The complexification and degree of a semi-algebraic set

The complexification of a semi-algebraic set is the smallest complex algebraic set containing S. Let S be defined by s polynomials of degrees less than d. We prove that the geometric degree of the complexification is less than . Received: 9 January 1997; in final form: 11 August 2000 / Published online: 17 May 2001     

Heterobimetallic Eu(iii)/Pt(ii) single-chain nanoparticles: a path to enlighten catalytic reactions

We introduce the formation and characterization of heterometallic single-chain nanoparticles entailing both catalytic and luminescent properties. A terpolymer containing two divergent ligand moieties, phosphines and phosphine oxides, is synthesized and intramolecularly folded into nanoparticles via a selective metal complexation of Pt(ii) and Eu(iii). The formation of heterometallic Eu(iii)/Pt(ii) nanoparticles is evidenced by size exclusion chromatography, multinuclear NMR (¹H, ³¹P{¹H}, ¹⁹F, ¹⁹⁵Pt) as well as diffusion-ordered NMR and IR spectroscopy. Critically, we demonstrate the activity of the SCNPs as a homogeneous and luminescent catalytic system in the amination reaction of allyl alcohol.

A bifunctional terpolymer containing two orthogonal ligand moieties was synthesized, giving way to the facile formation of heterometallic Eu(iii)/Pt(ii) single-chain nanoparticles, which display both catalytic and luminescent properties.     

Exploiting the regioselectivity of pyroglutamate alkylations for the synthesis of 6-azabicyclo[3.2.1]octanes and 4-azabicyclo[3.3.0]octanes

Depending on the N-protecting group of pyroglutamates, the reactivity can be directed to the formation of 6-azabicyclo[3.2.1]octanes or 4-azabicyclo[3.3.0]octanes, which are conformationally restricted glutamate analogues.     

Synthetic protocols and building blocks for molecular electronics

Simple and readily accessible aryl bromides are useful building blocks for thiol end-capped molecular wires. Thus, 4-bromophenyl tert-butyl sulfide and 1-bromo-4-(methoxymethyl)benzene serve as precursors for a variety of oligo(phenylenevinylene) and oligo(phenyleneethynylene) wires via efficient synthetic transformations as presented in this paper.     

On the temporal analysis of special GERT networks using a modified Markov renewal process

In this paper we show how to represent a STEOR network (GERT network with only nodes of the stochastic exclusive-or Type) by a modified Markov renewal process. The computation of the renewal kernel of this process yields the temporal analysis of the network.

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Zusammenfassung Es wird gezeigt, wie ein STEOR-Netzplan (GERT-Netzplan, der nur Knoten vom Typ stochastisch exklusiv-oder enthält) durch einen modifizierten Markowschen Erneuerungsprozeß beschrieben werden kann. Die Bestimmung der Erneuerungsfunktionen dieses Prozesses liefert die Auswertung des STEOR-Netzplans (im Sinne der Zeitplanung).

     

Electronic structure and reactivity of high-spin iron--alkyl- and--pterinperoxo complexes

The spectroscopic properties and electronic structure of the four-coordinate high-spin [FeIII(L3)(OOtBu)]+ complex (1; L3 = hydrotris(3-tert-butyl-5-isopropyl-1-pyrazolyl)borate; tBu = tert-butyl) are investigated and compared to the six-coordinated high-spin [Fe(6-Me3TPA)(OHx)(OOtBu)]x+ system (TPA = tris(2-pyridylmethyl)amine, x = 1 or 2) studied earlier [Lehnert, N.; Ho, R. Y. N.; Que, L., Jr.; Solomon, E. I. J. Am. Chem. Soc. 2001, 123, 12802-12816]. Complex 1 is characterized by Raman features at 889 and 830 cm-1 which are assigned to the O-O stretch (mixed with the symmetric C-C stretch) and a band at 625 cm-1 that corresponds to nu(Fe-O). The UV-vis spectrum shows a charge-transfer (CT) transition at 510 nm from the alkylperoxo pi v* (v = vertical to C-O-O plane) to a d orbital of Fe(III). A second CT is identified from MCD at 370 nm that is assigned to a transition from pi h* (h = horizontal to C-O-O plane) to an Fe(III) d orbital. For the TPA complex the pi v* CT is at 560 nm while the pi h* CT is to higher energy than 250 nm. These spectroscopic differences between four- and six-coordinate Fe(III)-OOR complexes are interpreted on the basis of their different ligand fields. In addition, the electronic structure of Fe-OOPtn complexes with the biologically relevant pterinperoxo ligand are investigated. Substitution of the tert-butyl group in 1 by pterin leads to the corresponding Fe(III)-OOPtn species (2), which shows a stronger electron donation from the peroxide to Fe(III) than 1. This is related to the lower ionization potential of pterin. Reduction of 2 by one electron leads to the Fe(II)-OOPtn complex (3), which is relevant as a model for potential intermediates in pterin-dependent hydroxylases. However, in the four-coordinate ligand field of 3, the additional electron is located in a nonbonding d orbital of iron. Hence, the pterinperoxo ligand is not activated for heterolytic cleavage of the O-O bond in this system. This is also evident from the calculated reaction energies that are endothermic by at least 20 kcal/mol.     

On Fundamental Domains and Volumes of Hyperbolic Coxeter–Weyl Groups

We present a simple method for determining the shape of fundamental domains of generalized modular groups related to Weyl groups of hyperbolic Kac?CMoody algebras. These domains are given as subsets of certain generalized upper half planes, on which the Weyl groups act via generalized modular transformations. Our construction only requires the Cartan matrix of the underlying finite-dimensional Lie algebra and the associated Coxeter labels as input information. We present a simple formula for determining the volume of these fundamental domains. This allows us to re-produce in a simple manner the known values for these volumes previously obtained by other methods.