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Summary. A method is described for computing the exact rational solution to a regular system Ax=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) of A for some prime p; (ii) using successive refinements to compute an integer vector ~ such that A,2-b (modp") for a suitably large integer m; and (iii) deducing the rational solution x from the p-adic approximation 2. For matrices A and b with entries of bounded size and dimensions n x n and n x 1, this method can be implemented in time O(n3(logn) 2) which is better than methods previously used.

Weekly settlement of red and purple sea urchins (Strongylocentrotus franciscanus and S. purpuratus) was documented at sites in northern and southern California (USA) for 4 settlement seasons from 1990 to 1993. Newly settled sea urchins (<0.5 mm) were collected from standardized settlement surfaces (scrub brushes), identified and counted. There was marked seasonality in settlement. For both species, most settlement occurred during one to several episodes between February and July of each year. Settlement tended to be more regularly annual in southern than in northern California and settlement densities generally were higher in the south. This pattern is consistent with greater retention of water in the Southern California Bight and more energetic offshore advection of water in northern California S. purpuratus showed geographic coherence in settlement whereas S. franciscanus did not. For purple sea urchns, northern sites formed a group, a second group was formed by sites near Santa Barbara plus a site in Orange County, and a third group was formed by 3 sites at San Diego.

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Growth and survival of the red sea urchin Strongylocentrotus franciscanus were studied at 18 sites from southern California to Alaska. USA. Growth was determined using tetracycline tagging and was modeled using the Tanaka growth equation. Survival rates were estirnated using size-frequency distributions and growth parameters. Using log-linear analysis, it was determined that growth transitions differed among sites (p G 0.001) but there was no north-south difference (p > 0.80). Parameters for the Tanaka growth function were estimated for all data combined (N = 2714). Residuals for sites showed no latitudinal trend and so results were consistent with the log-linear analysis. Relative jaw (demi-pyramid) size, measured as the allometnc exponent ß in jaw length as a function of test diameter, has been shown to b e responsive to available food. For red sea urchins, ß was negatively correlated with growth but there was no correlation of relative jaw size with latitude, which suggests that latitudinal differences in food availability do not exist. In contrast with annual growth rates, annual survival rates were correlated with latitude and were higher in the north. Mean annual survival probability was 0.93 yr-' from northern California to Alaska and 0.77 yr-' in southern California. Likely causes for changes in survival rate with latitude are disease and temperature-related Stress. This paper provides the basis for development of hypotheses for size and survival differences between northern and southern populations of red sea urchins and. potentially, for other marine species with planktonic larvae.

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