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.     

Design,synthesis, and evaluation of 6-carboxyalkyl and 6-phosphonoxyalkyl derivatives of 7-oxo-8-ribitylaminolumazines as inhibitors of riboflavin synthase and lumazine synthase

A series of 6-carboxyalkyl and 6-phosphonoxyalkyl derivatives of 7-oxo-8-D-ribityllumazine were synthesized as inhibitors of both Escherichia coli riboflavin synthase and Bacillus subtilis lumazine synthase. The compounds were designed to bind to both the ribitylpurine binding site and the phosphate binding site of lumazine synthase. In the carboxyalkyl series, maximum activity against both enzymes was observed with the 3'-carboxypropyl compound 22. Lengthening or shortening the chain linking the carboxyl group to the lumazine by one carbon resulted in decreased activity. In the phosphonoxyalkyl series, the 3'-phosphonoxypropyl compound 33 was more potent than the 4'-phosphonoxybutyl derivative 39 against lumazine synthase, but it was less potent against riboflavin synthase. Molecular modeling suggested that the terminal carboxyl group of 6-(3'-carboxypropyl)-7-oxo-8-D-ribityllumazine (22) may bind to the side chains of Arg127 and Lys135 of the enzyme. A hypothetical molecular model was also constructed for the binding of 6-(2'-carboxyethyl)-7-oxolumazine (15) in the active site of E. coli riboflavin synthase, which demonstrated that the active site could readily accommodate two molecules of the inhibitor.     

Thermal density fluctuations in amorphous polymers as revealed by small angle X-ray diffraction

Summary Thermal density fluctuations in amorphous polymethyl methacrylate, polycarbonate and polyethylene terephthalate were studied by means of small angle X-ray scattering. The measurements were performed in a temperature range from 20 °C up to about 50 °C above the glass transition temperatureT _g of the individual sample.AboveT _g the experimental values of the fluctuations are proportional to the isothermal compressibility and the temperature of the sample as predicted by the fluctuation theory for a system in thermodynamic equilibrium.At temperatures belowT _g this relation is no longer correct. The experimentally determined fluctuations are now proportional to the compressibility of the sample in the equilibrium state atT _g and to the actual temperature of the glassy sample. By considering a statistical ensemble with exchange of energy, particles and order between the systems of the ensemble an equation can be derived for the fluctuations of the number of particles per given volume which predicts the observed behaviour. The order parameter takes into account the fact that the properties of the glassy state depend on the way by which the state was produced.

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Zusammenfassung Es wurden thermische Dichtefluktuationen in amorphem Polymethylmethacrylat, Polycarbonat und Polyäthylenterephthalat mit Hilfe der Röntgenkleinwinkelstreuung untersucht. Die Messungen erstrecken sich über einen Temperaturbereich von 20 °C bis zu etwa 50 °C oberhalb der GlastemperaturT _g der jeweiligen Probe.In Übereinstimmung mit den Ergebnissen der Fluktuationstheorie sind die Fluktuationen oberhalb vonT _g proportional zur isothermen Kompressibilität und zur Temperatur der Probe.Unterhalb der Glastemperatur ist diese Beziehung nicht mehr erfüllt. Hier ist die Fluktuation proportional zur Kompressibilität der Probe im Gleichgewichtszustand beiT _g und zur Temperatur. Auf der Basis einer statistischen Gesamtheit mit der Austauschmöglichkeit von Energie, Partikel und Ordnung zwischen den Systemen der Gesamtheit kann eine Beziehung für die Dichtefluktuation abgeleitet werden, die mit den experimentellen Ergebnissen übereinstimmt. Der zusätzliche Ordnungsparameter berücksichtigt die Tatsache, daß die Eigenschaften im Glaszustand wegabhängig sind.

     

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.     

Analytical applications of infra-red spectroscopy : The behaviour of the oxinates of niobium,tantalum and associated elements

The behaviour of the oxinates of niobium, tantalum and associated metals in the infra-red region was studied and a method developed for the determination of niobium and tantalum. Vanadium caused no interference, but other heavy metals, such as molybdenum, manganese and cobalt, which interfered were removed by preliminary treatment when the method was applied to the determination of niobium and tantalum in steels.     

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.     

Bestimmung von Nichtmetallen in Metallen

Zusammenfassung Unter der Beschränkung auf die Nichtmetallspuren in den technisch wichtigen Metallen und Legierungen wird ein allgemeiner Überblick gegeben.Aus der Vielzahl der Analysenmethoden werden die in breiterem Umfang anwendbare Verbrennung für die Bestimmung von Kohlenstoff und Schwefel und die Heißextraktion für die Bestimmung von Wasserstoff, Sauerstoff und Stickstoff näher behandelt.Für die Bestimmung von Wasserstoff und Sauerstoff in Aluminium und Kupfer und für den Kohlenstoffgehalt der Nichteisenmetalle wird der heutige Wissensstand gegeben.Über Untersuchungen zur Probenhomogenität von AKP für Sauerstoff in Metallen und über eine Methode zur Herstellung von entsprechenden Primärstandards wird berichtet.Betrachtungen zur Frage von Aufwand und Nutzen analytischer Arbeit werden angestellt.

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Summary A general summary is given with a limitation to non-metal traces in metals and alloys of technical interest.From the great number of analytical methods those applicable on a larger scale, i.e. combustion-determination of carbon and sulphur — and vacuum fusion—determination of hydrogen, oxygen and nitrogen — are dealt with more closely.For the determination of hydrogen and oxygen in aluminium and copper as well as for the carbon contents of non-ferrous metals the present state of knowledge is described.Furthermore, investigations concerning homogenity of AKP-standard-samples for analysing the oxygen-contents in metals and a method for setting up respective primary standards are described.Finally, the problematic relation between expenses and usefulness of analytical work is discussed.

     

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.     

1-Alkoxy-2-azaallenyl-komplexe von chrom und wolfram

The successive reaction of (CO)₆M with Na[NCR₂¹] and [Et₃O]BF₄ yields (CO)₅M[C(NCR₂¹)OEt] (II: M = Cr; III: M = W; CR₂¹ = C(C₆H₄Br-p)₂ (a), CPh₂ (b), C(C₆H₄OMe-p)₂ (c), C(C₆H₄)₂O (d), CBu₂^tt (e)). Hexacarbonyltungsten, (CO)₆W, reacts with Na[NCPh₂] and MeOSO₂F to give (CO)₅W[C(NCPh₂) OMe] (IV). X-Ray analysis of IIe shows that: (1) the CNC fragment is almost linear (171.7°); (2) the two NC bond lengths are equal within experimental error; and (3) the O,C,Cr,N plane is perpendicular to the C(Me₃),C,N,C(Me₃) plane (90.0°). Therefore compounds II–IV are best described as 1-alkoxy-2-azaallenyl complexes.