Table of contents

For a complex scalar field ψ(x, y) in the plane, the flow lines (integral curves of the current field Im ψ*∇ψ) typically spiral slowly in or out of a phase vortex (where ψ = 0), with the distance between successive windings decreasing as 2πKr³ near the vortex at r = 0. The coefficient K depends on the derivatives of ψ at the vortex. In three dimensions, the flow spiral migrates slowly along the vortex line, in a helix whose pitch is proportional to r². For fields with well-defined orbital angular momentum, the flow lines can be determined explicitly not just near the vortex but also globally. The explicit forms of flow lines near phase extrema and saddles are also found.

We prove a lemma which allows one to extend results about the additivity of the minimal output entropy from highly symmetric channels to a much larger class. A similar result holds for the maximal output p-norm. Examples are given showing its use in a variety of situations. In particular, we prove the additivity and the multiplicativity for the shifted depolarizing channel.

We present the formal proof of a procedure to compute the phase-space volume of initial conditions for trajectories that, for a constant energy, escape or 'react' from a multi-dimensional potential well with one or several exit/entrance channels. The procedure relies on a phase-space formulation of transition state theory. It gives the volume of reactive initial conditions as the sum over the exit/entrance channels where each channel contributes by the product of the phase-space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel. An example is given to demonstrate the computational efficiency of the procedure.

The closed algebraic expressions of the determinants of some multivariate (multilevel) Vandermonde matrices and the associated Toeplitz/Karle–Hauptman matrices are worked out. The formula can usefully be applied to evaluate the determinant of the Karle–Hauptman matrix generated by a principal basic set of reflections, the knowledge of which determines the full diffraction pattern of an ideal crystal.

Many social and biological networks consist of communities—groups of nodes within which connections are dense, but between which connections are sparser. Recently, there has been considerable interest in designing algorithms for detecting community structures in real-world complex networks. In this paper, we propose an evolving network model which exhibits community structure. The network model is based on the inner-community preferential attachment and inter-community preferential attachment mechanisms. The degree distributions of this network model are analysed based on a mean-field method. Theoretical results and numerical simulations indicate that this network model has community structure and scale-free properties.

This paper is concerned with the group symmetries of the fourth Painlevé equation P_IV, a second-order nonlinear ordinary differential equation. It is well known that the parameter space of P_IV admits the action of the extended affine Weyl group . As shown by Noumi and Yamada, the action of as Bäcklund transformations of P_IV provides a derivation of its symmetric form SP₄. The dynamical system SP₄ is also equivalent to the isomonodromic deformation of an associated three-by-three matrix linear system (Lax pair). The action of the generators of on this Lax pair is derived using the Darboux transformation for an associated third-order operator.

A procedure for obtaining a 'minimal' discretization of a partial differential equation, preserving all of its Lie point symmetries, is presented. 'Minimal' in this case means that the differential equation is replaced by a partial difference scheme involving N difference equations, where N is the number of independent and dependent variables. We restrict ourselves to one scalar function of two independent variables. As examples, invariant discretizations of the heat, Burgers and Korteweg–de Vries equations are presented. Some exact solutions of the discrete schemes are obtained.

We consider generalizations of depolarizing channels to maps of the form with V_k being unitary and ∑_ka_k = a < 1. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions on V_k under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy can be proved for Φ ⊗ Ω with Ω arbitrary. We also show that the Holevo capacity need not equal log d − S_min(Φ) as one might expect for a convex combination of unitary conjugations.

We study the properties of quantum single-particle wave pulses created by sharp-edged or apodized shutters with single or periodic openings. In particular, we examine the following: the visibility of diffraction fringes depending on evolution time and temperature; the purity of the state depending on the opening-time window; the accuracy of a simplified description which uses 'source' boundary conditions instead of solving an initial value problem; and the effects of apodization on the energy width.

We exhibit long-lived resonances in scattering from two-dimensional soft cage potentials comprised of three and four Gaussian peaks. Specific low-energy resonances with very narrow width are shown to correspond to classical multiple-reflection events. These states have much larger probability densities inside the cage than outside and mimic bound states in the sense that the symmetry-breaking effect of the incident wave is relatively small. As a result, we have found that isolated states display the simple symmetry characteristics of bound states. Overlapping resonances exhibit a mixing of symmetry classes leading to wavefunctions of lower symmetry, like those of wider resonances at higher energy. We demonstrate that at energies below the lowest resonances of two-dimensional cages, where the distance across the entrance of the cage corresponds to less than half a wavelength, the wavefunction may still gain access to the interior region by squeezing its wavelength in the necessary direction at the expense of the kinetic energy in the direction normal to the opening. The resulting curvature of the wavefunction in the donor dimension corresponds to an imaginary wave number, curving away from the plane defined by zero amplitude. This mechanism for passing between obstacles may be relevant for electronic and optical devices having spatial structures with dimensions comparable to the wavelengths of the energy carriers.

We discuss the role that interactions play in the non-commutative structure that arises when the relative coordinates of two interacting particles are projected onto the lowest Landau level. It is shown that the interactions in general renormalize the non-commutative parameter away from the non-interacting value . The effective non-commutative parameter is in general also angular momentum dependent. A heuristic argument, based on the non-commutative coordinates, is given to find the filling fractions at incompressibilty, which are in general renormalized by the interactions, and the results are consistent with known results in the case of singular magnetic fields.

The exponential Hilbert space approach is used to link quantum mechanics in a small system with d-dimensional Hilbert space , with the collective quantum behaviour in a large system comprised of d coupled oscillators with Hilbert space H. In the large system, the expectation value of the mode position operator describes the location of a quantum state within the chain of d oscillators, and the expectation value of the mode momentum operator describes the change of the mode position with time. A continuum of modes (d → ∞) is also considered and in this case these operators obey the commutation relation . A consequence of this is that uncertainties in the location of a state and in its momentum obey an uncertainty relation. Displacements and squeezing in the mode phase space are discussed. For a certain Hamiltonian, the obey equations of motion which are very similar to those of a harmonic oscillator. If the system is in an entangled state, the formalism can be used to quantify concepts like the location of entanglement, and the speed with which the entanglement propagates.

We construct non-relativistic models of complex scalar bosons coupled to Chern–Simons gauge fields by using a Galilean covariant formulation based on the embedding of the (d, 1) Newtonian spacetime into a (d + 1, 1) Minkowski manifold with light-cone coordinates. We also examine various generalizations of the Carroll–Field–Jackiw three-dimensional Chern–Simons term for which the usual Lorentz covariance is broken. Models with cubic Chern–Simons term and non-Abelian gauge fields are briefly discussed. Our main result is the application of this covariant formalism to the investigation of the Aharonov–Bohm effect, for which we retrieve the invariant scattering amplitude up to one-loop.