The heat equation under conditions on the moments in higher dimensions

We consider the heat equation on the N‐dimensional cube (0, 1)^N and impose different classes of integral conditions, instead of usual boundary ones. Well‐posedness results for the heat equation under the condition that the moments of order 0 and 1 are conserved had been known so far only in the case of , for which such conditions can be easily interpreted as conservation of mass and barycenter. In this paper we show that in the case of general N the heat equation with such integral conditions is still well‐posed, upon suitably relaxing the notion of solution. Existence of solutions with general initial data in a suitable space of distributions over (0, 1)^N are proved by introducing two appropriate realizations of the Laplacian and checking by form methods that they generate analytic semigroups. The solution thus obtained turns out to solve the heat equation only in a certain distributional sense. However, one of these realizations is tightly related to a well‐known object of operator theory, the Krein–von Neumann extension of the Laplacian. This connection also establishes well‐posedness in a classical sense, as long as the initial data are L²‐functions.     

Hamilton connectedness and the partially square graphs

Let G be a κ-connected graph on n vertices. The partially square graphG^* of G is obtained by adding edges uv whenever the vertices u,v have a common neighbor x satisfying the condition N_G(x)⊂N_G[u]∪N_G[v]. Clearly G⊆G^*⊆G², where G² is the square of G. In particular G^*=G² if G is quasi-claw-free (and claw-free). In this paper we prove that a κ-connected, (κ?3) graph G is either hamiltonian-connected or the independence number of G^* is at least κ. As a consequence we answer positively two open questions. The first one by Ainouche and Kouider and the second one by Wu et al. As a by-product we prove that a quasi-claw-free (and hence a claw-free) graph satisfying the condition α(G²)<κ is hamiltonian-connected.     

Minimal regularity of the solution of some boundary value problems of Signorini's type in polygonal domains

We study the regularity in Sobolev spaces of the solution of transmission problems in a polygonal domain of the plane, with unilateral boundary conditions of Signorini's type in a part of the boundary and Dirichlet or Neumann boundary conditions on the remainder part. We use a penalization method combined with an appropriated lifting argument to get uniform estimates of the approximated solutions in order to obtain some minimal regularity results for the exact solution. The same method allows us to consider problems with thin obstacles. It can be easily extended to 3D problems. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular solution to Special Lagrangian Equations

We prove the existence of non-smooth solutions to three-dimensional Special Lagrangian Equations in the non-convex case.     

Über einfache Gruppen

Ohne Zusammenfassung     

Total synthesis,isolation, surfactant properties,and biological evaluation of ananatosides and related macrodilactone-containing rhamnolipids

Rhamnolipids are a specific class of microbial surfactants, which hold great biotechnological and therapeutic potential. However, their exploitation at the industrial level is hampered because they are mainly produced by the opportunistic pathogen Pseudomonas aeruginosa. The non-human pathogenic bacterium Pantoea ananatis is an alternative producer of rhamnolipid-like metabolites containing glucose instead of rhamnose residues. Herein, we present the isolation, structural characterization, and total synthesis of ananatoside A, a 15-membered macrodilactone-containing glucolipid, and ananatoside B, its open-chain congener, from organic extracts of P. ananatis. Ananatoside A was synthesized through three alternative pathways involving either an intramolecular glycosylation, a chemical macrolactonization or a direct enzymatic transformation from ananatoside B. A series of diasteroisomerically pure (1→2), (1→3), and (1→4)-macrolactonized rhamnolipids were also synthesized through intramolecular glycosylation and their anomeric configurations as well as ring conformations were solved using molecular modeling in tandem with NMR studies. We show that ananatoside B is a more potent surfactant than its macrolide counterpart. We present evidence that macrolactonization of rhamnolipids enhances their cytotoxic and hemolytic potential, pointing towards a mechanism involving the formation of pores into the lipidic cell membrane. Lastly, we demonstrate that ananatoside A and ananatoside B as well as synthetic macrolactonized rhamnolipids can be perceived by the plant immune system, and that this sensing is more pronounced for a macrolide featuring a rhamnose moiety in its native ¹C₄ conformation. Altogether our results suggest that macrolactonization of glycolipids can dramatically interfere with their surfactant properties and biological activity.

We show that macrolactonization of gluco- and rhamnolipids dramatically interfere with their surfactant properties and biological activity.     

A study of ACO capabilities for solving the maximum clique problem

This paper investigates the capabilities of the Ant Colony Optimization (ACO) meta-heuristic for solving the maximum clique problem, the goal of which is to find a largest set of pairwise adjacent vertices in a graph. We propose and compare two different instantiations of a generic ACO algorithm for this problem. Basically, the generic ACO algorithm successively generates maximal cliques through the repeated addition of vertices into partial cliques, and uses “pheromone trails” as a greedy heuristic to choose, at each step, the next vertex to enter the clique. The two instantiations differ in the way pheromone trails are laid and exploited, i.e., on edges or on vertices of the graph. We illustrate the behavior of the two ACO instantiations on a representative benchmark instance and we study the impact of pheromone on the solution process. We consider two measures—the re-sampling and the dispersion ratio—for providing an insight into the performance at run time. We also study the benefit of integrating a local search procedure within the proposed ACO algorithm, and we show that this improves the solution process. Finally, we compare ACO performance with that of three other representative heuristic approaches, showing that the former obtains competitive results.