Table of contents

Determining the number of limit cycles for a system is normally very difficult. The author presents some sufficient conditions which guarantee that the general predator-prey model of Huang (1988) has at least three limit cycles around the equilibrium point. The author's approach can be generalised to discuss the existence of 2n+1 limit cycles in the model.

Recently Ignatiev and Kuzmin (1988) developed a theoretical model to implement small violations of the Pauli principle. The authors remark that the Pauli principle is unique among discrete symmetries, and that in consequence any apparent violation actually signals new physical degrees of freedom, with no violation of the Pauli principle. They construct an algebraic model which incorporates the algebra of Ignatiev and Kuzmin and leads to the same apparent violations, yet preserves the Pauli principle. Their algebra is that of a Jordan pair, an algebraic concept which provides a natural framework for structures that do not assume bilinear commutation relations.

The authors present L operators both for the special case of Neumann's system and the Kowalewski-Chaplygin-Goryachev top. These are quantum systems integrable by the quantum inverse scattering (R-matrix) method. L operators, being 2*2 matrices, satisfy an algebra generated by an R matrix of the XXX type. The close connection between the two models is demonstrated. They carry out a non-obvious separation of variables and also give a dynamical group scheme for the eigenstate problem of Neumann's system. This separation differs from those in Euler angles and allows them to find eigenenergies in an effective way.

Dilute ferromagnetic spin systems are considered in the Griffiths phase below the percolation threshold. It is shown that, for relaxational dynamics, the spin autocorrelation functions are bounded both above and below by functions of the form exp(-A(1n t)^d(d-1)/) (Ising systems) and exp(-Bt¹2/) (Heisenberg systems), establishing these as the correct asymptotic forms.

The mean asphericities (A_d) of open and closed Gaussian polymer chains are computed exactly and analytically (up to simple quadratures) for arbitrary space dimensions d. Excellent agreement is found with existing simulation data for d=2, 3, and 4. The authors' technique can also be used to compute other averages of ratios of fluctuating variables as well as extended to include the effects of self-avoiding walk interactions.

Using an exact real-space renormalisation group technique. The authors show that self-avoiding trails (SAT) and self-avoiding walks (SAW) on the Sierpinski gasket obey the same critical exponent nu for the 'correlation length' and therefore belong to the same universality class. On the other hand, it is shown that SAT on branching Koch curves with maximum ramification number R_max>3 is in a different universality class from that of SAW.

The Bethe-ansatz equations for the anisotropic spin-S Heisenberg chain are analysed numerically. The position of the roots for the whole spectrum is calculated for the spin-1 model in a chain of width N=4. Some states are presented which violate the normally accepted string hypothesis.

It is conjectured that part of the limiting locus of partition function zeros for the hard hexagon model is an algebraic curve generated by a simple and rational algebraic equation. This equation arises from the same mathematical mechanism as that which determines algebraic-invariant properties of self-dual systems. A remarkable approximation to the critical point of the three-state Potts model (triangular lattice) is mentioned in passing.

The Harry Dym equation, which is related to the classical string problem, is derived in three different ways. An implicit cusp solitary-wave solution is constructed via a simple direct method. The existing connections between the Harry Dym and the Korteweg-de Vries equations are uniformised and simplified, and transformations between their respective solutions are carried out explicitly. Whenever possible, physical insights are provided.

It is shown that the problem of three equal-mass particles moving in one dimension, and interacting through anharmonic two-body forces, is exactly separable in hyperspherical coordinates. It is further proved, under mild conditions, that this is the most general 'well behaved' two-body force for which separation is possible. Some typical spectra are calculated numerically, and compared with analytic expressions based upon the semiclassical approximation.

An approach to the theory of unstable bound states of electrons in the nuclear field is suggested. This approach permits the determination of unstable bound states when a quasistationary approximation is unjustified and, therefore, methods normally used for this purpose in the atomic theory cannot be applied. The equations employed in the author's approach make it possible, without resorting to perturbation theory, to determine the energy distribution of excited states, the shape of the natural broadening of spectral lines, the amplitudes of the resonance scattering of photons and electrons over atomic systems, and to describe spontaneous radiation and autoionisation decay. Calculated results are given of the shape of the natural broadening of spectral lines in a three-level atomic system when energy levels overlap. In the quasistationary approximation standard expressions of the relativistic atomic theory for level energy shifts and radiation widths follow from the equations.

The problem of the numerical determination of eigenvalues of the one-dimensional Schrodinger equation with symmetrical potential V(x) is considered. The problem is reduced to the definition of an 'eigenvalue function' G(E) well defined for the given potential V. It is theoretically proved that the even-parity eigenvalues are given by G(E)=0, and the odd-parity eigenvalues are given by G(E)= infinity .The method is 'canonical' in the sense that it is independent of the eigenfunction; yet, along with the eigenvalue, it allows the determination of the eigenfunction initial values. This method is applied to the potentials V=x², V=x⁶-bx², V=x²+ lambda x²/(1++gx²), where exact eigenvalues are available. The numerical results, compared with the exact ones and with those of previous confirmed methods, show that the present method is accurate and efficient.

Though mixed crystals can form orientational glasses, the short-range model of the isotropic quadrupolar glass does not have a glass phase in three dimensions at non-zero temperature. A Monte Carlo investigation is presented which suggests that the glass phase is stabilised by cubic anisotropy.

A continuous expression for the spin-spin correlator of the 2D bond-dilute Ising model is considered. It is shown that its evaluation through straightforward use of RG techniques leads to inconsistent results. Connections with related results in the literature are discussed.

The author shows that for generic short-range classical lattice gas models the third law of thermodynamics holds-entropy goes to zero with temperature.

The percolation surface was calculated for a cubic lattice where each active site is occupied by a cube that only conducts along planes normal to one of the three Cartesian directions. The transport properties of the assembly of cubes, found using a resistor network approximation are more adversely affected by the presence of vacant sites than by the anisotropy of the different cubes. Studies of the correlation length and conductivity exponents indicate that the behaviour of the material is in general three dimensional, though systems comprised of voids and cells oriented in one direction have critical exponents corresponding to two-dimensional systems. An effective medium treatment gave simple analytical results for the percolation surface and effective conductivity that were quite accurate in some limits.

The author comments on a paper by Burzlaff (see ibid., vol.21, p.561 (1988)). His paper is devoted to the investigation of optical soliton generation for special input pulses within the framework of the non-linear Schrodinger equation. The author demonstrates that the main result by Burzlaff, namely the threshold condition for soliton generation, may be obtained in a general form for any initial pulse with a constant phase (and its Galilei transformation). In particular, the author's general result includes the Burzlaff results. The author notes also that the formula obtained is applicable to the sine-Gordon equation with a special type of initial condition.

As a consequence of an elementary quantification-theoretical analysis, the author maintains that formulations in the spirit of Caratheodory (1909) of the second law are dependent on the zeroth law. Any attempt to show the redundancy of the zeroth law on the basis of formulations of this kind of the second law is, therefore, bound to fail.

For original paper, see ibid., vol.22, p.341 (1989). It is pointed out that Walter's criticism of Caratheodory's principle is misplaced since the principle is irrelevant to the original argument; and the claim that the characterisation of the class of physical systems admitted for consideration already involves the notion of temperature is rejected.