Borel fibrations and G-spaces

Considering discrete groups G only, we present an elementary proof of the familiar equivalence of the category of G-spaces (with maps equivariant up to homotopy) and the category of Borel fibrations over BG.     

Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation

A rigorous analysis is given of the dynamics of the renormalization map associated to a discrete Schrödinger operatorH onl ²(), defined byH(n)=(n+1)+(n–1)+Vf(n)(n), whereV is a real parameter,f is a certain discontinuous period-1 function, and is the golden mean. The renormalization map forH is a diffeomorphism,T, of ³, preserving a cubic surfaceS _V . ForV8 we prove that the non-wandering set of the restriction ofT toS _v is a hyperbolic set, on whichT is conjugate to a subshift on six symbols. It follows from results in dynamical systems theory that the optimally approximating periodic operators toH have spectra which obey a global scaling law. We also define a set which we call the pseudospectrum of the operatorH. We prove it to be a Cantor set of measure zero, and obtain bounds on its Hausdorff dimension. It is an open question whether the pseudospectrum coincides with the spectrum ofH.     

Donor—acceptor interactions of substituted benzenes with molecular chlorine and carbon disulfide

CNDO molecular orbital calculations have been performed to analyze donor—acceptor interactions between molecular chlorine and benzene, toluene, mesitylene and hexamethylbenzene and the, as yet, unreported chlorine—hexafluorobenzene and carbon disulfide—benzene pairs. The stabilization energy and the dipole moment and its derivative (?p/?R_CICI) calculated for the benzene—chlorine complex are in good agreement with the estimated experimental values. The trends in the experimental stabilization energies and the Cl-Cl vibrational frequencies with increasing methyl substitution appear to be well reproduced by the calculations. The charge transferred from the benzene donor is polarized toward the outer chlorine atom or sulfur atom. For hexafluorobenzene-chlorine the direction of electronic charge polarization is reversed from that of the benzene and methylbenzene complexes. The calculated results are discussed within the framework of Muliiken's simplified resonance theory for complexes.     

Template synthesis in the nickel(II)–thiocarbohydrazide–propanone triple system

The complexing process proceeding in the Ni^II–thiocarbohydrazide (H₂N–H–NC(=S)–NH–NH₂)–propanone triple system in EtOH solution and nickel(II)hexacyanoferrate(II) gelatin-immobilized matrix has been studied. It has been found that in the first case, template synthesis leading, as a minimum, to formation of three coordination compounds of Ni^II with (N,N,S,S)-donor tetradentate ligands having NiL¹, NiL² and NiL³compositions where L¹ is 4,6,6-trimethyl-2,3,7,8-tetraazanonen-3-di(thiohydrazide)-1,9, L² is 4,6,6,12-tetrametyl-1,9-dithio-2,3,7,8,10,11-hexaazatridekadien-3,11-hydrazide-1 and L³ is 2,8,10,10,16-pentamethyl-5,13-dithio-3,4,6,7,11,12,14,15-octaazaheptadekatrien-2,7,15 is observed, whereas in the gelatin-immobilized matrix, a complexing process in the system considered does not occur.     

The direct determination of Brueckner orbitals with application to the H2 molecule

An iterative procedure is presented which allows for the direct determination of approximate Brueckner orbitals for small atomic and molecular systems. Starting from the Hartree-Fock determinant one first determines pair natural orbitals (PNOs) of independent electron pairs in the HF-field of the remaining electrons. The use of the generalized Brillouin-theorem then leads to an approximate Brueckner orbital for each electron pair. This procedure must be repeated up to self-consistency which is reached generally after 4–5 macroiterations. Applications to the ground state of H₂ show how important the use of Brueckner orbitals is to get good expectation values of one-electron operators and the correct asymptotic behaviour of the potential energy curve for large internuclear distances.

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Zusammenfassung Es wird eine Iterationsmethode beschrieben, die eine direkte Bestimmung genäherter Bruecknerorbitale für kleine atomare und molekulare Systeme gestattet. Ausgehend von der Hartree-Fock-Determinante bestimmt man zunächst Paar-NOs (PNOs) der unabhängigen Elektronenpaare im HF-Feld der übrigen Elektronen des Systems. Mit Hilfe des verallgemeinerten Brillouin-Theorems erhält man dann ein genähertes Bruecknerorbital für jedes Elektronenpaar. Dies Verfahren muß bis zur Selbstkonsistenz wiederholt werden, die man im allgemeinen nach 4–5 solcher Makroiterationen erreicht. Anwendung auf den Grundzustand des H₂-Moleküls zeigt, wie wichtig die Benutzung von Bruecknerorbitalen ist, wenn man gute Erwartungswerte von Einelektronenoperatoren und das richtige asymptotische Verhalten der Potentialkurve bei großen Abständen erhalten will.

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Résumé On propose un schéma itératif qui permet le calcul direct des orbitales de Brueckner approchées pour les atomes et les petites molécules. Partant du déterminant Hartree-Fock on calcule d'abord les PNO (pair natural orbitals) des paires indépendantes d'électrons dans le champ effectif des autres électrons dans l'approximation Hartree-Fock. Ensuite le théorème de Brillouin généralisé permet de calculer les orbitales de Brueckner approchées pour chaque paire d'electrons. On recommence le calcul des PNO, la self-consistance de cette méthode itérative s'établit en général au bout de 4 ou 5 macroitérations. Les applications à la molécule H₂ démontrent que l'utilisation d'orbitales de Brueckner s'impose si l'on s'intéresse à des bonnes valeurs moyennes d'opérateurs monoélectroniques ou au comportement asymptotique correct des courbes de potentiel aux grandes distances.

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Dedicated to the memory of Prof. K. H. Hansen.     

Indirect electrochemical methods of nitration: Novel nitroacetamidation of dienes

Solutions of nitronium tetrafluoroborate in acetonitrile prepared by anodic oxidation of nitrogen dioxide, efficiently nitrate aromatics, enolsilyethers, alkenes and conjugated dienes to afford respectively nitroaromatics, α-nitroketones, vicinal nitroamides and mixtures of nitroacetamides by 1,2- and 1,4-addition.     

The preparation of hot-pressed chalcogenide spinels. II. Stoichiometry and optical transparency in CdCr2S4 and CoCr2S4

The compounds CdCr₂S₄ and CoCr₂S₄ have been hot pressed into disks that are highly transparent in the infrared. Stoichiometry has been altered by varying the $\text{Cr}{}^{\mathrm{3+}}\text{M}{}^{\mathrm{2+}}$ ratio, where M²⁺ is Cd²⁺ or Co²⁺. The effects of nonstoichiometry on optical transmission were determined. Excess M²⁺ attenuates the transmission much more than excess Cr³⁺.     

Potassium addition to Fe/MgO, Ni/MgO and Ni/SiO2: Reducibility, dispersity and stabilization of the promoter by the support

The influence of potassium addition on the morphological properties of Fe/MgO and Ni/MgO has been investigated: KNO₃ addition to precursors results in a decrease of the reducibility and in a smaller dispersity of the metallic phase. A loss of potassium is observed at reduction temperatures in excess of 773 K. Silica-supported Ni behaves differently: silica inhibits the potassium volatilization and nickel reducibility is enhanced. This illustrates how promoter effects can be influenced by the nature of support.

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Fe/MgO Ni/MgO: KNO₃ . , 773 K. , : . .

     

Synthetic thermally reversible gel systems. V

Differential thermal analysis has been used to study the fusion of aqueous thermally reversible gels of gelatin and polyacrylylglycinamide (PAG). In the case of gelatin gels, endotherms close to the melting point are readily observed and these are sometimes preceeded by a small exothermic heat of gel reorganization. Calculations are presented to show that breaking of the gelatin gel network requires only a small fraction of the observed endothermic heat of fusion and that most of the heat is required for melting larger crystallites within gelatin aggregates and for perhaps a helix → coil transition. Failure to observe endotherms by DTA over the known temperature range of fusion of PAG gels is consistent with prior measurements and conclusions. The noncrystallinity of PAG gels and soluble aggregates together with a heat of crosslinking of only ?5 to ?10 kcal/mole of crosslinks places the heat of fusion of PAG gels outside the lower limits of DTA sensitivity.     

Water-Soluble imide-amide copolymers. II. Preparation and characterization of poly (acrylamide-co-p-maleimidobenzoic acid)

The comonomer required, p-maleimidobenzoic acid (MBA) was first prepared in good yield by refinements of published methods. p-Carboxysuccinanilic acid (CSA), and p-succinimidobenzoic acid (SBA), were also prepared to provide models useful for IR and NMR for spectroscopic assignments of the new copolymers. Polymerization of MBA with acrylamide in glacial acetic acid at 60°C gave copolymers with estimated viscosity average molecular weights of 60,000 to 90,000. Yields and viscosity average molecular weights decreased as the MBA to acrylamide monomer feed ratio was increased. The rate of incorporation of MBA into the copolymer rose from 7 to 23% when the mole ratio in the feed was raised from 5 to 20%. Decreasing the initiator concentration increased molecular weights by less than predicted and reduced the yield of copolymer for any given feed ratio of MBA to acrylamide. In all cases about 30–40% of the MBA units in the purified copolymers were hydrolyzed. A change to dimethyl sulfoxide solvent gave good, and poor yields of copolymer at 5 and 10 mol % MBA, respectively, and no copolymer at 20 mol % MBA. Viscosity average molecular weights of the copolymer products prepared in DMSO were somewhat lower than obtained for the copolymers prepared in acetic acid. Polymerization in a DMSO-water mixture gave a negligible yield of polymeric product. Instead, only hydrolysates of MBA precipitated when the coloured polymerization solutions were added to methanol.