Table of contents

If ( delta n/ delta t)_c is the coagulation rate of change of the distribution function n(v) for a system of particles, it is shown that integral ₀^infinity v( delta n/ delta t)_c dv is not necessarily zero. Rather, depending on the coagulation kernel and the behaviour of n(v) as v to infinity , it may be zero, non-zero and finite, or infinite. This has implications for the asymptotic behaviour of the solution of the steady state, time-independent coagulation equation. The results are applied to Brownian and gravitational coagulation.

The problem of long-range percolation in one dimension is proposed. The authors consider a one-dimensional bond percolation system with bonds connecting an infinite number of neighbours where the occupation probability for the nth nearest-neighbour bond p_n varies as p₁/n^s. Using the transfer-matrix method, they find that when s>2 only the short-range percolation exists; namely the system percolates only when p₁=1. A transition to long-range percolation is found at s=2 where the percolation threshold drops suddenly from the short-range value p₁^c=1 to the long-range value p₁^c=0.

The excess defect free energies of interface-like defects in Potts models on hierarchical lattices are calculated exactly. For q=2 (Ising model) the depinning transition takes place. For q>2 the free energies show anomalies which may be interpreted as the intrusion of the non-boundary phase into the interphase region, in agreement with recent Monte Carlo simulations.

The author studies site lattice animals using the cluster renormalisation (CR) approach previously applied to bond animals. The exponent v, characterising the asymptotic behaviour of the mean-square radius of gyration of site animals, and the fractal dimension d_f=1/v, are determined in two and three dimensions. He finds v=0.649+or-0.009 and v approximately=0.51 in two and three dimensions, respectively. In addition he studies restricted valence site animals on a triangular lattice using a single-parameter CR approach.

A direct evaluation of the lowest-order WKB integral is carried out for the linear plus Coulomb potential V(r)= mu r-( beta /r). The implicit relation for the bound-state energies defined by the WKB quantisation condition is expressed in terms of complete elliptic integrals. An approximate non-perturbative inversion of this relation provides an explicit analytic expression for the energy which reproduces known numerical results quite well. A criterion for the applicability of the approximation scheme is given.

The structure of possible single mass equations for an antisymmetric tensor-bispinor is examined. It is shown that all spin-³/₂ single mass equations are reducible.

The authors give the Lie point symmetry groups of a class of nonlinear diffusion equations.

Representations of Lie superalgebras may be realised as functions (superfields) on graded manifolds (suitably chosen onset spaces). The technique is illustrated for OSp(1/2) and OSp(2/2) with a review of well known results, and applied in the cases of OSp(3/2) and OSp(4/2) to construct some classes of irreducible representations.

There have been many analyses of Pearson random walks with equal step sizes. Several applications suggest the importance of these walks with unequal step sizes. Numerical comparisons are made of approximations obtained using the central limit theorem and the steepest descents method. These suggest that the latter generally, but not always, leads to more accurate results. The authors also derive an approximation that is extremely accurate at near-maximal extensions.

The perturbed ladder operators method is applied to the resolution of the perturbed harmonic oscillator wave equation for the case where the perturbation is expandable in a convergent series of Hermite polynomials H_k, when V(X)=b²X²+ Sigma _kC_kH_k(b¹2/X). It is found that the use of a Hermite polynomials basis, together with the use of binomial coefficient functions in the quantum number, greatly simplifies the determination of the perturbed ladder and factorisation functions. Thus, one obtains analytical expressions of the eigenenergies and eigenfunctions up to any order of the perturbation, without increasing intricacy. Thorough calculation has been given for a perturbing potential V(X) function even in X. As an illustrative application of the procedure, the resolution of the Schrodinger equation with a potential function V(X)=X²+ lambda X²/(1+gX²), g>0 is reinvestigated.

In certain situations the Feynman path integral 'collapses' into a countable sum over classical paths. This effect is sought in motion confined to a sector (in polar coordinates: r>or=0, 0<or= phi <or= alpha ). The exact space-time propagator is derived as a sum over a general nonclassical image set, with resulting structure radically dependent on number-theoretic properties of pi / alpha . It is shown that collapse occurs if and only if this number is an integer.

A manifestly gauge-invariant formulation of quantum mechanics is applied to a charged isotropic harmonic oscillator in a time-varying magnetic field in the magnetic dipole approximation. The energy operator for the problem is the sum of the kinetic and potential energies. The kinetic energy operator is the square of the gauge-invariant kinetic momentum operator divided by twice the mass. The energy eigenvalues and state probabilities are calculated and are shown to be the same in all gauges. In this problem there is no gauge in which the energy operator reduces to the unperturbed Hamiltonian, as there is in the electric dipole approximation. Consequently, eigenvalues of the unperturbed Hamiltonian and corresponding (gauge-dependent) state 'probabilities' are different from the gauge-invariant quantities in all gauges.

It is shown that positive quantum joint distributions in phase space lead to quantum mechanical operators depending on the state to be considered. It is also shown that a simple requirement of canonical invariance leads to the Weyl correspondence rule as the only allowed one.

The authors generalise previous work (Christensen and Duff 1979) on a simple set of arbitrary spin field equations for fields propagating on curved manifolds without torsion. As a prelude to an extensive study of index theorems and anomalies in the case of non-zero torsion, new arbitrary spin field equations are developed and the consistency conditions discussed. Some of the machinery needed for future investigations of arbitrary spin fields is developed using the two-component spinor formalism.

The author exhibits classical solutions of a graviton-graviphoton-graviscalar field theory which are antigravitating in the weak-field approximation. The theory itself is obtained by a Kaluza-Klein type reduction from five to four dimensions. The solutions are dyonic black holes with scalar charge. They share some similarities with the extreme Reissner-Nordstrom black holes of Einstein-Maxwell theory.

The author presents some old and many new exact solutions of the Einstein-Maxwell equations for the Bianchi type-III and Kantowski-Sachs space-times in a unique parametrisation. Solutions are given for the case of stiff matter and dust with (and without) a cosmological constant.

Multiple-time perturbation theory is applied to the Lorenz equations to obtain approximate solutions. These solutions are used to obtain the form of the mapping relating two successive maxima of one of the variables of the equations. Good agreement is obtained with the numerical results presented by Lorenz.

The author considers directed percolation on the square lattice, with probability p_H(p_V) for the horizontal (vertical) bonds to be unbroken. For p_H=1- epsilon ( epsilon small) and p_V> sigma ( epsilon ) the percolating cluster is asymptotically within a cone phi ₊< phi < phi ₊. The author calculates phi _+or- and the fluctuations of the boundaries at phi = phi _+or- as power series in epsilon , up to terms approximately epsilon ², showing that the transverse spread of the percolating cluster is random walk-like. For any given p_H the percolation threshold p_V,c( epsilon ), defined by phi ₊= phi _- at p_V=p_V,c is also calculated.

For each value of pi in (¹/₂,1), the author constructs a subgraph R( pi ) of the square lattice Z² such that the bond percolation process on R( pi ) has critical probability pi . It is shown that the critical probability pi (a) of the region ((x,y):0<or=y<or=a ln(x+1), 0<or=x< infinity ) depends in a non-trivial way upon the choice of the number a; pi (a) is a continuous, decreasing function of a and satisfies pi (a) to ¹/₂ as a to infinity and pi (a) to 1 as a down arrow 0. A family of 'one-dimensional' bond percolation processes is also constructed, whose critical probabilities range over the interval (0,1). All proofs are rigorous.

Critical percolation densities have been found numerically for various systems of lines uniformly distributed in the plane. The average number of intersections per line at percolation has also been found and varies only slightly over the cases considered. It may therefore provide a useful rule of thumb for deciding whether a system percolates. An estimate of the critical percolation density from the lattice percolation probability is presented. Possible extensions of the techniques described to three dimensions are discussed.

A physically evident requirement on asymptotic product states is formulated in a mathematically precise way and shown to fix the S-matrix uniquely for relativistic field theories of short-range interactions.

Phase transitions manifest themselves as strong softening in the equation of state. The collapse dynamics of neutron stars and stellar cores is considered. The transition of a neutron star to a pion-condensed star can be accompanied by an ejection of a small mass fraction for certain ultradense equations of state. The collapse outcome of stellar cores is not influenced by a phase transition in nuclear matter below nuclear density.

For both identical and distinguishable particles and for an arbitrary polynomial approximation to the realistic two-body forces, the A-body Hamiltonian may be converted into an infinite block-three-diagonal matrix in the properly arranged, translationally invariant, oscillator basis. As a consequence, the exact Green function and all the projections of eigenstates of the microscopic Schrodinger equation become expressible in terms of the matrix continued fractions. This generalises the recent reformulation of the one-dimensional A=1 anharmonic problem by Graffi and Greechi (1975). The quick convergence of this non-perturbative method of solving the many-body bound-state problem is demonstrated for the simplest three- and four-body examples. With the core-possessing type V(r)=-r²+r⁴ of the spin- and isospin-independent two-body interaction, only the 10^-3 error in the three-bosonic ground-state energy arises from restricting the formulae to 5*5 dimensional matrices.

The Fredholm integral equation method is applied to the scattering of electromagnetic waves by finite dielectric cylinders of circular cross section. The method is applied to a wide range of diameter/length ratios, including both rods and discs. Generally good agreement has been obtained with the experimental backscatter measurements of McCormick and Hendry.

Correct expressions for integrals given in a recent letter by Exton (ibid., vol.14, p.L75, 1981) are found.

One-dimensional site and bond percolation problems with bonds connecting Lth nearest neighbours are solved exactly by using an infinite cell to cell renormalisation group transformation. Both 'thermal' and magnetic critical exponents are obtained through the scaling relations. Although universality breaks down for site and bond percolation when L>or=2, Suzuki's weaker universality (1974) still holds.