14 research outputs found
Matrix positivity preservers in fixed dimension. II: positive definiteness and strict monotonicity of Schur function ratios
We continue the study of real polynomials acting entrywise on matrices of
fixed dimension to preserve positive semidefiniteness, together with the
related analysis of order properties of Schur polynomials.
Previous work has shown that, given a real polynomial with positive
coefficients that is perturbed by adding a higher-degree monomial, there exists
a negative lower bound for the coefficient of the perturbation which
characterizes when the perturbed polynomial remains positivity preserving. We
show here that, if the perturbation coefficient is strictly greater than this
bound then the transformed matrix becomes positive definite given a simple
genericity condition that can be readily verified. We identity a slightly
stronger genericity condition that ensures positive definiteness occurs at the
boundary.
The analysis is complemented by computing the rank of the transformed matrix
in terms of the location of the original matrix in a Schubert cell-type
stratification that we have introduced and explored previously. The proofs
require enhancing to strictness a Schur monotonicity result of Khare and Tao,
to show that the ratio of Schur polynomials is strictly increasing along each
coordinate on the positive orthant and non-decreasing on its closure whenever
the defining tuples satisfy a coordinate-wise domination condition.Comment: 26 pages, no figures, LaTe
Construction of ECT-B-splines, a survey
s-dimensional generalized polynomials are linear combinations of functions
forming an ECT-system on a compact interval with coefficients from R
.
ECT-spline curves in R
are constructed by glueing together at interval endpoints
generalized polynomials generated from different local ECT-systems
via connection matrices. If they are nonsingular, lower triangular and totally
positive there is a basis of the space of 1-dimensional ECT-splines consisting
of functions having minimal compact supports normalized to form a nonnegative
partition of unity. Its functions are called ECT-B-splines. One
way (which is semiconstructional) to prove existence of such a basis is based
upon zero bounds for ECT-splines. A constructional proof is based upon
a definition of ECT-B-splines by generalized divided differences extending
Schoenberg’s classical construction of ordinary polynomial B-splines. This
fact eplains why ECT-B-splines share many properties with ordinary polynomial
B-splines. In this paper we survey such constructional aspects of
ECT-splines which in particular situations reduce to classical results.
s
Key Words: ECT-systems, ECT-B-splines, ECT-spline curves, de-Boor algorithm
AMS Classification Number: 41A15, 41A0
Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines
open3siProvided that they are in appropriate configurations (tight data), given planar G1 Hermite data generate a unique cubic Pythagorean hodograph (PH) spline curve interpolant. On a given associated knot-vector, the corresponding spline function cannot be C1, save for exceptional cases. By contrast, we show that replacing cubic spaces by cubic-like sparse spaces makes it possible to produce infinitely many C1 PH spline functions interpolating any given tight G1 Hermite data. Such cubic-like sparse spaces involve the constants and monomials of consecutive degrees, and they have long been used for design purposes. Only lately they were investigated in view of producing PH curves and associated G1 PH spline interpolants with some flexibility. The present work strongly relies on these recent results.embargoed_20220415Ait-Haddou R.; Beccari C.V.; Mazure M.-L.Ait-Haddou R.; Beccari C.V.; Mazure M.-L