One proof of Minkowski's fundamental theorem on lattice points in an n-dimensional parallelopiped depends upon a multiple Fourier expansion. Its terms involve multiple integrals which are easily evaluated and so are shown to have positive values. Then the expansion takes only positive values and the desired proof follows at once. It seems of interest to note a multiple integral which assumes only positive values.

Let Q denote the closed unit cube in Rn. The elementary area A(f) of a Lipschitz function f on Q is given by the formula.

Let L1 denote temporarily the usual Lebesgue space over the circle group (equivalently: the additive group of real numbers modulo 2π), and let H1 denote the Hardy space comprised of those f in L1 whose complex Fourier coefficients vanish for all negative frequencies, so tha D. J. Newman has settled a conjecture by showing [1] that there exists no continuous projuction of L1 onto H1, i.e. that there exists in L1 no topological complement to H1.

Throughout this paper E, F and G denote separated locally convex spaces, F C G, the injection i: F → G being continuous (i.e. the topology on F is finer than that induced on it by the topology on G). E′, F′ and G′ denote the respective duals of E, F and G. i′ is the adjoint map of G′ into F', which is defined by restricting linear forms on G to F C G.

Given two topologies J1, J2 on a set X, J1 is said to be coarser than J2, written J1 ≦ J2, if every set open under J1 is open under J2. A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.

Let G be an n-generator group and let g = (g1, g2, …, gn) be an ordered set of n elements which generate G, then g is called a generating n-vector of G. Let denote the set of all generating n-vectors of G.

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), I shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n.

We consider a set of η first order simultaneous differential equations in the dependent variables y1, y2, …, yn and the independent variable x ⋮ No loss of gernerality results from taking the functions f1, f2, …, fn to be independent of x, for if this were not so an additional dependent variable yn+1, anc be introduced which always equals x and thus satisfies the differential equation

Huta [1], [2] has given two processes for solving a first order differential equation to sixth order accuracy. His methods are each eight stage Runge-Kutta processes and differ mainly in that the later process has simpler coefficients occurring in it.

A mathematical theory is developed which enables the wind-tunnel corrections to the lift and moment forces acting on an aerofoil in subsonic two-dimensional flow to be calculated. The usual “averaged” boundary condition for slotted walls is assumed and the corrections obtained by successive approximation from the open-jet tunnel.

We consider a single server queueing system in which customers arrive at the instants t0, t1, …, tm, …. We write τm = tm+1 − tm, m ≧ 0. There is a single server with distribution of service times B(x) given by where k is an integer not less than unity.

We consider the following problem. Calls arrive at a telephone exchange at the instants t0, t1, … tm …. The telephone exchange contains a denumerable infinity of channels. The holding times of calls are non-negative random variables distributed independently of the times at which calls arrive, independently of which channel a call engages and independently of each other with a common distribution function B(x). Takacs [3Τm = tm+1 − tm, m ≧ are identically and independently distributed non-negative random variables with common distribution function, A (x). Finch [1] has studied the transient behaviour in the case of a recurrent arrival process and exponential holding time, that is when the common distribution of holding time is given by In this paper we make no assumption about the arrival process {tm}. The underlying principle of this paper is the same as that of Finch [2]. We consider the instants of arrival t0, t1,…, tm,… as given and determine various probabilities of interest conditionally as functions of the inter-arrival intervals Τ1,…, Τm,… When the arrival process is a stochastic process we can then determine the relevant unconditional probabilities by integration.

A queue at which arrivals occur randomly in batches of fixed size r and for which the service times are independent negative exponential variates is considered. Expressions are obtained for the moments of the transient waiting time distribution and the distribution of the number of customers in the system just before the nth batch arrives. The distribution of the number of customers served in a busy period is also determined.

We consider a single server queue for which the interarrival times are identically and independently distributed with distribution function A(x) and whose service times are distributed independently of each other and of the interarrival times with distribution function B(x) = 1 − e−x, x ≧ 0. We suppose that the system starts from emptiness and use the results of P. D. Finch [2] to derive an explicit expression for qnj, the probability that the (n + 1)th arrival finds more than j customers in the system. The special cases M/M/1 and D/M/1 are considerend and it is shown in the general case that qnj is a partial sum of the usual Lagrange series for the limiting probability .