Based on his earlier work on the vibrations of 'drums with fractal boundary', the first author has refined M. V. Berry's conjecture that extended from the 'smooth' to the 'fractal' case H. Weyl'sâ€¦ (More)

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where eachâ€¦ (More)

Let QJ be a bounded open set of RDn (n > 1) with "fractal" boundary F. We extend Hermann Weyl's classical theorem by establishing a precise remainder estimate for the asymptotics of the eigenvaluesâ€¦ (More)

Tube formulas (by which we mean an explicit formula for the volume of an Îµ-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the subset.â€¦ (More)

In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi [7] on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques,â€¦ (More)

We construct a nonarchimedean (or p-adic) analogue of the classical ternary Cantor set C. In particular, we show that this nonarchimedean Cantor set C3 is self-similar. Furthermore, we characterizeâ€¦ (More)

We construct Dirac operators and spectral triples for certain, not necessarily selfsimilar, fractal sets built on curves. Connesâ€™ distance formula of noncommutative geometry provides a natural metricâ€¦ (More)

A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by the second author and H.Maier in terms of an inverse spectral problem for fractal strings. The inverse spectral problemâ€¦ (More)

We define a one-parameter family of geometric zeta functions for a Borel measure on the unit interval and a sequence which tends to zero. The construction of this family is based on that of theâ€¦ (More)