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Problems related to number theory: Sum-and-distance systems, reversible square matrices and divisor functions

Hill, Sally 2018. Problems related to number theory: Sum-and-distance systems, reversible square matrices and divisor functions. PhD Thesis, Cardiff University.
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Abstract

We say that two sets $A$ and $B$, each of cardinality $m$, form an $m+m$ \emph{sum-and-distance system} $\{A,B\}$ if the sum-and-distance set $A^*B$ comprised of all the absolute values of the sums and distances $a_i\pm b_j$ contains either the consecutive odd integers $\{1,3,5,\ldots 4m^2-1\}$ or with the inclusion of the set elements themselves, the consecutive integers $\{1,2,3,\ldots,2m(m+1)\}$ (an inclusive sum-and-distance system). Sum-and-distance systems can be thought of as a discrete analogue of the union of a Minkowski sum system with a Minkowski difference system. We show that they occur naturally within a traditional reversible square matrix, where conjugation with a specific orthogonal symmetric involution, always reveals a sum-and-distance system within the block structure of the conjugated matrix. Moreover, we show that the block representation is an algebra isomorphism. Building upon results of Ollerenshaw, and Br\'ee, for a fixed dimension $n$, we establish a bijection between the set of sum-and-distance systems and the set of traditional principal reversible square matrices of size $n\times n$. Using the $j$th non-trivial divisor function $c_j (n)$, which counts the total number of proper ordered factorisations of the integer $n= p_1^{a_1}\ldots p_t^{a_t}$ into $j$ parts, we prove that the total number of $n+n$ principal reversible square matrices, and so sum-and-distance systems, $N_n$, is given by \[ N_n = \sum_{j=1}^{\Omega(n)} \left( c_j(n)^2 +c_{j+1}(n)c_j(n) \right)=\sum_{j=1}^{\Omega(n)} c_j^{(0)}(n) c_j^{(1)}(n). \] \[=\sum_{j=1}^{\Omega(n)} \left(\sum^j_{i=1}(-1)^{j-i}{j \choose i} \prod_{k=1}^t {a_k +i -1 \choose i-1}\right ) \left ( \sum^j_{i=0}(-1)^{j-i}{j \choose i} \prod_{k=1}^t {a_k +i \choose i}\right), \] where $\Omega(n)=a_1 + a_2 + \ldots + a_t$ is the total number of prime factors (including repeats) of $n$. Further relations between the divisor functions and their Dirichlet series are deduced, as well as a construction algorithm for all sum-and-distance systems of either type. Superalgebra structures relating to the matrix symmetry properties are identified, including those for the reversible and most-perfect square matrices of those considered by Ollerenshaw and Br\'ee. For certain symmetry types, links between the block representation constructed from a sum-and-distance system, and quadratic forms are also established.

Item Type: Thesis (PhD)
Date Type: Completion
Status: Unpublished
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Funders: EPSRC
Date of First Compliant Deposit: 16 May 2018
Date of Acceptance: 14 May 2018
Last Modified: 24 Jul 2018 09:42
URI: http://orca-mwe.cf.ac.uk/id/eprint/111467

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