We develop a new combinatorial method to estimate Hausdorff measures of various self-similar sets. This method can be applied to the evaluation of Hausdorff measures which are induced by various Hausdorff functions including power functions. Moreover, a few examples for evaluations of the lower and upper bounds of Hausdorff measures of uniform Cantor sets are introduced.

The present paper gives a pair of adjoint classes of Riemann-Stieltjes integrable functions and two pairs of adjoint classes of generalized Stieltjes integrable functions.

In this paper we prove a uniform boundedness theorem and use it to show that if $fg$ is non-absolutely integrable on an interval in Euclidean space for each non-absolute integrable function $f$, then $g$ is almost everywhere a function of strongly bounded variation on $E$.

We give a nonstandard variant of Jordan's proof of the Jordan curve theorem which is free of the defects his contemporaries criticized and avoids the epsilontic burden of the classical proof. The proof is self-contained, except that the Jordan theorem for polygons is taken for granted.

Some special notions of approximate quasi-continuity and cliquishness are considered. Moreover, uniform, pointwise and transfinite convergence of sequences of such functions are investigated.

In this paper we introduce some modifications of the core topology on the plane. Here the role of the Euclidean topology on the real line on every or almost every direction is played by the density topology or the Hashimoto topology connected with the $\sigma$-ideal of null sets and the $\sigma$-ideal of meager sets. We demonstrate the proper inclusions between these families, we investigate the separation axioms and functions continuous with respect to these topologies.

One can think of the dominating number as the covering number for the ideal on the $\omega$-branching tree generated by finite branching subtrees. This paper is concerned with generalizations of this when "finite" is replaced by some other concept. A key example is obtained by thinking of the branching as being into the integers --- both positive and negative --- and replacing "finite" by "bounded either above or below". This notion was motivated by considerations related to decomposing functions of low Baire class into continuous functions.

Let $C(\Omega)$ be the algebra of continuous complex-valued functions on a topological space $\Omega$ and let $\rho$ be a function norm on $C(\Omega).$ We give necessary and sufficient conditions on the set $A_{\rho}=\{f\in C(\Omega)\:rho(f)<\infty\}$ to be an algebra. Also, we prove that every complete function norm is quasi-submultiplicative provided $A_{\rho}$ is an algebra and we give a characterization of the best multiplicative factor of $\rho.$ Finally we characterize the infinity norm and we prove that every quasi-submultiplicative function norm on $C(\Omega)$ is equivalent to the infinity norm.

We find the local characterizations of the Cantor intermediate value property and the strong Cantor intermediate value property. Moreover we show that there is no similar local characterization for the weakly Cantor intermediate value property.

In this paper we consider polynomial splines with equidistant nodes which may grow as O$|x|^s$. We present an integral representation of such splines with a distribution kernel where exponential splines are used as basic functions. By this means we characterize splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form the biorthogonal basis. We suggest a scheme of wavelet analysis in the spaces of growing splines based on integral representation of the splines. The key point of that scheme is the refinement equation for the exponential splines which contains only two terms. We construct the so called exponential wavelets. We establish conditions for a spline to be a basic wavelet which enable us to form a library of such wavelets. We give formulas for the decomposition of a spline into a weak orthogonal sum of the sparse-grid spline and an element of the corresponding wavelet space. Reconstruction formulas are presented which permit the use of arbitrary bases of spline and wavelet spaces.

The paper is closely related to an earlier paper of two authors of this paper (cf. [4]). In the paper radii of convergence of power series are investigated as values of a function defined on the Fréchet's space using the well--known formula of Cauchy and Hadamard.

In [MU] the concept of infinite conformal iterated function systems was introduced and developed. In this paper we consider 1-dimensional systems. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We extend to such systems the concept of scaling functions and we pay special attention to the real-analytic systems.

Let $\{I_n\}_{n=1}^\infty$ be a sequence of pairwise disjoint intervals tending to 0 from the right, such that putting $I_n=(a_n,b_n)$ we have $\lim\inf {\dfrac{|I_n|}{b_n}}=0$. We prove that for every Cantor set $C$ there exists some set $X$ of positive measure, such that for a.e. $x\in X$ there exists a $c\in C$ for which $(x+cI_n)\cap X=\emptyset$ for infinitely many $n$.

We show that no reasonable classical form of ``Fubini type theorems'' can hold for the $\sigma$-ideal of $\sigma$-porous sets in products of metric spaces (even in the plane). Then we prove that a ``Fubini type theorem'' in a weak decomposition form remains true also for this $\sigma$-ideal, and we illustrate how this fact may be applied to the study of the behavior of measures on small sets in product spaces. We also prove an analogical decomposition theorem for $\sigma$-directionally porous sets in Banach spaces; such sets arise naturally as exceptional sets in some questions concerning differentiability properties of Lipschitz functions on Banach spaces.

This paper is a continuation of [3], in which was introduced the Refinement-Ross-Riemann-Stieltjes $(R^3S)$ Integral, and in which some of its advantages were exhibited. After a brief summary of [3], this paper proves an integration by parts theorem which shows incidentally that if $f$ is $R^3S$-integrable with respect to $g$ then $g$ is $R^3S$-integrable with respect to $f$. Theorems on term-by-term integration of sequences analogous to the Helly-Bray Theorem are next proved, in a context of Wiener's functions of bounded generalized variation as developed by L. C. Young and me. In a similar context I prove also a theorem resembling the classical theorem of Riesz representing linear functionals by Stieltje.

In this paper we study the distinctness of the concepts of left transverse, right transverse and transverse in the space ${\cal H}[0, 1]$ of automorphisms on $[0, 1]$. We show that, in ${\cal H}[0, 1]$, the existence of a Borel probability measure left transverse and right transverse to a set $Y$ does not imply the set $Y$ is Haar null and we exhibit an uncountable disjoint collection of Borel sets that are not Haar null.

Let $A = \bigl \{b_0, b_1, \ldots,b_{k+\ell}; a_0,a_1, \ldots,a_{k+\ell} \bigr \}$ be a system of $2(k+\ell +1)$ real numbers such that $b_i \neq b_j$ for $i \neq j$, satisfying $\sum_{i=0}^{k+\ell} a_i b_i^p = 0$ for $p = 0,1,\ldots,k-1$ and $\sum_{i=0}^{k+\ell} a_i b_i^k = L \neq 0$. It is proved that if $f$ is measurable, and if $\sum_{i=0}^{k+\ell} a_i f(x+b_i h) = O(|h|^\lambda)$ as $h \to 0$, where $\lambda > k-1$, at each point $x$ on a measurable set $E$ then the Peano derivative $f_{([\lambda])}$ exists finitely $a.e.$ on $E$. This will extend a result of Ash [1]. It is further proved that if $p$ is a positive integer $\le k-1$ and if the upper and lower approximate Peano derivatives of $f$ of order $p$ are finite on a set $E$ then $f_{(p)}$ exists $a.e.$ on $E$.

Given $0 < s < u < v < 1$ and $s < t < v$, a Cantor set $Y$ is constructed in $[0, 1]$ with Hausdorff dimension $s$, packing dimension $t$, lower box dimension $u$ and upper box dimension $v$. In the sense that $t$ and $u$ are independent, so are the packing and lower box dimensions. Although $Y = \{0\} \cup \bigcup_{ \ell=0}^{ \infty} Y_\ell$, the lower and upper box dimensions of each $Y_\ell$ are respectively $s$ and $t$.

Hurewicz found connections between some topological notions and the combinatorial cardinals $\mathfrak{b}$ and $\mathfrak{d}$. Recɫaw gave topological meaning to the definition of the cardinal $\mathfrak{p}$. We extend the picture with a topological interpretation of the cardinal $\mathfrak{t}$. We compare our notion to the one related to $\mathfrak{p}$, and to some other classical notions. This sheds new light on the famous open problem whether $\mathfrak{p}=\mathfrak{t}$.

In the last two decades the Riemann generalized integral, having values in Banach spaces, has been increasingly studied. The development in this area concerns mainly with the Henstock-Kurzweil (see e.g.[1], [2]), and the Dushnik and the Young integrals (see e.g.[3]). Recently it has appeared in the literature many proper applications in this field (proper here considered in the sense that the results are not disguises of an essentially finite dimensional frame) (see e.g.[4],[5]). In this paper we give a result having its roots in this frame.

A symmetric measure-preserving system is one where the measure $Pr$ is preserved by two maps $T$ and $R$ where $R$ is self-inverse and $T\circ R = T$. We discuss the existence of such systems and some consequences, including when unimodal maps are conjugate to the symmetric tent map.

We show that a uniformly integrable, pointwise convergent sequence of Henstock-Kurzweil integrable functions converges in the Alexiewicz norm. In particular, this implies that a sequence satisfying the conditions in the Dominated Convergence Theorem is norm convergent.

We discuss two natural questions concerning some properties of\break quasiinvariant and invariant measures: the existence of nonmeasurable sets with respect to such measures and the existence of a nonzero $\sigma$-finite quasiinvariant measure on a proper Borel subgroup of the real line.

Let $\tau _1$ and $\tau _2$ be topologies in $X$ and let $\tau = \tau _1 \cap \tau _2$. Some conditions concerning the topologies $\tau $, $\tau _1$ and $\tau _2$ and describing the relations between the $\tau $-continuity (quasicontinuity) [cliquishness] and the $\tau _i$-continuity (quasicontinuity) [cliquishness], $i = 1,2$, of functions defined on $X$ are considered.

An integral is defined using the Moore-Smith limit and this new integral is compared to the Henstock integral.

Several kinds of convergence (including pointwise, monotone, a.c., uniform, $\ldots$) in the family of monotone functions are investigated

We offer an inequality involving upper and lower Darboux integrals of bounded real functions which implies a number of variations of the change of variables formula for Riemann integrals.

In the product $X\times Y$ of two uncountable complete separable metric spaces, not every $(s)$-set belongs to the $\sigma$-algebra generated by the products of $(s)$-sets in $X$ with $(s)$-sets in~$Y$. The construction makes use of the fact that the Boolean algebra $(s)/(s_0)$ is complete.