On the direct product of two weak uniquely completable partial latin squares

In this note we show by counter-example that the direct product of two weak uniquely completable partial latin squares is not necessarily a uniquely completable partial latin square. This counter-example rejects a conjecture by Gower (see [3]) on the direct product of two uniquely completable partial latin squares.

On the intersection problem for 1-factorizations and near 1-factorizations of K-v

The number of 1-factors (near 1-factors) that mu 1-factorizations (near 1-factorizations) of the complete graph K-v, v even (v odd), can have in common, is studied. The problem is completely settled for mu = 2 and mu = 3.

Automatic construction of explicit R matrices for the one-parameter families of irreducible typical highest weight (0(m)vertical bar alpha(n)) representations of U-q[gl(m vertical bar n)]

We detail the automatic construction of R matrices corresponding to (the tensor products of) the (O-m\alpha(n)) families of highest-weight representations of the quantum superalgebras Uq[gl(m\n)]. These representations are irreducible, contain a free complex parameter a, and are 2(mn)-dimensional. Our R matrices are actually (sparse) rank 4 tensors, containing a total of 2(4mn) components, each of which is in general an algebraic expression in the two complex variables q and a. Although the constructions are straightforward, we describe them in full here, to fill a perceived gap in the literature. As the algorithms are generally impracticable for manual calculation, we have implemented the entire process in MATHEMATICA; illustrating our results with U-q [gl(3\1)]. (C) 2002 Published by Elsevier Science B.V.

Link invariants associated with gauge equivalent solutions of the Yang-Baxter equation: The one-parameter family of minimal typical representations of Uq[gl(2|1)]

In this paper we investigate the construction of state models for link invariants using representations of the braid group obtained from various gauge choices for a solution of the trigonometric Yang-Baxter equation. Our results show that it is possible to obtain invariants of regular isotopy (as defined by Kauffman) which may not be ambient isotopic. We illustrate our results with explicit computations using solutions of the trigonometric Yang-Baxter equation associated with the one-parameter family of minimal typical representations of the quantum superalgebra U-q,[gl(2/1)]. We have implemented MATHEMATICA code to evaluate the invariants for all prime knots up to 10 crossings.

Decompositions of complete graphs into theta graphs with fewer than ten edges

A theta graph is a graph consisting of three pairwise internally disjoint paths with common end points. Methods for decomposing the complete graph K-nu into theta graphs with fewer than ten edges are given.

Large sets of large sets of Steiner triple systems of order 9

We describe a direct method of partitioning the 840 Steiner triple systems of order 9 into 120 large sets. The method produces partitions in which all of the large sets are isomorphic and we apply the method to each of the two non-isomorphic large sets of STS(9).

Computational properties of min/max autocorrelation factors

Minimum/maximum autocorrelation factor (MAF) is a suitable algorithm for orthogonalization of a vector random field. Orthogonalization avoids the use of multivariate geostatistics during joint stochastic modeling of geological attributes. This manuscript demonstrates in a practical way that computation of MAF is the same as discriminant analysis of the nested structures. Mathematica software is used to illustrate MAF calculations from a linear model of coregionalization (LMC) model. The limitation of two nested structures in the LMC for MAF is also discussed and linked to the effects of anisotropy and support. The analysis elucidates the matrix properties behind the approach and clarifies relationships that may be useful for model-based approaches. (C) 2003 Elsevier Science Ltd. All rights reserved.

Further results on strongbox secured secret sharing schemes

We extend our earlier work on ways in which defining sets of combinatorial designs can be used to create secret sharing schemes. We give an algorithm for classifying defining sets or designs according to their security properties and summarise the results of this algorithm for many small designs. Finally, we discuss briefly how defining sets can be applied to variations of the basic secret sharing scheme.

A numerical approach to modeling the catalytic voltammetry of surface-confined redox enzymes

A finite difference method for simulating voltammograms of electrochemically driven enzyme catalysis is presented. The method enables any enzyme mechanism to be simulated. The finite difference equations can be represented as a matrix equation containing a nonlinear sparse matrix. This equation has been solved using the software package Mathematica. Our focus is on the use of cyclic voltammetry since this is the most commonly employed electrochemical method used to elucidate mechanisms. The use of cyclic voltammetry to obtain data from systems obeying Michaelis-Menten kinetics is discussed, and we then verify our observations on the Michaelis-Menten system using the finite difference simulation. Finally, we demonstrate how the method can be used to obtain mechanistic information on a real redox enzyme system, the complex bacterial molybdoenzyme xanthine dehydrogenase.