Self-avoiding walks, lattice trees, and related geometrical models provide a link between the physics of polymers and the study of critical phenomena. In particular, these models in the presence of a surface provide insight into surface adsorption in dilute polymer systems in a good solvent. The theme of this review is the influence of polymer structure (topology) on the critical properties of these models. Emphasis is placed on recent results by rigorous methods, scaling theory, and conformal covariance theory. Numerical results that may be used to test the predictions of scaling of scaling and conformal covariance theories are also summarized. Related topics such as the adsorption of directed polymers, the semidilute regime, the theta point and theta solvents, and percolation (polymer gels) are briefly discussed in the final section.

An up-to-date review of the nucleon-nucleon scattering situation is presented in the intermediate-energy region (100 MeV up to a few GeV). Total cross-section measurements are discussed, but the main emphasis is on the spin physics. Technical advances for polarized beams and targets are presented. The most recent spin-dependent $\mathrm{NN}$ data are reviewed and their influence in phase-shift analyses is discussed. The direct reconstruction of the scattering amplitudes is studied and results are compared to those obtained by phase-shift analyses. Finally future plans and expected improvements are detailed.

A major goal in nuclear physics is to understand the stability, structure, and reactions of nuclei as a consequence of the interactions between individual nucleons. This colloquium describes one attempt to build a consistent picture of nuclear systems ranging in size from deuterons to neutron stars. The main ingredients are a nonrelativistic Hamiltonian containing two- and three-nucleon interactions, and correlated variational wave functions as approximate solutions of the many-body Schrödinger equation. A model that fits both two-body scattering data and nuclear masses will provide the best theoretical input for neutron star properties.

Over the past seven years, many examples of periodic crystals closely related to quasicrystalline alloys have been discovered. These crystals have been termed approximants, since the arrangements of atoms within their unit cells closely approximate the local atomic structures in quasicrystals. This colloquium focuses on these approximant structures, their description, and their relationship to quasicrystals.

Electrophotography is one means of arranging 100 million pigmented plastic particles on a sheet of paper to faithfully replicate an original. It is based on many diverse phenomena and employs many properties of matter. These include gaseous ionization in the charging step; photogeneration and charge transport through disordered solid-state materials in the latent-image-formation step; triboelectricity in the particle-charging step; mechanical, electrostatic, and magnetic forces to detach particles in the development and transfer steps; and the application and transfer of heat in the fixing step. In addition, it relies on a precise balance of thermorheological, chemical, and mechanical properties of large area films and small particles. This article reviews the physics of the latent-image formation and development steps.

The properties of the WKB method in the discrete representation are reviewed. The method provides the eigenvalues and the eigenvectors of the three-term recursion relations or, which is the same thing, the tridiagonal band matrices. Applications of the method to the splitting of the Rydberg atom levels in the external electric and magnetic fields are considered. Analytical treatment is given to the problem of the oscillator strength distribution in the quadratic Zeeman and the Stark-Zeeman spectra. In the case of the nonhydrogenic Rydberg atoms, the effect of the core on the pattern of the splitting is studied. Certain alternative applications of the discrete WKB method are considered in brief (the quasienergy spectra of nonlinear oscillators in resonant fields, rotational molecular spectra, calculation of infinite continued fractions).

Correlation functions are one of the key tools used to study the structure of the QCD vacuum. They are constructed out of the fundamental fields and can be calculated using quantum-field-theory methods, such as lattice gauge theory. One can obtain many of these functions using the rich phenomenology of hadron physics. They are also the object of study in various quark models of hadronic structure. This review begins with available phenomenological information about the correlation functions, with their most important properties emphasized. These are then compared with predictions of various theoretical approaches, including lattice numerical simulations, the operator product expansion, and the interacting instanton approximation.

A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential $\psi \left(\mathrm{x}\right)$. For rapidly decaying correlations of $\psi$, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.

This article reviews the physics of deep-inelastic lepton-hadron processes in the region of small $x$, where $x$ is the Bjorken scaling variable. The theoretical concepts concerning the Regge limit of deep-inelastic scattering are summarized and recent theoretical results on the small-$x$ limit of parton distributions in perturbative QCD are discussed. Presently available experimental data on the free- and bound-nucleon structure functions at small $x$ are reviewed in detail and their theoretical interpretations (including the low-$x$, low-$Q^2$ region) are discussed. QCD predictions are given for the deep-inelastic scattering structure functions $F_2$ and $F_L$ in the small-$x$ (${10}^{-5\mathrm{}}<x<\mathrm{}{10}^{-2\mathrm{}}$) and moderately large-$Q^2$ region relevant for HERA ($Q^2\sim 10$ Ge${\mathrm{V}}^2$).

We first show that, with the same input parameters, the standard solar models of Bahcall and Ulrich; of Sienkiewicz, Bahcall, and Paczyński; of Turck-Chièze, Cahen, Cassé, and Doom; and of the current Yale code all predict event rates for the chlorine experiment that are the same within ±0.1 SNU (solar neutrino units), i.e., approximately 1% of the total calculated rate. We then construct new standard solar models using the Yale stellar evolution computer code supplemented with a more accurate (exportable) nuclear energy generation routine, an improved equation of state, recent determinations of element abundances, and the new Livermore (OPAL) opacity calculations. We evaluate the individual effects of different improvements by calculating a series of precise models, changing only one aspect of the solar model at a time. We next add a new subroutine that calculates the diffusion of helium with respect to hydrogen with the aid of the Bahcall-Loeb formalism. Finally, we compare the neutrino fluxes computed from our best solar models constructed with and without helium diffusion. We find that helium diffusion increases the predicted event rates by about 0.8 SNU, or 11% of the total rate, in the chlorine experiment; by about 3.5 SNU, or 3%, in the gallium experiments; and by about 12% in the Kamiokande and SNO experiments. The best standard solar model including helium diffusion and the most accurate nuclear parameters, element abundances, radiative opacity, and equation of state predicts a value of 8.0±3.0 SNU for the ${}_{}{}^{37}{}_{}{}^{}\mathrm{Cl}$ experiment and ${132}_{-17\mathrm{}}^{+21\mathrm{}}$ SNU for the ${}_{}{}^{71}{}_{}{}^{}\mathrm{Ga}$ experiment. The quoted errors represent the total theoretical range and include the effects on the model predictions of $3\sigma$ errors in measured input parameters. All 15 calculations since 1968 of the predicted rate in the chlorine experiment given in this series of papers are consistent with both the range estimated in the present work and the 1968 best-estimate value of 7.5±2.3 SNU. Including the effects of helium diffusion and the other improvements in the description of the solar interior that are implemented in this paper, the inferred primordial solar helium abundance is $Y=0.273\mathrm{}$. The calculated depth of the convective zone is $R=0.707\mathrm{}{R}_{\odot }$, in agreement with the value of $0.713R_{\odot }$ inferred by Christensen-Dalsgaard, Gough, and Thompson from a recent analysis of the observed $p$-mode oscillation frequencies. Including helium diffusion increases the calculated present-day hydrogen surface abundance by about 4%, decreases the helium abundance by approximately 11%, and increases the calculated heavy-element abundance by about 4%. In the Appendix, we present detailed numerical tables of our best standard solar models computed both with and without including helium diffusion. In the context of the MSW (Mikheyev-Smirnov-Wolfenstein) or other weak-interaction solutions of the solar neutrino problem, the numerical models can be used to compute the influence of the matter in the sun on the observed neutrino fluxes.

The remarkable agreement of electroweak measurements with theory, places limits on the masses of the top quark and the $W$ boson. It is shown how these limits arise and what constraints each set of measurements provides within the context of the theory.

Protonated hydrogen, ${\mathrm{H}}_3^{+}$, composed of three protons and two electrons, is the simplest stable polyatomic system. It is the most abundant ionic species in molecular hydrogen plasmas, and it is assumed to play a central role in the chemical evolution of molecular clouds, the birthplace of stars. The existence of this species has been known since its discovery by J. J. Thomson in 1912, but its first spectrum was observed only in 1980. The infrared spectrum has since been extended to higher vibrational and rotational states. The existence of a large amount of ${\mathrm{H}}_3^{+}$ in Nature was first demonstrated in the serendipitous discovery of its infrared emission bands in the polar auroral regions of the Jupiter ionosphere. The spectrum is extremely intense and pure and can be used to monitor the morphology and temporal variation of the dynamic Jovian plasmas. Early this year, an identification of ${\mathrm{H}}_3^{+}$ emission was claimed in Supernova 1987A. In April strong ${\mathrm{H}}_3^{+}$ infrared emission was detected in Uranus. These developments are reviewed and the future is projected.

On the basis of current physical understanding, it is impossible to predict with confidence the interior constitution of neutron stars. Cooling of neutron stars provides a possible way of discriminating among possible states of matter within them. In the standard picture of cooling by neutrino emission developed over the past quarter of a century, neutron stars are expected to cool relatively slowly if their cores are made up of nucleons, and to cool faster if matter is in an exotic state, such as a pion condensate, a kaon condensate, or quark matter. This view has recently been called into question by the discovery of a number of other processes that could lead to copious neutrino emission and rapid cooling.

This is a tutorial review on knot invariants and their construction using the method of statistical mechanics. We begin with brief reviews of the elements of knot theory and relevant results in statistical mechanics. We then show how knot invariants, including those discovered recently, can be obtained by applying techniques used in solving lattice models in lattice statistics. Our approach is based on the consideration of solvable models with strictly local Boltzmann weights. The presentation, which is self-contained and elementary, is intended for a general readership. A table of polynomial invariants for knot and links containing up to six crossings is included in the Appendix.

This article describes recent technical developments that have made the total-energy pseudopotential the most powerful ab initio quantum-mechanical modeling method presently available. In addition to presenting technical details of the pseudopotential method, the article aims to heighten awareness of the capabilities of the method in order to stimulate its application to as wide a range of problems in as many scientific disciplines as possible.

This article describes the Electron Echo project, which used artificial electron beams to determine how natural particles are accelerated and lost from the trapping region in the earth's magnetosphere. The problems of injecting electron beams into the ionosphere, including vehicle charging and beam-plasma instabilities, are discussed, and images of the Echo beams both in space and in the laboratory are shown. The beam pulses were detected and analyzed after reflection from the conjugate hemisphere mirror points. Although the injected beams contained a wide spread of energies, the echoes displayed a monoenergetic peak, which together with the exact drift displacement during one bounce permitted a separate evaluation of electric and magnetic components of the transverse drifts in the distant magnetosphere. Thus distant electric fields were measured and found to have large turbulent fluctuations compared to the local ionospheric fields, and to differ in average value from the local fields by about 50 mV/m, indicating the existence of parallel potential drops. The measured bounce times and field-line lengths were compared with model magnetic geometries. The best fit was with the Tsyganenko-Usmanov 1982 version. Pitch-angle diffusion out of the loss cone resulted in a large loss in beam intensity during one bounce, showing the presence of important beam perturbations near the electron gyrofrequency range. Thus the Echo experiments evaluated many effects on the natural trapped particles, but were limited to a moderately disturbed condition of the magnetosphere when the magnetic morphology was stable.

The discovery of periodic conductance oscillations as a function of charge density in very small transistors has led to a new understanding of the behavior of electrons in such small structures. It has been demonstrated that, whereas a conventional transistor turns on only once as electrons are added to it, submicronsize transistors, isolated from their leads by tunnel junctions, turn on and off again every time an electron is added. This unusual behavior is primarily the result of the quantization of charge and the Coulomb interaction between electrons on the small transistor. However, recent experiments demonstrate that the quantization of energy is important as well.

Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond to Hamiltonians for which the velocity is a monotonic function of the canonical momentum. Twist maps have a Lagrangian variational formulation. One-parameter families of twist maps typically exhibit the full range of possible dynamics-from simple or integrable motion to complex or chaotic motion. One class of orbits, the minimizing orbits, can be found throughout this transition; the properties of the minimizing orbits are discussed in detail. Among these orbits are the periodic and quasiperiodic orbits, which form a scaffold in the phase space and constrain the motion of the remaining orbits. The theory of transport deals with the motion of ensembles of trajectories. The variational principle provides an efficient technique for computing the flux escaping from regions bounded by partial barriers formed from minimizing orbits. Unsolved problems in the theory of transport include the explanation for algebraic tails in correlation functions, and its extension to maps of more than two dimensions.

The random distribution of impurities in a semiconductor host lattice introduces potential fluctuations that allow energy levels within the forbidden energy gap. This statistical effect distorts the unperturbed density of states of the pure semiconductor, and, at high doping concentrations, substantial band tails appear. The changes in the density-of-states function are particularly important in determining the number of free carriers in a heavily doped semiconductor. Together with many-particle interactions, band tailing constitutes one of the most significant heavy-doping effects. Although the band-tailing phenomenon has been studied for many years, only a one-dimensional analytical model, which assumes a Gaussian whitenoise probability distribution of the potential fluctuation, exists. In this paper the different classes of theories that describe this band tailing of the density of states in heavily doped semiconductors are reviewed in detail.