Students encountering obstacles using a CAS

The paper describes a pilot study on the use of computer algebra at upper secondary level. A symbolic calculator was introduced in a pre-examination class studying for advanced pre-university mathematics. With the theoretical framework of Realistic Mathematics Education and Developmental Research as a background, the study focused on the identification of obstacles that students encountered while using computer algebra. Five obstacles were identified that have both a technical and a mathematical character. It is the author's belief that taking these barriers seriously is important in developing useful pedagogical strategies.This revised version was published online in September 2005 with corrections to the Cover Date.     

Derivation and Visualization of the Binomial Theorem

The binomial theorem presents us with the opportunity to weave many different mathematical strands into one lesson. It has a fascinating history — the study of which leads to a better understanding of how mathematics evolved. In this paper, we have involved computer graphics, geometry, algebra and combinatorics in the derivation of the binomial theorem. The study of functions with finite domains and ranges helps students understand some of the more subtle properties of functions which have the set of real numbers for their domain and range. These are the functions which they study to the exclusion of all others in high school and in their first two years in college. We believe that the lesson presented in this paper encourages students to express mathematical ideas in the vernacular, one of the major standards recommended by the National Council of Teachers of Mathematics (NCTM).This revised version was published online in September 2005 with corrections to the Cover Date.     

Some heuristics about the ecosystem of mathematics research data

Like in other sciences, research data play a growing role in mathematics, but in contrast to classical objects like measurements in physics they are much more heterogeneous. They may take the shape of abstract objects like the collection of integer sequences in OEIS, algorithms and their implementations as mathematical software, libraries of test problems or statistical data. From an infrastructure viewpoint, which aims at sustainable and connected data repositories which facilitate researchers to use existing information efficiently, it is essential to define an appropriate framework that allows not just storage but also connection and retrieval of the various types of data. Recently, there have been promising attempts to define standards for mathematical software, but the general task remains a big challenge, which is also addressed within the recently initiated GDML working group of the IMU. This is especially important in the fields of applied mathematics where research is often connected to research data originating from applications. The goal of this talk is a first attempt to analyse the diverse ecosystem of research data based on reference data from zbMATH. This approach has worked quite well for mathematical software, resulting in the formation of the swMATH database. Though reference data involve always a bias, the collected information of about 16 million reference data in zbMATH may be useful to identify the recent needs of researchers in different fields pertaining mathematical research data, and we discuss several aspects of such an analysis. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical model for the effect of malicious object on computer network immune system

In this paper, a nonlinear mathematical model is proposed and analyzed to study the effect of malicious object on the immune response of the computer network. Criteria for local stability, instability and global stability are obtained. It is shown that the immune response of the system decreases as the concentration of malicious objects increases, and certain criteria’s are obtained under which it settles down at its equilibrium level. This paper shows that the malicious objects have a grave effect on cyber defense mechanism. The paper has two parts – (i) in the first part a mathematical model is proposed in which dynamics of pathogen, immune response and relative characteristic of the damaged node in the network is investigated, (ii) in second part the effect of malicious object on the immune response of the network has been examined. Finally how and where to use this modeling is discussed.     

Counting systems and the First Hilbert problem

The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one to express different infinite numbers and to use these numbers for measuring infinite sets. Several counting systems are taken into consideration. It is emphasized in the paper that different mathematical languages can describe mathematical objects (in particular, sets and the number of their elements) with different accuracies. The traditional and the new approaches are compared and discussed.     

Computerized proof techniques for undergraduates

The use of computer algebra systems such as Maple and Mathematica is becoming increasingly important and widespread in mathematics learning, teaching and research. In this article, we present computerized proof techniques of Gosper, Wilf–Zeilberger and Zeilberger that can be used for enhancing the teaching and learning of topics in discrete mathematics. We demonstrate by examples how one can use these computerized proof techniques to raise students' interests in the discovery and proof of mathematical identities and enhance their problem-solving skills.     

Numerical computations and mathematical modelling with infinite and infinitesimal numbers

Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer—the Infinity Computer—able to work with all these types of numbers. The new computational tools both give possibilities to execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given.     

Mathematical notation and the use of functions (Poster Presentation)

In contrast to natural languages, mathematical notation is accepted as being exceptionally precise. It shall make mathematical statements unambiguous, it shall allow formal manipulation, it is model for programming languages, computer algebra systems and machine provers. However, what is traditional notation and is it indeed as precise as expected? We discuss some examples of notation which require caution. How are they adapted in computer algebra systems? Can we improve them somehow? What can we learn from functional programming? (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Information Algebras and Consequence Operators

We explore a connection between different ways of representing information in computer science. We show that relational databases, modules, algebraic specifications and constraint systems all satisfy the same ten axioms. A commutative semigroup together with a lattice satisfying these axioms is then called an “information algebra”. We show that any compact consequence operator satisfying the interpolation and the deduction property induces an information algebra. Conversely, each finitary information algebra can be obtained from a consequence operator in this way. Finally we show that arbitrary (not necessarily finitary) information algebras can be represented as some kind of abstract relational database called a tuple system. Mathematics Subject Classification (2000): Primary 03B22; Secondary 03G15 03G25 08A70 68Q99 94A99 03C950     

Dimensional analysis: an elegant technique for facilitating the teaching of mathematical modelling

Dimension analysis is promoted as a technique that promotes better understanding of the role of units and dimensions in mathematical modelling problems. The authors' student base consists of undergraduate students from the Science and Engineering Faculties who generally have one or two semesters of calculus and some linear algebra as part of their curriculum. Because of ‘In Service Training’ which is an integral part of their education, they have a reasonable understanding of the link between theory and practice in their particular industry, but manipulating mathematical formulae is not necessarily a strong point. Dimensional analysis involves both dimensionless products and linear algebra and, because of the latter, this branch of mathematical modelling was, until recently, beyond the reach of most undergraduates. However, it has been found that the skills of a good technologist can be blended with the use of computer algebra systems to successfully teach dimensional analysis to these undergraduates. This note illustrates the concept of dimensional analysis by examining the simple pendulum problem and shows how dimensionless products can lead to the discovery of the connection between the period of the pendulum swing and its length. Dimensional analysis is shown to lead to interesting systems of linear equations to solve, and can point the way to more quantitative analysis, and two student problems are discussed. It is the authors' experience that dimensional analysis broadens a student's viewpoint to include units and dimensions as an integral part of any physical problem. With this approach coupled with a computer algebra systems such as DERIVE, students can concentrate on understanding the model and the modelling process rather than the solution technique. Finally, it has been observed that students find dimensional analysis fun to do.     

Teaching mathematics using Steplets

This article evaluates online mathematical content used for teaching mathematics in engineering classes and in distance education for teacher training students. In the EU projects Xmath and dMath online computer algebra modules (Steplets) for undergraduate students assembled in the Xmath eBook have been designed. Two questionnaires, a compulsory student project and teaching in front of class show that using Steplets turn mathematics teaching from drill to understanding. The Steplets use algorithms developed for the Mathematica programming language.     

Curriculum, Classroom Practices, and Tool Design in the Learning of Functions Through Technology-Aided Experimental Approaches

The paper starts from classroom situations about the study of a functional relationship with help of technological tools as a ‘transposition’ of experimental approaches from research mathematical practices. It considers the limitation of this transposition in existing curricula and practices based on the use of non-symbolic software like dynamic geometry and spreadsheets. The paper focuses then on the potentialities of classroom use of computer algebra packages that could help to go beyond this shortcoming. It looks at a contradiction: while symbolic calculation is a basic tool for mathematicians, curricula and teachers are very cautious regarding their use by students. The rest of the paper considers the design and experiment of a computer environment Casyopée as means to contribute to an evolution of curricula and classroom practices to achieve the transposition in the domain of algebraic activities linked to functions.     

Beyond Cartesian limits: Leibniz's passage from algebraic to “transcendental” mathematics

This article deals with Leibniz's reception of Descartes' “geometry.” Leibnizian mathematics was based on five fundamental notions: calculus, characteristic, art of invention, method, and freedom. On the basis of methodological considerations Leibniz criticized Descartes' restriction of geometry to objects that could be given in terms of algebraic (i.e., finite) equations: “Descartes's mind was the limit of science.” The failure of algebra to solve equations of higher degree led Leibniz to develop linear algebra, and the failure of algebra to deal with transcendental problems led him to conceive of a science of the infinite. Hence Leibniz reconstructed the mathematical corpus, created new (transcendental) notions, and redefined known notions (equality, exactness, construction), thus establishing “a veritable complement of algebra for the transcendentals”: infinite equations, i.e., infinite series, became inestimable tools of mathematical research.     

Pupils learning algebra with DERIVE

This article deals with the use of computer algebra systems in mathematics secondary education. First, we present the global framework of a research project carried out in France with pupils of grades 9 to 12, from 1993 through 1995. Then we focus on a specific part of this project concerning two grade 9 classes of different mathematical ability taught by the same teacher. We briefly present the aims and organisation of DERIVE's use in the two classes, before embarking with more details on a specific topic: systems of linear equations. Finally, we present the data provided by the pupils' answers to a questionnaire taken at the end of the academic year; we use them for investigating pupils' assumptions about DERIVE's potential for mathematics learning, and compare these representations with those emerging from the entire population.     

A Program for Computing the Cohomology of Lie Superalgebras of Vector Fields

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. At present, since the required algebraic computations are very tedious, the cohomology is explicitly computed only in a few cases for different classes of Lie (super)algebras. That is why application of computer algebra methods is important for this problem. We describe an algorithm (and its C implementation) for computing the cohomology of Lie algebras and superalgebras. In elaborating the algorithm, we focused mainly on the cohomology with coefficients in trivial, adjoint, and coadjoint modules for Lie (super)algebras of the formal vector fields. These algebras have many applications to modern supersymmetric models of theoretical and mathematical physics. As an example, we consider the cohomology of the Poisson algebra Po(2) with coefficients in the trivial module and present 3- and 5-cocycles found by a computer. Bibliography: 6 titles.     

Algorithms for Computations in Local Symmetric Spaces

In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc.

In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.     

Opportunities for the use of computer algebra systems in middle secondary mathematics in England and Wales

This paper presents and discusses a number of ways in which mathematical attainment targets specified for pupils aged 11–16 in the revised National Curriculum can be achieved when teachers deploy computer algebra systems.