Table of contents

The authors prove that the differential calculi on quantum groups of A-series given in this letter and in previous papers are *-calculi.

m sets of covariant and contravariant q- bosonic spinors acting in the tensor product of m Fock spaces are constructed and their q-commutation relations determined. Both their transformation properties under the q-algebra su_q(2) and its dual matrix pseudogroup SU_q(2) are considered. It is proved that for m>1 the contravariant spinors are not related to the covariant ones through Hermitian conjugation. Similar results are also obtained for q-fermionic spinors.

The authors investigate the Lie symmetry vector fields of the wave equation Square Operator u+f(x₁, . . ., x_n, u)=0 where f is some nonlinear smooth function and n>or=2. The Painleve test is considered for the construction of explicitly space- and time-dependent integrable one-space-dimensional nonlinear wave equations.

From the exact solution of certain time-dependent classical harmonic oscillators in one dimension the authors investigate the behaviour of the adiabatic invariant J(t). For a subclass of such potentials Delta J identical to J(+ infinity )-J(- infinity )=0 whatever the regime. They show that this does not necessarily imply a breakdown of the commonly accepted asymptotic exponential law for Delta J.

The authors study the statistical properties of transition probabilities (or generalized intensities, equal to the squared matrix elements of operators having a classical limit). They first generalize the Shnirelman theorem to include semiclassically integrable states and use this to generalize the Feingold-Peres formula for the average value of generalized intensities. They perform an unfolding procedure to separate the smooth mean part of the intensities (as a function of frequency) from its fluctuating part by applying the generalized Feingold-Peres formula. (This formula relates the mean value of squared matrix elements to the power spectrum of the given observable over classical trajectories). Their approach is illustrated numerically by analysing the dipole transition probabilities in a family of billiards between integrability and chaos. The average values of the intensities as a function of frequency are excellently described by the generalized Feingold-Peres formula.

The scaling dimensions of the semiinfinite two-dimensional Ising model with smoothly inhomogeneous couplings are determined, in general, by the eigenvalues of a Schrodinger-type equation. The author obtains exact results for a marginal, two-parameter inhomogeneity that corresponds to a supersymmetric quantum mechanical potential. The results are compared with predictions of conformal invariance.

The author has investigated the critical behaviour of site-disordered Ising systems over a large range of dilution (0.6<or=p<or=1.0) investing about 10¹⁵ single spin flips with conventional local dynamics. Critical temperatures have been determined with high-precision using the cumulant method. The finite-size simulations lead to concentration-dependent exponents which seem to violate universality. The author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region. The author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder.

The critical dynamics of site-disordered three-dimensional Ising systems has been studied by conventional local dynamics simulations investing 10¹⁵ single spin flips. The author has verified that the correlation functions of disordered Ising systems obey dynamic scaling behaviour. The dynamic scaling functions have been calculated. The critical exponent z appears to be strongly dependent on the degree of disorder. The author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour. The dynamical exponent has been estimated to be z=2.4+or-0.1 which is surprisingly near to the renormalization group result in O( in 1/2).

A systematized renormalization-group approach is present to study systematically the physical properties of the one-dimensional systems arranged in the 1D sequence which are constructed by the inflation rule (A,B) to )A^m11B^m12, B^m21A^m22). A basic decimation rule is introduced, which is intuitive, simplified and exact, to investigate the electronic properties of the systems which contain crystals and quasicrystals. As typical examples, the spectra of global electronic bands of the 1D systems with (m₁₁,m₁₂,m₂₁,m₂₂)=(1,1,0,1). (1,1,1,1), and (1,1,2,1) are numerically calculated.

The author gives a simple upper and lower bound on the free energy density of the Hopfield model of size N with p stored patterns, in the limit where N and p tend to infinity with p/N to alpha <1. The two bounds coincide for alpha =0.

To illustrate the plausibility and difficulties involved in problems that require a modification of the winding number notion, the author studies the statistical model of a long chain entangling with a straight rod, and calculates the statistical rate for the chain to become disentangled using the reaction rate point of view. For the measure of entanglement the usual distinction between the clockwise and the anticlockwise winding numbers is no longer applicable because, as far as the mutual snaring is concerned, only the sum of these two numbers is relevant. The author performs the calculations for the equilibrium case with a small friction and the static case with a large friction. The results are consistent with the prediction of the existing phenomenological 'tube' model for polymers.

The authors calculate the lowest translationally invariant levels of the Z₃- and Z₄-symmetrical chiral Potts quantum chains, using numerical diagonalization of the Hamiltonian for N<or=12 and N<or=10 sites, respectively, and extrapolating n to infinity . In the high-temperature massive phase they find that the pattern of the low-lying zero momentum levels can be explained assuming the existence of n-1 particles carrying Z_n charges Q=1, . . ., n-1 (mass m_Q), and their scattering states. In the superintegrable case the masses of the n-1 particles become proportional to their respective charges: m_Q=Qm₁. Exponential convergence in N is observed for the single-particle gaps, while power convergence is seen for the scattering levels. They also verify that qualitatively the same pattern appears for the self-dual and integrable cases. For general Z_n they show that the energy-momentum relations of the particles show a parity nonconservation asymmetry which for very high temperatures is exclusive due to the presence of a macroscopic momentum P_m=(1-2Q/n) phi , where phi is the chiral angle and Q is the Z_n charge of the respective particle.

The authors discuss the critical behaviour of two-dimensional vesicles with osmotic pressure, but no bending energy, in the presence of an attractive boundary. The phase diagram is presented for the first time and exponents are calculated in the various regions of the phase diagram from an exact enumeration in conjunction with series analysis techniques. In particular special surface critical exponents are presented for the branched polymer phase.

A new approach to the solution of the quantum Yang-Baxter equation is presented and a complete SU(1,1)-invariant factorized unitary scattering matrix is constructed.

The author proposes a procedure for q-deformations of the real forms G of complex Lie (super-)algebras associated with (generalized) Cartan matrices. The procedure gives different q-deformations for the nonconjugate Cartan subalgebras of G. The author gives several illustrations, e.g., q-deformed Lorentz and conformal (super-)algebras. The q-deformed conformal algebra contains as a subalgebra a q- deformed Poincare algebra and as Hopf subalgebras two conjugate 11-generator q-deformed Weyl algebras. The q-deformed Lorentz algebra is a Hopf subalgebra of both Weyl algebras.

A geometric approach to the method of Lagrange multipliers is presented using the framework of the tangent bundle geometry. The nonholonomic systems with constraint functions linear in the velocities are studied in the first place and then, and using this study of the nonholonomic mechanical systems as a previous result, the holonomic systems are considered. The Lagrangian inverse problem is also analysed and, finally, the theory is illustrated with several examples.

New classes of exact solutions of the inhomogeneous Heisenberg ferromagnet equation are found by means of a geometrical technique. This technique consists of calculating one-parameter families of geodesics on surfaces in E³. The solutions correspond to geodesics on surfaces of rotation.

A Darboux transformation is developed for generating dark multi-soliton solutions of the MKdV equation. The Darboux matrices are found explicitly in recursive manner and a system of linear algebraic equations is derived for determining the dark N-soliton solutions. By means of the Binet-Cauchy formula an explicit expression of the dark N-soliton solution is obtained.

Making use of the gauge approach the author constructs the continuum model for the description of topological defects in planar elastic systems. The equations of motion for defect systems in the presence of both dislocation and disclination fields are presented. The exact vortex-like solution for static disclinations is obtained. The Schrodinger equation in the presence of a static disclination vortex is studied. The author shows that electrons (holes) can acquire a topological phase which depends on the disclination flux. When the interaction between long-wave electron fields and acoustic waves is present, the electrons with E<E_F are found to be localized. The scattered states (at E>E_F) acquire an additional phase shift due to the deformation potential.

The author discusses a conversion identity through which the series expansion of the Green's function pertaining to the scalar Helmholtz equation can be readily converted into a series expanded Green's dyadic. This expansion finds a simple application in the electromagnetic radiation theory leading to electric multipole and magnetic multipole terms of the vector potential and the resulting E and B fields. The dyadic is also used to obtain a series expansion of the magnetostatic vector potential. Finally, the role played by localized longitudinal and localized transverse currents in the generation of the electromagnetic field is briefly examined, leading to the conclusion that a localized longitudinal current is self-screening. That is, it does not produce any electromagnetic field outside the domain of its distribution.

The quantum mechanics of charged, massive, spin-1 bosons in the presence of a homogeneous magnetic field (HMF) is studied using a six-component wavefunction formalism. The energy eigenvalues are compared with those previously obtained via other formalisms, the equations of motion of certain operators are given, and the positive and negative energy eigensolutions are obtained by the use of a ladder operator method. The six-component current for the case of general external electromagnetic fields is also displayed and finally, the employment of the eigensolutions and current in a study of a spin-1 boson-antiboson plasma is discussed.

The authors propose an easy and effective variant of the realization of the dynamical symmetry of hydrogen-like atoms in two-dimensional space, based on the relationship between the Schrodinger equation for the isotropic harmonic oscillator in one-dimensional complex space and the Schrodinger equation for the one-electron atom in two-dimensional space, that permits them to use the operator method to solve the Schrodinger equation for the two-dimensional atomic system. For illustration of the algebraic method, they consider the problem of the two-dimensional hydrogenic donor in a magnetic field. By the use of the path integral they also establish the relationship between the two-dimensional Coulomb Green function and the Green function for the isotropic harmonic oscillator in one-dimensional complex space.

The diffeomorphism group of the D-dimensional space as the symmetry group for conformal quantum field theory models is proposed. The problem of calculation of critical exponents turns out to be reduced to the study of special diffeomorphism group representations. The general properties of these representations on the space of symmetric traceless tensors are investigated.

An arbitrary dimensional expression is given for the anomalous dimension of the fermion field in a model with a four-point interaction and a U(1) gauge field, at O(1/N²) within a large flavour expansion in the Landau gauge.

The authors reconsider the conjecture by Gepner (1991) that the fusion ring of a rational conformal field theory is isomorphic to a ring of polynomials in n variables quotiented by an ideal of constraints that derive from a potential. They show that, in a variety of cases, this is indeed true with one-variable polynomials.