Let

and let p denote any prime. The p-content vp(f) of f is denned by

In this paper we shall be concerned with the following problem. Let k1 ≤ k2 ≤…≤ ks be natural numbers, λ1,…, λs be nonzero real numbers, not all of the same sign. Is it then true that the values taken by

at integer points (x1,…, xs) ∈ ℤk are dense on the real line, provided at least one of the ratios λi/λj, is irrational? We shall refer to this, for brevity, as the inequality problem for k1,…, ks. Optimistically one may conjecture that the inequality problem is true whenever

A classical problem in the additive theory of numbers is the determination of the minimal s such that for all sufficiently large n the equation

is solvable in natural numbers xk. Improving on earlier results the author [2] has been able to prove that one may take s = 18. In a survey article W. Schwarz asked for an analogue for diophantine inequalities [6]. As a first contribution to this subject we prove

Theorem. Let λ2, …, λ23 be nonzero real numbers, λ2/λ3 irrational. Then the values taken by

at integer points ( x1, …, x22) are dense on the real line.

For each algebraic integer α, let ℤα denote the ring of integers of the algebraic number field ℚ(α). There has been continuing interest in finding ring-theoretic conditions characterizing when ℤα coincides with its subring ℤ[α] (cf.[15,18,1,13,12]). One way to extend such work is to consider the intermediate ring ℤ[α]+, the seminormalization (in the sense of [17]) of ℤ[α] in ℤα. Indeed, if we let Iα denote the conductor (ℤ[α]: ℤα), then it is easy to see (cf. Proposition 3.1) that ⅂[α] = ℤα, if, and only if, ℤ[α]+ = ℤα and Iα is a radical ideal of Zα. The condition ℤ[α]+ = ℤα seems worthy of separate attention in view of recent results (cf. [3]) that seminormal rings generated by algebraic integers are “often” automatically of the form ℤα. We show in Proposition 3.3 that the condition ℤ[α]+ = ℤα is equivalent to several universal properties, including notably that the canonical closed surjection Spec (ℤα) → Spec (ℤ[α]) be universally open, be universally going-down, or be a universal homeomorphism.

The main aim of this note is the proof of the following

Let −∞ ≤ a > b ≤ ∞ and let A ⊂ (a, b) be a measurable set such that λ((a, b)\A) = 0, where λ denotes Lebesgue measure on ℝ. Let f: A→ℝ be a measurable and midconvex function, i.e.

whenever. Then there exists a convex functionsuch that.

The main purpose of this article is to outline for non-compact Riemann surfaces the development of a theory of T-invariant algebras similar to that developed by T. W. Gamelin [7] in the case of the plane. The main idea is to introduce, using the global local uniformizer of R. C. Gunning and R. Narashimhan, a Cauchy transform operator for the Riemann surface which operates on measures and solves an inhomogeneous -equation. This, in turn, can be used, analogously to the Cauchy transform on the plane, to develop meromorphic (rational) approximation theory. We sketch the path of the development but omit most of the details when they are very much similar to the planar case. Our presentation follows closely that of [7]. The original motivation for this study was to obtain more information on Gleason parts useful in the study of Carleman (tangential) approximation theory (see [5]).

The purpose of this paper is to construct polynomials on ℂn which can approximate to the product of two holomorphic functions defined on a neighbourhood of any boundary point of a number of pseudoconvex domains in ℂn (called the “H-pseudoconvex domain”). It should be noted that we have only mentioned that the same conclusion holds true for a strictly pseudoconvex domain in the sense of Levi in [3, p. 109]. We shall begin with the definition of H-pseudoconvexity as follows, cf [3, p. 113].

When a weak rotlet and a circular cylinder rotate together in a viscous fluid at low Reynolds number R, the Stokes' flow solution indicates a uniform stream as the radial distance r tends to infinity. It is shown here, when R is distinctly non-zero, that the flow is modified to form a spiral motion in the domain where R In r = O(l), but is not damped until the more distant domain R2 In r = O(l).

We consider the second order linear differential equation

where p and q are real-valued members of with p(t)>0 for t ∈ [α, ∞). In particular we consider the following three questions dealing with the asymptotic behavior of solutions of (1.1).

In this paper I show that complex-valued multiplicative functions g which satisfy |g(n)|≤1 for all positive integers n, are generally well distributed in residue classes to small moduli.

Let μ, be a positive Radon measure with compact support in the euclidean n-space ℝn. Introducing the Fourier transform

and the averages over the spheres

we can write the α-energy, 0 < α < n, of μ as

where the positive constants c1 and c2 depend only on n and α. The second equality is based on the Plancherel formula and the fact that where .

We construct realizations of Dyck's regular map of genus three as polyhedra in ℝ3. One of these has one axis of symmetry of order three and three axes of symmetry of order two. The other polyhedra have three axes of symmetry. We show that a polyhedron realizing Dyck's regular map cannot have a symmetry group of order larger than six. Thus the symmetry groups of our realizations are maximal.

We construct a polyhedron with ten vertices of genus three which has three axes of symmetry. It is as symmetric as possible. Ten is the minimal number of vertices which a polyhedron of genus three can have. A modification of our polyhedron yields a symmetric polyhedral realization of Dyck's regular map.

J. E. Jayne and C. A. Rogers in [7] introduced the following notion.

Let X be a topological space and p be a metric defined on X × X. X is said to be fragmented by the metric p if, for every ε > 0 and each nonempty subset Y of X there is a nonempty relatively open subset U of Y such that ρ-diam (U)≤ ε.

While investigating Asplund spaces in [15], R. R. Phelps and the author noticed that weak* compact subsets of the duals of Asplund spaces (or equivalently, as it turned out, weak* compact subsets of dual Banach spaces with the Radon-Nikodým property) possessed many properties in common with weakly compact subsets of Banach spaces. The topological study of the spaces homeomorphic to the latter, the so-called Eberlein compact spaces, or EC spaces for short, had flourished and had already yielded a rich collection of results. Therefore it was natural to hope that a similar study of the former might also lead to interesting discoveries. In a series of letters with S. Fitzpatrick exchanged during the summer and the fall of 1981, we started to collect properties of compact spaces that are homeomorphic to weak* compact subsets of the duals of Asplund spaces, which we tentatively called “Asplund compact spaces“. However, as far as we are aware, Reynov's paper [16] is the first study in print of the topological properties of “Asplund compact spaces” or “compacta of RN type” as Reynov termed them.