We exhibit (§2) an example of §a compact Hausdorff space supporting a Radon probability measure μ and a continuous map ø : X → I, when I is the closed unit interval, for which the image measure ø(μ) is Lebesgue measur m with the properties:

(i) there exists an open set G ⊂ X for which ø(G) is not m-measurable;

(ii) μ is a non-atomic non-completion regular measure;

(iii) the measure algebras (X, μ) and (I, m) are isomorphic but for no choice c sets B ⊂ X, B′ ⊂ I of measure zero are and homeomorphic

(iv) there exists a selection p : I → X (i.e. p(t) ∊ ø−1(t) for all t ∊ I) which i Borel m-measurable, but there is no Lusin m-measurable selection.

In this paper is considered the average size of the 2-Selmer groups of a class of quadratic twists of each elliptic curve over ℚ with ℚ-torsion group ℤ2 × ℤ2. The existence is shown of a positive proportion of quadratic twists of such a curve, each of which has rank 0 Mordell-Weil group.

Improved upper and lower bounds of the counting functions of the conceivable additive decomposition sets of the set of primes are established. Suppose that where, ℝ′ differs from the set of primes in finitely many elements only and .

It is shown that the counting functions A(x) of ℐ and B(x) of ℬ for sufficiently large x, satisfy

The problem considered is that of determining the stress distribution in an infinite plate which contains a straight crack terminated at one end by a circular hole, and is in a state of plane strain or generalized plane stress under given loads. The problem has some relevance to the engineering practice of “stress relief”, in which holes are made at the ends of a crack with the object of reducing the concentration of stress. When the diameters of the holes are small compared with the length of the crack, the distribution of stress near one hole will not be greatly influenced by the presence of the other, and will then be approximately the same as in the simpler case considered here.

Throughout this paper, E will denote a finite-dimensional vector space over an ordered field . The real number field will be denoted by ℜ and its rational subfield by . Many of the basic notions in the theory of convexity (convex set, extreme point, hyperplane, etc.) can be defined in the general case just as they are when , but their behaviour may be different from that in the real case. By way of example, we consider the following theorem (due essentially to Minkowski), which is of fundamental importance both for geometric investigations and for the applications of convexity in analysis:

(1) Suppose . If K is a convex subset of E which is linearly closed and linearly bounded, then. K = con ex K; that is, K is the convex hull of its set of extreme points.

Answering a question of Füredi and Loeb [On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc.121(4) (1994), 1063–1073], we show that the maximum number of pairwise intersecting homothets of a $d$-dimensional centrally symmetric convex body $K$, none of which contains the center of another in its interior, is at most $O(3^{d}d\log d)$. If $K$ is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by $O(3^{d}\binom{2d}{d}d\log d)$. We establish analogous results for the case where the center is defined as an arbitrary point in the interior of $K$. We also show that, in the latter case, one can always find families of at least $\unicode[STIX]{x1D6FA}((2/\sqrt{3})^{d})$ translates of $K$ with the above property.

We give sufficient conditions for the spectra and essential spectra of certain classes of operators to be contained in or coincide with an interval of the form [μ, ∞).

Dörner [8] has recently proved a result for systems of additive equations over algebraic number fields, which we restate in a more specific setting.

It is well known that every convex polytope in d-dimensional euclidean space Ed can be approximated arbitrarily closely, in the Hausdorff sense, by convex polytopes whose faces are simplexes (see [2, Section 4.5]). In this paper we prove some generalizations of this result, investigating the possibility of approximating a given d-polytope (d-dimensional convex polytope) by polytopes whose facets (faces of d − 1 dimensions) are all of some prescribed type.

If θ is a real algebraic number of degree r ≥ 2, there is a computable number c = c(θ) > 0 such that

for all rational numbers p/q (q > 0). This follows directly from the definition of an algebraic number, as was shown by Liouville in 1843; and if r = 2 there is no more to be said. Axel Thue was the first to prove a stronger result when r ≥ 3; he showed that if

there are at most finitely many rational numbers p/q that satisfy

The starting point for the present paper is the following question, which asks whether points can be replaced by flats (translates of linear subspaces of arbitrary dimension) as the basic objects in a convexity structure on ℝd.

Problems of classifying and enumerating types of plane patterns, tilings, and other repeating geometrical structures have interested mathematicians, crystallographers and others for many years. Recently we have formulated the general principles that seem to underlie many of the published treatments of these topics, and so have been able to put on a mathematical basis classification criteria often justified mainly on intuitive grounds. In other words, we can now decide whether or not two given patterns are of the same “type”, at one of a number of different possible levels of classification, without relying on some vaguely expressed distinction based on a “feeling” as to whether the objects in question should be regarded as of distinguishable kinds, or not.

Let (X, ≺) denote a non-empty (partially) ordered set, or more generally a non-empty set X with an arbitrary transitive relation ≺ on it. The relation ≺ will be fixed throughout what follows, so to simplify the notation we often write (X, ≺) as X. A successor of x ∈ X is an element y ∈ X such that x ≺ y; thus x may or may not be a successor of itself. As usual, a subset A ⊂ X is cofinal if each x ∈ X has a successor in A. A partition of X is a family of (pairwise) disjoint nonempty subsets of X whose union is X.

In [1] Fröhlich considers the kernel D(Z(Γ)) of the map of class-groups C(Z(Γ)) → C(), Γ a finite abelian group, the maximal order in the rational group ring Q(Γ). We obtain under mild hypotheses a non-trivial lower bound for the cardinality k(Γ) of the finite group D(Z(Γ)) when Γ is the cyclic group of order 2pn, p an odd prime. In fact, let f be the smallest positive integer such that 2f ≡ 1 mod.pn, If 2|f then k(Γ) > 1 for pn ≠ 3, 32, 5 and k(Γ) is divisible by primes ≠ 2, p except possibly when pn is a Fermat prime or when pn = 32. The latter result contrasts with the fact that D(Z(Γ)) is a p-group if Γ is a p-group [1; Theorem 5].

It is well known that every irrational number θ possesses an infinity of rational approximations p/q satisfying

It is also well known that there is a wide class of irrational numbers which admit of no approximations which are essentially better, namely those θ whose continued fractions have bounded partial quotients. For any such θ there is a positive number c such that all rational approximations satisfy

Recent advances in the theory of exponential sums (see, for example, [6], [7], [8], [12]) have contributed to corresponding progress in our understanding of the solubility of systems of simultaneous additive equations (see, in particular, [1], [2], [3], [4]). In a previous memoir [11] we developed a version of Vaughan's iterative method (see Vaughan [8]) suitable for the analysis of simultaneous additive equations of differing degrees, discussing in detail the solubility of simultaneous cubic and quadratic equations. The mean value estimates derived in [11] are, unfortunately, weaker than might be hoped, owing to the presence of undesirable singular solutions in certain auxiliary systems of congruences. The methods of [12] provide a flexible alternative to Vaughan's iterative method, and, as was apparent even at the time of their initial development at the opening of the present decade, such ideas provide a means of avoiding altogether the aforementioned problematic singular solutions. The systematic development of such an approach having been described recently in [15], in this paper we apply such methods to investigate the solubility of pairs of additive equations, one cubic and one quadratic, thereby improving the main conclusion of [11].

We present a new characterization of σ-fragmentability and illustrate its usefulness by proving some results relating analyticity and crfragmentability. We show, for instance, that a Banach space with the weak topology is σ-fragmented if, and only if, it is almost Čech-analytic and that an almost Čech-analytic topological space is σ-fragmented by a lower-semicontinuous metric if, and only if, each compact subset of the space is fragmented by the metric.

§1. Introduction. In 1946, Davenport and Heilbronn [9] proved a result which opened up the study of Diophantine inequalities. Suppose that Q(x) is a diagonal quadratic form with non-zero real coefficients in s variables. We write

Unless stated otherwise all quadratic forms have rational integer coefficients and all representations are integral representations. For positive binary quadratic forms of the same discriminant it is known that two such forms are equivalent provided they represent the same integers. See, for instance, [Ki2], and for a sharper extension [W2]. On the other hand, in the quaternary case these value-sets are far from characterizing the forms even within a genus. It is therefore natural to ask for positive ternary forms the corresponding question, whose answer appears to be unknown.

Let denote the class of all compact convex sets in Euclidean n-dimensional space En, and let y be the collection of those members of k which are centrally symmetric. The topology in is that induced by the Hausdorff metric.