Distribution Of Extreme Points Research Articles

Properties of quasiperiodic functions

Today, quasiperiodic tilings are well known and have been studied in great detail since they are very useful to describe the properties of metallic and soft matter quasicrystals. A closely related topic are quasiperiodic functions which have also gained large interest recently. Different types of such functions and there interrelation will be presented here. The main topic will be quasiperiodic potentials generated by laser beams and their variability. The distribution of extremal points and local isomorphisms of quasiperiodic functions will also be addressed.

The distribution of extreme points in best complex polynomial approximation

LetK be a compact point set in the complex plane having positive logarithmic capacity and connected complement. For anyf continuous onK and analytic in the interior ofK we investigate the distribution of the extreme points for the error in best uniform approximation tof onK by polynomials. More precisely, if $$A_n (f): = \{ z \in K:|f(z) - p_n^* (f;z)| = \parallel f - p_n^* (f)\parallel _K \} ,$$ wherep n * (f) is the polynomial of degree≤n of best uniform approximation tof onK, we show that there is a subsequencenk with the property that the sequence of (n k +2)-point Fekete subsets of\(A_{n_k }\) has limiting distribution (ask→∞) equal to the equilibrium distribution forK. Analogues for weighted approximation are also given.

Near-equilibrium instabilities in closed chemical systems under nonideal conditions

The stability behavior of a binary system in the mass-closed condition is analyzed using an integrated form of second law. The integrated change in free energy is plotted against the reaction coordinate. The trajectory provides a distribution of extremal points as a function of system parameters. As the parameters of the system exceed the critical values implying marginal stability condition, the plot undergoes a sharp change in qualitative behavior. The minima and maxima of the plot reflect locally stable and unstable equilibria, respectively, and when multiple stable critical points coexist for same set of parameter values, this plot helps to predict the selection of a particular asymptotic regime by the system as a function of initial conditions. If the system is initially in the vicinity of an unstable equilibrium separating two stable regimes, a small bias or asymmetry in the perturbation will dictate its evolution to a particular final state.