Discrete groups of slow subgroup growth

It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least $n^{\frac{{\log n}}{{\log \log n}}} $ . In this paper we prove the existence of a finitely generated group whose subgroup growth is of type $n^{\frac{{\log n}}{{(\log \log n)^2 }}} $ . This is the slowest non-polynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth typen ^logn is also realized. The proofs involve analysis of the subgroup structure of finite alternating groups and finite simple groups in general. For example, we show there is an absolute constantc such that, ifT is any finite simple group, thenT has at mostn ^{c logn} subgroups of indexn.     

On the convergence of multipoint Padé-type approximants and quadrature formulas associated with the unit circle

We study the convergence of rational interpolants with prescribed poles on the unit circle to the Herglotz-Riesz transform of a complex measure supported on [–, ]. As a consequence, quadrature formulas arise which integrate exactly certain rational functions. Estimates of the rate of convergence of these quadrature formulas are also included.This research was performed as part of the European project ROLLS under contract CHRX-CT93-0416.     

Norm-graphs and bipartite turán numbers

For everyt>1 and positiven we construct explicit examples of graphsG with |V (G)|=n, |E(G)|c _t ·n ^2–1/t which do not contain a complete bipartite graghK _t,t !+1 This establishes the exact order of magnitude of the Turán numbers ex (n, K _t,s ) for any fixedt and allst!+1, improving over the previous probabilistic lower bounds for such pairs (t, s). The construction relies on elementary facts from commutative algebra.Research supported in part by NSF Grants DMS-8707320 and DMS-9102866.Research supported in part by Hungarian National Foundation for Scientific Research Grant     

Synthesis and structure of a dithioether-substituted butene-2-olide-4-ylidene-4 cobalt carbonyl complex

The complex (1) has been prepared and its structure determined. Complex1 is the first stable thioether derivative of a metal carbonyl containing only cobalt as metal. It crystallizes in the monoclinic P2₁/c space group, witha = 13.975( 4 ),b = 11.363( 3 ),c = 14.686(2)A, = 112.25(2)°,V = 2158.5(7)A³.Z = 4. The solution using direct methods and the anisotropic refinement of all nonhydrogen atoms led toR = 0.039,wR = 0.041 for 3296 reflections with F> 4.0(F) Analogous but less stable complexes are formed with 1,6-diphenyl-2,5-dithiahexane and the crown thioether 1,4,7-trithiacyclonane, the latter functioning either as a bidenlate or a tridentate ligand.     

Synthesis of Enantiopure Primary Amines by Stereoselective Ring Opening of Chiral Octahydro-1,3-benzoxazines by Grignard and Organoaluminum Reagents

Chiral 1,3-perhydrobenzoxazines 1, 2, and 9-14, prepared by condensation of 8-(benzylamino)menthol with different aldehydes, react with alkylmagnesium bromides and trimethylaluminum leading to the open amino alcohols 3a-d, 4a-d, and 15-20 in excellent chemical yields and good to excellent diastereomeric excess. The sequential elimination of the menthol appendage by heating with P(2)O(5) and the benzyl group by hydrogenolysis lead to primary amines 7a-d, 8a-d, and 27-30 in excellent chemical yields and ee. The addition of the alkyl group from the Grignard derivatives and the methyl group from the trimethylaluminum occurs from opposite sides of the heterocycle, yielding the final primary amines with the same stereochemistry.     

An Efficient and Simple Entry to N-Substituted beta-Enamino Acid Derivatives from 2-Alkyl-2-oxazolines and 2-Alkyl-2-thiazolines

Reaction of azaenolates of 2-alkyl-oxa(thia)zolines 6 with imidoyl chlorides 7 as electrophiles to furnish masked N-substituted beta-enamino acid derivatives 1-2 in 70-90% yield is described. Alternative routes are discussed. Compounds 1-2 generally appear in one tautomeric form, imino or enamino, depending on the nature of the imidoyl chloride. The configuration of the enamino moiety (Z) and the conformation (s-cis) of compounds 1-2 obtained were established by an NMR study and unequivocally set by nuclear Overhauser effect difference experiments. An X-ray structure of compound 1e is also reported, showing a strong intramolecular NH.N hydrogen bond. Ab initio calculations (HF/3-21G and HF/3-21+G) have been carried out on several representative examples (1e, 1p, and 1l) in an attempt to support and provide the correct geometry of these derivatives. Structural considerations among the possible isomers of compounds 1 are discussed. From these studies it was concluded that the theoretical calculations agree with the experimental results. In addition, a very simple one-pot procedure for the preparation of masked N-substituted alpha-alkylated beta-enamino acid derivatives 2 from 6, 7, and different alkyl halides (R(3)Y) is described.     

Protection of the Hydroxyphosphinyl Function of Phosphinic Dipeptides by Adamantyl. Application to the Solid-Phase Synthesis of Phosphinic Peptides

To develop solid-phase synthesis of phosphinic peptides, different FmocXaaPsi{PO(OAd)CH(2)}XaaOH building blocks have been prepared, where Fmoc is (fluorenylmethoxy)carbonyl. In this respect, the protection of the hydroxyphosphinyl function in these phosphinic dipeptides by the adamantyl group turns out to be convenient. The phosphinic adamantyl esters are completely stable in basic conditions and can be removed under relatively mild acidic conditions. Using these building blocks, despite the bulkiness of the adamantyl group, no particular problem of coupling was observed during the solid-phase synthesis of phosphinic peptides by the Fmoc strategy. The developed methodology is of particular interest to facilitate the development of potent inhibitors of zinc-metalloproteases.