Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k

Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble results for Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of arbitrary depth, including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations, we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking, we have sketched derivations of all results that have so far been proved.Comment: 19 pages, LaTe

Understanding what you are doing: A new angle on CAS?

Powerful Computer Algebra Systems (CAS) are often used only with reluctance in early undergraduate mathematics teaching, partly because of concerns that they may not encourage students to understand what they are doing. In this exploratory study, a version of a CAS that has been designed for secondary school students was used, with a view to considering the value of this sort of student learning support for first year undergraduate students enrolled in degree programs other than mathematics. Workshops were designed to help students understand aspects of elementary symbolic manipulation, through the use of the Algebra mode of an algebraic calculator, the Casio Algebra FX 2.0. The Algebra mode of this calculator allows a user to undertake elementary algebraic manipulation, routinely providing all intermediate results, in contrast to more powerful CAS software, which usually provides simplified results only. The students were volunteers from an introductory level unit, designed to provide a bridge between school and university studies of mathematics and with a focus on algebra and calculus. The two structured workshop sessions focussed respectively on the solution of linear equations and on relationships between factorising and expanding; attention focussed on using the calculators as personal learning devices. Following the workshops, structured interviews were used to systematically record student reactions to the experience. As a result of the study, the paper offers advice on the merits of using algebraic calculators in this sort of way

Graphics calculator use in examinations: accident or design?

As graphics calculators become more available, interest will focus on how to incorporate them appropriately into curriculum structures, and particularly into examinations. We describe and exemplify a typology of use of graphics calculators in mathematics examinations, from the perspective of people designing examinations, together with some principles for the awarding of partial credit to student responses. This typology can be used to help design examinations in which students are permitted to use graphics calculators as well as to interrogate existing examination practice

Graphics calculators in the mathematics curriculum: Integration or differentiation?

Graphics calculators are examples of powerful technologies that we want our students to learn to use well. However if we use them in our courses only for learning, students will not regard them with due importance because they are not integrated into the assessment. On the other hand, if graphics calculators are integrated into both learning and assessment there are risks associated with students becoming calculator dependent, issues of equity arise associated with calculator access and there may be problems with setting an appropriate examination. We discuss this dilemma in the light of our experiences and the reactions of our students over the last two years

Symbolic manipulation on a TI-92: New threat or hidden treasures?

The availability of hand held devices that can undertake symbolic manipulation is a recent phenomenon, potentially of great significance for both the algebra and calculus curriculum in the secondary and lower undergraduate years. The significance to date of symbolic manipulation for mathematics is described, and parallels drawn with the significance of arithmetic skills for the primary school. It is suggested that, while symbolic manipulation is central to mathematics, many students develop only a restricted competence with the associated mathematical ideas. The Texas Instruments TI-92 is used to suggest some potential beneficial uses of technology that involves symbolic manipulation

Graphics calculators and assessment

Graphics calculators are powerful tools for learning mathematics and we want our students to learn to use them effectively. The use of these hand held personal computers provides opportunities for learning in interactive and dynamic ways. However, it is not until their use is totally integrated into all aspects of the curriculum that students regard them with due importance. This includes their use in all kinds of assessment tasks such as assignments, tests and examinations as well as in activities and explorations aimed at developing students’ understanding. The incorporation of graphics calculators into assessment tasks requires careful construction of these tasks. In this paper, discuss issues of equity relating to calculator models, levels of calculator use and the purpose and design of appropriate tasks. We also describe a typology we have developed to assist in the design and wording of assessment tasks which encourage appropriate use of graphics calculators, but which do not compromise important course objectives

A slowly rotating perfect fluid body in an ambient vacuum

A global model of a slowly rotating perfect fluid ball in general relativity is presented. To second order in the rotation parameter, the junction surface is an ellipsoidal cylinder. The interior is given by a limiting case of the Wahlquist solution, and the vacuum region is not asymptotically flat. The impossibility of joining an asymptotically flat vacuum region has been shown in a preceding work.Comment: 7 pages, published versio