Bulletin of the London Mathematical Society - current issue

Automorphisms of doubly even self-dual binary codes

Gunther, A., Nebe, G. — Tue, 27 Oct 2009 09:10:40 PDT

The automorphism group of a binary doubly even self-dual code is always contained in the alternating group. On the other hand, given a permutation group G of degree n there exists a doubly even self-dual G-invariant code if and only if n is a multiple of 8, every simple self-dual F₂G-module occurs with even multiplicity in F₂ⁿ, and G is contained in the alternating group.

Rigidity of the Mori cone for Fano manifolds

Wisniewski, J. A. — Tue, 27 Oct 2009 09:10:41 PDT

The Mori cone is rigid in smooth connected families of Fano manifolds.

Finitary group cohomology and Eilenberg-MacLane spaces

Hamilton, M. — Tue, 27 Oct 2009 09:10:41 PDT

We say that a group G has cohomology almost everywhere finitary if and only if the nth cohomology functors of G commute with filtered colimits for all sufficiently large n. In this paper, we show that if G is a group in Kropholler's class lhF with cohomology almost everywhere finitary, then G has an Eilenberg–MacLane space K(G, 1) that is dominated by a CW-complex with finitely many n-cells for all sufficiently large n. It is an open question as to whether this holds for arbitrary G. We also remark that the converse holds for any group G.

Free centre-by-nilpotent-by-abelian groups

Johnson, M., Stohr, R. — Tue, 27 Oct 2009 09:10:41 PDT

We prove that the free centre-by-(nilpotent-of-class-(c – 1))-by-abelian groups F/[_c(F'), F] are torsion-free for c = 6. This is in startling contrast to the cases when c is a prime and when c = 4, where these relatively free groups contain non-trivial elements of finite order.

Automorphism invariance and identities

Khukhro, E. I., Klyachko, Ant. A., Makarenko, N. Yu., Melnikova, Yu. B. — Tue, 27 Oct 2009 09:10:41 PDT

If an outer (multilinear) commutator identity holds in a large subgroup of a group, then it holds also in a large characteristic subgroup. Similar assertions are valid for algebras and their ideals or subspaces. Varying the meaning of the word ‘large’, we obtain many interesting and useful facts. An example is produced showing that these results cannot be extended to arbitrary (non-multilinear) identities. As an application, a sharp estimate is given for the ‘virtual derived length’ of a (virtually solvable)-by-(virtually solvable) group.

Sum-product estimates for well-conditioned matrices

Solymosi, J., Vu, V. — Tue, 27 Oct 2009 09:10:41 PDT

We show that if A is a finite set of d x d well-conditioned matrices with complex entries, then the following sum–product estimate holds | A + A | x |A·A| = (|A| ^5/2).

Semigroups of chaotic operators

Bayart, F., Bermudez, T. — Tue, 27 Oct 2009 09:10:41 PDT

We prove the existence of chaotic semigroups of operators that do not contain any chaotic operator. In particular, we obtain a chaotic operator T such that T is not chaotic for some unimodular complex number .

Unconditional bases and strictly convex dual renormings

Smith, R. J., Troyanski, S. — Tue, 27 Oct 2009 09:10:41 PDT

We present equivalent conditions for a space X with an unconditional basis to admit an equivalent norm with a strictly convex dual norm.

Amalgams of designs and nets

McDonough, T. P., Mavron, V. C., Ward, H. N. — Tue, 27 Oct 2009 09:10:41 PDT

We present a procedure for amalgamating a net and a collection of designs into a single design. At first this amalgam is just point-regular, but it acquires additional regularities upon imposing restrictions on the ingredients. At its most regular, the amalgam is quasi-symmetric, and designs with the same parameters as those recently constructed by Bracken, McGuire and Ward appear. Along the way we discuss a class of designs generalising Hadamard designs, and we consider the problem of packing projective planes with disjoint line sets into the same point set.

A sharp combinatorial version of Vaaler's theorem

Ball, K. M., Prodromou, M. — Tue, 27 Oct 2009 09:10:42 PDT

In 1979 Vaaler proved that every d-dimensional central section of the cube [–1, 1]ⁿ has volume at least 2^d. We prove the following sharp combinatorial analogue. Let K be a d-dimensional subspace of Rⁿ. Then, there exists a probability measure P on the section [–1, 1]ⁿ K such that the quadratic form dominates the identity on K (in the sense that the difference is positive semi-definite).

Gorenstein dimension and proper actions

Bahlekeh, A., Dembegioti, F., Talelli, O. — Tue, 27 Oct 2009 09:10:42 PDT

We conjecture that a group G admits a finite-dimensional classifying space for proper actions if and only if the Gorenstein projective dimension of G is finite. We verify the one-dimensional case of this conjecture. Some evidence are given for the hypothesis that the Gorenstein projective ZG-modules are precisely Benson's class of cofibrant modules.

Positive solutions to a higher-order nonlinear delay boundary value problem on the half-line

Philos, Ch. G. — Tue, 27 Oct 2009 09:10:42 PDT

The main result of this paper establishes sufficient conditions that guarantee the existence of positive solutions to a boundary value problem on the half-line for nth-order (n > 2) nonlinear differential equations with positive delays. The application of the main result to the particular case of Emden–Fowler-type differential equations with constant delays as well as to the special case of linear differential equations with constant delays, is also presented. Moreover, some (general or specific) examples demonstrating the applicability of our result are included.

Reversibility in the group of homeomorphisms of the circle

Gill, N., O'Farrell, A. G., Short, I. — Tue, 27 Oct 2009 09:10:42 PDT

The group of orientation-preserving homeomorphisms of the circle is simple, and, because there are non-trivial involutions in this group, it must be generated by its involutions. We show that, in this group of homeomorphisms, each element can be expressed as a product of three involutions. We also characterise those elements of the group that can be expressed as a composite of two involutions, and perform a similar characterisation in the full group of homeomorphisms of the circle.

Vector product algebras

Darpo, E. — Tue, 27 Oct 2009 09:10:42 PDT

Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.

Eigenvalue decay of operators on harmonic function spaces

Bandtlow, O. F., Chu, C.-H. — Tue, 27 Oct 2009 09:10:42 PDT

Let be an open set in R^d (d > 1) and let h() be the Fréchet space of harmonic functions on . Given a bounded linear operator L : h () -> h(), we show that its eigenvalues _n, arranged in decreasing order and counting multiplicities, satisfy |_n| ≤ K exp(–cn^1/(d–1)), where K and c are two explicitly computable positive constants.

The structure of finite groups of conjugate rank 2

Dolfi, S., Jabara, E. — Tue, 27 Oct 2009 09:10:42 PDT

We give a structure theorem for the finite groups with three conjugacy class sizes. In particular, they are solvable groups with derived length at most 3 or nilpotent groups.

Decompositions of complete graphs into long cycles

Bryant, D., Horsley, D. — Tue, 27 Oct 2009 09:10:42 PDT

The problem of decomposing complete graphs into cycles of arbitrary specified lengths has attracted much attention, but remains largely unsolved. In this paper, the problem is settled in the case where the specified cycle lengths are each more than about half the order of the complete graph. The proof is based on a result that modifies certain existing cycle decompositions to produce new ones in which the lengths of two of the cycles are altered.

Generation of ray class fields by elliptic units

Jung, H. Y., Koo, J. K., Shin, D. H. — Tue, 27 Oct 2009 09:10:42 PDT

We show that a certain special value of a Siegel function generates the ray class field over the Hilbert class field for an imaginary quadratic field, from which we settle the Schertz's conjecture.

Harold Scott Macdonald Coxeter, FRS, 1907-2003

Roberts, S., Weiss, A. I. — Tue, 27 Oct 2009 09:10:42 PDT