We address the problem of setting the kernel bandwidth used by Manifold Learning algorithms to construct the graph Laplacian. Exploiting the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator, we set by optimizing the Laplacian’s ability to preserve the geometry of the data. Experiments show that this principled approach is effective an...

A bounded domain in C with connected Lipschitz boundary is pseudoconvex if the bottom of the essential spectrum of the Kohn Laplacian on the space of (0, q)-forms, 1 ≤ q ≤ n− 1, with L-coefficients is positive.

We prove Lp-bounds for the Riesz transforms d/ √ −∆, δ/ √ −∆ associated with the Hodge-Laplacian ∆ = −δd − dδ equipped with absolute and relative boundary conditions in a Lipschitz subdomain Ω of a (smooth) Riemannian manifoldM, for p in a certain interval depending on the Lipschitz character of the domain.

In this paper we prove that a function u ∈ C(Ω) is the continuous value of the Tug-of-War game described in [19] if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions

Recently the signless Laplacian matrix of graphs has been intensively investigated. While there are many results about the largest eigenvalue of the signless Laplacian, the properties of its smallest eigenvalue are less well studied. The present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless Laplacian.

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z) := ∑ k (z − λk) σ +. Here {λk} ∞ k=1 are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ Rd, and x+ := max(0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λk, on averages such as λ...

This paper focuses on an important aspect of cardiac surgical simulation, which is the deformation of mesh models to form smooth joins between them. A novel algorithm based on the Laplacian deformation method is developed. It extends the Laplacian method to handle deformation of 2-manifold mesh models with 1-D boundaries, and joining of 1-D boundaries to form smooth joins. Test results show tha...

The Mergelyan and Ahlfors-Beurling estimates for the Cauchy transform give quantitative information on uniform approximation by rational functions with poles off K. We will present an analogous result for an integral transform on the unit sphere in C2 introduced by Henkin, and show how it can be used to study approximation by functions that are locally harmonic with respect to the Kohn Laplacia...

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.