期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2004
卷号：101
期号：42
页码：15018-15022
DOI：10.1073/pnas.0405413101
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：We report 96 inequalities with common structure, all elementary to state but many not elementary to prove. If n is a positive integer, a = (a1,..., an) and b = (b1,..., bn) are arbitrary vectors in [IMG]f1.gif" BORDER="0">, and {rho}(mij) is the spectral radius of an n x n matrix with elements mij, then, for example: [IMG] fd2.gif" BORDER="0"> The second inequality is obtained from the first inequality by replacing min with max and x with + and by reversing the direction of the inequality. The third inequality is obtained from the first by replacing the summation by the spectral radius. The fourth inequality is obtained from the first by taking each summand as a coefficient in a quadratic form. The fifth inequality is obtained from the first by replacing both outer summations by products, min by x, x by +, and the nonnegative vectors a and b by nonnegative measurable functions f and g. The proofs of these inequalities are mysteriously diverse.
