Naslov
On the Total Positivity via Aissen-Schoenberg-Whitney Theorem

Autori
Martinjak, Ivica

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Croatian Combinatorial Days 2016, Book of Abstracts / - , 2016

Skup
1st Croatian Combinatorial Days

Mjesto i datum
Zagreb, Hrvatska, 29.09.2016. - 30.09.2016

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
totally positive matrix; Aissen-Schoenberg-Whitney theorem

Sažetak
Total positivity (TP) is a powerful concept that arises in various branches of mathematics. A matrix $M = [m_{; ; i, j}; ; ]_{; ; i, j \ge 0}; ; $ is {; ; \it totally positive}; ; if all its minors are non-negative real numbers. A remarkable spectral property of an $n \times n$ totally positive matrix $M$ is that $M$ has $n$ distinct positive eigenvalues. Any totally positive matrix can be relalized as a matrix of generalized complete homogeneous symmetric functions evaluated at non-negative real numbers. Total positivity of some types of triangular matrices, including {; ; \it Catalan triangles}; ; by Aigner and by Shapiro can be proved using the Aissen-Schoenberg-Whitney Theorem. On the base of these ideas, we demonstrate TP of some combinatorial matrices, indluding Stirling numbers of the first and the second kind. In addition, we present a class of totally positive matrices, defined by means of hypersequences that are recently introduced by Dil and Mez\H o.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA