harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{k=4}^\infty s_{b,k}\leq \sum_{k=4}^\infty \frac{b^{\left\lceil\frac{k-2}{2}\right\rceil}x_b}{b^{k-1}+1}\leq \sum_{k=4}^\infty \frac{b^{\left\lceil\frac{k-2}{2}\right\rceil}x_b}{b^{k-1}}= \sum_{k=2}^\infty \frac{2x_b}{b^k}= \frac{2x_b}{b(b-1)}.
I_{X2}\equiv\int\frac{d^4q}{(2\pi)^4}\,\Bigg\{\frac{2 q_\mu (q_\mu+k_\mu)}{\Big((k+q)^2+M^2\Big)^2\Big(q^2+M^2\Big)^2}- \frac{4 M^2}{\Big((k+q)^2+M^2\Big)\Big(q^2+M^2\Big)^3}\Bigg\}.\qquad
o(1)= \int_M |\nabla \Delta (u-u_k)|_g^2d\mu_g-\int_M f|u-u_k|^{2^\sharp}d\mu_g \geq \int_M |\nabla \Delta (u-u_k)|_g^2d\mu_g\Big[1-(\max_M f)(1+\epsilon)^{\frac{2^\sharp}{2}}Y_6(S^n)^{-\frac{2^\sharp}{2}}\big(\int_M |\nabla \Delta (u-u_k)|_g^2d\mu_g\big)^{\frac{6}{n-6}}\Big].
I_{1 0}= \int d^{1 0}x \sqrt{- g^{( 1 0 )}}\left[ e^{- 2 \phi}( R^{( 1 0 )}+ 4 ( D \phi )^{2}) - \frac{1}{2 \cdot 5 !}F^{( 1 0 ) 2}\right] ,
Q_0=-\sum_{\kappa=-\infty}^\infty\vert\kappa\vert\left\{N_\kappa^+-N_\kappa^--\left[\eta_{m\kappa}\left(0\right)-\eta_{-m\kappa}\left(0\right)\right]\frac{1}{\pi}+\left[\eta_{\infty\kappa}\left(\infty\right)-\eta_{-\infty\kappa}\left(\infty\right)\right]\frac{1}{\pi}\right\}\,.
Y_n(r) = \frac{1}{\pi}\frac{2^n (n-1)!}{r^n}\left[\sum_{k = 0}^N \frac{(n-k-1)!}{k!(n-1)!}\left( \frac{r}{2}\right)^{2k}+\mathcal O\left(\frac{1}{n^{N+1}}\right)\right] .
\|x^{AB}_m - ( x_0 + t_m v_0)\| = \|x^{AB}_m - (x_0 + t_m v_m^A) -t_m(v_0- v_m^A)\| &\leq \|x^{AB}_m -x^A_m\| + t_m\|v_0- v_m^A\| \le ( K+1) \left\| x^A_m -x^B_m\right\| +t_m \frac{\varepsilon}{2K+3}&\leq ( K+1)\cdot \frac{2\varepsilon}{2K+3}\, t_m + \frac{\varepsilon}{2K+3}\, t_m = \varepsilon t_m \, .
\begin{aligned}\|\partial_{x}^{k+1}u_{n}\|_{L^{2}_{t,x}}^{2}\le M_{k}\|\partial_{x}^{k}V_{n}\|_{L^{2}_{t,x}}+ M_{k}\|\partial_{x}^{k}\rho_{n}\|_{L^{2}_{t}L^{2}_{x}}+ M_{k}\|\partial_{x}^{k-1}V_{n}\|_{L^{2}_{t,x}}begin{align*}1ex] \le M_{k}+ M_{k}\|\partial_{x}^{k}\rho_{n}\|_{L^{2}_{t}L^{2}_{x}}. \end{aligned}
G_{010110}= G_{011100}+ \theta_3 G_{011000}= (G_{111000}- \theta_1 G_{110000})+ \theta_3(G_{110000}- \theta_1 G_{100000}) &= \theta_2 \theta_4 \theta_5 + \theta_2 \theta_4 \theta_6 + \theta_2 \theta_5 \theta_6+ 2 \theta_3 \theta_4 \theta_5 + 2 \theta_3 \theta_4 \theta_6 + 2 \theta_3 \theta_5 \theta_6+ \theta_4 \theta_5 \theta_6
0 \geq \nabla^2\log F & = \frac{\nabla^2 h_1}{h_1}-\frac{\nabla h_1\otimes \nabla h_1}{h_1^2}-\frac{\nabla^2 h_2}{h_2}+\frac{\nabla h_2\otimes \nabla h_2}{h_2^2}&=\frac{\nabla^2 h_1}{h_1}-\frac{\nabla^2 h_2}{h_2}.
\frac{\partial^{n}I(\rho)}{\partial\rho^{n}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\partial^{2n}g(y_{1},y_{2})}{\partial y_{1}^{n}\partial y_{2}^{n}}p(y_{1},y_{2})dy_{1}dy_{2}=E(\frac{\partial^{2n}g(y_{1},y_{2})}{\partial y_{1}^{n}\partial y_{2}^{n}})
\left( H\otimes \Delta \right) \Delta \left( h\right) =\phi \cdot \left(\Delta \otimes H\right) \Delta \left( h\right) \cdot \phi^{-1}, \left( \varepsilon \otimes H\right) \Delta \left( h\right) =\lambda^{-1}h\lambda ,\left( H\otimes \varepsilon \right) \Delta \left(h\right) =\rho^{-1}h\rho .
\begin{array}{c}{{\displaystyle{\sum_{\nu = 3 / 2}^{\infty}\nu^{- ( s - k - 1 )}}\ \Re \left\{\displaystyle{\frac{- e^{\displaystyle{- \imath \frac{\pi}{2}( s + 1 )}}}{\pi}}\, \displaystyle{\int_{- \imath z}^{\infty - \imath z}}t^{- s}\, \displaystyle{D_{T E}^{( k )}( t )}\d t \right\},}}\\ \end{array}
\|\widehat{q_2}f\|_{L^2 (\mathbb{R}^2)}^2 = \lambda^2 \|x_2 f\|_{L^2 (\mathbb{R}^2)}^2 + \frac{\theta^2}{4 \lambda^2}\| \xi_1 \widetilde{f}\|_{L^2 (\mathbb{R}^2)}^2.
G_0=\prod_{l=1}^n d_lG_i=\sum_{1\leq j_1<\ldots<j_i\leq n}\prod_{k\in\left\{j_1,\ldots,j_i\right\}}c_k\prod_{\substack{l\in\left\{1,\ldots,n\right\}l\notin\left\{j_1,\ldots,j_i\right\}}}d_l.
J= \sum_{n,m\geq0}\sum_{\substack{\mathfrak{a}\in \mathfrak{g}^{\otimes n}\mathfrak{b}\in \mathfrak{g}^{\otimes m}}}\sum_{a,b\in \mathfrak{g}}(\mathfrak{a}\otimes (a\otimes b-\varepsilon(a,b)b\otimes a)\otimes \mathfrak{b}- \alpha_{T}(\mathfrak{a})\otimes [a,b]_{\mathfrak{g}}\otimes \alpha_{T}(\mathfrak{b})).
[\begin{matrix}1&0&0&0\\ 0&2&0&0\\ 0&0&3&0\\ 0&0&0&4\\ 0&0&0&0\end{matrix}][\begin{matrix}0\\ 1\\ 0\\ 0\end{matrix}]=2[\begin{matrix}0\\ 1\\ 0\\ 0\\ 0\end{matrix}]
\begin{array}{ccccc}A_{1, s}\zeta(s)+A_{1, s-1}\zeta(s-1) +\dots+A_{1, 3}\zeta(3) = A_{1, 2}\zeta(2)+ A_1 +\theta_s A_{2, s-1}\zeta(s-1) +\dots+A_{2, 3}\zeta(3) = A_{2, 2}\zeta(2)+ A_2+\theta_{s-1}\dots\dots\dots\dots \dots\dots\dots\dots \dots\dots\dots\dots \dots\dots\dots\dots &A_{s-2, 3}\zeta(3) = A_{s-2, 2}\zeta(2)+ A_{s-2}+\theta_{3}\end{array},
\sum_{\substack{0<\gamma,\gamma'\le T 0<|\gamma-\gamma'|\le \frac{2\pi\lambda}{\log T}}}\!\!\!\!\!\!\!\! 1 \ = \ 2 \sum_{0<\gamma \le T}n(\gamma,\lambda) \ = \ 2 \sum_{0<\gamma_d \le T}m(\gamma_d) \, n(\gamma_d,\lambda) .
\lim_{n\rightarrow \infty}\int_{\mathbb{T}^d}\int_{\mathcal{Y}^d}f (w_n (x,y ) )\phi (x,y )dydx=\int_{\mathbb{T}^d}\int_{\mathcal{Y}^d}f (w (x,y ) )\phi (x,y )dydx.
\tilde\mu_2=\frac{\mu_2 (\mu_1^2-\mu_1\mu_2+\mu_2^2 )\,\mathbb E [(\xi_i-\eta_i)^3 ]}{3 (\mu_1^3-\mu_1^2\mu_2+\mu_1\mu_2^2-\mu_2^3 )\,\mathbb E [(\xi_i-\eta_i)^2 ]}.
{}0 \leq\displaystyle\sum_{n=1}^{\infty}(\widetilde{w}_{n}-w_{n})|A_{n}|^{2}\leq\displaystyle\sum_{n=2}^{\infty}\Big|\sqrt{\frac{g_{n-1}}{g_{n}}}\frac{A_{n}}{\sqrt\lambda_{n}}-\sqrt{\frac{g_{n}}{g_{n-1}}}\frac{A_{n-1}}{\sqrt\lambda_{n}}\Big|^{2}
d s_{5}^{2}= \Big [ 1 - ({\frac{\mu}{r}})^{2}\Big ]^{\frac{1}{3}}\Big [ - ( 1 - ({\frac{\mu}{r}})^{2}) d t^{2}+ ( 1 - ({\frac{\mu}{r}})^{2})^{- 2}d r^{2}+ r^{2}d \Omega_{3}^{2}\Big ] .
\begin{array}{c}{{N_{A}= \frac 1 4 \varepsilon_{A B C D E}p^{B}\overline{{{p}}}^{C}q^{D}\overline{{{q}}}^{E}\ ,}}\\{{p^{A}N_{A}= 0 \quad , \qquad N^{A}N_{A}= 1 .}}\\ \end{array}
\sum_b\Big\{\frac{\partial A^i_a}{\partial v^k_b}+\frac{1}{c^2}(x^k_a-x^k_b)v^j_b\,\frac{\partial A^i_a}{\partial x^j_b}+\frac{1}{c^2}(x^k_a-x^k_b)A^j_b\,\frac{\partial A^i_a}{\partial v^j_b}-\frac{1}{c^2}v^k_bv^j_b\,\frac{\partial A^i_a}{\partial v^j_b}\Big\}&+\frac{2}{c^2}v^k_aA^i_a+\frac{1}{c^2}A^k_av^i_a=0.
\sum_{k=1}^{n-1}(4^k+4^{n-k}) \zeta(2k,2n-2k)=\left(n+\frac43+\frac{4^n}6\right)\zeta(2n), \sum_{k=2}^{n-2}(2k-1)(2n-2k-1) \zeta(2k,2n-2k)=\frac34 (n-3)\zeta(2n).
\varphi_n(g)(p_0) =s_n \psi_n(g) s_n^{-1}(p_0)=s_n\psi_n(g)(p_0) &=s_n\psi(\rho_{t^n}(f), h)(p_0)=s_n\psi(\rho_{t^n}(f)) (p_h).
\tilde{L}({\rm D}(e_{0}^{n}u))=\sum_{N\geq2}(-1)^{N}\zeta^{\mathfrak{m}}(N)\tilde{L}({\rm Cut}_{(\{0\}^{N-1};\emptyset)}(e_{0}^{n}u))+\tilde{L}({\rm D}({\rm reg}_{0}(e_{0}^{n}u)))
\int_{\mathbb{D}}\bigg|\frac{d^n}{dz^n}(z^nf(z))\bigg|^{p'}\tau_{[n p']}(z)\,dA(z) =\int_{\mathbb{D}}\left|\int_{\mathbb{D}}f(\zeta)\overline{B_{z}^{\mu_{\nu}^{n}}(\zeta)}\nu(\zeta)\,dA(\zeta)\right|^{p'}\tau_{[n p']}(z)\,dA(z) =\|S^{\nu,\mu^{n}_{\nu}}_{f}(1)\|_{A^{p'}_{\tau_{[np']}}}^{p'}\lesssim\|h_{f}^{\nu}\|_{A^{p}_{\omega}\to A^{1}_{\nu}}^{p'}.
\mathcal{F}_{(N)}=\phi_{(1)}\circ...\circ\phi_{(N)}(\mathcal{F)}, & \mathcal{H}_{(N)}=\bigcup\nolimits_{1\leqslant k\leqslant N}\bigcup\nolimits_{1\leqslant j\leqslant m_{k}}\phi_{(j,k)}\circ\phi_{(k+1)}\circ...\circ\phi_{(N)}(\mathcal{F}),
\langle p_i,p_t\rangle \cap R =\left( \langle p_1,p_i,p_t\rangle\cap \langle p_2,p_i,p_t\rangle\right) \cap R &=\left( \langle p_1,p_i,p_t\rangle\cap R\right) \cap\left( \langle p_2,p_i,p_t\rangle\cap R\right) &=\langle x_{1i},x_{1t}\rangle\cap \langle x_{2i},x_{2t}\rangle=x_{it}.
y^{2}\sim x^{2 r}( x - \beta_{0 1})^{2}\cdots ( x - \beta_{0 , n_{c}- r - 1})^{2}( x - \gamma_{0}) ( x - \kappa_{0}) , \qquad r = 0 , 1 , 2 , \ldots , [ n_{f}/ 2 ] .
\Gamma^{0}= \left( \begin{matrix}{\gamma^{0}}&{0}\\{0}&{- \gamma^{0}}\\ \end{matrix}\right) \ , \Gamma^{1}= \left( \begin{matrix}{\gamma^{1}}&{0}\\{0}&{- \gamma^{1}}\\ \end{matrix}\right) \ , \Gamma^{2}= \left( \begin{matrix}{\gamma^{2}}&{0}\\{0}&{- \gamma^{2}}\\ \end{matrix}\right) \ ,
\Big ( \sum_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\|A^{\frac{s}{2}}\big ( \bar E(t_{n}-t) - B_{n-j}P_{h}\big ) \|^p \, \d s \Big )^{1/p}\leq c(t_n^{\frac{r\alpha}{2}+\gamma-\frac1{p'}}h^{2-s-r}+ t_n^{\max(\eta-1,0)}\tau^\mu),
1 = \sum_{n , n_{1}}\prod_{i = 1}^{n}\sum_{\underline{{{m}}}_{i}}\prod_{j = 1}^{n_{1}}\sum_{\underline{{{l}}}_{j}}\left| \tilde{\phi}^{( n , n_{1})}( \underline{{{m}}}_{i}, \underline{{{l}}}_{j}, \underline{{{K}}}- \sum_{i}\underline{{{m}}}_{i}- \sum_{j}\underline{{{l}}}_{j}) \right|^{2}\, .
\dim\mathbf{U}=\binom{c-b+3}{3}+\binom{c-a+3}{3}-\binom{b-a+3}{3}+\binom{c+a-e+3}{3}&-\binom{a+b-e+3}{3}-\binom{2a-e+3}{3}+\binom{b+c-e+3}{3}-\binom{2b-e+3}{3}+\delta(e,a,b,c)-3.
[ a_{0}, a_{n}] ={\frac{1}{A}}{\frac{1}{n}}a_{0}a_{n}\hbar +{\frac{1}{2 A}}\sum_{k \ne 0 , n}{\frac{\epsilon ( n )}{\sqrt{| k n ( n - k ) |}}}a_{k}a_{n - k}\hbar + O ( \hbar^{2}) , \, n \ne 0 ,
|u_m|+|u_{m-1}|=(u_{m-1}-u_m)+2u_m \leq (u_{m-1}-u_m)+\frac{2}{\ell}&\leq (u_{m-1}-u_m)+2(u_{m-1}-u_m) &=3(u_{m-1}-u_m) &\leq 3|u_m-u_{m-1}|.
{\frac{\lambda_{1}}{\lambda_{2}}}={\frac{X_{1}A_{3}}{A_{3}X_{2}}}, \quad{\frac{\lambda_{2}}{\lambda_{3}}}={\frac{X_{2}A_{1}}{A_{1}X_{3}}}, \quad{\frac{\lambda_{3}}{\lambda_{1}}}={\frac{X_{3}A_{2}}{A_{2}X_{1}}}
\binom{5k + 2}{k}=\frac{(5k+2)!}{(k)!(4k+2)!}= \frac{(p - 2k - 2)!}{(p - 6k - 4)!(4k + 2)!}= (-1)^{4k + 2}\binom{6k + 3}{2k + 1}= \binom{6k + 3}{2k + 1}
A^* A - AA^* = \left({\begin{array}{*{20}c}{- \left|{\lambda_3}\right|^2}{\lambda_3 \left({\overline \lambda_1 - \overline \lambda_2}\right)}{\overline \lambda_3 \left({\lambda_1 - \lambda_2}\right)}{\left|{\lambda_3}\right|^2}\end{array}}\right).
A_{1}( s ) = \frac{\zeta ( 2 s )}{\pi}\frac{R^{2 s - 1}}{c_{2}^{2 s}\, \Gamma ( 1 - s )}\! \int_{0}^{1}d \eta \, ( 1 - \eta^{2})^{- s}\! \int_{0}^{\infty}\! d k \, k^{- 2 s + 1}\left[ \frac{\partial}{\partial k}- \frac{\eta}{k}\frac{\partial}{\partial \eta}\right] D_{1}( k , \eta ) ,
\hbox{\rm (i)}\hskip.05truecm c_1a_1a_1^*d^\dag c_1a_1=c_1a_1, \hbox{\rm (ii)}\hskip.05truecm c_1a_1a_1^*d^\dag =(c_1a_1a_1^*d^\dag)^*, &\hbox{\rm (iii)}\hskip.05truecm c_1a_1-c_1a_1a_1^*d^\dag a_1=0, \hbox{\rm (iv)}\hskip.05truecm c_1a_1a_1^*d^\dag a_2=0.
\frac{y^{n}}{n^{r}( n + p )^{k}}= \int_{0}^{y}\Omega^{r - 1}\int_{0}^{x}\Omega^{k - 1}\Omega_{n}\, x^{p}\frac{d x}{x}= \int_{0}^{y}\Omega^{r - 1}\Omega_{p}\Omega^{k - 1}\Omega_{n}.
\left( \begin{matrix}{a^{\prime \prime}}&{b^{\prime \prime}}\\{c^{\prime \prime}}&{d^{\prime \prime}}\\ \end{matrix}\right) = \left( \begin{matrix}{a}&{b}\\{c}&{d}\\ \end{matrix}\right) + \left( \begin{matrix}{a^{\prime}}&{b^{\prime}}\\{c^{\prime}}&{d^{\prime}}\\ \end{matrix}\right)
[{\cal D}\mu] ={\cal D}\xi{\cal D}\pi_i{\cal D}A^\mu{\cal D}\rho{\cal D}\pi_\rho \prod_j \{\delta[ \Gamma_j (A^0+\xi, A^i, \pi_i, \rho) ] \}\mbox{det}\mid \{\widetilde{\Omega}_i, \Gamma_j \}\mid.
\textbf{b}^{(\ell)}(t_k)=\{b^{(\ell)}_{i_\ell}(t_k)\}\in \mathbb{R}^{n_\ell}b^{(\ell)}_{i_\ell}(t_k)= \int_{\mathbb{R}}\psi^{(\ell)}_{i_\ell}(x_\ell) e^{- t_k^2 x^2_\ell}\mathrm{d}x_\ell,
\aligned P_\nu(u,v)= |\nabla u|^2_2+|\nabla v|^2_2-\int_{\mathbb{R}^N}\bigl[(I_\mu*|u|^{2^*_\mu})|u|^{2^*_\mu}+(I_\mu*|v|^{2^*_\mu})|v|^{2^*_\mu}\bigr] &-\nu(\gamma_p+\gamma_q)\int_{\mathbb{R}^N}(I_\mu*|u|^{p})|v|^{q}=0.\endaligned
\pi_\beta(v_g^*)\lambda_\mu(g)(\xi\otimes \delta_t) =\pi_\beta(v_g^*)(\mu(t^{-1}g^{-1},g)\xi\otimes\delta_{gt}) &=\beta_{t^{-1}g^{-1}}(v_g^*)\mu(t^{-1}g^{-1},g)\xi\otimes\delta_{gt},
P X=\left(\frac{p}{2}X^{i}+ \lambda \frac{\sqrt{\Delta}}{2}Y^{i}-\frac{u(X)}{R}x^{i},\frac{p}{2}Y^{i}+ \lambda \frac{\sqrt{\Delta}}{2}X^{i}-\frac{u(X)}{R}y^{i},\left(\sigma_{p,q}Z^{j}-\frac{u(X)}{R}z^{j}\right)\right),
\|(x_I^{n}- x_I^*) - \mu ([\nabla F(x^n)]_I - [\nabla F(x^*)]_I)\|_2^2 & = \|x_I^{n}- x_I^*\|_2^2 + \mu^2 \|[\nabla F(x^n)]_I - [\nabla F(x^*)]_I\|_2^2 & - 2\mu \langle x_I^{n}- x_I^*, [\nabla F(x^n)]_I - [\nabla F(x^*)]_I \rangle & \leq (1-2\mu \alpha_{F,\varepsilon}+ \mu^2 L^2) \|x_I^n - x_I^*\|_2^2.
[\sum_{\alpha,\beta=1}^n g_{\alpha\beta}^j(\xi_j,t) \frac{\partial}{\partial \xi_j^{\alpha}}\wedge\frac{\partial}{\partial \xi_j^{\beta}},\sum_{\alpha,\beta=1}^n g_{\alpha\beta}^j(\xi_j,t) \frac{\partial}{\partial \xi_j^{\alpha}}\wedge\frac{\partial}{\partial \xi_j^{\beta}}]=0.
G(0;\Psi_1,I_+(\Psi_1)_o,f)\le\int_{\{\Psi<0\}}|\tilde f_0|^2\le\liminf_{j\rightarrow+\infty}\int_{\{\Psi_1<0\}\cap D_j}|\tilde f_j|^2 = \lim_{j\rightarrow+\infty}C_j.
\begin{bmatrix}\!X_1 X_2 X_3 X_4\end{bmatrix}^{-1}\!\!\!=\begin{bmatrix}X^{-1}-X^{-1}X_2X_4^{-1}-X_4^{-1}X_3X^{-1}X_4^{-1}X_3X^{-1}X_2X_4^{-1}+X_4^{-1}\!\end{bmatrix}.
\displaystyle\Lambda(h)=\nu(v)h, K(h)=\mathcal L^{+}(h)-\mathcal L^{\ast}(h), \mathcal L^{\ast}(h)=M [(hM)\ast\phi], begin{align*}2pt] \displaystyle \mathcal L^{+}(h)=\int_{\mathbb R^d\times\mathbb S^{d-1}}\phi(|v-v_{\ast}|) b(\cos\theta)\, [h^{\prime}M_{\ast}^{\prime}+h_{\ast}^{\prime}M^{\prime}] M_{\ast}\, dv_{\ast}d\sigma\,.
\frac{1-b(a)}{1-a}=\frac{b(a+1-a)-b(a)}{1-a}=b'(a)+O(1-a)\mbox{and}\lim_{a\uparrow1}\frac{1-b(a)}{1-a}=\lim_{a\uparrow1}b'(a)=\frac{1}{q}
M_{0}(1+\frac{\gamma-1}{2}M_{0}^{2})^{-\frac{\gamma+1}{2(\gamma-1)}}p_{t0}A_{0}=M_{4}(1+\frac{\gamma-1}{2}M_{4}^{2})^{-\frac{\gamma+1}{2(\gamma-1)}}p_{t4}A_{4}
\begin{aligned}S_{a,b,c,x',y',z'}(\psi_p,\psi_p' ;\tilde{c},w_{G_4}) < p^{-3\ell}(1-p^{-1})^{-3}\times & \sum_{(d,f,u',v',w')\in\mathcal{S}}\#(X_{a,b,c,x',y',z'}^{d,f,u',v',w'}(w_{G_4}\tilde{c})) S_{w_{G_4}}(\theta_{a,b,c,x',y',z'}^{d,f,u',v',w'};\ell),\end{aligned}
m^2_{\alpha}=V_0+m^2_{3/2}+m^2_{3/2}n_{\alpha}\left(1+\frac{1}{3}\tilde{\delta}_{GS}\frac{dP}{dy}\right)^{-2}|1-\frac{1}{3}\tilde{\delta}_{GS}\rho|^2 (T+\bar{T})^2\left|\hat{G}(T,\bar{T})\right|^2
\mathcal{S}(\widetilde A,u)\,:=\,-\frac{1}{2e^2}\|F_{\widetilde A}\|^2 + \frac{1}{2}\|{\rm d}^{\widetilde A}u \|^2 - \frac{e^2 \xi}{2}\|\mu_\tau^\kappa \circ u \|^2 \, ,
\int_{A_{k}}\left\vert \nabla u\right\vert^{p(x)}dx=\int_{A_{k}}h\left(x\right) (u-k)^{+}dx\leq \left\Vert h\right\Vert_{L^{N}(\Omega )}\left\Vert(u-k)^{+}\right\Vert_{L^{N^{\prime}}(A_{k})}.
S(\widehat x,\widehat y)=\frac{1}{(2\pi)^{2n}}\int e^{i(\Psi(\infty,\widehat x,\widehat\eta)-\langle\widehat y,\widehat\eta\rangle)}a(\infty,\widehat x,\widehat\eta)\big(1-\chi(\widehat\eta)\big)\big(1-\alpha(\eta_{2n})\big)d\widehat\eta.
\dfrac{p^{t}_{n}\ln p_{n}}{p^{s}_{n}-1+p^{t}_{n}}=\dfrac{\ln p_{n}}{p_{n}^{s-t}-{p^{-t}_{n}}+1}&<\dfrac{\ln p_{n}}{p_{n}^{s-t}}&<\dfrac{p_{n}-1}{p_{n}^{s-t}}\,\,\,\,(\mbox{use of}\,\ln x<1-x,\,x>0) &<\dfrac{1}{p_{n}^{s-t-1}}&<\dfrac{1}{n^{s-t-1}}\,\,\,\,(\mbox{use of}\,p_{n}>n).
n_{a\beta \gamma}^{b}=2\,d\omega_{a}^{b}(X_{\beta}, X_{\gamma})= X_{\beta}\omega_{a}^{b}(X_{\gamma})- X_{\gamma}\omega_{a}^{b}(X_{\beta}) - \omega_{a}^{b}\big{(}[X_{\beta}, X_{\gamma}] \big{)}.
k_5=\frac{1}{2}+\frac{k_1}{2}-\frac{k_2}{2k_3}+\sqrt{\left[\frac{1}{2}+\frac{k_1}{2}-\frac{k_2}{2k_3}\right]^2-\left[\frac{A}{k_3^2}+\frac{B}{k_3}+C\right]}.
(-1)^{k}G^{k}_m(x) = \int_{0}^{x}z(-1)^{k-1}G^{k-1}_m(z) dm(z) + x \int_{x}^{1}(-1)^{k-1}G^{k-1}_m(z) dm(z) \geq x \int_{x}^{1}(-1)^{k-1}G^{k-1}_m(z) dm(z).
\begin{array}{r l l l l l l}{{\omega_{2}}}&{{=}}&{{\omega_{3}}}&{{=}}&{{\omega_{5}}}&{{=}}&{{1 \; ,}}\\{{\mathrm{a n d}\hspace{1 0 m m}}}&{{}}&{{\omega_{1}^{2}}}&{{=}}&{{\omega_{4}}}&{{=}}&{{- ( q - q^{- 1})^{2}\; ,}}\\ \end{array}
\frac{d\tilde{Y}(\lambda)}{d\lambda}=\left(\begin{matrix}\tilde{A}\tilde{B}\tilde{C}-\tilde{A}\end{matrix}\right)\tilde{Y}(\lambda)=\left(\begin{matrix}\bar{A}+\bar{B}\bar{B}\bar{C}-\bar{B}-2\bar{A}-(\bar{A}+\bar{B})\end{matrix}\right)\tilde{Y}(\lambda),
\sum_{k,r=1}^{n,\nu}\gamma_{irk}\widetilde{\gamma}_{jpr}+\sum_{s,r=1}^{\nu}\widetilde{\gamma}_{irs}\widetilde{\gamma}_{jpr}=\sum_{r=1}^{\nu}\Bigl(\sum_{k=1}^{n}\gamma_{irk}+\sum_{s=1}^{\nu}\widetilde{\gamma}_{irs}\Bigr)\widetilde{\gamma}_{jpr}=\sum_{r=1}^{\nu}\widetilde{\gamma}_{jpr}
h_1(u, v)= (-t^{(k+r)/p}+\pi u)^p+\sum\limits_{i=0}^k\dbinom{k+r}{i+r}(t+\pi v)^{r+i}(-\pi v)^{k-i}= \pi^pu^p+(-t^{(k+r)/p})^p+t^{k+r}+\sum_{j=1}^{k+r}S_{k, r}(j)t^{k+r-j}\pi^jv^j = \pi^pu^p+\sum_{j=1}^{k+r}S_{k, r}(j)t^{k+r-j}\pi^jv^j,
\int \eta_i(x) f(\xi, \sigma^x\omega) d\nu_{\widehat \theta}^N (\xi,\omega) = \int \eta_j(x) e^{\vartheta_i(x/N) - \vartheta_j(x/N)}f(\xi, \omega) d\nu_{\widehat \theta}^N (\xi,\omega) \, .
\begin{array}{rl}\displaystyle \int\frac{d^4k_1}{(2\pi)^4}\int\frac{d^4k_2}{(2\pi)^4}D(k_1)D_m(k_1+k_2)\overline{D_m(k_2+p)}K_1 = \displaystyle 2(4\pi)^{-2}\int\frac{d^4k_2}{(2\pi)^4}\overline{D_m(k_2+p)}&\displaystyle \times\int_0^1 dx~K_0\left(\sqrt{x(m^2+(1-x)k_2^2)\tilde{p}^2}\right)\cos(xk_2\cdot\tilde{p}).\end{array}
I(x) ={\rm P}\int_1^\infty \frac{\coth(\pi x z)}{(z^2-1)^{3/2}}dz = - \frac{1}{2}\oint_{|z-1| = \epsilon}\frac{\coth(\pi x z)}{(z^2-1)^{3/2}}dz + \int_{1+\epsilon}^\infty\frac{\coth(\pi x z)}{(z^2-1)^{3/2}}dz.
|\phi_X(t)|^2 \geq \prod_{j=1}^\infty \left(p^2+(1-p)^2+2(1-p)p(1-\frac{t^2}{2^{2n+1}})\right) = \prod_{j=1}^\infty \left(1-(1-p)p\frac{t^2}{2^{2n}}\right).
\ell_{ji}(x)=\frac{1}{h^2}\begin{cases}\frac12(x-x_{j-2})(x-x_{j-1}), i=j-1, -(x-x_{j-1})(x-x_{j+1}), i=j, \frac12(x-x_{j+1})(x-x_{j+2}), i=j+1, 0, \end{cases}
\delta_{k_1,k_1'}\delta_{k_2,k_2'}=\delta_{n_1(k_1),n_1(k_1')}\delta_{n_2(k_2),n_2(k_2')}= \big\langle \psi_{n_1(k_1')}^{n_2(k_2')},\psi_{n_1(k_1)}^{n_2(k_2)}\big\rangle_{V_1}\big\langle \psi_{n_2(k_2')},\psi_{n_2(k_2)}\big\rangle_{V_2}.
{\frac{5}{4 8 l n ( 1 0 )}}( N_{c}^{2}- 1 ){\cal M}^{2}\leq{\frac{1}{1 6 \pi G_{i n d}}}\leq{\frac{2 5}{1 2 \pi^{2}}}( N_{c}^{2}- 1 ){\cal M}^{2}
\frac{\delta^{2}\mathcal{A}}{\delta x_{\mu}( s ) \delta x_{\mu}( s^{\prime})}= \frac{\delta^{2}\mathcal{A}}{\delta x_{1}( s ) \delta x_{1}( s^{\prime})}+ \frac{\delta^{2}\mathcal{A}}{\delta \vec{\phi}( s ) \delta \vec{\phi}( s^{\prime})}\; .
L^{\rho_+}(\xi_0 n^{2/3},0) \simeq \xi_0 n^{2/3}/(1-\rho_+)=\tfrac{1+\gamma}{\gamma}\xi_0 n^{2/3}+\tfrac{(1+\gamma)^2}{\gamma^2}\xi_0\kappa n^{1/3}+O(1)
\begin{array}{c}{{2 \pi{\cal E}= \int_{M_{3}}d ( \mathrm{t r}( \Phi_{A}H^{A}) ) + \int_{M_{3}}d ( \mathrm{t r}( \Phi_{B}K^{B}) )}}\\{{= \int_{S^{2}}\mathrm{t r}( \Phi_{A}H^{A}) + \int_{S^{2}}\mathrm{t r}( \Phi_{B}K^{B})}}\\ \end{array}
1 =0.95{e}^{\zeta^v(p_2)\sigma^2}(1+\zeta^v(p_2)p_2) &=0.95{e}^{\zeta^v(p_1)\sigma^2}(1+\zeta^v(p_1)p_1) &>0.95{e}^{\zeta^v(p_2)\sigma^2}(1+\zeta^v(p_1)p_1),
\beta'(A,\mathbf{v})(\tilde{A},\mathbf{h}) = \tilde{A}(v_1,v_2,\ldots,v_n) + \sum^n_{i=1}A(v_1,v_2,\ldots,v_{i-1},h_i,v_{i+1},v_{i+2},\ldots,v_n) .
t_{i}^{m}t_{k}^{n}\hat{\Phi}_{m n b d}^{p r a c}S ( t_{p}^{j}) S ( t_{r}^{l}) = S ( t_{b}^{s}) S ( t_{d}^{p}) \hat{\Phi}_{i k s p}^{j l m n}t_{m}^{a}t_{n}^{c}
W_{2^n-1}^2\left(\mu_{K(\gamma)}\right)=W_{2^n}^2\left(\mu_{K(\gamma)}\right)\frac{\mathrm{Cap}\left(\mu_{K(\gamma)}\right)}{a_{2^n}}\geq \frac{\sqrt{2}\mathrm{Cap}\left(\mu_{K(\gamma)}\right)}{a_{2^n}}
\begin{aligned}\int_x^{xD}\frac{\bar{\rho}_A(y)}{y}\mathrm{d}y = \int_x^{xD}\frac{\delta_n}{y^2}\int_0^{\sqrt[n]{n\delta_n^{-1}y}}r^{n-1}K_A(r) \mathrm{d}r \mathrm{d}y & = \int_{\sqrt[n]{n\delta_n^{-1}x}}^{\sqrt[n]{n\delta_n^{-1}xD}}\frac{n^2}{u^{n+1}}\int_0^{u}r^{n-1}K_A(r) \mathrm{d}r \mathrm{d}u.\end{aligned}
\begin{array}{c}t\longmapsto \left\vert \frac{\phi^{\prime}\left( t\right)}{\phi \left( t\right)}\right\vert \underset{t\rightarrow +\infty}{\lim}\left\vert \frac{\phi^{\prime}\left( t\right)}{\phi \left( t\right)}\right\vert =0.\end{array}
S_F(p)\Gamma_\nu (p, k) S_F(p+k) D^{\mu \nu}(k)=\int d^Dx \int d^Dy \,\,e^{i k\cdot x+i p\cdot y}\langle O\vert T\left[\psi(0) A_\mu(x) \bar \psi (y)\right]\vert O\rangle
P^{}_{n}(\lambda) = \frac{\prod_{j=n-r+1}^n (1-1/p^j) \prod_{i=1}^{\lceil (n-r)/2 \rceil}(1-1/p^{2i-1})}{p^{n(\lambda)+|\lambda|}\prod_{i \geq 1}\prod_{j=1}^{\lfloor m_i(\lambda)/2 \rfloor}(1-1/p^{2j})}.
U^{2 , \nu}{\frac{H \rightarrow 0 , \nu \rightarrow \infty}{H \nu = m}}\rightarrow P^{>}( m , 2 ) \qquad \mathrm{o r}\qquad U^{2 , - \nu}{\frac{H \rightarrow 0 , \nu \rightarrow \infty}{H \nu = m}}\rightarrow P^{<}( m , 2 ) .
la_1b_1r + la_2b_1r + la_3s_1^{-1}b_1r = 0, &la_1s_1b_2r + la_2s_1b_2r + la_3b_2r = 0, &la_1\cdot r + la_2 \cdot r + la_3s_1^{-1}\cdot r = 0, &la_1b_1r + la_1s_1b_2r + la_1 \cdot r = 0, &la_2b_1r + la_2s_1b_2r + la_2\cdot r = 0, &la_3s_1^{-1}b_1r + la_3b_2r + la_3s_1^{-1}\cdot r = 0.
\begin{array}{l}\,\,\,\,\,[f_1,f_2]_{*^{(2)}}(\xi)=[f_1,f_2]_{*_0}(\xi)+ +\hbar^2c_n\biggl(f_1(\xi)(1-{1\over2}\overleftarrow\partial\!\!_\alpha\xi^\alpha)\varepsilon^{[\beta]_n}[\partial_\beta]^nf_2(\xi)-f_1(\xi)[\overleftarrow\partial\!\!_\alpha]^n\varepsilon^{[\alpha]_n}(1-{1\over2}\xi^\beta\partial_\beta)f_2(\xi)\biggr).\end{array}
\sum_{j=1}^{\infty}\frac{1}{\mu_{j}^{p}(\Omega)}=\int_{\Omega}...\int_{\Omega}\frac{1}{2\pi}\ln\frac{1}{|y_{1}-y_{2}|}...\frac{1}{2\pi}\ln\frac{1}{|y_{p}-y_{1}|}dy_{1}...dy_{p}.
T(\phi) = i\pi \int_{\mathbb{R}^4}\mathcal{F}_1^2\mathcal{F}_2{\phi}(x,-\eta) \delta({x\cdot \eta})\mathop{\textup{sgn}}(\det(x \,\eta)) d\eta dx & = i\pi \int_{\mathbb{R}^4}\mathcal{F}_1 \widehat{\phi}(x,\eta) \delta({\eta \cdot x})\mathop{\textup{sgn}}(\det(\eta \,x)) dx d\eta.
F_{n}^{N}(z)=-\frac{1}{\sqrt{2\omega}}\int_{z}^{+\infty}d\zeta \zeta^{N}\big( L_{n}^{N}(\zeta) \big)^{2}\exp ( -\zeta ) =\frac{1}{\sqrt{2\omega}}\exp (-z) Q_{n}^{N}(z),
F ( T ) = - \Delta n^{2}\frac{3}{1 2 8}\frac{T}{R}\left( \alpha + \ln \frac{R T}{\hbar c}+ \frac{8 5 7}{3 4 5 6 0}\frac{c^{2}\hbar^{2}}{T^{2}R^{2}}\right) +{\cal O}( T^{- 2}) \,{,}
\displaystyle \int_{0}^{T}\Phi_{t}\circ dW_{t}=\displaystyle \mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{i=0}^{n-1}\frac{\Phi_{t_{i+1}}+\Phi_{t_{i}}}{2}(B_{t_{i+1}}-B_{t_{i}})
\gamma M_{r}^{2}(p_{j}^{n+1}-p_{j}^{n})-\rho_{j}^{n}(T_{j}^{n+1}-T_{j}^{n})+\lambda T_{j}^{n}(s^{n}\rho^{n}u^{n}+(1-s^{n})m^{n+1})|_{j-1/2}^{j+1/2}=0
\left\{x \in \mathcal{D}\colon \left(\tfrac{f}{\varphi}\right)(x) \ge \alpha \right\}= \left\{x \in \mathcal{D}\colon f(x) \ge \alpha \varphi(x) \right\}= \left\{x \in \mathcal{D}\colon F_{\alpha}(x) \ge 0 \right\}.
w_{k+n}(\cos(\alpha_{1}),...,\cos(\alpha_{k+n})|\rho)= \prod_{i_{1}\in\{-1,1\}}...\prod_{i_{k+n}\in\{-1,1\}}(1-2\rho\cos(\sum_{s=1}^{n+k}i_{s}\alpha_{s})+\rho^{2}).
\int_0^{2\pi}\frac{dp_2}{2\pi}\;(2\cos p_2)^m\;e^{inp_2}=\sum_{p+q=m}\frac{m!}{p!q!}\;\delta_{n,p-q}=\frac{m!}{\big(\tfrac{m+n}{2}\big)!\big(\tfrac{m-n}{2}\big)!}\; .
\mathbb{LF}_{(W,\mathcal{S})}=\bigcup_{t\in I}\mathbb{LF}^+_{\mathcal{S}_t}\times \{t\}\cup \bigcup_{t\in I}\mathbb{LF}^-_{\mathcal{S}_t}\times \{t\}\subset AdS^{n+1}\times I.
\omega^{(2)}_{r}=48 \rho^3 dt_0\int_{S^2_r}{\rho dx_0^i+d\rho\,(x-x_0)^i\over(t_0^2+\rho^2+ ({\bf x}-{\bf x}_0)^2)^4}\varepsilon_{ijk}dx^jdx^k.
a^\prime =-\frac{a_0^2}{2 b_0c_0}+ \frac{b_0}{2c_0}+ \frac{c_0}{2b_0}= : A_0 , b^\prime =\frac 12 \frac{a_0}{c_0}-\frac 12 \frac{b_0^2}{a_0c_0}+\frac 12 \frac{c_0}{a_0}= : B_0\,, c^\prime =\frac 12 \frac{a_0}{b_0}+\frac 12 \frac{b_0}{a_0}-\frac 12 \frac{c_0^2}{a_0b_0}=: C_0\,,
\Sigma_{+}\ : \ - 1 + \tilde{\epsilon}\left( \tilde{x}^{3}- \frac{3}{4}\tilde{u}_{0}^{4}\tilde{x}- \frac{1}{4}( \tilde{u}_{0}^{6}+ \tilde{u}_{1}) - \frac{1}{4}( \zeta + \frac{1}{\zeta}) \right) +{\cal O}( \tilde{\epsilon}^{2}) = 0 \ ,
\tilde{\textbf{U}}^n+\tau^\alpha\textbf{K}\tilde{\textbf{U}}^n=-\sum_{k=1}^{n-1}\omega^\alpha_k\tilde{\textbf{U}}^{n-k}+\sum_{k=0}^{n-1}\omega^\alpha_k\tilde{\textbf{U}}^0+\tau^\alpha\textbf{G}^n.
\rho_-(t) = \sup \{r: B_r(y) \textrm{is enclosed by}\hat{M}(t) \textrm{for some}y \in \mathbb{H}^{n+1}\}, \rho_+(t) = \inf \{r: B_r(y) \textrm{encloses}\hat{M}(t) \textrm{for some}y \in \mathbb{H}^{n+1}\}.
\|u\|_p^\gamma=2^{\tau\gamma}\sup_{|h|>2}\left(\frac{\|u\|_p}{|h|^\tau}\right)^\gamma=\frac{2^{\tau\gamma}}{4^{\gamma/p}}\left(\sup_{|h|>2}\left\|\frac{\delta_{h}^2u}{|h|^\tau}\right\|_p \right)^\gamma\leq \frac{2^{\tau\gamma}}{4^{\gamma/p}}[u]_{B^{\tau}_{p,\infty}}^\gamma.
0 =\sum_{s|r}\pi_{r,r/\ell_1}\pi_{r,s}^*(z_s) &=\ell_1\sum_{s|(r/\ell_1)}\pi_{r/\ell_1,s}(z_s)+\sum_{s\nmid(r/\ell_1)}\pi_{r/\ell_1,s/\ell_1}^*\pi_{s,s/\ell_1}(z_s) &=\ell_1\sum_{s|(r/\ell_1)}\pi_{r/\ell_1,s}(z_s)
\frac{{\cal E}_{\mathrm{H F}}}{N}= \frac{k_{\mathrm{F}}^{2}}{2 \pi}+ \frac{\tilde{S}_{1}^{2}}{4 \pi}\left[ \ln ( 1 6 k_{\mathrm{F}}^{2}\tilde{S}_{1}^{2}) - 1 \right] \ ,
\liminf \frac{c_{n}(\log(c_{n}))^{\epsilon}}{h_{n}}\geq\liminf \frac{(d(n-1)^{1+\alpha})^{\epsilon}}{n^{1+\epsilon}}\geq \liminf \frac{d^{\epsilon}(n-1)^{\epsilon + \alpha\epsilon}}{n^{1+\epsilon}}& \geq \liminf \frac{d^{\epsilon}(n-1)^{1+2\epsilon}}{n^{1+\epsilon}}= \liminf d^{\epsilon}\Big{(}1- \frac{1}{n}\Big{)}^{1+\epsilon}(n-1)^{\epsilon}= \infty.