Invariants Cohomologiques de Rost En Caractéristique PositiveRost''s Cohomological Invariants in Positive Characteristic

Let G/F be a semisimple algebraic group defined over a field F with characteristic . Let us denote by the Galois cohomology group introduced by Kato. If , we show that the p-primary part of Rost's invariant lifts in characteristic 0. This result allows to deduce properties of the Rost invariant in positive characteristic from known properties in characteristic 0. The case of Merkurjev–Suslin's invariant is specially interesting, i.e. if G/F=SL(D) for a central simple algebra D/F with degree p and class , one has and an element is a reduced norm if and only if the cup-product is trivial in ; one characterizes also in positive characteristic fields with p-dimension by the surjectivity of reduced norms.In a second part, we study Rost invariants when the base field is complete for a discrete valuation. As planned by Serre, invariants are then linked with Bruhat–Tits' theory, this yields a new proof of their nontriviality.     

Size of random Galois lattices and number of closed frequent itemsets

Given a sample of binary random vectors with i.i.d. Bernoulli(p) components, that is equal to 1 (resp. 0) with probability p (resp. 1−p), we first establish a formula for the mean of the size of the random Galois lattice built from this sample, and a more complex one for its variance. Then, noticing that closed α-frequent itemsets are in bijection with closed α-winning coalitions, we establish similar formulas for the mean and the variance of the number of closed α-frequent itemsets. This can be interesting for the study of the complexity of some data mining problems such as association rule mining, sequential pattern mining and classification.     

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.     

Quadratic characters in groups of odd order

We prove that in a finite group of odd order, the number of irreducible quadratic characters is the number of quadratic conjugacy classes.     

LC/ESI-MS n and 1H HR-MAS NMR analytical methods as useful taxonomical tools within the genus Cystoseira C. Agardh (Fucales; Phaeophyceae)

Species of the genus Cystoseira are particularly hard to discriminate, due to the complexity of their morphology, which can be influenced by their phenological state and ecological parameters. Our study emphasized on the relevance of two kinds of analytical tools, (1) LC/ESI-MSⁿ and (2) ¹H HR-MAS NMR, also called in vivo NMR, to identify Cystoseira specimens at the specific level and discuss their taxonomy. For these analyses, samples were collected at several locations in Brittany (France), where Cystoseira baccata, C. foeniculacea, C. humilis, C. nodicaulis and C. tamariscifolia were previously reported. To validate our chemical procedure, the sequence of the ITS2 has been obtained for each species to investigate their phylogenetic relationships at a molecular level. Our study highlighted the consistency of the two physico-chemical methods, compared to “classical” molecular approach, in studying taxonomy within the genus Cystoseira. Especially, LC/ESI-MSⁿ and phylogenetic analyses converged into the discrimination of two taxonomical groups among the 5 species. The occurrence of some specific signals in the ¹H HR-MAS NMR spectra and/or some characteristic chemical compounds during LC/ESI-MSⁿ analysis could be regarded as discriminating factors. LC/ESI-MSⁿ and ¹H HR-MAS NMR turned out to be two relevant and innovative techniques to discriminate taxonomically this complex genus.     

Integrable deformations of integrable Hamiltonian systems

We give new effective necessary conditions for the integrability of nonhomogeneous potentials which are sums of two homogeneous terms. These conditions were deduced from an analysis of the differential Galois group of variational equations. We apply the obtained result to polynomial perturbations of linear oscillator with two degrees of freedom.     

A general approach to fuzzy concepts

The paper proposes a flexible way to build concepts within fuzzy logic and set theory. The framework is general enough to capture some important particular cases, with their own independent interpretations, like “antitone” or “isotone” concepts constructed from fuzzy binary relations, but also to allow the two universes (of objects and attributes) to be equipped each with its own truth structure. Perhaps the most important feature of our approach is that we do not commit ourselves to any kind of logical connector, covering thus the case of a possibly non‐commutative conjunction too. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)