Selberg\u27s trace formula on the k-regular tree and applications

We survey graph theoretic analogues of the Selberg trace and pretrace formulas along with some applications. This paper includes a review of the basic geometry of a k-regular tree E (symmetry group, geodesies, horocycles, and the analogue of the Laplace operator). A detailed discussion of the spherical functions is given. The spherical and horocycle transforms are considered (along with three basic examples, which may be viewed as a short table of these transforms). Two versions of the pretrace formula for a finite connected k-regular graph ×□Γ\Ξ are given along with two applications. The first application is to obtain an asymptotic formula for the number of closed paths of length r in × (without backtracking but possibly with tails). The second application is to deduce the chaotic properties of the induced geodesic flow on × (which is analogous to a result of Wallace for a compact quotient of the Poincaré upper half plane). Finally, the Selberg trace formula is deduced and applied to the Ihara zeta function of X, leading to a graph theoretic analogue of the prime number theorem. © 2003 Hindawi Publishing Corporation. All rights reserved

On the digraph of a unitary matrix

Given a matrix M of size n, a digraph D on n vertices is said to be the digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, LD, is the digraph of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and LD is the digraph of a unitary matrix then LD is Hamiltonian. We conclude with some elementary observations. Among the motivations of this paper are coined quantum random walks, and, more generally, discrete quantum evolution on digraphs.Comment: 6 page

Fourier Analysis on Finite Groups and Applications

A friendly introduction to Fourier analysis on finite groups, accessible to undergraduates/graduates in mathematics, engineering and the physical sciences

Fourier analysis on finite groups and applications

A friendly introduction to Fourier analysis on finite groups, accessible to undergraduates/graduates in mathematics, engineering and the physical sciences

Harmonic analysis on symmetric spaces

This text explores the geometry and analysis of higher rank analogues of the symmetric spaces introduced in volume one. To illuminate both the parallels and differences of the higher rank theory, the space of positive matrices is treated in a manner mirroring that of the upper-half space in volume one. This concrete example furnishes motivation for the general theory of noncompact symmetric spaces, which is outlined in the final chapter. The book emphasizes motivation and comprehensibility, concrete examples and explicit computations (by pen and paper, and by computer), history, and, above all, applications in mathematics, statistics, physics, and engineering. The second edition includes new sections on Donald St. P. Richards’s central limit theorem for O(n)-invariant random variables on the symmetric space of GL(n, R), on random  matrix theory, and on advances in the theory of automorphic forms on arithmetic groups