Algorithms for zero-dimensional ideals using linear recurrent sequences

Inspired by Faug\`ere and Mou's sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing the annihilator of one or several such sequences.Comment: LNCS, Computer Algebra in Scientific Computing CASC 201

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A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces

International audienceWe consider the problem of constructing roadmaps of real algebraic sets. The problem was introduced by Canny to answer connectivity questions and solve motion planning problems. Given $s$ polynomial equations with rational coefficients, of degree $D$ in $n$ variables, Canny's algorithm has a Monte Carlo cost of $s^n\log(s) D^{O(n^2)}$ operations in $\mathbb{Q}$; a deterministic version runs in time $s^n \log(s) D^{O(n^4)}$. The next improvement was due to Basu, Pollack and Roy, with an algorithm of deterministic cost $s^{d+1} D^{O(n^2)}$ for the more general problem of computing roadmaps of semi-algebraic sets ($d \le n$ is the dimension of an associated object). We give a Monte Carlo algorithm of complexity $(nD)^{O(n^{1.5})}$ for the problem of computing a roadmap of a compact hypersurface $V$ of degree $D$ in $n$ variables; we also have to assume that $V$ has a finite number of singular points. Even under these extra assumptions, no previous algorithm featured a cost better than $D^{O(n^2)}$

Resolvent and Rational Canonical Forms of Matrices

The goal of this paper is to explain how to derive from the resolvent of a matrix the following classical rational canonical forms: the Dunford decomposition, the rational spectral decomposition, the Jacobson and the rational Jordan forms (with corresponding change of basis matrices). We don't pretend that this approach is more efficient than other algorithms classicaly used to build these forms, but we transform theoritical and non rational methods used in the context of pertubation matrices into effective rational algorithms well adapted to Computer Algebra. 1 Resolvent Let us collect some basic facts about the resolvent of a linear operator acting on a finite-dimensional vector space (the general and classical reference for the first paragraph is [7]). A will be a matrix belonging to the algebra Mn (k) of square matrices of order n with coefficients in the zero characteristic field k. We will note I the identity matrix of Mn (k) and K an algebraic closure of k. Let oe(A) be the s..

The prevalence of frailty and its associated factors in an Italian institutionalized older population: findings from the cross-sectional Alvise Cornaro Center Study

Background: While it is well established that frail older people have a higher risk of negative health outcomes, the prevalence of frailty and its associated factors in Italian older institutionalized population has never been investigated. Aims: The aims of this study were to assess the prevalence of frailty and to identify its associated factors in an Italian residential care home population. Methods: An observational cross-sectional study was designed to evaluate older people aged 70 or over of an Italian residential care home. A multidimensional assessment examining functional, geriatric, ophthalmic, and audiological domains was carried out to identify factors associated with frailty. Physical frailty was evaluated using Fried’s criteria. Results: Data analysis uncovered a 51.1% prevalence of pre-frailty and a 40.4% prevalence of frailty in the 94 eligible participants (64 females) whose data were complete. The multivariable analysis showed that a low education level (OR = 5.12, 95% CI 1.22–21.49), a low physical quality of life score (OR = 13.25, 95% CI 3.51–50.08), a low mental quality of life score (OR = 9.22, 95% CI 2.38–35.69), visual impairment (OR = 7.65, 95% CI 1.77–33.14), and hearing impairment (OR = 4.62, 95% CI 1.03–20.66) were independently associated with frailty. Conclusions: Frailty was found to be highly prevalent in the residential care home studied. Since frailty is a reversible condition, identifying the modifiable factors associated to it should be viewed as an important step in planning and implementing targeted, early prevention strategies