Institute of Mathematics AS CR, v. v. i.
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The rings whose torsionfree modules have injective dimension at most one
No full textsummary:The domains with torsion-free modules of injective dimension at most one have been examined by B. Olberding. A TF-projective module is one that is projective relative to all short exact sequences beginning with torsion-free modules. The rings, where each right ideal of is TF-projective, are precisely those rings whose torsionfree right modules have injective dimension at most one. The goal of this study is to comprehend the structure of the rings that B. Olberding recently studied. Along the way, we prove for a domain , that if each ideal of is TF-projective, then is a Noetherian ring with . Specifically, we prove that for a commutative domain , each ideal of is TF-projective if and only if is a Gorenstein Dedekind domain. A left -coherent ring all of its TF-projective left -modules are projective is precisely left PP ring. Furthermore, we demonstrate that any (cyclic) right -module of is TF-projective if and only if is a QF-ring
On solutions of a certain nonlinear differential-difference functional equation
No full textsummary:We investigate all the possible finite order entire solutions of the Fermat-type differential-difference functional equation , where , and , are nonzero polynomials. The results significantly improve some earlier findings, especially the results due to A. Banerjee and T. Biswas (2021). We also show that the equation does not have any non-entire meromorphic solution. We provide some examples to support the results
On boundedness, square integrability and uniform stability for neutral non autonomous third order differential equations with delay
No full textsummary:In the study of the solutions for a given class of neutral third order differential equations with delay, a suitable conditions are given based on the Lyapunov second method by considering convenable Lyapunov functional to guarantee the uniform asymptotic stability, the boundedness and the square integrability
Possibilities of developing pupils' logical thinking in mathematics lessons – Part 1
No full textsummary:Článek je rozdělen na dvě části. Nejprve představujeme základní teoretická východiska, která sloužila jako podklad pro vytvoření klasifikace vytvořené sbírky úloh s potenciálem rozvíjet logické myšlení. Ve druhé části se blíže věnujeme první ze čtyř kategorií – slovní úlohy, a uvádíme třináct ukázkových úloh. Ke každé úloze uvádíme krátký komentář s řešením a popisem vybrané metody řešení.summary:The article is divided into two parts. First, we introduce the fundamental theoretical foundations that served as the basis for creating a classification of the compiled collection of tasks with the potential to develop logical thinking. In the second part, we focus on the first of the four categories - word problems - and present thirteen sample tasks. For each task, we provide a brief commentary with a solution and a description of the selected solving method
Join or not join? Continuous versus discrete graphs
No full textsummary:Several studies indicate that students encounter significant difficulties when determining how to represent or interpret continuous data in a discrete manner and to represent or interpret discrete data in a continuous manner. In this paper, we focus on whether and when pre-service and in-service mathematics teachers represent functional dependence in a continuous or discrete manner. In pursuit of this objective, we employed two tasks from the research tool, which pre-service and in-service teachers solved, and subsequently scrutinized both their solutions and collaborative discussions concerning these tasks
On an Old Textbook. Teaching Mathematics “For Life” Two Hundred Years Ago
No full textsummary:V článku nejprve stručně načrtneme historii české triviální školy v Dýšině, zejména v 19. století. Dále přiblížíme zajímavé osudy rozvětveného učitelského rodu Rádlů a naznačíme obsah a význam nevelké učebnice počítání zpaměti Jana Rádla, která vyšla takřka před dvěma sty lety. Význam znalostí matematiky demonstruje zásadně na úlohách, v nichž se počítá s penězi, což bylo a stále je pro většinu populace vynikající motivací k učení
The sharp constant for truncated Hardy-Littlewood maximal inequality
No full textsummary:This paper focuses on the operator norm of the truncated Hardy-Littlewood maximal operator and the strong truncated Hardy-Littlewood maximal operator , respectively. We first present the -norm of , and then the -norm of is given. Our study may have some enlightening significance for the research on sharp constant for the classical Hardy-Littlewood maximal inequality
A survey and comparative analysis of different approaches to fuzzy differential equations modeling dynamics with uncertain parameters of deterministic character
Full text linksummary:Dynamics containing deterministic uncertainties can be modeled with fuzzy differential equations. Unlike classical differential equations, fuzzy differential equations lack a unified interpretation and theoretical foundation, as researchers adopt different approaches to fuzziness, solution concepts, and underlying mathematical structures. The main reason is whether the fuzzy function derivative is used in the equation in question and, if it is used, what meaning it carries. Researchers who do not involve a derivative of a fuzzy number-valued function either use the extension principle, an alternative concept of fuzzy function, or transform the problem into a differential inclusion. Various definitions have been used in studies involving the derivatives of fuzzy number-valued functions. The main reason is that none of the known derivatives can fully meet the requirements: either the fuzziness increases excessively, or it becomes impossible to solve higher-order equations, or unnatural assumptions must be made. In this study, we tried to classify almost all studies on fuzzy differential equations. We compared the results of studies conducted in relatively recent years, particularly in initial value and boundary value problems, using examples. We discussed the possible direction of future research on fuzzy differential equations
Science vs. Conspiracy: The question of falsifiability
No full textsummary:Přestože neexistují žádné věrohodné důkazy o existenci rozsáhlého tajného programu na rozprašování škodlivých chemikálií do atmosféry, konspirační teorie o tzv. chemtrails i nadále získává pozornost veřejnosti, zejména na sociálních sítích. V tomto dílu našeho seriálu si ji rozebereme a podíváme se na důležitou vlastnost teorií – falzifikovatelnost
On -universal injective modules and their applications in commutative rings
No full textsummary:Let be a commutative ring and be the -operation on . We introduce the concept of -universal injective modules and establish their fundamental properties. It is shown that the product of , where ranges over maximal -ideals of , is a \hbox {-universal} injective -module over , albeit not necessarily a universal injective \hbox {-module}. As applications, we characterize -IF rings and -coherent rings using -universal injective modules. Specifically, we demonstrate that is a -IF ring if and only if is -coherent and is a flat -module for every . These results extend existing results and provide deeper insights into the structure of -modules