The relationship between the modulation of electron beams at optical frequencies (the Schwarz-Hora effect) and at microwave frequencies is discussed. At optical frequencies the interaction between the modulating field and the electron beam must be described quantum mechanically, although the field itself may be described classically; in the microwave case the process may be described entirely classically. The interaction modifies the state functions of the individual electrons, but the observable modulation of the total electron beam results from the coherence of the modulating field. The main features of the Schwarz-Hora effect result from ‘single photon’ processes, but the beam modulation in the klystron is a ‘multi-photon’ process.

The exponential decay of the depth of modulation with distance from the interaction region, observable in the optical frequency case but not in the klystron, is not an inherently quantal effect. The periodic variation of the depth of modulation with distance along the beam, observed in the optical frequency case, is an essentially quantum mechanical effect, and is different, in its origins and in its dependence on the modulation frequency, from the space-charge waves which are observed on a klystron beam.

In this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.

Measurable selection theorems are proved, for a compact-valued measurable multifunction into a Hausdorff space that is the continuous image of a separable metric space, and for a closed-valued measurable multifunction from a suitable measure space to a regular Souslin space. The connection between Polish spaces and certain subsets of the real line is related to a measurable selection theorem for multifunctions into a Polish space.

The usual ‘closed-graph theorems’ are concerned with the question of which topological structures on sets X, Y will ensure equivalence of continuity and closed-graph conditions for certain mappings t: X → Y. In this paper some stronger closed-graph like conditions are introduced on a mapping between uniform spaces, by considering the ‘hypergraph’, or graph of the induced mapping between hyperspaces. The central result, included in Theorem 1, states that for arbitrary uniform spaces X, Y, t is continuous if its hypergraph is closed. Thus for topological vector spaces (and some others) the closed-hypergraph condition is equivalent to continuity. In section 2 some situations are found in which continuity is implied by certain intermediate conditions to closed-graph and closed-hypergraph conditions. In particular, a closed-hypergraph theorem is shown to hold for locally convex spaces, provided only that X is barrelled. Finally, in section 3, the hypergraph of a relation is studied, and the separatedness of the quotient space of an equivalence relation is shown to replace continuity with regard to closed-graph and closed-hypergraph conditions.

The propagation of scalar waves in a certain two-dimensional medium is considered. The incident field, which is due to the presence of a line source, is scattered by two coupled half-planes on each of which the impedance takes a constant value. The Wiener-Hopf technique is used to find a solution which is then examined asymptotically for high frequency. It is found that there is an illuminated region in which the solution is expressed in terms of geometrical optics rays, and a shadow region in which the solution is described by creeping modes. The point of impedance discontinuity may be regarded as producing secondary radiation. The nature of this secondary radiation is quite different according as the point of impedance discontinuity lies in the illuminated or shadow region of the geometrical optics field produced by the source.

For a certain class of operators in the direct product of two Hilbert spaces, two problems are solved: the inhomogeneous operator equation, and the eigenvalue problem. Illustrative examples are given.

Acenaphthenequinone (I) condensed with ethyl cyanoacetate in ethanol to give the unsaturated ester (IIIb) while condensation with malonic acid in toluene in the presence of diethylamine gave the hydroxy acid (IIa). Esterification of this acid gave an ester (IIb) which could also be obtained by condensation of acenaphthenequinone

with ethyl hydrogen malonate. The acid (IIa) on dehydration gave 2-oxo-Δ1,α-acenaphtheneacetic acid (IIIa) of m.p. 230°C. This result seems to be in conflict with that of Rodionov and Federova (1950) who reported a m.p. of 160° for the acid (IIIa) which they obtained directly by condensation of acenaphthenequinone with malonic acid in ethanol in the presence of ammonia. Our efforts to repeat their result gave only impure polymeric material and it seems unlikely, therefore, that their product was simply the geometric isomer of our acid.

α-Truxillic acid (1, R = CO2H) (Criegee and Höver 1960) underwent the Arndt-Eistert reaction to yield l,3-diphenylcyclobutane-2,4-diacetic acid (1, R = CH2·CO2H), cyclisation of which with hydrogen fluoride gave cis-trans-cis-5,6,6a,6b, 11,12,12a, 12b-octahydro-5,11-dioxodibenzo[a,g]biphenylene (11). Attempts to aromatise the compound by bromination with N-bromosuccinimide followed by treatment with pyridine or potassium acetate were unsuccessful. The ketone was therefore reduced with potassium borohydride to the corresponding diol (11, CO replaced by CHOH) in the hope that dehydration of the product followed by mild dehydrogenation would yield dibenzo[a,g]biphenylene, but the only product isolated was naphthalene.

The propagation of scalar waves in a certain stratified medium is studied; the field is due to a line source situated on an opaque plane boundary. The exact field can be expressed in terms of a Fourier integral involving Airy functions. By deforming the real axis, which is the contour of integration of the Fourier integral, only in the neighbourhood of the real axis, it is possible to give a simple but rigorous derivation of the asymptotic nature of the field. Two separate cases are considered: the point of observation lying in the illuminated region, when a steepest descents analysis is appropriate, or lying in the shadow region, when the asymptotic field is given by a residue series.

The suggestion is made that in certain cases the light particles emitted in ternary fission may be released otherwise than in the ground state—and, in particular, that low-lying particle-unstable states may be involved. This suggestion is examined quantitatively in respect of the release of 8Be in the first excited short-lived state of 2·9 MeV excitation, and is shown to provide a satisfactory interpretation of the observation of quaternary fission in the α,α mode recently reported by Kataria et al. (1973). It appears probable that the other modes of quaternary fission observed by these authors should be similarly interpreted.

The sharing of a common self-polar simplex by n−1 quadrics in [n] confers special features on their curve of intersection Γn. The three-dimensional locus Mn of chords of Γn. has certain singularities, some of which are described. In conclusion, a few comments refer to the case n = 4 when Mn is defined by a single equation.

Differential operators generated by the differential expression My(x) = —y″(x)+q(x)y(x) in L2(0, ∞) are considered. It is assumed that

is bounded for all x in [0, ∞) and some fixed ω > 0. The operators are shown to be bounded below and an estimate for the lower bound is obtained in terms of q(x). In the case where q(x) is LP (0, ∞) for some p ≧ 1, the results are compared with recent ones of W. N. Everitt. Some comments are made on the best-possible nature of the results.

This paper is devoted to a study of differential equations and inequalities of the form

and

The results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.

The analysis of the previous paper is extended, taking account of the spread of Q-values in the disintegration 5He→He+1n. Conclusions in respect of the experimental results of Cheifetz et al. (1972) remain essentially unchanged.

In this paper we determine the structure of some new types of ordered inverse semigroup in which the ordering need not be the natural ordering. In particular, we generalise some results of McFadden and O'Carroll on F-inverse semigroups.

The fourth-order equation considered is

Conditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).