Table of contents

Proves that a discrete solution exists for the critical inverse eigenvalue problem by direct application of Gohberg's theorem.

Directed bond percolation is shown to be in the same universality class as Reggeon field theory. The critical behaviour and critical exponents near the percolation threshold are thereby inferred.

A renormalisation group transformation is developed for two-dimensional site-percolation problems by using a scaling transformation in real space. A transition matrix A is defined for each cell, and the renormalised probability p'(p) of occupation of the cell is identified with the dominant eigenvalue R₁(p) of the transition matrix. A simple RG transformation has been applied on the square lattice up to cells of size 6*6 and the results for critical probability p_c and exponent v are given. A modified RG has been applied to the three planar lattices, and p_c and v have been calculated for the simplest choice of the cells. The modified RG transformation seems to yield better results as the coordination number of the lattice increases.

It is shown that in the Flory model (1942) of semi-flexible polymer chains, an assumption of random occupation of sites is not valid for estimating the excluded volume effects when the fraction of gauche bonds is small. Thus the model is never completely ordered except presumably at T=0, and the free energy is such that it cannot give rise to a melting transition from a state of zero configurational entropy at some finite temperature. This study also casts doubts on the conclusion of Gibbs and DiMarzio (1958) that the above model exhibits a second-order phase transition, i.e. a glass transition at a finite temperature in the super-cooled phase.

Presents a finite-size scaling theory for the non-linear relaxation of a simple relaxational model.

It is shown that the theory of dislocation mediated melting in two dimensions is equivalent to a coupled vector sine-Gordon field theory. A systematic renormalisation group method for studying the transition is given. Recursion relations are obtained to second order in the dislocation activity for the case of melting on a periodic substrate.

The subjoining of one compact Lie group H to another such group G is discussed with particular reference to the cases for which G=U(N) and H=U(n). It is shown that maximal subjoinings of these unitary groups are specified by means of the monomial symmetric functions. Subjoinings, which are defined in terms of mappings between weight spaces, are studied through the properties of characters of the irreducible representations. The branching rules corresponding to subjoinings are found to involve plethysms. Methods of evaluating the appropriate plethysms are illustrated, some of which make use of subjoining chains whilst others exploit the Weyl symmetry groups of G and H to obtain results more directly. The fact that maximal embeddings are special cases of non-maximal subjoinings is demonstrated and discussed.

The energy levels of boxed-in harmonic and inverted oscillators are constructed from the perturbative and asymptotic solutions that are valid in the limits of small and large sizes, respectively. In order to obtain expressions for the energy levels which are valid for boxes of any size, the authors use Pade approximants constructed as interpolations between the perturbative and asymptotic solutions. Special attention is paid to the lowest levels. The accuracy and range of validity of each type of solution are illustrated by comparing them with the exact solution which is obtained by constructing and diagonalising the matrix of the Hamiltonian of the system in the basis of eigenfunctions of the free particle in a box.

A method is proposed for finding a prolongation structure in the Wahlquist-Estabrook sense (1975) without using the concept of prolongation. The closure of this structure follows unambiguously from the analysis of a holonomy algebra for a connection in a fibre bundle associated with a given non-linear equation.

Associates a compact self-adjoint operator A((-2mE)¹2/), parametrised by the energy E<0, with the Schrodinger operator H=p²/2m+V. By setting all eigenvalues of A equal to unity, the discrete spectrum of H is obtained formally. It is determined by means of the approximation of A using finite Hermitian matrices which are calculated in the so-called Coulomb Sturmian basis. The method is illustrated by a few examples previously reported (Yukawa potential, (-r^{- alpha} ) potentials, 0< alpha <2, multiple centre potentials) and possible extensions are discussed.

The structure of the Poincare-covariant equations for free massive fields is analysed. It is assumed that the supplementary conditions follow from the field equations. By use of the notion of the commutant the general form of such equations is given. The necessary and sufficient conditions for their existence are found.

Searches for examples of particle trajectories which, approaching a naked singularity from infinity, make up for lost time before going back to infinity. In the Kerr-Newman metric the authors found a whole family of such trajectories, showing therefore that the causality violation is indeed a non-avoidable pathology.

For a general class of scalar-tensor theories of gravitation proposed by Nordtvedt (1970) it is shown that Birkhoff's theorem (1923) holds both in vacuum as well as in the presence of an electromagnetic field when the scalar field is time independent.

Using generalised mathematical considerations to inverse-square law of force is shown to imply specific spatial energy distributions relative to the interacting bodies. The retardation effects associated with energy redeployment when the bodies are in motion are examined. It is found that, as applied to the gravitational interaction between sun and planet and provided there is no discontinuity in the spatial energy distribution, retardation will give a law of motion conforming with Einstein's law of gravitation. A necessary condition is that the energy in transit in the field system is ineffective in determining force for a retardation period equal to the time required for a photon to travel from one body to the field and then return from the field to the other body. The implication is that gravitation could be a quantum interaction which assures causality and balance of action and reaction by this dual photon exchange interaction.

A description of a non-Riemannian geometry is given in which use is made of a pair of affine connections to characterise the manifold. This approach is used to discuss the 'Palatini' variational principles of general relativity and the non-symmetric unified theory. In the latter case, an identification of the metric tensor is generated by the variational principle. The implications of the adoption of this identification are discussed.

It is argued that the application of the dimensional regularisation technique to one-loop quantum gravity calculations is ambiguous. However, for the calculation of on-mass-shell S-matrix elements, this ambiguity can be resolved by requiring consistency with results obtained from other regularisation schemes. Some discussion is also given of the implications of this work for recent attempts to use higher derivative Lagrangians to solve the renormalisability problem in quantum gravity.

The scattering of two gravitating particles is studied, using a predictive system to first order in G. The centre of mass differential cross section is given. When one or both masses are taken to be zero, the scattering of light-scalar particles and light-light is obtained. All the results are in agreement with those based upon the quantised linearised theory.

A group structure hidden behind the usual manipulations involving cluster expansions, is brought to light through the recognition of the fundamental role played by Bell polynomials. The inversion problems are thereby reduced to matrix calculus. Simple derivations are given of the relations between cluster integrals and the virial coefficients.

The real space renormalisation group of Niemeijer and van Leeuwen (1974) is applied to a mixed-spin Ising model on a simple quadratic lattice. The motivation is the investigation of critical phenomena in Ising models with less than the usual translational symmetry. The models in question are relevant to the study of ferrimagnetism. Two calculations, characterised by different block constructions, are performed and compared. Exponent values are found to be in good agreement with those suggested by the universality hypothesis. The utility of the renormalisation group for dealing with ferrimagnetism is demonstrated, but the high degree of labour involved in such an exercise is indicated.

Presents some exact enumeration data and appropriate Pade mimic functions for the p dependence of the fraction of sites in finite clusters for the square lattice site problem having given valence. In addition, the authors report Monte-Carlo results on the corresponding quantities for sites in infinite clusters and use these data to investigate the degree of ramification of infinite clusters at and above the percolation threshold.

It is shown that generating function techniques provide an efficient means of enumerating the number of self-avoiding rings (polygons) on the square lattice. The techniques can be applied to a number of related problems in lattice statistics and statistical mechanics. The enumeration has been extended to polygons of up to 38 steps.

Uses high-field series expansions for the square lattice Ising model to investigate the physical singularity in the magnetisation as a function of the field. High-field series were obtained to order 35 at temperatures T approximately=0.5 T_c and T approximately=0.766T_c using series expansion techniques based on corner transfer matrices. At neither temperature is there any evidence of a spinodal line; the behaviour is consistent with the predictions of the droplet model, suggesting that the first-order transition line is a line of infinitely differentiable singularities.

The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma _iS_i^z+J_iS_i^xS_i+1^x) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N_JJ^NJ-1/J ₀^N(J) for 0<J<J₀ and P(J)=0 for J>J_O. A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta ²E/ delta Gamma ² which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.

Formulates an exact form of the virial theorem for a relativistic charged thermodynamic perfect fluid in curved spacetime in time-orthogonal coordinates and diagonal metric tensor. Its Newtonian limit leads to a generalisation of Chandrasekhar's tensor virial theorem in hydromagnetics. The author applies the exact form of the virial theorem in curved spacetime, to obtain equilibrium configurations in two cases.

Structures with isothermal cores have been discussed in great detail in the literature, but for all these structures the value of dP/d rho jumps at the core-envelope boundary. The authors have chosen cores with extreme relativistic conditions (dP/d rho =1 and ¹/₃). For such cores one can ensure the continuity of dP/d rho along with that of pressure, density, e^lambda and e^nu . Choosing polytropic envelopes, which have positive distribution function for all possible energies provided that dP/d rho <or=1, the central redshifts have been calculated. One can obtain any high value of central redshift. The structures are pulsationally stable for Z_c<or=1.43 when the polytropic index n=1. For n<1, one may obtain a maximum central redshift of 4.75 for pulsationally stable structures. Next, an envelope in which the density is a specific function of r is chosen. By assuming the surface density to be equal to 2*10¹⁴ g cm^-3, the mass of neutron stars has been calculated. The maximum mass of 4.7 M(.) is consistent with the results of other authors.

The maximum efficiency for idealised accretion is considered, and previous remarks justified.

An inhomogeneous magnetohydrostatic cosmological model has been derived which is of Petrov type 1D.

Uses the extension of the Ma RG approach (1973) to quantum functionals in the limit n to infinity to obtain non-perturbative information for quantum systems at T=O, restricting oneself to the n-vector paramagnon problem.