Table of contents

The authors demonstrate that the Lax representation for the cylindrical Korteweg de Vries equation can be obtained with the help of the Painleve property of this equation.

The Laplace transforms of the functions Ai(+or- eta ) and Bi(- eta ) are obtained using the integral definitions of these functions and a correction made to a recent derivation of the results of Ai(- eta ) and Bi(- eta ).

Using 1-1 mappings, the complete symmetry groups of contact transformations of general linear second-order ordinary differential equations are determined from two independent solutions of those equations, and applied to the harmonic oscillator with and without damping.

It is shown that the massless λϕ⁴ theory is structurally unstable, and quantum fluctuations cause symmetry breaking and the unfolding of the cusp catastrophe.

Analytical expressions of the atomic fine and hyperfine structure parameters in a space of constant curvature have been obtained by use of a ladder operator technique. It is found that the additional curvature contributions to the classical (flat) expressions increase with n.

It is argued that looking for a relationship between the exponent s and the cluster exponents may be more promising than looking for one between the resistor-insulator exponent t and the cluster exponents. The author proposes s= nu - beta /2.

An exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold. The authors use this approach to calculate exactly the first low-density expansion of L_BB(p) for arbitrary system dimensionality d, where L_BB(p) is the mean of backbone bonds and p is the bond occupation probability. Standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d; these disagree with the Sierpinski gasket model of the backbone. They also calculate the first low-density expansions of L_min(p) and L_red(p) which are, respectively, the mean number of bonds in the minimum path between i and j and the mean number of singly connected ('red') bonds.

The authors present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions. The results for the fractal support the Rammal and Toulouse proposition (1983) that dS(N)/dN varies as B(N)/S(N). The results for percolation are in excellent agreement with the Alexander and Orbach conjecture (1982) that the fracton dimensionality d=⁴/₃.

A particle performing a self-avoiding random walk is considered on a lattice in any number of dimensions d, which contains a fraction q of randomly distributed impurity sites. An impurity is assumed to trap the walker when stepped on. The author finds the average time to trapping to be lambda (1-q)/(q-(1-K_c)), where K_c and gamma are the inverse critical temperature and the critical index of the susceptibility for the classical Heisenberg model of magnetism with vanishing (n to 0) internal degrees of freedom.

The author presents a new approach for determining the Weyl invariants of unitary groups. Such an approach is based on the generating function of the basis of the representation for unitary groups built in an earlier work (Hage Hassan 1983). The author obtains two expressions for the Weyl invariants, both including the parameters involved in the generating function. Coupling coefficients appear in one of the expressions. The comparison of the development of the two expressions allows the calculation of the coupling coefficients. The application of this approach to the calculation of the coupling coefficients for the direct product ( lambda ¹0)(X)( lambda ² mu ²) of SU(3) leads to an expression of the isoscalar factor where no summation occurs in contradistinction with the known expression for the coupling coefficients which involve five summations.

A family of groups P^(m,n) parametrised by non-negative integers m>n is studied generalising the Poincare group P^(1,1). The action of finite-dimensional irreducible real representations of SL(2, C) is used to form semi-direct products. The authors define the complete list of unitary irreducible representations for each P^(m,n) by finding all sub-groups of SL(2, C) which are little groups; some of the subgroups do not occur in the Poincare group. The geometry of the SL(2, C) action and the classification of SL(2, C) invariant tensors is considered. These groups are appropriate symmetry groups for field theories in higher dimensions and generalise the notion of elementary relativistic quantum systems.

The authors consider the symmetry groups for Burgers-type and KdV-type equations, as well as their intertwining under the Hopf-Cole and Miura transformations. It is shown that Lie point symmetries of the one equation in the family may be mapped into integral or Backlund transformations of another equation in the family.

Some necessary and sufficient conditions for a critical point of the action integral to be locally or globally extremal are proved. Applications to systems with finite or infinite number of degrees of freedom are discussed.

An expression is derived for the spatially dependent Green function or response function in a solid occupying a half-space and having generally varying properties. The expression is valid in the same range as the WKB approximation, namely for slowly varying properties. Some generalisation is indicated and the time-dependent Green function or the resolvent is obtained as a Fourier transform.

The energy eigenvalues of harmonic oscillators in circular and spherical boxes are obtained through the Rayleigh-Schrodinger perturbative expansion, taking the free particle in a box as the non-perturbed system. The perturbative series is shown to be convergent for small boxes, and an upper bound for the radius of convergence is established. Pade-approximant solutions are also constructed for boxes of any size. Numerical comparison with the exact eigenvalues-which are obtained by constructing and diagonalising the Hamiltonian in the basis of the eigenfunctions of the free particle in a box-corroborates the accuracy and range of validity of the approximate solutions, particularly the convergence and the radius of convergence of the perturbative series.

The authors show that, contrarily to what Flessas and Anagnostatos claim (1982), the so-called Hill determinant method is perfectly suited to the numerical resolution of Schrodinger's equation.

The study of the evolution of the moments over a cluster of identical classical particles without mutual interactions is extended to the case where the motion of each particle is described by an arbitrary Hamiltonian. In the case of moments of the second and third order and ignoring contributions from moments higher than the third, differential equations are derived for the moments as well as explicit expansions. A distinction is made between the evolution of moments relative to the centroid and that of moments relative to a particle with initial position and velocity equal to those of the centroid. Although the former moments are more directly applicable and are independent of the choice of initial time, the latter have formal advantages. It is shown that these two types of moments differ in terms of order not less than four.

For pt.I see ibid., vol.15, p.409 (1982). The author continues the canonical formalism of paper I for the motion of a free particle. By the use of a computer the relevant matrices necessary for the computation of all Dirac brackets are established. An algorithm is described which allots numerical values to the canonical variables satisfying the constraints, and these values are used in the construction of the matrices. Algebraic expressions for Dirac brackets are predicted from a first set of numerical data, and these predicted expressions are confirmed by at least two further sets of data. Reduction to a sufficient set of variables is attempted and the reduced set expressed as a Lie algebra.

The problem of elastic wave propagation in a random layer overlying a homogeneous half-space is treated using an averaging method and the Born approximation. The wave is assumed to propagate normally to the layer so that the equations of motion are ordinary differential equations with random coefficients The two methods are shown to have different ranges of validity; the averaging method applies in the range where stochastic effects are important.

To avoid the internal contradiction in the pilot wave interpretation inherent in the assumption of a charged particle moving in a closed orbit without radiating, it is proposed that the wavefunctions, psi _n(r), corresponding to stationary states be real, implying that the particle is stationary. If the effect of radiation damping is included, the Hamiltonian contains a nonlinear term resulting in the stability of the stationary states.

The Mandelstam-Tamm time-energy inequality is exploited to obtain a transparent expression of the lifetime-energy uncertainty relation for decaying quantum systems along with some useful features of the quantum non-decay probability.

The differential geometry of admissible wavefronts of N-body scattering is investigated for each term of the multiple scattering series and within angular sectors of uniform asymptotic behaviour. The differential equations are dimension independent in form and thus investigated in R_n. A general theorem is proved stating that, if not spherical, the wavefront must be a (ruled) K=0 surface tangent to a sphere around the origin. For n=3 a rather simple geometrical description is given of the most general structure.

The exact space-time propagation for a general reflectionless potential is obtained in closed form involving error functions. It is indicated how bound states and transmission coefficients may be recovered from asymptotic behaviour of the propagator. A sum rule is derived that shows how the leading terms of the short-time form of the propagator can be used rigorously in a Feynman path integral formalism.

Following two recent papers by Bonnor (1980) and Raychaudhuri (1982) on the motion of charged dust in general relativity, the authors obtain exact solutions of the Einstein-Maxwell equations in both cylindrically symmetric and axially symmetric cases.

The authors show that a recently found non-minimal version of N=1 Poincare supergravity with (28+28) degrees of freedom is reducible to the non-minimal Breitenlohner set, in agreement with an earlier result that there exist at most five irreducible versions with auxiliary fields of spin up to one.

If a new criterion of reducibility for auxiliary field sets is employed, the authors show that the two new non-minimal sets for N=1 supergravity are reducible.

Using a method of dimensional reduction from an on-shell multiplet in ten space-time dimensions the authors give the supersymmetry transformation laws of a general irreducible representation of N=4 SUSY with central charges present. With more than one central charge an extra condition is used to give a finite number of components and they give the field transformations and Lagrangian for the case of two central charges.

The authors analyse the spectrum of multiplets with two central charges which vanish on-shell, and show that the general spin-reducing constraint still leaves an infinite number of propagating modes of the same mass. They perform an analysis by deriving field equations from actions involving integration over the central charge dimensions. The implications of the results for bypassing the N=3 barrier are briefly considered.

A new field-theoretic formulation of the Fokker-Planck approach to non-equilibrium statistical mechanics is presented. Starting with the nonlinear functional Fokker-Planck equation, a new generating functional is derived. No use of auxiliary conjugate fields or response functions is needed. The Feynman rules are deduced, and the renormalisation of the theory is carried out. Finally, the renormalisation group equation is solved, and scaling laws and critical exponents are calculated, which are in good agreement with previous results obtained through different formalisms.

A direct renormalisation method is used to calculate the index v which characterises the mean square displacement of a walk on a one-dimensional, anisotropic, homogeneous lattice: (R_n²)^1/3 approximately n^v for large number of steps n. The two-parameter, exact analysis is carried out for arbitrary choice of the cell size. The results are shown to be independent of the cell size.

The percolation problem for the complement of the union of randomly located, overlapping spheres is shown to be equivalent to a bond percolation problem on the edges of the Voronoi tesselation of the sphere centres. This result provides a convenient definition of cluster size, and therefore of the critical exponents, for this problem. It also provides an efficient algorithm for Monte Carlo computation of the percolation threshold and the critical exponents.

The recently recognised class of hierarchical lattices is examined through a number of examples. Definitions of length, intrinsic dimension and connectivity are made and used to discuss variations of exponents calculated on the Ising model. Generalisations of regular lattice results are found for exponents at discontinuity fixed points and in 1+ epsilon dimensions.

A method of obtaining the latent heat at a first-order transition from finite-lattice matrix elements is discussed, and demonstrated for the case of the (1+1)D Potts model. The method is shown to provide smoothly convergent finite-lattice sequences. The structure of the finite-lattice Hamiltonian eigenvalues is exhibited, and a characteristic signal of the first-order transition is proposed. The finite-size scaling behaviour at the transition is also discussed.

The author studies a recently developed, Pade-type method of series analysis which allows for the leading confluent singularity. A number of properties are described and by applying the method to various test series some pitfalls of the technique are revealed.

Exhibits, on the known solutions of the triangle relation for the exactly solved models, some simplifying methods of recovering their parametrisation in terms of algebraic varieties. The relation to the automorphy group, generated by the inverse and spatial symmetries of the model, is also analysed.

The authors investigate the asymptotic behaviour of the number of trees having n vertices, weakly embeddable in the d-dimensional hypercubic lattice, with restrictions on the number (n⁺) of vertices with degree greater than two. They show rigorously that if n⁺=o(n/log n) the growth constant is equal to the corresponding quantity for self-avoiding walks. For n⁺ increasing linearly with n they show that the growth constant still exists and present arguments indicating that it is strictly greater than that for self-avoiding walks.

Gives solutions of the polydisperse Smoluchowski coagulation equation delta f(x,t)/ delta t=¹/₂ integral ₀^xf(y,t)f(x-y,t)a(y,x-y)dy-f(x,t) integral ₀^infinity f(y,t)a(x,y)dy, f(x,0)=g(x) for arbitrary g(x) when the coagulation kernel a(x,y)=A+B(x+y)+Cxy. The solutions are given as recursions and infinite series and are practical for computation. For the given kernels the author also gives the gelation times t_g at which M₂(t)= integral ₀^infinity x²f(x,t)dx becomes infinite.

High-temperature series expansions are presented for the magnetisation M(K,h) of the spin-s Ising model on the face-centred cubic lattice. Series coefficients are derived to order 13 and 5 respectively in the variables K=J/kT and h=mH/kT, for ¹/₂<or=s<or=⁹/₂. The data are analysed for 2 Delta - gamma , where Delta and gamma denote the gap exponent and the susceptibility exponent respectively. The estimate 2 Delta - gamma =1.890+or-0.003 satisfies the data for all s.

This note discusses the possible use of perturbation methods in studying chaotic trajectories of ordinary differential equations, with particular focus on a recent paper on this topic by Rowlands (ibid., vol.16, p.585, 1983).

Series expansion analysis has been used to obtain estimates of the percolation probability exponent, beta , for directed site percolation on the square, triangular, simple cubic and body centred cubic lattices. These estimates are consistent with those obtained by Please (1977) for the corresponding bond problems and the results for site and bond problems on all lattices considered are summarised by beta =0.28+or-0.01 (d=2) and beta =0.59+or-0.02 (d=3).

The interdimensional scaling laws of second-order phase transitions are modified so as to apply to the percolation threshold. Interesting results are obtained for some of the geometric exponents as well as for some conductivity exponents in the case of a metal-insulator or superconductor-normal conductor composite or a random resistor network.