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169
Noncommutative Elimination in Ore Algebras Proves Multivariate Identities
 J. SYMBOLIC COMPUT
, 1996
"... ... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation. ..."
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Cited by 90 (9 self)
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... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation.
On the Theories of Triangular Sets
 J. SYMB. COMP
, 1999
"... Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent. ..."
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Cited by 85 (35 self)
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Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent.
Recent Developments on Direct Relative Orientation
, 2006
"... This paper presents a novel version of the fivepoint relative orientation algorithm given in Nister (2004). The name of the algorithm arises from the fact that it can operate even on the minimal five point correspondences required for a finite number of solutions to relative orientation. For the mi ..."
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Cited by 61 (0 self)
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This paper presents a novel version of the fivepoint relative orientation algorithm given in Nister (2004). The name of the algorithm arises from the fact that it can operate even on the minimal five point correspondences required for a finite number of solutions to relative orientation. For the minimal five correspondences the algorithm returns up to ten real solutions. The algorithm can also operate on many points. Like the previous version of the fivepoint algorithm, our method can operate correctly even in the face of critical surfaces, including planar and ruled quadric scenes. The paper
A computational algebra approach to the reverse engineering of gene regulatory networks
 Journal of Theoretical Biology
, 2004
"... This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is timediscrete deterministic dynamical systems, with a finite set of states for each of the variables. The simplest examples of such models are Boolean networks, in whi ..."
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Cited by 43 (9 self)
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This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is timediscrete 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 reverseengineering algorithm, avoiding costly enumeration strategies. To perform well, the algorithm requires wildtype together with perturbation time courses. This makes it suitable for small to mesoscale networks rather than networks on a genomewide 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.
Highlevel filtering for arrangements of conic arcs
 In Proc. ESA 2002
, 2002
"... Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for impleme ..."
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Cited by 33 (9 self)
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Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for implementing robust geometric algorithms is to use exact algebraic number types — yet this may lead to a very slow, inefficient program. In this paper we suggest a simple technique for filtering the computations involved in the arrangement construction: when constructing an arrangement vertex, we keep track of the steps that lead to its construction and the equations we need to solve to obtain its coordinates. This construction history can be used for answering predicates very efficiently, compared to a naïve implementation with an exact number type. Furthermore, using this representation most arrangement vertices may be computed approximately at first and can be refined later on in cases of ambiguity. Since such cases are relatively rare, the resulting implementation is both efficient and robust. 1
Degenerations Of Flag And Schubert Varieties To Toric Varieties
 Transformation Groups
, 1996
"... . In this paper, we prove the degenerations of Schubert varieties in a minuscule G=P , as well as the class of Kempf varieties in the flag variety SL(n)=B, to (normal) toric varieties. As a consequence, we obtain that determinantal varieties degenerate to (normal) toric varieties. Introduction In ..."
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Cited by 32 (4 self)
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. In this paper, we prove the degenerations of Schubert varieties in a minuscule G=P , as well as the class of Kempf varieties in the flag variety SL(n)=B, to (normal) toric varieties. As a consequence, we obtain that determinantal varieties degenerate to (normal) toric varieties. Introduction In this paper, we carry out the proof of the results announced in [21]. Let G be a semisimple, simply connected algebraic group defined over an algebraically closed field k. Fix a maximal torus T in G, a Borel subgroup B oe T . Let W be the Weyl group of G relative to T . Let Q ' B be a parabolic subgroup of classical type, say Q = T r i=1 P k i , where P k i , 1 i r, is a maximal parabolic subgroup of classical type (see [26] for the definition of a parabolic subgroup of classical type). Let W (Q) be the Weyl group of Q. For w 2 W=W (Q), let X(w)(= BwQ (mod Q) with the canonical reduced structure of a scheme) denote the Schubert variety in G=Q, associated to w. Given m = (m 1 ; : : : ; m ...