Table of contents

The source of a Kerr space-time is generally assumed to be reducible to a disc, although the singularity in the solution is in the form of a ring. This is because the maximally-extended exterior manifold is multiply-connected, being formed by two sheets joined through the ring. However, it is possible, in principle, to construct 'minimal' sources (reducible to the singularity) for space-times with such topology. This is illustrated in the Kerr case with zero mass constant, and some possibly general features are indicated.

A recent energy lower bound method of Carr and Post (see abstr. A62988 of 1971) incorporates the full permutation symmetry of the N particle system, but the resulting estimates for the ground state energies appear to be inferior to those given by method I of Hall and Post for small N (1967) and to method II of Hall for large N (1967).

The authors study a one dimensional model system of fermions which proves to be exactly soluble in the thermodynamic limit. It is shown that the thermodynamic functions for the model system are the same as for a system of noninteracting fermions in a harmonic oscillator well. As a special case of the model system, with a particular choice for the coupling constant, it reduces to a particular two-species system of fermions interacting via pairwise harmonic forces. For this case of the two-species system the free energy per particle, the pressure the specific heat at constant volume, and the isothermal compressibility are calculated in the canonical ensemble in closed form. It is shown that this system does not undergo a phase transition.

Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v¹⁶, for the square lattice to v²¹ and for the honeycomb lattice to v³², inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.

For pt. I see ibid., vol. 5, no. 5, 624. Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K for the simple cubic lattice to v¹⁷, the body-centred cubic lattice to v¹⁵, and the face-centred cubic lattice to v¹², inclusive. The coefficients are analysed by the ratio method and it is estimated that, subject to the implicit assumptions thereof, the critical temperatures correspond to v_f=0.21813+or-0.00001 (SC), v_f=0.15612+or-0.00003 (BCC) and v_f=0.10174+or-0.00001 (FCC). The symptotic behaviour of the coefficients is studied in detail; for the loose-packed lattices the decay of the even-odd oscillation in the ratios is found to be consistent with the assumption that the critical index of the high temperature specific heat is very close to 1/8.

New data for the number of selfavoiding walks and selfavoiding returns to the origin on two and three dimensional lattices are presented and studied numerically by the ratio method. Estimates for the critical attrition and critical indices are given. For a loose-packed lattice the selfavoiding walk generating function appears to have a singularity on the negative real axis. This singularity is at the same distance from the origin as the physical singularity, and is found to be cusp-like with an exponent of 1/2 in two dimensions and 3/4 in three dimensions. This behaviour enables a close analogy to be drawn between the behaviour of the Ising model high temperature susceptibility and the walk generating function.

The numbers of n step selfavoiding returns to the origin are given up to n=14 for the face-centred cubic lattice, n=16 for the body- centred cubic lattice, n=18 for the triangular lattice, n=20 for the simple cubic lattice and n=26 for the square lattice, inclusive. The technique used to obtain the data is described in outline.

For pt. I see abstr. A37614 of 1967. High temperature series expansions for the specific heat of the Ising model of a ferromagnet are given for the face-centred cubic, body-centred cubic, and simple cubic lattices. From a numerical study it is concluded that the critical index ( alpha ) is lattice independent and that in three dimensions alpha approximately=1/8. A numerical representation of the specific heat in the range T_C<or=T<or= infinity is given in each case, together with estimates of the critical energy, entropy, and free energy. Asymptotic forms for the specific heat and energy are given.

A particular model of the entanglement of two polymers is studied in which one is taken to have a solenoidal configuration whilst the other is a random llk. The various classes of configurations are analysed, and the probability distribution of the random flight polymer is calculated in detail when it lies in the same topological class as the axis of the solenoidal polymer.

A general method of deriving the generating function of the photocounts due to gaussian light of arbitrary spectral profile is presented. The eigenvalues of the basic integral equation corresponding to a general real-valued autocorrelation function are related to the usual infinite product representation of the generating function of the photocounts. The integral equation is solved by the Laplace transform technique. The eigenvalues are determined by imposing analyticity requirement on the Laplace transform and the eigenvalues are identified to be the zeros of an alternant. By appropriate parametrization and multiplication by an alternant the eigenvalues are directly related to the zeros of an entire function. By the use of the Hadamard factorization theorem the generating function is identified to be the reciprocal of the entire function evaluated at a chosen point. The method is extended to cover the superposition of two incoherent beams.

A theory is described which accounts for the dependence of the spectral distribution of amplified spontaneous emission on the length and inversion density of the active medium. For an inhomogeneously broadened line an unanticipated dip occurs in the plot of linewidth against length at lengths slightly greater than the critical length, but is explained by a simple physical analysis. Agreement occurs between the theory and that of Yariv and Leite (1963) in the regime over which their theory is expected to work. Observations by other workers on the spectral distribution of amplified spontaneous emission systems are discussed.

The (1,0)(+)(0,1) massive spin one field phi _{mu nu} (x) is quantized following the method of Takahashi and Umezawa (1953). The interaction Hamiltonian in the interaction picture is then calculated for such a field interacting in a simple way with Dirac fields. In contrast with a remark of Kyriakopoulos (1969) the generalized Matthews' theorem is found to apply to the calculation of S matrix elements. The analogous theory in terms of the Proca field is given in outline, and the two theories are found to be inequivalent. A discussion of how the inequivalence arises is given; then it is demonstrated how equivalent theories may be constructed, and the equivalence is proved.

Regge poles of local interactions with a 1/r singularity at the origin are determined from a polynomial equation which can be solved extremely rapidly on a computer. This equation is found to be particularly suitable for locating the poles at negative energies, but impractical at positive energies. A simple formula is derived for the residue at a pole.

A Veneziano model is applied to higher spin production processes in the near forward scattering region over an extended energy range, for the processes K⁺p to K⁰ Delta ⁺⁺ and K⁺p to K*⁺p. Absorption corrections are included to take some account of unitarity, and the U(6,6) symmetry scheme is used to determine the coupling constants. While the results for K*⁺p are an improvement on previous Regge calculations, the results for K⁰ Delta ⁺⁺ are similar and suffer from the fact that the N Delta coupling constant is too small.

A Veneziano model, which agrees to leading order in s with the Reggeized U(6,6) model, is applied to KN and KN charge-exchange scattering. Leading order 'duality-preserving' cuts are introduced and an improved agreement with low energy K⁺n to K⁰p data is obtained.

An efficient anticoincidence screen has been designed and operated in the Holborn underground laboratory at a depth of 61 hg cm^-2 below the surface. A series of experiments, using different types of counter inside this screen, has been performed to study the properties of the underground nonionizing radiation at energies greater than 5 MeV. It is established that the leakage of muons through the screen was very small. The rate of neutron knock-on events was found to be 1.8 l^-1 d^-1 for protons of energy greater than 12 MeV. The rate of events due to photons of energy greater than 10 MeV was about 15 l^-1 d^-1. The origin of these photons is discussed, and it is argued that most of them are due to the bremsstrahlung of electrons knocked on by muons. No evidence is found for any anomalous behaviour of the cosmic radiation underground.

Measurements on the three photon decay component in silicone fluids indicate that there is a detectable three photon yield contrary to what is observed in most other liquids. Degassing of the fluids increases the three photon yield. There is little or no effect on the three photon yield in silicone fluids with change in molecular weight.

Submillimetre-wave measurements of the power absorption coefficient alpha and refractive index n of room temperature solutions of p-difluorobenzene in nonpolar solvents are reported. A room temperature detector was used and careful precautions, which included phase modulation, have given very small estimated uncertainties: as low as +or-0.1% in alpha and +or-0.02% in n. Some variations with molar concentration are given.

Colgate (1968) has suggested that an exploding supernova should release an intense burst lasting a few microseconds of x and gamma ray photons. When it impinges on the upper atmosphere this burst produces a corresponding pulse of optical emission by fluorescence. A search has been made for such light pulses, using a system of coincident light receivers. Although many pulses having the appropriate temporal characteristics were observed, their other properties appear to rule out their having an origin in supernova explosions. The character and possible origin of these events are discussed and their intrinsic interest is pointed out.