We study the saturated state of an untapered free-electron laser (FEL) in the Compton regime, arising after exponential amplification of an initially low level of radiation by an initially monoenergetic, unbunched electron beam. The saturated state of the FEL is described by oscillations about an equilibrium state. Using the two invariants of the motion and certain assumptions motivated by computer simulations we provide approximate analytic descriptions of the radiation field and electron distribution in the saturation regime. We first consider a one-dimensional approximation and later extend our approach to treat an electron beam of finite radial extent. Of note is a result on the radiated power in the case of an electron beam with a small radius.

In the following, we obtain continua of localized wave solutions to the scalar homogeneous wave, damped wave, and Klein-Gordon equations. We do this by utilizing the fact that similar Ansätze (all of which involve a free-particle time-dependent Schrödinger-like equation) may be used to satisfy all three of these partial differential equations. This Schrödinger-like equation is reduced to an ordinary differential equation (ODE) using a dimensionless complex similarity transformation. A general solution to this ODE involving confluent hypergeometric functions is found. For an azimuthal dependence exp(iνφ),ν∈openR, this general solution includes many of the previously determined localized wave solutions as special cases.

A spherical-wave-function theory of bounded homogeneous elastic anisotropic media is developed. The anisotropic elastodynamic wave equations are solved exactly by using the method of plane-wave angular spectrum expansions. The series, integral representations, and addition theorems of the spherical wave functions of the first, second, third, and fourth kind for homogeneous elastic anisotropic media (HEAM) are presented. Weyl’s method for the scalar Green’s functions in isotropic media is generalized to the study of the dyadic Green’s functions in HEAM. The series representations of Green’s functions are of the form of separated variables. These representations are well suited to imposing a boundary condition when dealing with waves in spherically layered HEAM.

We study the validity of perturbational treatments of one-dimensional weakly discrete kink-bearing systems with the corresponding continuum model as the unperturbed state. We calculate the Peierls-Nabarro barrier height ${\mathrm{\Delta }}_{\mathrm{PN}}$, the Peierls-Nabarro frequency ${\mathit{f}}_{\mathrm{PN}}$, and the deviation of the kink creation energy from the continuum case. In the case of the sine-Gordon system and the ${\mathrm{\Phi }}^4$ system, we find that the dressing (change of the continuum kink shape due to discreteness) contributes to the values of ${\mathrm{\Delta }}_{\mathrm{PN}}$ and ${\mathit{f}}_{\mathrm{PN}}$ through all orders of perturbation, thereby making the whole perturbation scheme irrelevant. In contrast, the double-quadractic model (where the whole necessary nonlinearity is ‘‘hidden’’ in one nonanalytic point) can be treated by the perturbation scheme without restrictions. The results found in this paper put general limitations on analytical approaches to nonlinear phenomena in discrete systems and demonstrate a deep inherent difference between corresponding discrete and continuum models.

We derive cooling schedules for the global optimization of learning in neural networks. We discuss a two-level system with one global and one local minimum. The analysis is extended to systems with many minima. The optimal cooling schedule is (asymptotically) of the form η(t)=${\mathrm{\eta }}^{\mathrm{*}}$/lnt, with η(t) the learning parameter at time t and ${\mathrm{\eta }}^{\mathrm{*}}$ a constant, dependent on the reference learning parameters for the various transitions. In some simple cases, ${\mathrm{\eta }}^{\mathrm{*}}$ can be calculated. Simulations confirm the theoretical results.

We study the retrieval behaviors of neural networks which are trained to optimize their performance for an ensemble of noisy example patterns. In particular, we consider (1) the performance overlap, which reflects the performance of the network in an operating condition identical to the training condition; (2) the storage overlap, which reflects the ability of the network to merely memorize the stored information; (3) the attractor overlap, which reflects the precision of retrieval for dilute feedback networks; and (4) the boundary overlap, which defines the boundary of the basin of attraction, and hence the associative ability for dilute feedback networks. We find that for sufficiently low training noise, the network optimizes its overall performance by sacrificing the individual performance of a minority of patterns, resulting in a two-band distribution of the aligning fields. For a narrow range of storage level, the network loses and then regains its retrieval capability when the training noise level increases, and we interpret that this reentrant retrieval behavior is related to competing tendencies in structuring the basins of attraction for the stored patterns. Reentrant behavior is also observed in the space of synaptic interactions, in which the replica symmetric solution of the optimal network destabilizes and then restabilizes when the training noise level increases. We summarize these observations by picturing training noises as an instrument for widening the basins of attractions of the stored patterns at the expense of reducing the precision of retrieval.

We apply lattice dynamics to calculate the vibrational modes of a perfect A-DNA homopolymer crystal with sodium ions. A Green-function method is used to reduce the dimension of the matrix to be diagonalized. We concentrate on the biologically significant crystal modes with frequencies less than 100 wave numbers. 94% of the crystal modes were likely to be located in this frequency range. Characteristics of the crystal modes are analyzed. We find that the interhelical modes dominate in the frequency range 25–30 ${\mathrm{cm}}^{\mathrm{-}1}$ and the intrahelical modes dominate for frequencies greater than 80 ${\mathrm{cm}}^{\mathrm{-}1}$. The infrared absorption peaks at 35, 50, and 83 ${\mathrm{cm}}^{\mathrm{-}1}$ in our model. The former two bands are not counterion dependent and the latter one is. Our Raman scattering spectrum peaks around 30 ${\mathrm{cm}}^{\mathrm{-}1}$ and it is strongly counterion dependent. The counterion-dependency trend of this band is the same as that of the experimental 25-${\mathrm{cm}}^{\mathrm{-}1}$ feature, extrapolated to 0 relative humidity, observed in both DNA and RNA crystals. The characteristics of the experimental Raman scattering features at 25 and 35 ${\mathrm{cm}}^{\mathrm{-}1}$ are consistent with our results.

We apply a ‘‘cavity-type’’ method for the analysis of the learning ability of single- and multilayer perceptrons. We show that the mean-field equations obtained in this way, which are identical to the equations derived previously by the replica method, describe not only the properties of the optimal network, but also a learning process which leads to this network. We discuss the applicability of our ideas to the construction of learning algorithms. Our interpretation of the mean-field theory also leads naturally to a new concept, ‘‘flexibility,’’ which is a measure of the ability of the network to learn.

We propose a generalized Lévy walk to model fractal landscapes observed in noncoding DNA sequences. We find that this model provides a very close approximation to the empirical data and explains a number of statistical properties of genomic DNA sequences such as the distribution of strand-biased regions (those with an excess of one type of nucleotide) as well as local changes in the slope of the correlation exponent α. The generalized Lévy-walk model simultaneously accounts for the long-range correlations in noncoding DNA sequences and for the apparently paradoxical finding of long subregions of biased random walks (length ${\mathit{l}}_{\mathit{j}}$) within these correlated sequences. In the generalized Lévy-walk model, the ${\mathit{l}}_{\mathit{j}}$ are chosen from a power-law distribution P(${\mathit{l}}_{\mathit{j}}$)∝${\mathit{l}}_{\mathit{j}}^{\mathrm{-}\mathrm{\mu }}$. The correlation exponent α is related to μ through α=2-μ/2 if 2<μ<3. The model is consistent with the finding of ‘‘repetitive elements’’ of variable length interspersed within noncoding DNA.

We introduce a neural network architecture in which analog-valued neurons compete through a constraint on a sum of neuron outputs within localized clusters. Local competition is useful in feature-extraction and pattern-classification applications. We show that with continuous-time updating, these networks converge only to fixed points, while with discrete-time, parallel updating, they converge to either fixed points or period-two limit cycles. We derive a stability criterion guaranteeing that discrete-time networks converge only to fixed points when their cluster gains, which are related to the slopes of the neuron input-output transfer functions, are sufficiently small. Numerical tests are presented showing that image-processing networks incorporating local competition operate reliably and in agreement with the stability criterion. We also describe a simple competitive analog electronic circuit that demonstrates that these networks are easily implementable.

We analyze the attractors of associative-memory neural networks in which analog neurons compete locally. These networks are well suited for a variety of feature-extraction, pattern-classification, and data-compression tasks. For networks storing a finite number of patterns, we present bifurcation diagrams for the pattern overlaps. For networks storing an extensive number of patterns, we present phase diagrams showing attractor types as a function of pattern-storage fraction and neuron-transfer-function steepness. We also report results for the storage capacity of k-winner associative memories in the limit of infinite neuron gain. Numerical investigations of computer-generated networks confirm the phase diagrams.

In this paper, we study the quantum manifestations of classical chaos in phase space using Wigner and Husimi distribution functions. We test the claim that Husimi represents the correspondence better than Wigner does. The results show the claim is valid. We also use a quantum dissipation scheme empirically for classically damped motions often characterized by strange attractors. We believe quantum resemblance to classical distributions can be regarded as signatures of quantum chaos in phase space.

The transition from the regular to the chaotic regime in a simple model has been studied by two different methods: the stability matrix and the quasicrossing distribution. Good agreement between the two methods was obtained.

When a series of measurements is performed with increasingly coarse (or increasingly fine) precision, consecutive observations seem to be erratically distributed at first, and then organize themselves into cycles and patterns. The patterns, which arise because of roundoff errors, are related to a notion in number theory, the so-called Farey sequence.

Brownian-particle trapping is considered in the case when traps are gathered in ‘‘clouds’’ distributed in space in a noncorrelated manner. A general formula for the survival probability is derived. Its analysis shows that the trap-cluster formation leads to a process slowdown (meaning a slowdown in the trapping rate) in comparison with the case of noncorrelated traps. The slowdown is significant from the very beginning of the process or only at its final stage, depending on the cloud structure.

We derive an exact solution of the Smoluchowski equation for a Brownian particle moving in an arbitrary potential well with a general delocalized sink. The average rate constant for the general sink is expressed explicitly in terms of the corresponding rate constants for localized sinks with different initial conditions and sink positions. Simple analytical expressions are provided for diffusion in a harmonic potential well.

The interfacial growth with a conservation law is analyzed using the dynamic-renormalization-group method. The fixed point is found for the case of long-range spatially and temporally correlated noise. The scaling exponents χ and z are obtained. The result shows that long-range spatial correlations and temporal correlations roughen the interface. The results are compared with those of Medina et al. [Phys. Rev. A 39, 3053 (1989)] and Sun, Guo, and Grant [Phys. Rev. A 40, 6763 (1989)].

We have investigated numerically the onset of spatiotemporal intermittency in a one-dimensional lattice of coupled nonlinear maps. The phase boundary between the laminar and intermittent states has a complicated form, with several reentrant regions. Using finite-size scaling, we have determined the critical exponents associated with the transition to spatiotemporal intermittency for several values of the map’s coupling parameter. The exponents appear to be nonuniversal in the weak-coupling limit of the map.

We present three variants of an extremely simple stochastic model, one of which does not display a ‘‘resonant phenomenon’’ discussed recently [Ch. Doering and C. Gadoua, Phys. Rev. Lett. 69, 2318 (1992)], while the other two variants do.

In the presence of a small heat current and appropriate boundary conditions, ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ will reach the superfluid critical point without fine tuning the temperature. This provides a physical example of self-organized criticality via the mechanism of singular diffusion proposed by Carlson et al. [Phys. Rev. Lett. 65, 2547 (1990)].