Table of contents

A recently developed group theoretical approach to scattering is applied to the modified Coulomb problem and thus by a purely algebraic manipulation the S-matrix is derived.

New ansatze for spinor fields are suggested. Using these, the authors construct multiparameter families of exact solutions of the nonlinear many-dimensional Dirac and Dirac-Klein-Gordon equations, some solutions including arbitrary functions.

A finite-difference method for solving the Schrodinger eigenvalue equation is generalised in order to treat a larger number of potentials. Results are shown for asymmetrical two-well oscillators.

The authors analyse the effect of tip stability on the multifractality of interfacial patterns by applying the method of noise reduction to a random Laplacian growth on the square lattice. While the distribution of perimeter sites over the harmonic measure is smooth in the case of random fractal objects with tip splitting, separate peaks appear if the tips are stable. These peaks can be identified with the subset of growth sites corresponding to the tips. They demonstrate the consequences of such behaviour in the multifractal spectrum.

When bonds are removed one by one and at random from a finite-size resistor network, the conductance (or resistance) does not change continuously, but rather a sequence of conductance (resistance) jumps for various sizes occurs. The larger jumps arise from those bonds which carry a relatively large current just before they are cut. The authors report on numerical simulations of these jumps of a random resistor network on the square lattice. They also give a scaling argument to account for this phenomenon, which yields the number of conductance jumps larger than Delta G scaling as ( Delta G)^{- lambda (G)}, with lambda (G) =d nu /(d nu -t), where t is the conductivity exponent and nu is the percolation correlation-length exponent. Equivalently, the number of resistance jumps greater than Delta R scales as ( Delta R)^{- lambda (R)}, with lambda (R)=d nu /(d nu +t). These predictions account for the data on a qualitative level only, however, and they discuss some possible mechanisms for the quantitative discrepancies.

An anisotropic Ising model with three interaction parameters K₁, K₂, K₃ and a magnetic field H, formulated on the Kagome lattice, is solved exactly for an appropriate relation between K_i and H. When this relation is satisfied the system becomes equivalent to a free-fermion model.

The authors extend the Pirogov-Sinai theory (1975) to some class of quasiperiodic interactions and describe the phase diagram at low temperatures.

The consequences of the lack of isotropy of the momentum flux tensor of the Hardy-Pomeau-De Pazzis (1976) fluid are discussed. It is shown that this lack of isotropy is tantamount to introducing a force which is incompatible with a correct evolution of two-dimensional vortex configurations. In addition, a qualitative discussion is presented on the physical reasons why this problem can be cured by moving to the six-link lattice introduced by Frisch, Hasslacher and Pomeau (1986).

A numerical study of the role of topological point defects in the phase transition of the classical ferromagnetic Heisenberg model in three dimensions strongly suggests that the defects play an essential role in this phase transition. The authors find that the transition from the ordered to the disordered phase is accompanied by a proliferation and unbinding of pairs of oppositely charged defects. If configurations containing defects are not allowed, then the system appears to remain ordered at all temperatures.

The Q2R cellular automaton is used to simulate the growth of the liquid-vapour interface in the 3D Ising model. At the critical point the authors find that its width increases with time in a power law manner with an exponent of 0.34₅+or-0.01₅; below the critical temperature they study the size dependence of the equilibrium width.

In the non-perturbative regime, matrix models display a large-N phase transition. For finite but large N, the transition is anticipated by strong oscillations in some coefficients in the recurrence relations for the orthogonal polynomials that allow the calculation of the partition function. The author shows how to perform the limit, requiring the definition of different interpolating functions according to the parity of polynomials, in the cases of a single or two interacting matrices.

A new view of d=7 Clifford algebra is described in which, starting from a septet of mutually anticommuting Clifford operators, one introduces a septet of mutually commuting involutions in SO(8). These involutions can be seen to arise naturally from a study of ternary vector cross products in eight dimensions.

Derives several equations possessing O(2,1)⊗R(t,t^-1) as a prolongation algebra. An example of a Backlund transformation is given.

An asymptotic and numerical study is made of the singularity structure, in the complex t-plane, of the Duffing oscillator. The presence of logarithmic terms in the local psi-series expansion, of the form t⁴ ln t, leads to a multisheeted singularity structure of great complexity. This structure is built recursively from an elemental pattern which takes the form of four-armed 'stars' of singularities. This construction is deduced analytically from the properties of the mapping z=t⁴ ln t and is confirmed quite accurately numerically. A systematic resummation of the psi series, in terms of Lame functions, is developed. This series exhibits the same analytic structure at all orders and provides a 'semi-local' analytical representation of the solution which is apparently valid even in the chaotic regime.

The Klein-Gordon equation without dispersion, and with quadratic and cubic nonlinearities, has been studied in one and higher dimensions. Algebraic solitary wave solutions in all cases, as well as higher-order modes in higher dimensions (similar to nonlinear optics) have been shown to exist corresponding to specific initial values. While in the one-dimensional case, arbitrary initial values yield periodic solutions, asymptotically stable solutions are shown to exist in the higher-dimensional case. For both one- and higher-dimensional cases, solutions tending to zero with distance are shown to be achieved for other initial conditions by incorporating a small amount of 'saturating' fourth-order nonlinearity. Finally, it is shown how a general Klein-Gordon equation with dispersion and a forcing term may be reduced to the equation discussed in the paper.

It is shown how to generate both Lax equations and Backlund transformations for nonlinear evolution equations using the concept of pseudopotentials associated with the properties of the Riccati equation. Several examples are given: modified Korteweg-de Vries, Harry Dym, Kaup-Kupershmidt and Sawada-Kotera equations.

A new approximation to the scattering amplitude from an impenetrable sphere at short wavelengths is developed. In contrast with Fock's theory, it remains valid at large scattering angles, where it can be matched with the usual semiclassical approximations. The accuracy, both for the magnitude and for the phase of the scattering amplitude, is improved by one or more orders of magnitude over previously known approximations. The approximation remains accurate even at size parameters below unity, bridging the gap between short and long wavelengths. Large-angle diffraction can be interpreted as a tunnelling effect.

The concept of forward-nested multiplication is used to develop a unified power series-Hill determinant method for the calculation of Schrodinger-equation energy levels. A numerical application is made to the exponential cosine screened potential.

The authors examine the emission from the systems of one or two two-level atoms in an ideal cavity with one mode at multiphoton resonance. Exact results for the two-time dipole correlation function and the time-dependent spectra of multiphoton-induced fluorescence are presented.

The authors present a detailed statistical analysis of the Rosenbluth method (1955) of generating self-avoiding walks. This method became one of the standard methods for simulating long polymers. They show that this method, although very successful in yielding large samples, becomes exponentially poor with increasing chain length. This has to be taken into account for simulations and was not done yet. They describe a way to quantify the number of chains needed. However, when compared to direct simple sampling, the method still, carefully used, yields better results, especially in the vicinity of the theta point of polymers. Special care has to be taken for d=2. Some extensions to improve the situation are also discussed.

An analytical and numerical study of the scaling of Edwards random walks, defined as the zero-component limit (n to 0) of an n-component g₀ mod phi ² mod ² theory, is done in three dimensions. The Hausdorf dimension (1/ nu ) and the resistance exponent (x) is calculated using renormalisation group methods in (4- in ) dimensions. The numerical study is done by defining a Monte Carlo algorithm on the cubic lattice to generate ensembles of Edwards walks.

For pt.I see ibid., vol.20, p.5635, (1987). This is the second of three papers in which the authors discuss the applicability of series methods to interface wetting and layering transitions. Here they study the behaviour of a solid-on-solid interface, which is attracted to a surface by a local pinning potential. They find that, both for the standard and the restricted solid-on-solid models, the interface depins from the surface through an infinite sequence of layering transitions as the potential tends to zero. The applicability of the results to previous work on the Abraham model in three dimensions is discussed.

For pt.II see ibid., vol.21, p.159-171, (1988). In the third of this triad of papers which study interfacial phase transitions using series expansions the authors treat the wetting transition of an interface in the three-state chiral clock model. Previous work has shown that on a simple cubic lattice at low temperatures the interface wets through a large, possibly infinite, number of layering transitions. They extend the low-temperature series results to an arbitrary number of nearest neighbours and show that, in the mean-field limit of infinite coordination number, only two layering transitions are seen. This is in agreement with numerical solutions of mean-field equations. Hence the mean-field approximation in this case does not provide a correct description of interface behaviour in three dimensions.

The authors present a fractal model for a rough interface between an electrode and an electrolyte. They calculate that the complex surface impedance is Z=K(Z₀)^p where Z₀ is the impedance of a flat interface. If the fractal dimension, d_f, of the boundary is written as 2+ delta , where delta is small, then, to first order in delta , p=1-2 delta . For a purely capacitive interface, Z₀=1/i omega C, this gives an anomalous power-law frequency dependence as seen experimentally by Bottelberghs and Broers (1976) and by Armstrong and Burnham (1976). The authors explicitly calculate the prefactor K and the range of frequency for which this law is observed in terms of the range of lengths over which the interface is rough.

A cellular automata approach to non-equilibrium phase transitions in a surface reaction model is proposed. This surface reaction model describes a simple adsorption-dissociation-desorption on a catalytic surface. This model exhibits two second-order non-equilibrium phase transitions. The stationary critical exponents for the order parameters beta as well as dynamical critical exponents Delta , describing the critical slowing down, are found to be mean-field-like.

The low-temperature expansion of the checkerboard Potts model in a magnetic field is obtained up to order twelve from the so-called disorder solutions, various expansions and results known in the literature. It is shown that this expansion drastically simplifies on the dual of the disorder variety. The low-temperature expansion of the magnetisation is seen to become equal to one (up to order twelve) when it is restricted to the dual of the disorder variety. These results have to be seen as exact formal constraints on the analytic continuation of the low-temperature expansion of the partition function per site. Similar simplifications occur for the susceptibility and higher derivatives with respect to the magnetic field. These expansions are also analysed in the vicinity of these particular varieties.

The authors consider the one-dimensional Ising model which besides the interaction of nearest neighbours includes a long-range term. The latter is ferromagnetic while the nearest-neighbour interaction can have any sign. The crossover between short- and long-range interactions when varying the Hamiltonian parameters is analysed for such a model. For some values of the antiferromagnetic short-range interaction the transition from a ferro- to paramagnetic state is of first order. At the tricritical point where the transition order changes from second to first they obtain classical tricritical exponents. For a ferromagnetic short-range interaction the Curie temperature surpasses the transition temperature of the long-range model.

Monte Carlo methods are used to explore the universal configurational structure of two-dimensional spin- 1/2, spin-1 and border- phi ⁴ models. Comparison of spin- 1/2 and spin-1 data provides evidence that the magnetisation distribution (effectively the Helmholtz free-energy function) and its coupling derivative (effectively the internal-energy function) constitute readily accessible signatures of a universality class. It is shown that, when allowance is made for relatively large corrections-to-scaling effects, the behaviour of the border- phi ⁴ model may be satisfactorily matched to that of the other two models, substantiating the view that the border model does indeed belong to the Ising universality class.

The content-addressability of patterns stored in Ising-spin neural network models with symmetric interactions is studied. Numerical results from simulations on the ICL distributed array processor (DAP) involving systems with up to 2048 neurons are presented. Behaviour consistent with finite-size scaling, characteristic of a first-order phase transition, is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the Hopfield model and for systems using a local iterative learning algorithm designed to optimise the basins of attraction. Estimates are obtained for the critical minimum overlaps which an input pattern must have with a stored pattern in order to successfully retrieve it.

The typical fraction of the space of interactions between each pair of N Ising spins which solve the problem of storing a given set of p random patterns as N-bit spin configurations is considered. The volume is calculated explicitly as a function of the storage ratio, alpha =p/N, of the value kappa (>0) of the product of the spin and the magnetic field at each site and of the magnetisation, m. Here m may vary between 0 (no correlation) and 1 (completely correlated). The capacity increases with the correlation between patterns from alpha =2 for correlated patterns with kappa =0 and tends to infinity as m tends to 1. The calculations use a saddle-point method and the order parameters at the saddle point are assumed to be replica symmetric. This solution is shown to be locally stable. A local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist.

The authors calculate the number, p= alpha N of random N-bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter K. For each value of alpha and K, there is a minimum fraction f_min of wrong bits. They find a critical line, alpha _c(K) with alpha _c(0)=2. The minimum fraction of wrong bits vanishes for alpha < alpha _c(K) and increases from zero for alpha > alpha _c(K). The calculations are done using a saddle-point method and the order parameters at the saddle point are assumed to be replica symmetric. This solution is locally stable in a finite region of the K, alpha plane including the line, alpha _c(K) but there is a line above which the solution becomes unstable and replica symmetry must be broken.

It has been predicted and numerically shown that the spectrum of the Hamiltonian H(P)= Sigma _{n in -( infinity infinity )}(E(P,n) a_n^Dagger a_n+t(a_n+1^Dagger a_n+a_n-1^Dagger a_n)), in which E(P,n)=V cos (P2 pi n) and P is an irrational number, has a fractal distribution of eigenstates. Using a self-referential decomposition of a pertinent class of quadratic irrationals, it is shown here that such a conclusion is viable.