harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\partial (\varphi_{1},\mathellipsis ,\varphi_{m})}{\partial (x_{1},\mathellipsis ,x_{m})}=\frac{\partial (\varphi_{m+1},\mathellipsis ,\varphi_{m+n})}{\partial (y_{1},\mathellipsis ,y_{n})}=\frac{\partial (\varphi_{m+n+1},\mathellipsis ,\varphi_{m+n+p})}{\partial (z_{1},\mathellipsis ,z_{p})}=0.
q_{s_1,s_2}(x) = \begin{cases}\big[s_1^2 - (1/4)\big] x^{-2}, x \in (0, \varepsilon), begin{align*}1mm] \big[s_2^2 - (1/4)\big] (x - \pi)^{-2}, x \in (\pi - \varepsilon, \pi). \end{cases}
\phantom{=}\;\;(-1)^{pn}\big((n-1)[\omega]+n(p-1)\big)*\big((n-1)[\omega]+n(p-2)\big)*\cdots*(n-1)[\omega] = (-1)^{pn}\big((n-1)^p[\omega]^p- (n-1)[\omega]\big) = (-1)^{pn}(n-1) \big([\omega]^p-[\omega]\big).
\bar{q}_{\beta}(n,t)=\sum_{k=1}^{n}\underset{m_j\in\mathbb{N}}{\underset{m_{1}+m_{2}+\dots+m_{k}=n}{\sum}}\left(\frac{\lambda t^{\beta}(1-\rho)^{r}}{1-(1-\rho)^{r}}\right)^{k}\prod_{j=1}^{k}\rho^{m_{j}}\binom{r+m_{j}-1}{m_{j}}E_{\beta,k\beta+1}^{k+1}(-\lambda t^{\beta}).
{u}(t_{i}) = g(t_{i},{u}(t_{i})) + \int_{0}^{t_{i}}\frac{1}{\sqrt{t_{i}-s}}k_{1}(t_{i}, s,{u}(t_{i}),{u}(s))\mathrm{d}s+\int_{0}^{t_{i}}k_{2}(t_{i},s ,{u}(t_{i}),{u}(s))\mathrm{d}s,
SS^{\star}=\frac{1}{1+\frac{\Im f.{\frac{2}{R^2}}}{(1-\frac{1}{R^2})^2}}\simeq 1-\frac{2\Im f}{\frac{R^2}{(1-\frac{1}{R^2})^2}}\simeq 1-\frac{4\bigg(\frac{\sin\pi\beta}{R}\bigg)^2}{(1-\frac{1}{R^2})^2}
H_Y(y;t) = \sum_{m=0}^{+\infty}p(m;\mu t)\,F^{(m)}_X(y), ={\bf 1}_{\{y\geq 0\}}{\rm e}^{-\mu t}+\sum_{m=1}^{+\infty}p(m;\mu t)\,F^{(m)}_X(y), y\in{\mathbb R}, \;\; t\geq 0.
\theta_{1}( \tau ) = \sum_{n = - \infty}^{\infty}q^{( n + 1 / 2 )^{2}/ 2}, \quad \theta_{2}( \tau ) = \sum_{n = - \infty}^{\infty}( - 1 )^{n}q^{n^{2}/ 2}, \quad \theta_{3}( \tau ) = \sum_{n = - \infty}^{\infty}q^{n^{2}/ 2}.
\mathcal I := \prod_v \mathcal I_v^{\sharp}(\mathbf h,\breve{\mathbf g},\pmb{\phi})^{-1}= \prod_{p \mid 2N}\mathcal I_v^{\sharp}(\mathbf h,\breve{\mathbf g},\pmb{\phi})^{-1}= N \prod_{p\mid M}\frac{(p+1)}{2}= 2^{-\nu(M)}N \prod_{p\mid M}(p+1).
J_N(z, u) = \mathbf{E}_{z}^u \Big[\sum_{j=0}^{K-1}\Big(r\big(z^{u,N}(t_j), \nu_j\big) + \int_{t_j}^{t_{j+1}}\phi\big(z^{u,N}(s), \nu_j\big)\, ds\Big) +\psi\big(z^{u,N}(T)\big) \Big]\,
K ( z , \bar{z}) = z \bar{z}+ \sum{\frac{1}{L_{R}^{n - 2}}}K_{I_{1}. . I_{p}\bar{I}_{p + 1}. . \bar{I}_{n}}z^{I_{1}}. . z^{I_{p}}\bar{z}^{I_{p + 1}}. . \bar{z}^{n}
f_{R}(R)=\frac{\Gamma(\frac{1}{2}(\nu+p))}{\nu^{p/2}\pi^{p/2}\Gamma(\frac{1}{2}\nu)}\frac{2\pi^{p/2}R^{p-1}}{\Gamma(p/2)}(1+\frac{R^{2}}{\nu})^{-(\nu+p)/2}
r ( \tau ) = r_{o}+ \frac{4 ( 6 H^{2}r_{o}^{2}- 1 ) e^{- \sqrt{6 H^{2}r_{o}^{2}- 1}\; ( \tau - \tau_{o})}}{H ( e^{- \sqrt{6 H^{2}r_{o}^{2}- 1}\; ( \tau - \tau_{o})}- 4 H^{2}r_{o}^{2})^{2}- 4 H ( 6 H^{2}r_{o}^{2}- 1 )}.
T_{k}:=\sum_{\mathbf{q,l,j}\in\Lambda_{m}^{n}}C_{k}\left( \mathbf{q,l,j}\right) \left( \partial^{l_{n}}M_{b_{q_{n}}}\partial^{j_{n}}\right) \left(\partial^{l_{n-1}}M_{b_{q_{n-1}}}\partial^{j_{n-1}}\right) \dots\left(\partial^{l_{1}}M_{b_{q_{1}}}\partial^{j_{1}}\right) \mathbf{1}
X(\ell) = \iint \! \left| \frac{v}{v + w}\frac{\alpha}{u - \beta}\right|^s \, P_z(v)P_z(w) \, dv \, dw = \iint \! \left| \frac{\alpha v}{2vw - \beta(v + w)}\right|^s \,P_z(v)P_z(w) \, dv \, dw = \int \! \left|\frac{\alpha v}{2v - \beta}\right|^s \int \! \left| \frac{1}{w - \beta v (2v - \beta)^{-1}}\right|^s \,P_z(w)\, dw \, P_z(v) \, dv,
d_n(b_2c_2a_2 \dots b_2c_2a_2b_2 \otimes 1) = b_2c_2a_2 \dots b_2c_2a_2 \otimes b_2 + b_2c_2a_2b_2 \dots c_2a_2b_2a_1 \otimes b_1c_1
\partial_{t}P_{n}^{M}= - u_{n}\partial_{x}P_{n}^{M}+ \rho_{n}^{-1}\partial_{x}w_{n}- P_{n}^{M}p_{n}''(\rho_{n})|\rho_{n}^{2}\partial_{x}P_{n}^{M}|^{2}- 2p_{n}'(\rho_{n})\rho_{n}^{2}\left( \partial_{x}P_{n}^{M}\right)^{2}+ \rho_{n}p_{n}'(\rho_{n}) \partial_{x}^{2}P_{n}^{M}.
M = \frac{1}{8}\{k^{2}\not \! F ( A ) - \not \! F ( B ) - 4 \hat{\gamma}^{x}\not \! \partial k \}= \frac{1}{2}( \hat{\gamma}^{0}- \hat{\gamma}^{x}) ( \hat{\gamma}^{i}\partial_{\underline{{{i}}}}k ) \, .
\sum_{x_1\in T^d(r)}\sum_{x_2\in T^d(r)}P\big(\sum_{y:y\sim x_1}\eta_0^{d,p}(y)\geq d,\sum_{z:z\sim x_2}\eta_0^{d,p}(z)\geq d\big) \leq r^{2d}[P\big(\sum_{y:y\sim O}\eta^{d,p}_0(y)\geq d\big)]^2+2r^dd^2\exp\{-2dK(p)+o(d)\}= [r^dP\big(\sum_{y:y\sim O}\eta^{d,p}_0(y)\geq d\big)]^2+\exp\{d(\log r-2K(p))+o(d)\}.
\frac{f_9}{f_1}= \frac{f^{3}_{12}f_{18}}{f^{2}_{2}f_{6}f_{36}}+q\frac{f^{2}_4f_{6}f_{36}}{f^{3}_2f_{12}}, \frac{f_3}{f^{3}_1}= \frac{f^{6}_{4}f^{3}_{6}}{f^{9}_{2}f^{2}_{12}}+3q\frac{f^{2}_4f_{6}f^{2}_{12}}{f^{7}_2}, \frac{1}{f_1f_3}= \frac{f^{2}_{8}f^{5}_{12}}{f^{2}_{2}f_{4}f^{4}_{6}f^{2}_{24}}+q\frac{f^{5}_{4}f^{2}_{24}}{f^{4}_{2}f^{2}_{6}f^{2}_8f_{12}}.
\left[ \begin{matrix}{\varphi_{1 k_{-}}}\\{\varphi_{2 k_{-}}}\\ \end{matrix}\right] = \sqrt{{\frac{\kappa}{2}}}\left[ \begin{matrix}{e^{i k_{-}x^{-}}e^{i \pi / 2}}\\{e^{- i k_{-}x^{-}}e^{- i \pi / 2}}\\ \end{matrix}\right] .
\chi_{\gamma}\left( 2 \right) = ( n - 2 ) ! \ \sum_{i \in{\cal I}}\frac{D ( p_{1}, \ldots , p_{i - 1}, p_{i}- 2 , p_{i + 1}, \ldots , p_{m})}{p_{1}! \cdots p_{i - 1}! \ ( p_{i}- 2 ) ! \p_{i + 1}! \ \cdots p_{m}!},
\mathbf{Pr}\left(\hat{x}_{i,min,\gamma}> x_i + \epsilon\right) =\left[1- \frac{1}{K}\left(\gamma K\right) E\left\{F_\alpha\left(\left(\frac{\epsilon^\alpha}{\eta_{ij}}\right)^{1/(1-\alpha)}\right)\right\}\right]^M
P\left( (\mathfrak{B}_{n+1}, \mathfrak{R}_{n+1}) = (i+1,j) \mid (\mathfrak{B}_n, \mathfrak{R}_n) = (i,j) \right) = \frac{b(i)}{b(i)+r(j)},\ P\left( (\mathfrak{B}_{n+1}, \mathfrak{R}_{n+1}) = (i,j+1) \mid (\mathfrak{B}_n, \mathfrak{R}_n) = (i,j) \right) = \frac{r(j)}{b(i)+r(j)},\ \i,j\ge 0.
\begin{aligned}1-F(2\pi /s)+ F(-2\pi /s) \le \le \frac{2}{s a_0}\Bigl(\sum_{j=0}^k a_j \int_{0}^{s}\tt{Re}\, f(ju)du + \sum_{j=1}^k b_j \int_{0}^{s}\tt{Im}\, f(ju)du \Bigr).\end{aligned}
P_2(\mathbf W_i\in[h_i2^{-\ell}, (h_i+1)2^{-\ell}), ~1\leq i\leq m|\mathbf Y=\mathbf y) &= P_2(\mathbf W_{\ell,i}=h_i2^{-\ell}, ~1\leq i\leq m|\mathbf Y=\mathbf y) &= \psi_{\ell,h}(\mathbf y)= P''(\mathbf X''_i\in[h_i2^{-\ell}, (h_i+1)2^{-\ell}), ~1\leq i\leq m|\mathbf Y''=\mathbf y).
S_{2m}= m_K\frac{\phi(\mathfrak{m})^k}{|\mathfrak{m}|^{k+1}}(c_K\log R)^{k+1}|\mathcal{P}(N)|\left(\widetilde{I}_{2k}^{(m)}(F)+\widetilde{I}_{3k}^{(m)}(F)\right) + O\left(F_{\max}^2(\log R)^{k+1}\frac{(\phi(\mathfrak{m}))^{k}}{|\mathfrak{m}|^{k+1}}\frac{1}{D_0}|\mathcal{P}(N)|\right)
T_iY_i = ((t^{\frac12}-t^{-\frac12})+T^{-1}_i)Y_i=Y_{i+1}T_i + (t^{\frac12}-t^{-\frac12})Y_i, T_i Y_{i+1}= Y_i T^{-1}_i = Y_i (T_i - (t^{\frac12}-t^{-\frac12}))=Y_iT_i - (t^{\frac12}-t^{-\frac12})Y_i.
\frac{1}{\varepsilon_{k}^{n}}\int_{M}a(x)u_{k}\varphi_{k}d\mu_{g}=\int_{B(0,\rho)}a(\exp_{\xi_{k}}(\varepsilon_{k}z))\tilde{u}_{k}(z)\varphi(z)|g(\varepsilon_{k}z)|^{\frac{1}{2}}dz & =a(\xi_{k})\int_{B(0,\rho)}\tilde{u}_{k}(z)\varphi(z)dz+o(1) & =a(\xi)\int_{B(0,\rho)}\tilde{u}_{k}(z)\varphi(z)dz+o(1)
\Gamma ( p^{2}, m^{2}) = \frac{i \lambda^{2}\mu^{\epsilon}}{1 6 \pi^{2}\epsilon}- \frac{i \lambda^{2}\mu^{\epsilon}}{3 2 \pi^{2}}\left\{\gamma + \int_{0}^{1}d z \ln [ \frac{p^{2}z ( 1 - z ) - m^{2}}{4 \pi \mu^{2}}] \right\}.
\begin{array}{rcll}\partial_r^2 = 0 \mbox{\rm for}r\in [1,n-1], \partial_r\partial_t = \partial_t\partial_r \mbox{\rm for}r,t\in [1,n-1],~|r-t|>1, \partial_r\partial_{r+1}\partial_r = \partial_{r+1}\partial_r\partial_{r+1}\mbox{\rm for}r\in [1,n-2]. \end{array}
V_F(1/x) = \left[\begin{array}{cccc}F_0(\frac{1}{x_1}) F_1(\frac{1}{x_1}) \cdots F_{n-1}(\frac{1}{x_1}) F_0(\frac{1}{x_2}) F_1(\frac{1}{x_2}) \cdots F_{n-1}(\frac{1}{x_2}) \vdots \vdots \vdots F_0(\frac{1}{x_n}) F_1(\frac{1}{x_n}) \cdots F_{n-1}(\frac{1}{x_n})\end{array}\right],
\begin{array}{lll}V^{(1)}= \displaystyle{1 \over 4!}f\varphi^4+{1 \over 48}(\beta_f -4f \gamma_{\varphi}) \varphi^4\left( \log{\varphi^2 \over \mu^2}-{25 \over 6}\right) & \displaystyle-{1 \over 2}\xi R \varphi^2-{1 \over 4}(\beta_{\xi}-2\xi \gamma_{\varphi}) R \varphi^2\left( \log{\varphi^2 \over \mu^2}-3 \right),\end{array}
\chi_{(\mu,2\kappa,-s)}(\sigma^2)=\chi_{(\lambda,\kappa,0)}(v)\chi_{(\lambda,\kappa,0)}(v)\chi_{(\mu,2\kappa,-s)}(v^{-1})=q^{(\mu,\mu+2\rho_0)-2(\lambda,\lambda+2\rho_0)-2s(2\kappa+g)}
\sigma(s) =-\frac{s}{y(s)}-\frac{1}{y^2(s)}+\frac{\left(y'(s)\right)^2}{4y^3(s)}&=-\frac{r'(s)^2}{4 r(s)}-r(s)^2+s r(s), r(s) =-\frac{1}{y(s)}.
V(Z_\ell^{(k)})= \{v_i^j\colon i\in [\ell], j\in [k-1] \}E(Z_\ell^{(k)})= \{v_i^1v_i^2\cdots v_i^{k-1}v_{i+1}^j \colon i\in [\ell], j\in[k-1]\}\,,
x_{2k}= x_{2k+1}= r_-(\xi_1 + \cdots + \xi_{2k}) + r_+(\xi_{2k+1}+ \cdots + \xi_{n}) = r_+(\xi_1 + \cdots + \xi_{n}) + (r_- - r_+) (\xi_1 + \cdots + \xi_{2k}),
(z_0+z_2)^{k+\frac{j_2}{T}}Y_{M^3}(u,z_0+z_2)I(w_1,z_2)w_2 = (z_2+z_0)^{k+\frac{j_2}{T}}I(Y_{M^1}(u,z_0)w_1,z_2)w_2.
K_W := 2^{-1}W \sum_{i=1}^d (S_i^* +S_i) + 2^{-1}\sum_{i=1}^d (S_i^* +S_i) W, B_W := \sum_{i=1}^d U_i \tilde{W}(S_i^*-S_i) - \sum_{i=1}^d (S_i^*-S_i) \tilde{W}U_i,
\beta_{s,a^1}^1 =\frac{\bar{r}^1(s,a^1)-\bar{r}^1(s,a_s^1)}{\sum_{s'\in S}p_{s'}\bar{r}^1(s',a_{s'}^1)-\sum_{s'\in S}p^1(s'|s,a^1)\bar{r}^1(s',a_{s'}^1)}&=\frac{r^1(s,a^1,a_s^2)-r^1(s,a_s^1,a_s^2)}{\sum_{s'\in S}p_{s'}r^1(s',a_{s'}^1,a_{s'}^2)-\sum_{s'\in S}\left(\frac{\mu(s',s,a^1,a_s^2)}{||\mu||}+\delta(s,s')\right)r^1(s',a_{s'}^1,a_{s'}^2)}.
p_{\mathcal{G},T}(x)=\sum_{k=0}^{|B|}\bigg(\big(p_{G,r}(x)\big)^{|B|-k}\big(p_{G,\emptyset}(x)-xp_{G,r}(x)\big)^{k}\sum_{\substack{S\subseteq B |S|=k}}p_{H,S\cup T}(x)\bigg).
(2\lambda+Y+C-Z)^2-4(-BZ+\lambda^2+\lambda C+\lambda Y+YC) = 4\lambda^2+Y^2+C^2+Z^2+4\lambda Y+4\lambda C-4\lambda Z + &+2YC-2YZ-2ZC+4BZ-4\lambda^2-4\lambda C-4\lambda Y-4YC = Y^2+C^2+Z^2-4\lambda Z -2YC-2YZ-2ZC+4BZ
\begin{aligned}\delta_\pi(f)= \ [\pi, f]_{\Omega}= \ \Big(0,\ldots,0,\delta_\pi(f)^A_{\{1,\ldots,n+1\}}; \delta_\pi(f)^V_\emptyset,\ldots,\delta_\pi(f)^V_{\{2,\ldots,n+1\}}, 0\Big) \in \mathcal{C}_{\rm AssAct}^{n+1}(A,V) ,\end{aligned}
\tilde{E}(u,u) :={\cal E}^{\mu}(\mathbf{H}_{\partial\Gamma}u,\mathbf{H}_{\partial\Gamma}u), \tilde{F}:=\{u\in L^2(\partial\Gamma,\nu)\ ;\u=\tilde{g}\ \nu\mathchar`-a.e.\{\rm on}\ \partial\Gamma\{\rm for\some}\g\in\tilde{\cal F}^{\mu}\},
A=\frac{1}{(\beta-1)(\alpha+\beta-1)}\sum_{i}p^{\alpha+\beta-1}_{i}-\frac{1}{\alpha(\beta-1)}\sum_{i}p^{\alpha}_{i}q^{\beta-1}_{i}&B=\frac{1}{\alpha(1-\alpha-\beta)}\sum_{i}q^{\alpha+\beta-1}_{i}
S_1 = |\varpi|^{3}\left(\frac{|\varpi|^{n-1}}{(1-|\varpi|^{-1})^2}\mathcal{N}_1 -\frac{|\varpi|^{n-1}}{(1-|\varpi|^{-1})^2}\mathcal{N}_2\right)+O\left(|\varpi|^{n+3}\right).
\frac1T \sum \limits_{k=0}^{T-1}\langle g_{k+1}, x_k -x^* \rangle \leq \frac{\kappa}{\kappa +1}\frac{R_{0}^{\frac{1+\kappa}{\kappa}}}{\nu T}+ \frac{\nu^{\kappa}}{1+\kappa}\frac{1}{T}\sum \limits_{k=0}^{T-1}\|g_{k+1}\|^{1+\kappa}_q,
T_{{\rm SVC},n,\ell+1}^*=\sqrt{\frac{a_{n}D_2(\tau_C)}{v^*(D_1(\tau_C))^2}}\left(\sqrt{\frac{b_{n,\ell+1}}{s_{\ell+1}}}-\sqrt{\frac{b_{n,{\ell+2}}}{s_{\ell+2}}}\right) \geq \sqrt{\frac{a_{n}D_2(\tau_C)}{v^*(D_1(\tau_C))^2}}\left(\sqrt{\frac{b_{n,\ell}}{s_{\ell}}}-\sqrt{\frac{b_{n,{\ell+1}}}{s_{\ell+1}}}\right) &=T_{{\rm SVC},n,1}^*.
f \ast_g \pi(h) = \frac{1}{n!}\sum_{\sigma,\tau \in \Sigma_n}gf_{\sigma(1)}\wedge g^c h_{\tau(1)}\otimes \cdots \otimes g f_{\sigma(n)}\wedge g^c h_{\tau(n)}\pi(f \ast_g h) = \frac{1}{n!}\sum_{\sigma,\tau \in \Sigma_n}gf_{\sigma\tau(1)}\wedge g^c h_{\tau(1)}\otimes \cdots \otimes gf_{\sigma\tau(n)}\wedge g^c h_{\tau(n)}.
S + J S J^{-1}=\begin{pmatrix}E F -F E\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}I I i I -i I\end{pmatrix}\begin{pmatrix}E+iF 0 0 E-iF\end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix}I -i I I i I\end{pmatrix}
P_{0}T(W^{(\mathbf{\beta},\mathbf{\gamma})}(x_{1})...W^{(\mathbf{\beta}, \mathbf{\gamma})}(x_{n}))P_{0}=P_{0}T(W^{(\mathbf{0},\mathbf{0})}(x_{1})...W^{(\mathbf{0},\mathbf{0})}(x_{n}))P_{0}+\mathrm{divergences}
F(x) = H_1(1-x)+H_2(x)+H_3(1-x)+C_1, G(x) = H_1(1-x)+H_3(x)+C_2, H(x) = H_1(1-x)+H_2(1-x)+H_3(x)+C_3, K(x) = H_1(1-x)+H_2(x)+C_4,
P_{2r}(n_1,m_1,\dots,n_r,m_r) =1+\sum_{k\ge i_1\ge i_2\ge 1}m_{i_1}n_{i_2}+\sum_{k\ge i_1\ge i_2>i_3\ge i_4\ge 1}m_{i_1}n_{i_2}m_{i_3}n_{i_4}&\quad+\dots+m_{r}n_{r}m_{r-1}n_{r-1}\cdots m_1n_1,
\displaystyle{\operatornamewithlimits{\mbox{minimize}}_{\Theta}}\f_N^b(\Theta) \, \triangleq \, \displaystyle{\frac{1}{N}}\, \displaystyle{\sum_{i=1}^N}\, \ \left[ y_i \, m(x^i;\Theta) + b(m(x^i;\Theta)) \, \right],
\left.{\frac{\partial \Gamma_{n - 1}}{\partial \kappa}}\right|_{\Phi , K}= \left.{\frac{\partial W_{n - 1}}{\partial \kappa}}\right|_{J , K}= \left.{\frac{\partial_{r}< \chi_{n - 1}>_{J}}{\partial K_{A}}}\right|_{J}J_{A}( - 1 )^{\varepsilon_{A}}.
\mathcal{S}= \int{d^{4}x \sqrt{- \overline{{{g}}}}\left( \overline{{{\mathcal{R}}}}- \frac{3}{2}\left[ \overline{{{\nabla}}}\phi \right]^{2}- \frac{3}{2}m^{2}\left[ 1 - e^{- \phi}\right]^{2}\right)},
\int_{TM}\,\tilde{\mu}_P(x',y')\,\tilde{\lambda}_l(x'^\rho_i,y'^\rho_i;x^\nu_a,y^\nu_a)\,\tilde{x}'^\mu_k\,|x'^\nu_i,y'^\nu_i\rangle =\,-\imath \hbar \delta_{lk}|x^\nu_a,y^\nu_a\rangle &+\,\int_{TM}\,\tilde{\mu}_P(x',y')\,\tilde{\lambda}_l(x'^\nu_i,y'^\nu_i;x^\nu_a,y^\nu_a)\,\tilde{x}^\mu_k\,|x'^\nu_i,y'^\nu_i\rangle .
\begin{array}{c}{{\pi_{x}= p_{x}+ \frac{\lambda}{2}\left( x \left( x p_{x}+ y p_{y}\right) + \left( x p_{x}+ y p_{y}\right) x \right) ,}}\\{{\pi_{y}= p_{y}+ \frac{\lambda}{2}\left( y \left( x p_{x}+ y p_{y}\right) + \left( x p_{x}+ y p_{y}\right) y \right) .}}\\ \end{array}
\nabla^{2}=r^{1-n}\frac{\partial}{\partial r}r^{n-1}\frac{\partial}{\partial r}+r^{-2}\Delta_{S^{n-1}}=\frac{\partial^{2}}{\partial r^{2}}+\frac{n-1}{r}\frac{\partial}{\partial r}+r^{-2}\Delta_{S^{n-1}}
\frak R (x_1, x_2, x_3,v_1,v_2,v_3) &=\sum \limits_{j,k}\sum_{R=I\times J\times S\in \mathcal R^N_{\frak z}(j,k)}\int_{R}\psi_{j,k}(x_1-y_1, x_2-y_2, x_3-y_3) &\times(\psi_{j,k}(y_1-v_1, y_2-v_2, y_3-v_3) -\psi_{j,k}(x_I-v_1, x_J-v_2, x_S-v_3)) dy_1dy_2dy_3.
G_{\tilde\lambda}(l_1,l_2;t=1)=\langle l_2|\hat T|l_1\rangle ={\rm e}^{-\tilde\lambda (l_1+l_2)}\sum_{T:l_1\rightarrow l_2}1\equiv{\rm e}^{-\tilde\lambda (l_1+l_2)}\frac{1}{l_1+l_2}\Biggl({l_1 +l_2 \atop l_1}\Biggr).
\mathbb{S}_{f,g}= \frac{\sum_{w\in\mathfrak{S}_{n+1}}(-1)^{l(w)}\xi^{\langle 2\rho,w\bullet 0\rangle}}{d^n}\varepsilon(f)\varepsilon(g)(-1)^{\sum_{i=1}^{n}(k_{i-1}(f)+k_{i-1}(g))}S_{(\lambda_f,\mu_f),(\lambda_g,\mu_g)}.
g_{ij}=\begin{pmatrix}0 1 2 u^{3}1 -2 u^{3}u^{2}2 u^{3}u^{2}-4 u^{1}\end{pmatrix}, w_{12}= 0 ,w_{31}=-\frac{1}{\sqrt{u^{1}+u^{2}u^{3}+2 (u^{3})^{3}}},w_{23}=\frac{u^{3}}{\sqrt{u^{1}+u^{2}u^{3}+2 (u^{3})^{3}}}.
x_i^{(k+1)}= \frac{1}{1+\frac{h}{2}}\left[ -ha_{ii}^{-1}\sum_{j<i}a_{ij}x_j^{(k+1)}+ \left( 1-\frac{h}{2}\right) x_i^{(k)}- h a_{ii}^{-1}\sum_{j>i}a_{ij}x_j^{(k)}+ ha_{ii}^{-1}b_i \right].
=\sum_{R_{n}}b_{m}^{*}(R_{n})\int d^{3}r\varphi_{m}^{*}(r-R_{n})H_{at}(r)\psi_{m}(r)+\sum_{R_{n}}b_{m}^{*}(R_{n})\int d^{3}r\varphi_{m}^{*}(r-R_{n})\Delta U(r)\psi_{m}(r)
T^{L,\Xi,\widetilde{\mathbf{r}}}_{F,G,N}(f_1,\dots,f_{d}) := \frac{1}{N^{h-m}}\sum\limits_{\mathbf{n}\in \mathbb{Z}^h}\Big(\prod\limits_{j=1}^{d}f_j(\xi_j(\mathbf{n})+ \widetilde{\mathbf{r}}_j)\Big)F(\mathbf{n})G(L\mathbf{n}).
A =-\frac{\varphi(2)}{\Gamma_p(\frac{1}{2})}\sum_{j=0}^{p-2}\omega^j(x)~\pi^{(p-1)\ell_j}~\Gamma_p\left(\left\langle\frac{1}{4}-\frac{j}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac{3}{4}-\frac{j}{p-1}\right\rangle\right) &\times\Gamma_p\left(\left\langle\frac{1}{2}+\frac{j}{p-1}\right\rangle\right)\Gamma_p\left(\frac{j}{p-1}\right).
A\left( x\right) =\bigcap_{a\in G}B\left( T_{a}x,\frac{\lambda \delta\left( x\right)}{2}\right) , AA\left( x\right) =A\left( x\right) \cap \bigcap_{y\in A\left( x\right)}B\left( y,\frac{\lambda^{2}\delta \left( x\right)}{2}\right)
\left( \begin{array}{c}{{n_{e}}}\\{{n_{m}}}\\ \end{array}\right) \to \left( \begin{array}{c c}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{array}\right) \left( \begin{array}{c}{{n_{e}}}\\{{n_{m}}}\\ \end{array}\right) \ .
= \; C \; \times \exp \left( \sum_{b , b^{\prime}= 1}^{N}\sum_{j = 1}^{n_{b}}\sum_{j^{\prime}= 1}^{m_{b^{\prime}}}\tilde{\tilde{V}}( x_{j}^{( b )}- y_{j^{\prime}}^{( b^{\prime})}) \right)
V^{inst}= \frac{R}{2\pi}\sum_{n \ne 0}e^{in\theta_{\gamma_e}}K_0(2\pi R|nc|) A^{inst}= -\frac{R}{4\pi}\left( \frac{d c}{c}- \frac{d \overline{c}}{\overline{c}}\right) \sum_{n \ne 0}sign(n) \cdot e^{in\theta_{\gamma_e}}|c| K_1(2\pi R|nc|)
\sup_{k \le M}E(\eta_k) \le C_0, \sum_{k = 1}^M \tau \left\| \frac{\eta_k - \eta_{k - 1}}{\tau}\right\|_{L^2}^2 \le C_0, \sum_{k = 1}^M \tau R\left(\eta_{k - 1}, \tfrac{\eta_k - \eta_{k - 1}}{\tau}\right) \le C_0.
\begin{aligned}\| D_x u \|_{L_2 (Q_r)}\leq N (d, r) \delta^{-4}\sum_{k = 0}^{\infty}2^{-k}(|(-\Delta_x)^{1/3}u - ((-\Delta_x)^{1/3}u)_{Q_{1, 2^{k +2}}}|^2)_{Q_{1, 2^{k+2}}}^{1/2}, \end{aligned}
H^{(n)}(\xi) =K(\xi-d\Gamma(m))+d\Gamma(\omega)+\mu \varphi(v_n)\geq -\mu^2\lVert \omega^{-1/2}v_n \lVert^2\geq -\mu^2\lVert \omega^{-1/2}v \lVert^2 \Sigma_n(\xi) =\inf(\sigma(H^{(n)}(\xi)))
G_R(x,z;x',z')=-\int \frac{d^4 p}{(2\pi)^4}e^{ip_{\mu}(x^{\mu}-x'{}^{\mu})}\Biggl[\frac{e^{-2k(|z|+|z'|)}k}{{\bf p}^2-(\omega+i\epsilon)^2}+\int_0^{\infty}dm\, \frac{v_m(z) v_m(z')}{m^2+{\bf p}^2-(\omega+i\epsilon)^2}\Biggr],
A_{4F}=\int_{z_4}^{z_2}dz_3<O^{(1)}_{-\frac{1}{2}}(z_1)O^{(2)}_{-\frac{1}{2}}(z_2)V^{(3)}_{-\frac{1}{2}}(z_3)O^{(4)}_{-\frac{1}{2}}(z_4)>
\sup_{Q_{y}}N(\epsilon\lVert F\rVert_{L_{2}(Q_{y})}, \mathcal{F}, L_{2}(Q_{y}))=\sup_{Q_{y,x}}N(\epsilon\lVert F\circ\phi_{y}\rVert_{L_{2}(Q_{y,x})}, \mathcal{F}\circ\phi_{y}, L_{2}(Q_{y,x})),
G_{n+1}(X_{\delta}) =X_{\delta}X_{E_x^{n}}-X_{E_x^{(n-1)}}=(z+z^{-1})(z^n+z^{n-2}+\cdots +z^{-n})-(z^{n-1}+z^{n-3}+\cdots +z^{-(n-1)}) =z^{n+1}+z^{n-1}+\cdots t^{-n+1}+z^{-n-1}=X_{E_x^{(n+1)}}.
x_i \frac{\partial}{\partial x_i}: x^m \mapsto m_i \cdot x^m, i=1,...,n, & \frac{\partial e_{-\alpha}^{\lambda}}{\partial \lambda}_{\vert{\lambda = 0}}: x^m \mapsto ht_{-\alpha}(m) \cdot x^{m-\alpha}, \alpha \in R(N,\Sigma_{C(\Delta)}).
a_1(\nabla_X \bar{J})Y+a_2(\nabla_Y\bar{J})X+a_3(\nabla_{\bar{J}X}\bar{J})Y+a_4(\nabla_{\bar{J}Y}\bar{J})X+a_5(\nabla_{\bar{J}X}\bar{J})\bar{J}Y+ &+a_6(\nabla_{\bar{J}Y}\bar{J})\bar{J}X+a_7(\nabla_{X}\bar{J})\bar{J}Y+a_8(\nabla_{Y}\bar{J})\bar{J}X=0
0 = \frac{c_{\ell_0}}{nr_{0c_{\ell_0}}}\sum_{i \in I}\left ( \left ( -n + \frac{n}{c_{\ell_0}}\right ) r_{ic_{\ell_0}}+ \sum_{k \neq i}\frac{n}{c_{\ell_0}}r_{k c_{\ell_0}}\right ) = (-c_{\ell_0}+ |I|)\sum_{i \in I}\frac{r_{ic_{\ell_0}}}{r_{0c_{\ell_0}}}+ |I|.
\alpha_{( \cdot), (\cdot)}^{-1,1}= A_{( \cdot), (\cdot)}^{-1,1}+ \frac{1}{N^2}A_{(\cdot), (\cdot)}^{0,0}- \frac{1}{N}B_{(\cdot), (\cdot)}^{0,0}- \frac{1}{N}B_{(\cdot), (\cdot)}^{-1,1}.
\zeta(2n+1) = \frac{2 (-1)^n (2\pi)^{2n}}{(2n-1) 2^{2n}+ 1}\left( \sum_{1 \leq k < n}\frac{(-1)^{k-1}k}{(2n-2k+1)!}\frac{\zeta(2k+1)}{\pi^{2k}}+ \sum_{k \geq 0}\frac{(2k)!}{(2n+2k+1)!}\frac{\zeta(2k)}{2^{2k}}\right)
{\bf m}(t)=\ \Phi(t,0) m_0+ \int_0^t \Phi(t, s_1) h(s_1) ds_1 & -\gamma \int_0^t \Phi(t, s_2) \int_{s_2}^T \Psi(s_2,s_1)[K(s_1)h(s_1)+ (I-\Gamma)^T Q\eta] ds_1 ds_2 .
(3\alpha_1^2-1,\ldots, 3\alpha_k^2-1) + \sum_{i=1}^{\hat{k}}\mu_i (-1,\ldots,-1) + \lambda (1,\ldots,1)=0, \sum_{i=1}^{\hat{k}}\alpha_i=n, \alpha_1,\ldots, \alpha_{\hat{k}}\ge 1, \mu_1,\ldots, \mu_{\hat{k}}\ge 0, \sum_{i=1}^{\hat{k}}\mu_i \alpha_i=0
\langle f | g \rangle_{{\cal L}^{1 | 2}}= \langle f_{0}| g_{0}\rangle_{\cal L}^{s}+ \langle f_{1}| g_{1}\rangle_{\cal L}^{s + 1 / 2}+ \langle f_{2}| g_{2}\rangle_{\cal L}^{s + 1 / 2}+ \langle f_{3}| g_{3}\rangle_{\cal L}^{s + 1}\, ,
\begin{array}{r c l}{\psi_{2 , 1}}&{\cong}&{\rho_{1 , 1}= L_{- 1}\psi_{1 , 1}}\\{\phi_{2 , 1}}&{\cong}&{\phi_{1 , 1}^{\prime}= L_{- 1}\phi_{1 , 1}}\\{\xi_{2 , 1}}&{\cong}&{\phi_{1 , 1}= \frac{1}{2}L_{1}\rho_{1 , 1}.}\\ \end{array}
\sum_{i=m(l_0)}^\infty k_i(x)/n_i \le 4^{-l_0}/12 + \sum_{l=l_0+1}^\infty a(l)/n_{m(l)}\le 4^{-l_0}/12 + \sum_{l=l_0+1}^\infty 4^{-l-1}\le 4^{-l_0}/12 + 4^{-l_0}/12 < 4^{-l_0}/5.
x=\frac{1}{2 \pi r}(\alpha+2m+\ell), y=\frac{1}{2 \pi r}(\alpha+2m+2n-\ell), &u=\frac{1}{2 \pi r}(2\beta-\alpha+2n+\ell),v=-\frac{1}{2 \pi r}(\alpha+\ell+2 k),
a_\alpha^0 a_\alpha^0 = 0 &a_\alpha^0 a_{\alpha \beta}^1 = a_{\alpha \beta}^1 a_\beta^0 &a_{\alpha \gamma}^1 - a_{\alpha \beta}^1 a_{\beta \gamma}^1 =a_\alpha^0 a_{\alpha \beta \gamma}^2 + a_{\alpha \beta \gamma}^2 a_\gamma^0 .
\textrm{E}= 4 \pi \int_{0}^{\infty}d r r^{2}[{( \omega + e A_{0})}^{2}{\sigma}^{2}e^{- \nu}+ U + e^{- \lambda}{\sigma^{\prime}}^{2}+ ( 1 / 2 ){A_{0}^{\prime}}^{2}e^{- \nu - \lambda}] .
|t_m(n)| =|t_m(2n')|=\left|\sum_{j=0}^{\lfloor\frac{m}{2}\rfloor}{m\choose 2j}t_m(n'-j)\right|\leq\sum_{j=0}^{\lfloor\frac{m}{2}\rfloor}{m\choose 2j}|t_m(n'-j)| & <\sum_{j=0}^{\lfloor\frac{m}{2}\rfloor}{m\choose 2j}m(n')^{\frac{m}{2}}=m(2n')^{\frac{m}{2}}=mn^{\frac{m}{2}}.
(\tau_{i,j}\Pi )_{k,l}:= \begin{cases}\Pi_{k,l}, \displaystyle\frac{(\Pi_{i,j-1}+\Pi_{i-1,j})\Pi_{i+1,j}\Pi_{i,j+1}}{\Pi_{i,j}(\Pi_{i+1,j}+\Pi_{i,j+1})}.\end{cases}
\{h:x_i-y_1\leq h\leq x_{i}-1\}\cup \{h:x_i+y_i-1\leq h\leq 2x_{i}-3\}& \cup \bigcup_{j=1}^{i-1}\left(\{h:x_i-y_{j+1}\leq h\leq x_{i}-x_j-1\}\cup \{h:x_i+y_{j}-1\leq h\leq x_{i}+x_j-2\}\right)
\mathrm{\Large ~ \c h i ~}({C^{\prime}}) = X^{n - 4}( X^{2}-{\lambda_{1}^{\prime}}^{2}) ( X^{2}-{\lambda_{2}^{\prime}}^{2}) = X^{n}- ({\lambda_{1}^{\prime}}^{2}+{\lambda_{2}^{\prime}}^{2}) X^{n - 2}+{\lambda_{1}^{\prime}}^{2}{\lambda_{2}^{\prime}}^{2}X^{n - 4}.
(\begin{matrix}-c^{2}&c^{2}&b^{2}-a^{2}\\ -b^{2}&c^{2}-a^{2}&b^{2}\\ 1&1&1\end{matrix})\lambda=(\begin{matrix}d_{A}^{2}-d_{B}^{2}\\d_{A}^{2}-d_{C}^{2}\\ 1\end{matrix}).
\|u\|_{\widetilde{L}^q(\widetilde{B}^{s_1,s_2})}:=\sum_{j=-\infty}^{j_0}2^{js_1}\Vert \dot{\Delta}_j u\Vert_{L^q_T(L^2)}+ \sum_{j=j_0+1}^{+\infty}2^{js_2}\Vert \dot{\Delta}_j u\Vert_{L^q_T(L^2)}.
\sum_{\ell=0}^{k-1}\left[(f+g)(x_*)-(f+g)(x^{\ell+1})\right]\ge\frac{1}{2\beta}\left(\|x^{k}-x_*\|^2-\|x^0-x_*\|^2+2[(f+g)(x^{k})-(f+g)(x^{0})]\right) &\ge\frac{1}{2\beta}\left(\|x^{k}-x_*\|^2-\|x^0-x_*\|^2+2\left[(f+g)(x_*)-(f+g)(x^{0})\right]\right).
|H(s;\phi,k,l,\alpha)|\ll |k|^\varepsilon \alpha^\varepsilon l^{\frac{1}{2}+\varepsilon}(l,k_2^2)^{\frac{1}{2}}\sum_{\substack{h_0\mid \frac{k\cdot \textrm{rad}k}{(l,k)}p\mid h_0\Rightarrow p\mid (l,k_2)}}\sum_{\substack{h\mid k (h,l)=1}}|A(h_0h,1)|,
\mathbf{Q}_3=\left(\begin{array}{cc}a_1 \underline{a}_1 a_2 \underline{a}_2 b_1 \underline{b}_1 b_2 \underline{b}_2\end{array}\right),\mathbf{Q}_4=\left(\begin{array}{cc}a_1 0 a_2 0 0 b_1 0 b_2\end{array}\right)
[f(s|D|),x]-sf'(s|D|)\delta(x) =-s^2\int_{-\infty}^{\infty}\Big(\int_0^1\Big(\int_0^1\widehat{f''}(u)ve^{ius(1-vw)|D|}\delta^2(x)e^{iusvw|D|}dw\Big)dv\Big)du.
\int_{\tilde{\gamma}}\widetilde{d}G_{\Omega}(\cdot,z') =\int_{\gamma}\widetilde{d}G_{\Omega}(\cdot,z')+\int_{\gamma_2\left(\gamma_1\right)^{-1}}\widetilde{d}G_{\Omega}(\cdot,z') &=\int_{\gamma}\widetilde{d}G_{\Omega}(\cdot,z')+2\pi &=h(z').
f_k(\lambda_1,\dots,\lambda_n,\tilde{v},\tilde{w})=-\frac{1}{m|g|}\sum_{h,i=0}^{m-1}\sum_{\substack{t=1 (t\neq k)}}^n[\tilde{v}_i\tilde{w}_i]_{k,t}[\tilde{v}_{i-h-1}\tilde{w}_{i-h-1}]_{t,k}\frac{\lambda_t^{m-h-1}\lambda_k^h}{\lambda_t^m-\lambda_k^m}.
W_{E H}+ W_{c t}= -{\frac{4 ( d^{2}- 1 )}{d^{2}( d + 2 )}}\left( ( a - 1 )^{2}+ ( b - 1 )^{2}-{\frac{2}{( d - 1 )}}( a - 1 ) ( b - 1 ) \right) r^{d - 2}\Sigma_{d - 1}~ ~ .
P(x^m)P(x^{n+k\theta(n)}) =\beta(m)\beta(n+k\theta(n))x^{\theta(m)+\theta(n+k\theta(n))}&=\frac{1}{k+1}\beta(m)\beta(n)x^{\theta(m)+(k+1)\theta(n)}, (\hbox{by Eq.~\eqref{thetam+kthetam}and \eqref{betam+kthetam}})
\mathbb{E}^V \theta(N \zeta_{\alpha^{\prime}}(\mu)\zeta_{\alpha^{\prime}}(\nu))= \mathbb{E}^V \theta(\frac{N \eta}{\pi}\int_I \frac{\zeta_{\alpha^{\prime}}(\mu)\zeta_{\alpha^{\prime}}(\nu)}{(E-\lambda_{\alpha^{\prime}})^2+\eta^2}\mathcal{X}(E) dE )+o(1).
\psi_\rho^d=\sum_{l=0}^\infty\underbrace{\frac{\rho^d l^{d+1}}{l+n-1}\cdot e^{-\rho l}\cdot\frac{\lambda+l}{\lambda}}_{\widehat{\psi_\rho^d}(l)}\,\mathcal C_l^\lambda=\frac{\sqrt{\Gamma(2d)}}{2^d}\cdot\sum_{l=0}^\infty\frac{l}{l+n-1}\cdot \widehat{\Psi_\rho^d}(l)\,\mathcal C_l^\lambda .
\left(M\otimes_{\alpha_{X,Y}^{}}\left(F/L\right)^* \right)^* \stackrel{1}{=}\mathcal{L}_{X;Y^{\rm dual}}\left(M; \left(F/L \right)^{**}\right) \stackrel{1}{=}\mathcal{L}_{X;Y^{\rm dual}}\left(M;F/L \right).
g'''_0(x,a,b)=a\bigl(2F^{0,4}+12a^2F^{1,5}+24a^4F^{2,6}+16a^6F^{3,7}\bigr), &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots &g_0^{(l)}(x,a,b)=a\sum_{j=0}^lc_{j,l}a^{2j}F^{j,l+j+1}(x;a,b),