The theory is developed for the second moment of the first-passage-time variable of statistical flows with memory (non-Markovian dynamics) and applied to a model of free non-Markovian Brownian motion. More importantly, a novel quantity, called memory index $\sigma$, is introduced, giving a quantitative characterization of the influence on non-Markovian behavior. Thus, a theoretical or experimental evaluation of $\sigma$ allows one to assess the quality of widely used approximative (Markovian) Fokker-Planck-type descriptions of the dynamics.

Unitary representations of the infinite-parameter group $\mathrm{Diff}\left(R^3\right)$ are presented which describe particles with spin as well as tightly bound composite particles. These results support the idea that $\mathrm{Diff}\left(R^3\right)$ can serve as a "universal group" for quantum theory.

The spin and statistics of solitons in the (2+1)- and (3+1)-dimensional nonlinear $\sigma$ models is considered. For the (2+1)-dimensional case, there is the possibility of fractional spin and exotic statistics; for 3+1 dimensions the usual spin-statistics relation is demonstrated. The linking-number interpretation of the Hopf invariant and the use of suspension considerably simplify the analysis.

A generalization of the Einstein-Green-Kubo self-diffusion coefficient is obtained for a fluid in uniform shear flow. The associated nonlinear transport coefficients do not exist, because of a nonanalytic dependence on shear rate. The relationship of this anomalous nonlinearity to the long-time tails of the velocity autocorrelation function at zero shear rate is determined from mode-coupling theory. A universal function for the crossover between frequency and shear-rate dominated limits is found.

The authors have evaluated the coefficients of the first nonanalytic logarithmic term in the density expansion for the transport properties of a gas of hard spheres, which is due to long-range dynamical correlations between molecules. Their results resolve present uncertainties about the magnitude of the effect, appear to be consistent with available information from molecular-dynamics calculations, and satisfy bounds for the logarithmic contribution deduced from experimental viscosity data for real gases.

It is shown that the "geometrical phase factor" recently found by Berry in his study of the quantum adiabatic theorem is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem naturally defines a connection in such a bundle. This not only takes the mystery out of Berry's phase factor and provides calculational simple formulas, but makes a connection between Berry's work and that of Thouless et al. This connection allows the author to use Berry's ideas to interpret the integers of Thouless et al. in terms of eigenvalue degeneracies.

It is shown that a modulationally unstable pattern is selected and propagates into an initially unstable motionless state in the one-dimensional generalized Ginzburg-Landau equation. A further spatiotemporal transition occurs with a sharp interface from the selected unstable pattern to a stabilized pattern or a chaotic state. The distinct transition makes a coherent structure coexist with a chaotic state.

Renormalization-group methods are used to produce a novel Monte Carlo sampling algorithm which alleviates the problem of critical slowing down. The method is applied by means of a Kadanoff block-spin transformation to the one- and two-dimensional Ising models.

An exact solution of the Schrödinger equation for a charge $e$ in the potential of a flux tube along the $z$ axis and a Dirac magnetic monopole of strength $g$ at the origin is obtained. Single-valued solutions exist even with $2eg\ne \mathrm{integer}$. When $2eg=\mathrm{integer}$ and single-valuedness holds, solutions with $\left|J_z\mathrm{}\right|<\mathrm{}\left|eg\right|$ have a centrifugal barrier (and hence behavior at the origin) independent of flux, but solutions with $\left|J_z\mathrm{}\right|>\mathrm{}\left|eg\right|$ have a flux-dependent centrifugal barrier.

A phenomenological Landau model which describes a simple continuous transition from a high-symmetry phase to a phase associated with a nonmaximal isotropy subgroup—hitherto conjectured impossible—is constructed. A novel feature of the model is that a single order parameter describes successive simple continuous phase transitions between three or more phases of different symmetries. Effects of fluctuations are considered within a renormalization-group approach.

The tomographic transform is combined with Susskind-like methods for lattice fermions to yield a new formulation with no multiplicity of species and with the usual discrete chiral symmetry.

The polarization of ${\Xi }^0$ and $\Lambda$ hyperons produced by 400-GeV protons interacting with a beryllium target has been measured in the projectile fragmentation region. The ${\Xi }^0$ polarization agrees in sign, magnitude, and kinematic behavior with that of $\Lambda$. The target dependence of these ${\Xi }^0$ and $\Lambda$ polarizations was also investigated with use of Cu and Pb targets.

A quantum interference experiment is proposed which can unambiguously test for odd half-integer eigenvalues in the angular momentum spectra predicted for a spinless charged particle orbiting, but not penetrating, a solenoid with confined magnetic field or a superconducting vortex with odd integer number of flux quanta.

A phenomenological model of dendritic solidification incorporating interfacial kinetics, crystalline anisotropy, and a local approximation for the dynamics of the thermal diffusion field is proposed. The preliminary results are in qualitative agreement with natural dendritelike pattern formation.

The finite-element approach described in a previous paper approximates the operator Dirac equation on a lattice in such a way that the equal-time anticommutation relations are exactly maintained. Moreover, the approximation preserves unitarity (and chiral invariance in the massless theory) and there is no fermion doubling.

The claim that the Clauser-Horne inequality $S\left(\varphi \right)<~1$ is valid for all local theories is criticized. The criticism is based on the author's proof that this universality claim is incompatible with the conservation law of angular momentum. There exists an infinity of counterexamples to the universality claim made for $S\left(\varphi \right)<~1$. So Clauser and Horne offer no valid proof of the nonlocality of the quantum formalism; consequently experiments like Aspect's cannot be said to refute Einsteinian locality.

The Hamiltonian of a system containing decaying states is subjected to a nonunitary similarity transformation which introduces the wave functions of these decaying states as well as those of the "antidecaying" dual states explicitly into the representation. The dependence of the Green's-function pole on these wave functions is used to formulate a variational principle for their determination, yielding a non-Hermitian, nonlinear wave equation including self-energy contributions. Explicit solutions are given in a simple example.

An asymptotic solution to interacting walks in one dimension is presented. For repulsive and attractive interactions, respectively, the model describes aspects of uncorrelated diffusion on a lattice with sources or with randomly distributed traps. In the latter case, the average number of sites visited varies with the number of steps in the walk as $N^{\frac{1}{3}}$, while the survival probability decays as $N^{-\frac{1}{3}}\exp (-b{N}^{\frac{1}{3}})$, improving on previous predictions for diffusion with traps.

Percolation critical exponents are derived, for the first time, for a two-dimensional system of randomly distributed conducting sticks, which provides a very convenient model for the study of continuum percolation. In the present computer study it was found that the corresponding conductivity exponent, $t$, has the value of 1.24±0.03 and that the cluster exponents $\beta$, $\gamma$, and $\tau$ have the values 0.14±0.02, 2.3±0.2, and 2.0±0.1, respectively. These results, which are in excellent agreement with values derived for lattices, show that the conductivities of continuum systems and of lattice systems belong to the same universality class.

Results are presented from the analysis of solar system astrometric data, notably the range data to the Viking landers on Mars. A least-squares fit of the parameters of the solar system model to these data limits a simple time variation in the effective Newtonian gravitational constant to (0.2 ± 0.4) × ${10}^{-11\mathrm{}}$ ${\mathrm{yr}}^{-1\mathrm{}}$ and a rate of drift of atomic clocks relative to the implicit clock of relativistic dynamics to (0.1 ± 0.8) × ${10}^{-11\mathrm{}}$ ${\mathrm{yr}}^{-1\mathrm{}}$. The error limits quoted are the result of uncertainties in the masses of the asteroids.