Table of contents

Two cases are presented of bifurcation problems in the presence of a symmetry: it is shown that suitable group theoretical assumptions lead to the existence of Hopf bifurcation.

Two new idea are introduced in the case of the sine-Gordon 2 pi kink. The first refers to non-interacting phonon plus soliton which is found to be the correct picture in first-order theory. This justifies the additive paradigm of soliton plus oscillation plus translation and hence the Newtonian dynamics. The non-Newtonian behaviour is due to an inappropriate consideration of the soliton-phonon wavepacket interaction. The significance of the defective degenerate mode to the sG 2 pi kink in the presence of perturbing force fields and/or damping is discussed and a surprising result (the absence of the mode) is reported.

Suggests a generalisation of the Painleve test to include situations in which the leading order singularity is not determined by a simple singularity analysis. Such situation often arise in equations whose non-linearities are convective derivatives (e.g. the Hasegawa-Mima equation (1978) in plasma physics).

The authors find the eigenvalue spectrum of a particle constrained to 'hop' between the sites of a simple cubic lattice in the presence of a spherically asymmetric 'harmonic oscillator' potential. The eigenfunctions are found to be given in terms of the periodic Mathieu functions of period pi . At low kinetic energies of hopping the solutions differ considerably from those of the continuum theory: the particle localises itself in a given shell around the origin and the virial theorem is thereby violated. At high kinetic energies they obtain the usual equidistant levels of the harmonic oscillator on the continuous manifold. They discuss in some detail the spectrum of the three-dimensional case and the associated degeneracies. It is seen that the well-known accidental degeneracy of the harmonic oscillator wavefunctions is only partially lifted by the discretisation of space. Finally, mention is also made of the relevance of this solution to the physical problem of the inversion layer at the surface of a semiconductor when it is doped with a spatially constant electric charge density.

A fractal model is proposed to determine the critical behaviour of the low-field Hall effect in a three-dimensional metal-non-metal composite at the percolation threshold. The fractal lattice is constructed to imitate the geometric texture of three-dimensional percolation clusters at criticality and to realise the Hall problem on a three-dimensional discrete lattice. The exponents describing the power law dependence on scale length of the Hall and ohmic conductivities are found.

A variant of 'true self-avoiding walk' (TSAW) is studied. The walker chooses the most favourable direction as the next step in a lattice with random bonds. The process is deterministic and a numerical simulation in one dimension indicates that the scaling behaviour is the same as TSAW, i.e. x approximately t²3/, though there is no apparent mapping between the two models.

The authors analyse the results of kinetic gelation and propose a diagram compatible with the existing data. They introduce the concept of 'limited' and 'full' universality. The former refers to systems with an identical subset of critical exponents; the latter refers to systems with an identical set of critical exponents. The evidence suggests that kinetic gelation and random percolation are in the same limited universality class. It appears that the concentration of initiators plays the role of an active parameter, similar to that of the interplane coupling in quasi-two-dimensional models.

Direct real space renormalisation group procedures are constructed to calculate the self-avoiding walk exponent nu on a two-dimensional Penrose lattice. This lattice is neither translationally invariant nor hierarchical, but still self-similar, and therefore calls for a renormalisation group treatment. Within the accuracy of a small cell calculation the obtained values of nu are consistent with the universal value of nu in two dimensions which is believed to be close to nu =0.75.

Using the numerical results of the finite-size scaling study of the q-state Potts model by Blote and Nightingale (1982), the authors obtain a conjectured expression for the universal amplitude of the free energy density.

An algorithm for constructing canonical forms for any tensor representation of the classical compact Lie groups is given. This method is used to find a complete list of the symmetry breaking patterns produced by Higgs fields in the third-rank antisymmetric representations of U(n), SU(n) and SO(n) for n<or=7. A simple canonical form is also given for kth-rank symmetric tensor representations.

Group-subgroup (GH) transformation theory uses G transformation coefficients and GH transformation factors to describe the relationship between the various bases of representation space of any group decomposed into irreducible representation spaces of the group and its subgroups. The representation spaces may be formed by the processes of coupling, subduction or induction. The authors further develop this GH transformation theory with the study of the various symmetries (complex conjugation, transposition and associativity) of the GH transformation factor, emphasising the origins of each symmetry and a hierarchy within these symmetries. A method of calculation is given for certain transformation factors. The results and methods presented here generalise those of the Racah-Wigner coupling algebra.

Hilbert spaces of analytic functions are constructed which carry irreducible representations of the positive discrete series of Sp(6, R).

For pt.II see ibid., vol.18, p.1905 (1985). A complete set of tensor operators for SU(3) is given in a model space generated by two Bargmann vectors. Use is made of modified operator patterns to classify these tensors. Their tensorial properties are then discussed in the context of the SU(3) tensor algebra developed by Biedenharn and collaborators (1957). The authors also given analytical (semi-analytical) expressions for important classes of Wigner coefficients in Gel'fand (rotational) basis.

Mathematical and physical models for SU(3) are given in terms of unirreps of the so(n,2) and so*(2n) algebras. Vector coherent state representations are given for all the discrete series unirreps of SO(n,2) and SO*(2n). Traceless bosons are shown to arise naturally in these representations. A new and elegant realisation of the so*(8) approximately so(6,2) model for SU(3) is given and it is shown how to perform calculations in the so*(8) model using the powerful vector coherent state representations formalism.

The problem of an electron in a general central potential, subject to a constant external magnetic field, is known to have a degeneracy and to be invariant under SO(2). However, once the central potential has a symmetry group larger than SO(3), even in the presence of a magnetic field, residual accidental degeneracies may remain and a symmetry group larger than SO(2) may exist. In the present paper, the author investigates the case where the central potential is a harmonic oscillator and the magnetic field is strong, meaning that the Hamiltonian contains both a linear and a quadratic term in the magnetic field intensity H. In a recent work, Moshinsky et al. (1975) did claim that in such a case no accidental degeneracy is left. It is shown here that, although this assertion is valid for most values of H, there exist values for which there are residual accidental degeneracies. The analysis is based on a linear canonical transformation converting the Hamiltonian into that of an anisotropic oscillator in the absence of magnetic field. It is shown that the accidental degeneracies of the latter are either due to an SU(2), an SU(3) or an SU(2)*SU(2) symmetry group, or cannot be explained by the existence of any symmetry group. In particular, it is proved that accidental degeneracies, and a corresponding SU(2) symmetry group, may appear when the frequencies are irrationally related. Finally the accidental degeneracies and the associated symmetry group of the isotropic oscillator in a strong magnetic field (when it exists) are determined in terms of the ratio between the oscillator and cyclotron frequencies.

Shows that by adding supersymmetry breaking soft terms to N=4 super Yang-Mills, the E₈ gauge group can be spontaneously broken to SO₁₀, with the emergence of 4 conventional light generations of 16 plus mirror particles. However, this in only achieved after fine tuning, with further tuning required if a Higgs sector is to appear at low energies.

The Landau-Zener calculation, for avoided crossing of two energy levels, is generalised by putting in a third level, so that three levels approach one another. A three-level model of an atom, with two transitions driven by laser beams, can be transformed into this model by using the rotating-wave approximation. The resulting transition probabilities have a simple analytic form, like those of the original two-level model.

The attempt to reformulate quantum theory in the form of a theory of stochastic processes, which has recently met with considerable success, has failed up to now to describe the most typical quantum effect: the emission and absorption of field quanta. The authors explore the possibility of constructing a stochastic field theory based on the assumption that for any normal mode of a given classical field, the corresponding action variable becomes a stochastic variable assuming only positive integer values in units of h. The proposed method reproduces the known properties of simple quantum field theoretical models without recourse to probability amplitudes, which are quantities extraneous to classical probability theory. The probability transition rates, for simple states of the field, turn out to be those of simple Poisson processes.

Obtains expressions for Tr(J_lambda ^2p) in terms of the Brillouin function. Standard properties of Tr(J_lambda ^2p) are derived from them. Sum rules for the Bernoulli numbers and the Riemann zeta functions are deduced as corollaries.

The type of phase transition in a chain of Ising spins with multispin interaction is studied in a transverse field, using strong- and weak-coupling expansions. The transition is shown to be of first order if more than three spins are coupled. The critical exponents for the three-spin coupling model are estimated.

A detailed proof is presented of Vdovichenko's method (1965) giving a exact expression of the free energy of the Ising model on a two-dimensional lattice.

The authors solve exactly the one-dimensional random field Ising model for two classes of magnetic field distributions: symmetric exponential (model I) and non-symmetric exponential (model II). For both models, expressions for the free-energy at all finite temperatures are presented. The low-temperature region is examined in more detail; they obtain the zero-temperature energy and entropy in closed form; it is shown that the free energy of both models has an expansion in integer powers of temperature. Model I has a non-vanishing zero-point entropy for all values of the parameters as soon as randomness is diluted. In model II the zero-point entropy is zero except for a discrete sequence of values of one parameter. In some cases the zero-temperature magnetisation is positive whereas the average magnetic field is negative; the magnetisation may also change sign as a function of temperature.

As the Martinelli-Parisi expansion of recursive relations of the real space renormalisation group is not uniquely defined, the authors test a recently proposed criterion for choosing among the different possibilities. They deal with the Z(2) lattice gauge theory on a cubic lattice. Many alternatives are considered; all of these agree in the evaluation of the critical coupling constant and in excluding a second relevant operator at the non-trivial fixed point.

The Martinelli-Parisi systematic expansion is applied to the three-dimensional Z(2) system as a first application to gauge theory. To first order in perturbation, four kinds of new operators appear. The critical point is significantly improved, while the non-trivial fixed point an undesirable relevant eigenvalue.

It is generally accepted that integrable partial differential time evolution equations possess Lie-Backlund symmetries of arbitrarily high finite order. It has already been established that in (1+1) dimensions the full class of nonlinear Fokker-Planck (convection-diffusion) equations having this property consists of the known integrable examples. These are closely related either to the Burgers equations or to the Fokas-Yortsos-Rosen equation, both of which have been applied to unsaturated flow in porous media. Here the author shows that higher-order Lie-Backlund symmetries do not exist for any examples of the nonlinear Fokker-Planck equation in (1+2) (one time and two space) dimensions.

The exact solution is presented for the classical and quantum mechanical problem of a spinless, isospin-carrying test particle interacting with a singular monopole of the same large-distance asymptotic behaviour as the Prasad-Sommerfield 1-monopole. It is demonstrated that this problem has a dynamical O(4) symmetry. The classical mechanical equation of motion is traced back to a non-relativistic Coulomb problem. The long-range Higgs field results in monopole-test-particle bound states. The bound state spectrum is derived on a purely group theoretic basis using dynamical symmetry.

It is shown that the unique mass spin-1 wave equation proposed by Khalil (1979) is not derivable from an invariant Lagrangian and no charge can be defined for it.

Considers an exactly soluble directed travelling salesman problem, where the salesman is forbidden to move, during its visit to the cities, opposite to a particular direction. When the cities are randomly distributed, with concentration p, on the sites of a square lattice of linear size L, the optimised total contour length becomes (1/p)+(2(2p-1)/p²) (1/L) per city (p>0).

The dual of the Sierpinski gasket is argued not to be a Bethe lattice. This is a comment on a paper by Nencka-Ficek (1985).