Dimension theory and superpositions of continuous functions

The main result of this paper is the following: IfX is a compact two dimensional metric space, and {φ _i} _i ^{= 1/4} are four functions inC(X), then there exists a functionf inC(X) which cannot be represented in the form: $$f(x) = \sum\limits_{i = 1}^4 {g_\iota (\varphi _i (x))} $$ , with $$g_\iota \in C(R)$$ .     

Dimension,superposition of functions and separation of points,in compact metric spaces

In an earlier paper Berele and Regev associated to each p.i. algebraA a sequence of algebrasU ^k,t(A) which proved useful in studying the identities ofA. We now describeU ^k,t(A) as a universal object and describe how to recoverA from theU ^k,t(A).     

Uniformly continuous superposition operators in the space of bounded variation functions

Let I, J ? ? be intervals. The main result says that if a superposition operator H generated by a function of two variables h: I × J → ?, H (φ)(x) ? h (x, φ (x)), maps the set BV (I, J) of all bounded variation functions, φ: I → J into the Banach space BV (I, ?) and is uniformly continuous with respect to the BV ‐norm, then h (x, y) = a (x)y + b (x), x ∈ I, y ∈ J, for some a, b ∈ BV (I, ?) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dimension prints for continuous functions on the unit square

Acta Mathematica Hungarica - Recently, we have proved that the rectangular pointwise Lipschitz regularity of a continuous function on the unit square is directly related with the local suprema of...     

Uniformly continuous superposition operators in the Banach space of Hölder functions

Let I,J⊂R be intervals. One of the main results says that if a superposition operator H generated by a two place ,

H(φ)(x):=h(x,φ(x)),     

Decomposing symmetrically continuous and Sierpinski-Zygmund functions into continuous functions

In this paper we will investigate the smallest cardinal number such that for any symmetrically continuous function there is a partition of such that every restriction is continuous. The similar numbers for the classes of Sierpinski-Zygmund functions and all functions from to are also investigated and it is proved that all these numbers are equal. We also show that and that it is consistent with ZFC that each of these inequalities is strict.

     

Dimension functions of Cantor sets

We estimate the packing measure of Cantor sets associated to non-increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets.

     

On relations between usual continuous functions and generalized continuous functions

Summary I give here a theorem of realization of homomorphism in Cech homology theory, which improves some previous results of mine.     

Measurable functions and almost continuous functions

I show that if (X, ) is a Radon measure space and Y is a metric space, then a function from X to Y is -measurable iff it is almost continuous (=Lusin measurable). I discuss other cases in which measurable functions are almost continuous.Part of the work of this paper was done during a visit to Japan supported by the United Kingdom Science Research Council and Hokkaido University     

Hausdorff continuous interval-valued functions and quasicontinuous functions

In the paper it is shown that every Hausdorff continuous interval-valued function corresponds uniquely to an equivalence class of quasicontinuous functions. This one-to-one correspondence is used to construct the Dedekind order completion of C(X), the set of all real-valued continuous functions, when X is a compact Hausdorff topological space or a complete metric space.     

Axial continuous functions

A function from the plane to the plane is axial if it does not change one coordinate. We show that not every continuous function can be approximated by a superposition of continuous axial functions. This is a counterexample to a possible generalization of theorem of Eggleston about continuous bijections.     

Sequences of semicontinuous functions accompanying continuous functions

A space X is said to have property (USC) (resp. (LSC)) if whenever is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval [0,1] converging pointwise to the constant function 0 with the value 0, there is a sequence of continuous functions from X into [0,1] such that fn?gn (n∈ω) and converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepers' conjecture on properties S₁(Γ,Γ) and wQN.     

Completeness of systems of functions for classes of extended superposition

The problem of realization of Boolean functions by formulas of special type is considered. The notion of the supplement of systems of Boolean functions is defined. Criteria of completeness of the considered functional systems are obtained.     

Restrictions of continuous functions

Let α = {α₁, ...,α_k} be a finite multiset of non-negative real numbers. Consider the sequence of all positive integer multiples of all α _i ’s, and note the multiplicity of each term in this sequence. This sequence of multiplicities is the resonance sequence generated by {α ₁, ...,α_k}. Two multisets are combinatiorially equivalent if they generate the same resonance sequence. The paper is devoted to the classification of multisets up to combinatorial equivalence. We show that the problem of combinatorial equivalence of multisets is closely related to the problem of classification of systems of second order ordinary differential equations up to focal equivalence.