harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_g{}_{\nu\bar\nu}\langle O_2 | \biggl( \Sigma_{i,j=3}^{k}(g^\dagger \lambda^{[12]}g)^i_j(\nu^3_i{\bar\nu}^{4j}-\nu^4_i{\bar\nu}^{3j})\biggr)^{k-2}{| O_k\rangle}_{\nu\bar\nu}
\mathcal{M}\left\{e^{-i\rho{x}};s\right\}=\left(i\rho\right)^{-s}\Gamma(s)={\rho}^{-s}\Gamma(s)\left(\cos\left(\frac{\pi{s}}{2}\right)-i\sin\left(\frac{\pi{s}}{2}\right)\right);~~\rho\in{\mathbb{C}},
\sum_{m_1,\ldots,m_n \geq 0}c_{m_1}\cdots c_{m_n}a_{m_1+\ldots+m_n+r}\leq \sum_{m_1,\ldots, m_n \geq n}c_{m_1}\cdots c_{m_n}a_{m_1+\ldots+m_n+r}+ n\sum_{m=0}^{n-1}c_m \sum_{m_1,\ldots,m_{n-1}\geq 0}c_{m_1}\cdots c_{m_{n-1}}a_{m_1+\ldots+m_{n-1}+m+r}.
\frac{d}{dx}x^{k+1}=\frac{d}{dx}(x^{k}\cdot x)=x^{k}\cdot\frac{d}{dx}x+x\cdot\frac{d}{dx}x^{k}=x^{k}+x\cdot kx^{k-1}=x^{k}+kx^{k}=(k+1)x^{k}=(k+1)x^{(k+1)-1}
S_{\mathrm{m a s s i v e \I I A}}( C^{( 9 )}) \sim \int d^{1 0}x{\sqrt{| g |}}\left\{e^{- 2 \phi}\left[ R - 4 ( \partial \phi )^{2}\right] - \frac{1}{2 \times 1 0 !}( G^{( 1 0 )})^{2}+ \cdots \right\}\, .
\left( \partial_{\alpha}\partial^{\alpha}+ M^{2}\right) T^{( \beta \gamma ) \nu}= \frac{1}{4}\, J^{( \beta \gamma ) \nu}+ \frac{1}{4}J^{( \gamma \beta \nu )}+ \frac{1}{6 M^{2}}\partial_{\alpha}\partial^{\alpha}J^{\left( \gamma \beta \nu \right)},
P Q^{*}- Q P^{*}= P Q^{-1}(Q Q^{*}) - Q (P^{*}Q) Q^{-1}= P Q^{-1}(Q Q^{*}) - Q (Q^{*}P - 2{\rm i}I_{d}) Q^{-1}= ( \mathcal{A}+{\rm i}\mathcal{B}) \mathcal{B}^{-1}- \mathcal{B}^{-1}( \mathcal{A}+{\rm i}\mathcal{B}) + 2{\rm i}I_{d}= [\mathcal{A}, \mathcal{B}^{-1}] + 2{\rm i}I_{d},
I_{\mathrm{GMI}}= \sup_{a\in\mathbb{C},\theta<0}\Bigg(\theta\mathbb{E}_{x,\mathbf{z},\mathbf{h}}[|f(x,\mathbf{h},\mathbf{z})-ax|^2]- &\frac{\theta\mathbb{E}_{x,\mathbf{z},\mathbf{h}}[|f(x,\mathbf{h},\mathbf{z})|^2]}{1-\theta|a|^2\mathcal{E}_\mathrm{s}}\!+\!\log(1-\theta|a|^2\mathcal{E}_\mathrm{s})\Bigg).
(c - \delta/2) |\tilde{\xi}|^2 \leq \varepsilon \, \tilde{\xi}\cdot \tilde{\xi}+ (\varepsilon_k - \varepsilon) \, \tilde{\xi}\cdot \tilde{\xi}= \varepsilon_k \, \tilde{\xi}\cdot \tilde{\xi}\leq (c- \delta) |\tilde{\xi}|^2,
f_{d-1}(k) =\frac{(k+1)(d-1)^k(d-2)-[(d-1)^{k+1}-1]}{(d-2)^2}&\leq \frac{(k+1)(d-1)^k(d-2)-(d-1)^{k}(d-2)}{(d-2)^2}&= \frac{k(d-1)^{k}}{(d-2)}
(I-\mathbb{H})\frac{Z_{\alpha}}{(Z(\alpha,t)-z_j(t))^2}=- \partial_{\alpha}(I-\mathbb{H})\frac{1}{Z(\alpha,t)-z_j(t)}= -\partial_{\alpha}\frac{2}{(\Phi^{-1})_z(\Phi(z_j(t)))(\alpha-\Phi(z_j(t)))}= \frac{2}{(\Phi^{-1})_z(\Phi(z_j(t)))(\alpha-\Phi(z_j(t)))^2}.
\begin{aligned}I^{[k]}:I^{[\ell]}\ =\ (I_0^{[k]}:I^{[\ell]})+\!\!\!\!\! \sum_{\substack{p\in[s] \nu(I_p)\ge k-1}}\!\!\!\!(u_pI_p^{[k-1]}:I^{[\ell]}).\end{aligned}
\mathbf{Q}^{\prime}_\mathrm{z}=\mathbf{U}_{\mathbf{\tilde{A}}}\sqrt{\frac{1}{\tilde{\lambda}}\mathbf{\Lambda}_{\mathbf{\tilde{A}}}+\frac{1}{4}\mathbf{\Lambda}_{\mathbf{\tilde{A}}}^2}\mathbf{U}_{\mathbf{\tilde{A}}}^{\rm H}-\frac{1}{2}\mathbf{\tilde{A}}+(\tilde{\epsilon}-1)\mathbf{D}_0
\frac{(q^d;q^d)_{n-1}^d}{(1-q)^{dn-d}}=\prod_{1\leq m< n}\bigg(\frac{1-q^{md}}{1-q}\bigg)^d\equiv 0\pmod{\prod_{\substack{1< m<n m\mid n}}\Phi_m(q)^2}.
\left(\frac{\partial}{\partial \lambda}\right)^d\omega_\mu=\prod_{i=1}^n\left(\frac{\partial}{\partial\mu_i}\right)^{w_i}\omega_\mu , \left(\sum_{i=1}^na_{ij}\mu_i\frac{\partial}{\partial\mu_i}+c_j\right)\omega_\mu = 0.
\left( \begin{array}{c}{\vec{F}^{( 1 )}}{- \vec{F}^{( 2 )}}\end{array}\right) = \frac{1}{\sqrt{2}}\left( \begin{array}{c c}{1}{1}{- 1}{1}\end{array}\right) \left( \begin{array}{c}{\vec{F^{R}}}{\vec{F}^{L}}\end{array}\right) \, .
\begin{array}{ll}\mathcal S:\mathfrak d\times \mathfrak d\rightarrow \mathfrak n, \mathcal S(X,Y):= \nabla_X\nabla_Y-\nabla_{XY}\ \ \mathcal R:\mathfrak d\times \mathfrak d\rightarrow \mathfrak n, \mathcal R(X,Y):=\mathcal S(X,Y)-\mathcal S(Y,X)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}\end{array}
Li_{s}(z)=\frac{\Gamma(1-s)}{(2\pi)^{1-s}}[i^{1-s}\zeta(1-s,\frac{1}{2}+\frac{ln(-z)}{2\pi i})+i^{s-1}\zeta(1-s,\frac{1}{2}-\frac{ln(-z)}{2\pi i})],
\frac{\mathbf{P}\Big( \frac{1}{\sigma}(\hat{\theta}_n -\theta )\sqrt{\Sigma_{k=1}^n X_{k-1}^2}< - x \Big)}{1-\Phi \left( x\right)}=\exp\bigg\{\theta_2 C \frac{\ln n +x^{3}}{\sqrt{n}}\bigg \},
U^{(N-1)}=\begin{pmatrix}u_{11}^{(N-1)}\ldots u^{(N-1)}_{1(N-1)}0 u_{21}^{(N-1)}\ldots u^{(N-1)}_{2(N-1)}0 \vdots \vdots \vdots \vdots u_{(N-1)1}^{(N-1)}\ldots u^{(N-1)}_{(N-1)(N-1)}0 u_{N1}^{(N-1)}\ldots u^{(N-1)}_{N(N-1)}u^{(N-1)}_{NN}\end{pmatrix}.
D_{\mu \nu}(p,\xi) = \frac{1}{p^2}\left( \delta_{\mu \nu}- \frac{p_\mu p_\nu}{p^2}\right) + \xi \frac{p_\mu p_\nu}{p^4}= D_{\mu \nu}^T (p) +\xi \frac{p_\mu p_\nu}{p^4},
C^{\prime}_( B_) = \frac{1}{L}\sum_{j=1}^L (1 -\frac{\sum_{k=1}^{j-1}b_{k}^0}{M}) H( \frac{b_{j}^0}{M - \sum_{k=1}^{j -1}b_k}) & + \frac{\sum_{k=1}^{j-1}b_{k}^0}{M}, \forall k \in[2:L],
\hat{\cal R}= Q^{2 \left( \hat{\zeta}\otimes \hat{H}+ \hat{H}\otimes \hat{\zeta}\right)}\lambda^{2 \left( \hat{\zeta}\otimes\hat{H}- \hat{H}\otimes \hat{\zeta}\right)}\left( 1\!{\rm l}\otimes1\!{\rm l}+ \sigma Q^{\hat{\zeta}}\lambda^{\hat{\zeta}}\hat{X}_+\otimes Q^{-\hat{\zeta}}\lambda^{\hat{\zeta}}\hat{X}_- \right)
M_{11}(e_1)=M_{11}(e_1^{-1})= M_{12}(e_1^{-1})+M_{12}(e_2) = M_{21}(e_1)+M_{21}(e_2^{-1})= =M_{22}(e_1)+M_{21}(e_2^{-1})= M_{22}(e_1)+ M_{22}(e_2^{-1})+M_{22}(e_3).
\Psi_{( 1 )}= \sqrt{\frac{V}{2}}\left( \begin{matrix}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{matrix}\right) , \Psi_{( 2 )}= \sqrt{\frac{V}{2}}\left( \begin{matrix}{{\frac{1}{\tau}}}&{{0}}\\{{0}}&{{- \tau}}\\ \end{matrix}\right) .
g ( x_{+}) = \left( \begin{array}{c c}{{\cos ( \omega x_{+})}}&{{- \sin ( \omega x_{+})}}\\{{\sin ( \omega x_{+})}}&{{\cos ( \omega x_{+})}}\\ \end{array}\right) \quad ; \qquad J ( x_{+}) = \left( \begin{array}{c c}{{0}}&{{- \omega}}\\{{\omega}}&{{0}}\\ \end{array}\right) \, \, .
\int_{P^{i}(\zeta)}\left|\frac{\delta_{\Omega}(z)}{\delta_{\Omega}(\zeta)}\right|^{\gamma'-\varepsilon}\frac{d\lambda(z)}{\left|z-\zeta\right|}\lesssim_{\gamma'-\epsilon}\tau_{n}\left(\zeta,2^{i}\delta_{\Omega}(\zeta)\right)\prod_{j=1}^{n-1}\tau_{j}^{i}\left(\zeta,2^{i}\delta_{\Omega}(\zeta)\right),
\sum_{k=0}^a \binom{a}{k}\binom{b}{m+k}\binom{k}{c}\binom{m+k}{d}k^3 = \binom{a+b-c-d}{m+a-d}\binom{a}{c}\binom{b}{d}\cdot \frac{f_2(m,a,b,c,d)}{(a+b-c-d)(a+b-c-d-1)(a+b-c-d-2)}
\vartheta_{4}(x)[\frac{d}{dx}\vartheta_{3}(x)]-\vartheta_{3}(x)[\frac{d}{dx}\theta_{4}(x)]=\frac{1}{4x}\theta_{3}(x)\theta_{4}(x)[\theta_{3}(x)^{4}-\theta_{4}(x)^{4}]
\widetilde{F}^\delta\begin{pmatrix}z x \end{pmatrix}= \begin{pmatrix}z+ z^2 + a_3^\delta z^2 + \dots + a_l^\delta z^l +a_{l+1}^\delta(x)z^{l+1}+\dots b_0(x) + b_1^\delta(x)z+ b_2^\delta(x)z^2 + \dots \end{pmatrix},
w_e^t = w_e\mathbf{1}_{\{|w_e|\leq L\}}+ tw_e\mathbf{1}_{\{|w_e| > L\}}, \nu_x^t = \nu_x \mathbf{1}_{\{|\nu_x|\leq L\}}+ t\nu_x \mathbf{1}_{\{|\nu_x |> L\}},
\Phi_i: U_i\longrightarrow \mathbf{C}^n &[a]\longmapsto \Phi_i([a]):=\left(\frac{a_1}{a_i},\dots,\frac{a_{i-1}}{a_{i}},\frac{a_{i+1}}{a_{i}},\dots,\frac{a_n}{a_i}\right)
g_M(R^M(e_a',e_b')e_c',e_d')\eta^{ad}\eta^{bc}=e^{4\phi/3}g_N^\phi(R^{N\phi}(e_a,e_b)e_c,e_d)\eta^{ad}\eta^{bc}-\frac{3}{2}e^{8\phi/3}|G_2|^2 &=R^{N\phi}-\frac{3}{2}e^{8\phi/3}|G_2|^2 &=e^{2\phi/3}\left(R^N+6\Delta\phi-8|d\phi|^2\right)-\frac{3}{2}e^{8\phi/3}|G_2|^2.
|F_j(x_j)| \ll H^{2\delta}(\Psi(H)\, H^{n-m})^{-\frac1{m-1}}begin{align*}1ex] \le H^{2\delta}(H^{-(\ell_0+1)\delta}H^{n-m})^{-\frac1{m-1}}begin{align*}1ex] = H^{-\frac{n-m-\ell_0\delta-\delta(2m-1)}{m-1}}\qquad(2\le j\le m)\,.
F_{7 5}= \frac{1}{1 6 \cdot 2 ! 2 !}\partial^{i}l o g H \varepsilon_{i j_{1}. . . j_{4}}\Theta^{j_{1}}\wedge \Theta^{j_{2}}\wedge \bar{\Theta}^{j_{3}}\wedge \bar{\Theta}^{j_{4}}
{\cal L}= - \kappa \dot{B}\left( a \mp \frac{1}{\kappa}B^{\prime}\right) + i \psi^{\ast}\dot{\psi}- A_{0}( \kappa B^{\prime}+ \rho ) - \frac{1}{2 m}| ( \partial_{x}- i a ) \psi |^{2}- \frac{1}{2 m}B^{2}\rho
|\widetilde{T}^{1}_{B(0,r)}(x)| \leq\int_{B(0,2r)}|\widetilde{f}(y)|\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{Q}}dy = \sum_{k=-\infty}^{0}\int_{2^{k}r\leq|xy^{-1}|<2^{k+1}r}\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{Q}}|\widetilde{f}(y)|dy.
\mathcal{G}_{\mu\nu}=\eta_{MN}\partial_{\mu}X^M\partial_{\nu}X^N-2\eta_{MN}\partial_{\mu}X^M\overline{\theta}\Gamma^N\partial_{\nu}\theta+\eta_{MN}(\overline{\theta}\Gamma^M\partial_{\mu}\theta)(\overline{\theta}\Gamma^N\partial_{\nu}\theta)
{}\sum_{u=1}^{N}\left (t_{u}-t_{u-1}\right )({\bf a}^{(u)})^{\top}\bar{{\bf z}}^{(u)}=\sum_{u=1}^{N}\left (t_{u}-t_{u-1}\right )({\bf c}^{(u)})^{\top}\bar{{\bf d}}^{(u)}.
H_{\theta}= \Bigl ({\frac{\Omega_{\rho}}{\Omega^{2}}}\Bigr ){\frac{g_{a b}\pi_{0}^{a}\pi_{b}^{0}}{2}}+{\frac{\pi_{0}^{a}}{\Omega}}\int \rho^{- 2}g_{a b}\tilde{\pi}_{0}^{b}+ \tilde{H}
\frac{d(f^{\prime})^{-1}(z)}{dz}=\frac{dx}{dz}=\frac{dy}{dz}\frac{dx}{dy}=\frac{f^{\prime}(x)}{f^{\prime\prime}(x)}\frac{1}{f^{\prime}(x)}=\frac{1}{f^{\prime\prime}(x)}
\frac{(a^2+a)^3}{\bigl(ab^2+a^2b+x(a^2+a)\bigr)^3}=\frac{(a^2+a)^3}{\bigl(ab+\omega(a^3+a^2)+x(a^2+a)\bigr)^3}=\frac{1}{(b+\omega a+x)^3}
u(r,t)=\frac{P_{0}^{\prime}}{4\mu}(R^{2}-r^{2})+Re\{\sum_{n=1}^{N}\frac{iP_{n}^{\prime}}{\rho n\omega}[1-\frac{J_{0}(\Lambda_{n}\frac{r}{R})}{J_{0}(\Lambda_{n})}]e^{in\omega t}\}.
\prod_{a = 1}^{N}\overline{{{\psi}}}_{\alpha}^{( a )}( x^{( \pi ( a ) )}) = \mathrm{s i g n}( \pi^{- 1}) \prod_{a = 1}^{N}\overline{{{\psi}}}_{\alpha}^{( \pi^{- 1}( a ) )}( x^{( a )}) \; ,
\index{$\kappa_{\rm c}^{(\alpha)}(\ell)$}\kappa_{\rm c}^{(\alpha)}(\ell) := \Phi_{d,\ell}^{(\alpha)}\left(\frac{d-\alpha}{2}\right)=\frac{2^{\alpha}\Gamma \left(\frac{1}{4}(d+2\ell+\alpha)\right)^2}{\Gamma \left(\frac{1}{4}(d+2\ell-\alpha)\right)^2}.
X_{\pm}( s ) = \frac{\beta}{2}\sum_{n}\left[ E_{n}^{2}+ \frac{\lambda \phi^{2}}{3}+ \mu^{2}\pm 2 \mu E_{n}\pm \frac{\mu \lambda \phi^{2}}{3 E_{n}}\pm \frac{\lambda^{2}\phi^{4}}{1 4 4 \mu E_{n}}\mp \frac{\mu \lambda^{2}\phi^{4}}{3 6 E_{n}^{3}}\right]^{- \frac{s}{2}}\; .
\begin{array}{c c}{{{\cal B}{\cal M}{\cal B}^{- 1}= -{\cal M}^{\dagger}~}}&{{~{\cal B}= \left( \begin{array}{c c c}{{0}}&{{1}}&{{0}}\\{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- 1}}\\ \end{array}\right)}}\\ \end{array}
g^{2}\left( \begin{array}{c | c}{{- \frac{1}{\bar{D}^{2}}\delta_{\sigma}\nu + \frac{1}{\bar{D}^{2}}\bar{D}_{\sigma}\frac{1}{E}\bar{D}_{\nu}\frac{1}{\bar{D}^{2}}}}&{{- i \frac{1}{\bar{D}^{2}}\bar{D}_{\sigma}\frac{1}{E}}}\\{{i \frac{1}{E}\bar{D}_{\nu}\frac{1}{\bar{D}^{2}}}}&{{\frac{1}{E}- 1}}\\ \end{array}\right)
ABf(x) =\int_{\Omega}K_{A}(x,y) (Bf)(y)dy &=\int_{\Omega}K_{A}(x,y)\Big[\int_{\Omega}K_{B}(y,z)f(z)dz\Big]dy &=\int_{\Omega}\int_{\Omega}K_{A}(x,y) K_{B}(y,z)f(z)dzdy.
\sum_{m=0}^\infty C[a_1, \dots, a_r, \underbrace{N, \dots, N}_] (1 -Y)^m = &(1-Y) \sum_{m=0}^\infty \operatorname{SymC}[a_1, \dots, a_r, \underbrace{N, \dots, N}_] (1 - Y)^m + \quad\sum_{i=1}^r \operatorname{SymC}[a_1, \dots, \hat{a}_i , \dots, a_r, \underbrace{N, \dots, N}_] (1 - Y)^m.
\mathbb{E}_0\widetilde{\tau}^{(n)}_{M_n(a)}=\frac{1}{4}n\log n+O(n)+\frac{1+o(1)}{2c_n\pi_n(M_n)},_0\widetilde{\tau}^{(n)}_{M_n(a)}\asymp n^2+\frac{n}{c_n^2},
\big({\mathbb D}^2\big)^{A}{}_{B}= \,{\mathbb D}^{A}{}_{R}{\mathbb D}^{R}{}_{B}+ \,{\mathbb D}^{R}{}_{B}{\mathbb D}^{A}{}_{R},\, \langle{\mathbb D},{\mathbb D}\rangle = \,{\mathbb D}^{S}{}_{R}{\mathbb D}^{R}{}_{S},\,{\mathbb D}^2_o = \, \mathrm{tf}\big({\mathbb D}^2\big).
-\nabla\cdot w_{m}(qC-\theta_{s}S_{w}D\nabla C)+[w_{m}\theta_{s}S_{w}R\lambda C]_{par}+\sum\Omega_{ex}+Q_{c}=w_{m}[\frac{\partial(\theta_{s}S_{w}RC)}{\partial t}+\theta_{s}S_{w}R\lambda C]
\Big[ \frac{1}{p^{n -m + k}}A_{k, p}\Big]_0 =(2 \pi)^k \cdot k! \cdot \kappa_N^{-1}|_Y \cdot{\rm{Id}}_{{\rm{Sym}}^k N^{1, 0}_{y_0}}\otimes{\rm{Id}}_{F_{y_0}}.
\mathbb{H}_n=\{\,Z\in_n(\mathbb{C})\,\mid\, (Z)>0\,\}\;\;(), & \Lambda_n:=\{\,T=(t_{jl})\in_n(\mathbb{Q})\,\mid\,t_{jj}\in\mathbb{Z},\,2t_{jl}\in\mathbb{Z}\,\}.
G ( x_{1}, x_{2}, x_{3}, x_{4}) = ( x_{1 2}^{2})^{- \frac{1}{2}( \mu_{1}+ \mu_{2}- \mu_{3}+ \mu_{4})}( x_{1 3}^{2})^{- \frac{1}{2}( \mu_{1}- \mu_{2}+ \mu_{3}- \mu_{4})}\, ( x_{2 3}^{2})^{\frac{1}{2}( \mu_{1}- \mu_{2}- \mu_{3}+ \mu_{4})}( x_{3 4}^{2})^{- \mu_{4}}\, \tilde{G}( u , v )
\Pi ( k_{0}) = e^{2}{\frac{\mathrm{s e c h}^{2}{\frac{\beta m}{2}}}{1 + \tanh^{2}{\frac{\beta m}{2}}\tan^{2}{\frac{\beta k_{0}}{2}}}}{\frac{\tan{\frac{\beta k_{0}}{2}}}{k_{0}}}
d^{\star}\widetilde{\chi}=-\sum_{k,l=1}^{n}\left({h^{kl}}\frac{\partial \chi_k}{\partial x^l}-\sum_{r=1}^n\Gamma^r_{lk}\cdot \chi_r\right)=\sum_{k,l=1}^{n-1}\left({\overline{h}^{kl}}\frac{\partial \chi_k}{\partial x^l}-\sum_{r=1}^{n-1}\overline{\Gamma}^r_{lk}\cdot \chi_r\right)=
\frac{[bb_1,b_1/b]_\infty(q)_\infty^2}{[b,b_1,b_1]_\infty}=\sum_{k=-\infty}^\infty\frac{(-b)^kq^{\frac{k(k+1)}{2}}}{1-bq^k}-\frac{[b]_\infty}{b[b_1]_\infty}\sum_{k=-\infty}^\infty\frac{(-b_1)^{k+1}q^{\frac{k(k+1)}{2}}}{1-b_1q^k}.
P_{G_2}^{\Delta_S}(3|x)=P_{D_4}^{{\cal R}}(x)/x^2\quad ({\cal R}={\bf V},{\bf S},{\bf \bar{S}}),\quad P_{G_2}^{\Delta_L}(3|x)\left(P_{G_2}^{\Delta_S}(3|x)\right)^3= P_{D_4}^{\Delta}(x).
\Lambda(f \otimes \mathrm{Ad}(g),k) = 2^{k+1}N^{-1}\prod_{p\mid N}(p+1)^2 \frac{\langle f,f \rangle}{\langle h,h\rangle}\frac{|\langle F_{|\mathcal H \times \mathcal H}, g \times g\rangle|^2}{\langle g,g\rangle^2}.
\int_{\mathcal S}W(\zeta)d\zeta=1,~\int_{\mathcal S}\zeta \,W(\zeta)d\zeta=0,~\int_{\mathcal S}\zeta^2\,W(\zeta)d\zeta= p_1^2=-n\,a_{2,0}a_{2,2},a_{2,0}a_{2,2}<0,
\begin{array}{ll}\frac{d}{d\sigma}\left( \frac{1}{\sigma^{k+d/2}}\right) \exp\left( -\frac{1}{4\sigma}\right) =\left(-\frac{k}{\sigma^{k+1+d/2}}+\frac{1}{4\sigma^{k+2+d/2}}\right) \exp\left( -\frac{1}{4\sigma}\right)\stackrel{!}{=}0 \mbox{for}~\sigma=\frac{1}{4k}\end{array}
\left| \frac{\partial}{\partial y_l}\bar{H}_{}(\mathbf{0}) \right| = \left| \frac{1}{N}\mathbb{E}_{\mu_{}(dx|0)}\left[ \sum_{i, j \in B(l)}M_{ij}X_i + \sum_{i \in B(l)}\delta \psi ' (X_i)\right] \right| & \overset{Lemma~\ref{l_ce_moment_estimate}}{\lesssim}\frac{1}{N}\left( 2KR^2 + K \right) \sim \frac{1}{M}.
\sup_{T\leq1}(T^{\frac{1}{4}})^{2-2\alpha}\| \partial_s (g_{s \tau})_T + (\partial_s a_s \partial_1^2 w_{s}^{\tau})_T - [a_s , (\cdot)_T ] \partial_1^2 \partial_s w^{\tau}_{s}\| \lesssim [a_s]_{\alpha}[\partial_s w_{s}^{\tau}]_{2\alpha}+ \delta N (N_0 + N_0^{int}) + \delta N_0 + \delta N_0^{int},
G(s, z) = \frac{\Gamma(s - \tfrac{1}{2}+ z)\Gamma(s - \tfrac{1}{2}- z)}{\Gamma(s)\Gamma(s+k-1)}, \; \mathcal{Z}(s,w,z) = \frac{\zeta(s + w -\frac{1}{2}+ z)\zeta(s + w -\frac{1}{2}- z)}{\zeta^*(1+2z)}.
\begin{array}{lll}X^* = J^{(r)}\hat{C}C\hat{C}^{-1}J^{(r)}X^* + J^{(r)}\Big[(\hat{C}C\hat{C}^{-1}-I)\underline{v}+ \hat{C}(Dp - U(J^{(r)}X^*) - \mathbf{1}\frac{\beta}{2})\Big]. \end{array}
f_R(s) = \begin{cases}s_0^{-1/p}\mbox{if $s\in (0,s_0)$}, s^{-1/p}\mbox{if $s\in [s_0,R)$}, R^{-1/p}\max\{2- s/R, 0\}\mbox{if $s\geqslant R$}.\end{cases}
\begin{aligned}\vert\Phi_{j}\ast T(a_{j})(x)\vert =\left\vert \int_{\mathbb R^{n}}\Phi_{t}(x-y)T(a_{j})(y)dy\right\vert \leq t^{-n}\int_{B(x,t)}\vert T(a_{j})(y)\vert dy \leq\sup\limits_{y\in B(x,t)}\vert T(a_{j})(y)\vert.\end{aligned}
\displaystyle u_t^i-\Delta\left[\left(\beta_i+\sum_{j=1}^{m}\alpha_{ij}u^j\right)u^i\right]= \displaystyle\left(a_i-\sum_{j=1}^{m}b_{ij}u^j\right)u^i,\quad\mbox{in}\O\times(0,T),
\chi^{s'}(a, b) =~ s' (a) \cdot_{E'}s' (b) - s' (a \cdot b) =~ (\varphi s)(a) \cdot_{E'}(\varphi s) (b) - (\varphi s)(a \cdot b) =~ \varphi \big( s(a) \cdot_E s(b) - s(a \cdot b) \big) = \varphi (\chi^s (a, b) ) = \chi^s (a, b)
\Pi_{1}\left(\displaystyle \sum_{b \in T_{k}, r \in L(k)}r^{-1}\overline{N}(^{\ast}b)\left(\pi_{br}(x)\right)\right)= -\displaystyle \sum_{b \in T_{k}, r \in L(k)}r^{-1}\pi_{1}p \xi_{be_{k}}\pi_{br}(x)
\breve{\omega}_{i}d x^{i}\simeq \sum_{n = - \infty}^{\infty}{\frac{m_{0}a \{( x^{1}- n ) d x^{2}- x^{2}d x^{1}\}}{\{\breve{R^{2}}+ ( x^{1}- n )^{2}\}^{\frac{5}{2}}}}\,
(\alpha(\xi)|\alpha(\zeta)) =L(\alpha(\xi)\otimes\Omega_\phi)^\ast L(\alpha(\zeta)\otimes\Omega_\phi) &=L(U(\xi\otimes\Omega_\phi))^\ast L(U(\zeta\otimes\Omega_\phi)) &=L(\xi\otimes\Omega_\phi)^\ast U^\ast U L(\zeta\otimes\Omega_\phi) &=(\xi|\zeta).
\Psi_{r,\alpha,\beta}(x) \leq \Big(\frac{\log_+ (1/x)}{\beta\log r}+ \frac{1}{1-r^{-\alpha}}\Big)x \hbox{if $\beta=\alpha$,}\Psi_{r,\alpha,\beta}(x) \leq \Big(\frac 1{r^{\beta-\alpha}-1}+ \frac 1{1-r^{-\alpha}}\Big) x^{\alpha/\beta}\hbox{if $\beta>\alpha$.}
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}^+}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_{d,s,k}T R^d \left[\alpha +\frac{y}{\eta T}+ C_{d,\alpha}\left( \frac{\eta}{R}\right)^{d-1}+ C_{d,\alpha}\theta^{(d-1)/2}\right]\end{aligned}
\Pi_{K,l}:\boldsymbol{L}^{2}(K)\rightarrow\mathbb{P}_{l}(K)^{d}, \left(\mathbf{t}-\Pi_{K,l}(\mathbf{t}),\mathbf{v}\right)_{\boldsymbol{L}^{2}(K)}=0\quad\forall \mathbf{v}\in\mathbb{P}_{l}(K)^{d}.
x_{ij}x_{jk}=x_{ik}x_{ij}+x_{jk}x_{ik}-\beta x_{ik}, x_{ij}y_{jk}=y_{ik}x_{ij}+y_{jk}y_{ik}-\beta y_{ik}, x_{ik}y_{jk}=y_{jk}y_{ij}+y_{ij}x_{ik}+\beta y_{ij},y_{ik}x_{jk}=x_{jk}y_{ij}+y_{ij}y_{ik}+\beta y_{ij}
Z ( \kappa ) = \int \! [ d x ( t ) ] \exp \left[ - \int \! d \tau \Biggl ({\frac{1}{2}}\Bigl ({\frac{d x}{d \tau}}\Bigr )^{2}- x^{2}+{\frac{\kappa x^{4}}{2 \sqrt{2}}}+{\frac{1}{\sqrt{2}\kappa}}\Biggr ) \right] ,
\left\langle{^0t}, \left( w- w_{\Lambda}\right)x^{\alpha_1}x^{\alpha_2}\right\rangle = - \left({D(\Lambda)^{\alpha_1}}_{\beta_1}{D(\Lambda)^{\alpha_2}}_{\beta_2}- \delta^{\alpha_1}_{\beta_1}\delta^{\alpha_2}_{\beta_2}\right) c^{\beta_1 \beta_2}.
S=\frac{M_{3}}{2}\int d^{3}x\sqrt{-g}(R-2\Lambda)+\frac{1}{4\mu}\epsilon^{\lambda\mu\nu}\Gamma_{\lambda\sigma}^{\rho}(\partial_{\mu}\Gamma_{\rho\nu}^{\sigma}+\frac{2}{3}\Gamma_{\mu\alpha}^{\sigma}\Gamma_{\nu\rho}^{\alpha}),
\frac{1}{R^4}\sum_{a=0}^{2l}\sum_{b=0}^{2l}\sum_{m_a = -a}^{a}\sum_{m_b = -b}^{b}= \frac{1}{{\pi}^2}\int_{\Lambda}d^2\vec{p}_a \int_{\Lambda}d^2\vec{p}_b~,~{\Lambda}=\frac{2}{\theta},
A_n^-(n-1) =2^{n-3}+\sum_{j=2}^{n-3}a(n;n-1,j)+a(n-3) &=2^{n-3}+a(n-3)+\sum_{j=2}^{n-3}a(n-1;n-2;j)+(2^0+2^1+\cdots+2^{n-5}) &=2^{n-3}+a(n-3)+A_{n-1}^-(n-2)-2^{n-4}+2^{n-4}-1 &=A_{n-1}^-(n-2)+a(n-3)+2^{n-3}-1,
S_1M_1(0)\big(I_0(0)+S_0\big)^{-1}M_1(0)S_1 =S_1(X+iZ^*Z)\big(I_0(0)+S_0\big)^{-1}(X+iZ^*Z)S_1 &=S_1X\big(I_0(0)+S_0\big)^{-1}XS_1.
\lambda_1(t) = \lambda_1(\tau_1) \lambda_2(t) = \lambda_2(\tau_1) + \lambda_0 b_M (\tau_1 - t) \lambda_3(t) = \lambda_3(\tau_1) + \frac{1}{2}\lambda_0 b_M k_M (\tau_1 - t)^2 + (\lambda_0 b_E - a_M b_M k_M \lambda_1(\tau_1) + k_M \lambda_2(\tau_1)) (\tau_1 - t).
M_{\lambda,\mu}=\left\{m\in M \ \middle\vert\K_i\cdot m=q_i^{\langle\lambda,\alpha_i^\vee\rangle}m,\L_i\cdot m=q_i^{\langle\mu,\alpha_i^\vee\rangle}m,\ \forall 1\leq i\leq n\right\}.
( \gamma ) = \{S^{g + 1};{S_{2}}^{\prime},{S_{3}}^{\prime}, \cdots ,{S_{\mu}}^{\prime}\}[ \hspace{- 0 . 7 m m}[ S^{( g )}] \hspace{- 0 . 7 m m}] [ \hspace{- 0 . 7 m m}[ S^{( g - 1 )}] \hspace{- 0 . 7 m m}] \cdots [ \hspace{- 0 . 7 m m}[ S^{( 2 )}] \hspace{- 0 . 7 m m}] [ \hspace{- 0 . 7 m m}[ S^{( 1 )}] \hspace{- 0 . 7 m m}] ,
g_{l,l^{\prime}}^{j}\left( R\right) =\left( 2j+1\right) \sum_{m,m^{\prime}}\left( -1\right)^{m}\left(\begin{array}[c]{ccc}l l^{\prime}j m -m^{\prime}0\end{array}\right) G_{lm,l^{\prime}m^{\prime}}\left( R\mathbf{\hat{e}}_{z}\right) .
\left(\pi(1),\cdots,\pi(i_1-1),\pi(i_2),\cdots,\pi(j_2),\pi(j_1+1),\right. \left.\cdots,\pi(i_2-1),\pi(i_1),\cdots,\pi(j_1),\pi(j_2+1),\cdots,\pi(N)\right).
0 \longrightarrow \Omega^2_{\mathbb{P}^n}(d+2) \otimes \mathcal{I}_Z \longrightarrow \mathcal{I}_Z(d)^{\oplus \binom{n+1}{2}}\longrightarrow \Omega^1_{\mathbb{P}^n}(d+2) \otimes \mathcal{I}_Z \longrightarrow 0
V = - \Gamma ( 3 / 2 ) \frac{( \sin^{2}( \pi \epsilon ) + \sinh^{2}( \pi \nu ) )}{\cos ( \pi \epsilon ) \sqrt ( 8 \pi^{2}\alpha^{\prime})}( \frac{2 \pi \alpha^{\prime}}{r^{2}})^{3 / 2}.
x_{i,1,k}= 2m-3i+k+1, \tfrac{1}{2}S -x_{i,1,k}= 4m+3i-k, x_{i,2,k}= 4m+3i-k-1, \tfrac{1}{2}S - x_{i,2,k}= 2m-3i+k+2, x_{i,3,k}= 4m-3i+2k+1, \mbox{and}\tfrac{1}{2}S - x_{i,3,k}= 2m+3i-2k.
\left\{\phi ( x ) , \phi ( y ) \right\}= \phi^{\prime}( x ) \phi^{\prime}( y ) \epsilon ( x - y ) +{\frac{1}{\alpha}}\left\{\phi^{\prime}( y ) - \phi^{\prime}( x ) \right\}\delta ( x - y ) -{\frac{1}{\alpha^{2}}}\delta^{\prime}( x - y )
\phi ( z ) = \frac{c o n s t}{\sinh z}\exp \int_{0}^{\infty}\frac{d t}{t}\frac{\sinh \frac{1}{2}( 1 - \nu ) t \left( \cosh t ( \frac{1}{2}- z / ( i \pi ) ) - 1 \right)}{\sinh \frac{1}{2}\nu t \sinh t}
{\cal L}_{g , 4 A}= - \frac{1}{4}\sum_{n , m , l , k \neq ( 0 , 0 , 0 )}( \sum_{j = 0}^{N}a_{j n}a_{j m}a_{j l}a_{j k}) g_{L}f^{a b c}g_{L}f^{a d e}\tilde{A}_{\mu}^{n}\tilde{A}_{\nu}^{m}\tilde{A}^{l \mu}\tilde{A}^{k \nu},
{\cal L}=-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}-\,\frac{1}{12}\,H_{\mu\nu\rho}\,H^{\mu\nu\rho}\,+{m\over4}\,\epsilon^{\mu\nu\rho\lambda}\,B_{\mu\nu}F_{\rho\lambda}\,,
F_{2}( q^{2}) ={\frac{M e^{2}}{4 \pi^{2}}}\int_{0}^{1}d \alpha \int_{0}^{1}d x{\frac{m - x M}{\alpha ( 1 - \alpha ){\frac{1 - x}{x}}{\vec{q}}_{\perp}^{2}- M^{2}+{\frac{m^{2}}{x}}+{\frac{\lambda^{2}}{1 - x}}}}\ .
\left( 2 \pi i K_{+}^{2}- \frac 3 2 k_{0}K_{+}- \frac{k^{2}k_{0}}{3 2 \pi^{2}K_{+}}\right) - \left( 2 \pi i K_{-}^{2}- \frac 3 2 k_{0}K_{-}- \frac{k^{2}k_{0}}{3 2 \pi^{2}K_{-}}\right) = 0
\frac{\mathrm{d}}{\mathrm{d}s}\Phi_s\left(\frac{x-\mu}{\sigma}\right) =\frac{1}{2\Gamma(1/s)}\begin{cases}s^2\tilde\psi\left( s, (\frac{\mu-x}{\sigma})^s\right)+ \psi(1/s)\Gamma(1/s, (\frac{\mu-x}{\sigma})^s), x\leq\mu, -[s^2\tilde\psi\left( s, (\frac{x-\mu}{\sigma})^s\right)+ \psi(1/s)\Gamma(1/s, (\frac{x-\mu}{\sigma})^s)], x>\mu.\end{cases}
P=-\Delta_{\bf y}-2\lambda_{\alpha}H_{\alpha}^{\lambda_{\alpha}}({\bf y})\cdot\nabla_{\bf y}& +\big[\lambda_{\alpha}V_{\alpha}^{\lambda_{\alpha}}({\bf y})+\lambda_{\alpha}^2(K_{\alpha}^{\lambda_{\alpha}}({\bf y})-E)\big]
\int \frac{d^{d}k_{1}}{[ i \pi^{d / 2}]}( q k_{1}) P_{k_{1}, m_{1}}^{\nu_{1}+ 1}P_{k_{1}- q , m_{2}}^{\nu_{2}}= \nu_{2}q^{2}I_{\nu_{1}+ 1 \nu_{2}+ 1}^{( d + 2 )}.
R^{(1)}(z) = \sum_{j=1}^{m}\frac{1}{z+x_{j}}\mbox{Res}(J_{R}^{(1)}(s),s = -x_{j}), \mbox{for}z \in \mathbb{C}\setminus \bigcup_{j=1}^{m}\mathcal{D}_{-x_{j}}
\Upsilon_{1}= - 4 \pi i \left( \begin{matrix}{{\rho + j_{3}}}\\{{j_{1}+ i j_{2}}}\\ \end{matrix}\right) \quad , \quad \Upsilon_{2}= - 4 \pi i \left( \begin{matrix}{{j_{1}- i j_{2}}}\\{{\rho - j_{3}}}\\ \end{matrix}\right) \quad .
\left( T_{1}-T_{2}\right) h_{2m+1}^{+}=-\left( N\kappa-\kappa+m\right)h_{2m}^{-}, \left( T_{1}-T_{2}\right) h_{2m}^{+}=-\left( N\kappa-\kappa+m\right)h_{2m-1}^{-}, \left( T_{1}+T_{2}\right) h_{2m+1}^{+}=2\left( N\kappa+m+1\right)h_{2m}^{+}, \left( T_{1}+T_{2}\right) h_{2m}^{+}=\left( N\kappa-\kappa+m\right)h_{2m-1}^{+}.
V^TXU = \begin{bmatrix}X_1 X_2 X_3 \end{bmatrix}Q_{\perp}\hat{Q}= \begin{bmatrix}X_1 Q_{\perp}\hat{Q}0 X_3 Q_{\perp}\hat{Q}\end{bmatrix}= \begin{bmatrix}\hat{R}^{-1}0 B\hat{R}^{-1}\end{bmatrix}
6c^3 \le (d^3-d) +\left(\sqrt{3/2}c - d\right)\left(\sqrt{3/2}c - (d-1)\right)\left(\sqrt{3/2}c - (d-2)\right) &=\sqrt{27/8}c^3 - 9/2(d-1) c^2 + (3d^2-6d+2)\sqrt{3/2}c + (3d^2-3d).
\frac{\delta C_{Q}( \kappa )}{\delta \sigma ( x )}|_{\sigma = 0}= - 2 \lambda^{2}\sqrt{g ( x )}\Psi_{\kappa}( x ) - 2 \int d \mu C_{Q}( \mu ) \Psi_{\kappa}( x ) \Psi_{\mu}( x ) \, \frac{\kappa^{2}}{\kappa^{2}- \mu^{2}}\sqrt{g ( x )}
\mu_t^{*,t_0,\xi}:=\rho\big(-R^{-1}B\mathbb E\big[U(t,x_t^{*,t_0,\xi},\nu_t^{*,t_0,\xi})|\mathcal{F}_t^{W^0}\big]\big)\quad\nu_t^{*,t_0,\xi}:=\mathbb E\big[x_t^{*,t_0,\xi}|\mathcal{F}_t^{W^0}\big].
\widetilde{J_0}(q,Q) = \sum_{d \in \mathbb{Z}}f_d(q)Q^d \widetilde{J_1}(q,Q) = \sum_{d \in \mathbb{Z}}g_d(q)Q^d + \ell_q(Q) \sum_{d \in \mathbb{Z}}g_d(q)Q^d
\tilde{\pi}^{i j}= 2 \gamma^{1 / 2}\tilde{\gamma}^{- 1 / 3}e^{- \frac{3}{2}( \phi - \tilde{\phi})}\left[ \frac{\partial H}{\partial \tilde{h}_{i j}}- \frac{1}{3}\frac{\partial H}{\partial \tilde{h}_{k l}}\tilde{h}_{k l}\tilde{h}_{i j}\right] ,
\int \! dxdp~ f\star g \star h={1\over \hbar^2 \pi^2}\! \int \! d{\bf r}_1 d{\bf r}_2 d{\bf r}_3 f({\bf r}_1) g({\bf r}_2) h({\bf r}_3) ~\exp\! \left({-4i\over \hbar}A(r_3 , r_2,r_1 ) \right) .
\mathrm{D e t^{\prime}}_{P}^{-{\frac{1}{2}}}\Bigl [ -{\partial}_{\tau}^{2}+ 2 i e F{\partial}_{\tau}\Bigr ] ={\lbrack 4 \pi T \rbrack}^{-{\frac{D}{2}}}\mathrm{D e t^{\prime}}_{P}^{-{\frac{1}{2}}}\Bigl [{\bf I}- 2 i e F{{\partial}_{\tau}}^{- 1}\Bigr ] \quad
H^{(2m)}_n = \begin{cases}O(n^{-m(1-2\alpha)}) \mbox{if $-2+m(1-2\alpha)<-1$}, O(n^{-m(1-2\alpha)}\log n) \mbox{if $-2+m(1-2\alpha)=-1$}, O(n^{-1}) \mbox{if $-2+m(1-2\alpha)>-1$}, \end{cases}