arXiv subjects

Szymon Głab

Publications and source records attributed to Szymon Głab.

7 recordsLinked to original sources

arXiv2022

On $n$-saturated closed graphs

Geschke proved that there is clopen graph on $2^ω$ which is 3-saturated, but the clopen graphs on $2^ω$ do not even have infinite subgraphs that are 4-saturated; however there is $F_σ$ graph that is $ω_1$-saturated. It turns out that there is no closed graph on $2^ω$ which is $ω$-saturated. In this note we complete this picture by proving that for every $n$ there is an $n$-saturated closed graph on the Cantor space $2^ω$. The key lemma is based on probabilistic argument. The final construction is an inverse limit of finite graphs.

math.CO↗

arXiv2019

Cardinal functions of purely atomic measures

Let $μ$ be a purely atomic measure. By $f_μ:[0,\infty)\to\{0,1,2,\dots,ω,\mathfrak{c}\}$ we denote its cardinal function $f_μ(t)=\vert\{A\subset\mathbb N:μ(A)=t\}\vert$. We study the problem for which sets $R\subset\{0,1,2,\dots,ω,\mathfrak{c}\}$ there is a measure $μ$ such that $R$ is the range of $f_μ$. We are also interested in the set-theoretic and topological properties of the set of $μ$-values which are obtained uniquely.

math.CA↗

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↗

arXiv2016

Achievement sets of conditionally convergent series

Considering the sets of subsums of series (or achievement sets) we show that for conditionally convergent series the multidimensional case is much more complicated than that of the real line. Although we are far from the full topological classification of such sets, we present many surprising examples and catch the ideas standing behind them in general theorems.

math.FA↗

arXiv2016

On the set of limit points of conditionally convergent series

Let $\sum_{n=1}^\infty x_n$ be a conditionally convergent series in a Banach space and let $τ$ be a permutation of natural numbers. We study the set $\operatorname{LIM}(\sum_{n=1}^\infty x_{τ(n)})$ of all limit points of a sequence $(\sum_{n=1}^p x_{τ(n)})_{p=1}^\infty$ of partial sums of a rearranged series $\sum_{n=1}^\infty x_{τ(n)}$. We give full characterization of limit sets in finite dimensional spaces. Namely, a limit set in $\mathbb{R}^m$ is either compact and connected or it is closed and all its connected components are unbounded. On the other hand each set of one of these types is a limit set of some rearranged conditionally convergent series. Moreover, this characterization does not hold in infinite dimensional spaces. We show that if $\sum_{n=1}^\infty x_n$ has the Rearrangement Property and $A$ is a closed subset of the closure of the $\sum_{n=1}^\infty x_n$ sum range and it is $\varepsilon$-chainable for every $\varepsilon>0$, then there is a permutation $τ$ such that $A=\operatorname{LIM}(\sum_{n=1}^\infty x_{τ(n)})$. As a byproduct of this observation we obtain that series having the Rearrangement Property have closed sum ranges.

math.FA↗

arXiv2014

Large free sets in universal algebras

We prove that for each universal algebra $(A,\mathcal A)$ of cardinality $|A|\ge 2$ and an infinite set $X$ of cardinality $|X|\ge|\mathcal A|$, the $X$-th power $(A^X,\mathcal A^X)$ of the algebra $(A,\mathcal A)$ contains a free subset $\mathcal F\subset A^X$ of cardinality $|\mathcal F|=2^{|X|}$. This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family $\mathcal I\subset\mathcal P(X)$ of cardinality $|\mathcal I|=|\mathcal P(X)|$ in the Boolean algebra $\mathcal P(X)$ of subsets of an infinite set $X$.

math.LO↗

arXiv2011

Continuous dependence on parameters for second order discrete BVP's

Using min-max inequality we investigate the existence of solutions and thier dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle points to the Euler action functional.

math.CA↗