On factor refinement in number fields

Let be an order of an algebraic number field. It was shown by Ge that given a factorization of an -ideal into a product of -ideals it is possible to compute in polynomial time an overorder of and a gcd-free refinement of the input factorization; i.e., a factorization of into a power product of -ideals such that the bases of that power product are all invertible and pairwise coprime and the extensions of the factors of the input factorization are products of the bases of the output factorization. In this paper we prove that the order is the smallest overorder of in which such a gcd-free refinement of the input factorization exists. We also introduce a partial ordering on the gcd-free factorizations and prove that the factorization which is computed by Ge's algorithm is the smallest gcd-free refinement of the input factorization with respect to this partial ordering.

     

Digital Expansions with Respect to Different Bases

We consider the g-ary expansion N=∑ _k b _k (N, g)g ^k of non-negative integers N and prove various results on the distribution and the mean value of the k-th digit b _k (N, g) if g varies in an interval of the form 2≤g≤N ^η. As an application we also consider the average value of the sum-of-digits function s(N, g)=∑ _k b _k (N, g). Received 5 November 2001 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Improving conversion and enantioselectivity in hydrogenation by combining different monodentate phosphoramidites; a new combinatorial approach in asymmetric catalysis

The combination of monodentate ligands in the rhodium-catalysed enantioselective hydrogenation enables a new approach when searching for the optimal activity and enantioselectivity in catalysis.     

Application of monodentate secondary phosphine oxides,a new class of chiral ligands,in Ir(i)-catalyzed asymmetric imine hydrogenation

Secondary phosphine oxides were prepared from R(1)PCl(2) and R(2)MgBr, followed by hydrolysis. They were obtained in an enantiopure form by preparative chiral HPLC. These new monodentate ligands were tested in the iridium-catalyzed hydrogenation of imines at 25 bar. Enantioselectivities up to 76% were obtained at L/Ir = 2. Addition of pyridine (Pyr/Ir = 1:2) raised the ee to 83%. Using pyridine as an additive allowed reduction of the L/Ir ratio to 1 without reduction of ee. [reaction: see text]     

Mono- versus bidentate ligands in rhodium-catalyzed asymmetric hydrogenation. A comparative rate study

[reaction: see text] Bidentate chiral phosphines are no longer essential for achieving a fast and highly enantioselective hydrogenation of alpha- or beta-dehydroamino acid derivatives. In particular, a readily accessible and stable monodentate phosphoramidite can be highly effective in these asymmetric hydrogenations.     

Equilibrium, 1H and 13C NMR spectroscopy,and X-ray diffraction studies on the complexes Bi(DOTA)- and Bi(DO3A-Bu)

Several Bi(III) complexes are used in medicine as drugs. Bi(DO3A-Bu) has recently been proposed as a nonionic contrast agent in X-ray imaging (H(3)DO3A-Bu = 10-[2,3-dihydroxy-(1-hydroxymethyl)propyl]-1,4,7,10-tetraazacyclododecane-1,4,7,-triacetic acid). The solution equilibria and NMR structure and dynamics of Bi(DO3A-Bu) and of the similar Bi(DOTA)(-) have been investigated (H(4)DOTA = 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid). The stability constants were determined with the study of the competition equilibria between Br(-) ions and the ligands DOTA or DO3A-Bu for the Bi(III) by spectrophotometry. The stability constants, obtained for Bi(DOTA)(-) and Bi(DO3A-Bu), are very high, log K = 30.3 and 26.8, respectively. Potentiometric titrations indicated the dissociation of one of the protons among the three alcoholic OH groups in Bi(DO3A-Bu). The dissociation constant is log K = 7.53 (0.09) indicating that at physiological pH about 50% of the species possess -1 charge. It was shown by (1)H and (13)C NMR spectroscopy that the OH group attached to the middle carbon atom of the "butriol" side chain is coordinated to the Bi(III) and starts to deprotonate at pH > 5.5. The crystal structure of NaBi(DOTA).H(2)O shows an octacoordinated arrangement of the donor atoms around the Bi(III), with no water in the inner sphere. The crystals belong to the centrosymmetric space group C2/c. The temperature dependent (1)H and (13)C NMR spectra indicate that both Bi(DOTA)(-) and Bi(DO3A-Bu)(-) complexes are fluxional. For Bi(DOTA), the Delta(deltadeltadeltadelta) right harpoon over left harpoon Lambda(lambdalambdalambdalambda) fluxionality was identified, and on the basis of the activation parameters, a synchronous motion was suggested for the fluxional motion resulting in the change of ring conformation and of the helicity of the complex. The transition state is supposed to be more symmetrical than the initial state. The deprotonated Bi(DO3A-Bu) has a highly asymmetric NMR structure in solution, and its fluxional motion is slower than that of Bi(DOTA)(-).     

Regioselectivity of the tri-pi-methane rearrangement: mechanistic and exploratory organic photochemistry

The di-pi-methane rearrangement with two pi-groups attached to the central "methane carbon" of the reactant and which leads to a pi-substituted cyclopropane has been studied intensively. Our present research had the goal of elucidating the regioselectivity of the tri-pi-methane counterpart. The reactants with three pi groups attached to the central carbon mechanistically are capable of affording both di-pi-methane and tri-pi-methane photoproducts. In common with the di-pi-methane system, bridging of two of the pi-systems affords a cyclopropyldicarbinyl diradical intermediate that opens to an allyl carbinyl diradical. This diradical has the option of closing to a three-membered or a five-membered ring. It was found that the regioselectivity of the initial pi-pi-bridging step and the three-ring opening of the cyclopropyldicarbinyl diradical exhibit regioselectivity parallel to that of the di-pi counterpart. Both three-ring and five-ring photoproducts were formed with the ratio varying with conversion. Since the three-ring (i.e. di-pi-methane) photoproducts were found to ring expand to the five-ring (i.e. tri-pi-methane) products, kinetics were employed to determine to what extent the reaction proceeds in a two-step versus direct formation of the five-ring product. It was found that the direct route was the major one.     

Photochemical and thermal isomerization processes of a chiral auxiliary based donor-acceptor substituted chiroptical molecular switch: convergent synthesis,improved resolution and switching properties

A new type of chiroptical molecular switch is presented where irradiation employing different wavelengths of light induces a reversible helix inversion of a sterically overcrowded alkene bearing a second chiral entity in the form of a stereogenic center present in a pyrrolidine unit. The additional stereogenic center in the chiral auxiliary group has a distinct influence on the switching selectivity of this system and greatly facilitates the resolution of the different diastereoisomers, which is a considerable improvement compared with previously reported systems. In addition, the pyrrolidine stereogenic center causes small energetic differences between the various states of the switch system resulting in a small but significant directional preference in the helix inversion steps.     

On the correlation for Kac-like models in the convex case

The aim of this paper is to stu the behavior asm tends to of a family of measures exp[- ^(m)(x)]dx ^(m) on ^m , where ^(m) is a potential on ^m which is a perturbation in a suitable sense of the harmonic potential _j x _j ² .