We consider the Dirichlet Laplacian in the half-plane with constant magnetic field. Due to the translational invariance this operator admits a fiber decomposition and a family of dispersion curves, that are real analytic functions. Each of them is simple and monotonically decreasing from positive infinity to a finite value, which is the corresponding Landau level. These finite limits are thresh...

In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [28] of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. By [28] the local smoothing estimates are known to hold for small perturbations o...

The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this spectrum under local and global operations like motif doubling, graph joining or splitting. The eigenvalue 1 plays a particular role, and we therefore emphasize tho...

Content based music retrieval opens up large collections, both for the general public and music scholars. It basically enables the user to find (groups of) similar melodies, thus facilitating musicological research of many kinds. We present a graph spectral approach, new to the music retrieval field, in which melodies are represented as graphs, based on the intervals between the notes they are ...

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In...

This paper examines the system dynamics of a controlled networked multi-agent system, operating with a consensus-type algorithm, that is under the influence of attached node(s). We introduce an identification scheme, involving excitation and observation of the network by the attached node(s), that identifies the spectrum of the underlying system matrix in this case the influenced network’s modi...

On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy Eφ and conformal Laplacian ∆φ for a given conformal factor φ, based on the corresponding notions in Riemannian geometry in dimension n 6= 2. We derive a differential equation that describes the dependence of the effective resistances of Eφ on φ. We show that the spectrum of∆φ (Dirichlet or Neumann) has si...

We consider the three-dimensional Laplacian with a magnetic field created by an infinite rectilinear current bearing a constant current. The spectrum of the associated hamiltonian is the positive half-axis as the range of an infinity of band functions all decreasing toward 0. We make a precise asymptotics of the band function near the ground energy and we exhibit a semi-classical behavior. We p...

Let G be a graph with n vertices and m edges. The degree sequence of G is denoted by d1 ≥ d2 ≥ · · · ≥ dn. Let A(G) and D(G) = diag(di : 1 ≤ i ≤ n) be the adjacency matrix and the degree diagonal matrix of G, respectively. The Laplacian matrix of G is L(G) = D(G) − A(G). It is well known that L(G) is a symmetric, semidefinite matrix. We assume the spectrum of L(G), or the Laplacian spectrum of ...

A simple statistical model of azimuthal and temporal dispersion in mobile radio channels is proposed. The model includes the probability density function (pdf) of the delay and azimuth of the impinging waves as well as their expected power conditioned on the delay and azimuth. The statistical properties are extracted from macrocellular measurements conducted in a variety of urban environments. ...