The main object of this note is to show that a proof given by A. J. Macintyre [2] of a result on the overconvergence of partial sums of power series works more easily in the context of Dirichlet series. Applying this observation to the particular Dirichlet series Σane−ns, we can remove certain restrictions which Macintyre finds necessary in the direct treatment of power series.

In this paper two integrals involving E-functions are evaluated in terms of E-functions. The formulae to be established are:

where n is a positive integer,

and

where n is a positive integer,

and

the prime and the asterisk denoting that the factor sin {(s–s)π/2n} and the parameter βq+s–βq+s + 1 are omitted. The definitions and properties of MacRobert's E-function can be found in [1, pp. 348–352] and [3, pp. 203–206].

The set Dn of all n × n doubly-stochastic matrices is a semigroup with respect to ordinary matrix multiplication. This note is concerned with the determination of the maximal subgroups of Dn. It is shown that the number of subgroups is finite, that each subgroup is finite and is in fact isomorphic to a direct product of symmetric groups. These results are applied in § 3 to yield information about the least number of permutation matrices whose convex hull contains a given doubly-stochastic matrix.

A. Geddes [1, Theorem 3.3] has shown that the partial algebraic system which he has called a power-free group need not be cancellative. In other words, there exist power-free groups containing at least one element a with the property that ab can equal ac when b ≠ c. In the present paper we propose to study the structure of such non-cancellative power-free groups, and we shall in fact obtain a complete solution to this problem.

1.1. Let A = (aμν) be a normal triangular matrix, i.e., one for which aμμ ≠ 0 (μ ≥ 0), aμν = 0 (ν > μ).

where (i) 0≤m<n, (ii) Rμ>0 (μ≥0), (iii) K is a constant, depending on the matrix A and the sequence {Rμ}, but independent of m, n and the finite sequence {sν}.

Let H be any closed bounded convex set in En, and -H be its reflection in the origin. Then the vector sum K = H+ (−H) has the origin as centre and is called the difference set of H. Clearly every closed bounded convex set K with centre at the origin is the difference set of ½K. Excluding this trivial case, we define such a set K to be reducible if it is the difference set of some H which is not homothetic to K.

We wish to prove a theorem concerning the average values for the functions Ln(2u)e−u, 0 ≤x≤u < ∞, n=0,1,2, …, where Ln is the n-th Laguerre polynomial. Such functions will be called Laguerre functions with domain truncated at x.

By a permutation mapping in a finite projective plane π is meant a one-to-one mapping σ: P→l of the points of π onto the lines of π with the property that corresponding elements are incident. The simplest aspects of such mappings are discussed in this note.

In this paper solutions are given for two problems on the motion of a dusty gas, using the formulation of Saffman [1]. The gas containing a uniform distribution of dust, occupies the semi-infinite space above a rigid plane boundary. The motion induced in the dusty gas is considered in the two cases when the plane moves parallel to itself (i) in simple harmonic motion, and (ii) impulsively from rest with uniform velocity. In case (i) the change in phase velocity and the decay of oscillatory waves are noted as functions of the mass concentration of dust f. In case (ii) the problem is solved by use of the Laplace transform, some velocity distributions are calculated for f=0·2, and it is shown that the shear layer thickness is decreased by the factor (1+f)−1/2 at large times.

In this paper, solutions of the ordinary non-linear differential equation

are considered. This equation arises in the theory of both axisymmetric and two-dimensional viscous jets falling under gravity. In (1.1), y represents the first approximation to the velocity along the axis of symmetry, and x is a measure of the distance along this axis. Accordingly one of the conditions that the solution must satisfy is, that it cannot have a singularity at a finite value of x. The other condition to be imposed is that y must vanish at x = 0. For a derivation of this equation the reader is referred to Brown [1] or Clarke [2].

The dependence of the skin friction on the parameter β for the reversed flow solutions found by Stewartson of the Falkner-Skan equation f′′′ + ff″ + β (1−f′2)=0 is determined in the limit as β→0−.

Let A = {a1, …, an} and B={b1, …, bn} be two sets of n elements, and let R be a set of ordered pairs (ai, bi), or in other words a relation defined on A × B. By a map between A and B under R we mean a one-to-one correspondence between A and B such that if bi, corresponds to ai then (ai, bj) is one of the pairs in R.

1. Given a metric space (X, ρ) a family of subsets of X which includes the empty set Ø, and a non-negative function τ on with τ(Ø)=0, an outer measure μ* may be defined by

where empty infimums have value +∞. It is easily seen that μ* is a metric outer measure [i.e., if ρ(A, B)>0 then μ*(A∪B)=μ*(A)+μ*(B)] and from this it follows that all Borel sets in X are μ*-measurable.

Suppose, in n-dimensional Euclidean space, a sequence of disjoint closed spheres is packed into the open unit n-cube In in such a way as to ensure that the residual set has zero volume. Then, of course, is convergent and it can be shown, for n = 2, that is divergent. Let the exponent of convergence of the packing be the supremum of those real numbers t such that is divergent and let tn denote inf, where the infimum is taken over all packings which satisfy the above conditions. In recent work Z. A. Melzak [1] has been interested in finding estimates for in two-dimensional space. He has shown that t may take the value 2 and has produced some estimates for the Apollonius packing of disks. In this note we produce what is, perhaps, the most interesting estimate for tn by showing that tn is greater than n-1. Let sn denote the exact lower bound of the Besicovitch dimensions of every residual set which is formed by a packing of spheres in In.

A number of oscillating systems have been described in the literature which have proved useful in the study of elastico-viscous liquids. The most common type of system in use is the forced oscillating system of the coaxial-cylinder type; in this, the outer cylinder wall is made to oscillate about its axis with a prescribed frequency, and the resulting motion of the inner solid cylinder (constrained by a torsion wire) is recorded (see, for example, [1]). Another type of oscillating system— a, free oscillating system, also of the coaxial-cylinder type—was considered by the present authors in a previous paper [2]; this latter system is one which may prove to be simpler to design and to control in practice.

Professor Kneser has pointed out to me that the results proved in my paper [3] are not new. To be precise, my Theorem 1 is a special case of the result proved in [2], while the routine argument by which I deduced my Theorem 2 is given in substance in [1; 241]. Then my Theorem 3, though perhaps new, follows almost trivially.

By means of all functions regular in a domain †G ⋐ Cn Carathéodory [1,2] defined a metric in G. As usual, using this metric, we can define spherical shells in G. That is, if DG(t, q) denotes the Carathéodory metric between t, q∈G measured in G, then the Caratheéodory spherical shell S(t, q) passing through q and with centre t is defined by:

The object of this paper is to give a simple proof of Menger's famous theorem [1] for undirected and for directed graphs. Proofs of this theorem have been given by D. König [2], G. Nöbeling [3], G. Hajós [4], T. Gallai [5], P. Erdös [6] and 0. Ore [7]. The present proof is shorter, and formulated to apply to directed and undirected graphs equally. The term, graph is to be understood to mean either a finite undirected graph or a finite directed graph throughout. (A: B) denotes either an undirected edge between the two vertices A and B or a directed edge from A to B according to whether undirected or directed graphs are considered. G1 and G2 being two non-empty disjoint graphs, G1: G2-edge denotes an undirected edge between a vertex of G1 and a vertex of G2 or a directed edge from a vertex of G1 to a vertex of G2 as the case may be, and G1: G2-path denotes a path with one end-vertex in G1 and one in G2 having no intermediate vertex in G1 + G2, undirected or directed from the end in G1 to the end in G2, as the case may be. (A set of isolated vertices is a graph.)