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.

Throughout the paper, let

be forms with rational integer coefficients of degrees d1, …, ds all at least 2 with n ≥ 4. Let p be a prime and Q a box in ℝn,

Let (C1, C2, …) be a sequence of convex bodies in n-dimensional euclidean space En, and let υ(Ci) denote the volume and d(Ci) the diameter of Ci. It is shown that the conditions

imply that the sets Ci can be rearranged by the application of rigid motions so that the resulting sets form a packing in En of density 1. A corresponding result for coverings of En is also 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

The problem of illuminating the boundary of sets having constant width is considered and a bound for the number of directions needed is given. As a corollary, an estimate for Borsuk's partition problem is inferred. Also, the illumination number of sufficiently symmetric strictly convex bodies is determined.

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.

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.

The 2-ball property is shown to be transitive. Combining this with some results on the decomposability of convex bodies, we produce new examples of Banach spaces which contain proper semi-M-ideals. These semi-M-ideals are not hyperplanes, nor are they the direct sums of examples which are hyperplanes.

The concept of mixed invariant set is due to Bandt [1], Bedford [2], Dekking [3, 4], Marion [4] and Schulz [10]. An m-tuple B = (B1, …, Bm) of closed and bounded subsets Bi of a complete finitely compact (bounded and closed subsets are compact) metric space X is called a mixed invariant set with respect to contractions f1, …, fm and a transition matrix M = (mij), if, and only if,

for every i ∈ {1, …, m}. In the papers quoted an essential condition is that all mappings f1, …, fm be contractions. We will show that, under certain conditions, the construction of mixed invariant sets also works in cases where some of the mappings are isometries or even expanding mappings.

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.

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.

Let (R, ) be a commutative Noetherian local ring. We investigate conditions for a non-finitely generated R-module M to have a system of parameters. We prove that if

then any system of parameters for R/AnR (M) is a system of parameters for M. As an application we characterize by means of systems of parameters those balanced big Cohen–Macaulay R-modules M for which SuppR (M) = suppR (M).

We shall deal throughout with regular matrix methods of summability defined by a stochastic matrix A = (ank). Thus

means

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.

We examine the asymptotic form of a fundamental set of solutions of the third-order equation

as x → ∞, where the leading coefficient q is nowhere zero in some interval [a, ∞). The equation is self-adjoint in the case when p, r and iq are real. However, our analysis is not confined to this case, and we generally take the coefficients to be complex-valued.

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”.

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.

Sphere Theorems are employed to solve two asymmetric singularity-driven flows of a fluid exterior to a rigid spherical surface. The flows are generated by a rotlet and Stokeslet respectively whose axes are perpendicular to the sphere radius produced drawn through their locations. The flow details are analysed and the forces and couples acting on the sphere are calculated. The Stokeslet solution is also used to compute an approximation to the drag force experienced by a particle which sediments in the fluid in a direction perpendicular to a sphere radius.

In the application of electromagnetic methods to the non-destructive testing of electrically conducting materials for cracks or inclusions an electric current is applied to the specimen and the presence of a flaw is indicated by the perturbations it produces in the electromagnetic field. A number of different variants of the method can be used. The presence of a flaw may be observed by measuring either electric or magnetic field perturbations and the nature of the interrogating field will be sensitive to the choice of frequency chosen for the applied current. It is well known that when alternating current is applied to conductors the current tends to be confined to a surface layer whose depth, δ, is measured by the length 1/(ωσμ)½ where σ is the conductivity, μ is the magnetic permeability and ω the angular frequency. An important dimensionless parameter in the characterisation of the field perturbations is the ratio δ/l, where l is a length typical of the flaw dimensions. The electromagnetic field is described as a thin-skin or a thick-skin field according as this ratio is small or large respectively. In practical applications there is a need to model both thin and thick-skin fields. In the examination of surface fatigue cracks in large scale structures fabricated by welding together ferrous steel members surface fatigue cracks with depths of order 1–10 mm have been interrogated with currents at 5–6 KHz at which the skin depth is of order 0·1 mm (Dover, Collins and Michael [1]).

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,