Gap-Free Set Partitions

A set partition is called “gap-free” if its block sizes form an interval. In other words, there is at least one block of each size between the smallest and largest block sizes. Let B(n) and G(n), respectively, denote the number of partitions and the number of gap-free partitions of the set [n]. We prove that      

Dirichlet polyhedra for dihedral groups acting on complex hyperbolic space

We consider groups Γ generated by inversions in a pair of asymptotic complex hyperplanes in complex hyperbolic spaceH _? ⁿ . We show that there exists a Γ-invariant real hypersurfaceF ?H _? ⁿ such that the Dirichlet fundamental polyhedron for Γ centered at z₀ has two sides (resp. infinitely many sides) if and only ifz ₀ ∈F (resp.z ₀ ?F). The Dirichlet regions are determined explicitly in terms of coordinates on Γ-invariant horospheres and the geometry ofH _? ⁿ is developed in terms of these horospherical coordinates.     

The oxidative decarboxylation of polyaminocarboxylic acids

Summary The rates of oxidation of four chelating agents, NTA, EDTA, CDTA, and DTPA with Ce(IV), in sulfuric acid media, were determined spectrophotometrically by a stopped-flow technique. The reductive ability is in the order CDTA > EDTA > DTPA > NTA. The influence of varying the acidity of the medium was studied, and in each case a maximum in the rate constant vs. [H⁺] plot was observed. A possible interpretation of the reactivities and the influence of acidity is advanced.

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Oxydative Decarboxylierung von PolyaminocarbonsäurenII. Vergleichende kinetische Untersuchung der Oxydation von NTA, ÄDTA, CDTA und DTPA mit Ce(IV) in saurer Lösung

Zusammenfassung Die Oxydationsgeschwindigkeiten von 4 Chelaten (NTA, ÄDTA, CDTA und DTPA) mit Ce(IV) in saurer Lösung wurden spektrophotometrisch mit Hilfe der stopped-flow-Technik bestimmt. Die Reduzierfähigkeit nimmt in der Reihenfolge CDTA > ÄDTA > DTPA > NTA ab. Der Einfluß verschiedener Säuregehalte in der Lösung wurde untersucht, und in jedem Fall wurde ein Maximum in der graphischen Darstellung der Geschwindigkeitskonstante gegen [H⁺] beobachtet. Eine mögliche Erklärung des Reaktionsvermögens und des Säureeinflusses wird gegeben.

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Part I: Z. Anal. Chem. 246, 231 (1969).     

Stereochemical analysis of isopentenyl diphosphate isomerase type II from Staphylococcus aureus using chemically synthesized (S)- and (R)-[2-2H]isopentenyl diphosphates

[chemical reaction: see text]. To study the catalysis of isopentenyl diphosphate (IPP) isomerase type II from Staphylococcus aureus, which is a flavoprotein catalyzing the interconversion of IPP and dimethylallyl diphosphate, we have chemically synthesized (S)- and (R)-[2-2H]IPP and carried out stereochemical analysis of the reaction. Our results show that the C-2 deprotonation of IPP by this enzyme is pro-R stereospecific, suggesting a similar stereochemical course as the type I enzyme.     

Facile syntheses of unsolvated UI3 and tetramethylcyclopentadienyl uranium halides

In the course of comparing the reaction chemistry of (C5Me5)3U, 1, and its slightly less crowded analogue (C5Me4H)3U, 2, new syntheses of UI3, (C5Me4H)3U, (C5Me4H)3UCl, 3, and (C5Me5)3UCl, 4, have been developed. Additionally, (C5Me4H)3UI, 5, and (C5Me4H)2UCl2, 6, have been identified for the first time. A facile synthesis of unsolvated UI3 is reported that proceeds in high yield with inexpensive equipment from iodine and hot uranium turnings. Both UI3 and UI3(THF)4 react with KC5Me4H in toluene to make unsolvated (C5Me4H)3U in higher yield than in previous reports that involve reduction of tetravalent (C5Me4H)3UCl, 3. A more atom-efficient synthesis of complex 3 is also reported that proceeds from reduction of t-BuCl, PhCl, or HgCl2 by 2. Similarly, (C5Me4H)3U reacts with PhI or HgI2 to generate (C5Me4H)3UI. These studies also provided a basis to improve the synthesis of (C5Me5)3UCl from 1 by employing t-BuCl or HgCl2 as the halide source. Like (C5Me5)3UCl, the (C5Me4H)3UCl complex reacts with HgCl2 to form (C5Me4H)2 and (C5Me4H)2UCl2, 6, but unlike (C5Me5)3UX (X = Cl or I), the less substituted (C5Me4H)3UX complexes do not reduce t-BuCl or PhX. The synthesis of 6 from (C5Me4H)MgCl x THF and UCl4 is also included.     

Steric effects of substituted quinolines on lithium coordination geometry

The X-ray crystal structures of a series of lithium quinolates – lithium 8-hydroxyquinolinate (Liq), lithium 2-methyl-8-hydroxyquinolinate (MeLiq), and 2-phenyl-8-hydroxquinolinate (PhLiq), are compared. The substitution at the 2-position of the 8-hydroxyquinoline ligand has significant impact on the aggregation of the lithium complex in the crystalline state. Liq and MeLiq molecules crystallize as hexamers, whereas PhLiq crystallizes as a tetramer. The possible influence of crystal-packing forces on the preferred cluster structure was probed using density functional theory calculations on a systematically varied set of Liq, MeLiq, and PhLiq clusters. For Liq and MeLiq, the observed structures match the most stable computed structures. In the PhLiq case, the observed tetrameric structure is computed to be less stable (+1.2 kcal/mol/monomer) than the lowest energy structure, a hexamer. In this case, solid-state effects probably outweigh small differences in cluster stability.     

A simple reliable approximation for isotropic intermolecular forces. A critical test using HH(3Σu+) as a model

A simple semi-empirical approximation for the exchange energy (E_x), coupled with a tractable representation of the Coulomb energy (E_C), has been found to yield very accurate results for the isotropic part of the interaction energy (E_int = E_x + E_C) between two closed shell systems. The expression for E_int is based on the knowledge of the first order Coulomb energy and the first three terms in the asymptotic long range expansion of the second order Coulomb energy for the interaction and contains but one adjustable parameter which occurs in E_x. The usefulness of this approach for evaluating E_int is tested critically by using the non-bonded H(1s)H(1s) (³Σ_u⁺) interaction as a model (accurate values of the total interaction energy, the exchange energy, and various orders of Coulomb energies, are available for a wide range of R for this system). The results obtained for both E_int and (dE_int/dR) are inremarkable agreement with the exact results of Kotos and Wolniewicz for R > 3 a_o. Since the law of corresponding states for inert gas pairs holds equally well for the HH(³Σ_u⁺) interaction, our analysis of this simple system yields valuable information on the reliability of the approach for other van der Waals dimers.     

EVALUATION OF NEW BENZOPORPHYRIN DERIVATIVES WITH ENHANCED PDT EFFICACY*

Abstract— A first report on the biological evaluation of a series of isomerically pure benzoporphyrin derivatives ( cis- and frarcs-isomers) as methyl esters is described. In preliminary in vivo studies, the n- bexyl ether analogues of both cis- and trans -isomers of benzoporphyrin derivatives were found to be more active than the industrially prepared benzoporphyrin derivative, a mixture of monocarboxylic acids (BPDMA, Quadralogic Technologies, Vancouver). Further studies with 4-de-vinyl-4-(I-hexyloxyethyl) benzoporphyrin derivative showed that, like BPDMA, it had reduced residual skin phototoxicity compared in mice with Photofrin®. The uptake and clearance characteristics of BPDMA were also compared with the 4-(1-hexyloxyethyl)-derivative by in vivo reflection spectroscopy.     

Isomeric Schiff bases related by dual imino‐group reversals

Two isomeric pairs of Schiff bases, N,N′‐bis­(2‐methoxy­benzyl­idene)‐p‐phenyl­enediamine, C₂₂H₂₀N₂O₂, (I), and 2,2′‐dimeth­oxy‐N,N‐(p‐phenyl­enedimethyl­ene)dianiline, C₂₂H₂₀N₂O₂, (II), and (E,E)‐1,4‐bis­(3‐iodo­phen­yl)‐2,3‐diaza­buta‐1,3‐diene (alternative name: 3‐iodo­benzaldehyde azine), C₁₄H₁₀I₂N₂, (III), and N,N′‐bis­(3‐iodo­phen­yl)ethylenedi­imine, C₁₄H₁₀I₂N₂ [JAYFEV; Cho, Moore & Wilson (2005). Acta Cryst. E 61 , o3773–o3774], differ pairwise only in the orientation of their imino linkages and in all four individual cases occupy inversion centers in the crystal, yet all four compounds are found to assume unique packing arrangements. Compounds (I) and (II) differ substantially in mol­ecular conformation, possessing angles between their ring planes of 12.10 (15) and 46.29 (9)°, respectively. Compound (III) and JAYFEV are similar to each other in conformation, with angles between their imino linkages and benzene rings of 11.57 (15) and 7.4 (3)°, respectively. The crystal structures are distinguished from each other by different packing motifs involving the functional groups. Inter­molecular contacts between meth­oxy groups define an R₂²(6) motif in (I) but a C(3) motif in (II). Inter­molecular contacts are of the I⋯I type in (III), but they are of the N⋯I type in JAYFEV.