Quotients of uniform spaces

This paper develops the basic theory of quotients of uniform spaces via sufficiently nice group actions. We generalize and unify two fundamental constructions: quotients of topological groups via closed normal subgroups and quotients of metric spaces via actions by isometries. Basic results about inverse limits of topological groups are extended to inverse limits of group actions on uniform spaces, and notions of prodiscrete action and generalized covering map are introduced.     

The Riesz decomposition of finely superharmonic functions

This paper answers a question of Fuglede about minimal positive harmonic functions associated with irregular boundary points. As a consequence, an old and central problem of fine potential theory, concerning the Riesz decomposition, is resolved. Namely, it is shown that, on certain fine domains, there exist positive finely superharmonic functions which do not admit any positive finely harmonic minorant and yet are not fine potentials.     

Regular variation of order 1 nonlinear AR-ARCH models

We prove both geometric ergodicity and regular variation of the stationary distribution for a class of nonlinear stochastic recursions that includes nonlinear AR-ARCH models of order 1. The Lyapounov exponent for the model, the index of regular variation and the spectral measure for the regular variation all are characterized by a simple two-state Markov chain.     

A large deviation principle for 2D stochastic Navier–Stokes equation

In this paper one specifies the ergodic behavior of the 2D-stochastic Navier–Stokes equation by giving a Large Deviation Principle for the occupation measure for large time. It describes the exact rate of exponential convergence. The considered random force is non-degenerate and compatible with the strong Feller property.     

A non-isothermal Ginzburg–Landau model in superconductivity: Existence,uniqueness and asymptotic behaviour

In this paper we study a Ginzburg–Landau model which describes the behaviour of a superconducting material including thermal effects. We extend the traditional formulation of the problem, by introducing the temperature as an additional state variable. Accordingly, together with the Gor’kov–Eliashberg system, we introduce an evolution equation for the absolute temperature. We examine in detail the case which allows only variations of the concentration of superconducting electrons and of the temperature, neglecting the electromagnetic field. For this problem existence and uniqueness of the solution are shown. Finally we analyze the asymptotic behaviour of the solutions, proving that the system possesses a global attractor.     

Another independent self-dual basis for the trivial variety

We give a new independent self-dual 3-basis for the trivial variety with two binary operations. Received October 24, 2006; accepted in final form January 25, 2007.     

Fire containment in grids of dimension three and higher

We consider a deterministic discrete-time model of fire spread introduced by Hartnell [Firefighter! an application of domination, Presentation, in: 20th Conference on Numerical Mathematics and Computing, University of Manitoba in Winnipeg, Canada, September 1995] and the problem of minimizing the number of burnt vertices when a fixed number of vertices can be defended by firefighters per time step. While only two firefighters per time step are needed in the two-dimensional lattice to contain any outbreak, we prove a conjecture of Wang and Moeller [Fire control on graphs, J. Combin. Math. Combin. Comput. 41 (2002) 19-34] that 2d-1 firefighters per time step are needed to contain a fire outbreak starting at a single vertex in the d-dimensional square lattice for d?3; we also prove that in the d-dimensional lattice, d?3, for each positive integer f there is some outbreak of fire such that f firefighters per time step are insufficient to contain the outbreak. We prove another conjecture of Wang and Moeller that the proportion of elements in the three-dimensional grid P_n×P_n×P_n which can be saved with one firefighter per time step when an outbreak starts at one vertex goes to 0 as n gets large. Finally, we use integer programming to prove results about the minimum number of time steps needed and minimum number of burnt vertices when containing a fire outbreak in the two-dimensional square lattice with two firefighters per time step.     

The essential Lagrangian-Grassmannian and the homotopy type of the Fredholm Lagrangian-Grassmannian

Let H be a separable infinite dimensional Hilbert space endowed with a symplectic structure and let L₀⊂H be a Lagrangian subspace. Using the results of [A. Abbondandolo, P. Majer, Infinite dimensional Grassmannians, math.AT/0307192], we show that the Fredholm Lagrangian-Grassmannian FL₀(Λ) has the homotopy type of Gc(L₀), the Grassmannian of all Lagrangian subspaces of H that are compact perturbations of L₀. It is well known that the latter has the homotopy type of the quotient U(∞)/O(∞). As a corollary, we recover a result by B. Booss-Bavnbek and K. Furutani (see [B. Booss-Bavnbek, K. Furutani, Symplectic functional analysis and spectral invariants, Contemp. Math. 242 (1999) 53-83; K. Furutani, Fredholm-Lagrangian-Grassmannian and the Maslov index, J. Geom. Phys. 51 (2004) 269-331]) that the L₀-Maslov index is an isomorphism between the fundamental group of FL₀(Λ) and the integers.