The scattering process on multiloop infinite p + 1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group PGL(2,Qp). As the homogeneous spaces, they are, in fact, identical to p-adic multiloop surfaces. The Ihara–Selberg L-function is associated with the finite subgraph—the re...

A new bound (Theorem 4.4) for the duration of the chip-firing game with N chips on a n-vertex graph is obtained, by a careful analysis of the pseudo-inverse of the discrete Laplacian matrix of the graph. This new bound is expressed in terms of the entries of the pseudo-inverse. It is shown (Section 5) to be always better than the classic bound due to Björner, Lovász and Shor. In some cases the ...

Discrete Laplace–Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace–Beltrami operator are well-studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace–Beltrami o...

The convergence property of the discrete Laplace–Beltrami operators is the foundation of convergence analysis of the numerical simulation process of some geometric partial differential equations which involve the operator. In this paper we propose several simple discretization schemes of Laplace–Beltrami operators over triangulated surfaces. Convergence results for these discrete Laplace–Beltra...

agonal (scaling) preconditioning is used, the condition number of the preconditioned incremental unknowns matrix associated to the Laplace operator is p(d)O((logd h) ) for the first order incremental unknowns, and q(d)O(| logd h|) for the second order incremental unknowns. For comparison, the condition number of the nodal unknowns matrix associated to the Laplace operator is O(1/h). Therefore, ...

The models that are based of fractional derivatives should be highlighted among promising new models to describe turbulent fluid flows. In the present work, a steadystate flow in a duct is considered under the condition that the turbulent diffusion is governed by a fractional power of the Laplace operator. To study numerically flows in rectangular channels, finite-difference approximations are ...

In the machine learning community it is generally believed that graph Laplacians corresponding to a finite sample of data points converge to a continuous Laplace operator if the sample size increases. Even though this assertion serves as a justification for many Laplacianbased algorithms, so far only some aspects of this claim have been rigorously proved. In this paper we close this gap by esta...

The article consists of a survey on analytic and topological torsion. Analytic torsion is defined in terms of the spectrum of the analytic Laplace operator on a Riemannian manifold, whereas topological torsion is defined in terms of a triangulation. The celebrated theorem of Cheeger and Müller identifies these two notions for closed Riemannian manifolds. We also deal with manifolds with boundar...

Chebyshev polynomials of the first and the second kind in n variables z. , Zt , ... , z„ are introduced. The variables z, , z-,..... z„ are the characters of the representations of SL(n + 1, C) corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami op...

Many problems in image analysis, digital processing and shape optimization can be expressed as variational problems involving the discretization of the Laplace-Beltrami operator. Such discretizations have have been widely studied for meshes or polyhedral surfaces. On digital surfaces, direct applications of classical operators are usually not satisfactory (lack of multigrid convergence, lack of...