A new procedure for evaluating a function at infinity from its power series is analyzed. The limit in the procedure is shown to converge to the smallest singularity of the inverse function, which may or may not be the desired value.

It is shown by a calculation analogous to that carried out by Wallace and Zia for the pure Ising model that the lower critical dimension of the Ising model in a random field is 3 and not 2 as suggested by domain energy arguments. Further, the critical dimension for the roughening transition is shown to be 5 as compared to 3 for the pure Ising model.

A novel type of neutron interferometer was constructed and tested employing a split cylindrical zone plate with neutrons of 20 Å wavelength. Its performance and relative merits are discussed.

A very general cosmological solution to Einstein's equations exhibiting sensitive dependence on initial data is analyzed as a dynamical system. The metric and topologic entropies of the one-dimensional Poincaré map are determined and various results in number theory employed to calculate other invariants of the "chaotic" cosmological dynamics.

The exponents for large branched dilute polymers (which are also connected with the exponents of the lattice animals) are related in $D$ dimensions to the exponents of the Lee-Yang edge singularity of the Ising model in $D-2\mathrm{}$ dimensions. From the exact solution of the Ising model in zero and one dimension, one gets the polymer exponents in two and three dimensions, $\theta \mathrm{}(D=2\mathrm{})=1\mathrm{}$ and $\theta \mathrm{}(D=3\mathrm{})=\frac{3}{2}$.

The mean-field equations of the Ising model with competing interactions are studied by means of an iteration procedure. It is found that the phase diagram consists of regimes with stochastic behavior, indicating a complete "devil's staircase" with pinned configurations only, and analytic regimes where commensurate pinned phases are separated by incommensurate phases with Goldstone modes.

The critical behavior of the Yvon-Born-Green integral equation for fluids is analyzed by a moment expansion which yields a nonlinear differential equation accurately describing the long-range correlations. Phase plane analyses show that for dimensions $d<~4$ a critical point is characterized by $\eta =4-d$ with $g\left(\mathcal{r}\right)-1\mathrm{}$ negative for large distances, $\mathcal{r}$, in contrast to normal expectations. For $d>\mathrm{}4\mathrm{}$ the differential equation allows $\left[g\right(\mathcal{r})-1]>\mathrm{}0\mathrm{}$ and $\eta =0\mathrm{} or 4-d$. The compressibility never diverges if $d=1\mathrm{}$.

Similarity-type decay-mode solutions of the two-dimensional Korteweg-de Vries equation, ${(u_t+6u{u}_x+u_{\mathrm{xxx}})}_x+3\mathrm{}{\alpha }^2{u}_{\mathrm{yy}}=0\mathrm{}$, with $\alpha$ being constant, have been obtained explicitly in the simple analytic functions. Solutions of this decay mode have nonlinear superposition property similar to soliton solutions. Generalization of the customary Bäcklund transform enables one to produce multiple-soliton, multiple-decay-mode solutions.

It is shown that the acceleration of the sine-Gordon solitons by a constant torque increases as $t^2$ for small times $t$. For larger times, the velocity behaves like tanh $t^3$. Thus the soliton dynamical behavior is not Newtonian. Analytical arguments are given based on the linear perturbation theory which generalize this conclusion to any weak external perturbation of sine-Gordon solitons.

For the two-dimensional Ising model at arbitrary temperature, a system of quadratic difference equations is presented, involving the correlations of $n$ order variables and that of $n-2\mathrm{}$ order and two disorder variables. With suitable boundary conditions these equations specify the correlations.

A physically interesting nonlinear singular integro-differential equation which is an intermediary between the Korteweg-de Vries and Benjamin-Ono equations is considered via the inverse-scattering transform. Novel aspects of the theory and limits to the Benjamin-Ono equation are discussed.

From thermodynamic and causality considerations a general upper bound on the rate at which information can be transferred in terms of the message energy is inferred. This bound is consistent with Shannon's bounds for a band-limited channel. It prescribes the minimum energy cost for information transferred over a given time interval. As an application, a fundamental upper bound of ${10}^{15}$ operations/sec on the speed of an ideal digital computer is established.

It is shown that, for systems which enter chaos through period doubling bifurcations, the integrated noise power spectrum in the chaotic regime behaves as $N\left(\mathcal{r}\right)=N_0{(\mathcal{r}-{\mathcal{r}}_c)}^{\sigma }$, with $\sigma =1.5247\mathrm{}\dots$. Furthermore, the existence of a new universal constant which describes the scaling behavior of the average bandwidth in the strange attractor is reported. These results are directly applicable to experiments probing the onset of turbulence in physical systems.

The commutation of transfer matrices of $q$-state lattice models is studied and solutions which generalize both the $q=2\mathrm{}$ ferroelectric models and the special $q=3\mathrm{}$ models of Stroganov are found. For cases which are continuously connected to a shift operator, associated commuting Hamiltonians are given and free energies calculated.

A new approach to quantum gravity is proposed. The framework is sufficiently general to handle nontrivial space-time topologies required, e.g., in the black-hole formation and evaporation processes, and yet it incorporates notions such as energy-momentum, angular momentum, spin, mass, and discrete symmetries, normally associated with the Poincaré group.

A method is presented for carrying out Monte Carlo calculations for field theories with fermion degrees of freedom. As an example of this technique, results are given for a simple one-dimensional model.

If $\psi$ is a determinantal eigenfunction for the $N$-fermion Hamiltonian, $H$, with one- and two-body terms, then $e_0<~〈\psi ,H\psi 〉=E\left(K\right)\mathrm{}$, where $e_0$ is the ground-state energy, $K$ is the one-body reduced density matrix of $\psi$, and $E\left(K\right)\mathrm{}$ is the well-known expression in terms of direct and exchange energies. If an arbitrary one-body $K$ is given, which does not come from a determinantal $\psi$, then $E>~e_0$ does not necessarily hold. This Letter proves, however, that if the two-body part of $H$ is positive, then in fact $e_0<~e_{\mathrm{HF}}<~E\left(K\right)\mathrm{}$, where $e_{\mathrm{HF}}$ is the Hartree-Fock ground-state energy.

The notion of self-duality is extended to self-triality. One example from spin systems is given and completely solved by use of fermion variables. It is then shown that the O(8) Gross-Neveu model has self-triality: The Lagrangian $\mathcal{L}\left(\psi \right)=\mathcal{L}\mathrm{}\left(R\right)=\mathcal{L}\mathrm{}\left(L\right)\mathrm{}$, where $\psi$ is the original fermion while $R$ and $L$ are two types of kinks that occur dynamically. The anatomy of self-duality (triality) in the Ising and present examples is exposed as is the origin of the fermionic solutions.

A careful consideration of the propagation of null geodesics in a black-hole geometry modified by Hawking-radiation back reaction shows that the event horizon is stable and shifted only very slightly in radius from what is expected in a vacuum background, contrary to a recent claim in the literature.

At some special temperature $T_0$, the distribution functions of a two-dimensional one-component plasma are explicitly computed up to the four-body one. The correlations have a Gaussian falloff. The distribution functions at $T_0$ are used for building a temperature expansion around $T_0$.