Table of contents

An alternative generalized equation which incorporates the modified Korteweg-de Vries (MKdV) and KdV equations is proposed and from it a new integrable system of the MKdV type is found.

Consideration is given to the solution of the stochastic equation governing the coagulation of particles for the case of a size-independent coagulation probability. The corresponding expression for the expectation value of particle number (N) is developed in the form of an asymptotic series valid for N>>1, the first term being the solution of the relevant deterministic equation. The next term, giving the first-order correction to this result, is obtained explicitly, together with an estimate of the corresponding standard deviation of N.

Given the moments of the Hamiltonian of a system, the problem is to find a prescription for the matrix elements in the tridiagonal Lanczos representation in a form suitable for numerical calculation. In this letter a sequence of reductions of the moments is described, whereby the redundant information contained in the higher moments is progressively removed, leading to a numerically stable algorithm for the Lanczos matrix elements.

The author presents a method for using a gauge transformation to construct a bi-Hamiltonian structure of a finite-dimensional integrable Hamiltonian system reduced from a soliton equation. This is used to construct the bi-Hamiltonian structure for two systems which are related to the second-order polynomial spectral problem and its modified spectral problem, respectively.

The authors extend Sklyanin's formalism for constructing integrable open spin chains to the case of 'non-symmetric' (PT-invariant) R-matrices. They use this formalism to show that the Hamiltonian of an integrable open chain is invariant under gauge transformations of the R-matrix.

The time evolution and geometry of rough interfaces observed in experiments on immiscible displacement of viscous fluids in porous media are analysed using the concepts of dynamic scaling and self-affine fractal geometry. The authors find that the development of the interface is governed by dynamic scaling and they have determined the corresponding surface exponents alpha and beta . The values of the exponents calculated for the experimental patterns are different from those obtained for a variety of two-dimensional models of marginally stable interfaces.

The author summarizes an extensive numerical study of basin and attractor sizes among the 88 distinct elementary cellular automata (CA) rules. Based on this study and on previous work in discretized dynamical systems, he proposes a new classification of CA, complementary to that of Wolfram, in which attractor globality is important. With the use of fixed boundary conditions he finds global periodic attractors in CA for the first time. Not a single instance of attractor chaos is observed in this class of rules.

A recently derived theoretical method for modelling molecular beam epitaxy on stepped surfaces, which includes a nonlinear term for nucleation, has been extended so that large deviations from periodic step structure can be examined. The method is used in conjunction with Monte Carlo simulations to monitor the growth dynamics of a stepped surface with unequal terrace lengths and identify the stable configuration. The authors show that the equidistant step configuration is favoured even in growth regimes where nucleation on the terraces competes with atom incorporation at steps. Furthermore, they found a remarkable qualitative correspondence of the results obtained from the nonlinear diffusion equations and the simulations.

The authors present a new numerical method of obtaining critical temperatures and exponents for ferromagnetic spin models and dynamical transitions. This method is based on the spreading of damage in a temperature gradient. They illustrate the method on the two-dimensional Ising model and present new results on the +or- Ising spin glass in two dimensions. They also obtain new critical exponents, describing the behaviour of the width and the length of the damage front.

A very simple proof is presented for the assumption that if the potential in the Schrodinger equation is integrable and in the Rollnik class, then for all energies the scattering amplitude is trace class. Some consequences are discussed.

The authors give a complete description of q-deformation of SU(2) algebra in terms of symplectic geometry. The geometric meanings of such a deformation are manifestly shown at classical as well as at quantum level. As a model they study the classical and quantum dynamics of a particle with q-spin moving in electromagnetic fields in detail.

The authors have discussed the modular properties of characters in Feigin-Fuch representation for SU_k(2) WZW and minimal models, and they have given the explicit expressions of the S-matrix . The proof is given for a simple case.

Results are presented for both definite and indefinite integrals of certain products of two modified Bessel functions K_v. General recurrence relations are developed for these integrals which depend on both the order of the modified Bessel functions and various parameters. Explicit low-order formulae and special cases are given and many of these have application to mathematical and physical problems where the Green function for the Helmholtz operator in two dimensions (K₀) appears.

The operator-equivalent method was introduced by Stevens in 1952. This method enabled him to determine the quantum mechanical equivalent of a given spherical harmonic C_kq-the so-called Racah tensor operator C_kq-as an explicit function of the total angular momentum operator J within a constant J manifold. The method itself uses spherical harmonics in Cartesian coordinates and from each spherical harmonic the corresponding Racah tensor operator is calculated. This paper shows that it is useful to fix tau k- mod q mod as all spherical harmonics C_{mod q mod + tau , mod q mod} possess the same polynomial structure with coefficients showing a simple dependence on the absolute value of q. For the following Racah operators C_{mod q mod + tau , mod q mod} , which hold for all mod q mod , Stevens' operator-equivalent method has to be generalized. By an application of the Wigner-Eckart theorem to the operator-equivalents C_{mod q mod + tau} ,_{mod q}, _mod (J), the 3j-symbols with two identical js disclose its innermost functional dependence on the variables. To give an example the calculations are explicitly stated for tau =0, 1, 2, . . .5.

The authors discuss gravitational billiards, i.e. the two-dimensional motion of a point mass inside a hard boundary curve under the influence of a constant (e.g. gravitational) field. A parabolic boundary is shown to be a new example of integrable billiards. The system has a second integral of motion in addition to the energy, which is constructed analytically. Stability and bifurcation properties of the central periodic orbits are discussed. The results also shed new light on the known integrable case of elliptic non gravitational billiards.

The behaviour of water droplets condensed on ferromagnetic particles of submicron sizes, which are falling in a gaseous medium, is investigated. Under intensive lighting a number of droplets move in a magnetic field similarly to objects having magnetic charge. The mean value of this charge coincides with the value of the Dirac monopole charge.

Scattering of positive and negative energy Dirac particles moving under the action of vector and scalar point interaction potentials faced to an impenetrable wall has been discussed. The occurrence of a well defined resonance pattern is only found for scalar couplings.

In the theory of elastic waves for crystals of hexagonal, tetragonal and rhombic symmetries a pseudonormal vector is introduced by which the Green-Christoffel tensor can be represented in Kelvin form, i.e. as a sum of a diagonal tensor and a dyad. It is shown that this approach provides a number of new simple relations for specific directions in the aforementioned crystals. For crystals of trigonal, monoclinic and triclinic symmetries a more general representation of the Green-Christoffel tensor as a sum of a diagonal tensor and two dyads is suggested.

In this and the following two papers in this series it is shown how the radial part of non-relativistic and relativistic hydrogenic bound-state calculations involving the Green functions can be presented in a unified manner. The angular part of such calculations, being well understood, is performed in the standard way. In this, the first paper, it is shown how a suitable linear transformation of the two relativistic radial wavefunctions allows the pair of relativistic coupled differential equations to be written as two uncoupled second-order equations which are simple generalizations of the corresponding non-relativistic equation. This transformation is presented in a manner which allows for a simple extension to the Green function problem. The transformed relativistic wavefunctions are explicitly derived and the normalization is presented in a novel and simple way. A new derivation is given for the recursion relations for both non-relativistic and relativistic radial wavefunctions, some of which are new. These relations are required in the subsequent papers.

For pt.I see ibid p.79-94. This is the second in a series of three papers in which it is shown how the radial part of non-relativistic and relativistic hydrogenic bound-state calculations involving the Green functions can be presented in a unified manner. In this paper the nonrelativistic Green function is examined in detail; new functional forms are presented and a clear mathematical progression is shown to link these and most other known forms. A linear transformation of the four radial parts of the relativistic Green function is given which allows for the presentation of this function as a simple generalization of the non-relativistic Green function. Thus, many properties of the non-relativistic Green function are shown to have simple relativistic generalizations. In particular, new recursion relations of the radial parts of both the non-relativistic and relativistic Green functions are presented, along with new expressions for the double Laplace transforms and recursion relations between the radial matrix elements. A novel proof of the Sturmian form of the radial Green functions is given in an appendix.

A refined bound for the correlation information of an N-trial apparatus is developed via an heuristic argument for Hilbert spaces of arbitrary finite dimensionality. Conditional upon the proof of an easily motivated in equality the author then finds the optimal apparatus for large ensemble quantum inference, thereby solving the asymptotic-optimal state determination problem. In this way he identifies an alternative inferential uncertainty principle, which is then contrasted with the usual Heisenberg uncertainty principle.

The author investigates a family of solvable potentials related to the Jacobi polynomials. This one-dimensional potential family depends on three parameters and is restricted to the domain XO, so it can be interpreted as the radial part of a central potential in three dimensions (with l=0). Closed expressions are obtained for the bound state energy spectrum and the wavefunctions. The supersymmetric partner of this potential is also determined and it is found not to belong to the same potential family. It is shown that this potential family is a special subclass of the general six-parameter Natanzon potential class and similarities with another subclass, the Ginocchio potentials, are pointed out. Some aspects of supersymmetric quantum mechanics and shape invariance are also discussed in connection with the potential family under study.

The author determines the largest kinematical and dynamical invariance Lie superalgebras characterizing n-dimensonal harmonic oscillators when the spin-orbit supersymmetrization procedure is under study. These considerations essentially add new Heisenberg superstructures in the case of an arbitrary even number of spatial dimensions.

The classical scattering of a particle by an oscillating potential well is investigated. It is shown that the singularities of the scattering map S form a Cantor set for this system. This result is obtained from an analysis of the particle's return map, which contains a Smale's horseshoe and, consequently, a hyperbolic invariant set with chaotic dynamics. The Cantor set structure of the S-singularities involves arbitrarily complex scattering behaviour.

The authors investigate the localization properties of the eigenvectors of a banded random matrix ensemble, in which the diagonal matrix elements increase along the diagonal. They relate the results to a transition in the spectral statistics which is observed as a parameter is varied, and discuss the relevance of this model to the quantum mechanics of chaotic Hamiltonian systems.

Symmetric coupling of two critical circle maps near the golden mean rotation number is considered. On the basis of a renormalization group method the three universal types of interaction are found. The theoretical scaling predictions are confirmed by numerical calculations.

The finite-size energy spectrum of the anisotropic Heisenberg chain in an external magnetic field is calculated for free and twisted boundary conditions. As with periodic boundary conditions, it is found that the spectra exhibit non-analytical terms which do not fit into the form predicted on the basis of conformal invariance unless extra commensurability conditions between the size of the system and the external field are introduced. Taking these conditions into account some scaling dimensions for associated models are derived.

Non-intersecting (or vicious) random walker models in one dimension can be interpreted as models of domain walls in two dimensions. Three problems pertaining to vicious walker models are solved. The first is the exact evaluation of the partition function for the random turns model of vicious walkers on a lattice. In this model, at each tick of the clock, a randomly chosen walker must move one step to the left or one step to the right. The second problem is the calculation of the mean spacing between walls in terms of the chemical potential for a Brownian motion model of continuous domain walls in a strip, while the final problem solved is the calculation of the correlation between defects, which occur when two domain walls meet and end without crossing the whole system.

The deterministic Q2R cellular automation has been tested by many authors as a fast algorithm to simulate the Ising model in the microcanonical ensemble. However, the magnetic susceptibility curve, measured from the fluctuations of the magnetization, is found to be far below the expected results inside the ordered Ising phase. The non-ergodicity degree of this automaton seems to be inadequate to simulate the Ising dynamics. In this work, the authors introduce two modified automata, and test their performance for the square lattice. For the first one, they found improved results concerning the Ising transition. For the second modified automaton the Ising transition does not occur, the magnetization vanishing even for energies far below the normal critical threshold. On the other hand, another transition appears at a different critical energy value that seems to be the same as that of the periodic-chaotic transition already observed in the normal Q2R dynamics. The critical indices of this new transition are measured from finite-size scaling, the results indicating that it is at the same universality class as the Ising transition.

Power series in the time are constructed for various versions of Eden-like clusters. The use of an Euler transform gives series with smoothly varying coefficients of positive sign, indicating that the radius of convergence of the series in the new variable is the physically interesting singularity. The growth exponent obtained from the beginning terms of the series is larger than expected from the asymptotic behaviour of large compact clusters, indicating that the accurate determination of the asymptotic form requires very long series.

The low-energy excitation spectrum of an Ising quantum chain in a transverse field which is critical (h=J=1) for 1<or=n<or=L/2 and off-critical (h not=J) for L/2(n<or=L is calculated up to O(L^-1). Interface critical exponents are determined numerically in the odd sector and exactly in the even sector using finite-size scaling. One gets an ordinary surface transition when h)J and an extraordinary surface transition when h(J. The gap-exponent relation is verified, the mass gaps and the level degeneracy are in agreement with conformal invariance.

The authors study finite-size-scaling behaviour of a two-dimensional random tiling model. This model exhibits commensurate-incommensurate phase transitions of the Pokrovsky-Talapov type where anisotropic scaling appears. A simple scaling argument near this transition suggests an integral surface critical exponent which, in turn, implies a logarithmic singularity with a universal coefficient in the surface free energy. They confirm the existence of this singularity numerically employing the transfer matrix method in the case of free boundary conditions.

It is confirmed that there exists a Kosterlitz-Thouless (KT) like phase in the ferromagnetic 6-clock model on the square lattice through studies of the interfacial free energy estimated by Monte Carlo simulations. The authors find that lower and upper transition temperatures are T₁=0.75 and T₂=0.90 respectively, and that the critical exponent eta for the correlation function in the KT-like phase varies from 0.15 at T₁ to 0.26 at T₂. The correlation length zeta just above T₂ is estimated directly from the magnetization profile, and it is shown to behave as zeta approximately exp(bt^-12/), t=(T-T₂)/T₂ approaching the KT-like phase.

A theoretical treatment is developed of the relaxation of a spatially inhomogeneous system of particles simultaneously undergoing diffusion and coagulation. It is shown in general that the effect of coagulation is such as to induce interactions between otherwise independent diffusive modes characterized by wavenumbers k and k', leading to the spontaneous creation of modes with wavenumber K=k+or-k'. The consequences of this are investigated quantitatively for the situation when initially there exists a set of discrete modes, and numerical estimates are made of the effect.

The aim of the present work is to derive exact expressions for the second and third virial coefficients B/(T) and C(T) for fluids of molecules interacting according to the square-well potential of arbitrary well width and arbitrary dimensionality d. General expressions for the terms of the fourth virial coefficient D(T), where D(T)=D₁(T)+D₂(T)+D₃(T) are obtained when the width of the attractive well is equal to the radius of the hard sphere. For d=3 and 1, the values of D₁, D₂ are analytically obtained, whereas D₃ is computed numerically.

Finds and studies a group of transformations of the Parisi order parameter. The spin-glass free energy is invariant under these transformations. A classification of infinitesimal transformations is given.

The effects of external fields on the retrieval properties of highly dilute attractor neural networks with general classes of learning rules are examined. It can be shown that external fields increase basins of attraction making even perfect retrieval possible for relatively high loading. The application of different classes of noise distributions on the stimulus field indicates that certain first-order transitions occurring are peculiarities of the type of noise. Optimally adapted networks in the presence of external fields extend the critical loading above which perfect retrieval is impossible. In the presence of external fields in the high-temperature regime, Hebb networks also retrieve better than rules with more optimal performances at low temperatures.

The storage capacity of an autoassociative memory with extremely diluted connectivity and with threshold-linear elementary units is studied in its dependence on the graded structure and on the sparseness of the coding scheme, and on the form of the learning rule used. As the coding becomes sparse, more patterns can be stored, and the difference in capacity (measured for a given number of modifiable synapses per unit) between fully connected and highly diluted systems vanishes. Graded (non-binary) codings, especially when used with learning rules nonlinear in their post-synaptic factor, further increase the number of patterns that can be stored by making their retrieved representation even sparser.

The properties of an extremely diluted asymmetric network of neurons with a sigmoidal response function are investigated. It is shown that in the absence of thermal noise the storage capacity increases with decreasing gain of the response function. The information capacity of the network in the case of multistate patterns is discussed. It turns out that the network can process only a finite amount of information in spite of the unbounded amount of information that can be encoded in multistate patterns.

The authors use the collective-field method to discuss the weak- and strong-coupling phases of the one-plaquette U(N)-invariant model. In both phases they obtain a unified description in terms of collective fields and their collections. The ground-state energy including the next-to-leading-order term is finite owing to explicit cancellation of divergences except at the critical point of the phase transition.

The Darboux and Backlund transformations for the derivative nonlinear Schrodinger equation are derived by the same method for the nonlinear Schrodinger equation.

The authors analyse the scaling structure of power spectra arising from chaotic dynamical systems. The observation of anomalous scaling in spectral parameters can be understood by the use of multifractal analysis in the frequency domain. This analysis provides numerical tools for evaluating different chaotic behaviour. The frequency behaviour of oscillatory chaos seems to suggest the hypothesis of phase transition in the f( alpha )-spectrum.