If p is a prime, then a finite group is p-soluble if each of its composition factors is either a p-group or has order coprime to p. For example, soluble groups are p-soluble. However, there are many insoluble groups that are p-soluble. We shall prove the following result.

Let [ ]P2,p (where 2 < p) be the class of all finite nilpotent groups of class 2 and of exponent p with an additional predicate for a subgroup of the centre that contains the commutator subgroup. The Fraïssé limit D of this class exists. Non-forking is described for Th(D).

Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property $O$. It is deduced both that these groups are exact and satisfy Yu's Property $A$.

Given a first order elliptic partial differential equation we construct a solution which solves a given Riemann–Hilbert boundary value problem whose coefficients have singularities of the first kind at a finite number of some prescribed isolated points and are Holder–continuous outside those points while the free term has a finite number of integrable power singularities at some prescribed points. It is shown that the solution belongs to some weighted Sobolev space $W_{1, p}(D; \rho)$, where the weight function $\rho = \rho(z; \partial D)$ is the distance of the variable point $z$ from the boundary $\partial D$ raised to a certain power. The problem is solved by first reducing it to an analogous problem for holomorphic functions. The latter is then solved.

The purpose of this note is to establish a new version of the local Steiner formula and to give an application to convex bodies of constant width. This variant of the Steiner formula generalizes results of Hann [3] and Hug [6], who use much less elementary techniques than the methods of this paper. In fact, Hann asked for a simpler proof of these results [4, Problem 2, p. 900]. We remark that our formula can be considered as a Euclidean analogue of a spherical result proved in [2, p. 46], and that our method can also be applied in hyperbolic space.

For some remarks on related formulas in certain two-dimensional Minkowski spaces, see Hann [5, p. 363].

For further information about the notions used below, we refer to Schneider's book [9]. Let [Kscr ]n be the set of all convex bodies in Euclidean space IRn, that is, the set of all compact, convex, non-empty subsets of IRn. Let Sn-1 be the unit sphere. For K∈[Kscr ]n, let NorK be the set of all support elements of K, that is, the pairs (x, u)∈IRn×Sn−1 such that x is a boundary point of K and u is an outer unit normal vector of K at the point x. The support measures (or generalized curvature measures) of K, denoted by Θ0(K, ·), …, Θn−1(K, ·), are the unique Borel measures on IRn×Sn−1 that are concentrated on NorK and satisfy

formula here

for all integrable functions f[ratio ]IRn→IR; here λ denotes the Lebesgue measure on IRn. Equation (1), which is a consequence and a slight generalization of Theorem 4.2.1 in Schneider [9], is called the local Steiner formula. Our main result is the following.

The author of this paper has shown previously how a complex representation of a finite group can be split into a virtual sum of representations induced from one-dimensional representations of subgroups in a natural way (sometimes known as explicit Brauer induction). Here the modular case is treated, yielding an analogous result at the level of Brauer characters in general, and in the Green ring for trivial source modules.

We prove the existence of a power series having radius of convergence 0, whose partial sums have universal approximation properties on any compact set with connected complement that is contained in a finite union of circles centred at 0 and having rational radii, but do not have such properties on any compact set with nonempty interior. This relates to a theorem of A. I. Seleznev.

We generalise to Jordan–Banach algebras some results of Aupetit on perturbation by inessential elements. This enables us to show that the kh-socle is the largest inessential ideal, and to give some spectral characterisations of it.

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <[mid ]X[mid ], the stabilizer S{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory.

An example is provided of a space that does not have the homotopy type of an H-space, and yet has the same $n$ -type as a double loop space for each natural number $n$ . It is also locally homotopy equivalent to a double loop space at each prime $p$ .

Let S be an elementary operator on a C*-algebra with minimal length [lscr ](S)[ges ]1. Then S is completely positive if and only if S is n-positive for some natural number n[ges ]([lscr ](S)−1)/2.

Regenerative phenomena were introduced some forty years ago to address problems in the theory of continuous-time Markov processes. The early work in the theory left a number of difficult unsolved problems, in the classification of $p$-functions, oscillation and inequalities, the multiplicative theory, and the theory of unbounded semi-$p$-functions. Recent years have shown progress on all of these fronts, and this paper surveys these results, while drawing attention to significant problems that remain open.

Several results are proved, related to an old problem posed by G. R. MacLane, namely whether functions in the class [Ascr ] that are locally univalent can have arc tracts. In particular, a proof is given of an assertion of MacLane that if f ∈ [Ascr ] is locally univalent and has no arc tracts, then f′ ∈ [Ascr ].

In this note, two geometric properties of Banach spaces are proposed and discussed, related to the validity of the lemma of Riemann–Lebesgue in spaces of weakly integrable functions. The class of Banach spaces with the analytic Riemann–Lebesgue property is shown to be precisely the class of Banach spaces for which a weak form of a theorem of Ingham holds.

The paper introduces a new grading on the preprojective algebra of an arbitrary locally finite quiver. Viewing the algebra as a left module over the path algebra, the author uses the grading to give an explicit geometric construction of a canonical collection of exact sequences of its submodules. If a vertex of the quiver is a source, the above submodules behave nicely with respect to the corresponding reflection functor. It follows that when the quiver is finite and without oriented cycles, the canonical exact sequences are the almost split sequences with preprojective terms, and the indecomposable direct summands of the submodules are the non-isomorphic indecomposable preprojective modules. The proof extends that given by Gelfand and Ponomarev in the case when the finite quiver is a tree.

In this paper, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's specialization theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over $\mathbb{Q}$.