Table of contents

Consideration is given to the convergence properties of sums of identical, independently distributed random variables drawn from a class of discrete distributions with power-law tails, which are relevant to scale-free networks. Different limiting distributions, and rates of convergence to these limits, are identified and depend on the index of the tail. For indices ⩾2, the topology evolves to a random Poisson network, but the rate of convergence can be extraordinarily slow and unlikely to be yet evident for the current size of the WWW for example. It is shown that treating discrete scale-free behaviour with continuum or mean-field approximations can lead to incorrect results.

We calculate improved lower bounds for the connective constants for self-avoiding walks on the square, hexagonal, triangular, (4.8²) and (3.12²) lattices. The bound is found by Kesten's method of irreducible bridges. This involves using transfer-matrix techniques to exactly enumerate the number of bridges of a given span to very many steps. Upper bounds are obtained from recent exact enumeration data for the number of self-avoiding walks and compared to current best available upper bounds from other methods.

We consider a system of three equations, which will be called generalized Davey–Stewartson equations, involving three coupled equations, two for the long waves and one for the short wave propagating in an infinite elastic medium. We classify the system according to the signs of the parameters. Conserved quantities related to mass, momentum and energy are derived as well as a specific instance of the so-called virial theorem. Using these conservation laws and the virial theorem both global existence and nonexistence results are established under different constraints on the parameters in the elliptic–elliptic–elliptic case.

Wetting morphologies on solid substrates, which may be chemically or topographically structured, are studied theoretically by variation of the free energy which contains contributions from the substrate surface, the fluid–fluid interface and the three-phase contact line. The first variation of this free energy leads to two equations—the classical Laplace equation and a generalized contact line equation—which determine stationary wetting morphologies. From the second variation of the free energy we derive a general spectral stability criterion for stationary morphologies. In order to incorporate the constraint that the displaced contact line must lie within the substrate surface, we consider only normal interface displacements but introduce a variation of the domains of parametrization.

The complex scaling method provides scattering wavefunctions which regularize resonances and suggest a resolution of the identity in terms of such resonances, completed by the bound states and a smoothed continuum. But, in the case of inelastic scattering with many channels, the existence of such a resolution under complex scaling is still debated. Taking advantage of results obtained earlier for the two channel case, this paper proposes a representation in which the convergence of a resolution of the identity can be more easily tested. The representation is valid for any finite number of coupled channels for inelastic scattering without rearrangement.

For elementary numerical integration on a sphere, there is a distinct advantage in using an oblique array of integration sampling points based on a chosen pair of successive Fibonacci numbers. The pattern has a familiar appearance of intersecting spirals, avoiding the local anisotropy of a conventional latitude–longitude array. Besides the oblique Fibonacci array, the prescription we give is also based on a non-uniform scaling used for one-dimensional numerical integration, and indeed achieves the same order of accuracy as for one dimension: error ∼N⁻⁶ for N points. This benefit of Fibonacci is not shared by domains of integration with boundaries (e.g., a square, for which it was originally proposed); with non-uniform scaling the error goes as N⁻³, with or without Fibonacci. For experimental measurements over a sphere our prescription is realized by a non-uniform Fibonacci array of weighted sampling points.

Nonperturbative, oscillatory, winding number 1 solutions of the sine-Gordon equation are presented and studied numerically. We call these nonperturbative shape modes wobble solitons. Perturbed sine-Gordon kinks are found to decay to wobble solitons.

This paper provides explicit techniques to compute the exponentials of a variety of structured 4 × 4 matrices. The procedures are fully algorithmic and can be used to find the desired exponentials in closed form. With one exception, they require no spectral information about the matrix being exponentiated. They rely on a mixture of Lie theory and one particular Clifford algebra isomorphism. These can be extended, in some cases, to higher dimensions when combined with techniques such as Givens rotations.

In this paper we derive expressions for matrix elements (ϕ_i, Hϕ_j) for the Hamiltonian H = −Δ + ∑_qa(q)r^q in d ⩾ 2 dimensions. The basis functions in each angular momentum subspace are of the form . The matrix elements are given in terms of the Gamma function for all d. The significance of the parameters t and p and scale s are discussed. Applications to a variety of potentials are presented, including potentials with singular repulsive terms of the form β/r^α, α, β > 0, perturbed Coulomb potentials −D/r + Br + Ar², and potentials with weak repulsive terms, such as −γr² + r⁴, γ > 0.

For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O_α of the quantum mechanics based on H. In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT-symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h, the physical observables O_α, the localized states and the conserved probability density for the non-Hermitian PT-symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT-symmetric quantum mechanics and clarify its relationship with both conventional quantum mechanics and classical mechanics.

We describe some semiclassical spectral properties of Harper-like operators, i.e. of one-dimensional quantum Hamiltonians periodic in both momentum and position. The spectral region corresponding to the separatrices of the classical Hamiltonian is studied for the case of integer flux. We derive asymptotic formulae for the dispersion relations, the width of bands and gaps and show how geometric characteristics and the absence of symmetries of the Hamiltonian influence the form of the energy bands.

We obtain a class of parametric oscillation modes that we call K-modes with damping and absorption that are connected to the classical harmonic oscillator modes through the 'supersymmetric' one-dimensional matrix procedure similar to relationships of the same type between Dirac and Schrödinger equations in particle physics. When a single coupling parameter, denoted by K, is used, it characterizes both the damping and the dissipative features of these modes. Generalizations to several K parameters are also possible and lead to analytical results. If the problem is passed to the physical optics (and/or acoustics) context by switching from the oscillator equation to the corresponding Helmholtz equation, one may hope to detect the K-modes as waveguide modes of specially designed waveguides and/or cavities.

We present a class of mappings between the fields of the Cremmer–Sherk and pure BF models in 4D. These mappings are established by two distinct procedures. First, a mapping of their actions is produced iteratively resulting in an expansion of the fields of one model in terms of progressively higher derivatives of the other model fields. Second, an exact mapping is introduced by mapping their quantum correlation functions. The equivalence of both procedures is shown by resorting to the invariance under field scale transformations of the topological action. Related equivalences in 5D and 3D are discussed. The mapping in (2+1)D from the Maxwell–Chern–Simons to pure Chern–Simons models is investigated from a similar perspective.

We present a new conjecture for the SU_q(N) Perk–Schultz models. This conjecture extends a conjecture presented in our article (Alcaraz F C and Stroganov Yu G J. Phys. A: Math. Gen.35 6767–87).

The numerical values for p on lines 5 and 6 of page 1619 were given correctly to three significant digits. The values to ten significant digits are shown in the pdf, as is the corrected version of equation (41).