Multidimensional hyperbolic billiards

The theory of planar hyperbolic billiards is already quite well developed by having also achieved spectacular successes. In addition there also exists an excellent monograph by Chernov and Markarian on the topic. In contrast, apart from a series of works culminating in Sim\'anyi's remarkable result on the ergodicity of hard ball systems and other sporadic successes, the theory of hyperbolic billiards in dimension 3 or more is much less understood. The goal of this work is to survey the key results of their theory and highlight some central problems which deserve particular attention and efforts

Local Limit Theorem for the Lorentz Process and Its Recurrence in the Plane

For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Lai-Sang Young's axioms (which imply exponential decay of correlation and the CLT) a local CLT is proven. In fact, a unified version of the local CLT is found, covering among others the absolutely contionuous and the arithmetic cases. For the planar Lorentz process with a finite horizon this result implies a.) the local CLT and b.) the recurrence. For the latter case ($d=2$, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt, and J.-P. Conze, could already establish recurrence

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

Matching problems

We give necessary and sufficient conditions for coverability of parallelepipeds by a given figure. Two types of figures are considered: 1. parallelepiped, 2. figure consisting of two relatively fixed, not necessaryly connected, unit cubes, e.g. the field on the chess-board where the knight stands and a field attacked by the knight. © 1971 Academic Press, Inc

Számításigényes kutatás az alkalmazott matematikában = Computationally intensive research in applied mathematics

A pályázat fő célkitűzése az volt, hogy radikálisan új lehetőségeket építsen ki a BME Matematika Intézetében folyó változatos számítógépigényes kutatások számára. Külön hangsúlyt fektettünk arra, hogy ezekbe a kutatásokba minél szélesebb körben vonjuk be az Intézet hallgatóit és doktoranduszait. Ennek az intenzív hallgatói aktivitásnak is köszönhető, hogy a pályázat futamideje alatt a számítógépigényes kutatás területén egy a megelőzőnél jóval magasabb szintre léphettünk. A teljesség igénye nélkül megemlítünk néhány érdekesebb kutatási eredményt (további részletek olvashatók az OTKA adatbázisába feltöltött beszámolóban): - Hiperbolikus dinamikai rendszerek, statisztikus fizika: ismert fraktálok Hausdorff-mértékének becslése, belső állapotú bolyongások vizsgálata, instabilis lineáris oszcillátor visszacsatolással való stabilizálásának kérdése. - Kvantum-információ elmélet: kvantumrendszer állapotának becslése, kapcsolódó speciális mátrix-keresési feladatok a 4-szer 4-es mátrixok algebrájában. - Számítógépes algebra és határterületei: Gröbner bázisok és kombinatorikai problémák kapcsolata, nulladimenziós polinomideálok globális tulajdonságainak megértése, S-extremális halmazrendszerek számítógépes vizsgálata. - Reakció-diffúzió egyenletek: dinamikai rendszerek paramétereinek becslése többek között részletes egyensúly esetén, molbilitáskezelő algoritmusok, egyenletek közelítő megoldása, érzékenységvizsgálat. | Main objective of our proposal was to open up radically new perspectives for the extensive and colourful activity in the field of computationally sensitive research performed at the Institute of Mathematics of BME. Special emphasis was put on the participation of (both undergraduate and graduate) students of our Institute in this type of mathematical research. The intensive contribution of skilfull students played a principal role in our success; we managed to increase the level of our computationally sensitive research activity with several orders of magnitude during the years of OTKA support. Not aiming for completeness, some of the most interesting results are mentioned below (for more information see the detailed research report): - Hyperbolic dynamical systems and statistical physics: estimating the Hausdorff measure of known fractals, investigation of random walks with internal states, issue of feedback stabilization of unstable linear oscillators. - Quantum information theory: estimating the state of quantum systems, related matrix-search problems in the algebra of 4 by 4 matrices. - Computer algebra and related fields: Gröbner bases and their relation to combinatorics, global properties of zero dimensional polinom ideals, computer-aided study of S-extremal set systems. - Reaction-diffusion equations: estimating parameters eg. for systems with detailed balance, mobility management algorithms, approximate solutions, sensitivity analysis

Locally Perturbed Random Walks with Unbounded Jumps

In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if $d \ge 2$. The extension of their result to finite range random walks is straightforward. Here, however, we are interested in the situation when the random walk has unbounded range. Concretely we generalize the statement of \cite{SzT} to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law. We do this first: for diffusively scaled random walks on $\mathbf Z^d$ $(d \ge 2)$ having finite variance; and second: for random walks with distribution belonging to the non-normal domain of attraction of the normal law. This result can be applied to random walks with tail behavior analogous to that of the infinite horizon Lorentz-process; these, in particular, have infinite variance, and convergence to Brownian motion holds with the superdiffusive $\sqrt{n \log n}$ scaling.Comment: 16 page