arXiv subjects

Filip Turoboś

Publications and source records attributed to Filip Turoboś.

9 recordsLinked to original sources

arXiv2026

Why Do Students (Not) Choose Second-Cycle Mathematics Studies? Questionnaire and Graduate-Tracking Evidence from Poland

The Bologna Process has substantially reshaped higher education systems across Europe, including the structure of mathematical studies in Poland. One of the increasingly visible consequences of these transformations is the relatively low retention rate between first- and second-cycle studies. The aim of this paper is to investigate selected factors associated with students' willingness to continue mathematical education after obtaining a Bachelor's degree. The study combines questionnaire-based research conducted among mathematics students from 13 Polish higher education institutions with an auxiliary analysis of nationwide graduate-tracking data obtained from the Polish Graduate Tracking System (ELA). The survey investigated students' opinions on general and specialized courses, perceived labour-market usefulness of studies and future educational intentions. The results indicate that students willing to continue second-cycle studies evaluate both the substantive quality and practical usefulness of their studies more positively than students intending to leave mathematics or change institutions. Satisfaction with the chosen specialization emerged as one of the strongest differentiating factors between the analysed groups. At the same time, a substantial proportion of respondents expressed doubts regarding the professional utility of continuing mathematical education, despite administrative labour-market data suggesting several advantages associated with obtaining a Master's degree. The findings suggest that retention in mathematics is shaped not only by academic difficulty, but also by the perceived relationship between university curricula, specialization structures and labour-market expectations. We conclude with recommendations regarding curriculum design, cooperation with external stakeholders and other aspects of second-cycle mathematical education.

math.HO↗

arXiv2026

Applications of ball spaces theory: fixed point theorems in semimetric spaces and ball convergence

In the paper we apply some of the results from the theory of ball spaces in the semimetric spaces. This allowed us to obtain some fixed point theorems which we believe to be unknown to this day. We also show the limitations of the ball space approach to this topic. As a byproduct, we obtain the equivalence of some different notions of completness in semimetric spaces where the distance function is $1$-continuous. In the second part of the article, we generalize Caristi-Kirk results for for $b$-metric spaces. Additionally, we obtain characterization of semicompleteness for $1$-continuous $b$-metric space via fixed point theorem analogous to the result of Suzuki. In the epilogue, we introduce the concept of convergence in ball spaces, based on the idea that balls should resemble closed sets in topological sets. We prove several of its properties, compare it with convergence in semimetric spaces and pose several open questions connected with this notion.

math.GN↗

arXiv2025

Spectre operator, achievement sets and sets of P-sums in a hyperspace of compact sets

Let $(X,+,d)$ be an Abelian metric group and $A\subset X$. We investigate the spectre of a set $A$, defined as the set of all elements $z\in X$ such that for every $x\in A$ either $x+z \in A$ or $x-z \in A$. We consider the corresponding to this notion operator $S$ acting on the hyperspace of compact sets and examine its properties. Furthermore, we study the families of achievement sets and sets of $P$-sums in this hyperspace, as well as prove some properties of achievement sets in the plane.

math.GN↗

arXiv2023

How much can we extend the Assouad embedding theorem?

The celebrated Assouad embedding theorem has been known for over 40 years. It states that for any doubling metric space (with doubling constant $C_0$) there exists an integer $N$, such that for any $α\in (0.5,1)$ there exists a positive constant $C(α,C_0)$ and an injective function $F:X\to \mathbb{R}^N$ such that \[ \forall x,y\in X \quad C^{-1} d(x,y)^α \leq \|F(x)-F(y) \| \leq Cd(x,y)^α \] In the paper we use the remetrization techniques to extend the said theorem to a broad subclass of semimetric spaces. We also present the limitations of this extension -- in particular, we prove that in any semimetric space which satisfies the claim of the Assouad theorem, the relaxed triangle condition holds as well, which means that there exists $K\geq 1$ such that \[ \forall x,y,z\in X \quad d(x,z)\leq K\left( d(x,y)+d(y,z)\right). \]

math.GN↗

arXiv2021

Comparison of approximation algorithms for the travelling salesperson problem on semimetric graphs

The aim of the paper is to compare different approximation algorithms for the travelling salesperson problem. We pick the most popular and widespread methods known in the literature and contrast them with a novel approach (the polygonal Christofides algorithm) described in our previous work. The paper contains a brief summary of theory behind the algorithms and culminates in a series of numerical simulations (or "experiments"), whose purpose is to determine "the best" approximation algorithm for the travelling salesperson problem.

math.CO↗

arXiv2021

Approximate solutions to the Travelling Salesperson Problem on semimetric graphs

With the aid of the relaxed polygonal inequality (introduced by Fagin et al.) we strive to extend the applicability of Christofides approximation technique to the scope of all complete finite weighted graphs with positive weights. First section acquaints the Reader with the class of semimetric graphs and proves that every finite graph admits $γ$-polygon structure. Sections 2 and 3 establish the necessary notions from the graph and optimization theory to tackle the Traveling Salesperson Problem. In section 4 the minimal spanning tree method is introduced, while section 5 focuses on the analysis of this method through the lens of $γ$-polygon graphs. The final section of the paper adjusts the technique of Christofides by obtaining $\frac{3γ}{2}-$approximation for the TSP.

math.MG↗

arXiv2020

On functions preserving products of certain classes of semimetric spaces

In the paper we continue the research of Borsík and Doboš on functions which allow us to introduce a metric to the product of metric spaces. In this paper we extend their scope on broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving $b$-metric inequality. We provide analogues of the results of Borsík and Doboš, adjusted to the new, broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.

math.MG↗

arXiv2020

On characterization of functions preserving metric-type conditions via triangular and polygonal structures

Following the train of thought from our previous paper we revisit the theorems of Pongsriiam and Termwuttipong by further developing their characterization of certain property-preserving functions using the so-called triangle triplets. We develop more general analogues of disjoint sum lemmas for broader classes of metric-type spaces and we apply these to extend results of Borsìk an Doboš as well as those obtained by Khemaratchatakumthorn and Pongsriiam. As a byproduct we obtain methods of generating non-trivial and infinite strong $b$-metric spaces which are not metric.

math.MG↗

arXiv2018

A measure of non-compactness on $T_{3\frac 1 2 }$ spaces based on Arzelà-Ascoli type theorem

The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the $T_{3 \frac{1}{2}}$ space. Motivated by the generalized Arzelà-Ascoli theorem for Tichonoff space $T$ via Wallman compactifiaction $\operatorname{Wall}(T)$, we constuct a measure of non-compactness for the space $C^b(T)$. We also study some of the properties of this object and give another version of Darbo-type theorem, suitable for this particular case.

math.GN↗