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In this paper, we present a parallel method to evaluate functions on pairs of elements. It is a challenge to partition the Cartesian product of a set with itself in order to parallelize the function evaluation on all pairs. Our solution uses (a) replication of set elements to allow for partitioning and (b) aggregation of the results gathered for different copies of an element. Based on an execution model with nodes that execute tasks on local data without online communication, we present a generic algorithm and show how it can be implemented with MapReduce. Three different distribution schemes that define the partitioning of the Cartesian product are introduced, compared, and evaluated. Any one of the distribution schemes can be used to derive and implement a specific algorithm for parallel pairwise element computation.

Micellar casein and casein monomers in milk serum are in a dynamic equilibrium. At temperature below 15–20 °C a considerable amount of casein monomers, β-casein in particular, is released from the casein micelle into the aqueous serum phase. This study investigates the effects of added calcium and related variations of pH on this peculiar equilibrium in order to minimize the amount of caseins in the serum and to better understand the casein permeation during microfiltration. The pH was varied in the range of 6.3 to 7.3 and the content of calcium was increased up to 7.5 mM by adding CaCl2. Upon equilibration, the milk was separated by ultracentrifugation and the amounts of protein in the supernatant were analyzed. It was shown that the addition of low amounts of calcium shifts the equilibrium towards the micellar casein phase and can, thus, lower the serum casein content induced at low temperatures. Relative to that, the adjustment of pH separately from the CaCl2 addition had a minor effect on casein concentration and composition in the serum.

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