Table of contents

The authors propose the supersymmetric version of the Lie extended method in quantum physics. They point out the main ideas through the explicit example of the one-dimensional supersymmetric harmonic oscillator.

The general form of infinitesimal generators of completely positive nonlinear semigroups is discussed. It is argued that the condition of complete positiveness is too restrictive.

Models consisting of a complex scalar field in R₂ and featuring symmetry-breaking potentials, are proposed. These models possess topologically stable classical lumps. For a special subclass of these models, explicit solutions for these lumps can be found. The possibility of treating these lumps as solitons in (2+1 dimensions is briefly remarked on.

Proposes a population point process model of cavity radiation in which the process of spontaneous emission evolves as a multiphase birth process. The heterogeneity arising from the emissions in different phases lends itself to an interpretation of superposition of photons in different streams. Squeezing is shown to be a natural consequence of the heterophase evolution.

The authors present some surprising results of computer simulations on the driven diffusive motion of a 'polymer' in a 'sea of monomers'. The particles move on a lattice subject to a driving field that biases jump rates in a direction perpendicular to the polymer which occupies L sites (monomers occupy one site). This produces a stochastic asymmetric simple exclusion dynamics of the polymer-monomer lattice system. Simulations on a two-dimensional square lattice exhibit unexpected behaviour of the polymer velocity as a function of its length: the velocity V(L) first decreases and then increases. This may have relevance for the size segregation of particulate matter which results from the relative motion of different size particles induced by shaking (the Brazil nuts phenomenon). The location of the minimum of V(L) depends on the nature of the driving fields and on the density.

The route to chaos in quasiperiodically forced systems is investigated. It has been found that chaotic behaviour is obtained after the breaking of three-frequency torus, but strange non-chaotic attractors are present before three-frequency quasiperiodic behaviour occurs.

For a one-dimensional Hubbard model with a half-filled band and one extra hole the single-hole Green function is calculated analytically; the spectrum analysis shows that the N electrons tend to move so as to form identical and consecutive packets to satisfy rules fixed by arithmetical properties of the integer N.

A one-dimensional lattice of logistic maps is investigated in the case of strong nonlinearity and strong coupling. Although the dynamics may be classified as fully developed turbulence, spatio-temporal structure can be detected by computing time-delayed mutual information and two-point correlations. The correlation is found to be superior in detecting weak structure. An improved algorithm for mutual information is described.

It is shown that, for systems with repulsive (non-negative) interactions, the Yang-Lee zeros of the grand canonical partition function lie on the real z axis. As a result, a series of successively improving lower and upper bounds for the thermodynamic ratio of density to activity is obtained. In addition bounds on the transition activity and density are given in terms of the cluster coefficients.

The authors extend part of their previous work on autonomous second-order systems (Sarlet et al., 1987) to time-dependent differential equations. The main subject of the paper concerns the notion of adjoint symmetries: they are introduced as a particular type of 1-form, whose leading coefficients satisfy the adjoint equations of the equations determining symmetry vector fields. It is shown that all interesting properties of adjoint symmetries, known from the autonomous theory, have their counterparts in the present framework. Of particular interest is a result establishing that Lagrangian systems seem to be the only ones for which there is a natural duality between symmetries and adjoint symmetries. A number of examples illustrate how the construction of adjoint symmetries of a given system can be explored in a systematic way.

The quantum R matrix for Cartan's exceptional simple Lie algebra G₂ in its seven-dimensional (minimal) representation is presented. The R matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra U_q(G₂). This defines a new solvable seven-state 175-vertex model on the planar square lattice whose Boltzmann weights satisfy the Yang-Baxter equation. For the equivalent face model, the local state probability is obtained in terms of a relative of a level-1 string function related to the affine Lie algebra pair B₃⁽¹⁾ contains/implies G₂⁽¹⁾.

The relationship between the level of statistical errors in a measured intensity autocorrelation function and the relative size distribution width uncertainty is estimated. The results of the Fourier analysis of statistical errors existing in measured autocorrelation functions indicate that the individual error course can be similar to the systematic distortion. This has been also confirmed experimentally. The problem of the baseline error is also discussed. It is shown that the baseline error is compensated in dynamic light scattering data during normalisation.

The authors derive and study a hierarchy of nonlinear coupled evolution equations (among which is the coupled Korteveg-de Vries/Schrodinger equation) for which they prove that a mixed initial-boundary value problem is solvable. They give the method of solution together with the Backlund transformation and establish the infinite set of conserved densities. They finally discuss the applicability of such equations in plasma physics and hydrodynamics.

The problem of matching Green functions is studied for one or more coupled interfaces for those cases of physical interest, like quantum wells or superlattices, in which differential calculus is involved, e.g. Schrodinger equations. The Green functions are related to the transfer matrices of the constituent media and these can be evaluated by efficient numerical algorithms. One can then obtain the matched Green function of the composite system without having to directly evaluate any Green function or derivative thereof. The formulae for the matched wavefunction are also derived and practical aspects are discussed.

The on-lattice ballistic chain-chain aggregation model is studied by Monte Carlo simulations in one, two and three dimensions. The chains move along the lattice directions, with a velocity proportional to k^gamma , for a k-mer. The fractal dimension of the large polymers and the mass distribution are computed for different values of the mobility exponent gamma . In two and three dimensions, a long preasymptotic behaviour is observed, and in all dimensions the asymptotic behaviour is in agreement with the Smoluchowski theory.

For standard 2D bond percolation, the size of regions trapped by the infinite occupied cluster at bond density p is studied by Monte Carlo simulations. It is known that there is a transition at some density p_t>1/2 which determines the fractal behaviour of invasion percolation with trapping. The numerical results are that p_t approximately=0.520 and the critical exponents are those (( gamma =43/18 and v=4/3) of the usual percolation transition at p_c=1/2. Thus invasion percolation with trapping does not appear to belong to a new universality class.

The authors derive the operator content of the spin-S XXZ quantum chain with generalised toroidal boundary conditions compatible with the U(1) symmetry of the model. These results are derived by solving numerically the associated Bethe ansatz equations for finite chains and exploring the consequences of the conformal invariance of the infinite system. They also show that, as in the spin-1/2 case, the conformal anomaly and dimensions of general extended SU(2) algebras can be obtained from these spin-S chains by choosing properly the coupling constant and the boundary condition.

The system of a spinning particle in a general magnetic field is shown to be supersymmetric, having two supersymmetry generators. The author discusses the quantisation of this system in the case of a constant magnetic field, and describes the states by means of wavefunctionals, having a superfield as an argument.

The techniques of an invertible point transformation and the Painleve analysis can be used to construct integrable ordinary differential equations. The authors compare both techniques for anharmonic oscillators.

Makes some observations on a recent paper by Affleck et al. (ibid., vol.22, p.511, (1989)) about the singlet excitation in the spin-s=1 integrable model and logarithmic corrections for integrable spin models.

It is commented that the fact that the Edwards model has paths of Hausdorff dimension two does not contradict the conjecture that the exponent for root mean square distance for random walks should equal the reciprocal of the dimension of the sample paths.

In this comment, the authors discuss problems in the method recently proposed by Windwer (ibid., vol.22, p.L605, (1989)) for computing knot probability in self-avoiding walks on the simple cubic lattice.

The author replies to a comment by Sumners and Whittington (ibid., vol.23, p.1471, (1990)) on the detection of knots in self-avoiding walks on a simple cubic lattice.