PySSM : a Python module for Bayesian inference of linear Gaussian state space models

PySSM is a Python package that has been developed for the analysis of time series using linear Gaussian state space models (SSM). PySSM is easy to use; models can be set up quickly and efficiently and a variety of different settings are available to the user. It also takes advantage of scientific libraries Numpy and Scipy and other high level features of the Python language. PySSM is also used as a platform for interfacing between optimised and parallelised Fortran routines. These Fortran routines heavily utilise Basic Linear Algebra (BLAS) and Linear Algebra Package (LAPACK) functions for maximum performance. PySSM contains classes for filtering, classical smoothing as well as simulation smoothing.

Greek or not : the use of symbols and abbreviations in mathematics

The use of symbols and abbreviations adds uniqueness and complexity to the mathematical language register. In this article, the reader’s attention is drawn to the multitude of symbols and abbreviations which are used in mathematics. The conventions which underpin the use of the symbols and abbreviations and the linguistic difficulties which learners of mathematics may encounter due to the inclusion of the symbolic language are discussed. 2010 NAPLAN numeracy tests are used to illustrate examples of the complexities of the symbolic language of mathematics.

Computationally sound automated proofs of cryptographic schemes

Proving security of cryptographic schemes, which normally are short algorithms, has been known to be time-consuming and easy to get wrong. Using computers to analyse their security can help to solve the problem. This thesis focuses on methods of using computers to verify security of such schemes in cryptographic models. The contributions of this thesis to automated security proofs of cryptographic schemes can be divided into two groups: indirect and direct techniques. Regarding indirect ones, we propose a technique to verify the security of public-key-based key exchange protocols. Security of such protocols has been able to be proved automatically using an existing tool, but in a noncryptographic model. We show that under some conditions, security in that non-cryptographic model implies security in a common cryptographic one, the Bellare-Rogaway model [11]. The implication enables one to use that existing tool, which was designed to work with a different type of model, in order to achieve security proofs of public-key-based key exchange protocols in a cryptographic model. For direct techniques, we have two contributions. The first is a tool to verify Diffie-Hellmanbased key exchange protocols. In that work, we design a simple programming language for specifying Diffie-Hellman-based key exchange algorithms. The language has a semantics based on a cryptographic model, the Bellare-Rogaway model [11]. From the semantics, we build a Hoare-style logic which allows us to reason about the security of a key exchange algorithm, specified as a pair of initiator and responder programs. The other contribution to the direct technique line is on automated proofs for computational indistinguishability. Unlike the two other contributions, this one does not treat a fixed class of protocols. We construct a generic formalism which allows one to model the security problem of a variety of classes of cryptographic schemes as the indistinguishability between two pieces of information. We also design and implement an algorithm for solving indistinguishability problems. Compared to the two other works, this one covers significantly more types of schemes, but consequently, it can verify only weaker forms of security.

Planar difference functions

In 1980 Alltop produced a family of cubic phase sequences that nearly meet the Welch bound for maximum non-peak correlation magnitude. This family of sequences were shown by Wooters and Fields to be useful for quantum state tomography. Alltop’s construction used a function that is not planar, but whose difference function is planar. In this paper we show that Alltop type functions cannot exist in fields of characteristic 3 and that for a known class of planar functions, x^3 is the only Alltop type function.

Mutually unbiased bases as submodules and subspaces

Mutually unbiased bases (MUBs) have been used in several cryptographic and communications applications. There has been much speculation regarding connections between MUBs and finite geometries. Most of which has focused on a connection with projective and affine planes. We propose a connection with higher dimensional projective geometries and projective Hjelmslev geometries. We show that this proposed geometric structure is present in several constructions of MUBs.

Repeating patterns : strategies to assist young students to generalise the mathematical structure

This paper focuses on very young students' ability to engage in repeating pattern tasks and identifying strategies that assist them to ascertain the structure of the pattern. It describes results of a study which is part of the Early Years Generalising Project (EYGP) and involves Australian students in Years 1 to 4 (ages 5-10). This paper reports on the results from the early years' cohort (Year 1 and 2 students). Clinical interviews were used to collect data concerning students' ability to determine elements in different positions when two units of a repeating pattern were shown. This meant that students were required to identify the multiplicative structure of the pattern. Results indicate there are particular strategies that assist students to predict these elements, and there appears to be a hierarchy of pattern activities that help students to understand the structure of repeating patterns.

Modelling the information seeking user by the decision they make

The article focuses on how the information seeker makes decisions about relevance. It will employ a novel decision theory based on quantum probabilities. This direction derives from mounting research within the field of cognitive science showing that decision theory based on quantum probabilities is superior to modelling human judgements than standard probability models [2, 1]. By quantum probabilities, we mean decision event space is modelled as vector space rather than the usual Boolean algebra of sets. In this way,incompatible perspectives around a decision can be modelled leading to an interference term which modifies the law of total probability. The interference term is crucial in modifying the probability judgements made by current probabilistic systems so they align better with human judgement. The goal of this article is thus to model the information seeker user as a decision maker. For this purpose, signal detection models will be sketched which are in principle applicable in a wide variety of information seeking scenarios.

A family of Alltop functions that are EA-inequivalent to the cubic function

Sequences with optimal correlation properties are much sought after for applications in communication systems. In 1980, Alltop (\emph{IEEE Trans. Inf. Theory} 26(3):350-354, 1980) described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation. Subsequently these sequences were used to construct mutually unbiased bases (MUBs), a structure of importance in quantum information theory. The key feature of this cubic function is that its difference function is a planar function. Functions with planar difference functions have been called \emph{Alltop functions}. This paper provides a new family of Alltop functions and establishes the use of Alltop functions for construction of sequence sets and MUBs.

Perspectives on Reconceptualizing Early Mathematics Learning

This introductory section provides an overview of the different perspectives on reconceptualizing early mathematics learning. The chapters provide a broad scope in their topics and approaches to advancing young children’s mathematical learning. They incorporate studies that highlight the importance of pattern and structure across the curriculum, studies that target particular content such as statistics, early algebra, and beginning number, and studies that consider how technology and other tools can facilitate early mathematical development. Reconceptualizing the professional learning of teachers in promoting young children’s mathematics, including a consideration of the role of play, is also addressed. Although these themes are diffused throughout the chapters, we restrict our introduction to the core focus of each of the chapters.

Crunch time : how did the students perform?

Two newspaper numbers games based on simple arithmetic relationships are discussed. One is rather trivial, but very useful as an introduction to the second, whose potential to give students of elementary algebra practice in semi ad-hoc reasoning and to build general arithmetic reasoning skills was explored theoretically in an earlier paper. Preliminary results on the effectiveness of this general approach are presented, with student performance and feedback on an assignment task and formal examination included, and recommendations for future work.

Control of an underactuated-slender-hull unmanned underwater vehicle using Port-Hamiltonian theory

This paper presents a control design for tracking of attitude and speed of an underactuated slender-hull unmanned underwater vehicle (UUV). The control design is based on Port-Hamiltonian theory. The target dynamics (desired dynamic response) is shaped with particular attention to the target mass matrix so that the influence of the unactuated dynamics on the controlled system is suppressed. This results in achievable dynamics independent of uncontrolled states. Throughout the design, insight of the physical phenomena involved is used to propose the desired target dynamics. The performance of the design is demonstrated through simulation with a high-fidelity model.

The Number Crunch game : a simple vehicle for building algebraic reasoning skills

A newspaper numbers game based on simple arithmetic relationships is discussed. Its potential to give students of elementary algebra practice in semi-ad hoc reasoning and to build general arithmetic reasoning skills is explored.

Teaching basic finance with Excel, and without pain

The five quantities of interest in elementary finance problems are present value, future value, amount of periodic payment, number of periods and the rate of compound interest per period. A recursive approach to computing each of these five quantities in a modern version of Excel, for the case of ordinary annuities, is described. The aim is to increase student understanding and build confidence in the answer obtained, and this may be achieved with only linear relationships and in cases where student knowledge of algebra is essentially zero. Annuity problems may be solved without use of logarithms and black-box intrinsic functions; these being used only as check mechanisms. The author has had success with the method at Bond University and surrounding high schools in Queensland, Australia.

Spreadsheet Conditional Formatting Illuminates Investigations into Modular Arithmetic

Modular arithmetic has often been regarded as something of a mathematical curiosity, at least by those unfamiliar with its importance to both abstract algebra and number theory, and with its numerous applications. However, with the ubiquity of fast digital computers, and the need for reliable digital security systems such as RSA, this important branch of mathematics is now considered essential knowledge for many professionals. Indeed, computer arithmetic itself is, ipso facto, modular. This chapter describes how the modern graphical spreadsheet may be used to clearly illustrate the basics of modular arithmetic, and to solve certain classes of problems. Students may then gain structural insight and the foundations laid for applications to such areas as hashing, random number generation, and public-key cryptography.

Participatory Data Analytics: Collaborative interfaces for data composition and visualisation

This research proposes the development of interfaces to support collaborative, community-driven inquiry into data, which we refer to as Participatory Data Analytics. Since the investigation is led by local communities, it is not possible to anticipate which data will be relevant and what questions are going to be asked. Therefore, users have to be able to construct and tailor visualisations to their own needs. The poster presents early work towards defining a suitable compositional model, which will allow users to mix, match, and manipulate data sets to obtain visual representations with little-to-no programming knowledge. Following a user-centred design process, we are subsequently planning to identify appropriate interaction techniques and metaphors for generating such visual specifications on wall-sized, multi-touch displays.