Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations

A variety of numerical methods are available for determining the stability of a given solution of a partial differential equation. However for a family of solutions, calculation of boundaries in parameter space between stable and unstable solutions remains a major challenge. This paper describes an algorithm for the calculation of such stability boundaries, for the case of periodic travelling wave solutions of spatially extended local dynamical systems. The algorithm is based on numerical continuation of the spectrum. It is implemented in a fully automated way by the software package wavetrain, and two examples of its use are presented. One example is the Klausmeier model for banded vegetation in semi-arid environments, for which the change in stability is of Eckhaus (sideband) type; the other is the two-component Oregonator model for the photosensitive Belousov–Zhabotinskii reaction, for which the change in stability is of Hopf type.     

A note on some piecewise-linear difference equations with Mersenne-type periodic solutions

This paper studies solutions of some piecewise-linear difference equations. In two particular cases, a descent argument is used to show that all solutions are periodic with either prime period 3(2 ^k  ? 1) or 6(2 ^k  ? 1) for some k ≥ 1. The existence of solutions with such periods is also considered.     

Comparison between blue lasers and light‐emitting diodes for future solid‐state lighting

Solid‐state lighting (SSL) is now the most efficient source of high color quality white light ever created. Nevertheless, the blue InGaN light‐emitting diodes (LEDs) that are the light engine of SSL still have significant performance limitations. Foremost among these is the decrease in efficiency at high input current densities widely known as “efficiency droop.” Efficiency droop limits input power densities, contrary to the desire to produce more photons per unit LED chip area and to make SSL more affordable. Pending a solution to efficiency droop, an alternative device could be a blue laser diode (LD). LDs, operated in stimulated emission, can have high efficiencies at much higher input power densities than LEDs can. In this article, LEDs and LDs for future SSL are explored by comparing: their current state‐of‐the‐art input‐power‐density‐dependent power‐conversion efficiencies; potential improvements both in their peak power‐conversion efficiencies and in the input power densities at which those efficiencies peak; and their economics for practical SSL.     

Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the Stokes equation for the proliferation of tumor cells. For any positive radius R$R$, it is known that there exists a unique radially symmetric stationary solution. The proliferation rate μ$\mu$ and the cell-to-cell adhesiveness γ$\gamma$ are two parameters for characterizing “aggressiveness” of the tumor. We compute symmetry-breaking bifurcation branches of solutions by studying a polynomial discretization of the system. By tracking the discretized system, we numerically verified a sequence of μ/γ$\mu /\gamma$ symmetry breaking bifurcation branches. Furthermore, we study the stability of both radially symmetric and radially asymmetric stationary solutions.     

Numerically Computing Real Points on Algebraic Sets

Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach of Rouillier, Roy, and Safey El Din, which is based on a classical optimization approach of Seidenberg, to develop a homotopy based approach for computing at least one point on each connected component of a real algebraic set. Examples are presented demonstrating the effectiveness of this parallelizable homotopy based approach.     

Categorical Skew Lattices

Categorical skew lattices are a variety of skew lattices on which the natural partial order is especially well behaved. While most skew lattices of interest are categorical, not all are. They are characterized by a countable family of forbidden subalgebras. We also consider the subclass of strictly categorical skew lattices.     

Foam droplet separation for nanoparticle synthesis

A novel approach to nanoparticle synthesis was developed whereby foam bubble bursting produced aerosol droplets, an approach patterned after the marine foam aerosol cycle. The droplets were dried to remove solvent, leaving nanometer-sized particles composed of precursor material. Nanoparticles composed of sodium chloride (mean diameter, 100 nm), phosphotungstic acid ( 55 nm), and bovine insulin ( 5–30 nm) were synthesized. Foam droplet separation can be carried out at ambient temperature and pressure. The ‘soft’ nature of the process makes it compatible with a wide range of materials.     

New indications of potential rat intestinal α-glucosidase inhibition by Syzygium zeylanicum (L.) and its hypoglycemic effect in mice

Research on Chemical Intermediates - Natural α-glucosidase inhibitors (aGIs) have been considered effective agents for type 2 diabetes management. In this study, Syzygium zeylanicum (L.)...