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The equations governing the flow of a fluid with rigid, spherical substructure are summarized. A two-dimensional flow field is considered and applied to the geometry of a slider bearing. Order-of-magnitude arguments are used which reduce the governing equations to a system of coupled, linear, ordinary differential equations. The equations are solved subject to appropriate boundary conditions and the effects of substructure discussed with the help of a specific numerical example.

Gathering the harvest represents a complex managerial problem for agricultural cooperatives involved in harvesting and processing operations: balancing the risk of overinvestment with the risk of underproduction. The rate to harvest crops and the corresponding capital investment are critical strategic decisions in situations where poor weather conditions present a risk of crop loss. In this article, we discuss a case study of the Concord grape harvest and develop a mathematical model to control harvest risk. The model involves differentiation of a joint probability distribution that represents risks associated with the length of the harvest season and the size of the crop. This approach is becoming popular as a means of dealing with complex problems involving operational and supply chain risk. Significant cost avoidance, in the millions of dollars, results from practical implementation of the Harvest Model. Using real data, we found that the Harvest Model provides lower-cost solutions in situations involving moderate variability in both the length of season and the crop size as compared to solutions based on imposed risk policies determined by management.harvest risk, agriculture

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In order to make parallel analysis more accessible to researchers employing principal component techniques, regression equations are presented for the logarithms of the latent roots of random data correlation matrices with unities on the diagonal. These regression equations have as independent variables logarithms of: the single variable degrees of freedom; Bartlett-Lawley degrees of freedom; the next lowest ordered eigenvalue. The multiple correlation coefficients are at least 0.96 in all cases.