harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\omega_{m}=\frac{\Gamma^{2}\left( \gamma +\frac{1}{2}+D\right)}{\Gamma\left( 2D\right) \Gamma \left( 2\gamma +1\right)}\frac{\left( -2D\right)_{m}\left( -D+1\right)_{m}\left( \left( -D+\gamma +\frac{1}{2}\right)_{m}\right)^{2}}{\left( -1\right)^{m}m!\left( -D\right)_{m}\left( \left(-D-\gamma +\frac{1}{2}\right)_{m}\right)^{2}}
\sum_{k''=0}^{K_l}p_{K''|K'}(k'') = \!\sum_{k''=0}^{\infty}\!\! p_{K''|K'}(k'')\!-\!\!\sum_{k''=0}^{\infty}\!\! p_{K''|K'}(k''\!+\!K_l\!+\!1) &=\frac{1}{1-\xi}\bigg(1-B_{\xi}\big(1+K_l,1+K'\big)\frac{(1+K_l+K')!}{K_l!K'!}\bigg),
f''_{\varepsilon}(s) = \ \int_{-\infty}^\infty \widetilde{f}'' (s-\varepsilon \eta_{\delta}(s)t)\,(1 - \varepsilon \eta_\delta'(s) t )^2 \phi(t) \, dt \ - \varepsilon \int_{-\infty}^\infty \widetilde{f}' (s-\varepsilon \eta_{\delta}(s)t)\, \eta_\delta'' (s) t \phi(t) \, dt.
\mu_k =\alpha_k^2(1+\zeta_k^{-1}) \lambda_k =-\alpha_k^2(1+\zeta_k)(1+\delta^2 \zeta_k^{-1})M_{22}^{-1}\nu_k =\alpha_k(1+\zeta_k)-\lambda_k M_{21}\rho_k^2 =1-2\nu_k m+\lambda_k M_{11}+\zeta_k
Q_B^{-1}AQ_B =(Q_A^{-1}Q_B)^{-1}Q_A^{-1}AQ_A (Q_A^{-1}Q_B) =\begin{pmatrix}a b c d\end{pmatrix}^{-1}\begin{pmatrix}-1 0 0 1\end{pmatrix}\begin{pmatrix}a b c d\end{pmatrix}& =\frac{1}{ad-bc}\begin{pmatrix}-ad-bc -2bd 2ac ad+bc\end{pmatrix}=\frac{1}{ad-bc}\begin{pmatrix}0 -2bd 2ac 0\end{pmatrix}.
f_k=\sum_{\substack{(j_1,\dots, j_k)\in (\mathbb{Z}^*)^k j_1\le \dots \le j_k}}a_{j_1,\dots,j_{k}}^{(k)}:\mathfrak{Re}\langle \mathfrak{e}_{j_1},\cdot\rangle\dots \mathfrak{Re}\langle \mathfrak{e}_{j_{k}},\cdot\rangle :
\Delta = \sum_{j = 1}^{2}\sum_{p , q}\left( \frac{\partial^{2}}{\partial ( \mathrm{R e}\, \sigma_{p q}^{j j})^{2}}+ \frac{\partial^{2}}{\partial ( \mathrm{I m}\, \sigma_{p q}^{j j})^{2}}\right) + 4 \sum_{j \neq l}\sum_{p , q}\frac{\partial^{2}}{\partial \sigma_{p q}^{j l}\sigma_{p q}^{j l *}}\ .
(HG(f))(x)=(\sum_{n=1}^{N}\sum_{j \in J_{n}^{x}}\widehat{X}_{\sigma_{x}(j)}\widehat{T}_{w_{x}}(f))(w_{x}x)=(\sum_{j=1}^{k}\widehat{X}_{j}\widehat{T}_{w_{x}}(f))(w_{x}x).
\lim_{B/\varepsilon\to\infty}\Big{[}\inf_{\hat{\theta}}\sup_{\theta\in\mathcal{E}(\alpha_{0},B)}R(\theta,\hat{\theta})\Big{/}(\varepsilon /B)^{4\alpha_{0}/(2\alpha_{0}+1)}\Big{]}= c_{\mathrm{P}}(\alpha_{0})
\left|\frac{\overline{A}_{n_j}(\varepsilon_j)}{A_{n_j}}\right| = \frac{\pi \varphi^{n_j}(1+p_j) \left( 1+\mathcal{O}(\varphi^{2n_j}) \right)}{\pi \varphi^{n_j}\left( 1+\mathcal{O}(\varphi^{2n_j}) \right)}&= 1+p_j + \mathcal{O}(\varphi^{2n_j}) .
g = g_0^4 + r^2 h = g_0^4 + (g_1^2 + g_3^2 h)^2 h f = (f_1^2 + f_3^2 g)^2 g &= (f_1^2 + f_3^2 (g_0^4 + r^2 h))^2 (g_0^4 + g_1^4 h + g_3^4 h^3) &= ((f_1 + f_3 g_0^2)^4 + (f_3 r)^4 h^2) (g_0^4 + g_1^4 h + g_3^4 h^3) &= (f_1 g_0 + f_3 g_0^3)^4 + (f_1 g_1 + f_3 g_0^2 g_1 + f_3 g_3 r h)^4 h + (f_3 g_0 r)^4 h^2 + (f_1 g_3 + f_3 g_0^2 g_3+ f_3 g_1 r)h^3.
r(b) \ = \ \max\{j : c_{b+j}= 0\},~ \rho(b) \ = \Q ( (0,b+r(b),b+r(b)+1),(0,b+r(b)+1,1))= \frac{\varphi_C (1)}{\lambda_C \varphi_C(b+r(b)+1)}\mbox{, and}h (b,k) \ = \ \begin{cases}0 k < r (b)+1 1- \rho(b) k = r(b)+1 \rho (b) \lambda_C^{-(k-r(b)-2)}(1-\lambda_C^{-1}) k> r(b)+1.\end{cases}
U({\bf r}^\prime ,{\bf r}_2) =- \int_{-\infty}^{+\infty}d\omega \, 4\pi \alpha_1 ({\bf r}^\prime, i |\omega|)\alpha_2 ({\bf r}_2, i |\omega|) \,(D_{pj}^{(0)}({\bf r}^\prime ,{\bf r}_2 , i \omega))^2 .
\operatorname{Res}(P_1(0, 0, x_{3}, x_{4}), P_2(0, 0, x_{3}, x_{4})) = a_1^2 b_0 b_2-a_2 a_1 b_0 b_1-a_0 a_1 b_1 b_2 +a_2^2 b_0^2+a_0 a_2 b_1^2 & +a_0^2 b_2^2-2 a_0 a_2 b_0 b_2 = 1/16 \cdot \operatorname{Disc}_{a,b}(\operatorname{Disc}_{x_{3},x_{4}}( a \cdot P_{1}(0, 0, x_{3}, x_{4})+ b \cdot P_{2}(0, 0, x_{3}, x_{4}))).
d_{1+s}(p\|q):=\begin{cases}\frac{1}{s}\log(p^{1+s}q^{-s}+\bar{p}^{1+s}\bar{q}^{-s}), s\geq-1,s\neq0 p\log\frac{p}{q}+\bar{p}\log\frac{\bar{p}}{\bar{q}}, s=0\end{cases}
L_{n}(\phi_{n}) - L(\phi_{B}) \leq 2 \sum_{i=0}^{1}E \left[ \Big| p(\textbf{Z},i)P(Y=i \vert \textbf{Z}) - f_{i}(\textbf{Z}) \Big| \Bigg| D_{n}\right] + 2 \sum_{i=0}^{1}E \left[ \Big| (1-q(\textbf{X},i))P(Y=i \vert \textbf{X}) - g_{i}(\textbf{X}) \Big| \Bigg| D_{n}\right].
\Phi_\beta(G) :=\sum_{n\ge 1}\frac{(-1)^{n+1}}{n}\sum_{G_1,\ldots,G_n}\prod_{j=1}^nP_\beta(G_j), \Phi^s_\beta(G) :=\sum_{n\ge 1}\frac{(-1)^{n+1}}{n}\sum_{G_1,\ldots,G_n}\prod_{j=1}^nP^s_\beta(G_j),
U^{\epsilon}(t,x)= \int_0^1 G_t(x,y)\eta(y)dy + \sqrt{\epsilon}\int_0^t\int_0^1 G_{t-s}(x,y) \sigma(s, U(s))(y)W(dy, ds) & -\int_0^t\int_0^1\partial_yG_{t-s}(x,y)g(s, U(s))(y)dyds+\int_0^t\int_0^1G_{t-s}(x,y)f(s, U(s))(y)dyds.
\langle ~ \delta_{S}{\bar{Q}}_{\dot{\alpha}}({\cal I}^{+}) ~ \rangle ~ = ~ \epsilon^{\beta}~ \sum_{k}k_{\beta{\dot{\alpha}}}~ \sum_{n_{k}^{B}, n_{k}^{F}}~ \rho_{n_{k}^{B}n_{k}^{F}, n_{k}^{B}n_{k}^{F}}~ ( - )^{n_{k}^{F}}~ \left( ~ n_{k}^{B}~ + ~ n_{k}^{F}~ \right) ~ .
\|\omega\|_{L^4(M)}^2 + \|\nabla_A \omega\|^2 \leq C_M \left( \|D_A \omega\|^2 + \|D_A^* \omega \|^2 + \|\omega\|^2 \right) & \leq C_M \left( \|D_A^* \omega \|^2 + \|\omega\|^2 \right).
\displaystyle B(|v-v_{\ast}|, \cos\theta, z) = \phi(|v-v_{\ast}|)\, b(\cos\theta,z), \phi(\xi) = C_{\phi}\, \xi^{\gamma}, \,\gamma\in[0,1], begin{align*}4pt] \displaystyle \forall\eta\in[-1,1], |b(\eta,z)|\leq C_b, \, |\partial_{\eta}b(\eta,z)|\leq C_b, \, \,|\partial_z^k b(\eta, z)|\leq C_b^{\ast}, \, \forall\, 0\leq k\leq r\,.
\sum_{\lambda\in \mathcal{P}}\prod_{i\geq 1}\left[\lambda_i-\lambda_{i+2}+1 \atop \lambda_{i+1}-\lambda_{i+2}\right]_{t}q^{{\lambda_1\choose 2}+|\lambda|}=\prod_{i\geq 1}\frac{1}{(1-q^{2i-1})(1-tq^{2i})}.
{f^{( 1 2 , - 1 2 )}( \Omega , \bar{\Omega}) ={\frac{c}{\sqrt{\Omega_{2}}}}\left({\frac{\partial}{\partial \alpha}}\right)^{2 4}\sum_{m , n}\int{\frac{d t}{t}}t^{2 3 / 2}\left. e^{- t | m + n \Omega |^{2}+ \alpha ( m + n \bar{\Omega})}\right|_{\alpha = 0},}
\begin{aligned}W^{(n)}_{\lambda}(t;\zeta) = (-1)^{n(n-1)/2}\det_{1\leq a, b\leq n}\left(\partial_{(j_b)}^{\lambda_a+a-1}\right)\prod_{k=1}^n\psi^{(j_k)}(t;\zeta) &=a_\lambda(\partial)\prod_{k=1}^n\psi^{(j_k)}(t;\zeta)\ .\end{aligned}
|\eta+\xi|^{\alpha}- |\eta|^{\alpha}= \alpha \int_0^1 |\eta+\theta \xi|^{\alpha-2}(\eta+\theta \xi) d\theta \cdot \xi & = \alpha |\eta|^{\alpha-2}\eta \cdot \xi + \sum_{i,j=1}^d \tilde \sigma_{ij}(\xi,\eta) \xi_i \xi_j,
\int_{0}^{x}dt\psi_{n}^{2}(t;\omega ,N) = \Vert \psi_{n}\Vert^{2}-\int_{x}^{+\infty}dt\psi_{n}^{2}(t;\omega ,N) \hphantom{\int_{0}^{x}dt\psi_{n}^{2}(t;\omega ,N)}{}= \Vert \psi_{n}\Vert^{2}-\frac{1}{\sqrt{2\omega}}\int_{z}^{+\infty}d\zeta \zeta^{N}\big( L_{n}^{N}(\zeta) \big)^{2}\exp ( -\zeta ) .
[\nabla \widehat E^\alpha_{x,y}]_{a,b}= [ (\nabla \widehat E^\alpha_{x,y}\cdot T_{a,b}) T_{a,b}]_{a,b}&= \alpha e^{\alpha |G_{x,y}|^2}(G_{y,x}G_{x,b}\delta_{y,a}- G_{a,y}G_{y,x}\delta_{b,x}+G_{x,y}G_{y,b}\delta_{a,x}-G_{a,x}G_{x,y}\delta_{b,y}) \, .
\lefteqn{(-\partial_1^2-|\partial_1|^{-1}\partial_2^2)w}&+P\big(\sigma^2F+\sigma v\partial_2 Rw+\sigma w\partial_2 Rv+w\partial_2Rw\big) &+\frac{1}{2}\partial_2 R(w+\sigma v)^2-\frac{1}{2}P((w+\sigma v) \partial_1R(w+\sigma v)^2)=0
\int_{-1}^1 \frac{dy}{(y^2+z^2)^{\frac{2-\varepsilon}{2}}}=\frac{1}{z^{1-\varepsilon}}\int_{-1/z}^{1/z}\frac{dy}{(1+y^2)^{\frac{2-\varepsilon}{2}}}=\frac{2}{z^{1-\varepsilon}}( \int_0^1 \frac{dy}{(1+y^2)^{\frac{2-\varepsilon}{2}}}+ \int_1^{1/z}\frac{dy}{(1+y^2)^{\frac{2-\varepsilon}{2}}}) \sim \frac{1}{z^{1-\varepsilon}}
R \leq \min \begin{cases}2C, C+C_{0}+\frac{1}{2}\log \left(\frac{1- \rho_c^2 + (1-\rho) (1+ \rho - 2\rho_c^2) P}{1-\rho_c^2}\right), \frac{1}{2}\log \left(1 +2(1+ \rho) P \right), 2C_{0}+\frac{1}{2}\log \left( 1+2(1 + \rho - 2\rho_c^2 ) P \right),\end{cases}
S_1 =\left(1+o(1)\right)\frac{\varphi(VP_f)^kX(\log R)^k}{V(VP_f)^k}I_k(F), S_2 =\left(1+o(1)\right)\frac{\varphi(VP_f)^kX(\log R)^{k+1}}{V(VP_f)^k\log X}\sum_{m=1}^kJ^{(m)}_k(F),
C_3:= |R^N|_{L^\infty}+\frac{1+r}{2}(3|A|^2_{L^\infty}+\frac{3}{2}|B|^2_{L^\infty}), C_4:=\frac{1+r}{2}(3|C|^2_{L^\infty}+\frac{3}{2}|B|^2_{L^\infty}).
\mathcal{A}^{(3)}= -\bigg(v_n^{\delta+1},\psi_k''(w_n)\frac{\partial}{\partial x}w_n\bigg) \leq C(\delta,n)\int_{0}^{1}1\cdot\psi_k''(w_n)\frac{\partial}{\partial x}w_n\d x & \leq C(\nu,\delta,n)\int_{0}^{1}|w_n(x)|_+^2\d x+\frac{\nu}{4}\int_{0}^{1}\psi_k''(w_n(x))\bigg(\frac{\partial}{\partial x}w_n(x)\bigg)^2\d x.
D_r^BY_t =\tilde Y_t = D_r^B \xi + \int_t^T \tilde Y_s{{W}}(ds,B_s) +\int_r^T Y_s \nabla_x{{W}}(ds, B_s) -\int_t^T \tilde Z_s dB_s,\ 0\leq t\leq r\leq T.
\hat f_{i_0-i-1}(k) =g_{i_0-i,k}^{(n_{\tilde\varepsilon})}(1)=g_{i_0-i,k}^{(n_{\tilde\varepsilon}-1)}(g_{i_0-i,k}(1))=g_{i_0-i,k}^{(n_{\tilde\varepsilon}-1)}(\max\{\hat f_{i_0-i}(1),k\}) =g_{i_0-i,k}^{(n_{\tilde\varepsilon}-1)}(\max\{\hat f_{i_0-i}(1),k\})=\hat f_{i_0-i}^{(n_{\tilde\varepsilon}-1)}(\max\{\hat f_{i_0-i}(1),k\}).
\mu\prod_{j=1}^{n}P_j(\epsilon)r = \tilde{\pi}r + \epsilon n \tilde{\pi}\tilde{{P}}^{(1)}(I-\tilde{P}+\tilde{\Pi})^{-1}r - \epsilon \tilde{\pi}\tilde{P}^{(1)}(I-\tilde{P}+\tilde{\Pi})^{-2}r + O(\epsilon^{2}n^3)
\mathrm{LC}_{s=3/4}^0 \bigg[\frac{b_{1/2,0}(d,s)}{d^{1/2}\Gamma(s+1/4)}\bigg] = \frac{\pi}{\sqrt{|dD|}}\frac{1}{B(1)}\frac{\mathrm{Tr}_{d,D}((-3/\pi)\log(y|\eta(z)|^4))}{L_D(1)}&= \frac{3}{\sqrt{dD}L_D(1)}\mathrm{Tr}_{d,D}(-\log(y|\eta(z)|^4)).
|e_{ph}^0|_h = |p_h^0-\hat{p}_h^0|_h \le | p_h^0 - 0 |_h + |\hat{p}_h^0-p^0|_h + |p^0|_h \le \alpha_{20}\bigl( \| p_h^0 - 0 \|_0 + \|\hat{p}_h^0 - p^0\|_0 \bigr) + h\|p^0\|_1 & \le (2\alpha_{20}\alpha_{31}+ 1) h \| (u, p)^0 \|_{H^{k+1}\times H^k},
S_r(N) =\frac{1}{K}\,\int_{\mathbb{R}}\,V_1\left(\frac{\nu}{K}\right) \mathop{\mathop{\sum\sum}_{\substack{n, \,m =1}}^{\infty}}\delta_{n=m}\, A_{\pi}(n,r)m^{-it}\,\lambda_f(m)\,\left(\frac{n}{m}\right)^{i\nu}\,V_1\left(\frac{n}{N}\right)\,V_2\left(\frac{m}{N}\right)\,d\nu,
K_{2}'(x) = \lambda\int_{z=x}^{\infty}\int_{u=0}^{z-x}\int_{0}^{z-x-u}\bar{G}(u)V(z-u-s)f_{B}(s)f_n(u)\lambda e^{-\lambda(z-x)}ds du dz - V(x)\int_{z=x}^{\infty}\int_{u=0}^{z-x}\bar{G}(u)f_{B}(z-x-u)f_n(u)\lambda e^{-\lambda(z-x)}du dz.
(\Im c_m)^2-(\Im c_{m+1})^2 \, =(y_{m+1}-y_m)\big(2t_{n+1}-y_{m}-y_{m+1}+2\kappa\big) &>\, m(m+1)!\cdot\big(2t_{n+1}-t_{m+1}-t_{m+2}\big) &\ge\, m(m+1)!\cdot\big((n+1)!-n!\big)\,=\,mn(m+1)!\,n!,
F^{-1}(t) = \frac{(C(n, \alpha) +1) (t^{1/p})^{1-p}}{\psi \left((t^{1/p})^{- \frac{p}{n}}\right)^{n-1}}= \frac{(C(n, \alpha) +1) t^{\frac1p-1}}{\psi \left(t^{- \frac{1}{n}}\right)^{n-1}}
c_1=\frac{\frac{U_L^2+V_L^2}{2}- \frac{U_M^2+V_M^2}{2}}{U_L-U_M}\ \ \ \ \ \ \ \ \c_2=\frac{\frac{U_M^2+V_M^2}{2}- \frac{U_R^2+V_R^2}{2}}{-U_M-U_R},
U^{(v)}(t) = (-1)^{v}v! \left( \frac{\delta}{(t+n/2)^{1+v}}+\sum_{\ell=0}^{n-1}\frac{s+2}{(t+\ell+1/4)^{1+v}}+ \sum_{\ell=0}^{n-1}\frac{s+2}{(t +\ell +3/4)^{1+v}}-\sum_{\ell=0 \atop \ell \neq k}^{n}\frac{2s+4}{(t+\ell)^{1+v}}\right).
\int_{\Gamma_0}|\nabla_\Gamma f_N|^2\,\gamma\, d s=\int_{\Gamma_0}(\nabla_\Gamma f_N\nabla_\Gamma \gamma)\,(f-f_N)\, d s+\int_{\Gamma_0}(\nabla_\Gamma f_N\nabla_\Gamma f)\gamma\,\, d s.
\begin{array}{ccl}\varphi_{1\bar{1}1}= \varphi_{11\bar{1}}-i\varphi_{10}-W\varphi_{1}= i\varphi_{01}+iA_{11}\varphi_{\bar{1}}-W\varphi_{1}= iA_{11}\varphi_{\bar{1}}-W\varphi_{1}.\end{array}
\phi_1 := \phi_{21AB}(\tau, z) = 2 \zeta^{\pm 1}- 4 + (-4 \zeta^{\pm 2}+ 11 \zeta^{\pm 1}- 14) q + (2 \zeta^{\pm 3}- 14 \zeta^{\pm 2}+ 35 \zeta^{\pm 1}- 46)q^2 + O(q^3)
y_{j+1}\le A^{\alpha_j}(A^{\alpha_{j-1}}y_{j-1}^{\beta_{j-1}})^{\beta_j}=A^{\alpha_j+\alpha_{j-1}\beta_j}y_{j-1}^{\beta_{j-1}\beta_j}&\le \dots\le A^{\alpha_j+\alpha_{j-1}\beta_j + \alpha_{j-2}\beta_{j-1}\beta_j +\alpha_{j-3}\beta_{j-2}\beta_{j-1}\beta_j +\ldots+\alpha_0\beta_1\beta_2\ldots\beta_j}\cdot y_0^{\beta_0\beta_1\ldots\beta_j}.
\Delta_{\tilde{g}}u =\tilde{g}^{ij}\tilde{\nabla}_j\nabla_iu=\tilde{g}^{ij}\left( \nabla_j\nabla_iu - (\Gamma^{k}_{ij}(\tilde{g}) - \Gamma^{k}_{ij}(g))\nabla_ku \right), &=\Delta_{g}u + (\tilde{g}^{ij}-{g}^{ij})\nabla_j\nabla^{}_iu - \tilde{g}^{ij}(\Gamma^{k}_{ij}(\tilde{g}) - \Gamma^{k}_{ij}(g))\nabla_ku
\sum_{\substack{1\leq k \leq n_1, n_1+1\leq l \leq n_1+n_2}}\mathring{R}(\xi_{kl}, \xi_{kl})=\sum_{\substack{1\leq k \leq n_1, n_1+1\leq l \leq n_1+n_2}}R_{klkl}=0.
h_0 =H+\frac t{12}\left(2(\alpha_0-1)^2-\alpha_1^2+2\alpha_3^2-\alpha_4^2+6(\alpha_0-1)\alpha_3\right) &\quad+\frac{t-1}{12}\left(2(\alpha_0-1)^2-\alpha_1^2-\alpha_3^2+2\alpha_4^2+6(\alpha_0-1)\alpha_4\right).
\frac{d}{dt}\det(\mathbf{g}) = \det(\mathbf{g}) \; \mathrm{tr}\left(\mathbf{g}^{-1}\frac{d\mathbf{g}}{dt}\right) = \det(\mathbf{g}) (-2ug^{ij}h_{ij}) = -4 Hu \det(\mathbf{g}).
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|^m \geq\sqrt{\frac{n-{d+m-1 \choose m}}{{d+m-1 \choose m}(n-1)}}= \sqrt{\frac{n-(^m(\mathcal{X}))}{(^m(\mathcal{X}))(n-1)}}=\sqrt{\frac{1}{n-1}\left[\frac{n}{{d+m-1\choose m}}-1\right]}.
h_1=\, 1+a_1^4+a_1^2 b+a_1^4 b^2+a_1^2 b^5+b^8+(a_1^2 +a_1^6 +a_1^2 b^2 +a_1^2b^4 +a_1^2 b^6 )k &+(a_1^4 +a_1^6 b +a_1^4 b^4 )k^2+(a_1^6 +a_1^6 b^2 )k^3,
\phi(y)+\frac12|x-y|^2 \ge \phi(y)+\frac12\big( |x-x_0| - |x_0-y|\big)^2 &= \phi(y)+\frac12 |x-x_0|^2 +\frac12 |x_0-y|^2- |x-x_0||x_0-y| &\ge\phi(y)+\frac12 |y-x_0|^2 - |x-x_0||x_0-y|.
\log\left( \prod_{i=1}^\infty \frac{q^i}{q^i-1}\right) = \sum_{i=1}^\infty \log\left( 1 + \frac{1}{q^i-1}\right) \leq \sum_{i=1}^\infty \frac{1}{q^i-1}\leq \frac{1}{2}+ \frac{9}{8}\sum_{i=2}^\infty \frac{1}{q^i}= \frac{11}{16}.
\begin{gathered}G_i(y)=\sum_{j=0}^{\lfloor\frac{n+1-i}k\rfloor}y^je'_{n+1-kj-i}=\sum_{j=0}^{\lfloor\frac{n+1-i}k\rfloor}y^j[e_{n+1-kj-i}+re_{n-kj-i}] =rg_i(y)+\begin{cases}yg_{k-1}(y) i=0; g_{i-1}(y) i\geq 1.\end{cases}\end{gathered}
C_F: y^\ell+\sum_{r=1}^{\frac{\ell-1}{2}}a_{2r-1}(\mathfrak{F}_1\mathfrak{F}_2)^{\frac{\ell+1}{2}-r}y^{2r-1}-\mathfrak{F}_1\mathfrak{F}_2(\mathfrak{F}_1^{\ell-2}+\mathfrak{F}_2^{\ell-2})=0.
\tilde A\Psi_\Delta(x_1,x_2,x_3)=\tilde a\Psi_\Delta(x_1,x_2,x_3),\qquad\tilde{\cal C}\Psi_\Delta(x_1,x_2,x_3)=\Delta(1-\Delta)\Psi_\Delta(x_1,x_2,x_3),
\begin{aligned}|V\setminus (S_1\cup U_1\cup A_3\cup Z_0)| \ge N-|S_1|-|U_1|-|A_3|-|Z_0| & > 5N/8-(N/4-|U_1\setminus V_2|)-|Z_0| &\ge |U_1\setminus V_2|+N/8.\end{aligned}
\phi_{d \, ( - )}^{( 1 )}( t , \xi ) = - \frac{\sqrt{3}}{2}\frac{\sinh \xi}{\cosh^{2}\xi}\int_{R}d \xi^{\prime}\int_{0}^{t}d t^{\prime}j_{d}( t^{\prime}, \xi^{\prime}) \sin [ \sqrt{3}( t - t^{\prime}) ] \frac{\sinh \xi^{\prime}}{\cosh^{2}\xi^{\prime}}.
\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right] \frac{(q)_n \left(\frac{c}{d}\right)_n (-zd)^n q^\frac{n(n+1)}{2}}{(zq)_n (cq)_n}= \frac{z}{c}(c-d)\displaystyle\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right] \frac{(q)_n \left(\frac{zdq}{c}\right)_{n-1}(cq)_{N-n}(cq)^n}{(zq)_n (cq)_N}.
\vec{\nabla}\, \Bigl [ t r \left( Q^{\dagger}\, P_{1}^{- 1}( \vec{\nabla}Q ) \, P_{2}\right) \Bigr ] = t r \Bigl [ ( \vec{\nabla}Q^{\dagger}) \, P_{1}^{- 1}\, ( \vec{\nabla}Q ) \, P_{2}\Bigr ]
\Pi(i) =\prod_{j=0}^m\left(\prod_{k=1}^j(T^{r^j}-T^{r^{j-k}})\right)^{c_j}&=\prod_{k=1}^m(T^{r^k}-T)^{c_k+c_{k+1}r+\cdots}=\prod_{k=1}^m(T^{r^k}-T)^{[i/r^k]}\quad\hbox{($[\cdot]$ denotes the floor function.)}&=\prod_{k=1}^m(T^{r^k}-T)^{c_k+c_{k+1}r+\cdots+c_m r^{m-k}}
\hat{K}= \left( \begin{array}{c c c}{{\frac{B}{4}\delta^{\mu \nu \alpha \beta}}}&{{0}}&{{0}}\\{{0}}&{{- \frac{B}{1 6}}}&{{- \frac{B_{1}}{4}}}\\{{0}}&{{- \frac{B_{1}}{4}}}&{{\frac{B_{1}^{2}}{2 B}- A}}\\ \end{array}\right)
\xi^*(\psi\delta_{1,1}) = \pi_1^*(\psi_1), \xi^*(\delta_0\delta_{1,1}) = \pi_1^*(\delta_0) +\pi_2^*(\delta_0), \xi^*(\delta_{1,1}^2) = -\pi_1^*(\psi_2) - \pi_2^*(\psi), \xi^*(\delta_{1,1}\delta_{2,1}) = \pi_1^*(\delta_{0,\{1,2\}}) +\pi_2^*(\delta_{1,1}).
\int_{-\infty}^{\infty}\frac{S(e^y,\chi_{\phi})}{e^y}\exp(y\lambda) e^{-iy\xi}dy = \sum_{n=1}^{\infty}\frac{\chi(n)}{n^{i\phi}}\int_{\log n}^{\infty}\exp(-y(1-\lambda+i\xi)) dy = \frac{L(1-\lambda+i\phi+i\xi,\chi)}{1-\lambda+i\xi}.
\int_D \int_{B(x,\tau \delta_{\partial D}(x))}\frac{\lvert u(x)-u(y)\rvert^p}{\lvert x-y\rvert^{n+s p}}\,dy\,\delta_{\partial D}^\beta(x)\,dx\ge C\bigg(\int_D \lvert u(x)\rvert^q\delta_{\partial D}^{(q/p)(n-s p+\beta)-n}(x)\,dx\bigg)^{p/q}
\overline{\digamma}\left( \lambda \right) =\frac{\digamma \left( \lambda\right)}{h\left( \lambda \right)}=\overset{\infty}{\underset{n=0}{\sum}}\overline{\digamma}\left( \lambda_{n}\right) \frac{\varpi \left( \lambda\right)}{\left( \lambda -\lambda_{n}\right) \varpi^{\prime}\left(\lambda_{n}\right)}.
\ll\ =\ \preceq^\mathfrak{r},\preceq^*,\preceq^\mathfrak{c},\preceq^A,\preceq^\bullet\quad\mathcal{P}. \quad\mathcal{P}\ =\ \mathcal{I}\, \cap\, (A_\mathrm{n}\, \cup\, \mathfrak{r}\, \cup\, \mathfrak{r}^\perp\, \cup\, A_\mathrm{sa}A_+^\perp\, \cup\, A_+^\perp A_\mathrm{sa}),
E^+(\mathbb{Q}_{n,p}) := \lbrace P \in E(\mathbb{Q}_{n,p}) : \mathrm{Tr}_{n/m+1}(P) \in E(\mathbb{Q}_{m,p}) \textrm{for even}m \ (0 \leq m < n)\rbrace E^-(\mathbb{Q}_{n,p}) := \lbrace P \in E(\mathbb{Q}_{n,p}) : \mathrm{Tr}_{n/m+1}(P) \in E(\mathbb{Q}_{m,p}) \textrm{for odd}m \ (0 \leq m < n)\rbrace
[ J_{0}^{k}, \phi^{c}( w ) ] | \Psi^{a b}\rangle + \phi^{c}( w ) J_{0}^{k}| \Psi^{a b}\rangle = w^{- \Delta_{\phi}}C_{d}^{a b , c}f_{e}^{k d}| \phi^{e}\rangle + . . .
\frac{\partial}{\partial{t}}{U}_{k}(\rho)=\frac{1}{2}\int\frac{{d}^{d}{q}}{(2\pi)^{d}}\frac{\partial}{\partial{t}}{R}_{k}({q})\left(\frac{{N}-1}{{M}_0}+\frac{1}{{M}_1}\right)
G(q) =\frac 1{(q,q^4,q^6,q^9;q^{10})_\infty}=\frac{(-q,-q^9;q^{10})_\infty}{(q^4,q^6;q^{10})_\infty(q^2,q^{18};q^{20})_\infty}&=\frac{(q^8,q^{12};q^{20})_\infty\,\theta(-q;q^{10})}{(q^2;q^{2})_\infty}
dt_s^i= \dot{t}_s^i ds, \;\; d \dot{t}_s^i= \left[-H(t_s^i) \left(|\dot{t}_s^i|^2-1 \right) + \frac{3 \sigma^2}{2}\dot{t}_s^i \right]ds+ \sigma \sqrt{|\dot{t}_s^i|^2-1}d B^i_s.
Q_{C}= i \int d^{3}x \left[ \overline{{{c}}}^{a}\left( x \right) \nabla_{0}c^{a}\left( x \right) + g \overline{{{c}}}^{a}\left( x \right) f^{a b c}A_{0}^{b}\left( x \right) c^{c}\left( x \right) \right] ,
\left| \kappa(e^l) - \frac{1}{2}\sum_{k=1}^{q}\left| \left\langle L_n(\tau') - L_n(\tau), \nabla g_{k}^{H,\beta}(L_n(\tau))\right\rangle \right| - \frac{1}{2}\sum_{k=1}^{q}\left| \left\langle L_n(\sigma') - L_n(\sigma), \nabla g_{k}^{H,\beta}(L_n(\sigma))\right\rangle \right| \right| <C^\prime\varepsilon^2,
E = - \frac{1}{4 \pi \alpha^{\prime}}\int_{0}^{2 \pi}d \sigma [ \partial_{0}t + i f ( z^{i}\partial_{-}\bar{z}^{i}- \bar{z}^{i}\partial_{-}z^{i}+ 2 i \bar{\psi}_{R}^{i}\psi_{R}^{i}) ] = - \frac{1}{2}\frac{l}{\alpha^{\prime}}+ 2 f J_{R}~ .
\hat{\mathcal{O}}_{r+1:rT,1:d_0}:= \hat{\mathcal{O}}^-,\ \bar{\mathcal{O}}_{1:r(T-1),1:d_0}:= \bar{\mathcal{O}}^+,\ \bar{\mathcal{O}}_{r+1:rT,1:d_0}:= \bar{\mathcal{O}}^-.
\frac{\partial V}{\partial z}(r,0) =-d_s\int_{\{x:\,\psi(x_{1})>\psi^{\circledast}(r)\}}f^{\bigstar}(x)\, d\gamma(x) & =-d_s\int_{\Phi^{-1}(r)}^{\infty}f^{\circledast}(\Phi^{-1}(x_{1}))\, d\gamma(x) & =-d_s\int_{0}^{r}f^{\circledast}(\sigma)\,d\sigma,\quad\hbox{for}~r\in(0,\gamma(\Omega)).
\Big\Vert \varphi \Big\Vert_{\mathrm{L}^2(\mathbb{R})}^2 = \int_{-\infty}^T|\varphi|^2(\mathrm{x},t) dt = \alpha \delta^2 \int_{-\infty}^{T_{\delta}}|\varphi|^2(\delta \eta + \mathrm{z},\alpha \delta^2 \Tilde{\tau}) d\Tilde{\tau}= \alpha \delta^2 \Big\Vert \hat{\varphi}\Big\Vert_{\mathrm{L}^2(\mathbb{R})}^2
\det(A_{ij})^{n+1}_{i,j=1}= 2\prod^n_{i=1}(x^{\frac{1}{2}}_i-x^{-\frac{1}{2}}_i)\det(x^{\lambda_j+(n-j+\frac{1}{2})}_i-x^{-\lambda_j-(n-j+\frac{1}{2})}_i)^{n}_{i,j=1}.
n\log_2\begin{pmatrix}{\frac nt}\end{pmatrix}+2n\log_2\log_2\begin{pmatrix}{\frac nt}\end{pmatrix}<{\frac{3n}2}\log_2\begin{pmatrix}{\frac 1t}\end{pmatrix}\begin{pmatrix}1+{\frac{2\log_2\log_2(5n)}{\log_2(5n)}}\end{pmatrix}
\sigma\alpha_{i}=\alpha_{2m-i}\ (1\leq i\leq 2m-1), \sigma\alpha_{2m}=-\beta, &\sigma\alpha_{2m+i}=\alpha_{2m+2n-i}\ (1\leq i\leq 2n-1), \sigma=1\frak{g}_{\alpha_m}\frak{g}_{\alpha_{2m+n}}.
w=\omega+\int_{0}^{1}dt(1-t)(t\varpi^{\alpha\beta}z_{\alpha}z_{\beta}-ih^{\alpha\dot{\alpha}}z_{\alpha}\frac{\partial}{\partial y^{\dot{\alpha}}})C(-z_{\gamma}t,y_{\dot{\gamma}})e^{ity^{\delta}z_{\delta}}+...
\|E^su\|^2 =\int_0^1u(x)^2x^{-2s+\frac{N}{2}-1}dx &=-\frac{1}{\frac{N}{4}-s}\int_0^1u\cdot Du\cdot x^{-2s+\frac{N}{2}}dx &\leq\frac{1}{|\frac{N}{4}-s|}\|x^{-s}u\|\|x^{-s+1}Du\|
\tilde R_L^{(n)}(t,t';\{q\})\approx\frac{\tilde a_L^{(n)}(\{q\})+\tilde\alpha_L^{(n)}(\{q\})(\vartheta+\vartheta')}{(z-z_0)(z'-z_0)}-\frac{\tilde b_L^{(n)}(\{q\})\vartheta\vartheta'}{(z-z_0)(z'-z_0)}\Biggl[\frac{1}{z-z_0}-\frac{1}{z'-z_0}\Biggl]\,.
\zeta_{r}\left(n_{1},\dots,n_{r}\right)=\sum_{k_{1}>0,\dots,k_{r}>0}\frac{1}{k_{1}^{n_{1}}\left(k_{1}+k_{2}\right)^{n_{2}}\dots\left(k_{1}+\dots+k_{r}\right)^{n_{r}}}.
\gamma_M(A)=\sum_{\beta\in L+\kappa}e^{\pi i\sum_{j=1}^g\sum_{h=1}^g\tau_{jh}\langle a^j,\beta\rangle\langle a^h,\beta\rangle}=\sum_{\beta\in L+\kappa}e^{\pi i{}^t\widetilde{\beta}A\widetilde{\beta}},
(\begin{matrix}\sum k^{0}\\ \sum k^{1}\\ \sum k^{2}\\ \sum k^{3}\\ \sum k^{4}\\ \sum k^{5}\\ \sum k^{6}\end{matrix})=G_{7}(\begin{matrix}n\\n^{2}\\n^{3}\\n^{4}\\n^{5}\\n^{6}\\n^{7}\end{matrix}),
f(n,1,3) = \begin{cases}\displaystyle\sum_{i=n/2}^n{n \choose i}\mbox{if}n\mbox{is even,}{n-1 \choose \frac{n-1}{2}}+\displaystyle\sum_{i=(n+1)/2}^n{n \choose i}\mbox{if}n\mbox{is odd.}\end{cases}
Y(v,z){:=}{:}\left(\frac{1}{(n_1-1)!}\left(\frac{d}{dz}\right)^{n_1-1}\alpha_1(z)\right)...\left(\frac{1}{(n_k-1)!}\left(\frac{d}{dz}\right)^{n_k-1}\alpha_k(z)\right)Y(e^{\alpha},z){:},
\widehat{T}_{f_{+}}\widehat{T}_{g_{+}}+\widehat{T}_{f_{+}}\widehat{T}_{g_{-}}+\widehat{T}_{f_{-}}\widehat{T}_{g_{+}}+\widehat{T}_{f_{-}}\widehat{T}_{g_{-}}=\widehat{T}_{g_{+}}\widehat{T}_{f_{+}}+\widehat{T}_{g_{+}}\widehat{T}_{f_{-}}+\widehat{T}_{g_{-}}\widehat{T}_{f_{+}}+\widehat{T}_{g_{-}}\widehat{T}_{f_{-}}.
\sum_{k\geq 1}M^{pk}\alpha_k \leq\sum_{k\geq 1}(\rho M^p)^k\,+\,\sum_{k\geq 1}\sum_{i=0}^{k-1}\rho^{k-i}M^{p(k-i)}M^{pi}\beta_i &=\sum_{k\geq 1}2^{-k}+\left(\sum_{i\geq 0}M^{pi}\beta_i\right)\left(\sum_{j\geq 1}2^{-j}\right)\leq C.
\int_{A}^{\infty}\frac{1}{x\sqrt{2\pi\sigma^{2}}}e^{-\frac{(lnx-\mu)^{2}}{2\sigma^{2}}}xdx=\frac{1}{2}e^{\mu+\sigma^{2}/2}(1+erf(\frac{\mu+\sigma^{2}-lnA}{\sqrt{2}\sigma}))
\begin{aligned}\frac{(1+k)^{2}}{2k}\frac{1}{\eta(r_{0})}\left\{4 \pi \frac{(1-k)^{2}}{(1+k)^{2}}- \frac{1}{4}\int_{\Sigma}H^{2}+ \int_{\Sigma}\left( \frac{H}{2}-\eta(r_{0})|\nabla u| \right)^{2}\right\}\le 8\pi\left(\frac{\mathfrak{m}_{ADM}}{m}-1\right).\end{aligned}
\alpha_{i}^{\prime}={\cal{F}}( \alpha_{1}^{\prime \prime}, \ldots , \alpha_{N}^{\prime \prime}) \alpha_{i}^{\prime \prime}, \hspace{1 3 m m}\mathrm{w i t h}\hspace{1 0 m m}{\cal{F}}( \alpha_{1}, \ldots , \alpha_{N}) = \frac{\sum \alpha_{i}}{\sum \frac{\alpha_{i}}{m_{i}}}.
\delta u = \sum_{i = 1}^{3}d x^{i}\vee \frac{\partial u}{\partial x^{i}}- d t \vee \frac{\partial u}{\partial t}= d u + \sum_{i = 1}^{3}e_{i}\frac{\partial u}{\partial x^{i}}- e_{i}\frac{\partial u}{\partial t}.
\omega_{a^{\prime}b^{\prime}}= \left( \begin{array}{c c}{0}&{- 1}\\{1}&{0}\\ \end{array}\right) , X^{a^{\prime}b^{\prime}}= \left( \begin{array}{c c}{\frac{1}{2}}&{0}\\{0}&{- \frac{1}{q^{2}}}\\ \end{array}\right) .
\int_\gamma \frac{\Gamma(t - 1/4; \tau, \sigma)}{\Gamma(t + 1/4; \tau, \sigma)}\frac{\theta_0(t + 1/2; \tau)}{\theta_0(t + 1/4; \tau)}\frac{\theta_0(t + 1/2; \sigma)}{\theta_0(t + 1/4; \sigma)}dt = - (1 - i) \frac{\Gamma(3/4; \tau, \sigma)}{\Gamma(1/4; \tau, \sigma)}\frac{1}{(\tau; \tau)(\tau + 1/2; 2\tau)}\frac{1}{(\sigma; \sigma) (\sigma + 1/2; 2\sigma)},
\partial^{\beta'}B_{\alpha'}\left( \partial^{\beta_1 \cdot e_1}B_{\alpha_1 \cdot e_1}(f) \right)(x_1, x_2, \ldots, x_d) = \partial^{(\beta')^+}B_{(\alpha')^+}\left( \mu_{x_1}\right)(x_2, \ldots, x_d) ~~ \partial^{(\beta')^+}\mu_{x_1}(x_2, \ldots, x_d) = \partial^{\beta_1 \cdot e_1}B_{\alpha_1 \cdot e_1}(\lambda_{(x_2, \ldots, x_d)})(x_1).
U =(x_{0}+x_{m})(y_{0}+y_{m})+(y_{0}+y_{m})(z_{0}+z_{m})+(z_{0}+z_{m})(x_{0}+x_{m}), \Delta_{0}=x_{0}y_{0}+y_{0}z_{0}+z_{0}x_{0}, \Delta_{m}=x_{m}y_{m}+y_{m}z_{m}+z_{m}x_{m}, \Delta_{m,0}=x_{m}z_{0}+y_{m}z_{0}+y_{m}x_{0}+z_{m}x_{0}+x_{m}y_{0}+z_{m}y_{0}.
vw(n,m,j,\eta) =\eta+\sqrt{\frac{(2n+2)}{1-([\eta]_{m}-2^{-(j+1)})^2}}([\eta]_{m}-2^{-(j+1)})-\eta-\sqrt{\frac{(2n+2)}{1-([\eta]_{m}-2^{-j})^2}}([\eta]_{m}-2^{-j}) &=\sqrt{2n+2}\left(\frac{[\eta]_{m}-2^{-(j+1)}}{\sqrt{1-([\eta]_{m}-2^{-(j+1)})^2}}-\frac{[\eta]_{m}-2^{-j}}{\sqrt{1-([\eta]_{m}-2^{-j})^2}}\right).
\left( \frac{M}{M+1}- \alpha(M) \right) \log \left( \frac{\mu}{E_B}\right) - \sqrt{\frac{\lambda}{-\mu}}- \sqrt{\frac{\lambda}{\lambda - \mu}}- \alpha(M) \log\left( E_B \left( \frac{1}{\mu}- \frac{1}{\lambda}\right) \right) - \alpha(M) = 0