Exponential wealth distribution in a random market. A rigorous explanation

In simulations of some economic gas-like models, the asymptotic regime shows an exponential wealth distribution, independently of the initial wealth distribution given to the system. The appearance of this statistical equilibrium for this type of gas-like models is explained in a rigorous analytical way.Comment: 9 pages, 4 figure

Exponential wealth distribution: a new approach from functional iteration theory

Exponential distribution is ubiquitous in the framework of multi-agent systems. Usually, it appears as an equilibrium state in the asymptotic time evolution of statistical systems. It has been explained from very different perspectives. In statistical physics, it is obtained from the principle of maximum entropy. In the same context, it can also be derived without any consideration about information theory, only from geometrical arguments under the hypothesis of equiprobability in phase space. Also, several multi-agent economic models based on mappings, with random, deterministic or chaotic interactions, can give rise to the asymptotic appearance of the exponential wealth distribution. An alternative approach to this problem in the framework of iterations in the space of distributions has been recently presented. Concretely, the new iteration given by $f_{n+1}(x) = \int\int_{u+v>x}{f_n(u)f_n(v)\over u+v} dudv.$. It is found that the exponential distribution is a stable fixed point of the former functional iteration equation. From this point of view, it is easily understood why the exponential wealth distribution (or by extension, other kind of distributions) is asymptotically obtained in different multi-agent economic models.Comment: 6 pages, 5 figure

Non-linear response of single-molecule magnets: field-tuned quantum-to-classical crossovers

Quantum nanomagnets can show a field dependence of the relaxation time very different from their classical counterparts, due to resonant tunneling via excited states (near the anisotropy barrier top). The relaxation time then shows minima at the resonant fields H_{n}=n D at which the levels at both sides of the barrier become degenerate (D is the anisotropy constant). We showed that in Mn12, near zero field, this yields a contribution to the nonlinear susceptibility that makes it qualitatively different from the classical curves [Phys. Rev. B 72, 224433 (2005)]. Here we extend the experimental study to finite dc fields showing how the bias can trigger the system to display those quantum nonlinear responses, near the resonant fields, while recovering an classical-like behaviour for fields between them. The analysis of the experiments is done with heuristic expressions derived from simple balance equations and calculations with a Pauli-type quantum master equation.Comment: 4 pages, 3 figures. Submitted to Phys. Rev. B, brief report

Smoothening Functions and the Homomorphism Learning Problem

This thesis is an exploration of certain algebraic and geometrical aspects of the Learning With Errors (LWE) problem introduced in Reg05. On the algebraic front, we view it as a Learning Homomorphisms with Noise problem, and provide a generic construction of a public-key cryptosystem based on this generalization. On the geometric front, we explore the importance of the Gaussian distribution for the existing relationships between LWE and lattice problems. We prove that their smoothing properties does not make them special, but rather, the fact that it is infinitely divisible and l2 symmetric are important properties that make the Gaussian unique