Datasets:
prithivMLmods
/
Latex-KIE

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\begin{align*} 1=&K_2e_\lambda(I-R(\lambda,A)Le_\lambda)^{-1}e^{-\int_0^\cdot(\lambda+\mu(s))ds}\\=&K_2e_\lambda f\\=&\int_0^{+\infty}\beta_2(a)e^{-\lambda r}e^{-\int_0^a(\lambda+\mu(s)+e^{-\lambda r}\alpha(s))ds}da.\end{align*}
\begin{align*} f(a)+e^{-\int_0^a(\lambda+\mu(s))ds}\int_0^a\alpha(s)e^{-\lambda r}e^{\lambda s+\int_0^s\mu(\sigma)d\sigma}f(s)ds=g(a), \ a\geq 0,\end{align*}
\begin{align*} &R(\lambda,\mathcal {A}_{L,K_i,0})\\=&\left( \begin{array}{cc} R(\lambda,\mathfrak{A})+e_\lambda W_1 & e_\lambda W_2 \\ W_5& W_6 \\ \end{array}\right)\end{align*}
\begin{align*}f(x)= F'(x) g(F(x))\end{align*}
\begin{align*} (Du(x),Dv(y)) = (D_x s(x,y), D_y s(x,y))\end{align*}
\begin{align*} &- \int_{\Omega_{FEM}} \frac{\partial{\lambda_h}}{\partial t}(0) \bar{u}(x,0) ~dx - \int_{\Omega_{FEM_T}} \frac{\partial \lambda_h}{\partial t} \frac{\partial \bar{u}}{\partial t}~ dxdt \\&- \int_{\partial \Omega_{FEM}} \partial_n \lambda_h \bar{u} ~ dxdt + \int_{\Omega_{FEM_T}} ( a_h \nabla \lambda_h) (\nabla \bar{u}) ~ dxdt = 0.\end{align*}
\begin{align*}D_y s(x,y) = Dv(y).\end{align*}
\begin{align*}F(x) = D_y s(x,\bar y)\end{align*}
\begin{align*}u(x) := \sup_{y \in Y} s(x,y) - v(y)\end{align*}
\begin{align*}s\left( x,y\right) =\alpha \left( x \right) +\sigma\left( I\left( x\right) ,y\right).\end{align*}
\begin{align*}D\Phi_\epsilon(y,k) = \int_X (f_y,0) \phi_\epsilon(s_y-k) d{\mathcal H}^m +\frac{1}{\epsilon} \int_0^\epsilon dt \int_{X(y,k+t)} f \frac{(-s_{yy},1)}{|D_x s_y|} d{\mathcal H}^{m-1}.\end{align*}
\begin{align*}\int_{X(y,k)} V \cdot \hat n_{X_=} d{\mathcal H}^{m-1} = \int_{X_\le(y,k)} \nabla_X \cdot V d{\mathcal H}^m - \int_{\partial^* X \cap \overline{X_\le(y,k)}} V \cdot \hat n_X d{\mathcal H}^{m-1}\end{align*}
\begin{align*}\left|\int_{X(y,k)} V \cdot \hat n_{X_=} d{\mathcal H}^{m-1} \right|\le \|V\|_{W^{1,1}(X)} + \|V\|_{L^\infty(X)} {\mathcal H}^{m-1}(\partial^* X).\end{align*}
\begin{align*}h(y,k):=\mu[X_\le(y,k)]-\nu[(-\infty,y)], \end{align*}
\begin{align*}\int_{X_\le (y, s_y(x,y))} d\mu = \int_{-\infty}^y d\nu.\end{align*}
\begin{align*}V^\pm(x,y) := \mp \frac{(s_{yy})^{(1\pm 1)/2}}{|D_x s_y|} \hat n_{X_=}, \hat n_{X_=} (x,y):= \frac{D_x s_y}{|D_x s_y|},\end{align*}
\begin{align*}\left\Vert \bullet \right\Vert _{V_{h}}:=\left\Vert \bullet \right\Vert_{U^{0}},\end{align*}
\begin{align*}V^\pm_{\partial X}(x,y) := \frac{V^\pm \cdot \hat n_{X_=}}{\sqrt{1-(\hat n_X \cdot \hat n_{X_=})^2}} \hat n_\partial, \hat n_\partial(x,y) :=\frac{\hat n_{X_=} - (\hat n_{X_=} \cdot \hat n_X) \hat n_X}{\sqrt{1 - (\hat n_X \cdot \hat n_{X_=})^2}},\end{align*}
\begin{align*}\Phi(y,k) := \int_{X(y,k)} a(x,y) \cdot \hat n_{X_=}(x,y) d{\mathcal H}^{m-1} (x) \end{align*}
\begin{align*}\frac{\partial s}{\partial y}(x,y)=\frac{\partial \sigma}{\partial y}(I(x),y).\end{align*}
\begin{align*}D_x\frac{\partial s}{\partial y}(x,y) = \frac{\partial^2 \sigma}{\partial I \partial y}(I(x),y)D I(x),\end{align*}
\begin{align*}\mu \big [X_<(y_0,k^+(y_0)) \big ]=\nu[(0,y_0)] \leq \nu[(0,y_1)] =\mu \big[X_< (y_1,k^-(y_1)) \big ]\end{align*}
\begin{align*} \gamma(C)\cdot(1_C\otimes A)=C\otimes A . \end{align*}
\begin{align*} \varphi\bigl((\alpha(\eta)(\lambda(f)\otimes 1)\bigr)=\hat{\varphi}_{1}(\lambda(f))\varphi_{2}(\eta) =\varphi\bigl((\lambda(f)\otimes 1)(\alpha(\eta)\bigr)\end{align*}
\begin{align*} L=\rho_{i}(\beta_{\alpha_{h}(t)}(g_{1}))\xi\big(g\beta_{\alpha_{h}(t)}(g_{1}g_{2}), hg_{1}g_{2},t\big). \end{align*}
\begin{align*}A(u,v)=A(u,v;\tau):=e^{\pi i u}\sum_{n\in \mathbb{Z}} \frac{(-1)^nq^{\frac{n(n+1)}{2}}e^{2\pi i n v}}{1-e^{2\pi i u}q^n},\end{align*}
\begin{align*}\vartheta(z+n\tau)=(-1)^nq^{-\frac{n^2}{2}}w^{-n}\vartheta(z).\end{align*}
\begin{align*} M_{i,j}^{K} & = ( ~\varphi_i \circ F_K, \varphi_j \circ F_K)_K, \\ K_{i,j}^{K} & = ( a_h \nabla \varphi_i \circ F_K, \nabla \varphi_j \circ F_K)_K,\\ G_{i,j}^{\partial K} & = (\partial_n \varphi_i \circ F_K, \varphi_j \circ F_K)_{\partial K},\\ \end{align*}
\begin{align*}\vartheta' (0) =-2 \pi \eta^3 .\end{align*}
\begin{align*} b_{3,k}(z+\tau)=q^{-1}w^{-2}b_{3,k}(z). \end{align*}
\begin{align*}\ell_1(z+\tau)-q^8w^{16}\ell_1(z)=G_1(z)+G_2(z)-G_3(z)-G_4(z),\end{align*}
\begin{align*}L_2(z):=\lambda_2(z+\tau)-q^6w^{12}\lambda_2(z).\end{align*}
\begin{align*}\left(1-w^9 q^{9k}\right) =\left(1-w^3 q^{3k}\right)\left(1+w^3 q^{3k} +w^6 q^{6k}\right),\end{align*}
\begin{align*}\ell_1(z+\tau)-q^8w^{16}\ell_1(z)+q^2w^4\frac{i\eta^3}{\vartheta(2z)}L_2(z)=0.\end{align*}
\begin{align*}\mathcal{R}_1 (z+\tau)=-w^4q^8\mathcal{R}_2(z), \mathcal{R}_2 (z+\tau)=-wq^4\mathcal{R}_1(z).\end{align*}
\begin{align*}\vartheta' (\tau ) =-q^{-\frac{1}{2}} \frac{\partial}{\partial z} \left[ w^{-1} \vartheta (z) \right]_{z=0} = -q^{-\frac{1}{2}} \vartheta' (0)\end{align*}
\begin{align*}\|u\|^2= \|u\|_{L^2({\mathbb R})}+\int_{\mathbb R}|(-\Delta)^{\frac{1}{4}}u|^2dx.\end{align*}
\begin{align*}-(\nabla w)=0 \; \; \mathbb{R}\times \mathbb{R}_+,\;\; w\vline_{y=0}=v\frac{\partial w}{\partial \nu}=(-\Delta)^{1/2} v(x) \end{align*}
\begin{align*} \mathbf{u}^{k+1} = & (2 - \tau^2 (M^{L})^{-1} K)\mathbf{u}^k - \tau^2 (M^{L})^{-1} G \mathbf{u}^k -\mathbf{u}^{k-1}, \\ \boldsymbol{\lambda}^{k-1} = & - \tau^2 (M^{L})^{-1} G \boldsymbol{\lambda}^k + (2 - \tau^2 (M^{L})^{-1} K) \boldsymbol{\lambda}^k -\boldsymbol{\lambda}^{k+1}. \\\end{align*}
\begin{align*}\|w\|_{X_1}=\left(\int_{\mathbb R^2_+} |\nabla w|^2~dxdy\right)^{\frac{1}{2}}.\end{align*}
\begin{align*}\int_{I_R(x_0)} (e^{\alpha {{\tilde{w}(x,0)}}^2} -1)dx & \ge \frac{1}{\| w\|_{\mathcal{C}}} \int_{I_R(x_0)} \sum_{k=1}^{\infty} \frac{\alpha^k |w|^{2k}}{k!} dx \\&= \frac{1}{\|w\|_{\mathcal{C}}} \int_{I_R(x_0)} (e^{\alpha (w(x,0))^2} -1) dx.\end{align*}
\begin{align*}\tilde{I}(u)=\frac{1}{2}\int_{\mathbb {R}}|(-\Delta)^{\frac{1}{4}}u|^2dx+\frac{1}{2}\int_{\mathbb R}|u|^2 V(x) dx-\int_{\mathbb {R}}H(u)dx.\end{align*}
\begin{align*}I(w)=\frac{1}{2}\int_{\mathbb {R}^2_+}|\nabla w|^2dxdy+\frac{1}{2}\int_{\mathbb {R}}|w(x,0)|^2 V(x) dx-\int_{\mathbb {R}} H(w(x,0))dx.\end{align*}
\begin{align*}\int_{\mathbb {R}^2_+} \nabla w.\nabla \phi dxdy+\int_{\mathbb {R}} w(x,0)\phi(x,0) V(x) dx-\int_{\mathbb {R}}h(w(x,0))\phi(x,0) dx=0.\end{align*}
\begin{align*}I(w_k)=c+o(1) \;\;\textrm{and}\;\; I^{\prime}(w_k)=o(1).\end{align*}
\begin{align*} \psi_{k}(x,y) = \frac{1}{\sqrt{2\pi}}\left\{\begin{array}{lr}\sqrt{\log k}& 0\leq \sqrt{x^2+y^2}\leq {\frac{1}{k}}\\ \frac{\log {\frac{1}{\sqrt{x^2+y^2}}}}{\sqrt{\log k}} & \frac{1}{k}\leq \sqrt{x^2+y^2} \leq 1\\ 0 & \sqrt{x^2+y^2}\geq 1.\end{array}\right.\end{align*}
\begin{align*}c=\displaystyle \inf_{\gamma \in \Gamma } \displaystyle \max_{t\in[0, 1]}I(\gamma(t)).\end{align*}
\begin{align*}\displaystyle \sup_{t\geq 0}I(t\phi_k)=I(t_k\phi_k)\geq \frac{\pi}{2}\end{align*}
\begin{align*}{t_k^2\|\phi_k\|^2=\int_{\mathbb R} h(t_k\phi_k)t_k\phi_kdx.}\end{align*}
\begin{align*} g_h(x) = \int_{0}^T \nabla u_h \nabla \lambda_h dt + \gamma (a_h - a_0).\end{align*}
\begin{align*}(-\Delta)^{\frac{1}{2}} u+ V(x) u= \lambda u^{q} + h(u) , u> 0 \;\textrm{in } \;\mathbb R,\end{align*}
\begin{align*}\Gamma=\{\gamma:[0, 1]\rightarrow H^1_{0, L}(\mathcal C): \gamma \;\textrm{continuous on}\; [0, 1], \gamma(0)=0, \gamma(1)=e\}.\end{align*}
\begin{align*}\begin{array}{rlll}\Delta w &=0 \; \; \; \mathbb R^{2}_{+}\\\frac{\partial w}{\partial y}&= -V(x) u+ \frac{f(u)}{m(\|u\|_{X}^2)}\; \; \mathbb R.\end{array}\end{align*}
\begin{align*}\begin{array}{rlll}-m\left( \| w\|^2 \right)\Delta w &=0 \; \; \; \mathbb R^{2}_{+}\\-m\left( \| w\|^2 \right)\left(\frac{\partial w}{\partial y}+V(x)w(x,0)\right)&= f(w(x,0))\; \; \mathbb R\end{array}\end{align*}
\begin{align*}J(w)=\frac{1}{2} M\left(\| w\|^2\right) -\int_{\mathbb R} F(w(x,0))dx.\end{align*}
\begin{align*}M(t)\leq\left\{ \begin{array}{lr} a_0+a_1t+\frac{a_2}{\sigma+1} t^{\sigma+1},\; \mbox{if}\; \sigma\ne -1,\\ b_0+ a_1 t+a_2 \ln t\quad\quad\mbox{if}\; \sigma=-1, \end{array} \right.\end{align*}
\begin{align*} c_*=\inf_{\gamma\in \Gamma}\max_{t\in[0,1]}J(\gamma(t)). \end{align*}
\begin{align*} \rho_t + \div(\rho u) &=0\\ (\rho u)_t + \div (\rho u \otimes u) + \nabla p(\rho) &= \div(\sigma[ u]) + C_\kappa \rho \nabla \triangle \rho\, , \end{align*}
\begin{align*}\sigma[u]:= \lambda \div( u) \mathbb{I}+ \mu (\nabla u + (\nabla u)^T)\end{align*}
\begin{align*} \frac{\partial}{\partial t} \rho + \div(\rho u) &=0\\\frac{\partial}{\partial t} (\rho u) + \div (\rho u \otimes u) + \nabla (p(\rho) + C_\kappa \frac{\alpha}{2} \rho^2)&= \div(\sigma[ u]) + C_\kappa\alpha \rho \nabla c\\ c - \frac{1}{\alpha}\triangle c &= \rho\, , \end{align*}
\begin{align*} \int_0^{\pi/2}\cos^{\nu -1}x \cos ax dx={\pi \over 2^\nu \nu B\left ({\nu+a+1 \over 2}, {\nu-a+1 \over 2}\right )}\;,\end{align*}
\begin{align*} u_{l,j,m}^{k+1} = \tau^2 \Delta u_{l,j,m}^k + 2 u_{l,j,m}^k - u_{l,j,m}^{k-1},\end{align*}
\begin{align*}\frac{d}{dt} \int \tfrac{1}{2} \rho |u|^2 \, dx = - \int \Big (S : \nabla u + \zeta \rho |u|^2\Big ) \, dx \, .\end{align*}
\begin{align*}d S (\rho ; \psi ) = \frac{d}{d\tau} S (\rho + \tau \psi ) \Big |_{\tau = 0} = \big \langle \frac{\delta S}{\delta \rho }(\rho) , \psi \big \rangle \, ,\end{align*}
\begin{align*}K (\rho, m) = \int \frac{1}{2} \frac{ |m|^2}{\rho} dx\end{align*}
\begin{align*}\begin{aligned}s (\rho, q) &= \rho F_\rho (\rho, q) + q \cdot F_q (\rho, q) - F (\rho, q)\, ,\\r (\rho, q) &= \rho F_q (\rho, q)\, ,\\H(\rho , q) &= q \otimes F_q (\rho, q) \, .\end{aligned}\end{align*}
\begin{align*}S &= - \big ( p(\rho) -\tfrac{1}{2} |\nabla c|^2 - \tfrac{\beta}{2} c^2 - < \rho > c \big ) \mathbb{I} - \nabla c \otimes \nabla c \, .\end{align*}
\begin{align*} \frac{\partial}{\partial t} \rho + \div(\rho u) &=0\\\frac{\partial}{\partial t} (\rho u) + \div (\rho u \otimes u) + \nabla (p(\rho) + C_\kappa \frac{\alpha}{2} \rho^2)&= C_\kappa\alpha \rho \nabla c\\ c - \frac{1}{\alpha}\triangle c &= \rho\, , \end{align*}
\begin{align*}\mathcal{E}(\rho,c,\nabla c):= \int\Big( h(\rho) + \frac{C_\kappa \alpha}{2} (\rho - c )^2 + \frac{C_\kappa}{2}|\nabla c|^2\Big)\, dx\end{align*}
\begin{align*} \rho_t + \div(\rho u) &=0\\ \rho \frac{\operatorname{D} u}{\operatorname{D} t} &= - \rho \nabla \frac{\delta \mathcal{E}}{\delta \rho}\\ \frac{\delta \mathcal{E}}{\delta c} &= 0\, . \end{align*}
\begin{align*} &\mathcal{E}(\rho):= \int\Big( h(\rho) + \frac{C_\kappa \alpha}{2} \rho ^2 - \frac{C_\kappa \alpha}{2} \rho c\Big)\, dx\\ & c - \frac{1}{\alpha}\triangle c = \rho\, , \end{align*}
\begin{align*} \frac{\delta \mathcal{E}}{\delta \rho} = h'(\rho) + C_\kappa\alpha( \rho - c)\, ,\end{align*}
\begin{align*} \lambda_{l,j,m}^{k-1} = -\tau^2 (u - \tilde{u})_{l,j,m}^k z_{\delta} + \tau^2 \Delta \lambda_{l,j,m}^k + 2 \lambda_{l,j,m}^k - \lambda_{l,j,m}^{k+1},\end{align*}
\begin{align*} &\mathcal{E}(\rho)= \int\Big( h(\rho) + \frac{C_\kappa}{2} \nabla \rho \cdot \nabla c\Big) \, dx\\ & c - \frac{1}{\alpha}\triangle c = \rho\, . \end{align*}
\begin{align*} \begin{aligned} - \iint \left ( \frac{1}{2} \frac{|m |^2}{\rho} + F(\rho,\nabla\rho) \right ) \dot\theta(t) \le \int \left ( \frac{1}{2} \frac{|m |^2}{\rho } + F(\rho,\nabla\rho) \right ) \Big |_{t=0} \theta(0)dx \, , \\ \ \theta \in W^{1, \infty} [0, \infty) \ \mbox{compactly supported on $[0, \infty)$}, \end{aligned}\end{align*}
\begin{align*}\theta(\tau) := \begin{cases}1, &\hbox{for}\ 0\leq \tau < t, \\\frac{t-\tau}{\epsilon} + 1, &\hbox{for}\ t\leq \tau < t+\epsilon , \\0, &\hbox{for}\ \tau \geq t+\epsilon ,\end{cases}\end{align*}
\begin{align*} \left. \int_{\mathbb{T}^d}\left ( \frac{1}{2} \frac{|m|^2}{\rho} + F(\rho,\nabla\rho) \right)dx \right |^{t}_{\tau=0}\leq 0\, .\end{align*}
\begin{align*} \left. \int_{\mathbb{T}^d}\left ( \frac{1}{2} \frac{|\bar m|^2}{\bar \rho} + F(\bar \rho,\nabla\bar\rho) \right)dx \right |^{t}_{\tau=0}=0\, .\end{align*}
\begin{align*}F(\rho, \nabla\rho ) = h(\rho ) + \frac{1}{2}C_\kappa |\nabla \rho|^2,\end{align*}
\begin{align*} \rho h'(\rho) = p(\rho) + h(\rho) \, , p'(\rho) > 0.\end{align*}
\begin{align*}h(\rho) &= \frac{k}{\gamma -1} \rho^\gamma + o(\rho^\gamma) \, , \hbox{as}\ \rho\to +\infty\\| p'' (\rho ) | &\le A \frac{p'(\rho)}{\rho} = A h'' (\rho) \forall \rho > 0 \, .\end{align*}
\begin{align*}h(\rho \left | \bar \rho \right.) \geq \begin{cases}C_1 |\rho-\bar\rho|^2, &\hbox{for}\ 0\leq \rho\leq R_0,\ \bar\rho\in K, \\C_2 |\rho-\bar\rho|^\gamma, &\hbox{for}\ \rho> R_0,\ \bar\rho\in K.\end{cases}\end{align*}
\begin{align*}F(\rho, \nabla\rho\left | \bar \rho, \nabla\bar \rho \right. ) = h(\rho \left | \bar \rho \right. ) + \frac{1}{2}C_\kappa|\nabla \rho - \nabla\bar\rho|^2 \, .\end{align*}
\begin{align*}\frac{\partial u(x,t)}{\partial n} = -\frac{\partial u(x,t)}{\partial t} \end{align*}
\begin{align*}\sup_{t\in[0,T]}\varphi(t) = 0\, ,\end{align*}
\begin{align*} H(\rho , q) &= C_\kappa q \otimes q\, ;\\ s(\rho , q) & = h(\rho) + \tfrac{1}{2} C_\kappa |q|^2;\\ r(\rho , q) &= C_\kappa \rho q\, .\end{align*}
\begin{align*}F(\rho, q ) = h(\rho ) + \frac{1}{2} \kappa(\rho) |q|^2 \mbox{ with $\kappa (\rho) > 0$}\, .\end{align*}
\begin{align*}\sup_{t\in[0,T]}\Phi(t) = 0\, ,\end{align*}
\begin{align*} H(\rho , q) &= \kappa(\rho) q \otimes q\, , \\ s(\rho , q) & = p (\rho) + A(\rho) |q|^2 \, , \mbox{where $A(\rho) = \tfrac{1}{2} (\rho \kappa'(\rho) + \kappa (\rho) )$} \, , \\ r(\rho , q) &= B(\rho) q \, , \mbox{where $B(\rho) = \rho \kappa(\rho) $} \, ,\end{align*}
\begin{align*} F(\rho , q) = h(\rho) + \frac{1}{2 \rho} |q|^2 ,\end{align*}
\begin{align*}\begin{aligned}s ( \rho, q | \bar \rho , \bar q) &= p (\rho | \bar \rho) \, ,\\H ( \rho, q | \bar \rho , \bar q) &= \rho \Big ( \frac{q}{\rho} - \frac{\bar q}{\bar \rho} \Big ) \otimes \Big ( \frac{q}{\rho} - \frac{\bar q}{\bar \rho} \Big ) \, , \\ r ( \rho, q | \bar \rho , \bar q) &= 0\, . \end{aligned}\end{align*}
\begin{align*}\sup_{t\in[0,T]}\Psi(t) = 0\, ,\end{align*}
\begin{align*} A_5&:=\div (\bar m) \big( h'(\rho) - h'(\bar \rho) - h''(\bar \rho)( \rho-\bar \rho) \big)\, ,\\ A_6&:= \big( h'( \rho) - h'(\bar \rho)\big) \big( \div(m) - \div(\bar m) \big)\, . \end{align*}
\begin{align*} (\eta_{\mathrm{R}} - \eta) \Big( \rho, { m} | \bar \rho, \bar m \Big) = -h(\rho) + h(\bar \rho) + h'(\bar \rho)( \rho-\bar \rho) = - \int_{\bar \rho}^{\rho} ( \rho- s) h''(s)\operatorname{d} s\, . \end{align*}
\begin{align*} \frac{ u_{l,j,m}^{k+1} - u_{l,j,m}^k}{\tau} + \frac{ u_{l+1,j,m}^{k+1} - u_{l+1,j,m}^k}{\tau} - \\ \frac{ u_{l+1,j,m}^{k} - u_{l,j,m}^k}{ \delta x_1} - \frac{ u_{l+1,j,m}^{k+1} - u_{l,j,m}^{k+1}}{ \delta x_1} = 0,\end{align*}
\begin{align*} \sigma[u] : \nabla u = (\lambda + \frac{2}{3} \mu) (\div u)^2 + 2\mu \nabla^o u : \nabla^o u \geq 0\, ,\end{align*}
\begin{align*} \frac{\operatorname{d}}{\operatorname{d} t} \int_{\mathbb{T}^3}\eta^\alpha \operatorname{d} x= \int_{\mathbb{T}^3}\sum_{i=1}^6 A_i^\alpha \operatorname{d} x\, ,\end{align*}
\begin{align*} \frac{\operatorname{d}}{\operatorname{d} t} \int_{\mathbb{T}^3}\eta^\alpha_{\mathrm{R}} \operatorname{d}x = \int_{\mathbb{T}^3}\sum_{i=1}^8 A_i^\alpha \operatorname{d}x \, ,\end{align*}
\begin{align*}\begin{cases}x_i^* = x_i + h_k \delta_{ik} & i= 1, ..., d\, , \\\rho^* = \rho\end{cases}\end{align*}
\begin{align*} H(\rho , q) &= q\otimes F_q(\rho,q)\, ,\\ s(\rho , q) & = \rho F_\rho(\rho,q) + q\cdot F_q(\rho,q) - F(\rho,q)\, ,\\ r(\rho , q) & = \rho F_q(\rho,q)\, ,\end{align*}