harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( \Gamma_{}^{\left( 0 \right) 0}, \Gamma_{\Lambda R_{1}}^{\left( 2 \right) - 1}\right)^{a}+ \widetilde{V}^{a}\Gamma_{\Lambda R_{1}}^{\left( 2 \right) - 1}= \alpha_{R_{1}}^{\left( 2 \right) - 2 a}- \left( \Gamma_{}^{\left( 0 \right) 1}, \Gamma_{\Lambda R_{1}}^{\left( 2 \right) - 2}\right)^{a}
\lim_{t \rightarrow + \infty}\left\|\left(u(t) - S(t)(u_0^+, u_1^+), \partial_t u(t) - \partial_t S(t)(u_0^+, u_1^+)\right)\right\|_{H^{\sigma-\frac{1}{2},\frac{1}{2}}\times H^{\sigma-\frac{1}{2},-\frac{1}{2}}}= 0.
\oint_{\partial\Omega}\hat{n}(\epsilon\nabla\varphi)=\frac{\varphi_{i+1,j}-\varphi_{i,j}}{h_{i}^{x}}(\frac{h_{j}^{y}}{2}\epsilon_{i,j}^{x}+\frac{h_{j-1}^{y}}{2}\epsilon_{i,j-1}^{x})
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left(1- y^m z^n\right)^{\frac{m^1}{n^2}}= \left(\frac{1}{1-yz}\right)^{\frac{y}{1-y}}\exp\left\{- \frac{y}{(1-y)^2}\left(Li_2(z)- Li_2(yz)\right) \right\}.
V ( R ) = \frac{6 \pi}N m_{0}^{2}R + \frac{2 c_{L}}N m_{0}- \frac 2 N \int_{0}^{\infty}\cos ( k R ) \left[ \Delta_{( \lambda )}( k ) - \frac{1 2 \pi m_{0}^{2}}{k^{2}}\right] \frac{d k}{2 \pi}
P^\mu(\Sigma)=(1+\sigma_z)\left(J^\mu_{\Sigma}\right)^{-1}\left(J^\mu_{\Sigma}\right)^{-1\,\,T},\quad\mbox{with}\left(J^\mu_{\Sigma}\right)^{-1\,\,T}=\left(\begin{array}{ccc}1 0 \frac{-\sqrt{\mu\sigma_x}}{1+\sigma_z}0 1 \frac{-\sqrt{\mu}\sigma_y}{1+\sigma_z}0 0 \frac{1}{1+\sigma_z},\end{array}\right)
\eta \rhd c = t(c; \eta), \eta \rhd t(c; \tau_1, \ldots, \tau_r) = t(c; \eta, \tau_1, \ldots ,\tau_r) + t(c; \eta \rhd \tau_1 , \ldots ,\tau_r) + \ldots + t(c; \tau_1, \ldots , \eta \rhd \tau_r),
\eta_{M,N}(q\,|\,b) = \exp\Bigl(-\bigl(\mathcal{S}_N\log\Gamma_M\bigr)(0\,|\,b)\Bigr)\exp\Bigl(-\frac{1}{M!}\mathcal{S}_N\bigl(B^{(f)}_M(w)\,\log(w)\bigr)(q\,|\,b)+ & + \sum\limits_{k=0}^M \frac{B^{(f)}_k(0)\bigl(\mathcal{S}_N(-w)^{M-k}\bigr)(q\,|\,b)}{k!(M-k)!}\sum\limits_{l=1}^{M-k}\frac{1}{l}+O(q^{-1})\Bigr).
\lambda_{n}=2\left[ \left( \gamma_{n}+\gamma_{n+1}+\gamma_{n+2}\right)-\left( z^{2}+1\right) -\frac{n}{2}\right] \gamma_{n+1}, \tau_{n}=2\frac{h_{n+1}}{h_{n-2}}=2\gamma_{n+1}\gamma_{n}\gamma_{n-1}.
\iota_1(x,y) =\left(\frac{x_{0}}{x_{1}}, \frac{A_{-1}(\frac{x_{0}}{x_{1}})}{A_{1}(\frac{x_{0}}{x_{1}})\frac{y_{0}}{y_{1}}}\right) \iota_2(x,y)=\left(\frac{B_{-1}(\frac{y_{0}}{y_{1}})}{B_{1}(\frac{y_{0}}{y_{1}})\frac{x_{0}}{x_{1}}},\frac{y_{0}}{y_{1}}\right),
\Theta^{4}\Big [ \begin{array}{c}{0}\\{0}\\ \end{array}\Big ] ( \tau ) - \Theta^{4}\Big [ \begin{array}{c}{0}\\{1}\\ \end{array}\Big ] ( \tau ) - \Theta^{4}\Big [ \begin{array}{c}{1}\\{0}\\ \end{array}\Big ] ( \tau ) = 0 ,
|t_m(n)| =|t_m(2n'+1)|=\left|\sum_{j=0}^{\lfloor\frac{m-1}{2}\rfloor}{m\choose 2j+1}t_m(n'-j)\right|\leq\sum_{j=0}^{\lfloor\frac{m-1}{2}\rfloor}{m\choose 2j+1}|t_m(n'-j)| & <\sum_{j=0}^{\lfloor\frac{m-1}{2}\rfloor}{m\choose 2j+1}m(n')^{\frac{m}{2}}=m(2n')^{\frac{m}{2}}<mn^{\frac{m}{2}}.
\|T(f)(x)\|_{L_{|x|_h}^pL_\theta^{\bar{p}_2}(\mathbb H^n)}=\omega_Q^{\frac{1}{\bar{p}_2}-\frac{1}{\bar{p}_1}}\int_{\mathbb H^n}K(e,y)|y|_h^{-Q/p}dy\|f_j\|_{L_{|x|_h}^pL_\theta^{\bar{p}_2}(\mathbb H^n)}.
T^{(i)}_B (z) =\displaystyle\sum_{\mu =0}^{n-1}\phi_{(a, a+\bar{\varepsilon}_\mu )}(z)K^{(i)}\left( \left. a+\bar{\varepsilon}_\mu \begin{array}{c}a a \end{array}\right| z \right) \phi^{(a+\bar{\varepsilon}_\mu ,a)}(z)
D(0, \varepsilon) =\sigma_{\varepsilon}(I\diamond I\circ_{\ast}T)=\sigma_{\varepsilon}(\varphi(I)\diamond \varphi(I)\circ_{\ast}\varphi(T)) &=\sigma_{\varepsilon}(2\varphi(I)^{2}(\varphi(T)^{\ast}-\varphi(T))).
|Z - Z_{N,\Phi}^\mathrm{sample}|^2 = \left| \int_X \left(\exp \big(-\Phi(u) \big) - \exp \big(-\Phi_N(u) \big)\right) \mu_0(\mathrm{d}u) \right|^2 \leq 4\int_X (\Phi(u) - \Phi_N(u))^2 \mu_0(\mathrm{d}u).
f^{(d)}:=\left.\frac{\partial^{d}}{\partial\Theta^{d}}\,\left(f(\Upsilon_\Theta x)\right)\right|_{\Theta=0}=\sum_{l=0}^\infty\sum_{j=0}^{\left[\frac{d}{2}\right]}a_l^{2j+d_{2}}(f^{(d)})Y_l^{(2j+d_{2},0,\dots,0)}(x)
\mathcal{C}\left({\mathbf{u}, \mathbf{v}}\right)(\tau)=\begin{cases}\sum_{i=0}^{N-1-\tau}\omega_q^{u_{i}-v_{i+\tau}}, 0 \leq \tau \leq N-1, \sum_{i=0}^{N-1+\tau}\omega_q^{u_{i-\tau}-v_{i}}, -N+1 \leq \tau \leq-1, 0, |\tau| \geq N,\end{cases}
\frac{dA_{i}}{dt}=k_{B}T\sum_{j}[A_{i},A_{j}]\frac{dH}{dA_{j}}-\sum_{j}\lambda_{i,j}(A)\frac{dH}{dA_{j}}+\sum_{j}\frac{d\lambda_{i,j}(A)}{dA_{j}}+\eta_{i}(t).
\begin{aligned}0 \leq - \frac{\gamma_1 (A + M)}{2}\sum_i F^{ii}- \frac{\beta}{2n}F^{11}U_{11}+ \gamma_2 \sum_{U_{ii}> 0}F^{ii}U_{ii}&\leq - \frac{\gamma_1 (A + M)}{2}\sum_i F^{ii}- \frac{\beta}{8n C_3}(A + M) \sum_i F^{ii}+ \gamma_2 C_0 (1 + M) \sum_{i}F^{ii}\leq - \frac{\gamma_1 (A + M)}{2}\sum_i F^{ii},\end{aligned}
\psi_{1}\left( \hat{r}\right) =\left( \tilde{a}_{0}+\tilde{a}_{1}r^{2}\right) r^{\tilde{\nu}-1/2}e^{-\frac{1}{2}\left( \sqrt{a}r^{2}+\sqrt{| \tilde{d}|}r^{-2}\right)}e^{im\varphi},
\left(\frac{\delta \widetilde H}{\delta(\mathbf{a},\mathbf{b})}\right)_m=\left(\frac{\partial \widetilde H}{\partial(\mathbf{a},\mathbf{b})}\right)_m\!\!\!\!-\delta_m^-\left(\frac{\partial \widetilde H}{\partial\delta^+(\mathbf{a},\mathbf{b})}\right)_m\!\!\!\!-\delta_m^+\left(\frac{\partial \widetilde H}{\partial\delta^-(\mathbf{a},\mathbf{b})}\right)_m,
\frac{a_{p,q}(x_n)-a_{p,q}(x)}{x_n-x}=\frac{\pi_q(0^m1a_{n,m+2}a_{n,m+3}\cdots)}{\pi_p(0^m1a_{n,m+2}a_{n,m+3}\cdots)}\ge \left(\frac{p}{q}\right)^m\frac{\pi_q(10^{\infty})}{\pi_p(1^{\infty})}
f_{\gamma_{SCHCN}}(\gamma) =\mathcal{L}^{-1}\left[ \mathcal{L}_{\gamma_{SCHCN}}(s)\right]=\sum_{i=1}^{N_c+1}\frac{\alpha_i^*}{(i-1)!}\gamma^{i-1}e^{-\lambda_{0}\gamma}+ \sum_{j=1}^{N-N_c}\beta_j^* e^{-(1+\frac{j}{N_c})\lambda_{0}\gamma},
x_ks_{k,l}- x_ts_{t,p}=(s_k+ x_k l q_k) - (s_t + x_t p q_t) &=\sum_{i=t}^{k-1}(y_i+\varepsilon x_i) + x_k l q_k - x_t p q_t\ge\sum_{i=t}^{k-1}(y_i+\varepsilon x_i) - y_t &\ge \sum_{i=t+1}^{k-1}y_i.
E[\hat{R}_{1\ell i}|\mathcal{W}_{\ell}^{c}] =\int g(w,\hat{\gamma}_{\ell},\theta_{0})F_{0}(dw),E[\hat{R}_{2\ell i}|\mathcal{W}_{\ell}^{c}]=\int\phi(w,\hat{\gamma}_{\ell},\alpha_{0},\theta_{0})F_{0}(dw), E[\hat{R}_{3\ell i}|\mathcal{W}_{\ell}^{c}] =\int\phi(w,\gamma_{0},\hat{\alpha}_{\ell},\tilde{\theta}_{\ell})F_{0}(dw)=0,
\lambda_s+z\lambda_x=z_s+z\cdot z_x+A^0_x+\sum_{n=0}^{\infty}\frac{-nA^{n-1}A^0_x}{z^{n+1}}-\sum_{n=1}^{\infty}\frac{nA^{n-1}(z_s+z\cdot z_x)}{z^{n+1}}=
(\pounds_{\xi_i}{f})X \overset{\eqref{3.3B}}= \nabla_{\xi_i}({f}X) - \nabla_{{f}X}\,\xi_i -{f}(\nabla_{\xi_i}X - \nabla_{X}\,\xi_i) \ = (\nabla_{\xi_i}{f})X - \nabla_{{f}X}\,\xi_i +{f}\nabla_X\,\xi_i.
\gamma_{a_{2 k + 1}}^{( 1 )}\equiv \pi_{a_{2 k + 1}}- \pi_{a_{2 k + 1}}^{( 1 )}\approx 0 , \; \gamma_{a_{2 k + 1}}^{( 2 )}\equiv \pi_{a_{2 k + 1}}^{( 2 )}\approx 0 , \; k = 0 , \cdots , \Lambda ,
\left| \int_\Lambda f dx - \sum_{j=1}^N f(\xi_j) | I_j | \right| \leq \sum_{j=1}^N \left| \int_{I_j}[ f(x) - f(\xi_j) ]dx \right| \leq \tau \sum_j \int_{I_j}g(x) dx = \tau \int g(x) dx .
\begin{array}{ll}\displaystyle \int_{0}^{T}\Phi_{t}d B_{t}= \sum_{k=1}^{\infty}\int_{\tau_{k-1}\wedge T}^{\tau_{k}\wedge T}\Phi^{(k)}dB_{t}, \ \ (\tau_{0}=0) &=\displaystyle \sum_{k=1}^{\infty}\sum_{\substack{j\in [\tau_{k-1}\wedge T,\tau_{k}\wedge T]\bigcap \{J\setminus J_{K}\}}}\Phi^{(k)}(B_{t_{j+1}}-B_{t_{j}})\end{array}
(f^{(2)})^{-1}=\left(\begin{array}{cccc}0 \delta_{ij}-a_ia_j a_i 0 -\delta_{ij}+a_ia_j a_j\pi_i-a_i\pi_j -\pi_i a_i -2a_i -2\pi_i 0 -\delta_{ij}0 2 a_i \delta{ij}0 \end{array}\right)
m_{\nu}\!=\!P^{-1}m_{0,\nu}P=\mbox{\scriptsize $\left(\!\begin{array}{cccc}\!u_1\!-\!w_1\! -w_2 (v_1\!-\!w_1\!-\!w_2)/2 \!(u_1\!-\!u_2\!-\!w_1\!+\!w_2)/2\! -w_1 \!u_2\!+\!w_2\! \!(u_2\!-\!u_1\!+\!w_2\!-\!w_1)/2\! (v_2\!-\!w_1-\!w_2)/2 2w_1 0 u_1+w_1 w_1 0 2 w_2 w_2 u_2-w_2\end{array}\!\right)$}.
{\rm K}_{n,{2(m+1)\over 2m+1-n}}\!=\!{n! \over \pi}\left \{\int_{-\pi/2}^{\pi/2}\big |\cos (n+1)\varphi \big |^{2(m+1) \over n+1}\cos^{2m}\varphi d\varphi \right \}^{n+1 \over 2(m+1)}
\|x_n -x_{n+k}\| \leq \sum_{i=n}^{n+k-1}\|x_{i+1}-x_i\| \leq \sum_{i=n}^{n+k-1}\sigma\rho^i d(x_0) \leq \frac{\sigma d_C(x_0)}{1-\rho}\rho^n \to 0 n \to +\infty.
\Pi_{\mathrm{b}, \mu \nu}( p ) = 2 \kappa^{2}\int \frac{d^{3}k}{( 2 \pi )^{2}}\epsilon_{\mu \alpha \beta}\epsilon_{\sigma \rho \nu}\sin^{2}( k \wedge p ) D^{\alpha \rho}( k + p ) D^{\sigma \beta}( k )
V_{p}= - ( 1 + \frac{2 \mu}{z^{7 - p}})^{\alpha + \beta_{+}}( 1 + \frac{\mu}{z^{7 - p}})^{2 ( p - 8 ) / ( 7 - p )}+ \frac{( 8 - p ) ( 6 - p )}{4 z^{2}}- \mu^{2}\frac{( 7 - p )^{2}}{z^{2}( z^{7 - p}+ 2 \mu )^{2}}
[y_1, y_1]=y_5, [y_1, y_2]=y_4+\theta y_5, [y_2, y_1]=\frac{\alpha_3\alpha_6}{\alpha_2\gamma_1}y_5, [y_2, y_2]=\frac{\beta_2\alpha^2_3}{\alpha_2\gamma^2_1}y_5, [y_1, y_3]=\frac{\alpha_3\beta_4}{\alpha_2\gamma_1}y_5=-[y_3, y_1], [y_2, y_3]=y_4, [y_3, y_2]=\frac{\alpha^2_3\gamma_4}{\alpha_2\gamma^2_1}y_5.
- r^{2}d t^{2}+{\frac{d r^{2}}{r^{2}}}+ \cos^{2}\theta d \psi^{2}+ d \theta^{2}+ \sin^{2}\theta d \Omega_{3}^{2}+ r^{2}d y_{1}^{2}+{\frac{r^{2}}{1 + a^{4}r^{4}}}d y_{2}^{2}
\log\frac{p(\rho e^{i\theta})}{(\rho e^{\i\theta})^n}= \,b n^{1/2}+\frac{1}{2}\log\frac{c}{2\pi\sqrt{n}}-\frac{1}{2\xi}\,i\theta-\theta^2 n^{3/2}\gamma_n &+i\theta^3 n^2\gamma_n^\prime+O\bigl(\theta^4 n^{5/2}+n^{-1/2}\bigl),
\Vert (B^0-\zeta\overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant\vert f_0\vert^2 \Vert (\widetilde{B}^0-\zeta I)^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}&\leqslant \Vert f\Vert^2_{L_\infty}c(\phi)\vert\zeta\vert^{-1}.
\int_{t - \tau}^{t}\langle \tilde{a}_{ij}(\cdot) \partial_{j}u(s), \partial_{i}u(s)\rangle_{L^{2}(\Omega)}\mathrm{d}s = \tau \int_{0}^{1}\langle \tilde{a}_{ij}(\cdot) \partial_{j}z(s, t, \cdot), \partial_{i}z(s, t, \cdot) \rangle_{L^{2}(\Omega)}\mathrm{d}s.
\mathbb{E}\left[\nabla f(X_{t}^{x;l^\epsilon})\nabla_{x}X_{t}^{x;l^\epsilon}G\right]=\mathbb{E}\left[D_{s}f(X_{t}^{x;l^\epsilon})\nabla_{x}X_{\gamma^{\epsilon}_{s}}^{x;l^\epsilon}G\right].
|\int_{t+2(r+1)}^{\infty}\lambda''(s) r\left(\frac{1}{s-t}-\frac{1}{1+s-t}\right) ds| \leq r \sup_{x \geq t}\left(|\lambda''(x)|\right) \log(1+\frac{1}{2+2r}) &\leq C \sup_{x \geq t}\left(|\lambda''(x)|\right) r
( J_{2}- K_{1}- 1 ) ~ D_{x_{1}x_{2}, y_{1}y_{2}}^{J_{1}J_{2}, K_{1}K_{2}; 0}+ ( J_{2}- K_{2}- 1 ) ~ D_{x_{1}x_{2}, y_{1}y_{2}}^{J_{1}J_{2}, K_{1}K_{2}; J_{1}}.
\mathbb{E}\left[\left(B_{t_{i}}-B_{s}\right)\left(B_{t_{j}}-B_{t_{i}}\right)\right]=\frac{1}{2}\left[(t_{j}-s)^{2H}-(t_{j}-t_{i})^{2H}-(t_{i}-s)^{2H}\right]\leq\frac{1}{2}(t-s)^{2H},
J_3 = \int_\Omega\Big(f_0(V)-f_\epsilon(V) \Big) \psi^h_{\mu_j,\xi_j}dx = \epsilon\int_\Omega V^p\Big(\ln\ln(e+V)\Big)\psi^h_{\mu_j,\xi_j}dx +O\bigg( \epsilon^2 \Big( \ln\Big|\ln\frac{\epsilon}{|\ln\epsilon|^2}\Big|\Big)^2\bigg).
\gamma \frac{df}{d\gamma}\; + \; \sum_{n=1}^\infty\Big[ \lambda_n \frac{df}{d\lambda_n}+\kappa_n \frac{df}{d \kappa_n}+ \bar{\kappa}_n\frac{df}{d \bar{\kappa}_n}\Big] \; \; = \; \; f \; .
\omega' ( ( f_2 , \phi_2 ) , (f_1 , \phi_1 ) ) \ = \ \langle A(f_1) , \phi_2 \rangle + \langle B(\phi_1) , \phi_2 \rangle + \langle f_2 , D(\phi_1) \rangle = \ \langle f_1 ,{^tA}(\phi_2) \rangle + \langle{^tB}(\phi_2) , \phi_1 \rangle + \langle{^tD}(f_2) , \phi_1 \rangle .
\langle f_1,\dots,f_k \rangle = \langle f_j : j \in K \rangle + \langle f_j : j \notin K \rangle = (I_1 \cap \cdots \cap I_m) + \langle f_j : j \notin K \rangle = (I_1 + \langle f_j : j \notin K\rangle) \cap \cdots \cap (I_m + \langle f_j : j \notin K\rangle)
G_{R}=\sqrt{P_{R}/\mathrm{E}\left\{\left\Vert \mathbf{y}_{R}\right\Vert^{2}\right\}}=\sqrt{2NP_{R}/\left(P_{T}\bar{\alpha}_{sum}^{2}+2NN_{R}\sigma_{R}^{2}\right)}=\sqrt{P_{R}/3\left(P_{T}+N\sigma_{R}^{2}\right)},
\tfrac{1}{2}\left\{{\sum}_{i=1}^r({Z_j\bar{Z}_j+\bar{Z}_jZ_j}) -(\bar{L}_0+{L}_0)\right\}={\sum}_{i=1}^r{\bar{Z}_jZ_j}-L_0={\sum}_{i=1}^rZ_j\bar{Z}_j-\bar{L}_0.
\vec{\gamma}= \left( \begin{array}{c c}{{0}}&{{\vec{\sigma}}}\\{{- \vec{\sigma}}}&{{0}}\\ \end{array}\right) \; , \; \gamma_{5}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \; , \; \gamma_{0}= \left( \begin{array}{c c}{{0}}&{{- 1}}\\{{- 1}}&{{0}}\\ \end{array}\right)
\Phi = e^{- i ( m_{2}x_{2}^{0}+ p_{1 0}x_{1}^{0})}\psi ( \bf{x}) = e^{- i ( m_{1}+ m_{2}+ E ) X^{0}}e^{- i ( m_{1}- m_{2}) x^{0}/ 2}e^{- i E x^{0}/ 2}\psi ( \bf{x}) , x^{0}< 0 ,
\begin{array}{r l}{{s_{g a u g e}=}}&{{\beta_{1}\int_{0}^{T}d t \left( \frac{1}{S ( t )}\frac{d S ( t )}{d t}+ ( r_{1}- r_{b}) \right)^{2}+}}\\{{}}&{{+ \beta_{2}\int_{0}^{T}d t \left( \left( \frac{1}{C ( t )}\frac{d C ( t )}{d t}+ ( r_{2}- r_{b}) \right) - \alpha ( t ) \left( \frac{1}{S ( t )}\frac{d S ( t )}{d t}+ ( r_{1}- r_{b}) \right) \right)^{2}\ .}}\\ \end{array}
u^1_t - u^2_t = -\int_0^t \nabla \cdot e^{(t-s)\Delta}\left\{u^1_s F( K\ast u^1_s) - u^2_s F(K\ast u^2_s ) \right\}~ds &= -\int_0^t \nabla \cdot e^{(t-s)\Delta}\left\{(u^1_s-u^2_s)F( K\ast u^1_s ) +u^2_s (F(K\ast u^1_s) - F( K\ast u^2_s))\right\}~ds .
E=[\begin{matrix}\frac{\partial v_{1}}{\partial k_{1}}&0&0&0\\ 0&\frac{\partial v_{2}}{\partial k_{2}}&0&0\\ 0&0&\frac{\partial v_{3}}{\partial k_{3}}&0\\ 0&0&&\frac{\partial v_{4}}{\partial k_{4}}\\ \end{matrix}].
{\frac{t}{2 M}}= \ln{\frac{( r_{\infty}/ 2 M - 1 )^{1 / 2}+ \tan{( \eta / 2 )}}{( r_{\infty}/ 2 M - 1 )^{1 / 2}- \tan{( \eta / 2 )}}}+ \left({\frac{r_{\infty}}{2 M}}- 1 \right)^{1 / 2}\left[ \eta + \left({\frac{r_{\infty}}{4 M}}\right) ( \eta + \sin \eta ) \right]
\begin{array}{c}{\displaystyle ( C \Gamma^{\mu_{1}\dots \mu_{p}}) = \epsilon ( - 1 )^{( p - 1 ) ( p - 2 ) / 2}( C \Gamma^{\mu_{1}\dots \mu_{p}})^{T}\quad , \quad \mu = 0 , 1 , \dots , D - 1}\\{\displaystyle \epsilon = - \sqrt{2}\cos{\frac{\pi}{4}}( D + 1 ) \quad .}\\ \end{array}
\left(Q_A^{-1}Q_B\begin{pmatrix}\sqrt{-1}0 0 1 \end{pmatrix}Q_B^{-1}\right)(AB)\left(Q_B\begin{pmatrix}-\sqrt{-1}0 0 1\end{pmatrix}Q_B^{-1}Q_A\right)=\sqrt{-1}Q_A^{-1}AQ_A.
|-\frac{1}{r}\int_{t}^{t+6r}ds \int_{0}^{s-t}\frac{\rho d\rho}{(s-t)}\lambda''(s)\left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(-1-\rho^{2}+r^{2})^{2}+4r^{2}}}+F_{3}(r,\rho,\lambda(s))\right)| &\leq \frac{1}{r}\int_{t}^{t+6r}ds \int_{0}^{s-t}\frac{\rho d\rho}{(s-t)}|\lambda''(s)|\cdot 2 &\leq C r \sup_{x \geq t}|\lambda''(x)|
{\frac{\partial W}{\partial \theta}}= \int{\cal D}\bar{K}^{4 i}{\cal D}\bar{K}\raisebox{0 . 9 7 e x}{\mathrm{\scriptsize{~ 4 i ~}}}\raisebox{- 0 . 3 6 e x}{\mathrm{\scriptsize{~ , \! \theta ~}}}{\cal D}\bar{t}_{4}{\cal D}\bar{t}_{4 , \theta}{\frac{\partial \bar{W}}{\partial \theta}}
{\cal C}_{0}=i\,\int d{\bf{r}}\,{\textstyle{\cal X}^{\alpha}}({\bf r})\,j_{0}^{\alpha}({\bf r})\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\;{\bar{\cal C}}=i\,\int d{\bf{r}}\,\overline{{\cal Y}^{\alpha}}({\bf r})\,j_{0}^{\alpha}({\bf r})
\tilde{A}_{0,k_{i}}^{(e)}= \frac{1}{2^{k_{i}}(2k_{i})!\sqrt[4]{\pi}}\sqrt{2\pi}\frac{(2k_{i})!}{k_{i}!}= \frac{\sqrt[4]{\pi}\sqrt{(2k_{i})!}}{2^{k_{i}-1/2}k_{i}!}
\phi ( \theta , \varphi ) = a \left\{\cos^{2}\frac{\theta}{2}| 2 \lambda_{\phi}> - \frac{\sqrt{2}}{2}\sin \theta e^{- i \varphi}| 2 \lambda_{\phi}- \alpha > + \sin^{2}\frac{\theta}{2}e^{- 2 i \varphi}| 2 \lambda_{\phi}- 2 \alpha > \right\}\, .
P_{0}S^{*}P_{0}S P_{0}= P_{0}= P_{0}S P_{0}S^{*}P_{0}\quad \Longleftrightarrow \quad ( S_{0 0})^{*}= ( S_{0 0})^{- 1}\quad \mathrm{o n}\quad \mathcal{H}_{\mathrm{p h y s}}.
\sum_{\gamma\in\mathcal{B}_{n}}w(\gamma)=\sum_{\gamma\in\mathcal{B}_{n}}w(\gamma_{1})w(\gamma_{2})w(\gamma_{3})w(\gamma_{4})=a_{0}\sum_{\gamma\in\mathcal{B}_{n}}w(\gamma_2)w(\gamma_4)=a_{0}\sum_{k=0}^{n-2}A_{k}^{(1)}A_{n-k-2},
\nabla_A W_{BCD}=\nabla_{\underline{A}}W_{\underline{BCD}}-\frac{1}{4}\varepsilon_{AB}\nabla^E W_{ECD}-\frac{1}{4}\varepsilon_{AC}\nabla^E W_{EBD}-\frac{1}{4}\varepsilon_{AD}\nabla^E W_{EBC}
p_{\mathcal{G}\boxplus\{v_1,v_2\}}(X_1,X_1',X_2,X_2',X_3 \ldots, X_{m}) & = p_1 + p_2(X_1 + X_2+X_1'+X_2')+ p_{4}(X_1+X_1')(X_2+X_2')+p_5(X_1+X_2)(X_1'+X_2').
\frac{\log_2 \lvert K P_i(G) \cap Z : P_i(G) \cap Z \rvert}{\log_2 \lvert Z : P_i(G) \cap Z \rvert}= \frac{\log_2 \lvert K \varrho_k \cap Z \varrho_k \rvert}{\log_2 \lvert Z \varrho_k \rvert}.
\left| \alpha k^3 - k_1^3 - (n-k_1)^3 - \alpha (k- n )^3 \right| = |\mu k^3| \left| (\alpha - 1) \mu^2 + 3 \alpha( 1 - \mu) + 3 \mu_1 \mu_2 \right| \gtrsim |k^2|.
\int f(t) h(t/q) \left(\frac{1-t(1+q)}{q (1-q)t^2}u(t/q) +\frac{1}{(1-q)^2 t^2}\right)y(t) d_qt & =f(x/q) h(x/q) \left(y(x/q) u(x/q)- D_{q^{-1}}y(x)\right).
\frac{ab'+a'b}{b'+a'}= \frac{1}{2}\cdot\frac{(l^2-n^2)(m^2+n^2)+(l^2+n^2)(m^2-n^2)}{ln(m^2+n^2)+mn(l^2+n^2)}= \frac{1}{2}\cdot\frac{2(l^2m^2-n^4)}{n(lm+n^2)(l+m)}= \frac{lm-n^2}{(l+m)n}, \frac{ab'-a'b}{b'-a'}= -\left(\frac{ab'+a'b}{b'+a'}\right)^{-1}= -\frac{(l+m)n}{lm-n^2}.
2e(G) = \bigl[ \bigl( \tfrac{4}{(3r-2)^2}+\tfrac{4}{3r-2}\delta-\delta^2 \bigr) +\tfrac{3(r-2)}{3r-2}\delta+\tfrac{9(r-2)(r-3)+24(r-2)}{(3r-2)^2}-o(1)\bigr]n^2 &= \bigl( \tfrac{3r-5}{3r-2}+\delta-\delta^2-o(1)\bigr)n^2\,.
\|f_{p_n}\|_{L^q(0,1)}=\bigg(\int_{\Gamma_{p_n}}|\Psi_{p_n}|^q\,ds\bigg)^{1/q}\leq{\rm const\,}\|\Psi_{p_n}\|_{H^{1/2}(\Gamma_{p_n})}\leq{\rm const\,}\|\Psi_{p_n}\|_{H^{1}(D_1)}\leq{\rm const\,}
V = q\left({{\cal A}_0 \left({\bf y}\right) -{\cal A}_0 \left({{\bf y}^\prime}\right)}\right) = - \frac{{q^2}}{\pi}\frac{1}{{1 + e^2 a\left({1 - 12\alpha^2 g^2 \partial^2}\right)}}K_0 \left({\chi |{\bf y}-{\bf y}^\prime |}\right).
H\big(\frac{\partial\varphi_{ji}}{\partial x},\frac{\partial\varphi_{jm}}{\partial x}\big)-H\big(\frac{\partial\varphi_{j+1,i}}{\partial x},\frac{\partial\varphi_{j+1,m}}{\partial x}\big)=g_2^{im}(\Phi_j(x))-g_2^{im}(\Phi_{j+1}(x).
S_0\> '[X] \> = \>\int_0^T dt \int d\theta \> e(\theta \chi) \> \left ( - \frac{\kappa_0}{2}\right ) \> D X \cdot D^2 X\> = \>\int_0^T dt \> \left [ \> - \frac{\kappa_0}{2}\dot x^2+ i \zeta \cdot \dot \zeta + \frac{1}{T}\zeta \cdot \dot x \chi \> \right ] \>.
\mathcal{O}_{13}= (1,2),(1,3),(1,4),(2,3),(2,4),(3,4), \mathcal{O}_{15}= (1,2),(1,4),(1,3),(2,3),(2,4),(3,4), \mathcal{O}_{14}= (1,2),(1,3),(1,4),(2,4),(2,3),(3,4), \mathcal{O}_{16}= (1,2),(1,4),(1,3),(2,4),(2,3),(3,4).
|f(u)| =\bigg|\int_{[0,1]}\mathcal{E}_y'u'\bigg|=\big|\langle\mathcal{E}_y,u\rangle_{H^1}-\langle\mathcal{E}_y,u\rangle_{L^2}\big|\leq\big|\langle\mathcal{E}_y,u\rangle_{H^1}\big|+\big|\langle\mathcal{E}_y,u\rangle_{L^2}\big| &\leq\|\mathcal{E}_y\|_{H^1}\|u\|_{H^1}+\|\mathcal{E}_y\|_{L^2}\|u\|_{L^2}\leq2\|\mathcal{E}_y\|_{H^1}\|u\|_{H^1}\,.
L - H{\delta L \over \delta H}- I{\delta L \over \delta I}+[ H{d \over dt}+ H( 3H +DI) -\dot{H}]{\delta L \over \delta \dot{H}}+ [ I{d \over dt}+ I( 3H +DI) -\dot{I}]{\delta L \over \delta \dot{I}}\equiv{\cal D}L =0 .
\frac{\partial \Lambda_{k0}(x;\lambda,\alpha)}{\partial\lambda}=-\frac{1}{\lambda+\alpha}x+\frac{1}{\lambda+\alpha}\log\left(1-\frac{\alpha}{\lambda}\big(e^x-1\big)\right), & \frac{\partial L_k(x;\lambda)}{\partial\lambda}=\frac{1}{\lambda+\alpha}\log\left(1+(\lambda+\alpha)e^x\right).
\int_{\mathcal L_0 \cap \mathcal R_0}(cd,d)_p\chi_{\psi}(d)\chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh = \mathrm{vol}(\mathcal L_0 \cap \mathcal R_0) = (1-p^{-1})^2.
\sigma [\phi_1 ](h) = J_1(h) \Bigl\{\int_0^1 J_{t}(h)^{-1}{\bf V}( \phi_t (h) ){\bf V}( \phi_t (h) )^* (J_{t}(h)^{-1})^* dt\Bigr\}J_1(h)^*
\left(\frac{1+\zeta/\chi}{1-\zeta/\chi}\right)^{2}X(\zeta)R_{p}(\chi)-\left(\frac{1+\chi/\zeta}{1-\chi/\zeta}\right)^{2}R_{p}(\chi)X(\zeta) =2(p+1)D\delta(\zeta/\chi)R_{p+1}(\chi)-2\delta(\zeta/\chi)D_{\chi}R_{p+1}(\chi).
V(t,Z(t,x)) = - \frac 3 8 \left(1+u_0(t)\right)^3 + \sum_{\substack{i \in K_{N_0}1 \le i < N_0}}V_i(t) x^i + O\left(x^{N_0}\right) \mbox{as}\, x \searrow 0 \, \mbox{and}\, t > 0,
\begin{matrix}{{s \hat{e}^{\pm}= \hat{\psi}^{\pm}, \quad}}&{{\quad s \hat{\psi}^{\pm}= 0 , \quad}}&{{\quad s \skew 5 \hat{\bar{\psi}}= \hat{A}, \quad}}&{{\quad s \hat{A}= 0 .}}\\ \end{matrix}
\frac{\partial}{\partial r}\Big(\frac{\partial X}{\partial \mu}\Big) =A'_X(X)\frac{\partial X}{\partial \mu}, \frac{\partial X}{\partial \mu}(0,\mu)=(X_0)'_\mu(\mu), \frac{\partial}{\partial r}\Big(\frac{\partial Y}{\partial \mu}\Big) =B'_Y(Y)\frac{\partial Y}{\partial \mu}, \frac{\partial Y}{\partial \mu}(0,\mu)=(Y_0)'_\mu(\mu).
\theta_{n+1}^{(\epsilon)}= \theta_n^{(\epsilon)}+ \frac{\alpha_{t_n}}{\gamma}(AX_{t_n}^{(\epsilon)})^{\rm T}\left( (X_{t_{n+1}}^{(\epsilon)}-X_{t_n}^{(\epsilon)}) - \theta_n^{(\epsilon)}A X_{t_n}^{(\epsilon)}\Delta t \right).
\theta_{*}(E,x) = \liminf_{\rho \rightarrow 0^{+}}\frac{\mathcal{H}^{n}( E \cap B_{\rho}(x))}{\omega_{n}\rho^{n}}, \mbox{and}\theta^{*}(E,x) = \limsup_{\rho \rightarrow 0^{+}}\frac{\mathcal{H}^{n}( E \cap B_{\rho}(x))}{\omega_{n}\rho^{n}},
\theta_{N + 3}\equiv \frac{N}{\lambda_{0}}\phi_{0}+ \frac{1}{2 L}\sum_{a}\int_{- L}^{L}d x^{-}( \psi_{R}^{a \dag}\psi_{L}^{a}+ \psi_{L}^{a \dag}\psi_{R}^{a}) \approx 0 .
E\left[T^2\right] - E\left[T_1^2\right] = E\left[T^2 - T_1^2\right] = E\left[\left(T + T_1\right) \left(T - T_1\right)\right] &\leq E\left[\left(2T_1 + T_2\right)^r\right]^{\frac{1}{r}}E\left[\left(T - T_1\right)^s\right]^{\frac{1}{s}},
E_i^2 E_{i+1}- [2] E_i E_{i+1}E_i + E_{i+1}E_i^2 = 0 i,i+1 \neq m, E_{i+1}^2 E_{i}- [2] E_{i+1}E_{i}E_{i+1}+ E_{i}E_{i+1}^2 = 0 i,i+1 \neq m, F_i^2 F_{i+1}- [2] F_i F_{i+1}F_i + F_{i+1}F_i^2 = 0 i,i+1 \neq m, F_{i+1}^2 F_{i}- [2] F_{i+1}F_{i}F_{i+1}+ F_{i}F_{i+1}^2 = 0 i,i+1 \neq m,
\left|\frac{\mu}{\rho(a,t)}\mathbf{z}_x(a,t)\right| =\left|\mathbf{w}_t(a,t)-\frac{\mathbf{b}_x(a,t)}{\rho(a,t)}\right| \leq C+C |\mathbf{b}_x(a,t)|,\hbox{\rm where}~a=0~\hbox{\rm or}~a=1.
\frac{\partial \Omega}{\partial z^{\alpha}}= \frac{1}{n !}\frac{\partial h ( x )}{\partial z^{\alpha}}\epsilon_{{\mu}{\nu}\cdots{\rho}}d x^{\mu}d x^{\nu}\cdots d x^{\rho}+ \frac{1}{( n - 1 ) !}h ( x ) \epsilon_{{\mu}{\nu}\cdots{\rho}}\frac{\partial d x^{\mu}}{\partial z^{\alpha}}d x^{\nu}\cdots d x^{\rho}.
\begin{array}{rl}\sigma_i\displaystyle\frac{\rm d}{{\rm d}t}\theta_i h_i =\sigma_i\mathfrak{M}_i(t,h), \sigma_i\displaystyle\frac{\rm d}{{\rm d}t}\theta_i h_i v_{a\,i}=-{\cal K}(h)|\boldsymbol{v}_i| v_{a\,i}. \end{array}
\binom{k+(i+j)p}{m+ip}=\sum_{l=0}^{k}\binom{(i+j)p}{m-l+ip}\binom{k}{l}&=\binom{i+j}{j}\binom{k}{m}+\sum_{l=1}^{m}\binom{(i+j)p}{l+ip}\binom{k}{m-l}+\sum_{l=1}^{k-m}\binom{(i+j)p}{l+jp}\binom{k}{m+l}.
K_{Hh_1H}\circ K_{Hh_2H}(xH, A)= \int_X K_{Hh_2H}(w, A)\> K_{Hh_1H}(xH, dw) &=\int_G (\delta_y*\omega_H*\delta_{h_2}*\omega_H)(p_G^{-1}(A)) \> d(\delta_x*\omega_H*\delta_{h_1}*\omega_H)(y) &=(\delta_x*\omega_H*\delta_{h_1}*\omega_H*\delta_{h_2}*\omega_H)(p_G^{-1}(A))
\gamma_{\pm n}=\frac{\mp 1}{\kappa d_n\mp\gamma_{\pm (n+1)}}=\frac{\mp1}{\kappa d_n+\displaystyle\frac{1}{\kappa d_{n+1}+\displaystyle\frac{1}{\ddots}}}\,\,\mbox{for}\,\,n\geq1.
\frac{1}{\sqrt{T}}\sum_{t=k+1}^{T}\big \{1[\mathbf{y}_{t,k}^{\ast}\leq\hat{\mathbf{q}}_{t,k}^{\ast}(\tau)]-E\left[ F_{\mathbf{y}|\mathbf{x}}^{(k)}(\mathbf{q}_{t,k}(\tau)|\mathbf{x}_{t,k})\right] \big \}=\mathbb{V}_{T,k}(\tau)^{\ast}+\mathbb{V}_{T,k}(\tau) & +\nabla G^{(k)}(\tau)^{\top}\sqrt{T}\{\hat{\beta}^{\ast}(\tau)-\beta(\tau)\}+o_{p}(1).
q^{-\lambda_i}t^{-(v_\lambda(i)-1)+\frac12(n-1)}(E_\lambda, E_\mu)_{q,t}= (Y_i E_\lambda, E_\mu)_{q,t}= ( E_\lambda, Y^{-1}_i E_\mu)_{q,t}= ( E_\lambda, q^{\mu_i}t^{(v_\mu(i)-1)\frac12(n-1)}E_\mu)_{q,t}&= \overline{q^{\mu_i}t^{(v_\mu(i)-1)-\frac12(n-1)}}( E_\lambda, E_\mu)_{q,t}= q^{-\mu_i}t^{-(v_\mu(i)-1)+\frac12(n-1)}( E_\lambda, E_\mu)_{q,t}.
\mathbf E[C_k(n)] = C_k(n_0)\cdot \prod_{m=n_0}^{n-1}\frac{m+kp^{k-1}}{m}\stackrel{n\to\infty}\sim \frac{C_k(n_0)\cdot\Gamma(n_0)}{\Gamma(n_0+kp^{k-1})}n^{kp^{k-1}}.
\mathsf{B}_p(x^*):=\sqrt{3\mathsf{B}}C_q\mathsf{\tilde G}_p(x^*)\left[(1+L\hat\alpha)^2+(3+2L\hat\alpha)\sqrt{\mathsf{A}}\mathsf{\tilde G}_p(x^*)+2\mathsf{A}\mathsf{\tilde G}_p(x^*)^2\right].
{_{0}\mathcal{F}_0}^{(d)}\left( \begin{matrix}\end{matrix};\mathbf{1}+\mathbf{z},\mathbf{u}\right) = e^{E_{0}(\mathbf{z})}{_{0}\mathcal{F}_0}^{(d)}\left( \begin{matrix}\end{matrix};\mathbf{z},\mathbf{u}\right) ={_{0}\mathcal{F}_0}^{(d)}\left( \begin{matrix}\end{matrix};\mathbf{z},\mathbf{u}\right) e^{|\mathbf{u}|}.
\|f\|_{{F_{rt}^s}(K)}= \|q^{sj}\Delta_j f \|_{L^r(\ell^t(K))}&= \Bigg\|\Bigg\{\sum_{j=0}^{\infty}|q^{sj}\Delta_j f|^t \Bigg\}^{\frac{1}{t}}\Bigg\|_{L^r(K)}&= \Bigg( \int_{K}\Bigg\{\sum_{j=0}^{\infty}q^{sjt}| \Delta_j f|^t \Bigg\}^{\frac{r}{t}}dx \Bigg)^{\frac{1}{r}}.
\frac{de_\alpha}{dt}=\frac{\alpha(a)}{\alpha(s)}e_\alpha^\prime+\sum_{\beta\in\Delta\backslash\{-\alpha\}}\frac{\beta(a)}{\beta(s)}e_{-\beta}[e_\beta,e_\alpha]\,\,,\,\,\,\,\alpha\in\Delta\,,