Institute of Mathematics AS CR, v. v. i.
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Sufficient conditions to determine the linear dependency of two meromorphic functions
No full textsummary:We comprehensively explore the generalized concept of sharing sets to establish conditions for the linear dependency of two meromorphic functions. By applying this approach, we significantly extend and enhance the existing results related to URSM (unique range set of meromorphic functions). It is well known that URSMs can be represented as zeros of specific polynomials. However, our findings demonstrate that the concept of URSM can be understood from a broader perspective, where it can be characterized as a special case of the zero sets of two interconnected polynomials. Such investigations have not been conducted before, thus the text breaks the barriers of the traditional definition of URSM
On common index divisors and monogenity of septic number fields defined by trinomials of type
No full textsummary:Let be a septic number field generated by a root of an irreducible polynomial . In this paper, we explicitly characterize the index of . More precisely, for all and , we show that . Our results answer completely to Problem 22 of W. Narkiewicz's book (2004) for these families of number fields. In particular, we provide sufficient conditions for which is not monogenic. We illustrate our results by some computational examples
Existence of regular time-periodic solutions for a class of non-Newtonian double-diffusive convection system
No full textsummary:We investigate a system of partial differential equations that models the motion of an incompressible double-diffusion convection fluid. The additional stress tensor is generated by a potential with -structure. In a three-dimensional periodic setting and , we employ a regularized approximation scheme in conjunction with the Galerkin method to establish the existence of regular solutions, provided that the forcing term is properly small. Furthermore, we demonstrate the existence of periodic regular solutions with period when the external force exhibits periodicity in time with the same period
Publishing LaTeX Documents on the Web Using TeX4ht and GitHub Actions
No full textsummary:Článek představí sadu šablon pro nástroj TeX4ht, který slouží k převodu LaTeXových dokumentů do HTML. Tyto šablony výrazně usnadňují publikaci různých typů dokumentů na webu a přinášejí možnosti zpracování a automatizace. První šablona je určena pro převod knižních dokumentů do webové podoby. Umožňuje rozdělení textu do jednotlivých kapitol s automaticky generovanou navigací a podporou responzivního designu, takže je výsledek dobře čitelný i na mobilních zařízeních. Druhá šablona slouží k tvorbě staticky generovaných blogů. Každý příspěvek je psán jako samostatný LaTeXový dokument, který je pomocí TeX4ht převeden do HTML. Následně jsou tyto články zpracovány statickým generátorem webů, jako je například Jekyll, který se postará o sestavení celého blogu, vytvoření rozcestníků, archivů a další navigace. Třetí šablona je zaměřena na převod prezentací vytvořených v prostředí Beamer do formy tzv. handoutů – přehledových materiálů pro posluchače. Výsledkem je čitelný a dobře strukturovaný webový dokument vhodný pro sdílení po přednášce. Všechny šablony jsou navrženy tak, aby fungovaly v rámci GitHub Actions. To znamená, že dokumenty mohou být automaticky zkompilovány a publikovány online pokaždé, když dojde ke změně v repozitáři.summary:The article presents a set of templates for the TeX4ht tool, which serves to convert LaTeX documents into HTML. These templates greatly simplify the publication of various types of documents on the web and bring modern capabilities for processing and automation. The first template is intended for converting book-style documents into a web format. It allows the text to be divided into individual chapters with automatically generated navigation and support for responsive design, making the result easily readable even on mobile devices. The second template is used for creating statically generated blogs. Each post is written as a separate LaTeX document, which is converted to HTML using TeX4ht. Subsequently, these articles are processed by a static site generator such as Jekyll, which takes care of assembling the entire blog, creating indexes, archives, and other navigation elements. The third template focuses on converting presentations created in the Beamer environment into so-called handouts – overview materials for the audience. The result is a readable and well-structured web document suitable for sharing after the lecture. All templates are designed to work within GitHub Actions. This means that the documents can be automatically compiled and published online whenever a change is made in the repository. This approach ensures that the web version of the document is always up to date
Typesetting Source Code in LaTeX Using the Minted Package
No full textsummary:Tento článok poskytuje základný prehľad LaTeXového balíka minted, ktorý slúži na zvýrazňovanie syntaxe zdrojového kódu v odborných dokumentoch. Balík využíva externú pythonovú knižnicu Pygments, čo umožňuje farebné formátovanie kódu v rôznych programovacích jazykoch s vysokou presnosťou a prispôsobiteľnosťou. Článok detailne popisuje základné použitie balíka, možnosti formátovania a metódy vkladania kódu. Záverečná časť analyzuje aktuálny vývoj balíka so zameraním na verziu 3 a jej bezpečnostné vylepšenia.summary:This article provides a comprehensive overview of the LaTeX package minted, which is used for syntax highlighting of the source code in scholarly documents. The package uses the Python library Pygments, enabling color formatting of code in various programming languages with high accuracy and customization options. The article details the basic usage of the package, formatting options, and methods for inserting code. The final section analyzes the current development of the package, focusing on version 3 and its security enhancements
Combinatorial spectra using polynomials
No full textsummary:In this paper, we would like to introduce some new methods for studying magic type-colorings of graphs or domination of graphs, based on the combinatorial spectrum on polynomial rings. We hope that this concept will be potentially useful for the graph theorists
Fitted norm preconditioners for the Hodge-Laplacian in mixed form
No full textsummary:We use the practical framework for abstract perturbed saddle-point problems recently introduced by Hong et al. to analyze the mixed formulation of the Hodge-Laplace problem on a Hilbert complex. We compose two parameter-dependent norms in which the uniform continuity and stability of the problem follow. This not only guarantees the well-posedness of the corresponding variational formulation on the continuous level, but also of related compatible discrete models. We further simplify the obtained norms and, in both cases, arrive at the same norm-equivalent preconditioner that is easily implementable. The efficiency and uniformity of the preconditioner are demonstrated numerically by the fast convergence and uniformly bounded number of preconditioned MinRes iterations required to solve various instances of Hodge-Laplace problems in two and three space dimensions
Some results on -number of monomial ideals
No full textsummary:We investigate the v-number of various classes of monomial ideals. First, we consider the relationship between the v-number and the regularity of the mixed product ideal , proving that . Next, we investigate an open conjecture on the v-number: if a monomial ideal has linear powers, then for all , We prove that if a monomial ideal with linear powers and (for any ) has no embedded associated primes, then Additionally, we calculate the \hbox {v-number} of ordinary power and square-free power of edge ideal. Finally, we propose a conjecture that the v-number of ordinary powers of line graph is equal to the v-number of square-free powers
Logarithmic and Exponential Functions in STEM Education
No full textsummary:Cílem příspěvku je učiteli matematiky nabídnout z palety aplikací logaritmické a exponenciální funkce takové, které by tyto nástroje přiblížily žákům a studentům jako praktické nástroje napříč obory STEM. Záměrně se vyhýbáme čistě matematickým tématům a předkládáme několik oddělených kapitol s příklady, které mohou sloužit jako doplňkový materiál, ze kterého si pedagog může vybrat dle schopností a zájmu svých žáků.summary:This article aims to provide teachers of mathematics with a range of applications of both logarithms and exponential functions based in the fields of STEM. Such examples reveal the functions as practical and necessary tools. We intentionally avoid pure mathematics and present several separate chapters with problems that can be used as support material for pedagogues, who can pick them according to the abilities and interests of their students