Bernd Sing

An Analysis of Low- and High-Frequency Summer Climate Variability around the Caribbean Antilles

This study contrasts the pattern of low-frequency (LF) and high-frequency (HF) climate variability in the eastern Caribbean. A low-pass Butterworth filter is used to study oscillations in rainfall and regional SST on time scales of greater and less than 8 yr in the period 1901‐2002. The results show that the southern and northern Antilles are dominated by HF variability, whereas rainfall fluctuations in the eastern Antilles oscillate at quasi-decadal periods over the 102-yr record. In the southern Antilles, the HF rainfall signal derives from a late-summer responseto the ENSO phase: warm and dry versus cool and wet. In the northern Antilles, the HF signal relates to a combination of an ENSO and North Atlantic Oscillation (NAO) phase: a warm ENSO and a negative NAO bring wetter conditions, while a cool ENSO and a positive NAO bring drier conditions. The early rainfall LF signal in SST is characterized by a dipole between the North Atlantic and South Atlantic and is associated with cross-equatorial winds that promote convection in the Caribbean. The study analyzes the upper-ocean structure—in particular, a low (high) salinity signal in the tropical North Atlantic (North Pacific) that relates to LF (HF) climate variability.

Kolakoski-(3,1) is a (deformed) model set

Unlike the (classical) Kolakoski sequence on the alphabet {1,2}, its analogue on {1,3} can be related to a primitive substitution rule. Using this connection, we prove that the corresponding bi-infinite fixed point is a regular generic model set and thus has a pure point diffraction spectrum. The Kolakoski-(3,1) sequence is then obtained as a deformation, without loosing the pure point diffraction property.

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 inRd, 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 l. 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 l-adic internal space, we add an approach to “visualize” such internal spaces.

Diffraction Spectrum of Lattice Gas Models above Tc

The diffraction spectra of lattice gas models on ℤd with finite-range ferromagnetic two-body interactions above Tc or with certain rates of decay of the potential are considered. We show that these diffraction spectra almost surely exist, are ℤd-periodic and consist of a pure point part and an absolutely continuous part with continuous density.

Deformed model sets and distorted Penrose tilings

Abstract In this article, we show how the mathematical concept of a deformed model set can be used to gain insight into the diffraction pattern of quasicrystalline structures. We explain what a deformed model set is, what its characteristic features are and how it relates to certain disorder phenomena in solids. We then apply this concept to distorted Penrose tilings, i.e., Penrose tilings where we apply size-effect-like distortions. While the size effect in crystals only operates on the diffuse scattering, there is also an intensity transfer on the Bragg peaks in distorted Penrose tilings. The persistence of pure point diffraction in distorted Penrose tilings can be explained by interpreting such tilings as deformed model sets.

Deformed Penrose tilings

Monte Carlo (MC) simulation of a model quasicrystal (2D Penrose rhomb tiling) shows that the kinds of local distortions that result from size-effect-like relaxations are in fact very similar to mathematical constructions called deformed model sets. Of particular interest is the fact that these deformed model sets are pure point-diffractive, i.e. they give diffraction patterns that have sharp Bragg peaks and no diffuse scattering. Although the aforementioned MC simulations give diffraction patterns displaying some diffuse scattering, this can be attributed to the fact that the simulations include a certain amount of unavoidable randomness. Examples of simple deformed model sets have been constructed by simple prescription and hence contain no randomness. In this case the diffraction patterns show no diffuse scattering. It is demonstrated that simple deformed model sets can be constructed, based on the 2D Penrose rhomb tiling, by using deformations which are defined in terms of the local environment of each vertex. The resulting model sets are all topologically equivalent to the Penrose tiling (same connectedness), are perfectly quasicrystalline but show an enormous variation in the Bragg peak intensities. For the examples described, which are based on nearest-neighbour environments, 12 deformation parameters may be chosen independently. If more distant neighbours are taken into account further sets of parameters may be defined. One example of these simple deformed tilings, which shows great similarity to the size-effect-distorted tiling, is discussed in detail.

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 the studies of Ray and Sahai (1980), Sahai and Ray (1980), A. Sahai and A. Sahai (1985), and Singh and Espejo (2003).The bias and mean squared 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 squared 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.

Generalizing Krawtchouk Polynomials Using Hadamard Matrices

We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged in a square matrix; in particular, the case where this matrix is a Hadamard matrix is considered. Orthogonality relations and recurrence relations are established, and coefficients for the expansion of any polynomial in terms of m-polynomials are obtained. We conclude this paper by an implementation of m-polynomials and some of the results obtained for them in Mathematica.

Manganese oxide tunnel structures determined with TEM

Complex manganese oxides AxMnO2 (x < 1) show a tendency to form complex tunnel structures. The major structural unit of the tunnel walls is a rutile-type chain of edge sharing MnO6 octahedra. A variety of structures arises due to the ability of these chains to form double and triple chain blocks by sharing octahedral edges. We will present the structure determination of three such tunnel compounds with approximately the same A/Mn ratio, SrMn3O6 [1], CaMn3O6 and Sr0.9Mn3(O,F)6, where the shape of the channels and the ordering pattern of the A-cations vary. We have solved their structures with transmission electron microscopy (TEM) and then refined them from XRD and/or NPD data. The HREM images taken along the direction of the tunnel propagation clearly show that the three compounds have different shapes of tunnels. In the case of SrMn3O6 the tunnel structure is of a ’’figure-of-eight’’ shape. The electron diffraction patterns of SrMn3O6 show satellite reflections due to an incommensurate modulation caused by the ordered Sr-distribution over the available positions in the tunnels, only 67% of the tunnel sites are filled with cations. This ordering also affects the positions of the oxygen, making one Mn-O distance substantially longer, resulting in a pyramidal instead of an octahedral coordination for some of the Mn-cations. Local areas were found with a range of different modulations, corresponding to slightly different Sr-contents. Although it has nominally the same composition as SrMn3O6, the CaMn3O6 structure contains smaller six-sided tunnels comparable to those of CaFe2O4. Each tunnel comprises a single string of the Ca-cations in which every third A-position is vacant. The empty sites in neighbouring Ca-strings are shifted relative to each other by one repeat period along the c-axis of the CaFe2O4 subcell giving rise to a monoclinic distortion. The analysis of the Mn-O distances allows to speculate that there is Mn +3 /Mn +4 charge ordering with the Mn +4 cations located near the cation vacancies in the A-sublattice. Introducing fluorine into the anion sublattice drastically changes the connectivity scheme of the octahedral strings. The Sr0.9Mn3(O,F)6 compound shows large todorokite-type 3x3 channels in which four Sr strings are placed. The electron diffraction patterns show satellite reflections indicating a composite structure with two different repeat periods for the Sr- and octahedral sublattices due to the ordering of Sr in the tunnels. Replacing Sr by Ca or adding F to the compound thus results in different structures, which will be compared and discussed.