Table of contents

We have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint. Using the series we numerically find simple (algebraic) exact solutions for the generating functions. In all cases the size exponent ν = 1.

An initially empty (no edges) graph of the order of M is assumed to evolve by adding one edge at a time. This edge can connect either two linked components and form a new component of a larger order (coalescence of graphs) or increase (by one) the number of edges in a given linked component. The evolution equation for the generating functional of the probability to find in the system a given set of occupation numbers (the numbers of graphs of the order of g having exactly ν edges) at time t is formulated and solved exactly. The expression for the graph composition spectrum is derived and analysed in the limit of large M.

According to the droplet picture of spin glasses, the low-temperature phase of spin glasses should be replica symmetric. However, analysis of the stability of this state suggested that it was unstable and this instability lends support to the Parisi replica symmetry breaking picture of spin glasses. The finite-size scaling functions in the critical region of spin glasses below T_c in dimensions greater than 6 can be determined and for them the replica-symmetric solution is unstable order by order in perturbation theory. Nevertheless the exact solution can be shown to be replica symmetric. It is suggested that a similar mechanism might apply in the low-temperature phase of spin glasses in less than six dimensions, but that a replica symmetry broken state might exist in more than six dimensions.

We introduce a new class of models for interacting particles. Our construction is based on Jacobians for the radial coordinates on certain superspaces. The resulting models contain two parameters determining the strengths of the interactions. This extends and generalizes the models of the Calogero–Moser–Sutherland type for interacting particles in ordinary spaces. The latter ones are included in our models as special cases. Using results which we obtained previously for spherical functions in superspaces, we obtain various properties and some explicit forms for the solutions. We present physical interpretations. Our models involve two kinds of interacting particles. One of the models can be viewed as describing interacting electrons in a lower and upper band of a quasi-one-dimensional semiconductor. Another model is quasi-two-dimensional. Two kinds of particles are confined to two different spatial directions, the interaction contains dipole–dipole or tensor forces.

We investigate the detection dynamics of the parallel interference canceller (PIC) for code-division multiple-access (CDMA) multiuser detection, applied to a randomly spread, fully synchronous base-band uncoded CDMA channel model with additive white Gaussian noise (AWGN) under perfect power control in the large-system limit. It is known that the predictions of the density evolution (DE) can fairly explain the detection dynamics only in the case where the detection dynamics converge. At transients, though, the predictions of DE systematically deviate from computer simulation results. Furthermore, when the detection dynamics fail to converge, the deviation of the predictions of DE from the results of numerical experiments becomes large. As an alternative, generating functional analysis (GFA) can take into account the effect of the Onsager reaction term exactly and does not need the Gaussian assumption of the local field. We present GFA to evaluate the detection dynamics of PIC for CDMA multiuser detection. The predictions of GFA exhibit good consistency with the computer simulation result for any condition, even if the dynamics fail to converge.

Based on the recently introduced orthonormal Hermite–Gaussian-type modes, a general class of sets of non-orthonormal Gaussian-type modes is introduced, along with their associated bi-orthonormal partner sets. The conditions between these two bi-orthonormal sets of modes have been derived, expressed in terms of their generating functions, and the relations with Wünsche's Hermite two-dimensional functions and the two-variable Hermite polynomials have been established. A closed-form expression for Gaussian-type modes is derived from their derivative and recurrence relations, which result from the generating function. It is shown that the evolution of non-orthonormal Gaussian-type modes under linear canonical transformations can be described by the same mechanism as used for the evolution of orthonormal Hermite–Gaussian-type modes, when, simultaneously, the associated bio-orthonormal modes are taken into account.

We develop a uniform semiclassical trace formula for the density of states of a three-dimensional isotropic harmonic oscillator (HO), perturbed by a term . This term breaks the U(3) symmetry of the HO, resulting in a spherical system with SO(3) symmetry. We first treat the anharmonic term for small in semiclassical perturbation theory by integration of the action of the perturbed periodic HO orbit families over the manifold which is covered by the parameters describing their four-fold degeneracy. Then, we obtain an analytical uniform trace formula for arbitrary which in the limit of strong perturbations (or high energy) asymptotically goes over into the correct trace formula of the full anharmonic system with SO(3) symmetry, and in the limit (or energy) →0 restores the HO trace formula with U(3) symmetry. We demonstrate that the gross-shell structure of this anharmonically perturbed system is dominated by the two-fold degenerate diameter and circular orbits, and not by the orbits with the largest classical degeneracy, which are the three-fold degenerate tori with rational ratios ω_r:ω_φ = N:M of radial and angular frequencies. The same holds also for the limit of a purely quartic spherical potential V(r) ∝ r⁴.

The Boltzmann–Shannon information entropy of linear potential wavefunctions is known to be controlled by the information entropy of the Airy function Ai(x). Here, the entropy asymptotics is analysed so that the first two leading terms (previously calculated in the WKB approximation) as well as the following term (already conjectured) are derived by using only the specific properties of the Airy function.

Non-perturbative, exact analytical results are obtained for the harmonically driven two-level system with the use of a Floquet–Green operator formalism. From this operator, we find the quasi-energies and the Fourier components of the Floquet eigenstates which we use to construct the time-evolution operator of the system. As an application of our formalism, we study dynamic localization and high-order harmonic generation in this system. Our results show the existence of an initial condition that leads to an emission spectrum with only hyper-Raman lines for any frequency and strength of the driving field. We show the important role of the initial condition in determining the emission spectrum of this system. In the non-perturbative regime, the emission spectrum is found to be, in general, very sensitive to the amplitude of the driving field. We also derive an expression for the (coherent) emission spectrum which shows its dependence on the degree of localization of the system.

The amplitude-phase method is generalized to coupled Schrödinger scattering states with a common angular momentum quantum number. A pair of exponential-type amplitude-phase solutions u^(±)_j(r)exp[±iϕ_j(r)] for each channel is obtained, containing a common complex scalar phase function ϕ_j(r) and two (column) vector amplitudes u^(±)_j(r). The amplitude functions satisfy certain nonlinear generalized Milne equations and the scalar product of the two amplitudes determines the derivative of the common phase function. Fundamental amplitude-phase matrix solutions that are proportional to Jost-like Schrödinger matrix solutions are constructed. It is shown how a generalized amplitude-phase S-matrix formula can be derived from Wronskian relations involving the two amplitude-phase matrix solutions and a regular matrix solution.

We write the SU(2) lattice gauge theory Hamiltonian in d dimensions in terms of pre-potentials which are SU(2) fundamental doublets of harmonic oscillators. The Hamiltonian in terms of pre-potentials has SU(2) ⊗ U(1) local gauge invariance. These pre-potentials enable us to solve the SU(2) Gauss law and characterize the SU(2) gauge invariant Hilbert space in terms of a set of integers. We discuss the consequences of the additional U(1) gauge invariance. The extension to SU(N) lattice gauge theory is discussed.

It is expected that the implementation of minimal length in quantum models leads to a consequent lowering of Planck's scale. In this paper, using the quantum model with minimal length of Kempf et al (1994 J. Math. Phys.35 4483; 1995 Phys. Rev. D 52 1108; 1997 J. Phys. A30 2093; 1996 J. Math. Phys.37 2121), we examine the effect of the minimal length on the Casimir force between parallel plates.

A solution of Love's integral equation (Love E R 1949 Q. J. Mech. Appl. Math.2 428), which forms the basis for the analysis of the electrostatic field due to two equal circular co-axial parallel conducting plates, is considered for the case when the ratio, τ, of distance of separation to radius of the plates is greater than 2. The kernel of the integral equation is expanded into an infinite series in odd powers of 1/τ and an approximate kernel accurate to is deduced therefrom by terminating the series after an arbitrary but finite number of terms, N. The approximate kernel is rearranged into a degenerate form and the integral equation with this kernel is reduced to a system of N linear equations. An explicit analytical solution is obtained for N = 4 and the resulting analytical expression for the capacity of the circular plate condenser is shown to be accurate to . Analytical expressions of lower orders of accuracy with respect to 1/τ are deduced from the four-term (i.e., N = 4) solution and predictions (of capacity) from the expressions of different orders of accuracy (with respect to 1/τ) are compared with very accurate numerical solutions obtained by solving the linear system for large enough N. It is shown that the approximation predicts the capacity extremely well for any τ ⩾ 2 and an approximation gives, for all practical purposes, results of adequate accuracy for τ ⩾ 4. It is further shown that an approximate solution, applicable for the case of large distances of separation between the plates, due to Sneddon (Sneddon I N 1966 Mixed Boundary Value Problems in Potential Theory (Amsterdam: North-Holland) pp 230–46) is accurate to for τ ⩾ 2.