Morrey–Sobolev Extension Domains

We show that every uniform domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 2\) is a Morrey–Sobolev \({\mathscr {W}}^{1,\,p}\)-extension domain for all \(p\in [1,\,n)\), and moreover, that this result is essentially the best possible for each \(p\in [1,\,n)\) in the sense that, given a simply connected planar domain or a domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 3\) that is quasiconformal equivalent to a uniform domain, if it is a \({\mathscr {W}}^{1,\,p} \)-extension domain, then it must be uniform.     

Charging and discharging at the nanoscale: Fermi level equilibration of metallic nanoparticles

The redox properties of metallic nanoparticles are discussed, in particular the relationships between excess charge, size and the Fermi level of the electrons. The redox potentials are derived using simple electrostatic models to provide a straightforward understanding of the basic phenomena. The different techniques used to measure the variation of Fermi level are presented. Finally, redox aspects of processes such as toxicity, electrochromicity and surface plasmon spectroscopy are discussed.     

Sobolev mappings with integrable dilatations

We show that each quasi-light mapping f in the Sobolev space W ¹ⁿ (, R ⁿ ) satisfying ¦Df(x)¦ ⁿ K(x, f)J(x, f) for almost every x and for some KL ^r (), r>n-1, is open and discrete. The assumption that f be quasilight can be dropped if, in addition, it is required that f W ^1p (, R ⁿ ) for some p > = n + 1/ (n-2). More generally, we consider mappings in the John Ball classes Axxx _p,q (), and give conditions that guarantee their discreteness and openness.     

Intrinsic Geometry and Analysis of Diffusion Processes and L ∞-Variational Problems

The aim of this paper is twofold. First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (a) for all n ≧ 1, the diffusion matrix A is weak upper semicontinuous on Ω if and only if the intrinsic differential and the local intrinsic distance structures coincide; (b) if n = 1, or if n ≧ 2 and A is weak upper semicontinuous on Ω, the intrinsic distance and differential structures always coincide; (c) if n ≧ 2 and A fails to be weak upper semicontinuous on Ω, the (non-)coincidence of the intrinsic distance and differential structures depend on the geometry of the non-weak-upper-semicontinuity set of A. Second, for an arbitrary diffusion matrix A, we show that the intrinsic distance completely determines the absolute minimizer of the corresponding L ^∞-variational problem, and then obtain the existence and uniqueness for given boundary data. We also give an example of a diffusion matrix A for which there is an absolute minimizer that is not of class C ¹. When A is continuous, we also obtain the linear approximation property of the absolute minimizer.     

Understanding the eighteen-electron rule

It is pointed out that the preferred closed-shell electron structures, such as those in typical high-symmetry 18-electron systems, are driven both by the bonding contributions to the centre and by the kinetic-energy (nodal-structure) terms in the ligand subsystem, L_n. The latter imposes a filling order s < p < d, even for the ML_n complex. Then the 18e principle can be right for the right reasons even without any np contributions at the central atom.     

Quasihyperbolic boundary conditions and Poincaré domains

We prove that a domain in whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient is a (q,p)-\Poincare domain for all p and q satisfying and , where denotes the Sobolev conjugate exponent. An elementary example shows that the given ranges for p and q are sharp. The proof makes use of estimates for a variational capacity. When p=2 we give an application to the solvability of the Neumann problem on domains with irregular boundaries. We also discuss the relationship between this growth condition on the quasihyperbolic metric and the s-John condition. Received: 2 May 2000 / Published online: 17 June 2002     

An interactive algorithm to find the most preferred solution of multi-objective integer programs

In this paper, we develop an interactive algorithm that finds the most preferred solution of a decision maker (DM) for multi-objective integer programming problems. We assume that the DM’s preferences are consistent with a quasiconcave value function unknown to us. Based on the properties of quasiconcave value functions and pairwise preference information obtained from the DM, we generate constraints to restrict the implied inferior regions. The algorithm continues iteratively and guarantees to find the most preferred solution for integer programs. We test the performance of the algorithm on multi-objective assignment, knapsack, and shortest path problems and show that it works well.     

Molecular tweezers for hydrogen: synthesis, characterization, and reactivity

The first ansa-aminoborane N-TMPN-CH2C6H4B(C6F5)2 (where TMPNH is 2,2,6,6-tetramethylpiperidinyl) which is able to reversibly activate H2 through an intramolecular mechanism is synthesized. This new substance makes use of the concept of molecular tweezers where the active N and B centers are located close to each other so that one H2 molecule can fit in this void and be activated. Because of the fixed geometry of this ansa-ammonium-borate it forms a short N-H...H-B dihydrogen bond of 1.78 A as determined by X-ray analysis. Therefore, the bound hydrogen can be released above 100 degrees C. In addition, the short H...H contact and the N-H...H (154 degrees) and B-H...H (125 degrees) angles show that the dihydrogen interaction in N-TMPNH-CH2C6H4BH(C6F5)2 is partially covalent in nature. As a basis for discussing the mechanism, quantum chemical calculations are performed and it is found that the energy needed for splitting H2 can arise from the Coulomb attraction between the resulting ionic fragments, or "Coulomb pays for Heitler-London". The air- and moisture-stable N-TMPNH-CH2C6H4BH(C6F5)2 is employed in the catalytic reduction of nonsterically demanding imines and enamines under mild conditions (110 degrees C and 2 atm of H2) to give the corresponding amines in high yields.     

Sequential action of two flavoenzymes, PgaE and PgaM, in angucycline biosynthesis: chemoenzymatic synthesis of gaudimycin C

Tailoring steps in aromatic polyketide antibiotic biosynthesis are an important source of structural diversity and, consequently, an intriguing focal point for enzymological studies. PgaE and PgaM from Streptomyces sp. PGA64 are representatives of flavoenzymes catalyzing early post-PKS reactions in angucycline biosynthesis. This in vitro study illustrates that the chemoenzymatic conversion of UWM6 into the metabolite, gaudimycin C, requires multiple closely coupled reactions to prevent intermediate degradation. The NMR structure of gaudimycin C confirms that the reaction cascade involves C12- and C12b-hydroxylation, C2,3-dehydration, and stereospecific ketoreduction at C6. Enzymatic 18O incorporation studies verify that the oxygens at C12 and C12b derive from O2 and H2O, respectively. The results indicate that PgaM deviates mechanistically from flavoprotein monooxygenases, and suggest an alternative catalytic mechanism involving a quinone methide intermediate.