On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas

We analyse the process of energy exchanges generated by the elastic collisions between a point-particle, confined to a two-dimensional cell with convex boundaries, and a `piston', i.e. a line-segment, which moves back and forth along a one-dimensional interval partially intersecting the cell. This model can be considered as the elementary building block of a spatially extended high-dimensional billiard modeling heat transport in a class of hybrid materials exhibiting the kinetics of gases and spatial structure of solids. Using heuristic arguments and numerical analysis, we argue that, in a regime of rare interactions, the billiard process converges to a Markov jump process for the energy exchanges and obtain the expression of its generator.Comment: 23 pages, 6 figure

An evolution model for polygonal tessellations as models for crack networks and other natural patterns

We introduce and study a general framework for modeling the evolution of crack networks. The evolution steps are triggered by exponential clocks corresponding to local micro-events, and thus reflect the state of the pattern. In an appropriate simultaneous limit of pattern domain tending to infinity and time step tending to zero, a continuous time model, specifically a system of ODE is derived that describes the dynamics of averaged quantities. In comparison with the previous, discrete time model, studied recently by two of the present three authors, this approach has several advantages. In particular, the emergence of non-physical solutions characteristic to the discrete time model is ruled out in the relevant nonlinear version of the new model. We also comment on the possibilities of studying further types of pattern formation phenomena based on the introduced general framework.Comment: 30 pages, 7 figure

Financial Literacy. Who, whom and what are they Training for? Comparative Analysis 2016–2020

The research assessed whether progress has been made in the education and development of financial awareness in Hungary since 2016. We reviewed the changes in the state’s role in the development of financial literacy, and in a questionnaire survey we examined what kind of organizations, who and in what topics are involved in trainings outside the public education. It was found that between 2016 and 2020, there was an increasing focus on developing the financial literacy, while the vast majority of the trainings continues to target the most easily reachable school age group. The National Core Curriculum identified economic and financial education as a goal for schools. However, outside of vocational high schools, such knowledge is not taught as a compulsory subject. In 2017, the Government adopted a strategy to improve the financial awareness of the population, and the first accredited financial literacy textbooks were published. The results of non-public education organisations show that the number of training programs and their participants has tripled. The average duration of trainings has become longer, multi-day trainings mainly for adults appeared. Knowledge transfer continued to focus on individual frugality and financial awareness, financial self-knowledge, attitude, and behaviour. The education of investment and entrepreneurship knowledge still isn’t a priority. Most of the trainings do not take into account the income situation and social background of the target groups and do not pay attention to the development of financially vulnerable groups

Hiperbolikus dinamikai rendszerek: attraktorok és korreláció lecsengés. = Hyperbolic dynamical systems: attractors and correlation decay.

A team kivételesen eredményes munkát végzett a periódus során. Young (Annals of Mathematics, 1998) síkbeli szóró biliárdokra vonatkozó torony konstrukciójának többdimenzióra való kiterjesztésével már 1998-tól foglalkoztunk. Bálint és Tóth friss és gondolatgazdag eredménye az első áttörés. Szász és Varjú a síkbeli konstrukciót alkalmazzák a Brown mozgás dinamikai elméletének modelljeire, nevezetesen a Lorentz folyamat sztochasztikájára. Véges, sőt végtelen horizont esetén is igazoltak lokális határeloszlástételeket, és rekurrenciát. Utóbbi esetben már globális határeloszlástételük is Bleher nevezetes, 1992-es sejtésének első szigorú bizonyítása. Dolgopyattal közös eredményeik Erdős-Taylor ill. Darling-Kac bolyongásokra vonatkozó tételeinek kiterjesztései periódikus Lorentz folyamatra, ezekkel Sinai 1981-es mély sejtésére szellemes és technikás bizonyítást adnak. Solomyak (Annals of Mathematics, 1995) végtelen Bernoulli konvolúciók abszolút folytonosságára vonatkozó, áttörést jelentő cikkét általánosította Tóth és Simon, majd Tóth. Solomyak eredménye olyan mértékek egy paraméteres családjára bizonyított abszolút folytonosságot, melyek stacionáriusak az egyenesen értelmezett IFS-ek egy paraméteres családjára. Ezen IFS-ek két lineáris -1/2-nél nagyobb - rátájú kontrakcióból állnak és azonos valószínűséggel alkalmazzuk mind két függvényt. Simon és Tóth ezt a tételt kiterjesztette tetszőlegesen sok függvényre és Tóth részeredményeket ért el a különböző kontrakciók esetén. | The team had an exceptionally successfull research period. We started the work on the multidimensional extension of Young's tower construction for planar dispersing billiards, published in 1998 in Annals of Mathematics. The fresh result of Bálint and Tóth, rich in ideas, is the first breakthrough here. Szász and Varjú applied the planar construction to dynamical models of Brownian motion: to stochastic properties of the planar Lorentz process. They obtained local limit theorems, and recurrence as well, in case of finite and even infinite horizon. In the latter case their - weaker - global limit theorem in itself provides the first rigorous verification of Bleher's 1992 conjecture. Their results, joint with Dolgopyat, extend classical results for random walks of Erdős-Taylor and of Darling-Kac to the periodic Lorentz process, which made it possible for them to give a technical and witty proof for Sinai's 1981 deep conjecture. Simon and Tóth (2006) and Tóth (2008) generalized a breakthrough result of Solomyak (Annals of Mathematics 1995). Solomyak considered a one parameter family of Iterated Function System (IFS) that consists of two linear contractions with the same (> 1/2) ratio of contraction which are applied with the same probability. Simon and Tóth generalized this result for arbitrary (but finite) number of contractions. Tóth obtained partial results about the generalization of the Solomyak's theorem for different probabilities