m

F Nous contacter


0

2020 - Sem 2 - Tichy - Rivat  | enregistrements trouvés : 55

O

-A +A

P Q

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Post-edited  Interview at Cirm : Robert Tichy
Tichy, Robert (Personne interviewée) | CIRM (Editeur )

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 ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.
Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Multi angle  Normal and non-normal numbers
Madritsch, Manfred (Auteur de la Conférence) | CIRM (Editeur )

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Multi angle  Metric discrepancy theory
Tichy, Robert (Auteur de la Conférence) | CIRM (Editeur )

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a < q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place for the so-called modular form of Zaremba's conjecture.
Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a 1/2 takes place for the so-called modular form of Zaremba's conjecture....

11A55 ; 11J70 ; 11B30 ; 20G05 ; 20G40

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

I present the basics and numerical result of two (or three) concrete applications of quasi-Monte-Carlo methods in financial engineering. The applications are in: derivative pricing, in portfolio selection, and in credit risk management.

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

This is joint work with Jörg Thuswaldner from University of Leoben.

A linear recurrent number system is a generalization of the $q$-adic number system, where we replace the sequence of powers of $q$ by a linear recurrent sequence $G_{k+d}=a_1G_{k+d-1}+\cdots+a_dG_k$ for $k\geq 0$. Under some mild conditions on the recurrent sequence every positive integer $n$ has a representation of the form \[n=\sum_{j=0}^k \varepsilon_j(n)G_j.\]

The $q$-adic number system corresponds to the linear recursion $G_{k+1}=qG_k$ and $G_0=1$. The first example of a real generalization is due to Zeckendorf who showed that the Fibonacci sequence $G_0=1$, $G_1=2$, $G_{k+2}=G_{k+1}+G_k$ for $k\geq0$ yields a representation for each positive integer. This is unique if we additionally suppose that no two consecutive ones exist in the representation. Similar restrictions hold for different recurrent sequences and they build the essence of these number systems.

In the present talk we investigate the representation of primes and almost primes in linear recurrent number systems. We start by showing the different results due to Fouvry, Mauduit and Rivat in the case of $q$-adic number systems. Then we shed some light on their main tools and techniques. The heart of our considerations is the following Bombieri-Vinogradov type result
\[\sum_{q < x^{\vartheta-\varepsilon}}\max_{y < x}\max_{1\leq a\leq q} \left\vert\sum_{\substack{n< y,s_G(n)\equiv b\bmod d\\ n\equiv b\bmod q}}1 -\frac1q\sum_{n < y,s_G(n)\equiv b\bmod d}1\right\vert \ll x(\log 2x)^{-A},\]
which we establish under the assumption that $a_1\geq30$. This lower bound is due to numerical estimations. With this tool in hand we are able to show that \[ \left\vert\{n\leq x\colon s_G(n)\equiv b\bmod d, n=p_1\text{ or }n=p_1p_2\}\right\vert\gg \frac{x}{\log x}.\]
This is joint work with Jörg Thuswaldner from University of Leoben.

A linear recurrent number system is a generalization of the $q$-adic number system, where we replace the sequence of powers of $q$ by a linear recurrent sequence $G_{k+d}=a_1G_{k+d-1}+\cdots+a_dG_k$ for $k\geq 0$. Under some mild conditions on the recurrent sequence every positive integer $n$ has a representation of the form \[n=\sum_{j=0}^k \varepsilon_j(n)G_j.\]

The $q$-adic ...

11A63 ; 11L07 ; 11N05

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Let $q=p^r$, where $p$ is a prime number and $ ß=(\beta_1 ,\ldots ,\beta_r)$ be a basis of $\mathbb{F}_q$ over $\mathbb{F}_p$.
Any $\xi \in \mathbb{F}_q$ has a unique representation $\xi =\sum_{i=1}^r x_i \beta _i$ with $x_1,\ldots ,x_r \in \mathbb{F}_p$.
The coefficients $x_1,\ldots ,x_r$ are called the digits of $\xi$ with respect to the basis $ß$.
The analog of the Rudin-Shapiro function is $R(\xi)=x_1x_2+\cdots + x_{r-1}x_r$. For $f \in \mathbb{F}_q [X]$, non constant and $c\in\mathbb{F}_p$, we obtain some formulas for the number of solutions in $\mathbb{F}_q$ of $R(f(\xi ))=c$. The proof uses the Hooley-Katz bound for the number of zeros of polynomials in $\mathbb{F}_p$ with several variables.

This is a joint work with László Mérai and Arne Winterhof.
Let $q=p^r$, where $p$ is a prime number and $ ß=(\beta_1 ,\ldots ,\beta_r)$ be a basis of $\mathbb{F}_q$ over $\mathbb{F}_p$.
Any $\xi \in \mathbb{F}_q$ has a unique representation $\xi =\sum_{i=1}^r x_i \beta _i$ with $x_1,\ldots ,x_r \in \mathbb{F}_p$.
The coefficients $x_1,\ldots ,x_r$ are called the digits of $\xi$ with respect to the basis $ß$.
The analog of the Rudin-Shapiro function is $R(\xi)=x_1x_2+\cdots + x_{r-1}x_r$. For $f \in ...

11A63 ; 11T23 ; 11T30

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

In this talk, we derive a novel non-reversible, continuous-time Markov chain Monte Carlo (MCMC) sampler, called Coordinate Sampler, based on a piecewise deterministic Markov process (PDMP), which can be seen as a variant of the Zigzag sampler. In addition to proving a theoretical validation for this new sampling algorithm, we show that the Markov chain it induces exhibits geometrical ergodicity convergence, for distributions whose tails decay at least as fast as an exponential distribution and at most as fast as a Gaussian distribution. Several numerical examples highlight that our coordinate sampler is more efficient than the Zigzag sampler, in terms of effective sample size.
[This is joint work with Wu Changye, ref. arXiv:1809.03388]
In this talk, we derive a novel non-reversible, continuous-time Markov chain Monte Carlo (MCMC) sampler, called Coordinate Sampler, based on a piecewise deterministic Markov process (PDMP), which can be seen as a variant of the Zigzag sampler. In addition to proving a theoretical validation for this new sampling algorithm, we show that the Markov chain it induces exhibits geometrical ergodicity convergence, for distributions whose tails decay at ...

62F15 ; 60J25

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

In the second part we show how the classical result can be used also for SDEs with drift that may be discontinuous and diffusion that may be degenerate. In that context I will present a concept of (multidimensional) piecewise Lipschitz drift where the set of discontinuities is a sufficiently smooth hypersurface in the multi-dimensional euclidean space. We discuss geometric properties of the set of discontinuities that are needed to transfer the convergence result from the Lipschitz case to the piecewise Lipschitz case.
In the second part we show how the classical result can be used also for SDEs with drift that may be discontinuous and diffusion that may be degenerate. In that context I will present a concept of (multidimensional) piecewise Lipschitz drift where the set of discontinuities is a sufficiently smooth hypersurface in the multi-dimensional euclidean space. We discuss geometric properties of the set of discontinuities that are needed to transfer the ...

65C05 ; 91G60 ; 60H10

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Some number-theoretic problems have led to the study of some infinite series that show multifractal behaviour, which means that their Hölder pointwise regularity may widely change from point to point. Reviewing some examples such as lacunary trigonometric series, Davenport series, Brjuno-type functions, I will put emphasis on the methods encountered in the literature to compute the pointwise Hölder exponent of such functions.

11A55 ; 26A15 ; 26A30 ; 28A80

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Exponential Diophantine equations, say of the form (1) $u_{1}+...+u_{k}=b$ where the $u_{i}$ are exponential terms with fixed integer bases and unknown exponents and b is a fixed integer, play a central role in the theory of Diophantine equations, with several applications of many types. However, we can bound the solutions only in case of k = 2 (by results of Gyory and others, based upon Baker’s method), for k > 2 only the number of so-called non-degenerate solutions can be bounded (by the Thue-Siegel-Roth-Schmidt method; see also results of Evertse and others). In particular, there is a big need for a method which is capable to solve (1) completely in concrete cases.
Skolem’s conjecture (roughly) says that if (1) has no solutions, then it has no solutions modulo m with some m. In the talk we present a new method which relies on the principle behind the conjecture, and which (at least in principle) is capable to solve equations of type (1), for any value of k. We give several applications, as well. Then we provide results towards the solution of Skolem’s conjecture. First we show that in certain sense it is ’almost always’ valid. Then we provide a proof for the conjecture in some cases with k = 2, 3. (The handled cases include Catalan’s equation and Fermat’s equation, too - the precise connection will be explained in the talk). Note that previously Skolem’s conjecture was proved only for k = 1, by Schinzel.
The new results presented are (partly) joint with Bertok, Berczes, Luca, Tijdeman.
Exponential Diophantine equations, say of the form (1) $u_{1}+...+u_{k}=b$ where the $u_{i}$ are exponential terms with fixed integer bases and unknown exponents and b is a fixed integer, play a central role in the theory of Diophantine equations, with several applications of many types. However, we can bound the solutions only in case of k = 2 (by results of Gyory and others, based upon Baker’s method), for k > 2 only the number of so-called ...

11D41 ; 11D61 ; 11D79

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

This talk will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential equations. Unfortunately, only particular parameter constellations allow for closed form solutions such that in general one needs to rely on numerical methods. Instead of studying these associated integro-differential equations, we adapt the problem in a way that allows us to apply deterministic numerical integration algorithms such as QMC rules.
This talk will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential ...

91B30 ; 91G60 ; 60J25 ; 65R20

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

We consider two common pseudorandom number generators constructed from iterations of linear and Möbius maps
$x \mapsto gx$ and $ x \mapsto (ax+b)/(cx+d)$
over a residue ring modulo an integer q ≥ 2, which are known as congruential and inversive generators, respectively. There is an extensive literature on the pseudorandomness of elements $u_{n}, n=1,2,...$, of the corresponding orbits. In this talk we are interested in what happens in these orbits at prime times, that is, we study elements $u_{p}$, $p = 2, 3, . . .$, where $p$ runs over primes.
We give a short survey of previous results on the distribution of $u_{p}$ for the above maps and then:
- Explain how B. Kerr, L. Mérai and I. E. Shparlinski (2019) have used a method of N. M. Korobov (1972) to study the congruential generator on primes modulo a large power of a fixed prime, e.g. $q=3^{\gamma }$ with a large $\gamma$. We also give applications of this result to digits of Mersenne numbers $2^{p}-1$.
- Present a result of L. Mérai and I. E. Shparlinski (2020) on the distribution of the inversive generator on primes modulo a large prime, q. The proof takes advantage of the flexibility of Heath-Brown’s identity, while Vaughan’s identity does not seem to be enough for our purpose. We also pose several open questions and discuss links to Sarnak’s conjecture on pseudorandomness of the Möbius function.
We consider two common pseudorandom number generators constructed from iterations of linear and Möbius maps
$x \mapsto gx$ and $ x \mapsto (ax+b)/(cx+d)$
over a residue ring modulo an integer q ≥ 2, which are known as congruential and inversive generators, respectively. There is an extensive literature on the pseudorandomness of elements $u_{n}, n=1,2,...$, of the corresponding orbits. In this talk we are interested in what happens in these ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Virtualconference  Poisson-generic points
Weiss, Benjamin (Auteur de la Conférence) | CIRM (Editeur )

I will discuss a criterion for randomness of sequences of zeros and ones which is strictly stronger than normality, butholds for almost every sequence generated by i.i.d. random variables with distribution {1/2, 1/2}. Briefly put, the idea is count the number of times blocks of length n appear in the initial block of length $2^n$. I will also discuss an extension of this idea to toral automorphisms. (joint work with Yuval Peres)

11K16 ; 37D99 ; 60F99

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

We show a connection between Fourier series and a celebrated theorem of G. Pick on the number of integer points in an integer polygon. Then we discuss an Euler-Maclaurin formula over polygons.

00-02

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

This talk will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential equations. Unfortunately, only particular parameter constellations allow for closed form solutions such that in general one needs to rely on numerical methods. Instead of studying these associated integro-differential equations, we adapt the problem in a way that allows us to apply deterministic numerical integration algorithms such as QMC rules.
This talk will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential ...

91B30 ; 91G60 ; 60J25 ; 65R20

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

This talk will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential equations. Unfortunately, only particular parameter constellations allow for closed form solutions such that in general one needs to rely on numerical methods. Instead of studying these associated integro-differential equations, we adapt the problem in a way that allows us to apply deterministic numerical integration algorithms such as QMC rules.
This talk will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential ...

91B30 ; 91G60 ; 60J25 ; 65R20

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

The recent papers Gajek-Kucinsky (2017), Avram-Goreac-LiWu (2020) investigated the control problem of optimizing dividends when limiting capital injections by bankruptcy is taken into consideration. The first paper works under the spectrally negative Levy model; the second works under the Cramer-Lundberg model with exponential jumps, where the results are considerably more explicit.
The first talk extends, exploiting the W-Z scale functions, results of Gajek-Kucinsky (2017) to the case when a final penalty is taken into consideration as well. This requires the introduction of new scale and Gerber-Shiu functions.
The second talk illustrates the fact that quite reasonable approximations of the general problem may be obtained using the exponential particular case studied in Avram-Goreac-LiWu (2020). We start by experimenting with de Vylder type approximations for the scale function $W_q(x)$; this amounts essentially to replacing our process by one with exponential jumps and cleverly crafted parameters based on the first three moments of the claims. We show that very good approximations may be obtained for two fundamental objects of interest: the growth exponent $\Phi_q$ of the scale function $W_q(x)$, and the (last) global minimum of $W_q'(x)$, which is fundamental in the de Finetti barrier problem. Turning then to the dividends and limited capital injections problem, we show that a new exponential approximation specific to this problem achieves very good results: it consists in plugging into the objective function for exponential claims the exact "non-exponential ingredients" (scale functions and, survival and mean functions) of our non-exponential examples.
The recent papers Gajek-Kucinsky (2017), Avram-Goreac-LiWu (2020) investigated the control problem of optimizing dividends when limiting capital injections by bankruptcy is taken into consideration. The first paper works under the spectrally negative Levy model; the second works under the Cramer-Lundberg model with exponential jumps, where the results are considerably more explicit.
The first talk extends, exploiting the W-Z scale functions, ...

60G40 ; 60J35 ; 60J75

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

In this talk I will report on recent progress on two different problems in discrepancy theory. In the first part I will present a recent extension of the notion of jittered sampling to arbitrary partitions of the unit cube. In this joint work with Markus Kiderlen from Aarhus, we introduce the notion of a uniformly distributed triangular array. Moreover, we show that the expected Lp-discrepancy of a point sample generated from an arbitrary equi volume partition of the unit cube is always strictly smaller than the expected Lp-discrepancy of a set of N uniform random samples for p > 1.
The second part of the talk is dedicated to greedy energy minimization. I will give a new characterisation of the classical van der Corput sequence in terms of a minimization problem and will discuss various related open questions.
In this talk I will report on recent progress on two different problems in discrepancy theory. In the first part I will present a recent extension of the notion of jittered sampling to arbitrary partitions of the unit cube. In this joint work with Markus Kiderlen from Aarhus, we introduce the notion of a uniformly distributed triangular array. Moreover, we show that the expected Lp-discrepancy of a point sample generated from an arbitrary equi ...

11K38 ; 11K31 ; 52C25

Z