harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[\nabla_t,D^k]\nabla_t\psi =\sum_{l=0}^{k-1}D^l(R^{SM}(\partial_t,.))\star D^{k-1-l}\nabla_t\psi+\sum_{l=0}^{k-1}D^{k-l}\dot{g}\star D^{l+1}\nabla_t\psi &\quad+\sum_{\sum l_i+\sum{m_j}=k-1}{}^{G}\nabla^{l_1}R^P\star\underbrace{D^{m_1+1}\phi\star\ldots\star D^{m_{l_1}+1}\phi}_{l_1-}\star D^{l_2}\nabla_t\phi\star D^{l_3+1}\phi\star D^{l_4}\nabla_t\psi
\left\langle \left\langle g^{4}F_{\mu_{1}\nu_{1}}^{a_{1}}F_{\mu_{1}\nu_{1}}^{a_{1}}F_{\mu_{2}\nu_{2}}^{a_{2}}F_{\mu_{2}\nu_{2}}^{a_{2}}\right\rangle \right\rangle = \mathcal{N}\left( \left\langle g^{2}F F \right\rangle \right)^{2},
\lim_{x\to +\infty}\Bigl| \Phi^{\emph{r}}_{\xi}(x;g)-\sum_{\emph{w}\in W}C^{\emph{r}}(\emph{w}\xi;g) e^{\langle \emph{w}\xi, x\rangle}\Bigr| =0\quad\quad\emph{Re}(\xi)=0.
\tilde{f}_{ijk}^{\alpha a}=\sum_{\tiny\begin{matrix}\beta+\gamma=\alpha, b + c = a |\beta|+|b|,|\gamma|+|c|<|\alpha|+|a| \end{matrix}}C_{\alpha}^\beta C_a^b\nabla_k(\nabla_iV^{(\beta,b)}\nabla_jV^{(\gamma,c)}-\nabla_iH^{(\beta,b)}\cdot \nabla_jH^{(\gamma,c)}),
m(s,z) = - k - \int_0^1 V(s+t) e^{-2kt}\,dt + \frac 1k \int_0^1 \int_0^{t_1}e^{-2kt_1}(1 -e^{-2kt_2}) V(s+t_1) V(s+t_2)\,dt_2\,dt_1 + O(\lvert k\rvert^{-2})
T_f = \left (\frac{1}{h^2}\right )^{x/(1-qy)}\left ( \frac{\sqrt{m_{Dp}}m^{d_\perp-q-1}}{aB V_p}\right )^{y/(1-qy)}\sim\left (\frac{1}{V_p}\right)^{\frac{x+y/2}{1-qy}}
\int_{0}^{T}\int_{\Omega}\int_{0}^{1}\int_{Y^{\ast}}-u_{1}(x,t,y,s)v_{1}(x)v_{2}(y)c_{1}(t)\partial_{s}c_{2}(s) +A(y,s)(\nabla u(x,t)+\nabla_{y}u_{1}(x,t,y,s)) \cdot v_{1}(x)\nabla_{y}v_{2}\left( y\right) c_{1}(t)c_{2}\left( s\right)dydsdxdt=0
|L(s,i_0,\dots,i_{d},\ell_0, \ell_1, \ldots, \ell_s)|\le \prod_{j=0}^s \left( |E_{i_{\ell_j}}|\prod_{t=\ell_j+1}^{\ell_{j+1}-1}\min\{|E_{i_t}|, q^j\}\right),
(1\otimes\tau)(E_{P_1}{\otimes}E_{P_2})((1\otimes\xi)(\alpha\otimes\beta)(1\otimes\zeta)) =E_{P_1}(\alpha)\tau(\beta\zeta\xi) &=E_{P_1}(\alpha)\tau(\beta E_{P_2}(\zeta\xi)) &=E_{P_1}((\alpha\otimes\beta)E_{P_2}(\zeta\xi)).
C_{1}\left[ J_{\lambda}\left( \frac{\sqrt{4 \alpha c_{\epsilon}}}{\epsilon x^{\frac{\epsilon}{2}}}\right) + \frac{\sqrt{4 \alpha c_{\epsilon}}}{x^{\frac{\epsilon}{2}}}J_{\lambda}^{\prime}\left( \frac{\sqrt{4 \alpha c_{\epsilon}}}{\epsilon x^{\frac{\epsilon}{2}}}\right) \right]_{x = a}= 0 .
\hbar \dot{\rm C}_{\rm F}(t) = -i{\rm C}_{\rm F}(t)(\varepsilon_{{\rm F},+}(t) + \varepsilon_{{\rm F},-}(t)) - \hbar{\rm C}_{\rm F}(t)\langle{\rm F},A(t)|\frac{\partial}{\partial t}|{\rm F},A(t) \rangle.
\mathbf{A}=\mathbf{S}^{(3)}+ \mathbf{U}^{(3)}\left( \mathbf{S}^{(2)}+ \mathbf{U}^{(2)}\mathbf{K}^{(2)}\mathbf{V}^{(2)^\mathrm{T}}\right) \mathbf{V}^{(3)^\mathrm{T}}
\alpha_{-2}:= -\frac{2t}{3}a_1 - \frac{4t}{3}(a_2 - a_{-2}+ a_3 - a_{-3}) -\frac{1}{3}v_{(2,3)}+ \frac{8}{3}(a_2 \cdot v_{(1,3)}+ a_3 \cdot v_{(1,2)}) \beta_{-2}:= - \frac{4t}{3}(a_2 - a_{-2}+ a_3 - a_{-3}) + \frac{8}{3}(a_2 \cdot v_{(1,3)}- a_3 \cdot v_{(1,2)}).
\sum_{n=0}^{\infty}q^{n^{2}/2}\left(-t\right)^{n}S_{n}\left(xq^{n};q\right)=\left(tq^{1/2};q\right)_{\infty}\sum_{n=0}^{\infty}\frac{q^{5n^{2}/2}\left(xt\right)^{n}}{\left(q;q\right)_{n}\left(tq^{1/2},tq^{3/2};q^{2}\right)_{n}}.
\mathbb{E}^{0}(\varphi(\tau_{x}){\bf 1}_{\tau_{x>t}}L_{T}|\mathcal{F}_{t}^{Q})= \mathbb{E}^{0}[\varphi(\tau_{x}){\bf 1}_{\tau_{x>t}}] + \int_{0}^{t}\mathbb{E}^{0}[{\bf 1}_{\tau_{x>u}}L_{u}h(X_{u}) \varphi(\tau_{x-X_{u}})|\mathcal{F}_{u}^{Q}]dQ_{u}.
S_{v}p=[\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&\frac{1}{s}\end{matrix}][\begin{matrix}p_{x}\\p_{y}\\p_{z}\\ 1\end{matrix}]=[\begin{matrix}p_{x}\\p_{y}\\p_{z}\\ \frac{1}{s}\end{matrix}],
L_{s}= \frac{q^{2}}{1 + q^{2}}e^{- 1}( \dot{x}_{\mu})^{2}+ \frac{e}{1 + q^{2}}m^{2}+ b \dot{e}+ \frac{b^{2}}{1 + q^{2}}+ \dot{\bar{c}}\dot{c}.
\delta c(g_{1}, \ldots, g_{p+1}) = g_{1}.c(g_{2}, \ldots, g_{p+1}) + \sum_{i=1}^{p}(-1)^{i}c(g_{1}, \ldots, g_{i}g_{i+1}, \ldots, g_{p+1}) & + (-1)^{p+1}c(g_{1}, \ldots, g_{p}).
(\pi_{Q_S}\circ\sigma_V)(T) =\pi_{Q_S}(h\cdot \sigma(\tau_{h^{-1}}(T)))=\tau_h(\pi_{Q_S}(\sigma(\tau_{h^{-1}}(T)))) &=\tau_h(\tau_{h^{-1}}(T))=T,T\in\mathcal{U}_V,
(u+\epsilon)(x) (u+\epsilon)(y)(\varphi_\epsilon(x)-\varphi_\epsilon(y))^2 \leq \left[u(y)(v(x)-v(y))-v(y)(u(x)-u(y))\right](\varphi_\epsilon(x)-\varphi_\epsilon(y)) &+\epsilon(v(x)-v(y))(\varphi_\epsilon(x)-\varphi_\epsilon(y)),
\max_{1\le i\le m}\left\{\sum_{j\in E_{\mathrm{ss}}[i]}\alpha_{ij}^* + \sum_{k\in E_{\mathrm{sa}}[i]}\alpha_{ik}^* + \sum_{j\in E_{\mathrm{ss}}[i]}|\alpha_{ij}^*|\right\}&\quad\le \max_{1\le i\le m}v(i) = \sqrt{2f(U^{(0)},V^{(0)})}\max_{1\le i\le m}\sqrt{4|E_{\mathrm{ss}}[i]|+|E_{\mathrm{sa}}[i]|},
c ( x , \lambda ) = - \kappa x^{4}\lim_{y \rightarrow \infty}\lim_{y^{\prime}\rightarrow \infty}e^{4 \phi ( y^{\prime})}\partial_{y^{\prime}}\int_{- \lambda}^{y}d y_{1}\int_{y_{1}}^{\infty}d y_{2}\int_{- \lambda}^{y_{2}}d y_{3}F ( x , y_{1}, y_{2}, y_{3}, y^{\prime}, \lambda ) \, ,
\int_{{\bf p}}\frac{p_{\mu}}{\left(p^{2}+2p\cdot P-M^{2}\right)^{\alpha}}=\frac{-i\pi^{n/2}P_{\mu}}{(2\pi)^{d}\Gamma(\alpha)\left(-P^{2}-M^{2}\right)^{\alpha-n/2}}\Gamma\left(\alpha-\frac{n}{2}\right) \;\;,
\Big{(}\mathcal{D}_t(u-U_h)^{\frac12},v_h\Big{)}-\gamma(\nabla (\sigma-\Sigma_h)^{\frac12},\nabla v_h)+(\nabla (u-U_h)^{\frac12},\nabla v_h) &+(f^{\frac12}(u)-\frac12\mathfrak{F}(U_{h}^{1},u_{H}^{1})-\frac12 f(U_{h}^{0}),v_h)=0,
b'(p) = (1-\frac{1}{k})^{-\frac{p}{2k}}[ 1 + \frac{p-k(k+1)}{2k}\log(1-\frac{1}{k})^{-1}] = (1-\frac{1}{k})^{-\frac{p}{2k}}[ 1 + \frac{p-k(k+1)}{2k}(\log k - \log(k-1))] \geq (1-\frac{1}{k})^{-\frac{p}{2k}}[ 1 + \frac{2-k(k+1)}{2k}(\log k - \log(k-1))] \geq (1-\frac{1}{k})^{-\frac{p}{2k}}[ 1 + \frac{2-k(k+1)}{2k}\frac{1}{k-1}],
\frac{1}{2}{L_A}({\left|{\nabla u}\right|^2}) = Trace\left({A{\mkern 1mu}o{\mkern 1mu}hes{s^2}\left( u \right)}\right) + \left\langle{\nabla u,\nabla ({\Delta_A}u)}\right\rangle + Ri{c_A}(\nabla u,\nabla u)
\delta{\cal B}_{0 1}= - \frac{m}{\Delta_{m , n}^{1 / 2}}\, \frac{Q_{1}}{r^{6}}, \delta{\cal C}_{0 1}= - \frac{n}{\Delta_{m , n}^{1 / 2}}\, \frac{Q_{1}}{r^{6}}~ ~ .
\mathfrak{d}^{\rm (in)}_{n,d}= \mathfrak{d}^{\rm (in)}_{n-2,d+2}+ \mathfrak{d}^{\rm (out)}_{n-1,d-1}+ 2 \cdot \mathfrak{d}^{\rm (out)}_{n-2,d}+2\sum\limits_{i=0}^{n-2}\sum\limits_{j=0}^{d-2}\tilde{s}_{i,j}\cdot \mathfrak{d}^{\rm (in)}_{n-2-i,d-2-j}.
\ll X^{1/2}\left(\frac{3}{\lambda}\right)^{N}\int_{\substack{\vert s\vert>T \Re(s)=c}}\frac{\vert ds\vert}{\vert s\vert^{N+1}}\ll\frac{X^{1/2}}{N}\left(\frac{3}{\lambda T}\right)^{N}.
\begin{array}{r c l}{\varphi_{1}^{+}\varphi_{2}^{+}\varphi_{4}^{-}}{=}{\varphi_{4}^{-}\varphi_{1}^{+}\varphi_{2}^{+}+ D_{2 4}\varphi_{1}^{+}+ D_{1 4}\varphi_{2}^{+},}\end{array}
\begin{array}{l l}{T_{+ +}^{L}= ( T_{0 0}^{L}+ T_{0 1}^{L}) / 2}\\{T_{--}^{L}= ( T_{0 0}^{L}- T_{0 1}^{L}) / 2}\\{T_{+ -}^{L}= T_{- +}^{L}= 0}\\ \end{array}
P_t^\lambda (f)(x)-f(x_0) =\int_0^\infty P_t^\lambda (x,y)[f(y)-f(x_0)]dy+\left(\int_0^\infty P_t^\lambda (x,y)dy-1\right)f(x_0) &=:I_1(x,t)+I_2(x,t),x,t\in (0,\infty ).
\Omega = \frac{1}{\sqrt{2}}\left( \begin{array}{c c c c}{{1}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{1}}&{{- 1}}&{{0}}\\{{0}}&{{1}}&{{1}}&{{0}}\\{{- 1}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right)
\begin{aligned}\mathcal{L}V(x) = C\big[\beta(\beta+2-N)|x|^{-\beta-2}-\vartheta|x|^{-\beta-2s}\big] &\leq -C\vartheta|x|^{-\beta-2s}\leq -C\vartheta|x|^{-\alpha}\quad,\end{aligned}
\Phi'= \frac{(2\varepsilon(\hat{\bf p}))^{-1/2}}{(\varepsilon(\hat{\bf p})+mc^2)^{\frac{1}{2}}}\left((\varepsilon(\hat{\bf p})+mc^2)\Phi(\varepsilon(\hat{\bf p})+mc^2)+ c^2\vec\sigma{\bf\hat p}\Phi\vec\sigma{\bf\hat p}\right) \frac{(2\varepsilon(\hat{\bf p}))^{-1/2}}{(\varepsilon(\hat{\bf p})+mc^2)^{\frac{1}{2}}}\,;
\frac{\| U^1 \|_2^{N+\gamma-(N-2)p}\| \nabla U^1 \|_2^{Np-(N+\gamma)}}{\iint \frac{|U^1(x)|^p |U^1(y)|^p}{|x-y|^{N-\gamma}}dxdy}=C_{GN}^{-1}= \frac{\| Q \|_2^{N+\gamma-(N-2)p}\| \nabla Q \|_2^{Np-(N+\gamma)}}{\iint \frac{|Q(x)|^p |Q(y)|^p}{|x-y|^{N-\gamma}}dxdy}.
\phi(\{\sum_ia_ie_1\lambda_i\}^*) = \sum_i{\lambda_i}^*f{a_i}^*~~~~(\textrm{since}~\phi(e_1)= f~\textrm{and}~{\phi}|_M = id) = \{\sum_ia_i f\lambda_i\}^* = \{\phi(\sum_ia_i e_1 \lambda_i)\}^*.
-\frac{(84n^2 - 56kn + 108n + 4k^2 - 20k + 21) \bigl(\frac{1}{2}+ n-k\bigr)^2}{64n^2 (2n-k+2)(2n-k+1)}&\quad- \frac{(84n^2 - 56nk + 4k^2 + 52n - 12k + 5) \bigl(n+k - \frac{1}{2}\bigr) \bigl(2n-k+2)}{64n^2 (2n-k+1)(2n-k+2)}&= \frac{\bigl(\frac{1}{2}+ n \bigr) \bigl(n+k - \frac{1}{2}\bigr) \bigl(\frac{1}{2}+ n-k \bigr)^2}{16n^2 (2n-k+2)(2n-k+1)}-1.
J_{r}^{( n )}( t ; \{q \}; L ) \approx \frac{\hat{S}_{r}^{( n )}( \{q \}; L ) + \hat{\Sigma}_{r}^{( n )}( \{q \}; L ) \vartheta}{z - z_{0}}\equiv \frac{\hat{J}_{r}^{( n )}( \{q \}; L ; \vartheta )}{z - z_{0}}\, .
x = 1 / L , \, \, \, y = 1 / \gamma_{L}, \, \, \, \a = \frac{\Lambda}{M_{7}^{5}}, \, \, \, b = \frac{Q^{2}}{M_{7}^{5}}, \, \, \, \, d = \frac{2 \alpha^{\prime}}{M_{7}^{5}}.
ind(\overline{\partial}_A) = p_{1,B}( \mathfrak{g}_P )[X] + \frac{1}{2}dim(G)( 1 - b^1_B(M) + b^+_B(M) ) + \sum_j \frac{1}{m_j}\sum_{k=1}^{m_j-1}\frac{\chi_{\mathfrak{g}_P,j}(k) - dim(G)}{det(1 - g_k |_{T^*_{x_j}X})}
a_1= \frac{\left( 1+K \right)m\mu}{\left( m+\mu K \right)\overline{\gamma}}, a_2=\frac{\left( 1+K \right)m\mu}{\left( m+\mu K\left( 1+\Delta \right) \right)\overline{\gamma}}, a_3= \frac{\left( 1+K \right)m\mu}{\left( m+\mu K\left( 1-\Delta \right) \right)\overline{\gamma}}, a_4=\frac{\left( 1+K \right)\mu}{\overline{\gamma}}.
exp\left(\imath\,\vartheta_{kP}\left((\varphi^{-1}_{k}(x),z_k)\right)\right)=\,\exp\left(\left(-\frac{\imath x^\mu z_\mu}{\hbar}\right)\right), & n_{kP}(\varphi^{-1}_k(x),z_k)\propto \,\delta(z_k-P).
\mathcal{C}_{\mathrm{d}}^{1}(\mathbf{R}, 2(N+1)) & =\left\{\nu_{N}: \mathbf{R}\to \mathbf{R}_{\geq 0}\left| \, \nu_{N}\mathop{\mathrm{supp}}\nu_{N}, \ \int_{-\infty}^{\infty}\nu_{N}(x) \, \d x = 2(N+1)\right.\right\}
\begin{aligned}\lim_k \bigl( \|XJ_{\theta,1}^{-1}U_\mu^{n_k}p\|^2 -\|pXJ_{\theta,1}^{-1}\chi^{-n_k}\|^2\bigr)= 0 \ \ \lim_k \bigl(\|XJ_{\theta,1}^{-1}U_\mu^{-n_k}p\|^2-\|pXJ_{\theta,1}^{-1}\chi^{-n_k}\|^2\bigr)=0 \end{aligned}
\int_{-1}^{1}P_{n}^{(-2,\beta)}(x)P_{m}^{(-2,\beta)}(x)w_{-2,\beta}(x)dx=\frac{2^{\beta-1}\Gamma(n-1)\Gamma(n+\beta+1)}{n!(2n+\beta-1)\Gamma(n+\beta-1)}\delta_{n,m}\quad(n,m\geq2).
f_{W}(0,z) =1+\dfrac{a_{1}(0)}{z}+\dfrac{a_{2}(0)}{z^{2}}+\cdots=1+\varphi_{W}\left( z\right) f_{W}^{\prime}(0,z) =-z-a_{1}(0)+\dfrac{a_{1}^{\prime}(0)-a_{2}(0)}{z}+\cdots & =-z-\psi_{W}\left( z\right) -a_{1}(0)\left( 1+\varphi_{W}\left( z\right)\right)
f(\mathbf{u},\mathbf{v})\triangleq \mathbb{P}_{p^n_{W|U,V}}\Big((\mathbf{u},\mathbf{v},\mathbf{W})\notin\mathcal{A}_{\epsilon^{(1)}_{\alpha,\delta_1},\epsilon^{(2)}_{\alpha,\delta_2}}\Big|\mathbf{U}=\mathbf{u},\mathbf{V}=\mathbf{v}\Big),
\begin{array}{l}{[}U_{a}, U^{b}{]}= S_{a}^{b}\in g^{0}{[}U_{a}, U_{b}{]}= K_{ab}\in g^{+2}{[}U^{a}, U^{b}{]}= K^{ab}\in g^{-2}{[}S_{a}^{b}, U_{c}{]}= U_{(abc)}\in g^{+1}\end{array}
\sum_{\substack{q_1, \ldots, q_\ell q_i q_i < R}}\prod_{j=1}^\ell \frac{\log^{a_j}q_j}{q_j^{a_j}\log^{a_j}R}\ \ll \ \sum_{\substack{q_1, \ldots, q_\ell q_i < R}}\prod_{j=1}^\ell \frac{\log^{a_j}q_j}{q_j^{a_j}\log^{a_j}R}\ \ll \ \prod_{j=1}^\ell \left[ \sum_{q_j < R}\frac{\log^{a_j}q_j}{q_j^{a_j}\log^{a_j}R}\right].
u(0):=u_{0}=\dfrac{\dot{a}_{0}}{a_{0}}; \;v(0):=v_{0}=\dfrac{\dot{b}_{0}}{b_{0}};\, \rho(0)=\rho_{0}\; ;\psi(0):=\psi_{0}=\dot{\phi}_{0}; \; \phi(0)=\phi_{0}.
\underset{\bar{{\bf w}}\in \mathcal{W}}\min ~ \frac{1}{2}\| \bar{{\bf w}}- \bar{{\bf w}}' \|_2^2, \underset{{{\bf s}}\in \mathcal{S}}\min ~ \frac{1}{2}\|{{\bf s}}-{{\bf s}}' \|_2^2, \underset{{{\bf y}}\in \bigtriangleup_M}\min ~ \sum\limits_{m=1}^{M}{y}_m \log\frac{{y}_m}{{y}'_m}- \sum\limits_{m=1}^{M}({y}_m -{y}'_m)
{\bf \Omega}( x_{1}, \ldots , x_{4 i - 4}, x_{4 i - 3}^{+}, x_{4 i - 2}^{+}, x_{4 i - 1}^{+}, x_{4 i}^{+}, x_{4 i + 1}, \ldots , x_{N}) ={\bf \Omega}( x_{1}, \ldots , x_{N}) \, , \nonumber
\int_0^T \|W_l f_\phi\|^2_{H^{\frac{s}{4(4+s)}}_xL^2_v}dt\lesssim\eta^{-8}\|f|_{t=0}\|_{L^2}^2+\eta^{2s}\|f_\phi\|^2_{L^2([0,T]; L^2_xH^s_v)}+\eta^{-8}(\|g\|_{L^2([0,T]; L^2_xH^{-1}_v)}^2+\|f\|_{L^2([0,T]; L^2)}^2).
\Omega = \sqrt{\frac{Q^{2}}{4 \pi \epsilon_{0}m r_{c}^{2}}}={\frac{e \pi Q^{2}}{4 \pi \epsilon_{0}J_{1}^{3}}}{\frac{1}{{\bar{\sigma}}^{2}}}={\frac{\pi m Q^{4}}{2 \epsilon_{0}^{2}J_{1}^{3}}}.
S_{\mathrm{off}}^{c<p^2}:=\sum_{d_1,\,d_2<D}\,\, \,\ \sum_{Id_1, \,Jd_2<D}x_{Id_1}\bar{x}_{Jd_2}\sqrt{IJ}\sum_{L, M\geq 1}\frac{G_k(LMd_1d_2\slash p)}{\sqrt{LM}}\times\left(2 \pi i^{2k}\sum_{\substack{p|c c <p^2}}\frac{\mathcal{S}(IL, JM, c )}{c}\,J_{2k-1}\left(\frac{4\pi \sqrt{ILJM}}{c}\right)\right)
\left(X_{b_{i}},X_{s_{i}},X_{b_{j}}\right)\left(\tilde{\omega}\right) =\left(\omega_{i},X_{s_{i}}\left(\omega\right),\omega_{j}\right)\quad \left(X_{b_{i}},X_{s_{i}},X_{b_{j}}\right)\left(\tilde{\omega}^{\prime}\right) =\left(\omega_{i},X_{s_{i}}\left(\omega^{\prime}\right),\omega_{j}\right).
H(u_1|u_2):=\int_\Omega \left\{\bigg[f_1\ln\Big(\frac{f_1}{f_2}\Big)-(f_1-f_2)\bigg]+ \cfrac{b}{c}\bigg[ g_1 \ln\Big(\frac{g_1}{g_2}\Big) - (g_1-g_2) \bigg] \right\}\,\mathrm{d}x\,,
f_Y(y) = \frac{k}{l}\, \left( \frac{e^{y}}{l}\right)^{k-1}\, e^{-\left( \frac{e^{y}}{l}\right)^{k}}\, \left\lvert e^{y}\right \rvert & = \frac{k}{l^k}\, e^{yk}\, e^{- \frac{1}{l^k}e^{yk}}& = b \, q \, e^{b y}\, e^{-q \, e^{b y}}
\left.\frac{\mathrm{d}C(0,\tau_2,1-\tau_2)}{\mathrm{d}\tau_2}\right|_{\tau_2=\tau^\prime}=0, \frac{\mathrm{d}^2C(0,\tau_2,1-\tau_2)}{\mathrm{d}\tau_2^2}\geq 0, \quad\forall \tau_2\in[0,1],
S_{\mathrm{free}}\;=\; S^{(\xi)}_{\mathrm{free}}+S^{(\psi)}_{\mathrm{free}}\, , S_{\mathrm{int}}\;=\; S^{(\xi)}_{\mathrm{int}}+S^{(\psi)}_{\mathrm{int}}+Q_{\mathrm{int}}^{(\psi,\xi)}\, .
{\mathcal{K}}( \hat{M}_{f}, \hat{M}_{b}) = \mathrm{c o n s t}\int{\mathcal{D}}{\mathcal{F}}{\mathcal{D}}{\mathcal{B}}\exp \left\{-{\mathcal{W_{F}}}-{\mathcal{W_{B}}}-{\mathcal{W_{F B}}}\right\}
F_{s,k}(p):= \sum_{m=1}^{s-1}c_m |p|^{2m}\sum_{\substack{n_0+\ldots+n_m=s n_i\ge1}}\{\{\ldots\{\{f_{n_0}+f_{n_0}^*,G^0_{n_1}\},G^0_{n_2}\}\ldots\},G^0_{n_m}\}\,;
(a-b)(h(a)-h(b)) =(a-b)\int_{b}^{a}h'(\tau)d\tau &=(a-b)\int_{b}^{a}(H'(\tau))^2d\tau &\geq\left(\int_{b}^{a}H'(\tau)d\tau\right)^2 ~ &=[H(a)-H(b)]^2
\frac{\partial^{a_1}}{\partial s_{1}^{a_1}}\frac{\partial^{a_3}}{\partial s_{2}^{a_3}}\Theta(s_1+s_1+1,h) = \frac{d^{a_1+a_3}}{dz^{a_1+a_3}}\Theta(z,h) \Big|_{z=s_1+s_2+1}.
\frac{d}{dt}y_{j}(t) = -2\pi \Im \Psi_{D_t}(z_{j}(t),\xi(t)), \frac{d}{dt}x_{j}(t) = -2\pi \Re \Psi_{D_t}(z_{j}(t),\xi(t)), \frac{d}{dt}x^{r}_{j}(t) = -2\pi \Re \Psi_{D_t}(z^{r}_{j}(t),\xi(t)).
\begin{array}{l}\frac{\Theta'}{2 \Theta}U_0 + \frac{\partial U_0}{\partial r}= 0 4 i r(x,y) \{(\frac{k+1}{r(x,y)}+ \frac{\Theta'}{2 \Theta})U_{k+1}+ \frac{\partial U_{k + 1}}{\partial r}\}= \Delta_y WU_k.\end{array}
P\{m_n = 1 \, | \, \mathcal{F}_{n-1}\}= P\{m_n = 1 \, | \, X_{n-1}\}=P\{3 X_{n-1}+ \, \xi_n \equiv 2\ \; 4 \, | \, X_{n-1}\}= \frac{1}{2}
\Sigma_{n}(E) = \frac{\eta}{\pi\hbar}\sum_{m}|\langle m|q|n\rangle|^{2}\int_{0}^{\Lambda}\frac{(E^{\prime}- E_{m})(\frac{E^{\prime}- E_{m}}{\hbar})^{2}\theta(E^{\prime}- E_{m})dE^{\prime}}{E^{\prime}- E -i\epsilon}
\lim_{s \to 0}\| \nabla [y_j (s) - b] \|_{L^p(\Omega)^{3 \times 3}}\leq \lim_{s \to 0}\| \nabla [y_j (s) - y_0 (s)] \|_{L^p(\Omega)^{3 \times 3}}+ \lim_{s \to 0}\| B^{\frac{1}{2}}[e^{- s B}b - b] \|_{L^p (\Omega)^{3 \times 3}}= 0.
(\mathbf{J}_\nu(\, \underline{u}\,(\underline{v}+\underline{w})^\intercal ))_{kn}= \sum_{s=-\infty}^\infty\sum_{t=0}^\infty \frac{(-1)^t2^{-2t-s}}{t!(t+s)!}J_{\nu-s}( u_kv_n ) (u_kw_n)^{2t+s},
\tilde{A}_{1}= C_{1},\ \tilde{A}_{2}= C_{2}+ \frac{z(c_{0})^{2}}{1- t_{2}}E, \tilde{B}= ( -t_{1}+ c +\alpha z) C_{1}+ (1- t_{2}) C_{2}+ z (c_{0})^{2}E.
g_t(\hat{x}_1+z,\hat{x}_1+z+t^{-1/d}(\hat{x}_2+y_2-\hat{x}_1),\hdots,\hat{x}_1+z+t^{-1/d}(\hat{x}_{l+1}+y_{l+1}-\hat{x}_1)) & =g_1(\hat{x}_1,\hat{x}_2+y_2,\hdots,\hat{x}_{l+1}+y_{l+1}),
t^{1-\alpha}\eta_{2xt}-\frac{k+c}{x}\eta_{2xx}+\frac{k+c}{x^2}\eta_{2x}-\frac14m(k+1)x^k(-(1-\alpha)t^{-1-\alpha}\tau &+(1-\alpha)t^{-\alpha}\tau_t-t^{1-\alpha}\tau_{tt})+\frac12mkx^{k-1}t^{1-\alpha}\sigma_{1t}-\frac12m(k+c)(k-2)x^{k-3}\sigma_1=0.
\tilde{\Theta}UA\varphi =\tilde{\Theta}US(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\varphi &=-\tilde{\Theta}SU(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\varphi &=\tilde{\Theta}S(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)U\varphi=-A\tilde{\Theta}U\varphi,
m_{p}= \sqrt{\alpha_{G}}\sim 8 . 1 \times 1 0^{- 2 3}\approx 1 . 7 6 \times 1 0^{- 3 0}\, \mathrm{k g}\approx 0 . 0 0 1 0 5 \, m_{p \ \mathrm{e x p}}\, .
J_{g}(u)=\frac{1}{m}\Vert u\Vert_{g,a,b}^{2}\leq\liminf_{k\rightarrow\infty}\frac{1}{m}\Vert u_{k}\Vert_{g,a,b}^{2}=\lim_{k\rightarrow\infty}J_{g}(u_{k})=\tau.
\left. \frac{d}{d\tau}\right|_{\tau=0}x^{\alpha}(g_{A}(\tau)g)=\left.\frac{\partial x^{\alpha}(g_{A}(\tau)g)}{\partial x^{M}(g_{A}(\tau))}\right|_{g_{A}=e}\cdot\left.\frac{d x^{M}(g_{A}(\tau))}{d \tau}\right|_{\tau=0}=R^{\alpha}_{M}(g)\delta^{M}_{A}=R^{\alpha}_{A}(g).
\sum \limits_{j=1, j \neq i}^{m-n}sec(U_i, U_j) = \sum \limits_{j=1, j \neq i}^{m-n}\hat{s}ec(U_i, U_j) - \lambda^2 (m-n-1) \| \nabla (f + \log \lambda)\|^2 &+ \frac{\lambda^4}{4}(m-n-1) \| grad_{\mathcal{H}}\frac{1}{\lambda^2}\|^2 -\lambda^2 (m-n-1) g\left(\nabla f, \nabla_{\mathcal{H}}\frac{1}{\lambda^2}\right),
g_{m-1+m\tau}(x)=\, (x-1)\sum_{\substack{0\le i_0\le m-1\cr 0\le i_1\le m}}(-1)^{i_0+i_1}\binom{m-1}{i_0}\binom m{i_1}\frac{x^{r(i_0+i_1)}-1}{x^r-1}\cr \cdot\Bigl[\sum_{l=0}^{r-1}\binom{i_0+m-1+(l-m)k}{m-1}\binom{i_1+m-lk}mx^l\cr -\sum_{l=0}^{m-1}\binom{i_0+m-1+(l-m)k}{m-1}\binom{i_1+m-lk}{m-1}x^l\Bigr]-1.
v: = \sum_{n \in \Lambda}f_M(y_n)e_n + \sum_{n \in \mathbb{N}\setminus \Lambda}a_ne_n &= \sum_{n=1}^\infty f_M(y_n) e_n +\sum_{n \in B\setminus \Lambda}(a_n - f_M(y_n))e_n
n(2\ln n)^{\frac{1}{2}}\int_ID_n(\sqrt{4-y^2}/S(I) \cdot G_n(x)/2)dy = n(2\ln n)^{\frac{1}{2}}\int_{-I}D_n(\sqrt{4-y^2}/S(-I) \cdot G_n(x)/2)dy \to M(-I)e^{c_0-x}=M(I)e^{c_0-x},
\bold{r}_{l_0}=\left(\begin{array}{cccccccccccccccc}x_{1}x_{2}\cdots \cdots x_{k+m}x_{k+m+1}x_{k+m+2}\cdots x_{1}x_{2}\cdots \cdots x_{k+m}ax_{k}x_{k+m+1}\cdots \end{array}\right. \left.\begin{array}{ccccc}\cdots x_{r+1}x_{r+2}\cdots x_n \cdots x_{r}x_1 \cdots x_1\end{array}\right),
H^*(X_{D}) =\mathbb{Z}[x_1,x_2,x_3]/\langle x_1^{m_1}, x_2^{m_2}, x_3(-x_1+x_2+x_3)(x_2+x_3)^{m_3-2}\rangle, H^*(X_{\widetilde{D}}) =\mathbb{Z}[y_1,y_2,y_3]/\langle y_1^{\widetilde{m}_1}, y_2^{\widetilde{m}_2}, y_3(-y_1+y_2+y_3)(y_2+y_3)^{\widetilde{m}_3-2}\rangle.
\Delta = \frac{1}{2}\Delta_{h}'' (r-r_{h})^{2}+\frac{1}{6}\Delta_{h}'''(r-r_{h})^{3}+\frac{1}{24}\Delta_{h}^{(iv)}(r-r_{h})^{4}+\frac{1}{120}\Delta_{h}^{(v)}(r-r_{h})^{5}+O(r-r_{h})^{6}.
S\rho_1^{\gamma_0}(T)\rho_1(\rho_2(A))=S\rho_1^{\gamma_0}(T\rho_2(A))=S\rho_1^{\gamma_0}(\rho_2'(A)T)=S\rho_1(\rho_2'(A))\rho_1^{\gamma_0}(T)=\rho_1'(\rho_2'(A)) S\rho_1^{\gamma_0}(T).
E_b(W_n(w)) =\{v\in W_n(w)\mid \lvert v\rvert_S>n_b(W_n(w))\}&=\{v\in W_n(w)\mid \lvert v\rvert_S>\max \{\lvert v\rvert_S\mid v\in W_n(w), \tau_b(v)\neq v\}\}.
(N_i)_{v,v'}=\frac{d_vd_{v'}}{r_v}\epsilon_v\bullet\epsilon_{v'}=\begin{cases}d_vd_{v'}~~ ~~v\textup{a child of}v'\textup{in}\mathcal{T}_i, m_{v'}d_vd_{v'}~~ ~~v\textup{parent of}v'\textup{in}\mathcal{T}_i, -d_vd_{P(v)}-\sum_{w\textup{child of}v\textup{in}\mathcal{T}_i}m_wd_vd_w~~ ~~v=v', 0~~ ~~\textup{otherwise},\end{cases}
M'_{\varphi_R}(t) = -\int_0^\infty m^s \int \Delta^2 \varphi_R |u_m(t)|^2 dx dm + 4\sum_{j,k=1}^d \int_0^\infty m^s \int \partial^2_{jk}\varphi_R \partial_j \overline{u}_m(t) \partial_k u_m(t) dx dm &\mathrel{\phantom{= -\int_0^\infty m^s \int \Delta^2 \varphi_R |u_m(t)|^2 dx dm}}- \frac{2\alpha}{\alpha+2}\int \Delta \varphi_R |u(t)|^{\alpha+2}dx,
\ell_{j}\left( s\right) \left( \mathsf{S}_{j}\left( s\right)\varphi,w\right) =\left\langle \left( \gamma_{\operatorname*{D};j}\left(s\right) \right)^{\prime}\varphi,\overline{w}\right\rangle_{\mathbb{R}^{3}}\quad\forall w\in H^{1}\left( \mathbb{R}^{3}\right) ,
\mathrm{SINR}=\frac{h_{\mathrm{11}}X_{\mathrm{11}}^{-\alpha}}{\sum_{i\in\Phi_{b}^{'}}h_{i}D_{i}^{-\alpha}\mathbf{1}\left(D_{i}\geq\max(X_{11},X_{11}^{\frac{\mu}{2}}\left(\frac{P}{N}\right)^{\frac{2-\mu}{2\alpha}}\left(\frac{1}{M}\right)^{\frac{1}{2\alpha}})\right)+\frac{N}{P}},
\int_D \hat{u}m \,dx = \tau \int_D \hat{u}^2\,dx + (1-\tau) \int_D \int_{\underline\alpha}^{\overline\alpha}u^2 \,d\alpha dx &\geq \left( \tau + \frac{1-\tau}{\overline\alpha - \underline\alpha}\right) \int_D \hat{u}^2\,dx &\geq c_0 \int_D \hat{u}^2\,dx &\geq \frac{c_0}{|D|}\left(\int_D \hat{u}\,dx\right)^2 = \frac{c_0}{|D|}\|u\|_{L^1({\Omega})}^2,
Q_{F_1,V_8}^{Maxwell}= \int_{V_8}*_s j_{D2}^{Maxwell}= \frac{T_{F1}}{2}\left( 1 - \cos \theta_0 \ \{+ F_2 \{{\rm flux \above \brane}}\atop{-F_2 \{\rm flux \below \brane}}}\right),
\left( \Omega_{\alpha \beta}^{g_{1}g_{2}, g_{3}}\right) \, \left( g_{3}^{*}\Omega_{\alpha \beta}^{g_{1}, g_{2}}\right) \: = \: \left( \Omega_{\alpha \beta}^{g_{1}, g_{2}g_{3}}\right) \, \left( \Omega_{\alpha \beta}^{g_{2}, g_{3}}\right) \, \left( \omega_{\alpha}^{g_{1}, g_{2}, g_{3}}\right) \, \left( \omega_{\beta}^{g_{1}, g_{2}, g_{3}}\right)^{- 1}
\begin{gathered}\int^{\tau}_0 \int^{t_n}_0 \ldots \int^{t_2}_0 a(t_1,\ldots,t_n) (dw(t_1)-f(w(t_1),t_1,t_n)dt_1)\ldots \ldots (dw(t_{n-1})-f(w(t_{n-1}),t_{n-1},t_n)dt_{n-1})dw(t_n),\end{gathered}
\Pr(Y = k) = \binom{n}{k}q^k(1-q)^{n-k}=(1+o(1)) \frac{1}{\sqrt{2 \pi}}\sqrt{\frac{n}{k(n-k)}}\left(\frac{n q}{k}\right)^k \left(\frac{n(1-q)}{n-k}\right)^{n-k}.
\gamma_F(ab^2, \psi) = \gamma_F(a, \psi), \gamma_F(a b, \psi) = \gamma_F(a, \psi) \cdot \gamma_F(b, \psi) \cdot (a,b)_F, \gamma_F(a, b \psi) = \gamma_F(a,\psi) \cdot (a,b)_F, \gamma_F(a, \psi)^2 = (-1, a)_F, \gamma_F(a, \psi)^4 = 1, \gamma_F(\psi)^2 = \gamma_F(-1, \psi)^{-1}, \gamma_F(\psi)^8 = 1
\frac{d p^{\pm}}{dt}-F(t,p^{\pm})=(0,0,\dots,0),\ \ \frac{d \varphi^{\pm}}{d t}-D F(t,p^{\pm})\varphi^{\pm}=\lambda_{\pm}\varphi^{\pm}, &p^{\pm}(\cdot+T)=p^{\pm}(\cdot),\ \varphi^{\pm}(\cdot+T)=\varphi^{\pm}(\cdot).
(-\Delta)^s R_{2n}^\lambda(x) =a_n^\lambda\, \sum_{k=0}^{n}\frac{(-n)_{k}(n+\lambda)_{k}}{(\lambda+\frac{1}{2})_{k}k!}(-\Delta)^s \bigg\{\frac{1}{\left(1+x^2\right)^{k+\frac{\lambda+1}{2}}}\bigg\},
-\frac{1}{4\pi i}\int^{i\infty}_{-i\infty}\mathrm{Re}\Big[\frac{\Gamma'}{\Gamma}\Big(\frac{s}{2}\Big)\Big]\hat{g}(s)ds=-\frac14\hat{g}(0)-\frac{1}{4\pi i}\int_{(\frac12)}\mathrm{Re}\Big[\frac{\Gamma'}{\Gamma}\Big(\frac{s}{2}\Big)\Big]\hat{g}(s)ds.
i\hbar\frac{\partial\psi_{c}(t,R)}{\partial t}=(\frac{\hbar^{2}}{2M}\nabla^{2}-G\int d^{3}R^{\prime}\int d^{3}y\int d^{3}z\frac{|\psi_{c}(t,R^{\prime})|^{2}\rho_{c}(y)\rho_{c}(z)}{|R-R^{\prime}-y+z|})\psi_{c}(t,R),
\mathcal L_{\rm mass}=\frac{1}{2}\left(\psi^1_{\mu L}\ \psi^2_{\mu L}\right)\sigma^{\mu\nu}\left(\begin{array}{cc}\omega/R i\,p_5 -i\,p_5 \omega/R\end{array}\right)\left(\begin{array}{c}\psi^1_{\nu L}\psi^2_{\nu L}\end{array}\right)+h.c.
K_{m}^{\nu}(z,w)=\frac{\nu}{\pi}\left( \sum\limits_{k=0}^{m-1}\frac{(\nu z \overline{w})^{k}}{k!}+\frac{(z \overline{w})^{m}}{\Gamma^{2}(m+1)}\,{_2F_2}\left(\begin{array}{c}1, 1 m+1,m+1 \end{array}\bigg | \nu z \overline{w}\right)\right) .
\|T (f_1, \dots, f_m)\|_{L^p(w)}\le \zeta^{-1}(1-r)^{-(m+1)}[\vec{w}]_{A_{\vec{p},\vec{r}}}^{\frac1{1-r}}\prod_{i=1}^{m+1}\|f_i\|_{L^{p_i}(w_i)},
\langle v,w\rangle_{L_g}=\int_0^\infty g(s)\langle \partial_{x}^{k}v(s),\partial_{x}^{k}w(s)\rangle ds \mbox{\ \and\ \}\|v(s)\|_{L_g}=\left(\int_0^\infty g(s)\|\partial_{x}^{k}v(s)\|^2ds\right)^\frac{1}{2},
B_t(P(\cdot))(s):=-\int_t^s \nabla_x^2 V_\rho(\tau,q(\tau,t))\left[\int_t^\tau A(p(\sigma,t))P(\sigma) d\sigma\right] d\tau, &R_0(s):=-\int_t^s \nabla_x^2 V_\rho(\tau,q(\tau,t))d\tau.
p^{-1}(p(\Gamma g))\cap \Omega =\{\Gamma c_1 g,\ldots, \Gamma c_n g\}, & p^{-1}(p(\Gamma c_i^{-1}g))\cap \Omega =\{\Gamma c_1c_i^{-1}g,\ldots, \Gamma c_nc_i^{-1}g\}, & p^{-1}(p(\Gamma c_i g))\cap \Omega =\{\Gamma c_1 c_i g,\ldots, \Gamma c_n c_i g\}.
\mathcal{K}=\frac{1}{2\pi i}\oint_{|z|=1}\big( \log{h(z;\boldsymbol{\xi})}\big)\big( \log{h(z;\boldsymbol{\xi})}\big)^{*}\frac{dz}{z}=\|\log h(z;\boldsymbol{\xi})\|^2_{H^2}.
\begin{aligned}-16L\mathcal Q(t) = ~{}2(9+\tilde \sigma) \int |\varphi'| f^2 + (-3+ 5\tilde\sigma) \int |\varphi'| f_x^2 & ~{}+4 (-1 + \tilde\sigma) \int |\varphi'| f_{xx}^2 + (1+\tilde \sigma)\int |\varphi'| f_{xxx}^2 .\end{aligned}