The state of knowledge of fast neutron polarisation due to elastic scattering is reviewed in relation to the optical model of the process. There is clearly a substantial lack of polarisation data at neutron energies and concerning nuclei appropriate to the optical model description. The situation for the extremes of small and of large angle scattering is also considered.

Let m(λ) denote the Titchmarsh–Weyl m-function associated with the differential equation

with Neumann boundary condition at 0. In the case q ∊ CM[0, ε) for some integer M and ε > 0 we prove the following result (Theorem 2.1): If for some integer ŋ ≧ 3 we know that Im {λ(ŋ + l)/2m0(λ)} = 0 on the ray λ = |λ| eiπ/ŋ, then q(v) (0) = 0 for all v in the set {0, 1, … M} for Which ≢ ŋ − 2 mod ŋ.

This paper is concerned with the problem of showing uniform stability and equiasymptotic stability of thezero solution of functional differential equations with either finite or infinite delay. The investigations are based on Liapunov's direct method and attention is focused on those equations whose right-hand sides are unbounded for bounded state variables.

We consider the class of Noetherian UFN-rings, that is, Noetherian prime rings such that every non-zero ideal contains a non-zero normal element and such that the monoid of non-zero normal elements is a unique factorisation monoid. We ask whether this class of rings, a generalisation of the commutative Noetherian unique factorisation domains (UFD), is closed under polynomial extensions. The general question is apparently difficult and remains open. However, we give positive answers in special cases, in particular, for algebras over an infinite field and for domains.

Upper and lower bounds for the solutions of a nonlinear Dirichlet problem are given and isoperimetric inequalities for the maximal pressure of an ideal charged gas are constructed. The method used here is based on a geometrical result for two-dimensional abstract surfaces.

Sufficient conditions for oscillation of solutions to third order hyperbolic characteristic initial value problems are established. The results generalize known oscillation criteria for second order hyperbolic problems.

By using a mild hypothesis on the acoustic tensor, it is shown that the disturbances in a nonlinear elastic body travel with a finite speed. Moreover, a uniqueness theorem for the displacement problem of nonlinear elastodynamics is proved. No assumption is made on the extension of the body.

Two non-singular conies ω and α are said to be related by a pentagram if there exist pentads ofdistinct points {Oi} on ω and {Ai} on α (1 ≦ i ≦ 5) such that A1 ≡ O2O4. O3O5, A2 ≡ O3O5. O1O4, A3 ≡ O1O4. O2O5, A4 ≡ O2O5. O1O3 and A5 ≡ O1O3. O2O4. It is shown that relation by a pentagram is a poristic property; and a necessary condition on their mutual projective invariants that two nonsingular conies be so related is derived. Some ramifications are discussed.

Page 58: Lemma 7.1 is incorrect. It is claimed that Ao =cocl for c =k−l when in fact the result is proved for c = (dim H)−1. Thus while Corollaries 7.4 and 7.6 still stand (as does Corollary 7.3 by alternative arguments) the remaining results in Section 7 must be withdrawn.

A description is given of the rational functions A(X), B(X) over a field Ω for which A(B(X)) is an nth power in Ω(X).

The location of the trajectory of a charged nuclear particle can be determined by a number of devices whose operation depends on characteristics of gas counters operated in the region of proportional energy response. Such devices have advantages in precision, speed or available size over other types of position-sensitive counters. The events may be located by observing pulses on one of a set of closely spaced anode wires in a common gas volume, by measuring the drift time of primary electrons to a particular anode wire, or by determining the location of an electron avalanche along the length of a single anode. The operation of the different types of systems and the factors affecting their precision are described. Precision to less than 1 mm can be obtained in a number of ways,

The properties of convex representatives of polyconvex functions W : R2×2 → R are studied. The structure of the largest convex representative ϕW : R5 → R is given, and this is used to derive a representation formula for ϕW when W is C1 and satisfies a mild growth condition. The regularity of ϕW is discussed when W is C1; in particular, when W is C1, grows super-quadratically and is strictly polyconvex, it is shown that W has a C1 convex representative.

The equation {r(x)y'}' = q(x)y is considered with the coefficients q and rtaking complex values. Conditions are obtained for there to be solutions with the asymptotic form , . These involve the coefficients and their derivatives up to some order N.

Some second order semilinear elliptic boundary value problems of the Ambrosetti-Prodi-type are studied. Existence and multiplicity of solutions is proved in dependence on a parameter. Constructing a global strongly increasing fixed point operator in a suitable function space, observing - under appropriate conditions, which are in some sense optimal–that the fixed point operator has some properties similar to a strongly positive linear endomorphism, one unifies and improves the treatment of such problems, whether the nonlinearity is dependent on the gradient or not, and obtains some new results.

The question of the existence of true transition resistance at the surface of certain electrodes in given liquids has been in debate for many years, but has never yet been conclusively settled. It is admitted universally that in certain cases such resistance does occur, as, e.g., when a non-conducting oxide is formed on the surface of the metal; but the point in dispute is whether or not it occurs when no direct chemical action takes place between the metal and the products of electrolysis, or between it and substances dissolved in the liquid.

At long intervals notices of circular crystals have appeared before this Society. In 1853 Sir David Brewster read a paper on the subject, which followed one by Mr Fox Talbot in 1836, and which again had been preceded by one from Sir David Brewster about twenty years before. It is not easy to account for these long intervals, unless they may be attributed to difficulty and uncertainty in manipulation, for except in very few instances the crystals observed by Sir David Brewster are of microscopic size, and, he remarks, require the perfection of optical appliances for their observation, and naturally so when crystals of the 200th of an inch in diameter are looked upon as of respectable size.

The author commenced by stating that although these researches lead him to believe in the existence of two planets revolving in orbits external to that of Neptune, and although there was a great deal of evidence to show that he had actually determined the elements of the orbits, yet the latter point, being dependent on a coincidence of probabilities only, cannot be considered a certainty until the planets are observed.

The author accepts the theory of cometary orbits which supposes that these bodies, wandering through space, are attracted by the sun into the solar system so as to pursue parabolic orbits, and that some of these, in passing a planet, may have their velocity diminished, in which case they will afterwards describe an ellipse in a definite period.

Let B(ℤ)* be the Banach dual of the space of all bounded complex-valued functions on ℤ. For each n ε ℤ, let Ln be the translation operator on B(ℤ) and Tn be its adjoint operator on B(ℤ)*. This paper concerns itself with equations of the form

where (an)nεℤ is a sequence of complex numbers.

In a paper communicated to this Society last session, I pointed out that the proof of Euc. I. 5, given by Mr Mill, is unsound; endeavouring, at the same time, to show that this is no mere accident, but that it is impossible to give a mathematically correct analysis of the processes of Synthetic Geometry on any theory that holds figures to be merely illustrative, and does not admit that intuition in the Kantian sense—i.e., actual looking at a single engraved or imaginary figure—may be a necessary and sufficient step in a demonstration perfectly general.

Almost-everywhere convergence of wavelet transforms of Lp-functions under minimal conditions on wavelets was proved by Rao et al. in 1994. However, results on convergence almost everywhere do not provide any information regarding the exceptional set (of Lebesgue measure zero), where convergence does not hold. We prove that if a wavelet ψ satisfies a single additional condition xψ(x) ∈ L1 (R), then, instead of almost-everywhere convergence, we have a more sophisticated result, i.e. convergence of wavelet transforms everywhere on the entire Lebesgue set of Lp-functions. For example, wavelets with compact support, used frequently in applications, obviously satisfy this extra condition. Moreover, we prove that our conditions on wavelets ensure the Riemann localization principle in L1 for the wavelet transforms.