We shall introduce a p-local geometry Δp(G) for the pair (G, p) of a finite group G and a prime divisor p of the order of G, which is constructed by ‘maximal parabolic like’ subgroups of G. Under the natural hypothesis, Δp(G) behaves very much like the building associated with a group of Lie type in characteristic p. We shall also show that, under some hypothesis, Δp(G) is homotopy equivalent to the subgroup complex [Bscr ]cenp(G) which is a more essential part of the p-radical complex [Bscr ]p(G). Some of the p-local sporadic geometries can be understood well in our system.

In [16] and [17], Tan looks at the principal blocks of the symmetric groups [Sfr ]9, [Sfr ]10, [Sfr ]11, [Sfr ]12 over a field of characteristic three, and highlights the contrast with blocks of small abelian defect. Here we continue by determining the Ext-quivers of the blocks of [Sfr ]13 over the same field; this in particular provides examples of a [4:1]-pair and a [3:2]-pair with non-abelian defect.

Soit {pk(n)}w(n)k=1 la suite croissante des facteurs premiers distincts d'un entier n. Nous donnons, lorsque k → ∞, une approximation uniforme de la loi de répartition limite de la fonction arithmétique n [map ] pk(n), précisant ainsi un résultat classique d'Erdős. Deux applications en sont déduites, relatives à la médiane de cette loi et à celle de la fonction “ nombre de facteurs premiers ”.

Let ∥x∥ denote distance to a nearest integer. Heilbronn's theorem in Diophantine approximation [8] may be stated as follows.

Let G be a group, and let 〈A[mid ]R〉 be a finite group presentation for G with [mid ]R[mid ][ges ][mid ]A[mid ]. Then there exists a finite semigroup presentation 〈B[mid ]Q〉 for G such that [mid ]Q[mid ]- [mid ]B[mid ] = [mid ]R[mid ]- [mid ]A[mid ]. Moreover, B is either the same generating set or else it contains one additional generator.

Let G be a compact, simply-connected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simply-connected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories [Cscr ], hocolim[Cscr ]BGI where GI are certain connected maximal rank subgroups of G.

In two previous papers [7, 10] the algebraic and transcendental values of the Gauss hypergeometric series

F(a, b; c; z) = 1+abcz1!+a(a+1)b(b+1)c(c+1)z22!+… (1·1)

were investigated, for various real rational parameters a; b; c and algebraic and rational values of z ∈ (0, 1), by applying the singular values of the complete elliptic integral of the first kind K(k) to certain classical F transformation formulae, where k denotes the modulus. Our main aim in the present paper is to use similar methods to determine the special values of (1·1) for the case a = 112, b = 712 and c = 23.

Let G be a finite group; by an algebraic 2-complex over G we mean an exact sequence of Z[G]-modules of the form:

E = (0 → J → E2 → E1 → E0 → Z → 0)

where Er is finitely generated free over Z[G] for 0 [les ] r [les ] 2. The module J is determined up to stability by the fact of appearing in such an exact sequence; we denote its stable class by Ω3(Z); E is said to be minimal when rkZ(J) attains the minimum possible value within Ω3(Z).

Let A be an Artin algebra. Then there are finitely many non-isomorphic simple A-modules. Suppose S1, S2, …, Sn form a complete list of all non-isomorphic simple A-modules and we fix this ordering of simple modules. Let Pi and Qi be the projective cover and the injective envelope of Si respectively. With this order of simple modules we define for each i the standard module Δ(i) to be the maximal quotient of Pi with composition factors Sj with j [les ] i. Let Δ be the set of all these standard modules Δ(i). We denote by [Fscr ](Δ) the subcategory of A-mod whose objects are the modules M which have a Δ-filtration, namely there is a finite chain

0 = M0 ⊂ M1 ⊂ M2 ⊂ … ⊂ Mt = M

of submodules of M such that Mi/Mi−1 is isomorphic to a module in Δ for all i. The modules in [Fscr ](Δ) are called Δ-good modules. Dually, we define the costandard module ∇(i) to be the maximal submodule of Qi with composition factors Sj with j [les ] i and denote by ∇ the collection of all costandard modules. In this way, we have also the subcategory [Fscr ](∇) of A-mod whose objects are these modules which have a ∇-filtration. Of course, we have the notion of ∇-good modules. Note that Δ(n) is always projective and ∇(n) is always injective.

Every isogeny class over an algebraically closed field contains a principally polarized abelian variety ([10, corollary 1 to theorem 4 in section 23]). Howe ([3]; see also [4]) gave examples of isogeny classes of abelian varieties over finite fields with no principal polarizations (but not with the degrees of all the polarizations divisible by a given non-zero integer, as in Theorem 1·1 below). In [17] we obtained, for all odd primes [lscr ], isogeny classes of abelian varieties in positive characteristic, all of whose polarizations have degree divisible by [lscr ]2. We gave results in the more general context of invertible sheaves; see also Theorems 6·1 and 5·2 below. Our results gave the first examples for which all the polarizations of the abelian varieties in an isogeny class have degree divisible by a given prime. Inspired by our results in [17], Howe [5] recently obtained, for all odd primes [lscr ], examples of isogeny classes of abelian varieties over fields of arbitrary characteristic different from [lscr ] (including number fields), all of whose polarizations have degree divisible by [lscr ]2.

We investigate a generalization of the method introduced by Kouchnirenko to compute the codimension (colength) of an ideal under a certain non-degeneracy condition on a given system of generators of I. We also discuss Newton non-degenerate ideals and give characterizations using the notion of reductions and Newton polyhedra of ideals.

Suppose that (X,ω) is a symplectic manifold and that there exists a Liouville vector field V defined in a neighbourhood of and transverse to M = ∂X. Then V induces a contact form α = ιVω[mid ]M on M which determines the germ of ω along M. One should think of the contact manifold (M,ξ = ker α) as controlling the behaviour of ω ‘at infinity’. If V points out of X along M then we call (X,ω) a convex filling of (M,ξ), and if V points into X along M then we call (X,ω) a concave filling of (M,ξ).

In this paper we formulate a new conjecture and introduce methods to verify it in many cases.

Using twistor techniques we shall show that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1). The zero section is totally geodesic and the Obata connection restricts to the given connection on the zero section.

We also prove an analogous result for vector bundles: any vector bundle with real-analytic connection whose curvature is of type (1, 1) over X can be extended to a hyperholomorphic bundle over a neighbourhood of the zero section of TX.

We introduce order conserving embeddings as a more general form of order preserving embeddings between finite dimensional nest algebras. The structure of these embeddings is determined, in terms of order indecomposable decompositions, and they are shown to be determined up to inner conjugacy by their induced maps on K0. Classifications of direct systems and limit algebras are obtained in terms of dimension distribution groups.

Let X be a finite CW-complex. We show that the image of the homotopy groups of X under suspension have an exponent at every prime. As a corollary we recover Long's result that finite H-spaces have exponents at all primes. We show that the stable homotopy groups of X have an exponent at p if and only if X is rationally equivalent to a point. This allows us to construct many examples of spaces with infinite dimensional rational homotopy groups and without an exponent at any prime. These examples further support Moore's conjecture.

Let Si: ℝd → ℝd for i = 1, …, n be contracting similarities, and let (p1, …, pn) be a probability vector. Let K and μ be the self-similar set and the self-similar measure associated with (Si,pi)i. For q ∈ ℝ and r > 0, define the qth covering moment and the qth packing moment of μ by

[formula here]

where the infimum is taken over all r-spanning subsets E of K, and the supremum is taken over all r-separated subsets F of K. If the Open Set Condition (OSC) is satisfied then it is well known that

[formula here]

where β(q) is defined by [sum ]ipqirβi(q) = 1 (here ri denotes the Lipschitz constant of Si). Assuming the OSC, we determine the exact rate of convergence in (*): there exist multiplicatively periodic functions πq, Πq: (0,∞) → ℝ such that

[formula here]

where ε(r) → 0 as r[searr ]0. As an application of (**) we show that the empirical multi-fractal moment measures converges weakly:

[formula here]

where, for each positive r, Er is a (suitable) minimal r-spanning subset of K and Fr is a (suitable) maximal r-separated subset of K, and [Hscr ]q,β(q)μ and [Pscr ]q,β(q)μ are the multifractal Hausdorff measure and the multifractal packing measure, respectively.

Mackey's imprimitivity theorem characterizes the unitary representations of a locally compact group G which have been induced from representations of a closed subgroup K; Rieffel's influential reformulation says that the group C*-algebra C*(K) is Morita equivalent to the crossed product C0(G/K)×G [14]. There have since been many important generalizations of this theorem, especially by Rieffel [15, 16] and by Green [3, 4]. These are all special cases of the symmetric imprimitivity theorem of [11], which gives a Morita equivalence between two crossed products of induced C*-algebras.

The Kuhn–Schwartz non-realizability theorem states the following: if the mod p cohomology of a topological space is finitely generated as a module over the Steenrod algebra [Ascr ] then it is finite. We generalize this result to the category of G-spaces, where G is a compact Lie group, by considering the equivariant cohomology of a G-space as an object in the category of all [Ascr ]-modules with a compatible H*(BG; [ ]p)-module structure.

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.