For appropriate sets of regular cardinals $a$, we show that for every cardinal $\lambda$ there is a subset $c$ of $a$ that generates $J_{< \lambda^+}(a)$ over $J_{< \lambda}(a)$.

For a bounded domain, we consider the L^\infty-functional involving a nonnegative Hamilton function. Under the continuous Dirichlet boundary condition and some assumptions of Hamiltonian H, the uniqueness of absolute minimizers for Hamiltonian H is established. This extendes the uniqueness theorem to a larger class of Hamiltonian $H(x,p)$ with $x$-dependence. As a corollary, we confirm an open question on the uniqueness of absolute minimizers posed by Jensen-Wang-Yu. Our proofs rely on geometric structure of the action function induced by Hamiltonian H(x,p), and the identification of the absolute subminimality with convexity of the associated Hamilton-Jacobi flow.  

In the classical Polya’s urn process, there are balls of different colors in the urn, and one step of the process consists of taking out a random ball and it putting back together with one more ball of the same color. ("Ask a friend whether he’s using is A or B, and buy the same".)

It also can be modified by saying that the reinforcement probability is proportional not to the number of balls of a given color, but to its power $\alpha$, and the asymptotic behaviour for $\alpha=1$, $\alpha>1$ and for $\alpha<1$ are quite different.

The talk will be devoted to the model of interacting urns : at each moment, there is a competition for reinforcement between randomly chosen subset of colours; for a real-life analogue, one can consider companies competing on different markets (one company produces toys and computers, another sells computers and cars, etc.).

We will describe possible types of the limit behaviour of such model for different values of $\alpha$; it turns out that what happens for $\alpha>>1$ is quite different from $\alpha=1$, and both are quite interesting (this is a joint work with Christian Hirsch and Mark Holmes).