Manuel Morán

Porosity, σ-porosity and measures

We show that given a σ-finite Borel regular measure μ in a metric space X, every σ-porous subset of X of finite measure can be approximated by strongly porous sets. It follows that every σ-porous set is the union of a σ-strongly porous set and a μ-null set. This answers in the positive the question whether a measure which is absolutely continuous with respect to the σ-ideal of all σ-strongly porous sets is absolutely continuous with respect to the σ-ideal of all σ-porous sets. Using these results, we obtain a natural decomposition of measures according to their upper porosity and obtain detailed information on values that upper porosity may attain almost everywhere.

Singularity of self-similar measures with respect to Hausdorff measures

Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base-p expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets) self-similar measures? and Hausdorff measures3. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-sitnilar meas;ures with respect to those measures. We show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the Law of the Iterated Logarithm.

Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle

We state a self-similar tiling principle which shows that any open subset of a self-similar set with open set condition may be tiled without loss of measure by copies under similitudes of any closed subset with positive measure. We use this method to get the optimal coverings and packings which give the exact value of the Hausdorff-type and packing measures. In particular, we show that the exact value of these measures coincides with the supremum or with the infimum of the inverse of the density of the natural probability measure on suitable classes of sets. This gives criteria for the numerical analysis of the measures, and allows us to compare their complexity in terms of computability.

General dynamics in overlapping generations models

Abstract In this article we analyse the dynamics generated by the equilibrium solution in an overlapping generations model with production. We adopt an inverse approach to characterise and construct the class of economies that generates any twice continuously differentiable dynamics. To do this we introduce a technique based on the theory of partial differential equations.

Lipschitz continuous dynamic programming with discount II

We construct an alternative theoretical framework for stochastic dynamic programming which allows us to replace concavity assumptions with more flexible Lipschitz continuous assumptions. This framework allows us to prove that the value function of stochastic dynamic programming problems with discount is Lipschitz continuous in the presence of nonconcavities in the data of the problem. Our method allows us to treat problems with noninterior optimal paths. We also describe a discretization algorithm for the numerical computation of the value function, and we obtain the rate of convergence of this algorithm. c

Problems on Self-similar Geometry

Abstract. We discuss some results and open questions in the field of classical self-similar constructions: Boundaries of self-similar sets with open set condition; the dimension of a self-similar set with a big overlapping; the singularity of self-similar measures with respect to Hausdorff and packing measures and a variational property of self-similar measures which plays a role in multifractality. The multidimensional Legendre Transform is shown to satisfy the multifractal formalism for intersections of several multifractal layers relative to different self-similar measures.

Potential Collapse in Fisheries with Increasing Returns and Stock-dependent Costs

Abstract We develop a bioeconomic model to analyze a fishery with fixed costs as well as a within-season continuous cost function for the generalized Schaefer production function with increasing marginal returns to effort level. We analyze the consequences of the combined effects of increasing marginal returns and fixed costs. We find that regardless of the magnitude of the fixed costs, cyclical policies are optimal. We also demonstrate that the danger of potential collapse increases with increasing fixed costs. This result is quite counterintuitive, as higher costs are usually considered to have a conservative effect on resources. JEL Classification Codes: Q20, Q22, Q57

An algorithm for computing the centered Hausdorff measures of self-similar sets ☆

Abstract We provide an algorithm for computing the centered Hausdorff measures of self-similar sets satisfying the strong separation condition. We prove the convergence of the algorithm and test its utility on some examples.

Degrees of Freedom of a Time Series

We give a formal proof that if f is a smooth dynamics on a d-dimensional smooth manifold and μ is an ergodic and exact dimensional measure with Hausdorff dimension dim μ>d−1, then the number d of degrees of freedom of the dynamics can be recovered from the observation of an orbit. We implement, with this purpose, an algorithm based on the analysis of the microstructure of μ. We show how a correct estimation of d permits the computation of the Liapunov spectrum with a high accuracy avoiding the issue of the spurious exponents.

Computability of the packing measure of totally disconnected self-similar sets

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called SSC and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show both, the accuracy of the algorithm for the most regular cases and the possibility of using the additional information provided by it to obtain formulas for the packing measure of certain self-similar sets. For example, we are able to obtain a formula for the packing measure of any Sierpinski gasket with contractio factor in the interval