On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Windmill polynomials over fields of characteristic two

Chambers andSmeets [3] have designed a windmill arrangement of linear feedback shift registers (LFSRs) to generate pn-sequences overGF(2) with high speed. When the windmill hasv vanes, the associated minimal feedback polynomial (having degreen, relatively prime tov) can be taken to have the shapef ₁(x ^v )+x ⁿ f ₂(x ^–v ), where the polynomialsf ₁ andf ₂ have degree [n/v]. Their numerical evidence, whenv is divisible by 4, suggests that, surprisingly, there areno such windmill polynomials which are irreducible ifn±3 (mod 8), while about twice as many irreducible and primitive windmill polynomials as they expected occur ifn±1 (mod 8). A discussion of this behaviour is presented here with proofs. The brief explanation is that the Galois group of the underlying generic windmill polynomial overGF (4) is equal to the alternating groupA _n .     

Geometric techniques in representation theory

A discussion of results and conjectures, focussed around the extension of the modular representation theory for finite Lie-type groups to more general groups that act on building-like geometries.Based on a lecture given at the conference Groups and Geometries-Finite and Algebraic, on 26 March 1986 at Noordwijkerhout, Netherlands (a NATO Advanced Research Workshop).Partially supported by NSF Grant MCS 83-00855.     

A convenient synthesis of a selective gelatinase inhibitor as an antimetastatic agent

Compound 1, 2-(4-phenoxyphenylsulfonylmethyl)thiirane, is a potent and selective inhibitor for human gelatinases (J. Am. Chem. Soc. 2000, 122, 6799-6800), enzymes implicated in a number of diseases, including cancer. This compound is showing excellent promise in animal trials in a number of disease models. Large quantities of this compound were necessary for these studies. A convenient four-step synthetic route for compound 1 is described herein. The synthesis is amenable to scale-up to tens of grams and gives an overall yield of 57% for this important compound.     

Preparation of highly branched poly(vinyl acetate) by transfer to allylic carbonate comonomers

Copolymerization of vinyl acetate with allyl carbonates that contain isopropyl groups yields highly branched poly(vinyl acetates).     

Strong, low-energy two-photon absorption in extended amine-terminated cyano-substituted phenylenevinylene oligomers

Three quadrupolar oligophenylenevinylenes with five rings in the conjugated backbone, terminal donor groups, and various acceptors and/or donors along the backbone were synthesized and their two-photon spectroscopic properties investigated. These chromophores exhibit large two-photon absorption cross sections over a wide wavelength range and two distinct peaks, the strongest of which (deltamax > 3600 GM) is observed at 960-970 nm, a wavelength close to twice the value of the linear absorption maximum (2lambda(1)max). The findings on these chromophores are compared with those for analogous molecules with shorter conjugation length, for which the main two-photon band is at significantly shorter wavelength than 2lambda(1)max.     

N-α-Benzyloxyacetyl derivatives of (S)-4-benzyl-5,5-dimethyloxazolidin-2-one for the asymmetric synthesis of differentially protected α,β-dihydroxyaldehydes

α-Dibenzylamino- and α-benzyloxy- derivatives of N-acetyl-(S)-4-benzyl-5,5-dimethyloxazolidin-2-one readily undergo highly stereoselective boron mediated syn-aldol reactions with a range of aromatic and aliphatic aldehydes, generating the syn-aldol products in good to excellent yields as single diastereoisomers after purification. In the α-dibenzylamino series, deprotection of the functionalised aldol fragments to the corresponding α-amino-β-hydroxy methyl ester or α-amino-β-hydroxyaldehyde proved problematic, with a range of N- and O-protecting groups giving mixtures of products arising from endocyclic and exocyclic cleavage pathways. However, in the α-benzyloxy series, O-silyl protection of the aldol products, and subsequent DIBAL reduction gives stereoselectively the corresponding N-1′-hydroxyalkyloxazolidin-2-ones, which undergo base promoted fragmentation to the desired highly functionalised and differentially protected α,β-dihydroxyaldehydes in good yields and without loss of stereochemical integrity.     

Coal solubilization by enhanced enzyme activity in organic solvents

Applied Biochemistry and Biotechnology - Both oxidative and reductive enzymes can be utilized to enhance coal solubilization in aqueous and organic media. Aerobic solubilization was carried out...     

Cover Picture

The cover picture shows in the background the whole cell of a methanotrophic bacterium on which are superimposed components of methane monooxygenase (the structure of the hydroxylase component (top), one of the two four-helix bundles that house the catalytic diiron centers (left)) and a schematic diagram of the catalytic cycle by which the enzyme converts dioxygen and methane into methanol and water. More about this unusual enzyme system is reported by Lippard et al. on p. 2782 ff.