For a strictly totally positive M × N matrix A we show that the ratio IlAxllp/llxllp has exactly R = min{ M, N ) nonzero critical values for each fixed p ~ (1, ~ ) . Letting ~ denote the i th critical value, and x ~ an associated critical vector, we show that ~l > " " " > An > 0 and x i (unique up to multiplication by a constant) has exactly i 1 sign changes. These critical values are generaliz...

In this note we give necessary and sufficient conditions in the frequency domain for rational matrices to be strictly positive real. Based on this result, the matrix form of the Lefschetz-KalmanYakubovich lemma i s proved, which gives necessary and sufficient conditions for strictly positive real transfer matrices in the state-space realization form.

In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are act...

In this paper we seek for inner product dependent strictly positive definite kernels on subsets of C. We present separated necessary and sufficient conditions in order that a positive definite kernel on C be strictly positive definite. One emphasis is on strictly positive definite kernels on the unit circle. Since positive definite kernels on the circle were already characterized in [1], the st...

We prove results concerning the interlacing of eigenvalues of principal submatrices of strictly totally positive matrices. §

We begin by revisiting the idea of using a universe of types to write generic programs in a dependently typed setting by constructing a universe for Strictly Positive Types (SPTs). Here we extend this construction to cover dependent types, i.e. Strictly Positive Families (SPFs), thereby fixing a gap left open in previous work. Using the approach presented here we are able to represent all of Ep...

This paper presents sufficient conditions for the design of strictly positive real (SPR), fixed-order dynamic compensators. The primary motivation for designing SPR compensators is for application to positive real (PR) plants. When an SPR compensator is connected to a PR plant in a negative feedback configuration, the closed loop is guaranteed stable for arbitrary plant variations as long as th...

We establish a natural translation from word rewriting systems to strictly positive polymodal logics. Thereby, the latter can be considered as a generalization of the former. As a corollary we obtain examples of undecidable strictly positive normal modal logics. The translation has its counterpart on the level of proofs: we formulate a natural deep inference proof system for strictly positive l...

We study strictly positive definite functions on the unit circle in the Euclidean space of dimension two. We develop several conditions pertaining to the determination of such functions. The major result is obtained by considering the set of real numbers as a vector space over the field of rational numbers and then applying the Kronecker approximation theorem and Weyl’s criterion on equidistrib...

Strict positive realness (SPR) is an important concept in absolute stability theory, adaptive control, system identification, etc. This paper characterizes the strictly positive real (SPR) regions in coefficient space and presents a robust design method for SPR transfer functions. We first introduce the concepts of SPR regions and weak SPR regions and show that the SPR region associated with a ...