arXiv subjects
Publications and source records attributed to Peter Hellekalek.
In this paper, we discuss the well known 3x+1 conjecture in form of the accelerated Collatz function T defined on the positive odd integers. We present a sequence of quotient spaces and an invertible map that are intrinsically related to the behavior of T. This approach allows to express the 3x+1 conjecture in form of equivalent problems, which might be more accessible than the original conjecture.
In this paper, we study quasi-Monte Carlo (QMC) integration in weighted Sobolev spaces. In contrast to many previous results the QMC algorithms considered here are of open type, i.e., they are extensible in the number of sample points without having to discard the samples already used. As the underlying integration nodes we consider randomized Halton sequences in prime bases $\boldsymbol{p}=(p_1,...,p_s)$ for which we study the root mean square (RMS) worst-case error. The randomization method is a $\boldsymbol{p}$-adic shift which is based on $\boldsymbol{p}$-adic arithmetic. The obtained error bounds are optimal in the order of magnitude of the number of sample nodes. Furthermore we obtain conditions on the coordinate weights under which the error bounds are independent of the dimension $s$. In terms of the field of Information-Based Complexity this means that the corresponding QMC rule achieves a strong polynomial tractability error bound. Our findings on the RMS worst-case error of randomized Halton sequences can be carried over to the RMS $L_2$-discrepancy. Except for the $\boldsymbol{p}$-adic shift our results are fully constructive and no search algorithms (such as the component-by-component algorithm) are required.
The starting point of this paper is the interplay between the construction principle of a sequence and the characters of the compact abelian group that underlies the construction. In case of the Halton sequence in base $\mathbf b=(b_1, \ldots, b_s)$ in the $s$-dimensional unit cube $[0,1)^s$, which is an important type of a digital sequence, this kind of duality principle leads to the so-called $\mathbf b$-adic function system and provides the basis for the $\mathbf b$-adic method, which we present in connection with hybrid sequences. This method employs structural properties of the compact group of $\mathbf b$-adic integers as well as $\mathbf b$-adic arithmetic to derive tools for the analysis of the uniform distribution of sequences in $[0,1)^s$. We first clarify the point which function systems are needed to analyze digital sequences. Then, we present the hybrid spectral test in terms of trigonometric-, Walsh-, and $\mathbf b$-adic functions. Various notions of diaphony as well as many figures of merit for rank-1 quadrature rules in Quasi-Monte Carlo integration and for certain linear types of pseudo-random number generators are included in this measure of uniform distribution. Further, discrepancy may be approximated arbitrarily close by suitable versions of the spectral test.
For bases $\mathbf{b}=(b_1,..., b_s)$ of $s$ not necessarily distinct integers $b_i\ge 2$, we prove a version of the inequality of \etk \ for the hybrid function system composed of the Walsh functions in base $\bfb^{(1)}=(b_1,..., b_{s_1})$ and, as second component, the $\bfb^{(2)}$-adic functions, $\bfb^{(2)}=(b_{s_1+1},..., b_s)$, with $s=s_1+s_2$, $s_1$ and $s_2$ not both equal to 0. Further, we point out why this choice of a hybrid function system covers all possible cases of sequences that employ addition of digit vectors as their main construction principle.
In this paper, we study the different possibilities to add two vectors of digits of a given length $m$. Our results show that there are at least $2^{m-1}$ different additions of such vectors, while there exist only two types of addition that we may employ, addition with carry and addition without carry. The proofs of our results are elementary.