Table of contents

A two-parametric solution of the constant Yang-Baxter equation is used for the construction of structures corresponding to quantized groups. The corresponding Hopf algebra is generated by five generators.

As a good introduction to and example of the differential calculus on quantum vector spaces pioneered by Wess and Zumino (1990) the author considers two-parameter quantum deformation of the plane and its calculus. The main structures are derived anew without an R matrix and their generalizations to multiparameter deformation and to quantum superspace are briefly discussed in an arbitrary number of dimensions.

A new proof of the Kochen-Specker theorem uses 33 rays, instead of 117 in the original proof. If the number of dimensions is increased from 3 to 4, only 24 rays are needed.

Using a q-deformation of the usual Clifford algebra, one constructs a realization of the quantum exceptional algebras E₆, E₇, E₈ and F₄ in terms of q-fermionic operators.

The authors show that every translation invariant valence bond state on a one-dimensional quantum spin chain arises as the unique ground state of a certain family of finite-range interactions. For each interaction in this family they show the existence of a nonzero spectral gap above the ground state energy. A special example of this structure is a state recently studied by Affleck et al. (1987). For the Hamiltonian studied by these authors they estimate the gap, and prove that it lies between 1/3 and 10/27.

The authors have measured the roughening exponent chi at the interface between a pair of growing Eden clusters. Their numerical results suggest that chi differs from the corresponding exponent for a freely growing interface, and that it depends on the initial geometry of the clusters, which grow into a vacant sector with an angle theta . For theta <180' they find chi =1/3, and their results are consistent with chi =2/3 for theta =180' and chi =1 for theta >180'.

Using resummation techniques, the author extracts estimates for the critical exponents 1/v and eta of the three-dimensional O(3) sigma model from the known O(1/N²) and O(1/N³) field theory corrections, respectively, and compare the former with recent Monte Carlo results.

The domain growth after a quench is controlled by an exponent which generally has very broad universal properties. The authors investigate the alternating bond Ising chain with Glauber dynamics, and find that its growth exponent is universal, even though the dynamical critical exponent z is non-universal. They show that usual scaling theory can be reformulated to account for the above findings.

The Langmuir isotherm, essentially an equilibrium relation but often used as a boundary condition in modelling adsorption kinetics, is introduced and discussed. Starting from a more fundamental, and therefore more generally applicable, boundary condition the authors justify mathematically the use of the isotherm relation in this context, showing under what conditions it may be considered a valid approximation to the more fundamental relation.

The classical Cover results on linear separability of points in R^d are a milestone in neural network theory. Nevertheless they are not valid for digital input networks because in this case the points are not, in general position, vertices of a d-dimensional hypercube. The author shows here that for large d all Cover findings can be extended to this case. It is also shown that for n<O((d+1)³2/) the number of linear separations of n random hypercube vertices tends to that of n points in general position.

Two hierarchies of thermodynamic evolution equations leading to continued-fraction expansions for generalized frequency- and wavevector-dependent transport coefficients are cast in Hamiltonian form. In the nonlinear regime, the Jacobi identity implies several requirements on the transport coefficients which had not been previously explored and which are beyond the scope of the thermodynamic restrictions coming from the second law.

The authors consider a two-level quantum mechanical system driven by an analytic time-dependent Hamiltonian of the form H( in t). In the adiabatic limit, in <<1, the transition probability P(+,-) from one energy level (labelled by -) at time t=- infinity to the other (labelled by +) at time t=+0OI is known to behave as P(+,-)=exp(- alpha _-1 in ^-1) exp( alpha ₀)(1+O( in )). Using a simple iterative procedure generating Hamiltonians H₀=H,H₁,. . .H_N+1, they compute the full asymptotic expansion of the transition probability ln P(+,-)=- alpha _-1 in ^-1+ alpha ₀+_j=0^N alpha _j in ^j+O ( in ^N+1) For all N>or=0.

The effect of initial fluctuations on the dynamics of dark solitons is studied in the framework of the nonlinear Schrodinger equation with nonzero boundary conditions. The perturbation method based on the inverse scattering technique is developed to obtain statistical characteristics of soliton parameters. The possibility of creation of new solitons due to perturbation is considered and a criterion for that to occur is stated. The features of the soliton dynamics in comparison with those of bright solitons are discussed.

The author studies diffusion by using a specific type of cellular automaton, where particles move in an environment of scatterers. The deterministic collision rules, used in the model, give rise to certain difficulties related to nonergodicity. The class of models, introduced by Gunn and Ortuno (1985), also contains the model studied by Gates (1972). Previously developed theory, including the Boltzmann approximation, the ring and repeated-ring approximations and the effective medium approximation (EMA), is generalized for this model. The validity is further investigated using computer simulations. In the presence of reflecting scatterers EMA yields low-density results that show the breakdown of the Boltzmann equation, while they agree quantitatively with the low-density simulations. EMA also agrees with a phase transition that occurs for some parameter choices.

For pt.I see ibid., vol.24, p.787-806 (1991). As a model for diffusion, a lattice version of the Lorentz gas is studied, using specular collision rules. A previously developed theory, containing the molecular chaos approximation, several ring approximations and the effective medium approximation (EMA), is applied and compared with simulations. The correction to the Boltzmann equation result, resulting from effective medium approximation (EMA) is opposite to the simulation results. This indicates that other diagrams than ring diagrams should be considered, which is peculiar if one considers the successes of the EMA for other models. This is investigated by using variants of the model. The diffusion tensor is in general nonisotropic; the corresponding features are qualitatively described by theory.

The authors have used transfer matrix techniques to study a model of polymer adsorption and collapse. They consider the conventional model of a self-avoiding walk with attractive nearest-neighbour interactions on the square lattice and introduce adsorbing boundaries. The numerical results are consistent with the existence of a multicritical point where the adsorption and collapse transitions coincide and suggest that the adsorption transition of a collapsed polymer chain in first order. There is also clear evidence that the collapse transition lowers the adsorption temperature. Estimates of the surface scaling dimensions at the ordinary- Theta transition support the conjecture of Seno and Stella (1988) that the surface critical behaviour at the Theta and Theta ' points belong to different universality classes.

The average connectivity constant mu of self-avoiding walks (SAWs) is obtained from exact enumeration of SAWs on Monte Carlo generated percolating clusters in a randomly diluted square lattice. For averages over the (infinite) percolating cluster, mu decreases almost linearly with bond dilution (1-p), where p is the bond occupation concentration. The authors find mu (p_c)=1.31+or-0.03 at the percolation threshold p_c and could not detect any significant difference between mu (p_c) and p_c mu (1). The variation of theta -point for SAWs on the same lattice with dilution is also estimated, analysing the partition function zeros. Within the limited accuracy of their analysis, its variation with dilution is observed as being quite weak and the theta -point increases somewhat (compared to pure lattice value) near p_c; they find a non-vanishing theta point (K_theta (p_c) equivalent to 0.59, where K₀=J/k theta ) on the square lattice percolation cluster at p_c.

The context of the renormalization group applied to the lattice polymer statistical mechanics with the nearest-neighbour triplet interaction in addition to the usual nearest-neighbour pair interaction yields a two-variable quasi-scaling relation. It asserts that a reduced moment is described by two arbitrary reduced moments. The validity of the two-variable quasi-scaling has been examined for lattice polymers with polymerization degree up to 1000 in the poor solvent regime and was confirmed. It is well known that single-variable scaling fails in this solvent regime. Linear relations connecting three reduced moments which are valid near the respective Gaussian points and considered to be universal are presented. The results, however, are not consistent with the premise employed in the continuous polymer model. The working RG is nonlinear. The consistency may be attained by polymers with polymerization degree much larger than 1000 for which the linearized RG will work. The authors' basic equation involves no particular reference point such as the Theta -point. This is in contrast to previous scaling simulations and experiments.

A method is given for the rapid evaluation of the Madelung constants for ionic compounds having hexagonal structures.

The convex hull of any subset of vertices of an n-dimensional hypercube contains no other vertex of the hypercube. This result permits the application of some theorems of n-dimensional geometry to digital feed-forward neural networks. Also, the construction of the convex hull is proposed as an alternative to more traditional learning algorithms. Some preliminary simulation results are reported.

Shows how to obtain the kernel for arbitrary fermion sectors in supersymmetric quantum mechanics by applying standard coordinate space functional integral techniques to the coherent state fermionic path integral.

The authors study the equivalent Schrodinger equation for the bounce configuration. They get the lowest eigenvalue in the limit of small external currents using the Landau-Lifshitz approach.