A group is said to be (2,3)-generated if it can be generated by an element of order 2 and an element of order 3. The class of (2,3)-generated groups seems to be rather extensive. From results of Schupp [7] and Mason and Pride [3], it is known that there are 2ℵ0 isomorphism classes of (2,3)-generated groups, and moreover that every countable group can be embedded in a (2,3)-generated group. All finite non-abelian simple groups are (2,3)-generated, with the exception of some groups of low rank in characteristics 2 and 3 (see Wilson [13] or Sanchini and Tamburini [6]).

We study finite centralizing extensions A⊂H of noetherian Hopf algebras. Our main results provide necessary and sufficient conditions for the fibres of the surjection spec H[Rarr ]spec A to coincide with the X-orbits in spec H, where X denotes the finite group of characters of H that restrict to the counit of A. In particular, all of the fibres are X-orbits if and only if the fibre over the augmentation ideal of A is an X-orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.

We give a formula for the Betti numbers of 3-Sasakian manifolds or orbifolds which can be obtained as 3-Sasakian quotients of a sphere by a torus. This answers a question of Galicki and Salamon about the topology of 3-Sasakian manifolds.

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.

We deal with weighted inequalities of the type

formula here

where:

T is either the Hardy–Littlewood maximal operator or a singular integral operator; G is any measurable subset of ℝn; ƒ is any measurable function, vanishing outside G, such that Tƒ is well-defined; v and w are weights, that is, nonnegative locally integrable functions on G;

p, q ∈(1, ∞).

We define a Henstock-type integral on measure spaces with metric topologies, and give an example of a function which is integrable but whose absolute value is not.

It is proved that the infinitesimal generator A of a strongly continuous semigroup of linear operators on a Hilbert space also generates a strongly continuous group if and only if the resolvent of −A, (λ+A)−1, is also a bounded function on some right-hand-side half plane of complex numbers, and converges strongly to zero as the real part of λ tends to infinity. An application to a partial differential equation is given.

In this paper, the behaviour of the positive eigenfunction $\phi$ of $\Lp u=\la |u|^{p-2}u$ in $\Om$, $u_{|\p \Om} =0$, $p>1$, is studied near its critical points. Under some convexity and symmetry assumptions on $\Om$, $\phi$ is seen to have a unique critical point at $x=0$; also, the behaviour of both $\phi$ and $\nabla\phi$ is determined nearby. Positive solutions $u$ to some general problems $\Lp u=f(u)$ in $\Om$, $u_{|\p \Om} =0$, are also considered, with some convexity restrictions on $u$.

The class of nilpotent groups of class $c$ and prime exponent $p\,{>}\,c$ with additional predicates $P_c\,{\subseteq}\,P_{c-1}\,{\subseteq}\,\ldots\,{\subseteq}\,P_1$ for suitable subgroups has the amalgamation property. Hence the Fraïssé limit $D$ of the finite groups of this class exists. $\langle 1\rangle\,{\subseteq}\,P_c(D)\,{\subseteq}\,\ldots\,{\subseteq}\,P_2(D)\,{\subseteq}\,P_1(D)\,{=}\,D$ is the lower and the upper central series of $D$. In this extended language, $D$ is ultrahomogeneous. The elementary theory of $D$ allows the elimination of quantifiers and it is $\aleph_0$-categorical. For $c\,{=}\,2$ this was proved by Baudisch in Bull. London Math. Soc. 33 (2001) 513–519.

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 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.

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 ].

An attempt is made to generalise Bloch and Kato's calculation of some of the local Tamagawa factors appearing in their conjecture, from the crystalline case to the semi-stable case. Success is attained only under severe conditions, but the result applies to the symmetric powers of an elliptic curve with multiplicative reduction.

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.

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.

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.

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 $A$.

Shelah has recently proved that an uncountable free group cannot be the automorphism group of a countable structure. In fact, he proved a more general result: an uncountable free group cannot be a Polish group. A natural question is: can an uncountable $\aleph _{1}$-free group be a Polish group? A negative answer is given here; indeed, it is proved that an $\aleph _{1}$-free group cannot be a homomorphic image of a Polish group. In fact, a stronger result is proved, involving a non-commutative analogue of the notion of separable group.