Every closed three-manifold occurs as a spacelike hypersurface of a vacuum space-time. For most of these three-manifolds, however, the vacuum space-times that contain them have no maximal slice. For asymptotically flat manifolds there are no restrictions on which three-manifolds can occur obeying the local energy conditions $\rho >{\left(J_a{J}^a\right)}^{\frac{1}{2}}$ and the space-times that contain them in most cases have no maximal slice.

An analytical method to determine the dimensionality of strange attractors in two-dimensional maps is introduced. In this method, the geometric structures of an attractor are obtained from a procedure developed previously. Such structures often appear to be the Cartesian product of a curve and a Cantor set. From the geometric structures, we determine the Hausdorff dimension first for the Cantor set, and then for the attractor. The results compare well with numerical results obtained for the Hénon, Zaslavskii, and Kaplan-Yorke maps.

Dimension and correlation entropy are measured for various settings of our hybrid optically bistable device. Measured dimension is found to be significantly less than dimensions consistent with the Kaplan-Yorke conjecture. The standard method of determining correlation entropy is shown to yield more than one value.

The effect of an unstable continuum on the time evolution of an embedded state is examined. It is found that the decay law remains exponential, but that the lifetime is modified. Under suitable conditions, the initial state may survive longer than would be the case if the continuum were stable.

The average lifetime of a chaotic transient versus a system parameter is studied for the case wherein a chaotic attractor is converted into a chaotic transient upon collision with its basin boundary (a crisis). Typically the average lifetime $T$ depends upon the system parameter $p$ via $T\sim {|p-p_c|}^{-\gamma }$, where $p_c$ denotes the value of $p$ at the crisis and we call $\gamma$ the critical exponent of the chaotic transient. A theory determining $\gamma$ for two-dimensional maps is developed and compared with numerical experiments. The theory also applies to critical behavior at interior crises.

An explicit realization of $\mathrm{U}\mathrm{}\left(n\right)\mathrm{}$ quantum holonomy is presented in the context of nonrelativistic (2 + 1)-dimensional electrodynamics and used to construct the adiabatic effective action. It is shown that the U(1) subgroup of the $\mathrm{U}\mathrm{}\left(n\right)\mathrm{}$ holonomy gives rise to an effective Chern-Simons topological mass term and its relationship with the fractional quantum Hall effect is discussed.

The problem of calculating the eigenvalues of the Dirac equation by the finite-basis expansion method is studied. Bounds for the eigenvalues are obtained explaining the numerical results on the spectrum that have been observed previously. It is argued that the problem of variational collapse can be avoided by finding the minimum over the wave-function large component of the maximum over the wave-function small component of the energy functional. A numerical example is discussed.

It is proved that charge transport induced by adiabatic variations of an incommensurate potential acting on noninteracting electrons is quantized. The corresponding integers are topological invariants (Chern numbers of various kinds) which label the gaps. Similar results are obtained for electrons in a periodic potential, under the influence of an irrational magnetic field.

A mapping of the unit interval of all realizations of the disorder for a random multiplicative process allows us to show that (a) the flucatuations due to different realizations of the disorder are characterized by a multifractal spectrum and (b) the fluctuations in space for a given realization of the disorder do not possess properties of self-similarity. These results have relevant implications for various physical problems with a multiplicative structure.

It is shown that in simple three-dimensional flows there are uncountably many different codimension-2 scenarios for the onset of chaos. We analyze these scenarios via one-dimensional Poincaré maps, and show how to construct renormalization groups to obtain new universal results. Comparisons with numerical studies are given.

It is shown that energy should be quantized if there exist cosmic spinning strings. Quantization of energy arises naturally from the fact that the wave function must be single valued in the gravitational field of a spinning string. This effect arises when the vortexlike vacuum configurations carry constant nonzero angular momentum per unit length. Spinning strings occur naturally in spontaneously broken grand unified theories. A limit on $J$ coming from the upper limit of the photon mass is presented. The analogy with quantization of energy in the presence of gravitational magnetic mass is established.

Recently, Berry recognized in quantum mechanics a topological phase factor arising from the adiabatic transport of a system around a closed circuit, which is essentially the Aharonov-Bohm effect in parameter space. Here we consider manifestations of this phase factor for a photon in a state of adiabatically invariant helicity. An interferometer is suggested to see this phase. Also, an effective optical activity for a helical optical fiber is predicted. These effects emerge on a classical level as topological features of Maxwell's theory.

We report the first experimental verification of Berry's topological phase. The key element in the experiment was a single-mode, helically wound optical fiber, inside which a photon of a given helicity could be adiabatically transported around a closed path in momentum space. The experiment confirmed at the classical level that the angle of rotation of linearly polarized light in this fiber gives a direct measure of Berry's phase. The topological nature of this effect was also verified, i.e., the rotation was found to be independent of deformations of fiber path if the solid angle of the path in momentum space stayed constant.

I consider a connected self-avoiding polymer network made of identical long chains, with fixed topology. Using renormalization theory and conformal invariance, I conjecture in 2D, and give in $d=4-\epsilon$, to order $O\left(\epsilon \right)$, the exact value of its critical exponent $\gamma$ as a function of the topological invariants. In 2D, the exact result fits with recent numerical data for three- and four-leg stars by Lipson et al.

A recent exact multidimensional inverse-scattering method, developed for Schrödinger's equation, establishes a linear integral equation which determines the wave field at all space-time points from the scattering data. We present a new derivation of this integral equation which depends only on very general features of the scattering process such as causality, linearity, and the far-field decay of the wave field. Consequently, this integral equation is shown to hold for a wide class of wave equations (including, for example, the acoustic wave equation).

It is conjectured that the component wave functions of the Rabi Hamiltonian in Bargmann's Hilbert space are terminating series of spheroidal wave functions and generalized spheroidal wave functions of Leitner and Meixner. Numerical calculations strongly support the conjecture.

We study the statistical mechanics of two-dimensional surfaces of fixed connectivity embedded in $d$ dimensions, as exemplified by hard spheres tethered together by strings into a triangular net. Without self-avoidance, entropy generates elastic interactions at large distances, and the radius of gyration $R_G$ increases as ${\left(\ln L\right)}^{\frac{1}{2}}$, where $L$ is the linear size of the uncrumpled surface. With self-avoidance $R_G$ grows as $L^{\nu }$, with $\nu =\frac{4}{\mathrm{}(d+2\mathrm{})\mathrm{}}$ as obtained from a Flory theory and in good agreement with our Monte Carlo results for $d=3\mathrm{}$.

At a critical point the golden-mean Kolmogrov-Arnol'd-Moser trajectory of Chirikov's standard map breaks up into a fractal orbit called a cantorus. The transition describes a pinning of the incommensurate phase of the Frenkel-Kontorowa model. We find that the fractal dimension of the cantorus is $D=0\mathrm{}$ and that the transition from the Kolmogorov-Arnol'd-Moser trajectory with dimension $D=1\mathrm{}$ to the cantorus is governed by an exponent $\overline{\nu }=0.98\mathrm{}\dots$ and a universal scaling function. It is argued that the exponent is equal to that of the Lyapunov exponent.

The influence of the boundary conditions on the finite-size dependence of the mass gap in the 1D Ising model with a transverse field is studied by means of combined exact results and large-size numerical calculations. The well-known exponential form (for transverse fields smaller than the critical one) is only recovered in a limited range of parameters. A power-law behavior for the mass gap is even found in the case of antiperiodic boundary conditions.

It is shown that no continuous process in which the energy tensor of accreted matter remains bounded and satisfies the weak energy condition in a neighborhood of the apparent horizon can reduce the surface gravity of a black hole to zero within a finite advanced time. This gives a precise expression to the third law of black-hole mechanics.