Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models

Let be a nonnegative, smooth function with , supported in , symmetric, , and strictly increasing in . We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation

We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as : they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments.

     

Generic Fredholm alternative-type results for the one dimensional p-Laplacian

In this paper we give a generic necessary and sufficient condition for the solvability of? Received November 1999     

Nature of C60 and C70 fullerene encapsulation in a porphyrin- and metalloporphyrin-based cage: Insights from dispersion-corrected density functional theory calculations

The search for efficient synthetic hosts able to encapsulate fullerenes has attracted attention with regard to the purification and formation of ordered supramolecular architectures. This study of a porphyrin-based cage as an extension of the well-described ExCage⁶⁺ and BlueCage⁶⁺, involving viologen as sidearms, provides an interesting scenario where the oblate C₇₀ fullerene is preferred in comparison to the spherical C₆₀. Our results expose the nature of the fullerene-cage interaction involving ∼50% of dispersion-type interactions evidencing the strong π⋯π surface stacking, with a complementary contribution by the electrostatic and orbital polarization character produced by a charge reorganization with a charge accumulation facing the porphyrin macrocycles and a charge depletion along the equator formed by the viologens sidearms. Interestingly, the central N₄H₂ ring from each porphyrin contributes to the dispersion term via N-H⋯π interactions, which is decreased when the metallate N₄Zn is evaluated. Thus, the formation of stable and selective fullerene encapsulation can be achieved by taking into account two main driving forces, namely, (a) the extension of the π⋯π and X-H⋯π stacking surface and (b) charge reorganization over the fullerene surfaces, which can be used to control fine tuning of the encapsulation thanks to the introduction of more electron-deficient and electron-rich groups within the host cage.     

Synthesis and catalytic properties of nanoparticulate intermetallic Ga-Pd compounds

A two-step synthesis for the preparation of single-phase and nanoparticulate GaPd and GaPd(2) by coreduction of ionic metal-precursors with LiHBEt(3) in THF without additional stabilizers is described. The coreduction leads initially to the formation of Pd nanoparticles followed by a Pd-mediated reduction of Ga(3+) on their surfaces, requiring an additional annealing step. The majority of the intermetallic particles have diameters of 3 and 7 nm for GaPd and GaPd(2), respectively, and unexpected narrow size distributions as determined by disk centrifuge measurements. The nanoparticles have been characterized by XRD, TEM, and chemical analysis to ensure the formation of the intermetallic compounds. Unsupported nanoparticles possess high catalytic activity while maintaining the excellent selectivity of the ground bulk materials in the semihydrogenation of acetylene. The activity could be further increased by depositing the particles on α-Al(2)O(3).     

Direct Arylation in the Presence of Palladium Pincer Complexes

Direct arylation is an atom-economical alternative to more established procedures such as Stille, Suzuki or Negishi arylation reactions. In comparison with other palladium sources and ligands, the use of palladium pincer complexes as catalysts or pre-catalysts for direct arylation has resulted in improved efficiency, higher reaction yields, and advantageous reaction conditions. In addition to a revision of the literature concerning intra- and intermolecular direct arylation reactions performed in the presence of palladium pincer complexes, the role of these remarkably active catalysts will also be discussed.     

Silk‐Ion Jelly: a novel ion conducting polymeric material with high conductivity and excellent mechanical stability

In this work, a thin, flexible and mechanically stable polymer conducting material (Silk‐Ion Jelly) was developed though application of Ion Jelly on to silk fabrics. Ion Jelly was prepared through jellification of a room temperature ionic liquid, 1‐butyl‐3‐methyl‐imidazolium dicyanamide ([bmim][dca]) using gelatin and water and applied to silk fabrics using two different processes: impregnation and in‐situ. Various parameters influencing ionic conductivity such as Ion Jelly composition (ratio of [bmim][dca], water and gelatin) and incorporation as well as the type of application process were thoroughly investigated. It was observed that the Ion Jelly compositions containing lower gelatin and water ratio as well as application through in‐situ process at high temperature (200 °C) led to considerable improvement in conductivity, mainly due to increased [bmim][dca] concentration, structural flexibility and reduced silk crystallinity. Silk‐Ion Jelly prepared using optimized conditions showed excellent mechanical stability and possessed high room temperature conductivity (2.9 × 10^?3 S. cm^?1), similar to [bmim][dca], and therefore, this novel ion conducting material may find potential applications in electrochemical devices due to its eco‐friendly preparation route using biomaterials and green solvents. Copyright © 2012 John Wiley & Sons, Ltd.     

3-Aryl-4-Isoquinolinone Derivatives An Efficient Oxidative Preparation

A comparative study of the oxidation of 3-aryl-4-hydroxytetrahydroisoquinolines has been carried out. A modification of the Jones conditions turn out to be the best methodology for the regioselective preparation of target 4-isoquinolinone derivatives.     

Theoretical approach to cationic polymerization of alkenylfurans. II. Ab initio and semiempirical study of relevant steps in the reaction mechanism

SCF ab initio and AM1 semiempirical calculations with full geometry optimization have been carried out to model the initial steps of alkenylfuran cationic polymerization. Semiempirical procedures have been used to model propagation steps. Proton affinities of the relevant neutral species as well as the heats of formation of probable dimers were calculated. The results allow the explanation of the experimental behavior. The capabilities of the employed methods to describe these type of processes is discussed. It is confirmed that competition between vinyl and ring site attacks is the most important feature in these systems. © 1992 John Wiley & Sons, Inc.     

Nature of mercury inclusion in intermediate 6-valence electron [M@Au8Hgx(PPh3)8]n+ (M = Au,Pd, Pt; x = 0-2) protected gold superatoms: Insights from relativistic density functional theory calculations

Superatomic clusters offer useful templates displaying distinctive physical and chemical characteristics. Here, we explore the [M@Au₈(PPh₃)₈]ⁿ⁺ (M = Au, n = 3; Pd, Pt, n = 2) robust framework to gain an understanding of the nature of the inclusion of mercury atoms at Au₄ faces, leading to [M@Au₈Hg_x(PPh₃)₈]ⁿ⁺ (x = 1, 2). Our results show a weak interaction of about 25 kcal mol⁻¹ per Hg atom, which is mainly of electrostatic character, followed by orbital and London dispersion-type interactions. This weak interaction can be understood as the formation of host-guest species, for which the inherent electronic and optical properties of the [M@Au₈(PPh₃)₈] cluster along the series do not vary to a large extent. This demonstrates that, in [M@Au₈Hg_x(PPh₃)₈], each Hg can be considered an inclusion atom rather than a dopant element, where the parent cluster is able to act as a Lewis acid host. Furthermore, the viable formation of such species can serve as useful examples to stimulate future experimental characterization of inclusion complexes involving related superatomic structures with available open faces.