Table of contents

Taking into consideration that the Fock space of a parabose oscillator may be described by bilinear commutation relations involving creation and annihilation operators, we construct the coherent states for odd and even order parabosons.

We study the online dynamics of learning in fully connected soft committee machines in the student - teacher scenario. The locally optimal modulation function, which determines the learning algorithm, is obtained from a variational argument in such a manner as to maximize the average generalization error decay per example. Simulations results for the resulting algorithm are presented for a few cases. The symmetric phase plateaux are found to be vastly reduced in comparison to those found when online backpropagation algorithms are used. A discussion of the implementation of these ideas as practical algorithms is given.

The general expression for the local matrix of a quantum chain with the site space in any representation of su(3) is obtained. This is made by generalizing from the fundamental representation and imposing the fulfilment of the Yang - Baxter equation. Then, a non-homogeneous spin chain combining different representations of su(3) is solved by developing a method inspired in the nested Bethe ansatz. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations. A conjecture about the solution of a chain with the site states in different representations of su(n) is presented. The thermodynamic limit of the ground state is calculated.

On the basis of long simulations of a binary mixture of soft spheres just below the glass transition, we make an exploratory study of the activated processes that contribute to the dynamics. We concentrate on statistical measures of the size of the activated processes.

All classical equations of kinetic coefficients in physics give only the first response to the external forces and fields. We have constructed new equations for kinetic coefficients, with all responses of the fields and forces, by using the perturbation method, which, customarily, was a mathematical tool for the approximate solution of the equations. Naturally, the recurrent equations which were obtained lead to recurrent solutions, which were found by the Green's function technique. The exact analytical formulae, produced by this method, play the same role for the calculations of kinetic coefficients as the Kirchhoff system of equations does for conductivity. In the framework of this solution one has obtained the upper boundary for fields when the solution is yet converged. We have considered the Hall and Seebeck coefficients and the elastic moduli (Skal A S 1997 Physica A to be published), and suggested that all other kinetic and transport coefficients may be rewritten in this way. On the basis of the formulae obtained, numerical calculations of the Hall and Seebeck coefficients are presented. The new universality classes for the Hall and Seebeck coefficients and the upper bond of the critical Hall conductivity exponent for all orders of a magnetic field contributions are obtained.

The diagonalization of the uncertainty matrix and the minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In the case of canonical observables, this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). The group-related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are a particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1,1), su(2), and sp(N,R) algebras. Unlike the group-related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N,R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators (q>0) and of any k power of are positive definite and can be diagonalized by symplectic linear transformations.

In this paper we continue the investigation of the structural phase transition in a quantum anharmonic crystal. We calculate the influence of breaking the symmetry of the interaction potential on the behaviour of fluctuations and their critical exponents. More precisely, it is shown that when the low-momentum asymptotics of the phonon spectrum (i.e. the Fourier transform of the interaction matrix) have a quartic asymptotic in one direction, the critical exponents for the momentum and position fluctuations increase on the critical line. The quantum nature of the fluctuations in T = 0 is, however, preserved.

Guidelines and hints are given on how to introduce an uncertainty treatment in conformity with recent international recommendations into the Monte Carlo simulation of, for instance, radiation particle transport processes in order to establish confidence in the results. The main problem is how to calculate sufficiently accurately, at a justifiable computational expense, the sensitivity coefficients of the quantities of interest with respect to numerous input quantities involved, data and uncertainties of which are given. This problem is solved using identical random-number sequences together with the input data slightly varied in the range determined by the associated uncertainties. Simple examples of application are treated in detail.

Tensor products of irreducible representations of the Jordanian quantum algebras and are considered. For both the highest weight finite-dimensional representations of and the lowest weight infinite-dimensional ones of , it is shown that tensor product representations are reducible and that the decomposition rules to irreducible representations are exactly the same as those of corresponding Lie algebras.

The path decomposition expansion for the path integral is used to derive the method of images.

Self-dual p-forms like first-order systems are investigated at classical, as well as at path integral level. Converting the self-dual system into a second-order gauge theory without introducing extrafields, we subsequently prove that: (i) the gauge theory, massive abelian p-form gauge fields and self-dual p-forms describe the same dynamics on the stationary surface of the field equations for the last model; (ii) self-dual p-forms and massive abelian p-form gauge fields represent a first-, respectively second-order BRST gauge-fixed version of the gauge system. The connection with the case of introducing extrafields is briefly addressed.

In this paper, we obtain three types of Bäcklund transformations on surfaces with in three-dimensional Minkowski space , where and are principal curvatures, and l(>0) are arbitrary constants. Using these transformations, one can construct families of surfaces with in from known ones.

It is shown that the presence of Lie point symmetries of (non-Hamiltonian) dynamical systems can ensure the convergence of the coordinate transformations which take the dynamical system (or vector field) into Poincaré - Dulac normal form.

In this paper we extend the notion of Lichnerowicz - Poisson cohomology to Jacobi manifolds. We study the relation of the so-called Lichnerowicz - Jacobi cohomology with the basic de Rham cohomology and the cohomology of the Lie algebra of functions relative to the representation defined by the Hamiltonian vector fields. A natural pairing with the canonical homology is constructed. The relation between the Lichnerowicz - Poisson cohomology of a quantizable Poisson manifold and the Lichnerowicz - Jacobi cohomology of the total space of a prequantization bundle is obtained. Particular cases of cosymplectic, contact and locally conformal symplectic manifolds are discussed. Finally, the Lichnerowicz - Jacobi cohomology of a non-transitive example is studied.

The dynamical behaviour caused by dry friction is studied for a spring-block system pulled with constant velocity over a surface. The dynamical consequences of a general type of phenomenological friction law (stick-time-dependent static friction, velocity-dependent kinetic friction) are investigated. Three types of motion are possible: stick - slip motion, continuous sliding, and oscillations without sticking events. A rather complete discussion of local and global bifurcation scenarios of these attractors and their unstable counterparts is present.

A classical theory is developed for the time evolution and scattering of minimally coupled massive scalar fields on closed spacetimes that evolve from initial to final static states. The time evolution is obtained by reformulating the field equation as an abstract Cauchy problem on a Hilbert space. Semigroup theory is used to prove the existence of a two-parameter family of evolution operators, and the field solution is obtained as a mapping of Cauchy data. The scattering theory is also formulated on a Hilbert space, and the wave operators and scattering operator are constructed from the evolution operators. It is shown that this approach most readily applies to spacetimes that undergo contraction.

In this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption . Those spatial forms for which the equation can be reduced to an ordinary differential equation are studied. The symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by Bluman and Cole and are unobtainable by Lie classical method.

Using the generalized regular representation, an explicit construction of the unitary irreducible representations of the (2 + 1)-Poincaré group is presented. A detailed description of the angular momentum and spin in 2 + 1 dimensions is given. On this base the relativistic wave equations for all spins (including fractional) are constructed.

A full (triangular) quantum deformation of so(3,2) is presented by considering this algebra as the conformal algebra of the (2 + 1)-dimensional Minkowskian spacetime. Non-relativistic contractions are analysed and used to obtain quantum Hopf structures for the conformal algebras of the 2 + 1 Galilean and Carroll spacetimes. Relations between the latter and the null-plane quantum Poincaré algebra are studied.

In this paper, we derive an -deformation of the algebra and its quantum Miura transformation. The vertex operators for this -deformed algebra and its commutation relations are also obtained.

A generalized three-level Jaynes - Cummings model (JCM) which includes various ordinary JCMs is shown explicitly to have an SU(3) structure: the Hamiltonian can be treated as a linear function of the generators of the SU(3) group. Based on this algebraic structure, the exact algebraic solutions of the Schrödinger equation, as well as eigenvalues and eigenstates of the Hamiltonian, are obtained by an algebraic method. Thus the three-level JCM is completely solved algebraically. The SU(N) structure of the N-level JCM is also constructed explicitly and can be solved by the same method.

On the basis of graded RTT formalism, the defining relation of the super-Yangian is derived and its oscillator realization is constructed.

We use the Fock-space representation of the quantum affine algebra of type to obtain a description of the global crystal basis of its basic level 1 module. We formulate a conjecture relating this basis to decomposition matrices of spin symmetric groups in characteristic 2n+1.

Motivated by Haldane's exclusion statistics, we construct creation and annihilation operators for g-ons using a bosonic algebra. We find that g-ons appear due to the breaking of a discrete symmetry of the original bosonic system. This symmetry is intimately related to the braid group and we demonstrate a link between exclusion statistics and fractional statistics.

By using the non-symmetric Hermite polynomials and a technique based on the Yangian Gelfand - Zetlin bases, we decompose the space of states of the Calogero model with spin into irreducible Yangian modules, construct an orthogonal basis of eigenvectors and derive product-type formulae for norms of these eigenvectors.