The theory of bifurcation from equilibria based on center-manifold reduction and Poincaré-Birkhoff normal forms is reviewed at an introductory level. Both differential equations and maps are discussed, and recent results explaining the symmetry of the normal form are derived. The emphasis is on the simplest generic bifurcations in one-parameter systems. Two applications are developed in detail: a Hopf bifurcation occurring in a model of three-wave mode coupling and steady-state bifurcations occurring in the real Landau-Ginzburg equation. The former provides an example of the importance of degenerate bifurcations in problems with more than one parameter and the latter illustrates new effects introduced into a bifurcation problem by a continuous symmetry.

The introduction of a powerful new microwave source, the free-electron laser, provides new opportunities for novel heating and current-drive schemes to be used in toroidal fusion devices. This high-power, pulsed source has a number of technical advantages for these applications, and its use is predicted to lead to improved current-drive efficiencies and opacities in reactor-grade fusion plasmas in specific cases. The Microwave Tokamak Experiment at the Lawrence Livermore National Laboratory will provide a test for some of these new heating and current-drive schemes. Although the motivation for much of this research has derived from the application of a free-electron laser to the heating of a tokamak plasma at a frequency near the electron cyclotron frequency, the underlying physics, i.e., the highly nonlinear interaction of an intense, pulsed, coherent electromagnetic wave with an electron in a magnetized plasma including relativistic effects, is of general interest. Other relevant applications include ionospheric modification by radio-frequency waves, high-energy electron accelerators, and the propagation of intense, pulsed electromagnetic waves in space and astrophysical plasmas. This review reports recent theoretical progress in the analysis and computer simulation of the absorption and current drive produced by intense pulses, and of the possible complications that may arise, e.g., parametric instabilities, nonlinear self-focusing, trapped-particle sideband instability, and instabilities of the heated plasma.

The problem of measuring the viscosity of fluids under shock-loading conditions is discussed. The authors examine in detail the method of Sakharov et al. (1965) and Zaidel' (1967) for measuring shear viscosity from the decay of perturbations on a corrugated shock front. The relevance of initial conditions, finite shock amplitude, bulk viscosity, and the sensitivity of the measurements to the shock boundary conditions are discussed. The validity of the viscous perturbation approach is examined by numerically solving the second-order Navier-Stokes equations. These numerical experiments indicate that shock instabilities may occur even when the Kontorovich-D'yakov stability criteria are satsified. The corrugated shock front induces mixing of the shocked sample. This mixing is particularly vigorous in viscous materials and may be responsible for the rapid rate of some shock-induced chemical reactions. The experimental results for water at 15 GPa are discussed, and several possibilities are considered to explain why the viscosity obtained by these experiments is so different from those obtained by other methods. Two possible reasons are favored: (1) the analytic method may be inappropriate because it ignores possible complications at the onset of the shock perturbations, and (2) the large effective viscosity determined by this method may reflect the existence of ice VII on the Rayleigh path of the Hugoniot. The latter interpretation reconciles the experimental results with estimates and measurements obtained by other means and is consistent with pressure-volume-temperature Hugoniot data and the phase diagram of ${\mathrm{H}}_2$O.

Very-long-baseline interferometry (VLBI) is a novel observing technique for measuring the relative positions of widely separated points on the surface of the Earth with centimeter-level accuracy. Such accuracy is two or three orders of magnitude better than was available with classical techniques only a few decades ago. This enormous improvement in accuracy has opened up for study a broad new spectrum of geophysical phenomena. The new measurements allow direct observation of the tectonic motions and deformations of the Earth's crustal plates, observations of unprecedented detail of the variations in the rotation of the Earth, and direct measurement of the elastic deformations of the Earth in response to tidal forces. These new measurements have placed significant constraints on models of the interior structure of the Earth; for example, measurements of the variations in the Earth's nutation have been shown to be particulary sensitive to the shape of the core-mantle boundary. The VLBI measurements, coupled with other space-based geodetic observing techniques such as the Global Positioning System, allow construction of a global reference frame accurate at the centimeter level. Such a frame will be essential to studying long-term global changes, especially those changes related to sea-level variations as recorded by tide gauge measurements.

Magnetic materials research has entered a new and exciting period with the advent of the ternary rare-earth-iron-boron compounds, ${\mathrm{R}}_2$${\mathrm{Fe}}_{14}$B. From the fundamental physics perspective the ${\mathrm{R}}_2$${\mathrm{Fe}}_{14}$B series and its isostructural relatives comprise a rich, fascinating area for the investigation of many intrinsic properties, including magnetic structures, magnetocrystalline anisotropy, and rare-earth-transition-metal exchange interactions. Intense interest in the technological aspects of these compounds has been ignited by the fact that energy products eclipsing all previous values have been realized in practical magnets based on ${\mathrm{Nd}}_2$${\mathrm{Fe}}_{14}$B, the prototypical representative; these magnets also feature economic advantages over the earlier samarium-cobalt materials. Both facets of the ${\mathrm{R}}_2$${\mathrm{Fe}}_{14}$B systems are considered in this review.

The electronic transport theory of semiconductors is not, from a first-principles point of view, as well understood as is that of metals, where the degeneracy of the Fermi system leads to a simplified but comprehensive theory. In the case of semiconductors, degeneracy usually plays no simplifying role at all. However, in many transport problems of current interest one is effectively dealing with the equivalent of a single particle interacting with an environment, e.g., a heat bath or a random potential. In view of this, the author presents a simple formalism for the quantum dynamics of a single continuous degree of freedom. The quantum-statistical description is in terms of the density matrix, and the Feynman rules for a standard treatment of the density matrix are presented and illustrated by applications to problems of current interest. It is shown that such an effect as, for example, the intracollisional field effect, which in the past has been dealt with using complicated formalisms, in the present treatment is described in an elementary way. The single-particle approach conveniently displays the interference aspect of quantum-mechanical transport, as is discussed in a treatment of the weak localization effect in disordered conductors. The real-space representation of quantum transport is stressed, as is appropriate for a proper discussion of mesoscopic physics. The author treats the connection between the linear-response formalism and the Landauer approach by expressing the conductance in terms of the scattering properties of a sample. He also discusses the conductance fluctuations of mesoscopic samples.

At the thirty-year anniversary of the introduction of the technique of computer-generated random-dot stereograms and random-dot cinematograms into psychology, the impact of the technique on brain research and on the study of artificial intelligence is reviewed. The main finding—that stereoscopic depth perception (stereopsis), motion perception, and preattentive texture discrimination are basically bottom-up processes, which occur without the help of the top-down processes of cognition and semantic memory—greatly simplifies the study of these processes of early vision and permits the linking of human perception with monkey neurophysiology. Particularly interesting are the unexpected findings that stereopsis (assumed to be local) is a global process, while texture discrimination (assumed to be a global process, governed by statistics) is local, based on some conspicuous local features (textons). It is shown that the top-down process of "shape (depth) from shading" does not affect stereopsis, and some of the models of machine vision are evaluated. The asymmetry effect of human texture discrimination is discussed, together with recent nonlinear spatial filter models and a novel extension of the texton theory that can cope with the asymmetry problem. This didactic review attempts to introduce the physicist to the field of psychobiology and its problems—including metascientific problems of brain research, problems of scientific creativity, the state of artificial intelligence research (including connectionist neural networks) aimed at modeling brain activity, and the fundamental role of focal attention in mental events.

We compute all the three-dimensional quasicrystallographic space groups with $n$-fold axial point groups and standard lattices by a method that treats crystals and quasicrystals on an equal footing. We do not rely on projecting higher-dimensional crystallographic space groups, our results are valid for arbitrary $n$, and our analysis is elementary. We regard space groups as a scheme for classifying diffraction patterns to be carried out in three-dimensional reciprocal space. The familiar three-dimensional crystallographic space groups with axial point groups emerge simply and directly as special cases of the general $n$-fold three-dimensional quasicrystallographic treatment with $n=3,4,\mathrm{and} 6$. We give a general discussion of extinctions in quasicrystals and give a simple (three-dimensional) geometrical specification of the extinctions for each axial space group. The paper is intended both for people trying to systematize quasicrystal diffraction patterns and for people interested in a simple alternative approach to the computation of crystallographic or quasicrystallographic space groups.

The basic physics regarding self-trapping of light particles in simple fluid hosts is reviewed pedagogically. Electron and positronium self-trapping in fluid helium is taken as a historical starting point. The theoretical context consists of simplified continuum models with averaged interactions, but required improvements are discussed. Experimental examples are chosen to illustrate bulk, surface, and impurity effects. Equilibrium and dynamical aspects of the field are illustrated. In noting applications to more complex systems, reference is made to recent developments using path-integral and computer simulation methods. The article spans certain aspects of studies in this fascinating area over the last 30 years.

A systematic review of theoretical results for the longitudinal and transverse impedances obtained by different methods is presented. Definitions, general theorems, modal analysis, diffraction model, and analytical results comprise the content of the paper. Several new results are included. In particular, necessary and sufficient conditions for the independence of the impedance on the beam longitudinal direction are given. The impedances of two basic simple structures—that of a cavity and that of a step—are studied in detail. The transition from the regime of a cavity to the regime of a step is explained, an approximate formula describing this transition is given, and the criterion for determining the applicability of each regime is established. The asymptotic behavior of the impedance for a finite number $M$ of periodically arranged cavities as a function of $M$ is studied. The difference in the behavior of the impedance for a single cavity and that for an infinite number of cavities is explained as the result of the interference of the diffracted waves. A criterion for determining the transition in the impedance behavior from small $M$ to large $M$ is presented.

The concept of boson realization (or mapping) of Lie algebras appeared first in nuclear physics in 1962 as the idea of expanding bilinear forms in fermion creation and annihilation operators in Taylor series of boson operators, with the object of converting the study of nuclear vibrational motion into a problem of coupled oscillators. The physical situations of interest are quite diverse, depending, for instance, on whether excitations for fixed- or variable-particle number are being studied, on how total angular momentum is decomposed into orbital and spin parts, and on whether isotopic spin and other intrinsic degrees of freedom enter. As a consequence, all of the semisimple algebras other than the exceptional ones have proved to be of interest at one time or another, and all are studied in this review. Though the salient historical facts are presented in the introduction, in the body of the review the progression is (generally) from the simplest algebras to the more complex ones. With a sufficiently broad view of the physics requirements, the mathematical problem is the realization of an arbitrary representation of a Lie algebra in a subspace of a suitably chosen Hilbert space of bosons (Heisenberg-Weyl algebra). Indeed, if one includes the study of odd nuclei, one is forced to consider the mappings to spaces that are direct-product spaces of bosons and (quasi)fermions. Though all the methods that have been used for these problems are reviewed, emphasis is placed on a relatively new algebraic method that has emerged over the past decade. Many of the classic results are rederived, and some new results are obtained for odd systems. The major application of these ideas is to the derivation, starting from the shell model, of the phenomenological models of nuclear collective motion, in particular, the geometric model of Bohr and Mottelson and the more recently developed interacting boson model of Arima and Iachello. A critical discussion of those applications is interwoven with the theoretical developments on which they are based; many other applications are included, some of practical interest, some simply to illustrate the concepts, and some to suggest new lines of inquiry.

Non-neutral plasmas, like electrically neutral plasmas, exhibit a broad range of collective properties, such as plasma waves and instabilities, and the ability to support long-lived, large-amplitude coherent structures. This paper reviews the equilibrium and stability properties of intense non-neutral electron flow in crossed electric and magnetic fields. Following a description of equilibrium properties for magnetically insulated electron flow in planar geometry, extraordinary-mode stability properties are investigated for relativistic non-neutral electron flow between planar conductors. Particular emphasis is placed on the magnetron and diocotron instabilities, and detailed stability behavior is shown to exhibit a sensitive dependence on the self field intensity (as measured by the dimensionless parameter $s_e=\frac{{\gamma }_e^0{\omega }_{\mathrm{pe}}^2}{{\omega }_{\mathrm{ce}}^2}$) as well as on the shape of the equilibrium profiles. The influence of cylindrical effects (such as the centrifugal and Coriolis accelerations of an electron fluid element) on stability behavior is then investigated for rotating electron flow in cylindrical geometry. Finally, the properties of large-amplitude coherent structures in non-neutral plasmas with circulating electron flow are investigated. Topics covered in this area include particle-in-cell computer simulations of dense ($s_e\sim 1$) electron flow in relativistic magnetrons which show large-amplitude spoke formation in the circulating electron density, and application of a cold-fluid guiding-center model to investigate large-amplitude vortex structures in low-density ($s_e\ll 1$) non-neutral plasma. The accessibility and stability of such stationary structures (in the rotating frame) remain important topics for future investigation.

Recent experimental progress in the search for atomic electric dipole moments (EDMs) $d_A$ of cesium and thallium leads in particular to a substantially increased sensitivity to a possible electron EDM $d_e$ compared with existing upper bounds. Further considerable improvement in the measurement of $d_{\mathrm{Tl}}$ is likely. After a brief synopsis of the theory of atomic EDMs, the authors discuss—in view of the expected experimental sensitivity to $d_e$—the predictions for the electron EDM in various models of $\mathrm{CP}$ violation.

This article is a review of recent developments in the phenomenological description of unconventional superconductivity. Starting with the BCS theory of superconductivity with anisotropic Cooper pairing, the authors explain the group-theoretical derivation of the generalized Ginzburg-Landau theory for unconventional superconductivity. This is used to classify the possible superconducting states in a system with given crystal symmetry, including strong-coupling effects and spin-orbit interaction. On the basis of the BCS theory the unusual low-temperature properties and the (resonant) impurity scattering effects are discussed for superconductors with anisotropic pairing. Using the Ginzburg-Landau theory, the authors study several bulk properties of such superconductors: spontaneous lattice distortion, upper critical magnetic field, splitting of a phase transition due to uniaxial stress. Two possible mechanisms for ultrasound absorption are discussed: collective modes and damping by domain-wall motion. The boundary conditions for the Ginzburg-Landau theory are derived from a correlation function formulation and by group-theoretical methods. They are applied to a study of the Josephson and proximity effects if unconventional superconductors are involved there. The magnetic properties of superconductors that break time-reversal symmetry are analyzed. Examples of current and magnetic-field distributions close to inhomogeneities of the superconducting order parameter are given and their physical origin is discussed. Vortices in a superconductor with a multicomponent order parameter can exhibit various topological structures. As examples the authors show fractional vortices on domain walls and nonaxial vortices in the bulk. Furthermore, the problem of the possible coexistence of a superconducting and a magnetically ordered phase in an unconventional superconductor is analyzed. The combination of two order parameters that are almost degenerate in their critical temperature is considered with respect to the phase-transition behavior and effects on the lower and upper critical fields. Because heavy-fermion superconductors—which are possible realizations of unconventional superconductivity—have been the main motivation for the phenomenological studies presented here, the authors compare the theoretical results with the experimental facts and data. In particular, they emphasize the intriguing features of the compound U${\mathrm{Pt}}_3$ and consider in detail the alloy ${\mathrm{U}}_{1-x}{\mathrm{Th}}_x{\mathrm{Be}}_{13}$.

A number of examples are reviewed wherein profitable use has been made of idealizations that involve infinite or infinitesimal quantities. Contexts range from pendulums and reacting chemicals via diffusion and convection theory to duck-shaped limit cycles and economics.