There are speculations that grand unified theories (GUT's) may arise from higher-dimensional models of gravity in which the extra dimensions are spontaneously compactified. The GUT predictions for proton decay and ${{sin}^2\theta }_w$ can be significantly modified when nonrenormalizable interactions, scaled by inverse powers of the compactification scale $M_c$, are added to the standard GUT Lagrangian. For example, the decay rate for $p\to e^{+}{\pi }^0$ in minimal SU(5) can be lowered by one (two) orders of magnitude if $M_c$ is on the order of ${10}^{17}$ GeV. However, ${{sin}^2\theta }_w$ then decreases by 0.005 (0.01). The SO(10) model is also discussed.

Simulations and effective-medium calculations were performed on the absorption of impurity particles by static sinks in three dimensions. The theory shows no qualitative effects due to two-sink clusters and agrees well with simulations. The conclusion is drawn that effective-medium theory is adequate for time-dependent impurity concentrations $C\mathrm{}\left(t\right)\mathrm{}$ above the lower bound recently demonstrated by several authors. Crossing of the concentration curves occurs at a time such that $\frac{C\mathrm{}\left(t\right)\mathrm{}}{C\mathrm{}\left(0\right)\mathrm{}}={10}^{-67\mathrm{}}$ or less.

The theory of transport in nonlinear dynamics is developed in terms of "leaky" barriers which remain when invariant tori are destroyed. A critical exponent for transport times across destroyed tori is obtained which explains numerical results of Chirikov. The combined effects of many destroyed tori lead to power-law decay of correlations observed in many computations.

A new type of "crisis" is found in coupled logistic maps. In this crisis several chaotic attractors undergo cyclic collisions with their basin boundaries and merge into a bigger one, whose chaotic dynamical behavior exhibits cyclic transitions among its components with a well defined overall period.

The role of symmetry is examined in systems displaying period-doubling instabilities. It is found that symmetric orbits will not undergo period doubling except in extraordinary cases. Such exceptional cases cannot occur in a large class of systems, including the sinusoidally driven damped pendulum and the Lorenz equations.

Sinai billiard systems are studied numerically and analytically. Let $h$ be the Kolmogorov-Sinai entropy, $〈\tau 〉$ the mean free time of the point mass, and $\epsilon$ the scaling factor of the size of convex scatters (so that $\epsilon \to 0$ implies vanishing scatters). The following universal behavior is conjectured for any periodic Lorentz model in $d(>~2)$-space: ${\lim }_{\epsilon \to 0}h\frac{〈\tau 〉}{(-\ln \epsilon )}=d.\mathrm{}$

The quantum mechanical expression for the current density is a function of the external vector potential. As a result, the sources of $\overrightarrow{\mathrm{A}}\left(\overrightarrow{\mathrm{r}}\right)$ can induce spurious sources of $\overrightarrow{\mathrm{j}}\left(\overrightarrow{\mathrm{r}}\right)$, if the wave functions are not carefully chosen. The ideal infinite solenoid is given as an example, and boundary conditions which enforce momentum conservation across the solenoid wall are introduced. As a consequence of these boundary conditions, the wave function in the exterior region is not, in general, single valued; it is a single-valued function multiplied by the well known Dirac phase factor.

The inertial frame of reference in the neighborhood of a test body provided by the construction of Fermi normal coordinates is generalized to include the effect of the body's gravitational field. The metric obtained provides a simple physical description of relativistic corrections to the orbital motion of a satellite of the Earth. The main correction is the nonlinear Schwarzschild field of the Earth; in these coordinates there are also three much smaller terms arising from the solar tidal influence.

Time dependence of the fundamental "constants" is examined within the framework of Kaluza-Klein theories. Relationships among low-energy couplings and masses are derived. It is suggested that detection of a time variation in any of these parameters may provide evidence for extra space dimensions. Experimental bounds are reviewed and new measurements advocated.

An upper bound is given for the maximum number, $N_c$, of negative particles (fermions or bosons or a mixture of both) of charge $-e$ that can be bound to an atomic nucleus of charge $+\mathrm{ze}$. If $z$ is integral then $N_c<~2z$. In particular, this is the first proof that ${\mathrm{H}}^{--}$ is not stable. For a molecule, $N_c<~2Z+K-1$, where $K$ is the number of atoms in the molecule and $Z$ is the total nuclear charge.

With use of a novel Monte Carlo method suited for the study of surface phenomena, the authors have investigated a three-dimensional Ising model with modified surface exchange. The results verify the scaling behavior of the magnetization profile and confirm and extend renormalization-group predictions. The surface-layer magnetization exponent is ${\beta }_1=0.78\mathrm{}\pm 0.02$ except at the "special" (multicritical) transition where ${{\beta }_1}^m=0.175\mathrm{}\pm 0.025$ and the crossover exponent is $\varphi =0.56\mathrm{}\pm 0.04$.

The Einstein equations describing gravitational fields in vacuum are written as a compact exterior system of spinor-valued forms. A second system of equations is given, such that their integrability conditions are satisfied by virtue of the Einstein equations. This suggests the possibility of integrating the field equations by means of an inversetype procedure.

"Five-mode" massive aluminum gravitational radiation detectors cooled to millikelvin temperatures are considered for the purpose of approaching quantum sensitivity over a large bandwidth. A five-mode detector with a center frequency at 800 Hz would have a two-sided usable bandwidth of the order of 400 Hz. A five-mode detector with a center frequency at 1660 Hz would have a usable two-sided bandwidth of the order of 1000 Hz.

It is found that the level fluctuations of the quantum Sinai's billiard are consistent with the predictions of the Gaussian orthogonal ensemble of random matrices. This reinforces the belief that level fluctuation laws are universal.

A reflecting convex obstacle is uniquely defined by the scattering amplitude $f(k_0,{\nu }_0,n)$ known at a fixed frequency $k_0$, for a fixed direction ${\nu }_0$ of the incident wave, and for all directions $n$ of the scattered waves in a solid angle. Earlier the uniqueness theorem was proved under the assumption that $f(k,{\nu }_0,n)$ is known for $a<~k<~b$, $b>a$, and all $n$.

The authors study a symmetric two-state system with "bare" tunneling frequency ${\Delta }_0$, dimensionless Ohmic dissipation coefficient $\alpha$, and heat-bath cutoff ${\omega }_c$. Defining ${\Delta }_r\equiv {\Delta }_0{\left(\frac{{\Delta }_0}{{\omega }_c}\right)}^{\frac{\alpha }{(1-\alpha )}}$ for $\alpha <1\mathrm{}$ and ${\Delta }_r\equiv 0$ for $\alpha >1\mathrm{}$, they find to lowest order in $\frac{{\Delta }_r}{{\omega }_c}$, $\frac{\mathrm{kT}}{\hslash {\omega }_c}$ (a) for all $\alpha \mathrm{kT}\gg \hslash {\Delta }_r$, incoherent relaxation at a rate $\left(\frac{{{\Delta }_0}^2}{{\omega }_c}\right)\left(\frac{\sqrt{\pi }}{2}\right)\left[\frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha +\frac{1}{2})}\right]\times {\left(\frac{\pi \mathrm{kT}}{\hslash {\omega }_c}\right)}^{2\alpha -1\mathrm{}}$; (b) for $T=0\mathrm{}$, $\frac{1}{2}<\alpha <1\mathrm{}$, incoherent relaxation at a rate $\sim {\Delta }_r$; and (c) for $T=0\mathrm{}$, $0<\mathrm{}\alpha <\frac{1}{2}$, damped oscillations with frequency $\sim {\Delta }_r$ and $Q$ factor $\frac{1}{2}cot\left[\frac{\left(\frac{\pi }{2}\right)\alpha }{(1-\alpha )}\right]$ plus a power-law background.

It is shown that the existence of a potential is not a generic property of a continuous nonlinear dissipative system but requires the complete integrability of an associated Hamiltonian system.

Numerical evidence of a percolation phase transition on infinitely ramified exact fractals is presented. The percolation transition is studied by a real-space renormalization-group technique on a family of exact fractal lattices with fractal dimensionality $\overline{d}$ between 1 and 2. The critical exponents for percolation depend strongly on the geometry of the fractals.

The gravitational field produced by an infinite static cylinder with a net spin polarization along its symmetry axis is considered, in the context of the Einstein-Cartan theory of gravitation. An exact solution is obtained, which shows that the cylinder's spin angular momentum gives rise to magneticlike components in the gravitational field outside.

A new computational approach to time-dependent transition amplitudes in multistate quantum systems is developed. The method, which avoids explicit computation of matrix eigenvectors, is based upon the recursive generation of residues of Green's functions. The residues are computed from the eigenvalues of several tridiagonal matrices. Numerical results are presented for multiphoton excitation of overtones in a system with over 3000 states.