The ACA (APPLICATIONS OF COMPUTER ALGEBRA) conference has taken place for
several years now:
1995 in Albuquerque, New Mexico, USA; 1996 in Linz, Austria;
1997 in Maui, Hawaii, USA; 1998 in Prague, Czech Republic;
1999 in El Escorial, Spain; 2000 in St. Petersburg, Russia;
2001 in Albuquerque, New Mexico, USA;
and this year will be in Volos, Greece (June 25--28, 2002).
From 1996 onwards, we have chaired a session (with a slightly changing title) devoted to ``Nonstandard Applications of Computer Algebra''. The session traditionally collects contributions that, while using Computer Algebra techniques and/or Computer Algebra Systems, could not be easily allocated in the `standard' sessions. Examples of topics treated in papers presented in previous editions of this conference are: Verification of Expert Systems, Development of Expert Systems, Railway Traffic Control, Artificial Intelligence, Thermodynamics, Molecular Dynamics, Statistics, Electrical Networks, Logic, Robotics, Sociology, ...
We're organizing again in 2002 the session:
NONSTANDARD APPLICATIONS OF COMPUTER ALGEBRA
Submissions of works in this area are welcome. In case you are interested in submitting a paper, please send an email message to both of us, including a tentative title and a 1 to 2 page long abstract with a general description of your work.
Although the general proceedings are published `only' in electronic form, articles corresponding to the presentations in this session in 1996--1997 and 1998--1999 have been published in ``Mathematics and Computers in Simulation'' (Transactions of IMACS) journal (volumes 45/1--2 and 51/5, respectively). We'll try to have the articles accepted in this session compiled in another special volume too.
Sincerely,
Dr. Eugenio Roanes-Lozano, Universidad Complutense de Madrid
(eroanes@mat.ucm.es)
Dr. Michael Wester, Cotopaxi and University of New Mexico
(wester@math.unm.edu)
Talks:
José L. Valcarce
Dept. of Applied Mathematics, University of Vigo, Pontevedra, Spain
and
Francisco Botana
Dept. of Applied Mathematics, University of Vigo, Pontevedra, Spain
fbotana@uvigo.es
Current dynamic geometry environments such as CABRI, The Geometer's Sketchpad or Cinderella allow users to get the envelopes of some families of plane lines by tracing one of the lines while it is moved, or, in a more automated way, by asking for the locus of the line, which depends on a point, while this point moves along a predefined path. Thus, the obtained envelope is merely a graphic object on the screen: no analytic knowledge is delivered, and, even being dynamic if generated as a locus in Cabri or Cinderella, the envelope is a special object in the environment, since no point can be constructed on it. Furthermore, the interactive approaches taken by these systems disallow the generation of envelopes of lines which depend on points not bound to an element, but constraining them.
The talk will describe an improvement of Lugares, a symbolic-dynamic geometry environment written in Visual Prolog, which allows to obtain the equations and plots of the envelopes of families of plane lines. In this system, the envelopes are full-dynamic objects in the sense that they respond to dragging and accept the construction of points on them. The prototype pursues the cooperation between dynamic geometry environments and computer algebra systems. Once a line depending on a point is created in the geometric environment, a computer algebra system (Mathematica and/or CoCoA) is launched, the inputs being the polynomials describing the family and the path of the point. After some symbolic processing, the symbolic engine returns the equation of the envelope to the dynamic component, where it is graphed.
As an illustration, it will be shown how this approach allows the discovery of new envelopes. Guzmán has recently generalized the Simson-Steiner's theorems stating that given a triangle ABC, a point X on its plane, three projection directions not equal nor parallel to the sides of the triangle, and their projections M, N, P on the sides, the locus of X such that the oriented area of the triangle MNP is a constant, is a conic. Constructing a point X on this conic when there is no area, the program easily finds the equation and the plot of the envelope of lines MNP (Guzmán's lines).
Eugenio Roanes-Lozano(1), Eugenio Roanes-Macias(1), Matilde Villar-Mena(2)
(1) Dept. Algebra, Universidad Complutense de Madrid (Spain) (2) School of Computer Science, Universidad Politecnica de Madrid (Spain)
Dynamic Geometry Systems (DGSs) are devoted to "rule and compass" Geometry. The elementary operations are: select an object, draw a line through two points, draw a circle with centre a certain point through another point... Moreover, the user can: construct a segment given its endpoints, construct the parallel line to a given line through a given point, construct the intersection point(s) of two objects...
The "constructive steps" of a geometric construction can be stored in all main DGSs and are the equivalent to a procedure in a usual language. They are denoted "scripts" in The Geometer's Sketchpad v.3, "macros" in Cabri-Geometry II, "construction texts" in Cinderella... Clearly, the input are geometric objects instead of numeric or algebraic data.
Computer Algebra Systems (CASs) as well as Dynamic Geometry Systems are very powerful tools, but have evolved independently. Although Euclidean Geometry packages exist in some CASs, CASs have incorporated neither mouse drawing capabilities nor dynamic capabilities. Meanwhile, the well-known DGSs do not provide algebraic facilities.
A connection between both kinds of systems could allow users to obtain equations (depending on free-coordinates) directly from sketches. This could save time on many occasions and could make automatic theorem proving and automatic discovery far more attractive.
Observe that the processes implemented in DGSs are constructive. The "constructive steps" give a sound description of the sketch that describes the problem or theorem or guess... Our key idea is to translate the description in the "constructive steps" into CAS-acceptable syntax and to write a package in the CAS that allows the CAS to understand these steps. For instance the following part of a The Geometer's Sketchpad 4 JavaSketch HTM file:
{1} Point(339,143);
{2} Point(339,162);
{3} Circle(1,2)[color(0,128,0)];
would be translated to Maple code as:
A:=point(A_x,A_y);
B:=point(B_x,B_y);
a:=circumCP(A,B);
(points {1} and {2} are free and are the so called "parents" of object {3}). Procedure circumCP is adequately defined in Maple.
We have studied this connection for some time now. The state-of-the-art can be found below:
An overview of the results of the whole project will be presented at ACA'2002. Now "parametric Geometry" packages for Maple 7 and Derive 5 are available. Moreover, an external windows-style translator that can take as input either:
- .TXT versions of The Geometer's Sketchpad v.3 scripts
- .HTM JavaSketch files of The Geometer's Sketchpad v.4 sketches
and produce either
- Maple 7 code
- Derive 5 code
has been developed.
The presentation will be illustrated with examples of well known theorems.
Acknowledgments:
This work is partially supported by project TIC2000-1368-C03-03 (Ministry of Science and Technology, Spain).
References
Raya Khanin
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Cambridge, UK
R.Khanin@damtp.cam.ac.uk
My talk deals with doing automatic dimensional analysis, scaling and ordering of terms in equations using Computer Algebra. Dimensional analysis is a preliminary analysis of the model or equation ([1]). Alone it can never give a complete solution of a problem. It can, however, be used as a quick check for the dimensional sanity of formulas. It may also yield some significant insight into the problem and show the effect of particular parameters. The application of the main theorem of dimensional analysis, the Pi-theorem, usually reduces the number of essentially independent variables to their minimum possible number revealing some dependencies between them.
All modern Computer Algebra Systems have more or less comprehensive packages dealing with units and unit conversion (for example, Derive ([2]), Maple7 and Mathematica). At present, however, only Macsyma has a package which implements functions for automatic dimensional analysis in addition to the unit conversion.
I have developed a Mathematica package DimensionalAnalysis. The basic features of the package (introduction of a concept of dimension, dimensional formulae for principal physical quantities, named dimensionless parameters) have been discussed elsewhere ([3]) and will be briefly mentioned. The package has now been extended to include tools for checking dimensional homogenuity of differential/algebraic expressions, doing nondimensionalisation of equations, and yielding scaling laws for modelling. I will also talk about possible automation of term ordering in the model equations. The package can be used in different areas of science, engineering and even practical life situations, like cooking. Several examples will be considered.
References:
Laureano Lambán, Vico Pascual and Julio Rubio
Universidad de La Rioja
Luis de Ulloa s/n Edificio Vives
26004 Logroño (La Rioja), Spain
{lalamban,mvico,jurubio}@dmc.unirioja.es
Bill Pletsch
Albuquerque Technical Vocational Institute
Albuquerque, New Mexico, USA
Let us consider a deuteron colliding with another deuteron ignoring charge and spin. In the case where after the collision two deuterons are returned there are only two possible reactions. Either nothing happened or a particle was exchanged. Generalizing this simple problem from scattering theory results in an excursion into Polya's theory of counting and the theory of double cosets.
Until the advent of computer algebra, the theory of double cosets has been restricted to a few elegant but computationally impossible theorems. Impossible in the sense that in principle the calculation can be done but it will take ten thousand years.
Today, using Computer Algebra much can be calculated quickly. Using Macsyma and Maple in the special case of Young group generated double cosets, we will see just how valuable Computer Algebra can be. Some surprising and stimulating patterns emerge after a just few computer algebra experiments.
We will expand upon and extend the results presented last year.
Piotr Zarzycki
Institute of Mathematics, University of Gdañsk, Gdañsk,
Poland
and
Adam Marlewski
Institute of Mathematics, Poznañ University of Technology,
Poznañ, Poland
AMARLEW@math.put.poznan.pl
First, we recall basic definitions and theorems on continued fractions. Next, we present some instructive examples, DERIVE built-in and utility functions. At last, we construct new DERIVE functions and we apply them to investigate the expansions of numbers and functions. In particular, we obtain the periodical continued fraction of the expression SQRT(k^2-1) and we apply it to answer the question of number of solutions in certain Diophantine equations.
Go to: ACA'2002 main page, Conferences on Applications of Computer Algebra main page.