The complex of maximal lattice free simplices

The simplicial complexK(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form (x: Axb), withA a fixed generic (n + 1) ×n matrix. The topological space associated withK(A) is shown to be homeomorphic to ⁿ , and the space obtained by identifying lattice translates of these simplices is homeorphic to then-torus.Corresponding author.The first author was partially supported by Hungarian NSF grants 1907 and 1909, and also by U.S. NSF grant CCR-9111491. The research of the second author was supported by DMS9103608 and the third author by NSF grant SES9121936.     

On integer points in polyhedra: A lower bound

Given a polyhedronP we writeP _I for the convex hull of the integral points inP. It is known thatP _I can have at most135-2 vertices ifP is a rational polyhedron with size . Here we give an example showing thatP _I can have as many as ( ^n–1) vertices. The construction uses the Dirichlet unit theorem.The results of the paper were obtained while this author was visiting the Cowles Foundation at Yale University     

Variegated micelle surfaces: correlating the microstructure of mixed surfactant micelles with bulk solution properties

Electron paramagnetic resonance, viscosity, and small-angle neutron scattering (SANS) measurements have been used to study the interaction of mixed anionic/nonionic surfactant micelles with the polyampholytic protein gelatin. Sodium dodecyl sulfate (SDS) and the nonionic surfactant dodecylmalono-bis-N-methylglucamide (C12BNMG) were chosen as "interacting" and "noninteracting" surfactants, respectively; SDS micelles bind strongly to gelatin but C12BNMG micelles do not. Further, the two surfactants interact synergistically in the absence of the gelatin. The effects of total surfactant concentration and surfactant mole fraction have been investigated. Previous work (Griffiths et al. Langmuir 2000, 16 (26), 9983-9990) has shown that above a critical solution mole fraction, mixed micelles bind to gelatin. This critical mole fraction corresponds to a micelle surface that has no displaceable water (Griffiths et al. J. Phys. Chem. B 2001, 105 (31), 7465). On binding of the mixed micelle, the bulk solution viscosity increases, with the viscosity-surfactant concentration behavior being strongly dependent on the solution surfactant mole fraction. The viscosity at a stoichiometry of approximately one micelle per gelatin molecule observed in SDS-rich mixtures scales with the surface area of the micelle occupied by the interacting surfactant, SDS. Below the critical solution mole fraction, there is no significant increase in viscosity with increasing surfactant concentration. Further, the SANS behavior of the gelatin/mixed surfactant systems below the critical micelle mole fraction can be described as a simple summation of those arising from the separate gelatin and binary mixed surfactant micelles. By contrast, for systems above the critical micelle mole fraction, the SANS data cannot be described by such a simple approach. No signature from any unperturbed gelatin could be detected in the gelatin/mixed surfactant system. The gelatin scattering is very similar in form to the surfactant scattering, confirming the widely accepted picture that the polymer "wraps" around the micelle surface. The gelatin scattering in the presence of deuterated surfactants is insensitive to the micelle composition provided the composition is above the critical value, suggesting that the viscosity enhancement observed arises from the number and strength of the micelle-polymer contact points rather than the gelatin conformation per se.     

Chemical and Biological Studies of Dichloro(2-((dimethylamino)methyl)phenyl)gold(III)

Several new organogold(III) derivatives of the type [AuX(2)(damp)] (damp = o-C(6)H(4)CH(2)NMe(2)) have been prepared [X = CN, SCN, dtc, or X(2) = tm; dtc = R(2)NCS(2) (R = Me (dmtc) or Et (detc)); tm = SCH(CO(2))CH(2)CO(2)Na] together with [AuCl(tpca)(damp)]Cl (tpca = o-Ph(2)PC(6)H(4)CO(2)H), [Au(dtc)(damp)]Y (Y = Cl, BPh(4)) and K[Au(CN)(3)(damp)]. The (13)C NMR spectra of these and previous derivatives have been fully assigned. In [Au(dtc)(2)(damp)] and K[Au(CN)(3)(damp)], the damp ligand is coordinated only through carbon, as shown by X-ray crystallography and/or NMR. [Au(detc)(2)(damp)] has space group C2/c, with a = 29.884(4) ?, b = 13.446(2) ?, c = 12.401(2) ?, beta = 99.45(3)(o), V = 4915 ?(3), Z = 8, and R = 0.057 for 1918 reflections. The damp and one detc ligand are monodentate, the other detc is bidentate; in solution, the complex shows dynamic behavior, with the detc ligands appearing equivalent. The crystal structure of [Au(dmtc)(damp)]BPh(4) [Pna2(1), a = 26.149(5) ?, b = 11.250(2) ?, c = 11.921(2) ?, V = 3507 ?(3), Z = 4, R = 0.073, 1772 reflections] shows both ligands to be bidentate in the cation, but the two Au-S distances are nonequivalent. The crystal structure of [Au(tm)(damp)] has also been determined [P2(1)/n, a = 18.267(7) ?, b = 9.618(3) ?, c = 18.938(4) ?, beta = 113.45(3)(o), V = 3053 ?(3), Z = 8, R = 0.079, 1389 reflections]. The tm is bound through sulfur and the carboxyl group which allows five-membered ring formation. In all three structures, the trans-influence of the sigma-bonded aryl group is apparent. [AuCl(2)(damp)] has been tested in vitroagainst a range of microbial strains and several human tumor lines, where it displays differential cytotoxicity similar to that of cisplatin. Against the ZR-75-1 human tumor xenograft, both [AuCl(2)(damp)] and cisplatin showed limited activity.     

Reactions of 2,2′,5′,2′-terfuran

Formylation of 2,2′,5′,2′-terfuran ( 1 ) with N-methylformanilide and phosphorus oxychloride gave 5-formyl-2,2′,5′,2′-terfuran ( 2 ) and 5,5′-diformyl-2,2′5′,2′-terfuran ( 3 ). Reduction of 2 and 3 afforded 5-hydroxymethyl-2,2′,5′,2′-terfuran ( 4 ) and 5,5′ dihydroxymethyl-2,2′,5′,2′-terfuran ( 5 ), respectively. Terfuran 1 reacted with phenylmagnesium bromide to give 5-(phenylhydroxymethyl)-2,2′,5′,2′-terfuran ( 6 ), and was carbonated to 5-carboxy 2,2′,5′,2′-terfuran ( 7 ) and 5,5′-dicarboxy-2,2′,5′,2′-terfuran ( 8 ). Bromination of 1 with N-bromosuccinimide gave 5,5′-dibromo 2,2′,5′,2′-terfuran ( 9 ).     

The determination of ring junction stereochemistry in steroids using mass-analysed ion kinetic energy spectrometry

The unimolecular mass-analysed ion kinetic energy (MIKE) spectra of 9 pairs of hydrocarbon and ketone steroid isomers, differing only in the stereochemistry at the A/B and C/D ring junctions, have been measured and are discussed with a view to unambiguous structural identification. Reproducible differences in the MIKE spectra are observed, which are large enough in certain instances to suggest that MIKE spectrometry may be used for determining the stereochemistry of the A/B and C/D ring junctions in steroidal isomers, even if the second isomer is not available. This fortunate situation is rarely observed in conventional mass spectrometry of stereoisomeric steroids. Furthermore, these differences in the MIKE spectra may be correlated with differences in strain energy between configurational isomers. The sensitivity of MIKE spectrometry to differences in strain energies makes it a potentially powerful stereochemical probe.