harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_n ={(1+\gamma_n) \Delta^2 + 6 \beta_n \sqrt{1+\gamma_n}\Delta +{24 (\sqrt{p \log{n}}+ \log{n}) \over n-1}\over \sqrt{4(1-{1\over n}) (1-\gamma_n) \Delta^2 +{16p \over n-1}(1-\theta_n) - 16 \beta_n \sqrt{1-\gamma_n}\Delta}}.
\begin{array}{lll}S_1 = \{a_1, \ldots, a_m \}, S_2 = S(v) \setminus S_1, E_2 = \{c \in E(v) \vert ca_i \in I, \mbox{for all}i = 1, \ldots m \}, E_1 = E(v) \setminus E_2. \end{array}
\int\limits_{\gamma(t_{j},\varepsilon)}\sum_{k\in\mathbb{T}(v,j)}a_{k}(t)\Psi_{k,t}(x)dt=\sum_{k\in\mathbb{T}(v,j)}\int\limits_{\gamma(t_{j},\varepsilon)}a_{k}(t)\Psi_{k,t}(x)dt.
\left[ J_{ab},\bar{P}_{c}\right] =\frac{1}{\sqrt{2}}\left[ J_{ab},P_{c}\right] +\frac{1}{\sqrt{2}}\left[ J_{ab},Z_{c}\right] , & =\frac{1}{\sqrt{2}}f_{ab,c}^{d}P_{d}+\frac{1}{\sqrt{2}}f_{ab,c}^{d}Z_{d}, & =f_{ab,c}^{d}\bar{P}_{d},
\left\langle a\right| N\left( \psi_{L}^{\dagger}\psi_{L}\psi_{R}^{\dagger}\psi_{R}\right) \left| B,a\right\rangle =\frac{1}{4\pi^{2}\left| z\right|^{2}}\left( \frac{\sinh^{2}\left( \frac{\alpha}{2}\log\left| z\right| \right)}{\sinh^{2}\left( \log \left| z\right| \right)}- \frac{\alpha^{2}}{4}\right) .
\begin{cases}p_\theta \in C(0,\infty)\cap C^1(0,\infty), \lim_{v\to \infty}p_\theta(v)=0 p_\theta v \in (0,\infty) p_\theta(v)\leq p_0(1+v^{-\Gamma}), \rho\geq 0,\end{cases}
\sum_{n=1}^{\infty}x_nt^{n-1}=\frac{1}{\lambda_1-\lambda_2}\left[ (x_2-\lambda_2x_1)\sum_{n=1}^{\infty}\lambda_1^{n-1}t^{n-1}-(x_2-\lambda_1x_1)\sum_{n=1}^{\infty}\lambda_2^{n-1}t^{n-1}\right] &=\sum_{n=1}^{\infty}\left[\frac{1}{\lambda_1-\lambda_2}\left( (x_2-\lambda_2x_1)\lambda_1^{n-1}-(x_2-\lambda_1x_1)\lambda_2^{n-1}\right)\right]t^{n-1}.
\Big\{g,\partial_{\alpha_1 \cdots\alpha_k}g, \partial_{\alpha_1 \cdots \alpha_{k-1}}A_{\alpha_k}^a\Big\}\rightarrow \Big\{g, \partial_{(\alpha_1 \cdots \alpha_{k-1}}A_{\alpha_k)}^a, D^{(\Theta)}_{\alpha_1 \cdots\alpha_{k-1}}I^a_{\alpha_k}\Big\}.
f \circ \ell_\gamma\ :=\ \left(\left(\cdots \left(\left(f\circ \ell_{\omega^{\beta_k}}\right)\circ\ell_{\omega^{\beta_{k-1}}}\right)\circ\cdots\right)\circ \ell_{\omega^{\beta_2}}\right)\circ\ell_{\omega^{\beta_1}}.
-2\int_{B_\rho}T_2(\nabla\varphi,\nabla\Delta\varphi)d\mu_g = -\frac{(n^2-28)(n-4)(n-6)^2}{12n(n-1)(n+2)}|W(p)|^2\omega_{n-1}\int_0^\rho r^{n+3}\frac{u_\epsilon^2}{(\epsilon^2+r^2)^4}[(n+2)\epsilon^2+4r^2]dr &+\int_{B_\rho}\frac{O(r^3)u_\epsilon^2}{(\epsilon^2+r^2)^2}dx.
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=\frac{\alpha\eta}{\tau}-\beta\rho^2\leqslant 0,\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=\frac{\alpha\eta}{\rho}-\beta\tau^2 \leqslant 0,
\mathbf{y}_l = \mathbf{H}_l^{\mathrm{T}}\mathbf{W}^{\mathrm{RF}}\mathbf{x}^{\mathrm{BB}}_l + \mathbf{n}_l = \left(\mathbf{H}^{\mathrm{BB}}_l\right)^{\mathrm{T}}\mathbf{x}^{\mathrm{BB}}_l + \mathbf{n}_l,
\Gamma_{F}[ \varphi_{0}, 0 ] \, = \, \int d^{3}x \, \frac{4}{\pi^{2}L^{3}}\sum_{n = - \infty}^{\infty}\frac{1}{n^{4}}[ \cos ( n e L \varphi_{0}) - ( - 1 )^{n}] \; ,
4 \int_{2}^{\infty}\frac{d y}{2 \pi}\frac{1}{\sqrt{y^{2}- 4}}\frac{y \left( \sqrt{1 + \sigma_{0}}\sqrt{1 + \sigma_{1}}- \coth 2 x_{0}\right) + ( \sigma_{0}+ \sigma_{1})}{y^{2}+ 2 y \sqrt{1 + \sigma_{0}}\sqrt{1 + \sigma_{1}}+ 2 ( \sigma_{0}+ \sigma_{1})}
\overline{\phi}=\limsup \limits_{\xi \rightarrow-\infty}\phi(\xi),\underline{\phi}=\liminf\limits_{\xi \rightarrow-\infty}\phi(\xi), begin{align*}0.2cm] \overline{\psi}=\limsup \limits_{\xi \rightarrow-\infty}\psi(\xi),\underline{\psi}=\liminf\limits_{\xi \rightarrow-\infty}\psi(\xi),
\lim_{\epsilon\rightarrow 0}\int_{D\setminus B(Q, \epsilon)}\mathrm{div}\nabla\big(\mathcal{D}_{m}(Q, X)\big)dX =\lim_{\epsilon\rightarrow 0}\int_{D\setminus B(Q, \epsilon)}\Delta\mathcal{D}_{m}(Q, X)dX &=\int_{D}\Delta\mathcal{D}_{m}(Q, X)dX.
v(n) = (-1)^g \sum_{k_g = 1}^{n_g}\cdots \sum_{k_1 = 1}^{n_1}(-1)^{k_1 + \cdots + k_g}\binom{n_1}{k_1}\cdots \binom{n_g}{k_g}\frac{k_1 p_1}{k_1 p_1 + k_2 p_2 + \cdots + k_g p_g}.
c_{j_{q}k_{{s}_1}}^{h_{{t}_1}}c_{h_{{t}_1}k_{{s}_2}}^{h_{{t}_2}}\ldots c_{h_{{t}_{n-2}}k_{{t}_{n-1}}}^{h_{{t}_{n-1}}}c_{h_{{t}_{n-1}}k_{{s}_n}}^{i_{p}}g^{k_{{s}_1}}g^{k_{{s}_2}}\ldots g^{k_{{s}_{n-1}}}g^{k_{{s}_n}}dg^{j_{q}}\; ,
h_{n, \epsilon m}(k,\nu) = \frac{1}{4(k+h^\vee)}(( \epsilon m (k+h^\vee)-n)^2 - (k+1)^2 + 2(\nu |\nu +2\rho^{\natural})),\, h_{m,\gamma}(k,\nu) = \frac{1}{4(k+h^\vee)}((2(\nu + \rho^{\natural}|\gamma)+2m(k+h^\vee))^2 -(k+1)^2 + 2(\nu |\nu +2 \rho^{\natural}) ) \, .
d s_{B}^{2}= a \prod_{i = 1}^{3}( z - z_{i})^{- \frac{2}{3}}\prod_{i = 1}^{3}( \bar{z}- \bar{z_{i}})^{- \frac{2}{3}}d z d \bar{z}+ a^{\prime}\prod_{i = 1}^{3}( z^{\prime}- z_{i}^{\prime})^{- \frac{2}{3}}\prod_{i = 1}^{3}( \bar{z^{\prime}}- \bar{z_{i}^{\prime}})^{- \frac{2}{3}}d z^{\prime}d \bar{z^{\prime}}
\begin{array}{l}{y^{0}= \displaystyle a \frac{1 + x x^{\prime}}{1 - x x^{\prime}}, i y^{n + 1}= a \frac{2 x^{n + 1}}{1 - x x^{\prime}},}\\{y^{i}= a \displaystyle \frac{2 x^{i}}{1 - x x^{\prime}}, ( i = 1 , 2 , \cdots , n ) .}\\ \end{array}
2 \varphi(m) \frac{\varphi(l)^2}{l}\prod_{\substack{p \mid l}}\left( 1 - \frac{1}{(p-1)^2}\right) \sum_{\substack{1 \leq c \leq ln, \gcd(c,n^*) =1}}w^*\left( \frac{c}{ln}\right) \prod_{\substack{p \mid \gcd(l,c)}}\left(1 + \frac{1}{p-2}\right) .
{\mathbb C}{\rm ov}(\bar{S}_{N_{\beta,\omega}}, G_{N_{\beta,\omega}}) = - \varphi'(0)\left(\frac{1}{\beta^2}- \frac{1}{(\beta+\omega)^2}\right) + \frac{\psi'(\beta)}{(\psi(\beta))^2}- \frac{\psi'(\beta+\omega)}{(\psi(\beta+\omega))^2}.
\begin{array}[c]{cl}-\Delta Z=0 \Omega, \frac{\partial Z}{\partial\mathbf{n}}-\frac{\operatorname*{i}}{k}\operatorname*{div}_{\Gamma}T_{k}\nabla_{\Gamma}Z=\left\langle L_{\Omega}\mathbf{r},\mathbf{n}\right\rangle -\frac{\operatorname*{i}}{k}\operatorname*{div}_{\Gamma}T_{k}^{\operatorname*{low}}\mathbf{r}_{T}\Gamma.\end{array}
G=\left\{\begin{pmatrix}\alpha 0 0 \alpha\end{pmatrix}, \ \begin{pmatrix}\alpha 0 0 -\alpha\end{pmatrix}, \ \begin{pmatrix}0 \alpha \alpha 0 \end{pmatrix}, \ \begin{pmatrix}0 \alpha -\alpha 0 \end{pmatrix}\ \middle| \ \alpha\in C^\times\right\}.
\left(\frac{z}{z-\beta}\right)^{c}= e^{N\phi_{A}(z)}\left(\frac{z-\beta}{z-a}\right)^{c}\frac{\sqrt{2\pi}(a^2-1)^c}{a\Gamma(c)N^{\frac{1}{2}-c}(z-\beta)}, a>1, \left(\frac{z}{z-a}\right)^{c}= e^{N\phi_{A}(z)}\frac{a(1-a^2)^{c-1}}{N^{1-c}\Gamma(c)(z-a)}, a<1.
\lim_{j\rightarrow+\infty}\mathcal{F}_{\varepsilon_{j}}^{\xi_{i}}((\hat{u}_{j})_{z}^{\xi_{i}},\Omega_{z}^{\xi_{i}})=\liminf_{j\rightarrow+\infty}\mathcal{F}_{\varepsilon_{j}}^{\xi_{i}}((u_{j})_{z}^{\xi_{i}},\Omega_{z}^{\xi_{i}})~.
\begin{array}{c}{{u^{\alpha}= u_{0}^{\alpha}+ \xi^{\alpha}s - \frac 1 2 \left( \stackrel{s}{\Gamma}_{\beta \gamma}^{\alpha}\right)_{0}\xi^{\beta}\xi^{\gamma}s^{2}-}}\\{{\frac 1{3 !}\left( \Gamma^{\alpha}{}_{\beta \gamma \delta}\right)_{0}\xi^{\beta}\xi^{\gamma}\xi^{\delta}s^{3}- \cdots}}\\ \end{array}
\mbox{Ric}(E_p,E_q)=f\mbox{Ric}(e_p,e_q)=f\sum_{j=1}^n\tilde{g}( R(e_j,e_p)e_q,e_j)=\frac{1}{f}\sum_{j=1}^n\tilde{g}( R(E_j,E_p)E_q,E_j),
\langle \mathbf{H}, \hat{\mathbf{H}}\rangle_{\mathbb{R}^{(k \times d) \times (k \times d)}}\equiv \mathbf{H}: \hat{\mathbf{H}}:=\sum_{i, I = 1}^{k}\sum_{j, J = 1}^{d}H_{ijIJ}\hat{H}_{ijIJ}\mathbf{H}, \hat{\mathbf{H}}\in \mathbb{R}^{(k \times d) \times (k \times d)}.
\|\Delta f\|^{2}_{L^2(\mathbf{R}^3)}= \|\Delta_r f\|^{2}_{L^2(\mathbf{R}^3)}+ \bigg\| \sum_{j=1}^3 L_j^2f \bigg\|_{L^2(\mathbf{R}^3)}^2 + 2 \mathrm{Re}\bigg\langle \Delta_r f, \sum_{j=1}^3 L_j^2f \bigg\rangle.
( m , \tilde{m}= m + \frac{i k}{( k , 2 k^{\prime})}, n = 1 , p = k^{\prime}- \frac{i k^{\prime}}{( k , 2 k^{\prime})}\mathrm{o r}2 k^{\prime}- \frac{i k^{\prime}}{( k , 2 k^{\prime})}) .
\bar{\partial}^{\Psi}\! \partial^{\Psi}\! : \! \psi \! : = : \! \left( \psi_{; \nu \mu}+ i \psi^{; \eta}C_{\eta \nu \mu}\right) \partial^{\Psi}\! X^{\nu}\bar{\partial}^{\Psi}\! X^{\mu}\! : + O ( \epsilon^{4}) .
\theta_{( 3 )}^{( 1 , 0 )}( \tau ) = \sum_{( m_{1}, m_{2}) \in{\bf Z}^{2}}q^{\frac{1}{9}+ m_{1}+ 3 ( m_{1}^{2}+ m_{2}^{2}- m_{1}m_{2})}= : q^{\frac{1}{9}}{\tilde{\theta}_{( 3 )}^{0}}( \tau ) ,
\mathrm{rec}_S(\iota_v(\beta_v)\widetilde{\mathcal{U}}_S\overline{k^{\times}}) =\sigma_v^{h_k}[\widetilde{G}_S,\widetilde{G}_S] , \mathrm{rec}_S(\iota_v(\alpha_v)\widetilde{\mathcal{U}}_S\overline{k^{\times}}) =\tau_v^{h_k}[\widetilde{G}_S,\widetilde{G}_S] .
= \frac{1}{\pi^{2}}\int d^{2}p \frac{1}{p^{2}+ m_{d}^{2}}\frac{1}{p^{2}}\sin^{2}( p_{1}L ) \bigg [ \frac{p_{2}}{p_{1}}e^{i p_{2}L}- \frac{p_{1}}{p_{2}}\Big ( 1 - e^{i p_{2}L}\Big ) \bigg ] = 0 .
\mathrm{\cal{A}}({\mathbf{u}}) = \int d^{1 + n}x \frac{( 1 +{\mathbf{u}}^{\dag}{\mathbf{u}}) \partial^{\mu}{\mathbf{u}}^{\dag}\partial_{\mu}{\mathbf{u}}- \partial^{\mu}{\mathbf{u}}^{\dag}{\mathbf{u}}{\mathbf{u}}^{\dag}\partial_{\mu}{\mathbf{u}}}{( 1 +{\mathbf{u}}^{\dag}{\mathbf{u}})^{2}},
\nu_1(n-1) = \sum_{s=1}^{n-1}\sum_{j=0}^{m-1}(a^j\star a^j)_s = \sum_{j=0}^{m-1}\sum_{s=1}^{n-1}(a^j\star a^j)_s = \sum_{j=0}^{m-1}p_1^2(a^j)-\sum_{j=0}^{m-1}p_2(a^j) = \sum_{j=0}^{m-1}p_1^2(a^j)-\nu_0.
\sum_{m = 1}^{\infty}\frac{\sin^{2}( \pi m y )}{m^{2}- n^{2}/ y^{2}}\frac{\sqrt{x + \sqrt{x^{2}+ m^{2}}}}{\sqrt{x^{2}+ m^{2}}}= - \frac{1}{2 x^{3 / 2}}\mathrm{R e}\int_{1}^{\infty}\frac{d z}{\sqrt{z^{2}- 1}}\frac{\sqrt{1 + i \sqrt{z^{2}- 1}}}{z^{2}+ n^{2}/ ( x^{2}y^{2})}\frac{1}{F ( x , y , z )}
\int_{C}{d^{4}}{\eta}D_{c}^{( 4 )}( \eta ,{\eta}) = \int_{\partial C}d^{3}\eta K - 6 a \int_{\partial C}d^{3}\eta K K^{\prime}- \frac{2}{3}\frac{1}{a^{2}}\pi^{- \frac{1}{2}}\Gamma ( \frac{5}{2}) \int_{S_{4}}d^{4}\eta
D(\hat{h})=\frac1\pi \sum_{t=3}^\infty \frac{L(1,\chi_d)}{l}\prod_{p\mid l}\left(1+\left(p-\chi_d(p)\right)\frac{\left(p^\infty,l\right)-1}{p-1}\right)\hat{h}\!\left(\frac{1}{\pi}\log\!\left(\frac{t+\sqrt{t^2-4}}{2}\right)\right) &+\sum_{n=1}^\infty\frac{\Lambda(n)}{n}\hat{h}\!\left(\frac{\log n}{\pi}\right)
\sum_{j=1}^{J}\mathcal{V}_r(\gamma_j^\ell)\leq J^{\frac1{r'}}\Big(\sum_{j=1}^{J}\mathcal{V}^r_r(\gamma_j^\ell)\Big)^{\frac1r}\lesssim_r \|F\|_{\ell^r}^{r-1}\Big(\sum_{j=1}^J \mathcal{V}_r^r(\gamma_j^\ell)\Big)^{\frac 1r}=\frac{\|F\|_{\ell^r}^{r-1}}2\Big(\sum_{n\in\mathbb{J}}\log\Big(\frac{1+|F_n|}{1-|F_n|}\Big)^r\Big)^{\frac 1r}.
\mu_{R,1}(k,\ell,m,n):=\sum_{(A,B,C,D)\in\mathcal{X}_{R}}\int\overline{\mathcal{K}_{A}(x,k)}\mathcal{K}_{B}(x,\ell)\overline{\mathcal{K}_{C}(x,m)}\mathcal{K}_{D}(x,n)\,dx, \mathcal{X}_{R}:= \left\{(A_{0},A_{1},A_{2},A_{3}) \in \{S,R\}^4 \, : \,\exists \, j=0,1,2,3 \,:\, A_{j}=R\right\},
S_{m}(n)=\frac{1}{m+1}\sum_{k=0}^{m}(\begin{matrix}m+1\\k\end{matrix})B_{k}^{+}n^{m+1-k}=m!\sum_{k=0}^{m}\frac{B_{k}^{+}n^{m+1-k}}{k!(m+1-k)!},
\langle \Psi | \phi^{2}( x ) \cdots \phi^{2}( x ) \pi^{2}( x ) ( \mathrm{o r}\;{\phi^{\prime}}^{2}( x ) ) | \Psi \rangle = \langle \Psi | \phi^{2}( x ) | \Psi \rangle \cdots \langle \Psi | \phi^{2}( x ) | \Psi \rangle \langle \Psi | \pi^{2}( x ) ( \mathrm{o r}\;{\phi^{\prime}}^{2}( x ) ) | \Psi \rangle .
{\cal L}_{\rm YM}=-{1\over 4 g^2_{\rm YM}}G_{\mu\nu}^a G^{a\mu\nu}+{\theta\over 32\pi^2}G_{\mu\nu}^a{\widetilde G}^{a\mu\nu}= -{1\over 4}{\rm Re}\left( \tau\left[G_{\mu\nu}^a G^{a\mu\nu}+i G_{\mu\nu}^a{\widetilde G}^{a\mu\nu}\right]\right)~.
\MoveEqLeft \int_{0}^{b}(g'(\nu) - g'(-\nu)) \left( \int_{\sqrt{\xi}\nu}^\infty e^{i\mu^2}d\mu \right) d\nu= \int_0^b (g'(\nu) - g'(-\nu)) \left( \frac{\sqrt \pi}{2}e^{i \pi/4}+ O( \sqrt{\xi}\nu) \right) d\nu & = \frac{\sqrt \pi}{2}e^{i \pi/4}(g(b) - g(-b)) + O \left( \sqrt{\xi}\int_{|\nu| \le b}|\nu g'(\nu)| d\nu \right)
\dim AR(f)_{d-2}= h^0(T\langle C\rangle(d-3)) &=h^1(T\langle C\rangle(d-3))+\chi(T\langle C\rangle(d-3)) &=h^1(T\langle C\rangle(d-3))- \left[{d-1 \choose 2}- \tau(C)\right].
\Gamma [ g ] \, = \,{\frac{1}{1 2 \pi}}\epsilon^{\mu \nu \rho}\, t r \, \int_{M}d^{3}x \Big ( g^{- 1}\partial_{\mu}g g^{- 1}\partial_{\nu}g g^{- 1}\partial_{\rho}g \, \Big ) \, \, ,
\mathrm{I = \frac{2 \Gamma ( 4 - \o m e g a - \s i g m a )}{( 4 \p i )^{2}}\int_{0}^{1}d x x^{- \s i g m a}\int_{0}^{x}d G \int_{G}^{1 - x + G}d y ~ [ ~ H_{0}^{( 1 )}~ ]^{\o m e g a + \s i g m a - 4}+ F . T .}
\begin{matrix}\Psi(x_{1},x_{2},t)&=(e^{ik_{1}n_{1}^{2}t}sinn_{1}x_{1})(e^{ik_{2}n_{2}^{2}t}sinn_{2}x_{2})\\ &=e^{i(k_{1}n_{1}^{2}+k_{2}n_{2}^{2})t}sinn_{1}x_{1}sinn_{2}x_{2}\end{matrix}
\left\|\sum_{i=1}^N A_ix_i^{\hat{K}+1}+y^{\hat{K}+1}-b\right\|^2 = \frac{1}{\beta(\epsilon)^2}\|\lambda^{\hat{K}+1}-\lambda^{\hat{K}}\|^2=\frac{\mu(\epsilon)^2}{\beta(\epsilon)^2}\|y^{\hat{K}+1}-y^{\hat{K}}\|^2\leq \frac{1}{9}\theta_{\hat{K}}=O(\epsilon^4).
\langle \frac{\phi_\alpha}{z-\bar{\psi}},1,\tau,\ldots,\tau\rangle_{0,n+2,\mathfrak D}^B=|G_\sigma|\langle \frac{\phi_\alpha\otimes 1_{\hat b}}{z-\bar{\psi}},1_{\tilde{b}},\tau,\ldots,\tau\rangle_{0,n+2,\mathfrak D}^{\mathcal P_\sigma}
F_{\nu}^{\prime}(\alpha)=\lim_{\varepsilon\downarrow0}\Re\left[\frac{F_{\nu}(\alpha+i\varepsilon)-F_{\nu}(\alpha)}{i\varepsilon}\right]=F_{\nu}(\alpha)^{2}\int_{\mathbb{R}}\frac{d\nu(t)}{(\alpha-t)^{2}}
\tilde{\beta}^{2}= \frac{\lambda \, 2^{n}\, \rho \, ( \rho - 1 )}{( n + 1 ) ( 1 + \rho )^{n}}\, \mathrm{{A p p e l l}}F_{1}[ n + 1 , 2 , - n , n + 2 , 1 - \rho , \frac{1 - \rho}{2}] .
R_{1 p + 1}( z ) R_{2 p + 1}( z \tau^{2}) \cdots R_{p p + 1}( z \tau^{2 p - 2}) v^{( i_{1}\cdots i_{p})}\otimes v_{i}= \prod_{j = 1}^{p - 1}b ( z \tau^{2 j}) v^{( i_{1}\cdots i_{p})}\otimes v_{i}.
\sum_{n_k=i}a_k\overline{\varphi}(\partial[v_0^k,\ldots,v_{i+1}^k])[v_{i+1}^k,\ldots,v_d^k]=(-1)^{i+1}\sum_{n_k=i}a_k\varphi([v_0^k,\ldots,v_i^k,\widehat{v_{i+1}^k}])[v_{i+1}^k,\ldots,v_d^k]
\|S_{k,\theta}^{\frac12}(P^{(k,\alpha)}_t-P^{(k)}_t) S_{k,\theta}^{\frac12}\|_{HS}=\|\bar{f}\otimes f-\bar{g}\otimes g\|_2 \leq\|(\bar{f}-\bar{g})\otimes f\|_2+\|\bar{g}\otimes(f-g)\|_2 =\|f-g\|_2(\|f\|_2+\|g\|_2).
\lim\limits_{q\to\infty}\dfrac{k_4(P^\ast(q))}{q^4}=\frac{1}{1536}+3\times \lim\limits_{q\to\infty}{_{3}}F_{2}\left(\begin{array}{ccc}\chi_4, \chi_4, \chi_4^3 & \varepsilon, \varepsilon\end{array}| 1\right).
L_{{\bf{k}},{\bf{q}}}= d^{\dagger}_{(1/2)({\bf{k}}+{\bf{q}}/2)}({\bf{k}}+{\bf{q}}/2)d_{(1/2)({\bf{k}}-{\bf{q}}/2)}({\bf{k}}-{\bf{q}}/2) = b^{\dagger}_{{\bf{k}}+{\bf{q}}/2}exp(-i\mbox{}X_{0})exp(i\mbox{}X_{0})b_{{\bf{k}}-{\bf{q}}/2}= b^{\dagger}_{{\bf{k}}+{\bf{q}}/2}b_{{\bf{k}}-{\bf{q}}/2}
\delta_{\alpha\beta}=F_{\alpha\overline{\beta}}& =X_{\alpha}\overline{X_{\beta}}F-\left( \nabla_{X_{\alpha}}\overline{X_{\beta}}\right) F & =\left\langle \nabla_{X_{\alpha}}\nu,\overline{X_{\beta}}\right\rangle\chi & =\Pi\left( X_{\alpha},\overline{X_{\beta}}\right) \chi,
\mathrm{t r}( \gamma^{[ A}\tilde{\gamma}^{B}\gamma^{C ]}\tilde{\gamma}_{[ D}\gamma_{E}\tilde{\gamma}_{F ]}) = 4 \epsilon^{A B C}{}_{D E F}- 2 4 \delta_{~ D}^{[ A}\delta_{~ E}^{B}\delta_{~ F}^{C ]}
\frac{\gamma_{1^n}}{\gamma_{1^n}^{s_i}}= \frac{\gamma_{1^n}/\gamma_{\sigma_i}}{(\gamma_{1^n}/\gamma_{\sigma_i})^{s_i}}= -\frac{x_i-x_{i+1}+\Delta_{i,i+1}}{x_{i+1}-x_i+\Delta_{i,i+1}}.
1 - \prod_{k = k'}^{j-1}\left( 1- \frac{|R(I,k)|}{|D(I,j)|^2}\right) \ge 1- \prod_{k=k'}^{j-1}\left(\frac{1}{n^2}- \frac{1}{|D(I,k)|}\right) \ge 1 - \left( 1- \frac{1}{2n^2}\right)^{j-k'}.
(\begin{matrix}p_{m}&p_{m-1}&\cdot\cdot\cdot&p_{n}&\cdot\cdot\cdot&p_{1}&p_{0}&0&\cdot\cdot\cdot&0\\ 0&\cdot\cdot\cdot&0&q_{n}&\cdot\cdot\cdot&q_{1}&q_{0}&0&\cdot\cdot\cdot&0\end{matrix}).
\frac{d}{dr}\tilde{\theta}(r) = \frac{1}{\log^2 2}\cdot \frac{1}{r}\left[ \int_{2r}^{4r}\frac{\theta(s)}{s}ds - \int_{r}^{2r}\frac{\theta(s)}{s}ds \right] & = \frac{1}{\log^2 2}\int_1^2 \frac{\theta(2rt) - \theta(rt)}{rt}\, dt \geq 0.
\alpha_1 \alpha_2 = 0, \alpha_2 \alpha_3 = 0, \alpha_3 \alpha_1 = 0, \beta_1 \beta_3 = 0, \beta_3 \beta_2 = 0, \beta_2 \beta_1 = 0, (\alpha_1 \beta_1)^{m_1}= (\beta_3 \alpha_3)^{m_3}, (\beta_1 \alpha_1)^{m_1}= (\alpha_2 \beta_2)^{m_2}, (\beta_2 \alpha_2)^{m_2}= (\alpha_3 \beta_3)^{m_3}.
P_{ed}=\frac{\frac{1}{1+\rho(T,\delta)}-\frac{1}{1+\rho(T,\delta)+\rho( S_{tm},1)}}{1-\frac{1}{1+\rho( S_{tm},1)}}=\frac{1+\rho( S_{tm},1)}{(1+\rho(T, \delta))(1+\rho(T,\delta)+\rho( S_{tm},1))}
\psi_\lambda(g)=\psi_\lambda(kk'^{-1}a_t)=\sum^r_{i=0}c_i(t)\langle\tau(kk'^{-1})f^\Lambda_i, f^{\Lambda^\vee}_{r-i}\rangle=\sum^r_{i=0}\tilde{c}_i(t)\langle\tau(kk'^{-1})f^\Lambda_i, h^\Lambda_i\rangle.
\partial_s\Big(\frac{1}{z}\Big)(s) =-\frac{z'(s)}{z(s)^2}&=-\frac{h^2\beta_h'(s)+\frac{h^2}{2}\overline{\beta_h'(s)}+\frac{\sigma^2}{2}\overline{\beta_\sigma'(s)}}{\big(2\sigma^2+\mathcal{O}(h^2+s\sigma^4)\big)^2}&\sim\frac{i\sigma^4}{4\sigma^4}
\tilde{\mathcal{S}}_i^{\mathrm{deal}}=\left\{(\; u_i^{\mathrm{a}}(\hat{\boldsymbol{\tau}}_i), u_i^{\mathrm{s}}(\hat{\boldsymbol{\tau}}_i) \;)|\; \hat{\boldsymbol{\tau}}_i \in \mathcal{A}_{i}\right\}
\left\Vert z\right\Vert_{H_{0}^{\alpha/2}\left( \Omega\right)}^{2}=\sum_{k=1}^{\infty}a_{k}^{2}\rho_{k}^{\alpha/2}=\left\Vert (-\Delta)^{\alpha/4}z\right\Vert_{L^{2}\left( \Omega\right)}^{2},
S_F(f):\Gamma(TM_0) \longrightarrow \Gamma(\pi^*TM) , S_F(f) \hat{X}={}^c\nabla_{\hat{X}}(\nabla \varphi)-g(\nabla\varphi,\varrho\hat{X})\nabla\varphi-\frac{1}{n}(\Delta\varphi-\|\nabla\varphi\|^2) \varrho \hat{X},
\lim_{k \- \infty}\int_{\Omega_{j,k}}c_{n}\vert \nabla w_{k}\vert^{2}_{g_{0}}+R_{g_{0}}w_{k}^{2}\leq c_{n}n(n+2)\lim_{k\-\infty}\int\varphi_{j,k}^{\frac{4}{n-2}}w_{k}^{2}.
\Delta^2 f(z) =f(z+2)-2f(z+1)+f(z) &=\frac{f(z+1)^3-(2f(z+1)-f(z))B(z)}{f(z)f(z+1)-B(z)}-2f(z+1)+f(z) &=\frac{(\Delta f(z))^2f(z+1)}{f(z)f(z+1)-B(z)}=A(z)f(z+1),
\{Q_\alpha,Q_\beta\}= (C\Gamma^\mu)_{\alpha\beta}P_\mu + (C\Gamma_\mu\Gamma_{11})_{\alpha\beta}{\hat Z}^\mu + (C\Gamma_{\mu\nu})_{\alpha\beta}{\hat Z}^{\mu\nu}+ (C\Gamma_{11})_{\alpha\beta}{\hat Z}\quad .
w_{s u s t r}={\frac{1}{2 ( 2 \pi )^{D - 1}}}\ \sum_{s t a t e s \S}{\frac{( e_{1}+ e_{2}) E}{\epsilon}}\sum_{k = 1}^{\infty}( - )^{( k + 1 ) ( a_{S}+ 1 )}\Bigl ({\frac{\vert \epsilon \vert}{k}}\Bigr )^{D / 2}\exp{\Bigl ( -{\frac{\pi k m_{S}^{\ 2}}{\vert \epsilon \vert}}\Bigr )},
\frac{1}{\tau}\int_0^\tau x h(x)\, dx \leq - \frac{1}{\tau}\int_0^\tau \Bigl[x G'(x)U(G(x))+xG(x)G'(x)U'(G(x))\Bigr]\, dx + G(t)U(G(\tau)) = - \frac{1}{\tau}\int_0^\tau xG'(x)\Bigl[U(G(x))+G(x)U'(G(x))\Bigr]\, dx + G(\tau)U(G(\tau))\, .
\begin{array}{c}\Gamma_{\rho \nu \lambda}^{\left( 2\right)}(p_{1,}k_2,q_2)=g_{\rho \nu}(p_1+k_2)_\lambda +g_{\nu \lambda}(q_2-k_2)_\rho -g_{\lambda \rho}(q_2+p_1)_\nu \Gamma_{\sigma \mu \lambda}^{(2)}(p_2,k_1,q_2)=g_{\mu \sigma}(p_2+k_1)_\lambda -g_{\mu \lambda}(k_1+q_2)_\sigma +g_{\lambda \sigma}(q_2-p_2)_\mu\end{array}
\langle{\cal L}( 0 ) \tilde{q}^{k}( 1 ) q^{k}( 2 ) q^{k}( 3 ) \tilde{q}^{k}( 4 ) \rangle = \Theta^{2 2 2 2}\F^{( k - 2 )}( x , u ) + O ( \theta \bar{\theta}) \; .
\partial_1 \, Q_k^n(x,y) = (n-k) \left[P_k^{n-1}(x,y) + k\,b_{k-1}(\beta)\, P_{k-2}^{n-3}(x,y)\right], \partial_2 \, Q_k^n(x,y) = k \left[P_{k-1}^{n-1}(x,y) + (n-k)\,b_{n-k-1}(\alpha) \,P_{k-1}^{n-3}(x,y)\right].
\|q_1^* - q_0^*\|_{\alpha}\lesssim \|u_1^* - u_0^*\|_{\alpha}+ \| a^*_1 - a^*_0\|_{\alpha}(\displaystyle\max_{i=0,1}\|u_i^*\|_{\alpha}+ N_0^{int}) + \delta N_0^{int}.
\mathcal{C}:=\frac{1}{\sqrt{2\omega(D)A(D,\mathtt{m})}}\left( \begin{matrix}A(D,\mathtt{m}) -\tfrac{1}{2}\mathtt{m}p'(\mathtt{m}) -\tfrac{1}{2}\mathtt{m}p'(\mathtt{m}) A(D,\mathtt{m}) \end{matrix}\right)\,,
\hat{\Psi}_\alpha(x) = \left(\mu_\alpha^{\frac{n-2}{2}}\hat{u}_\alpha (x) \right) \mu_\alpha^{\frac{n-2}{2}}\hat{\psi}_\alpha(x) +\frac{n-2}{2n}\hat{\varphi}_\alpha(x) \left(\mu_\alpha^{\frac{n-2}{2}}\hat{u}_\alpha (x) \right)^2.
N^2\pi_Y^i\circ \pi_Y^j = \Gamma_p\circ \pi_X^i \circ ({}^t\Gamma_p \circ\Gamma_p)\circ \pi^j_X\circ{}^t\Gamma_p & = \Gamma_p\circ ({}^t\Gamma_p \circ \Gamma_p)\circ \pi^i_X\circ\pi^j_X \circ{}^t\Gamma_p & = N \Gamma_p\circ \pi_X^i\circ\pi_X^j \circ{}^t\Gamma_p & =\begin{cases}N^2\pi_Y^i, i=j\,; 0, i\neq j.\end{cases}
d_{\rm TV}(D_n, Po(\lambda_n))\leq \frac{1-e^{-\lambda_n}}{\lambda_n}\sum_{i=1}^{n-1}p_{i,n}^2 \leq \min\{1, \frac{1}{\lambda_n}\}\sum_{i=1}^{n-1}p_{i,n}^2.
=(A_{0}(\tilde{\textbf{u}}_{1}+\bar{\textbf{u}})-A_{0}(\tilde{\textbf{u}}_{2}+\bar{\textbf{u}}))\partial_{t}{\textbf{u}}_{1}+\sum_{j=1}^{d}(A_{0}A_{j}(\tilde{\textbf{u}}_{1}+\bar{\textbf{u}})-A_{0}A_{j}(\tilde{\textbf{u}}_{2}+\bar{\textbf{u}}))\partial_{x_{j}}\tilde{\textbf{u}}_{1}.
\bar{\mathbf{G}}_i =\operatorname{diag}(\mathbf{g}_1(i),\cdots\mathbf{g}_i(T))\in\mathbb{R}^{T\times M_1T}, \bar{\mathbf{H}}_{ij}=\operatorname{diag}(h_{ij}(1),\cdots,h_{ij}(T))\in \mathbb{R}^{T\times T}, \bar{\mathbf{F}}_j =\operatorname{diag}(\mathbf{f}_j(1),\cdots,\mathbf{f}_j(T))\in\mathbb{R}^{M_2T\times T},
u(t) = e^{-(t-t_0) \nu_1 (-\Delta)^{\alpha}}u_0 - \int_{t_0}^t e^{-(t-s) \nu_1 (-\Delta)^{\alpha}}{\mathbb P}_H \div (u(s) \otimes u(s)-B(s)\otimes B(s)) \d s, B(t) = e^{-(t-t_0) \nu_2 (-\Delta)^{\alpha}}B_0 - \int_{t_0}^t e^{-(t-s) \nu_2 (-\Delta)^{\alpha}}{\mathbb P}_H \div (B(s) \otimes u(s)-u(s)\otimes B(s)) \d s,
\bar{\Psi}^{\prime \rho}{}_{r}( z^{\prime}) = \Omega ( z ; g )^{- \eta}D^{\rho}{}_{\sigma}( L^{- 1}( z ; g ) ) D_{r}{}^{s}( U^{- 1}( z ; g ) ) \bar{\Psi}^{\sigma}{}_{s}( z ) \, .
P^{b}(x_{1}, x_{2}, x_{3}) = \left ( \begin{array}{ccc}0 1 0 begin{align*}-0.2cm]-1 0 -\dfrac{b_{3}}{b_{1}}x_{1}begin{align*}-0.2cm]0 \dfrac{b_{3}}{b_{1}}x_{1}0 \end{array}\right ).
\int_{0}^{\infty}\frac{e^{-x}}{1+x}dx=\int_{0}^{1}\frac{dx}{1-lnx}=\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{2}{1+\frac{2}{1+\frac{3}{1+3/\cdot\cdot\cdot}}}}}}
\eta^{I J}P_{I}^{x}P_{J}^{x}= \eta^{I J}\big ( h_{u v}k_{I}^{u}k_{J}^{v}- m_{I}^{A}m_{J}^{B}\rho_{A B}\big ) = \eta^{I J}h_{u v}k_{I}^{u}k_{J}^{v}\, ,
S \, K_{[ p - 2 m , m ]}\, T = d_{[ p - 2 m , m ]}\sum_{n = m}^{[ p / 2 ]}e_{p - 2 n , n - m}\, S \bigl ( K^{\otimes ( p - 2 n )}\otimes K_{[ 0 , 1 ]}^{\otimes n}\bigr ) T
(P^U,P^\rho,P^S) = - A_U f_{J_{2,l}}^{(l)}D_r(U,\rho,S) - \big( A_r f_{J_{2,l}}^{(l)},(D_r(A_U f_{J_{2,l}}^{(l)}) + \tfrac{n-1}{r}A_U)\rho, 0 \big)
\vec{\sigma}\cdot \vec{p}\psi_{R(2)}+ e \vec{\sigma}\cdot (\vec{t}\wedge \hat{r}) A(r) \psi_{R(2)}- \frac{1}{2}v \psi_{L(2)}+ \frac{3 v}{2}\vec{t}\cdot \hat{r}\psi_{L(2)}\phi(r) = 0,
\lVert x\rVert_{L_p}=\sup\Bigl\{\,\Bigl|\int_Ex(t)y(t)\,dt\Bigr|:\,\lVert y\rVert_{L_q}\le1\,\Bigr\}, \lVert y\rVert_{L_q}=\sup\Bigl\{\,\Bigl|\int_Ex(t)y(t)\,dt\Bigr|:\,\lVert x\rVert_{L_p}\le1\,\Bigr\}.
V = \left( \begin{matrix}{{v}}\\{{- v}}\\ \end{matrix}\right) , \quad \bar{M}V = i \left( \begin{matrix}{{A}}&{{B}}\\{{B}}&{{A}}\\ \end{matrix}\right) \left( \begin{matrix}{{v}}\\{{- v}}\\ \end{matrix}\right) = i \lambda V .
(\lambda_{1}\ell_{\bf i}+\lambda_{2}\ell_{\bf j}+\lambda_{3}\ell_{\bf k}+c_{0}u_{0}+c_{1}u_{\bf i}+c_{2}u_{\bf j}+c_{3}u_{\bf k}+d_{1}[u_{0},\,u_{\bf i}]+d_{2}U_{\bf j}+d_{3}U_{\bf k}) &=(\lambda_{1}\ell_{\bf i}+\lambda_{2}\ell_{\bf j}+\lambda_{3}\ell_{\bf k})=0.
\left(\begin{matrix}x=0 x=1 x=\infty \, \left( \frac13 \right) \begin{matrix}0 \theta^0_1 \theta^0_2 \end{matrix}\begin{matrix}0 0 \theta^1 \end{matrix}\overbrace{\begin{matrix}t^{\frac13}\theta^\infty_1/3 \omega t^{\frac13}\theta^\infty_1/3 \omega^2 t^{\frac13}\theta^\infty_1/3 \end{matrix}}\end{matrix}\right) ,
m(l):=\sum_{k_1,\ldots,k_l=0}^{\infty}\sum_{i_1,\ldots,i_l=1}^n 2^{(k_1+\ldots+k_l)r}d_{i_1}^r\ldots d_{i_l}^r\Pr(B_{i_1,k_1,\ldots,i_l,k_l}).
a^{ij}(x',x_{n+1}) =a^{ij}(x',-x_{n+1}),i,j=1,\dots,n, a^{n+1,j}(x',x_{n+1}) =-a^{n+1,j}(x',-x_{n+1}),j=1,\dots,n, a^{n+1,n+1}(x',x_{n+1}) =a^{n+1,n+1}(x',-x_{n+1}),
\begin{array}{l}\lim\limits_{r\rightarrow 0}\frac{d(-r+\sqrt{r^2\!-\!x_{nj}})}{dr}=(\frac{B_nP_n}{B_mP_m})^{\frac{1}{\alpha_n}}(R_{mk})^{\frac{\alpha_m}{\alpha_n}}\frac{-\cos(\theta_{mk})}{R_{mk}}-1=-\frac{R_n^{lb}}{R_{mk}}\cos(\theta_{mk})-1\end{array}
\frac{d}{dt}(\phi_X)_{_\Sigma}(0) = \mu\frac{\beta_{_\Sigma}^2-\alpha_{_\Sigma}^2}{2}, \frac{d^2}{dt^2}(\phi_X)_{_\Sigma}(0) = \frac{\alpha_{_\Sigma}^3-\beta_{_\Sigma}^3}{3}.
(P_h\delta_t v^{m+1/2}, v^{m+1/2}) = \left(P_h \frac{1}{\tau}(v^{m+1}- v^m) , \frac{1}{2}(v^{m+1}+ v^m)\right) & = \frac{1}{2\tau}\left( (P_h v^{m+1}, v^{m+1}) - (P_h v^{m}, v^{m}) \right) &= \frac{1}{2\tau}( \|v^{m+1}\|_P^2 - \|v^{m}\|_P^2 )
[0,n-1][1+k,n-2-k] = [0,1+k][n-2-k,n-1] - [0,n-2-k][1+k,n-1] begin{align*}2mm]= [0,1+k][n-2-k,n-1] - [0,n-2-k][1+k,n-1] + [1+k,n-1][n-2-k,n-1] - [1+k,n-1][n-2-k,n-1] begin{align*}2mm]= [n-2-k,n-1] \Big([0,1+k] + [1+k,n-1] \Big) - [1+k,n-1] \Big( [0,n-2-k] + [n-2-k,n-1] \Big) begin{align*}2mm]= -[n-2-k,n-1] q_{2+k}+ [1+k,n-1] q_{n-k-1},
\int \frac{d \, p_{k}}{( 2 \pi )^{D}}\exp \left\{i \left[ p_{k}\frac{x_{k}- x_{k - 1}}{\Delta \tau}-{\cal H}( \lambda_{k}, \chi_{k}, \overline{{{x}}}_{k}, p_{k}) \right] \Delta \tau \right\}\, ,