Bulletin of the London Mathematical Society - Advance Access

Diophantine approximation with arithmetic Functions, II

Alkan, E., Ford, K., Zaharescu, A. — 2009-07-03

We prove that real numbers can be well approximated by the normalized Fourier coefficients of newforms.

A priori analysis of initial data for the Riccati equation and asymptotic properties of its solutions

Chernyavskaya, N. A., Schiff, J., Shuster, L. A. — 2009-07-03

We obtain two main results for the Cauchy problem where x₀, y₀ R, r > 0, q ≥ 0, 1/r L₁^loc(R), q L₁^loc(R) and (1) For given initial data x₀, y₀ and functions r and q, we give a condition that can be used to determine whether the solution of the problem can be continued to the whole of R. (2) When the solution is defined on an infinite interval, we study its asymptotic properties as the argument tends to infinity.

Morris's pigeonhole principle and the Helly theorem for unions of convex sets

Eckhoff, J., Nischke, K.-P. — 2009-06-14

In 1973, H. C. Morris devised a combinatorial scheme, a ‘generalized pigeonhole principle’, which he used to prove a conjecture of Grünbaum and Motzkin from 1961. This conjecture proposed, in an abstract setting, a Helly-type theorem for certain families of disjoint unions of sets. A geometric instance dealing with disjoint unions of convex sets in R^d was proved in a special case by Larman in 1968 and in the general case by Amenta in 1996. Also covered by the conjecture is a topological extension of Amenta's theorem obtained by Kalai and Meshulam in 2008.

Morris's proof of the generalized pigeonhole principle is extremely involved, and the validity of some of his arguments is open to dispute. In the present paper, the principle is placed on a sound basis and established in a relatively short and transparent manner. This includes a particular case, left open by Morris, which is applied here to families of disjoint unions of boxes in R^d.

A direct proof of Z-stability for approximately homogeneous C*-algebras of bounded topological dimension

Dadarlat, M., Phillips, N. C., Toms, A. S. — 2009-06-12

We prove that a unital simple approximately homogeneous C*-algebra with no dimension growth absorbs the Jiang–Su algebra tensorially without appealing to the classification theory of these algebras. Our main result continues to hold under the slightly weaker hypothesis of exponentially slow dimension growth.

A brief note on the spectrum of the basic Dirac operator

Habib, G., Richardson, K. — 2009-06-03

In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation (M, F) with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O’Neill tensor and the first eigenvalue of the Dirac operator on M. We discuss examples and also define a new version of the basic Laplacian whose spectrum does not depend on the choice of bundle-like metric.

Automorphisms of doubly even self-dual binary codes

Gunther, A., Nebe, G. — 2009-05-27

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.

A Cauchy integral formula in superspace

De Bie, H., Sommen, F. — 2009-05-22

In previous work the framework for a hypercomplex function theory in superspace was established and amply investigated. In this paper a Cauchy integral formula is obtained in this new framework by exploiting techniques from orthogonal Clifford analysis. After introducing Clifford algebra-valued surface- and volume-elements, a purely fermionic Cauchy formula is proved. Combining this formula with the already well-known bosonic Cauchy formula yields the general case. Here the integration over the boundary of a supermanifold is an integration over the even as well as the odd boundary (in a formal way). Finally, some additional results such as a Cauchy–Pompeiu formula and a representation formula for monogenic functions are proved.

Area of small disks

Croke, C. B. — 2009-05-22

This paper considers Riemannian metrics on 2-dimensional disks where all geodesics are minimizing. A sharp reverse isoperimetric inequality is proved. This in turn yields near optimal bounds for the area of disks as well as near optimal upper bounds on the first non-zero Neumann eigenvalue of the Laplacian in terms only of the radius.

Zeros and the universality for the Euler-Zagier-Hurwitz type of multiple zeta-functions

Nakamura, T. — 2009-05-22

In this paper, we show relations between the zero-free region and the universality for the Euler–Zagier–Hurwitz type of multiple zeta-functions. Roughly speaking these relations imply that we can obtain the universality for the Euler–Zagier–Hurwitz type of multiple zeta-functions by their zero-free property, and vice versa. Moreover, we obtain the non-trivial zeros, joint denseness and functional independence for the Euler–Zagier–Hurwitz type of multiple zeta-functions.

Free centre-by-nilpotent-by-abelian groups

Johnson, M., Stohr, R. — 2009-05-22

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.

Non-linear factorization of linear operators

Johnson, W. B., Maurey, B., Schechtman, G. — 2009-05-22

We show, in particular, that a linear operator between finite-dimensional normed spaces, which factors through a third Banach space Z via Lipschitz maps, factors linearly through the identity from L([0, 1], Z) to L₁([0, 1], Z) (and thus, in particular, through each L_p(Z), for 1 ≤ p ≤ ) with the same factorization constant. It follows that, for each 1 ≤ p ≤ , the class of L_p spaces is closed under uniform (and even coarse) equivalences. The case p = 1 is new and solves a problem raised by Heinrich and Mankiewicz in 1982. The proof is based on a simple local–global linearization idea.

A geometric proof of the Karpelevich-Mostow's theorem

Di Scala, A. J., Olmos, C. — 2009-05-12

In this paper we give a geometric proof of Karpelevich's theorem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non-compact type, has a totally geodesic orbit. In fact, this is equivalent to a well-known result of Mostow about the existence of compatible Cartan decompositions.

The ring of reciprocal polynomials and rank varieties

Woodcock, C. — 2009-05-07

Let p be a prime and let G be a finite p-group. In a recent paper we introduced a commutative graded Z-algebra R_G (which classifies the so-called convolutions on G). Now let K be an algebraically closed field of characteristic p and let M be a non-zero finitely generated K[G]-module. A general rank variety W_G(M) is constructed quite explicitly as a determinantal subvariety of the variety of K-valued points of the spectrum of R_G. Further, it is shown that the quotient variety W_G(M)/G is inseparably isogenous to the usual cohomological support variety V_G(M).

The Riesz energy of the Nth roots of unity: an asymptotic expansion for large N

Brauchart, J. S., Hardin, D. P., Saff, E. B. — 2009-05-04

We derive the complete asymptotic expansion in terms of powers of N for the Riesz s-energy of N equally spaced points on the unit circle as N -> . For s ≥ – 2, such points form optimal energy N-point configurations with respect to the Riesz potential 1/r^s, s != 0, where r is the Euclidean distance between points. By analytic continuation we deduce the expansion for all complex values of s. The Riemann zeta function plays an essential role in this asymptotic expansion.

A note on the Coates-Sinnott conjecture

Aoki, M. — 2009-04-28

Let K be a finite abelian extension of a totally real number field. The Brumer conjecture asserts that the Stickelberger element annihilates the ideal class group of K. In this article, we will prove under some assumptions that the conjecture implies the Coates–Sinnott conjecture which is an analogue of the Brumer conjecture for higher K-groups.

Linear groups with many two-generator soluble subgroups

Wilson, J. S. — 2009-04-17

Let G be a linear group of degree n over a field and let S be any generating set. It is proved that (a) if every pair of products of at most elements of S S^{– 1} generates a soluble group then G is soluble and (b) if G is soluble and every pair of products of at most elements of S S^–1 generates a nilpotent group then G is locally nilpotent.