On Superalgebras of Matrices with Symmetry Properties

It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for square matrices give rise to similar superalgebra structures, pointing to novel symmetry types in their complementary parts. In particular, this provides a unifying framework for the composite `most perfect square' symmetry and the related class of `reversible squares'; moreover, the semi-magic square algebra is identified as part of a 2-gradation of the general square matrix algebra. We derive explicit representation formulae for matrices of all symmetry types considered, which can be used to construct all such matrices.Comment: 25 page

On the structure of additive systems of integers

A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an analogous concept where only sums of elements are considered. We establish a bijection between sum systems and sum-and-distance systems of corresponding size, and show that sum systems are equivalent to principal reversible cuboids, which are tensors with integer entries and a symmetry of ‘reversible square’ type. We prove a structure theorem for principal reversible cuboids, which gives rise to an explicit construction formula for all sum systems in terms of joint ordered factorisations of their component set cardinalities

Some properties and applications of non-trivial divisor functions

The jth divisor function dj , which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative jth non-trivial divisor function cj , which counts the ordered factorisations of a positive integer into j factors each of which is greater than or equal to 2, is rather less well studied. Additionally, we consider the associated divisor function c(r)j , for r≥0 , whose definition is motivated by the sum-over divisors recurrence for dj . We give an overview of properties of dj , cj and c(r)j , specifically regarding their Dirichlet series and generating functions as well as representations in terms of binomial coefficient sums and hypergeometric series. Noting general inequalities between the three types of divisor function, we then observe how their ratios can be expressed as binomial coefficient sums and hypergeometric series, and find explicit Dirichlet series and Euler products for some of these. As an illustrative application of the non-trivial and associated divisor functions, we show how they can be used to count principal reversible square matrices of the type considered by Ollerenshaw and Brée and so sum-and-distance systems of integers

Structural-Properties Of Amorphous Hydrogenated Carbon .1. A High-Resolution Neutron-Diffraction Study

The structure of samples of amorphous hydrogenated carbon, prepared from acetylene and propane precursors, containing 35 and 32 at.% hydrogen, respectively, was investigated by time-of-flight neutron diffraction in the range 0.2-50 angstrom-1 using the ISIS spallation source. The large dynamic range of the data ensures a real-space resolution sufficient to reveal directly the proportions of sp2 and sp3 hybridized carbon. The results show that, in these hard carbon materials, the carbon-atom sites are predominantly sp2 bonded, and the carbon-carbon single bond:carbon-carbon double bond ratio is about 2.5:1. The detailed information on atomic correlations thus provided is used to discuss current structural models, and in particular the data are used to show that these models require significant modification

A spectroscopic study of the structure of amorphous hydrogenated carbon

A range of amorphous hydrogenated carbon (a-C:H) samples have been studied using inelastic neutron spectroscopy (INS) and Fourier transform infrared (FTIR) spectroscopy. Using these complementary techniques, the bonding environments of both carbon and hydrogen can be probed in some detail, with the INS data providing not only qualitative but also quantitative information. By comparing the data from each of the samples we have been able to examine the effects of different deposition conditions, i.e. precursor gas, deposition energy and deposition method, on the atomic-scale structure of a-C:H

Optical Properties of Single-Crystal Zinc

Measurements were made of the absorptivity of single crystals of zinc from 0.1 to 3.0 eV at 4.2K. Polarized radiation was used with the electric field vector parallel or perpendicular to the c axis of the crystal. New structure was found for E⃗ ⊥c^at about 0.15 eV; no low-energy structure was observed for E⃗ ∥c^. The low-energy interband transition is attributed to transitions near the point K in the Brillouin zone. The structure in the near-infrared and visible spectra is highly temperature dependent. Comparisons of the absorptivity and the conductivity are made between our 4.2-K data and the data of Rubloff for 77 and 300 K and with the calculations of Kasowski. Thermomodulation measurements on basal-plane samples are described which show the conductivity doublet as originating from two distinct infrared-absorption structures. The low-energy absorptivity is found to be nearly constant for E⃗ ∥c^. An optical effective-mass component m∗∥of 2.0m0 is computed from theories of the anomalous skin effect.This article is from Physical Review B 5 (1972): 2829, doi:10.1103/PhysRevB.5.2829. Posted with permission.</p