Table of contents

Critical finite-size scaling functions for the order-parameter distribution of the two- and three-dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite-size scaling functions has been derived by employing a scaling limit which differs from the traditional finite-size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite-size scaling function for the order-parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.

Properties of certain q-orthogonal polynomials are connected to the q-oscillator algebra. The Wall and q-Laguerre polynomials are shown to arise as matrix elements of q-exponentials of the generators in a representation of this algebra. A realization is presented where the continuous q-Hermite polynomials form a basis of the representation space. Various identities are interpreted within this model. In particular, the connection formula between the continuous big q-Hermite and the continuous q-Hermite polynomials is thus obtained, and two generating functions for these last polynomials are derived algebraically.

In the context of the fractal percolation process, the possibility of directed percolation is raised: Is there any non-trivial value of the parameters such that there is 'oriented percolation' or, perhaps 'stiff percolation' in this model? Somewhat surprisingly, the answer is no.

We consider the limit of strong Coulomb attraction for generalized Hubbard models with eta -symmetry. In this limit these models are equivalent to the ferromagnetic spin- 1/2 Heisenberg quantum spin chain. In order to study the behaviour of the superconducting phase in the electronic model under perturbations which break the eta -symmetry we investigate the ground state of the ferromagnetic non-critical XXZ-chain in the sector with fixed magnetization. It turns out to be a large bound state of N magnons. We find that the perturbations considered here lead to the disappearance of the off-diagonal long-range order.

We perform a numerical spectral analysis of a quasi-periodically driven spin- 1/2 system, the spectrum of which is singular continuous. We compute fractal dimensions of spectral measures and discuss their connections with the time behaviour of various dynamical quantities, such as the moments of the distribution of the wavepacket. Our data suggest a close similarity between the information dimension of the spectrum and the exponent ruling the algebraic growth of the 'entropic width' of wavepackets.

With use of the exact recursion equations of the generating functions, we determine the asymptotic behaviour of bond percolation on branching Koch curves and obtain the critical exponents alpha , beta , gamma and nu . The scaling law 2- alpha = gamma +2 beta =d_f nu is obtained; this is similar to the scaling law for a regular lattice, except that the fractal dimension d_f is replaced by the Euclidean dimension d.

We study a frustrated spin-glass model on recursive diamond-shaped lattices. We prove the existence of a mean-field-type phase transition and analyse the high- and low-temperature regions.

Results from high-temperature series for a class of discrete, classical spin models on the square lattice suggest that the phase boundary is, at least in part, given by a simple heuristic principle. It is shown how this principle can be extended to triangular and hexagonal lattices by making use of duality and the star-triangle transformation. For the Potts model, the exact phase transition is recovered, not only for integer number of states, but also in the percolation limit. For more complicated models, a number of phase diagrams are derived, which are presumably exact except in regions where partially broken symmetry phases or massless phases intervene. The principle can also be applied to quenched bond-diluted spin models. The results for the Potts model are obtained for all three lattices and shown to be exact in certain limiting cases. The M to 1 limit of the wreath product model S(M):S(M), which corresponds to 'double' percolation, is studied to provide an example for which the principle is completely incorrect.

We consider the six-vertex model with anti-periodic boundary conditions across a finite strip. The row-to-row transfer matrix is diagonalized by the 'commuting transfer matrices' method. From the exact solution we obtain an independent derivation of the interfacial tension of the six-vertex model in the anti-ferroelectric phase. The nature of the corresponding integrable boundary condition on the XXZ spin chain is also discussed.

We investigate families of generalized mean-field theories that can be formulated using the Peierls-Bogoliubov inequality. For test Hamiltonians describing mutually non-interacting subsystems of increasing size, the thermodynamics of these mean-field-type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean-field critical points. Finite-size scaling analysis of this mean-field critical behaviour allows us to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki (1987), which is thus given a clear interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls-Bogoliubov inequality. In the case of the 2D Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity in the thermodynamic functions.

A perceptron can classify a set of random patterns into two groups. The maximal gap between the two groups is calculated using replica theory. The replica-symmetric solution is shown to be unstable and gives results which are qualitatively different from the one-step replica-symmetry breaking solution (RSB1). The results of a calculation without the replica method are given, yielding an exact upper bound for the gap which almost coincides with the RSB1 solution. Finding the classification of a set of patterns with a maximal gap is a difficult combinatorial optimization problem. An algorithm is developed which gives large gaps.

In this paper we study the geometrical properties of the high-lying eigenfunctions (200 000 and above) which are deep in the semiclassical regime. The system we are analysing is the billiard system inside the region defined by the quadratic (complex) conformal map w=z+ lambda z² of the unit disc mod z mod <or=1 as introduced by Robnik (1983), with the shape parameter value lambda =0.15, so that the billiard is still convex and has KAM-type classical dynamics, where regular and irregular regions of classical motion coexist in the classical phase space. By inspecting 100 and by showing 36 consecutive numerically calculated eigenfunctions we reach the following conclusions: (i) Percival's (1973) conjectured classification in regular and irregular states works well: the mixed-type states 'living' on regular and irregular regions disappear in the semiclassical limit. (ii) The irregular (chaotic) states can be strongly localized due to the slow classical diffusion, but become fully extended in the semiclassical limit when the break time becomes sufficiently large with respect to the classical diffusion time. (iii) Almost all states can be clearly associated with some relevant classical object such as the invariant torus, cantorus or periodic orbits. This paper is largely qualitative but deep in the semiclassical limit and as such it is a prelude to our next paper which is quantitative and numerically massive but at about ten times lower energies.

Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra U_p,q(gl(2)) to the quantum group GL_p,q(2), we show how the (2j+1)-dimensional representations of GL_p,q(2) can be obtained by 'exponentiating' the well known (2j+1)-dimensional representations of U_p,q(gl(2)) for j=1, 3/2, ...; j= 1/2 corresponds to the defining two-dimensional T-matrix. The earlier results on the finite-dimensional representations of GL_q(2) and SL_q(2) (or SU_q(2)) are obtained when p=q. Representations of U_q,q(2)(q in C mod R) and U_q(2)(q in R mod (0)) are also considered. The structure of the Clebsch-Gordan matrix for U_p,q(gl(2)) is studied. The same Clebsch-Gordan coefficients are applicable in the reduction of the direct product representations of the quantum group GL_p,q(2).

Reduction of the left regular representation of the quantum algebra sl_q(3) is studied and q-difference intertwining operators are constructed. The irreducible representations correspond to a q-deformation of the spaces of local sections of certain line bundles over the flag manifold.

Deformations of the Lie algebras so(4), so(3,1) and e(3) that leave their so(3) subalgebra undeformed and preserve their coset structure are considered. It is shown that such deformed algebras are associative for any choice of the deformation parameters. Their Casimir operators are obtained and some of their unitary irreducible representations are constructed. For vanishing deformation, the unitary irreducible representations of the deformed algebras transform into those of the corresponding Lie algebras that contain each of the so(3) unitary irreducible representations at most once. It is also proved that similar deformations of the Lie algebras su(3), sl(3,R) and of the semidirect sum of an Abelian algebra t(5) and so(3) do not lead to associative algebras.

For zero-curvature representations (ZCRs) A_t-B_x-AB+BA=0 of evolution equations u_t=f(x, u, u_x, ..., u_x...x), we develop a description which is invariant under gauge transformations A'=SAS^-1-S_xS^-1 and B'=SBS^-1-S_tS^-1, where A, B and S are matrix functions of x, u, u_x, u_xx, ... . We prove that every fixed matrix A of any dimension and order (in u_x...x) determines a continual class of evolution equations which admit ZCRs with this A. Then we quote examples illustrating how a dependence of A on an essential parameter restricts classes of represented equations. One of our examples shows that some non-integrable systems can admit parametric Lax pairs and infinitely many non-trivial conservation laws.

A class of fully nonlinear third-order partial differential equations (PDES) is considered. This class contains several examples which have recently appeared in the literature and for which rather unusual travelling-wave solutions have been given. These solutions consist essentially of sums of exponentials; we trace the occurrence of these exponentials back to the existence of a linear subequation which appears as a factor in the travelling-wave reduction. In addition, we consider the Painleve analysis of this set of equations, both for the original PDE and also for reductions to ordinary differential equations (ODES). No equation in the class considered survives the combination of PDE and ODE tests. Also, an equation in the class considered which is known to be integrable is shown to possess only the weak Painleve property. Our analysis, therefore, confirms the limitations of the Painleve test as a test for complete integrability when applied to fully nonlinear PDES.

We extend the BRS and anti-BRS symmetry to the two-point space of Connes' non-commutative model building scheme. The constraint relations are derived and the quantum Lagrangian constructed. We find that the quantum Lagrangian can be written as a functional of the curvature for symmetric gauges with the BRS, anti-BRS auxiliary field finding a geometrical interpretation as the extension of the Higgs scalar.

One-dimensional wave propagation near the marginal state of modulational instability in an infinitely long, straight and homogeneous nonlinear elastic tube filled with an incompressible, inviscid fluid is considered. Using the reductive perturbation method, the amplitude modulation of weakly nonlinear waves is examined. It is shown that the amplitude modulation of these waves near the marginal state is governed by a generalized nonlinear Schrodinger equation (GNLS). Some exact solutions including oscillatory and solitary waves of the GNLS equation are presented.

Symmetries in the Lagrangian formalism of arbitrary order are analysed with the help of the so-called Anderson-Duchamp-Krupka equations. For the case of second-order equations and a scalar field we establish a polynomial structure in the second-order derivatives. This structure can be used to clarify the form of a general symmetry. As an illustration we analyse the case of Lagrangian equations with Poincare invariance or with universal invariance.

We discuss the motion of a quantum-mechanical system constrained to move on an arbitrary submanifold M of its configuration space Rⁿ by a real confining potential. A complete perturbative expansion for the Hamiltonian describing the dynamics of the system is obtained in terms of the quantities characterizing the intrinsic and extrinsic geometric properties of the constraint's surface M. In accordance with Heisenberg's principle the zeroth-order term of the expansion represents strong fluctuations of the system in directions normal to M, whereas the first-order term is naturally interpreted as the Hamiltonian describing the effective dynamics along the constraint's surface. The effective motion on M results in coupling with Abelian/non-Abelian gauge fields and quantum potentials. The rest of the perturbative expansion allows one to take into account further interactions between normal and effective degrees of freedom. As a concrete example we consider the dynamics of an electron constrained on a circle by a hypothetical semiconductor device. Dunham's expansion for the rotovibrational energy of a rigid diatom is also obtained.

We introduce a new variational characterization of Gaussian diffusion processes as minimum uncertainty states. We then define a variational method constrained by kinematics of diffusions and Schrodinger dynamics to seek states of local minimum uncertainty for general non-harmonic potentials.

We investigate the behaviour of resonances as a function of the coupling strength between bound and unbound states on the basis of a simple S-matrix model. Resonance energies and widths are calculated for well isolated, overlapping and strongly overlapping resonance states. The formation of shorter and longer time scales (trapping effect) is traced. We illustrate that the cross section results from an interference of all resonance states in spite of the fact that their lifetimes may be very different.