The purpose of this note is to bring into attention an apparently forgotten result of C. M. Petty: a convex body has minimal surface area among its affine transformations of the same volume if, and only if, its area measure is isotropic. We obtain sharp affine inequalities which demonstrate the fact that this “surface isotropic” position is a natural framework for the study of hyperplane projections of convex bodies.

The paper shows that no origin-symmetric convex polyhedron in R3 is the intersection body of a star body. It is shown also that every originsymmetric convex body in Rd, for d = 3 and 4, can be seen as the intersection body of a star-shaped set whose radial function satisfies conditions related to suitable non-integer Sobolev classes.

An origin-symmetric convex body K in ℝn is called an intersection body if its radial function ρK is the spherical Radon transform of a non-negative measure µ on the unit sphere Sn−1. When µ is a positive continuous function, K is called the intersection body of a star body. The notion of intersection body was introduced by Lutwak [L]. It played a key role in the solution of the Busemann-Petty problem, see [G1], [G2], [L], [Z1] and [Z2]. Koldobsky [K] showed that the cross-polytope is an intersection body. This indicates that the statement in [Z3] that no origin-symmetric convex polytope in ℝn (n > 3) is an intersection body is not correct. This paper will prove the weaker statement that no origin-symmetric convex polytope in ℝn (n > 3) is the intersection body of a star body.

Hilbert's inequality asserts that

for arbitrary complex numbers ar. The constant π was first obtained by Schur [5], and is best possible. Following a suggestion of Selberg, Montgomery and Vaughan [4] showed that

where the γr are distinct real numbers and

Let k be a positive integer and g(k) be the least integer n > k + 1 such that all prime factors of are >k. We prove that

where c is an absolute positive constant. We also establish a new theorem on the distribution of fractional parts of a smooth function.

Investigations concerning the gaps between consecutive prime numbers have long occupied an important position on the interface between additive and multiplicative number theory. Perhaps the most famous problem concerning these gaps, the Twin Prime Conjecture, asserts that the aforementioned gaps are infinitely often as small as 2. Although a proof of this conjecture seems presently far beyond our reach (but see [5] and [10] for related results), weak evidence in its favour comes from studying unusually short gaps between prime numbers. Thus, while it follows from the Prime Number Theorem that the average gap between consecutive primes of size about x is around log x, it is now known that such gaps can be infinitely often smaller than 0–249 log x (this is a celebrated result of Maier [12], building on earlier work of a number of authors; see in particular [7], [13], [3] and [11]). A conjecture weaker than the Twin Prime Conjecture asserts that there are infinitely many gaps between prime numbers which are powers of 2, but unfortunately this conjecture also seems well beyond our grasp. Extending this line of thought, Kent D. Boklan has posed the problem of establishing that the gaps between prime numbers infinitely often have only small prime divisors, and here the latter divisors should be small relative to the size of the small gaps established by Maier [12]. In this paper we show that the gaps between consecutive prime numbers infinitely often have only small prime divisors, thereby solving Boklan's problem. It transpires that the methods which we develop to treat Boklan's problem are capable also of detecting multiplicative properties of more general type in the differences between consecutive primes, and this theme we also explore herein.

The following Bernstein-type theorem in hyperbolic spaces is proved. Let ∑ be a non-zero constant mean curvature complete hypersurface in the hyperbolic space ℍn. Suppose that there exists a one-to-one orthogonal projection from ∑ into a horosphere. (1) If the projection is surjective, then ∑ is a horosphere. (2) If the projection is not surjective and its image is simply connected, then ∑ is a hypersphere.

Let ν be a rank 1 henselian valuation of a field K having unique extension ῡ to an algebraic closure of K. For any subextension L/K of /K, let G (L), Res (L) denote respectively the value group and the residue field of the valuation obtained by restricting ῡ to L. If a∈\K define

This note presents a classification of commutative, cancellative monoids S by flatness properties of their associated S-acts.

For the optimal approximation of convex bodies by inscribed or circumscribed polytopes there are precise asymptotic results with respect to different notions of distance. In this paper we derive some results on optimal approximation without restricting the polytopes to be inscribed or circumscribed.

It is shown that the regular n-dimensional simplex has a non-convex cross-section body for all n ≥ 4.

It is shown that the cross-section body of a convex body K ⊂ ℝ3, that is the symmetric body which has for radial function in the direction u the maximal volume of a section of K by an hyperplane orthogonal to u, is a convex body in ℝ3.

It is shown that every compact convex set K which is centrally symmetric and has a non-empty interior admits a packing of Euclidean 3-space with density at least 0.46421 … The best such bound previously known is 0.30051 … due to the theorem of Minkowski-Hlawka. It is probable that there is such a lower bound which is significantly greater than the one shown in this note, since there is a packing of congruent spheres which has density

We consider the spectral function

associated with the Sturm-Liouville equation

in situations where q(x) →−∞ as x → ∞ and (1.1) is in the Weyl limit-point case at ∞. As usual, q is real-valued and locally integrable in [0, ∞], and our particular concern is where q(x) has the form

where c (>0) is a parameter, s and p are non-negative on [0, ∞], p(x) → ∞ and p(x) = 0{s(x) } as x → ∞. As the boundary condition at x = 0, we take the Dirichlet condition y(0) = 0 for convenience: we can equally take the Neumann condition y′ (0) = 0 or generally a1y(0) + a2y′ (0) = 0 with real a1 and a2.

It is well known that boundary value problems for hyperbolic equations are in general “not well posed” problems. This paper is concerned with the uniqueness of solutions to boundary value problems for the hyperbolic equation uxx − Qu = utt. Here Q is a function of the variable x alone, and satisfies the following conditions:

(a) Q:[0, ∞) → ℝ;

(b) Q is Lebesgue integrable on any compact subinterval of [0, ∞);

(c) Q(x)→ ∞ as x → ∞.

We prove a concentration inequality for δ-concave measures over ℝn. Using this result, we study the moments of order q of a norm with respect to a δ-concave measure over ℝn. We obtain a lower bound for q∈ ]−1, 0] and an upper bound for q∈ ]0,+ ∞[ in terms of the measure of the unit ball associated to the norm. This allows us to give Kahane-Khinchine type inequalities for negative exponent.

Let be a class of structures for a first-order language and let n be a positive integer. A structure A for the language is said to be n-residually if, given elements a1, … an and b1, … bn of A with ai≠bi for 1 ≤i≤n, there exist B ∈ and an epimorphism φ:A → B such that φ(ai≠φ(bi for 1 ≤i ≤ n. We abbreviate 1 -residually to residually , and A is said to be fully residually if it is n-residually for all n ≤ 1. We shall only be using two cases. One is where the language is the first-order language of groups, {·−1,1}, and A and all members of are groups. In this case we may take all the bi in the definition to be equal to 1. However, we are mainly interested in the first-order language of rings, {+,−·,0,1}. Again if A and all members of are rings, we may take all bi to be zero in the definition.

Let p(n) be the usual partition function. Let l be an odd prime, and let r (mod t) be any arithmetic progression. If there exists an integer n ≡ r (mod t) such that p(n) ≢ 0 (mod l), then, for large X,

In the paper [5] bounds are found for the sum,

for a suitable Dirichlet character χ mod r, and real functionf(x). The proofs in that paper use Bombieri and Iwaniec's method [1], one formulation of which has as part of its first step the estimation of S in terms of a sum of many shorter sums of the form,

where e(x) = exp (2πix), mi∈ [M, 2M], and each mi, lies in its own interval, of length N ≥ M/4, that is disjoint from those of the others. This paper addresses a problem springing from above: to bound the numbers of ‘similar’ pairs, Si+, Si+, satisfying both

and

where ‖x‖ = min{|x − n|: n ∈ ℤ}. Lemma 5.2.1 of [3] (partial summation) shows that each sum in a similar pair is a bounded multiple of the other.

A universal algebra is called congruence compact if every family of congruence classes with the finite intersection property has a non-empty intersection. This paper determines the structure of all right congruence compact monoids S for which Green's relations ℐ and ℋ coincide. The results are thus sufficiently general to describe, in particular, all congruence compact commutative monoids and all right congruence compact Clifford inverse monoids.