The existence and the properties of the solutions of a nonlinear differential system describing a diffusion system with concentration-dependent diffusion coefficients are investigated. It is shown that there exists at least one solution which is monotonie decreasing and asymptotically tends to the prescribed value.

This paper deals with the nature of movable singularities of solutions of Emden's equation

at which the solution becomes infinite. If m = 1 + 2/p with p > 1 an integer, then the solution becomes infinite at a given point x = c as

By the general theory of P. Painlevé on movable poles of solutions of non-linear second order differential equations this ‘pseudo-pole’ cannot actually be a pole of order p. Instead of a bona fide Laurent series at x = c we obtain a series expansion of the form

where Pn(t) is a polynomial in t of degree at most [n/(2p + 2)]. The object of this paper is to derive these series and to prove convergence for p = 2. In this case deg [P6m] is strictly equal to m. For other values of p, see Section 8, Addenda.

In this paper we consider the effect of the passage of shallow water waves over vertical walled objects on a flat seabed. The effect of reflections at the successive walls is taken into account when determining the place of breaking of the wave. The results are obtained by the introduction of special reflection and transmission coefficients at each wall and recurrence relations are formulated connecting their values at successive walls. The effect of this is to reduce the calculation of the place of breaking of the wave over a stepped seabed to an equivalent one for a wave over a flat seabed, the result of which is well known.

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.

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.

In this note it will be proved that any small relatively bounded perturbation of a self-adjoint operator with discrete spectrum also has discrete spectrum.

1-Nitro-, 1-bromo-, and 1-acetamidofluoranthene undergo bromination to yield 4,9-dibrominated products. 1-Acetamidofluoranthene also yields 1-acetamido-4,8,9-tribrofluoranthene and as a side product 1-acetamido-4,8,9,x-tetrabromofluoranthene, from the former of which 1,4,8,9-tetrabromofluoranthene is obtained and found to be identical with the product resulting from the interaction of 1-bromofluoranthene with three molecules of bromine.

The asymptotic behaviour of double integrals over a portion of the plane is investigated when the integrand contains an exponential factor with a large parameter. The exponent can have a stationary point which may or may not be close to the boundary of the domain of integration. Results are first derived for a rectangular region with a particularly simple exponent in the integrand and shown to be uniformly valid under certain conditions. In some circumstances the asymptotic terms can be evaluated by means of a universal function. The theory is then generalised to cover more complicated exponents and arbitrary wedge-shaped domains; it is found that the asymptotic behaviour can still be expressed in terms of the same universal function.

In this paper we discuss the structure semigroup of the L1-algebra of an Arens-Singer semigroup. Arising from this study we provide a complete and rather unexpected description of a nontrivial structure semigroup. We then link the above ideas with that of the almost periodic compactification of the semigroup.

The suggestion is made, and rendered plausible by detailed calculation, that the short-range α-particles of fission, identified by Kugler and Clarke [1], are emitted, in competition with ‘prompt’ γ-rays, from highly excited post-neutron-emission fragments of even neutron number N and, predominantly, of mass number A in the range 140 ≦ A ≧ 145. A similar process is energetically forbidden in relation to triton emission.

This paper presents a generalisation of earlier results on the dimension of the space of integrable-square solutions of the ordinary linear differential equation Su = λTu, where S and T are formally symmetric ordinary differential operators and λ is a spectral parameter.

It is the purpose of this paper to make a study of the solutions of the following k-formally self-adjoint differential equations

where ar, br), r = 1, 2, …, k, denote k semi-open intervals in which ar is finite and br is arbitrary and the λs, s = 1, 2, …, k, are spectral parameters.

The main theme of the paper is that of extending the Hermann Weyl limit-point, limit-circle theory to the multi-parameter case. That is we consider under which circumstances there exist, for each r, one or two solutions yr(xr) of (*) which are square integrable in a suitably defined Hilbert space Hr. This is then generalised to consider the problem of investigating the possibility of the product

of solutions of (*) being square integrable in H, the tensor product of the separate spaces Hr. The analyticky of the corresponding generalised Hermann Weyl coefficients mr(λ1, λ2,…, λk), r = 1,…, k, is also investigated. Some examples illustrating the theory are given and an alternative formulation of the problem is suggested.

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.

There is now strong evidence, from observations of directional correlation between the α-particles and neutrons of fission, for the release of 5He as a neutron-unstable light particle with a frequency of a few per 104 low-energy fission events (Cheifetz et al. 1972).

Because the energy spectrum of the neutrons liberated in the disintegration of 5He particles travelling with a fixed kinetic energy is a rectangular spectrum (assuming isotropy of disintegration directions in the centre-of-mass space), it is in principle possible to deduce the kinetic energy spectrum characterising the 5He particles of ternary fission at the moment of disintegration from the energy spectrum of the liberated neutrons.

Detailed calculations have been made in relation to the limitations of actual experimental arrangements with regard to this possibility. These limitations are shown to be severe—not least in respect of statistical accuracy. Indeed, it is shown that considerable difficulty attaches to the ‘simple’ evaluation of the 5He/4He ratio (which Cheifetz et al. estimate as 1 :(8 ± 2) for 252Cf, for α-particles of energy greater than 9 MeV).

An instrumental cause is suggested for the maximum reported by Cheifetz et al. in the energy spectrum ascribed to the 5He disintegration neutrons.

The propagation of sound across a vortex sheet separating two fluids, with the same density and sound speed, which are in supersonic relative motion is examined for both harmonic and impulsive sources. It is found that instability waves are present when M < 2√2, M being the Mach number, and it is suggested that these may account for the phenomena observed in certain experiments. For M > 2 there are also neutral stability waves and these lie in regions in the supersonic stream which are otherwise silent when the excitation is an impulsive source.

The curve Γ common to two quadric surfaces has 24 principal chords; they are the sides of six skew quadrilaterals each of which has for its two diagonals a pair of opposite edges of that tetrahedron S which is self-polar for both quadrics. These three pairs of quadrilaterals serve to identify the three pairs of quadrics through Γ that are mutually apolar.

The vertices of the quadrilaterals lie four on each edge of S. Both nodes of the plane projection of Γ from such a vertex are biflecnodes.

The propagation of sound across a vortex sheet separating two fluids in relative subsonic motion is examined for both harmonic and impulsive sources, the sound source being in the moving fluid. It is found that instability waves arise in a certain region depending on M, the Mach number.

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, ∞).

The first part of this paper is concerned with the elastic response of an isotropie infinite plane containing a central circular inhomogeneity to a deformation such that in the homogeneous infinite plane the associated elastic displacement is the gradient of a harmonic function. Based upon the Papkovich stress-function approach, it is shown that it is possible to represent all solutions to such problems by means of only one family of equations. The second part of the paper is concerned with the elastokinetics of two bonded dissimilar half-planes subjected to any uniformly moving body force. It is again shown that there exist simple relations between components of the elastic field in the composite plane and those of a particular field in the homogeneous infinite plane.

It is shown that, from an operational point of view, the production and detection of spin-polarised electrons by Mott scattering is equivalent to the performance of a Stern-Gerlach type experiment with heavy spin-one-half particles. The argument is based on standard quantum mechanical formalism using density matrices and Stokes vectors for the description of polarised assemblies of particles.