Table of contents

An so(3) tensor realisation for the Lie algebra sp(3, R) of the symplectic nuclear collective model is given. In this realisation, a complete classification of all subalgebras of sp(3, R) that contain the physical angular momentum algebra so(3) is obtained.

The authors show that for period doubling bifurcations in the map 1-Cx^{2 mu} , in the limit mu >>1, C_infinity (the limit point of the bifurcations) and the universal exponents alpha and delta have non-analytic behaviour in mu .

A new series of confluent hypergeometric functions is investigated. It is shown to have the Voigt lineshape function as its sum. Kummer transformation is applied on this series to obtain a much faster converging series for the Voigt lineshape function. The well known asymptotic behaviour of confluent hypergeometric functions is made use of to also obtain an asymptotic series for the Voigt lineshape function.

It is shown that the light reflection laws on a moving mirror are formally identical with the kinematical properties of the Compton effect.

The geometry of crumpled paper balls is investigated. It is shown that these systems are fractals and their properties are studied.

A general Monte Carlo renormalisation group transformation for the static as well as the kinetic properties of aggregates is formulated and applied to different growth models: particle aggregation (DLA), cluster aggregation (ClCl) and invasion percolation (IP). The leading critical exponent is determined numerically. For DLA, the first correction to scaling exponent has been estimated as well.

Based on an equivalence between the q-state Potts model and conformally invariant field theories, the author speculates that all fractal dimensionalities of two-dimensional percolation are given by the formula D=(100-x²)/48, where x is an integer. The same formula should also apply to self-avoiding walks and lattice animals. This approach is extended to cluster-weighted percolation. His conjecture seems to agree with previous exact and numerical results except for the backbone exponent, and possibly for lattice animals.

The authors compute the probability that a person will survive a shootout. The shootout involves N persons randomly placed in a d-dimensional space, each firing a single shot and killing his nearest neighbour with a probability p. They present a formulation which gives P$ dN(p), the probability that a given person will survive, as a polynomial of p containing a finite number of terms. The coefficients appearing in the polynomial are explicitly evaluated for d=1 and d=2 in the limit of N to infinity to yield exact expressions for P_infinity (p). In particular, P_infinity (1) gives the probability that a given particle is not the nearest neighbour of any other particle in a classical ideal gas and the authors further determine P_infinity (1) for d=3, 4 and 5 using Monte Carlo simulations.

The authors calculate the absorption probability distribution for particles diffusing onto perfectly absorbing boundaries. The boundaries studied are rough but not fractal; nevertheless, non-classical behaviour is evident in the singularities of the measure and their distribution.

The authors present an approach to dilute Ising and Potts models, based on the Fortuin-Kasteleyn random cluster representation, which is simultaneously rigorous, intuitive and surprisingly simple. Their analysis yields, with no dimensional restrictions or other caveats, the following asymptotic form of the phase boundary. For the regular dilute model in which bonds have constant ferromagnetic coupling J with probability p and are vacant with probability 1-p, the critical temperature scales as exp(-J/(kT_c(p))) approximately mod p-p_c mod , implying that the crossover exponent is Phi =1. If the constant couplings are replaced by a distribution F(J) with mass near J=0, quite different crossover behaviour is observed. For example, if F(J) approximately J^alpha then, for p near p_c, T_c(p) approximately mod p-p_c mod ¹ alpha /.

It is shown that the amplitude fluctuations of vector iterates in the transfer-matrix approach to the statistics of the one-dimensional random Ising chain correspond to an exact strange set characterised by a multifractal spectrum. The temperature and disorder dependence of the spectrum are also investigated.

Near the directed percolation threshold, p_d, the wetting velocity upsilon scales as (1- upsilon ) approximately (p_d-p)^theta . The authors extrapolate from calculations on strips of finite width to show that, on the two-dimensional square lattice theta =1.73+or-0.02. This agrees with a scaling argument by Barma and Ray (Phys. Rev. B., vol.34, p.3403, 1986) which predicts theta = nu /sub ///.

Explicit expressions, which determine the boundaries of the regions containing temperature zeros of the partition function of the two-dimensional Ising model on an anisotropic triangular lattice, are obtained, along with a general formula for the density of zeros valid everywhere in regions containing zeros.

Using computer simulation, a random walk motion is studied in a random percolating system at percolation threshold in the presence of a random field. The RMS displacement is found to decrease systematically as a function of field strength and shows a non-universal power law dependence on it. The long time behaviour seems similar to the critical slowing down recently predicted in an analogous magnetic system. The spectral dimensionality varies with the field strength and is found to be lees than.

A p sequence is an infinite sequence of 0 and 1 generated by the production rule p( alpha , omega , j)=((j+1)/ alpha + omega )-(j/ alpha + omega ), j in N, depending upon two parameters: alpha , omega . The properties of quasiperiodic p sequences ( alpha irrational) under deflation and inflation transformations are investigated. For this purpose, a set of so-called simple replacement rules (SRR) is defined, and it is shown that a p sequence is always transformed into a p sequence by repeated application of SRR. The parameters of the transformed sequence are calculated explicitly and the conditions are given under which a p sequence transforms into itself (self-similarity). The theory is applied to the construction of a one-dimensional diatomic quasicrystal, whose diffraction spectrum is calculated.

Analytical inversion symmetry of the biorthogonal systems of SU₄ contains/implies SU₂ × Su₂, SU_n contains/implies SO_n and Sp₄ contains/implies U₂ bases for two-parametric (covariant and mixed tensor) irreducible representation is discovered. This symmetry relates the dual isoscalar factors or resubducing coefficients. It allowed us to invert, by means of a special analytical continuation procedure, the non-orthogonal isofactors of SU_n contains/implies SO_n for couplings (p0) × (0q) to (λ 0 μ) and (p₁0) × (p₂0) to (λ ν 0), as well as the resubducing coefficients (transformation brackets) for expansion of the SU₃ contains/implies SO₃ Elliott basis states and Sp₄ contains/implies U₂ Smirnov and Tolstoy basis states in terms of the corresponding canonical basis states. New expressions for bilinear combinations of SU_n contains/implies SO_n special isofactors are obtained. Expansion of SU₃ canonical basis states in terms of SU₃ contains/implies SO₃ Elliott states is found. Isofactors for coupling of two Elliott states are given.

The induction coefficient is defined as the element of a matrix reducing the induced representation into irreducible constituents. The authors give the algebra of these coefficients and a closed analytic expression in terms of matrix elements of coset representatives. The consequences of this choice are derived and the importance of some of these results for the Racah-Wigner coupling algebra of the symmetric and unitary groups noted.

For pt.I see ibid. vol.18, p.3319-25, 1985. The convergence of a discretisation procedure for path integrals associated with a class of parabolic second-order differential equations with time-dependent coefficients is shown. The proof is based on a generalisation of the product formula.

A linear operator non-polynomial differential equation relativistically generalising the Airy equation and functions is considered. A fundamental system of two solutions of that equation representing generalised Airy functions of the first and second kind is obtained. Also, the general solution of another differential equation closely connected with the one considered is found.

The authors use the Green function method to solve the problem of steady one-dimensional flow of an incompressible, viscous, electrically conducting fluid through a pipe with sector cross section, in the presence of an applied transverse uniform magnetic field. They obtain an analytic solution in this case which other methods have not given before.

For the general case of an inhomogeneous anisotropic and gyrotropic medium a differential tensor equation, expressing the evolution of the tangential component of the field vectors of an electromagnetic wave is obtained. A fundamental solution of this equation is given by a multiplicative integral. A plane-stratified system of anisotropic and gyrotropic layers is considered. By means of the characteristic matrix of such a medium Fresnel's reflection and transmission operators are derived. These operators have wide utility because they describe exactly the interaction of light with any plane-stratified gyroanisotropic structure. The conservation of the normal component of the Poynting vector in such a structure allows the authors to find a correlation between the operators of reflection and transmission. The operator dispersion equation of the multilayer gyroanisotropic waveguide is presented. All the calculations in this paper are based on the direct manipulation of tensors and their invariants, eliminating the use of coordinate systems. This facilitates solutions and provides results of great generality which are suitable for computer use.

Application of gauge transformation for generating explicit auto-Backlund relations is demonstrated for a number of integrable systems, e.g. KdV, SG, NLS, DNLS, mixed DNLS, modified DNLS, LLE, etc. This method turns out to be much simpler, effective and straightforward. Using Backlund relations all conserved quantities are expressed in a novel derivative-free form, which simplifies the calculation of important soliton characteristics.

For pt.I see ibid., vol.20, p.397-409, 1987. The spectrum of an ellipsoidal box of prolate and oblate symmetry are calculated by an extract diagonalisation and by semiclassical methods. The influence of a separatrix in phase space is analysed with the help of the WKB phases derived in an earlier paper. Differences between the prolate the oblate L_z=0 states are explained using this separatrix. It is found that the EBK method, improved by the uniform approximation, leads to a spectrum in very good agreement with the wave mechanical results. The spectrum of a two-dimensional elliptical membrane ('billiard' box) is also studied with similar conclusions.

The supersymmetric version of the one-dimensional harmonic oscillator is studied by taking into account its conformal properties. The largest superalgebra of symmetries and supersymmetries is derived as Osp(2/2) Square Operator Sh(1), the semidirect sum of Osp(2/2) and the Heisenberg superalgebra. Through a one-to-one correspondence between the nonrelativistic free case and the harmonic oscillator description, the authors deduce the (expected) supersymmetries of the Schrodinger equation. The above structure appears as the largest spectrum-generating superalgebra of the harmonic oscillator and its representation within an energy basis is given. The physical three-dimensional case is also considered when the maximal set of (super)symmetries is required and this case is compared with recent work.

The eigenvalue problem of a N-level system coupled to a bosonic degree of freedom is solved without using RWA. For that purpose, the bosonic degree of freedom is transformed to Bargmann's Hilbert space of analytical functions. In this representation the Schrodinger equation is a system of N coupled linear differential equations of first order. Using a discrete symmetry, these equations are simplified by a suitable transformation of the independent variable. Starting from the simplified equations, the authors develop a method to solve the eigenvalue problem of the N-level system. In addition, they present a simple approximate treatment and compare it with the exact results. The approximation turns out to be quite good up to outer level resonance and can be used to explain the differently structured regions in the energy spectra.

The generalisation of the quantum inverse scattering method to the study of direct and inverse problems for the multicomponent non-linear Schrodinger model of bosons or fermions with repulsive coupling is made. Two sets of Yang-Baxter equations are solved to obtain commutation relations between the scattering state operators. The eigenfunctions have been constructed for the infinite number of conserved quantities and the eigenvalues of the first three conserved quantities-number of particles, momentum and energy-are obtained. The global Izergin-Korepin relations and the relations between the quantum Jost functions are derived, and from them, the quantum Gel'fand-Levitan equations are established. Finally, the series expansion for the field operators in terms of the scattering state operators is written out explicitly.

In scalar phi ⁴ quantum field theory the instanton solution which gives the minimal action is a spherically symmetric solution due to the proof of the Sobolev inequality for scalar functions. The value of the minimal action is related to the Sobolev constant of the Sobolev inequality. The authors generalise this to matrix valued phi ⁴ field theory by generalising the Sobolev inequality to matrix valued functions.

The radial distribution functions of randomly distributed one-dimensional spheres are studied. The differences between equilibrium fluids and sequentially constructed systems are discussed in terms of the corresponding configuration spaces and numerical simulations. It is shown that for the sequentially constructed systems, which correspond to the car parking problem, a 'memory effect' occurs, such that the sequential spheres are temporally distinguishable. Exact solutions for some simple cases and numerical simulations for large systems are shown, discussed and compared with the equilibrium fluid results.

The authors consider the effect of prescribed spatially periodic temperatures at the bounding walls of a 2D Boussinesq fluid on 1D pattern forming transitions. They determine the normal-form equation for the slowly varying pattern amplitude. Quasiperiodic behaviour is found when one takes into account the deviation of the external wavelength modulation from the critical value for the onset of classical Rayleigh-Benard convection.

The authors have studied the self-avoiding walks (SAW) on a family of finitely ramified fractals. The first member (b=2) of the family is the two-dimensional Sierpinski gasket, while the last member (b= infinity ) appears to be a wedge of the homogeneous triangular lattice. By means of the exact renormalisation group transformations they have calculated the critical exponents alpha , nu and gamma , and the connectivity constant mu , of SAW on each member of a sequence (2<or=b<or=8) of the studied fractal family. The obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two-dimensional lattice.

The authors introduce and study the kinetics of 'cluster eating', in which a cluster of mass i and a cluster of mass j react to form a lighter cluster of mass mod i-j mod . They write the rate equations for this process, which describe the kinetics in the mean-field limit, where spatial fluctuations in cluster density and in cluster shape are neglected. An asymptotic solution to these equations is derived for the particular case in which the reaction rate is independent of the masses of the reacting clusters. At long times, they find that the density of clusters of mass k, c_k(t) decays as A_k/( tau log^k-1 tau ), where tau is proportional to the time, A_k=(N-1)!/(N-k)! and N is the largest cluster mass in the initial state. This very unusual behaviour is checked by numerical simulations. A more general situation where the reaction matrix depends on the parities of the masses of the two incident clusters is also discussed briefly and a wide variety of possible kinetic behaviours is delineated. Finally, the authors study cluster eating below the upper critical dimension, where fluctuations in cluster density give rise to a non-classical kinetic behaviour.

The simulated annealing algorithm for optimisation problems such as the travelling salesman problem is reviewed. The concept of the autocorrelation function for cost functions is introduced and it is shown how numerical experiments to measure this quantity can provide criteria as to how rapidly a system can be annealed close to equilibrium. From this the authors obtain an optimum annealing schedule of general applicability.

A relation between the shape of Eden clusters and the number of perimeter sites per unit area of the surface is derived which is analogous to the Wulff construction of equilibrium shapes in thermodynamic systems. New data are presented for the surface width and the surface skewness of Eden clusters grown on a square lattice. The width depends on the average orientation of the surface with respect to the underlying lattice. Its corrections to scaling are discussed. The skewness has unexpected changes of sign.

Frustration effects due to competing interactions between magnetic atoms in quasicrystals are studied in an exactly soluble example: the Ising chain in a quasiperiodic two-valued magnetic field. At zero temperature the model exhibits an infinity of pure phases, characterised by a modulation of the order parameter which reflects the aperiodic structure of the system, up to a larger and larger length scale.

The authors have derived a low-temperature expansion appropriate to the 'superantiferromagnetic' or (2*1) ordered phase of the Ising model with first- and second-neighbour interactions on a square lattice. The critical exponent beta shows a non-universal variation along the critical line, which is in reasonable agreement with the variation expected on the basis of scaling.

For original paper see Commun. Math. Phys., vol.104, p.457-70, 1986. The author links the recently proposed unification of the boson and fermion stochastic calculus with the general problem of boson-fermion equivalence (duality, reciprocity, etc.) for quantum fields. Even if via the Fock construction the common Fock space for bosons and fermions can be introduced, it still does not allow for the unrestricted boson-fermion equivalence for field theory models. All local fermion field theory models thus have boson equivalents (violating the weak local commutativity condition for space dimension three). The reverse statement is not valid: not all boson models admit a pure fermion reconstruction.

Methods recently devised for anharmonic oscillator calculations are used to investigate a longstanding problem for the rotating displaced oscillator.

The authors reconsider the renormalisation treatment of the model introduced by Shnidman (Phys. Rev. Lett., vol.56, p.201, 1986) to explain the observed non-universal behaviour of micellar solutions. They show that his derivation of the recursion relations would introduce non-universality even for a model which obviously belongs to the Ising universality class. This argument casts some doubt on the validity of Shnidman analysis.

Using the node-link picture of the infinite cluster near the percolation threshold, a crossover of the critical behaviour of the self-avoiding walks on the random lattice near the percolation threshold (or just above threshold) is suggested and a scaling function of this crossover is proposed. The author compares the result of this qualitative argument with a Monte Carlo simulation result.

Comments on the dynamic critical exponents recently reported by Leyvraz and Jan (ibid., vol.19, p.603, 1986), in particular that for the spin-exchange kinetic Ising model in one dimension, z=3. This result is at variance with rigorous inequalities that z cannot be less than five for this model, i.e z>or=5. The source of the discrepancy appears to lie in the Monte Carlo algorithm which subsumes a critically slow dynamical process, leading to an apparently faster dynamics in the critical region. In addition, the author extends the finding for one dimension to the quasilinear (non-branching) fractal Koch curve, concluding z=3d_f, where d_f is the fractal dimension. He discusses the physical factors comprising the lower bound to the dynamic exponent in one dimension, z=5. He then obtains the generalised lower bound for spin-exchange dynamics of the non-branching Koch curve, z=3d_f+d_w=5d_f, where d_w is the random walk dimension.

As pointed out by Luscombe (ibid., vol.20, p.1299-302, 1987), the claim, made by the authors, that z=3 for the one-dimensional Ising model with conserved order parameter is erroneous. The authors show that the origin of the error lies in the fact that this model is not capable of relaxing to equilibrium at the critical temperature T=0.

The authors provide an example of a lattice on which infinite AB percolation clusters can occur, answering a question of Halley. They identify parameter values for which infinite AB percolation occurs and does not occur on the lattice.

We give here the missing conclusions of our paper with the same title where we gave the critical exponents for the six- and eight-state models with cubic symmetry for several values of the coupling constant. From the finite-size correction to the ground-state energy, we find that the central charges are c ≈ 1.25 for the six-state model and c ≈ 1.30 for the eight-state model for all the considered values of the coupling constant. We show that for a special value of the coupling constant the six-state model exhibits N = 1 superconformal invariance.