On the Curvature of the Central Path of Linear Programming Theory

We prove a linear bound on the average total curvature of the central path of linear programming theory in terms of the number of independent variables of the primal problem, and independent of the number of constraints.     

Newton's method on Riemannian manifolds: covariant alpha theory

In this paper, Smale's theory is generalized to the contextof intrinsic Newton iteration on geodesically complete analyticRiemannian and Hermitian manifolds. Results are valid for analyticmappings from a manifold to a linear space of the same dimension,or for analytic vector fields on the manifold. The invariant is defined by means of high-order covariant derivatives. Boundson the size of the basin of quadratic convergence are given.If the ambient manifold has negative sectional curvature, thosebounds depend on the curvature. A criterion of quadratic convergencefor Newton iteration from the information available at a pointis also given.     

Newton's method on Riemannian manifolds and a geometric model for the human spine

To study a geometric model of the human spine we are led tofinding a constrained minimum of a real valued function definedon a product of special orthogonal groups. To take advantgeof its Lie group structure we consider Newton's method on thismanifold. Comparisons between measured spines and computed spinesshow the pertinence of this approach.     

Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions

The role of surface energy in the behavior of colloidal particles in liquid crystalline phases is investigated. When the surface energy dominates, a hedgehog defect is formed and, according to an electrostatic analogy, the distortions around the particles exhibit a dipolar character. By contrast, for weaker anchoring, the configuration becomes quadrupolar as evidenced by the structure of latex clusters in lyotropic systems and the observation of defects reminiscent of Saturn rings in thermotropic systems. Received 21 June 1999     

Severi varieties on blow-ups of the symmetric square of an elliptic curve

We prove that certain Severi varieties of nodal curves of positive genus on general blow-ups of the twofold symmetric product of a general elliptic curve are nonempty and smooth of the expected dimension. This result, besides its intrinsic value, is an important preliminary step for the proof of nonemptiness of Severi varieties on general Enriques surfaces.     

Etude des fragmentations en spectrographie de masse de dithiannes-1,3 substitués en 2 par des groupements insaturés

The mass spectra of the title compounds always show an intense molecular ion. Some fissions involve loss of the ethylenic group, while others are related to cleavage of the dithiane ring, eventually followed by recyclizations involving the ethylenic chain stabilized by the resonance possibilities. If a phenyl group is present in the ethylenic chain, the spectra are modified by the formation of the very stable benzothiopyrylium ion.     

Computing the real roots of a polynomial by the exclusion algorithm

We describe a new algorithm for localizing the real roots of a polynomialP(x). This algorithm determines intervals on whichP(x) does not possess any root. The remainder set contains the real roots ofP(x) and can be arbitrarily small.