harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
h(x)=\frac{x\big((x-1)^2+2t(x-1)(x-\lambda^{\prime})+t^2(x-\lambda^{\prime})^2\big)}{\frac{\lambda^{\prime}}{\lambda}(x-1)^2+2t\sqrt{\frac{\lambda^{\prime}}{\lambda}}(x-1)(x-\lambda^{\prime})+t^2(x-\lambda^{\prime})^2}
\lim_{x \to +\infty}\frac{\psi(x)}{\varphi_2'(x)}= 1, \lim_{x \to +\infty}\frac{\psi'(x)}{\varphi_2''(x)}= 1, \lim_{x \to +\infty}\frac{\psi''(x)}{\varphi_2'''(x)}= 1.
\dot{x}_{\ell,i}^{(1)}= x_{\ell,i}^{(1)}\prod_{k=2}^d \frac{x_{j_{k,i},i}^{(k)}}{a_{j_{k,i},i}^{(k)}}, \ell = 1,2,\ldots,n_1,\; i=1,2,\ldots,r, \dot{x}_{\ell,i}^{(k)}= a_{j_{k,i},i}^{(k)}\frac{x_{\ell,i}^{(k)}}{x_{j_{k,i},i}^{(k)}}, \ell=1,2,\ldots,n_k,\; i=1,2,\ldots,r,\; k=2,3,\ldots,d,
\sum_{i=0}^1\widetilde{v}^{(i)}{}^*\widetilde{v}^{(i)}=\begin{bmatrix}1_{(A\otimes A\otimes V)_\omega}0 0 0\end{bmatrix},\sum_{i=0}^1\widetilde{v}^{(i)}\widetilde{v}^{(i)}{}^*=\begin{bmatrix}0 0 0 1_{(A\otimes A\otimes V)_\omega}\end{bmatrix}
{}_2F_1(a,b,2b; z) =\frac{\Gamma(2b)}{2\Gamma^2(b)}\int_{-1}^1(\frac{1+t}{2})^{b-1}(\frac{1-t}{2})^{b-1}(1-z\frac{t+1}{2})^{-a}dt &=\frac{\Gamma(2b)}{2^{2b-1}(1-\frac{z}{2})^a\Gamma^2(b)}\int_{-1}^1(1-t^2)^{b-1}(1-\frac{z}{2(1-\frac{z}{2})}t)^{-a}dt.
\sum_{k = 1}^D \theta_{1k}R_{\alpha,\beta- e_k}+ (1-\delta_{|\beta|,M})(\beta_1+1)R_{\alpha,\beta+e_1}-\sum_{i,j=1}^D\frac{\tilde{C}_{ij}(\beta)}{2\rho}\theta_{ij}R_{\alpha,0}=0.
\begin{aligned}|a_{j,\alpha ,\beta}| \lesssim h^{|\alpha +\beta|}(\alpha !\beta !)^{-\frac 1{4\sigma}},\ \forall h>0,\j=1,2, begin{align*}1ex]|a_{0,\alpha ,\beta}| \lesssim e^{-c(|\alpha |+|\beta |)},\ \forall c>0,\end{aligned}
\bar{V}:=\frac{\hbar^2}{2M}K*\left(\frac{\|\nabla\bar\rho\|^2}{2\bar{D}^2}-\frac{\nabla^2 \sqrt{\bar{D}}}{\sqrt{\bar{D}}}\right)\,,\widehat{H}_\textit{eff}:=\widehat{H}_e - \frac{\hbar^2}{2M}\left[K*\left(\bar{D}^{-1}\nabla\bar{\rho}\right)\right]\,.
L \{{{\bf B}^{\prime}},{{\bf E}^{\prime}}\}= \frac{\epsilon_{0}}{2}\int d^{3}x \left( c^{2}{{\bf B}^{\prime}}^{2}-{{\bf E}^{\prime}}^{2}\right) - \rho_{m}\, \psi -{\bf j}_{m}\cdot{\bf W}
\delta_a (T(\mathbf y))(\nabla k) = k^{-1}\left[ \mathbf y_1^{-1}D(T)(\mathbf y_1,\mathbf y_2) \right] \cdot ( (\nabla k) \delta_a (k)) \,\,\,\,\, \, - k^{-1}D(T)(\mathbf y_2, \mathbf y_1) \cdot (\delta_a (k) \nabla k) .
\Bigl(\int_{B(0,M)}|(\Gamma(t,0)-I)h(y)|^2\,{\rm d}y\Bigr)^{\frac{1}{2}}\lesssim_M\|(\Gamma(t,0)-I)(1_{S_{1}(0,M)}h)\|_{L^2}+\sum_{j=2}^\infty e^{-c\frac{4^{j}}{t}}\|h\|_{L^2},
\frac{d}{dt}F_1 = - \int_{\Omega}\sum_{i=1}^m c_i^{-1}|\nabla_x c_i|^2dx -\int_{\Omega}|\Delta \psi|^2dx \leq 0, \frac{d}{dt}F_2 =- \int_{\Omega}\left( \sum_{i=1}^m q_i^2 c_i\right) |\nabla_x \psi|^2 dx-\int_{\Omega}|\Delta \psi|^2dx \leq 0.
E_{M}\, = \, \frac{l}{4}\sum_{n = 1}^{\infty}\left[ \left( 1 \, + \, \frac{r^{2}k_{n 1}^{2}}{l^{2}}\right) \omega_{n 1}^{2}A_{n 1}^{2}\, + \, \left( 1 \, + \, \frac{r^{2}k_{n 2}^{2}}{l^{2}}\right) \omega_{n 2}^{2}A_{n 2}^{2}\right]
\mathrm{P}\left(U_1\in \mathcal{S}_1, E\right) = 2 \left(1- F\left( \phi_1\right)\right)F\left(\tilde{\tau}\right) +\frac{1}{2}\sum^2_{i=1}\left(1 - \left[F\left( \phi_i\right)\right]^2\right) - \sum^2_{i=1}F\left(\tilde{\tau}\right)\left(1 - F\left(\phi_i\right)\right),
\left\|\sum\limits_{j \in \mathbb{J}}L_jA_jh\right\|^2=\sum\limits_{j \in \mathbb{J}}\|A_jh\|^2=\sum\limits_{j \in \mathbb{J}}\sum\limits_{k \in \mathbb{L}_j}|\langle h, u_{j,k}\rangle|^2; ~\left\|\sum\limits_{j \in \mathbb{J}}L_j\Psi_jh\right\|^2=\sum\limits_{j \in \mathbb{J}}\sum\limits_{k \in \mathbb{L}_j}|\langle h, v_{j,k}\rangle|^2.
\begin{aligned}d\widehat X_i(t) = b_i(\widehat Z(t))\, dt+\sqrt{\widehat X_i(t)}\sum_{k=1}^{n+m}\sigma_{ik}(\widehat Z(t))\, d\widehat W_k(t),\quad\forall\, t>0, d\widehat Y_l(t) = e_l(\widehat Z(t))\, dt+\sum_{k=1}^{n+m}\sigma_{n+l,k}(\widehat Z(t))\, d\widehat W_k(t),\quad\forall\, t>0,\end{aligned}
\begin{array}{cccc}\left\{P_{\phi}^\tau ,X_{\phi}^\tau \right\}=1, \left\{P^r_{\phi},X_{\phi}^s \right\}=-\delta^{rs}, \left\{P_{\phi}^r,X_{\phi}^\tau \right\}=0, \left\{P_{\phi}^\tau ,X_{\phi}^r\right\}=0,\end{array}
T(z) = \mathcal{O}\begin{pmatrix}1 z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) 1 z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) 1 z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z)\end{pmatrix}.
x_{i-1,j-1}+ x_{i-1,j}+ x_{i,j-1}+ x_{i,j}= S \mbox{and}x_{m+2-i,n+2-j}+ x_{m+2-i,n+1-j}+ x_{m+1-i,n+2-j}+ x_{m+1-i,n+1-j}= S,
|\frac{\partial x}{\partial\lambda}|=\sqrt{h_{ki}h_{kj}\frac{\partial q^{i}}{\partial\lambda}\frac{\partial q^{j}}{\partial\lambda}}=\sqrt{g_{ij}\frac{\partial q^{i}}{\partial\lambda}\frac{\partial q^{j}}{\partial\lambda}}=\sqrt{h_{i}^{2}(\frac{\partial q^{i}}{\partial\lambda})^{2}}
g_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 -1)^2}- \frac{\log x}{x}\right),
\bigl|c_{1,1}^{(z)}\bigr| \left( q_2^2\,\bigl|c_{2,2}^{(z)}\bigr| -(p_1-p_2)p_2 \right) \leqslant \frac{p_1}{q_1}\left( q_2^2 \cdot \frac{p_2}{q_2}-(p_1-p_2)p_2 \right) = p_1p_2,
Z_{\scriptscriptstyle SD}= \frac{1}{N_{\scriptscriptstyle SD}}\int\limits_{\Omega^{\scriptscriptstyle 1}(M)}\mathcal{D}f \hskip 4pt e^{-i \hskip3pt \bigl\langle f, \hskip4pt (\mathrm{1\mkern-5muI\!}\hskip2pt + \ast d_{\scriptscriptstyle 1})f \bigr\rangle}
\Delta m^{2}( \beta ) - \Delta m^{2}( \infty ) = \frac{\mu^{D - 2}\lambda}{2 ( 2 \pi )^{D / 2}}\sum_{n = 1}^{\infty}\biggl ( \frac{m}{\mu^{2}\beta n}\biggr )^{\frac{D}{2}- 1}K_{\frac{D}{2}- 1}( m n \beta )
v(\psi_0(a)) = v\left(\sum_{j=1}^{n}\psi_j(a)\varphi(a)^j \right) & = v(\varphi(a)) + v\left(\sum_{j=1}^{n}\psi_j(a)\varphi(a)^{j-1}\right) & \geq v(\varphi(a)) > 0
F_\gamma({\cal P}_\gamma(x)) - F_\gamma(\bar x) = F_\gamma({\cal P}_\gamma(x)) - F_\gamma(y) \le \langle \nabla F_\gamma(y),{\cal P}_\gamma(x) - y\rangle + \frac{L_f}2\|{\cal P}_\gamma(x) - y\|^2 = \frac{L_f}2\|{\cal P}_\gamma(x) - y\|^2 = \frac{L_f}2[{\rm dist}({\cal P}_\gamma(x),{\cal X})]^2,
{\frac{d}{d t}}\ln R = \mathrm{R e}( P ) + b_{\scriptscriptstyle R}\left( \frac{1}{R}\nabla^{2}R - ( \nabla S )^{2}\right) -{\frac{b_{\scriptscriptstyle I}}{R^{2}}}\nabla \cdot ( R^{2}\nabla S ) ~ ~ ~
a_{k,n}^{\left[ j\right]}=\left( -i\right)^{k}k!\sum_{m=-j}^{j}\left(V^{-1}\left[ j\right] \right)_{k+1,j-m+1}\frac{1}{2\pi}\int_{-\pi}^{+\pi}g_{n}^{\left[ j\right]}\left( \theta\right) e^{im\theta}d\theta\ .
I_1 = \int_{B_{R}(0)}\frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \leq{}\frac{C}{(1+|x|^2)^{(n-\alpha)/2}}\int_{B_{R}(0)}\frac{1}{|y|^{2q\theta_2 + \sigma_{1}}}\,dy \leq{}\frac{C}{(1+|x|^2)^{(n-\alpha)/2}};
C_\theta(\tau) = \langle{\cal A}(\tau){\cal B}\rangle \; - \;\langle{\cal A}\rangle \lim_{\tau^\prime \rightarrow \infty}\langle{\cal U}_{\tau^\prime}({\scriptstyle{\cal B}}){\cal B}\rangle \stackrel{\tau \rightarrow \infty}{\longrightarrow}0 \; ,
\sum_{k=0}^{n-r}\frac{(-1)^kq^{2nk}(q^{-(n-r)})_k(-q^{-(n-r)})_k}{(q)_k(q^{2r+1})_k}= \frac{(1+q^{2r})(q)_{2r}(-1)_{2n}}{2(q^2;q^2)_{n+r}},
\sum_{\ell_{M-1}<j \leq \ell_M}(-1)^{p(\eta,j)-1}i_{\eta \cup j}= U\left(\prod_{r=m+1}^{M-1}{D_{j_{r}-1}}\right)\left(D_{\ell_M-1}-\sum_{\ell_{M-1}<j<\ell_M}{x_{j}}\right) &= U\left(\prod_{r=m+1}^{M-1}{D_{j_{r}-1}}\right)D_{\ell_{M-1}}&= U\cdot B(M-1).
\left( -\frac{d^{2}}{dr^{2}}+\omega^{2}r^{2}+\frac{\mu +A_{\ell,j,m,n}-\frac{1}{4}}{r^{2}}-E_{k,\ell,j,m,n}\right) F_{k,\ell,j,m,n}(r)=0.
\dot{x}(t)=[\begin{matrix}-d_{3}&-d_{2}&-d_{1}&-d_{0}\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{matrix}]x(t)+[\begin{matrix}1\\ 0\\ 0\\ 0\\ \end{matrix}]u(t)
\Sigma ( z ) = \frac{\int_{- \infty}^{z}\sqrt{- g^{( 4 )}}d z^{\prime}\left( T_{t}^{t}- T_{z}^{z}- T_{x}^{x}- T_{y}^{y}\right) \int d x d y}{\sqrt{g^{( 2 )}}\int d x d y}.
\int_{x_2=0^-}\left[\mathbf{R}\mathbf{U}(x_1'-x_1,0^-)\cdot \mathbf{t}(x_1,0^-) - \mathbf{R}\mathbf{\Sigma}(x_1'-x_1,0^-)\cdot \mathbf{u}(x_1,0^-)\right]\mathrm{d}x_1=0.
y^{\delta}\tilde{\Gamma}^{\alpha}{}_{\beta \delta}\tilde{B}^{\beta}|_{1}= y^{\delta}( \Gamma^{\alpha}{}_{\beta \delta}+ \Gamma^{\alpha}{}_{\beta \delta , \sigma}x^{\sigma}/ 2 ) [ B^{\beta}+ ( B^{\beta}{}_{, \sigma}+ \Gamma^{\beta}{}_{\sigma \tau}B^{\tau}) d^{\sigma}/ 2 ]
\int_u^v \frac{(qx/u,qx/v,abrx;q)_{\infty}}{(ax,bx,cx;q)_{\infty}}d_qx =\frac{(1-q)v(q,u/v,qv/u;q)_{\infty}(acuv,bcuv,abr/c;q)_{\infty}}{(au,av,bu,bv,cu,cv;q)_{\infty}}& \times{}_3\phi_2\left( \begin{gathered}cu,cv,cuv/r acuv, bcuv \end{gathered};\,q, \frac{abr}{c}\right).
S_{1}(q,P,t)=-\frac{B}{2A}q^{2}+ \frac{\sqrt{\kappa}}{2 A}\Bigl[q\sqrt{\frac{A P}{\kappa}-q^{2}}+\frac{A P}{\kappa}\sin^{-1}\Bigl(\sqrt{\frac{\kappa}{A P}}q\Bigr)\Bigr]+s_{1}(P,t),
\sum_{n=1}^{\infty}d(n, N)q^n + \sum_{n=1}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{q^{n(n+1)/2}}{1-q^n}\frac{(-q)_n}{(q)_n}= \frac{1}{2}\left\{\frac{(-q)_N}{(q)_N}- 1\right\}+ \sum_{n=1}^{N}\frac{(-q)_n}{(q)_n}\frac{q^n}{1-q^n}
G_{k_1,k_2}(z_{12})=(z_{12})^{-h_1-h_2}\left(D_{(h_1;k_1)(h_2;k_2)}- \int^{z_{12}}\!\frac{{\rm d}\zeta}{\zeta^{1-h_1-h_2}}[ G_{k_1-1,k_2}(\zeta)+G_{k_1,k_2-1}(\zeta)]\right)\,.
\frac{1}{\Gamma(2-\alpha)}\int_z^x{(x-t)^{1-\alpha}}(t-z)^{\alpha - 2}E_{\alpha, \alpha-1}(\lambda (t-z)^{\alpha}) dt & = \frac{1}{\Gamma(2-\alpha)}\int_0^{x-z}{(x-(\xi + z))^{1-\alpha}}\xi^{\alpha - 2}E_{\alpha, \alpha-1}(\lambda \xi^{\alpha}) d\xi.
I_{m,n}(k) = \left(\prod_{i=0}^{k-1}C_{m,n}(i)\right)I_{m,n}(0) = \left(\prod_{i=0}^{k-1}C_{m,n}(i)\right)\left(\begin{array}{c}I_{m,n}^{(1)}(0) I_{m,n}^{(2)}(0)\end{array}\right),
\begin{psmallmatrix}x(\beta) s(\beta) w(\beta) y(\beta)\end{psmallmatrix}= \beta \mu e + (1 - \beta) \begin{psmallmatrix}x^k s^k w^k y^k\end{psmallmatrix}+ \beta^2 \begin{psmallmatrix}\Delta x \Delta s \Delta w \Delta y\end{psmallmatrix}\, .
\gamma_F(\frac{1}{2}\psi \circ q_9) = \gamma_F(\frac{1}{2}\psi)^2 \cdot \gamma_F(-J_1, \frac{1}{2}\psi) \cdot (J_2, \frac{\nu}{ab - cd J})_F = \gamma_F(J_1, \frac{1}{2}\psi)^{-1}\cdot (J_1, (ab - cd J) \nu)_F.
v(0,0) \le\mathbf{E}\bigg[\int_0^1e^{-\frac{1}{2}\int_0^t\|\beta_s\|^2ds}\langle \alpha^*_t(\beta)+c\beta_t,\beta_t\rangle\,dt+e^{-\frac{1}{2}\int_0^1\|\beta_t\|^2dt}f(X_1^\beta)\bigg] & = J_f[\alpha^*(\beta)+c\beta,\beta]
\textbf{U}^n =[U^n_{11},\ldots,U^n_{M_{x}-1,1},U^n_{12},\ldots,U^n_{M_{x}-1,M_{y}-1}]^T, \textbf{G}^n =[G^n_{11},\ldots,G^n_{M_{x}-1,1},G^n_{12},\ldots,G^n_{M_{x}-1,M_{y}-1}]^T,
\Psi(x) = \left( f(x) +{\cal F}^\mu(x) \widehat{b}_\mu^+ + \frac{1}{2!}\varepsilon_{\mu\nu\lambda}{\tilde{\cal F}}^\mu(x) \widehat{b}^{+\nu}\widehat{b}^{+\lambda}+ \frac{1}{3!}\tilde{f}(x) \varepsilon_{\mu\nu\lambda}\widehat{b}^{+\mu}\widehat{b}^{+\nu}\widehat{b}^{+\lambda}\right) |0\rangle.
\begin{aligned}\| \Delta w \|_{L^2(I,L^2)}\leq C' \bigg(\frac{1+\| a\|_{{W^{1}_\infty}}}{\min({\rho(a)},1)}\bigg)^5 \bigg(\frac{1+\| b\|_{{W^{1}_\infty}}}{\min({\rho(b)},1)}\bigg)^5 \|a-b\|_{{W^{1}_\infty}}. \end{aligned}
\frac{1}{i\tau}\{\overline{p_\Phi},p_\Phi \}=\frac{1}{i\tau}\{\Re p_{\Phi}-i\Im p_{\Phi}, \Re p_{\Phi}+i\Im p_{\Phi}\}&=\frac{1}{\tau}\{\Re p_{\Phi},\Im p_{\Phi}\}-\frac{1}{\tau}\{\Im p_{\Phi},\Re p_{\Phi}\}= \frac{2}{\tau}\{\Re p_{\Phi},\Im p_{\Phi}\}.
||\omega \lambda(t)^{2}(T(y_{1})-T(y_{2}))(t)||_{L^{2}(\rho(\omega \lambda(t)^{2})d\omega)}&\leq C ||y_{1}-y_{2}||_{Z}\frac{\log^{-1}(t)+\log^{-\epsilon+2b}(t)+\log^{-\frac{\epsilon}{2}+b}(t)}{t^{3}\log^{\frac{\epsilon}{2}+b}(t)}
\mbox{I}:= \sum_{i_{\ell_1}\ge i_{0}}^\infty\sum_{i_{\ell_2}\ge i_{\ell_1}}^\infty \cdots\sum_{i_{\ell_k}\ge i_{\ell_{k-1}}}^\infty \left( \prod_{t=1}^k 2^{\left(\frac{(d+1)(d-\ell_t+t)}{d(k+1)}-\ell_{t+1}+\ell_t\right) i_{\ell_t}}\right)
y(t)=\int_{-\infty}^{\infty}e^{j\pi.u^{2}/k}.e^{j2\pi.(u+f_{0})t}.du=e^{j2\pi.f_{0}t}\int_{-\infty}^{\infty}e^{j\pi.u^{2}/k}.e^{j2\pi.ut}.du
{\cal R}_{\alpha\beta}{\eta}^{\alpha}{\eta}^{\beta}= \Gamma^{\alpha}_{n+1\alpha,n+1}-\Gamma^{\alpha}_{n+1\,n+1,\alpha}+\Gamma^{\beta}_{n+1\,\alpha}\Gamma^{\alpha}_{\beta\, n+1}-\Gamma^{\alpha}_{n+1\,n+1}\Gamma^{\beta}_{\alpha\beta}= \kappa^{2}-\dot{h}
\mathcal L_\mu^{(i)}(\delta\mu,\delta\b u_i)=\begin{pmatrix}2\nabla\cdot(\delta\mu\;\varepsilon(\b u_i)) + 2\nabla\cdot(\mu\;\varepsilon(\delta\b u_i))-\rho(\delta\b u_i)_{tt}\delta \b u_i\end{pmatrix}.
\begin{aligned}d_2^m \cdot (0 \oplus \pi_1)\left(\Theta_2(h \cdot (g_1,g_2)^m)\right) = d_2^m (0, \pi_1 \Theta_1(h_2g_2^m)) & = (0,\pi_1(\Theta_1(h_2))).\end{aligned}
cd_p(U_{i}\cap \widehat{G}_v)= cd_p(U_{i_0}\cap \widehat{G}_v)=cd_p\big(\displaystyle\bigcap_{i\geq i_0}{(U_i\cap \widehat{G}_v)}\big)\leq cd_p\big(\displaystyle\bigcap_{i\geq 1}{U_i}\big)\leq 2,
\iint \left(F_{1, h}(x) \cdot F_{1, h}\circ T^{ak}\right)(x)\left(F_{2, h}\cdot F_{2, h}\circ T^{bk}\right)(y) K_{b-a}(x, y)\, d\mu(x) d\mu(y) &= \iint f_1 \otimes f_2(x, y) K_{b-a}(x, y)\, \left[{f_1 \otimes f_2}(U^h(x, y)) \,{f_1 \otimes f_2}(U^k(x, y)) \, f_1 \otimes f_2(U^{h+k}(x, y)) \right]\,d\mu \otimes \mu(x, y),
(\sum\limits_{i =1}^{j}w_i) \chi - \sum\limits_{i=1}^{j-1}\chi_i -\sum\limits_{i = j+1}^{n-1}k_i + (j-1)r \leq \chi_j \leq (\sum\limits_{i =1}^{j}w_i) \chi - \sum\limits_{i=1}^{j-1}\chi_i + \sum\limits_{i=1}^{j}k_i + (j-1)r,
\lim_{r \to \infty}\frac{1}{r^n}\sum_{B \in P : B \cap rC \neq \emptyset}\lambda(B) \leq \lim_{r \to \infty}\frac{\lambda(|P| \cap (r+r_0)C)}{r^n}= \lim_{r \to \infty}\frac{\lambda(|P| \cap rC)}{(r-r_0)^n}= \rho(|P|).
E(\iota(g_1,g_2),s_{m,2n},\Phi_{\varphi \otimes \overline{\varphi}}) = \frac{\kappa_{m,2n}}{\mathrm{vol}\, [H]}\int_{[H]}\theta_{\varphi}(g_1,h) \, \overline{\theta_{\varphi}(g_2,h)}\, dh
L_{i n t}= B_{k}H_{k}^{( e x t )}= \frac{1}{2 a}\varepsilon_{k m n}\left( F_{k m}^{a}\phi^{a}- \frac{1}{e a^{2}}\varepsilon_{a b c}\phi^{a}D_{k}\phi^{b}D_{m}\phi^{c}\right) H_{n}^{( e x t )}.
\big|A_N\bigl(e^{i\lambda}\bigr)\bigl(1-e^{i\lambda\mu}\bigr)^{n}\lambda^{2n}g(\lambda)+\lambda^{2n}C^{\mu,0}_{N}\bigl(e^{i\lambda}\bigr)\big|^2 &=\alpha_1|\lambda|^{2n}\big|1-e^{i\lambda\mu}\big|^{2n}\bigl(f^0(\lambda)-f_1(\lambda)\bigr)\bigl(f^0(\lambda)+\lambda^{2n}g(\lambda)\bigr)^2
\zeta=\left\{\zeta_0, \cdots, \zeta_{f^{n}-1},\zeta_{f^{n}}, \cdots ,\zeta_{f^{n-1}-1}\cdots, \zeta_{f^{k+1}}, \cdots, \zeta_{f^k-1}, \cdots,\zeta_{f^{1}},\cdots , \zeta_{f^{0}-1}\right\}
L_{V, A}= -{1\over 2(p+1)!}\, \left(\,\partial_{\,[\,\mu_1}V_{\mu_2\dots\mu_{p+1}\,]}+ m\,A_{\mu_1\dots\mu_{p+1}}\, \right)^2 -{1\over 2(p+2)!}F^2{}_{\mu_1\dots\mu_{p+2}}(A)
q(p^{-1}(T_2 \cdot [\phi_{6,u}])) = (T_2 \cdot [x(ux + y)]) \cup (T_2 \cdot [x^2 - (ux + y)^2]) \cup (T_2 \cdot [(x^2 + (ux + y)^2])= &= \{x(ux + ty)\}\cup \{x^2 - (ux + ty)^2\}\cup \{x^2 + (ux + ty)^2\},
{T_{\varepsilon_i}^{\vec{v}}}^T = \begin{pmatrix}x_i -\alpha_1y_i-\alpha_0z_i -\alpha_0y_i y_i x_i+\alpha_2y_i -\alpha_0z_i z_i y_i+\alpha_2z_i x_i+\alpha_2y_i+\alpha_1z_i\end{pmatrix}.
\left|\lambda_{\min}(T_n)-\lambda_{\min}(T)\right|= \left|\inf_{\|f\|_2=1}\langle f,T_nf\rangle-\inf_{\|f\|_2=1}\langle f,Tf\rangle\right|\leq \sup_{\|f\|_2=1}|\langle f,T_nf\rangle-\langle f,Tf\rangle| \to 0
0\leq\|u(t+T)\|_2^2 \leq \limsup_{\tau \to t+}\|u(\tau)\|_2^2 - 2\nu\int_{t}^{t+T}\|\nabla u(\tau)\|_2^2\, d\tau + 2\int_{t}^{t+T}(f,u)\, d\tau &\leq \nu^2 \kappa_0^{-1}G^2 - \nu\int_{t}^{t+T}\|\nabla u(\tau)\|_2^2\, d\tau + \frac{1}{\nu}\int_{t}^{t+T}\|f\|_{H^{-1}}^2\, d\tau.
\int_{{\cal F}\times S^{2}}\omega \ ( p , \zeta ) \; \langle p^{\prime}, \zeta^{\prime}| p , \zeta \rangle \langle p , \varsigma | p^{\prime \prime}, \zeta^{\prime \prime}\rangle = c \langle p^{\prime}, \zeta^{\prime}| p^{\prime \prime}, \zeta^{\prime \prime}\rangle
f(u,v)=-\tilde{\Gamma}^1_{22}+(\tilde{\Gamma}^2_{22}-2\tilde{\Gamma}^1_{12})u+(2\tilde{\Gamma}^2_{12}-\tilde{\Gamma}^1_{11})u^2+\tilde{\Gamma}^1_{11}v^2+\tilde{\Gamma}^2_{11}(u^2-v^2)u, &g(u,v)=\tilde{\Gamma}^2_{22}v+2\tilde{\Gamma}^2_{12}uv+\tilde{\Gamma}^2_{11}(u^2-v^2)v,
V_{-}( x , a_{0}) = \frac{( \omega_{0}- \omega_{1}) ( 3 \omega_{0}+ \omega_{1})}{4{( \omega_{0}+ \omega_{1})}^{2}x^{2}}- \frac{\omega_{1}}{2}+ \frac{{( \omega_{0}+ \omega_{1})}^{2}x^{2}}{1 6}~ .
\mathsf{G}_{\ell}(i,j)=\begin{cases}\Psi_{i,j}^{m_{i,j}}\ell=0 \frac{\Gamma(\ell+m_{i,j})}{\ell!\Gamma(m_{i,j})}\left( \frac{\Omega_{i,j}}{m_{i,j}}\right)^{\ell}\Psi_{i,j}^{m_{i,j}+\ell}\ell>0\end{cases}
\left(\begin{array}{c}\sigma^{i}[\omega_{0}^{+0i}\tilde{\epsilon}_{2}-i F^{+0i}\tilde{\epsilon}^{1}] \sigma^{i}[-\omega_{0}^{+0i}\tilde{\epsilon}^{1}-i F^{+0i}\tilde{\epsilon}_{2}]\end{array}\right)=0\, ,
\dfrac{1}{2}=\dfrac{1}{2}\left(\mathrm{FPdim}(x)^2-\sum_{y\in\Gamma_1}c_{x,x^\ast}^y\mathrm{FPdim}(y)^2\right)-\sum_{z\in \Gamma_2}c_{x,x^\ast}^z\mathrm{FPdim}(z)^2
T_{\beta \gamma}( w ) = -{\frac{3 / 2}{( w - q_{1})^{2}}}-{\frac{\partial \psi_{1}^{*}( q_{1})}{w - q_{1}}}-{\frac{1}{8}}\Delta_{1}^{2}( q ) -{\frac{1}{3 2}}S_{1}^{2}( q ) + O ( w - q_{1}) .
(p_n-q_n)((p_n-p_n\wedge q_n)\vee(q_n-p_n\wedge q_n)) = ((p_n-p_n\wedge q_n)-(q_n-p_n\wedge q_n))((p_n-p_n\wedge q_n)\vee (q_n-p_n\wedge q_n)) = (p_n-p_n\wedge q_n)-(q_n-p_n\wedge q_n)=p_n-q_n.
\underbrace{d_{s_i,1}d_{s_j,s_j}d_{s_i,s_i}\cdots d_{s_{j'}s_{j'}}}_{m_{ij}}=\underbrace{d_{s_j,s_j}\cdots d_{s_{i'}s_{i'}}d_{s_{j'}s_{j'}}d_{s_{i'},1}}_{m_{ij}}
wt(C_{4})=\begin{cases}(p-1)p^{e-2}, \qquad\ \biggl(\frac{c^{2}-4ai}{p}\biggr)=-1, (p-1)p^{e-2}+2p^{m+d-1}, \qquad\ \biggl(\frac{c^{2}-4ai}{p}\biggr)=1.\end{cases}
S_p(\mathbf{r},\mathbf{s},p)=\sideset{}{'}\sum_{\substack{\mathbf{d}|\mathbf{r},\mathbf{e}|\mathbf{s}d_i,e_i|p\forall i (d_ie_i,p)=1\forall i\ne m}}\frac{p\mu(d) \mu(e) d e}{[d_m,e_m,p] \prod_{i\ne m}[d_i,e_i]}=\begin{cases}(p-1)^2, p|(r_m,s_m), -(p-1), p|r_ms_m, p\nmid (r_m, s_m), 1, p\nmid r_m s_m.\end{cases}
V ( \sigma ) = \frac{1}{2 \lambda_{0}}\sigma^{2}- i \, \mathrm{t r}\ \displaystyle \int_{0}^{\sigma}s d s \frac{1}{L}\sum_{n = - \infty}^{\infty}\int \frac{d^{D - 1}k}{( 2 \pi )^{D - 1}}\frac{1}{k^{2}- s^{2}}\, .
\sum_{n}^{\infty}{\frac{a^{n}}{n !}}\; H_{n}( x ) H_{n}( y ) ={\frac{1}{\sqrt{1 - 4 a^{2}}}}\; \mathrm{e x p}\left[{\frac{4 a x y - 4 t^{2}( x^{2}+ y^{2})}{1 - 4 a^{2}}}\right] \;
\frac{\hat{S}_N(\rho)}{M_{\hat\mu_{\rho},1}}= \left(1-\frac{\rho}{\frac1{\hat\gamma(\rho)}\frac{1-\rho}{1-(1-\rho)c}+\rho}\right) \frac1n\sum_{i=1}^n z_iz_i^* + \frac{\rho}{\frac1{\hat\gamma(\rho)}\frac{1-\rho}{1-(1-\rho)c}+\rho}I_N.
B^zA^z-Y^z = \sum_{k=1}^n P_k\left(V_{k,z}-\int_{\mathbb{R}}(\lambda_kA)^{is}V_{k,z}Y^{-is}\widehat{g}_z(s)\,ds\right) = \sum_{k=1}^n P_kV_{k,z}-\int_{\mathbb{R}}\left(\sum_{k=1}^n P_k(\lambda_kA)^{is}V_{k,z}\right)Y^{-is}\,\widehat{g}_z(s)\,ds.
{\cal C}\left[ \begin{array}{c}{\alpha}_1\ \cdots\ \alpha_g{\beta}_1\ \cdots\ \beta_g \end{array}\right]={\cal C}\left[ \begin{array}{c}{\alpha}_1{\beta}_1 \end{array}\right]{\cal C}\left[ \begin{array}{c}{\alpha}_2{\beta}_2 \end{array}\right] \cdots{\cal C}\left[ \begin{array}{c}{\alpha}_g{\beta}_g \end{array}\right]
\begin{cases}\displaystyle J_{\infty}(B_{i.j}) \twoheadrightarrow gr^{i,j}(J_{\infty}(B_{i.j-1})) \mbox{if}j-1\geq 1 J_{\infty}(B_{i.j}) \twoheadrightarrow gr^{i.j}(J_{\infty}(B_{i-1.i-1})) \mbox{if}j=1 \end{cases},
\bar{U}^{n+1}_j=\bar{U}^{n}_j-\lambda(\hat{F}(\tilde{U}^{-}_{j+\frac{1}{2}},\tilde{U}^{+}_{j+\frac{1}{2}})-\hat{F}(\tilde{U}^{-}_{j-\frac{1}{2}},\tilde{U}^{+}_{j-\frac{1}{2}}).
\mathrm{Tr}(x\Phi(a)) = \mathrm{Tr}(\Phi^{\dagger}(x)a) & = \frac{1}{n}\mathrm{Tr}(\Phi^{\dagger}(x))\mathrm{Tr}(a) & = \mathrm{Tr}(x\Phi(I)\frac{\mathrm{Tr}(a)}{n})
ds_{SU(2)\times U(1)}^2 = M d{\bf x}^2 +{16\pi^2 \over M}d\chi^2 + k \left[{db^2 \over b}+ b \left( d\alpha^2 + \sin^2\alpha\, d\beta^2 + (d \gamma + \cos\alpha\, d\beta)^2 \right) \right]
P_{z, z^*}^2((y_i)_{i\in I}) =P_{z, z^*}\left(\sum_{i\in J}z_i^*(y_i)z_i\right) = \sum_{i\in J}z_i^*\left( z_i^*(y_i)z_i \right) z_i &= \sum_{i\in J}z_i^*(y_i) z_i^*( z_i ) z_i = \sum_{i\in J}z_i^*(y_i)z_i = P_{z, z^*}((y_i)_{i\in I}).
f_{a,c,\gamma}(z) = a \left( 4! \sum_{j_0, j_1 = 0}^{1, 1}\left[ \prod_{i = 0}^{1}(-\gamma_i)^{1-j_i}\binom{1}{j_i}\right] \frac{z^{2+j_0+j_1}}{2 + j_0 + j_1}\right) + c.
d s^{2}= ( 1 - \frac{N^{4}}{r^{4}}) ( \frac{r}{8 N})^{2}( d \tau + 4 N \cos \theta d \phi )^{2}+ ( 1 - \frac{N^{4}}{r^{4}})^{- 1}d r^{2}+ \frac{1}{4}r^{2}d \Omega^{2}.
u(x,t) = \frac{1}{(2\pi)^{n}}\int_{\mathbb R^n}e^{i x\cdot \xi}\cos (t |\xi|) \widehat f(\xi) d\xi+ \frac{1}{(2\pi)^{n}}\int_{\mathbb R^n}e^{i x\cdot \xi}\sin (t |\xi|) \frac{\widehat g(\xi)}{|\xi|}d\xi.
\tilde{A}_{\tilde{\mathcal{N}}_{i}, \tilde{M_{i}}, \tilde{\lambda}_{i}}\equiv \tilde{\mathcal{N}}_{i}e^{- \bar{\eta}\tilde{M}_{i}\eta - \bar{\eta}\tilde{\lambda}_{i}}, \quad \tilde{\mathcal{N}}_{i}= g ( \tau_{i}, p_{i}) e^{\bar{\epsilon}_{i}H ( \tau_{i}, p_{i}) - \bar{\epsilon}_{i}F ( \tau_{i}) \epsilon_{i}}, \, \tilde{M}
Z(x) = K(m, H_0) \left\{\int_{-\infty}^\infty dB_{\xi_1}\int_{-\infty}^{\xi_1}dB_{\xi_2}\ldots \int_{-\infty}^{\xi_{m-1}}dB_{\xi_m}\int_0^x \prod_{i=1}^m (s- \xi_i)^{H_0 - \frac{3}{2}}{\bf 1}_{(\xi_i < s)}\, ds \right\}\,\, ,
A = (a_{ij})_{n \times n}:= L(u^a) \big[R(u^a)_t - a^2 R(u^a)_{xx}+ f'(u^a) R(u^a)_x + R(u^a)_{tt}\big], &B = (b_i)_{n \times 1}:=L(u^a)\big[2 a^2 R(u^a)_x W_x - E_1 - E_2 + Q - 2 R(u^a)_t W_t\big].
\eta^{\nu \alpha}\xi_{\sigma}\nabla_{\alpha}\xi^{\sigma}= - \eta^{\nu \alpha}\delta \xi_{\alpha}, \qquad \xi^{\sigma}\eta^{\nu \alpha}\eta_{\sigma}{^{\rho}}( \nabla_{\alpha}\xi_{\rho}+ \nabla_{\rho}\xi_{\alpha}) = - \xi_{\sigma}\delta \eta^{\nu \sigma},
\left( \begin{array}{c c c}{{( \kappa - \ln \tilde{q}) \eta ( \tilde{\tau})^{2}}}&{{- \Lambda_{1 , 2}( \tilde{\tau})}}&{{- \eta ( \tilde{\tau})^{2}}}\\{{- \Lambda_{1 , 2}( \tilde{\tau})}}&{{0}}&{{0}}\\{{- \eta ( \tilde{\tau})^{2}}}&{{0}}&{{0}}\\ \end{array}\right)
\varphi (g_2 g^{n}_1 g_3) = \varphi \big( (g^{j}_1g_3) (g^{i}_1 g_2) \big) \mbox{or}\varphi (g_2 g^{n}_1 g_3) = \varphi \big( (g^{i}_1 g_2) (g^{j}_1 g_3) \big) \,.
\mathcal{R}_{2,P}(R_3')= \left\{\begin{aligned}(R_1,R_2): R_1,R_2\geq 0,\R_2\leq \min\{I(U_1;Y_2|U_2),\I(U_1,U_2;Y_2)-I(U_2;Y_3)\}\triangleq r_c, R_1+R_2 \leq I(X_1;Y_1|U_1)+ \min\{I(U_1;Y_2|U_2), I(U_1,U_2;Y_2)-I(U_2;Y_3)\}\triangleq r_d \end{aligned}\right\}.
\phantom{={}}C_{1,1}^{p/q}C_{1,2}\int_0^1 |f(y)|^p (1-y)^{ap}(1-y)^{1+(p/q)-ap-(m-b)p-u-bp}y^k \, dy &= C_{1,1}^{p/q}C_{1,2}\int_0^1 |f(y)|^p (1-y)^{-w}y^k \, dy &= C_{1,1}^{p/q}C_{1,2}\|f\|^p_{L^p(x^k(1-x)^{-w}\, dx)}
R_{1 1 2}^{2}= R_{1 1 3}^{3}= \frac{1}{g^{2}}\left( \frac{h^{\prime \prime}}{h}- \frac{h^{\prime}}{h}\frac{g^{\prime}}{g}\right) \quad ; \quad R_{2 3 2}^{3}= \frac{1}{h^{2}}\left( 1 - \left( \frac{h^{\prime}}{g}\right)^{2}\right)
\frac{f^{-1}\big(\frac{y}{n^2}\big)}{f^{-1}\big(\frac{y}{(n+1)^2}\big)}= \frac{2\frac{y}{n^2}\big(1 + O\big(\frac{1}{n}\big)\big)}{2\frac{y}{(n+1)^2}\big(1 + O\big(\frac{1}{n}\big)\big)}= 1 + O\left(\frac{1}{n}\right).
\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) &\geq \bar{G}\int \Pi_{j=1}^{m}\phi_1\left( x^j\right) \int \phi_m(w) \, Q(x,w) dP_0^m(w) dP_0^m (x).
\left ( \begin{array}{c}(\mathcal{F}f_{1})(\xi) \vdots (\mathcal{F}f_{N})(\xi) \end{array}\right ) = \mathcal{A}(z) \left ( \begin{array}{c}(\mathcal{F}f_{1})(\frac{\xi}{2}) \vdots (\mathcal{F}f_{N})(\frac{\xi}{2}) \end{array}\right ) .
\frac{\partial w^{\varepsilon}}{\partial t}- L^{\varepsilon}w^{\varepsilon}=\frac{\partial u^0}{\partial t}(x^\varepsilon,t) - \Theta \cdot \nabla \nabla u^0(x^\varepsilon,t) \ + \ \phi^\varepsilon(x^\varepsilon,t) =\phi^\varepsilon(x^\varepsilon,t), w^{\varepsilon}(x,0) = \varphi(x) + \psi^\varepsilon(x)
I_{\nu}\left({\nu z}\right) =\frac{c_{1}\left( \nu \right)}{\left({1+z^{2}}\right)^{1/4}}\exp \left\{{\sum\limits_{s=1}^{n-1}{\frac{\tilde{{E}}_{s}\left( p\right) -k_{s}}{\nu^{s}}}}\right\}\left\{{e^{\nu \xi}+\varepsilon_{n,1}\left({\nu ,\xi}\right)}\right\},
\pi(\theta | \mathbf{y})=\frac{f(\mathbf{y}| \boldsymbol{\theta})\pi(\boldsymbol{\theta}) n^{-p/2}| I(\boldsymbol{\theta}^{*})|^{1/2}}{f(y | \boldsymbol{\theta}^{*})\pi(\boldsymbol{\theta}^{*})(2 \pi)^{p/2}}\{1+O_{p}(n^{-1})\},
\frac{dT_2}{dt}=\frac{d}{dt_2}\bigg[\frac{1}{\alpha_2}((\frac{t_2}{\beta_2})^{\nu_2}- (\frac{\beta_2}{t_2})^{\nu_2})\bigg] &= \frac{1}{\alpha_2}\frac{d}{dt_2}\bigg[(\frac{t_2}{\beta_2})^{\nu_2}- (\frac{\beta_2}{t_2})^{\nu_2}\bigg] &=\frac{\nu_2}{\alpha_2\beta_2}\bigg[(\frac{t_2}{\beta_2})^{\nu_2-1}-(\frac{\beta_2}{t_2})^{\nu_2+1}\bigg].
\sum_{i=1}^dE\left[\frac{\partial\varphi}{\partial x_i}(\sqrt{1-t}F+\sqrt{t}Z)\frac{Z_i}{\sqrt{t}}\right]=\sum_{i,j=1}^dE\left[\frac{\partial^2\varphi}{\partial x_i\partial x_j}(\sqrt{1-t}F+\sqrt{t}Z)\mathfrak{C}(i,j)\right].