For various classes of right noetherian rings it is shown that projective right modules are either finitely generated or free.

In the first part of the paper a criterion is given for two self-adjoint operators T, S in a Hilbert space to have the same essential spectrum, S being given in terms of T and a perturbation P. If P is a symmetric operator and the operator sum T+P is self-adjoint, then S = T+P. Otherwise, T is assumed to be semi-bounded and S is taken to be the form extension of T+P defined in terms of semi-bounded sesquilinear forms. In the case when S = T+P, the result obtained generalises the results of Schechter, and Gustafson and Weidmann for Tm- compact (m> 1) perturbations of T. In the second part of the paper a detailed study is made of the Dirichlet integral

associated with the general second-order (degenerate) elliptic differential expression in a domain Conditions under which t is closed and bounded below are established, the most significant feature of the results being that the restriction of q to suitable subsets of Ω can have large negative singularities on the boundary of Ω and at infinity. Lastly some examples are given to illustrate the abstract theory.

A large class of self-adjoint fourth-order differential equations has the property that if one solution is oscillatory then all solutions are oscillatory. This paper establishes necessary and sufficient conditions for this property to hold for a corresponding class of non-self-adjoint differential equations.

Optimal switching curves are investigated for some second-order control systems with random disturbances, saturating control, and a mean-square-error performance index.

It is first shown that the optimal switching curves are asymptotic at infinity to the optimal switching curves for corresponding deterministic systems.

The systems are then discretised in time and state, and ordinary dynamic programming is applied to the resulting Markov chain models. The discretisation techniques are described in some detail.

The same techniques are applied to a deterministic system for which the optimal switching curve is known, and it is found that the resulting discretisation errors are considerable.

Thus it turns out that ordinary discrete dynamic programming gives only a rough approximation to the optimal switching curve. It seems desirable to tackle the equations of continuous dynamic programming by techniques more refined than crude discretisation. One such technique is suggested.

The paper presents a list of unsolved problems about operators on Hilbert space, accompanied by just enough definitions and general discussion to set the problems in a reasonable context. The subjects are: quasitriangular matrices, the resemblances between normal and Toeplitz operators, dilation theory, the algebra of shifts, some special invariant subspaces, the category (in the sense of Baire) of the set of non-cyclicoperators, non-commutative(i.e. operator) approximation theory, infinitary operators, and the possibility of attacking invariance problems by compactness or convexity arguments.

This paper develops the general theory of the propagation of Lipschitz discontinuities in first- and second-order partial derivatives of the initial data for a conservative quasi-linear hyperbolic system with discontinuous coefficients. After establishing that such weak discontinuities propagate along characteristics the appropriate transport equations are derived. The effect on this form of wave propagation of the strong discontinuity associatedwith the discontinuous coefficients is then studied and the transmission and reflection characteristics of the resulting waves are analysed. In conclusion, an application of this general theory is made to the propagation of plane shear waves through two different continuous hyperelastic solids to determine the transmitted and reflected waves.

A widely accepted model of nerve conduction in the squid axon is the systemof four non-linear partial differential equations developed by Hodgkin and Huxley. Under space clamp and current clamp conditions these equations are reduced to a system of ordinary differential equations.

We find that under appropriate assumptions on the functions and parameters in the resulting fourth order Hodgkin-Huxley equations there occurs a bifurcation of periodic solutions from the steady state. This bifurcation takes place as the current parameter, I, passes through a critical value.

The diffraction of a line source by an absorbing semi-infinite plane in the presence of a subsonic fluid flow is examined. Expressions for the total far field for the leading edge (no wake present) and the trailing edge (wake present) situation are given.

It is found that the presence of fluid flow attenuates the sound level in the shadow region of the absorbent screen. There is greater attenuation of the sound level in the trailing edge situation than there is for the leading edge situation.

Sound level attenuation on the source side of the absorbent screen is found to depend more critically on the absorptive properties of the screen than on the fluid flow. The radiated sound intensity, in the half space in which the source is located, can be considerably reduced by a suitable choice of the absorption parameter.

A d-dimensional zonotope Z in Ed which is the vector sum of n line segments is linearly equivalent to the image of a regular n-cube under some orthogonal projection. The zonotope in En-d which is the image of the same cube under projection on to the orthogonal complementary subspace is said to be associated with Z. In this paper is proved a conjecture of G. C. Shephard, which asserts that, if Z tiles Ed by translation, with adjacent zonotopes meeting facet against facet, then tiles En-d in the same manner. A number of conditions, conjectured by Shephard and H. S. M. Coxeter to be equivalent to the tiling property, are also proved.

This paper arises from an attempt to generalise the following result of [1].

THEOREM (Armitage-Fröhlich). If K is a cyclic extension of Q of degree lr, where 2 has even order mod l, then

In this paper we continue our investigation of the topological filtration on the complex representation ring R(G) of a finite group, see [4] and [5]. To recall the basic definitions from (1): let

map a k-dimensional representation ζ to the (flat) vector bundle over the classifying space BG associated to the universal G-bundle by the G-structure on Ck. Then, if denotes the (2k − l)-skeleton of BG,

Let x, y, Q denote large real numbers, with x > y > Q. The object of this paper is to prove the following result

for arbitrary A > 0, with a larger value of Q than hitherto. The symbol denotes as usual the suppression of an absolute constant, π(N; q, a) denotes the number of prime numbers up to N which are congruent to a (mod q), and ϕ(q) denotes Euler's function.

A centrally symmetric d-polytope (d-dimensional convex polytope) P in Euclidean space Ed is called k-neighbourly provided every subset of k vertices of P, which does not contain two opposite vertices of P, is the set of vertices of a (k − 1)-simplex which is a face of P. Contrasting the situation of neighbourly polytopes without the symmetry assumption (see, e.g., Griinbaum [1; chap. 7]), it appears that the possible neighbourliness properties of centrally symmetric polytopes are rather restricted. For d ≥ 2 and n ≥ 1, let k(d, n) denote the greatest integer k, such that there exists a k-neighbourly, centrally symmetric d-polytope with 2(d + n) vertices. McMullen and Shephard [4] have shown that , and for n ≥ 3. They conjectured that

Let K be an algebraic number field. By a. full module in K [l,p.83] we mean a finitely-generated (necessarily free) subgroup M of the additive group of K whose rank is equal to the degree [K : ℚ] of K over the rational field ℚ. The intersection of M with ℤK, the ring of integers of K, is also a full module I, and we shall concern ourselves chiefly with the latter, in that we wish to count the number of rational integers in a given interval which can be expressed as the norms of elements of I. More precisely, we shall adapt the methods of [2] to prove the following

We shall prove the following

THEOREM 1. Let α1, …, αn be any positive algebraic numbers and let u1…, un, ν be positive integers, relatively prime in pairs, such that ν ≥ 2 and ui > v for at least one i (1 ≤ i ≤ n). Then for every ε > 0 there are only a finite number of positive integers v such that the inequality

is satisfied, where for real α we understand by ‖α‖ the distance of α from the nearest integer.

Let q be a power of an odd prime, [q] denote the Galois field GF(q) and write X(x) = xq − x. Let f(x) be a polynomial, having no linear factors, over [q], of positive degree, and write . Consider the continued fraction expansions

and

where the Ai(x) and aj,(x) are polynomials over [q] of degree ≥ 1 (if i ≥ 1, j ≥ 1). Plainly A0(x) = ao(x). Suppose that n = nf is the integer denned uniquely as the largest m such that

In this note we derive an implicit representation of the solution to the problem of plane, inviscid, irrotational flow from a symmetric nozzle of arbitrary wall shape. For the case in which the nozzle wall has a slope which is everywhere much less than unity, we are able to convert this implicit representation into an explicit one in an asymptotic sense (based upon the smallness of the wall slope). Particular attention is paid to the contraction ratio of the jet. This work is complementary to that of Lesser [2] and Larock [l].

Let K, K′ be two centrally symmetric convex bodies in En, with their centres at the origin o. Let Vr denote the r-dimensional volume function. A problem of H. Busemann and C. M. Petty [1], see also, H. Busemann [2] asks:—

“If, for each (n − 1)-dimensional subspace L of En,

does it follow that

If n = 2 or, if K is an ellipsoid, then Busemann [3] shows that it does follow. However we will show that, at least for n ≥ 12, the result does not hold for general centrally symmetric convex bodies K, even if K′ is an ellipsoid. We do not construct the counter example explicitly; instead we use a probabilistic argument to establish its existence.

Davies and Rogers [5] constructed a compact metric space Ω which is singular for a certain Hausdorff measure μh, in the sense that all subsets of Ω have μh-measure zero or infinity and μh(Ω) = ∞. (For a further study of this example see Boardman [3]). The interest lies in its extremely good descriptive character, which was lacking in the earlier examples given by Besicovitch [2] (a plane set singular for linear measure) and Choquet [4] (a plane set singular for any Hausdorff measure for which a segment has positive measure).