We present an exact analytic computation of the Lyapunov dimension of the universal attractor of the complex Ginzburg-Landau partial differential equation for a finite range of its parameter values. We obtain upper bounds on the attractor's dimension when the parameters do not permit an exact evaluation by our methods. The exact Lyapunov dimension agrees with an estimate of the number of degrees of freedom based on a simple linear stability analysis and mode-counting argument.

The Davey-Stewartson I equation is a nonlinear evolution equation originally derived in the context of multidimensional, weakly nonlinear water waves. It has recently been exactly solved by the classical inverse-scattering method for localized potentials, and also possesses nonlocal soliton solutions. We have calculated Poisson-bracket relations for elements of the scattering matrix, as well as corresponding quantum commutation relations. Commutation relations are found that are a (2+1)-dimensional generalization of a Yang-Baxter algebra.

Reversible cellular automata are used to investigate the thermodynamic behavior of large systems. Additive conserved quantities are regarded as the energy of these models. By the consideration of a large system as the sum of a subsystem and a heat bath, it is numerically shown that a canonical distribution is realized under certain conditions concerning the conserved quantities.

We describe a mechanism for generic $\frac{1}{f}$ noise in nonintegrable Hamiltonian systems. The phenomenon is observed in the velocity fluctuations of a particle in a 2D periodic potential and is associated with anomalously enhanced deterministic diffusion. It is explained in terms of a renewal process and trapping in a hierarchy of nested cantori.

We generalize to a local relativistic quantum field theory a proposal of Ignatiev and Kuzmin for a single oscillator which has small violation of the Pauli principle and thus provide a theoretical framework which, for the first time, allows quantitative tests of the Pauli principle. Our theory provides a continuous interpolation between fully hindered parafermi statistics of order 2 ($\beta =0\mathrm{}$), which is equivalent to Fermi statistics, and ordinary parafermi statistics of order 2 ($\beta =1\mathrm{}$). We suggest two types of experiments which can place bounds on $\beta$.

An explicit parametrization is obtained of the set of all space-time solutions of Einstein's equations which are globally hyperbolic and contain a compact spatial hypersurface with constant mean curvature. This parametrization is based upon the conformal treatment of the initial-value problem for Einstein's equations, which is studied by the method of sub and super solutions for quasilinear elliptic partial differential equations. The Yamabe-Aubin-Trudinger-Schoen classification of conformal classes of Riemannian metrics plays a key role.

The local complex action is found for a closed, interacting lattice string whose configurations generate the low-temperature graphs of the three-dimensional Ising model. This action depends upon Ising degrees of freedom on the surface, as well as its extrinsic curvature.

A linearization scheme is proposed for the general Hamiltonian of an interacting fermion system, consisting of a mean-field approximation in which pairing and umklapp play dominant roles. A variety of models emerge, characterized by a hierarchy of spectrum-generating algebras and superalgebras, of which some are supersymmetric. The superconducting phase transition is shown to be related to supersymmetry breaking.

The problem of finding the best path in a random medium is investigated. If the random medium is allowed to undergo slow drifts, the best path can be drastically different. The scaling of the excited states is also discussed. A host of new exponents are found for the 2d problem. The implications for the growing surfaces are also pointed out.

We develop a singular perturbation approach to the problem of first-passage times for non-Markovian processes driven by Gaussian colored noise near the white-noise limit. In particular we treat the problem of overdamped tunneling in a bistable quartic potential in the presence of Gaussian noise of short correlation time τ. The correct treatment of the absorbing boundary yields a lowest-order correction proportional to ${\mathrm{\tau }}^{1/2}$ with a proportionality constant involving the Milne extrapolation length for the Fokker-Planck equation, given in terms of the Riemann ζ function.

Light propagation in a discrete waveguide is the classical-optics analog of the quantum dynamics of a kicked oscillator. A specific model is introduced which shows subdiffusive beam spreading in the ray optical limit. At finite wavelength, the beam is localized in both position and momentum space through wave interference effects. It is argued that the localization phenomenon is accessible to a laboratory experiment using conventional laser systems.

We present general-relativistic solutions of self-similar spherical collapse of an adiabatic perfect fluid. We show that if the equation of state is soft enough (Γ-1≪1), a naked singularity forms. The singularity resembles the shell-focusing naked singularities that arise in dust collapse. This solution increases significantly the range of matter fields that should be ruled out in order that the cosmic-censorship hypothesis will hold.

A Monte Carlo simulation of two-dimensional vesicles, modeled as closed, planar, self-avoiding tethered chains is presented. The effects of an internal pressure increment, Δp, and a bending rigidity, κ, on the sizes and shapes is studied. For κ=0 scaling behavior is found, which, in the asymptotic deflated regime, Δp<0, matches the behavior of branched polymers. For κ>0 various nontrivial equilibrium shapes or cytotypes appear and dynamical behavior reminiscent of nonlinear flickering in real vesicles is found.

The effects of transverse electric fields on the conductance fluctuations in an Sb loop have been studied. We show that the electric field can be used to tune the position (or phase) of Aharonov-Bohm oscillations as well as to alter the aperiodic conductance fluctuation patterns. We disucss two mechanisms which might cause the observed dependence of the fluctuation pattern on transverse electric field. The first is the electrostatic Aharonov-Bohm effect, and the second is the spatial shifting of the electron trajectories by the electric field.

Replica-symmetric solutions of the graph-bipartitioning problem with finite connectivity are presented. With the constraint ${sum}_{\mathrm{i}=1\mathrm{}}^{\mathrm{N}}$${\mathrm{S}}_{\mathrm{i}}$=0 strictly enforced, another solution can be obtained, which gives a lower cost function than that given by the spin-glass solution. It is believed that this new solution is currently the best candidate for the true solution of the graph-bipartitioning problem.

We show in this Letter exact results for the Ising model on the two-dimensional Kagomé lattice with nearest- and next-nearest-neighbor interactions ${\mathrm{J}}_1$ and ${\mathrm{J}}_2$. In some regions of phase space, we find a nonzero critical temperature despite a finite zero-point entropy. For a narrow range of ${\mathrm{J}}_2$/${\mathrm{J}}_1$ we find successive transitions with a reentrance at low temperature. We studied the nature of ordering by Monte Carlo method and found that in these regions one sublattice remains disordered below the transition and down to zero temperature except in the reentrant region. Thus disorder can coexist with order at equilibrium.

We present experimental evidence for deviations from universality in the transition to chaos from quasiperiodicity in a nonlinear dynamical system. The dimension of the quasiperiodic set at the transition is 0.795±0.005, not 0.87 as predicted by the standard model based on the circle map. We also deviate from predictions of the model at the golden mean: The power spectrum, tongue convergence rate, and spectrum of critical exponents, f(α), all differ from the theory.

Leggett and Garg have argued that the phenomenon of quantum flux tunneling oscillations will, according to the predictions of quantum mechanics, contradict the pair of assumptions ‘‘macroscopic realism’’ and ‘‘noninvasive measurability.’’ It is argued here that there can be no contradiction of realism in such a case, and that the contradiction is between quantum mechanics and noninvasive measurability.

The interrelation between random matrices and physical matrix Hamiltonians is established by introduction of a construction scheme whereby a single matrix is projected onto a subensemble of smaller matrices. Exact analytical results are obtained for random matrices and compared with numerical calculations on a model Hamiltonian.

A recent study by Zaslavskii et al. demonstrates that under certain conditions a charged particle may be accelerated by an arbitrarily small electrostatic wave packet in a magnetic field to arbitrary energy by the process of Arnol’d diffusion in 1(1/2 dimensions. A relativistic calculation shows, however, that the particles can only be accelerated up to a critical energy by a wave packet above a critical amplitude.