Table of contents

A new theoretical approach to graph processing, based on the transformation of a topological graph into a time-dependent periodic melody, is proposed. By using a self-tuning oscillatory dynamical network, graphs can be uniquely reconstructed from their melodies. This opens a perspective for a content-addressable memory invariant under large topology-preserving distortions.

Explicit coefficients of normal forms for generalised Hopf bifurcations are calculated to the seventh (but, to save space, they are given only to the fifth) order near the critical point. The real part of the third-order coefficient reproduces, at the critical point, the third-order focal value of Andronov et al (1973).

The results of the Painleve analysis and of a search for constants of the motion of the complex Lorenz model are presented.

The author first presents a non-linear Schrodinger equation that describes wave propagation in fluids and plasmas with sharp boundaries and dissipation. Then the author shows that an exact solution can be found.

The scattering amplitudes for all currently known supersymmetric shape-invariant potentials are calculated by analytically continuing the explicit wavefunctions obtained via supersymmetric operator techniques. The procedure is illustrated in detail for the superpotential W(x)=A tanh alpha x+B sech alpha x, for which the S matrix has not been previously calculated.

The annealed random map model is one of the simplest models of statistical mechanics with stochastic dynamics. For this model, the authors define valleys by saying that two configurations submitted to the same stochastic forces belong to the same valley at time t if their trajectories have met before time t. They compute in the long-time limit the probability distribution of the number and the sizes of these valleys. They find a structure very reminiscent of the valley structure of the mean-field spin glasses with sample-to-sample fluctuations. Interpreting the annealed random map model as an aggregation model, they obtain non-self-averaging effects for the number of macroscopic clusters and for their sizes.

Numerical and analytical calculations of the electronic properties of a one-dimensional quasiperiodic lattice are presented. The lattice consists of two types of bonds A and B whose distribution follows the recursion formula S_l+1=S^m_l-1Sⁿ_l where m, n=1, 2, 3, . . . and S_l is a building block sequence for l>or=2. An exact evaluation of the wavefunctions yields the scaling for several pairs of values of m and n. Numerical calculations of the wavefunction and the resistance reveal a rich self-similar structure as well as localisation for one of the quasiperiodic sequences studied.

The previously introduced method of mean-field renormalisation has provided a unified approach to bulk and surface critical behaviour. The method is used for the computation of the critical exponent associated with the order parameter at corners. Applications to the square and triangular Ising model are presented. The angle dependence of the corner critical exponent is qualitatively reproduced. The results for the bulk critical exponents are more accurate than in previously employed mean-field renormalisation schemes.

The author shows that for almost all interactions, that is, generically, ground states of classical lattice gas models must have either long-range order or be completely uniform.

In bond percolation on lattice with p>p_c, there are unoccupied bonds which, if occupied, would immediately join the backbone. In general, when such a bond is occupied, several 'tag end' bonds may join the backbone as well. The author defines n_i to be the number of bonds that would join the backbone if such a bond i were occupied and n to be the average of n_i taken over all such bonds on the lattice. The author derives a formula for n and finds that for all lattices, near p_c, n= gamma _Bp_c(p-p_c)^-1, where gamma _B is the exponent for the divergence of the backbone fraction. Identical formulae apply to site percolation.

Based on the works of Li, Zhu and co-workers (1985), a general Lie algebraic structure is given for symmetries of integrable systems which are generated by hereditary symmetries.

It is shown that the complete set of polynomial zeros of degree one of the Racah coefficients can be obtained only from the full eight-parameter solution of the multiplicative Diophantine equation: xyz=uvw subject to the constraint z=x+y+u+v+w. All other parametric solutions obtained are shown to represent only proper subsets of the complete set.

The authors show by exploiting the properties of the coherent state and the integration within the ordered product technique that it is possible to derive an explicit expression for the class operator of the SU(2) group which has not been obtained before.

The authors present an algorithm for recursively calculating coupling, recoupling, induction and similar coefficients in terms of the primitive coefficients. The algorithm is shown to be complete for any compact group. They use the 6-j symbol as an example for describing their algorithm, but the algorithm applies to a wide class of group theoretic transformation factors.

Series of gamma functions can usually be expressed as generalised hypergeometric functions of one or more variables, providing a tool for classifying series transformations. While most studies of these functions concern the multivariable type, the authors have derived new transformations and extended known results for single-variable functions. The derivations of the new identities are straightforward and the results are presented in their most general form.

A Lax pair for the Goryachev-Chaplygin top (GCT) is presented. The pair is obtained from that for the Kowalewski top found by Reyman and Semenov-Tian-Shansky (1987). Explicit formulae are constructed in terms of theta functions for the dynamical variables.

A technique based on the Campbell-Baker-Hausdorff (CBH) formula is introduced to calculate the effective Hamiltonian for kicked dynamics, classical and quantum. An integrable example is exactly solved by this method. A non-integrable kicked spin dynamics is treated approximately up to seventh order in a perturbation parameter. The CBH expansion is evaluated in a transparent way with the help of a REDUCE program, thereby illustrating that the CBH expansion is asymptotic.

Applying the thermodynamic formalism to mixing repellers for regular maps, the author obtains some rigorous relations between the dimensions of the repellers and the dynamic properties of the transformation on them. For a class of one-dimensional expanding maps, these quantities can be approximated to any desired accuracy by means of the corresponding variables for linear expanding maps, which can be computed exactly. Finally, a mathematical foundation and some properties of the generalised dimensions are given.

The conservation laws for the process of reflection and transmission of electromagnetic waves on a plane interface of isotropic transparent media are determined. Using these laws, relations have been established between the transverse shift (TS) of a centre of gravity of reflected and transmitted wavepackets, the change of the normal component of the intrinsic Minkowski angular momentum of the electromagnetic field and the Abraham transverse momentum (or the transverse electromagnetic power flow (TPF)). The previous investigations of the TS and TPF phenomena are discussed from the point of view of conservation laws.

Approximate master equations for the Friedrichs model are discussed. The approximate Zwanzig equation predicts behaviour very close to the exact one, but presents only minor simplifications of necessary calculations, while an approximate convolutionless equation usually gives a reasonably good account of the behaviour of exact solutions and simplifies the calculations significantly. It is also shown that the van Hove and Markov limits give incorrect asymptotics in a physically interesting case.

The author applies Darboux transformations to the Schrodinger equation in three dimensions and derives the general form of the potentials and potential differences which can be treated by these transformations. In particular the author establishes the existence of a new class of potentials whose solutions are related to each other by these transformations.

Levinson's theorem for a Schrodinger equation with both local and non-local symmetric potentials is studied in terms of the Sturm-Liouville theorem. A new convention for the phase shifts is applied instead of the usual one. It is proved that the usual Levinson theorem holds for the case with both local potential and non-local symmetric cutoff potential, which is not necessarily separable. The problems related to the positive-energy bound states and the physically redundant solutions are also discussed.

The charged particle scattering amplitude on the magnetic field of a toroidal solenoid is obtained in the first Born and high-energy approximations. It is shown that multiconnectedness of the accessible space, non-triviality vector potentials in it and the single-valuedness of the wavefunctions used are not enough for the existence of the Aharonov-Bohm effect. Criteria are given for this. Surrounding the toroidal solenoid with barriers of different geometrical forms, the author observes that the magnetic field contribution to the scattering amplitude depends crucially both on the barrier height and its form. The latter could hardly be explained by particle penetration into the H not=0 region. The author proposes an addendum to Tonomura's experiments which probably could clear up any doubts about the existence of the AB effect. The gedanken experiment is discussed which shows that the effect of inaccessible fields could be observed even in a simply connected space.

An analysis is given of high-temperature series expansions for the susceptibility of an O(2)-symmetric spin model with discretely valued spin-spin interaction on triangular and square lattices. This analysis does not yield strong evidence for a phase transition in either case, although it is weakly consistent with this possibility. The authors also comment on the high-temperature series for the free energy and specific heat.

A model for a solid sphere undergoing Brownian motion in a viscoelastic (Maxwell) fluid is described in terms of a non-Markovian Langevin equation. By solving this equation exactly, the particle's density-density and current-current time correlation functions are calculated. From the former, the time-dependent self-diffusion coefficient, D(t), is evaluated for an ensemble of Brownian particles. Then a numerical estimation of D(t) is performed for typical values of the sphere's parameters and for two viscoelastic fluids describable by Maxwell's model. A comparison of this result with the corresponding expressions for D(t) for a Newtonian fluid in the Stokes and Boussinesq-Basset approximations for the drag, shows that for the Maxwell fluid the behaviour of D(t) is analytic and similar to that of the Newtonian fluid in the Stokes regime. The authors find that elasticity has a minor influence on D(t) and that persistent correlations (long-time tails) in diffusion do not occur for this model. They also compare their results with other related works.

The authors calculate the surface absorption probabilities for particles diffusing onto perfectly absorbing substrates. The 'f( alpha )' formalism purporting to represent the multifractal properties of the absorption probabilities is discussed in the context of some simple test cases. The f( alpha ) universality classes of several similar surfaces are studied and it is found that the f( alpha ) curves for surfaces with the same fractal dimension are different.

The authors evaluate the site spontaneous magnetisations on the general 4-8 or bathroom tile lattice using mappings to checkerboard, free-fermion and Union Jack Ising models. Their results verify the conjecture of Lin et al (1987), who proposed their formula from twelfth-order low-temperature series expansions.

The zero-temperature physical properties of an Ising chain in a Markovian field taking only two values with non-zero mean are evaluated and a related discrete stochastic mapping with the theory of finite Markov chains is investigated. A discontinuous behaviour of the magnetisation and the residual entropy, dependent on both mean field and exchange, is found which can be related to flips of microscopic spin clusters. For non-zero temperature the mapping is characterised by the fractal properties of its attractor and by the Lyapunov exponent. An explicit expression for the measure in the nth iteration of the Chapman-Kolmogorov equation is obtained.

The author presents a dodecagonal quasiperiodic tiling of the plane in terms of three kinds of tiles: a square, a regular hexagon and a thin rhombus whose acute inner angles are 30 degrees . The dodecagonal tiling is self-similar with respect to a dilatation by 2+ square root 3. It is associated with a dodecagonal quasiperiodic lattice which is obtained by a projection from a hyperhoneycomb lattice in four dimensions. The hyperhoneycomb lattice is a non-Bravais lattice composed of four Bravais sublattices and the author must assign 'windows' with different orientations to different sublattices. The author discusses in detail the relationships of the present method of obtaining a dodecagonal tiling to other methods.

The critical behaviour of Dyson's hierarchical Ising model is studied in the presence of a random field. Simple scaling relations determine the exponents eta , eta exactly for random fields with either short- or long-range correlations. The Schwartz-Soffer inequality eta <or=2 eta is satisfied as an equality provided the random-field correlations are not too long ranged. For short-range random fields, the exponent v is calculated to order in = lambda -4/3, where lambda is the 'range parameter' of the Dyson model. The exponent v is also computed to O( in ²) with in =2 lambda -3, for the hierarchical O(n) model in zero field: the results reveal striking similarities with the exponents of the corresponding one-dimensional model with long-range interactions.

The authors derive an inequality between the number of trees and the number of lattice animals with exactly c cycles, a_n(c), for all positive c. If they assume that a_n(c) approximately n^{- theta c} lambda ⁿ_c, n to infinity , c fixed, they use this to show that theta _c= theta _o-c where theta ₀ is the corresponding exponent for trees.

Starting from a model for CsOH.H₂O the authors investigate the phase diagram of a two-dimensional Ising model on the Kagome lattice with antiferromagnetic nearest-neighbour and competing ferromagnetic second- and third-nearest-neighbour interactions by Monte Carlo simulations. They obtain two Kosterlitz-Thouless transitions as in the six-state clock model and a crossover to second-order transitions where the critical exponents are consistent with those of the three-state Potts model. They also determine the ground state for all values of the coupling constants.

A Hopfield-type neural network that can store ultrametrically organised patterns with a finite magnetisation, or bias, is studied. Both the patterns and their ancestors are remembered in one network. A homogeneous field (or threshold) is included in the model. In zero field the (replica symmetric) mean-field equations map onto those of the Hopfield model, implying that transition temperatures, storage capacity, etc. are the same. By varying appropriately the field and/or a constant added to all the synaptic strengths, it is possible to climb up and down the hierarchical tree or to focus on a certain level in the hierarchy and thereby increase the capacity slightly.

The spectrum of a bosonic string is affected by interaction processes where the string undergoes splitting and rejoining. For low-lying states, it is possible to construct effective Lagrangians and Hamiltonians which describe the effects of the interaction, at least in the ultraviolet limit, i.e. when the splitting and rejoining process proceeds very rapidly. The derivation is performed by considering the partition function of the system consisting of an open bosonic string and integrating some of the ultraviolet degrees of freedom.

The Green function for a spinless charged particle in the presence of an external plane-wave electromagnetic field is calculated by algebraic techniques in terms of the free-particle Green function.

A new approach to electromagnetic field representations in isotropic chiral media is presented. It is shown that by a simple transformation of the fields the constitutive relations of the isotropic chiral medium (which include differential operators) can be transformed into those of a special form of bianisotropic media. It is finally found that the electromagnetic field can be represented by two scalar Hertz potentials which are solutions of a system of second-order differential equations. The Green functions of this system are calculated.

Lowest-order cranking about a principal axis breaks down for a system with non-zero (J_z) along its symmetry axis. Such a system should be cranked about an axis at an angle to the symmetry axis. This can be understood in terms of the Berry phases, which acts like the field of a magnetic monopole. Possible applications are discussed.

An explicit formula for the spectral dimension of the Sierpinski type fractal family studied by Borjan et al (1987) is obtained using the method of images. For large values of their parameter b, the spectral dimension d(b)=2-(log log b)/log b+terms of order (1/log b).

Numerical calculations based on Kirchhoff laws are used to calculate the resistance of a random mixture of conductors and superconductors on the Sierpinski gasket in two dimensions. Using modified finite-size scaling arguments the authors obtain for the superconductivity exponent s=0.27+or-0.03, which is not predicted by any known critical exponent relations. Their method is confirmed by re-obtaining the exact known result for the conductivity exponent mu for the problem of a conductor-insulator mixture on the gasket.

The author discusses transport equations on a curved interface which is semipermeable. The moving interface has finite thickness and separates two media with different physical properties. The author introduces a thermodynamical field theory with the material properties of such an inhomogeneous system.