Sums of regular Cantor sets of large dimension and the Square Fibonacci Hamiltonian

We show that under natural technical conditions, the sum of a $C^2$ dynamically defined Cantor set with a compact set in most cases (for almost all parameters) has positive Lebesgue measure, provided that the sum of the Hausdorff dimensions of these sets exceeds one. As an application, we show that for many parameters, the Square Fibonacci Hamiltonian has spectrum of positive Lebesgue measure, while at the same time the density of states measure is purely singular.Comment: 13 page

Formal GAGA for good moduli spaces

We prove formal GAGA for good moduli space morphisms under an assumption of "enough vector bundles" (which holds for instance for quotient stacks). This supports the philosophy that though they are non-separated, good moduli space morphisms largely behave like proper morphisms.Comment: 16 pages (updated to match published numbering

Absolutely Continuous Convolutions of Singular Measures and an Application to the Square Fibonacci Hamiltonian

We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.Comment: 28 pages, to appear in Duke Math.

Linear State Space Modeling of Gamma-Ray Burst Lightcurves

Linear State Space Modeling determines the hidden autoregressive (AR) process in a noisy time series; for an AR process the time series' current value is the sum of current stochastic ``noise'' and a linear combination of previous values. We present preliminary results from modeling a sample of 4 channel BATSE LAD lightcurves. We determine the order of the AR process necessary to model the bursts. The comparison of decay constants for different energy bands shows that structure decays more rapidly at high energy. The resulting models can be interpreted physically; for example, they may reveal the response of the burst emission region to the injection of energy.Comment: 5 pages, 2 figures, AIPPROC LaTeX, to appear in "Gamma-Ray Bursts, 4th Huntsville Symposium," eds. C. Meegan, R. Preece and T. Koshu

Stability, creation and annihilation of charges in gauge theories

We show how to construct physical, minimal energy states for systems of static and moving charges. These states are manifestly gauge invariant. For charge-anticharge systems we also construct states in which the gauge fields are restricted to a finite volume around the location of the matter fields. Although this is an excited state, it is not singular, unlike all previous finite volume descriptions. We use our states to model the processes of pair creation and annihilation.Comment: Six EPS figures. Version 2: minor typos correcte

Spectral Properties of Schr\"odinger Operators Arising in the Study of Quasicrystals

We survey results that have been obtained for self-adjoint operators, and especially Schr\"odinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus our attention on the one-dimensional case, and in particular on several key examples. The most prominent of these is the Fibonacci Hamiltonian, for which much is known by now and to which an entire section is devoted here. Other examples that are discussed in detail are given by the more general class of Schr\"odinger operators with Sturmian potentials. We put some emphasis on the methods that have been introduced quite recently in the study of these operators, many of them coming from hyperbolic dynamics. We conclude with a multitude of numerical calculations that illustrate the validity of the known rigorous results and suggest conjectures for further exploration.Comment: 56 page