Extended GCD and Hermite normal form algorithms via lattice basis reduction

Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small integer multipliers x(1), ..., x(m) for the equation s = gcd (s(1), ..., s(m)) = x(1)s(1) + ... + x(m)s(m), where s1, ... , s(m) are given integers. The method generalises to produce small unimodular transformation matrices for computing the Hermite normal form of an integer matrix.

Life-history consequences of divergent selection on egg size in Drosophila melanogaster

Life histories are generally assumed to evolve via antagonistic pleiotropy (negative genetic correlations) among traits, and trade-offs between life-history traits are typically studied using either phenotypic manipulations or selection experiments. We investigated the trade-off between egg size and fecundity in Drosophila melanogaster by examining both the phenotypic and genetic relationships between these traits after artificial selection for large and small eggs, relative to female body size. Egg size responded strongly to selection in both directions, increasing in the large-egg selected lines and decreasing in the small-egg selected lines. Phenotypic correlations between egg size and fecundity in the large-egg selected lines were negative, but no relationship between these traits occurred in either the control or small-egg selected lines. There was no negative genetic correlation between egg size and fecundity. Total reproductive allocation decreased in the small-egg selected lines but did not increase in the large-egg lines. Our results have three implications. First, our selection procedure may have forced females selected for large eggs into a physiological trade-off not reflected in a negative genetic correlation between these traits. Second, the lack of a negative genetic correlation between egg size and number suggests that the phenotypic trade-off frequently observed between egg size and number in other organisms may not evolve over the short term via a direct genetic trade-off whereby increases in egg size are automatically accompanied by decreased fecundity. Finally, total reproductive allocation may not evolve independently of egg size as commonly assumed.