harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{i=1}^n (f_i(x)-f_i(y))^2 \le \frac{l^2}{n^2}\sum_{i=1}^n \biggl(\sum_{j=1}^n (p_{ij}(x)-p_{ij}(y))\biggr)^2 &\le \frac{l^2}{n}\|P(x)-P(y)\|^2_{}\, .
|T^{D}_{\Delta,k}[V_{k+1}](x_k, r_k)-T_k[V_{k+1}](x_k, r_k)| \leq|V_{k}(x_k,r_k^{(\tau)})- &V_{k}(x_k,r_k)|+|\bar{T}^{D}_{\Delta,k}[V_{k+1}](x_k, r^{(\tau)}_k)-T_k[V_{k+1}](x_k,r^{(\tau)}_k)|.
z(\zeta, \boldsymbol{\nu}^i) = \z(\zeta, \boldsymbol{\nu}^{i-1}) \exp \left ( - \int_{-\infty}^{\nu_i}A_i(t) dt \right ) &\ + \left( \phi(\Phi^{-1}(\zeta)) \right )^{-1}\left ( 1 - \exp \left ( - \int_{-\infty}^{\nu_i}A_i(t) dt \right ) \right ).
\mathcal{L}(f)(x) = \frac{1}{\alpha}\bigg( -\langle x ; \nabla(f)(x)\rangle + \sum_{k=1}^d \partial_k D^{\alpha-1}_{k}(f)(x)+\frac{1}{p_\alpha(x)}\sum_{k=1}^d\partial_k R_k^\alpha(p_\alpha,f)(x) &\quad+\sum_{k=1}^d\left(\frac{\partial_k(p_\alpha)(x)}{p_\alpha(x)}D_k^{\alpha-1}(f)(x)+\partial_kD_k^{\alpha-1}(f)(x)\right)\bigg),
c^2+1 = \frac{M_2}{M_1^2}= \frac{-2 \, (1/\lambda^*) \, \boldsymbol{\pi}\, C^{-1}{\mathbf 1}}{(1/\lambda^*)^2}= 2 \boldsymbol{\pi}C{\mathbf 1}\boldsymbol{\pi}C^{-1}{\mathbf 1}.
\aligned(- \Delta u )(\phi) = \int_D \nabla u\cdot \nabla \phi \, dvol & = \lim_{s\to \infty}\int_D \alpha_s'(u) \nabla u\cdot\nabla \phi \, dvol & = \lim_{s\to\infty}(-\Delta\alpha_s(u)) (\phi))\endaligned
\langle\phi^{\epsilon}_\alpha\,,\,\mathfrak{Op}^\epsilon(\mathfrak h^\epsilon)\phi^{\epsilon}_\beta\rangle_{\mathcal{H}}=\Lambda^\epsilon(\alpha,\beta)\langle\phi^{\epsilon}_{\alpha-\beta}\,,\,\mathfrak{Op}^\epsilon(\mathfrak h^\epsilon)\phi^{\epsilon}_0 \rangle_{\mathcal{H}}\,.
\big(\mathbb{E}\|P_k(\varepsilon)\|_X^q\big)^{1/q}\leq \int_{0}^1\big(\mathbb{E}\|P(t\varepsilon) p_{k+1}^{(m)}(t) \|_X^q\big)^{1/q}d t\leq B^m \int_{0}^1\big(\mathbb{E}\|P(t\varepsilon) \|_X^q\big)^{1/q}d t.
\left|{\bf E\,}\left[ \frac{\sum_{i=1}^{n}\omega_{i}}{\sqrt{n}}\cdot\left| \frac{\sum_{i=1}^{n}\zeta_{i}}{\sqrt{n}}\right| \right] \right| \leq\frac{A}{\left( 1-\rho^{2}\right)^{3/2}\sqrt{n}}\left({\bf E\,}|\omega|^{3}+{\bf E\,}|\zeta |^{3}\right) .
\left\{Q_{r}, Q_{s}\right\}= \left( C^{( 1 3 )}\, \Gamma_{M N}^{( 1 3 )}\right)_{r s}\, Z_{\bf 0}^{M N}+ \left( C^{( 1 3 )}\, \Gamma_{M_{1}M_{2}M_{3}}\right)_{r s}Z_{\bf 0}^{M_{1}M_{2}M_{3}}\, ,
\sum_{j=1}^\infty \|b_j f\|_{D(\mu_{l+1})}^2 \leq \frac{2}{|\Phi(\zeta_{l+1}))|^2}\Big[\|f\|_{D(\mu_{l+1})}^2 + 8 \|M_E\|^2 \|f\|_{D(\mu_{l+1})}^2 \Big] & \leq \frac{1}{\varepsilon^2}\Big(2 +16\|M_E\|^2 \Big) \|f\|_{D(\mu_{l+1})}^2,
\alpha^\mu_{-n}M_{\mu\nu}\widetilde\alpha^\nu_{-n}\rightarrow\alpha^\alpha_{-n}M_{\alpha\beta}\widetilde\alpha^\beta_{-n}-\alpha^\alpha_{-n}M_{\alpha j}\widetilde\alpha^j_{-n}+\alpha^j_{-n}M_{j\alpha}\widetilde\alpha^\alpha_{-n}-\alpha^i_{-n}M_{ij}\widetilde\alpha^j_{-n}
\small\sum \limits_{j=1, j \neq i}^{m-n}sec(U_i, U_j) = \sum \limits_{j=1, j \neq i}^{m-n}\frac{g(R(U_i, U_j)U_j, U_i)}{g(U_i, U_i)g(U_j, U_j) - g(U_i, U_j)^2}&= \frac{1}{(m-n-1)}Ric(U_i, U_i).
S_2 = \left( 2P_1(1)P_3(1) + P_3(1)^2 + \frac{1}{\theta_3}\int_0^1 P_3'(x)^2 dx + \kappa + \lambda + o(1) \right)\sum_{q}\Psi \left( \frac{q}{Q}\right) \frac{q}{\varphi(q)}\varphi^+(q),
d(H^{(i)}|H^{(i-1)}):= \begin{cases}\frac{|E(H^{(i)})\cap \mathcal{K}_i(H^{(i-1)})|}{|\mathcal{K}_i(H^{(i-1)})|}|\mathcal{K}_i(H^{(i-1)})|>0, 0 .\end{cases}
\gamma^{i}= - i \left( \begin{array}{c c}{{0}}&{{\sigma_{i}}}\\{{- \sigma_{i}}}&{{0}}\\ \end{array}\right) \, , \quad \gamma^{4}= \left( \begin{array}{c c}{{0}}&{{{\bf 1}_{2}}}\\{{{\bf 1}_{2}}}&{{0}}\\ \end{array}\right) \, ,
\lim_{A\to\infty}\limsup_{x \to \infty}\frac{\int_{A}^{x-A}g(x-u)g(u)du}{g(x)}&=\lim_{A\to\infty}\limsup_{x \to \infty}\frac{2\int_{A}^{x/2}g(x-u)g(u)du}{g(x)}\le 8\lim_{A\to\infty}\int_A^{\infty}u^{-2}du=0
\beta_u=\lim_{x \rightarrow\infty}\left(\sum_{p\leq x, p\in \mathcal{P}_u}\frac{1}{p}-\alpha_u\log \log x\right)\beta_u=\gamma_u-\sum_{p\in \mathcal{P}_u,}\sum_{k\geq 2}\frac{1}{k p^k},
\mathcal{S}_a:=\{u\in H^1(\mathbb{R}^N):|u|_2=a\},E(u)=\frac12|\nabla u|^2_2-\frac{1}{22^*_\mu}\int_{\mathbb{R}^N}(I_\mu\ast|u|^{2^*_\mu})|u|^{2^*_\mu}.
k_{1}k_{2}= \frac{y_{1}- 3}{x_{1}- 4}\cdot \frac{y_{2}- 3}{x_{2}- 4}= \frac{y_{1}y_{2}- 3 ( y_{1}+ y_{2}) + 9}{x_{1}x_{2}- 4 ( x_{1}+ x_{2}) + 1 6}= \frac{1}{2}
\mathbb{V}(\overline{T}_n)=\frac{1}{n^2}\sum_{1\leq i \leq n}\mathbb{E}(X_i^2X_{i+1}^2)+\frac{2}{n^2}\sum_{1\leq i<j \leq n}\mathbb{E}(X_iX_{i+1}X_jX_{j+1})-\frac{1}{n^2}\mathbb{E}^2\Big(\sum_{1\leq i \leq n}X_iX_{i+1}\Big).
(a \xi)(s) = \int_G a(t) \beta_t( \xi(t^{-1}s)) \, dt, (\xi a)(s) = \int_G \xi(t) \alpha_t( a(t^{-1}s)) \, dt, \langle \xi, \eta \rangle(s) = \int_G \alpha_t( \langle \xi(t), \eta(ts) \rangle) \, dt,
\left|b^*\left(\frac{m_j}{n_j}\sum_{k\in A_1}x_k\right)\right| \leqslant \frac{1}{p}\sum_{i\in S}(\#S_i)\frac{m_j}{n_j}C = \frac{Cm_j}{pn_j}\#A_1\leqslant \frac{Cm_j}{pn_j}K & = m_j\frac{C(K/n_j)}{s(b^*)}.
\sum_{s}\sum_{n=1}^{\infty}N^{rs}_{m_1,n}n N^{st}_{n,m_2}= \delta^{rt}\frac{1}{m_1}\oint \frac{dy}{2 \pi i}\xi_t(y)^{-m_1}\frac{1}{m_2}\oint \frac{dz}{2 \pi i}\xi_t(z)^{-m_2}\frac{\xi_t'(z)\bar{\xi}_t'(y)}{(1 - \xi_t(z)\bar{\xi}_t(y))^2}
u(x) =O\left( \left(\frac{1}{|x|}\right)^{n-2}\right) v(x) = \begin{cases}O\left( \left(\frac{1}{|x|}\right)^{n-2}\right), \beta\leq n-(n-2)\sigma o\left( (\frac{1}{|x|})^{\frac{n-2}{n}[(n-2)\sigma+\beta]}\right), \beta>n-(n-2)\sigma. \end{cases}
(-1)^{k-1}+ \displaystyle \sum_{r=1}^{k-1}(-1)^{r+k+1}(r+1)! \genfrac{\{}{\}}{0pt}{}{k}{r+1}=(-1)^{k-1}\biggl(1+\displaystyle \sum_{r=1}^{k-1}(-1)^r (r+1)! \genfrac{\{}{\}}{0pt}{}{k}{r+1}\biggr)
2 n^{1/2}(p^{n,1}_+(\eta) - p^{n,1}_-(\eta)) = n^{1/2}(1 - n^{-1/2}\eta) - n^{1/2}(1 + n^{-1/2}\eta)(1 - 2 \beta n^{-1/2}\eta) & = -2 (1 - \beta) \eta + 2 \beta n^{-1/2}\eta^2.
\int_{X}{\rm II}_{x}d\eta(x)=-\sum_{s\in G}\mu(s)\int_{Y}\int_{\pi^{-1}(y)}H\left(\xi_{n-1}^{s^{-1}.x}\right)d\eta^{y}(x)\frac{ds\nu}{d\nu}(y)d\nu(y)=-\int_{X}H(\xi_{n-1}^{x})d\eta(x).
\begin{array}{cl}\Xi_{2}^{h}= \{(\pm e_{a}\pm e_{b}\pm e_{c}\pm e_{d})/2: a,b,c,d\mbox{distinct}, e_{a}(e_{b}(e_{c}e_{d}))=\pm e_{h}\mbox{if exactly one of}a,b,c,d= 0\mbox{or}h, e_{a}(e_{b}(e_{c}e_{d}))=\pm 1 \mbox{otherwise}\}.\end{array}
|\Phi| \leq l^Lq^{3l^2L^2-L(n-1)}|G|^d \leq l^{n/(9l^2)}q^{3l^2\cdot n^2/(81l^4)-n/(9l^2)\cdot n/2}|G|^d &\leq q^n\cdot q^{n^2(1/(27l^2)-1/(18l^2))}|G|^d = q^{n-n^2/(54l^2)}|G|^d = (q^{n^2})^{1/n-1/(54l^2)}|G|^d &\leq (q^{n^2})^{1/(216l^2)-1/(54l^2)}|G|^d = (q^{n^2})^{-1/(72l^2)}|G|^d \leq |G|^{d-1/(72l^2)},
=e^{-\lambda t}\sum_{k=0}^\infty I_{2k}\Bigl(\frac{2\lambda}{c_1+c_2}\sqrt{(c_1t-\beta)(c_2t+\beta)}\Bigr) \Biggl(\sqrt{\frac{c_2t+\beta}{c_1t-\beta}}\Biggr)^{2k}
T_2=\int_{\mathcal Q_t}(\partial_{t}u + u \cdot \nabla u ) \cdot w = \left[ \int_{\mathcal F (s)}u(\cdot,s) \cdot w(\cdot,s) \right]_{s=0}^{s=t}- \int_{\mathcal Q_t}(\partial_t w + u \cdot \nabla w) \cdot u\,.
\sum_{i=1}^{r}\bigg(\frac{1}{\sigma_{i}}-\frac{1}{\widetilde{\sigma}_{i}}\bigg)^{2}+\sum_{i=r+1}^{s}\frac{1}{\widetilde{\sigma}_{i}^{2}}\leq\|B^{\dagger}-A^{\dagger}\|_{F}^{2}\leq\sum_{i=1}^{r}\bigg(\frac{1}{\sigma_{i}}+\frac{1}{\widetilde{\sigma}_{i}}\bigg)^{2}+\sum_{i=r+1}^{s}\frac{1}{\widetilde{\sigma}_{i}^{2}}.
{\frac{1}{\sqrt{\pi}}}\int_{0}^{\infty}{\frac{d s}{s^{3 / 2}}}\exp \left( -{\frac{n^{2}}{s}}-{\frac{\beta^{2}{\cal E}^{2}}{4}}s \right) ={\frac{e^{- \beta n |{\cal E}|}}{n}}, \qquad n > 0
\alpha=\beta(\mu_U-\mu_D)/2, \;\;\cosh\theta_C=\frac{1}{\sqrt{2}}\left[\frac{S_{UD}(p)}{\sqrt{S_{UD}^2(p)-R_{UD}^2(p)}}+1\right]^{1/2},\;\;\sinh\theta_C=\frac{1}{\sqrt{2}}\left[\frac{S_{UD}}{\sqrt{S_{UD}^2(p)-R_{UD}^2(p)}}-1\right]^{1/2}
D_{\mu\nu}=\frac{1}{p^2+M^2}(\epsilon_{\mu\nu\lambda}p_\lambda+\frac{p_\mu p_\nu}{M}+M\delta_{\mu\nu})\;,{}~~~~D^{-1}_{\mu\nu}= -\epsilon_{\mu\nu\lambda}p_\lambda +M\delta_{\mu\nu}\;.
\begin{aligned}G-c_4 ={}\{v_1:[v_2, v_5, v_4, v_5], v_2:[v_1, v_3, v_4, v_3], v_3:[v_2, v_5, v_2, v_4], v_4:[v_1, v_5, v_2, v_3],v_5:[v_1, v_4, v_1, v_3]\}. \end{aligned}
T((y_1, \rho_1) \cdot (y_2,\rho_2))= T( (y_1y_2, \rho_1^{y_2}\rho_2{{\nu}}(y_1,y_2))) = (y_1y_2,[\zeta_{y_1y_2}(\rho_1^{y_2}\rho_2{{\nu}}(y_1,y_2))])
\widehat{H}(s) = (2\pi)^{-s - 1}\frac{\Gamma\left(\frac{L^2}{2}+ 1\right)^2 \Gamma\left(K + \frac{L^2}{2}\right)}{\Gamma(L^2 + 1) \Gamma\left(K - \frac{L^2}{2}- 1\right)}\frac{\Gamma\left(\frac{K - L^2 + s - 1}{2}\right)}{\Gamma\left(\frac{K + L^2 - s + 1}{2}\right)}.
\sum_{i=j-2}^{j+2}(2^j\rho)^d\int_{S^{d-1}}\sup_{r\in [2^{j}\rho,2^{j+1}\rho]}|\mathbf{K}_{i,n,l}(r\theta)|\lesssim2^{j\alpha}~~\forall~~l\geq -4.
\begin{array}{c c c}{\psi^{a +}}&{= - \frac 1{\sqrt{2}}\left( \chi^{a}+ \overline{{\chi}}^{a}\right)}&{}\\{\psi^{a -}}&{= - \frac i{\sqrt{2}}\left( \chi^{a}- \overline{{\chi}}^{a}\right)}&{}\\ \end{array}
\beta_1^h(x) = \frac{\hat y_1^h(x) - x_1}{h^2}- u^h(x_1) + x_2\frac{R^h_{21}(x_1)}{h}+ x_3\frac{R^h_{31}(x_1)}{h}\,; \beta_i^h(x) = \frac{1}{h}\left(\frac{\hat y_i^h(x) - hx_i}{h}- v_i^h(x_1) - hw^h(x_1)x_i^\bot\right)\,,i=2,3\,.
K_{\rho,2,\varphi_p}(z)=\frac{1}{\rho^{2-pk_p}}\sum^\infty_{j=0}\frac{2j+2-pk_p}{2\pi{\rho^{2j}}}|z|^{2j}=\frac{1}{\rho^{2-pk_p}}\frac{2-pk_p+pk_p|z|^2/\rho^2}{2\pi(1-|z|^2/\rho^2)^2}.
Q_l^{(\lambda)}(\xi)= -\frac{d}{d \lambda}\left[P_l^{(\lambda)}(\xi)\right] = \sum_{j=0}^{[\frac{l}{2}]}\sum_{k=0}^{l-j-1}(-1)^{j+1}\frac{\Gamma(l-j+\lambda)}{(\lambda+k)\Gamma(\lambda)j!(l-2j)!}(2\xi)^{l-2j},
f_\lambda(t){}=\cosh\Bigl(\frac{t\sqrt{-c}}{2}\Bigr)-\frac{2\lambda}{\sqrt{-c}}\sinh\Bigl(\frac{t\sqrt{-c}}{2}\Bigr), g_\lambda(t){}=\Bigl(\cosh\Bigl(\frac{t\sqrt{-c}}{2}\Bigr)-1\Bigr)\Bigl(1+2\cosh\Bigl(\frac{t\sqrt{-c}}{2}\Bigr)-\frac{2\lambda}{\sqrt{-c}}\sinh\Bigl(\frac{t\sqrt{-c}}{2}\Bigr)\Bigr).
\sum_{j=1}^N \sum_{i=1}^{M-1}\Biggl[(a_{iH}- a^0_{iH})+\sum_{k=1}^{H-1}\{(a_{ik}-a_{iH})b_{kj}- (a^0_{ik}-a^0_{iH}) b^0_{kj}\}\Biggr]^2 &=\sum_{j=1}^N \sum_{i=1}^{M-1}\Biggl[(a_{iH}- a^0_{iH})+\sum_{k=1}^{H-1}(a_{ik}b_{kj}{-}a^0_{ik}b^0_{kj}) \Biggr]^2.
[ \delta_{\kappa_{1}}, \delta_{\kappa_{2}}] z^{M}= \delta_{b}z^{M}= - \mathrm{i}( b_{\nu}{\widetilde B}^{\nu}) z^{M}, \qquad b^{\mu}= - \mathrm{i}{\overline{{\kappa}}}_{1}\Gamma^{\mu}\kappa_{2},
\sup_{\genfrac{}{}{0pt}{}{\scriptstyle{x,\bar x\in \overline{B},\, t,\bar t\in[s,s+1],}}{{\scriptstyle{x \not= \bar x,\, t \not= \bar t,\, s\in[-1,1]}}}}\frac{|v_n(x,t_n+t)-v_n(\bar x,t_n+\bar t)|}{|x-\bar x|^\gamma+|t-\bar t|^{\frac{\gamma}{2}}}< K,
d_{\mathcal{P},\pm}=\left[b (\mathsf{M}^{\mathbb{R}}_{1}+ \mathsf{M}^{\mathbb{Z}}) \mp \frac{1}{2\pi}\,\partial_{1}\right]+\left[ b\, \mathsf{M}^{\mathbb{R}}_{2}\pm \frac{1}{2\pi}\,\partial_{2}\right]
r_{3}:= -b_{1}^{2}b_{2}+ b_{2}d_{2}+ 3d_{2}^{2},r_{4}:= -b_{1}^{3}b_{2}+b_{1}^{2}b_{3}+ 5b_{1}b_{2}d_{2}- b_{1}b_{2}^{2}+ b_{2}b_{3}- 6b_{3}d_{2},
\lim_{t\rightarrow+\infty}\frac{y^{(t)}}{x_{1}^{(t)}}=\begin{cases}\frac{\gamma}{\gamma_1}\mbox{if}\gamma_{2}=0,\delta_{1}\neq0 \frac{\gamma x_{1}^{(t_{0})}+\delta x_{2}^{(t_{0})}}{\gamma_1x_{1}^{\left(t_{0}\right)}}\mbox{if}\gamma_{2}=\delta_{1}=0, +\infty \mbox{if}\gamma_{2}=0,\delta_{1}\neq0.\end{cases}
\int_{\mathcal{D}\times\mathcal{D}}(\delta_h^{\alpha}\otimes \delta_h^{\alpha}) k(x,y)\varphi(x,y)dxdy = \int_{\mathcal{D}}k(x,y)(\delta_h^{\alpha}\otimes \delta_h^{\alpha})^*\varphi(x,y)dxdy
\sum_{k=1}^{n}|x_{k}y_{k}|\le(\sum_{k=1}^{n}|x_{k}|^{p})^{\frac{1}{p}}(\sum_{k=1}^{n}|y_{k}|^{q})^{\frac{1}{q}}forall(x_{1},...,x_{n}),(y_{1},...,y_{n})\in\mathbb{R}^{n}or\mathbb{C}^{n}.
\begin{cases}D_{r_k}h_4(z_*)=\sin \frac{2\pi(k-1)}{n}, D_{x_k}h_4(z_*)=-\frac{\sin \frac{\pi}{n}+\sin \frac{\pi(2k-1)}{n}}{2 \sin \frac{\pi}{n}}, \k=1,2,\ldots,n. \end{cases}
P^{(j_1+1/2,j_2+1/2)}_{K-j_1-j_2)/2}(\xi,\eta)=\!\sum^N_{m=0}\!\Big({n+j_1+1/2 \atop m}\Big)\Big({n+j_2+1/2 \atop n-m}\Big)(\xi^2)^m (\eta^2)^{n-m}(-1)^{n-m}.
( \sigma_{\mu}{}^{\rho})_{\alpha \beta}\, [ \mathrm{A{\footnotesize \#}.{\footnotesize \#}}]_{\nu \rho}{}^{\alpha \beta}= b_{1}^{2}\sum_{k = 0}^{\infty}\frac{f_{\mathrm{A \# . \#}}( k )}{( k ! )^{2}}\, [ \mathrm{a{\footnotesize \# ~^{\prime}~}~ o r ~ b{\footnotesize \# ~^{\prime}~}}]_{\mu \nu}( k )
\Phi(x;s|-a,b,-q^{1/2}a,q^{1/2}b|q)=\sum_{l,m\geq 0}{(a^2;q)_m(q^l s;q)_m\over (q;q)_m(q^{l+1}s/a^2;q)_m}(q x^2/a^2)^m &\times{(b/a;q^{1/2})_l(s/a^2;q^{1/2})_l (s;q)_l\over (q^{1/2};q^{1/2})_l (q^{1/2}s/ab;q^{1/2})_l (s/a^2;q)_l}(q^{1/2}x/b)^l.
E\big[|\textit{\textbf{I}}^{\prime}_{1}(t)|^2\big] = (1-\lambda) E\big[|\textit{\textbf{I}}_{1}(t)|^2\big] \approx \frac{\alpha(1-\lambda)^2}{\lambda}\cdot \frac{M^2}{\tau}\cdot \beta_{1,k}^2.
\left. \begin{array}{l l}a(\textbf{x}) = \frac{1}{2}\frac{(y_2-y_3)R_1^2 + (y_3-y_1)R_2^2 + (y_1-y_2)R_3^2}{x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)}b(\textbf{x}) = \frac{1}{2}\frac{(x_3-x_2)R_1^2 + (x_1-x_3)R_2^2 + (x_2-x_1)R_3^2}{x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)}.\end{array}\right.
J_n(c^{-1}nz){H_n^{(1)}}'(nz) =\frac{4e^{2\pi i/3}}{z}\left(\frac{(1-z^2)\zeta(c^{-1}z)}{\zeta(z)(1-c^{-2}z^2)}\right)^{1/4}\left(\frac{Ai(n^{2/3}\zeta(c^{-1}z))}{n^{1/3}}+\O{}(Ei(5/3,7/3)(c^{-1}z)\right) &\quad\quad\quad\left(\frac{A'_-(n^{2/3}\zeta(z))}{n^{2/3}}+\O{}(E_-(8/3,4/3)(z))\right)
I_q^{p;k;h}+(-1)^{k\left(u(p,p^*)+h_q^p+h_q^{p^*}\right)}I_q^{p^*;k;h}= \sigma^{\tfrac 12(1-\varepsilon)}\left(I_q^{s;k;h'}+(-1)^{k\left(u(s,s^*)+h_q^s+h_q^{s^*}\right)}I_q^{s^*;k;h'}\right)
Put~~ u=(\frac{t_2}{\beta_2})^{\nu_2}-(\frac{\beta_2}{t_2})^{\nu_2}&du=\frac{\nu_2}{\beta_2}\bigg[(\frac{t_2}{\beta_2})^{\nu_2-1}+(\frac{\beta_2}{t_2})^{\nu_2+1}\bigg]dt_2.
\int_{\mathbb R^2}\sum_{k\in \Bbb Z}2^{2k}\big\vert \big(f(\cdot,\cdot,\cdot)\ast_1\psi^{(1)}_{j}(y_1)\big) \ast_{2,3}\psi^{(2)}_{j,k}(y_2,y_3)\big\vert^2 w_{y_1,2^{-j}}(R_{2^{-k},2^{-j-k}}(y_2,y_3) ){dy_2\,dy_3}&=\int_{\mathbb R^2}\big\vert S_{j}(F_{y_1,2^{-j}}) (y_2,y_3)\big\vert^2 w_{y_1,2^{-j}}(y_2,y_3 ){dy_2\,dy_3},
\lefteqn{\int_{[u,t]}\{(t-u)\wedge w - (w-u)\}\mu(dw)}&=\int_{[u,t-u]}\{(t-u)\wedge w - (w-u)\}\mu(dw)+\int_{(t-u,t]}\{(t-u)\wedge w - (w-u)\}\mu(dw) &=\int_{[u,t-u]}u\mu(dw)+ \int_{(t-u,t]}(t-w) \mu(dw).
\int_0^T\int_{U}\left ( \frac{1}{2}\langle A \nabla \left(\rho P_{\cdot}g_n\right), \nabla \varphi \rangle + (\rho P_{\cdot}g_n) \langle \beta, \nabla \varphi \rangle -(\rho P_{\cdot}g_n) \partial_t\varphi \right ) dxdt=0.
\Psi(sp)= \Psi(0)+\frac{s}{t}\cdot\big(\Psi(tp)-\Psi(0)\big) = \Omega(0)\Phi(0)+\frac{s}{t}\cdot\big(\Omega(tp)\Phi(tp)-\Omega(0)\Phi(0)\big)\equiv\Omega(sp)\Phi(sp)\pmod{p^2}.
\int_0^1 \frac{\sqrt{1-x^2}\ln (x)}{\left(2-x^2\right)^{3/2}}\, dx = \frac{1}{4}\int_0^1 \left(\sum_{n = 0}^{\infty}\left(-\frac{1}{2}\right)^n \binom{n}{\left\lfloor \frac{n}{2}\right\rfloor}(n+1) u^{n+\frac{1}{2}}\ln (1-u)\right) \, du.
u^N_t(x)=e^{t\Delta}u^N_0(x) - \int_0^t \nabla \cdot e^{(t-s)\Delta}\langle \mu_s^N, V^N ( x-\cdot) F\big( K\ast u^N_s (\cdot)\big) \rangle \ds - \frac{1}{N}\sum_{i=1}^N \int_0^t e^{(t-s)\Delta}\nabla V^N (x-X_s^{i,N})\cdot dW^i_s,
A(\theta) =m_A{\rm e}^{\theta}+C_A -\int_{-\infty}^{\infty}\frac{d\theta'}{2\pi}\frac{1}{\cosh(\theta-\theta')}\ln(1+ B(\theta')^2), B(\theta) =m_B{\rm e}^{\theta}+C_B -\int_{-\infty}^{\infty}\frac{d\theta'}{2\pi}\frac{1}{\cosh(\theta-\theta')}{\rm e}^{-A(\theta')}
Er[f_{k}(t),k\in\{N+1,...\}]=\sum_{k=N+1}^{\infty}\int_{a}^{b}\int_{a}^{b}K_{X}(s,t)f_{k}(t)f_{k}(s)dsdt-\beta_{k}(\int_{a}^{b}f_{k}(t)f_{k}(t)dt-1)
\mu : \Omega_{k(k+1)k}\twoheadrightarrow \Pi\Omega_{k}^\xi, \begin{cases}\xi_{k+1}^i \otimes_{k+1}\xi_{k+1}^j \mapsto 0, \xi_{k+1}^i \otimes_{k+1}\xi_{k+1}^j s_{k+1,k+1}\mapsto (-1)^{i+j}Y^\xi_{i+j,k},\end{cases}
K^0(\varepsilon ;\zeta):=([\Lambda^\varepsilon ] \mathbf{D}+[\widetilde{\Lambda}^\varepsilon ])(B^0-\zeta \overline{Q_0})^{-1}=([\Psi^\varepsilon ]\nabla +[\widetilde{\Lambda}^\varepsilon ])(B^0-\zeta \overline{Q_0})^{-1}.
\Pi^1=R_{vac}\frac{\Psi^1}{\Psi^0},\quad \Pi^2=R_{vac}\frac{\Psi^2}{\Psi^0},...,\Pi^i=R_{vac}\frac{\Psi^i}{\Psi^0},...,\Pi^N=R_{vac}\frac{\Psi^N}{\Psi^0},
d s^{2}= H_{1}^{3 / 4}H_{2}^{3 / 4}H_{3}^{1 / 4}\left[ H_{1}^{- 1}H_{2}^{- 1}H_{3}^{- 1}( - d t^{2}+ d s^{2}) + H_{1}^{- 1}d{\vec{x}}^{2}+ H_{2}^{- 1}d{\vec{y}}^{2}\right] ~ ,
A = \left( \begin{array}{c c c}{{2 \omega_{\eta}}}&{{- t}}&{{i v}}\\{{- t}}&{{2 \omega_{t}}}&{{- i r}}\\{{- i v}}&{{i r}}&{{2 \omega_{b}}}\\ \end{array}\right) \; \; B = \left( \begin{array}{c c c}{{0}}&{{- i t}}&{{u}}\\{{- i t}}&{{0}}&{{s}}\\{{u}}&{{s}}&{{0}}\\ \end{array}\right) \; ,
\partial_s \mathcal{N}(t, s,\xi,v,\overline v) = e^{(t- s)\mathcal{L}}\mathcal{C}[f, \mathcal{L}] \left(e^{s\mathcal{L}}v, e^{s\mathcal{L}}e^{\xi\mathcal{A}}\overline v \right).
\begin{gathered}- \Delta_\theta := (\Delta_z) |_{\Gamma_\theta}, \ \ \ \x_\theta := z|_{\Gamma_\theta}, Q_{\epsilon, \theta}: = - \Delta_\theta - i \epsilon x_\theta^2 , \ \ \ \P_{\epsilon, \theta}:= Q_{\epsilon , \theta}+ V . \end{gathered}
\hat{I}_{\mathrm{D}}= - \int_{W}d^{p + 1}\sigma \sqrt{- \det{\hat{g}}_{\mu \nu}}\ ; \qquad \hat{g}_{\mu \nu}= \partial_{\mu}X^{\hat{M}}\partial_{\nu}X^{\hat{N}}\eta_{\hat{M}\hat{N}}\ .
\sigma(\Omega)=n_{p}e^{2}a^{2}\frac{\pi^{\frac{1}{2}}t^{2}(1-e^{\frac{\hbar\Omega}{k_{B}T}})}{2\hbar^{2}\Omega(E_{a}k_{B}T)^{1/2}}exp(-\frac{(\hbar\Omega-4E_{a})^{2}}{16E_{a}k_{B}T})
Var[X]=\sigma_{\kappa}^{2}=\frac{1}{\beta^{2}}\frac{2(1-4\kappa^{2})^{2}-(1-\kappa^{2})^{2}(1-9\kappa^{2})}{(1-4\kappa^{2})^{2}(1-9\kappa^{2})}
\mu^{*2n}(g)=\sum_{h\in G}\mu^{*n}(gh^{-1})\mu^{*n}(h)\leq \sqrt{\sum_{h\in G}\mu^{*n}(gh^{-1})^2\sum_{h\in G}\mu^{*n}(h)^2}= \sum_{h\in G}\mu^{*n}(h)^2=\sum_{h\in G}\mu^{*n}(h^{-1})\mu^{*n}(h)=\mu^{*2n}(e),
{\rm vol}_{\phi}[\iota_{t}]_{x}= \rho_{\phi}[\iota_{t}] (x) e^{1}(t) \wedge \cdots \wedge e^{n}(t), \rho_{\phi}[\iota_{t}] (x) =\sqrt{({\rm vol}_{g})_{\iota_{t}(x)}((\iota_{t})_{*}e_{1}(t), \cdots, (\iota_{t})_{*}e_{n}(t), -\xi_{\iota_{t}(x)}, \phi (\iota_{t})_{*}e_{1}(t), \cdots, \phi (\iota_{t})_{*}e_{n}(t))}.
X_{l , m}^{( L )}\equiv ( \partial_{A_{1}}\cdots \partial_{A_{l + 2 m}}\omega^{L}) K_{[ l , m ]}^{A_{1}\dots A_{l + 2 m}, B_{1}\dots B_{l + 2 m}}( \partial_{B_{1}}\cdots \partial_{B_{l + 2 m}}\omega^{\prime L}) .
[c, z_1] [z_2, z_3, z_4] = - [c, z_1] \bigl[ z_4, [z_2, z_3] \bigr] = - [c, z_1] [z_4, z_2, z_3] + [c, z_1] [z_4, z_3, z_2] = \ [c, z_4] [z_1, z_2, z_3] - [c, z_4] [z_1, z_3, z_2] = [c, z_4] \bigl[ z_1, [z_2, z_3] \bigr] = - [c, z_4] [z_2, z_3, z_1] ,
\begin{aligned}I_{1}\leq \|\,|x|^{r_1}u\|_{L^{\infty}_{T}L^2_{x\,y}}\|\partial_x u\|_{L^{1}_{T}L^\infty_{xy}}&\leq \mu_2^{T}(u)\;T^\frac12\|\partial_x u\|_{L^2_{T}L^\infty_{xy}}&\leq T^\frac12 \mu_{1,1}^{T}(u)\mu_{2,1}^{T}(u).\end{aligned}
-2\left(\sqrt{|\alpha|+1}+\sqrt{|\alpha+2\beta|+1}-2\sqrt{|\alpha+\beta|+1}\right) \leq 2\left(\sqrt{|\alpha+\beta|+1}-\sqrt{|\alpha|+1}\right) &=\frac{2|\beta|}{\sqrt{|\alpha+\beta|+1}+\sqrt{|\alpha|+1}}&\leq 2|\beta|.
{\cal Z}_{\mathrm{3 D{\,}A H M}}= \int D A_{\mu}D \theta^{\mathrm{s i n g .}}D \theta^{\mathrm{r e g .}}\exp \left\{- \int d^{3}x \left[ \frac{1}{4 e^{2}}{\cal F}_{\mu \nu}^{2}+ \frac{\eta^{2}}{2}\left( \partial_{\mu}\theta - A_{\mu}\right)^{2}\right] \right\}.
\int_{D_R}|(i\nabla + A_{\mathbf e})\tilde\varphi_{a_{n_\ell}}|^2 dx = \lambda_{a_{n_\ell}}\,|a_{n_\ell}|^2 \int_{D_R}|\tilde \varphi_{a_{n_\ell}}|^2 dx -i \int_{\partial D_R}\big( (i\nabla +A_{\mathbf e})\tilde \varphi_{a_{n_\ell}}\cdot \nu \big) \overline{\tilde \varphi_{a_{n_\ell}}}\,ds.
\begin{aligned}x \psi_n(x)=P_n(z)\psi_n(x)\ , \frac{1}{N}\frac{\partial}{\partial x}\psi_n(x) =-Q_n(z)^T\psi_n(x)\ , y \chi_n(y)=Q_n(z)\chi_n(y)\ , \frac{1}{N}\frac{\partial}{\partial y}\chi_n(y) =-P_n(z)^T\chi_n(y)\ , \end{aligned}
\langle D\Phi_{t}(x;h),l\rangle_{\bar{\mathcal{H}}}=J_{t}(x;h)\cdot\int_{0}^{t}J_{s}^{-1}(x;h)\cdot V(\Phi_{s}(x;h))dl_{s},\ \ \ \mathrm{for\all}\l\in\bar{\mathcal{H}},
\mathcal{E}_\varepsilon=\mathcal{H}_{\varepsilon}[u_{\varepsilon}] +\int_{0}^{2\pi}\frac{1}{2}|u_{\varepsilon}|^{2}+ \bar{c}|\mathcal{J}_{\varepsilon}u_{\varepsilon}|^{4\sigma+2}\dx.
\frac{3}{2}n \lambda_{n}\widehat{Q}_{\beta}^{1}\psi_{0}= - P_{0}\bigl ( Q_{\beta}^{1}\overline{{{P}}}_{0}\psi_{n}+ Q_{\beta}^{2}\psi_{n - 1}+ \ldots + Q_{\beta}^{n + 1}\psi_{0}\bigr ) \; , \nonumber
\big(v\nabla(-\Delta)^{-1}w +w\nabla(-\Delta)^{-1}v\big)_{m}=(-\Delta)\Big{\{}\big{(}(-\Delta)^{-1}v\big{)}\big{(}\partial_{m}(-\Delta)^{-1}w\big{)}\Big{\}}+2\nabla\cdot\Big{\{}\big{(}(-\Delta)^{-1}v\big{)}\big{(}\partial_{m}\nabla(-\Delta)^{-1}w\big{)}\Big{\}}+\partial_{m}\Big{\{}\big{(}(-\Delta)^{-1}v\big{)}w\Big{\}}.
\sum_{a=2}^{n-2}\sum_{j=0}^{n-2-a}\sum_{i=1}^{a-1}\binom{n-2-a}{j}C_{n-2-j,i}=\sum_{j=0}^{n-4}\sum_{i=1}^{n-3-j}C_{n-2-j,i}\sum_{a=i+1}^{n-2-j}\binom{n-2-a}{j}&\quad=\sum_{j=0}^{n-4}\sum_{i=1}^{n-3-j}\binom{n-2-i}{j+1}C_{n-2-j,i}
\| \prod_{i=1}^k \mathcal E_\lambda^\gamma f_i \|_{L^q(d\mu)}\le C \langle \mu\rangle_\alpha^{\,\,\frac1{q}}L^{-\frac{k^2-k}{2q}}\lambda^{-\frac{\alpha-d+k}{q}}\prod_{i=1}^k\| f_i \|_p
- \frac{1}{2}\mathrm{T r}[ \frac{M^{2}}{- \partial^{2}+ M^{2}{\tilde{\omega}}}\alpha \frac{M^{2}}{- \partial^{2}+ M^{2}{\tilde{\omega}}}\alpha ] = -{\frac{1}{4 8 \pi}}\int d^{2}x{\frac{1}{\omega^{2}}}\partial_{\mu}\omega \partial^{\mu}\omega \; .
\phi_j (g) = \begin{cases}s_1^{a_1}\cdots s_j^{D-A}\cdots s_{n-1}^{a_{n-1}}u \mbox{if}A \le D-1 \mbox{and}a_j = 0 s_1^{a_1}\cdots s_j^{0}\cdots s_{n-1}^{a_{n-1}}u \mbox{if}A=D \mbox{and}a_j > 0 g \mbox{if}A \le D-1 \mbox{and}a_j > 0 g \mbox{if}A=D \mbox{and}a_j = 0. \end{cases}
ds^2_B=a \prod_{i=1}^{3}(z-z_i)^{-\frac{2}{3}}\prod_{i=1}^{3}(\bar{z}-\bar{z_i})^{-\frac{2}{3}}dz d\bar{z}+a' \prod_{i=1}^{3}(z'-z'_i)^{-\frac{2}{3}}\prod_{i=1}^{3}(\bar{z'}-\bar{z'_i})^{-\frac{2}{3}}dz' d\bar{z'}
d_{\Omega}(\varphi(0), \phi(0)) = \lim_{n \rightarrow \infty}d_{\Omega_n}(\varphi_n(0), \phi_n(0)) & = \lim_{n \rightarrow \infty}d_{\Omega_n}(\varphi_n(0), \sigma_n(m_n)) + d_{\Omega_n}(\sigma_n(m_n), \phi_n(0)) &= \infty
g(s)= \begin{cases}s^{g_0-1+\delta}, 0<s<2 begin{align*}1mm] s_{2k+1}^{-\delta}s^{g_1-1}, s_{2k}\leq s < s_{2k+1}begin{align*}1mm] s_{2k+2}^{\delta}s^{g_0-1}, s_{2k+1}\leq s < s_{2k+2}\end{cases}.
\sum_{n=0}^{\infty}\frac{q^{3n^{2}+(2m+1)n}\left(-1\right)^{n}}{\left(q^{2},-q^{m+1},-cq^{m+2};q^{2}\right)_{n}}=\frac{(-1)^{m}q^{-\binom{m}{2}}a_{m}(q)}{(q,q^{4};q^{5})_{\infty}}-\frac{(-1)^{m}q^{-\binom{m}{2}}b_{m}(q)}{(q^{2},q^{3};q^{5})_{\infty}},
\sum_{s=0}^{A}\left|1-2\varphi(s)\right|= \sum_{n=0}^{(A-2)/2}\left\{\left|1-2\varphi(2n)\right|+\left|1-2\varphi(2n+1)\right|\right\}+|1-2\varphi(A)| = \sum_{n=0}^{(A-2)/2}1+\left|1-2\varphi(A)\right|=\frac{A}{2}+\left|1-2\varphi(A)\right|
\Gamma{\cal H}^{(1)}_{\eta',x_1,\dots,x_m;q}(v,w) \cong \begin{pmatrix}\Gamma{\cal S}^{(1)}_{\eta',x_1,\dots,x_m;q}(v,w) \Gamma{\cal S}^{(1)}_{\eta',x_1,\dots,x_m;q}(v-2f,w) \Gamma{\cal S}^{(1)}_{\eta',x_1,\dots,x_m;q}(v,w-2f) \Gamma{\cal S}^{(1)}_{\eta',x_1,\dots,x_m;q}(v-2f,w-2f) \end{pmatrix}\!,
P_{w,r}^V(V_{\tau_s^R}> x + y | V_1 \geq x) \leq\begin{cases}C_1(1 + y^{2/3}x^{-1/3}) e^{-C_2 y^{2/3}x^{-1/3}}~,~ \mbox{if}y \leq x C_1(1 + y^{1/3}) e^{-C_2 y^{1/3}}~,~ \mbox{if}y \geq x.\end{cases}
G_0(\lambda) = \frac{c_n e^{{\rm i}\sqrt{\lambda}|x|}}{|x|^{n - 2}}\int_0^\infty e^{-t}t^{\frac{n - 3}{2}}\Big( \frac{t}{2}-{\rm i}\sqrt{\lambda}|x| \Big)^{\frac{n - 3}{2}}{\rm d}t,
X^{u+v+w+x}X^v= qX^{u+v+x}X^{v+w}= qX^{u+v+x}X^{v+w}+X^vX^{u+v+w+x}-X^vX^{u+v+w+x}= qX^{u+v+x}X^{v+w}+X^vX^{u+v+w+x}-q^{-1}X^{u+v}X^{v+w+x};
\mathbf{h}(q,k)=\sum_{z=1}^{Z}\beta_{z}(\mathbf{a}_{\mathrm{r}}(\mu^{\mathrm{r}}_{z})\otimes\mathbf{a}_{\mathrm{t}}(\mu^{\mathrm{t}}_{z})\exp(jq\gamma_{z}-jk\eta_z)
\Sigma(Z_1,Z_2,T) =\sigma_c +{\left(e^{2h}-1\right)^2 \left(3+(2+\sqrt{7})e^{2h}\right)^2\over 20\cdot 2^{1/3}e^{4h}\left(1+e^{2h}\right)^2}\left[{{\Phi(Z_1,T)-\Phi(Z_2,T)}\over{Z_1 - Z_2}}\right] \epsilon^{1/3}+{\cal O}(\epsilon^{2/3})
G^{\alpha\beta}=\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}=[\begin{matrix}0&-B_{x}&-B_{y}&-B_{z}\\B_{x}&0&E_{z}/c&-E_{y}/c\\B_{y}&-E_{z}/c&0&E_{x}/c\\B_{z}&E_{y}/c&-E_{x}/c&0\end{matrix}]
\left\langle l_1, m_1 \right| \cos(n \theta) \left| l_2 , m_2\right\rangle=\sum\limits_{l=|l_1-l_2|}^{l_1+l_2}\sqrt{\left( l_1+\frac{1}{2}\right)\left( l_2+\frac{1}{2}\right)}(-1)^{m_1}
\beta^{(1)}_{\alpha,\epsilon_1}=(\alpha - 1) \big( I(U;W) + \epsilon_1 - d_{\alpha}(p_{U,W}, p_U p_W) \big), \beta^{(2)}_{\alpha,\epsilon_2}=(\alpha - 1) \big( I(U,V;W) + \epsilon_2 - d_{\alpha}(p_{U,V,W}, p_{U,V}p_W) \big),
J_i ={1\over 2}\int d^3x \langle x|{\rm tr}\, \gamma_5 \gamma_i{1\over \gamma\cdot D + M}| x \rangle = -{1\over 2}\int d^3x \langle x|{\rm tr}\, \gamma_5 \gamma_i (\gamma\cdot D){1 \over -(\gamma\cdot D)^2 +M^2}| x \rangle