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Documents  Tichy, Robert | enregistrements trouvés : 4

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Outreach;Mathematics Education and Popularization of Mathematics

Robert Franz Tichy is an Austrian mathematician and professor at Graz University of Technology.

He studied mathematics at the University of Vienna and finished with a Ph.D. thesis on uniform distribution under the supervision of Edmund Hlawka. He received his habilitation at TU Wien in 1983. Currently he is a professor at the Institute for Analysis and Number Theory at TU Graz. Previous positions include head of the Department of Mathematics and Dean of the Faculty of Mathematics, Physics and Geodesy at TU Graz, President of the Austrian Mathematical Society, and Member of the Board (Kuratorium) of the FWF, the Austrian Science Foundation.

His research deals with Number theory, Analysis and Actuarial mathematics, and in particular with number theoretic algorithms, digital expansions, diophantine problems, combinatorial and asymptotic analysis, quasi Monte Carlo methods and actuarial risk models. Among his contributions are results in discrepancy theory, a criterion (joint with Yuri Bilu) for the finiteness of the solution set of a separable diophantine equation, as well as investigations of graph theoretic indices and of combinatorial algorithms with analytic methods. He also investigated (with Istvan Berkes and Walter Philipp) pseudorandom properties of lacunary sequences.
In 1985 he received the Prize of the Austrian Mathematical Society. Since 2004 he has been a Corresponding Member of the Austrian Academy of Sciences. In 2017 he received an honorary doctorate from the University of Debrecen. He taught as a visiting professor at the University of Illinois at Urbana-Champaign and the Tata Institute of Fundamental Research. In 2017 he was a guest professor at Paris 7; currently (until February 2021) he holds the Morlet chair at the Centre International de Rencontres Mathématiques in Luminy (https://www.chairejeanmorlet.com/2020...​)
Robert Franz Tichy is an Austrian mathematician and professor at Graz University of Technology.

He studied mathematics at the University of Vienna and finished with a Ph.D. thesis on uniform distribution under the supervision of Edmund Hlawka. He received his habilitation at TU Wien in 1983. Currently he is a professor at the Institute for Analysis and Number Theory at TU Graz. Previous positions include head of the Department of Mathematics ...

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Research talks

We fix a positive integer $q\geq 2$. Then every real number $x\in[0,1]$ admits a representation of the form

$x=\sum_{n\geq 1}\frac{a_{n}}{q^{n}}$,

where $a_{n}\in \mathcal{N} :=\{0,1,\ .\ .\ .\ ,\ q-1\}$ for $n\geq 1$. For given $x\in[0,1], N\geq 1$, and $\mathrm{d}=d_{1}\ldots d_{k}\in \mathcal{N}^{k}$ we denote by $\Pi(x,\ \mathrm{d},\ N)$ the frequency of occurrences of the block $\mathrm{d}$ among the first $N$ digits of $x$, i.e.

$\Pi(x, \mathrm{d},N):=\frac{1}{N}|\{0\leq n< N:a_{n+1}=d_{1}, . . . a_{n+k}=d_{k}\}$

from a probabilistic point of view we would expect that in a randomly chosen $x\in[0,1]$ each block $\mathrm{d}$ of $k$ digits occurs with the same frequency $q^{-k}$. In this respect we call a real $x\in[0,1]$ normal to base $q$ if $\Pi(x,\ \mathrm{d},\ N)=q^{-k}$ for each $k\geq 1$ and each $|\mathrm{d}|=k$. When Borel introduced this concept he could show that almost all (with respect to Lebesgue measure) reals are normal in all bases $q\geq 2$ simultaneously. However, still today all constructions of normal numbers have an artificial touch and we do not know whether given reals such as $\sqrt{2},$ log2, $e$ or $\pi$ are normal to a single base.
On the other hand the set of non-normal numbers is large from a topological point of view. We say that a typical element (in the sense of Baire) $x\in[0,1]$ has property $P$ if the set $S :=${$x\in[0,1]:x$ has property $P$} is residual - meaning the countable intersection of dense sets. The set of non-normal numbers is residual.
In the present talk we will consider the construction of sets of normal and non-normal numbers with respect to recent results on absolutely normal and extremely non-normal numbers.
We fix a positive integer $q\geq 2$. Then every real number $x\in[0,1]$ admits a representation of the form

$x=\sum_{n\geq 1}\frac{a_{n}}{q^{n}}$,

where $a_{n}\in \mathcal{N} :=\{0,1,\ .\ .\ .\ ,\ q-1\}$ for $n\geq 1$. For given $x\in[0,1], N\geq 1$, and $\mathrm{d}=d_{1}\ldots d_{k}\in \mathcal{N}^{k}$ we denote by $\Pi(x,\ \mathrm{d},\ N)$ the frequency of occurrences of the block $\mathrm{d}$ among the first $N$ digits of $x$, i.e. ...

11K16 ; 11A63

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Research School;Number Theory;Probability and Statistics

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Research talks

This is a survey on progress in metric discrepancy theory and probabilistic aspects in harmonic analysis. We start with classical limit theorems of Salem and Zygmund as well as with the work of Erdoes and Gaal and of Walter Philipp. A focus lies on laws of the iterated logarithm for discrepancy functions of lacunary sequences. We show the connection to certain diophantine properties of the underlying lacunary sequences obtaining precise asymptotic formulas. Different phenomena for subexponentially growing, for exponentially growing and for superexponentially growing sequences are established. Furthermore, relations to arithmetic dynamical systems and to Donald Knuth`s concept of pseudorandomness are discussed. Recent results are contained in joint work with Christoph Aistleitner and Istvan Berkes and it is planed to publish parts of it in a Jean Morlet Springer lecture Notes volume. This is a survey on progress in metric discrepancy theory and probabilistic aspects in harmonic analysis. We start with classical limit theorems of Salem and Zygmund as well as with the work of Erdoes and Gaal and of Walter Philipp. A focus lies on laws of the iterated logarithm for discrepancy functions of lacunary sequences. We show the connection to certain diophantine properties of the underlying lacunary sequences obtaining precise ...

11K38 ; 11J83 ; 11K60

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