In this paper we first establish the relation between the zeta-determinant of a Dirac Laplacian with the Dirichlet boundary condition and the APS boundary condition on a cylinder. Using this result and the gluing formula of the zetadeterminant given by Burghelea, Friedlander and Kappeler with some assumptions, we prove the adiabatic decomposition theorem of the zeta-determinant of a Dirac Lapla...

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere S. The spaces of symmetric eigentensors are explicitly given. Mathematical Subject Classification (2000):53B21, 53B50, 58C40

Explicit sharp estimates for the Green function of the Laplacian in C domains were completed in 1986 by Zhao [42]. Sharp estimates of the Green function of Lipschitz domains were given in 2000 by Bogdan [6]. Explicit qualitatively sharp estimates for the classical heat kernel in C domains were established in 2002 by Zhang [41]. Qualitatively sharp heat kernel estimates in Lipschitz domains were...

Abstract. Subordination of a killed Brownian motion in a domain D ⊂ R via an α/2-stable subordinator gives rise to a process Zt whose infinitesimal generator is −(− |D), the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Zt when D is either a bounded C1,1 domain or an exterior C1,...

Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied using the Laplace transform with respect to time and the Fourier ...

We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1–parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded ...

Let G be a simple graph with adjacency matrix A (= AG). The eigenvalues and the spectrum of A are also called the eigenvalues and the spectrum of G, respectively. If we consider a matrix Q = D + A instead of A, where D is the diagonal matrix of vertex–degrees (in G), we get the signless Laplacian eigenvalues and the signless Laplacian spectrum, respectively. For short, the signless Laplacian ei...

The Kirchoff Matrix Tree Theorem states that the number of spanning trees in a graph G is equal to the absolute value of any cofactor of the Laplacian matrix of G. As the theory of simplicial complexes is a generalization of the theory of graphs one would suspect that there is a generalization of the notion of spanning trees to simplicial complexes, such that the number of spanning trees in a g...

We note that building a magnetic Laplacian from the Markov transition matrix, rather than the graph adjacency matrix, yields several benefits for the magnetic eigenmaps algorithm. The two largest benefits are that the embedding becomes more stable as a function of the rotation parameter g, and the principal eigenvector of the magnetic Laplacian now converges to the page rank of the network as a...

A graph can be associated with a matrix in several ways. For instance, by associating the vertices of the graph to the rows/columns and then using 1 to indicate an edge and 0 otherwise we get the adjacency matrix A. The combinatorial Laplacian matrix is defined by L = D − A where D is a diagonal matrix with diagonal entries the degrees and A is again the adjacency matrix. Both of these matrices...