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Displaying 141 – 160 of 342

Irregularities of continuous distributions

Michael Drmota (1989)

Annales de l'institut Fourier

This paper deals with a continuous analogon to irregularities of point distributions. If a continuous fonction $x : [0, 1] \to X$ where $X$ is a compact body, is interpreted as a particle’s movement in time, then the discrepancy measures the difference between the particle’s stay in a proper subset and the volume of the subset. The essential part of this paper is to give lower bounds for the discrepancy in terms of the arc length of $x (t)$ , $0 \leq t \leq 1$ . Furthermore it is shown that these estimates are the best possible despite of...

Irregularities of Point Distributions Relative to Homothetic Convex Bodies I.

Gyula Károlyi (1995)

Monatshefte für Mathematik

Kriterien in der Theorie der Gleichverteilung.

Rudolf J. Taschner (1978/1979)

Monatshefte für Mathematik

$L^{2}$ discrepancy

Oto Strauch (1994)

Mathematica Slovaca

$L_{2}$ discrepancy of generalized Zaremba point sets

Henri Faure, Friedrich Pillichshammer (2011)

Journal de Théorie des Nombres de Bordeaux

We give an exact formula for the $L_{2}$ discrepancy of a class of generalized two-dimensional Hammersley point sets in base $b$ , namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base $b$ . The Zaremba point set introduced by White in 1975 is the special case where the $b$ shifts are taken repeatedly in sequential order, hence needing at least $b^{b}$ points to obtain the optimal order of $L_{2}$ discrepancy. On the...

$L_{p}$ - and $S_{p, q}^{r} B$ -discrepancy of (order 2) digital nets

Lev Markhasin (2015)

Acta Arithmetica

Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the $L_{p}$ -discrepancy. We prove this for arbitrary prime base b with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range, and enlarge that range for order 2 digital nets. The discrepancy function in Triebel-Lizorkin and Sobolev spaces with dominating mixed...

$L^{p}$ -discrepancy and statistical independence of sequences

Peter J. Grabner, Oto Strauch, Robert Franz Tichy (1999)

Czechoslovak Mathematical Journal

We characterize statistical independence of sequences by the $L^{p}$ -discrepancy and the Wiener $L^{p}$ -discrepancy. Furthermore, we find asymptotic information on the distribution of the $L^{2}$ -discrepancy of sequences.