Recently Cusick [4] presented a very elegant and short proof of the fact that a pair of fundamental units of a totally real cubic or quartic number field can be found by taking ‘successive minima’ of the function tr(ε2), where ε runs through the group of units and tr denotes the absolute trace. Hidden in Cusick's proof there is a general theorem, which shows how strictly convex functions can be used to find lattice-vectors extensible to a basis of a given geometrical lattice, and which we state and prove in § 3. A result analogous to Cusick's for some families of functions related to tr (ε2) is given in Theorem 1, thereby improving results of Brunotte and Halter-Koch [2], [5]. For a survey of unit groups of rank 2 and more literature the reader is also referred to [2].

In problems of additive number theory one frequently needs to obtain a non-trivial estimate for the absolute value of a trigonometric sum of the type

where f(X) ε ℤ [X] and 1 ≤ m ε ℤ. The general procedure is first to reduce the estimation to the case where m is a prime power, by means of the Chinese Remainder Theorem. The case m = pr (p prime) can often be reduced to that of a lower power of p, by a substitution of the type x = u + vps (where 0 ≤ u < ps and 0 ≤ v < pr-s), followed by the use of a p-adic Taylor expansion f(u+psv) = f(u) +psvf′(u) +…. Frequently this gives T(f, pr) = 0 when r ≥ 2, or at least allows one to reduce to the case m = p. In the latter case an appeal to Weil's estimate

usually gives a good estimate for (0.1), at least if deg f = o(√p).

In this paper we prove that, for any graph G, there exists a graph F such that G is a subgraph of F and

We may define a mobility problem to be one of the type: ‘Given a (fully or partlys pecified) collection of bodies in ℝn, assumed not to interpenetrate, can they be moved in some (fully or partly specified) fashion?’ Early examples of mobility problems nclude ‘Chinese puzzles’ and Sam Loyd's ‘15 puzzle’ (to which could be added the more recent Rubik's Cube!) Less combinatorial examples include the ‘piano mover’ problem (see, e.g. [2, 3, 9]) and the several recent papers on collision avoidance, such as [1, 5, 6, 7, 8].

In an earlier paper [3], we introduced a new notion of completely 0-simple semigroup of quotients. The definition is motivated by a connection between Artinian simple rings and completely 0-simple semigroups. Given an Artinian simple ring Q, one obtains a completely 0-simple multiplicative sub-semigroup by taking the union of the subsets eiQej(i, j = l,…, n), where 1 = e1 + … + en and the ei are primitive orthogonal idempotents. If Q is the classical ring of left quotients of R, then is a semigroup of left quotients of in the sense that every element q in can be written as q = a-1b for some elements a, b in with a2 ≠ 0.

Chatters has asked whether a unique factorization domain (UFD) R, equipped with an automorphism σ transitive on the height 1 prime ideals of R, is necessarily Dedekind. If R contains an uncountable field, then Chatters observed that the answer is affirm ative. In this note was show:

Theorem. Let R be a local UFD equipped with an automorphism σ which is transitive on the height 1 prime ideals of R. Suppose that σ induces an automorphism of finite order on the residue field κ of R (for example, if κ is a global or finite field), Then R is Dedekind.

Let A be a (commutative Noetherian) local ring (with identity) having maximal ideal m and positive dimension. This note is concerned with, among other things, a complex of A -modules which was studied in [8] and which involves modules of generalized fractions derived from A and subsets of systems of parameters for A; in ([8], 3·5), the complex was shown to have connections with local cohomology. The complex is described as follows.

The cohomology of finitely generated metabelian groups has been studied in a series of papers by Bieri, Groves, and Strebel [2, 3, 4]. In particular, Bieri and Groves [2] have shown that every metabelian group of type (FP)∞ is of finite rank, and so is virtually of type (FP). The purpose of the present paper is to provide a generalization of this result and to use our methods to prove the existence of a pathological class of finitely generated soluble groups.

Let G be a group such that

(1) G is finitely presented with deficiency one,

(2) G/G' is infinite cyclic, with a distinguished generator t.

In this paper we present a proof of G. W. Whitehead's conjecture about symmetric products of the sphere spectrum S. Our methods are based on the transfer, the Steinberg module, and the structure of the Hecke algebra. Our results are valid for all primes and extend those of the first author for p = 2 [7]. As originally stated, the conjecture asserts that

is zero on p-primary components in positive degrees [11]. By considering the quotients L(k) = Σ-kSPpkS/SPk-1-S, this is seen to be equivalent to the exactness of

on homotopy groups, where ∂k is the connecting map and ε is the inclusion of the bottom cell. Here and throughout all spaces and spectra are localized at p.

We work in the DIFF category. A knot K = (Sn+2, Sn) is a ribbon knot if Sn bounds an immersed disc Dn+1 →Sn+2 with no triple points and such that the components of the singular set are n-discs whose boundary (n – l)-spheres either lie on Sn or are disjoint from Sn. Pushing Dn+1 into Dn+3 produces a ribbon disc pair D = (Dn+3, Dn+1), with the ribbon knot (Sn+2, Sn) on its boundary. The double of a ribbon (n+1)-disc pair is an (n + l)-ribbon knot. Every (n+l)-ribbon knot is obtained in this manner.

Let Mω(X) be a weighted hypergroup algebra, and Lω(X) be the Banach algebra of measures μ ε Mω(X) such that the function x ↦ (1/ω(x))δx* |μ| is norm continuous. We characterize compact multipliers on Lω(X). This extends the characterization of compact multipliers on weighted group algebras and some classes of weighted semigroup algebras.

There are various ways of using the affine geometry of a compact convex set K to topologize its extreme boundary ∂eK (see [2, 7, 10] and the references quoted therein). The best-known method is to take as closed sets the traces on ∂eK of the closed split faces of K. This topology is known as the facial topology [1, 2, 4], and is significant because the facially continuous functions correspond to the order-bounded linear operators on A(K) and to the multipliers in A(K) [1, 3, 20]. Typically, split faces are scarce, so that the facial topology is coarse. The Choquet topology [7, 15, 18], whose closed sets are the traces of compact extremal subsets of K, is finer, but is usually less easily recognized. The Bishop–de Leeuw theorem was extended in [5] by showing that a maximal measure on K induces a measure on a σ-algebra on ∂eK containing the Choquet closed sets as well as the traces of the Baire subsets of K. The maximal topology is finer even than the Choquet topology, and the Bishop-de Leeuw theorem was further extended in [18] to a σ-algebra including the maximally closed sets. Although the definition of the maximal topology in [18] made use of integral representation theory, a geometrical description of it will be given in Proposition 2·1 below. The Choquet and maximally continuous functions on ∂eK were characterized in [18] in terms of the geometry and integral representation theory of K, respectively. It will be shown in Theorem 2·4 that the Choquet and maximally continuous functions coincide if is a union of faces of K, even though the topologies may differ.

Let F(eis) denote a homeomorphism of the positively oriented unit circle onto a convex curve Γ and let f (eit) = F(eiΦ(t)), where Φ(t) is a non-decreasing function such that Φ(2π) – Φ(0) ≤ 2πN (N a positive integer). If f (eit) has Fourier coefficients cn, we show that is either constant or an N -valent analytic function in {|z| < 1}. We prove that where d is the distance from 0 to Γ and δ(N) > 0 depends only on N. This settles affirmatively a conjecture of H. S. Shapiro.

Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such that

Random equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].

In Eagleson and Weber [2] a central limit theorem for weakly exchangeable arrays is given as a consequence of a reverse martingale central limit theorem. As noted in their remarks, a direct application of this is a central limit theorem for the classical U-statistics. Here we give a corollary to the functional law of the iterated logarithm of Scott and Huggins [4] and use this to obtain laws of the iterated logarithm for weakly exchangeable arrays and hence for U-statistics under a finite (2 + δ)th moment condition.

Given a sequence of non-negative independent and identically distributed random variables, we determine conditions on the common distribution such that the sum of appropriately normalized and centred upper kn extreme values based on the first n random variables converges in distribution to a normal random variable, where kn → ∞ and kn/ n → 0 as n → ∞. The probabilistic problem is motivated by recent statistical work on the estimation of the exponent of a regularly varying distribution function. Our main tool is a new Brownian bridge approximation to the uniform empirical and quantile processes in weighted supremum norms.

A strong approximation for randomly indexed U-statistics is derived. The special case when the indices are stopping times based upon the jackknife estimator of the variance is discussed.

A point H is known to lie on a given bounded interval ℋ. A searcher wishes to locate H by making successive guesses g1, g2, …, each with the knowledge of whether the previous guesses were too high or low, or exactly right. Under these circumstances it is easy to devise a search strategy which ensures the convergence of the gi to H. One such strategy is the ‘halving’ strategy which always guesses the midpoint of the interval on which H is currently known to lie. The problem becomes well defined, and more difficult, if the searcher has to minimize a given cost function which in some way measures the speed of convergence of the gi to H.