Steady states in the twin-well potential oscillator: Computer simulations and approximate analytical studies

The paper is focused on the phenomena of various steady-state oscillations exhibited by the twin-well potential system. Regions of existence of different attractors in the system parameter domain are examined and a picture book of different steady states for fixed damping and forcing is presented: 20 different combinations of single or coexisting, small orbit or large orbit, periodic and chaotic attractors are displayed. Computer simulations are followed by an approximate analytical analysis: A study of various forms of instability of periodic solutions gives close form approximate criteria for occurrence of T-periodic small orbit and large orbit oscillations, and for cross-well chaos.     

An analysis of mean life and lifetime of unstable elementary particles

A theoretical analysis of the concept of lifetime and mean life of unstable elementary particles is presented. New analytic formulas for lifetime and mean life as a function of decay width and the mass of unstable particle are derived for Breit-Wigner and Matthews-Salam energy distributions. It is demonstrated that, for unstable particles with a larger width or decay energy threshold, the deviation from the generally accepted mean life ^m = ^–1 is significant. The behavior of the decay law P(t) for small times is analyzed, and it is shown that the Breit-Wigner distribution violates the condition P(t = 0) = 0, whereas the Matthews-Salam distribution satisfies it.     

De Sitter Quasigroups

The Cayley–Klein parameters for the de Sitter groups SO(4, 1) and SO(3, 2) are introduced, and in an extension of the earlier investigation of quasigroups connected with Clifford groups, quasigroups connected with the SO(4, 1) and SO(3, 2) groups are determined. It is shown that these quasigroups have eight-dimensional, double-valued irreducible cracovian representations. The covariance of a five-dimensional form of the Dirac equation with respect to the quasi-rotations forming quasigroups connected with the groups SO(4, 1) and SO(3, 2) is demonstrated. An analogy is drawn between Weyl's hidden symmetry group and a quasigroup.     

Optical and microstructural properties of self-assembled InAs quantum structures in silicon

The InAs quantum structures were formed in silicon by sequential ion implantation and subsequent thermal annealing. Two kinds of crystalline InAs nanostructures were successfully synthesized: nanodots (NDs) and nanopyramids (NPs). The peaks at 215 and 235 cm^?1, corresponding to the transverse optical (TO) and longitudinal optical (LO) InAs single-phonon modes, respectively, are clearly visible in the Raman spectra. Moreover, the PL band at around 1.3 µm, due to light emission from InAs NDs with an average diameter 7±2 nm, was observed. The InAs NPs were found only in samples annealed for 20 ms at temperatures ranging from 1000 up to 1200°C. The crystallinity and pyramidal shape of InAs quantum structures were confirmed by HRTEM and XRD techniques. The average size of the NPs is 50 nm base and 50 nm height, and they are oriented parallel to the Si (001) planes. The lattice parameter of the NPs increases from 6.051 to 6.055 Å with the annealing temperature increasing from 1100 to 1200°C, due to lattice relaxation. Energy dispersive spectroscopy (EDS) shows almost stoichiometric composition of the InAs NPs.     

Duality transformations in electrodynamics

Electrodynamics admitting a duality transformation group is considered. For such an electrodynamics an extension of the classical Rainich-Misner-Wheeler theory is presented.On leave of absence from the University of Warsaw, Warsaw, Poland.     

On the universality of Einstein equations

It is proved that a Lagrangian field theory based on a linear connection in space-time is equivalent to Einstein's general relativity interacting with additional matter fields.     

The Weierstrass–Mandelbrot Process Revisited

We derive a functional central limit theorem for quasi-Gaussian processes. In particular, we prove that the limit of the Mandelbrot–Weierstrass process is a complex fractional Brownian motion.     

On the two-vector invariants of point groups

The paper is devoted to group-theoretical analysis of a function of two vectors, invariant under some point group (special attention is paid to the O_h group). In particular, the invariants in the direct product of spaces transforming according to the lth and l'th irreducible representations of the rotation group are studied. A compact formula determining the number of such invariants for the group O_h is found. It is shown that all possible invariants in the considered product space can be constructed from all possible scalar products of vector functions of both vectors transforming according to complex conjugate irreducible representations. The addition theorem for these functions is proved.