Type: Seminar

Distribution: World

Expiry: 2 Dec 2010

Auth: carberry@60-241-127-220.static.tpgi.com.au (carberry) in SMS-WASM

Next week Thursday 2nd December we have 3 speakers: Robert Tichy (TU Graz, Austria), Igor Shparlinski (Macquarie University) and Johann Brauchart (UNSW). The talks will be held at Quad room G032 from 2-5pm. The titles and abstracts of the talks are below. Anyone is welcome to attend. Some talks may be interesting not only to applied mathematicians. After the talks we take the speakers out for dinner at a local restaurant in Kingsford. You are welcome to join. Please let me know (josef.dick@unsw.edu.au) by Monday 29th of November if you would like to do so. ------------------ Title: Probabilistic results on the Discrepancy of Sequences: Theory and Applications Speaker: Robert Tichy (TU Graz) Abstract: In the first part of the talk we present new results concerning limit theorems and laws of the iterated logarithm for the discrepancy and related quantities. This extents a classical result of Walter Philipp concerning exponentially growing sequences. In particular sub-exponential sequences are considered. Furthermore, the involved constants can be determined in various important special cases. The proofs make use on methods from probability theory as well as on recent quantitative results from diophantine approximation. In a second part we discuss applications of discrepancy theory in ruin models. ------------------ Title: Distribution of Points on Modular Hyperbolas. Speaker: Igor Shparlinski (Macquarie University) Abstract: We’ll give a survey of various interesting results about the distribution of the set of points on the modular hyperbola xy = a (mod p) for a prime p. These results show that this curve is not a typical curve f(x,y) = 0 (mod p) and also have many surprising applications to other, seemingly unrelated areas. Some of these properties can also be extended to points satisfying the congruence xy = a (mod m) for a composite m, where they become even more special. ------------------- Title: Discrete (Riesz) Energy, Discrepancy and Digital Nets Speaker: Johann S. Brauchart (UNSW) Abstract: A famous problem in physics concerns the distribution of electrons and resulting electrostatic energy on conducting spheres. With this classical model in mind one defines the Riesz $s$-energy and logarithmic energy of an $N$-point configuration of unit point charges interacting through a potential $1/r^s$ ($s\neq0$) or $\log(1/r)$ ($s=0$). Here, $r$ is the Euclidean distance in the ambient space. See Hardin and Saff [Notices Amer. Math. Soc. 51 (2004)]. A sequence of optimal $s$-energy $N$-point configurations on the unit sphere is asymptotically uniformly distributed in the sense that each spherical cap gets a fair share of points (that is, the spherical cap discrepancy tends to zero) as $N$ goes to infinity if $s > - 2$. Stolarsky’s invariance principle [Proc. Amer. Math. Soc. 41 (1973)] shows that the sum of distances (the $(-1)$-energy) and the spherical cap $\mathbb{L}_2$-discrepancy are intimately related. Read differently it expresses the worst-case error of an equal weight numerical integration rule for functions from the unit ball in a certain reproducing kernel Hilbert space setting in terms of the $\mathbb{L}_2$-discrepancy and vice versa. We show that this principle can be also derived using reproducing kernel Hilbert spaces. % Interestingly, the generalized discrepancy, which measures the uniform distribution of a point set with respect to the functions from a certain Sobolev space, introduced by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997)] for the purpose of obtaining a Koksma-Hlawka like inequality on the $2$-sphere can be interpreted as a worst case error for numerical integration (Womersley and Sloan [Adv. Comput. Math. 21 (2004)]) and, essentially, reduces to the sum of distances in our setting. (Our result is valid for $d\geq2$.) Digital Nets provide a very efficient method to generate point sets in the $d$-dimensional unit cube with desirable properties like small ’discrepancy’ used, for example, for quasi-Monte Carlo rules. Such nets can be lifted to the unit sphere in $\mathbb{R}^{d+1}$ my means of an area preserving map. Some of the ’good’ properties of digital nets should also carry over to the sphere. We present results regarding the discrete energy, the discrepancy and the worst-case error for numerical integration of such nets on the $d$-sphere. This talk is based on joint work with Rob Womersley and Josef Dick.

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