On commuting mappings

The cardinality of the centralizer of an element in infinite symmetric semigroups is studied.     

On derivable mappings

A linear mapping δ from an algebra A into an A-bimodule M is called derivable at c∈A if δ(a)b+aδ(b)=δ(c) for all a,b∈A with ab=c. For a norm-closed unital subalgebra A of operators on a Banach space X, we show that if C∈A has a right inverse in B(X) and the linear span of the range of rank-one operators in A is dense in X then the only derivable mappings at C from A into B(X) are derivations; in particular the result holds for all completely distributive subspace lattice algebras, J-subspace lattice algebras, and norm-closed unital standard algebras of B(X). As an application, every Jordan derivation from such an algebra into B(X) is a derivation. For a large class of reflexive algebras A on a Banach space X, we show that inner derivations from A into B(X) can be characterized by boundedness and derivability at any fixed C∈A, provided C has a right inverse in B(X). We also show that if A is a canonical subalgebra of an AF C^∗-algebra B and M is a unital Banach A-bimodule, then every bounded local derivation from A into M is a derivation; moreover, every bounded linear mapping from A into B that is derivable at the unit I is a derivation.     

On context-free and Szilard languages

Szilard languages of context-free grammars are studied. Especially, classical pumping, generalized pumping, Sokolowski's criterion, and semilinearity are considered as possible distinguishing properties between context-free and Szilard languages.This work was supported by the Academy of Finland.     

On monotone and quasicompact mappings

In this paper, some properties of monotone mappings and quasi-compact mappings have been studied.     

On intermediate factorial languages

We prove that factorial languages defined over non-trivial finite alphabets under some natural conditions have intermediate complexity functions, i.e., the number of words in such a language grows faster than any polynomial but slower than any exponential function.     

On some auxiliary mappings generated by nonexpansive and strictly pseudo-contractive mappings

Starting by a finite family of mappings, we define the concept of procedure with Lipschitzian dependence of the coefficients. We give seven concrete examples of such procedures and prove the strong convergence of two viscosity methods.     

On mappings preserving pentagons

Summary We introduce a new characterization of linear isometries. More precisely, we prove that if a one-to-one mapping f:<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿ(2<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>≦n<<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>∞) maps every regular pentagon of side length a> 0 onto a pentagon with side length b> 0, then there exists a linear isometry I :<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿup to translation such that f(x) = (b/a) I(x).     

On pseudomonotone set-valued mappings

In this paper a common generalization of the algebraic and topological pseudomonotonicity for set-valued maps is introduced. Existence results for variational inequalities governed by such maps are given.     

On some nonself mappings

Let X be a Banach space, let K be a non–empty closed subset of X and let T : K → X be a non–self mapping. The main result of this paper is that if T satisfies the contractive–type condition (1.1) below and maps ?K (?K the boundary of K) into K then T has a unique fixed point in K.     

On finite-to-one open mappings

We investigate open finite-to-one (mainly, k-to-one) mappings of topological spaces. The notion of a rigid space is introduced. It is shown that the class of rigid spaces contains all Euclidean spaces and all zero-dimensional Lindelöf spaces. Examples are constructed and problems are formulated.     

On cubic-linear polynomial mappings

In the field of the Jacobian conjecture it is well-known after Dru?kowski that from a polynomial ‘cubic-homogeneous’ mapping we can build a higher-dimensional ‘cubic-linear’ mapping and the other way round, so that one of them is invertible if and only if the other one is. We make this point clearer through the concept of ‘pairing’ and apply it to the related conjugability problem: one of the two maps is conjugable if and only if the other one is; moreover, we find simple formulas expressing the inverse or the conjugations of one in terms of the inverse or conjugations of the other. Two nontrivial examples of conjugable cubic-linear mappings are provided as an application.     

On quadratic symplectic mappings

Dedicated to M. Hénon on the occasion of his 60th birthday     

On mean quasiconformal mappings

The research was partly supported by the International Science Foundation (Grant U 96 000).     

On surjective quadratic mappings

In the paper, quadratic mappings acting from one finite-dimensional space to another are studied. Sufficient conditions for the stable surjectivity of a quadratic surjective mapping (i.e., for the condition that every quadratic mapping sufficiently close to a given one is also surjective) are obtained. The existence problem for nontrivial zeros of a surjective quadratic mapping acting from Rⁿ to Rⁿ is studied. For n = 3, the absence of these zeros is proved.     

On approximation of approximately linear mappings by linear mappings

Assume A is a normed linear space, B is a Banach space, and f: A → B is a mapping “approximately linear.” We solve the following Ulam problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.”     

On locally quasiconformal mappings

A homeomorphism w=f(z) of a domain D is called a locally quasiconformal mapping, if for each subdomain D' of D with 'D, the restriction of f(z) on D' is a quasiconformal mapping. We give some conditions for a measurable function μ(z) on the unit disc to be the complex dilatation of a locally quasiconformal mapping f which can be homeomorphically extended to the closed unit disc.