Random walks in compact groups

Let X_1,X_2,... be independent identically distributed random elements of a compact group G. We discuss the speed of convergence of the law of the product X_l*...*X_1 to the Haar measure. We give poly-log estimates for certain finite groups and for compact semi-simple Lie groups. We improve earlier results of Solovay, Kitaev, Gamburd, Shahshahani and Dinai.Comment: 35 pages, no figures, revision based on referee's report, results and proofs unchange

Diophantine property in the group of affine transformations of the line

We investigate the Diophantine property of a pair of elements in the group of affine transformations of the line. We say that a pair of elements g_1,g_2 in this group is Diophantine if there is a number A such that a product of length l of elements of the set {g_1,g_2,g_1^{-1},g_2^{-1}} is either the unit element or of distance at least A^{-l} from the unit element. We prove that the set of non-Diophantine pairs in a certain one parameter family is of Hausdorff dimension 0.Comment: 12 pages, no figures, reference to [ABRS] update

Random walks in Euclidean space

Consider a sequence of independent random isometries of Euclidean space with a previously fixed probability law. Apply these isometries successively to the origin and consider the sequence of random points that we obtain this way. We prove a local limit theorem under a suitable moment condition and a necessary non-degeneracy condition. Under stronger hypothesis, we prove a limit theorem on a wide range of scales: between e^(-cl^(1/4)) and l^(1/2), where l is the number of steps.Comment: 62 pages, 1 figure, revision based on referee's report, proofs and results unchange

Gráfok, geometria, véletlen, algoritmusok = Graphs, geometry, randomness, algorithms

Jelentős eredményeket értünk el a gráfelmélet, geometria, sztochasztika és algoritmusok kérdésköreiben, sokszor a területek közös elméletét gyarapítottuk. Kiemelünk néhány karakterisztikus eredményt: Hozzájárultunk annak megértéséhez ''hogyan viselkednek'' végtelen tranzitív gráfok minimális elvágó élhalmazai kvaziizometriák mellett (Babson és Benjamini két kérdésének megválaszolásával). Gráfokhoz rendeltünk egy geometria jellegű slope paramétert, amely különböző változatai több kutatást indítottak el. Gráfok pakolásainak központi megoldatlan kérdésével, az Bollobás-Eldridge-sejtéssel, kapcsolatban több részeredményt értünk el. Az OTKA résztvevői sok társszerzővel dolgoztak együtt. Az OTKA pályázat segítségévél született munkában társszerzőink között vannak: Pavel Valtr, Oded Schramm, Bezdek András, Yuval Peres, Bollobás Béla, Turán György, Jiri Matousek, Alexandr Kostochka, T. Sós Vera a témaköreink nemzetközileg elismert nagyságai. A szegedi kombinatorika szeminárium munkája a kutatás mellett a fiatal diákok érdeklődését is felkelti. Szakdolgozatok mellett két phd értekezés is születendőben van és további phd hallgatók kutatnak kombinatorika témában. A szeminárium honlapja: http://www.math.u-szeged.hu/~hajnal/seminars/kombszem/kombszem.htm | We have achieved several important results in graph theory, geometry, probability theory and algorithm theory, often connecting these central fields. We underline a few characteristic results. We contributed to understanding how minimal cut sets in infinite transitive graphs are behaving under quasiisometries (we have answered two questions of Babson and Benjami). We have introduced and investigated the geometrical notion, the slope parameter of a graph. This notion motivated further research. We made major steps in the topics of graph packing, strongly related to the Bollobas-Eldridge conjecture. For example if H is a bipartite graph on n vertices, with maximal degree D, then for large enough n H is a spanning subgraph of G a graph on n vertices with minimal degree at least D/(D+1). Our participants worked with several co-authors, among others, with Pavel Valtr, Oded Schramm, Andras Bezdek, Yuval Peres, Bela Bollobas, Gyorgy Turan, Jiri Matousek, Alexandr Kostocka, Vera T. Sos. The Combinatorics Seminar in Szeged is not only a research center, but it plays important role in education. Several students have written their diploma thesis in combinatorics, two phd dissertations is about to be submitted and other phd students are working strongly connected to our seminar. The homepage of the seminar is http://www.math.u-szeged.hu/~hajnal/seminars/kombszem/kombszem.ht

Hypertriglyceridemia-induced acute pancreatitis: A prospective, multicenter, international cohort analysis of 716 acute pancreatitis cases

Background Hypertriglyceridemia is the third most common cause of acute pancreatitis (AP). It has been shown that hypertriglyceridemia aggravates the severity and related complications of AP; however, detailed analyses of large cohorts are inadequate and contradictory. Our aim was to investigate the dose-dependent effect of hypertriglyceridemia on AP. Methods AP patients over 18 years old who underwent triglyceride measurement within the initial three days were included into our cohort analysis from a prospective international, multicenter AP registry operated by the Hungarian Pancreatic Study Group. Data on 716 AP cases were analyzed. Six groups were created based on the highest triglyceride level (Peer reviewe