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

 2017 <<  2018 <<  2019 <<  2020-21  >> 2022

 23. 1. 2020 Peter Eliaš – Ordered vector spaces in probability theory 6. 2. 2020 Roman Frič – Lifting and sagging observables Abstract: Important constructions in probability theory can be schematized via two simple commutative triangle diagrams. We shall deal with joint experiments and conditional probability. 20. 2. 2020 Ján Haluška – Generalized complex numbers Abstract: An unified model of generalized complex numbers is defined via an operation of multiplication in the form of a Toeplitz matrix. 5. 3. 2020 Emília Halušková – On discrete properties of monotone functions Abstract: A correspondence between monotone functions with respect a linear order and monounary algebras which consist of at most four types of components will be described. Further, several properties of monotone functions with respect a partial order will be given. 25. 6. 2020 Ján Haluška – Bitopology on $\mathbb{E}_4$ equipped with a skew circulated multiplication Abstract: An operation of multiplication on $\mathbb{E}_4$ is introduced via a skew circulated matrix, it is associative, commutative and distributive. The resulting algebra $\mathbb{W}$ over $\mathbb{R}$ is isomorphic to $\mathbb{C}\times\mathbb{C}$ and with partially invertible elements. The related algebraic, geometrical, and topological properties are given. There are sub-planes of $\mathbb{W}$ isomorphic to the Gauss and Clifford complex number planes. A topology on $\mathbb{W}$ are given via a norm which is sum of two non-equivalent seminorms. 1. 10. 2020 Michal Hospodár – Right and left quotients on subclasses of convex languages Abstract: We study the state complexity and nondeterministic state complexity of right quotient and left quotient on the classes of prefix-, suffix-, factor-, and subword-free, -closed, and -convex regular languages, and on the classes of right, left, two-sided, and all-sided ideal languages. We get tight upper bounds in all cases except for state complexity of left quotient of all-sided ideals and subword-closed languages by a regular language, and nondeterministic state complexity of left quotient on subword-convex languages. 30. 9. 2021 Jozef Pócs – Aggregation functions and their connection to universal algebra Abstract: Aggregation functions form a composition closed family, i.e., a clone. We show that any generating set of this clone enables to define an algebraic structure, such that aggregation functions form free algebra in the variety generated by the mentioned algebraic structure. 14. 10. 2021 Irena Jadlovská – Kneser oscillation theorem for second-order half-linear delay differential equations Abstract: A sharp extension of the classical Kneser oscillation theorem to a class of second-order half-linear delay differential equations is established, using a novel method of iteratively improved monotonicity properties of a nonoscillatory solution. 28. 10. 2021 Miroslav Repický – Ideals on $\omega$ determined by a sequence of capacities Abstract: A capacity on $\omega$ is a function $\nu:\mathcal P(\omega)\to[0,\infty]$ with the following properties: (i) $\nu(\emptyset)=0$; (ii) $a\subseteq b\subseteq\omega$ implies $\nu(a)\le\nu(b)$; (iii) $\nu(a)<\infty$ and $\nu(\omega\setminus a)=\infty$ for every $a\in[\omega]^{<\omega}$; (iv) $\nu(a)=\lim_{n\in\omega}\nu(a\cap n)$ for every $a\subseteq\omega$. Let $\mathrm{Fin}(\nu)=\{a\subseteq\omega:\nu(a)<\infty\}$ and $\mathrm{Fin}^*(\nu)=\{a\subseteq\omega:\nu(a)<\infty$ and $\nu(\omega\setminus a)=\infty\}$. For a lower semi-continuous submeasure $\mu$ on~$\omega$ the family $\mathrm{Exh}(\mu)=\{a\subseteq\omega:\lim_{n\in\omega}\mu(a\setminus n)=0\}$ is a proper ideal on $\omega$ whenever $\omega\notin\mathrm{Exh}(\mu)$. Obviously, if $\mu(\omega)=\infty$, then $\mu$ is a capacity and $\mathrm{Fin}(\mu)=\mathrm{Fin}(\mu^*)$ is an ideal. We prove that if $\omega\notin\mathrm{Exh}(\mu)$, then there exists an increasing sequence of capacities $\{\nu_k:k\in\omega\}$ such that $\mathrm{Exh}(\mu)=\bigcap_{k\in\omega}\mathrm{Fin}(\nu_k)$ and $\mathrm{Fin}(\nu_k)=\mathrm{Fin}^*(\nu_k)$ for all $k\in\omega$. We ask for a simple description of ideals $I$ for which there is a sequence of capacities $\{\nu_k:k\in\omega\}$ such that $I=\bigcap_{k\in\omega}\mathrm{Fin}(\nu_k)$. These ideals include the $F_\sigma$ ideals and the analytic $P$-ideals and every such ideal is $F_{\sigma\delta}$. 11. 11. 2021 Viktor Olejár – State complexity of the cut operation in subclasses of convex languages Abstract: We study the state complexity of the cut operation assuming that both operands belong to some, possibly different, subclass of convex languages. We consider the classes of ideal, closed, and free languages, and for all 144 pairs of classes, we get the exact state complexity of the cut operation. Assuming that $K$ and $L$ have state complexity at most $m$ and $n$, respectively, there are nine possible values of the state complexity of their cut: $m$ if $K$ is a right ideal, $m+n-1$ or $m+n-2$ if $K$ is prefix-closed or prefix-free, $mn-n+m$, $mn-n+1$, or $mn-n-m+2$ if $K$ is left ideal or suffix-closed, and $mn-2n+m$, $mn-2n+2$, or $mn-2n-m+4$ if $K$ is suffix-free. To get lower bounds, we use languages described over a fixed alphabet of size at most three, except for three cases when the languages are described over an alphabet of size linearly growing with $m$.