harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f(\bar x_k) - f^\star(\delta) \le \frac{\| \lambda_1 \|_2^2}{2 \alpha k}+ \frac{\alpha \sigma_g^2}{2}+ \frac{\alpha}{2k}\sum_{i=1}^k \| \delta_i - \delta \|_2^2 + \frac{\alpha}{k}\sum_{i=1}^k (\delta_i - \delta)^T (g(x_i) + \delta) & + \frac{1}{k}\sum_{i=1}^k 2 \| \epsilon_i \|_2 \| g(x_i) + \delta \|_2
1 \leq \frac1Nz_n^* \left( \frac{1-\rho_n}{1-(1-\rho_n)c_N}\frac1n\sum_{i=1}^{n-1}z_iz_i^* + ({\hat{\gamma}}(\rho_n)+\ell)\rho_n I_N \right)^{-1}z_n \triangleq \hat{e}_n.
q_{a}^{2}= \frac{\displaystyle \prod_{b = 1}^{N}( \bar{\sigma}_{a}^{2}- \lambda_{b})}{\displaystyle \prod_{b = 1 , b \neq a}^{N}( \bar{\sigma}_{a}^{2}- \bar{\sigma}_{b}^{2})}= \frac{\Lambda ( \bar{\sigma}_{a}^{2})}{A^{\prime}( \bar{\sigma}_{a}^{2})}
\frac{d}{dt}\mathbb{E}[u_t] \bigg|_{t=0}= \int_U \left( \langle W_u (u^{-1}(y)), v(y) \rangle + W_\xi (u^{-1}(y)) : [\nabla_y v](y) [\nabla_x u](u^{-1}(y)) \right) dy &= \int_U \left( \langle W_u (u^{-1}(y)), v(y) \rangle + W_\xi (u^{-1}(y)) [\nabla_x u]^t (u^{-1}(y)) : [\nabla_y v](y) \right) dy.
f_S (x)= - \frac{\log x}{2(x-1)}+ \frac{1}{2}\sum_{\substack{p\in S p^{1/e}< x}}\left((1 - \frac{1}{q^{n_p}}) \frac{q\log{p}}{e(q^2 -1)}- \frac{\log x}{x}\right)
\mathbf{x}^{(A)}= \mathbf{p}_{\mathrm{0}}^{(A)}s_{\mathrm{0}}+\mathbf{p}_{\mathrm{c}}^{(A)}s_{\mathrm{c}}^{(A)}+ \sum_{k=1,2}\mathbf{p}_k^{(A)}s_{k}^{(A)}, \mathbf{x}^{(B)}= \mathbf{p}_{\mathrm{0}}^{(B)}s_{\mathrm{0}}+\mathbf{p}_{\mathrm{c}}^{(B)}s_{\mathrm{c}}^{(B)}+ \sum_{k=1,2}\mathbf{p}_k^{(B)}s_{k}^{(B)},
\Gamma_{[ s ]}( p ) = Z \gamma_{[ r ]}\delta_{s}^{r}- i g^{2}\int \bar{d}^{D}\! \! q \gamma_{[ r ]}\frac 1{\gamma . q}\Gamma_{[ s ]}( q ) \frac 1{\gamma . q}\gamma_{[ r^{\prime}]}\Delta^{r r^{\prime}}( p - q ) .
[\bigoplus_{I}\sum_{p}(p + \iota_{E}(\omega)) \eta_{I,p}] = L_{\omega}[\bigoplus_{I}\eta_{I}] = \nabla_{\omega}(\iota_{E}([\bigoplus_{I}\eta_{I}])) = \nabla_{\omega}[\iota_{E}(\bigoplus_{I}\eta_{I})].
\left\{\prod_{l_i^{(k)}}\left(\prod_{i=0}^{l_i^{(k)}-1}(\Theta_j-i)\right)-\prod_{i=1}^{|l_0^{(k)}|}(i-|l_0^{(k)}|-\Theta_0)\prod_{l_i^{(k)},j\neq 0}\left(\prod_{i=0}^{|l_i^{(k)}|-1}(\Theta_j+|l_i^{(k)}|-i)\right)x_k\right\}\Pi(x)=0
\begin{aligned}\varphi_{\prescript{L}{}{j}}(v)v_{\widehat{G}}(\widehat{j}(t))\varphi_{\prescript{L}{}{j}}(v)^{-1}=\widehat{j}(v_{\widehat{S}}(t)) \varphi_{\prescript{L}{}{j}}(vw) =\varphi_{\prescript{L}{}{j}}(v)v_{\widehat{G}}(\varphi_{\prescript{L}{}{j}}(w)),\end{aligned}
M_4:=\sum_{m,n,p,q\geqslant 1}\Big[ \frac{1}{2N}V^{(1)}_{mnpq}a^*_ma^*_na_pa_q+\frac{1}{2N}V^{(2)}_{mnpq}b^*_mb^*_nb_pb_q &+\frac{1}{N}V^{(12)}_{mnpq}a^*_ma_pb^*_nb_q\Big].
\partial_{t}\varphi_{t}+\frac{1}{2}|\nabla\varphi_{t}|^{2}-\big[\varphi_{t}[e^{\prime}]^{-1}(-\varphi_{t})+e([e^{\prime}]^{-1}(-\varphi_{t}))\big]m^{\prime}(\rho_{t})=0.
{\mathcal{W_{F B}}}= \frac{1}{N}\sum_{k , l}\left[{\bf{\chi}}_{k}^{\dag}({\bf{s}}_{l}\otimes{\bf{p}}_{l}^{\dag}) \phi_{k}+ \phi_{k}^{\dag}({\bf{p}}_{l}\otimes{\bf{s}}_{l}^{\dag}) \chi_{k}\right]
(\mathcal{E}_{k}^{\alpha}) = \sum_{j=1}^{+\infty}\langle \mathcal{E}_{k}^{\alpha}\phi_j,\phi_j\rangle_{L^2}= \sum_{j=1}^{+\infty}\sum_{i}\lambda_i \langle\partial^{\alpha}\phi_i,\phi_j\rangle_{L^2}^2 = \sum_{i=1}^{+\infty}\lambda_i ||\partial^{\alpha}\phi_i||_{L^2}^2
K_m \le \dfrac{\sum_{i=1}^{2^m}\mathbb{E}\left(|X_i|\mathbf{1}(b_{2^{m-1}}<|X_{i}|\le b_{2^{m}})\right)}{b_{2^{m-1}}}&\le \dfrac{\sum_{i=1}^{2^m}b_{2^m}\mathbb{P}(|X_{i}|>b_{2^{m-1}})}{b_{2^{m-1}}}&\le C 2^m \mathbb{P}(|X|>b_{2^{m-1}})\to 0 m\to\infty.
||f_S + X_S \alpha_S - (f_T + X_T \alpha_T)||^2 &= ||f_S - f_T||^2 + 2\langle f_S - f_T, X_S\alpha_S - X_T\alpha_T \rangle + ||X_S\alpha_S - X_T\alpha_T||^2 &> 2\langle f_S - f_T, X_S\alpha_S - X_T\alpha_T \rangle,
\begin{aligned}R_(\mathbf{H}) = \max\limits_{d_t,\mathbf{A}_c \in \mathbb{Z}^{K \times K}}\min\limits_{m \in \mathcal{K}}\frac{1}{2}\log^+ \left( \frac{P}{\|\mathbf{F}_c \, \mathbf{a}_{c,m}\|^2}\right) (\mathbf{A}_c)=K \end{aligned}
\sum_{\alpha,\beta\in \mathbb{F}_{p^m}}S(\alpha,\beta)^2=\sum_{\alpha,\beta\in\mathbb{F}_{p^m}}\left(T(\alpha,\beta)^2+2T(\alpha,\beta)T(\pi^{\frac{p^k+1}{2}}\alpha,-\pi \beta)+ T(\pi^{\frac{p^k+1}{2}}\alpha,-\pi \beta)^2\right).
\sum_{k=1}^{N}\big|b_{k}^2\big|\le C\Delta^{\frac12-\kappa}(1+|x|_{L^{\max(p,8q)}})^{K+2}\int_{0}^{T}\bigl(1+\frac1{t^{3\kappa}}\bigr)\bigl(1+\frac{1}{(T-t)^{1-\kappa}}\bigr)dt.
\underset{n\rightarrow +\infty}{\lim}(\Phi^{\prime}(u_{n},v_{n}),(u_{n}-u,v_{n}-v))=\underset{n\rightarrow +\infty}{\lim}(\Phi^{\prime}(u_{n},v_{n})-\Phi^{\prime}(u,v),(u_{n}-u,v_{n}-v))=0.
[\Delta_1(\epsilon_1), \Delta_{-1}(\epsilon_2) ]g= [ \int_{x_{0}}^x \partial_\mu (g \epsilon_1 g^{-1})dy^\mu, \epsilon_2]g+ g [ \int_{x_{0}}^x \partial_\mu (g^{-1}\epsilon_2 g) dy^\mu,\epsilon_1] = [g, \epsilon_{12}]
\chi(q)=\sum_{n\ge0}\frac{q^{n^{2}}}{\prod_{1\le i\le n}(1-q^{i}+q^{2i})}=\frac{1}{2\prod_{n>0}(1-q^{n})}\sum_{n\in Z}\frac{(-1)^{n}(1+q^{n})q^{n(3n+1)/2}}{1-q^{n}+q^{2n}}
1+6\sum^{\infty}_{n=1}\frac{nq^{2n}}{1-q^{2n}}-30\sum^{\infty}_{n=1}\frac{nq^{10n}}{1-q^{10n}}=\frac{\left(x^{2}+22 q^2 x y +125 q^4y^2\right)^{1/2}}{\left(xy\right)^{1/6}},
{h_1}(m) =\frac{{\Gamma{N_0}W}}{{- D\ln (1 - \varepsilon ) r_0^\alpha}}\cdot \frac{{\exp \big(({2^{\frac{b}{W}}}- 1)\lambda{}_0\pi (R_0^2 - MR_s^2 + mR_s^2)\big) - 1}}{{\pi (R_0^2 - MR_s^2 + mR_s^2)}}
\begin{aligned}\widetilde{\Pi}(u) =\frac{up(u)}{\sigma(u)}+\kappa\int_{u-2}^u\frac{p(t+1)}{\sigma(t)}\,dt, \widetilde{\Xi}(u) =\frac{uq(u)}{\sigma(u)}-\kappa\int_{u-2}^u\frac{q(t+1)}{\sigma(t)}\,dt.\end{aligned}
{\mathcal{L}_{SD}}=\beta^2 g\left( m^2 A^2/2\beta^2 \right) - \frac{\beta^2}2 h\left( m^2 B^2/2\beta^3\right)-\frac{\chi\theta}{2}\varepsilon^{\mu \nu \rho\lambda}\partial_\mu B_{\nu \rho}A_\lambda\; .
\frac{d}{dt}\bigg|_{t=0}\int_M \sqrt{\det(g-F)}= \left< \sqrt{\det(I-g^{-1}F_0)}g \left(I - (g^{-1}F_0)^2\right)^{-1}g^{-1}F_0 , dA \right>_{L^2}& = \left< d^* \left( \sqrt{\det(I-g^{-1}F_0)}g\left(I - (g^{-1}F_0)^2\right)^{-1}g^{-1}F_0 \right), A \right>_{L^2}.
\int_{\Omega}u_n^{1-\gamma}dx \leq\int_{\Omega}u^{1-\gamma}dx+\int_{\Omega}|u_n-u|^{1-\gamma}dx \leq\int_{\Omega}u^{1-\gamma}dx+C\|u_n-u\|_{L^2(\Omega)}^{1-\gamma}=\int_{\Omega}u^{1-\gamma}dx +o(1)
\bar{G}^2 \int \Pi_{j=1}^{m}\phi_1\left( w^j\right) \Pi_{j=1}^{m}\phi_1\left( x^j\right) \, Q(x,w) (dP_0^m \times P_0^m)(x,w) \qquad>\int \int \phi_m \left(x\right) \phi_m(w) \, Q(x,w) dP_0^m(w) dP_0^m (x),
\bigl(D_z^+F\bigr)^2 =\bigl(f(\eta+\delta_z)-f(\eta)\bigr)^2=f(\eta+\delta_z)^2-f(\eta)^2-2f(\eta+\delta_z) f(\eta)+2f(\eta)^2 =D_z^+F^2-2f(\eta)\bigl(f(\eta+\delta_z)- f(\eta)\bigr)=D_z^+F^2-2 F D_z^+F
t_k(u)=Q\left(\textstyle u-\frac{(k+1)\pi}{2(p+1)}\right) P\left(\textstyle u+\frac{(k+1)\pi}{2(p+1)}\right)- Q\left(\textstyle u+\frac{(k+1)\pi}{2(p+1)}\right) P\left(\textstyle u-\frac{(k+1)\pi}{2(p+1)}\right)
-\nabla \cdot \left( D_p^2 L( \nabla u,\cdot) \nabla z \right) = \nabla \cdot \left( \int_{0}^1 \left( D_p^2 L( \nabla u{+}t\nabla z,\cdot){-}D_p^2 L( \nabla u,\cdot) \right)\,dt \, \nabla z \right).
j^{(a - 1)}=\sum_{i = 0}^{a - 2}j_i2^i = \sum_{i = n - k - 1}^{n - k + b - 3}j_i2^i = 2^{n - k - 1}\sum_{i = 0}^{b - 2}d_i2^i = 2^{n - k - 1}d^{(b - 1)}
I\left( y\right) =\int_0^\infty \frac{QdQ}{Q^2+\left(2\pi Ty\right)^2}\frac 1{\exp \left( Q/T\right) -1}=\frac12\Theta[{\rm Re}(y)]\left(\ln y-\frac 1{2y}-\psi\left(y\right)\right)+(y\leftrightarrow-y),
g_{v , 2}( x ) = \frac{k}{2}C_{1}^{2}({\hat{h_{v}}}+ 4 \Omega_{s p h}^{2})^{- 1}\bigg [ \{( 5 + 4 k^{2}) - 9 k^{2}s n^{2}( x ) \}s n ( x ) c n^{2}( x ) d n ( x ) \bigg ] .
B_3(\widehat{\phi},\widehat{w}_h)=\int_{\widehat{K}}\widehat{\phi}\frac{\partial (\widehat{w}_h-\widehat{I}_{\widehat{K}}\widehat{w}_h)}{\partial \xi_1}\;\mathrm{d}\xi_1\mathrm{d}\xi_2\mathrm{d}\xi_3,
\lambda^{2}I_{2}-A_{3}A_{1}=\left[\begin{array}[c]{cc}\left( \frac{5}{2}-\frac{1}{2}\sigma_{3}\right) \sigma_{1}-\frac{1}{2}\sigma_{1}-1 4\left( \sigma_{3}-1\right) \sigma_{1}\frac{1}{2}+\left( \frac{1}{2}\sigma_{3}-\frac{5}{2}\right) \sigma_{1}\end{array}\right] ,
\forall \varphi,\psi \in C_c^{\infty}(\mathcal{D}),\D^{\alpha,\alpha}k(\varphi \otimes\psi) = \int_{\mathcal{D}\times\mathcal{D}}\ \k(x,y)\partial^{\alpha}\varphi(x)\partial^{\alpha}\psi(y)dxdy = \int_{\mathcal{D}\times\mathcal{D}}k_{\alpha}(x,y)\varphi(x)\psi(y)dxdy = T_{k_{\alpha}}(\varphi \otimes\psi)
M_q(\Gamma^{\,*}_m)\leqslant\sum\limits_{i=0}^{N_0}M_q(\Gamma_{im})\leqslant\sum\limits_{i=1}^{N_0}\frac{\omega_{n-1}}{I_i^{q-1}}+\frac{\omega_{n-1}}{I_0^{p-1}}:=C_0\,, m=1,2,\ldots\,.
\left\Vert x-x_{k-1}\right\Vert_{A^{T}A}^{2}-\left\Vert x-x_{k}\right\Vert_{A^{T}A}^{2}=\phi_{k}^{2}\left({1+\ell_{k-1}^{(w)}}-\ell_{k}^{(w)}\right).
h_k(W) =h_k(W_1e_{j_1}W_2e_{j_2}W_3\cdots W_se_{j_s}W_{s+1}) &=W''_1e'_{j_1}h(W_2)e'_{j_2}h(W_3)\cdots h(W_s)e'_{j_s}W''_{s+1}
\begin{tabular}{lll}$x_{1}=e_{1}^{T}\Gamma \bar{w}_{2^{k}}\left( t\right) $, $x_{0}=1$, $\dot{x}_{0}=0$ $x_{2}=\tau e_{1}^{T}P\Gamma \bar{w}_{2^{k}}\left( t\right) $ $x_{0}=0$ $\dot{x}_{0}=1$\end{tabular}
\mathbb{P}\left(d_{J_{1}}\left(\left(A_{\cdot}^{n},D_{\cdot}^{n}\right),f_{\cdot}^{n}\right)>\epsilon|A_{\delta',\tilde{T}}^{n}\cap B_{\delta,\tilde{T}}^{n},\left(A_{\cdot}^{n},D_{\cdot}^{n}\right),\left\{E_{T}^{n}>\tilde{T}\right\}\right)=0.
\mathrm{DTE}_{(p,q)}(X)=\mu+\sigma\frac{\Gamma\left((m+1)/2\right)\sqrt{m}\left[\left(1+\frac{\xi_{p}^{2}}{m}\right)^{-(m-1)/2}-\left(1+\frac{\xi_{q}^{2}}{m}\right)^{-(m-1)/2}\right]}{\Gamma(m/2)(m-1)\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},
i_{n,mn}(s_{n,((e,\mu),1)}) = \sum_{\tau \in s(e) E^{<mn}, [\tau]_n = \mu}s_{mn,((e,\tau),1)}i_{n,mn}(p_{n,(\mu,1)}) = \sum_{\tau \in E^{<mn}, [\tau]_n = \mu}p_{mn,(\tau,1)},
\vert\vert E(\lambda)-E(1)\vert\vert^2 =\sum_{n=1}^N\frac{\vert \lambda^n-1\vert^2}{n^{2\alpha+1}}+\sum_{n=N+1}^{+\infty}\frac{\vert \lambda^n-1\vert^2}{n^{2\alpha+1}}&:=\mathcal{S}_1(\lambda)+\mathcal{S}_2(\lambda)
log_{b}\sum_{i=0}^{N}a_{i}=log_{b}a_{0}+log_{b}(1+\sum_{i=1}^{N}\frac{a_{i}}{a_{0}})=log_{b}a_{0}+log_{b}(1+\sum_{i=1}^{N}b^{(log_{b}a_{i}-log_{b}a_{0})})
L_{F}= \bar{\psi}i \gamma^{\mu}( \partial_{\mu}- i e A_{\mu}) \psi - m \bar{\psi}\psi +{\frac{g}{2}}\mid \bar{\psi}\gamma^{\mu}\psi \mid^{2}-{\frac{p e^{2}}{4}}\mid A_{\mu \nu}\mid^{2}+{\frac{q e^{2}}{2}}\epsilon_{\mu \nu \lambda}A^{\mu}A^{\nu \lambda}+ J_{\mu}A^{\mu}.
\int_{- \infty}^{\infty}\left( \frac{1}{2}\phi^{2}+ U ( \phi ) \right) d y \rightarrow \int_{- \infty}^{\infty}\left( \frac{- 3 \xi^{2}}{8 \kappa^{2}}+ \frac{3 \xi}{2 \kappa^{2}}\delta ( y ) \right) d y .
\begin{array}{lll}\omega(T\Gamma)=T\omega(\Gamma), \rho_1(T\Gamma)=T\rho_1(\Gamma), \rho_2(T\Gamma)=T\rho_2(\Gamma), begin{align*}0.05in]\rho_{2,s}(T\Gamma)=T\rho_{2,s}(\Gamma), T\xi_{\Gamma,i}=\xi_{T\Gamma,i}, T\mathcal{B}_\Gamma=\mathcal{B}_{T\Gamma}.\end{array}
T_0 = t_0 - \frac{(x_1 -a)(x_1-c)}{x_1^2-q}(1-s_0), T_i = t - \frac{t x_i - x_{i+1}}{x_i-x_{i+1}}(1-s_i) (i=1,\ldots,n-1), T_n = t_n - \frac{(bx_n -1)(dx_n-1)}{1-x_n^2}(1-s_n).
{\left\{\ \sum_{j}A^{j}\kappa_{j}^{\alpha}\equiv \sum_{j}A^{j}\left( \eta_{j}^{\alpha}- \xi_{j}^{i}u_{i}^{\alpha}\right) \ \right\}}_{\Big \vert \displaystyle{u^{\alpha}= U^{\alpha}( z )}}= 0 .
{K}^{(1)}_i(Y)= \sum_{j \in I_p}\sum_{l \in I_p}K_{ijl}^{(1)}Y_{j}Y_{l},{K}^{(3)}_i(Z)= \sum_{j \in J_p}\sum_{l \in J_p}K_{ijl}^{(3)}Z_{j}Z_{l},
-\mathrm{tr}\ln\left(\bar{D}^2\right)-\frac{1}{2}\mathrm{tr}\ln E+\frac{1}{2}\mathrm{tr}\left(G_0F\right)+\frac{1}{4}\mathrm{tr}\left(G_0FG_0F\right)+\frac{1}{6}\mathrm{tr}\left(G_0FG_0FG_0F\right)+\ldots
\delta s^{2}= [ \pm d \chi^{2}- d \lambda^{2}- d \theta^{2}+ ( 1 + \epsilon 4 \tan^{- 1}\left[ e^{\pm t}\right] ) \frac{b ( \lambda )}{a ( \lambda )}( \hat{{\delta}}t )^{2}- \sin^{2}\theta ( \delta \varphi )^{2}] ,
\frac{1}{2}\frac{d}{dt}\int_\Omega|T|^2dxdydz+\int_\Omega|\nabla_HT|^2dxdydz = -\int_\Omega\left[v\cdot\nabla_HT_2-\left(\int_{-h}^z\nabla_H\cdot vd\xi\right)\left(\partial_zT_2+\frac{1}{h}\right)\right]Tdxdydz.
c^{( 0 )}= \frac 1 6 ~ , ~ c^{( 1 / 2 )}= -{\frac{N^{( 1 / 2 )}}{1 2}}~ , ~ c^{( 1 )}={\frac{N^{( 1 )}}{6}}- 1 ~ , ~ c^{( 3 / 2 )}= -{\frac{N^{( 3 / 2 )}}{1 2}}~ , ~ c^{( 2 )}={\frac{N^{( 2 )}}{6}}- ( d + 2 ) ~ .
\int_{0}^{t/2}dy \frac{e^{-y}\sin(a\tan^{-1}(\frac{\pi}{2(\log(t)-\log(y))}))}{y((\log(t)-\log(y))^{2}+\frac{\pi^{2}}{4})^{a/2}}=a\frac{\pi}{2}\int_{0}^{t/2}dy \frac{e^{-y}}{y\log^{a+1}(t/y)}+ \int_{0}^{t/2}dy
F_{x}=\frac{ee^{\prime}}{r^{2}}(1+\frac{ra_{r}^{\prime}}{c^{2}})[Acos(rx)(1-\frac{3ra_{r}^{\prime}}{2c^{2}})+A(\frac{ra_{x}^{\prime}}{2c^{2}})-B(\frac{u_{x}u_{r}}{c^{2}})-C(\frac{ra_{x}^{\prime}}{c^{2}})]
\Lambda_0^1 = \frac{1}{\tilde{\Delta}_{(m,n)}^{1/2}}\left(\begin{array}{cc}m \qquad - n \,e^{-\phi_0}+ \chi_0 \,(m - \chi_0 n) \,e^{\phi_0}n\qquad (m - \chi_0 n)\, e^{\phi_0}\end{array}\right),
= \sum_{r_1,\dots,r_{m-k+1}}\frac{m!}{r_1! \cdots r_{m-k+1}!}\left(\frac{x_1}{1!}\right)^{r_1}\cdots \left(\frac{x_{m-k+1}}{(m-k+1)!}\right)^{r_{m-k+1}}
\theta^0=-2\varepsilon^{-1},\theta^\infty_1=\tilde{\theta}^\infty_1+2\varepsilon^{-1},t_2=\varepsilon \tilde{t}_2,H_{t_2}=\varepsilon^{-1}\tilde{H}_{2}, q_2=-\tilde{q}_2,p_2=-\tilde{p}_2-\frac{1}{\varepsilon \tilde{q}_2}, Y=x^{-\varepsilon^{-1}}\tilde{Y}.
D_{p,q}=\left\{Z\in\mathcal{M}_{p,q}\left(\mathbb{C}\right);I_{q}-\overline{Z}^{t}Z> 0\right\},S_{p,q}=\left\{Z\in\mathcal{M}_{p,q}\left(\mathbb{C}\right);I_{q}-\overline{Z}^{t}Z= 0\right\},p>q.
\mathcal{M}_{\mathcal{R},\alpha}^{(k)}(\vec{f})(x):=\sup_{\substack{R \ni x, R\in\mathcal R, R\le 2^k}}\prod_{i=1}^m \frac{1}{|R|^{1-\frac{\alpha}{mn}}}\int_R f_i(y_i)dy_i.
\sigma = \frac{H_n(A_n(A,B), B_n(A,B))^{2d^n}}{\rho_{d^n}(A_n(A,B), B_n(A,B))^{d_n}}&= \frac{H_n(A_n(A,B), B_n(A,B))^{2d^n}}{\rho_d(A,B)^{Nd_n}}\textrm{(By Lemma \ref{resultant}})
x_i=\frac{(S_{i+1}-S^{-}_{i+2})(S^{-}_i-S^{-}_{i+1})}{(S^{-}_i-S_{i+1})(S^{-}_{i+1}-S^{-}_{i+2})}, y_i=\frac{(S^{-}_{i+1}-S_{i+1})(S^{-}_{i+2}-S_{i+2})(S^{-}_i-S^{-}_{i+1})}{(S^{-}_{i+1}-S_{i+2})(S^{-}_i-S_{i+1})(S^{-}_{i+1}-S^{-}_{i+2})}.
\sum_{\substack{d_1,\dots,d_m \geq 0 d_2+\dots+d_m \leq b_1}}|a(j)_{d_1,d_2,\dots, d_m}| F^{m}p^{\sum_i \frac{\sum_{k=i}^{m}d_k (\sum_{k=i}^{m}d_k-1)}{2}}\leq \sum_{\substack{d_1,\dots,d_m \geq 0 d_2+\dots+d_m \leq b_1}}E p^{-b_1 d_1 -\frac{d_1(d_1+1)}{2}}F^{m}p^{\sum_i \frac{\sum_{k=i}^{m}d_k (\sum_{k=i}^{m}d_k-1)}{2}}.
\begin{aligned}a_{1i}=\sigma_i,\a_{2i}=\sigma_i,\a_{3i}=\mu_i\xi_1-\theta_i\xi_2,\a_{4i}=\theta_i\xi_1+\mu_i\xi_2, & a_{5i}=-\mu_i\xi_1+\theta_i\xi_2,\a_{6i}=-\theta_i\xi_1-\mu_i\xi_2,\a_{7i}=\sigma_i,\a_{8i}=\sigma_i.\end{aligned}
\sigma^+_1 + \sigma^+_{-1}= \sigma_1^- + \sigma^-_{-1}\sigma^+_{1}- \sigma^+_{-1}= \frac{1-g'(s)\delta}{1+g'(s)\delta}\left( \sigma_1^- - \sigma_{-1}^- \right)= (1-2c) \left( \sigma_1^- - \sigma_{-1}^- \right).
\begin{array}{lll}C^* = \overline{\delta^{-1}(C'\setminus m_S)}=\overline{\delta^{-1}\alpha^{-1}(C\setminus m_R)}=\overline{\beta^{-1}\gamma^{-1}(C\setminus m_R)}= \beta^{-1}(\overline{\gamma^{-1}(C\setminus m_R)})\mbox{since $\beta$ is quasi finite}= \beta^{-1}(\tilde C) \end{array}
|\cot2\phi_{k}- \cot2\psi_{k}| = \frac{|\sin 2(\phi_k -\psi_k)|}{|\sin 2\phi_k \sin 2\psi_k|}\geq |\sin 2(\phi_k -\psi_k)| & = 2|\cos (\alpha_k +\beta_k)|\cos (\phi_k -\psi_k) > \left(2-4.937284\delta_k^2\right)\cos (\phi_k -\psi_k) &> 1.9999\cos(\phi_k -\psi_k), 1\leq k\leq 5,
K = L n ( \frac{n ( n - 2 )}{4 n}) + L = L [ \frac{n^{2}- 2 n + 4}{4}] = 2 \times \frac{n ( n + 1 )}{2}( \frac{n - 3}{4}) [ \frac{n^{2}- 2 n + 4}{4}] .
^{( 3 )}R = - 2 \frac{\left( \theta^{\prime}\left( 1 - \frac{3 M}{2 r}\right) - \frac{1}{r \tan \theta}\right)^{2}+ \left( \frac{\sqrt{3}M}{2 r}\theta^{\prime}\right)^{2}}{1 + r^{2}( \theta^{\prime})^{2}\left( 1 - \frac{2 M}{r}\right)}
B(x) = \frac{3\lambda^2x^2}{4!}F(t)^{-2}+ 2\lambda x \left(\xi - \frac{1}{6}\right) F(t)^{-4/3}+ 8 \left( \xi - \frac{1}{6}\right)^2 F(t)^{-2/3}- \frac{1}{135},\ \ \ \F(t) \equiv 1- \frac{3\lambda t}{(4\pi)^2}.
L_{1 1}^{( \pm )}L_{1 2}^{( + )}= q^{\mp 1}L_{1 2}^{( + )}L_{1 1}^{( \pm )}, L_{2 2}^{( \pm )}L_{1 2}^{( + )}= q^{\pm 1}L_{1 2}^{( + )}L_{2 2}^{( \pm )}
\|w^n_1 - w^{n-1}_1\|_{y_{k+1}}+ \|w^n_{-1}- w^{n-1}_{-1}\|_{y_{k+1}}& \le c (2\kappa \delta)^{n-1}\left( k\sum_{j=1}^{n-1}j^k +1 \right) \le c (2\kappa \delta)^{n-1}(k (n-1)^{k+1}+1) & \le c (2\kappa \delta)^{n-1}(k+1) (n-1)^{k+1}.
\left\| y^{\frac{1-2s}{2}}_{n+1}w \right\|_{L^2(B_{r}^+)}^2\lesssim \ \left\| V \right\|_{C^1(M)}\tau^{2-2s}\left( e^{\tau A}\left\| y^{\frac{1-2s}{2}}_{n+1}w \right\|_{L^2(B_{r/2}^+)}^2 \right. & \left. + \e^{-\tau B}\left\| y^{\frac{1-2s}{2}}_{n+1}w \right\|_{L^2(B_{2r}^+)}^2 \right).
\Gamma_a(p,q)_{\mu}=-i \int \frac{d^3k}{(2 \pi^3)}\frac{\bar u(p-q) \gamma_{\nu}(p \!\!\! /-q \!\!\! / -k \!\!\! +m) \gamma_{\mu}(p \!\!\! / -k \!\!\! / +m) \gamma_{\lambda}u(p)}{[(p-k)^2-m^2][(p-k-q)^2-m^2]}G(k)^{\nu \lambda}
N_3 [\epsilon,v,w] = n_3 (1-\chi_2) \chi_4 = \underbrace{n_3 (\chi_{2,{\sf a}}-\chi_2)}_{={\sf t}_7}+ \underbrace{n_3 (\chi_4-\chi_{3,{\sf e}})}_{={\sf t}_8}+ \underbrace{n_3 (1-\chi_{2,{\sf a}}) \chi_{3,{\sf e}}}_{={\sf t}_9}.
\bar h^{(-)}\stackrel{d}{=}\psi\left(\sum_{k=1}^\infty (\tfrac{\xi_k}{\xi_k+S_k})^2 \,\psi\left( |z|^2 \bar h^{(-)}+ \sum_{j=1}^\infty (\tfrac{\xi^{(k)}_j}{\xi_k+S_k})^2h_j^{(-)}\right) \right) h^{(-)}\stackrel{d}{=}\frac{\bar h^{(-)}}{1- |z|^2\bar h^{(-)}\psi\left( \sum_{k=1}^\infty \zeta_k^2h_k^{(-)}\right)},
P(S_n(\kappa_n(0))\leq\lfloor\kappa_n(0)\rfloor-a_n) =P\left(\frac{S_n(\kappa_n(0))}{a_c^{(n)}}\leq o(1)\right) &\geq P\left(S_n(\kappa_n(0))=0\right)=(1-\pi_n(\kappa_n(0)))^{n-a_n}.
\delta x^{\alpha \alpha^{\prime}}= -{\frac{1}{2}}( \epsilon_{a}^{\alpha}\theta^{\alpha^{\prime}a}+ \epsilon^{\alpha^{\prime}a}\theta_{a}^{\alpha}) \ , \delta \theta_{a}^{\alpha}= \epsilon_{a}^{\alpha}\ , \delta \theta^{\alpha^{\prime}a}= \epsilon^{\alpha^{\prime}a}\ .
C(\alpha)\int_0^\infty \left(e^{u\xi}- 1\right)e^{-\xi}\xi^{-\alpha}d\xi = 1 - (1-u)^{\alpha-1}\intertext{for all $u \le 1$, and}C(\alpha)\int_0^\infty e^{-\xi}\xi^{1-\alpha}d\xi = \alpha - 1.
\dfrac{\partial \lambda_i}{\partial \eta}=-\dfrac{f_\eta(\lambda_i)}{f_\lambda(\lambda_i)},\ \ \frac{\partial^2 \lambda_i}{\partial \eta^2}=-\frac{f_{\eta\eta}(\lambda_i)+2f_{\lambda\eta}(\lambda_i)\lambda_{i,\eta}+f_{\lambda\lambda}(\lambda_i)\lambda_{i, \eta}^2}{f_\lambda(\lambda_i)}.
T \equiv \beta_H^{-1}=\left.\frac{\tilde{A}'}{4\pi\sqrt{\tilde{A}\tilde{B}}}\right |_{r_+}=\frac{\tilde{d}}{4\pi r_+}\left[1-\left(\frac{r_-}{r_+}\right)^{\tilde{d}}\right]^{(\tilde{d}-2)/2\tilde{d}},
-\partial_q W = \frac{4\pi \hbar}{T_{BH}}\int_{-\infty}dz\,r^2 F\,\biggl[\frac{\partial_q F}{F}\,\langle T^{\tau}_{\tau}+T^{z}_{z}\rangle+2\frac{\partial_q r}{r}\, \langle T^{\theta}_{\theta}+T^{\phi}_{\phi}\rangle\biggr]+\cdots
{\rm div}(J_{M_{g,0}}^{*}\chi_{g}^{8})|_{\Omega_{\Lambda}^{0}}=({\mu}_{\Lambda}^{0})^{*}{\rm div}(j^{*}\chi_{g}^{8})|_{{\frak M}_{g}}=2^{4}({\mu}_{\Lambda}^{0})^{*}{\frak M}'_{g}.
U^{ab}= \left(\delta^{ab}\partial + f^{abc}J_{1}^{c}\right) \sigma_{1}=\left(\begin{array}{cc}0 \delta^{ab}\partial + f^{abc}J_{1}^{c}\noalign{\vspace{6pt}}\delta^{ab}\partial + f^{abc}J_{1}^{c}0\end{array}\right)\;.
\Pi_{\alpha-1}^{(i)}(\eta_i,t) = \Phi_{\alpha-1}(t) \, e^{- v_i^{(i)}(\ell_i) \, \eta_i}, \Pi_0^{(i)}(\eta_i,t) =\Phi_0(t) \, e^{- v_i^{(i)}(\ell_i) \, \eta_i}, \Pi_{\alpha}^{(i)}(\eta_1,t) = \left(\Phi_{\alpha}(t) - \frac{\partial_t{\Phi}_{\alpha-1}(t)}{v_i^{(i)}(\ell_3)}\, \eta_1 \right) e^{-v_i^{(i)}(\ell_i) \, \eta_i}
\Theta_{j}=\sum_{k=1}^{n}\gamma_{ipk}\gamma_{kpj}+\sum_{r=1}^{\nu}\widetilde{\gamma}_{ipr}\gamma_{irj}\;\mbox{and}\;\widetilde{\Theta}_{u}=\sum_{k=1}^{n}\gamma_{ipk}\widetilde{\gamma}_{kpu}+\sum_{r=1}^{\nu}\widetilde{\gamma}_{ipr}\widetilde{\gamma}_{iru}
E_{1,n}:0=(n+2)(n+1)\rho_{0}^{2}\rho_{n+2}+..... E_{2,n}:0=(n+2)(n+1)\rho_{0}^{3}\varphi_{n+2}+..... E_{3,n}:0=(n+2)(n+1)g_{0}\rho_{0}^{3}\theta_{n+2}+....
\int \prod_{i>j}{\cal D}c^{ij}{\cal D}\bar{c}^{ij}{\cal D}A_{\mu}^{ij}e^{-S_{\rm off-diag}}= \exp \left\{{1 \over 8 \pi}\int_{\cal M}d\mu R \sum_{i>j}\log |\lambda_i -\lambda_j| \right\}~.
\varphi^{\prime}\colon \Lambda^{q+1}\cong (\Lambda \oplus P)^{\ast}\longrightarrow \ker(d_{n-1}) = H_{n-1}(\widetilde{K}^{[n-1]}) \xrightarrow[]{\h_{n-1}^{-1}\}\pi_{n-1}(\widetilde{K}^{[n-1]}) x \longmapsto h_{n-1}^{-1}\big( [d_{n}(x)] \big)
C_1{}= \left\{(f,g) \in \mathrm{L}^2(X,m;\mathbb{R}^2) : f \leq g\right\}, C_{2,\alpha}{}= \left\{(f,g) \in \mathrm{L}^2(X,m;\mathbb{R}^2) : |f-g| \leq \alpha\right\}.
\sum_{m=2}^{\infty}\frac{\tau_{A_{s_1}\smallsetminus U_{s_1}\cup (V_{s_2})^{-}}(p^{m}) \tau_{B_{s_2}\smallsetminus V_{s_2}\cup (U_{s_1})^{-}}(p^{m})}{p^{m}}\ll \frac{1}{p^{2-\varepsilon}}
c h a n g e d ( b_{0}+ \tilde{b}_{0}) \int_{\pi}^{\infty}d s^{\prime}e^{- s^{\prime}( L_{0}+ \tilde{L}_{0})}| \mathcal{B}\rangle = \frac{b_{0}+ \tilde{b}_{0}}{L_{0}+ \tilde{L}_{0}}e^{- \pi ( L_{0}+ \tilde{L}_{0})}| \mathcal{B}\rangle
K_{l+\nu}(r'',r';T)=\int_{r(0) = r'}^{r(T)=r''}{\cal D}r(t) \,\; \mu_{l+ \nu}[r^{2}]\exp \left(\frac{i}{\hbar}\int_{0}^{T}dt\left\{\frac{M}{2}\, \left[ \stackrel{.}{r}(t) \right]^{2}-V(r(t))\right\}\right)\;
\sum_{n\ge0}q^{n^{2}}f_{n}\cdot(cz)^{n}=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}e^{-t^{2}/2}[F(e^{t\sqrt{2log(q)}}\cdot cz)+F(e^{-t\sqrt{2log(q)}}\cdot cz)]dt.
\begin{array}{lllll}r_1:=auv+bvu+cw^2, r_2:=avw+bwv+cu^2, r_3:=awu+buw+cv^2, r_4:=bu'v'+av'u'+cw'^2, r_5:=bv'w'+aw'v'+cu'^2, r_6:=bw'u'+au'w'+cv'^2, r_7:=uu'-u'u, r_8:=vu'-u'v, r_9:=wu'-u'w, r_{10}:=uv'-v'u, r_{11}:=vv'-v'v, r_{12}:=wv'-v'w, r_{13}:=uw'-w'u, r_{14}:=vw'-w'v, r_{15}:=ww'-w'w.\end{array}
D^\dagger := \arg\min_{D \in \mathbb{N}}\left\{\tilde C(\alpha,\beta) \varepsilon^2 \sqrt{\sum_{j=1}^{D \wedge M}X_j^{-4}}+ (7+4\sqrt{x_{\alpha/2}}) \left[\sigma^2 \ln^{3/2}(1/\sigma) \vee a^{-2}_{D \wedge M}\right]\right\},
\mathbb{P}[\Psi^{\phi}(h) \in d\omega_2 \vert \Psi^{\phi}(0)=\omega_1]=2\frac{j(\omega_2-h)}{j(\omega_1)}\times \int_{\omega_1}^{\omega_2}\int_{\infty}^{0}(r^{h})'(\omega)\Phi(\omega-h,y,\omega_2-h)\tilde{\Phi}(-\omega,y,-\omega_1)dy d\omega
\mathbb{E}\big(e^{it\eta^\omega_{p,E}(I))}\big) =\sum_{m=0}^\infty e^{itm}\mathbb{P}\big(\eta^\omega_{p,E}(I))=m\big) &=1+\mathbb{E}(\eta^\omega_{p,E}(I))[e^{it}-1]+R_{L_n}
d \, s^{2}= ( 1 - \frac{r_{g}}{2 r_{f}})^{2}d t^{2}- ( 1 + \frac{r_{g}}{2 r_{f}})^{2}d r^{2}- r^{2}( \sin^{2}\theta d \, \varphi^{2}+ d \, \theta^{2}) ,
R_s^u= \log_2\left(1+\frac{\mathbf{w}_2^H\mathbf{h}_{ss}\mathbf{h}_{ss}^H\mathbf{w}_2}{\sigma^2+\mathbf{w}_1^H\mathbf{h}_{ps}\mathbf{h}_{ps}^H\mathbf{w}_1}\right).
d\lambda_i(t)=2h(\lambda_i(t))g(\lambda_i(t))d\beta_i(t)+\left(b(\lambda_i(t))+\alpha\sum_{k=1}^{N}\lambda_k(t)+2\sum_{k\ne i}^{}\frac{G\left(\lambda_i(t),\lambda_k(t)\right)}{\lambda_i(t)-\lambda_k(t)}\right)dt,
\left\langle \mathbf{L}_{\mu}\Psi|\varphi\right\rangle =-\left\langle \Psi|\mathbf{L}_{\mu}\varphi\right\rangle,\left\langle \mathbf{R}_{\mu}\Psi|\varphi\right\rangle =-\left\langle \Psi|\mathbf{R}_{\mu}\varphi\right\rangle.
I_1 = (\Pi_{23})^2 \int d^4x_{4,5}d^4u_{4,5}d^4\theta^+_{4,5}\; \Pi_{14}\Pi_{42}\Pi_{35}\Pi_{51}\; (D^+_4)^4\left({\delta^8(\theta_4-\theta_5)\over x^2_{45}}\right) \delta^{(-2,2)}(u_4,u_5) \ .
\mathring{R}_{ij,kk}=\mathring{R}_{ik,jk}=\mathring{R}_{ki,kj}+\mathring{R}_{li}R_{lkjk}+\mathring{R}_{kl}R_{lijk}&=\mathring{R}_{kk,ij}+\mathring{R}_{li}R_{lkjk}+\mathring{R}_{kl}R_{lijk}&=\mathring{R}_{li}R_{lkjk}+\mathring{R}_{kl}R_{lijk},
\sum\limits_{\substack{(A,B)\in \mathbb{Z}^2}}\Gamma\left(\frac{A}{X^4}\right)\Gamma\left(\frac{B}{X^6}\right) \mu^2\left(\Delta_{A,B}\right)=\sum\limits_{k=1}^\infty \mu(k) S(X;k^2)=:S(X),
{\cal L}_0=(\partial^\mu+ig_0 A^\mu)\phi\,(\partial_\mu-ig_0 A_\mu){\bar{\phi}}-m_0^2\phi\,{\bar{\phi}}-\frac{\lambda_0}{4}(\phi\,{\bar{\phi}})^2-\frac{1}{4\Omega_{D-2}}\,F_{\mu\nu}F^{\mu\nu}.
\frac{W_2^2(\mu,\nu)}{2}=\sup_{(\varphi,\psi)}\bigg\{\int_X \varphi \,d\mu -\int_X \psi \,d\nu \,\bigg|\, \varphi(x)-\psi(y) \le \frac{d^2(x,y)}{2}\bigg\}
a_{2}= [ a_{0}, a_{1}] + b_{0}b_{1}^{t}- b_{1}b_{0}^{t}= 2 g_{0}g_{1}J^{3}+ 2 \epsilon^{x y z}\bar{a}_{0}^{x}\bar{a}_{1}^{y}\bar{J}^{z}+ \frac{1}{2}\vec{\beta}_{0}\cdot \vec{\beta}_{1}( J^{3}+ \bar{J}^{3}) = 0
i(\bar\partial_B \partial_B^* + \partial_B^* \bar\partial_B) = \bar\partial_B [\epsilon(\kappa_B^{0,1}),\Lambda] + [\epsilon(\kappa_B^{0,1}),\Lambda]\bar\partial_B, i(\partial_B \bar\partial_B^* + \bar\partial_B^* \partial_B) = -\partial_B [\epsilon(\kappa_B^{1,0}),\Lambda] - [\epsilon(\kappa_B^{1,0}),\Lambda]\partial_B.