Table of contents

We propose a mechanism which produces periodic variations of the degree of predictability in dynamical systems. It is shown that even in the absence of noise when the control parameter changes periodically in time, below and above the threshold for the onset of chaos, stochastic resonance effects appear. As a result one has an alternation of chaotic and regular, i.e. predictable, evolutions in an almost periodic way, so that the Lyapunov exponent is positive but some time correlations do not decay.

We discuss the energy-density distribution for a gas of q-bosons with complex deformation parameter by exploiting a new q-complex oscillator algebra in which both the number operator and the energy eigenvalues are real. The corresponding Planck distribution generalizes the results obtained for a real q-boson gas, including Wien's and Stephan's laws, and the role of an effective Planck constant depending on the parameter modulus.

An orthogonality relation on the full real line for the Rogers-Szego polynomials is discussed. It is argued that Fourier transformation with the standard exponential kernel exp(ixy) relates the Rogers-Szego and Stieltjes-Wigert functions.

It is shown that the quantum version of the constrained Ablowitz-Kaup-Newell-Sogur (AKNS) flows from a completely integrable system and the corresponding Lax pair can be constructed. The quantum inverse problem for the system is then formulated following the technique of Jurgo(1989) and Gaunoulis(1987). Finally the algebraic Bethe ansatz is explicitly constructed, thereby determining the energy eigenvalue of the excited states.

We illustrate a new systematic diagonalization procedure through its application to the well known Jaynes-Cummings model. Our method, starting from the consideration of peculiar invariance properties of the Hamiltonian, singles out specific constants of motion by which we succeed in introducing appropriate unitary transformations accomplishing a canonical reduction of the Hamiltonian into a diagonal form. The potentialities of the treatment is briefly discussed.

An exact solution of the Milburn equation is given for the two-photon Jaynes-Cummings model of atom-field interaction with non-trivial dynamics. It is shown that the intrinsic decoherence in the atom-field interaction is responsible for the deterioration of the revivals of the atomic inversion.

Applying the formal series symmetry theory to a differential-difference system, two-dimensional Toda equation, we found that as with the continuous Toda field theory, there exist two sets of infinitely many symmetries for the discrete Toda system. Each set of symmetries constitutes a generalized W_infinity algebra.

The modifications that must be introduced within the framework of standard, classical N-body statistical mechanics techniques, in order to deal with the recently proposed generalized scenario of non-extensive statistical mechanics are discussed. A generalized version of the equipartition theorem is shown to hold. The ideal gas is revisited as a simple application.

A Riemann space associated with a correlated walk (cw) model is investigated by a numerical method, which is applicable to many other stochastic processes. The behaviour of the Riemann curvature is shown to have a close relation to the stability of the cw system. The Riemann space is spanned by two parameters representing jump probabilities of the cw model. The cw manifold evolves through some characteristic eras: it starts at a point, and changes to a line, and then to a homogeneous spherical surface of the scalar curvature R= 1/2 . The homogeneous space immediately transforms to an inhomogeneous space. The inhomogeneity gradually grows: a region of the inhomogeneous space violently oscillates in time and another region expands fast. As a whole, the curvature of the space decreases and soon becomes negative. The oscillations have already started to fade away. The negative curvature goes on decreasing and finally the space converges to a homogeneous saddle surface of R=-1. Such a behaviour of the curvature is shown to be well understood by the terms of 'stability' and 'order parameter' of stochastic processes. In other words, the inhomogeneity, the oscillation, and the decrease behaviour of the curvature are closely related to stability and orderliness of the correlated system.

Zamolodchikov's tetrahedron equations, which were derived by considering the scattering of straight strings, can be written in three different labelling schemes: one can use as labels the states of the vacua between the strings, the states of the string segments, or the states of the particles at the intersections of the strings. We give a detailed derivation of the three corresponding tetrahedron equations and also show how the Frenkel-Moore equations fits in as a non-local string labelling. We then discuss how an analogue of the Wu-Kadanoff duality can be defined between each pair of the above three labelling schemes. It turns out that there are two cases, for which one can simultaneously construct a duality between all three pairs of labellings.

Starting from a set of assumptions on the long-time limit behaviour of the nonequilibrium relaxation of mean-field models in the thermodynamic limit, we derive analytical results for the long-time relaxation of the Sherrington-Kirkpatrick model, starting from a random configuration. The system never achieves local equilibrium in any fixed sector of phase space, but remains in an asymptotic out-of-equilibrium regime. We clearly state and motivate the assumptions made. For the study of the out-of-equilibrium dynamics of spin-glass models, we propose as a tool, both numerical and analytical, the use of 'triangle relations' which describe the geometry of the configurations at three (long) different times.

We investigate numerically the critical lines and the critical properties of the fluid-dimer and the Neel-dimer transitions of the S= 1/2 antiferromagnetic XXZ chain with next-nearest interactions, and we confirm that the universality class of this model belongs to the quantum sine-Gordon model, as is expected from the bosonization. The method which we use in this paper to calculate the critical lines is free from the logarithmic corrections on the Kosterlitz-Thouless (K-T)-type transition line, which have made the K-T critical point difficult to obtain. By the use of this method, it is possible to determine the K-T critical line with high precision from small size data, and to identify the universality class.

Both 'high-' and 'low-temperature' perturbation series are used to locate and characterize the first-order transition in the three-state Potts model in (3+1) dimensions on the simple-cubic, body-centred cubic and face-centred cubic lattices. Estimates are presented for the vacuum energy, 'latent heat', magnetization, susceptibility and mass gap at the transition. The results show a remarkable degree of universality between the different lattices, and a 'law of corresponding states' is followed very closely.

For paper II, see ibid, vol. 27, p.1503 (1994). The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the 3-state Potts model on the simple cubic lattice to order z⁴³ and the high-temperature expansion of the partition function to order v²¹. We use the numerical data to show that the transition is first-order, and estimate the latent heat, the discontinuity in the magnetization, and a number of other critical parameters.

In this paper we study the complete phase diagram of a model of interacting branched polymers. The model we consider is a lattice animal one, where the collapse transition can be driven both by a contact fugacity between two occupied nearest neighbours and by a fugacity related to each occupied edge. Using a Potts model formulation of the problem we conjecture the existence of two different universality classes for the theta transitions (with thermal exponents, nu and phi , equal to ( 1/2 , 2/3) and (8/15, 8/15) Separated by a higher-order percolation point. We also present convincing numerical evidence for these exponent values using a transfer-matrix approach. We discuss the possibility of a collapse-collapse transition and we predict the behaviour of our model when an adsorbing surface is included.

We consider several lattice models of branched polymers in dilute solution in which the polymer molecule is modelled as a tree or animal. In general the thermodynamic properties of the polymer are determined by monomer-monomer and monomer-solvent interactions. We examine a two-variable model which incorporates both types of interaction and discuss its relationship to other models which have previously been investigated. In particular, we discuss the collapse transition in these models.

Based on a commonly used model for the time evolution of extensive variables and on the basis of the analogies between the nonlinear stochastic theory and imaginary time quantum mechanics in curved space, we set up a canonical operator formulation of non-equilibrium thermodynamics which emphasizes the fundamental role played by the Boltzmann constant k in fluctuations. The k to 0 of this theory yields the classical description of non-equilibrium thermodynamics. The new formulation is technically useful and provides new insights which have important consequences, namely the thermodynamic uncertainty principle and the quantization of the entropy production rate for stationary states.

We calculate the direct average ((N)) of the number of metastable states in the projection rule neural network by means of the saddle-point method. ((N)) is obtained as a function of the Hamming distance g to a given pattern, stability gamma and energy E. The critical storage capacity of the model is calculated. The examination of the energy dependence of ((N)) leads to lower bounds for the energies of the spin-glass states.

A replica-symmetric mean-field theory approach is presented to the multi-neuron interaction model introduced by de Almeida and Iglesias (1990 Phys. Lett. 146A 239). Fixed-point equations are derived for the relevant order parameters of the model, extended to include biased patterns, without truncating the interaction. The capacity-bias and the temperature-capacity phase diagrams are discussed. Compared with the truncated version of the model, it is found that the capacity at zero temperature is infinite and that the retrieval states satisfy the de Almeida-Thouless stability condition.

Using an asymmetric associative network with synchronous updating, it is possible to recall a sequence of patterns. To obtain a stable sequence generation with a large storage capacity, we introduce a threshold that eliminates the contribution of weakly correlated patterns. For this system we find a set of evolution equations for the overlaps of the states with the patterns to be recognized. We solve these equations in the limit of the stationary cycle, and obtain the critical value of the capacity as a function of the threshold and temperature. Finally, a numerical simulation is made, confirming the theoretical results.

A comparative study is presented on quantum transports in weakly opened circle and stadium billiards in the perpendicular magnetic field B. While in the circle the magnetoconductance g(B) shows grossly regular oscillations, in the stadium it exhibits a transition from mild to violent undulations with increase of B. The rich fluctuation features of g(B), characterized by gradients of the cusp-like central peaks in autocorrelation functions, are attributed to the stability and instability of phase space in the underlying classical dynamics.

A 'dilute' generalization of the Birman-Wenzl-Murakami algebra is considered. It can be 'Baxterized' to a solution of the Yang-Baxter algebra. The D_n+1⁽²⁾ vertex models constitute a series of solvable lattice models which realize this algebraic structure. They can be regarded as a dilute version of the B_n⁽¹⁾ vertex models.

We study the relationships among the various forms of the q-oscillator algebra and consider the conditions under which it supports a Hopf structure. We also present a generalization of this algebra together with its corresponding Hopf structure. Its multimode extensions are also considered.

It is shown that deformed oscillator algebras with a non-standard behaviour, similar to that observed for the quon algebra, arise quite naturally when imposing covariance properties under the transformations of some quantum group. By non-standard behaviour, we mean that there may not be enough rules available to order creation operators and that some Fock states with positive squared norm for generic q-values disappear for q to 1. Two examples are given for the quantum groups O_q(3,R) and SU_q(2)*SU_q(2) approximately=O_q(4,R), respectively.

The coupling (Wigner-Clebsch-Gordan) coefficients of the unitary quantum algebras u_q(n) are considered. The tensorial properties of the generator powers and their ordered products, used in the explicit projectors and weight lowering procedures, are established. Different expressions for the multiplicity-free isofactors of u_q(n) coupling coefficients (those coupling an arbitrary and symmetric representations) are derived. Explicit expression of the arbitrary isofactors in terms of their boundary values are proposed. Proportionality of the semistretched isofactors to the stretched q-recoupling coefficients of u_q(n-1) (q-analogues of 9j-symbols) is demonstrated. The stretched isofactors of u_q(3) are expressed in terms of Clebsch-Gordan coefficients of u_q(2).

For pt. 1, see ibid., vol. 27, p. 1, (1994). The framework developed in part I for defining Lie and covariant differentiation of spinors is investigated to show its compatibility with tensor calculus. The special role played by conformal Killing vectors and conformal connections is also determined.

Equilibrium crystal shapes are defined uniquely by the Wulff construction. The classical kinematic theory of crystal growth, due mainly to Frank and Chernov, provides a mathematically equivalent prescription for the limiting growth shape. To connect these two well studied states, we derive a local geometric growth model and examine the transient shape evolution of an equilibrium form containing both facets and rough regions. Our model is appropriate to the weakly driven growth of a two-dimensional single crystal with n-gonal symmetry and arbitrary closed initial shape. The model links disparate kinetic processes determined by the local interfacial structure to the isotropic growth drive, and reproduces the experimentally observed transition from a partly rounded equilibrium shape to a highly faceted crystal which we term 'global kinetic faceting'. We solve for the transient shape dynamics globally, and locally, and in the latter case present a curvature evolution equation valid for any local growth law. Both approaches show that, during kinetic faceting, rough orientations grow out of existence with decreasing curvature.

The key point of the Hamiltonian formalism of Toda molecules is the commutativity of the Hamiltonians (tr y^k, tr y^l)=0, where y in GL(n) and (,) is a Poisson bracket associated with the classical r-matrix. To quantize the Toda molecule, we have to consider the q-analogue of the above formula. In this paper, we show the commutativity of the quantized first- and higher-order Hamiltonians (tr_q X^m, tr_q X)=0, where X is a matrix of quantum group GL_q(n).

We study the angular properties of plane waves backscattered from random fractal curves and surfaces (diffractals). For this purpose we apply the physical optics (Kirchhoff) approximation without resorting to the high-frequency limit. The fractals of interest are described by Gaussian random processes possessing power-law spectra with and without a large-scale cutoff. We show that if the wavelength of the incident field is of order of the topothesy, small variations of its value may alter the angular pattern of the backscattered power qualitatively. The most important prediction is a peak which Kirchhoff diffractals exhibit under specific conditions at a particular angle of incidence. The position of the peak is linked to the dimension of the topothesy and provides a good possibility for experimental measurement of these intrinsic parameters of the fractal surfaces. The effect of large-scale cut-off which determines the RMS height of the surface is also studied.

In this paper, the q-deformed quantum harmonic oscillators in the supersymmetric quantum systems are introduced. The q-supercoherent states associated with the q-deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated in detail. It is shown that the q-supercoherent states are completeness. The uncertainty relations for q-supercoherent states are discussed as well.

An analytical study of the temperature effects on the vibron soliton of the Takeno model for energy transport in protein is presented, using Davydov's thermally averaged Hamiltonian method. The conditions of the existence of vibron solitons in the model system are found and ideal protein parameters are used to show whether these conditions can be satisfied.

We study the renormalization-group flow properties of the Wess-Zumino-Witten model in the region of couplings between g²=0 and g²=4 pi /k by evaluating the two-loop Zamolodchikov c-function. We also discuss the region of negative couplings.

It is demonstrated that the replica symmetric saddle point is unstable when the distribution of aligned fields displays a gap.