Table of contents

One-dimensional maps have chaotic bands which may merge at critical values of the control parameter, a. Just after the merger, one can define a band-to-band hopping time, which diverges as mod a-a_c mod ^-1z/ for a map with a zth-order maximum. Numerical data are presented to support this conclusion.

It is shown that the retarded Maxwell field produced by a flat space-time isolated system consisting of two mutually interacting charged particles does not peel in the asymptotic past.

It is shown that a Monte Carlo renormalisation-group technique can be employed for direct renormalisation of a system on a continuum, i.e. without restriction to a periodic lattice. For the problem of overlapping discs the authors find the critical area fraction s_c=0.688+or-0.005 and the correlation-length critical exponent nu =1.33+or-0.07. The latter result indicates the regular-irregular lattice universality.

A number of results are obtained for Potts models on an arbitrary planar lattice. It is shown that the dual of a dilute Potts model is a graph-generating function and that the dual of a constrained dilute Potts model is an undiluted Potts model. It is also shown that the dilute Potts model generates a correlated site-bond percolation. For uncorrelated site-bond percolations the analysis determines the percolation threshold from a knowledge of the critical point of a Potts model. This generalises a recent result of Kondor (1980) who uses a star-triangle transformation to derive this relationship for the honeycomb and triangle lattices.

Critical exponents of the two-dimensional, q-state Potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q. Results for the temperature exponent agree accurately with the conjecture of den Nijs (1979). The magnetic exponent is found to behave in accordance with the conjecture of Nienhuis et al. (1980).

The asymptotical (k to infinity ) behaviour of zeros of the polynomials g_k^m( nu ) encountered in the treatment of direct and inverse problems of scattering in neutron transport as well as radiative transfer theories is investigated in terms of the amplitude w_k of the kth Legendre polynomial needed in the expansion of the scattering function. The parameters w_k describe the anisotropy of scattering of the medium considered. In particular, it is shown that the asymptotical density of zeros of the polynomials g_k^m( nu ) is an inverted semicircle for the anisotropic non-multiplying scattering medium.

It is shown for gaseous systems that the internal energy function can be uniquely constructed solely on the basis of a set of properties of the final temperature reached in the equilibration process due to thermal contact between two gaseous systems with a fixed density. A formulation of a theory of measurement is used, in which an additive quantity, such as mass or length, can be defined as a unique numerical representation of a certain algebraic structure equipped with two operations addition (+) and qualitative comparison (an order relation <or=).

Necessary and sufficient conditions of invariance on (first- and second-rank) spinors under the Poincare group are established through an infinitesimal method. The group E(3) and its subgroups are then specifically considered and their invariant first-rank spinors are derived. Second-rank spinors and four-vectors are also connected through invariance arguments, and their consideration enlightens some geometrical concepts such as Lie derivatives of spinor fields.

It is shown that the symmetric group S_n may be usefully embedded in the orthogonal group O_n, and that this embedding leads directly to an n-independent 'reduced' notation for both the spin and ordinary representations of S_n. Making use of this embedding, together with the properties of Q-functions (or Hall-Littlewood functions), branching rules for O_n (spin down) S_n are developed and the general rule for the decomposition of spin representations under S_n (spin down) S_n-1 is obtained. Simple methods are given for calculating all possible Kronecker products involving the spin and ordinary representations of S_n and the resolution of Kronecker squares into their symmetric and antisymmetric parts. The spin representations of S_n are systematically classified as to their orthogonal, symplectic or complex characters. The emphasis throughout is on obtaining results that obviate the need for explicit character tables and presenting results in an n-independent manner as much as possible.

The reduced matrix elements of a coupled tensor between (i) uncoupled kets and (ii) coupled kets are derived in a group-subgroup basis using the phase choices of Butler (1975).

The reflection coefficient R produced by an isotropic medium is expressed in terms of the reflection coefficient r produced by a second isotropic medium known as the 'carrier' medium. At the same time, the field W in the first medium is expressed in terms of an integral equation involving the field w in the carrier medium. The expansion of W and R as power series in terms of a parameter alpha gives rise to an interesting theorem: the coefficient of alpha ⁿ⁺¹ in R contains a large number of multiple integrals of order n+1, and the values of these integrals are mostly equal in pairs. This pairing of equal contributions to R is a kind of reciprocity between pairs of perturbation terms of order alpha ⁿ⁺¹, the two contributions in each pair being produced by n+1 re-radiative processes taking place in differing arrangements.

The exact analytical solution of inhomogeneous differential equations, resulting from the improvement of the known methods due to Post (1956) and Hall (1967), is presented. As an illustration, the energy lower bounds of N-fermion systems with oscillator and gravitational interaction are calculated and the results compared with the earlier data.

The eigenvalues of the Hamiltonian H=-d²/dx²+x²+ lambda x²/(1+gx²), with lambda and g as parameters, are studied. A simple numerical procedure where the input data are known exactly and the requirements on the computer memory are not stringent is presented. The results agree very well with the available calculations reported earlier.

Using the transfer matrix method, the solution of the scattering problem on the finite periodic chain of equal symmetric potentials is found in terms of the partial reflection and transmission coefficients. A simple algebraic result is presented showing that the transmission coefficient oscillates rapidly between one and some lower value when the energies lie in the allowed Bloch band. A special case of delta potentials is discussed using the momentum representation. The explicit form for the transmission amplitude can also be used to determine positions of the bound states.

For the class of generalised Hulthen s-wave potentials V_GH^(p)(r)= Sigma _i=1^p g_ia_i exp(- gamma _ir)/(1- Sigma _j=1^p a_j exp(- gamma _jr), p>or=1, the exact solution is presented for both the Schrodinger and Klein-Gordon equations. This simultaneous solubility represents a new non-perturbative treatment of the kinematical relativistic corrections and is based on a non-standard generalisation of the hypergeometric function.

The global structure of a class of Einstein spaces is investigated. These spaces have algebraically special Weyl tensors and contain homogeneous hypersurfaces. It is found that they display simple examples of a variety of interesting configurations involving horizons and singularities.

From covariant Dirac theory in curved space-time, dynamical equations for the motion of the spin and the spin-induced non-geodesic behaviour of the particle trajectories are deduced. This is done for arbitrary space-times in a generally covariant and observer-independent way. The procedure is thereby based on a WKB scheme and a Gordon decomposition of the Dirac probability four-current. A complete correspondence between the quantum mechanical equations of motion and the classical equations for extended isolated bodies or pole-dipole particles is found. This can as well be taken as a confirmation that to the first order of a WKB approximation the gyro-gravitational factors of the classical angular momentum and of the intrinsic quantum mechanical spin agree.

The authors describe a detailed numerical investigation of the spectrum ( lambda _k, lambda ) of the Rayleigh singular integral operator A_gamma governing the one-dimensional test-particle gas at mass ratio gamma (Rayleigh's piston). Results confirm that the discrete spectrum lambda _k( gamma ) is empty apart from the equilibrium eigenvalue, lambda ₀=0, in the range gamma >0.28..., but acquires new points in a regular manner as gamma decreases. Thus the discretum interval lambda _k in (0,1) is gradually filled to ever-increasing density as the Brownian motion regime gamma <<1 is reached. Spectra and eigenfunctions, together with derived results for the velocity autocorrelation function and complex admittance for charged test particles, are used to illustrate the effectiveness of the Rayleigh-Fokker-Planck approximation based on the artificial assumption gamma to 0.

Series analysis techniques are used to form estimates of the local ( gamma ₁₁) and layer exponents ( gamma ₁) for self-avoiding walks on the tetrahedral lattice, confined to a half space. The estimates are compared with predictions of surface scaling and with a renormalisation group calculation of Bray and Moore (1977). In addition, a new upper bound on gamma ₁₁ is derived for the self-avoiding walk problem.

Using a continuous class of real-space renormalisation transformations the authors study the critical behaviour of the Ising model in a variety of two-dimensional lattices. They use the simplest possible cluster, with two cells, and a single renormalised coupling. Certain of these transformations-well separated from the majority spin rule-produce sharply more accurate critical properties than the rest. From this study they note an optimal set of conditions which characterise the successful transformations. The optimal transformations appear to have a certain 'locality' property: renormalised couplings beyond nearest neighbour are especially small.

Transfer matrix methods are used to obtain the partition function and correlation length of Ising chains at low temperatures. For the infinite-spin chain, asymptotically exact eigenvalues and eigenvectors are obtained from an iteration scheme, and from a secular equation method. It is also shown how the Ising chain for general spin S (including S to infinity ) may be treated using a 2*2 transfer matrix. For low temperatures (K identical to J/K_BT>>1) the inverse correlation length, in units of the spin spacing, is xi ^-1=4S e^-2K+O(e^-4K) for finite S, and xi ^-1=8K e^-2K(ln 4K+ gamma +O((lnK)/K)) for infinite spin where gamma is Euler's constant.

The topological classification of solutions to the Euclideanised SO(3) Einstein-Yang-Mills equations is discussed. As well as the usual Pontriagin index, two new two-valued indices, associated with the second Stiefel-Whitney class, are required to specify the bundles. The Charap-Duff solution is classified in this framework.

The authors identify a class of symmetries for Lagrangian systems, called higher-order Noether symmetries, which yield a corresponding first integral without further integrations. First, a version of Noether's theorem is recalled in which Noether symmetries are considered to be symmetries of the two-form d theta , theta being the Cartan form. nth-order Noether symmetries are then defined by the requirement L_Yⁿd theta =0. The picture is further generalised by exploring various ways in which the computation of repeated Lie derivatives of d theta can lead to the identification of first integrals. A number of illustrative examples are discussed.

It is shown that in the Thirring model there exists a more general definition of the current than the usual one, which leads to a one-parameter family of renormalised solutions of the model. The set of vector currents obtained in this way yields many possible quantisations of the model, and in particular a quantisation with a finite renormalisation. It is also shown that with such a definition of the current the conformal dimension of the two-point function is not fixed.

The authors analyse a stochastic linearised extension of the Yukawa-Takabayasi-Feynman bilocal oscillator model and show that: (a) the external Poincare group P commutes with an internal extension of the Lorentz group, i.e. U(1)(X)SO(6,2); (b) the corresponding internal fundamental spinor representation of the associated D₄ algebra yields eight quarks and eight leptons, which correspond to heuristic proposals of Nambu and Salam.

Within the framework of classical electrodynamics, the authors present a quantitative analysis of the intensity-dependent distribution in frequency and angle of the spontaneous incoherent radiation that arises from the collision of a beam of relativistic electrons with a very intense, linearly polarised plane electromagnetic wave. As a useful corollary, this investigation establishes a uniform asymptotic expansion of an important class of functions that frequently occurs in the literature under the name of 'generalised Bessel functions', but for which to date only qualitative results are employed for the most interesting range of large values of its variables.

The scaling behaviour of a superconducting system with another ordering in a magnetic field is considered by the renormalisation-group approach. Exact recursion relations up to the order O( epsilon ) are obtained and analysed. The usual bicritical and tetracritical fixed points do not appear for physically interesting values of the symmetry indices of the order parameters. This phenomenon is associated with the strong influence of the vector-potential fluctuations on the critical behaviour of the system.

Operators are derived for the vortex density and the vortex-antivortex pair density of s=¹/₂ models on the square and triangular lattices. At T=0 the finite-lattice method estimates of the number of vortices and antivortices per plaquette for the XY model are 0.025+or-0.002 for the square lattice and 0.052+or-0.008 for the triangular lattice. High-temperature series expansions on the triangular lattice allow the vortex and pair densities of the XY model to be plotted against inverse temperature.

The authors point out a logical fallacy in Robinson's analysis (ibid., vol.13, p.877, 1980) of a thought experiment purporting to show violation of Heisenberg's uncertainty principle. The real problem concerning the interpretation of Heisenberg's principle is precisely stated.

In the previous paper by M S Marinov, the maximal torus was described wrongly, and some equalities contained a false factor. In fact, it is the maximal torus H of the universal covering group, that is an r-simplex (the Weyl alcove)