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.

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.

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

A well-known result of Ehrenfeucht states that a difference polynomial f(X)-g(Y) in two variables X, Y with complex coefficients is irreducible if the degrees of f and g are coprime. Panaitopol and Stefǎnescu generalized this result, by giving an irreducibility condition for a larger class of polynomials called “generalized difference polynomials”. This paper gives an irreducibility criterion for more general polynomials, of which the criterion of Panaitopol and Stefǎnescu is a special case.

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

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, ∞).

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.

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.

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.

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.

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.

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 $\mathbb{F}_{q}(T)[\sqrt{D}]$.

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.

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

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 consider positive-definite ternary quadratic forms with integer coefficients. Such a form, f, can be written in matrix notation as

Here x′ is the transpose of the column vector x = {x1, x2, x3) and a,ij = aji is the coefficient of xixj in f. Clearly det A is positive and even and so

is a negative integer.

Let N(ρ; ω) be the number of points of a d-dimensional lattice Γ. where d≥2, inside a ball of radius ρ centred at the point ω. Denote by (ρ) the number N(ρ; ω) averaged over ω in the elementary cell Ω of the lattice Γ. The main result is the following lower bound for for dimensions d ≅ l(mod 4):

We study a fast method for computing potentials of advection–diffusion operators $-{\rm\Delta}+2\mathbf{b}\boldsymbol{\cdot }{\rm\nabla}+c$ with $\mathbf{b}\in \mathbb{C}^{n}$ and $c\in \mathbb{C}$ over rectangular boxes in $\mathbb{R}^{n}$. By combining high-order cubature formulas with modern methods of structured tensor product approximations, we derive an approximation of the potentials which is accurate and provides approximation formulas of high order. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. The action of volume potentials on the basis functions allows one-dimensional integral representations with separable integrands, i.e. a product of functions depending on only one of the variables. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Since only one-dimensional operations are used, the resulting method is effective also in the high-dimensional case.

For a stationary random closed set Ξ in ℝd it is well known that the first-order characteristics volume fraction VV, surface intensity SV and spherical contact distribution function Hs(t) are related by

(1 – VV) Hs′(0) = SV.