harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{S}\langle\beta_{\tilde g^{(4)}}h,Y\rangle_{\tilde g^{(4)}}\cdot u\cdot dvol_{g_S}=\int_{S}\langle h,\delta_{\tilde g^{(4)}}^*Y+\frac{1}{2}(\delta_{\tilde g^{(4)}}Y)\tilde g^{(4)}\rangle_{\tilde g^{(4)}}u\cdot dvol_{g_S}.
\arraycolsep=1.4pt\def\arraystretch{1.3}\left. \begin{array}{cc}K_1 = |A_1|^2-|A_2|^2, K_2 = |A_1|^2-|A_3|^2, H = A_1A_2A_3+\overline{A_1A_2A_3},\end{array}\right.
\gamma_{sp}(B(G_1,G_2,G_3)) = \gamma_{sp}(B(H,G_3)) &\geq \gamma_{sp}(H)+\gamma_{sp}(G_3) -1 &= \gamma_{sp}(B(G_1,G_2))+\gamma_{sp}(G_3) -1 &\geq \gamma_{sp}(G_1)+\gamma_{sp}(G_2)+\gamma_{sp}(G_3) -2.
{\xi_{k,{l_k},{j_k}}}={\rho_k}{\alpha_{{k,{\rm{R}}},{l_k}}}{\alpha_{{\rm{T}},k,{j_k}}}{\bf{a}}_{{\rm{S,}}k}^H\left({\Theta_{{k,{\rm{R}}},{l_k}}^{\rm{D}}}\right){{\bf{a}}_{{\rm{S}},k}}\left({\Theta_{{\rm{T}},k,{j_k}}^{\rm{A}}}\right).
\log\left( \frac{f_k f_{k+2}}{f_{k+1}^2}\right) = \log\left(\frac{h_k f_k}{g_k^2}\right) - 2 \log\left(\frac{h_k f_{k+1}}{|g_k| |g_{k+1}|}\right)+\log\left(\frac{h_k f_{k+2}}{g_{k+1}^2}\right).
\lim_{p\rightarrow\infty}u_{p}(x)=\lim_{p\rightarrow\infty}1-\left(\frac{\left\vert x\right\vert}{R}\right)^{\frac{p-N}{p-1}}=1-\frac{\left\vert x\right\vert}{R}=\frac{\rho(x)}{\left\Vert \rho\right\Vert_{\infty}};0\leq\left\vert x\right\vert \leq R.
\tilde{u}:=\lim_{k\to \infty}\sum_{j=1}^\infty \Big( \frac{\alpha^j_k}{1-\big(\alpha^0_k+\beta^0_k\big)}f^j_k - \frac{\beta^j_k}{1-\big(\alpha^0_k+\beta^0_k\big)}g^j_k \Big);
\dfrac{\pi}{2}=\lim_{q\rightarrow 1^-}(1-q)\prod_{n=1}^\infty\dfrac{(1-q^{2n})^4}{(1-q^n)^2}=\lim_{q\rightarrow 1^-}(1-q)\dfrac{(q^2;q^2)_\infty^4}{(q;q)_\infty^2}.
\sin(2\overline{v}_{1})\partial_{R}\overline{v}_{1}-\sin(2\overline{v}_{2})\partial_{R}\overline{v}_{2}= \sin(2\overline{v}_{1})\left(\partial_{R}(\overline{v}_{1}-\overline{v}_{2})\right) + \left(\sin(2\overline{v}_{1})-\sin(2\overline{v}_{2})\right)\partial_{R}\overline{v}_{2}
\frac{L_{\Gamma}'(s,\rho)}{L_{\Gamma}(s,\rho)}= - \frac{d}{ds}\sum_{\overline{\gamma}\in \overline{\Gamma}_{p}}\sum_{k=0}^{\infty}\sum_{m=1}^{\infty}\frac{1}{m}\chi_{\rho}(\gamma^{m})e^{-(s+k)m\ell(\gamma)}&= \sum_{\overline{\gamma}\in \overline{\Gamma}_{p}}\sum_{k=0}^{\infty}\sum_{m=1}^{\infty}\ell(\gamma)\chi_{\rho}(\gamma^{m})e^{-(s+k)m\ell(\gamma)}.
q(\zeta)=\mathbf a_-(\zeta)\begin{pmatrix}1 \bar{\zeta}\zeta 1\end{pmatrix}=\begin{pmatrix}1 0 \zeta 1\end{pmatrix}\begin{pmatrix}\mathbf a_-(\zeta) 0 0 \mathbf a_-(\zeta)^{-1}\end{pmatrix}\begin{pmatrix}1 \bar{\zeta}0 1\end{pmatrix}\in SU(1,1).
{S_L}\left[\phi,\hat{g}\right]={1\over{8\pi}}\int{d^2}\xi\sqrt{\hat{g}}\left({1\over{2}}\phi\hat{\Delta}\phi+Q\hat{R}\phi\right)+{Q\over{4\pi}}\oint{d}\hat{s}{k_{\hat{g}}}\phi+2{\mu_2^2}\int{d^2}\xi\sqrt{\hat{g}}{e^{\alpha\phi}}+{\lambda_2}\oint{d}\hat{s}{e^{\alpha\phi/2}}.
g_p = (1-\rho^{p+1})(1-\rho^n)^{n-2-p}\left( \frac{\rho}{1-\rho^{n+1}}\right)^{n-1-p}&=\frac{\left( \sum_{m=0}^p \rho^{m+1}\right) \left( \sum_{\ell=0}^{n-1}\rho^{\ell+1}\right)^{n-2-p}}{\left( \sum_{k=0}^{n}\rho^k \right)^{n-1-p}},\;\;\;( 0 \leq p \leq n-1 ).
\setlength{\abovedisplayskip}{4 pt}\setlength{\belowdisplayskip}{4 pt}E_{c}=\max\frac{1}{{{T}}}\sum\limits_{k = 1}^{{T}}\sum\limits_{i \in \mathcal{U}}\sum\limits_{n_{ik}\in{{{\mathcal{C}_c}}\mathord{\left/{\vphantom{{{C_c}}{{C_k}}}}\right. \kern-\nulldelimiterspace}{{\mathcal{C}_i}}}}{{{p_{kin_{ik}}}E_{k,i}(\theta_{i,n_{ik},k}^A)}}.
\mathbb{E}= \sqrt{\frac{\varepsilon_{av}}{\mu_{av}}}\widetilde{\mathbb{E}}, \mathbb{H}=\sqrt{\frac{\varepsilon_{av}}{\mu_{av}}}\widetilde{\mathbb{H}},{\rm where}\varepsilon= \frac{\widetilde{\varepsilon}}{\varepsilon_{av}}, \mu =\frac{\widetilde{\mu}}{\mu_{av}}, k_* = \sqrt{\varepsilon_{av}\mu_{av}}\widetilde{k_*},
\Vert r-\Pi_{V_2}r\Vert_{L^2(\mathbb{P}^{X_2})}&\leq \sum_{k=1}^{q-1}\Vert r_{k}-\Pi_{V_{2k}}r_{k}\Vert_{L^2(\mathbb{P}^{X_{2k}})}\leq \sum_{k=1}^{q-1}C_3\left( h_{1k}(x_1)+\sum_{j=1,j\neq k}^{q-1}\Vert r_{j}\Vert_{L^2(\mathbb{P}^{X_{2j}})}\Vert h'_{jk}\Vert\right) d_{2k}^{-\beta}.
W^{M}[ V ] = \epsilon n^{( 0 ) M}n^{( 0 ) N}n^{( 0 ) L}V_{N ; L}- e_{\mu}^{( 0 ) M}q^{( 0 ) \mu \nu}n_{N}^{( 0 )}{\cal L}_{V}e_{\nu}^{( 0 ) N}.
ds^2_{\rm asym}= \frac{1}{2}M_{ij}d{\bf x}_i\cdot d{\bf x}_j+\frac{g^4}{2(4\pi)^2}(M^{-1})_{ij}(d\xi_i+{\bf W}_{ik}\cdot d{\bf x}_k)(d\xi_j+{\bf W}_{jl}\cdot d{\bf x}_l) \,\, .
{\sinh\left({\theta\over2}-{i\pi\over2 n}\alpha\right)\sinh\left({\theta\over2}-{i\pi\over2 n}(2-\alpha)\right)\over\sinh\left({\theta\over2}+{i\pi\over2 n}\alpha\right)\sinh\left({\theta\over2}+{i\pi\over2 n}(2-\alpha)\right)}\ ,
T_{A_{n}}(f) \leqslant\mathbb{E}\left( \left\vert \mathbb{G}_{n}^{(N)}(f)\right\vert (d_{0}+\varepsilon)1_{A_{n}}\right) =(d_{0}+\varepsilon) \sqrt{\mathbb{V}\left(\mathbb{G}_n^{(N)}(f)\right)}\mathbb{E}\left( \left\vert \mathcal{N}(0,1)\right\vert \right) \leqslant \sqrt{\frac{2}{\pi}}(d_{0}+\varepsilon)\sigma_{f}.
U_{k_1, k_{1 z}; k_2, k_{2 z}; I_z, I_z^\prime}= \sum_{m, I_z}\langle k, m; I, I_z | k_2, k_{2 z}\rangle D^{k_1}_{m, k_{1z}}(\alpha, \beta, \gamma) D^I_{I_z, I_z^\prime}(\alpha_I,\beta_I, \gamma_I) | I, I_z^\prime \rangle.
\Pi(x)=\sum_{n\le x}\Lambda(n)\int_{n}^{x}\frac{dt}{t\log^2 t}+\frac{1}{\log x}\sum_{n\le x}\Lambda(n)=\frac{\psi(x)}{\log x}+\int_{2}^{x}\frac{\psi(t)}{t\log^2 t}dt.
C_r^n = \psi^{\alpha_1,n}(g_1) \cdots \psi^{\alpha_s,n}(g_s)\psi^{\alpha_{s+1},n}(f_1) \cdots \psi^{\alpha_{r},n}(f_{r-s}), D_s^n = \psi^{\beta_{1},n}(f_{r-s+1}) \cdots \psi^{\beta_{s},n}(f_r).
d = 0, \beta = 0.95, \mathbf{R}^0 =\mathbf{0}, \mathbf{R}^1 = \begin{bmatrix}0.7221 0.9685 0.1557\end{bmatrix}, \mathbf{P}= \begin{bmatrix}0.8061 0.1574 0.0365 0.1957 0.0067 0.7976 0.1378 0.5959 0.2663\end{bmatrix}.
S = \left( \begin{array}{l l l l}{{\partial_{1}X^{0}}}&{{\partial_{1}X^{1}}}&{{. . .}}&{{\partial_{1}X^{p}}}\\{{\partial_{2}X^{0}}}&{{\partial_{2}X^{1}}}&{{. . .}}&{{\partial_{2}X^{p}}}\\{{. . .}}&{{. . .}}&{{. . .}}&{{. . .}}\\{{G^{0 \nu}n_{\nu}}}&{{G^{1 \nu}n_{\nu}}}&{{. . .}}&{{G^{p \nu}n_{\nu}}}\\ \end{array}\right) \ .
\sum_{n\leq x}f(n)=\left(\frac{1}{e^{\gamma \tau}\Gamma (\tau )}+o(1)\right)\frac{x}{\log x}\prod_{p\leq x}\left(1+\frac{f(p)}{p}+\frac{f\left(p^2\right)}{p^2}+\cdots\right) .
\ddot{\mu}_{H_r,\epsilon}(\rho) = \frac{\sinh(\rho)\cosh^{\frac{2(n-r)}{r}-1}(\rho)\left(nH_r\frac{\cosh^n(\rho)}{\sinh^r(\rho)}-(n-r)(nH_rJ_{n,r,\epsilon}(\rho))\right)}{r(nH_rJ_{n,r,\epsilon}(\rho))^{\frac{r-1}{r}}\left(\cosh^{\frac{2(n-r)}{r}}(\rho)-(nH_rJ_{n,r,\epsilon}(\rho))^{\frac{2}{r}}\right)^{\frac{3}{2}}}.
\|\sum_{j=0}^\infty\tilde S_{j+\ell}f \tilde S_j g \|_{B^s_{p, \infty}(\real^{d_s})}\le C \sum_{j=0}^{\infty}\sup_{k\ge 0}2^{ks}\|\tilde S_k(\tilde S_{k+j+\ell}f \tilde S_{k+j}g) \|_{L_p(\real^{d_s})}\, .
< \Psi_{1}| \Psi_{2}> = N \int_{- \infty}^{\infty}\frac{d^{2}\vec{p}}{p^{0}}\int_{0}^{2 \pi}\frac{d \varphi}{( p , n )}\overline{{{\Psi_{1}( p , \varphi )}}}\Psi_{2}( p , \varphi ) ,
(1-it)^{-z}=\frac{y}{y+z}\,_2F_1(z,y+z;y+z+1;it) + \frac{z}{y+z}(1-it)^{-z}\,_2F_1(1,y;y+z+1;it) &= \frac{y}{y+z}(1-it)^{1-z}\,_2F_1(1,y+1;y+z+1;it) & + \frac{z}{y+z}(1-it)^{-z}\,_2F_1(1,y;y+z+1;it),
\beta(x,y)=\begin{cases}\chi(\delta_{0}^{-1}(y-f(x)-5\delta_{0})+1), \mbox{if $ f(x)+5\delta_0\le y$;}1, \mbox{if $6\delta_{0}<y<f(x)+5\delta_0$;}1-\chi(\delta_{0}^{-1}(y-5\delta_{0})+1), \mbox{if $y\le 6\delta_{0}$}\end{cases}
|\Omega| \le \sum_{x\in \Omega}e^{(1-\rho)\left(a - \log\frac{1}{P_X(x)}\right)}\le \sum_x P_X(x)^{1-\rho}e^{(1-\rho)a}= e^{(1-\rho)a + \phi(\rho;P)},
\mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{j=0}^{n-1}X(u_{j}) (t_{j+1}-t_{j})=\mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{\substack{j\in J\setminus J_{K}}}X(u_{j}) (t_{j+1}-t_{j}).
\sum_{\xi_{\Delta}}\int_{\Omega}c_{\Delta}(\omega, \xi_{\Delta})f(\omega)g(\xi_{\Delta}\omega_{\Delta^c})\mu(d\omega) = \sum_{\zeta_{\Delta}}\int_{\Omega}\hat{c}_{\Delta}(\omega, \zeta_{\Delta})f(\zeta_{\Delta}\omega_{\Delta^c})g(\omega)\mu(d\omega),
g(b,n)=\sum_{\substack{c_i,m_i \ge1 c_0+c_1+c_2+c_3=b+2 m_0+m_1+m_2+m_3=n}}m_0f(c_0,m_0)f(c_1,m_1)f(c_2,m_2)f(c_3,m_3).
\| \boldsymbol{r}_k \| = \min_{p \in \mathbb{P}_k, p(0) = 1}\| p(A C^{(\ell)}) \boldsymbol{r}_0 \| \leq \kappa(T) \min_{p \in \mathbb{P}_k, p(0) = 1}\max_{1 \leq i \leq s}\| p(J_i) \| \| \boldsymbol{r}_0 \|,
\begin{aligned}\| g_\infty -g_n \| \leq \sum_{k \ge n}\| g_{k+1}-g_k \| \le \sum_{k\geq n}\sqrt 2(1-\sqrt{\mu_{k+1}})^{1/2}\lesssim \sum_{k\geq n}\big( k^{1+ s}(1-\sqrt{\mu_{k+1}})^{1/2}\big) \cdot \frac{1}{k^{1+ s}}\lesssim \frac{1}{n^{(1+ 2s)/2}}\, , \end{aligned}
\int_{\theta_1\in[0,2\pi]}\log\big|e^{-m_1u_1+im_1\theta_1}\cdot\dots\cdot e^{-m_nu_n+im_n\theta_n}-1\big|\d\theta_1=-2\pi\sum_{j=1}^k\log\frac{|\alpha_j|}{e^{-m_1u_1}}
a_0(t)=-t, a_1(t)=-t(2\cdot 5+5^2t), a_2(t)=-t(11\cdot 5+2\cdot 5^3 t+5^4t^2), &a_3(t)=-t(28\cdot 5 +11\cdot 5^3 t+ 2\cdot 5^5 t^2+5^6t^3), &a_4(t)=-t(7\cdot 5^2+ 28\cdot 5^3t+11\cdot 5^5 t^2+2\cdot 5^7t^3+5^8t^4).
\begin{cases}u_{1,\infty}= \frac{M_{13}M_{14}}{M}, u_{3,\infty}= M_{13}-\frac{M_{13}M_{14}}{M}= \frac{M_{13}M_{23}}{M}, u_{2,\infty}= \frac{M_{23}M_{24}}{M}, u_{4,\infty}= M_{14}-\frac{M_{13}M_{14}}{M}= \frac{M_{14}M_{24}}{M}. \end{cases}
\delta_{f}=\delta_{\widetilde{f}}=\lim_{n\to \infty}\max\{\rho((g^{n})^{*}|_{N^{1}(A)}), \rho((f_{T}^{n})^{*}|_{N^{1}( \overline{T})})\}^{1/n}=\max\{\delta_{g},\delta_{f_{T}}\}.
P=\sum_{s_{zN}}\cdot\cdot\cdot\sum_{s_{z2}}\sum_{s_{z1}}\int_{V_{N}}\cdot\cdot\cdot\int_{V_{2}}\int_{V_{1}}|\Psi|^{2}d^{3}r_{1}d^{3}r_{2}\cdot\cdot\cdot d^{3}r_{N}
Z= \int \prod_{n=1}^{\mathcal{N}}D[P_n]D[\pi^{ij}] D[h_{ij}] \delta(\chi)\delta(\chi^1)\delta(\chi^2)|{\rm Det}\{\chi^\mu, H_\nu\}|\delta(\frac{H_i}{\sqrt{h}})\delta(\frac{H}{\sqrt{h}}) e^{i S}.
C_{2}= \frac{1}{2}\tilde{A}_{\mu}^{\nu}\tilde{A}_{\nu}^{\mu}= \frac{P ( N + P ) ( N - 1 )}{2 N}, \tilde{A}_{\mu}^{\nu}- \frac{\delta_{\mu}^{\nu}}{N}\sum_{\lambda}A_{\lambda}^{\lambda}.
f\star_{\vec{c}}^{n}g={\sum_{l_{1},l_{2}=0}^{n}\sum_{m_{1}=-l_{1}}^{l_{1}}\sum_{m_{2}=-l_{2}}^{l_{2}}}\phi_{l_{1}m_{1}}\gamma_{l_{2}m_{2}}Y_{l_{1}}^{m_{1}}\star_{\vec{c}}^{n}Y_{l_{2}}^{m_{2}}\ ,
(1+z)^{2p}\cdot\Phi(p)\equiv \Psi_1(p)+z^{2p}\cdot\Psi_2(p)+2z^{p}\cdot\Phi(0)-\frac{z\cdot\Psi_1(0)+z^{2p}\cdot\Psi_2(0)}{1+z}= \Psi_1(p)+z^{2p}\cdot\Psi_2(p)-\frac{z(z^{p-1}-1)^2\cdot\Psi_1(0)}{1+z}\equiv \Psi_1(p)+z^{2p}\cdot\Psi_2(p)\pmod{p^2}.
d V \epsilon_{\alpha \beta \gamma \delta}\Phi_{\alpha}\frac{\partial{( \Phi_{\beta}, \Phi_{\gamma}, \Phi_{\delta})}}{\partial ( x , y , z )}= 3 d \eta d \xi d \phi m n \frac{d}{d \eta}\biggl ( \frac{1}{1 + f^{2}( \eta )}\biggr ) ,
(\rho_+-\rho_-)(\widehat{\sigma^{B}})^{2}-2(\rho_+u_+-\rho_-u_-)\widehat{\sigma^{B}}+\rho_+u_+^{2}-\rho_-u_-^{2}-B(\frac{1}{\rho_+^\alpha}-\frac{1}{\rho_-^\alpha})=0.
\left(\int_{B_{r_0/2}}f^{1/p}\right)^p=\left(\int_{B_{r_0/2}}e^{(\alpha_+^\ast-2\varepsilon)z_1}\right)^p \geq a\left(\int_{B_{r_0/2}}|x|^{-\frac{(\alpha_+^\ast-2\varepsilon)(m_+(x_0)-m_-(x_0))}{2\pi}}\right)^p=+\infty
\gamma^{2}=(\begin{matrix}0&0&0&-i\\ 0&0&i&0\\ 0&i&0&0\\ -i&0&0&0\end{matrix}),\gamma^{3}=(\begin{matrix}0&0&1&0\\ 0&0&0&-1\\ -1&0&0&0\\ 0&1&0&0\end{matrix}).
i \Delta_{+}( x ) = \langle \, 0 \, | \, \varphi ( x ) \, \varphi ( 0 ) \, | \, 0 \, \rangle = \frac{1}{4 \pi}\int d \theta e^{- \tau m \cosh \theta}= \frac{1}{2 \pi}K_{0}( m \tau )
\sum_{i\in[n]}c'(v_i) = \tfrac{n}{k}+ \mu \sum_{i\in[n]}\left(c(v_i) - \tfrac{1}{k}\right) &= \sum_{i \in [n]}c(v_i) + (\mu - 1) \left( \sum\nolimits_{i\in[n]}c(v_i) - \tfrac{n}{k}\right) < \sum_{i \in [n]}c(v_i).
\Sigma_{1, n+1}^* L_{1, n+1}= \Sigma_{1, n+1}^* \big[ v_{n+1}u_n^* + H_{1, n+1}L_{2, n+1}\big] \widetilde H_{1,n}^{-1}= \Sigma_{1, n+1}^* v_{n+1}u_n^* \widetilde H_{1,n}^{-1}.
\left(\sigma_i^2(AW) - \sigma_k^2(A) \right) \alpha^{(k)}_i + \sigma_i(AW) y^{(i)\intercal}\beta^{(k)}= 0, \Rightarrow \alpha_i^{(k)}= \frac{-\sigma_i(AW)}{\sigma_i^2(AW) - \sigma_k^2(A)}y^{(i)\intercal}\beta^{(k)}.
\int_{E^c}|f_n-f|\chi_{A_n}\,dm_n< \frac{\varepsilon}{6}\cdot \frac{m_n(E^c )}{m(E^c)+1}< \frac{\varepsilon}{6}\cdot \frac{m(E^c)+1}{m(E^c)+1}=\frac{\varepsilon}{6}.
\mu^{\ast}(\delta([\nu^{-k}\rho, \nu^{l}\rho]) \rtimes \sigma) = \sum_{i = -k-1}^{l}\sum_{j=i}^{l}\sum_{\tau, \sigma'}\delta([ \nu^{-i}\rho , \nu^{k}\rho]) \times \times \delta([\nu^{j+1}\rho, \nu^{l}\rho]) \times \tau \otimes \delta([\nu^{i+1}\rho, \nu^{j}\rho]) \rtimes \sigma'.
| \hat \mu_n (A) - \mu_\infty (A) | \ = \ |\hat \mu_n (A_M) - \mu_\infty (A_M) | + \hat \mu_n (\{M+1,\dots\}) + \mu_\infty (\{M+1,\dots \}) \ \le\ |\hat \mu_n (A_M) - \mu_\infty (A_M) | + \hat\mu (\{M+1,\dots\})+ \mu_\infty (\{M+1,\dots\}).
\lim_{t \to 0^+}F^{-1}(t) = \lim_{t \to 0^+}\frac{t^{\frac1p-1}}{\psi \left(t^{- \frac{1}{n}}\right)^{n-1}}= \lim_{t \to 0^+}\phi(1)^{1-n}t^{\frac{n-1}{n}+ \frac1p -1}=0.
\left|\int_{M}a(x)\left(\left(W_{\delta_{\varepsilon}(t),\eta}+\Phi_{\delta_{\varepsilon}(t),\eta}\right)^{2^{*}-\varepsilon}-\left(W_{\delta_{\varepsilon}(t),\eta}\right)^{2^{*}-\varepsilon}-f_{\varepsilon}(W_{\delta_{\varepsilon}(t),\eta})\Phi_{\delta_{\varepsilon}(t),\eta}\right)d\mu_{g}\right|=o(|\varepsilon|).
\left\lbrace\begin{array}{ll}\phi''-\Delta\phi =-(e, AU) \hbox{in}(0,T) \times \Omega, \partial_\nu \phi + \lambda\phi = (e, DH) \hbox{on}(0,T) \times \Gamma, t=0:~\phi=0, ~ \phi'=0 \hbox{in}\Omega.\end{array}\right.
\widehat{J}_1(\tau) := \cos(\tau \widehat{\mathcal{A}}^{1/2}) - \cos(\tau (\widehat{\mathcal{A}}^0)^{1/2}), \widehat{J}_2(\tau) := \widehat{\mathcal{A}}^{-1/2}\sin(\tau \widehat{\mathcal{A}}^{1/2}) - (\widehat{\mathcal{A}}^0)^{-1/2}\sin(\tau (\widehat{\mathcal{A}}^0)^{1/2}).
e_{\theta}\cdot \mathbb{P}u\cdot \nabla u =e_{\theta}\cdot(h-\nabla \Phi) &=e_{\theta}\cdot(h-e_{r}\partial_r \Phi-e_{z}\partial_z \Phi) &=v\cdot \nabla u^{\theta}+\frac{u^{r}}{r}u^{\theta}.
0=\,[\hat{x}^\mu_{k},\hat{p}_{yl\rho}]=\,\left[x^\mu_k(\hat{\tilde{x}}),\widehat{\frac{\partial \tilde{x}^{\mu}_l}{\partial x^\rho_l}}\,\hat{\tilde{p}}_{y l \mu}\right]=\,\widehat{\frac{\partial \tilde{x}^{\mu}_l}{\partial x^\rho_l}}\,\left[x^\mu_k(\hat{\tilde{x}}),\hat{\tilde{p}}_{y l \mu}\right].
L_0 = -\frac{1}{2}\left( \begin{array}{rr}1 0 0 -1 \end{array}\right)\,, \qquad L_+ = \frac{1}{\sqrt{2}}\left( \begin{array}{rr}0 0 1 0 \end{array}\right)\,, \qquad L_- = \frac{1}{\sqrt{2}}\left( \begin{array}{rr}0 1 0 0 \end{array}\right)
\tau \sum_{j=1}^{n}\| B_{n-j}P_{h}\|_{\mathcal{L}_2^0}^p \leq \tau \sum_{j=1}^{n}\|A_h^{\frac{s}{2}}B_{n-j}\|^p\|A_h^{-\frac{s}{2}}P_hA^{\frac{s}{2}}\|^p\|A^{-\frac{s}{2}}\|_{\mathcal{L}_2^0}^{p}&\leq c\tau \sum_{j=1}^nt_{n-j+1}^{((1-\frac{s}{2})\alpha+\gamma -1)p}\| A^{-\frac{s}{2}}\|_{\mathcal{L}_2^0}^p< \infty.
(u_{n}^{\prime},v_{n}) =(u_{n}^{\prime},f_{\varepsilon}^{\prime}(u_{n}))+(u_{n}^{\prime},K\ast(1-2u_{n})) & =\frac{d}{dt}\left( \int_{\Omega}f_{\varepsilon}(u_{n})+\int_{\Omega}\int_{\Omega}K(x-y)u_{n}(x)(1-u_{n}(y))\right)
\mathbf{E}[r_{j_1}\cdot r_{j_2}] = \sum_{j=1}^\kappa \frac{1}{\kappa}\cdot \mathbf{E}[q_{j_1}(X_j) \cdot q_{j_2}(X_j)] = \sum_{j=1}^\kappa \frac{1}{\kappa}\cdot \langle v_{j_1}, v_{j_2}\rangle=\sum_{j=1}^\kappa \frac{1}{\kappa}\cdot \langle h_{j_1}, h_{j_2}\rangle = \mathbf{E}[p_{j_1}\cdot p_{j_2}].
D_{i,i+j}(\xi_{1}^{i}) :=\sum_{u_{i+1}^{k},v_{i+1}^{k}\in\{0,1\}^{k-i}}\mathcal{Q}(u_{i+1}^{k},v_{i+1}^{k}|1\xi_{1}^{i-1}1,1\xi_{1}^{i-1}0){\bf1}\{u_{i+j}\neq v_{i+j}\}&=\mathcal{Q}(u_{i+j}\neq v_{i+j}|1\xi_{1}^{i-1}1,1\xi_{1}^{i-1}0), \mbox{for}j=0,\ldots,k-i,
U_{n+1}=2 \sum_{k=0}^{n+1}R^k_{B,N}= R_{B,N}\Big(2 \sum_{k=0}^{n}R_{B,N}^k\Big) + 2{\rm I}_{N}= R_{B,N}U_{n}+2{\rm I}_{N}, U_{0}=2{\rm I}_{N}\, .
\Lambda_n(f_1,\dots,f_n) :=\textup{p.v.}\int_{(\mathbb{R}^{n-1})^n}\bigg(\prod_{j=1}^n\widehat{f_j}(\xi_j )\bigg)\det\begin{pmatrix}1 \!\cdots\! 1 \xi_1 \!\cdots\! \xi_n \end{pmatrix}^{-1}d\xi_1\cdots d\xi_n.
\begin{cases}\varphi_t-\varphi_{5x}+\varphi_{3x}+\varphi_x=0, \ \ (0,T)\times (0,2\pi), \partial_x^{j}\varphi(t,0)=\partial_{x}^{j}\varphi(t,2\pi), \ \ \ (0,T), \varphi(T,x)=\varphi_T(x), \ \ \ (0,2\pi),\end{cases}
\Gamma_{W Z}[ \widetilde{\phi}_{1}\widetilde{\phi}_{2}] \ = \ \Gamma_{W Z}[ \widetilde{\phi}_{1}] + \Gamma_{W Z}[ \widetilde{\phi}_{2}] - \frac{i}{4 \pi}\int_{\Sigma}T r ( \phi_{1}^{- 1}d \phi_{1}\, \phi_{2}d \phi_{2}^{- 1}) \ .
g_{i}({\bf p};\pi,Q^{\epsilon}) \leq \frac{\delta_{i}+ \beta_{i}\pi_{i}p_{i}^{2}+ \gamma_{i}}{\delta_{i}+ \beta_{i}\pi_{i}+ \gamma_{i}}= p_{i}+ \frac{\delta_{i}+ \beta_{i}\pi_{i}p_{i}^{2}+ \gamma_{i}- (\beta_{i}\pi_{i}+ \gamma_{i}+ \delta_{i})p_{i}}{\delta_{i}+ \beta_{i}\pi_{i}+ \gamma_{i}}.
|\mu_{x,y}(f(X_k^{-m},\ldots,X_k^m))-\nu_y(f(X_k^{-m},\ldots,X_k^m))| \quad\le(4m+2)\|f\|_\infty e^{-k}\max_{w=-m,\ldots,m}\sum_{v\in\mathbb{Z}}e^{\|(k,w)-(0,v)\|}D_{(k,w)(0,v)}&\le(4m+2)\|D\|_*\|f\|_\infty e^{-k}\le(8m+4)\|f\|_\infty e^{-k}.
k_{1}= \left( \begin{array}{c}{{p}}\\{{p}}\\ \end{array}\right) k_{2}= \left( \begin{array}{c}{{e}}\\{{- p}}\\ \end{array}\right) k_{3}= \left( \begin{array}{c}{{- p}}\\{{- p}}\\ \end{array}\right) k_{4}= \left( \begin{array}{c}{{- e}}\\{{p}}\\ \end{array}\right) .
\lim \limits_{P_0 \rightarrow \infty}P_s(E_{12})=\frac{1}{2}\sum\limits_{k=0}^R\frac{\bar{\gamma}_k^{R}}{\prod \limits_{\substack{j=0 j\neq k}}^R (\bar{\gamma}_k-\bar{\gamma}_j)}\left\lbrace 1-\sqrt{\frac{\bar{\gamma}_k |d_{\mathrm{min}}|^2}{2+\bar{\gamma}_k |d_{\mathrm{min}}|^2}}\right\rbrace
\widehat Q_j^\mu(x)= \frac{\tau_j}{x_j+\tau_j}\sum_{l=0}^{N-2}d_l^\mu \hat \varrho_{l+1,j}\,\varphi_l(x) + \frac{1}{x_j+\tau_j}\times \sum_{l=0}^{N-2}d_l^\mu\hat \varrho_{l+1,j}\,\bigg\{\frac{l+2-\mu}{l+2}a_{l+1}\varphi_{l+1}(x)+b_l \varphi_{l}(x)+ \frac{l+1}{l+1-\mu}c_{l-1}\varphi_{l-1}(x)\bigg\},
\exists \lim_{\epsilon \to 0}\frac{1}{\epsilon^2}K^\phi(f_\alpha^{(n)}||f_\rho^{(n)}) = \frac{1}{2}I(\tilde{f_\alpha}) \geq \frac{n}{2\alpha (1-\alpha)}- \frac{\sqrt{n}}{\alpha (1-\alpha)}+ O \left( 1 \right)
\widehat{E}_{3}=I+HH^{\ast}E_{S_{A}}-(I+F_{S_{A}}K^{\ast}K)^{-1}S_{A}^{\dagger}-HH^{\ast}E_{S_{A}}-S_{A}S_{A}^{\dagger}+(I+F_{S_{A}}K^{\ast}K)^{-1}S_{A}^{\dagger}&=E_{S_{A}}.
f(x,w)=\frac{w_{0}e^{-w^{2}/2}}{\sqrt{2\pi}}[w_{0}-\zeta(\frac{1}{2})-\sum_{n=1}^{\infty}\frac{G_{-n}(w_{0})}{2nQ_{n}}+\sum_{n=1}^{\infty}S_{n}(w_{0})G_{n}(w)e^{-\sqrt{n}z}]
T^{\mu\nu}_0=-\left(\frac{g_{str}\alpha'}{2}\right)^2\prod_u\frac{\delta(\sqrt{\alpha'}k_u)}{\sqrt{\tau_2}}\left( \frac{\eta^{\mu\nu}}{2\pi\alpha'}\prod_o \frac{\delta(\sqrt{\alpha'}k_o)}{\sqrt{\tau_2}}X_{1,1}+\frac{k_\mu k_\nu}{4}2^{-(d+1)}X_{1,0}\right)
\pi_k \textbf{End}^0_{S,gr}(X) = \pi_k \textbf{End}_{S,gr}(X) &= Hom_{A-Mod^{gr}}(M[k+1],Sym_A(M[1]) \otimes_A B) &= Hom_{A-Mod^{gr}}(M,\wedge^{k+1}M \otimes_A B) &= Hom_{A-Mod^{gr}}(M \otimes_A (\wedge^{k+1}M)^\vee, B)
\mu\left(\bigcup_{\sigma \in \mathbf{C_n^{\infty}}}[\mathbf{I^\infty_{\sigma,n}}] - X^c\right) = \mu\left(\bigcup_{\sigma \in \mathbf{C_n^{\infty}}}[\mathbf{I^\infty_{\sigma,n}}] - \left(\bigcap_{i \in \mathbb{N}}X_i\right)^c\right) \leq \sum_{i=0}^{\infty}2^{-n-i-1}= 2^{-n}.
Z = 2 \alpha^{2}\, \ln \mu \, \mathrm{D i a g o n a l}\Bigl ( - 1 / 3 \, , \, \, \frac{1}{1 2}( - \sqrt \Theta + 1 7 + 1 6 N ) , \, \, \frac{1}{1 2}( \sqrt \Theta + 1 7 + 1 6 N ) \Bigr ) \, .
\textup{hspec}^{\mathcal{P}}_{\trianglelefteq}(G) = \textup{hspec}^{\mathcal{D}}_{\trianglelefteq}(G) = \textup{hspec}^{\mathcal{F}}_{\trianglelefteq}(G) = [0,1/3] \cup \{1\}\textup{hspec}^{\mathcal{L}}_{\trianglelefteq}(G) = [0,1/5] \cup \{3/5\}\cup \{1\}
\sum_{t=1}^T \ell(\hat y_t, y_t) - \inf_{f \in \mathcal{L}_L^d}\sum_{t=1}^T \ell\bigl(f (\b x_t), y_t\bigr) \leq M \left(3+ L \right) \sqrt{N_T T}& \leq M(3+L) \Bigl(\sqrt{T}+ 2(3d)^{\frac{d}{2(d+2)}}T^{\frac{d+1}{d+2}}\Bigr) \,.
\Sigma_{1}^{i}( z - v ) = \frac{- 3 f_{l}^{2}}{1 6 ( 4 \pi^{2})}\left( 2 \frac{Y_{L}^{( e )}}{Y_{R}^{( e )}}L \delta_{i ( e )}+ \frac{Y_{R}^{( e )}}{Y_{L}^{( e )}}R \right) / \partial \delta^{4}( z - v ) .
c_{4}|\begin{matrix}p_{v}-p_{u_{1}}&(p_{v}-p_{u_{1}})\cdot\lambda_{4,u_{1}}\\p_{v}-p_{u_{2}}&(p_{v}-p_{u_{2}})\cdot\lambda_{4,u_{2}}\\p_{v}-p_{u_{3}}&(p_{v}-p_{u_{3}})\cdot\lambda_{4,u_{3}}\end{matrix}|=0.
Q=-2\left( \begin{array}{ccc}N_1-N_2 N-N_1+\frac 12N_2 N-N_1+\frac 12N_2 N-N_1+\frac 12N_2 N_1 N_1-N N-N_1+\frac 12N_2 N_1-N N_1-2N\end{array}\right)
\left\|{A}\xi- \hat{A}\xi \right\|_\alpha^\alpha =\int_{-\pi}^{\pi}\left| A(e^{i\theta}) -{h}(\theta) \right|^{\alpha}f(\theta)d \theta + \int_{-\pi}^{\pi}\left|{h}(\theta) \right|^{\alpha}g(\theta)d \theta.
P(i,t)(\{j\}\times U) = M_{ij}\mu_{j}(U), \forall i,j\in\underline{N},\;\forall t\in\mathbb{R}_{+},\;\forall U\in\mathfrak{B}, \nu_{1}(\{j\}\times U) = p_{j}\mu_{j}(U), \forall j\in\underline{N},\;\forall U\in\mathfrak{B}.
\bar p(z)=\sum_{j=0}^{d-1}{d-1\choose j}\bar s_{j+1}z^j, \bar q(z)=\sum_{j=0}^{d-1}{d-1\choose j}\bar s_jz^j,\bar r(z)=\frac{\bar p(z)}{\bar q(z)},g(z)=\bar r (r(z)).
[\mathbf{f}(a),\mathbf{g}(1)]=-[[\mathbf{h}(\bar{a},1),\mathbf{f}(1)],\mathbf{g}(1)]=-[\mathbf{h}(\bar{a},1),\mathbf{h}(1,1)]+[\mathbf{f}(1),\mathbf{g}(a)]=[\mathbf{f}(1),\mathbf{g}(a)],
\varphi(q)= \frac{1}{2}\int_{\mathbb{R}_{+}^{n}}\Delta v_{q}v_qdzdt-\frac{(n-6)(n-2)\omega_{n-1}I_{n-1}^{n}}{4(n-1)^{2}(n-4)}\|\pi(q)\|^{2}\le0.
- \; \frac{1}{e}\; \frac{\lambda^{2}}{( x_{1}^{2}+ \lambda^{2})^{\frac{3}{2}}}\; \left[ \mathrm{a r c t a n}\left( \frac{x_{2}}{\sqrt{x_{1}^{2}+ \lambda^{2}}}\right) \; - \; \pi \Big ( \frac{1}{2}+ m \Big ) \right] \; .
x_1 \cdot \frac{\partial f}{\partial x_1},..., x_n \cdot \frac{\partial f}{\partial x_n}, & w_{-\alpha}(f) := \sum\limits_{m \in \Delta \cap M}ht_{-\alpha}(m) \cdot a_m \cdot x^{m-\alpha}, \alpha \in R(N,\Sigma_{C(\Delta)}).
\delta L_{T G}= \varepsilon^{\mu \nu \rho}( R_{\beta \mu \nu}^{\alpha}\delta \Gamma_{\alpha \rho}^{\beta}- \bar{R}_{\beta \mu \nu}^{\alpha}\delta \bar{\Gamma}_{\alpha \rho}^{\beta}) - \partial_{\mu}\, [ \varepsilon^{\mu \nu \rho}N_{\beta \nu}^{\alpha}( \delta \Gamma_{\alpha \rho}^{\beta}+ \delta \bar{\Gamma}_{\alpha \rho}^{\beta}) ]
Z_N(\zeta)=\left(\prod_{i=1}^mK_{N,L_i}\right)\mathrm{det}\left[[b_{j,k}(\zeta)]_{\substack{j=1,\ldots,(N+1)/2 k=1,\ldots,(N-1)/2}}[\mu_{2j-1}]_{j=1,\ldots,(N+1)/2}\right].
r_{N/\{w_1, w_2\}}(z) = r(z \cup \{w_1,w_2\}) - r(\{w_1 , w_2\}) = \rho(\langle z, w_1,w_2 \rangle) - \rho(\langle w_1, w_2 \rangle) = \rho(W) - \rho(W) = 0.
\frac{1}{2}\int_{\epsilon^{2}R^{2}}^{\infty}\frac{d t}{t^{3}}\sum_{k = - \infty}^{\infty}e^{- k^{2}\pi t / R^{2}}= R^{- 4}{\cal I}( \epsilon^{2}, \alpha = - 2 ) = \frac{1}{R^{4}}\left\{\frac{3}{4 \pi^{2}}\zeta ( 5 ) + \frac{1}{5}\epsilon^{- 5}\right\}
f(t,x) =\sum_{n=0}^{\left\lfloor\frac N2\right\rfloor}\sum_{\sigma\in\mathfrak{C}_{2n}^N}a(\sigma)\exp\left( s_{\sigma(1)}+\cdots+s_{\sigma(2n)}\right), g(t,x) =\sum_{n=0}^{\left\lfloor\frac{N-1}2\right\rfloor}\sum_{\sigma\in\mathfrak{C}_{2n+1}^N}a(\sigma)\exp\left( s_{\sigma(1)}+\cdots+s_{\sigma(2n+1)}\right).
A_{\mathcal{H}_{3}}(\underline{X}) =\left[ \begin{array}{cccccc}X_4 X_5 X_6 \noalign{\medskip}X_4 X_5 X_6 \noalign{\medskip}X_4 X_5 X_6 \noalign{\medskip}-X_1 -X_2 -X_3 \noalign{\medskip}-X_1 -X_2 -X_3 \noalign{\medskip}-X_1 -X_2 -X_3\end{array}\right],
{H_{{\varphi_n}{\sf E}_n}}=\sum_{j\in\mathbb{J}_n}E_j^{(n)}\langle \varphi_j^{(n)}, \cdot\rangle\,\varphi_j^{(n)}=\sum_{j=1}^{n+1}E_j^{(n)}\langle \varphi_j^{(n)}, \cdot\rangle\,\varphi_j^{(n)},{\sf E}_n=\{E_j^{(n)}, j\in\mathbb{J}_n\}.
\ell(\psi_-) = v_D(\psi_-) \quad \ell_j(\psi_+) = \begin{cases}v_D\bigl(\Im(\psi_+)\bigr), v_D\bigl(\Im(\psi_+)\bigr) < 2j; +\infty, v_D\bigl(\Im(\psi_+)\bigr) \geq 2j . \end{cases}
\textnormal{confluence}\left(\widetilde{J^{K\textnormal{th}}}\right)(z,Q) = \sum_{i=0}^N \left( 1-P^{-1}\right)^i \sum_{\substack{0 \leq a, b \leq N a + b = i}}\frac{1}{a !}\left( \frac{\log(Q)}{z}\right)^a g_b(z,Q)
e^{ia_1+\dots+ia_m}=(e^{a_1/N}e^{a_2/N}\cdots e^{a_m/N}e^{c_N})^N & =\prod_{k=1}^M(e^{ix_k},e^{iy_k})\cdot e^{a_1}e^{a_2}\cdots e^{a_m}e^{Nc_N}.
\sum_{b=1}^{\frac{n}{M^22^{2M+5}}}\binom{2n}{b}\binom{2n}{2b}\exp\left(-\frac{3b\eta}{16M}\right) \leq \sum_{b=1}^{\infty}\left(2n\exp\left(1-\frac{\eta}{16M}\right)\right)^{3b}.
\Sigma_n(z_1, z_2) = (2n^2)^{m-s-1}\sum_{j = 0}^{n-1}\frac{q_j^{(n)}( 1 - \frac{z_1}{2n^2}) q_j^{(n)}(1 - \frac{z_2}{2n^2})}{\frac{d_{j}^{(n)}d_{j+1}^{(n)}k_j^{(s)}h_{j-m}^{(s)}}{k_{j-m}^{(s)}}},
\epsilon^{- 1}( y ) \delta ( y ) \bar{\Psi}_{\mu}^{A}( \delta - \gamma_{5}\sigma_{3})_{A}^{\; B}\gamma_{5}= - \delta ( y ) \alpha_{0}\bar{\Psi}_{\mu}^{A}( \delta + \gamma_{5}\sigma_{3}) ( \sigma_{1})_{A}^{\; B}\ .