On an obstacle problem for degenerate elliptic operators involving the critical Sobolev exponent

We establish the existence of a solution to the variational inequality (the obstacle problem) (1.1) which involves the critical Sobolev exponent. This result is also extended to an obstacle problem with a lower order perturbation. Dedicated to Professor F. Browder on the occasion of his 80-th birthday     

The Howe Duality for the Dunkl Version of the Dirac Operator

The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|². A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. ?rsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. K?hler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš to recall our long lasting cooperation on many topics in mathematics The work presented here is a part of the research project MSM 0021620839 and was supported also by the grant GA ČR 201/05/2117. Received: October, 2007. Accepted: February, 2008.     

On a class of almost hypoelliptic operators in generalized Sobolev spaces

The paper considers differential operators in the generalized Sobolev spaces. Differring necessary and sufficient conditions on the polyhedron ? are found, under which the function h _? ^s satisfies the Beurling condition for s great enough. They coincide (become necessary and sufficient) for polyhedrons in ? ₊ ² . The regularity of a class of differential equations is investigated.     

The necessary and sufficient conditions for the qualified convergence of difference methods for approximate solution of the ill-posed Cauchy problem in a Banach space

We study properties of finite-difference methods for approximate solution of the ill-posed Cauchy problem for a homogeneous equation of the first order with a sectorial operator in a Banach space. We obtain the necessary and sufficient conditions for the qualified (with respect to the step of grid) uniform (on a segment) convergence of approximations to an exact solution of the problem. These conditions represent a priori data about the segment, where a solution exists, or about the sourcewise representation of a certain value of the desired solution.     

On the gluing problem for Dirac operators on manifolds with cylindrical ends

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.     

A fixed point result in Menger spaces using a real function

The main result of this paper is a fixed point theorem of self-mappings in Menger spaces which satisfy certain inequality. This inequality involves a class of real functions which we call Φ-functions. As a corollary we obtain a result in the corresponding metric spaces. The result is supported by an example. The class of real functions we have used is the conceptual extension of altering distance functions used in metric fixed point theory.      

A certain boundary-value problem with shifts in a four-dimensional space

We consider a problem with shifts in boundary conditions for the Bianchi equation in a four-dimensional space. We establish sufficient conditions which allow one to evaluate its unique solution in terms of the Riemann function.     

Quasi-interpolants from spline interpolation operators

For bi-infinite Toeplitz matrices, it is easy to see that thekth partial sum of the Neumann series reproduces polynomials of orderk There is no guarantee, however, that the spectral radius is less than 1. A principal result of this paper is to show that for the spline interpolation Toeplitz case the spectral radius is less than 1 whenA is invertible and the main diagonal is the central diagonal. This is not true for all totally positive Toeplitz matrices as shown by an example in Section 2.Communicated by Charles A. Micchelli.     

Topologies on function spaces

Function spaces play an important role in complex analysis, in the theory of differential equations, in functional analysis and in almost every other branch of modern mathematics. In this paper we give and study the notion of clopen convergence. Also, we study the notion of clopen continuity and define new topologies on function spaces. These results generalize basic results of R. Arens, J. Dugundji and A. Di Concilio (see [1], [4], [2] and [3]).     

Growth Estimates for Cauchy Integrals of Measures and Rectifiability

We show that if μ is a finite Borel measure on the complex plane such that

A note on holomorphic approximation in Banach spaces

Given a complex Banach space X and a holomorphic function f on its unit ball B, we discuss the problem whether f can be approximated, uniformly on smaller balls, by functions g holomorphic on all of X. Research partially supported by NSF grant DMS0700281.     

On the topology of Hankel multipliers and of Hankel convolution operators

The topologies of simple convergence and of bounded convergence are shown to coincide on the spaces of Hankel multipliers and of Hankel convolution operators. The properties of these spaces being bornological, nuclear, Montel, and reflexive are established.     

A nonstationary group pursuit problem

We consider a linear nonstationary problem of conflict interaction of controlled objects, where the number of pursuers equals ν and the number of evaders equals µ. All participants are assumed to have equal dynamic abilities. The purpose of the pursuers is to catch all evaders, while the purpose of the latter is to avoid being caught for at least one of them. We establish sufficient solvability conditions for the local evasion problem.     

Continuous operators on asymmetric normed spaces

If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖⁺‖, q(y) = ‖y ⁺‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases. The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.     

The Gellerstedt problem for a parabolic-hyperbolic equation of the second kind

In this paper we consider the Gellerstedt problem for a parabolic-hyperbolic equation of the second kind. We prove the unique solvability of this problem by means of a new representation for a solution to the modified Cauchy problem in a generalized class R.     

The Leray separating operator in the theory of the Cauchy problem

The paper is devoted to the presentation of Leray’s approach to the Cauchy problem for strictly hyperbolic operators. In the first section we give the main definitions of strictly hyperbolic operators and separating operators corresponding to them. We present the plan of derivation of the a priori estimates necessary for the proof of solvability of the Cauchy problem. In the second section we generalize the Leray approach to some classes of PDO which are not hyperbolic.     

Strong Cauchy completeness in uniform frames

We show that for uniform frames, with the underlying frame being Boolean, uniform paracompactness and strong Cauchy completeness are equivalent conditions. Certain aspects of uniform paracompactness are also considered. We then introduce the pointfree notion of preparacompactness and show that the completion of a preparacompact uniform frame is strongly Cauchy complete. We also formulate pointfree filter characterizations of Lindelöfness for regular frames in analogy to their classical topological counterparts.     

On a pythagorean-type functional equation with exponent 3 involving finite fields

We solve the functional equationf(x ³ +y ³) =f(x)³ +f(y)³ for maps of a finite field into itself.     

An index theorem for open manifolds whose ends are warped products

We give an index formula for the Dirac operator acting onL2-sections of the spinor bundle of an open spin manifold with a co-compact cylindrical end over which the metric is a warped product.     

The operad of finite labeled tournaments

We study the operad of finite labeled tournaments. We describe the structure of suboperads of this operad generated by simple tournaments. We prove that a suboperad generated by a tournament with two vertices (i.e., the operad of finite linearly ordered sets) is isomorphic to the operad of symmetric groups, and a suboperad generated by a simple tournament with more that two vertices is isomorphic to the quotient operad of the free operad with respect to a certain congruence. We obtain this congruence explicitly.