We present discrete forms of the Painlevé transcendental equations ${\mathit{P}}_{\mathrm{III}}$,${\mathit{P}}_{\mathrm{IV}}$, and ${\mathit{P}}_{\mathrm{V}}$ that complement the list of the already known ${\mathit{P}}_{\mathrm{I}}$ and ${\mathit{P}}_{\mathrm{II}}$. These, most likely integrable, nonautonomous mappings go over to the usual Painlevé equations in the continuous limit, while in the autonomous limit we recover discrete system that belong to the integrable family of Quispel et al. Finally, we show that the discrete Painlevé mappings satisfy the same reduction relations as the continuous Painlevé transcendents, namely, ${\mathit{P}}_{\mathrm{V}}$→{${\mathit{P}}_{\mathrm{III}}$, ${\mathit{P}}_{\mathrm{IV}}$}→${\mathit{P}}_{\mathrm{II}}$→${\mathit{P}}_{\mathrm{I}}$.

We report here on the first successful construction and operation of a Michelson interferometer for x rays. Its decisive strategem is the mixing of Laue- and Bragg-case beams diffracted by only one set of Bragg planes throughout the instrument. We illustrate the operability of the new interferometer by measuring the copper K${\mathrm{\alpha }}_1$K${\mathrm{\alpha }}_2$ doublet in Fourier-spectrometric mode. A total of more than 750 orders were scanned.

We consider random sequential adsorption on a lattice. We use analytical results on the Bethe lattice and cactus as references to develop systematic perturbationlike expansions which are very rapidly convergent. The latter produces the jamming density of a square lattice with an accuracy within ${10}^{\mathrm{-}5}$. This expansion is based on both physical and mathematical considerations and is not restricted to random sequential adsorption.

We present an exact solution of a one-dimensional sandpile model for which sand is dropped along the wall and N=2 grains of sand fall over the neighboring downhill sites when the critical slope is exceeded. The slopes of N consecutive sites organize into a local state. The time evolution of the local states along the spatial direction shows a natural tree structure. As a result, various multifractals can be identified. The spatial two-point correlation function decreases exponentially with a correlation length of the order of the lattice spacing.

We prove the rigorous lower bound ${\mathit{z}}_{\exp }$≥α/ν for the dynamic critical exponent of a broad class of multilevel (or ‘‘multigrid’’) variants of the Swendsen-Wang algorithm. This proves that such algorithms do suffer from critical slowing down. We conjecture that such algorithms in fact lie in the same dynamic universality class as the stanard Swendsen-Wang algorithm.

We describe a class of nonlinear birational representations of groups generated by a finite number of involutions. These groups are symmetries of the Yang-Baxter equations and their higher-dimensional generalizations. They provide discrete dynamical systems with a variety of behaviors, from chaotic to integrable, according to the number of invariants of the representation.

We present numerical results on a model in which quantization of chaos (instead of suppression of the diffusion) leads to a diffusive or even faster excitation with infinite conductance in the corresponding solid-state model. We suggest a possible mechanism for this effect.

We describe a new type of soliton-impurity interaction and demonstrate that the soliton can be totally reflected by an attractive impurity if its initial velocity lies in certain resonance ‘‘windows.’’ This effect has an analogy with the resonance phenomena in kink-antikink collisions [Campbell, Schonfeld, and Wingate, Physica (Amsterdam) 9D, 1 (1983)], and it can be explained by a resonant energy exchange between the soliton and the impurity mode. Taking the sine-Gordon and ${\mathrm{\phi }}^4$ models as examples, we find a number of resonance windows by numerical simulations and develop a collective-coordinate approach to describe the effect analytically.

A dissipative Fermi system is defined and the low-temperature thermodynamics is examined. Below a critical density the system behaves as a Boltzmann gas with Curie behavior of the susceptibility. Above the critical density, conventional Fermi behavior reappears, with Pauli behavior of the susceptibility. The diamagnetic susceptibility is smooth through the transition region which thus represents a type of metal-to-metal transition. In contrast, a dissipative ideal Bose system is Boltzmann-like for any density at low temperature.

We show that regular and irregular spectral statistics have direct, distinctive, and observable time-dependent manifestations in the behavior of the survival probability P(t)=‖〈ψ(0)‖ψ(t)〉${\mathrm{\Vert }}^2$, averaged over Hamiltonian ensembles and initial conditions. Specifically, systems exhibiting energy-level repulsion display characteristically strong decorrelations at short times. The proof relies solely on Liouville spectral properties of ensembles of bound quantum systems.

The effects of Gaussian quasimonochromatic (narrow band) noise have been investigated with an analog electronic circuit model. The escape probability and the reciprocal mean time to reach the top of a potential barrier were found to differ exponentially strongly. An extremely steep statistical distribution was observed and is discussed. For systems fluctuating in a symmetric single-well potential the distribution was found to be independent of the shape of the potential.

The concept of ‘‘fractional statistics’’ is reformulated as a generalization of the Pauli exclusion principle, and a definition independent of the dimension of space is obtained. When applied to the vortexlike quasiparticles of the fractional quantum Hall effect, it gives the same result as that based on the braid-group. It is also used to classify spinons in gapless spin-1/2 antiferromagnetic chains as semions. An extensive one-particle Hilbert-space dimension is essential, limiting fractional statistics of this type to topological excitations confined to the interior of condensed matter. The new definition does not apply to ‘‘anyon gas’’ models as currently formulated: A possible resolution of this difficulty is proposed.

Completing Ginibre’s work we determine the joint probability density of eigenvalues in a Gaussian ensemble of real asymmetric matrices, which is invariant under orthogonal transformations. The symmetry parameter τ may vary from -1 (antisymmetric ensemble) through 0 (completely asymmetric ensemble) to +1 (symmetric ensemble). The elliptic law for the average density of eigenvalues in the limit of large dimension is recovered. Matrices of the type considered appear in models for neural-network dynamics and dissipative quantum dynamics.

The basins of attraction of period-2 attractors can be subdivided into two domains, one for each of the stable trajectories which are one drive cycle apart. This generalizes to period-n attractors. Periodic drive signals can be replaced by certain chaotic signals which results in the elimination of multiple domains of attraction. The attractors are similar to the original ones, but two systems on the same attractor cannot get out of phase with each other.

Novel solutions of the Yang-Baxter relation for a series of multistate vertex models are presented in the braid-group limit. The obtained family of solutions gives a sequence of ‘‘colored’’-braid-group representations. Possible generalizations of the multivariable Alexander polynomial are discussed.

We present a new formulation of Feynman’s path integral, based on Voronin’s theorems on the universality of the Riemann zeta function. The result is a discrete sum over ‘‘paths,’’ each given by a zeta function. A new measure which leads to the correct quantum mechanics is explicitly given.

The wave functions of Sinai-billiard-shaped microwave cavities are experimentally studied. Some of the general features observed are parity breaking in the lowest eigenstates, ‘‘bouncing-ball’’ states, and states with quasirectangular or quasicircular symmetry. The above features are associated with nonisolated periodic orbits. Some states are observed which can be associated with isolated periodic orbits, leading to scars. This work represents the first direct experimental observation of scarred eigenfunctions. At high frequencies the eigenfunctions are very complex, and are yet to be classified in a general scheme.

Practical application of the generalized Bell’s theorem in the so-called key distribution process in cryptography is reported. The proposed scheme is based on the Bohm’s version of the Einstein-Podolsky-Rosen gedanken experiment and Bell’s theorem is used to test for eavesdropping.

Chaos introduces essential complications into semiclassical mechanics and the conventional wisdom maintains that the semiclassical time-dependent Green’s function fails to describe the quantum dynamics once the underlying chaos has had time to develop much finer structure than a quantum cell (h). We develop a method to evaluate the semiclassical approximation and test it for the first time under these circumstances. The comparison of the quantum and semiclassical dynamics of the stadium billiard shows remarkable agreement despite the very intricate underlying classical dynamics.

Low-energy spectral fluctuations are analyzed for higher-dimensional separable Hamiltonian systems including rectangular-well, harmonic-oscillator, and anharmonic spectra. Both nearest-neighbor spacing distribution and spectral rigidity via the ${\mathrm{\Delta }}_3$ statistics are studied. It is found that higher-dimensional statistics based on a few hundreds of states are very likely to reveal a level repulsion typical for chaotic systems. Especially critical in this sense are systems close to harmonic oscillators. Anharmonic effects tend to improve the convergence rate.