harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
m_{0,\mu}(\xi - \eta, \eta) = -\frac{\xi \cdot (\xi-\eta)}{2 |\xi||\xi-\eta|}\big( (\xi - \eta) \times \eta \big),m_{\mu, \nu}(\xi - \eta, \eta) = \frac{c_{\mu}}{4 | \xi |}\frac{1}{|\xi - \eta|}\big( (\xi - \eta) \times \eta \big)^{2},
\widehat{B}(k) \sim\frac{(1-\lambda)^{1/4}}{\left(\lambda(1+\lambda)\right)^{1/12}}\frac{\sqrt{2}}{\sqrt{\frac{1+\lambda}{1-\lambda}}\left(\frac{1+\lambda}{1-\lambda}-\frac{1-\lambda}{1+\lambda}\right)^{1/4}}\frac{Ai(0)}{n^{1/3}}=\frac{1-\lambda}{\left(\lambda(1+\lambda)\right)^{1/3}}\frac{1}{3^{2/3}\Gamma(2/3)}\frac{1}{n^{1/3}},
\int_1^\infty \tau^{2s}\left| \real \Big(\tilde T_{2j}(i\tau/2) + \sum_{l = 0}^k (-1)^{j+l}i^{l-1}\tau^{-(2j-1+l)}T_{2j}^l\Big) \right| d\tau \lesssim \frac{1}{|\sin (\pi s)|}\| (u,v) \|_{\dot H^s}^2\|(u,v)\|_{l^2_1 DU^2}^{2j-2}.
\rho_{atom}(r,\theta,\phi)=P_{core}\rho_{core}(r)+P_{val}\kappa^{3}\rho_{val}(\kappa r)+\sum_{l=0}^{l_{max}}\kappa^{\prime3}R_{l}(\kappa^{\prime}r)\sum_{m=-l}^{l}P_{lm\pm}Y_{lm\pm}(\theta,\phi)
\lim_{n \rightarrow \infty}\frac{x_{n+\rho-1}}{x_n}= \lim_{n \rightarrow \infty}\frac{\lambda_1^{n+\rho-1}\left[\left(B-a\lambda_2\right)-\left(B-a\lambda_1\right)\left(\frac{\lambda_2}{\lambda_1}\right)^{n+\rho-1}\right]}{\lambda_1^{n-1}\left[\left(B-a\lambda_2\right)-\left(B-a\lambda_1\right)\left(\frac{\lambda_2}{\lambda_1}\right)^{n-1}\right]}= \lambda_1^{\rho}.
(\widetilde{F}_{\eta}(v_g,\omega)-i)^{-1}-(d\Gamma(\omega)-i)^{-1}= \eta (\widetilde{F}_{\eta}(v_g,\omega)-i)^{-1}W(v_g,-1)(d\Gamma(\omega)-i)^{-1}= \eta^2(\widetilde{F}_{\eta}(v_g,\omega)-i)^{-1}W(v_g,-1)(d\Gamma(\omega)-i)^{-1}W(v_g,-1)(d\Gamma(\omega)-i)^{-1}&+\eta(d\Gamma(\omega)-i)^{-1}W(v_g,-1)(d\Gamma(\omega)-i)^{-1}.
\hat{\theta}_2(t) = \Big( 1 - C \exp \big\{- \beta z(t) \big\}\Big) \Phi \Big( \frac{\beta}{2}\sqrt{1 - \alpha}\sqrt{t}- \frac{\beta \alpha}{2 \sqrt{1- \alpha}}\sqrt{t}- \frac{z(t)}{\sqrt{1 - \alpha}\sqrt{t}}\Big)^2 \to 1
C_{a,b}\left(k\right) D^{\frac{n\pi i}{\log a}}\frac{\left( -z\right)^{\ell}}{\ell!}\prod\limits_{j=1}^{\ell+k}\left( k\frac{\log |b|}{\log a}+\frac{\left(2n+l_{Q,k}\right) \pi i}{\log a}-j\right),
R_{I}:=\left\{\exp(s):s\in\mathfrak{a},\alpha_{1}(s)\le0,\alpha_{2}(s)<0\mbox{for all}\alpha_{1}\in\Delta\mbox{and}\alpha_{2}\in\Delta\setminus I\right\}, D_{I}:=R_{I}\cap A_{I}=\left\{\exp(s):s\in\mathfrak{a}_{I},\alpha(s)<0\mbox{for all}\alpha\in\Delta\setminus I\right\}.
T_{n}(x):=\frac{(2n)!!}{(2n-1)!!}\sum_{k=0}^{n}\binom{n-\frac{1}{2}}{n-k}\binom{n-\frac{1}{2}}{k}\left(\frac{x-1}{2}\right)^{k}\left(\frac{x+1}{2}\right)^{n-k},
P\left(\left\lvert \frac{1}{N}\sum_{i=1}^{N}\phi(x^{t+1}_i,x_i) - \mathbb{E}\left[\phi(\eta_t(X+\tau^tZ),X)\right] \right\vert \geq \epsilon \right) \leq K_t e^{-\kappa_t n \epsilon^2},
\Phi(\sigma_{(a,0)}((b,j)))=\Phi((b,j+1))=(3b,j+1)=\sigma'_{(3a,0)}((3b,j))=\sigma'_{\Phi((a,0))}\Phi((b,j)), &\Phi(\sigma_{(a,1)}((b,0)))=\Phi((b+3,1))=(3b+1,1)=\sigma'_{(3a,1)}((3b,0))=\sigma'_{\Phi((a,1))}\Phi((b,0)), &\Phi(\sigma_{(a,1)}((b,1)))=\Phi((b+1,0))=(3b+3,0)=\sigma'_{(3a,1)}((3b,1))=\sigma'_{\Phi((a,1))}\Phi((b,0)).
\sum_{j = 0}^{\infty}r_j \int_{B_{r_j}}r^{3-n}|\nabla u|^2 \leq \bigg(\sum_{j = 0}^{\infty}r_j \bigg) \int_{B_{1}}r^{3-n}|\nabla u|^2 = 2 \int_{B_{1}}r^{3-n}|\nabla u|^2.
\mathcal{W}\left(if\right)\left(x_1,x_2\right)=if_0\left(x_1,x_2\right)+iif_1{\left(x_1,x_2\right)+ijf}_2\left(-x_1,x_2\right)+ikf_3\left(-x_1,x_2\right) &=i\left(f_0\left(x_1,x_2\right)+if_1{\left(x_1,x_2\right)+jf}_2\left(-x_1,x_2\right)+kf_3\left(-x_1,x_2\right)\right) &=i\mathcal{W}\left(f\right)\left(x_1,x_2\right)
\xi_{\alpha \beta}(\mu_{1}^{2},\mu_{2}^{2},\mu_{3}^{2},p^{2},q^{2})=\int_{0}^{1}dz\int_{0}^{1-z}dy\frac{z^{\alpha}y^{\beta}}{Q(y,z,\mu_{1}^{2},\mu_{2}^{2},\mu_{3}^{2},p^{2},q^{2})},
\lim_{N\rightarrow \infty}\limsup_{M\rightarrow \infty}\rho(\mu^*(t),\mu_{k^{N,M}}^{N,M})=0 \ \ \ \ \ \Rightarrow \ \ \ \ \ \lim_{i\rightarrow \infty}\limsup_{j\rightarrow \infty}\rho(\mu^*(t),\mu_{k^{N_i,M_{i,j}}}^{N_i, M_{i,j}})=0.
m_H(W\setminus \underline{W}) =m_H(W)-m_H(\underline{W}) &=m_H(W)-m_H(\Gamma_{H,H^{*}}^{-1}(W^*)) &=m_H(W)-(\Gamma_{H,H^{*}})_{\ast}(m_H)(W^*) &=m_H(W)-m_{H^*}(W^*)
\left(\frac{(13)}{(x^2_{13})^{\frac{D-2}{2}}}\right)^{m-j-k}\left(\frac{(23)}{(x^2_{23})^{\frac{D-2}{2}}}\right)^{n-j-k}\left[\frac{(13)}{(x^2_{13})^{\frac{D-2}{2}}}\ \frac{(23^-)}{(x^2_{23})^{\frac{D-2}{2}}}- 1 \leftrightarrow 2 \right]^j \ .
a_*(x) :=a(x,u_*,p_*) b_*(x) :=a_u(x,u_*,p_*).(u_*)_{xx}+D_uf(x,u_*,p_*) c_*(x) :=a_p(x,u_*,p_*).(u_*)_{xx}+D_pf(x,u_*,p_*).
\sigma^{0}( x , k ; \lambda ) = \sum_{l = 0}^{\infty}\sigma_{l}^{0}( x , k ; \lambda ) , \quad \sigma_{l}^{0}( x , \epsilon k ; \epsilon^{2}\lambda ) = \epsilon^{- 2 ( 1 + l )}\sigma_{l}^{0}( x , k ; \lambda ) ,
(A-5I)^{3}x_{3}=(\begin{matrix}0&0&0&14\\ 0&0&0&-4\\ 0&0&0&3\\ 0&0&0&-1\end{matrix})(\begin{matrix}x_{31}\\x_{32}\\x_{33}\\x_{34}\end{matrix})=(\begin{matrix}14x_{34}\\ -4x_{34}\\ 3x_{34}\\ -x_{34}\end{matrix})=(\begin{matrix}0\\ 0\\ 0\\ 0\end{matrix})
K=\left[ \begin{array}{c}D^{[1:J^*,:]}I_{d}-I_{d}p^{\prime}-p^{\prime}\end{array}\right] ,~g=\left[ \begin{array}{c}e^{[1:J^*]}\rho \cdot \mathbf{1}_{d}\rho \cdot \mathbf{1}_{d}0 0\end{array}\right] ,~\tau =\left[ \begin{array}{c}\mathbf{1}_{J^*}\mathbf{0}_{d}\mathbf{0}_{d}0 0\end{array}\right] .
E\Big[\prod_{i=1}^p |S_{s_i}-y_i|_+^{-2}\Big]= E\Big[\prod_{i=1}^{p-1}|S_{s_i}-y_i|_+^{-2}|S_{s_p}- S_{s_{p-1}}-( y_p-S_{s_{p-1}})|_+^{-2}\Big] \le C | s_p-s_{p-1}|_+^{-1/2}E\Big[\prod_{i=1}^{p-1}|S_{s_i}-y_i |_+^{-2}| S_{s_{p-1}}-y_p |_+^{-1}\Big]\,.
C_1 :=\sum_{2<p\le 23}b_p f(p)= \sum_{2< p\le 23}\frac{\pi}{4}\frac{M}{m_p^2}e^\gamma{\rm d}({\rm L}_p) = \frac{\pi}{4}\,M e^\gamma\cdot 0.39012\cdots = 0.5463\cdots
\kappa_\rho=\sup_{0<t<R-\rho}\frac{-g(t)}{t}, \lambda_\rho =\sup \{t\in [0,R-\rho): \kappa_\rho+g'(t)<0\}, \tilde\theta_\rho=\frac{\kappa_\rho}{2-\kappa_\rho},
\psi_0(x)=\alpha e^x,\psi_i(x)=\frac{1}{\lambda}(e^x-1),\phi_{i0}(x)=\frac{(\lambda-\alpha)e^x}{1-\lambda\alpha e^x}, \intertext{and}\psi_k(x)=(\lambda+\alpha)e^x,\phi_{k0}(x)=\frac{\lambda^{-1}(e^x-1)}{1-\lambda^{-1}\alpha(e^x-1)}.
R^{\mu}{}_{\nu, \lambda\rho}= \partial_{\lambda}\Gamma^{\mu}{}_{\nu, \rho}- \partial_{\rho}\Gamma^{\mu}{}_{\nu, \lambda}+ \Gamma^{\mu}{}_{\sigma, \lambda}\Gamma^{\sigma}{}_{\nu, \rho}- \Gamma^{\mu}{}_{\sigma, \rho}\Gamma^{\sigma}{}_{\nu, \lambda},
\Lambda_1(r)=\frac{\int_{\partial B_1}(|\nabla_\theta u_{(r)}|^2+r^2 u_{(r)}^{p+1}v_{(r)}^{p+1})}{\int_{\partial B_1}u_{(r)}^2},\Lambda_2(r)=\frac{\int_{\partial B_1}(|\nabla_\theta v_{(r)}|^2+r^2 u_{(r)}^{p+1}v_{(r)}^{p+1})}{\int_{\partial B_1}u_{(r)}^2}
E_{C}( a , T ) = \frac{5 \xi^{2}}{3 2 a \pi}+ \frac{8 a^{3}\pi^{3}\xi^{2}{T^{4}}}{4 5}+ \frac{5 1 2 a^{5}\pi^{5}\xi^{2}{T^{6}}}{9 4 5}+{\cal O}( T^{7})
\mathfrak{u}_{\mathfrak{R},K}=\sum_{k\neq 0}\sum_{|l|\leq K}e^{ilt}\left( \begin{array}{c}e^{i(k-\frac{1}{2})\theta}U^+_{k,l}e^{i(k+\frac{1}{2})\theta}U^-_{k,l}\end{array}\right)-\sum_{l\neq 0; |l|\leq K}e^{ilt}\left( \begin{array}{c}u^+_{l}\frac{1}{\sqrt{z}}u^-_l\frac{1}{\sqrt{\bar{z}}}\end{array}\right).
\Big\langle J^{(1)}_{\mu_1}(x_1) J^{(1)}_{\mu_2}(x_2) .....J^{(1)}_{\mu_n}(x_n) J^{(I_1)}_{\nu_1}(y_1)J^{(I_2)}_{\nu_2}(y_2) ..... J^{(I_k)}_{\nu_k}(y_k) \Big\rangle_0^\theta
H^{s,m}= \{f \in \mathcal{S}' \ ; \|f\|_{H^{s,m}}<\infty\}, \|f\|_{H^{s,m}}=\|\langle x \rangle^m \mathcal{F}^{-1}\langle \xi \rangle^s{{\mathcal F}}f\|_{L_{x}^{2}},
F ( r ) ={\frac{1}{3 l^{2}( r^{2}- n^{2})^{2}}}\left[ 3 r^{6}+ ( l^{2}- 1 5 n^{2}) r^{4}- 3 n^{2}( 2 l^{2}- 1 5 n^{2}) r^{2}- 6 m r l^{2}- 3 n^{4}( l^{2}- 5 n^{2}) \right]
\begin{matrix}X_1 = 2\theta_4a_1+\theta_4^2+\theta_1^2 X_2 = 2\theta_4a_2+\theta_4^2+\theta_2^2 X_3 = 2\theta_4a_3+\theta_4^2+\theta_3^2\end{matrix}\ \ \ \ \ \Y=4\theta_4(b_1c_2-b_2c_1)
\det (S) = P_{A_{\alpha}(G_2)}(x-\alpha n_1)\cdot \left(1-(1-\alpha)^2\Gamma_{A_{\alpha}(G_1)}(x-\alpha n_2)\cdot\mathbf{1}_{n_2}^T\left((x-\alpha n_1)I_{n_2}-A_{\alpha}(G_2)\right)^{-1}\mathbf{1}_{n_2}\right) &= P_{A_{\alpha}(G_2)}(x-\alpha n_1)\cdot \left(1-(1-\alpha)^2\Gamma_{A_{\alpha}(G_1)}(x-\alpha n_2) \Gamma_{A_{\alpha}(G_2)}(x-\alpha n_1)\right).
\Vert (B_\varepsilon (\vartheta )+\lambda_0 Q_0^\varepsilon )^{-1}-(B^0(\vartheta)+\lambda_0 \overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \widehat{C}_1\varepsilon , 0< \vartheta \leqslant 1,\; 0 < \varepsilon \leqslant \vartheta^{-1},
\varepsilon\delta T_{\delta}\left( T_{\varepsilon}(|\nabla v|)\right)=\delta T_{\delta}(\varepsilon T_{\varepsilon}(|\nabla v|))=\delta T_{\delta}(|\nabla_{y}T_{\varepsilon}(v)|)=|\nabla_{z}T_{\delta}\left( T_{\varepsilon}(v)\right) |.
p_{-2}= 0, p_{-1}= 1,\quad p_n = a_n p_{n - 1}+ p_{n - 2}\textrm{for}n \geq 0, \textrm{and}q_{-2}= 1, q_{-1}= 0,\quad q_n = a_n q_{n - 1}+ q_{n - 2}\textrm{for}n \geq 0,
Y_{n}(x)\stackrel{\rm def}{=}\left[ \sum_{j\leq \eta_{1}}K\left( \frac{x-X_{j}}{h_{n}}\right) -{\bf E\,}K\left( \frac{x-X}{h_{n}}\right) \right]\left/ \sqrt{{\bf E\,}K^{2}\left( \frac{x-X}{h_{n}}\right)}\right. .
P_{X_1}^{\epsilon}*P_{X_2}^{\epsilon}=(\gamma_{K}-\epsilon D^3\gamma_{K-\delta})*(\sum_{j=0}^J\epsilon^{j}D^{3j}\gamma_{L-j\delta}) &=\sum_{j=0}^J\epsilon^{j}D^{3j}\gamma_{K+L-j\delta}-\sum_{j=0}^J\epsilon^{j+1}D^{3(j+1)}\gamma_{K+L-(j+1)\delta}&=\gamma_{K+L}-\epsilon^{J+1}D^{3(J+1)}\gamma_{K+L-(J+1)\delta}.
\delta_R (L) := \delta_R(\Sigma (L), \iota, \mathfrak{t}_L), \ \ \ \bar{\delta}_R (L) := \overline\delta_R(\Sigma (L), \iota, \mathfrak{t}_L),\ \ \ \underline{\delta}_R (L) := \underline\delta_R(\Sigma (L), \iota, \mathfrak{t}_L),
\delta G^{A B}( \partial_{A}\partial_{B}\varphi - \overline{\Gamma}_{A B}^{C}\partial_{C}\varphi) + \overline{G}^{A B}(\partial_{A}\partial_{B}\chi - \overline{\Gamma}_{A B}^{C}\partial_{C}\chi - \delta \Gamma_{A B}^{C}\partial_{C}\varphi) + \frac{\partial V}{\partial\varphi}\chi =0,
\mathbf{A}_{\mathrm{b2u}}=\mathrm{diag}\left[\sqrt{a_{\mathrm{b2u},1}}\mathbf{I}_{N_{\mathrm{ue}}}\, \cdots \, \sqrt{a_{\mathrm{b2u},K}}\mathbf{I}_{N_{\mathrm{ue}}}\right]
m_{1}^{*}(x_{e})-m_{1}^{*}(x_{b}) \,=\, \Delta t \, \lambda^{2}\, \zeta \, (\frac{1}{2}-\sigma_{1}) \, \left[ \frac{\partial \rho}{\partial x}(x_{e})-\frac{\partial \rho}{\partial x}(x_{b}) \right] +{\rm{O}}(\Delta x^{3}) .
\frac{d}{dt}\int_{\mathbb{T}_{\alpha}}(|\omega^{\nu}|^{2}-|\bigtriangledown\psi^{\nu}|^{2})dxdy=-2\nu\int_{\mathbb{T}_{\alpha}}(|\bigtriangledown\omega^{\nu}|^{2}-|\omega^{\nu}|^{2})dxdy,
{\mathrm{I n d}}D = \frac{1}{8 \pi^{2}}\int [ R^{a b}\mathrm{\tiny ~ \wedge ~}R_{a b}+ 2 ( T^{a}\mathrm{\tiny ~ \wedge ~}T_{a}- R_{a b}\mathrm{\tiny ~ \wedge ~}e^{a}\mathrm{\tiny ~ \wedge ~}e^{b}) ] ,
M(\vec{X},Y)=[\begin{matrix}\begin{matrix}\mu_{1}(\Sigma_{11})^{-1}\\ -(\Sigma_{11})^{-1}\\ \Sigma_{21}(\Sigma_{11})^{-1}\end{matrix}&\begin{matrix}\mu_{2}-\mu_{1}(\Sigma_{11})^{-1}\Sigma_{12}\\ (\Sigma_{11})^{-1}\Sigma_{12}\\ \Sigma_{22}-\Sigma_{21}(\Sigma_{11})^{-1}\Sigma_{12}\end{matrix}\end{matrix}]
JX_{1}=-\frac{2R_{+}R_{-}}{r}\partial_{r}, \;\; J\partial_{r}=\frac{r}{2R_{+}R_{-}}X_{1}, &JX_{2}=\frac{R_{+}}{R_{-}}X_{5}, \;\;\; JX_{5}=-\frac{R_{-}}{R_{+}}X_{2}, &JX_{3}=\frac{R_{+}}{R_{-}}X_{6}, \;\;\; JX_{6}=-\frac{R_{-}}{R_{+}}X_{3}, &JX_{4}=\frac{R_{+}}{R_{-}}X_{7}, \;\;\; JX_{7}=-\frac{R_{-}}{R_{+}}X_{4}.
{}^{M\!L\!\!}{B}_{p}(x,1-y) = \sum_{n=0}^{\infty}\frac{(y)_n}{n!}\int_{0}^{1}t^{x+n-1}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right) dt =\sum_{n=0}^{\infty}\frac{(y)_n}{n!}{}^{M\!L\!\!}{B}_{p}(x+n,1)
\lim_{v\rightarrow 0+}u_{2e}(v) = \lim_{v\rightarrow 0+}\frac{1-e^{-\frac{v}{2\theta^2}}}{v^{2h}}= \lim_{v\rightarrow 0+}\frac{\frac{1}{2\theta^2}e^{-\frac{v}{2\theta^2}}}{2hv^{2h-1}}=\frac{1}{2\theta^2},
\sum_{i_{1}= 0}^{n_{1}}\dotsm \sum_{i_{r}= 0}^{n_{r}}\sum_{j = 0}^{\infty}\frac{(q^{m + 1 + (i_{1}+ \dotsm + i_{r}) + j})_{\infty}(b)_{j}(q^{-n_{1}})_{i_{1}}\dotsm (q^{-n_{r}})_{i_{r}}}{(a q^{(i_{1}+ \dotsm + i_{r}) + j})_{\infty}(q)_{j}(q)_{i_{1}}\dotsm (q)_{i_{r}}}x^{j}x_{1}^{i_{1}}\dotsm x_{r}^{i_{r}}.
[ \delta_{L_{1}}, \delta_{L_{2}}] e{^a}_{\mu}= \beta{^a}_{b}e{^b}_{\mu}+{\frac{\kappa^{4}}{4}}\varepsilon^{a b c d}\bar{\psi}{^j}\gamma_{5}\gamma_{d}\psi{^j}( \partial_{\mu}\beta_{b c}) ,
\sum_{i=1}^m\sum_{j=1}^n\nabla_x\varphi(t_j,x_i)\big(\Delta_j\ell^{x_{i+1}}(t_{j+1}) - \Delta_i\ell^{x_{i}}(t_{j+1})\big) = - \sum_{i=1}^m\sum_{j=1}^n\nabla^2_x\varphi(t_j,y_i)\Delta_j\ell^{x_{i+1}}(t_{j+1})(x_{i+1}- x_i)
M =\left(\begin{array}{ccc|ccc|c}0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \hline 0 0 0 0 0 0 0 0 0 0 0 m_{5,5}m_{5,6}m_{5,7}0 0 0 0 0 0 0 \hline0 0 0 0 0 0 0\end{array}\right).
h(E(A))\geq \prod_{p}\left(1+\sum_{i=2}^{a(n)}\frac{u(i)-u(i-1)}{p^i}-\frac{1}{p^{a(n)+1}}+\frac{1}{p^{a(n)+k}}-\frac{1}{p^{a(n)+k+1}}\right)
\alpha \left|k^3 - k_1^3 - (n-k_1)^3 - \alpha(k-n)^3\right| =| k^3|\left|\alpha - \mu_1^3 - (1 - \mu_1 + \mu_2)^3 +\alpha \mu_2^3 \right| &= |k^3| \left| 1- \alpha + \mathcal{O}(\epsilon)\right| \gtrsim |k^3|.
0 = \sum_{i=0}^s\sum_{j=0}^i(2(s+1)(i-j)j + (s-i)i + 2(s-i) + (s+1)i)\frac{x_{i-j}}{(i-j)!}\frac{x_j}{j!}\sum_{\ell=0}^s (s+\ell)!\frac{B_{s-i, \ell}(x_1, x_2, \ldots)}{(s-i)!}& = \sum_{i=0}^s\sum_{j=0}^i(2(s+1)(j+1)(i-j+1) - (i+1)(i+2))\frac{x_{i-j}}{(i-j)!}\frac{x_j}{j!}\sum_{\ell=0}^s (s+\ell)!\frac{B_{s-i, \ell}(x_1, x_2, \ldots)}{(s-i)!}
Z[C]=\int D\phi\exp\{-\int dx^3 [\frac{g^2}{32 \pi^2}\, (\partial_\mu \phi-2\pi j_\mu(x)) (\partial^\mu \phi-2\pi j_\mu(x)) + \frac{M^2_\gamma g^2}{16 \pi^2}\, (1-\cos \phi)] -M_WL\}\,\,\, .
\sum_{n\geq1}\chi(n)n\Gamma\left(-\frac12;2n^2t\right)e^{n^2t}=\sum_{r=-1}^m\frac{b_{\frac r2}}{\left(\frac r2\right)!}t^{\frac r2}+O\left(t^{\frac{m+1}2}\right)
\Pi_{_h \, \rho ,}^{\rho}\,^{\mu \nu}\left( k \right) = - \kappa \; \Gamma_{_h \,}^{\mu \nu}= - \kappa^{2}\omega \frac{\pi^{2}T^{4}}{1 8 0}\left( 4 \, \eta^{\mu 0}\eta^{\nu 0}- \eta^{\mu \nu}\right) ,
\Phi(\mathcal{E})=(\mathcal{E}\otimes_{\mathbb{C}}\mathrm{Sym}(\mathfrak{g}), d_{\Phi(\mathcal{E})}), \d_{\Phi(\mathcal{E})}(u\otimes v)=d_{\mathcal{E}}(u) \otimes v +\sum_{i=1}^k (u \cdot e_i^{\vee})\otimes (e_i \cdot v)
A^{-1}=\begin{pmatrix}1 c_2+c_3+\cdots+c_N \cdots c_2c_3\cdots c_N 1 c_1+c_3+\cdots+c_N \cdots c_1c_3\cdots c_N \vdots \vdots \vdots 1 c_1+c_2+\cdots+c_{N-1}\cdots c_1c_2\cdots c_{N-1}\end{pmatrix}.
\frac{W_{+}( z ) W_{+}( z^{\prime})}{e^{i \hbar \gamma^{2}h^{+}( z - z^{\prime})}}- \frac{W_{+}( z^{\prime}) W_{+}( z )}{e^{- i \hbar \gamma^{2}h^{-}( z - z^{\prime})}}= \frac{i \hbar}{2 \gamma^{2}}\left( \eta^{2}- 1 + \frac{\gamma^{2}\hbar}{\pi}\right) : \! e^{2 i \gamma \varphi ( z )}e^{2 i \gamma \varphi ( z^{\prime})}\! : \! \partial_{z}\delta_{2 \pi}( z - z^{\prime}) ,
\left( \begin{array}{c c}{M_{1 1}}&{M_{1 2}}\\{M_{2 1}}&{M_{2 2}}\\ \end{array}\right) \left( \begin{array}{c}{W_{2 2}^{0 0}}\\{W_{0 0}^{0 0}}\\ \end{array}\right) = \omega^{2}\left( \begin{array}{c}{W_{2 2}^{0 0}}\\{W_{0 0}^{0 0}}\\ \end{array}\right) ,
L_{s}^{m}= \; \frac{q^{2}}{1 + q^{2}}\; e^{- 1}\; ( \dot{x}_{\mu}+ q^{1 / 2}i \chi \psi_{\mu})^{2}+ \; \frac{e}{1 + q^{2}}\; m^{2}+ \frac{i}{2}( \psi_{\mu}\; \dot{\psi}^{\mu}- \psi_{5}\dot{\psi}_{5}) - i \chi \psi_{5}m .
\frac{\mu + n\xi}{\beta_n}\mathbb{E}(S \mid X_{0}=(1,n)) = \frac{\mu}{\beta_{n}}\mathbb{E}(S_{1,n-1}) + \frac{n\xi}{\beta_{n}}\mathbb{E}(S_{1,n-1}) = \frac{\mu}{\beta_{n}}\mathbb{E}(S_{1,n-1}) + \frac{n\xi}{\beta_{n}}\left( \frac{1}{n}\times 0 + \frac{n-1}{n}\mathbb{E}(S_{1,n-1}) \right),
\mathop{{\rm Res}}_{s=\frac{m+1}{2}}\xi_{n,m}(s)=\tfrac{1}{4}\cdot\frac{\xi\left(\tfrac{n-m}{2}\right)}{\xi\left(\tfrac{n+m+2}{2}\right)}\, \frac{\prod_{\substack{1\leq j\leq [\frac{n}{2}] j\ne\frac{n-m}{2}}}\xi(n-m-2j)}{\prod_{j=1}^{[\frac{n}{2}]}\xi(n+m+2-2j)}.
\frac{d}{dt}\sum^m_{n=1}\| g_n \|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2 + \sum^m_{n=1}\| \partial_y g_n\|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2 + \epsilon \sum^m_{n=1}\| \partial_x g_n\|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2 & \le C_2(\sum^m_{n=1}\| g_n\|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2 + \|{w}\|_{H^m_{k+\ell}(\mathbb{R}^2_+)}^2),
\frac{(\mathcal X_s)}{(\mathbb B_0^{(d)}(r))}\le \frac{(\mathbb B_0^{(d-1)}(r)) \frac{\delta r}{\sqrt{d}}}{(\mathbb B_0^{(d)}(r))}= \frac{\delta}{\sqrt{\pi d}}\frac{\Gamma(\frac{d}{2}+1)}{\Gamma(\frac{d}{2}+\frac{1}{2})}\le \frac{\delta}{\sqrt{\pi d}}\cdot \sqrt{\frac{d}{2}+\frac{1}{2}}\le \delta
\hat{A}=\sum_{j_{1}=1}^{n_{1}}\sum_{j_{2}=1}^{n_{2}}\cdot\cdot\cdot\sum_{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},...,j_{d}}e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes\cdot\cdot\cdot\otimes e_{j_{d}}^{d},
\sum_{i=1}^{n-1}\left( \psi_n(a_i) \psi(b_i) - \frac{1}{2}\left( \psi_{n-1}(a^0_i)\psi(b^0_i) + \psi_{n-1}(a^1_i)\psi(b^1_i) \right) \right) \leq (4 \sqrt{c_1 c_2}+2) a_n b_n,
\big(\partial_{x_1}^{\gamma}f\big)(x_1) =\sum_{\underset{\gamma_1+\cdots+\gamma_s=\gamma}{1\le m\le|\gamma|}}c_{m,\gamma}\chi_{i}^{(m)}(g(x_1)) (\partial^{\gamma_1}_{x_1}g)(x_1)\cdot\ldots\cdot(\partial^{\gamma_s}_{x_1}g)(x_1)\,,
(T_u^{*}H_uT_u - T_v^{*}H_vT_v)(z) = \left(T_u^{*}\left(H_uT_u - H_vT_v\right)\right)(z) + \left(\left(T_u^{*}- T_v^{*}\right)H_vT_v\right)(z) = \left(T_u^{*}\left(H_uT_u - H_vT_v\right)\right)(z) + \mathcal{O}\left(|z|^{-3}\right)|z| = \left(T_u^{*}\left(H_uT_u - H_vT_v\right)\right)(z) + \mathcal{O}\left(|z|^{-2}\right).
\tilde{K}_m(t):=\sup_{0<s<t}s^{1/2}\lVert \nabla \tilde{V}_m(t)\rVert_{L^\infty_H L^p_z},\tilde{H}_m(t):=\sup_{0<s<t}\lVert \tilde{V}_m(t)\rVert_{L^\infty_H L^p_z}.
\lefteqn{[(p\otimes p)e_+]}=[e_+]-\sum_{i=0}^n [(T_iT_i^*\otimes T_iT_i^*)e_+]-\sum_{i=0}^{n-1}[P_i-\sum_{j=i+1}^nP_iP_j]-[P_n] =-n[e_+] -\sum_{i=0}^n [P_i]+\sum_{i<j}[P_iP_j].
\left( \begin{matrix}{\tilde{\mathcal{F}}^{+}}\\{\tilde{G}_{+}}\\ \end{matrix}\right) = \mathcal{S}\left( \begin{matrix}{\mathcal{F}^{+}}\\{G_{+}}\\ \end{matrix}\right) = \left( \begin{matrix}{A}&{B}\\{C}&{D}\\ \end{matrix}\right) \left( \begin{matrix}{\mathcal{F}^{+}}\\{G_{+}}\\ \end{matrix}\right) .
\int d^{1 0}\hat{x}\sqrt{| \hat{\jmath}|}\, \left\{{\textstyle \frac{1}{4 \cdot 5 !}}\left( \hat{G}_{( 5 )}^{+}\right)^{2}+{\textstyle \frac{1}{4 \cdot 5 !}}\left( \hat{G}_{( 5 )}^{-}\right)^{2}+{\textstyle \frac{1}{4 ! \cdot 5 !}}\hat{\epsilon}\left[ \hat{G}_{( 5 )}^{+}\hat{\cal H}\hat{C}_{( 2 )}^{-}+ \hat{G}_{( 5 )}^{-}\hat{\cal H}\hat{C}_{( 2 )}^{+}\right] \right\}\, .
\frac{\partial}{\partial t}\frac{\delta l^\mathcal{L}}{\delta\mathcal{U}}-\operatorname{div}^{\Gamma_0}\frac{\delta l^\mathcal{L}}{\delta\Gamma}=-\operatorname{ad}^*_{\mathcal{U}}\frac{\delta l^\mathcal{L}}{\delta\mathcal{U}}+\operatorname{Tr}\left(\operatorname{ad}^*_{\sigma^2}\frac{\delta l^\mathcal{L}}{\delta\Gamma}\right).
e^{\widetilde{\alpha}(z,\bar{z})}= \frac{e^{\alpha(z,\bar{z})}}{|r(z,\bar{z})|^2}, \ \ \e^{\widetilde{\alpha}}=\frac{1}{2}\langle \widetilde{r}_z, \widetilde{r}_{\bar{z}}\rangle, \ \e^\alpha = \frac{1}{2}\langle r_z, r_{\bar{z}}\rangle.
\begin{cases}\sum_j A_j u_x(1,0)=0, \sum_j A_j u_{xx}(2,0)=0, \sum_j A_j u_j(l,x_1^*) = \sum_k B_k w_{lk}(1,x_1^*)+v_l(x_1^*), \sum_j A_j u_{j}(3-l,x_1^*) = g(3-l,x_1^*), \sum_k B_k w_{lk}(,x_2^*)+v_{3-l}(x_2^*)=g(l,x_2^*), \sum_j A_j (u_j)_{x}(l,x_1^*) = \sum_k B_k (w_{lk})_{x}(1,x_1^*)+(v_l)_{x}(x_1^*) \end{cases}
\mathcal{P}_c(\theta_c, \tau) = \sum_{k \in \mathcal{K}}\mathcal{A}_k \int_{0}^{\infty}\exp(- (N_S + N_0) \zeta_{k}(r)) \mathcal{L}_{I_B}(\zeta_{k}(r)) & \times \mathcal{L}_{I_U}(\zeta_{k}(r)) \mathcal{L}_{I_J}(\zeta_{k}(r)) f_{D_k}(r) dr ,
\begin{aligned}\#\{(\mu,\mu'): |\langle\mu-\mu',\alpha\rangle|\leq \rho\}\leq\sum_{\mu}\#\{\mu' : \mu'\in T_\mu\}+\sum_{\mu}\#\{\mu' : \mu'\in S_\mu\}\leq2\cdot\#\{(\mu,\mu'): \mu,\mu'\in T\}+\sum_{\mu}\#\{\mu': \mu'\in S_\mu\},\end{aligned}
[x]_F = [x]_F^{*G}= \bigcap_n \Big[\bigcup_m A_{n, m}\Big]^{*G}= \bigcap_n \bigcap_V \Big[\bigcup_m A_{n, m}\Big]^{\Delta V}= \bigcap_n \bigcap_V \bigcup_m A_{n, m}^{\Delta V}
\eta= \left(n-2-\left(k+\frac{-h^2+3h+4}{2}-i\right),n-2-(h-2+i_{h-4}),\ldots,n-2-(3+i_1)\right) = \left(\frac{n+d+h^2-3h-9}{2}+1,n-h-i_{h-4},\ldots,n-5-i_1\right),
\left| \theta_{\alpha}\cdot (x_{\pm}(h) - x_{\pm}(\tilde{h}) \right|^{2}= \left|\theta_{\alpha}\cdot \left( \left(h - \tilde{h}\right) \theta_{C}+ \left( \psi(x) - \psi(\tilde{x}) \right) \theta_{C}^{\perp}\right) \right|^{2}& \le |h - \tilde{h}|^{2}+ |\psi(x) - \psi(\tilde{x})|^{2}
q_{11}^{\xi}(t,k,l,\xi,\eta)=\frac{\varepsilon_0e^{-5(|k-l|+|\xi-\eta|)}}{t^2}\cdot\frac{\eta\cdot A_k(t,\xi)-\xi\cdot A_l(t,\eta)}{\sqrt{A_k(t,\xi)A_l(t,\eta)}}.
M_t^Gf(x)=\mathcal F^{-1}(m^G(t\xi)\mathcal Ff)(x)=\frac{1}{\Gamma(\alpha)}\int_t^{\infty}\frac{t}{u}\bigg(1-\frac{t}{u}\bigg)^{\alpha-1}\mathcal P_{u}^{\alpha}f(x)\frac{{\rm d}u}{u}.
\exp \bigg( 2 \sum_{m=1}^{M_1}Z_m^{(W)}(t) \bigg) \ll \exp \bigg( 4 \sum_{m=1}^{M_1}\sigma_{m,W}^2 \bigg) \frac{1}{M_1^4}\ll \frac{e^{2 M_1}}{M_1^4}.
\begin{array}{l l l}{\displaystyle \left( \partial / \partial \theta \right) A}&{}&{\mathrm{f o r \l e f t \d e r i v a t i v e},}\\{\displaystyle \partial A / \partial \theta}&{}&{\mathrm{f o r \r i g h t \d e r i v a t i v e}.}\\ \end{array}
\int_{- T / 2}^{T / 2}d s_{i}\simeq \int_{- T / 2}^{T / 2}d t_{i}\sqrt{1 + h_{1 1}[ x ( t_{i}) ]}\simeq T + \frac{1}{2}\int_{- T / 2}^{T / 2}d t_{i}h_{1 1}[ x ( t_{i}) ]
\Delta L = -{\frac{1}{1 2 \kappa^{2}}}\int d^{1 1}x \left( \partial_{1 1}C_{A B C}-{\frac{\sqrt{2}}{1 6 \pi}}\left({\frac{\kappa}{4 \pi}}\right)^{2 / 3}\overline{{{\chi}}}\Gamma_{A B C}\chi \ \delta ( x^{1 1}) \right)^{2}\ ,
f^{1}( u ) \equiv f^{( + 1 , 0 )}( u ) = f^{i}u_{i}^{1}+ g^{( i j ) k}u_{i}^{1}u_{j}^{1}u_{k}^{4}+ h^{i j k}u_{1}^{1}u_{j}^{2}u_{k}^{3}+ \ldots .
\langle{\cal O}_{W_{2}}^{n_{2}}{\cal O}_{W_{3}}^{n_{3}}\cdots{\cal O}_{W_{D}}^{n_{D}}\rangle = d^{n_{2}}\langle{\cal O}_{W_{3}}^{n_{3}}\cdots{\cal O}_{W_{D}}^{n_{D}}\rangle
|Q(\eta_\delta W) - Q(W)| \leq \int_{\delta^2\leq |z|\leq \delta}|\eta_\delta(d W)^N + (Wd\eta_\delta)^N|^2+\int_{|z|\leq \delta}|(d W)^N |^2+\int_S (1-\eta_\delta^2)|W|^2|k|^2 = \int_{\delta^2\leq |z|\leq \delta}|\eta_\delta(d W)^N + (Wd\eta_\delta)^N|^2+ O(\delta^2),
ch_{\Lambda;\,0}^{(0)}=ch^{-}_{\Lambda}, \;\; ch^{(\frac{1}{2})}_{\Lambda;\,0}= ch^{+}_{\Lambda},\;\; ch^{(0)}_{\Lambda;\,\frac{1}{2}}=t_{\xi}ch^{-}_{\Lambda}, \;\; ch^{(\frac{1}{2})}_{\Lambda;\,\frac{1}{2}}= t_{\xi}ch^{+}_{\Lambda},
\begin{aligned}g_1 g_2 \neq g_2 g_1 \mbox{and}g_1 (g_1 g_3) \neq (g_1 g_3) g_1 \,, \varphi (g_1 g_2) = \varphi (g_1) \varphi (g_2) \mbox{and}\varphi (g_1 (g_1 g_3)) = \varphi (g_1 g_3) \varphi (g_1) \,, \end{aligned}
u(\rho)= \Bigl(-\int_0^\rho m'(y)g(y)dy\Bigr)m'(\rho)\int_0^\rho\frac{dy}{m'(y)^2}&+ \Bigl(\int_0^\rho m'(y) (\int_0^y \frac{dz}{m'(z)^2}) g(y)dy \Bigr)m'(\rho).
\varphi_\tau : \begin{pmatrix}u -\overline{v}v \overline{u}\end{pmatrix}\mapsto \begin{pmatrix}u\cdot e^{\imath \tau}-\overline{v}\cdot e^{-\imath \tau}v\cdot e^{\imath \tau}\overline{u}\cdot e^{-\imath \tau}\end{pmatrix}.
L\partial_{tt}\left(\frac1N\right)-2M\partial_{ts}\left (\frac1N\right)+N\partial_{ss}\left(\frac 1N\right) +\frac{A_{2}}N\partial_t\left(\frac 1N\right)+\frac{A_{1}}N\partial_s\left(\frac1N\right) =\frac{A_0}{N^2}.
\begin{pmatrix}n_1 n_2 m_1 m_2 \vdots m_h \vdots m_{(n-1)/2}\end{pmatrix}= \begin{pmatrix}\frac{1}{2}- \frac{1}{2}\displaystyle{\sum_{i=n+3}^{2n}}c^{n+3-2i}P_{i-n-3,-i}\frac{3}{2}+ \frac{1}{2}\displaystyle{\sum_{i=n+3}^{2n}}c^{n+3-2i}P_{i-n-3,-i}2 2 \vdots 2 \end{pmatrix}.
\int_{0}^{n}\frac{\sqrt{n^{2}-x^{2}}}{2+e^{\frac{1}{e}}}dx\le\int_{0}^{n}\frac{\sqrt{n^{2}-x^{2}}}{2+x^{-x}}dx\le\int_{0}^{n}\frac{\sqrt{n^{2}-x^{2}}}{2}dx,
\dot x_\text s(t) =Ax_\text s(t)+B_uu(t)-B_uD_0\bigl(y(t)-C_yx_\text s(t)\bigr) \dot x_\text a(t) =Ax_\text a(t)+B_uu(t),x_\text a(t_i)=x_\text s(t_i) u(t) =D_0C_yx_\text a(t)+\eta(t)
\sum_{i=1}^{N}L_iv\frac{v_i}{u_{i,\infty}}= - \frac{1}{2}\sum_{r=1}^{R}k_ru_{\infty}^{y_r}\left(\sum_{i=1}^{N}(y_{r,i}' - y_{r,i})\frac{v_i}{u_{i,\infty}}\right)^2 v\in \mathbb R^N.
\begin{aligned}\dot{x}= -y + a_{02}y^2, \dot{y}= ( b_{02}+ b_{12}x )y^2 + ( b_{11}x+b_{21}x^2 )y+ (b_{20}x^2 +b_{30}x^3+x) & = x + b_{20}x^2 + b_{11}x y + b_{02}y^2 + b_{30}x^3 + b_{21}x^2 y+ b_{12}x y^2, \end{aligned}
\det(M_{\bar{b},\bar{a}})=(\prod_c \frac{1}{d_c})\frac{d_b}{\sqrt{\pi_b}}\sqrt{\pi_a}\det(L_{\bar{b},\bar{a}}), \det(M_{\overline{\{v,b\}},\overline{\{u,a\}}})=(\prod_c \frac{1}{d_c})\frac{d_b}{\sqrt{\pi_b}}\frac{d_v}{\sqrt{\pi_v}}\sqrt{\pi_a}\sqrt{\pi_u}\det(L_{\overline{\{v,b\}},\overline{\{u,a\}}}).