Relations, residuals, regular interiors, and relative regular equivalence

Given a relation α (a binary sociogram) and an a priori equivalence relation π, both on the same set of individuals, it is interesting to look for the largest equivalence πo that is contained in and is regular with respect to α. The equivalence relation πo is called the regular interior of π with respect to α. The computation of πo involves the left and right residuals, a concept that generalized group inverses to the algebra of relations. A polynomial-time procedure is presented (Theorem 11) and illustrated with examples. In particular, the regular interior gives meet in the lattice of regular equivalences: the regular meet of regular equivalences is the regular interior of their intersection. Finally, the concept of relative regular equivalence is defined and compared with regular equivalence.

Ecological Relations Between the Herring and the Plankton off The North-East Coast of England

I. 430 plankton samples, which were taken by several herring drifters using the Continuous Plankton Recorder in the Shields fishing area during the summer seasons of 1931 to 1933, are analysed to show the main changes in the plankton during those seasons. 2. A comparison is made between the proportions of the different zooplankton organisms found in the plankton and the proportions of these recorded by Savage (1937) in the stomachs of herring obtained from drifters working in the same area and during the same time. The comparisons are made for 29 ten-day periods in the seasons 1931 to 1933, and in addition, for 6 ten-day periods relating to a single drifter which obtained both plankton and stomach samples at the same time in 1932. 3. The comparisons in 2 provide evidence that the herring feeds by selecting certain organisms by individual acts of capture and not by swimming open-mouthed to strain out the plankton indiscriminately: (a) Calanus and Temora in the stomachs either correspond fairly closely to the proportions in the plankton or they may be in very much higher proportions. The latter is always true regarding Anomalocera. (b) Acartia, Oithona, Cladocera and Lamellibranch larvae are always in larger proportions in the plankton than in the stomachs; this applies also to Centropages with two insignificant exceptions. (c) There is a close correspondence between the numbers of Limacina and Sagitta in the plankton and stomachs in the latter half of the 1931 season, but not during 1932 and 1933, when the numbers in the stomachs were insignificant ; during the former period there was a great scarcity of Calanus in the plankton.