harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T_2 =\mathrm{P}\left(x_i<z_i<\frac{\frac{\epsilon_{1,i}}{\rho}}{\alpha_{i,x}^2-\beta_{i,x}^2\epsilon_{1,i}}\right) =\mathrm{P}\left(x_i<z_i<\frac{\frac{\epsilon_{1,i}}{\rho}}{1-\max\left\{0,\frac{x_i-\frac{\epsilon_{1,i}}{\rho}}{x_i(1+\epsilon_{1,i})}\right\}(1+\epsilon_{1,i})}\right).
\begin{cases}S({\bf{v}})\partial_t{\bf{v}}+{\bf{A(v)}}\partial_x{\bf{v}}+f(\partial_y{\bf{v}})-{\bf{B(v}})\partial_y^2{\bf{v}}=g({\bf{v}}), (\tilde{u}, \partial_y\tilde{q})|_{y=0}={\bf{0}},\tilde{\theta}|_{y=0}=0, \lim\limits_{y\to\infty}{\bf{v}}(t, x, y)={\bf{0}},{\bf{v}}|_{t=0}=(\tilde{u}_0, \tilde{\theta}_0, (\tilde{h}_0)^2/2)^T(x, y),\end{cases}
\aleph^{a b c}= C \; \left( 1 + \Sigma_{s}{\frac{m_{s}}{| \vec{x}- \vec{x}_{s}|}}\right)^{- 1}\bar{\epsilon}_{0}\; \gamma^{a b c}\epsilon_{0}\ , \qquad a = 0 , 1 , \dots 9 ,
\mathrm{MTCE}_{\boldsymbol{q}}(\mathbf{X}) =\mathrm{E}\left[\mathbf{X}|\mathbf{X}>VaR_{\boldsymbol{q}}(\mathbf{X})\right] &=\mathrm{E}[\mathbf{X}|X_{1}>VaR_{q_{1}}(X_{1}),\cdots,X_{n}>VaR_{q_{n}}(X_{n})],
\sum_{l=N+1}^{n^{1-\varepsilon}}\mathbf{P}(\tau_x>l-1,\nu_n>l-1) \le \sum_{l=N+1}^{n^{1-\varepsilon}}\mathbf{P}(\tau_x>l-1,\nu_l>l-1) &\le \sum_{l=N+1}^{\infty}e^{-Cl^\varepsilon}.
0= \frac{1}{2}\mathcal{L}(|h|^{2}+|H-\xi|^{2}) \leq -\frac{1}{2}(|h|^{2}-|H|^{2})(3|h|^{2}-2|H|^{2}+\langle H,H-\xi\rangle) &+|h|^{2}-|H|^{2}+2(2-\langle H,H-\xi\rangle).
\varphi_{m,(a^{\prime},b^{\prime},c^{\prime})}(x,y,z)= \dfrac{\mathfrak{c}(m-4)}{2}\left(a^{\prime}\left(x-\dfrac{d}{\mathfrak{c}}\right)^{2}+b^{\prime}\left(y-\dfrac{d}{\mathfrak{c}}\right)^{2}+c^{\prime}\left(z-\dfrac{d}{\mathfrak{c}}\right)^{2}\right).
\phi_{k}(r)=\begin{cases}\frac{1}{2}\left(r \csc^{2}(\sqrt{k}r)-\frac{\cot (\sqrt{k}r)}{\sqrt{k}}\right) k>0 \frac{r}{3}k=0 \frac{1}{2}\left(\frac{\coth (\sqrt{-k}r)}{\sqrt{-k}}-r\, \mathrm{csch}^{2}(\sqrt{-k}r)\right) k<0.\end{cases}
(D_ld)_{\Lambda_1\cup\Lambda}=(D_l)_{\Lambda_1\cup\Lambda}d=-(D_l)_{\Lambda_1\cup\Lambda}(x_l^*-\overline{x}_l)=(-D_l(x_l^*-\overline{x}_l))_{\Lambda_1\cup\Lambda}=(-(D_lx_l^*-c_l))_{\Lambda_1\cup\Lambda}.
Z_{H,\lambda}^{k}(t):={\int_{\mathbb{R}^k}^{'}\int_{0}^{t}\prod_{i=1}^{k}\bigl({(s-y_i)_{+}^{d-1}}e^{-\lambda(s-y_i)_{+}}\bigr)\,ds\,B(dy_1)\ldots B(dy_k)},
\hat{k}_j=\underset{k\in\left\lbrace 1,2,\cdots,K \right\rbrace}{\mathrm{argmin}}\sqrt{\beta^2(\tau_j-\tau_k)^2+(\theta^\mathrm{A}_j-\theta^\mathrm{A}_k)^2+(\theta^\mathrm{E}_j-\theta^\mathrm{E}_k)^2}
\aligned \widehat{\mathcal U}({\rm mor};3,1;\alpha_1,\alpha_3)^{\boxplus\tau}&=\bigcup_{\alpha_2 \in \frak A_2}\widehat{\mathcal U}({\rm mor};2,1;\alpha_1,\alpha_2)^{\boxplus\tau}\times_{R_{\alpha_2}}\widehat{\mathcal U}({\rm mor};3,2;\alpha_2,\alpha_3)^{\boxplus\tau}.\endaligned
E_{\lambda}(\Omega_{\infty};u,v,w) =(uvw^{2})^{k-1}E_{\lambda}(((B^*\times T_{0}) \setminus Y)_{\infty};u,v,w) &=(uvw^{2})^{k-1}(E_{\lambda}((B^*\times T_{0})_{\infty};u,v,w)-E_{\lambda}(Y_{\infty};u,v,w)) &=-(uvw^2)^{k-1}E_{\lambda}(Y_{\infty};u,v,w).
\frac{\partial^{2}y}{\partial x_{i}\partial x_{j}}=\sum_{k}(\frac{\partial y}{\partial u_{k}}\frac{\partial^{2}u_{k}}{\partial x_{i}\partial x_{j}})+\sum_{k,l}(\frac{\partial^{2}y}{\partial u_{k}\partial u_{l}}\frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{l}}{\partial x_{j}}).
G_{T_{2}}^{\ominus}=G_{T_{1}}^{\ominus}-S_{T_{1}}^{\ominus}(T_{2}-T_{1})-T_{2}\int_{T_{1}}^{T_{2}}\frac{C_{p}^{\ominus}}{T}dT+\int_{T_{1}}^{T_{2}}C_{p}^{\ominus}dT
L\left[ 2x\left( \phi-1\right) P_{n+1}P_{n}\right] =\left[ \gamma_{n}+\gamma_{n+1}+\gamma_{n+2}-\left( z^{2}+1\right) \right] h_{n+1}, L\left[ 2x\left( \phi-1\right) P_{n+1}P_{n-2}\right] =h_{n+1},
\Psi = ( 1 -{\frac{\rho}{4}}x^{+}x^{-})^{\frac{1}{2}}\left( \begin{array}{c}{e^{-{\frac{\theta}{2}}}e^{\imath v}p ( x^{-})}\\{e^{{\frac{\theta}{2}}}e^{\imath u}f ( x^{+})}\\ \end{array}\right)
\int{\cal D}v \;\delta^{n}\left(\int v d\tau \right) F_{q G}[v,I]=\int{\cal D}v \;\delta^{n}\left(\int \phi d\tau \right)F_{q G}[\phi,I] \;{\rm Det}\;\frac{\delta \phi_{\tau}(v)}{\delta v(\tau^{\prime})}\; ,
\sum_{k=1}^{n}k^{\ell-1}E_k\ll\begin{cases}n^{\ell+3/8},~ ~t\geqslant n^{1/2}-n^{1/4}n^{\ell+1/4}\sqrt{n^{1/2}-t},~ t<n^{1/2}-n^{1/4}.\end{cases}
\vert J_1,\gamma_1;\; J_2,\gamma_2\rangle_q = \frac{1}{\sqrt{E_q(J_1,J_2)}}\sum_{n_1,n_2 = 0}^\infty \frac{J_1^{\frac{n_1}{2}}J_2^{\frac{n_2}{2}}\exp{-i(\gamma_1 [n_1]_q + \gamma_2 [n_2]_q)}}{\sqrt{[n_1]_q![n_2]_q!}}\vert n_1,n_2\rangle_q ,
\left( \begin{array}{c}{{{\tilde{t}}_{1}}}\\{{{\tilde{t}}_{2}}}\\ \end{array}\right) = 2 \pi i \left( \begin{array}{c c}{{1}}&{{1 / 2}}\\{{0}}&{{- 1 / 2}}\\ \end{array}\right) \left( \begin{array}{c}{{t_{1}}}\\{{t_{2}}}\\ \end{array}\right) \, \, \, ,
\langle \prod_{i=1}^{N-2+(N-k)d}{\cal O}_{e}(z_{i}) \rangle = \int_{{\cal M}_{0,d}^{M_{N}^{k}}}\bigwedge_{i=1}^{N-2+(N-k)d}\phi^{*}_{i}(e) =\int_{{\cal M}^{CP^{N-1}}_{0,d}}c_{(N,k)}\wedge (\bigwedge_{i=1}^{N-2+(N-k)d}\varphi^{*}_{i}(e))
_cD_{0t}^{\alpha}\left(\int_0^x (x-t)^{\alpha-1}E_{\alpha, \alpha}[\lambda(x-t)^{\alpha}]f(t)dt\right) = f(0)E_{\alpha, 1}(\lambda x^{\alpha}) + \int_0^xf'(z)E_{\alpha, 1}(\lambda (x-z)^{\alpha})dz.
nH_{\psi}= nH -\frac{u'(r)}{W}\langle\nabla^P \psi, \nabla^P r\rangle = nH -\psi'(r)\frac{u'(r)}{W}={\rm div}^P_{2\psi}\bigg(\frac{u'(r)}{W}\nabla^P r\bigg)\cdot
\frac{\partial Z_{R}}{\partial t}=-K^{\prime}(\theta)\psi^{\prime}(\theta)\frac{\partial\theta}{\partial z}-K(\theta)\psi^{\prime\prime}(\theta)\frac{\partial\theta}{\partial z}-K(\theta)\psi^{\prime}(\theta)\frac{\partial^{2}\theta/\partial z^{2}}{\partial\theta/\partial z}+K^{\prime}(\theta)
\nu_{\Lambda,2,m}^{(y,\eta,t)}(df)=\prod_{p=1}^{n-1-k}\left|\frac{\partial^2_{y^{j_{2p-1}}y^{j_{2p}}}f(y)}{\Lambda^2}\right|^2 \mu_\Lambda(f)G_{\Lambda}(y,\eta,t)^*(\delta_{t,\frac{1}{m}}^{n+1})(f)df,
x^{T}Ax=\sum_{u\in V}x_{u}\sum_{u^{\prime}\in N(u)}x_{u^{\prime}}\ge\sum_{k=0}^{r-1}|V_{k}|w_{k}[w_{k-1}+(d-1)w_{k+1}]-(d-1)|V_{r-1}|w_{r-1}w_{r},
\begin{cases}u(0,t)=h_0(t),\,\,u(L,t)=h_1(t),\,\,u_{x}(L,t)=h_2(t), v(0,t)=g_0(t),\,\,v(L,t)=g_1(t),\,\,v_{x}(L,t)=g_2(t),\end{cases}
V_0^{1/2}(F_{h_1}-F_{h_2}) = V_0^{1/2}\left(\frac{1}{2}r\cdot g(6(x+h_1)-1)-g(6(x+h_2)-1)\right) &= \frac{1}{2}r\cdot V_0^{1/2}(g(6(x+h_1)-1)-g(6(x+h_2)-1)) &= \frac{1}{2}r\cdot V_{-1}^2(g(x+h_1)-g(x+h_2)) &= \frac{1}{2}r\cdot V_{-1+h_2}^{1+h_2}(g_{h_1-h_2}-g) &= \frac{1}{2}r\cdot 2=r.
\int_{\mathbb{R}^{d}}\int_{-\infty}^{T}|\mathcal{T}g(t,x)|^{2}dtdx =T^{1+\frac{\alpha d}{2}}\int_{\mathbb{R}^{d}}\int_{-\infty}^{1}|\mathcal{T}\tilde{g}(t,x)|^{2}dtdx \leq NT^{1+\frac{\alpha d}{2}}\int_{\mathbb{R}^{d}}\int_{-\infty}^{1}|\tilde{g}(t,x)|^{2}dtdx =N\int_{\mathbb{R}^{d}}\int_{-\infty}^{T}|g(t,x)|^{2}dtdx.
\begin{array}{c c c c c}{\pi_{\tau \tau}\approx 0 ,}&{\pi_{\tau \sigma}\approx 0 ,}&{\pi_{\sigma \sigma}\approx 0 ,}&{\chi_{1}\left( P , X^{\prime}\right) \approx 0 ,}&{\chi_{2}\left( P , X^{\prime}\right) \approx 0 .}\\ \end{array}
{\cal T}_{{\cal S}_+^p}(\overline{M}_+) =\{B\in{\cal S}^p\, |\, B = \Pi_{{\cal S}_+^p}^{\prime}(\overline{M}_+;B)\}=\{B\in{\cal S}^p\, |\, [P_\beta \P_\gamma]^T B[P_\beta \P_\gamma] \succeq 0\}\, .
p_{n+2}=-12\sum_{j=0}^{n}(\begin{matrix}2n+2\\ 2j+2\end{matrix})p_{n-j}\sum_{k=0}^{j}(\begin{matrix}2j+1\\ 2k+1\end{matrix})p_{k}p_{j-k},p_{0}=1,p_{1}=0.
\int_{0}^1 \Lambda(x) dx = \sum_{n=1}^\infty |I_n|c_n = \alpha^{-1}\sum_{n=1}^\infty \lambda_n - \lambda_{n+1}=\lim_{M \to \infty}\alpha^{-1}\sum_{n=1}^M \lambda_n - \lambda_{n+1}= \lim_{M \to \infty}\alpha^{-1}(\lambda_1 - \lambda_{M+1}) = \alpha^{-1}\lambda_1
\|u\|_\infty\leq C(q)\left(1+\|u\|_{BMO}(\ln (\|u\|_{W^{1,q}}+\|u\|_\infty))^{\frac12}\right) \|\nabla b\|_\infty\leq C(q)\left(1+\|\nabla b\|_{BMO}(\ln (\|\nabla b\|_{W^{1,q}}+\|u\|_\infty))^{\frac12}\right)
\left[ Z_{ab},\bar{P}_{c}\right] =\frac{1}{\sqrt{2}}\left[ Z_{ab},P_{c}\right] +\frac{1}{\sqrt{2}}\left[ Z_{ab},Z_{c}\right] , & =\frac{1}{\sqrt{2}}f_{ab,c}^{d}Z_{d}-\frac{1}{\sqrt{2}}f_{ab,c}^{d}P_{d}, & =-f_{ab,c}^{d}\bar{Z}_{d},
\begin{vmatrix}mb_0 (m-1)b_1 (m-2)b_2 b_0 b_1 b_2 0 mb_0 (m-1)b_1 \end{vmatrix}\geqslant 0, \quad \frac{b_1^2}{b_0b_2}\geqslant\frac{2 m}{m-1}>2,
\left[\frac{\partial^{2}}{\partial r^{2}}+(\frac{1}{r}-4i\omega\Omega\theta r)\frac{\partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+(q^{2}-4i\omega\Omega\theta)\right]f(r,\theta)=0.
L_{S}=-\frac{1}{G}[\frac{1}{2}g^{\alpha\beta}(\frac{\partial_{\alpha}G\partial_{\beta}G}{G^{2}}+\frac{\partial_{\alpha}\mu\partial_{\beta}\mu}{\mu^{2}}-\partial_{\alpha}\omega\partial_{\beta}\omega)+\frac{V_{G}(G)}{G^{2}}+\frac{V_{\mu}(\mu)}{\mu^{2}}+V_{\omega}(\omega)]\sqrt{-g},
\epsilon_i =\begin{pmatrix}1 0 0 -1\end{pmatrix}\otimes \begin{pmatrix}1 0 0 -1\end{pmatrix}, f_i(0) =1\otimes \begin{pmatrix}0 1 -1 0\end{pmatrix},\;f_i(1)= \rho(u)\otimes \begin{pmatrix}0 1 0 0\end{pmatrix}+\rho(u)^{-1}\otimes \begin{pmatrix}0 0 -1 0\end{pmatrix}.
\begin{array}{c}{u = u_{0}+ \frac{p^{+}}2 \left[ ( \sigma^{+}+ f ( \sigma^{+}) ) + ( \sigma^{-}- g ( \sigma^{-}) ) \right]}{v = v_{0}+ \frac{p^{-}}2 \left[ ( \sigma^{+}- f ( \sigma^{+}) ) + ( \sigma^{-}+ g ( \sigma^{-}) ) \right]}\end{array}
G_{1}({\bf{q}},{\bf{k}}^{'},{\bf{q}}^{'}; n) = (\frac{1}{-i \mbox{}\beta})\frac{(1-f_{n}^{*}(-{\bf{q}}) ) \Lambda_{{\bf{k}}^{'}}(-{\bf{q}})\delta_{{\bf{q}},{\bf{q}}^{'}}}{(1- f_{n}^{*}(-{\bf{q}}) - f_{n}({\bf{q}}) ) ( i\omega_{n}- \omega_{{\bf{k}}^{'}}({\bf{q}}) )}
\bigl\langle\Xi(X_1),Y\bigr\rangle =\bigl\langle\Xi(X_1),[Y_1,b]\bigr\rangle =-\bigl\langle\Xi(Y_1),[b,X_1]\bigr\rangle-\bigl\langle\Xi(b),[X_1,Y_1]\bigr\rangle =\bigl\langle\Xi(Y_1),X\bigr\rangle\,,
I_M = <-\frac{1}{3!}m^2 a_{mnp}a^{mnp}- \frac{1}{2!}m^2\psi^2 + \frac{1}{4!}m\epsilon^{mnpq}\psi G_{mnpq}+ \frac{1}{3!}m\epsilon^{mnpq}(a_{mnp}- C_{mnp})\partial_q\phi>.
\nu\bigl((\psi\otimes\operatorname{id})(\Delta((\operatorname{id}\otimes\omega)[(1\otimes\tilde{y}c)(\Delta a)]))\bigr) &=\psi\bigl((\operatorname{id}\otimes\omega)[(1\otimes\tilde{y})(1\otimes(\nu\otimes\operatorname{id})((1\otimes c)E))(\Delta a)]\bigr) &=\psi\bigl((\operatorname{id}\otimes\omega)[(1\otimes\tilde{y}c)(\Delta a)]\bigr).
Z ( \beta ) = \pi^{\zeta / 2}\, \beta^{- \zeta / 2}\, d e t ( \widetilde{T})^{- 1 / 2}\, \prod_{k = 1}^{N}d e t ( \widetilde{T}_{k}^{*}\widetilde{T}_{k})^{\frac{1}{2}( - 1 )^{k - 1}}
\mathfrak{P}_\nu^{2}(P,Q;z):= \bigg\{f\in \mathcal{H}_k^*(N): \bigg|\sum_{\substack{P<p\leq Q p\nmid N}}\frac{\lambda_f(p^\nu)}{p}\bigg|>\left(\frac{96(\nu+1)^2z}{\log_2^2 (kN)P}\right)^{1/2}\bigg\}.
b_3 = -\frac{1}{2 X_{1+2+3}}\left( -8 a_1^2 \left( X_{1+2}+ X_{2+3}+ X_{1+3}+ 3X_{1+2+3}\right) \right. &\left. + 12 a_2 \left( X_{1+2+3}\right) + 8a_1^2 \left( X_{1+2}+ X_{2+3}+ X_{1+3}\right) \right) &=12 a_1^2-6a_2.
{{\dot \partial}_j}(g(X,l)) ={{\dot \partial}_j}({g_{km}}{X^k}{l^m})= 2{C_{kmj}}{X^k}{l^m}+{g_{km}}{{\dot \partial}_j}{X^k}{l^m}+{g_{km}}{X^k}{{\dot \partial}_j}{l^m}={g_{kl}}{X^k}(\frac{{\delta_j^m -{l^m}{l_j}}}{F}) = \frac{{{X_j}- g(X,l){l_j}}}{F}.
[\begin{matrix}x^{\prime}\\y^{\prime}\\z^{\prime}\\ 1\end{matrix}]=[\begin{matrix}1-2a^{2}&-2ab&-2ac&-2ad\\ -2ab&1-2b^{2}&-2bc&-2bd\\ -2ac&-2bc&1-2c^{2}&-2cd\\ 0&0&0&1\end{matrix}][\begin{matrix}x\\y\\z\\ 1\end{matrix}]
\begin{aligned}\lambda \|D^{m}_v u\|_{L_p (Q_r)}+ \| D^{m+1}_v u \|_{L_p (Q_r)}\leq N \| D^{m-1}_v D_x u \|_{L_p (Q_{r_1})}+N \| D^{m}_v u \|_{L_p (Q_{r_1})}.\end{aligned}
\varphi (\theta )=-\frac{2\cosh \theta \sin \frac{\pi}{\lambda}}{\sinh^{2}\theta +\sin^{2}\frac{\pi}{\lambda}}\, \sim \, -4\sin \frac{\pi}{\lambda}\, \mathrm{e}^{-\left| \theta \right|}+O\left( \mathrm{e}^{-2\left| \theta \right|}\right)\,,
\widetilde{w}_{r_1}\tilde{v}\widetilde{w}_{r_1}^{-1}=\widetilde w(\widetilde w_{r_1}\tilde{v_1}\widetilde w_{r_1}^{-1}) \widetilde w^{-1}\in \tilde{U}_{[w_{r_1}(r_2)]}
|E|\zeta_{n}(s,t)|^{2}|\leq 2 \sum_{k=1}^{\lfloor nt \rfloor}(\lfloor nt \rfloor -k) |\gamma_{k}(s)|+2\{nt\}\sum_{k=1}^{\lfloor nt \rfloor}|\gamma_{k}(s)|+(\lfloor nt \rfloor +\{nt\}^2)|\gamma_{0}(s)|.
(Q_{ij}) =\frac{1}{(A_1A_2A_3)^\frac{1}{3}}\left(\begin{array}{ccc}A_1 0 0 0 A_2 0 0 0 A_3\end{array}\right)=\left(\begin{array}{ccc}f_1 0 0 0 f_2 0 0 0 f_3\end{array}\right)
\kappa(\epsilon) = \mu(I-e^{-\alpha}\tilde{{P}})^{-1}r + \epsilon e^{-\alpha}\mu \sum_{k=0}^{\infty}a_{k+1,1}e^{-\alpha k}\tilde{P}^k \tilde{P}^{(1)}(I-e^{-\alpha}\tilde{P})^{-1}r + O(\epsilon^2 (\log(1/\epsilon))^w)
\mathrm{vol}_{\omega^k}\bigl( P \varphi(x_0) + r P d\varphi(x_0) (B) \bigr) \leq \mathrm{vol}_{\omega^k}\Bigl( \bigcup_{j=1}^N \bigl( P x_j + (1-\epsilon)r (B\cap \mathbb{H}_0) \bigr) \Bigr) \leq N \mathrm{vol}_{\omega^k}\bigl( (1-\epsilon)r (B\cap \mathbb{H}_0) \bigr) = N \pi^k (1-\epsilon)^{2k}r^{2k}.
\sum_{\pi'\atop{\pi'^S \cong \pi^S}}\mathrm{tr}(\pi'(\Phi^S))RT(\pi'(\Phi^1_S\mathrm{ch}_{K^S_U}))= 2 \mathrm{tr}(\Pi(f^S))TRT(\Pi(f_S^1\mathrm{ch}_{K^S})).
\frac{\partial^2}{\partial t^2}\left[ \log(\cosh(t)^2+1) \right] = \frac{\partial}{\partial t}\frac{2\sinh(2t)}{\cosh(2t)+3}& = \frac{4\cosh(2t)\,(\cosh(2t)+3) - 4\sinh(2t)\sinh(2t)}{(\cosh(2t)+3)^2}& = \frac{4+12\cosh(2t)}{(\cosh(2t)+3)^2}> 0.
\int_{0}^{T}\int_{\mathbb{T}}v(gu)dxdt =\lim_{N\rightarrow\infty}\int_{0}^{T}\int_{\mathbb{T}}v\nabla f_{N}dxdt=\lim_{N\rightarrow\infty}-\int_{0}^{T}\int_{\mathbb{T}}f_{N}\nabla vdxdt =-\int_{0}^{T}\int_{\mathbb{T}}f\nabla vdxdt=\int_{0}^{T}\int_{\mathbb{T}}v\nabla fdxdt
\langle \tau |=\sum_{h_{1},...,h_{\mathsf{N}}=0}^{p-1}\prod_{a=1}^{\mathsf{N}}\hat{Q}_{\tau ,a}^{(h_{a})}\prod_{1\leq b<a\leq \mathsf{N}}(X_{a}^{(h_{a})}-X_{b}^{(h_{b})})\langle h_{1},...,h_{\mathsf{N}}|,
\frac{\partial}{\partial t}\left(1-h^2\right)= \Delta\left(1-h^2\right)+2|\nabla h|^2{}-2h\left(\nabla_{k}L\nabla_{\bar{k}}h+\nabla_{\bar{k}}L\nabla_{k}h\right) = \Delta\left(1-h^2\right)+2|\nabla h|^2+\nabla_{k}L\nabla_{\bar{k}}\left(1-h^2\right)+\nabla_{\bar{k}}L\nabla_{k}\left(1-h^2\right).
\check{N}_{i,0}(\xi) =\begin{cases}1 \mbox{if}\xi_{i}\leq \xi \leq \xi_{i+1}0 \mbox{otherwise}\end{cases}, N_{i,p}(\xi) = \frac{\xi-\xi_{i}}{\xi_{i+p}-\xi_{i}}N_{i,p-1}(\xi) + \frac{\xi_{i+p+1}-\xi}{\xi_{i+p+1}-\xi_{i+1}}N_{i+1,p-1}(\xi),
A_hu(x,y) = (2\pi)^{-n}h^{-n/2-1/2}& \int e^{i (\frac{x-x'}{x^3}\frac{\xi_{\mathrm{1c}}}{h}+ \frac{y-y'}{x}\frac{\eta_{\mathrm{1c}}}{h^{1/2}})}a(x,y,\xi_{\mathrm{1c}},\eta_{\mathrm{1c}},h) u(x',y') \frac{dx'dy'}{(x')^{n+2}}d\xi_{\mathrm{1c}}d\eta_{\mathrm{1c}};
\sum_{j=1}^\infty \frac{1}{Z_j^2}= \frac{1}{2}+\frac{1}{x}\qquad\mbox{and}\sum_{j=1}^\infty \frac{1}{Z_j^4}= \frac{1}{6}\left(1+\frac{4}{x}+\frac{6}{x^2}\right).
[\ell^2_\mu,\,q\cdot\!\nu]=-i\left[2(\mu\cdot\nu)\, \ell_\mu+\sum_{\rho\in\Delta_+}g_{|\rho|}(\rho\cdot\mu)(\rho^\vee\!\cdot\nu) \frac{\ell_\mu^2-\ell^2_{s_\rho(\mu)}}{\ell_\mu-\ell_{s_{\rho}(\mu)}}\check{s}_\rho\right].
N=209,\ \varepsilon=1 :\ \ \ \ \ \ \ \ \ \ \ \ \ \ \{20, 87, 44, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1\}N=211,\ \varepsilon=1 :\ \ \ \ \ \ \ \ \ \ \ \ \ \ \{20, 91, 39, 1, 1, 0\}
k(t,t) = \sum_{i\in I}\mu_i e_i^2(t) + \sum_{j\in J}\tilde e_j^2(t) \geq \sum_{i\in I}\mu_i e_i^2(t) + \tilde e_{j_0}^2(t) > \sum_{i\in I}\mu_i e_i^2(t) = k(t,t)\, ,
\frac{\partial f_k}{\partial[\tilde{v}_i]_{j,\alpha}}=-\frac{[\tilde{w}_i]_{\alpha,k}}{m|g|}\sum_{h=0}^{m-1}[\tilde{v}_{i+h+1}\tilde{w}_{i+h+1}]_{k,j}\frac{\lambda_j^{m-h-1}\lambda_k^h}{\lambda_j^m-\lambda_k^m}.
\lim_{x\rightarrow1^{-}}(1-x)(1+x)^{\beta+3}f^{\prime\prime\prime}(x)\overline{g}^{\prime\prime}(x) = \lim_{x\rightarrow1^{-}}(1+x)^{\beta+3}\cdot\lim_{x\rightarrow1^{-}}(1-x)f^{\prime\prime\prime}(x)\cdot\lim_{x\rightarrow1^{-}}\overline{g}^{\prime\prime}(x) = 0.
\tilde\gamma_1 \mapsto \widetilde{(M\tilde\gamma)}_1 = \tilde\gamma_1 e^* - \tilde\gamma_2c^* + (B^Tc^*e^*B)_0^TA \tilde\gamma_2 \mapsto \widetilde{(M\tilde\gamma)}_2 = -\tilde\gamma_1b^* + \tilde\gamma_2a^* + (Aa^*b^*A^T)_0^TB^T,
\lim_{t\rightarrow \infty}\frac{||\mu^{\boxtimes t}||}{t}=\lim_{t\rightarrow \infty}\frac{1}{t h_t(\alpha_t)}=\lim_{t\rightarrow \infty}\left( \frac{1}{t \cdot \alpha_t}\cdot \exp[-(t-1)u(\alpha_t)] \right) =e\cdot \int_0^\infty\frac{s^2+1}{s^2}\,d\rho(s) =eV.
\sigma = (1) (2,3) (4,5,6) (7,8,9,10) \cdots \left( \frac{n^2-n}{2}+1 , \frac{n^2-n}{2}+1 , \frac{n^2-n}{2}+1 , \ldots, \frac{n^2-n}{2}+n , \right) \cdots
\lambda[\frac{1+t_{j}^{\prime}}{1+t_{1}^{\prime}}-1]=\frac{\mu\theta U_{j}}{f}[\frac{\partial^{2}U}{\partial L\partial x_{j}}\cdot\frac{1}{U_{j}}-\frac{\partial^{2}U}{\partial L\partial x_{1}}\cdot\frac{1}{U_{1}}]=\frac{\mu\theta U_{j}}{f}\frac{\partial}{\partial L}(lnU_{j}-lnU_{1}),
\omega^{(3)}= \left(\omega^{(3)}- \sum_{\beta \le i_3 < \alpha^\prime+N-m-1-i_1-i_2}\omega^{(3)}_{i_3}x^{i_3}\right) + \sum_{\beta \le i_3 < \alpha^\prime+N-m-1-i_1-i_2}\omega^{(3)}_{i_3}x^{i_3}
L_j^n c_{j-1}^{n+1}+M_j^n c_{j}^{n+1}+N_j^n c_{j+1}^{n+1}=\sum\limits_{l=1}^{n}(b_{n-l}-b_{n+1-l})(a_1c_{j-1}^{l}+a_2c_{j}^{l}+a_1c_{j+1}^{l}){}+b_n(a_1c_{j-1}^{0}+a_2c_{j}^{0}+a_1c_{j+1}^{0}),
g(y)=\frac{e^{ay/c}}{\left( c_{1}e^{ay/c}+1\right)^{2}}\,\Big(c_{1}\frac{a^{2}}{c_{3}}y-\frac{2ac_{1}}{c_{2}}\ln \left( c_{1}e^{ay/c}+1\right) +c_{2}\Big).
\int_{\tilde{X}}fd\tilde{\nu}=\int_{G/\Gamma}\int_{X}f_{z}d\phi_{\tau(z)}dm_{G/\Gamma}(z) =\int_{G/\Gamma}\int_{G/P}\int_{X}f_{z}d\boldsymbol{\beta}_{\nu}(w)d\nu_{P}(\tau(z).w)dm_{G/\Gamma}(z).
\ell_1\circ h_i \circ \mu_1 =x^rP^n(x), \ell_1\circ \hat h_i\circ \mu_2=x^n \mu_1^{(-1)}\circ h_{i+1}\circ \ell_2 =x^n , \mu_2^{(-1)}\circ \hat h_{i+1}\circ \ell_2=x^rP(x^n)
V_{l o n g}= - \int \frac{d s}{s^{3 / 2}}\frac{e^{- b^{2}s}}{8 \sqrt{\pi}c_{1}s c_{2}s}( c_{1}^{2}- c_{2}^{2})^{2}s^{4}( \frac{\sqrt{\pi}}{L_{1}s^{1 / 2}}) = - \frac{( c_{1}^{2}- c_{2}^{2})^{2}}{8 c_{1}L_{1}c_{2}b^{2}}.
\langle T_{i}^{k}\rangle^{( b )}= \frac{\zeta \delta_{i}^{k}}{\pi \xi^{2}}\int_{0}^{\infty}d \omega \, \frac{A - B \omega / a}{A + B \omega / a}e^{- 2 \omega \ln ( a / \xi )}\tilde{F}_{0}^{( i )},
|\partial_{t}\left(\frac{16}{\lambda(t)^{2}}\int_{t}^{\infty}ds \lambda''(s) \left(K_{1}(s-t,\lambda(t))-\frac{\lambda(t)^{2}}{4(1+s-t)}\right)\right)| &\leq \frac{C}{t^{3}\log^{b+1}(t)}+ C \sup_{x \geq t}|e'''(x)|
\!\!\!\!{\cal L}_{DSM}=\frac 1{1+4{\Phi}^2}\left(\phantom{\frac II}\!{\textstyle \frac 12}\left(\delta^{ij}+4\Phi^i\Phi^j\right) \partial_\mu \Phi^i\partial^\mu\Phi^j-\varepsilon^{\mu \nu}\varepsilon^{ijk}\Phi^i\partial_\mu \Phi^j\partial_\nu \Phi^k \right)~ .
\begin{aligned}C(\{s, v\}, v) = c_{sv}(0) \forall v \in V, v \neq s C(S, v) = \min_{\substack{u \in S u \neq s, v}}C(S \setminus \{v\}, u) + c_{uv}(C(S \setminus \{v\}, u)) \forall S \subseteq V, v \in S. \end{aligned}
-\int_0^s \Theta( u ,x) f'(u) \mathrm{d}u+ f(s){\Theta}(s,x)-f(0){\Theta}(0,x) =-\alpha\int_0^{s}\int_0^1 f(u)A(x,y)\Theta(u,y)\mathrm{d}y\mathrm{d}u+\alpha^2\beta^2 \int_0^s f(u)\mathrm{d}\xi_1(u,x).
\phi_{n,\lambda}(x) =\frac{1}{e^{x}}\sum_{k=0}^{\infty}\frac{(k)_{n,\lambda}}{k!}x^{k}=\frac{1}{e^{x}}\bigg(x\frac{d}{dx}\bigg)_{n,\lambda}e^{x},(n\ge 1).
H^{k}_{s}= \overline{C^{\infty}_{c}(G,\Omega^{1/2}_{s})}^{\langle \cdot | \cdot \rangle_{k,s}}, \langle \omega \ | \ \eta \rangle_{k,s}= \langle (1 + P^{2})^{k}\star \omega \ | \ \eta\rangle_{s}\in C(M).
| - \frac{1}{2}\rangle = -\frac{1}{\sqrt{6(1+2x^2)}}[(2 x + 2) |\downarrow \uparrow \downarrow \rangle + (2 x -1) |\downarrow \downarrow \uparrow \rangle + (2 x - 1) |\uparrow\downarrow \downarrow \rangle ]
\left|\frac{\psi(x)-x}{x}\right| \leq A(x_0,\sigma)\cdot \left(B_2r\right)^{5 - 2\sigma}\exp\left(\frac{B_2(5 - 8\sigma)}{3}r\right) &= A(x_0,\sigma)\cdot \left(B_2\frac{\log^{3/5}x}{(\log\log x)^{1/5}}\right)^{5 - 2\sigma}\exp\left(\frac{B_2(5 - 8\sigma)}{3}\frac{\log^{3/5}x}{(\log\log x)^{1/5}}\right).
j = \left( \begin{array}{c c c c c c}{\theta_{1}}&{}&{}&{}&{}&{}\\{}&{\ddots}&{}&{}&{}&{}\\{}&{}&{\theta_{q + 1}}&{}&{}&{b}\\{}&{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{}&{\theta_{p}}&{}\\{}&{}&{c^{t}}&{}&{}&{0}\\ \end{array}\right) .
\big\langle \mu^r(h ?)\psi, x \big\rangle = \mu^r(hx')\psi(x'') = \mu^r(h'x')\psi\big(S^{-1}(h''')h''x''\big) = \psi\big(S^{-1}(h''')\mu^r(h'x')h''x''\big) &= \psi\big(S^{-1}(h'') \mu^r ((h'x)' )(h'x)''\big) = \mu^r(h'x) \psi\big(S^{-1}(h'')\big),
\beta(g) = - 2 (g^* - g) \left\{1 +{1\over N}\left[{2\over \ln \Theta}\left(1 +{l(\Theta)\over \ln \Theta}\right) +{5\over 2 \ln^2 \Theta}+ O\left({l(\Theta)^2\over\ln^3 \Theta}\right) \right] \right\},
\big[S^{(4)}(x_1,x_2,x_3)\big]^{2}+S^{(4)}(x_1,x_2,x_3) \qquad{}= L_1^2+L_2^2+L_3^2+\big(x_1^2+x_2^2+x_3^2\big)\left(\frac{\mu_1(\mu_1-R_1)}{x_1^2}+\frac{\mu_2(\mu_2-R_2)}{x_2^2}+\frac{\mu_3(\mu_3-R_3)}{x_3^2}\right),
n_{j}(\psi) = \begin{dcases}2\sum_{i=0}^{\ell/2}q^i - q^{\ell/2}, \ell\leq 2j 2\sum_{i=0}^{(\ell-1)/2}q^i = 2\frac{q^{(\ell+1)/2}- 1}{q-1}, \ell\leq 2j 2\sum_{i=0}^{j-1}q^i + (\ell-2j+1)q^j, \ell > 2j . \end{dcases}
0=F_l^{\prime\prime}+\frac{2}{r}F_l^\prime \left[1+rL_-^\prime(kr)\right] -\frac{s^\prime}{s-\frac{\omega}{m}}\left[F_l^\prime+F_l L_-^\prime(kr)\right] +W(r,m) F_l -\left[K,F_l\right]
\begin{aligned}\mathbb{E}^{u_1,u_2}[\tilde{H}_{1{v_1}}(t,x,y,z,z_1,z_2,u_1,u_2;q_1,k_1,k_{11},k_{21},p_1,Q_{11},Q_{21})(v_1-u_1(t))|\mathcal{F}_t^1]\geq 0, \mathbb{E}^{u_1,u_2}[\tilde{H}_{2{v_2}}(t,x,y,z,z_1,z_2,u_1,u_2;q_2,k_2,k_{12},k_{22},p_2,Q_{12},Q_{22})(v_2-u_2(t))|\mathcal{F}_t^2]\geq 0,\end{aligned}
\partial_t\int_0^\infty P_t^\lambda (x,y)u(y,r)dy =\partial_t\int_0^\infty P_t^\lambda (x,y)y^\lambda y^{-\lambda}u(y,r)dy &=\partial_t\int_0^\infty P_t^\lambda (x,y)y^\lambda [y^{-\lambda}u(y,r )-x^{-\lambda}u(x,r)]dy &=\partial_t\int_0^\infty P_t^\lambda (x,y)y^\lambda \int_x^y\partial_z[z^{-\lambda}u(z,r )]dzdy,x,t,r \in (0,\infty ).
\mathrm{Bl}_{\mathcal{C}}(\mathcal{S})\,{:=}\, \mathrm{Bl}_{\mathcal{C}}(\varSigma_{\mathcal{S}};\mathbb{F}_{\ell_{\mathrm{in}}}(\mathsf{b}_{\mathrm{in}}), \mathbb{F}_{\ell_{\mathrm{out}}}(\mathsf{b}_{\mathrm{out}})) \,,
\Delta_{\varepsilon , \pm 1 , \overline{{\varepsilon}}}= \frac{\sigma^{2}}2 \cdot \frac{m - i \varepsilon \widehat{p}}{2 m}\cdot \frac{m - i \overline{{\varepsilon}}\overline{{p}}}{2 m}\cdot \frac 1 2 \sigma_{p}\left( \sigma_{p}\pm 1 \right) = \Psi_{\varepsilon , \pm 1 , \overline{{\varepsilon}}}\cdot \overline{{\Psi}}_{\varepsilon , \pm 1 , \overline{{\varepsilon}}},
P=\frac{nRT}{V}(1+\frac{\overline{m}^{2}B}{V-\overline{m}B})-\frac{\overline{m}^{2}A}{V(V+\overline{m}B)}-\frac{nRT}{V}[\sum_{i}n_{i}(m_{i}-1)\beta\frac{g^{\prime}(\beta)}{g(\beta)}]
\sum_{k=1}^{N_D}U_k^* U_k = \sum_{k=1}^{N_D}S_{d_k}S_{d_k}^* S_{c(k)}^*S_{c(k)}S_{d_k}S_{d_k}^* = \sum_{k=1}^{N_D}S_{d_k}S_{d_k}^* =P_D, \sum_{k=1}^{N_C}T_l^* T_l = \sum_{l=1}^{N_C}S_{c_l}S_{c_l}^* S_{d(l)}^*S_{d(l)}S_{c_l}S_{c_l}^* = \sum_{l=1}^{N_C}S_{c_l}S_{c_l}^* =P_C.
\cos^{2}\alpha = \frac{4}{9}\cos^{2}\beta = \frac{4}{9}( 1 - \sin^{2}\beta ) = \frac{4}{9}- \frac{1}{9}\sin^{2}\alpha = \frac{4}{9}- \frac{1}{9}( 1 - \cos^{2}\alpha ) = \frac{1}{3}+ \frac{1}{9}\cos^{2}\alpha
\mathcal{Q}=s^2\chi_0\left(\begin{array}{c}\frac{q^+(t)}{\sqrt{z}}\frac{q^-(t)}{\sqrt{\bar{z}}}\end{array}\right)\mbox{or}\mathcal{Q}=s^2\chi_0\left(\begin{array}{c}\frac{q^+(t)}{\sqrt{\bar{z}}}\frac{q^-(t)}{\sqrt{z}}\end{array}\right)
\begin{aligned}(28) \Leftrightarrow \frac{\partial}{\partial y}\int dx \x \b(x,y,t) = \frac{\partial}{\partial y}\left(y \alpha(y,t)\right)+\frac{1}{4}\frac{\partial^{2}}{\partial y^{2}}\alpha(y,t)-\frac{\partial}{\partial t}\alpha(y,t)\end{aligned}
{0=\Omega_i - \Omega_j =}\frac{dg_{ij}}{g_{ij}}+ \lambda \frac{d f_{ij}}{f_{ij}}+ \frac{1}{f_j^k}\frac{df_{ij}}{f_{ij}^{k+1}}+ \left( \frac{1}{f_{ij}^k}- 1 \right) \frac{df_j}{f_j^{k+1}}
( \psi_{\omega}^{( \lambda )}, \psi_{\sigma}^{( \lambda )}) ={\frac{1}{( \omega - \sigma )}}\left[ ( \psi_{\sigma}^{( \lambda )}, H ( \lambda ) \psi_{\omega}^{( \lambda )})^{*}- ( \psi_{\omega}^{( \lambda )}, H ( \lambda ) \psi_{\sigma}^{( \lambda )}) \right] ~ ~ ~
\zeta(4)=\frac{\pi^{4}}{90},\:\zeta(6)=\frac{\pi^{6}}{945},\:\zeta(8)=\frac{\pi^{8}}{9450},\:\zeta(10)=\frac{\pi^{10}}{93555},\:\zeta(12)=\frac{691\pi^{12}}{638512875}
\left({\dot{\hat{a}_0}\over\hat{a}_0}+{\dot{\hat{c}}\over\hat{c}}\right)^2={{8\pi\hat{G}_4}\over 3}\hat{\varrho}+{\hat{c}^2\over 3}\hat{\Lambda}_{\rm eff}+{\hat{\cal C}\hat{c}^2\over\hat{a}^4_0}-{{\hat{k}\hat{c}^2}\over\hat{a}^2_0},
\pi(\l')^{-1}S_{g,\L}\, \pi(\l') f = \sum_{\l \in \L}\langle \pi(\l') f, \pi(\l) g \rangle \, \pi(\l')^{-1}\pi(\l) g & = \sum_{\l \in \L}\langle f, \pi(\l')^{-1}\pi(\l) g \rangle \, \pi(\l')^{-1}\pi(\l) g & = \sum_{\l \in \L}\langle f, \pi(\l-\l') g \rangle \, \pi(\l-\l') g & = S_{g,\L}f.