Michel FN

We study the average number of intersecting points of a given curve with random hyperplanes in an n-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan, this problem is closely linked to finding the average number of real zeros of random polynomials. They showed that a real polynomial of degree n has on average log n + O(1) real zeros (M. Kac's theorem).This result leads us to the following problem: given a real sequence (αk )k N, study the averagewhere ρ(fn) is the number of real zeros of fn(X) = α0+α1X+ … + αnXn. We give theoretical results for the Thue—Morse polynomials and numerical evidence for other polvnomials

This paper sheds light on some unexpected consequences of health insurance regulation that may pose a big challenge to insurers' risk management. Because mandated uniform contributions to health insurance trigger risk‐selection efforts, risk adjustment (RA) schemes become necessary. A good deal of research into the optimal RA formula has been performed. A recent proposal in Switzerland has been to add ‘Hospitalization exceeding three days during the previous year’ as an indicator of high risk. Applying the new formula to an individual Swiss health insurer, its payments into the RA scheme are predicted to increase substantially, reaching up to 13% of premium income. Its mistake had been to implement Managed Care successfully, resulting in low rates of hospitalization. The expected risk management response is to extend hospital stays beyond three days, contrary to stated policy objectives.