harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
p_\tau = \frac{\partial L_\tau}{\partial \dot{t}_\tau}= -\frac{f(t_\tau)}{2m}(p_{1,\tau}^2 + p_{2,\tau}^2) - \frac{m\omega^2(t_\tau)f^{-1}(t_\tau)}{2}(x_{1,\tau}^2 + x_{2,\tau}^2).
\begin{array}{ll}\|\alpha_2\|_{\mathcal{C}_0^{k-2}}=\|\frac{u^s_{yyt}}{u^s_y}\|_{\mathcal{C}_0^{k-2}}\leq C_{\ell}\|\frac{u^s_{yyt}}{u^s_y}\|_{\mathcal{C}_{\ell}^{k-2}}\leq C\|\alpha\|_{\mathcal{C}_{\ell}^{k-1}}\leq C(T).\end{array}
\frac{|A|^2}{(2k)^{2l+3}}\leq \sum_{u,v}{1_{wA^{\sigma_{s_1}}}(u) 1_{A^{-\sigma_{s_2}}y^{-1}}(v)1_{Az^{-1}}(uv^{-1})}& = \langle 1_{wA^{\sigma_{s_1}}}, 1_{Az^{-1}}\ast 1_{A^{-\sigma_{s_2}}y^{-1}}\rangle_{\ell_2(G)}\leq \|1_{wA^{\sigma_{s_1}}}\|_{\ell_2(G)}\| 1_{Az^{-1}}\ast 1_{A^{-\sigma_{s_2}}y^{-1}}\|_{\ell_2(G)}.
\frac{\rho^2R''(\rho|\rho_{\rm tx})}{R(\rho|\rho_{\rm tx})}+\frac{\rho R'(\rho|\rho_{\rm tx})}{R(\rho|\rho_{\rm tx})}-\frac{\rho^2 T'(t|t_0)}{DT(t|t0)}=\frac{\Phi''(\varphi|\varphi_{\rm tx})}{-\Phi(\varphi|\varphi_{\rm tx})}\overset{(a)}{=}\alpha,
\lim_{h\to 0^+}\int_{x-h}^{x+h}\frac{\vert f(x)-f(t) \vert dt}{\vert x-t \vert}\geq \lim_{h\to 0^+}\int_{x-h}^{x+h}\frac{\vert f(x)-f(t) \vert dt}{2h}=0.
C_{1}(\lambda,b,\mathbf{m}) := \displaystyle\frac{\Lambda_{1}(\lambda,b)-b\Omega_{\mathbf{m}}^{\pm}(\lambda,b)-bI_{1}(\lambda b)K_{1}(\lambda b)}{\mathbf{m}}, C_{2}(b,\mathbf{m}) := \displaystyle -\frac{b}{\mathbf{m}}.
\limsup_{t \rightarrow \infty}\frac{1}{t}\sum_{\tau=0}^{t-1}\mathbb{E}p_k(\tau) \geq \limsup_{t \rightarrow \infty}\frac{1}{t}\sum_{\tau=0}^{t-1}\sum_{\omega \in \Omega}\mathcal{P}_{i^{*}}(\omega)\Phi_k(\tau) - \Delta_{\pi,\mathcal{P}_{i^{*}}},
\tilde{C}= - \frac{3}{2}g^{2}\delta^{a b}\int \frac{d^{4}k}{( 2 \pi )^{4}}d^{2}\theta d^{2}{\overline{\theta}}\frac{1}{( k^{2}+ m^{2})}\left\{V^{a}( p , \theta ,{\overline{\theta}}) V^{b}( - p , \theta ,{\overline{\theta}}) \right\}.
\frac{n+1}{4\pi}\int_{S^{2}}{\widetilde{W_{\vec{c}}^{j}}(P)}\W_{\vec{c}}^{j}(\dot{A})\dS=\frac{n+1}{4\pi}\int_{S^{2}}{W_{\vec{c}}^{j}(P)}\star_{\vec{c}}^{n}W_{\vec{c}}^{j}(\dot{A})\dS
\left(\sum_{j=1}^n f_j(\tau_j)\right)^2 =(\operatorname{Tra}(S_{f,\tau}))^2=\left(\sum_{k=1}^d \lambda_k\right)^2\leq d \sum_{k=1}^d \lambda_k^2 =d\operatorname{Tra}(S^2_{f,\tau})=d\sum_{j=1}^n\sum_{k=1}^nf_j(\tau_k)f_k(\tau_j).
\frac{(1-c^2b_1^hA_{j+1}^v)(1-c^2b_1^hA_{j-1}^v)+(c^2b_1^ha_j^va_{j-1}^v)^2}{(1-c^2b_1^hA_{j}^v)\big(1-c^2b_1^h(A_{j+1}^v-(a_{j-1}^v)^2)\big)}=1,
P_{1,n+1}^*(0) - P_{1,n+1}^*(a) = -a v_{n}^* H_{4,n}^{-1}H_{4,n}R_{n}^*(a) L_{1, n+1}^* \Sigma_{1, n+1}= -a v_{n}^* H_{4,n}^{-1}R_n(a)u_{4,n}v_{n+1}^* R_{n+1}^*(a) \Sigma_{1, n+1}= -a v_{n}^* H_{4,n}^{-1}u_{n}P_{1,n+1}^*(a),
-\int_{B_2}S\cdot (\nabla\phi\otimes (0,1))\, dV_t^{(m)}(\cdot,S)=-\int_0^1 (1+|\partial_{x_1}f_1^{(m)}|^2)^{-\frac12}\partial_{x_1}f_1^{(m)}\partial_{x_1}\phi\,dx_1.
\sum_{m=0}^{\infty}\sum_{\ell_1=0}^{\min\{n_1,m\}}\sum_{\ell_2=0}^{\min\{n_2,m\}}A_{m,\ell_1,\ell_2}=\sum_{\ell_1=0}^{n_1}\sum_{\ell_2=0}^{n_2}\sum_{m=\max\{\ell_1,\ell_2\}}^{n_1}A_{m,\ell_1,\ell_2}+\sum_{\ell_1=0}^{n_1}\sum_{\ell_2=0}^{n_2}\sum_{m=n_1+1}^{n_2}A_{m,\ell_1,\ell_2}+\sum_{\ell_1=0}^{n_1}\sum_{\ell_2=0}^{n_2}\sum_{m=n_2+1}^{\infty}A_{m,\ell_1,\ell_2},
\MoveEqLeft \triangle_k^N(f(k) g(k) h(k)) =\sum_{m_1+m_2+m_3=N}C_{m_1, m_2,m_3}\triangle_k^{m_1}f(k) \triangle_k^{m_2}g(k+m_1) \triangle_k^{m_3}h(k+m_1+m_2),
-\eta_{2}=\sigma\alpha_1=\frac{1}{\sigma\alpha_2+\eta_3}=\frac{1}{\sigma\alpha_2-\frac{1}{\sigma\alpha_3+\eta_4}}=\frac{1}{\sigma\alpha_2-\frac{1}{\sigma\alpha_3-\frac{1}{\sigma\alpha_4-\dots}}},
j_S(z) = \begin{pmatrix}0 1 -1 0 \end{pmatrix}, z\in \gamma, j_S(z) = \begin{pmatrix}1 1 0 1 \end{pmatrix}, z\in (4, \infty), j_S(z) = e^{\pi i \alpha \sigma_3}, z\in (-\infty,0), j_S(z) = \begin{pmatrix}1 0 1 1 \end{pmatrix}, z \in \Sigma^\gamma.
\|v\|_{L^{q}(t_1,t_2\,;\,L^{p}(K))}:=\begin{cases}\displaystyle \left(\int_{t_1}^{t_2}\|v(t)\|_p^{q}\,dt \right)^{1/q}(1 \leq q<\infty) \displaystyle \sup_{t_1 <t < t_2}\|v(t)\|_p (q=\infty),\end{cases}
\mathbb{E}[C(j+1)] = \frac{2|\textbf{F}_u(j)|\cdot[1+|\textbf{X}_{u,v}(j)|\cdot p]}{\sum_{v\in [n]}F_v(j)}&\sim \frac{2|\textbf{F}_u(j)|}{\sum_{v\in [n]}F_v(j)}&= \frac{2[f(t_{j})\pm g_f(t_{j}))]}{n[f(t_{j})\pm g_f(t_{j}))]}&\subseteq \frac{2}{n}\Bigl[1\pm \frac{3g_f(t_{j})}{f(t_{j})}\Bigr] &\subseteq \frac{1}{n}\Bigl[2\pm \frac{K}{2}g_f(t_{j})\Bigr],
a_{2m}B^{(+)}_ms^{(+)}_m(y^{(+)}) =B^{(+)}_{m-1}\big[\tfrac{n}{2J}\delta^h \d+\tfrac 14 b_{2m}\big]s^{(+)}_{m-1}(y^{(+)}) +\tfrac{1}{16}c_{2m}B^{(+)}_{m-2}s^{(+)}_{m-2}(y^{(+)}),
Y^{(1)}= - \frac{2}{\sqrt{-g}}(\kappa v^{i}\bar{\tilde{\cal H}}^{j}\gamma_{i j k}\gamma^{(4)}E^{k}+ \bar{\kappa}v^{i}\tilde{\cal H}^{j}\gamma_{i j k}\gamma^{(4)}\bar{E}^{k}) ,
(|\nabla \rho|^q)_t+(|\nabla \rho|^qu)+(q-1)|\nabla \rho|^qu = -q |\nabla \rho|^{q-2}(\nabla \rho)^\top D( u) (\nabla \rho)-q \rho|\nabla \rho|^{q-2}\nabla \rho \cdot \nabla u.
-2^NK_N(H_N,e_1)^{-1}H_N^Ne_1= 2K_N(H_N,e_1)^{-1}\left(\sum\limits_{i=0}^{N-1}T_{N,i}H_N^i e_1\right)= 2\begin{bmatrix}T_{N,0}\vdots T_{N,N-1}\end{bmatrix}
\mathfrak{X}(L_i, L_i) =(L_{i+1}, L_0), \mathfrak{X}^{2n-4-i}(L_{i+1}, L_0) =(L_{2n-3}, L_{2n-4-i}), \mathfrak{X}(L_{2n-3}, L_{2n-4-i}) =(L_{2n-3-i}, L_{2n-3-i}), \mathfrak{X}(L_{2n-3-i}, L_{2n-3-i}) =(L_{2n-2-i}, L_{0}), \mathfrak{X}^{i-1}(L_{2n-2-i}, L_{0}) =(L_{2n-3}, L_{i-1}), \mathfrak{X}(L_{2n-3}, L_{i-1}) =(L_{i}, L_{i}).
J_{\varepsilon}(u):=\frac{1}{2}\int_{M}|\nabla_{g}u|^{2}+au^{2}d\mu_{g}+\frac{1}{2}\int_{\partial M}\varepsilon\gamma u^{2}d\sigma-\frac{(n-2)^{2}}{2(n-1)}\int_{\partial M}\left(u^{+}\right)^{\frac{2(n-1)}{n-2}}d\sigma.
\widetilde{V}_n(x) = \begin{cases}\displaystyle{\sum_{m=0}^{\frac{n-1}{2}}\frac{n(n-m-1)!}{m!(n-2m)!}x^{n-2m-1}}n: , \displaystyle{\sum_{m=0}^{\frac{n}{2}-1}{n-m-1 \choose m}x^{n-2m-2}}n: .\end{cases}
T_{V/N^2}(P_{V/N^2}(0,r_k))=NT_{V}\Big(N^2P_{V/N^2}(0,r_k)\Big)=NT_V( P_V(0,Nr_k)), begin{align*}3pt]0<\Delta_N(r_1,r_2)=N\Delta(Nr_1,Nr_2)\leqslant C_0\frac{r_1(r_2-r_1)N^{3-5/4}}{\big(r_1-\rho_N\big)^{5/4}},
\begin{aligned}\mathcal{R}_{\beta_0,\beta_1}(z):= (\mathcal{A}_0 -zI)^{-1}- \left(I-z \mathcal{A}_0^{-1}\right)^{-1}\Pi(\overline{\beta_0 + \beta_1 M(z)})^{-1}\beta_1\bigl(\Pi^*\left( I - z \mathcal{A}_0^{-1}\right)^{-1}\bigr) begin{align*}0.3em] = (\mathcal{A}_0 -zI)^{-1}-S(z)(\overline{\beta_0 + \beta_1 M(z)})^{-1}\beta_1S^*(\overline{z}).\end{aligned}
E_+(X^0)=2N\int d\tilde{\sigma}\frac{e^{3H(X^0-\tilde{\sigma})}+H^2L^2e^{-H(X^0-\tilde{\sigma})}}{\sqrt{e^{4H(X^0-\tilde{\sigma})}+e^{2H(X^0-\tilde{\sigma})}+H^2L^2}}.
\{d_\beta\}_{\beta\in \Delta^+}<\{d'_\beta\}_{\beta\in \Delta^+}\Longleftrightarrow \exists\ \gamma\in \Delta^+\ \mathrm{s.t.}\d_\gamma<d'_\gamma\ \mathrm{and}\d_\beta=d'_\beta\ \mathrm{for\all}\ \beta<\gamma.
\mathcal{A}=h_n/2\bigg(\sum \limits_{i=1}^{n}r_{i}+\sum \limits_{i=1}^{n-1}r_{i}\bigg)+h_{n-1}/2\bigg(\sum \limits_{i=1}^{n-1}r_{i}+\sum \limits_{i=1}^{n-2}r_{i}\bigg)+ \cdots +1/2r_1h_1.
j_{\mu}^{3}= \frac{\frac{\strut \displaystyle M}{\strut \displaystyle \cos \theta_{W}}}{1 + \strut \displaystyle \Bigl ( \frac{\strut \displaystyle M}{\strut \displaystyle M^{\prime}}\Bigr )^{2}\tan^{2}\theta_{W}}Z_{\mu}
\left\langle Q(\Pi),\frac{H}{1-tE(y)H}\right\rangle =\left\langle Q(\Pi)[X+1],\frac{1}{1-tE(y)H}\right\rangle \ref{l-spconst}) =\sum_{k=0}^{\infty}\left\langle Q(\Pi)[X+1],E(y)^{k}H^{k}\right\rangle t^{k}=\sum_{k=0}^{\infty}\Theta_{y,k}(Q(\Pi)[X+1])t^{k}\ref{l-scalprodh}).
z_0^{-1}\delta(\frac{z_1-z_2}{z_0})(\frac{z_1-z_2}{z_0})^{\frac{j_1}{T}}Y_{M^3}(u,z_1)I(w_1,z_2)w_2 - e^{\frac{j_1}{T}\pi i}z_0^{-1}\delta(\frac{-z_2+z_1}{z_0})(\frac{z_2-z_1}{z_0})^{\frac{j_1}{T}}I(w_1,z_2)Y_{M^2}(u,z_1)w_2 = z_1^{-1}\delta(\frac{z_2+z_0}{z_1})(\frac{z_2+z_0}{z_1})^{\frac{j_2}{T}}I(Y_{M^1}(u,z_0)w_1,z_2)w_2
\mathcal{S}^\mathcal{C}_n=\begin{pmatrix}\sqrt{e_0}0 \ldots 0 0 \sqrt{d_1/e_0}\sqrt{e_1}\ldots 0 0 \vdots \vdots \ddots \vdots \vdots 0 0 \ldots \sqrt{e_{n-2}}0 0 0 \ldots \sqrt{d_{n-1}/e_{n-2}}\sqrt{e_{n-1}}\end{pmatrix},
\times\sum_{{\bf{q}}_{1}\neq{\bf{q}},{\bf{0}}}n_{{\bf{k}}-{\bf{q}}/2 +{\bf{q}}_{1}}e^{i\mbox{}\theta({\bf{k}}+{\bf{q}}_{1}/2, -{\bf{q}}+{\bf{q}}_{1})}e^{-i\mbox{}\theta({\bf{k}}-{\bf{q}}/2 +{\bf{q}}_{1}/2,{\bf{q}}_{1})}
\delta_{*}(t)=\frac{1}{t(1+t^{2})}\int_{1}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)_{*}}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds
\sqrt{\omega}\lambda(t) T(y)(t,\omega) = -\lambda(t) \int_{t}^{\infty}\sin((t-x)\sqrt{\omega})\left(F_{2}(y)(x,\omega)-\mathcal{F}(\sqrt{\cdot}F(x,\cdot \lambda(x)))(\omega \lambda(x)^{2})-\mathcal{F}(\sqrt{\cdot}F_{3}(x,\cdot\lambda(x)))(\omega \lambda(x)^{2})\right) dx
< \mathrm{t r}[ A_{\mu}, A_{\nu}]^{2}> \simeq 2{< \mathrm{t r}[ X_{\mu}, \tilde{A}_{\nu}] [ X_{\mu}, \tilde{A}_{\nu}] > - < \mathrm{t r}[ X_{\mu}, \tilde{A}_{\nu}] [ X_{\nu}, \tilde{A}_{\mu}] >}.
\mu_{-}= \left( \begin{array}{c c}{{I_{n}}}&{{0}}\\{{\mu_{- 2 1}}}&{{I_{k}}}\\ \end{array}\right) , \qquad \mu_{+}= \left( \begin{array}{c c}{{I_{n}}}&{{\mu_{+ 1 2}}}\\{{0}}&{{I_{k}}}\\ \end{array}\right) ,
B=\sinh(ka/2)\cosh(ka/2)\left[-\frac{a}{2}+\sinh^{2}(ka/2)\left(\frac{2(a/2)}{3}-\frac{1}{2(a/2)k^{2}}+\frac{\sinh(ka/2)\cosh(ka/2)}{2(a/2)^{2}k^{3}}\right) \right]^{-1/2}
\epsilon_{2}= \frac{(K_{sp}+1)}{K_{sp}\overline{\gamma}_{sp}}\int_{y=0}^{\delta^{2}}\frac{1}{\pi \sqrt{1-(1-y)^{2}}}dy &= \frac{(K_{sp}+1)}{K_{sp}\overline{\gamma}_{sp}}\left( \frac{1}{\pi}\mbox{sin}^{-1}(\delta^{2}-1) + \frac{1}{2}\right).
\int_{B_{N}}|f_{1}f_{2}|^{\frac{s}{2}}\leq C_{s,\epsilon}r^{2\cdot (-6+{\frac{3s}{2}})}N^{{s(\gamma_{lin}+\epsilon)}(\frac{3}{4}-\frac{2}{2^M})}\biggl(\int_{B_{N+r+r^2+r^4}}\prod_{i=1}^{2}\|f_{i}\|_{L^2({w}_{B_{r^4}(x+b_{i})})}^{\frac{s}{2}}\,dx \biggr)^{\frac{1}{4}}\prod_{i=1}^{2}\|f_{i}\|_{L^{s,N}(\mathbb{R}^{6})}^{\frac{s}{4}(1+\frac{1}{2})}.
z =\frac{1}{\sqrt{1-a +\tau^2 (1+a)}}\left( \frac{\sqrt{1-a}\, \cos (t)}{\sqrt{1+a}}+ \frac{i\,\tau\,\sqrt{1+a}\, \sin (t)}{\sqrt{1-a}}\right), w = \frac{1}{\sqrt{1-a +\tau^2 (1+a)}}\left( \frac{\sqrt{1-a}\, \sin (t)}{\sqrt{1+a}}- \frac{i\,\tau\,\sqrt{1+a}\, \cos (t)}{\sqrt{1-a}}\right),
\left( \begin{array}{l}{t^{{\vec{\Lambda}}_{1}}= p^{{\vec{\Lambda}}_{1}}+ \mathrm{i}\, q_{{\vec{\Lambda}}_{1}}\,}{{\bar{t}}_{{\vec{\Lambda}}_{1}}= p^{{\vec{\Lambda}}_{1}}- \mathrm{i}\, q_{{\vec{\Lambda}}_{1}}\,}\end{array}\right)
( f^{'}\cos \theta \cos 2 \theta + \frac{2}{r}f \sin \theta \sin 2 \theta ) \frac{\partial \gamma^{2}}{\partial x^{2}}+ ( f^{'}\sin \theta \sin 2 \theta + \frac{2}{r}f \cos \theta \cos 2 \theta ) \frac{\partial \gamma^{1}}{\partial x^{1}}
|V-\theta_{1}^{(2)}U| \geq\frac{1}{2}(|V-\theta_{1}^{(1)}U|+|V-\theta_{1}^{(2)}U|) & \geq\frac{1}{2}|U||\theta_{1}^{(1)}-\theta_{1}^{(2)}|=|U|\left\vert\sqrt{\frac{c+2}{c}}\right\vert \geq|U|\sqrt{\frac{|c|-2}{|c|}},
\rho(r)=\frac{C(\Phi_{max}-\Phi(r))^{\frac{3+\alpha}{2}}}{(\Phi_{max}-\Phi_{min})^{\frac{\alpha}{2}}},C=\frac{6m\pi2^{5/2}B(1+\frac{\alpha}{2},\frac{3}{2})}{(2\pi\hbar/m)^{3}}
V(z)={{D-2}\over{16}}\left[(D-6)\left({{{\cal C}^{\prime}}\over{\cal C}}\right)^2+4{{{\cal C}^{\prime\prime}}\over{\cal C}}\right]=-{{(1+\Delta)\Lambda}\over{2\Delta(1+\bar{K}|z|)^2}}+{{2\bar{K}}\over{\Delta+2}}\delta(z).
B ( \omega_{n^{\prime}}, p ) = \frac{\alpha}{\pi}T \sum_{n = - \infty}^{\infty}\int_{- \infty}^{\infty}\frac{d k B ( \omega_{n}, k )}{\omega_{n}^{2}+ k^{2}+ m^{2}( T )}\int_{0}^{\infty}\frac{d x \exp ( - l^{2}x / 2 )}{( \omega_{n}- \omega_{n^{\prime}})^{2}+ ( k - p )^{2}+ x},
\begin{cases}\Delta^{m}u=0,\,\,in\,\, D, \frac{\partial}{\partial N}\Delta^{j}u=g_{j},\,\,on\,\,\partial D, \nabla\big(u-\mathcal{M}_{1}\widetilde{g}_{0}\big)\in L^{p}(D) \end{cases}
\gamma_{\mathrm{STBC}}= \frac{\gamma_0}{MR_c}\frac{\mathbf{h}^{\dagger}\times \mathbf{h}}{1+\sum_{k=1}^{K}|g_k|^2\mathcal{P}_k/\mathcal{P}_w}=\frac{\gamma_0}{MR_c}\frac{\sum_{m=1}^{M}|h_m|^2}{1+\sum_{k=1}^{K}|g_k|^2\zeta_k}.
\Re\lambda_j^A = \Re\lambda_j^A |v_j|^2 = \Re\left\langle v_j,Av_j\right\rangle \geqslant \Re\left\langle v_j,Av_j\right\rangle -\frac{|b|}{2+b}|Av_j| \geqslant \gamma_A\frac{2}{2+b}>0,
\sum_{s_{j}}\sum_{s_{k}}\frac{( N_{i}- 2 s_{i}) ! N_{j}! N_{k}!}{2^{s_{j}+ s_{k}}s_{j}! s_{k}! t_{i j}! t_{i k}! t_{j k}!}= \frac{( N_{j}+ N_{k}) !}{2^{( N_{j}+ N_{k}- N_{i}+ 2 s_{i}) / 2}( \frac{N_{j}+ N_{k}- N_{i}+ 2 s_{i}}{2}) !}
j-1 =a_1+10a_2+\dots+10^{J-1}a_J, i_{k-1}-1 =a_2+10a_3+\dots+10^{J-1}a_{J+1}, \vdots i_1-1 =a_k+10a_{k+1}+\dots+10^{J-1}a_{J+k-1}, i-1 =a_{k+1}+10a_{k=2}+\dots+10^{J-1}a_{J+k}.
\frac{\binom{K-2}{s-1}}{\binom{K-2}{s-1}+\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^iN}\geq \frac{\binom{K-2}{s-2}}{\binom{K-2}{s-2}+\sum_{i=0}^{K-s}\binom{K-1}{s+i-1}(N-1)^iN}.
(\Delta - \lambda_w)\wedge^a E_w(z) = (\Delta-\lambda_w)\left(H(a-y)\cdot (y^w + c_w y^{1-w})\right) &= (y^2 \frac{\partial^2}{\partial y^2}- w(w-1))\left(H(a-y)\cdot (y^w + c_w y^{1-w})\right) &=y^2\left(\delta'_a\cdot(y^w+c_wy^{1-w})-2\delta_a\cdot(wy^{w-1}+(1-w)c_wy^{-w})\right).
\sigma=[\begin{matrix}-p+\frac{\mu J_{m}(1+\gamma^{2})}{J_{m}-\gamma^{2}}&\frac{\mu J_{m}\gamma}{J_{m}-\gamma^{2}}&0\\ \frac{\mu J_{m}\gamma}{J_{m}-\gamma^{2}}&-p+\frac{\mu J_{m}}{J_{m}-\gamma^{2}}&0\\ 0&0&-p+\frac{\mu J_{m}}{J_{m}-\gamma^{2}}\end{matrix}]
v\in C([0,T];D^1_0\cap D^2)\cap L^2([0,T];D^3)\cap L^{p_0}([0,T];D^{3,q}),\v_t\in L^2([0,T];D^1_0), &t^{\frac{1}{2}}v\in L^\infty([0,T];D^3),\t^{\frac{1}{2}}v_t\in L^\infty([0,T];D^1_0) \cap L^2([0,T];D^2), & tv\in L^\infty([0,T];D^{3,q}),\tv_t\in L^\infty([0,T];D^2),\tv_{tt}\in L^2([0,T];D^1_0),\v(0,x)=u_0.
=\frac{\beta}{2\pi}(k^{\mu}k^{\nu}-k^2g^{\mu \nu})\left[\frac{1}{\sqrt{k^2}}\left(1+\frac{4m^2}{k^2}\right)\log \left(\frac{1-\sqrt{\frac{4m^2}{k^2}}}{1+\sqrt{\frac{4m^2}{k^2}}}\right)+\frac{4m}{k^2}\right]\,\,,
\ln \frac{w_{2}}{w_{1}}( e^{\frac{2}{3}t_{1}+ \frac{1}{3}t_{2}}) + 2 \pi i = r \left[ \ln \frac{w_{3}}{w_{2}}( e^{\frac{2}{3}t_{1}+ \frac{1}{3}t_{2}}) - 2 \pi i \right] , r \in{\bf R}.
\displaystyle{\frac{m_{\pi}^2f_{\pi}j_{\pi}(q^2)}{m_{\pi}^2-q^2}=\partial^{\mu}J_{\mu}(q)=\frac{q^2f_{\pi}j_{\pi}(q^2)}{m_{\pi}^2-q^2}+\partial^{\mu}\widehat J_{\mu}(q).}
\mu^{\circ}+ \ell - 2q + 1 \geq \mathcal{C}_{-1}(\alpha, \nu, q + w', \lbrace 1, \ldots, q-1 \rbrace, \lbrace q, \ldots, \ell \rbrace \setminus \lbrace q+w' \rbrace) + 1 = \left \lceil \frac{\nu_{q+w'}- \sum_{i_0=1}^{q-1}m_{q+w', i_0}+ 1 + \sum_{i_0 = q+1}^{\ell}m_{q+w', i_0}}{\alpha_{q + w'}}\right \rceil.
V(\lambda,\mu)(3|3) = V_{\bar 0}\oplus V_{\bar 1}, V_{\bar 0}= \operatorname{Vect}_F\langle v_0,v_1R_y,v_0R_y^2\rangle,V_{\bar 1}= \operatorname{Vect}_F\langle v_1,v_0R_y,v_1R_y^2\rangle,
\tau'' c=\begin{cases}\tau c \tau c\in C'', \tau \theta\sigma \tau c \tau c\in \{a,\theta a,\sigma a, \theta \sigma a\}\tau \theta\sigma \tau c\in C'', \tau(\theta \sigma \tau)^2 c \tau c,\tau\theta\sigma\tau c\in \{a,\sigma a, \theta a, \sigma \theta a\}, \end{cases}
| v_{\pm}( \lambda ) \rangle = \sqrt{\frac{\omega ( \lambda ) \mp M}{2 \omega ( \lambda )}}\left( \begin{array}{c}{| v_{1}( \lambda ) \rangle}\\{\displaystyle{\frac{C}{\omega ( \lambda ) \mp M}}| v_{1}( \lambda ) \rangle}\\ \end{array}\right) ,
{\cal J}_{i}= \varepsilon_{i k l}\sum_{a}\sum_{s = 0}^{\infty}D^{s}[ x_{a}^{k}( \delta_{l}^{j}+ v_{a}^{j}\varphi_{a l}\varphi_{a t}^{- 1}) ] \frac{\partial}{\partial x_{a}^{j ( s )}},
F[P(\mu_1 ,\mu_2 );\gamma_* ]\left[{\begin{array}{*{20}c}{v_1 (\gamma_* )}{v_2 (\gamma_* )}\end{array}}\right] = s_* \left[{\begin{array}{*{20}c}{u_1 (\gamma_* )}{u_2 (\gamma_* )}\end{array}}\right],
\Gamma_{R_{1}}^{( 1 )}= -{\frac{1}{4}}\ln ( \mu^{2}r_{+}r_{-}) +{\frac{1}{2}}\ln ( r_{+}/ r_{-}) +{\frac{1}{2}}\ln \Bigr [ 1 - ( r_{-}/ r_{+})^{2}\Bigr ] .
\partial_y\Big[\big(\partial_y + 2\,\ell^{-1}\big)\,h + \frac{(1-\gamma)^{-1}\,\ell}{3\,M_*^3}\,e^{-2y/\ell}\,T_{55}(y)\Big]=\frac{(1-\gamma)^{-1}\,\ell}{3\,M_*^3}\partial_\mu T^{\mu 5}\,.
\frac{\partial \widehat W_{M_n,n}}{\partial t}+ \overline b(t) (-\Delta )^\beta \widehat W_{M_n,n}= a_0{\bf \overline R}_{n,\beta}\widehat W_{M_n,n}-\overline F_{Q_n}(\widehat W_{M_n,n}(x,t)) -\overline g_{M_n,n}(x,t),
H^{(3)}_4=\frac{1}{2}\{\{H_2, F^{(3)}\}, F^{(3)}\}+\{H_3, F^{(3)}\}+H_4=\frac{1}{2}\{H_{3, \le 1}, F^{(3)}\}+\{H_{3}^{(3)}, F^{(3)}\}+H_4
-\frac{\partial L_{d}D_{\alpha\beta}I(p\|q)}{\partial q_{j}}=-\frac{1}{\alpha}\left[\left(\frac{a-1}{a-b}\right)(X.Y)^{a-2}-\left(\frac{b-1}{a-b}\right)(X.Y)^{b-2}\right]\left[M-N\right]
\lim_{t\searrow0}\phi\big[\big(\exp(\frac{t}{2}B)\exp(tA)\exp(\frac{t}{2}B)\big)^\frac{1}{t}\big] = \ \phi\big[\lim_{t\searrow0}\big(\exp(\frac{t}{2}B)\exp(tA)\exp(\frac{t}{2}B)\big)^\frac{1}{t}\big] = \ \phi[\exp(A+B)],
\bar D^{Opt}{\setlength\arraycolsep{0.3pt}=}\begin{cases}\bar D_{ct}-\frac{1}{\lambda \bar N_l}(\bar D_{ct}-\bar D_{lt}) \frac{1}{\bar N_l}+\frac{1}{\bar N_c}\ge \lambda,\frac{1}{\bar N_l}< \lambda, \bar D_{lt}\frac{1}{\bar N_l}\ge \lambda, infeasible \frac{1}{\bar N_l}+\frac{1}{\bar N_c}< \lambda. \end{cases}
N^{2}=\sum_{k=1}^{t-1}\left(\sum_{j=k+1}^{t}2^{n_{j}-n_{k}}\right)\,2^{2(n_{k}+1)}+\sum_{k=1}^{t}\left(1-\sum_{j=k+1}^{t}2^{n_{j}-n_{k}+1}\right)\,2^{2 n_{k}}.
\qquad \qquad \qquad \qquad \qquad \frac{d v}{d x}= - \frac{c \varepsilon^{2}H Z^{4}n}{m \nu_{0}^{2}v^{3}}\left({0 . 9 6 + \log \frac{h \nu_{0}v^{4}}{8 \pi c \varepsilon^{2}H Z^{4}}}\right) ~ ~ . \qquad \qquad \qquad \qquad ( 1 7^{\prime})
\sqrt{\alpha}\frac{\sqrt{R-b}-\sqrt{R-a}}{\sqrt{R-b}+\sqrt{R-a}}\frac{\sqrt{b}+\sqrt{a}}{\sqrt{b+Q}-\sqrt{a+Q}}=1, \frac{\sqrt{ab}+\sqrt{(a+Q)(b+Q)}-Q}{2}+\sqrt{(R-a)(R-b)}=1.
\begin{array}{l}{{( m , \tilde{m}, n = 0 , p ) = ( m + \frac{k}{( k , 2 k^{\prime})}, \tilde{m}- \frac{k}{( k , 2 k^{\prime})}, n = 0 , p - \frac{2 k^{\prime}}{( k , 2 k^{\prime})}) ,}}\\{{( m , \tilde{m}, n = 1 , p ) = ( m + \frac{k}{( k , 2 k^{\prime})}, \tilde{m}- \frac{k}{( k , 2 k^{\prime})}, n = 1 , p + \frac{2 k^{\prime}}{( k , 2 k^{\prime})})}}\\ \end{array}
|\Psi(u)\rangle=\,\frac{1}{\sqrt{N}}\,\sum^N_{k=1}\,e^{\imath\,\vartheta_{k\Psi}(\varphi^{-1}_{k}(x),z_k)}\,n_{k\Psi}(\varphi^{-1}_{k}(x),z_k)\,|\varphi^{-1}_{k}(x),z_k\rangle,
\mathbb{P}( F^+, \gamma, z) = \sum_{n=0}^{k-2}\frac{\overline{L(f, e^{-2 \pi i d /c}, n+1 )}}{(k-2-n)!}(-2 \pi i )^{k-2-n}\left(\frac{c z + d}{c}\right)^{k-2-n}.
\vert \nu_{A}\rangle ={\frac{1}{\sqrt{1 + \vert \nu_{A}\vert^{2}}}}\left( \begin{array}{c}{- \nu_{A}}\\{1}\\ \end{array}\right) \quad , \quad \langle \nu_{A}\vert ={\frac{1}{\sqrt{1 + \vert \nu_{A}\vert^{2}}}}\Bigl ( - \nu_{A}^{*}\quad 1 \Bigr ) .
\displaystyle K^{<\epsilon}(f,a)= \int_{H_{\overline{Q}}\left[<\epsilon \sigma(a)\right]}\int_{aU_Q\left[<\epsilon \sigma(a) \right]a^{-1}}f\left(a^{-1}h_{\overline{Q}}h(u_Q) a\right) \xi\left(h_{\overline{Q}}h(u_Q)\right) j(u_Q) du_Q dh_{\overline{Q}}
\chi^{i \alpha}\rightarrow \chi^{i \alpha}+ \epsilon g^{i \bar{j}}( \Omega_{2})_{\bar{j}\bar{k}}\bar{\chi}_{\alpha}^{\bar{k}}\qquad \qquad \bar{\chi}_{\alpha}^{\bar{i}}\rightarrow \bar{\chi}_{\alpha}^{\bar{i}}+ \epsilon g^{\bar{i}j}( \Omega_{2})_{j k}\chi^{k \alpha}
\tilde{G}( z , w , x_{i}) = \frac{\langle \partial_{z}X ( z , \bar{z}) \partial_{w}\bar{X}( w , \bar{w}) \sigma_{+}( x_{1}) \sigma_{-}( x_{2}) \sigma_{+}( x_{3}) \sigma_{-}( x_{4}) \rangle}{\langle \sigma_{+}( x_{1}) \sigma_{-}( x_{2}) \sigma_{+}( x_{3}) \sigma_{-}( x_{4}) \rangle}
{\cal L}_{\eta}| \Phi_{p h}\rangle = \Bigl ( \eta \hat{\partial}+ \frac{1}{2}\hat{\partial}^{I}\eta^{J}M^{I J}+ \frac{1}{2}\eta^{I}M^{z I}+ \hat{\partial}^{+}\eta^{I}\alpha^{I}( \frac{\partial^{J}}{\partial^{+}}\bar{\alpha}^{J}- \frac{s + d - 4}{\hat{\partial}^{+}}\bar{\alpha}^{z}) \Bigr ) | \Phi_{p h}\rangle \, .
p T_{k}\sum_{l=1}^{k}T_l \leq p \beta \left( \frac{c_1}{pn}\right)^{k-1}\frac{1}{p^2n}\frac{T_1}{1-(c_1/pn)}<\beta^2 \left( \frac{c_1}{pn}\right)^{k}\frac{1}{p^2n}.
Im\,I_{1_\beta}\approx \frac{-g^2}{8}\int\frac{d^3l}{(2\pi)^2 E_1 E_2}2\delta(E_1-E_2) \Bigg(\frac{2}{e^{\beta E_1}-1}+\frac{2}{(e^{\beta E_1}-1)^2}\Bigg).
\int_\Omega \overline U_+^2(x,t)dx=\int_\Omega \overline U_+^2(x,t_0)dx+\int_{t_0}^t \Big(\frac{d}{dt}\int_{\Omega}\overline U_+^2(x,\tau)dx\Big)d\tau\forall\,\, t\in (t_0,t_0+T_{\max}).
[\begin{matrix}\dot{p}_{3}(t)\\ \dot{q}_{6}(t)\end{matrix}]=[\begin{matrix}-\frac{R_{2}}{I_{3}}&-\frac{r}{C_{6}}\\ \frac{r}{I_{3}}&-\frac{1}{R_{7}\cdot C_{6}}\end{matrix}][\begin{matrix}p_{3}(t)\\q_{6}(t)\end{matrix}]+[\begin{matrix}1\\ 0\end{matrix}][e_{1}(t)]
\phi_0(x) =-\dfrac{1}{2\varepsilon}\arctan\left(\cosh(\beta(\lambda+\varepsilon\theta(x)))\cosh(\lambda+\varepsilon \theta(x))(\tanh(\beta(\lambda+\varepsilon \theta(x)))+v\sqrt{c}\tanh(\lambda+\varepsilon \theta(x)))\right) & +\dfrac{1}{2\varepsilon}\arctan\left(\cosh(\beta\lambda)\cosh(\lambda)(\tanh(\beta\lambda)+v\sqrt{c}\tanh(\lambda))\right),
1 \geq \lim_{n \rightarrow \infty}\dfrac{\ell(H^2(I_n;M))}{\ell(R/I_n)}\geq \lim_{n \rightarrow \infty}\dfrac{t}{t+(3n^{d-2}+n^{d-3}+\cdots+n^2)+(4(2n)^{d-2})^d}= 1.
{\cal A}_k^{(s,t)}:=\left(\begin{array}{cccc}Q_1^{(s,t)}1 Q_1^{(s,t)}E_1^{(s,t)}Q_2^{(s,t)}+E_1^{(s,t)}\ddots &\ddots \ddots 1 & Q_{k-1}^{(s,t)}E_{k-1}^{(s,t)}Q_k^{(s,t)}+E_{k-1}^{(s,t)}\end{array}\right).
J^{*}[\psi^{(k)}]_{T,t}=\hbox{\vtop{\offinterlineskip\halign{\hfil#\hfil\cr{\rm l.i.m.}\cr$\stackrel{}{{}_{p\to \infty}}$\cr}}}\sum\limits_{j_1, \ldots j_k=0}^{p}C_{j_k \ldots j_1}\zeta_{j_1}^{(i_1)}\ldots\zeta_{j_k}^{(i_k)}
\|r_h^+ u - r_h u\|_{0,2}^2 = a_h^+(r_h^+ u - r_h u, w) &= a_h^+(r_h^+ u - r_h u, w - w_h^+) + a_h^+(r_h^+ u - r_h u, w_h^+) &= a_h^+(r_h^+ u - r_h u, w - w_h^+) + a_h^+(u - r_h u, w_h^+) &= a_h^+(r_h^+ u - r_h u, w - w_h^+) + a_h^+(u - r_h u, w_h^+ - w_h) &\;\;\; + a_h^+(u - r_h u, w_h) - a_h(u-r_h u, w_h) &=: T_1 + T_2 + T_3,
\Gamma^{5}=\left( \begin{array}{cc}0 1_{2}1_{2}0\end{array}\right);\quad \Gamma^{i}=\left( \begin{array}{cc}0 \sigma^{i}-\sigma^{i}0\end{array}\right);\quad \Gamma^{0}=\left( \begin{array}{cc}1_{2}0 0 -1_{2}\end{array}\right);\quad \Gamma^{4}=-i\Gamma^{5}=\left( \begin{array}{cc}0 i_{2}i_{2}0\end{array}\right).
\Psi^{[M,\, m,\, s;\,\epsilon]}_{a,\, b;\,\epsilon^{\prime}}(\tau,z_1,z_2,t) = q^{\frac{(m+1)ab}{M}}e^{\frac{2\pi i (m+1)}{M}(bz_1+az_2)}\Phi^{[m;\,s]}(M\tau, z_1+a\tau+\epsilon, z_2+b\tau+\epsilon, \frac{t}{M}),
\mathbb{S}\{E_{i}[n,m]\}=\frac{1}{MN}\sum_{v=0}^{N-1}\sum_{u=0}^{M-1}FFT^{2}\{E_{i}(n,m)\}\cdot\tilde{K}(u,v)\cdot\tilde{L}(u,v)\cdot e^{i\frac{2\pi}{M}um}\cdot e^{i\frac{2\pi}{N}vn}
\bigl|f([x_\alpha, y_\alpha] - [x'_\alpha,y'_\alpha])\bigr| \ =\ \bigl| [x_\alpha, y_\alpha - y'_\alpha]_f + [x_\alpha - x'_\alpha, y'_\alpha]_f\bigr| \ \leq\ \|x_\alpha\|_f \|y_\alpha - y'_\alpha\|_f + \|x_\alpha - x'_\alpha\|_f \| y'_\alpha \|_f.
\tilde \varphi_{n,i}(x) = \varphi_{n,i}(x) + \sum_{k = 1}^{\infty}(\tilde Q_{n,i; k,0}(x) \varphi_{k,0}(x) - \tilde Q_{n,i; k,1}(x) \varphi_{k,1}(x)), n \ge 1, \, i = 0, 1,
0 = \delta^2 N^2 ( u^s_{j-1}-2u^s_j + u^s_{j+1}) - V'(u^s_j) \lambda^s_k \phi_j^{(k)}= -\delta^2 N^2 ( \phi^{(k)}_{j-1}-2\phi^{(k)}_j + \phi^{(k)}_{j+1}) +V''(u^s_j)\phi^{(k)}_j,
\Bigl ( V ( \psi \otimes \bar{w}, z ) \; \left( | \chi \rangle_{l}\otimes | \bar{u}\rangle_{l}\right) \Bigr )_{m}: = \lim_{\bar{z}\rightarrow 0}\bar{z}^{\bar{\Delta}_{j}+ \bar{\Delta}_{l}- \bar{\Delta}_{m}}\; \Bigl ( \phi ( \psi \otimes \bar{w}; z , \bar{z}) \; \left( | \chi \rangle_{l}\otimes | \bar{u}\rangle_{l}\right) \Bigr )_{m},
\left[ J_{ab},\bar{Z}_{c}\right] =\frac{1}{\sqrt{2}}\left[ J_{ab},P_{c}\right] -\frac{1}{\sqrt{2}}\left[ J_{ab},Z_{c}\right] , & =\frac{1}{\sqrt{2}}f_{ab,c}^{d}P_{d}-\frac{1}{\sqrt{2}}f_{ab,c}^{d}Z_{d}, & =f_{ab,c}^{d}\bar{Z}_{d},
\mathcal{S}_1 = \left\{\begin{bmatrix}a b 0 b c d 0 d e \end{bmatrix}: (a,b,c,d) \in \mathbb{R}^4 \right\}\mathcal{S}_2 = \left\{\begin{bmatrix}a a b a a b b b c \end{bmatrix}: (a,b,c) \in \mathbb{R}^3 \right\}.
\lim_{n\to\infty}\frac{a_n a_{n-1}}{b_n}=0, \lim_{n\to\infty}\frac{a_n a_{n-1}(a_{n-1}+a_{n-2})}{b_n b_{n-1}}=0, \lim_{n\to\infty}\frac{a_n a_{n-1}}{b_nb_{n-1}}\left[ \frac{a_{n-1}^{2}}{b_n}+ \frac{a_{n-2}(a_{n-2}+ a_{n-3})}{b_{n-2}}\right]=0,
-\frac{p_{d-1}^{(h)}}{p_{d}^{(h)}}=-\frac{p'_h(0)}{p_h(0)}=\sum_{j=1}^dx_j^{-k},~-\frac{p_{1}^{(h)}}{p_{0}^{(h)}}=-\frac{p'_{h,{\rm rev}}(0)}{p_{h,{\rm rev}}(0)}=\sum_{j=1}^dx_j^{k}~{\rm for}~k=2^h.
\chi^{4}= : \! \chi^{4}\! : + 6 \,{\frac{1}{2 \pi}}\ln{\frac{\Lambda_{0 , \mathrm{e f f}}}{\Lambda}}\, \chi^{2}+ 3 \, \bigl ({\frac{1}{2 \pi}}\ln{\frac{\Lambda_{0 , \mathrm{e f f}}}{\Lambda}}\Bigr )^{2}.