K-theory for the leaf space of foliations by Reeb components

The K-theory of the C¹-algebra $\text{C}{}^1\text{(V, F)}$ associated to C^∞-foliations (V, F) of a manifold V in the simplest non-trivial case, i.e., dim V = 2, is studied. Since the case of the Kronecker foliation was settled by Pimsner and Voiculescu (J. Operator Theory4 (1980), 93–118), the remaining problem deals with foliations by Reeb components. The K-theory of $\text{C}{}^1\text{(V, F)}$ for the Reeb foliation of S³ is also computed. In these cases the C¹-algebra $\text{C}{}^1\text{(V, F)}$ is obtained from simpler C¹-algebras by means of pullback diagrams and short exact sequences. The K-groups $\text{K}{}_1\text{(C}{}^1\text{(V, F))}$ are computed using the associated Mayer-Vietoris and six-term exact sequences. The results characterize the C¹-algebra of the Reeb foliation of $\text{T}$² uniquely as an extension of C(S¹) by C(S¹). For the foliations of $\text{T}$² it is found that the K-groups count the number of Reeb components separated by stable compact leaves. A C^∞-foliation of $\text{T}$² such that K₁(C¹($\text{T}$², F)) has infinite rank is also constructed. Finally it is proved, by explicit calculation using (M. Penington, “K-Theory and C¹-Algebras of Lie Groups and Foliations,” D. Phil. thesis, Oxford, 1983), that the natural map $\text{μ: K}{}_1\text{,}{}_{\mathrm{\tau }}\text{(BG) → K}{}_1\text{(C}{}^1\text{(V, F))}$ is an isomorphism for foliations by Reeb components of $\text{T}$² and S³. In particular this proves the Baum-Connes conjecture (P. Baum and A. Connes, Geometric K-theory for Lie groups, preprint, 1982; A. Connes, Proc. Symp. Pure Math.38 (1982), 521–628) when V = $\text{T}$².     

Dirichlet Series and Curious infinite Products

By number-theoretical methods we give a result generalizingthose of J. O. Shallit, one particular case of which reads asfollows: ...=2^–;, where a(n) is equal to 1 if the sum of the digits of n in basetwo is even, and – 1 if this sum is odd. Moreover we provea conjecture of Shallit concerning a way of approximating 2^–     

Reductive cleavage of axially disymmetric tertiary amines and quaternary ammonium salts by lithium aluminium hydride. Synthesis of new 1,1′-binaphthyl substituted amines

Axially disymmetric tertiary amines or quaternary ammonium salts $\text{A}$ are synthesized by double alkylation of primary or secondary amines with racemic or optically pure 2,2′-bis (bromomethyl)-1,1′-binaphthyl. Their reductive cleavage by lithium aluminium hydride in refluxing THF leads to chiral secondary or tertiary amines $\text{B}$, substituted by a binaphthyl unit, with high yields and absence of racemization.     

A beta-glucosidase from Sclerotinia sclerotiorum: biochemical characterization and use in oligosaccharide synthesis

The filamentous fungus Sclerotinia sclerotiorum, grown on a xylose medium, was found to excrete one beta-glucosidase (beta-glu x). The enzyme was purified to apparent homogeneity by ammonium sulfate precipitation, gel filtration, anion-exchange chromatography, and high-performance liquid chromatography (HPLC) gel filtration chromatography. Its molecular mass was estimated to be 130 kDa by HPLC gel filtration and 60 kDa by sodium dodecyl sulfate polyacrylamide gel electrophoresis, suggesting that beta-glu x may be a homodimer. For p-nitrophenyl beta-d-glucopyranoside hydrolysis, apparent Km and Vmax values were found to be 0.09 mM and 193 U/mg, respectively, while optimum temperature and pH were 55-60 degrees C and pH 5.0, respectively. beta-Glu x was strongly inhibited by Fe2+ and activated about 35% by Ca2+. beta-Glu x possesses strong transglucosylation activity in comparison with commercially available beta-glucosidases. The production rate of total glucooligosaccharides (GOSs) from 30% cellobiose at 50 degrees C and pH 5.0 for 6 h with 0.6 U/mL of enzyme preparation was 80 g/L. It reached 105 g/L under the same conditions when using cellobiose at 350 g/L (1.023 M). Finally, GOS structure was determined by mass spectrometry and 3C nuclear magnetic resonance spectroscopy.     

Facile Preparation of (±)-12-Epiprostaglandins from 7-Oxabicyclo[2.2.1]hept-5-en-2-one via an all-cis-formyllactone related to Corey lactone

The bicyclic monoselenoacetal 7 , easily obtained from (±)-7-oxabicyclo[2.2.1]hept-5-en-2-one ( 6 ) via a radical addition-acyl migration sequence, was converted to racemic 12-epiprostaglandins 3 and 4 . The key intermediate was the all-cis-formyllactone 2b related to Corey lactone (see 12 ; Scheme 1). The presence of a (tert-butyl)-dimethylsilyl protective group for the 11-OH substituent (prostaglandin numbering) was found to be crucial in avoidingβ -elimination and epimerization during the Wittig-Horner reaction (Scheme 2). Epimerization at C(12) at the formyllactone stage (see 2b ) was also possible and gave the known precursor 1b of naturally occurring prostaglandins and analogs.     

Photochemical reactivity of α,β-unsaturated δ-lactams: Synthesis of seco-steroids. Photochemical reactions. XIII [1]

The UV. irradiation of 17 β-hydroxy-2-aza-4-androsten-3-one (1) , N-methyl-17 β-hydroxy-2-aza-4-androsten-3-one (3) , 17 β-hydroxy-4-aza-5 β-androst-1-en-3-one (2) and N-methyl-17 β-hydroxy-4-aza-5 β-androst-1-en-3-one (4) , gives rise to 1,10-seco (from 1 and 3 ) and 5, 10-seco (from 2 and 4 ) steroids.     

Two Highly Populated Conformations at Room Temperature of Monterine and Granjine, Antitumor Bisbenzylisoquinoline Alkaloids: Origin and Tridimensional Structures

Monterine 1 as well as granjine 3, 1R,1'S configured biphenylic bisbenzylisoquinoline alkaloids, generate two highly populated conformers. The interconversion of two forms was detected by saturation tranfer in (1)H NMR NOEs experiments. Tridimensional structure of the conformers was determined on the basis of (1)H NMR analysis of anisotropic shielding protons, by NOEs measurements and vicinal proton coupling constants of CH1-CH(2)alpha and CH1'-CH(2)alpha'. The structures established from NMR data were further refined to observe the mobility of 3D conformations by molecular dynamics simulation in vacuo. The highly populated conformers, monterine 1a and 1b, as well as granjine 3a and 3b, are interconvertible by rotation about the C1'-Calpha', Calpha'-C9', and C11'-C11 bonds and inversion of the benzyl D ring by reference to CH(2)alpha'. The slow exchange system was investigated by dynamic NMR spectroscopy: DeltaG()(c) 77.9 KJ/mol and k(c) 200 s(-)(1) for monterine 1; DeltaG()(c) 77.7 KJ/mol and k(c) 211 s(-)(1) for granjine 3. Natural and synthetic biphenylic bisbenzylisoquinolines displayed, in vitro, cytotoxic activities against human prostate and breast cancer cell lines.     

Mononuclear and dinuclear copper(I) complexes with a particularly bulky phosphine ligand. Unusual synthesis and crystal structure characterizations

Summary The addition of NPm (diphenylaminodiphenylphosphinomethane) to CuI or the addition of KI to (NPm)₂CuNO₃ leads to the same P-bonded Cu^I complex, (NPm)₂CuI, presenting a trigonal geometry around the Cu atom. The reaction of this new complex (or of its chloro analogue) with a Cu^II salt yields dinuclear species of general formula [(NPm)Cu(-X)₂Cu(NPm)] (X = Cl or I). X-ray analysis of these complexes show that they are isostructural and retain the trigonal geometry around the metal atom. The Cu···Cu distances are 2.775(1)Å for X = Cl and 2.642(1) Å for X = I. The Cu-(-X)-Cu angle is more acute for the iodide [61.48(3)°] than for the chloride [74.17(8)°] complex. These values are discussed in terms of Cu···Cu interactions induced by the electron donor ability of X and the bulk of the phosphine L.     

Synthesis of 15N-enriched pseudouridine derivatives

[reaction: see text]. A procedure for the chemical synthesis of [3-15N]-labeled pseudouridine and a methylated derivative was developed. A suitably protected pseudouridine precursor was nitrated at N3 followed by treatment with 15NH4Cl to afford the 15N-labeled product in six steps with a 20% yield. This methodology will allow for the production of RNAs with [3-15N]pseudouridine and [3-15N-methyl]pseudouridine at specific locations.