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Solutions to two problems on permanents
- Source :
- Linear Algebra Appl. 436 (2012), no. 1, 53-58
- Publication Year :
- 2011
-
Abstract
- In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. Bapat conjectured that certain quotients of permanents, which generalize symmetric function means, are concave. We prove this conjecture by using concavity properties of hyperbolic polynomials. Motivated by problems on random point processes, Shirai and Takahashi raised the problem: Determine all real numbers $\alpha$ for which the $\alpha$-permanent (or $\alpha$-determinant) is nonnegative for all positive semidefinite matrices. We give a complete solution to this problem by using recent results of Scott and Sokal on completely monotone functions. It turns out that the conjectured answer to the problem is false.<br />Comment: 6 pages, to appear in Linear Algebra and its Applications
- Subjects :
- Mathematics - Rings and Algebras
Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Journal :
- Linear Algebra Appl. 436 (2012), no. 1, 53-58
- Publication Type :
- Report
- Accession number :
- edsarx.1104.3531
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.laa.2011.06.022