Table of contents

Examples of bound states are presented, for a system of three identical particles as well as for a system of particles on a lattice. The particles may be bosons as well as fermions. The interaction between them is given by a pair potential, which does not allow two-particle bound states, has positive scattering length and is not too different from realistic interatomic potentials.

With a view to connecting random mutation on the molecular level to punctuated equilibrium behaviour on the phenotype level, we propose a new model for biological evolution, which incorporates random mutation and natural selection. In this scheme the system evolves continuously into new configurations, yielding non-stationary behaviour of the total fitness. Furthermore, both the waiting time distribution of species and the avalanche size distribution display power-law behaviours with exponents close to two, which are consistent with the fossil data. These features are rather robust, indicating the key role of entropy.

An antiferromagnetic spin chain is equivalent to the two-flavour massless Schwinger model in a uniform background charge density in the strong coupling regime. The gapless mode of the spin chain is represented by a massless boson of the Schwinger model. In a two-leg spin ladder system the massless boson aquires a finite mass due to inter-chain interactions. The gap energy is found to be about when the inter-chain Heisenberg coupling is small compared with the intra-chain Heisenberg coupling. It is also shown that a cyclically symmetric -leg ladder system is gapless or gapful for an odd or even , respectively.

We present some simple computer simulations that indicate that short-time aging is realized in a simple model of binary glasses. It is interesting to note that modest computer simulations are enough to prove this effect. We also find indications of a dynamically growing correlation length.

We present a method for determining the globally optimal on-line learning rule for a soft committee machine under a statistical mechanics framework. This rule maximizes the total reduction in generalization error over the whole learning process. A simple example demonstrates that the locally optimal rule, which maximizes the rate of decrease in generalization error, may perform poorly in comparison.

In learning under external disturbance, it is expected that some tolerance in the system will optimize the learning process. In this paper, we give one example of this in learning from stochastic rules by the Gibbs algorithm. Using the replica method, we show that for the case of output noise, there exists an optimal temperature at which the generalization error is a minimum. This temperature exists even in the limit of large training sets and is determined by the stable replica symmetric solution. On the other hand, for other types of noise no such temperature exists and the asymptotic behaviour is determined by the one-step replica symmetric breaking solution. Further, the asymptotic expressions for learning curves are derived. They are precisely the same as those for the minimum-error algorithm.

The properties of recurrence and ergodicity breaking in a discrete model that simulates the generic Hamiltonian motion are studied. These properties are respectively characterized by the distribution of orbit periods and the division in sectors of phase space. Despite its simplicity, the model can exhibit an intricate structure and, in fact, is able to mimic both regular and chaotic evolution. This complexity is also revealed in the appearance of power-law decays in the period distribution.

The ferromagnetic q-states Potts model on a square lattice is analysed, for q > 4, through an elaborate version of the operatorial variational method. In the variational approach proposed in this paper, the duality relations are exactly satisfied, involving at a more fundamental level, a duality relationship between variational parameters. Besides some exact predictions, the approach is very effective in the numerical estimates over a wide range of temperature and can be systematically improved.

For the three-dimensional gonihedric Ising models defined by Savvidy and Wegner the bare string tension is zero and the energy of a spin interface depends only on the number of bends and self-intersections, in antithesis to the standard nearest-neighbour three-dimensional Ising action. When the parameter , weighting the self-intersections, is small the model has a first-order transition and when it is larger the transition is continuous. In this paper we investigate the scaling of the renormalized string tension, which is entirely generated by fluctuations, using Monte Carlo simulations for , 0.1, 0.5 and 1.0. The scaling of the string tension allows us to obtain an estimate for the critical exponents and using both finite-size scaling and data collapse for the scaling function. The behaviour of the string tension when the self-avoidance parameter is small also clearly demonstrates the first-order nature of the transition in this case, in contrast to larger values. Direct estimates of are in good agreement with those obtained from the scaling of the string tension. We have also measured .

Information theory is used to perform a thermodynamic study of nonequilibrium anisotropic radiation. We limit our analysis to a second-order truncation of the moments, obtaining a distribution function which leads to a natural closure of the hierarchy of radiative transfer equations in the so-called variable Eddington factor scheme. Some Eddington factors appearing in the literature can be recovered as particular cases of our two-parameter Eddington factor. We focus our attention on the study of the thermodynamic properties of such systems and relate it to recent nonequilibrium thermodynamic theories. Finally, we comment on the possibility of introducing a nonequilibrium chemical potential for photons.

Reaction-diffusion systems which include processes of the form or are characterized by the appearance of `imaginary' multiplicative noise terms in an effective Langevin-type description. However, if `real' as well as `imaginary' noise is present, then competition between the two could potentially lead to novel behaviour. We thus investigate the asymptotic properties of the following two `mixed noise' reaction-diffusion systems. The first is a combination of the annihilation and scattering processes , , and . We demonstrate (to all orders in perturbation theory) that this system belongs to the same universality class as the single species annihilation reaction . Our second system consists of competing annihilation and fission processes, and , a model which exhibits a transition between active and absorbing phases. However, this transition and the active phase are not accessible to perturbative methods, as the field theory describing these reactions is shown to be non-renormalizable. This corresponds to the fact that there is no stationary state in the active phase, where the particle density diverges at finite times. We discuss the implications of our analysis for a recent study of another active/absorbing transition in a system with multiplicative noise.

Set inversion is applied to recent collision-induced scattering data concerning gaseous . It makes it possible to approximate the set of all vectors of independent components of the dipole - quadrupole and dipole - octopole polarizability tensors. Numerical analysis shows that short-range effects must be taken into account in the high-frequency range of each dipole - multipole contribution to the CIS isotropic spectrum of . It also demonstrates the agreement between experiment and recent ab initio calculations.

We consider a one-dimensional solid-on-solid growth model in which the nearest-neighbour height differences are restricted to take the values . Deposition occurs at local minima with probability f, while diffusion moves of single adatoms within a layer occur with probability 1 - f. Interlayer transport and detachment from step edges is suppressed. For the stationary distribution of the model is known, hence the growth-induced surface current can be computed analytically for small diffusion rates. In the opposite, diffusion-dominated limit, , a description in terms of step flow is possible for slopes larger than a critical slope . For smaller slopes the surface phase separates into regions of slope . The stationary domain size diverges for as , where . We suggest that the large-scale behaviour in this limit can be described by the noisy Kuramoto - Sivashinsky equation in its noise-dominated regime.

A perceptron with N random weights can store of the order of N patterns by removing a fraction of the weights without changing their strengths. The critical storage capacity as a function of the concentration of the remaining bonds for random outputs and for outputs given by a teacher perceptron is calculated. A simple Hebb-like dilution algorithm is presented which, in the teacher case, reaches the optimal generalization ability.

We studied a random heteropolymer chain with a Gaussian distribution of types of monomers. Long-range correlations between types of monomers were introduced. The mean-field analysis of such a heteropolymer indicates the existence of an infinite energy barrier between the heteropolymer random coil and the frozen states. Thus, the frozen state is kinetically unavailable for the random heteropolymer with power-law correlations in the sequence of the monomer. The relationship between our results and some obtained earlier for the DNA intrones sequences are discussed.

The link invariant associated with the Izergin - Korepin 19-vertex model is deduced using the method of statistical mechanics. It is shown that the Izergin - Korepin model leads to an invariant which is precisely the 3-state Akutsu - Wadati polynomial, previously known only for 2- and 3-braid knots. We give a table of the invariant for all knots and links up to seven crossings.

The classical `Kirchhoff's theorem' (the energy density of the radiation at equilibrium at high temperature, T, is a function of T only) is used to obtain the Casimir energy at zero temperature without recourse to regularization. The validity of `Kirchhoff's theorem' at the high-temperature limit for the case at hand is confirmed. The Casimir entropy is defined and its temperature dependence is displayed. The Casimir entropy at high temperatures is shown to approach a positive geometry-dependent but temperature-independent constant.

A two-temperature lattice gas model with repulsive nearest-neighbour interactions is studied using Monte Carlo simulations and dynamical mean-field approximations. The evolution of the two-dimensional, half-filled system is described by an anisotropic Kawasaki dynamics assuming that the hopping of particles along the principal directions is governed by two heat baths at different temperatures and . The system undergoes an order - disorder phase transition as is varied for sufficiently low fixed . The non-equilibrium phase transition remains continuous and the critical behaviour belongs to the Ising universality class. The measure of violation of the fluctuation - dissipation theorem can be controlled by the value of the fixed temperature. We have found an exponential decay of spatial correlations above the critical region in contrast to the two-temperature model with attractive interactions.

We study a kinetic neural network in which the intensity of synaptic couplings varies on a timescale of order compared with that for neuron variations. We describe some exact and mean-field results for . This includes, for example, the Hopfield model with random fluctuations of synapse intensities such that neurons couple each other, on average, according to the Hebbian learning rule. The consequences of such fluctuations on the performance of the network are analysed in detail for some specific choices of the rate and fluctuation distribution, including the case in which couplings are asymmetric.

Under some integrability conditions we derive raising and lowering differential recurrence relations for polynomials orthogonal with respect to a weight function supported in the real line. We also derive a second-order differential equation satisfied by these polynomials. We discuss the Lie algebra generated by the generalized creation and annihilation operators. From the differential equations, Plancherel - Rotach type asymptotics are derived. Under certain conditions, stated in the text, an Airy function emerges.

A solvable model for non relativistic quantum scattering in three-dimensional space of a particle off several interactions centred at n arbitrary points is given. The interaction at one centre can be very different from the interaction at another. Each interaction centred at a given point can be effective on any partial wave with respect to this point. Each centred interaction is a sum of projectors on a finite number of partial waves relative to the centre. Elementary applications are given.

The concept of parameter-space size adjustment is proposed in order to enable successful application of genetic algorithms to continuous optimization problems. Performance of genetic algorithms with six different combinations of selection and reproduction mechanisms, with and without parameter-space size adjustment, were severely tested on eleven multiminima test functions. An algorithm with the best performance was employed for the determination of the model parameters of the optical constants of Pt, Ni and Cr.

We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel straight strips or planar layers coupled by a finite number N of windows in the common boundary. If the windows are small enough, there is just one isolated eigenvalue. We find upper and lower asymptotic bounds on the gap between the eigenvalue and the essential spectrum in the planar case, as well as for N = 1 in three dimensions. Based on these results, we formulate a conjecture on the weak-coupling asymptotic behaviour of such bound states.

The diagrammatic cumulant expansion for the periodic Anderson model with infinite Coulomb repulsion is considered here for an hypercubic lattice of infinite dimension . The nearest neighbour hopping of the uncorrelated electrons is described exactly by a conduction band, while two different models of hybridization are treated as a perturbation. The same type of simplifications obtained by Metzner for the cumulant expansion of the Hubbard model in the limit of , are also shown to be valid for the periodic Anderson model. The derivation of these properties had to be modified because of the exact treatment of the conduction band.

A quantum theory is developed for the scattering of massive scalar fields on a class of non-globally hyperbolic spacetimes represented by foliations of Minkowski spacetime with a fixed compact set removed from each Cauchy surface. The field is restricted to the exterior of this set (exterior domain). At the classical level, the boundary value problem is recast as an abstract Cauchy problem in a Hilbert space, and the field solution is obtained as a unitary mapping of Cauchy data. The scattering theory is treated using a two Hilbert space approach, and the wave operators are constructed and shown to be asymptotically complete. At the quantum level, a field operator is constructed yielding a representation of the CCRs on a Fock space. Representations for the `in' and `out' asymptotic fields are developed, and a scattering operator is constructed and shown to be unitarily implementable.

In this paper we give a complete analysis for general tridiagonal matrix inversion for both non-block and block cases, and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or Toeplitz tridiagonal matrices.

We consider an interaction between superconformal fields of the same gradation. This entails the construction of a supersymmetric Poisson tensor for these fields, generating a new class of Hamiltonian systems. The Lax representation is found for one of them by supersymmetrizing the Lax operator for the Hirota - Satsuma equation. The supersymmetric equation is not reducible to the classical Hirota - Satsuma case. We show that our generalized system can be reduced to the the supersymmetric KdV equation (a = 4). Surprisingly the integrals of motion are not reduced to integrals of motion of the supersymmetric KdV equation.

We numerically investigate the structure of the basins of attraction and the nature of the spurious attractors of the pseudo-inverse and the optimal weights attractor neural networks. We show that the number of attractors in the optimal weights model increases as the margin parameter increases, and that the basins of attraction of the stored patterns are not significantly enlarged by . Moreover, the number of attractors is smaller than in the pseudo-inverse model.

We investigate the nature of particular solutions to the ultradiscrete Painlevé equations. We start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions. For the ultradiscrete Painlevé equations II and III we present special solutions reminiscent of the Casorati determinant ones which exist in the continuous and discrete cases. Finally we analyse the discrete Painlevé equation I and show how it contains both the continuous and the ultradiscrete ones as particular limits.

In the present paper we study transitions induced by squeezed light generated by an optical device as a degenerate parametric amplifier. We adopt the dipole approximation and approach the problem by path integral methods. The light variables do not appear in the final propagator as they are integrated over. Using perturbation theory, we calculate the transition probability from the ground state 1s of the hydrogen atom to the state 3d. Further we obtain the equation obeyed by a free electron in squeezed light.

From the point of view of the Young superdiagram method, an analytic Bethe ansatz is carried out for Lie super algebra . For the transfer matrix eigenvalue formulae in dressed-vacuum form, we present some expressions, which are quantum analogues of Jacobi - Trudi and Giambelli formulae for Lie superalgebra . We also propose transfer-matrix functional relations, which are Hirota bilinear difference equations with some constraints.

In a recent paper `The capacity of the Hopfield model', Feng and Tirozzi claim to prove rigorous results on the storage capacity that are in conflict with the predictions of the replica approach. We show that their results are in error and that their approach, even when the worst mistakes are corrected, does not give any mathematically rigorous results.