Existence of 3D-Vortices in Abelian Gauge Theories

We consider the Abelian gauge theory related to the interaction between the solitary waves of the nonlinear Klein-Gordon equation and the electromagnetic field. We analyze the existence of electromagnetostatic solutions and, in particular, of vortices. Dedicated to the memory of Professor Aldo Cossu     

Approximations of relations by continuous functions

Let X be a Tychonoff space, C(X) be the space of all continuous real-valued functions defined on X and CL(X×R) be the hyperspace of all nonempty closed subsets of X×R. We prove the following result. Let X be a countably paracompact normal space. The following are equivalent: (a) dimX=0; (b) the closure of C(X) in CL(X×R) with the Vietoris topology consists of all F∈CL(X×R) such that F(x)≠∅ for every x∈X and F maps isolated points into singletons; (c) each usco map which maps isolated points into singletons can be approximated by continuous functions in CL(X×R) with the locally finite topology. From the mentioned result we can also obtain the answer to Problem 5.5 in [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] and to Question 5.5 in [R.A. McCoy, Comparison of hyperspace and function space topologies, Quad. Mat. 3 (1998) 243-258] in the realm of normal, countably paracompact, strongly zero-dimensional spaces. Generalizations of some results from [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] are also given.     

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-Banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-Banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderón-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented. The author is supported by the National Science Foundation under grant DMS 0099881. The author is supported by KBN Grant 1 P03A 013 26.     

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.     

A topological aspect of Sobolev mappings

Let be a connected open set, . We give a sufficient condition for a mapping , , to have the property that sgn is almost everywhere of one sign. Following the work of Müller, Spector, and Tang [MST], we give an application of our results to the theory of non-linear elasticity. Received: 13 October 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001     

A mixed directed-undirected data structure for a parallel implementation of a domain decomposition algorithm

The choice of data structures influences the parallelization, efficiency and the manageability of a mesh refinement program. We introduce a mixed directed-undirected graph that combines both communication and scheduling needs. An inverted index is maintained for the directed graph to improve code performance and readability.This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.     

Algebraic properties of the 1 + 1-dimensional Heisenberg spin field model

The Estabrook-Wahlquist prolongation method is applied to the (compact and noncompact) continuous isotropic Heisenberg model in 1 + 1 dimensions. Using a special realization (an algebra of the Kac-Moody type) of the arising incomplete prolongation Lie algebra, a whole family of nonlinear field equations containing the original Heisenberg system is generated.     

Castelnuovo-Mumford regularity by approximation

The Castelnuovo-Mumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means. We apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups.     

Uniform persistence for nonautonomous and random parabolic Kolmogorov systems

The purpose of this paper is to investigate uniform persistence for nonautonomous and random parabolic Kolmogorov systems via the skew-product semiflows approach. It is first shown that the uniform persistence of the skew-product semiflow associated with a nonautonomous (random) parabolic Kolmogorov system implies that of the system. Various sufficient conditions in terms of the so-called unsaturatedness and/or Lyapunov exponents for uniform persistence of the skew-product semiflows are then provided. Among others, it is shown that if the associated skew-product semiflow has a global attractor and its restriction to the boundary of the state space has a Morse decomposition which is unsaturated or whose external Lyapunov exponents are positive, then it is uniformly persistent. More specific conditions are discussed for uniform persistence in n-species, particularly 3-species, random competitive systems.     

An analytical proof of Hardy-like inequalities related to the Dirac operator

We prove some sharp Hardy-type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.