Table of contents

An N-dimensional integral evaluated by K Aomoto (1988) is shown to represent the density matrix for an impurity particle in the 1/r² quantum many-body problem on a line. The value of the N-dimensional integral representing the same density matrix in periodic boundary conditions is conjectured, as too is the value of an N-dimensional integral which represents a two-point correlation function in the system. Also, the partition function of a related classical Hamiltonian is evaluated by formulating a conjecture which asserts that the sum of Jacobians of a certain change of variables in N-dimensions is a constant.

For the N-Coulomb particle Schrodinger operator with central charge Z there is a well known condition of stability N(2Z+1 obtained by Lieb (1984). This estimate is extended to operators with slowly decreasing potentials.

The boundary conditions at the singular point for the Schrodinger equation with delta '-interaction in one dimension are proposed. It is also shown that the boundary conditions adopted by Albeverio et al (1988) are irrelevant to the delta '-interaction.

It is shown that a relativistic spin-1/2 neutral particle with an anomalous magnetic moment moving in an x-y plane in a two-dimensional electrostatic field: E=(E_x(_x, _y), E_y(_x, _y), O) has (N-1)-fold degeneracy of the ground state. The degree of degeneracy of the ground state is defined by the value of the anomalous magnetic moment and a linear charge density of the filament creating this electrostatic field.

Excitations in the integrable model with two- and three-particle interactions are calculated on the basis of the Bethe ansatz equations obtained in a previous paper. It is shown that the model exhibits two kinds of excitations, one connected with the massless particle-hole excitations, the other with massive excitations corresponding to the collective motion of the pseudoparticles.

The thermally weighted average end-to-end distance (R_N) of interacting self-avoiding walks (SAWs) are obtained enumerating all the (finite) N-stepped SAW configurations on the infinite percolation cluster of bond diluted square lattice at the percolation threshold. Averaging over 250 percolation clusters and enumerating all the possible SAWs on them for N up to 31, (R_N) is fitted to a scaling form (R²_N) approximately N^{2 nu theta} f(N^phi tau )=(T-^theta )/^theta is the temperature interval away from the ^theta -point, phi is the crossover exponent and nu theta is the tricritical size exponent. The best fit is obtained for ^theta approximately=0.71 (compared to ^theta ₀ approximately=1.54 on a pure square lattice), nu approximately=0.74 (compared to nu ^theta _phi =4 mod 7 approximately=0.57 in two dimensions) and phi approximately=0.20. They also obtain an estimate for of SAW size exponent nu ^C for collapsed phase on the percolation cluster.

Self-avoiding walks with a curvature-dependent energy are studied with renormalization group methods on some fractal lattices. Fixed points corresponding to universal and non-universal behaviours are generally present. However initial conditions of the renormalization group recursions can prevent non-universality. When universality holds the persistence length is found to diverge much faster than in the periodic lattice as the curvature energy increases.

The author studies the effect of randomness in the initial conditions on the deterministic diffusion equation with nonlinear terms. Physically, this describes, among other things, the time development of a system quenched from a high temperature to the vicinity of the critical point, in the approximation where the effects of thermal noise are neglected. He considers the case of a non-conserved order parameter with O(n) symmetry, and shows that the nonlinearities are irrelevant for the large time behaviour for dimension d)2. The model is investigated for d(2 using the renormalization group and in -expansion. It is found, to all orders in in , that the local fluctuations in the order parameter scale like t^-12/, and have a universal distribution. The time dependence of the response function, describing the dependence on the initial condition, is characterized by another exponent which is computed to O( in ²). These results are checked in the exactly soluble cases of n to infinity and d=0.

The finite lattice method of series expansion is used to extend the enumeration of self-avoiding polygons on the triangular lattice through to rings of 35 steps. The authors also give the enumeration of these triangular polygons grouped by perimeter and area of up to 21 unit triangles for perimeters up to 24 steps.

The author considers solutions of the Yang-Baxter equation such that the logarithmic derivative of the transfer matrix yields a quantum spin Hamiltonian which is isotropic in spin space, i.e. SU(2)-invariant. Four such solutions are known for each value of the spin S. (For S=1/2 they degenerate into the same solution, and for S=1 they only give three different solutions). For S<or=6 he shows that these are the only solutions which are SU(2)-invariant, except for S=3 when there is a fifth solution.

The authors compute the fluctuations of the order parameter in the Curie-Weiss version of a site-dilute antiferromagnet. Their results show: (i) Gaussian fluctuations away from criticality or at a first-order critical point with sample and thermal fluctuations contributing in same order; (ii) Non-Gaussian fluctuations with critical exponents modified by the presence of dilution at the second-order critical point. In this case sample-induced fluctuations are enhanced to dominate over the thermal ones. Critical exponents are the same as in the Curie-Weiss random field Ising model.

Main chain polymer liquid crystals have been modelled as worms. In reality the stiffness is not distributed in this manner-rods are connected to each other by spacers. The authors examine the limits of this problem, that is worm and jointed-rod models, to see when each is applicable. Hairpins, found naturally in the worm problem, also exist for jointed systems but their scaling is quite different.

Neural networks with symmetric couplings which have an intermediate form between the Hebb learning rule and the pseudo-inverse one, storing strongly correlated patterns, are studied. Signal-to-noise analysis is made and replica-symmetric thermodynamic calculations are performed. Both approaches show that both in the Hopfield model limit and in the pseudo-inverse model limit the maximal capacity of the order of (2p/ln(1/p)^-1) where p<<1 is the average neural activity) can be achieved by appropriate adjustment of the threshold term of the Hamiltonian.

Neural networks with multi-state neurons are studied in the case of low loading. For symmetric couplings satisfying a certain positivity condition, a Lyapunov function is shown to exist in the space of overlaps between the instantaneous microscopic state of the system and the learned patterns. Furthermore, an algorithm is derived for zero temperature to determine all the fixed points. As an illustration, the three-state model is worked out explicitly for Hebbian couplings. For finite temperature the time evolution of the overlap is studied for couplings which need not be symmetric. The stability properties are discussed in detail for the three-state model. For asymmetric couplings limit-cycle behaviour is shown to be possible.

The authors conducted a comparative study of the density distribution of metastable states in analogue neural networks and the Boltzmann machine by evaluating number densities of the attractors of the networks as functions of storage capacity, analogue gain or temperature and pattern overlap. The analysis is based on the fact that the Boltzmann machine and the analogue neural network can be described by the Thouless-Anderson-Palmer equations with and without the Onsager reaction term, respectively. They found the remarkable result that the spurious-state density around spin glass equilibrium states is much larger for the Boltzmann machine than for the analogue neural network for a reasonably wide range of analogue gain or temperature, which leads to an expectation that the analogue neural network should possess a much better potential for memory retrieval than the Boltzmann machine.

A one-dimensional model is used to study the tilt/no-tilt transition in the liquid condensed phase of a lipid monolayer at the air/water interface. The head groups are modelled by hard rods of length b and the alkane chains by rigid tails of length a(a>>b). The interaction between these model lipid molecules is purely repulsive with a soft, short range, repulsion allowed between the tilting tails. The model is aimed at highlighting the excluded volume role in the tilt/no-tilt transition. The model is solved analytically and, in the limit of the temperature T to 0, the equation of state exhibits-at most-three distinct phases; isotropic, 'tilting' and 'nontilting'. At finite temperatures the transition from one phase to the other is continuous but, at low temperatures, still sharp.

The first Lyapunov exponent in a period window for a weak-coupled map lattice is calculated. Within the windows the behaviour of the coupled map lattice could be recovered by considering a small number of modes. The depth of the windows is well defined.

The quantum superalgebra U_q(osp(2,1)) is embedded into the Zⁿ-algebra and its cyclic representations are presented in the nongeneric case.

Using the Krichever-Novikov bases and the operator product expansions, the author constructs the N=2 superconformal algebra on a genus-g Riemann surface and the BRST charge corresponding to the superconformal algebra. He also checks the nilpotency of the BRST charge, and obtains the critical dimension of spacetime as D=2 for the N=2 superconformal theory on the higher-genus Riemann surface.

Tensor operators transforming under finite dimensional irreducible representations of the quantum group U_q(n) are defined. Using them tensor operators transforming under representations of the quantum algebra U_q(u(n)) are introduced. The Wigner-Eckart theorem on matrix elements of tensor operators defined is derived.

Based on the definitions of quantum group SU_q(2) and the two-dimensional quantum plane a la Woronowicz (1987) and Manin (1988), the covariant spinor calculus, very similar to the classical SU(2) group, is presented. New features arise from the noncommutation among the entries of the quantum matrix and among the 'coordinates' of the quantum plane. q-deformed Pauli matrices are defined and their applications are illustrated.

The construction of the adiabatic connection is studied in the case where the symmetry of a Hamiltonian is broken explicitly by a slowly varying perturbation. The type time variation of the perturbation corresponds to the one generated by the symmetry group of the unperturbed Hamiltonian. It is proven that the adiabatic connection for this type of system is completely determined by the group structure, up to a set of reduced matrix elements: systems with the same symmetries will have adiabatic connections differing at most in these reduced matrix elements. Several examples are detailed.

The authors consider linear problems associated with the Kadomtsev-Petviashvili equation. They prove that the linear problems are (1+1)-dimensional Hamiltonian systems under the symmetry constraints. Moreover, they find that the Hamiltonian flows of the linear problems are commutative.

The transcendental method for finding the exact analytical closed-form solution to the linear unidimensional integral equation of neutron slowing down with energy-dependent cross-section in an infinite homogeneous medium is studied in some detail. An original method of genesis of the isomorphic integral form of the process, and genesis of the general form of the analytical solution is applied. This, together with the exact solution of the transcendental equation of order one also determined, constitutes the exact solution of the problem. The numerical results obtained for different magnitudes of absorption rates and for different moderator masses show agreement with more conventional solutions, like those of Teichmann (1961) and Sengupta (1974). The conditions for the existence of the exact solutions are discussed.

For a strongly interacting boson system, it has been proposed that the concept of off-diagonal long-range ordering (ODLRO) is equivalent to Bose-Einstein condensation for a free boson system. Therefore, the existence of ODLRO in the ground state implies superfluidity. The author checks the validity of this proposition from another point of view. He rigorously shows that, for a hard-core lattice-boson system with a short-ranged interaction ODLRO is suppressed when a charged excitation gap develops and, hence, the boson system becomes a Mott-insulator. Finally, by using his theorem, he shows some interesting properties of the antiferromagnetic Heisenberg model, which can be taken as a hard-core lattice-boson system.

The author provides analytical and nonperturbative expressions for the effective coupling constant of QED in the presence of slowly varying background fields. His results agree with previous numerical calculations but, for strong magnetic fields, he observes some deviations from the expected logarithmic increase of the fine structure constant. These effects tend to reduce the effective charge, thereby providing further evidence against the existence of a new, strong-coupling phase of QED in heavy-ion collisions.

The author has earlier defined the quantum two-time localization problem as the minimizing of a particle's position spread about specified points at two distinct times. In the present article optimum localization is found for relativistic massive free particles. For short time intervals, spreading necessarily occurs at the speed of light while for long times the previously found diffusion-like behaviour is recovered. In defining relativistic localization, use is made of the work of Newton and Wigner (1949); in particular, their restriction to the positive energy hyperboloid is found to be necessary to recover the nonrelativistic limit of wavepacket spreading.

The mean-square forces that result from the zero-point fluctuations of quantized fields are calculated when acting on spheres and hemispheres of variable sizes. For the Maxwell field the boundary conditions of a perfectly conducting surface are imposed; the scalar field is investigated for Neumann and Dirichlet boundary conditions. The force is averaged over a finite time T; small and large objects are distinguished on the scale of cT. The results for the sphere and the hemisphere are compared with those for a piston that is embedded in an infinite plane. A small hemisphere and a small piston are found to have fluctuations of the same order of magnitude, while on a small sphere the fluctuations are by two orders of magnitude smaller because of correlations of fluctuations on the two sides of the sphere. Large spheres are shown to fit into the picture of large objects being composed of many patches each with the fluctuations impinging as on a large piston.

For pt.I see ibid., vol.25, p.3015-37 (1992). The zero-point fluctuations of a scalar field with Neumann boundary conditions are investigated on a flat circular disk. The Helmholtz equation is separated in oblate spheroidal coordinates; the mean-square force acting perpendicularly on the disk is calculated by picturing the disk as an oblate spheroid with zero eccentricity. The force is averaged over a time T, and the disk is taken small compared with cT. The results show that the fluctuations on a disk are roughly equal to those on a small sphere. For a one-sided disk divergences arise from the sharp edge, but the mean-square force has the same power dependence on T as for a hemisphere.

The classification of symmetric catastrophes is studied to obtain Landau potentials for statistical models. Potentials for symmetric models with two order parameters are thoroughly discussed. The double-cusp catastrophe is used for illustration. Its various symmetries and corresponding statistical models are revealed. As an important example, the three-state Potts model is studied in detail. Emphasis is placed on connections with exact results from conformal field theories describing 2D symmetric models.

The definite integral over I₀(z) evaluated by Bakulev (1991) is shown to be a special case of a tabulated integral.