harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
U_{E,T},U_{G_i,S}(i=1,2,3,4) U_{G_1,S}U_{G_4,S}\equiv 1,U^{2}_{G_2,S}\equiv 1,U^{2}_{G_3,S}\equiv 1, &U_{G_1,S}U_{G_3,S}U_{G_4,S}\equiv 1.
\gamma_{r}= \left( \begin{array}{c c}{{0}}&{{e^{- i \theta}}}\\{{e^{i \theta}}}&{{0}}\\ \end{array}\right) , \qquad \gamma_{\theta}= \left( \begin{array}{c c}{{0}}&{{- i e^{- i \theta}}}\\{{i e^{i \theta}}}&{{0}}\\ \end{array}\right) .
M:= \begin{pmatrix}(g_1, h_{1}^{(1)}h_{1}^{(2)})_{\Omega}\dots (g_m, h_{1}^{(1)}h_{1}^{(2)})_{\Omega}\vdots \vdots \vdots (g_1, h_{m}^{(1)}h_{m}^{(2)})_{\Omega}\dots (g_m, h_{m}^{(1)}h_{m}^{(2)})_{\Omega}\end{pmatrix}
( \mathrm{M a s s})^{2}={\frac{1}{4 \rho_{2}^{( 0 )}}}\left[{\frac{1}{2}}( \hat{\alpha}_{L})^{2}+ N_{L}- 1 \right] ={\frac{1}{4 \rho_{2}^{( 0 )}}}\left[{\frac{1}{2}}( \hat{\alpha}_{R})^{2}+ N_{R}- \delta \right] ~ .
=\left(1-q^{-n}+2q^{-1+(n-1)s_1}-q^{n^2-ns_1-s_2}-q^{n^2-n-ns_1-s_2}\right)\cdot \frac{(1-q^{-1})(1-q^{-n})q^{n^2+1-(2n-1)s_1-s_2}}{(1-q^{n^2+1-(2n-1)s_1-s_2})(1-q^{n^2-ns_1-s_2})},
\begin{aligned}f(0) = \, |x(0)-z_1(0)|+ |\bar{y}(0)-z_2(0)| = \, |x(0)-x_a(0)+\gamma \epsilon_1(\eta_a(0),0)| &\,+ |\bar{y}(0)-\bar{y}_a(0)+\gamma \epsilon_2(\eta_a(0),0)| = \, |x(0)-x_a(0)|+ |\bar{y}(0)-\bar{y}_a(0)|.\end{aligned}
\sum_{i=1}^N\|x^{k+1}_i-x^*_i\|^2 \leq \sum_{i=1}^N \|x^k_i-x^*_i\|^2 +N\tilde C\alpha_k^2 +4\alpha_k MN\sum_{i=1}^N \bar L_i \|\hat v_i^k - y^k\| \cr \; - 2 \alpha_k\sum_{i=1}^N\left(F_i(x^k_i,Ny^k) - F_i(x^*_i,\bar{x}^*)\right)^T(x_i^k-x^*_i).
A = 2^4 (\tilde{c}_4 - 6 \tilde{\Delta}) = 2^4 \tilde{A}B = 2^8 (\tilde{c}_4^2 + 6 \tilde{c}_4 \tilde{\Delta}^{1/3}+ (6 \tilde{\Delta}^{1/3})^2) = 2^{8}\tilde{B}
\delta_{\alpha{\mathbf i}}{\cal R}^{\mu}= 1 6 i \left\{\frac{1}{1 9 2}\left[ \frac{3}{2}J_{\alpha{\mathbf i}}^{\mu}+ \sigma_{\alpha}^{\mu \nu \beta}J_{\nu \beta{\mathbf i}}\right] - \frac{3}{4}( 8 + a ) \partial^{\mu}D_{\alpha{\mathbf i}}T | + \frac{1}{2}( a - 2 4 ) \sigma_{\alpha}^{\mu \nu \beta}\partial_{\nu}D_{\beta{\mathbf i}}T | \right\}.
\begin{split}Z^{(1)}_t = Z_0 + \int_0^t \int_{\mathbb{D}\times \mathbb{D}\times \Xi}\alpha(Z_s^{(0)}, v_s, \xi) d{\tilde{N}}_{\mu_{0}}& + \int_0^t \int_{\mathbb{D}\times \mathbb{D}\times \Xi}\hat{\alpha}(Z^{(0)}_s, v, \xi) \beta(|X_s^{(0)}- y_s|)d\mu_0 Q(d\xi) ds \end{split}
\partial_r a (r) = \frac{1}{(\sinh r)^{n-1}}\int_0^r (\sinh s)^{n-1}ds \leq \frac{1}{(\sinh r)^{n-1}}\int_0^r (\sinh s)^{n-2}(\cosh s) ds = \frac{1}{n-1}= \frac{1}{2\rho}.
\begin{array}{ll}\sum\limits_{\alpha,\beta \in \mathbb{F}_{p^m}}S(\alpha,\beta) =2p^m+2p^{\frac{m}{2}}( n_{1,1,0}-n_{-1,-1,0}+2(\sqrt{q^*}+1)(n_{1,1,1}-n_{-1,-1,1})+ 2(\sqrt{q^*}-1)(n_{1,-1,1}-n_{-1,1,1})+(p^d+1)(n_{1,1,2}-n_{-1,-1,2})+(p^d-1)(n_{1,-1,2}-n_{-1,1,2})), \end{array}
i_4(M) < j_4(S_t) \sum_{\lambda}\sum_{l_1+2l_2+4l_4=t}i_{4, \lambda}(Cl_m(q))^{l_1}i_{2, \lambda}(Cl_m(q))^{l_2}|Cl_m(q)|^{3l_4+l_2}&< |S_t|^\frac{3}{4}|Cl_m(q)|^{\frac{3}{4}(l_1+2l_2+4l_4)}q^{c''n}&< |M|^\frac{3}{4}q^{c'''n}.
{\frac{M}{M_{P l}}}={\frac{( P_{k}^{S})^{1 / 4}}{2 \sqrt{L}u^{1 / 2}[ ( 1 + u^{2}) ( 1 + 3 u^{2}) ]^{1 / 8}}}\approx{\frac{2 . 0 \times 1 0^{- 4}\left( 1 0^{1 0}P_{k}^{S}\right)^{1 / 4}\left({6 0 / L}\right)^{1 / 2}}{u^{1 / 2}[ ( 1 + u^{2}) ( 1 + 3 u^{2}) ]^{1 / 8}}};
- \partial_{q}W = \frac{4 \pi \hbar}{T_{B H}}\int_{r_{+}}d r \, r^{2}\: \biggl [ \frac{\partial_{q}F}{F}\biggr ] \, \langle T_{\alpha}^{\alpha}\rangle \; + \frac{4 \pi \hbar}{T_{B H}}\int_{r_{+}}d r \, r^{2}\: \biggl [ \frac{4 \partial_{q}r}{r}- \frac{2 \partial_{q}F}{F}\biggr ] \, \langle T_{\theta}^{\theta}\rangle + \cdots .
\lim_{n\to\infty}\,\frac{1}{t_n^{-1}\lambda^1_n}\int_{-t_n^{-1}\lambda^1_n}^{t_n^{-1}\lambda^1_n}\int_{\left|x-\frac{c^1_n}{t_n}\right|<Mt_n^{-1}\lambda^1_n}\,\left|\partial_t\widetilde{u}_n+\ell_1\cdot\nabla \widetilde{u}_n\right|^2\,dxdt=0.
\alpha(v)= \frac{8|v|^{4}+(1-\overline{v}^2)(1-|v|^2)(v^2+|v|^2)}{2|v|^2(1-\overline{v}^2)(|v|^2-v^2)}&=\frac{4\overline{v}}{(1-\overline{v}^2)(\overline{v}-v)}+\frac{(1-|v|^2)(v+\overline{v})}{2|v|^2(\overline{v}-v)},
\begin{gathered}\mu_{0}^{n,k}=1,\mu_{l+1}^{n,k}=-(n-k+l+1)\mu_{l}^{n,k}; \begin{cases}r_{0,l+1}^{n,k}=-(n-k+l+1)r_{0,l}^{n,k}, l>0, r_{m,l+1}^{n,k}=r_{m-1,l}^{n,k}-(n-k+l+1)r_{m,l}^{n,k}, 1\leq m\leq l-1, r_{l,l+1}^{n,k}=n-k+l+1.\end{cases}\end{gathered}
\begin{aligned}\tilde v_\ell^+(x_1,X_2)=u_\ell(x_1-\ell,X_2)~\text{on}~\Omega_\ell^+\,, \tilde v_\ell^-(x_1,X_2)=u_\ell(x_1+\ell,X_2)~\text{on}~\Omega_\ell^-\,.\end{aligned}
P(x,t|x^{\prime},t^{\prime})=\sqrt{\frac{\theta}{2\pi D(1-e^{-2\theta(t-t^{\prime})})}}exp[-\frac{\theta}{2D}\frac{(x-x^{\prime}e^{-\theta(t-t^{\prime})})^{2}}{1-e^{-2\theta(t-t^{\prime})}}]
\|P\|_{4}^{4}= \sum_{|\gamma| = 2m}{\left| \sum_{|\alpha|=m}{c_{\alpha}c_{\gamma - \alpha}}\right|^{2}}\leq \sum_{|\gamma|=2m}{\left( \sum_{\substack{|\alpha|=m \alpha \leq \gamma}}{|c_{\alpha}|^{2}|c_{\gamma - \alpha}|^{2}}\right) \kappa(\gamma,m)}.
\frac{\tau^{2}/ r + 2 r}{( \tau^{2}- r^{2})^{5 / 2}}\qquad , \qquad \frac{3 \tau^{2}+ 2 r^{2}}{( \tau^{2}- r^{2})^{7 / 2}}\qquad \mathrm{e t c .}
\int d^{3}q \int_{0}d x x^{\alpha^{\prime}q^{2}- \alpha^{\prime}s - 1}= \pi^{3 / 2}( \alpha^{\prime})^{- 3 / 2}\int_{0}d x x^{- 1 - \alpha^{\prime}s}[ \ln ( 1 / x ) ]^{- 3 / 2}\ ,
m_n(\mu)=e^{n L\kappa_1(\mu)}\cdot\displaystyle\sum_{\substack{\Gamma \textup{simple chain in}NC(n) \Gamma=(\pi_0,\ldots,\pi_{|\Gamma|}),\pi_0=0_n}}\frac{1}{|\Gamma| !}\displaystyle\prod_{i=1}^{|\Gamma|}L\kappa\Big[K_{\pi_i}(\pi_{i-1})\Big]\left(\mu \right).
\| \phi_j \phi_j^* - \phi_l \phi_l^*\|_{H.S.}^2 = 2\left(1- tr( \phi_j \phi_j^* \phi_l \phi_l^* )\right) &= 2\left(1- |\langle \phi_j, \phi_l \rangle|^2 )\right).
K_{t}(w; z_{\ell(1)}, \ldots , z_{\ell(t)})=\frac{g(w, z_{\ell(1)})g(w, z_{\ell(t)})}{g(z_{\ell(t)}, z_{\ell(1)})}\prod_{s=1}^{t-1}g(z_{\ell(s+1)}, z_{\ell(s)})
\begin{aligned}\psi^{+0}_{n,p}=(n-p+a_{12}-a_{123}+j^{(3)})\,\prod_{j=-\infty}^pu_{n,p-j}\ , \rho^{0+}_{n,p}=(n-p+a_{12}-a_{123}-j^{(3)})\,\prod_{k=-\infty}^nu_{n-k,p}\ ,\end{aligned}
M_{p l}^{2}={\frac{M_{X}^{3}}{k}}e^{- 2 k y_{1}- 2 c}\left({\frac{1}{2}}e^{- 2 k ( y_{2}- 2 y_{1})}- e^{- 2 k ( y_{2}- y_{1})}+ 1 -{\frac{1}{2}}e^{- 2 k ( \pi \rho - y_{2})}\right) ~ . ~ \,
e^{\textstyle -J\alpha_-\Phi(\sigma, \tau )}=c_J \>\sum_{m=-J}^J \,{(-1)^{J-m}\, \lambda_m^{(J)}(\varpi)\over\sqrt{\lfloor \varpi\rfloor}\sqrt{\lfloor \varpi+2m\rfloor}}\>\psi_m^{(J)}(x_+)\,{\overline \psi_{m}^{(J)}}(x_-)
G^-=\{x\in\mathbb{R}^2\, | \ \begin{bmatrix}0 0 -\beta_2v_{12}-\beta_2v_{22}-\beta_1v_{11}-\beta_1v_{21}\end{bmatrix}\begin{bmatrix}x_1 x_2\end{bmatrix}\succeq_{\mathcal{L}^3}\begin{bmatrix}-\eta -\beta_2\alpha_2 -\beta_1\alpha_1\end{bmatrix}\}.
{\rm Hilb}({\rm ACYB}(B_2),t)=\big(1-4t+2t^2\big)^{-1},{\rm Hilb}({\rm ACYB}(B_3),t)=\big(1-9t+16t^2-4t^3\big)^{-1},{\rm Hilb}({\rm ACYB}(B_4),t)=\big(1-16t+64t^2-60t^3+9t^4\big)^{-1},{\rm Hilb}({\rm ACYB}(D_4),t)=\big(1-12t+18t^2-4t^3\big)^{-1}.
\langle J'(u,v), (u,v)\rangle= 2a-2^*(s_1)\big[a-\kappa(\alpha+\beta)c\big]-\kappa(\alpha+\beta)^2c = \big[2-2^*(s_1)\big]a+\kappa\big[2^*(s_1)-\alpha-\beta\big](\alpha+\beta)c \leq \big[2-2^*(s_1)\big]a.
p_{X_{1I}X_{2I}Y_IU_IQ}(x_1,x_2,y,u,q) =p_{X_{1I}X_{2I}}(x_{1},x_{2})p_{Q|X_{1I}X_{2I}}(q|x_1,x_2)p_{Y|X_{1}X_{2}}(y|x_{1},x_{2})p_{U|Y}(u|y).
|\phi^{N,i}(t_0,\boldsymbol{\xi})-\Psi^{N,i}(t_0,\boldsymbol{\xi})|\leq\frac{C}{N}\Big(1+\frac{1}{N}\underset{j=1}{\overset{N}\sum}|\xi^{i}-\xi^{j}|^2+\frac{1}{N^2}\underset{i,j=1}{\overset{N}{\sum}}|\xi^{i}-\xi^{j}|^2\Big)^{\frac{1}{2}}.
d^{3}_{r}B_{\mathfrak a}(f\wedge g)(x,y,z,t)=B\left( d^2f(x,y,z),g(t)\right) + B\left(d^2_{c}f(x,y,t),g(z) \right)+B\left(d^2_{c}f(x,z,t),g(y) \right)+B\left(d^2_{c}f(y,z,t),g(x) \right) + B\left( (f\circ \alpha)\wedge d^1g \right)(x,y,z,a)
f(t) = (-1)^m \alpha \left( \prod_{n=1}^{m-1}e^{\lambda (\bar{t}_n - \bar{t}_{n-1})}( \lambda G_{k_n}- G_{k_n}\lambda) \right) e^{\lambda(\bar{t}_m - \bar{t}_{m-1})}\lambda G_{k_m}{\bf 1},
{\cal L}={\bar{\nu}}_{e}\left( i \not \partial - m_{e}\right) \nu_{e}+{\bar{\nu}}_{\mu}\left( i \not \partial - m_{\mu}\right) \nu_{\mu}- m_{e \mu}\left({\bar{\nu}}_{e}\nu_{\mu}+{\bar{\nu}}_{\mu}\nu_{e}\right) .
W^{(s)}_C(Z) = Z^{n-k}(1-Z)^k \sum_{i=0}^s \frac{(-1)^{s-i}}{[s]_s}q^{{s-i \choose 2}}{s \brack i}T_C( \frac{1+(q^i-1)Z}{1-Z}, \frac{1}{Z}),
[ T_{\vec{m}_{1}}, \cdots , T_{\vec{m}_{M}}] = \frac{- i}{2 \pi \Lambda}\sin \bigl ( 2 \pi \Lambda ( \vec{m}_{1}, \cdots , \vec{m}_{M}) \big ) T_{\vec{m}_{1}+ \cdots + \vec{m}_{M}}
\begin{array}{lcl}\partial_{\overline z}\left<T\left(z,\overline z\right)\right> = -{1\over16\pi}\,{1\over{3!}}\,\partial_z^3h^{zz}\left(z,\overline{z}\right)\int_0^{+\infty}{1\over2}\,dr^2\delta\left(r^2-a^2\right)\int_0^{2\pi}d\theta \ = -{1\over96}\,\partial_z^3h^{zz}\left(z,\overline{z}\right)\;.\end{array}
M(u) =\sum_{1\leq k<\infty}\sigma_k^2\cdot \dim F_u(\sigma_k), Q(u)^2 =\sum_{1\leq k\leq \infty}\ell_k, |J(u)|^2 =\sum_{1\leq k\leq \infty}(Q+\sigma_k^2)\ell_k.
I(\tau) \approx \tau S(t_n+\tau\xi_n)^{-1}\left [| S(t_n+\tau\xi_n)v(t_n) |^2 S(t_n+\tau\xi_n)v(t_n)\right] &=\tau S(t_n+\tau\xi_n)^{-1}\left [( S(t_n+\tau\xi_n)v(t_n) )^2 \overline{S(t_n+\tau\xi_n){v}(t_n)}\right]=:J(\tau)
s_1=\ln\sqrt{w}+\int\frac{F(\tilde{x}^0)}{2}d\tilde{x}^0, s_2=\frac{1}{4w^2}\frac{dw}{d\tilde{x}^0}+\frac{1}{8}\int d\tilde{x}^0[\frac{1}{w^3}(\frac{dw}{d\tilde{x}^0})^2+\frac{F^2}{w}+\frac{2}{w}\frac{dF}{d\tilde{x}^0}], \cdots\cdots
\Delta E(\vartheta )=-2\cos \left( \vartheta \right) \left| \frac{\det 'M}{\det M_{0}}\right|^{-1/2}\left( \frac{S_{E}\left[ \varphi_{\mathrm{inst}}\right]}{2\pi}\right)^{1/2}e^{-S_{E}\left[ \varphi_{\mathrm{inst}}\right]}\,,
\prod_{i=1}^p \frac{1-t_i}{1-t_i/t_{p+1}}-\prod_{i=1}^p Q_{i}^N =\prod_{i=1}^p \frac{1-z_{p+1}}{1-z_{p+1}/z_i}\left(\underbrace{1-\prod_{i=1}^p\left(1- \frac{(1-z_i)^{N+1}}{(1-z_{p+1})^{N+1}}\frac{z_{p+1}}{z_i}\right)}_{\spadesuit}\right).
{}^C D_{a^+}^\alpha f(t) = D_{a^+}^\alpha\left( f(t) - \sum_{k=0}^{m-1}\frac{(t-a)^{k}}{k!}f^{(k)}(a^+)\right) = D_{a^+}^\alpha f(t)-\sum_{k=0}^{m-1}\frac{(t-a)^{k-\alpha}}{\Gamma(k+1-\alpha)}f^{(k)}(a^+).
\Delta_{L}^{M}\equiv \delta^{M}_{L}(\partial^{2}_{T}-\partial^{2}_{\sigma})+2\varphi^{P}_{,T}\Gamma^{M}_{PQ}(\varphi)\partial_{T}+\varphi^{P}_{,T}\varphi^{Q}_{,T}\partial_{L}\Gamma^{M}_{PQ}
\hat h = \frac{\mu}{\mu(q)}\hat h^{(0)}+ \frac{C_2-C_1}{\mu c^2}\frac{\alpha^2}{q^2}+ \frac{\hbar^2}{2}\frac{\alpha}{\mu^2c^2}C_1\left(\frac{{\bf q}}{q^3}\frac{\partial}{\partial{\bf q}}+ 4\pi\lambda\delta({\bf q})\right)\,,
{\textbf{a}^{(1)}}= \textbf{a}^n + \Delta t \mathfrak{L}(\textbf{a}^n,t_n),{\textbf{a}^{*}}= \textbf{a}^{(1)}+ \Delta t \mathfrak{L}(\textbf{a}^{{1}},t_{n+1}), \textbf{a}^{n+1}= \frac{1}{2}\textbf{a}^n + \frac{1}{2}\textbf{a}^{*}.
\begin{aligned}\frac{1}{t_1}-\frac{2}{n}\le \frac{1}{r}+ \frac{q-2}{N}- \frac{2}{n}&= \frac{1}{r}+\frac{q}{N}- \left[ \frac{2}{N}+\frac{2}{n}\right] &= \frac{1}{r}+\left[\frac{q}{N}-1\right].\end{aligned}
\limsup_{{\mathcal N}\ni n\to\infty}\phi\left(\frac{1}{n}\sum_{k=1}^n |\alpha_{k,n}|\right) \le \limsup_{{\mathcal N}\ni n\to\infty}C_n^{\phi}(|\alpha_{1,n}|,\ldots,|\alpha_{n,n}|) < \infty.
(\phi^2+\phi*\phi)(a) = c^2(1+J'(\alpha,\alpha))\alpha^2(a)+\bar c^2(1+J'(\bar\alpha,\bar\alpha))\bar\alpha^2(a) +2(1+J(\alpha,\bar\alpha)) = c^2(1+J'(\alpha,\alpha))\bar\alpha(a)+\bar c^2(1+J'(\bar\alpha,\bar\alpha))\alpha(a).
n(2\ln n)^{\frac{1}{2}}\sup_{y\in I}\Big(\mathbb{P}(\xi^{(n)}([y,y+G_n(x)/S(I)])=0)-D_n(\sqrt{4-y^2}/S(I) \cdot G_n(x)/2)\Big) \leq n(2\ln n)^{\frac{1}{2}}O((n\ln n)^{-1})=O((\ln n)^{-1/2})\to 0,\,\,\,n\to+\infty,
\left( \prod_{s=1}^{l}\prod_{p=1}^{i}(u - \hbar'(p - 1) + a_s) \right) A_i(u) \xi_{\Lambda}= \left( \prod_{s=1}^{l}\prod_{p=1}^{i}(u - \nu_{i,p}^{(s)}) \right) \xi_{\Lambda},
\mathcal{A}^{0}\varphi(\rho_{-})=b^{0}(\rho_{-})\varphi'(\rho_{-}) +\int_{\rho_{-}}^{\infty}(\varphi(\rho_{+})-\varphi(\rho_{-}))f^{0}(\rho_{-},\rho_{+})d\rho_{+}
W^{j}_{4}=-\varepsilon\int_{\mathbb{T}^{d}}(\partial_{x_{j}}w^{1}-\partial_{x_{j}}w^{2})\left(\Theta_{p}(t,x,\mu^{2},Dw^{2})\mu^{1}_{x_{j}}-\Theta_{q}(t,x,\mu^{2},Dw^{2})\mu^{2}_{x_{j}}\right)\dx,
\frac{\partial\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_i}= 4a_i\sum_{l=1}^{M-1}\sum_{q=l+1}^{M}\Re\left[\hat{\phi}^{\left(l,i\right)^*}\hat{\phi}^{\left(q,i\right)}\right]-2 \Re{\left\{\hat{\phi}^{\left(i,i\right)}\right\}}+4a_i\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,i\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_i\rVert^2\right).
\prod_{i=2}^n W(x_i|x_{i-1}) = \prod_{(x^\prime,x) \in c \backslash c^\prime}W(x|x^\prime) \prod_{(x^\prime,x) \in c^\prime}W(x|x^\prime) \le \prod_{(x^\prime,x) \in c \backslash c^\prime}W(x|x^\prime) e^{- (k-j) H_\infty^W(X)}.
\int_0^T |F_{T-t}(\overline{X}(t)-X(t))-F_{T-t}(\overline{X^n}(t)-X(t))|^2 dt &=\int_0^T|P( X(t) > \overline{X}(t)- \overline{X(T-t)}- P(X(t)> \overline{X^n}(t)- \overline{X(T-t)})|^2 dt
G_{\varepsilon_{k}}(W_{\varepsilon_{k},\xi})=\frac{1}{\varepsilon_{k}^{3}}\int_{B_{g}(\xi,r)}W_{\varepsilon_{k},\xi}^{2}\psi(W_{\varepsilon_{k},\xi})d\mu_{g}=\int_{\mathbb{R}^{3}}U^{2}(z)\chi^{2}(|\varepsilon_{k}z|)\tilde{\psi}_{\varepsilon_{k}}|g(\varepsilon_{k}z)|^{\frac{1}{2}}dz.
\Bigr\lvert\sum_{\substack{n\in M'' 1\leq n\leq N}}\mu(i)\Bigr\rvert\leq \epsilon N+\epsilon N+\sum_{j=0}^{N/Lk}\sum_{a=0}^{k-1}\Bigr\lvert \sum_{\substack{m\in[z+jLk,z+(j+1)Lk) m\equiv a \bmod{k}}}\mu(m) \Bigr\rvert\leq 3 \epsilon N.
-\frac{1}{2m}\big( \frac{\partial f}{\partial y}p_xp_y+af p_yp_z+ay\frac{\partial f}{\partial y}p_yp_z \big)=p_x \frac{\partial f}{\partial t}+p_z y\big( \frac{\partial a}{\partial t}f+a\frac{\partial f}{\partial t}\big),
H^{1}( d P_{9},{\cal O}_{d P_{9}}( 3 \sigma |_{d P_{9}}- 6 F ) ) |_{b_{- 6}}\stackrel{m_{3}}{\rightarrow}H^{1}( d P_{9},{\cal O}_{d P_{9}}( 6 \sigma |_{d P_{9}}- F ) ) |_{c_{- 3}}.
\frac{\gamma_{i,j}(x)(\mu_i\nu_{i,j})}{\sum_{i',j'}\gamma_{i',j'}(x)(\mu_{i'}\nu_{i',j'})}= \left(\frac{\sum_{j'}(\nu_{i,j'}\gamma_{i,j'}(x))\mu_i}{\sum_{i'}\sum_{j'}(\nu_{i',j'}\gamma_{i',j'}(x))\mu_{i'}}\right)\left(\frac{\gamma_{i,j}(x)\nu_{i,j}}{\sum_{j'}\nu_{i,j'}\gamma_{i,j'}(x)}\right),
s_{(2)}[s_{(1^2)}] = h_3[e_2] = h_2(x_1x_2,x_1x_3,\ldots)=x_1^2x_2^2 + x_1^2x_2x_3 + 3x_1x_2x_3x_4+\cdots = s_{2,2}(x_1,x_2,x_3,\ldots) + s_{1,1,1,1}(x_1,x_2,x_3,\ldots),
\lim\limits_{n\to \infty}\frac{1}{n}\sum\limits_{i=0}^{n-1}g_1(S_i(w)) =\int_{\Omega}g_1(w)d\mu, \lim\limits_{n\to \infty}\frac{1}{n}\sum\limits_{i=0}^{n-1}g_2(S_i(w)) =\int_{\Omega}g_2(w)d\mu,
\int_{-\infty}^{\infty}S_{j}\left(q^{2\alpha-1/2};q\right)S_{k}\left(q^{2\alpha-1/2};q\right)q^{2\alpha^{2}}d\alpha=\sqrt{\frac{\pi}{\log q^{-2}}}\frac{q^{-j}\delta_{j,k}}{\left(q;q\right)_{j}}.
y_1=x, \, y_2=\frac{1}{\sqrt{2}}(y+z), \, y_3=\frac{1}{\sqrt{2}}(y-z); [y_1,y_2]=-\frac{1}{\sqrt{2}}y_1, \, [y_2,y_3]=\frac{1}{\sqrt{2}}(y_2+y_3), \, [y_3,y_1]=-\frac{1}{\sqrt{2}}y_1.
U ( p )^{6}+ \beta_{5}( p ) U ( p )^{5}+ \beta_{4}( p ) U ( p )^{4}+ \beta_{3}( p ) U ( p )^{3}+ \beta_{2}( p ) U ( p )^{2}+ \beta_{1}( p ) U ( p ) + \beta_{0}( p ) = 0 .
\bullet\; (B_0,B_+) \colon H\to H_0\times\cdots \times H_{m^+}\mbox{is bijective}, \bullet\; \ker B_+\subset \ker B_{m+1}, \bullet\; \mbox{for all}x\in H, \mathcal Q(x)=\mathcal Q_+(B_0x)-\mathcal Q_-(B_{m+1}x).
C_{v}= \frac{3 k \langle N \rangle}{2}\left[ 1 + \frac{\langle N \rangle}{2 V}\lambda_{T}^{3}\delta ( q ) - 2 \frac{\langle N \rangle^{2}}{V^{2}}\lambda_{T}^{6}\left( 4 \delta^{2}( q ) - \frac{\Gamma ( q )}{1 2}\right) + . . . \right] ,
R_n(e^{2\pi i a/n}) =\sum_{d\mid n}(-1)^{n-d}\mu(n/d)e^{\frac{2\pi i a}{n}(\binom{n}{2}-\frac{n}{d}\binom{d}{2})}P_d(e^{2\pi i a/d}) &=\sum_{d\mid n}(-1)^{(n-d)(a-1)}\mu(n/d)\binom{2(a,d)-1}{(a,d)-1}&=\sum_{d\mid n}(-1)^{(n-n/d)(a-1)}\mu(d)\binom{2(a,n/d)-1}{(a,n/d)-1},
\frac{d}{dt}\tilde{\xi}_{i,1}-2^{-1}m \tilde\xi_{i,1}+\beta_i\tilde\xi_{i,2}=\tilde{\mathcal{E}}_{i,1}, \frac{d}{dt}\tilde\xi_{i,2}-2^{-1}m \tilde\xi_{i,2}-\beta_i\tilde\xi_{i,1}=\tilde{\mathcal{E}}_{i,2},
\begin{array}[c]{l}A_{n,k}^{m}\left({\gamma^{2}}\right) a_{n,k-1}^{m}\left({\gamma^{2}}\right) +\left\{{\lambda_{n}^{m}\left({\gamma^{2}}\right) +B_{n,k}^{m}\left({\gamma^{2}}\right)}\right\}a_{n,k}^{m}\left({\gamma^{2}}\right) +C_{n,k}^{m}a_{n,k+1}^{m}\left({\gamma^{2}}\right) =0,\end{array}
\overline{{\hat\sigma}_{12}^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})}:=\frac{1}{(2\pi)^2}\iint_{[0,2\pi]\times[0,2\pi]}d\phi_1d\phi_2\;{\hat\sigma}_{12}^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}}).
\nabla_{R}\cdot \left[ \frac{\vec{R}- i \frac 1 2 \vec{\Gamma}}{\left[ \left( \vec{R}- i \frac 1 2 \vec{\Gamma}\right)^{2}\right]^{\frac 3 2}}\right] = - \frac{\delta \left( R - \frac 1 2 \Gamma \right)}{R^{2}}\delta \left( \cos \theta \right)
S = - \frac{1}{2}\int_{X}d^{6}\xi \sqrt{- \gamma}[ \gamma^{i j}\partial_{i}x^{M}\partial_{j}x^{N}\eta_{M N}+ \frac{1}{2}\gamma^{i l}\gamma^{j m}\gamma^{k n}F_{i j k}F_{l m n}- 4 ]
\mathbf U(t) = \mathbf S(t) \mathbf u_0 + i \int_0^{t \wedge \tau_R}\mathbf S(t-s) \mathbf N(\mathbf u(s)) \, ds + i \int_0^t \mathbf S(t-s) \mathbf M(\mathbf U(s)) \, dW(s)
\begin{aligned}(\widetilde{X_P})_{+}(u^{(1)},...,u^{(r-1)})= &=\sum_{l=0}^{r-1}\left( \begin{matrix}r-1 l \end{matrix}\right) (\widetilde{X_P})_{+}(u^{(1)}_+,...,u_+^{(l)},u_-^{(l+1)},...,u_-^{(r-1)})\ . \end{aligned}
\rho^+(n)=\sum_{m h=n}a_m\rho(h,z) =\sum_{m h=n}a_m\rho(h,\omega)-u_1(n)-w_1(n) &=\sum_{m h=n}a_m\rho(h,\omega)+\sum_{j=1}^{\lfloor 2\log{x}\rfloor}(-1)^j(v_j(n)+w_j(n)).
\lim_{t\rightarrow \infty}\int_{0}^{q_{i}\left( t\right)}\left[h_{i}\left( t,s,x\left( \eta \left( s\right) \right) ,y\left( \eta \left(s\right) \right) ,z\left( \eta \left( s\right) \right) \right) -h_{i}\left(t,s,u\left( \eta \left( s\right) \right) ,v\left( \eta \left( s\right)\right) ,w\left( \eta \left( s\right) \right) \right) \right] ds=0,
M\equiv[\begin{matrix}0&1&1&0&0&0&0&1\\ 0&1&0&1&0&0&0&1\\ 0&1&0&0&0&1&2&2\\ 0&0&0&0&0&2&2&2\\ 0&0&0&1&0&1&0&1\\ 0&0&0&0&1&0&2&1\\ 0&0&0&0&0&0&3&1\\ 0&0&0&0&0&0&0&1\end{matrix}]
E_{N}= \; 4\pi (N-1) \frak{a}+ e_\Lambda \frak{a}^2 - \frac{1}{2}\sum_{p\in\Lambda^*_+}\left[ p^2+8\pi \frak{a}- \sqrt{|p|^4 + 16 \pi \frak{a}p^2}- \frac{(8\pi \frak{a})^2}{2p^2}\right] + \mathcal{O}(N^{-1/4}) \, .
{\cal L}_{eff}^{nm}= \partial_{+}A_i \partial_{-}A_i - \frac 12 (\partial_i A_i + \frac{1}{\partial_{-}}\ast J_{nm}^{+})^2 - \frac 1 4\left( \partial_i{A}_j - \partial_j{A}_i\right)^2 + A_i J^i_{nm}\ .
\tau_{(A(\Gamma),d_*)}(g) =\lim_{n\to\infty}\frac{d_*(g^n,1)}n=\lim_{n\to\infty}\frac{\|g^n\|_*}n, \tau_{(\Gamma^e,d)}(g) =\lim_{n\to\infty}\frac{d(v^{g^n},v)}n,
\mathbb{L}^\prime[\varphi_0]\varphi^\prime=\varphi^\prime_{tt}-\mu\varphi^\prime_{xx}-\left(2\mathbb H[\phi_{0,x}\phi^\prime_x]-\left[\phi^\prime;\mathbb H\right]\phi_{0,xx}-\left[\phi_0;\mathbb H\right]\phi^\prime_{xx}\right)_x\,,\phi^\prime:=\mathbb H[\varphi^\prime]\,,
\Delta_A \mathcal{H}_{\mathcal{J}}(\sigma) = - 2 \cdot \sum_{e =\{x, y\}\in E : x\in \partial A, y \in \partial^{ext}A}J_{e}\sigma^{(A)}_x \sigma^{(A)}_y = 2 \cdot\sum_{e =\{x , y\}\in E: x\in\partial A, y \in \partial^{ext}A}J_{e}\sigma_x \sigma_y.
0 = p_A (v) = \Big[A_{34}- \dfrac{tB_0}{2a_0'}A_{23}-\dfrac{sB_0}{2a_0}A_{14}- st\dfrac{D_0 + B_0^2}{4a_0a_0'}A_{12}\Big] + &+ \sqrt{D_0}\Big[ \dfrac{t}{2a_0'}A_{23}+ \dfrac{s}{2a_0}A_{14}+ st \dfrac{B_0}{2a_0a_0'}A_{12}\Big].
\widehat{g}_{\sigma_{1},\sigma_{2},\sigma_{3}}:=\left(\frac{\widehat{e}_{\sigma_{3}-\sigma_{1}}\widehat{e}_{\sigma_{2}}}{\widehat{e}_{\sigma_{2}-\sigma_{1}}\widehat{e}_{\sigma_{3}}}\cdot\widehat{f}_{1},\cdots,\frac{\widehat{e}_{\sigma_{3}-\sigma_{1}}\widehat{e}_{\sigma_{2}}}{\widehat{e}_{\sigma_{2}-\sigma_{1}}\widehat{e}_{\sigma_{3}}}\cdot\widehat{f}_{j}\right).
\Upsilon_i(\epsilon_i,\nu_i) = \frac{n}{2k}+\frac{\epsilon_i}{4}\displaystyle{\sum_{i=n+3}^{2n}}c^{\frac{n+3-2i}{2}}P_{i-n-3,-i}+ \frac{\nu_i}{4}\displaystyle{\sum_{i=n+3}^{2n}}\zeta^{\frac{2n+3-2i}{2}}c^{\frac{n+3-2i}{2}}P_{i-n-3,-i}
\begin{array}{c}{{G_{T}( z^{\prime \prime}) = G_{T}( z_{1}^{\prime \prime}) G_{T}( z_{s}( z^{\prime}; u ) ) \, ,}}\\{{{}}}\\{{G_{T}( z^{\prime}) = G_{T}( z_{1})^{- 1}G_{T}( z ) \, ,}}\\ \end{array}
\eta_{j, R}\partial_t\tilde{\phi}_j = \eta_{j, R}\left[-(-\Delta)^s\tilde{\phi}_j + pU_j^{p-1}\tilde{\phi}_j + pU_j^{p-1}\psi + S_{\mu, \xi, j}\right]\mathbb{R}^n\times (t_0, \infty),
a(s,s') :=\frac{z(s)}{1 + 4isz(s)}+\frac{\widetilde z(s')}{1 - 4is'\widetilde z(s')}, b(s,s') :=\frac{\zeta(s)}{1 + 4i sz(s)}+\frac{\widetilde\zeta(s')}{1 - 4i s'\widetilde z(s')}, c(s,s') :=\frac{is}{1 + 4i sz(s)}\zeta(s) \cdot \zeta(s)+\frac{is'}{1 - 4i s'\widetilde z(s')}\widetilde\zeta(s') \cdot \widetilde\zeta(s')+\gamma(s)+\widetilde\gamma(s').
\mathcal{G}_{X^{\ast}X}(f)(x) =\int_{X^{\ast}}f([y^{\ast},0][x,0])dy^{\ast}&=\int_{X^{\ast}}\psi(\langle y^{\ast}, x\rangle) f(y^{\ast})dy^{\ast}&=\mathcal{F}'_{X^{\ast}X}(f)(x).
\begin{array}{ccc}I_u = (x_a, x_bx_c), I_v = (x_b, x_ax_c), I_w = (x_c, x_ax_b), I_{uv}= (x_a, x_b, x_w), I_{uw}= (x_a, x_c, x_v), I_{vw}= (x_b, x_c, x_u).\end{array}
F_i=K_i^{\oplus{c-1 \choose c-1}}\oplus K^{\oplus{c \choose c-1}}_{i-2}\oplus K^{\oplus{c+1 \choose c-1}}_{i-4}\oplus\cdots \oplus K^{\oplus{c+j-1\choose c-1}}_{i-2j}\oplus\cdots
{\frac{1}{2 m_{p + 1}}}{\frac{\partial S_{0}}{\partial \sigma_{\mu_{1}\dots \mu_{p + 1}}}}{\frac{\partial S_{0}}{\partial \sigma^{\mu_{1}\dots \mu_{p + 1}}}}- \frac{m_{p + 1}}{2}p ={\frac{\partial S_{0}}{\partial \Omega_{p + 1}}}\ .
S = \,_{1}\langle \,_{2}\langle I | | \Phi \rangle_{1}Q^{( 2 )}| \Phi \rangle_{2}+ \frac{2}{3}\,_{1}\langle \,_{2}\langle \,_{3}\langle V | | \Phi \rangle_{1}| \Phi \rangle_{2}| \Phi \rangle_{3},
\begin{aligned}\Phi(t,\tau) := \sum_{j\le j_0}\sum_{j'\le j+4}2^{j'(\frac{N}{2}+1)}\Vert \dot\Delta_{j'}u(\tau) \Vert_{L^2}\psi_j(t,\tau) & +\sum_{j\le j_0}\sum_{j'>j-4}2^{(j-j')(\frac{N}{2}+\bar{s}+ \delta s \alpha)}2^{j'(\frac{N}{2}+1)}\Vert \dot\Delta_{j'}u(\tau) \Vert_{L^2}\psi_j(t,\tau) ,\end{aligned}
\frac{\partial^{2}\varphi_{E}( \tau )}{\partial \tau^{2}}+ 3 \Bigl ( \frac{\partial a ( \tau ) / \partial \tau}{a ( \tau )}\Bigr ) \frac{\partial \varphi_{E}( \tau )}{\partial \tau}- \frac{\delta^{2}V ( \hat{\phi})}{\delta \hat{\phi}^{2}}\varphi_{E}( \tau ) = 0 .
\renewcommand{\arraystretch}{1.3}\begin{array}{rrc}T^k T^k \left| a\right> = t (t+1) \left| a\right>, \,\,T^3 \left| a\right> = t^z \left| a\right>, \, \,F\left| a\right> = f \left| a\right>, Q \left| a\right> = q \left| a\right>, \end{array}
\{\eta_{A B}\}= \left( \begin{array}{c c}{g_{a b}}&{0}\\{0}&{g_{i j}}\\ \end{array}\right) , \quad \{\eta^{A B}\}= \left( \begin{array}{c c}{g^{a b}}&{0}\\{0}&{g^{i j}}\\ \end{array}\right) ,
\mathcal{R}(x)= \sum_{\rho}\lim_{s \rightarrow \frac{k-\rho}{2}}\frac{\left( s- \frac{k-\rho}{2}\right) \Gamma(s)}{\zeta(k-2s)}x^{-s}= -\frac{1}{2}\sum_{\rho}\frac{\Gamma(\frac{k-\rho}{2})}{\zeta'(\rho)}x^{-\frac{k-\rho}{2}},
{\cal G}_{4 4}\left(\rho,z\right)={\displaystyle\frac{1}{4\pi A{\cal C}_{66}}\sum_{l=1}^{4}{\frac{\Lambda_{\rm{4}}\left(a=-A_l\right)}{\sqrt{A_l\rho^2+z^2}\prod_{j=1,(j\neq l)}^{4}{\left(A_j-A_l\right)}}}}
\alpha_0=a_0-a_4, \alpha_1=a_1-a_5, \alpha_2=a_2-a_6, \alpha_3=a_3-a_7, \gamma_0=a_0+a_4, \gamma_1=a_1+a_5, \gamma_2=a_2+a_6, \gamma_3=a_3+a_7,
S_{i,s_1}^{j,s_2}=\{u_{i'}(s') \colon i'\in \{0,\dotsc l+1\}; s'\in S\}\cup\,\{p_2 \colon (p_1,p_2)\in \hat h_i(s_1)\sqcap \hat h_{i'}(s') i'\in [k], s'\in S, p_1\in M \}\cup\,\{p_2 \colon (p_1,p_2)\in \hat h_j(s_2)\sqcap \hat h_{i'}(s') i'\in [k], s'\in S, p_1\in M \}.
\sum\limits_{n = 0}^{\infty}d_{o}(n)q^{n}\sum_{n = 0}^{\infty}q^{n(n+1)/2}\equiv \sum\limits_{n = 1}^{\infty}\frac{q^{2n}}{1 - q^{2n}}- \sum\limits_{n = 1}^{\infty}\frac{q^{4n}}{1 - q^{4n}}\pmod{2}
\frac{S_{n\delta,n\delta}(\vec{t})}{S_{n\delta,n\delta}(\vec{\tau})}=\frac{S_{\delta}(\vec{\tau})^{n}\Psi(t_{1},\dots,t_{d})(s_{n\delta},n)}{S_{\delta}(\vec{t})^{n}\Psi(\tau_{1},\dots,\tau_{d})(s_{n\delta},n)}=\frac{S_{\delta}(\vec{\tau})^{n}}{S_{\delta}(\vec{t})^{n}},
F_1(h_1(y),y)h_1'(y)=\frac{h_1}{x_1}yh_1', &F_2(x_2-h_2(y)+h_1(y),y)(h_2'(y)-h_1'(y))=\frac{h_2-h_1}{x_2-x_1}y(h_2'-h_1'), &F_3(x_2-h_2(y)+y,y)(1-h_2'(y))=\frac{y-h_2}{1-x_2}y(1-h_2'),
W_{X}= ( \eta_{A}- \mathrm{\frac{1}{2}}m^{2}( P_{+})_{A}^{B}\bar{\phi}_{B}) \phi^{A}- \phi_{A a}^{*}\pi^{A a}- \bar{\phi}_{A}( \lambda^{A}- \mathrm{\frac{1}{2}}m^{2}( P_{-})_{B}^{A}\phi^{B}) .
\frac{\mathit{n}(v,t)}{\mathit{gl}(v,t)}= \sum_{v^*}t^{-\sum_{k<l}\langle v^{(k)}, v^{(l)}\rangle}\prod_{k\ge 1}\frac{H(v^{(k)}, v^{(k+1)}, t^{-1})}{\mathit{gl}(v^{(k)},t)},