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.

A study is made of the length L(h, k) of the Euclidean algorithm for determining the g.c.d. of two polynomials h, k in [X], a finite field. We obtain exact formulae for the number of pairs with a fixed length N which lie in a given range, as well as the average length and variance of the Euclidean algorithm for such pairs.

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.

One construction used to produce a random number table is to take a smooth function F(x) taking values between 0 and 1, to evaluate it at N points spaced 1/M apart, and to ignore the first t decimal digits. With T = 10t this corresponds to taking the fractional part of

where T>M>N. The grounds for assuming this sequence to be random are that it is so difficult to prove anything about it.

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.

This theoretical work shows formally that unsteady interactive boundary layers can break up within a finite time by encountering a nonlinear localized singularity. The theory is an extension of, and is guided to a large extent by, Brotherton-Ratcliffe and Smith's (1987) work on a special case. Two major types of singularity are proposed, a “moderate” type yielding a singular pressure gradient and a “severe” type associated with a pressure discontinuity. Each type produces a singular response in the skin friction in the case of wall-bounded flows. The present finite-time singularity applies to any unsteady interactive flow, e.g., incompressible or compressible boundary layers, internal flows, wakes, in two or three dimensions; and the singularity and its associated change in flow structure have numerous repercussions, which are discussed, physically and theoretically, concerning boundary-layer transition in particular.

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