Synthesis of a new polyfluorinated vinylogue of tetrathiafulvalene through 1,3-dipolar cycloaddition of ethyl beta-bromoperfluorodithiocrotonate with dimethyl acetylenedicarboxylate

The ethyl ester of beta-bromoperfluorodithiocrotonic acid reacts with dimethyl acetylenedicarboxylate to give 1,4-difluoro-2,3-bis(trifluoromethyl)-but-2-ene-1,4-diylidene-2,2'-bis(4',5'-dicarbomethoxy-1',3'-dithiole) (4), a new type of vinylogue of tetrathiafulvalene. The thermal transformations of this compound lead, depending on the temperature, to the formation of the cyclization products: 11,14-difluoro-2,3,8,9-tetra(carbomethoxy)-12,13-bis(trifluoromethyl)-1,4,7,10-tetrathiadispiro[4.0.4.4]tetradeca-2,8,11,13-tetraene (8) or 5,8-difluoro-6,7-bis(trifluoromethyl)-2,3-bis(carboxymethyl)-1,4-benzodithiine (11).     

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn–Hilliard equation.     

Convex Sobolev inequalities and spectral gap

This Note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux, and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities including in the limit case corresponding to the logarithmic Sobolev inequalities. To cite this article: J.-P. Bartier, J. Dolbeault, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Design and FPGA-realization of a self-synchronizing chaotic decoder in the presence of noise

This paper describes requirement for wireless transmission of Chaotic Code Division Multiplexed Access (Chaotic CDMA) and it focuses on real-time synchronization algorithm embedded into electronic programmable device. CDMA with quasi-orthogonal codes is used to allow multi-users to transmit simultaneously in the same channel. Since the channel is shared between all users, the receiver system has to cope with channel noise and overall with interference from other users. As a result, one of the main problems of communication with quasi-orthogonal chaotic codes is to implement a real time decoder in presence of noise. Even if set-membership algorithm are efficient in real time synchronization of chaotic discrete generators in the presence of noise, these algorithms require a large memory resource. In this paper, we propose an evolution of set-membership algorithm toward genetic algorithm to be implemented into electronic programmable device. The advantage of genetic algorithm compared with set-membership algorithm is that they require a fixed size of memory.     

Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combination of the theory introduced in Global smooth solution curves using rigorous branch following (van den Berg et al., Math. Comput. 79(271):1565–1584, 2010) and in Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs (Gameiro and Lessard, J. Diff. Equ. 249(9):2237–2268, 2010). Using this method, one can obtain proofs of existence of global smooth solution curves of equilibria for large (continuous) parameter ranges and about local uniqueness of the solutions on the curve. As an application, we compute several smooth branches of equilibria for the three-dimensional Cahn–Hilliard equation.     

Gradient bounds for solutions of semilinear parabolic equations without Bernstein's quadratic condition

We establish gradient estimates for bounded solutions of semilinear parabolic equations, where the nonlinearity only satisfies one-sided quadratic upper growth assumptions, instead of the classical (two-sided) Bernstein's condition. This extends a recent work of Al. and Ar. Tersenov (Indiana Univ. Math. J. 50 (2001) 1899–1913), where results of this kind were obtained for radial solutions in a ball, by a different technique. To cite this article: J.-Ph. Bartier, Ph. Souplet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).