# Thomas BritzUNSW Sydney | UNSW · School of Mathematics and Statistics

Thomas Britz

PhD

## About

53

Publications

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Introduction

Thomas Britz currently works at the School of Mathematics and Statistics, UNSW Sydney. Thomas does research mostly in Combinatorics and applications thereof.

Additional affiliations

Education

July 1998 - June 2002

## Publications

Publications (53)

We present new bounds on the (normalised) closeness centrality $\bar{\mathsf{C}}_C$ of connected graphs. The main result is the fundamental and elegant new bound $\bar{\mathsf{C}}_C\bar{l}\geq 1$ relating the closeness centrality to the mean distance $\bar{l}$ of a graph. Combining this bound with known upper bounds on the mean distance, we find te...

In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod 3). We also show that 4-GDDs of type 2t8s exist for all but a finite number of feasible values of s and t. The largest unknown case has type 24818 and has 152 points. A number of 4-GDDs with at most 50 points are also constructed. These include one of t...

In this paper we provide a 4‐GDD of type 2 2 5 5 ${2}^{2}{5}^{5}$, thereby solving the existence question for the last remaining feasible type for a 4‐GDD with no more than 30 points. We then show that 4‐GDDs of type 2 t 5 s ${2}^{t}{5}^{s}$ exist for all but a finite specified set of feasible pairs ( t , s ) $(t,s)$.

Extending previous results in the literature, random colored substitution networks are introduced and are proved to be scale-free under natural conditions. Furthermore, the asymptotic node degrees, arc cardinalities and node cardinalities for these networks are derived. These results are achieved by proving stronger results regarding stochastic sub...

In this paper we provide a $4$-GDD of type $2^2 5^5$, thereby solving the existence question for the last remaining feasible type for a $4$-GDD with no more than $30$ points. We then show that $4$-GDDs of type $2^t 5^s$ exist for all but a finite specified set of feasible pairs $(t,s)$.

We extend and provide new proofs of the Wei-type duality theorems, due to Ducoat and Ravagnani, for Gabidulin–Roth rank-metric codes and for Delsarte rank-metric codes. These results follow as corollaries from fundamental Wei-type duality theorems that we prove for certain general combinatorial structures.

For trace class operators $A, B \in \mathcal{B}_1(\mathcal{H})$ ($\mathcal{H}$ a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form \[ {\det}_{\mathcal{H}} ((I_{\mathcal{H}} - A) (I_{\mathcal{H}} - B)) = {\det}_{\mathcal{H}} (I_{\mathcal{H}} - A) {\det}_{\mathcal{H}} (I_{\mathcal{H}} - B). \]...

A frequency square is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only binary frequency squares of order $n$ with $n/2$ zeros and $n/2$ ones in each row and column. Two such frequency squares are orthogonal if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally o...

It’s very likely that you, dear Reader, have heard of the Fibonacci numbers, and it’s likely that you like them. But it’s also likely that you don’t know much about these numbers: most people don’t, and there is a lot to know about them. It is therefore my pleasure here to present to you some less-known facts about these Fibonacci numbers.

A \emph{frequency square} is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only {\em binary} frequency squares of order $n$ with $n/2$ zeroes and $n/2$ ones in each row and column. Two such frequency squares are \emph{orthogonal} if, when superimposed, each of the 4 possible ordered pairs of ent...

This is a 2019 calendar with each month being represented by a bridge from around the world. Each bridge has historical, cultural - and mathematical relevance, and each is given a (very) simple real-life maths problem for readers to enjoy and solve.

Simple mathematical expressions are given for the betweenness centrality of nodes in trees, forests and cycles. As application, a centrality test is given for when a network might be a forest.

Objectives:
To investigate the organisation and characteristics of general practice in Australia by applying novel network analysis methods to national Medicare claims data.
Design:
We analysed Medicare claims for general practitioner consultations during 1994-2014 for a random 10% sample of Australian residents, and applied hierarchical block m...

Objectives: To investigate the organisation and characteristics of general practice in Australia by applying novel network analysis methods to national Medicare claims data. Design: We analysed Medicare claims for general practitioner consultations during 1994-2014 for a random 10% sample of Australian residents, and applied hierarchical block mode...

Mubayi's Conjecture states that if $\mathcal{F}$ is a family of $k$-sized subsets of $[n] = \{1,\ldots,n\}$ which, for $k \geq d \geq 2$, satisfies $A_1 \cap\cdots\cap A_d \neq \emptyset$ whenever $|A_1 \cup\cdots\cup A_d| \leq 2k$ for all distinct sets $A_1,\ldots,A_d \in\mathcal{F}$, then $|\mathcal{F}|\leq \binom{n-1}{k-1}$, with equality occurr...

The current paper aimed to investigate the effectiveness of five law enforcement interventions in disrupting and dismantling criminal networks. We tested three law enforcement interventions that targeted social capital in criminal networks (betweenness, degree and cut-set) and two interventions that targeted human capital (actors who possess money...

Restoring missing ecological interactions by reintroducing locally extinct species or ecological surrogates for extinct species has been mooted as an approach to restore ecosystems. Australia's apex predator, the dingo, is subject to culling in order to prevent attacks on livestock. Dingo culling has been linked to ecological cascades evidenced by...

The critical exponent of a matroid is one of the important parameters in
matroid theory and is related to the Rota and Crapo's Critical Problem. This
paper introduces the covering dimension of a linear code over a finite field,
which is analogous to the critical exponent of a representable matroid. An
upper bound on the covering dimension is conjec...

We prove that each maximal partial Latin cube must have more than 29.289% of its cells filled and show by construction that this is a nearly tight bound. We also prove upper and lower bounds on the number of cells containing a fixed symbol in maximal partial Latin cubes and hypercubes, and we use these bounds to determine for small orders n the num...

Just like any other cultural group, mathematicians like to tell stories. We tell heroic stories about famous mathematicians, to inspire or reinforce our cultural values, and we encase our results in narratives to explain how they are interesting and how they relate to other results. We also tell stories to convince others that our results are valid...

It is shown that each chain of linear codes has an associated demi-matroid, a combinatorial structure that extends the notion of a vector matroid of a linear code. These demi-matroids are proven to determine important properties of the chain, and it is shown that linear code chain duality is represented by demi-matroid duality in a natural and yet...

Quasi-uniform random vectors have probability distributions that are uniform over their projections. They are of fundamental interest because a linear information inequality is valid if and only if it is satisfied by all quasi-uniform random vectors. In this paper, we investigate properties of codes induced by quasi-uniform random vectors. We prove...

We present a construction of demi-matroids, a generalization of matroids, from linear codes over finite Frobenius rings, as well as a Greene-type identity for rank generating functions of demi-matroids. We also prove a MacWilliams-type identity for Hamming support enumerators of linear codes over finite Frobenius rings. As a special case, these res...

We present several fundamental duality theorems for matroids and more general combinatorial structures. As a special case,
these results show that the maximal cardinalities of fixed-ranked sets of a matroid determine the corresponding maximal cardinalities
of the dual matroid. Our main results are applied to perfect matroid designs, graphs, transve...

It is proved that the set of higher weight enumerators of a linear code over a finite field is equivalent to the Tutte polynomial associated to the code. An explicit expression for the Tutte polynomial is given in terms of the subcode weights. Generalizations of these results are proved and are applied to codeword m -tuples. These general results a...

Quasi-uniform random variables have probability distributions that are uniform over their supports. They are of fundamental interest because a linear information inequality is valid if and only if it is satisfied by all quasi-uniform random variables. In this paper, we investigate properties of codes induced by quasi-uniform random variables.We pro...

Wei's celebrated Duality Theorem is generalized in several ways, expressed as duality theorems for linear codes over division rings and, more generally, duality theorems for matroids. These results are further generalized, resulting in two Wei-type duality theorems for new combinatorial structures that are introduced and named {\em demi-matroids}....

This paper presents new connections between designs and matroids. We generalize the Assmus-Mattson theorem and another coding-theoretical theorem with respect to matroids, and thereby present new sufficient conditions for obtaining $t$-designs from matroids. These conditions may be relaxed for some self-dual matroids. We use our results to prove ne...

We consider the coboundary polynomial for a matroid as a generalization of the weight enumerator of a linear code. By describing properties of this polynomial and of a more general polynomial, we investigate the matroid analogue of the MacWilliams identity. From coding-theoretical approaches, upper bounds are given on the size of circuits and cocir...

We present new constructions of t-designs by considering subcode supports of linear codes over finite fields. In particular, we prove an Assmus-Mattson type
theorem for such subcodes, as well as an automorphism characterization. We derive new t-designs (t ≤ 5) from our constructions.

The purpose of this paper is to provide links between matroid theory and the theory of subcode weights and supports in linear codes. We describe such weights and supports in terms of certain matroids arising from the vector matroids associated to the linear codes. Our results generalize classical results by Whitney, Tutte, Crapo and Rota, Greene, a...

A matrix is free, or generic, if its nonzero entries are algebraically independent. Necessary and sufficient combinatorial conditions are presented for a complex free matrix to have a free Moore-Penrose inverse. These conditions extend previously known results for square, nonsingular free matrices. The result used to prove this characterization rel...

An efficient algorithm is presented for calculating higher weight enumerators of linear codes given generator matrices. By this algorithm, the higher weight enumerators of the unique doubly-even, self-dual code are calculated. The algorithm is based on a previously shown relationship between Tutte polynomials and higher weight enumerators.

Jang and Park asked in [On a MacWilliams type identity and a perfectness for a binary linear (n, n - 1, j)-poset code, Discrete Math. 265 (2003) 85-104] whether, for each poset P = {l...., n}, the P-weights and P-distances satisfy the inequalities w(P)(x) - w(P)(y) <= d(P) (x, y) <= w(P) (x) + w(P) (y) - w(P) (xy) for all vectors x, y is an element...

The Critical Theorem, due to Henry Crapo and Gian-Carlo Rota, has previously been extended or generalised in a number of different ways. The main result of the present paper is a general form of the Critical Theorem that encompasses many of these results. Applications include generalisations of a theorem by Curtis Greene that describes how the weig...

We show by example that the covering radius of a binary linear code is not generally determined by the Tutte polynomial of the matroid. This answers Problem 361 (P.J. Cameron (Ed.), Research problems, Discrete Math. 231 (2001) 469–478).

Bipartite graphs are used todescribe the generalized Schur co mplements o f real matrices having nosquare submatrix with twoo r mo re no nzerodiago nals. Fo r any matrix A with this property, including any nearly reducible matrix, the sign pattern of each generalized Schur complement is shown to be determined uniquely by the sign pattern of A. More...

The Moore–Penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs. For such a matrix, the sign of every entry of the Moore–Penrose inverse is shown to be determined uniquely by the signs of the matrix entries; i.e., the matrix has a signed generalized inverse. Necessary and su...

An n is spectrally arbitrary if given any self-conjugate spectrum there exists a matrix realization of with that spectrum. If replacing any nonzero entry (or entries) by zero destroys this property, then is a minimal spectrally arbitrary sign pattern. For n 3, several families of nn spectrally arbitrary sign patterns are presented, and their minima...

The novel concept of a cyclic sequence of a digraph that has precisely one factor is defined, and is used to characterize the entries of the inverse of a matrix with such a digraph. This leads to a characterization of a strongly sign-nonsingular matrix in terms of cyclic sequences. Nonsingular nearly reducible matrices are a well-known class of mat...

The novel concept of a cyclic sequence of a digraph that has precisely one factor is defined, and is used to characterize the entries of the inverse of a matrix with such a digraph. This leads to a characterization of a strongly sign-nonsingular matrix in terms of cyclic sequences. Nonsingular nearly reducible matrices are a well-known class of mat...

An n,n sign pattern A is spectrally arbitrary if given any self-conjugate spectrum there exists a matrix realization of A with that spectrum. If replacing any nonzero entry (or entries) of A by zero destroys this property, then A is a minimal spectrally arbitrary sign pattern. For n 3, several families of n n spectrally arbitrary sign patterns are...

The mathematical analysis of the consistency of Feldberg's simple BDF start in electrochemical digital simulation is presented. The method leads to rather accurate results compared with the more obvious rational start in which the BDF order is worked up from a two-point start, increasing the number of points in time until the desired number is reac...

this article, we bring together the main results in the subject, along with a number of ramifications and corollaries. The main idea is to develop the theory---from scratch and with complete proofs---by consistently applying the poset-theoretic viewpoint. In this spirit, we begin our presentation in Section 2 by stating the fundamental theorem of C...

We present generalisations of several MacWilliams type identities, including those by Klve and Shiromoto, and of the theorems of Greene and Barg that describe support weight enumerators of the code. One of our main tools is a generalisation of a decomposition theorem due to Brylawski. 1

The inverse matrix M−1 of a non-singular free matrix M need not be free. In this paper, we present several necessary and sufficient conditions for M−1 to be free.

Throughout the history of mathematics, the notion of an equivalence relation has played a fundamental role. It dates back at least to the time when the natural numbers first were introduced: a non-negative integer may be thought of as a representative of the equivalence class of sets with the same cardinality. To express such a simple and “obvious”...

this article, we bring together the main results in the subject, along with a number of ramifications and corollaries. The main idea is to develop the theory---from scratch and with complete proofs---by consistently applying the poset-theoretic viewpoint. In this spirit, we begin our presentation in Section 2 by stating the fundamental theorem of C...

Bipartite graphs and digraphs are used to describe algebraic operations on a free matrix, including Moore-Penrose inversion, finding Schur complements, and normalized LU factorization. A description of the structural properties of a free matrix and its Moore-Penrose inverse is proved, and necessary and sufficient conditions are given for the Moore-...

## Projects

Projects (2)

The project aims to examine through the current research outcomes on centrality measures. It is a thesis project under the supervision of Dr. Thomas Britz in UNSW Australia. Thanks to his kind and ground academic support, this project is able to grow and flourish.
Our research detects various mathematical properties of centrality measures and investigate their behaviours on different graph structures which offers a comprehensive vision of the inner link between centrality measures and their underlying graph structures.