Exploration and reduction of data using principal component analysis

In a data set with two variables only, a scatterplot between the two variables can be easily plotted to represent the data visually. When the number of variables in the data set is large, however, it is more difficult to represent visually. The method of principal component analysis (PCA) can sometimes be used to represent the data faithfully in few dimensions (eg. three or less), with little or no loss of information. This reduction in dimensionality is best achieved when the original variables are highly correlated, positively or negatively. In this case, it is quite conceivable that 20 or 30 original variables can be adequately represented by two or three new variables, which are suitable combinations of the original ones, and which are called principal components. Principal components are uncorrelated between themselves, so that each component describes a different dimension of the data. The principal components can also be arranged in descending order of their variance. The first component has the largest variance, and is the most important, followed by the second component with the second largest variance, and so on. The first two components can then be evaluated for each case in the data set and plotted against each other in a scattergraph, the score for the first component being plotted along the horizontal axis, the score of the second component being plotted on the vertical axis. This scatterplot is a parsimonious two-dimensional picture of the variables and cases in the original data set. We illustrate the method by applying it to simulated datasets, and to a dataset containing national track record times for males and females in various countries.peer-reviewe

Francesco Azopardi : a Maltese musician and theorist : new evidence about his career and didactic writings

Francesco Azopardi (1748-1809) lived in Malta during the second half of the eighteenth century. He was professionally trained in Naples for four years and remained in this city working as a composer, teacher and conductor until 1774. The Cathedral authorities subsequently employed Azopardi in Malta as an organist. His duties included composition, playing the organ and teaching singers. Azopardi's workload increased significantly following the retirement of his predecessor Benigno Zerafa. Azopardi was a prolific composer and he composed new music regularly for the liturgical services of the Cathedral Church. In addition, he taught many students including Nicolo lsouard and produced two important composition treatises for his students.peer-reviewe

Measuring reliability and consistency in contingency tables

The association between two categorical variables is very often assessed by making a cross-tabulation and calculating the x2 statistic for that table. However there are many other related parameters which can be used to assess subtle patterns in the table. In this article we will discuss parameters which can be fruitfully used in situations such as the test-retest method for the reliability of questions in a pilot questionnaire ; the measurement of the change of people’s attitude with time ; the comparison of two medical diagnoses of a given patient ; the prediction of heart disease status using an independent risk scale.peer-reviewe

The evolving link between learning and assessment : from 'transmission check' to 'learning support'

Learning and assessment are now considered as two sides of the same coin we simply cannot speak of one without also referring to the other. This paper, which traces the evolution of the link between learning and assessment, explores what led to our shift in understanding of the learning process from the behaviourist to the constructivist model, and the implications that this 'revolution ' has had for assessment. Making assessment at the service of learning is subsequently identified as the challenge ahead for the educational community.peer-reviewe

Square compactness and the filter extension property

We show that the consistency strength of κ being 2κ-square compact is at least weak compact and strictly less than indescribable. This is the first known improvement to the upper bound of strong compactness obtained in 1973 by Hajnal and Juhasz

Using inquiry-based learning to support the mathematical learning of students with SEBD

This paper, which draws on action research methodology, explores the use of inquirybased learning (IBL) in the teaching of mathematics to students with social, emotional and behavioural difficulties (SEBD). The year-long study was conducted in a Form 3 secondary class that grouped 13 male students with SEBD in a Maltese secondary school. After first creating an IBL-friendly classroom environment in the initial months, the actual implementation of IBL pedagogy in class began in the second term and spread over a 15 week period. The data included teacher observations that were recorded in a reflective research journal, two sessions of in-depth interviews with students, student journal writing, samples of students’ work and student marks in the school-based halfyearly and annual mathematics examinations. The findings indicate that the use of IBL in the mathematics classroom can benefit students with SEBD in a number of ways. These include infusing a sense of enjoyment during lessons, improved student behaviour and motivation to learn, and facilitating the learning of mathematics which generally translated in higher achievement levels.peer-reviewe

The Tamano theorem in MAP

In this paper we continue with the study of paracompact maps introduced in [1]. We give two external characterizations for paracompact maps including a characterization analogous to The Tamano Theorem in the category TOP (of topological spaces and continuous maps as morphisms). A necessary and sufficient condition for the Tychonoff product of a closed map and a compact map to be closed is also given.peer-reviewe

Sustainable development and building design in Malta

This paper spans across the three pillars of sustainability and outlines how a sustainability index can be applied in the context of the building industry, using Malta as a case study. Sustainability criteria are identified which can help promote a culture change in the way most of our clients view a project as different from a contractor’s point of view. Design considerations are also dealt with pointing one way forward if Malta really wants to reduce the environmental impact of buildings. Highlighted issues touch on the visual impact (aesthetics), materials, embodied energy, lighting, space heating and finally ending with an epilogue on land use and waste management, being only the tip of the ‘mountain’ in Malta.peer-reviewe

A category of continuous maps

The study of General Topology is usually concerned with the category TOP of topological spaces as objects, and continuous maps as morphisms. The concepts of space and map are equally important and one can even look at a space as a map from this space onto a singleton space and in this manner identify these two concepts. With this in mind, a branch of General Topology which has become known as General Topology of Continuous Maps, or Fibrewise General Topology, was initiated. This field of research is concerned most of all in extending the main notions and results concerning topological spaces to those of continuous maps. In this way one can see some well-known results in a new and clearer light and one can also be led to further developments which otherwise would not have suggested themselves. The fibrewise viewpoint is standard in the theory of fibre bundles, however, it has been recognized relatively recently that the same viewpoint is also as important in other areas such as General Topology.peer-reviewe