Overview
Discrete Wigner distributions have gained a lot of importance due to their applications in Quantum Information. In this article we examine the Wigner functions of a system whose coordinate and momenta take values over a finite field with prime characteristic. We follow an algebraic approach which approaches the problem by defining a Kernel and taking its square root. The properties of the Displacement operators are used to get an elegant way of constructing the phase point operators. The choice of signs which arise in taking the square root are fixed by considering the marginals property of the phase point operators. We find that, the Wigner functions obtained is consistent with those obtained by Wootters et al, in spite of the fact that our method is very different from that described by them. The method also posses the advantage that one need not have explicit knowledge of the mutually unbiased basis in order to be able to construct the Wigner functions.

Overview
We present details of an agent-based model that is capable of reproducing almost all the observed stylized facts of financial markets. We extend the results discussed in the article titled "A Mean-Field Model of Financial Markets: Reproducing long tailed distributions and volatility correlations" (see below). The model consists of two rules according to which each agent in the market trades. We are currently analysing how these simple set of rules give rise to the observed long-tailed distributions for the return and the volume of trade distribution to help us come up with a semi-analytical theory of the same.

Abstract
A model for financial market activity should reproduce the several stylized facts that have been observed to be invariant across different markets and periods. Here we present a mean-field model of agents trading a particular asset, where their decisions (to buy or to sell or to hold) is based exclusively on the price history. As there are no direct interactions between agents, the price (computed as a function of the difference between the numbers of buyers and sellers at a given time) is the sole mediating signal driving market activity. We observe that this simple model reproduces the long-tailed distribution of price fluctuations (measured by logarithmic returns) and trading volume (measured in terms of the number of agents trading at a given instant), that has been seen in most markets across the world. By using a quenched random distribution of a model parameter that governs the risk avoidance nature of an agent, we obtain quantitatively accurate exponents for the two distributions. In addition, the model exhibits volatility clustering, i.e., correlation between periods with large price fluctuations, remarkably similar to that seen in reality. To the best of our knowledge, this is the simplest model that gives a quantitatively exact description of financial market behavior.