| 23. 1. 2020 |
Peter Eliaš – Ordered vector spaces in probability theory |
| 6. 2. 2020 |
Roman Frič – Lifting and sagging observables |
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Abstract: Important constructions in probability theory can be schematized
via two simple commutative triangle diagrams. We shall deal with joint experiments
and conditional probability.
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| 20. 2. 2020 |
Ján Haluška – Generalized complex numbers |
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Abstract: An unified model of generalized complex numbers is defined
via an operation of multiplication in the form of a Toeplitz matrix.
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| 5. 3. 2020 |
Emília Halušková – On discrete properties of monotone functions |
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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.
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| 25. 6. 2020 |
Ján Haluška – Bitopology on $\mathbb{E}_4$ equipped with a skew circulated multiplication |
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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.
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| 1. 10. 2020 |
Michal Hospodár – Right and left quotients on subclasses of convex languages |
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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.
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| 30. 9. 2021 |
Jozef Pócs – Aggregation functions and their connection to universal algebra |
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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.
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| 14. 10. 2021 |
Irena Jadlovská – Kneser oscillation theorem for second-order half-linear delay
differential equations |
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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.
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| 28. 10. 2021 |
Miroslav Repický – Ideals on $\omega$ determined by a sequence of capacities |
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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 $F_{\sigma\delta}$.
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| 11. 11. 2021 |
Viktor Olejár – State complexity of the cut operation in subclasses of convex languages |
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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 $m$.
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