ERCIM News No.22 - July 1995 - CWI

**by Henk Nieland**

**The Research Institute for Applications of Computer Algebra RIACA was established in 1993 by the Mathematical Centre Foundation (to which CWI belongs), the Foundation Computer Algebra Nederland CAN - which is responsible for one of the twenty odd expertise centres founded in the late 1980s by the Dutch Research Council NWO - and the Gödel School GmbH for Symbolic Computation, Linz, Austria. RIACA functions as an international R&D centre for applications of computer algebra in science and technology.
**

Whereas the CAN foundation focuses on making available computing facilities, distributing software, providing information and support, consultancy, and education in The Netherlands, RIACA's first concern is to act as a research centre on an international scale. Contacts were established with several institutions (mainly in Europe), and one subgoal is to initiate cooperation on a larger scale within ERCIM. So far, attention focused on three projects: Human Interaction for Symbolic Computation (HISC), Computer Algebra and Geometrical Optics (CAGO), and a smaller `backup' project Algebra.

In HISC new research directions are explored to simplify the use of symbolic computation tools. In the long-term RIACA/CWI project ACELA a software environment is developed for writing interactive mathematical books. In particular work is underway on such a book on Lie algebras, offering full hypertext facilities and including some comprehension of the syntactical and semantical structure of the formulas. Research items in the project include:

- efficient generation and visualization of implicitly defined surfaces
- 'intelligent plotting', for example automatic labelling of extrema, zeros, etc.
- integration of numeric and symbolic computation software, together with visualization packages, and related graphical front-ends
- hypertext and multimedia authoring and reading systems for Mathematics
- visualization of combinatorial objects
- implementation of the `Chains of Recurrences' method within Maple and Mathematica.

Geometrical optics concerns the study of the behaviour of light rays before and after refraction or reflection at one or more surfaces, in particular the aberrations. There are several ways to study aberrations. In particular the method of eikonals and the closely related method of Lie transformations (a way of describing trajectories for any object, called `contact transformations' in the classical literature) can benefit from computer algebra support. Research in the CAGO project addressed, among other things, the following items:

- computation of third-order aberrations in simple electron-optical systems
- development of Lie-theoretical methods for the determination of the path of a light ray in GRIN-media and the computation of the `root map'
- aberration theory for imaging at a curved surface
- further development of the package MexLie for the computation of optical aberrations
- development of a user interface for Lie Optics, which computes numerical expressions for third- and fifth-order aberrations for an arbitrary number of refracting surfaces.

A workshop on the subject attracted quite a number of opticians from practice.

Topics studied in the smaller Algebra project included:

- algorithms to compute conservation laws for nonlinear partial differential equations, for example the variable coefficient Korteweg-de Vries equation
- algorithms for finite-dimensional Lie algebras
- computation of invariants of SL2(R).

Arjeh Cohen - RIACA/Technical University Eindhoven

Tel: +31 40 47 4270

E-mail: amcwin.tue.nl