harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f\cdot g (x) =\sum_{k}\big(f \ast \varphi_k \cdot g \ast \psi_k \big) \ast \psi_k(x) +\sum_{k}\big(f \ast \psi_k \cdot g \ast \varphi_k \big) \ast \psi_k(x) +\sum_{k}\big(f \ast \psi_k \cdot g \ast \psi_k \big) \ast \varphi_k(x) &=\sum_{k}Q_k(P_kf \cdot Q_k g)(x)+\sum_{k}Q_k(Q_kf \cdot P_k g) +\sum_{k}P_k(Q_kf \cdot Q_k g).
| f(g_1(x)) - f(g_2(x))|=| f(g_2(x) - (g_2-g_1)(x)) - f(g_2(x))| & = \left|\int_0^1\frac{d}{dt}f(g_2(x)-t(g_2-g_1)(x)) \, dt \right| &\qquad= \left|\int_0^1(\nabla f)(g_2(x)-t(g_2-g_1)(x))\cdot (g_2-g_1)(x) \, dt \right| &\leq |(g_2-g_1)(x)| \int_0^1 |(\nabla f)(g_2(x)-t(g_2-g_1)(x)) | \, dt.
p_{\mu}\; = \; q^{- 3 / 2}\Bigl ( \frac{\partial L_{0}^{m}}{\partial \dot{x}^{\mu}}\Bigr ) \equiv ( \dot{x}_{\mu}+ q^{1 / 2}i \chi \psi_{\mu}) [ ( \dot{x}+ q^{1 / 2}i \chi \psi )^{2}]^{- 1 / 2}\; m ,
\begin{aligned}F^{ii}\nabla_{j}u \nabla_{iij}u \geq \, F^{ii}\nabla_{j}u (\nabla_{j}U_{ii}- \nabla_{j}A^{ii}) - C |\nabla u|^2 \sum F^{ii}\geq \, (\psi_{p_{l}}- F^{ii}A^{ii}_{p_l}) \nabla_{j}u \nabla_{jl}u - C |\nabla u|^2 \sum F^{ii}- C |\nabla u|^2.\end{aligned}
{\rm Prob}(a< S_B(x_1)<b) \ = \ \frac{2}{\sqrt{\pi}}\frac{\Gamma(n/2+1/2)}{\Gamma(n/2)}\sum_{i=1}^\infty \int_{a B^{-i}}^{b B^{-i}}(1-x_1^2)^{n/2-1}\dx_1.
\lim_{n \to \infty}\langle\tilde{u}_n(t), P_n \psi\rangle_{\mathrm{H}}- \langle\tilde{u}(t), \psi\rangle_{\mathrm{H}}& = \lim_{n \to \infty}\langle\tilde{u}_n(t) - \tilde{u}(t), P_n \psi\rangle_{\mathrm{H}}+ \lim_{n \to \infty}\langle\tilde{u}(t), P_n \psi - \psi\rangle_{\mathrm{H}}= 0 \tilde{\mathbb{P}}
P_{(m,0,\ldots, 0)}= \sum_{i=1}^n t^{n-i}\Big(\frac{1-q^m}{1-q^mt}\Big)\Big(\frac{1-q^m t}{1-q^m t^2}\Big)\cdots \Big( \frac{1-q^mt^{n-i-1}}{1-q^mt^{n-i}}\Big)E_{m\varepsilon_i}= \sum_{i=1}^n t^{n-i}\Big(\frac{1-q^m}{1-q^mt^{n-i}}\Big) E_{m\varepsilon_i}.
\begin{array}{ll}D_{\mu}\phi = (\nabla_{\mu}- i \frac{q}{2\sqrt{\beta}}H_{\mu})\phi, D_{\mu}\sigma = (\nabla_{\mu}+ i \frac{e}{2}A_{\mu}-i \frac{\alpha e}{2 \sqrt{\beta}}H_{\mu}) \sigma,\end{array}
\int_{L(s)}h_\emptyset (s)(t)dt= \int h_\emptyset (s)(t) \chi_{L(s)}(t)dt= \lim_{|\sigma|\rightarrow \infty, s\in K_\sigma'}\int h_\emptyset (s)(t) \chi_{L_\sigma(s)}(t)dt\geq 1-4\varepsilon
\left|B^n_R\right| = \frac{1}{n!}\left\{R^n +n\sum_{i=0}^{p}\tilde\alpha_i \sum_{j=0}^{p-i}\frac{2^j (p-i)!}{(p-i-j)!}R^{2(p-j)-1}\chi_{i+j+1}(R)\right\},
\frac{\det\left( A_{G\backslash\left\{j\right\}}-\lambda I\right)}{\det\left( A_{G\backslash\left\{m\right\}}-\lambda I\right)}=\left(\frac{\det\left( A-\lambda I\right)_{\operatorname{row}j=b}}{\det\left(A-\lambda I\right)_{\operatorname{row}m=b}}\right)^{2}
a_{0,1}= r_0-a_{0,0}-a_{0,2}=2a_{0,0}+ 2a_{0,2}+ r_{0}a_{1,1}= 2a_{1,0}+2a_{1,2}+ r_{1}a_{2,1}= 2a_{0,0}+2a_{1,2}+ z_{0}a_{3,1}= 2a_{1,0}+a_{0,2}+ z_{1}
\left\lvert V(G) \right\rvert \leq \left\lvert V(T_S) \right\rvert (d+1)(2 \ell +1)^w \leq (2k-1)(d+1)(\ell + 1)(2\ell + 1)^{2w}(d+1)(2 \ell +1)^w = \mathcal{O}(kd^2(2\ell+1)^{3w+1}).
y(s,t)-x-(t-s)v(\xi) &=q(t,s;x,\eta(t,s))-x-(t-s)v(p(t,s;x,\eta(t,s))) &=\int_s^t \left[v\left(\xi+\int_\tau^t \nabla_x V_\rho(\sigma,q(\sigma,s;x,\eta(t,s)))d\sigma\right)-v(\xi)\right] d\tau.
S = \int_{0}^{1}\left[ - \frac{1}{2 e}\left( \dot{x}^{\mu}- i \psi^{\mu}\chi + i \varepsilon^{\mu \nu \rho \zeta}b_{\nu}\psi_{\rho}\psi_{\zeta}+{\frac{\alpha}{2}}b^{\mu}\right)^{2}- i \psi_{\mu}\dot{\psi}^{\mu}\right] d \tau ,
|D\ell^{k}| (J) = \sum_{i=0}^{N_{k}}| \ell (z_{i+1}^{k}) - \ell (z_{i}^{k})| &= | \ell(z_{1}^{k}) - \ell(a) | + \sum_{i=1}^{N_{k}-1}| \ell (z_{i+1}^{k}) - \ell (z_{i}^{k})|+| \ell(b) - \ell(z_{N_{k}}^{k})|.
B=[\begin{matrix}1&a_{12}&a_{13}&\cdot\cdot\cdot&a_{1n}\\ 0&1&a_{23}&\cdot\cdot\cdot&a_{2n}\\ \vdots&\vdots&\vdots&&\vdots\\ 0&0&0&\cdot\cdot\cdot&1\end{matrix}]
a_1^p=[a_1,b_1], \; a_2^p=[a_1,b_1^{\lambda_1}b_2^{\lambda_2}],\;b_1^p=[a_2,b_1b_2], & b_2^p=[a_2,b_2], \; [a_1,a_2]=[b_1,b_2]=1.
\Gamma = \Big\langle a_1,b_1,\ldots, a_g,b_g, e_1,\ldots, e_r, \, p_1,\ldots, p_s, h_1,\ldots,h_t \ \Big\vert\e_1^{m_1}= \ldots = e_r^{m_r}= \big(\prod_{j=1}^g [a_j,b_j]\big) e_1\cdots e_r p_1\cdots p_s h_1\cdots h_t = 1 \Big\rangle
\|u\|_{2}^2 \leq \int_{\{|u|\leq M\}}u^2\, dx +\frac{2}{\alpha}\int_{\{|u|\geq M\}}F(u)\, dx &\le M^2T+\frac{2}{\alpha}\int_0^T F(u)\, dx &\le M^2T+\frac{2}{\alpha}\biggl( J_q(u)+q\|u'\|_{2}^2\biggr) &\le M^2T+\frac{2qC}{\alpha}.
\textbf{S}(t,z)(z\partial z + \mathfrak{E}_{(\partial)}+ \mu)\alpha = \textbf{S}(t,z) \left( \frac{(\alpha)}{2}-\frac{\textnormal{dim}_\mathbb{C}(X)}{2}\right) \alpha =\left( \frac{\alpha}{2}-\frac{\textnormal{dim}_\mathbb{C}(X)}{2}\right) S(t,z) (\alpha)
\left|P_{3}\right|\le\frac{\epsilon^{3}}{4}\sum_{i=1}^{N}\int_{G_{N}}\left|\nabla_{i}f(x)\right|M_{\epsilon,i}^{2}(x)dx\le CN^{3}\left[1+\left(\frac{\mathcal{D}_{N}(f)}{N}\right)^{\frac{3}{4}}\right]\epsilon^{\frac{7}{2}}
\sum_{k=1}^{3}\binom{8}{2k}\binom{2k+1}{3}\frac{B_{2k-2}}{2k+1}=\frac{28}{3}B_{0}+140 B_{2}+140B_{4}&= \frac{28}{3}+ \frac{70}{3}- \frac{14}{3}=28=\frac{1}{2}\binom{8}{3},
L^{n,2}_\tau = n^{2\alpha+ 1}\int^{\tau}_0\gamma_s (\sigma'\sigma)^2(X_{\eta_n(s)}) (s-\eta_n(s))^\alpha (W_s-W_{\eta_n(s)}) \left( \int_0^{\eta_n(s)}\psi_{n,1}(u,s) dW_u \right) ds,
\mathcal{T}_D(f)(z) = -\frac{1}{2\pi}\iint_D \left(\frac{f(\zeta)}{\zeta}\frac{\zeta+z}{\zeta - z}+ \frac{\overline{f(\zeta)}}{\overline{\zeta}}\frac{1+z\overline{\zeta}}{1-z\overline{\zeta}}\right)\, d\xi\,d\eta,
R({\mathcal{H}_{1}}) = \left(1-{P}_{\mathrm{d}}\right) \frac{{T}_{\mathrm{f}}-{T}_{\mathrm{s}}}{{T}_{\mathrm{f}}}\sum_{i = 1}^{N}P\left({\mathcal{H}}_{1,i}\right) C\left({\mathcal{H}}_{1,i}\right),
V(a) = - \frac{\kappa_{(4)}^{2}}{6}\rho_{X0}a^{-3w_{X}+2}- \frac{\kappa_{(4)}^{2}}{12\lambda}\rho_{X0}^{2}a^{-6w_{X}+ 2}- \frac{\sigma_{0}^2}{6}a^{-4}- \frac{\Lambda_{(4)}}{6}a^{2}- \frac{{\cal U}_0}{\lambda \kappa_{(4)}^{2}}a^{-2}+ \frac{k}{2}
0= -2c||\xi^\perp||X_j^- 0= -c(||\xi^\perp||+1)g(X_j,\xi_a) +2c\eta_a(\xi)g(X_j,\xi^\perp) 0= -c(||\xi^\perp||+1)g(X_j,\phi\xi_a)+2c\eta_a(\xi)g(X_j,\phi\xi^\perp) 0= c(2||\xi^\perp||+1)(||\xi^\perp||-1)\left\{g(X_j,\phi\xi^\perp)^2+g(X_j,\xi^\perp)^2\right\}.
\widetilde z = (r^2 t + t_0, r^3 x + x_0 - r^2 t v_0, r v + v_0),\widetilde u (z) = u (\widetilde z), & Y = \partial_t - v \cdot D_x, \widetilde P_0 = \partial_t - v \cdot D_x - a^{ij}( r^2 t + t_0 ) D_{v_i v_j}.
\begin{gathered}R_{0}+ \tilde{R}_{0}> H(Y_{0}|Y_{j}Z_{j}), R_{j}+ \tilde{R}_{j}> H(Y_{j}|Y_{0}Z_{j}), R_{0}+ \tilde{R}_{0}+ R_{j}+ \tilde{R}_{j}> H(Y_{0}Y_{j}|Z_{j}), \end{gathered}
\left(U^T\left(\frac{3n+2}{6}- \int_0^\infty Tr[\gamma e^{-2y \gamma}] e^{-y \gamma}dy\right)U\right)^{ij}=\left(\frac{1}{3}+\frac{\lambda_i}{2}\sum_l \frac{1}{2\lambda_l+\lambda_i}\right) \delta^{ij}.
u_{i}= g(t_{i}, u_{i}) + \underbrace{\sum_{j=0}^{i}w_{i,j}k_{1}(t_{i}, t_{j},{u}_{i},{u}_{j})}_{}+ \underbrace{\sum_{j=0}^{i}\omega_{j}k_{2}(t_{i}, t_{j},{u}_{i},{u}_{j})}_{}, i=0, 1, \cdots, n,
H^2\,\frac{Q^2+m^2-M^2}{Q^2}\stackrel{\eqref{eq:N0}}{=}H^2 N_0^2\stackrel{\eqref{eq:prop'}}{=}4\left(\nu(N)\right)^2\stackrel{\eqref{def:m}}{=}4\frac{m^2}{r^4}\,.
f_0(z) = 1+2q+2q^4+2q^9+2q^{16}+2q^{25}+2q^{36}+2q^{49}+2q^{64}+2q^{81}+\cdots, f_3(z) = q^{-3}-248q+26572q^4-85995q^5+1707264q^8-4096248q^9+\cdots, f_4(z) = q^{-4}+492q+143376q^4+565760q^5+18473000q^8+51180012q^9+\cdots, f_7(z) = q^{-7}-4119q+8288256q^4-52756480q^5+5734772736q^8+\cdots.
{\rm ch}[W(\Lambda_0)] \ \_{=}^{?}\sum_{n_1,n_2,n_3,n_4 \geq 0}\frac{q^{n_1^2+n_2^2+n_3^2+n_4^2+n_1 n_2+n_1 n_4}}{(q)_{n_1}(q)_{n_2}(q)_{n_3}(q)_{n_4}}
({\cal E}_{\tau}^{\underline{{D - 1}}})^{2}+ \sum_{i = 1}^{i = D - 2}({\cal E}_{\tau}^{i})^{2}= ({\cal E}_{\tau}^{\underline{0}})^{2}\equiv ( \partial_{\tau}X^{\underline{0}}- i \partial_{\tau}\bar{\Theta}\Gamma^{\underline{0}}\Theta )^{2},
\frac{\delta_{\Omega}(\zeta)}{2^{i}\delta_{\Omega}(\zeta)}\frac{1}{\prod_{j=1}^{n}\tau_{j}^{2}\left(\zeta,\delta_{\Omega}(\zeta)\right)}\tau_{m}\left(\zeta,\delta_{\Omega}(\zeta)\right)\tau_{l}\left(\zeta,\delta_{\Omega}(\zeta)\right)\frac{1}{\left|z-\zeta\right|},
\sum_{x, y, z \in S}(1_G(x, y) 1_G(y,z) 1_G(z,x) - p^3) = \sum_{x,y,z \in S}(1_G(x,y) - p) 1_G(y,z) 1_G(z,x) + & \sum_{x,y,z \in S}p (1_G(y,z) - p) 1_G(z,x) + \sum_{x,y,z \in S}p^2 (1_G(z,x) - p).
\sup_{b_l\leq (1-\theta)}|\tilde{u}_l| \leq C(n, p, \theta, V_M)\Big[I_{\tilde{u}_l}^{(l)}(1)\Big]^{\frac{1}{2}}\leq C(n, p, \theta, V_M, \alpha)\Big[I_{\tilde{u}_l}^{(l)}(\frac{1}{2})\Big]^{\frac{1}{2}}&= C(n, p, \theta, V_M, \alpha) \sup_{b_l\leq (1-\theta)}|\nabla \tilde{u}_l|_{g_l}\leq C(n, p, \theta, V_M, \alpha)
\partial_{t}^{m}u=F(t,x,\partial_{x}^{\alpha}\partial_{t}^{k}u)=\sum_{\alpha\in\mathbb{N}_{0}^{n},0\le k\le m-1,|\alpha|+k\le m}A_{\alpha,k}(t,x)\partial_{x}^{\alpha}\partial_{t}^{k}u,
\int_0^\infty \frac{d \kappa^2}{\lambda}\rho_{ii}\left( \frac{\kappa^2}{\lambda}\right)H(\kappa^2) \leq C \int_0^\infty \frac{d\kappa^2}{\lambda}\rho_{ii}\left(\frac{\kappa^2}{\lambda}\right) G(\kappa^2).
288\cdot V^{2}=|\begin{matrix}0&1&1&1&1\\ 1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\\ 1&d_{12}^{2}&0&d_{23}^{2}&d_{24}^{2}\\ 1&d_{13}^{2}&d_{23}^{2}&0&d_{34}^{2}\\ 1&d_{14}^{2}&d_{24}^{2}&d_{34}^{2}&0\end{matrix}|
\frac{1}{\sqrt{- i}}\left( \frac{x^{-}}{x^{+}}\right)^{2}F ( x^{+}, x^{-}, 0 ) = \sum_{\alpha}\frac{1}{2 ( 2 \pi )^{5 / 2}}\frac{M_{\alpha}^{9 / 2}}{\sqrt{r}}K_{9 / 2}( M_{\alpha}r ) \frac{| \langle u | \alpha \rangle |^{2}}{l K^{3}| N_{u}|^{2}}
\inf_{P_{X},Q_{X}:\left|P_{X}-Q_{X}\right|\geq\epsilon}D_{1+s}(P_{X}\|Q_{X}) =\inf_{P_{X},Q_{X}:\left|P_{X}-Q_{X}\right|=\epsilon}D_{1+s}(P_{X}\|Q_{X}) =\inf_{q\in[0,1-\epsilon]}d_{1+s}(q+\epsilon\|q),
T_{a b}= \frac{1}{4 \pi \alpha^{\prime}}\, ( \partial_{a}X^{\mu}\partial_{b}X_{\mu}-{\frac 1 2}\, g_{a b}\, \partial_{c}X^{\mu}\partial^{c}X_{\mu}) - \frac{1}{4 \pi k}\, ( R_{a b}-{\frac 1 2}\, g_{a b}\, R ) .
\frac{(q^5;q^6)_\infty(q^7;q^6)_\infty}{(q^2;q^6)_\infty(q^4;q^6)_\infty}=\frac{(q;q^2)_\infty(q^6;q^6)_\infty}{(1-q)\,(q^2;q^2)_\infty(q^3;q^6)_\infty}.
A = \langle (1,0,0,0)\rangle B = \langle (0,1,0,0)\rangle C = \langle (0,0,1,0)\rangle D = \langle (0,0,0,1)\rangle W = \langle (0,1,1,1)\rangle X = \langle (1,0,1,1)\rangle Y = \langle (1,1,0,1)\rangle Z = \langle (1,1,1,0)\rangle.
E_{C}^{( 2 ) \; r e g}( R \to \infty , \Lambda ) = - \frac{\Lambda}{2 \pi}\ln \left[ \sinh \left( R \sqrt{\Lambda^{2}+ \omega_{0}^{2}}\right) \right] + \frac{1}{2 \pi}\int_{0}^{\Lambda}d y \left( R \sqrt{y^{2}+ \omega_{0}^{2}}- \ln 2 \right) - \frac{\omega_{0}^{2}}{4}.
\left | \iint_{\begin{subarray}{l}\xi=\xi_{1}+\xi_{2}\tau=\tau_{1}+\tau_{2}\end{subarray}}| 5\delta \xi^{4}+3\gamma \xi^{2}+2 \xi_1 |^{\frac{1}{4}}\hat{f}(\xi, \tau)\hat{g}(\xi_1, \tau_1)\hat{h}(-\xi_2,-\tau_2) \right | \lesssim \| f\|_{Y^{0,\frac14,1}}\|g\|_{X^{0,\frac38,1}}\|h\|_{X^{0,\frac14,1}}.
d_0([g_1\otimes \cdots \otimes g_n]) = g_1 \cdot [g_2\otimes \cdots \otimes g_n], d_i([g_1\otimes \cdots \otimes g_n]) = [g_1\otimes \cdots \otimes g_i g_{i+1}\otimes \cdots \otimes g_n], \ 1\leq i \leq n-1, d_n([g_1\otimes \cdots \otimes g_n]) = [g_1\otimes \cdots \otimes g_{n-1}].
\mathcal{B}_\pm(-U, \varGamma)= \ker (\varGamma \pm 1) \cap \ker (-C + 1)= \mathcal{T}_\mp(U,\varGamma), & \mathcal{T}_\pm(-U, \varGamma)= \ker (\varGamma \mp 1) \cap \ker (-C - 1)= \mathcal{B}_\mp(U,\varGamma),
\left[ A^{-}, A^{+}\right]_{q \left( x \right)}= - \alpha \left( x \right) \cdot \sinh{}\frac i 2 \frac d{d x}\left[ \alpha \left( x \right) \cdot \sinh \left( 2 \rho_{\frac s 2}\left( x \right) \right) \right]
1+f\left(\sum_{i=1}^{N-1}D(u_i,u_{i+1})\right)-f(D(x,y)) &=1-\frac{1}{\displaystyle\sum_{i=1:N-1,\,u_{i+1}\neq u_i}\exp\left(|u_{i+1}-u_i|\right)}+\frac{1}{\exp\left(|x-y|\right)}&\geq 1-1+ \frac{1}{\exp\left(|x-y|\right)}&\geq 0.
\langle f, \sqrt{W(x-y)}(1-\Delta_x)^{\delta-1}(1-\Delta_y)^{\delta-1}\sqrt{W(x-y)}f\rangle = \int \overline{f(x,y)}\sqrt{W(x-y)}G(x-x') G(y-y') \sqrt{W(x'-y')}f(x',y') dxdydx'dy' \le \int \frac{W(x-y) |G(x-x')|^2 |f(x',y')|^2 + W(x'-y') G(y-y')|^2 |f(x,y)|^2}{2}& = C_\delta \|W\|_{L^1}\langle f, f \rangle
\left(J_a^\alpha f_n\right)(x)= & \sum_{k=0}^n (-\alpha)^kC_n^k (x-a)^{n-k}\left[\frac{1}{k+1}\left(\frac{x-a}{\alpha}\right)^{k+1}E_1\left(\frac{x-a}{\alpha}\right)-\frac{k!}{k+1}e_k\left(\frac{x-a}{\alpha}\right)e^{\frac{-(x-a)}{\alpha}}+1\right]
\begin{aligned}\log \mathcal{H}(k) =-k\sum_{5\leq p\leq k^{2/3}}\log\left(1-\frac{1}{p}\right)+\sum_{p>k^{2/3}}\log\cosh\left(\frac{k}{p}\right)+ O\left(k^{2/3}\right) &=-k\sum_{5\leq p\leq k}\log\left(1-\frac{1}{p}\right)+\sum_{k^{2/3}<p<k^{4/3}}f\left(\frac{k}{p}\right)+ O\left(k^{2/3}\right),\end{aligned}
\Phi_{L . W .}^{( 3 , 0 )}= \Phi_{0 , 3}^{( 3 , 0 )}= V^{( 3 , 0 )}\Phi_{1 , 0}^{( 3 , 0 )}= ( a_{2 2}^{*}a_{1 1}^{*^{3}}- a_{1 2}^{*}a_{1 1}^{*^{2}}) V^{( 3 , 0 )}
\int_{B_{9/10}}\left|\nabla v(x)-q\right|^p\,dx \le\; 2^{p-1}\left(\int_{B_{9/10}}\left|\nabla u(x)-q\right|^p\,dx+\int_{B_{9/10}}\left|\nabla v(x)-\nabla u(x)\right|^p\,dx\right) \le\; C_2\varepsilon^pa^p+C_1\varepsilon^{p+\delta}+C\sigma(a^p+1),
|Y(t) - y| = \left| J_{k-1}(t - T_{k-1}) + Y_{k-1}- y \right| \leq t - T_{k-1}+ \left| \sum_{i=1}^{k-1}(Y_i - Y_{i-1}) \right| & \leq t- T_{k-1}+ T_{k-1}= t.
\frac{1}{4 j}+ \sum_{m=1}^i \frac{E_{2 m}}{(4 m)}\binom{2 j}{2 m}\frac{m}{j}= 1 + \sum_{m=1}^i E_{2m}\binom{2 j}{2 m}= \sum_{m=0}^j E_{2m}\binom{2 j}{2 m}
\mathcal{V}_{L}(z) = \sum_{i=1}^{n}\mathcal{V}_{L,i}= \sum_{i=1}^{n}\left(-z + (1+\alpha-w_i) \log{(z-w_i)}+ (z+w_i \log{(z-w_i)}) f_i(w_1,\ldots, w_n)\right),
\mathbb{E}_{, k}[x_{i_1}, \cdots, x_{i_n}] = \frac{(n-k)!}{n!}\sum_{j_1 \neq j_2 \neq \cdots \neq j_k}x_{j_1}^* \cdots x_{j_k}^*x_{j_k}\cdots x_{j_1}.
\mathbb{E}(Q_i) = \frac{k\theta(2+\lambda\theta+3k\lambda\theta)}{2\lambda}+2q^k\left(\frac{1-k^2\lambda\theta}{\lambda^2}\right) +q^{k+1}\left(\frac{k\theta(1+k\lambda\theta+2k)}{\lambda}\right)-\frac{1}{\lambda^2}q^{2k}-\frac{k\theta}{\lambda}q^{2k+1}.
\Gamma_{4 V}^{[ r , 0 ]}\ = \ - ( C_{1}^{[ r ]})_{2 V , 2 V}\ - ( C_{2}^{[ r ]})_{2 V , 2 V}\ - ( C_{3}^{[ r ]})_{2 V , 2 V}\ + V_{4 V}^{[ r , 0 ]},
\frac{\int_{0}^{t}ROI(\tau)d\tau}{ROI(t)}=(-U_{n}^{T}K^{-1}Q+V_{p})\frac{\int_{0}^{t}C_{p}(\tau)d\tau}{ROI(t)}+\frac{U_{n}^{T}K^{-1}A}{U_{n}^{T}A+V_{p}C_{p}}
sum_{j=1}^k (R-\rho)^{j}f_j(\rho) \leq \epsilon \sum_{j=1}^k (R-\rho)^j f_j(R) + C_\epsilon (R-\rho)^{k+\alpha}[\nabla^k\sigma]_{C^{\alpha}(B^{g_P}(p,R))}+ C_\epsilon \|\sigma\|_{L^\infty(B^{g_P}(p,R))}.
\Psi = \psi + \psi_{\alpha A}\lambda_{\alpha A}+ \frac{1}{2}\ \psi_{\alpha A \beta B}\lambda_{\alpha A}\lambda_{\beta B}+{\dots}+ \frac{1}{\Lambda !}\ \psi_{\alpha_{1}A_{1},\dots , \alpha_{\Lambda}A_{\Lambda}}\lambda_{\alpha_{1}A_{2}}\dots \lambda_{\alpha_{\Lambda}A_{\Lambda}}\ .
\exp\Bigl(-\bigl(\mathcal{S}_N \log\Gamma_M\bigr)(q\,|\,a,b)\Bigr) = \prod\limits_{n_1,\cdots ,n_M=0}^\infty \Bigl[\frac{q+b_0+\Omega}{\prod\limits_{j_1=1}^N q+b_0+b_{j_1}+\Omega}\star & \star\frac{\prod\limits_{j_1<j_2}^N q+b_0+b_{j_1}+b_{j_2}+\Omega}{\prod\limits_{j_1<j_2<j_3}^N q+b_0+b_{j_1}+b_{j_2}+b_{j_3}+\Omega}\cdots\Bigr],
\sum_{n=1}^{\infty}\frac{n(-1)^{n-1}q^{n(n+1)/2}}{(q)_n(1+q^n)}+(q)_{\infty}\sum_{k=1}^{\infty}\frac{q^{k(k+1)}}{(q)_k^{2}(1-q^k)}F(0;q^k;-q^k) &=\frac{1}{4}\left(1-\frac{(q)_{\infty}}{(-q)_{\infty}}\right)+\frac{1}{2}\frac{(q)_{\infty}}{(-q)_{\infty}}\sum_{n=1}^{\infty}\frac{(-q)_n}{(q)_n}\frac{q^n}{1-q^n}.
E=\begin{pmatrix}x_{1,1}x_{1,2}\cdots x_{1,n-A(\delta-1)}x_{2,1}x_{2,2}\cdots x_{2,n-A(\delta-1)}\vdots \vdots \ddots \vdots x_{k,1}x_{k,2}\cdots x_{k,n-A(\delta-1)}\end{pmatrix}=(E_1|E_2|\dots|E_A),
\biggr|\frac{\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^{k+2}\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^k-\big(\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^{k+1}\big)^2}{\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^{k+1}\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^k}\biggr|=
\eta_{s,t}^{l,n+1}:=\int_{s}^{t}h\left( y_{u}^{n+1},y_{u}^{n}\right) \eta_{s,u}^{l,n}dx_{u}\eta_{s,t}^{l,l}:=\int_{s}^{t}h\left(y_{u}^{l},y_{u}^{l-1}\right) dx_{u}l\geq 1\eta_{s,t}^{0,0}:=f\left( \xi \right) \left( x_{t}-x_{s}\right)
\,\varphi_{\underline{d}',\underline{d}}(V_{\underline{d}'})= r+1-\,V_{\underline{d}'}^{X_{2}^{c},0}= r+1-(\,V_{\underline{d}''}^{X_1,0}-\,V_{\underline{d}''}^{X_{3}^{c},0}).
{\mathsf W}_{- +}^{( 2 1 0 )}= \frac{1}{4 !}h_{a b}^{^{( 2 1 0 )}}< \widehat{\Psi}_{( - ) a}^{*}| B \Gamma_{[ \mu}\Gamma_{\nu}\Gamma_{\rho}\Gamma_{\lambda ]}| \widehat{\Psi}_{( + ) b}> \Phi_{\mu \nu \rho \lambda}
\{\eta_1\wedge\eta_2,\omega_1\}=-\big(\{\omega_1,\eta_1\}\eta_2+(-1)^{|\omega_1|}\eta_1\{\omega_1,\eta_2\}\big) &=(-1)^{|\omega_1|}\{\eta_1,\omega_1\}\eta_2+\eta_1\{\eta_2,\omega_1\}&=(-1)^{|\omega_1|}(\iota_{\eta_1}\omega_1)\eta_2+\eta_1(\iota_{\eta_2}\omega_1),
|x_k-Q|\,\le\,|x_1-Q|+\sum_{j=1}^{k-1}|x_{j+1}-x_j|\,\le\, \frac{\kappa r}{24}+ \sum_{j=1}^{\infty}r^{1-\gamma}\rho_Q(x_j)^{\gamma}&\le \frac{\kappa r}{24}+r^{1-\gamma}\sum_{j=1}^{\infty}(1-\delta_*^2)^{(j-1)\gamma/\beta_1}K^{-\gamma/\beta_1}r^{\gamma}\,=\frac{\kappa r}{24}+ r K^{-\gamma/\beta_1}\,\frac{1}{1-(1-\delta_*^2)^{\gamma/\beta_1}}.
g_t =\begin{cases}g_0 M_0 \setminus B(p_0, 2 t^{1/4}) (1 - \chi_{t}(r) )\phi_{t}^* g_0 + \chi_{t}(r) t \cdot g_{cal}(1/2) t^{-1/4}< r < 2 t^{-1/4}t \cdot g_{cal}r < (1/2) t^{-1/4}\end{cases}
\left[ - \nabla_{\widehat{\mathrm{\boldmath \scriptstyle{\xi}}}, D}^{2}- \widehat{{\mathcal W}}^{({D})}( \widehat{\mathrm{\boldmath \xi}}) \right] \widehat{\Phi}( \widehat{\mathrm{\boldmath \xi}}) = \widehat{\eta}\widehat{\Phi}( \widehat{\mathrm{\boldmath \xi}}) .
f(6 \cdot 2^{2m}) = f(3)\,f(2^{2m+1}) = 3\,f(2^{2m+1}) * = f\big( (2^m)^2 + (2^m)^2 + 2^2(2^m)^2 \big) = 2\,f(2^{2m}) + f(2^{2m+2})
\begin{gathered}I_{0,1}^d=\iota(u^{m,n}_y)=\frac{u_y^{m,n}}{u_{m,n}},I_{0,2}^d=\iota(u_{yy}^{m,n})=\frac{u_{yy}^{m,n}}{u_{m,n}}, I_{1,1}^d=\iota(u_{xy}^{m,n})=\frac{u_{m+1,n+1}\, u_{m,n}-u_{m+1,n}\, u_{m,n+1}}{u_{m,n}^2\,u_{m+1,n}\, \Delta x_m\,\delta y_n}= \frac{u_{m,n}\, u_{xy}^{m,n}- u_x^{m,n}\, u_y^{m,n}}{u_{m,n}^2\, u_{m+1,n}}.\end{gathered}
\aligned \frac{1}{2}\mathcal{L}\sum_{i,j,k,p}(h_{ijk}^{p^{\ast}})^{2}= H^{2}\bigl[H^{2}-2S+\frac{1}{2}H^{4}-H^2\lambda^2-KH^{2}-K^{2}\bigl] &+H^{2}(S+2-\frac{3}{2}H^{2}-\lambda_{1}^{2}-\lambda_{2}^{2}-2\lambda^2)(\lambda_{1}^{2}+\lambda_{2}^{2}+2\lambda^2).\endaligned
\alpha = (\underbrace{1,\ldots,1}_n,\underbrace{r_n}_{\neq 1},r_{n+1},r_{n+1}\ldots) \mapsto s(\alpha) = (\underbrace{1,\ldots,1}_{n+1},0,0,\ldots):\Gamma_{\log}\to\Psi_{\log}\subseteq\Gamma_{\log}
\Pi_{2 , 2}^{0 , \beta}( m_{0}, i \Omega , \vec{k}= 0 ) = - \frac{\lambda^{2}~ I_{\beta}^{\epsilon}( m_{0})}{2 ( 4 \pi )^{2}}\left[ \frac{1}{\tilde{\epsilon}}- \ln \left( \frac{m_{0}^{2}}{\mu^{2}}\right) + 2 \right] + F ( \Omega^{2}) ,
\gamma_{k+1}^{\delta}- \gamma_k^{\delta}\leq \mu \delta\|F(u_k^{\delta})-v^{\delta}\|^{p-1}+ \bigg[ 2^{p+q-1}\beta_k^q \frac{G_q}{q}\frac{p}{C_p}+ \beta_k\bigg( \frac{(p-1)\epsilon_2^{\frac{p}{p-1}}}{C_p}+\frac{\epsilon_2^{-p}}{C_p}-(1+r_k) \bigg)\bigg]\rho^2.
\ln\|(z-T)^{-1}x_r\|_p \approx\ln e^{\frac{1}{|z|}}+\ln \big|\frac{1}{z}+\frac{z^{m-1}}{r^{m}}\big(e^{\frac{r}{z}}-\sum\limits_{n=0}^{m}\frac{1}{n!}\frac{r^n}{z^n}\big)\big|.
d\overline{V}_t \le -2{a}\lambda_2\tilde{\gamma}\overline{V}_tdt-2\kappa \tilde{\gamma}\overline{V}_t dt+\frac{m\tau^2\tilde{\gamma}^2 (N-1)}{2N}dt+\frac{\tau}{N}\tilde{\gamma}\sum_{i=1}^N{dB_{i,t}}^T e_{i,t}.
L_p(f,g,j+1)=4^{-1}(-1)^{k'+1}\Gamma(j+1)\Gamma(j-k')^{*}G(\varepsilon_f^{-1}) G(\varepsilon_g^{-1})\frac{L'_{\{p\}}(f,g,j+1)}{\Omega_\infty(j+1)}\Omega_p(j+1,\frac{1-p^{-1}\varphi^{-1}}{1-\varphi}(-t)^{-j-1}\underline{v}),
1+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{5+\frac{1}{2+\frac{1}{1+\frac{1}{3+\frac{1}{1+\frac{1}{1+\frac{1}{390+\cdot\cdot\cdot}}}}}}}}}}}}}}
\theta\sum_{i=1}^n \Bigl\|I_i\sum_{j=1}^m x_j\Bigr\|_{T_\theta}\leqslant \theta \sum_{i=1}^n\sum_{j\in A_i}\|x_j\|_{T_\theta}+ \sum_{i=1}^n \sum_{j\in B_i\setminus A_i}\|x_j\|_{T_\theta}\leqslant \theta a m + 2an.
\delta_i=\det(A_i)=\prod_{j=0}^{i-1}\frac{\binom{r+s-2i+j}{s-i}}{\binom{s-i+j}{s-i}}=\prod_{j=0}^{i-1}\prod_{k=0}^{s-i-1}\frac{r+s-2i+j-k}{s-i+j-k}.
a_s \ = \ \frac{f(s,X_s^{t,x},\hat Y_s^{t,x},\hat Z_s^{t,x})-f(s,X_s^{t,x},\check Y_s^{t,x},\hat Z_s^{t,x})}{\hat Y_s^{t,x}-\check Y_s^{t,x}}1_{\{\hat Y_s^{t,x}-\check Y_s^{t,x}\neq0\}}, t \leq s \leq T
\|\sigma_r\ast\phi_{\varepsilon_0}-\phi_{\varepsilon_0}\|_1 =\|\int_{S^n_1}(\phi_{\varepsilon_0}(rh^{-1}g)-\phi_{\varepsilon_0}(g))d\sigma_1(h)\|_1 &\leq \int_{S^n_1}\|\phi_{\varepsilon_0}(rh^{-1}g)-\phi_{\varepsilon_0}(g)\|_1d\sigma_1(h)\leq C_{\phi_{\varepsilon_0}}r.
\big[\varphi_2(B_1) - \varphi_2(C_1)\big] \cdot \hat{b}_1 = 0, & \big[\varphi_2(C_2) - \varphi_2(B_1)\big] \cdot \hat{b}_1 = 0, & \big[\varphi_2(B_2) - \varphi_2(C_2)\big] \cdot \hat{b}_1 = 0, & \big[\varphi_2(C_1) - \varphi_2(B_2)\big] \cdot \hat{b}_1 = 0,
\Delta_{\mathbf{R}_{0}(\tau)}(\mathcal{I})=\left(1+\tau\cdot\mathbf{1}^{\mathtt{T}}\cdot\left(\mathbf{r}_{0}(0)_{\kappa(\mathcal{I})}^{\kappa(\mathcal{I})}\right)^{-1}\cdot\mathbf{1}\right)\cdot\Delta_{\mathbf{R}_{0}(0)}(\mathcal{I})=\Delta_{\mathbf{R}_{0}(0)}(\mathcal{I})\in\mathbb{C}
\gamma_{\textrm{NHF}}^{\textrm{NOMA}}=\frac{P_\textrm{F}}{P_\textrm{N}+{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^2}/{|h_{\textrm{BF}}|^2}}+\frac{\beta_{\textrm{N}}\eta P_\textrm{B}|h_{\textrm{BN}}|^2 |h_{\textrm{NF}}|^2}{d_{\textrm{BN}}^{\alpha}d_{\textrm{NF}}^{\alpha}\sigma_{\textrm{F}}^2}.
F_p=\; p^2(\sigma_p^2+\gamma_p^2)+\widehat V(p/N^\beta)(\sigma_p+\gamma_p)^2 G_p=\; 2p^2 \sigma_p\gamma_p+\widehat V(p/N^\beta)(\sigma_p+\gamma_p)^2+\frac{1}{N}\sum_{q\in \Lambda_+^*}\widehat V((p-q)/N^\beta)\eta_q,
F_{( e )}^{\mu \nu}= \left( \begin{array}{c c c c}{0}&{- E_{x}}&{- E_{y}}&{- E_{z}}\\{E_{x}}&{0}&{- B_{z}}&{B_{y}}\\{E_{y}}&{B_{z}}&{0}&{- B_{x}}\\{E_{z}}&{- B_{y}}&{B_{x}}&{0}\\ \end{array}\right) .
\begin{aligned}\psi\bigl(f, \iota_{N,n}(g)\bigr) = \|\zeta' \circ \theta_1(f) - \iota_{N,n}(g)\| &\leq \|\zeta' \circ \theta_1(f) - \zeta' \circ \theta_2(g)\| + \|\zeta' \circ \theta_2(g) - \iota_{N,n}(g)\| &< \varphi_{\theta_1, \theta_2}(f, g) + \delta, \end{aligned}
\theta(1/z;q)(qtz;q)_\infty =\frac{\theta(1/z;q)}{(-qtz;q)_\infty}\,(q^2t^2z^2;q^2)_\infty & = \frac{\theta(1/z;q)}{(q^{1/2}tz;q)_\infty}\,(q^{1/2}tz;q^{1/2})_\infty=\frac{\theta(1/z;q)}{(q^{3/2}tz;q)_\infty}\,(qtz;q^{1/2})_\infty
\mathbf{y}_1(t)= \mathbf{H}_{11}\mathbf{x}_1(t)+\mathbf{H}_{12}\mathbf{x}_2(t)+\mathbf{z}_1(t), \mathbf{y}_2(t)= \mathbf{H}_{22}\mathbf{x}_2(t)+\mathbf{z}_2(t),
u(t)= e^{-A(t-t_0)}u_0 -\chi\int_{t_0}^t e^{-A(t-s)}\nabla\cdot ( u(s) \nabla A^{-1}u(s))ds &+\int_{t_0}^t e^{-A(t-s)}u(s)\Big(1+ a_0(s,\cdot)-a_1(s,\cdot)u(s)-a_2(s,\cdot)\int_{\Omega}u(s)\Big)ds.
D(p)=\left\{\frac{p^2}{4\pi n\kappa}[\zeta F(p)-4\kappa^2z]-(z\zeta-\frac{p^4}{4\pi^2 n^2})\sqrt{F(p)}\right\}\left[\zeta^2F(p)-\frac{\kappa^2 p^4}{\pi^2 n^2}\right]^{-1}
\sum_{j=1}^{\log_2(\frac{X}{T_0})}\frac{1}{\log (2^{-j}X)}\prod_{2^{-j}X < p' < T}\left(1 - \frac{|h(p')|}{p'}\right) \ll \frac{1}{(\log T)^{\alpha_f/H}}\sum_{j=1}^{\log_2(\frac{X}{T_0})}\frac{1}{(\log X - j \log 2)^{1-\alpha_f/H}}\ll_{\alpha_f}\frac{(\log X)^{\alpha_f/H}}{(\log T)^{\alpha_f/H}}\ll_{\alpha_f}1,
1 = \int_{supp \, \hat g}|\hat g(\xi)|^2 \, d\xi = \int_E |\hat g(\xi)|^2 \, d\xi + \int_{supp \, \hat g - E}|\hat g(\xi)|^2 \, d\xi &< m(E)+m(supp \, \hat g - E)=m(supp \, \hat g) = 1,
\pi(XK_{\lambda})_{j}^{i}c_{k}^{j}\pi(K_{\lambda^{\prime}}Y)_{\ell}^{k}=\pi(X)_{j}^{i}\pi(K_{\lambda}K_{\lambda^{\prime}})_{j}^{j}c_{k}^{j}\pi(Y)_{\ell}^{k}=\pi(XK_{\lambda}K_{\lambda^{\prime}})_{j}^{i}c_{k}^{j}\pi(Y)_{\ell}^{k}.
\langle \lambda_i^*e_N\xi, \lambda_i^*e_N\eta \rangle = \langle e_N\lambda_i\lambda_i^*e_N\xi, \eta \rangle = \langle E_N(\lambda_i\lambda_i^*)e_N \xi, \eta \rangle = \langle e_N\xi, \eta \rangle &= \langle e_N\xi, e_N\eta \rangle.
\hat P_n^{(\alpha,\beta)}(x) =M\begin{bmatrix}\frac{p_{n+1}^{(\alpha,\beta)}(x)-(n+1+\alpha)\rho^{(\alpha,\beta)}_np_n^{(\alpha,\beta)}(x)}{x}0 0 \frac{p_{n+1}^{(\alpha,\beta)}(x)-(n+1+\beta)\rho^{(\alpha,\beta)}_np_n^{(\alpha,\beta)}(x)}{x-1}\end{bmatrix}M^{-1}.
\exists \beta_1,\beta_2>0, \inf_{v\in V}\sup_{w\in W}\frac{a(v,w)}{\|v\|_{V}\|w\|_{W}}= \beta_1,\ \inf_{w\in W}\sup_{v\in V}\frac{a(v,w)}{\|v\|_{V}\|w\|_{W}}= \beta_2.
\langle L(u,p_{3},p_{4}) \rangle = \left[\begin{array}{ll}c_{3}c_{4}- b_{3}b_{4}u -u(b_{3}d_{4}-c_{3}a_{4}) \vspace{-2mm}d_{3}b_{4}- a_{3}c_{4}d_{3}d_{4}- a_{3}a_{4}u\end{array}\right].