ICM 2006 Mathematical Software Abstracts · ICM 2006 – Mathematical Software. Abstracts...

ICM 2006

Mathematical Software

Abstracts

ICM 2006 – Mathematical Software. Abstracts

Hyperbolic Workshop

Judit Abardia

Department of Mathematics, Universitat Autonoma de Barcelona, [email protected]

2000 Mathematics Subject Classification. 97U70, 51M09, 51M15

Using the mathematical software The Geometer’s Sketchpad, we created the nec-essary tools to work with the Upper Half-Plane model of Hyperbolic Geometry inthe same way that the Euclidian Geometry. That is, we made sketches to constructnot only the basic objects as lines, segments or rays but circles with a given centerand point, center and radius or the angle bisector as well as the parallelism angle,perpendicular line, horocycle,... and some of the triangle centers.

All the constructions were made using the environment that Sketchpad pro-vides. That is, we construct the tools only using synthetic properties and thedescription of the lines and isometries in the Upper Half-Plane model. For in-stance, to construct a hyperbolic line that passes through two given points we usethat this line is an euclidian circle perpendicular to the boundary line.

Once we had the basic tools implemented we began to use them by showingthat the equivalent statements of the fifth Euclidian postulate are not fulfilled.

For example, using the tools it is possible to illustrate that in the Hyperbolicplane, given three different points, does not always exist a circumference meetingthe three points. As the Sketchpad makes dynamic constructions we can plot threeany points and then drag them. The circumference will be drawn only in the casesit exists so that we can find the existence conditions for the relation between thevertices.

As an example of application we made a second sketch that allow to workwith triangles. Once we know how construct segments we can easily plot a hyper-bolic triangle and then using the hyperbolic bisector, perpendicular bisector andperpendicular line tools to construct some of the remarkable points and lines.

We also made the isometries of the Upper Half-Plane and used the tools to con-struct tessellations. The isometries allow to see how the Euclidian and Hyperboliclength differ but the angles do not.

The tools are of public domain although the program The Geometer Sketch-pad is not. They can be found in the web page of the The Geometer Sketchpad,http://www.keypress.com/sketchpad/general resources/advanced sketch gallery/index.php, in the topic Beyond Euclid under the title Half Plane Model of Hyper-bolic Geometry and also in my homepage (http://mat.uab.es/∼juditab/HypGeom.htm) where it is explained step by step how they are constructed.

These tools can be useful to study and see the differences between Euclidean andHyperbolic Geometry in an easier and interactive way so that they are appropriatedto be used in the beginning of the study of the Hyperbolic Geometry. These toolsmake a class more dynamic in the sense that the teacher can make constructionsand demonstrations with the program in an easy and quick way without having todraw again the picture to explain a similar but different case. Moreover, students

ICM 2006 – Madrid, 22-30 August 2006 1


can state hypothesis that the can be verified instantaneously in a visual way and,if necessary, make changes without having to clean the picture. However, they cannot only be considered as mathematical education software applied in Geometrybut they can be used in other areas as well specially in the Analysis area.

[1] Reventos, A., Goemetria Axiomatica. Institut d’Estudis Catalans, Barcelona, 1993.[2] Steketee, S., Jackiw, N., Chanan, S., The Geometer Sketchpad. Dynamic Geome-

try Software for Exploring Mathematics. Reference Manual. Key Curriculum Press,Emeryville, CA, 2001.

2 ICM 2006 – Madrid, 22-30 August 2006


CoCoA: Computations in Commutative Algebra

John A. Abbott, Anna M. Bigatti*

Dipartimento di Matematica, Universita di Genova, Via Dodecaneso 35 - 16146Genova, [email protected]

2000 Mathematics Subject Classification. 13P05, 13P10, 13P99, 68W30

What is CoCoA?CoCoAis a special-purpose system for doing Computations in Commutative Algebra.It runs on all common platforms. ([1])

CoCoA is one of the products of a research team in Computer Algebra basedin Genoa (Italy) with the cooperation of researchers and students in Europe andNorth America.

CoCoA is freely available software for research and educational purposes: thelatest version is CoCoA 4.6 (May 2006) which can be obtained from

http://cocoa.dima.unige.itas well as from the mirror sites

http://www.reed.edu/mirrors/cocoaThe system offers a textual interface, an Emacs mode, and a graphical user

interface common to most platforms.

The main features of CoCoAAn important purpose of the CoCoA program is to provide a “laboratory”

for studying computational commutative algebra: it together with Singular andMacaulay 2 form an elite group of highly specialized systems having as their mainforte the capability to calculate Grobner bases. Although a number of generalpurpose symbolic computation systems (e.g. REDUCE and Maple) do offer thepossibility to compute Grobner bases, their non-specialist nature implies a numberof severe compromises which make them far less suitable to act as a laboratory:e.g. relatively poor execution speed and limited control over the algorithm param-eters.

Aside from computing Grobner bases CoCoA’s particular strengths includeideal/module operations (such as syzygies and minimal free resolutions, intersec-tions, divisions, the radical of an ideal, . . . ), polynomial factorization, exact lin-ear algebra, computing Hilbert functions, and computing with zero-dimensionalschemes and toric ideals.

The usefulness of these technical skills is enhanced by the mathematically nat-ural language for describing computations. This language is readily learned bystudents, and enables researchers to explore and develop new algorithms withoutthe administrative tedium necessary when using “low-level” languages.

The users of CoCoACurrently CoCoA is used by researchers in several countries. Most of them are

Commutative Algebraists and Algebraic Geometers, but also people working in



different areas such as Analysis and Statistics have already benefitted from oursystem.

CoCoA places great emphasis on being easy and natural to use, so it is also usedas the main system for teaching advanced courses in several universities in Europeand North America. It is mentioned in some of the most widely used text booksin Computational Algebra, and plays a major role in the book Kreuzer-Robbiano“Computational Commutative Algebra” (Springer).

The future of CoCoALately the CoCoA project has entered a new phase: completely rebuilding and

restructuring the program employing more modern algorithms where available. Theaims of the new version include offering better flexibility and performance whileretaining the simplicity of use for which CoCoA has become widely appreciated.

A crucial design decision was the passage from C to C++ as the implemen-tation language: the improved expressivity of C++ allows the source code to bemore readable while offering better run-time performance. We expect concomitantbenefits for future maintenance of the source. The new design is expressly as alibrary; a server (communicating via OpenMath) and a standalone interactivesystem will be built on top of this library.

CoCoALib, being the core of the project, is also its most evolved part, and isthe part that we shall look at most closely. In keeping with the theme of readyaccessibility the software is to be free and open in the sense of the GPL.

Philosophy of CoCoALibOur aim is for CoCoALib to offer reference implementations of the principal

algorithms in computational commutative algebra. The development of the libraryand the other components requires an enormous investment of time and resources.So that this investment is worthwhile we want to ensure that the software is widelyavailable and will live for a long period of time. Consequently our implementationshave to satisfy various design criteria:∗ the code must exhibit good run-time performance∗ the source code must be clear and well designed∗ the source code must be well documented (both for users and maintainers)∗ the source code must be clean and portable∗ the code must be easy and natural to use

One implication of these criteria is that the design should reflect the underlyingmathematical structure since this will ensure that the library is natural to use.

Another implication is that we regard clear and comprehensible code as beinggenerally more desirable than arcanely convoluted code striving to achieve theutmost in run-time performance because clear code is easier to maintain and shouldlive longer. Our experience is showing that this emphasis on cleanliness is alsoproviding quite good run-time performance.

[1] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra,Available at http://cocoa.dima.unige.it



A Maple package for the computation of realzeros of hypergeometric functions

Alfredo Deano*, Amparo Gil, Javier Segura

Departamento de Matematicas, Univ. Carlos III. Avda. Universidad, 30. 28911,Leganes. Madrid, Spain.Departamento de Matematicas, Estadıstica y Computacion, Univ. de Cantabria. Avda.de los Castros s/n. 39005, Santander, [email protected], [email protected], [email protected]

2000 Mathematics Subject Classification. 33C05, 33C10, 33C15

We present a set of Maple routines for the computation of real zeros of classi-cal hypergeometric functions. The algorithms are based on a fixed point methodconstructed for ratios of special functions, and is based on general properties ofthe hypergeometric differential equations and of first order systems of difference-differential equations (DDEs) that relate two functions of the same family withtheir derivatives.

The problem of computation of real zeros of hypergeometric functions arises inmany different contexts, from physical applications to problems in numerical anal-ysis, such as the computation of Gaussian quadrature rules. Several algorithms,such as matrix methods [6], Newton-type iterations and asymptotic approxima-tions are proposed in the literature, but a general algorithm suitable for differentfamilies of hypergeometric functions seems to be missing.

Our fixed point method is applicable to a wide spectrum of classical specialfunctions, given that the basic ingredients are common to many of them. The casesconsidered include classical orthogonal polynomials of Jacobi, Laguerre and Her-mite type, Coulomb wave functions, Bessel functions, confluent (Kummer) hyper-geometric functions 1F1(a; c; x) and Gauss hypergeometric functions 2F1(a, b; c; x).

Let y(1)(x), y(2)(x) and w(1)(x), w(2)(x) be two fundamental sets of solu-tions of two ODEs of hypergeometric type:

y′′(x) + Ay(x)y(x) = 0, w′′(x) + Aw(x)w(x) = 0.

As explained in [8], [9] both the pair y(1)(x), w(1)(x) and y(2)(x), w(2)(x)satisfy a unique system of difference-differential equations (DDEs):

y′(x) = a(x)y(x) + d(x)w(x)w′(x) = e(x)y(x) + b(x)w(x) (1)

Under quite general hypothesis, the coefficients a(x), b(x), d(x) and e(x) aredifferentiable functions of x in an interval I, and this implies [10] that the zeros of



y(x) and w(x) in I are real, simple and interlaced. With suitable normalizationsof the functions y(x) and w(x) and the following change of variable:

z(x) =∫ x √

|d(t)e(t)|dt (2)

we obtain a transformed system of DDEs:

˙y(z) = −η(z)y(z) + w(z)˙w(x) = −y(z) + η(z)w(z)

(3)

Here the dots indicate differentiation with respect to z, and:

η(z) =b(z)− a(z)

2x(z) +

14

d

dz

∣∣∣d(z)e(z)

∣∣∣. (4)

If we define the ratio of functions:

H(z) = sign(d)y(z)w(z)

, (5)

then given an initial data z0 the fixed point scheme:

zn = zn−1 − arctan(H(zn−1)), n = 1, 2, . . . (6)

defines a globally and quadratically convergent method for the computation of thezeros of H(z), which are the zeros of y(z) and are the transformed zeros of y(x) bymeans of the change of variable (2). See [10]. This information can be translatedto the original variable x by undoing the change of variable.

Apart from the basic structure of the method and its numerical properties, wehighlight the following aspects of the algorithm:

• The user can select the interval where zeros are to be computed. This resultsin higher efficiency when only some zeros are needed.

• The program combines symbolic and numeric computations, and it has beendeveloped in Maple, which gives the possibility of using arbitrary precision.

• Maple provides internal routines for the computation of zeros of a few specialfunctions. Comparative tests with these routines have been performed.

• The computation of the function H(z) is a relevant issue in the developmentof the package. This function admits one or several continued fraction ex-pansions, that can be obtained from [7] or from the three term recurrencerelations available for instance in the reference [1].

• Maple can also perform a direct computation of these ratios H(z), but com-putations using continued fractions seem to be more efficient than Maplesubroutines in some cases, particularly when the parameters are large. Ac-curacy and timings tests have been carried out in this direction.



The following procedures are included in the package: zerosLegendre, ze-rosHermite, zerosLaguerre, zerosGegenbauer, zerosJacobi, zerosBessel,zerosCoulomb, zerosHyperConflu, zerosHyperGauss.

The following example illustrates a calling sequence and the corresponding out-put of the procedure zerosLaguerre for the computation of zeros of the general-ized Laguerre polynomials L

(α)n (x). The input sequence corresponds to the values

of the parameters n, α and the interval [a, b] for finding the zeros, respectively:

> zerosLaguerre(20,1,.1,1000);[0.174906752386614628075567068440, 0.587303080638268563979546393077,1.23822510183423625411479251087, 2.13139626007693336644411704962,3.27213313351698717505156378361, 4.66749446588835560306521304258,6.32653619767383619740234944518, 8.26067095201372850080575980794,10.4841673812082284988888303266, 13.0148487721526284337148512282,15.8750870127847548478429019406, 19.0932519076062506143281617151,22.7058938881731318996858612990, 26.7611702293794136557692799390,31.3245161370075262786233336111, 36.4887033461490472895610529486,42.3934227457744878495931594863, 49.2688138498684854341708713949,57.5544209713148040923628461683, 68.3770378145522808165199409351]

[1] M. Abramowitz, I. Stegun (Eds). Handbook of Mathematical Functions, with formu-las, graphs and mathematical tables. Dover, 1972.

[2] A. Deano, A. Gil, J. Segura. A Maple package for the computation of real zeros ofspecial functions. In preparation.

[3] A. Gil, J. Segura. A combined symbolic and numerical algorithm for the computationof zeros of orthogonal polynomials and special functions. J. Symb. Comput. 35. 2003(465–485).

[4] A. Gil, J. Segura. A combined symbolic and numerical algorithm for the computationof zeros of orthogonal polynomials and special functions. J. Symb. Comput. 35. 2003(465–485).

[5] A. Gil, W. Koepf, J. Segura. Computing the real zeros of hypergeometric functions.Numer. Algorithms. 36. 2004 (113–134).

[6] G.H. Golub, J.H. Welsch. Calculation of Gauss quadrature rules. Math. Comput. 231969 (221–230).

[7] L. Lorentzen, H. Waadeland. Continued fractions with applications. North Holland,1992.

[8] I. Marx. On the structure of recurrence relations. Willow Run Research, Univ. ofMichigan. 1954 (45–50).

[9] I. Marx. On the structure of recurrence relations II. Willow Run Research, Univ. ofMichigan. 1954 (99–103).

[10] J. Segura. The zeros of special functions from a fixed point method. SIAM. J. Numer.Anal. 40 (1). 2002 (114–133).



A Mathematica Package For StudyingRiemannian Geometry of Geodesic Spheres

Jose Carlos Dıaz-Ramos*, Eduardo Garcıa-Rıo

Department of Geometry and Topology, University of Santiago de Compostela, 15782Santiago de Compostela, [email protected], [email protected]

2000 Mathematics Subject Classification. 53C25, 53C30

Let (Mn, g) be a Riemannian manifold of dimension n. For each point m ∈ M wedenote by TmM the tangent space of M at m and by expm the exponential mapat m, that is, expm(v) = γv(1) where v ∈ TmM and γv is the geodesic with initialconditions γv(0) = m and γ′v(0) = v. The geodesic sphere with center m and radiusr is the set Gm(r) = expm(Sn−1(r)), where Sn−1(r) = x ∈ TmM : g(x, x) = r2is the Euclidean sphere of radius r centered at the origin of TmM . Since theexponential map is a local diffeomorphism in a neighborhood of 0 ∈ TmM , theradius r is assumed to be sufficiently small so that Gm(r) is a hypersurface. In thiscase, Gm(r) = p ∈ M : d(m, p) = r, where d is the Riemannian distance.

Geometric properties of geodesic spheres cannot be calculated in general. In-stead of providing explicit formulas, one approximates them by a power seriesexpansion as a function of the radius, but even in this case, calculations are farfrom simple. Of special relevance when studying the geometry of geodesic spheresare the so-called scalar curvature invariants, that is, traces of the tensor productof the curvature tensor and its covariant derivatives. A total scalar curvature asso-ciated with a scalar curvature invariant f is, by definition, the function F definedby

F(m, r) =∫

Gm(r)

f = rn−1

∫Sn−1

(fθm)(expm(ru))du, (1)

where θm is the volume density function. Note that, with f = 1 we retrieve thevolume of a geodesic sphere, which was first studied in [4].

The package we present here is written in Mathematica and is fully documented(contact the authors to get a copy). Henceforth, we briefly explain how to use thispackage and what kind of results can be obtained with it.

To load the package one writes

Needs["RiemannianGeometry‘GeodesicSpheres‘"];

The objective of this package is to provide power series expansions of somegeometric objects defined on geodesic spheres that have been used to characterizeRiemannian manifolds. The classical approach is to write this power series expan-sion in such a way that the coefficient of each term is expressed in terms of thecurvature of the ambient manifold, thus relating its geometry with the geometryof the geodesic sphere. Hence, each function of this package must be given someinformation about the ambient manifold. Then, by writing



info = Sequence[Dimension -> n, Delta->δ, Curvature -> R,RicciCurvature -> ρ, ScalarCurvature -> τ,CovariantDerivative -> ∇, CovariantLaplacian -> ∆,Indexes->i,j,k,l];

one saves in the variable info the dimension of the manifold, the name of theKronecker delta, and the symbols for the curvature, the Ricci tensor, the scalarcurvature, the covariant derivative and the Laplacian of the ambient manifold. Thisvariable info will then be passed as an argument to each function of this package.The command Indexes->i,j,k,l specifies which indexes to use when summa-tions appear in forthcoming formulas. Additionally, we may write the sentenceGSNotation[info] to use a notation for covariant tensors similar to the classicalnotation in Differential Geometry.

In order to calculate the second fundamental form of a geodesic sphere ourpackage uses the Ledger formula [1], implemented as

σ[u_, r_][a_, b_]= Separate[n][Ledger[u, r, info][a, b]/r]

(the function Separate just arranges the power series expansion in a suitable way).Now, we get the mean curvature of geodesic spheres and the volume density

function. See [1] for details.

h[u_, r_] = TensorSimplify[∑n

a=2 σ[u, r][a, a], u, info]

θ[u_, r_] = Separate[n][E^(∫(h[u,r]-(n-1)/r)dr)]

The function TensorSimplify[expr, u, vars, opts] simplifies the expressionexpr assuming that expr is a tensor depending on the variables vars defined on ageodesic sphere at a point exp(ru), where u ∈ TM is a unit vector. The informationabout the ambient manifold is passed as an option as in the example. Note thatthe mean curvature h is a function on a geodesic sphere, and hence a 0-tensor; forthis reason the argument vars is omitted in its calculation.

The curvature tensor of a geodesic sphere is obtained from the second funda-mental form using the Gauss equation:

R[u_, r_][a_, b_, c_, d_] = Separate[n][RadialSeries[R[a, b, c, d], u, r, 4, info]+σ[u, r][a, c]σ[u, r][b, d] - σ[u, r][a, d]σ[u, r][b, c]]

(the function RadialSeries gives the power series expanssion of a tensor definedin the ambient manifold).

The first terms of the power series expansion of the volume of a geodesic spherewere given in [4] after long and tedious calculations. It is so hard to achieve sucha formula because the curvature tensor has many symmetries and hence it is noteasy to interpret a combination of all its components. Gray and Vanhecke suc-ceeded in writing the coefficients of this power series expansion in terms of thescalar curvature invariants of the ambient manifold, giving a relatively short andunderstandable formula. Our package has a function, SphereIntegrate[φ, u,n], which integrates a tensor on a sphere of dimension n−1. We get the expansionfor the volume with our software just by typing



FormatIntegral[n][TensorSimplify[rn-1 SphereIntegrate[θ[u, r], u, n], info]]

(see (1) with f = 1). Here, FormatIntegral just formats an expression obtainedafter integrating, and TensorSimplify does the hard work of reducing the verycomplicated formula obtained after integration to the well-known formula in [4].

We have illustrated this package with a classical problem concerning the volumeof geodesic spheres. There are many other typical examples that can be obtainedwith it. This package may also be used to obtain other results. For example, whenwe move our attention from the volume to total scalar curvatures, the associatedpower series expansions can be calculated with our package. In this case, a majordifference with the volume arises. Gray and Vanhecke conjectured [4] that if thevolume of each geodesic celestial sphere is the same as the one of a Euclidean sphereof the same radius, then the manifold is flat. This conjecture has been solved fora few special cases but the general problem remains open. Similar conjecturesmay be stated for the other two-point homogeneous spaces. When one considersother total scalar curvatures, for example

∫Gm(r)

‖R‖, the answer to the analogousconjecture is affirmative: “Let (M, g) be a Riemannian manifold and (M, g) a two-point homogeneous space. Assume that the holonomy group of M is containedin the holonomy group of M and that, for each m and each sufficiently small r,∫

Gm(r)‖R‖ is the same as the corresponding one in M . Then, M is locally isometric

to M”. See [2] for details.The concept of geodesic sphere can be extended to the Lorentzian setting,

obtaining the so-called geodesic celestial spheres [3]. Using this package, we canobtain a power series expansion for the volume of geodesic celestial spheres. Thispower series allows us to characterize locally isotropic Lorentzian manifolds asthose manifolds whose volume of geodesic celestial spheres is independent of theinfinitesimal observer [3].

Acknowledgement: This package has been awarded first prize in the Spanishcontest on mathematical software SOFMAT-04, Addlink Software Cientıfico.

[1] Chen, B.-Y., Vanhecke, L., Differential geometry of geodesic spheres, J. Riene Angew.Math. 325 (1981), 28–67.

[2] Dıaz-Ramos, J. C., Garcıa-Rıo, E., Hervella, L. M., Total scalar curvatures of geode-sic spheres associated to quadratic curvature invariants, Ann. Mat. Pura Appl. 184(2005), 115–130.

[3] Dıaz-Ramos, J. C., Garcıa-Rıo, E., Hervella, L. M., Comparison results for the vol-ume of geodesic celestial spheres in Lorentzian manifolds, Differential Geom. Appl.23 no. 1, (2005), 1–16

[4] Gray, A., Vanhecke, L., Riemannian geometry as determined by the volumes of smallgeodesic balls, Acta Math. 142 (1979), 157–198.



Teaching and Learning Calculus using WIRIStechnology in Moodle environment

Ramon Eixarch1, M. Rosa Estela2, Monica Blanco2, Marta Ginovart2

Eusebi Jarauta2, Sebastia Xambo3, Jaume Franch4, Narciso Roman4

1Maths for More, S. L., Rambla de Prat 21, 1-1, 08012 Barcelona, Spain.2Departament de Matematica Aplicada III, Universitat Politecnica de Catalunya, Spain;3Departament de Matematica Aplicada II, Universitat Politecnica de Catalunya, Spain.4Departament de Matematica Aplicada IV, Universitat Politecnica de Catalunya, [email protected], [email protected], [email protected],

[email protected], [email protected], [email protected],

[email protected], [email protected]

2000 Mathematics Subject Classification. 20K

This project focuses on the teaching and learning of mathematical topics at differ-ent engineering schools of the Universitat Politecnica de Catalunya (UPC).

The field of application of the project are the mathematics taught in the first yearof university studies, with special regard to Calculus. The engineering schools in-volved in the project are Escola Tecnica Superior d’Enginyers de Camins, Canalsi Ports de Barcelona (UPC), Facultat de Matematiques i Estadıstica (UPC) andEscola Universitaria d’ Enginyeria Tecnica Agrıcola de Barcelona (UPC). Duringthis year the former two are running a pilot course and the latter is about to im-plement a similar one.

As teachers have to ensure that technology is actually an efficient learning resource,a mean to acquire the technical attitudes and skills required to tackle a problemsuccessfully, and not just an optional software module. Of course the use of tech-nology redefines the teaching and learning process. Our project works in this sense.

In this course we use virtual tools aiming at the improvement of the learningprocess. These virtual tools use WIRIS technology integrated in Moodle (Modu-lar Object-Oriented Dynamic Learning Environment). WIRIS is a software familyof products dedicated to mathematical calculation and formulas designing mostlyused as education tools for learning mathematics. Students and teachers have freeaccess to WIRIS tools through education portals.

Moodle is a course management system, a software package designed to help edu-cators create quality online courses and manage learner outcomes. The design anddevelopment of Moodle are based upon a particular philosophy of learning, a wayof thinking that is referred to in shorthand as a “social constructionist pedagogy”.Constructivism claims that people actively construct new knowledge as they inter-act with their environment. When the constructivist point of view is extended toa social group that collaboratively creates a small culture of shared artifacts with



shared meanings, then we are talking about social constructivism.

Learners can access our course through almost any browser, including InternetExplorer, Mozilla and Firefox. The course comprises a combination of learning ac-tivities for students, such as Assignment, Chat, Choice, Forum, Glossary, Journal,Lesson, Wiki and Workshop.

WIRIS is an Internet platform which, on the one hand, performs general math-ematical computations asked by its users and, on the other hand, supports thecreation of Web-accessible interactive documents and materials. Basic philosophyand briefly expressed, is that in the programming of a system for doing mathemat-ics by computer “mathematical language should be the ruler” (at least as much aspossible). This is not easy, because mathematical language is quite complex, butthe point behind the thought is that the syntax and semantics of mathematicalexpressions, including those of a logical nature, have evolved trough the centuriesinto a very expressive, concise, and reliable language. Hence, substantial advan-tages are gained by making as much of this language as intelligible as possible toa computer program.

In order to apply WIRIS technology to teaching and learning mathematics at ourengineering schools we have designed and developed a first project called EVAM(Virtual Tool for Mathematics Learning), which was followed by a second projectcalled BasicMatWeb. The main features of both projects are outlined below.

1. EVAM [http://wiris.upc.es/EVAM/]: EVAM is a virtual tool which helps rein-forcing the mathematical background of students entering an engineering school,namely, basic linear algebra (matrices, determinants, systems of linear equations),trigonometry, single variable functions (basic concepts, limits and continuity, rulesand techniques of differentiation, maxima and minima, Taylor expansions, basictechniques of integration) and plane geometry.

2. BasicMatWeb [http://wiris.upc.es/basicmatweb/]: The creation of BasicMatWebcan be envisaged as the continuation of the previous tool. This virtual tool aims atthe teaching and self-learning of the basic mathematical topics taught during thefirst year of engineering studies, including, among others, linear algebra (algebraicstructures, real vector spaces, linear functions), multivariable functions (geometricrepresentations, limits and continuity, partial differentiation, maxima and minima,Taylor series), and ordinary differential equations (general properties, analyticalmethods for solving some types of first-order ordinary differential equations).

Some advantages for users and for the academic community is that only a standardWeb browser with Java is required on the part and on the end user. In other words,users need no complementary software. The interface and the computational en-gine can also be adjusted according to the specifications of the school involved.In any case, the architecture of WIRIS enhances and excellent adjustment to the



computer and the communications facilities available.

Expected conclusions: The methodology we have just sketched in our project willfit perfectly in the framework of the European Credit Transfer System. As a pro-fessional engineer-to-be, this methodology aids the student gain competence inworking both independently and in team, managing time effectively and usingcomputer resources appropriately.

[1] Estela, M. R., Ginovart, M., Jarauta, E., Xambo, S., EVAM, Eina Virtual d’Aprenentatgede les Matematiques. (http://wiris.upc.es/EVAM), 2003.

[2] Estela, M. R., Ginovart, M., Jarauta, E., Xambo, S., BASICMATWEB, Web d’ensenyamenti autoaprenentatge en xarxa de matematiques. (http://wiris.upc.es/basicmatweb), 2004.

[3] Estela, M, R., Fonaments de Calcul. Edicions UPC, Barcelona, Second Edition 2005.[4] Moodle For Teachers, Trainers And Administrators. GNU General Public License

Version 2, June 1991.[5] Xambo, S., Eixarch, R., Marques, D., WIRIS: An Internet platform for the teaching

of mathematics in large educational communities. Contributions to Science, 2 (2):269-276, 2002.



PFPK: a tool for working with perfect forms andthe geometry of the Voronoı space

Philippe Elbaz-Vincent

Institut de Mathematiques et de Modelisation de Montpellier, Universite Montpellier II,Case courrier 051, 34095 Montpellier cedex 5, [email protected]

2000 Mathematics Subject Classification. 11H

Let N ≥ 2 be an integer. Denote by CN the space of definite and positive realquadratic forms of rank N . Given h ∈ CN , there is only a finite number of minimalvectors of h, i.e., the non zero vectors v ∈ ZN such that h(v) is minimal. We willdenote it by m(h). A form (or equivalently a lattice) h ∈ CN is said perfect if itis caracterised by its minimum on ZN − 0 and by m(h). Denote by Γ either thegroup GLN (Z) or SLN (Z). Voronoı has shown [4] (Thm., p.110) that modulo theaction of Γ and up to scalar multiplication by positive real numbers, there is onlya finite number of perfect forms.

Denote by C∗N the space of positive real quadratic forms on RN such that the

kernel is generated by a (strict) subspace of QN . Let X∗N be the quotient of C∗

N

by positive homotheties, π : C∗N → X∗

N the quotient map, XN = π(CN ) and∂X∗

N = X∗N −XN . The group Γ acts on C∗

N , and on X∗N , by the action

h · γ = γthγ, γ ∈ Γ, h ∈ C∗N ,

where γt denotes the transpose of γ.If v ∈ ZN −0, we can consider the form v ∈ C∗

N , defined as v(x) = (v|x)2. Givena finite subset B of ZN − 0, the convex hull of B is the subset of X∗

N image byπ of the subset ∑

j

λj vj , vj ∈ B, λj ≥ 0

of C∗N . If h is a perfect form, we denote by σ(h) ⊂ X∗

N the convex hull of the setm(h) of its minimal vectors. Voronoı has shown [4] (§§8-15) that the cells σ(h) andtheir intersections, when h runs through the set of perfect forms, give a cellulardecomposition (in the sense of algebraic topology) of X∗

N , compatible with theaction of Γ. We endow X∗

N of the corresponding CW-structure. If τ is a closed cellof X∗

N and if h is a perfect form such that τ ⊂ σ(h), we denote by m(τ) the setof vectors v of m(h) such that v is in τ . The cell τ is the convex hull of m(τ) andm(τ) ∩m(τ ′) = m(τ ∩ τ ′).The space X∗

N is called the Voronoı space of the perfect forms of rank N . It iscontractible and gives a compactification of the symmetric space GLN (R)/ON (R).As a consequence, the space X∗

N is of prime importance, and it is useful to be ableto compute it and also to compute X∗

N/Γ.For this purpose, we have developped a C library called LibPFPK, which is underthe GNU Public licence (a free software licence), in order to compute the full



geometry of X∗N/Γ and the main part of the geometry of X∗

N . Furthermore, withthis software, we can compute the differentials associated to this complex, somemass formula and also some eutactic properties. With an associated program calledpfpk, we can use the library via a command line mode and for instance, computestabilisers of cells, or action of element of Γ on a cell, etc. The library LibPFPK alsouses PARI[3] for the linear algebra part and is multithreads (POSIX threads). Thislibrary (and program) is only available under Unix (mainly tested and developpedunder GNU/Linux) at the current time.Example of computations done with LibPFPK

Denote by Σn, 0 ≤ n ≤ d(N) = N(N + 1)/2− 1, a set of representatives, modulothe action of Γ, of the cells σ of dimension n in the Voronoı complex and such thatno elements of their stabilizer in Γ change their orientation. For N ≤ 6 we havecomputed such sets Σn and we have the results below (the subscript (−)op meansthat we work modulo the cells such that the orientation is changed by an elementof its stabiliser).Proposition 1. (Elbaz-Vincent/Gangl/Soule, 2002).The cardinality of Σn is given as follows (empty slots denote zeros):

n 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 totalGL5(Z) 5 10 16 23 25 23 16 9 4 3 136

GL5(Z)op 0 0 0 1 7 6 1 0 2 3 20GL6(Z) 3 10 28 71 162 329 589 874 1066 1039 775 425 181 57 18 7 5634

GL6(Z)op 0 0 0 0 3 46 163 340 544 636 469 200 49 5 0 0 2455SL6(Z) 3 10 28 71 163 347 691 1152 1532 1551 1134 585 222 62 18 7 7576

SL6(Z)op 0 3 10 18 43 169 460 815 1132 1270 970 434 114 27 14 7 5486

Answering a question of Martinet, we have shown that there are, up to equivalence,exactly 21 strongly eutactic forms and 6 strongly semi–eutactic forms in rank 6.We also have confirmed the computations of Berge and Martinet in rank 5.

Euler characteristic (a.k.a. “mass formula”): From the data we can verifythat if N = 5, 6, χ(SLN (Z)) =

∑σ∈E(−1)dim(σ) 1

|Γσ| = 0, E family of represen-tatives of cells of the Voronoı complex of rank N and Γσ is the stabiliser of σ inSLN (Z). More precisely, for N = 6, we get:

45047

1451520− 10633

11520+

6425

576− 12541

192+

7438673

34560− 3841271

8640+

9238

15− 266865

448+

14205227

34560

−14081573

69120+

830183

11520− 205189

11520+

61213

20736− 1169

3840+

17

1008− 1

2880= 0 .

We have performed recently (2005) similar computations for N = 7 (but re-quires several months of computation for the full geometry of the Voronoı space).



For instance, the mass formula for N = 7 is as follows:

290879

107520− 13994381

103680+

31815503

13824− 1362329683

69120+

6986939119

69120

−7902421301

23040+

340039739981

414720− 174175928729

120960+

132108094091

69120− 27016703389

13824

+13463035571

8640− 14977461287

15360+

22103821919

46080− 8522164169

46080+

17886026827

322560

−1764066533

138240+

101908213

46080− 12961451

46080+

10538393

414720

−162617

103680+

721

11520− 43

32256

= χ (SL7(Z)) = 0

For N = 6, 7, neither the mass formulas, nor the geometry of the Voronoı spacewere known.

Important applications are the computations of the cohomology of Γ and theK-theory of Z which are detailled in [1, 2] and in a forthcoming paper (in particularfor N = 7) with H. Gangl and C. Soule.

[1] Elbaz-Vincent, Ph., Gangl, H., and Soule, C., Quelques calculs de la cohomologie deGLN (Z) et de la K-theorie de Z, C.R. Acad. des Sc. Paris, Serie I, 335, (2002),321–324.

[2] Elbaz-Vincent, Ph., Perfects Lattices, Homology of Modular groups and AlgebraicK-theory (based on joint work with H. Gangl and C. Soule), Oberwolfach Reports(OWR), Volume 2, Issue 1, 2005.

[3] Batut, C., Belabas, K., Bernardi, D., Cohen, H., Olivier, M., The PARI/GP package,1989-2006, Laboratoire A2X, Universite Bordeaux I. Primary ftp site: ftp://megrez.math.u-bordeaux.fr/pub/pari, Home Page: http://www.parigp-home.de.

[4] G. Voronoı, Nouvelles applications des parametres continus a la theorie des formesquadratiques I, J. Crelle 133 (1907), 97–178.



DamFlow Project: a postprocessing tool forparallel hydrodynamical simulations

M. J. Castro Dıaz†, A. M. Ferreiro Ferreiro?, J. A. Garcıa Rodrıguez†

† Dpto. de Analisis Matematico, Universidad de Malaga, [email protected],Spain?Dpto. de Ecuaciones Diferenciales y Analisis Numerico, Universidad de Sevilla,[email protected], [email protected]

2000 Mathematics Subject Classification. 68U20, 76M12

Introduction

This article deals with the development of efficient post-processing tools for thevisualization and analysis of the results obtained with the one and two dimensionalmodels developed in the context of the DamFlow Project [3].

The DamFlow Project focuses in the development of efficient parallel imple-mentations of Finite Volume solvers on unstructured grids to solve 2d hyperbolicsystems of conservation laws with source terms and nonconservative products.Problems of this nature arise, for example, in Computational Fluid Dynamics.

We are mainly interested in the application of these systems to the study ofgeophysical flows: models based on Shallow Water systems are useful for the simu-lation of rivers, floodings, dambreak problems, ocean currents, estuarine systems,sediment transport problems, etc. Due to the huge spatial and temporal scales inwhich these phenomena take place, they give place to very long lasting simulations(in terms of CPU time). Extremely efficient implementations are required to beable to analyze these problems in small computational time.

In [2] an efficient implementation of the first order well-balanced numericalscheme for a general system of balance laws with nonconservative products forcoupled systems of balance laws was carried out. The scheme was parallelized ina PC cluster by means of domain decomposition techniques and MPI. Domaindecomposition algorithms are used to break the domain into pieces that are thensent to each node of the cluster. Computations are performed, for each piece of thedomain, in its corresponding node and then at each time step the results are sentto each neighbour domain to continue the calculus. Very good speed-up resultswere obtained and the scheme was assessed with numerical and experimental data.Currently several problems in two dimensional domains have been implementedand parallelized: shallow water models, sediment transport models and pollutanttransport problems.

The development of these highly efficient numerical methods for realistic pro-blems has a great complexity. But, once the results are obtained, another difficultyappears when analizing the results. In some cases the results of the simulations canhave several gigabytes of information with long temporal series of data referring toseveral fluid properties like pressure, velocities, flows module, etc. Efficient tools are



required to analize these temporal series and to interactively select the informationneeded at each region: for instance, users may want to extract temporal series ofany magnitude at selected points or sections, store movies or images with colorbarsto analize the obtained results, etc.

Te development of bidimensional or three dimensional post-processing softwaredevelopment is a difficult task, that requieres a long training period and a higheffort. However, the use of third party software has several drawbacks: the way inwhich results are stored has to be adapted to the data structure that the programreads. It is also necessary to use the operating system in which the software mustrun. Finally another important drawback comes from the fact that the softwarecannot be easily adapted to meet users necessities.

Therefore, we have decided to develop our own visualization tool called PostDF(DamFlow Post-Processor), specifically designed to analize the results obtainedby the algorithms developed in the DamFlow [3] project. It runs on any commonOperating System like Linux, Unix, OSX or Windows. It is installed in our clusters,so that the simulations can be analized as they are produced, without generatinguseless network traffic.

Details of the implementation

PostDF has been developed using only open source software. It is mainly basedin the work of Phrabu Ramanchandran, author of Mayavi [1].

The implementation of PostDF has been carried out using Python as mainlanguage. Python is a powerful object oriented language developed by Guido VanRostrum, whose main characteristic is that it allows to implement relatively bigprojects with a rather short and clear code.

The GUI (Graphical User Interface) is implemented using Tkinter (TCL/Tkextension for Python) as windows Toolkit. The tool is composed of two differentparts: one devoted to 2D visualizations and the other one to 3D visualizations.The chosen 3D visualization kernel is VTK (“The Visualization Toolkit”, see [2])which is an open source, high level toolkit developed in C++ on top of OpenGL. Itsupports hundreds of algorithms for scientific visualization and image processing.The 2D kernel is based in Matplotlib (and consequently in the vector graphicslibrary Antigrain). The quality of these open source tools can compete with thebest propietary software.

There are several modules implemented in PostDF to analize different prop-erties of the simulation: visualization of vector properties (like fluid flows andvelocity), visualization of scalar properties associated to the fluid (temperature,quantity of pollutant, velocity modules, etc.), extracting 2D graphics for values atparticular points of sections, configuration and visualization of cuts with a plane,interactively selected by the user, analyzing temporal evolution of selected vari-ables in one point of the domain, plotting 2D graphics of the result, saving timesof one animation in format .jpg in a directory selected by the user and generatingvideos in format .mpeg or .avi to present the temporal evolution of the results.

Currently, the developed modules focuss on shallow water problems for one



and several layers of fluids, sediment transport problems and polluntant transportproblems. But new modules can be easily added to study another kind of problems.

Main difficulties of the implementation

The main difficulty comes from the fact that several software tools are used:Scipy is used for numerical calculations, MatPlotLib is used for the 2D stuff andVTK for the 3D visualizations the GUI is developed using Tkinter and Pmw(Python Mega Widgets), and there are third party modules from Mayavi andcontributors (lightning configuration modules, modules to generate cuts on thesolution fly, etc.)

Another difficulty comes from the fact that, as it is well known, Python is notefficient for performing intensive numerical tasks and in the visualization tool thereare several ones: the filling of VTK structures in the temporal evolution, interpo-lation of data, generation of sections, etc. All these tasks have been implementedin an efficient way using C++. As result, Python implementation has been mixedwith some particular modules developed in C++: to do this we have used somewrappers developed with Swig and Weave (a module of Scipy).

Another fact to consider is that the postprocessing tool is tailored for visualiz-ing numerical results obtained from parallel simulations. As mentioned above, thenumerical methods are implemented in a PC cluster using domain decompositiontechniques and MPI (Message Passing Interface). To improve the performance ofthe results of each computing node are stored in different files in an asynchronousway. So, the postprocessing tool is adapted to read results stored of n files (onefor each computing node of the cluster). The results of the n files must be joinedto show the results in the whole computational domain. It is necessary thus, tointerpolate the obtained results in order to recover the simulation in the completedomain. More details of the implementation can be seen at [5].

We are working currently in automatizing preprocessing tasks, to develop anintegrated and easy tool to generate meshes starting from the CAD data file of thedomain, and to study and to impose boundary and initial conditions in an easyway.

[1] Mayavi www.mayavi.org. Raphu Ramachandran.[2] VTK (The Visualization Toolkit) www.vtk.org. Kitware.[3] DamFlow project, www.damflow.org.[4] M. J. Castro, J. A. Garcıa, J. M. Gonzalez, C. Pares. “A parallel 2D finite volume

scheme for solving systems of balance laws with nonconservative products: applica-tion to shallow flows”. Accepted in Computer Methods in Applied Mechanics andEngineering, 2006.

[5] Ana Marıa Ferreiro Ferreiro. “Desarrollo de tecnicas de post-proceso de flujos hidrodinamicos,modelizacion de problemas de transporte de sedimentos y simulacion numerica medi-ante tecnicas de volumenes finitos”. Phd Thesis, March 2006.



DamFlow Project: an efficient 2D finite volumesolver for hydrodynamical fluids

M. J. Castro†, J. A. Garcıa †, J. M. Gonzalez?, C. Pares†

†Departamento de Analisis Matematico, Universidad de Malaga, Spain?Departamento de Matematica Aplicada, Universidad de Malaga, [email protected]

2000 Mathematics Subject Classification. 76M12, 65M60, 65Y05

Introduction

This article deals with the development of efficient implementations of FiniteVolume solvers on non-structured grids for 2d hyperbolic systems of conservationlaws with source terms and nonconservative products.

We are concerned in particular with the simulation of one or two layer fluidsthat can be modeled by the shallow water systems, formulated under the formof a conservation law with source terms or balance law. We are mainly interestedin the application of these systems to geophysical flows: models based on sha-llow water systems are useful for the simulation of rivers, channels, dambreakproblems, etc. Simulating this phenomena leads to very long lasting simulations inbig computational domains, so extremely efficient solvers are needed to solve andanalyze these problems in small computational time.

In this work we present an efficient implementation of a first order Roe-typewell-balanced numerical scheme for a general system of balance laws with noncon-servative products and source terms. This scheme is parallelized in a PC clusterby means of a domain decomposition technique and the use of MPI. Even if verygood speed-up results are obtained, the nature of our targeted problems demandsmore computing power: for example, 2d tidal simulations of months or years in theStrait of Gibraltar can lead to days or even weeks of CPU time in a PC cluster.Most of this CPU time is spent on performing a huge number of small matrixcomputations, similar to those carried out in multimedia software. Modern CPU’sare provided with specific SIMD units devoted to these purposes. We introduce atechnique to develop a high level C++ small matrix library that takes advantageof SIMD registers, hiding the difficulties related to the use of very low level coding.

Equations and numerical scheme

We consider a general problem consisting of a system of conservation laws withnon conservative products and source terms given by:

∂W

∂t+

∂F1

∂x1(W ) +

∂F2

∂x2(W ) = B1(W ) · ∂W

∂x1+ B2(W ) · ∂W

∂x2(1)

+S1(W )∂H

∂x1+ S2(W )

∂H

∂x2,

where W (x, t) : D × (0, T ) 7→ Ω ⊂ RN , being D a bounded domain of R2; x =(x1, x2) denotes an arbitrary point of D; Ω is an open convex subset of RN . Finally



Fi : Ω 7→ RN , Bi : Ω 7→ MN , Si : Ω 7→ RN , i = 1, 2, are regular functions, andH : D 7→ R is a known function. Observe that if B1 = B2 = S1 = S2 = 0, (1) is asystem of conservation laws; and if B1 = B2 = 0, (1) is a system of conservationlaws with source term or balance law. The one and two layer shallow water systemis a particular case of this general problem (see [2]).

The discretization of (1) is performed by means of a finite volume scheme. Thecomputational domain is divided into discretization cells or finite volumes. At theedges of each cell, a projected one dimensional Riemann Problem in the normaldirection to the edge is considered, whose solution is approached by a Roe-typefirst order well-balance numerical scheme (see [3] for details).

Brief details of the implementation

As mentioned above, a domain decomposition technique is used to break thedomain into pieces that are sent to each node of the cluster. For each piece of thedomain, the computations are performed in its corresponding node of the cluster,and a communication procedure among the nodes is performed at the end of eachtime step. In order to reduce the the data transfer, a specific buffer structure hasbeen developed. The implementation has been carried out with MPI and C++.

In order to reduce the computational cost, we have developed a high level C++small matrix library that takes advantage of multimedia SIMD instructions, presentin modern processors (like Pentium processors), hiding the difficulties related tothe use of very low level coding (mostly assembler) needed to develop an efficientSIMD implementation. This has been implemented in a 8 dual Intel Xeon EM64Tcluster. Each Xeon EM64T processor has 16 SSE2 registers that provide to theprocessors with an SIMD parallel architecture. To develop a high level C++ librarywithout loosing the efficiency of a low level SSE implementation, several high leveltechniques have been used: templates, operator overloading, function inlining, etc.(see [3] for details).

Numerical performance of the matrix library

We consider a shallow layer of water flowing in a rectangular channel of 1 m. widthand 10 m. long with a bump placed at the middle of the domain given by thedepth function H(x1, x2) = 1−0.2 e−(x1−5)2 . Three meshes of the domain are con-structed with 2590, 5162 and 10832 volumes respectively. The initial condition isq(x1, x2) = 0, and:

h(x1, x2) =

H(x1, x2) + 0.7 if 4 ≤ x1 ≤ 6,

H(x1, x2) + 0.5 other case.(2)

The numerical scheme is run in the time interval [0, 10] with CFL = 0.9. Wallboundary conditions q · η = 0 are considered. Table (3) shows the CPU time foreach run. As it can be seen in Figure 1, the speed-up of the parallelization usingSSE noticeably diminishes for meshes 1 and 3 in the one layer case, with respectto the case in which they are not used. This phenomena is due to the fact that,due to the great efficiency of the SSE parallelization, the calculus time for eachiteration in each node is very small compared to the time spent in communications.



mesh 1 mesh 2 mesh 3CPUs. SSE NON-SSE SSE NON-SSE SSE NON-SSE

1 0m18.507s 4m52.201s 0m51.764s 14m16.735s 3m5.985s 50m21.319s2 0m10.685s 2m32.606s 0m29.066s 7m6.800s 1m38.830s 25m25.037s4 0m6.876s 1m17.556s 0m17.078s 3m38.655s 0m53.459s 12m43.717s8 0m4.340s 0m40.120s 0m10.032s 1m51.360s 0m29.315s 6m26.135s

Table 1: Calculus time: meshes 1, 2 and 3.

(a) Mesh 1: SIMD speed-up. (b) Mesh 3: SIMD speed-up.

Figure 1: Speed-up for meshes 1 and 3: one layer model.

The efficiency of mixing both kinds of parallelism increases with the mesh size. Toshow this behaviour, we consider a fourth mesh much finer than mesh number 3(mesh4, with 244.163 volumes) to compute again test 1 and compare the speed-up(see Table 2).

N. CPUs. 1 2 4 8Time for mesh 4 25m 26.436s 12m 53.427s 6m34.203s 3m24.476s

Speed-up for mesh 3 1 1.8818 3.4790 6.3443Speed-up for mesh 4 1 1.9736 3.8722 7.4651

Table 2: Speed-up: meshes 3 and 4.

[1] M. J. Castro, C. Pares. “On the Well-Balance Property of Roe’s Method for Non-conservative Hyperbolic Systems. Applications to Shallow-Water Systems”. ESAIM-Math. Model. Num., 38(5): 821–852, 2004.

[2] M. J. Castro, J. A. Garcıa, J. M. Gonzalez, C. Pares. “A parallel 2D finite volumescheme for solving systems of balance laws with nonconservative products: applica-tion to shallow flows”. Accepted in Computer Methods in Applied Mechanics andEngineering, 2006.

[3] Jose A. Garcıa Rodrıguez. “Paralelizacion de esquemas de volumenes finitos: apli-cacion a la resolucion de sistemas de tipo aguas someras”. Phd Thesis, Universidadde Malaga, June 2005.



New features of Singular 3.0 and applications tosingularity theory, algebraic geometry, grouptheory, and general relativity.

Gert-Martin Greuel* and Gerhard Pfister

Department of Mathematics, University of Kaiserslautern, Erwin–Schrodinger–Str.,D-67663 Kaiserslautern, [email protected], [email protected]

2000 Mathematics Subject Classification. 14Q20, 13P10, 68W30

Singular1 is a Computer Algebra system for polynomial computations withemphasis on the special needs of commutative algebra, algebraic geometry, and sin-gularity theory. However, Singular is not restricted to these areas, and it has beenused in various non-mathematical applications. Singular’s main computationalobjects are ideals and modules over a large variety of baserings. The baserings arepolynomial rings or localizations thereof over a field (e.g., finite fields, the rationals,arbitrary length floating points, algebraic extensions, transcendental extensions) orquotient rings with respect to an ideal.

Singular features one of the fastest and most general implementations of var-ious algorithms for computing Grobner, respectively standard bases, with respectto arbitrary monomial orderings, developed by the authors. The implementationincludes Buchberger’s algorithm (if the ordering is a well ordering) and Mora’salgorithm (if the ordering is a tangent cone ordering) as special cases. Further-more, it provides polynomial factorizations, resultant, characteristic set and gcdcomputations, syzygy and free-resolution computations, parametrization of curves,resolution of arbitrary singularities, ring normalization and computation of invari-ant rings, and many more related functionalities. Using triangular sets, Singularprovides also symbolic-numerical algorithms for solving polynomial systems.

Moreover, the system extension SINGULAR:PLURAL provides several algo-rithms related to Grobner basis calculations in “polynomial like” non-commutativerings. In order not mix with the commutative world, PLURAL has its own descrip-tion.

Based on an easy-to-use interactive shell and a C-like programming language,Singular’s internal functionality is augmented and user-extendible by librarieswritten in the Singular programming language. A general and efficient implemen-tation of communication links allows Singular to make its functionality availableto other programs.

The original motivation for the authors to develop a computer algebra systemwas the need to compute invariants of ideals and modules in local rings. In thesequel, the development of Singular was always influenced by concrete problemscoming either from mathematics or from applications.

1The Richard D. Jenks Memorial Prize for Excellence in Software Engineering for ComputerAlgebra was awarded to the Singular team in 2004.



In the presentation we show some typical applications of Singular to mathe-matics and physics: singularity theory, algebraic geometry, theory of finite groupsand general relativity.

The equisingularity stratum of a plane curve singularity

Applications of computer algebra to algebraic geometry are mainly concernedwith computing certain invariants of mathematical objects. For many invariantsalgorithms and implementations are available. However, in mathematical researchquite often families of varieties are studied and the problem arises to determine thelocus in the parameter space where these invariants are constant. These problemsare usually much harder and often algorithms are not known. Here we explain anew algorithm and its implementation where a complete answer exists. Namely thecomputation of the equisingularity stratum in a family of plane curve singularities.It computes the locus of parameters where the toplogical type of the plane curvesingularities stays constant. Computations with Singular do nicely illustrate thefact, proved by J. Wahl, that equisingularity stratum is smooth but in general nota linear subspace in the parameter space of the semiumiversal deformation of agiven plane curve singularity.

Resolution of singularities

The problem of existence and construction of a resolution of singularities is oneof the central tasks in algebraic geometry. In its shortest formulation it can bestated as: Given a variety X over a field K, a resolution of singularities of X is aproper birational morphism π : Y → X such that Y is a non–singular variety.

Although the problem of the existence of a resolution of singularities in charac-teristic zero was already proved by Hironaka in the 1960s and although algorithmicproofs of it have been given independently by the groups of Bierstone and Milmanand of Encinas and Villamayor in the early 1990s, the explicit construction of a re-solution of singularities of a given variety is still a very complicated computationaltask.

A modification of the algorithm of Encinas and Villamayor is implemented inSingular.

Characterizing finite solvable groups

In this part we describe an application of computer algebra and algebraic ge-ometry to group theory. For any word w in X, Y,X−1, Y −1 define inductivelyU1 = w and Un+1 = [xUnX−1, Y UnY −1]. B. Plotkin conjectured that there ex-ists a word w such that a finite group G is solvable if and only if there is ann such that Un(x, y) = e for all x, y ∈ G, e ∈ G the neutral element. Theproof is based on Thompson’s classification of the minimal finite non–solvablegroups (PSL(2, p), PSL(2, 2n), PSL(2, 3n), PSL(3, 3), Suzuki(2n)). Let G be oneof these groups then it is enough to prove that the polynomial equations defined byU1(x, y) = U2(x, y) have a non–trivial solution. For G = PSL(2, p) we may assumex =

(0 −11 t

), y =

(1 bc 1+bc

). The entries of U1 − U2 define an ideal I in Z[b, c, t]. It

has to be proved that V (I) has a rational point in F3p for every prime number

p. Here we can use the Hasse–Weil theorem for absolutely irreducible curves to



estimate the number of Fp–rational points in V (I). Grobner basis computationswere needed to prove that V (I) is an asolutely irreducible curve. The other groupsexcept the Suzuki groups can be treated similarly. In case of the Suzuki groupsthe corresponding V (I) is a surface. To prove that V (I) is absolutely irreducibleis much more difficult. The theorem of Hasse–Weil has to be replaced here by theLefschetz–Weil–Grothendieck–Verdier–trace formula.

Heavy Singular computations were necessary to verify the assumptions andto compute the etale cohomology groups for this formulae.

SINGULAR and general relativity

Finally we describe an application of Singular to the study of solutions ofvacuum Einstein equations. These equations are equivalent to the Ernst equationsfor a complex valued function E = f + ib. The associated space-time metric isthen obtained by solving certain ODEs derived from E. Those ODEs are singularat the so-called E-ergosurface f=0. The question is what happens to the metric atf=0. Examples are known where the metric is singular there, and others where it issmooth. By Taylorizing E and comparing coefficients on can derive necessary andsufficient conditions for the metric to be smooth along f=0. Assuming that E hasa zero of order ≤ 2 and looking at the Taylor terms of degree 1 and 2 shows indeedthat they lead to a smooth space-time metric. For zeroes of order k ≥ 3, however,there are more equations than variables. It was suspected by physicists that theequations are independent which would imply that f has no zeroes of higher orderwhich in turn would imply that the metric is smooth. However, they were not ableto prove or disprove this conjecture by commercially available computer algebrasystems. Computing a Grobner basis using Singular showed indeed that theequations were not independent for k=3,4,5 (a few seconds for k=3,4 about 50 minfor k=5 on a standard laptop). However, the Groebner basis was too big to giveany insight. It turned out that Singular could compute (in almost the same time)the minimal associated prime ideals which were suprisingly simple. From these wecould read off the solutions and it was easy to guess how the solutions should looklike for general k. Indeed, it could be proved (without using a computer) that thesolutions suggested by Singular were all solutions and that the correspondingspace-time metric was smooth along the E-ergosurface.

[1] G.-M. Greuel, G. Pfister: A SINGULAR Introduction to Commutative Algebra. SpringerVerlag, Berlin, Heidelberg, New York (2002), 605 pages.

[2] A. Fruhbis–Kruger, G. Pfister: Algorithmic resolution of singularities, In: C. Lossen,G. Pfister: Singularities and Computer Algebra. Cambridge University Press (2006),157-183

[3] T. Bandman, G.-M. Greuel, F. Grunewald, B. Kunyavskii, G. Pfister, E. Plotkin:Engel-Like Identities Characterizing Finite Solvable Groups. Preprint,http://arXiv.org/abs/math.GR/0303165, (2003). To be published in Compositio Math.(2006).

[4] P. Chrusciel, G.-M. Greuel, R. Meinel, S. Szybka: The Ernst equation and ergosur-faces. Preprint, http://arxiv.org/abs/gr-qc/0603041, (2006), 1-23.

[5] Singular homepage: http://www.singular.uni-kl.de



Developer of LiveTeXmacs

Chu-ching HuangCenter of general education, Chang-gung University, 259, wen-hua 1st Rd. Kwei-shan,Tao-yuan, [email protected]

2000 Mathematics Subject Classification. 97-xx

For the purpose of arising the learning efficiency of freshmen in Mathematic andcomputer, we build a scientific computing environment, Live TeXmacs and itsvariants for different learning purposes. Live TeXmacs is directly bootable Linux-based cdrom/dvdrom developed based on famous Knoppix Linux integrated withmany innovatory technologies in recent computer science. We remove several non-necessary packages and adds the packages that needs in mathematical teachingand learning environment. The additional applications in Live TeXmacs include:

1. scientific publishing system and CAS;

2. web applications for supporting web mathematical calculation and buildingcontent management system;

3. enhanced packages for making courseware.

Live TeXmacs is a open sources solution for mathematical learning and containstwo implementations: iContent and iTraining.

iContent: The core of this implementation is TeXmacss. It is not only a TeX-ompliance system with GUI but it also owns many cutting-edge technologies: nativeplotting capacity, external application plugin, multiple formats for documentationimport/export etc. Specially, the plugin functionality supports many open-sourceComputer Algebra System (CAS). In other words, we can execute the CAS sessionand get the answer directly on the Texmacs document. Plenty of CAS are includedin this platform, for instance:

• Symbolic calculation systems: Maxima, M2, Axiom etc.

• Numerical calculation systems: Octave, Mupad (Light version) etc.

• Scientific plotting systems: Gnuplot, Maxima, Axiom etc.

• Scientific Visualization systems: actually python programming environmentwith numeric, visual, pygame and wx modules.

CAS pre-installed in Live TeXmacs can be used in the traditional mathematicalcourses, e,g, calculus, linear algebra, geometry, differential equation etc. Some lec-tures are also developed by technologies of introducing above, for example: gametechnology applied to visualization for mathematical model etc. iContent plays therole for building Sientific Publishing System.

iTraining: This part is developed under the consideration to extend the learningbeyound the class and arise the learning efficiency. Up to now, it supports:



1. Web-based CAS: Without double, no open CAS owns web-based calculationcapacity. But incorporated with the WMI group, (Web Mathematics Inter-active), Maxima, Gnuplot, Octave and Mupad also can make service throughthe internet. WMI also supports the GUI for the people without any expe-rience about CAS. Even novice can utilize web-based CAS very well. Testbank related to WMI supported is also supported.

2. CMS: python-based CMS, Zope, is introduced as the Courseware CMS. Withthe simple and unified user interface, it allows a group of developers to worksin/within same project. All the developed courseware are put on this servers.Wih web-based interface, it is very easy for tutors to manage aor update thecourseware even they are novices.

3. Schemes for Resources transformation: LiveTeXmacs supports the elemen-tary output format for coursewares, e.g. pdf, html and tm etc. Moreover,it contains some packages for developers with the capacity of screen cap-tion/record and output standard HTML or SCORM 1.2 formats. Even record-ing through internet also works. These schemes can make coursewares morerapidly then ever.

4. Analyzer System: To get the best effect of mathematical e-Learning, we alsoset up the basic user log analyzer to study which user need and when theyvisit. These study already helps us easily to determine which part wants toimprove and when the server can be stop for maintenance. Also we also setup the test bank in order for different topic according to the analyzed result.

Despite the above schemes described above, we also develop a web-based self-quiz system with simple user interface such that designer can enlarge test bank withadditional quiz pre-made by TeXmacs (exported in XHTML or MathML format).As browsing the quiz site, server will generate a test sheet randomly according tothe chosen topic. System will also evaluate the result and correct the answer if anywrong after quiz is over. We welcome the tester answer the sheet together with thehelp of CAS in LiveTeXmacs or WMI, introduced above.

Till now, we have introduced CAS into traditional Mathematical courses, fromelementary course, Calculus,Linear algebra to advance course, differential equationetc. Example that we describes here is a graduate level 3-credit course: ?’StochasticProcesses: Stochastic Differential Equations?’, for management college in Chang-Gung University, Taiwan.

As well-known, this lecture needs the background involving probability theory,differential equation (both ODE and PDE) and real analysis etc. For the graduatepupils without enough mathematical training, it is be a hard work to step in. Inthe schedule, we use the following methods to low down the learning curve:

• plenty examples and graphics to describe the theory of probability;

• CAS for evaluating integration, solving differential equations, evaluating stochas-tic differential etc.;



• Simulation and visualization for the Models, Brownian motion and stochasticdifferential equation.

Whatever new technology we use can not replace wholly traditional mathe-matical education, but the schemes in LiveTeXmacs actually do help to pupils tounderstand the mathematical theory and think deeply.

LiveTeXmacs suggests a solution for mathematical e-Learning and is still inprogress. The core contains two parts: iTraining solution as platform for e-Learningand iContent solution as framework for making coursewares for e-Learning. Allthe coursewares are developed under cross-platform compliant and support thestandard HTML, SCORM etc. formats. All the implement uses open sources andthis makes the solution is more effective and more competitive than proprietarysolutions. LiveTeXmacs applies the cut-in point how to introduce such technologyinto mathematical teaching and learning. Setup more complicated mathematicale-Learning environment than here introduced will be not a big problem if followsour solution.

Our web site:

• Our mathematical Portal: blue http://math.cgu.edu.tw:8080/Calculus

• Web Mathematic Interactive: blue http://math1.cgu.edu.tw/wmi

• Download site :

1. images of platform: blue ftp://math.cgu.edu.tw/pub/KNOPPIX

2. Lecture demo: blue ftp://math.cgu.edu.tw/pub/KNOPPIX/StochasticIntegral

[1] Eddelbuettel, D., Quantian: A Scientific Computing Environment, Proceeding of the3rd International Workshop on Distributed Statistical Computing, Vienna, Austria,2003.

[2] Sangwin, C. J., Encouraging higher level mathematical learning using computer aidedassement ICM2002 Satellite Conference on Mathematical Education, Lhasa, Tibet,August 2002.

[3] Watson A. and Mason J., Student-generated examples in the learning of mathematics,Canadian Journals for Science, Mathematics and Technology Education, 2(2), (2002).

[4] Kendall, W. S., Symbolic Ito Calculus in AXIOM: A Nongoing Story, Research Report,University of Warwick 327, (1999).



Smith Normal Form can be computed usingGrobner bases

Manuel A. Insua*, Manuel Ladra

Departamento de Algebra, Universidad de Santiago de Compostela, Spain


It is well known that it is possible to decompose a module using Grobner bases,on the other hand, Smith Normal Form of a matrix can be reached from a moduledecomposition; then, it seems that there must be a connection between Grobnerbases and Smith Normal Form and this connection was our first objective in [1].

In http://algo.inria.fr/chyzak/Mgfun/faq.html somebody asks Frederic Chyzak“Is it possible to compute a generalized Smith normal form for a matrix overlinear operators using Grobner bases?”. His answer is that it is possible to makerows operations but he doesn’t know whether this extends to a diagonalizationprocedure. This fact encouraged us in our objective of seeking the relation betweenGobner bases and Smith Normal Form.

In [1] we prove that it is possible to compute Smith Normal Form of a matrixusing Grobner bases. The point key to achieve it is that computing a Grobnerbase performs rows operations in a matrix. This fact plus the theory of Grobnerbases when the coefficient ring is a principal ideal domain (p.i.d.)[2, 3] allowedus to programme SmithGroebner package. SmithGroebner is a functions packagewriting in Mathematica which can obtain, using Grobner bases, Smith NormalForm of a matrix with entries in any p.i.d. which we want.

The default rings which we can work are:1

Z Z[i] Q R C Zp

Fq Z[ 1+√−19

2 ] Q[x] R[x] C[x] Zp[x]Fq[x]

SmithGroebner is able to calculate:

1. Smith Normal Form of a matrix A with entries over any principal idealdomain (p.i.d.) of the previous list and the transforming matrices P and Q.

2. Frobenius canonical form, Second Canonical form, Jacobson canonical formand Jordan canonical form of a matrix with entries over any field of theprevious list.

3. It is well known that every finitely generated module M over a p.i.d. Ddecomposes in the following way

M = Dz1 ⊕ · · · ⊕Dzs

1Fq = finite field of characteristic p and q = pr elements.



such that

(0 : z1) ⊇ (0 : z2) ⊇ · · · ⊇ (0 : zs); (0 : zk) 6= D, k = 1, . . . , s

SmithGroebner calculates zi and the generators of (0 : zi) i = 1, . . . , s too.

The main characteristic of SmithGroebner is that it was programmed thinking inany p.i.d. To make calculations over any new p.i.d., which is not in the previous list,we must add only a piece of Mathematica code in the function CargaFunciones. Inthis code we must programme the algorithm to obtain the greater common divisorof two elements of the p.i.d. because this algorithm is different one p.i.d. to another.One time that the package “knows” how to calculate the greater common divisorof two elements we don’t need programme any more to obtain Smith Normal Formof a matrix.

[1] Insua Hermo, Manuel A., Varias perspectivas sobre las Bases de Grobner: FormaNormal de Smith, Algoritmo de Berlekamp y Algebras de Leibniz, Tesis Doctoral,Servicio de Publicaciones, Universidad de Santiago de Compostela (2005).

[2] Moller, H., On the construction of Grobner basis Using Syzygies, J. Symbolic Com-putation 6 (1988), 345–359.

[3] Pan, L., On D-bases of Polynomial Ideals Over Principal Ideal Domains, J. SymbolicComputation 7 (1988), 55–69.



Formalization of the Jordan Curve Theorem inMizar

Artur Korni lowicz

Institute of Computer Science, University of Bialystok, ul. Sosnowa 64, 15-887Bialystok, [email protected]

2000 Mathematics Subject Classification. 68T15

The aim of this work is a brief presentation of the formalization (mechanization)of the proof of the Jordan curve theorem done in Mizar - a state-of-the-art proofchecker.

The year 2005 was the 100th anniversary of the first correct proof of the Jordancurve theorem given by Oswald Veblen. The theorem was formulated by CamilleJordan in his textbook “Cours d’Analyze de l’Ecole Polytechnique” in 1887. Nowa-days there are different formulations of the theorem, but all are equivalent to thestatement: any simple closed curve in the plane separates the plane into two disjointregions, the inside and the outside.

Since Jordan’s times many different proofs of the theorem have arisen. Someof them were false (for example the original one), some had small gaps. They useddifferent mathematical backgrounds and techniques to reach the target, like thetheory of planar graphs, the Brouwer fixed point theorem, or the technique ofapproximations of a given curve.

The year 2005 was, in some sense, decisive is the history of proofs of the the-orem. Its two different formalizations, entirely verified by computers, were com-pleted that year. First, in January, Thomas C. Hales completed his formaliza-tion in HOL. All files containing his formalization are available at his homepagehttp://www.math.pitt.edu/~thales/. Second, in September, the author of thiswork finished a formalization in Mizar, see [2], (relevant files are available at theMizar homepage, http://mizar.uwb.edu.pl). Both works confirm once againthat practical formalization of nontrivial mathematics with existing proof assis-tants is possible.

From now on we focus on the Mizar formalization only. The Mizar formal-ization of the Jordan curve theorem follows, with small modifications, the proofpresented in [1]. The statement is formulated as:

for C being Simple_closed_curve holds C is

Jordan;

where, the property Jordan is defined as:

definition

let S be Subset of TOP-REAL 2;

attr S is Jordan means

:: JORDAN1:def 2

S‘ <> &



ex A1, A2 being Subset of TOP-REAL 2 st

S‘ = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 &

for C1, C2 being Subset of (TOP-REAL 2)|S‘ st C1 = A1 & C2 = A2 holds

C1 is_a_component_of (TOP-REAL 2)|S‘ & C2 is_a_component_of (TOP-REAL 2)|S‘;

end;

The formalization was done in strong international cooperation, especially be-tween University of Bialystok, Poland and Shinshu University, Japan. Originally,we started a formalization of the Jordan curve theorem following [4], with the ideaof external and internal approximations of a given curve. Using this machinerya so called “easy case” of the theorem (about curves containing only horizontaland vertical intervals) was formalized, see [3]. Next, a proof of the existence ofa bounded component of the complement of a given curve was completed, see[5]. In the meantime, when the uniqueness of the component was to be proved,Korni lowicz and Shidama formalized the Brouwer fixed point theorem in Mizar,which created a new possibility to complete the formalization of the Jordan curvetheorem faster and with a shorter proof (no proof of the easy case, which simplybecome a trivial consequence of the general case, is required) following [1]. Thenwe decided to reorganize our formalization.The idea of the proof is to affinically transform a given simple closed curve to putit into a fixed rectangle, to construct a particular point lying inside a boundedcomponent of the complement of the transformed curve, that is the existence of abounded component, and referring to the Brouwer fixed point theorem, the Tietzeextension theorem and the Fashoda meet theorem (about intersecting of north-south and west-east paths) to show the uniqueness of the bounded component.The structure of the proof is presented in Figure 2.

Fundamental Group of the Circle

Brouwer Fixed Point Theorem

Urysohn Lemma

Tietze Extension TheoremFashoda Meet Theorem

Jordan Curve Theorem

?

XXXXXXXXz ?

?

9

Figure 2: Proof of the Jordan curve theorem

Figure 3 presents some quantitative data about our formalization. Only mainfiles have been taken into account with the calculations.

At the end we will present the Mizar system itself. The Mizar project startedaround 1973 as an attempt to reconstruct mathematical vernacular in a computer-oriented environment. Preliminary experiments resulted in a system implementedin 1976, see [6]. Proofs were (and still are) written in Jaskowski natural deductionstyle. The next step in the development of the project, started in 1989, was theestablishment of the Mizar Mathematical Library - the easily accessible reposi-tory of formally checked mathematical texts. The library is based on the TarskiGrothendieck set theory. Recently, the Mizar Mathematical Library is considered



Theory Articles Lines Bytes Zip BytesAlgebraic topology 7 16 364 560 856 105 172Brouwer 1 1 900 65 679 13 319Urysohn 3 7 132 248 030 37 599Tietze 1 1 569 59 566 13 083Fashoda 1 1 451 65 524 14 501Jordan 1 6 801 229 257 41 077total 14 35 217 1 228 912 224 751

Figure 3: Numeric data

the biggest such a repository in the world. Its version 4.53.937 includes 937 articleswritten by 180 authors with 7926 definitions, 42150 theorems, 724 schemes and6856 registrations. More data can be found at http://merak.pb.bialystok.pl/.

1. Gauld D., Brouwer’s Fixed Point Theorem and the Jordan Curve Theorem.http://aitken.math.auckland.ac.nz/~gauld/750-05/section5.pdf.

2. Korni lowicz A., Jordan curve theorem. Formalized Mathematics, 13(4):481–491, 2005.3. Korni lowicz A., Properties of left and right components. Formalized Mathematics,

8(1):163–168, 1999.4. Takeuchi, Y. and Nakamura Y., On the Jordan curve theorem. Technical Report

19804, Dept. of Information Eng., Shinshu University, 500 Wakasato, Nagano city,Japan, April 1980.

5. Trybulec A., Preparing the internal approximations of simple closed curves. Formal-ized Mathematics, 10(2):85–87, 2002.

6. Trybulec A., The Mizar-QC/6000 Logic Information Language. ALLC Bulletin,6(2):136–140, 1978.



Fermat, Computer Algebra System

Robert H. Lewis

Mathematics Department, Fordham University, Bronx, New York [email protected]


Fermat is an interactive system for mathematical experimentation. It is a supercalculator – computer algebra system, in which items being computed can be inte-gers (of arbitrary size), rational numbers, real numbers, complex numbers, modu-lar numbers, finite field elements, multivariable polynomials, rational functions, orpolynomials modulo other polynomials. The main areas of application are multi-variate rational function arithmetic and matrix algebra over rings of multivariatepolynomials or rational functions. Fermat does not do simplification of transcen-dental functions or symbolic integration.

A session with Fermat usually starts by choosing rational or modular “mode”to establish the ground field (or ground ring) F as Z or Z/n. On top of this maybe attached any number of symbolic variables t1, t2, . . . , tn, thereby creating thepolynomial ring F [t1, t2, . . . , tn] and its quotient field. Further, some polynomialsp, q, . . . can be chosen to mod out with, creating the quotient ring F (t1, t2, . . .) / <p, q, . . . >. Finally, it is possible to allow Laurent polynomials, those with negativeas well as positive exponents. Once the computational ring is established in thisway, all computations are of elements of this ring. The computational ring can bechanged later in the session.

In an earlier version, called FFermat, the basic number type is real (or complex)numbers or “floats” of 18 digits. That version allows for numerical computingtechniques, has extensive graphics capabilities, no sophisticated polynomial g.c.d.algorithms, and is available only for Mac OS.

Fermat runs on Mac OSX, OS9, Linux, Unix, and Windows95/98/NT/XP etc.Fermat was originally written in Pascal for a DEC Vax, then for Mac OS during1985 - 1996. It was ported to Windows in 1998. In 2003 it was translated into Cand ported to Linux (Intel machines) and Unix (Sparc/Sun). It is about 87000lines of C code.

Fermat has extensive built-in primitives for array and matrix manipulations,such as submatrix, sparse matrix, determinant, normalize, column reduce, rowechelon, Smith form, and matrix inverse. It is consistently faster than some wellknown computer algebra systems, especially in multivariate polynomial g.c.d. It isalso space efficient.

The basic data item in Fermat is a multivariate rational function or quolyno-mial. The numerator and denominator are polynomials with no common factor.Polynomials are implemented recursively as general linked lists, unlike some sys-tems that implement polynomials as lists of monomials. To implement (most) finitefields, the user finds an irreducible monic polynomial in a symbolic variable, sayp(x), and commands Fermat to mod out by it. Low level data structures are set up



to facilitate arithmetic and g.c.d. over this newly created ground field. Two specialfields, GF (28) and GF (216), are more efficiently implemented at the bit level.

To help implement the Dixon resultant technique [7], special features have beenadded to the determinant function.

Fermat provides a complete programming language. Programs and data can besaved to an ordinary text file that can be examined as such, read during a latersession, or read by some other software system.

I envision the users of Fermat to be rather sophisticated in both mathemat-ics and programming. (However, you don’t have to do any programming to useFermat.) At many places in the design and implementation of Fermat I had tobalance the conflicting goals of flexibilty and safety. That is, whether to allow theuser certain freedoms or language features that might perhaps be abused, or tocircumscribe the user in the name of safety. Since I regard the users as sophisti-cated, I have usually chosen freedom. It is easy to interrupt a long computation,examine data, save to a file, make changes, and resume the computation.

Fermat is shareware. Executable binaries and manuals can be downloaded fromthe web site, http://www.bway.net/ lewis. Pages there compare Fermat to otherCA systems on tests such as determinant, rational function arithmetic, and g.c.d. ofmultivariate polynomials. Some of these tests have been conducted by independentresearches.

Fermat has been crucial to the success of several projects. The authors wereunable to make progress with large well known computer algebra systems. [3],[9] and [10] were pure mathematics applications in which matrix normalizationsor characteristic polynomial were important. [1], [8], [11], and [12] needed to solvesystems of polynomial equations, and used the Dixon resultant method [2], [7] withFermat. [4], [6] and [6] use Fermat’s very efficient rational function arithmetic. [13]used Fermat’s row reduction of integer matrices.

[1] Bozoki, Sandor and Robert H. Lewis, Solving the least squares method problem inthe AHP for 3 x 3 and 4 x 4 matrices. Central European Journal for OperationsResearch, 13 (2005) p. 255 – 270.

[2] L. Buse, M. Elkadi, and B. Mourrain, Generalized resultants over unirational alge-braic varieties. J. Symbolic Comp. 29 (2000), p. 515-526.

[3] Brumer, Armand. The Rank of Jo(N). Asterisque 228 (1995) p. 41–68.[4] Chetyrkin K. G., M. Faisst, C. Sturm, and M. Tentyukov. ε-Finite basis of master in-

tegrals for the integration-by-parts method. (2006) 28pp. http://arxiv.org/pdf/hep-ph/0601165.

[5] Czakon, M. The four-loop QCD beta-function and anomalous dimensions. DESY-04-223, SFB-CPP-04-62 (2004). in Nucl.Phys.B710: p. 485–498, 2005.

[6] Czakon, M, J. Gluza, T. Riemann. Master integrals for massive two-loop BHABHAscattering in QED. DESY-04-222, SFB-CPP-04-61, Dec 2004. 21pp. Published inPhys.Rev. D71:073009, 2005.

[7] D. Kapur, T. Saxena, and L. Yang, Algebraic and geometric reasoning using Dixonresultants. In: Proc. of the International Symposium on Symbolic and Algebraic Com-putation. A.C.M. Press (1994).

[8] Lewis, Robert H. and Stephen Bridgett, Conic tangency equations and Apolloniusproblems in biochemistry and pharmacology. Mathematics and Computers in Simu-lation 61(2) (2003) p. 101–114.



[9] Lewis, Robert H. and Guy D. Moore. Computer search for nilpotent complexes.Experimental Mathematics 6:3 (1997) p. 239–246.

[10] Lewis, Robert H. and Sal Liriano. Isomorphism classes and derived series of almost-free Groups. Experimental Mathematics 3 (1994) p.255–258.

[11] Lewis, Robert H. and Peter F. Stiller. Solving the recognition problem for six linesusing the Dixon resultant. Mathematics and Computers in Simulation 49 (1999) p.205–219.

[12] Little, John. Solving the Selesnick-Burrus filter design equations using computationalalgebra and algebraic geometry. Advances in Applied Mathematics, 31 (2003), p.463–500.

[13] Yuen, David S. Siegel modular cusp forms (2004). Personal communication.



ActiveMath: Advanced Mathematics on the Web

The ActiveMath Group

DFKI GmbH and University of Saarland,[email protected]|dfki.de

2000 Mathematics Subject Classification. 97U70

Presenting mathematics in Web browsers is a challenge. But presentation is notthe only task on the Web, an open network of loosely connected resources andservices. The new wave on the Web is based on semantics where one declares anduses objects by their meanings so as to promote interoperability. Mathematicalobjects can be expressed semantically using the OpenMath language. Documentsto write about mathematical symbols can be defined in OMDoc.

ActiveMath is an intelligent learning environment on the Web. It uses thesemantics of the OpenMath and OMDoc languages to provide high-quality Webpresentations of mathematical documents, intelligent selection of content-items toachieve learning goals, search for text and mathematical objects, copy and pasteof formulæ, and interactive exercises with learner inputs evaluated by classicalcomputer algebra systems.

ActiveMath is a learning environment that is free and open-source for publiceducational institutions. It is made as a Java Web application and serves 30-50learners simultaneously. More information about the software, the underlying re-search, and access to a demo can be obtained from http://www.activemath.org/.

1 Advanced Mathematics on the Web

For mathematical documents to satisfy the promises of the Web they need at least: (1)to be accessible by several paradigms including the simple book or search paradigms,(2) to be accessed using simple http requests so as to enable access by automated pro-cesses such as search engines or bookmark-checking and, of course, (3) to be accessibleto contemporary Web browsers manipulated by humans where they are presented in ahighly readable fashion, respecting notation customs and supporting the readers, as wellas (4) to be authored and managed using easily accessible authoring tools, and (5) to beencoded in a way that promotes interoperability where the mathematical semantics offragments and operators are agreed upon and are made available to other programmes

Moreover, the following requirements apply for a mathematics learning environmenton the Web: (a) interactive exercises should be provided allowing input of formulæ andmathematical evaluation, (b) learner modelling should be achieved based on the (au-thored) domain-concepts’ structure, (c) content selection and advice should be availableto guide learners through the available content.

These requirements are among the foundations of the ActiveMath learning environ-ment, a mathematics learning environment on the Web. We present how ActiveMathsatisfies these needs.



2 Semantic Mathematics and its Usage

As hinted in the requirements, ActiveMath relies on a semantic encoding of mathe-matics. The OMDoc language [1] has been chosen for this: it provides the structure ofmathematical documents with content-elements such as definition, proof, or example.Each of these items contains a metadata and textual part. The metadata is a place tospecify information about the item, such as the relation of an item of being requiring an-other or the typical learning-time. The textual part is the content that will be presentedto the user; it is made of text fragments with links and formulæ. Formulæ are encoded inOpenMath, a European standard to encode mathematical objects [2]. OpenMath ex-presses the objects as terms and function applications of mathematical symbols; the latterare the semantic hooks, which have a description in content-dictionaries or, in OMDoc, insymbol elements. The latter can be attributed one or several definitions. ActiveMathuses these ingredients to support the following features:

Tutorial Component Based on this structure of symbols and other conceptualitems, as well as their relations, a domain model is built and used as a basis for thebeliefs about the learner’s competencies. The metadata and element types allow the tu-torial component [3] to generate a book matching the learners’ goals and desired learningscenario.

Access to Content in ActiveMath is done in the well-known paradigm of books.Books can be re-arranged which allows appropriation of the content by the learner.Accessing the content is also possible through a search interface, which allows to search fortexts, formulæ, or item-characteristics. The semantic nature of the formulæ in OpenMathis a fundamental requirement of the mathematical search.

Interactive Exercises Semantic mathematical formulæ can be computed by enginesthat understand them. Using this feature and a carefully designed user-interface, Active-Math offers an exercise system where learners can input values of multiple-choices orformulæ blanks and request evaluations by a computer algebra system (currently, Yacas,Wiris, Mathematica, and Maxima). These systems allow the (constrained) evaluationof the learners’ input which leads to feedback, for further steps maybe, and providesdiagnosis to the learner-model for updating its current beliefs.

High-Quality Formulæ Rendering on the Web Rendering mathematics onthe Web is a challenge because of the diversity of the browser capabilities. ActiveMathmostly still uses HtML and css, which is most widespread. It also offers xHtML+MathML,which better supports difficult mathematical layout, and offers pdf and svg, using theLaTeX engine. Experience has proved that HtML is best suited for interactivity, followedby xHtML+MathML and svg. The layout quality of LaTeX is still unsurpassed.All these presentation formats are produced using xsl stylesheets. The content is madeof authored OMDoc documents, including new symbols. Notations are also authored, in-cluding notations for arbitrary formulæ patterns.

Input of Formulæ Finally, in order for the learner to input mathematical formulæ,OpenMath is used again: the learner can use either a textual syntax, imitating classicalsyntaxes or a graphical input editor: all the inputs are translated to OpenMath which issent to the computer algebra systems for evaluation or to the search engine as query.



Because the learners’ input of formulæ is often very close to the formulæ presentedin the documents, the ability to drag-and-drop a term from the presentation to the inputeditor is provided.

3 Software

ActiveMath is a Java Web application. It is distributed free of charge and open-sourcefor public academic institutions or individuals and runs on any contemporary operating-system with Java 1.4 and more than 512 Mb of RAM. It is designed to serve well aclassroom of 30-50 learners simultaneously and has been experimented to run with 200students. The architecture of ActiveMath uses several xml-rpc Web services whichallows loose coordination of components and usage of client components, see [4].

A demo of ActiveMath can be seen at:

http://www.activemath.org/demo.php.

References

1. Kohlhase, M.: OMDoc: Towards an openmath representation of mathematical doc-uments. Seki Report SR-00-02, Fachbereich Informatik, Universitat des Saarlandes(2000) See also http://www.mathweb.org/omdoc.

2. Buswell, S., Caprotti, O., Carlisle, D., Dewar, M., Gaetano, M., Kohlhase, M.: Theopenmath standard, version 2.0. Technical report, The OpenMath Society (2004)Available at http://www.openmath.org/.

3. Ullrich, C.: Course Generation Based on HTN Planning. In Jedlitschka, A., Brand-herm, B., eds.: Proceedings of 13th Annual Workshop of the SIG Adaptivity and UserModeling in Interactive Systems, Saarbrucken, Germany (2005) 74–79

4. Melis, E., Goguadze, G., Homik, M., Libbrecht, P., Ullrich, C., Winterstein, S.:Semantic-Aware Components and Services of ActiveMath. British Journal of Ed-ucational Technology 37 (2006) 405–423



Complex structures on indecomposable6-dimensional nilpotent real Lie algebras

Magnin, L.

Institut Mathematique de Bourgogne UMR CNRS 5584, Universite de Bourgogne, BP47870, 21078 Dijon, [email protected]

2000 Mathematics Subject Classification. 17B30, 53C15

This talk is at the interplay between mathematical software systems and advancedpure mathematical research. In practically all fields of Mathematics, modern com-puter technology can be of great aid to pure research. Simulation, working outof examples, conjecture testing, discovering of specific numerical formulas: all thisso called experimental Mathematics can eventually lead to deep insights into newtheoretical developments, and is also currently used to find out which theoremscan be proved theoretically. But resort to computers is really crucial to some un-solved theoretical problems, where due to the present lack of an effective construc-tion scheme, one has to face computations which simply cannot be done by handeven by the most skilled mathematician: typically, a large set of non-linear equa-tions which require repeated steps each involving human inspection to select sometractable unknown and then computer computations, which can eventually resultin huge expressions (sometimes more than a page for a single matrix element). Insuch cases, computer algebra systems with associated programming code are of theutmost importance.

Outline of the talk. In this talk, we apply that method to solve a non trivialproblem that is accessible to a broad mathematical audience: the computation ofall integrable complex structures on real 6-dimensional indecomposable nilpotentLie algebras. To be specific, let g be any real 6-dimensional nilpotent Lie algebraand Xg the set of integrable complex structures on g (see definition below). Anupper bound has been given in [6] for the dimension of Xg. We give for each g withXg 6= ∅ an explicit description of Xg, along with a complete system Gg of rep-resentatives of the various equivalence classes under the action of Aut g. No suchovertly explicit results were known, and we get that the bounds of [6] are sharpexcept for one case. We also prove the rather striking fact that Xg is a smoothsubmanifold of the space of 6x6 real matrices. In the last moments of the talk,for each J ∈ Gg we equip the simply connected real Lie group G0 associated tog with the structure of left invariant complex manifold G = (G0, J) defined by J.We compute a global holomorphic chart on G, proving these objects to be holo-morphically equivalent to C3. We also give the expression of the multiplication inthat chart and discuss various non standard complex Heisenberg groups. Theseresults are to appear in [1], [2]. Finally, one should mention that methods devel-oped, and possibly results obtained, in the present work could have applicationsin modern mathematical physics. Indeed, in Superstring Theory, one usually adds



to the 4-dimensional space time of relativistic theories a 6-dimensional compactmanifold (e.g. a 3-dimensional complex Calabi-Yau manifold). A complete studyof these manifolds is likely to require very advanced mathematical machinery, buton one hand it is not impossible that methods similar to those used could proveuseful in such applications even though all complex varieties constructed here areholomorphically equivalent to C3. And on the other hand, should a quantization ofsuch manifolds prove necessary, their systematic study as Lie algebras and groupscould be a useful intermediary step. In any case, there is a point in studying moresystematically 6-dimensional Lie algebras as we do here. We here are interestedonly in indecomposable Lie algebras, though direct products could be processed inthe same way.

Labeling the algebras. There are 22 indecomposable nonisomorphic nilpotentreal 6-dimensional Lie algebras in the Morozov classification, labeled M1-M22.Types M14 and M18 are splitted into M14±1 and M18±1. Over C, types M14and M18 are not splitted and types M5 and M10 do not appear since then M5 ∼=n × n where n denotes the 3-dimensional complex Heisenberg Lie algebra andM10 ∼= M8. We label the algebras according to [3], except for M5, M10, M14 andM18.

Complex structures. Let g be any finite dimensional real Lie algebra. An al-most complex structure on g is a linear map J : g → g such that J2 = −1. Thealmost complex structure J is said to be integrable if it satisfies the condition

[JX, JY ]− [X, Y ]− J [JX, Y ]− J [X, JY ] = 0 ∀X, Y ∈ g.

From the Newlander-Nirenberg theorem, that condition simply means that theconnected simply connected real Lie group G0 associated to g can be given thestructure of a complex manifold with the same underlying real structure and suchthat the canonical complex structure on G0 is the left invariant almost complexstructure J associated to J. By a complex structure on g, we will mean an integrablealmost complex structure on g. If the complex structure J satisfies the strongercondition

J [X, Y ] = [X, JY ] ∀X, Y ∈ g,

then (g, J) is a complex Lie algebra with C acting via J on g.Let J a complex structure on g. The complexification gC of g splits as gC =

g(1,0)⊕g(0,1) where g(1,0) = X−iJX; X ∈ g, g(0,1) = X+iJX; X ∈ g. We willdenote g(1,0) by m. The integrability of J amounts to m being a complex subalgebraof gC. In that way the set of complex structures on g can be identified with the setof all complex subalgebras m of gC such that gC = m⊕m, bar denoting conjugationin gC. This is the algebraic approach. Our approach is more trivial since we simplyfix a basis of g and compute all possible matrices in that basis for a complexstructure.

Finally, the automorphism group Aut g of g acts on the set Xg of all complexstructures on g by J 7→ Φ−1 J Φ ∀Φ ∈ Aut g. Two complex structures J1, J2

on g are said to be equivalent if they are on the same Aut g orbit.



Presentation of results. We consider only indecomposable 6-dimensional nilpo-tent real Lie algebras with Xg 6= ∅. We first give the commutation relations of thebasis (xj)16j66 of g we use, and the matrices J = (Jk

j ) = (ξkj ) in that basis of the el-

ements of Xg. These matrices have been obtained by writing specific programs withthe computer algebra system Reduce by A. Hearn. The programs solve simultane-ously the equation J2 = −1 and the torsion equations ij|k (16 i, j, k66) obtainedby projecting on xk the equation [Jxi, Jxj ]− [xi, xj ]− J [Jxi, xj ]− J [xi, Jxj ] = 0.These programs, and the files containing as data the matrices J are downloadablein [4]. They are indexed in [4] according to the sections of [1]. Let’s simply say thatthe equations are semilinear in the sense that they can be solved in a succession ofsteps, each of which consists in solving some equation of degree 1 in some variable.All matrix elements Jk

j ’s are rational functions under some condition. It is a facthowever that some of them are huge. We then refer to [5] and to the appropriatefile matJ∗∗.red in [4]. We will explicitly consider in the talk some examples ofprograms, discuss and run them, and examine their outputs.

For all the Lie algebras we consider, we prove that Xg is a smooth submanifoldof R36. The dimension of Xg is equal to the upper bound given in [6] except in thecase of M10. Then we give the automorphism group of g, representatives of thevarious equivalence classes, and the commutation relations of the correspondingalgebra m = g(1,0) in terms of the basis (xj)16j66 with xj = xj − iJxj . Finally weconsider the Lie group aspects with the left invariant complex manifold G = (G0, J)defined by J, J ∈ Gg, G0 the simply connected real Lie group associated to g. Thespace HC(G) of complex valued holomorphic functions on G is comprised of allcomplex smooth functions f on G0 which are annihilated by any antiholomorphicvector field. Introducing second kind coordinates on G0, we solve the equationsto compute a global holomorphic chart on G, and then express the multiplicationin G terms of that chart. There, everything can be expressed by polynomials instandard complex variables and their conjugates. If times permits it, we will discussnon standard complex Heisenberg groups.

[1] Magnin, L., Complex Structures on Indecomposable 6-dimensional Nilpotent Real LieAlgebras, International Journal of Computation and Algebra, accepted and to appear.(Preprint available at http://www.u-bourgogne.fr/monge/l.magnin)

[2] Magnin, L., Left Invariant Complex Structures on Nilpotent Simply Connected In-decomposable 6-dimensional Real Lie Groups, International Journal of Computationand Algebra, accepted and to appear. (Preprint available at http://www.u-bourgogne.fr/monge/l.magnin)

[3] Magnin, L., Adjoint and Trivial Cohomology Tables for Indecomposable Nilpotent LieAlgebras of Dimension ≤ 7 over C, online book (PostScript .ps file), (906 pages +vi), 1995, accessible at http://www.u-bourgogne.fr/monge/l.magnin

[4] Magnin, L., http://www.u-bourgogne.fr/monge/l.magnin/CS/CSindex.html[5] Magnin, L., Technical Report for Complex Structures on Indecomposable 6-dimensional

Nilpotent Real Lie Algebras, online report (PostScript .ps file) (382 pages), 2004, ac-cessible at http://www.u-bourgogne.fr/monge/l.magnin

[6] Salamon, S.M., Complex Structures on Nilpotent Lie Algebras, J. Pure Appl. Algebra,157, 2001, 311–333.



WIRIS OM Tools a Semantic Formula Editor

Daniel Marques, Ramon Eixarch, Gloria Casanellas, Bruno Martınez

Maths for More S.L., Rambla de Prat 21 1r 1a 08012 Barcelona, [email protected]

2000 Mathematics Subject Classification. 20 68-04

Abstract

The increasing use of automatic computer processing of information re-quests a new approach in mathematical formula editors. We are used toWYSIWYG editors that produce beautiful presentation of formula. The newICT services, such as database searching or calculation web-services, requirestoring the semantic information behind a formula. This can only be donewith a new generation of formula editors. We present WIRIS OM Tools [1] asemantic oriented formula editor which addresses these concerns. It is basedon the OpenMath language and a suitable transformation process betweenOpenMath and MathML expressions. Additionally, this approach comes upwhich new features for the users such as error, type and syntax checking.Our editing system is currently being used in the LeActiveMath project.

1 The goal

Currently available formula editors accomplish the task of editing formulas when theyhave to be human readable. For automatic computer processing, the resulting formulasturn out to be ambiguous or, simply, there is no interpretation to them.

It is expected that a single formula editor could give support to a broad spectrum ofComputer Algebra System [6], [8], search repositories with mathematics [5], sharing mathcontent over the web or other tools. Currently used editors don’t fulfil the requirementsrequested for such tasks.

The solution consists in developing formula editors that output suitable markup lan-guage for software components. Such markup expressions should contain unambiguoussemantic information. Both MathML content markup and OpenMath [4] serve to thispurpose.

We propose a solution based on OpenMath due to the extensibility offered by thislanguage compared to MathML content fixed number of symbols. WIRIS Editor [9], amature presentation editor, has been adopted to produce content markup.

2 State of the art

All general purpose WYSIWYG formula editors are presentation oriented, for exampleMathType, WebEQ, Formulator, MathCast and TechExplorer. Other editors appear in-tegrated in Computer Algebra Systems, word processors or scientific applications and arenot the focus of this article.

Some projects attempt to obtain content formulas using a parser or XSLT transfor-mations to convert presentation to content markup. That is in general a difficult task and



there is probably no general solution at the end of the road. Moreover, errors detected inthe input formula by the parser can not be displayed back in the original editor.

Content oriented editors are difficult to find and do not exist at the ready to use stage.Some examples are JOME [2] or IBM MathML Expression Editor [7], a TechExplorerapplication. The latest uses templates to create content MathML. We refer to templatebased editors to editors where the user interface has a palette with atomic formulas.

Once one of such atomic formulas is selected, this is added to the currently buildformula. Now, only the blanks are editable. Formulas can be built up recursively withnew atomic formulas. However, this kind of edition provides a bad experience to the user.Our approach adopts a presentation style of edition to improve the user experience.

3 WIRIS OM Tools

Our main goal was developing a content formula editor. Some of the system requirementswere

1. Keyboard input for numbers and basic operators. This includes constructions likefractions, roots, big operators and matrices which can be typed using the linearinput.

2. A configurable palette with atomic formulas reminds the user which constructionsare available.

3. Support OpenMath extensibility. That allows the edition of new symbols and is akey feature for a content oriented editor.

4. Error, type and syntax checking in real time. This feature is missing in presentationformula editors and only is possible with semantic information.

5. Web enabled and multiplatform supported, achieved using the Java plugin tech-nology.

6. Extra standard edition features like copy, paste, undo and redo. Not only insidethe editor but also with the web applications where the editor is embedded.

7. Embed true type fonts. To get high quality formulas is necessary to provide goodsupport for font typesetting.

In order to achieve such requirements, Maths for More, as a subcontractor in LeAc-tiveMath [3] project, has developed a suite of software components to edit OpenMath inWYSIWYG mode. We have named it WIRIS OM Tools. We have also defined an XMLlanguage called Domain that defines the editor behaviour.

The main components of WIRIS OM Tools are

Formula Editor Full solution for editing OpenMath in a WYSIWYG mode.

FormulaComponent Low level component for editing formulas and scriptMathML.

Phrasebook OpenMath to/from MathML translator.

Domain editor WYSIWYG editor of Domain files.

Font editor Basic font typesetting editor in TrueType format.

The main steps in the editor workflow are the following. First of all, editing mathe-matics in a WYSIWYG mode requires the use of some presentation markup language, wehave used MathML. Since we want to have OpenMath expressions we need to translate



MathML to and from OpenMath. The translation process is done by the Phrasebookcomponent and defined by the Domain file. Finally, to have a good user experience wealso need icons, organized in toolbars, to ease the editing process. The Domain languagealso includes the toolbar definition.

The translation of the formula representation to OpenMath is enhanced to makesyntax checking in real time possible. This means that, at the same time the user istyping a formula, its correctness is tested in background and the errors are highlighted.The arity of the functions is also checked using the STS signature for OpenMath symbol.

4 Conclusions and future work

WIRIS OM Tools provide an editor for content (OpenMath) as friendly as the currentlyused representation editors. In fact, a user will never notice that he is using a contentoriented editor.

Considering solely the edition aspects, content oriented editors offer the benefit ofsyntax and type error checking at real time. The extensibility of the editor will permitcreate specialized user interfaces for any mathematics topic. Highly extensible and con-figurable editors are needed when an operator or expression has a meaning different fromthe usual arithmetic practice.

The current content editor lacks of desired presentation features. More precisely, whenrendering a formula, symbols are represented depending on the current configuration ofthe editor. For example, divisions could be rendered as fractions ( 1

2) or use the opera-

tor ‘/’ (1/2); but a single formula can not contain both fractions and the ’/’ operator.Features like text attributes (colours, font size and style) are also missing. The solutionimplies providing a standard set of new symbols that defines the presentation aspects ofOpenMath.

Acknowledgements. We are grateful to the LeActiveMath project for having trusted inour development team and for the support and testing work in the project development.

1. WIRIS OM Editor. Test page at http://www.wiris.com/demo/omeditor/

2. Laurent Dirat, Michel Buffa, Jean-Marc Fedou and Peter Sander. JOME: OpenMathon the Web. Available athttp://www.tech.plym.ac.uk/maths/ctmhome/ictmt4/P55 Dira.pdf

3. LeActiveMath. Home page at http://www.leactivemath.org

4. OpenMath. Home page at http://www.openmath.org

5. Paul Libbrecht and Erica Melis. Semantic Search in LeActiveMath. Proceedings ofthe WebALT 2006 Conference. Available athttp://www.activemath.org/publications/Libbrecht-etal-SemanticSearch-

WebALT2006.pdf

6. Research Institute for Applications of Computer Algebra. Phrasebooks web pagehttp://www.mathdox.org/products/phrasebooks

7. Samuel S. Dooley. Bringing MathML Content and Presentation Markup to the Webwith the IBM MathML Expression Editor. Available athttp://www.mathmlconference.org/2002/presentations/dooleyxml/

8. The MONET Project. Home page at http://monet.nag.co.uk

9. Wiris. Home page at http://www.wiris.com



GBLA LC: Grobner Basis by Linear Algebra andLinear Codes

Borges-Quintana, M.(a); Borges-Trenard, M. A.(a);

Martınez-Moro, E.(b)(*)

(a) Departamento de Matematica, Facultad de Matematica y Computacion, Universidadde Oriente, Santiago de Cuba, Cuba.[[email protected]], [email protected]

(b) Departamento de Matematica Aplicada, Universidad de Valladolid, Castilla, [email protected]

2000 Mathematics Subject Classification. 68W30, 13P10, 94B05

GBLA LC “Grobner Basis by Linear Algebra and Linear Codes” is a collectionof programs and procedures in GAP 4.3 [4]. Its primary goal is to compute withthe structure of linear codes by means of the results and techniques in [1, 2, 3].In those papers, we have associated a binomial ideal with the structure of a linearcode such that the difference of two vectors belongs to the code if and only if thecorresponding binomial belongs to the ideal. The Grobner basis of this ideal for aconvenient ordering (i.e. any total degree compatible ordering for a binary code)allows us to deal with some combinatorics problems related to the code (decoding,permutation equivalence of codes, codewords of minimal weight, minimal code-words, etc).The main routines that GBLA LC performs are:

• The computation of a reduced basis (a reduced Grobner basis for binarycodes) for the ideal associated to a given linear code.

• A function that computes a set of coset leaders and, a matrix structurematphi (related with the border basis for the ideal of the code) that allowsa faster decoding procedure in contrast with the use of the clasical reductionprocedure by means of Grobner basis.

• Given two binary codes and corresponding Grobner bases, some proceduresare given that allows to explore the problem of the equivalence of the twocodes.

• As an application of the results of some functions, it is possible to obtainsome valuable information related with the code, like the set of codewordsof minimal weight, a set of minimal codewords that it is enough to decode,a minimal cycle basis for binary codes associated with the set of cycles in agraph.

Mathematical Background of the system Finite fields, linear codes andGrobner bases.



The functions that require knowledge of Grobner bases, monomial orderingsand the like can be used as a black box, although probably in some cases somebasic notions would be need it in order to understand their input and output.Software desings The collection of programs is organized by the topics involvedfinite fields, linear codes, Grobner bases, although there are functions immersein more than one of them. There are also several internal functions that wereneeded for creating the necessary settings for the external functions. We shouldsay that there are probably some auxiliary functions or objects that there may bebuilt in GAP. For example, by using the GAP package GUAVA and analyzing theconnection with our settings some functions and objects could be simpler and eventhey may disappear.

In the following address http://www.math.arq.uva.es/~edgar/gbla.htmlinterested users will get access to the source code in GAP, a manual in rtf formatwhere for each function it is given its name, a brief description and the syntax, apdf manual where comments and more description are given, and also examples ofthe use of GBLA LC, a txt file “Examples.txt” with a pretty complete runningexample to allow a quit starting point, it is given also “results.txt” with the outputscorresponding to the example. Access to author’s works related with the subjectare also provided.

Let us start a running example, first we introduce the (lenth n = 10, dimenssionk = 4) binary linear code CB-1 defined by the check matrix given below by theglobal variable H.

gap> n:=10;;k:=4;;m:=1;;p:=2;;F:=GF(2);; gap>alpha_prim:=RootOfDefiningPolynomial(GF(2)); Z(2)^0 gap>alpha_ext:=RootOfDefiningPolynomial(F); Z(2)^0 gap> a:=alpha_ext; Z(2)^0 gap>R:=PolynomialRing(Rationals,n*m);;x:=IndeterminatesOfPolynomia\ Ring(R);; ###Loading the package. gap>Read("D://Documentos//mijail//IMU-Info//Mathsoft//gbla_lc.txt");;

If H is not defined before loading the package you may get some warning message.But in this example we want to show that there is a function matestandarinterthat converts a matrix with coefficient in a finite field in polynomial notation intothe internal GAP representation.

Let us define the parity check matrix for the code CB-1.

gap> H:=[ [ 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ], [ 1, 0, 1, 1, 0, 1, 0, 0,\>0, 0 ],[ 1, 1, 0, 1, 0, 0, 1, 0, 0, 0 ], [ 1, 1, 1, 0, 0, 0, 0, 1, 0,\>0 ], [ 1, 1, 1, 1, 0, 0, 0, 0, 1, 0 ], [ 1, 1, 1, 1, 0, 0, 0, 0, 0, 1\> ] ];;gap> H:=matestandarinter(H); ### Output removed. gap>order:=MonomialGrLOrder;;Gr:=Greduce1(H,m,n,k,p); [ [ 1, x_1, x_2, x_3, x_4,x_5, x_6, x_7, x_8, x_9, x_10, x_1*x_2, x_1*x_3,

x_1*x_4, x_1*x_5, x_1*x_6, x_1*x_7, x_1*x_8, x_1*x_9, x_1*x_10,x_2*x_3, x_2*x_4, x_2*x_7, x_2*x_8, x_2*x_9, x_2*x_10, x_3*x_4,x_3*x_8, x_3*x_9, x_3*x_10, x_4*x_9, x_4*x_10, x_5*x_9, x_5*x_10,x_6*x_9, x_6*x_10, x_7*x_9, x_7*x_10, x_8*x_9, x_8*x_10, x_9*x_10,



x_1*x_2*x_3, x_1*x_2*x_4, x_1*x_2*x_7, x_1*x_2*x_8, x_1*x_2*x_9,x_1*x_2*x_10, x_1*x_3*x_4, x_1*x_3*x_8, x_1*x_3*x_9, x_1*x_3*x_10,x_1*x_4*x_9, x_1*x_4*x_10, x_1*x_5*x_9, x_1*x_5*x_10, x_1*x_6*x_9,x_1*x_6*x_10, x_1*x_7*x_9, x_1*x_7*x_10, x_1*x_8*x_9, x_1*x_8*x_10,x_1*x_9*x_10, x_2*x_3*x_8, x_5*x_9*x_10 ],

[ [ 1, [ x_1^2, 1 ], [ x_2^2, 1 ], [ x_3^2, 1 ], [ x_4^2, 1 ],[ x_5^2, 1 ], [ x_6^2, 1 ], [ x_7^2, 1 ], [ x_8^2, 1 ],[ x_9^2, 1 ], [ x_10^2, 1 ] ],

[ 2, [ x_2*x_5, x_1*x_6 ], [ x_2*x_6, x_1*x_5 ], [ x_3*x_5, x_1*x_7 ],[ x_3*x_6, x_2*x_7 ], [ x_3*x_7, x_1*x_5 ], [ x_4*x_5, x_1*x_8 ],[ x_4*x_6, x_2*x_8 ], [ x_4*x_7, x_3*x_8 ], [ x_4*x_8, x_1*x_5 ],[ x_5*x_6, x_1*x_2 ], [ x_5*x_7, x_1*x_3 ], [ x_5*x_8, x_1*x_4 ],[ x_6*x_7, x_2*x_3 ], [ x_6*x_8, x_2*x_4 ], [ x_7*x_8, x_3*x_4 ] ],

[ 3, [ x_2*x_3*x_4, x_9*x_10 ], [ x_2*x_3*x_9, x_4*x_10 ],[ x_2*x_3*x_10, x_4*x_9 ], [ x_2*x_4*x_9, x_3*x_10 ],[ x_2*x_4*x_10, x_3*x_9 ], [ x_2*x_7*x_9, x_8*x_10 ],[ x_2*x_7*x_10, x_8*x_9 ], [ x_2*x_8*x_9, x_7*x_10 ],[ x_2*x_8*x_10, x_7*x_9 ], [ x_2*x_9*x_10, x_3*x_4 ],[ x_3*x_4*x_9, x_2*x_10 ], [ x_3*x_4*x_10, x_2*x_9 ],[ x_3*x_8*x_9, x_6*x_10 ], [ x_3*x_8*x_10, x_6*x_9 ],[ x_3*x_9*x_10, x_2*x_4 ], [ x_4*x_9*x_10, x_2*x_3 ],[ x_6*x_9*x_10, x_3*x_8 ], [ x_7*x_9*x_10, x_2*x_8 ],[ x_8*x_9*x_10, x_2*x_7 ] ],

[ 4, [ x_1*x_2*x_3*x_8, x_5*x_9*x_10 ],[ x_1*x_5*x_9*x_10, x_2*x_3*x_8 ] ] ], 1 ]

Note that Gr is a list of three components. The first one gives a set of cosetleaders for the linear code, the second one corresponds to a Grobner basis forthe corresponding ordering given as a set of pairs ([maximal term,canonical form])ordered by the length of their maximal term (first a sublist of all pairs with maximalterms of length 1, second a sublist of all pairs of maximal terms of length 2, . . .), thethird one is the error-correcting capability of the code. The function Gbasepoly(Gr)returns the Grobner basis as a set of binomials.gap> Gpol:=Gbasepoly(Gr);;

Having the received vector y = [0, 1, 0, 0, 0, 0, 1, 1, 1, 0], the corresponding mono-mial is y = x2x7x8x9, let us decode by using the Grobner basis Gpol.gap> y:=x[2]*x[7]*x[8]*x[9];;error:=reducepolyGbase(Gpol,y);x_10

Note that the code is 1 error-correcting, so error = x10 means that the corre-sponding codeword is [0, 1, 0, 0, 0, 0, 1, 1, 1, 1]. Then, as it is expected, the syndromeof this vector is the all zero vector.gap>vectestandarTOinter([0,1,0,0,0,0,1,1,1,1])*TransposedMat(H);

[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ]We invite readers to continue the study of this example at the URL address.

The rest of the example is related with the equivalence of linear codes.



[1] M. Borges-Quintana, M. Borges-Trenard and E. Martınez-Moro. On a Grobner basesstructure associated to linear codes. To appear in Journal of Discrete MathematicalSciences & Cryptography. http://arxiv.org/abs/math.AC/0506045

[2] M. Borges-Quintana, M. Borges-Trenard, and E. Martınez-Moro. A general frameworkfor applying FGLM techniques to linear codes. AAECC 16, Lecture Notes in ComputerScience, 3857:76–86, 2006.

[3] M. Borges-Quintana, M. Borges-Trenard, P. Fitzpatrick and E. Martınez-Moro. Ona Grobner bases and combinatorics for binary codes. Submitted to Appl. AlgebraEngrg. Comm. Comput. http://arxiv.org/abs/math.CO/0509164

[4] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.3 ; 2002,(http://www.gap-system.org).



TUGlab: A cooperative game theory toolbox

Miras Calvo, Miguel Angel∗ and Sanchez Rodrıguez, Estela

Departamento de Matematicas, Universidad de Vigo, Rua Leonardo da Vinci (s/n),36310 Vigo, Pontevedra, [email protected]

Departamento de Estadıstica e Investigacion Operativa, Universidad de Vigo, RuaLeonardo da Vinci (s/n), 36310 Vigo, Pontevedra, [email protected]

2000 Mathematics Subject Classification. 91A12, 97-04

The package TUGlab (Transferable Utility Games laboratory) is a Matlab toolbox,see [3], specifically designed to emphasize the geometrical aspects of cooperativegame theory so, as a consequence, the procedures implemented are restricted toany given 3 or 4 persons TU game. Though originally thought to be used byinstructors and students as a complement to the books and other materials usedin introductory courses on cooperative game theory, see, for instance, [1], [2], [4]and [5], TUGlab has grown to cover topics of interest for researches.

The TUGlab toolbox is very easy to install and use, it runs on any implemen-tation of the later releases of the Matlab product: Matlab 6 and Matlab 7 on Unix,PC or Macintosh. It includes a comprehensive user’s guide, some tutorials, helpmenus, several animations and a library with many interesting examples of well-known games. The whole package can be downloaded from the web page of theSantiago Game Theory (SaGaTh) Group: http://eio.usc.es/pub/io/xogos ordirectly from the site

http://webs.uvigo.es/matematicas/campus_vigo/profesores/mmiras/TUGlabWeb/TUGlab.html

Among the topics cover by TUGlab we can highlight: checking monotonicity,convexity or balancedness of a game; computing value solutions as the Shapleyvalue, the nucleolus or the τ -value; drawing the imputation set, the Weber set andthe core of a game; returning the Harsanyi dividends, the multi-linear extensionand the normalization of any given game,...

[1] Curiel, I., 1997. Cooperative game theory and applications. Kluwer Academic Publish-ers, Dordrecht. The Netherlands.

[2] Driessen, T. 1988. Cooperative games, solutions and applications. Kluwer AcademicPublishers. Wiley.

[3] Mathworks, 2005. http://www.mathworks.com.[4] Owen, G. 1995. Game Theory. Academic Press. San Diego.[5] Rafels i Pallarola, C., Izquierdo i Aznar, J. M., Marın Solano, J., Martınez de Albeniz

Salas, F. J., Nunez Oliva, M., Ybern Carballo, N., 1999. Jocs cooperatius i aplicacionsecononomiques. Edicions de la Universitat de Barcelona, Barcelona.



Computation of modular endomorphism algebras

Jordi Quer

Departament de Matematica Aplicada II, Universitat Politecnica de Catalunya, JordiGirona, 1-3, 08034-Barcelona, [email protected]

2000 Mathematics Subject Classification. 11F11,11G18

Introduction. Let f =∑

anqn ∈ Sk(N, ε) be a newform of weight k ≥ 2, levelN and Nebentypus character ε. The Fourier coefficients an are algebraic integersand generate a number field E.

For weight k = 2, to such a form Shimura attached an abelian variety Af

defined over Q of dimension [E : Q], that may be constructed as a quotient of thejacobian J1(N) of the modular curve X1(N) classifying generalised elliptic curvestogether with a point of order N . The variety Af has the property that the algebraEndQ(Af )⊗Q of endomorphisms defined over Q is isomorphic to the field E. Thevarieties Af are known as modular abelian varieties, and the generalised Shimura-Taniyama conjecture asserts that every abelian variety A defined over Q for whichEndQ(A)⊗Q is a number field whose degree over Q equals the dimension of A ismodular. This conjecture is known to be true when A has complex multiplication(proof due to Shimura) and also when A is an elliptic curve: this one-dimensionalcase is a recent result by Wiles and others that became famous because it impliesFermat’s Last Theorem.

The variety Af is simple over Q and its decomposition up to isogeny over thealgebraic closure Q is determined by the full algebra of endomorphisms Xf =End(Af )⊗Q; this is one of the reasons for the interest in computing that algebra.If Af has no complex multiplication (the CM case is easier), then the algebra Xf

is a matrix algebra, either over a totally real field F , or otherwise over a totallyindefinite quaternion algebra over a totally real field F . In [4] a construction ofthis algebra is given, as a crossed product algebra, using some operators built frominner twists of the modular form f .

For larger weight k > 2, Stoll attached to every newform a Grothendieck motiveMf , and in [1] a similar crossed product algebra realisation is given for its fullendomorphism algebra Xf = End(Mf ).

Computational aspects. From the computational point of view, the theory ofmodular symbols introduced by Manin is the basic tool (this theory is not only atool for computing: it has also many theoretical applications). Probably the firstapplication of modular symbols (and one of the first intensive uses of computersin Number Theory) was the elaboration of the Antwerp tables of modular ellipticcurves over Q of conductor N ≤ 200 by Birch and Swinnerton-Dyer (Lecture Notesin Math vol. 476). The Antwerp tables have been an extremely useful tool for



several generations of number theorists (today we use Cremona’s tables: availableon the internet, last year reached the level N = 70000).

In the past years an excellent package for computing with modular symbolshas been developed in Magma by William Stein. The software Magma1 is one of themost powerful tools for doing computations in Number Theory and ArithmeticGeometry, although unfortunately it is not free. William Stein’s implementation2

of modular symbols computations is the only such implementation that works forforms on any weight and Nebentypus, and has many more functions and capabilitiesthan any other. Recently Stein is promoting a new software Sage (Software forAlgebra and Geometry Experimentation), completely free and open source, andhe plans to add to it all the modular symbols computing facilities, but presentlyit is still in an early stage of development for modular symbols computations.

Computation of modular endomorphism algebras. The objective of thecomputational work described here is (mainly) to implement functions for comput-ing the endomorphism algebra Xf for every newform f . Among other applications,we are interested in building a table of such algebras for all forms with small weightand level: there are several finiteness conjectures and feelings about the algebrasone should expect to appear. We also want to check other properties of the abelianvarieties Af for which we need that information.

For implementing these computations the author got a co-developer licencefrom Magma. The functions are built using most of the facilities of William Stein’spackage. It has also been necessary to program several computations that were notin Magma, especially the computation of Hilbert symbols for number fields otherthan Q.

The main functions have as input a newform f and give as output informationabout Xf : its center, a totally real number field F ; its Brauer class, an element oforder dividing 2 in the Brauer group Br(F ), given as the (even, finite) list of placesof F in which the algebra ramifies; the subalgebra of endomorphisms defined overQ (note that Xf has a natural Galois action), which is the number field generatedby the Fourier coefficients of f ; etc.

For doing these computations we use the theory and the formulas given in [2](weight 2) and [1] (larger weight) for the Brauer class of Xf in terms of productsof Hilbert symbols and of obstructions to the existence of a square root for theNebentypus character ε.

Currently the package is being tested, improved and documented, and in thefuture it will be (probably) included in the standard Magma distribution. A versionof it and also a table of modular endomorphism algebras computed using thisversion can be examined in http://www-ma2.upc.es/~quer/Madrid06/ (warning:both the software and the table are working versions, and NOT in final form).

An application. In [3] Ribet discusses the fields of definition up to isogeny of Q-Hilbert-Blumenthal abelian varieties: an abelian variety B defined over a number

1http://magma.maths.usyd.edu.au/2See his web page http://modular.ucsd.edu/ for more information and downloading.



field is a Hilbert-Blumenthal variety (or Real Multiplication abelian variety) ifits endomorphism algebra End(B) ⊗ Q is a totally real number field F of degree[F : Q] = dim A. It is a Q-variety if it is equivariantly isogenous to all its Galoisconjugates.

The obstruction to descend up to isogeny the field of definition of a Q-Hilbert-Blumenthal abelian variety B to a given number field L is computed in [3] as theinflation InfGL

GQof an element c in

H2(GQ, F ∗)[2] ' Hom(GQ, F ∗/F ∗2)×H2(GQ, ±1).

Let c = (cδ, c±) be the decomposition of that obstruction corresponding to the

given cartesian product. The main result in [3] is that when B is of odd dimension,the fields L that trivialise the component cδ also trivialise the component c±.This fact was already known in the case of dimension 1: for the so-called Q-curvesElkies proved it using very different arguments. In the even dimensional case, Ribetremarks that his proof cannot be applied and raises the question of whether theparity of the dimension is really necessary for the property to be true.

The main source of Q-Hilbert-Blumenthal abelian varieties (conjecturally thisis in fact the unique source) are the varieties Af : when Xf has trivial Brauer classthen Af factors over the algebraic closure Q up to isogeny as a power Bn

f of such avariety. The functions we implemented for computing Xf need as very importantingredients the computation of the elements cδ and c±, so this computation is partof our package.

Now, looking at the tables one immediately sees that Ribet’s result cannot begeneralised to the even dimensional case. The simplest counterexamples are the Q-Hilbert-Blumenthal surfaces Bf whose square is isogenous to 4-dimensional abelianvarieties Af corresponding to certain weight 2 newforms f of level N = 110 (cf.page 90 of the tables). In these examples the field Q already trivialises cδ but oneneeds to go to a quadratic field in order to kill the c± component of the obstruction.

[1] Ghate, E., Gonzalez-Jimenez, E., Quer, J., On the Brauer class of modular endomor-phism algebras. Int. Math. Res. Not. 2005, no. 12, 701–723.

[2] Quer, J., La classe de Brauer de l’algebre d’endomorphismes d’une variete abeliennemodulaire, C. R. Acad. Sci. Paris Ser. I Math. 327 (1998), no. 3, 227–230.

[3] Ribet, K., Fields of definition of abelian varieties with real multiplication. Arithmeticgeometry (Tempe, AZ, 1993), 107–118, Contemp. Math., 174, Amer. Math. Soc.,Providence, RI, 1994.

[4] Ribet, K. A. Twists of modular forms and endomorphisms of abelian varieties. Math.Ann. 253 (1980), no. 1, 43–62.



Computing Spectral Sequences with the Kenzosystem

A. Romero1, J. Rubio1, F. Sergeraert2

1Departamento de Matematicas y Computacion, Universidad de La Rioja, Spain2Institut Fourier. Universite Joseph Fourier. Grenoble, [email protected], [email protected],

[email protected]

2000 Mathematics Subject Classification. 18-04, 55-04, 18G40, 55T10, 55T20

Spectral Sequences are a useful tool in Algebraic Topology providing informationon homology groups by successive approximations from the homology of appro-priate associated complexes. However, they are not real algorithms except in veryparticular cases. On the contrary, the effective homology method provides actualalgorithms for the computation of homology groups of complicated spaces. TheKenzo system [1] is based in this method, allowing us to reach in particular manyhomology groups of various loop spaces otherwise so far not reachable.

The effective homology method can also be used for the computation of spectralsequences. Following these ideas, a set of programs enhancing the Kenzo systemhas been developed, allowing the user to calculate the whole set of components ofspectral sequences associated to filtered complexes.

Preliminaries

A Spectral Sequence ([2]) is a family of “pages” Erp,q, d

r of differential bigradedmodules, each page being made of the homology groups of the preceding one.Therefore if we know the stage r in the spectral sequence (Er, dr) we can buildthe bigraded module at the stage r + 1, Er+1, but this cannot define the nextdifferential dr+1 which therefore must be independently defined too.

In some cases, it is in fact a matter of computability: the higher differentialsof the spectral sequence are mathematically defined, but their definition is notconstructive, that is, the differentials are not computable with the provided infor-mation. In the case of spectral sequences associated to filtered complexes a formalexpression for the groups Er

p,q is known (see [2]), but it is not sufficient to computethem because the subgroups that appear in this formula cannot be obtained ingeneral, only in very simple cases. This means that a spectral sequence is not analgorithm that a machine can compute automatically: at each level r some ex-tra information is needed and obviously a machine is not “able” to obtain thisinformation.

On the contrary, the effective homology method (based on the concept of “ob-ject with effective homology”), provides real algorithms for the computation ofhomology groups in many common situations. Some basic ideas about effectivehomology (that can be found in [3]) are included in the following definitions.



Definition. A reduction ρ between two chain complexes A and B (denoted byA ⇒ B) is a triple ρ = (f, g, h) where (a) The components f and g are chaincomplex morphisms f : A → B and g : B → A; (b) The component h is ahomotopy operator h : A → A (a graded group homomorphism of degree +1); (c)The following relations are satisfied: (1) fg = idB ; (2) gf + dAh + hdA = idA;(3) fh = 0; (4) hg = 0; (5) hh = 0.

Remark. If A ⇒ B, then A = B⊕C, with C acyclic, which implies that H∗(A) ∼=H∗(B).

Definition. A (strong chain) equivalence between the complexes A and B (A ⇐⇒B) is a triple (D, ρ, ρ′) where D is a chain complex, ρ = (D ⇒ A) and ρ′ = (D ⇒B).

Definition. An object with effective homology is a triple (X, EC, ε) where ECis an effective chain complex (that is, a free chain complex whose groups in eachdegree are finitely generated, so an elementary algorithm can compute its homologygroups) and C(X) ε⇐⇒ EC.

Obviously H∗(X) ∼= H∗(EC), which means that it is possible to compute thehomology groups of X by means of those of EC.

The Kenzo program

Kenzo [1] is a symbolic computation system for Algebraic Topology, developedby Francis Sergeraert and some coworkers. It was written in the programminglanguage Common Lisp and has succeeded in computing homology groups of somecomplicated spaces that had not been determined before.

The notion of object with effective homology is the fundamental idea of Kenzo.To compute the homology groups of a complex X, the system consider two cases:if X is effective, then Hi(X) can be determined by means of elementary operationswith differential matrices. Otherwise, Kenzo tries to use the effective homologytheory.

For instance, this system can compute the homology groups of the space X =Ω(Ω(Ω(P∞R/P 3R) ∪4 e4) ∪2 e2). For dimension n = 5, we obtain

> (homology X 5)Homology in dimension 5 :Component Z/16ZComponent Z/8Z...Component Z/2Z

which means H5X = Z16 ⊕ Z8 ⊕ Z232 .

A new module for Kenzo

A new module for Kenzo (in Common Lisp) has been developed, allowing compu-tations of spectral sequences of filtered complexes.



The new methods work in a way similar to the mechanism of Kenzo for com-puting homology groups: for effective complexes, the groups Er

p,q can be computedthrough elementary methods with integer matrices; otherwise, the effective homol-ogy is needed to compute the Er

p,q by means of an analogous spectral sequencededuced of an appropriate filtration on the associated effective complex.

This new module allows us to compute:

1. The groups Erp,q, with their generators.

2. The differential maps drp,q.

3. The convergence level: the minimal r such that Erp,q = E∞

p,q.4. The filtration of Hp+q through the E∞

p,q.

Examples

The programs developed allow us to compute the two most classical examples ofspectral sequences: Serre and Eilenberg-Moore. In the case of the Serre spectralsequence (associated to a fibration F → E → B), it is not difficult to prove thatit is isomorphic after level t = 2 to the spectral sequence of the effective complexof the total space of the fibration, E = F ×τ B. The Eilenberg-Moore spectralsequence associated to the loop space of a simplicial set, ΩX, is isomorphic to thespectral sequence of the corresponding effective complex after t = 1, that is, forevery level.

We present here as an example the E∞p,q of the Eilenberg-Moore spectral se-

quence between the space X = ΩS3 ∪2 e3 and its loop space ΩX = Ω(ΩS3 ∪2 e3).Up to our knowledge, this spectral sequence has not appeared in the literature.

q

12 Z52 Z7

2 Z2

11 0 Z62 Z4

2

10 Z2 ⊕ Z10 ⊕ Z Z2 Z42 Z2

9 0 0 Z32 Z3

2

8 0 0 Z6 Z22 Z2

7 0 0 0 Z22

6 0 Z Z2 Z2

5 0 0 Z2

4 0 Z Z2

3 0 0

2 0 Z2

1 0

0 Z

0 1 2 3 4 5 6 p



[1] X. Dousson, J. Rubio, F. Sergeraert, and Y. Siret, The Kenzo program (InstitutFourier, Grenoble, 1999) http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/.

[2] S. Mac Lane, Homology (Springer, 1963).[3] J. Rubio and F. Sergeraert. Constructive Algebraic Topology, Bulletin des Sciences

Mathematiques 126 (2002) 389-412.



Statistical Inference Package SIP

Esa Uusipaikka

Department of Statistics, University of Turku, Assistentinkatu 7, 20014 Turun yliopisto,[email protected]

2000 Mathematics Subject Classification. 62F

“The fundamental problem towards which the study of statistics is addressed,is that of inference. Some data are observed and we wish to make statements,inferences, about one or more unknown features of the physical system which gaverise to these data.” [1]

Statistical inference is always based on observations from the phenomenon un-der consideration. The set of observations is the first necessary component of sta-tistical evidence on which the inference relies. The second necessary componentis a statistical model. The statistical model is based on the assumption that theobservations contain random variation, that is, can be considered to have arisenfrom some probability distribution.

Statistical inference concerns some characteristic or characteristics of the phe-nomena from which the observations have arisen. The characteristics of the phe-nomena under consideration are some functions of the unknown parameter of thestatistical model and are called the parameter functions of interest. Statisticalinference consists of statements concerning the unknown value(s) of the interestfunction(s). Statistical inference differs from other possible modes of inferences inthat it always gives measures of uncertainties of the statements made.

SIP makes it easy for the user to do classical likelihood based statistical infer-ence. It contains procedures for maximum likelihood estimation, likelihood ratiotests of general hypotheses concerning parameters, and profile likelihood based con-fidence intervals for general interest functions of parameters. SIP contains largecollection of discrete and absolutely continuous univariate distributions and alsomultivariate distributions. It gives user possibility to form complicated modelsfrom the simpler ones. SIP contains many sophisticated statistical models suchas univariate/multivariate linear/non-linear regression model, logistic regressionmodels, Poisson regression models, multinomial regression models etc. SIP usesa new method for calculation of profile likelihood based confidence intervals forgeneral parameter functions of interest in general parametric statistical models [2].

[1] O’Hagan, A., Kendall’s Advanced Theory of Statistics Volume 2B, Bayesian Inference.Edward Arnold, London, 1994.

[2] Uusipaikka, E. I., A New Method for Construction of Profile Likelihood Based Confi-dence Intervals, 1996 Proceedings of the Biometrics Section of the American StatisticalAssociation (1996), 244–249.



GNU TeXmacs

Joris van der Hoeven

Departement de Mathematiques (bat. 425) et CNRS, Universite Paris-Sud, 91405 OrsayCedex, [email protected]

2000 Mathematics Subject Classification. 68U15, 68U35

Ordinary users of computers have the choice between several office suits, like Mi-crosoft Office, Open Office, or Star Office, for common desktop tasks,like text editing, drawing pictures or data administration. A major challenge is toprovide a similar software for scientists, with more specific support for things likemathematical formulas, complex computations, presentations from a laptop andso on.

Currently, in the fields of mathematics, physics and computer science, peopleuse different tools for each task. For instance, mathematical texts are usually writ-ten with TEX/LATEX, computations are done with numerical computation softwareor computer algebra systems, and presentations are done using Power Point ordirectly from a Pdf file.

Several of these tools, and in particular TEX/LATEX, are quite user unfriendly. Atany rate, it is not straightforward to combine several tools, like including graphicsor computer algebra output in a paper. Moreover, different tools use different dataformats, which are often not very compatible. For instance, it remains complicatedto simply put a mathematical text on the web (in a non-graphical format), becausethe “web-standard” MathML for mathematics is still poorly supported.

The GNU TEXmacs project [1, 2] aims to provide a solution to the aboveproblems. The software includes a user friendly and “wysiwyw” (what you seeis what you want) mathematical text editor, with a similar typesetting quality asTEX, but not directly based on TEX/LATEX. All TEXmacs documents are structuredand can be saved in either TEXmacs, Scheme or Xml format. The structure of thedocuments can be exploited when editing and new style files can be created by theuser. TEXmacs also provides Scheme as an extension language, which allows theuser to create new structured editing features.

TEXmacs admits interfaces with many other systems, like computer algebrasystems (Axiom, Macaulay 2, Maxima, Maple, Mupad, Pari-GP), scien-tific computation software (GNU Octave and Scilab), and other programs likeGNU R, Dr. Geo, Qcl, etc. Converters exist for several standard formats, likeTEX/LATEX and Html/MathML. Recent additions are a presentation mode anda tool for drawing technical pictures.

Another important aspect of TEXmacs is that is free software in the sense ofthe Free Software Foundation. We think that this is particularly important forscientific software, because it allows other scientists to access to the source code,adapt the software to new purposes and freely redistribute the (possibly improved)software to colleagues. In particular, one of the design goals of TEXmacs is to help



other scientists who wish to create free academic software by providing a good userinterface.

The main design goal of providing a flexible, structured and integrated editingplatform for scientists is also the major difficulty from the implementation pointof view. Indeed, our requirements include wysiwygness, scriptability, extensiblemarkup, interactive styling, sufficiently high reactivity, user friendliness, etc.

Although it is not necessarily hard to develop a program which satisfies one,two or three of these requirements, the main point here is to simultaneously fulfillall of them. This required the development of new algorithms and data structuresfor styling and typesetting, as well as the implementation of a new paradigm formodular programming. Some of the new ideas and techniques are described inthe documentation of the source code and [2]. A more complete overview will bewritten as soon as the implementation of the TEXmacs core will be finished.

[1] J. van der Hoeven. Gnu TEXmacs. http://www.texmacs.org, 1998–2006.[2] J. van der Hoeven. Gnu TEXmacs: A free, structured, wysiwyg and technical text

editor. In Daniel Flipo, editor, Le document au XXI-ieme siecle, volume 39–40, pages39–50, Metz, 14–17 mai 2001. Actes du congres GUTenberg.


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