Seminar of the Extension of the Mathematical Institute SAS in Košice

··· 2023 • 2024 • 2025 • 2026

next:
7. 5. 2026

previous:
8. 1. 2026	Simon Dieck (Delft University of Technology) – Automata learning algorithms obtained from Myhill-Nerode-style theorems
	link to slides
22. 1. 2026	Irena Jadlovská – A note on solving 3D discrete weakly delayed systems
	Abstract: We discuss the effective application of Putzer’s algorithm to the solution of three-dimensional discrete weakly delayed (WD) systems, as defined in [1]. In particular, we show how the powers of the augmented $(3m+1)\times(3m+1)$ matrix $\mathcal A$ can be computed through a simple recursion involving only $3\times3$ matrices. [1] Diblík, Josef, et al. "General solutions of weakly delayed discrete systems in 3D." Advances in Nonlinear Analysis 14.1 (2025): 20250121.
12. 2. 2026	Galina Jirásková – The boundary of free convex languages
	Abstract: We present a complete solution of the problem of finding the deterministic state complexity of the boundary operation on the classes of prefix-free, suffix-free, bifix-free, factor-free, and subword-free languages. link to photo with the solution
26. 2. 2026	Miroslav Repický – Combinatorial properties of ideals
	Abstract: We deal with the cardinal characteristics $\mathrm{cov}^(\mathcal{I})$ for the ideal $\mathcal{Z}$ of zero-density sets, the summable ideal $\mathcal{I}_{1/n}$, and for the generalizations $\mathcal{I}(\bar I)$ and $\mathcal{S}(\bar I)$ of these ideals dependent on an interval partition $\bar I$ of the set of natural numbers. We have characterizations of the inclusions $\mathcal{Z}(\bar I)\subseteq\mathcal{Z}$, $\mathcal{Z}(\bar I)\supseteq\mathcal{Z}$, $\mathcal{S}(\bar I)\subseteq\mathcal{I}_{1/n}$, $\mathcal{S}(\bar I)\supseteq\mathcal{I}_{1/n}$ using simple properties of the partition $\bar I$, and we show that the relational structures related to the cardinal characteristics of $\mathrm{cov}^(\mathcal{I})$ for these ideals have their equivalent expressions in the sense of Tukey-connections in the form of pseudo-localizations of functions.
12. 3. 2026	Jozef Pócs – Characterization of Archimedean $t$-norms, which are 2-D linear splines.
	Abstract: It is known that every continuous Archimedean $t$-norm can be described using an additive generator. It is also known that if the additive generator is a so-called 1-D linear spline, then the resulting $t$-norm is a 2-D linear spline. We will show that the converse implication also holds, i.e., if an Archimedean $t$-norm is a 2-D linear spline, then the corresponding additive generator is a 1-D linear spline.
26. 3. 2026	Peter Eliaš – Matematical taxonomy, set representations of orthoposets, and free orthomodular posets over orthoposets
	Abstract: In [1], the following problem was formulated: to describe all "classifications", that is lists of properties that allow to classificate individuals into $n$ classes or species. In [2], this question was addressed combinatorially and all classifications for up to 7 classes were found. Later it was proved that the number of optimal classifications of $n$ species corresponds to the number of trees (unoriented acyclic graphs) with $n$ vertices.
	Our observations for small number of elements revealed that the number of connected orthoposets with $2n$ elements corresponds to the number of trees with $n$ vertices. The connection to "classifications" motivated us to formulate and prove the following results:
	1. If $X$ is a nonempty set and $\mathcal{S}\subseteq\mathcal{P}(X)$ is a complement-closed family of subsets then $(\mathcal{S},\subseteq,{}^c)$ is an orthoposet. 2. If $P$ is an orthoposet and $\mathcal{I}$ is the family of all ideals on $P$ then $P$ is isomorphic to the family $\mathcal{S}_P=\{\{I\in\mathcal{I}\colon x\notin I\}\colon x\in P\}$. 3. If $P$ is an orthoposet then the free orthomodularny poset over $P$ is isomorphic to the closure of $\mathcal{S}_P$ under disjoint unions (while preserving the property of being closed under complements).
	[1] Wexler, P. J., On the number of taxonomies, or the odds of 'structuralism', American Anthropologist, vol. 73 (1971), 1258. [2] Wexler P. J, Fremlin D. H., The number of classifications up to seven classificanda, Classification Society Bulletin, vol. 4, no. 3, (1979).
23. 4. 2026	Ján Haluška – From the system 12-TET to a linear variety of ordered commutative algebras $\mathscr{W}_{12}$ for pipe organs
	Abstract: All objects in this paper are objects of a great complex nature (psycho-acoustical, mathematical, physical, fyzikálnej, material, spiritual, historical, musician, etc.). They hold mathematicaly with respect to any certainty in advance, which does does not turn out of the predetermined compromise corridors. A very satisfying fact confirming the correctness of the theory is the so-called practice criterion. Concerning organ stops, there are three psychological phenomena on which is based the present tone system 12-tone equal temperament (12-TET). They are: Pythagorean Comma, Octave Equivalence, and Equal Timbre of all tones in a stop.