It is shown that the Laplace transform of the diffusion-controlled bimolecular rate kernel can be directly obtained as the solution of a Riccati equation with the annihilation distance as the independent variable. This provides an efficient alternative to solving the Smoluchowski equation. The solution is expanded in powers of the square root of the Laplace variable; for the first time, the question of convergence is discussed and settled, and terms other than the first two are given explicitly and recursively by quadratures.

It is shown that quantum gravity with a cosmological constant in a purely affine picture is a power-counting-renormalizable (unitary) theory. The relevance of such a result is based on the equivalence, proved by Kijowski, of the affine formulation of gravity with the standard affine-metric picture.

It has been found that some charged, rotating black holes are unstable to the growth of a small, perturbing, massless scalar field. Additionally, there are strong indications that some uncharged black holes will be unstable to gravitational and electromagnetic perturbations as well.

A modified form of percolation theory is solved exactly on a Cayley tree of arbitrary coordination number. In this model, cluster growth proceeds "dynamically" by invasion along a path of least resistance. Although there are clear parallels to ordinary "static" percolation at the percolation threshold, there are also significant differences. In particular, the scaling function which describes the shape of large clusters is different in the two models.

Phase-locked solitary waves are shown to be the stable fixed points of an infinite-dimensional map obtained from a bistable optical ring cavity.

Particles diffusing in $d$-dimensional space among a random distribution of stationary spherical traps are considered. Given a particle at the origin at time $t=0\mathrm{}$, it is shown that the density of particles at the origin as $t\to \infty$ must decay at least as fast as $\exp [-t^{\frac{d}{\mathrm{}(d+2\mathrm{})\mathrm{}}}]$. The density here is obtained by averaging the diffusive field for a given configuration of traps over all configurations. The present upper bound coincides with the lower bound recently derived by Grassberger and Procaccia.

Neutron stars for which the ratio $\frac{T}{\mathrm{}\left|W\right|\mathrm{}}$ of kinetic to gravitational potential energy exceeds ∼ 0.08 are likely to be unstable and to spin down by gravitational-wave—driven oscillations. A class of pulsars with nearly identical frequencies may thus result from the collapse of accreting, rapidly rotating white dwarfs. It is likely that the fast pulsar, PSR 1937 + 214, rotates more slowly than this, but the question remains open partly because of uncertainty in the equation of state.

New effects in heat-conducting nonequilibrium fluids near critical points are predicted theoretically: The heat conductivity gains a new divergent contribution and the heat diffusivity becomes strongly nonisotropic.

A systematic weak-coupling expansion is developed for the Liouville quantum field theory on a periodic spatial interval. Matrix elements of various Liouville operators are computed to order $g^8$ by perturbation in nonzero modes about the exact solution of the zero-mode problem. To this order the results agree with the explicit operator solution given previously by Braaten, Curtright, and Thorn, in which the Liouville field was expressed in terms of a free pseudoscalar field by means of an operator Bäcklund transformation.

A method for solving the inverse problem of certain hyperbolic as well as elliptic systems of $n$ equations in the plane is given. This result can be used to linearize the initial-value problem of several physically significant nonlinear evolution equations in two spatial and one temporal dimension.

A general method is given for solving certain inverse problems in the plane. The results can be used to construct the solution to the initial-value problems of related nonlinear evolution equations in two spatial and one temporal dimension. The method also allows one to compute lumps, i.e., multidimensional solitons tending to zero in all spatial directions.

An improved Coulomb potential is presented for use in the relativistic Schrödinger equation. This potential takes into account contributions of all two-photon exchange diagrams and gives the lowest-order relativistic corrections to the static Coulomb potential for arbitrary masses of the two particles. The cases spin 0-spin 0, spin 0-spin ½, and spin ½-spin ½ are considered. When the mass of one of the particles goes to infinity, then the well-known corrections to the binding energy and to the phase shifts as obtained with either the Klein-Gordon or the Dirac equation are reproduced.

Spontaneous compactification of $d=11\mathrm{}$ supergravity on the squashed seven-sphere ($S^7$) yields a spontaneously broken phase of the $N=8\mathrm{}$ theory on the round $S^7$ with SO(8) broken to SO(5) ⊗ SU(2) and with $N=1\mathrm{}$ or $N=0\mathrm{}$ supersymmetry according as the squashing is left or right handed. A feature unique to Kaluza-Klein supergravity is that the massless gravitino of the $N=1\mathrm{}$ phase comes from a massive $N=8\mathrm{}$ supermultiplet, as do the Higgs scalars. A solution is found with squashing and torsion but only in the right-squashed ($N=0\mathrm{}$) case.

Self-diffusion in a Lorentz gas on a triangular lattice is studied both analytically and numerically. A simple estimate for the diffusion coefficient, based on the idea of a random walk between traps, is found to be in good agreement with the numerical results.

The long-time behavior of a system of two nonlinear oscillators interacting through a linear continuous string is investigated numerically. The string is treated exactly (no spatial lattice). It is found that such classical systems display many features normally associated with quantum mechanics, such as a phase space which breaks up into an infinite number of finite-size cells of ergodic motion.

A quantitative formulation of the uncertainty principle, based on Deutsch's idea of using entropy as a measure of uncertainty to remove the inadequacies of the standard treatment, is presented. This uncertainty in general involves the resolution of the measuring device.

The authors investigate possible candidates of configuration spaces for the description of the rotational degrees of freedom of quantum spinning particles (including half-integer spin states), and point out their connections in relation to the set of physical observables and superselecting sectors present in the theory.

In the two-dimensional Liouville quantum field theory a perturbation theory may be defined for which space-translation invariance is spontaneously broken and space-time (semi)compactifies spontaneously.

A new method of solution is proposed for a Boltzmann equation describing the slowing down of charged particles in solids. It seems as if it would be more generally applicable. It can be considered as a multigroup method with the energy groups varying from point to point in a natural way dictated by the local spectrum. The energy moments of the density of moving particles are calculated at each point in the target, and at each point an extremal (in energy) density of particles is constructed. The extremal density allows for particles being stopped, and, hence, removed from the beam.

The general scale-invariant Lagrangian, $-\frac{1}{4}\int d^{4}x{\mathrm{}(-g)\mathrm{}}^{\frac{1}{2}}[\alpha c^{\mu \nu \lambda \sigma }{c}_{\mu \nu \lambda \sigma }+\beta R^2]$, with $\alpha \beta >~0$, is considered. Although this theory admits linearized solutions with negative energy, it is shown that all exact solutions representing isolated systems have precisely zero energy. This result also holds in the presence of arbitrary matter. It can be understood as resulting from a confinement of energy. The implications of this result for quantum gravity are discussed.