We study the minimal length of faithful nuclear representations of operators acting between finite-dimensional Banach spaces and the related minimal number of contact points of the John ellipsoid of maximal volume contained in the unit ball of a finite-dimensional Banach space. In both cases the classical upper estimates, which follow from the Caratheodory theorem, are shown to be exact. Related isometric characterizations of are proved.

Let α be a real number and k a positive integer. We shall be interested in integer values of n for which ║αnk║ is small. For the case k = 1 we have Dirichlet's Theorem. For any N ≥ 1 there exists n ≤ N with

In this paper we continue the investigation begun in [6] concerning the number of solutions of the inequality

for almost all α (in the sense of Lebesgue measure on ℝ), where β is a given real number, , and both m and n are confined to sets of numbertheoretic interest. Our aim is to extend existing results ([7], [8], [5] for example), where only n is restricted. Here we shall prove the following result where, as elsewhere in this paper, p denotes a prime, and a square-free integer may be positive or negative.

Let K be any field of characteristic 0 and let T and X be algebraically independent over K. For n ≥ 1 let k(n) ≥ 2 be an integer and let fn(X, T) = xk(n) + T ε K [X, T]. We shall regard T as a “parameter” and X as a “variable”. We put F1(X, T) = f1(X, T) and define, for n ≥ 1,

Let f(x) be a polynomial of degree d over Fq, the finite field with q = pn elements. Let V(f) denote the number of distinct values of f(x), xєFq. Then, it is easy to see that

where [x] denotes the greatest integer ≤x. A polynomial for which equality is achieved in (1) is called a minimal value set polynomial. Minimal value set polynomials have been studied in [1] and [3].

For an ideal A of a commutative ring R with identity and a unitary R-module E the notion of an E-sequence of length d in A can be extended as follows. For d = 0 the E-sequence is empty, and for d = 1 it is a subset {ai|i ∈ I} = α ⊆ A such that . For d > 1 we may define, inductively, an E-sequence of length d in A as a sequence

of subsets of A such that a, is an E-sequence of length 1 in A and α2,…, αd is a -sequence of length d − 1 in A. Thus in the standard notion of E-sequence the sets αj, are singletons, and, in effect, the extended notion due to Hochster [1] and Northcott [3] the sets a; are finite. Many of the standard results concerning E-sequences when E is Noetherian extend to the above generalization when the Noetherian condition is dropped. For example it follows from the results of the present note that every maximal E-sequence has the same length (which may be infinite) and every E-sequence can be extended to a maximal E-sequence. This maximal length is inf which we call the homological grade of E in A and denote by hgrR (A; E). So 0 ≤ hgrR (A; E) ≤ ∞, hgrR (A; E) = 0, if, and only if, 0:EA≠0 and hgrg (A; E) = ∞, if, and only if, for all nєℤ.

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.

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.

Let

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

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.

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.

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 .

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.

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).

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.