harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{k=1}^{\infty}\int\limits_{\left( B_{2^{-n+k+3}|\Lambda |}(M_0)\setminus B_{2^{-n+k+2}|\Lambda |}(M_0)\right)\setminus\widetilde{\Omega}_{n-2}}\, \frac{|\Phi_n(M)|}{\rho^2_{M_0}(M)}\, dm_3(M)\leq C\cdot 2^{n}\cdot \omega (2^{-n}).
Pf(x) =\frac{d_x^+g(|x| +1)+d_x^0g(|x|)+\lambda d_x^-g(|x|-1)}{d_x^++d_x^0+\lambda d_x^{-}}\, \ge \frac{(d_x^+ + d_x^0) g(|x|+1)+ \lambda d_x^-g(|x|-1)}{d_x^++d_x^0+\lambda d_x^{-}}.
f - q|_g = \L(p|_{L^2(xy)}L(z)) + p|_{L^2(xy)}L^2(z) - L^2(p|_{L(xL(y))}z) - L^2(p|_{xL^2(y)}z) \equiv \ -L(p|_{L(xL(y))}L(z)) -L(p|_{xL^2(y)}L(z)) - p|_{L(xL(y))}L^2(z) - p|_{xL^2(y)}L^2(z) &\ + L(p|_{L(xL(y))}L(z)) + p|_{L(xL(y))}L^2(z) + L(p|_{xL^2(y)}L(z)) + p|_{xL^2(y)}L^2(z) \equiv \ 0.
F(x^k) - \eta^{k+1}\le F(x^k) - \eta^k - \frac{1-\beta}{2}\bar{\mu}_k\big(F(x^k) - \eta^k\big) = \biggl(1-\frac{1-\beta}{2}\bar{\mu}_k\biggr)\big(F(x^k) - \eta^k\big).
\left[ \frac{\partial^{2}}{\partial r^{2}}+ ( \frac{1}{r}- 4 i \omega \Omega \theta r ) \frac{\partial}{\partial r}+ \frac{1}{r^{2}}\frac{\partial^{2}}{\partial \theta^{2}}+ ( q^{2}- 4 i \omega \Omega \theta ) \right] f ( r , \theta ) = 0 .
\mu^{\rtimes n}(\phi(G^n,\bar b)) = \int_{a\in Z}\mu^{\rtimes n-i}(\phi(a,G^{n-i},\bar b)) \, \mathrm{d}\mu^{\rtimes i}+ \int_{a\not\in Z}\mu^{\rtimes n-i}(\phi(a,G^{n-i},\bar b)) \, \mathrm{d}\mu^{\rtimes i}& \le \int_{a\in Z}1\, \mathrm{d}\mu^{\rtimes i}+ \int_{a\not\in Z}\frac{r}{2}\, \mathrm{d}\mu^{\rtimes i}\le \mu(Z) + \frac{r}{2}
\frac{\tau_{k}}{\gamma_{k}}=\left[1+\delta_{k}(1+\delta_{k-1}(1+\dots+\delta_{2}(1+\delta_{1})\dots))\right]=\frac{\left\Vert p_{k}\right\Vert^{2}}{\left\Vert r_{k}\right\Vert^{2}}.
|v|_{1/2,g}^2 :=\int_g\int_g \frac{|v(x)-v(y)|^2}{|x-y|^{n}}\,ds_x\,ds_y, \|v\|_{1/2,g}^2 := h_{K(g)}^{-1}\|v\|_{L^2(g)}^2 + |v|_{H^{1/2}(g)}^2,
\det R(z) =|\exp \big(\int_0^1tr \omega_R(\gamma(s)\big)| &=\exp\big(Re\frac{1}{4\pi}\int_0^{2\pi}\log (1-z_1^2-z_2^2-2z_1z_2\cos\theta)d\theta\big) &=\exp\big(\frac{1}{4\pi}\int_0^{2\pi}\log |1-z_1^2-z_2^2-2z_1z_2\cos\theta|d\theta\big).
\dot{y}_R(t)= 2 \Im \int_{\mathbb{R}^N}\nabla \varphi_R(x) \cdot \nabla u \bar{u}dx = 2R \Im \int_{\mathbb{R}^N}\nabla \varphi \left( \frac{x}{R}\right) \cdot \nabla u \bar{u}dx,
\Vert \alpha_\gamma(\nu_n)-\nu_n\Vert=\Vert \xi_\gamma*\mu_n-\xi*\mu_n\Vert\leq\Vert\xi_\gamma*\mu_n-\mu_n\Vert+\Vert\mu_n-\xi*\mu_n\Vert\rightarrow 0, \gamma\in \Gamma.
\mu_{\rm wkb}=2^{-1/2}\biggl( \frac{n\epsilon}{2\pi}\biggr)^{-1/2}n^{-1/2}b_1^{-1/4}\vert \eta \vert^{-1/2}\exp \biggl[\mp ib_1^{1/2}\biggl(\frac{n\epsilon}{4\pi}\biggr)n\biggl(\vert \eta \vert^2-\vert \eta_{\rm i}\vert^2\biggr)\biggr].
\mathrm{(II)}~ \mathrm{MDTCov}_{(\boldsymbol{a},\boldsymbol{b})}(\mathbf{Y})=\mathbf{\Sigma}^{\frac{1}{2}}\left[ \frac{\boldsymbol{\Omega}}{F_{\mathbf{Z}}(\boldsymbol{\xi_{a}},\boldsymbol{\xi_{b}})}-\frac{\boldsymbol{\delta}\boldsymbol{\delta}^{T}}{F_{\mathbf{Z}}^{2}(\boldsymbol{\xi_{a}},\boldsymbol{\xi_{b}})}\right] \mathbf{\Sigma}^{\frac{1}{2}},
c(\xi) =c(\mathfrak d_{\xi})\chi(\mathfrak f_{\xi})a(\mathfrak f_{\xi,N}) \prod_{p\mid \mathfrak f_{\xi,0}}(a(p^{e_p}) - p^{k-1}\chi_{-\xi}(p)a(p^{e_p-1})) = c(\mathfrak d_{\xi})\chi(\mathfrak f_{\xi})a(\mathfrak f_{\xi,N}) \mathfrak f_{\xi,0}^{k-1/2}\prod_{p\mid \mathfrak f_{\xi,0}}\Psi_p(\xi;\alpha_p).
\tilde{\Theta}_\pm A\varphi =\tilde{\Theta}_\pm \gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=-\gamma^{(2n+1)}\tilde{\Theta}_\pm (P_{\rm F}-D_aP_{\rm F}D_a)\varphi=-A\tilde{\Theta}_\pm \varphi.
\psi^{0}( z ) = \sum_{m \in{\bf Z}}d_{m}^{0}z^{- m - \frac{1}{2}}, \quad \widetilde{\psi}^{0}( \overline{{z}}) = \sum_{m \in{\bf Z}}d_{m}^{0}\overline{{z}}^{- m - \frac{1}{2}},
\frac{r^2}{\beta_{X^4}-1}= 1, \frac{q^2}{\beta_{X^4}-1}+1 = \frac{q^2}{\beta_{X^4}-1}+2 r^2, r q+\frac{p q}{\beta_{X^4}-1}= 2 r q+\frac{p q}{\beta_{X^4}-1},
M_{0}(v) \coloneqq v-\frac{1}{v^2}\int_0^vv'^2dv'=\frac{2}{3}v, M_{1}(v) \coloneqq \frac{1}{v^2}\int_0^v\left(1-e^{-\int_{v'}^vA(v'')dv''}\right)v'^2dv', M_{2}(v) \coloneqq -\frac{A(v)}{v}\int_0^ve^{-\int_{v'}^vA(v'')dv''}v'^2dv',
S[\Psi^*,\Psi]\ =\{1\over 2\pi\rho}\int_{{\bf R}\times S^1}dtd\varphi\, [{i\over 2}\rho(\Psi^*{\dot \Psi}-{\dot\Psi}^*\Psi)+i\Psi^*\partial_\varphi\Phi ]
(a^{1/k}+b^{1/k})^{k}=\sum_{i=0}^{k}(\begin{matrix}k\\i\end{matrix})a^{(k-i)/k}b^{i/k}=a\sum_{i=0}^{k}(\begin{matrix}k\\i\end{matrix})a^{-i/k}b^{i/k}=a\sum_{i=0}^{k}(\begin{matrix}k\\i\end{matrix})(\frac{b}{a})^{i/k}
(1,0)_{n}^- \cap (AP_{n}// \P_2) = F^1_{n}(^+(0,0)_{n-1}^- ){^-(0,1)_{n}}\cap (AP_{n}// \P_2) = F^1_{n}(^-(0,0)_{n-1}^+ ) (1,1)_{n}\cap (AP_{n}// \P_2) = (-1)^{n}F^1_{n}((1,0)_{n-1}^+) (id +(-1)^{n}\phi_{n})((1,0)_{n}^+) \cap k (AP_{n}// \P_2) = F^1_{n}(^+(0,0)^+_{n-1}\cap ( AP_{n-1}// \P_1) )
\bigg(\frac{\mathrm{d}}{\mathrm{d}r}\bigg)^2 \bigg(\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}\bigg)^{k-1}\left(r^{2k-1}\phi(r)\right) = \bigg(\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}\bigg)^{k}\left(r^{2k}\frac{\mathrm{d}\phi}{\mathrm{d}r}(r)\right)
\mathcal{C}_{q}(u)\mathcal{C}_{q}(v)+q^{1/2}\mathcal{S}_{q}(u)\mathcal{S}_{q}(v) ={}_{3}\phi_{3}\left[\begin{matrix}0, u^{-1}vq^{1/2}, uv^{-1}q^{1/2}q^{1/2}, -q^{1/2}, -q \end{matrix}\ ;q, -uvq^{1/2}\right]
\mathrm{P}\left( \bar{E}_{m,2}\right) =\prod^{m-1}_{n=1}\left[1-\frac{\gamma\left(M-n+1,\max\left\{\xi_n, \frac{\epsilon_{2,n}}{\rho \beta_n^2}\right\}\right)}{(M-n)!}\right] \times \frac{\left[\gamma\left(M-m+1,\frac{\epsilon_{2,m}}{\rho \beta_m^2}\right) - \gamma(M-m+1,\xi_m)\right]}{(M-m)!},
\sum_{k=3}^\infty \frac{k^{k-2}}{k!}\Bigg[k-1- k\beta\Bigg]\big(\beta e^{-\beta}\big)^{k-2}< \frac{2e^2}{3^{5/2}\sqrt{2\pi}}\sum_{k=3}^\infty \big(\beta e^{1-\beta}\big)^{k-2}&= \frac{2e^2}{3^{5/2}\sqrt{2\pi}}\frac{\beta e^{1-\beta}}{1 - \beta e^{1-\beta}}.
\begin{aligned}\qquad\nabla^2 U=0 s.t.\qquad U(0,t) =0 U(1/2,t) =0 U(\theta,0) =0 U(\theta,M) =\int_{0}^{\infty}\left[\frac{1}{x^{1/2+\theta}}-\frac{1}{x^{1/2-\theta}}\right]\frac{\sin\left(M\log x\right)}{\left(e^{x}+1\right)}dx \end{aligned}
aba^*b^* =a_1b_1(w_1w_2^*z_2^*(a{\bar a}^*)z_2w_2w_1^*)(w_1w_2^*(b{\bar b}^*)w_2w_1^*))w_1w_2^* =a_1b_1(x_1a^*_1x_1^*)(x_2^*b^*_1x_2) w_1w_2^*
C(j_1,j_2,\ldots ,j_s)-C(j'_1,j_2,\ldots,j_s) & = \sum_{i_1+\cdots+i_s+i_{s+1}+l=k}(j^{2i_1}_1-j'^{2i_1}_1)j_2^{2i_2}\cdots j_s^{2i_s}b_{i_1}b_{i_2}\cdots b_{i_s}a_{i_{s+1}}a^{(2k)}_l.
{\cal L}_\eta|\Phi_{ph}\rangle=(\eta\hat{\partial}+\frac{1}{2}\hat{\partial}^I\eta^JM^{IJ}+\frac{1}{2}\eta^IM^{zI})|\Phi_{ph}\rangle-\hat{\partial}^+\eta^I\alpha^I(\bar{\alpha}^-|\Phi\rangle)\Bigr|_{\alpha^+=0}^{\vphantom{5pt}}\,,
F_{{\hat{M}}{\hat{N}}{\hat{P}}{\hat{Q}}{\hat{R}}}{\hat{\Gamma}}^{{\hat{M}}{\hat{N}}{\hat{P}}{\hat{Q}}{\hat{R}}}= 5 e^{3 a \varphi}F_{[ 4 ]}\cdot \Gamma \, \Gamma^{9}( 1 - \hat{\Gamma}^{1 1}) ,
\begin{aligned}\mu_{1,k}^T(u)= \|u\|_{L_T^\infty H^s}+\|D_x^{s}\,\partial_x u\|_{L_x^\infty L_{yT}^2}+\|D_y^{s}\,\partial_x u\|_{L_x^\infty L_{yT}^2}&+\|\partial_x u\|_{L_x^\infty L_{yT}^2}+\|u\|_{L_T^{\frac{3k}{2}+}L_{xy}^\infty}+\|\partial_xu\|_{L_T^{\frac{3k}{k+2}}L_{xy}^\infty}+\|u\|_{L_x^{\frac{k}{2}}L_{yT}^{\infty}},\end{aligned}
|u|^{n'}\leq |v|^{n'}+\frac{n'2^{n'-1}A(u)^{n}}{n}|v|^{n'}+2^{n'-1}+n'2^{n'-1}A(u)^{n'}= \left(1+\frac{2^{n'-1}|A(u)|^{n}}{n-1}\right)|v|^{n'}+C_{4},
e(\alpha) = \frac{\Gamma(n+2)}{\Gamma(x+ 1) \Gamma(n-x+1)}\int_0^1 p^{x+1}(1-p)^{n-x}{\rm d}p= \frac{\Gamma(n+2) \Gamma(x + 2)}{\Gamma(n+3) \Gamma(x + 1)}.
\mathcal{R}_{k(-j)j}^{\nu}(\alpha_{2,\nu})=\{\xi\in\mathcal{O}_{\nu}:|\langle k,\omega_{\nu}(\xi)\rangle+ \bar{\Omega}_{\nu,(-j)}(\xi)-\bar{\Omega}_{\nu,j}(\xi)|<\alpha_{2,\nu}\frac{|j|}{\langle k\rangle^{\tau}}\}.
\varphi_{ij}(v)=\sum_{S\subseteq N}\frac{(|S|-1)!(n-|S|)!}{n!}(v(S)-v(S\backslash\{i\})-v(S\backslash\{j\})+v(S\backslash\{i,j\}))\sum_{t=|S|}^{n}\frac{1}{t}
=\frac{1}{(n+1)^2 [\exp\left\{\frac{i\pi x}{n+1}\right\}-1]}\sum_{j,j'=0}^n\Big[\Phi\left(\frac{j-1}{n+1}, \frac{j'}{n+1}\right)- \Phi\left(\frac{j}{n+1}, \frac{j'}{n+1}\right)\Big] &\times \exp\left\{\frac{2 i\pi jx}{n+1}\right\}\exp\left\{\frac{2 i\pi j'x}{n+1}\right\}
\lambda_{n,r,d}= 2^{r-1}\frac{d+r-1}{\ell-r+1}\binom{n-r}{n-\ell}\Big/ \binom{n}{\ell}= 2^{r-1}\frac{d+r-1}{\ell-r+1}\frac{(n-r)!\,\ell!\,(n-\ell)!}{(n-\ell)!\,(\ell-r)!\,n!}= 2^{r-1}\frac{(n-r)!}{n!}\frac{\ell!\,(d+r-1)}{(\ell-r+1)!},
F^*(z\,|\,\overline\omega_1,\omega_2)\cdot F^*(z\,|\,\overline\omega_1,-\omega_2)= \prod_{k\geq 0}\big(1-x_2 (q_2 \widetilde{q}_2)^{k/2}\big)\cdot \prod_{k\geq 1}\big(1-x_2^{-1}(q_2 \widetilde{q}_2)^{k/2}\big)^{-1},
A(f,g)(x,\omega) = e^{\pi i x \cdot \omega}V_gf(x,\omega) = e^{\pi i x \cdot \omega}e^{-2 \pi i x \cdot \omega}V_{\widehat{g}}\widehat{f}(\omega, -x) & = e^{-\pi i x \cdot \omega}V_{\widehat{g}}\widehat{f}(\omega, -x) = A(\widehat{f}, \widehat{g})(\omega, -x).
\frac{S}{6}-W_{1212}&= \frac{1}{2}\left( R_{11}+ R_{22}\right) - R_{1212}&= \frac{1}{2}\left(R_{1313}+ R_{1414}+ R_{2323}+R_{2424}\right) &= \frac{1}{2}\left(R_{1212}+R_{1313}+ R_{1414}+ R_{2323}+R_{2424}+R_{3434}\right) - \frac{1}{2}\left(R_{1212}+R_{3434}\right) &= \frac{S}{4}- \frac{1}{2}\left(R_{1212}+R_{3434}\right).
\mathrm{d}x\left( t\right) =f\left( t,x\left( t\right) ,r\left(t\right) ,u\left( t\right) \right) \mathrm{d}t+g\left( t,x\left(t-d\left( t,r(t)\right) \right) ,r\left( t\right) ,u\left( t\right)\right) \mathrm{d}w\left( t\right) ,t\in J,
( \overline{{{\bf 4}}},{\bf 1},{\bf 2}) ( - 1 ) = ( \overline{{{\bf 3}}},{\bf 1}) ( \textstyle{-{\frac{1}{3}},{\frac{1}{2}}, - 1}) \oplus ( \overline{{{\bf 3}}},{\bf 1}) ( \textstyle{-{\frac{1}{3}}, -{\frac{1}{2}}, - 1}) \oplus ({\bf 1},{\bf 1}) ( \textstyle{1 ,{\frac{1}{2}}, - 1}) \oplus ({\bf 1},{\bf 1}) ( \textstyle{1 , -{\frac{1}{2}}, - 1})
\left( \begin{array}{c}{z^{\prime}}\\{\bar{z}^{\prime}}\\{\bf 1}\\ \end{array}\right) ={\bf M}\left( \begin{array}{c}{z}\\{\bar{z}}\\{\bf 1}\\ \end{array}\right) \equiv \left( \begin{array}{c c c}{u}&{0}&{t}\\{0}&{\bar{u}}&{\bar{t}}\\{0}&{0}&{{\bf 1}}\\ \end{array}\right) \left( \begin{array}{c}{z}\\{\bar{z}}\\{\bf 1}\\ \end{array}\right) \ .
r_2(\xi,\eta,\rho) = O \Big( \frac{|\xi-\rho|^{3/2}+ |\rho-\eta|^{3/2}}{|\eta|^{3/2}+ |\xi|^{3/2}}\mathbf{1}_{[2^2,\infty)}(|\eta|/|\xi-\rho|) \Big) ,
\delta\mathcal{R}^{(1)}_s=\varrho_{s\chi\eta}[e_1(s\chi\eta_t\partial_z+s\chi \bar{\eta}_t\partial_{\bar{z}}) &+e_2(s\chi_{\bar{z}}\bar{\eta}\partial_z-s\chi_z\bar{\eta}\partial_{\bar{z}}) &+e_3(-s\chi_{\bar{z}}\eta\partial_z+s\chi_z\eta\partial_{\bar{z}})]
\lim_{\epsilon\to0}\int_{D_j}(v''_{\epsilon}(\Psi))|f|^2e^{-u(-v_{\epsilon}(\Psi))-\Psi}= \int_{D_j}\frac{1}{B}\mathbb{I}_{\{-t_{0}-B<\Psi<-t_{0}\}}|f|^2e^{-u(-v_{t_0,B}(\Psi))-\Psi}\leq \left(\sup_{D_j}e^{-u(-v_{t_0,B}(\Psi))}\right)\int_{D_j}\frac{1}{B}\mathbb{I}_{\{-t_{0}-B<\Psi<-t_{0}\}}|f|^2e^{-\Psi}< +\infty.
F^\theta_s = f(s, Y_s, Z_s)- \theta f^\prime (s, Y_s^\prime, Z^\prime_s) -\rho_s \delta_\theta Y_s, =\theta \delta f_s + \big(f(s, Y_s, Z_s) - f(s, Y_s^\prime, Z_s)\big) + \big(f(s, Y_s^\prime, Z_s)- \theta f(s, Y^\prime_s, Z^\prime_s)\big) -\rho_s \delta_\theta Y_s.
P_1 = 1 P_5 = -(C - B) \cdot B^{8}P_2 = - B P_6 = - B^{12}\cdot (C^{2}- B + C) P_3 = - B^{3}P_7 = B^{16}\cdot (C^{3}- B^{2}+ B C) P_4 = C \cdot B^{5}P_8 = C \cdot B^{21}\cdot (B C^{2}- 2 B^{2}+ 3 B C - C^{2})
\phi^{( 0 )}\ = \ \left[ \begin{array}{c l}{{\displaystyle 0 \ ,}}&{{r \le R_{g}}}\\{{\displaystyle v \left( 1 - \frac{K_{0}( m_{H}r )}{\ln ( 2 / m_{H}R_{g})}\right) ,}}&{{r \gg R_{g}.}}\\ \end{array}\right.
\alpha_k = \alpha_{k,K^{*}}\leq \frac{1}{\binom{k-1}{l-1}}\sum_{\{L_i \subseteq K^{*},\; |L_i|=l\}}\alpha_{l,L_i}\leq \frac{\binom{k}{l}}{\binom{k-1}{l-1}}\alpha_l = \alpha_l \cdot \frac{k}{l}.
M_a(\lambda) \asymp \sum_{\nu_1=1}^\infty\cdots\sum_{\nu_r=1}^\infty \frac{\prod_{k=1}^r\nu_k^{2ma}}{\left[1+ \lambda\prod_{k=1}^r\nu_k^{2m}(1+\nu_r^2)^{-1}\right]^{2}}& \asymp \int_1^\infty\int_1^\infty\cdots\int_1^\infty\left[1+\lambda x_1^{b}\cdots x_{r-1}^{b}x_r^{b(m-1)/m}\right]^{-2}dx_1\cdots dx_{r-1}dx_r,
\lim_{t \to \infty}F^{-1}(t) = \lim_{t \to \infty}\frac{t^{\frac1p-1}}{\psi \left(t^{- \frac{1}{n}}\right)^{n-1}}=\lim_{t \to \infty}t^{\frac1p-\frac1n}\left (\frac{t^{- \frac1n}}{\psi \left(t^{- \frac{1}{n}}\right)}\right)^{n-1}\ge \lim_{t \to \infty}\frac{t^{\frac1p-\frac1n}}{\phi (1 )^{n-1}}= \infty.
1+x+\frac{x^{2}}{2!}(1-\frac{1}{n})+\cdot\cdot\cdot+\frac{x^{m}}{m!}(1-\frac{1}{n})(1-\frac{2}{n})\cdot\cdot\cdot(1-\frac{m-1}{n})\le t_{n}.
\left| \langle \Pi_0 (\partial_{z_j}^T) , \nabla_{E_i}E_i \rangle \right| = \left| \langle v^T , \nabla_{E_i}E_i \rangle \right| = \left| \langle \Pi (v) , \nabla_{E_i}E_i \rangle \right| \leq \delta \, |v| \, |A| \leq 2 \, \delta \, |A| \, .
D^{2}_{D}\sim \sum_{i = 1}^{N}\mathcal{E}_{i}\mathcal{E}_{i}^{*}\otimes (\gamma_{i}\gamma_{i}^{*}+ q^{-2}\gamma_{i}^{*}\gamma_{i}) - q^{-1}(q - q^{-1})\sum_{i = 1}^{N}\sum_{j = i + 1}^{N}q^{i - j}\mathcal{E}_{j}\mathcal{E}_{j}^{*}\otimes \gamma_{i}^{*}\gamma_{i}.
\int_{\Theta_{d}}f(\theta )g(\theta )\mathrm{d}(\mathrm{F}_{\ast}\mathrm{P})\left( \theta \right) =\int_{\Theta_{d}}f(\theta )\mathrm{d}(\mathrm{F}_{\ast}\mathrm{P})\left( \theta \right) \int_{\Theta_{d}}g(\theta )\mathrm{d}(\mathrm{F}_{\ast}\mathrm{P})\left( \theta \right) \ .
c_s(\nu) = \frac{\chi(\nu)}{10},\ \ \bar{c}_s(\nu) = \hat{n}\left( \frac{\chi(\nu)}{10}+ 1\right), \ \ \chi(\nu) = \begin{cases}16 \nu \quad \nu \leq 2, 16 \cdot 2 + 13 (\nu - 2) .\end{cases}
\left\|P\int_0^T U_0(t-s)f(s,x)ds\right\|_{L_t^{\theta_p}(L^{p,2}_{D})}=\left\|U_0(t)P\int_0^T U(s)^*f(s,x)ds\right\|_{L_t^{\theta_p}(L^{p,2}_{D})}.
\frac{\partial f_j}{\partial \lambda_k}=-\frac{1}{m|g|}\sum_{h,i=0}^{m-1}[\tilde{v}_i\tilde{w}_i]_{j,k}[\tilde{v}_{i-h-1}\tilde{w}_{i-h-1}]_{k,j}\frac{\partial}{\partial\lambda_j}\frac{\lambda_j^{h+1}\lambda_k^{m-h-2}}{\lambda_j^m-\lambda_k^m}.
\Big(\mathbb{G}^{(\mathrm{k})}- \mathbb{G}^{(0)}\Big)(\mathrm{x},\mathrm{y}) = a |\mathrm{x}-\mathrm{y}|^2\log{|\mathrm{x}-\mathrm{y}|}+ b |\mathrm{x}-\mathrm{y}|^2 + \mathcal{O}(|\mathrm{x}-\mathrm{y}|^3),
\begin{aligned}g_1 (g_1 g_2) \neq (g_1 g_2) g_1 \mbox{and}g_1 g_3 \neq g_3 g_1 \,, \varphi (g_1 (g_1 g_2)) = \varphi (g_1) \varphi (g_1 g_2) \mbox{and}\varphi (g_1 g_3) = \varphi (g_3) \varphi (g_1) \,, \end{aligned}
Z_{\phi_1}(T, v, k) \geq Z_{\phi_1}(S_{n-s},z_{n-s}, s+k) + s\phi_1(1, s+t+k)\prod_{i=1}^t Z_{\phi_1}(S_{|T_i|},z_{|T_i|}, 1) &\geq (n-s-1)c^{(n-s)+(s+k)}+1+sc^{(s+t)+k+1}\prod_{i=1}^t ((|T_i|-1)c^{|T_i|+1}+1)
\langle x, y_0^*\rangle - F(x) = \langle x, y_0^*-x^*\rangle + F^*(x^*), =F^*(y_0^*) - \left( F^*(y_0^*)- F^*(x^*) - \langle x, y^*_0-x^*\rangle \right), \leq F^*(y_0^*) - (F^*(y_0^*) - F^*(x^*)- \sup_{z \in \partial F^*(x^*)\cap K}\langle z, y^*_0 - x^*\rangle.
\frac{1}{z_0}= \frac{\int_{i+2}^{i}\eta^4d\tau}{\int_{i+2}^{i+1}\eta^4d\tau}= 1+\frac{\int_{i+1}^{i}\eta^4d\tau}{\int_{i+2}^{i+1}\eta^4d\tau}= 1 + \chi(T^{-1}) = 1 + e^{-2\pi i/6},
[E(z),F(w)]=\sum_{k=0}^{n-1}s_3^{-1}a_{n,k}\delta(q^{2n-2k}s_1 z/w)T_k(q^{n-k}s_1 z)+\sum_{k=0}^{n-1}s_3a_{n,k}\delta(q^{2n-2k}s_1 w/z)\bar{T}_{k}(q^{n-k}s_1 w)\,,
\check{F}_{k-1,x}=-\displaystyle\frac{1}{w}\sum_{l=0}^{k-1}\check{F}_{l-1,x}\check{G}_{k-1-l}+\sum_{l=0}^{k+1}\left(2\check{F}_{l-1}\check{G}_{k-l}+\frac{2u}{w}\check{F}_{l-1}\check{H}_{k-l}\right),
p \equiv \frac{2 B_3 (B_1 - \alpha B_2)}{( \alpha - \beta ) (1 + B_1^2 - B_2^2)}, \hspace{0.3cm}q \equiv \frac{2 B_3 (B_1 - \beta B_2)}{( \alpha - \beta ) (1 + B_1^2 - B_2^2)}, \hspace{0.3cm}a \equiv \frac{(1 + B_1^2 - B_3^2)}{(1 + B_1^2 - B_2^2)}.
a_1(0)\left| \int_{\Gamma_1}(a\phi_{k\ell})^2d\sigma \right| =\frac{1}{\sqrt{\lambda_{k\ell}}}\left| w(a) ((a_1\otimes a_2)\phi_{k\ell})\right| &\hskip 5 cm\le \frac{1}{\sqrt{\lambda_{k\ell}}}\|w(a)\|_{V'}\|(a_1\otimes a_2)\phi_{k\ell}\|_V,
\int_{\Omega\times\mathbb{R}^N}f(x,y,\frac{\mathrm{d}\mu[v]}{\mathrm{d}|\mu[v]|}(x,y))\,\mathrm{d}|\mu[v]|(x,y) =\int_{\Omega}\int_{\mathbb{R^N}}f(x,y,\frac{\mathrm{d}Dv}{\mathrm{d}|Dv|}(x))\,\mathrm{d}|\lambda_x|(y)\,\mathrm{d}|Dv|(x) &=\int_{\Omega}\int_0^1f(x,v^{\theta},\frac{\mathrm{d}Dv}{\mathrm{d}|Dv|}(x))\,\mathrm{d}\theta\,\mathrm{d}|Dv|(x)
2\xi_n M_1(t,\xi_n)\begin{pmatrix}0 e^{-i\xi_n t}r_{N+1}(t)\end{pmatrix}& =\begin{pmatrix}\varphi_1(t)r_{N+1}(t)-e^{2i\xi_n t}\varphi_1(0) r_{N+1}(t) - r_{N+1}(t)\int_0^t e^{i 2\xi_n(x-t)}\varphi_1'(x)\,dx 0\end{pmatrix}.
L_D = \int_{\cal M}\varepsilon^{ijk}\bigg( \dots +{k \over 8 \pi}f_{abc}A^a_i A^b_j B^c_k +{1 \over 2 \pi}\chi^a_i \partial_j B_{ka}+{1 \over 4 \pi}f_{abc}\chi^a_iB^b_j B^c_k + \dots\bigg).
L(e_{ii})-L(e_{jj}) =\sum_{x<i}C_{xi}^{ii}e_{xi}+\sum_{x\in X}C_{xx}^{ii}e_{xx}+\sum_{y>i}C_{iy}^{ii}e_{iy}&-\sum_{x<j}C_{xj}^{jj}e_{xj}-\sum_{y\in X}C_{yy}^{jj}e_{yy}-\sum_{y>j}C_{jy}^{jj}e_{jy}.
\prod_{b = 1}^{l}\left[ - G_{1 2}\Big ( ( - L , 0 ) , ( L , 0 ) ; \varphi \Big ) G_{2 1}\Big ( ( - L , 0 ) , ( L , 0 ) ; \varphi \Big ) \right] \prod_{c = l + 1}^{N}G_{1 2}\Big ( ( - L , 0 ) , ( L , 0 ) ; \varphi \Big )
|\partial_{t}v_{1}(t,r)| \leq \frac{C r}{t^{2}\log^{b+1}(t)}\int_{0}^{1}B_{3}(r(1-\sigma)) d\sigma + \frac{C r B_{3}(r)}{t^{2}\log^{b+1}(t)}\leq \frac{C r}{t^{2}\log^{b+1}(t)}
P = \frac{1}{2 \alpha^{\prime}\dot{t}}\; \frac{\dot{R}^{2}- R^{2}}{( 1 + \frac{K}{4}R^{2})^{2}}= \frac{1}{2 \sqrt{b}\alpha^{2}}[ \frac{\dot{r}^{2}}{1 - K r^{2}}- r^{2}] = \frac{b \alpha^{2}- 2 r^{2}}{2 \sqrt{b}\alpha^{2}}.
\sqrt{\frac{3}{N}}\frac{1}{\pi}=\sum_{k=0}^\infty \frac{\left(\frac{1}{2}\right)_k\left(\frac{1}{3}\right)_k\left(\frac{2}{3}\right)_k}{(k!)^3}\left(\widehat{b}^\dagger_N k+\widehat{a}^\dagger_N\right)\left(\widehat{X}^\dagger_N\right)^k ,
dr=\sum_{i}\frac{\partial r}{\partial x_{i}}dx_{i}=\sum_{i}|\frac{\partial r}{\partial x_{i}}|\frac{\frac{\partial r}{\partial x_{i}}}{|\frac{\partial r}{\partial x_{i}}|}dx_{i}=\sum_{i}|\frac{\partial r}{\partial x_{i}}|dx_{i}\hat{x}_{i},
\dfrac{\partial^{(k)}}{\partial a_j}\left(a_{i_1}\otimes \ldots \otimes a_{i_{m}}\right) =\left(-1\right)^{|a_j|(|a_{i_1}|+\ldots |a_{i_{k-1}}|)}\delta_j^{i_k}\left(a_{i_1}\otimes \ldots \otimes\widehat{a_{i_k}}\otimes \ldots \otimes a_{i_{m}}\right)
\frac{|Q| - 2 + |I_u \cap I_v|}{t}= \frac{|S|}{t}< \omega \big( (G-uv)-S \big) = \omega(G-S) + 1 = \big( |I| - |I_u \cup I_v| + 1 \big) + 1 = |I| - |I_u \cup I_v| + 2 \,.
D(s, \varphi, \tilde \varphi, \Phi) \ = \L(s, \pi \times \tilde \pi) \cdot \prod_{v \in S}\frac{\Psi_v (s, W_v, \tilde W_v, \Phi_v)}{L_v (s, \pi_v \times \tilde \pi_v)}, \Re(s) \gg 0.
\Gamma_{\phi}^{i j}( \nu ) = \frac{\omega^{4}}{\sigma \nu ( \nu^{2}- \omega^{2})}\left( \delta_{i j}\cot \left( \pi \frac{\nu}{\omega}\right) + ( 1 - \delta_{i j}) \csc \left( \pi \frac{\nu}{\omega}\right) \right) \, .
\begin{array}{c}\left\{\left(i \mu \frac{\partial}{\partial x_1}+ \frac{\eta x_2}{2 \mu}\right)^2 -\left(i \mu \frac{\partial}{\partial x_2}- \frac{\eta x_1}{2 \mu}\right)^2\right. \left. -48 \exp \left[ -2 \sqrt{3}\left(\lambda x_2 + \frac{i \theta}{2 \lambda}\frac{\partial}{\partial x_1}\right) \right]\right\}\psi(x_1,x_2)=0\end{array}
\begin{aligned}\nu(t_1,t_2)= \int_{\{|x-y|=t_1, |z-y|=t_2\}}\,d\mu(x)\,d\mu(y)\,d\mu(z) = \int\omega_{t_1}(x-y)\,\omega_{t_2}(z-y)\,d\mu(x)\,d\mu(y)\,d\mu(z) = <T_{t_1}\circ T_{t_2}1,1 >_\mu\end{aligned}
( f \circ_i g) \circ_{i+j-1}h =~ f \circ_i ( g \circ_j h), ~ 1 \leq i \leq m,~ 1 \leq j \leq n, (f \circ_i g) \circ_{j+n-1}h =~ ( f \circ_j h) \circ_i g, ~ 1 \leq i < j \leq m,
\lim_{t\to 0^+}\frac 1t \, \left[ 1- \int_0^1 e^{-tv(1-v) \lambda_k}\,dv \right] = \int_0^1 v(1-v) \lambda_k \,dv = \frac 16 \, \lambda_k & = \frac 16 \, \lim_{t\to 0^+}\frac 1t \, \left[ 1- \int_0^1 e^{-t\lambda_k}\,dv \right].
\langle R_H (z) u , v \rangle = \langle R_{H_0}(z) u , v \rangle - \langle Z R_H (z) u , Y R_{H_0}(\bar{z}) v \rangle = \langle R_{H_0}(z) u , v \rangle - \langle Y R_{H_0}(z) u , Z R_H (\bar{z}) v \rangle .
\tilde\vartheta_2^2=\tilde\vartheta_3^2{\vartheta_4^2\over\vartheta_3^2}\left(1-4{\vartheta_2^4\over\vartheta_3^4}\right),\;\tilde\vartheta_4^2=\tilde\vartheta_3^2{\vartheta_2^2\over\vartheta_3^2}\left(1-4{\vartheta_4^4\over\vartheta_3^4}\right).
S ={\frac{1}{4 G}}A_{\S}- \int_{\Sigma}\left( 8 \pi a_{1}R + 4 \pi a_{2}R_{\mu \nu}n_{i}^{\mu}n_{i}^{\nu}+ 8 \pi a_{3}R_{\mu \nu \lambda \rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}\right) .
H_+^{\prime\mspace{2mu}\textsc{cs}}\equiv \frac{1}{2}\sum_{j=1}^N (z_j \, \partial_{z_j})^2 + \frac{\beta}{2}\sum_{i<j}^N \frac{z_i + z_j}{z_i - z_j}\, \bigl(z_i \, \partial_{z_i}- z_j \, \partial_{z_j}\bigr) \, ,
p(t,x)=\frac{1}{\sqrt{2\pi\sigma^{2}t}}\{exp[-\frac{(x-x_{0}-\nu t)^{2}}{2\sigma^{2}t}]-e^{2\nu(\alpha-x_{0})/\sigma^{2}}exp[-\frac{(x+x_{0}-2\alpha-\nu t)^{2}}{2\sigma^{2}t}]\}
(\nabla_X\theta_a)Y= (\nabla_X\phi)\phi_aY+\phi(\nabla_X\phi_a)Y-g(\nabla_X\xi_a,Y)\xi-\eta_a(Y)\nabla_X\xi \nabla_X\phi\xi_a= (\nabla_X\phi)\xi_a+\phi\nabla_X\xi_a.
U(s,t)(\omega) = \sqrt{4st(1-s)(1-t) \ln\left(\ln\left(\frac{st}{(1-s)(1-t)}\right)\right)}\frac{W\left(\frac{s}{1-s}, \frac{t}{1-t}\right)(\omega)}{\sqrt{4 \frac{st}{(1-s)(1-t)}\ln\left(\ln\left(\frac{st}{(1-s)(1-t)}\right)\right)}}\to 0
K_{i j}^{0}( \, p_{i j 0}^{\prime}, \, p_{i j 0}, \, P_{i j 0}) \, = \beta_{i}\beta_{j}\, \Lambda_{i j}^{+ +}\, \beta_{i}\beta_{j}\, K_{i j}^{T}( \, s_{i j}, \, s_{i j}, \, P_{0}- h_{k}\, ) \, \Lambda_{i j}^{+ +}
\rho_t = \rho^R_{t-s}\left( \rho_s \right) + \int_s^t \rho^R_{t-u}\left( \left[ \left( \alpha_R \left( u \right) a^\dagger - \overline{\alpha_R \left( u \right)}a \right)+\left( \overline{\beta_R \left( u \right)}\sigma^{-}- \beta_R \left( u \right) \sigma^+ \right) ,\rho_u \right] \right) du ,
(q;q)_\infty (-q^m;q^m)_\infty \sum_{n=0}^\infty a_k(n)q^n &=(-q^{m};q^{m})_\infty(q^{k};q^{k})_\infty\sum_{n=1}^{\infty}\frac{q^{kn}+ 2q^{2kn}+\dots+ (k-1)q^{(k-1)kn}}{1 + q^{kn}+ q^{2kn}+\dots+ q^{(k-1)kn}},
g_{S_{N}}(s) =\widetilde{g}_{N}\left( g_{x_{1}}(s)\right) =\exp(\left(g_{x_{1}}(s)-e\right) g) & =\exp(\left( e-p+sp-e\right) g)=\exp(\left( -p+sp\right) g) & =\exp(-pg+spg)=\exp(-\left( 1-s\right) pg).
\alpha_1 =\frac{1}{2}\left(4 n^2+4 n \alpha+3 \alpha^2+4n\lambda+2 \alpha \lambda +3\lambda^2\right), \beta_1 =-\frac{i}{2}(\alpha-\lambda )(\alpha+\lambda) (2 n+\alpha+\lambda ), \gamma_1 =\frac{1}{16}(2n+\alpha -\lambda) (2n-\alpha +\lambda) (2n+3\alpha +\lambda) (2n+\alpha+3 \lambda).
\langle f,g\rangle_{U}\equiv\int_{SO(3)}\langle\Pi(R)f,\Pi(R)g\rangle dg=\frac{1}{8\pi^{2}}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{2\pi}\langle\Pi(R)f,\Pi(R)g\rangle sin\theta d\phi d\theta d\psi,f,g\in V,
\begin{aligned}\|\underline{f}_{s-1}\|^2_{L^{2}}(2\rho_s, \tau_0) \leq C(\|\underline{\beta}_{s-1}, \underline{y}\|^2_{L^{2}(N_{s-1}\cap B_{4\rho_{s-1}})}+ \|\underline{f}_{s-1}\|_{L^2}^2(2\rho_{s-1}, \tau_0) &\leq 2C\|\underline{\beta}_{s-1}, \underline{y}\|^2_{L^{2}(N_{s-1}\cap B_{4\rho_{s-1}})}\end{aligned}
2n =n_1 +n_2 +\sum_{h=1}^{\frac{n-1}{2}}2m_h, \sum_{i=1}^{2n}\zeta^{\frac{(2n+3-2i)k}{2}}=n_1 +n_2 +\sum_{h=1}^{\frac{n-1}{2}}2m_h\cos (2\pi hk/n)\mbox{for}1\leq k\leq \widetilde{m}= (n-1)/2, -1 - \sum_{i=n+3}^{2n}c^{n+3-2i}P_{i-n-3,-i}=n_1 -n_2 \quad\psi_c^+.
Z^\flat(s) =\frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}L(s+s',\chi)H(s')ds'-\delta_\chi \frac{\varphi(q)}{q}H(1-s) &=\frac{1}{2\pi i}\int_{-\infty}^\infty H(it)L(s+it,\chi)dt &\ll \int_{-\infty}^\infty \frac{|L(s+it,\chi)|}{1+t^2}dt.
\begin{aligned}\Delta(x_1^{\pm1})=x_1^{\pm1}\otimes x_1^{\pm1}, &\Delta(x_i)=x_i\otimes x_1+x_1\otimes x_i,~1<i\leq n, &\varepsilon(x_1)= 1,~ \varepsilon(x_i)=0,~1<i\leq n, &S(x_1)=x_1^{-1},~S(x_i)=-x_1^{-1}x_ix_1^{-1},~1<i\leq n,\end{aligned}
H_{\beta \gamma}=\beta \delta_{\beta, \gamma},\quad D_{\beta \gamma}= \delta_{\beta, \gamma}D_{\beta},\quad D_{\beta}=-ig_l\left(z(\beta\cdot q) +\sum_{\kappa\in\Delta_l,\ \kappa\cdot\beta=1}z(\kappa\cdot q)\right),
\partial(x_{1}\wedge\cdot\cdot\cdot\wedge x_{p+1})=\frac{1}{p+1}\sum_{j<l}(-1)^{j+l+1}[x_{j},x_{l}]\wedge x_{1}\wedge\cdot\cdot\cdot\wedge\hat{x}_{j}\wedge\cdot\cdot\cdot\wedge\hat{x}_{l}\wedge\cdot\cdot\cdot\wedge x_{p+1}.
P_{b}(a,b,b)=\frac{-1}{8\pi^{2}b^{3}}\sum_{l=0}^{\infty}(2l+1)\int_{0}^{\infty}dz\frac{I_{\nu}^{(a)}(az)/I_{\nu}^{(b)}(bz)}{K_{\nu}^{(a)}(az)I_{\nu}^{(b)}(bz)-K_{\nu}^{(b)}(bz)I_{\nu}^{(a)}(az)},
R_{c}\simeq \frac{1}{4}\ \bigg [ \frac{8 ( J_{x}^{2}- J_{z}^{2})}{J_{x}( J_{x}- J_{y})}\bigg ]^{\frac{1}{2 - 2 \eta}}\quad \qquad \Big ( \eta = \frac{1}{\pi}\arccos ( J_{z}/ J_{x}) \Big )
\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}.