The Eden model in many dimensions is studied by an exact enumeration method. Our result including the first-order $\frac{1}{d}$ correction has the asymptotic behavior $〈{R_n}^2〉\sim 2 \ln \mathrm{}\left(n\right)\mathrm{}(1+\frac{3}{2d})$, which does not agree with the naive expectation $R\sim n^{\frac{1}{d}}$ [or $d(n^{\frac{1}{d}}-1)$]. This suggests that when $n$ is large, $〈{R_n}^2〉$ might be a singular function of $d$. We also present a result on the Cayley tree.

It is shown that the Zamolodchikov model on the cubic lattice of $N$ sites is equivalent to a spin model on a lattice of $2N$ sites with just two-spin and three-spin interactions. Together with the known transfer-matrix commutativity, this implies that the partition function $Z$ (for $N$ large) factors into a product of single-variable functions. These functions, and hence $Z$, are calculated by using symmetry relations. The model appears to be a critical free-fermion model.

We introduce a noncanonical ("new-time") transformation which exchanges the roles of a coupling constant and the energy in Hamiltonian systems while preserving integrability. In this way we can construct new integrable systems and, for example, explain the observed duality between the Hénon-Heiles and Holt models. It is shown that the transformation can sometimes connect weak- and full-Painlevé Hamiltonians. We also discuss quantum integrability and find the origin of the deformation $-\frac{5}{72}{\hslash }^2{x}^{-2\mathrm{}}$.

When Einstein-Brillouin-Keller quantization is possible, it applies to all conserved dynamical variables (not only to the Hamiltonian) and in particular to the time average of any dynamical variable. Thus, for an integrable system of $n$ degrees of freedom, the eigenvalues of $n$ independent constants of motion form a locally regular $n$-dimensional lattice, in the limit $h\to 0$. Failure of doing that may be an indication of quantum chaos.

Quantum chaos of a single wave function is defined in terms of a path correlation function, and used to show how chaos sets in, in the particle-in-a-stadium problem. The criterion is shown to be equivalent to the requirement of a random distribution of the wave-function values.

Certain unstable periodic orbits are shown to permanently scar some quantum eigenfunctions as $\hslash \to 0$, in the sense that extra density surrounds the region of the periodic orbit.

The size distribution $n_i\mathrm{}\left(t\right)\mathrm{}$ for systems with Smoluchowski aggregation kinetics is shown to have the dynamic scaling form $n_i\mathrm{}\left(t\right)\mathrm{}\sim t^{-\theta }g\left(\frac{1}{t^z}\right)$, with $\theta =2\mathrm{}$ and $z=1\mathrm{}$. This gives a light-scattering intensity $〈I〉\sim t^z$, and an effective hydrodynamic cluster radius $〈R〉\sim t^{\frac{z}{D}}$, where $D$ is the cluster fractal dimension. We find that human immunoglobulin (IgG) monomers exhibit Smoluchowski aggregation kinetics with $z=1\mathrm{}$, and $D=2.56\mathrm{}\pm 0.3$, upon heating to 47-62°C.

Following continuous-time random-walk dynamics, we consider particle diffusion and trapping on fractals, and study the interplay of the spatial ($\alpha$) and temporal ($\gamma$) stochastic aspects. At long times and for broad waiting-time distributions the mean squared displacement behaves as $〈r^2\mathrm{}\left(t\right)\mathrm{}〉\sim t^{\alpha \gamma }$, whereas the particle decay is algebraic, $\Phi \mathrm{}\left(t\right)\mathrm{}\sim t^{-\gamma }$.

An example is presented of a model of an amorphous quantum mechanical system, a liquid of quantal Ising spins, which can be solved exactly within certain many-body theories. Analytical solutions of the model in mean-field theory are shown to reveal a decrease in the extent of the ferromagnetic region (compared to an equivalent classical system) and the occurrence of some degree of quantum localization. Both phenomena are analyzed as a competition between quantum mechanics and the condensed phase.

We investigate the classical stability of Freund-Rubin-type solutions $M^{\mathrm{pqr}} \left[\mathrm{SU}\mathrm{}\right(3)\mathrm{}\otimes \mathrm{SU}\mathrm{}(2)\mathrm{}\otimes \frac{\mathrm{U}\mathrm{}\left(1\right)\mathrm{}}{\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}}\otimes \mathrm{U}\mathrm{}(1)\mathrm{}\otimes \mathrm{U}\mathrm{}(1\left)\right]$, $Q^{\mathrm{pqr}} \left[\mathrm{SU}\mathrm{}\right(2)\mathrm{}\otimes \mathrm{SU}\mathrm{}(2)\mathrm{}\otimes \frac{\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}}{\mathrm{U}\mathrm{}\left(1\right)\mathrm{}}\otimes \mathrm{U}\mathrm{}(1\left)\right]$, and $N^{\mathrm{pqr}} \left[\mathrm{SU}\mathrm{}\right(3)\mathrm{}\otimes \frac{\mathrm{U}\mathrm{}\left(1\right)\mathrm{}}{\mathrm{U}\mathrm{}\left(1\right)\mathrm{}}\otimes \mathrm{U}\mathrm{}(1\left)\right]$ against relative dilatations between the coset directions. It is shown that $M^{\mathrm{pqr}}$ is stable only for $\frac{98}{243}<~\frac{p^2}{q^2}<~\frac{6358}{4563}$, $Q^{\mathrm{pqr}}$ is stable only for a certain region of $\frac{p^2}{r^2}$ and $\frac{q^2}{r^2}$, while $N^{\mathrm{pqr}}$ is stable for any $\frac{p^2}{q^2}$ against these small fluctuations.

Quantum effects of the transition to chaos of an (almost) classical dynamical system are studied by renormalization-group methods. Planck's constant is a relevant variable which determines the time $t_h\propto {\hslash }^{-\frac{1}{\gamma }}$ at which the system crosses over to quantal behavior. The exponent $\gamma$ is 6.039 for the limit of period-doubling bifurcations and is 3.04 for the disappearance of the final Kolmogorov-Arnold-Moser trajectory in the standard map. For the standard map the scaling of the energy diffusion coefficient is also calculated.

We solve for the internal structure of $d$-dimensional percolation clusters. We find that if we randomly choose two points, then the backbone connecting these two points can be described as a randomly constructed "necklace" whose building blocks are $s$-site volatile fractals (they are not stable under a length scale change). We find the blob-size distribution function and calculate the relevant parameters numerically, thereby obtaining the first accurate estimates for the fractal dimension of the backbone in $d=2,3\mathrm{}$.

It may look surprising that direct application of linear system theory to the Laplace transform of the time-dependent Schrödinger equation does not yield the evolution operator or density matrix in a physically reliable form, but requires a further "reduction procedure," earlier introduced in the frame of the so-called determinantal formalsim. It is shown that this procedure can in fact be viewed as a factorization of determinants intended so as to remove unexpected terms from the result.

An optical field produced by intracavity four-wave mixing is shown to exhibit the following nonclassical features: photon antibunching, squeezing, and violation of Cauchy-Schwarz and Bell's inequalities. These intrinsic quantum mechanical effects are shown to be associated with the nonexistence of a positive normalizable Glauber-Sudarshan $P$ function.

A class of models for relaxation in strongly interacting glassy materials is suggested. Degrees of freedom are divided into a sequence of levels such that those in level $n+1\mathrm{}$ are locked except when some of those in level $n$ find the right combination to release them, this representing the hierarchy of constraints in real systems. The Kohlrausch anomalous relaxation law, $\exp [-{\left(\frac{t}{\tau }\right)}^{\beta }]$, emerges naturally, and a maximum time scale is found which exhibits a Vogel-Fulcher-type temperature dependence.

Measurements of chaotic vibrations of a forced nonlinear oscillator with a two-well potential suggest that the frequency-amplitude boundary between periodic and chaotic motions may be fractal. The fractal dimension of this curve is 1.26. The experiment consisted of an elastic beam and magnets which were used to create two stable equilibrium positions. Analysis has shown that this system possesses strange-attractor solutions for a single-mode model. However, a fractal boundary suggests that the role of higher modes may be important.

By means of the Feynman path-integral method the experiment proposed by Silverman is analyzed under the classical approximation. The results give a new verification of the Aharonov-Bohm effect. It is pointed out that the multiple-valued wave functions are reasonable for describing the experiment. Furthermore, it is verified that the eigenvalues of the angular momentum of a charged, spinless particle-solenoid composite are flux dependent.

An experiment is proposed for measuring the earth's gravitomagnetic field by monitoring its effect on the plane of swing of a Foucault pendulum at the south pole ("dragging of inertial frames by earth's rotation"). With great effort a 10% experiment in a measurement time of several months might be achieved.

Compactification of twelve-dimensional space-time with geometry the product of Minkowski space-time with an eight-sphere requires a gauge group which contains Spin(8). This procedure leads to left-right asymmetric gauge couplings to fermions. A model with gauge group Spin (18) has four chiral families.

The finite-temperature excitation spectrum of the classical Toda lattice is obtained by numerical solution of the Bethe Ansatz. Bethe-Ansatz thermodynamics agrees exactly with known transfer-integral results. A soliton-phonon phenomenology is developed and its validity assessed with the Bethe-Ansatz spectrum as an exact standard. The phenomenology's intrinsic limitations become evident in a transition region, where a distinction between phonon and (nontopological) soliton is impossible.