Kronecker Sequences Research Articles

Ergodicity Of Fuzzy Markov Chains Based On Simulation Using Sequences

As shown in [1], we reduce periodicity of fuzzy Markov chains using the Halton quasi-random generator. In this paper, we employ two different quasi-random sequences namely Faure and Kronecker to generate the membership values of fuzzy Markov chain. Using simulation it is revealed that the number of ergodic fuzzy Markov chain simulated by Kronecker sequences is more than the one obtained by Faure sequences.

Van der Corput Sequences and Linear Permutations

Abstract This extended abstract is concerned with the irregularities of distribution of one-dimensional permuted van der Corput sequences that are generated from linear permutations. We show how to obtain upper bounds for the discrepancy and diaphony of these sequences, by relating them to Kronecker sequences and applying earlier results of Faure and Niederreiter.

Probabilistic diophantine approximation and the distribution of Halton–Kronecker sequences

By a Halton–Kronecker sequence we mean a sequence in the s+t-dimensional unit-cube which is the combination of an s-dimensional Halton sequence and a t-dimensional Kronecker sequence ({n⋅α})n=0,1,… with α∈Rt. The investigation of such ‘hybrid sequences’ for their use in Monte Carlo and quasi-Monte Carlo methods first was motivated by Spanier (1995) [20]. By suitably adapting techniques of Jozsef Beck on probabilistic diophantine approximation, developed in Beck (1994) [2], we can show that for almost all α∈Rt for the discrepancy DN of a Halton–Kronecker sequence we have DN=O((logN)s+t+ϵN) for all ϵ>0, which most probably essentially is the best possible metrical result for this type of sequences.

Generation of parallel modified Kronecker sequences

The generation of appropriate parallel and high-quality quasirandom sequences (low-discrepancy sequences) is crucial to the success of quasi-Monte Carlo methods. The Kronecker sequence is well known to be one of the special types of low-discrepancy sequences, and one of its important advantages is that the Kronecker sequence is easy to implement due to its definition via the fractional parts of multiples of irrationals. However, the original Kronecker sequence suffers from correlations for different dimensions. These correlations result in poorly distributed two-dimensional projections. An approach to this is to find a modified Kronecker sequence via generalizing golden ratio and generate parallel sequences. This paper presents a new algorithm for finding a modified Kronecker sequence within special choices of irrationals. This modified sequence is numerically tested and shown empirically to be superior to the other widely used quasirandom sequences. In addition, based on analysis and insight into the correlations between dimensions of the Kronecker sequence, we illustrate why our algorithm is efficient for breaking these correlations.

DISCREPANCY ESTIMATES FOR ROTATION SEQUENCES AND OSCILLATION SEQUENCES

Two kinds of sequences, which are of interest in problems of uniform distribution and dynamical systems, are considered. The rotation sequences (or Kronecker sequences) (KS) are closely related to the orbits of the rotation map and the oscillation sequences (OS) are a discrete-time form of orbits of the simplest oscillators. The discrepancy of these sequences, which is a measure of deviation of the empirical distribution of a sequence from the ideal uniform distribution, is studied.

ON HYBRID SEQUENCES BUILT FROM NIEDERREITER–HALTON SEQUENCES AND KRONECKER SEQUENCES

AbstractWe discuss the distribution properties of hybrid sequences whose components stem from Niederreiter–Halton sequences on the one hand, and Kronecker sequences on the other. In this paper, we give necessary and sufficient conditions on the uniform distribution of such sequences, and derive a result regarding their discrepancy. We conclude with a short summary and a discussion of topics for future research.

Metrical results on the discrepancy of Halton–Kronecker sequences

By a Halton–Kronecker sequence we mean a sequence in the (s+t)-dimensional unit-cube which is the combination of an s-dimensional Halton-sequence and a t-dimensional Kronecker sequence. The distribution of such sequences was studied for the first time quite recently by Niederreiter. In this paper we obtain metrical results for the discrepancy of Halton–Kronecker sequences which are similar to results for the pure Kronecker sequences obtained by Khintchine and by W.M. Schmidt.

Kronecker Product of Two-dimensional Arrays

Kronecker sequences constructed from short sequences are good sequences for spread spectrum communication systems. In this paper we study a similar problem for two-dimensional arrays, and we determine the linear complexity of the Kronecker product of two arrays. Our result shows that similar good property on linear complexity holds for Kronecker product of arrays.

Equidistribution of Kronecker sequences along closed horocycles

It is well known that (i) for every irrational number α the Kronecker sequence mα (m = 1,...,M) is equidistributed modulo one in the limit $$ \rightarrow\infty $$ , and (ii) closed horocycles of length $$ \ell $$ become equidistributed in the unit tangent bundle $$ T_1\mathcal{M} $$ of a hyperbolic surface $$ \mathcal{M} $$ of finite area, as $$ \ell\rightarrow\infty $$ . In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant $$ \nu > 0 $$ the Kronecker sequence embedded in $$ T_1\mathcal{M} $$ along a long closed horocycle becomes equidistributed in $$ T_1\mathcal{M} $$ for almost all α, provided that $$ $$ . This equidistribution result holds in fact under explicit diophantine conditions on α (e.g. for α = √2) provided that $$ \nu < 1 $$ , $$ \nu < 2 $$ with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for $$ \nu = 2 $$ , our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence n 2α modulo one.

Multidimensional declustering schemes using Golden Ratio and Kronecker Sequences

We propose a new declustering scheme for allocating uniform multidimensional data among parallel disks. The scheme, aimed at reducing disk access time for range queries, is based on Golden Ratio Sequences for two dimensions and Kronecker Sequences for higher dimensions. Using exhaustive simulation, we show that, in two dimensions, the worst-case (additive) deviation of the scheme from the optimal response time for any range query is one when the number of disks (M) is at most 22; its worst-case deviation is two when M /spl les/ 94; and its worst-case deviation is four when M /spl les/ 550. In two dimensions, we prove that whenever M is a Fibonacci number, the average performance of the scheme is within 14 percent of the (generally, unachievable) strictly optimal scheme and its worst-case response time is within a multiplicative factor three of the optimal response time for any query, and within a factor 1.5 of the optimal for large queries. We also present comprehensive simulation results, on two-dimensional as well as on higher-dimensional data, that compare and demonstrate the advantages of our scheme over some recently proposed schemes in the literature.

On the Independence of Hartman Sequences

Given Open image in new window and Open image in new window, we define Open image in new window by setting Open image in new window if and only if ({α1n},..., {αkn}), where {α} denotes the fractional part of α, i.e. α is considered as an element of the torus Open image in new window. If the topological boundary of A has Haar measure 0, then Open image in new window is called a Hartman sequence, which is a generalisation of Kronecker and Beatty sequences. In this article we answer a question of Winkler by showing explicitly for which sets Open image in new window and vectors Open image in new window we have Open image in new window. The main tool of the proof is Weyl’s theorem on uniform distribution.

Ergodic Group Rotations, Hartman Setsand Kronecker Sequences

Let Open image in new window be a homomorphism with dense image Open image in new window in the compact group C. If M ⊆ C is a continuity set, i.e. its topological boundary has Haar measure 0, then Open image in new window is called a Hartman set. If M is aperiodic then S contains the essential information about (C, ι) or, equivalently, about the dynamical system (C, T) where T is the ergodic group rotation Open image in new window. Using Pontryagin’s duality the paper presents a new method to get this information from S: The set S induces a filter Open image in new window which is an isomorphism invariant for (C, T) and turns out to be a complete invariant for ergodic group rotations. If one takes Open image in new window, one gets the interesting special case of Kronecker sequences (nα) which are classical objects in number theory and diophantine analysis.

Corrigendum to the paper "On the distribution of s-dimensional Kronecker sequences" Acta Arith. 51 (1988), 335-347

Corrigendum to the paper "On the distribution of s-dimensional Kronecker sequences" Acta Arith. 51 (1988), 335-347

Kronecker sequences for spread-spectrum communication

The paper considers the correlation properties of sets of sequences in which each sequence is constructed by taking the Kronecker product (or tensor product) of two short sequences. It is shown that if the short sequences are carefully chosen then the correlation properties of such sets are nearly as good as those of sequences of comparable length that have been obtained by other means. Furthermore, it is much easier to optimise the phases of these sequences because of the structure that they inherit from the Kronecker product operation. This structure also enhances the ease of synchronisation for such sequences. If the signal/noise ratio is large, the time required to synchronise to Kronecker sequences is significantly less than the time required to synchronise to other sequences of comparable lengths. Thus, the Kronecker sequences allow one to achieve rapid acquisition at the expense of a very small deterioration in some of the correlation properties.