1. Introduction. In the study of diophantine equations in two variables, it is often necessary to consider rational functions on a curve with prescribed zeros and poles. Although it is well known that such functions can, in principle, always be effectively constructed, the detailed proof does not appear to have been given. The purpose of the present paper is to give the complete proof of such a construction. Our method, and the statement of our results, are motivated by the applications to diophantine equations which we have in mind. In particular, our results will play an important role in a subsequent paper (1), in which explicit bounds will be established for the integer points on any curve of genus 1.

In this paper, by extending the results of Yoshikawa (8), we obtain local a priori inequalities for hypoelliptic pseudo-differential operators. Using these inequalities we then show how the results of Hormander ((3), Theorem 8·7·2), on the solvability of the adjoint operator of a principally normal operator can be extended to the adjoint operator of a hypoelliptic pseudo-differential operator. Finally, we consider a class of operators which satisfy more particular a priori inequalities and we show that these operators are hypoelliptic. This class of operators was studied by Egorov (1), and he shows them to be ‘of principal type’. They include elliptic operators and also the subelliptic operators of Hormander (4).

One of the classical models of applied probability is that which may be described as the covering of a line with a random collection of intervals of random length. It appears as such in ((9), pp. 23–25), but also as the problem of Type II counters with random dead time (1, 4, 10), as a model for a pedestrian crossing a busy road(5), and as the queue M/G/∞. If an excuse is required for returning to a problem which has been the object of so much research in the past, it is to be found in the fact, noted by Kendall (5), that the model gives rise to the general stable infinitely divisible p-function, and that the theory of the semigroup P of standard p-functions (6) requires deeper information than previous, practically motivated, studies have provided.

A decomposition of a topological vector space E is a sequence of non-trivial subspaces of E such that each x in E can be expressed uniquely in the form , where yi∈Ei for each i. It follows at once that a basis of E corresponds to the decomposition consisting of the one-dimensional subspaces En = lin{xn}; the theory of bases can therefore be regarded as a special case of the general theory of decompositions, and every property of a decomposition may be naturally denned for a basis.

In a paper (1) published in 1922 in these Proceedings, I conjectured that a curve of genus greater than one contained only a finite number of rational points, but this has never been proved. A particular case is given by the equation of genus three,

and so any results for such equations may be of interest.

It has been known for some time that one can construct a proof of the spectral theorem for a normal operator on a Hilbert space by applying the Gelfand representation theorem to the Abelian von Neumann algebra generated by the normal operator, and using the fact that the maximal ideal space of an Abelian von Neumann algebra is extremely disconnected. This, in fact, is the spirit of the monograph (8). On the other hand, it is difficult to find in print accounts of the spectral theorem from this viewpoint and, in particular, the treatment in (8) uses a considerable amount of measure theory and does not have the proof of the spectral theorem as its main objective.

Let M1 denote the set of Borel probability measures on the real line and M1 the space of their Fourier transforms, or ‘characteristic functions’. It is well known that the natural correspondence between M1 and M1 is in fact a homeomorphism, provided M1 is endowed with the usual weak topology relative to bounded continuous functions (which topology can be metrized by the construction of Lévy) and that M1 is given the topology (also metric) appropriate to locally uniform convergence, which we call the ‘compact-open’ topology. In particular, if

where μn ∈ M1 (and so φn ∈ M1), then φn(λ)→φ0(λ) uniformly on all compact sets if and only if μn ⇒ μ0 (weak convergence). However, the continuity theorem for characteristic functions implies that even if it is only known that φn(λ)→φ0 pointwise (for all λ), the conclusion that μn ⇒ μ0 is still valid. This raises a question as to whether the pointwise (i.e. product) topology for M1 might not, in fact, be equivalent to the compact-open one, since the corresponding notions of sequential convergence do coincide. The purpose of this note is to show that the answer is negative, even in a much more general setting.

1. Introduction: The well-known result that every function f(x) ∈ L2(–∞, ∞) can be expanded in L2(– ∞,∞) by where and was recently followed by an expansion theorem for generalized functions. In papers by Wildlund(8), Giertz(4) and Zemanian(9) it was shown that a tempered distribution f∈S′ can be expanded by Hermite functions that is

for all ψ∈S (the space of rapidly decreasing test functions).

Istrăţescu (2) has introduced a class of operators on a Banach space called operators of class (N, k). An operator T (a bounded linear transformation) on a Banach space X is said to be an operator of class (N, k), k = 2, 3,…, if ‖Tx‖k ≤ ‖Tkx‖ for all x ∈ X such that ‖x‖ = 1. If k = 2, such an operator is called an operator of class (N)(3). If X is a Hilbert space, then the class of operators of class (N) on X is an extension of the class of hyponormal operators on X (3). The object of this note is to generalize to operators of class (N, k) on a Banach space X, some results which are known to be true for normal operators on a Hilbert space, particularly with regard to their ascent and descent.

Suppose that λ > − 1 and that

The generalized function (x + iO)λ is defined by

for λ ╪ -1, -2, … and (x + iO)λ is denned by

for n = 1, 2,…, see (l).

The system of equations here to be analysed is

The method of solution is by way of appeal to the theory of singular integral equations of Cauchy type. Apart from the self-reciprocal series of Titchmarsh(l) no comparable systems appear to have been discussed in the literature, the order of the matrix element falling foul of the restrictions otherwise there imposed.

In order to study the convergence of a series arising from the use of Martin's method in potential scattering a framework consisting of a modifidation to Schwartz's formulation of distribution theory is set up. This modification makes essential use of norms on the spaces of test functions and linear functionals. By making the norm on the space of test functions satisfy extra axioms a bound is obtained on the norm of a generalized convolution of two linear functionals, and this bound is used to prove the convergence of the series. The proof of convergence is generalized to cover the multi-channel problem. The method used may also be applied to many other physical problems.

1. Introduction: Consider the system of ordinary differential equations

where the unknown x(t) is a complex m-vector, t is a real variable, D is the operator d/dt and a0, …, an are complex m × m matrices whose elements are continuous functions of t, x, Dx, …, Dn−1x. Furthermore, det a0 ╪ 0. In the special case when a0, …, an are constant matrices the trivial solution x = 0 is asymptotically stable if and only if all the roots of the characteristic equation det f(ζ) = 0 have negative real parts, where

1. Introduction: In what follows, X will be a real inner-product linear space (inner product (x, y), norm |x| = (x, x)½); En will be n-dimensional real Euclidean space. We shall be dealing with sets {as: 1 ≤ s ≤ rk; r ≥ 1, k ≥ 2} of elements of X; we write a for the vector (a1; …, ark), and put . We suppose that |as| ≤ ∈ (< 0) for every s.

A result due to Titchmarsh(1) is that corresponding to the series

is the reciprocal formula

We consider here the supersonic inviscid flow past a wing-body combination consisting of a semi-infinite plane wing symmetrically placed about an infinite convex cylinder. When the cylinder is circular in cross-section a formal solution for the Laplace transform of the velocity potential exists (Neilson(l), and Stewartson, 1951, unpublished). This solution is given in the form of a series which is only slowly convergent near the boundary surface separating the disturbed and the undisturbed regions. However, this slow convergence has been overcome by Waechter(4) using the Poisson Summation formula on the series solution. This enables the solution to be expressed in terms of integrals which can be evaluated as a convergent residue series, taking on different forms in the different regions of the interaction.

By a generalization of the Infeld-Hull Factorization Kernels we find it possible to factorize a wide family of second-order ordinary differential equations at specified points of their spectrum.

In (general) elastico-viscous liquids the response to stress at any instant will depend on previous rheological history, the equations of state needed to describe the rheological properties of a typical material element at any instant t being expressible in the form of a (properly invariant†) set of integro-differential equations relating its deformation, stress and temperature histories (as defined by a metric tensor (of a convected frame of reference), a stress tensor and the temperature measured in the element as functions of previous time t'( < t)) together with the time lag (t – t') and physical constant tensors associated with the element (1). Thus in any type of oscillatory motion a rheological history will necessarily be a function of the frequency of the forcing agent, the rheological history of a number of different types of elastico-viscous liquids in some simple shearing oscillatory flows being a rather simple oscillatory history (see, for example, (2–4)). It is, therefore, to be expected that a liquid with elastic properties will behave somewhat differently from any inelastic viscous liquid when subjected to any kind of oscillatory motion, and it is for this reason that oscillatory motions have been used extensively to detect and measure the elastic properties of liquids (see, for example, (2–5)).

1. Introduction. A Markov Renewal Process (MRP) with m(<∞) states is one which records at each time t, the number of times a system visits each of the m states up to time t, if the system moves from state to state according to a Markov chain with transition probability matrix P0 = [pij] and if the time required for each successive move is a random variable whose distribution function (d.f.) depends on the two states between which the move is made. Thus, if the system moves from state i to state j, the holding time in the state i has Fij(x) as its d.f. (i, j = 1,2, …, m).