Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold (M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of ₁ (M) on a finite dimensional vector space to a representation on aA-Hilbert moduleW of finite type whereA is a finite von Neumann algebra. If (M,W) is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, theL ₂-analytic andL ₂-Reidemeister torsions are equal.The first three authors were supported by NSF. The first two authors wish to thank the Erwin-Schrödinger-Institute, Vienna, for hospitality and support during the summer of 1993 when part of this work was done.     

Hexacoordinate Phosphorus. 7. Synthesis and Characterization of Neutral Phosphorus(V) Compounds Containing Divalent Tridentate Diphenol Imine, Azo, and Thio Ligands

The reactions of bis(trimethylsilyl)ated forms of the Schiff base ligands N-(2-hydroxyphenyl)salicylideneamine {(HO)C(6)H(4)N(CH)C(6)H(4)(OH)}, N-(4-tert-butyl-2-hydroxyphenyl)salicylideneamine {(HO)((t)Bu)C(6)H(3)N(CH)C(6)H(4)(OH)}, N-(2-hydroxy-4-nitrophenyl)salicylideneamine {(HO)(O(2)N)C(6)H(3)N(CH)C(6)H(4)(OH)}, and the structurally related ligand 2,2'-azophenol with halogeno- and (trifluoromethyl)halogenophosphoranes yield a series of neutral hexacoordinate phosphorus(V) compounds by means of trimethylsilyl halide elimination. In all of these cases the ligands chelate in a meridional conformation in which bicyclic five- and six-membered chelate rings are formed through structures containing two phenolic P-O bonds and one N-P bond. The hexacoordinate nature of these compounds is evidenced by their high-field (31)P NMR chemical shifts and their characteristic J(PF) coupling patterns and is further substantiated by the crystal structures of {O((t)Bu)C(6)H(3)N(CH)C(6)H(4)O}PCl(3) and {OC(6)H(4)N=NC(6)H(4)O}PF(3). Crystal data for {O((t)Bu)C(6)H(3)N(CH)C(6)H(4)O}PCl(3): triclinic, space group P&onemacr; (No. 2), a = 11.167(1) ?, b = 15.684(1) ?, c = 17.047(2) ?, V = 2840(1) ?(3), Z = 2. Final R and R(w) values were 0.051 and 0.079, respectively. Crystal data for {OC(6)H(4)N=NC(6)H(4)O}PF(3): monoclinic, space group P2(1)/c (No. 14), a = 6.9393(8) ?, b = 12.450(2) ?, c = 13.907(2) ?, V = 1190.7(6) ?(3), Z = 4. Final R and R(w) values were 0.045 and 0.056, respectively. The molecular structures of {O((t)Bu)C(6)H(3)N(CH)C(6)H(4)O}PCl(3) and {OC(6)H(4)N=NC(6)H(4)O}PF(3) show that in both cases the Schiff base ligand chelates occupy the meridional plane about the six-coordinate phosphorus atom. In the case of {OC(6)H(4)N=NC(6)H(4)O}PF(3) the equivalent nitrogen atoms in the chelate rings are disordered to form half-occupancy pairs. The silylated form of the related thiobis(phenol), 2,2'-thiobis(4,6-tert-butylphenol), reacted similarly with pentavalent halides to form the six-coordinate complex [{2-O-3,5-((t)Bu)(2)C(6)H(2)}(2)S]PCl(3) which was also verified by a crystal structure. Crystal data for [{2-O-3,5-((t)Bu)(2)C(6)H(2)}(2)S]PCl(3): monoclinic P2(1)/n, a = 13.989(2), b = 13.594(2), c = 16.483(2) ?, beta = 97.98(2) degrees, V = 3104(2) ?(3), Z = 4; final R and R(w)() values were 0.039 and 0.052, respectively. In contrast to the above six coordinate complexes, this compound possesses a facial structure in which two phenoxy substituents form planar chelates centered on the bridging sulfur and intersecting at the P-S axis. The P-S bond length, 2.331(1) ?, is slightly shorter than has been previously observed in the example wherein the ligand possesses two tert-butyl groups and the phosphorus carries three OCH(2)CF(3 )substituents indicating stronger interaction between P and S in the present case.     

Identification of minor C18 triene and conjugated diene isomers in hydrogenated soybean oil and margarine by GC-MI-FT-IR spectroscopy

Vegetable oils are partially hydrogenated in order to produce palatable products of suitable plasticity. The constituents of these new dietary products are complex mixtures of fatty acid isomers with different nutritional properties. A rapid method is described for separating and identifying fatty acid methyl ester (FAME) isomers of linolenic (octadecatrienoic, 18:3) acid and of conjugated octadecadienoic (18:2) acid, minor species found in hydrogenated soybean oil and margarine, by capillary gas chromatography-matrix isolation-Fourier transform-infrared (GC-MI-FT-IR) spectroscopy. FAMEs of 18:3 acid isomers in margarine, soybean oil hydrogenated in our laboratory, and isomerized linolenic acid were identified by this method, and MI-FT-IR spectra of FAME geometric isomers of octadecatrienoic and conjugated octadecadienoic acids are reported for the first time. Five major C18 triene GC peaks are found in chromatograms of isomerized methyl linolenate, representing species with tri-cis and tri-trans configurations and three species with cis-trans mixed configurations. FAME isomers with these configurations are also found for a hydrogenated soybean oil having an iodine value of 111. Methyl linolenate (tri-cis) is no longer found when soybean oil is further hydrogenated to an iodine value of 96. IR spectra characteristic of a tri-trans isomer are obtained for two test samples with iodine values of 111 and 96. Besides methyl linolenate, only isomers with a mono-trans di-cis configuration are found for the margarine analyzed. Conjugated cis-trans and trans-trans 18:2 FAME isomers are also found in all the hydrogenated soybean oil and margarine analyzed.     

Facile pyrazolylborate ligand degradation at lanthanide centers: X-ray crystal structures of pyrazolylborinate-bridged bimetallics

Adventitious hydrolysis of a number of different complexes with the molecular formula Ln(Tp(Me2))(2)X [Tp(Me2) = (HB(dmpz)(3)), where X is a basic anionic ligand] in various solvents, yielded crystals of highly insoluble dimers of the general formula [Ln(Tp(Me2))(mu-BOp(Me2))](2) (1) [Ln = La, Ce, Sm; BOp(Me2) = (HBO(dmpz)(2))(2)(-); dmpzH = 3,5-dimethylpyrazole]. The results of several single-crystal X-ray determinations are reported. One metal nitrogen distance, that lying across from the two negatively charged bridging oxygen atoms, is 0.06 A longer than the others, suggesting an unusual trans influence at a lanthanide center. The formation of 1 is proposed to involve the intermediacy of Ln(Tp(Me2))(2)OH formed by protonolysis with adventitious water.     

Organolanthanide-catalyzed intramolecular hydroamination/cyclization of amines tethered to 1,2-disubstituted alkenes

[reaction: see text] This contribution reports the organolanthanide-catalyzed intramolecular hydroamination/cyclization of amines tethered to 1,2-disubstituted alkenes to afford the corresponding mono- and disubstituted pyrrolidines and piperidines by using coordinatively unsaturated complexes of the type (eta(5)-Me(5)C(5))(2)LnCH(TMS)(2) (Ln = La, Sm), [Me(2)Si(eta(5)-Me(4)C(5))(2)]NdCH(TMS)(2), [Et(2)Si(eta(5)-Me(4)C(5))(eta(5)-C(5)H(4))]NdCH(TMS)(2), and [Me(2)Si(eta(5)-Me(4)C(5))((t)()BuN)]LnE(TMS)(2) (Ln = Sm, Y, Yb, Lu; E = N, CH) as precatalysts. [Me(2)Si(eta(5)-Me(4)C(5))((t)BuN)]LnE(TMS)(2) mediates intramolecular hydroamination/cyclization of sterically demanding amino-olefins to afford disubstituted pyrrolidines in high diastereoselectivity (trans/cis = 16/1) and in good to excellent yield.     

Gallium halide induced heterocycle expansion of dihalodiphosphadiaryldiazanes [(XPNR)2] to the corresponding triphosphatriazanes [(XPNR)3

Reactions of the cyclic diphosphadiazanes (XPNR)(2) (X = Cl, Br; R = 2,6-dimethylphenyl = Dmp, 2,6-diisopropylphenyl = Dipp) with GaX(3) followed by 4-(dimethylamino)pyridine (DMAP) give the corresponding trimers (XPNR)(3). An unusual cyclophosphazanium tetrachlorogallate salt [(DippN)(3)P(3)Cl(2)][GaCl(4)] has been isolated from the reaction of (ClPNDipp)(2) with GaCl(3) and represents an intermediate in the disproportionation process. Dissociation of the gallate ion on reaction of [(DippN)(3)P(3)Cl(2)][GaCl(4)] with DMAP releases a halide ion, which associates with the dicoordinate phosphenium center to give (ClPNDipp)(3). The observations indicate that the presence of medium-sized substituents at nitrogen (R) thermodynamically destabilize the dimer with respect to the trimer, without offering sufficient stabilization of the monomer, as observed for MesNPX (Mes* = 2,4,6-tri-tert-butylphenyl) (Mes* > Dipp > Dmp). Nevertheless, lability of the N-P bond in these derivatives of (XPNR)(2) allows for transformations between dimer and trimer that may include transient existence of the corresponding monomer. Manipulation of substituent steric strain to modify the relative stability of phosphazane oligomers provides a new methodology for diversification of phosphazane chemistry.     

Synthesis amd characterization of [Et4,N]2[Mo(CO)4(SR)2] (R = Ph,Bz): New,potentially chelating bis-thiolate species

The new complexes [Et₄N]₂ [Mo(CO)₄(SR)₂] (R = Ph, Bz) have been prepared by reaction of [Et₄N] [SR] with (norbornadiene)Mo(CO)₄ at low temperature. The IR spectra and electrochemical behavior of these two species are different, perhaps implicating different conformational isomers with respect to the thiolate ligands. These complexes may prove to be valuable reagents for the synthesis of new heterometallic compounds, by virtue of their cis-monodentate thiolate ligands.     

Syntheses, structures, and stabilities of [PPh(4)][WS(3)(SR)](R=(i)Bu,(i)Pr,(i)Bu, benzyl, allyl) and [PPh(4)][MoS(3)(S(t)Bu)

Intermediates in the condensation process of [MS(4)](2)(-) (M = Mo, W) to polythiometalates, in the presence of alkyl halides, had not been reported prior to our communication of [PPh(4)][WS(3)(SEt)] (Boorman, P. M.; Wang, M.; Parvez, M. J. Chem. Soc., Chem. Commun. 1995, 999-1000). We now report the isolation of a range of related compounds, with 1 degrees, 2 degrees, and 3 degrees alkyl thiolate ligands, including one Mo example. [PPh(4)][WS(3)(SR)] (R = (i)Bu (1), (i)Pr (2), (t)Bu (3), benzyl (5), allyl (6)) and [PPh(4)][MoS(3)(S(t)Bu)] (4) have been isolated in fair to good yields from the reaction of [PPh(4)](2)[MS(4)] with the appropriate alkyl halide in acetonitrile and subjected to analysis by X-ray crystallography. Crystal data are as follows: for 1, triclinic space group P1 (No. 2), a = 11.0377(6) A, b = 11.1307(5) A, c = 13.6286(7) A, alpha = 82.941(1) degrees, beta = 84.877(1) degrees, gamma = 60.826(1) degrees, Z = 2; for 2, monoclinic space group P2(1)/c (No. 14), a = 9.499(6) A, b = 15.913(5) A, c = 18.582(6) A, beta = 99.29(4) degrees, Z = 4; for 3, monoclinic space group P2(1)/n (No. 14), a = 10.667(2) A, b = 17.578(2) A, c = 16.117(3) A, beta = 101.67(1) degrees, Z = 4; for 4, monoclinic space group P2(1)/n (No. 14), a = 10.558(3) A, b = 17.477(3) A, c = 15.954(3) A, beta = 101.18(2) degrees, Z = 4; for 5, monoclinic space group P2(1)/n (No. 14), a = 16.2111(9) A, b = 11.0080(6) A, c = 18.1339(10) A, beta = 111.722(1) degrees, Z = 4; for 6, triclinic space group P1 (No. 2), a = 9.4716(9) A, b = 10.4336(10) A, c = 14.4186(14) A, alpha = 100.183(2) degrees, beta = 90.457(2) degrees, gamma = 91.747(2) degrees, Z = 2. Structures 3 and 4 are isomorphous, and 1 exhibits disorder about the tertiary carbon. 6 has been shown to exhibit fluxionality in solution by variable-temperature (1)H NMR studies, and an allyl migration mechanism is implicated in this process. The kinetics for the reaction of [WS(4)](2)(-) and EtBr were measured and suggest an associative nucleophilic substitution (S(N)2) mechanism. The decomposition of the [WS(3)(SEt)](-) ion is shown to be second order with respect to this ion, suggesting the formation of a transient binuclear intermediate. M-S bond cleavage is the predominant step in decomposition of 1-6 to yield alkyl sulfides, alkyl thiols, and polythiometalates such as [PPh(4)](2)[M(3)S(9)]. In contrast, reactions of [PPh(4)](2)[WO(x)()S(4)(-)(x)()] (x = 1, 2) with (t)BuBr result in the additional decomposition product of isobutene, presumably by C-S bond cleavage and beta-hydrogen transfer. Interestingly, the reaction of [PPh(4)](2)[WOS(3)] with BzCl yields 5 as the only isolable W thiolate species.     

Removal of trace quantities of cadmium from aqueous iodide solutions by ion flotation

Summary Cadmium ions react with the collector, ethylhexadecyldimethylammonium bromide (EHDABr) to form a surface active sublate which can be removed from aqueous iodide solutions by ion flotation. A typical ion flotation procedure involves passing air through a 250-ml solution containing 5 ppm Cd²⁺, 0.120 MI^– and 1×10^–3M EHDABr at a flow rate of 20 ml/min for 2 hrs. The procedure is simple and efficient. Chromium, copper and zinc ions do not interfere under the experimental conditions.

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Entfernung von Cadmiumspuren aus wärigen Jodidlösungen durch lonenflotation

Zusammenfassung Cadmiumionen reagieren mit Äthylhexadecyldimethylammoniumbromid (EHDABr) unter Bildung eines oberflächenaktiven Sublates, das durch Ionenflotation aus wäßrigen Jodidlösungen entfernt werden kann. Einen typischen Ionenflotationsvorgang erzielt man beim Durchleiten von Luft durch 250 ml einer Lösung von 5 ppm Cd²⁺ in 0,120 M Jodid und 1·10^–3M EHDABr mit einer Geschwindigkeit von 20 ml/min durch 2 Stunden. Die Arbeitsweise ist einfach und wirkungsvoll. Cr, Cu und Zn stören unter den angegebenen Arbeitsbedingungen nicht.

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Taken partly from the thesis submitted to the Graduate School for the Master of Science in Chemistry.