Institute of Mathematics AS CR, v. v. i.
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Overview of mathematical competitions for lower and upper secondary school students
No full textsummary:Matematické soutěže mohou být zajímavým a účinným nástrojem, jak zvýšit zájem studentů o matematiku. Celkem představíme 35 matematických soutěží, které jsou v tuto chvíli dostupné pro žáky a studenty základních a středních škol. V tomto článku je uvedeno 9 matematických soutěží určených žákům 2. stupně ZŠ i studentům SŠ. Toto je třetí část ze série článků pojednávajících o matematických soutěžích.summary:Mathematical competitions can be a very useful tool for increasing interest in mathematics among pupils and students. In a series of papers we present a list and a description of 35 mathematical competitions that are currently available for primary and secondary school students. In this article, we present 9 of them, that are intended for elementary and also for secondary school students
Minimizing risk probability for infinite discounted piecewise deterministic Markov decision processes
Get PDFsummary:The purpose of this paper is to study the risk probability problem for infinite horizon piecewise deterministic Markov decision processes (PDMDPs) with varying discount factors and unbounded transition rates. Different from the usual expected total rewards, we aim to minimize the risk probability that the total rewards do not exceed a given target value. Under the condition of the controlled state process being non-explosive is slightly weaker than the corresponding ones in the previous literature, we prove the existence and uniqueness of a solution to the optimality equation, and the existence of the risk probability optimal policy by using the value iteration algorithm. Finally, we provide two examples to illustrate our results, one of which explains and verifies our conditions and the other shows the computational results of the value function and the risk probability optimal policy
Higher-dimensional Auslander-Reiten sequences
No full textsummary:Zhou and Zhu have shown that if is an -angulated category and is a cluster tilting subcategory of , then the quotient category is an -abelian category. We show that if has Auslander-Reiten -angles, then has Auslander-Reiten -exact sequences
Area differences under analytic maps and operators
No full textsummary:Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping and that of , we study various norms for , where is the Toeplitz operator with symbol . In Theorem \ref {thm:Transitivity}, given polynomials and we find a symbol such that . We extend some of our results to the polydisc
Semiholonomic jets and induced modules in Cartan geometry calculus
Get PDFsummary:The famous Erlangen Programme was coined by Felix Klein in 1872 as an algebraic approach allowing to incorporate fixed symmetry groups as the core ingredient for geometric analysis, seeing the chosen symmetries as intrinsic invariance of all objects and tools. This idea was broadened essentially by Elie Cartan in the beginning of the last century, and we may consider (curved) geometries as modelled over certain (flat) Klein’s models. The aim of this short survey is to explain carefully the basic concepts and algebraic tools built over several recent decades. We focus on the direct link between the jets of sections of homogeneous bundles and the associated induced modules, allowing us to understand the overall structure of invariant linear differential operators in purely algebraic terms. This allows us to extend essential parts of the concepts and procedures to the curved cases
Cartan geometry, supergravity and group manifold approach
Get PDFsummary:We make a case for the unique relevance of Cartan geometry for gauge theories of gravity and supergravity. We introduce our discussion by recapitulating historical threads, providing motivations. In a first part we review the geometry of classical gauge theory, as a background for understanding gauge theories of gravity in terms of Cartan geometry. The second part introduces the basics of the group manifold approach to supergravity, hinting at the deep rooted connections to Cartan supergeometry. The contribution is intended, not as an extensive review, but as a conceptual overview, and hopefully a bridge between communities in physics and mathematics
Overleaf Community Edition: Installation and Comparison with Overleaf Professional
No full textsummary:Webový editor Overleaf umožňuje pohodlné písanie dokumentov pomocou LaTeXu. Veľkou výhodou Overleafu je, že umožňuje spoluprácu viacerých používateľov v reálnom čase. V tomto článku popíšeme postup inštalácie bezplatnej verzie Overleaf Community Edition na vlastné zariadenie a porovnáme ho s platenou verziou Overleaf Professional.summary:The Overleaf web editor enables convenient writing of documents using LaTeX. A significant advantage of Overleaf is that it enables collaboration among multiple users in real-time. In this article, we describe the process of installing the free version of Overleaf Community Edition on your own device and compare it with the paid version of Overleaf Professional
Quantized semisimple Lie groups
Get PDFsummary:The goal of this expository paper is to give a quick introduction to -deformations of semisimple Lie groups. We discuss principally the rank one examples of , , and related algebras. We treat quantized enveloping algebras, representations of , generalities on Hopf algebras and quantum groups, -structures, quantized algebras of functions on -deformed compact semisimple groups, the Peter-Weyl theorem, -Hopf algebras associated to complex semisimple Lie groups and the Drinfeld double, representations of , the Plancherel formula for . This exposition is expanding the material treated in a series of lectures given by the second author at the CaLISTA CA 21100 Training School, “Quantum Groups and Noncommutative Geometry in Prague” in 2023
On the hyper-order of analytic solutions of linear differential equations near a finite singular point
Get PDFsummary:We study the hyper-order of analytic solutions of linear differential equations with analytic coefficients having the same order near a finite singular point. We improve previous results given by S. Cherief and S. Hamouda (2021). We also consider the nonhomogeneous linear differential equations
