It was shown by Davenport and Roth [7] that the values taken by

at integer points ( x1, …, x8) ∈ ℤ8 are dense on the real line, providing at least one of the ratios λi/λj, is irrational. Here and throughout, λi denote such nonzero real numbers. More precisely, Liu, Ng and Tsang [8] showed that for all the inequality

has infinitely many solutions in integers. Later Baker [1] obtained the same result in the enlarged range . In this note we improve this further, the progress being considerable.

In [2], Fabrikant and his colleagues obtain a closed form solution to a generalized potential problem for a surface of revolution. This they specialize to solve three electrostatic problems for a spherical cap, including one for which the boundary conditions are not axisymmetric. In all three the solutions are expressed in terms of elementary functions.

Let g(n) be a complex-valued multiplicative function which satisfies |g(n)|≤ 1 for all n. The aim of this paper is to establish the following theorem.

In order for an indefinite integral ternary quadratic form to have class number exceeding one, its discriminant must be divisible by the cube of at least one odd prime, or by a sufficiently large power of 2 (see [4], [1]). More generally, for such a form to have class number 2t, t> 1, it is necessary not only that the discriminant be divisible by at least t distinct primes, but also that these primes interact with each other in rather specific ways. Consequently, the minimal absolute value ∆(t) of the discriminant of an indefinite integral ternary quadratic form of class number 2' increases rapidly as a function of the natural number t.

In this paper we show that every finite connected graph G = (V, E), without loops and for which its spanning trees are the blocks of a balanced incomplete block design on E containing more than one block (E is the set of edges), is vertex 2–connected.

The Hausdorff dimension has been used for many years for assessing the sizes of sets in Euclidean and other metric spaces, see, for example, [1,2,5,6,8,10]. However, different sets with the same Hausdorff dimension may have very different characteristics, for example, a straight line segment in ℝ2 and the Cartesian product in ℝ2 of two suitably chosen Cantor sets in ℝ will both have Hausdorff dimension 1. In this paper we develop a measure-theoretic method of distinguishing between the sets of such pairs.

Suppose x1, x2Є[0,1] and α = log 3/ log 4. Then

The relation is readily seen to be satisfied with equality for both of X1, x2 equal to any of the values 0, ½, 1 so that the value of α is “best possible”.

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