Considered here is a class of theories of gravity characterized by a set of equations which represent the gravitational and electromagnetic structure of the theories in a spherically symmetric and static gravitational field. If one demands that the weak equivalence principle (WEP) and the principle of universality of gravitational red shift (UGR) be satisfied, it is found that the theories under investigation must be metric. This result lends support to the current version of Schiff's conjecture that WEP+UGR→EEP, where EEP refers to the Einstein equivalence principle.

Percolation theory is extended to the case of circuit elements with nonlinear $I-V$ characteristics, particularly the special cases $V=I^{\alpha }\mathcal{r}$, which form the universality classes. Near the percolation threshold the "conductance" $\frac{I}{V^{\frac{1}{\alpha }}}$ vanishes like ${(p-p_c)}^t$, where $t$ depends on $\alpha$ and dimensionality. The Skal-Shklovskii-De Gennes model gives $t\left(\alpha \right)=(d-1\mathrm{})\mathrm{}\nu +\frac{(\varphi -\nu )}{\alpha }$, where $\varphi$ is only weakly dependent on $\alpha$ and $d$, and approaches unity at $d=6\mathrm{}$. Renormalization methods are used to study the $\alpha$ dependence of $t$ in two dimensions.

A one-dimensional quantum mechanical model defined by the "quadratic mapping Hamiltonian" which is almost periodic is presented. The model admits a Cantor spectrum of Lebesgue measure zero with a singular continuous measure and produces extended states displaying an unexpected chaotic behavior at large distances.

Let $G$ be the time-dependent quantum mechanical propagator for a particle that is free except for a half-plane barrier with a straight edge. A closed-form exact expression for $G$ is presented in terms of the Fresnel integral for the case where the initial and final positions lie in a plane perpendicular to the edge. $G$ is not expressible as a sum over classical paths.

Analytic results have been obtained for the thermodynamics of the two-dimensional $S=\frac{1}{2}$ $X-Y$ model using the simplest form of the generalized Trotter formula for the partition function. The model exhibits a phase transition of unconventional nature. It is claimed that this is one of the best descriptions given so far of the critical thermodynamics of the $S=\frac{1}{2}$ two-dimensional $X-Y$ model. Central in the present approach is the mapping of a family of two-dimensional $S=\frac{1}{2}$ models on staggered eight-vertex models having unconventional weights.

Irrational decimation schemes are constructed for functional integrals with the very interesting feature of preserving several distinct site Hamiltonians into the fixed-point limit. This method is applied to the quasiperiodicity transition to turbulence in order to compute the effects of external noise.

A periodically kicked quantum rotator is related to the Anderson problem of conduction in a one-dimensional disordered lattice. Classically the second model is always chaotic, while the first is chaotic for some values of the parameters. With use of the Anderson-model result that all states are localized, it is concluded that the local quasienergy spectrum of the rotator problem is discrete and that its wave function is almost periodic in time. This allows one to understand on physical grounds some numerical results recently obtained in the context of the rotator problem.

The Brownian motion of a general classical anharmonic oscillator is studied in the lowviscosity limit for a general non-Markoffian interaction with a heat bath. Memory effects are shown to have a profound influence on the rate of energy accumulation and relaxation.

The rotational quantum mechanics of a new analytic model for a hindered rotor is presented, and rotational-state distributions of the hindered rotor are given in terms of unhindered rotor states.

The size dependence of slow relaxation processes in the infinite-range Ising spin-glass is investigated by computer simulation. Below the transition temperature, relaxation of variables which do not change sign under inversion of the spins is complete by a time $\tau$, where ln $\tau \propto N^{\frac{1}{4}}$ and so diverges as $N$, the number of spins, tends to infinity. The "ergodic time" ${\tau }_{\mathrm{eg}}$ satisfies ln ${\tau }_{\mathrm{eg}}\propto N^{\frac{1}{2}}$. These results are consistent with a physical picture where barriers between free-energy minima in phase space have a height proportional to the square root of the number of spins to be flipped.

It is shown, from microscopic considerations based on the Fokker-Planck equation, that the boundary condition (used in conjunction with the diffusion equation) in which the particle density is set to zero on a perfectly absorbing surface is untenable, and, for the first time, the boundary condition for any plane (partially or perfectly) absorbing surface is derived.

Experiments on the Belousov-Zhabotinskii reaction in a stirred flow reactor reveal behavior that is strikingly similar to that generated by one-dimensional maps with a single extremum. In particular, a period-doubling sequence is observed that leads to a regime containing both chaotic and periodic states. Within the experimental resolution the ordering of the periodic states is in accord with the theory of one-dimensional maps.

It is shown rigorously that the one-dimensional Ising model with long-range antiferromagnetic interactions exhibits a complete devil's staircase.

A simple inequality relating the root mean square deviations of position and osmotic velocity for a diffusion process is presented and, in the framework of Nelson's stochastic mechanics, is related to the Heisenberg position-momentum uncertainty relations.

Expressions are given for the friction force and velocity autocorrelation function of a hard-spherical Brownian particle with "stick" boundary conditions in a fluid with internal rotational degrees of freedom. The velocity autocorrelation function is found to have a $t^{-\frac{3}{2}}$ tail. Comparison with recent experiments indicates that rotational transport coefficients may be measurable by means of light-scattering experiments on Brownian particles. Preliminary estimates are given for rotational transport coefficients of water.

An exact Monte Carlo method for calculating thermodynamic properties of quantum spin systems is described. Results for one-dimensional ferromagnetic and antiferromagnetic systems and for three-dimensional ferromagnetic systems show that the method can be used to study quantum spin systems as extensively as classical spin systems are studied with conventional Monte Carlo methods.

The linear-polarization correlation of pairs of photons emitted in a radiative cascade of calcium has been measured. The new experimental scheme, using two-channel polarizers (i.e., optical analogs of Stern-Gerlach filters), is a straightforward transposition of Einstein-Podolsky-Rosen-Bohm gedankenexperiment. The present results, in excellent agreement with the quantum mechanical predictions, lead to the greatest violation of generalized Bell's inequalities ever achieved.

It is shown experimentally that a bistable system is driven by a suitable external modulation to a region of random intermittency which displays a low-frequency power-law divergence. This low-frequency divergence is associated with a topological alternation between two strange attractors in phase space, either unsymmetric or fully symmetric depending on whether the two potential valleys are differently or equally located. This picture seems sufficiently general to apply to most cases of low-frequency noise currently reported.

Space-dependent extrema of the one-loop effective action in quantum field theory are found with use of scattering variables in the angular momentum. The $N$-component $g{\left({\overrightarrow{\varphi }}^2\right)}^2$ theory in three space dimensions is studied in detail. These instantons are classified in "homotopy classes" by "Regge poles." A comprehensive picture of them is obtained from exact results in the limits $g\to 0^{+}$ and $g\to \infty$ and from variational calculations. The large-order behavior of the $\frac{1}{N}$ series is obtained.

It is proven that when correlations in an equilibrium classical system in $\nu$ dimensions, $\nu >~2$, containing charges decay faster than ${\left(\mathrm{distance}\right)}^{-(l+\nu )}$ then the charge cloud surrounding particles has no multipole moments of order $k<~l$. This yields $l+1\mathrm{}$ sum rules with $l=\infty$ when the decay is exponential. This extends previous results for $l<~2$ sum rules and also generalizes them to systems containing fixed dipoles (or higher multipoles). Some consequences are described.