6 papers
Integer matrix factorisations, superalgebras and the quadratic form obstruction
Nicholas J. Higham, Matthew C. Lettington, Karl Michael Schmidt
We identify and analyse obstructions to factorisation of integer matrices into products or of matrices with rational or integer entries. The obstructions arise as qua…
Divisor Functions and the Number of Sum Systems
Matthew C. Lettington, Karl Michael Schmidt
Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions which for non-neg…
Some Properties and Applications of Non-trivial Divisor Functions
S. L. Hill, M. N. Huxley, M. C. Lettington +1
The th divisor function , which counts the ordered factorisations of a positive integer into positive integer factors, is a very well-known arithmetic function; in part…
On the Structure of Additive Systems of Integers
M. N. Huxley, M. C. Lettington, K. M. Schmidt
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…
On Block Representations and Spectral Properties of Semimagic Square Matrices
S. L. Hill, M. C. Lettington, K. M. Schmidt
Using the decomposition of semimagic squares into the associated and balanced symmetry types as a motivation, we introduce an equivalent representation in terms of block-structured…
On Superalgebras of Matrices with Symmetry Properties
S. L. Hill, M. C. Lettington, K. M. Schmidt
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…