Table of contents

Some group integrals which appear in the lattice QCD with Susskind fermions for zero and finite temperatures in the framework of strong coupling approximation are calculated.

The gauge transformations in the extended (to potentials and their derivatives) space are shown to lead to the Lie-Backlund tangent transformation group.

Another interesting example of how mathematical methods in physics can be applied to the area of ecology is presented. By using a theorem which first appeared in a Russian paper in 1958 for a generalised Lienard system, a uniqueness theorem of limit cycles of a predator-prey system, which includes Lotka-Volterra, Gause and other systems as special cases, is obtained. Several examples show that the theorem is very useful in dealing with the uniqueness problem of limit cycles for certain ecological systems.

The authors study dilute spin systems with finite connectivity ( alpha ) at low temperature. Near percolation ( alpha =1+ in ) and for small admixtures of antiferromagnetic bonds they find a solution with continuous components for the global order parameter (in contradistinction with an ansatz proposed by Mezard and Parisi (1985,1987) and by Kanter and Sompolinsky (1987) that only contains delta functions). The authors relate it to an ansatz proposed by Morita (1984) and by Katsura (1986) for spins on a Bethe lattice with a +or-J bond distribution. They show that such a solution is stable in the longitudinal sector.

The authors study mutual synchronisation in a model of interacting limit cycle oscillators with random intrinsic frequencies. It is shown rigorously that the model exhibits no long-range order in one dimension, and that in higher-dimensional lattices, large clusters of synchronised oscillators necessarily have a sponge-like structure. Surprisingly, the phase-locking behaviour of the mean-field model is completely different from that of any finite-dimensional lattice, indicating that d= infinity is the upper critical dimension for phase locking.

It is proved that on the basis of the second law of thermodynamics, the Nernst theorem and the conclusion that the heat capacities tend to zero as the temperature approaches absolute zero can be derived from each other. In the process of the proof, it is not necessary to appeal to the unattainability of absolute zero. Therefore, the conclusion that the heat capacities tend to zero as the temperature approaches absolute zero is an equivalent theorem of the Nernst theorem.

Using the results on one-loop corrections to the effective Lagrangian in QED for constant prescribed electromagnetic fields, the authors demonstrate a new way to derive the trace anomaly in spinor and scalar QED.

The usefulness of finite point transformations is emphasised, as a systematic tool for studying the eventual linearisation of ordinary second-order differential equations.

The collective field is applied to treat higher-order terms in the 1/N expansion of the ground-state energy in the Calogero model and related O(N), U(N) and Sp(N) invariant matrix models. All these models share the common structure in the large-N limit and have a unified description in terms of collective fields.

A classification of ODE from the point of view of singular point analysis is suggested. The general formula for the Painleve resonances is derived and the resonance analysis of low-order dominant truncations is performed. The resonance formulae for the Painleve chains are presented and the significance of the Painleve chains for the classification is explained.

General time-dependent two-point correlation functions for simple ring polymers in the presence of both self-avoiding and hydrodynamic interactions are calculated to O( in ) ( in identical to 4-d, d being the spatial dimensionality) using renormalisation group techniques. Results are presented in universal functional form to O( in ).

The authors prove that, for the general dynamics of a quantum spin, there exists a stochastic process in an extended phase space which at each time allows them to compute correlations between the different components of the spin. The scheme is limited to integer spins, but some possibilities of extension are discussed as well as the connection with Nelson's stochastic dynamics (1985).

An elementary introduction of quantum-state-valued Markovian stochastic processes (QSP) for N-state quantum systems is given. It is pointed out that a so-called master constraint must be fulfilled. For a given master equation a continuous and, as a new alternative possibility, a discontinuous QSP are derived. Both are discussed as possible models for state reduction during measurement.

The authors propose an investigation of the properties of SU(2) nonlinear Hamiltonian flows when a dissipative, gradient-like dynamics is superimposed. The corresponding flows are analysed and a method is designed to compute the boundaries of the basins of attraction to any desired accuracy. The shape and general features of these domains are examined in relation to the relative size of the dissipative component of the motion.

Considers Alfven waves propagating obliquely in an atmosphere, subject to a uniform magnetic field of arbitrary direction, in the presence of viscous stresses and electrical resistance. This problem is fundamental to theories of atmospheric heating by dissipation of Alfven waves, on which there is a relatively substantial literature. The Alfven wave equation is deduced for an atmosphere with non-uniform diffusivities and propagation speed. The wave equation is solved exactly in the case of an isothermal atmosphere, for which the Alfven speed increases exponentially on twice the scale height and the dynamic viscosity increases exponentially on the scale height; the rate of ionisation is assumed uniform, leading to a constant electrical diffusivity. The exact solution includes, as particular cases, those obtained before for Alfven waves in an isothermal atmosphere, in the non-dissipative case with vertical (Ferraro and Plumpton, 1958) and oblique (Schwartz et al., 1984) magnetic field, and in the case of resistive dissipation along (Campos, 1983). The wave fields are expressed at all altitudes in the terms of hypergeometric functions, which are used to plot the amplitudes and phases for several combinations of wave frequency, horizontal wavenumber, inclination of the magnetic field to the vertical and viscous and resistive diffusivities. It is shown that, for certain ranges of values of the parameters, intense localised dissipation of waves can occur.

Schrodinger pointed out the paradoxical fact that a skilful experimenter can steer, without any interaction, a distant particle (that is correlated with a nearby one on account of past interactions) into any of a wide set of states. He gave a sufficient condition for the range of this distant steering. In this paper a necessary and sufficient condition, enlarging Schrodinger's set in some cases, is derived. The significance of Schrodinger's approach to quantum non-separability (i.e. to quantum distant correlations) is discussed, and previous work along these lines is put into relation with distant steering.

An expression for the high-energy Delbruck amplitude at large scattering angles is derived. This expression is exact in the parameter Z alpha . The consideration is based on the use of the relativistic electron Green's function in a Coulomb field.

The factorisation method is used to analyse the motion of wavepackets in symmetric, confining, one-dimensional reflectionless potentials. The authors calculate the quantal time advance and compare it to the classical limit. It is proven that the broadening of wavepackets is delayed when they move over the potential well.

It has been shown by simulation by Baumgartner and Muthukumar (1987), and by a theoretical model by Edwards and Muthukumar (1988), that disorder localises the locus of a random polymer. This work is extended here to include volume effects and the authors suggest a model of how the statistics of a polymer lying on a surface is affected by the roughness of the surface. It is shown that an appropriate equation for the mean size of the polymer R is derived from an entropy R²/L+L/R²+w(L/R)²- nu L ln R corresponding to the three-dimensional form R²/L+L/R²+wL²/R³- nu L/R where w is the excluded volume, nu is the 'scattering power' of the disorder and L is the length of the polymer. For w small the polymer localises but, however small w is, for large enough L the excluded volume becomes dominant. The localised radius is independent of L and is proportional to nu ^-1/2 on a surface and nu ^-1 in three dimensions. A remarkable intermediate case arises in which R=wL/ nu for small w.

A recently developed renormalisation group approach to interface pinning problems is generalised to deal with the depinning from the defect at a finite distance z removed from the surface in the planar Abraham model. The phase diagrams and the incremental defect free energies over the whole range of temperatures and distances z are obtained. The agreement with exact results, when available, is quite satisfactory.

A numerical random-walk method for simulating pore-size radial displacement of oil from wetting porous media under varying conditions of pore-size distribution, wettability, mobility ratio and capillary numbers is described. The algorithm involves Monte Carlo decision making, random walks and percolation theory. The likelihood of having a walker start from a peripheral site, or from the origin, is determined by the viscosity ratio, M. Sticking probabilities, however, depend on the interfacial tension, gamma , and the solid-liquid contact angle, theta . Three limiting behaviours are identified in terms of viscosity ratio and capillary number: viscous fingering, plug flow and invasion percolation. Numerical experiments are performed for M=13 ( gamma =18 mN m^-1, theta =50 degrees ), and for M=7.6*10^-5 ( gamma =66 mN m^-1, theta =70 degrees ) at flow rates spanning four decades on a porous network of pores and sites having a log-normal size distribution. Typical runs last about 5-10 min. Preliminary evidence of partially dendritic growth at high capillary number is discussed. Agreement with previously reported experiments is excellent.

The authors study the retrieval phase of spin-glass-like neural networks. Considering that the dynamics should depend only on gauge-invariant quantities, they propose that two such parameters, characterising the symmetry of the neural net's connections and the stabilities of the patterns, are responsible for most of the dynamical effects. This is supported by a numerical study of the shape of the basins of attraction for a one-pattern neural network model. The effects of stability and symmetry on the short-time dynamics of this model are studied analytically, and the full dynamics for vanishing symmetry is shown to be exactly solvable.

The four-state Potts quantum chain with a toroidal boundary condition leaving a global Z₃ symmetry of the Hamiltonian is investigated. At the critical point, the infinite system shows SU(2) Kac-Moody symmetry. The conjectured operator content is confirmed by numerical finite-size calculations.

The persistent appearance and use of an incorrect form of the Darwin term appearing in the non-relativistic approximation of the Dirac equation justifies an examination of when its use leads to no errors, as well as an examination of the reasons for its appearance.

The authors have found an infinite number of exact solutions for the hydrogenic atom in the external potential V(r)=gr+λr², not only for an s-wave state but for higher waves as well, from supersymmetric considerations. The general Schrodinger equation has been treated by the shifted 1/N expansion method. The eigenvalues obtained from the shifted 1/N expansion are compared with those obtained by Bessis et al (1987) and also with the supersymmetric exact values.

Shows that the partition function per site K_m(K_x, K_tau ) of the square lattice Ising model with m-spin interaction K_x and two-spin interaction K_tau satisfies the inverse relation K_m(K_x, K_tau )K_m(-K_x, K_tau +or-i pi /2)=+or-2i sinh 2K_tau for all m values.