Non-deterministic inductive definitions

We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call “non-deterministic inductive definitions”. We give applications to formal topology as well as a predicative justification of this principle.     

Equivalents of the finitary non-deterministic inductive definitions

We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs. We also show that the finitary NID is equivalent to its binary fragment and that the elementary NID is equivalent to a variant of NID based on the notion of biclosed subset. Our result suggests that proving these statements in constructive topology requires genuine extensions of CZF with the elementary or finitary NID.     

Almost strictly totally positive matrices

A determinantal identity, frequently used in the study of totally positive matrices, is extended, and then used to re-prove the well-known univariate knot insertion formula for B-splines. Also we introduce a class of matrices, intermediate between totally positive and strictly totally positive matrices. The determinantal identity is used to show any minor of such matrices is positive if and only if its diagonal entries are positive. Among others, this class of matrices includes B-splines collocation matrices and Hurwitz matrices.This author acknowledges a sabbatical stay at IBM T.J. Watson Research Center in 1990, which was supported by a DGICYT grant from Spain.     

On positive strictly singular operators and domination

We study the domination problem by positive strictly singular % operators between Banach lattices. Precisely we show that if E and %F are two Banach lattices such that the norms on E' and F are %order continuous and E satisfies the subsequence splitting property, %and %0S T : E F are two positive operators, then T strictly %singular implies S strictly singular. The special case of %endomorphisms is also considered. Applications to the class of %strictly co-singular (or Pelczynski) operators are given too.     

The inductive wedge product of positive currents

We define the wedge product of positive currents. Our main issue is about studying the convergence of dd^c_gj∧T$d{d}^c{g}_j\wedge T$ where _gj$g_j$ is a given sequence of plurisubharmonic functions and T is a positive current.     

Unitarily invariant strictly positive definite kernels on spheres

We present a Fourier characterization for the continuous and unitarily invariant strictly positive definite kernels on the unit sphere in \({\mathbb {C}}^{q}\), thus adding to a celebrated work of I. J. Schoenberg on positive definite functions on real spheres.     

Some extremal problems for strictly totally positive matrices

For a strictly totally positive M × N matrix A we show that the ratio $\text{∥A}\text{x}\text{∥}{}_{\mathrm{p}}\text{∥}\text{x}\text{∥}{}_{\mathrm{p}}$ has exactly R = min{ M, N} nonzero critical values for each fixed p? (1, ∞). Letting λ_i denote the ith critical value, and xⁱ an associated critical vector, we show that λ₁ > … > λ_R > 0 and xⁱ (unique up to multiplication by a constant) has exactly i ? 1 sign changes. These critical values are generalizations to l^p of the s-numbers of A and satisfy many of the same extremal properties enjoyed by the s-numbers, but with respect to the l^p norm.     

The robust property of rational strictly positive real matrices

It is shown that not every strictly positive real matrix has robustness, and the necessary and sufficient conditions for a rational strictly positive real matrix with real coefficients to be robust are derived.Institute of Systems Science, Academia Sinica     

On the characterization of almost strictly totally positive matrices

A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations.     

Cohomogeneity one Riemannian manifolds of even dimension with strictly positive curvature, I

We classify compact positively curved Riemannian manifolds of even dimension acted on by a semisimple (but not simple) group with a codimension one orbit. Received: 19 September 2000; in final form: 29 October 2001/ Published online: 29 April 2002     

Characterizations of rectangular totally and strictly totally positive matrices

An n×m real matrix A is said to be totally positive (strictly totally positive) if every minor is nonnegative (positive). In this paper, we study characterizations of these classes of matrices by minors, by their full rank factorization and by their thin QR factorization.     

On the characterization of almost strictly totally positive matrices

A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations. Both authors were partially supported by the DGICYT Spain Research Grant PB93-0310     

Wellfoundedness proofs by means of non-monotonic inductive definitions II: First order operators

In this paper, we give two proofs of the wellfoundedness of a recursive notation system for Π_N-reflecting ordinals. One is based on distinguished classes, and the other is based on -inductive definitions.     

On strictly positive solutions for some semilinear elliptic problems

Let B be a ball in ${\mathbb{R}^{N}}$ , N ≥ 1, let m be a possibly discontinuous and unbounded function that changes sign in B and let 0 < p < 1. We study existence and nonexistence of strictly positive solutions for semilinear elliptic problems of the form ${-\Delta u=m(x) u^{p}}$ in B, u = 0 on ?B.