Total Positivity: Tests and Parametrizations

Total Positivity: Tests and Parametrizations

Regulators and Total Positivity

The classical regulator R of a number field K is given [F1] by a rapidly convergent series of the form R w = ∞

Total Positivity and Convexity Preservation

It is well established that systems of totally positive blending functions, such as the Bernstein and B-spline bases, preserve monotonicity and convexity and are generally ‘shape preserving’ [9 ]. In this paper we show that total positivity is equivalent to the preservation of all orders of convexity. By a system we understand a sequence of functions (u0, . . . , un) defined on an interval [a, ...

Total Positivity, Grassmannians, and Networks

The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The boundary measurements of networks give...

Total positivity and cluster algebras

This is a brief and informal introduction to cluster algebras. It roughly follows the historical path of their discovery, made jointly with A. Zelevinsky. Total positivity serves as the main motivation. Mathematics Subject Classification (2000). Primary 13F60, Secondary 05E10, 05E15, 14M15, 15A23, 15B48, 20F55, 22E46.