We assign names and new generators to the transitive groups of degree up to 15, reflecting their structure.

We describe the “Deep Thought” algorithm, which can, among other things, take a commutator presentation for a finitely generated torsion-free nilpotent group G, and produce explicit polynomials for the multiplication of elements of G. These polynomials were first shown to exist by Philip Hall, and allow for “symbolic collection” in finitely generated nilpotent groups. We discuss various practicalissues in calculations in such groups, including the construction of a hybrid collector, making use of both the polynomials and ordinary collection from the left.

This paper provides details of a Magma computer program for calculating various homotopy-theoretic functors, defined on finitely presented groups. A copy of the program is included as an Add-On. The program can be used to compute: the nonabelian tensor product of two finite groups, the first homology of a finite group with coefficients in the arbirary finite module, the second integral homology of a finite group relative to its normal subgroup, the third homology of the finite p-group with coefficients in Zp, Baer invariants of a finite group, and the capability and terminality of a finite group. Various other related constructions can also be computed.

This paper considers a number of related problems concerning the computation of eigenvalues and complex resonances of a general self-adjoint operator H. The feature which ties the different sections together is that one restricts oneself to spectral properties of H which can be proved by using only vectors from a pre-assigned (possibly finite-dimensional) linear subspace L.

In this paper we prove a conjecture due to R. Carter [2], concerning the action of the finite general linear group GLn(q) on a cuspidal module. As an application of this result, we work out the caseGL4(q).

We give an algorithm that takes as input a transitive permutation group (G, Ω) of degree n={m\choose 2}, and decides whether or not Ω is G-isomorphic to the action of G on the set of unordered pairs of some set Γ on which G acts 2-homogeneously. The algorithm is constructive: if a suitable action exists, then one such will be found, together with a suitable isomorphism. We give a deterministic O(sn logcn) implemention of the algorithm that assumes advance knowledge of the suborbits of (G, Ω). This leads to deterministic O(sn2) and Monte-Carlo O(sn logcn) implementations that do not make this assumption.

We describe a mechanically verified proof of correctness of the floating point multiplication, division, and square root instructions of the AMD-K7 microprocessor. The instructions are implemented in hardware and represented here by register-transfer level specifications, the primitives of which are logical operations on bit vectors. On the other hand, the statements of correctness, derived from IEEE Standard 754, are arithmetic in nature and considerably more abstract. Therefore, we begin by developing a theory of bit vectors and their role in floating point representations and rounding. We then present the hardware model and a rigorous proof of its correctness. All of our definitions, lemmas and theorems have been formally encoded in the ACL2 logic, and every step in the proof has been mechanically checked with the ACL2 prover.