Group law computations on Jacobians of hyperelliptic Curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form.

Fast formulas for computing cryptographic pairings

The most powerful known primitive in public-key cryptography is undoubtedly elliptic curve pairings. Upon their introduction just over ten years ago the computation of pairings was far too slow for them to be considered a practical option. This resulted in a vast amount of research from many mathematicians and computer scientists around the globe aiming to improve this computation speed. From the use of modern results in algebraic and arithmetic geometry to the application of foundational number theory that dates back to the days of Gauss and Euler, cryptographic pairings have since experienced a great deal of improvement. As a result, what was an extremely expensive computation that took several minutes is now a high-speed operation that takes less than a millisecond. This thesis presents a range of optimisations to the state-of-the-art in cryptographic pairing computation. Both through extending prior techniques, and introducing several novel ideas of our own, our work has contributed to recordbreaking pairing implementations.

The dynamics of cellular automata on 2-manifolds is affected by topology

In this paper, we demonstrate that the distribution of Wolfram classes within a cellular automata rule space in the triangular tessellation is not consistent across different topological general. Using a statistical mechanics approach, cellular automata dynamical classes were approximated for cellular automata defined on genus-0, genus-1 and genus-2 2-manifolds. A distribution-free equality test for empirical distributions was applied to identify cases in which Wolfram classes were distributed differently across topologies. This result implies that global structure and local dynamics contribute to the long term evolution of cellular automata.

On the effect of topology on cellular automata rule spaces

This thesis presents an empirical study of the effects of topology on cellular automata rule spaces. The classical definition of a cellular automaton is restricted to that of a regular lattice, often with periodic boundary conditions. This definition is extended to allow for arbitrary topologies. The dynamics of cellular automata within the triangular tessellation were analysed when transformed to 2-manifolds of topological genus 0, genus 1 and genus 2. Cellular automata dynamics were analysed from a statistical mechanics perspective. The sample sizes required to obtain accurate entropy calculations were determined by an entropy error analysis which observed the error in the computed entropy against increasing sample sizes. Each cellular automata rule space was sampled repeatedly and the selected cellular automata were simulated over many thousands of trials for each topology. This resulted in an entropy distribution for each rule space. The computed entropy distributions are indicative of the cellular automata dynamical class distribution. Through the comparison of these dynamical class distributions using the E-statistic, it was identified that such topological changes cause these distributions to alter. This is a significant result which implies that both global structure and local dynamics play a important role in defining long term behaviour of cellular automata.

Dating of divergences within the Rattus genus phylogeny using whole mitochondrial genomes

The timing and order of divergences within the genus Rattus have, to date, been quite speculative. In order to address these important issues we sequenced six new whole mitochondrial genomes from wild-caught specimens from four species, Rattus exulans, Rattus praetor, Rattus rattus and Rattus tanezumi. The only rat whole mitochondrial genomes available previously were all from Rattus norvegicus specimens. Our phylogenetic and dating analyses place the deepest divergence within Rattus at ∼3.5 million years ago (Mya). This divergence separates the New Guinean endemic R. praetor lineage from the Asian lineages. Within the Asian/Island Southeast Asian clade R. norvegicus diverged earliest at ∼2.9 Mya. R. exulans and the ancestor of the sister species R. rattus and R. tanezumi subsequently diverged at ∼2.2 Mya, with R. rattus and R. tanezumi separating as recently as ∼0.4 Mya. Our results give both a better resolved species divergence order and diversification dates within Rattus than previous studies.

Characterisation of candidate nuclear genes for species delineation in the genus Cherax

Understanding the evolutionary history and phylogenetic relationships between rare and common species is necessary for the effective management of rare species. The genus Cherax, a group of freshwater crayfish species, is of interest in this regard as a number of species are rare or have restricted distributions while other species are common and widespread. Here we describe the characterisation of three novel nuclear genes of the haemocyanin superfamily for phylogenetic reconstruction of the genus. All novel markers developed in this study amplified consistently in species from three divergent clades of the genus Cherax. The level of polymorphism found in these markers was consistently higher than that found in other nuclear genes previously used in invertebrate systematics, such as NaK ATP-ase. In combination, these markers will be useful to delineate phylogenetic relationships between rare and common Cherax species.

Exploring the fundamental theorem of arithmetic in excel 2007

This paper discusses how fundamentals of number theory, such as unique prime factorization and greatest common divisor can be made accessible to secondary school students through spreadsheets. In addition, the three basic multiplicative functions of number theory are defined and illustrated through a spreadsheet environment. Primes are defined simply as those natural numbers with just two divisors. One focus of the paper is to show the ease with which spreadsheets can be used to introduce students to some basics of elementary number theory. Complete instructions are given to build a spreadsheet to enable the user to input a positive integer, either with a slider or manually, and see the prime decomposition. The spreadsheet environment allows students to observe patterns, gain structural insight, form and test conjectures, and solve problems in elementary number theory.

Exposure to bioaerosols in the school environment in Brisbane, Australia

Introduction: Exposure to bioaerosols in indoor environments has been linked to various adverse health effects, such as airway disorders and upper respiratory tract symptoms. The aim of this study was to assess exposure to bioaerosols in the school environment in Brisbane, Australia. Methods: Culturable fungi and endotoxin measurements were conducted in six schools between October 2010 and May 2011. Culturable fungi (2 indoor air and 1-2 outdoor air samples per school) were assessed using a Biotest RCS High Flow Air Sampler, with a flow rate of either 50L/min or 20L/min. A rose pengar agar was used for recovery, which was incubated prior to counting and partial identification. Endotoxins were sampled (8h, 2L/min) using SKC glass fibre filters (4 indoor air samples per school) and analysed using an endpoint chromogenic LAL assay. Results: The arithmetic mean for fungi concentration in indoor and outdoor air was 710 cfu/m3(125- 1900 cfu/m3) and 524 cfu/m3 (140-1250 cfu/m3), respectively. The most frequently isolated fungal genus from the outdoor air was Cladosporium (over 40 %), followed by isolated Penicillium (21%) and Aspergillus (12%). The percent of Penicillium, Cladosporium and Aspergillus in indoor air samples was 32%, 32% and 8%, respectively. The aritmetic mean of endotoxin concentration was 0.59 EU/m3 (0-2,2 EU/m3). Discussion: The results of the current study are in agreement with previously reported studies, in that airborne fungi and endotoxin concentrations varied extensively, and were mostly dependent on climatic conditions. In addition, the indoor air mycoflora largely reflected the fungal flora present in the outdoor air, with Cladosporium being the most common in both outdoor and indoor (with Penicillium) air. In indoor air, unusually high endotoxin levels, over 1 EU/m3, were detected at 2 schools. Although these schools were not affected by the recent Brisbane floods, persistent rain prior to and during the study perios could explain the results.

Revisiting Coptotermes (Isoptera: Rhinotermitidae): a global taxonomic road map for species validity and distribution of an economically important subterranean termite genus

Coptotermes Wasmann (Isoptera: Rhinotermitidae) is one of the most economically important subterranean termite genera and some species are successful invaders. However, despite its important pest status, the taxonomic validity of many named Coptotermes species remains unclear. In this study, we reviewed all named species within the genus and investigated evidence supporting the validity of each named species. Species were systematically scrutinized according to the region of their original description: Southeast Asia, India, China, Africa, the Neotropics, and Australia. We estimate that of the currently 69 named species described by accepted nomenclatural rules, only 21 taxa have solid evidence for validity, 44 names have uncertain status, and the remaining species names should be synonymized or were made unavailable. Species with high degrees of invasiveness may be known under additional junior synonyms due to independent parochial descriptions. Molecular data for a vast majority of species are scarce and significant effort is needed to complete the taxonomic and phylogenetic revision of the genus. Because of the wide distribution of Coptotermes, we advocate for an integrative taxonomic effort to establish the distribution of each putative species, provide specimens and corresponding molecular data, check original descriptions and type specimens (if available), and provide evidence for a more robust phylogenetic position of each species. This study embodies both consensus and contention of those studying Coptotermes and thus pinpoints the current uncertainty of many species. This project is intended to be a roadmap for identifying those Coptotermes species names that need to be more thoroughly investigated, as an incentive to complete a necessary revision process.