Table of contents

It is shown that higher-order Noether symmetries of Lagrangian systems, as introduced by Sarlet and Cantrijn (1981), can be transformed to ordinary Noether symmetries by an approximate type (1,1) tensor field.

Motivated by recent field-theory approaches to the Eden model of cluster growth, the authors study a truncated version with a maximal number of three nearest-neighbour particles. Numerical simulations on cubic lattices show that the modification is irrelevant to the asymptotic exponent and affects only the short-distance behaviour. Beyond a crossover scale, which increases with the dimensionality, the total mass of the clusters scales as the volume of a d-dimensional sphere with asymptotic filling factors of approximately 0.7 and approximately 0.5 in d=2 and d=3, respectively.

A dispersive system describing a vector multiplet interacting with the Korteweg-de Vries field is shown to be a number of a bi-Hamiltonian integrable hierarchy.

The authors have studied the distribution of cluster sizes n_s in an irreversible kinetic gelation model using Monte Carlo simulations on a simple cubic lattice. They find quite surprising, pronounced oscillations in the distribution n_s with cluster size s. The variation of n_s with s and the fraction p of bonds formed is described by unconventional scaling behaviour.

The equilibrium solution of the stochastic coagulation-fragmentation equations, for a spatially homogeneous system containing M monomers, is studied. If the coagulation-fragmentation rates K_ij and F_ij satisfy the condition F_ij/K_ij= lambda a_ia_j/a_i+j (needed to guarantee detailed balance) with a_k varies as Ak^{- delta} z_c^-k and 2< delta <3, then the ensemble average size distribution (x_k) has a clearly bimodal character in the gel phase. The distribution of gel species is sharply peaked near k=Mg, where g is the gel fraction for the infinite system, with a width varies as M¹( delta -1/). The scaling function describing its precise shape is determined analytically. This function is highly non-symmetric; its second moment diverges.

The growth of wetting layers is studied as a function of time t in the framework of effective interface models. The thickness of such layers is found to grow as t^1/4 and t^1/5 for three-dimensional systems which are governed by non-retarded and retarded van der Waals forces. In the fluctuation regimes, a universal growth law t^psi with psi =(3-d)/4 is found where d is the bulk dimensionality. It is also shown that the dynamic critical exponent z is super-universal: z=2 holds both in the mean field and in the fluctuation regimes.

Exact scaling treatments are given of anisotropic diffusion in pure and diluted chains, and in a fractal. A dynamic decimation method is used to obtain detailed dynamic scaling descriptions. The results for chains show crossover to drift or localised behaviour, induced by anisotropy or dilution. The fractal results include bias-induced crossover, with exponent unity, from anomalous diffusion behaviour (exponent log₂ 5) to drift behaviour and scaling corrections (exponent log₂ 5/3) arising from rotational anisotropy.

A decimation transformation is applied to the homogeneous equations of motion for creation and annihilation operators of kink-like excitations of the transverse Ising and XY chains at zero temperature. The authors obtain the exact dispersion relations and their dynamic scaling form. The static behaviour is derived as the zero frequency limit of the recursion relations.

The authors examine a problem arising in the determination of the disorder solution using the method of exact decimation for spin models in a field. The problem concerns the validity of the change of boundary conditions, a crucial step essential to the decimation approach. They study this validity question for a general checkerboard Ising lattice which has crossing and multispin interactions as well as magnetic field. Their study leads to an equation determination the disorder solution and a sufficient condition ensuring the validity of the solution. The nearest-neighbour model is re-examined in light of these discussions.

By a Monte Carlo simulation of a grand canonical ensemble for a lattice tree model of branched polymers in three dimensions the author obtains the estimates theta =1.501+or-0.043, nu =0.495+or-0.013 for the critical exponents. These estimates strongly support the exact conjecture theta =3/2, nu =1/2 by Parisi and Sourlas (1981) obtained using dimensional reduction. The attrition constant mu is also estimated.

The author introduces a new random walk which can be used as a model for the theta polymer. The walk belongs to a different universality class from the usual SAW. Extensive Monte Carlo calculations have been performed to calculate the exponents gamma and nu . The author finds in two dimensions nu -0.535 and gamma =1.025 and in three dimensions nu =0.50 and gamma =1.0. This result strongly suggests that the upper critical dimension is equal to three.

Two models of fractal objects are presented. The first model represents a new class of fractals which is intermediate to statistical fractals and exact fractals in that each configuration of the random ensemble has an exact (and adjustable) fractal dimension. The second model has a continuously adjustable backbone dimensions and can be used to directly generate conducting backbones.

The authors have considered the effective resistance of resistor networks which can be mapped onto a checkerboard geometry. Each square in the board is represented by four resistors of the same magnitude, R_i, where R_i=R₁ or R₂, with probabilities p₁ and p₂=1-p₁. The configuration of the four resistors in a square can be chosen naturally in four different ways. For each of these they have calculated the effective medium theory (EMT) result for the effective resistance and compared it with the result of a numerical calculation for a large random network of the appropriate configuration. The agreement between EMT and our simulation is very good. It is worth noticing that the effective resistance falls outside the corresponding Hashin-Shtrikman bounds to the effective resistance of a continuous two-phase material. From EMT the authors have obtained percolation thresholds, which contain transcendental numbers (e.g. 1/ pi ).

The Hahn and Meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter. This leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure. The Meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the Pollaczek polynomials. The orthogonality relation for the Hahn polynomials with respect to a continuous measure is apparently new. A close connection between the Hahn polynomials of an imaginary argument and representations of the Lorentz group SO(3,1) is considered.

Young supertableaux are introduced for finite-dimensional representations of orthosymplectic superalgebras and used to study their content.

Based on the published reduction of Kronecker products, the Clebsch-Gordon coefficients for irreducible representations of the nonsymmorphic space group of garnets at the symmetry points in the representation domain are calculated.

Vector-coupling or Clebsch-Gordan coefficients for irreducible representations of the space group D_6h⁴ have been calculated at symmetry points of the hexagonal Brillouin zone. All arms of the wavevector stars and all wavevector selection rules are enumerated.

Several exact representations (as an integral and as an infinite series) for the partial derivative delta zeta (z,q)/ delta z mod _z=-1 of the generalized Riemann zeta function zeta (z,q) are given.

Osterwalder-Schrader positivity is shown to be fulfilled for stochastically quantised lattice gauge theories, spin models and P( phi ) interactions bounded from below. Problems arising in the stochastic quantisation of bottomless P( phi ) models are discussed. A stochastic equation is derived to stabilise the quantum Einstein gravity. It is shown that the stochastic quantisation of the Yang-Mills leads to a well defined semiclassical expansion.

A variational correction procedure is derived for evaluating the solutions of sets of coupled differential equations. The procedure is compared and contrasted with the Kohn correction method and with perturbation theory; and illustrated by considering a simple exactly soluble problem. The procedure has particular applications in scattering theory.

The author shows a very efficient solution of the equation of Saito's orthogonality-condition model (OCM) for bound and resonant states by means of a separable expansion of the potential (PSE method). He derives some simplifications of the published formulae of the PSE method, which facilitate its application to the OCM and may be useful in solving the Schrodinger equation as well.

The author has found an orthogonal set of solutions of the Dirac equation for the electron interacting with a quantised electromagnetic plane wave. The orthogonality of the wavefunctions is proved and the physical interpretation of the solutions is discussed.

The few-particle Schrodinger equation does not define the symmetry of the wavefunction, which must be chosen to match the symmetry of the particles. It is shown, by reference to the S-states of a three-particle system, that the symmetry does constrain degrees of freedom associated with normalisation of the exact wavefunction. The first particle is treated as infinitely massive, and distinguishable. The systems where the second and third particles are (i) distinguishable, (ii) indistinguishable with a symmetric wavefunction (bosons) and (iii) indistinguishable with an antisymmetric wavefunction (fermions) may be treated as special cases of a continuous description of particle symmetry. Cases (ii) and (iii) are opposite extremes in the analysis.

The eigenvalue problem of a Hamiltonian which has been used as a model in exciton dynamics and in the theory of nonradiative transitions is formulated in Bargmann's Hilbert space of analytical functions (1962). For particle combinations of the parameters of the Hamiltonian one finds exact solutions which are elementary transcendental functions. In the general case the eigenvalues and the eigenfunctions are determined by matrix continued fractions. The eigenfunctions can be represented accurately by Neumann expansions with very few terms.

A recently developed droplet theory of low-dimensional Ising systems is generalised to describe the q-state Potts universality class, and thence the percolation problem realised in the q to 1 limit. An integral equation for the Potts model free energy is derived and used to obtain the generating function for the cluster size probability distribution in the percolation problem, above the percolation threshold. A closed equation for the distribution is obtained and shown to yield scaling behaviour. The form of the distribution is determined analytically in the regimes of small and large cluster sizes yielding, respectively, power law decay and Kunz-Souillard exponential decay. The form of the distribution in the regime of intermediate cluster sizes is determined numerically in space dimension d=2.

New consistency equations involving the internal energy and-when possible-the order parameter are proposed for lattice models; these equations are shown to be very efficient in classical examples such as the two- and three-dimensional Ising model. However, their peculiarity is that they can also be applied to models where the order parameter is unknown and, consequently, any mean-field approach is ruled out. Applications to the self-avoiding walk and surface models are shown to be successful.

The dynamical critical exponent z is calculated using the finite size scaling method for the two-dimensional three-state Potts model with nonconserved dynamics. The value of z is found to be between 2.1-2.3, with the most likely value being 2.2.

The general Z₄ and Z₅ models on the simple quadratic lattice, and the pair triplet Ising model on the triangular lattice are employed to illustrate an extension of the finite size scaling technique which obtains the qualitative structure of the phase equilibrium surface in terms of multiple phase coexistence.

Different numerical methods for accurate calculation of low-lying eigenvalues of lattice Hamiltonians are proposed and critically compared. A dynamical procedure, called basis vector importance sampling, is shown to select the relevant subspace of the Hilbert space very effectively. This method is used to compute the mass gap of O(2) symmetric quantum chains up to a length of nine sites. Kosterlitz-Thouless type freezing transitions of Z(p) symmetric chains are also studied via the spectrum of quantum kinks.

The author investigates the dynamics of Monte Carlo simulations of the kinetic Ising model involving multi-spin-flip updating schemes as opposed to the conventional random single-spin-flip updating schemes normally used.

It is shown that the proper-time method can be adapted to make the calculation of higher derivative terms in the one-loop effective Lagrangian easy.

The renormalisation group approach is used for exploring the critical behaviour of bosonised models in the presence of a long-range correlated random field. The zero-temperature regime is studied in some detail for different types of random field correlation functions. As a result, the random critical exponents are not derivable from the pure ones with appropriate dimensional shifts.

The authors show that the existence of zeros of the energy gap for finite quantum chains is related to a non-vanishing wavevector. Finite-scaling ansatze are formulated for incommensurable and oscillatory structures. The ansatze are verified in the one-dimensional XY model in a transverse field.

Computer simulations confirm Dhar's prediction (ibid., vol.17, p.L257, 1984) that the asymptotic exponent for the RMS distance against time relation is zero right at the percolation threshold, if a particle diffuses under the influence of a constant force in a random medium. For intermediate times and weak force, the exponent for the mean displacement (as a function of time) in the direction of the force confirms the prediction of Ohtsuki and Keyes (Phys. Rev. Lett., vol.52, p.1177, 1984).

In an attempt to search for evidence that gravity is not quantised, Page and Geilker (1981) have looked for those components of the wavefunctions in an Everett-type interpretation of quantum mechanics which are usually assumed unobservable, in an experiment involving gravitational attraction between two lead balls. It is pointed out that they use one of several possible interpretations of this type, an interpretation the author claims to be untenable, as it is not consistent with a very wide range of experimental evidence.

It has been shown that the angular correlation range is infinite for the directed self-avoiding walk problem on the square lattice. This result implies a one-dimensional-like critical exponent, namely nu =1.