Seminar of the Extension of the Mathematical Institute SAS in Košice

···   2019   •   2020-21   •   2022   •   2023   •   2024
19. 1. 2023 Jan Makovský (Institute of Philosophy of the Czech Academy of Sciences) – Leibniz's path to the infinitesimal
Abstract: What do the world vortex and souls, collisions of bodies, ether, atoms and infinity have in common? The struggle for the independence of geometrically constructed nature takes place within the concept of movement: in the abyss of the "science of the point", between the divisible and the indivisible, quantity and its limit.
26. 1. 2023 Peter Eliaš – Functors, monads, adjunctions
Abstract: We will make a quick overview of the basic concepts of category theory: functor, natural transformation, monad, adjunction. We will illustrate the concepts based on the results of work [Jenča, G. Orthomodular posets are algebras over bounded posets with involution. Soft Comput 26, 491–498 (2022)] and indicate the possibilities of solving the problem formulated there.
9. 2. 2023 Ján Haluška – Mathematical model of organ sound generated with a set of constant mensure stops
Abstract: Concerning organ stops, there are three psychological phenomena on which is based the present tone system 12-tone equal temperament. They are: Pythagorean Comma, Octave Equivalence, and Equal Timbre of all tones in a stop.
Essential fact is a finding that musical structures the fifth circle and transposition can be used to define operations "multiplication" $\otimes$ and "addition" $\oplus$ in 12-TET $\mathbb{T}_{12}$. Based on this, we construct a generalized 12-TET (G12-TET), the ordered tempered Hilbert vector algebra $\mathscr{W}_{12}$ over $\mathbb{R}$, where the classical 12-TET is a subset.
We can summarize that: An organ sound in G12-TET system $\mathscr{W}_{12}$, which is produced with a pipe set of constant mensured stops, is a linear variety over $\mathbb{R}$ associated with sound of the Principal stop.
23. 2. 2023 Emília Halušková – Algebras with easy direct limits and sets $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$
Abstract: In universal algebra, the construction of a direct limit is one of the basic methods for building new algebras from given algebras of fixed type.
Let $A$ be an algebra. We will focus on the investigation of direct limits of families of algebras in which all algebras are isomorphic to $A$. If every such direct limit is isomorphic to some retract of the algebra $A$, then we say that the algebra $A$ has easy direct limits, in short $A$ has EDL.
Let us consider the commonly used operations of addition and multiplication of real numbers. Then the additive group of rational numbers and the rings of whole, rational and real numbers have EDL. The multiplicative group of rational numbers, the additive group and the multiplicative monoid of integers do not have EDL. Questions if the additive and multiplicative group of real numbers, the additive, multiplicative group and the field of complex numbers have EDL are open.
16. 3. 2023 Michal Hospodár – Operations on subregular languages (continuation)
Abstract: We study the nondeterministic state complexity of basic regular operations on subregular language classes. In particular, we focus on the classes of singleton, finite, and symmetric definite languages, and consider the operations of intersection, union, concatenation, power, and complementation. We get the exact complexity in all cases, except for complementation of singleton languages where we only have a lower bound $\sqrt{n}$ and an upper bound $n$. The complexity of intersection on singleton languages is $\min\{m, n\}$, and for all other operations, except for complementation, the known upper bounds for finite languages are met by singleton languages. The known lower bound $2^{n-1}$ for complementation on left ideals is tight also for symmetric definite languages. In all the remaining cases, the nondeterministic state complexity of all considered operations is the same as in the regular case, although sometimes we need to use a larger alphabet to describe the corresponding witnesses.
30. 3. 2023 Galina Jirásková – Undecidable problems for deterministic biautomata
Abstract: We introduce the concept of deterministic biautomaton and its three submodels. These four models correspond to the four models of grammars that are known from the literature, and the language classes corresponding to them form a chain of strict subclasses. We show that it is undecidable whether a language accepted by a nondeterministic biautomaton belongs to the class corresponding to a given model of deterministic biautomata. In the proof, we use Greibach's theorem and closure properties of considered classes.
13. 4. 2023 Viktor Olejár – Closure properties of subregular languages under operations
Abstract: A class of languages is closed under a given operation if the resulting language belongs to this class whenever the operands belong to it. We examine the closure properties of various subclasses of regular languages under basic operations of intersection, union, concatenation and power, positive closure and star, reversal, and complementation. We consider the following classes: definite languages and their variants (left ideal, finitely generated left ideal, symmetric definite, generalized definite and combinational), two-sided comets and their variants comets and stars, and the classes of singleton, finite, ordered, star-free, and power-separating languages. We also give an overview about subclasses of convex languages (classes of ideal, free, and closed languages), union-free languages, and group languages. We summarize some inclusion relations between these classes. Subsequently, for all pairs of a class and an operation, we provide an answer whether this class is closed under this operation or not.
11. 5. 2023 Jozef Pócs – Sugeno Integrals on Bounded Lattices
Abstract: Discrete Sugeno integral on a bounded distributive lattice L can be characterized as a uniquely determined compatible aggregation function that extends a given L-valued capacity. It is shown that this unique extension property of an L-valued capacity to a compatible function is equivalent to the distributivity of the lattice L. This result gives a new characterization of bounded distributive lattices.
25. 5. 2023 Irena Jadlovská – Comparison theorems in oscillation theory
Abstract: We revise Kondrat’ev’s and Chanturia’s comparison results for oscillation/Property A of higher-order ordinary linear differential equations, which can be seen as a generalization of famous Sturm’s comparison and separation theorems for second-order differential equations. Extensions for delay differential equations are discussed and further open problems are indicated.
28. 9. 2023 Miroslav Repický – Partial orders of ideals
Abstract: For ideals $I$ on the set of natural numbers $\omega$ let
  • $\mathfrak{b}_I= \min\{|B|:B\subseteq{}^\omega\omega\text{ and } (\forall f\in{}^\omega\omega)(\exists g\in B)\ \{n:g(n)\not\le f(n)\}\notin I\},$
  • $\mathfrak{d}_I= \min\{|D|:D\subseteq{}^\omega\omega\text{ and } (\forall f\in{}^\omega\omega)(\exists g\in D)\ \{n:f(n)\not\le g(n)\}\in I\}$
(the bounding number and the dominating number). Let
  • $F(\mathrm{Fin})=\{r\in{}^\omega\omega:\forall n$ $|r^{-1}[\{n\}]|<\omega\}$
(the set of all finite-to-one functions). The Rudin-Blass and the Katìtov-Blass partial (quasi) orderings of ideals $I$ and~$J$ are defined by
  • $I\le_\mathrm{RB} J\equiv(\exists r\in F(\mathrm{Fin}))\ I=\{a\subseteq\omega:r^{-1}[a]\in J\}$,
  • $I\le_\mathrm{KB} J\equiv(\exists r\in F(\mathrm{Fin}))\ I\subseteq \{a\subseteq\omega:r^{-1}[a]\in J\}$,
  • $I\le_\overline{\mathrm{KB}} J\equiv(\exists r\in F(\mathrm{Fin}))\ \{a\subseteq\omega:r^{-1}[a]\in I\}\subseteq J$
($\le_\overline{\mathrm{KB}}$ is a~modification of $\le_\mathrm{KB}$ not studied before). We consider the following generalizations of these partial orderings:
  • $I\le_\mathrm{RB}^\sqcup J\equiv (\exists E\in[F(\mathrm{Fin})]^{<\omega})\ I=\{a\subseteq\omega:\bigcup_{r\in E}r^{-1}[a]\in J\}$,
  • $I\le_\mathrm{RB}^\sqcap J\equiv (\exists E\in[F(\mathrm{Fin})]^{<\omega})\ I=\{a\subseteq\omega:\bigcap_{r\in E}r^{-1}[a]\in J\}$,
  • $I\le_\mathrm{KB}^\sqcup J\equiv (\exists E\in[F(\mathrm{Fin})]^{<\omega})\ I\subseteq\{a\subseteq\omega:\bigcup_{r\in E}r^{-1}[a]\in J\}$,
  • $I\le_\mathrm{KB}^\sqcap J\equiv (\exists E\in[F(\mathrm{Fin})]^{<\omega})\ I\subseteq\{a\subseteq\omega:\bigcap_{r\in E}r^{-1}[a]\in J\}$,
  • $I\le_\overline{\mathrm{KB}}^\sqcup J\equiv (\exists E\in[F(\mathrm{Fin})]^{<\omega})\ \{a\subseteq\omega:\bigcup_{r\in E}r^{-1}[a]\in I\}\subseteq J$,
  • $I\le_\overline{\mathrm{KB}}^\sqcap J\equiv (\exists E\in[F(\mathrm{Fin})]^{<\omega})\ \{a\subseteq\omega:\bigcap_{r\in E}r^{-1}[a]\in I\}\subseteq J$.
By considering finite-to-finite relations instead of functions we get another nine definitions of partial orderings distinguished by the suffix -$\mathrm{r}$ in the notation. We prove some inclusions and equalities between these partial orderings and we show that this collection of partial orderings has these two maximal elements with respect to the inclusion: ${\le_{\overline{\mathrm{KB}}\textrm{-}\mathrm{r}}}={\le_{\overline{\mathrm{KB}}\textrm{-}\mathrm{r}}^\sqcap}={\le_{\overline{\mathrm{KB}}\textrm{-}\mathrm{r}}^\sqcup}$ a ${\le_\mathrm{KB}^\sqcap}={\le_{\mathrm{KB}\textrm{-}\mathrm{r}}^\sqcap}$. We prove that the invariantt $\mathfrak{b}_I$ is ${\le_{\mathrm{KB}\textrm{-}\mathrm{r}}^\sqcap}$- and ${\le_{\overline{\mathrm{KB}}\textrm{-}\mathrm{r}}^\sqcup}$-increasing and the invariant $\mathfrak{d}_I$ is ${\le_{\mathrm{KB}\textrm{-}\mathrm{r}}^\sqcap}$- and ${\le_{\overline{\mathrm{KB}}\textrm{-}\mathrm{r}}^\sqcup}$-decreasing.
12. 10. 2023 Peter Eliaš – Construction of the free orthomodular poset over a given orthoposet
Abstract: We describe a functor which is left adjoint to the forgetful functor from the category of orthomodular posets into the category of bounded posets with an involution. This functor is obtained as a composition of two functors: the first one goes from the category of bounded posets with an involution into the category of orthoposets, and the second goes from the category of orthoposets into the category of orthomodular posets. While the first functor is idempotent and maps some elements into the least or the greatest element of the poset, the second functor corresponds to the construction of the free orthomodular poset over a given orthoposet. The elements of the free orthomodular poset are defined as the equivalence classes on the set of all well-formed terms over the given orthoposet.
26. 10. 2023 Ján Haluška – Mathematical structures of the generalized 12-TET
Abstract: From a scientific point of view, tones in music are the material, live, psychological objects of great complexity. Therefore, all musical concepts expressed mathematically are approximate and applied with respect to a pre-specified accuracy which indicates many pre-determined compromise corridors: psycho-acoustical, mathematical, physical, material, spiritual, art-historian, natural-scientific, etc. Concerning all tuned European instruments, there are three basic psychological phenomena, factors when creating tone systems. In the present time, the prevailing tone system is 12-tone equally tempered tone system (shortly: 12-TET). His fundamental principles are:
  1. Pythagorean Comma approximation;
  2. Octave Equivalence;
  3. Suitable Tuning triangle for pipe organs.
In this paper there is shown which structures arise when the 12-TET based arithmetic operations $\oplus$, $\otimes$ are introduced on the space of tone Fourier decomposition sequences and the dynamic keyboard is used. In detail, the following mutually compatible structures are naturally derived from the generalized 12-TET:
  1. Cyclic and linear orders $\lll$, $\ll$;
  2. Ring operation multiplication $\otimes$ and addition $\oplus$;
  3. Vector operations;
  4. Scalar product structures;
  5. Subalgebras;
  6. Invertible elements (if they exist);
  7. Hilbert space structure;
  8. Affine structures.
23. 11. 2023 Emília Halušková – On Bernoulli shift mapping
Abstract: The Bernoulli shift is a mapping that is paradigmatic in the theory of dynamical systems. We will look at its properties through the structure of the corresponding monounary algebra.
This is the joint work with R. Schwartzová.