Frame Wavelets with Compact Supports for L^2（R^n）

The construction of frame wavelets with compact supports is a meaningful problem in wavelet analysis. In particular, it is a hard work to construct the frame wavelets with explicit analytic forms. For a given n × n real expansive matrix A, the frame-sets with respect to A are a family of sets in R^n. Based on the frame-sets, a class of high-dimensional frame wavelets with analytic forms are constructed, which can be non-bandlimited, or even compactly supported. As an application, the construction is illustrated by several examples, in which some new frame wavelets with compact supports are constructed. Moreover, since the main result of this paper is about general dilation matrices, in the examples we present a family of frame wavelets associated with some non-integer dilation matrices that is meaningful in computational geometry.     

Endohedral nickel and palladium atoms in metal clusters: analogy to endohedral noble gas atoms in fullerenes in polyhedra with five-fold symmetry

Nickel and palladium atoms with their closed-shell d(10) electronic configurations are encapsulated in the icosahedral clusters [Ni@Ni(10)E(2)(CO)(18)](4-)(E = Sb, Bi, Sb[rightward arrow]Ni(CO)(3), CH(3)Sn and n-C(4)H(9)Sn) and the geometrically related pentagonal antiprismatic cluster Pd@Bi(10)(4+) found in Bi(14)PdBr(16). Such endohedral d(10) atoms in pentagonal antiprismatic clusters are donors of zero skeletal electrons and interact only weakly with the atoms in the surrounding polyhedron so that they may be regarded as analogous to endohedral noble gases in fullerenes such as He@C(60). On the other hand, endohedral nickel and palladium atoms in 10- and 11-vertex flattened deltahedral bare metal clusters of group 13 metals without five-fold symmetry, such as Ni@E(10)(10-) found in Na(10)NiE(10)(E = Ga, In) and Pd@Tl(11)(7-) found in A(8)Tl(11)Pd (A = Cs, Rb, K), interact significantly with the cluster atoms, particularly those at the flattened vertices of the deltahedron. The role of endohedral d(10) atoms Ni and Pd in polyhedra with five-fold symmetry as "pseudo-noble-gases" can be related to their positions at the "composite divide" of the "Metallurgists' Periodic Table" proposed by H. E. N. Stone on the basis of alloy systematics as well as the equivalence of the five d orbitals in polyhedra with five-fold symmetry.     

The oxidation of water by cerium(IV) catalysed by nanoparticulate RuO2 on mesoporous silica

Mesoporous silicates are prepared by templating on the hexagonal (H1) mesophase of surfactant bipyridine complexes of ruthenium(II) using a true liquid-crystal templating approach. On calcination, the surfactant template is removed except for the central metal ion that is oxidised, forming nanoparticles of RuO2 that deposit within the pores. RuO2 is a known oxidation catalyst and, despite its anhydrous nature in these silicates, is found to be very active in catalyzing the oxidation of water by acidic CeIV.     

Chemical bonding topology of ternary transition metal-centered bismuth cluster halides: from molecules to metals

The bismuth polyhedra in ternary transition metal-centered bismuth cluster halides may form discrete molecules or ions, infinite chains, and/or infinite layers. The chemical bonding in many of these diverse structures is related to that in deltahedral boranes exhibiting three-dimensional aromaticity by replacing the multicenter core bond in the boranes with two-center two-electron (2c-2e) bonds from the central transition metal to the nearest neighbor bismuth vertices. Examples of discrete molecules or ions include octahedral MBi(6)(micro-X)(12)(z)()(-) (X = Br, I; M = Rh, Ir, z = 3; M = Ru, z = 4) with exclusively 2c-2e bonds and pentagonal bipyramidal RhBi(7)Br(8) with a 5c-4e bond in the equatorial pentagonal plane indicative of M?bius aromaticity. The compound Ru(3)Bi(24)Br(20) contains a more complicated discrete bismuth cluster ion Ru(2)Bi(17)(micro-Br)(4)(5+), which can be dissected into a RuBi(5) closo octahedron and a RuBi(8) nido capped square antiprism bridged by a Ru(2)Bi(4)(micro-Br)(4) structural unit. In RuBi(4)X(2) (X = Br, I), the same Ru(2)Bi(4)(micro-Br)(4) structural unit bridges Bi(4) squares similar to those found in the known Zintl ion Bi(4)(2)(-) to give infinite chains of Ru(2)Bi(4) octahedra. The electron counts of the RuBi(5), RuBi(8), and Ru(2)Bi(4) polyhedra in these structures follow the Wade-Mingos rules. A different infinite chain structure is constructed from fused RhBi(7/2)Bi bicapped trigonal prisms in Rh(2)Bi(9)Br(3). This Rh(2)Bi(9)Br(3) structure can alternatively be derived from alternating Rh(2/2)Bi(4) octahedra and Rh(2/)(2)Bi(5) pentagonal bipyramids with electron counts obeying the Wade-Mingos rules. Related chemical bonding principles appear to apply to more complicated layer structures such as Pt(3)Bi(13)I(7) containing Kagomé nets of PtBi(8/2) cubes and Ni(4)Bi(12)X(3) containing linked chains of NiBi(6/3)Bi capped trigonal prisms.     

Argentomctric coulomelric titration of thioacetamide

The coulometric determination of thioacetamide (TAA) with electrogenerated silver is described. The titration is done in a solution 0.1M in both ammonia and sodium hydroxide, and the end-point is detected potentiometncally with a silver-silver sulphide electrode. On repeat analyses of approx. 2-mg samples of TAA an average error of -04% (relative standard deviation 0.25%) was obtained. Important steps in the procedure include cleaning the silver generating electrode in nitric acid before each titration, purging well with nitrogen to remove oxygen, and not using too large a sample.     

Stereoselective reductase-catalysed deoxygenation of sulfoxides in aerobic and anaerobic bacteria

Direct and indirect evidence, of unexpected stereoselective reductase-catalysed deoxygenations of sulfoxides, was found. The deoxygenations proceeded simultaneously, with the expected dioxygenase-catalysed asymmetric sulfoxidation of sulfides, during some biotransformations with the aerobic bacterium Pseudomonas putida UV4. Stereoselective reductase-catalysed asymmetric deoxygenation of racemic alkylaryl, dialkyl and phenolic sulfoxides was observed, without evidence of the reverse sulfoxidation reaction, using anaerobic bacterial strains. A purified dimethyl sulfoxide reductase, obtained from the intact cells of the anaerobic bacterium Citrobacter braakii DMSO 11, yielded, from the corresponding racemates, enantiopure alkylaryl sulfoxide and thiosulfinate samples.