Seminar of the Extension of the Mathematical Institute SAS in Košice
| 26. 1. 2017 | Ján Borsík – Points of porouscontinuity |
| Abstract: Given a real function $f$ and $r\in[0,1)$, let us define $P_r(f)$ as the set of all $x$ for which there exists a set $A$ containing $x$ such that porosity at $x$ of the complement of $A$ is greater than $r$, and the restriction of $f$ to the set $A$ is continuous at $x$. Similarly we define $M_r(f)$ with the requirement that the porosity of the complement is greater or equal to $r$. Further we define $S_r(f)$ as the set of all $x$ such that for each $\varepsilon > 0$ there exists a set $A$ containing $x$ such that its complement has porosity at $x$ greater that $r$ and $f(A)\subseteq(f(x)-\varepsilon, f(x)+\varepsilon)$. Similarly we define $N_r(f)$ with $\ge$ instead of $\gt$. In the lecture we give complete characterization of sets $P_r(f)$, $S_r(f)$, $M_r(f)$, and $N_r(f)$. | |
| 9. 2. 2017 | Peter Eliaš – Closed families of sets and functions with respect to the Galois connection determined by uniform approximability |
| Abstract: Let $C(X,Y)$ denote the metric space of all continuous maps from a topological space $X$ to a metric space $Y$ equipped with the supremum metric. Let $\textit{CL}(X)$ denote the family of all closed subsets of $X$. Given $\Phi\subseteq C(X,Y)$, let us consider the Galois connection between subsets of $C(X,Y)$ and subsets of $\textit{CL}(X)$, determined by the relation $(f,E)\in R$ $\Leftrightarrow$ “there exists a sequence of pairwise distinct functions from $\Phi$ unifomly converging to the function $f$ on the set $E$ ”. Families of functions and sets closed under corresponding closure operators form a complete lattice. In the case that $\Phi$ is the set of all constant functions (the set of all linear functions), the corresponding lattice is isomorphic to the lattice of all partitions of the real line (the real line with one point removed, respectively) into closed sets,\ and each element of the lattice is generated by a finite or a countable set of functions. | |
| 23. 2. 2017 | Roman Frič – Independence (link to the presentation) |
| 16. 3. 2017 | Emília Halušková – EKP for monounary algebras |
| Abstract: We say that an algebra $A$ has EKP if for every congruence $r$ on the algebra $A$ there exists an endomorphism of $A$ whose kernel is the congruence $r$. For monounary algebras with an injective operation, finite monounary algebras, as well as two subfamilies of the family of all connected monounary algebras, we described all monounary algebras possessing EKP. We also showed that every monounary algebra with EKP has interesting podalgebras possessing EKP. (link to the presentation) | |
| 30. 3. 2017 | Ivana Krajňáková – The square operation on deterministic, alternating, and boolean automata |
| Abstract:
We study the square operation on deterministic, alternating, and
boolean finite-state automata.
We investigate the complexity of the square of the languages accepted by $n$-state
deterministic automata with $k$ final states.
For any such $n$ and $k$ we describe binary language such that the minimal
deterministic automaton for the square has $(n − k) 2^n + k · 2^{n−1}$
states. We focus on languages represented by deterministic automata with only one non-final state. Using our language which is hard for the square operation on deterministic automata, we define a binary language accepted by an $n$-state alternating automaton and such that every alternating automaton for its square has at least $2^n +n+ 1$ states. By generalizing this result we show the tightness of the upper bound $2^m + n + 1$ for the compelxity of concatenation of langauages represented by alternating automata with $m$ and $n$ states. This solves an open problem of of state complexity of concatenation, formulated by Fellah, Jürgensen, Yu [1990, Internat. J. Computer Math. 35, 117–132]. |
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| 20. 4. 2017 | Michal Hospodár – On the Magic Number Problem of the Cut Operation |
| Abstract: We investigate the state complexity of languages resulting from the cut operation of two regular languages represented by deterministic finite automata with $m$ and $n$ states, respectively. We study the magic number problem of the cut operation and show that the entire range of complexities, up to the known upper bound, can be produced in case of binary alphabets. Moreover, we prove that in the unary case only complexities up to $2m-1$ and between $n$ and $m+n-2$ can be produced, while complexities within the interval $2m$ up to $n-1$ cannot be reached—these non-producible numbers are called “magic.” | |
| 8. 6. 2017 | Galina Jirásková |
| 22. 6. 2017 | Miroslav Ploščica |
| 13. 7. 2017 | Peter Mlynárčik – Nondeterministic Complexity of Operations on Free and Convex Languages (link to the presentation) |
| 5. 10. 2017 | Jozef Pócs – On $*$-associated comonotone functions |
| Abstract:
We give a positive answer to two open problems stated by Boczek and Kaluszka. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone rightcontinuous operation. More precisely, the comonotonicity of functions is equivalent not only to $+$-associatedness of functions (as proved by Boczek and Kaluszka), but also to their $*$-associatedness with $*$ being an arbitrary strictly monotone and right-continuous binary operation. The second open problem deals with an existence of a pair of binary operations for which the generalized upper and lower Sugeno integrals coincide. Using a fairly elementary observation we show that there are many such operations, for instance binary operations generated by infima and suprema preserving functions. |
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| 19. 10. 2017 | Miroslav Repický – Egorov's Theorem and Generalized Egorov's Theorem for Ideal Convergencies |
Abstract:
By analyzing Pinciroli's result about Egorov's generalized theorem, Korch abstracted
two conditions for a pair of convergence of sequences of functions that suffice to
Pincoroli's arguments can be applied to this pair of convergences.
Following Korch's results, we investigated classes of ideals on $\omega$, for
which Egorov's theorem or a generalized Egorov's theorem holds
between ideal pointwise and ideal uniform convergence.
We investigated closure properties of these families of ideals under the following operations:
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| 23. 11. 2017 | Ján Haluška |
| 7. 12. 2017 | Dušana Štiberová – Probability integral as a linearization |