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.

Connecting with the horizon : developing teachers' appreciation of mathematical structure

A professional learning program for teachers of junior secondary mathematics regarding the content and pedagogy of senior secondary mathematics is the context for this study of teachers’ mathematical and pedagogical knowledge. The analysis of teachers’ reflections on their learning explored teachers’ understanding of mathematical connections and their appreciation of mathematical structure. The findings indicate that a professional learning program about senior secondary mathematics can enable practicing teachers to deepen and broaden their knowledge for teaching junior secondary mathematics and develop their practice to support their students’ present and future learning of mathematics. Further research is needed about professional learning approaches and tasks that may enable teachers to imbed and develop awareness of structure in their practice.

Letters & Numbers : a vehicle to illustrate mathematical and computing fundamentals

The television quiz program Letters and Numbers, broadcast on the SBS network, has recently become quite popular in Australia. This paper explores the potential of this game to illustrate and engage student interest in a range of fundamental concepts of computer science and mathematics. The Numbers Game in particular has a rich mathematical structure whose analysis and solution involves concepts of counting and problem size, discrete (tree) structures, language theory, recurrences, computational complexity, and even advanced memory management. This paper presents an analysis of these games and their teaching applications, and presents some initial results of use in student assignments.

First year pre-service teachers' mathematical content knowledge : methods of solution to a ratio question

In this article, pre-service teachers' mathematics content knowledge is explored through the analysis of two items about ratio from a Mathematical Competency, Skills and Knowledge Test. Pre-service teachers' thinking strategies, common errors and misconceptions in their responses are presented and discussed. Of particular interest was the range and nature of common incorrect responses for one whole-whole ratio question. Results suggested pre-service teachers had difficulty interpreting a worded multi-step, ratio (scale) question, with errors relating to ratio and/or conversion of measurement knowledge. These difficulties reveal underdeveloped knowledge of mathematical structure and mathematical connections as well as an inability to deconstruct key components of a mathematical problem. Most pre-service teachers also lacked knowledge of standard procedures and methods of solutions.

Adapting the task : what preservice teachers notice when adapting mathematical tasks

This paper reports on an in-depth study that explores preservice teachers’ pedagogical adaptations to a rich mathematical task. Data were collected from six elementary preservice teachers working in pairs to first solve a mathematics problem and then design adaptations to make the problem more accessible and more challenging for diverse learners. Results indicate that preservice teachers are able to draw upon a range of strategies to vary the mathematical content, the context, and the question asked. However, they also did not notice or attend to how their adaptations changed the mathematical structure of the problem. This study provides insights into what is involved in learning to adapt classroom mathematical tasks as an important pedagogical practice.

Structure of potentials with N higgs doublets

Extensions of the standard model with N Higgs doublets are simple extensions presenting a rich mathematical structure. An underlying Minkowski structure emerges from the study of both variable space and parameter space. The former can be completely parametrized in terms of two future lightlike Minkowski vectors with spatial parts forming an angle whose cosine is -(N-1)(-1). For the parameter space, the Minkowski parametrization enables one to impose sufficient conditions for bounded below potentials, characterize certain classes of local minima, and distinguish charge breaking vacua from neutral vacua. A particular class of neutral minima presents a degenerate mass spectrum for the physical charged Higgs bosons.

Structure of trajectories of complex-matrix eigenvalues in the Hermitian-non-Hermitian transition

The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.

Genome characterization through a mathematical model of the genetic code: an analysis of the whole chromosome 1 of A. thaliana

The objective of this work is to characterize the genome of the chromosome 1 of A.thaliana, a small flowering plants used as a model organism in studies of biology and genetics, on the basis of a recent mathematical model of the genetic code. I analyze and compare different portions of the genome: genes, exons, coding sequences (CDS), introns, long introns, intergenes, untranslated regions (UTR) and regulatory sequences. In order to accomplish the task, I transformed nucleotide sequences into binary sequences based on the definition of the three different dichotomic classes. The descriptive analysis of binary strings indicate the presence of regularities in each portion of the genome considered. In particular, there are remarkable differences between coding sequences (CDS and exons) and non-coding sequences, suggesting that the frame is important only for coding sequences and that dichotomic classes can be useful to recognize them. Then, I assessed the existence of short-range dependence between binary sequences computed on the basis of the different dichotomic classes. I used three different measures of dependence: the well-known chi-squared test and two indices derived from the concept of entropy i.e. Mutual Information (MI) and Sρ, a normalized version of the “Bhattacharya Hellinger Matusita distance”. The results show that there is a significant short-range dependence structure only for the coding sequences whose existence is a clue of an underlying error detection and correction mechanism. No doubt, further studies are needed in order to assess how the information carried by dichotomic classes could discriminate between coding and noncoding sequence and, therefore, contribute to unveil the role of the mathematical structure in error detection and correction mechanisms. Still, I have shown the potential of the approach presented for understanding the management of genetic information.

A Mathematical Model for Simplifying Representations of Objects in a Geographic Information System

The study of operations on representations of objects is well documented in the realm of spatial engineering. However, the mathematical structure and formal proof of these operational phenomena are not thoroughly explored. Other works have often focused on query-based models that seek to order classes and instances of objects in the form of semantic hierarchies or graphs. In some models, nodes of graphs represent objects and are connected by edges that represent different types of coarsening operators. This work, however, studies how the coarsening operator "simplification" can manipulate partitions of finite sets, independent from objects and their attributes. Partitions that are "simplified first have a collection of elements filtered (removed), and then the remaining partition is amalgamated (some sub-collections are unified). Simplification has many interesting mathematical properties. A finite composition of simplifications can also be accomplished with some single simplification. Also, if one partition is a simplification of the other, the simplified partition is defined to be less than the other partition according to the simp relation. This relation is shown to be a partial-order relation based on simplification. Collections of partitions can not only be proven to have a partial- order structure, but also have a lattice structure and are complete. In regard to a geographic information system (GIs), partitions related to subsets of attribute domains for objects are called views. Objects belong to different views based whether or not their attribute values lie in the underlying view domain. Given a particular view, objects with their attribute n-tuple codings contained in the view are part of the actualization set on views, and objects are labeled according to the particular subset of the view in which their coding lies. Though the scope of the work does not mainly focus on queries related directly to geographic objects, it provides verification for the existence of particular views in a system with this underlying structure. Given a finite attribute domain, one can say with mathematical certainty that different views of objects are partially ordered by simplification, and every collection of views has a greatest lower bound and least upper bound, which provides the validity for exploring queries in this regard.

The sphynx's new riddle : how to relate the canonical formula of myth to quantum interaction

We introduce Claude Lévi Strauss' canonical formula (CF), an attempt to rigorously formalise the general narrative structure of myth. This formula utilises the Klein group as its basis, but a recent work draws attention to its natural quaternion form, which opens up the possibility that it may require a quantum inspired interpretation. We present the CF in a form that can be understood by a non-anthropological audience, using the formalisation of a key myth (that of Adonis) to draw attention to its mathematical structure. The future potential formalisation of mythological structure within a quantum inspired framework is proposed and discussed, with a probabilistic interpretation further generalising the formula

Time scales of the stick-slip dynamics of the peeling of an adhesive tape

The stick-slip dynamics of the peeling of an adhesive tape is characterized by bifurcations that have been experimentally well studied. In this work, we investigate the time scale in which the the stick-slips happen leading to the bifurcations. This is fundamental to understanding the triboluminescence and acoustic emissions associated with the bifurcations. We establish a relationship between the time scale of the bifurcations and the inherent mathematical structure of the peeling dynamics by studying a characteristic time quantity associated with the dynamics.

Scalable Intelligence for Scheduling Systems

A personalização é um aspeto chave de uma interação homem-computador efetiva. Numa era em que existe uma abundância de informação e tantas pessoas a interagir com ela, de muitas maneiras, a capacidade de se ajustar aos seus utilizadores é crucial para qualquer sistema moderno. A criação de sistemas adaptáveis é um domínio bastante complexo que necessita de métodos muito específicos para ter sucesso. No entanto, nos dias de hoje ainda não existe um modelo ou arquitetura padrão para usar nos sistemas adaptativos modernos. A principal motivação desta tese é a proposta de uma arquitetura para modelação do utilizador que seja capaz de incorporar diferentes módulos necessários para criar um sistema com inteligência escalável com técnicas de modelação. Os módulos cooperam de forma a analisar os utilizadores e caracterizar o seu comportamento, usando essa informação para fornecer uma experiência de sistema customizada que irá aumentar não só a usabilidade do sistema mas também a produtividade e conhecimento do utilizador. A arquitetura proposta é constituída por três componentes: uma unidade de informação do utilizador, uma estrutura matemática capaz de classificar os utilizadores e a técnica a usar quando se adapta o conteúdo. A unidade de informação do utilizador é responsável por conhecer os vários tipos de indivíduos que podem usar o sistema, por capturar cada detalhe de interações relevantes entre si e os seus utilizadores e também contém a base de dados que guarda essa informação. A estrutura matemática é o classificador de utilizadores, e tem como tarefa a sua análise e classificação num de três perfis: iniciado, intermédio ou avançado. Tanto as redes de Bayes como as neuronais são utilizadas, e uma explicação de como as preparar e treinar para lidar com a informação do utilizador é apresentada. Com o perfil do utilizador definido torna-se necessária uma técnica para adaptar o conteúdo do sistema. Nesta proposta, uma abordagem de iniciativa mista é apresentada tendo como base a liberdade de tanto o utilizador como o sistema controlarem a comunicação entre si. A arquitetura proposta foi desenvolvida como parte integrante do projeto ADSyS - um sistema de escalonamento dinâmico - utilizado para resolver problemas de escalonamento sujeitos a eventos dinâmicos. Possui uma complexidade elevada mesmo para utilizadores frequentes, daí a necessidade de adaptar o seu conteúdo de forma a aumentar a sua usabilidade. Com o objetivo de avaliar as contribuições deste trabalho, um estudo computacional acerca do reconhecimento dos utilizadores foi desenvolvido, tendo por base duas sessões de avaliação de usabilidade com grupos de utilizadores distintos. Foi possível concluir acerca dos benefícios na utilização de técnicas de modelação do utilizador com a arquitetura proposta.

Zwischen Inszenierter Wirklichkeit und realem Traum : Rolle und Problematik des Erzählers in den phantastischen Prosawerken von Leo Perutz und Alexander Lernet-Holenia

Souvent laissée pour contre par les théoriciens, longtemps associée à la littérature populaire, la littérature fantastique a su depuis se tailler une place parmi les grands genres de la littérature. Depuis le milieu du XXe siècle, les chercheurs et le public découvrent ou redécouvrent un genre qui a toujours fait parti du paysage littéraire. Une œuvre majeure contribuera à la redécouverte scientifique du genre : Écrite en 1970 l’œuvre de Tzvetan Todorov Introduction à la littérature fantastique donne tant au néophyte qu’au chercheur un ouvrage qui trouve encore des échos aujourd’hui. Sa définition du fantastique, son approche du rôle du narrateur et du lecteur sont une référence, un point de départ et surtout un incontournable pour ceux et celles qui désirent comprendre et apprendre le fantastique. Dans ce mémoire, nous nous concentrerons particulièrement sur le rôle et la problématique du narrateur dans l’œuvre de deux représentants majeurs de la littérature fantastique de langue allemande du début du XXe siècle : Leo Perutz (1882-1957) et Alexander Lernet-Holenia (1897-1976). Le narrateur semble jouer un rôle prédominant dans la littérature fantastique. En effet, par son discours souvent présenté au « je », il semble créer une dynamique très particulière : il manipule son propre discours, il agit sur le lecteur en étant narrateur et personnage dans le récit, il crée une tension entre les différentes couches narratives par son état instable. Bref, il semble contribuer à l’apparition du fantastique dans le texte et également provoquer une certaine hésitation chez le lecteur. Le texte joue également un rôle : Leo Perutz produit un modèle de texte qui semble marquer aussi son collègue Alexander Lernet-Holenia. La structure presque mathématique des textes de Perutz rappelle que le fantastique peut jouer même à l’intérieur de paramètres rigides avec l’ordre et le chaos, les frontières entre le réel et le rêve. Nous étudierons des œuvres de Leo Perutz et d’Alexander Lernet-Holenia qui ont été publiées entre 1915 et 1937. Les théories qui serviront à appuyer mon travail sont tirées entre autre des études de Tzvetan Todorov et Uwe Durst sur la littérature fantastique et de Gérard Genette sur le narrateur.

Some last thoughts on compositional data analysis

One of the disadvantages of old age is that there is more past than future: this, however, may be turned into an advantage if the wealth of experience and, hopefully, wisdom gained in the past can be reflected upon and throw some light on possible future trends. To an extent, then, this talk is necessarily personal, certainly nostalgic, but also self critical and inquisitive about our understanding of the discipline of statistics. A number of almost philosophical themes will run through the talk: search for appropriate modelling in relation to the real problem envisaged, emphasis on sensible balances between simplicity and complexity, the relative roles of theory and practice, the nature of communication of inferential ideas to the statistical layman, the inter-related roles of teaching, consultation and research. A list of keywords might be: identification of sample space and its mathematical structure, choices between transform and stay, the role of parametric modelling, the role of a sample space metric, the underused hypothesis lattice, the nature of compositional change, particularly in relation to the modelling of processes. While the main theme will be relevance to compositional data analysis we shall point to substantial implications for general multivariate analysis arising from experience of the development of compositional data analysis…

Analyzing shapes as compositions of distances

We propose to analyze shapes as “compositions” of distances in Aitchison geometry as an alternate and complementary tool to classical shape analysis, especially when size is non-informative. Shapes are typically described by the location of user-chosen landmarks. However the shape – considered as invariant under scaling, translation, mirroring and rotation – does not uniquely define the location of landmarks. A simple approach is to use distances of landmarks instead of the locations of landmarks them self. Distances are positive numbers defined up to joint scaling, a mathematical structure quite similar to compositions. The shape fixes only ratios of distances. Perturbations correspond to relative changes of the size of subshapes and of aspect ratios. The power transform increases the expression of the shape by increasing distance ratios. In analogy to the subcompositional consistency, results should not depend too much on the choice of distances, because different subsets of the pairwise distances of landmarks uniquely define the shape. Various compositional analysis tools can be applied to sets of distances directly or after minor modifications concerning the singularity of the covariance matrix and yield results with direct interpretations in terms of shape changes. The remaining problem is that not all sets of distances correspond to a valid shape. Nevertheless interpolated or predicted shapes can be backtransformated by multidimensional scaling (when all pairwise distances are used) or free geodetic adjustment (when sufficiently many distances are used)