harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(z-1)^n\frac{(1+\xi_t(z))^{2n}}{\xi_t^n(z)}= (z-1)^n\sum_{k = -n}^n\binom{2n}{n+k}\xi_t^k & = \binom{2n}{n}(z-1)^n + \sum_{k = 1}^n\binom{2n}{n+k}\left\{\frac{(z-1)^{n+k}}{(1+z)^k}e^{ktz}+ (z-1)^{n-k}(1+z)^ke^{-ktz}\right\}& = \binom{2n}{n}(z-1)^n + \sum_{k \in [-n..n]\setminus \{0\}}\binom{2n}{n+k}\frac{(z-1)^{n+k}}{(1+z)^k}e^{ktz}
K\dfrac{\varepsilon^2h_{00}^{m+1}}{(2W_0)^{2m}}{h^i}_l =^{h}{{{R_0}^i}_{0l}}+ 2\Phi^i_{\parallel l}- \Phi^i_{l \parallel 0}+ 2\Phi^r{{\Phi_r}^i}_l - \Phi^r_l\Phi^i_r.
f\left(\frac{\partial}{\partial p_i}\right)=\left(g^{ij}-\frac{p^ip^j+y^iy^j}{F^2+K^2}\right)\frac{\partial}{\partial y^j}+\frac{p^iy_j-y^ip_j}{F^2+K^2}\frac{\partial}{\partial p_j}
I_{\beta \mu \lambda \rho}( q ) = \int d^{d}k \frac{2 k_{\beta}k_{\mu}( q_{\nu}p_{\rho}+ q_{\nu}q_{\rho}) + k_{\beta}( q_{\nu}q_{\rho}p_{\mu}- q_{\nu}q_{\mu}p_{\rho})}{( k + p )^{2}( k - q )^{2}k^{2}}\; .
\Psi_{k,l,r}=(-1)^{r_{k}-1}\sum_{i\in\Omega_{k,l}}\prod_{j\ne k}(\begin{matrix}i_{j}+r_{j}-1\\i_{j}\end{matrix})(\frac{1}{\sigma_{j}^{2}}-\frac{1}{\sigma_{k}^{2}})^{-(r_{j}+i_{j})},
(\boldsymbol{v},\boldsymbol{q}) = (\boldsymbol{y}- \frac{1}{\sqrt{\alpha}}\boldsymbol{p}- \frac{1}{\sqrt{\alpha}}\boldsymbol{p}^\perp, \boldsymbol{p}+ \sqrt{\alpha}\boldsymbol{y}- \sqrt{\alpha}\boldsymbol{y}^\perp)
\delta_{\epsilon}B = -{\frac{1}{2}}\bar{\epsilon}\Gamma_{M N}\theta ( d X^{M}d X^{N}+{\frac{2}{3}}\bar{\theta}\Gamma^{M}d \theta d X^{N}+{\frac{1}{1 5}}\bar{\theta}\Gamma^{M}d \theta \bar{\theta}\Gamma^{N}d \theta )
{\omega^\beta}_{\beta\mu}= \frac{\partial_\beta\partial_\mu D(n^{(\beta)}||p_{\beta,\mu}) - \partial_\beta\partial_\mu D(n||p_{\beta,\mu})}{\partial_\beta D(n^{(\beta)}||p_{\beta,\mu})}&= \frac{\sum_i n_i^{(\beta)}- \sum_i n_i}{\partial_\beta D(n^{(\beta)}||p_{\beta,\mu})}&= 0.
\Big( V_{g_{{}_0}}W \Big)_{q}= \Big( U^{-1}U^{L}_{g_{{}_0}}U W \Big)_{q}= \sqrt{\lambda(q, g_{{}_0})}L_{{}_{h'_{{}_{q,g_{{}_0}}}}}W_{{}_{q'_{{}_{q,g_{{}_0}}}}};
f_{\Sigma,\mu}(x)= &\det(2\pi\boldsymbol\Sigma)^{-\frac{1}{2}}\, \exp \left( -\frac{1}{2}(\mathbf{x}- \boldsymbol\mu)^{{{\!\mathsf{T}}}}\boldsymbol\Sigma^{-1}(\mathbf{x}- \boldsymbol\mu) \right).
|\nabla_s(t^{\frac{d}{2}}u(tx))|_2^2 =\int_{\mathbb{R}^d\times \mathbb{R}^d}\frac{t^d|u(tx)-u(ty)|^2}{|x-y|^{d+2s}}dxdy &=t^{2s}\int_{\mathbb{R}^d\times \mathbb{R}^d}\frac{|u(tx)-u(ty)|^2}{|tx-ty|^{d+2s}}d(tx)d(ty) &=t^{2s}|\nabla_su|_2^2.
Z_{S D M}[ 0 ] ={\cal N}\int{\cal D}a_{\mu}e x p - \int \Bigl ({\frac{m}{2}}a_{\mu}a^{\mu}-{\frac{1}{2}}a_{\mu}\epsilon^{\mu \nu \rho}\partial_{\nu}a_{\rho}) \Bigr ) d^{3}x .
K =(a^2,b^2,c^2,d^2,abx,cdx), L =(a^2,b^2,c^2,d^2,ab,cd), V_1 =H^2+H(a,b)(c,d)+ab(c^2,d^2)+(a^2,b^2)cd, V_2 =H^2+H(a,c)(b,d)+ac(b^2,d^2)+(a^2,c^2)bd.
\limsup_{| x | \rightarrow \infty}| x |^{\min\{\zeta,n\}- 2 +k}\left| D^k\left( u ( x ) - \left( \frac{1}{2}x^{\prime}A x + \beta \cdot x + \gamma \right) \right) \right| < \infty,\ \ \forall k=0,1,\cdots,m+1.
L_{V_{\alpha}}\beta_+ - (k+1) \lambda_{\alpha}\beta_+ = \frac{1}{2}\left((-1)^{k+1}\beta_0 \cdot \alpha_+ - \alpha_+ \cdot \beta_0 +(-1)^{k+1}\beta_{\mp}\cdot \alpha_+ + \alpha_+ \cdot \beta_{\mp}+ \beta_+ \cdot \alpha_0 - \alpha_0 \cdot \beta_+ \right) &+ k \cdot \alpha_{\mp}\beta_+ -(k+1) \cdot \alpha_{\mp}\beta_+.
z_j^-\Omega_\lambda= \begin{cases}d_{j}^-(\lambda)\prod_{\ell=j+1}^{n}(\lambda_j-\lambda_\ell-j+\ell)\Omega_{\lambda-\epsilon_j}, \lambda- \epsilon_j\in \mathbb{P}\ell(\lambda - \epsilon_j)\leq p, 0, .\end{cases}
\left(P^2-m^2 -\frac{e}{2}\sigma^{\mu\nu}F_{\mu\nu}\right)\phi(x)=0\;,\;\;\;F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\;,\;\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]\;.
\mathbf{\mu}_7^{(k+1)}= \mathbf{\Sigma}_7^{(k+1)}\left[ \begin{array}{c}\tilde{K}_1 \tilde{K}_2 + \mathbf{V}_{\mathbf{x}_{k}}^{\downarrow -1}\mathbf{m}_{\mathbf{x}_{k}}^{\downarrow}\end{array}\right].
\begin{aligned}\mathcal{g}^{\Delta^{(s)}}(V,W) := \mathcal{W}( \nabla^{(s)}_{V}\nabla^{(s)}_{W}(\Delta^{(s)})^{-n-2}) = 2^{m}\mathcal{g}^{\Delta}(V,W), \mathcal{G}^{\Delta^{(s)}}(V,W) := \mathcal{W}( \nabla^{(s)}_{V}\nabla^{(s)}_{W}(\Delta^{(s)})^{-n}) = 2^{m}\mathcal{G}^{\Delta}(V,W).\end{aligned}
\sigma^{+}= \ \left( \begin{array}{c c}{0}&{1}\\{0}&{0}\\ \end{array}\right) \ , \ \sigma^{-}= \ \left( \begin{array}{c c}{0}&{0}\\{1}&{0}\\ \end{array}\right) \ , \ \sigma^{0}= \ \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) \ .
0 =\left( \left( \mathbf{I}+\bar{K}(\nu,0)\right) r_{0},r_{0}\right) & \ +\left( \left( \bar{K}(\nu,\varepsilon)-\bar{K}(\nu,0)\right) \left[Z^{\perp}(\nu,\varepsilon)+\mathbf{I}\right] r_{0},r_{0}\right) ,
\mathbb P\begin{bmatrix}\begin{Vmatrix}{\frac 1k}\sum_{s\in{\hat S}}{\frac 14}f\cdot\tau(s)\end{Vmatrix}_{L^2}\leq{\frac 14}||f||_{L^2}\end{bmatrix}\geq 1-{\frac{2^{\frac{n+4}2}}{n!\eta}}\exp\begin{pmatrix}{\frac{-k}{16\ln 2}}\end{pmatrix}
\partial_{t}(\tau N) =\partial_{t}(\partial_{s}B) &= \partial_{s}(\partial_{t}B) &= \partial_{s}\Big[-\kappa_{s}T+F(s,t)N\Big] &= -\kappa_{ss}T-\kappa_{s}\kappa N+F_{s}(s,t)N-\kappa F(s,t) T+\tau F(s,t) B &= \Big[-\kappa_{ss}-\kappa F(s,t)\Big]T - \Big[\kappa_{s}\kappa+F_{s}(s,t)\Big]N+\tau F(s,t)B.
\int_{\{\Psi<0\}}|f|^2_he^{-\Psi}= \int_{\{\Psi<0\}}|f|^2_he^{-\Psi+a\Psi}+\int_{0}^{+\infty}\left(\int_{\{\Psi<-\frac{l}{a}\}}|f|^2_he^{-\Psi+a\Psi}\right)e^{l}dl \ge \left(1+\int_0^{+\infty}e^{-\frac{-1+q'}{a}l}dl\right)\frac{1}{K_{\Psi,f,h,a}(z_0)}= \frac{a+q'-1}{q'-1}\cdot\frac{1}{K_{\Psi,f,h,a}(z_0)}
K_{11}= 2c\phi_2+c(\tau-2)\phi_1^2, K_{12}= 4c\phi_3 + 4 c^2\phi_1\phi_2 +2c(\tau-2)\phi_1(c\phi_1^2+\phi_2) K_{22}= 8c\phi_4 +4c^2\phi_2^2 + 16 c^2\phi_1\phi_3 +8 c^3\phi_1^2\phi_2+4c(\tau-2)(c\phi_1^2+\phi_2)^2.
X_{1}=~\{x, T^{*3}x ,\ldots, T^{*3(3j-n+1-r)}x, y, T^{*3}y ,\ldots, T^{*3(3j-n-2r)}y\}X_{2}=~\{T^{*3(3j-n+2-r)}x ,\ldots,T^{*3j}x, T^{*3(3j-n-2r+1)}y ,\ldots, T^{*3(j-1-r)}y, z,\ldots, T^{*3(n-2j-2+r)}z\}.
\mathbb{S}(R^{\bigstar})=\frac{2}{d^{n+1}}\left[\frac{d_1d_2\dots d_n}{n+1}+ \frac{W^{(n)}_1}{n}+\dots+\frac{W^{(n)}_s}{n-s+1}+\dots+\frac{W^{(n)}_{n-1}}{2}\right]
Z_{\alpha}^{- 1}d ( P_{\mu}) Z_{\alpha}= \alpha^{- 1}d ( P_{\mu}) ; \hspace{. 3 c m}Z_{\alpha}^{- 1}d ( \hat{P}_{\mu}) Z_{\alpha}= \alpha \hspace{. 1 c m}d \hspace{. 1 c m}( \hat{P}_{\mu})
I_{\tiny \mathrm{r e n}}\equiv I ( G_{\tiny \mathrm{r e n}}, \Lambda_{\tiny \mathrm{r e n}}, c_{\tiny \mathrm{r e n}}^{i}) = I ( G_{B}, \Lambda_{B}, c_{B}^{i}) + W_{\tiny \mathrm{d i v}}, \hspace{0 . 5 c m}W_{\tiny \mathrm{r e n}}= W - W_{\tiny \mathrm{d i v}}.
\begin{array}{rcl}f(x)\!\!\! \!\!\!>\!\!\! \!\!\!Q_{12}(x)\!=\!8 x^2\!-\!1\!+\!\underline T_{10}^{\cos,0}(4x)\!+\!8x^2 \underline T_{10}^{\cos,0}(2x)\!+\!P_7(x) \overline T_1^{\sin,0}(2x),\end{array}
g(\rho,\theta;\vec{u};\vec{p}) = \prod_{m=1}^s (\rho e^{i \theta}- p_m) \; \prod_{\alpha=1}^r \frac{- u_\alpha \rho e^{i\theta}}{(\rho e^{i\theta}- u_\alpha) (q_1 q_2 \rho e^{i\theta}- u_\alpha)}.
\begin{aligned}y^* \in \mathcal N_{\mathcal T_D(g(\bar x))}(\nabla g(\bar x)u)\cap\ker\nabla g(\bar x)^*, z^* \in \mathcal T_{\mathcal N_{\mathcal T_D(g(\bar x))}(\nabla g(\bar x)u)}(y^*),\end{aligned}
{\cal W}^{(1)}=\frac{1}{2\,{\cal Z}^{(1)}}\sum_{n_1\neq n_2}\frac{(n_1-n_2-1)}{(n_1-n_2)}\exp\left [-\frac{g^2 A}{2}(n_1^2+n_2^2)+\frac{g^2 A}{2}(n_1+n_2-\frac{1}{2})-g^2A_1n_1\right].
\begin{bmatrix}u_{1,k}^1 u_{2,k}^1\end{bmatrix}=-\begin{bmatrix}D_1^{11}-I -I D_2^{11}\end{bmatrix}^{-1}\begin{bmatrix}C_1^1 0 0 C_2^1\end{bmatrix}\begin{bmatrix}x_{1,k}x_{2,k}\end{bmatrix}.
_{2}\phi_{2}\left(b,bq;c,cq;q^{2},\frac{c^{2}}{b^{2}}\right)=\frac{\left(c/b;q\right)_{\infty}}{\left(c;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}}{\left(q;q\right)_{n}}\left(\frac{c}{b}\right)^{n}A_{q}\left(cq^{n-1}\right)
N(f)=q^{m-1}+\tau\eta(-1)^{\frac{r-1}{2}}\eta(h-c)q^{m-\frac{r+1}{2}}, \mbox{if $r$ is odd, $q^{r-1}+\tau\eta(-1)^{\frac{r-1}{2}}\eta(h)q^{\frac{r-1}{2}}$ times.}
\int_{\Sigma \cap \{|x_{n+1}| \leq L\}}e^{-\frac{|x|^2}{4}}\, |\nabla^\Sigma u|^2 &\leq C \int_{\Sigma \cap \{|x_{n+1}| \leq L\}}e^{-\frac{|x|^2}{4}}u^2 + C L^{-1}\int_{\Sigma \cap \{|x_{n+1}|=L\}}e^{-\frac{|x|^2}{4}}\, u^2.
\hat I_t^{-1}=\hat I_{t-1}^{-1}-\hat I_{t-1}^{-1}X_{t-m}^{t-1}[1+(X_{t-m}^{t-1})^T\hat I_{t-1}^{-1}X_{t-m}^{t-1}]^{-1}(X_{t-m}^{t-1})^T \hat I_{t-1}^{-1},
\|y^{k+1}-z\|^2 =(1+\varphi)\|x^{k+1}-z\|^2-\varphi\|x^{k}-z\|^2+\varphi(1+\varphi)\|x^{k+1}-x^k\|^2 &=(1+\varphi)\|x^{k+1}-z\|^2-\varphi\|x^{k}-z\|^2+\frac{1}{\varphi}\|y^{k+1}-x^k\|^2.
\Big(-2^{-n}A^{1,0}a_0\Big)_0 = -2^{m-n}n^m P_0(m) \Big(-2^{-n}A^{1,0}a_0\Big)_1 = 2^{1+m-n}n^{1+m}P_1(m) \Big(-2^{-n}A^{1,0}a_0\Big)_2 = -\Big(2^{2+m-n}n^{2+m}P_2(m) + 2^{m-n}n^m P_0(m) m \alpha_p(3)\Big).
T_h(\lambda)(u)({x}) =\frac{\eta_0^1}{4\pi}\int_{B_1}\frac{\exp({i\sqrt{\lambda}h|x-y|)}}{|x-y|}u(y)dy +\frac{\eta_0^2}{4\pi}\int_{B_2}\frac{\exp({i\sqrt{\lambda}h|x-y|)}}{|x-y|}u(y)dy
K_{i}\equiv \frac{i}{2}\partial_{k}g_{i \bar{j}}\bar{\chi}^{\bar{j}\beta}\chi_{\beta}^{k}\qquad \qquad \bar{K}_{\bar{j}}\equiv - \frac{i}{2}\partial_{\bar{l}}g_{i \bar{j}}\bar{\chi}^{\bar{l}\beta}\chi_{\beta}^{i}
\big[u,\partial_t u\big]= \sum_{|k|\leq N_0}\exp(\lambda_k t)\sum_{p=0}^{m_k-1}c_{k,p}\sum_{l=0}^p\frac{t^{p-l}}{(p-l)!}W_{k,l}+\sum_{|n|> N_0}c_{n}\exp(\lambda_n t)\varphi_{n}\,.
\Phi_{- \frac 1 2 + i \varepsilon , m , \bar{m}}\approx \left( \frac 1{2 i \varepsilon}+ \frac \phi{\alpha_{+}}\right) \gamma^{- \frac 1 2 + i \varepsilon - m}\bar{\gamma}^{- \frac 1 2 + i \varepsilon - \bar{m}}e^{- \frac \phi{\alpha_{+}}}+
\log \frac{1}{\|X_0(\varphi(re^{i\theta}))\|}= \log \sqrt{1 + \sum_{i=1}^n|\varphi_i(re^{i\theta})|^2}\le \sum_{i=1}^n\log^+|\varphi_i(re^{i\theta})| + \log(\sqrt{1+n})
\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}= \frac{1}{\tau_{n}}\exp\left\{-(n-1)\beta^2 + \beta J\right\}\exp\left\{- \frac{\beta}{\sqrt{n}}\sum_{i=1}^{n}A_{i,i}-\beta J'\right\}Z_{n}(\beta),
\int_{B_{N}}\prod_{i=1}^{2}\|f_{i}\|_{L^2({B_{N^{{1}/{2}}}}(x))}^{\frac{s}{2}}\, dx \lesssim_{\epsilon,\nu}N^{\epsilon+\frac{3s}{4}}\prod_{i=1}^{2}\|f_{i}\|_{L^2(\mathbb{R}^{6})}^{\frac{s}{4}}\prod_{i=1}^{2}\|f_{i}\|_{L^{s,N}(\mathbb{R}^{6})}^{\frac{s}{4}}.
\dfrac 1 2 \|v_n(s_n)\|_{H^1_0(\Omega)}^2 = J(v_n(s_n)) + \dfrac{\lambda_m}m \|v_n(s_n)\|_{L^m(\Omega)}^m \leq J(v_n(0)) + \dfrac{\lambda_m}m \|v_n(s_n)\|_{L^m(\Omega)}^m \stackrel{\eqref{c:vn:Lm}}{\to}J(\phi) + \dfrac{\lambda_m}m \|\phi\|_{L^m(\Omega)}^m = \dfrac 1 2 \|\phi\|_{H^1_0(\Omega)}^2
q(1)\langle \partial_x (\chi_\lambda(x/t)^3(\phi'(x/t))^2e^{it\phi(x/t)}),\mathbf{u}^v_\lambda \rangle =q(1)\langle (\chi_\lambda(x/t)\phi'(x/t))^3e^{it\phi(x/t)},\mathbf{u}^v_\lambda \rangle &+q(1)t^{-1}\langle (3\chi_\lambda(x/t)^2\chi_\lambda'(x/t)(\phi'(x/t))^2+2\chi_\lambda(x/t)^3\phi'(x/t)\phi''(x/t)e^{it\phi(x/t)}),\mathbf{u}^v_\lambda \rangle,
\mathrm{Spec}(U|_\mathcal{L})\setminus \{\pm 1\}= \{e^{\pm i{\arccos \mu}}| \mu\in \mathrm{Spec}(T)\setminus\{\pm 1\}\}, \mathrm{ker}(e^{\pm i{\arccos \mu}}I -U|_\mathcal{L}) = (d_A^\ast - e^{\pm i{\arccos \mu}}d_B^\ast )\mathrm{ker}\left(\mu I-T\right).
i \bullet j = \begin{cases}-1 \textrm{if}i \textrm{or}j \in I' \textrm{and}|i-j|=1 1 \textrm{if}i \textrm{or}j \in I'' \textrm{and}|i-j|=1 2 \textrm{if}i \in I' \textrm{and}i=j -2 \textrm{if}i \in I'' \textrm{and}i=j 0 \textrm{if}|i-j| >1 0 \textrm{if}i=j=m.\end{cases}
J_4=\int_0^t\!\int_{\Gamma_{t}(2 \delta)}(\zeta \circ d_\Gamma)\mathbf{w}|_\Gamma\theta_{0}^{\prime}(\rho)(\mathbf{n}-\varepsilon \nabla_\tau h_{\varepsilon})(\mu_{\varepsilon}-\mu_A) \,\mathrm{d}x\,\mathrm{d}\varsigma \triangleq\; J_{41}+J_{42},
NLP = \int_0^T \int \langle D \rangle^{N-1}u_0^2 \langle D \rangle^{N-1}\partial_y \Delta^{-1}\left[ \partial_i (u^j_0 \partial_j u^i_0) + \partial_i( u^j_{\neq}\partial_j u^i_{\neq})_0 \right] & = NLP_0 + NLP_{\neq}.
\times \exp \left\{- \frac{g_{m}^{2}}{2}\left[ \int d^{3}x d^{3}y \rho ({\bf x}) D_{0}({\bf x}-{\bf y}) \rho ({\bf y}) - \sum_{{a , b = 1 \atop a \ne b}}^{N}D_{m}({\bf z}_{a}-{\bf z}_{b}) \right] \right\}.
R_f^-(\hat k)= K_f^-(\hat k)_{class}\left[1-\frac{i\beta^2}{16\pi}\sinh\theta \left( \frac{1}{\cosh\theta+1}-\frac{1}{\cosh\theta}\right) +\frac{i \beta^2}{8\pi}\frac{\theta}{\cosh\theta}\right].
\hat{E}_{k;n}(\hat{S})=\sum_{j_1=0}^{n}\sum_{j_2=0}^{n-j_1}...\sum_{j_{{N_\rho}-1}=0}^{n-\sum_{r=1}^{{N_\rho}-2}j_{r}}\tilde{E}_{k;j_1,j_2,...,j_{{N_\rho}-1},n-\sum_{r=1}^{{N_\rho}-1}j_{r}}\biggl(\prod_{d=1}^{{N_\rho}-1}\rho_{d}^{j_d}\biggr) \rho_{N_\rho}^{n-\sum_{r=1}^{{N_\rho}-1}j_{r}}.
A(x,1) =\frac{(v''-1)(v'-1)(v'v''-v'-v''+2)x^2}{(x-1)v'v''+v'+v''-x-1}, B(x,1) =\frac{-(xv''-v''+1)(xv'-v'+1)(xv'+xv''-3x+1)}{(x-1)v'v''+v'+v''-x-1},
\begin{array}{rl}\theta_6 + \theta_{18}- \theta_{-18}- \theta_{-6}= -i ~q^{-{1\over 12}}{\eta^3\over \eta(\tau/2) \eta(2\tau)}\theta_{-2}+ \theta_{10}- \theta_{22}- \theta_{-14}= q^{-{1\over 12}}{\eta^3\over \eta(\tau/2) \eta(2\tau)}. \end{array}
\frac{\partial^\gamma}{\partial t^\gamma}u(x,t)=\begin{cases}\frac{1}{\Gamma(1-\gamma)}\int\limits_0^t \frac{\partial u(x,s)}{\partial s}(t-s)^{-\gamma}ds,0<\gamma <1 \frac{\partial u(x,s)}{\partial s},\qquad\qquad\qquad\qquad\qquad\gamma=1.\end{cases}
\tilde{\cal H}= \frac{1}{2}\pi_{a}^{2}+ \frac{1}{2}{\sigma^{\prime}}_{a}^{2}- ( \sigma_{a}\pi^{a}) \theta + \frac{1}{2}\lambda \bigl ( \sigma_{a}^{2}- 1 \bigr ) + \frac{1}{2}\sigma_{a}^{2}\theta^{2}.
\nabla^\alpha\mathfrak{C}_{\alpha\beta}{_{|\mathcal{I}^0}}=0,\;\left(\partial_0 (R_{ij}-\Lambda_{ij})-\overline{\nabla}_{i}\mathfrak{C}_{j0}-\overline{\nabla}_{j}\mathfrak{C}_{i0}\right){_{|\mathcal{I}^0}}=0.
\int_{\Pi^{\xi}}\int_{\mathbb{R}}\int_{\mathbb{R}}g(z+s\xi,z+t\xi)|t-s|^{n-1}dsdtd\mathcal{H}^{n-1}(z) & =\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}g(y-\tau\xi,y)|\tau|^{n-1}d\tau dy~.
|\zeta-\zeta_0+z| = \big||\zeta-\zeta_0|\mathrm{e}^{\mathrm{i}\arg(\zeta-\zeta_0)}- |z|\mathrm{e}^{\mathrm{i}\theta}\big| = \big||\zeta-\zeta_0|- |z|\mathrm{e}^{\mathrm{i}\theta- \mathrm{i}\arg(\zeta-\zeta_0)}\big| \geqslant |z|\cdot|\sin(\theta-\arg(\zeta-\zeta_0))| \geqslant |z|\sin\phi_1.
\left[-\frac{1}{N^2}\,\lambda_i\,\frac{\partial^2}{\partial \lambda_i^2}-\frac{1}{N^2}\,\sum_{j\neq i}\lambda_j\,\frac{1}{\lambda_j-\lambda_i}\,\left(\frac{\partial}{\partial \lambda_j}- \frac{\partial}{\partial \lambda_i}\right) +\frac{1}{N}\,(2\eta -1)\,\frac{\partial}{\partial \lambda_i}+1\right]Z(\lambda )=0.
\Psi_{2}(1/X, 1/Y) = Y^{-3}- \left (\frac{x_{1}x_{2}+ x_{1}x_{3}+ x_{2}x_{3}}{x_{1}x_{2}x_{3}}\right ) Y^{-2}+ \left ( \frac{x_{1}+ x_{2}+ x_{3}}{x_{1}x_{2}x_{3}}\right )Y^{-1}- \frac{1}{x_{1}x_{2}x_{3}}.
\hat{F}_{\mu \nu}^{i j}( z ) = \int_{T^{4}}d^{4}x \int_{T^{4}}d^{4}x^{\prime}\, \psi^{i}{}^{\dagger}( x ; z ) \sigma_{\mu}( D_{z}^{\dagger}D_{z})^{- 1}( x , x^{\prime}; z ) \sigma_{\nu}^{\dagger}\psi^{j}( x ; z ) - [ \mu \leftrightarrow \nu ] .
\lbrack (\bar P+\bar K)_{\bar l}, \chi^{i\alpha}\rbrack =\lbrack (\bar P+\bar K)_{\bar l}, \bar \chi_{i}^{\alpha}\rbrack = 0\qquad\qquad\{\chi^{i\alpha}, \bar\chi_{k\beta}\}= \delta^i_k \delta^{\alpha}_{\beta}
\sup\limits_{B_{\frac{\delta R}{2}}(\bar x^\prime)\cap \partial\Omega}u_{\beta\beta}\le \epsilon \sup\limits_{B_{\frac{3\delta R}{4}}(\bar x^\prime)\cap \Omega}|D^2u| + C_\epsilon (1+ \sup\limits_{B_{\frac{3\delta R}{4}}(\bar x^\prime)\cap \partial\Omega}\sup\limits_{|\tau|=1,\tau\cdot \nu=0}|u_{\tau\tau}|) + \frac{C}{(\delta R/ 4)^2},
P^{(1)}(z) = \begin{cases}E_{in}^{(1)}(z) \Phi_{\alpha}(n^3 f(z))D_0^n(z) \begin{pmatrix}1 0 0 0 z^{\beta}0 0 0 z^{\beta}\end{pmatrix}, z \in D(0,r_n) E_{out}^{(1)}(z) N(z) = N(z), z \in A(0; r_n, R),\end{cases}
\left| \nabla_y^\beta \partial_{z_d}q_\lambda(y',y_d,z_d)\right| \leq \frac{Ce^{-c|\lambda|^{\frac12}z_d}}{(y_d+z_d+|y'|)^{d-1+\beta}}\Big ( |\lambda|^\frac12 + \frac{1}{y_d+z_d+|y'|}\Big ).
\int_0^r s_3^{-(2N-2)}\Big( \int_{0}^{s_3}s_2^{2N-2}\Big( \int_0^{s_2}s_1^{-(2N-2)}\Big(\int_{B(x, s_1)}U_{N-k}(y)dy \Big) ds_1\Big) ds_2 \Big) ds_3 = r^{-(2N-2)}\int_{\partial B(x, r)}(-\Delta)^{N-k-2}u d\sigma + \omega_{2N-1}(-\Delta)^{N-k-2}u(x) & + \frac{\omega_{2N-1}}{2 (2N-1)}(-\Delta)^{N-k-1}u(x) r^2.
S_b = \sum_{f_b \in \mathcal F_{|b|}}(-1)^{|f_b^{-1}(2)|}ck^{|b|-|f_b^{-1}(1)|+2 \mathbf{1}_{f_b \equiv 1}}&= ck^{|b|}\left[ k^{2-|b|}- k^{-|b|}+ \sum_{f_b \in \mathcal F_{|b|}}(-1)^{|f_b^{-1}(2)|}k^{-|f_b^{-1}(1)|}\right] &= ck^{|b|}\left[ k^{2-|b|}- k^{-|b|}+ \left( \frac{1}{k}-1\right)^{|b|}\right] &= c\left[ (1-k)^{|b|}+ k^2-1 \right].
{\tilde U}_1^{t^0_1,y_1^0,y_2^0,\delta}:=\{(x_1,y_1): 0\le y_1-y^0_1<\rho, 0\le x_1-t^0_1+y_1(y_1-y_2^0)<\rho^2\delta\},{\tilde U_2}^{x^0_2,y_2^0,\delta}:=\{(x_2,y_2): 0\le y_2-y_2^0< \rho(1\wedge\delta),\,0\le x_2-x^0_2+y_2^0(y_2-y_2^0)< \rho^2\delta \}.
V = M^{2}- \gamma^{k}M_{, k}+ \frac 1 4 \bar{R}+{\frac{1}{1 6}}F^{j k}F_{j k}- \left( e A_{0}+ \frac i 8 F \right)^{2}+{\frac{e}{2}}({\cal F}- 2 \breve{\gamma}^{0}\gamma^{k}\nabla_{k}A_{0}) ~ .
(n-\delta-N) R_n(x) - (n+\gamma+1) R_{n+1}(x) =- \frac{(N-x)(x+\gamma+\delta+N+1)}{N}\hat R_{n}(x), (n+1) \hat R_{n}(x) -(n-N+1) \hat R_{n+1}(x) = N R_{n+1}(x).
g_a^m(x)=\frac{a^m}{\Sigma_n}\sum_{l=0}^{m+1}l!\left(\alpha_l^m+\frac{\alpha_l^{m+1}}{\lambda}\right)e^{-al}\,\frac{C_l^\lambda(\cos\chi)}{|x-r\hat{e}|^{l+2\lambda}},
D_c:=\left [\begin{array}{ccccc}I_{m}0 0 0 0 0 \Sigma_c 0 0 0 0 0 ({\widehat \Theta_c})_{(\alpha,\alpha)}0 0 0 0 0 ({\widehat \Theta_c})_{(\beta,\beta)}0 0 0 0 0 2I_{|\alpha||\beta|}\end{array}\right ],
L_{\Sigma}=10log_{10}(\frac{p_{1}^{2}+p_{2}^{2}+...+p_{n}^{2}}{p_{0}^{2}})dB=10log_{10}[(\frac{p_{1}}{p_{0}})^{2}+(\frac{p_{2}}{p_{0}})^{2}+...+(\frac{p_{n}}{p_{0}})^{2}]dB.
\eta_j=\min_{\substack{V\subset \mathcal{H}^2_{0,D}(\Omega){\rm dim}V=j}}\max_{\substack{v\in V u\ne 0}}\frac{\mathcal Q_{\sigma}(v,v)}{\int_{\partial\Omega}\left(\frac{\partial v}{\partial\nu}\right)^2d\sigma},
\begin{array}{| c | c | c | c |}\hline{{\mathrm{o r d e r}}}&{{N_{c}= 2}}&{{N_{c}= 3}}&{{N_{c}\longrightarrow \infty}}\\ \hline{{\epsilon}}&{{0 . 5 3 9}}&{{0 . 5 0 5}}&{{0 . 4 8 0}}\\ \hline{{\epsilon^{2}}}&{{0 . 6 2 0}}&{{0 . 6 0 1}}&{{0 . 5 8 5}}\\ \hline{{\epsilon^{3}}}&{{1 . 0 3}}&{{0 . 9 3 3}}&{{0 . 8 8 0}}\\ \hline{{}}\\ \end{array}
P_{DH}(p)=\prod_{\Phi_{Q^u}\cup \Phi_s^+}\frac{\kappa(\alpha,p)}{\kappa(\alpha,\rho)}=\prod_{\Phi_{Q^u}\cup \Phi_s^+}\frac{\kappa(\alpha,\alpha)}{2\kappa(\alpha,\rho)}p(\alpha^{\vee})
\tilde \Psi_{n_1,n_2}(r,\alpha,\theta,\beta)=\frac{F_{k,\ell,j,m,n_1,n_2}(r)}{r^{5/2}}\frac{G_{\ell,j,m,n_1,n_2}(\alpha)}{\sin^2 \alpha}\frac{\Theta_{j,m,n_1,n_2}(\theta )}{\sin\theta^{3/2}}\frac{H_{m,n_1,n_2}(\beta)}{\sqrt{\sin 2 \beta}}\ .
\frac{d^2R_{n\ell}(s)}{dr^2}+\frac{2}{s}\frac{dR_{n\ell}(s)}{dr}+\left[\frac{\left(\frac{2\mu E}{\hbar^2}\right)s^2+\left(\frac{2\mu Ze^2}{\hbar^2}\right)s+(-\ell(\ell+1))}{s^2}\right]R_{n\ell}(s)=0,
{\frac{d^{2}x^{\rho}}{d \lambda^{2}}}+ \Gamma_{\mu \nu}^{\rho}{\frac{d x^{\mu}}{d \lambda}}{\frac{d x^{\nu}}{d \lambda}}+{\cal W}^{- 1}g^{\rho \sigma}\partial_{y}({\cal W}g_{\sigma \mu}){\frac{d x^{\mu}}{d \lambda}}{\frac{d y}{d \lambda}}= 0 ,
\lambda ( p^{\mu}) \, \equiv \left( \begin{array}{c}{{\left( \zeta_{\lambda}\, \Theta_{[ j ]}\right) \, \phi_{_L}^{\ast}( p^{\mu})}}\\{{\phi_{_L}( p^{\mu})}}\\ \end{array}\right) \, \, , \quad \rho ( p^{\mu}) \, \equiv \left( \begin{array}{c}{{\phi_{_R}( p^{\mu})}}\\{{\left( \zeta_{\rho}\, \Theta_{[ j ]}\right)^{\ast}\, \phi_{_R}^{\ast}( p^{\mu})}}\\ \end{array}\right) \, \, .
A_1 = R_1[x_1,y_1]/\bigl(x_1y_1 -(b_1c_1), (x_1)^2 -(-b_1d_1+a_1x_1+b_1y_1), (y_1)^2 -(-b_1c_1+c_1x_1+d_1y_1)\bigr).
\nabla\varphi=-\frac{q}{4\pi\epsilon_{0}}\frac{1}{|r-r_{s}|^{2}(1-n_{s}\cdot\beta_{s})^{3}}[n_{s}(1-{\beta_{s}}^{2}+(r-r_{s})\cdot\dot{\beta}_{s}/c)-\beta_{s}(1-n_{s}\cdot\beta_{s})]
(I_\rho \otimes T)\sigma_{\varrho, \rho}=\rho^{\gamma_0}(T) V^*\varrho(V)=V^*(\rho_{\widetilde \gamma})^{\gamma_0}(T) \varrho(V)=V^*T\varrho(V)=V^*\varrho'(V)T=\sigma_{\varrho', \rho}(T \otimes I_\rho).
\begin{aligned}\int_{u_0}^{2 T - u_0}\int_{S^2}(L(r \phi))^2 d \omega d v = \int_{u_0}^{2 T - u_0}\int_{S^2}r^2 (L \phi)^2 + L(r \phi^2) d v = \int_{u_0}^{2 T - u_0}\int_{S^2}r^2 (L \phi)^2 d v + \int_{S^2}r \phi^2 (2 T - u_0,u_0,\omega) d \omega - \int_{S^2}r \phi^2 (u_0,u_0,\omega) d \omega. \end{aligned}
(K_{il}^{X_{3}^{c},0}+K_{il}^{X_{2}^{c},0}) =\,V_{X_2}(-(i+1)A-(l+1)B)+\,V_{X_3}(-(d-l+1)B) &= \,V_{X_2}(-(i+1)A-lB)-1+\,V_{X_3}(-(d-l+1)B) &=\,K_{il}^{X_1,0}-1.
T_{n_v}E_{o_-}(w) \in\left(\sum_{c\in W_{0,P},\mu\in \Lambda(1)}C[q_s^{\pm 1}]E_{o_-}(\mu)T_{n_c}\right)\cap \mathcal{H}& = \left(\sum_{c\in W_{0,P},\mu\in \Lambda(1)}C[q_s^{\pm 1}]E_{o_-}(\mu n_c)\right)\cap \mathcal{H}& = \sum_{c\in W_{0,P},\mu\in \Lambda(1)}C[q_s]E_{o_-}(\mu n_c).
\mathbf{E}\left[\left\|\mathcal{X}^{t}-\mathcal{X}^{\star}\right\|_{F}^{2}|\mathcal{X}^{0}\right]\leq\left(1-\mathop{\min}_{k=1,2,\cdots,l}\frac{\lambda_{\min}\left(\widehat{\mathcal{A}}_{(k)}\widehat{\mathcal{A}}_{(k)}^{H}\right)}{\|\widehat{\mathcal{A}}_{(k)}\|_{F}^{2}}\right)^{t}\|\mathcal{X}^{0}-\mathcal{X}^{\star}\|_{F}^{2}.
| \boldsymbol{i}| = n( \ell) = n( \ell - 1) + n (\ell) - n( \ell -1) \leq n( \ell -1) + y_k \preccurlyeq k / \log (1/ r_{\sup}) + c_3 \log k \preccurlyeq k / \log (1/ r_{\sup}) .
\alpha = \sqrt{\frac{N_{k}}{\hbar \tau_{2}}}\left[ - i \tau \hat{A}_{1}+ i \hat{A}_{2}\right] , \alpha^{\dagger}= \sqrt{\frac{N_{k}}{\hbar \tau_{2}}}\left[ i \bar{\tau}\hat{A}_{1}- i \hat{A}_{2}\right] .
\lim_{n\rightarrow \infty}\left( (\eta_0,w_0), S^*(\tau_n)(\varphi_{\tau},\psi_{\tau})\right)_{\overline{X}_2}=\left( (\eta_0,w_0), S^*(\tau)(\varphi_{\tau},\psi_{\tau})\right)_{\overline{X}_2}.
\sum_{l=1}^M\sum\limits_{\substack{j=1 j\neq l}}^M\mathbb{E}\left[y_l^* y_j|\hat{\mathbf{H}}\right]=\sum_{l=1}^M\sum\limits_{\substack{j=1 j\neq l}}^M\left(a_c^2\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}+\sum_{q=1}^M a_q^2\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}\right)
R_{2}^{1}= - 2 i \frac{d g_{1}\wedge d \overline{{{g}}}_{1}}{( 1 - g_{1}\overline{{{g}}}_{1})^{2}}, R_{4}^{3}= + 2 i \frac{d g_{2}\wedge d \overline{{{g}}}_{2}}{( 1 + g_{2}\overline{{{g}}}_{2})^{2}},
\left< u^{n+1}, \varphi v \right> - \Delta t \left< \frac{\left(u^{n+1/2}\right)^{2}}{2}, (\varphi v)_{x}\right> + \Delta t \left< \mathcal{H}\left(u^{n+1/2}\right)_{x}, (\varphi v)_{x}\right> = \left< u^{n}, \varphi v \right>,
[\mathbf{H}_n,\mathbf{H}_m]=\mathbf{k}n\delta_{n+m,0}\quad,\quad [\mathbf{H}_n,\mathbf{E}^{\pm}_m]=\pm\sqrt 2\mathbf{E}^{\pm}_{m+n}\quad,\quad [\mathbf{E}^+_n,\mathbf{E}^-_m]=\mathbf{k}n\delta_{n+m,0}+ \sqrt2\mathbf{H}_{n+m}\quad
{\cal E}_{\mathrm{l o g}}= - V ( t )^{2}\int \! \frac{\mathrm{d^{3}}k}{( 2 \pi )^{3}}\frac{1}{1 6 ( \omega_{k}^{0})^{3}}= - \frac{\lambda^{2}\mu^{\epsilon}}{2 5 6 \pi^{2}}\left( \phi ( t )^{2}- \phi ( 0 )^{2}\right)^{2}\left\{\frac{2}{\epsilon}+ \ln \frac{4 \pi \mu^{2}}{m_{0}^{2}}- \gamma \right\}\; .
\lfloor\frac{m}{n}\rfloor+\lfloor\frac{2m}{n}\rfloor+...+\lfloor\frac{(n-1)m}{n}\rfloor=\lfloor\frac{n}{m}\rfloor+\lfloor\frac{2n}{m}\rfloor+...+\lfloor\frac{(m-1)n}{m}\rfloor.
\begin{array}{r c l l}{{{\cal C}_{+}}}&{{=}}&{{\sigma_{2}\otimes \sigma_{1}\otimes \sigma_{2}\otimes \sigma_{1}\otimes \ldots \qquad}}&{{\eta = + 1 \nonumber}}\\{{{\cal C}_{-}}}&{{=}}&{{\sigma_{1}\otimes \sigma_{2}\otimes \sigma_{1}\otimes \sigma_{2}\otimes \ldots \propto{\cal C}_{+}\Gamma_{*}\qquad}}&{{\eta = - 1 \, .}}\\ \end{array}
\tau_{k}^\star = \frac{\sqrt{1 + \iota_{k}}\tilde{\mathbf{h}}_{k}^H\mathbf{w}_{k}}{\sum_{p \in \mathcal{K}}|\tilde{\mathbf{h}}_{k}^H\mathbf{w}_{p}|^2 + \sigma_{k}^2}, \forall k \in \mathcal{K}.
\hat{p}(X_1,\ldots, X_i, X_i', \ldots, X_j, X_j', \ldots,X_m) & = p_1 + p_2(X_i+X'_i+X_j+X'_j) + p_{4}(X_i+X'_i)(X_j+X'_j) + p_{5}(X_i+X_j)(X'_i+X'_j)
S_C^{o,h}(i_{\kappa},\kappa,j) = \frac{w_{j,i_{\kappa}}}{h}\sum\limits_{k\in[h]}\Bigl(\pi_{i_{\kappa},k}^{o,\kappa,h}(n_{\kappa}) n_{\kappa}+\pi_{i_{\kappa},k}^{o,\kappa,h}(n_{\kappa}+1)\bigl(n_{\kappa}+1\bigr)\Bigr)= C_j^0 - \frac{o(h)}{h},
\omega_{k,n}= \frac{\omega_k}{\sum_{j=n+1}^\infty \omega_j}= \begin{cases}\frac{\tilde{\omega}_k}{\sum_{j=n+1}^\infty \tilde{\omega}_j}~ k \ge 1, \frac{\omega_0}{(1-\omega_0) \sum_{j=n+1}^\infty \tilde{\omega}_j}~ k =0 .\end{cases}
\psi_{\varphi\varphi\pi\varphi\ldots\pi_N}(\stackrel{\rightharpoonup}{x_1},\ldots, \stackrel{\rightharpoonup}{x_N})\rightarrow (-1)^{\#(\pi-{\rm indices})}\psi_{\varphi\varphi\pi\varphi\ldots\pi_N}(-\stackrel{\rightharpoonup}{x_1},\ldots, \stackrel{\rightharpoonup}{x_N}) \ \ \ .