Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as DiffusionMaps and Laplacian Eigenmaps, are often used for manifold learning and nonlinear dimensionality reduction. It was previously shown by Belkin&Niyogi (2007, Convergence of Laplacian eigenmaps, vol. 19. Proceedings of the 2006 Conference on Advances in Neural Information Processing System...

We consider overdetermined boundary value problems for the ∞-Laplacian in a domain Ω of Rn and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞-Laplacian, the normalized or game-theoretic ∞-Laplacian and the limit of the p-Laplacian as p→∞ are considered and provide different answers. Mathematics Subject Classification (2000). 35R35, 4...

Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. Laplace’s equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial optimization, computer vision, computer graphics, and machine le...

Laplacian matrices of graphs arise in large-scale computational applications such as semi-supervised machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits; and elliptic partial differential equations discretized on unstructured grids with finite elements. A Lean Algebraic Multigrid (LAMG) solver of the symmetric l...

Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class to which Laplacian matrices belong) in provably nearly-linear time. While these algorithms are highly interesting from a theoretical perspective, there are n...

Suppose μ1, μ2, ... , μn are Laplacian eigenvalues of a graph G. The Laplacian energy of G is defined as LE(G) = ∑n i=1 |μi − 2m/n|. In this paper, some new bounds for the Laplacian eigenvalues and Laplacian energy of some special types of the subgraphs of Kn are presented. AMS subject classifications: 05C50

in this paper, we present a computational method for solving boundary integral equations with loga-rithmic singular kernels which occur as reformulations of a boundary value problem for the laplacian equation. themethod is based on the use of the galerkin method with cas wavelets constructed on the unit interval as basis.this approach utilizes the non-uniform gauss-legendre quadrature rule for ...

In this paper, we consider the doubly critical coupled systems involving fractional Laplacian in ℝ n with partial singular weight: ( − Δ ) s u γ 1 | x ′ 2 = ∗ β + η α v , (0.1)where ∈ (0, 1), 0 ≤ α, < 2s n, m × η1, η2 > 1, : / s, γ1, γ2 γH, and H is some explicit constant. By establishing new embedding results partially weighted Morrey norms product space ˙ ), provide sufficient conditions unde...

The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. Barriers to transport, which mitigate mixing, describe a skeleton about which possibly turbulent flow evolves. We propose a novel, direct geometric method to identify subsets of phase space that remain strongly coherent over a finite time durati...

For a graph, the least signless Laplacian eigenvalue is the least eigenvalue of its signless Laplacian matrix. This paper investigates how the least signless Laplacian eigenvalue of a graph changes under some perturbations, and minimizes the least signless Laplacian eigenvalue among all the nonbipartite graphs with given matching number and edge cover number, respectively.