Mathematics and Statistics Research Collection

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  • Publication
    Constructions of new q-cryptomorphisms
    In the theory of classical matroids, there are several known equivalent axiomatic systems that define a matroid, which are described as matroid cryptomorphisms. A q-matroid is a q-analogue of a matroid where subspaces play the role of the subsets in the classical theory. In this article we establish cryptomorphisms of q-matroids. In doing so we highlight the difference between classical theory and its q-analogue. We introduce a comprehensive set of q-matroid axiom systems and show cryptomorphisms between them and existing axiom systems of a q-matroid. These axioms are described as the rank, closure, basis, independence, dependence, circuit, hyperplane, flat, open space, spanning space, non-spanning space, and bi-colouring axioms.
      4Scopus© Citations 9
  • Publication
    Iterates of a compact holomorphic map on a finite rank homogeneous ball
    (University of Szeged, 2019-06-01) ;
    We study iterates, fn, of a fixed-point free compact holomorphic map f : B → B where B is the open unit ball of any JB∗-triple of finite rank. These spaces include L(H,K), H,K Hilbert, dim(H) arbitrary, dim(K) < 1, or any classical Cartan factor or C∗-algebra of finite rank. Apart from the Hilbert ball, the sequence of iterates (fn)n does not generally converge (locally uniformly on B) and little is known of accumulation points. We present a short proof of a Wolff theorem for B and establish key properties of the resulting f-invariant subdomains. We define a concept of closed convex holomorphic hull, Ch(x), for x ϵ ∂B and prove the following. There is a unique tripotent u in ∂B such that all constant subsequential limits of (fn)n lie in Ch(u). As a consequence we also get a short proof of the classical Hilbert ball results.
      6Scopus© Citations 1
  • Publication
    Infinite Mixtures of Infinite Factor Analysers
    (International Society for Bayesian Analysis, 2020-09) ; ;
    Factor-analytic Gaussian mixtures are often employed as a model-based approach to clustering high-dimensional data. Typically, the numbers of clusters and latent factors must be fixed in advance of model fitting. The pair which optimises some model selection criterion is then chosen. For computational reasons, having the number of factors differ across clusters is rarely considered. Here the infinite mixture of infinite factor analysers (IMIFA) model is introduced. IMIFA employs a Pitman-Yor process prior to facilitate automatic inference of the number of clusters using the stick-breaking construction and a slice sampler. Automatic inference of the cluster-specific numbers of factors is achieved using multiplicative gamma process shrinkage priors and an adaptive Gibbs sampler. IMIFA is presented as the flagship of a family of factor-analytic mixtures. Applications to benchmark data, metabolomic spectral data, and a handwritten digit example illustrate the IMIFA model’s advantageous features. These include obviating the need for model selection criteria, reducing the computational burden associated with the search of the model space, improving clustering performance by allowing cluster-specific numbers of factors, and uncertainty quantification.
      10Scopus© Citations 16
  • Publication
    Is fear DEIS chun chainnte: An initiative to support teaching for robust understanding in post-primary mathematics classrooms
    This paper outlines a research project which aims to support the pedagogical practices of Mathematics teachers in socioeconomically underprivileged post-primary schools. In Ireland, the socioeconomic background of a student continues to determine how likely they are to experience high-quality post-primary education and make the transition to further or higher education. This is particularly relevant in mathematics, which remains a gateway subject in accessing third-level education. This research aims to address such inequity by supporting teachers to incorporate student-centred practices in their mathematics pedagogy. The project is undertaken in two phases: First, case studies of high-quality mathematics teaching and learning will be conducted in four schools. The findings from the case studies will inform the design of an intervention, which will involve 10 pilot schools across Ireland in the 2023-24 academic year. The research team (and research Mathematicians) will collaborate with the schools and support the teachers to reflect on and reform their teaching by engaging in school-based Lesson Study. This paper provides an outline of the project and initial findings which will contribute to research on the teaching and learning of mathematics in Ireland.
      16
  • Publication
    Supporting teachers to engage with Structured Problem Solving in their Junior Cycle classrooms –developing Educative Curriculum Materials for use with Lesson Study
    Successive curriculum reforms in Ireland have led to an increased emphasis on problem-solving approaches to teaching mathematics. However, there is little research to suggest that classroom practices have changed significantly. In this paper we outline the design of a set of Educative Curriculum Materials (ECMs) as part of a professional development intervention, which aims to support teachers to incorporate Structured Problem Solving into their classroom practice. These materials are designed to be used in Lesson Study as part of a professional development intervention. We begin by describing Structured Problem Solving and the challenge it poses for teachers, before outlining the role professional development, and specifically ECMs, can play in supporting teachers with this approach. Finally we highlight some key features of the ECMs currently being developed.
      15