Asymptotic aspects of Schreier graphs and Hanoi Towers groups

We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior. To cite this article: R. Grigorchuk, Z. ?unik´, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On n ‐contractive fuzzy logics

It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL‐chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL‐chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak cancellation defined in [31]. The algebraic counterpart of these logics is studied characterizing the subdirectly irreducible, the semisimple, and the simple algebras. Finally, some important algebraic and logical properties of the considered logics are discussed: local finiteness, finite embeddability property, finite model property, decidability, and standard completeness. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complete growth functions of hyperbolic groups

. We study a generalization of the growth functions of finitely generated groups, namely the growth functions Σ _{g ∈ G} gz ^{| g |} with coefficients in the group ring ℤ[G]. Rationality and methods of computation of such functions are discussed, in particular for hyperbolic groups. The complete growth functions of surface groups are explicitly computed. The operator and geodesic growth functions are also studied. Oblatum 20-IX-1996 & 13-I-1997     

Combinatorics of the subshift associated with Grigorchuk’s group

We study combinatorial properties of the subshift induced by the substitution that describes Lysenok’s presentation of Grigorchuk’s group of intermediate growth by generators and relators. This subshift has recently appeared in two different contexts: on the one hand, it allowed embedding Grigorchuk’s group in a topological full group, and on the other hand, it was useful in the spectral theory of Laplacians on the associated Schreier graphs.     

Local solvability of equations of magnetohydrodynamics on the whole space

Using the technique of expanding domains, we prove the existence of a weak, local in time solution to the equations of magnetohydrodynamics, derived from the equations for viscous, compressible and heat-conducting fluids, on the whole space under special assumptions on pressure and entropy. Compared with the same approach for barotropic compressible fluids, we show how to overcome loss of the global integrability of temperature and velocity fields in corresponding spaces.     

Intrinsic Flat and Gromov-Hausdorff Convergence of Manifolds with Ricci Curvature Bounded Below

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature bounded below and diameter bounded above, Gromov-Hausdorff convergence agrees with intrinsic flat convergence. In particular, the limiting current is essentially unique, has multiplicity one, and mass equal to the Hausdorff measure. Moreover, the limit spaces satisfy a constancy theorem.     

Dynamics of heat conducting fluids inside heat conducting domains

The existence of a solution is proved for a model of compressible, heat conducting fluid inside a heat conducting domain. Boundary conditions for the temperature involving a radiation term are assumed. Compared to other approaches, the scheme used for the proof requires lower regularity of the domain boundary in the case of its application to barotropic compressible fluids.     

Stereoselective synthesis of cobalt(III) anionic complexes with chiral pyruvylaminoacid oximes and metal–ligand interactions in aqueous solution

A series of cobalt(III) anionic complexes with hydroxyimino analogs of dipeptides – oximes of pyruvylglycine, pyruvyl-l-alanine, pyruvyl-l-methionine and pyruvyl-l-phenylalanine (H₂pamaco, amac = amino acid) of composition Cat₃[Co(H₋₁pamaco)₂] · nH₂O (Cat – a monovalent metal cation) has been synthesized and investigated by UV–Vis, CD and ¹H NMR spectroscopy. It was established that for the two latter ligands, non-racemic mixtures of two diastereomeric complexes were formed as a result of synthesis. In comparison with the complex containing pyruvyl-l-alanine, in which the relative content of R (C₂) and S (C₂) isomers is approximately equal, the presence of bulky substituents in the molecules of Met- and Phe-containing ligands results in a significant discrimination in formation of the S (C₂) isomer. This conclusion was drawn upon analysis of the CD and NMR spectral data as well as from molecular modeling. The relative content of the diastereomers was evaluated on the basis of NMR data. The protonation constants of the ligands have been determined in aqueous solution by glass electrode potentiometry, whereas their interactions with Co(II) were studied both under argon and oxygen atmosphere.     

Influence of medium acidity upon the luminescence properties of 2,5-diphenyl-1,3,4-oxadiazole and 2,5-diphenyl-1,3-oxazole

The luminescence properties of 2,5-diphenyl-1,3,4-oxadiazole and 2,5-diphenyl-1,3-oxazole in aqueous solutions of sulfuric acid (pH 7 to H_o − 10) were studied. The spectral parameters are essentially acidity dependent, which is due to the acid-base equilibria of these heterocycles both in the ground and in the first excited singlet states. The difference between these two compounds is governed by their dissimilar solvation. The basicity constants of 2,5-diphenyl-1,3,4-oxadiazole in the S₀ state (pK_BH+ = − 2.49) and 2,5-diphenyl-1,3-oxazole in the S₀ and S₁ states (pK_BH+ = −1.93, pK_BH+^* = 1.95) were experimentally obtained. The enthalpies of formation, electron charge density, and geometry of the bases and corresponding conjugate acids in the S₀ and S₁ states were calculated by the MNDO method.