Stephen A. Linton

Distance-transitive representations of the sporadic groups

A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free. In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.

HPC-GAP: engineering a 21st-century high-performance computer algebra system

Symbolic computation has underpinned a number of key advances in Mathematics and Computer Science. Applications are typically large and potentially highly parallel, making them good candidates for parallel execution at a variety of scales from multi‐core to high‐performance computing systems. However, much existing work on parallel computing is based around numeric rather than symbolic computations. In particular, symbolic computing presents particular problems in terms of varying granularity and irregular task sizes that do not match conventional approaches to parallelisation. It also presents problems in terms of the structure of the algorithms and data. This paper describes a new implementation of the free open‐source GAP computational algebra system that places parallelism at the heart of the design, dealing with the key scalability and cross‐platform portability problems. We provide three system layers that deal with the three most important classes of hardware: individual shared memory multi‐core nodes, mid‐scale distributed clusters of (multi‐core) nodes and full‐blown high‐performance computing systems, comprising large‐scale tightly connected networks of multi‐core nodes. This requires us to develop new cross‐layer programming abstractions in the form of new domain‐specific skeletons that allow us to seamlessly target different hardware levels. Our results show that, using our approach, we can achieve good scalability and speedups for two realistic exemplars, on high‐performance systems comprising up to 32000 cores, as well as on ubiquitous multi‐core systems and distributed clusters. The work reported here paves the way towards full‐scale exploitation of symbolic computation by high‐performance computing systems, and we demonstrate the potential with two major case studies.

Qualitative modelling via constraint programming

Qualitative modelling is a technique integrating the fields of theoretical computer science, artificial intelligence and the physical and biological sciences. The aim is to be able to model the behaviour of systems without estimating parameter values and fixing the exact quantitative dynamics. Traditional applications are the study of the dynamics of physical and biological systems at a higher level of abstraction than that obtained by estimation of numerical parameter values for a fixed quantitative model. Qualitative modelling has been studied and implemented to varying degrees of sophistication in Petri nets, process calculi and constraint programming. In this paper we reflect on the strengths and weaknesses of existing frameworks, we demonstrate how recent advances in constraint programming can be leveraged to produce high quality qualitative models, and we describe the advances in theory and technology that would be needed to make constraint programming the best option for scientific investigation in the broadest sense.

Interfacing Coq + SSReflect with GAP

We report on an extendable implementation of the communication interface connecting Coq proof assistant to the computational algebra system GAP using the Symbolic Computation Software Composability Protocol (SCSCP). It allows Coq to issue OpenMath requests to a local or remote GAP instances and represent server responses as Coq terms.

The primitive distance-transitive representations of the Fischer groups

We classify the primitive distance-transitive representations of the Fischer sporadic simple groups and their automorphism groups. It turns out that the only primitive distance-transitive representationsof thesegroups are their rank 3 representations. In the process of our work, we also classify and study the primitive multiplicity-free permutation representations of these Fischer groups. Our methods, which we describe in some detail, demonstrate the use of computational and randomized techniques in the classification of distance-transitive graphs and the study of very large permutation representations.

Product Replacement in the Monster

We show that the product replacement algorithm can be used to produce random elements of the Monster group. These random elements are shown to have the same distribution of element orders as uniformly distributed random elements after a small number of steps.

Towards the Calculation of Casimir Forces for Inhomogeneous Planar Media

Casimir forces arise from vacuum fluctuations. They are fully understood only for simple models, and are important in nano- and microtechnologies. We report our experience of computer algebra calculations toward the Casimir force for models involving inhomogeneous dielectrics. We describe a methodology that greatly increases confidence in any results obtained, and use this methodology to demonstrate that the analytic derivation of scalar Green’s functions is at the boundatry of current computer algebra technology. We further demonstrate that Lifshitz theory of electromagnetic vacuum energy can not be directly applied to calculate the Casimir stress for models of this type, and produce results that indicate the possibility of alternative regularizations. We discuss the relative strengths and weaknesses of computer algebra systems when applied to this type of problem, and suggest combined numerical and symbolic approaches toward a more general computational framework.

Casimir forces for inhomogeneous planar media.

Casimir forces arise from vacuum fluctuations. They are fully understood only for simple models, and are important in nano- and microtechnologies. We report our experience of computer algebra calculations towards the Casimir force for models involving inhomogeneous dielectrics. We describe a methodology that greatly increases confidence in any results obtained, and use this methodology to demonstrate that the analytic derivation of scalar Greens functions is at the boundary of current computer algebra technology. We further demonstrate that Lifshitz theory of electromagnetic vacuum energy can not be directly applied to calculate the Casimir stress for models of this type, and produce results that have led to alternative regularisations. Using a combination of our new computational framework and the new theory based on our results, we provide specific calculations of Casimir forces for planar dielectrics having permittivity that declines exponentially. We discuss the relative strengths and weaknesses of computer algebra systems when applied to this type of problem, and describe a combined numerical and symbolic computational framework for calculating Casimir forces for arbitrary planar models.

GAP 4 at Twenty-one - Algorithms, System Design and Applications

The first public beta release of GAP 4[6] was made on July 18 1997. Since then the system has been cited in over 2400 publications, and its distribution now includes over 130 contributed extension pack- ages. This tutorial will review the special features of computational abstract algebra and how they are reflected in the system design; some areas of current algorithmic development, and some recent achievements.

An efficient programming model for memory-intensive recursive algorithms using parallel disks

In order to keep up with the demand for solutions to problems with ever-increasing data sets, both academia and industry have embraced commodity computer clusters with locally attached disks or SANs as an inexpensive alternative to supercomputers. With the advent of tools for parallel disks programming, such as MapReduce, STXXL and Roomy --- that allow the developer to focus on higher-level algorithms --- the programmer productivity for memory-intensive programs has increased many-fold. However, such parallel tools were primarily targeted at iterative programs. We propose a programming model for migrating recursive RAM-based legacy algorithms to parallel disks. Many memory-intensive symbolic algebra algorithms are most easily expressed as recursive algorithms. In this case, the programming challenge is multiplied, since the developer must re-structure such an algorithm with two criteria in mind: converting a naturally recursive algorithm into an iterative algorithm, while simultaneously exposing any potential data parallelism (as needed for parallel disks). This model alleviates the large effort going into the design phase of an external memory algorithm. Research in this area over the past 10 years has focused on per-problem solutions, without providing much insight into the connection between legacy algorithms and out-of-core algorithms. Our method shows how legacy algorithms employing recursion and non-streaming memory access can be more easily translated into efficient parallel disk-based algorithms. We demonstrate the ideas on a largest computation of its kind: the determinization via subset construction and minimization of very large nondeterministic finite set automata (NFA). To our knowledge, this is the largest subset construction reported in the literature. Determinization for large NFA has long been a large computational hurdle in the study of permutation classes defined by token passing networks. The programming model was used to design and implement an efficient NFA determinization algorithm that solves the next stage in analyzing token passing networks representing two stacks in series.