On a metric for the class of compact convex sets

A distance function, defined in [12], for the class of compact convex sets inn-space is introduced in a new way, and some of its properties are developed. This concept is compared with some traditional distance functions for convex sets.     

On the Topology of Real Algebraic Plane Curves

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.     

Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model

In the present Note, we study the asymptotic behavior of the distribution density of the stock price process in the Hull–White model. The leading terms in the asymptotic expansions at zero and infinity are found for such a density and the corresponding error estimates are given. Similar problems are solved for time averages of the volatility process, which are also of interest in the study of Asian options. To cite this article: A. Gulisashvili, E.M. Stein, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Group-wise sufficient dimension reduction with principal fitted components

Sufficient dimension reduction methodologies in regressions of Y on a p-variate X aim at obtaining a reduction \(R(X) \in {\mathbb R}^{d}, d \le p\), that retains all the regression information of Y in X. When the predictors fall naturally into a number of known groups or domains, it has been established that exploiting the grouping information often leads to more effective sufficient dimension reduction of the predictors. In this article, we consider group-wise sufficient dimension reduction based on principal fitted components, when the grouping information is unknown. Principal fitted components methodology is coupled with an agglomerative clustering procedure to identify a suitable grouping structure. Simulations and real data analysis demonstrate that the group-wise principal fitted components sufficient dimension reduction is superior to the standard principal fitted components and to general sufficient dimension reduction methods.     

On dominant poles and model reduction of second order time-delay systems

The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the system?s input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.     

Synthese von optisch aktiven,natürlichen Carotinoiden und strukturell verwandten Naturprodukten. V. Synthese von (3R, 3′R)-, (3S, 3′S)- und (3R,3′S; meso)-Zeaxanthin durch asymmetrische Hydroborierung. Ein neuer Zugang zu Optisch aktiven Carotinoidbausteinen. Vorläufige Mitteilung

Synthesis of Optically Active Natural Carotenoids and Structurally Related Compounds. V. Synthesis of (3R, 3′R)-, (3S, 3′S)- and (3R,3′S; meso)-zeaxanthin by Asymmetric Hydroboration. A New Approach to Optically Active Carotenoid Building Units The synthesis of (3R, 3′R)-, (3S, 3′S)- and (3R,3′S; meso)-zeaxanthin ( 1 ), ( 19 ) and ( 21 ) is reported utilizing asymmetric hydroboration as the key reaction. Thus, safranol isopropenylmethylether ( 4 ) is hydroborated with (+)- and (?)-(IPC)₂BH to give the optically pure key intermediates 5 and 7 resp., which are transformed into the above-mentioned C₄₀-compounds.