Sharkovsky’s program for the classification of triangular maps is almost completed

For a continuous map of the interval, there are more than 50 conditions characterizing zero topological entropy. Some are applicable to the class of triangular maps (x,y)?(f(x),g_x(y)) of the square, but only a few of them are equivalent in this more general setting. In 1989, A.N. Sharkovsky posed the problem of proving or disproving all possible implications between them. During last 20 years, 32 conditions were considered, and most of the work was done. Only 45 relations out of 992 remained not clear. In this paper we give a survey of known results, provide two new examples disproving another 26 possible implications, and spell out the remaining 19 open problems; all but one concern distributional chaos.     

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C¹ cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.     

Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations

In this paper, two Chebyshev-like third order methods free from second derivatives are considered and analyzed for systems of nonlinear equations. The methods can be obtained by having different approximations to the second derivatives present in the Chebyshev method. We study the local and third order convergence of the methods using the point of attraction theory. The computational aspects of the methods are also studied using some numerical experiments including an application to the Chandrasekhar integral equations in Radiative Transfer.     

A nonsmooth Lyapunov function and stability for ODE’s of Caratheodory type

The problem of Lyapunov stability for systems of ODE’s of Caratheodory type is considered. It is proved that without the necessity of calculating Dini or Clarke generalized gradients, the locally Lipschitz Lyapunov function can follow different kinds of stability.     

Some new nonlinear Volterra–Fredholm-type discrete inequalities and their applications

Some new explicit bounds on solutions to a class of new nonlinear Volterra–Fredholm-type discrete inequalities are established, which can be used as effective tools in the study of certain sum–difference equations. Application examples are also indicated.     

Facet degradation of high-power diode laser arrays

Micro-Raman facet temperatures of high-power diode lasers with different waveguide architectures are compared. For regular operation conditions, the thermal behavior of ‘unaged’ arrays emitting in the 808-nm wavelength region with different architectures is similar, however, with an increased load thermal behaviors differ significantly and exhibit failure events at facet temperatures typically between 150 and 450 °C. From various experiments, among them facet temperature measurements for ultrahigh-power operation as well as by preparative failure analytics, we provide evidence that in arrays the front facets are significantly affected by device operation and influence the failure behavior of the whole high-power diode laser also in cases when the device failure is accompanied by dislocation creation inside the device. Received: 3 October 1999 / Accepted: 9 November 1999 / Published online: 8 March 2000     

Temperature-induced phase transition in phlogopite revealed by Raman spectroscopy

Raman study of a natural hydrous phlogopite was carried out at temperatures up to 500 °C for the first time. Evolution of four well-resolved Raman modes at wavenumbers 196, 278, 322, and 682 cm⁻¹ was followed in detail with temperature increase. The analysis of data reveals linear decrease of vibrational wavenumbers in the studied temperature range, with small but experimentally significant discontinuities occurring at a temperature of 365±15 °C. Although the overall appearance of Raman spectra remains intact on crossing this temperature, the presence of discontinuities, as well as a marked difference between Gruneisen parameters calculated for high- and low-temperature ranges, signifies the presence of a temperature-induced phase transformation. By combining and correlating the results of the present Raman study with the high-temperature X-ray work performed by Tutti et al. [High-temperature study and thermal expansion of phlogopite, Phys. Chem. Miner. 27 (2000) 599-603] we arrive at the interpretation of a temperature-induced structural phase transformation in phlogopite without a significant symmetry change, with an underlying microscopic mechanism involving deformation of Mg octahedra and rotation of tetrahedral grid from ditrigonal toward hexagonal at the transition temperature.     

Implicit meanflow–multigrid algorithms for Reynolds stress model computation of 3‐D anisotropy‐driven and compressible flows

The present paper investigates the multigrid (MG) acceleration of compressible Reynolds‐averaged Navier–Stokes computations using Reynolds‐stress model 7‐equation turbulence closures, as well as lower‐level 2‐equation models. The basic single‐grid SG algorithm combines upwind‐biased discretization with a subiterative local‐dual‐time‐stepping time‐integration procedure. MG acceleration, using characteristic MG restriction and prolongation operators, is applied on meanflow variables only (MF–MG), turbulence variables being simply injected onto coarser grids. A previously developed non‐time‐consistent (for steady flows) full‐approximation‐multigrid (s–MG) is assessed for 3‐D anisotropy‐driven and/or separated flows, which are dominated by the convergence of turbulence variables. Even for these difficult test cases CPU‐speed‐ups r_CPUSUP∈[3, 5] are obtained. Alternative, potentially time‐consistent approaches (unsteady u–MG), where MG acceleration is applied at each subiteration, are also examined, using different subiterative strategies, MG cycles, and turbulence models. For 2‐D shock wave/turbulent boundary layer interaction, the fastest s–MG approach, with a V(2, 0) sawtooth cycle, systematically yields CPU‐speed‐ups of 5±½, quasi‐independent of the particular turbulence closure used. Copyright © 2008 John Wiley & Sons, Ltd.