Table of contents

Using symbolic computations, the unique metric in the space of fields, required to describe self-dual SU(3) Yang-Mills equations by a harmonic map, is determined. Moreover the complete Lie algebra of Killing fields for this metric is established.

The authors have applied the Painleve Test to the coupled nonlinear system advocated by Hirota and Satsuma (1981). They have also made a Lie point symmetry analysis of these equations and have shown that the reduced ordinary nonlinear equations are not members of the Painleve class. Also the Painleve test itself is seen to fail, so that on both counts the equations are not completely integrable in the usual sense.

Finds a closed spatially homogeneous solution of the nonlinear Kac's model (1956), in 1+1 dimensions (velocity v and time t). Choosing for the even velocity part of the distribution function the Bobylev-Krook-Wu mode, (1975-6), the author adds an odd velocity part. The author finds the possibility of the existence of the Tjon relaxation effect, when the time t is increasing. This depends on both the initial condition and the cross section.

It is shown that the symbolic binomial expansions for the Clebsch-Gordan and Racah coefficients are exact for n=1 (where n+1 indicates the number of terms in the series expansions). When exact, these binomial forms reveal polynomial zeros of degree one, which are trivial structure zeros, hitherto considered as 'non-trivial' zeros, along with polynomial zeros of degree >or=2.

The square-root iteration x_t+1+=1-2 mod x_t mod ^1/2 has a linear invariant density, and a characteristic Liapunov exponent lambda =¹/₂.

Solves the recently obtained large-n differential renormalisation-group equations for the Ginsburg-Landau-Wilson and time-dependent Ginsburg-Landau models. The author determines the fixed points of the solutions and makes brief comparisons with solutions determined previously from generating functions of nonlinear scaling fields.

The diffusion of a particle on a percolation network is studied in the presence of an external field which makes the particle more likely to move along the field than otherwise. It is shown that while in weak fields, less than a critical value, the mean displacement is linear in time; in the presence of strong fields, the particle has zero drift velocity but the mean displacement varies as t^alpha for large times t, where alpha is a non-universal field-dependent exponent less than one.

The authors introduce a method of obtaining information about percolation clusters in which the clusters are 'burned' in different ways. This method yields the backbone and the elastic backbone, their number of loops, shortest path and other quantities. It works in any dimension and can be applied also to other clusters. They obtain accurate estimates for the backbone exponents in two and three dimensions.

On the basis of scaling assumptions, the critical behaviour of the distribution (number) of dead (dangling) ends in the infinite cluster of percolating systems is investigated. Critical exponents describing the dead-end distribution are introduced and scaling relations between them are derived. The exponents are also expressed in terms of other exponents nu , beta and beta _B for percolation and their explicit values are evaluated from known estimates for nu , beta and beta _B.

The authors consider the problem of spiral self-avoiding walks as recently introduced by Privman (1983). They prove that the number of n-step spiral self-avoiding walks is given by s_n=exp(2 pi (n/3)¹2/)/(n⁷4/c)(1+O(1/ square root n)) where c= pi /(4.3⁵4/). Similar results for various subsets of these walks are also obtained.

The authors give by explicit counterexample numerical evidence that critical exponent universality fails for the two-dimensional, continuous-spin, Ising model. This result is contrary to what has previously been assumed.

The mean field renormalisation group approach is applied to the bond diluted transverse Ising model. The critical surface in the temperature-transverse field-concentration space is obtained for the two- and three-dimensional models and estimates of critical exponents are also presented.

Geometrical arguments are given which suggest that the Alexander-Orbach conjecture (1982) does not hold for lattice animals in d=2 and for the diffusion-limited aggregates for large dimensions.

Finite-size scaling in the diffusion-limited aggregation model of Witten and Sander (1981, 1983) is studied in two dimensions via Monte-Carlo simulations on semi-infinite strips of width n=2-20. Due to a crossover between self-avoiding walk-like growth for small n and dendritic growth for large n, wide strips (n=8-20) are needed to deduce the Witten-Sander fractal dimension D approximately ⁵/₃ from the finite-size scaling analysis.

The critical dimensions x_CH(q) of the chiral and x_CB(q) of the cubic operator in the two-dimensional q-state Potts model satisfy the extended scaling relations x_CH=(n²-y²)/(4x)+x and x_CB=(m²-y²)/(4x)+x with x+y=2, 2 cos( pi y/2)= square root q, n an odd and m an even integer; n=1 and m=0 give the leading exponents. At q=3 x_CH is relevant, but takes the special value x_T+1=9/5. The crossover exponent at the percolation point in the bond-diluted random Ising model, determined by x_CB at q=1, is equal to one. Along the Baxter line in the Ashkin-Teller, eight-vertex and ANNNI model x_CH=x_T/4+1/x_T.

From one point of view, the Baxter model (1971) is a model of two coupled 2D Ising lattices. It is known that the specific heat exponent alpha in this model is proportional to the coupling between the lattices g, for small coupling: alpha approximately=4g/ pi <<1. The critical behaviour of the specific heat of the Baxter model with impurity lattice bonds is studied and it is found that the Harris criterion (1974) does not hold in this case. In particular for alpha approximately g<0 the critical specific heat of the pure model C_pure( pi ) approximately - mod tau mod ^4(g) pi /( tau =(T-T_c)/T_c ) changes to a function with stronger cusp singularity C_imp( tau ) approximately -(ln ln 1/ mod tau mod )^-1, while according to the Harris criterion it should not change in this case. For alpha approximately g>0 the change is from C_pure( tau ) approximately mod tau mod ^-4g pi / to C_imp( tau ) approximately ln ln 1/ mod tau mod .

For the Ising Sherrington-Kirkpatrick spin glass (1975), the author studies the constraints imposed on the distributions P(q,(J_ij)), P(U,M,(J_ij)) for the overlap q, energy U and magnetisation M of the underlying metastable states, by the general nature of current replica symmetry breaking schemes. He confirms that whilst P(q,(J_ij)) shows significant sample to sample variations the thermodynamic energy and field cooled magnetisation are well behaved. For both the Edwards-Anderson order parameter and the Minimum overlap the case is less clear.

The groups SO( nu -1), SO( nu ), SO( nu +1), SO( nu +1, 1) and SO( nu +2, 2) ( nu =1, 2, 4, 8) and their spin representations are described in terms of the division algebras R, C, H and O.

The problem of optimal analytical extrapolation of holomorphic functions from a finite set of interior data points to another interior point is completely solved in the general case of data known with unequal errors. Simple and easy to handle algorithms are obtained.

The potential energy V is placed into the unperturbed Hamiltonian H₀ and the kinetic energy K remains in the perturbation H₁. It is shown that a convergent N/D method, through standard Fredholm determinants, provides suitable matrix elements of an operator such as H₁+H₁(E-H)^-1H₁.

The problem of determining all standard classical Hamiltonians in two dimensions with Euclidean metric which admit constants of motion quadratic in the momenta is resolved. Several general results are given which make it obvious that the systems found do possess such integrals.

The concept of combinants introduced in the formulation of the generating function for probabilities is analysed, demonstrating the fact that they play the same role in computing cumulants as probabilities do in computing moments. The mathematical framework of Bell polynomials is used to relate combinants and probabilities. The effective use of combinants in branching processes is brought out. Also the coupled differential equations governing the combinants yield direct coupled equations for cumulants. The concept of mixed combinants is developed. This will be explored in later contributions.

The evolution operator for the Dirac equation is expressed as a sequential Feynman path integral. The usual approximations in terms of integrals over finite-dimensional spaces have been explicitly calculated and written in terms of translation operators. This expression should lend itself well to numerical approximations.

When a theory containing a divergent level shift is renormalised, all energy moments above and including the second (essentially (0 mod H² mod 0)) continue to diverge. A mathematical model of this situation is considered and it is shown that the decay of a state in time is unambiguous and well behaved, despite the divergences. The only novelty is that the usual t² decay term at small times is replaced by t² log t.

The authors calculate the average of the quantum-dynamical evolution operator of a harmonic oscillator linearly coupled to a stochastic field with Lorentzian line shape of arbitrary bandwidth (Ornstein-Uhlenbeck process). In so doing they develop some new techniques to cope with the non-commutativity of the boson operators. They also compare the exact results with van Kampen's second-order cumulant expansion and find that the agreement is good if the bandwith is large.

For pt.II see ibid., vol.16, no.7, p.1397-408 (1983). It is shown how quantum mechanical wavefunctions can be obtained from a sequence of simple canonical transformations, which map the given system onto a simple reference system. The resulting wavefunctions are at least uniformly valid up to order h(cross). Under some more restrictive conditions for the individual transformation steps the authors even find the exact wavefunctions. The essential point of the paper is to enlarge the conventional coordinate-momentum phase space by taking time and energy as an additional conjugate pair. In this extended space they exploit the possibility of using transformations which intermix energy and time with position coordinates and momenta. Compared with transformations in the conventional position-momentum phase space, they gain the advantage that scattering states and bound states can be treated in a unified way. Therefore this method is appropriate for systems with mixed spectra. In addition it allows for more flexibility in choosing the individual transformation steps. The practicability of the method is demonstrated by several examples.

A generic family of plane billiards has been discovered recently. The shape of the boundary is given by the quadratic conformal image of the unit circle, and is thus real analytic. For small deformations of the unit disc the billiard is a typical KAM system, but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point. The Kolmogorov entropy has been calculated, and it increases with the deformation of the boundary. The author studies aspects of the quantum chaos for this billiard. He solves numerically the eigenvalue problem for the Laplace operator with Dirichlet's boundary condition. He examines the spectrum, and inspects the avoided crossings at which mixing of nearby states occurs. The variation of the nodal structure and of the localisation properties of the eigenfunctions is studied. In analysing the level spacing distribution he finds a continuous transition from the Poisson distribution towards the Wigner distribution. The exponent in the level repulsion law varies continuously along with a generic perturbation. For small perturbations it seems to be proportional to the square root of the perturbation parameter.

The Onsager-Machlup (OM) function is introduced for a spatially uniform system evolving in accordance with a discrete time Markovian law. Circulation of probability flow is expressed in terms of the OM function. Equivalence is established between the two conditions: circulation identical to 0 and the fulfilment of the detailed balance.

The authors show that the dependence of the critical temperature T_c on the space dimension d of the q-state Potts model can be fitted by straight lines, with slopes predicted by a long-range interaction mean-field theory with broken permutational symmetry: S_q contains/implies S_m(X)S_q-m. By means of phenomenological arguments they 'reconstruct', in terms of the above mean field, the (d, T_c) plots with a high degree of accuracy. The model suggests a factorisation of the partition function near or at the critical point. Predictions, implied by this ansatz, are given on the behaviour of the specific heat exponent alpha and the latent heat L which are corroborated by 'data' presently available in the literature: q=4 with 2.5<or=d<or=6 for alpha , and q>or=4 with 1<or=d<or=2 for L.

The authors study the spectrum of (minus) the Laplacian on a random one-dimensional lattice. It extends monotonically throughout the positive real axis with a continuum limit behaviour for small values. The authors find as expected localisation effects throughout the whole range. They use a variety of techniques which are shown to be consistent and in agreement with a Monte Carlo simulation. They include small and large disorder expansions, direct integral equations for probability distributions as well as the replica method. The latter is also used to investigate Green functions.

In the problem of harmonic chains with random masses the characteristic function is the analytic continuation into the complex frequency plane of the accumulated density of states and the exponential growth rate. A scheme is developed for the calculation of its asymptotic expansion in powers of the frequency. It is found that it changes sign under the unusual transformation to - omega , ((m^k)) to (-1)^k-1((m^k)). Its first nine Taylor coefficients are presented in a table. With the first twelve of these coefficients, a two-point Pade approximant for a related function is used for the calculation of the derivative of the specific heat, without making use of the spectral density. These calculations are carried out for several families of mass distributions.

It is shown that a graded extension of the space group of a (generalised) simple cubic lattice exists in any space dimension, D. The fermionic variables which arise admit a Kahlerian interpretation. Each graded space group is a subgroup of a graded extension of the appropriate Euclidean group, E(D). The relevance of this to the construction of lattice theories is discussed.

Three-particle Coulomb asymptotic states are derived. The knock-out reaction amplitude with distorted waves in the initial and final states is extracted using the Faddeev equations. The conditions are discussed under which this amplitude can be approximated by the distorted wave impulse amplitude. The influence of the Coulomb final state interaction on the differential quasielastic knock-out cross section is investigated. To this end the main singular part of the DWIA amplitude is singled out and compared with conventional approximations.

This contribution deals with the tau -continuum limit of anisotropic plane Ising lattice and in arriving at the quantum dual Hamiltonian by a route different to that of Fradkin and Susskind (1978), one sees the necessity of effecting the exchange of coupling strengths in duality transformations. Construction prescribed by Savit's (1980, 1982) procedure to arrive at classical dual is also seen to imply this. Implication of this exchange features is also discussed.

The heat capacity of the S= ∞ Ising ferromagnet arrayed on a simple cubic lattice is investigated by analysis of Monte Carlo data for the internal energy. A function is least-squares fitted to the energy data close to the critical temperature T_c. The function is consistent with the expected singularity of the heat capacity C(t) = A^± | t |^-α(1 + D^± | t |^Δ₁) + B^±, where t = (T - T_c)/T_c and the ± superscripts refer to t > 0 and t < 0, respectively. The results of the data analysis show that the relation B⁺ = B^- is supported and that A⁺/A^- accords with the expected universal value for a system with Ising symmetry. The ratio of the confluent singularity amplitudes D⁺/D^- is somewhat lower than the expected universal value. The amplitudes of the leading and confluent singularities of the order parameter are also reported.

The purpose of this comment is to discuss critically the energy operator introduced by Yang and collaborators (1982, 1983). The authors also present a critique of the gauge independent transition amplitudes defined by these authors which, they believe, obscures the physical interpretation of transition amplitudes.