Vector Bäcklund transformations which relate solutions of the vacuum Einstein equations having two commuting Killing fields are introduced. Such transformations generalize those found by Pohlmeyer in connection with the nonlinear $\sigma$ model. A simple algebraic superposition principle, which permits the combination of Bäcklund transforms in order to get new solutions, is given. The superposition preserves the asymptotic flatness condition, and the whole scheme is manisfestly O(2,1) invariant.

The phenomenon of dimensional reduction in quantum theories of gravity is studied. In the five-dimensional Kaluza-Klein model, the one-loop effective potential as a function of $g_{55}$ is computed. When the distance around the fifth dimension is larger than the Planck length, the loop expansion is reliable. The result leads to a force tending to make the fifth dimension contract to a size on the order of the Planck length. This can be interpreted as a gravitational version of the Casimir effect in electrodynamics.

It is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal lattices of low lacunarity. These lattices are special examples of Sierpinski carpets. Their being of low lacunarity means that they are arbitrarily close to being translationally invariant. The claim is substantiated for the Ising model in $D=1+\epsilon$ dimensions, and for resistor network models with $1<D<\mathrm{}2\mathrm{}$.

A quantum action variable is defined in the context of a quantum Hamilton-Jacobi theory. The action variable can be used to find the exact bound-state energy levels of a quantum system without solving an equation of motion for the system wave functions.

The large-scale stochasticity threshold for Hamiltonians with two degrees of freedom with only one primary resonance can be analytically estimated because of the quite steep growth of the stochastic layer of this resonance. For Hamiltonians without primary resonance, the threshold is computed by available techniques after canonical transformations. Thus the first analytical estimate of the threshold is obtained for the Hénon-Heiles Hamiltonian and an example of Walker and Ford.

A breather soliton of the sine-Gordon equation is shown to behave stochastically in the presence of an external oscillating field, which leads to random pair creations of kink and antikink solitons.

It is shown that $\int {\left|V\right(\overrightarrow{\mathrm{x}}\left)\right|}^{\frac{3}{2}}{d}^{3}x>\left(\frac{{\pi }^2}{4}\right){\left(\frac{3{\hslash }^2}{2m}\right)}^{\frac{3}{2}}$ is a necessary condition for one or more normalizable bound-state solutions to the nonrelativistic Schrödinger energy-eigenvalue equation for any potential function $V\left(\overrightarrow{\mathrm{x}}\right)$ such that the integral is finite. Moreover, for potentials such that $\int {\left|V\right(\overrightarrow{\mathrm{x}}\left)\right|}^{\frac{3\gamma }{2}}{d}^{3}x$ is finite for a value or values of $\gamma >1\mathrm{}$, the magnitude of a negative energy eigenvalue is related to the latter integral by a general inequality derived here.

Power spectra and phase-space trajectories for a sine-wave-driven Josephson-junction analog are measured, showing period-doubling bifurcations and transition to chaos. The total noise power follows closely a universal power law with a critical exponent σ and the average bandwidth in the chaotic regime scales with a universal number $\beta$, both in good agreement with theoretical predictions. Measured values of other universal numbers are reported.

It is shown that for systems developing stationary periodic patterns there exists at most one stable wavelength state if a supercritical region is connected to a subcritical one by the imposition of a slow spatial variation of the external parameters. In nonpotential systems the selected wavelength depends on the particular combination of parameters that vary but not on the (slow) rate of spatial variation. Suitable parameter variations force the system into a dynamic state.

Molecular-dynamics calculations for interacting waterlike particles are used to estimate values of percolation exponents. Results are consistent with accepted values for ordinary random-bond percolation for a three-dimensional lattice. Study of a correlated-site percolation problem, defined by the connectivity of those water molecules that are four-coordinated, shows no evidence for different exponents. Thus there is no obvious basis for the frequent criticisms against the application of lattice percolation models to describe continuum correlated systems such as polymer gels.

The authors have studied whether numerically generated sequences from the logistic parabola $f_b\mathrm{}\left(x\right)=4\mathrm{bx}(1-x)\mathrm{}$ with $b$, $x\in \mathrm{}[0,1]\mathrm{}$, for values of $b$, above the Feigenbaum critical value $b_{\infty }$, are truly chaotic or whether they are periodic but with exceedingly large periods and very long transients. Using the logistic parabola the authors calculate via Monte Carlo simulation the average walk length for trapping on a one-dimensional lattice with a centrosymmetric trap. Comparison with exact results suggests that the only "truly chaotic" sequence is the one for which $b=1\mathrm{}$.

Correlations of linear polarizations of pairs of photons have been measured with time-varying analyzers. The analyzer in each leg of the apparatus is an acousto-optical switch followed by two linear polarizers. The switches operate at incommensurate frequencies near 50 MHz. Each analyzer amounts to a polarizer which jumps between two orientations in a time short compared with the photon transit time. The results are in good agreement with quantum mechanical predictions but violate Bell's inequalities by 5 standard deviations.

An experiment has been performed to demonstrate a new source-independent null test of the inverse square law of gravitation. A single-axis superconducting gravity gradiometer was rotated into three orthogonal orientations to measure the Laplacian of the gravitational potential produced by a 1600-kg lead pendulum at an average distance of 2.3 m. The result is that if one assumes a potential of the form $\varphi \left(\mathcal{r}\right)=-\left(\frac{\mathrm{GM}}{\mathcal{r}}\right)[1+\alpha \exp (-\mu \mathcal{r}\left)\right]$, the value of $\alpha$ is + 0.024±0.036 at ${\mu }^{-1\mathrm{}}=1\mathrm{}$ m.

The connectivity properties of liquid water are studied with molecular dynamics. It is found that the hydrogen-bonded network includes tiny spatially correlated "patches" of four-bonded molecules, and that the local density near a "patch" is lower than the global density. The diameter of the finite patches increases with decreasing temperature, and is comparable to the characteristic length scale of density fluctuations recently measured by small-angle x-ray scattering.

It is shown that the entropy of the radiation evaporated by an uncharged, nonrotating black hole into vacuum in the course of its lifetime is approximately $\frac{4}{3}$ times the initial entropy of this black hole. Also considered is a thermodynamically reversible process in which an increase of black-hole entropy is equal to the decrease of the entropy of its surroundings. Implications of these results for the generalized second law of thermodynamics and for the interpretation of black-hole entropy are pointed out.

The influence at criticality of a "far" wall on the order parameter close to a "near" wall as discussed by Fisher and de Gennes is recast in a form more suitable for microscopic analysis. An argument beyond ordinary scaling is presented which yields the perturbation exponent $d^{*}=d$ for longitudinal perturbations and ${d_{\perp }}^{*}=d-1\mathrm{}$ for transverse perturbations. These results are consistent with explicit renormalization-group computations to $O(\epsilon =4-d)$ which involve solution of the nonlocal equation of state.

The intermittent-type chaos occurring in rf- and dc-current-driven Josephson junctions is investigated. A simple physical model is proposed and is used for an analytic calculation of the power spectrum. The latter has a broadband part which decays algebraically at high frequencies. Comparison with numerical results shows good agreement. Further application and generalizations of this approach are outlined.

The discovery of new integrable two-dimensional Hamiltonian systems is reported. The analytic structure of the solutions makes necessary the generalization of the Painlevé conjecture, a widely used integrability criterion. Such a generalization is presented, which the authors believe should replace the usual conjecture for two-dimensional Hamiltonian systems. It is indeed compatible with all the systems already found and, in addition, leads to still new integrable cases.

The relativistic corrections to the Newtonian tidal accelerations generated by a rotating system are studied. The possibility of testing the relativistic theory of gravitation by measuring such effects in a laboratory in orbit around the Earth is considered. A recent proposal to measure a rotation-dependent tidal acceleration as an alternative to the Stanford gyroscope experiment is critically examined and it is shown that such an experiment does not circumvent the basic difficulties associated with the gyroscope experiment.

A quantum mechanical particle which moves in a symmetric double-well potential, and whose interaction with the environment is described in the classical regime by a phenomenological friction coefficient $\eta$, is considered. It is shown that, provided $\frac{\eta {q_0}^2}{\hslash }$ exceeds a critical value of order unity ($\pm q_0$ are the locations of the potential minima), the mean rate of tunneling between the degenerate minima decreases with temperature, leading at $T=0\mathrm{}$ to spontaneous symmetry breaking.