73,995 research outputs found
The Lee-Yang and P\'olya-Schur Programs. II. Theory of Stable Polynomials and Applications
In the first part of this series we characterized all linear operators on
spaces of multivariate polynomials preserving the property of being
non-vanishing in products of open circular domains. For such sets this
completes the multivariate generalization of the classification program
initiated by P\'olya-Schur for univariate real polynomials. We build on these
classification theorems to develop here a theory of multivariate stable
polynomials. Applications and examples show that this theory provides a natural
framework for dealing in a uniform way with Lee-Yang type problems in
statistical mechanics, combinatorics, and geometric function theory in one or
several variables. In particular, we answer a question of Hinkkanen on
multivariate apolarity.Comment: 32 page
Stable Complete Intersections
A complete intersection of n polynomials in n indeterminates has only a
finite number of zeros. In this paper we address the following question: how do
the zeros change when the coefficients of the polynomials are perturbed? In the
first part we show how to construct semi-algebraic sets in the parameter space
over which all the complete intersection ideals share the same number of
isolated real zeros. In the second part we show how to modify the complete
intersection and get a new one which generates the same ideal but whose real
zeros are more stable with respect to perturbations of the coefficients.Comment: 1 figur
Stability inequalities for Lebesgue constants via Markov-like inequalities
We prove that L^infty-norming sets for finite-dimensional multivariatefunction spaces on compact sets are stable under small perturbations. This implies stability of interpolation operator norms (Lebesgue constants), in spaces of algebraic and trigonometric polynomials
Cosmographic analysis with Chebyshev polynomials
The limits of standard cosmography are here revised addressing the problem of
error propagation during statistical analyses. To do so, we propose the use of
Chebyshev polynomials to parameterize cosmic distances. In particular, we
demonstrate that building up rational Chebyshev polynomials significantly
reduces error propagations with respect to standard Taylor series. This
technique provides unbiased estimations of the cosmographic parameters and
performs significatively better than previous numerical approximations. To
figure this out, we compare rational Chebyshev polynomials with Pad\'e series.
In addition, we theoretically evaluate the convergence radius of (1,1)
Chebyshev rational polynomial and we compare it with the convergence radii of
Taylor and Pad\'e approximations. We thus focus on regions in which convergence
of Chebyshev rational functions is better than standard approaches. With this
recipe, as high-redshift data are employed, rational Chebyshev polynomials
remain highly stable and enable one to derive highly accurate analytical
approximations of Hubble's rate in terms of the cosmographic series. Finally,
we check our theoretical predictions by setting bounds on cosmographic
parameters through Monte Carlo integration techniques, based on the
Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic
data, using the JLA supernovae sample and the most recent versions of Hubble
parameter and baryon acoustic oscillation measurements. We find that
cosmography with Taylor series fails to be predictive with the aforementioned
data sets, while turns out to be much more stable using the Chebyshev approach.Comment: 17 pages, 6 figures, 5 table
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