Algorithms for Lie algebras of algebraic groups

In this thesis we present several new algorithms for dealing with simple algebraic groups and their Lie algebras. These groups and algebras have been studied for a long time, first in a theoretical sense and later with regards to effective calculations on the computer, including implementations in the GAP and Magma computer algebra systems. We build in particular on work by Arjeh Cohen, Willem de Graaf, Sergei Haller, Scott Murray, and Don Taylor. The work is partly stimulated by the matrix group recognition project: an international project which is aimed at the algorithmic analysis of problems with matrix groups over finite fields. Many algorithms that have been previously developed in this branch of research, however, apply only to groups and algebras over fields of characteristic 0 or at least 5. For instance, Cohen and Murray, and, independently, Ryba recently gave an algorithm for computing a split maximal toral subalgebra of a Lie algebra in all characteristics except 2 and (to a certain extent) 3. Unfortunately, not only their proofs but also their algorithms do not work in the excluded cases. Similarly, the algorithm for computing a Chevalley basis of a Lie algebra, when given a split toral subalgebra, is straightforward in almost all characteristics, and has consequently been implemented in major computer algebra systems such as GAP and Magma. In characteristics 2 and 3, however, the algorithm is much more involved. This thesis starts with an extensive introduction to the mathematical objects occurring in this thesis, such as root data, algebraic groups, and Lie algebras. The new results in this thesis are a heuristic algorithm for computing split maximal toral subalgebras of Lie algebras of split simple algebraic groups over fields of characteristic 2, and an algorithm for computing Chevalley bases of Lie algebras of split simple algebraic groups over any field. The latter algorithm is proved to be polynomial in the case where the field is finite. These algorithms are applied to the problem of recognizing these Lie algebras among all Lie algebras, and they help in the analysis of the associated algebraic groups. We also apply these algorithms in the computer aided proof that there is no graph on which a certain group acts distance transitively. All of the algorithms presented in this thesis have been implemented in the Magma computer algebra system

Pedagogical conditions of training future managers of foreign economic activity for cross-cultural communication

The study reveals the pedagogical conditions of training future managers of foreign economic activity for cross-cultural communication. The analysis of the scientific research in this field has shown that the essence of the pedagogical conditions has not been generalized or systemized yet. On the base of the invariable vectors of the pedagogical process, the pedagogical conditions of training future managers of foreign economic activity for cross-cultural communication are outlined. They are: creating multicultural surrounding in the process of training managers for gaining experience in cross-cultural communication (organization); adaptive gradual management of the educational activity with due regards for personal, professional, communicative qualities of future managers of foreign economic activity (management); subject and subject interaction, directed at the optimal management of cross-cultural conflicts (communication)

Shunting passenger trains : getting ready for departure

In this paper we consider the problem of shunting train units on a railway station. Train units arrive at and depart from the station according to a given train schedule and in between the units may have to be stored at the station. The assignment of arriving to departing train units (called matching) and the scheduling of the movements to realize this matching is called shunting. The goal is to realize the shunting using a minimal number of shunt movements. For a restricted version of this problem an ILP approach has been presented in the literature. In this paper, we consider the general shunting problem and derive a greedy heuristic approach and an exact solution method based on dynamic programming. Both methods are flexible in the sense that they allow the incorporation of practical planning rules and may be extended to cover additional requirements from practice

Automated proofs using bracket algebra with Cinderella and OpenMath

This paper describes the results of a project intended to make it possible to put forward geometrical theorems by pointing and clicking, and then obtain a proof for that theorem automatically. This goal was achieved by adding various options to Cinderella [1], a computer program with which one can create geometrical configurations. Its internal ‘Randomized prover’ is able to discover theorems automatically. In the project the functionality was added to find proofs for these theorems with the aid of the computer algebra package GAP [9]. Communication between these two programs and the various steps in generating the proof is done by means of OpenMath [5, 7]. The proofs are represented by bracket calculations as proposed in [8]

Algorithms for Lie algebras of algebraic groups

In this thesis we present several new algorithms for dealing with simple algebraic groups and their Lie algebras. These groups and algebras have been studied for a long time, first in a theoretical sense and later with regards to effective calculations on the computer, including implementations in the GAP and Magma computer algebra systems. We build in particular on work by Arjeh Cohen, Willem de Graaf, Sergei Haller, Scott Murray, and Don Taylor. The work is partly stimulated by the matrix group recognition project: an international project which is aimed at the algorithmic analysis of problems with matrix groups over finite fields. Many algorithms that have been previously developed in this branch of research, however, apply only to groups and algebras over fields of characteristic 0 or at least 5. For instance, Cohen and Murray, and, independently, Ryba recently gave an algorithm for computing a split maximal toral subalgebra of a Lie algebra in all characteristics except 2 and (to a certain extent) 3. Unfortunately, not only their proofs but also their algorithms do not work in the excluded cases. Similarly, the algorithm for computing a Chevalley basis of a Lie algebra, when given a split toral subalgebra, is straightforward in almost all characteristics, and has consequently been implemented in major computer algebra systems such as GAP and Magma. In characteristics 2 and 3, however, the algorithm is much more involved. This thesis starts with an extensive introduction to the mathematical objects occurring in this thesis, such as root data, algebraic groups, and Lie algebras. The new results in this thesis are a heuristic algorithm for computing split maximal toral subalgebras of Lie algebras of split simple algebraic groups over fields of characteristic 2, and an algorithm for computing Chevalley bases of Lie algebras of split simple algebraic groups over any field. The latter algorithm is proved to be polynomial in the case where the field is finite. These algorithms are applied to the problem of recognizing these Lie algebras among all Lie algebras, and they help in the analysis of the associated algebraic groups. We also apply these algorithms in the computer aided proof that there is no graph on which a certain group acts distance transitively. All of the algorithms presented in this thesis have been implemented in the Magma computer algebra system

Open Math in SCIEnce : SCSCP and POPCORN

In this short communication we want to give an overview of how OpenMath is used in the European project "SCIEnce" [12]. The main aim of this project is to allow unified communication between different computer algebra systems (CASes) or different instances of one CAS. This may involve one or more computers, clusters, and even grids. The main topics are the use of OpenMath to marshal mathematical objects for transport between different CASes, an alternative textual OpenMath representation more suitable for human reading and writing, and finally the publicly released Java Library developed for the project