Abstract: A look at connections between partial differential equations and special functions from an historical and personal viewpoint, with some observations about special functions in general and Meijer G-functions in particular.

Abstract: Tournament graphs are directed graphs with an (asymmetric) adjacency ma­trix D which summarizes the result of regular round-robin tournaments between N players: Every player plays against all the others, if i wins against j then D_i,j = 1 and D_j,i = 0. Clearly D_i,i = 0. If N is odd then a tournament can be regular each player wins exactly half the times.
To construct regular tournaments we introduced a random walk in the space of the tournament adjacency matrices which will be argued to be ergodic.
The spectrum of D for regular tournaments consists of one point on the real axis, the rest are in the complex plane, all with real part = 1/2. The spectral statistics on the "critical lime" will be studied using a trace formula for the spectral density. For large N it will be shown that the mean spectral density approaches the semi-circle law. Moreover, numerical simulations supported by theoretical arguments derived by using the trace formula, show that the spectral statistics is consistent with the predictions of the Gaussian Unitary Ensemble of random matrices.

Abstract: We model constitutions by effectivity functions. We assume that the constitution is common knowledge among the members of the society. However, the preferences of the citizen are private information. We investigate whether there exist decision schemes (i. e., functions that map profiles of (dichotomous) preferences on the set of outcomes to lotteries on the set of social states), with the following properties: i) The distribution of power induced by the decision scheme is identical to the effectivity function under consideration; and ii) the (incomplete information) game associated with the decision scheme has a Bayesian Nash equilibrium in pure strategies. If the effectivity function is monotonic and superadditive, then we find a class of decision schemes with the foregoing properties.
When applied to n-person games in strategic form, a decision scheme d is a mapping from profiles of (dichotomous) preferences on the set of pure strategy vectors to probability distributions over outcomes ( or, equivalently, over pure strategy vectors). We prove that for every feasible and individually rational payoff vector of the strategic game there exists a decision scheme that yields the payoff vector as a (pure) Nash equilibrium payoff in the game induced by the strategic game and the decision scheme. This can be viewed as a kind of purification result.
This is a joint work with Shmuel Zamir.

Abstract: A drumhead of which shape produces the lowest possible sound among all drumheads of given area? An answer to this simplest problem of geometric optimization of eigenvalues is given by the Rayleigh-Faber-Krahn inequality, stating that the optimal shape here is the disc. Different problems of geometric optimization for eigenvalues of the Laplacian on Euclidean domains and of its cousin, Laplace-Beltrami operator on surfaces, will be discussed.

Abstract: An overview is given of the utilization of strategic market games in the development of a game theory based theory of money and financial institutions.
The accompanying discussion paper is available from the Cowles foundation repository.