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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 106 (30 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.
On triangular decompositions of algebraic varieties
 Presented at the MEGA2000 Conference
, 1999
"... We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifti ..."
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Cited by 75 (34 self)
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We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifting property for calculations modulo regular chains. This allows a good management of the intermediate computations, as confirmed by several implementations and applications of this work. Our algorithm is also well suited for parallel execution.
Computing Cylindrical Algebraic Decomposition via Triangular Decomposition
, 2009
"... Cylindrical algebraic decomposition is one of the most important tools for computing with semialgebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ R[y1,..., yn] we apply comprehensiv ..."
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Cited by 42 (14 self)
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Cylindrical algebraic decomposition is one of the most important tools for computing with semialgebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ R[y1,..., yn] we apply comprehensive triangular decomposition in order to obtain an Finvariant cylindrical decomposition of the ndimensional complex space, from which we extract an Finvariant cylindrical algebraic decomposition of the ndimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.
Algorithms for Computing Triangular Decomposition of Polynomial Systems
, 2011
"... We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we reg ..."
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Cited by 25 (17 self)
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We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we regard as essential to the recent success and for future research directions in the development of triangular decomposition methods.
Properness defects of projections and computation of one point in each connected component of a real algebraic set
, 2003
"... ..."
Properness defects of projections and computation of at least one point in each connected component of a real algebraic set
, 2004
"... ..."
Algebraic approaches to stability analysis of biological systems
 MATH. COMPUT. SCI
, 2008
"... In this paper, we improve and extend the approach of Wang and Xia for stability analysis of biological systems by making use of Gröbner bases, (CADbased) quantifier elimination, and discriminant varieties, as well as the stability criterion of Liénard and Chipart, and showing how to analyze the st ..."
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Cited by 11 (1 self)
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In this paper, we improve and extend the approach of Wang and Xia for stability analysis of biological systems by making use of Gröbner bases, (CADbased) quantifier elimination, and discriminant varieties, as well as the stability criterion of Liénard and Chipart, and showing how to analyze the stability of Hopf bifurcation points. The stability and bifurcations for a class of selfassembling micelle systems with chemical sinks are analyzed in detail. We provide experimental results with comparisons for 15 biological models taken from the literature.
Structure of carbamoyl phosphate synthetase: A journey of 96 Å from substrate to product,
 Biochemistry
, 1997
"... ABSTRACT: Carbamoyl phosphate synthetase catalyzes the production of carbamoyl phosphate from bicarbonate, glutamine, and two molecules of MgATP. As isolated from Escherichia coli, the enzyme has a total molecular weight of ∼160K and consists of two polypeptide chains referred to as the large and s ..."
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Cited by 11 (3 self)
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ABSTRACT: Carbamoyl phosphate synthetase catalyzes the production of carbamoyl phosphate from bicarbonate, glutamine, and two molecules of MgATP. As isolated from Escherichia coli, the enzyme has a total molecular weight of ∼160K and consists of two polypeptide chains referred to as the large and small subunits. Here we describe the Xray crystal structure of this enzyme determined to 2.8 Å resolution in the presence of ADP, Mn 2+ , phosphate, and ornithine. The small subunit is distinctly bilobal with the active site residues located in the interface formed by the NH 2 and COOHterminal domains. Interestingly, the structure of the COOHterminal half is similar to that observed in the trpGtype amidotransferase family. The large subunit can be envisioned as two halves referred to as the carboxyphosphate and carbamoyl phosphate synthetic components. Each component contains four distinct domains. Strikingly, the two halves of the large subunit are related by a nearly exact 2fold rotational axis, thus suggesting that this polypeptide chain evolved from a homodimeric precursor. The molecular motifs of the first three domains observed in each synthetic component are similar to those observed in biotin carboxylase. A linear distance of ∼80 Å separates the binding sites for the hydrolysis of glutamine in the small subunit and the ATPdependent phosphorylations of bicarbonate and carbamate in the large subunit. The reactive and unstable enzyme intermediates must therefore be sequentially channeled from one active site to the next through the interior of the protein. Carbamoyl phosphate synthetase (CPS) catalyzes one of the most remarkable reactions ever described in biological chemistry. This enzyme assembles carbamoyl phosphate from bicarbonate, glutamine, and two molecules of MgATP coupled with the production of glutamate, phosphate, and two molecules of MgADP. On the basis of isotopic labeling studies and the discovery of three additional partial reactions, Anderson and Meister (1) postulated that carbamoyl phosphate is constructed within the active site(s) of this enzyme by a series of four separate chemical reactions as illustrated below: There are thus a minimum of three discrete, highly reactive, and unstable intermediates in this reaction scheme: carboxyphosphate, carbamate, and ammonia. The existence of these intermediates and the order of events have been supported by steady state and rapid mixing kinetic analyses The synthesis of carbamoyl phosphate is critical for the initiation of two separate biosynthetic pathways. In one, carbamoyl phosphate is coupled to aspartate in a reaction catalyzed by aspartate transcarbamoylase. The carbon and nitrogen nuclei of carbamoyl phosphate are ultimately incorporated in the aromatic moieties of pyrimidine nucleotides such as UMP. In the second pathway, carbamoyl phosphate is condensed with ornithine at the start of the urea cycle and is utilized for the detoxification of ammonia and the biosynthesis of arginine. As would be anticipated, CPS therefore displays a complex pattern of metabolic regulation. The precise set of allosteric ligands depends on the particular biological and physiological niche of the cell where CPS resides. With the CPS from Escherichia coli, where there is a single enzyme for both pathways, the protein is allosterically activated by ornithine and inhibited by UMP.
Efficient computations of irredundant triangular decompositions with the regularchains library
 Proc. of CASA2007
, 2007
"... Abstract. We present new functionalities that we have added to the RegularChains library in Maple to efficiently compute irredundant triangular decompositions. We report on the implementation of different strategies. Our experiments show that, for difficult input systems, the computing time for remo ..."
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Cited by 10 (6 self)
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Abstract. We present new functionalities that we have added to the RegularChains library in Maple to efficiently compute irredundant triangular decompositions. We report on the implementation of different strategies. Our experiments show that, for difficult input systems, the computing time for removing redundant components can be reduced to a small portion of the total time needed for solving these systems. Since testing the inclusion between two quasicomponents can be as expensive as a radical membership test, and many pairs of quasicomponents may need to be compared, we believe that we have obtained an efficient solution.