In this note we prove that every convex Borel set in a finite dimensional real Banach space can be obtained, starting from the compact convex sets, by the iteration of countable increasing unions and countable decreasing intersections. This question was first raised by V. Klee [1, p. 451]. It was answered affirmatively by Klee for R2 in [2, pp. 109–111] and for R3 by D. G. Larman in [4]. C. A. Rogers has given an equivalent formulation of the question for Rn in [6].

The idea of obtaining bounds on the order of approximation of algebraic functions by rational functions apparently goes back to Maillet [8]. It is very easy, using an argument analogous to that used by Liouville for algebraic numbers, to see that if w(z) is any algebraic function of degree n ≥ 2 over Q[z] then for all pairs of polynomials r(z) and s(z) with s(z) ≢ 0 we have

This paper gives an algorithm for generating all the solutions in integers x0, x1…, xn of the inequality

where 1, α1, …,αn are numbers, linearly independent over the rationals, in a real algebraic number field of degree n + 1 ≥ 3 and c is any sufficiently large positive constant. It is well known [2, p. 79] that if c is small enough, then (1) has no integer solutions with x1…, xn not all zero.

Central to the study of electromagnetic induction in turbulently moving conductors is the evolution of the ensemble averaged field, and so the subject of “mean field electrodynamics” has been born. It has provided a particularly fruitful approach to the dynamo problem, that is the study of the self-excitation of magnetic field by a moving fluid. It has been shown by Steenbeck, Krause and Rädler, in studies dating from 1966, that regeneration of field is efficient when the motions are sufficiently vigorous and lack mirror-symmetry. Using a commonly accepted approximation of the subject (the neglect of third order cumulants) which is valid when the correlation length and/or the correlation time of the turbulence are sufficiently small, and paying proper regard to a theorem due to Bochner, it is shown here that mirror-symmetric homogeneous turbulence cannot be regenerative.

This paper is concerned with the determination of the small deflexions and stresses in a thin isotropic homogeneous semi-circular plate subject to hydrostatic pressure distributed over the entire plate. The semi-circular plate is clamped along the bounding diameter and elastically restrained, against rotation along the circular edge, according to a general boundary condition including the simply supported and rigidly clamped boundaries as particular cases. The boundary conditions around the circular edge and the condition of zero deflexion on the straight edge are exactly satisfied while the remaining condition of zero slope on the bounding diameter is approximately satisfied. Calculations have been carried out corresponding to three cases of boundary constraint; namely, those for which the edge is fully restrained, the edge is simply supported and an intermediate case. Positions and magnitudes of maximum deflexions are determined. Numerical results are presented in tabular and graphical forms.

The flow in the wake of a finite rotating disc was recently investigated by Leslie [1], who showed that the structure of the wake was very similar to that found for the two-dimensional flow past the trailing edge of a flat plate by Goldstein. Here, we investigate the inner region where the Goldstein wake joins onto the flow over the plate itself. Because there is no pressure jump across the rim at the edge of the Ekman layer, no third deck is necessary, so this inner region has width O (E½) and contains just two decks, the lower thickness O(E⅔) and the upper with thickness O(E½); E is the Ekman number for the flow. The resultant differential equation in the lower deck for the radial and axial velocities is uncoupled from that for the azimuthal velocity, and has exactly the same form as the equation found by Stewartson [4] for the lower deck in the flat plate problem cited above.

Let a1 …, ar be elements of a commutative ring R with identity. When the R-length of

is finite and R is Noetherian the multiplicity e(a1 …, ar\R) can be defined in one of three ways. Starting either from the earliest definition by means of Hilbert functions (see Samuel [4]) or from the latest by induction on r(see Wright [5]) it can be shown that e(a1 …, ar\R) depends only on the ideal

and the number r of generators. The other definition due to Auslander and Buchsbaum [1] is as the Euler-Poincare characteristic of a Koszul complex. The main purpose of this note is to investigate the extent to which the separate homology modules and the Koszul complex itself are independent of the sequence a1 …, ar. First it is shown that when A = (a1, …, ar) and B = (b1 …, br) are related by a reversible R-linear transformation the two Koszul complexes are isomorphic. A special case of the final theorem states that if A and B are minimal generating sets of the same ideal and R is a local ring then again the Koszul complexes are isomorphic.

The main purpose of this paper is to evaluate the Whitehead group K1 (Zπ) of the quaternion group π of order 8. We show that the natural mapping of the units ± π of Zπ into K1(Zπ) induces an isomorphism between K1(Zπ) and ± V, where V, Klein's 4-group, is the commutator quotient group of π.

For a cell compex M decomposing the closed orientable 2-manifold Pg of genus g let pi (M) and νj(M) denote the number of i-gonal cells (faces, countries) and j-valent vertices of the graph (= 1 – skeleton) of M, respectively. It will be supposed that i, j ≥ 3. From Euler's formula follows

and

is even, where e is the number of edges of M. As seen, the relation above does not impose restrictions on the numbers p4 (M), ν4(M). In an attempt to characterize the vectors {pi(M)}, {νi(M)}which partially determine the combinatorial structure of the cell-decompositions, the first step could involve answering the question: Given sequences p = (p3, p5, …), ν = (ν3, ν5, …) of non-negative integers satisfying conditions

and

does there exist a cell-decomposition M of Pg, for which pi (M) = pi, νj(M) = νj for all i, j ≠ 4? (If so, the sequences p, ν are called realizable in the sequel. M itself is a realization of p, ν. For brevity's sake we will often use the word map instead of cell-decomposition in the sequel.)

A reduction method due to R. Rado [7] yields an elegant proof of the Halls' theorems on transversals of a family of sets (see for example [4]). We use here this method to give simple new proofs of some basic theorems on common transversals of a pair of families of sets.

Let τ(n; k, h) denote the number of divisors of n which are congruent to h (mod k), and τ(n; k) the number of divisors of n prime to k, so that

Let

Erdős [1] proved that, if ε and η are fixed arbitrary positive numbers, then for almost all integers n ≤ x, we have

provided

Hall and Sudbery [2] showed that it is sufficient that

and apart from the ε, this upper bound for k is best possible, for it is clear that k must not exceed the normal order of τ(n). For n ≤ x, Hardy and Ramanujan [3] showed that this normal order is (log x)log 2 = 2 log log x.

It has been known for many years that a laminar boundary layer adjoining a supersonic stream is in principle capable of spontaneously evolving from an unseparated to a separated form in which the boundary layer is detached from the wall and in between a slowly moving reversed flow is set up. The phenomenon arises because any thickening of the boundary layer tends to induce an adverse pressure gradient which in turn accelerates the thickening of the boundary layer and hence its evolution from the normal (Blasius) form.

For any subsets E, F of a group G, written additively, we write

and