UNSW Applied Math Seminars: Tichy, Shparlinski, Brauchart

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.