Smart Geodesic Tracing in GRworkbench

We have developed a new tool for numerical work in General Relativity: GRworkbench. We discuss how GRworkbench's implementation of a numerically-amenable analogue to Differential Geometry facilitates the development of robust and chart-independent numerical algorithms. We consider, as an example, geodesic tracing on two charts covering the exterior Schwarzschild space-time.     

Latin Squares without Orthogonal Mates

In 1779 Euler proved that for every even n there exists a latin square of order n that has no orthogonal mate, and in 1944 Mann proved that for every n of the form 4k + 1, k ≥ 1, there exists a latin square of order n that has no orthogonal mate. Except for the two smallest cases, n = 3 and n = 7, it is not known whether a latin square of order n = 4k + 3 with no orthogonal mate exists or not. We complete the determination of all n for which there exists a mate-less latin square of order n by proving that, with the exception of n = 3, for all n = 4k + 3 there exists a latin square of order n with no orthogonal mate. We will also show how the methods used in this paper can be applied more generally by deriving several earlier non-orthogonality results.     

Block transitive Steiner systems with more than one point orbit

For all ‘reasonable’ finite t, k, and s, we construct a t‐(?₀, k, 1) design and a group of automorphisms which is transitive on blocks and has s orbits on points. In particular, there is a 2‐(?0, 4, 1) design with a block‐transitive group of automorphisms having two point orbits. This answers a question of P. J. Cameron and C. E. Praeger. The construction is presented in a purely combinatorial way, but is a by‐product of a new way of looking at a model‐theoretic construction of E. Hrushovski. © 2004 Wiley Periodicals, Inc.     

Isosteres of peptides: boron analogs as dipolar forms of α‐amino acids – a theoretical study

Modification of peptides to produce peptidomimetics is of great interest, with the aim of designing potent, selective, and metabolically stable analogs having certain conformational properties. Organoboranes have been reported in the literature with a wide range of therapeutic applications. One of the therapeutically important class of molecules is amine‐carboxyboranes derived from amino acids by replacement of the Cα atom of an amino acid/peptide by boron. The conformational preferences of these peptides, with respect to backbone ω, ?, and ψ torsion angles, have been investigated by theoretical calculations. The amide bond in these molecules has the same geometry in the ground and transition states as the natural peptides. However, the boron isosteres of glycine and alanine peptides are less structured than their natural derivatives, but are distinguished by structures with a positive value for the ? angle, which is normally disfavored for natural peptides. This property could be used to build peptides with a geometry not usually seen in natural peptides. Copyright © 2007 John Wiley & Sons, Ltd.     

The effect of rapid rotation on a vertical bridgman furnace at large Rayleigh number

In quasi-steady operation, convection currents in a Bridgmandevice, used for producing a semi-conductor crystal, createinhomogeneities that may make the crystal unusable. It has oftenbeen suggested that additional forces due to rotation or magnetismmight be efficacious in reducing the segregation of the elementsof the alloy. It has been found that, over a wide range of rotationrates, there is no improvement in performance due to rotationabout the vertical axis. However, numerical results that havebeen obtained previously (Lee & Pearlstein, J. Crys. Growth240, 2002) indicate that, when effects of centrifugal buoyancyare introduced, a substantial reduction in segregation is achieved.In the work reported here, by contrast, in which we extend previouslarge-Rayleigh-number asymptotic analysis to include centrifugalbuoyancy, we find no improvement in radial segregation, butrather increasing segregation with increasing rotation rate.     

A note on an SOR-like method for augmented systems

Golub et al. (2001, BIT, 41, 71–85) gave a generalizedsuccessive over-relaxation method for the augmented systems.In this paper, the connection between the SOR-like method andthe preconditioned conjugate gradient (PCG) method for the augmentedsystems is investigated. It is shown that the PCG method isat least as accurate (fast) as the SOR-like method. Numericalexamples demonstrate that the PCG method is much faster thanthe SOR-like method.