Repetitive Delone Sets and Quasicrystals

This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set whose patch-counting function N(T), for radius T, is finite for all T is called repetitive if there is a function M(T) such that every ball of radius M(T)+T contains a copy of each kind of patch of radius T that occurs in the set. This is equivalent to the minimality of an associated topological dynamical system with R^n-action. There is a lower bound for M(T) in terms of N(T), namely N(T) = O(M(T)^n), but no general upper bound. The complexity of a repetitive Delone set can be measured by the growth rate of its repetitivity function M(T). For example, M(T) is bounded if and only if the set is a crystal. A set is called is linearly repetitive if M(T) = O(T) and densely repetitive if M(T) = O(N(T))^{1/n}). We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, i.e. the associated topological dynamical system is strictly ergodic. It follows that such sets are diffractive. In the reverse direction, we construct a repetitive Delone set in R^n which has M(T) = O(T(log T)^{2/n}(log log log T)^{4/n}), but does not have uniform patch frequencies. Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets, and we propose considering them as a notion of "perfectly ordered quasicrystal".Comment: To appear in "Ergodic Theory and Dynamical Systems" vol.23 (2003). 37 pages. Uses packages latexsym, ifthen, cite and files amssym.def, amssym.te

Diffraction from visible lattice points and k-th power free integers

We prove that the set of visible points of any lattice of dimension at least 2 has pure point diffraction spectrum, and we determine the diffraction spectrum explicitly. This settles previous speculation on the exact nature of the diffraction in this situation, see math-ph/9903046 and references therein. Using similar methods we show the same result for the 1-dimensional set of k-th power free integers with k at least 2. Of special interest is the fact that neither of these sets is a Delone set --- each has holes of unbounded inradius. We provide a careful formulation of the mathematical ideas underlying the study of diffraction from infinite point sets.Comment: 45 pages, with minor corrections and improvements; dedicated to Ludwig Danzer on the occasion of his 70th birthda

Entropy and diffraction of the kk-free points in nn-dimensional lattices

We consider the $k$th-power-free points in $n$-dimensional lattices and explicitly calculate their entropies and diffraction spectra. This is of particular interest since these sets have holes of unbounded inradius.Comment: 27 pages, 2 figures; revised version with new references [7,8,23]; latest version with new Theorem 6 and updated reference [7

Multiple planar coincidences with N-fold symmetry

Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions.Comment: 13 pages, two figures. For previous related work see math.MG/0511147 and math.CO/0301021. Minor changes and references update

On right conjugacy closed loops of twice prime order

The right conjugacy closed loops of order 2p, where p is an odd prime, are classified up to isomorphism.Comment: Clarified definitions, added some remarks and a tabl

Local Complexity of Delone Sets and Crystallinity

This paper characterizes when a Delone set X is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. Let N(T) count the number of translation-inequivalent patches of radius T in X and let M(T) be the minimum radius such that every closed ball of radius M(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a `gap in the spectrum' of possible growth rates between being bounded and having linear growth, and that having linear growth is equivalent to X being an ideal crystal. Explicitly, for N(T), if R is the covering radius of X then either N(T) is bounded or N(T) >= T/2R for all T>0. The constant 1/2R in this bound is best possible in all dimensions. For M(T), either M(T) is bounded or M(T) >= T/3 for all T>0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has M(T) >= c(n)T for all T>0, for a certain constant c(n) which depends on the dimension n of X and is greater than 1/3 when n > 1.Comment: 26 pages. Uses latexsym and amsfonts package

Smoking Duration, Respiratory Symptoms, and COPD in Adults Aged ≥45 Years with a Smoking History

BACKGROUND: The purpose of this study was to assess the relationship of smoking duration with respiratory symptoms and history of chronic obstructive pulmonary disease (COPD) in the South Carolina Behavioral Risk Factor Surveillance System survey in 2012. METHODS: Data from 4,135 adults aged ≥45 years with a smoking history were analyzed using multivariable logistic regression that accounted for sex, age, race/ethnicity, education, and current smoking status, as well as the complex sampling design. RESULTS:The distribution of smoking duration ranged from 19.2% (1-9 years) to 36.2% (≥30 years). Among 1,454 respondents who had smoked for ≥30 years, 58.3% were current smokers, 25.0% had frequent productive cough, 11.2% had frequent shortness of breath, 16.7% strongly agreed that shortness of breath affected physical activity, and 25.6% had been diagnosed with COPD. Prevalence of COPD and each respiratory symptom was lower among former smokers who quit ≥10 years earlier compared with current smokers. Smoking duration had a linear relationship with COPD (P\u3c0.001) and all three respiratory symptoms (P\u3c0.001) after adjusting for smoking status and other covariates. While COPD prevalence increased with prolonged smoking duration in both men and women, women had a higher age-adjusted prevalence of COPD in the 1–9 years, 20–29 years, and ≥30 years duration periods. CONCLUSION:These state population data confirm that prolonged tobacco use is associated with respiratory symptoms and COPD after controlling for current smoking behavior

Monarch butterflies do not place all of their eggs in one basket: oviposition on nine Midwestern milkweed species

Over the past two decades, the population of monarch butterflies east of the Rocky Mountains has experienced a significant decline in overwintering numbers. Habitat restoration that includes planting milkweeds is essential to boost monarch numbers within the breeding range. Milkweeds are the only host plants for larval monarch butterflies, but female oviposition preference for different milkweed species, especially those with overlapping ranges, is not well documented. We examined the relative inclination to lay eggs on nine milkweed species native to Iowa (no choice), and oviposition preference (choice) among the four most commonly occurring Iowa species (Asclepias incarnata, Asclepias syriaca, Asclepias tuberosa, and Asclepias verticillata). In both experiments, eggs were counted daily for four days. The milkweeds tested were Asclepias exaltata (poke milkweed), Asclepias hirtella (tall green milkweed), A. incarnata (swamp milkweed), Asclepias speciosa (showy milkweed), Asclepias sullivantii (prairie milkweed), A. syriaca (common milkweed), A. tuberosa (butterfly milkweed), A. verticillata (whorled milkweed), and Cynanchum laeve (honeyvine milkweed). When females were given only a single species on which to lay eggs, there were significant differences among milkweed species in the average number of eggs laid; A. incarnata had the highest average egg count. When females were given a choice among A. incarnata, A. syriaca, A. tuberosa, and A. verticillata, there were also differences among milkweed species in the number of eggs laid; again, A. incarnata had the highest average number of eggs laid. Additionally, females laid more total eggs when four plants of different milkweed species were available than when there were four plants of a single milkweed species. Our results show that monarch butterflies will lay eggs on all nine milkweeds, but that there are clear preferences for some milkweed species over others

Intron variants of the p53 gene are associated with increased risk for ovarian cancer but not in carriers of BRCA1 or BRCA2 germline mutations

Two biallelic polymorphisms in introns 3 and 6 of the p53 gene were analysed for a possible risk-modifying effect for ovarian cancer. Germline DNA was genotyped from 310 German Caucasian ovarian cancer patients and 364 healthy controls. We also typed 124 affected and 276 unaffected female carriers with known deleterious BRCA1 or BRCA2 germline mutation from high-risk breast-ovarian cancer families. Genotyping was based on PCR and high-resolution gel electrophoresis. German ovarian cancer patients who carried the rare allele of the MspI restriction fragment length polymorphism (RELP) in intron 6 were found to have an overall 1.93-fold increased risk (95% confidence internal (CI) 1.27–2.91) which further increased with the age at diagnosis of 41–60 years (odds ratio (OR) 2.71, 95% CI 1.10–6.71 for 41–50 and OR 2.44, 95% CI 1.12–5.28 for 51–60). The 16 bp duplication polymorphism in intron 3 was in a strong linkage to the MspI RFLP. In BRCA1 or BRCA2 mutation carriers, no difference in allele frequency was observed for carriers affected or unaffected with ovarian cancer. Our data suggest that intronic polymorphisms of the p53 gene modify the risk for ovarian cancer patients but not in carriers with BRCA1 or BRCA2 mutations. © 1999 Cancer Research Campaig

Microgeographical, inter-individual, and intra-individual variation in the flower characters of Iberian pear Pyrus bourgaeana (Rosaceae)

Flower characteristics have been traditionally considered relatively constant within species. However, there are an increasing number of examples of variation in flower characteristics. In this study, we examined the variation in attracting and rewarding flower characters at several ecological levels in a metapopulation of Pyrus bourgaeana in the Doñana area (SW Spain). We answered the following questions: what are the variances of morphological and nectar characters of flowers? How important are intra-individual and inter-individual variance in flower characters? Are there microgeographical differences in flower characters? And if so, are they consistent between years? In 2008 and 2009, we sampled flowers of 72 trees from five localities. For six flower morphological and two nectar characteristics, we calculated coefficients of variation (CV). The partitioning of total variation among-localities, among-individuals, and within-individuals was estimated. To analyze differences among localities and their consistency between years, we conducted generalized linear mixed models. The CVs of nectar characters were always higher than those of morphological characters. As expected, inter-individual variation was the main source of variation of flower morphology, but nectar characters had significant variation at both intra- and inter-individual levels. For most floral traits, there were no differences among localities. Our study documents that variation is a scale-dependent phenomenon and that it is essential to consider intra- and inter-individual variance when investigating the causes and consequences of variation. It also shows that single year studies of floral characters should be viewed with caution