We give a lower bound on the number of small positive eigenvalues of the p-form Laplacian in a certain type of collapse with curvature bounded below.

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.

In this work we consider the nonlinear graph p-Laplacian and the set of eigenvalues and associated eigenvectors of this operator defined by a variational principle. We prove a unifying nodal domain theorem for the graph p-Laplacian for any p ≥ 1. While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p = 2), the behavior changes...

2 Eigenvalues of graphs 5 2.1 Matrices associated with graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The largest eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Adjacency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3...

For one-dimensional p-Laplacian with weights in Lγ : L 0, 1 ,R 1 ≤ γ ≤ ∞ balls, we are interested in the extremal values of the mth positive half-eigenvalues associated with Dirichlet, Neumann, and generalized periodic boundary conditions, respectively. It will be shown that the extremal value problems for half-eigenvalues are equivalent to those for eigenvalues, and all these extremal values a...

Let $G=(V,E)$, $V={1,2,ldots,n}$, $E={e_1,e_2,ldots,e_m}$,be a simple connected graph, with sequence of vertex degrees$Delta =d_1geq d_2geqcdotsgeq d_n=delta >0$ and Laplacian eigenvalues$mu_1geq mu_2geqcdotsgeqmu_{n-1}>mu_n=0$. Denote by $Kf(G)=nsum_{i=1}^{n-1}frac{1}{mu_i}$ and $t=t(G)=frac 1n prod_{i=1}^{n-1} mu_i$ the Kirchhoff index and number of spanning tree...

In this paper an efficient analytical method is presented for calculating the eigenvalues of special matrices related to finite element meshes (FEMs) with regular topologies. In the proposed method, a skeleton graph is used as the model of a FEM. This graph is then considered as the Cartesian product of its generators. The eigenvalues of the Laplacian matrix of the entire graph are then easily ...

We study theoretical and practical aspects of high-precision computation of Maass forms. First, we compute to over 1000 decimal places the Laplacian and Hecke eigenvalues for the first few Maass forms on PSL(2,Z)\H. Second,we give an algorithm for rigorously verifying that a proposed eigenvalue together with a proposed set of Fourier coefficients indeed correspond to a true Maass cusp form. We ...

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ±∞ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this...

In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains X R . This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as ...