Table of contents

It is shown that an operator (in general non-local) commutes with the Hamiltonian describing the finite XX quantum chain with certain non-diagonal boundary terms. In the infinite-volume limit this operator gives the `topological' charge.

It is argued that systems whose elements are renewed according to an extremal criterion can generally be expected to exhibit long-term memory. This is verified for the minimal extremally driven model, which is first defined and then solved for all system sizes N⩾2 and times t⩾0, yielding exact expressions for the persistence R(t) = [1 + t/(N-1)]^-1and the two-time correlation functionC(t_w + t,t_w) = (1-1/N)(N + t_w)/(N + t_w + t-1). The existence of long-term memory is inferred from the scaling ofC(t_w + t,t_w)~f(t/t_w), denoting aging. Finally, we suggest ways of investigating the robustness of this mechanism when competing processes are present.

Numerical studies of the Anderson transition are based on the finite-size scaling analysis of the smallest positive Lyapunov exponent. We prove numerically that the same scaling holds also for higher Lyapunov exponents. This supports the one-parameter scaling theory of localization. We found the critical disorder 16.50⩽W_c⩽16.53 and the critical exponent 1.50⩽ν⩽1.54 from numerical data for quasi-one-dimensional systems up to the system size 24²×∞. The finite-size effects and the role of irrelevant scaling parameters are discussed.

We study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs. We use two different algorithms to efficiently simulate the frontier, including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice. We find that ϕ_c = 0.676 339±0.000 004, thus providing an extra significant digit of accuracy to this constant.

The wavefunctions corresponding to the zero-energy eigenvalue of a one-dimensional quantum chain Hamiltonian can be written in a simple way using quadratic algebras. Hamiltonians describing stochastic processes have stationary states given by such wavefunctions and various quadratic algebras have been found and applied to several diffusion processes. We show that similar methods can also be applied for equilibrium processes. As an example, for a class of q-deformed O(N) symmetric antiferromagnetic quantum chains, we give the zero-energy wavefunctions for periodic boundary conditions corresponding to momenta zero and π. We also consider free and various non-diagonal boundary conditions and give the corresponding wavefunctions. All correlation lengths are derived.

The density of states for the three-dimensional Ising model is calculated with high precision by means of multicanonical simulations. This allows us to estimate the leading partition function zeros for lattice sizes up to L = 32. We have evaluated the critical exponent ν and the correction to scaling through an analysis of a multi-parameter fit and of the Bulirsch-Stoer (BST) extrapolation algorithm. The performance of the BST algorithm is also explored in case of the 2D Ising model, where the exact partition function zeros are known.

We present a general scheme to calculate within the independent interval approximation generalized (level-dependent) persistence properties for processes having a finite density of zero crossings. Our results are especially relevant for the diffusion equation evolving from random initial conditions - one of the simplest coarsening systems. Exact results are obtained in certain limits, and rely on a new method to deal with constrained multiplicative processes. An excellent agreement of our analytical predictions with direct numerical simulations of the diffusion equation is found.

We give the 1/d-expansion, to order 1/d⁵, for the limiting reduced free energy of a weakly embedded two-variable site animal model of branched polymers. Hence, we are able to derive expansions for the free energy of related one-variable models and the growth constant of weakly and strongly embedded trees, again to order 1/d⁵. We argue that the free energy expansions are asymptotically correct for a small range of fugacities only. Exact results show that animals with a compact hypercubic structure make the overwhelming contribution to the total number of weakly embedded site animals, especially for large d. This result is not reflected in the free energy 1/d-expansions, where trees make the dominant contribution.

We also derive new exact enumeration data for lattice animals using intermediate calculations in the derivation of the 1/d-expansions. Thus, we give partition functions for the general d-dimensional hypercubic lattice for one to 13 sites.

A model of self-interacting columns, related to models of partially directed walks and to histogram polygons, is considered. The generating function of this model is found in terms of q-deformed Bessel functions using a functional recursion scheme. A transition, related to a deflation-inflation transition seen in staircase polygons, or to a rough-smooth transition in a solid-on-solid model, is found, and its scaling exponents are found in the context of a tricritical scaling analysis. If the columns are also interacting with the horizontal axis, then the inflated or smooth phase is also found to undergo an adsorption transition. A special point exists where the model is critical with respect to both an adsorption transition and a deflation-inflation transition.

An ansatz is proposed for the time correlation C(t) = ⟨exp (-it)⟩₀ of a non-conserved phase function , where is the Liouville operator and the brackets denote an equilibrium canonical average. The new expression decays exponentially as t→∞ and reduces to a series in powers of t² as t→0, agreeing with a finite, arbitrarily chosen, number of terms in an exact t-expansion. The ansatz takes the form of a series shown to converge absolutely and uniformly. Its time integral can be fitted to experiment or to a computer simulation by adjusting the decay constant characterizing the long-time behaviour. If the decay constant is sufficiently large, the new model yields results of an order of magnitude consistent with the often-used exponential decay model.

Recently, a class of Hermitian matrix ensembles was proposed by Mézard et al in which the matrix elements depend on the Euclidean separation of coordinates drawn from a random distribution. Matrix ensembles of this kind appear in a variety of physical contexts. In this paper, we formulate and investigate the spectral properties of these Euclidean random matrix ensembles within the framework of a supersymmetric statistical field theory. We discuss applications of these results to various model systems.

Schwinger's approach gives a fresh look on Tamm's problem (charge, being initially at rest, exhibits an instant acceleration, moves with a finite velocity, and, after an instant deceleration, goes to the state of rest). Schwinger's angular and frequency distributions are compared with Tamm's ones which in turn are compared with exact distributions. Criteria for the validity of Tamm's formulae are checked by numerical calculations.

The question of what information about a quantum state may be inferred from a sequence of measurements made on it, is addressed. The main result is that maximum-likelihood estimation gives an arguably natural optimum approach in quantum theory. It singles out a particular choice of basis, in which the state is expressed. Various incompatible and incomplete observations made on the ensemble are treated as a complete generalized measurement. Consequently, the maximum likelihood (relative entropy) is the best measure for relating experimental data with theoretical predictions of quantum theory.

In this paper we use the method of characteristic curves for solving linear partial differential equations to study the invariant algebraic surfaces of the Rikitake system

= -µ x + y(z + β)       = -µ y + x(z-β)       ż = α-xy.

Our main results are the following. First, we show that the cofactor of any invariant algebraic surface is of the form rz + c, where r is an integer. Second, we characterize all invariant algebraic surfaces. Moreover, as a corollary we characterize all values of the parameters for which the Rikitake system has a rational or algebraic first integral.

In the two-dimensional isotropic parabolic potential barrier V(x,y) = V₀-mγ²(x² + y²)/2, though it is a model of an unstable system in quantum mechanics, we can obtain the stationary states corresponding to the real energy eigenvalue V₀. Further, they are infinitely degenerate. For the first few eigenstates, we will find the stationary flows round a right angle that are expressed by the complex velocity potentials W = ±γz²/2. From the hydrodynamical point of view vortex structures for the general solutions are also studied.

A further analogue of Newton's binomial formula was introduced in the (q,h)-deformed quantum plane by Benaoum, leading to a more generalized analysis, called a (q,h)-analysis. The purpose of this paper is to establish some further interesting properties for (q,h)-analysis. The (q,h)-analogue of the multinomial formula, the (q,h)-reciprocal formula and (q,h)-analogues of Chu's and Vandermonde's identities are obtained.

We comment on serious errors in a recently published paper (El-Sayed A M A 1999 J. Phys. A: Math. Gen.32 8647-54).