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

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

Generation of high-order harmonics with tunable photon energy and spectral width using double pulses

This work theoretically investigates high-order harmonic generation in rare gas atoms driven by two temporally delayed ultrashort laser pulses. Apart from their temporal delay, the two pulses are identical. Using a single-atom model of the laser-matter interaction it is shown that the photon energy of the generated harmonics is controllable within the range of one eV -- a bandwidth comparable to the photon energy of the fundamental field -- by varying the time delay between the generating laser pulses. It is also demonstrated that high-order harmonics generated by double pulses have advantageous characteristics, which mimick certain properties of an extreme ultraviolet (XUV) monochromator. With the proposed method, a simpler setup at a much lower cost and comparatively higher spectral yield can be implemented in contrast to other approaches.Comment: 7 pages, 6 figures, after peer-review, corrected typo in author lis

A central limit theorem for time-dependent dynamical systems

The work [8] established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case). In fact, the present paper is the first one where non-random, i. e. specific examples are also found, which are not small perturbations of a given map. Our approach uses martingale approximation technique in the form of [9]

Véletlen jelenségek térben és időben = Randomness in Space and Time

A statissztikus fizika és kvantummechanika matematikai problémáival foglalkozunk. A hidrodinamika mikroszkopikus modelljeinek tág osztályait vizsgáltuk, különös tekintettel hiperbolikus skálázású rendszerek makroszkopikus viselkedésére a lökéshulllámok tartományában. Levezettük a rugalmasságtan nemlineáris egyenleteit. Áttörést jelentő eredmények születtek aszimmetrikus modellek 1/3 kitevős skálázásával kapcsolatban (szubdiffúzív viselkedés). Erdős-Rényi-Barabási tipusú ömszervező strukturák kritikus viselkedését, a gráf komponenseinek aszimptokus méretét tisztáztuk. Nem-Markov bolyongások határeloszlását meghatározva meglepő különbséget tapasztaltunk az irányított és nem irányított élek eseteinél. Leírtuk véletlen mátrixok sajátértékeinek pomtfolyamatát. Sinai biliárdok és általánosabb kaotikus dinamikai rendszerek ergodikus viselkedését tisztáztuk, eredményesen tárgyaltuk a sikbeli Lorentz folyamat rekurrenciájának és Brown-közelítésének problémáját. Meghatároztuk iterált függvényrendszerek attraktorainak Hausdorff dimnenzióját, véletlen Cantor halmazok különbségét. Tisztáztuk a kvantummechanikai állapottér geometriájának kérdéseit. A kvantumrendszerek állapotbecsléseit vizsgálva azok megbízhatóságát jellemeztük. Algoritmust adtunk n-szintű rendszer "constrained" becsléseire, a becslési stratégia számítógépes vizsgálatára is sor került. A folytonos optimalizáció különféle algoritmusit konstruáltuk meg, tisztáztuk azok hatékonyságát. | Mathematical problems of statistical physics and quantum mechanics are investigated. Various microscopic models of hydrodynamics are introduced, existence of hyperbolic scaling limits is prroven including the derivation of the equations of nonlinear elastodynamics. Fairly deep results have been obtained on the 1/3 exponent scaling of asymmetric systems (subdiffusive behaviour of the tagged particle). In case of Erdős-Rényi-Barabási type self-organized systems the size of the graph components has been determined. Investigating the limit distribution of non-Markovian random walks, a considerable difference in the behavior of models with directed and non-directed bonds has been observed. Characterization of the point process of eigenvalues of random matrices was presented. Ergodic behavior of Sinai billiards and of more general dynamical systems has been described, discussion of recurrence and Brownian approximation of the planar Lorentz gas has been completed. Hausdorff dimension of attractors of iterated maps has been calculated, difference of random Cantor sets was also studied. Geometry of quantum mechanical and reliability of estimators for the state of quantum mechanical systems has been investigated. Algorithms for constrained estimation for n-level quantum systems have been introduced and investigated by computer simulations. Various methods of continuous optimalization and their effectiveness was also studied

All-Optical Experimental Control of High-Harmonic Photon Energy

We generate high-order harmonics in gaseous medium with tunable photon energy using time domain interferometry of double pulses in a non-collinear generation geometry. The method is based on the fact that the generated harmonics inherit certain spectral properties of the driving laser. The two temporally delayed ultrashort laser pulses, identical in all parameters, are produced by a custom-made split-and-delay unit utilizing wave front splitting without a significant energy loss. The arrangement is easy to implement in any attosecond pulse generation beamline, and is suitable for the production of an extreme ultraviolet source with simply and quickly variable central photon energy, useful for a broad range of applications.Comment: 6 pages, 5 figures, after peer-revie

Generation of high-order harmonics with tunable photon energy and spectral width using double pulses

This work theoretically investigates high-order harmonic generation in rare-gas atoms driven by two temporally delayed ultrashort laser pulses. Apart from their temporal delay, the two pulses are identical. Using a single-atom model of the laser-matter interaction it is shown that the photon energy of the generated harmonics is controllable within the range of one eV-a bandwidth comparable to the photon energy of the fundamental field-by varying the time delay between the generating laser pulses. It is also demonstrated that high-order harmonics generated by double pulses have advantageous characteristics, which mimick certain properties of an extreme ultraviolet monochromator. With the proposed method, a simpler setup at a much lower cost and comparatively higher spectral yield can be implemented in contrast to other approaches

The characteristics and prognostic role of acute abdominal on-admission pain in acute pancreatitis: A prospective cohort analysis of 1432 cases

Introduction Pain is the most common symptom in acute pancreatitis (AP) and is among the diagnostic criteria. Therefore, we aimed to characterize acute abdominal pain in AP. Methods The Hungarian Pancreatic Study Group prospectively collected multicentre clinical data on 1435 adult AP patients between 2012 and 2017. Pain was characterized by its intensity (mild or intense), duration prior to admission (hours), localization (nine regions of the abdomen) and type (sharp, dull or cramping). Results 97.3% of patients (n = 1394) had pain on admission. Of the initial population with acute abdominal pain, 727 patients answered questions about pain intensity, 1148 about pain type, 1134 about pain localization and 1202 about pain duration. Pain was mostly intense (70%, n = 511/727), characterized by cramping (61%, n = 705/1148), mostly starting less than 24 h prior to admission (56.7%, n = 682/1202). Interestingly, 50.9% of the patients (n = 577/1134) had atypical pain, which means pain other than epigastric or belt-like upper abdominal pain. We observed a higher proportion of peripancreatic fluid collection (19.5% vs. 11.0%; p = 0.009) and oedematous pancreas (8.4% vs. 3.1%; p = 0.016) with intense pain. Sharp pain was associated with AP severity (OR = 2.481 95% CI: 1.550-3.969) and increased mortality (OR = 2.263, 95% CI: 1.199-4.059) compared to other types. Longstanding pain (>72 h) on admission was not associated with outcomes. Pain characteristics showed little association with the patient's baseline characteristics. Conclusion A comprehensive patient interview should include questions about pain characteristics, including pain type. Patients with sharp and intense pain might need special monitoring and tailored pain management. Significance Acute abdominal pain is the leading presenting symptom in acute pancreatitis; however, we currently lack specific guidelines for pain assessment and management. In our cohort analysis, intense and sharp pain on admission was associated with higher odds for severe AP and several systemic and local complications. Therefore, a comprehensive patient interview should include questions about pain characteristics and patients with intense and sharp pain might need closer monitoring

Spectrally tunable ultrashort monochromatized extreme ultraviolet pulses at 100 kHz

We present the experimental realization of spectrally tunable, ultrashort, quasimonochromatic extreme ultraviolet (XUV) pulses generated at 100 kHz repetition rate in a user-oriented gas high harmonic generation (GHHG) beamline of the Extreme Light Infrastructure - Attosecond Light Pulse Source (ELI ALPS) facility. Versatile spectral and temporal shaping of the XUV pulses are accomplished with a double-grating, time-delay compensated monochromator accommodating the two composing stages in a novel, asymmetrical geometry. This configuration supports the achievement of high monochromatic XUV flux (2.8e10+/-0.9e10 photons/s) combined with ultrashort pulse duration (4.0+/-0.2 fs using 12.1+/-0.6 fs driving pulses) and small spot size (sub-100 um). Focusability, spectral bandwidth, and overall photon flux of the produced radiation were investigated covering a wide range of instrumental configurations. Moreover, complete temporal (intensity and phase) characterization of the few-femtosecond monochromatic XUV pulses - a goal that is difficult to achieve by conventional reconstruction techniques - has been realized using ptychographic algorithm on experimentally recorded XUV-IR pump-probe traces. The presented results contribute to in-situ, time-resolved experiments accessing direct information on the electronic structure dynamics of novel target materials.Comment: 20 pages, 8 figure