Abstract: In this paper higher-order convexity properties of real functions are characterized in terms of a Dinghas-type derivative. The main tool used is a mean value inequality for Dinghas-type derivatives.

Abstract: We generalize Bourgain's theorem on the Kakeya maximal function in the plane by proving norm estimates with respect to measures satisfying certain conditions. We use this to extend the classical result of Davies on the Hausdor dimension of Kakeya sets in the plane.

Abstract: In this work we provide a characterization of C^(k,1) functions on R^n (that is k times differentiable with locally Lipschitz k-th derivatives) by means of (k+1)-th divided differences and Riemann derivatives. In particular we prove that the class of C^(k,1) functions is equivalent to the class of functions with bounded (k+1)-th divided difference. From this result we deduce a Taylor's formula for this class of functions and a characterization through Riemann derivatives.

Abstract: A set $E\subseteq \mathbb{R}^n $ is $s$-straight for $s>0$ if $E$ has finite Method II outer $s$-measure equal to its Method I outer $s$-measure. If $E$\ is Method II $s$-measurable this means $E$ has finite Hausdorff $s$-measure equal to its Hausdorff $s$-content. Here we make a first study of such sets, following their 1995 introduction by Foran. Primary facts are proved about subsets, intersections, unions, and some mappings of $s$-straight $s$-sets. Basic examples of $1$-straight and countable unions of $1$-straight $1$-sets are constructed from line segments. It is noted that self-similar $s $-sets are $s$-straight. Verifying a conjecture of Foran, the circle is proved to be the countable union of perfect $1$-straight $1$-sets along with a set of Hausdorff $1$-measure zero. Such perfect sets are then further examined. Also examined are subsets of $1$-straight sets $E$ maximal in the sense that their Hausdorff $1$-measure equals the diameter of $E$.

Abstract: Let f be a real (or complex) function with domain Df containing the positive integers. We introduce the functional sequence {fσn(x)} as follows:fσn(x) = xf(n) + Pn k=1(f(k) − f(x + k)) and say that the function f limit summable at the point x0 if the sequence {fσn(x0)} is convergent, (fσn(x0) → fσ(x0)) as n → ∞, and we call the function fσ(x) as the limit summand function (of f). In this article, we first give a necessary condition for the limit summability of functions and present some elementary properties. Then we prove some tests about limit summability of functions and consider the relation between f(x) and fσ(x). One of the main theorems in this paper gives a uniqueness conditions for a function to be a limit summand function. Finally, as a consequence of this theorem we deduce a generalization of a result due to Bohr-Mollerup [1].

Abstract: The partition–based and integral–based Hentschel–Procaccia multifractal spectra in separable metric spaces are introduced and examined. It is shown that they satisfy the basic theorems one would expect of a spectrum S of a finite Borel measure μ in a space X: • For μ-almost all x ∈ X −d+S(1) 6 lim r→0 log μB(x, r) log r 6 lim r→0 log μB(x, r) log r 6 −d−S(1), where d±S(1) denote the right and left derivatives of S at 1. • For each α within certain interval, dimH { x ∈ X : lim r→0 log μB(x, r) log r = α } 6 S(α), where dimH is Hausdorff dimension, and S L(α) denotes the Legendre transform of S.

Abstract: Some versions of Dieudonn e theorems are given for set functions, not necessarily positive, taking values in Dedekind complete (l)-groups, relatively to the \(D)-convergence".

Abstract: We present a method of integration along the lines of the Henstock{ Kurzweil integral. All L r -derivatives are integrable in this method.

Abstract: We deal with two types of chaos: the well known chaos in the sense of Li and Yorke and $\omega$-chaos which was introduced by S. Li in 1993. In this paper we prove that every bitransitive map $f \in C(I,I)$ is conjugate to $g \in C(I,I)$, which satisfies the following conditions, 1. there is a $c$-dense $\omega$-scrambled set for $g$, 2. there is an extremely LY-scrambled set for $g$ with full Lebesgue measure, 3. every $\omega$-scrambled set of $g$ has zero Lebesgue measure.

Abstract: In this paper, give a new, full characterization of the primitive of a Kurzweil-Henstock integrable function in multidimensional space.

Abstract: We give characterizations of sets $E\subset[0,1]$ for which the local monotonicity of each function $f:[0,1]\to\mathbb{R}$ from a given class $\mathcal{F}$, at all points $x\in E$, implies the global monotonicity of $f$ on $[0,1]$. We consider as $\mathcal{F}$ -- the families of continuous functions, differentiable functions, absolutely continuous functions, functions of class $C^n$ ($n=1,2,...,\infty$), real analytic functions and polynomials.

Abstract: In this paper we prove that for any modulus of continuity on [0 , ∞ ) there exists a concave majorant that is infinitely differentiable on (0 , ∞ ) and satisfies an additional inequality. This extends the results of Stechkin and Korneychuk obtained previously without the requirement that ma-jorants be differentiable.

Abstract: It is shown that every cliquish function f mapping a pseudometrizable space X into a separable metric space Y can be expressed as the quasiuniform limit of a sequence of quasicontinuous functions f k .

Abstract: We show that there is a continuous map χ of the unit interval into itself of type 2∞ which has a trajectory disjoint from the set Rec(χ) of recurrent points of χ, but contained in the closure of Rec(χ). In particular, Rec(χ) is not closed. A function ψ of type 2∞, with nonclosed set of recurrent points, was found by H. Chu and J. Xiong [Proc. Amer. Math. Soc. 97 (1986), 361–366]. However, there is no trajectory contained in Rec(ψ) \\Rec(ψ), since any point in Rec(ψ) is eventually mapped into Rec(ψ). Moreover, our construction is simpler. We use χ to show that there is a continuous map of the interval of type 2∞ for which the set of recurrent points is not an Fσ set. This example disproves a conjecture of A. N. Sharkovsky et al., from 1989. We also provide another application of χ.

Abstract: Given a power series f ( x ) = P ∞ n =1 a n x n with nonnegative coef-ficients satisfying P ∞ n =1 a n = 1 we give sufficient conditions on the sequence ( a n ) to guarantee that the coefficients of the Taylor series of h ( x ) = 1 / (1 − f ( x )) form a nonincreasing sequence. This type of result is useful when one wishes to apply Tauberian theorems.

Abstract: This paper is a step by step account of the Riesz approach to the Lebesgue integral. Besides, motivating the use of ``almost everywhere'' tools, we eliminate unnecessary equivalences, and we give a simple representation of a complete space of integrable functions, usually missing from classical treatises.