harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\{n \}}\Psi_{P , \{n \}}( z^{\prime}) \overline{{\Psi_{P , \{n \}}( z )}}= \sum_{\{n \}}\Psi_{P , \{n \}}( z^{\prime}) \Psi_{P , \{n \}}( \tilde{z}) = < z^{\prime}, \tilde{z}>^{P}\: ,
\hat{I}_{\varepsilon,j}^1 + \Delta\big(\varphi_jc\big) = \psi_j\mathcal{L}^{\mathbb{R}^n_{\gamma_j}}_\varepsilon\big(\varphi_jc\big) + \Delta\big(\varphi_jc\big) = \psi_j\Big(\mathcal{L}^{\mathbb{R}^n_{\gamma_j}}_\varepsilon\big(\varphi_jc\big) + \Delta\big(\varphi_jc\big)\Big).
\begin{cases}\partial_{t}\rho - \epsilon \Delta \rho = F D \times \bigcup_{k=0}^{\frac{T}{2\tau}-1}(t_{2k},t_{2k+1}] \partial_{t}\rho = 0 D \times \bigcup_{k=0}^{\frac{T}{2\tau}-1}(t_{2k+1},t_{2k+2}] \frac{\partial \rho}{\partial n}= 0 \partial D \times [0,T] \rho(0,x)=\rho_{0}D\end{cases}
\{\cdot , \cdot\}_{(\lambda)}\Omega = -\frac{1}{F(\lambda)}d\cdot \wedge \, d\cdot \wedge\big(\sigma_{(\lambda)}+ \frac{g_{(\lambda)}}{r-1}\Theta\big) \wedge \frac{\Theta^{r-2}}{(r-2)!}\wedge dF^1(\lambda)\wedge\ldots \wedge dF^k(\lambda),
\| Du \|_{\ell^{2}_{\mathcal{N}}(\L \setminus B_{R})}= \bigg( \sum_{|\ell| > R}\sum_{\rho\in \mathcal{N}(\ell) - \ell}\big|D_\rho u(\ell)\big|^2 \bigg)^{1/2}< \varepsilon.
\langle A((-\Delta)^\alpha)u,v\rangle = \left\langle \alpha_1, ((-\Delta)^{a_1}u)v \right\rangle + \left\langle \alpha_2, ((-\Delta)^{a_2}u)v \right\rangle + \cdots + \left\langle \alpha_p, ((-\Delta)^{a_p}u)v\right\rangle.
\frac{\det(V_{F})}{\det((V_{\chi})_F)}= \det(I+(M_{\chi})_FV_F)=\det(I+M_{\sqrt{\chi}}VM_{\sqrt{\chi}}) = \begin{vmatrix}I_F V_F -(M_{\chi})_F I_F\end{vmatrix}=1+\sum\limits_{A\subset F,A\neq \phi}\prod\limits_{x\in A}\chi(x)V_A.
\beta(h) = \zeta\left(e^{i \widehat{p}_j\otimes \widehat{x}_j}e^{-i \widehat{p}_0\otimes \widehat{x}_0}(1\otimes h) e^{i\widehat{p}_0 \otimes \widehat{x}_0}e^{-i \widehat{p}_j\otimes \widehat{x}_j}\right)
\begin{array}{l}\tilde{Z}(\tilde{z}^{\prime})=G(g)^{-1}\tilde{Z}(\tilde{z})\tilde{K}(\tilde{z};g)~~~{}\bar{\tilde{Z}}(\tilde{z}^{\prime})=\bar{\tilde{K}}(\tilde{z};g)\bar{\tilde{Z}}(\tilde{z})G(g)\end{array}
\iint\psi_{v}^{\prime*}\psi_{e}^{\prime*}\mu_{e}\psi_{e}\psi_{v}d\tau_{e}d\tau_{n}\approx\int\psi_{v}^{\prime*}\psi_{v}d\tau_{n}\int\psi_{e}^{\prime*}\mu_{e}\psi_{e}d\tau_{e}.
D_1 \mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)(\zeta_- +{\mathfrak a}_-) - D_1^\flat\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)(\zeta_- +{\mathfrak a}_-) = D_1^\flat\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)\circ D\mathcal{J}_-(\phi_-,\phi_+,T)(\zeta_-+{\mathfrak a}_-).
\lambda^{\#,(n,k)}_h = i \, R_1 - h^{\frac 23}a'_n e^{-\pi i/3}+ h\, (2k-1)\, \frac{e^{-\pi i/4}}{\sqrt{2R_1}}& - h^{\frac 43}\frac{e^{-\pi i/6}}{a'_n}\left(\hat \kappa - \frac{1}{2R_1}\right) + O(h^{\frac 53})\,.
\int_{\frac{w}{2\lambda(t)}}^{\infty}\frac{R dR}{(1+R^{2})^{3}}\frac{4 R^{2}\lambda(t)^{2}}{2w\left(\sqrt{(1+w^{2}-R^{2}\lambda(t)^{2})^{2}+4R^{2}\lambda(t)^{2}}+1+w^{2}-R^{2}\lambda(t)^{2}\right)}&\leq \int_{\frac{w}{2\lambda(t)}}^{\infty}\frac{dR}{R^{5}}\frac{R^{2}\lambda(t)^{2}}{w}\leq \frac{C\lambda(t)^{4}}{w^{3}}, w \geq 1
(r^*)^*(X) =r^*(E-X)+||X||_{r^*}-r^*(E) &=r(E-(E-X))+||E-X||_r-r(E)+||X||_{r^*}&{\text ~~~~~~~~~}-r(E-E)-||E||_r+r(E) &=r(X)+\sum_{x\in X}(\lambda_P(\{x\})-r(\{x\})) &=r^\flat(X).
\mbox{\rm e}^{\mu\nu}\equiv g^{\mu\nu}-\frac{\eta\cdot k}{\mbox{\tt D}}\left(\eta^\mu k^\nu+\eta^\nu k^\mu\right)+\frac{k^2}{\mbox{\tt D}}\eta^\mu \eta^\nu + \frac{\eta^2}{\mbox{\tt D}}k^\mu k^\nu\ ,
Z_{\Psi}[ \eta ] = \int [ d \Phi^{A}] \exp \frac{i}{\hbar}\left\{W \left[ \Phi^{A}, \Phi_{A}^{*}= \frac{\partial \Psi}{\partial \Phi^{A}}\right] + \, \eta_{A}\Phi^{A}\, \right\}\, ,
[y_1, y_1]=y_4+\theta_1y_5, [y_1, y_2]=\frac{\alpha_3}{\alpha_1}y_4+\theta_2y_5, [y_2, y_1]=\frac{\alpha_6\gamma_3}{\alpha_1\beta_6}y_5, [y_2, y_2]=\frac{\beta_2\gamma_3}{\alpha_1\beta_6}y_5, [y_1, y_3]=\frac{\beta_4}{\beta_6}y_5, [y_3, y_1]=y_5, [y_2, y_3]=\frac{\gamma_1}{\gamma_3}y_4+\theta_3y_5, [y_3, y_2]=y_4.
G_{\sigma}^{( n )}( \zeta_{1}, \cdots , \zeta_{2 n}) = \sum_{\varepsilon_{j}= \pm \atop \varepsilon_{1}+ \cdots + \varepsilon_{2 n}= 0}v_{\varepsilon_{1}}\otimes \cdots \otimes v_{\varepsilon_{2 n}}G_{\sigma}^{( n )}( \zeta_{1}, \cdots , \zeta_{2 n})^{\varepsilon_{1}\cdots \varepsilon_{2 n}}, ~ ( \sigma = \pm ) .
\Theta = \left\{(\theta, z), ~ \theta=[\theta_1,\cdots,\theta_k] \in \mathbf{R}^{d \times k}, ~ \Delta \le \min_{g \neq h}\|\theta_g - \theta_h\|, \right. \left. z: [n] \to [k], ~|\{i \in [n], z_i = u\}| \ge \alpha n, \forall u \in [k] \vphantom{\int_1^2}\right\}.
\Phi_h ((a,u) \cdot_H (b,v)) =~ \Phi_h (ab, a \cdot v + u \cdot b + H(a,b)) =~ (ab, a \cdot v+u \cdot b + H(a,b) - h (ab)) =~ (ab, a \cdot v + u \cdot b + H(a,b) - a \cdot h(b) - h(a) \cdot b + (\delta h)(a,b) ) =~ (a , u - h(a)) \cdot_{H + \delta h}(b, v - h (b)) =~ \Phi_h (a,u) \cdot_{H + \delta h}\Phi_h (b,v).
\tau_g(b)^* =Q\circ \pi((c(g),b)^*)=Q\circ \pi(c(g)^{-1},b^*) &=Q\circ \pi(c(g)^{-1}c(g^{-1})^{-1}c(g^{-1}),b^*)=Q\circ \pi((c(g^{-1})c(g))^{-1}c(g^{-1}),b^*) &=Q\circ \pi(c(g^{-1}),\overline{\gamma(c(g^{-1})c(g))}b^*)=Q\circ \pi(c(g^{-1}),\overline{\gamma(\sigma(g^{-1},g))}b^*),
\frac{A_{nh}5\pi\sqrt{n}}{2\cdot 5^n\exp\bigl(-\frac{5h^2}{4n}\bigr)}= \frac{25h\sqrt{5\pi}\exp(\alpha^2)}{4n^2}G(\alpha)\Bigl(1+ O\Bigl(\frac hn +\frac{h^4}{n^3}+\frac{\log n}{n^{1/2-\varepsilon}}\Bigr)\Bigr).
\overline{G}_{\sigma}^{(n)}(\zeta', x^{2}\zeta_{2n})^{\varepsilon'\,\varepsilon_{2n}}=\overline{G}_{\sigma}^{(n)}(\zeta_{2n},\zeta')^{\varepsilon_{2n}\,\varepsilon'}\prod_{j=1}^{2n-1}\frac{\zeta_{2n}}{\zeta_j}.
Z = \int \big [{\frac{( \Pi_{i = 0}^{2}\mathrm{d e t}\Delta_{0 , i}^{( - 1 )^{i}})^{2}( \Pi_{i = 0}^{2}\mathrm{d e t}\Delta_{i , 0}^{( - 1 )^{i}})^{2}}{( \Pi_{i = 0}^{2}\mathrm{d e t}\Delta_{i , 1}^{( - 1 )^{i}})^{- 1}( \Pi_{i = 0}^{2}\mathrm{d e t}\Delta_{1 , i}^{( - 1 )^{i}})^{- 2}( \Pi_{i = 0}^{2}\mathrm{d e t}\Delta_{0 , i}^{( - 1 )^{i}})^{2}}}\big ]^{\frac{1}{4}}.
2a_j = |T_{x_j = 1}|+|F_{x_j = 0}| = \left( |T_{x_i = 1, x_j = 1}| + |T_{x_i = 0, x_j = 1}|\right) + \left( |F_{x_i = 0, x_j = 0}| + |F_{x_i = 1, x_j = 0}|\right) = \left( |T_{x_i = 1, x_j = 1}| + |F_{x_i = 1, x_j = 0}|\right) + \left( |T_{x_i = 0, x_j = 1}| + |F_{x_i = 0, x_j = 0}| \right).
H_n(x)\geq \frac{1}{2}-\frac{1}{2\cos(x/2)}\geq \frac{1}{2}-\frac{1}{2\cos(1.34)}=-1.68...>-1.72...=-\frac{0.29 \cdot 4^{11}}{{22\choose 11}}=-\frac{0.29}{b_{22}}.
\langle \theta_{\varphi^{\vee}}f , \theta_{\varphi^{\vee}}f \rangle = \frac{\mathrm{vol}\, [H]}{\kappa_{m,2n}}\int_{[G \times G]}E(\iota(g_1,g_2),s_{m,2n},\Phi_{\varphi \otimes \overline{\varphi}}) \, \overline{f(g_1)}\, f(g_2) \, dg_1 \, dg_2 \ .
\frac{1}{( z^{2}- t^{2})^{\kappa + 1}}= \frac{1}{z^{2 \kappa + 2}}\sum_{k = 0}^{\infty}\left( \begin{array}{c}{\kappa + k}\\{k}\\ \end{array}\right) \left( \frac{t}{z}\right)^{2 k}.
2(1-a)\Biggl\{\sum_{i=0}^{k-2}(i+1)a^{i}- \binom{k}{2}a^{k-2}\Biggr\}<(b-1)\Biggl\{(k-1)k+\sum_{i=0}^{k-2}(k-1-i)a^{i}\Biggr\}\,.
\alpha^{\prime}\left( \frac{c}{\hbar}\right) m^{2}= 2 4 \sum_{n > 0}\frac{2 n^{2}- H^{2}m^{2}\frac{\alpha^{\prime 2}}{c^{2}}}{\sqrt{n^{2}- H^{2}m^{2}\frac{\alpha^{\prime 2}}{c^{2}}}}+ 2 N
A(e_1,p_1;e_2,p_2;e_3,p_3)=\frac{1}{\alpha^{'}G^2_0}\langle:cV_{-1}(p_1,x_1)::cV_{-1}(p_2,x_2)::cV_{0}(p_3,x_3):\rangle+1\leftrightarrow 2
E_H:=\big\{\left. (F_i,F_\ell)\in V_H \times V_H \;\right| \; (F_i,v_k,F_\ell) \textnormal{is an alternating}\{F_i,v_k\}\in E\setminus \mathcal{M},\{v_k,F_\ell\}\in\mathcal{M}\big\}.
\begin{array}{l}(x_1+vx_2)(x_3+vx_4):=x_1x_3+\mu(x_2x_4^\theta)^\theta+v(x_1^\theta x_4+(x_2^\theta x_3^\theta)^\theta), begin{align*}4pt]\overline{x_1+vx_2}:=x_1-vx_2^\theta\end{array}
L_{eff}=\int d^{2}x\;[(\partial_{t}\mid\phi\mid)^{2}+ (\partial_{t}\mid\chi\mid)^{2}]+\delta L_{eff}+ \kappa\int d^{2}x\;[\frac{B_{2}}{q_{1}}\dot{\omega}_{1}+ \frac{B_{1}}{q_{2}}\dot{\omega}_{2}] \;\;.
A_0\otimes 1 |m_1,l_1,m_2,l_2\rangle = (m_1 + l_1)|m_1,l_1,m_2,l_2\rangle, \Delta(A_0)|m_1,l_1,m_2,l_2\rangle = (m_1 + m_2 + l_1 + l_2)|m_1,l_1,m_2,l_2\rangle.
~^{ABR}~_{0}D^\alpha [x^{\nu-1}E^\sigma_{\alpha, \nu}(\lambda x^\alpha ) ]=\frac{B(\alpha)}{1-\alpha}\frac{d}{dx}[x^{\nu}E^{1+\sigma}_{\alpha, 1+\nu}(\lambda x^\alpha) ]=\frac{B(\alpha)}{1-\alpha}x^{\nu-1}E^{1+\sigma}_{\alpha, \nu}(\lambda x^\alpha)
\mathrm{e}^{\lambda\sum_{i=1}^d (y_i-x_i)-dn\ln(1+\lambda)}\le \mathrm{e}^{\lambda d(y_1-x_1) + d^2 - nd\lambda+nd\frac{\lambda^2}{2}}\le C_d \mathrm{e}^{d\frac{(y_1-x_1-n)}{\sqrt n}}.
\mathbb{P}\lim_{n\rightarrow \infty}\limsup_{m\rightarrow \infty}\sup_{t\in \lbrack 0,T]}\left\vert \sum_{i\in I_{n}([nt]+1)}\left(f(t_{i})-f(u_{[nt]})\right) \xi_{i,m}\right\vert =0.
m_1 :=\frac{|A|}{K_1}=\frac{|B|}{K_1}, r := \gamma m_1, r_1 := \gamma_1 m_1, r_2:= 96g^3K_1r, r_3 := \frac{rK_1}{L}, r^\diamond := r_1 +r_2 +r -(Lf -1)r_3, m_2 := \frac{|A|}{K_2}=\frac{|B|}{K_2}, D_4 :=D_1-2(Lfr_3+7r^\diamond).
||f||_{X}=_{t \geq T_{0}}\left(|f(t)| b \log^{b}(t) \sqrt{\log(\log(t))}+ |f'(t)| t \log^{b+1}(t) \sqrt{\log(\log(t))}+|f''(t)| t^{2}\log^{b+1}(t) \sqrt{\log(\log(t))}\right)
d_j D_s^i u_1(s_0) + e_j D_s^i v_1(s_0) + \nu_1 D_s^i u_1(s_0) D_t^j u_2(t_0) & + \nu_2 D_s^i u_1(s_0) D_t^j v_2(t_0) + \nu_3 D_s^i v_1(s_0) D_t^j u_2(t_0) + \nu_4 D_s^i v_1(s_0) D_t^j v_2(t_0) = D_s^i D_t^j f(s_0,t_0),
|f_{\lambda}'(z^{(n)}_k(\lambda))| =\frac{1}{d}\frac{\|\tilde{f}_{\lambda}(\tilde{z}^{(n)}_k(\lambda))\|^2}{\|\tilde{z}^{(n)}_k(\lambda))\|^2}|(\det(D\tilde{f}_{\lambda})(\tilde{z}^{(n)}_k(\lambda))|,
\left\|\frac{Pf}{\|Pf\|}-f\right\|^2 =1+1-\left \langle \frac{Pf}{\|Pf\|}, f\right\rangle -\left \langle f, \frac{Pf}{\|Pf\|}\right\rangle=2-\left \langle \frac{P^2f}{\|Pf\|}, f\right\rangle -\left \langle f, \frac{P^2f}{\|Pf\|}\right\rangle &=2-2\|Pf\|.
e^{itH_0}J_a^* E_\pm(t-s) =e^{itH_0}J_a^* J_a e^{-i(t-s)H_0}\tilde P_\pm &=e^{itH_0}(J_a^* J_a-I) e^{-itH_0}e^{isH_0}\tilde P_\pm+e^{isH_0}\tilde P_\pm.
|\nabla \overline{\eta}(t)|\le [(1-t)u^p+tv^p]^{(1-p)/p}[(1-t)u^{p-1}|\nabla u|+tv^{p-1}|\nabla v|] &= [(1-t)u^p+tv^p]^{(1-p)/p}[(1-t)^{1/p'}u^{p-1}(1-t)^{1/p}|\nabla u|+t^{1/p'}v^{p-1}t^{1/p}|\nabla v|] &\le [(1-t)u^p+tv^p]^{(1-p)/p}[(1-t)u^p+tv^p]^{1/p'}[(1-t)|\nabla u|^p+t|\nabla v|^p]^{1/p}~\mbox{a.e. in}~ \Omega
\Lambda^{(3)}_{n,\delta}=\exp \left( i\mu n^{\alpha+\frac 12}\int_0^{t-\delta}(t-s)^\alpha \xi^n_s dW_s +i \lambda n^{\alpha+\frac 12}\int_{t- \delta}^{\eta_n(t)}\psi_{n,1}(s,\eta_n(t)) dW_s \right) \times \left[ \exp \left(i\mu n^{\alpha+\frac 12}\int_{t-\delta}^t (t-s)^\alpha \xi^n_s dW_s \right)-1 \right]
\hat{x}^\mu_k |x^\mu_l,y^\nu_l\rangle=\,\sum_{l}\,\delta_{kl}\,x^\mu_l\,|x^\mu_l,y^\nu_l\rangle, \hat{y}^\nu_k |x^\mu_l,y^\nu_l\rangle=\,\sum_{l}\delta_{kl}\,y^\nu_l\,|x^\mu_l,y^\nu_l\rangle,
\aligned \frac{2^*_\mu-1}{22^*_\mu}S^{\frac{2^*_\mu}{2^*_\mu-1}}_{H,L}>m_{\nu_n}(a,b)=J_{\nu_n}(u_n,v_n)-\frac{1}{22^*_\mu}P_{\nu_n}(u_n,v_n) \geq \bigl(\frac12-\frac1{22^*_\mu}\bigr)(|\nabla u_n|^2_2+|\nabla v_n|^2_2)-C\nu_n=\bigl(\frac12-\frac1{22^*_\mu}\bigr)(|\nabla u_n|^2_2+|\nabla v_n|^2_2)+o_n(1).\endaligned
- \, S_{\, n}^{[ r ]}( p_{1}) \ = \ - \frac{1}{u_{r , 2 n + 1}}\, \frac{t ( p_{1})}{p_{1}{}^{2}+ u_{r , 2 n}\Lambda^{2}}\qquad ( \, n = 1 \ldots r \, ) \, ,
U = \Sigma_{m}\left[ \left(\bigcup_{j=0}^{s+t}V(j,s+t-j,0) \right)\cap Z( g_{1},\dots,g_{r}) \right] = \bigcup_{j=0}^{s+t}\Sigma_{m}\left( V(j,s+t-j,0) \cap Z( g_{1},\dots,g_{r}) \right) =: \bigcup_{j=0}^{s+t}U_{j}.
\sigma \left(\frac{L^S(r,\pi \boxtimes \chi, \varrho_{5})}{(2\pi i)^{3r}G(\chi)^3 C(\pi)}\right) = \frac{L^S(r,{}^\sigma\!\pi \boxtimes{}^\sigma\!\chi, \varrho_{5})}{(2\pi i)^{3r}G({}^\sigma\!\chi)^3 C({}^\sigma\!\pi)},
{\| \alpha_j(\xi,\eta,\rho)\cdot \varphi_{k_1}(\xi)\varphi_{k_2}(\eta)\varphi_{k_3}(\rho-\xi)\varphi_{k_4}(\rho-\eta)\|}_{S^\infty}\lesssim (2^{5k_3/2}+ 2^{5k_4/2}){(2^{k_1}+ 2^{k_2})}^{-1}, \mbox{for}\alpha_j(\xi,\eta,\rho) := a_j(\xi,\rho) - a_j(\xi+\eta-\rho,\eta) , j=1,\dots,4 ,
Z_{k}(\lambda_{1}^{2},\lambda_{2}^{2},q^{2};\lambda^{2})=\int_{0}^{1}dzz^{k}ln\left( \frac{q^{2}z(1-z)+(\lambda_{1}^{2}-\lambda_{2}^{2})z-\lambda_{1}^{2}}{(-\lambda^{2})}\right) .
\{H_{0},G_{n_0}\}= -\Big(f_{n_0}+f_{n_0}^*+\sum_{l=1}^{n_0-1}c_{l}\sum_{\substack{h_0+\ldots+h_{l}=n_0 h_i\ge1}}\{\{\ldots\{\{f_{h_0}+f_{h_0}^*,G_{h_1}\},G_{h_2}\}\ldots\},G_{h_l}\}\Big)\,,
\lim_{\vec{p}\rightarrow 0}\frac{C_3^{*}\left(\left|\vec{p}\right| \right)}{4p_0V}\sim \lim_{\vec{p}\rightarrow0}\frac{C_3^{*}\left(\left| \vec{p}\right| \right) \left|\vec{p}\right|^3}{4p_0}=0.
S=\int d^{4}x\sqrt{-g}[\frac{\mathcal{R}}{16\pi G}-\frac{1}{4\pi}((D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)+V(\Phi)+\xi\Phi^{\dagger}\Phi\mathcal{R})-\frac{1}{4\pi}\frac{1}{4}\mid F\mid^{2}].
\Phi_{n,m}\leq 2 e^{-2n (1/2 - ( i' - 1 )/n)^2}& = 2 e^{-2n (1/2 - ( \lceil (n-m)/2 \rceil - 1)/(n) )^2}& \leq 2 e^{-2n(\frac{1}{2}- \frac{(n-m)/2}{n})^2}& \leq 2 e^{-m^2/(2n)}
\Leftrightarrow \bar{\xi}={\frac{1}{2^{\frac{1}{4}}}}\left( \begin{matrix}{{0}}&{{- \Psi^{T}}}\\ \end{matrix}\right) , \, \hspace{2 m m}\xi \equiv{\frac{1}{2^{\frac{1}{4}}}}\left( \begin{matrix}{{\Psi}}\\{{0}}\\ \end{matrix}\right)
0\leq \left\lceil\frac{n}{n-d}\right\rceil-1-d =\left\lceil1+\frac{d}{n-d}\right\rceil-1-d &=\left\lceil\frac{d}{n-d}\right\rceil-d &\leq \left\lceil\frac{d}{2}\right\rceil-d &=\begin{cases}-\frac{d}{2}, \mbox{if}d\mbox{is even}\frac{1-d}{2}, \mbox{if}d\mbox{is odd}\end{cases}&<0,
l_1\Big(t_2(v_1, \,v_2)\Big) + l_2\Big(t_1(v_1),\, t_1(v_2)\Big) = \,t_1\left(\Phi_{t_1(v_1)}v_2 + (-1)^{|v_1|\,|v_2|}\Phi_{t_1(v_2)}v_1 + m_2(v_1,v_2)\right) + t_2\Big(m_1(v_1), v_2)\Big) + (-1)^{|v_1|}t_2\Big(v_1,m_1(v_2)\Big).
\overset{\cdot}{z}\left( t\right) =\lambda_{1}\left( \alpha \right)z\left( t\right) +\psi_{1}\left( \alpha \right) \left( 0\right) f\left( \left[ S_{\alpha}\left( t\right) \phi \right] \left( 0\right) ,\left[S_{\alpha}\left( t\right) \phi \right] \left( -r\right) ,\alpha \right) ,
\nu(g(V)) = \sum_i \nu(g(U_i)) \leq \sum_i (1+\epsilon)^3 |g'(\xi_i)|^2 \nu(U_i) \leq (1+\epsilon)^3 \sum_i \int_{U_i}(|g'(\xi)|^2 + \epsilon) d\nu(\xi) = (1 + \epsilon)^3 \left(\int_V |g'(\xi)|^2 d\nu(\xi) + \epsilon \right)
Q(F_a, 0)\le c_3^{r+1}\,\biggl(\frac{1}{m\sqrt{p(0)\,\beta_{r,m}(M^*,0)}}+\frac{(r+1)^{5r/2}}{(p(0)\,\beta_{r,m}(M^*, 0))^{(r+1)/2}}\biggr),\quad\hbox{if}\tau=0,
c \le h ( b ) \equiv \frac{1}{2 1 6}\left[ 9 + 1 5 0 b + 9 7 b^{2}+ H ( b )^{1 / 3}+ \frac{( 3 + b )^{2}\left( 9 + 6 b - 2 5 9 1 b^{2}\right)}{H ( b )^{1 / 3}}\right] ,
\lim_{n\to \infty}\|\psi_n(t\otimes f_{e,g})(a u_k)\psi_n(t\otimes f_{h,e})-\delta_{g,kh}\psi_n(t\otimes f_{e,e})\alpha_{g^{-1}}(a)\psi_n(t\otimes f_{e,e})\|= 0.
{\left<{\Phi^{\ast}}_{0 1 2 3}^{- 1}( \overline{{\phi}}_{0 1 2 3}) , \Phi_{0 1 2 3}( \psi_{0 1 2 3}) \right>}^{\tilde{}}= \left< \overline{{\phi}}_{0 1 2 3}, \psi_{0 1 2 3}\right> ,
\lim_{t\downarrow 0}\left<\tau_\varphi,[V(tD)]-[e_1]\right>=\lim_{t\downarrow 0}{\rm Tr}^\chi_s(\Pi(t))=\frac{1}{(2\pi)^{\dim M}}\int_{T^*M}\chi(x){\rm tr}_s(\sigma_{t^{-1}}(\Pi(t)))(x,\xi) dxd\xi,
(1) h(xy) = xh(y) + yh(x). (2) h(1-x) \geq x - x^2/2 - x^3/2 \;\;\; 0 \leq x \leq 1. (3) h(1-x) \leq x - x^2/2 \;\;\;\; 0 \leq x \leq 1.
\begin{array}{l}v=v^1e_1 + v^2e_2 + v^3e_3, begin{align*}4pt]v^1=-\{c_1\cosh x^3 - c_2\sinh x^3\}x^1+\{c_2\cosh x^3 -c_1\sinh x^3\}x^2 + \sinh x^3, begin{align*}4pt]v^2=\{c_2\cosh x^3 -c_1\sinh x^3\}x^1-\{c_1\cosh x^3 - c_2\sinh x^3\}x^2 + \cosh x^3, begin{align*}4pt]v^3=c_3,\end{array}
\|[Z_p,Z_q]\|_{p+q}=\bigg\|\bigg[\sum_{n=1}^{d_1^p}\big[X_{n1}^{(p)},\dots,X_{np}^{(p)}\big], \sum_{m=1}^{d_1^q}\big[X_{m1}^{(q)},\dots,X_{mq}^{(q)}\big]\bigg]\bigg\|_{p+q}\leq\sum_{n=1}^{d_1^p}\sum_{m=1}^{d_1^q}\big\|\big[\big[X_{n1}^{(p)},\dots,X_{np}^{(p)}\big], \big[X_{m1}^{(q)},\dots,X_{mq}^{(q)}\big]\big]\big\|_{p+q}.
\left( N_{q}f\right) \left( u;v\right) =\frac{\left( q;q\right)_{\infty}}{\left( 1-q\right)^{n}}\sum_{k\in\mathbb{Z}}\frac{q^{k}f\left( q^{k}\right)}{\left( -\dfrac{v}{u};q\right)_{k+1}}.
\mathcal A\left(\mathbb{H}_{\mathbf{f}}\left([y_{\textup{f}}],[x_{\textup{f}}]\right)\right)(\kappa,s)=\mathcal A\left(\mathbb{H}_{\mathbf{f}}\left([x_{\textup{f}}],[y_{\textup{f}}]\right)\right)(\kappa,\kappa-k-s),
|T(f)(q_{l_n})|= |\int fd\mu_n|= |\int f_nd\mu_n+\int\sum_{m\in B\backslash \{n\}}f_m|\geq |\int f_nd\mu_n|- |\int\sum_{m\in B\backslash \{n\}}f_m|\geq \varepsilon - \varepsilon/3=2\varepsilon/3.
\mathcal{A}_{0}:= \mathcal{A}_{0}[\phi],\mathcal{A}_{1}:= c - f'(\phi) - \mathcal{K}* + k\partial_{\theta}\mathcal{K}_{1}*,\mathcal{A}_{2}:= -\frac{1}{2}k\partial_{\theta}\mathcal{K}_{2},
- \ddot{\tilde{\chi}}+ \tilde{\chi}^{\prime \prime}- \frac{1}{r}\tilde{\chi}^{\prime}- m_{e f f}^{2}\tilde{\chi}+ \kappa^{2}\frac{q^{2}c_{1}^{2}}{m_{H}^{2}}K_{0}^{2}( k_{0}r ) \cos ( 2 \omega_{0}t + 2 \delta ) \tilde{\chi}= 0 ,
\begin{array}l Q_{n+1}(z) = \big[(1+ic_{n+1})z+(1-ic_{n+1})\big] Q_{n}(z) - 4 d_{n+1}z Q_{n-1}(z), begin{align*}2ex] R_{n+1}(z) = \big[(1+ic_{n+1})z+(1-ic_{n+1})\big] R_{n}(z) - 4 d_{n+1}z R_{n-1}(z), \end{array}n \geq 1,
\begin{cases}\|e^{\pm,4}_k\|_{s,X}\le C\delta^2\theta_k^{L_4(s)}\Delta_k, s_0\le s\le s_1-2\cr\|\overline{e}_{\pm,k}^{(4)}\|_{s,X}\le C\delta^2\theta_k^{L_5(s-\frac 12)}\Delta_k, s_0+\frac 32\le s\le s_1-\frac 72\cr\|\tilde{e}_{k}^{(4)}\|_{H^s(\omega_X)}\le C\delta^2\theta_k^{L_5(s)}\Delta_k, s_0+1\le s\le s_1-4\end{cases}
B_{n,i}^* = \prod_{j=1}^{n-1}\frac{(3j-1)(2j-1)!(6j-3)!}{(4j-2)!(4j-1)!}\sum_{r=1}^{i-1}(-1)^{i+r-1}\frac{\binom{2n+r-2}{2n-1}\binom{4n-r-1}{2n-1}}{\binom{4n-2}{2n-1}}, i=1,\ldots,2n+1.
P\left(\bigcup_{i=1}^N A_i\right)\ge \sum_{i=1}^N\left\{\left[\frac{1}{\lfloor\frac{\gamma_i}{\alpha_i}\rfloor}-\frac{\frac{\gamma_i}{\alpha_i}-\lfloor\frac{\gamma_i}{\alpha_i}\rfloor}{(1+\lfloor\frac{\gamma_i}{\alpha_i}\rfloor)(\lfloor\frac{\gamma_i}{\alpha_i}\rfloor)}\right]\alpha_i\right\},
\frac{f(\phi_{\tau_{i}+R^{-1}(p_{i})t}(p))}{|R^{-1}(p_{i})t|}= \frac{f(\phi_{\tau_{i}+R^{-1}(p_{i})t}(p))-f(\phi_{\tau_i}(p))}{|R^{-1}(p_{i})t|}+\frac{f(x_i)}{|R^{-1}(p_{i})t|}= \frac{|\int_{R^{-1}(p_{i})t}^0|\nabla f|^2ds|}{|R^{-1}(p_{i})t|}+\frac{f(x_i)}{|R^{-1}(p_{i})t|}\to R_{\max}(1+\frac{C_0}{|t|}),~{\rm as}~i\to \infty.
vv_{n-i}^{*}v_{k}v_{k}^{*}=vv^{*}v_{n-i-1}^{*}v_{n-i-1}v_{n-i-1}^{*}v_{k}v_{k}^{*}& =v_{n-i-1}^{*}v_{n-i-1}vv^{*}v_{n-i-1}^{*}v_{k}v_{k}^{*}& = v_{n-i-1}^{*}v_{n-i}v_{n-i}^{*}v_{k}v_{k}^{*}=v_{n-i-1}^{*}v_{\max\{n-i,k\}}v_{\max\{n-i,k\}}^{*},
g_1(q) := \frac{\left(q^6;q^6\right)_\infty}{4\left(q^2;q^2\right)_\infty\left(q^3;q^3\right)_\infty}f(q) =: \sum_{n=0}^\infty a(n)q^n,g_2(q) := \frac{3\left(q^3;q^3\right)^3_\infty}{4(q;q)_\infty\left(q^2;q^2\right)_\infty\left(q^6;q^6\right)_\infty}.
{\mathrm d}X_{t,i}\, =\, \frac{\,1\,}{\,n\,}\sum_{\stackrel{j=1}{j\neq i}}^{n}h \big( X_{t,i}, X_{t,j}\big){\mathrm d}t +{\mathrm d}W_{t,i}\, ; t \ge 0 \, , i \, =\, 1, \ldots , n \, .
\max (|k-k_1|, |k- k_2|, |k-k_3|) \geq 21, & \max (|k_1 - k_3| , |k_2-k_3|) \geq 5 \mbox{and}\min (k_1,k_2) \geq -\frac{48m}{100}\mbox{and}k_3 \leq -\frac{49m}{100}.
\begin{array}{l}v_{(0)}= e,v_{(1)}= s_4,v_{(2)}= v_{(3)}=s_5s_4,v_{(4)}= s_1s_5s_4,v_{(5)}= s_4s_1s_5s_4, begin{align*}2mm]v_{(6)}= v_{(7)}=v_{(8)}=s_2s_4s_1s_5s_4,v_{(9)}= v_{(10)}=v_{(11)}=v_{(12)}=v_{(13)}=v_{(14)}=v.\end{array}
\{\tilde\alpha_k , \alpha_\ell \}= (\ell-k) \alpha_{k+\ell}\{\tilde\alpha_k , \tilde\alpha_\ell \}= (\ell-k) \tilde\alpha_{k+\ell}\{\tilde\alpha_k , \beta_\ell \}= (2k+\ell+1) \beta_{k+\ell}\{\tilde\alpha_k , \tilde\beta_\ell \}= (2k+\ell+1) \tilde\beta_{k+\ell}
\mathbb{P}(E_x) \le{n \choose \lfloor x \rfloor}\frac{B_{n-\lfloor x \rfloor}}{B_n}\le \left(\frac{en}{x}\right)^x (1 + o(1))\sqrt{\frac{r_n}{r_{n-x}}}e^{-x(r_n - 1 + 1/r_n)}\le (1 + o(1)) \left(\frac{en}{x}\right)^x \left( \frac{n}{er_n}\right)^{-x}= (1 + o(1)) e^{-x(\ln x - \ln r -2)}.
\frac{1}{n}\sum_{k=1}^{n}\left[ e^{t\Delta}\right](x_k) = \left\langle e^{t\Delta}f, \frac{1}{n}\sum_{k=1}^{n}\delta_{x_k}\right\rangle = \left\langle f, \frac{1}{n}\sum_{k=1}^{n}e^{t\Delta}\delta_{x_k}\right\rangle.
p(xy)= \sum_{\delta_r+\gamma_s=\gamma}\tau(x'_rk_{\delta_r},y_s)\tau(x_r,S(k_{-\gamma_s}y'_s))k_{\delta_r-\gamma_s}= k_{-\gamma}\left(\sum_{\delta_r+\gamma_s=\gamma}\tau(x'_r,y_s)\tau(x_r,S(y'_s))k_{2\delta_r}\right),
\mathcal{J}_{\lambda}'(u)(\varphi) = \int_{\Omega}\langle \nabla u, \nabla \varphi\rangle \, dx + \alpha \iint_{\mathbb{R}^{2n}}\dfrac{(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\, dxdy &- \lambda \int_{\Omega}u(x)\varphi(x) \, dx- \int_{\Omega}f(x,u(x))\varphi(x)\, dx \textrm{for every}\varphi \in \mathbb{X}(\Omega)\,.
\begin{cases}\frac{\partial x}{\partial s_1}=\Re(f_1(0,0,s_3)+f_2(0,0,s_3)) \frac{\partial y}{\partial s_1}=\Im(f_1(0,0,s_3)-f_2(0,0,s_3)) \frac{\partial t}{\partial s_1}=\Re g(0,0,s_3),\end{cases}
\begin{aligned}\dot{x}= -a(\theta,t,\lambda) x^N + f_{N}(x,y,\theta,t,\lambda) + f_{\geq N+1}(x,y,\theta,t,\lambda) \dot{y}= x^{N-1}B(\theta,t,\lambda) y + g_{N}(x,y,\theta,t,\lambda) + g_{\geq N+1}(x,y,\theta,t,\lambda) \dot{\theta}= \omega + h_P(x,y,\theta,t,\lambda) + h_{\geq P+1}(x,y,\theta,t,\lambda),\end{aligned}
\omega(z)=\frac{i}{2\pi}\partial \overline{\partial}\log\bigl(\frac 1{1-|z|^2}\bigr)=\frac{1}{(1-|z|^2)^2}\sum^{n}_{j,k=1}[(1-|z|^2)\delta_{j,k}+z_k\overline{z_j}] \frac{i}{2\pi}dz_j\wedge d\overline{z_k}\ .
\rho ( b - c ) \sqrt{\frac{( 1 / 2 - d ) ( 1 / 2 - a )}{( 1 / 2 - c ) ( 1 / 2 - b )}}= ( a - c ) ( c - d ) ( K ( q ) - F ( \nu , q ) ) - \rho^{2}( E ( q ) - E ( \nu , q ) ) .
da_d(n-2) \leq a_d(n-1), -\binom{d}{2}a_d(n-3) \leq -d^3 \binom{d}{2}a_d(n-6), da_d(n-4) \leq d(d+1)^2 a_d(n-6), -d^2a_d(n-5) \leq -d^3a_d(n-6).
\sum_{\alpha >1}\left( \sum_{i}\mathring{\lambda}_{i}\mathring{h}^{\alpha}_{i i}\right)^{2}\le \sum_{\alpha >1}\left( \sum_{i}\mathring{\lambda}_{i}^{2}\right) \left( \sum_{i}\left( \mathring{h}^{\alpha}_{i i}\right)^{2}\right) =P_{1}Q_{1}.
\kappa_1^2 := \sum\limits_{i=1}^m \int_{\mathbb{R}^n}(\Pi\nabla \mathcal{L}\xi_i)\cdot (a\Pi\nabla \mathcal{L}\xi_i)\, d\mu < +\infty\,, \kappa_2^2 := \sum\limits_{1 \le i,j \le m}\int_{\mathbb{R}^n}(\Pi\nabla A_{ij})\cdot (a\Pi\nabla A_{ij})\, d\mu < +\infty\,.
\|{\mathbb{T}}\|^\ell_{\dot{B}_{2,1}^{\frac n2+2}}\ge c_0\big(\|{\mathbb{T}}\|^\ell_{\dot{B}_{2,1}^{\frac n2}}\big)^{\frac{1}{1-\theta_{1}}},\theta_1=\frac{4}{n-2\sigma+2}\in(0,1).
\mathbf{G}_{j}\mathbf{\perp \!\!\!\perp}_{\mathcal{G}}\left[ \overline{\mathbf{B}}_{j-1}\mathbf{\backslash}\overline{\mathbf{G}}_{j-1}\right] \mathbf{|}\overline{\mathbf{G}}_{j-1},\overline{\mathbf{A}}_{j-1}j=1,\dots ,p.
\mathbb{E}^{\tilde{\mathbb{P}}}\left[\left(\Pi-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi;\tilde{A}_{0}\right]\right)\left(\Pi-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi;\tilde{A}_{0}\right]\right)^{\top};\tilde{A}_{0}\right]=Q_{c},
0 = (V - E) \psi(1) + A^*\psi(2), 0 = A \psi(1) + (V - E) \psi(2) + A^*\psi(3), 0 = A \psi(2) + (V - E) \psi(3) + A^*\psi(4), & \vdots 0 = A \psi(n-1) + (V(k_x) - E) \psi(n) + A^*\psi(n+1).
\implies\begin{cases}a\left(r^k-\dfrac{1}{r^k}\right)=a\left(r^k+\dfrac{(-1)^{k+1}}{r^k}\right)=P_k \left(r^k+\dfrac{1}{r^k}\right)=\left(r^k+\dfrac{(-1)(-1)^{k+1}}{r^k}\right)=Q_k\end{cases}
\omega_1=2\mathbf{i}\int_{-\infty}^{e_3}\frac{d\mathbf{u}}{\sqrt{g_{2}\mathbf{u}+g_{3}-4\mathbf{u}^{3}}}\quad \omega_2=2\int_{e_1}^{+\infty}\frac{d\mathbf{u}}{\sqrt{4\mathbf{u}^{3}-g_{2}\mathbf{u}-g_{3}}}.
i_{abc}(t)=\sqrt{\frac{2}{3}}[\begin{matrix}1&0&\frac{1}{\sqrt{2}}\\ -\frac{1}{2}&\frac{\sqrt{3}}{2}&\frac{1}{\sqrt{2}}\\ -\frac{1}{2}&-\frac{\sqrt{3}}{2}&\frac{1}{\sqrt{2}}\\ \end{matrix}][\begin{matrix}i_{\alpha}(t)\\i_{\beta}(t)\\i_{\gamma}(t)\end{matrix}].
\begin{aligned}\|\nabla w-I-S\|_{L^{p(\cdot)}(Q')}=\|\nabla z-S\|_{L^{p(\cdot)}(Q')}\le C\|ez\|_{L^{p(\cdot)}(Q')}&= C\|ew-I\|_{L^{p(\cdot)}(Q')}\le C\|d(\nabla v,SO(n))\|_{L^{p(\cdot)}(Q)},\end{aligned}
( \dfrac{12bc}{ad}+ 20-\dfrac{c^3}{ad^2}-\dfrac{b^3}{a^2d}-\dfrac{12c^2}{bd}-\dfrac{12b^2}{ac}-\dfrac{3ac}{b^2}-\dfrac{3bd}{c^2}) = \dfrac{4 (n+1)^2 [- (n+5) k^2 + (n^2+4n-5)k- n^2+1 ]}{k^2 (k+1) (k+2) (n-k+1) (n-k) (n-k-1)^2}.
\begin{multlined}Z(\mathbf{o})=\sum_{\mathbf{c}}e^{-\beta\sum_p H_p(\mathbf{c})}\mathbf{O}(\mathbf{o}|\mathbf{c}) =\sum_{\mathbf{c}}\prod_{p_W\in P_W}\prod_{p_O\in P_O}W_{p_W}(c_{{p_W}^{(0)}},c_{{p_W}^{(1)}},\ldots) \cdot O_{p_O}(c_{{p_O}^{(0)}},c_{{p_O}^{(1)}},\ldots)\;.\end{multlined}
h_2 \left( I + \begin{pmatrix}a 0 0 d \end{pmatrix}p \right) h_2^{-1}= I + \frac{\begin{pmatrix}a \alpha^2 + d \beta \gamma a \alpha \beta - d \alpha \beta a \alpha \gamma - d \alpha \gamma a \beta \gamma - d \alpha^2 \end{pmatrix}p}{-\alpha^2 - \beta \gamma}.