harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
a_{\beta,2}={\pi \over 3\alpha}(1-\alpha^2) \int_{\Sigma}\left[\frac 16 R +{1+\alpha^2 \over 60 \alpha^2}(2R_{\mu\nu\lambda\rho}n^{\mu}_in^{\lambda}_in^{\nu}_jn^{\rho}_j-R_{\mu\nu}n^{\mu}_i n^{\nu}_i) \right]~~~.
\Phi(z,s,a)\Gamma(s)=\sum_{n=0}^{\infty}\frac{z^{n}}{(n+a)^{s}}\int_{0}^{\infty}x^{s}e^{-x}\frac{dx}{x}=\sum_{n=0}^{\infty}\int_{0}^{\infty}t^{s}z^{n}e^{-(n+a)t}\frac{dt}{t}
K \varphi (\xi) \ = \ \int\limits_{\mathbb R^d}a (\xi-q) \mu (\xi, q) \varphi (q) \dq \ = \ \int\limits_{\mathbb T^d}\hat a (\xi-\eta) \mu (\xi, \eta) \varphi(\eta) \d\eta, \varphi \in L^2(\mathbb T^d),
\overline{g}(\mathrm{div}^{\nabla'}(T_{r}),Y)=\sum_{j=1}^{r}\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{R}(T_{r-1}Z_{i},\widehat{W})(-\mathcal{A}_{\widehat{W}})^{j-1}Y,Z_{i}).
\sum\limits_{n = -\infty}^{\infty}(-1)^{n}q^{n(3n + 1)/2}\cdot \sum\limits_{n = 0}^{\infty}d_{e}(n)q^{n}= \sum\limits_{n = -\infty}^{\infty}(-1)^{n}q^{2n(3n + 1)}\sum\limits_{n = 1}^{\infty}\sigma_{0}(n)q^{4n}.
\overline{F}_\epsilon(u)=\frac{{\epsilon}^{2s-1}}{2}\int_{\Omega}{\int_{\Omega}{\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy}}+\frac{1}{\epsilon}\int_{\Omega}{\overline{W}(u)\,dx}
\Omega^{(\alpha\beta)}_{jk}\,=\, \frac{2 R^{(\alpha\beta)}_j R^{(\alpha\beta)}_k - (R^{(\alpha\beta)})^2 \delta_{jk}}{(R^{(\alpha\beta)})^4}\;\;\;,\;\;\; R^{(\alpha\beta)}\equiv R^{(\alpha)}-R^{(\beta)}\;.
u_{\underline{{m}}}^{\underline{{a}}}\equiv ( 1 / 2 ( u_{\underline{{m}}}^{+}+ u_{\underline{{m}}}^{-}) , u_{\underline{{m}}}^{i}, 1 / 2 ( u_{\underline{{m}}}^{+}- u_{\underline{{m}}}^{-}) ) \; .
\alpha_A (\varphi_0) \simeq \alpha_3 (\varphi_0) +{\partial\widehat\sigma \over \partial\varphi_0}\left({B\over M}\right)_A+{\partial\widehat\delta \over \delta\varphi_0}\left({D\over M}\right)_A+ a_3{\partial\alpha\over \partial\varphi_0}\left({E\over M}\right)_A\ ,
a_{\bf i}= \left(\odot_{\ell=1}^d \overline{\bf u}^{(\ell)}_{i_\ell}\right) \xi^T + \sum_{\nu \in{\cal L}({\bf i})}c_\nu{\bf U}_\nu({\bf i}), \mbox{at the expense}O(d R + 2 d \gamma R_0).
\frac{1}{2 \pi}\mathcal{P}\! \! \int_{- \infty}^{\infty}\! d p_{\perp}^{\prime}\frac{1}{p^{2}+ p_{\perp}^{2}+ M^{2}}= \frac{\Theta ( p^{2}+ M^{2})}{2 \sqrt{p^{2}+ M^{2}}}
B^{N}(t) =B(t_{k-1}^N)+\frac{\Delta_N B_k}{\Delta_N}(t-t_{k-1}^N)t^N_{k-1}\le t\le t^N_k, \Delta_N B_k =B(t_{k}^N)-B(t^N_{k-1}),\Delta_N=\frac{T}{N},t_k^N=\frac{kT}{N}.
\tilde{\gamma}:=\gamma_{|\mathcal{A}}+(\pi^{1},P_{\Psi,\tau}\circ\pi^{1})_{\#}\bigg(\gamma_{|\overline{\Omega}\times\overline{\Omega}\backslash\mathcal{A}}\bigg)+(P_{-\Psi,\tau}\circ\pi^{2},\pi^{2})_{\#}\bigg(\gamma_{|\overline{\Omega}\times\overline{\Omega}\backslash\mathcal{A}}\bigg).
|u_k|^{\alpha(x)}|v_k|^{\beta(x)}-|\tilde{u}_{k}|^{\alpha(x)}|\tilde{v}_{k}|^{\beta(x)}\leqslant \big[\varepsilon|\tilde{v}_{k}|^{\alpha(x)}+ c(\varepsilon) |u|^{\alpha(x)}\big]|v_k|^{\beta(x)}&+ |\tilde{u}_{k}|^{\alpha(x)}\big[\varepsilon|\tilde{v}_{k}|^{\beta(x)}+c(\varepsilon)|v|^{\beta(x)}\big].
I [ \, \phi \, ] \, = \,{\frac{1}{2}}\int d^{n + 1}x \, \partial_{\mu}\Big ( ( x^{0})^{- n + 1}\, \phi \partial_{\mu}\phi \Big ) \, - \,{\frac{1}{2}}\int d^{n + 1}x \, \phi \, \partial_{\mu}\Big ( ( x^{0})^{- n + 1}\, \partial_{\mu}\phi \Big ) \, .
|J_{5}|\leq\frac{\epsilon}{N}\stackrel[i=1]{N}{\sum}\mathrm{trace}((|x_{i}^{k}|^{2}+|u^{k}|^{2}(t,x_{i}))R_{N}(t))\leq2\mathcal{E}_{1}(t)+\epsilon||u||_{L^{\infty}L^{\infty}}^{2}.
\frac{1}{x}\frac{\partial \psi(x, S, Q)}{\partial x}= \frac{2 \ln (2x) + \frac{1}{2}S^2 + 2 Q}{x}\le \frac{4x - 2 + \frac{1}{2}S^2 + 2 Q}{x}\le 4+\frac{S^2 + 4 Q-4}{\exp\left(\frac{1}{4}S^2 - Q\right)}\: .
d_{ABC}\,h^Ah^Bh^C = 3\Big\{h^1\,\big(h^2\big)^2 -h^1\,\big(h^\mu\big)^2 -h^2\,\big(h^i\big)^2+\gamma_{\mu ij}\,h^\mu\, h^i\,h^j \Big\}\ .
\textstyle{\int}\, dx \, (\sigma \, C_2)^{\ast}\, C_1\vert_{y=-\infty}^{y=+\infty}= \textstyle{\int}\, d \tilde{x}\, dy \, (p_y \, C_2)^{\ast}\, C_1\vert_{q=-q_0}^{q=+q_0}= 0 \; .
D_1^{BS,(p\to p)}(\lambda) =-(2\lambda-n-2)(\lambda-p)D_1^{(p\to p)}(n-\lambda), D_2^{BS,(p\to p)}(\lambda) =-(2\lambda-n-4)(\lambda-n+p-1)(\lambda-p-1)_2D_2^{(p\to p)}(n-\lambda), D_3^{BS,(p\to p)}(\lambda) =3(2\lambda-n-6)(2\lambda-n-4)(\lambda-p-2)_3(\lambda-n+p-2)_2 D^{(p\to p)}_3(n-\lambda).
| \rho \cdot \alpha |_{\mathrm{m i n}}+ | \rho \cdot \alpha |_{\mathrm{m a x}}< h^{\vee}, \qquad | \rho^{\vee}\cdot \alpha |_{\mathrm{m i n}}+ | \rho^{\vee}\cdot \alpha |_{\mathrm{m a x}}< h .
\langle{T^{j}_{\k}}\rangle-\langle T^{\,j}_{_{\ominus}k}\rangle\sim{\pi M^2\over 2}\Phi^2\Big( (2c^{-2}-c_{_{\rm S}}^{-2}) u^j u_k -\gamma^j_{_k}\Big)\,\ln \Big\{{w_{\!_\odot}\over w}\Big\}\ .
D(G_n(W)\|G_n(W')) \le 8\int_{\ell\in[0,1]^n}\sum_{i,j=1}^n (W({\ell_i},{\ell_j})- W'({\ell_i},{\ell_j}))^2d\ell &=8\sum_{i,j=1}^n \int_{\ell\in[0,1]^n}(W(\ell_i,\ell_j)- W'(\ell_i,\ell_j))^2 d\ell &\le 8n^2\int_{[0,1]^2}(W(x,y)- W'(x,y))^2dxdy &= 8n^2\|W-W'\|_2^2
\int_{0}^{\infty}\partial_y f(0,z,t,0)\tilde{J}(0,z)dz=\tilde{j}(-t)\int_{-\infty}^{0}\mathbb{P}[\tilde{S}^{\downarrow}(0) \in dz \vert \tilde{S}^{\downarrow}(-t)=0]=\tilde{j}(-t)
\sum_{i=1}^{n}\left[\frac{(p+x_{i})(x_{i}-x_{0}-1)}{(1+2x_{i}) (p-x_{0}-n-1)}\prod_{j\neq i}^{n}\mathbf{f}(x_{i},x_{j})+\frac{(p-x_{i}-1)(x_{i}+x_{0}+2)}{(1+2x_{i}) (p-x_{0}-n-1)}\prod_{j\neq i}^{n}\mathbf{h}(x_{i},x_{j})\right] = n\,.
{\cal J}F_\varepsilon(\vartheta,\mu,y^I,z^I)=\left[\begin{array}{cccc}\displaystyle \sum_{i=1}^p [y^I]_i\nabla^2c_i(x) A^T{\cal J}c(x)^T 0 begin{align*}4pt]A I 0 0 begin{align*}4pt]{\cal J}c(x) 0 I -I begin{align*}4pt]0 0{\cal J}_{y^I}\Psi_{\varepsilon}(y^I,z^I){\cal J}_{z^I}\Psi_{\varepsilon}(y^I,z^I).\end{array}\right].
U_m^{(\ell)}(t)=\sum_{k=0}^{\min[m-2,\ell]}\binom{\ell}{k}\left(\partial_t^k u_{m-k}(t)\right)\ ,V_m^{(\ell)}(t)=\sum_{k=0}^{\min[m-2,\ell]}\binom{\ell}{k}\left(\partial_t^k v_{m-k}(t)\right)\ .
\underset{\tau\in(0,\infty)}\sup~ \bigl(\frac{1}{(1+\tau)^n}-e^{-n\tau}\bigr)=\frac{1}{(1+\tau_n)^n}-e^{-n\tau_n}=\frac{\tau_n}{(1+\tau_n)^{n+1}}\le \frac{\tau_n}{(n+1)\tau_n}\le \frac{1}{n+1}.
\sum_{i=1}^{n-1}(v^2-v^{-2})^2\left(v^{4n+2-4i}F_iK_{\varpi_{i+1}}E_iK_{\varpi_i}+ v^{-4n+2+4i}F_iK_{-\varpi_i}E_iK_{-\varpi_{i+1}}\right)+ (v-v^{-1})^2(F_nK_{-\varpi_n}E_n+v^2F_nE_nK_{\varpi_n})+ c^2v^2F_n^2K_{-\varpi_n}E_n^2K_{\varpi_n}.
{\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}{\cal D}_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}{\cal D}_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}={\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}D_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}D_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}\, ,
B_{t}^{(n)}(\omega)=\begin{cases}\sum_{i=0}^{2^{n}-1}F_{i}^{t,(n)}\left(\omega_{(i+1)2^{-n}t}-\omega_{i2^{-n}t}\right), 0<t\leq1, 0,\vphantom{\sum_{i=0}^{2^{n}-1}}t=0.\end{cases}
\big[ a_{ij}\big] + \big[ b_{ij}\big] = \big[ (a_{ij}+b_{ij}) \bmod{p^{i}}\big], \big[ a_{ij}\big] \cdot \big[ b_{ij}\big] = \left[ \left( \sum_{k=1}^{m}a_{ik}b_{kj}\right) \bmod{p^{i}}\right].
g_{s}^{1 / 2}\, d s^{2}= H_{2}^{- 5 / 8}H_{6}^{- 1 / 8}\eta_{\mu \nu}d x^{\mu}d x^{\nu}+ H_{2}^{3 / 8}H_{6}^{7 / 8}( d r^{2}+ r^{2}d \Omega ) + V^{1 / 2}H_{2}^{3 / 8}H_{6}^{- 1 / 8}d s_{K 3}^{2}
G^{(n),1}_m(s):=p^{(n)}\left(s_m\right), G^{(n),5}_{m}(s;x,y) :=\sum_{i\in\mathcal{I}_{m}}\sum_{j\leq i}d^{(n)}_{ij}\left(s\right)f_i(x)f_j(y), &G^{(n),3}_m(s):=r^{(n)}\left(s_m\right), G^{(n),6}_{m}(s;x,y) :=\sum_{i\in\mathcal{I}_{m}}\mu_i^{(n)}\left(s\right)f_i(x).
(-1)^n\frac{\widehat c_{n,\rho,q}^{(k)}(z)}{n!}=(-\rho)^n\underbrace{\int_0^1\cdots\int_0^1}_k\binom{(-x_1\cdots x_k+z)/\rho}{n}d_q x_1\dots d_q x_k &=\rho^n\underbrace{\int_0^1\cdots\int_0^1}_k\binom{(x_1\cdots x_k-z)/\rho+(n-1)}{n}d_q x_1\dots d_q x_k\,.
D_{ij}v=\frac{D_{ij}w_{11}}{w_{11}}-\frac{D_iw_{11}D_jw_{11}}{w^2_{11}}+\eta'(D_{ik}uD_{jk}u+D_kuD_{ijk}u)+\eta''D_kuD_{ik}uD_{l}uD_{jl}u+bD_{ij}\phi \leq 0,
E_{\hat{S}_1}-E_{S_1^*}=E_{W_t}+E_{W_t,S_1^*}, E_{\hat{S}_1,\hat{S}_3}-E_{S_1^*,S_3^*}=E_{W_t,S_3^*}-E_{W_t}-E_{W_t,S_1^*}-{|Q_t|}-d_{t,S_1^*}, E_{\hat{S}_1,V_1}-E_{S_1^*,V_1}=E_{W_t,V_1}, E_{\hat{S}_2,V_1}-E_{S_2^*,V_1}=d_{t,V_1}.
\mathbb{P}[S^{\downarrow}(t) \in dy ]=\frac{\mathbb{P}[\tau_{0}=\infty \vert S(t)=y]}{\partial_x \mathbb{P}[\tau_{0}=\infty \vert S(s)=x]_{|x=0}}\partial_x f(\omega_1,0,t,y)dy
\Pr_{\rho\sim P_\epsilon}[ \rho\neq \tilde{D}_{1}(H\rho^\intercal)] \leq 2e^{-\frac{l^2}{3\epsilon N}}+(2l+1)\mathop{\max}_{\substack{w,w'\in\{\epsilon N\pm l\}w'\leq w}}\left\{\Pr_{\rho\sim \lambda_{w,w'}}\big[ |\Omega_\rho^{\{w,w'\}}|>1\big||\rho|=w \big]\right\}.
\frac{1}{81\xi(\mathcal{R}_H(\mathbf{x}))}\leq\frac{x_1^{(1)}x_2^{(1)}}{3\xi(\mathcal{R}_H(\mathbf{x}))}=\frac{1}{3x_3^{(1)}}\leq\frac{x_3^{(n+1)}}{x_3^{(1)}}=\prod\limits_{k=1}^{n}\frac{x_3^{(k+1)}}{x_3^{(k)}}\leq 2^n,
A_{\mu \nu \gamma}\Pi^{\mu \mu^{\prime}}\Pi^{\nu \nu^{\prime}}\Pi^{\gamma \gamma^{\prime}}\left( i \epsilon_{\mu^{\prime}\nu^{\prime}\gamma^{\prime}\alpha}( X \partial^{\alpha}X^{\dagger}- X^{\dagger}\partial^{\alpha}X ) \phi^{N}\right)
\operatorname*{l i m}_{x_{0}\to 0}{\frac{x_{0}^{d - 2 \nu}}{( x_{0}^{2}+ |{\bf x}-{\bf x}^{\prime}|^{2})^{d - \nu}}}= c_{d , \nu}\delta^{( d )}({\bf x}-{\bf x}^{\prime}) ,
e^{\frac 4 3 \phi}\simeq \frac{\left( 3 \, r_{3}^{2}+ 1 0 \, r_{3}\, r_{4}+ 3 \, r_{4}^{2}\right)^{2}\, \left( 5 \, r_{3}^{2}+ 6 \, r_{3}\, r_{4}+ 5 \, r_{4}^{2}\right)}{1 2 8 \, \left( r_{3}+ r_{4}\right)}\, \, \sqrt{u_{0}+ 2 \, \kappa \, | u |^{2}}\; \kappa \, x + O ( x^{2}) \, ,
\partial f_1 = e_2^{-1}e_4^{-1}e_1,\hskip 5mm\partial f_2 = e_3^{-1}e_5^{-1}e_2,\hskip 5mm\partial f_3 = e_1^{-1}e_6e_3,\hskip 5mm\partial f_4 = e_4e_5 e_6^{-1}.
\frac{1}{4}(\alpha_\eta \beta_{1-\eta'}+\alpha_{1-\eta}\beta_{\eta'})(1+\cos x) (x-y)^2 &= \frac{1}{4}\left( -\left(\frac{\sin (\eta - \frac{1}{2})x}{\sin \frac{1}{2}x}\right)^2 + \left( \frac{\cos (\eta - \frac{1}{2}) x}{\cos \frac{1}{2}x}\right)^2 \right) \cos^2(\frac{1}{2}x) (x-y)^2 + O(|x-y|^2(|\eta - \eta'|+ |x-y|) ).
65346534=2\times 3^3\times 11^2\times 73\times 137 =594\times 110011 =6534\times(10^4+1) &653406534=2\times 3^3\times 11^3\times 9091 =594\times 1100011 =6534\times(10^5+1) &6534006534=2\times 3^3\times 11^2\times 101\times 9901 =594\times 11000011 =6534\times(10^6+1)
\int_B \exp(\alpha_0 |t_j M_j|^{\frac n{n-1}}) \geq \int_B \exp(\alpha_n |M_j|^{\frac n{n-1}}) dx &=\int_{B\setminus B_{\log j/j}(0)}\exp(\alpha_n |M_j|^{\frac n{n-1}}) dx + \int_{B_{\log j/i}(0)}\exp(\alpha_n |M_j|^{\frac n{n-1}})dx.
Q_t(z)= \sum_{\alpha\in \Delta_{>0}}(\pi_1(x_\alpha)(z)+t^{2\alpha(x_0)}\pi_2(x_\alpha)(z))\psi_{\alpha}^*(z) &+\sum_{\alpha\in \Delta_{>0}}\Phi_{\alpha}(z)\psi_{\alpha}^*(z)-\frac{1}{2}\sum_{\alpha,\beta,\gamma\in \Delta_{>0}}c_{\alpha,\beta}^\gamma\psi_{\alpha}^*(z)\psi_{\beta}^*(z)\psi_{\gamma}(z),
\Lambda(n)^{\min}((\lambda,\lambda'),(\mu,\mu')) = \big\{((\alpha,\tau), (\beta,\tau)) \mathbin{\big|}{}(\alpha,\beta) \in \Lambda^{\min}(\lambda,\mu), \quad\tau \in s(\alpha)\Lambda^{<n}, [\alpha\tau] = \lambda' [\beta\tau] = \mu'\big\}.
|\Phi\rangle_{\nu}=|\Phi_{0}\rangle_{\nu}\oplus|\Phi_{1}\rangle_{\nu}\oplus|\Phi_{2}\rangle_{\nu}\oplus\cdot\cdot\cdot=b|0\rangle\oplus\sum_{i}b_{i}|\phi_{i}\rangle\oplus\sum_{ij}b_{ij}|\phi_{i},\phi_{j}\rangle_{\nu}\oplus\cdot\cdot\cdot
\partial_t g_{ij}=-2 g^{pq}h (A_{pq}, A_{ij}), \partial_t g^{ij}= 2g^{pq}g^{ik}g^{jl}h(A_{pq}, A_{kl}), \partial_t d\mu = -|H|^2 d\mu, \partial_t \Gamma_{ij}^k =-g^{kl}\left( \nabla_ih( H,A_{jl})+ \nabla_jh( H,A_{il})- \nabla_lh( H,A_{ij}) \right).
e^{- i \sum_{j = 1}^2 (P_j \cdot x_j - Q_j \cdot y_j)}= e^{- i X \cdot \sum_{j = 1}^2 (P_j - Q_j)}e^{- i \sum_{j = 1}^2 (P_j \cdot x_j' - Q_j \cdot y_j')}
\exp(P_{X}(u)) = \prod_{n=1}^{\infty}Z_{X}(u^{n})^{{\mu(n)}/{n}}&= \left( \prod_{n=1}^{\infty}(1-u^{2n})^{-{\mu(n)}/{n}}\right)^{\epsilon-\nu}\left( \prod_{n=1}^{\infty}(1-u^{n})^{-{\mu(n)}/{n}}\right) \prod_{n=1}^{\infty}g_{X}(u^{n})^{-{\mu(n)}/{n}}&= \exp( (\epsilon-\nu) u^{2}+u ) \prod_{n=1}^{\infty}g_{X}(u^{n})^{-{\mu(n)}/{n}},
\frac{2}{2-t}(1-t)^{x}=\sum_{k=0}^{\infty}\mathcal{E}_{k,\lambda}(x)\frac{1}{k!}\big(\log_{\lambda}(1-t)\big)^{k}&=\sum_{k=0}^{\infty}\mathcal{E}_{k,\lambda}(x)\sum_{n=k}^{\infty}S_{1,\lambda}(n,k)\frac{(-1)^{n}t^{n}}{n!}&=\sum_{n=0}^{\infty}\bigg(\sum_{k=0}^{n}(-1)^{n}S_{1,\lambda}(n,k)\mathcal{E}_{k,\lambda}(x)\bigg)\frac{t^{n}}{n!}.
\|C\|_2^2 \ge \sum_n \|P_n C P_{n+1}\|_2^2 + \|P_{n+1}C P_n\|_2^2 &\ge \sum_n \frac{1}{n+1}\left(\|P_n C P_{n+1}\|_1^2 + \|P_{n+1}C P_n\|_1^2\right) &\ge \sum_n \frac{1}{2(n+1)}\left(\|P_n C P_{n+1}\|_1 + \|P_{n+1}C P_n\|_1\right)^2 &\ge \sum_n \frac{1}{2(n+1)}\left(1 - \frac{1}{m}\binom{n+2}{2}\right)^2.
(|m_{k'+1}-m_{k'}|+\mathrlap{|f(b)-m_{k'+1}|)\mathbf{1}_{\ge}+|f(b)-m_{k'}|\mathbf{1}_{<}}=(2|m_{k'+1}-f(b)|+|f(b)-m_{k'}|)\mathbf{1}_{\ge}+|f(b)-m_{k'}|\mathbf{1}_{<}=2|m_{k'+1}-f(b)|\mathbf{1}_{\ge}+(-1)^{k'+\alpha+1}[(f(b)-f(\tau_{k'}))\mathbf{1}_{k'\ge1}+(f(b)-f(a))\mathbf{1}_{k'=0}]+\lambda\mathbf{1}_{k'\ge1}.
\phi^{-1}\rho = (\tau,(\phi_{\tau^{-1}1}^{-1}\alpha_1,\phi_{\tau^{-1}2}^{-1}, \phi_{\tau^{-1}3}^{-1}\alpha_3)) &= (\tau,((\phi_{\tau^{-1}1}^{-1}\alpha_1,\varepsilon, \phi_{\tau^{-1}3}^{-1}\alpha_3))= \psi \in \prescript{\tau}{}N_l^A
[\mathcal{A},\mathcal{B}]:=[\mathcal{A}_1,\mathcal{B}_1]_1+\sum_{\tau=i,j,k}[\mathcal{A}_1,\mathcal{B}_\tau]_\tau+\sum_{\tau=i,j,k}[\mathcal{A}_\tau,\mathcal{B}_1+\mathcal{B}_\tau]_\tau,\mbox{~~for all ~~}\mathcal{A},\mathcal{B}\in\mathfrak{h}_{24}.
\rho_{j_{k},j_{l}}^{[n]}=\frac{G_{R}^{2}P_{T}}{G_{R}^{2}\sigma_{R}^{2}\left[\mathbf{F}_{m}^{H}\mathbf{\tilde{W}}^{\left(J\right)}\left(\mathbf{\tilde{W}}^{\left(J\right)}\right)^{H}\mathbf{F}_{m}\right]_{n,n}+\sigma_{S}^{2}\left[\mathbf{V}_{j_{l},j_{k}}^{H}\mathbf{V}_{j_{l},j_{k}}\right]_{n,n}},
\eta(q(\tau_i))=\frac{u_i}{(2\pi)^{3/2}\sqrt{\beta}}e^{ik_i\cdot q(\tau_i)}\equiv\frac{u_i}{(2\pi)^{3/2}\sqrt{\beta}}e^{i\left[{\bf k}_i{\bf q}(\tau_i)+(\frac{2\pi}{\beta})(n_i+\frac12) q_4(\tau_i)\right]}.
\int_{\mathbb{R}}R^{\theta}_{i,j}({\rm d}y)\mathbf{P}_{0,j}(I \in A{\rm e}^{\alpha(t-y)})=\int_{\mathbb{R}}\mathbf{P}_{0,j}(I\in{\rm d}v)R^\theta_{i,j}(t - \alpha^{-1}\log v, t-\alpha^{-1}\log v + \alpha^{-1}\log \lambda).
E(X_{j})=\mu=(\mu_{1},\mu_{2})^{T},\Sigma=[\begin{matrix}\sigma_{11}&\sigma_{12}\\ \sigma_{21}&\sigma_{22}\end{matrix}]=[\begin{matrix}\sigma_{1}^{2}&\rho\sigma_{1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{matrix}],
\mathcal U_\varepsilon^o - \mathcal L\bigl( \dfrac t \varepsilon \bigr) V^o = \mathcal L \bigl( \dfrac t \varepsilon \bigr) \bigl( V_\varepsilon^o - V^o \bigr) \rightarrow 0 ~~ ~ L^\infty(0,T;H^1(\mathbb T^2)).
Y =\{(x,y)\in X \mid y\geq 0\}, Z =\{(x,y)\in X \mid y\leq 0\}, A =\{(x,y)\in \mathbb R^2 \mid x\geq 0,\ |y|\leq x\}, B =\{(x,y)\in A \mid y\geq 0\}, C =\{(x,y)\in A \mid y\leq 0\}.
\mathcal D_{i_0}(\phi_{i_0}(n,k)U_{n,k}) =C_{i_0}\phi_{i_0+1}(n,k)U_{n,k}= 0, \mathcal D_{i_0}(\psi_{i_0}(n,k)U_{n,k}) =C_{i_0}\psi_{i_0+1}(n,k)U_{n,k}=0.
T^{2}=\frac{\left( 1-4\alpha \right) |\delta|^{2\alpha-2}}{2w^{4\alpha+1}}\left[ 2\kappa v^{3}+\delta \,\acute{}\,v^{2}+2\delta^{2}\left( \kappa-\lambda \right) v+\delta^{2}\delta \,\acute{}\, \right] .
\int_{B_{r}(x_0)}|x|^{-\sigma_i}f_{i}(x,u_{1}(x),{}\ldots,u_{L}(x)) \varphi_{r}(x) \,dx ={}u_{i}(x_0) + \sum_{j=1}^{k}\int_{\partial B_{r}(x_0)}u_{ij}(x)\frac{\partial}{\partial n}\Big[(-\Delta)^{k-j}\varphi_{r}(x)\Big]\,ds.
\nu(u,z_0):=\lim_{r\to 0}\frac{\frac{1}{2\pi}\int_0^{2\pi}u(z_0+re^{it})dt}{\log r}=\lim_{r\to 0}\frac{\max_{|\lambda|=r}u(z_0+\lambda)}{\log r}=\triangle u(\{z_0\}).
\int_{\{z_j: (1-|z_j|^2)^{L_j}\geq\varepsilon_{j}\}}(1-|z_j|^2)^{mL_j-2}\ \frac{dm(z)}{\pi}=\int_0^{(1-\epsilon_j^{1/L_j})^{1/2}}(1-r^2)^{mL_j-2}2r\dr =\int_{\epsilon_j^{1/L_j}}^1 s^{mL_j-2}ds=\frac 1{mL_j}\bigl[1+\textrm O(\frac 1{mL_j})\bigr]
\lim\limits_{T\rightarrow\infty}\psi(T,\tau) = \lim\limits_{T\rightarrow\infty}\frac{2}{T}\int\limits_0^{T}\bigl(\rho^2(u)+\rho(u+\tau )\rho(u-\tau)\bigr)du & = \lim\limits_{T\rightarrow\infty}\bigl(\rho^2(T)+\rho(T+\tau)\rho(T-\tau )\bigr)=0.
\begin{aligned}|\nabla\varphi(x)-O| \le |\nabla\varphi(x)-B|+d(B,SO(n))\le 2|\nabla\varphi(x)-B|+d(\nabla\varphi(x),SO(n)) &\le C\|d(\nabla v,SO(n))\|_{L^{p(\cdot)}(Q)}^\frac{1}{2}+d(\nabla\varphi(x),SO(n))\end{aligned}
\stackrel{( - 1 , 0 )}{\Phi}_{\alpha_{L - 2}}^{* ( 1 )}Z_{\alpha_{L - 1}}^{\alpha_{L - 2}}= \delta_{1}\left( \stackrel{( - 2 , 0 )}{\eta}_{\alpha_{L - 1}}^{* ( 1 1 )}- \stackrel{( - 2 , 0 )}{\gamma}_{\alpha_{L}}A_{\alpha_{L - 1}}^{\alpha_{L}}\right) .
\mathfrak{G}/G_1 \ni [x] \mapsto \Big( \mathfrak{J}_L \big( \mathfrak{J}^L f \big)_{{}_{[x]}}, g_{{}_{[x]}}\Big)= \Big( \mathfrak{J}_L \big( \mathfrak{J}^L f \big)_{{}_x}, g_{{}_x}\Big)
\hat{\gamma}= \frac{\exp\left(-\frac{1}{2}\sum_{i,\,j=1}^{2n}\left(\hat{R}^i-r^i\right)h_{ij}\left(\hat{R}^j-r^j\right)\right)}{\mathrm{Tr}\exp\left(-\frac{1}{2}\sum_{i,\,j=1}^{2n}\left(\hat{R}^i-r^i\right)h_{ij}\left(\hat{R}^j-r^j\right)\right)}\;,
D_s L(x) D_s^{-1}= \left( \prod_{i=1}^{m}(P^{-i+1}D_s P^{i-1}) Q_i (P^{-i}D_s P^{i})^{-1}\right) \cdot (P^{-m}D_s P^{m}) P^k D_s^{-1}.
\hat{\bf B}_{L (R)}^{(\tau)}({\bf x},{\bf u}; x_0) = \sum_{\xi, \, \zeta = \pi, \, \sigma}{\bf P}_\xi ({\bf x}, x_0) \hat{B}_{L (R)}^{(\tau) ( \xi, \, \zeta )}({\bf x},{\bf u}; x_0){\bf P}_\zeta ({\bf u}, x_0) .
\frac{1}{|\mathbb{S}^{n-1}|}\int_{\mathbf{x}\in\mathbb{S}^{n-1}}f( \mathbf{x}) d \sigma(\mathbf{x}) = \frac{1}{| \mathbb{X}|}\sum_{\mathbf{u}\in \mathbb{X}}f(\mathbf{u})
Z[A_{+}^{\prime}] = \int \! D \bar{\psi}^{\prime}D \psi^{\prime}e^{-\int \! d^2 x\,[ \psi_{-}^{\prime \dag}i \partial_{+}\psi_{-}^{\prime}+ \psi_{+}^{\prime \dag}i \partial_{-}\psi_{+}^{\prime}- A_{+}^{\prime}j_{-}^{\prime}]}\quad.
\tilde{V}_{\theta}= - \tilde{\omega}^{2}\left( \tilde{f}- 1 \right) +{\frac{1 5}{4 r^{2}}}-{\frac{4 \operatorname{s i n}^{4}\theta R_{\theta}^{8}{h}^{2}}{r^{1 0}f^{4}}}.
\begin{aligned}(\nabla^2f)_g-\frac{(\Delta f)_g}{n}g=(\nabla^2 f)_{g_\kappa}-\frac{(\Delta f)_{g_\kappa}}{n}g_\kappa+ +df\otimes d(\log h)+d(\log h)\otimes df-\frac{2}{n}g_\kappa((\nabla f)_{g_\kappa},(\nabla\log h)_{g_\kappa})g_\kappa. \end{aligned}
\mathcal{K}_n(\boldsymbol{\mathcal{A}},A) := \{p(\boldsymbol{\mathcal{A}})*A \hbox{s.t.}deg(p) \leq n-1 \}, \mathcal{K}_n^{D}(B^D,\boldsymbol{\mathcal{A}}) := \{B^D * p^{D}(\boldsymbol{\mathcal{A}}) \hbox{s.t.}deg(p^D) \leq n-1 \}.
_{p}F_{q}\left( \left.\begin{array}{c}a_{1},\ldots ,a_{p}b_{1},\ldots b_{q}\end{array}\right\vert z\right) =\sum_{k=0}^{\infty}\frac{\left( a_{1}\right)_{k}\cdots \left( a_{p}\right)_{k}}{\left( b_{1}\right)_{k}\cdots \left(b_{q}\right)_{k}}\frac{z^{k}}{k!},
N_0^{-1}D_x (-\Delta_x)^{-s}u (x) = \int D_x u (x-y) \frac{1}{|y|^{d-2s}}\, dy & = - \lim_{\varepsilon \downarrow 0}\int_{|y| > \varepsilon}D_y u (x-y) \frac{1}{|y|^{d-2s}}\, dy = -(d-2s) \lim_{\varepsilon \downarrow 0}\int_{|y| > \varepsilon}u (x-y) \frac{y}{|y|^{d-2s+2}}\, dy,
S = \int ~ d^{4}x \sqrt{- g}~ \left[ ~ \frac{\tilde{R}( g , T )}{\kappa}- \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- \frac{1}{2}\tilde{H}_{\mu \nu \lambda}\tilde{H}^{\mu \nu \lambda}+ \frac{1}{\sqrt{\kappa}}T^{\mu \nu \lambda}\tilde{H}_{\mu \nu \lambda}~ \right]
g_{0 0}^{2}= 4 ( h_{0}+{\frac{q_{0}}{r}}) \left( ( h_{1}+{\frac{p_{1}}{r}}) ( h_{2}+{\frac{p_{2}}{r}}) ( h_{3}+{\frac{p_{3}}{r}}) + a ( h_{3}+{\frac{p_{3}}{r}})^{3}\right) ,
= 4 \pi \Biggl [ \left( \frac 1 c \left( -{\bf j}_{m}\times{\bf E}\right) + \rho_{m}{\bf B}\right) \otimes \varepsilon^{1}\vee \varepsilon^{1}+ \left( \frac 1 c \left({\bf j}_{e}\times{\bf B}\right) + \rho_{e}{\bf E}\right) \otimes \varepsilon^{2}\vee \varepsilon^{2}+
e T_{\mu \nu}= e \Theta_{\mu \nu}+ \frac{1}{2}g_{\mu \nu}\left({\tilde{\overline{{\Psi}}}}_{L}\frac{\delta{S}^{-}}{\delta{\tilde{\overline{{\Psi}}}}_{L}}+{\tilde{\Psi}}_{L}\frac{\delta{S}^{-}}{\delta{\tilde{\Psi}}_{L_{0}}}\right) .
u^\infty(\hat x,d)=\int_{\partial{B_R}}\left[{\frac{\partial G^\infty(\hat{x},y)}{\partial\nu(y)}}u^s(y,d)-\frac{\partial u^s(y,d)}{\partial\nu(y)}G^{\infty}(\hat{x},y)\right]ds(y),\hat x \in \mathbb S^1_+\cup \mathbb S^1_-,
\int_{\Omega}\langle\varphi, \partial_{i_1}\cdots \partial_{i_p}\mathcal{L}\eta\rangle\, d\mu = \prod_{j=1}^p\lambda_{i_j}^{-1}\int_{\Omega}\langle \varphi \circ\Phi, \partial_{i_1}\cdots \partial_{i_p}\eta\rangle\, d\mu.
\nabla_{\mu}e_{\nu}^{\! A a}\equiv \partial_{\mu}e_{\nu}^{\! A a}- \Gamma_{\mu \nu}^{\rho}e_{\rho}^{\! A a}+ \omega_{\mu B}^{\! A}e_{\nu}^{\! B a}+ \omega_{\mu \! b}^{\! a}e_{\nu}^{\! A b}= 0 ,
\left( \begin{array}{c c c c}{0}&{0}&{\delta^{3}( x - y )}&{0}\\{0}&{0}&{0}&{\nabla^{- 2}}\\{- \delta^{3}( x - y )}&{0}&{0}&{0}\\{0}&{- \nabla^{- 2}}&{0}&{0}\\ \end{array}\right) ,
\|\psi_{n,t}(b)^*\psi_{n,t}(b) +(\psi_{n,e}(c)+\alpha1_{k_n})^*(\psi_{n,e}(c)+\alpha1_{k_n})-\|b\|^21_{k_n}\| &\leq \|\psi_{n,t}(b)^*\psi_{n,t}(b)-\psi_{n,e}(b^*b)\|+\|\psi_{n,e}(c)^*\psi_{n,e}(c)-\psi_{n,e}(c^*c)\| &= \|\psi_{n,t^{-1}}(b^*)\psi_{n,t}(b)-\psi_{n,e}(b^*b)\|+\|\psi_{n,e}(c^*)\psi_{n,e}(c)-\psi_{n,e}(c^*c)\|<2\varepsilon_n
E_j^2=\underset{g,h}{\sum}\frac{[X_{ij}(g),X_{ki}(h)]}{(z_i-g\cdot z_j)(z_k-hg\cdot z_j)}+\frac{-[X_{ij}(g),X_{ki}(h)]}{(z_i-h^{-1}\cdot z_k)(z_k-hg\cdot z_j)}+\frac{[X_{ij}(g),X_{ki}(h)]}{(z_i-h^{-1}\cdot z_k)(z_k - h\cdot \infty )}.
\mathfrak{a}= \mathfrak{a}_1\oplus\cdots\oplus\mathfrak{a}_{r}, \begin{cases}\mathfrak{a}_1 := \mathrm{span}\big\{X_1,\ldots,X_m\}, \mathfrak{a}_k := [\mathfrak{a}_1,\mathfrak{a}_{k-1}]; [\mathfrak{a}_1,\mathfrak{a}_{r}] = \{0\}. \end{cases}
C_{\mu}^{a b}= 2 i g \overline{{\psi_{A}^{a}}}g^{A B}\psi_{C}^{b}{A_{\mu B}}^{C}+ \left( \overline{{\psi_{A}^{a}}}g^{A B}( \partial_{\mu}\psi_{B}^{b}) + \psi_{B}^{b}g^{A B}( \partial_{\mu}\overline{{\psi_{A}^{a}}}) \right) ,
|R_\ell|^2 \sim{1\over v}\;{|\Gamma (\ell + 1+i\eta)|^2\over (\Gamma(2\ell+1))^2}\;(2kr)^{2\ell}e^{-\pi\eta}={m\over \hbar}\;{2^{2\ell}\over(\Gamma(2\ell+1))^2}\; e^{-\pi\eta}k^{2\ell-1}r^{2\ell}|\Gamma (\ell + 1+i\eta)|^2
\left(\mu\circ\left(1\otimes{_i\Delta_s}\right)\right)(f\otimes g)(x)= \mu\left(f\otimes{_i\Delta_s}(g)\right)(x) = f\cdot x\left({_i\Delta_s}(g)\right) = f\cdot x\left(\sum_{j=0}^{|s|-1}\frac{s^j(g)}{s^j(\ell^i_s)}\right).
d s^{2}= - \left( 1 - \frac{r_{0}^{2}}{r^{2}}\right) d u d v + \frac{p ( u )}{r^{2}}d u^{2}+ \left( 1 - \frac{r_{0}^{2}}{r^{2}}\right)^{- 2}d r^{2}+ r^{2}d \Omega_{3}.
\begin{array}{ll}\d C_1:=\frac{1}{2}\min \left[\{2^{p+q}AC(5N)^{p+q}N^{-1}\}^{-1/(p+q)},\right. &\quad\{2^{p+q}AE\|u_t^0\|_\infty^p(5N)^qN^{-1}\}^{-1/q}, &\quad\{2^{p+q}AE\|u^0\|_\infty^q(5N)^pN^{-1}\}^{-1/p}, &\left.\quad\{2^rBC(5N)^rN^{-1}\}^{-1/(r+1)}\right],\end{array}
H_k(\eta ,y,\zeta ,w,(z,v)^{\eta})=\sum_{(x,\kappa ,u)\in\eta}\left((1-{\bf 1}_{D_{y,w}}(x){\bf 1}_{[0,\zeta ]}(z))\delta_{(x,\kappa ,u)}+{\bf 1}_{D_{y,w}}(x){\bf 1}_{[0,\zeta ]}(z)\delta_{(y+wv,\kappa_{u^{*}},u)}\right),
h_{ij}(z_{ij}+1) - h_{ij}(z_{ij}) = \frac{1}{n}\log \Big\langle e^{b \sigma_i \sigma_j}\Big\rangle_t = \frac{1}{n}\log \Big\{\cosh(b) \big[1 + \tanh(b) \langle \sigma_i \sigma_j \rangle_t \big] \Big\}\,,
\frac{1}{m_{l}}=\frac{1}{\hbar^{2}}\sum_{m}\cdot\frac{\partial^{2}E_{c}(k)}{\partial k_{l}\partial k_{m}}\approx\frac{1}{m}+\frac{2}{E_{g}m^{2}}\sum_{m,n}\langle u_{c,0}|p_{l}|u_{n,0}\rangle\langle u_{n,0}|p_{m}|u_{c,0}\rangle
H_\mathrm{BH}=\frac{\mu_+-\mu_-}{2}\big(a^*_1a_2+a^*_2a_1\big)+\frac{\lambda\, w_{1111}}{2(N-1)}\big(a^*_1 a^*_1 a_1a_1+a^*_2 a^*_2 a_2a_2\big).
r=(\begin{matrix}p_{x}&p_{y}&p_{z}\end{matrix})(\begin{matrix}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{matrix})(\begin{matrix}q_{x}\\q_{y}\\q_{z}\end{matrix})=p_{i}T_{ij}q_{j}
\int \eta_i(x) f(\sigma^x\xi, \omega) d\nu_{\widehat \theta}^N (\xi,\omega) = \int \eta_j(x) e^{\vartheta_i(x/N) - \vartheta_j(x/N)}f(\xi, \omega) d\nu_{\widehat \theta}^N (\xi,\omega),
u_{M}^{N}(x):=\frac{1}{N}\sum_{i=1}^N\left[ g(X^{x,i}_{M})+\frac{1}{d}\sum\limits_{k= 1}^M \widetilde{r}^2(X^{x,i}_{k-1})f\left(X^{x,i}_{k-1}+\widetilde{r}(X^{x,i}_{k-1})Y^i\right)\right], x\in D.
M_2 \big(\psi_{n_1}^{n_2}\otimes \psi_{n_2}\big) = c_{n_2-1}\psi_{n_1}^{n_2-1}\otimes \psi_{n_2-1}+c_{n_2}\psi_{n_1}^{n_2}\otimes \psi_{n_2}+c_{n_2+1}\psi_{n_1}^{n_2+1}\otimes \psi_{n_2+1}.
r\ge 0, i_1>\dots >i_r\ge 0, a_1,\ldots ,a_r\ge 1, \tau\in P, t\ge 0, k_1>\dots >k_t\ge 0, c_1,\ldots ,c_t\ge 1, c_1+\dots +c_t-(a_1+\dots +a_r)=2(n-m).
z^* -\bar{z}^* = \lim\limits_{(z,\mu)\rightarrow (z^*,0)}(x-z) = \lim\limits_{(z,\mu)\rightarrow (z^*,0)}\sum\limits_{i\in \mathcal{I}^*}\cfrac{\mu^2}{A_i x-b_i}( -A_i^T)
\alpha_{i_1,\dots, i_k}^{(k)}=\frac{\ell_1!\dots \ell_r!}{k!}\,a_{\underbrace{j_1,\dots,j_1}_{\ell_1\ \mathrm{times}},\dots,\underbrace{j_r,\dots, j_r}_{\ell_r\ \mathrm{times}}}^{(k)}.