Table of contents

The authors examine some of the standard features of primary fields in the framework of a q-deformed conformal field theory. By introducing a q-OPE between the energy-momentum tensor and a primary field, they derive the q-analogue of the conformal Ward identities for correlation functions of primary fields. They also obtain solutions to these identities for the two-point function.

A building principle working for both atoms and monoatomic ions is proposed. This principle relies on the q-deformed 'chain' SO(4))SO(3)q.

The authors present three real forms of quantum superalgebra U_q(OSp(1/2)). By defining suitable contraction limits they describe the q-deformations of D=2 superPoincare and D=2 superEuclidean algebras as Hopf bialgebras.

The peculiarities of the Hamiltonian description of 2D hydrodynamics and magnetohydrodynamics are discussed in connection with recent attempts to construct a thermodynamical picture for 2D turbulence. The unconstrained Hamiltonian variables are displayed in both cases and the role of topology of the flow in tentative statistical equilibria is discussed. As a by-product the Clebsch-type variables for 2D magnetohydrodynamics are obtained describing flows with initial zero vorticity.

The authors consider a system of identical particles distributed in two clusters and described by harmonic oscillator wavefunctions. A set of non-spurious two-cluster states, whose centre-of-mass is at rest, is constructed. It is characterized by the cluster Yamanouchi symbols as well as by its overall permutational symmetry, angular momentum and total energy. Applications to nuclear cluster models, to the evaluation of nuclear spectroscopic factors, to the description of nuclei as quark-clusters, and to the study of interacting rare-gas clusters are pointed out.

The planar differential cross section for the Mott scattering of two identical charge anyons is obtained in a closed form. This expression exhibits a marked asymmetry between forward and backward angles, and is especially simple for the semion-semion scattering. At low energies, a pronounced dip in the differential cross section due to planar Coulomb-anyonic interference appears at the forward angles, whose implication is discussed.

A modified version of the random sequential adsorption model for the deposition of spherical particles onto a plane is presented. Successive particles are added via randomly positioned vertical trajectories and, if they do not stick to the plane at first contact, follow the path of steepest descent on the previously deposited particles before being adsorbed. All particles reaching a stable position in contact with three previously adsorbed particles are removed. The limiting coverage is found to be 0.610 56+or-0.000 05 (larger than that of the standard RSA model) and is shown to be reached exponentially with the number of trials. The size distribution of connected clusters of spheres is analysed and is found to exhibit an exponential decay for large cluster sizes.

The author studies the winding angle Theta of planar Levy flights around a given point, say the origin O. In particular, the author calculates the variance ( theta ²) and show that, after a long time t, it is proportional to In t (for Brownian random walks ( theta ²) varies as (In t)²). In the same conditions, the author shows that the variable x= theta /In t is approximately distributed according to a Gaussian law. Finally, the authors compare results with computer simulations.

The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent.

The antiperiodic 3D spherical model is studied on a hypercube lattice, two sides of which are finite and the other infinite. The universal correlation length finite-size scaling amplitude in the conjectured relationship xi /L=A/x is calculated exactly and found to be A=0.1361 . . ., which should be compared with the antiperiodic 3D Ising model value of A approximately=0.12.

A class of mean field models for the patterns formed by growing and coalescing droplets on a line is introduced and solved. All the models are found to have a scale-invariant regime where the patterns are self-similar in time.

The ground state of the one-dimensional Falicov-Kimball model (1969) is considered for large values of the interaction strength U. If the ion density rho _i=p/q (p and q relatively prime) is rational and equals the electron density rho _e the author proves that for U)U^hom_c(q) the ion configurations is the most homogeneous one. If 0( rho _e( rho _i and U)U^seg_c( rho _e/ rho _i) w prove that all the ions condense into one large cluster. This last result proves a recent conjecture known as the segregation principle.

Using a renormalization group calculation the authors show that the most probable value of the static correlation function of the Ising model on a SAW decays as a stretched exponential (with an exponent equal to the dimension D of the SAW). They prove that the thermal exponent v_T equals D^-1. On the basis of extensive Monte Carlo simulations they conclude that the spin-spin autocorrelation function also decays as a stretched exponential. They find evidence for a breakdown of dynamical scaling.

An attempt is made to apply the mathematical theory of random jump processes with complete connections to the physically oriented description of very long memory effects. A new stochastic renormalization approach is introduced. Starting from a stochastic process with infinite memory, the author attaches to each step a very small probability that the memory is lost. As a result of this change the memory acts over a variable number of steps, forming 'blocks' of different sizes. This renormalization procedure leads to the self-similarity of the probability of the number of steps over which the memory is kept. The model yields a new explanation for the fractal time.

Dynamics of a kink in the phi ⁴ model with additional terms accounting for localized or spatially random inhomogeneities are analyzed by means of the perturbation theory. First, conditions for the capture of the free kink by a local defect are found. Next, emission of radiation by the kink colliding with the local defect, and collision-induced generation of the kink's shape (internal) mode and the defect-sustained impurity mode are considered. Finally, the non-dissipative braking of the moving kink in a randomly inhomogeneous medium on account of the excitation of the shape mode is analyzed.

Q2R cellular automata are shown to attain thermal equilibration in the two-dimensional Edwards-Anderson model (1975) of a spin glass. Some problems associated with equilibration at low temperatures are identified as finite-size effects. The authors use a thermalization test due to Bhatt and Young (1988). Their results indicate the possibility of calculating Gibbs averages in this model by means of the Q2R algorithm. A rough efficiency comparison with the conventional (Metropolis) Monte Carlo algorithm is also made.

The authors perform a detailed investigation of the storage properties of a model for neural networks that exhibits the same organization into clusters as Dyson's hierarchical model (1969) for ferromagnetism, combined with Hebb's learning algorithm for an extensive number of stored patterns p= alpha N, where N is the size of the network. The authors first perform a signal-to-noise analysis, obtaining a succession of critical storage capacities for the 'ancestor' and its 'descendants' that are below the Hopfield value. Afterwards they apply the statistical mechanics formulation of Amit. Gutfreund and Sompolinsky (1987), to obtain also in this case a succession of critical storage capacities that are below the corresponding value for Hopfield's model. In both cases they consider the ratio of the critical storage capacity for the 'ancestor' to the same quantity as evaluated in Hopfield's model, and they prove rigorously that the signal-to-noise method provides a lower bound for this ratio, that is bounded from above by unity.

The authors have investigated the multifractal scaling of conductance jumps in a hierarchical percolation lattice, resulting from cutting the current carrying bonds in the lattice. Due to the iterative nature of the model, exact renormalization group (RG) equations are obtained and used to extract the minimum conductance jump of the lattice. They find an asymptotically analytic expression for the minimum conductance jump, Delta g_min approximately=exp(-c(log L)²) decreasing faster than any power law. They observed slow convergence to the asymptotic behaviour due to the importance of the irrelevant terms in the RG equations at low generations of the lattice. Numerical calculations are performed in order to validate the analytic results and to calculate the f- alpha spectrum to confirm left-sided multifractality as proposed by Lee and Stanley (1988).

The authors study the multiple scattering process of light particles in random media in the small-angle approximation. Particular consideration is given to the reflection of particles impinging under a glancing angle on a surface. The particle flux is described by a linear integro-differential equation. Results for the space, angle and path-length-dependent particle flux are derived using the method of eigenfunctions. The properties of the solution are discussed for various power-law scattering potentials, and the differences from the well known diffusion case, corresponding to Coulomb scattering, are emphasized. For the special case of an inverse-square interaction potential, a rigorous solution of the transport equation is derived, and compared to two approximations which have been employed in the literature: deviations occur in particular for small path lengths travelled (small energy loss). The influence of the boundary condition at the surface on the solution is investigated. Applications of the theory to energy loss spectra of reflected particles are discussed.

The maximum capacity for storage of temporal sequences in a time-summating neural network is analysed within a statistical-mechanical framework. Each neuron of such a network maintains an activity trace consisting of a decaying sum of all previous inputs to that neuron. The maximum storage capacity is expressed as a function of the temporal correlations of the local fields of the neurons and evaluated using a perturbation expansion in powers of the decay rate.

A neural-network approach is presented for making short-term predictions on time series. The neural network does better at short-term predictions of a chaotic signal than does an optimum autoregressive model. Also, the neural network is clearly capable of distinguishing between chaos and additive noise.

An associative algebra is defined on the set of certain irreducible representation spaces of the algebra U_q(sl(n)) when q is a root of unity. This associative algebra is shown to be connected with some algebras defined by the decomposition rule of tensor products of the irreducible representations of certain finite groups.

It is shown that a recursive use of the transformation for a terminating ₃F₂(1) series used by Weber and Erdelyi (1952), which belongs, as shown by Whipple (1925), to a set of equivalent ₃F₂(1) functions obtained by Thomae (1979), results in a 72-element group associated with 18 terminating series. The generators, conjugacy classes, invariant subgroups, characters and dimensions of irreducible representations for this group are presented.

The authors solve the one-dimensional diffusion equations for the electromagnetic cascade shower in a spinorial representation.

The effects of time-dependent unitary transformations on the geometrical phase are investigated in the general non adiabatic setting. A detailed study considering both paths of an interference experiment shows that even though the geometry of the fibre bundle: Hilbert space to space of states changes. The measured relative phase, geometrical phase and dynamical phase are all invariant under these transformations.

The authors point out that the very notion of space reflection is ill-defined when physical states are defined on a fibre bundle describing a charge-Dirac monopole system. In other words, there is no lift of the space reflection operator to the total space of such a non-trivial bundle. They construct a well defined transposition operator of two dyons on the non-trivial two-dyon bundle; consequently, they can correctly define its action on local sections. It is shown that symmetric wavefunctions defined on this bundle cannot be transformed into antisymmetric ones by a gauge transformation, in contradiction to the well known statement first pointed out in connection with the dyon spin problem.

A new method for the numerical evaluation of slowly convergent or even divergent series involving the Hermite functions is presented. The author considers series with either of the forms where c_m decays algebraically as m to infinity . The first series is a Fourier-Hermite series, while the second arises in the representation of Green functions for problems whose eigenfunctions involve the Hermite functions. By use of the Poisson summation formula, the author derives rapidly convergent asymptotic expansions for the remainders of these series after a sufficiently large number of terms. The series can then be evaluated as a partial sum plus an asymptotic approximation to its remainder. The asymptotic expansion for the remainder of G(x,y) also reveals the nature of the possible singular behaviour of this series near x=y.

Dynamical systems may possess, in addition to symmetries that leave the equations of motion invariant, reversing symmetries that invert the equations of motion. Such dynamical systems are called (weakly) reversible. Some consequences of the existence of reversing symmetries for dynamical systems with discrete time (mappings) are discussed. A reversing symmetry group is introduced and it is shown that every (weakly) reversible mapping L can be decomposed into two mappings K₀ and K₁ of the same order 2^l (limit l to infinity included) such that K₀² K₁²=I. Some applications are discussed briefly.

A nonlinear model of heat propagation is presented, from which a new heat conduction equation is derived. An exact similarity solution in closed form of this equation is obtained, which reveals the travelling wave characteristics for the transient temperature distribution. It is shown that the temperature disturbances propagate with finite velocity, which is a monotonically decreasing function of time.

Hamiltonian theories with fermions are proved to be equivalent to hierarchies of ordinary Hamiltonian theories. The corresponding Poisson brackets are defined in terms of the original super-Poisson structure, while Hamiltonian functions are simply the coefficients in the expansion of the super-Hamiltonian function as a formal power series in Grassman generators. Fermion extensions of the KdV equation are considered to illustrate the general result; its space-supersymmetric extensions are used to show in particular how supersymmetry transformations can be recast as ordinary Hamiltonian symmetries.

Self-adjoint extensions of the operator- Delta with the domain C₀^infinity (R³) in the space C^k(+)L₂(R³) are described. Such operators are interpreted as Hamiltonians of a point interaction of a quantum particle with the vacuum. Bound states and scattering objects of these Hamiltonians are investigated.

The zero point energy associated with a Hermitian massless scalar field in the presence of perfectly reflecting plates in a 3D flat space-time is discussed. A new technique to unify two different methods-the zeta function and a variant of the cut-off method-used to obtain the so-called Casimir energy is presented, and the proof of the analytic equivalence between both methods is given.

The Noether theory of infinitesimal point symmetry transformations is revisited and generalized under the broad scope of the general theory of transformations of Lagrangian mechanics. The basic generalized point symmetries of a Lagrangian function are thus obtained and briefly discussed. It is proved that (under very general and physically reasonable provisions) the point symmetry group of the Lagrangian is necessarily a finite Lie group, and a rather simple technique is then introduced for the explicit calculation of the associated Lie algebra. Next, the algebra obeyed by the set of basic Noether quantities (which also includes the traditional Noether constants of motion) is examined. In this way it is shown that not all the generalized point symmetries of a Lagrangian yield an associated conservation law. The author presents three simple examples of this (not so well known) aspect of the Noether theory.

The authors revise their formula for the main overlap evolution at t=3 generated by the parallel dynamics in the symmetric extremely diluted neural network. They show that for t>or=3 correlations between the Gaussian and the Bernoulli's memory-like noises have to be taken into account. To derive the exact formula for t=3 they use their previously developed truncated dynamics method.

The author shows that the technique of integration within ordered product (IWOP) can also provide one with a simple approach to antinormally ordering some multimode exponential operators both in boson and fermion cases.