The total positivity interval

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 of Riordan arrays

An infinite matrix is called totally positive if its minors of all orders are nonnegative. A nonnegative sequence (an)n≥0 is called log-convex (logconcave, resp.) if aiaj+1 ≥ ai+1aj ( aiaj+1 ≤ ai+1aj , resp.) for 0 ≤ i < j . The object of this talk is to study various positivity properties of Riordan arrays, including the total positivity of such a matrix, the log-convexity of the 0th column an...

Sign variation, the Grassmannian, and total positivity

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 for cominuscule Grassmannians

In this paper we explore the combinatorics of the nonnegative part (G/P )≥0 of a cominuscule Grassmannian. For each such Grassmannian we define Γ -diagrams — certain fillings of generalized Young diagrams which are in bijection with the cells of (G/P )≥0. In the classical cases, we describe Γ -diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can...