harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\Vert E_2\right\Vert_{H^1(\mathcal G)}\leq \sum_{j=1}^{\infty}\left\vert \lambda_j^{(2)}\right\vert \left\Vert a_j^{(2)}-\Pi_l(G_j^{(2)},\tilde G_j^{(2)})\right\Vert_{H^1(\mathcal G)}\leq \varepsilon \sum_{j=1}^{\infty}\left\vert \lambda_j^{(2)}\right\vert & \leq \left(\varepsilon C_0\right)^{2}\left\Vert F \right\Vert_{H^1(\mathcal G)}.
\widetilde \nu ( x^{\mu}) = +{\frac{1}{2}}( \psi_{\downarrow}( x^{\mu}) + \psi_{\uparrow}^{c}( x^{\mu}) ) +{\frac{\gamma^{5}}{2}}( \psi_{\uparrow}( x^{\mu}) - \psi_{\downarrow}^{c}( x^{\mu}) ) \, .
\Lambda(f \otimes \mathrm{Ad}(g), k)^{\mathrm{alg}}= 2^{2k+\nu(N/M)}\tilde{C}(N,M,\chi) \cdot \frac{|D|^{k-1/2}\Lambda(f,D,k)}{c^+(f)|c_h(|D|)|^2}\cdot \frac{|\langle (\mathrm{id}\otimes U_M)F_{\chi|\mathcal H \times \mathcal H}, g \times g \rangle|^2}{\langle g,g \rangle^4},
{\bf E}\,\exp \left\{\lambda |\xi_{n}|\right\}\le 4\,\exp \left\{\sum_{m=2}^{\infty}\left( \frac{720\,e\lambda \kappa}{\log m}\right)^{m}\left( \Omega^{m/2}(n,B)+\frac{1}{n^{m/2-1}}\,\Omega (n,B)\right)\right\},
[\nabla \widehat E^\alpha_{x,y}]_{a,b}= [(\nabla \widehat E^\alpha_{x,y}\cdot S_{a,b}) S_{a,b}+ (\nabla \widehat E^\alpha_{x,y}\cdot T_{a,b}) T_{a,b}]_{a,b}&= \alpha e^{\alpha |G_{x,y}|^2}(G_{y,x}G_{x,b}\delta_{y,a}- G_{a,y}G_{y,x}\delta_{b,x}+G_{x,y}G_{y,b}\delta_{a,x}-G_{a,x}G_{x,y}\delta_{b,y}) \, .
\| \sum_{i=1}^{N}f_i \ast \hat \varphi_i |_{S}\|_2^2 = \sum_{i}\|f_i \ast \hat \varphi_i |_{S}\|_2^2+ \sum_{i \neq j}\int f_i \ast \hat \varphi_i \overline{f_j \ast \hat \varphi_j}d\sigma.
C^{-1}\left\Vert \left\{a_{j}\right\}_{j=1}^{\infty}\right\Vert_{\ell^{2}\left( \mu\right)}^{2}\leq\left\Vert \sum\nolimits_{j=1}^{\infty}a_{j}\varphi_{z_{j}}\right\Vert_{B_{2}}^{2}\leq C\left\Vert \left\{a_{j}\right\}_{j=1}^{\infty}\right\Vert_{\ell^{2}\left( \mu\right)}^{2}
K_{ij}:=\sum_{k=1}^{H_0-1}(a_{ik}b_{kj}-a^0_{ik}b^0_{kj})+a_{iH_0}b_{H_0j}-a^0_{iH_0}b^0_{H_0j}+\sum_{k=H_0+1}^{H-1}a_{ik}b_{kj}+ a_{iH}b_{Hj}, L_j:= \sum_{k=1}^{H_0-1}(a_{Mk}b_{kj}-a^0_{Mk}b^0_{kj})+a_{MH_0}b_{H_0j}-a^0_{MH_0}b^0_{H_0j}+\sum_{k=H_0+1}^{H-1}a_{Mk}b_{kj}+ a_{MH}b_{Hj},
Q_{1,\lambda}\left( \varepsilon \right) =\frac{1}{2\varepsilon}\left[B+\left( B^{2}+4\varepsilon A_{\lambda}\right)^{\frac{1}{2}}\right] ,\text{}Q_{2,\lambda}\left( \varepsilon \right) =\frac{1}{2\varepsilon}\left[B-\left( B^{2}+4\varepsilon A_{\lambda}\right)^{\frac{1}{2}}\right] .
\beta F = - \frac{N}{\beta^3}\int d \theta d \phi \int_{r_+ + h}^L dr \frac{\sqrt{g_4}}{( - g^{'}_{tt})^2}= - N \int_0^\beta d \tau \int d \theta d \phi \int_{r_+ + h}^L dr \sqrt{g_4}\frac{1}{\beta_{local}^4},
a_{i_1}A^3 = \frac{2 a_{i_1}a_{\zeta}(a_{\xi}+ a_{i_1})}{a_{i_0}a_{i_1}a_{i_2}a_{i_3}a_{\xi}a_{\zeta}}= \frac{2 (a_{\xi}+ a_{i_1})}{a_{i_0}a_{i_2}a_{i_3}a_{\xi}}= \frac{a_{\xi}+ a_{i_1}}{a_{i_2}a_{i_3}a_{\xi}\bar{a}_{\zeta}}.
\left( \bigg| \frac{1}{lk}\left( \underset{i=1}{\overset{k}{\sum}}\underset{j=1}{\overset{l}{\sum}}(B^j_i - p_i) \right)^2 - \mu_p' + \frac{l-1}{k}\left( \underset{i=1}{\overset{k}{\sum}}(p_i - \frac{1}{2}) \right)^2 + \zeta \bigg| \leq c_{\alpha,n}\right),
ds^2= -\left(1-\frac{r_{+}^2}{r^2}\right)d{\hat t}^2 +\left(1-\frac{r_{-}^2}{r^2}\right)d{\hat x}^2 +{\left(1-\frac{r_{+}^2}{r^2}\right)}^{-1}{\left(1-\frac{r_{-}^2}{r^2}\right)}^{-1}\frac{l^2 dr^2}{r^2}
\begin{array}{rcl}u_{\varepsilon,R,a}(x) = \displaystyle u_{\varepsilon,R}(x)+\left(\frac{n-2k}{k}u_{\varepsilon,R}(x)+ |x| \partial_ru_{\varepsilon,R}(x)\right)a\cdot x \displaystyle+O''\left(|a|^2\varepsilon^{\frac{n-2k}{2k}}R^{\frac{2k-n}{2k}}|x|^2\right).\end{array}
\begin{cases}(\rho u A(x))_x = 0, ((\rho u^2+p)A(x))_x = A'(x)p, \big((e+\frac{1}{2}u^2+\frac{p}{\rho})\rho u A(x)\big)_x =q_0A(x)\rho\phi(T)Z, (\rho u Z A(x))_x = -A(x)\rho\phi(T)Z.\end{cases}
e^{zL_{1}}e^{z_0L_{-1}}=e^{(1-zz_0)^{-1}z_0L_{-1}}(1-zz_0)^{-2\deg}e^{(1-zz_0)^{-1}zL_{1}}=e^{(1-zz_0)^{-1}z_0L_{-1}}e^{z(1-zz_0)L_{1}}(1-zz_0)^{-2\deg},\ \ \ \ \ \ (1-zz_0)^{-2\deg}= e^{-z(1-zz_0)L_{1}}e^{-(1-zz_0)^{-1}z_0L_{-1}}e^{zL_{1}}e^{z_0L_{-1}}.
\left\{\frac{1}{8}, \frac{1}{8}, \frac{1}{8}, \frac{1}{8}, \frac{1}{8}, \frac{1}{8}, \frac{1}{8}, \frac{1}{8}\right\}\mbox{\boldmath $M$}= \left\{0, \frac{2}{3}, \frac{1}{3}, 0, 0, 0, 0, 0\right\}
W\left[ C\right] = e^{-TE_{class}}\frac{\det\left( -\Delta_F -\frac{1}{4}R^{(2)}+1\right)}{\det\left( -\Delta +2\right)\det^{\frac{1}{2}}\left( -\Delta +4 - R^{(2)}\right) \det^{\frac{5}{2}}\left( -\Delta\right)}.
\Phi(\omega,s,\alpha)=\sum_{n=0}^{\infty}\frac{\omega^{n}}{(n+\alpha)^{s}}=\sum_{m=0}^{q-1}\sum_{n=0}^{\infty}\frac{\omega^{qn+m}}{(qn+m+\alpha)^{s}}=\sum_{m=0}^{q-1}\omega^{m}q^{-s}\zeta(s,\frac{m+\alpha}{q})
\langle V_{\mu}( x ) V_{\nu}( y ) \rangle_{0}= e^{2}\int \frac{d^{4}k}{( 2 \pi )^{4}}\left( g_{\mu \nu}+ \frac{k_{\mu}k_{\nu}}{m_{V}^{2}}\right) \frac{1}{k^{2}+ m_{V}^{2}}e^{i k \cdot ( x - y )},
C_{I,L}(N)g^{2L-2}_{YM}{\dot U^I\over U^{3L+2(I-2)}}=C_{I,L}(N){\dot U^2\over g^2_{YM}}\left ({g^2_{YM}\dot U^2\over U^{7}}\right )^L\left ({\dot U\over U^2}\right )^{I-2L-2}.
\widetilde{G}=- \left( \kappa + \mathrm{i}\omega \right) a^\dagger a - \frac{2 \kappa \, \overline{\alpha_0}}{\kappa - \mathrm{i}\omega}a+ \left\vert \alpha_0 \right\vert^2 \left( \frac{1}{\kappa - \mathrm{i}\omega}- \frac{2 \kappa}{\kappa^2 +\omega^2}\right) I .
L(\Phi)=\frac{d+2}{4-d}\omega Q(\Phi)+\frac{d-1}{4-d}\mathbf{c}\cdot \mathbf{P}(\Phi),\ \ \ \ \ \N(\Phi)=-\frac{2}{4-d}\left(2\omega Q(\Phi)+\mathbf{c}\cdot \mathbf{P}(\Phi)\right).
( 2 a )^{- s}s \sum_{k = 1}^{\infty}\frac{( - 1 )^{k}}{( k \pi )^{2}}\int_{0}^{\infty}d u \, e^{- u}\left[ \zeta \left( s + 1 , \frac 3 2 + \frac{c}{2 a}- \frac{i u}{2 k \pi}\right) + \zeta \left( s + 1 , \frac 3 2 + \frac{c}{2 a}+ \frac{i u}{2 k \pi}\right) \right] =
\begin{aligned}\mathbb{P}\big[\mathcal{B}_{\ell}^{(2)}\big] \leqslant \mathbb{P}\big[\mathcal{B}_{\ell}^{(2)}\cap \mathcal{X}_{\ell}\big]+\mathbb{P}\big[\,\overline{\mathcal{X}_{\ell}}\,\big].\end{aligned}
u(\xi) \sim \dfrac{1}{\mu}\sim\begin{cases}B_{1}e^{\omega_{1}\xi}+B_{2}e^{\omega_{2}\xi}+\dfrac{1}{\mu}, (D>0), (B_{3}\xi+B_{4})e^{\omega\xi}+\dfrac{1}{\mu},(D=0), e^{-\frac{\mu c}{2}\xi}\bar{Z}(\xi) +\dfrac{1}{\mu},(D<0),\end{cases}
0 = (\eta_1^2(y_1)+ \rho_1)\eta_2^2(y_2) = \eta_1^2(y_1)\eta_2^2(y_2)-\rho_1\rho_2 = (\eta_1(y_1)\eta_2(y_2) + \rho_{1,2}/2)(\eta_1(y_1)\eta_2(y_2) - \rho_{1,2}/2).
\begin{aligned}H_{i}(\cdot)\triangleq b(t,x,u_1,u_2)q_i(t)+\sigma(t,x,u_1,u_2)k_i(t)+\sum_{j=1}^2[\sigma_j(t,x)k_{ji}(t)+h_j(t,x)Q_{ji}(t)] - [f(t,x,y,z,z_1,z_2,u_1,u_2)-\sum_{j=1}^2h_j(t,x)z_j(t)]p_i(t)+l_i(t,x,y,z,z_1,z_2,u_1,u_2), \end{aligned}
\left|p_n\right|\leq\left(\frac{m}{4}+\frac{1}{2}\right)\left|p_n^{(1)}\right|+\left(\frac{m}{4}-\frac{1}{2}\right)\left|p_n^{(2)}\right|\leq2\left(\frac{m}{4}+\frac{1}{2}\right)+2\left(\frac{m}{4}-\frac{1}{2}\right)=m,\;n\geq1.
\int_{\tau_n^-}^{\tau_n^+}\Delta_1(x)(x-\bar{\tau})^j\ \d{x}=\begin{cases}0 j=0, \textrm{or}j\textrm{is odd}; \frac{j}{2^{j+2}(j+1)(j+2)}\big(\tau_n^+-\tau_n^-\big)^{j+2}j>0, \textrm{and}j \textrm{is even}, \end{cases}
I_1 \leq \ -\rho\lambda e^{\lambda u}\Bigl( \lambda - 2c\Bigr) |\nabla u|_g^2\varphi v -3\left(\frac{f'(u)}{f(u)}\right)^2|\nabla u|_g^2 \varphi v &+ 2c \left|\frac{f'(u)}{f(u)}\right| \varepsilon^{-1}\varphi v^3 + \frac{\varepsilon'}{(1-\varepsilon')}\rho^2\lambda^2 e^{2\lambda u}|\nabla u |^2_g \varphi v.
\mathcal I(tu) \leq t^{p_2^+}\left[1- M (C(Y))^{p_2^+}\|u\|^{p_2^+}\,\right]+ |\Omega| C_M =t^{p_2^+}\left[1- M (C(Y))^{p_2^+}\,\right]+ |\Omega| C_M.
I_{\upsilon}(q;x)=\frac{\left( \frac{x}{2}\right)^{\upsilon}2^{2\upsilon+q-\frac{1}{2}}}{\sqrt{\pi}\ \Gamma \left( \upsilon +\frac{1}{2}\right)}\int_{0}^{1}t^{\upsilon +q-1}\left( 1-t\right)^{\upsilon -\frac{1}{2}}\exp(2xt)\dt.
x(t_{n+1}) = x(t_n+h) = x(t_n) + \sum_{k=1}^{p}\frac{1}{k!}\frac{d^k x}{dt^k}(t_n) h^k + O(h^{p+1}) &= x(t_n) + \sum_{\tau \in \mathcal{T}(p)}\frac{h^{r(\tau)}}{\sigma(\tau) \gamma(\tau)}F(\tau)(x(t_n)) + O(h^{p+1})
\max \{\left|A_{k,m}(z)\right|, \left|B_{k,m}(z)\right|\}\leq (1+\delta)^m\max \{\left|A_k(z^{d^m})\right|, \left|B_k(z^{d^m})\right|\}\prod_{j=0}^{m-1}\widetilde{P}(\left|z\right|^{d^j}).
\left(\begin{matrix}x=0 \, \left( \frac12 \right) x=\infty \, \left( \frac12 \right) \overbrace{\begin{matrix}0 0 1 \theta^0/2 -1 \theta^0/2 \end{matrix}}\overbrace{\begin{matrix}\sqrt{t}\theta^\infty_1/2 -\sqrt{t}\theta^\infty_1/2 0 \theta^\infty_2 \end{matrix}}\end{matrix}\right),
K^{(j)}_n=\left|\begin{array}{cccc}1 1 \cdots 1 u_{j}u_{j+1}\cdots u_{j+n-1}u_{j+1}u_{j+2}\cdots u_{j+n}\vdots \vdots \vdots u_{j+n-2}u_{j+n-1}\cdots u_{j+2n-3}\end{array}\right|,\ \K^{(j)}_0=1.
\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{NDF=1,NDN=0}\right\rbrace \to\frac{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^{2}\left(2^{R}-1\right)(1+k)\left(k-2^{\textrm{R}}\right)}{\lambda_{\textrm{BN}}\left(k-\left(2^{\textrm{R}}-1\right)\right)}\frac{1}{P_{\textrm{B}}}.
L\rightarrow\tilde{L}=L(\dot{Q}_{(d-1)}^{i},Q_{(d-1)}^{i},\cdots,Q_{(2)}^{i},Q_{(1)}^{i},Q_{(0)}^{i})+\sum_{\alpha=1}^{d-1}\lambda^{(\alpha)}_{i}\left(Q_{(\alpha)}^{i}-\dot{Q}_{(\alpha-1)}^{i}\right),
x^{\lambda-1+k}f(x) =\frac{l_k}{(k-1)!}x^{k-1}+\frac{b_k}{(\lambda)_k}x^{\lambda+k-1}&{}+x^{\lambda+k-1}\int_0^{\infty}M_1(u)u^{\lambda+k-2}e^{-xu}\, du,
\begin{array}{rl}E_{8}^{+}= \{\pm e_{a}\}\cup \{(\pm e_{a}\pm e_{b}\pm e_{c}\pm e_{d})/2: a,b,c,d\mbox{distinct}, \; e_{a}(e_{b}(e_{c}e_{d}))=\pm 1\}, & a,b,c,d\in\{0,...,7\}. \end{array}
\int_{-\infty}^{\infty}\frac{d\epsilon}{2\pi}\frac{1}{2!}(\frac{\epsilon}{2\pi})^{2}\{\frac{1}{1+e^{\beta(\epsilon-\mu)}}-\theta(-\epsilon)\}=\frac{1}{3!}(\frac{\mu}{2\pi})^{3}+(\frac{\mu}{2\pi})\frac{T^{2}}{4!},
\begin{aligned}(x, p, q) \rightarrow \left( \left(\bar x, p \right), \left(m^{-1}\circ \pi \circ q, m^{-1}\circ \pi \circ{}_{i}x \circ \iota\right) \right) & ((\bar x, p), (r, \gamma)) \rightarrow (x, p, \pi_{\ker x_i}\circ m \circ \; r),\end{aligned}
\widehat{\mathbf{\Sigma}}\mathbf{=}\frac{\mathrm{Median}\left( \mathbf{d}\left( \widehat{\mathbf{\mu}}\mathbf{,}\widetilde{\mathbf{\Sigma}}\right)\right)}{\mathrm{Median}\left( \chi_{p}^{2}\right)}\widetilde{\mathbf{\Sigma}}.
\partial_{t}V_{n}+ \left(u_{n}+ \frac{\lambda_{n}(\rho_{n})}{\rho_{n}^{2}}\partial_{x}\rho_{n}\right) \partial_{x}V_{n}- \frac{\lambda_{n}(\rho_{n})}{\rho_{n}}\partial_{x}^{2}V_{n}= - \frac{(\lambda_{n}'(\rho_{n})\rho_{n}+ \lambda_{n}(\rho_{n}))}{(\lambda_{n}(\rho_{n}))^{2}}V_{n}^{2}.
\lim_{t \to \infty}\frac{1}{t}\log \Vert \mathbf{X}_{i_1}\cdots \mathbf{X}_{i_t}\Vert=\lim_{t \to \infty}\frac{1}{t}\log \Vert \boldsymbol{\alpha}_0 \mathbf{X}_{i_1}\cdots \mathbf{X}_{i_t}\Vert\,.
A_{(\alpha\beta\gamma)\delta\cdot\cdot\cdot}=\frac{1}{3!}(A_{\alpha\beta\gamma\delta\cdot\cdot\cdot}+A_{\gamma\alpha\beta\delta\cdot\cdot\cdot}+A_{\beta\gamma\alpha\delta\cdot\cdot\cdot}+A_{\alpha\gamma\beta\delta\cdot\cdot\cdot}+A_{\gamma\beta\alpha\delta\cdot\cdot\cdot}+A_{\beta\alpha\gamma\delta\cdot\cdot\cdot})
\mu_{l}^{i}=\max_{\mathbf{r}_{\mathcal{C}_{l}l}}\left(\hat{U}_{l}\left(\mathbf{r}_{\mathcal{C}_{l}l},r_{lP_{l}}=i\right)+\sum_{k\in\mathcal{C}_{l}}\mu_{k}^{r_{kl}}\right),
\pi'_1(T_\xi)^*\pi_1'(T_\eta) = \pi'_1(T^*_\xi T_\eta)) = \pi'_1(\langle \xi, \eta\rangle_A) = \tilde{P}\tilde\varphi^\infty(\alpha_A(\langle \xi, \eta\rangle_A))\tilde{P}.
\eta(2^k i) =e^{-2^{k+1}\pi/24}\prod_{\ell=1}^\infty(1-e^{-2^{k+1}\pi\ell})=e^{-2^{k+1}\pi/24}\prod_{\ell=1}^\infty(1-e^{-2^{k}\pi\ell})\prod_{\ell=1}^\infty(1+e^{-2^{k}\pi\ell}) &=:e^{-2^k\pi/24}\cdot g_k\cdot \eta(2^{k-1}i),
\mathcal{R}(e_1)= e_1 + a_{4}e_4 + a_{5}e_{5},\ \mathcal{R}(e_2)=b e_3,\ \mathcal{R}(e_3)= \mathcal{R}(e_5) = 0, \mathcal{R}(e_4)= \frac{-b}{a_{5}}e_{2},
-\mathcal{L}_v^0 w := -\partial_x \left( a_{11}(v)\ \partial_x w + a_{13}(v)\ \partial_\eta w \right)-\partial_y \left( a_{22}(v)\ \partial_y w + a_{23}(v)\ \partial_\eta w \right) & -\partial_\eta \left( a_{31}(v)\ \partial_x w + a_{32}(v)\ \partial_y w+ a_{33}(v)\ \partial_\eta w \right)\ ,
Q_2= \frac{1}{|Dw|^2\log|Dw|^2}\bigg[\big[2(1-G_u)fu_1v-4G_{uu}u_1-4G_{ux_1}\big]\sum_{1\leq k\leq n}w_ku_{k1}&-4v^2\sum_{2\leq i\leq n}G_{ux_i}\sum_{1\leq k\leq n}w_ku_{ki}-4(1-G_u)(G_{uu}u_1^2+2G_{ux_1}u_1+G_{x_1x_1})u_{11}&-4(1-G_u)(v^2+1)\sum_{2\leq i\leq n}(G_{ux_i}u_1+G_{x_1x_i})u_{1i}-4(1-G_u)v^2\sum_{2\leq i\leq n}G_{x_ix_i}u_{ii}\bigg];
L_T K_T =1 + \int_{0}^{T}{\bf 1}_{u\leq t}L_u S_u \rho_u dQ_u + \int_{0}^{T}K_u L_u h(X_u)dQ_u + \int_{0}^{T}{\bf 1}_{u\leq t}S_u L_u \rho_u h(X_u)du
U ( t , t_{0}) ={\bf 1}- \int_{t_{0}}^{t}d t_{2}\int_{t_{0}}^{t_{2}}d t_{1}\left( 2 i \delta ( t_{2}- t_{1}) H_{I}( t_{1}) + H_{I}( t_{2}) U ( t_{2}, t_{1}) H_{I}( t_{1}) \right) .
\Phi_w^{\prime}(t)=t\|w\|^2 -\lambda t^{p+q-1}\int_{-1}^1f(x)|w_1(x, 0)|^p|w_2(x,0)|^q dx &-\int_{-1}^1 g_1(tw(x, 0))w_1(x, 0) dx-\int_{-1}^1 g_2(tw(x, 0))w_2(x, 0) dx=0.
tlh[\frac{1}{2}aclh(x)]^{2}=\frac{1}{2}\sqrt{2-2\sqrt{2}x\sqrt{\sqrt{x^{4}+1}-x^{2}}}=(2x^{2}+2+2\sqrt{x^{4}+1})^{-1/2}(\sqrt{\sqrt{x^{4}+1}+1}-x)
\beta(s)=\frac{1}{2^{s}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+\frac{1}{2})^{s}}=\frac{1}{(-4)^{s}(s-1)!}[\psi^{(s-1)}(\frac{1}{4})-\psi^{(s-1)}(\frac{3}{4})]
F_{k,-m,0}(z) = P_{k,-m}(z,1-k/2) &= \varphi_{k,-m}(z,1-k/2) + \sum_{(-1)^{\lambda_k}n \equiv 0,1 (4)}b_{k,-m}(n,1-k/2)\mathcal{W}_{k,n}(y,1-k/2)e^{2\pi inx}&= q^{-m}-\frac{\Gamma(1-k,4\pi my)}{\Gamma(1-k)}q^{-m}+ \sum_{(-1)^{\lambda_k}n \equiv 0,1 (4)}b_{k,-m}(n,1-k/2)\mathcal{W}_{k,n}(y,1-k/2)e^{2\pi inx}.
\mathrm{E}(X^n)=\int_{-\infty}^\infty x^n g(x) \mathrm{d}x=\frac{1}{\sigma\mathrm{B}(\alpha,\beta)}\sum_{j=0}^{\infty}(-1)^j\binom{\beta-1}{j}\,\int_{-\infty}^\infty x^n\,h_j^{(s)}\mathrm{d}x.
H_D = \lambda (\tau) H (\tau)- \vec{\lambda}(\tau) \vec{H}_p (\tau) +\int{d^3\sigma \Big[ \mu_\tau (\tau, \vec{\sigma}) \pi^\tau (\tau, \vec{\sigma})- A_\tau (\tau, \vec{\sigma}) \Gamma (\tau, \vec{\sigma}) \Big]},
s(H){-}s(H'){=}\left|d_i{-}\frac{rm}{n}\right|{+}\left|d_j{-}\frac{rm}{n}\right|{-}\left|(d_i{+}1){-}\frac{rm}{n}\right|{-}\left|(d_j{-}1){-}\frac{rm}{n}\right| &{=}\left(\frac{rm}{n}{-}d_i\right)\!\!+\!\!\left(d_j{-}\frac{rm}{n}\right)\!\!-\!\!\left(\frac{rm}{n}{-}(d_i{+}1)\right)\!\!-\!\!\left((d_j{-}1){-}\frac{rm}{n}\right)\!=\!2.
v_{y_0}(y) =- \frac{4}{27 a^2(x_0)}\left( \left(\frac{\nu_{x_0}\cdot A(x_0)\nu_{x_0}}{(\nu_{x_0})_n}y_n \right)^3 \right. & \left. - 3 \left( \frac{\nu_{x_0}\cdot A(x_0)\nu_{x_0}}{(\nu_{x_0})_n}(a^{n+1,n+1}(x_0)) \right) y_ny_{n+1}^2 \right) &-g(y_0)y_n + y_{n}\frac{(y''-y_0)\cdot \nu_{x_0}''}{(\nu_{x_0})_n},
g_{1}([X,Y],V) =g_{1}(\varphi V,\mathcal{A}_{Y}\mathcal{B}X+\mathcal{V}\nabla^{^{M_1}}_{X}\mathcal{B}Y)-g_{2}(\pi_{\ast}(\mathcal{C}Y),\pi_{\ast}(\nabla^{^{M_1}}_{X}\varphi V)) &-g_{1}(\varphi V,\mathcal{A}_{X}\mathcal{B}Y+\mathcal{V}\nabla^{^{M_1}}_{Y}\mathcal{B}X)-g_{2}(\pi_{\ast}(\mathcal{C}X),\pi_{\ast}(\nabla^{^{M_1}}_{Y}\varphi V)).
S(0,a_1,\ldots , a_n)\ =\ (-1)^n (n-1)\,a_1\,{\mathcal R}_{Q,Q'}+ (-1)^n n\,a_1\,{\mathcal R}_{Q,Q'}\ =\ (-1)^n (2n-1)\, a_1\,{\mathcal R}_{Q,Q'}
\int_{0}^{\infty}\frac{\mathrm{d}q}{\sqrt{a^{2}+ q^{2}}}\frac{1}{\mathrm{e}^{\sqrt{a^{2}+ q^{2}}}+ 1}= - \frac{1}{2}\ln \frac{a}{\pi}- \frac{1}{2}\mathrm{C}+ O ( a^{2}) \ .
A_{3}^{\mathrm{T}}P_{13}^{\mathrm{T}}+P_{13}A_{3}=\left[\begin{array}[c]{cc}0 0 0 2\left( \sigma_{3}-1\right)\end{array}\right] p_{134}, A_{1}^{\mathrm{T}}P_{13}+P_{13}^{\mathrm{T}}A_{1}=\left[\begin{array}[c]{cc}0 \sigma_{3}-4\sigma_{1}\sigma_{3}-4\sigma_{1}2\left( 1-\sigma_{1}\right)\end{array}\right] p_{134}.
\int_{B_R}(x\cdotp\nabla\bar{w})Ke^{2m\bar{w}}dx=\int_{B_R}(x\cdotp\nabla\bar{w})(-\Delta)^m\bar{w}dx - t\alpha\int_{B_1}(x\cdotp\nabla\bar{w})(-\Delta)^mu_0 dx,
S_\gamma^* S_\gamma = \sum_{\eta \in E_Z}Z^G(\gamma,\eta) S_{\eta}S_{\eta}^* & = \sum_{\eta \in E_Z}Z^G(\gamma,\eta) (P_B S_{\eta}S_{\eta}^* P_B + P_A S_{\eta}S_{\eta}^* P_A) & = \sum_{\eta \in E_Z}Z^G(\gamma,\eta) S_{(B,\eta)}S_{(B,\eta)}^* + \sum_{\eta \in E_Z}Z^G(\gamma,\eta) S_{(A,\eta)}S_{(A,\eta)}^*.
\sup_{(b,d)\in \mathcal H_m}\bigg( \Phi\big(a^{2,i},b,c_{2,i},d,Du_m(y,t_1),D^2u_m(y,t_1)\big)+ru_m(y,t_1)\bigg) &\leq \partial_t u_m(y,t_1)+\frac{1}{k}
A_3(s, t) = \frac{1}{96\pi^2}\left( \frac{-t}{s}\right)^2\!\int\limits_0^{\infty}\! dx A_B(s,\sqrt{-t}x)\!\!\int\limits_0^{\infty}\! dx' A_B(s, \sqrt{-t}x') &\times \!\!\int\limits_0^{\infty}\! dx'' A_B(s,\sqrt{-t}x'') \,G(x, x', x'') \;.
\gamma^{'}=(1+\sigma_{\theta}^2g(\mathbf{a})) \ln\left[\gamma(1+\sigma_{\theta}^2g(\mathbf{a}))\right] g(\mathbf{a}) =\mathbf{a}^H\mathbf{H}^H\mathbf{C}_w^{-1}\mathbf{H}\mathbf{a}\; .
\Lambda_{\pm}(q_0)=\Lambda(q_0)\pm\frac{2\kappa\bigg(\frac{2}{\pi}\bigg)^{\frac{1}{2}}}{[\frac{1}{2}(q_0-1)]!}\bigg(\frac{1+k}{1-k}\bigg)^{-\frac{\kappa}{k}}\bigg(\frac{8\kappa}{1-k^2}\bigg)^{\frac{1}{2}q_0}\bigg[1+O\bigg(\frac{1}{\kappa}\bigg)\bigg]
\begin{array}{l}2m \; \partial_{t}\left\langle \phi^{'}\left( \frac{|x|}{m}\right) \frac{x_j}{|x|}v, - i \, \partial_{x_j}v \right\rangle = \int (- \triangle \triangle a) \, |v|^{2}\, + \, 4 \partial^{2}_{x_j x_{k}}a \,\Re \left( \overline{\partial_{x_j}v}\partial_{x_k}v \right) \, + \, 2 \triangle a H(|v|^{2}) \; dx,\end{array}
\int_{\mathcal{C}}d \hat{\mu}(\zeta) = 1 = \mathcal{\hat{M}}[1] \mbox{and}\int_{\mathcal{C}}\zeta^{n}d \hat{\mu}(\zeta) = \hat{\mu}_{-n}= \mathcal{\hat{M}}[z^n], \ \n =\pm 1,\pm 2, \ldots\, .
N_{\mathbf{x}\mathfrak{m}}(\mathbf{a}) = \left( \begin{array}{c}N_{x_1 \mathfrak{m}}(\mathbf{a}) N_{x_2 \mathfrak{m}}(\mathbf{a}) \vdots N_{x_n \mathfrak{m}}(\mathbf{a}) \end{array}\right) = \mathcal{D}_\mathbf{a}( N_\mathfrak{m}(\mathbf{a}) ).
\begin{pmatrix}a_1 a_2 a_3 1 b_1 b_2 b_3 1 c_1 c_2 c_3 1 d_1 d_2 d_3 1\end{pmatrix}\begin{pmatrix}e_1' e_2' e_3' r'\end{pmatrix}=\begin{pmatrix}-a_1^2-a_2^2-a_3^2 -b_1^2-b_2^2-b_3^2 -c_1^2-c_2^2-c_3^2 -d_1^2-d_2^2-d_3^2\end{pmatrix}
A_2 = \left[\begin{array}{ccc}\frac{2}{5}-\frac{\cos \phi}{2}0 -\frac{\cos \phi}{2}-\frac{2}{5}0 \cos \phi 0 -\frac{\cos \phi}{2}-\frac{2}{5}0 \frac{2}{5}-\frac{\cos \phi}{2}\end{array}\right],
k > \frac{N-8\delta_m^3 \cdot m^3}{m}-(r^2+r+1) & \ge \frac{K_1-8\delta_m^3 \cdot m^3}{m}-(r^2+r+1) & = \frac{K_1-8\delta_m^3 \cdot m^3}{m}-\frac{m^2}{36}-\frac{3}{4}\ge 0 .
\frac{1}{B-s+1}\left(\sum_{k=0}^{\frac{n}{2}- s}\binom{n}{k}\right)^2< S_{n,s}< \binom{n}{\lfloor{\frac{n}{2}}\rfloor-s}\sum_{j=0}^{\lfloor{\frac{n}{2}}\rfloor-s}\binom{n}{j}<2^n e^{-\frac{2s^2}{n}}\binom{n}{\lfloor{\frac{n}{2}}\rfloor-s},
H(t, M, \kappa_{\tau}) = \rho^{\tau [\frac{d - \frac{m}{2}}{2}+ 1]}\;\; H(\frac{t}{\rho^{2 \tau}}, \frac{M}{\rho^{2 \tau [\frac{d - \frac{m}{2}}{2}- 1]}}, \frac{\kappa_{\tau}}{\rho}) .
\begin{array}{rcl}\delta A_{m}= \sigma_{mn}A_{n}+ m \lambda_{m}\, , & & \delta V_{mn}={\cal D}\sigma_{mn}\, , & & \delta B_{m}= 2\partial\lambda_{m}+ 2\partial \sigma_{mn}A_{n}+ m \sigma_{mn}B_{n}\, , \end{array}
\pm(e_{i}-e_{j}) 1\leq i <j\leq p, \pm(f_{i}-f_{j}) 1\leq i <j\leq q, \pm(e_{i}-f_{j}) 1\leq i\leq p,1\leq j\leq q, e_{i}+f_{j}1\leq i\leq p,1\leq j\leq q.
\|\mathfrak R \chi_{E^j}\|_{L^2(d\sigma)}\leq \sum_{k=1}^K \|\mathfrak R \chi_{E^j_k}\|_{L^2(d\sigma)}\leq C_{\epsilon}|E^j|^{1/K}(|E^j|^{C\gamma^{K-1}})^{\epsilon}\|\chi_{E^j}\|_{L_x^{q_0}(\mathbb R^n ; L_t^{r_0}(\mathbb R))}
\begin{aligned}\frac{M^{p-2}}{2^{p(n+3)}}\int_{\widetilde{K}_n}(u-\tilde{k}_n)_-^2\,dx+\underset{\widetilde{Q}_n}{\iint}\frac{|(u-\tilde{k}_n)_{-}(x,t)-(u-\tilde{k}_n)_{-}|^p}{|x-y|^{N+sp}}\,dx \,dy\,dt\leq\boldsymbol\gamma \frac{2^{n(N+2p)}}{\theta\varrho^{sp}}M^{p}|A_n|.\end{aligned}
(A!_\lambda B)^{-1}-(A\nabla_\lambda B)^{-1}=\lambda(1-\lambda)(B-A)(A\sharp B)^{-2}(A\nabla_\lambda B)^{-1}(B-A) & (A!_\lambda B)^{-1}(A\nabla_\lambda B)-I=\lambda(1-\lambda)(A-B)A^{-1}B^{-1}(A-B).
L_{N,j}\Lambda Q_{c_j}=-\sum_{k=1}^N\frac{\partial\lambda_k}{\partial c_j}H_k'(Q_{c_j})=-\sum_{k=1}^N\frac{\partial\lambda_k}{\partial c_j}(-c_j)^{k-1}Q_{c_j}=-\left(\prod_{k=1,k\neq j}^N(c_k-c_j)\right)Q_{c_j}.
|\partial_{R}(v_{corr}(x,R\lambda(x)))| \leq C\lambda(x) \begin{cases}\frac{\log(\log(x))}{x^{2}\log^{b+1}(x)}, R \lambda(x) \leq \log^{N}(x) \frac{1}{x^{2}\log^{b}(x)}, \log^{N}(x) \leq R\lambda(x) \leq \frac{x}{2}\frac{1}{\sqrt{x}}, R \lambda(x) > \frac{x}{2}\end{cases}
\begin{aligned}E_{D^2}(e^{2\pi i t}) = E_{D^2}\left(M^{-1}\circ M(e^{2\pi i t}) \right) &= M^{-1}\circ Q\left(M(e^{2\pi i t})\right) &= \hat{P}(e^{2 \pi i t}) = e^{2 \pi i P(t)},\end{aligned}
W^{(n,m,i,\kappa)}(y) =y^{\kappa}(1-y)^{n-1+\kappa}\Psi^{n,m,i}_0(y) \, T^{(\kappa)}\, (\Psi^{n,m,i}_0(y))^* D^{(n,m,i,\kappa)}=y(1-y)\partial_y^2+\partial_y(C+\kappa-y(U+2\kappa))-(V+\kappa U+\kappa(\kappa-1)),
y_{n+1}^{1-2s}\hat{w}'' + (1-2s)y_{n+1}^{-2s}\hat{w}' - y_{n+1}^{1-2s}|\xi|^2 \hat{w}= 0, -\lim\limits_{y_{n+1}\rightarrow 0}y_{n+1}^{1-2s}\hat{w}' = \hat{g}.
\mu_1 =\max_{f:\sum_{v\in V}\deg v\cdot f(v)^2=1}\sum_{h\in H}\biggl(\sum_{v_i h}f(v_i)-\sum_{v^j h}f(v^j)\biggr)^2 &\leq\max_{f:\sum_{v\in V}\deg v\cdot f(v)^2=1}\sum_{h\in H}\biggl(\sum_{v_i \in h:f(v_i)>0}f(v_i)-\sum_{v^j \in h:f(v^j)<0}f(v^j)\biggr)^2,
\begin{cases}\Delta_{t}u(\gamma_{t}) + \sigma (t)\Delta_{xx}u(\gamma_{t}) + f(\gamma_{t}, u(\gamma_{t}),-\Delta_{x}u(\gamma_{t}))=0, \ \ \gamma_{t}\in \Lambda_{t}, \t\in[0,T), u(\gamma) = g(\gamma), \ \ \gamma\in \Lambda_{T}, \end{cases}
\varphi\left(\begin{array}{c c}a b 0 d \end{array}\right) =\left(\begin{array}{c c c}a^{\ell_1}0 c_2 \,a^{p^{e_2}}\, b^{p^{e_2}}\, d^{- \ell_3}0 d^{- \ell_2}c_1 \, a^{p^{e_1}}\, b^{p^{e_1}}\, d^{- \ell_3}0 0 d^{- \ell_3}\end{array}\right) .
W [{\psi}] = \int_{- \infty}^{\infty}d x^{+}\int_{0}^{L}d x^{-}{\psi}_{+}^{\star}i{\partial}^{-}{\psi}_{+}- \frac{i}{2}m \int_{- \infty}^{\infty}d x^{+}\int_{0}^{L}d x^{-}\int_{0}^{L}d y^{-}{\psi}_{+}^{\star}( x^{-}, x^{+}) \epsilon ( x^{-}- y^{-}){\psi}_{+}( y^{-}, x^{+})
\vartheta(\tau, z) = q^{1/8}(\zeta^{1/2}- \zeta^{-1/2}) \prod_{n=1}^{\infty}(1 - q^n \zeta)(1 -q^n)(1-q^n \zeta^{-1}) &= \sum_{n=1}^{\infty}\left( \frac{4}{n}\right) q^{n^2 / 8}(\zeta^{n/2}- \zeta^{-n/2})
Q( w^\prime \, c^l \, (c^*)^m) = Q( u \, (a a^*) \, v \, c^l \, (c^*)^m) = Q( u \, (1 - q^2 c c^*) \, v \, c^l \, (c^*)^m) = Q( u \, v \, c^l \, (c^*)^m) - q^2 Q( u \, (c c^*) \, v \, c^l \, (c^*)^m) = Q( u \, v \, c^l \, (c^*)^m) - q^r Q( u \, v \, c^{l+1}\, (c^*)^{m+1}) = 0 - 0 = 0.
C\|f\|_{L^{\infty}(\mathcal G)}\geq{1\over |B|}\int_B\left|J_4(g,\tilde g)\right|dg &={1\over |B|}\int_B\left|\big(b(g)-b_B\big)R_j(f_2)(\tilde g)\right|dg &=\left({1\over |B|}\int_{B}\big|b(g)-b_B\big|dg\right)\left|\int_{(r_oB)^c}K_j\big (\tilde g, g'\big)f\big(g'\big)dg'\right|.
\lim_{\epsilon \downarrow 0}\int_a^b \left\vert \det Y(x \pm i\epsilon) - \det Y_\pm(x) \right\vert{\rm d}x \qquad{}=\lim_{\epsilon \downarrow 0}\int_a^b \left\vert Y_{11}Y_{22}(x \pm i\epsilon) - ( Y_{11}Y_{22})_\pm (x) - \left(Y_{12}Y_{21}(x \pm i\epsilon) - ( Y_{12}Y_{21})_\pm (x) \right)\right\vert{\rm d}x = 0.
D_{\nu,\theta}(f) =-\int_{\mathbb{R}^3\times \mathbb{R}^+}A_{\nu,\theta}\big\{\mathcal{M}_{\nu,\theta}(f)-f\big\}\ln f dvdI\cr =A_{\nu,\theta}\int_{\mathbb{R}^3\times \mathbb{R}^+}\big\{f-\mathcal{M}_{\nu,\theta}(f)\big\}H^{\prime}(f)vdI\cr \geq A_{\nu,\theta}\big\{H(f)-H(\mathcal{M}_{\nu,\theta})\big\}.
\begin{aligned}\mathop{\max}\limits_{\left\{{{n_k}}\right\}_{k = 1}^{{N_{\rm{R}}}}}\sum\limits_{k = 1}^{{N_{\rm{R}}}}{{{\log}_2}\left({{{1 +{{\bar C}_k}\left({1 -{2^{- 1 - ({x_k}+n_k)}}}\right)}}}\right)}{\rm{s.t.}}\quad{n_k}\in{{\mathbb Z}^+}&\sum\limits_{k = 1}^{{N_{\rm{R}}}}{{n_k}}= 1.\end{aligned}
q_\eta ({\bf{x}}) =S(\eta)^{\frac{1}{1 - \beta}}M_{\eta,\beta}\big[ 1 + c_\beta S(\eta)^{-1}\lbrace \rm{vec}^\top (\boldsymbol{\Sigma}^{-1}) \rm{vec}({\bf{x}}{\bf{x}}^\top) - 2 (\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu})^\top{\bf{x}}\rbrace\big]^{\frac{1}{1 - \beta}},
\begin{aligned}\Vert \langle y\rangle^{l+1}\eta_i\Vert_{L^\infty}\leq 2\vartheta_0^{-1}\sigma_0\leq 2\vartheta_0^{-2}, &M(x)\leq \vartheta_0^{-1}(C(u_e,h_e,u_b)+\vartheta_0^{-1}+\sigma_0)\leq 5 \vartheta_0^{-2}.\end{aligned}
lim_{\lambda\rightarrow\infty}\int_{0}^{\pi/2}f(x)sin(\lambda x)dx=0\Rightarrow lim_{\lambda\rightarrow\infty}\int_{0}^{\pi/2}\frac{sin(\lambda x)}{x}dx=lim_{\lambda\rightarrow\infty}\int_{0}^{\pi/2}\frac{sin(\lambda x)}{sin(x)}dx.
Y_{\mathbf{a}}\left(\mathbf{n}\right)=\int_{\left[1,+\infty\right)^{r}}\frac{d\mathbf{x}}{\left(x_{1}+a_{1}\right)^{n_{1}}\left(x_{1}+a_{1}+x_{2}+a_{2}\right)^{n_{2}}\dots\left(x_{1}+a_{1}+\dots+x_{r}+a_{r}\right)^{n_{r}}}