Table of contents

For large n, graphs of the modulus of the sum exhibit self-similar structure. S_n(x) can be closely approximated by a theta function near its natural boundary. An exact renormalization enables this function to be calculated efficiently, and an approximate arithmetic renormalization explains the self-similarity. Atomic diffraction experiments could enable the self-similarity to be detected in the laboratory.

An isomorphism of the Lie algebras L₁₁ admissible by the full Boltzmann kinetic equation with an arbitrary differential cross section and by the Euler gas dynamics system of equations with a general state equation is set up. The similarity is also proved between extended algebras L₁₂ admissible by the same equations for specified power-like intermolecular potentials and for polytropic gas. This allows the solution of the problem of classification of the full Boltzmann equation invariant H-solutions using an optimal system of subalgebras known for the Euler system. Representations of essentially different H-solutions of the spatially inhomogeneous Boltzmann equation with one and two independent invariant variables in the explicit form are obtained on this basis.

We study Lagrangian trajectories and scalar transport statistics in decaying Burgers turbulence. We choose velocity fields solutions of the inviscid Burgers equation whose probability distributions are specified by Kida's statistics. They are time-correlated, and neither time-reversal invariant nor Gaussian. We discuss in some detail the effect of shocks on trajectories and transport equations. We derive the inviscid limit of these equations using a formalism of operators localized on shocks. We compute the probability distribution functions of the trajectories although they do not define Markov processes. As physically expected, these trajectories are statistically well defined but collapse with probability one at infinite time. We point out that the advected scalars enjoy inverse energy cascades. We also make a few comments on the connection between our computations and persistence problems.

The benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem. The performance of this new method, called extremal optimization (EO), is compared with simulated annealing (SA) in extensive numerical simulations. While generally a complex (NP-hard) problem, the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold. At this threshold, the relative error of SA for large graphs is found to diverge relative to EO at equalized runtime. On the other hand, EO, based on the extremal dynamics of self-organized critical systems, reproduces known results about optimal partitions at this critical point quite well.

We describe the changes and the destruction of islands of stability in four dynamical systems: (a) the standard map, (b) a Hamiltonian with a cubic nonlinearity, (c) a Hamiltonian with a quartic nonlinearity and (d) the Sitnikov problem. As the perturbation increases the size of the island increases and then decreases abruptly. This decrease is due to the joining of an outer and an inner chaotic domain. The island disappears after a direct (supercritical) or an inverse (subcritical) bifurcation of its central periodic orbit C. In the first case, when C becomes unstable, a chaotic domain is formed near C. This domain is initially separated from the outer `chaotic sea' by KAM curves. But as the perturbation increases the inner chaotic domain grows outwards, while the outer `chaotic sea' progresses inwards. The last KAM curve is destroyed by forming a cantorus and the two chaotic domains join. But even then the escape of orbits through the cantorus takes a long time (stickiness effect). In the inverse bifurcation case the island around the central orbit is limited by two equal period unstable orbits. As the perturbation changes these two orbits approach and join the central orbit, that becomes unstable. Then the island disappears but no cantori are formed. In this case the stickiness is due to the delay of deviation of an orbit from the unstable periodic orbit when its eigenvalue is not much larger than 1.

A new algorithm for the derivation of low-density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems. Numerical evidence shows that the computational complexity grows exponentially, but with a growth factor < , which is much smaller than the growth factor = of the previous best algorithm. For bond (site) percolation on the directed square lattice the series has been extended to order 171 (158). Analysis of the series yields sharper estimates of the critical points and exponents.

We consider the dynamics of non-interacting Brownian particles which are driven by correlated (non-independent) noise sources. In simple confining potentials the particles tend to aggregate as the noise correlation is increased. If two particles are subject to the same noise they will coalesce and remain together ever after. We show that complete aggregation of the particles can be expected even in the case of a disordered potential which does not confine the individual particle trajectories. Finally, we examine the case of correlation in the noises which depends on the separation of the two particles.

Zyczkowski, Horodecki, Sanpera and Lewenstein (ZHSL) recently proposed a `natural measure' on the N-dimensional quantum systems, but expressed surprise when it led them to conclude that for N = 2 × 2, disentangled (separable) systems are more probable (0.632±0.002) in nature than entangled ones. We contend, however, that ZHSL's (rejected) intuition has, in fact, a sound theoretical basis, and that the a priori probability of disentangled 2 × 2 systems should more properly be viewed as (considerably) less than 0.5. We arrive at this conclusion in two quite distinct ways, the first based on classical and the second, quantum considerations. Both approaches, however, replace (in whole or part) the ZHSL (product) measure by ones based on the volume elements of monotone metrics, which in the classical case amounts to adopting the Jeffreys' prior of Bayesian theory. Only the quantum-theoretic analysis - which yields the smallest probabilities of disentanglement - uses the minimum number of parameters possible, that is N<sup>2</sup>-1, as opposed to N<sup>2</sup>+N-1 (although this `over-parametrization', as recently indicated by Byrd, should be avoidable). However, despite substantial computation, we are not able to obtain precise estimates of these probabilities and the need for additional (possibly supercomputer) analyses is indicated - particularly so for higher-dimensional quantum systems (such as the 2 × 3 ones, which we also study here).

The problem of a relativistic `free' Dirac particle in a one-dimensional box, i.e., at the box, but not confined to the box, is considered. A four-parameter family of self-adjoint extensions of the momentum operator P = -i1₂(d/dx) is obtained, as well as sub-families of boundary conditions for which this operator transforms as a vector. Physical conditions (self-adjointness and not spontaneously broken CT symmetry in the subspace of positive energies) imposed upon the Hamiltonian operator, which is a function of the momentum operator, give the physical Hamiltonian operator for this problem. The physical self-adjoint extension of H corresponds to the periodic boundary condition.

This paper is a continuation of our previous work (Boos H E and Mangazeev V V 1999 J. Phys. A: Math. Gen.32 3041-54). We obtain two more functional relations for the eigenvalues of the transfer matrices for the sl(3) chiral Potts model at q² = -1. This model, up to a modification of boundary conditions, is equivalent to the three-layer three-dimensional Zamolodchikov model. From these relations we derive the Bethe ansatz equations.

Applying the Pasquier-Gaudin procedure we construct Baxter's Q operator for the homogeneous XXX model as an integral operator in the standard representation of SL(2). The connection between the Q operator and the local Hamiltonians is discussed. We show that Lipatov's duality symmetry operator arises naturally as the leading term of the asymptotic expansion of the Q operator for large values of the spectral parameter.

By replacing the parametrization of a domain with polyhedral approximations we give optimal extensions of theorems of Gauss, Green and Stokes'. Permitted domains of integration range from smooth submanifolds to structures that may not be locally Euclidean and have no tangent vectors defined anywhere. One may still calculate divergence and curl over a domain, and flux across its boundary which itself may have no normal vectors defined anywhere.

The covariant single-time equations of the quantum field theory are formulated in the relativistic configurational representation. The explicit formulae for the Green functions corresponding to the scattering states are calculated in this representation. Using the derived nonhomogeneous equations the scattering problem is solved exactly for certain potentials (combinations of zero-range potentials). The equations and their solutions are studied in the non-relativistic limit. The conditions of total reflection, available for such potentials, are investigated.

The properties of the set of extended Jordanian twists for algebra sl(3) are studied. Starting from the simplest algebraic construction - the peripheric Hopf algebra - we explicitly construct the complete family of extended twisted algebras {(sl(3))} corresponding to the set of four-dimensional Frobenius subalgebras {L()} in sl(3). It is proved that the extended twisted algebras with different values of the parameter are connected by a special kind of Reshetikhin twist. We study the relations between the family {(sl(3))} and the one-dimensional set {(sl(3))} produced by the standard Reshetikhin twist from the Drinfeld-Jimbo quantization (sl(3)). These sets of deformations are in one-to-one correspondence: each element of {(sl(3))} can be obtained by a limiting procedure from the unique point in the set {(sl(3))}.

We study the problem of the existence of a local quantum scalar field theory in a general affine metric space that in the semiclassical approximation would lead to the autoparallel motion of wavepackets, thus providing a deviation of the spinless particle trajectory from the geodesics in the presence of torsion. The problem is shown to be equivalent to the inverse problem of the calculus of variations for the autoparallel motion with the additional conditions that the action (if it exists) has to be invariant under time reparametrizations and general coordinate transformations, while depending analytically on the torsion tensor. The problem is proved to have no solution for a generic torsion in four-dimensional spacetime. A solution exists only if the contracted torsion tensor is a gradient of a scalar field, while the traceless part is zero. The corresponding field theory describes coupling of matter to the dilaton field.

The dynamics of wavepackets in a relativistic Dirac oscillator (DO) is compared with that of the Jaynes-Cummings model. The strong spin-orbit coupling of the DO produces the entanglement of the spin with the orbital motion similar to that observed in the model of quantum optics. The resulting collapses and revivals of the spin extend to a relativistic theory our previous findings on a nonrelativistic oscillator where they were known as spin-orbit pendulum. The Foldy-Wouthuysen transformation can be performed exactly for the DO. It produces the well known smoothing effect over the Compton wavelength. Thus, after this transformation, zitterbewegung disappears just as the components of the WP associated to negative energy states.

An integrable Kondo problem in the one-dimensional supersymmetric extended Hubbard model is studied by means of the boundary graded quantum inverse scattering method. The boundary K-matrices depending on the local moments of the impurities are presented as a non-trivial realization of the graded reflection equation algebras in a two-dimensional impurity Hilbert space. Further, the model is solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.