We consider the class of graph-directed constructions which are connected and have the property of finite ramification. By assuming the existence of a fixed point for a certain renormalization map, it is possible to construct a Laplace operator on fractals in this class via their Dirichlet forms. Our main aim is to consider the eigenvalues of the Laplace operator and provide a formula for the s...

The purpose of this supplement is to provide a more complete account of the mathematics underlying our analyses in the main text. In particular, the order complex and clique topology are described more precisely here. The order complex of a matrix is analogous to its Jordan Form, in that it captures features that are invariant under a certain type of matrix transformation. Likewise, the clique ...

We establish universality of local eigenvalue correlations in unitary random matrix ensembles 1 Zn | det M | 2α e −ntr V (M) dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x| 2α e −nV (x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local ...

Consider the Laplacian in a bounded domain in Rd with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are ‘quasi-orthogonal’ on the boundary with respect to a certain norm. Boundary orthogonality is proved asymptotically within a narrow eigenvalue window of width o(E1/2) centered about E, as E → ∞. For the special case of Dirichlet boundary conditions, the norm...

We investigate the problem of defining propagating constants and modes in metallic waveguides of an arbitrary cross section, filled with a homogeneous bi-isotropic material. The approach follows the guidelines of the classical theory for the isotropic, homogeneous, lossless waveguide: starting with the Maxwell system, we formulate a spectral problem where the square of the propagation constant ...

In this paper, we study a nonlinear time-fractional Volterra equation with nonsingular Mittag-Leffler kernel in Hilbert spaces. By applying the properties of functions and method eigenvalue expansion, give mild solution our problem. Our main tool here is using some Sobolev embeddings.

Spectral clustering methods are based on solving eigenvalue problems for the identification of clusters, e.g., the identification of metastable subsets of a Markov chain. Usually, real-valued eigenvectors are mandatory for this type of algorithms. The Perron Cluster Analysis (PCCA+) is a well-known spectral clustering method of Markov chains. It is applicable for reversible Markov chains, becau...

A coherent mathematical framework for the psychophysics of contrast perception emerges when contrast sensitivity is posed as an eigenvalue problem. This more general mathematical theory is broad enough to encompass Fourier analysis as it is used in vision research. We present a model of space-variant contrast detection to illustrate the main features of the theory, and obtain a new contrast sen...

In this paper, we consider the numerical discretization of elliptic eigenvalue problems by Finite Element Methods and its solution by a multigrid method. From the general theory of finite element and multigrid methods, it is well known that the asymptotic convergence rates become visible only if the mesh width h is sufficiently small, h ≤ h0. We investigate the dependence of the maximal mesh wi...