Table of contents

Using generating function techniques, the authors construct integrity bases for O(3) scalar operators in the enveloping algebra of G, where G is one of the subgroups of SU(6) in the three dynamical chains for the interacting boson model.

A Ward identity in a broken supersymmetric model involving a supersymmetry current is studied using analytical regularisation. The authors show, in the one-loop approximation, that this Ward identity is satisfied if the amplitudes are regularised by the analytic regularisation method.

Two first-order differential operators are introduced to generate recursion formulae for hypergeometric functions. These operators, on the one hand, factorise the associated second-order operator and, on the other hand, reproduce the equations resulting from the action of the supersymmetry generators on the positive-energy solution of the Coulomb field. The results for the confluent hypergeometric functions also provide a natural basis for constructing ladder operator recursion relations for the Coulomb Green function with the outgoing wave boundary condition.

The analysis of a sum of modified remnant functions, P_{sigma , tau} ^alpha (z)=¹/₂(R_{sigma , tau} ^alpha (Z)+R_{sigma , tau} ^{- alpha} (z)), where tau =0, sigma is half-odd integer, 0<or= alpha <1 and z complex, is presented. An analytical expression for z not=0 and mod arg z mod < pi is obtained as well as a power expansion valid for all z. The behaviour of P_{sigma , tau} ^alpha (z) as mod z mod to infinity and z to 0 is also obtained.

The trace method which has been proposed by Wadati and Sawada (1980) is applied to the sine-Gordon equation. The N soliton solution is derived.

Reciprocal Backlund transformations are introduced for nonlinear integro-differential equations in (2+1) dimensions. Invariance under such transformations is investigated.

Numerical simulation of two-dimensional bond percolation at the percolation threshold shows that configurations with no singly connected bonds (SCB) appear with a finite probability. Nevertheless, the average minimal width of the percolation channels is small. This result confirms an assumption used to derive the critical exponent of the critical current density in superconductor-insulator composites. Higher-order bonds are shown to play the role of SCB in configurations with small numbers of SCB. Implications of the properties of low-connectivity bonds on the physical properties of percolating systems are discussed.

The crossover between invasion percolation (IP) and the Eden model is studied in one-dimensional. Although the mean run length diverges in IP, it converges in the Eden model and for a broad class of perturbations to IP. In addition, IP has anomalous fluctuation effects which are absent in the Eden model. By studying a specific family of models which interpolate between the IP ( alpha =0) and Eden ( alpha =¹/₂) limits, we show that the behaviour of the variance crosses over to that found in the Eden model for any alpha >0.

A model of cavity radiation is proposed in which the stimulated emission is modelled as an age-dependent birth process. The photon population statistics, under certain special conditions, are shown to possess thermal characteristics to second order.

Species live in an environment where nutritious and poisonous food is randomly distributed. A survival criterion is analysed based on how poor on average the food can be.

The critical exponent zeta _k of the kth moment of the current distribution for random diode insulator networks on the square and simple cubic lattices is calculated for a range of k using low density series expansion techniques. It is also shown that zeta ₁= nu /sub /// where nu /sub /// is the critical exponent of the parallel connectedness length for directed percolation The values of zeta _k for the simple cubic lattice are well fitted by a simple exponential formula with ( zeta _k-1)/( zeta _k+1-1)=1/2. The Skal=Shklovskii scaling relation for the conductivity is generalised to the kth moment and it follows that the exponent K describing the divergence of flicker noise is given by kappa approximately=(d-1) nu _{perpendicular to} + nu /sub ///+ zeta ₄-2 zeta ₂. The authors estimate this for both lattices.

The authors apply finite-size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic, directed cyclic and undirected bond percolation on a triangular lattice. The results are in good agreement with known estimates, and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases, as well as the distinction between locally directed and fully isotropic problems. In addition, the authors draw tentative comments on the extension, to directed problems, of the relation A= pi eta (where A is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point), known to hold for various isotropic systems.

The authors evaluate numerically the percolation threshold of a 3D assembly of widthless monodisperse discs (radius R, number per unit volume N) in terms of the quasi-invariant NR. The value obtained is consistent with previous estimates. This model should apply to the permeability problem of fractured rocks.

It is shown that the disorder solutions exhibited by two-dimensional statistical lattice systems verify the star-triangle relation. These solutions of the star-triangle relation are of a different type than those corresponding to completely integrable models.

The authors present new results which indicate that the leading correction-to-scaling exponent in the mean squared end-to-end distance in two dimensions is the analytic term. A potential source of the various correction-to-scaling terms reported in the literature is pointed out.

The authors obtain the zeros of the partition function for the isotropic triangular lattice three-state Potts model in a finite lattice. The distribution exhibits new points on the real axis which are well fitted by an algebraic equation deduced from the inversion relation and the symmetries of the anisotropic model. They identify a symmetry approximately relating all six transition points.

The authors have obtained the diagrams and weights for the thirteenth and fourteenth terms of the high temperature multigraph expansion for the free energy of general Ising systems. As an application we have computed the coefficients through fourteenth order for the square lattice with nearest- and next-nearest-neighbour interactions.

A bond-diluted Ising model is used to simulate a process from a Cantor set to a translationally invariant lattice. The free energy function shows that there is no singularity, which seem to imply that no transition exists in this process.

Expositions of classical thermodynamics frequently include the so-called zeroth law amongst its 'fundamental principles'. It is shown here that, given only the first law and the second law (the latter in a formulation manifestly free of any explicit or implicit reference to temperature), the transitivity of the relation 'is in diathermic equilibrium with' can be deduced. The zeroth law which is an assertion of just this transistivity is therefore redundant. The existence of the absolute temperature function of course emerges directly, i.e. without appeal to a prior empirical temperature.

A heat exchanger consists of a helicoid and the surface of revolution obtained by rotating a profile curve about the axis of the helix. The surfaces are rigidly welded together where they meet. This limits the number of profile curves for which the heat exchanger may be infinitesimally deformed. These profiles curves and the deformations they allow are determined by two arbitrary antiperiodic functions.

The author shows that the conditions of application of Opechowskis theorem for double groups of subgroups of O(3) are directly associated to the structure of their commutator groups. Some characteristics of the structure of classes are also discussed.

The authors describe a method, analogous to the Elliott method (1958) for O(3) contained in/implied by U(3), for resolution of the multiplicity in the decomposition into irreducibles of the tensor product of two irreducible representations of a compact semisimple Lie group G. The method is based on a decomposition of highest weight vectors in the product representation into direct product states, focusing on components of the form e(X)e₊^mu with e a weight vector and e₊^mu a highest weight vector. The weight of the vector e corresponds to the 'shift' weight in a tensor operator formulation of the problem. A result from an earlier paper (Gould et al. 1984) based on the fact that a tensor product representation can be generated cyclically from the product of a highest and a lowest weight vector is used to give an explicit characterisation of the space of shift weight vectors e that can appear in the decomposition of a highest weight state. This characterisation is in terms of lowering operators in the complexified Lie algebra L of G, and closely parallels Verma's well known enveloping algebra characterisation (1968) of the highest weight states of finite-dimensional irreducible representations of complex semisimple Lie algebra.

For pt.I see ibid., vol.19, p.1523 (1986). Methods introduced in previously (Edwards et al. 1986) for resolving the multiplicity of irreducible subrepresentations occurring in the decomposition of the tensor product of two irreducible representations of a compact semisimple Lie group are illustrated by application to U(3), U(4) and general U(n). For U(3) the authors rederives very simple the known multiplicity structure for an irreducible tensor operator of fixed shift weight in terms of the decomposition of tensor product highest weight vectors into certain direct product states. The use the method to illustrate structural parallels between the Clebsch-Gordan problem for general U(n) and the U(3) case, Finally, they study in detail the multiplicity structure for a specific U(4) irreducible operator showing both the similarities and differences with U(3).

For pt.II see ibid., vol.19, p.1531 (1986). This paper is the third in a series directed towards a projection-based solution to the Clebsch-Gordan multiplicity problem for semisimple (compact) Lie groups. The authors show that the project states for the Clebsch-Gordan problem approach orthogonality in the classical limit of large quantum numbers in a manner analogous to that of Elliot's (1962) well known solution to the U(3) contains/implies O(3) state labelling problem.

For pt.I see J. Math. Phys. vol.12 p.45. The statistical distribution of the angular momentum states J of a many-particle system j^N may be approximated by a Wigner-type distribution involving two parameters. It is shown that these two parameters may be determined group theoretically to yield simple expressions involving just j and N. A connection with the compositions of integers in number theory is noted.

The convergence of the Trotter formula for the one-dimensional harmonic oscillator is studied. Pade approximants improve the convergence.

Hass, Velicky and Ehrenreich (1984) have shown that Green function values can be obtained appreciably faster by performing the calculation at complex energies with large imaginary parts, and then returning to real energies by an analytic continuation algorithm. However, analytic continuation is numerically unstable and thus an error analysis is essential if one is to have confidence in the results of the continuation. Error estimates are obtained for the power series method of Hass et al. and for a method based upon Cauchy's theorem introduced herein. These estimates can be used to select appropriate values for the parameters of the continuation algorithms.

The radial part of many scattering waves can be expressed in terms of Whittaker functions M_{kappa , mu} and/or W_{kappa , mu} , and integrals over products of these waves are required in the analysis of many processes. Integrals over n Whittaker functions of the first kind are generalised hypergeometric series which often have variables outside the range of convergence, or are such that the series are slowly converging. Using matrix series for the Whittaker functions, analytic continuations of the resulting generalised hypergeometric series have been obtained which always result in convergent series apart from a few special cases involving degeneracy. In addition, an asymptotic matrix series for the integral from some radius R to infinity of products of whittaker functions of the second kind is given. As an example, the Dirac-Coulomb matrix elements arising in bremsstrahlung are evaluated.

Partial radiative capture cross sections for very low energy (<100 eV) electrons and protons are calculated via the classical method of Fourier components. Comparisons of these cross sections are made with quantum mechanical calculations.

The rate of convergence of the operator method iterative series as a function of the variational parameter is investigated numerically. The optimal method of calculation of this parameter is proposed.

A complete system of three-particle hyperspherical harmonics (HH) with orbital momenta L<or=4 is constructed in an explicit form with separated rotational degrees of freedom. Formulae for any L values are given. Permutational and other properties of HH and a method of HH orthonormalisation are considered.

The problem of first passage time is mathematically equivalent to the tunnelling of a particle out of a potential well and hence can be treated by the multi-dimensional WKB technique. In general, the most probable tunnelling path (MPTP) is curved and is difficult to determine. The authors discuss a model potential for which the curved MPTP can be solved analytically. It is shown that the rate of tunnelling calculated along the straight path as usually assumed is unreliable. Numerical results of the mean switching time for the bistable two-mode laser and comparison with experiments and other theories are also discussed.

Recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem. The averaged cost function is calculated and agrees well with numerical results. The problem bears close resemblance to that of spin glasses. The authors find a spin glass transition in the system, and the low temperature phase space has an ultrametric structure. This sheds light on the nature of hard computation problems.

Nonrelativistic fermions whose motion is constrained by an infinite potential step (hard wall) are considered. Relative to the bulk, a net fermionic charge is induced at the wall due to the vanishing of the many-particle wavefunctions. The charge, linear charge density and the fluctuations from their mean values are calculated.

The authors consider a vector-valued stochastic process, which is multiplicatively driven by the Markov jump process. They obtain a closed expression for the average of the vector process by solving the Burshtein equation for the marginal average. It is shown that the solution for t>t₀ requires knowledge of an initial correlation operator, due to the finite correlation time of the jump process. They derive the equation for the steady-state solution, which is applied to evaluate the stationary correlation functions. Then some specific limits of the jump process, are discussed, which arise if one takes the correlation time to be zero or infinite. For the special case L(x)=A+xB they derive from the Burshtein equation a recurrence relation between the moments of the vector process. This relation is solved and it is shown how all initial moments at t₀ determine the moments for t>or=t₀. The recurrence relation is applied to solve the two-state and the three-state process more explicitly. It is pointed out that the occurring initial correlations cannot be neglected in general.

Two models of anisotropic spiral self-avoiding random walks recently proposed by Manna (1984) have been investigated by the method of exact series expansions. The number of such n-step walks, c_n, appears to behave like c_n approximately constant* mu ⁿn^beta exp( alpha square root n) where both mu , which is known exactly, and the constant factor are model dependent but alpha approximately=0.14 and beta approximately=0.9 appear to be model independent. The mean square end-to-end distance exponent nu =0.855+or-0.02 for both models.

For pt.I see ibid. vol.19 p.411 (1985). A graph theoretic analysis is made of the m-spin correlation functions of the lambda -state Potts model. In paper I, the correlation functions for m=2 were expressed in terms of rooted mod- lambda flow polynomials. The authors introduce a more general type of polynomial, the partitioned m-rooted flow polynomial, which plays a fundamental role in the calculation of the multispin correlation functions. The m-rooted equivalent transmissivities of Tsallis and Levy (1981) are interpreted in terms of percolation theory and are expressed as linear combinations of the above correlation functions.

The authors present Monte Carlo simulations of the spreading of bond percolation on three- and four-dimensional simple (hyper-)cubic lattices. The algorithm is the same as that applied previously in two dimensions, with the spreading proceeding from a hyperplane. They find the spreading dimension to be d=1.82+or-0.02 (three-dimensional) and d=1.88+or-0.03 (four-dimensional). Also they obtain values of the percolation probability and of the static exponents with errors at least comparable to the best values from the literature.

The critical behaviour of a three-dimensional Ising model, bounded by two plane surfaces meeting at an angle alpha , is studied using the Migdal-Kadanoff and two-terminal cluster renormalisation methods. The authors obtain expressions for the edge fixed point and for the independent edge exponent y_he at the various transitions. They find that y_he is not completely universal but depends on the angle alpha , and that it shows some unexpected features.

The Ising model on the centred square lattice (Union Jack lattice) is studied by Vdovichenko's method, for a more general case than Vaks et al. (1966). The reentrant transition of the system is interpreted in terms of effective interactions. A method of associating a factor -1 to some corners of a loop in Vdovichenko's method (1965) is adopted, by which one can obtain the expression of the free energy by calculating a finite number of determinants which consist only of elements of integers.

An attempt to generalise the combinatorial solution of the two-dimensional (2D) nearest-neighbour (NN) Ising model to three dimensions is reported. A generating function, which fully exposes the symmetry of the lattice, is derived for the 2D square Ising model. A 'natural' extension to the 3D simple cubic (SC) lattice is shown to be false. A certain circumstance for the 2D case leads to the conjecture that tanh (J/kT_c)=( square root 5-2) cos( pi /8) for the SCNN Ising model.

Within a real space renormalisation group (RG) scheme, the authors study the criticality of the ferromagnetic Z(4) model on an anisotropic square lattice. They use an RG cluster which has already proved to be very efficient for the Potts model on the same lattice. The establishment of the RG recurrence relations is greatly simplified through the break-collapse method. The phase diagram (exhibiting ferromagnetic, paramagnetic and nematic-like phases) recovers all the available exact results, and is believed to be of high precision everywhere. If the model is alternatively thought of as being associated with a particular hierarchical lattice rather than with the square lattice, then it is exact everywhere.

The authors introduce a biased diffusion model of aggregation at a surface, which reduces to a ballistic model in one limit. They characterise the structure of the aggregates by a variety of properties and find that it is a strong function of the parameter governing the diffusion process. For thin films there is a crossover between a regime where there are several highly ramified pseudo-one-dimensional clusters and another regime where there is a single cluster which spans the lattice in both directions.

The singular travelling wave solution of a linearly damped double sinh-Gordon system has been obtained. It is shown that the solution is linearly stable.

The (1+3)- and (1+1)-dimensional Dirac equation with both scalar-like and vector-like potentials is discussed. The authors prove that if the scalar-like potential is just equal to the vector-like potential, the confinement is impossible, i.e. there must be scattering states. Two exact solutions with linear potential and harmonic oscillator potential in this condition are given.

Using the Metropolis Monte Carlo scheme the authors have analysed and compared the behaviour of the two-dimensional classical planar and step models. In particular, they analyse the energy, specific heat, correlation function and susceptibility on a range of finite lattices. While both models show similar numerical difficulties, it is argued that in the case of the step model a phase transition is unlikely.

A new class of planar fractals called the Pascal-Sierpinski gaskets is described, of which the well known Sierpinski gasket is a special case. Some of these gaskets are true Mandelbrot fractals possessing non-integral dimensions as well as self-similarity; the remaining ones are not self-similar, but appear to have nonintegral dimensions.