Institute of Mathematics AS CR, v. v. i.
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On the disk-cyclic linear relations
Full text linksummary:The study of linear dynamical systems for linear relations was initiated by C.-C. Chen et al. in (2017). Then E. Abakumov et al. extended hypercyclicty to linear relations in (2018). We extend the concept of disk-cyclicity studied in M. Amouch, O. Benchiheb (2020), Z. Z. Jamil, M. Helal (2013), Y.-X. Liang, Z.-H. Zhou (2015), Z. J. Zeana (2002) for linear operators to linear relations
Notes on number of one-troughed travelling waves in asymmetrically supported bending beam
No full textsummary:We study the boundary value problem for nonlinear fourth-order partial differential equation with jumping nonlinearity which can serve, e.g., as a model of an asymmetrically supported bending beam. We focus on a special type of solutions, the so-called one-troughed travelling waves. The main goal of this paper is to show the existence of at least two different one-troughed travelling waves for particular wave speeds and input parameters of the studied problem. We present the upper bounds for the maximal number of one-troughed solutions together with a visualisation of obtained results and corresponding solutions. Finally, we list several open questions regarding this topic
Geometric “place”, or a locus
No full textsummary:V první části uvádíme historický pohled na nahrazování tradičního pojmu geometrické místo bodů „moderním“ termínem množina bodů. K tomuto nahrazování vymezujeme naše stanovisko. Po formulaci stanoviska následuje deset řešených úloh, kde hledáme geometrické místo bodů. Uvedené úlohy dokládají, že ve škole můžeme užívat nejen termín množina všech bodů dané vlastnosti, ale i termín geometrické místo bodů.summary:In the first part of the article we present a historical perspective on the replacement of the traditional term locus by the "modern" term set of points. We define our position on this substitution of the term. The formulation of our position is followed by ten solved problems where we search for the locus. These problems illustrate that in school we can use not only the term set of all points of a given property, but also the term locus
An example Ginsburg said in 1984 he was “unable to find” and a forbidden subposet characterization of subsets of regular posets
Full text linksummary:In 1984, Ginsburg wrote, ``We have been unable to find an example of an ordered set having the properties of [being complete, densely ordered, with no antichain other than and that is a cutset] and in which all antichains are countable.'' In this very brief note, such an example is shown. Posets that can be embedded in regular posets are characterized as posets that do not contain or its dual as a subposet. Any such poset can be embedded in a regular poset that can be embedded in any other regular poset containing
Intuitionistic-like unsharp implication and negation defined on a poset
Full text linksummary:The aim of the present paper is to show that the concepts of the intuitionistic implication and negation formalized by means of a Heyting algebra can be generalized in such a way that these concepts are formalized by means of a bounded poset. In this case it is not assumed that the poset is relatively pseudocomplemented. The considered logical connectives negation, implication or even conjunction are not operations in this poset but so-called operators since they assign to given entries not necessarily an element of the poset as a result but a subset of mutually incomparable elements. We show that these operators for negation and implication can be characterized by several simple conditions formulated in the language of posets together with the operator of taking the lower cone. Moreover, our implication and conjunction form an adjoint pair. We call these connectives ``unsharp'' or ``inexact'' in accordance with the existing literature. We also introduce the concept of a deductive system of a bounded poset with implication and prove that it induces an equivalence relation satisfying a certain substitution property with respect to implication. Moreover, the restriction of this equivalence to the base set is uniquely determined by its kernel, i.e., the class containing the top element
The origin and developments of Kurzweil's generalized Riemann integral
No full textsummary:The paper describes to origin and motivation of Kurzweil in introducing a Riemann-type definition for generalized Perron integrals and his further contributions to the topics
The topology of the space of integrable functions in
No full textsummary:It is known that there is no natural Banach norm on the space of -dimensional Henstock-Kurzweil integrable functions on . We show that the space is the uncountable union of Fréchet spaces . On each space, an -norm is defined. A -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an -norm is also defined for a -continuous linear operator. Hence, many important results in functional analysis hold for the space. It is well-known that every control-convergent sequence in the space always belongs to a space. Hence, results in functional analysis can be applied to the space. Compact linear operators and the existence of solutions to integral equations are also given. The results for the one-dimensional case have been discussed in V. Boonpogkrong (2022). Proofs of many results for the -dimensional and the one-dimensional cases are similar
Ergodicity of increments of the Rosenblatt process and some consequences
No full textsummary:A new proof of the mixing property of the increments of Rosenblatt processes is given. The proof relies on infinite divisibility of the Rosenblatt law that allows to prove only the pointwise convergence of characteristic functions. Subsequently, the result is used to prove weak consistency of an estimator for the self-similarity parameter of a Rosenblatt process, and to prove the existence of a random attractor for a random dynamical system induced by a stochastic reaction-diffusion equation driven by additive Rosenblatt noise
A novel kernel function bridging iteration bounds in interior-point algorithms for linear programming
Full text linksummary:Kernel functions play an important role in designing and analyzing interior-point methods. They are not only used for determining search directions but also for measuring the distance between the given iterate and the -center in the algorithms. Currently, interior-point methods based on kernel functions are among the most effective methods for solving different types of optimization problems and are very active research area in mathematical programming. Therefore, in this work, we introduce a novel kernel function that bridges the gap between the iteration bounds for large-update and small-update methods. To the best of our knowledge, we are the first to achieve these results