The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

Random Walks And The Colored Jones Function

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones function: one in terms of the permanent of a matrix associated to a chord diagram, and another in terms of counting paths of intersecting chords. Electronic supplementary material to this article is available at and is accessible to authorized users. * S. G. was partially supported by an NSF and by an Israel-US BSF grant. † M. L. was partly supported by GAUK 158 grant and by the Project LN00A056 of the Czech Ministry of Education.     

Fonctions constructibles et intégration motivique II

We extend the formalism of an earlier Note to a global setting for which a theorem on fiber integrals and a Fubini theorem are obtained. We compare our formalism to the previous constructions given by Denef and Loeser. To cite this article: R. Cluckers, F. Loeser, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

JET: Preparing the future in fusion

JET (Joint European Torus) is the largest tokamak in the world and the only fusion facility able to operate with Tritium, the fusion fuel, and Beryllium, the ITER first wall material. JET also features the most complete remote handling equipment for invessel maintenance. As a multinational research center, JET provides logistic experience in preparing for operation of the global facility, tokamak ITER.Experiments on JET are focused on ITER-relevant studies, in particular on detailing the operational scenarios (EL My H-modes and advanced regimes), on enhancing the heating systems, on developing diagnostics for burning plasmas etc. Pioneering real-time control techniques have been implemented that maximize performance and minimize internal disturbances of JET plasmas. In helium plasmas, ion cyclotron heating (ICRH) created fast α-particles, mimicking their populations in future burning plasmas. The recent successful Trace Tritium campaign provided important new data on fuel transport. Current enhancements on JET include a new ITER-like ELM-resilient high power ICRH antenna (7 MW) and over twenty new diagnostics that will further extend the JET scientific capabilities and push the facility even closer to the ITER parameters.A special mention is given to the involvement of the fusion experts from Association EURATOM-IPP.CR, who have been actively participating in the collective use of JET facility for more than three years.     

Corrigendum: The complexity of counting graph homomorphisms

We close a gap in the proof of Theorem 4.1 in our paper “The complexity of counting graph homomorphisms” [Random Structures Algorithms 17 (2000), 260–289]. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Equivalence between mixed finite element and multi-point finite volume methods

We consider the lowest-order Raviart–Thomas mixed finite element method for elliptic problems on simplicial meshes in two or three space dimensions. This method produces saddle-point type problems for scalar and flux unknowns. We show how to easily eliminate the flux unknowns, which implies an equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. We describe the stencil of the final matrix and give sufficient conditions for its symmetry and positive definiteness. We present a numerical example illustrating the performance of the proposed method. To cite this article: M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Vibration analysis of planar serial-frame structures

A hybrid analytical/numerical method is proposed that permits the efficient dynamic analysis of planar serial-frame structures. The method utilizes a numerical implementation of a transfer matrix solution to the equation of motion. By analyzing the transverse and longitudinal motions of each segment simultaneously and considering the compatibility requirements across each frame angle, the undetermined variables of the entire frame structure system can be reduced to six which can be determined by application of the boundary conditions. The main feature of this method is to decrease the dimensions of the matrix involved in the finite element methods and certain other analytical methods.     

Des équations de Dirac et de Schrödinger pour la transformation de Fourier

Dyson has associated with the Fredholm determinants of the even (resp. odd) Dirichlet kernels a Schrödinger equation on the half-axis and has used, in tandem, the Gel'fand–Levitan and Marchenko methods of inverse scattering theory to study the asymptotics of these determinants. We have proposed following our unearthing of the conductor operator to seek to realize the Fourier transform itself as a scattering, and we obtain here to this end two Dirac systems on the entire real axis which are intrinsically associated, respectively, to the cosine and to the sine transforms. To cite this article: J.-F. Burnol, C. R. Acad. Sci. Paris, Ser. I 336 (2003).