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, ∞).

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.

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].

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.

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.

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.

In this paper we investigate the moments and the distribution of $L(1,\unicode[STIX]{x1D712}_{D})$, where $\unicode[STIX]{x1D712}_{D}$ varies over quadratic characters associated to square-free polynomials $D$ of degree $n$ over $\mathbb{F}_{q}$, as $n\rightarrow \infty$. Our first result gives asymptotic formulas for the complex moments of $L(1,\unicode[STIX]{x1D712}_{D})$ in a large uniform range. Previously, only the first moment has been computed due to the work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of $L(1,\unicode[STIX]{x1D712}_{D})$ is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of $L(1,\unicode[STIX]{x1D712}_{D})$, which is not present in the number field setting. We also obtain $\unicode[STIX]{x1D6FA}$-results for the extreme values of $L(1,\unicode[STIX]{x1D712}_{D})$, which we conjecture to be the best possible. Specializing $n=2g+1$ and making use of one case of Artin’s class number formula, we obtain similar results for the class number $h_{D}$ associated to $\mathbb{F}_{q}(T)[\sqrt{D}]$. Similarly, specializing to $n=2g+2$ we can appeal to the second case of Artin’s class number formula and deduce analogous results for $h_{D}R_{D}$, where $R_{D}$ is the regulator of $\mathbb{F}_{q}(T)[\sqrt{D}]$.

Let H be any closed bounded convex set in En, and -H be its reflection in the origin. Then the vector sum K = H+ (−H) has the origin as centre and is called the difference set of H. Clearly every closed bounded convex set K with centre at the origin is the difference set of ½K. Excluding this trivial case, we define such a set K to be reducible if it is the difference set of some H which is not homothetic to K.

We calculate the minimum numbers of k-dimensional flats and cells of any Euclidean d-arrangement of n hyperplanes. The bounds are obtained by calculating lower bounds for the values of the doubly indexed Whitney numbers of a basepointed geometric lattice of rank r with n points. Additional geometric results concern the minimum number of cells of a Euclidean or projective arrangement met by a subspace in general position and the minimum number of non-Radon partitions of a Euclidean point set. We include remarks on the relationship between Euclidean arrangements and basepointed geometric lattices and on the minimum numbers of cells of arrangements with a bounded region.

Given a convex function u, defined in an open bounded convex subset Ω of ℝn, we consider the set

where η is a Borel subset of Ω,ρ is nonnegative, and ∂u(x) denotes the subgradient (or subdifferential) of u at x. We prove that Pp(u; η) is a Borel set and its n-dimensional measure is a polynomial of degree n with respect to ρ. The coefficients of this polynomial are nonnegative measures defined on the Borel subsets of Ω. We find an upper bound for the values attained by these measures on the sublevel sets of u. Such a bound depends on the quermassintegrals of the sublevel set and on the Lipschitz constant of u. Finally we prove that one of these measures coincides with the Lebesgue measure of the image under the subgradient map of u.

An arithmetic function f(n) is said to be additive, if it satisfies the relation f(ab) = f(a) + f(b), for every pair of coprime integers a and b; and stronglya dditive if, in addition, f(pm) = f(p) for every prime-power pm.

Let K be an algebraic number field. By a. full module in K [l,p.83] we mean a finitely-generated (necessarily free) subgroup M of the additive group of K whose rank is equal to the degree [K : ℚ] of K over the rational field ℚ. The intersection of M with ℤK, the ring of integers of K, is also a full module I, and we shall concern ourselves chiefly with the latter, in that we wish to count the number of rational integers in a given interval which can be expressed as the norms of elements of I. More precisely, we shall adapt the methods of [2] to prove the following

In 1973, Montgomery [12] introduced, in order to study the vertical distribution of the zeros of the Riemann zeta function, the pair correlation function

where w(u) = 4/(4 + u2) and γjj = 1, 2, run over the imaginary part of the nontrivial zeros of ζ(s). It is easy to see that, for T → ∞,

uniformly in X, and Montgomery [12], see also Goldston-Montgomery [7], proved that under the Riemann Hypothesis (RH)

uniformly for X ≤ T ≤ XA, for any fixed A > 1. He also conjectured, under RH, that (1) holds uniformly for Xε ≤ T ≤ X, for every fixed ε > 0. We denote by MC the above conjecture.

We show that if a polytope K1, in ℝd can be partitioned into a finite number of sets, and these sets can be moved by isometries in a locally discrete group to form a convex body K2, then K2 is a polytope and a similar partition can be made where the sets involved are simplices with disjoint interiors. This gives partial answers to questions of Tarski, Sallee and Wagon.

The object of the present note is to prove the following theorem. †

Theorem. Let P be a convex d-polytope in d-dimensional real vector space Rd (d ≥ 2). If each d-polytope which is the intersection of two translates of P has the same number of vertices as P, then P is a direct linear sum ‡ of simplices.

Extending classical results of Nair and Tenenbaum, we provide general, sharp upper bounds for sums of the type where x,y,u,v have comparable logarithms, F belongs to a class defined by a weak form of sub-multiplicativity, and the Qj are arbitrary binary forms. A specific feature of the results is that the bounds are uniform within the F-class and that, as in a recent version given by Henriot, the dependency with respect to the coefficients of the Qj is made explicit. These estimates play a crucial rôle in the proof, published separately by the authors, of Manin’s conjecture for Châtelet surfaces.

Let X and Y be separable metrizable spaces, and f: X→Y a function. It is wished to recover f from its values on a small set via a simple algorithm. It is shown that this is possible if f is Baire class one, and in fact a characterization is obtained. This leads to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space.

Suppose K is a convex body in Euclidean n-space En and that all the orthogonal projections of K onto p-dimensional linear subspaces have the same p-dimensional volume, that is K has constant outer p-measure. If p = 1, this means that K is of constant width; if p = n − 1, this means that K has constant brightness. It is known that, when the boundary of K is smooth enough to admit principal radii of curvature R1, …, Rn-1 as functions of the outer normal u, and if we define Fp(u) to be the p-th elementary symmetric function of these radii, then, when K has constant width,

In this paper it is proved that, for x sufficiently large, every integer m with

can be written as m = Σ1≤n≤xεn/n, where εi, = 0 or 1.

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$ -transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\,\text{mod}\,1$ . Further, define

$$\begin{eqnarray}W_{y}(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{x\in [0,1]:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\}\end{eqnarray}$$

$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\},\end{eqnarray}$$

$\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{{>}0}$

$\unicode[STIX]{x1D6F9}(n)\rightarrow 0$

$n\rightarrow \infty$