A theorem of Brady and Meier states that a right-angled Artin group is a duality group if and only if the flag complex of the defining graph is Cohen–Macaulay. We use this to give an example of a RAAG with the property that its outer automorphism group is not a virtual duality group. This gives a partial answer to a question of Vogtmann. In an appendix, Brück describes how he used a computer-assisted search to find further examples.

The aim of this note is to present a new, elementary proof of a result of Baas and Madsen on the mod p cohomology of certain quotients of the spectrum BP.

Let K be a quadratic number field with 2-class group of type (2,2). Thus if Sk is the Sylow 2-subgroup of the ideal class group of K, then Sk = ℤ/2ℤ × ℤ/2ℤ Let

K ⊂ K1 ⊂ K2 ⊂ K3 ⊂…

the 2-class field tower of K. Thus K1 is the maximal abelian unramified extension of K of degree a power of 2; K2 is the maximal abelian unramified extension of K of degree a power of 2; etc. By class field theory the Galois group Ga1 (K1/K) ≅ Sk ≅ ℤ/2ℤ × ℤ/2ℤ, and in this case it is known that Ga(K2/Kl) is a cyclic group (cf. [3] and [10]). Then by class field theory the class number of K2 is odd, and hence K2 = K3 = K4 = …. We say that the 2-class field tower of K terminates at K1 if the class number of K1 is odd (and hence K1 = K2 = K3 = … ); otherwise we say that the 2-class field tower of K terminates at K2. Our goal in this paper is to determine how likely it is for the 2-class field tower of K to terminate at K1 and how likely it is for the 2-class field tower of K to terminate at K2. We shall consider separately the imaginary quadratic fields and the real quadratic fields.

§ 1. Introductory. The formula to be proved is

where n is zero or a positive integer, R(m)>0, R(αs-l)>0, s = 1, 2, …, p, p≧q.

In § 2 a product of two modified Bessel Functions of the Second Kind is expressed as an integral with a function of the same type as a factor of the integrand. In § 3 an integral involving a product of these functions, regarded as functions of their orders, is evaluated in terms of another function of this kind. These results were suggested by a study of Mellin's inversion formula.

We present a ready to compute trace formula for Hecke operators on vector-valued modular forms of integral weight for SL2(ℤ) transforming under the Weil representation. As a corollary, we obtain a ready to compute dimension formula for the corresponding space of vector-valued cusp forms, which is more general than the dimension formulae previously published in the vector-valued setting.

Let E[τ] be a locally convex Hausdorff topological vector space. An extended decomposition of E[τ] is a family {Ea}α∈A of closed subspaces of E such that, for each x in E and each α in A, there exists a unique point xα in Eα, with Here convergence will have the following meaning. Let Ф denote the set of all finite subsets of A. The sum is said to be convergent to x if for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф containing φ0. It follows that is Cauchy if and only if, for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф disjoint from φ0.

We are interested in two parameter eigenvalue problems of the form

subject to Dirichlet boundary conditions

The weight function 5 and the potential q will both be assumed to lie in L2[0,1]. The problem (1.1), (1.2) generates eigencurves

in the sense that for any fixed λ, ν(λ) is the nth eigenvalue ν, (according to oscillation indexing) of (1.1), (1.2). These curves are in fact analytic functions of λ and have been the object of considerable study in recent years. The survey paper [1] provides background in this area and itemises properties of eigencurves.