Table of contents

The cyon model for anyon systems is explicitly analysed using the quantum electrodynamics of particles carrying anomalous magnetic-dipole moments in (2+1) dimensions. Statistical interaction appears together with a further interaction term whose cut-off dependence reflects the unrenormalizable character of the theory.

Starting with the affine super-algebra Osp(2,2), we have shown how different choices of Heisenberg subalgebras in conjunction with the dressing operator approach of Wilson (1958) leads to different super-symmetric bi-Hamiltonian systems. Subsequently, the explicit form of the symplectic operators are deduced with the help of the trace identity. Lastly a possible relation with the tau -function is indicated.

High-order generators of Lie algebras are defined and their basic properties are discussed in this letter. In 'classical' cases, the high-order generators are simply the enveloping algebras of Lie algebras, while, in 'quantum' cases, we can define the high-order central extensions of (infinite-dimensional) Lie algebras and can see that the high-order generators are no longer the enveloping algebras of Lie algebras. As an example, we calculate the first-high-order central extensions of Virasoro algebras and find that the first-order and usual-order central extensions are closely related to each other.

New types of two-dimensional quasi-exactly solvable models are found. They are obtained from one-dimensional finite-difference equations with the help of the generating function which is composed from a set of functions obeying simple recurrence equations. In particular, this set can consist of some kinds of classical orthogonal polynomials. It is shown that the models found are described in terms of SO(2,1) and Heisenberg-Weyl dynamical algebras.

The Prelle-Singer procedure for determining elementary first integrals of two-dimensional autonomous systems of ordinary differential equations is introduced and how it can be generalized to higher dimensions is discussed. The application of this procedure in several dynamical systems is reported.

The writhe of a self-avoiding walk in a three-dimensional space is the average over all projections onto a plane of the sum of the signed crossings. We compute this number using a Monte Carlo simulation. Our results suggest that the average of the absolute value of the writhe of self-avoiding walks increases as n^alpha , where n is the length of the walks and alpha approximately=0.5. The mean crossing number of walks is also computed and found to have a power-law dependence on the length of the walks. In addition, we consider the effects of solvent quality on the writhe and mean crossing number of walks.

We consider the classical Liouville field theory on the Riemann surface with h>1 genus. In terms of the uniformization theorem of the Riemann surface, we show explicitly the classical exchange algebra (CEA) for the chiral components of the Liouville field. We find that this classical exchange algebra is just the same as that on a Riemann sphere with 2h punctures.

If the physical picture inherent in Kolmogorov's description of turbulence is given a mathematical implementation, then a finite theory of turbulence yielding the Kolmogorov spectrum can be constructed. We discuss how this can be done in a screening approximation which exploits the viscoelastic effect.

I review the state of knowledge about random surface ensembles in continuous embedding spaces and their possible role in defining strings in arbitrary dimensions. The application of rigorous statistical mechanics, approximate calculation and numerical simulation is described.

Following ideas of Gull, Skilling and MacKay (1992), we develop and explore a statistical-mechanics framework through which one may assign values to the parameters of a model for a 'rule' (instanced, here, by the noisy linear perceptron), on the basis of data instancing the rule. The 'evidence' which the data offers in support of a given assignment, is likened to the free energy of a system with quenched variables (the data): the most probable (MAP) assignments of parameters are those which minimize this free-energy; tracking the free-energy minimum may lead to 'phase transitions' in the preferred assignments. We explore the extent to which the MAP assignments lead to optimal performance.

We consider a generalization of the infinite-range Sherrington-Kirkpatrick spin-glass model with arbitrary spin S and the inclusion of crystal-field effects. For integer S, replica-symmetric calculations have shown the presence of both continuous and discontinuous transitions and a tricritical point. For S=1, we report a detailed numerical analysis of the replica-symmetric solutions. We locate the first-order boundary and clarify some inconsistencies of the previous analyses. Some analytic asymptotic expansions are used to support the numerical findings.

Inversion relations are obtained for the standard scalar q-state triangular Potts model with two- and three-spin interactions, generalizing previously known results for two-spin interaction models. It is shown that these inversion relations generate a group of symmetries of the model which is naturally represented in terms of birational transformations in a four dimensional parameter space. This group of birational transformations is generically a very large one, namely a hyperbolic Coxeter group. In this framework of very large groups of symmetries, a remarkable situation pops out: the one for which q corresponds to Tutte-Beraha numbers.

A relation between the anisotropic t-J model and a generalized six-vertex model has been established. On the basis of this relation, we obtain the exact Bethe ansatz solution for the ground state energy of anisotropic t-J model for arbitrary spin and special values of coupling parameters. We consider also the anisotropic generalization of t-J model with localons.

The critical behaviour of a long-range, one-dimensional non-equilibrium system where the desorption rate decreases with distance as some power approximately r^{- alpha -d} is studied by means of the coherent anomaly method. The alpha -dependence of the critical parameter lambda _c and the critical exponent beta is determined.

The D-dimensional ferromagnetic Ising model with weak Gaussian random fields is considered. In dimensions D<3 due to rare large-scale thermal excitations (large-spin clusters with magnetization opposite to the ferromagnetic background) in the low-temperature region h₀²<<T<<1 (where h₀ is the characteristic value of the field) the free energy is shown to contain a non-analytic contribution of the form exp(-(const/2h₀²)(h₀²/T)^(3-D/2)).

We study the critical exponents for the susceptibility and the correlation length for three continuous-spin models, which we call the border model, the double-well model and the single-well model, by the method of high-temperature series. We observe variations from the predictions of universality, but in some cases longer series may be required before firm conclusions can be drawn.

The average probability density P(r,t) of random walks on fractals is revisited within the continuous-time random walks formalism. Corrections to the accepted asymptotic stretched Gaussian decay of P(r,t) of the form r^alpha are discussed. It is shown that P(r,t) obeys a diffusion equation with a fractional time derivative asymptotically, and predictions about the value of alpha are presented.

The thermodynamic and retrieval properties of fully connected Q-Ising networks are studied in the replica-symmetric mean-field approximation. In particular, capacity-gain parameter and capacity-temperature phase diagrams are derived for Q=3, 4 and Q= infinity and different distributions of the stored patterns. Furthermore, the optimal gain function is determined in order to obtain the best performance. Where appropriate, the results are compared with the diluted and layered versions of these models.

Despite significant work since the original paper by H Furstenberg(1963), explicit formulae for Lyapunov exponents of infinite products of random matrices are available only in a very few cases. In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random 2*2 real matrices. All these products are constructed using only two types of matrices, A and B, which are chosen according to a stochastic process. The matrix A is singular, namely its determinant is zero. This formula is derived by using a particular decomposition for the matrix B, which allows us to write the Lyapunov exponent as a sum of convergent series. The key point is the computation of all the integer powers of B, which is achieved by a suitable change of frame. The computation then follows by looking at each of the special types of B (hyperbolic, parabolic and elliptic). Finally, we show, with an example, that the Lyapunov exponent is a discontinuous function of the given parameter.

We give field-theoretic realizations of both the q-Heisenberg and the q-Virasoro algebra. In particular, we obtain the operator product expansions among the current and the energy-momentum tensor obtained using the Sugawara construction.

A double rotor vector coherent state construction of the SU(4) contains/implies SU(2)*SU(2) Wigner supermultiplet leads to a simple matrix representation of this scheme. Matrix elements of the vector coherent state realizations of the group generators are given by very simple expressions in terms of ordinary spin (S) and isospin (T) Wigner coefficients with intrinsic projection labels K_S and K_T. The K-matrix technique is used to effectively elevate these labels to the status of good quantum numbers. The KK matrices are given in analytic form for a number of important irreducible representations.

We show that some examples in the literature of non-standard symmetry reductions of ordinary differential equations can be understood using the concept of a solvable structure of one-forms.

The path integral on the single-sheeted hyperboloid, i.e. in D-dimensional imaginary Lobachevsky space, is evaluated. A potential problem which we call the 'Kepler problem', and the case of a constant magnetic field are also discussed.

A method is proposed of obtaining (2+1)-dimensional nonlinear equations with nonanalytic dispersion relations. Bilocal formalism is shown to make it possible to represent these equations in a form close to that for their counterparts in 1+1 dimensions.

We show that a large class of physical theories which has been under intensive investigation recently, share the same geometric features in their Hamiltonian formulation. These dynamical systems range from harmonic oscillations to WZW-like models and to the KdV dynamics on Diff₀S¹. To the same class belong also the Hamiltonian systems on groups of maps. The common feature of these models are the 'chiral' equations of motion for so-called chiral decomposition of the phase space.

We study the motion of neutral spinning particles in static inhomogeneous magnetic fields, where the orbital motion and spin dynamics do not factorize. Using the phase-space representation of quantum mechanics, the Moyal propagator is obtained for the case of a linearized field. As an example, the evolution of particles in coherent rotational states is calculated. In order to interpret the Wigner state function in a classical mechanical way, we calculated the Husimi functions.

Luttinger and Lu(1980) have derived a variational method for representing the Yukawa interaction of bosons and fermions in non-relativistic field theories by a general potential between fermions and fictitious particles ('shadows'). In the present work, this is considerably improved by applying Jensen's inequality only once and representing the kinetic term in the path-integral average as a quantum-mechanical N-body problem using the replica technique. Various weak- and strong-coupling approximations as well as variational bounds for this term are discussed.

A relativistic equation is derived for a slowly varying potential by suitably approximating the one-dimensional Dirac equation. This equation is shown to be akin to the Schrodinger equation with an effective potential and effective eigenvalues. An iterative procedure for solving this equation is indicated. As an application, the relativistic treatment of the Mathieu potential on the basis of this equation is considered and results are compared with those obtained by solving the exact one-dimensional Dirac equation. These results are likely to take adequate account of the relativistic effects on electrons near Fermi levels in metals.

Coherent states for a family of isospectral oscillator Hamiltonians are derived from a suitable choice of annihilation and creation operators. The Fock-Bargmann representation is also considered.

It is shown that the solvable structure of the stability group of a massless field yields, in a simple and short way, useful information about the physical polarization states, and the structure of the Hilbert space for such a field. It is also shown that such fields are necessarily gauge fields; the exact form of the gauge transformation follows from the structure of the equivalence classes implied by the group properties. Some examples for spinor and tensor fields in general dimensions are worked out and the structure of the Hilbert space for gravitational radiation is discussed in some detail.

For original paper see Milosevic et.al., ibid., vol.26, p.1061 (1993) The kinetics of random sequential adsorption of k-mers on a one-dimensional lattice, occupied initially by point impurities with a random distribution, is solved exactly. The solution of the process on a continuum substrate is also derived from the discrete case.