Table of contents

The NP-complete problems of partitioning and colouring of random graphs, with p partitions and colours respectively, are mapped onto the statistical mechanical problem of p-state Potts glasses. An estimate of the cost functions of these optimisation problems has been derived using the Potts glass mean-field theory. This estimate applies to dense graphs in the thermodynamic limit. An exact expression for the cost function in the large-p limit is given.

All the polynomials in symmetric ((2)) and antisymmetric ((11)) u(n) bosons are constructed in u(n) bases and u(n)-reduced matrix elements for the bosons between polynomial basis states are computed. Applications to representation theory of Lie groups, paired-fermion and boson physics are briefly discussed.

Ternary vector cross products are studied in their own right. Results include a new proof of Hurwitz's theorem and a 'principle of duplicity'. Upon breaking the symmetry in eight dimensions, by choosing a preferred axis, this last principle implies the well known triality principle for octonions and SO(8) transformations. In displaying canonical forms it helps to put the eight basis vectors in correspondence with the eight points of the three-dimensional affine geometry over F₂.

The author presents the replica symmetric solution of the graph bipartitioning problem. He points out the possibility of many solutions, even with the replica symmetry assumption. He also comments on the possibility of constructing a model with an arbitrary finite number of transition temperatures.

The authors employ a general space and time transformation to find a class of second-order differential equations with the maximum number (eight) of Lie symmetry generators. They apply this transformation to obtain the symmetry generators for the harmonic oscillator and for a non-linear equation, introduced by Leach (1985), starting from the symmetry generators of the free-particle equation.

The master equation for vibrational relaxation of harmonic oscillators is solved in the case when both vibration-vibration and vibration-translation mechanisms are involved in the process. It is shown that for the Landau-Teller transition probabilities the derived master equation has the same general solution as that recently published for an incomplete vibrational relaxation. Some other possible applications of the solution are also discussed.

The Aubry model with site energy V cos(Qn+ phi ) is analysed. The author has proved that the zero-eigenenergy state (ZES) exists only if the system contains an odd number of sites and if phi =+or-1/2 pi . The eigenfunction of the ZES has a definite symmetry. In the vicinity of the critical value V=2, the characteristic features of this wavefunction have been demonstrated.

The authors have studied the spectral dimension d48T of an infinite class of fractals. The first member (b=2) of the class is the two-dimensional Sierpinski gasket, while the last member (b= infinity ) appears to be a wedge of the ordinary triangular lattice. By studying the electric resistance of the fractals they have been able to calculate exact values of d for the first 200 members of the class. An analysis of the obtained data reveals that for large b the spectral dimension should approach the upper limit of 2 according to the formula d approximately=2-constant (ln b)^beta , where beta is not larger than one. This result implies, among other things, that the scaling exponents of the resistivity and diffusion constant should logarithmically vanish at the fractal-lattice crossover.

Non-symmetric spin glasses are spin models for which the pair interactions between the spin are random but not symmetric. By studying the time evolution of two configurations in a mean-field model, one finds a transition temperature T₀. For T>T₀, two different initial conditions end up by becoming identical after an infinite time. This means that the thermal noise is strong enough to eliminate the memory of the initial conditions. For T<T₀, two different initial conditions never become identical.

The remanent magnetisation and energy at zero temperature are investigated numerically for the infinite-range Ising spin glass. Three different relaxation processes are considered: sequential, random and maximum spin flip. It is shown that most of the metastable states are unstable against one spin flip only.

Generalisations, of order K>or=2, of the Pascal triangle are used to construct generalised Pascal-Sierpinski gaskets of orders (K, L>or=2). It is shown that all such gaskets are self-affine fractals, but when K=2 and L is prime then the gasket is rigorously self-similar and possesses a similarity dimension. The evolutionary morphology of the gaskets of orders (K, L prime) bears a resemblance to the growth of pyrolitic graphite films and other material structures.

An analytic derivation is given of rules exactly describing the hierarchical band clustering in the spectrum of Harper's equation for various incommensurate systems. These rules, involving transformations of the incommensurability parameter phi , are shown to apply exactly for arbitrary coupling strength. The approach also provides energy scale factors and hence the fractal dimension D_f(=0.50+or-0.01) of the self-similar spectrum occurring when phi takes a fixed point value phi _g (the golden mean).

To ensure large basins of attraction in spin-glass-like neural networks of two-state elements xi _i^mu =+or-1. The authors propose to study learning rules with optimal stability Delta , where delta is the largest number satisfying Delta <or=( Sigma _j J_ij xi _j^mu ) xi _i^mu ; mu =1. . . . .p: i=1. . . . .N (where N is the number of neurons and p is the number of patterns). They motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix (J_ij). In addition, numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns.

The time evolution operator U(t, s) of a spinless nonrelativistic N-body quantum system in Euclidean space with real analytic time-dependent scalar interaction v(x, t) is studied. A complete formal asymptotic expansion involving simple connected graphs is derived for the full coordinate, and mixed coordinate-momentum representation propagators. The derivation is based on Dyson's series for U(t, s), and the combinatorics involved in the cluster expansion of the classical grand partition function. These results provide an efficient means of generating nonperturbative propagator expansions in the physical variables: mass m, Planck's constant h, time displacement t-s. The structural bridge between the WKB and Wigner-Kirkwood expansions is sketched for mixed representations of U(t, s). In the heat equation context the graphical expansions are found for mixed representations of the density operator e^{- beta H}. Finally, an explicit differential formula is obtained for Wigner's distribution function f(x, p; beta ).

The real Clifford algebras have been classified for a long time. The author recalls the classification and uses it to describe all their representations over the reals.

In this paper the group corresponding to the Virasoro algebra is constructed. The so-called 'descent equation' is used to derive the 2-cocycle for the group.

The supercharacter of OSp(M/N) associated with an arbitrary Young diagram is defined. The distinction is made between OSp(M/N) standard and non-standard supercharacters. The corresponding modification rule which may be used to express a non-standard supercharacter in terms of standard supercharacters is presented, exemplified and proved. This rule involves the removal of N/2+1 boundary strips from the Young diagram. In the case N=0 the rule reduces to the well known rule appropriate to O(M). For a non-standard supercharacter corresponding to a typical irreducible representation of OSp(M/N) the modification yields a single typical standard supercharacter. On the other hand, for a non-standard supercharacter corresponding to an atypical irreducible representation the rule yields a linear combination of atypical standard supercharacters.

The supercharacter of U(M/N) associated with an arbitrary composite Young diagram is defined. The distinction is made between standard and non-standard supercharacters. A modification rule is presented which may be used to express any non-standard supercharacter in terms of standard supercharacters. In the case N=0 the rule reduces to the rule already known to be appropriate to U(M).

A similarity analysis of the non-linear two-dimensional non-stationary ideal MHD equations is presented. In the case of a magnetic field perpendicular to the isentropic motion of the plasma, the authors establish the complete Lie algebra of infinitesimal symmetries. The laws of conservation are mentioned. The similarity method for partial differential equations as a procedure for reducing the number of independent variables is applied repeatedly. Finally, they obtain systems of ordinary differential equations for similarity solutions of the MHD equations considered.

The symbolic method of Kramers (1930) is used to obtain an expression for a 9-j symbol in terms of multiple products of spinor invariants. This technique is generalised from the unitary (compact) symplectic group Sp(2) to Sp(2n), and a generating function is found for a class of multiplicity-free 9-( sigma ) symbols, where ( sigma ) denotes an irreducible representation of Sp(2n) of the form ( sigma ₁ sigma ₂ . . sigma _n) for which sigma _i=0 (i>1). Schwinger's generating function for a 9-j symbol is recovered by setting n=1. A specialisation to 6-( sigma ) symbols is made by setting one of the nine irreducible representations equal to the scalar (00. . .0), and the method is checked by working out a sample 6-( sigma ) symbol previously obtained by an aufbau approach.

A complete set of scalar effective operators acting within the configurations f^N has been found. Through the use of Lie groups this set is resolved into orthogonal operators which act on N electrons simultaneously. A direct group theoretic correspondence is established between N-body electron operators and N-body nuclear states. This connection is utilised to expedite the electron classification scheme. Unlike the case of the d shell, little simplification occurs as N increases. In addition to the 21 previously established operators with N<or=3, one has 65 four-body, 107 five-body, 182 six-body and 50 seven-body operators. Also, it is shown how the Hermiticity of each operator can be established by examining its transformation properties under the Lie group Sp(14).

Intermittency behaviour occurs and can be exactly calculated for iterated asymmetric tent maps. A typical trajectory exhibits long regular phases with monotonic growth (according to a power law) which are interrupted by short irregular bursts at apparently random times. The lengths of the laminar phases in a long trajectory turn out to be geometrically distributed and to be independent. They are governed by their own ergodic theorem. The results can be applied to a larger class of one-hump maps.

Using Seeley's (1969) method a systematic determination of G(x, y; omega ) for the equation ( Del _x²+ omega ²)G(x,y; omega )= delta (x-y), (x,y in Gamma ), is carried out for Dirichlet and Neumann boundary conditions. The region Gamma is taken to be two dimensional with an arbitrary smooth boundary delta Gamma . An asymptotic expansion of Tr G^B for large omega is obtained, where G^B is the boundary contribution to G. Using these results an earlier disagreement between Durhuus et. al. (1982) and McKean and Singer (1967) is resolved and errors in certain coefficients of Tr G^B obtained by Pleijel (1954) are noted.

The scaling behaviour of the period doubling sequences, perturbed by correlated noises, is investigated using a renormalisation group. The renormalisation transformation is achieved by a path integral formalism which allows a necessary generalisation of the random perturbation notion. For a stationary and weakly correlated (in a sense specified in this paper) Gaussian random perturbation it is proved that the scaling behaviour depends only on the same universal constant K=6.619. . . as for uncorrelated perturbations.

A study is made of the efficiency of an iterative procedure devised for solving a system of non-linear equations F(x)=0, in which the choice of an iterative scheme randomly selected from a set of schemes is generally based on satisfying the criterion of a smaller norm //F// in the subsequent iteration. The practicality of using such a procedure is illustrated by solving a set of n=5 equations.

By generalising earlier results, the authors prove that whenever a Lagrangian dynamical system can be associated with a (1,1)-type tensor field which is left invariant by the dynamics, satisfies the Nijenhuis condition and is compatible with the tangent bundle structure, the system necessarily decomposes into a collection of lower-dimensional mutually non-interacting Lagrangian subsystems.

A Robnik billiard R_lambda is the image in the complex w plane of the unit disc in the z plane under the quadratic conformal map w=z+ lambda z². The parameter range is 0( lambda (1/2. For lambda =1/4 the billiard is strictly convex, its boundary is analytic and it has a zero curvature at one point of the boundary. A rigorous demonstration is given that the billiard R_1/4 has invariant curves in the surface of section. Thus it is not ergodic. The proof is based in KAM theory and makes substantial use of computer algebra. The authors give the first elements of a cascade of bifurcations whose properties show several similarities with numerical results given by Benettin et al. (1980) for conservative dynamical systems and by d'Humieres et al. (1982) in the experimental study of the forced pendulum.

A renormalisation group method for one-dimensional iterated maps displaying period doublings is studied. The renormalisation transformation involves variable rescaling, similar to the doubling transformation of Feigenbaum (1978, 1979), but is defined on the space of parameters of a given mapping. Numerical calculations are carried out for maps with different analytic dependence near the extremum.

Jet fields are defined on fibred manifolds and represented as type (1,1) tensor fields corresponding to Cartan-Ehresmann connections. When defined on the first jet bundle they may be used to generalise the second-order differential equation fields used in theoretical mechanics and to extend several existing results to field theories.

In this paper the problem of obtaining the equations of motion for Hamiltonian systems with constraints is considered. Conditions are given which ensure that the phase space points satisfying the primary and secondary constraints form a symplectic manifold, on which the resulting equations of motion are Hamiltonian and uniquely determined.

For linear wave propagation one is often interested more in the local distribution of the wavevectors than in the global spectral distribution (i.e. the Fourier transform). A function that may act as such a local frequency spectrum is the real-valued Wigner distribution function. In this paper this concept of local frequency spectra is generalised to non-linear wave propagation which is governed by a class of non-linear wave equations. This class includes such well known equations as the non-linear Schrodinger equation, the Korteweg-de Vries equation and the Burgers equation. Furthermore the derivation of a transport equation for these local frequency spectra is given on the basis of the dispersion relation for the linearised wave equation. By taking local moments of this transport equation with respect to the frequency variable, an infinite hierarchy of so-called balance equations is constructed. For the non-linear Schrodinger equation the successive conservation laws (in principle, infinitely many) have been calculated straightforwardly from these balance equations.

In this paper, several new solutions of the evolution equation for Kelvin-Helmholtz waves near direct resonance are obtained by Darboux transformation.

A theoretical investigation has been made on the field induced rotation of a sphere immersed in a dielectric liquid and subjected to electric fields at various intensities and frequencies. The regions, in a bidimensional parameter space, with qualitatively different dynamics and the explicit expression of the angular frequency of the sphere were found.

A method of partial-wave analysis is used to extend the treatment of Aharonov and Bohm (1959) for the scattering from an unscreened solenoid of infinitesimal radius to include screening and non-zero radius. The method yields a scattering amplitude which bears a formal resemblance to that appropriate to modified Coulomb scattering. The scattering amplitude and the momentum transfer cross section retain a dependence on the enclosed flux as the screening barrier becomes infinite, resulting in a component of force, periodic in the enclosed flux, being exerted on the barrier. The resulting implications concerning the locality of electromagnetic interactions are discussed. The force is shown to persist when the conditions of strict impenetrability are relaxed and the possibility of directly observing this quantum force is examined.

One-dimensional spin chains are investigated by constructing a mapping of the anisotropic Heisenberg model in the limit Delta to 1 onto the non-linear Schrodinger model (a gas of bosons with delta function interactions). Three applications of this mapping are given. First it is shown that the Bethe ansatz solutions of the anisotropic spin-1/2 Heisenberg model go over to the continuous Bethe ansatz solutions of the non-linear Schrodinger model. Then bound-state energies for arbitrary spin are calculated and as a third application a nearly identical mapping of the anisotropic spin-1/2 Heisenberg model in the limit Delta to 0 onto a non-linear Schrodinger model for fermion fields is given. Finally the authors construct an anisotropic SU(N) model. The continuum limit of this model is shown to be the (N-1)-component non-linear Schrodinger model, which can be solved by means of the nested Bethe ansatz.

The quantum Zeno paradox is examined within the many-worlds and relative states interpretations of quantum mechanics. In the many-worlds interpretation the effect is predicted to persist. The possibility of recombining worlds is not expected to be relevant. In the simplest form of relative states interpretation the effect may be avoided but this form of interpretation experiences difficulties in coping with conventional problems of quantum theory. The more complex version of the relative states interpretation, which takes account of the correlations of system with apparatus, predicts the occurrence of the paradox.

Recurrence formulae determining radial matrix elements of r^q between quasirelativistic hydrogenic wavefunctions are derived. In the diagonal case they are a generalisation of the well known Kramer's relations. The formulae are also transformed to the basis of the Dirac wavefunctions.

The main objective of this paper is to find an analytic expression for the maximum eigenvalue of a special matrix A of order n, with n in the range 2<or=n( infinity . This will be done using a new result which states that if one knows any entry C_kl(j) of any real positive square matrix C^j (where j denotes the number of iterations), then, after taking the thermodynamic limit (lim j to infinity ), the maximum eigenvalue of this matrix is known. The analysis that is presented is of particular interest, because the structure of the above-mentioned matrix A, appears in several problems related to physical and biophysical systems in one dimension, i.e. the one-dimensional fluid model, denaturation of DNA (where loop entropy is taken into account), in the study of the homogeneous island model, etc. As a particular application of this method, the author derives the Takahashi equation of state, which is the more general solution for a one-dimensional fluid model when it is assumed that the interaction potential between nearest-neighbour particles is arbitrary.

Recently Vlad et al. (1984) introduced a new stochastic description of memory effects, based on a system of age-dependent master equations (ADME). In this paper the ADME approach is applied to physical processes obeying a phenomenological master equation. The joint probabilities and the correlation functions of the ages of different fluctuation states are computed. The suggested method is extended to systems for which the transition rates are age dependent. A physical interpretation of the main results is given by means of a system of age-dependent integral equations. The author proves that the ADME formalism is a generalisation of the well known continuous time random walk theory (CTRW).

The author establishes relationships between Bose, Fermi and spin-1/2 systems on lattices of arbitrary dimension and arbitrary but finite size which are defined to live in the (common) state space of the boson system. The underlying systems display an essential affinity by being described by precisely the same form of Hamiltonians, which are the polynomial functions of canonical operator variables: boson, fermion or spin-1/2 variables, respectively. For large enough, densely populated finite volume lattices it entails the computation of partition and correlation functions for lattice fermions in terms of bosons only. Albeit an approximation, its accuracy increases as the continuum limit is approached.

A new technique for calculating the shapes of random walks is presented. The method is used to derive an exact analytical expression for the asphericity of an unrestricted closed or ring walk embedded in d spatial dimensions. A graphical procedure is developed to systematise a 1/d series expansion for the individual principal radii of gyration and their respective probability distribution functions P(R²_i)(1<or=i<or=d). The average principal radii of gyration are calculated to O(1/d²) for both open and closed walks and selected terms in the 1/d expansion are summed to all orders in 1/d in the determination of P(R²_i). This leads to an explicit analytical form for P(R²_i) for open walks. The distribution of the largest eigenvalue is compared with a distribution obtained from numerical simulations of walks in three dimensions. The agreement between the two is extremely good. Other predictions for various parameters that characterise the average shape of open and closed walks in three dimensions are also found to agree remarkably well with the results of simulations, the error being of the order of 5%.

The self-directed walk is studied in three dimensions. Monte Carlo simulations on the simple cubic lattice provide an estimate of the radius of gyration exponent nu =0.67+or-0.01 in agreement with the authors' recent Flory-like theory leading to nu =2/d when 2<or=d<or=4.

Questions concerning the universality between different two-dimensional Ising-like systems have recently been raised, with evidence for the existence of several different subclasses of Ising-like systems being presented. Some new facts concerning similarity between border model and low-temperature spin-1 Ising model series analyses are noted. For the spin-1 case an alternative determination of the critical temperature determines unequivocally that this model is in the Ising universality class and it is suggested that a similar independent result could lead to a similar conclusion in the border model case. The possibility of an analytic correction to scaling in the spin-1 model is proposed.

The discrete cubic model for six states of the lattice variables is investigated at one special point of the phase diagram where a number of contradicting results are known concerning the order of the phase transition and the critical exponents. A new implementation of the MCRG method is used for determining the critical point with great accuracy and for calculating the critical exponents. A continuous phase transition is found. Following the recent results for six-state self-dual quantum chains with cubic symmetry where for a special point superconformal invariance is discovered, the critical exponents obtained are explained in the framework of this theory indicating a superconformal point with a conformal anomaly of c=1.25.

The eigenstates of a six-site cluster with spins of S=1/2 coupled by Heisenberg interactions are studied exactly. The authors have investigated various criteria (variational, perturbational and related to a stability of 'spin waves') to distinguish a local energy minimum, if any, from ordinary quantum states. They have found that a sensible definition of a local energy minimum is similar to the classical one. It involves investigating the stability of the system against infinitesimal spin rotations. They give an example of a six-spin system with three minima which are stable against single-spin rotations of which two seem to be stable against simultaneous many-spin rotations.

A generalisation of the Hopfield model which includes interactions between p()2) Ising spins is considered. The exact storage capacity behaves as N^p-1/2(p-1)! ln N when the number of nodes, N, is large. In the limit p to infinity , the thermodynamics of the model can be solved exactly without using the replica method; at zero temperature, a solution which is completely correlated with the input pattern exists for alpha < alpha _c where alpha _c to infinity as p to infinity and this solution has lower energy than the spin-glass solution if alpha < alpha ₁=1/4 ln 2 where the number of patterns n=(2 alpha /p!)N^p-1. For finite values of p, the correlation with the input pattern is not complete; for p=3 and 4, approximate values of alpha _c and alpha ₁ are obtained and for p to infinity the replica symmetric approximation gives alpha _c approximately p/4 ln p.

The author uses the technique of invasion percolation to compute the critical points and critical exponents of percolation on a random lattice which is the dual of a Voronoi network.

The author considers the one-parameter lattice models of classical statistical mechanics from a simple algebraic viewpoint. The way in which the limiting locus of partition function zeros emerges through a sequence of semi-infinite (m* infinity ) lattice sections is considered. Convergence to the limiting locus C_infinity is obtained through two sequences of algebraic curves. For all arbitrarily large but finite m the partition function per site is a branch of an algebraic function Lambda ₁ defined by an irreducible polynomial and possesses only a finite number of algebraic singular points. In the limit of m to infinity infinitely many branch points accumulate on C_infinity ; an algebraic basis to the universality hypothesis is presented in terms of the accumulation of branch points at the critical point. It is argued that for many two-dimensional lattice models the critical exponents alpha and nu will be of the form alpha =2/s and nu =(s-1)/s, where s is an integer >or=3.

Applications of the mathematical formalism put forward in the previous paper (see ibid., p.3471 (1987)) are made to a number of well known lattice models in statistical mechanics. Sequences of algebraic curves are obtained from the branches of an algebraic function Lambda ₁, where Lambda ⁺₁ at real temperatures is the branch of Lambda ₁ which is the partition function per site of an m* infinity lattice section. These sequences are viewed as approximations to parts of the limiting locus C_infinity of partition function zeros. Approximations to critical points can be obtained in a natural extension of these curves beyond the branch points of Lambda ⁺₁.

A Monte Carlo renormalisation group approach using a scaling transformation in real space is applied to the critical properties for the site percolation problem on the simple cubic lattice. The authors find a sequence of estimates for the critical concentration p_c and the eigenvalue lambda at various values of the rescaling length b. Extrapolation of these sequences to the limit b to infinity yields the site percolation threshold p_c=0.3115+0.0004/-0.0003 and the connectedness length exponent nu =0.88+or-0.04.

The self-replicating properties of the Julia sets J_c(p) for the iterative processes z implies z^p+c are examined for integers p)1. It is shown that the corresponding Mandelbrot sets M(p) contain (p-1)-fold symmetries, which results in J_c(p) having p-fold symmetries.

The Koch curve evolves from a base equilateral triangle by the trisection of each side and the replication of the original triangle on the mid-section, the process being repeated ad infinitum by the addition of sets of successively smaller triangles. The process is generalised to replace the trisectioning by (2k+1)-sectioning. It is shown that a square is the only other regular polygon on which the (2k+1)-sectioning procedure can be implemented. The Koch curves thus generated are strictly self-similar, their fractal dimensions being similarity dimensions and enclose simply connected areas. Randomisation of the generating procedure is also discussed.

The action and pseudocharge of a free-space electromagnetic wave are shown to be identically zero. These results are Lorentz and gauge invariant.

In a recent letter Lyklema and Evertsz (see ibid., vol.19, p.895-900 (1986)) found a value of nu for the eta =1 Laplacian random walk which differs from the value the authors obtained for the diffusion-limited self-avoiding walk (DLSAW). They show that this results from different boundary conditions at the ends of these chains and not from a failing in our Monte Carlo studies of the DLSAW.

Imperfections are pointed out in the arguments of Nemeth (See ibid., vol.20, p.2211, (1987)) on the possibilities of straight phase boundaries in the phase diagram of Ising spin glasses.