Bryant W. York

Constructing Permutation Representations for Matrix Groups

New techniques, both theoretical and practical, are presented for constructing permutation representations for computing with matrix groups defined over finite fields. The permutation representation is constructed on a conjugacy class of subgroups of prime order. We construct a base for the permutation representation, which in turn simplifies the computation of a strong generating set. In addition, we present an elementary test for checking the simplicity of the permutation image. The theory has been successfully tested on a representation of the sporadic simple groupLy, discovered by Lyons(1972). With noa prioriassumptions, we find a permutation representation of degree 9606125 on a conjugacy class of subgroups of order 3, find the order of the resulting permutation group, and verify simplicity. A Monte Carlo variation of the algorithm was used to achieve better space and time efficiency. The construction of the permutation representation required four CPU days on a SPARCserver 670MP with 64 MB. The permutation representation was used implicitly in the sense that the group element was stored as a matrix, and its permutation action on a “point” was determined using a pre-computed data structure. Thus, additional computations required little additional space. The algorithm has also been implemented using the MasPar MP-1 SIMD parallel computer and 8 SPARC-2s running under MPI. The results of those parallel experiments are briefly reviewed.

Generalized Stone-Wales Transformations

Abstract We introduce a large class of transformations allowing rearrangements of the fullerene surface. We call this set of rearrangements generalized Stone-Wales transformations (gSW) because the well-known Stone-Wales (or pyracylene) flip may be seen as the simplest representative of the novel gSW family. The interconversion between the two C28 fullerene isomers is presented as a simple example of gSW application. Further examples involving the complete generation of C 60 and C 70 isomer spaces are provided. In both cases, just one fullerene molecule is used as seed data. Some considerations on gSW energetics are also reported.

Constructing permutation representations for large matrix groups

New techniques, both theoretical and practical, are presented for constructing a permutation representation for a matrix group. We assume that the resulting permutation degree, n, can be 10,000,000 and larger. The key idea is to build the new permutation representation using the conjugation action on a conjugacy class of subgroups of prime order. A unique signature for each group element corresponding to the conjugacy class is used in order to avoid matrix multiplication. The requirement of at least n matrix multiplications would otherwise have made the computation hopelessly impractical. Additional software optimizations are described, which reduce the CPU time by at least an additional factor of 10. Further, a special data structure is designed that serves both as a search tree and as a hash array, while requiring space of only 1.6n log2 n bits. The technique has been implemented and tested on the sporadic simple group Ly, discovered by Lyons [9], in both a sequential (SPARCserver 670MP) and parallel SIMD (MasPar MP-1) version. Starting with a generating set for Ly as a subgroup of GL(111,5) [5], a set of generating permutations for Ly acting on 9, 606, 125 points is constructed as well as a base for this permutation representation. The sequential version required four days of CPU time to construct a data structure which can be used to compute the permutation image of an arbitrary matrix. The parallel version did so in 12 hours. Work is in progress on a faster parallel implementation.

Applications of Bicrystallography: Revealing Generic Similarities in Coincidence Site Lattice Boundaries of all Holohedral Cubic Materials and Facilitating the Design of 3D Printed Models of such Grain Boundaries

1. Nano-Crystallography Group, Department of Physics, Portland State University, Portland, OR 97207-0751 2. Crystal Solutions, LLC, Portland, OR 97205 3. Department of Computer Science, Portland State University, Portland, OR 97207-0751 4. 3D Systems Corporation, Wilsonville, OR 97070 5. Department of Chemistry, University of Washington, Seattle, WA 98195 6. Chemical and Materials Sciences Division, Pacific Northwest National Laboratory, Richland, WA 99352

Matrix inversion in O (log n ) on a scan-enhanced reconfigurable mesh computer

With the arrival ofthe current generation offast processor chips and improved interconnect technology, low-cost 3-dimensional reconfigurable mesh computers have become more feasible. They could present an attractive price/performance option to large supercomputers and small clusters of workstations. In 1976 Csanky introduced a parallel algorithm for matrix inversion which executed in O(log n) steps on n4 processors. Csankys algorithm was designed for a CREW PRAM. More recently, Leighton produced an implementation of Csankys algorithm for n meshes of trees which achieves O(log n) steps on 4n4 3n3 processors.