We discuss the limitations of some low-frequency approximations commonly used in the theory of laser-assisted electron-atom collisions resulting in the excitation of the target.

It is shown in this Comment that the claim of Svozil that his paper [Phys. Rev. A 39, 2222 (1989)] demonstrates that nonvanishing spacelike and timelike contributions to the Feynman propagator do not lead to a violation of microscopic causality as claimed in the work of Hegerfeldt [Phys. Rev. D 10, 3320 (1974); Phys Rev. Lett. 54, 2395 (1985)] and Rubin [Phys. Rev. D 35, 3836 (1987)] is incorrect, and that his calculation is irrelevant to the issues discussed by these authors.

A clarification concerning the solution of the initial-value problem for the propagation of light pulses in dispersive media is given. In a proper interpretation, this solution corresponds to two single pulses traveling in opposite directions. Furthermore, a critical review of important research related to Rubin’s arguments but not mentioned by him in the preceding Comment [Phys. Rev. A 41, 1727 (1990)] is presented.

It is pointed out that the valence-electron Schrödinger equation used by Kostelecký, Nieto, and Truax [Phys. Rev. A 38, 4413 (1988)] had already been presented and studied by Simons [J. Chem. Phys. 55, 756 (1971)], and it is shown that expectation values calculated using its eigenfunctions are unreliable.

The preceding Comment by Goodfriend [Phys. Rev. A 41, 1730 (1990)] contains three criticisms of our model for analytical quantum-defect wave functions vis-á-vis the atomic Fues potential of Simons [J. Phys. Chem. 55, 756 (1971)]. We rebut the first two criticisms explicitly. This makes the third criticism moot. We stand by our results.

A systematic construction of exactly solvable models of interactions between the measuring apparatus and the particle whose position is to be measured is shown. The two types of root-mean-square errors are calculated for these models and many new models with better statistics than the standard von Neumann model are continuously found. The best among them is the model which has been shown to circumvent the standard quantum limit for the position monitoring. A feature of the models similar to Bondurant’s improvement of the interferometric position measurement is also pointed out.

The decay of the ${\mathrm{He}}^{+}$(3p) state has been observed downstream from a thin amorphous carbon foil bombarded by a helium beam of velocity ranging from 2.6 to 4.7 (in units of the Bohr velocity $v_0$). The data analysis has been performed in the framework of a model that allows one to determine whether the bound state of an emergent ion may have been formed inside the solid target or only at emergence. We show that the ${\mathrm{He}}^{+}$(3p) state can be bound inside solid carbon if the ion velocity is high enough and we have observed the velocity threshold.

We have observed a bound-free photodissociation of ${\mathrm{H}}_3^{+}$ caused by a single-photon absorption from highly excited rovibrational levels of the ${\mathrm{H}}_3^{+}$ ground state into the continuum of the first-excited singlet electronic state. The apparent threshold energy of the dissociation has been measured to be ∼2.5 eV, in agreement with the calculated one by Talbi and Saxon [D. Talbi and R. Saxon, J. Chem. Phys. 89, 2235 (1988)].

We present the results of a calculation of high-order harmonic generation by a classical hydrogen atom, submitted to an intense, single-mode, radiation field. The spectra showing these harmonics associated with different kinds of trajectories as well as with an ensemble average of trajectories are discussed.

We present a scheme of interpreting the generalized ε expansion encountered in the renormalization-group analysis of D-dimensional self-avoiding tethered manifolds embedded in d-dimensional external space. Owing to the nonlinear nature of the parameter ε, O(ε) results of previous studies of the ε expansion are somewhat ambiguous. Our scheme resolves ambiguities and gives optimal numerical results to every order in ε. The O(ε) optimal values of the radius-of-gyration exponent ν are calculated for polymers in two and three dimensions, and for membranes in various dimensions. The optimized polymer exponents are better than the O(ε) results obtained from the traditional ε expansion. The exponent for two-dimensional membranes embedded in high-dimensional space, i.e., for D=2 and d→∞, is 4/d in agreement with the Flory exponent. We further explore the ε-expansion properties in the limit d,D→0 and find multiple-body excluded-volume interactions to be relevant. We show that these interactions cannot be ignored in the physically relevant case of two-dimensional membranes in three-dimensions.

The adiabatic switching technique is adapted to the calculation of expectation values for chaotic systems. Semiclassical results obtained in this manner are compared to accurate quantum expectation values for the Hénon-Heiles system at high energy. Although good agreement is found for most states, the incidence of discrepancies grows as the energy increases. Almost all such discrepancies can be attributed to either quantum trapping effects for states with extreme values of quantum numbers or tunneling effects associated with the avoided crossings of energy curves as a function of a parameter in the Hamiltonian. As the energy is raised and the system becomes more chaotic, the increased strength of the avoided crossings leads to more frequent and stronger tunneling. The possible generality of this result and its implications for statistical quantum behavior are discussed.

We consider the problem of prediction and system identification for time series having broadband power spectra that arise from the intrinsic nonlinear dynamics of the system. We view the motion of the system in a reconstructed phase space that captures the attractor (usually strange) on which the system evolves and give a procedure for constructing parametrized maps that evolve points in the phase space into the future. The predictor of future points in the phase space is a combination of operation on past points by the map and its iterates. Thus the map is regarded as a dynamical system and not just a fit to the data. The invariants of the dynamical system, the Lyapunov exponents and optimum moments of the invariant density on the attractor, are used as constraints on the choice of mapping parameters. The parameter values are chosen through a constrained least-squares optimization procedure, constrained by the values of these invariants. We give a detailed discussion of methods to extract the Lyapunov exponents and optimum moments from data and show how to equate them to the values for the parametric map in the constrained optimization. We also discuss the motivation and methods we utilize for choosing the form of our parametric maps. Their form has a strong similarity to the work in statistics on kernel density estimation, but the goals and techniques differ in detail. Our methodology is applied to "data" from the Hénon map and the Lorenz system of differential equations and shown to be feasible. We find that the parameter values that minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system. The maps that do reproduce the values of the invariants are not optimum in the least-squares sense, yet still are excellent predictors. We discuss several technical and general problems associated with prediction and system identification on strange attractors. In particular, we consider the matter of the evolution of points that are off the attractor (where few or no data are available), onto the attractor where long-term motion takes place. We find that we are able to realize maps that give a least-squares approximation to the data with rms variation over the attractor of 0.5% or less and still reproduce the dynamical invariants to 5% or better. The dynamical invariants are the classifiers of the dynamical system producing the broadband time series in the first place, so this quality of the maps is essential in representing the correct dynamics.

Using the scanning simulation method we study self-attracting trails (with energy ɛ per intersection) terminally attached to an impenetrable linear boundary on a square lattice, at their tricritical collapse transition. We obtain (with 95% confidence limits) that the partition function exponents are ${\gamma }_{1t}$=0.634±0.025 (for trails with one end attached to the boundary) and ${\gamma }_{11t}$=-0.44±0.02 (both ends are attached). These results differ significantly from the exact values (8/7≃1.143 and (4/7≃0.571, respectively, derived by Duplantier and Saleur (DS) [Phys. Rev. Lett. 59, 539 (1987)] for self-avoiding walks (SAW’s) on a dilute hexagonal lattice at the FTHETA’ point. Our values are within the error bars of the numerical estimates of Seno and Stella for SAW’s on the same lattice at the FTHETA point. The crossover exponent is calculated in several ways and the various results approximately converge to ${\phi }_t$≃0.71 if corrections to scaling are taken into account. This value is significantly larger than the DS exact value for the FTHETA’ point φ=(3/7≃0.43; it is also larger than recent numerical estimates of ${\phi }_t$ for SAW’s at the FTHETA point. These results suggest that tricritical trails are in a different universality class than SAW’s in both the FTHETA and the FTHETA’ point. The results for the shape exponent ${\nu }_t$ are consistent with the DS value (4/7≃0.571. As expected the results for the tricritical temperature -ε/$k_B$$T_t$=1.086±0.004 and the growth parameter ${\mu }_t$=2.9902±0.003 are equal to those obtained by the same method for trails on a square lattice without a boundary; these values are slightly smaller than their analytical upper bounds ln3≃1.0986 and 3, respectively.

We show that the structure of unstable orbits in strange attractors resulting from a class of differentiable flows is universal. The structure is particularly simple near a bistable point in parameter space, and possible orbits lie on a ‘‘complete’’ binary tree. We suggest experimental systems that would exhibit this universality. This is part of an attempt to classify low-dimensional chaotic systems.

Clusters grown with the dielectric-breakdown model (DBM) in cylinder geometry show two growth phases: a scaling regime for cluster heights smaller than the cylinder circumference and a subsequent steady state, which is translationally invariant in the main growth direction. By studying a very simple, exactly solvable growth model involving arrays of sticks, we arrive at an affine framework, suitably describing the scale invariance of its clusters in the scaling regime. Within this framework we derive new relations between various exponents, which are expected to also hold for DBM clusters. Numerical work on DBM in the scaling regime is presented, indicating that the clusters are indeed self-affine in this phase.

The Willshaw model of associative memory, implemented in a fully connected network with stochastic asynchronous dynamics, is studied. In addition to Willshaw’s learning rule, the network contains uniform synaptic inhibition, of relative strength K, and negative neural threshold -θ,θ>0. The P stored memories are sparsely coded. The total number of on bits in each memory is Nf, where f is much smaller than 1 but much larger than lnN/N. Mean-field theory of the system is solved in the limit where C==exp(-$f^2$P) is finite. Memory states are stable (at zero temperature), as long as C>$h_0$==K-1+θ and $h_0$>0. When C<$h_0$ or $h_0$<0, P retrieval phases, highly correlated with the memory states, exist. These phases are only partially frozen at low temperature, so that the full memories can be retrieved from them by averaging over the dynamic fluctuations of the neural activity. In particular, when $h_0$<0 the retrieval phases at low temperatures correspond to freezing of most of the population in a quiescent state while the rest are active with a time average that can be significantly smaller than the saturation level. These features resemble, to some extent, the observed patterns of neural activity in the cortex, in experiments of short-term memory tasks. The maximal value of P for which stable retrieval phases exist, scales as $f^{\mathrm{-}3}$/‖lnf‖ for f≫1/lnN, and as $f^{\mathrm{-}2}$ln(Nf/‖lnf‖) for f≪1/lnN. Numerical simulations of the model with N=1000 and f=0.04 are presented. We also discuss the possible realization of the model in a biologically plausible architecture, where the inhibition is provided by special inhibitory neurons.

We study box counting on finite fractal sets and investigate how to obtain the generalized dimensions and the spectrum of scaling indices with highest possible accuracy. As a model we use a simple one-dimensional Cantor set for which the f(α) spectrum may be found analytically—the exact result is compared with the box-counting solution on the finite levels. There is a connection between the q value and the size of the boxes giving the most accurate result for the f(α) spectrum on any finite level.

Nonlinear model systems based on Eigen’s theory of macromolecular self-organization are investigated stochastically. Nonequilibrium phases are usually achieved through delicate transient processes from an initial condition. The minimum initial values required for a secure relaxation to the expected phase are estimated for the case of internal fluctuation and for the system perturbed by multiplicative white noises. These values set the criterion for the stochastic transition from the survival phase to extinction phase. It is found that, over a narrow critical region preceding the noise-induced transition point, the relaxation time diverges abruptly while the effective domain for a secure relaxation shrinks drastically. Numerical results also demonstrate that Stratonovich calculi are favored over Ito calculi in describing the dynamic systems.

A new class of cellular automata is observed to evolve to a self-organized critical state. These ‘‘vector automata’’ obey a threshold relaxation condition that depends on the gradient of a scalar field, which is locally conserved except at the system boundaries. Both square and triangular lattices are studied and lead to virtually identical statistics; however, a particular (and natural) choice of boundary shape for the square lattice leads to trivial dynamics characteristic of a minimally stable configuration.

We use Floquet theory to study the dynamical properties of a Fokker-Planck equation with time-dependent coefficients describing a nonlinear Brownian rotor driven by a time-periodic angle-dependent external force consisting of two traveling sine waves with amplitudes ${\epsilon }_1$ and ${\epsilon }_2$. For the case ${\epsilon }_2$=0, the Fokker-Planck equation is separable (in the sense that it has two well-defined eigen-numbers), and the nearest-neighbor spacing distribution appears to be Poisson random for large ${\epsilon }_1$. For both ${\epsilon }_1$≠0 and ${\epsilon }_2$≠0, we find evidence of nonlinear resonance and level repulsion in the Floquet spectrum and first-passage time, and the spectrum exhibits level repulsion and universal random matrix-type behavior. The long-time state is quasiperiodic in time and angle and is affected by an increasing number of modes of the system as the external field amplitude is increased.