Daniel Lazard

CAD and topology of semi-algebraic sets

Although Cylindrical Algebraic Decomposition (CAD) is widely used to study the topology of semi-algebraic sets (especially algebraic curves), there are very few studies of the topological properties of the output of the CAD algorithms. In this paper three possible bad topological properties of the output of CAD algorithms are described. It is shown that these properties may not occur after a generic change of coordinates and that they may be avoided, in dimension not greater than three, with a modification of the CAD algorithm. As this modification of the CAD algorithm requires to solve some polynomial systems, it is also shown that the computation with real algebraic numbers may be advantageously replaced, in all the variants of the CAD algorithm, by solving systems of polynomial equations and inequalities.

Using Computer Algebra to Certify the Global Convergence of a Numerical Optimization Process

Abstract.The basic objective of blind signal separation is to recover a set of source signals from a set of observations that are mixtures of the sources with no, or very limited knowledge about the mixture structure and source signals. To extract the original sources, many algorithms have been proposed; among them, the cross-correlation and constant modulus algorithm (CC-CMA) appears to be the algorithm of choice due to its computational simplicity.An important issue in CC-CMA algorithm is the global convergence analysis, because the cost function is not quadratic nor convex and contains undesirable stationary points. If these undesirable points are local minimums, the convergence of the algorithm may not be guaranteed and the CC-CMA would fail to separate source signals.The main result of this paper is to complete the classification of these stationary points and to prove that they are not local minimums unless if the mixing parameter is equal to 1.This is obtained by using the theory of discriminant varieties to determine the stationnary points as a function of the parameter and then to show that the Hessian matrix of the cost function is not positive semidefinite at these stationnay points, unless if the mixing parameter is 1.

Molecular and Functional Characterization of Angiotensin II AT2 Receptor in Neuroblastoma N1E-115 Cells

Since the discovery in 1989 of a second subtype (AT2) of angiotensin II (Ang II) receptors (1,2), its functional significance has remained a matter of debate. Expression of AT2 receptors was detected in adrenals, uterus, ovary, heart, brain, and at highest levels in the developing fetus. This subtype has been associated with reduction of intracellular cGMP levels (3–5), modulation of tyrosine phosphorylation (4, 6–9), inhibition of a T-type calcium channel (10, 11), opening of potassium channels (12, 13), release of Prostaglandin (14) and production of arachidonic acid (15). Recent molecular cloning of the AT2 receptor cDNA in rat (6, 16), mouse (17) and human (18, 19) revealed a structural organization typical of G-protein coupled receptors, but did not provide yet a clear understanding of transduction pathways and physiological functions associated with this subtype.