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Seminar of the Extension of the Mathematical Institute SAS in Košice

···   2022   •   2023   •   2024   •   2025
 
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23. 1. 2025 Irena Jadlovská – On the Lambert W function and its applications
Abstract: In the talk, we discuss numerous applications of the Lambert W function - the multivalued inverse of the function $w \to w \exp^w$. Particular attention is devoted to its role in both the quantitative and qualitative analysis of delay differential equations.
5. 2. 2025 Ahmed Ibrahim Abosaied – Hardy-type inequalities
link to slides
20. 2. 2025 Peter Mlynárčik – Kuratowski algebra and nondeterministic complexity
Abstract: My presentation is divided to two parts.
  1. Common work with Galina Jirásková and Michal Hospodár.
  2. Essential part of my work is teaching at university and elementary school, so I present interesting problems with wizard, elfs and hats.
The first part. Kazimierz Kuratowski (1896-1980) was interested in the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. In 1922 he published that at most 14 distinct sets is possible to obtain. We investigate similar cases with regular languages and instead of closure operation and complement in topological space we consider positive Kleene closure and complement: $$L^+=\bigcup\limits_{i=1}^{\infty}L^i,\ L^c=\Sigma^*\setminus L$$ With such two operations and their repeatedly applying on the set $L$ it is possible to obtain at most 10 distinct languages. Next we consider nondetermnistic complexity of language $L$ with $\mathrm{nsc}(L)=n$ and nondeterministic complexities of 10 possible obtained languages. Some of them have complexities $n$, $2^n$, and upper bounds $2^{2^n}$, Dedekind number $M(n)$.
The second part. We will look at several tasks where the main role is played by a wizard and elves. The wizard has the elves in his power and amuses himself by casting spells colored hats to the elves, gives them some rules and clues and elves must guess the color of their own hat... In a way of solutions of such tasks we can find really nice mathematics concerning to epistemic logic, algebra, probability...
6. 3. 2025 Viktor Olejár – Descriptional and Computational Complexity of Regular Languages
Abstract: We review some selected results obtained during the past few years of our PhD study. We mostly focus on summarizing outcomes that fall into one of three key topics of our thesis: closure properties, nondeterministic state complexity, and computational complexity related to decision problems. The talk serves as preparation and feedback for our thesis defense.
20. 3. 2025 Jozef Pócs – On representation of OFWA operators
Abstract: Ordered functional weighted averaging operators (OFWA operators) represent a generalization of OWA operators that are commonly used in decision making. It is shown that the class of OFWA operators is identical to the class of all intermediate functions, i.e. functions between min and max. The involved representation of OFWA operators determines a mapping between the set of all functions defined on the unit real interval and the family of all intermediate functions. The main aim is to describe some properties of this representation, in particular from a topological and algebraic point of view.
3. 4. 2025 Miroslav Repický – Two-parameter cardinal invariants of ideals
Abstract: Given ideals $\mathcal{I}$ and $\mathcal{J}$ on an infinite set $X$ we define \begin{align*} &\mathrm{add}^\mathcal{J}(\mathcal{I})=\min\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{I}\ \textrm{a}\ \forall I\in\mathcal{I}\ \exists A\in\mathcal{A}\ I\nsupseteq^\mathcal{J} A\},\\ &\mathrm{cof}^\mathcal{J}(\mathcal{I})=\min\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{I}\ \textrm{a}\ \forall I\in\mathcal{I}\ \exists A\in\mathcal{A}\ I\subseteq^\mathcal{J} A\}, \end{align*} where $A\subseteq^\mathcal{J} B$ means $A\setminus B\in\mathcal{J}$. We investigate estimations of these cardinal invariants for pairs of several standard ideals either on a countable set or on the real line. We also study these invariants in the case when at least one of the ideals $\mathcal{I}$ and $\mathcal{J}$ is a maximal ideal.
15. 5. 2025 Peter Eliaš – Constructing free orthomodular poset over a given orthoposet
Abstract: We describe an algorithm for finding the free orthomodular poset over a given orthoposet. We can prove its correctness, but we cannot yet prove that the algorithm always finds an answer.
We don't know the answers to these questions:
  1. Is it true that if $\tau$ and $\sigma$ are terms over an orthoposet $P$ and there exists an OP-morphism $f\colon P\to Q$ from $P$ to some orthomodular poset $Q$ such that $\mathrm{val}_f(\tau)\nleq\mathrm{val}_f(\sigma)$, then there exists such a morphism also for $Q=\{0,1\}$?
  2. Is it true that if an orthoposet $P$ is finite then also the free orthomodular poset over $P$ is finite?
29. 5. 2025 Ján Haluška – On the ordered Hilbert space called normal principal stop
Abstract:
26. 6. 2025 Michal Hospodár – Square and other operations on star-free languages
Abstract: The state complexity of most basic regular operations on star-free languages is the same as in the regular case. A notable exception is the reversal where a known lower bound $2^n-1$ with an example of alphabet size $n-1$ from [J. A. Brzozowski, B. Liu: Quotient complexity of star-free languages, Int. J. Found. Comput. Sci. 23 (2012) 1261–1276] is shown to be tight. For the square operation, we show an upper bound $(n-1)2^{n}-2(n-2)$ and a similar lower bound which is smaller by 18 in case of $n=6$.