The number of non-isomorphic models in quasi-varieties of semigroups

Define \( n_K (\lambda )\) tobe eitherω, or the number of non-isomorphic algebras in \(K\) ] having cardinality λ, whichever cardinal is larger. It is proved here that if \(K\) ] is a quasi-variety (universal Horn class) of semigroups, then \(n_K\) is one of four functions. Each of these functions satisfies: \(n_K (\omega ) = \omega\) or \(n_K (\omega ) = 2^\omega\) . If \(n_K (\lambda )< 2^\lambda\) for some infinite λ then \(K\) ] is a residually finitevariety.     

Aggregation control by homologous tripodal tetradentate amino acid ligands in oxo-bridged diiron(III) aquo complexes

Isoelectronic oxo-bridged diiron(III) aquo complexes of the homologous tripodal tetradentate amino acid ligands, N,N'-bis(2-pyridylmethyl)-3-aminoacetate (bpg(-)) and N,N'-bis(2-pyridylmethyl)-3-aminopropionate (bpp(-)), containing [(H(2)O)Fe(III)-(mu-O)-Fe(III)(H(2)O)](4+) cores, oligomerise, respectively, by dehydration and deprotonation, or by dehydration only, in reversible reactions. In the solid state, [Fe(2)(O)(bpp)(2)(H(2)O)(2)](ClO(4))(2) (1(ClO(4))(2)) exhibits stereochemistry identical to that of [Fe(2)(O)(bpg)(2)(H(2)O)(2)](ClO(4))(2) (2(ClO(4))(2)), with the ligand carboxylate donor oxygen atoms and the water molecules located cis to the oxo bridge and the tertiary amine group trans to it. Despite their structural similarity, 1(2+) and 2(2+) display markedly different aggregation behaviour in solution. In the absence of significant water, 1(2+) dehydrates and dimerises to give the tetranuclear complex, [Fe(4)(O)(2)(bpp)(4)](ClO(4))(4) (3(ClO(4))(4)), in which the carboxylate groups of the four bpp(-) ligands act as bridging groups between two [Fe(2)(O)(bpp)(2)](2+) units. Under similar conditions, 2(2+) dehydrates and deprotonates to form dinuclear and trinuclear oligomers, [Fe(2)(O)(OH)(bpg)(2)](ClO(4)) (4ClO(4)) and [Fe(3)(O)(2)(OH)(bpg)(3)](ClO(4)) (5(ClO(4))), related by addition of 'Fe(O)(bpg)' units. The trinuclear 5(ClO(4)), characterised crystallographically as two solvates 5(ClO(4)).3H(2)O and 5(ClO(4)).2MeOH, is based on a hexagonal [Fe(3)(O)(2)(OH)(bpg)(3)](+) unit, formally containing one hydroxo and two oxo bridges. The different aggregation behaviour of 1(ClO(4))(2) and 2(ClO(4))(2) results from the difference of one methylene group in the pendant carboxylate arms of the amino acid ligands.     

Protein‐free ligand screening: simplification of chiral chromatographic development via novel adaptation of NMR screening methodologies

We demonstrate here a promising NMR method that provides evidence for chiral compound selector interaction as a first‐pass screening method. A novel adaptation of commonly used protein‐based screening technologies, this approach relies upon ligand‐to‐stationary phase interaction wherein the stationary phase is tethered to sepharose beads. At only minutes per experiment, this methodology significantly reduces the time required for chiral separation methodology development and complements currently available chromatographic purity technologies.Copyright © 2009 John Wiley & Sons, Ltd.     

Star-linear equational theories of groupoids

We prove that there are precisely six equational theories E of groupoids with the property that every term is E-equivalent to a unique linear term. Presented by J. Berman. Received November 11, 2004; accepted in final form March 12, 2006. The first and third authors were supported by the Ministry of Science and Environment of Serbia, grant no. 144011; the second and fifth authors were supported by MŠMT, research project MSM 0021620839, and by the Grant Agency of the Czech Republic, grant #201/02/0594; the fourth author was supported by the NSF grant #DMS-0245622.     

Next‐Generation D2‐Symmetric Chiral Porphyrins for Cobalt(II)‐Based Metalloradical Catalysis: Catalyst Engineering by Distal Bridging

Novel D₂‐symmetric chiral amidoporphyrins with alkyl bridges across two chiral amide units on both sides of the porphyrin plane (designated “HuPhyrin”) have been effectively constructed in a modular fashion to permit variation of the bridge length. The Co^II complexes of HuPhyrin, [Co(HuPhyrin)], represent new‐generation metalloradical catalysts where the metal‐centered d‐radical is situated inside a cavity‐like ligand with a more rigid chiral environment and enhanced hydrogen‐bonding capability. As demonstrated with cyclopropanation and aziridination as model reactions, the bridged [Co(HuPhyrin)] functions notably different from the open catalysts, exhibiting significant enhancement in both reactivity and stereoselectivity. Furthermore, the length of the distal alkyl bridge can have a remarkable influence on the catalytic properties.     

Einstein metrics and quaternionic Kähler manifolds

The research is partially suported by a grant from the Natural Sciences and Engineering Research Council of Canada