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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Study of the permanent conjecture and some of its generalizations. II


Author: O. S. Rothaus
Journal: Trans. Amer. Math. Soc. 232 (1977), 143-154
MSC: Primary 15A48
MathSciNet review: 0447290
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Abstract: In this paper we investigate in a more systematic manner some of the topics initiated in part I of the paper with same title [5]. More specifically, we study in greater detail the properties of the function $ E(y)$ defined in [5] attached to convex polytopes, whose properties in the special case of the space of doubly stochastic matrices are connected with the permanent conjecture. Some close links with Perron-Frobenius theory are developed, and we obtain as a by-product of our study what is, I believe, a new expression for the maximum eigenvalue of a nonnegative matrix, which leads to some new estimates of the same. A final section of the paper investigates some purely algebraic properties of $ E(y)$, and we obtain some very interesting information connecting a doubly stochastic matrix and its transversals.

In order to keep this paper as self-contained as possible, facts used here drawn from part I are stated with as much explicit detail as possible.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0447290-9
PII: S 0002-9947(1977)0447290-9
Keywords: Permanent conjecture, Perron-Frobenius theory
Article copyright: © Copyright 1977 American Mathematical Society