Table of contents

It is shown that the (1+1)-dimensional SO(2, 1) invariant non-linear sigma model possesses the Painleve (P) property. Consequently the nature of the Backlund transformation is discussed.

The propagator for an arbitrary quadratic system with memory in one dimension is calculated using the Schwinger action principle. The propagator has the Pauli-Van Vleck form.

Using a transfer matrix technique and finite size scaling, the authors calculate the exponent nu _t of two-dimensional polymers at the theta point. They find nu _t=0.55+or-0.01 by two slightly different calculations on the square lattice. This value is compared with those which had been previously proposed in the literature.

Estimates of the fractal dimension phi for 'chemical distance' (shortest-path distance) between points on a percolation cluster are inferred from computations of the first-passage velocity nu (p) on square (d=2) and simple cubic (d=3) bond lattices with bonds randomly assigned time delay b with probability p, otherwise time delay a>>b. The computations, implemented on strips in a manner analogous to the transfer matrix for conductivity, yield estimates of phi based on a new scaling law, nu (p_c) approximately a^-1(a/b)¹ phi /, where p_c is the percolation threshold for b-bonds. For d=2, the authors obtain phi =1.021+or-0.005, which is significantly lower than previous estimates. For d=3, they obtain phi =1.26+or-0.06, in agreement with the Havlin-Nossal proposal phi =d-(1+ beta )/ nu . Their results do not exclude the possibility that phi =1 for d=2, indicating that the chemical distance may be non-fractal below some lower critical dimension between two and three.

A seven-parameter Ising model with crossing and four-spin interactions on a checkerboard lattice is mapped by a duality transformation onto a general eight-vertex model, which has only five independent parameters. In a six-dimensional subspace of the Ising model (for which the corresponding eight-vertex model satisfies the free fermion condition) an exact solution is found.

The authors present a particularly simple model of deterministic classical motion in a two-dimensional random environment. As the parameters of the model are varied, a transition occurs from all trajectories being localised to some being extended. They construct a mean-field theory for this transition, and relate the model exactly to percolation models in particular parameter ranges. They point out that it is a member of a new class of site percolation analogues of direct bond percolation.

The authors introduce a new class of statistical growth fractals which is of interest because the chemical dimension d_l is continuously tunable. They also study other exponents characterising these fractals.

The authors consider the percolation of a surface without holes in a system of randomly occupied plaquettes. They show that duality can relate not only the critical points but also the correlation length exponents of this problem to the usual random bond percolation. The generalisation to hypersurfaces in higher dimensions is discussed.

The authors have determined the minimum energy configuration of N point charges confined to the interior of a circle. The minimisation problem in the multi-dimensional configuration space is solved by a technique based on the simulated-annealing method. They observe striking effects, which could lead to a better understanding of phenomena such as crystallisation, symmetry breaking, commensurate-incommensurate transition, etc. Moreover, an experimental verification of their results appears to be possible.

Methods for a graphical representation of determinants are described in detail. Three different graphs are discussed and in each case an efficient graphical method of matrix element evaluation is given. The graphs store information about large determinantal basis sets in a very compact form and allow for some insight into the structure of such basis sets. A computer representation and some algorithms for the graphs are described in the appendices.

Upper bounds for multiplicities of irreducible representations of a semisimple compact Lie group in the tensor product of representations and multiplicities of irreducible representations of a subgroup in an irreducible representation of a group are derived. Pairs of groups and subgroups appear from the representation theory of non-compact groups.

The proof of convergence of a discretisation procedure for path integrals associated with parabolic second-order differential equations is presented.

The applicability of the Painleve test of the complete integrability of the one-component nonlinear Klein-Fock-Gordon equations in an arbitrary Riemannian space, in the formulation of Weiss et al. (J. Math. Phys., vol.25, p.13, 1984) is discussed. Three infinite series of these equations are found in the flat two-dimensional space which possess the Painleve property and include, as a special case, the Liouville, sine-Gordon, and Dodd-Bullough equations. It is pointed out that the approach of Weiss et al. to select integrable nonlinear equations is not sufficiently reliable and needs some strengthening.

The Lagrangian description of a system is analysed from a geometric viewpoint in order to find a concept for equivalence of singular Lagrangians generalising that of the regular case. Geometric and gauge equivalence of singular Lagrangians are studied and the authors also give some conditions in which second-order differential equations exist satisfying the dynamical equation on the final constraint submanifold.

A relativistic effect involving the local number density of particles on a rotating contour is predicted from the requirement that the total number of particles is constant. The effect is conjugate to that resulting from the invariance of the action or phase integral and is observable through the phenomenon of unipolar induction.

It is well known that conservation laws impose limitations on the measurability of quantum mechanical observables. In particular, it has been shown that predictable and repeatable position measurements are impossible due to momentum conservation. The authors provide evidence that the assumption of repeatability may even be dropped. The consequences of this result for the concept of observable and of quantum theoretical reality are discussed elsewhere.

A simple numerical method for calculating eigenvalues and corresponding eigenvectors of the Schrodinger equation for a perturbed oscillator is described. The derived results are compared with previously derived numerical data and with available exact values.

The relativistic analogue of the space-translation method is derived. Using this method the generalisation of the Kroll-Watson formula is obtained for the scattering of an arbitrary charged particle (e.g. mesons, hyperons, quarks, etc.). The separation of the background and resonant parts of the scattering amplitude is predicted.

Ordering in the antiferromagnetic Potts model on a simple cubic lattice with q=3 or 4 states per ion is investigated by Monte Carlo computer experiments. The results are analysed by forming the ion-pair correlation function, a technique permitting a more reliable identification of long-range order. In agreement with Banavar et al. (1982), we find a continuous phase transition to a two-sublattice ordered phase. The Monte Carlo results are used to test the mean-field approximation, which is found to yield qualitatively correct results for q=3 and 4, although exaggerating the ordering tendencies.

The generalised Euler formula is derived from Poisson's summation formula. In special cases, it reduces to the ordinary Euler formula and Walfisz formula in one dimensions. As an application, the authors use the generalised Euler formula to calculate the expression for the number of quantum states of a single non-relativistic particle in a rectangular three-dimensional box of finite size.

Gravitational effect in colloidal suspensions is examined both theoretically and experimentally by light scattering. In contrast to the previous theory the present theory predicts the cube of the nearest-neighbour distance to vary linearly as a function of height of the suspension. The position of the first peak in the static structure factor of the suspension having liquid-like order is used to obtain average nearest-neighbour distance. The experimental data fit well to the present theory. The bulk modulus of the liquid order estimated for the first time by this method is found to evolve as a function of time. The time taken for the colloidal suspension to reach gravitational equilibrium as well as deionisation equilibrium is obtained. The time to reach gravitational equilibrium is found to be much less than earlier theoretical estimates based on a simple model. A possible mechanism for this is proposed. Concentration dependence of the saturation bulk modulus is obtained and discussed.

The authors investigate the automorphic properties of the partition function of an anisotropic but symmetric Ashkin-Teller model on a square lattice. They discuss the structure and the location of critical and solubility lines of the isotropic model in the present framework and comment on their algebraic and/or non-algebraic character.

Random-walk recursion relations are developed on a simple example of hierarchical lattice and are found to be the generating functions for first-passage walks. With exact renormalisation of a general harmonic model the spectral dimension, F, for the bond hierarchy class is shown to be F=2ln(g)/(ln(g)+ln( lambda _r)), where g and lambda _r are respectively the aggregation number and resistance eigenvalue. The behaviour of F across families of hierarchies is discussed. It is noted that the renormalisation group has limitations in its abilities to give results for the above models on hierarchies and that F so found may not maintain on hierarchies its usual significance for random-walk statistics.

The upper bound on the critical temperature in the Ising model, recently derived from Callen's correlation equalities by Sa Barreto and O'Carroll (1983) and improved by Monroe (1984), is further improved by a more economical use of Messager and Miracle-Sole inequalities, starting from a different Callen correlation equality.

The hidden symmetries in various integrable models are derived by applying a newly developed method that uses the Riemann-Hilbert transform in various reductions of the linearisation systems. The method is extended to linearisation systems with higher algebras and with supersymmetry.

The main technical apparatus of the harmonic superspace approach to extended SUSY, the calculus of harmonic variables on homogeneous spaces of the SUSY automorphism groups is presented in detail for N=2, 3, 4. The authors construct the basic harmonics for the coset manifolds G/H with G=SU(2), H=U(1); G=SU(3), H=SU(2)*H(1) and H=U(1)*U(1); G=SU(4), H=SU(3)*U(1), H=SU(2)*SU(2)*U(1), H=SU(2)*U(1)*U(1) and H=U(1)*U(1), G=USp(2), H=SU(2)*SU(2), H=SU(2)*U(1) and H=U(1)*U(1) and tabulate a number of useful relations between them.

It is shown that, when only two turning points are important, the generalised Bohr-Sommerfeld condition for Regge-pole positions can be derived from the behaviour of the semiclassical wavefunctions in a region far from the turning points. The analysis proves that the poles can be accurately determined from the Bohr-Sommerfeld formula also when the turning points lie close together. The implications of the present analysis as to certain asymptotic properties of the semiclassical pole positions and the corresponding pole residues are discussed.

The one-loop correction to the Maxwell Lagrangian with constant external field is obtained by taking into account the anomalous magnetic and electric moments of the electron. The imaginary part of the effective Lagrangian is found. It defines the probability of creating electron-positron pairs. The asymptotics of the effective Lagrangian for weak, (H, E<<m²/e), and very strong, (H, E>>m²/e), fields taking into account the dependence of anomalous moments of the electron on E, H are studied.

The Monte Carlo simulation for the one-dimensional one-component plasma model is carried out. The authors obtain the pair distribution function G₂(x) of a periodic structure, confirming the Kunz prediction that the one-dimensional one-component plasma remains in a crystalline state for any value of the plasma parameter. The Monte Carlo results are then compared with those obtained from the hypernetted-chain equation. Although the correlation energies calculated from the two approaches agree fairly well, the hypernetted chain g₂(x) never reproduces the periodic structure, demonstrating that this scheme does not apply to a crystalline state.

The t dependence of the average of the end-to-end distance (R_t²) of t-step random walks on self-avoiding walks is studied by small cell real-space renormalisation. The results are in qualitative agreement with earlier Monte Carlo data.