Volume 29 (2021), Issue 1 (Special Issue: Ostrava Mathematical Seminar)

Editors: Diana Barseghyan & Pasha Zusmanovich

1. Rota-Baxter operators and Bernoulli polynomials

Vsevolod Gubarev.
We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the latter.

2. Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

Daniel Uzcátegui Contreras ; Dardo Goyeneche ; Ondřej Turek ; Zuzana Václavíková.
It is known that a real symmetric circulant matrix with diagonal entries d ≥ 0, off-diagonal entries ±1 and orthogonal rows exists only of order 2d + 2 (and trivially of order 1) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries d ≥ 0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with d different from an odd integer is n = 2d + 2. We also discuss a similar problem for symmetric circulant matrices defined over finite rings ℤ m . As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.

3. Metric groups, unitary representations and continuous logic

Aleksander Ivanov.
We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find Lω 1 ω -axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan’s property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.

4. Entropy in Thermodynamics: from Foliation to Categorization

Radosław A. Kycia.
We overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on a set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.

5. Crystallographic actions on Lie groups and post-Lie algebra structures

Dietrich Burde.
This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in 2017.

6. Jets and the variational calculus

David J. Saunders.
We review the approach to the calculus of variations using Ehresmann’s theory of jets. We describe different types of jet manifold, different types of variational problem and different cohomological structures associated with such problems.

7. A look on some results about Camassa–Holm type equations

Igor Leite Freire.
We present an overview of some contributions of the author regarding Camassa–Holm type equations. We show that an equation unifying both Camassa–Holm and Novikov equations can be derived using the invariance under certain suitable scaling, conservation of the Sobolev norm and existence of peakon solutions. Qualitative analysis of the two-peakon dynamics is given.

8. The inverse problem in the calculus of variations: new developments

Thoan Do ; Geoff Prince.
We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of n second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for n = 2. We then examine a new class of solutions in arbitrary dimension n and give some non-trivial examples in dimension 3.

9. On the Mathematical Theory of Records

Alexei Stepanov.
In the present work, we briefly analyze the development of the mathematical theory of records. We first consider applications associated with records. We then view distributional and limit results for record values and times. We further present methods of generation of continuous records. In the end of this work, we discuss some tests based on records.