Quantitative genetic diversity is a fundamental component of the interaction between natural populations and their environment. In breeding programmes, quantitative genetic studies on tropical trees have so far focused on fast-growing, light-demanding species, but no information exists on shade-tolerant, slow-growing species. For this study, 27 3-year-old open-pollinated families of the Neotropical shade-tolerant rainforest tree Sextonia rubra were measured in semicontrolled conditions for 20 morphological, growth, and photosynthesis traits; the effect of genetic relatedness, habitat of provenance, and mother tree status on seedling traits was analysed. Nine traits displayed significant genetic effects, while mother tree status and habitat effects were not significant (P>0.05) for an y trait. Estimated heritability varied between 0.14 and 0.28, with growth-related traits having the highest values. Additive genetic variation correlated positively with nonheritable variation, suggesting that ecological-evolutionary factors increasing or decreasing additive genetic variance may also affect nonheritable variation in the same direction. Our results suggest that quantitative genetic variability should be taken into account in ecological studies on, and in the management of, natural tropical rainforests; further research is needed to investigate genetic × environment interactions, in particular from the point of view of the genetic response of shade-tolerant plant species to variations in light availability.

Given a real number α, we aim at computing the best rational approximation with at most k digits and with exactly k digits at the numerator (denominator). Our approach exploits Farey sequences. Our method turns out to be very fast in the sense that, once the development of α in continued fractions is available, the required operations are just a few and their number remains essentially constant for any k (in double precision finite arithmetic). Estimations of error bounds are also provided.

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a b s t r a c tWe study a continuous model which describes the action of a velocity dependent drag on an oscillatory system. The solution of the resulting differential equation turns out to be asymptotic to a function which is expressible by means of the gamma and the hypergeometric functions. Among the applications, we consider that to the free oscillations around the fixed point which are ruled by van der Pol's equation. A section is devoted to computational procedures and simulations.

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