Table of contents

The authors have obtained the four different classes of symmetries for the equation p_t+ delta _u³p+ delta _v³ delta _vp-3 delta _u(p delta _u delta _v^-1p)-3 delta _v(p delta _v delta _u^-1p)=0 which is an analogue of the KdV equation in three spacetime dimensions. Of these four types of symmetries, two depend explicitly on the coordinates. It is then demonstrated that these generators of symmetries close on an infinite-dimensional Kac-Moody algebra.

The Klein inequality is translated into a dynamical mechanism embedded in the quantum-Liouville equation For all t in R. The results can be stated for arbitrary convex (concave) measure functions with a certain entropy function as a special case.

It is shown that there exist only two black-and-white Bravais quasilattices (P_F53m and I_P53m) with icosahedral point symmetry; the former is related to a division of the primitive icosahedral quasilattice (P53m) into two face-centered ones (F53m) and the latter to a division of the body-centered quasilattice (I53m) into two primitive ones. It is shown also that an order-disorder transformation of an icosahedral quasicrystal can be a second-order transition, if the ordering is associated with either of the two sublattice divisions.

The construction of generating functions for the multiplicities of the irreducible representation Gamma in the decomposition of sth supersymmetrical power gamma (s), where Gamma and gamma are the representations of some supergroup G, is discussed. A number of examples are considered. The definitions are given for general groups, but the derived results are for the simplest specific cases.

Expansion coefficients of the coherent states in the basis of Hamiltonian eigenstates contain information about the local character of motion of a quantum system. The author analyses three quantities: the minimal number of relevant eigenvectors, the sum of moduli of coefficients and the Shannon entropy of a coherent state, and show that all of them might be used as indicators of quantum chaos. They also allow one to distinguish between three known universality classes. Results obtained are confirmed by a numerical study of kicked tops.

The quantum mechanical amplitudes for a free, non-relativistic particle moving on a finite-dimensional, homogeneous space S/R are derived by applying first class constraints to a particle moving on the Lie group S, where the amplitudes are already known exactly. This requires gauge fixing, and gauge invariance of the final result is ensured by invoking BRST symmetry and employing ghosts.

An attempt has been made to derive an expression for the energy for the hydrogen atom in a uniform magnetic field having a dependence on only one variable as conjectured by Feneuille (1982). The Schrodinger equation is transformed by a complex Kustaanheimo-Stiefel transformation to that for a four-dimensional oscillator with a constraint. The equivalent oscillator problem is solved approximately to obtain an expression for the quantity n²E (n being the principal quantum number, E the energy) as a function of a single variable beta _c=n³w_c (w_c being the cyclotron frequency). Agreement with experimental observations is discussed.

A new method is proposed for the non-perturbative quantization of certain nonlinear field theories (group-bundle theories), based on a generalization of the idea of a promeasure from vector spaces to infinite-dimensional Lie groups. The quantum theory is not automatically finite, but there is a natural way of imposing a momentum cut-off, leading to the possibility of renormalization. The method relies on the geometrical structure of the classical theory and so may provide clues for the quantization of gravity.

The authors consider 2D random packings of hard discs under gravity. They address the question of the nature of the typical packing of identical discs in the thermodynamic limit. They present numerical results on packings built by gravitational deposition of discs with random radii. Their results suggest that, in the limit of zero randomness: (i) the convergence of the system towards its thermodynamic limit becomes extremely slow; (ii) the limit packing fraction seems to increase drastically. They show in particular that if round-off errors are the only source of randomness of the system, the asymptotic regime cannot be reached because of the huge number of discs needed (at least 10¹⁶ discs using single precision floating-point arithmetic).

Monte Carlo simulations of polymer chains containing up to fifteen hard-sphere repeat units have been carried out, the principal interest being the density distribution of the spheres in the neighbourhood of a rigid boundary. The density discontinuity recently reported by Croxton (1986), in the direction normal to the boundary and at a distance of one sphere diameter from it, is confirmed, although there are considerable differences in the calculated densities very close to the boundary. It is shown that the amplitude of the discontinuity is largely independent of the self-avoiding (excluded volume) condition imposed on the growing chain, but is approximately halved when the rigid boundary restraint is removed. The discontinuity originates mainly in the distribution of the second sphere of the chain sequence. The details of the configurational features of the chains are largely in agreement with Croxton's findings, although trains consisting of more than two spheres have been found. The exponent gamma in the expression (R_N²) varies as (N-1)^gamma , where (R_N²) is the mean square end-to-end length of a chain containing N spheres, is evaluated at approximately 1.37 for 3<or=N<or=15.

The authors consider the dynamics of the n-component Ginzburg-Landau model with nonconserved order parameter (model A) following a quench from a high-temperature equilibrium state to zero temperature. The two-time correlation function of the order-parameter field is found in the 1/n expansion to have the asymptotic scaling form Ck(t,t')=t'^d2/(t/t')^lambda 2/f (k²t,k²t') for t>>t', with f(0,0)=constant. The form of the new exponent lambda (which is a non-trivial function of n and d) was given explicitly to O(1/n) in a recent letter. The purpose of this study is to present a more detailed account of the calculation leading to the O(1/n) form for lambda . They also examine the role of thermal fluctuations in the ordered phase and the effect of long-range initial correlations on the ordering process.

For pt. I see ibid., vol.60, p.1, (1990). By using a novel Monte Carlo algorithm which uses non-local moves to decrease the critical slowing-down, the authors simulate two-dimensional self-avoiding walks (SAWs) in a variable-length fixed-endpoint ensemble. This allows one to determine with reasonable accuracy the critical exponent alpha _sing. As a byproduct, they obtain also accurate measurements of the exponent nu and the connective constant mu . They thus get a direct check of the hyperscaling relation dv=2- alpha _sing. Estimates of alpha _sing and mu are obtained by a maximum-likelihood fit which combines data generated at different fugacities.

A new Monte Carlo method for the determination of relaxation time constants of classical spin systems is presented. The method is applied to a dynamical finite-size scaling calculation.

A simple finite-size scaling method for two-dimensional surface growth models with restricted step heights is proposed. A master equation is written for the time evolution of the surface configuration probabilities in which enter model-dependent interconfiguration rates. These equations are numerically solved for finite systems of increasing sizes and the exponents governing the time evolution of the surface thickness are extracted using finite-size scaling analysis. The method is used to study crossovers in realistic models for ballistic deposition.

The authors use Monte Carlo techniques to study the antiferromagnetic Ising model on the two-dimensional Penrose lattice. Two types of interaction are included, resulting in subtle frustration effects and a rather complex phase diagram.

Describes the development of high-temperature series expansions for general mixed spin-S-spin-S' Ising models on simple loose-packed lattices. It extends previous work on the simpler special case S'=1/2. Coefficients of the series for the initial susceptibility and specific heat of these models on the square, SC and BCC lattices are presented. A preliminary analysis of the results for the particular case S=1, S'=3/2 is given, and the critical ratio K_c and the susceptibility exponent gamma are estimated.

A model consisting of atoms on a triangular lattice, with one degree of freedom, interacting with harmonic forces up to second neighbours and with a phi ⁴ on-site potential is studied. The system's free energy is calculated using an independent site approximation and a quantum variational 'ansatz'. Phase diagrams are constructed by minimizing the free energy, using the simulated annealing Monte Carlo method. They consist of 1D and 2D stable superstructures between which discommensurations intervene. The model is relevant to one degree of freedom systems as, e.g., free to rotate molecular groups.

Presents a complete classification of special points of five-dimensional (5D) Bravais lattices which yield with the cut-and-projection method (2+1)-reducible quasilattices in 3D; the quasilattices are periodic along the c axis but quasiperiodic only along the plane perpendicular to it. There exist five Bravais classes of 5D lattices associated with the (2+1)-reducible quasilattices, namely primitive octagonal, decagonal and dodecagonal lattices, the centred octagonal lattice and the pentagonal lattice. The author also discusses the special points of the reciprocal lattices of these 5D lattices.

An explicit expression is obtained for the perimeter and area generating function G(y, z)= Sigma _n>or=2 Sigma _m>or=1 c_n,myⁿZ^m, where c_n,m is the number of row-convex polygons with area m and perimeter n. A similar expression is obtained for the area-perimeter generating function for staircase polygons. Both expressions contain q-series.

Calculates the exponent theta _S governing the growth of the projected area of the three-dimensional self-avoiding polygons in a plane. The series result suggests that theta _S<2v_SAP, where v_SAP is the correlation length exponent of the three-dimensional self-avoiding polygons.

Using results of the anisotropic surface tension of the two-dimensional Ising model on a square lattice, the exact location of a new class of transition between different cluster shapes is obtained at all temperatures below T_c, in systems with conserved total magnetization and periodic boundary conditions. The authors study the general cases of anisotropic couplings and rectangular system geometries. Monte Carlo simulations confirm the findings. Finite-size effects relevant to the simulations are also analysed. Generalizations to other system topologies, and the saddle point associated with the transitions, are discussed.

In many situations of physical interest, the dissipative fluxes may be considered as a sum of an infinite number of terms or an integral. The formalism of extended irreversible thermodynamics (EIT) is generalized to cover these situations. Several kinds of memory functions of special physical interest, such as the stretched exponential or the potential ones, find in the present formulation of EIT a natural place. As simple illustrations, the formal development is applied to charged suspensions, porous media and monatomic ideal gases.

The authors present the results of an analytical study of certain thermodynamic properties of an ideal relativistic Bose gas confined to the curved space S³*R^d-3, with 3<or=d<or=4. As d approaches the marginal dimensionality 4, the amplitudes pertaining to these properties display a singularity similar to that displayed by the correlation length xi _c of a spherical model of ferromagnetism in the flat-space geometry L^d-d'* infinity ^d', with 2(d<or=4 and d'(2. These results are in sharp contrast with Cardy's (1985) generalization of xi _c for a system in the space S^d-1*R¹, with 2(d(4.

A neural network capable of retrieval of sequential patterns is analysed. The authors treat the network proposed by Buhmann and Schulten (1987). Although they concluded that the thermal noise induces retrieval of sequential patterns, the authors show that the network has limit cycle solutions even at zero temperature. Behaviour of the network at high temperatures is also analysed. The transition temperatures between a high-temperature fixed point phase and a limit cycle phase are calculated numerically to draw a phase diagram. The characteristic features of the phase diagram are discussed. The reentrant transition temperatures between the limit cycle and low-temperature fixed point phases are calculated analytically.

The capacity for storing random patterns in a diluted neural network is determined following the method of Gardner (1989). Two types of dilution are considered. In the quenched case, the broken couplings are chosen at random and are independent of the stored patterns. By contrast, in the annealed case, the disconnected couplings are selected in order to optimize the storage of the patterns. By the same token, the vanishing couplings are strongly correlated with the stored patterns. The authors also determine the distribution of the synaptic strengths. This distribution illustrates the difference between quenched and annealed dilution most clearly.

By adapting an attractor neural network to an appropriate training overlap, the authors optimize its attractor overlap, and subsequently the storage capacity, when retrieval noise (temperature) is present in the system. The training overlap is determined self-consistently by the optimal attractor overlap. The phase diagram of the optimal attractor overlap in the temperature-storage space is found. A novel co-existence phase of strong and weak retrievers is present. The maximum storage capacity deviates from the storage capacity of the maximally stable network on increasing temperature, and in the high-temperature regime (T>or=0.38 for Gaussian noise), the Hopfield network yields the maximum storage capacity. This analysis demonstrates the principles of specialization and adaptation in neural networks.

The dependence of the residual energy e_res on the cooling rate gamma is investigated numerically for a one-dimensional chain of classical particles with anharmonic competing interactions. Due to the complex landscape of the potential energy of the system, with exponentially many barriers and valleys, e_res( gamma ) shows a non-trivial behaviour. For large cooling rates e_res is independent of gamma . In the intermediate gamma -range some plateaux are found which can be understood by means of a simple double well potential and for small gamma the authors find a power-law behaviour for e_res( gamma ), which supports a conjecture by Grest et al. (1987). This power-law behaviour can be explained analytically by means of a kinetic Ising model and the correspondence of the exponents from the analytical theory and those from the simulation is fair for a certain range of the potential parameters.

A gauge invariant interaction between a symmetric second-rank tensor field and a scalar field is examined in detail in exactly two dimensions. The quantum dynamics of the tensor field are determined solely by the gauge fixing term in the effective action. The divergence structure of this model is analysed. All divergences can be removed by renormalizations of the scalar field wavefunction (both linear and nonlinear), and the non-polynomial interaction functions of the scalar field. The authors illustrate how operator regularization can be used to compute radiative effects to one-loop order.

The perimeter generating function derived recently by Brak et al (1990) for row-convex polygons on the square lattice is generalized to the rectangular lattice.

Motivated by some further examples of the use of Feynman integrals which arise in perturbation calculations of the equilibrium properties of a magnetic model of phase transitions, a generalization of the familiar H function of Fox (1961) was proposed recently by Inayat-Hussain (1987). A brief discussion of contour selection and convergence conditions of the integrals involved is presented. It is also shown how some recent work of Buschman (1990) can be applied to derive various recurrence relations for the general H function. Finally, some relevant comments are made on the validity and novelty of two summation formulae for the Clausenian hypergeometric function, which were derived by Inayat-Hussain.

The authors consider a constrained fermionic model in a non-trivial topological sector and compute the conformal charge showing its dependence on the topological charge.

The authors show that the beta model cannot be used to describe the enstrophy cascade in two-dimensional fully developed turbulence, in contrast with some recent claims. Moreover they point out that the scaling law for the energy spectrum cannot be deduced by the scaling of the velocity difference, because of the non-locality of the interactions in the Fourier space.

Examines the integral for the correction to the soliton position in the adiabatic approximation. Using the modified conservation laws for the mass and the energy, the general correction to the soliton position is obtained. It is easily seen that this correction is due to the formation of the shelf.