Integration of Einstein's inertial-mass induction to the particle horizon in Robertson-Walker space yields an equation for the scale factor of the universe. A generalized Milne-McCrea theorem is used, and $G$ is allowed to vary slowly in order to preserve the weak equivalence principle. Among the results given is that at the present time $\frac{d\left(\frac{G}{G_0}\right)}{\mathrm{dt}}=-0.22\mathrm{}{H}_0$ for $q_0=0.03\mathrm{}$. The age is somewhat less than for Friedman models. These results differ minimally from general relativity.

The initial value problem for a system of two relativistic scalar fields in two space-time dimensions is solved. One of them has a self-interaction of the sine-Gordon type, while the other is massless and moves in a background geometry which has a dynamical evolution of its own. The system has soliton solutions and is shown to be completely integrable using the inverse scattering method. The appropriate linear equations are found by thinking of the problem as the isometric embedding of a surface in a three-dimensional Euclidean space.

There is a connection between nonlinear partial differential equations that can be solved by the inverse scattering transform and nonlinear ordinary differential equations without movable critical points (e.g., Painlevé transcendents). We exploit this connection to reduce the second equation of Painlevé to a linear integral equation. We also describe a class of nonlinear ordinary differential equations that can be exactly linearized by this method.

The algebra of the weakly vanishing Hamiltonian generators of localized translations, rotations, and supersymmetry transformations of supergravity theory is found. I argue that supergravity is the square root of ordinary general relativity in the same way as the Dirac equation is the square root of the Klein-Gordon equation and the Ramond-Neveu-Schwarz model is the square root of the spinless string.

We employed repeated-ring kinetic theory incorporating the equilibrium correlations in a dense fluid to study Brownian motion. It is demonstrated that the theory predicts the Stokes-Einstein law relating the diffusion coefficient of a Brownian particle to its radius and the fluid shear viscosity. In addition, the correct long-time behavior ($t^{-\frac{3}{2}}$) of the velocity autocorrelation function is obtained.

Although the time-dependent Foldy-Wouthuysen (FW) transformation of the "Schrödinger" equation $\frac{i\partial \psi }{\partial t}=H\psi$ yields the transformed equation $\frac{i\partial {\psi }^{\prime }}{\partial t}=H^{\prime }{\psi }^{\prime }$, where ${\psi }^{\prime }=U\psi$ and $H^{\prime }=UH{U}^{-1\mathrm{}}-\frac{\mathrm{iU}\partial \left(U^{-1\mathrm{}}\right)}{\partial t}$, the expectation values (${\psi }_1, H{\psi }_j$) of $H$ are not equal to (${{\psi }_i}^{\prime }, H^{\prime }{{\psi }_j}^{\prime }$), but rather, to (${{\psi }_i}^{\prime }, \mathrm{UH}{U}^{-1\mathrm{}}{{\psi }_j}^{\prime }$). But one still has $({{\psi }_i}^{\prime }, {{\psi }_j}^{\prime })=({\psi }_i, {\psi }_j)$. I discuss implications for perturbation calculations, scattering amplitudes, electrodynamics, and the external-field $\pi N$ ambiguity, and look at exact special cases.

The surface exponents associated with critical phenomena in semi-infinite systems are derived exactly in terms of bulk exponents. Results are ${\gamma }_{1,1}=\nu -1\mathrm{}$, ${\gamma }_1=\nu +\frac{(\gamma -1)}{2}$, ${\beta }_1=\frac{(3-\alpha )}{2}-\nu$, ${\Delta }_1=\frac{(1-\alpha )}{2}$, ${\eta }_{\parallel }=\frac{1}{\nu }$, and ${\eta }_{\perp }=1-\frac{(\gamma -1)}{2\nu }$, where $\nu$, $\gamma$, $\alpha$, etc., are the standard bulk exponents. The bulk-surface crossover exponent is ${\varphi }_s=1-\nu$. For $\nu >1\mathrm{}$ no surface phase can exist.

Exact solutions of similarity type are obtained for Boltzmann equations describing the temporal relaxation to equilibrium of an $N$-component gas with elastic collision cross sections of special form.

An exactly solvable model of the crystal-vacuum interface is constructed which exhibits a roughening transition. The model is obtained as a special limit of a ferromagnetic Ising model and it is isomorphic to the symmetric six-vertex model. Some of the thermodynamic properties of the system are discussed.

For both $T>\mathrm{}{T}_c$ and $T<\mathrm{}{T}_c$, we report exact expressions for the $n$-spin correlation function of the two-dimensional Ising model suitable for studying large separations between spins. In particular, the scaling limit of these correlation functions can be shown to exist and yields $n$-point Schwinger functions of some relativistic quantum field theories.

We present numerical data and scaling theories for the critical behavior of random resistor networks near the percolation threshold. We determine the critical exponents of a suitably defined resistance correlation function by a Padé analysis of low-concentration expansions as a function of dimensionality. We verify that $d=6\mathrm{}$ is the critical dimensionality for the onset of mean-field behavior. We use the coherent-potential approximation to construct a mean-field scaling function for the critical region.

The $n$-component Ginzburg-Landau-Wilson model for a semi-infinite system is solved exactly at $T=T_c$ in the limit $n\to \infty$. In the scaling regime the spin-spin correlation function is $G(\overrightarrow{\rho }, z, z^{\prime }, T_c)=\mathrm{const}{\{{[{\rho }^2+{(z-z^{\prime })}^2]}^{-1\mathrm{}}-{[{\rho }^2+{(z+z^{\prime })}^2]}^{-1\mathrm{}}\}}^{\frac{\mathrm{}(d-2\mathrm{})\mathrm{}}{2}}$, for dimensionalities $d$ in the range $2<d<\mathrm{}4\mathrm{}$, where $z$, $z^{\prime }$ are the distances of the two spins from the surface and $\overrightarrow{\mathrm{p}}$ is their separation parallel to the surface. The critical exponents ${\eta }_{\perp }$ and ${\eta }_{\parallel }$ are $\frac{\mathrm{}(d-2\mathrm{})\mathrm{}}{2}$ and ($d-2\mathrm{}$), respectively.

A unified geometric formulation of gravitation and supergravity is presented. The action for these theories is constructed out of the components of the curvature tensor for bundle spaces with four-dimensional Lorentz base manifold and structure groups Sp(4) for gravity and OSp(1,4) for supergravity. The requirement of invariance under reflections, local Lorentz transformations, and general coordinate transformations uniquely determines the action and ensures the existence of local supersymmetry in supergravity.

The spectrum of the mass operator on the soliton sectors of the ${\left[\lambda {\left(\overrightarrow{\varphi }·\overrightarrow{\varphi }\right)}^2\right]}_2$ theory with explicitly broken $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ invariance ($N>~1$) in the two-phase region is studied. It is shown rigorously that, for small $\lambda >0\mathrm{}$, the mass gap $m_s$ is positive and $m_s\equiv m_s\left(\lambda \right)\sim {\lambda }^{-1\mathrm{}}$, as $\lambda \to 0^{+}$. It is argued that $m_s$ is separated from the rest of the mass spectrum by an upper gap, so that the soliton is a stable particle. Our methods apply to other models with spontaneously broken, discrete symmetries.

The possible local counterterms in supergravity are investigated to all loop orders. Supersymmetry implies that (1) supergravity-matter coupling is one-loop nonrenormalizable, with a specific counterterm; (2) pure supergravity is renormalizable at both one and two loops; (3) it fails at three loops; (4) extended supergravity models may avoid the three-loop catastrophe, and have no dangerous local counterterms to any order. In that case, the nonleading divergences could be removed by field redefinitions, which would establish renormalizability for these systems.

An exactly solvable model for relativistic two-body interaction, consisting of two harmonic oscillators coupled via a massless scalar field is studied. It is shown that prescribing the asymptotic field does not lead to a normal Cauchy problem for the particles, whereas the formulation as a Cauchy problem for particle and field is consistent. Radiation damping of the system is also discussed.

A 680-kg, Weber-type, cylindrical, gravitational-wave detector has been operated at liquid helium temperature. The average vibrational energy in the lowest longitudinal mode at 1315.3 Hz was found to be consistent with the level of thermal noise at the antenna temperature. An effective noise temperature of 0.39 K for pulse excitation was measured. This is a factor of 20 below the lowest published values for room-temperature detectors.

The Zakharov-Shabat theory of integrable systems with more than one spatial dimension breaks down when certain resonances between solitons occur. Remarkably, the resonance condition is precisely the same as that for the strong interaction of three weakly coupled linear dispersive waves. This provides a mechanism for the creation of new particles by collision.

The implications of the weak equivalence principles are investigated in detail for electromagnetic systems in a general framework. In particular, I show that the universality of free-fall trajectories {Galileo weak equivalence principle (WEP[I])} does not imply the validity of the Einstein equivalence principle (EEP). However, WEP[I] plus the universality of free-fall rotation states (WEP[II]) does imply EEP. To test WEP[II] and EEP, I suggest that Eötvös-type experiments on polarized bodies be performed.

A Lagrangian is obtained by deriving the path-integral representation of the diffusion process. It can be applied, e.g., to nonequilibrium thermodynamics and to quantized motion in general relativity. In second quantization it is shown to lead to a particularly well-behaved energy-momentum tensor as a source of gravity.