Irregularities of distributions: dispersion and discrepancy
| školitel: | prof. RNDr. Jan Vybíral, Ph.D. |
| e-mail: | zobrazit e-mail |
| typ práce: | dizertační práce |
| zaměření: | MI_MM, MI_AMSM, MINF |
| klíčová slova: | Discrepancy, dispersion, randomized algorithms, numerical integration |
| popis: | In many applications (like sampling, numerical integration, function recovery and many others) it is necessary to construct a set of points, which are uniformly distributed across a given domain. There are many ways, how to measure the quality of the uniform distribution of a point set. We will study two of then, dispersion and discrepancy. Dispersion of a point set is the volume of the largest box, which avoids the given point set. The more classical notion of discrepancy measures the deviation of the number of points in any rectangle from its normalized volume. We shall review the existing lower and upper bounds of dispersion and discrepancy, using the methods from combinatorics, harmonic analysis, functional analysis, and probability theory. The student will try to attack some of the old problems on the field as well as some of the modern modifications - like the spherical discrepancy and dispersion, line dispersion, or star discrepancy. We will be interested in both upper bounds, which usually require a construction of a good point set, as well as lower bounds, which often rely on some abstract tools. |
| literatura: | [1] J. Matoušek, Geometric discrepancy, Springer-Verlag, Berlin, 2010 [2] M. Ullrich and J. Vybiral, An upper bound on the minimal dispersion, J. Complexity 45 (2018), 120–126. [3] B. Bukh and T. Chao, Empty axis-parallel boxes, Int. Math. Res. Notices 18 (2022), 13811–13828 [4] D. Bilyk and M. T. Lacey, On the small ball inequality in three dimensions, Duke Math. J. 143 (2008), no. 1, 81–115. |
| naposledy změněno: | 02.04.2025 21:59:23 |
za obsah této stránky zodpovídá:
Ľubomíra Dvořáková | naposledy změněno: 12.9.2011