Table of contents

A two-parameter deformation of the universal algebra of osp(4/2) is carried out, yielding a Z₂-graded Hopf algebra with a bijective antipode. This Hopf algebra depends on the extra parameter in both its algebraic and coalgebraic structures, and also admits nontrivial finite-dimensional irreps at arbitrary deformation parameters.

A new path integral representation of the quantum mechanical propagator is proposed, which is even more intimately connected with classical mechanics than the Feynman path integral. The emergence of the semiclassical propagator is discussed in detail. It is argued that those paths which actually contribute in the new path integral have a very simple relation with the classical dynamical trajectories in configuration space.

The geometric phase and the geometric distance function are intimately related via length of the curve (a concept the author introduces) for any parametric evolution of the quantum system. He offers an interpretation of the nonadiabatic Berry phase as the integral over a difference between the differentials of two geometric quantities, which enables him to say that the geometric phase is just (half) the integral of the contracted length of the curve that the system traverses during a cyclic excursion.

As a function of a parameter characterizing the degree of mixing of site values, the density of nonzero sites of some one-dimensional cellular automata is shown to exhibit a sequence of period-doubling bifurcations and to behave chaotically when the degree of mixing is sufficiently large. The automata network rules which are considered appear to be useful to model complex systems, as in epidemiology, in which the motion of the individuals is believed to play an important role.

The author considers the finite-size (FS) scaling behaviour of the transverse correlation lengths xi /sub ///⁽ⁿ⁾ for an interface confined in a strip geometry exactly at a second-order fluctuation-dominated wetting transition. He shows that the dimensionless amplitude ratios A/sub ///⁽ⁿ⁾= Sigma L²/k_BT xi /sub ///⁽ⁿ⁾ (with Sigma the surface stiffness coefficient) are related to the short-distance expansion critical exponent theta by A/sub ///⁽ⁿ⁾= pi ²n(n+ theta -1)/2 for all fluctuation-dominated wetting transitions, provided the binding potential V(y) in the capillary-wave model is conformally mapped from the semi-infinite plane. He argues that the existence of this amplitude-exponent relation reflects the conformal invariance of the capillary-wave model Lagrange density and lists a number of one-point functions which can be shown to exhibit this invariance explicitly.

Using an interfacial potential approach the authors study capillary condensation within an infinitely long rectangular fissure in an otherwise planar surface. They demonstrate how the groove fills sharply, but continuously, with the wetting liquid as the chemical potential and temperature are varied. This is contrasted to the situation for an infinitely deep slit, where the transition is first order.

The mean field renormalization group (MFRG) method is applied to the spin-1/2 XXZ model. The phase diagram is obtained for various lattice structures in two and three dimensions, as well as estimates for the critical exponents. The results of the original MFRG method are quite good, but the more sophisticated approach that also includes surface exponents is not suitable for application to the small clusters used here.

A constrained monomer-dimer (CMD) model on decorated hypercubic lattices, which is exactly solvable in any dimensionality d, is studied. The model is critical in the limit of maximal packing of the dimer configurations allowed by the constraint. For d=2 it is equivalent to the rooted-tree model on the square lattice considered by Duplantier and David (1988). The validity of two finite-size scaling predictions in higher dimensionalities is studied: (1) the existence of logarithmic corrections in the free energy due to corners, and (ii) the amplitude-exponent relation for the pair correlation function. The logarithmic finite-size corrections are obtained for arbitrary d, for boundary conditions which are periodic for d'>or=0 dimensions and free in the remaining d-d' dimensions. Dimer-dimer correlation functions are studied at d=3 for a system finite in two dimensions and infinite in the third. It is shown that the amplitude-exponent relation for the CMD model holds for d=2 and breaks down for d=3.

The author considers the one-dimensional gas of rods with finite range interaction. Using the transfer matrix method he obtains an integral equation that describes the gas of rods. The integral equation has a nonzero solution, if pressure and chemical potential satisfy a condition that is only the equation of state. In particular, he discusses the case of continuum piecewise linear next-neighbour interaction potential for rods.

The partition function of a six-vertex model with domain wall boundary conditions is considered on the finite lattice. The authors show that the partition function satisfies a recursive relation. They solve the recursion relation by a determinant formula. This gives a determinant representation for the partition function. They use the Quantum Inverse Scattering Method (QISM).

The author gives a complete and rigorous description of the ground states of the Hubbard model on the Kagome lattice for electron densities n>or=5/3 and U>0. If 11/6>n>or=5/3 the system shows a ferromagnetic behaviour at zero temperature. If n is above 11/6 the system is paramagnetic. The proof of these results uses some graph-theoretic methods. The results are applicable to all line graphs of planar lattices, of which the Kagome lattice is an example.

The authors examine the distribution of metastable states in the p-state Potts model. The random-bond Potts model on a chain as well as the uniform bond model on a self-similar lattice are studied. Analytic results are obtained for arbitrary continuous distributions of nearest-neighbour bonds for the chain. When negative bonds are present, the ground state for p>2 is frustrated, in contrast to the Ising model which has p=2. The results for p>2 show similarity to the discrete random-field Ising model.

A continuous-time approximation for the learning process in single-layer perceptrons is presented. Both statics and dynamics are treated by the replica method within the framework of the replica symmetric theory. The Adaline rule extended by a decay factor is investigated in detail, especially for linearly-separable problems. It is shown that the poor generalization ability of this rule near alpha =1, the so called overfitting, can be cured by an appropriate decay factor or by an appropriate training time.

This paper investigates the ability of a single-layer, time-summating neural network to associate and store temporal sequences. In particular, the associative learning of temporal sequences is reformulated as an equivalent classification task involving static patterns. This leads to a generalization of the perceptron learning rule and convergence theorem to the case of temporal sequences. Using geometrical arguments based on linear separability it is shown how a time-summating network can handle temporal features such as ordering and coarticulation effects. Such an ability is a consequence of the fact that the time-summating network develops an activity trace consisting of a decaying sum of all previous inputs to the network. On the other hand, such an activity trace may also lead to an accumulation of errors in the presence of noisy inputs. This motivates a modification of the perceptron learning rule involving the introduction of a stability parameter that guarantees a certain level of robustness to noise. The performance of the network in the presence of random input sequences is then analysed using statistical-mechanical techniques. Finally, it is shown how, with small modifications, the time-summating network can be trained to store and recall complex sequences.

The problem of obtaining the Sp(6,R) irreducible representations that arise from the symmetrized products of A copies of the fundamental representation of Sp(6,R) is considered. A simple method is developed that rapidly leads to the determination of the Sp(6,R) content up to a prescribed cut-off. In practical calculations A is identified with the nucleon number. It is shown that for a given excitation energy cut-off, n omega , and sufficiently large A, the number and multiplicities of the Sp(6,R) irreducible representations is fixed and the content can be expressed in an A-independent manner. Complete results are given for n<or=8 and all A>or=16. There is no difficulty in principle in going to higher excitations and details are given for determining the Sp(6,R) content for A<or=16.

New types of reductions of (2+1)-dimensional integrable systems which are associated with mixed potential-eigenfunction constraints are discussed. Necessary and sufficient conditions for the admissibility of such constraints are presented. Classification results for integrable equations will be used for applying these conditions. Several typical examples are considered, including the KP, mKP, (2+1)-dimensional Savada-Kotera-Kaup-Kupershmidt, Harry Dym and Nizhnik-Veselov-Novikov equations.

The authors study the role of (1,1) graded tensor field T in the analysis of complete integrability of dynamical systems with fermionic variables. They find that such a tensor T can be a recursion operator if and only if T is even as a graded map, namely, if and only if p(T)=0. They clarify this fact by constructing an odd tensor for two examples, a supersymmetric Toda chain and a supersymmetric harmonic oscillator. They explicitly show that it cannot be a recursion operator since it does not allow new constants of motion to be built from the first two, in contrast to what usually happens with ordinary, i.e. nongraded systems.

Partially invariant solutions of a general class of nonlinear Schrodinger equations, involving four arbitrary functions of the modulus rho of the solution and its derivative rho _x, are obtained. The modulus rho ( xi ) is assumed to depend on a symmetry variable xi , whereas the phase omega (x,t) depends on both independent variables. Both rho and omega are obtained explicitly, as are the conditions on the coefficients in the equation, necessary for such solutions to exist.

There exist super-potentials which result in Lienard-Wiechert potentials. The physical meaning of super-potentials is the coordinate of the particle which produces the Lienard-Wiechert potentials. In this paper, the super-potentials in far fields are considered analytically and numerically. It is shown that the super-potentials can be calculated from the far electric fields produced by the particle, if the motion is periodic and nonrelativistic. This means that one can estimate the particle trajectory from the far electric fields.

The Lecanda-Roman-Roy geometric constraint algorithm for presymplectic Lagrangian systems is applied to a mechanical model of singular field theories coupled to time independent external fields. The simple, yet intrinsic structure of the algorithm allows the influence of the external field to be traced through the constraint analysis, showing clearly where pathologies arise-namely in the second generation nondynamical constraints arising from the stability of the first generation compatibility constraints. Using a coordinate independent geometric algorithm provides a more systematic tool for investigating singular field theories than the usual ad hoc manipulation of the field equations; where the essential structure is often obscured by the details of the representation of the model and the complexity of the algebra.

If two potentials V₁(x) and V₂(x) are ordered V₁<V₂, then the comparison theorem tells the author that the corresponding Schrodinger eigenvalues are ordered E₁<E₂. He presents some simple conditions which allow him to predict such spectral ordering for the ground state, even when the graphs of the potentials cross over. As illustrations, the truncated quartic oscillator and the Yukawa potential are studied. By allowing Coulomb 'tangents' to cross over the Yukawa in various ways, he is able to improve earlier energy upper bounds which he had obtained by representing the Yukawa potential as a smooth transformation of the Coulomb potential, and also to augment these results with lower bounds.

For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.

In the presence of an idealized potential on two parallel planes represented by two one-dimensional delta -functions at x₃=-d/2 and x₃=+d/2 the authors discuss the Feynmann propagators for relativistic scalar and spinor fields. These propagators take into account bound states, scattering states and resonances. The Casimir energy for this configuration is calculated. For massive fields the Casimir force decreases exponentially with rising distances. In the scalar case they find an attractive force and in the spinor case a repulsive force. An attempt to treat the same problem for a massive scalar field using nonrelativistic quantum field theory leads to a vanishing Casimir force.