A simple mathematical model is given which shows how phase locking, bistability, period-doubling bifurcations, and chaos may result from periodic stimulation of nonlinear oscillators. A new fixed-point theorem, which extends the classic results of Arnold, is used in the analysis.

A powerful method is presented for solving systems of nonlinear equations such as those occurring in the Landau theory of phase transitions, in the Higgs mechanism of spontaneous symmetry breaking, and in renormalization-group studies of critical phenomena. As an illustration a ferroelectric phase transition in perovskites is considered with the most general free energy of sixth degree. A special case is a fourth-degree potential which corresponds, e.g., to an SO(7), adjoint representation, Higgs problem.

Exact solutions to the Feigenbaum renormalization-group recursion relation, and the associated eigenvalue equations describing deterministic as well as stochastic perturbations, are found for the case of intermittency. These solutions are generated by a reformulation of the one-dimensional iterated map that exploits its topological equivalence to a translation. Direct resummation of series expansions gives the same results.

The Onsager reciprocity theorem has been since its discovery in 1931 one of the most prominent successes of nonequilibrium statistical mechanics. Its interpretation and application suffer from several limiting assumptions made in its derivation. The most notable of these is the linear relation between the currents and the affinities. A generalization of the Onsager theorem has been obtained, which is comparatively free of ad hoc assumptions.

If the proton is considered to be infinitely massive, no bound state of a system made up of a positron and a hydrogen atom can exist. In this Letter a new method is introduced for taking into account finite nuclear mass. By using this method it is shown that the inclusion of the finite mass of the proton does not result in the appearance of a bound state.

Quantum mechanical Hamiltonian models of Turing machines are constructed here on a finite lattice of spin-½ systems. The models do not dissipate any energy and they operate at the quantum limit in that the system (energy uncertainty)/(computation speed) is close to the limit given by the time-energy uncertainty principle.

The occurrence of sudden qualitative changes of chaotic (or "turbulent") dynamics is discussed and illustrated within the context of the one-dimensional quadratic map. For this case, the chaotic region can suddenly widen or disappear, and the cause and properties of these phenomena are investigated.

The projection method is a constructive proof of complete integrability of classical many-body systems. This method is extended to the quantum case. The wave functions and propagators of the $n$-particle Calogero systems are given in the form of integrals involving the wave functions and propagators of free motion in space of $n^2$ dimensions. Quantization of angular momentum in the large space yields solutions for discrete values of the coupling constant.

In this experiment, an electron- and optical-holographic technique is employed with small toroidal ferromagnets each forming a magnetic-flux closure. The holographic interferometry proves that a phase difference between two electron beams having passed through the field-free regions agrees well with the fundamental relation known as the Aharonov-Bohm effect. It is also confirmed from the same hologram that flux leakage from the toroids does not affect the conclusion.

Three subclasses of stationary axially symmetric diagonal Weyl metrics satisfying the empty-space Einstein equations are presented in an explicit form. These solutions generalize the $C$ metric (mass with constant acceleration) and contain six free parameters. Four of these parameters appear as exponents playing a role similar to $\delta$ in the Tomimatsu-Sato solution (mass with quadrupole moment). For integer values of the exponents these metrics are constructed from rational functions only. Special cases include various metrics of physical interest.

A force-measuring system composed of a macroscopic harmonic oscillator, mechanical-electrical transducer, and linear amplifier is described and the sensitivity is calculated for a "back-action evasion" mode of operation. It is shown that it is possible to surpass the standard amplifier limit with this device, and the specific requirements to do so are given. These results may also apply to a quantum-limited amplifier and thus give the requirements to perform quantum-nondemolition measurements.

Molecular dynamics simulations have been used to compute the thermal conductivity of the strongly coupled, nearly classical hydrogen plasma. The relaxation of a suitably defined heat current is significantly faster than the decay of the microscopic electric current. Electrical and thermal conductivities are not related by a simple Wiedemann-Franz law in the dense plasma.

A model of spin-½ statistics that explains the observed frequencies on the basis of the validity of the principle of locality is proposed. The model is based on the observation that certain density conditions on the unit sphere correspond with the observed frequencies while the resulting expectation values violate Bell's inequality.

A description of the decay into the bulk of surface perturbations in a system at its critical point is presented. The description, justified by detailed renormalization-group calculations, yields asymptotic order-parameter decay, $z^{-\frac{(d-2+\eta )}{2}}$, along with power-law behavior connected with surface exponents under suitable conditions. Comparison with the phenomenological theories of Widom and of Fisher and de Gennes is also made.

In the framework of stochastic quantization the stochastic process is derived for the harmonic oscillator in equilibrium with a thermal bath. This allows one to treat the quantum and thermal fluctuations within the same scheme of stochastic differential equations. As an application of the theory the quantum-classical crossover is computed in the dynamical critical exponent of the Van Hove model.

The accuracy of an $L^2$ discretization approximation as a means of incorporating effects of three-particle breakup on two-body scattering amplitudes has been tested in a soluble three-particle model. The coupled-reaction-channel Ansatz for the Schrödinger wave function was found to respond better to discretization than any of the connected-kernel, three-particle scattering formalisms.

It is shown that a general class of transverse electromagnetic waves with $\overrightarrow{\mathrm{E}}\parallel \overrightarrow{\mathrm{B}}$ exists. These waves possess magnetic helicity. In the case of plasma, both a high-frequency branch with ${\omega }^2={{\omega }_p}^2+k^2{c}^2$ and a low-frequency branch with $\omega \approx 0$ are allowed. The zero-frequency branch corresponds to the force-free magnetic field $\nabla \times \overrightarrow{\mathrm{B}}=k\overrightarrow{\mathrm{B}}$. These waves also exist in magnetized plasmas over a wide frequency range.

The eigenvalues, $E_{\mathrm{nl}}$, of the radial Schrödinger equation are approximated by $c_k(E=E_{\mathrm{nl}})=0\mathrm{}$ ($k=2, 3, \dots \infty$), where $E$ is the energy and the $c_k$ are the power-series coefficients of $f\left(\mathcal{r}\right)=\frac{\psi \left(\mathcal{r}\right)}{\left[{\mathcal{r}}^{l+1\mathrm{}}w\right(\mathcal{r}\left)\right]}$ with $\psi \left(\mathcal{r}\right)$ the wave function and $w\left(\mathcal{r}\right)$ a weight function. This is an extension of the method used to solve certain potentials exactly where $c_k(E=E_{\mathrm{nl}})=0\mathrm{}$ terminates a power series.

It is shown that the directed percolation on certain two-dimensional lattices, in which the occupation probability is unity along one spatial direction, is related to a random-walk problem, and is therefore exactly solvable. As an example, the case of the triangular lattice is solved. It is also shown that the square-lattice solution obtained previously by Domany and Kinzel can be derived using Minkowski's "taxicab geometry."

An exact vacuum solution of Einstein's field equations is presented, describing two isolated bodies balanced by their gravitational spin-spin interaction.