The Experts below are selected from a list of 47535 Experts worldwide ranked by ideXlab platform
Gabriele Grillo - One of the best experts on this subject based on the ideXlab platform.
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the fractional porous medium equation on the Hyperbolic Space
Calculus of Variations and Partial Differential Equations, 2020Co-Authors: Elvise Berchio, Debdip Ganguly, Matteo Bonforte, Gabriele GrilloAbstract:We consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the Hyperbolic Space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $$L^p$$ Spaces or to larger (weighted) Spaces determined either in terms of a ground state of $$\Delta _{\mathbb {H}^{N}}$$ , or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $$L^1-L^\infty $$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.
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the fractional porous medium equation on the Hyperbolic Space
arXiv: Analysis of PDEs, 2020Co-Authors: Elvise Berchio, Debdip Ganguly, Matteo Bonforte, Gabriele GrilloAbstract:We consider the nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the Hyperbolic Space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $L^p$ Spaces or to larger (weighted) Spaces determined either in terms of a ground state of $\Delta_{\mathbb{H}^n}$, or of the (fractional) Green's function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1-L^\infty$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.
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improved lp poincare inequalities on the Hyperbolic Space
Nonlinear Analysis-theory Methods & Applications, 2017Co-Authors: Elvise Berchio, Debdip Ganguly, Lorenzo Dambrosio, Gabriele GrilloAbstract:Abstract We investigate the possibility of improving the p -Poincare inequality ‖ ∇ H N u ‖ p p ≥ Λ p ‖ u ‖ p p on the Hyperbolic Space, where p > 1 and Λ p : = [ ( N − 1 ) / p ] p is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincare–Hardy inequality, namely an improvement of the best p -Poincare inequality in terms of the Hardy weight r − p , r being geodesic distance from a given pole. Certain Hardy–Maz’ya-type inequalities in the Euclidean half-Space are also obtained.
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radial fast diffusion on the Hyperbolic Space
Proceedings of The London Mathematical Society, 2014Co-Authors: Gabriele Grillo, Matteo MuratoriAbstract:We consider positive radial solutions to the fast diffusion equation ut = ∆(u) on the Hyperbolic Space H for N ≥ 2, m ∈ (ms, 1), ms = N−2 N+2 . By radial we mean solutions depending only on the geodesic distance r from a given point o ∈ H . We investigate their fine asymptotics near the extinction time T in terms of a separable solution of the form V(r, t) = (1 − t/T )1/(1−m)V (r), where V is the unique positive energy solution, radial w.r.t. o, to −∆V = c V 1/m for a suitable c > 0, a semilinear elliptic problem thoroughly studied in [29], [7]. We show that u converges to V in relative error, in the sense that ‖u(·, t)/V(·, t)− 1‖∞ → 0 as t→ T−. In particular the solution is bounded above and below, near the extinction time T , by multiples of (1− t/T )1/(1−m)e−(N−1)r/m. Solutions are smooth, and bounds on derivatives are given as well. In particular, sharp convergence results as t → T− are shown for spatial derivatives, again in the form of convergence in relative error.
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radial fast diffusion on the Hyperbolic Space
arXiv: Analysis of PDEs, 2013Co-Authors: Gabriele Grillo, Matteo MuratoriAbstract:We consider radial solutions to the fast diffusion equation $u_t=\Delta u^m$ on the Hyperbolic Space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate their fine asymptotics near the extinction time $T$ in terms of a separable solution of the form ${\mathcal V}(r,t)=(1-t/T)^{1/(1-m)}V^{1/m}(r)$, where $V$ is the unique positive energy solution, radial w.r.t. $o$, to $-\Delta V=c\,V^{1/m}$ for a suitable $c>0$, a semilinear elliptic problem thoroughly studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to ${\mathcal V}$ in relative error, in the sense that $\|{u^m(\cdot,t)}/{{\mathcal V}^m(\cdot,t)}-1\|_\infty\to0$ as $t\to T^-$. In particular the solution is bounded above and below, near the extinction time $T$, by multiples of $(1-t/T)^{1/(1-m)}e^{-(N-1)r/m}$.
Kayhan Batmanghelich - One of the best experts on this subject based on the ideXlab platform.
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semi supervised hierarchical drug embedding in Hyperbolic Space
Journal of Chemical Information and Modeling, 2020Co-Authors: Shyam Visweswaran, Kayhan BatmanghelichAbstract:Learning accurate drug representations is essential for tasks such as computational drug repositioning and prediction of drug side effects. A drug hierarchy is a valuable source that encodes knowledge of relations among drugs in a tree-like structure where drugs that act on the same organs, treat the same disease, or bind to the same biological target are grouped together. However, its utility in learning drug representations has not yet been explored, and currently described drug representations cannot place novel molecules in a drug hierarchy. Here, we develop a semi-supervised drug embedding that incorporates two sources of information: (1) underlying chemical grammar that is inferred from chemical structures of drugs and drug-like molecules (unsupervised) and (2) hierarchical relations that are encoded in an expert-crafted hierarchy of approved drugs (supervised). We use the Variational Auto-Encoder (VAE) framework to encode the chemical structures of molecules and use the drug-drug similarity information obtained from the hierarchy to induce the clustering of drugs in Hyperbolic Space. The Hyperbolic Space is amenable for encoding hierarchical relations. Both quantitative and qualitative results support that the learned drug embedding can accurately reproduce the chemical structure and recapitulate the hierarchical relations among drugs. Furthermore, our approach can infer the pharmacological properties of novel molecules by retrieving similar drugs from the embedding Space. We demonstrate that our drug embedding can predict new uses and discover new side effects of existing drugs. We show that it significantly outperforms comparison methods in both tasks.
Matteo Bonforte - One of the best experts on this subject based on the ideXlab platform.
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the fractional porous medium equation on the Hyperbolic Space
Calculus of Variations and Partial Differential Equations, 2020Co-Authors: Elvise Berchio, Debdip Ganguly, Matteo Bonforte, Gabriele GrilloAbstract:We consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the Hyperbolic Space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $$L^p$$ Spaces or to larger (weighted) Spaces determined either in terms of a ground state of $$\Delta _{\mathbb {H}^{N}}$$ , or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $$L^1-L^\infty $$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.
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the fractional porous medium equation on the Hyperbolic Space
arXiv: Analysis of PDEs, 2020Co-Authors: Elvise Berchio, Debdip Ganguly, Matteo Bonforte, Gabriele GrilloAbstract:We consider the nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the Hyperbolic Space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $L^p$ Spaces or to larger (weighted) Spaces determined either in terms of a ground state of $\Delta_{\mathbb{H}^n}$, or of the (fractional) Green's function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1-L^\infty$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.
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classification of radial solutions to the emden fowler equation on the Hyperbolic Space
Calculus of Variations and Partial Differential Equations, 2013Co-Authors: Matteo Bonforte, Filippo Gazzola, Gabriele Grillo, Juan Luis VazquezAbstract:We study the Emden–Fowler equation −Δu = |u| p−1 u on the Hyperbolic Space $${{\mathbb H}^n}$$ . We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is p = (n + 2)/(n − 2) as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers (Bhakta and Sandeep, Poincare Sobolev equations in the Hyperbolic Space, 2011; Mancini and Sandeep, Ann Sci Norm Sup Pisa Cl Sci 7(5):635–671, 2008) consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.
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classification of radial solutions to the emden fowler equation on the Hyperbolic Space
arXiv: Analysis of PDEs, 2011Co-Authors: Matteo Bonforte, Filippo Gazzola, Gabriele Grillo, Juan Luis VazquezAbstract:We study the Emden-Fowler equation $-\Delta u=|u|^{p-1}u$ on the Hyperbolic Space ${\mathbb H}^n$. We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is $p=(n+2)/(n-2)$ as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers \cite{mancini, bhakta} consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.
Chao Xia - One of the best experts on this subject based on the ideXlab platform.
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isoperimetric type problems and alexandrov fenchel type inequalities in the Hyperbolic Space
Advances in Mathematics, 2014Co-Authors: Guofang Wang, Chao XiaAbstract:Abstract In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the Hyperbolic Space H n , by using quermassintegral preserving curvature flows. As a byproduct, we obtain Hyperbolic Alexandrov–Fenchel inequalities.
Shyam Visweswaran - One of the best experts on this subject based on the ideXlab platform.
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semi supervised hierarchical drug embedding in Hyperbolic Space
Journal of Chemical Information and Modeling, 2020Co-Authors: Shyam Visweswaran, Kayhan BatmanghelichAbstract:Learning accurate drug representations is essential for tasks such as computational drug repositioning and prediction of drug side effects. A drug hierarchy is a valuable source that encodes knowledge of relations among drugs in a tree-like structure where drugs that act on the same organs, treat the same disease, or bind to the same biological target are grouped together. However, its utility in learning drug representations has not yet been explored, and currently described drug representations cannot place novel molecules in a drug hierarchy. Here, we develop a semi-supervised drug embedding that incorporates two sources of information: (1) underlying chemical grammar that is inferred from chemical structures of drugs and drug-like molecules (unsupervised) and (2) hierarchical relations that are encoded in an expert-crafted hierarchy of approved drugs (supervised). We use the Variational Auto-Encoder (VAE) framework to encode the chemical structures of molecules and use the drug-drug similarity information obtained from the hierarchy to induce the clustering of drugs in Hyperbolic Space. The Hyperbolic Space is amenable for encoding hierarchical relations. Both quantitative and qualitative results support that the learned drug embedding can accurately reproduce the chemical structure and recapitulate the hierarchical relations among drugs. Furthermore, our approach can infer the pharmacological properties of novel molecules by retrieving similar drugs from the embedding Space. We demonstrate that our drug embedding can predict new uses and discover new side effects of existing drugs. We show that it significantly outperforms comparison methods in both tasks.
