A four-parametric set of solutions to $G_{\mu \nu }=0\mathrm{}$ is presented; for discrete values of three of the parameters, the metric is constructed from rational functions. The set has the structure of an Abelian group, with flat space-time as group identity, and contains many solutions of physical interest.

The Boltzmann equation considered by Bobylev, and Krook and Wu (BKW) is rewritten in the form of a stochastic equation, similar in form to the kinetic equations of Tjon and Wu, and Ernst. A new class of models, which reflects the transformation of the BKW model to other dimensionalities, is constructed, and its equilibrium distributions, non-equilibrium solutions (corresponding to the BKW mode), and $H$ theorems are found.

The special relativistic transformation of temperature is not known with certainty and some authors believe it to be arbitrary within a certain range of possibilities. A procedure is proposed here which can in principle, and for the first time, lead to an experimental attack on this problem. Even though the experimental difficulties may be prohibitive, if the principle of the method is granted, then a definite transformation must exist.

Conservative Hamiltonian systems with two degrees of freedom are discussed where a typical trajectory fills the whole surface of constant energy. The trace of the quantum mechanical Green's function is approximated by a sum over classical periodic orbits. This leads directly to Selberg's trace formula for the motion of a particle on a surface of constant negative curvature, and, when applied to the anisotropic Kepler problem, yields excellent results for all the energy levels.

It is shown that in the turbulent regime of systems with period-doubling subharmonic bifurcations, the maximum Lyapunov characteristic exponent behaves like $\overline{\lambda }={\overline{\lambda }}_0{(\mathcal{r}-{\mathcal{r}}_c)}^t$, with $t$ a universal exponent which is calculated to be $t=0.449\mathrm{}8069\dots$. This result is in agreement with the available data on $\overline{\lambda }$ for a number of dynamical systems.

The interferometers now being developed to detect gravitational vaves work by measuring small changes in the positions of free masses. There has been a controversy whether quantum-mechanical radiation-pressure fluctuations disturb this measurement. This Letter resolves the controversy: They do.

In this Letter, a simple model of disordered systems—the random-energy model—is introduced and solved. This model is the limit of a family of disordered models, when the correlations between the energy levels become negligible. The properties are qualitatively the same as those of the Sherrington-Kirkpatrick model. Moreover, this random-energy model looks like a simple approximation to any spin-glass model.

The flight time of neutrons through diffracting crystals has been studied experimentally and is found in good agreement with theoretical prediction. The existence of an anomalous drift velocity along lattice planes has been demonstrated and the drift neutrons are shown to exhibit diminished absorption (anomalous Borrmann transmission) in the crystal.

The Clifford algebra $C_4$ is used to construct a theory of fermions that is calculationally identical to the single-particle Dirac spinor theory. The anomalies in the conservation laws of the spinor theory are removed. A verifiable prediction is made that massless fermions cannot have charge or magnetic moment.

The inverse-square distance dependence of the gravitational force has been tested over a range of approximately 2 to 5 cm, by use of a test mass suspended from a torsion balance to probe the gravitational field inside a mass tube. The result supports an inverse-square law. Assuming a force deviating from inverse square by a factor [$1+\epsilon \ln \gamma$ (cm)] it is found that $\epsilon =(1\mathrm{}\pm 7)\times {10}^{-5\mathrm{}}$.

It is shown that in unified theories containing only fermions and gauge fields and in which scale invariance is spontaneously broken, radiative corrections induce an order-$R$ term in the vacuum action functional which is uniquely determined by the flat-space-time theory.

The method of Newton for the solution of the inverse-scattering problem is modified with the assumption that the potential is known for radial distances $\mathcal{r}>{\mathcal{r}}_0$. The modified method is improved by applying a least-squares method.

An exactly solvable modification of the planar Ising ferromagnet is proposed which has a roughening transition below the Curie temperature. The computation confirms the de Gennes-Fisher scaling theory of correlations with homogeneous surface fields, giving an exponent value ${\Delta }_1=\frac{1}{2}$.

The equation governing four-wave interactions in a nonlinear dispersive system is studied. It is shown that a nonlinear steady-state plane wave can bifurcate into nonplanar steady-state solutions. In the case of an isotropic medium, the bifurcation is degenerate and the bifurcated solutions may preserve or break the symmetry. An example is given of a symmetry-breaking solution for deep-water gravity waves and its stability is discussed.

For a certain class of three-manifolds, the angular momentum of an asymptotically flat quantum gravitational field can have half-integral values.

Differential renormalization equation for the $q$-state Potts model are proposed, and the critical behavior of the model near $q=q_c$ discussed. The equations give rise to critical and tricritical fixed points which merge at $q=q_c$ when a dilution field becomes marginal, to an essential singularity in the latent heat as a function of $q=q_c$, in accordance with the exact result of Baxter, and, for $q=q_c$, to a logarithm correction to the power-law behavior of the free energy as a function of $T-T_c$.

An Ising model on a square lattice is studied, where one row of horizontal bonds has an energy ${E_1}^{\prime }$ different from all other horizontal bonds. The correlation of two spins is calculated in this row, resulting in exponents $\beta$ and $\eta$ which depend on ${E_1}^{\prime }$. The long-distance behavior of the correlation for fixed $T\ne T_c$ is found to have different forms depending upon the value of ${E_1}^{\prime }$.

A slow-neutron interferometer system has been used to test a nonlinear variant of the Schrödinger equation $i\hslash \frac{\partial \psi (\overrightarrow{\mathrm{r}}, t)}{\partial t}=\left[\right(-\frac{{\hslash }^2}{2m}){\nabla }^2+U(\overrightarrow{\mathrm{r}}, t\left)\right]\psi -b\ln \left(a^3{\left|\psi \right|}^2\right)\psi$. If this equation were correct, then, as Shimony has suggested, repositioning an attenuating plate downstream in a neutron beam would produce a phase modification. No measurable phase shift beyond experimental uncertainty was found and an upper limit of 3.4×${10}^{-13\mathrm{}}$ eV for the energy constant $b$ was established.

The velocity autocorrelation function is measured directly for the first time for one Brownian test particle in nitrogen at two different densities. The persistence of the autocorrelation is more pronounced than predicted on the basis of the modified Langevin equation. The measured distribution for the Brownian displacement shows a considerable departure from the Gaussian distribution at both densities.

The determination of the geometrical structure of the universe through the magnitude-vs-redshift relation in standard cosmology has not been very successful, mainly because of the intrinsic insensitivity of the $m-\mathrm{vs}-z$ relation to the deceleration parameter ${\overline{q}}_0$ which determines the spatial curvature and therefore the geometry. By relaxing the assumption usually made, i.e., the identity of gravitational and atomic clocks, sufficient sensitivity is achieved for the determination of ${\overline{q}}_0$. Existing observational evidence then leads one to conclude that the universe is open.