Table of contents

It is shown that a G-graded algebra in which xy is a scalar multiple of yx is just a twisted version of a Z₂-graded superalgebra.

The contribution to the spectral density from all the repetitions of a single classical closed orbit can have singularities that are not eigenenergies of the system. Only by summing over all different topologies of orbits do these false singularities get replaced by delta peaks at the true eigenenergies. The authors illustrate this with an example where the expansion and sum over repetitions are exact rather than asymptotic.

The singularity-structure aspects of the Davey-Stewartson equation is investigated and it is shown that the system possesses the generalised Painleve property in the sense of Weiss, Tabor and Carnevale (1983). Further, the associated Backlund transformation in constructed and Hirota's bilinearisation is obtained straightforwardly from the Painleve analysis.

The control of the distance between the quantum path and a given target state in the state space of an (N+2) tight-binding isolated system prepared in a non-stationary initial state is studied as a function of an asymmetry introduced by a change in only one of the coupling constants. When N>6, a bistable-like characteristic is found for the variation of the minimum distance between the path and the target as a function of this asymmetry. Optimised systems are proposed to reach the best control of this distance by the asymmetry.

The authors give the boson realisation of Virasoro and Kac-Moody algebras without central terms and construct their indecomposable representations from some representations of the Heisenberg-Weyl algebra. The boson realisation of the Kac-Moody algebra SU(2) associated to the Lie algebra SU(2) is discussed in detail.

The authors reconsider the problem of diffusion on a self-avoiding walk with local bridges. Levy flight arguments are presented to estimate the spectral dimension, based on known results on the statistics of loops in a SAW. Agreement with ESR experiments on proteins and with numerical simulations is found to be satisfactory.

The blackish area M_u of crumpled paper balls of dimension D=2.5 obtained through the immersion of such objects in a black solution or with the aid of a ballistic painting technique is investigated. Among other scaling relations the authors find that in the first case M_u approximately L^1.39+or-0.16 and in the second case M_u approximately L^1.24+or-0.12 where L is the unfolded size.

By Monte Carlo simulation the author investigated the relaxation of an isolated droplet in the two-dimensional pure Ising model below the ordering temperature T_c. For system sizes 180² and 240² the author observed a stretched exponential decay, namely C(t) approximately exp(-(t/ tau )¹2/), of the temporal spin autocorrelation function C(t) in the intermediate and long time regimes (of the order of 10⁴ Monte Carlo steps spin^-1). The author also computed the correlation function C(t) for the two-dimensional Ising model with quenched random site disorder out to 10⁵ Monte Carlo steps spin^-1 for system sizes up to 180². The relaxation is much slower in the latter model than in the pure Ising model.

A deterministic growth process, based on the solution of the lattice Laplace equation for the temperature field, is introduced for modelling pattern formation in dendritic solidification. By a simple parameterisation of the physical and environmental conditions during crystal growth a wide variety of dendritic patterns, similar to regular snowflakes, are produced. The fractal scaling of the dendritic patterns is investigated.

The authors demonstrate that a tunable family of patterns resembling a range of experimentally observed snowflakes is obtained from a simple but somewhat ad hoc non-deterministic growth model-an extension of the Niemeyer-Pietronero-Wiesmann eta model in which they (i) use 'noise reduction' to enable the asymptotic behaviour to become more readily apparent from simulations of reasonable size, (ii) replace the eta model boundary conditions by the boundary conditions of the diffusion-limited aggregation (DLA) model and (iii) include a surface tension parameter. They measure the fractal dimension of some real snowflakes and obtain quantitative agreement with certain cases of the model. They conclude that the model, although somewhat ad hoc, gives patterns that are in both qualitative and quantitative agreement with a range of real snowflake patterns.

The authors introduce a new definition of a cluster's accessible external perimeter, which is different from the usual definition of the 'hull'. This external perimeter can be measured by probing (adsorbent) particles coming from the cluster's exterior. The size of these particles determines which perimeter sites are accessible from the outside and hence one can define different 'external' or 'accessible' perimeters according to this size. Using a Monte Carlo simulation they measured the fractal dimensions of these accessible perimeters, D_e, for percolation clusters (at the threshold) on the square and the triangular lattices. The fractal dimension of the accessible perimeter is found to depend on the size of the probing particles. For several sizes which are larger than some (lattice-dependent) threshold they find D_e approximately=4/3, which is significantly smaller than the 'hull dimension', D_h=7/4 (found with smaller probing particles).

The author introduces an heuristic method to obtain a large number of low-energy local minima of the SK model. With this method and some assumptions, the author presents a simple explanation of the ultrametricity of local minima of the SK model.

A generalised mixed-spin model on a checkerboard square lattice is solved exactly along a trajectory in the parameter space by the use of an exact decimation method. A closed-form expression of the free energy per site is obtained and a criterion determining the validity of the solution is established.

The problem of packing spheres of a maximum radius on the surface of a four-dimensional hypersphere is considered. It is shown how near-optimal solutions can be obtained by packing soft spheres, modelled as classical particles interacting under an inverse power potential, followed by a subsequent hardening of the interaction. In order to avoid trapping in high-lying local minima, the simulated annealing method is used to optimise the soft-sphere packing. Several improvements over other work (based on local optimisation of random initial configurations of hard spheres) have been found. The freezing behaviour of this system is discussed as a function of particle number, softness of the potential and cooling rate. Apart from their geometric interest, these results are useful in the study of topological frustration, metallic glasses and quasicrystals.

A manufacturing process is proposed for calculating matrix elements of families of basic functions Q_t(x) between the eigenfunctions Psi ^m_j(x) of factorisable equations. This procedure, well adapted for computer algebra, relies on the well known property that solutions of factorisable equations are also solutions of a couple of first-order difference-differential equations. As a particular case, it is applied to the determination of closed form expressions of the 'curved' hydrogenic pseudoradial integrals which are needed when studying space-curvature effects in atomic structure calculations. Several other applications are pointed out.

Among the simple Lie superalgebras (LSA), the basic classical ones are well known, together with many of their finite-dimensional representations. This is particularly true for the typical representations which are known to be completely reducible. It is well known that reducible but indecomposable representations of the basic LSA are not completely characterised by their highest weight (HW) or the eigenvalues of their Casimir operators. In fact, there may be several different finite representations characterised by the same HW. Such representations are called non-typical or atypical. The author establishes a complete characterisation of the typical and non-typical representations by means of properly defined Young supertableaux (YST). Therefore, the definition the author uses for the author's YST differs slightly from the standard definitions. The examples given help the illustration of this new interpretation for the YST which makes possible the complete classification of the indecomposable representations characterised by a generalised notion of HW.

Indecomposable representations of Lie superalgebras are studied on quotient spaces of the universal enveloping algebra of the Heisenberg-Weyl superalgebra (b⁺_i,b_i,f⁺_u,e) by boson-fermion realisation. These representations are constructed from certain types of indecomposable representations of the Heisenberg-Weyl superalgebra and induce usual irreducible representations on invariant subspaces of a quotient space. As a physically significant example, the explicit form of the boson-fermion realisation of indecomposable representations of the classical Lie superalgebra SU(2/1) are obtained and discussed.

Following the geometrical concept of pseudo-Euclidean Hurwitz pairs, the authors given their systematic classification according to the relationship with real Clifford algebras. Next, a generalisation of the pairs to the curvilinear case is given and it is shown that they provide a convenient framework for the Kaluza-Klein theories.

Using a comoving representation the authors reformulate the path integral method for the chiral anomaly and show that the anomaly comes from the accumulating effect in argument change of diagonalised matrix elements for non-Hermitian operators. The analysis of the Wess-Zumino consistency condition reveals further that it is the contribution of zero modes which needs to be considered. Thus a new regularisation scheme is proposed which leads to a unified expression for the anomaly in 2n dimensions both for Abelian and non-Abelian cases.

The geometric quantisation scheme is applied to the compact Kaehler orbit manifolds of the modified Kepler problem. Thus the authors obtain the quantisation of the magnetic charge and energy spectrum of the corresponding quantum problem. A new regularisation of the standard Kepler problem is presented.

The semiclassical density of states depends, according to the periodic-orbit sum formula, on the linear stability of the orbits. This means, however, that contributions from the marginally stable or 'resonant' orbits, which necessarily accompany stable ones, diverge unphysically. The remedy for a system of two degrees of freedom is found to lie in the classical non-linear normal forms for periodic orbits, which describe how satellite periodic orbits coalesce with the central one as resonance is approached ( in to 0). Through these forms the resonant contributions are expressed as diffraction integrals (the first few being 'diffraction catastrophes') uniformly valid in in and h(cross), and finite even for in to 0 provided h(cross)=0. An extension is proposed to incorporate, jointly, multiple resonances found in repetitions of orbits.

The author shows that the rigid spherically symmetric model of the electron is causal in the sense that (i) an external force acting at a time t does not influence the motion before t, and (ii) the motion before a certain time t₀ and the system of forces determine uniquely the motion after t₀. The author also makes some comments concerning the runaway and pre-acceleration phenomena in connection with previous results. The author shows that the non-uniqueness of the general solution, as well as the apparent existence of pre-acceleration for certain values of the radius, are due to the existence of non-trivial solutions for the free case.

The author studies the solutions of the equation of motion for a classical extended charge in the presence of a linear force. In very general conditions, the author obtains the qualitative behaviour of the solutions, which for radii greater than a critical value do not present runaways, and display two different terms: one decaying with time and one that oscillates with bounded and not decreasing amplitude. For a slightly more restricted class corresponding essentially to radii not too close to the critical one, but much smaller than the length travelled by the light in a period of the linear force, the author shows that the non-radiating oscillations do not exist and the solution looks very similar to that of the Abraham-Lorentz model. The author also analyses some aspects concerning the interaction of the oscillator with Rayleigh-Jeans radiation and shows that its effect is important even upon the phase-space trajectory unless the electromagnetic mass is much smaller than the mechanical one. The results extend those of a previous paper (see ibid., vol.20, p.5885 (1987)) dealing with a Yukawa charge distribution to a wide class of charge distributions.

Certain exact odd and even wavefunctions for the potential x²+ λx²/(1+gx²), x in (-∞,∞) are obtained. The coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code REDUCE. The chief use of these results is to provide a check for the effectiveness of numerical algorithms. The results supplement those of Flessas (1982) for the same potential on xε(0,∞).

The model of a quantum damped oscillator based on the Caldirola-Kanai equation is investigated by exploiting its coherent state basis. It is shown how these coherent states facilitate the derivation of fluid-dynamical equations corresponding to this model in the classical limit. From the conservative form of these equations it is concluded that the system represented by the Caldirola-Kanai equation cannot be considered as the quantum analogue of the classical damped oscillator. This is contrasted with the case of a harmonic oscillator interacting with a heat bath.

For the non-linear ordinary differential equation in the radial component of the probability density discussed by Kostin (1971), a general solution is obtained. This solution is a homogeneous function of two linearly independent functions satisfying a certain associated linear homogeneous equation.

The pair-cutoff real-time Green function method with the coherent state approach is used to investigate the (1+1)-dimensional ϕ⁶ field theory. The topological and non-topological soliton solution as well as their elementary excitation spectra at finite temperature are found. Two critical temperatures, one corresponding to the topological soliton disappearance and the other to the symmetry breaking restoration, are given.

The winding number of a random walk in the plane is the net angle through which the walker encircles the origin. The author discusses the moments and asymptotic form of the winding number distribution for walks of constant step size l. The author shows that the root mean square winding number theta _rms grows logarithmically with the number of steps N as N to infinity , in contrast with the N^1/2 growth of the radial distribution. The corresponding diffusion process, however, has infinite theta _rms, and does not provide a good approximation for the random walk distribution. Instead, a diffusion process considered recently by Rudnick and Hu (1987), where the area surrounding the origin has been removed out to a radius R, provides the correct asymptotic distribution. The author finds that the optimum radius for convergence of the finite step and diffusion distributions is precisely R=e^-2l.

The authors present a simple fractal model of the interface between oil and water in viscous fingering or the cluster boundary in diffusion-limited aggregation (DLA). The continuum equations for immiscible displacement of Newtonian fluids with no surface tension in a porous medium are solved analytically to find the flow velocity on the boundary. The moments of the velocity distribution scale with the size of the system yielding a family of exponents. The scaling depends on the geometry of the interface and the oil/water viscosity ratio. For an infinite viscosity ratio, which corresponds to DLA, the form of the scaling is in qualitative agreement with the numerical results of Amitrano et al. (1986). The method can be easily extended to study other systems with a fractal boundary condition.

The authors discuss properties of the set of scattering singularities in regions of irregular scattering. The authors show how a symbolic organisation of the set can be used to determine the fractal dimension and the scaling function. These yields information on the distribution of Lyapunov exponents of bounded orbits. The specific model studied is the motion of a particle in a plane, elastically reflected by three circular discs centred on the corners of an equilateral triangle.

Monte Carlo simulations are performed on the two-dimensional square-lattice Ising model to calculate the intrinsic structure, in the sense of Bricmont et al. (1981) of the phase-separation line. Based on the calculated temperature dependence and the system-size dependence, the critical behavior of the intrinsic structure is discussed.

The Sherman theorem on closed paths for the two-dimensional Ising model on a square with + boundary conditions is generalised to arbitrary boundary conditions. The refinement allows a rigorous cluster expansion for boundary observables (in particular for the surface tension) in terms of open random trajectories on the lattice. As an application the authors discuss the grand canonical cluster expansion for the surface tension and prove its convergence to the canonical cluster expansion and to the SOS limit.

The authors study the critical frontiers and the high- and low-temperature asymptotic behavior of the Potts model two-site correlation function for several Sierpinski carpets with fractal dimension lying between one and two. They simulate the Sierpinski carpets by means of appropriate hierarchical lattices, obtaining from their basic cells the two graphs (among others) that arise from the anisotropic bond-moving schemes. They correspond to the asymptotic high- and low-temperature dominant terms for the hierarchical lattice two-site correlation function exact result. By taking the infinite limit size of the basic cell these asymptotic behaviors tends to those of the Sierpinski carpets. It is then possible to compare them with similar results obtained from analytical extension to d<2 of the hypercubic lattice. Their results indicate that these asymptotic behaviors do not always coincide, suggesting that possibly they may not identify the Sierpinski carpets exhibited by Gefen et al. (1984) as a correct implementation of the analytical extension of hypercubic lattices.

Within the real space renormalisation group framework, the authors discuss the critical point and exponent nu of the Potts ferromagnet in b-sized Migdal-Kadanoff-like hierarchical lattices, Both b to infinity and b to 1 limits are exhibited. The important discrepancies that might exist between the exact results for d-dimensional hierarchical lattices and d-dimensional Bravais lattices are illustrated.

The method presented permits us to systematically compute physical magnitudes for large but finite size in theories solvable by the nested Bethe ansatz. The value of the central charge c and the surface tension are explicitly calculable in this way. For the fundamental vertex and spin models associated to simply laced Lie algebras, c turns out to be equal to the rank of the algebra.

Systems with classical as well as quantal degrees of freedom are described. This is done by means of a relative Hamiltonian and a derivation satisfying a compatibility relation. Various equivalent equilibrium conditions are studied.

The author considers self-avoiding walks (SAW) on the dilute lattices on which the correlation function of the probability of occupied bonds (sites) obeys a power law r^-a for large separation r. The SAW is studied by a renormalisation-group expansion in epsilon =4-d and delta =4-a. The author finds the extended Harris criterion, for n=0 n-vector model or SAW, is that the disorder is irrelevant if av-2>0, which does not depend on the relation between a and d. If the disorder is relevant, the SAW has a new fixed point which has a correlation-length exponent nu =2/a.

The author simulates the continuum percolation of discs in two dimensions and calculates estimates for the threshold parameter and some critical exponents. The author uses a cluster growth method which relies on their recent discovery that the continuum percolation of discs can be formulated as a bond percolation problem on a planar triangulated random lattice. The algorithm is very efficient.

The scanning method for the simulation of linear chain is extended to general models of branched polymers without loops. A branched chain grows in 'time' (namely, a number of steps from the origin). Therefore (i) in contrast to other simulation techniques, which are of a relaxation type, the chains are statistically independent and the statistical error can reliably be estimated, (ii) the probability of a chain is known and hence the entropy, and (iii) the scanning construction enables one to study geometrical properties which depend on time. For self-avoiding trees on a square lattice, the author obtains the relatively accurate estimates for the static critical exponents, nu =0.640+or-0.004 and theta =1.003+or-0.02 and for the connective constant mu =5.1419+or-0.003. The author also obtains critical exponents gamma _t approximately=1.26 and nu _t approximately=0.83, which characterise the growth in time of the number of bonds and the gyration radius respectively. Application of the scanning method to more complex branched polymers is discussed.

Monte Carlo simulation of the AB percolation model on large triangular lattices gives indications that this problem has the same set of critical exponents as ordinary percolation although a critical amplitude ratio, which is thought to be universal, appears to be significantly different from accepted values for two-dimensional percolation. In addition, finite-size scaling functions for both the percolation probability and susceptibility seem to be significantly different from those for ordinary percolation in two dimensions. The author also shows that this problem is closely connected to that of the percolation hull and in fact is dual to the complement of the hull problem.

The author considers phase diagrams for a class of layered systems. Diagrams are studied by means of an inductive argument based on the low-temperature expansions technique. The author shows that the layered structure of the ground state enhances the emergence of an infinite set of phases. As examples, the author considers the ANNNI and the three-state chiral Potts models. The author also discusses briefly a new version of the three-state Potts model which exhibits the phase diagram of the devil's staircase type.

The first-rank tensor field coupled to an external electromagnetic field is reduced to a constrained mechanical model. By using the constrained mechanical model the authors trace the origin of both the non-positive definiteness of the commutators and the non-causal modes of propagation. They find that these diseases have a common origin inherent in the constrained dynamical systems.

The authors discuss the supersymmetric tachyon and its applications. Both unitary and non-unitary representations for the subalgebra are examined. If they abandon the standpoint that any elementary particle in relativistic quantum theory must be described by unitary irreducible representations of the Poincare algebra or the superalgebra, then they can construct the supersymmetric invariant action for supersymmetric tachyons. The scalar neutrino's mass is lighter than the photino's mass if the neutrino is the tachyon and the photon is a massless particle in the simplest supersymmetry-breaking model. There is a possibility that the cold dark matter consists of scalar neutrinos.

It is shown that non-bijective quadratic transformation generated by the Kelly matrix changes the problem of a five-dimensional hydrogen atom into the problem of an eight-dimensional isotropic oscillator.

Various authors have shown that the experimentally observed fractal dimension of a random interface may depend drastically on the size of the probe used for the measurements. The authors propose a general explanation for such dependence that they can explicitly calculate for a simple model.

Zero-temperature finite-size Bethe ansatz calculations of the central charge for the Hubbard chain are compared with the finite-temperature Bethe ansatz calculations of the same charge.

A renormalization group method is presented to analyse the multifractal structure of the growth probability in the diffusion-limited aggregation (DLA). A renormalisation group transformation is derived for the probability distribution of the growth bond's conductance. After repeated scaling, an 'invariant' distribution is reached as a fixed-point distribution. The growth probability assigned to each growth bond is represented by a random multiplicative process of the cell's growth probability with fixed-point distribution. A hierarchy of generalised dimensions D(q) is calculated and the alpha -f spectrum is found.

The MFRG method is used to study the critical behaviour of the semi-infinite three-dimensional q-state Potts model. Estimates of Delta _c, the critical enhancement at the special transition, and the phase diagram are obtained for different values of q and compared with results from other techniques.

The authors extend to the S=1 Ising model a simple procedure which was proposed recently for the q-state Potts model. The approximate critical conditions are obtained for the ferromagnetic model on the square and simple cubic lattice.

The scaling behaviour of Koch curves with a rather general branched structure is studied by means of the exact renormalisation method and the general formula of spectral dimension is obtained. It is proved that the spectral dimension is only dependent on the topology of the curve. The Einstein relation between the random walk dimension, fractal dimension and size-scaling exponent of DC conductivity is also checked.

Assuming that two molecules of a fluid are connected (i.e. belong to the same cluster) when their centres are closer than a given distance d, the authors present, in explicit form, the exact pair-connectedness function for a one-dimensional fluid of hard rods of length sigma such that sigma <d<2 sigma and for densities which are compatible with the existence of a region of translational invariance in the system.