Table of contents

It is shown that multifractal Bernoulli fluctuations appear at morphological phase transition from monofractality to multifractality. This type of fluctuation is studied in detail and it is shown that the multifractal fluctuations of wavefunctions at disorder-induced localization - delocalization transitions can be identified as multifractal Bernoulli fluctuations (both in two- and three-dimensional cases).

We study the geometrical properties of the domain structure that results when a random Ising ferromagnet is quenched to zero temperature. We define a novel connectivity transition between like spins in the frozen configuration, and show that it is in a new universality class, distinct from that of random percolation. The scaling is robust and independent of model details. The relevance of this transition to spinodal decomposition is discussed.

The properties of a trapped ideal Bose gas in n-dimensional space are studied. General analytic expressions of the critical temperature of Bose-Einstein condensation, the jump of heat capacity at and the fraction of condensation at temperatures below have been derived. How these physical quantities depend on the external potential, particle characteristics and space dimensionality are discussed. We find that when a proper external potential is taken, Bose-Einstein condensation may occur in any dimensional space.

The adsorption transition in the phase diagram of a self-interacting lattice polygon is examined. The polygon has a nearest-neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall. The partition function of this model is , where is the number of polygons with c nearest-neighbour contacts, v visits to the wall, and n edges (and counted up to translations parallel to the wall). The limiting free energy of this model is , and it is known to be a non-analytic function of y for each . The non-analyticity is at , and this corresponds to an adsorption transition of the polygon on the wall. In this paper it is proved that for all .

In the variational approach to quantum statistics, a smearing formula efficiently describes the consequences of quantum fluctuations upon an interaction potential. The result is an effective classical potential from which the partition function can be obtained by a simple integral. In this work, the smearing formula is extended to higher orders in the variational perturbation theory. An application to the singular Coulomb potential exhibits the same fast convergence with increasing orders that has been observed in previous variational perturbation expansions of the anharmonic oscillator with quartic potential.

The method is to use a computer to find a function for the process that always has negative expectation by considering all relevant configurations of 0's and 1's at the boundary of a finite process in one dimension. It is shown that a branching annihilating random walk will die out if the diffusion parameter is greater than 0.176. The method may also be applied to attractive processes, and is used for the contact process in one dimension obtaining the same values as Ziezold and Grillenberger (1988) for up to 10 places in from the boundaries.

The open nonlinear Schrödinger model with spin degrees of freedom is considered. We find two types of integrable open boundary conditions. We study the Bethe ansatz states of the model at the infinitely strong coupling limit. For some boundary conditions, the spins of the Bethe ansatz wavefunctions are all aligned up or down. For the other type of boundary conditions all the spin configurations are degenerate.

We consider the motions on the unit sphere , in a potential , which have been shown earlier to derive from a completely integrable Hamiltonian. The separation of variables is completed in terms of an independent variable u which differs from the canonical time t: where m and are the two integrals of motion, S is the action, and is the integration constant (constant along each trajectory). Finally a Lax pair is deduced from a separable form of the system (Lax P D 1968 Commun. Pure Appl. Math. 21 467).

A model for the diffusion of vector fields driven by external forces is proposed. Using the renormalization group and the -expansion, the dynamical critical properties of the model with Gaussian noise for dimensions below the critical dimension are investigated, and new transport universality classes are obtained.

We consider matrices transforming between the standard Young-Yamanouchi basis of the symmetric group and bases adapted to the product subgroups (the split basis). We derive closed formulae for transformation coefficients for b = 3, which includes the first cases when a choice of multiplicity separation is required. We discuss considerations which can be applied to obtain a simple form for the multiplicity separation. We show that the combinatorial and algebraic structure of the Littlewood-Richardson rule, also known as the Biedenharn-Louck pattern calculus, does not assist with finding a simple multiplicity separation.

Using computer experiments on a simple three-state system and an NP-complete system of permanents we compare different proposed simulated annealing schedules in order to find the cooling strategy which has the least total entropy production during the annealing process for given initial and final states and fixed number of iterations. The schedules considered are constant thermodynamic speed, exponential, logarithmic, and linear cooling schedules. The constant thermodynamic speed schedule is shown to be the best. We are actually considering two different schedules with constant thermodynamic speed, the original one valid for near-equilibrium processes, and a version based on the natural timescale valid also at higher speeds. The latter one delivers better results, especially in case of fast cooling or when the system is far from equilibrium. Also with the lowest energy encountered during the entire optimization (the best-so-far-energy) as the indicator of merit, constant thermodynamic speed is superior. Finally, we have compared two different estimators of the relaxation time. One estimator is using the second largest eigenvalue of the thermalized form of the transition probability matrix and the other is using a simpler approximation for small deviations from equilibrium. These two different expressions only agree at high temperatures.

All bi-algebra structures for centrally extended Galilei algebra are classified. The corresponding Lie-Poisson structures on centrally extended Galilei group are found.

Solitary wave solutions are determined analytically for two forms of nonlinear Schrödinger equations with saturation effects in the denominators of their nonlinearity terms. Pertinent parameters, being explicitly stated, make the results applicable to soliton phenomena.

Equation (2.6 c) was incorrectly printed. The correct version is

The following references were incorrect. The correct versions are

[3] Yuen H P and Lax M 1973 IEEE Trans. IT 19 740

[4] Holevo A S 1979 Rep. Math. Phys. 16 385