By Gerhard Larcher, Friedrich Pillichshammer, Arne Winterhof, Chaoping Xing

Harald Niederreiter's pioneering study within the box of utilized algebra and quantity thought has resulted in very important and tremendous breakthroughs in lots of parts. This choice of survey articles has been authored by way of shut colleagues and major specialists to mark the party of his seventieth birthday. The booklet offers a latest evaluation of alternative study components, overlaying uniform distribution and quasi-Monte Carlo tools in addition to finite fields and their functions, particularly, cryptography and pseudorandom quantity new release. Many effects are released right here for the 1st time. The publication serves as an invaluable start line for graduate scholars new to those parts or as a refresher for researchers eager to stick with fresh traits.

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Comb. Theory Ser. A 117, 668–682, 2010. [18] Y. Zheng and X. M. Zhang, On plateaued functions. IEEE Trans. Inf. Theory 47, 1215–1223, 2001. 3 Applications of geometric discrepancy in numerical analysis and statistics Josef Dick The University of New South Wales, Sydney Dedicated to Harald Niederreiter on the occasion of his 70th birthday. Abstract In this paper we discuss various connections between geometric discrepancy measures, such as discrepancy with respect to convex sets (and convex sets with smooth boundary in particular), and applications to numerical analysis and statistics, such as point distributions on the sphere, the acceptancerejection algorithm and certain Markov chain Monte Carlo algorithms.

1) for Hölder conjugates 1 ≤ p, q ≤ ∞, with the obvious modifications for p or q = ∞. 1) is a variant of an inequality due to Koksma [30]. From these considerations, one obtains the L q discrepancy as a quality criterion for the point set PN ,1 : 1/q q L B (PN,1 ) = [0,1] | q PN ,1 (t)| dt for 1 ≤ q ≤ ∞, again with the obvious modifications for q = ∞. As above, B is the set of boxes anchored at the origin. There is a natural generalization of the above approach to dimensions s > 1 by using partial derivatives of f .

Soc. 13, 497–506, 1976. G. Larcher, F. Pillichshammer, A. Winterhof and C. P. Xing 19 [13] H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Am. Math. Soc. 84, 957–1041, 1978. [14] H. Niederreiter, Dyadic fractions with small partial quotients. Monatsh. Math. 101, 309–315, 1986. [15] H. Niederreiter, Point sets and sequences with small discrepancy. Monatsh. Math. 104, 273–337, 1987. [16] H. Niederreiter, Rational functions with partial quotients of small degree in their continued fraction expansion.

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