The Jacobian and the Ginzburg-Landau energy

We study the Ginzburg-Landau functional for , where U is a bounded, open subset of . We show that if a sequence of functions satisfies , then their Jacobians are precompact in the dual of for every . Moreover, any limiting measure is a sum of point masses. We also characterize the -limit of the functionals , in terms of the function space B2V introduced by the authors in [16,17]: we show that I(u) is finite if and only if , and for is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if then the Jacobians are again precompact in for all , and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation of the Jacobian measure is a lower bound for the limit of the Ginzburg-Landau functional. Received: 15 December 2000 / Accepted: 23 January 2001 / Published online: 25 June 2001     

Lévy processes and their subordination in matrix Lie groups

Lévy processes in matrix Lie groups are studied. Subordination (random time change) is used to show that quasi-invariance of the Brownian motion in a Lie group induces absolute continuity of the laws of the corresponding pure jump processes. These results are applied to several examples which are discussed in detail.     

On the Log-Laplace Equation for Nonlocal Operators Generating Sub-Markovian Semigroups

We consider the semilinear Cauchy problem for a class of pseudo-differential operators generating sub-Markovian semigroups. Solutions of such problems with negative definite nonlinearity play an important role in constructing branching measure-valued processes. We establish local existence and uniqueness of solutions in the context of the Dirichlet space associated to the problem. Comparison and global properties of solutions are also studied. Accepted 29 August 2001. Online publication 17 December 2001.     

The quantum de Rham complexes associated with SL h (2)

Quantum de Rham complexes on the quantum plane and the quantum group itself are constructed for the nonstandard deformation of Fun(SL(2)). It is shown that in contrast to the standardq-deformation of SL(2), the above complexes are unique for SL _h (2). Also, as a byproduct, a new deformation of the two-dimensional Heisenberg algebra is obtained which can be used to construct models ofh-deformed quantum mechanics.     

Global solution to a phase transition model based on a microforce balance

Our aim in this article is to prove the existence and the uniqueness of a global solution to a nonisothermal Ginzburg-Landau (Allen-Cahn) system. This system is obtained by considering, in addition to the fundamental laws of thermodynamics, a balance law for internal microforces proposed by M. Gurtin.     

The Hochschild class of the Chern character for semifinite spectral triples

We provide a proof of Connes’ formula for a representative of the Hochschild class of the Chern character for (p,∞)-summable spectral triples. Our proof is valid for all semifinite von Neumann algebras, and all integral p?1. We employ the minimum possible hypotheses on the spectral triples.     

On essential extensions of reduced rings and domains

A ring is said to be a left essential extension of a reduced ring (domain) if it contains a left ideal which is a reduced ring (domain) and intersects nontrivially every nonzero twosided ideal of the ring. We prove that every ring which is a left essential extension of a reduced ring is a subdirect sum of rings which are essential extensions of domains, but the converse implication does not hold. We give some applications of this result and discuss several related questions.Received: 6 January 2003     

Spectral Properties of Image Measures Under the Infinite Conflict Interaction

We introduce the conflict interaction with two positions between a couple of image probability measures and consider the associated dynamical system. We prove the existence of invariant limiting measures and find the criteria for these measures to be a pure point, absolutely continuous, or singular cotinuous as well as to have any topological type and arbitary Hausdorff dimension.