1.
An Lp differentiable non-differentiable function. - Ash, J. Marshall
There is a set $E$ of positive Lebesgue measure and a function nowhere differentiable on $E$ which is differentiable in the $L^p$ sense for every positive $p$ at each point of $E$. For every $p\in(0,\infty]$ and every positive integer $k$ there is a set $E=E(k,p)$ of positive measure and a function which for every $q< p$ has $k$ $L^q$ Peano derivatives at every point of $E$ despite not having an $L^p$ $k$th derivative at any point of $E$.
3.
Hausdorff measure of p-Cantor sets. - Cabrelli, C.; Molter, U.; Paulauskas, V.; Shonkwiler, R.
In this paper we analyze Cantor type sets constructed by the removal of open intervals whose lengths are the terms of the $p$-sequence, $\{k^{-p}\}_{k=1}^\infty$. We prove that these Cantor sets are $s$-sets, by providing sharp estimates of their Hausdorff measure and dimension. Sets of similar structure arise when studying the set of extremal points of the boundaries of the so-called random stable zonotopes.
4.
Dynamical systems generated by functions with connected G? graphs. - ?iklová, Michaela
In 2001, Cs\" ornyei, O'Neil and Preiss proved that the composition of any two Darboux Baire-1 functions $[0,1]\rightarrow [0,1]$ possesses a fixed point, solving a long-standing open problem. In 2004 Szuca proved that this result can be generalized to any $f$ in the class $\cal J$ of functions $[0,1]\rightarrow [0,1]$ with connected $G_\delta$ graph. As a consequence, he proved that for such functions the Sharkovsky theorem is satisfied. As the main result of this paper we prove that as for continuous maps of the interval, any $f$ in $\cal J$ has positive topological entropy if and only if it has...
5.
Sets of statistical cluster points and ?-cluster points. - Sleziak, Martin; Toma, Vladimír; ?in?ura, Juraj; alát, Tibor
Let $\I$ be an admissible (i.e., proper and containing all finite subsets of $\N$) ideal of subsets of the set $\N$ of positive integers. The concept of $\I$-convergence of sequences in metric spaces generalizes the concept of statistical convergence and also the usual concept of convergence of sequences. In this paper we investigate some problems concerning the sets of $\I$-cluster points and, in particular, the sets of statistical cluster points of sequences in metric spaces which are known to be closed sets. In the first part of the paper we give a sufficient condition on a sequence $x=(x_n)_{n=1}^\infty$ in a...
6.
A characterization of the GHk integral. - Das, A. G.; Kundu, Sarmila
A concept of a general derivative and a notion of bounded variation have been produced leading to the presentation of the Fundamental Theorem of Calculus for the $GH_k$ integral with a characterization of the integral.
9.
Dirichlet forms on fractal subsets of the real line. - Freiberg, Uta
Measure theoretic Dirichlet forms on compact subsets of the real line are introduced. Using the technique of Dirichlet--Neumann--bracketing, estimates of the eigenvalue counting functions of the associated measure geometric Laplacians are obtained.
10.
Convergence of sequences of functions having some generalized Pawlak properties. - Grande, Zbigniew
A function $f:\mathbb R \to \mathbb R$ has the property ${\cal M}_1$ (${\cal M}_2$) if the restricted function $f\rest D(f)$ ($f\rest D_{ap}(f)$) is monotone. ($D(f)$ [$D_{ap}(f)$] denotes the set of all discontinuity points [the set of all approximate discontinuity points] of $f$.) In this article I investigate the uniform, pointwise and transfinite limits of sequences of functions with the property ${\cal M}_i$, $i = 1,2$.
11.
On the convergence of sequences of integrally quasicontinuous functions. - Grande, Zbigniew
A function $f:\mathR ^n \to \mathR$ satisfies condition $(Q_i(x))$ (resp. $(Q_s(x))$, [$Q_o(x)$]) at a point $x$ if for each real $r > 0$ and for each set $U \ni x$ belonging to the Euclidean topology in $\mathR ^n$ (resp.~to the strong density topology [to the ordinary density topology]) there is an open set $I$ such that $I \cap U \neq \emptyset $, $f$ is Lebesgue integrable on $I\cap U$ and $$\left|\frac{1}{\mu (U\cap I)}\int_{U \cap I} f(t)dt - f(x) \right| < r.$$ These notions are modifications of quasicontinuity or approximate quasicontinuity. In this article we investigate the limits of sequences of...
12.
On Green's theorem and Cauchy's theorem. - Greenlee, W. M.
Green's Theorem is proved using only the geometric (or physical) definition of curl, without the use of partial derivatives. The curl free (conservative) case can then be used to prove Cauchy's Theorem.
13.
Transversal mappings between manifolds and non-trivial measures on visible parts. - Järvenpää, Esa; Järvenpää, Maarit; Niemelä, Juho
This paper has two aims. On the one hand, we generalize the notion of sliced measures by means of transversal mappings and study dimensional properties of these measures. On the other hand, as an application of these results, we explain in what sense typical visible parts of a set with large Hausdorff dimension are smaller than the set itself. This is achieved by establishing a connection between dimensional properties of generalized slices and those of visible parts.
14.
Triangular maps non-decreasing on the fibers. - Ko?an, Zden?k
There is a list of about 50 properties which characterize continuous maps of the interval with zero topological entropy. Most of them were proved by A.\ N.\ Sharkovsky [cf., e.g., Sharkovsky et al., Dynamics of One-Dimensional Mappings, Kluwer 1997]. It is also well known that only a few of these properties remain equivalent for continuous maps of the square. Recall, e.g., the famous Kolyada's example of a triangular map of type $2^\infty$ with positive topological entropy. In 1989 Sharkovsky formulated the problem to classify these conditions in a special case of triangular maps of the square. The present paper is...
15.
On some properties of essential Darboux rings of real functions defined on topological spaces. - Korczak, Ewa; Pawlak, Ryszard J.
This paper deals with rings of real Darboux functions defined on some topological spaces. Results are given concerning the existence of essential, as well as prime Darboux rings. We prove that, under some assumptions connected with the domain $X$ of the functions, the equalities: $D(X)=S_{lf}(X), S(X)=\dim(\Re )$ hold, where $D(X)$ is a ${\cal D}$-number of $X$, $S(X)$ ($S_{lf}(X)$) denotes the Souslin (lf-Souslin) number of $X$ and $\dim(\Re )$ is a Goldie dimension of an arbitrary prime Darboux ring $\Re$.
16.
Compactness of families of convolution operators with respect to convergence almost everywhere. - Kostyukovsky, Sergey; Olevskii, Alexander
For a given sequence of measures $\mu_n$ on the circle $\mathbb{T}$ weakly convergent to the Dirac measure, we ask, is it possible to extract a subsequence $n(j)$ such that for any $f$ in the space $L^1 (L^2 ,L^{\infty })$ the convolutions $f\ast\mu_{n(j)}$ converge to $f$ almost everywhere. We show that it is crucial whether the measures are absolutely continuous, discrete or singular (non-atomic).
18.
Extensions of real and vector functions of one variable which preserve differentiability. - Nekvinda, A.; Zají?ek, L.
Let $f: F \to X$ be a locally bounded function from a closed set $F \subset \R$ to a normed linear space. Then there exists its extension $f^* : \R \to X$ which is differentiable at all points at which $f$ is differentiable. Moreover, $f^*$ is Lipschitz if $f$ is Lipschitz and, in the case $X = \R$, the extension ``preserves Dini derivatives''. The paper partly extends results proved by V. Jarn\'\i k (1923), G. Petruska and M. Laczkovich (1974) and J. Ma\v r\'\i k (1984).
19.
On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class. - Samko, Natasha
We study quasi-monotonic functions of the Zygmund-Bary-Stechkin class $\Phi$ with the main emphasis on properties of the index numbers of functions in this class. A special attention is paid to functions whose lower and upper index numbers do not coincide with each other (non-equilibrated functions). It is proved that the bounds for functions in $\Phi$ known in terms of these indices, are exact in a certain sense. We also single out some special family of none-equilibrated functions in $\Phi$ which oscillate in a certain way between two power functions. Given two numbers $0<\al\le \bt<1$, we explicitly construct examples of functions...
20.
A summability factor theorem for infinite series. - Sava\c{s}, Ekrem
We obtain sufficient conditions for the series $\sum\nolimits a_{n}\lambda_n$ to be absolutely summable of order $k$ by a triangular matrix.
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