Table of contents

A recent mapping of the interface roughening problem to directed polymers by Lässig and Kinzelbech has shown that the upper critical dimension of the Kardar - Parisi - Zhang equation is less than or equal to four. By combining the mode-coupling technique with a small- (roughening exponent) expansion, we show that the upper critical dimension is four. The validity of this conclusion is obviously limited by the applicability of the mode-coupling technique to the strong-coupling regime.

The learning dynamics of an on-line algorithm for principal component analysis is described exactly in the thermodynamic limit by means of coupled ordinary differential equations for a set of order parameters. It is demonstrated that learning is delayed significantly because existing symmetries among student vectors have to be broken. A closely related effect is the perfect or partial loss of initial knowledge in the course of learning. The analysis shows that different learning rates for the student vectors improve the performance of the algorithm drastically.

Using methods of conformal field theory, we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k, in the limit when k is large.

Using a histogram Monte Carlo simulation method, we calculate the existence probability for bond percolation on simple cubic (sc) and body-centred cubic (bcc) lattices, and site percolation on sc lattices with free boundary conditions. The spanning rule considered by Reynolds, Stanley, and Klein is used to define percolating clusters. We find that for such systems has very good finite-size scaling behaviour and the value of at the critical point is universal and is about .

We study the long-range effective drift and diffusivity of a particle in a random medium moving subject to a given molecular diffusivity and a local drift. The local drift models the effect of a random electrostatic field on a neutral but polarizable molecule. Although the electrostatic field is assumed to obey Gaussian statistics the induced statistics of the drift velocity field are non-Gaussian.

We show that a four-loop perturbation theory calculation of the effective diffusivity is in rather good agreement with the outcome of a numerical simulation for a reasonable range of the disorder parameter. We also measure the effective drift in our simulation and confirm the validity of the `Einstein relation' that expresses the equality of the renormalization factors, induced by the random medium, for the effective drift and effective diffusivity, relative to their molecular values. The Einstein relation has previously only been confirmed for Gaussian random drift fields. The simulation result, for our non-Gaussian drift model, is consistent with a previous theoretical analysis showing the Einstein relation should remain true, independently of the precise character of the statistics of the drift velocity field.

We study numerically the dynamic and spectral properties of a one-dimensional quasi-periodic system, where site energies are given by with denoting the kth quasiperiodic lattice site. When is given by the reciprocal lattice vector with n and m being successive Fibonacci numbers, the variance of the wavepacket is found to grow quadratically in time, regardless of the potential strength V. For other values of f, there exists a critical value of V beyond which the growth of the wavepacket variance is bounded. In particular an anomalous diffusion takes place for corresponding to with generic integers m and n. The level-spacing distribution is also examined, and the corresponding exponent is observed to decrease with V.

We study the problem of two crosslinked polymer chains in a good solvent, modelled by two mutually crossing self-avoiding walks situated on fractals that belong to the Sierpinski gasket (SG) family (whose members are labelled by an integer b, ). By applying the Monte Carlo renormalization group (MCRG) method, we calculate the critical exponent y associated with the number of crossings of the two self-avoiding-walk paths, for a sequence of SG fractals with . For the problem under study, we find that our MCRG approach provides results that are virtually rigorous, that is, results with exceptionally small deviations (at most 0.07%) from the available exact renormalization group results. We discuss our set of MCRG data for y as a function of the fractal parameter b, and compare its behaviour with the finite-size scaling predictions.

We determine the central extensions of a whole family of Lie algebras, obtained by the method of graded contractions from , N arbitrary. All the inhomogeneous orthogonal and pseudo-orthogonal algebras are members of this family, as well as a large number of other non-semisimple algebras, all of which have at least a semidirect structure (in some cases two or more). The dimensions of their second cohomology groups and the explicit expression of their central extensions are given.

Evolution equations like the heat or diffusion equation or the Schrödinger equation can be associated with stochastic processes. In this paper we study the relation between equations of the form and Lévy processes (i.e. quantum stochastic processes with independent and stationary increments) on quantum groups and braided groups. Solutions of such equations are calculated as Appell systems. Wigner distributions of these processes are defined and it is proven that they satisfy a Fokker-Planck equation.

In this paper, a differential-difference KdV equation is considered. We present two Bäcklund transformations and corresponding nonlinear superposition formulae for it. As an application of the obtained results, N-soliton solutions first obtained by Ohta and Hirota are rederived. Furthermore, a sequence of rational solutions are also obtained. Thus the integrability of the differential-difference KdV equation is further confirmed.

We study geodesic flows on Lie groups with the left-invariant non-holonomic constraint. In the case of the existence of an invariant measure, we find new integrable non-Hamiltonian systems on SO(4) and other six-dimensional Lie groups.

A phase-space approach to finite-dimensional systems is developed from basic principles. For a system describable by a Hilbert space of dimension d we define a one-to-one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d. The properties fulfilled by this correspondence and its uniqueness are examined. This formalism is applied to the number difference and phase difference of a two-mode field. This case is compared with the marginal distribution for these variables arising from a two-mode Wigner function for number and phase.

A rather general form of the conventional cut and project scheme is used to define quasicrystals as point sets in real n-dimensional Euclidean space. The inflation or, equivalently, the self-similarity properties of such quasicrystals are studied here assuming only the convexity of the acceptance window. Our result is a description of inflation centres of all types in a quasicrystal and a proof that our description is complete: there are no other inflation centres. For any chosen quasicrystal point (`internal inflation centre') u, its inflation properties are given as a set of scaling factors. It turns out that the scaling factors form a one-dimensional quasicrystal with a u-dependent acceptance window (`scaling window'). The intersection of the scaling windows associated with all points of a quasicrystal is the one-dimensional quasicrystal of universal (`internal') scaling symmetries. Its acceptance window is the interval . External inflation centres of a cut and project quasicrystal are those which are not among quasicrystal points. Their complete description is given analogically to the description of the internal ones imposing some additional requirements on the scaling factors. Between any two adjacent quasicrystal points one finds a countable infinity of external inflation centres. The scaling factors belonging to any such centre u form an infinite u-dependent subset of points of the quasicrystal with acceptance window containing .

It has recently been established that a product bundle, composed of two gauge structures, under some circumstances, possesses a geometry which does not split. Here we provide an educated extension of the above idea to products of many vector bundles with a distinct group structure associated with each factor fibrespace in the splice. The model employs connection 1-forms with values in a space product of Lie algebras, and therefore interlaces the various gauge structures in a non-trivial manner. Special attention is given to the structure of the geometric ghost sectors and the super-algebra they possess.

The level-1 irreducible highest-weight modules of the quantum affine algebra are decomposed into irreducible components with respect to the level-0 -action previously defined in Saito et al 1998. The components of the decomposition are found to be the so-called tame representations of parametrized by the skew Young diagrams of the border-strip type. This result verifies a recent conjecture due to Kirillov et al.

The connection between the singular manifold method (Painlevé expansions truncated at the constant term) and symmetry reductions of two members of a family of Cahn-Hilliard equations is considered. The conjecture that similarity information for a nonlinear partial differential equation may always be fully recovered from the singular manifold method is violated for these equations, and is thus shown to be invalid in general. Given that several earlier examples demonstrate the connection between the two techniques in some cases, it now becomes necessary to establish when such a relationship exists - a question related to a deeper understanding of Painlevé analysis. This issue is also briefly discussed.

Representation theory for the Jordanian quantum algebra is developed. Closed form expressions are given for the action of the generators of on the basis vectors of finite-dimensional irreducible representations. It is shown how representation theory of has a close connection to the combinatorial identities involving summation formulae. A general formula is obtained for the Clebsch-Gordan coefficients of . These Clebsch-Gordan coefficients are shown to coincide with those of su(2) for , but for they are, in general, a non-zero monomial in .