It is proved that there are two minimal $\Pi_1^0$ classes that are not automorphic.

All finite groups G are determined that admit a subgroup K of index pa, p a prime, under the assumption that G has an irreducible and faithful GF(q)-module in characteristic p whose dimension over GF(q) is at most a. As an application to the theory of permutation groups, the maximal transitive subgroups of the primitive affine permutation groups are determined. The above-mentioned classification is generalized by dropping the assumption that p|q. In both cases surprisingly nice results are obtained.

Functionals of higher derivative type are linear combinations of functionals of the form f→f(n)(ζ), where n[ges ]2 and 0<[mid ]ζ[mid ]<1. The paper shows that, if L is a functional of higher derivative type and f is a function in the class S of univalent functions that maximises Re{L} over S, then L(f)≠0. In addition, if the function f is a rational function, then it must be a rotation of the Koebe function k(z)=z(1−z)−2. These results are applied to establish several cases of the two-functional conjecture for functionals of higher derivative type.

Let T = {T(t)}t[ges ]0 be a C0-semigroup on a Banach space X. The following results are proved.

(i) If X is separable, there exist separable Hilbert spaces X0 and X1, continuous dense embeddings j0[ratio ]X0 → X and j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that j0 ∘ T0(t) = T(t) ∘ j0 and T1(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0.

(ii) If T is [odot ]-reflexive, there exist reflexive Banach spaces X0 and X1 , continuous dense embeddings j[ratio ]D(A2) → X0, j0[ratio ]X0 → X, j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that T0(t) ∘ j = j ∘ T(t), j0 ∘ T0(t) = T(t) ∘ j0 and T(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0, and such that σ(A0) = σ(A) = σ(A1), where Ak is the generator of Tk, k = 0, [emptyv ], 1.

Let f be analytic and univalent in Δ={z:[mid ]z[mid ]<1}. We write

formula here

for 0[les ]r<1 and θ∈[0, 2π], and remark that L(r, θ) is the length of the image in f(Δ) of the ray joining 0 to reiθ in Δ. We shall be concerned with the growth of L(r, θ), as a function of r, for univalent functions in Δ and, more particularly, for functions in the subclass of close-to-convex univalent functions in Δ.

A negative answer to the Kuroš–Černikov Question 21 in [7], whether a group satisfying the normalizer condition is hypercentral, was given by Heineken and Mohamed in 1968 [6]. They constructed groups G satisfying:

(i) G is a locally finite p-group for a prime p,

(ii) G/G′≅Cp∞ and G′ is countable elementary abelian,

(iii) every proper subgroup of G is subnormal and nilpotent,

(iv) Z(G)={1},

(v) the set of normal subgroups of G contained in G′ is linearly ordered by set inclusion, see [3, p. 334],

(vi) KG′ is a proper subgroup in G for every proper subgroup K of G, see [6, Lemma 1(a)].

Let G be a group, and let Fn[G] be the free G-group of rank n. Then Fn[G] is just the natural non-abelian analogue of the free ℤG-module of rank n, and correspondingly the group Φn(G) of equivariant automorphisms of Fn[G] is a natural analogue of the general linear group GLn(ℤG). The groups Φn(G) have been studied recently in [4, 8, 5]. In particular, in [5] it was shown that if G is not finitely presentable (f.p.) then neither is Φn(G), and conversely, that Φn(G) is f.p. if G is f.p. and n≠2. It is a common phenomenon that for small ranks the automorphism groups of free objects may behave unstably (see the survey article [2]), and the main aim of the present paper is to show that this turns out to be the case for the groups Φ2(G).

The Beurling algebras $l^1({\cal D},\omega)\;({\cal D}={\bb N},{\bb Z})$ that are semi-simple, with compact Gelfand transform, are considered. The paper gives a necessary and sufficient condition (on $\omega$ ) such that $l^1({\cal D},\omega)$ possesses a uniform quantitative version of Wiener's theorem in the sense that there exists a function $\phi:]0,+\infty[\longrightarrow ]0,+\infty[$ such that, for every invertible element $x$ in the unit ball of $l^1({\cal D},\omega)$ , we have \[ \|x^{-1}\|\le \phi(r(x^{-1}))\quad r(x^{-1})\hbox{ is the spectral radius of }x^{-1}. \]

Inspired by the work of Bloch and Kato in [2], David Burns constructed several ‘equivariant Tamagawa invariants’ associated to motives of number fields. These invariants lie in relative $K$-groups of group-rings of Galois groups, and in [3] Burns gave several conjectures (see Conjecture 3.1) about their values. In this paper I shall verify Burns' conjecture concerning the invariant $T\Omega^{\rm loc}( N/{\bf Q},1)$ for some families of quaternion extensions $N/{\bf Q}$. Using the results of [9] I intend in a subsequent paper to verify Burns' conjecture for those families of quaternion fields which are not covered here.

The first example of a non-finitely based variety of centre-by-abelian-by-nilpotent groups was found in 1995. This example has an infinite exponent but it can be seen that the method used to construct it also gives a variety that has an exponent of 128. Later it was proved that there exists a variety with similar properties but with an exponent of 32. The paper constructs a non-finitely based variety of centre-by-abelian-by-nilpotent groups which has an exponent of 8.

Let Y be a topological space; and suppose that the circle group, S1, which will be denoted by G throughout this paper, is acting on Y. Then the equivariant cohomology H*G(Y) of Y is defined to be H*(YG), where YG=(EG×Y)/G, the Borel construction. Because of the bundle map YG→BG, the cohomology H*G(Y) is an H*(BG)-module. With rational coefficients H*(BG; ℚ)=ℚ[u], where deg(u)=2. And, when Y is a reasonably nice finite-dimensional space, for example, a finite-dimensional G-CW-complex, then the localized module H*G(Y; ℚ)[u−1] is isomorphic to H*G (YG; ℚ)[u−1] via the restriction homomorphism, where YG⊆Y is the fixed point set. (See, for example, [2].) That this very useful Localization Theorem does not generalize to infinite dimensional spaces can be seen by taking Y=EG, for example.

In this paper, we prove the following theorems. (i) Let G be a graph of minimum degree δ[ges ]5. If G is embeddable in a surface σ and satisfies (δ−5)[mid ]V(G)[mid ]+6χ(σ)[ges ]0, then G is edge reconstructible. (ii) Any graph of minimum degree 4 that triangulates a surface is edge reconstructible. (iii) Any graph which triangulates a surface of characteristic χ[ges ]0 is edge reconstructible.

We discuss Gaussian generalized random fields indexed by smooth sections of vector bundles with respect to Markov properties. We propose a new set-up which is suitable for the present question and within which new phenomena are detected naturally. In particular, we give a counterexample to the belief that locality in the RKHS implies the germ Markov property. We also prove the close connection between the Markov property and cokernels of local operators. Furthermore, we prove the Markov property for a very degenerate Gaussian random field.

The close relationship between the notions of positive forms and representations for a C*-algebra A is one of the most basic facts in the subject. In particular the weak containment of representations is well understood in terms of positive forms: given a representation π of A in a Hilbert space H and a positive form φ on A, its associated representation π φ is weakly contained in π (that is, ker π φ ⊃ ker π) if and only if φ belongs to the weak* closure of the cone of all finite sums of coefficients of π. Among the results on the subject, let us recall the following ones. Suppose that A is concretely represented in H. Then every positive form φ on A is the weak* limit of forms of the type x [map ] [sum ]ki=1 〈ξi, xξi〉 with the ξi in H; moreover if A is a von Neumann subalgebra of ℒ(H) and φ is normal, there exists a sequence (ξi)i [ges ] 1 in H such that φ (x) = [sum ]i [ges ] 1 〈ξi, xξi〉 for all x.

Finite-dimensional algebras over an algebraically closed field are divided into two disjoint classes, called tame and wild respectively, by Drozd's tame and wild dichotomy (see [5] and [2]). A tame algebra, roughly speaking, has its n-dimensional indecomposable modules parametrized by finitely many one-parameter families, for all natural numbers n, but a wild algebra has more indecomposable modules and it is considered hopeless to classify them. In [2], Crawley-Boevey showed that all but finitely many n-dimensional indecomposable modules over a tame algebra are τ-invariant, for all natural numbers n, and conjectured that the converse would be true, where τ := DTr is the Auslander–Reiten translation (see [1]) and we call an indecomposable module X τ-invariant if X ≅ = τX.

If the sum of a series of positive numbers is not convergent and the series satisfies certain regularity conditions, a probability measure can be found whose support contains no interval, yet whose Fourier series decreases faster than the given series.

It is proved that the word problem of the direct product of two free groups of rank 2 can be recognised by a 2-tape real-time but not by a 1-tape real-time Turing machine. It is also proved that the Baumslag–Solitar groups $B$(1, $r$) have the 5-tape real-time word problem for all $r\neq 0$.

Let $\Lambda$ be a connected representation finite selfinjective algebra. According to $G$ . Zwara the partial orders $\le_{\rm ext}$ and $\le_{\rm deg}$ on the isomorphism classes of $d$ -dimensional $\Lambda$ -modules are equivalent if and only if the stable Auslander–Reiten quiver $\Gamma_{\Lambda}$ of $\Lambda$ is not isomorphic to ${\bb Z}D_{3m}/\tau^{2m-1}$ for all $m\ge 2$ . The paper describes all minimal degenerations $M\le_{\rm deg} N$ with $M\not\leq_{\rm ext} N$ in the case when $\Gamma_\Lambda\cong {\bb Z}D_{3m}/\tau^{2m-1}$ for some $m\ge 2$ .

Extensions of the concept of asymptotic independence which are useful in the study of abelian C$^*$-algebras generated by the union of two of their C$^*$-subalgebras are examined. Non-trivial examples are also given in terms of classes of bounded continuous functions defined on a locally compact group.

Weakly almost periodic compactifications have been seriously studied for over 30 years. In the pioneering papers of de Leeuw and Glicksberg [4] and [5], the approach adopted was operator-theoretic. The current definition is more likely to be created from the perspective of universal algebra (see [1, Chapter 3]). For a discrete group or semigroup S, the weakly almost periodic compactification wS is the largest compact semigroup which (i) contains S as a dense subsemigroup, and (ii) has multiplication continuous in each variable separately (where largest means that any other compact semigroup with the properties (i) and (ii) is a quotient of wS). A third viewpoint is to envisage wS as the Gelfand space of the C*-algebra of bounded weakly almost periodic functions on S (for the definition of such functions, see below).

In this paper, we are concerned only with the simplest semigroup (ℕ, +). The three approaches described above give three methods of obtaining information about wℕ. An early striking result about wℕ, that it contains more than one idempotent, was obtained by T. T. West using operator theory [13]. He considered the weak operator closure of the semigroup {T, T2, T3, …} of iterates of a single operator T on the Hilbert space L2(μ) for a particular measure μ on [0, 1]. Brown and Moran, in a series of papers culminating in [2], used sophisticated techniques from harmonic analysis to produce measures μ that permitted the detection of further structure in wℕ; in particular, they found 2[cfr ] distinct idempotents. However, for many years, no other way of showing the existence of more than one idempotent in wℕ was found.

The breakthrough came in 1991, and it was made by Ruppert [11]. In his paper, he created a direct construction of a family of weakly almost periodic functions which could detect 2[cfr ] different idempotents in wℕ. His method was very ingenious (he used a unique variant of the p-adic expansion of integers) and rather complicated. Our main aim in this paper is to construct weakly almost periodic functions which are easy to describe and so appear more ‘natural’ than Ruppert's. We also show that there are enough functions of our type to distinguish 2[cfr ] idempotents in wℕ.