Study of a model for the distribution of wealth

An equation for the evolution of the distribution of wealth in a population of economic agents making binary transactions with a constant total amount of "money" has recently been proposed by one of us (RLR). This equation takes the form of an iterated nonlinear map of the distribution of wealth. The equilibrium distribution is known and takes a rather simple form. If this distribution is such that, at some time, the higher momenta of the distribution exist, one can find exactly their law of evolution. A seemingly simple extension of the laws of exchange yields also explicit iteration formulae for the higher momenta, but with a major difference with the original iteration because high order momenta grow indefinitely. This provides a quantitative model where the spreading of wealth, namely the difference between the rich and the poor, tends to increase with time.Comment: 12 pages, 2 figure

Iron K-alpha Emission from X-ray Reflection: Predictions for Gamma-Ray Burst Models

Recent observations of several gamma-ray burst (GRB) afterglows have shown evidence for a large amount of X-ray line emitting material, possibly arising from ionized iron. A significant detection of an X-ray spectral feature, such as that found in the Chandra observation of GRB 991216, may provide important constraints on the immediate environment of the burst and hence on progenitor models. The large Fe K-alpha equivalent widths inferred from the X-ray observations favor models in which the line is produced when the primary X-ray emission from the source strikes Thomson-thick material and Compton scatters into our line of sight. We present such reflection spectra here, computed in a fully self-consistent manner, and discuss the range of ionization parameters that may be relevant to different models of GRBs. We argue that the presence of a strong hydrogen-like K-alpha line is unlikely, because Fe-XXVI photons would be trapped resonantly and removed from the line core by Compton scattering. In contrast, a strong narrow emission line from He-like Fe-XXV is prominent in the model spectra. We briefly discuss how these constraints may affect the line energy determination in GRB 991216.Comment: 8 pages, 3 figures, Ap.J. Letters accepte

Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

We introduce a family of hybrid discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid approximation) measured in suitable norms are derived, showing optimal orders of convergence

Free Fermionic Elliptic Reflection Matrices and Quantum Group Invariance

Elliptic diagonal solutions for the reflection matrices associated to the elliptic $R$ matrix of the eight vertex free fermion model are presented. They lead through the second derivative of the open chain transfer matrix to an XY hamiltonian in a magnetic field which is invariant under a quantum deformed Clifford--Hopf algebra.Comment: 9 pages, Late