main-content

## Über dieses Buch

This volume contains the outstanding collection of invited papers and refereed papers selected for the Second International Congress on Mathematical Software, ICMS 2006, held in Castro Urdiales, Spain, September 1-3, 2006. We cordially invite you to visit the ICMS 2006 website http://www.icms2006.unican.es where you can find all relevant information about this interesting event. ICMS 2006 was the second edition of this congress, which follows up the successful ICMS 2002 held in Beijing, China. Since its inception, this congress has been a satellite event of the International Congress of Mathematicians - ICM, the world’s largest conference on mathematics, celebrated every four years since the edition of 1900 in Paris, where David Hilbert presented his 23 famous problems. For the first time, this 2006 edition of ICM is held in Spain (see: http://www.icm2006.org for details), and so is ICMS 2006. This congress was devoted to all aspects of mathematical software, whose appearance is — in our opinion — one of the most important events in mathematics. Mathematical software systems are used to construct examples, to prove theorems, and to find new mathematical phenomena. Conversely, mathematical research often motivates developments of new algorithms and new systems. Beyond mathematics, mathematical software systems are becoming indispensable tools in many branches of science and technology.

## Inhaltsverzeichnis

### A General Computational Scheme for Testing Admissibility of Nilpotent Orbits of Real Lie Groups of Inner Type

One of the most fundamental problems in the field of Representation Theory is the description of all the unitary representations of a given group. For non-compact real reductive Lie groups, there is evidence that new unitary representations can be obtained from data provided by their

nilpotent orbits. In this paper, we describe a general scheme for determining the admissibility of a given real nilpotent orbit. We implement some parts of the scheme using the software system

LiE

. We give a detailed example and study the complexity of the algorithms.

Alfred G. Noël

### Efficient Implementation of Polynomial Arithmetic in a Multiple-Level Programming Environment

The purpose of this study is to investigate implementation techniques for polynomial arithmetic in a multiple-level programming environment. Indeed, certain polynomial data types and algorithms can further take advantage of the features of lower level languages, such as their specialized data structures or direct access to machine arithmetic. Whereas, other polynomial operations, like Gröbner basis over an arbitrary field, are suitable for generic programming in a high-level language.

We are interested in the integration of polynomial data type implementations realized at different language levels, such as

Lisp

,

C

and

Assembly

. In particular, we consider situations for which code from different levels can be combined together within the same application in order to achieve high-performance.

We have developed implementation techniques in the multiple-level programming environment provided by the computer algebra system

AXIOM

. For a given algorithm realizing a polynomial operation, available at the user level, we combine the strengths of each language level and the features of a specific machine architecture. Our experimentations show that this allows us to improve performances of this operation in a significant manner.

Xin Li, Marc Moreno Maza

### Development of a Maple Macro Package Suitable for Drawing Fine ${\rm\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}$ -Pictures

The authors have developed a Maple macro package:

KETpic

which generates

${\rm L\kern-.36em\raise.3ex\hbox{\sc a}\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}$

source codes for clear drawings. KETpic enables us to draw every kind of complicated figure easily. Users are simply required to command Maple, with KETpic loaded, to plot graphs, to create

${\rm L\kern-.36em\raise.3ex\hbox{\sc a}\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}$

source codes by KETpic commands, and to embed them into

${\rm L\kern-.36em\raise.3ex\hbox{\sc a}\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}$

source files. Desired figures are obtained by the usual

${\rm L\kern-.36em\raise.3ex\hbox{\sc a}\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}$

compilation. Two versions of KETpic for Macintosh and Windows are provided, both of which are available for Linux users. Figures finally obtained, either on a PC display or on printed matter, are clear and possess the highest accuracy. KETpic does not require an expensive printer. Carefully prepared figures are significantly advantageous for mathematical education because they facilitate students’ understanding of difficult mathematical notions. In this paper, we describe the advantages of KETpic with typical examples.

Masayoshi Sekiguchi, Satoshi Yamashita, Setsuo Takato

### Matlab-Based Problem-Solving Environment for Geometric Processing of Surfaces

In this paper a new problem-solving environment (PSE) for geometric processing of surfaces is introduced. The PSE has been designed to be responsive to the needs of our collaboration with an industrial partner, the Spanish company CANDEMAT S.A., devoted to build moulds and dies for the automotive industry. The PSE has been implemented in

Matlab

and is aimed to support the full range of activities carried out by our partner in the field of geometric processing of surfaces for the automotive industry. Firstly, the paper describes the architecture of the system and some implementation details. Then, some examples of its application to critical problems in the automotive industry – such as the computation of the intersection curves of surfaces, the generation of tool-path trajectories for NC machining and the visualization of geometric entities stored in industrial files of several formats – are briefly described. The PSE has shown to provide our partner with accurate, reliable solutions to these and other problems and to serve as a communication channel for exchange of geometrical data as well as a platform for trial and research support.

A. Gálvez, A. Iglesias

### A Mathematica Notebook for Computing the Homology of Iterated Products of Groups

Let

G

be a group which admits the structure of an iterated product of central extensions and semidirect products of abelian groups

G

i

(both finite and infinite). We describe a

Mathematica 4.0

notebook for computing the homology of

G

, in terms of some homological models for the factor groups

G

i

and the products involved. Computational results provided by our program have allowed the simplification of some of the formulae involved in the calculation of

H

n

(

G

). Consequently the efficiency of the method has been improved as well. We include some executions and examples.

V. Álvarez, J. A. Armario, M. D. Frau, P. Real

### GCLC — A Tool for Constructive Euclidean Geometry and More Than That

We present

gclc

/

Wingclc

— a tool for visualizing geometrical (and not only geometrical) objects and notions, for teaching/studying mathematics, and for producing mathematical illustrations of high quality.

gclc

uses a language

gc

for declarative representation of figures and for storing mathematical contents of visual nature in textual form. In

gclc

, there is a build-in geometrical theorem prover which directly links visual and semantical geometrical information with deductive properties and machine–generated proofs.

Predrag Janičić

### jReality, jtem, and Oorange — A Way to Do Math with Computers

This paper presents our approach to the question of how to code mathematics (mostly experimantal and motivated from geometry) in Java. We are especially interested in the question how the development of mathematical software and the mathematics itself influence each other and how the design of programming tools and code can support this interrelationship.

Tim Hoffmann, Markus Schmies

Starting with MuPAD Pro 3.0, we introduced a new framework for 2D and 3D graphics including animations in computer algebra. Based around the concept of graphical objects that are fully manipulable from the programming level as well as interactively, the framework has proven to be well-designed and flexible. We will present both the users’ and the developers’ perspective, including how to implement new graphical primitives and a discussion of current limitations.

Christopher Creutzig

### An Efficient Implementation for Computing Gröbner Bases over Algebraic Number Fields

In this paper we discuss Gröbner basis computation over algebraic number fields. Buchberger algorithm can be executed over any computable field, but the computation is often inefficient if the field operations for algebraic numbers are directly used. Instead we can execute the algorithm over the rationals by adding the defining polynomials to the input ideal and by setting an elimination order. In this paper we propose another method, which is a combination of the two methods above. We implement it in a computer algebra system Risa/Asir and examine its efficiency.

Masayuki Noro

### Tree Checking for Sparse Complexes

We detail here the sparse variant of the algorithm sketched in [2] for checking if a simplicial complex is a tree. A full worst case complexity analysis is given and several optimizations are discussed. The practical complexity is discussed for some examples.

Massimo Caboara, Sara Faridi, Peter Selinger

### The SARAG Library: Some Algorithms in Real Algebraic Geometry

In this paper we present

SARAG

, which is a software library for real algebraic geometry written in the free computer algebra system Maxima.

SARAG

stands for “Some Algorithms in Real Algebraic Geometry” and has two main applications: extending the capabilities of Maxima in the field of real algebraic geometry and being part of the interactive version of the book “Algorithms in Real Algebraic Geometry” by S. Basu, R. Pollack, M.-F. Roy, which can be now freely downloaded.

The routines of the library deal with: theory of signed sub-resultants, linear algebra, gcd computation, real roots counting, real roots isolation, sign determination, Thom encodings, study of the topology of curves. At the moment

SARAG

is being used as a tool to develop, implement and tune algorithms coming from new research results, e.g. an algorithm for faster gcd computation, an algorithm for the study of the topology of curves over non-Archimedian real closed fields.

Fabrizio Caruso

### Algebraic Computation of Some Intersection D-Modules

Let

D

C

n

be a locally quasi-homogeneous free divisor (e.g. a free hyperplane arrangement),

$\cal E$

an integrable logarithmic connection with respect to

D

and

${\cal L}$

the local system of horizontal sections of

${\cal E}$

on

X

D

. Let

${\rm IC}_X({\cal E})$

be the holonomic regular

${\cal D}_{X}$

-module whose de Rham complex is the intersection complex

${\rm IC}_X({\cal L})$

of Deligne-Goresky-MacPherson. In this paper we show how to use our previous results on the algebraic description of

${\rm IC}_X({\cal E})$

in order to obtain explicit presentations of it. Concrete examples for

n

= 2 are included.

F. J. Calderón Moreno, L. Narváez Macarro

### Plural, a Non–commutative Extension of Singular: Past, Present and Future

We describe the non–commutative extension of the computer algebra system

Singular

, called

Plural

. In the system, we provide rich functionality for symbolic computation within a wide class of non–commutative algebras. We discuss the computational objects of

Plural

, the implementation of main algorithms, various aspects of software engineering and numerous applications.

Viktor Levandovskyy

### Development of NZMATH

NZMATH is a system oriented to calculations of number theory, based on Python. Currently, it has several basic data types and several modules for number theoretic computations. NZMATH has two key visions 1) user / developer fusion and 2) speed of development, and the system has been growing along the lines. The development is of open source by nature, and we are making effort to be as agile as possible. There are many areas to be developed, especially a module for algebraic numbers is awaited. Some experimental user interface construction is also discussed.

Matsui Tetsushi

### KASH: Recent Developments

In recent years the computer algebra system KASH/KANT for number theory has evolved considerably. We present its new features and introduce the related components, QaoS (Querying Algebraic Objects System) and GiANT (Graphical Algebraic Number Theory).

Sebastian Freundt, Aneesh Karve, Anita Krahmann, Sebastian Pauli

### Making Change and Finding Repfigits: Balancing a Knapsack

We will discuss knapsack problems that arise in certain computational number theory settings. A common theme is that the search space for the standard real relaxation is large; in a sense this translates to a poor choice of variables. Lattice reduction methods have been developed in the past few years to improve handling of such problems. We show explicitly how they may be applied to computation of Frobenius instances, Keith numbers (also called “repfigits”), and as a first step in computation of Frobenius numbers.

Daniel Lichtblau

### Robust HGCD with No Backup Steps

Subquadratic divide-and-conquer algorithms for computing the greatest common divisor have been studied for a couple of decades. The integer case has been notoriously difficult, with the need for “backup steps” in various forms. This paper explains why backup steps are necessary for algorithms based directly on the quotient sequence, and proposes a robustness criterion that can be used to construct a “half-gcd” algorithm without any backup steps.

Niels Möller

### The Design of CoCoALib

We describe some of the more important aspects of the design of CoCoALib, a new C++ library for effecting

Co

mputations in

Co

mmutative

A

lgebra. Special effort has been invested in making the code clean and portable while not neglecting run-time performance; one of the primary goals is to offer freely available reference implementations of the most important algorithms in the field.

J. Abbott

### Generation of Oriented Matroids Using Satisfiability Solvers

Oriented matroids are a combinatorial abstraction of finite sets of points in ℝ

n

. They have been used to study various problems in discrete and computational geometry (for more material on oriented matroids, see [1,2]). A number of methods to generate oriented matroids have been proposed (for instance in [4,8-10]) as these methods can be used as a building block for the algorithmic treatment of some hard problems: Algorithms to generate oriented matroids have for instance been used to decide whether certain 4-polytopes exist [6,10,14]. For these questions it is important to have effective algorithms for the generation of oriented matroids. We propose to use satisfiability solvers to generate oriented matroids. We have adapted this approach to generate oriented matroids that satisfy certain geometric constraints. Even though one can use the generated oriented matroids as a first step to find realizations (see for instance [2,6]), we will only focus on non-realizability results.

Lars Schewe

### Flexible Object Hierarchies in Polymake

Initially polymake [1,2,3] was conceived as a collection of tools for studying convex polyhedra only. The early versions of polymake had a very primitive data management, built around a single data type for polyhedra. However, as the time passed, more and more different discrete mathematical structures like graphs and simplicial complexes came along. This gave rise to a properly typed object hierarchy, which was strict enough to support established object-oriented (OO) software techniques, but, on the other side, flexible enough to allow for continuous extensions without breaking the compatibility.

Ewgenij Gawrilow, Michael Joswig

### A Presentation of the Gfan Software

Gfan [8] is a software package for computing Gröbner fans and tropical varieties of polynomial ideals. The Gröbner fan of an ideal

I

⊂ ℚ[

x

1

,...,

x

n

] is a polyhedral complex defined in [9]. For a homogeneous ideal the Gröbner fan is a complete fan and the normal fan of a polytope. Its cones are in bijection with the various initial ideals of

I

. In particular, the full dimensional cones are in bijection with the monomial initial ideals and thereby also in bijection with the reduced Gröbner bases of

I

. In [3] the local basis change of Gröbner bases was introduced. This method allows us to go from one Gröbner basis in the fan to a neighboring one, giving an effective algorithm for computing the Gröbner fan by traversing its maximal cones. The method can be refined by applying the reverse search technique [1]. This works even in the non-homogeneous case [5].

Anders N. Jensen

### Parallel Homotopy Algorithms to Solve Polynomial Systems

Homotopy continuation methods to compute numerical approximations to all isolated solutions of a polynomial system are known as “embarrassingly parallel”, i.e.: because of their low communication overhead, these methods scale very well for a large number of processors. Because so many important problems remain unsolved mainly due to their intrinsic computational complexity, it would be embarrassing not to develop parallel implementations of polynomial homotopy continuation methods. This paper concerns the development of “parallel PHCpack”, a project which started a couple of years ago in collaboration with Yusong Wang, and which currently continues with Anton Leykin (parallel irreducible decomposition) and Yan Zhuang (parallel polyhedral homotopies). We report on our efforts to make PHCpack ready to solve large polynomial systems which arise in applications.

2000 Mathematics Subject Classification.

Primary 65H10. Secondary 14Q99, 68W30.

Anton Leykin, Jan Verschelde, Yan Zhuang

### DEpthLAUNAY

This paper describes the software DEpthLAUNAY. The main goal of the application is to compute Delaunay depth layers and levels of a planar point set [ACH]. Some other geometric structures can be computed as well (convex hull, convex layers and levels, Voronoi diagram and Voronoi levels, Delaunay triangulation, Delaunay empty circles, etc.) The application has been developed using CGAL [CGAL].

Manuel Abellanas, Alfredo de las Vegas

### iB4e: A Software Framework for Parametrizing Specialized LP Problems

Given a polytope

P

, the classical linear programming (LP) problem asks us to find a point in

P

which attains maximal inner product with a given real objective vector

c

. When the objective is a vector of unknown parameters, the LP problem amounts to computing certain information about the polytope

P

, such as its vertices and normal fan.

Peter Huggins

### Primal-Dual Enumeration for Multiparametric Linear Programming

Optimal control problems for constrained linear systems with a linear cost can be posed as multiparametric linear programs (pLPs) and solved explicitly offline. Several algorithms have recently been proposed in the literature that solve these pLPs in a fairly efficient manner, all of which have as a base operation the computation and removal of redundant constraints. For many problems, it is this redundancy elimination that requires the vast majority of the computation time. This paper introduces a new solution technique for multiparametric linear programs based on the primal–dual paradigm. The proposed approach reposes the problem as the vertex enumeration of a linearly transformed polytope and then simultaneously computes both its vertex and halfspace representations. Exploitation of the halfspace representation allows, for smaller problems, a very significant reduction in the number of redundancy elimination operations required, resulting in many cases in a much faster algorithm.

Colin N. Jones, Jan M. Maciejowski

### A Parallel, Asynchronous Method for Derivative-Free Nonlinear Programs

Derivative-free optimization algorithms are needed to solve real-world engineering problems that have computationally expensive and noisy objective function and constraint evaluations. In particular, we are focused on problems that involve running cumbersome simulation codes with run times measured in hours. In such cases, attempts to compute derivatives can prove futile because analytical derivatives are typically unavailable and noise limits the accuracy of numerical approximations. Furthermore, the objective and constraint functions may be inherently nonsmooth, i.e., because the underlying model is nonsmooth.

Joshua D. Griffin, Tamara G. Kolda

### Convergent SDP-Relaxations for Polynomial Optimization with Sparsity

We consider a polynomial programming problem

P

on a compact basic semi-algebraic set

K

⊂ ℝ

n

, described by

m

polynomial inequalities

g

j

(

X

)≥0, and with criterion

f

∈ ℝ[

X

]. We propose a hierarchy of semidefinite relaxations that take sparsity of the original data into account, in the spirit of those of Waki et al. [7]. The novelty with respect to [7] is that we prove convergence to the global optimum of

P

when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the

running intersection property

in graph theory.

Jean B. Lasserre

### Algorithm and Software for Integration over a Convex Polyhedron

We present a new efficient algorithm for numerical integration over a convex polyhedron in multi-dimensional Euclidian space defined by a system of linear inequalities. The software routines which implement this algorithm are described. A numerical example of calculating an integral using these routines is given.

Mark Korenblit, Efraim Shmerling

### A Matlab Implementation of an Algorithm for Computing Integrals of Products of Bessel Functions

We present a Matlab program that computes infinite range integrals of an arbitrary product of Bessel functions of the first kind. The algorithm uses an integral representation of the upper incomplete Gamma function to integrate the tail of the integrand. This paper describes the algorithm and then focuses on some implementation aspects of the Matlab program. Finally we mention a generalisation that incorporates the Laplace transform of a product of Bessel functions.

Joris Van Deun, Ronald Cools

### Computation of the Real Zeros of the Kummer Function M(a;c;x)

An algorithm for computing the real zeros of the Kummer function

M

(

a

;

c

;

x

) is presented. The computation of ratios of functions of the type

M

(

a

+1;

c

+1;

x

)/

M

(

a

;

c

;

x

),

M

(

a

+1;

c

;

x

)/

M

(

a

;

c

;

x

) plays a key role in the algorithm, which is based on global fixed-point iterations. We analyse the accuracy and efficiency of three continued fraction representations converging to these ratios as a function of the parameter values. The condition of the change of variables appearing in the fixed point method is also studied. Comparison with implicit Maple functions is provided, including the Laguerre polynomial case.

Alfredo Deaño, Amparo Gil, Javier Segura

### Towards Reliable Software for the Evaluation of a Class of Special Functions

Special functions are pervasive in all fields of science. The most well-known application areas are in physics, engineering, chemistry, computer science and statistics. Because of their importance, several books and a large collection of papers have been devoted to the numerical computation of these functions. But up to this date, even environments such as Maple, Mathematica, MATLAB and libraries such as IMSL, CERN and NAG offer no routines for the reliable evaluation of special functions. Here the notion reliable indicates that, together with the function evaluation a guaranteed upper bound on the total error or, equivalently, an enclosure for the exact result, is computed.

We point out how limit-periodic continued fraction representations of these functions can be helpful in this respect. The newly developed (and implemented) scalable precision technique is mainly based on the use of sharpened a priori truncation error and round-off error upper bounds for real continued fraction representations of special functions of a real variable. The implementation is reliable in the sense that it returns a sharp interval enclosure for the requested function evaluation, at the same cost as the evaluation.

Annie Cuyt, Stefan Becuwe

### Multimedia Prototype of a Bilingual Model Within Technology Based Learning Environment: An Implementation of a Mathematics Learning Framework

In response to the current globalization in the educational arena, and to the new policy of change in the medium of instruction for teaching mathematics and science in English as implemented by the Malaysian Government in 2003, we introduce the e-learning Bilingual Model which has been designed and used at the university. The multimedia prototype of the model consists of text-based content for first-year mathematics subjects with exact forms available in both the English language and the native language. Content is identified to provide descriptions of core concepts dynamically using, audio, video and graphics, and also constructed to provide bilingual glossaries. The combination of Information Communication Technologies (ICT) such as Short Messaging Service (SMS), MOODLE e-learning, and Freeware Online-portals for Group-websites (FrOG) become the setting for an integrated framework for Technology Based Learning Environment, and work as instructional delivery tools in addition to the traditional method of teaching mathematics.

Zur’aini Dahlan, Noraimi Shafie, Rozeha A. Rashid

### Methods to Access and Retrieve Mathematical Content in ActiveMath

This article describes how mathematical content items and formulæ are processed, retrieved, and accessed in

ActiveMath

. Central to the retrieval and access is a search tool which allows for searching text, attributes, relations and formulæ, and presenting items. The search tool has been evaluated according to the standard measures of precision and recall as well as for usability. We report results of these evaluations.

Paul Libbrecht, Erica Melis

### Logiweb – A System for Web Publication of Mathematics

Logiweb is a system for electronic publication and archival of machine checked mathematics of high typographic quality. It can verify the formal correctness of

pages

, i.e. mathematical papers expressed suitably. The present paper is an example of such a Logiweb page and the present paper is formally correct in the sense that it has been verified by Logiweb. The paper may of course contain informal errors like any other paper. Logiweb is neutral with respect to choice of logic and choice of notation and can support any kind of formal reasoning.

Logiweb uses the World Wide Web to publish Logiweb pages and Logiweb pages can be viewed by ordinary Web browsers. Logiweb pages can reference definitions, lemmas, and proofs on previously referenced Logiweb pages across the Internet. When Logiweb verifies a Logiweb page, it takes all transitively referenced pages into account.

Klaus Grue

### Interfacing with the Numerical Homotopy Algorithms in PHCpack

PHCpack

implements numerical algorithms for solving polynomial systems using homotopy continuation methods. In this paper we describe two types of interfaces to

PHCpack

. The first interface

PHCmaple

originally follows OpenXM, in the sense that the program (in our case Maple) that uses

PHCpack

needs only the executable version

phc

built by the package

PHCpack

. Following the recent development of

PHCpack

,

PHCmaple

has been extended with functions that deal with singular polynomial systems, in particular, the deflation procedures that guarantee the ability to refine approximations to an isolated solution even if it is multiple. The second interface to

PHCpack

was developed in conjunction with MPI (Message Passing Interface), needed to run the path trackers on parallel machines. This interface gives access to the functionality of

PHCpack

as a conventional software library.

Anton Leykin, Jan Verschelde

### Computational Construction of a Maximum Equilateral Triangle Inscribed in an Origami

We present an origami construction of a maximum equilateral triangle inscribed in an origami, and an automated proof of the correctness of the construction. The construction and the correctness proof are achieved by a computational origami system called

Eos

(E-origami system). In the construction we apply the techniques of geometrical constraint solving, and in the automated proof we apply Gröbner bases theory and the cylindrical algebraic decomposition method. The cylindrical algebraic decomposition is indispensable to the automated proof of the maximality since the specification of this property involves the notion of inequalities. The interplay of construction and proof by Gröbner bases method and the cylindrical algebraic decomposition supported by

Eos

is the feature of our work.

Tetsuo Ida, Hidekazu Takahashi, Mircea Marin, Fadoua Ghourabi, Asem Kasem

### A System for Interfacing MATLAB with External Software Geared Toward Automatic Differentiation

MATLAB is commonly considered to be an attractive, high-productivity programming environment by many computational scientists and engineers. So-called MEX-files are dynamically linked subroutines produced from, say, C or Fortran source code that, when compiled, can be run directly from within MATLAB as if they were MATLAB built-in functions. When applying automatic differentiation to a MATLAB program that calls external software via MEX-files, code is mechanically generated for the MATLAB part and for the external part in two separate phases. These resulting code fragments need to be put together via new MEX-files. This work introduces a novel software tool called automatic differentiation mexfunction generator that automatically generates MEX interface functions for gluing these automatically generated code fragments.

H. Martin Bücker, Atya Elsheikh, Andre Vehreschild

### KNOPPIX/Math: Portable and Distributable Collection of Mathematical Software and Free Documents

We propose a new computer environment for mathematicians that can be set up easily and quickly.

Tatsuyoshi Hamada, Kuniyasu Suzaki, Kengo Iijima, Arimitsu Shikoda

### Stability of Parametric Decomposition

We deal with ideals generated by polynomials with parametric coefficients, and introduce “stabilities on ideal structures” based on stability of forms of Gröbner bases. Then, we extend those stabilities to radicals and irreducible decompositions and show the computational tractability on those computations by integrating existing techniques.

Kazuhiro Yokoyama

### On the GAP Package sgpviz

This is a brief description of the

GAP

package

sgpviz

, a package designed to visualize finite semigroups through their

$\mathcal{D}$

-classes or Cayley graphs, as well as to make friendlier the usage of

GAP

when dealing with finite semigroups.

### Making Research on Symmetric Functions with MuPAD-Combinat

We report on the 2005 AIM workshop “Generalized Kostka Polynomials“, which gathered 20 researchers in the active area of

q

,

t

-analogues of symmetric functions. Our goal is to present a typical use-case of the open source package MuPAD-Combinat in a research environment.

Francois Descouens

### Calculating Cocyclic Hadamard Matrices in Mathematica: Exhaustive and Heuristic Searches

We describe a notebook in

Mathematica

which, taking as input data a homological model for a finite group

G

of order |

G

| = 4

t

, performs an exhaustive search for constructing the whole set of cocyclic Hadamard matrices over

G

. Since such an exhaustive search is not practical for orders 4

t

≥28, the program also provides an alternate method, in which an heuristic search (in terms of a genetic algorithm) is performed. We include some executions and examples.

V. Álvarez, J. A. Armario, M. D. Frau, P. Real

### An Interactive User Interface for Division Algorithms and the Buchberger Algorithm

Our objectives of building the interactive user interface are as follows:

(1) To select reducers in division algorithms and S-pairs in the Buchberger algorithm interactively.

(2) To visualize division algorithms and the Buchberger algorithm and to understand the algorithms intuitively.

(3) To create a user interface of division algorithms and the Buchberger algorithm without using computer algebra system languages.

Objective (1) has a mathematical background.We have studied and implemented division algorithms and the Buchberger algorithm in the ring of differential operators with rational function coefficients whose denominators do not vanish at the origin,

${\mathcal{D}_{alg}}$

([5], [3]). In the ring of polynomials and the local ring of that, methods of efficiently computing a remainder and a Gröbner basis have been studied in detail ([1], [2], [4]). However, in the ring

${\mathcal{D}_{alg}}$

, methods of those have not been studied in detail. As far as we have known, no system has satisfied our objectives. Therefore, we have designed an interactive user interface as a tool to understand and improve these algorithms. This system is a tool for us to study algorithms, however it may be useful for educational purposes.

Hiromasa Nakayama

### Experiment of Multithreading Symbolic and Algebraic Computations with OpenMP

This paper describes the current status of a project for multithreading algebraic computations, which aims at the utilization of today’s high-spec PCs with hyperthreading or dual-core technologies. Our effort is done by applying OpenXM with minimal cost of development, and includes memory management in multithreaded environment. Our empirical results show that the performance gain can be attained in numeric cases and in some cases of purely symbolic computations.

Hirokazu Murao

### Links to Projects. Mathematical Software, icms2006—Developer’s Meeting

This document is a collection of links and descriptions of projects related to icms2006—developer’s meeting.

Andrés Iglesias, Nobuki Takayama

### Backmatter

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