Constructing Numerical Semigroups of a Given Genus

Let n_g denote the number of numerical semigroups of genus g. Bras-Amoros conjectured that n_g possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were based on analyzing the semigroup tree. We offer a new, simpler approach to counting numerical semigroups of a given genus. Our method gives direct constructions of families of numerical semigroups, without referring to the generators or the semigroup tree. In particular, we give an improved asymptotic lower bound for n_g.Comment: 11 pages, 3 figures, 2 tables; accepted by Semigroup Foru

Mortality tempo versus removal of causes of mortality

We propose an alternative way of dealing with mortality tempo. Bongaarts and Feeney have developed a model that assumes a fixed delay postponing each death. Our model, however, assumes that changes take place with the removal of a given cause of mortality. Cross-sectional risks of mortality by age and expectations of life therefore are not biased, contrary to the model of the two authors. Treating the two approaches as two particular cases of a more general process, we demonstrate that these two particular cases are the only ones that have general properties: The only model enjoying a decomposable expression is the removal model and the only model enjoying the proportionality property is the fixed delay model.cause of death, causes of mortality, cross-sectional life-table, fictitious life-table, life tables, mortality tempo, multiple decrement life-table, reference life-table

Towards a Better Understanding of the Semigroup Tree

In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which of the two types takes place in particular for well-known classes of semigroups. Also we study the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof. We conclude with some thoughts that show how this study of the semigroup tree may help in solving the conjecture of Fibonacci-like behavior of the number of semigroups with given genus.Comment: 17 pages, 2 figure