Circulants,displacements and decompositions of matrices

In this paper are suggested new formulas for representation of matrices and their inverses in the form of sums of products of factor circulants, which are based on the analysis of the factor cyclic displacement of matrices. The results in applications to Toeplitz matrices generalized the Gohberg-Semencul, Ben-Artzi-Shalom and Heinig-Rost formulas and are useful for complexity analysis.     

An extension of Edelstein's theorem to a uniform space

Summary The paper contains fixed point theorems of Edelstein's type for contractive mappings in (,)-chainable uniform spaces.     

Versions of separability in bitopological spaces

We study selective versions of separability in bitopological spaces. In particular, we investigate these properties in function spaces endowed with the topology of pointwise convergence and the compact-open topology.     

Duality in Function Spaces

In this paper we use Bartle’s technique to study duality between a topological space and a function space. Normally such a duality forms an essential part of Functional Analysis. We introduce several new topologies such as the topology of even convergence T_e, the closed-cocompact topology T_{k ′}, the (strong) local proximal convergence. We explore the topological groups of self-homeomorphisms of a topological space and shed light on the earlier work of Arens, Dieudonné, Di Concilio. We also study the concepts such as evenly equidistant, functionally equicontinuous, due to Bouziad-Troallic and topologically equicontinuous due to Royden. In memory of Professor Enrico Meccariello who made a considerable contribution to this work and who suddenly passed away before his time     

Infinite analogues of block toeplitz matrices and related orthogonal functions

This paper concerns a class of infinite block matrices that are analogous to finite block Toeplitz matrices. Also studied are corresponding matrix-valued functions that are orthogonal for a matrixvalued inner product. An appendix presents basic results on orthogonalization in a Hilbert module.     

On the density of the space of continuous and uniformly continuous functions

For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent.     

A Real Analyticity Result for Symmetric Functions of the Eigenvalues of a Domain-Dependent Neumann Problem for the Laplace Operator

Let Ω be an open connected subset of for which the imbedding of the Sobolev space W ^1,2(Ω) into the space L ²(Ω) is compact. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset (Ω) of , where is a Lipschitz continuous homeomorphism of Ω onto (Ω). Then we prove a result of real analytic dependence for symmetric functions of the eigenvalues upon variation of . This paper represents an extension of a part of the work performed by P.D. Lamberti in his PhD Thesis at the University of Padova under the guidance of M. Lanza de Cristoforis.     

Selective separability and its variations

A space X is said to be selectively separable (=M-separable) if for each sequence {Dn:n∈ω} of dense subsets of X, there are finite sets Fn⊂Dn (n∈ω) such that ?{Fn:n∈ω} is dense in X. On selective separability and its variations, we show the following: (1) Selective separability, R-separability and GN-separability are preserved under finite unions; (2) Assuming CH (the continuum hypothesis), there is a countable regular maximal R-separable space X such that X² is not selectively separable; (3) c{0,1} has a selectively separable, countable and dense subset S such that the group generated by S is not selectively separable. These answer some questions posed in Bella et al. (2008) [7].     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

Perturbation of the mapping and solvability theorems in Banach space

Here we consider perturbations of continuous mappings on Banach spaces, and investigate their images under various conditions. Consequently we study the solvability of some classes of equations and inclusions. For these we start by the investigation of local properties of the considered mapping and local comparisons of this mapping with certain smooth mappings. Moreover, we study different mixed problems.     

Exponential Stability for Discrete Time Linear Equations Defined by Positive Operators

In this paper the problem of exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear positive operators acting on an ordered finite dimensional Hilbert space is investigated. The class of linear equations considered in this paper contains as particular cases linear equations described by Lyapunov operators or symmetric Stein operators as well as nonsymmetric Stein operators. Such equations occur in connection with the problem of mean square exponential stability for a class of difference stochastic equations affected by independent random perturbations and Markovian jumping as well us in connection with some iterative procedures which allow us to compute global solutions of discrete time generalized symmetric or nonsymmetric Riccati equations. The exponential stability is characterized in terms of the existence of some globally defined and bounded solutions of some suitable backward affine equations (inequalities) or forward affine equations (inequalities).     

When is a Volterra space Baire?

In this paper, we study the problem when a Volterra space is Baire. It is shown that every stratifiable Volterra space is Baire. This answers affirmatively a question of Gruenhage and Lutzer in [G. Gruenhage, D. Lutzer, Baire and Volterra spaces, Proc. Amer. Math. Soc. 128 (2000) 3115-3124]. Further, it is established that a locally convex topological vector space is Volterra if and only if it is Baire; and the weak topology of a topological vector space fails to be Baire if the dual of the space contains an infinite linearly independent pointwise bounded subset.     

Multi-valued contraction mappings in metric spaces

Summary Some fixed point theorems for multi-valued contraction mappings in metric spaces, related to a fixed point theorem due to T. Zamfirescu [6], [5], are presented.     

An extension of A. Ostrowski's Theorem on the round-off stability of iterations

Summary A. M. Ostrowski established the stability of the procedure of successive approximations for Banach contractive maps. In this paper we generalize the above result by using a more general contractive definition introduced by F. Browder. Further, we study the case of maps on metrically convex metric spaces and compact metric spaces, obtaining results relative to fixed point theorems of D. W. Boyd and J. S. W. Wong, and M. Edelstein. Finally, as a by-product of our basic lemma, we extend a recent result of T. Vidalis concerning the convergence of an iteration procedure involving an infinite sequence of maps.     

Some recent results in metric fixed point theory

This is a survey of recent results on best approximation and fixed point theory in certain geodesic spaces. Some of these results are related to fundamental fixed point theorems in topology that have been known for many years. However the metric approach is emphasized here. Dedicated to Edward Fadell and Albrecht Dold on the occasion of their 80th birthdays     

On strongly replete lattices,support of A measure,and the wallman remainder

LetX be an abstract set andL a lattice of subsets ofX. To eachL-regular measure on the algebra generated byL, there are associated two measures on appropriate algebras of the Wallman space. In terms of these measures, we can obtain characterization for-smoothness,-smoothness, and tightness of the original measure. In particular, tight regular measures and their properties are investigated.     

Some remarks on measures of noncompactness and retractions onto spheres

Some ways to obtain upper and lower bounds for measures of noncompactness of retractions onto spheres in infinite-dimensional normed spaces are discussed. Moreover, relations with 0-epi maps are revealed and the extension of condensing maps on spheres is discussed. As an application, some results of Birkhoff-Kellog type and Nussbaum's fixed point theorem on spheres are obtained.