harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{d}\log \Xi(t)={\mathcal{I}}\big(T-t,\mathfrak{X}(t)\big)\mathrm{d}t+\mathcal{H}(t,\mathfrak{X})\, \mathrm{d}W(t)-\frac{1}{\,2\,}\left( \mathcal{H}\mathcal{H}'\right) (t,\mathfrak{X})\, \mathrm{d}t \le \mathrm{d}\log \left( L(t)Z^{v,\pi}(t)\right)\,,
\int \phi P \phi\dv = \int \Big\{\frac{n-6}{n-2}(\Delta \phi)^2 + \frac{4}{n-2}|\nabla^2 \phi|^2 + \frac{(n-2)^2 + 4}{n-2}\sigma_1(A)|\nabla \phi|^2 + \frac{n-4}{2}Q \phi^2 \Big\}\dv.
A = 2\left ( 1-\frac{1}{K(2N-1)}\right )|E_1|^2 + \left ( 1-\frac{1}{K(2N-1)}\right )\frac{E_1+E^*_1}{\cos \frac{\pi}{2N-1}}, B = \frac{\lambda_0}{K(2N-1)}\cdot \left \{2 |E_1|^2+ \frac{E_1+E^*_1}{\cos \frac{\pi}{2N-1}}\right \}.
D_{xy}(x_{k})\approx\frac{1}{N_{k}}\sum_{j=1}^{N_{t}}\sum_{i=0,x_{i}\in S(x_{k},r)}^{N_{s}-1}\frac{(x_{i+1}^{j}-x_{i}^{j})(y_{i+1}^{j}-y_{i}^{j})}{2\Delta t}.
\frac{\zeta'(z)}{\zeta(z)-a}=\sum_{|\rho_a-s_0| \leq r-\delta}\frac{1}{z-\rho_a}+O\bigg(\frac{1}{\delta^2}(\log M_{r}(s_0)+N_{r-\delta}(s_0))(\log 1/\delta+1) \bigg).
\begin{cases}\displaystyle u_y+(uu_x-vv_x)=f(u,v)+\varepsilon u_{xx}-\frac{2\varepsilon u_xv_x}{v}, begin{align*}10pt] \displaystyle v_y+(uv_x-vu_x)=g(u,v)+\varepsilon v_{xx}-\frac{2\varepsilon v_x^2}{v}.\end{cases}
\overline{g}((\nabla'_{Z_{i}}\mathcal{A}_{\widehat{W}})Y, T_{r-1}Z_{i})= \overline{g}((\nabla'_{Y}\mathcal{A}_{\widehat{W}})Z_{i},T_{r-1}Z_{i})+\overline{g}(\overline{R}(Y,Z_{i})T_{r-1}Z_{i},\widehat{W}) =\overline{g}(T_{r-1}(\nabla'_{Y}\mathcal{A}_{\widehat{W}})Z_{i},Z_{i})+\overline{g}(\overline{R}(\widehat{W},T_{r-1}Z_{i})Z_{i},Y),
\aligned\frac{4(n-1)}{n-2}\Delta \varphi+R\varphi= -\frac{n-1}{n}t^2\tau^{2a}\varphi^{N-1}+\left(|\sigma+LW|^{2}+k^{2}\right)\varphi^{-N-1}, -\frac{1}{2}L^{*}LW= \frac{n-1}{n}t\varphi^{N}d\tau^a.\endaligned
\cos \phi \, p_{0,\ell}(\phi)-p_{0,\ell+1}(\phi) = \sin \phi \sin \psi_{0,\ell}\sum_{k=0}^\infty (-1)^k h_0(2k) s_{0,2k}(\ell,\phi) &\;\;+\sin \phi \cos \psi_{0,\ell}\sum_{k=0}^\infty (-1)^k h_0(2k+1) s_{0,2k+1}(\ell,\phi) &\;\;+\phi^{-1/2}O(\ell^{-3/2})
2A=\sum_{i=1}^{n}(x_{i}y_{i+1}-x_{i+1}y_{i})=\sum_{i=1}^{n}x_{i}y_{i+1}-\sum_{i=1}^{n}x_{i+1}y_{i}=\sum_{i=1}^{n}x_{i-1}y_{i}-\sum_{i=1}^{n}x_{i+1}y_{i}
\log\left(1+\frac{g(p)}{p^s-1}\right) - g(p)p^{-s}= \left(\log\left(1+\frac{g(p)}{p^s-1}\right) - \frac{g(p)}{p^{s}-1}\right) + \left(\frac{g(p)}{p^s-1}- \frac{g(p)}{p^s}\right) &= -\frac{g(p)^2}{(p^{s}-1)^2}\sum_{l \geq 2}(-1)^{l}\frac{1}{l}\left(\frac{g(p)}{p^s-1}\right)^{l-2}+ \frac{g(p)}{p^s(p^s-1)}.
\widetilde{\mathfrak{S}}_{t}f(x)=\Pi_{x}^{h}\left[e_{q}(t)f(\xi^{h}_{t})\right]+\Pi_{x}^{h}\left[\int_{0}^{t}q(\xi^{h}_{s})e_{q}(s)\pi^{h}(\xi^{h}_{s},\widetilde{\mathfrak{S}}_{t-s}f){\rm d}s\right], f\in\mathcal{B}^{+}_{b}(E).
\left( I_{d\times d}+ \theta_j h^{\beta - 1}\nabla g_0 (g_0^{-1}(x)) \nabla \psi_j \left(\frac{x - x_j}h \right) \right)^{-1}&= \sum\limits_{k=0}^\infty \theta_j^k h^{k\beta - k}\left( -\nabla g_0 (g_0^{-1}(x)) \nabla \psi_j \left(\frac{x - x_j}h \right) \right)^k.
\lim_{m \to \infty, m}\frac{1}{m}\sum_{j = 0}^{m-1}\frac{e^{2 \pi i \frac{j + \theta_1/2}{m}}}{1 + \exp \left\{2 \pi i \frac{j + \theta_1 /2}{m}\right\}+ \frac{z}{m}}= (-1)^{\theta_1 +1}\frac{e^{- z}}{1 - (-1)^{\theta_1}e^{-z}}.
\tilde{\psi}((x_{ij})^{t}) =\frac{1}{2}\left(\psi((x_{ji})) +\overline{\psi((x_{ji}))}\right)=\frac{1}{2}\left(\sum_{i,j=1}^{n}x_{ji}\psi(E_{ij}) +\sum_{i,j=1}^{n}x_{ji}\overline{\psi(E_{ij})}\right) &=\frac{1}{2}\left(\sum_{i=1}^{n}x_{ii}(\psi(E_{ii})+\psi(E_{ii})^{t})+\sum_{1\leq i\neq j \leq n}x_{ji}(\psi(E_{ij}) + \psi(E_{ji}))\right).
\mathbb{R}^{2n}\buildrel \tau^{B[0]}_{k_0,\varepsilon_0}\over \rightarrow\mathbb{R}^{2n}\buildrel \tau^{B[1]}_{k_{1},\varepsilon_{1}}\over\rightarrow\cdots\buildrel \tau^{B[T-1]}_{k_{T-1},\varepsilon_{T-1}}\over \rightarrow\mathbb{R}^{2n},
\|\mathcal{D}_{J,1}^{m_1}\mathcal{D}_{J,2}^{m_2}\widehat{f}_{j}\|^2_{L^2(\mathcal{W}^+_J)}\le C(m_1,m_2)\sum_{\substack{a_1+b_1=m_1 a_2+b_2=m_2}}\| \partial^{(a_1+a_2,b_1+b_2)}\widehat{f}_{j}\|^2_{L^2(\mathcal{W}^+_J)}
\left. \begin{array}{r}h(a,c)A+l(a,b,c)\bar A\bar B h(b,c)\bar B+l(a,b,c)BA e^c+j^2.f(c,a)A\bar C+f(c,b)C\bar B+j^2.g(c,a,b)(\bar A\bar B\bar C+j.CBA)\end{array}\right)
[x^{2^{i-1}},y] \equiv [x,y]^{2^{i-1}}[x,y,x]^{\binom{2^{i-1}}{2}}\, \cdots \, [x, y, x, \overset{i-2}\ldots, x]^{\binom{2^{i-1}}{i-1}}\gamma_{i+1}(G_k).
\Delta^{a}= \int d^{2}\theta \frac{\partial^{2}\delta_{L}}{\partial \theta^{2}\delta \Phi^{A}( \theta )}\theta^{a}\frac{\delta}{\delta \bar{\Phi}_{A}( \theta )}= ( - 1 )^{\epsilon_{A}}\frac{\delta_{L}}{\delta \phi^{A}}\frac{\delta}{\delta \phi_{A a}^{*}}.
\Phi^{\prime}\ = \ \Phi ( \chi^{\prime}, \chi^{*}) \ = s u m a_{m_{1}m_{2}n_{1}n_{2}}{\chi^{\prime}}_{1}^{* m_{1}}{\chi^{\prime}}_{2}^{* m_{2}}{\chi^{\prime}}_{1}^{n_{1}}{\chi^{\prime}}_{2}^{n_{2}}\ .
\Big((n-3)F_{e}^{(1)}g^{ef}G_f^0+2k\big(F^{(2)}+\sum_{i=1}^{n}F_{e}^{(1)}g^{ef}G_f^iF^{(1)}_{a_i,b_i}\big)\Big)F^{(k+1)}\sim 0,
F_{c}[ G ] = \int \left[ d u_{\alpha}( x ) \, d u_{\beta}^{*}( x ) \, \delta \left( u_{\gamma}^{*}( x ) u_{\gamma}( x ) - 1 \right) \right] \exp \left[ - 2 \int d^{2}x \, u^{*}E^{i}D_{i}u \right]
\{{\cal Q}, \omega_{d}\}= ( - 1 )^{g}i_{v_{1}}\cdots i_{v_{g}}d \Omega_{d + g}+ \sum_{k = 1}^{g}( - 1 )^{k + 1}{\cal L}_{v_{k}}i_{v_{1}}\cdots i_{v_{k - 1}}i_{v_{k + 1}}\cdots i_{v_{g}}\Omega_{d + g}= d \omega_{d}.
\widehat{R}= \exp \left( \frac{\hbar}{2}\sum_{\mu_1, \mu_2}\sum_{k_1, k_2}\Delta^{\mu_1 \mu_2}_{k_1 k_2}\frac{\partial}{\partial t^{\mu_1}_{k_1}}\frac{\partial}{\partial t^{\mu_2}_{k_2}}\right),
lim_{|i-j|\rightarrow\infty}lim_{L\rightarrow\infty}\langle\psi_{0}^{\pm}|\sigma_{i}^{z}\sigma_{j}^{z}|\psi_{0}^{\pm}\rangle-\langle\psi_{0}^{\pm}|\sigma_{i}^{z}|\psi_{0}^{\pm}\rangle\langle\psi_{0}^{\pm}|\sigma_{j}^{z}|\psi_{0}^{\pm}\rangle>0.
\int ( \sum_j |(P_j f)(x)|^2)^2 dx \gtrsim \sum_l ( \sum_{j<l}a_j^2 \lambda_j)^2 \cdot (\frac{\rho_0}{\lambda_l}-\frac{\rho_0}{\lambda_{l+1}}) & \gtrsim \sum_{l}( \sum_{j<l}j^{-1-2\delta}j^{\epsilon})^2 \cdot \frac 1{l^{1+\epsilon}}& \gtrsim \sum_l l^{2(\epsilon-2\delta)}\cdot \frac 1{l^{1+\epsilon}}=\infty.
g_{1}([X,Y],W) =g_{1}(\varphi\nabla^{^{M_1}}_{X}Y,\phi W)+g_{1}(\varphi\nabla^{^{M_1}}_{X}Y,\omega W)+\eta(Y)g_{1}(X,\omega W) &-g_{1}(\varphi\nabla^{^{M_1}}_{Y}X,\phi W)-g_{1}(\varphi\nabla^{^{M_1}}_{Y}X,\omega W)-\eta(X)g_{1}(Y,\omega W)
\sigma^l = \pi \Omega_{n+1}\left[{\Gamma({n\over 2}+1)\over \Gamma({n\over 2}+l+1)\Gamma({n+2l\over 2b})}\right]^2{2l+n\over n}{l+n-1\choose l}S\left({T\over b}\right)^{{n+2l\over b}-1}\left({\omega\over 2}\right)^{{n+2l\over b}+2l-1}.
\sum^{p}_{j=1}|\partial_{j}\mathcal{L}(x)|\lesssim L_1(p),\sum^{p}_{j,k=1}|\partial_{jk}\mathcal{L}(x)|\lesssim L_2(p),\sum^{p}_{j,k,l=1}|\partial_{jkl}\mathcal{L}(x)|\lesssim L_3(p),
\Pi_{\beta}(d\beta\mid Y,\sigma^{2}):=\mathrm{e}^{-\|Y-X\beta\|^{2}/(2\sigma^{2})}\pi(\beta)d\beta \bigg{/}\int \mathrm{e}^{-\|Y-X\tilde{\beta}\|^{2}/(2\sigma^{2})}\pi(\tilde{\beta})d\tilde{\beta}
\mathcal{F}\big(F_{n;\underline{k}}(f_{\underline{k}})\big)(\xi):=\int_{\mathbb{R}^{n+1}}M'_{n+1}(\xi;\underline{\eta})\varphi_{k_1}(\eta_1)\ldots \varphi_{k_{n+1}}(\eta_{n+1}) &\times\widehat{f_{\underline{k}}}(\xi-\eta_1-\ldots-\eta_{n+1})\cdot\widehat{h}(\eta_1)\cdot\ldots\cdot\widehat{h}(\eta_{n+1})\, d\underline{\eta},
{\cal V}^{I}(\Phi)={\frac{m^2}{\beta^2}}+{\frac{\mu^2-{\cal M}^2}{8\pi}}-{\frac{m^2}{\beta^2}}({\frac{\mu^2}{{\cal M}^2}})^{{\frac{\beta^2}{8 \pi}}}\cos(\beta \Phi) \;.
(a,\, 0) S^{-1}A(\varepsilon_0) S\left(\begin{matrix}\mathfrak{a}\mathfrak{b}\end{matrix}\right)= (\mathfrak a'\,,\mathfrak b') S^{-1}A(\varepsilon_0) S\left(\begin{matrix}d\alpha-b\beta 0\end{matrix}\right)
kh = \left( I + \begin{pmatrix}0 \beta \gamma 0 \end{pmatrix}\right) \left( \begin{pmatrix}w 0 0 z \end{pmatrix}+ \begin{pmatrix}a 0 0 d \end{pmatrix}p \right) = \begin{pmatrix}w 0 0 z \end{pmatrix}+ \begin{pmatrix}a \beta z \gamma w d \end{pmatrix}p
\delta_{c_2}\circ \delta_{c_2}= b_1 \delta_{c_1}+ b_2 \delta_{c_2}+ b_3 \delta_{c_3}+ b_4 \delta_{c_4}, \delta_{c_3}\circ \delta_{c_3}= b_4 \delta_{c_1}+ b_3 \delta_{c_2}+ b_2 \delta_{c_3}+ b_1 \delta_{c_4},
(\widehat{C_E},\|\cdot\|_{\widehat{C_E}})=\overline{(C_E,\|\cdot\|_{C_{\widehat{E}}})}^{\|\cdot\|_{C_{\widehat{E}}}}\subseteq (C_{\widehat{E}},\|\cdot\|_{C_{\widehat{E}}})
|Kl_p(\psi_p,\psi_p';c,w_{G_n})| \leq C_n \cdot \min(p^{\sigma +a_2+\cdots+ \varrho/2+\frac{n(n-1)}{2}m},p^{\ell/2+ 2 \sigma+(n-3)\varrho +a_2+\cdots+a_{n-2}-\ell +\frac{n(n-1)}{2}m}).
\nabla_{X_1^h}Y_1^{h}=(\nabla{\hskip -0.2cm^{^{^{1}}}}_{X_1}Y_1)^h+f_{_{2}}^vB{\hskip -0.1cm_{_{f\!_{_{1}}}}}\!\!(X_1,Y_1)^hgrad(f_{_{2}}^v)
\left\langle \nabla P,g\right\rangle =\left\langle P,\operatorname{div}g\right\rangle =\left\langle P,\operatorname{div}\left( a\nabla P_{g}\right)\right\rangle & =\left\langle \operatorname{div}\mathcal{Q}f,P_{g}\right\rangle=\left\langle \mathcal{Q}f,\nabla P_{g}\right\rangle ,
T_j(pg_{L+1})(\mathbf{x}) =\sum_{\ell=0}^{d}\sum_{\|\alpha\| \leq \ell}d_{\ell,\alpha}T_j \circ T^{\alpha}(g_{L+1+\ell})(\mathbf{x}) &=\sum_{\ell=0}^{d}\sum_{\|\alpha\| \leq \ell}d_{\ell,\alpha}T^{\alpha+e_j}(g_{L+1+\ell})(\mathbf{x})=\sum_{\ell=0}^{d+1}\sum_{\|\alpha\| \leq \ell}d_{\ell,\alpha}'T^{\alpha}(g_{L+\ell})(\mathbf{x}).
\mathcal{X}= \frac{1}{\sqrt{2}}\left(\begin{array}{cc}0 a^{\dagger}+b^{\dagger}a+b 0 \end{array}\right), \mathcal{P}= -\frac{i}{\sqrt{2}}\left(\begin{array}{cc}0 a^{\dagger}-b^{\dagger}-a+b 0 \end{array}\right).
-\mu'(t) \geq |\partial\{u_1 > t\}|^2_M \left( \int_{\{u_1 = t\}}|\nabla u_1| \right)^{-1}& \geq \alpha^2 A(|u_1 > t|_M)^2 \left( \int_{\{u_1 = t\}}|\nabla u_1| \right)^{-1}&= \alpha^2 A(\mu(t))^2 \left( \int_{u_1 > t}-\Delta u_1 \right)^{-1}&= \alpha^2 A(\mu(t))^2 \left( \lambda_1 \int_0^{\mu(t)}\mu^{-1}(\sigma) d\sigma \right)^{-1},
\frac{d}{ds}\int_{I(s)}|\tilde{w}(\xi,s)|^2\rho_d\,d\xi \le -2\int_{I(s)}|\tilde{w}(\xi,s)|^2\rho_d\,d\xi+O(e^{-d\theta_*s})+O(e^{-\frac{\theta}{4}s}) =-2\int_{I(s)}|\tilde{w}(\xi,s)|^2\rho_d\,d\xi+O(e^{-2\theta' s})
I=(\frac{6}{\pi^{2}})^{4}[16 x_{24}^{2}(\frac{1}{2s}-1) D_{4455}+ \frac{64}{9}\frac{x_{24}^{2}}{x_{13}^{2}}\frac{1}{s}D_{3355}+ \frac{16}{3}\frac{x_{24}^{2}}{x_{13}^{4}}\frac{1}{s}D_{2255}
B^{\top}(t)\bar{\bf w}^{(\epsilon)}(t) =B^{\top}(t)\mbox{\boldmath $\rho$}_{\epsilon}(t)\geq{\bf a}(t)-\mbox{\boldmath $\epsilon$}+\int_{t}^{T}K^{\top}(s,t)\mbox{\boldmath $\rho$}_{\epsilon}(s)ds & \geq{\bf a}(t)-\mbox{\boldmath $\epsilon$}+\int_{t}^{T}K^{\top}(s,t)\bar{\bf w}^{(\epsilon)}(s)ds.
\sum_{d \geq 0}\sum_{g \geq 2}\mathsf{N}_{g, (1,1,d)}u^{2g-2}q^{d}= -K^2(z,\tau)|_{y=-e^{2\pi i z}, q=e^{2\pi i\tau}}= (y + 2 + y^{-1}) \prod_{m \geq 1}\frac{(1 + y q^m)^2 (1 + y^{-1}q^m)^2}{(1-q^m)^4}\,.
\textstyle \sum_v(-1)^{w-v}{j+w-v-1\choose w-v}{r-\alpha+j\choose v}{r-\alpha+j-v \choose j-v}\textstyle ={r-\alpha+j\choose j}\sum_v(-1)^{w-v}{j+w-v-1\choose w-v}{j \choose v}\textstyle = \delta_{w = 0}{r-\alpha+j\choose j}.
H_{}(x) = \sum_{l=1}^{M}\left( \sum_{i \in B(l)}\left( \psi (x_i) + s_i x_i + \frac{1}{2}\sum_{\substack{j \in B(l), 1 \leq |j-i| \leq R}}M_{ij}x_i x_j \right) \right) = \sum_{l=1}^{M}H_K (x^{B(l)}).
(f\star_{\vec{c}}^{n}g)_{lm}=\rho_{lm}& =(-1)^{n+m}\sqrt{n+1}\sum_{l_{1},l_{2}=0}^{n}\sum_{m_{1}=-l_{1}}^{l_{1}}\left[\begin{array}[c]{ccc}l_{1}l_{2}l m_{1}m_{2}-m\end{array}\right] \!\!{[j]}\ \frac{c_{l}^{n}}{c_{l_{1}}^{n}c_{l_{2}}^{n}}\ \phi_{l_{1}m_{1}}\gamma_{l_{2}m_{2}}
x_0 = f(x_1, \ldots, x_n) \Leftrightarrow \Leftrightarrow l(x_0) \oplus \ldots \oplus l(x_n) = 0 \mbox{and}x_0 \oplus \ldots \oplus x_n \oplus \lambda(l(x_1), \ldots, l(x_n)) = 0,
1 \le C\left(\mu_i \|u_i\|_{\lambda_i}^2 + \mathop{\sum_{j=1}^k}_{j\neq i}b_{ij}\|u_j\|_{\lambda_j}^2\right)\le 4C\left(\mu_i + \mathop{\sum_{j=1}^k}_{j\neq i}b_{ij}\right)I(u_1,\dots,u_d) &\le 4C\left(\mu_i + \mathop{\sum_{j=1}^k}_{j\neq i}b_{ij}\right) b^{-1}c(\lambda_i;0,\dots,0;1,\dots,1),
\int_y^{\varphi(t,y)}\frac{h(0)-\left(\frac 1 p -1\right) F'(s)}{F(s)}ds \leq -\frac{h(0)}{r-1}\left(y^{1-r}- \varphi(t,y)^{1-r}\right)+ C = h(0)t + C.
H_{ee}=\frac{1}{2}\sum_{n,m,\sigma}\langle n_{1}m_{1},n_{2}m_{2}|\frac{e^{2}}{|r_{1}-r_{2}|}|n_{3}m_{3},n_{4}m_{4}\rangle c_{n_{1}m_{1}\sigma_{1}}^{\dagger}c_{n_{2}m_{2}\sigma_{2}}^{\dagger}c_{n_{4}m_{4}\sigma_{2}}c_{n_{3}m_{3}\sigma_{1}}
\widetilde{\eta}^{1 / 2}= p \, \left\{1 + C_{1}p^{- 2 / 3}+ \frac{\epsilon}{4}\left[ - 2 c + g^{( 0 )}- \ln \left| \frac{E}{\mu^{2}}\right| \right] + o ( \epsilon ) \right\}\; ,
g(W,\gamma,\theta) =\frac{1}{T}\sum_{t=1}^{T}D(X_{t})\pi(a(X_{t},\theta,\gamma_{2},\gamma_{3}))[Y_{2t}-\Lambda(a(X_{t},\theta,\gamma_{2},\gamma_{3}))], \pi(a) =\frac{\Lambda_{a}(a)}{\Lambda(a)[1-\Lambda(a)]},\Lambda_{a}(a)=\frac{d\Lambda(a)}{da}.
f (x)=\int_{-\infty}^\infty \frac{d \xi}{2 \pi}{\rm e}^{i(1 - x) \, \xi}\langle \widetilde{H}(p) |W_{\scriptstyle\Pi}({v \!\cdot\! n}\, \xi \mu -i0) | \widetilde{H}(p) \rangle\, ,
\kappa_{00}=\sum_{n\geq 0}q_1(0)^{n-1}\prod_{k=0}^{n-1}q_{(01)^k00(1)}+\sum_{n\geq 1}q_1(0)^{n-2}\prod_{k=0}^{n-1}q_{(01)^k00(1)}\kappa_{11}=\sum_{n\geq 0}q_1(0)^{n}\prod_{k=1}^{n-1}q_{(01)^k1(1)}+\sum_{n\geq 1}q_1(0)^{n}\prod_{k=1}^{n-1}q_{(01)^k1(1)},
\begin{aligned}r_-(x)-\hat r_-(x)=\frac D2\lambda(l+1)(\gamma+x^2)^{\frac{l-3}2}\times &\Big[\frac{2c_1}{D\lambda(l+1)}(\gamma+x^2)^{\frac{3-l}2}(\gamma_1+x^2)^l-(l+1)\lambda x^2(\gamma+x^2)^{\frac{l+1}2}-\gamma-lx^2\Big].\end{aligned}
[B,A^{\frac12}] = [|D_0|^{-\frac13},a^2] = -|D_0|^{-\frac13}[|D_0|^{\frac13},a^2]|D_0|^{-\frac13}= -[|D_0|^{\frac13},|D_0|^{-\frac13}a^2|D_0|^{-\frac13}].
\begin{aligned}0 = - \frac{c}{C^m_n}\sum_i S_{n - m - 1;i}(X^{-1}) (X^{i\bar i})^2 X_{i\bar i l}+ b_t f_l S_n (X^{-1}) - b_t S_n (X^{-1}) f \sum_i X^{i\bar i}X_{i\bar il},\end{aligned}
v_{\omega}\left( \sigma^{-}\left( y^{-}\right) \right) \approx{\frac{1}{\sqrt{2 \omega}}}\, e^{{\frac{i \omega}{\lambda}}\ln{\left( - \lambda^{2}\Delta y^{-}\right)}}\ , \hspace{1 c m}0 < - \lambda y^{-}\ll 1 \; .
{g'}_{\mu\nu}(x,l) = [f(l)^{\star}g_{\mu\nu}(l)] (x) = \frac{\partial{x'}^{\alpha}(x,l)}{\partial x^{\mu}}g_{\alpha\beta}(x'(x,l),l) \frac{{x'}^{\beta}(x,l)}{\partial x^{\nu}}
\Theta_i:=\prod_{j=i-1}^{1}Q_{j,i}\prod_{j=n}^{i+1}Q_{i,j}, \bar \Theta_{i}:= \prod_{j=i+1}^{n}Q_{j,i}\prod_{j=1}^{i-1}Q_{i,j}, 1 \le i \le n,
D_\mu\delta q^i\equiv{\cal D}_\mu\delta q^i-{(j_u)_\mu}^\nu{\cal D}_\nu\delta q^j{(J_u)_j}^i-{(j_u)_\mu}^\nu\partial_\nu q^j{\cal D}_k{(J_u)_j}^i\delta q^k=0.
W = W^{\mathrm{c l}}\frac{H}{H_{0}}\frac{3 n \left( 1 - \beta^{2}\right)}{1 0}\left( \frac{1 + \zeta}{2}\frac{n + \mu - 1}{n + \mu}+ \frac{1 - \zeta}{2}\right) \left( \frac{5}{6}l_{2}^{2}+ \frac{1}{6}l_{3}^{2}\right) .
\int_0^1 \int_0^1 ve^{ius(1-vw)|D|}\delta^2(x)e^{iusvw|D|}\,dwdv = \int_0^1 \int_{w}^1 e^{ius(1-w)|D|}\delta^2(x)e^{iusw|D|}\,dvdw = \int_0^1 (1-w)e^{ius(1-w)|D|}\delta^2(x)e^{iusw|D|}\,dw.
\mathrm{d}Z_t = \dot \beta_t(X_t\mathrm{d}t - Z_t \mathrm{d}t- (1/\mu)(\nabla f(X_t) \mathrm{d}t+ \sigma(X_t, t) \mathrm{d}B_t)) \mathrm{d}X_t = \dot \beta_t (Z_t - X_t)\mathrm{d}t
J(q)=\int_{0}^{I_{\Phi\left(\frac{q-\mu}{\sigma}\right)}(\alpha,\beta)}\,Q(u)\mathrm{d}u=\sum_{i=0}^{\infty}\frac{f_i\,\,\,I_{\Phi\left(\frac{q-\mu}{\sigma}\right)}(\alpha,\beta)^{(i/\alpha+1)}}{i/\alpha+1}.
E_1(x)= \dfrac{1}{4}(4+4+\dfrac{1}{4}+\dfrac{1}{4})=\dfrac{17}{8}, E_2(x)= \dfrac{1}{6}(4\times 4+4\times 4\times\dfrac{1}{4}+\dfrac{1}{4}\times\dfrac{1}{4})=\dfrac{107}{32}, E_3(x)= \dfrac{1}{4}(2\times 4\times 4\times\dfrac{1}{4}+2\times 4\times\dfrac{1}{4}\times\dfrac{1}{4})=\dfrac{17}{8}, E_4(x)= 4\times 4 \times\dfrac{1}{4}\times\dfrac{1}{4}=1,
\mathcal{X}= \left\{t\in \mathfrak{M}\left(L,\eta^{-1},\eta^{-(2+\epsilon)}\right) :\min\left\{|\widehat{B}(t)|,|\widehat{B_1}(s+t)|\right\}\leq c_1\sigma^{(m+1)/2}U^{-m/2}\right\}
|\begin{matrix}e_{x}&e_{y}&e_{z}\\ -x_{A}&0&0\\ 0&R_{0}&0\end{matrix}|+|\begin{matrix}e_{x}&e_{y}&e_{z}\\L-x_{A}&0&0\\ 0&R_{B}&0\end{matrix}|=-x_{A}R_{0}e_{z}+(L-x_{A})R_{B}e_{z}=0.
\widetilde R^m_0(\lambda; r,r') = G_{m,0}(r,r') - \delta_{m, k(\alpha)}\, \frac{|\lambda|^{\mu(\alpha)}\, g_1(r,r')}{\sin(\mu(\alpha) \pi) \, (i-\cot(\mu(\alpha) \pi))}+ o(|\lambda|^{\mu(\alpha)}) \lambda\to 0- .
m_{i,k}= \delta_{i,k}\int_0^{+\infty}x^{k}G_{q,n}^{n,0}\left(x\biggl|\begin{array}{cccc}b_1, b_2, \ldots, b_q a_1, a_2, \ldots, a_n \end{array}\right) dx = \delta_{i,k}\frac{\prod_{\alpha=1}^n \Gamma\left(a_\alpha +k+1 \right)}{\prod_{\alpha=1}^q \Gamma\left(b_\alpha +k+1 \right)}
h^\theta_{l+1}(T) = \frac{(2l+1+\theta)(l+1)(l+\theta)}{(2l-1+\theta)(l-1)(l+\theta-1)}e^{-(l+\theta/2)T}h^\theta_l(T) \leq 5e^{-(l+\theta/2)T}h^\theta_l(T) < h^\theta_l(T),
\frac{1}{(2\pi)^{n/2}}\int\cdot\cdot\cdot\int|\prod_{r}\frac{2(x,r)}{(r,r)}|^{\gamma}e^{-(x_{1}^{2}+\cdot\cdot\cdot+x_{n}^{2})/2}dx_{1}\cdot\cdot\cdot dx_{n}=\prod_{j=1}^{n}\frac{\Gamma(1+d_{j}\gamma)}{\Gamma(1+\gamma)}
\mu_{m}^{n(I)}(J) =\mu_{m}^{n(I_{b(I)})}(J)\stackrel{\eqref{e:frost-case-1}}{<}\frac{1}{2}\mu_{m}^{n(I_{b(I)})+1}(J) \stackrel{\eqref{e:frost-case-2}}{=}\frac{1}{2}\mu_{m}^{n(I_{b(I)+1})}(J)<\cdots \cdots < \frac{1}{2^{b(J)-b(I)}}\mu_{m}^{n(I_{b(J)})}(J)=\frac{1}{2^{b(J)-b(I)}}\mu_{m}^{m}(J)=\frac{\ell(J)^{d}}{2^{b(J)-b(I)}}.
A:=\inf_{\varphi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\partial_r\varphi)^2\sinh^ndr}{\int_{0}^{\infty}\varphi^2\sinh^{n-2}dr}=\inf_{\varphi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\partial_s\varphi)^2(1+s^2)^{1/2}s^nds}{\int_{0}^{\infty}\varphi^2(1+s^2)^{-1/2}s^{n-2}ds},
\left|\phi(A)\right| \leq \max_{1\leq i\leq n}{\left\|A\xi_i\right\|}\leq \left(\sum_{i=1}^{n}{\left\|A\xi_i \right\|^2}\right)^{\frac{1}{2}}=\|(A\xi_i)_{i=1}^{n}\|_2.
I_{D}( p ) ={\frac{1}{2 \kappa_{D}^{2}}}\int d^{D}x \sqrt{- g}[{\cal R}-{\frac{1}{2}}( \partial \phi )^{2}-{\frac{1}{2 ( p + 2 ) !}}e^{- a ( p ) \phi}F_{p + 2}^{2}] ,
r \sum_{i \in C}x_i - \frac{r(r+1)}{2}-\sum_{\{i,j\}\subseteq C}x_ix_j &= rp - \frac{r(r+1)}{2}- \frac{p(p-1)}{2}= \frac{p-r -(p-r)^2}{2}\leq 0,
\|\Xi^V_{\tau,\tau_1}\|_{\xi,\zeta}\le \frac{8}{(\xi-1)}+ \bigl\||\Pi{V_\beta}|\bigr\|_{\xi,\zeta,1,1}+ \bigl\||\Pi{V_\beta}|\bigr\|_{\xi,\zeta,1,1; 2}=\delta_{\xi,\zeta}(V)
\begin{aligned}\mathrm{Re}\W_Y(z) =\mathrm{Re}\f(w(z))+f(-w(z))\;,z\in[z_-,z_+]\;, f(-w(z)) =-\frac{1}{2}\int_{z_-}^{z_+}\frac{\mathrm{d}z'}{\sqrt{z'-\delta_U}}\frac{\rho_Y(z')}{\sqrt{z-\delta_U}+\sqrt{z'-\delta_U}}\;.\end{aligned}
\| \Psi(x_1; \omega(k))\cdots\Psi(x_N; \omega(k))\|_H^2 = \int_{[-1, 1]^N}{\prod_{i=1}^N{|\Psi(x_i; \omega(k))|^2}\Delta_N(x)^2\prod_{i=1}^N{\left((1-x_i)^a(1+x_i)^b dx_i\right)}}
\mathbb E \langle F\star \bar{\mu}, G \star \bar{\mu}\rangle = \sum_{k,l=1}^d \langle x_k, x_l^*\rangle\mathbb E\bigl (F^k\star \bar{\mu}\cdot G^l\star \bar{\mu}\bigr)=\sum_{k,l=1}^d \langle x_k, x_l^*\rangle \mathbb E(F^k G^l)\star \nu =\mathbb E\Bigl(\sum_{k,l=1}^d \langle x_k, x_l^*\rangle F^k G^l\Bigr)\star \nu = \mathbb E\langle F, G\rangle\star \nu.
\phi(m^*m)\phi(n^*n) \pi_\phi(w^*w) =\pi_\phi ((d n^*m d)^*(d n^*m d))=\pi_\phi(d^*d)^2 \pi_\phi(m^*nn^*m) &= \phi(m^*nn^*m) 1_{D'_{A^\delta}/J_\phi}=\phi(m^*m)\phi(n^*n)1_{D'_{A^\delta}/J_\phi},
\lim\limits_{\lambda_s^{}\to 1}{\pi_s(0)}=\lim\limits_{\lambda_s^{}\to 1}\frac{\mu_s-\lambda_s^{}}{\mu_s-\lambda_s^{}\cdot \tau^{B_s}}&=\lim\limits_{\tau \to \infty}\frac{\mu_s-1}{\mu_s-\tau^{B_s}}=0,
\delta{\cal J}( x ) = \left( a^{m}+ \lambda_{M}^{m n}x_{n}+ \lambda_{D}x^{m}+ ( x^{2}\Lambda_{K}^{m}- 2 x^{m}x \cdot \Lambda_{K}) \right){\frac{\partial}{\partial x^{m}}}\;{\cal J}( x )
W_{A D S}= ( N_{c}+ 1 - N_{f}) \; \omega_{N_{c}+ 1 - N_{f}}( \frac{2^{N_{c}- 1}\Lambda_{N = 1}^{3 ( N_{c}+ 1 ) - N_{f}}}{\mathrm{P f}M})^{\frac{1}{N_{c}+ 1 - N_{f}}}
\eta_{\mu^0}(\lambda) = \frac{\Gamma_{p, d}(d (n+1) / 2)}{\Gamma_{p, d}(d p / 2)}\frac{\Gamma_{p, d}(\frac{1}{2}(\lambda - \rho + k + d p))}{\Gamma_{p, d}(\frac{1}{2}(\lambda + \rho + k))}.
G(x,x')=\frac1{(2\pi)^3}\sum_{n=-\infty}^\infty e^{in(\theta-\theta')}\int_{-\infty}^\infty dk e^{ik(z-z')}\int_{-\infty}^{\infty}d\omega e^{i\omega (\tau-\tau')}g_n(r,r') \ .
\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}) .
\|x-z\| \le \|x-Ty\|+ \|Ty - z\| \le \|T(x -y)\| + \sum_{n=1}^N\lambda_n\big|1 - \|Ty_n\|\big| < (1+ \eta)\gamma + \max_{1 \le n \le N}{\big|1 - \|Ty_n\|\big|}\le (1+ \eta)\gamma + \eta < \alpha.
\int_C\big( \mathbf{e}^{\frac{1}{\mathbf{l}(\mathbf{z})\mathbf{log}\, \mathbf{z}}}\big)^{d\mathbf{z}}=\frac{\mathbf{log}\, (a,\pi )}{\mathbf{log}\, (a,-\pi )}=e^{2i\arctan \frac{\pi}{\ln a}}.
\bar{S}_{f r e e}^{( 0 )}[ \phi ] = -{\frac{1}{2}}\mathrm{T r}\left( [ \hat{x}_{\mu}^{\prime}, \phi ( \hat{x}) ] [ \hat{x}_{\mu}^{\prime}, \phi ( \hat{x}) ] \right) ={\frac{1}{2}}\mathrm{T r}( \hat{\partial}_{\mu}\phi ( \hat{x}) )^{2}.
\inf_{k \in [m]_0}\left\{\inf_{x \in \mathcal{X}}\left\{\Gamma \theta^k(x) + \sum_{i \in [m]}\max \{0, x_i - \overline{b}_i, \max\{0,x_i - \overline{b}_i + \Delta b_i\}- \theta^k(x)\}\right\}\right\}
[Q,\theta_0(x_a)]=\sum_{b\in \Delta_-\sqcup I,\alpha\in \Delta_+}c_{\alpha,a}^b \theta_0(x_b)(1\*x_\alpha^*)-1\*\sum_{\beta,\gamma\in \Delta_+}c_{a,\beta}^{\gamma}\chi(x_{\gamma})x_{\beta}^* [Q,1\* x_{\alpha}^*]=-1\*\frac{1}{2}\sum_{\beta,\gamma\in\Delta_+}c_{\beta,\gamma}^{\alpha}x_{\beta}^*x_{\gamma}^*
E=\gamma(v)m_{0}c^{2}=\frac{m_{0}c^{2}}{\sqrt{1-v^{2}/c^{2}}}\approx m_{0}c^{2}(1+\frac{v^{2}}{2c^{2}}+...)=m_{0}c^{2}+\frac{1}{2}m_{0}v^{2}+...
{\rm Prob}(a<x_1<b,x\in S^1) \ = \ \frac{\int_a^b\frac{1}{\sqrt{1-x_1^2}}S_{0}(\sqrt{1-x_1^2}) dx_1}{S_1(1)}&\ = \ \frac{1}{\pi}\int_a^b\frac{1}{\sqrt{1-x_1^2}}\dx_1 &\ = \ \frac{\arcsin(b)-\arcsin(a)}{\pi},
\begin{array}{r l r l r l r l r l}{{y_{2}y_{4}}}&{{y_{8}y_{1 0}}}&{{y_{1}y_{4}}}&{{y_{8}y_{9}}}&{{y_{4}y_{1 1}}}&{{y_{3}y_{1 2}}}&{{y_{5}y_{8}}}&{{y_{6}y_{7}}}&{{y_{6}y_{2}}}&{{y_{1 2}y_{1}}}\\{{y_{2}y_{3}}}&{{y_{7}y_{1 0}}}&{{y_{1}y_{3}}}&{{y_{7}y_{9}}}&{{y_{1 1}y_{8}}}&{{y_{1 2}y_{7}}}&{{y_{4}y_{5}}}&{{y_{3}y_{6}}}&{{y_{5}y_{2}}}&{{y_{1 1}y_{1}}}\\ \end{array}
f'' = \frac{a''}{a}+ 2n \frac{b''}{b}= - 2n \frac{a^2}{b^4}+ \frac{4n(n+1)}{b^2}- 4n \frac{a'b'}{ab}- 2n(2n-1)\left(\frac{b'}{b}\right)^2 + \left(\frac{a'}{a}+ 2n \frac{b'}{b}\right) f'.
\frac{1}{R^{4}}\sum_{a = 0}^{2 l}\sum_{b = 0}^{2 l}\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},
S_{s t r}={\frac{1}{2}}\int_{{\cal M}^{2}}d^{2}\xi \sqrt{- g}g^{m n}( \xi ) \partial_{m}\hat{X}^{\underline{{m}}}( \xi ) \partial_{n}\hat{X}^{\underline{{n}}}( \xi ) \eta_{\underline{{m}}\underline{{n}}}.
\varphi_{\delta,\sigma}(t,x,s,y) =\varphi(t,x,y)+|t-\bar{t}|^2+|x-\bar{x}|^4+|y-\bar{y}|^4+\frac{|t-s|^2}{\delta}&+|t-t_\delta|^2+|x-x_\delta|^4+|y-y_\delta|^4+a_\sigma t+\zeta_\sigma\cdot x-b_\sigma s-\xi_\sigma\cdot y
B_{\alpha}^{(i,j)}\tau((i,j))-\tau((i,j)) B_{(i,j) \alpha}^{(j,i)}=B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}\tau((i,j)) -\tau((i,j)) B_{(i,j) \alpha+\varepsilon_{j}-\varepsilon_{i}}^{(j,i)}.
\left( \begin{array}{l l l l l}{1}&{x_{1 1}}&{\cdots}&{\cdots}&{x_{1 n}}\\{}&{1}&{\ddots}&{}&{\vdots}\\{}&{}&{\ddots}&{\ddots}&{\vdots}\\{}&{}&{}&{\ddots}&{x_{n n}}\\{0}&{}&{}&{}&{1}\\ \end{array}\right) .
E_2\, =1+a_1+a_1 b^2+b^4+a_1^3 k+a_1^4 k^2, E_1\, =1+a_1+a_1 b+a_1 b^2+a_1 b^3+b^4+a_1^3 k+a_1^3 b k+a_1^4 k^2, E_0\, =k+a_1 k+a_1^3 b k+a_1 b^2 k+b^4 k+a_1^3 k^2+a_1^4 k^3,
f_{\omega^{\prime}l m}^{b}= \int d \tilde{\omega}d \theta_{0}d \eta_{0}( \alpha_{\omega \omega^{\prime}l m}^{b}p_{\omega}+ \beta_{\omega \omega^{\prime}l m}^{b}\bar{p}_{\omega}+ \mathrm{t e r m s i n v o l v i n g}q_{\omega}) ,
\mathcal{M}_{f_{Y}}(s)=\Gamma^m\left(\frac{s}{2}\right)\prod_{i=1}^m\frac{(\sqrt{2}\sigma_i)^s}{\sqrt{2\pi}\sigma_i}\prod_{i=m+1}^n\frac{\Gamma(a_i+b_i)\Gamma(a_i-1+s)}{\Gamma(a_i)\Gamma(a_i+b_i-1+s)}.
\left( \hat{A}_{\mu}^{a}\hat{A}_{\mu}^{a}\right) \left( \hat{A}^{a \mu}\hat{A}^{a \mu}\right) - \left( \left( \hat{A}_{\mu}^{a}\hat{A}_{\mu}^{a}\right) \hat{A}^{a \mu}\right) \hat{A}^{a \mu}= 0 .
(L(x),x)_V =\sum\limits_{v\in V}\delta_V(v)[\mathcal{Q}(x)(v)-n(v)x(v))]x(v) =\sum\limits_{v\in V}\delta_V(v)[(avg^*\circ avg)(x)(v)-n(v)x(v))]x(v) =\sum\limits_{v\in V}\sum\limits_{e\in E_v}\frac{\delta_E(e)}{|e|}[(avg(x))(e)-x(v))]x(v).