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SEMINARS
Totally Nonnegative Matrices and Stable Polynomials
时间  Datetime
2017-09-14 14:00 — 15:00 
地点  Venue
Middle Lecture Room
报告人  Speaker
Mohammad Adm
单位  Affiliation
University of Konstanz, Germany and University of Regina, Canada
邀请人  Host
Mikhail Tyaglov
报告摘要  Abstract

A real matrix is called totally nonnegative and totally positive if all of its minors are 
nonnegative and positive, respectively. Such matrices arise in a remarkable variety of ways in 
mathematics and many areas of its applications. It has a long time been conjectured by Juergen 
Garloff, that each matrix in a matrix interval with respect to the checkerboard partial order is 
nonsingular totally nonnegative if the two corner matrices are so. The Garloff’s Conjecture was 
affirmatively answered for the totally positive matrices and some subclasses of the totally nonnegative 
matrices.
 
In this talk, we present a condensed form of the Cauchon Algorithm which provides an efficient criterion 
for total nonnegativity of a given matrix and gives an optimal determinantal test for total nonnegativity. 
We briefly report on the way in which Garloff’s Conjecture was settled. These and related results evoke 
the (open) question whether the interval property holds for general nonsingular sign regular matrices. 
I will conclude my talk by reporting on the usefulness of such matrices in studying real stable polynomials, 
i.e., polynomials having their zeros in the open left half of the complex plane, and presenting a sufficient 
condition for an interval family of polynomials to be stable by using intervals of totally nonnagtive matrices.