Table of contents

Bioconvection is a fluid instability common to many biological organisms including swimming bacteria, alga and protozoa. The statistics of bioconvective pattern formation is tested against percolation models for space-filling. A percolation threshold is found (p=0.63) and compared to theoretical point distributions for random tesselations. Simulations reveal that a percolation backbone defines a complete path across the observation window but remains incomplete as an equal partitioning grid (Voronoi diagram). The generic development of incomplete Voronoi grids and their yet unknown statistical properties captures some interest as an alternative to traditional point-based lattices.

The asymmetric simple-exclusion model with parallel dynamics (the deterministic cellular automaton 184) is extended to take into account the exchange of cars between different lanes on a multi-lane roadway. The traffic flow model is presented by the one-dimensional asymmetric exclusion model with injection or extraction of particles. The system is driven asymptotically into a steady state exhibiting a self-organized criticality. The typical interval (s) between consecutive jams scales as (s) approximately=L^v with v=0.62+or-0.04 where L is the system size. It is shown that the jam-interval distribution n_s(L) satisfies the finite-size scaling form n_s(L) approximately=L^{- beta} f(s/L^v) with beta =2v.

In a previous paper we gave enumerations and results of the enumeration of directed animals of three different forms (sites, bonds, and bonds without loops) on a variety of two-dimensional lattices. This letter adds results for site animals without loops, and gives an exact solution for the sites (no loops) Kagome lattice, which is actually a pair of directed self-avoiding walks.

The su( nu ) Calogero model confined in the harmonic potential is considered. We show that the model has Yangian symmetry Y(su( nu )). Unifying with the creation-annihilation operators which constitute the su( nu ) loop algebra, we reveal the algebraic structure of this system. The relation with the energy spectrum is also discussed.

Relativistic quantum mechanics suffers from structural problems which are traced back. To the lack of a position operator x, satisfying (,p)=inl with the ordinary momentum operator p, in the basic symmetry group-the Poincare group. In this letter we provide a finite-dimensional extension of the Poincare group containing only one more generator pi (in l+l dimensions), satisfying the commutation relation (k, pi )=inl with the ordinary boost generator k. The unitary irreducible representations are calculated and the carrier space proves to be the set of Shapiro's wavefunctions. The generalized equations of motion constitute a simple example of exactly solvable finite-difference set of equations associated with infinite-order polarization equations.

The process of random sequential adsorption in one dimension is generalized so that the rate at which a particle is placed in an empty space is dependent on the size of the space being destroyed and on the sizes of the two spaces created on either side of the particle. We write down the equation for this process involving p(z), the probability of choosing to deposit a particle of size z, and F(x,y mod z), the rate at which spaces of length x and y are created from a space of length x+y+z, given that a particle of size z is deposited. A scaling theory is given for this process and solutions are presented for two special cases. The results of a numerical simulation of a third case are presented.

In the frame of the generalized master equation (GME) approach, an analysis of the asymptotic spin relaxation is made for different j values. The fully coupled (FC) model and the rotating-wave approximation (RWA) for the system-reservoir coupling are considered and it is seen that they give rise to structurally different GMEs. In particular, we examine the evolution of the diagonal and off-diagonal matrix elements of the density operator, the mean values of the relevant observables and the equilibrium solution for both coupling models and display their significant differences. Particularly the spin j=1 case is examined.

The dissipative properties of the Vlasov equation are investigated for a model system of A fermions in a harmonic oscillator plus a two-body interaction. The dissipation is quantified in terms of relaxation rates towards the final equilibrium which is a Boltzmann distribution in phase space. We look particularly for the dependence on the initial state and on the strength of the two-body interaction.

The Abelian sandpile models feature a finite Abelian group G generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of G as a product of cyclic groups G=Z(d₁)*Z(d₂)*Z(d₃)...*Z(d_g), where g is the least number of generators of G, and d_i is a multiple of d_i+1. The structure of G is determined in terms of the toppling matrix Delta . We construct scalar functions, linear in the height variables of the pile, that are invariant under toppling at any site. These invariants provide convenient coordinates to label the recurrent configurations of the sandpile. For an L*L square lattice, we show that g=L. In this case, we observe that the system has non-trivial symmetries, transcending the obvious symmetries of the square, namely those coming from the action of the cyclotomic Galois group Gal_L of the 2(L+1)th roots of unity (which operates on the set of eigenvalues of h). These eigenvalues are algebraic integers, the product of which is the order mod G mod . With the help of Gal_L we are able to group the eigenvalues into certain subsets the products of which are separately integers, and thus obtain an explicit factorization of mod G mod . We also use Gal_L to define other simpler sets of toppling invariants.

We have studied the survival probability f(N,R) of an N-step square lattice self-avoiding walk (SAW), with one end at the centre of an absorbing circular cavity of size R. Our numerical results clearly indicate that the form of f(N,R) depends on the method by which the surviving SAW are generated.

We find and discuss the scaling dimensions of the branch 0 manifold of the Nienhuis O(n) loop model on the square lattice, concentrating on the surface dimensions. The results are extracted from a Bethe ansatz calculation of the finite-size corrections to the eigenspectrum of the six-vertex model with free boundary conditions. These results are especially interesting for polymer physics at two values of the crossing parameter lambda . Interacting self-avoiding walks on the Manhattan lattice at the collapse temperature ( lambda = pi /3) and Hamiltonian walks on the Manhattan lattice ( lambda = pi /2) are discussed in detail. Our calculations illustrate the importance of examining both odd and even strip widths when performing finite-size correction calculations to obtain scaling dimensions.

A family of generalized Fibonacci lattices, which are locally isomorphic and exhibit a peculiar type of self-similarity, have been found to exist by shifting the position of the strip (initial phase) in the projection method. The necessary and sufficient conditions of generating such self-similar quasiperiodic lattices are established. The power-law growth behaviour of the wavefunction at E=0 in the off-diagonal model defined on some of these lattices has been analysed. It is shown that, although various structures resulting from different initial phases are in a local isomorphism class, they lead to a variety of maximum exponents of power for the scaling of the wavefunction.

We study the phase diagram of statistical systems of closed and open interfaces built on a cubic lattice. Interacting closed interfaces can be written as Ising models, while open surfaces can be written as Z(2) gauge systems. When the open surfaces reduce to closed interfaces with few defects, the gauge model can be also written as an Ising spin model. We apply the lower bound renormalization group (LBRG) transformation introduced by Kadanoff (1975 Phys. Rev. Lett. 34 1005) to study the Ising models describing closed and open surfaces with few defects. In particular, we have studied the Ising-like transition of self-avoiding surfaces between the random isotropic phase and the phase with broken global symmetry at varying values of the mean curvature. Our results are compared with previous numerical work. The limits of the LBRG transformation in describing regions of the phase diagram where non-ferromagnetic ground states are relevant are also discussed.

The interaction algebra of a discrete classical spin model with transitive symmetry group is generated by the s independent matrices corresponding to its set of minimal edge-transitive graphs. If this algebra is non-Abelian, its centre is at least the Potts model interaction algebra. For a spin model on the square lattice with four different interactions, the existence of an Abelian interaction algebra implies the existence of four 3s-dimensional disorder solutions. In the non-Abelian case, there are at least four (2s+2)-dimensional disorder solutions. As examples, the Potts and Ashkin-Teller models (Abelian) and the F(6)-model (non-Abelian) are discussed. The possibility of an expansion around a disorder solution in terms of a class of subgraphs of the shadow lattice is indicated. Inversion relations for spin models on the checker-board lattice are derived using an explicit form for the inverse of the transfer matrix.

We present new series data for both high- and low-density bond and site percolation on the square lattice. The series have been obtained by the finite-lattice method, and in all cases extend pre-existing series. An analysis of these series gives refined estimates of critical points, critical exponents and amplitudes for bond and site animals, and for the percolation probability and mean-size exponents.

The master equation of one-dimensional three-species reaction-diffusion processes is mapped onto an imaginary-time Schrodinger equation. In many cases the Hamiltonian obtained is that of an integrable quantum chain with known properties. Within this approach we search for three-state integrable quantum chains with known spectra and which are related to diffusive-reactive systems. Two integrable models are found to appear naturally in this context: the U_qSU(2)-invariant model with external fields and the three-state U_qSU(P/M)-invariant Perk-Schultz models with external fields. A non-local similarity transformation which brings the Hamiltonian governing the chemical processes to the known standard forms is described, leading in the case of periodic boundary conditions to a generalization of the Dzialoshinsky-Moriya interaction for N-state Hamiltonians (N>2).

A one-dimensional lattice-gas model with order preservation is considered where the occupation probabilities of sites correspond to Bose statistics as a consequence of the prescribed dynamics. The master equation for the particle-cluster dynamics at the sites is formulated. The corresponding continuum nonlinear diffusion equation is derived for the space- and time-dependent concentration fluctuations. The equation can be regarded, in the presence of a drift force, as the Burgers equation when terms irrelevant in the sense of renormalization-group ideas are neglected. Collective centre-of-mass and tagged-particle diffusion are investigated by numerical simulations and the results agree with the analytical derivations. Subdiffusive behaviour of the mean-square displacement of tagged particles and normal collective and centre-of-mass diffusion are observed when no bias is present. The dispersion of the centre-of-mass displacement exhibits superdiffusive behaviour in the case of mean drift of the particles. Discrepancies of about 20% between the numerically determined superdiffusion coefficients and the predictions of the mode-coupling theory are found and discussed.

A new 'non-standard' quantization of the universal enveloping algebra of the split (natural) real form so(2,2) of D₂ is presented. Some (classical) graded contractions of so(2,2) associated to a Z₂*Z₂ grading are studied, and the automorphisms defining this grading are generalized to the quantum case, thus providing quantum contractions of this algebra. This produces a new family of 'non-standard' quantum algebras. Some of these algebras can be realized as (2+1) kinematical algebras; we explicitly introduce a new deformation of Poincare algebra, which is naturally linked to the null plane basis. Another realization of these quantum algebras as deformations of the conformal algebras for the two-dimensional Euclidean, Galilei and Minkowski spaces is given, and its new properties are emphasized.

A basis-independent criterion for the classification of irreducible group representation, into potentially-real, pseudo-real and essentially-complex representations, is given for an arbitrary group which may also possess infinite-dimensional representations. These considerations are applied, in particular, to the finite- and infinite-dimensional representations D(j₀,c) of the orthochronous proper Lorentz group SO(3,1) and it is shown that the irreps which are neither unitary nor pseudo-unitary are essentially-complex. Further, among the unitary and pseudo-unitary irreps of SO(3,1), those irreps with a half-odd-integer j₀ are shown to be pseudo-real, while the others with an integer j₀ (including zero) are potentially-real.

For pt. I see ibid., vol. 28, p. 957 (1995). General conditions are determined under which a decomposable representation of a group admits invertible C-matrices. Applying these results, the C-matrices admitted by the decomposable representations of the orthochronous proper Lorentz group SO(3,1) are obtained and the representations are classified into three reality types.

For pt. II see ibid., vol. 28, p. 967 (1995). The bilinear metrics and the C-matrices admitted by the irreducible representations (irreps) of the orthochronous and the homogeneous Lorentz groups are determined and the irreps are classified into three reality types.

The coupling (Wigner-Clebsch-Gordan) coefficients and their isofactors for the unitary quantum algebras u_q(n) with the repeating irreducible representations in the coproduct decomposition are considered. Generalizing the U(n) case, the biorthogonal systems of the u_q(n) isofactors with the dual multiplicity labels are constructed by means of the recoupling technique in terms of the isofactors with simpler multiplicity structure. A first construction, correlated with the inverted Littlewood-Richardson rules, gives the bilinear combinations of isofactors after applying the proportionality of the q-recoupling (Racah) coefficients to the boundary q-isofactors. An alternative recursive construction gives the nonorthogonal q-isofactors satisfying the most elementary boundary conditions and proportional to the u_q(n-1) recoupling coefficients for some less restricted values of parameters. Some multiplicity-free and more general u_q(n) recoupling coefficients are found, the blocks (bilinear combinations) of which (equal to the resubducing coefficients of the complementary chains of q-algebras) are proposed to use for the orthonormalization of some u_q(n) biorthogonal isofactors, including the general u_q(3) case.

We consider a problem which may be viewed as inverse to the Schwinger realization of Lie algebra, and suggest a procedure for deforming the algebra so obtained. We illustrate the method through a few simple examples extending Schwinger's su(1,1) construction. As results, various q-deformed algebras are (re-)produced as well as their undeformed counterparts. Some extensions of the method are briefly pointed out.

Let D be the set of differential operators on functions defined on R³, Let alpha ^(p), p=0,1,2 and 3 be scalar, vector, pseudovector and density fields on R³, respectively. Then alpha ^(p) can be regarded as a multilinear map D^p to D. Let r₀,...,r_p in R³ and let (r₀,...,r_p) be the Euclidean p-simplex having these vertices. Then the integral of alpha ^(p) over (r₀,...,r_p) is a function F( alpha ^(p)) (r₀,...,r_p) which satisfies F^{d alpha} = delta _ASF^alpha . Here d alpha ^(p) means grad alpha ⁽⁰), curl alpha ⁽¹) or div alpha ⁽²), and delta _AS is a natural cohomological operator (the usual Alexander-Spanier co-boundary operator). For any function F(r₀,...,r_p) there is a natural map Phi ^F:D^p to D which satisfies Phi ( delta _ASF)= delta _H Phi ^F Where delta _H is the Hochschild co-boundary for co-chains Phi ^F on D. Thus, when alpha ^(p) is regarded as being the cochain Phi (F alpha ^(p)) on D, grad, curl and div all become Hochschild co-boundary operators: grad alpha ⁽⁰)(H) is the commutator of the operator H with the function alpha ⁽⁰) and curl alpha ⁽¹)(H₁,H₂) measures the amount by which alpha ⁽¹) fails to be a derivation on D. If div alpha ⁽²)=0 then alpha ⁽²) provides a deformation of the composition product on D. This new viewpoint of fields as operator-valued maps of p-tuples of operators has implications in several areas of physics and mathematics. One consequence is that the Hamiltonian in quantum mechanics may be regarded as its own probability current density operator. Another is that Maxwell's equations describe the algebraic character of the electric and magnetic fields E and B regarded as co-chains on D. We give some explicit formulae for alpha ^(p)(H₁,...,H_p).

We calculate the boundary S-matrix for the open antiferromagnetic spin- 1/2 isotropic Heisenberg chain with boundary magnetic fields. Our approach, which starts from the model's Bethe-ansatz solution, is an extension of the Korepin-Andrei-Destri method. Our result agrees with the boundary S-matrix for the boundary sine-Gordon model with beta ² to 8 pi and with 'fixed' boundary conditions.

For pt. II see ibid., vol. 27, p. 5941 (1994). The general method for defining the Lie and covariant derivatives of a spinor field, presented in parts I and II, is compared with that provided by Buchdahl's 'invariant formalism'. The results are found to be essentially in agreement (where the invariant formalism is applicable).

Two- and three-dimensional networks of a columnar type are described, which result from partitions based on Poisson point distributions. The metric and topological properties of such laminated Poisson networks are derived and the applicability of the Lewis and Aboav-Weaire laws to them is tested. The 2D Poisson network contains cells with i>or=4 (i is the number of sides) and both laws are obeyed. The 3D Poisson networks are of various types and have F>or=6 (F is the number of faces in a cell) and F=14. In one particular type of 3D Poisson network the two laws are again exactly obeyed. In another type, the laws show large deviations at low F but are asymptotically obeyed when F tends to infinite.

We give an analytical description of the locus of the two-gap elliptic potentials associated with the corresponding flow of the Calogero-Moser system. We start with the description of Treibich-Verdier two-gap elliptic potentials. The explicit formulae for the covers, wavefunctions and Lame polynomials are derived, together with a new Lax representation for the particle dynamics on the locus. We then consider more general potentials within the Weierstrass reduction theory of theta functions to lower genera. The reduction conditions in the moduli space of the genus-2 algebraic curves are given. This is a subvariety of the Humbert surface, which can be singled out by the condition of the vanishing of some theta constants.

The possible boundary conditions consistent with the integrability of the classical sine-Gordon equation are studied. A boundary value problem on the half-line x<or=0 with local boundary condition at the origin is considered. The most general form of this boundary condition is found such that the problem be integrable. For the resulting system an infinite number of involutive integrals of motion exist. These integrals are calculated and one is identified as the Hamiltonian. The results found agree with some recent work of Ghoshal and Zamolodchikov(1994).

We prove the absence of the absolutely continuous spectrum for the operator -(d²/dx²)+ Sigma _{j epsilon z} alpha delta '(x-j)+fx, f>0 and alpha not=0 by means of the crystal momentum representation and the Howland's criterion for Floquet-type operators.

We present an inverse scattering construction of generalized point interactions (GPIs), these being point-like objects with non-trivial scattering behaviour. The construction is developed for single centre S-wave GPI models with rational S-matrices, and starts from an integral transform suggested by the scattering data. The theory of unitary dilations is then applied to construct a unitary mapping between Pontryagin spaces which extend the usual position and momentum Hilbert spaces. The GPI Hamiltonian is defined as a multiplication operator on the momentum Pontryagin space and its free parameters are fixed by a physical locality requirement. We determine the spectral properties and domain of the Hamiltonian in general, and construct the resolvent and Moller wave operators thus verifying that the Hamiltonian exhibits the required scattering behaviour. The physical Hilbert space is identified. The construction is illustrated by GPI models representing the effective range approximation. For negative effective range we recover a known class of GPI models, whilst the positive effective range models appear to be new. We discuss the interpretation of these models, along with possible extensions to our construction.

We present an approximate analytic solution to the time-dependent Schrodinger equation of a charged harmonic oscillator in the presence of a strong laser field, based upon the technique advanced recently by Frasca (1992). The resulting wavefunction is then used to derive expressions for the survival probability of the ground state and the probability of transition to an arbitrary higher state.

The semiclassical propagator in configuration space for a particle with one-dimensional motion in a potential well of infinite depth is investigated. It is shown that, if the motion's period and its derivative are asymptotically decreasing functions of the energy, the large-energy classical paths give the same contribution (in modulus) to the propagator, and are therefore asymptotically equiprobable. This extends the results previously obtained for the quartic anharmonic oscillator.

We propose a new model with the parity conservation of the total number of particles and show that this model belongs to the directed percolation universality class.