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Abstract: We investigate amenability, weak amenability and -amenability of the -algebra of polynomial hypergroups, and derive from these properties some applications for the corresponding orthogonal polynomials. [Copyright &y& Elsevier]

Lag windows whose corresponding spectral windows are Jacobi polynomials or sums of Jacobi polynomials are introduced. The bias and variance of their spectral density estimators are investigated and their window bandwidth and characteristic exponent are determined. [Copyright &y& Elsevier]

Abstract: Loss of Hes7 function leads to irregular somite formation demonstrating that Hes7 is a crucial component of the segmentation clock during somitogenesis. Experiments revealed that not only the repressor functionality but also the half-life of the protein is crucial for oscillatory expression of Hes7 and regular somite formation. Numerical integration of a delay equation system supported this finding. However, in a recent paper it was shown that the number of binding sites is also decisive for damped or undamped oscillations. It was shown that for more than one binding site the Hill coefficient increases. This leads to a completely different behavior. The oscillations are undamped and thus the mathematical model can no longer explain the results observed in the experiments. In this paper we propose a more sophisticated model for the Hes7 oscillator. Since Hes7 is degraded by the ubiquitin-proteasome pathway we include Michaelis-Menten kinetics for the ubiquitination of Hes7. We identify the Michaelis-Menten constant as an additional model parameter for oscillatory behavior. By increasing the Michaelis-Menten constant we found damped oscillations even if the Hill coefficient is increased. [Copyright &y& Elsevier]