On rank, underdetermined systems and consistency

The concepts of rank, underdetermined systems and consistency in linear algebra are discussed in the context of a puzzle. The article begins with a specific example, moving on to a generalization of the example and then to the general n x n case. As well as providing a solution of the puzzle, the article aims to provide students with a greater understanding of these abstract ideas in linear algebra through the study of the puzzle.

Innovation clusters: key concepts

Innovation is easier to describe than it is to systematically analyse, and easier to analyse than it is to effectively promote. Part of the problem, of course, is the imprecise way in which the activity of innovation itself is conceptualised. To achieve more precision, the logic of analysis suggests that innovation should be should be systematically analysed and then divided into rough categories to produce a working taxonomy based on a number of key dimensions. A major part of the purpose of this paper is to develop such a working taxonomy.

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

Key concepts in development geography

Key Concepts in Develpment Geography is an introductory text that provides students with the core concepts that form contemporary research and ideas within the development geography discipline. The book covers the meanings and measurement of development; its theory and practice; work, employment and development; people, culture and development; and contemporary issues in development.

Does training improve understanding of core concepts in cognitive behaviour therapy by people with intellectual disabilities? A randomized experiment

Background. People with intellectual disabilities (ID) experience similar or even higher rates of mental health problems than the general population and there is a need to develop appropriate treatments. Cognitive behaviour therapy (CBT) is effective for a wide range of disorders in the general population. However, there is some evidence that people with ID may lack the cognitive skills needed to take part in CBT. Aims. To test if people with ID can learn skills required for CBT, specifically the ability to distinguish between thoughts, feelings, and behaviours and to link thoughts and feelings (cognitive mediation). Method. A randomized independent groups design was used to examine the effect of training in CBT on two tasks measuring CBT skills. Thirty-four adults with ID were randomly allocated to the experimental condition ðN ¼ 18Þ or to the control condition ðN ¼ 16Þ. CBT skills were assessed blind at baseline and after the intervention. Results. The training led to significant improvements in participants’ ability to link thoughts and feelings, and this skill was generalized to new material. There was no effect of training on participants’ ability to distinguish amongst thoughts, feelings, and behaviours. People with ID can, therefore, learn some skills required for CBT. This implies that preparatory training for CBT might be useful for people with ID. The results might be applicable to other groups who find aspects of CBT difficult.

Connections between concepts revealed by the electronic structure of carbon monoxide

Different models for the electronic structure of carbon monoxide are suggested in influential textbooks. Therefore, this electronic structure offers an interesting subject in teaching because it can be used as an example to relate seemingly conflicting concepts. Understanding the connections between ostensibly different methods and between different concepts, related or conflicting, is important in academic studies. The related reactivities of CO, O2, and N-2 and the notations of molecular orbitals are topics of interest and are discussed in detail.

Correlations between two sets of angular relation equations

It is shown here that the angular relation equations between direct and reciprocal vectors are very similar to the angular relation equations in Euler's theorem. These two sets of equations are usually treated separately as unrelated equations in different fields. In this careful study, the connection between the two sets of angular equations is revealed by considering the cosine rule for the spherical triangle. It is found that understanding of the correlation is hindered by the facts that the same variables are defined differently and different symbols are used to represent them in the two fields. Understanding the connection between different concepts is not only stimulating and beneficial, but also a fundamental tool in innovation and research, and has historical significance. The background of the work presented here contains elements of many scientific disciplines. This work illustrates the common ground of two theories usually considered separately and is therefore of benefit not only for its own sake but also to illustrate a general principle that a theory relevant to one discipline can often be used in another. The paper works with chemistry related concepts using mathematical methodologies unfamiliar to the usual audience of mainstream experimental and theoretical chemists.

An integrated view of data quality in Earth observation

Data quality is a difficult notion to define precisely, and different communities have different views and understandings of the subject. This causes confusion, a lack of harmonization of data across communities and omission of vital quality information. For some existing data infrastructures, data quality standards cannot address the problem adequately and cannot fulfil all user needs or cover all concepts of data quality. In this study, we discuss some philosophical issues on data quality. We identify actual user needs on data quality, review existing standards and specifications on data quality, and propose an integrated model for data quality in the field of Earth observation (EO). We also propose a practical mechanism for applying the integrated quality information model to a large number of datasets through metadata inheritance. While our data quality management approach is in the domain of EO, we believe that the ideas and methodologies for data quality management can be applied to wider domains and disciplines to facilitate quality-enabled scientific research.