This paper is a survey of a new family of Banach spaces \(\mathcal{B}\) that provide the same structure for the Henstock-Kurzweil (HK) integrable functions as the \(L^p\) spaces provide for the Lebesgue integrable functions. These spaces also contain the wide sense Denjoy integrable functions. They were first use to provide the foundations for the Feynman formulation of quantum mechanics. It has recently been observed that these spaces contain the test functions \(\mathcal{D}\) as a continuous dense embedding. Thus, by the Hahn-Banach theorem, \(\mathcal{D}' \subset \mathcal{B}'\). A new family that extend the space of functions of bounded mean oscillation \(BMO[\mathbb{R}^n]\), to include the HK-integrable functions are also introduced.

In my talk I presented results from previous papers on the existence of hypercyclic algebras for convolution operators acting on the space of entire functions.

Let \(f:ℝ \to ℝ\) be continuous. We examine the relationship between the so-called “big Lip” and “little lip” functions: Lip \(f\) and lip \(f\).

We define and discuss a pure mathematics formulation of an approach proposed in the physics literature to analysing anistropy of fractal sets.

We introduce the continued logarithm representation of real numbers and prove results on the occurrence and frequency of digits with respect to this representation.

In this article we will prove that if the continuous closed curve \(\gamma : [0, 1] \rightarrow \mathbb{R}^2\) has finite \(p\)-variation with \(p < 2\), then \begin{equation*} (\iint\limits_{\mathbb{R}^2}|\eta(\gamma, (x, y))|^q \,dx \,dy)^{1/q} \le (\frac{1}{2})^\frac{1}{q}(\zeta(\frac{2}{pq})-1)(||\gamma||_{p, [0, 1]})^{\frac{2}{q}} \end{equation*} for all \(q \in [1, \frac{2}{p})\), where \(\eta(\gamma, (x, y))\) is the winding number of \(\gamma\) at \((x, y), \zeta\) is the Reimann zeta function, and \(||\gamma||_{p, [0, 1]}\) is the \(p\)-variation of \(\gamma\) on the interval \([0, 1]\). Our main contribution is that we have explicitly given a bound by known constants, and we have found this by an elementary proof. We are going to be using a method introduced by L.C. Young in 1936.

A topological space \(X\) is called {\em Piotrowski} if every quasicontinuous map \(f:Z\to X\) from a Baire space \(Z\) to \(X\) has a continuity point. In this paper we survey known results on Piotrowski spaces and investigate the relation of Piotrowski spaces to strictly fragmentable, Stegall, and game determined spaces. Also we prove that a Piotrowski Tychonoff space \(X\) contains a dense (completely) metrizable Baire subspace if and only if \(X\) is Baire (Choquet).

The representation of a given quantity with less information is often referred to as ‘quantization’ and it is an important subject in information theory. In this paper, we have considered absolutely continuous probability measures on unit discs, squares, and the real line. For these probability measures the optimal sets of \(n\)-means and the \(n\)th quantization errors are calculated for some positive integers \(n\).

We prove that all negative generalized Boole transformations are conservative, exact, pointwise dual ergodic, and quasi-finite with respect to Lebesgue measure on the real line. We then provide a formula for computing the Krengel, Parry, and Poisson entropy of all conservative rational functions that preserve Lebesgue measure on the real line.

The main purpose of the paper is to show that differentiability properties of a measurable function defined in Euclidean space can be described using square functions which involve its second symmetric divided differences. Classical results of Marcinkiewicz, Stein and Zygmund describe, up to sets of Lebesgue measure zero, the set of points where a function \(f\) is differentiable in terms of a certain square function \(g(f)\). It is natural to ask for the behavior of the divided differences at the complement of this set, that is, on the set of points where \(f\) is not differentiable. In the nineties, Anderson and Pitt proved that the growth of the divided differences of a function in the Zygmund class obeys a version of the classical Kolmogorov’s Law of the Iterated Logarithm (LIL). A square function, which is the conical analogue of \(g(f)\) will be used to state and prove a general version of the LIL of Anderson and Pitt as well as to prove analogues of the classical results of Marcinkiewicz, Stein and Zygmund. Sobolev spaces can also be described using this new square function.

In this paper we study magic sets for certain families \(\mathcal{H}\subseteq {^\mathbb{R}\mathbb{R}}\) which are subsets \(M\subseteq\mathbb{R}\) such that for all functions \(f,g\in\mathcal{H}\) we have that \(g[M]\subseteq f[M]\Rightarrow f=g\). Specifically we are interested in magic sets for the family \(\mathcal{G}\) of all continuous functions that are not constant on any open subset of \(\mathbb{R}\). We will show that these magic sets are stable in the following sense: Adding and removing a countable set does not destroy the property of being a magic set. Moreover, if the union of less than \(\mathfrak{c}\) meager sets is still meager, we can also add and remove sets of cardinality less than \(\mathfrak{c}\) without destroying the magic set.

Then we will enlarge the family \(\mathcal{G}\) to a family \(\mathcal{F}\) by replacing the continuity with symmetry and assuming that the functions are locally bounded. A function \(f:\mathbb{R}\to\mathbb{R}\) is symmetric iff for every \(x\in\mathbb{R}\) we have that \(\lim_{h\downarrow 0}\frac{1}{2}\left(f(x+h)+f(x-h)\right )=f(x)\). For this family of functions we will construct \(2^\mathfrak{c}\) pairwise different magic sets which cannot be destroyed by adding and removing a set of cardinality less than \(\mathfrak{c}\). We will see that under the continuum hypothesis magic sets and these more stable magic sets for the family \(\mathcal{F}\) are the same. We shall also see that the assumption of local boundedness cannot be omitted. Finally, we will prove that for the existence of a magic set for the family \(\mathcal{F}\) it is sufficient to assume that the union of less than \(\mathfrak{c}\) meager sets is still meager. So for example Martin’s axiom for \(\sigma\)-centered partial orders implies the existence of a magic set.

Consider the random open balls \(B_n(\omega):=B(\omega_n,r_n)\) with their centers \(\omega_n\) being i.u.d. on the \(d\)-dimensional unit cube \([0,1]^d\) and with their radii \(r_n\sim cn^{-\frac{1}{d}}\) for some constant \(0<c<(\beta(d))^{-\frac{1}{d}}\), where \(\beta(d)\) is the volume of the \(d\) dimensional unit ball. We call \([0,1]^d-\bigcup_{n=1}^{\infty} B_n(\omega)\) a random cutout set. In this paper, we present an exposition of the Zähle cutout model by a detailed study of such a random cutout set for the purpose of teaching and learning. We show that with probability one the Hausdorff dimension of such random cut-out set is at most \(d(1-\beta(d)c^d)\) and frequently equals \(d(1-\beta(d)c^d)\).

We answer a question posed by Miklós Laczkovich on the Minkowski sum of two curves.

In terms of a group \(G\) of isometries of Euclidean space, it is given a necessary and sufficient condition for the uniqueness of a \(G\)-measure on the Borel \(\sigma\)-algebra of this space.

We show that there exists a derivative \(f\colon [0,1]\to[0,1]\) such that the graph of \(f\circ f\) is dense in \([0,1]^2\), so not a \(G_\delta\)-set. In particular, \(f\circ f\) is everywhere discontinuous, so not of Baire class 1, and hence it is not a derivative. %neither of Baire class 1 nor a derivative.

A proof of the Luzim-Menchoff theorem.

An answer to the question of the title is given.