Based on a one-dimensional model, a perturbation expansion is carried out to solve the equations describing a weakly nonlinear laser pulse in a plasma in which the electrons are treated relativistically and the plasma frequency is much less than the laser frequency. To lowest order, the expansion yields two coupled equations for the vector and scalar potentials. For a pulse which is long compared with a plasma wavelength, the coupled equations reduce to the nonlinear Schrödinger equation with well-known soliton solutions. An initial pulse of hyperbolic-secant shape which is short compared with a plasma wavelength broadens and acquires a characteristic asymmetric shape with a steep trailing edge and a much broader, gently sloping front portion, and has a frequency and wave-number shift which vary from a positive value at the front to a negative value at the rear of the pulse. The peak and rear part of a short pulse are strongly influenced by nonlinear effects, whereas the front is governed primarily by linear dispersion. The average pulse frequency continually decreases as energy is lost to the plasma wake. The wake-field phase velocity is shown to be approximately equal to the velocity of the pulse peak.

Electron-energy distribution functions (EEDF), electron-transport parameters, and rate coefficients have been calculated by solving the time-dependent Boltzmann equation in weakly ionized plasmas in ${\mathrm{N}}_2$ and ${\mathrm{H}}_2$, for a wide range of the field frequency going from ω≪${\nu }_{\mathit{e}}$ up to ω≫${\nu }_{\mathit{m}}^{\mathit{e}}$, where ${\nu }_{\mathit{e}}$ and ${\nu }_{\mathit{m}}^{\mathit{e}}$ are two characteristic relaxation frequencies for energy and momentum transfer, respectively. The effects produced by the electron-vibration superelastic collisions have been taken into account in the Boltzmann equation and their influence in the electron transport has been analyzed in detail. Both the temporal behavior and the time-averaged value of the EEDF and of several macroscopic quantities have been obtained and discussed for different values of the field frequency. Among several other aspects, it is shown that the instantaneous power absorbed from the field and the one dissipated in electron collisions are not in phase, although, obviously, they exactly compensate each other on the average.

The stopping power of an extended charge distribution at any velocity by a free-electron gas is calculated in the random-phase approximation at any degeneracy. Asymptotic expressions are given in the low- and high-velocity range for any charge distribution with spherical symmetry. We pay attention to the determination of the critical distances of pointlike behavior for an extended charge and uncorrelated behavior for two separated charges. The particular cases of randomly orientated dicluster and atomic ion with extended charges distribution are given particular attention.

The partition functions are calculated for gases consisting of Ln and X atoms and LnX molecules and their singly charged ions (Ln≡Dy,Ho,Tm and X≡I,Br).

High-temperature (${\mathit{T}}_{\mathit{e}}$>150 eV), small-diameter (∼200 μm) plasma columns have been efficiently generated by very fast (13 ns rise time, 28 ns full width at half maximum) pulsed discharge excitation of capillary channels filled with preionized gas. Discharges in argon-filled capillaries at currents between 20 and 60 kA produced plasmas with Ar x–AR xiv line emission, in which the degree of ionization was controlled by the magnitude of the current pulse. The characteristics of these plasmas differ from those created by vacuum discharges in the same capillaries and approach those necessary for soft-x-ray amplification in low-Z elements.

The electron density and temperature of the plasma produced by a capillary discharge with open geometry have been characterized. Peak densities of 1×${10}^{19}$ ${\mathrm{cm}}^{\mathrm{-}3}$ and electron temperatures of 2.5 eV are observed. The plasma is uniform over distances on the order of 1 cm. The time evolution of both the electron density and temperature profiles has been studied over the duration of the 5-μs-long discharge.

The coherent effect of undulator radiation is investigated using relativistic photoelectron beam (RPE) produced by the fourth harmonics of a Q-switched Nd-doped yttrium aluminum garnet (Nd-YAG) laser. The Q-switched laser pulse contains a burst of 30 ps micropulses superposed with a weak 8 ns pulse. The current and shape of the RPE were controlled by changing the incident laser intensity. The temporal structure of the RPE was measured using the Cherenkov radiation emitted by relativistic electrons impinged on an optical fiber. When the irradiated laser intensity is stronger than 5 MW/${\mathrm{cm}}^2$, the wave form of the RPE does not show any microstructure, which is due to the saturation of the RPE current density by space-charge effect. The measured radiation power from the RPE having no microstructure was near the noise level of a microwave diode, even though the current was much larger than that of the RPE having micropulses. With irradiated intensity less than 1 MW/${\mathrm{cm}}^2$, the temporal structure of the RPE closely follows that of the laser pulse. The measured power of the undulator radiation generated by micropulses was more than ${10}^3$ times stronger than that of the theoretically predicted incoherent radiation and tends towards a quadratic dependence on the electron-beam current. The enhancement and tendency is due to coherent radiation emitted by electron-beam micropulses whose typical bunch lengths are comparable to the radiation wavelength (7.4 mm).

We describe a scheme for obtaining very short wavelengths (λ∼10 Å) in recombination lasers. The rapid cooling rates necessary to achieve population inversion during recombination are attained by adiabatic expansion of submicrometer spheres. The lasing region is made up of many such spheres. The spheres are heated impulsively by a powerful picosecond laser. First, they ionize, then as they expand, they cool and recombine. We have calculated the optimum sphere size and initial temperature for maximum gain in the n=3 to 2 transition of hydrogenlike ions of elements with atomic numbers, Z, between 10 and 30. Gain of about 250 ${\mathrm{cm}}^{\mathrm{-}1}$ is calculated in aluminum at 38.8 Å. Gain rapidly decreases with Z so that gain in titanium at 13.6 Å is about 10 ${\mathrm{cm}}^{\mathrm{-}1}$. We have calculated the required pump-laser intensity and found it to be attainable with current lasers. The propagation of the pump through the ‘‘gas’’ of spheres is considered and the problems arising from pump scattering by the spheres are discussed.

This paper is the second of a series. The first [G. E. Hahne, Phys. Rev. A 45, 7526 (1992)] developed a general theory of the transition operator approach to diffraction of time-harmonic electromagnetic waves from fixed obstacles, such that the response of the obstacle, denoted by Ω, to an impinging electromagnetic signal with wave number ${\mathit{k}}_0$ was simulated by nonlocal, homogeneous Leontovich (i.e., impedance) boundary conditions on the obstacle’s surface, which surface is called ∂Ω. Moreover, the exterior region, called ${\mathrm{\Omega }}^{\mathrm{ex}}$, was presumed to be unbounded empty space, and has an electromagnetic response that can be expressed in terms of the so-called radiation impedance operator, denoted Z${\mathrm{̆}}_{\mathit{k}0}^{+}$; Z${\mathrm{̆}}_{\mathit{k}0}^{+}$ is a certain invertible, linear functional operator that maps the space of complex tangent-vector fields on ∂Ω into itself.

The matching of the limiting tangential electric and magnetic fields on ∂Ω yielded a functional-operator expression for the transition operator and thereby a formal reduction to quadratures of the entire direct-scattering problem. This paper is intended to serve as an illustration and elaboration of the formalism presented in the first paper. After a brief recapitulation of the theory, the following topics are dealt with. First, an infinite sum in terms of vector spherical harmonics is obtained for Z${\mathrm{̆}}_{\mathit{k}0}^{+}$ for ∂Ω a sphere. Second, a quasiplanar approximation, based directly on the exact result for planar ∂Ω, is obtained for Z${\mathrm{̆}}_{\mathit{k}0}^{+}$ when ‖${\mathit{k}}_0$‖ is large compared to the local surface curvatures of ∂Ω and attention can be restricted to small neighborhoods on ∂Ω; when augmented by the method of stationary phase, this approximation is shown to lead to the familiar physical optics method for smooth, convex surfaces. Third, the latter method, supported by further interventions of the method of stationary phase, is applied in a well-established manner to secure the results of geometrical optics for the complete Green’s function for the time-harmonic Maxwell field in the presence of a smooth-surfaced, convex, perfectly electrically conducting obstacle.

One feature of the latter computations is that the original source currents from which the free-space Maxwell Green’s function is constructed are presumed to generate ordinary outgoing (‘‘causal’’) electromagnetic waves for electric current sources, and purely ingoing (‘‘anticausal’’) waves for magnetic current sources (i.e., sinks) of electromagnetic radiation. Fourth, a formal construction is derived for mapping the Leontovich boundary conditions on an inner surface into a set of Leontovich boundary conditions on an outer surface, in the circumstance that a layer comprised of a material medium, which has possibly nonuniform and nonsymmetric tensors representing its constitutive properties, fills the domain between the surfaces; the construction presumes that a Green’s function is available for the corresponding Maxwell equations in the medium when the medium is extended appropriately to fill all space. The main body of the paper concludes with a brief discussion of directions of possible future work and applications. The first appendix shows how to apply the method of stationary phase in the present context, in particular for the mixed case that the original radiation source is anticausal and the response currents generated in the obstacle are causal sources of radiation. A second appendix develops an acoustic (Helmholtz equation) analog to the fourth topic mentioned above, and exhibits an interplay of the theory with symplectic transformations in the case that the principle of reciprocity holds; it is noted that a symplectic connection also exists in the electromagnetic case when the propagation medium’s constitutive properties are such that reciprocity holds.

A formalism of electromagnetism covariant under the complex rotation group is first described and then discussed thoroughly, in the case of two-dimensional electromagnetic fields, in homogeneous, isotropic chiral media. As a result, we obtain the generalized Descartes-Snell laws, the Fresnel relations for reflection and refraction, as well as the Brewster condition for plane waves incident on a surface of discontinuity between chiral and achiral media.

The effect of spatially periodic on-site potentials on solitons of the (1+1)-dimensional nonlinear Schrödinger equation depends not only on the amplitude of the perturbation but also on two length ratios—between the wavelength of the potential and either the spatial width of the soliton or its phase-modulation wavelength. For solitons which are slowly moving or at rest, only the first ratio is important: Solitons which are narrow compared to the wavelength of the potential move like particles in an effective potential; wide solitons move like ‘‘renormalized’’ particles; and for competing length scales, solitons are easily destabilized by the perturbation and break up into smaller localized excitations and radiation. For initially rapidly moving solitions in short-wavelength potentials, a different resonance effect is described which can inhibit soliton propagation. Away from this resonance the solitons can move in a ‘‘dressed’’ form radiating very slowly. Furthermore, it is shown numerically that two such ‘‘dressed’’ solitons reemerge after collision, essentially unchanged. We also comment on the structural stability of completely integrable dynamics under spatially periodic perturbations. Finally, we remark on the relevance of our results for the dynamical behavior of large polarons and bipolarons in disordered ionic lattices.

In this paper we examine the influence of different input-output relations on the generalization ability of a single-layer perceptron. The input-output relations can be linear, binary, or sigmoid. With this choice we take into account most of the cases which are of present interest. The generalization problem will be realizable or unrealizable if the input-output relations for teacher and student are identical or not. We show that sometimes it can have a positive effect on the generalization ability, if one learns with errors.

We investigate analog neural networks. They have continuous state variables that depend continuously on time. Although they all have an energy function, not all can have their dynamics derived from a Hamiltonian. Some necessary conditions are given for the network to have Hamiltonian dynamics. We give an example and, using symplectic transformations, describe a whole class of neural networks with Hamiltonian dynamics.

The retrieval dynamics of an extremely diluted network of three-state neurons designed by the projection rule is investigated. Depending on the parameters of the model four different fixed points with different stability can exist. Only one of them corresponds to full retrieval. It is discussed how the parameters of the model must be chosen in order to get perfect retrieval and large basins of attraction. Finally it is shown that a network of three-state neurons storing binary patterns has better retrieval properties than a network of binary neurons storing the same patterns.

We have studied the avalanche fluctuations resulting from the grain-by-grain quiescent perturbation of real sandpiles of various sizes. Contrary to previous reports, we find the attribute of self-organization, namely, a power-law distribution of avalanche sizes, to be the generic behavior, independent of system size. However, as the system size is increased, the power-law behavior is supplemented by uniformly large, periodic avalanches that carry away the dominant fraction of the total avalanched mass. Additionally, we find that a scale invariance in pile-mass fluctuations depends on the ratio of pile to grain diameters.

Candidates for type-II Pomeau-Manneville intermittency are reported in a hydrodynamic system. Two time series are studied with return maps, power spectra, statistical features of the laminar period lengths, and correlation dimensions. Four degrees of freedom only are seemingly necessary to describe the dynamics.

We show a simple procedure which allows us to generate a strange nonchaotic trajectory. It can explain properties of strange nonchaotic attractors.

The Nosé-Hoover equation has recently been introduced to simulate, in a deterministic and reversible way, the equilibrium properties of a system at constant temperature. However, for many one-dimensional potentials, such as the harmonic oscillator, the Nosé-Hoover scheme is not adequate since the dynamics is not sufficiently ergodic. We present modifications of the Nosé-Hoover equation in which the kinetic energy and the virial are treated in an equivalent manner and which explicitly include the virial within a canonical framework. We show that these modifications can yield an adequate statistical description for one-dimensional potentials such as the double-well and harmonic oscillator.

We investigate the phenomenon of velocity selection for traveling wave fronts in a class of coupled-map lattices, derived by discretizations of the Fisher equation [Ann. Eugenics 7, 355 (1937)]. We find that the velocity selection can be understood in terms of a discrete analog of the marginal-stability hypothesis. A perturbative approach also enables us to estimate the selected velocity accurately for small values of the discretization mesh sizes.

A different pattern-selection regime is shown to exist in regions of high nonlinearity in the diffusively coupled logistic lattice with nearest-neighbor interaction. For this regime, the phase diagram of several prominent attractors is determined. The results indicate that the spatiotemporal chaos in some of the regions so far believed to be areas of fully developed spatiotemporal chaos is of a transient nature.