Restructuring ordered binary trees

We consider the problem of restructuring an ordered binary tree T, preserving the in-order sequence of its nodes, so as to reduce its height to some target value h. Such a restructuring necessarily involves the downward displacement of some of the nodes of T. Our results, focusing both on the maximum displacement over all nodes and on the maximum displacement over leaves only, provide (i) an explicit tradeoff between the worst-case displacement and the height restriction (including a family of trees that exhibit the worst-case displacements) and (ii) efficient algorithms to achieve height-restricted restructuring while minimizing the maximum node displacement.     

Optimal Algorithms for Probabilistic Solitude Detection on Anonymous Rings

Probabilistic algorithms are developed for a basic problem in distributed computation, assuming anonymous, asynchronous, unidirectional rings of processors. The problem, known as Solitude Detection, requires that a nonempty subset of the processors, calledcontenders, determine whether or not there is exactly one contender. Monte Carlo algorithms are developed that err with probability bounded by a specified parameter and exhibit either message or processor termination. The algorithms transmit an optimal expected number of bits, to within a constant factor. Their bit complexities display a surprisingly rich dependence on the kind of termination exhibited and on the processors' knowledge of the size of the ring. Two probabilistic tools are isolated and then combined in various ways to achieve all our algorithms.     

Subregular Representations of {\mathfrak{s}}{\mathfrak{l}}_n and Simple Singularities of Type A n−1

Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type A.     

Establishing order in planar subdivisions

A planar subdivision is the partition of the plane induced by an embedded planar graph. A representation of such a subdivision isordered if, for each vertexv of the associated graphG, the (say) clockwise sequence of edges in the embedding ofG incident withv appears explicitly.The worst-case complexity of establishing order in a planar subdivision, i.e., converting an unordered representation into an ordered one, is shown to be (n + log (G)), wheren is the size (number of vertices) of the underlying graphG and (G) is (essentially) the number of topologically distinct embeddings ofG in the plane.This work was supported by the National Science and Engineering Research Council of Canada under Grant A3583. A preliminary version of this paper appeared in theProceedings of the Third Annual ACM Symposium on Computational Geometry.     

Enolization of acetone in superheated water detected via radical formation

Muoniated free radicals have been detected in muon-irradiated aqueous solutions of acetone at high temperatures and pressures. At temperatures below 250 degrees C, the radical product is consistent with muonium addition to the keto form of acetone. However, at higher temperatures, a different radical was detected, which is attributed to muonium addition to the enol form. Muon hyperfine coupling constants have been determined for both radicals over a wide range of temperatures, significantly extending the range of conditions under which these radicals and the keto-enol equilibrium have been studied.     

Reactions of N-heterocyclic carbenes (NHCs) with one-electron oxidants: possible formation of a carbene cation radical

One-electron oxidation of N-heterocyclic carbenes (NHCs) has been carried out using oxidising agents such as tetracyanoethylene (TCNE) and ferrocenium [Cp(2)Fe](+); the formation of carbene radical cations is postulated.     

The structure of malabricol

A single crystal X-ray analysis has confirmed the constitution and established the stereochemistry of the triterpene, malabaricol.     

Synthesis of the ABC ring system of manzamine A

A synthesis of the core ABC ring system of the manzamine alkaloids is described, starting from arecoline. The key steps involve a Claisen rearrangement to set up a 4-substituted-3-methylenepiperidine and a stereoselective azomethine ylide dipolar cycloaddition reaction. Condensation of the aldehyde 6 and sarcosine ethyl ester hydrochloride salt gives an intermediate azomethine ylide, which undergoes an intramolecular cycloaddition reaction to set up two new rings and three new chiral centers stereoselectively. The aldehyde 6 was not a suitable substrate for related azomethine ylide cycloaddition reactions with other amines. However, the related dimethyl acetal 26 could be condensed with a variety of amines to give the desired tricyclic products. The cycloaddition reaction with N-methyl or N-allyl glycine ethyl ester gave almost exclusively the exo adduct, whereas cycloaddition with glycine ethyl ester gave the endo adduct.