Some contributions to the theory of cyclic quartic extensions of the rationals

K is a cyclic quartic extension of Q iff $\text{K = Q((rd + p d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>)</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where d > 1, p and r are rational integers, d squarefree, for which p² + q² = r²d for some integer q. Following a paper of A. A. Albert we show that the absolute discriminant, $\text{d(}\text{K}\text{Q}\text{)}$, of the general cyclic quartic extension is given by $\text{d(}\text{K}\text{Q}\text{) = (W}{}^2\text{d}{}^2\text{)}$ for an explicitly computable rational integer W. We next find that the relative discriminant, $\text{d(}\text{K}\text{F}\text{)}$, is given by $\text{d(}\text{K}\text{F}\text{) = (W d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where $\text{F = Q (d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is K′s uniquely determined quadratic subfield. We use this last result in conjunction with Corollary 3, page 359, of Narkiewicz's “Elementary and Analytic Theory of Algebraic Numbers” (PWN-Polish Scientific Publishers, 1974) to establish the following Theorem 1: If the (wide) class number of$\text{F = Q(d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$is odd then every cyclic quartic extensionKofQcontainingFhas a relative integral basis overF. We give a second, more organic, proof of Theorem 1 which also allows us to prove the following converse result, namely Theorem 2: Suppose the quadratic fieldFis contained in some cyclic quartic extension ofQand suppose thatFhas even (wide) class number. There then is a cyclic quartic extensionKofQcontainingFsuch thatKhas no relative integral basis overF.     

Controlled assembly of carbon nanotubes by designed amphiphilic Peptide helices

Carbon nanotubes have properties potentially useful in diverse electrical and mechanical nanoscale devices and for making strong, light materials. However, carbon nanotubes are difficult to solubilize and organize into architectures necessary for many applications. In the present paper, we describe an amphiphilic alpha-helical peptide specifically designed not only to coat and solubilize carbon nanotubes, but also to control the assembly of the peptide-coated nanotubes into macromolecular structures through peptide-peptide interactions between adjacent peptide-wrapped nanotubes. The data presented herein show that the peptide folds into an amphiphilic alpha-helix in the presence of carbon nanotubes and disperses them in aqueous solution by noncovalent interactions with the nanotube surface. Electron microscopy and polarized Raman studies reveal that the peptide-coated nanotubes assemble into fibers with the nanotubes aligned along the fiber axis. Most importantly, the size and morphology of the fibers can be controlled by manipulating solution conditions that affect peptide-peptide interactions.     

Syntheses and oxygenation of iron(II) “strapped” porphyrin complexes

A general route to ‘strapped’ porphyrins which are bridged between diagonally opposite meso-positions is described. Studies on the oxygenation and carbonylation of their iron(II) complexes have shown that they are inferior to ‘capped’ porphyrins as models for the oxygen-carrying haemoproteins. The results reported herein suggest that strapping structures which do not enforce well-defined and rigid cavities for oxygen binding afford little protection against irreversible oxidation of (dioxygen) (porphyrin) iron(II) complexes.     

Pressure dependence of viscosity

We reanalyze the pressure dependence of viscosity of liquids of constant composition under isothermal conditions. Based exclusively on very general considerations concerning the relationship between viscosity and "free volume," we show that, at moderate values of pressure, viscosity increases, as a rule, with increasing pressure, provided the liquid is in stable or metastable (undercooled) equilibrium states. However, even if the behavior of the viscosity is governed by free volume effects, deviations from a positive pressure dependence are possible, when the liquid's thermal expansion coefficient is negative. We derive an equation that allows one to quantitatively determine the pressure dependence of viscosity, which requires, in the simplest case, only the knowledge of the temperature dependence of viscosity at constant pressure, the thermal expansion coefficient, and the isothermal compressibility of the liquid. As an example, the negative pressure dependence of water in the range of temperatures 0-4 degrees C and of several silicate liquids, such as albite, jadeite, dacite, basalts, etc., could be explained in such a way. Other glass-forming liquids initially (for moderate pressures) show a positive pressure dependence of viscosity that changes to a negative one when subjected to high (approximately GPa) isostatic pressure. A detailed analysis of water and already mentioned silicate melts at GPa pressures shows that, in addition to free volume effects, other pressure induced structural transformations may have to be accounted for in a variety of cases. By this reason, the theoretical analysis is extended (i) in order to describe the pressure dependence of viscosity for systems that are in frozen-in thermodynamic nonequilibrium states (glasses, i.e., undercooled liquids below the glass transition temperature Tg) and (ii) to systems which undergo, in addition to variations of the free volume, pressure induced changes of other structural parameters. In such cases a decrease of viscosity with increasing pressure may occur, in principle, even if the thermal expansion coefficient is positive. In this way, the present analysis grants a general tool to estimate the pressure dependence of viscosity and supposedly settles the controversy in the current literature.     

Valence bond treatment of systems involving orbital degeneracy

Ground and excited electronic states of the allyl cation and anion, the butadiene radical cation and anion, and of the pentadienyl cation and anion have been calculated by a semiempirical valence bond method. The results of such calculations are in essential agreement with the known properties of such systems. A comparison of the results of the valence bond treatment with those expected on the basis of the qualitative resonance theory yields serious discrepancies which show up the limitations inherent in the latter formalism.

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Zusammenfassung Der Grundzustand und die elektronisch angeregten Zustände der folgenden Systeme wurden mittels einer semiempirischen Valence-Bond-Methode berechnet: Allyl Kation und Anion, Butadien Radikal Kation und Anion, Pentadienyl Kation und Anion. Die Ergebnisse stehen in vernünftiger Übereinstimmung mit den bekannten experimentellen Daten. Ein Vergleich der Resultate, die mittels der Valence-Bond-Methode erhalten werden, mit jenen, die man aufgrund der qualitativen Resonanz-Theorie erwarten würde, ergibt wesentliche Unterschiede, die auf die Grenzen hinweisen, die dem letztgenannten Formalismus eigen sind.

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Résumé L'état fondamental et les états électroniques excités du cation et de l'anion allyl, du cation radical et de l'anion radical du butadiène et du cation et de l'anion pentadiényl ont été calculés par la méthode semiempirique valence bond. Les résultats de ces calculs correspondent raisonnablement aux données expérimentales connues pour ces systèmes. Une comparaison des résultats théoriques avec les prédictions de la théorie de résonance donne des différences qui démontrent les limites de ce formalisme qualitatif.

     

Diallylphosphines as New Bidentate Ligands in Iridium Complexes

Abstract

Addition of four equivalents of tButyldiallylphosphine to a solution of one equivalent of [(COE)₂Cl]₂ in CHCl₃ at low temperature produced two isomers of the metallated complex 2, formed by C-H activation. 2 evolve at 40°C to 3, by a hydride transfer from iridium to the cyclooctene (COE) ligand.     

A generalized nonautonomous SIRVS model

A nonautonomous SIRVS model with time‐dependent parameters is considered. The global dynamics is investigated, and conditions for the permanence, extinction, and disease‐free equilibrium are studied. Substitute quantities that replace the basic reproduction number are presented. Numerical simulations illustrate the dynamic behavior of the proposed model. Copyright © 2012 John Wiley & Sons, Ltd.