harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_{x_1}\Big[E_{x_2}\Big[F\Big(\mathcal Z_{h_1}(x_1,\cdot)+ \mathcal Z_{h_2}(x_2,\cdot)\Big)\Big]\Big] \stackrel{*}{=}E_x\big[F\big(\mathcal Z_{\mathbf{s}(h_1,h_2)}(x,\cdot)\big)\big],
\begin{array}{rl}W = x^{a}, W = x_{1}^{a_{1}}x_{2}+ x_{2}^{a_{2}}x_{3}+ \dots + x_{n}^{a_{n}}x_{1}, W = x_{1}^{a_{1}}x_{2}+ x_{2}^{a_{2}}x_{3}+ \dots + x_{n}^{a_{n}}.\end{array}
\theta_i^2\frac{Z_i^*(u)}{e_i^*(u)}+\frac{T_i^{\infty}}{e_i^*(u)^2}=\theta_i^2 \frac{T_i^{\infty}-T_i(u)}{e^{2\rho_i(u)}}&=\theta_i^2\int_0^{+\infty}e^{2\rho_i(u+v)-2\rho_i(u)}dv.
\dim_K(H^{d}(X_{\Gamma},{\mathcal{M}}^{*,{\Gamma}}\otimes_K\Omega^{\bullet}_{X_{\Gamma}})) =\dim_K(H^{d}(X_{\Gamma},{\mathcal{M}}^{{\Gamma}}\otimes_K\Omega^{\bullet}_{{X}_{\Gamma}})){}=\dim_K(\check{F}_{red}^{d-i})+\dim_K(F_{red}^{i+1}).
0 = g a_0 + a_0^3 + 2 a_0 \Sigma_2 + 2 \Sigma_2 a_0 + \sum_{n > 0}{1\over n}\left( a_n^\dagger a_0 a_n + a_n a_0 a_n^\dagger - a_0 \right)+6 \Sigma_3 \; .
s =\int_{\{g(x)\geq h\}}f(x)\sqrt{1+(\nabla g(x))^2}\,dx &=\int_{\partial B_{n-1}}\int_0^{\sqrt{1-h^2}}f(ru)\sqrt{1+(\nabla g(ru))^2}\,r^{n-2}\,dr\,\mu_{\partial B_n}(du).
\vartheta_{\mathcal L}^{(\ell)}(z) = \sum_{s=0}^{n-\ell}2^s\binom{\ell +s}{s}z^{sq}\sum_{r=0}^{q-1}z^r\left( \frac{1}{(1-z^q)^{n-\ell}}\sum_{h\geq0}N_{\mathcal L}^{\mathrm{red}}(hq+r,\ell +s) z^{hq}\right) &= \frac{1}{(1-z^q)^{n-\ell}}\sum_{s=0}^{n-\ell}2^s\binom{\ell +s}{s}z^{sq}\sum_{k\geq0}N_{\mathcal L}^{\mathrm{red}}(k,\ell +s) z^{k},
\left\|y_n^k\partial_{y_n}^{k'}w_n^l\partial_{w_n}^{l'}\partial_{\xi'}^\alpha\partial_{x}^\beta q_{z}(x,\xi',\cdot,\cdot)\right\|_{X_2}=\left\|y_n^k\partial_{y_n}^{k'}w_n^l\partial_{w_n}^{l'}\partial_{\xi'}^\alpha\partial_{x}^\beta \left(h(z,\xi',\cdot,\cdot)\right)\right\|_{X_2}\leq C\langle\xi'\rangle^{m+1-k+k'-l+l'-|\alpha|}
d_{n, i}= \frac{2}{\sqrt{n}}\begin{cases}\lfloor nt \rfloor - \lfloor ns \rfloor i \le \lfloor ns \rfloor - 1 \lfloor nt \rfloor - i + 1 \lfloor ns \rfloor - 1 < i \le \lfloor nt \rfloor - 1 0 \end{cases}~,
f_{\sigma}^{(j)}(\kappa_i) = f_{\sigma}^{(j)}( \sigma(\lambda_i)) = \sum_{k=j}^n (k)_j \sigma( c_k) \sigma(\lambda_i)^{k-j}& = \sigma\left( \sum_{k=j}^n (k)_j c_k \lambda_i^{k-j}\right) = \sigma\left( f^{(j)}(\lambda_i) \right).
NT^{KN(KN+KM)}+\min\{(K-1)N,M\}(T+1)^{KN(KN+KM)}+MN(T_{\sf d}+1)^{KM(KN+KM)}&=NT^{KN(KN+KM)}+\min\{(K-1)N,M\}(T+1)^{KN(KN+KM)}+N\frac{\lambda_2}{\lambda_1}T^{KN(KN+KM)}&\leq \frac{M}{\lambda_1}(T+1)^{KN(KM+KN)},
\alpha\big((\ell_E)_{E\in \mathbf{E}}, (n_E)_{E\in \mathbf{E}}\big)={}\big((n_E)_{E\in \mathbf{E}},(0)_{1\le i\le r}, ((-1)^{\delta_{v,v'_E}}\ell_E)_{v\in E\in \mathbf{E}}\big), \beta\big((n_E)_{E\in \mathbf{E}}\big)={}\big((n_E)_{v\in E\in \mathbf{E}}\big).
\Psi_{16}\rightarrow exp\left(i\frac{\alpha_a}{2}\Sigma^a(1 -\Gamma_{11})\right)\Psi_{16},\quad C_{10}\Psi_{\overline{16}}\rightarrow exp\left(-i\frac{\alpha_a}{2}\left[\Sigma^a(1-\Gamma_{11})\right]^*\right)C_{10}\Psi_{\overline{16}}
\mathrm{d}_{\partial\mathcal{M}^{\mathcal{B}}_{\infty}}(A^{+}_{\infty},B^{+}_{\infty})={\lim}_{n\rightarrow\infty}\mathrm{d}_{\partial\mathcal{M}^{\mathcal{B}}_{n}}(A^{+}_{n},B^{+}_{n}).
\left(f_0,\frac{\xi_1}{\xi_2},\frac{\xi_2}{\xi_3},\frac{\xi_3}{\xi_4},\frac{\xi_4}{\xi_5},\frac{\xi_5}{\xi_6},f_6\right)=(0,1,\omega^{-1},1,\omega^{-1},1,0).
R(X,Y)Z = \frac{5}{12}\left(g(Y,Z)X - g(X,Z)Y\right) + \frac{1}{3}\left(g(AY,Z)AX - g(AX,Z)AY + g(BY,Z)BX - g(BX,Z)BY\right) +[S_{JX},S_{JY}] Z
n^{-2}\sum_{k=1}^{K_n}\sum_{\ell=0}^{L_n-1}\left( c^{(1)}_{\ell}\right)^m n^{-2}\sum_{k=1}^{K_n}k^{-1}\left( \sum_{j=1}^k c^{(1)}_{k+L_n-j}\right)^m
\begin{aligned}L_3(q) = \sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{8n^2 - n +1 - 4j^2 - j}+ q^{8n^2 + n +1 - 4j^2-j}& + \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{8n^2 + 7n + 2 - 4j^2 - 3j}+ q^{8n^2 + 9n + 3 - 4j^2 -3 j}\end{aligned}
\tau(L_1,L_2,L_3)= \mu_c^{+}(Y_1,Y_2,Y_3)-\mu_c^{-}(Y_1,Y_2,Y_3)=\nu_c^{+}(Y_1,Y_2,Y_3)-\nu_c^{-}(Y_1,Y_2,Y_3).
\log \frak{H}(\omega; \mathbb{A}) = \log \frak{A}(\omega; \mathbb{A}^{-1})^{-1}& = - \log \frak{A}(\omega; \mathbb{A}^{-1}) & \leq- \log \frak{G}_{E}(\omega; \mathbb{A}^{-1}) & =\log \frak{G}_{E}(\omega; \mathbb{A}).
\mathrm{exp}{\left( \frac{2\pi i t}{8L}\right)}\cdot \xi^{(l, \pm)}(\Phi_t(y)) = \mathrm{exp}{\left( \frac{2\pi i 0}{8L}\right)}\cdot \xi^{(l, \pm)}(\Phi_0(y)) = \xi^{(l, \pm)}(y)
G\bigl(z,\rho(z,\bar{z})\bigr) =A\Bigl(F\bigl(z,\rho(z,\bar{z})\bigr),\bar{F}\bigl(\bar{z},\bar{\rho}(\bar{z},z)\bigr)\Bigr) +B\Bigl(\bar{F}\bigl(\bar{z},\bar{\rho}(\bar{z},z)\bigr),\bar{F}\bigl(\bar{z},\bar{\rho}(\bar{z},z)\bigr)\Bigr) + O(3)
\begin{array}{r l}\Delta_g u+hu = fu^{q-1}+\frac{\rho_1+\vert\Psi+\rho_2\mathcal{L}_gW\vert^2_g}{u^{q+1}}\quad-\frac{b}{u}-c\langle\nabla u,Y\rangle \left(\frac{d}{u^2}+\frac{1}{u^{q+2}}\right)-\frac{\langle\nabla u,Y\rangle^2}{u^{q+3}}\end{array}
\begin{cases}u_t-u_{5x}+u_{3x}+u_x+uu_x=f(x)v(t), \ \ (0,T)\times (0,2\pi), \partial_x^{j}u(t,0)=\partial_x^{j}u (t,2\pi), \ \ \ (0,T), u(0,x)=u_0(x), \ \ \ (0,2\pi),\end{cases}
\hat{\Theta}^{0 0}\, = \, - \hat{\Theta}^{0 1}\, = \, - \hat{\Theta}^{1 0}\, = \, \hat{\Theta}^{1 1}\, = \, \frac{1}{2}\, : ( \partial_{1}\hat{\phi}) ( \partial_{1}\hat{\phi}) : \, \, .
\widetilde{\Delta}\circ \widetilde{\varphi}(x,m) =\widetilde{\Delta}(\varphi_L(x),\varphi_M(m)) &=\Delta(\varphi_L(x))+\rho_{\beta}(\varphi_M(m))- \tau \circ (\rho_{\beta}(\varphi_M(m))) .
\frac{1}{\sqrt{m}}E\left[J_{s,t}^m\right] = -\int_s^t E\left[(-\nabla_q V(r,q_r)-\partial_r \psi(r,q_r)+\tilde F(r,q_r))\cdot \left(\int (\nabla_z\chi)(r,q_r,z) h(r,q_r,z)dz\right)\right]dr &-\int_s^tE\left[ \int \left((\nabla_q\chi)(r,q_r,z )\cdot z\right) h(r,q_r,z)dz \right]dr+O(m^{1/2}),
\|\phi^{}_{{\rho,\varepsilon}}\|_\omega\leq \Big\|\frac{z\overline{a}e^{-i\varepsilon}-1}{z\overline{a}e^{-i\varepsilon}-1-\rho}\Big\|_\omega\Big\|\frac{z\overline{b}e^{i\varepsilon}-1}{z\overline{b}e^{i\varepsilon}-1-\rho}\Big\|_\omega= \Big\|\frac{z-1}{z-1-\rho}\Big\|_\omega^2.
\begin{array}{c c c c c}{{L_{1}}}&{{L_{1 / 2}}}&{{L_{0}}}&{{L_{- 1 / 2}}}&{{L_{- 1}}}\\{{Z_{r s}}}&{{Q_{r}}}&{{R_{r s}}}&{{S_{r}}}&{{\widetilde{Z}_{r s}}}\\ \end{array}\, ,
\Bigl(\mathbb E \Bigl\| \sum^n_{j=1}\varepsilon_j d_j\Bigr\|^p\Bigr )^{\frac 1p}\leq \beta \Bigl(\mathbb E \Bigl \| \sum^n_{j=1}d_j\Bigr\|^p\Bigr )^{\frac 1p}.
\Omega_jf=\lim_{\mbox{\scriptsize supp}\psi_a\downarrow\mathcal{T}_a}\lim_{t\to\infty}e^{itH}\varphi(H^a)(\psi_{aj}(H^a)-P^a_j)P_a^{{\varepsilon}_a}(t)e^{-itH}f.
E_{(x,T)}\left[f\left(X_{\sigma+\tau}^{(x,T)}\right)\left|\right.\Sigma^\sigma\right]=E_{\left(X^{(x,T)}_{\sigma},T-\sigma\right)}\left[f\left(X^{\left({X^{(x,T)}_{\sigma}},T-\sigma\right)}_\tau\right)\right]
D_n(x)-T_n(x)=D^+_{n+1}(x)-T^+_{n+1}(x)=\frac{1}{x}\left(T^-_{n+2}(x)-D^-_{n+2}(x)\right)=x^{\frac{n+1}{2}}U_{n+1}\left(\frac{1}{2\sqrt{x}}\right).
I(\phi)\;=\;\frac{1}{2}\int_{\cal G}d^{d+1}y\,\partial_{\mu}\,(\sqrt{g}\,g^{\mu\nu}\,\phi\,\partial_{\nu}\phi)\;=\;\lim_{\epsilon\rightarrow 0}\;\frac{1}{2}\int_{\bar{\cal G}\cap T_{\epsilon}}d^{d}{\bf y}\,\sqrt{h}\,\phi\,{\bf n}\cdot\nabla\,\phi
\int_{B}|f_0 (w_\delta)-f_0(W_\delta)|^\frac{(p+1)(1+\gamma)}{p}=\begin{cases}O(\delta^{\frac{p+1}{p}}|\log \delta|^\frac{1}{3}) n=4,\ \gamma=0 O(\delta^{\frac{p+1}{p}}) n\geq 5,\ \gamma=0, O\left(\delta^{\frac{p+1}{p}\left(1-\frac{n\gamma}{2}\right)}\right) n\geq 4,\ \gamma>0.\end{cases}
F_3= 1+ \sum_{k=1}^{s-1}\frac{(-1)^{k}\|x\|^{2k}}{k!\prod_{i=1}^k(2i-n)}\varrho^k - c_n(-1)^{s}\|x\|^n\varrho^{s}\left( \sum_{k=1}^\infty\frac{(-1)^{k}\|x\|^{2k}}{k!\prod_{i=1}^k(2i+n)}\hat{q}_k \varrho^k\right).
I\left(f_{\omega_{\pi_\nu},s}^\circ,v^\circ_{\pi_{\nu}}\right)(I_m)=\int \limits_{Z_n(k_\nu)\backslash T^-}\mu(\underline{t})\left|\det \underline{t}\right|^{ms+\frac{m}{2}}\int\limits_{K_{n,\nu}}\pi_{\nu}(A\underline{t})v^\circ_{\pi_{\nu}}dA d\underline{t}.
Z := \int{\cal D}A_{\mu\nu}{\cal D}\Lambda_\mu{\cal D}C_\mu{\cal D}\bar C_\mu{\cal D}d{\cal D}\bar d{\cal D}N{\cal D}P{\cal D}B^{(1)}\delta(\partial^\nu A_{\mu\nu}-\partial_\mu N)\exp \left[ i \int d^4x{\cal L}_{tot}' \right] ,
Z_{n+1,x}= \begin{cases}\displaystyle \frac{X_{n+1,x}}{W_{n,x}}\geq \frac{X_{n+1,x}}{X_n}, \mbox{if}W_{n,x}\geq t^{\ast}, \displaystyle \frac{X_{n+1,x}- t^{\ast}}{W_{n,x}+t^{\ast}}+ \frac{t^{\ast}}{\Tilde{t}}\geq \frac{X_{n+1,x}- t^{\ast}}{\Tilde{t}}+ \frac{t^{\ast}}{\Tilde{t}}\geq \frac{X_{n+1,x}}{X_n}, \mbox{if}W_{n,x}< t^{\ast}.\end{cases}
t^\lambda_+ = (\lambda+1)^{-1}\partial_t t_+^{\lambda+1}&=(\lambda+1)^{-1}\partial_t\left\{Y(t) + \sum_{j=1}^\infty \frac{1}{j!}(\lambda+1)^j(\log t_+)^j\right\}&=(\lambda+1)^{-1}\delta(t) + \sum_{j=1}^\infty \frac{1}{j!}(\lambda+1)^{j-1}\partial_t(\log t_+)^j,
\frac{1}{2\pi i}\int_{b-i \tau}^{b+i \tau}\left(\sum_{n\le x}\left(\frac{\Phi_k(n)}{n^\beta}\right)^{-z}\right)\frac{y^z}{z}\,dz = \frac{x}{2\pi i}\int_{b-i \tau}^{b+i \tau}R_{k, \beta}(z) \frac{(\alpha x^\delta)^{z}}{z(1-z+\delta z)}dz & +O\left(x^{1-\varepsilon+\delta \varepsilon}y^{\varepsilon}\exp{(- \sqrt{f_k^\prime \log x \log\log x}})\right).
0 = \mathcal L_\partial(c_S) = \mathcal L_\partial(\theta_*(\omega_S)) = \theta_*(\underset{=\mathrm{div}(\partial) \omega_S}{\underbrace{\mathcal L_\partial(\omega_S)}}) + \lambda \theta_*(\omega_S) = (\lambda+\mathrm{div}(\partial)) c_S\,.
\Pi_{\left(\frac{1}{1-\left|\frac{1}{\mathbf{p}}\right|_{\leq s}};p_1^* ,\ldots,p_s^*,1\right)}^{s+1}(E_1,\ldots,E_{s+1};\mathbb{K})=\mathcal{L}(E_1,\ldots,E_{s+1};\mathbb{K})
\omega(k) = \Big( T(k)\varphi_k, \varphi_k \Big)= \Big(T_0\varphi_k, \varphi_k \Big) +\tfrac{1}{\ln k}\Big (T_1 \varphi_k,\varphi_k \Big) + O\Big(\tfrac{1}{\ln^2k}\Big),k \rightarrow 0+.
\begin{array}{rclcc}F = -2 \, \frac{k}{\sqrt{a^2+1}}\, \frac{1}{r^3}\left(1+\frac{k}{r}\right)^{-2}dt \, \wedge \,{\vec x}\, \cdot \,{\vec x}; \mbox{electric $0$-brane}{F}= - 2 \, \frac{k}{\sqrt{a^2+1}}\, \frac{1}{r^3}\, \epsilon_{abc}\, x^a \, dx^b \, \wedge \, dx^c ; \mbox{magnetic solitonic $0$-brane}\end{array}
\gamma =\{c_1,\dots,c_r\}\,,\, \mathbf{c}=(c_1,\dots,c_r)\in[1,n_1]_<^r, \gamma' =\{c'_1,\dots,c'_s\}\,,\, \mathbf{c'}=(c'_1,\dots,c'_r)\in[1,n_1]_<^s, \delta =\{d_1,\dots,d_r\}\,,\, \mathbf{d}=(d_1,\dots,d_r)\in[n_1+1,n_2]_<^r, \delta =\{d'_1,\dots,d'_t\}\,,\, \mathbf{d'}=(d'_1,\dots,d'_t)\in[n_1+1,n_2]_<^t.
{\rm Alt:}{\mathcal D}^{p,p}(V^*)\longrightarrow \Lambda^{2p}(V^*) &\omega\mapsto{\rm Alt}(\omega)(v_1,...,v_p,v_{p+1},...v_{2p}) &=\frac{1}{(2p)!}\sum_{\sigma\in S_{2p}}\varepsilon(\sigma)\omega(v_{\sigma(1)}\wedge...\wedge v_{\sigma(p)},v_{\sigma(p+1)}\wedge ...\wedge v_{\sigma(2p)}).
\mathcal{C}_{22}:= \left[\frac{2A+1}{\log q}-iq,\frac{2A+1}{\log q}-i\right]\cup \left[\frac{2A+1}{\log q}-i,\frac{1}{4}-i\right] \cup \left[\frac{1}{4}-i,\frac{1}{4}+i\right] \hskip80pt \cup \left[\frac{1}{4}+i,\frac{2A+1}{\log q}+i\right] \cup \left[\frac{2A+1}{\log q}+i,\frac{2A+1}{\log q}+iq\right],
\begin{array}{l}\dot L_{jk}=\frac{1}{2}\coth((q_{j}-q_{k})/2)(\dot q_{k}-\dot q_{j})L_{jk}\quad\quad\quad+\sum_{l\neq j,k}\frac{1}{2}(\coth((q_{j}-q_{l})/2)-\coth((q_{l}-q_{k})/2))L_{jl}L_{lk}\end{array}
f : \{1,\dots, (q^n-1)/(q^k-1)\}\longrightarrow \mathcal G_{q^k}(1,m) i \longmapsto \langle (\underbrace{0, \dots ,0}_{\epsilon(i)},1, \phi_i^{-1}(i- \sum_{j=0}^{\epsilon(i) -1}q^{jk}) )\rangle .
x_{2}=\frac{|\begin{matrix}a_{1}&c_{1}\\a_{3}&c_{2}\end{matrix}|}{|\begin{matrix}a_{1}&a_{2}\\a_{3}&a_{4}\end{matrix}|}=\frac{a_{1}c_{2}-c_{1}a_{3}}{a_{1}a_{4}-a_{2}a_{3}}
\mathbb{E}[M_{ij}]=\begin{cases}i-j \mbox{for}i,j\geq1,i\geq j \frac{1}{a_{i0}}\sum_{x=i}^{n_{i}}x\cdot B\left(n_{i},x,1-\epsilon\right) \mbox{for}i\geq1,j=0.\end{cases}
|A(k,\sigma)| = \int_{B_\sigma \cap \{u > k\}}1\,dx \leq \int_{A(h,\sigma)}\left|\frac{u-h}{k-h}\right|^{p(x)}\,dx= \int_{A(h,\sigma)}\left(\frac{\sigma}{k-h}\right)^{p(x)}\left|\frac{u-h}{\sigma}\right|^{p(x)}\,dx &\leq \frac{\sigma^{p^-}}{(k-h)^{p^+}}\int_{A(h,\sigma)}\left|\frac{u-h}{\sigma}\right|^{p(x)}\,dx.
(\delta f)(a_1, \, a_2, \, \ldots \, , a_{n+1}) = a_1 f(a_2, \, \ldots\, , a_{n+1}) \, + & \Big ( \sum_{i=1}^n (-1)^i f(a_1, \, a_2, \, \ldots \, , a_i a_{i+1}, \,\ldots \, , a_n) \Big) + (-1)^{n+1}f(a_1, \, a_2, \, \ldots \, , a_n)a_{n+1}.
\begin{aligned}\int_{\Omega\setminus\Sigma_r}|h|^p dx = \int_0^r\int_{S_t=\Lambda_{t}(\partial\Omega)}|h(y)|^p dS_t(y)dt = \int_0^r\int_{\partial\Omega}|h(\Lambda_t(z))|^p |\nabla\Lambda_{t}|dS(z)dt \leq Cr\int_{\partial\Omega}|\mathcal{M}(h)|^p dS \leq Cr\int_{\partial\Omega}|(h)^{*}|^p dS,\end{aligned}
D(P_{X}\|Q_{X}) :=\sum_{x\in\mathrm{supp}(P_{X})}P_{X}(x)\log\frac{P_{X}(x)}{Q_{X}(x)}*D_{1+s}(P_{X}\|Q_{X}) :=\frac{1}{s}\log\sum_{x\in\mathrm{supp}(P_{X})}P_{X}(x)^{1+s}Q_{X}(x)^{-s},
T_1 =\frac{\rho a_M}{\ln 2}\int^{\infty}_{0}\frac{1}{1+x\rho a_M}\left(1- \frac{1}{\mathcal{R}_D^M}\underset{k_0+\cdots+k_N=M}{\sum}\right. \left.{M \choose k_0, \cdots,k_N}\left(\prod^{N}_{n=0}b_n^{k_n}\right)e^{-\sum_{n=0}^{N}k_nc_nx}\right) dx,
\sum_{n\le x}n^{\mu-1}= \int_{1-}^xu^{\mu-1}d[u]=\int_{1}^xu^{\mu-1}du-\int_{1-}^xu^{\mu-1}d\rho(u) = \int_{0}^xu^{\mu-1}du+(\mu-1)\int_1^xu^{\mu-2}\rho(u)du+O(1)=\int_{0}^xu^{\mu-1}du+O(1).
\int \mid \alpha , \lambda > < \alpha , \lambda \mid{\frac{1}{2 \pi}}K_{\lambda + \frac{1}{2}}( \mid \alpha \mid ) I_{\lambda + \frac{1}{2}}( \mid \alpha \mid ) d^{2}\alpha = \sum_{m = 0}^{\infty}\mid m , \lambda > < m , \lambda \mid ={\bf 1},
\int_{Q}\int_{Q}\frac{|\Psi_a(x)-\Psi_a(y)|^{p-2}\,(\Psi_a(x) - \Psi_a(y)) - |u_s(x)-u_s(y)|^{p-2}(u_s(x) - u_s (y))}{|x-y|^{N+sp}}&\times(\varphi(x)-\varphi(y))\,dx\,dy\leq C\,[\varphi]_{W^{s,p}(Q)}
a_{p}(t)=-i\int_{-\infty}^{t}(E_{XUV}(t^{\prime})+E_{IR}(t^{\prime}))\cdot d[p+A_{XUV}(t^{\prime})+A_{IR}(t^{\prime})]e^{+i(I_{P}t^{\prime}-S(t,t^{\prime}))}dt^{\prime}(1.6)
\sqrt{\ell}\sum_{\nu\geq \ell/k}\frac{1}{\sqrt{\nu}(\nu+\ell)}\cdot\frac{1}{\nu^2\min(k^2/\ell^2)}= \frac{\ell^{5/2}}{k^2}\sum_{\nu\geq \ell/k}\frac{1}{\nu^{5/2}(\nu+\ell)}=\mathcal{O}\left(\frac{1}{\sqrt{k}}\right).
\sum_{\substack{\mathbf{r}\in\mathcal{D}'_k r_m=1}}\frac{(y_{\mathbf{r}}^{(m)})^2}{\phi_\omega(r)^2}\Bigl(\prod_{p|r}(\phi_L(p)-1)\Bigr)\sum_{A|r}\Bigl(\prod_{p|A}\frac{-1}{\phi_L(p)-1}\Bigr)\sum_{\substack{\mathbf{s}\in\mathcal{D}'_k A(\mathbf{r},\mathbf{s})=A}}1,
\varphi(\gamma^{1/2}, q^{d_1})=\varphi(q^{d_1},\gamma^{1/2})=q^{-1/2},\ \varphi(\psi^-_i(z),q^{d_2})=q^{-1},\ \varphi(q^{d_2},\psi^+_i(z))=q,
\left( D_{0}\right)_{\; \; b}^{a}= \delta_{\; \; b}^{a}\partial_{0}+ f_{b c}^{a}H_{0}^{c}, \; \left( D_{0}\right)_{a}^{\; \; b}= \delta_{a}^{\; \; b}\partial_{0}- f_{a c}^{b}H_{0}^{c}.
\ll \sum_{\substack{D \leq \exp((\log \log T)^2) \gcd(D,Q)=1}}\frac{\bar f_i(D)}{D}\frac{T}{Q}\frac{Q}{\phi(Q)}\frac{1}{\log T}\prod_{\substack{p\leq T p \nmid Q}}\left(1 + \frac{|f_i(p)|}{p}\right).
e^{n^\alpha}-2 \sum_{j=1}^n \alpha j^{\alpha-1}e^{j^\alpha}\varepsilon_j \leq \sum_{j=1}^n\varphi(a_j(x))\leq e^{n^\alpha}-2 \sum_{j=1}^n \alpha j^{\alpha-1}e^{j^\alpha}\varepsilon_j.
S_{q , g}^{e x t}={\frac{1}{4 \sqrt{\pi}}}\int_{\epsilon^{2}/ l^{2}}^{\infty}{\frac{d s}{s^{3 / 2}}}k_{H}( s ) \Theta_{\Sigma_{g}}( s{\frac{l^{2}}{l_{1}^{2}}}) e^{-{s / 4}}~ ~
\frac{f(R_n\rho e_1)}{f(R_ne_1)}\to \rho^{-\alpha}\,, \ \ \prod_{i=1}^{2k-\ell-1}\frac{f \bigl( ||R_n \rho \theta + y_i|| e_1 \bigr)}{f(R_ne_1)}\to \rho^{-\alpha(2k-\ell-1)}\,, \ \ \n \to \infty
\int_a^b e^{F \circ \gamma_{y_1, \ldots, \hat y^i, \ldots, y_n}}\,dt = \int_a^b e^{f \circ \psi}\Big(g_{kl}(\psi) \frac{\partial\psi^k}{\partial y^i}\frac{\partial\psi^l}{\partial y^i}\Big)^{1/2}\Big|_{y = \gamma_{y_1, \ldots, \hat y^i, \ldots, y_n}}\,dt
\begin{array}{l}{\displaystyle\dot{x}= - \frac{1}{2}x^2 + x y + \frac{1}{2N}y^2+ \frac{1}{2}W_1(z)}\vspace{2mm}{\displaystyle\dot{y}= \frac{N}{2}x^2 + \frac{1}{2}y^2- \frac{1}{2}W_2(z)}\vspace{2mm}\dot{z}= - z (y + N x)\end{array}
\int_0^h \left[DE(\eta(t)) + D_2R(\eta(t), \partial_t \eta(t))\right] \langle \varphi(t)\rangle\,dt = \int_0^h \langle \sigma(t),\varphi(t)\rangle \,dt + \int_0^h \langle f(t), \varphi(t) \rangle_{L^2}\,dt
\sup_{f\in M}\|f\|_{Y_q(\tilde \Omega_{[\hat t(n)]})}\stackrel{(\ref{nu2}), (\ref{emb_nu})}{\underset{\mathfrak{Z}_0}{\lesssim}}2^{((1‐\lambda)\mu_*‐\lambda \alpha_*)k_*\hat t(n)}\stackrel{(\ref{mtpr0eq})}{=}n^{‐\hat \nu}.
P_c(\Psi_t)(x)=[\nabla_i\Psi_t(x)\nabla_i\nabla_j\Psi_t(x)-\nabla_l\Psi_t\Gamma_{ij}^l(x) \nabla_k\nabla_k\Psi_t(x)-\nabla_l\Psi_t\Gamma_{kk}^l(x)-\frac{\sum_k(\nabla_k\nabla_k\Psi_t-\nabla_l\Psi_t\Gamma_{kk}^l)}{n}(x)]^T
\gamma \left( r \right) = \frac{\exp \left\{\sqrt{\frac{q}{k}}\log \frac{r^{2}}{\rho^{2}\sqrt{q}}\right\}}{\sqrt{q}\exp \left\{\sqrt{\frac{q}{k}}\log \frac{r^{2}}{\rho^{2}\sqrt{q}}\right\}+ 1},
\begin{array}[c]{cl}\displaystyle \sum \limits_{i=1}^{n}-\psi_{x_i}(\bar{X}(t_1),\bar{X}(t_2),\cdots,\bar{X}(t_n))y(t_{i}) = \displaystyle \int \limits_{0}^{T}\big{[}f_x(\bar{X}{(t)},\bar{u}(t))y(t)+p(t)^{}(b(\bar{X}{(t)},u^{\varepsilon}(t))-b(\bar{X}{(t)},\bar{u}(t)))\big{]}dt.\end{array}
\frac{d}{dt}\|w^n (t)\|_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}^2 + \|\partial_y w^n(t) \|_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}^2 + \epsilon \|\partial_x w^n(t) \|_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}^2 &\le B^{n-1}_T\|{w}^n(t)\|^2_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}+D^{n-1}_T\| w^n\|^{m+2}_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}\, .
\binom{n_{1}}{2}n_{k}a_{1}^{2}a_{k}+\binom{n_{k}}{2}n_{1}a_{k}^{2}a_{1}=\frac{1}{2}n_{1}n_{k}a_{1}a_{k}\left( n_{1}a_{1}+n_{k}a_{k}-a_{1}-a_{k}\right) & \leq\frac{1}{2}n_{1}n_{k}a_{1}a_{k}\left( n_{1}a_{1}+n_{k}a_{k}-2\sqrt{a_{1}a_{k}}\right) & \leq n_{1}n_{k}a_{1}a_{k}\left( lb-\sqrt{a_{1}a_{k}}\right) .
\delta_c (m) (a) :=~ x \cdot_1 m - m \cdot_1 x = x \cdot_2 m - m \cdot_2 x, m \in C^0_{com}(\mathfrak{g}, M) x \in \mathfrak{g}, \delta_c (h_1, \ldots, h_n ) :=~ ( \delta_1 h_1, \ldots, \underbrace{\delta_1 h_i + \delta_2 h_{i-1}}_{i-}, \ldots, \delta_2 h_n),
\omega (g,h) = \lambda_g \omega (g,h)=0, \forall h\in H_4 , &\omega (x,x) = \lambda_x \omega (1_H,x)+\lambda_g \omega (x,x)=\lambda_x^2 , & \omega (x,xg) = \lambda_x \omega (1_H,xg) +\lambda_g \omega (x,xg)=-\lambda_x^2 ,
\rho \ddot{u}_i = \big(A_{iJKj}u_{j,K}- \beta_{Ji}\dot{\tau}-(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L}- B_{iJKj}\dot{u}_{j})_{,K}\big)_{,J}a \ddot{\tau}=-\beta_{Ki}\dot{u}_{i,K}+ m_{IJ}\dot{\tau}_{,IJ}+ M_{jLKI}u_{j,LKI}+ K_{IJ}\tau_{,IJ}.
(l+1)(l+2+2\sigma_n) p'_{\sigma, l+1}+ 2s(\sigma_n + 1) p'_{\sigma+\bar n, l}+ (\sigma_i +1)(\sigma_i + 2) p'_{\sigma + 2\bar \imath, l-1}+ \tilde{c}_{\sigma l}^{\mu m}p'_{\mu m}= 0 \forall (\sigma, l)
I_{x_0}(F_n\mathrm{Sub}_\sigma)+I_{x_0}(F_n\mathrm{Sh}(\Omega(C)))=f_{\sigma,x_0}(F_n\mathrm{Sh}(\mathrm H_C^{\mathrm{dR}})\otimes\mathcal O(C))+I_{x_0}(F_n\mathrm{Sh}(\Omega(C))),
\textbf{err}^{(2)}:= \displaystyle\frac{1}{\alpha_\mathrm{m}}\int_{0}^t\int_{\partial\Omega}\gamma^{\textbf{int}}_{1,\mathrm{y}}\Phi^{\textbf{e}}(\xi,t;\mathrm{y},\tau)\gamma^{\textbf{int}}_{0}\mathrm{U}_{\mathrm{i}}(\mathrm{y},\tau)d\sigma_{\mathrm{y}}d\tau.
\rho\big((z, \theta), (\bar{z}, \bar{\theta})\big) =\Big(\mathop{\operatorname{ess\,sup}}_{0 \leq t \leq T}\big\|\bar{D}^{1}\big(z - \bar{z}\big)(t, \cdot)\big\|_{L^{2}(\Omega)}^{2}+\mathop{\operatorname{ess\,sup}}_{0 \leq t \leq T}\big\|(\theta - \bar{\theta})(t, \cdot)\big\|_{H^{1}(\Omega)}^{2}\Big)^{1/2}
W=(\begin{matrix}n\\ 1\end{matrix})|A_{1}|-(\begin{matrix}n\\ 2\end{matrix})|A_{1}\cap A_{2}|+\cdot\cdot\cdot+(-1)^{p-1}(\begin{matrix}n\\p\end{matrix})|A_{1}\cap\cdot\cdot\cdot\cap A_{p}|+\cdot\cdot\cdot
\mu_{1}(\varphi) =\frac{1}{1-\cos^{2}\varphi}\Big((1-4\cos^{2}\varphi)\mu_{1}(\frac{\pi}{2}) &+\frac{3}{4}(\cos\varphi+2\cos^{2}\varphi)\mu_{1}(\frac{\pi}{3}) &+\frac{3}{4}(2\cos^{2}\varphi-\cos\varphi)\mu_{1}(\frac{2\pi}{3})\Big),
w_{0 r}(r) = R^\alpha \delta^{-\alpha-1}\alpha \left( - (\alpha + 3) \left(\tfrac{r}{\delta}\right) + (\alpha + 2) \left(\tfrac{r}{\delta}\right)^2\right) \begin{cases}\leq -R^\alpha \delta^{-\alpha-2}r \alpha, \geq -R^\alpha \delta^{-\alpha-2}r \alpha(\alpha + 3) \end{cases}
| z > = e^{- \frac{1}{2}| z |^{2}}\, e^{z a^{\dagger}}\, | 0 > \ \ \ , \ \ \ | q , p > = e^{- \frac{i}{\hbar}q \hat{p}}\, e^{\frac{i}{\hbar}p \hat{q}}\, | 0 > \ ,
p\left( \mathbf{b}\right) :=\#\left\{i:b_{i}=\theta\right\},\text{}q\left( \mathbf{b}\right) :=\#\left\{i:b_{i}=\theta^{\ast}\right\}, \ell\left( \mathbf{b}\right) :=\sum_{i=1}^{k}\left( 1_{b_{i}=\theta^{\ast}}-1_{b_{i}=\theta}\right) =q\left( \mathbf{b}\right)-p\left( \mathbf{b}\right) .
\delta_{\kappa}Z^{\underline{M}}E_{\underline{M}}^{\underline{a}}= 0 , \quad \delta_{\kappa}Z^{\underline{M}}E_{\underline{M}}^{\underline{\alpha}}= ( 1 + \bar{\Gamma})_{\underline{\beta}}^{\underline{\alpha}}\kappa^{\underline{\beta}}( \xi ) ,
\begin{aligned}\lim_{n\to\infty}\sum_{j=1}^{n-1}\sum_{l=1}^k\frac{\binom{c_nkn}l(k)_l(k(j-1))_{c_nkn-l}}{(kn)_{c_nkn}}\frac k{k(n-j+1)-l}\frac{E_{n,k}(\tau(A^{(n)}_{c_nkn,j,l})|\mathcal{WA}^{(n,k)}_{M_n,j,l})}{nk}= &(1-c)\frac{p_c^{(k)}(1)}{k+1}.\end{aligned}
b\left(4\frac{C^1_{2,1}}{C^1_1}-3\frac{C^1_1}{\mu}\right)\omega_1\wedge\omega_2=-c\omega_2\wedge\theta_{56}, &c\left(4\frac{C^1_{2,1}}{C^1_1}-3\frac{C^1_1}{\mu}\right)\omega_1\wedge\omega_2=-b\omega_2\wedge\theta_{56}, &\omega_1\wedge\theta_{56}=0,
\begin{aligned}\left\{\varsigma_{Q_j,i}=1\right\}\subseteq\left\{\varsigma_{Q_j,m}=1\right\}, i \le m, &\left\{\varsigma_{Q_j,i}=0\right\}\subseteq\left\{\varsigma_{Q_j,r}=0\right\}, r \le i.\end{aligned}
\left\{\bigcup_{i=1}^{n_1n_2}\{\sum_{j\neq i}^{r n_1}\left| g_{ij}\right| \geq \epsilon_2 \}\right\}\leq 4 n_1^2 n_2^2 \exp{\left( - \frac{M \epsilon^2}{128 c_0^2 r^2 n_1^2}\right)}.
q_1 = x_{n-1}x_n, begin{align*}2mm]q_2 = x_1 x_n + x_{n-1}x_{n+1}, begin{align*}2mm]q_3 = x_2 x_n + x_{n-1}x_{n+2}, begin{align*}2mm] \ \ \vdots begin{align*}2mm]q_{n-1}= x_{n-2}x_n + x_{n-1}x_{2n-2}.
[f,g](x) =W_{\alpha,\beta, m}(x)(1-x^2)(f'(x)\overline{g}(x)-f(x)\overline{g}'(x)) &=\frac{(1-x)^{\alpha+1}(1+x)^{\beta+1}}{\left(P_m^{(-\alpha-1,\beta-1)}(x)\right)^2}(f'(x)\overline{g}(x)-f(x)\overline{g}'(x)) (x\in (-1,1))\,
S_2 = \sum_{k \le 2(h - 1) \le 2k}(-1)^h \, \alpha_{h - 1,k}\, r^{2h-k-1}\, F_{N+2h}(r) &= \sum_{k + 1 \le 2h \le 2(k + 1)}(-1)^h \, \alpha_{h - 1,k}\, r^{2h-k-1}\, F_{N+2h}(r).
P^{(N_f=1,\nu=1)}(\lambda,m)=\frac{\lambda}{2}e^{-\lambda^2/4}\frac{1}{mI_1(m)}\left[tI_2(\lambda)I_3(t)-\lambda I_2(t)I_3(\lambda) \right]_{t=\sqrt{\lambda^2+m^2}}\:.
p^k_n=\sum\limits_{w_{1}+\cdots+w_{k+1}= n}\frac{n!}{w_{1}!\times w_{2}!\times\cdots\times w_{j}!\times\cdots\times w_{k+1}!}\begin{bmatrix}(1)\times(1)\times\cdots \times J^0_{w_{j}}\times\cdots\times (1)\end{bmatrix}
\begin{array}{c c c}{\left[ B_{a}, B_{b}\right]}&{=}&{C_{a b}^{c}B_{c}}\\{}&{}&{}\\{\left[ B_{a}, Q_{\beta}\right]}&{=}&{C_{a \beta}^{\gamma}Q_{\gamma}}\\{}&{}&{}\\{\left\{Q_{\alpha}, Q_{\beta}\right\}}&{=}&{C_{\alpha \beta}^{c}B_{c}}\\ \end{array}
E =\int_0^{2 \pi}d \phi \, r\left\{\frac{1}{r^2}\frac{\partial f^*}{\partial \phi}\frac{\partial f}{\partial \phi}- \mu^2 f^* f+ \lambda (f^* f)^2\right\}=2 \pi r\left\{\frac{1}{r^2}\alpha^2 v^2- \mu^2 v^2+ \lambda v^4\right\}.
\int_{0}^{2 \pi}d \varphi \int_{0}^{r ( \varphi )}{\frac{r d r}{2 ( \rho_{i}^{2}- r^{2})}}={\frac{1}{4}}\int_{0}^{2 \pi}d \varphi \ln{\frac{\rho_{i}^{2}}{\rho_{i}^{2}- r^{2}( \varphi )}}\, .
\int_{\mathbb{Z}_p}f(x) d \mu_{-1}(x) = \lim_{N \rightarrow \infty}\sum_{x=0}^{p^N-1}f(x) \mu_{-1}(x+ p^N \mathbb{Z}_p) = \lim_{N \rightarrow \infty}\sum_{x=0}^{p^N-1}f(x) (-1)^x, (\textnormal{see}\,\, [8]).
-\frac{1}{4}\sqrt{-G}\,F_{MN}{}^k F^{MN}{}_k=-\frac{1}{4}\sqrt{-g}\,\Phi^{\frac{1}{D-2}}[B_{\mu\nu}{}^k B^{\mu\nu}{}_k+\frac{2}{\Phi}g^{\mu\nu}D_\mu A^k \cdot D_\nu A_k],
\sum_{\substack{0 \leq m \leq y/x x/2 < n_1 \le x}}\Bigg|\sum_{n_2 \in I(m, n_1) - mx}\Bigg(\prod_{j = 0}^{r-1}\Lambda(n_2 + jr! n_1 + mx) - \alpha_r\Bigg) \Bigg|^2 \Lambda(n_1).
\mathcal{E}_{i}(t):= G(\mathcal{E}(t),\Psi_{i})\ \textup{for}\i\in \mathbb{Z}\setminus\{0\}, &\mathcal{W}_j(t):= G(\mathcal{E}(t),\Upsilon_j), \ \overline{\mathcal{W}}_j(t):=G(\mathcal{E}(t),\overline{\Upsilon}_j) \ \textup{for}\ 1\leq j\leq J.
J_{N}^{i}( \sigma ) = J_{0}^{i}+ \sum_{n = 1}^{\infty}\cos n \sigma ( J_{n}^{i}+ J_{- n}^{i}) \quad \mathrm{a n d}\quad J_{D}^{i}( \sigma ) = i \sum_{n = 1}^{\infty}\sin n \sigma ( J_{n}^{i}- J_{- n}^{i}) .
\mathcal{C}_{1,2,\dots,k+1}^{n}\left(z_{1},\dots,z_{k+1}\right)=\frac{\left(\mathcal{C}_{1,2,\dots,k}\left(z_{1},\dots,z_{k}\right)+\mathcal{B}_{k+1}+z_{k+1}\right)^{n}}{n}.
\sum_{s_j}\sum_{s_k}\frac{(N_i-2s_i)!N_j!N_k!}{2^{s_j+s_k}s_j!s_k! t_{ij}!t_{ik}!t_{jk}!}=\frac{(N_j+N_k)!}{2^{(N_j+N_k-N_i+2s_i)/2}(\frac{N_j+N_k-N_i+2s_i}{2})!}