Let F be a multifunction with values in Lₚ(Ω, X). In this note, we study which regularity properties of F are preserved when we consider the decomposable hull of F.

It is shown to be consistent that every function of first Baire class can be decomposed into ${\aleph }_1$ continuous functions yet the least cardinal of a dominating family in ${}^{\omega }\omega$ is ${\aleph }_2$. The model used in the one obtained by adding ${\omega }_2$ Miller reals to a model of the Continuum Hypothesis.

Using the idea of the optimal decomposition developed in recent papers (Edmunds-Krbec, 2000) and in Cruz-Uribe-Krbec we study the boundedness of the operator Tg(x) = ∫x1 g(u)du / u, x ∈ (0,1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser's lemma, O'Neil's convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs...

The main purpose of this paper is to present a natural method of decomposition into special cubes and to demonstrate how it makes it possible to efficiently achieve many well-known fundamental results from quasianalytic geometry as, for instance, Gabrielov's complement theorem, o-minimality or quasianalytic cell decomposition.

The aim of this paper is to show that every Hausdorff continuous interval-valued function on a completely regular topological space X corresponds to a Dedekind cut in C(X) and conversely.

Six different functions measuring the defect of a quasi-copula, i. e., how far away it is from a copula, are discussed. This is done by means of extremal non-positive volumes of specific rectangles (in a way that a zero defect characterizes copulas). Based on these defect functions, six transformations of quasi-copulas are investigated which give rise to six different partitions of the set of all quasi-copulas. For each of these partitions, each equivalence class contains exactly one copula being...

For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which $limin{f}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|=0$ and $limsu{p}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|>0$. We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible definitions...

It is well known that one can often construct a star-product by expanding the product of two Toeplitz operators asymptotically into a series of other Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin-Toeplitz quantization. We show that one can obtain in a similar way in fact any star-product which is equivalent to the Berezin-Toeplitz star-product, by using instead of Toeplitz operators other suitable mappings from compactly supported smooth functions...

El objeto de esta nota es presentar una noción del grado topológico para funciones reales convexas sci (semicontinuas inferiormente) basándose en la teoría del grado introducida por F. Browder.

In this paper we define the derivative and the Denjoy integral of mappings from a vector lattice to a complete vector lattice and show the fundamental theorem of calculus.

In a previous paper we defined a Denjoy integral for mappings from a vector lattice to a complete vector lattice. In this paper we define a Henstock-Kurzweil integral for mappings from a vector lattice to a complete vector lattice and consider the relation between these two integrals.

According to A. Lasota, a continuous function $f$ from a real compact interval $I$ into itself is called generically chaotic if the set of all points $(x,y)$, for which ${lim\; inf}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|=0$ and ${lim\; sup}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|>0$, is residual in $I\times I$. Being inspired by this definition we say that $f$ is densely chaotic if this set is dense in $I\times I$. A characterization of the generically chaotic functions is known. In the paper the densely chaotic functions are characterized and it is proved that in the class of piecewise monotone maps with finite number of pieces the...