2-abelian complexity of the Thue-Morse sequence

Scientific conference (2012, December 14)

Let k be an integer. Two words u and v of the same length are k-abelian equivalent, if they have the same prefix (resp. suffix) of length k-1 and if, for all words x of length k, the numbers of ... [more ▼]

Let k be an integer. Two words u and v of the same length are k-abelian equivalent, if they have the same prefix (resp. suffix) of length k-1 and if, for all words x of length k, the numbers of occurrences of x in u and v are the same. This notion has received some recent interest, see the works of Karhumäki et al. The k-abelian complexity of an infinite word x maps an integer n to the number of k-abelian classes partitioning the set of factors of length n occurring in x. The Thue-Morse word is a well-known and extensively studied 2-automatic sequence. It is trivially abelian periodic and its (1)-abelian complexity takes only two values. The aim of this talk is to explain how to compute the 2-abelian complexity a(n) of the Thue-Morse word, showing in particular that it is unbounded. We conjecture that a(n) is 2-regular in the sense of Allouche and Shallit. This question can be related to a recent work of Madill and Rampersad where the (1)-abelian complexity of the paper folding word is shown to be 2-regular. We will also explain why the usual logical framework introduced by Büchi and popularized by Bruyère (indeed, automatic sequences can be characterized in an extension of the Presburger arithmetic) and the recent work of Charlier et al. about "automatic theorem-proving" of properties of automatic sequences cannot be applied to these questions related to abelian complexity. [less ▲]

2-Alkyl-3-alkylamino-2H-benzo- and pyridothiadiazine 1,1-dioxides: from K+ATP channel openers to Ca++ channel blockers?

in Biological Chemistry (2002), 383

2-Aminobenzamide and 2-Aminobenzoic Acid as New MALDI Matrices Inducing Radical Mediated In-Source Decay of Peptides and Proteins

in Journal of the American Society for Mass Spectrometry (2012), 23

(+/-)-2-Benzoyl-8-ethyl-1,2-dihydro-isoquinoline-1-carbonitrile: an original Reissert compound

in Acta Crystallographica Section E-Structure Reports Online (2007), 63(Part 7), 3161-3073

The title compound, C19H16N2O, is a Reissert compound. The heterocyclic fragment of the molecule exhibits a 1,3-diplanar conformation. The phenyl ring is connected to the isoquinoline ring system via an ... [more ▼]

The title compound, C19H16N2O, is a Reissert compound. The heterocyclic fragment of the molecule exhibits a 1,3-diplanar conformation. The phenyl ring is connected to the isoquinoline ring system via an amide bond that adopts an anti conformation with respect to the adjacent C-N bond in the adjacent heterocyclic ring. Intra-and intermolecular C-H center dot center dot center dot O hydrogen bonds are present in the crystal structure. [less ▲]

2-N-alkylation of 3-alkylamino-substituted 2H-benzo- and 2H-pyridothiadiazine 1,1-dioxides: from K+ATP channel openers to Ca++ channel blockers?

Poster (2001, November 17)

2-oxepane-1,5-dione: a precursor of a novel class of versatile semicrystalline biodegradable (co)-polyesters

Poster (2003, July 01)

2-oxepane-1,5-dione (OPD) was synthesized by Baeyer-Villiger oxidation of 1,4-cyclohexanedione and ring-opening polymerized by 1-phenyl-2-propanol / tin octoate with formation of a semicrystalline ... [more ▼]

2-oxepane-1,5-dione (OPD) was synthesized by Baeyer-Villiger oxidation of 1,4-cyclohexanedione and ring-opening polymerized by 1-phenyl-2-propanol / tin octoate with formation of a semicrystalline polyester of high melting temperature (Tm: 147°C). A crystallographic study showed that the polyester is orthorombic in the P212121 space with two molecules extended along the c-axis as is the case for poly(ε-caprolactone) (PCL). Dibutyltin dimethoxide and 2,2-dibutyl-2-stanna-1,3-dioxepane (DSDOP) proved to be efficient initiators for the random copolymerization of ε-caprolactone (εCL) and OPD. Random distribution of the comonomer units was confirmed by DSC, 1H and 13C NMR analyses. A single Tm was observed, which varies regularly and monotonically with the OPD content as result of a cocrystallisation phenomenon. The single glass transition temperature (Tg) obeys the Fox equation. It is noteworthy that a copolymer containing 30 mol % OPD exhibits a Tm of 90°C. It is stable up to 140°C under nitrogen for one hour, such that processing by extrusion is quite possible. The OPD decomposition is maximum at 223°C for a heating rate of 10°C/min. The mechanism of thermal degradation was investigated and cleared up. The ketone increases the copolymer hydrophilicity and thus sensitivity to hydrolysis. The mechanical properties are quite reminiscent of PCL of comparable molecular weight. Quite interestingly, OPD imparts sensitivity to photo-cross-linking to this novel class of materials. [less ▲]

2-oxepane-1,5-dione: A precursor of a novel class of versatile semicrystalline biodegradable (Co)polyesters

in Macromolecules (2002), 35(21), 7857-7859

A 2-step methodology for the generation of accurate FE models of nonlinear systems,

in ASME Internaional Mechanical Engineering Congress, Anaheim, 2004 (2004, November)

Model updating and validation is currently a central issue in the fields of computational structural mechanics and dynamics. The vast majority of applications however concerns linear structures. On the ... [more ▼]

Model updating and validation is currently a central issue in the fields of computational structural mechanics and dynamics. The vast majority of applications however concerns linear structures. On the other hand, updating nonlinear models is something the structural dynamicist prefers to avoid mainly because tools such as modal analysis are no longer available. The objective of the present study is to propose a two-step methodology for dealing with nonlinear systems. Its most appealing feature is that it decouples the estimation of the linear and nonlinear parameters. A numerical application consisting of an aeroplane-like structure is used to assess the efficiency of the procedure. [less ▲]

2. Importance des corrélations nucléaires dans l’étude de la matrice densité à un corps d’un noyau

Conference (1988)

2. Implantation des cultures

in Bodson, Bernard; Destain, Jean-Pierre (Eds.) Livre Blanc - Céréales (2013, February 27)