1,490 research outputs found
Tropical totally positive matrices
We investigate the tropical analogues of totally positive and totally
nonnegative matrices. These arise when considering the images by the
nonarchimedean valuation of the corresponding classes of matrices over a real
nonarchimedean valued field, like the field of real Puiseux series. We show
that the nonarchimedean valuation sends the totally positive matrices precisely
to the Monge matrices. This leads to explicit polyhedral representations of the
tropical analogues of totally positive and totally nonnegative matrices. We
also show that tropical totally nonnegative matrices with a finite permanent
can be factorized in terms of elementary matrices. We finally determine the
eigenvalues of tropical totally nonnegative matrices, and relate them with the
eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of
FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author
is sported by the French Chateaubriand grant and INRIA postdoctoral
fellowshi
Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials
Generalizing recent results of Egge and Mongelli, we show that each diagonal
sequence of the Jacobi-Stirling numbers \js(n,k;z) and \JS(n,k;z) is a
P\'olya frequency sequence if and only if and study the
-total positivity properties of these numbers. Moreover, the polynomial
sequences \biggl\{\sum_{k=0}^n\JS(n,k;z)y^k\biggr\}_{n\geq 0}\quad \text{and}
\quad \biggl\{\sum_{k=0}^n\js(n,k;z)y^k\biggr\}_{n\geq 0} are proved to be
strongly -log-convex. In the same vein, we extend a recent result of
Chen et al. about the Ramanujan polynomials to Chapoton's generalized Ramanujan
polynomials. Finally, bridging the Ramanujan polynomials and a sequence arising
from the Lambert function, we obtain a neat proof of the unimodality of the
latter sequence, which was proved previously by Kalugin and Jeffrey.Comment: 17 pages, 2 tables, the proof of Lemma 3.3 is corrected, final
version to appear in Advances in Applied Mathematic
The inverse eigenvalue problem for symmetric anti-bidiagonal matrices
The inverse eigenvalue problem for real symmetric matrices of the form
0 0 0 . 0 0 *
0 0 0 . 0 * *
0 0 0 . * * 0
. . . . . . .
0 0 * . 0 0 0
0 * * . 0 0 0
* * 0 . 0 0 0
is solved. The solution is shown to be unique. The problem is also shown to
be equivalent to the inverse eigenvalue problem for a certain subclass of
Jacobi matrices.Comment: 6 pages; miscalculation corrected; acknowledgments adde
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