Loss of regularity for supercritical nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse, namely that there is a loss of regularity, in the spirit of the result due to G. Lebeau in the case of the wave equation. As a consequence, the Cauchy problem for energy-supercritical equations is not well-posed in the sense of Hadamard. We reduce the problem to a supercritical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functional à la Brenier.     

Ideal secret sharing schemes whose minimal qualified subsets have at most three participants

One of the main open problems in secret sharing is the characterization of the access structures of ideal secret sharing schemes. Brickell and Davenport proved that every one of these ideal access structures is related in a certain way to a unique matroid. Specifically, they are matroid ports. In addition to the search of general results, this difficult open problem has been studied in previous works for several families of access structures. In this paper we do the same for access structures with rank 3, that is, structures whose minimal qualified subsets have at most three participants. We completely characterize and classify the rank-3 access structures that are matroid ports. We prove that all access structures with rank three that are ports of matroids greater than 3 are ideal. After the results in this paper, the only open problem in the characterization of the ideal access structures with rank three is to characterize the rank-3 matroids that can be represented by an ideal secret sharing scheme. A previous version of this paper appeared in Fifth Conference on Security and Cryptography for Networks, SCN 2006, Lecture Notes in Computer Science 4116 (2006) 201–215.     

Density functional methods in the study of oxygen transfer reactions

The oxygen transfer reactions from heterocyclic compounds such as oxaziridine, methyldioxirane and ethylene oxide to ethylene and phosphine have been studied using different computational methods, and the results have been compared to those obtained with CASPT2 and CBS-QB3 methods. The results show that the generalized gradient approximation (GGA) functionals BLYP and PBEPBE, highly underestimate the barriers. On the other hand, the hybrid meta-GGA functionals (BB1K, M05 and M05-2X) and BHandHLYP tend to overestimate the barriers. Finally, the B3LYP and OPBE functionals provide reasonable barriers. Nevertheless, none of the tested functionals describes all the studied reactions with the same accuracy.     

Bicyclic Guanidine Promoted Mechanistically Divergent Depolymerization and Recycling of a Biobased Polycarbonate

We here report the organocatalytic and temperature-controlled depolymerization of biobased poly(limonene carbonate) providing access to its trans-configured cyclic carbonate as the major product. The base TBD (1,5,7-triazabicyclo[4.4.0]dec-5-ene) offers a unique opportunity to break down polycarbonates via end-group activation or main chain scission pathways as supported by various controls and computational analysis. These energetically competitive processes represent an unprecedented divergent approach to polycarbonate recycling. The trans limonene carbonate can be converted back to its polycarbonate via ring-opening polymerization using the same organocatalyst in the presence of an alcohol initiator, offering thus a potential circular and practical route for polycarbonate recycling.