A note on attractors with finite fractal dimension

Conditions for the finite fractal dimension of precompact invariantsets are formulated and exponential attractors for discreteand continuous dynamical systems are constructed.     

Dimension estimate of the exponential attractor for the chemotaxis‐growth system

We consider a chemotaxis‐growth model which takes into account diffusion, chemotaxis, production of chemical substance, and growth. We present estimates from above and below of the fractal dimension dim?? of the exponential attractor ?? in terms of the coefficients of the system. Comparisons are made between the sizes of the global and exponential attractors. Numerical simulations are presented which confirm the analytical results obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

The effect of an external mirror on the characteristics of an N2 laser-pumped distributed feedback dye laser

The effect of an external mirror on the DFB dye-laser parameters was investigated experimentally. When a 100% mirror was used, the output energy quadrupled, the divergence decreased to the diffraction limit. Besides the study of the improvement of laser parameters, the temporal behaviour of the amplified spontaneous emission and the DFB dye-laser action were simultaneously investigated.     

Upper and Lower Bounds for the Kolmogorov Entropy of the Attractor for the RDE in an Unbounded Domain

The long-time behaviour of bounded solutions of a reaction-diffusion system in an unbounded domain ⁿ , for which the nonlinearity f(u, _x u) explicitly depends on _x u is studied. We prove the existence of a global attractor, fractal dimension of which is infinite, and give upper and lower bounds for the Kolmogorov entropy of the attractor and analyze the sharpness of these bounds.     

Tunable distributed-feedback laser based on dye doped microporous quartz glass

The first distributed feedback (DFB) dye laser on the basis of microporous quartz glass (MQG) is reported. MQG consists of a specially processed quartz sample doped with a dye. As a pumping source a second harmonic YAG: Nd³⁺ laser with 12.5 pps repetition rate was used. Stable generation of the narrow line tuned within a spectral range of 562–584 nm and 633–663 nm with a linewidth of 0.04 nm was obtained. A high photochemical stability of the dye in the quartz matrix and reliability of MQG as an active medium for the tunable DFB laser were experimentally demonstrated.     

Geometrical properties of nonlinear maps and their application, Part II: Nonlinear Riemann-Hilbert problems with closed boundary data for multiply connected domains

We use the degree of quasiruled Fredholm maps on quasicylindrical domains developed in Part I to prove the existence of at least two homotopically different solutions of classical nonlinear Riemann-Hilbert problems for multiply connected domains.     

Nonlinear Riemann - Hilbert Problems without Transversality

Nonlinear Riemann - Hilbert problems (RHP) generalize two fundamental classical problems for complex analytic functions, namely: 1. the conformal mapping problem, and 2. the linear Riemann - Hilbert problem. This paper presents new results on global existence for the nonlinear (RHP) in doubly connected domains with nonclosed restriction curves for the boundary data. More precisely, our nonlinear (RHP) is required to become ?at infinity”?, i.e., for solutions having large moduli, a linear (RHP) with variable coefficients. Global existence for q-connected domains was already obtained in [9] for the special case that the restriction curves for the boundary data ?at infinity”? coincide with straight lines corresponding to linear (RHP)-s with special so-called constant - coefficient transversality boundary conditions. In this paper, the boundary conditions are much more general including highly nonlinear conditions for bounded solutions in the context of nontransversality. In order to prove global existence, we reduce the problem to nonlinear singular integral equations which can be treated by a degree theory of Fredholm - quasiruled mappings specifically constructed for mappings defined by nonlinar pseudodifferential operators.     

On the stability of a space vehicle riding on an intense laser beam

The Breakthrough Starshot Initiative is suggested to develop the concept of propelling a nanoscale spacecraft by the radiation pressure of an intense laser beam. In this project, the nanocraft is a gram‐scale robotic spacecraft comprising two main parts: StarChip and Lightsail. To achieve the goal of the project, it is necessary to solve a number of scientific problems. One of these tasks is to make sure that the nanocraft position and orientation inside the intense laser beam column are stable. The nanocraft driven by intense laser beam pressure acting on its Lightsail is sensitive to the torques and lateral forces reacting on the surface of the sail. These forces influence the orientation and lateral displacement of the spacecraft, thus affecting its dynamics. If unstable, the nanocraft might be expelled from the area of laser beam. In choosing the models for nanocraft stability studies, we are using several assumptions: (i) configuration of nanocraft is treated as a rigid body; (ii) flat or concave shape of circular sail; and (iii) mirror reflection of laser beam from surface of the Lightsail. We found conditions of position stability for spherical and conical shapes of the sail. The simplest stable configurations require the StarChip to be removed from the sail to make the distance to the center of mass of the nanocraft bigger than the curvature radius of the sail. Stability criteria do not require the spinning of the nanocraft. A flat sail is never stable. Copyright © 2016 John Wiley & Sons, Ltd.     

Synthesis of polymerizable oligomers via cationic polymerization of α-oxides

A series of new polymerizable oligomers of the general formula , where R = CH₃, CH₂Cl, n = 5?40, was synthesized via cationic polymerization of propylene oxide and epichlorohydrin in the presence of tin tetrachloride and methacrylic acid or its anhydride as chain transfer agents. It was shown that the chain length of oligomers is determined by the initial monomer/regulator ratio, a relation that makes it possible to prepare oligomers with the desired molecular mass. The synthesized oligomers readily dissolve in organic solvents and polymerize and copolymerize with unsaturated monomers to give networks or branched polymers.