Table of contents

It is shown that reciprocal Backlund transformations may be used to reduce an important class of nonlinear boundary value problems for higher-order nonlinear diffusion equations to linear canonical boundary value problems.

The ground-state problem for discrete one-dimensional systems with interactions between n-nearest neighbours is connected with 2n-dimensional maps. Fixed points of these maps correspond to periodic system configurations. For a general class of systems it is explicitly shown for the first time that fixed points with some elliptic character cannot represent ground states.

The magnetic-field scaling relation between the coordinates, momenta and the energy in the Hamiltonian for the Kepler motion in a uniform magnetic field, i.e. H( gamma ^-13/p, gamma ²3/r; gamma ¹3/m, gamma =1)= gamma ^-23/H(p,r; m, gamma ), where gamma =B/B₀ is the normalised field strength and m the constant value of the angular moment,um B-parallel component is shown to ensure a representation of the semiclassical energy spectrum: E=(n+¹/₂)^-2f( gamma ¹3/(n+¹/₂), gamma ¹3/m), m=0, +or-1, ...; n=0, 1, .... The authors discuss how the function f, in the two-dimensional approximation (z=p_z=0, z//B), can be constructed.

Monte Carlo studies in three dimensions confirm the hypothesis of Toulouse and Rammal that the number of distinct sites visited by a random walk on a random network at its percolation threshold varies as (time)²3/.

The author analyses the chain generating function for the self-avoiding walk problem on the honeycomb lattice. She uses the exact critical point value of Nienhuis and finds exponent values of gamma approximately 1.344 and Delta ₁ approximately 0.93 for the dominant and first confluent exponents. The former value is in agreement with Nienhuis' exact result. Second and third confluent exponents, Delta ₂ approximately 1.2 and Delta ₃ approximately 1.5, are also identified, these exponents being consistent with Monte Carlo results and Nienhuis' exact irrelevant eigenvalue, respectively.

The authors consider the scaling behavior of the radius of gyration for a system of dilute linear polymers. They focus on rho _N, the mean-square end-to-end distance of an N-step self-avoiding walk for which an additional two terms were recently calculated for the close-packed triangular lattice. Combining several extrapolation methods, they find that a consistent description of the scaling behaviour exists if and only if the correction-to-scaling exponent Delta is roughly half as large as commonly believed. They conclude that all data are consistent with the equation rho _N=AN^{2 nu} (1+B/N^Delta +C/N) where nu =³/₄, Delta equivalent to ²/₃, A equivalent to 1/ square root 2, AB equivalent to 0.21 and C<10^-1.

The authors introduce the cumulant method into the context of position-space renormalisation group. The advantage of this method over previous procedures is that one need not systematically evaluate all 2^N states of each Kadanoff cell, where N is the number of elements (sites or bonds) in the cell. They illustrate this method by calculating, for bond percolation with an eight-bond cell, the critical exponents characterizing the following quantities: (i) the mean number of backbone bonds, (ii) the mean number of bonds in the minimum and maximum paths along the backbone, (iii) the mean number of singly connected ('red') bonds, and (iv) the mean number of bonds in all the self-avoiding walks connecting the two extreme points. They derive previously obtained results of the conductivity and resistivity.

A generalisation of the star-triangle transformation is used to establish a correspondence between a nonplanar Ising lattice with two interaction constants and the standard triangular Ising model. The critical point of the nonplanar lattice is obtained exactly for various values of the ratio of the interaction constants. The singularity in the zero-field free energy of the nonplanar lattice is investigated. It is shown to be of the same form as that displayed by the corresponding standard Ising model.

The structure of a recently introduced droplet in the q-state Potts model is analysed. The authors derive exact relations from which it follows that the incipient infinite droplet at the critical temperature is a self-similar fractal made of links and blobs, just as recently found in random percolation. The number of links N_links between two points separated by a distance of the order b is given by N_links varies as b¹ nu B(q)/ where nu _B(q) is the connectedness length exponent when T=T_c(q) and p_B is used as an independent variable. This result and the available calculations of nu _B(q) indicate that the number of links decreases as q increases. An analysis of the structure of the usual clusters made of nearest-neighbour sites in a given Potts site configuration is also made. In particular, it shows that for d=2 the incipient infinite cluster is made only of blobs and no links.

Using Monte Carlo methods the author studies the two-dimensional three-stages Potts model with antiferromagnetic nearest neighbours and ferromagnetic next-nearest neighbours couplings on a square lattice. The author focuses on the case with only antiferromagnetic coupling; from the analysis of the data gathered, there is evidence that this system shows critical behaviour only at T=0.

Jennison has described an historic experiment in which he succeeds in bringing an electromagnetic wave to rest in the laboratory. It is here pointed out that the electrostatic field could not be produced by static charge in the laboratory; the concept of the formation of such charge from the original wave therefore appears to be somewhat misdirected.

By examining the branching rules for all irreducible representations of the classical groups U(k), SU(k), SO(2k+1), Sp(2k) and SO(2k) on restriction to U(1)*U(1)*U(1), standard Young tableaux are specified for each of these groups. It is shown that these tableaux determine the corresponding characters of the irreducible representations. The rules for constructing these tableaux are derived and in this way the determination of weight multiplicities is reduced to a simple combinatorial exercise. General formula for such weight multiplicities are given encompassing the most difficult case: namely that of SO(2k). Illustrative examples are provided, including some yielding the explicit k-dependence of weight multiplicities.

Generalising results by Reed/Simon and Thirring, the author proves for a large class of Hamiltonians (among others, Hamiltonians of Coulomb systems) which can be written in the form H( alpha )=H₀+ alpha H' that their eigenvalues decrease with increasing alpha . The author applies this result to Coulomb systems in which the distances between the infinitely heavy particles are varying and also use it to obtain a completion and simplification of Adamowski et al.'s proof for the stability of the biexciton.

The even- and odd-parity eigenvalues of the bounded pure λx^2m oscillators are obtained as the roots of two equations derived explicitly.

For pt.I see ibid., vol.15, no.9, p.2785-79 (1982). The implications of intrinsic symmetries on the structure of rotational secular equations in the angular momentum projection formalism are studied for both the exact and approximate versions. The symmetry group of the rigid rotor, D₂, is used to illustrate formal results.

Exact centre-of-mass separation is performed for the problem of a charged two-body system in a homogeneous magnetic field. Common eigenstates of three exact and approximate constants of motion are constructed in the creation and annihilation operator formalism. Their expression as an infinite series in coordinate representation is obtained using two real linear canonical transformations. An expansion of the wavefunction in this eigenstate basis provides an infinite system of differential equations for the relative motion. This system of equations differs from the infinite nucleus mass approximation by the existence of small terms which couple states with different values of the magnetic quantum number. The usual limitation of the number of electron Landau states leads to a finite number of coupled equations.

Some simple closed-form approximations are derived for the oscillatory part of the Wigner function F for a system of degenerate fermions completely filling the first M shells of an oscillator potential in N dimensions. The formalism for a general dimensionality is retained since various properties of the physical case (N=3) may be related to similar systems in one dimension higher or lower. In particular the magnitude of the oscillations increases with N, but it is shown that if the last shell is only half filled the structure is decreased to that obtained for a closed-shell system in one dimension lower. The structure is shown to decrease slowly in the 'classical' or 'macroscopic' limit of large particle number, when the density is shown to tend to its Thomas-Fermi value. The formula for the full-shell case is sufficiently simple to generalise to the problem of almost degenerate fermions at a finite temperature T, where it is seen that substantial damping of the oscillations may be obtained for any particle number. An explicit expression for the damping is derived and this implies that for deep-inelastic nuclear reactions at a temperature of around 2 to 3 MeV, considerable structure may still be present in F. A simple expression relating the Fermi energy and the particle number of the system at finite temperature is given.

The time-evolution operator associated with an atom in a circularly polarised radiation field can be factorised as U(t)=exp(-i omega J₃t)exp(-iHt), where J₃ is the component of the total angular momentum orthogonal to the two field components and H can explicitly be given in terms of the atomic Hamiltonian. The author establishes for a one-electron model the self-adjointness of H and in addition he develops a complex dilation theory for this operator. The purpose of this approach is to obtain an analytic continuation theory for multiphoton processes so that the imaginary parts of the complex eigenvalues of the complex dilated Hamiltonian H( zeta ) can be related to the various rate constants that govern such processes.

A general treatment of the quantal harmonic oscillator with variable mass is given. Various examples additional to those obtained by Colegrave and Abdalla for which a closed form solution is possible are given.

A real symplectic flow may be viewed as being pseudo-unitary, provided there exists a symplectic complex structure J which commutes with it. The author determines the class of dynamical systems for which J exists. Such systems may be quantised algebraically but the state space may have indefinite metric. This procedure admits a larger class of classical systems than does Segal's algebraic quantisation but there are problems in interpreting the indefinite metric and these are listed and discussed.

A method proposed recently by Singh (1981) of finding lower bounds to the eigen-energies of Schrodinger's equation is reviewed and compared with the methods of Bazley and Fox (1961) and of Lowdin (1965). Numerical results for an anharmonic oscillator suggest that the Singh method is less accurate than the others, but unlike the Lowdin single bracketing function technique does not suffer from any eigenvalue ordering problems.

For pt.II see ibid., vol.11, p.1501 (1978). Suggests constructing an inverse of the infinite-dimensional matrix Q by means of its recurrent algebraic decomposition into the easily invertible two-diagonal factors. The merits and feasibility of the method are illustrated on the five-diagonal matrices. Quartic and decadic-decadic anharmonic propagators are chosen as examples of the application.

Formulae for calculating CAM pole positions and corresponding S-matrix residues are derived within the framework of the phase-integral method of Froman and Froman (1965, 1974). The formulae allow for the use of higher-order approximations with rigorous error estimates. The contributions from two and three isolated tuning points are considered, which allows applications in the study of rainbow and diffraction features as well as orbiting phenomena in elastic scattering processes.

By introducing a physically acceptable, supplementary condition for the field functions in unified field theory, the electric and magnetic monopole solutions free from singularities everywhere were deduced. This indicates that the divergence difficulties of a point particle can be removed by utilising a reasonable Boal-Moffat condition (1975).

Let M be a space-time which is asymptotically flat at past null infinity I^-, satisfies the dominant energy condition and contains a trapped surface T. The authors show that if T can be connected to I^- by means of a nonsingular null-hypersurface N, then m²>or=A/16 pi where m is the Bondi mass with respect to N and A in the area of T.

Discusses the canonical treatment of general relativity in a vierbein formulation. The authors derive the primary constraints, and find that they satisfy an algebra the same as that of the generators of local Poincare transformations, namely the Poincare algebra.

Series expansions are used to estimate the critical exponent delta for several two- and three-dimensional mixed spin Ising models. The evidence supports the conjecture of extended spin independence. This suggests that mixed models and the standard 'single spin' models share the same value of delta .

A new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system. The model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces, but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance. The name invasion percolation is proposed for this new process. Similarities to, and differences from, ordinary percolation theory are discussed.

The traditional approach using Carnot cycles to the problem of establishing the existence of absolute temperature and entropy and to deriving d'Q=T dS is shown to remain valid for systems whose entropy is non-extensive as well as for systems whose entropy is extensive. However, it is seen that the analytical approach to this problem is valid only for systems whose entropy is extensive. Two methods for resolving this difficulty are presented.

Explicit expressions for the vector potential associated with a DC current on a saddle-shaped Helmholtz coil have been evaluated. The Cartesian and cylindrical components of the magnetic field in a vacuum are derived from the vector potential. The solution for the components of the magnetic field can be written in closed form using elliptic integrals of the first and second kinds. The expressions may serve as a basis for the optimisation of various coil types used in high-resolution NMR spectroscopy and NMR tomography.

A class of self-similar solutions are rederived for the nonlinear thermal conduction problem. Variational methods applied to the heat flux provide an expression valid over a broad range of physical parameters. The results are particularly useful for the understanding of heat flux in the plasmas of internal confinement fusion. They describe both rapid laser and radiant heating of matter. The developments are shown to correspond well to several classical examples.

In the melting of a solid crystalline medium, the shear modulus changes abruptly to zero. This behaviour can be explained firstly by representing the stress tensor in terms of the correlation function and the potential of interaction of the system and secondly by using an extension of Ruelle's criterion (1977) for thermodynamic stability.