harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gg \frac{\prod_{p \leq C}(p^4 - 3p^3 - 2p^2)}{{\mathcal P}\varphi({\mathcal P})^3}\frac{X^2}{\log^3 X}= \prod_{p \leq C}\left(\frac{p^4 - 3p^3 - 2p^2}{p(p-1)^3}\right) \frac{X^2}{\log^3 X}&= \prod_{p \leq C}\left(\frac{(p-1)^3-5p+1}{(p-1)^3}\right) \frac{X^2}{\log^3 X}\gg \frac{X^2}{\log^3 X}
\left(r\tilde{\rho}\right)''-\dfrac{f^{2}}{2r}\tilde{\rho}=-\dfrac{1}{4}r\left(A-B\right)^{2}\tilde{\rho}+\dfrac{\lambda}{2}r\tilde{\rho}\left(\tilde{\rho}^{2}-\dfrac{2\mu^{2}}{\lambda}\right)
g(\Delta x_{0}, \Delta p_{0}) := 4 \frac{\Delta x_{0}\Delta p_{0}}{\hbar}\frac{2 \Delta x_{0}\Delta p_{0}+ \sqrt{4 (\Delta x_{0}\Delta p_{0})^2 +{\hbar}^2}-\hbar}{2 \Delta x_{0}\Delta p_{0}+\sqrt{4 (\Delta x_{0}\Delta p_{0})^2 +{\hbar}^2}+\hbar}
|\nabla G_t(x)|^2 \leq \sup_{t>0}|\nabla G_t(x)|^2 = \sup_{t>0}\dfrac{|x|^2}{64\pi^2 d^4t^4}\mathrm{e}^{-|x|^2/2dt}&= \dfrac{1}{|x|^6}\sup_{u>0}\dfrac{u^4}{4\pi^2}\mathrm{e}^{-u},
\sum_{b=0}^{h_{n}-1}|\sum_{i=r_{n}-k_{n}+2}^{r_{n}}\mu(T^{k_n(h_n+c_n)}(I_{n,b}^{[i]}) \cap B) - \mu(I_{n,b}^{[i]})\mu(B)| \leq \sum_{y=0}^{h_{n+1}-1}|\mu(T^{h_{n+1}}(I_{n+1,y}) \cap B) - \mu(I_{n+1,y})\mu(B)|
2 \mathrm{Re}\bigg\langle \Delta_r f, \sum_{j=1}^3 L_j^2f \bigg\rangle \geq -\frac{3}{4}\bigg( \frac{1}{\delta}\bigg\| \frac{f}{|x|^2}\bigg\|_{L^2(\mathbf{R}^3)}^2 + \delta \bigg\| \sum_{j=1}^3 L_j^2f \bigg\|_{L^2(\mathbf{R}^3)}^2 \bigg)
S_{k+1/2}^{+,new}(4NM,\chi;f \otimes \chi) := \{h \in S_{k+1/2}^+(4NM,\chi): h|T_{p^2}= a_{\chi}(p)h p\nmid 2N \}\subset S_{k+1/2}^{+,new}(4NM,\chi),
a^2,b^2=\frac{1}{2}\left(\frac{\alpha^2+A^2}{l^4}+\frac{1}{L^2}\pm\sqrt{\left(\frac{\alpha^2+A^2}{l^4}+\frac{1}{L^2}\right)^2-\frac{4A^2\alpha^2}{l^8}}\right)
U_{J,z}\langle \alpha \rangle=\sum_{\beta:\mathrm{admissible\,\,orbit\,\,set\,\,with\,\,}[\beta]=\Gamma}\# \{\,u\in \mathcal{M}_{2}^{J}(\alpha,\beta)/\mathbb{R})\,|\,(0,z)\in u\,\}\cdot \langle \beta \rangle.
\mathrm{vol}(N_d(\mathrm{CH}(S))\cap B(x, \rho)) =\mathrm{vol}(N_d(\mathrm{CH}(\gamma S))\cap B(\gamma x, \rho)) \lesssim e^{a\rho\beta}V\left(1+\frac{1}{a}\log A(1+AC')\right)\lesssim e^{a\rho\beta},
J_k(v):=\frac{1}{2\tau_k}\int_\Omega |v|^2\,\d x+\frac{1}{2}\int_\Omega |\nabla v|^2\,\d x- \left\langle \dfrac{u_{k-1}}{\tau_k}+ f_k ,v \right\rangle_V \mbox{for}\v \in V
\psi_1'(a) =\xi''(a)\Big[\frac{(1+z_1+z_2)q}{z_1}-a-\frac{q\xi'(q)(1+z_2) (1+z_1+z_2)}{z_1[(1+z_2)\xi'(q)+z_1\xi'(a)]}\Big] = \frac{\xi''(a)[\xi'(a)[(1+z_1+z_2)q-z_1a] -\xi'(q)(1+z_2)a]}{(1+z_2)\xi'(q)+z_1\xi'(a)}.
\frac{{\rm d}Y}{{\rm d}x}=\left(\sum_{\nu=1}^n\sum_{k=0}^{r_{\nu}}\frac{A_{\nu}^{(k)}}{(x-u_{\nu})^{k+1}}+\sum_{k=1}^{r_{\infty}}A_{\infty}^{(k)}x^{k-1}\right)Y,A_*^{(k)}\in M(m, \mathbb{C})
\sum_{\substack{q_2\sim Q_2 (q_2,10)=1}}\,\sum_{\substack{a< d q_1q_2 (a,d q_1q_2)=1}}\, \sum_{\substack{|\eta|\le E/Y (\eta+a/d q_1q_2)Y\in\mathbb{Z}}}F_Y\Bigl(\frac{a}{q_1q_2d}+\frac{b}{d}+\eta\Bigr) &\ll E^{27/77}\Bigl(D^{27/77}(Q_1Q_2^2)^{1/21}+Q_1Q_2^2D\Bigl(\frac{Y}{D E}\Bigr)^{-10/21}\Bigr).
\begin{aligned}|I_1(t)| \leq C\bigg\{|\delta^1 \widehat{X}(t)|\cdot |\delta^1 X(t)|+|\delta^1 \widehat{X}(t)|^2+|\delta^1 \widehat{X}(t)|\cdot |\delta^1 Y(t)| \quad+|\delta^1 \widehat{X}(t)| |\delta^1Z(t)-p^0(t)\delta\sigma(t,v(t))\mathbf{1}_{E_\varepsilon}(t)|\bigg\}.\end{aligned}
h=g_1x_1^{\varepsilon_1}y_1^{-\varepsilon_1}g_1^{-1}g_2x_2^{\varepsilon_2}y_2^{-\varepsilon_2}g_2^{-1}\cdots g_nx_n^{\varepsilon_n}y_n^{-\varepsilon_n}g_n^{-1}
\int_0^T e_1(U_{n+j})(t)\,dt \le D(T) \Big(e_1(U_{n+j})(0) + \int_0^T (|u_{n-1}|^2 + |u_n|^2+\sum_{k=1}^{j-1}|u_{n+k}|^2)\,dt\Big)\,.
x^-_{\theta}(1)=y_{1\,2}^-(1)=[\,x_2^-(0),\ldots, x_n^-(0),\,x_{1\,n-1}^-(1)\,]_{(s^2,\,r^{-1},\,\ldots,r^{-1})}, x_{\theta}^+(-1)=\tau(x^-_{\theta}(1))=y_{1\,2}^+(-1) =[\,[\,[\,x_{1\,n-1}^+(-1),\,x_n^+(0)\,]_{r^2},\,x_{n-1}^+(0)\,]_{s^{-1}},\ldots, x_2^+(0)\,]_{s^{-1}}.
g\mapsto F_g,\;\;\; F_g(f) = \mathbb E \Bigl(\sum_{k=0}^{n}\langle f_k,g_k\rangle\Bigr)\;\;\; \bigl(f\in H^{s_q^n}_p(X), g\in H^{s_{q'}^n}_{p'}(X^*)\bigr).
(B_\varepsilon (\vartheta)-\zeta Q_0^\varepsilon)^{-1}=f^\varepsilon (\widetilde{B}_\varepsilon (\vartheta )-\zeta I )^{-1}(f^\varepsilon )^*,\zeta \in \mathbb{C}\setminus \mathbb{R}_+, &(B^0(\vartheta )-\zeta \overline{Q_0})^{-1}=f_0(\widetilde{B}^0(\vartheta)-\zeta I)^{-1}f_0 ,\zeta \in \mathbb{C}\setminus \mathbb{R}_+.
\xi^2 = \frac{\rho_k-\theta_k}{(1+\rho_{k+1})\nu_k-\theta_k}&=\frac{\sqrt{L/\ell}-1-(\sqrt{L/\ell}-1)/(\sqrt{L/\ell}+1)}{(\sqrt{L/\ell}-1)\cdot(\sqrt{L/\ell}-1)/(\sqrt{L/\ell}+1)}&=1+\frac{1}{\sqrt{L/\ell}-1}.
\begin{aligned}S_{k}(2m)= m^{k}+ \sum_{j=1}^{m}((m-j)^k+(m+j)^k)= m^{k}+2\sum_{j=1}^{m}\sum_{i=0}^{[\frac{k}{2}]}{\binom{k}{2i}}m^{k-2i}j^{2i}= m^{k}+2\sum_{i=0}^{[\frac{k}{2}]}{\binom{k}{2i}}m^{k-2i}S_{2i}(m). \end{aligned}
\begin{array}{ll}SU(n)^3: n-4+k = 0 SU(2)^2 U(1)_R: (n+2k-4)(n-2)+(k-2)k=0{U(1)_R}^3: n (n-1) (n+2k-4)^3 + 2kn(k-2)^3-k(k-1)n^3=0 \end{array}
\sum_{K}\prod_{1\le j,\ell\le N}\frac{\|H\|_{\infty}^{k_{j,\ell}}\prod\limits_{i=1}^{N}r_i!}{k_{j,\ell}!}=\sum_{K}\prod_{i=1}^{N}\binom{r_i}{k_{i,1},\dotsc,k_{i,N}}\,\|H\|_{\infty}^{\sum\limits_{\ell=1}^{N}k_{i,\ell}}=\prod_{i=1}^{N}\sum_{r_i\ge0}\left(N\|H\|_{\infty}\right)^{r_i}=\left(1-N\|H\|_{\infty}\right)^{-N}
\frac{d^{2}x^{a}}{d \tilde{\tau}^{2}}+ \Gamma_{b c}^{a}\frac{d x^{b}}{d \tilde{\tau}}\frac{d x^{c}}{d \tilde{\tau}}= \Omega{\cal F}^{a}\hspace *{1 m m}_{b}\frac{d x^{b}}{d \tilde{\tau}},
* G_{\it 3}= \frac{\sqrt{- g}}{3 !}\epsilon_{m n p q s t}g^{q q^{\prime}}g^{s s^{\prime}}g^{t t^{\prime}}G_{q^{\prime}s^{\prime}t^{\prime}}d y^{m}\wedge d y^{n}\wedge d y^{p}\wedge g Z ( r ) \chi_{\it 4}\
\frac{\frac{1 N}{2 2 . 4}}{\frac{1 7 V}{2 2 . 4}+ \frac{V}{7 0 0}\cdot 1 0 0 0}= \frac{1 7}{1 7 + \frac{2 2 4}{7}}= \frac{1 1 9}{3 4 3}\approx 3 4 . 6 9 \%
I({U}_1,\cdots,{U}_l;{Y}_1,\cdots,{Y}_l) = H(U_1,\cdots,U_l)-\sum_{i=1}^l h_b(d_i), H({X}|{U}_1,\cdots,{U}_l) =1+\sum_{i=1}^l h_b(P_i)-H(U_1,\cdots,U_l),
F\left ( \frac{\|\Phi(A)^*U x\|}{\|A^*x\|}A^*x \otimes W h(x,f) \right)= F\left(\frac{\|\Phi(A)^*U x\|}{\|A^*x\|}W^* A^*x \otimes h(x,f) \right)= F\left(A^*x \otimes f\right).
S ={\pi^3 R^5\over 2 \kappa^2_{10}}\left[ \; \int\limits_{0}^{1}\!du\, d^4x\, \sqrt{-g}\left({\cal R}- 2\Lambda\right) + 2 \int\! d^4x\, \sqrt{-h}K + a \int\! d^4x\, \sqrt{-h}\; \right]\,.
- e_{A}E_{t}\frac{\kappa}{r}+ \frac{e_{A}}{r}( \vec{\Theta}\gamma^{t s}\vec{\Theta}) E_{s}+ \frac{E_{t}}{2 r}\left( \vec{\Theta}_{\hat{\alpha}A}( \vec{\Theta}_{\hat{\alpha}}\cdot \vec{e}) - ( \vec{\Theta}_{\hat{\alpha}}\cdot \vec{e}) \vec{\Theta}_{\hat{\alpha}A}\right) .
\|\sum_{n=1}^{\infty}\frac{1}{n!}\,\|ad^{n}S_{I_{m; i+r-m}}(V^{(m, i+r-m-1)}_{I_{r-j;i}})\|\leq C\cdot t \cdot \frac{64}{(m+1)^2}\,t^{\frac{m-1}{3}}\frac{t^{\frac{r-j-1}{3}}}{(r-j+1)^2}\,,
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left|u(x)-u(y)\right|^{p-2}\,\left(u(x)-u(y)\right)}{|x-y|^{N+s\,p}}\, (\varphi(x)-\varphi(y))\, dx\,dy &= C(\beta)\, \int_{\mathbb{R}^N}u^{p-1}\, |x|^{-s\,p}\, \varphi(x)\, dx, \mbox{for all}\varphi\in C^\infty_0(\mathbb{R}^N\setminus\{0\}),
\pi_1(\Upsilon_n) = \langle\, c_1,\hdots,c_n\,|\, c_1^2c_2^2\cdots c_n^2 = 1\ \rangle \pi_1(\Upsilon_{2g+1}) \cong \langle\, a_1,b_1,\hdots,a_g,b_g,c\,|\, [a_1,b_1]\cdots[a_g,b_g]c^2 = 1\ \rangle \pi_1(\Upsilon_{2g+2}) \cong \langle\, a_1,b_1,\hdots,a_{g},b_{g},c,d\,|\,[a_1,b_1]\cdots[a_{g},b_{g}]cdcd^{-1}= 1\ \rangle
\varepsilon_k =0 b^k =\frac{1}{c_k}(x^k-y^k)-Ax^k+\frac{\alpha_k}{c_k}(x^k-x^{k-1}), v^k =Ay^k+b^k r^k =c_kv^k+y^k-x^k-\alpha_k(x^k-x^{k-1}).
\begin{aligned}\mathcal{E}_{u}:=\{(n,m)\rightarrow (n+1,m+1) : (n,m)\in\mathcal{V}\}\mathcal{E}_{\ell}:=\{(n,m)\rightarrow (n+1,m) : (n,m)\in\mathcal{V}\}\mathcal{E}_{d}:=\{(n,m)\rightarrow (n+1,m-j) : (n,m)\in\mathcal{V},\,\,\,1\leq j\leq p\}.\end{aligned}
\lambda{\psi}(z) - \frac{1}{2}\triangle_v{\psi}(z) - v \cdot D_x{\psi}(z) - F(z) \cdot D_v{\psi}(z) = g(z) \, = \lambda{\psi}(z) - \frac{1}{2}\mathrm{Tr}\big(Q D^2{\psi}(z) \big) - \langle Az, D{\psi}(z)\rangle - \langle B(z), D{\psi}(z)\rangle
(\begin{matrix}ct^{\prime}\\x^{\prime}\\y^{\prime}\\z^{\prime}\end{matrix})=(\begin{matrix}\gamma&-\gamma\beta&0&0\\ -\gamma\beta&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix})(\begin{matrix}ct\\x\\y\\z\end{matrix})
\Sigma_1=\sup_{\beta\in\mathbb{R}}\sum_{\substack{|\eta|\le E/Y Y(\eta+\beta)\in\mathbb{Z}}}F_{E'}\Bigl(D' V^2\beta+D' V^2\eta\Bigr)\le\sup_{\beta'\in\mathbb{R}}\sum_{a\le 2E}F_{E'}\Bigl(\beta'+\frac{D' V^2 a}{Y}\Bigr).
S = - \int_{\tau_{1}}^{\tau_{2}}\! \biggl \{\gamma \! \int_{\sigma_{0}( \tau )}^{\sigma_{3}( \tau )}\! \! \sqrt{- g}d \sigma + \sum_{i = 1}^{3}m_{i}\sqrt{V_{i}^{2}( \tau )}\biggr \}d \tau .
H_{2,1}(F_{f}/2) =\frac{1}{192}\left(-3\tau^4_{1}+4(1-\tau^2_{1})\tau^2_{1}\tau_{2}-4(1-\tau^2_{1})(3+\tau^2_{1})\tau^2_{2}\right. &\left.+16\tau_{1}\tau_{3}(1-\tau^2_{1})(1-|\tau^2_{2}|)\right).
\dot{\epsilon}_{\delta\sigma}=\epsilon_{\delta\sigma}\left[3{\dot{a}\over a}(1+w)+2{\dot{c}\over c}{{1+\epsilon_{\delta\sigma}}\over{1+\epsilon}}+\left({\dot{G}_4\over G_4}-4{\dot{c}\over c}\right)(1+\epsilon_{\delta\sigma})\right].
\lambda^{-1}(x-J^A_\lambda x)=_X\frac{\gamma}{\lambda\gamma}(x-J^A_\gamma x)=_X\frac{1}{\gamma}\left(\frac{\gamma}{\lambda}x+\left(1-\frac{\gamma}{\lambda}\right)J^A_\lambda x-J^A_\lambda x\right).
\mathcal{Z}_{sym}(a^-,a^+)=\{{z}\in\mathcal{S}_{sym}(a^-,a^+)\;:\;\forall{v}\in\mathcal{S}_{sym}(a^-,a^+),\, \mathfrak{E}_{W}({z})\leq \mathfrak{E}_{W}({v})\}.
\frac{p^{s_F^\ell(L^{ra_i};x)}-1}{a_i/(\ell+1)}\ge \frac{p^{s_F^\ell(L^{md_i+j};x)}-1}{a_i/(\ell+1)}\ge \frac{p^{ie}-1}{a_i/(\ell+1)}= \frac{p^e-1}{m/(\ell+1)}\cdot \frac{d_im}{\bigl\lceil (md_i + j)/r \bigr\rceil},
A_i = \{j = 1, \dotsc, r-1 \mid \alpha^{-1}(j-1) < i \alpha^{-1}(j) \geq i \}, B_i = \{j = 1, \dotsc, r-1 \mid \alpha^{-1}(j-1) \geq i \alpha^{-1}(j) < i \}.
d_{q-1}\tilde \varphi(x_0,\ldots,x_q) =\sum_{i=0}^q(-1)^i\varphi(0,x_0,\ldots,\hat x_i,\ldots,x_q) &=\sum_{i=1}^{q+1}(-1)^{i+1}\varphi(0,x_0,\ldots,\hat x_{i-1},\ldots,x_q)+d_q\varphi(0,x_0,\ldots,x_q) &=\varphi(x_0,\ldots,x_q)
\beta_{o}^{\mu}=\sum_{e\neq 0}\beta_{e}^{\mu}U_{-e,o}\,\,,\quad\beta_{o}^{b}=\sum_{e\neq 0}\beta_{e}^{b}U_{-e,o}\,\,,\quad\beta_{o}^{c}=\sum_{e\neq 0}U_{o,-e}^{-1}\beta_{e}^{c}\,\,.
(i) \ \c_p(t_1, \ldots, t_p), \p>1, \ \ \ \sum_{i=1}^p t_i=0. (ii) \ \c_p(t_1, \ldots, t_p)=0 \ \ \p>1 \ \ \ \ \sum_{i=1}^p t_i\neq 0. (iii) \ \c_1(0)=1 \ \ \ \c_1(t)=0 \ \ \ \t\neq 0.
P^{}_{,R}= \mathrm{Pr}\Big\{|H^R|^2 < \max \big\{\frac{(2^{\Tilde{R}_T}-1)\sigma^2_0}{\big[c^2_T-(2^{\Tilde{R}_T}-1)c^2_R\big]P_{BS}}, \frac{(2^{\Tilde{R}_R}-1)\sigma^2_0}{c^2_RP_{BS}}\big\}\Big\},
\chi=\frac{4}{1-\sqrt{\beta}}\left\{\sqrt{\beta}\log\left(\frac{\sqrt{\beta(1-\alpha)}+\sqrt{\beta-\alpha}}{\sqrt{(1-\beta)\alpha}}\right)-\log\left(\frac{\sqrt{1-\alpha}+\sqrt{\beta-\alpha}}{\sqrt{1-\beta}}\right)\right\}.
\left[ f^{^{\prime \prime}}+f^{^{\prime}2}+\frac{h_{m}^{^{\prime \prime}}+2h_{m}^{^{\prime}}f^{^{\prime}}}{h_{m}}+E-V\left( \hat{r}\right) -\frac{m^{2}-1/4}{r^{2}}\right] R_{m}\left( \hat{r}\right) =0.
\log \bigg( \frac{N \mu^{k^* + 1}e^{sk^* t^*}}{s^{k^* + 1}k^*!}\bigg) = \log N - (k^* + 1) \log \bigg( \frac{s}{\mu}\bigg) + s k^* t^* - \log k^*! &= - (k^* + 1 - k_N) \log \bigg( \frac{s}{\mu}\bigg) + s k^* t^* - k^* \log k^* + k^* + o(k_N).
v_{0}=2\sqrt{\frac{\mathfrak{S}_{0}}{\rho_{0}^{3}}}\rho_{1}=2J_{2}\sqrt{\frac{\mathfrak{S}_{0}}{\rho_{0}}}\rho_{0}=\frac{1}{2h}(4J_{2}^{2}\mathfrak{S}_{0}-J_{1})h\neq0.
\begin{aligned}v_1(A) \geq \frac{1}{\liminf_{n \rightarrow \infty}\frac{h_2(r_n)}{h_1(r_n)}}\nu_2(A) & = \limsup_{n \rightarrow \infty}\frac{h_1(r_n)}{h_2(r_n)}\nu_2(A) & \geq \liminf_{n \rightarrow \infty}\frac{h_1(r_n)}{h_2(r_n)}\nu_2(A).\end{aligned}
\left(1-\frac{x}{v(1-v)}\right)A'\Big(\frac{x}{v},v\Big) =-\frac{x}{v^2(1-v)}A'(x,1)+\frac{x(1+v)}{v^2}\big(A(x,1)-1\big)-\frac{x^2}{v^2}A(v,1).
y_{jk}(1,\l) = -i b_j e^{i b_k \l}\int_{0}^1 e^{i (b_j-b_k) \l s}\left(q_{jk}(1-s) +\sum_{l=1}^n q_{jl}(1-s)\rho_{lk}(1-s,\l)\right)ds.
V(|x+y|) - V(|x|) - DV(|x|)\cdot y = \frac{1}{|x+y|^{2}}- \frac{1}{|x|^{2}}+ \frac{2 x\cdot y}{|x|^{4}}\le K \frac{|y|^{2}\vee | x \cdot y |}{|x|^{4}},
| \varphi > \to \Psi [ \phi ; \varphi ,{\cal P}, f ] = N_{f}e x p \{i \int{\cal P}_{x}\phi_{x}-{\frac{1}{2}}\int_{x , y}( \phi_{x}- \varphi_{x}) f_{x y}( \phi_{y}- \varphi_{y}) \}\; ,
{\cal H}={\frac{u}{\cal R}}\sqrt{\left({\frac{u}{\cal R}}\right)^{2}{P_{u}}^{2}+ \vec{L}^{2}+ \alpha^{\prime}\, m^{2}\, L^{2}g_{s}^{\frac{- 2}{D - 3}}e^{- 2{\frac{D - 4}{D - 3}}\phi}}- q A_{t}\, ,
D =\sum_{x,y,z\in\mathbb{F}_{q}^{\times}}\theta(x^dz)\theta(axz)\theta(bz)\theta(-zy^2) &=\frac{1}{(q-1)^4}\sum_{l,m,n,k=0}^{q-2}G_{-m}G_{-l}G_{-n}G_{-k}T^l(a)T^n(b)T^k(-1) &\times\sum_{x\in\mathbb{F}_{q}^{\times}}T^{l+md}(x)\sum_{y\in\mathbb{F}_{q}^{\times}}T^{2k}(y)\sum_{z\in\mathbb{F}_{q}^{\times}}T^{l+m+n+k}(z).
\left[\left(\frac{(m\tau_1+n)(t^{-1}-t)-2m\tau_2}{|m\tau+n|^2}\right)\frac{1+(t_{+}^{m,n})^2}{(t-t_{+}^{m, n})^2}-\frac{(t^{-2}+1)}{m\xi^0+n}\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\right]\Big|_{m=0}=\frac{2}{t|n|}\,.
G^{(k;\epsilon_1\cdots\epsilon_k)}_{00,00}=\frac{{\bf a}^2}{4}\sum_{n_1,n_2}\left(\!\frac{h_1}{4}\!\right)^{2n_1}\!\left(\!\frac{h_2}{4}\!\right)^{2n_2}\frac{(2(n_1\!+\!n_2))!}{n_1!^2n_2!^2}\;F_{k;\epsilon_1\cdots\epsilon_k}(n_1,n_2;\phi)\; ,
\begin{array}{l}{{c ( p^{4}h + h p^{4}- \frac{1}{2}S p ( p^{2}) ( p^{2}h + h p^{2}) ) +}}\\{{( p^{3}h p + p h p^{3}- S p ( p^{2}) p h p ) + ( 1 + c ) G_{4}h = 0}}\\ \end{array}
u(\mathbf x)=\int_{\mathbb S^{1}}e^{\mathrm{i}k\xi\cdot \mathbf{x}}g (\xi)\mathrm d\xi,\ \xi \in \mathbb S^{n-1},\ \mathbf{x}\in \mathbb{R}^n,\g \in L^{2}(\mathbb S^{n-1}), n= 2 \mbox{or}3,
Z=\left(\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}q^{m^{2}+5n^{2}}\right)^{2}, \mathbf{Z}= \left ( \sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}q^{2m^{2}+ 2m n+ 3n^{2}}\right )^{2}.
\mathcal{H}^{2k}(B_{4r}(a)\cap (\cup^m_{j=1}A_{i_j})) =\mathcal{H}^{2k}(\cup^m_{j=1}(B_{4r}(a)\cap A_{i_j}))=\sum^m_{j=1}\mathcal{H}^{2k}(B_{4r}(a)\cap A_{i_j}) &\geq \sum_{j=1}^{m}\Omega(2k)r^{2k}=m\Omega(2k)r^{2k}
u_n^T J_n(\theta) u_n = \begin{pmatrix}\frac{n^{2/ \alpha}}{n^{2/ \alpha_0}}I_n^{11}(\theta) \frac{n^{1/ \alpha}}{n^{1/ \alpha_0}}v_n^T r_n^T(\theta) I_n^{21}(\theta) v_n^T r_n^T(\theta) I_n^{22}(\theta) r_n(\theta) v_n + v_n^T R_n(\theta) v_n\end{pmatrix}
\begin{array}{c c c}{{( \rho_{1}\rho_{2}\rho_{3})^{*}}}&{{\Longrightarrow}}&{{\rho_{3}^{*}\circ ( \rho_{1}\rho_{2})^{*}}}\\{{\Downarrow}}&{{}}&{{\Downarrow}}\\{{( \rho_{2}\rho_{3})^{*}\circ \rho_{1}^{*}}}&{{\Longrightarrow}}&{{\rho_{3}^{*}\circ \rho_{2}^{*}\circ \rho_{1}^{*}}}\\ \end{array}
(\overline{B(\lambda_i(x_0),\frac{\beta}{2})}\setminus B(\lambda_i(x_0),\frac{\beta}{4}))\bigcap Eig(\bar{A}(x))=\emptyset,\forall x\in(x_0-\alpha_{i,\beta}(x_0),x_0+\alpha_{i,\beta}(x_0))
\frac{1}{2}\int_\Omega e^{-2g}u(y,s)^2\,dy\biggr|_{s=0}^{s=t}+(1-\mu)\int_0^t\int_\Omega e^{-2g}H(\nabla u)^2\,dy\,ds \qquad \le \int_0^t\int_\Omega e^{-2g}u^2\left[\mu^{-1}H(\nabla g)^2-\partial_t g\right]\,dy\,ds
D_t^2q= \sum_{j=1}^2\frac{\lambda_j i}{2\pi}\frac{D_t^2\zeta-\ddot{z}_j(t)}{(\zeta(\alpha,t)-z_j(t))^2}-\sum_{j=1}^2\frac{\lambda_j i}{\pi}\frac{(D_t\zeta)^2-2D_t\zeta \dot{z}_j}{(\zeta(\alpha,t)-z_j(t))^3}-\sum_{j=1}^2\frac{\lambda_j i}{\pi}\frac{\dot{z}_j^2}{(\zeta(\alpha,t)-z_j(t))^3}.
{\frac{\dot{\bar{G}}_{4}}{\bar{G}_{4}}}= e^{2 m_{0}( R_{0}- R )}{\frac{R_{\tau}\left[ 3 ( \Omega_{R}^{2}- 1 ) R_{\tau \tau}+ 2 m_{0}\Omega_{R}^{2}( c^{2}+ R_{\tau}^{2}) \right]}{( \Omega_{R}^{2}- 1 ) ( c^{2}+ R_{\tau}^{2})}}.
{\cal R}( x ) = \sum_{m = 0}^{\infty}\frac{b_{m}}{m !}\Lambda^{m}( x ) = \left( \frac{e^{\Lambda ( x )}- 1}{\Lambda ( x )}\right)^{- 1}, \qquad \Lambda_{j}^{i}( x ) = x^{k}f_{k j}^{i},
\begin{aligned}\ (-1)^S \Bigl[D_{\sigma(m+n-1)-1}\circ\dots\circ D_{\sigma(t+1)-1}\circ D_{\ell-1}\circ D_{\sigma(t-2)}\circ\dots D_{\sigma(m)}(x_m)\Bigr]\ \otimes \Bigl[D_{\sigma(m-1)-1}\circ\dots\circ D_{\sigma(J)-1}\circ D_{\sigma(J-1)}\circ \dots\circ D_{\sigma(0)}\circ F_{\ell-J}(y_n)\Bigr]\end{aligned}
\begin{aligned}f_{0}^{(1)}\coloneqq \frac{c_{0}}{d_{0}^2}f_{1}^{(1)}\coloneqq \frac{c_{1}d_{0}-2 c_{0}d_{1}}{d_{0}^3}f_{2}^{(1)}\coloneqq \frac{3c_{0}d_{1}^{2}+c_{2}d_{0}^{2}-2c_{0}d_{0}d_{2}-2c_{1}d_{0}d_{1}}{d_{0}^4}.\end{aligned}
D_{k}({\vec{x}},{\vec{x}^{\prime}}) = \sum_{n = - \infty}^{\infty}\int_{0}^{\infty}{\frac{q d q}{2 \pi \alpha}}{\frac{e^{i n ( \theta - \theta^{\prime}) / \alpha}}{q^{2}+ k_{4}^{2}}}J_{\left| n / \alpha \right|}( q r ) J_{\left| n / \alpha \right|}( q r^{\prime}) ,
c_n= (-1)^{\frac{n-1}{4}}2^{-2n-2}\cdot 3^{-1}\frac{\left(\tfrac{n+3}{2}\right)!(n+1)!}{\left(\tfrac{n-1}{2}\right)!}\frac{B_{\frac{n-1}{2}}}{B_{\frac{n+3}{2}}B_{n+1}B_{n-1}}.
\rho_N= \sum_{n=1}^N\frac{r'_n(x)}{(2i\lambda)^n}-\sum_{n=0}^{N-1}\frac{r_{n+1}(x)}{(2i\lambda)^n}& + \varphi_1\sum_{n=1}^N\Big(\sum_{k+ l=n}r_kr_l\Big)\frac{1}{(2i\lambda)^n}+ \varphi_1\sum_{n=N+1}^{2N}\Big(\sum_{k+l= n}r_kr_l\Big)\frac{1}{(2i\lambda)^n}-\varphi_2
\|\tilde{f}^{L_s}\|_2^2 \geq \|f\|_2^2/2 -R^2 2^{-2L_s s}\gtrsim C_0\sqrt{\log\log (n)}\rho_s^2 & = \frac{C_02^{3L_s/2}\sqrt{\log\log n}}{2n (\frac{b}{\log(n)}\wedge 2^{L_s})}\gtrsim \frac{C_02^{L_s}\sqrt{\log\log n}}{n b' \frac{m'}{m}},
\displaystyle{E^{T E}( a ) = \left. \frac{\hbar \, c}{a}\, \sum_{\nu = 3 / 2}^{\nu_{0}}\nu \, \left[ \zeta_{\nu}^{T E}( s , 0^{+}) - \zeta_{\nu}^{T E}( s , x ) \right] \right|_{s \rightarrow - 1}}.
\mathbb{E}\big[ \mathbb{E}\big[ X_{T\wedge (n,K)}\, | \, \mathcal{F}_0 \big] \, \big|\, \mathcal{S}_0\big]=\mathbb{E}\big[ X_{T\wedge (n,K)}\, | \, \mathcal{S}_0 \big] \leqslant 2 \mathbb{E}\big[ X_0 \, | \, \mathcal{S}_0\big].
Z(X_1+X_2|Y^2) = 2\sum_{y^2}\sqrt{P_{X_1+X_2,Y^2}(0,y^2)P_{X_1+X_2,Y^2}(0,y^2)}= 2\sum_{y^2}\sqrt{P_{X_1Y_1}(0,y_1)P_{X_1Y_1}(1,y_1)P_{X_2Y_2}(1,y_2)P_{X_2Y_2}(0,y_2)}&\cdot\sqrt{f(y_1)^2+f(y_2)^2-4},
\frac{4\pi}{k}\mathbf{q}_{ij}= \frac{(\partial\mathbf{r}/\partial s)_{i}\times(\partial^{2}\mathbf{r}/\partial s^{2})_{i}}{|(\partial\mathbf{r}/\partial s)_{i}|^{3}}\log(\frac{1}{\epsilon})+O(1)
\sum_{n=0}^{\infty}\frac{q^{2n^2+2n}}{(q/z;q^2)_{n+1}(zq;q^2)_{n+1}}=\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}b^{2'}_\omega(m,n)z^mq^n =\sum_{n=0}^{\infty}\frac{z^nq^n}{(q/z;q^2)_{n+1}}.
g_{\mathbb{K}_{2}}(s):=\frac{15}{8}\left( 1-s^{2}\right) \left(1-5s^{2}\right) \mathbf{1}_{\left\{0\leq s<1\right\}},g_{\mathbb{K}_{3}}(s):=\frac{35}{16}\left( 1-s^{2}\right)^{2}\left( 1-7s^{2}\right) \mathbf{1}_{\left\{0\leq s<1\right\}}.
\sum_{k=0}^{\infty}\bigg(\frac{1}{2}\bigg)^{k-1}\frac{(a)_k(1-a)_k}{(1)_{k}(b)_{k}}\sum_{i=1}^k\frac{1}{(a-1+i)(-a+i)}=\frac{\Gamma(\frac{b}{2})\Gamma(\frac{1+b}{2})}{\Gamma(\frac{a+b}{2})\Gamma(\frac{1-a+b}{2})}begin{align*}1mm] \:\:\times \frac{\psi(\frac{1-a+b}{2})-\psi(\frac{a+b}{2})}{1-2a}.
\left\langle\Psi,\left(N^{-1}\sum_{k=1}^Nq_{k}^{\varphi}\right) \left(\widehat{w^{\varphi}}\right)^2\Psi\right\rangle =\left\langle\Psi,\left(\widehat{n^{\varphi}}\right)^{2}\left(\widehat{w^{\varphi}}\right)^2\Psi\right\rangle=\left\Vert \widehat{w^{\varphi}}\widehat{n^{\varphi}}\Psi\right\Vert^2_2
\frac{C}{(\log(\log(t)))^{2}t^{2}\log^{b+1}(t)}+ \frac{C}{(\log(\log(t)))^{2}\log^{b+1}(t) t^{2}}+ \frac{C \log(\log(t))}{t^{2}\log^{b+2}(t)}+ \frac{C}{(\log(\log(t)))^{3/2}t^{2}\log^{b+1}(t)}&+ \frac{C}{t^{2}\log^{b+2}(t) (\log(\log(t))^{5/2}}+ \frac{C}{(\log(\log(t))^{5/2}t^{3}\log^{b+2}(t)}
\frac{\P(X_1>s/t_1)}{\P(X_1>s/t_2)}< \frac{t_1^2}{t_2^2}\textrm{and}\lim_{s\to \infty}\frac{\P(X_1>s/t_1)}{\P(X_1>s/t_2)}=\frac{t_1^2}{t_2^2}.
\lambda=\lambda\,\int_\Omega v\, \frac{u_1^2}{v}=\int_\Omega \left\langle \nabla v,\nabla\left(\frac{u_1^2}{v}\right)\right\rangle\, dx\le \int_\Omega |\nabla u_1|^2\, dx=\lambda_1(\Omega),
\mathsf{h}=\left[\begin{array}{lll}0\; b\; p 0\; 0\; q 0\; 0\; 0\end{array}\right]\qquad\mbox{or as}\tilde{\mathsf{h}}=\left[\begin{array}{lll}0\; 0\; p c\; 0\; q 0\; 0\; 0\end{array}\right].
2{\frac{\mu}{R^{2}}}u = \operatorname{a r c t a n}{\frac{\mu r^{2}/ R^{2}- E^{2}R^{2}/ 2 \mu}{\sqrt{( E^{2}-{\frac{\mu^{2}}{R^{2}}}) r^{2}-{\frac{\mu^{2}}{R^{4}}}r^{4}+{\frac{\mu^{2}M}{R^{2}}}}}}.
\begin{cases}\displaystyle\phantom{\Big[}\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}+a(x,M(t))f(t,x)\,=\,0, \phantom{\Big[}N(t)\,:=\,f(t,x=0) =\int_{0}^\infty a(x, M(t))\,f(t,x)\,dx, \end{cases}
{\cal Z}= \left( \frac{1}{1 - z^{2}}\right)^{9}( 1 + 4 z^{3}+ 1 0 z^{6}+ 1 2 z^{7}+ 1 5 z^{8}+ 2 4 z^{9}+ 3 0 z^{1 0}+ 2 4 z^{1 1}+ 8 z^{1 2})
\int_{B_r}\int_{B_r}\left|\log\left(\frac{\delta+u(x)}{\delta+u(y)}\right)\right|^p \frac{1}{|x-y|^{N+s\,p}}\, dx\,dy & \le C\, r^{N-s\,p}\left\{\delta^{1-p}\, r^{s\,p}\,\int_{\mathbb{R}^N\setminus B_{2\,r}}\frac{u_-(y)^{p-1}}{|y-x_0|^{N+sp}}\,dy+ 1\right\},
R^{2}- 1 +{\frac{8 \pi R^{2}}{3 M_{\mathrm{P}}^{2}}}\left( V ( f ) +{\frac{n^{2}}{8 \pi^{4}f^{2}R^{6}}}-{\frac{f^{2}}{2}}\right) +{\frac{1}{R^{2}H_{0}^{2}}}( R^{2}- 1 )^{2}= 0 \ .
\begin{gathered}(A\otimes B)_V= A_V \sqcup B_V\;, (A\otimes B)_\psi= A_\psi \hat\sqcup B_\psi\;, (A\otimes B)_A(l_1\sqcup l_2)=A_A(l_1) \otimes B_A(l_2)\;.\end{gathered}
E_1 =\{g\in\mathcal E:\b(g)\geq m_b(F_{k_0}) \},E_2=\{g\in\mathcal E:\b(g)\leq m_b(F_{k_0}) \}; F_1 =\{g'\in F_{k_0}:\b(g')\geq m_b(F_{k_0}) \},F_2=\{g'\in F_{k_0}:\b(g')\leq m_b(F_{k_0}) \}.
| J_{a}={\epsilon_{a}}, \; J_{b}={\epsilon_{b}}, \; D_{a}= \epsilon_{a}\delta , \; D_{b}= - \epsilon_{b}\delta \rangle \equiv{\frac{1}{4}}| a_{\epsilon_{a}}b_{\epsilon_{b}}\rangle +{\frac{\delta}{4 \sqrt{3}}}| a_{\epsilon_{a}}b_{\epsilon_{b}}9 \rangle
\frac{\Gamma ( - \alpha^{\prime}s ) \Gamma ( - \alpha^{\prime}t )}{\Gamma ( 1 - \alpha^{\prime}s - \alpha^{\prime}t )}= \frac{1}{{\alpha^{\prime}}^{2}s t}- \frac{\pi^{2}}{6}- \zeta ( 3 ) ( s + t ) \alpha^{\prime}+{\cal O}({\alpha^{\prime}}^{2}) .
\mathbb{T}^{[M', M]}(z)\otimes \mathbb{T}^{[M', M]}(w)=\sum_{a,b=0}^{r}\sum_{c, d=0}^{r}\left(\mathbb{T}^{[M', M]}(z)_{ab}\mathbb{T}^{[M', M]}(w)_{cd}\right)\otimes E_{ab}\otimes E_{cd}.
y^5 + (54t^4+18t^3+34t^2+18t+39)y^3 + (5t^8+23t^7+44t^6+20t^5+35t^4+30t^3+17t^2+33t+21)y +(57t^{10}+18t^9+24t^8+58t^7+14t^6+9t^5+41t^4+17t^3+38t^2+48t+44) =0
F-F^* = \frac{2}{\pi}\int_0^{\infty}[D,\frac{1}{D^2+f^2+t^2}] dt = \frac{2}{\pi}\int_0^{\infty}\frac{1}{D^2+f^2+t^2}[D^2+f^2+t^2,D]\frac{1}{D^2+f^2+t^2}dt.
\frac{d}{dt}E_1(u) =\int_\Omega \Phi'(u)u_t\,dx=\int_\Omega \Phi'(u)u_t\,dx-\int_\Omega \Phi'(\lambda)u_t\,dx &=\int_\Omega \big(\Phi'(u)-\Phi'(\lambda) \big) u_t\,dx &=\int_\Omega \big(\Phi'(u)-\Phi'(\lambda) \big) \big(u-\lambda \big ) u(1-u)\,dx.
{\hat Y}^I_l{\hat Y}^J_m = \sum_{K=0}^N(-)^{N+K}\sqrt{(2I+1)(2J+1)}\Bigl\{\begin{array}{ccc}I J K{N\over 2}{N\over 2}{N\over 2}\ \end{array}\Bigr\}C^{IJK}_{lmn}{\hat Y}_n^K
\begin{aligned}\max~ ~\|e\|_2^2 ~ ~\mathcal X:=\arg\min_{x}\{\|x\|_{q}^{q}:\|{A}x-y\|_2\leq\sigma\}, &~x^*\in\mathcal X, &~y={A}x_0+w, &~\|w\|_2\leq\epsilon,\|x_0\|_0=N.\end{aligned}
\langle T^\mu_\mu \rangle_R - (T_\mu^\mu)_0 = 2\pi m^2\int_{-\infty}^{+\infty}\frac{d\theta}{2\pi}\left[\frac{e^{-m R \cosh\theta}}{1+ e^{-mR \cosh\theta}}+2 \frac{e^{-2 m R \cosh\theta}}{(1+e^{-m R \cosh\theta})^2}+\frac{9}{2}\frac{e^{-3 m R \cosh\theta}}{(1+e^{-m R \cosh\theta})^3}+\cdots \right]\,\,\, ,
\eta_{(j_1-1)2^{{r\over 2}-1}+k}= \theta_{j_1}+ \lambda_k^+, \eta_{[{m_1\over2}]2^{{r\over 2}-1}+l}= \lambda_l^+ , \eta_{[{m_1\over2}]2^{{r\over 2}-1}+2^{{r\over 2}-2}+(j_2-1)2^{{r\over 2}-1}+k}= \theta_{j_2}'+ \lambda_k^- , \eta_{[{m_1\over2}]2^{{r\over 2}-1}+2^{{r\over 2}-2}+[{m_2\over 2}]2^{{r\over 2}-1}+l}= \lambda_l^- ,
C_{\mathbb{R},2,4}\geq \frac{2}{\sqrt{3}}>1.1546 C_{\mathbb{R},2,8}\geq \frac{2^{\frac{5}{4}}}{1.892}>1.2570 C_{\mathbb{R},2,p}\geq \frac{2^{\frac{2mp+2m-p-2m^{2}}{mp}}}{1.9836}>1.3591p=1+\log_{9/10}1/19 C_{\mathbb{R},2,p}\geq \frac{2^{\frac{2mp+2m-p-2m^{2}}{mp}}}{1.9999}>1.4105p=1+\log_{99/100}1/199.
\delta_G u(k_1,k_2,k_3)(x_1,(a_2,x_2)) = u(k_1,k_2,k_3)(x_1x_2) u(k_1,k_2,k_3)(x_1)^{-1}= \widehat{F}(k_1,k_2)\left( \frac{F(k_3, x_1x_2)}{F(k_3,x_1)}\right)
\frac{q^{\sum_{\alpha<\beta}2r_{\beta}+\sum_{\alpha}r_{\alpha}}}{\prod_{j\geq1}(q)_{\nu_j}}= \frac{q^{\sum_{\beta=1}^{\ell}(2\beta-1)r_{\beta}}}{\prod_{j\geq1}(q)_{\nu_j}}= \frac{q^{\sum_{m=1}^{p-2}N_m^2}}{(q)_{N_1-N_2}(q)_{N_2-N_3}\dots},
\frac{\sin (\eta - \frac{1}{2})x \sin (\eta' -\frac{1}{2})y}{2 \sin \frac{1}{2}x \sin \frac{1}{2}y}- \frac{\sin^2 (\eta - \frac{1}{2})x}{2 \sin^2 \frac{1}{2}x}= O(|\eta - \eta'|+ |x-y| )