Upper bounds for the diameter and height of graphs of convex polyhedra

Let Δ(d, n) be the maximum diameter of the graph of ad-dimensional polyhedronP withn-facets. It was conjectured by Hirsch in 1957 that Δ(d, n) depends linearly onn andd. However, all known upper bounds for Δ(d, n) were exponential ind. We prove a quasi-polynomial bound Δ(d, n)≤n ^{2 logd+3}. LetP be ad-dimensional polyhedron withn facets, let ϕ be a linear objective function which is bounded onP and letv be a vertex ofP. We prove that in the graph ofP there exists a monotone path leading fromv to a vertex with maximal ϕ-value whose length is at most . This research was supported in part by a BSF grant, by a GIF grant, and by ONR-N00014-91-C0026.     

Cycloaddition of methyl 2-(2,6-dichorophenyl)-2H-azirine-3-carboxylate to electron-rich 2-azadienes

tert-Butyldimethylsililoxy-2-aza-1,3-butadienes react with 2H-azirine 3 leading to Diels-Alder cycloadducts in moderate yields. The reactions are endo- and regioselective with the azirine being added by its less hindered face. There is only one product in the case of 1b, 4b. There are two isomers (4 and 5) from 1a, 1c and 1d. A different result was obtained with the diene 1e. Diene 1e formed products 4e and 8. Some of compounds 4 and 5 have been hydrolysed leading to functionalised aziridines 7. Compound 8 gave aziridine 9.     

Biosynthesis and antioxidant activity of 4NRC β-glycoside

This Letter describes glycosylation of 4-Nerolidylcatechol (4-NRC) the major secondary metabolite from Pothomorphe peltata and Pothomorphe umbellata showing remarkable antioxidant, antimalarial, anti-inflammatory, and anti-HIV activities. One step biosynthesis was catalyzed by Cunninghamella echinulata ATCC 9245 and the reaction was undertaken in PDSM medium at 27 °C, 200 rpm for 96 h. After purification by silica gel flash column chromatography the 4-NRC β-glycoside was identified by ultrahigh resolution mass spectrometry and ¹H NMR. The antioxidant activity was evaluated by differential pulse voltammetry.     

Note on generalized Wiener integrals

We establish a change of path formula for generalized Wiener integrals and develop a rotation theorem for Wiener measure with respect to Gaussian paths.     

Synthesis of (22R, 25R)-3β,26-Dihydroxy-5α-furostan-6-one #

The synthesis of a plant growth promoter furostanol which bears the characteristic functionality of teasterone on rings A and B is described.

     

Asymmetric Diels–Alder cycloadditions of d-erythrose 1,3-butadienes to achiral t-butyl 2H-azirine 3-carboxylate

Two d-erythrose 1,3-butadienes were reacted with electrophilic achiral t-butyl 2H-azirine 3-carboxylate giving cycloadducts with good yields and moderate selectivity. The isomers could be separated to give the major (R)-isomers at C-2 in approximately 50% yield in both cases. Alternatively LACASA-DA methodology was applied to one of the reactions leading to homochiral (R)- and (S)-products by changing the chiral nature of an extra chiral BINOL inductor used.     

Synthesis of the bicyclo[6.2.1]undecane ring system by a solvent-free Diels–Alder reaction

A bicyclo[6.2.1]undecane model compound of the core structure of the biologically active furanoheliangolide sesquiterpene was synthesized. This new and short route was developed by using a solvent-free Diels–Alder reaction between cyclopentadiene and 3-nitro-2-cyclohexenone, followed by simple transformations. Theoretical calculations were performed in order to understand reactivity aspects of the cycloaddition.