55 research outputs found
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
(, 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
SzámĂtásigĂ©nyes kutatás az alkalmazott matematikában = Computationally intensive research in applied mathematics
A pályázat fĹ‘ cĂ©lkitűzĂ©se az volt, hogy radikálisan Ăşj lehetĹ‘sĂ©geket Ă©pĂtsen ki a BME Matematika IntĂ©zetĂ©ben folyĂł változatos számĂtĂłgĂ©pigĂ©nyes kutatások számára. KĂĽlön hangsĂşlyt fektettĂĽnk arra, hogy ezekbe a kutatásokba minĂ©l szĂ©lesebb körben vonjuk be az IntĂ©zet hallgatĂłit Ă©s doktoranduszait. Ennek az intenzĂv hallgatĂłi aktivitásnak is köszönhetĹ‘, hogy a pályázat futamideje alatt a számĂtĂłgĂ©pigĂ©nyes kutatás terĂĽletĂ©n egy a megelĹ‘zĹ‘nĂ©l jĂłval magasabb szintre lĂ©phettĂĽnk. A teljessĂ©g igĂ©nye nĂ©lkĂĽl megemlĂtĂĽnk nĂ©hány Ă©rdekesebb kutatási eredmĂ©nyt (további rĂ©szletek olvashatĂłk az OTKA adatbázisába feltöltött beszámolĂłban): - Hiperbolikus dinamikai rendszerek, statisztikus fizika: ismert fraktálok Hausdorff-mĂ©rtĂ©kĂ©nek becslĂ©se, belsĹ‘ állapotĂş bolyongások vizsgálata, instabilis lineáris oszcillátor visszacsatolással valĂł stabilizálásának kĂ©rdĂ©se. - Kvantum-informáciĂł elmĂ©let: kvantumrendszer állapotának becslĂ©se, kapcsolĂłdĂł speciális mátrix-keresĂ©si feladatok a 4-szer 4-es mátrixok algebrájában. - SzámĂtĂłgĂ©pes algebra Ă©s határterĂĽletei: Gröbner bázisok Ă©s kombinatorikai problĂ©mák kapcsolata, nulladimenziĂłs polinomideálok globális tulajdonságainak megĂ©rtĂ©se, S-extremális halmazrendszerek számĂtĂłgĂ©pes vizsgálata. - ReakciĂł-diffĂşziĂł egyenletek: dinamikai rendszerek paramĂ©tereinek becslĂ©se többek között rĂ©szletes egyensĂşly esetĂ©n, molbilitáskezelĹ‘ algoritmusok, egyenletek közelĂtĹ‘ megoldása, Ă©rzĂ©kenysĂ©gvizsgálat. | 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
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