Norms and Generating Functions in Clifford Algebras

Generating functions are commonly used in combinatorics to recover sequences from power series expansions. Convergence of formal power series in Clifford algebras of arbitrary signature is discussed. Given , powers of u are recovered by expanding (1 − tu)⁻¹ as a polynomial in t with Clifford-algebraic coefficients. It is clear that (1 − tu)(1 + tu + t ² u ² + ...) = 1, provided the sum (1 + tu + t ² u ² + ...) exists, in which case u ^m is the Cliffordalgebraic coefficient of t ^m in the series expansion of (1 − tu)⁻¹. In this paper, conditions on for the existence of (1 − tu)⁻¹ are given, and an explicit formulation of the generating function is obtained. Allowing A to be an m × m matrix with entries in , a “Clifford-Frobenius” norm of A is defined. Norm inequalities are then considered, and conditions for the existence of (I − tA)⁻¹ are determined. As an application, adjacency matrices for graphs are defined with vectors of as entries. For positive odd integer k > 3, k-cycles based at a fixed vertex of a graph are enumerated by considering the appropriate entry of A ^k . Moreover, k-cycles in finite graphs are enumerated and expected numbers of k-cycles in random graphs are obtained from the norm of the degree-2k part of tr(1 − tu)⁻¹. Unlike earlier work using commutative subalgebras of , this approach represents a “true” application of Clifford algebras to graph theory.      

Synthesis, characterization, and reactivity of iron trisamidoamine complexes that undergo both metal- and ligand-centered oxidative transformations

Functional systems that combine redox-active metals and noninnocent ligands are no longer rare chemical oddities; they are instead emerging as significant components of catalytic and enzymatic reactions. The present work examines the synthetic and functional aspects of iron compounds ligated by a family of new trisamidoamine ligands of the type [(RNC6H4)3N]3- (L1). When R is the electron-rich 4-t-Bu-Ph moiety, the ligand can undergo oxidative rearrangement and store oxidizing equivalents under specific conditions. Starting ferrous complexes of the general formula [(L1)FeIIsolv]- (solv=CH3CN, dimethylformamide) can be easily oxidized (a) by dioxygen to afford the corresponding [(L1)FeIIIOH]- complexes, featuring several cases of terminal hydroxo units, and (b) by organochlorides (R-Cl) to provide [(L1)FeIIIsolv] congeners and coupled R-R products. Efforts to synthesize [(L1)FeIII-O-FeIII(L1)]2- by using [Cl3FeIII-O-FeIIICl3]2- indicate that intrinsic FeIIICl units can oxidatively rearrange the ligand to afford [(L1re)(Cl)FeII][Et4N]2, although the oxidizing equivalent is not retained. Compound [(L1re)(Cl)FeII][Et4N]2 can be further oxidized to [(L1re-2)(Cl)FeIII][Et4N] by CH2Cl2. Finally, oxidation of [(L1)FeIIIsolv] by FeCl3 affords [(L1reH)(Cl)FeII(micro-Cl)2FeII(Cl)(L1re-2H)], which features a similar ligand rearrangement that also gives rise to a diamagnetic, doubly oxidized moiety. These results underscore the complexity of chemical transformations available to systems in which both the metal and the ligand are redox-active entities.     

Heterocarbenoids of germanium and tin and their polyhedral oxidation products: The case for thermodynamic product control in Group 14 chalcogenides

The polycyclic Group 14 amides [P(μ-N^tBu)₂P(^tBuN)₂]M, M = Ge (4), Sn (5) were synthesized from cis-[P(μ-N^tBu)₂P(^tBuNLi · THF)₂] and GeCl₂ · dioxane or SnCl₂, respectively. Oxidation of these heterocarbenoids or of the analogous diazastannylene [MeSi(μ-^tBuN)₂SiMe(^tBuN)₂]Sn with O₂, S₈ and Se_n furnished the chalcogenides {[P(μ-N^tBu)₂P(^tBuN)₂]GeO}₂ (6), {[P(μ-N^tBu)₂P(^tBuN)₂]SnE}₂, E = O (7), S (8), Se (9), {[SP(μ-N^tBu)₂P(^tBuN)₂]SnS}₂ (10), and {[MeSi(μ-^tBuN)₂SiMe(^tBuN)₂]SnE}₂, E = S (11), Se (12), respectively. All products (6-12) were shown by single-crystal X-ray methods to consist of dimeric molecules with central (M-E)₂ rings, M = Group 14 element, E = chalcogen. The exclusive formation of dimeric compounds with bridging M-E-M bonds, vs. alternative monomeric structures with terminal ME bonds, is rationalized in terms of the thermodynamic favorability of the dimers. The case is made that most, if not all, currently known Group 14 chalcogenides, even those labeled “kinetically stabilized”, are really thermodynamic products.     

Tandem free-radical addition/substitution chemistry and its application to the preparation of novel AT1 receptor antagonists

Benzothiophene and benzoselenophene analogues of the thiophene-containing antihypertensives milfasartan and eprosartan were prepared and tested for AT(1) receptor antagonist properties. While the sulfur-containing systems were prepared following existing methodology, the selenium-containing analogues required the development of novel, tandem free-radical chemistry involving addition of aryl radicals to alkynes, followed by intramolecular homolytic substitution at the higher heteroatom. All four compounds prepared proved to be excellent AT(1) receptor antagonists, with pK(B) estimates of 7.2-9.5.     

Synthesis and characterization of a tetramethyl furanone functionalized diiminedioxime,a potential ligand for Cu radiopharmaceuticals,and its copper(II) and nickel(II) complexes

As part of our on-going effort to develop ⁶⁴Cu-based radiopharmaceuticals for PET (positron emission tomography) imaging of multidrug resistance in cancer, we prepared a tetramethylfuranone-functionalized diiminedioxime ligand, TMFPreH (TMFPreH = 4-[3-(4-hydroxyimino-2,2,5,5-dimethyl-dihydro-furan-3-ylideneamino)-propylimino]-2,2,5,5-tetramethyl-dihydrofuran-3(2H)-one oxime) and its Cu(II) and Ni(II) complexes. When the copper(II) complex was prepared from Cu(ClO₄)₂ in ethanol, it was isolated as a Cu(II)-bridged dimer, but when it was prepared from Cu(OAc)₂ and heated in acetone, an unusual example of an acetone adduct of the ligand is formed by reduction of one of the imine double bonds by the solvent. The Ni(II) complex is square pyramidal with the perchlorate counterion at the apex.     

Analysis of the crystal structures of 1,3‐di‐tert‐butyl‐2,3‐dihydro‐1H‐1,3,2‐diazasilol‐2‐ylidene and 1,3‐di‐tert‐butyl‐2,2‐dichloro‐1,3‐diaza‐2‐sila‐4‐cyclopentene

The crystal structures of the title compounds, 1,3‐di‐tert‐butyl‐2,3‐dihydro‐1H‐1,3,2‐diazasilol‐2‐ylidene, C₁₀H₂₀N₂Si ( 1 ) and 1,3‐di‐tert‐butyl‐2,2‐dichloro‐1,3‐diaza‐2‐sila‐4‐cyclopentene, C₁₀H₂₀N₂SiCl2 ( 3 ) were solved and are reported. Compound ( 1 ) crystallized in space group P mmn and each molecule has a mirror plane, which bisects the C‐C backbone of the N‐C‐C‐N framework. Compound ( 1 ) was also found to have a 2‐fold twin component. In compound ( 3 ) the space group P 2₁/m results with the mirror plane passing through the N‐C‐C‐N backbone. We compare these structures with the gas phase determination previously reported for ( 1 ) and the incomplete single crystal data for ( 3 ). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Untersuchung von Rohstoffen und Mineralien

Ohne Zusammenfassung     

2,2′‐Bi[benzo[b]thiophene]: an unexpected isolation of the benzo[b]thiophene dimer

The crystal structure of 2,2′‐bi[benzo[b]thiophene], C₁₆H₁₀S₂, at 173 K has triclinic (P) symmetry. It is of interest with respect to its apparent mode of synthesis, as it is a by‐product of a Stille cross‐coupling reaction in which it was not explictly detected by spectroscopic methods. It was upon crystal structure analysis of a specimen isolated from the mother liquor that this reaction was determined to give rise to the title compound, which is a dimer arising from the starting material. Two independent half‐molecules of this dimer comprise the asymmetric unit, and the full molecules are generated via inversion centers. Both molecules in the unit cell exhibit ring disorder, and they are essentially identical because of their rigidity and planarity.