On majorization and Schur products

Some majorization inequalities induced by Schur products in Euclidean Jordan algebras

“…The results of this paper are strongly motivated by the matrix theory works of Bapat and Sunder [5], Ando [2,3], and presentations found in Zhan ([25], Section 2.2) and Bhatia ([7], Chapters 4 and 5).…”

An update on Minc’s survey of open problems involving permanents

“…In this figure, a line links two partitions α and β to indicate that β α if β is lower down the page than α. This diagram is complete except that it has not yet been established which, if any, of the partitions (4, 2), (3 2 ) or (2 3 ) is dominated by (5,1). It can be seen that in every case the combination of two parts increases the partition and that the only relationship in Fig.…”

“…He also showed for arbitrary n that if A ∈ n and per(A) 1 2 then per(A k ) per(A) for all k 1. per(A(j |i)).…”

“…1. For proofs of these relationships, the reader should consult [1,2,66,78]. It is also worth noting that Problem 2 in Minc's catalogue asks for the resolution of a specific case of the permanental dominance conjecture.…”

An update on a few permanent conjectures

“…In the author's opinion, there is still a long way to go for a complete solution of the general conjecture. Cheon [3] that is weaker than the POT conjecture and explains that the weaker one may be true for a physical reason (quantum theory). The counterexample provided in [52] settles POT conjecture negatively but does not work for the Bapat & Sunder conjecture.…”