arXiv subjects

Marek Balcerzak

Publications and source records attributed to Marek Balcerzak.

32 records · Page 2Linked to original sources

arXiv2021

Another characterization of meager ideals

We show that an ideal $\mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of $\mathcal{I}$-limit points is comeager and, in addition, every accumulation point of $x$ is also an $\mathcal{I}$-limit point (that is, a limit of a subsequence $(x_{n_k})$ such that $\{n_1,n_2,\ldots,\} \notin \mathcal{I}$). The analogous characterization holds also for $\mathcal{I}$-cluster points.

math.GN↗

arXiv2021

A novel, Fourier series based method of control optimization and its application to a discontinuous capsule drive model

This paper presents a novel, Fourier series based numerical method of open-loop control optimization. Due to its flexible assumptions, it can be applied in a large variety of systems, including discontinuous ones or even so-called black boxes, whose equations are not fully known. This aspect is particularly important in mechanical systems, where friction or impact induced discontinuities are common. The paper includes a mathematical background of the new method, a detailed discussion of the algorithm and a numerical example, in which control function of a discontinuous capsule drive is optimized. It is expected that the proposed method can facilitate research in all areas where control of non-smooth, discontinuous or black box systems is crucial. In particular, authors hope that the presented algorithm is going to be used for control optimization of other capsule drives.

math.OC↗

arXiv2021

Cutting sets of continuous functions on the unit interval

For a function $f\colon [0,1]\to\mathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $f\colon [0,1]\to\mathbb R$ and a Cantor set $D\subset [0,1]$ with $\{0,1\}\subset D$, we obtain conditions equivalent to the conjunction $f\in C[0,1]$ (or $f\in C^\infty [0,1]$) and $D\subset E(f)$. This generalizes some ideas of Zabeti. We observe that, if $f$ is continuous, then $E(f)$ is a closed nowhere dense subset of $f^{-1}[\{ 0\}]$ where each $x\in \{0,1\}\cap E(f)$ is an accumulation point of $E(f)$. Our main result states that, for a closed nowhere dense set $F\subset [0,1]$ with each $x\in \{0,1\}\cap E(f)$ being an accumulation point of $F$, there exists $f\in C^\infty [0,1]$ such that $F=E(f)$.

math.CA↗

arXiv2020

Convergent subseries of divergent series

Let $\mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $\sum_n x_n$ is divergent. For each $x \in \mathscr{X}$, let $\mathcal{I}_x$ be the collection of all $A\subseteq \mathbf{N}$ such that the subseries $\sum_{n \in A}x_n$ is convergent. Moreover, let $\mathscr{A}$ be the set of sequences $x \in \mathscr{X}$ such that $\lim_n x_n=0$ and $\mathcal{I}_x\neq \mathcal{I}_y$ for all sequences $y=(y_n) \in \mathscr{X}$ with $\liminf_n y_{n+1}/y_n>0$. We show that $\mathscr{A}$ is comeager and that contains uncountably many sequences $x$ which generate pairwise nonisomorphic ideals $\mathcal{I}_x$. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.

math.CA↗

arXiv2020

The Baire Category of Subsequences and Permutations which preserve Limit Points

Let $\mathcal{I}$ be a meager ideal on $\mathbf{N}$. We show that if $x$ is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of $x$ which preserve the set of $\mathcal{I}$-cluster points of $x$ is topologically large if and only if every ordinary limit point of $x$ is also an $\mathcal{I}$-cluster point of $x$. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. \textbf{263} (2019), 221--229]. As an application, if $x$ is a sequence with values in a first countable compact space which is $\mathcal{I}$-convergent to $\ell$, then the set of subsequences [resp. permutations] which are $\mathcal{I}$-convergent to $\ell$ is topologically large if and only if $x$ is convergent to $\ell$ in the ordinary sense. Analogous results hold for $\mathcal{I}$-limit points, provided $\mathcal{I}$ is an analytic P-ideal.

math.GN↗

arXiv2019

Families of feebly continuous functions and their properties

Let $f\colon\mathbb{R}^2\to\mathbb{R}$. The notions of feebly continuity and very feebly continuity of $f$ at a point $\langle x,y\rangle\in\mathbb{R}^2$ were considered by I. Leader in 2009. We study properties of the sets $FC(f)$ (respectively, $VFC(f)\supset FC(f)$) of points at which $f$ is feebly continuous (very feebly continuous). We prove that $VFC(f)$ is densely nonmeager, and, if $f$ has the Baire property (is measurable), then $FC(f)$ is residual (has full outer Lebesgue measure). We describe several examples of functions $f$ for which $FC(f)\neq VFC(f)$. Then we consider the notion of two-feebly continuity which is strictly weaker than very feebly continuity. We prove that the set of points where (an arbitrary) $f$ is two-feebly continuous forms a residual set of full outer measure. Finally, we study the existence of large algebraic structures inside or outside various sets of feebly continuous functions.

math.GN↗

arXiv2019

A Tauberian theorem for ideal statistical convergence

Given an ideal $\mathcal{I}$ on the positive integers, a real sequence $(x_n)$ is said to be $\mathcal{I}$-statistically convergent to $\ell$ provided that $$ \textstyle \left\{n \in \mathbf{N}: \frac{1}{n}|\{k \le n: x_k \notin U\}| \ge \varepsilon\right\} \in \mathcal{I} $$ for all neighborhoods $U$ of $\ell$ and all $\varepsilon>0$. First, we show that $\mathcal{I}$-statistical convergence coincides with $\mathcal{J}$-convergence, for some unique ideal $\mathcal{J}=\mathcal{J}(\mathcal{I})$. In addition, $\mathcal{J}$ is Borel [analytic, coanalytic, respectively] whenever $\mathcal{I}$ is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if $\mathcal{I}$ is the summable ideal $\{A\subseteq \mathbf{N}: \sum_{a \in A}1/a<\infty\}$ or the density zero ideal $\{A\subseteq \mathbf{N}: \lim_{n\to \infty} \frac{1}{n}|A\cap [1,n]|=0\}$ then $\mathcal{I}$-statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if $\mathcal{I}$ is maximal.

math.FA↗

arXiv2019

A lower density operator for the Borel algebra

We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the $σ$-ideal of countable sets, for an uncountable Polish space, is equivalent to the Continuum Hypothesis.

math.GN↗

arXiv2018

Ideal convergent subseries in Banach spaces

Assume that $\mathcal{I}$ is an ideal on $\mathbb{N}$, and $\sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(\mathcal{I}):=\left\{t \in \{0,1\}^{\mathbb{N}} \colon \sum_n t(n)x_n \textrm{ is } \mathcal{I}\textrm{-convergent}\right\}$. In the category case, we assume that $\mathcal{I}$ has the Baire property and $\sum_n x_n$ is not unconditionally convergent, and we deduce that $A(\mathcal{I})$ is meager. We also study the smallness of $A(\mathcal{I})$ in the measure case when the Haar probability measure $λ$ on $\{0,1\}^{\mathbb{N}}$ is considered. If $\mathcal{I}$ is analytic or coanalytic, and $\sum_n x_n$ is $\mathcal{I}$-divergent, then $λ(A(\mathcal{I}))=0$ which extends the theorem of Dindoš, Šalát and Toma. Generalizing one of their examples, we show that, for every ideal $\mathcal{I}$ on $\mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $λ(A(Fin))=0$ and $λ(A(\mathcal{I}))=1$.

math.FA↗

arXiv2017

On the Relationship between Ideal Cluster Points and Ideal Limit Points

Let $X$ be a first countable space which admits a non-trivial convergent sequence and let $\mathcal{I}$ be an analytic P-ideal. First, it is shown that the sets of $\mathcal{I}$-limit points of all sequences in $X$ are closed if and only if $\mathcal{I}$ is also an $F_σ$-ideal. Moreover, let $(x_n)$ be a sequence taking values in a Polish space without isolated points. It is known that the set $A$ of its statistical limit points is an $F_σ$-set, the set $B$ of its statistical cluster points is closed, and that the set $C$ of its ordinary limit points is closed, with $A\subseteq B\subseteq C$. It is proved the sets $A$ and $B$ own some additional relationship: indeed, the set $S$ of isolated points of $B$ is contained also in $A$. Conversely, if $A$ is an $F_σ$-set, $B$ is a closed set with a subset $S$ of isolated points such that $B\setminus S\neq \emptyset$ is regular closed, and $C$ is a closed set with $S\subseteq A\subseteq B\subseteq C$, then there exists a sequence $(x_n)$ for which: $A$ is the set of its statistical limit points, $B$ is the set of its statistical cluster points, and $C$ is the set of its ordinary limit points. Lastly, we discuss topological nature of the set of $\mathcal{I}$-limit points when $\mathcal{I}$ is neither $F_σ$- nor analytic P-ideal.

math.FA↗

arXiv2016

Ideal convergent subsequences and rearrangements for divergent sequences of functions

Let $\I$ be an ideal on $\N$ which is either analytic or coanalytic. Assume that $(f_n)$ is a sequence of functions with the Baire property from a Polish space $X$ into a complete metric space $Z$, which is divergent on a comeager set. We investigate the Baire category of $\I$-convergent subsequences and rearrangements of $(f_n)$. Our result generalizes a theorem of Kallman. A similar theorem for subsequences is obtained if $(X,μ)$ is a $σ$-finite complete measure space and a sequence $(f_n)$ of measurable functions from $X$ to $Z$ is $\I$-divergent $μ$-almost everywhere. Then the set of subsequences of $(f_n)$, $\I$-divergent $μ$-almost everywhere, is of full product measure on $\{ 0,1\}^\N$. Here we assume additionally that $\mathcal I$ has property (G).

math.CA↗

arXiv2013

Uniform openness of multiplication in Banach spaces $L_p$

We show that multiplication from $L_p\times L_q$ to $L_1$ (for $p,q\in [1,\infty]$, $1/p+1/q=1$) is a uniformly open mapping. We also prove the uniform openness of the multiplication from $\ell_1\times c_0$ to $\ell_1$. This strengthens the former results obtained by M. Balcerzak, A. Majchrzycki and A. Wachowicz.

math.FA↗

arXiv2013

Nonseparable spaceability and strong algebrability of sets of continuous singular functions

Let CBV denote the Banach algebra of all continuous real-valued functions of bounded variation, defined in [0,1]. We show that the set of strongly singular functions in CBV is nonseparably spaceable. We also prove that certain families of singular functions constitute strongly c-algebrable sets. The argument is based on a new general criterion of strong c-algebrability.

math.FA↗

arXiv1996

Ideals without ccc

Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F subseteq P(X) of size c, consisting of Borel sets which are not in I. Condition (M) states that there is a function f:X-> X with f^{-1}[{x}] notin I for each x in X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B notin I and a perfect set P subseteq X for which the family {B+x: x in P} is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) => (M) => (B) => not-ccc can hold. We build a sigma-ideal on the Cantor group witnessing''(M) and not (D)'' (Section 2). A modified version of that sigma-ideal contains the whole space (Section 3). Some consistency results deriving (M) from (B) for''nicely'' defined ideals are established (Section 4). We show that both ccc and (M) can fail (Theorems 1.3 and 4.2). Finally, some sharp versions of (M) for invariant ideals on Polish groups are investigated (Section 5).

math.LO↗