The aim of this paper is to prove an inequality of Golod-Shafarevich type for metabelian Lie algebras and to show that this inequality is best possible up to a constant factor. Investigations of this kind were started in [4] in connection with the solution of the class field tower problem. It was shown that if there is a finite p-group which may be presented as a pro-p-group with d ≥ 2 generators and r relations then the inequality r > ¼(d − 1)2 holds; and Vinberg [8] improved this result by showing

The inequality (1) also holds for presentations of nilpotent Lie algebras (see [6]) (with the exception (d, r) = (2, 1)) and nilpotent associative algebras (see [4, 8, 6]). If the relations have of degree at least m ≥ 3 then more relations are needed. More precisely, Koch [5] has shown that the inequality

holds for finite p-groups, and the corresponding results hold for associative algebras and Lie algebras.

Throughout this paper ∈ denotes an arbitrary positive number. For real α, ‖α‖ denotes the distance from α to the nearest integer. For natural numbers k we write K = 2k−1. In 1948 Heilbronn [8] showed that for any real α and N > C1(∈)

This theorem has since been generalized in many ways. In particular, results of the following type have been proved for natural numbers k ≥ 2, h = 1,2 and s.

Let denote a sequence of integers exceeding 1, and let τ(n, ) be the number of those divisors of n which belong to . We say that is a Behrend sequence if

where, here and in the sequel, we use the notation p.p. to indicate that a relation holds on a set of asymptotic density one.

Let ℚp and ℤp respectively denote the p-adic numbers and the p-adic integers for a fixed prime number p. For any subset Y of and n ∈ ℕ we denote by Nn(Y) the cardinality of the image of Y under the natural map . (Thus we have Nn(Y)≤ pnr.)

Let R be a ring (all rings considered here are commutative with identity and Noetherian), M a finitely generated R-module, and I an ideal of R. The jth local cohomology module of M with support in I is defined by

In this paper, we prove a uniform version of a theorem of Brodmann about annihilation of local cohomology modules. As a corollary of this, we deduce a generalization of a theorem of Hochster and Huneke about uniform annihilation of Koszul homology.

A special monoid is a monoid presented by generators and defining relations of the form w = e, where w is a non-empty word on generators and e is the empty word. Groups are special monoids. But there exist special monoids that are not groups. Special monoids have been extensively studied by Adjan[1] and Makanin[7] (see also [2]).

The classification of all finite primitive distance-transitive graphs is basically divided into two cases. In the one case, known as the almost simple case, we have an almost simple group acting primitively as a group of automorphisms on the graph. In the other case, known as the affine case, the vertices of the graph can be identified with the vectors of a finite-dimensional vector space over some finite field. In this case the automorphism group G of the graph Γ contains a normal p-subgroup N which is elementary Abelian and acts regularly on the set of vertices of Γ. Let G0 be the subgroup of G that stabilizes a vertex. Identifying the vertices of Γ with G0-cosets in G, one obtains a vector space V on which N acts as a group of translations, G0, stabilizes 0 and, as Γ is primitive, G0 acts irreducibly on V.

We study bitangency properties of space curves in general position from a global viewpoint. As a consequence we obtain some results on their total numbers of bitangent osculating planes and cross tangents, and prove that in the absence of both, the number of zero torsion points of a curve in general position is a multiple 4.

Let F = (F1, …, Fn-1): (ℝn, 0)→(ℝn-1, 0) and G:(ℝn, 0)→(ℝ, 0) be germs of analytic mappings, and let X = F-1(0). Assume that 0 ∈ ℝn is an isolated singular point in X, i.e. 0 ∈ ℝn is isolated in {x ∈ X|rank[DF(x)] < n-1}. Hence a germ of X/{0} at the origin is either void or a finite disjoint union of analytic curves. Let b denote the number of branches, i.e. connected components, of X/{0} and let b+ (resp. b-, b0) denote the number of branches of X/{0} on which G is positive (resp. G is negative, G vanishes). The problem is to calculate the numbers b, b+, b-, b0 in terms of F and G.

In [5], Pleijel proved an identity relating the area A of a convex plane domain and the length L of its boundary (of class C1). In particular, it contains the isoperimetric inequality L2 –4πA ≥ 0.

It is proved that if ƒ is a homeomorphism of the two-sphere with an invariant set V of cardinality N = 4, then either ƒ has periodic orbits of all periods or it belongs to one of a small number of algebraically finite isotopy classes relative to V. For N ≤ 4, the second case always holds. On the other hand, for each N ≥ 7 we give examples of pseudo-Anosov homeomorphisms of the sphere, relative to a set of N points, for which not all periods occur.

In a series of papers [13, 14, 15], Hilton introduced the terminology relative group to denote a group epimorphism ∈:G↠Q, and relative space to denote a map ƒ:E→B between connected spaces inducing an epimorphism of fundamental groups. He pointed out the desirability of relativizing the theory of P-localization of nilpotent groups and spaces developed in [17], and carried out the algebraic part of this project in [14, 16]. The homotopy-theoretic part was settled by Llerena in [18, 19].

The following result is due to J. H. Grace (see [4], p. 356, also see [9], §3).

Theorem A. if p is a polynomial such that p(–1) = p(1) then the derivative p' has a zero in each of the half-planes {Rez ≤0} and {Rez ≥0}.

This article aims to study the relationship between the universal enveloping C*-algebra C*(M) and the universal enveloping von Neumann algebra W*(M), when M is a JW-algebra. In our main result (Theorem 2·7) we show that C*(M) can be realized as the C*-subalgebra of W*(M) generated by M.

In this article, we study the amenability of Banach algebras in general, and that of Lipschitz algebras in particular. After introducing an alternative definition of amenability, we extend a result of [5], thereby proving a new characterization of amenability for Banach algebras. This characterization relates the amenability of a Banach algebra A to the space of bounded homomorphisms from A into another Banach algebra B (Theorem 4). This result allows us to solve the problem of amenability for virtually all Lipschitz algebras (of complex or Banach algebra valued functions), a class of algebras which has been studied in [2], [4] and [5].

This article generalizes some spectral inequalities for non-negative matrices (see [2] and [3]) to compact integral operators with non-negative kernels defined on Banach function spaces. The spectral radius of a sum of such operators is estimated under certain conditions and a generalization of this inequality is given. An inequality for the spectral radius of a compact integral operator with the kernel equal to a weighted geometric mean of non-negative kernels is also proved.

This paper may be read independently of Baxter and Williams [1], hereinafter denoted by [BW1].

As in [BW1], we are mainly concerned with discounted additive functionals. We find that the large-deviation behaviour of the average depends on the precise average used. We derive, in certain cases, a link (but not equality) between the Cesàro average and Abel average limits, and would expect that other averages would produce other limiting behaviours. We have focused on the exponentially discounted (Abel average) case, both because of its tractability and because of its frequent appearance in decision/control problems and models of financial markets. We do give in subsection (e) the promised ‘excursion’ treatment of symmetry characterizations of the type announced in [BW1]; and this new treatment is simpler, more illuminating and more general. First, however, we focus attention on a different kind of asymptotic behaviour from that studied in [BW1], and on differential equations for exact results.

Let S(N(t)) be defined by

where {N(t), t ≥ 0} is a Poisson process with intensity parameter 1/μ > 0 and {Xi i ≥ 1} is a family of independent random variables which are also independent of {N(t), t ≥ 0}.