Table of contents

A Frenet-Serret like equation is discussed in superspace. It is shown that new nonlinear supersymmetric equations can be generated which are completely integrable. As a particular case, the super sine-Gordon equation also belongs to this class.

The author explicitly builds exponential-type bi-solitons of L_qK= Sigma _i+j=1^i+j=qa_ij delta _bc^i+jK=constant*K^N-1K_x N integer>or=2, b=xⁱ and C=t^j or equivalently L_qG=constant (G_x)^N for the potentials G_x=K. The author assumes both that their denominators have no soliton couplings and that L_q are either factorised linear operators or germs of factorised operators. K^N and K^N-1K_x nonlinearities with associated factorised linear operators belong to a class of non-integrable equations sharing similar properties.

The authors finish a line of argument started in an earlier paper and show that the application of the Hill determinant and the associated analytic continued fraction method for the calculation of eigenvalues of anharmonic oscillators is of dubious validity and may thus lead to erroneous results. In this context some theorems concerning the eigenvalues of the lambda x⁴ and the doubly anharmonic oscillators are proved.

The time-dependent Schrodinger equation is solved exactly for an electron constrained to a circular orbit concentric with a cylinder of smaller radius containing magnetic flux changing in time. The average kinetic angular momentum and energy are changed by the induced electric field in such a way as to satisfy Ehrenfest's theorem.The kinetic angular momentum and energy eigenvalues depend on the instantaneous enclosed flux.

Gives conditions under which a certain, very general, version of the ice model of lattice statistics can be solved. Special cases of this model include the eight-vertex model, the three-colour problem and the staggered ice model soluble by the Pfaffian method. All of these special cases are shown to satisfy the authors' criteria of solubility.

Presents a discussion of the appearance of one-dimensional (1D) effects in narrow strips of percolating systems. Using both scaling and probabilistic arguments, rather than specific models of the infinite percolating cluster, the authors predict that 1D effects are observable at strip lengths L_x>L_y exp(L_y/ xi ₂)^D(A) where D(A) is the fractal dimensionality of very large clusters below the percolation threshold.

Introduces a transfer-matrix formulation to compute the conductance of random resistor networks. The authors apply this method to strips of width up to 40, and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions, t=1.28+or-0.03.

In a recent letter (S. Simons, 1982) to the editor, criticisms were raised against an argument that the authors used in previous publications (1980, 1981, 1982) to justify a modification in the definition of the nonequilibrium entropy. These criticisms call for some comments that are developed in the present note.

Parisi and Sourlas (1981) have argued that the isotropic branched polymer problem in dimension d+1 is related to the Lee-Yang edge singularity problem in dimension d-1. To test if there is a relation with the directed branched polymer problem, the authors calculate the generating functions for both site and bond directed branched polymers for arbitrary dimension d to order s_max=10 and b_max=8. Their analysis lends support to the proposal that theta (d+1)-1= theta _D(d)= sigma _LY(d-1)+1, where theta and theta _D are the critical exponents for lattice animals and directed lattice animals, and sigma _LY is the Lee-Yang edge singularity exponent. They also obtain expansions for the growth parameter in the variable sigma ^-1=(2d-1)^-1 for bond and site directed lattice animals.

Explicit expressions are obtained for the infinitesimal operators of the degenerate representations of the groups SL(n,C), SO(n,C) and Sp(n,C) in a discrete basis. They are used to obtain the infinitesimal operators of unitary representations of the group K(X)K in a K basis, where K is one of the groups SU(n), SO(n), Sp(n). The subgroup K is diagonally embedded into K(X)K. Matrix elements (generalised Wigner d functions) of the degenerate representations of GL(n,C) and U(n)(X)U(n) are evaluated. Clebsch-Gordan series are derived for the tensor product of irreducible representations of K which are given by one non-zero integer. The infinitesimal operators are applied to obtain recurrence relations for the Clebsch-Gordan coefficients of this tensor product. It is remarkable that they connect Clebsch-Gordan coefficients corresponding to different resulting representations.

Studies a nonlinear heat equation in a finite interval of space subject to a white noise forcing term. The equation without the forcing term exhibits several equilibrium configurations, two of which are stable. The solution of the complete forced equation is a stochastic process in space and time that has a unique stochastic equilibrium. The authors study this process in the limit of small noise, and obtain lower and upper bounds for the probability of large fluctuations. They then apply these estimates to calculate the transition probability between the stable configurations (tunnelling). This model problem can be interpreted as a rigorous version of some recent attempts to describe Euclidean quantum systems in terms of stochastic equilibrium states of a nonlinear stochastic differential equation in infinite dimensions. However, its significance goes beyond this situation and the authors' methods may be applicable to models in other areas of natural science.

Studies the intersection of a normal form for the conic umbilic catastrophe of general codimension K with planes in control space having all but two control variables set to zero, and display in a set of figures and a table the geometry in which strata of the bifurcation set intersect these planes, and the singularity types occurring on them. Five distinct curve forms arise. The authors compare these sections with those of the cuspoids and discuss their occurrence as optical caustics.

Isotropic random flights, where the number of individual flights N is random, are studied. N is taken to be governed by a Poisson distribution and also by a negative binomial distribution, each with mean (N). The probability density function of the length of the vector sum is shown to be mixed, in that it contains impulse components (Dirac delta functions) as well as the absolutely continuous component. The limiting density functions are also obtained, and in the negative binomial case lead to the random flight version of the K-density function introduced by Jakeman and collaborators (1976, 1978). Finally, the moments about the origin are explicitly evaluated for both fixed N and random N.

A class of bivariational functionals is derived whose stationary points are pairs of solutions of the single-particle Schrodinger equation (or Dirac equation, respectively) subject to so-called 'complementary boundary conditions'. The formulation of the boundary value problem is sufficiently general to include matching conditions and Bloch conditions as well as scattering conditions. It is shown how bivariational translational techniques may be applied to problems with three- and two-dimensional translational symmetry (calculation of complex band structures and propagation matrices, scattering problems).

A system of dispersion equations for quasi-periodic solutions of the multidimensional sine-Gordon equation is discussed. This system of algebraical equations determines the parameters appearing in the solution which involves abstract theta-functions. In the case of the two-phase quasi-periodic solutions, it is shown that the form involving theta-functions represents a broader class than the class of solutions given by the expression 4 tan^-1 fg. The condition for the equivalence of both classes is also reported.

Motivated by considerations arising from many-body quantum physics the authors consider the moment problem in the general case where the moments are finite real numbers. They present a well defined analytic procedure for the construction of an infinite set of exact solutions to the above problem and discuss several special cases.

As a contribution to a tentative formulation of atomic physics in a curved space, the determination of atomic fine structure energies in a space of constant curvature is investigated. Starting from the Dirac equation in a curved space-time, the analogue of the Pauli equation in a general coordinate system is derived. When particularising these results to the model of a spherical three-space with a Coulombic field, one obtains the 'curved' form of the one-electron fine structure Hamiltonian, i.e. the curved form of the Lande spin-orbit interaction and of the relativistic correction of the kinetic energy as well as some additional terms which vanish at the traditional flat limit. The theoretical curvature induced shifts and splittings of the fine structure energy levels are put in evidence and examined for the particular case of the hydrogenic n=2 levels.

The free electromagnetic field is canonically quantised in a gauge-invariant way by interpreting the Fourier coefficients of the magnetic induction field B as generalised coordinates, and the coefficients of the electric field E as their conjugate momenta. The usual commutation relations among the components of E and B are obtained. A canonical transformation, corresponding to a rotation in generalised phase space, is made on the Fourier coefficients. This transformation is shown to give a duality transformation on the electric and magnetic fields. The free-field Maxwell equations and the commutation relations are invariant under duality transformations. However, if interactions are introduced, the invariance under duality transformations is broken, and the original canonical theory should be used.

Numerical upper bounds are found for the total spin-flip cross section when the total cross section and the forward slope are known and the partial waves are unitary.

A theory of a self-interacting scalar field and gravitation is discussed in the context of a Robertson-Walker metric. The calculations are efficiently reformulated using a metric compatible connection with torsion, although the relation to the Brans-Dicke theory is explicitly displayed. New exact solutions are derived and their relevance to recent cosmological models pointed out.

The electrovacuum Ernst equations are formulated as a nonlinear sigma -model on the symmetric (Kahler) space SU(1,2)/S(U(1)*U(2)). It is shown, using this formulation, that a generalised Robinson-type identity for the electrovacuum Ernst equations may be derived. A special role played in the derivation of this identity by the hidden symmetry group SU(1,2) is established. A theorem is proven that the only possible exterior solution for a (pseudo-) stationary, rotating, electrovacuum black hole with non-degenerate event horizon is the Kerr-Newman solution with m²-a²-P²-Q²>0.

The authors have set up the Klein-Gordon equation in the background of the Schwarzschild curved space-time and studied the scattering of radial tardyons and tachyons from a black hole. They also show that black holes of mass below 7*10¹⁴ g may contain bound states of tardyons of pion mass which will be unstable on account of the presence of an attractive r^-4 term.

Derives the BRS identities for quantum gravity in the axial gauge and uses them to explain why the one-loop counterterm is not generally covariant.

Studies the non-equilibrium corrections to the classical Landau-Lifshitz formulae for the fluctuations of the heat flux and of the viscous pressure. The authors' analysis is based both on a non-equilibrium entropy and on a microscopic model. The results are of the same order but not coincident.

The authors have derived series for weakly and strongly embeddable trees in d-dimensional simple hypercubic lattices for arbitrary integral d. For d=2,3,...,9 they present series evidence that such trees are in the same universality class as lattice animals. In addition they have derived expansions in inverse powers of sigma =2d-1 for the growth parameters for bond and site trees and compare these with the corresponding results for animals.

The quantum Hamiltonian analogue of the two-dimensional ANNNI model is investigated by finite-lattice mass gap methods. By using lattice sizes capable of simulating systems of varying modulation, the authors are able to show the existence of a modulated phase between the paramagnetic and (2,2) antiphase regions. The modulation on the incommensurate to paramagnetic boundary is shown to vary and this variation is calculated as a function of the anisotropy. In addition, they find evidence for an XY-like transition from the incommensurate to the paramagnetic phase and perhaps a non-universal transition from the paramagnetic phase to the antiphase.

A new method of calculating critical indices from series expansions is given. The method involves rational approximation with a denominator chosen so that the effect of singularities other than the critical one is minimised. The method suggests that the index for the low-temperature spontaneous magnetisation in the spin-¹/₂ Ising model may be 0.323+or-0.003 for both BCC and FCC lattices.

In a spin glass with Ising spins, the problems of computing the magnetic partition function and finding a ground state are studied. In a finite two-dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice. In contrast to this fact, the same problems are shown to belong to the class of NP-hard problems, both in the two-dimensional case within a magnetic field, and in the three-dimensional case. NP-hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it.

Studies the Boltzmann equation for a Lorentz gas with scattering on stationary hard spheres in the presence of a constant field E. The exact initial and asymptotic time evolutions are given and compared with numerical calculations. Starting with an initial equilibrium velocity distribution the authors study the influence of the initial temperature T on the drift velocity of the Lorentz gas. The drift velocity quickly reaches a maximum and then decreases slowly towards zero. In particular an upper bound, close to 0.8 E^1/2 lambda ^1/2, exists for the drift velocity. Here lambda =( pi a²n)^-1 is the mean free path, related to the density n and radius a of the scatterers. In an initially cool gas the drift velocity slows down as t^-1/2 soon after the maximum is passed. In an initially hot gas, however, there are two asymptotic regimes. After a time of order lambda ^1/2E^-1/2 the drift velocity stays constant for a time interval whose length is proportional to T^3/2, and eventually decays as t^-1/2.

For pt.III see ibid., vol.15, no.9, p.2947-71 (1982). At a 'semiclassical' level changing from translation-invariant to scale-covariant measures introduces the likelihood of instability. The authors present two mechanisms whereby, because of the quantum effects, instability is avoided.

For pt.IV see ibid., vol.15, no.10, p.3273-83 (1982). In the large-N limit the ultraviolet singularities due to the self-interaction and the 'hard-core' change of measure are additive. For d>4 dimensions the hard core is the less singular. The theory can be renormalised in the large-N limit for d=5 dimensions when scale covariance is obligatory (and also in d=4,3 dimensions).

The critical exponent omega ₅, which is a correction to scaling absent in the Ising model, corresponding to insertions of the operator phi ⁵ in a phi ⁴ theory near four dimensions, is calculated to third order in 4D.

The limit distribution of a random Ising model (site disorder) with an exchange interaction of infinite range is calculated exactly. It is shown that randomness changes the analytic properties of the limit distribution at the critical point considerably. At criticality this system cannot be described by a Hamiltonian which is a polynomial in the spin variables.

For pt.I see ibid., vol.14, no.10, p.2817-27 (1981). The problem of the existence of the Stark-Wannier ladder for the Bloch electrons in homogeneous electric fields is considered. If the direction of the electric field coincides with one of the reciprocal lattice vectors, as is well known, the Hamiltonian of the problem can be written as a direct integral of one-dimensional-like Hamiltonians. For these Hamiltonians, the existence of Stark-Wannier ladders of well separated resonances is proved. The wavefunctions corresponding to these resonances are shown to decay exponentially along the field direction.

The q-state two-dimensional ferromagnetic Potts model has a first-order transition for q>4, its spontaneous magnetisation having a jump discontinuity. The magnitude of this discontinuity is calculated exactly for the square, triangular and honeycomb lattices: it depends only on q and is the same for all three lattices.

The authors estimated that the ratio a/m between the specific angular momentum 'a' and the total mass 'm' (both geometrised units) of the cores of massive main-sequence stars exceeds the same ratio for neutron stars by about four orders of magnitude. They consider the question whether efficient mechanisms exist which damp away the excess amount of a/m during stellar evolution.

Presents a study of evolution of helicity-0 waves (density waves) on the FRW cosmological background by using a two-time scale method for solving the perturbed field equation. The kinetic description is adopted by means of the collisionless Liouville equation self-consistently coupled with Einstein's equations. The wave packets obtained are solutions of the shift type. The effects of the expansion of the background (geometrical effects) are contained in these solutions and lead to a power law for the scale factor instead of an exponential law in time. The rate of growth is obtained from dispersion relations which are studied in the case of a cold gravitational plasma. For large wavenumber q the Newtonian dispersion relation is recovered as is the time behaviour of the Newtonian solution, and for vanishing q the behaviour of the relativistic hydrodynamics for the pressureless case is also recovered. It is found that in a relativistic treatment of the cold gravitational plasma, the smaller q becomes the faster the instability grows.

A two-parameter Backlund transformation for the Boussinesq equation, u_tt-u_xx-3(u²)_xx-u_xxxx=0, was obtained by using Hirota's bilinear operator. One can use the two arbitrary parameters in this Backlund transformation to derive the soliton solution, nonlinear superposition formula and infinitely many conservation laws.

Based on the third-order operator factorisation, the resonance interaction of solitons for the nonlinear string equation with a positive dispersion term is considered and operators for creation and annihilation of a resonance triad are constructed.

It is shown that the claims of Home and Sengupta (1981) and of Singh (1981) to have discovered fallacies in the author's analysis of a proposed, technically feasible test of Heisenberg's uncertainty principle are based on misunderstandings.