# Seminar of the Extension of the Mathematical Institute SAS in Košice

 2018 <<   2019 <<   2020-21 <<   2022   >> 2023,

 12. 5. 2022 Ján Haluška – An ensemble of 6 ordered vector algebras, epimorphic to 6 generalized complex numbers is a generalization of 12-TET Abstract: A $12$-dimensional linearly ordered vector algebra $\mathscr{W}_{12}$ over $\mathbb{R}$ is inspired with 12-Tone Equal Temperament Tuning system (12-TET) which is generally known in mathematical acoustics. In the paper introduced definition of multiplication $\otimes$ is based on a chain of 85 vectors of the Pythagorean approximation cycle and the notion of octave equivalence. From the operation $\otimes$, it is derived an operation $\oplus$ (operation of transposition, shift of vectors). The operation $\otimes$ is associative, commutative and distributive with respect to $\oplus$. In the paper, there are considered first 6 sub-algebras of $\mathscr{W}_{12}$; $\mathscr{W}_{1} \subset \mathscr{W}_{2} \subset \mathscr{W}_{3} \subset \mathscr{W}_{6} \subset \mathscr{W}_{12}$, $\mathscr{W}_{2} \subset \mathscr{W}_{4} \subset \mathscr{W}_{12}$. We can compute all possible invertible vectors (if they exist) in all ordered sub-algebras of $\mathscr{W}_{12}$. We describe only some interconnections among various substructures of $\mathscr{W}_{12}$, neither its numerous non-mathematical applications. According to results of the paper, we can claim that each tonal European musical composition is algebraically isomorphic with a sequence of lineals (=chords) of vectors (=tones) in $\mathscr{W}_{12}$ in time. 26. 5. 2022 Peter Eliaš – On tensor product Abstract: We compare construction of tensor product in the category of vector space, where morphisms are linear maps, and the category of complete lattices, where morphisms are join-preserving maps. Tensor product of vector spaces $V$, $W$ is a vector space $T=V\otimes W$, together with a map $\otimes\colon V\times W\to T$, possessing a universal property with respect to all bilinear maps from the cartesian product $V\times W$ into arbitrary vector space $U$. Tensor product of complete lattices $K$, $L$ (in the category where morphisms are join-preserving maps) is the lattice $G$ of all Galois connections between $K$, $L$, together with an appropriate map $\varphi\colon K\times L\to G$. This map possess a universal property with respect to all maps defined of $K\times L$ that preserve joins in each variable separately. 9. 6. 2022 Michal Hospodár – Operations on subregular languages and nondeterministic state complexity Abstract: We study the nondeterministic state complexity of basic regular operations on subregular language families. In particular, we focus on the classes of combinational, finitely generated left ideal, group, star, comet, two-sided comet, ordered, and power-separating languages, and consider the operations of intersection, union, concatenation, power, Kleene star, reversal, and complementation. We get the exact complexity in all cases, except for complementation of group languages where we only have an exponential lower bound. The complexity of all operations on combinational languages is given by a constant function, except for the $k$-th power where it is $k+1$. For all considered operations, the known upper bounds for left ideals are met by finitely generated left ideal languages. The nondeterministic state complexity of the $k$-th power, star, and reversal on star languages is $n$. 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. 23. 6. 2022 Emília Halušková – On discrete properties of several real functions Abstract: Monounary algebras are the most simple algebraic structures. They are defined as a pair $(A, f)$, where $A$ is a nonempty set and $f$ is a mapping from $A$ to $A$. They are represented by oriented graphs in which exactly one edge emerges from each vertex. They are interdisciplinary in nature. We will illustrate this in Sharkovsky's theorem (1964) from the theory of dynamical systems and Łojasiewicz's theorem (1951) from the theory of iterative roots. Next, we present the classification of monounary algebras defined on an interval of real numbers with a continuous strictly monotone function according to decompositions into connected components, more precisely according to the types of non-isomorphic connected components that occur in these algebras. Monounary algebras defined intervals of real number with a continuous strictly monotone operation contain at most 4 mutually nonisomorphic components. If the interval is closed and bounded, then 8 combinations of these components are possible. For other intervals, 10 combinations are possible. The bijectivity of the function, the set of fixed points of the function or its second iteration and the bounding of the function on the maximal subset of the domain that does not contain the fixed points of the function play a role in the criteria. 8. 9. 2022 Małgorzata Jastrzębska (UPH Siedlce) – Lattices in associative rings Abstract: The aim of the talk is to present the methods of lattice theory in studying the properties of associative rings. The lattices of ideals, the lattices of one-sided ideals and the lattices of one-sided annihilators will be discussed. I will present the relationships between these lattices and their selected properties. I will also present a few theorems characterizing the properties of rings with the help of mentioned lattices. 22. 9. 2022 Viktor Olejár – Selected topics in quantum finite automata Abstract: Quantum finite automata literature offers an alternative mathematical model for studying quantum systems with finite memory. Many different models have been proposed with varying computational powers. Moore-Crutchfield quantum finite automaton (MCQFA) is one of the earliest proposed models which is obtained by replacing the transition matrices of the classical finite automata by unitary operators. Despite the fact that they are weaker than their classical counterparts in terms of their language recognition power, for certain tasks MCQFAs have been shown to be more succinct. We explore two such examples: the language $\mathtt{MOD}_{p}=\{a^{j} \mid j \equiv 0 \mod p\}$ where $p$ is a prime number, and the promise problem $\mathtt{EVENODD}^k = (\mathtt{EVENODD}^k_{\text{yes}}, \mathtt{EVENODD}^k_{\text{no}})$ for a fixed positive integer $k$. We also review some past activities concerning research and education in related quantum computing fields: relevant organizations, workshops (first a second), summer schools, etc. 6. 10. 2022 Galina Jirásková – Multiple concatenation and state complexity Abstract: We describe witnesses for the concatenation of $k$ languages over an alphabet of $k + 1$ symbols with a significantly simpler proof then that from the literature. Then we use slightly modified Maslov’s automata to get witnesses over a $k$-letter alphabet which solves an open problem stated by Caron et al. [2018, Fund. Inform. 160, 255–279]. We prove that for $k = 3$, the ternary alphabet is optimal. We also obtain lower bounds $n_1 - 1 + (1/2^{2k-2})2^{n_2+···+n_k}$ and $(1/2^{2k-2})n_1 2^{n_2+···+n_k}$ in the binary and ternary case, respectively. Finally, we show that an upper bound for unary cyclic languages is $n_1n_k/d + n_1 + ··· + n_k - k +1+ d$ where $n_1 \le···\le n_k$ and $d = \gcd(n_1,...,n_k)$. 20. 10. 2022 Ján Brajerčík (University of Prešov) – Problems of global variational geometry Abstract: In the talk we introduce basic concepts concerning global variational geometry such as smooth manifold, fibered manifold, jet, Lagrangian, Euler-Lagrange equation. Examples of problems solved by methods of global variational geometry are discussed. Reference: The Lepage Research Institute 3. 11. 2022 Irena Jadlovská – Canonical representation of disconjugate operators Abstract: We recall Polya and Trench’s classical results on factorizations of disconjugate linear operators. The usefulness of Trench’s canonical representation in the study of binomial differential equations is demonstrated. Several recently obtained related results are discussed, including explicit formulas for canonical representation of n-th order noncanonical disconjugate operators, general relations between solutions of canonical and noncanonical two-term functional differential equations, and extensions of the canonical classification scheme of possible nonoscillatory solutions from linear to differential equations with power-type nonlinearity or dynamic equations on time scales. 24. 11. 2022 Jozef Pócs – Join dense subset of aggregation functions Abstract: Aggregation functions defined on a bounded lattice form a bounded lattice with respect to the point-wise operations of join and meet. We describe a general way of finding join dense subsets in lattices of functions and also discuss the minimality of these sets. The obtained results are applied to the lattice of aggregation functions and to the lattice of idempotent aggregation functions. 8. 12. 2022 Miroslav Repický – Slalom numbers Abstract: For ideals $I$ and $J$ on $\omega$ let \begin{align*} &\mathfrak{sl}_e(I,J)= \min\{|S|:S\subseteq{}^\omega I\text{ and } (\forall\varphi\in{}^\omega\omega)(\exists s\in S)\ \{n:\varphi(n)\in s(n)\}\in J^+\},\\ &\mathfrak{sl}_t(I,J)= \min\{|S|:S\subseteq{}^\omega I\text{ and } (\forall\varphi\in{}^\omega\omega)(\exists s\in S)\ \{n:\varphi(n)\in s(n)\}\in J^\text{d}\}, \end{align*} where $J^+=\mathcal{P}(\omega)\setminus J$ and $J^\text{d}=\{\omega\setminus a:a\in J\}$ (and where a slalom $s$ is evaded or traversed by a function $\varphi$ with respect to $J$). These cardinal invariants are monotone in parameters with respect to Katětov-Blass partial order of ideals or with respect to a reverse modification of this partial order. As a consequence we get some restrictions on possible values of these invariants for ideals with the Baire property.