On the convergence of subdifferentials of convex functions

Research partially supported by NSF grant DMS-9001096. This author would like to thank l'équipe d'analyse convexe of U.S.T.L. Montpellier for its hospitality.     

Self-dual codes over the Kleinian four group

We introduce self-dual codes over the Kleinian four group K=Z ₂×Z ₂ for a natural quadratic form on K ⁿ and develop the theory. Topics studied are: weight enumerators, mass formulas, classification up to length 8, neighbourhood graphs, extremal codes, shadows, generalized t-designs, lexicographic codes, the Hexacode and its odd and shorter cousin, automorphism groups, marked codes. Kleinian codes form a new and natural fourth step in a series of analogies between binary codes, lattices and vertex operator algebras. This analogy will be emphasized and explained in detail.     

Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part

Let be a compact set with interior G. Let L ¹ (G,dx), >0 dx-a.e. on G, and m:=dx. Let A=(a _ij ) be symmetric, and globally uniformly strictly elliptic on G. Let be such that ; f, , is closable in L ² (G,m) with closure ( ^r ,D( ^r )). The latter is fulfilled if satisfies the Hamza type condition, or _i L ¹ _loc (G,dx), 1id. Conservative, non-symmetric diffusion processes X _t related to the extension of a generalized Dirichlet form where satisfies are constructed and analyzed. If G is a bounded Lipschitz domain, H ^1,1 (G), and a _ij D( ^r ), a Skorokhod decomposition for X _t is given. This happens through a local time that is uniquely associated to the smooth measure 1_{{ Tr ()>0}} d, where Tr denotes the trace and the surface measure on G.This research has been financially supported by TMR grant HPMF-CT-2000-00942 of the European Union. Mathematics Subject Classification (2000): 60J60, 60J55, 31C15, 31C25, 35J25     

The Berkeley Center for Structural Biology at the Advanced Light Source

The Berkeley Center for Structural Biology (BCSB) operates and develops a suite of protein crystallography beamlines at the Advanced Light Source (ALS) located at Lawrence Berkeley National Laboratory (LBNL). Although the ALS was conceived as a low-energy (1.9-GeV), third-generation synchrotron source of vacuum ultraviolet (VUV) and soft X-ray radiation, it was realized during the development of the facility in the mid-1990s that a multipole wiggler coupled with brightness-preserving optics would result in a beamline whose performance in the energy range of 5 to 15 keV would be sufficient for most protein crystallographic experiments. Later, the hard X-ray capabilities of the ALS were expanded by the addition of three superconducting bending magnets, resulting in additional protein crystallography facilities at the ALS [1 A.A. MacDowell, J Synchrotron Radiation 11(6), 447–55 (2004).[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]].     

On Convergence to Infinity

 By a metric mode of convergence to infinity in a regular Hausdorff space X, we mean a sequence of closed subsets of X with and , and a sequence (or net) in X is convergent to infinity with respect to provided for each contains eventually. Modulo a natural equivalence relation, these correspond to one-point extensions of the space with a countable base at the ideal point, and in the metrizable setting, they correspond to metric boundedness structures for the space. In this article, we study the interplay between these objects and certain continuous functions that may determine the metric mode of convergence to infinity, called forcing functions. Falling out of our results is a simple proof that each noncompact metrizable space admits uncountably many distinct metric uniformities. (Received 2 March 1999)     

How important are temperature effects for cluster polarizabilities?

State-of-the-art first-principle all-electron density functional theory calculations on small sodium clusters are performed to study the temperature dependency of their polarizabilities. For this purpose Born-Oppenheimer molecular dynamics simulations with more than 100,000 time steps (>200 ps) are recorded employing gradient corrected functionals in combination with a double-zeta valence polarization basis set. For each cluster 18 trajectories between 50 and 900 K are collected. The cluster polarizabilities are then calculated along these trajectories employing a triple-zeta valence polarization basis set augmented with field-induced polarization functions. The analysis of these calculations shows that the temperature dependency of the sodium cluster polarizabilities varies strongly with cluster size. For several clusters characteristic changes in the polarizability per atom as a function of temperature are observed. It is shown that the inclusion of finite temperature effects resolves the long-standing mismatch between calculated and measured sodium cluster polarizabilities.     

Spectroscopic characterization of molybdenum dinitrogen complexes containing a combination of di- and triphosphine coligands: 31P NMR analysis of five-spin systems

The three molybdenum-N2 complexes [Mo(N2)(dpepp)(depe)] (1), [Mo(N2)(dpepp)(dppe)] (2), and [Mo(N2)(dpepp)(1,2-dppp)] (3), all of which contain a combination of a bi- and a tridentate phosphine ligand, were prepared and investigated by vibrational and (31)P NMR spectroscopy. As a tridentate ligand bis(2-diphenylphosphinoethyl)phenylphosphine (dpepp) has been employed. The three different bidentate ligands are 1,2-bis(diethylphosphino)ethane (depe), 1,2-bis(diphenylphosphino)ethane (dppe), and R-(+)-1,2-bis(diphenylphosphino)propane (1,2-dppp). N-N as well as metal-N vibrations of 1-3 are identified and interpreted in terms of the geometric and electronic structures of the complexes. (31)P NMR spectra are recorded and fully analyzed. Moreover, correlation spectroscopy (COSY)-45 measurements are performed to determine the relative signs of coupling constants. Special attention is directed to a detection of different isomers and their (31)P NMR, as well as vibrational spectroscopic properties. The implications of the results for the area of synthetic nitrogen fixation with phosphine complexes are discussed.     

Reactivity of 2,2′‐Bis(2N‐(1,1′,3,3′‐tetramethyl‐guanidino))diphenylene‐amine with CuI and [Cu(MeCN)4][PF6]: Benzimidazole Formation vs. Cu Oxidation

The reaction of 2,2′‐Bis(2N‐(1,1′,3,3′‐tetramethyl‐guanidino))diphenylene‐amine (TMG₂PA) ( 1 ) with CuI in MeCN results in the formation of [Cu^II(TMG₂PA^amid)I] ( 2 ) indicatingthat Cu^I is the target of an oxidative attack of the N‐H proton of the ligand which itself is converted to molecular hydrogen. In contrast, if [Cu(MeCN)₄][PF₆] is used as the Cu^I source, [Cu^I₂(TMGbenz)₂][PF₆]₂ ( 3 ) is obtained instead. The use of the non‐coordinating counterion [PF₆]^– apparently prevents Cu^I from oxidation but induces itself a cyclisation reaction within the ligand which results in the formation of a benzimidazole‐guanidine ligand.     

The Gaussian and augmented-plane-wave density functional method for ab initio molecular dynamics simulations

A new algorithm for density-functional-theory-based ab initio molecular dynamics simulations is presented. The Kohn–Sham orbitals are expanded in Gaussian-type functions and an augmented-plane-wave-type approach is used to represent the electronic density. This extends previous work of ours where the density was expanded only in plane waves. We describe the total density in a smooth extended part which we represent in plane waves as in our previous work and parts localised close to the nuclei which are expanded in Gaussians. Using this representation of the charge we show how the localised and extended part can be treated separately, achieving a computational cost for the calculation of the Kohn–Sham matrix that scales with the system size N as O(NlogN). Furthermore, we are able to reduce drastically the size of the plane-wave basis. In addition, we introduce a multiple-cutoff method that improves considerably the performance of this approach. Finally, we demonstrate with a series of numerical examples the accuracy and efficiency of the new algorithm, both for electronic structure calculations and for ab initio molecular dynamics simulations. Received: 15 December 1998 /Accepted: 18 February 1999 /Published online: 14 July 1999