By Emil Grosswald, Marvin Isadore Knopp, Mark Sheingorn

Emil Grosswald was once a mathematician of significant accomplishment and memorable breadth of imaginative and prescient. This quantity can pay tribute to the span of his mathematical pursuits, that is mirrored within the wide variety of papers accrued right here. With contributions via prime modern researchers in quantity concept, modular capabilities, combinatorics, and comparable research, this ebook will curiosity graduate scholars and experts in those fields. The top of the range of the articles and their shut connection to present examine traits make this quantity a needs to for any arithmetic library

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Fl(m)) V (i,j) G F ( i i < l ) , . . (m))]. , 4 ( m » y = 1]. EXAMPLE. The Borda social welfare function is also nondictatorial. (7) Independence of Irrelevant Alternatives (IIA) DEFINITION. A SWF is IIA if and only if the group choice between alternatives a and b depends only on how the individuals feel about a and b, not on their choices regarding other alternatives. (fc) «*- (j,i) G 5

A more combinatorial and discrete aspect of economics is the theory of social welfare functions 22 , 2 3 . Its most famous result won Kenneth Arrow a Nobel prize in 1973 for work done in 1951. f. in terms of Boolean matrices rather than the usual relation-theoretic approach. For this, see Sen 31 . Sen also won a Nobel prize in 1998 for his excellent contributions to welfare economics including social choice theory, welfare and poverty indices, and studies of famine. In 1994, John Harsanyi won a Nobel prize for his game theory solutions, and Reinhard Selten won a Nobel prize for his perfect equilibrium concept in game theory; John Nash won the same year for his solution concepts in game theory such as Nash equilibrium.

S. Brams and A. Taylor, Fair Division: From Cake-Cutting to Dispute Resolution, Cambridge Univ. Press, Cambridge, 1996. 6. -Q. Cao, K. H. Kim, and F. W. Roush, Incline Algebra and Applications, John Wiley, New York, 1984. 7. -P. -C. Falmagne, Knowledge Spaces, Springer, 1999. 8. -C. Falmagne, Elements of Psychophysical Theory, Oxford, 1985. 9. P. Fishburn, Decision theory and discrete mathematics, Discrete Applied Mathematics 68(1996),209-221. 10. P. Fishburn, Acyclic sets of linear orders: a progress report, preprint, AT& T Shannon Laboratory, 2000.