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.     

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.     

N-labelling studies of the reaction of 2,6-di-t-butyl-4-methyl-4-nitrocyclohexa-2,5-dienone with nitrogen dioxide; A revised mechanism

Reaction of the ¹⁵N-4-nitrodienone (1) with excess nitrogen dioxide in benzene for 16 h gives the 4,5,6-trinitro ketone (2), labelled (¹⁵NO₂) at C6 but not at C4, and recovered 4-nitrodienone (1) in which extensive loss of ¹⁵NO₂ has occurred. The mechanistic implications of these and related results are discussed.     

Multiple nitrene insertions into metal-sulfur bonds of dithiocarbamate complexes: synthesis of sulfido-amido and zwitterionic tetraamido complexes

The iodine(III) reagent, PhI[double bond, length as m-dash]NTs, acts as a source of the nitrene fragment NTs, which undergoes facile insertion into the metal-sulfur bonds of a range of dithiocarbamate complexes. Addition of two equivalents of PhI=NTs to [M(S(2)CNR2)2] affords sulfido-amido complexes [M{SC(NR2)SNTs}2](M=Ni, Cu), which insert two further nitrene fragments to afford zwitterionic tetraamido complexes [M{TsNSC(NR2)SNTs}2](M=Co, Ni, Cu). Crystallographic studies have been carried out on both types of complex allowing possible resonance hydrids of the new ligand types to be assessed.     

High-performance liquid chromatographic analysis of free long chain fatty acids produced during lipolysis by anaerobic digestor sludge

The development of sample extraction techniques in conjunction with application of a modified version of an established HPLC technique allowed the rapid estimation of concentrations of long chain fatty acids (LCFA) produced by the lipolytic activity of bacteria in anaerobic digestor sludges or experiments using the sludges as inocula. It was established that free long chain fatty acids will preferentially partition into certain phases in the sludge or experimental cultures. These data, and application of the developed techniques for monitoring of LCFA, may lead to the avoidance of toxicity and failure of full-scale anaerobic digestors treating high lipid content waste in the future.     

Structure and decompression melting of a novel, high-pressure nanoconfined 2-D ice

Molecular dynamics (MD) simulations of water confined in nanospaces between layers of talc (system composition Mg(3)Si(4)O(10)(OH)(2) + 2H(2)O) at 300 K and pressures of approximately 0.45 GPa show the presence of a novel 2-D ice structure, and the simulation results at lower pressures provide insight into the mechanisms of its decompression melting. Talc is hydrophobic at ambient pressure and temperature, but weak hydrogen bonding between the talc surface and the water molecules plays an important role in stabilizing the hydrated structure at high pressure. The simulation results suggest that experimentally accessible elevated pressures may cause formation of a wide range of previously unknown water structures in nanoconfinement. In the talc 2-D ice, each water molecule is coordinated by six O(b) atoms of one basal siloxane sheet and three water molecules. The water molecules are arranged in a buckled hexagonal array in the a-b crystallographic plane with two sublayers along [001]. Each H(2)O molecule has four H-bonds, accepting one from the talc OH group and one from another water molecule and donating one to an O(b) and one to another water molecule. In plan view, the molecules are arranged in six-member rings reflecting the substrate talc structure. Decompression melting occurs by migration of water molecules to interstitial sites in the centers of six-member rings and eventual formation of separate empty and water-filled regions.