Universal elements in some classes of mappings and classes of G-spaces

In the present paper it is proved the existence of topologically universal elements in some classes of mappings and classes of G-spaces generalizing some results from [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Math. Stud., vol. 198, Elsevier, 2005] and [S.D. Iliadis, Universal Borel mappings and Borel actions of groups, Topology Appl., in press, doi:10.1016/j.topol.2007.05.029].     

Universal mappings for certain classes of operators and polynomials between Banach spaces

A well‐known result of J. Lindenstrauss and A. Pełczyński (1968) gives the existence of a universal non‐weakly compact operator between Banach spaces. We show the existence of universal non‐Rosenthal, non‐limited, and non‐Grothendieck operators. We also prove that there does not exist a universal non‐Dunford–Pettis operator, but there is a universal class of non‐Dunford–Pettis operators. Moreover, we show that, for several classes of polynomials between Banach spaces, including the non‐weakly compact polynomials, there does not exist a universal polynomial.     

On the global theory of some classes of mappings

This paper is devoted to the study of the global theory of certain mappings between Riemannian manifolds. We generalize results by Vilms, Yano and Ishihara, and study in detail projective, umbilical and harmonic maps.The work of the author was supported by RBRF, grant 94-01-01595 (Russia).     

Linear mappings derivable at some nontrivial elements

Suppose that A is an algebra and M is an A-bimodule. Let A be any element in A. A linear mapping δ from A into M is said to be derivable at A if δ(ST)=δ(S)T+Sδ(T) for any S,T in A with ST=A. Given an algebra A, such as a non-abelian von Neumann algebra or an irreducible CDCSL algebra on a Hilbert space H with dimH?2, we show that there exists a nontrivial idempotent P in A such that for any Q∈PAP which is invertible in PAP, every linear mapping derivable at Q from A into some unital A-bimodule (for example, A or B(H)) is derivation.     

On some classes of bounded univalent mappings in several complex variables

Let B ⁿ be the Euclidean unit ball in C ⁿ . In this paper, we study several properties of strongly starlike mappings of order α (0 < α < 1) and bounded convex mappings on B ⁿ . We prove that K-quasiregular strongly starlike mappings of order α on B ⁿ have continuous and univalent extensions to ${\overline{B}^n}$ . We show that bounded convex mappings on B ⁿ are strongly starlike of some order α. We give a coefficient estimate for K-quasiregular strongly starlike mappings of order α on B ⁿ . Finally, we give examples of strongly starlike mappings of order α and bounded convex mappings on B ⁿ .     

Regularizability of some classes of mappings that are inverses of integral operators

We study the regularizability of mappings that are inverses of integral operators acting fromC(0,1) toL ₂(0,1) and possessing noninjective continuation toL ₂(0,1). We construct classes of such operators with regularizable as well as nonregularizable inverses for which the continuation of the operators toL ₂(0,1) has an infinite-dimensional kernel. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 222–229, February, 1999.     

Discontinuous wave equations and a topological degree for some classes of multi-valued mappings

The Leray-Schauder degree is extended to certain multi-valued mappings on separable Hilbert spaces with applications to the existence of weak periodic solutions of discontinuous semilinear wave equations with fixed ends.     

Radical classes and weak radical mappings of GMV-algebras

We use the concept of generalized MV-algebra (GMV-algebra, in short) in the sense of Galatos and Tsinakis; the main tool in their investigation was a truncation construction. The relations between radical classes of GMV-algebras and radical classes of lattice ordered groups are investigated in the present paper. Further, we apply the truncation construction for dealing with weak retract mappings of GMV-algebras. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence — Physics of Information (Grant I/2/2005).     

Linear mappings on symmetry classes of tensors

In this paper we give a survey of results concerning linear mappings on symmetry classes of tensors that preserve decomposable elements and its related topic about linear mappings on spaces of matrices that preserve a fixed rank.     

Linear mappings on symmetry classes of tensors

In this paper we give a survey of results concerning linear mappings on symmetry classes of tensors that preserve decomposable elements and its related topic about linear mappings on spaces of matrices that preserve a fixed rank.     

Borel fibrations and G-spaces

Considering discrete groups G only, we present an elementary proof of the familiar equivalence of the category of G-spaces (with maps equivariant up to homotopy) and the category of Borel fibrations over BG.     

Topologizations of G-spaces

For a G-space X, we put and explore a question whether X admits a non-discrete Hausdorff G-invariant topology.     

The Roper-Suffridge extension operator and classes of biholomorphic mappings

In this paper we give a survey about the Roper-Suffridge extension operator and the developments in the theory of univalent mappings in several variables to which it has led. We begin with the basic geometric properties (most of which now have a number of different proofs) and discuss relations with the theory of Loewner chains and generalizations and modifications of the operator, some of which are very recent. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday