Orthogonal designs II

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1,k) and ordern exist for everyk < n whenn = 2 ^t+2?3 andn = 2 ^t+2?5 (wheret is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n. Coupled with some results of earlier work, this means that theweighing matrix conjecture ‘For every ordern ≡ 0 (mod 4) there is, for eachk ?n, a square {0, 1, ? 1} matrixW = W(n, k) satisfyingWW ^t =kIn’ is resolved in the affirmative for all ordersn = 2^t+1?3,n = 2^t+1?5 (t a positive integer). The fact that the matrices we find are skew-symmetric for allk < n whenn ≡ 0 (mod 8) and because of other considerations we pose three other conjectures about weighing matrices having additional structure and resolve these conjectures affirmatively in a few cases. In an appendix we give a table of the known results for orders ? 64.     

Hadamard matrices of order 28m, 36m and 44m

We show that if four suitable matrices of order m exist then there are Hadamard matrices of order 28m, 36m, and 44m. In particular we show that Hadamard matrices of orders 14(q + 1), 18(q + 1), and 22(q + 1) exist when q is a prime power and q ≡ 1 (mod 4).Also we show that if n is the order of a conference matrix there is an Hadamard matrix of order 4mn.As a consequence there are Hadamard matrices of the following orders less than 4000: 476, 532, 836, 1036, 1012, 1100, 1148, 1276, 1364, 1372, 1476, 1672, 1836, 2024, 2052, 2156, 2212, 2380, 2484, 2508, 2548, 2716, 3036, 3476, 3892.All these orders seem to be new.     

Menthyloxycarbonyl phosphonium chlorides: new derivatives for determining the enantiomeric excess of chiral tertiary phosphines

The direct alkoxycarbonylation of tertiary phosphines with alkyl chloroformates has been examined and the products characterised by NMR and X-ray crystallographic analysis. The derivatisation of representative chiral tertiary phosphines with enantiopure menthyl chloroformate allows the determination of the enantiomeric excess of scalemic mixtures by ³¹P NMR spectroscopic analysis.     

HETEROCYCLIC CATIONS WITH DELOCALISED PI-SYSTEMS AND SHORT INTRAMOLECULAR SULFUR… SULFUR CONTACTS: THE STRUCTURES OF TWO DERIVATIVES OF THE 2-(1,3-DITHIOL-2-YLIDENEMETHYL)-1,3-DITHIOLIUM CATION

Abstract

The effective size of bonded divalent sulfur in S…S contacts is a function of the orientation of the groups, with the shortest contacts occurring when the groups are coplanar. A lower limit for the loner is indicated by the structures of the 2-(1,3-benzodithiol-2-ylidenemethyl)-1,3-benzodithiolium and 2-(1,3-dithiolan-2-ylidenemethyl)-1,3-dithiolanium cations, 2 and 3. These show almost mm symmetry with all four sulfur atoms involved in stabilisation of the positive charge. Short intramolecular sulfur… sulfur contact distances, 0.5–0.7 Å within the sum of traditional van der Waals radii, and maximised by in-plane angular distortions, indicate a lower limit to the effective size of the bonded divalent sulfur atom of ca. 1.45 Å.     

Tracking accuracy of a semi‐Lagrangian method for advection–dispersion modelling in rivers

There is an increasing need to improve the computational efficiency of river water quality models because: (1) Monte‐Carlo‐type multi‐simulation methods, that return solutions with statistical distributions or confidence intervals, are becoming the norm, and (2) the systems modelled are increasingly large and complex. So far, most models are based on Eulerian numerical schemes for advection, but these do not meet the requirement of efficiency, being restricted to Courant numbers below unity. The alternative of using semi‐Lagrangian methods, consisting of modelling advection by the method of characteristics, is free from any inherent Courant number restriction. However, it is subject to errors of tracking that result in potential phase errors in the solutions. The aim of this article is primarily to understand and estimate these tracking errors, assuming the use of a cell‐based backward method of characteristics, and considering conditions that would prevail in practical applications in rivers. This is achieved separately for non‐uniform flows and unsteady flows, either via theoretical considerations or using numerical experiments. The main conclusion is that, tracking errors are expected to be negligible in practical applications in both unsteady flows and non‐uniform flows. Also, a very significant computational time saving compared to Eulerian schemes is achievable. Copyright © 2006 John Wiley & Sons, Ltd.     

Squares,clique graphs,and chordality

We study the squares and the clique graphs of chordal graphs and various special classes of chordal graphs. Chordality conditions for squares and clique graphs are given. Several theorems concering chordal graphs are generalized. © 1996 John Wiley & Sons, Inc.     

Rotational predissociation of extremely weakly bound atom-molecule complexes produced by Feshbach resonance association

We study the rotational predissociation of atom-molecule complexes with very small binding energy. Such complexes can be produced by Feshbach resonance association of ultracold molecules with ultracold atoms. Numerical calculations of the predissociation lifetimes based on the computation of the energy dependence of the scattering matrix elements become inaccurate when the binding energy is smaller than the energy width of the predissociating state. We derive expressions that represent accurately the predissociation lifetimes in terms of the real and imaginary parts of the scattering length and effective range for molecules in an excited rotational state. Our results show that the predissociation lifetimes are the longest when the binding energy is positive, i.e., when the predissociating state is just above the excited state threshold.     

Radial oscillations of neutron stars in strong magnetic fields

The eigen frequencies of radial pulsations of neutron stars are calculated in a strong magnetic field. At low densities we use the magnetic BPS equation of state (EOS) similar to that obtained by Lai and Shapiro while at high densities the EOS obtained from the relativistic nuclear mean field theory is taken and extended to include strong magnetic field. It is found that magnetized neutron stars support higher maximum mass whereas the effect of magnetic field on radial stability for observed neutron star masses is minimal.     

Laser cooling of antihydrogen

We present a short review of the essential techniques of cooling free atoms by resonant laser radiation. The different contributions to the light forces are explained and their application to the problem of damping the thermal motion of free atoms is described. Due to quantum mechanical fluctuations of the light force there exists a limit temperature for a given atomic transition. Deceleration of atomic beams by the radiation pressure demands techniques to maintain the resonance condition while the Doppler shift of the decelerated atom is rapidly changing. Radiation forces may serve to compress and deflect slow atomic beams as well as to trap cold atoms. The possible use of pulsed laser radiation is discussed.