Table of contents

The relationship of the group SL(2,R) and the NLS^- equation is presented. As a consequence, the SL(2,R) gauge equivalence between the NLS^- and the M-HF model is proved, which provides a new example in geometrically explaining dynamical properties of soliton equations by the SL(2,R) structure.

A closed-plane meander of order n is a closed self-avoiding curve intersecting an infinite line 2n times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on transfer matrix methods, for the enumeration of plane meanders. While the algorithm has exponential complexity, its rate of growth is much smaller than that of previous algorithms. The algorithm is easily modified to enumerate various systems of closed meanders, semi-meanders, open meanders and many other geometries.

We study the diffusion of Brownian particles on the surface of a sphere and compute the distribution of solid angles enclosed by the diffusing particles. This function describes the distribution of geometric phases in two-state quantum systems (or polarized light) undergoing random evolution. Our results are also relevant to recent experiments which observe the Brownian motion of molecules on curved surfaces such as micelles and biological membranes. Our theoretical analysis agrees well with the results of computer experiments.

We have substantially extended the series for the number of self-avoiding walks and the mean-square end-to-end distance on the simple cubic lattice. Our analysis of the series gives refined estimates for the critical point and critical exponents. Our estimates of the exponents γ and ν are in good agreement with recent high-precision Monte Carlo estimates, and also with recent renormalization group estimates. Critical amplitude estimates are also given. A new, improved rigorous upper bound for the connective constant µ<4.7114 is obtained.

In this paper we study the phase diagram of two Ising planes coupled by a standard spin-spin interaction with bond randomness in each plane. The whole phase diagram is analysed with the help of Monte Carlo simulations and field theory arguments.

By adapting Feynman's sum over paths method to a quantum mechanical system whose phase space is a torus, a new proof of the Landsberg-Schaar identity for quadratic Gauss sums is given. In contrast to existing non-elementary proofs, which use infinite sums and a limiting process or contour integration, only finite sums are involved. The toroidal nature of the classical phase space leads to discrete position and momentum, and hence discrete time. The corresponding `path integrals' are finite sums whose normalizations are derived, which are shown to intertwine cyclicity and discreteness to give a finite version of Kelvin's method of images.

The Poincaré-Cartan (PC) form of a Lagrangian on the bundle J² = J²(N,M) is, as a general rule, defined on J³ thus leading to a non-equivalence between Euler-Lagrange and Hamilton-Cartan equations. This naturally leads to the problem of determining what Lagrangians have a PC form projectable onto J², as they will then admit a second-order Hamiltonian formalism. There are specific examples of this phenomenon in field theory. This paper provides an explicit classification of such Lagrangians.

We consider ballistic electron motion in a quasi-one-dimensional ring under the uniform high-frequency electric field induced by an electromagnetic field. The electron satisfies a nonlinear equation of motion which is formally identical to that for a pendulum with a vibrating suspension point. The averaging method of Kapitza is used. The electromagnetic emission spectrum is calculated. The spectrum consists of low-frequency radiation, scattered radiation at the incident radiation frequency and combination scattered radiation; the intensities and frequencies of all components depend nonlinearly on the incident radiation frequency. At a certain value of that intensity the spontaneous symmetry breakdown occurs. As a result, the system acquires some static electric dipole moment.

Twin observables are opposite subsystem observables A₊ and A_- if they are indistinguishable in measurement in a given mixed or pure state ρ in the sense that the measurement of one amounts to the same as the measurement of the other (direct measurement and so-called distant measurement, respectively). It is pointed out that twin observables may reveal quantum correlations that give rise to the disappearance of interference in the two-slit experiment. Twin observables in general states are investigated in detail algebraically and geometrically. It is shown that there is a far-reaching correspondence between the detectable (in ρ) spectral entities of the two operators. Twin observables are state-dependently quantum-logically equivalent, and the direct subsystem measurement of one of them ipso facto gives rise to the indirect (i.e. distant) measurement of the other. Existence of nontrivial twins requires a singularity of ρ. Systems in thermodynamic equilibrium do not admit subsystem twins. These observables may enable one to simplify the matrix representing ρ.

The coherent states for a particle on a sphere are introduced. These states are labelled by points of the classical phase space, i.e. the position on the sphere and the angular momentum of a particle. As with the coherent states for a particle on a circle discussed in Kowalski et al (1996 J. Phys. A: Math. Gen. 29 4149), we deal with a deformation of the classical phase space related to quantum fluctuations. The expectation values of the position and the angular momentum in the coherent states are regarded as the best possible approximation of the classical phase space. The correctness of the introduced coherent states is illustrated by an example of the rotator.

Representations and convergence criteria for infinite m-dimensional lattice sums of generalized hypergeometric functions {_pF_{p + 1}} are deduced by appealing to the principle of mathematical induction. In particular, we show that such a lattice sum may be expressed essentially as a finite sum of Mellin transforms of products of Bessel functions of order ½(m-2) and the functions {_pF_{p + 1}} in the lattice sum. In addition, a direct derivation for the three-dimensional case is provided. Moreover, we construct a countably infinite class of null-functions on increasingly larger open intervals which are parametrically independent of the functions {_pF_{p + 1}} generating them.

We derive an exact analytic solution to a Klein-Gordon equation for a step potential barrier with cutoff plane wave initial conditions, in order to explore wave evolution in a classical forbidden region. We find that the relativistic solution rapidly evanesces within a depth 2x_p inside the potential, where x_p is the penetration length of the stationary solution. Beyond the characteristic distance 2x_p, a Sommerfeld-type precursor travels along the potential at the speed of light, c. However, no spatial propagation of a main wavefront along the structure is observed. We also find a non-causal time evolution of the wavefront peak. The effect is only an apparent violation of Einstein causality.