Table of contents

Abel's integral equation relates the line-of-sight radiance to the emission coefficient distribution function of an extended, cylindrically symmetric transparent radiation source. In the presence of absorption this equation is modified. The author presents two new inversion formulae for this attenuated Abel equation for the case of constant absorption throughout the source. As one of them is completely derivative-free and the other requires differentiation of a weighted integral over the radiance rather than a differentiation of the radiance function itself, they are particularly well suited for use with measured radiance data, where differentiation invariably results in a very large amplification of the random experimental error inherent in the data. An illustrative example is also given.

Considers two nonlinear lattice equations and studies some geometrical features. The author demonstrates that two nonlinear partial differential equations are deduced from the fact that the SL(2,R) connection has zero curvature.

A simple position space renormalisation group scheme is used to find the fractal dimension of Sutherland type aggregation clusters grown in low-dimensional space. Clusters which move along paths of fractal dimension d_W=0, 1, 2 are considered and a systematic variation in cluster fractal dimension matching that of recent computer simulation is found.

The authors define the 'skeleton' of a cluster aggregate as the set of all sites belonging to the shortest paths connecting a chosen site with the Lth chemical shell surrounding that site. The fractal properties of skeletons of percolation clusters at criticality have been studied, and it is inferred that the mass of the skeleton M_s scales with the chemical distance l (for l<<L) as M_s approximately (l_l^s), where (d_l^s)=1 is universal for l<or=d<or=6. Numerical evidence which supports this conclusion is presented for d=2, and an analytical proof is given for d=6.

The exact values of the universal amplitude A relating the correlation length xi approximately L/A to the strip width L at the critical point of Ising and 3-state Potts models are obtained for the case of antiperiodic, or twisted, boundary conditions. Predictions are also made for the finite-size scaling behaviour of the interfacial tension.

The Flory approximation (1969) for the self-avoiding chain problem is compared with a conventional perturbation theory expansion. While in perturbation theory each term is averaged over the unperturbed set of configurations, the Flory approximation is equivalent to the perturbation theory with the averaging over the stretched set of configurations. This imposes restrictions on the integration domain in higher-order terms and they can be treated self consistently. The accuracy delta nu / nu of the Flory approximation for self-avoiding chain problems is estimated to be 10^-1-10^-2 for 1<d<4.

Evidence is given that the principle of the impossibility of packing rigid chains on a lattice to high density in a disordered array is limited to regions of linear size comparable to the chain length, which has the consequence that for finite chain length, the ground state of a dense system of semiflexible lattice chains does not exhibit a long-ranged orientational order, but is highly degenerated with non-vanishing entropy. This is shown by simulating various systems at concentrations >0.95 on the square and the cubic lattice.

The detailed topological or 'connectivity' properties of the clusters formed in diffusion limited aggregation (DLA) and cluster-cluster aggregation (CCA) are considered for spatial dimensions d=2, 3 and 4. Specifically, for both aggregation phenomena the authors calculate the fractal dimension d_min= nu ^-1 defined by l approximately R^d(min) where l is the shortest path between two points separated by a Pythagorean distance R. For CCA, they find that d_min increases monotonically with d, presumably tending toward a limiting value d_min=2 at the upper critical dimensionality d_c as found previously for lattice animals and percolation. For DLA, on the other hand, they find that d_min=1 within the accuracy of the calculations for d=2, 3 and 4, suggesting the absence of an upper critical dimension. They also discuss some of the subtle features encountered in calculating d_min for DLA.

The authors list all possible representations of the semisimple groups SU(m₁)*SU(m₂)* . . . *SU(m_k) for finite N=2 supersymmetric Yang-Mills theories.

The Fulton-Gouterman transformation (1961) diagonalises special electron-phonon systems with respect to the electronic subsystem, but the transformation operator cannot be written in a simple exponential form. An alternative unitary transformation, displaying a simple exponential form, is presented.

Fourier transforms are unreliable near discontinuities because of the Gibbs phenomenon. Before using the fast Fourier transform technique to evaluate Hilbert transforms, it is desirable to remove any discontinuities by smoothing, as suggested by Papoulis (1962). This is especially true if one wishes to use Fourier transforms to find the Hilbert transform h(t) of a function f(t) which has an infinite discontinuity: it is then necessary to smooth f(t) as well as the Hilbert transform kernel. An alternative to smoothing f(t) is to remove a discontinuity at a point t=d by multiplying f(t) by the factor (t-d): this has the advantage of having better asymptotic behaviour. A numerical example is given and here the two methods perform about equally.

In the literature, a symmetry requirement in the mixed, second partials appearing in the Lagrangian is needed before conservation laws for second-order variational problems can be obtained. In this paper this assumption is removed and an example of a modified, linear Korteweg-de Vries equation is analysed.

A principle of uniform density of periodic orbits in the phase space of a Hamiltonian system with bound classical motion is proposed and used to obtain information about the semiclassical quantum eigenvalue spectrum. It supplies a more refined statistic than the 'one state per Planck cell' rule for the average semiclassical density of states, namely the limiting behaviour of a certain correlation function of the density of states. Unlike the average, this correlation shows markedly different behaviour for systems with integrable and ergodic classical motion.

The simplest illustration of the recently suggested asymptotic-perturbative approach to the band-matrix Hamiltonians is found in the harmonic oscillator complemented by the non-polynomial anharmonicity lambda r²/(1+gr²). In the paper, the detailed construction of the effective Hamiltonian is given and the convergence of its fixed-point expansion is shown.

For pt.I see ibid., vol.17, p.3441, 1984. Constructs the asymptotic power-series expansions of the Green function and wavefunctions for the lambda r²/(1+gr²) anharmonicity oscillator. As a new form of the fixed-point perturbative formalism, the method may be extended to any band-matrix Hamiltonian in principle-the results illustrate its algebraic flexibility as well as a quick numerical convergence.

Discusses the application of WKB theory to Harper's equation psi _n+1+ psi _n-1+2 alpha cos(2 pi beta n+ delta ) psi _n=E psi _m, in the case in which beta is very close to a rational number, p/q. The WKB wavefunction for this system is a vector valued quantity, proportional to an eigenvector u of a matrix H(x,p), which is parametrised by the phase space coordinates x and p. The complex phase of u is determined by a non-holonomic connection rule; when transported around a cycle and in phase space, u is multiplied by a phase factor e^{i gamma (c)}. This phase change manifests itself as a modification of the Bohr-Sommerfeld quantisation conditions.

The h(cross) to 0 semiclassical expansion of Wigner (1932) and Kirkwood (1933) is obtained for quantum systems of finitely many particles. In a d-dimensional Euclidean space R^d without boundaries quantum systems defined by a Hamiltonian, H, given as the sum of the negative Laplacian perturbed by a potential nu (x) are considered. The semiclassical behaviour of the kernels of the semigroup family of operators (e^-zH:Rez>or=0) is determined in terms of an asymptotic expansion in the variable h(cross). The order of the expansion is proportional to the number of bounded derivatives nu (x) supports. The expansion is uniform in R^d*R^d and accompanied by explicit bounds for the error term. The results are obtained for potentials nu (x) that are Fourier images of complex bounded measures.

In several branches of physics the Hamiltonian of a many-body system can be reduced to a rational Jacobi matrix (i.e. a Jacobi matrix whose elements are rational functions of the suffix) by means of a method of the Lanczos type. Here it is shown that one can calculate in a simple analytical way the asymptotic eigenvalue density of all these matrices by means of its moments without solving the corresponding eigenvalue problem. The method is applied to a large class of quantum mechanical models of Hamiltonians.

For the O(2) anharmonic oscillator with negative anharmonicity and one-dimensional double-well potential U(x)=(x²-R²)²/8R², R is a large parameter for both problems, conventional Rayleigh-Schrodinger expansions in power series of 1/R² for the energy eigenvalues agree. The authors studied asymptotic expansions for the eigenvalues which contain, in addition to the perturbation series, exponentially small terms in R² due to the tunnelling through corresponding potential barriers. Numerical calculations are also performed and comparisons with the asymptotic formulae are given.

The Monte Carlo and Langevin dynamical methods of simulating the thermodynamics of physical systems are compared by calculating relaxation times according to the two dynamics for a system which is analytically tractable, namely a single (planar) spin in a potential which has either a single minimum or two minima separated by a barrier. With no restriction on the maximum allowed spin reorientation per Monte Carlo step the Langevin method is faster than the Monte Carlo method for a single minimum potential. However a careful choice of restriction can make the Monte Carlo method as efficient as the Langevin method. For the double-well potential the Monte Carlo method with no restriction is the most efficient. One is forced to use a finite-time step size when numerically solving the Langevin equation and the departures this produces from the equilibrium Boltzmann distribution are studied.

It is generally argued that the energy dissipation of three-dimensional turbulent flow is concentrated on a set with non-integer Hausdorff dimension. Recently, in order to explain experimental data, it has been proposed that this set does not possess a global dilatation invariance: it can be considered to be a multifractal set. The authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta -model.

The authors show that an instability for q>q_c of the Migdal-Kadanoff recursion relations for the two-dimensional Potts model is related to the appearance of a discontinuity fixed point in the Martinelli-Parisi expansion (1981). Consequently, at first order in this expansion, they can evidence a crossover to discontinuous transition, without explicit introduction of vacancies. Through numerical derivation of the free energy, they simulate remarkably well the discontinuous jump in the internal energies.

The authors have studied the behaviour of two- and three-dimensional self avoiding walks confined to a wedge of wedge angle alpha . Series have been obtained and analysed for the (angular dependent) critical exponents characterising various edge susceptibilities. In terms of a general scaling form for the edge free energy, f_e approximately t^{(d-2) nu} psi _e(ht^{-y0 nu} , h₁t^{-y1 nu} , h₂t^{-y2 nu} ), they find for the two-dimensional case the following scaling indices: y0=91/48, y1=3/8, y2( alpha )=-5 pi /8 alpha . They argue that these results are exact, from which follow all exponents for the bulk, surface and edge problem. In three dimensions the authors obtain y0 approximately=2.488, y1=0.65+or-0.02, y2( alpha )= alpha +b pi / alpha where a=0.51+or-0.04, b=-0.847+or-0.017, which, for y2, is precisely of the functional form given by mean-field theory, y2( alpha )=¹/₂- pi /2. They argue that a=¹/₂ for all three-dimensional O(N) models. This simple angular dependence of y2 is different from that suggested by Cardy's one-loop epsilon -expansion, (1983). For the square lattice, they have also studied the case in which the wedge is rotated through an angle of pi /4, and find that the various exponents are unchanged. For the three-dimensional Ising model in a wedge, analogy with the SAW results, plus mean-field results in conjunction with RG and series work yield y0 approximately=2.485, y1=0.71+or-0.02 and y2=a+b pi / alpha with a=¹/₂ and b=-0.79+or-0.02.

A real space renormalisation group method has been applied to find the critical temperature of the Ising model on cubic and octahedral lattices. The result obtained for a cubic lattice is in good agreement with that obtained from high-temperature series expansion and is better than the results of other papers. A modification of the transformation tested for a simple cubic lattice allowed the authors to find the critical temperature for an octahedral lattice. The latter has not so far been studied by any version of the real space renormalisation group method.

A solid-on-solid (SOS) model in a field h conjugate to the orientation of the surface is exactly solved with the aid of Pfaffians. The free energy (h) directly gives the equilibrium shape of a finite crystal. The phase diagram exhibits rough and smooth phases, corresponding to rounded and flat portions of the crystal surface. The solid-on-solid model undergoes transitions of the Pokrovsky-Talapov type (1979) characterised by a specific heat exponent alpha =¹/₂. One special point of the phase diagram corresponds to the appearance of a facet via an alpha =0 transition. Height-height correlations are derived along a special line in the phase diagram. With the aid of the known equivalence of this SOS model with an Ising model, several exponents can be translated from one model to the other. This enables one to derive the topology of the phase diagram of the antiferromagnetic triangular Ising model with first- and second-neighbour couplings in a field.

A simple computer simulation for one-dimensional, classical, disordered systems with symmetric hopping rates is presented, which is well suited for vectorised programming. The data for the frequency-dependent conductivity are compared to analytical results for various classes of distributions of the transfer rates. In particular, for singular distributions of the form rho (W) approximately W^{- alpha} , 0<or=W<or=1, it is demonstrated that the well known effective medium theory is an excellent approximation for -1<or= alpha <or=0.9.

The inclusion of transverse vertex corrections in the gauge technique, needed for the restoration of gauge covariance, results in a self-consistent equation for the source propagator spectral function which agrees with perturbation theory when expanded to order e⁴. The authors have managed to solve the equation in the infrared and ultraviolet limits, for scalar and spinor sources, in an arbitrary covariant gauge. The gauge covariance at asymptopia is thus established.

The authors carefully analyse the RG results of directed site lattice animals on a square lattice obtained from a new RSRG method proposed by them in 1984. Here, asymptotic forms for ν_||(b + 1, b) and ν_⊥(b + 1, b), the values obtained from cell-to-cell transformation of linear size b + 1 to b, are proposed in the large b limit. By fitting the data, they find ν_|| = 0.796 and ν_⊥ = 0.507. These results are in very good agreement with the known values.

The authors show that the number of spiral self-avoiding walks on the square lattice is directly determined by the number of partitions of the integers. The next term in the asymptotic expansion is found, and numerous typographical errors in the original letter of Guttmann and Wormald (1984) are corrected.