We have determined a new Rydberg value, $R_{\infty }=109\mathrm{}737.31476\left(32\right)\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, from an absolute wavelength measurement of the $2S_{\frac{1}{2}}-3\mathrm{}{P}_{\frac{1}{2}}$ component of the hydrogen Balmer-$\alpha$ line. Polarization spectroscopy of a mild helium-hydrogen discharge with a cw dye laser provided a threefold higher accuracy than the only previous Doppler-free measurement. Another twofold improvement can be obtained by a more precise wavelength measurement of a near-coincident ${}_{}{}^{127}{}_{}{}^{}\mathrm{I}_2^{}{}_{}{}^{}$ reference line.

Using a molecular-dynamics technique, we studied some nonequilibrium properties of the sine-Gordon chain in one space dimension. We found a $t^{\frac{4}{3}}$ long-time tail in the mean-square displacement, and self-diffusion is traced back to the motion of kink solitons. It is shown that the propagation of these kinks gives rise to a new excitation branch.

The radius of gyration $R_s$ of two-dimensional clusters containing $s$ sites each is found to vary as $s^{\rho }$ for $10\lesssim s\lesssim 100$, with $\rho \simeq 0.66$ for percolation clusters below the threshold and for random animals, $\rho \simeq 0.54$ at the percolation threshold, and $\rho \simeq \frac{1}{2}$ for percolation clusters above threshold and for growing animals. The density profiles indicate that the percolation clusters have a relatively narrow surface layer (like raindrops) above $p_c$ but not below $p_c$.

A quantum-mechanical Hamiltonian formulation of lattice spin systems is used to search for phase transitions in the O(2), O(3), and O(4) Heisenberg models in two dimensions. Strong-coupling expansions which have been calculated through eighth order indicate a phase transition at nonzero coupling for the O(2) model, but the non-Abelian models are predicted to exist only in the strong-coupling phase.

A Bäcklund transformation for the Ernst equation arising in general relativity in connection with several physical problems is derived, using the pseudopotential method of Wahlquist and Estabrook. A prolongation structure is also constructed, using a method of writing the equations in terms of differential forms, and an equation in the spirit of Lax is constructed, somewhat different from that given by Maison. Possible uses of the Bäcklund transformation to generate new solutions are mentioned.

There exist functions $F\left(x\right)\mathrm{}$ whose asymptotic expansions, when computed at fixed $x$ and optimal order, seem to converge to a numerical value which deviates significantly from $F\left(x\right)\mathrm{}$ itself. Several examples are given, showing that this phenomenon is not exceptional, and should occur in quantum theory. In particular, the semiclassical expansion, at large quantum numbers, of levels of the quartic oscillator $V=x^4$ presents such discrepancies, which we explain quantitatively as contributions from classical trajectories for which space and time coordinates become complex.

We present a renormalization-group approach for treating the percolation properties of the nearest-neighbor triangular Ising model. We obtain exponents for the line of percolation transitions $T_c<~T<~\infty$. In particular, we find a possibly exact result for the connectedness-length exponent ${\nu }_p=\frac{\ln \sqrt{3}}{\ln \left(\frac{3}{2}\right)}$ in the "pure" percolation ($T=\infty$) limit, which holds for $T_c<T<~\infty$. At $T=T_c$ we find the connectedness-length exponents of the percolation problem to be identical to the correlation-length exponents for the thermal problem.

This paper presents results of a numerical study of general relativistic gravitational collapse of infinite cylinders and discusses their implication for possible astrophysical collapse. It is found that almost no gravitational waves are generated during the initial infall period. However, up to 65% of the rest-mass energy is emitted in the form of gravitational waves during the bounce and subsequent oscillations of the cylinder.

The energy loss due to the emission of gravitational radiation during small-angle scattering according to general relativity is computed by means of a fast approximation method. The result differs by a factor of about 2.3 from the one based on the assumption that the Einstein quadrupole formula applies to energy changes in the material source.

It is shown that certain types of position-space, or "cell," renormalization-group transformations from an Ising-model object Hamiltonian onto another Ising-model image Hamiltonian are well defined and smooth in the thermodynamic limit, provided the object Hamiltonian has a large enough magnetic field. In certain other cases there is evidence (though not a rigorous proof) that no thermodynamic limit exists, or, if it exists, the transformation is not smooth.

We examine the effect of an assumed variation in Planck's constant $\hslash$ on the element abundances produced by nucleosynthesis in the standard hot big-bang model of the universe. In order to be consistent with current estimates of the primordial helium and deuterium abundances, we find that at the epoch of nucleosynthesis (a red shift $z\approx {10}^8-{10}^{10}$, $t\approx {10}^{-1\mathrm{}}-{10}^3$ sec after the initial event), $0.2\lesssim \frac{{\hslash }_{\mathrm{nucleosynthesis}}}{{\hslash }_{\mathrm{present}}}\lesssim 3$. This result supports the applicability of the local laws of physics to the earliest epoch of the universe examined to date.

We propose a method by which it is possible to narrow the energy distribution of relativistic protons. The method is based on the conversion of relativistic hydrogen ions into neutral hydrogen atoms, consequent velocity-selective excitation of relativistic hydrogen atoms by the $1S-2P$ transition, and photoionization of the selectively excited atoms. The method can be realized using existing lasers on present accelerators.

A linear eigenvalue problem in the spirit of Lax is constructed for the stationary, axially symmetric Einstein equations. It is conjectured that this entails the complete integrability of the system.

The method of Bäcklund transformations is employed to derive in 2 + 1 and 3 + 1 dimensions exact solutions of the sine-Gordon equation $[{\nabla }^2-c^{-2\mathrm{}}(\frac{{\partial }^2}{\partial t^2}\left)\right]\chi =sin\chi$. A formula is developed in 3 + 1 dimensions which permits us to generate without additional quadratures an infinite class of new time-dependent solutions.

A simple proof is given of a general result which shows how to construct solutions of a coupled system of linear self-adjoint partial differential equations once one has succeeded in deriving a decoupled equation in the manner described below. In the case of electromagnetic and gravitational perturbations of algebraically special vacuum space-times, this procedure yields the formulas of Cohen and Kegeles and of Chrzanowski.

We define coherent states for general potentials, requiring that they have the physically interesting properties of the harmonic-oscillator coherent states. We exhibit these states for several solvable examples and show that they obey a quantum approximation to the classical motion.

We extend the rigorous theory of complex scaling to atoms in constant electric field. This allows one to give a precise mathematical definition of resonance and leads to several results about the perturbation series: Borel summability at nonreal field and a relation between the asymptotics of the perturbation coefficients for large $n$ and the width of the resonance for small field.

The deviation of a resistor's $I-V$ curve from linearity as a result of Joule heating and its temperature coefficient of resistivity are shown to be sufficient to predict the low-frequency limiting spectral density of resistance fluctuations caused by spontaneous enthalpy fluctuations. The prediction agrees with previous experiments. Previous predictions that substantially exceeded measured noise in metal film resistors are shown to rely on incorrectly normalized current-jump data.

A new real-space renormalization method is developed which leads to differential renormalization-group equations. For a two-dimensional triangular Ising lattice these equations can be solved exactly and yield and exact critical properties.

For stationary axially symmetric vacuum metrics, we give a series of transformations ${\beta }^{\mathrm{}\left(k\right)\mathrm{}}$ which automatically preserve asymptotic flatness. We show how to generate the Kerr metric from the Schwarzschild, using ${\beta }^{\mathrm{}\left(0\right)\mathrm{}}$. We also show, using ${\beta }^{\mathrm{}\left(k\right)\mathrm{}}$, that the Tomimatsu-Sata (TS) class of metrices must be larger than previously realized, and for $\delta =2\mathrm{}$ there is a five-parameter TS metric. As an example, we present a two-parameter metric from this family, which we claim to be a new, physically realistic, asymptotically flat, rotating vacuum solution.