Presumably the simplest model of chemical turbulence is proposed, for which the spatial degree of freedom plays an important role. The concentration, as well as space and time, is discretized. As in the real experiment a turbulent phase is found sandwiched by two ordered phases. In addition, the model predicts a solitonlike phase. Thus the model can exhibit a variety of behaviors encountered in partial differential equation systems. It is also closely related to cellular automata.

The evolution of a distribution function according to the linearized relativistic Boltzmann equation is shown to be causal for a wide class of scattering cross sections.

A particle in a chaotic region of phase space can spend a long time near the boundary of a regular region since transport there is slow. This "stickiness" of regular regions is thought to be responsible for previous observations in numerical experiments of a long-time algebraic decay of the particle survivial probability, i.e., survival probability $\sim t^{-z}$ for large $t$. This paper presents a global model for transport in such systems and demonstrates the essential role of the infinite hierarchy of small islands interspersed in the chaotic region. Results for $z$ are discussed.

An operational procedure is given for determining experimentally, in principle at least, the Wigner function for an ensemble of particles. This manner of "measuring" a quantum state, whether pure or mixed, via its Wigner function, seems the simplest possible, and closely parallels the method one might use in classical mechanics to determine a (true) phase-space probability density.

Classical transport coefficients in a $d$-dimensional medium with a potential $V\left(\mathrm{x}\right)$ and/or conductivity $a\left(\mathrm{x}\right)$ are found to vary discontinuously as functions of the "wavelengths" of the inhomogeneities. For example, with a potential depending on one direction only, say $V\left(\mathrm{x}\right)=cos2\pi x_1+cos2\pi k{x}_1$, the effective diffusion coefficient $D\left(k\right)\mathrm{}$ has the same value $D^{*}$ for all irrational $k$, but differs from $D^{*}$ and depends on $k$ for $k$ rational. Thus $D\left(k\right)\mathrm{}$ is discontinuous at rational $k$. Moreover, $D\left(k\right)\mathrm{}$ is continuous at irrational $k$. This pathology is reflected in the time scales on which the diffusion approaches its limiting behavior.

By means of the quantum-potential interpretation we show that there is no need for a break or "cut" in the way we regard reality between quantum and classical levels.

We study a class of local probabilistic growth processes that includes the kinetic-growth algorithm for generating percolation clusters. The shapes of the growing clusters are controlled by $p$, the probability of growth. For $p>\mathrm{}{p}_c$, the shapes are scale invariant with time and show interesting morphological features including both smoothly curved sections and straight facets. The facets are shown to be related to the problem of directed percolation and disappear below the directed-percolation threshold. A simple random-walk model for computing the shapes of our clusters is described.

It is shown that a straightforward version of a classical Kaluza-Klein theory predicts the existence of classical vacuum polarization and that if the $g_{55}$ field carries a mass than the theory admits centrally symmetric solutions for which the Schwarzschild singularity becomes a naked point singularity with no event horizon and for which the electric field is everywhere finite.

The traditional version of the JWKB connection formula at a linear turning point is incorrect. The correct version follows from the Borel-based summability of the Airy-function asymptotic expansions. A key point is that the classically forbidden region lies on a Stokes line. The real exponentially growing solution has an explicitly complex JWKB expansion, while the explicitly real exponentially growing JWKB expansion represents a complex solution. Inconsistencies in applications of the traditional formula, such as in transmission through a barrier, are eliminated.

The necessary and sufficient conditions under which fully probabilistic cellular-automata (PCA) rules possess an underlying Hamiltonian (i.e., are "reversible") are established. It is argued that, even for irreversible rules, continuous ferromagnetic transitions in PCA with "up-down" symmetry belong in the universality class of kinetic Ising models. The nonstationary (e.g., periodic) states achieved for asymptotically large times by certain PCA rules in the (mean field) limit of infinite dimension are argued to persist in two and three dimensions, where fluctuations are strong.

A new Monte Carlo method for quantum spins is described. The spin Hamiltonian is mapped onto a model of bosons, which may then be studied by a path-integral formalism. The power of this mapping comes from the fact that, by letting the system evolve through unphysical spin states between time slices, we obtain simple expressions for the needed matrix elements. The method is illustrated by measuring the energy and correlation functions for a spin-½ ring as well as the specific heat for the two-dimensional $\mathrm{XY}$ model.

A general expression for the response function is derived by the method of recurrence relations. Memory effects appear as corrections to the dynamic random-phase-approximation form. The dynamic structure for the three-dimensional electron gas is calculated to third order and compared with Al data at a large wave vector. Also shown is the dynamic local-field term.

Ionization of hydrogen atoms with principal quantum number $n=32\mathrm{}$, 40, and 51-74 by a 9.92-GHz electric field $\mathrm{F}\mathrm{}\left(t\right)=\hat{z}{F}_{0}cos\omega t$ was studied with a superimposed static electric field ${\overline{F}}_s=0, 2, 5, \mathrm{and} 8$ V/cm. The measured field strengths $F_0\mathrm{}(10\%)\mathrm{}$ at which 10% of the atoms were ionized are in excellent agreement with classical calculations in both one and two spatial dimensions. Covering finer detail as well as gross structure of the $n$ dependence of $F_0\mathrm{}(10\%)\mathrm{}$, the agreement supports the application of classical dynamics to the analysis of this strongly perturbed quantum system.

An interface attracted by weaker bonds to a wall can be depinned by quenched impurities in the bulk as well as by thermal fluctuations. Exact calculations in two dimensions indicate that the depinning transition with randomness is characterized by a discontinuous specific heat and a quadratic divergence of the localization length. Numerical simulations indicate delocalization by randomness in cases where there is no analogous depinning by thermal fluctuations, e.g., as in an interface bound to a strip of weak bonds in the bulk, or for two attracting interfaces.

In contrast to previous treatments, a new analysis of two-dimensional many-electron systems subject to periodic boundary conditions in a magnetic field leads to a fully two-dimensional structure of the quantum numbers at rational Landau-level filling. The structure of the new symmetry analysis has an intrinsically many-particle character. Full agreement between numerical studies of quantized-Hall-effect systems in periodic and spherical geometries is achieved, and the problem of ground-state degeneracy is clarified.

The structure of the group corresponding to a Kac-Moody algebra is analyzed. The two-cocycle for the group is constructed in terms of local coordinates and the transition functions between different coordinate systems are computed. The results are extended to the group $\mathrm{Map}(S^3, \mathfrak{G})$ which arises in a (3+1)-dimensional Yang-Mills theory.

Certain first-order nonlinear ordinary differential equations exemplified by strongly damped, quasiperiodically driven pendula and Josephson junctions are isomorphic to Schrödinger equations with quasiperiodic potentials. The implications of this equivalence are discussed. In particular, it is shown that the transition to Anderson localization in the Schrödinger problem corresponds to the occurrence of a novel type of strange attractor in the pendulum problem. This transition should be experimentally observable in the frequency spectrum of the pendulum or Josephson junction.

It is shown that the one-loop infinities cancel between diagrams for $M=5\mathrm{}$ open external lines, and any $M$ for suitably constrained kinematics, in the O(32) superstring theory when one employs a principal-part prescription for regularization.

It is shown that the symmetry algebra of the Kadomtsev-Petviashvili equation can be related to an infinite-dimensional subalgebra of the loop algebra [$\mathrm{SL}(5, R)\otimes R(t, t^{-1\mathrm{}})\oplus \left[R\right(t, t^{-1\mathrm{}}\left)\frac{d}{\mathrm{dt}}\right]$. The algebra is used to generate new classes of solutions of the Kadomtsev-Petviashvili equation, depending on several arbitrary functions.

A simple model of restructuring for loopless clusters is presented in which single-bonded particles may escape from the cluster and diffuse in space until they reach the cluster again. In its steady-state regime, this model defines a new statistics of clusters at equilibrium with a fractal dimension $D=1.54\mathrm{}\pm 0.08$ in two dimensions estimated by direct simulations. Comparisons are made with out-of-equilibrium diffusion-limited aggregation and with at-equilibrium statistics of animals.