Produktintegration mit nicht-äquidistanten Stützstellen

Summary For the numerical evaluation of , 0<<1 andx smooth, product integration rules are applied. It is known that high-order rules, e.g. Gauss-Legendre quadrature, become normal-order rules in this case. In this paper it is shown that the high order is preserved by a nonequidistant spacing. Furthermore, the leading error terms of this product integration method and numerical examples are given.

     

On the singular graph and the Weyr characteristic of anM-matrix

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA _ii irreducible. LetS be the set of indices such that the diagonal blockA is singular. We define the singular graph ofA to be the setS with partial order defined by > if there exists a chain of non-zero blocksA _i, A_ij, , A_l.Let ₁ be the set of maximal elements ofS, and define thep-th level _p ,p = 2, 3, , inductively as the set of maximal elements ofS \( _{1 p-1}). Denote by _p the number of elements in _p . The Weyr characteristic (associated with 0) ofA is defined to be (A) = ( ₁, ₂,, _h ), where ₁ + + _p = dim KerA ^p ,p = 1, 2, , and _h > 0, _h+1 = 0.Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show that(A) = ( ₁, , _h) if and only if certain submatricesD _p,p+1 ,p = 1, , h – 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for –A. We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS have(A) = ( ₁, , _h). This condition is satisfied ifS is a rooted forest.The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.     

Efficient nickel-mediated intramolecular amination of aryl chlorides

The use of an in situ generated Ni(0) catalyst associated with 2,2'-bipyridine or N,N'-bis(2,6-diisopropylphenyl)dihydroimidazol-2-ylidene (SIPr) as a ligand and NaO-t-Bu as the base for the intramolecular coupling of aryl chlorides with amines is described. The procedure has been applied to the formation of five-, six-, and seven-membered rings. [reaction: see text]     

Platinum complexes of oxopurines: cis‐bis(theophyllinato‐N7)bis(triphenylphosphine)platinum(II) and cis‐chloro(theobrominato‐N1)bis(triphenylphosphine)platinum(II) ethanol hemisolvate

The syntheses and structures of two mixed‐ligand complexes of platinum(II) with deprotonated oxopurine bases and tri­phenyl­phosphine are reported, namely the theophyllinate complex cis‐bis(1,2,3,6‐tetra­hydro‐1,3‐di­methyl­purine‐2,6‐dionato‐κN⁷)­bis(tri­phenyl­phosphine‐κP)­platinum(II), [Pt(C₇H₇N₄O₂)₂(C₁₈H₁₅P)₂], (I), and the theobrominate complex cis‐chloro(1,2,3,6‐tetrahydro‐3,7‐dimethylpurine‐2,6‐dionato‐κN¹)­bis(tri­phenyl­phosphine‐κP)­platinum(II) ethanol hemisolvate, [PtCl(C₇H₇N₄O₂)(C₁₈H₁₅P)₂]·0.5C₂H₅OH, (II). In (I), the coordination geometry of Pt is square planar, formed by the two coordinating N atoms of the theophyl­linate anions in a cis arrangement and two P atoms from the tri­phenyl­phosphine groups. In (II), there are two crystallographically independent mol­ecules. They both exhibit a square‐planar coordination geometry around Pt involving one Cl atom, the coordinating N atom of the theobrominate anion and two P atoms from the tri­phenyl­phosphine groups. The two tri­phenyl­phosphine groups are arranged in a cis configuration in both structures. The heterocyclic rings are rotated with respect to the coordination plane of the metal by 82.99 (8) and 88.09 (8)° in complex (I), and by 85.91 (16) and 88.14 (18)° in complex (II). Both structures are stabilized by intramolecular stacking interactions involving the purine rings and the phenyl rings of adjacent tri­phenyl­phosphine moieties.     

Struktur-Reaktivitäts-Beziehungen bei koordinativ-ungesättigten Chelatkomplexen. VI. Synthese,Adduktbildung, Redoxpotentiale und photochrome lodderivate von Eisen(II)-Komplexen Schiffscher Basen mit elektronenziehenden Substituenten

Structure Reactivity Correlations in Coordinatively Unsaturated Chelate Complexes. VI. Synthesis, Adduct Formation, Redox Potentials, and Photochromic Iodine Derivatives of Iron(II) Complexes with Schiff Base Ligands Possessing Electron-Withdrawing Substituents Iron(II) complexes of the type 1 have been prepared by different synthetic methods. In contrast to similar chelates of the “saloph” and “salen” types, the high-spin complexes 1 form stable high-spin diadducts ( 1a – 1d ) and an unusual triadduct ( 1e ) with pyridine. The oxidation potentials of the Fe^II/Fe^III couple as measured by cyclic voltammetry are dependent on the solvent as well as the equatorial ligand substituents. The potentials are more positive in pyridine than in DMF, indicating a stabilization of Fe^II by pyridine. The redox potentials are discussed with respect to those of other metals in the same ligand environment. The complexes form iodine derivatives which show photochromic behaviour in THF solution. The rate of the reaction with dioxygen in the solid state as well as in pyridine solution decreases in the order 1f > 1a ≈ 1b > 1c ≈ 1d > 1e and correlates with the increasing oxidation potentials.     

The infrared spectrum of Au-.CO2

The Au-.CO2 ion-molecule complex has been studied by gas phase infrared photodissociation spectroscopy. Several sharp transitions can be identified as combination bands involving the asymmetric stretch vibrational mode of the CO2 ligand. Their frequencies are redshifted by several hundred cm(-1) from the frequencies of free CO2. We discuss our findings in the framework of ab initio and density-functional theory calculations, using anharmonic corrections to predict vibrational transition energies. The infrared spectrum is consistent with the formation of an aurylcarboxylate anion with a strongly bent CO2 subunit.     

Neutron Diffraction Analysis of H3Co2[C5H2(t-Bu)3]2, a Molecule with a Triply Hydrogen-Bridged Metal–Metal Bond: Some Comments on Structural Patterns in M(μ-H)nM Systems (n=1, 2, 3, 4)

The structure of H₃Co₂[C₅H₂(t-Bu)₃]₂ has been analyzed by low-temperature single-crystal neutron diffraction techniques, and shown to consist of two CoCp moieties with three hydride ligands bridging the central Co–Co bond. Despite a fairly extensive twinning problem, the structure could be solved and successfully refined to a final R factor of 9.2% for 2024 reflections. Average molecular parameters in the H₃Co₂ core of the molecule are as follows: Co–Co=2.275(21) Å, Co–H=1.637(16) Å, HH=2.050(20) Å, Co–H–Co=88.0(9)°, H–Co–H=77.0(7)°. Also included in this paper is a discussion on the molecular dimensions of symmetric hydride-bridged dinuclear systems (M(-H)_nM, n=1, 2, 3, 4) that have been studied to date by neutron diffraction.     

1H magic angle rotation NMR in polymer studies

The principles of the method of NMR line narrowing by measurement with spinning of the sample about the magic axis (MAR-NMR) are introduced, with particular emphasis on the effects of internal motion upon the possibilities and limitations of the method. The applications of the method in ¹H-NMR studies of polymer structure and dynamics are then reviewed. Due to both theoretical and experimental limitations, narrowing of dipolar broadened NMR lines by MAR can be observed in ¹H NMR spectra only in those cases where internal motion is anisotropic, or in heterogeneous systems where line width is limited by differences of magnetic susceptibility. In polymers, both solid and liquid, the method makes possible differentiation between isotropic and anisotropic internal motion. In systems with anisotropic internal motion, MAR-NMR makes possible a characterization of motional codes which normally are obscured by residual dipolar interactions, as well as of geometrical restrictions upon these motions.     

Sequence-specific binding of DNA to liposomes containing di-alkyl peptide nucleic acid (PNA) amphiphiles

We present a method to covalently attach peptide nucleic acid (PNA) to liposomes by conjugation of PNA peptide to charged amino acids and synthetic di-alkyl lipids ("PNA amphiphile," PNAA) followed by co-extrusion with disteroylphosphatidylcholine (DSPC) and cholesterol. Attachment of four Glu residues and two ethylene oxide spacers to the PNAA was required to confer proper hydration for extrusion and presentation for DNA hybridization. The extent of DNA oligomer binding to 10-mer PNAA liposomes was assessed using capillary zone electrophoresis. Nearly all PNAs on the liposome surface are complexed with a stoichiometric amount of complementary DNA 10-mers after 3-h incubation in pH 8.0 Tris buffer. No binding to PNAA liposomes was observed using DNA 10-mers with a single mismatch. Longer DNA showed a greatly attenuated binding efficiency, likely because of electrostatic repulsion between the PNAA liposome double layer and the DNA backbone. Langmuir isotherms of PNAA:DSPC:chol monolayers indicate miscibility of these components at the compositions used for liposome preparation. PNAA liposomes preserve the high sequence-selectivity of PNAs and emerge as a useful sequence tag for highly sensitive bioanalytical devices.     

The solubility of an hydroxyaluminosilicate

Hydroxyaluminosilicates (HAS) are critical secondary mineral phases in the biogeochemical cycle of aluminium. They are formed from the reaction of silicic acid (Si(OH)₄) with an aluminium hydroxide template and act as a geochemical control of the biological availability of Al. There are two main forms of HAS which we have called HAS_A and HAS_B and which of these will predominate will depend upon the Si(OH)₄ to Al ratio in any one environment. In all but the most heavily weathered environments or those undergoing a progressive acidification Si(OH)₄ will be present in significant excess to Al and HAS_B will be the dominant secondary mineral phase. We have tried to determine the solubility of HAS_B(s) so that its contribution to Al solubility control might be compared with other secondary minerals such as Al(OH)_3(gibbsite). In preliminary experiments, the dissolution of HAS_B(s) was found to be non-congruent with almost no Al being released during 18 months ageing. We then demonstrated that HAS_B(s) was significantly less soluble than Al(OH)_3(s) prepared under identical experimental conditions. We have used this information to describe a solubility expression for HAS_B(s) at a predefined quasi-equibrium and to calculate a solubility constant.

K^*Al₂Si₂O₅(OH)₄=[Al₂O⁴⁺][SiO₂]²[OH^-]⁴