Computing Ideals of Points

Abbott, John; Bigatti, Anna Maria; Kreuzer, Martin; Robbiano, Lorenzo

doi:10.1006/jsco.2000.0411

Cited by 55 publications

(77 citation statements)

References 11 publications

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“…The classical Padé approximation theory for univariate polynomials says that for any polynomials f, g ∈ F[x], where F is any field and g has degree t > 1, and for any positive integers t 1 and t 2 with t 1 + t 2 = t + 1, there are polynomials a ∈ F[x] of degree < t 1 and b ∈ F[x] of degree < t 2 so that (1) b · f ≡ a (mod g), and the ratio a/b is unique for all the solutions a and b. Furthermore, the extended Euclidean algorithm can be used to find a minimal solution a and b.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Gröbner bases and generalized Padé approximation

Farr

Gao

2005

Math. Comp.

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Abstract. It is shown how to find general multivariate Padé approximation using the Gröbner basis technique. This method is more flexible than previous approaches, and several examples are given to illustrate this advantage. When the number of variables is small compared to the degree of approximation, the Gröbner basis technique is more efficient than the linear algebra methods in the literature.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Gröbner bases and generalized Padé approximation

Farr

Gao

2005

Math. Comp.

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“…oller algorithm (Abbott et al, 2000). The complexity of this step is Oðm 2 ðgðm; nÞ þ mÞðlog pÞ 2 þ m 2 n 2 Þ as reported in Abbott et al (2000), where gðm; nÞ is Oðm ðnÀ1Þ=n Þ (Berman, 1981). Since g is sublinear in m; the worst case complexity of this step is quadratic in the number of nodes and cubic in the number of time points (Robbiano, 1998).…”

Section: Complexity Of the Algorithmmentioning

confidence: 97%

Section: Complexity Of the Algorithmmentioning

confidence: 99%

A computational algebra approach to the reverse engineering of gene regulatory networks

Laubenbacher

Stigler

2004

Journal of Theoretical Biology

231

253

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This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is time-discrete deterministic dynamical systems, with a finite set of states for each of the variables. The simplest examples of such models are Boolean networks, in which variables have only two possible states. The use of a larger number of possible states allows a finer discretization of experimental data and more than one possible mode of action for the variables, depending on threshold values. Furthermore, with a suitable choice of state set, one can employ powerful tools from computational algebra, that underlie the reverse-engineering algorithm, avoiding costly enumeration strategies. To perform well, the algorithm requires wildtype together with perturbation time courses. This makes it suitable for small to meso-scale networks rather than networks on a genome-wide scale. An analysis of the complexity of the algorithm is performed. The algorithm is validated on a recently published Boolean network model of segment polarity development in Drosophila melanogaster. r

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“…We have calculated the Ideal of table 4.1 with the function IdealOfPoints [62] in ApCoCoA [61] and the lexicographic order. In our case the ideal is: z 2 − 3z + 2, y 2 − 3y + 2, x 2 − 3x + 2 , and its corresponding Gröbner basis is: G = z 2 − 3z + 2, y 2 − 3y + 2, x 2 − 3x + 2 .…”

Section: Theoremmentioning

confidence: 99%

“…Calculate the reduced Gröbner Basis G (this can be also calculated with the function IdealOfPoints [62] in ApCoCoA).…”

Section: Define a Term Ordering τ (For Example Lexicographic)mentioning

confidence: 99%

Intelligent Management of Sepsis in the Intensive Care Unit

Ripoll

Rodríguez

Vellido

2012

Medical Applications of Intelligent Data Analysis

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Sepsis is a transversal pathology and one of the main causes of death in the Intensive Care Unit (ICU). It has in fact become the tenth most common cause of death in western societies. Its mortality rates can reach up to 60% for Septic Shock, its most acute manifestation. For these reasons, the prediction of the mortality caused by Sepsis is an open and relevant medical research challenge. This problem requires prediction methods that are robust and accurate, but also readily interpretable. This is paramount if they are to be used in the demanding context of real-time decision making at the ICU. In this brief contribution, three different methods are presented. One is based on a variant of the well-known support vector machine (SVM) model and provides and automated ranking of relevance of the mortality predictors while the other two are based on logistic-regression and logistic regression over latent Factors. The reported results show that the methods presented outperform in terms of accuracy alternative techniques currently in use in clinical settings, while simultaneously assessing the relative impact of individual pathology indicators.

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Computing Ideals of Points

Cited by 55 publications

References 11 publications

Gröbner bases and generalized Padé approximation

Gröbner bases and generalized Padé approximation

A computational algebra approach to the reverse engineering of gene regulatory networks

Intelligent Management of Sepsis in the Intensive Care Unit

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