Numerical Simulations of Random Walk in Random Environment

This paper is concerned with the numerical simulation of a random walk in a random environment in dimension d = 2. Consider a nearest neighbor random walk on the 2-dimensional integer lattice. The transition probabilities at each site are assumed to be themselves random variables, but fixed for all time. This is the random environment. Consider a parallel strip of radius R centered on an axis through the origin. Let X _R be the probability that the walk that started at the origin exits the strip through one of the boundary lines. Then X _R is a random variable, depending on the environment. In dimension d = 1, the variable X _R converges in distribution to the Bernoulli variable, X = 0, 1 with equal probability, as R . Here the 2-dimensional problem is studied using Gauss-Seidel and multigrid algorithms.     

Polymeric and mixed carboxylate compounds: Crystal structure of [Zn(tiglate)0.54 (benzoate)1.46]x

The crystal structure of [Zn(O₂CC(CH₃)CHCH₃)_0.54(O₂CC₆H₅)_1.46]_x has been determined by single crystal X-ray diffraction. The compound crystallizes in the monoclinic system, space group P2₁/c witha=10.311(1),b=13.170(1),c=19.000(2) Å,=91.895(7)°,V=2578.7 ų,Z=4,D _x=1.525g cm^–3, (Mo K)=0.71073 Å,=1–95 mm^–1, F(000)=1206,T=295K,R=0.046 for 4548 unique reflections. The two different carboxylates are randomly scrambled along a one dimensional polymer, made up of binuclear units connected by a syn-anti carboxylate bridge. Each binuclear unit contains three syn-syn carboxylates which bridge two zinc atoms. Vibrational data indicate that [Zn(O₂CC(CH₃)CHCH₃)₂]_x has the same polymeric structure; orientation of tiglate is the same as for the mixed ligand compound. Recrystallization of mixed carboxylate species produces either an alteration in carboxylate ligand ratio, or a total separation of the two carboxylates, giving two different compounds.     

Structure and properties of Ge2.5PSx glasses

Ge_2.5PS_x glasses were studied with a combination of Raman spectroscopy, nuclear magnetic resonance, and neutron diffraction. From these experiments the distribution of bonding configurations was determined, and used to explain the compositional dependence of the index of refraction and the glass transition temperature. On reducing the sulfur content of these glasses below the stoichiometric amount, the sulfur deficit is accommodated by the progressive loss of the non-bridging sulfur of SPS_3/2 groups, followed by the conversion of the resultant PS_3/2 groups into species such as P₄S₃ characterized by P-P bonding. The presence of metal-metal bonds involving germanium, found in samples with the lowest sulfur content, was found to be the most important structural feature in determining the optical response.     

Molecular structures of M(tfac)3 (M=Al, Co) and Cu(H2O)(fod)2: Examples of unusual supramolecular architecture

The molecular structures of Al(tfac)₃ (1), Co(tfac)₃ (2) (H-tfac = 1,1,1-trifluoroacetylacetone) and Cu(H₂O)(fod)₂ (3) (H-fod = 1,1,1,2,2,3,3-hepta-fluoro-7,7-dimethyloctane-4,6-dione) have been determined. The metal coordination spheres in compounds 1 and 2 are essentially the same as the respective M(acac)₃ derivatives. Despite the isomorphous nature of the structures of compounds 1 and 2, the identity of the nearest intermolecular van der Waals contacts are altered by minor changes in the metal coordination sphere. The geometry about copper in compound 3 is close to that of an ideal square bipyramid with the -diketonate ligands occupying the basal plane. The water ligand in each molecule of compound 3 is hydrogen bonded to an oxygen of a -diketonate ligand on an adjacent molecule resulting in the formation of dimers, which form rods along the y-axis due to weak C–F···Cu interactions. Crystal data: (1) orthorhombic, Pca2₁, a = 14.949(3), b = 19.806(4), c = 13.624(3) Å, V = 4033(1) ų, and Z = 8, and (2) orthorhombic, Pca2₁, a = 14.930(3), b = 19.620(4), c = 13.540(3) Å, V = 3966(1) ų, and Z = 8,; (3) monoclinic, P2₁/c, a = 12.447(3), b = 10.486(2) c = 21.980(4) Å, = 102.65(3)°, V = 2799(1) ų, and Z = 4.     

Synthesis of 2,6-disubstituted piperazines by a diastereoselective palladium-catalyzed hydroamination reaction

A highly diastereoselective intramolecular hydroamination is the key step in a modular synthesis of 2,6-disubstituted piperazines. The requisite hydroamination substrates were prepared in excellent yields by nucleophilic displacement of cyclic sulfamidates derived from amino acids. A variety of alkyl and aryl substituents at the 2-position were tolerated. The stereochemistry of the piperazines was determined to be trans by X-ray crystallography, which also showed the preferred conformation of the 2,6-disubstituted piperazine to be a twist-boat due to A(1,3) strain.     

Sintokamides A to E, chlorinated peptides from the sponge Dysidea sp. that inhibit transactivation of the N-terminus of the androgen receptor in prostate cancer cells

The new chlorinated peptides sintokamides A to E (1-5) have been isolated from specimens of the marine sponge Dysidea sp. collected in Indonesia. Their structures were elucidated by a combination of spectroscopic and single-crystal X-ray diffraction analyses. Sintokamide A (1) is an inhibitor of N-terminus transactivation of the androgen receptor in prostate cancer cells.     

Platinum-catalyzed enantioselective tandem alkylation/arylation of primary phosphines. Asymmetric synthesis of P-stereogenic 1-phosphaacenaphthenes

Enantioselective tandem alkylation/arylation of primary phosphines with 1-bromo-8-chloromethylnaphthalene catalyzed by Pt(DuPhos) complexes gave P-stereogenic 1-phosphaacenaphthenes (AcePhos) in up to 74% ee. Diastereoselective formation of four P-C bonds in one pot with bis(primary) phosphines gave C2-symmetric diphosphines, including the o-phenylene derivative DuAcePhos, for which the rac isomer was formed with high enantioselectivity. These reactions, which appear to proceed via an unusual metal-mediated nucleophilic aromatic substitution pathway, yield a new class of heterocycles with potential applications in asymmetric catalysis.     

Discovery of novel cathepsin S inhibitors by pharmacophore-based virtual high-throughput screening

The cysteine protease cathepsin S (CatS) is involved in the pathogenesis of autoimmune disorders, atherosclerosis, and obesity. Therefore, it represents a promising pharmacological target for drug development. We generated ligand-based and structure-based pharmacophore models for noncovalent and covalent CatS inhibitors to perform virtual high-throughput screening of chemical databases in order to discover novel scaffolds for CatS inhibitors. An in vitro evaluation of the resulting 15 structures revealed seven CatS inhibitors with kinetic constants in the low micromolar range. These compounds can be subjected to further chemical modifications to obtain drugs for the treatment of autoimmune disorders and atherosclerosis.     

Inheritance of hyper-duality in imprimitive Bose–Mesner algebras

We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose–Mesner algebras associated with a hyper-dual pair of imprimitive Bose–Mesner algebras. Let and denote Bose–Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to . Also suppose that is imprimitive with respect to a subset of Hadamard idempotents, so is dual imprimitive with respect to the subset of primitive idempotents, where is the formal duality associated with ψ. Let denote the block Bose–Mesner algebra of on the block containing p, and let denote the quotient Bose–Mesner algebra of with respect to . Then there is a hyper-duality from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to .     

Discrete behavior of Seshadri constants on surfaces

Working over , we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constants for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this result is of practical value, which we demonstrate by giving significantly improved explicit lower bounds for Seshadri constants on and new results about ample divisors on blow ups of at general points.