Iterated Function Systems in Mixed Euclidean and p-adic Spaces

We investigate graph-directed iterated function systems in mixed Euclidean and p-adic spaces. Hausdorff measure and Hausdorff dimension in such spaces are defined, and an upper bound for the Hausdorff dimension is obtained. The relation between the Haar measure and the Hausdorff measure is clarified. Finally, we discus an example in {Bbb R}\times{\Bbb Q}\sb 2 and calculate upper and lower bounds for its Hausdorff dimension.Comment: 10 pages, 2 Figures; Proceedings of the Conference "Fractal 2006" held in Vienna, Austria, from February 12 to February 15, 200

More Kolakoski Sequences

Our goal in this article is to review the known properties of the mysterious Kolakoski sequence and at the same time look at generalizations of it over arbitrary two letter alphabets. Our primary focus will here be the case where one of the letters is odd while the other is even, since in the other cases the sequences in question can be rewritten as (well-known) primitive substitution sequences. We will look at word and letter frequencies, squares, palindromes and complexity.Comment: 17 pages, 3 tables, 1 figur

A two parameter ratio-product-ratio estimator using auxiliary information

We propose a two parameter ratio-product-ratio estimator for a finite population mean in a simple random sample without replacement following the methodology in Ray and Sahai (1980), Sahai and Ray (1980), Sahai and Sahai (1985) and Singh and Ruiz Espejo (2003). The bias and mean square error of our proposed estimator are obtained to the first degree of approximation. We derive conditions for the parameters under which the proposed estimator has smaller mean square error than the sample mean, ratio and product estimators. We carry out an application showing that the proposed estimator outperforms the traditional estimators using groundwater data taken from a geological site in the state of Florida.Comment: 13 pages, 2 figures, 4 table

Computing modular coincidences for substitution tilings and point sets

Computing modular coincidences can show whether a given substitution system, which is supported on a point lattice in $R\!\!\!\! R^d$, consists of model sets or not. We prove the computatibility of this problem and determine an upper bound for the number of iterations needed. The main tool is a simple algorithm for computing modular coincidences, which is essentially a generalization of the Dekking coincidence to more than one dimension, and the proof of equivalence of this generalized Dekking coincidence and modular coincidence. As a consequence, we also obtain some conditions for the existence of modular coincidences. In a separate section, and throughout the article, a number of examples are given

Kolakoski-(2m,2n) are limit-periodic model sets

We consider (generalized) Kolakoski sequences on an alphabet with two even numbers. They can be related to a primitive substitution rule of constant length ell. Using this connection, we prove that they have pure point dynamical and pure point diffractive spectrum, where we make use of the strong interplay between these two concepts. Since these sequences can then be described as model sets with ell-adic internal space, we add an approach to ``visualize'' such internal spaces.Comment: 15 pages, 3 figures; updated references, corrected typo

Modulated crystals and almost periodic measures

Abstract Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyze these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similar to de Bruijn’s analysis, we find stability with respect to almost periodic modulations

Cryptic speciation of benthic Prorocentrum (Dinophyceae) species and their potential as ecological indicators

The response of marine ecosystems to rapid climate changes has been well recognized but not studied extensively. Benthic microalgae, in contrast to the phytoplankton that is able to be transported by currents, have limited dispersal ability and thus are a better ecological indicator to climate changes. Here we performed sampling in the Yellow Sea, the East China Sea and South China Sea and established twenty-six strains of benthic Prorocentrum for detailed morphological and molecular examinations. Five Prorocentrum species, including P. concavum, P. fukuyoi, P. mexicanum, P. tsawwassenense, and P. cf. sculptile, were identified. Both P. concavum and P. fukuyoi displayed marked intraspecific divergences in large subunit (LSU) ribosomal RNA gene sequences, corresponding to their geographical origins. In contrast, P. mexicanum strains shared identical LSU sequence. Prorocentrum tsawwassenense and P. cf. sculptile are not suitable ecological indicators as they were rarely observed. Prorocentrum mexicanum is not recommended either as it is present across the region. In contrast, P. concavum and P. fukuyoi have advantages as ecological indicators for climate changes in the Western Pacific as they comprise several ribotypes with differentiated biogeography. Toxin analysis was also performed on all five species except P. fukuyoi by liquid chromatography coupled to tandem mass spectrometry, but okadaic acid was not detectable