilon -5 \varepsilon \varepsilon \varepsilon -6 \varepsilon \varepsilon -5 \varepsilon\end{bmatrix},
|\int_{t}^{t+\lambda(t)^{1-\alpha}}\frac{ds}{(s-t)}\int_{0}^{s-t}\frac{\rho d\rho \lambda''(s)}{r}\partial_{r}F_{3}(r,\rho,\lambda(s))|\leq C \int_{t}^{t+\lambda(t)^{1-\alpha}}\frac{ds}{(s-t)}\int_{0}^{s-t}\frac{\rho d\rho}{r}|\lambda''(s)| r \lambda(s)^{2\alpha -2}&\leq \frac{C}{t^{2}\log^{b+1}(t)}
-\int_0^T \int_\Omega \big(\tfrac{1}{2}\bar \rho |\bar u|^2+h_1(\bar \rho) + h_2(\bar \rho) \big) \dot{\theta}(t) \dxdt = \int_\Omega \big( \tfrac{1}{2}\bar \rho |\bar u|^2+h_1(\bar \rho) + h_2(\bar \rho) \big) \big|_{t=0}\theta(0) \dx \ ,
{\cal A}_{V_{\propto}+ \mathrm{N O}}= -{\frac{i g^{2}}{( 2 \pi )^{2 6}}}2^{1 2}\pi \int_{0}^{1}{\frac{d q^{\prime}}{q^{3}}}\int_{0}^{1}d \nu_{1 2}^{\prime}\, \left[ \, f ( - q^{2}) \, \right]^{- 2 4}\left[ \, F \left( \nu_{1 2}^{\prime}, - q^{2}\right) \, \right]^{- 2},
\operatorname{tr}(B(T_\alpha\otimes\operatorname{id}_m-\,T\otimes\operatorname{id}_m)A) = \operatorname{tr}(B((T_\alpha-T)\otimes\operatorname{id}_m)A)=\sum\nolimits_{i,j=1}^m \operatorname{tr}(B_{ij}(T_\alpha-T)(A_{ji}))
\begin{array}{l}A=\left(1-b\zeta G_0^{(1)}(0+,0)\right),B=-b\eta G_0^{(1)}(0+,0), C=(\zeta+\eta)\left(bG^{(12)}(0+,0-)-a G_0^{(1)}(0+,0)\right), D=-b\zeta G_0^{(1)}(0-,0),E=\left(1-b\eta G_0^{(1)}(0-,0)\right), F=(\zeta+\eta)\left(bG_0^{(12)}(0-,0+)-a G_0^{(1)}(0-,0)\right), \Delta=1-b\left(\zeta G_0^{(1)}(0+,0)+\eta G_0^{(1)}(0-,0)\right).\end{array}
\frac{(\beta)_{m_1}}{(\beta)_{m_q+1}}=\left(\prod_{i=1}^{q-1}\frac{(\beta)_{m_i+c^{'}_{i}}}{(\beta)_{m_{i+1}}}\right)\left(\prod_{i=1}^{q-1}\frac{1}{(m_i+\beta)^{c^{'}_{i}}}\right)\frac{1}{m_q+\beta},
W_{q,p}^l:=\vert \left\{I_p \in \mathcal{E}_{q, p}\vert \S(I_p)=\{1, 2,\cdots, l\}\right\}\vert =\sum_{\substack{a_1+\cdots + a_l=p, a_1 \geq 2,\cdots a_l \geq 2}}\frac{p!}{(a_1)! \cdots (a_l)!},
\begin{array}{ll}\delta F^{\nu}=F^{\nu}-F^0\ast_{sp}G_{\nu}=\sum_{i=1}^dv_iF^{\nu}\ast G_{\nu,i}+\sum_{j=1}^{2d}W^S_j(F^{\nu},F^{\nu})\ast G_{\nu}=\sum_{j=1}^dP^S_j(F^{\nu},F^{\nu})\ast G_{\nu,v_j}+\sum_{j=d+1}^{2d}W^S_j(F^{\nu},F^{\nu})\ast G_{\nu},\end{array}
1 - S = \left( \begin{array}{c c}{{\frac{1}{2 V_{o o}}\sqrt{1 - 3 V_{o o}}\left[ \sqrt{1 + V_{o o}}- \sqrt{1 - 3 V_{o o}}\right]}}&{{0}}\\{{0}}&{{\frac{1}{2 V_{e e}}\sqrt{1 - V_{e e}}\left[ \sqrt{1 + 3 V_{e e}}- \sqrt{( 1 - V_{e e})}\right]}}\\ \end{array}\right) \, ,
\frac{\partial^{p_{1}+\cdot\cdot\cdot+p_{n}}}{\partial^{p_{1}}x_{1}\cdot\cdot\cdot\partial^{p_{n}}x_{n}}f=\sum_{k=1}^{\infty}\frac{1}{2^{k}M_{k}}\frac{\partial^{p_{1}+\cdot\cdot\cdot+p_{n}}f_{k}}{\partial^{p_{1}}x_{1}\cdot\cdot\cdot\partial^{p_{n}}x_{n}}
\hskip -2mm9\sum_{k=0}^{n-1}(2k+1)W_k^2 &=n\sum_{s=0}^{n-1}{n+s\choose 2s}{2s\choose s}\frac{n-s}{s+1}\sum_{m=s}^{2s}\sum_{i=0}^s{m\choose i}{s\choose i}{i\choose m-s}\frac{81}{(2i-3)(2m-2i-3)}.
A(z) = A^{(n)}(z) = \left(\begin{matrix}A_{11}^{(n)}(z) A^{(n)}_{12}(z) A_{21}^{(n)}(z) A^{(n)}_{22}(z)\end{matrix}\right), B(z)= \left(\begin{matrix}B_{11}(z) B_{12}(z) B_{21}(z) B_{22}(z)\end{matrix}\right)
\partial_tX_t\cdot\nu(X_t) =\frac{-\dot{x}(t)\cdot\nabla u(t,x(t))}{\sqrt{1+|\nabla u(t,x(t))|^2}}+\frac{\dot{x}(t)\cdot\nabla u(t,x(t))}{\sqrt{1+|\nabla u(t,x(t))|^2}}+\frac{\partial_tu(t,x(t))}{\sqrt{1+|\nabla u(t,x(t))|^2}}&=\frac{\partial_tu(t,x(t))}{\sqrt{1+|\nabla u(t,x(t))|^2}}.
[n]_q := 1 + q + \cdots + q^{n-1}[n]!_q := [n]_q [n-1]_q \cdots [1]_q{n \brack a_1, \dots , a_r}_q := \frac{[n]!_q}{[a_1]!_q \cdots [a_r]!_q}{n \brack a}_q := \frac{[n]!_q}{[a]!_q [n-a]!_q}.
{S_{1}}[ \chi , \phi ,{g_{c}}] ={\frac{1}{2 \gamma}}\int{d^{2}}\xi \sqrt{g_{c}}{{g_{c}}^{a b}}{\partial_{a}}\chi{\partial_{b}}\chi -{\frac{1}{2 \gamma}}{\eta_{c}}\int{d^{2}}\xi \sqrt{g_{c}}{\chi^{2}}-{\frac{1}{4 \gamma}}{k_{c}}\oint{d}{s_{c}}{\phi^{2}}.
\gamma_{i-1}\alpha_i^2=(1-\alpha_i)\alpha_{i-1}^2\gamma_i+\mu\alpha_i\gamma_i\gamma_{i-1}\geq(1-\alpha_i)\mu\gamma_{i-1}\gamma_i+\mu\alpha_i\gamma_i\gamma_{i-1}=\mu\gamma_i\gamma_{i-1},
\|P\|_B=\max\limits_{f_j=\pm 1}\left[R \left(\sum_{i=1}^n\left(\sum_{j=1}^{n+1}f_jl_{ij}\right)^2\right)^{1/2}+\left|\sum_{j=1}^{n+1}f_j\lambda_j(x^{(0)})\right|\right].
p \ = \ \sum_{j =0}^{\infty}\mu_j x^j + \sum_{l=0}^{n-1}\sum_{j=1}^{\infty}\lambda_{l, j}(l+x)^{-j}, p' \ = \ \sum_{j =0}^{\infty}\mu'_j x^j + \sum_{l=0}^{n-1}\sum_{j=1}^{\infty}\lambda'_{l,j}(l+x)^{-j}
I_{1}=(\tfrac{3}{4}r+\tfrac{s}{4}-\tfrac{1}{16}(s-r),\tfrac{3}{4}r+\tfrac{s}{4}+\tfrac{1}{16}(s-r)), I_{2}=(\tfrac{3}{4}s+\tfrac{r}{4}-\tfrac{1}{16}(s-r),\tfrac{3}{4}s+\tfrac{r}{4}+\tfrac{1}{16}(s-r)),
a \left( \eta \right) = a_{\mathrm{m i n}}\, \sqrt{1 + \gamma^{2}\, \eta^{2}}\; \; , \; \; a_{\mathrm{m i n}}= \sqrt{\frac{3 \, Q^{2}\, \omega_{4}}{1 6 \, M}}\; , \; \gamma = \frac{4 \, M}{\sqrt{3}\, Q}\, \, .
\begin{array}{ll}N:= \d2^{p+q-1}\sum_{\gamma=0}^1AE(\|u_t^0\|_\infty^{p-\gamma}\|u_{tx}^0\|_\infty^\gamma\|u_0\|_\infty^q+\|u_t^0\|_\infty^p\|u^0\|_\infty^{q-\gamma}\|u_x^0\|_\infty^\gamma) &\d+\sum_{\gamma=0}^12^{r-\gamma}BE\|u^0\|_\infty^{r-\gamma}\|u_x^0\|^\gamma,\end{array}
\begin{array}{r c l}{{A}}&{{=}}&{{\displaystyle < 0 | \varphi_{1}\varphi_{2}N \left( D_{3 4}\star \varphi_{4}\star \varphi_{4}+ \varphi_{4}\star D_{3 4}\star \varphi_{4}+ \varphi_{4}\star \varphi_{4}\star D_{3 4}\right) | 0 > .}}\\ \end{array}
J_i \leq \big[ C M_q(R)(\sigma R)^{-2}K^{\frac{-4}{n}}\big]^{\frac{(1+\kappa)^i -1}{\kappa}}B^{\frac{(1+\kappa)^i -1}{\kappa^2}-\frac{i}{\kappa}}J_0^{(1+\kappa)^i}\mbox{for all}i =0, 1,\dots
\alpha\xi^{m}+\beta \eta^{m}+q_{m}(\xi,\eta)= \eta^{m}\left(\frac{\alpha\xi^{m}}{\eta^{m}}+\beta +\frac{q_{m}(\xi,\eta)}{\eta^{m}}\right) \geq \frac{\beta}{2}\eta^{m}>c|\eta|.
V_j^{-1}V_Fc^F\left( V_{0F}^{-1}V_F\Gamma{\cal P}^{-1}\xi_E\right) V_F^{-1}V_j=\left(\frac{dV_{0j}^{-1}V_F\Gamma{\cal P}^{-1}\xi_E}{dV_{0F}^{-1}V_F\Gamma{\cal P}^{-1}\xi_E}\right)^{-1}c^j\left( V^{-1}_jV_F\Gamma{\cal P}^{-1}\xi_E\right) \ .
a_1 \wedge m_*(b) = a_2 \wedge m_*(b) \implies m(a_1) \wedge b = m(a_1) \wedge m \circ m_*(b) = m(a_2) \wedge m \circ m_*(b) = m(a_2) \wedge b \iff (m(a_1), m(a_2)) \in \Phi_b\iff (a_1, a_2) \in m^{-1}(\Phi_b).
\Gamma^{\star}[\star_{\widetilde{g}}(d_{A}\sigma)]=\star_{\Gamma^{\star}\widetilde{g}}\Gamma^{\star}(d_{A}\sigma),\ \Gamma^{\star}\widetilde{g}=\lambda^{2}g,\ \textrm{and}\ \star_{\lambda^{2}g}=\lambda^{5}\star_{g}\ (\textrm{see Remark}\ \ref{rmk linear algebra}),
\overline{\rho}\left(\overline{\pi}\left(\frac{z-\alpha}{c}\right)\right)=\overline{\pi}\left(\frac{w-\alpha}{c}\right) \iff\overline{\pi}\left(\rho\left(\frac{z-\alpha}{c}\right)\right)=\overline{\pi}\left(\frac{w-\alpha}{c}\right) \iff\rho\left(\frac{z-\alpha}{c}\right)-\frac{w-\alpha}{c}\in M_U.
\left[ A_{\kappa \rho}, A^{\lambda \sigma}\right] = \delta_{\kappa}^{\lambda}{M^{\sigma}}_{\rho}- \delta_{\kappa}^{\sigma}{M^{\lambda}}_{\rho}- \delta_{\rho}^{\lambda}{M^{\sigma}}_{\kappa}+ \delta_{\rho}^{\sigma}{M^{\lambda}}_{\kappa}.
e^{S_{I_{k;q}}}\,(G_{I_{k;q}}+tV^{(k,q-1)}_{I_{k;q}})\,e^{-S_{I_{k;q}}}=G_{I_{k;q}}+t\sum_{j=1}^{\infty}t^{j-1}(V^{(k,q-1)}_{I_{k;q}})^{diag}_j
\sqrt{- g}g^{\mu \nu}={\frac{1}{1 - \lambda_{+ +}\lambda_{--}}}\left( \begin{array}{c c}{{2 \lambda{--}}}&{{1 + \lambda_{+ +}\lambda_{--}}}\\{{1 + \lambda_{+ +}\lambda_{--}}}&{{2 \lambda_{+ +}}}\\ \end{array}\right)
I_{2}= - \frac{d}{d s}\left( s g ( s ) \int_{\Gamma}\frac{d z}{2 \pi i}z^{s}\left( \frac{\tanh ( \omega - z ) - \sigma}{\omega^{\prime}- 2 z}+ \frac{\sigma}{\omega^{\prime}- 2 z}\right) \right)_{s = 0}\, .
\lambda_s^{-p-i_s}a_s^{i_s}a_{s+1}^qxa_s^pw =(\lambda_s^{-1}a_s)^{i_s}a_{s+1}^qx(\lambda_s^{-1}a_s)^p(u\otimes f_0) & = (\lambda_s^{-1}a_s)^{i_s}a_{s+1}^qx(u\otimes f_p) & =(\lambda_s^{-1}a_s)^{i_s}a_{s+1}^qe_{i_1,\dotsc,i_{s-1},q}& =(\lambda_s^{-1}a_s)^{i_s}e_{i_1,\dotsc,i_{s-1},0}& =e_{i_1,\dotsc,i_s}.
\frac{\partial^{2}z}{\partial t^{2}}=(\frac{\partial}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial}{\partial y}\frac{\partial y}{\partial t})(\frac{\partial}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial}{\partial y}\frac{\partial y}{\partial t})z
i_{221}=-D_{1,1}D_{2,1}x_{3,1}, i_{222}=-D_{1,1}D_{2,1}x_{3,2}, i_{123}=-x_{1,1}D_{2,1}D_{3,2}i_{213}=-D_{1,1}x_{2,1}D_{3,2}, i_{211}=D_{1,1}x_{2,1}x_{3,1}, i_{212}=D_{1,1}x_{2,1}x_{3,2}, i_{121}=x_{1,1}D_{2,1}x_{3,1}, i_{122}=x_{1,2}D_{2,1}x_{3,2}, i_{112}=-x_{1,1}x_{2,1}x_{3,2}, i_{111}=-x_{1,1}x_{2,1}x_{3,2}.
G_{0}= \frac{1}{4 \pi^2 x^2}\, \rightarrow \, \frac{\Gamma(\frac{d}{2})}{(d - 2) 2 \pi^\frac{d}{2}}\, \biggl(\frac{1}{x^2}\biggr)^{(\frac{d}{2}- 1)}= \frac{1}{4 \pi^2}\, \frac{\Gamma(1 + \epsilon)}{\pi^\epsilon}\, \biggl(\frac{1}{x^2}\biggr)^{(1 + \epsilon)}
P \left( \max_{c \log n \le k \le n/4}\frac{S_k}{h(k/n)}> \lambda \right) \le \sum_{m=1}^{\infty}P \left( F_m > \lambda h( e^{-m}) \right) \le \sum_{m=1}^{\infty}\exp \left( - (\lambda m^p - E F_m )^2/ 4 c'' m \right).
\mathcal{Li}_{K,\mathfrak{s}}({\bf z}):=\sum_{i_1>i_2>\cdots>i_r>0}\frac{z_1^{q^{i_1}}z_2^{q^{i_2}}\cdots z_r^{q^{i_r}}}{(\theta^{q^{i_1}}-t)^{s_1}(\theta^{q^{i_2}}-t)^{s_2}\cdots (\theta^{q^{i_r}}-t)^{s_r}}
l_N^{-1}r_N(x,\xi') =\frac{1}{4\sqrt{1-r^{-2}c^{2}}}\log \left|\frac{\sqrt{1-r^{-2}c^2}-\aleph\sqrt{c^2-r^{-2}c^2}}{\sqrt{1-r^{-2}c^2}+\aleph \sqrt{c^2-c^2r^{-2}}}\right|^2 &=\frac{c^{-1}r}{2\sqrt{c^{-2}r^2-1}}\log \left|\frac{\sqrt{c^{-2}r^2-1}-\aleph\sqrt{r^2-1}}{\sqrt{c^{-2}r^2-1}+\aleph \sqrt{r^2-1}}\right|.
\Big\Vert \nabla \mathrm{H}\Big\Vert_{\mathbb{L}^2(\Omega)}^2 = \sum_{\substack{\mathrm{n}\in \mathbb{N}\mathrm{i}=1,2,3}}\Big|\Big\langle \nabla \mathrm{H};\mathrm{e}^{(\mathrm{i})}_{\mathrm{n}}\Big\rangle_{\mathbb{L}^{2}(\Omega)}\Big|^2.
\frac{\partial Y}{\partial x}=\left( \frac{A_{t_1}}{x-t_1}+\frac{A_{t_2}}{x-t_2}+A_{\infty\,1}+N x \right)Y , \frac{\partial Y}{\partial t_1}=\left(N_1-\frac{A_{t_1}}{x-t_1}\right)Y ,\frac{\partial Y}{\partial t_2}=\left(N_2-\frac{A_{t_2}}{x-t_2}\right)Y .
\|\hat g\|_{L^q(U)}\ge \frac18\gamma^{-1}\bigg(\int_{e+1}^{K-1}z^{-q/2}(\log z)^{-2q}(z+1)^{\beta-1}\,dz\bigg)^{1/q}&\ge \frac3{16}(1+\log K)^{-3/2}(1+\log K)^{-2}\bigg(\int_{e+1}^{K-1}(z+1)^{-q/2+\beta-1}\,dz\bigg)^{1/q}&= \frac3{16}(1+\log K)^{-7/2}\Big(\beta-\frac q2\Big)^{-1/q}(K^{\beta-q/2}-(e+2)^{\beta-q/2})^{1/q}.
(\partial_t + \nabla_\infty^* \nabla_\infty) b = \nabla_\infty a \times b + a \times \nabla_\infty b + a\times a \times b + F_\infty \times b + R_{\Sigma}\times b + b\times \mu(u) + a \times L_u (\partial_t u) + a \times (\partial_t \mu(u)) + \nabla_\infty (L_u (\partial_t u)) + \nabla_\infty (\partial_t \mu(u))
\mathrm{Var}(\alpha;n,h) := \frac{1}{\left| \mathcal{M}_n \right|}\sum_{f_0 \in \mathcal{M}_n}\left| \sum_{f \in I(f_0,h)}\alpha(f) - q^{h+1}\langle \alpha \rangle_{\mathcal{M}_n}\right|^2,
i\dot{c_N}= 2 \alpha_{NNNN}c_N + \alpha_{NNNN}e^{-2i\alpha_{NNNN}t}\, \overline{c_N}& i\dot{c_{k}}= 2 \alpha_{kNkN}c_{k}+ \alpha_{2N-k,k,N,N}e^{-2i \alpha_{NNNN}t}\,\overline{c_{2N-k}}\mbox{if $k \in \{0 \dots N-1\}\cup \{N+1 \dots 2N\}$}& i\dot{c_{k}}= 2 \alpha_{kNkN}c_{k}\mbox{if $k \geq 2N+1$},
R^{- 1}( j_{1}\hat{y_{1}}+ \frac{j_{2}- \tau_{1}j_{1}}{\tau_{2}}\hat{y_{2}}) R = j_{1}^{\prime}\hat{y_{1}}+ \frac{j_{2}^{\prime}- \tau_{1}j_{1}^{\prime}}{\tau_{2}}\hat{y_{2}}
\pi_{\varphi^\omega}((u_n)^\omega) p = \Pi_{\varphi^\omega, \psi^\omega}(\pi_{\psi^\omega}((u_n)^\omega) r) &= \Pi_{\varphi^\omega, \psi^\omega}(r\pi_{\psi^\omega}((u_n)^\omega) r) &= (\Pi_{\varphi, \psi}(r \pi_\psi(u_n) r))^\omega &= (p \pi_\varphi(u_n) p)^\omega \in \mathcal U(\mathcal P).
\dim L\wedge L + \dim \mathrm{Im}\gamma^{\prime}_{2}\leq \dim L\wedge L+ \sum\limits_{i=2}^{c}\dim \ker \alpha_{i}\cr =\dim L/ L^{2}\wedge L/ L^{2}+ \sum\limits_{i=2}^{c}\dim (L^{i}/L^{i+1}\otimes_{mod}(L/Z(L))^{ab}).
V\left( \frac{g_2g_3u}{HX}\right) = \frac{1}{2\pi i}\Psi\Big( \frac{u}{XQ^{\vartheta}}\Big) \int_{(\varepsilon)}\left(\frac{XH}{g_2g_3 u}\right)^{s_3}\widetilde{V}(s_3) \,ds_3.
\begin{cases}P( (-\Delta)^s)(u^{(\ell)}_1-u^{(\ell)}_2) + (F^{(\ell)}_1-F^{(\ell)}_2)u^{(\ell)}_1+F^{(1)}_2(u^{(\ell)}_1-u^{(\ell)}_2)=0 \Omega, \mathcal{L}(u^{(\ell)}_1-u^{(\ell)}_2)=0 \Omega^c, u^{(\ell)}_1-u^{(\ell)}_2=0 \Omega^c. \end{cases}
\sigma_{e_\beta}(z_1) = \begin{cases}1, \;\; \mbox{if}\;\; \beta = \alpha_1 \; \mbox{and}\; \gamma_1 = s_1, -z_1^2, \;\; \mbox{if}\;\; \beta = \alpha_1 \; \mbox{and}\; \gamma_1 = e, 0, \;\; \mbox{if}\;\; \beta \neq \alpha_1,\end{cases}
(L_0.(v^{(n)})^{\dag}).v^{(m)}= (v^{(n)})^{\dag}.L_0.v^{(m)}= h \delta_{n,m}+ \delta_{n,m-1}= (h (v^{(n)})^{\dag}+ (v^{(n+1)})^{\dag}).v^{(m)}.
\begin{aligned}\max_{\mathbf{\Lambda}}~ R = \sum_{k=1}^K w_k \log_2 \left( \left| \mathbf{I}_{N_S}+ \frac{P\mathbf{\Lambda}_k (\mathbf{\Sigma}_k^{(N_S)})^2}{\sigma^2KN_S}\right| \right) s.t.~ \mathrm{trace}\{\mathbf{\Lambda}\}= KN_S, &\mathbf{\Lambda}^{(n, n)}\ge 0,~\mathrm{for}~n = 1, \cdots, KN_S,\end{aligned}
\frac{R_{(1,0)}(2, \frac{m}{2}-3;q)\cdot R_{(2,-2)}(\frac{m}{2}-2,2;q)}{R_{(2,-2)}(\frac{m}{2}-3,3;q)}= \prod_{i=5}^{\frac{m}{2}}\theta(q^i;q^m)\cdot \frac{(q^m;q^m)^2_\infty}{(q)^2_\infty}\prod_{i=1}^4 \theta(q^i;q^m) &= \frac{(q^m;q^m)_\infty}{(q)_\infty}&= \prod_{t=1}^\infty \frac{(1-q^{mt})}{(1-q^t)}.
Q(X_j,z+t_0(X_j),\ldots) = Q(X_j,z+t_0(X_j), D_t \tilde h(X_j,z+t_0(X_j) ) ) Q(X_j,t_0(X_j),\ldots) =Q(X_j,t_0(X_j), D_t \tilde h(X_j,t_0(X_j) ) ) .
\sum_{\varphi\in [\varphi]}f_{e,[\varphi]}(s_{u,v}\sigma_{\varphi}(u)\sigma_{\varphi}(v))=\sum_{(\sigma_{\varphi}(u))_u\in \{\pm 1\}^{S_1}}\prod_{(i,j)\in E(S_1)}f_{e,[\varphi]}(s_{u,v}\sigma_{\varphi}(u)\sigma_{\varphi}(v))
M = \begin{pmatrix}1 x_1 x_2 x_1^2 x_1x_2 x_1^2x_2 1 x_2 x_1 x_2^2 x_1x_2 x_1x_2^2 1 x_3 x_2 x_3^2 x_2x_3 x_2x_3^2 1 x_1 x_3 x_1^2 x_1x_3 x_1^2x_3 1 x_2 x_3 x_2^2 x_2x_3 x_2^2x_3 1 x_3 x_1 x_3^2 x_1x_3 x_1x_3^2\end{pmatrix}.
|P_1^{2^*-2}\bigl( u_0 +\sum_{j=2}^k P_j)|\le C U_1^{2^*-2}\le\lambda^{-\frac{N-2}2}\frac{\lambda^{\frac{N+2}2}}{(1+ \lambda|y-\xi_1|)^{\frac{N+2}2+\tau}},y\in S.
\begin{gathered}\forall b \in \Gamma , \ \forall 0 < r < \widetilde{r}, X_{\sigma}\cap B ( b , r ) A_{\pm}( b , r ) = \varphi^{- 1}\big( \{- y_{-}^{2}+ y_{+}^{2}< 0 \}\cap \{\pm y_{-}> 0 \}\big) \cap B ( b , r ) .\end{gathered}
(\mathbf{L}_{1}^{-1})_{b}^{a}=\delta_{b}^{a}+\omega_{1b}^{a}\mathbf{m},\mathbf{L}_{1b}^{a}=\delta_{b}^{a}-\omega_{1b}^{a}\mathbf{m}, \mathbf{x}_{1}^{a}=(_{c}\mathbf{L}^{-1})_{b}^{a}\mathbf{x}^{b}.
\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{m}{m-2}L_{2}\right),~m>2,
\int_0^\infty \frac{1}{(w+b)^2(w+c)}e^{-w}dw= \int_0^1+\int_1^\infty= = \frac{1}{bc}\int_0^{1/b}\frac{e^{-bu}du}{(1+u)^2(1+\frac{b}{c}u)}+\int_0^\infty\frac{e^{-(w+1)}dw}{(w+b+1)^2(w+c+1)}\approx \frac1{bc}+\frac1{1+c}.
\partial_t\int_MG(x,t;y,0) dv_{g(t)}x =\int_M [\partial_tG(x,t;y,0)-R(x,t)G(x,t;y,0)] dv_{g(t)}x =\int_M [2P(x,t)-R(x,t)] G(x,t;y,0) dv_{g(t)}x,
\left.\frac{d{P}^{x,i,\uparrow}}{d{P}^{x,i}}\right|_{\mathcal{F}_t}=\frac{U^-_{J_t}(\xi_t)}{U^-_i(x)}\mathbf{1}_{(t<{\tau}^-_0)}, t\geq 0, i\in E, x>0,
(A_{+,n}- z I)^{-1}= \big[I + (A_- - z I)^{-1}\chi_n(A_-) \phi \chi_n(A_-)\big]^{-1}(A_- - z I)^{-1}, & (A_+ - z I)^{-1}= \big[I + (A_- - z I)^{-1}\phi\big]^{-1}(A_- - z I)^{-1},
{\cal L}( J ) = \frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi + \frac{J}{2}\phi^{2}+ \frac{\lambda \mu^{2 \varepsilon}}{4 !}\phi^{4}+{\cal L}_{\mathrm{C T}}( J ) - \mu^{- 2 \varepsilon}\frac{\zeta}{2}\; J^{2}.
L \left( s_{\rho}( p ) , s_{\rho}( q ) \right)_{\mu \nu}= L \left( p , q \right)_{\mu^{\prime}\nu^{\prime}}, \M \left( s_{\rho}( q ) \right)_{\mu \nu}= M ( q )_{\mu^{\prime}\nu^{\prime}}, \quad \mu^{\prime}\equiv s_{\rho}( \mu ) , \ \nu^{\prime}\equiv s_{\rho}( \nu ) ,
J:= \frac{1}{k\pi e^{\frac{1}{\sqrt{k}}}}\big(-\alpha_{1,k}+ 2i{k\pi}+ O(1)\big)e^{-\alpha_{1,k}-i(2k\pi +\alpha_{2,k})+O(k^{-\frac{1}{2}})}\times e^{-\frac{1}{2}-i\sqrt{k\pi}+O(k^{-\frac{1}{2}})},
\sup_{(t,s)\in [0,T]\times \mathcal{S}}|u(t,s)| \leq \sum_{i=1}^{[\frac{T}{\delta}]+1}(2K_2)^i|F_{k-1}|_{0,T}\leq (2K_2)^{[\frac{T}{\delta}]+2}|F_{k-1}|_{0,T}&\leq\Bigl(2K_2\Bigr)^{T\big( 2C_{12}K_2(1+|v|_0)\vee T \vee 1\big)+2}|F_{k-1}|_{0,T},
\MoveEqLeft e^{itH}[P_\delta(\tfrac{x-y}{t})]e^{-itH}e^{itH}[P_\delta(\tfrac{x-y}{t})]e^{-itH}= &e^{itH}e^{-itH_0}e^{itH_0}[P_\delta(\tfrac{x-y}{t})]e^{-itH_0}e^{itH_0}[P_\delta(\tfrac{x-y}{t})]e^{-itH}
\omega(q) = 1+ \sum_{n=1}^{\infty}\frac{q^{2n^{2}+2n}}{(1-q)^{2}(1-q^{3})^{2}\cdots (1-q^{2n+1})^{2}}&= 1 + 2q + 3q^{2}+ 4q^{3}+ 6q^{4}+ 8q^{5}+ 10q^{6}+14q^{7}+\dots
e_{1}=(\begin{matrix}1\\ 0\\ 0\\ 0\end{matrix});e_{2}=(\begin{matrix}0\\ 1\\ 0\\ 0\end{matrix});e_{3}=(\begin{matrix}0\\ 0\\ 1\\ 0\end{matrix});e_{4}=(\begin{matrix}0\\ 0\\ 0\\ 1\end{matrix}),
\sum_{\mathfrak{a}\in \mathcal{E}_k}\overline{\langle W,E_{\mathfrak{a}}(.,1/2+it) \rangle}\rho_{\theta*E_{\mathfrak{a}}(.,1/2+it)}(m)= \sum_{\mathfrak{a}^{\prime}\in \mathcal{E}^{\prime}_{4k}}\langle \theta*W, E_{\mathfrak{a}^{\prime}}(.,1/2+it/2)\rangle \rho_{E_{\mathfrak{a}^{\prime}}(.,1/2+it/2),\infty}(m)
[T^{p^2}\textbf{a}]_i\equiv\sum_{j=0}^{p^2}\binom{p^2}{j}\textbf{a}_{i+j}\equiv\sum_{j=0}^{p}\binom{p^2}{jp}\textbf{a}_{i+jp}\equiv \sum_{j=0}^{p}\binom{p}{j}\textbf{a}_{i+jp}\pmod{p^2},
\Delta\left(h(f_0);f\right)=\left\|{A}\xi- \hat{A}\xi \right\|_\alpha^\alpha =\int_{-\pi}^{\pi}\left|\left(C^0(e^{i\theta}) \right)^{<\frac{1}{\alpha - 1}>}(f_0(\theta))^{\frac{-1}{\alpha - 1}}\right|^{\alpha}f(\theta) d \theta.
\frac{\partial^2 H^c}{\partial p_n\partial q_l}\Big|_{z^\bullet}=\frac{\partial^2 H^c}{\partial p_n\partial p_l}\Big|_{z^\bullet}=0\quad\frac{\partial^2 H^c}{\partial q_n\partial q_l}\Big|_{z^\bullet}=\frac{\partial^2 H^c}{\partial q_n\partial p_l}\Big|_{z^\bullet}=0.
g^2\int \frac{d^6p}{(2\pi)^6}\,\frac{d^6k}{(2\pi)^6}frac{e^{-i(\theta_{12}(p_1\, k_2-p_2\, k_1)+\theta_{34}(p_3\, k_4-p_4\, k_3))}}{p^2\, k^2\,( p+k)^2}=g^2 \frac{1}{\theta_{12}+\theta_{34}}\, \frac{1}{\theta_{12}\, \theta_{34}}.
\prod_{i=1}^n x_i^{-m_i + \sum_{j=1}^n [-b_{ij}]_+m_j}= (1 + \hat y_k)^{-m_k}\prod_{i=1}^n (x_i')^{-m'_i + \sum_{j=1}^n [-b'_{ij}]_+m'_j},
\sum_{i \in \mathcal{I}}{\!\! \pi_i(d_i^*,\boldsymbol{d}_{-i}^* )}\! =\! \sum_{i \in \mathcal{I}}{\left\{d_i^* \!-\! d_i^*\gamma \sum_{j \in \mathcal{I}}d_j^* \!-\! \epsilon_i d_i^{*2}\right\}}& = \frac{\gamma\left(\sum_{j \in \mathcal{I}}\frac{1}{\epsilon_j}\right)^2}{4\left( 1+ \gamma\sum_{j \in \mathcal{I}}\frac{1}{\epsilon_j}\right)^2}.
\frac{t}{d}= \oint \frac{d z}{2 \pi \imath}U_{e f f}^{\prime}\left( z + \frac{R ( t )}{z}\right) = \left( \frac{m_{0}^{2}}{d}- \Gamma_{2}\right) R ( t ) - \sum_{n > 2}\frac{( 2 n ) !}{( n ! )^{2}}\Gamma_{2 n}R ( t )^{n}
{u}(x,t,\varepsilon,\rho)= \frac{1}{\Lambda^+_0 - \Lambda^-_0}\int\limits_{-\infty}^{\infty}\Gamma\left( \frac{x-\rho s}{\varepsilon}, \frac{t}{\varepsilon}\right) \Lambda'(s)\, ds +O\left(\left(\frac{\rho}{\varepsilon}\right)^{1/4}\right)
{\cal{L}}^{pv}= q (det g^{\eta\eta'})[-\overline{\psi}\gamma_{\mu}\sigma_{\lambda\omega}\gamma_{5}\psi H_{\gamma\rho\delta}+g_{\alpha\delta}\overline{\psi}\gamma^{\mu}\sigma_{\omega\nu}\gamma_{5}\psi H_{\mu\lambda\rho}-g_{\alpha\delta}\overline{\psi}\gamma_{\mu}\sigma_{\omega\nu}\gamma_{5}\psi H_{\beta\lambda\rho}]
V(f_{n},\tau_{n})=[\begin{matrix}cosh(\tau_{n}\sqrt{-f_{n})}&1/\sqrt{-f_{n}}sinh(\tau_{n}\sqrt{-f_{n}}\\ \sqrt{-f_{n}}sinh(\tau_{n}\sqrt{-f_{n}})&cosh(\tau_{n}\sqrt{-f_{n}})\end{matrix}]f_{n}<0(3b)
{x_{t}^{( 1 / 3 )}}^{\prime}( u ) x_{t}^{( 1 / 3 )}( v ) -{x_{t}^{( 1 / 3 )}}^{\prime}( v ) x_{t}^{( 1 / 3 )}( u ) = x_{t}^{( 1 / 3 )}( u + v ) [ \wp^{( 1 / 3 )}( v ) - \wp^{( 1 / 3 )}( u ) ] ,
\mathrm{PD}(\omega) = \bigg( \int_{\{k^0_x \}\times S^1}\omega \bigg)[\overline{\alpha}_x]- \bigg( \int_{S^1 \times \{k^0_y \}}\omega \bigg)[\overline{\alpha_y}]-\sum_{i = 1}^{n-1}\bigg(\int_{\partial \overline{D}_i}\omega\bigg)[\overline{\alpha}_i]
\phi: \frac{F_{p^r}[x]}{\langle f(x) \rangle}\longrightarrow \frac{F_{p^r}[x]}{\langle F(x) \rangle}\oplus \frac{F_{p^r}[x]}{\langle F^{*}(x) \rangle}\cong \frac{F_{p^r}[x]}{\langle{x^N}-1 \rangle}P(x) \mapsto \phi(P(x))=(P(x),0)
\sigma = \sin(\theta)\,\cos(\varphi)\,\sigma_1 +\sin(\theta)\,\sin(\varphi)\,\sigma_2 +\cos(\theta)\,\sigma_3 \bar{\sigma}= \cos(\theta)\cos(\varphi)\,\sigma_1 +\cos(\theta)\sin(\varphi)\,\sigma_2 -\sin(\theta)\,\sigma_3 \hat{\sigma}= -\sin(\varphi)\,\sigma_1+\cos(\varphi)\,\sigma_2
=\frac{x\rho}{a(t)r}\left[ \left( \ddot{a}(t)-\frac{\xi^{2}}{a(t)^{3}}\right) r+K\dot{f}(s)+\frac{2\pi}{s}{\displaystyle\int\limits_{0}^{s}}e^{f(\tau)}\tau d\tau\right]
\zeta^{\prime}( s \mid D_{0}D_{0}^{\dagger}) = 2 | k | \left\{- \left( \ln \frac{4 \pi | k |}{| \tau |}\right) \left( \frac{| \tau |}{4 \pi | k |}\right)^{s}\zeta_{R}( s ) + \frac{| \tau |^{s}}{( 4 \pi | k | )^{s}}\zeta_{R}^{\prime}( s ) \right\},
{\cal L}= \frac{\pi \alpha}{1 2}\langle \theta^{2}\rangle [ F_{\mu \nu}^{0}F^{0 \nu \eta}F_{\eta \rho}^{0}F^{0 \rho \mu}- ( F_{\mu \nu}^{0}F^{0 \mu \nu})^{2}] \, \, \, .
\Psi^{(p,g)}(\iota_{p,g}(\theta_{x,y})) = \Psi_{(p,g)}(x\otimes g) \Psi_{(p,g)}(y\otimes g)^* &=\psi_p(x) U_g U_g^* \psi_p(y)^* \ (\eqref{E:DefPsi}) &= \psi_p(x)\psi_p(y)^* = \psi^{(p)}(\theta_{x,y}).
\displaystyle{\prod_{i=1}^{d}}\big{[}Y_{e+i}\big{]}^{r}\times\big{[}F|(\bigcup_{1\leq i<\theta}Y_{e^{'}\!+i})\big{]}^{r,f^{'}}\subset\big{[}F|(\bigcup_{1\leq i<\theta}Y_{e+i})\big{]}^{r,f^{'}\!\!+d}.
\frac{X_{R}^{\mu}\left( \stackrel{i + 1}{\tau_{R}+ \triangle \tau_{R}}\right) - X_{R}^{\mu}( \stackrel{i}{\tau_{R}})}{\triangle \tau_{R}}= \Pi_{R}^{\mu}\cdot 2 \pi \alpha^{\prime}= \frac{2 p^{\mu}2 \pi \alpha^{\prime}}{\triangle \tau_{R}}
[\mathrm{Sh}_n(\sigma(\mathrm H_C^{\mathrm{dR}}))|\Omega(C)]=\mathrm{Sh}_{n-1}(\sigma(\mathrm H_C^{\mathrm{dR}}))\oplus [\mathrm{Sh}_{n-1}(\sigma(\mathrm H_C^{\mathrm{dR}}))|d\mathcal O(C)]\subset\mathrm{Sub}_\sigma.
\Psi \left( x_{\parallel}, x_{\perp}\right) \to \sum_{m = 0}^{\infty}\frac{\phi_{+ , +}\psi_{m}^{( R )}( x_{\parallel}) + \phi_{+ , -}\psi_{m}^{( L )}( x_{\parallel})}{\sqrt{2 \pi m !}l}\left( \frac{\bar{z}}{\sqrt{2}l}\right)^{m}\exp \left( - \frac{z \bar{z}}{4 l^{2}}\right) ,
2n \left ( \binom{2n}{n}\frac{1}{4^n}\right )^2 = \frac{1}{2}\prod_{j=2}^n \left (1 + \frac{1}{4j(j-1)}\right ), (2n+1) \left ( \binom{2n}{n}\frac{1}{4^n}\right )^2 = \prod_{j=1}^n \left ( 1 - \frac{1}{4j^2}\right )
\varphi (g_1 g_2) \varphi (g_1 g^{-1}_2) \overset{(\ref{eq:var-noncomm})}{=}\varphi (g_1) \varphi (g_2) \varphi (g^{-1}_2) \varphi (g_1) \overset{{\bf A2}}{=}\varphi (g_1) \varphi (g_1) \overset{{\bf A3}}{=}\varphi (g^{2}_1) \,.
-\langle\frac{dE}{dt}\rangle=\frac{32G^{4}m_{1}^{2}m_{2}^{2}(m_{1}+m_{2})}{5c^{5}a^{5}(1-e^{2})^{7/2}}(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4})
= \int_{\mathbb{R}}\Bigg\{\frac{1}{2}\partial_{t}A_{0}\partial_{x}\textbf{w}- \partial_{x}A_{0}\partial_{t}\textbf{w}- \frac{1}{2}\partial_{x}A_{1}\partial_{x}\textbf{w}\Bigg\}\cdot \partial_{x}\textbf{w}+ (\partial_{x}(A_{0}D)\textbf{w}) \cdot \partial_{x}\textbf{w}~ dx.
z ( 1 - z ){\frac{d^{2}\tilde{\Psi}}{d z^{2}}}+ ( 1 - z ){\frac{d \tilde{\Psi}}{d z}}+ \left({\frac{A_{1}}{z}}+{\frac{s / 4}{1 - z}}+ B_{1}\right) \tilde{\Psi}= 0 ,
E_\chi\colon y^2 + \pi_K^r a_1xy + \pi_K^{3r}a_3 y = x^3 + \pi_K^{2r}(a_2 +\gamma a_1^2)x^2 +\pi_K^{4r}a_4x + \pi_K^{6r}(a_6 +\gamma a_3^2),
( A_{1}(\textbf{w}+\bar{\textbf{u}})\partial_{x}{\textbf{w}},{\textbf{w}}) = \frac{1}{2}\partial_{x}(A_{1}(\textbf{w}+\bar{\textbf{u}}){\textbf{w}},{\textbf{w}}) - \frac{1}{2}(\partial_{x}A_{1}(\textbf{w}+\bar{\textbf{u}})\bar{\textbf{w}},{\textbf{w}}).
\phi_{\varepsilon}^{\prime}\left( t\right) =\frac{1}{\varepsilon}d\left(t,\phi_{\varepsilon}\left( t\right) \right) \gamma \left( t,\phi_{\varepsilon}\left( t\right) \right) +\psi^{\prime}\left( t\right),\phi_{\varepsilon}\left( 0\right) =\psi \left( 0\right) ,
{\cal L}_{\mathrm{m a s s}}= \frac{m^{\prime}}{2}( \overline{{{N}}}_{R}j N_{R}- i \overline{{{N}}}_{R}j N_{R}i ) + \frac{M}{2}\left[ \overline{{{N}}}_{R}( N_{R})^{c}- i \overline{{{N}}}_{R}( N_{R})^{c}i + \mathrm{H . c .}\right] .
\prod_{(i,j)\in \Delta_m}\frac{1-R_{ij}}{1+R_{ij}}= \sum_{s=2}^{m}(-1)^s \frac{1-R_{1s}}{1+R_{1s}}\prod_{(i,j) \in \Delta_m \atop{i, j \in \{1,\dots,m\}\backslash\{1, s\}}}\frac{1-R_{ij}}{1+R_{ij}},