What is the probability that a random integral quadratic form in nn variables has an integral zero?

We show that the density of quadratic forms in $n$ variables over $\mathbb Z_p$ that are isotropic is a rational function of $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. When real quadratic forms in $n$ variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite). As a consequence, for each $n$, we determine an exact expression for the probability that a random integral quadratic form in $n$ variables is isotropic (i.e., has a nontrivial zero over $\mathbb Z$), when these integral quadratic forms are chosen according to the GOE distribution. In particular, we find an exact expression for the probability that a random integral quaternary quadratic form has an integral zero; numerically, this probability is approximately $98.3\%$.Comment: 17 pages. This article supercedes arXiv:1311.554

One-loop Effective Actions in Shape-invariant Scalar Backgrounds

The field-theoretic one-loop effective action in a static scalar background depending nontrivially on a single spatial coordinate is related, in the proper-time formalism, to the trace of the evolution kernel (or heat kernel) for an appropriate, one dimensional, quantum-mechanical Hamiltonian. We describe a recursive procedure applicable to these traces for shape-invariant Hamiltonians, resolving subtleties from the continuum mode contributions by utilizing the expression for the regularized Witten index. For some cases which include those of domain-wall-type scalar backgrounds, our recursive procedure yields the full expression for the scalar or fermion one-loop effective action in both (1+1) and (3+1)-dimensions.Comment: 11 pages, LaTeX2

EQUITY-PREMIUM PUZZLE: EVIDENCE FROM BRAZILIAN DATA

This paper uses 1992:1-2004:2 quarterly data and two diferent methods (approximation under lognormality and calibration) to evaluate the existence of an equity- premium puzzle in Brazil. In contrast with some previous works in the Brazilian literature, I conclude that the model used by Mehra and Prescott (1985), either with additive or recursive preferences, is not able to satisfactorily rationalize the equity premium observed in the Brazilian data. The second contribution of the paper is calling the attention to the fact that the utility function calculated under the discrete-state approximation may not exist if the data (as it is the case with Brazilian time series) implies the existence of states in which high negative rates of consumption growth are attained with relatively high probability.