Seminar of the Extension of the Mathematical Institute SAS in Košice

2019 <<   2020-21 <<   2022 <<   2023
september 2023 Miroslav Repický
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.