An Extremal Problem Concerning the Derivatives of a Polynomial at Its Roots

Let w be a monic polynomial of degree n +1 with roots x_j in the interval [−1, 1]. We consider the problem of finding the roots x_j for which the minimum of ¦ w '( x_j )¦, for 0≤ j ≤ n , is as large as possible. We prove that the Clenshaw–Curtis points cos( j π/ n ) are the only solution when n is even and that they get asymptotically close to the solution for odd values of n , as n goes to infinity. Our problem is related to the problem of minimizing the norm of inverse Vandermonde matrices.     

Symmetry transformations with noncommutative and nonassociative parameters

We study generalized symmetry transformations which involve nonassociative and noncommutative parameters. The structure underlying the group gradings is determined and examples are given. Graded algebras beyond Grassmann algebras are also presented. Nontrivial examples relevant for graded extensions beyond supersymmetry are given which resemble several features of quarks and might lead to a connection between the external and internal symmetries of the phenomenological models. Lie groups of transformations involving nonassociative and noncommutative parameters are obtained together with their corresponding graded Lie algebraic structures.     

Dynamics of the modified Michaelis-Menten system

The theory of cooperative and competitive systems is used to study the dynamical behaviour of the modified Michaelis-Menten system. Conditions for convergence to equilibria and existence of stable closed orbits are given, improving previous results obtained in [R.A. Smith, Some modified Michaelis-Menten equations having stable closed orbits, Proc. Roy. Soc. Edinburgh Sect. A 109 (1988) 341-359].     

A functorial construction of moduli of sheaves

We show how natural functors from the category of coherent sheaves on a projective scheme to categories of Kronecker modules can be used to construct moduli spaces of semistable sheaves. This construction simplifies or clarifies technical aspects of existing constructions and yields new simpler definitions of theta functions, about which more complete results can be proved. Dedicated to the memory of Joseph Le Potier.     

Paradoxical vocal fold movement: A case report

We present a case of one patient with respiratory stridor that was resolved by speech therapy. Paradoxical vocal fold movements were observed by flexible fiberoptic videolaryngoscopy in this patient during episodes of wheezing and dyspnea. Otherwise, normal vocal fold movement was observed in normal conditions of breathing (out of the crisis) and during phonation. Many different terms have been used to describe this entity in the literature, and it is crucial that clinicians recognize the subtle signs of functional stridor. Paradoxical vocal fold motion has to be considered as an important cause of respiratory disease. Its recognition and treatment are discussed in this report.     

On the probability distribution of data at points in real complete intersection varieties

We show several estimates on the probability distribution of some data at points in real complete intersection varieties: norms of real affine solutions, condition number of real solution of real systems of multi-variate polynomial equations and convergence radius of Newton's operator for under-determined system of multi-variate polynomial equations.     

Self-Assembly of Supramolecular Architectures Using Chlorotetra(Pyrrole-2-Carboxylato)Diruthenium Molecules as Building Blocks

Two new complexes, based on the unit Ru₂Cl(μ-O₂CC₄H₄N)₄ (1) (O₂CC₄H₄N = pyrrole-2-carboxylate), Ru₂Cl(μ-O₂CC₄H₄N)₄(H₂O)·4H₂O [1(H ₂ O)·4H ₂ O], and Ru₂Cl(μ-O₂CC₄H₄N)₄(Me₂CO) [1(Me ₂ CO)], are synthesized and structurally characterized. The physical properties of these complexes are studied and compared with those previously reported for Ru₂Cl(μ-O₂CC₄H₄N)₄(thf)·thf·H₂O [1(thf)·thf·H ₂ O]. The nature of the solvent molecule bonded to the axial position of the dimetallic unit determines the supramolecular interactions leading to different arrangements in the solid state. The presence of NH groups in the pyrrolic rings favours the existence of hydrogen bond interactions that are present in the three complexes. In addition, complex 1(Me ₂ CO) shows π–π stacking interactions through pyrrolic rings of different dimetallic units. Dedicated to the memory of Prof. F. Albert Cotton.     

A Multivalued Approach to the Planar Isosceles Three Body Problem

The isosceles three body problem consists of three point masses located on the vertices of an isosceles triangle on the plane. The two masses on the asymmetric edge are equal. This problem has been extensively studied but not as a perturbation of the Kepler problem. In this case we arrive at a differential inclusion as a natural formulation when we regularize the problem. We also derive an extension of the vectorfield that allows us to consider orbits across singular sets.