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.     

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.     

Zur Kenntnis der Zustandsdiagramme SECl3/MCl2 (SE = La,Sm, Gd,Yb; M = Sr,Ba)

On Phase Diagrams of Rare Earth Trichlorides/MCl₂ (Rare Earth = La, Sm, Gd, yb; M = Sr, Ba) The systems LaCl₃(SmCl₃, GdCl₃, YbCl₃)? SrCl₂(BaCl₂) were determined by difference thermal analysis. The phase diagrams contain compounds of the formula BaLnCl₅, M₂LnCl₇, Ba₃LnCl₉ and probably Sr₄LnCl₁₁, which decompose peritectoidally resp. peritectically. Sr₂SmCl₇ and Sr₂GdCl₇ are isotypic, and as indicated by the very similar reflection patterns of the Guinier photographs related to the structure of the compound Sm₃Cl₇. The phase diagrams are comparable to the systems LnCl₃? LnCl₂ with similar radii quotients of the cations. The enthalpies of mixing of the systems LaCl₃(GdCl₃, YbCl₃)? SrCl₂ were measured calorimetrically. The values are exothermic, the minima were found at approximately 65 mol-% SrCl₂.     

A Convenient Preparation of Malonoylperoxides

Cyclic Malonoyl peroxides, first reported in 1971 (1) are an important class of compounds for the study of α-lactones (2). These peroxides were prepared by treating a substituted malonic acid with 98% hydrogen peroxide in methanesulfonic acid. A serious disadvantage of this method is the use and the storage of concentrated hydrogen peroxide as well as the remote control device recommended (3) for the operation of this explosive reagent.     

Separation of Aliphatic and Aromatic Carboxylic Acids by Conventional and Ultra High-Performance Ion-Exclusion Chromatography

An ion-exclusion chromatography (IELC) comparison between a conventional ion-exchange column and an ultra high-performance liquid chromatography (UHPLC) dynamically surfactant modified C18 column for the separation of an aliphatic carboxylic acid and two aromatic carboxylic acids is presented. Professional software is used to optimize the conventional IELC separation conditions for acetylsalicylic acid and the hydrolysis products: salicylic acid and acetic acid. Four different variables are simultaneously optimized including H₂SO₄ concentration, pH, flow rate, and sample injection volume. Thirty different runs are suggested by the software. The resolutions and the time of each run are calculated and feed back to the software to predict the optimum conditions. Derringer’s desirability functions are used to evaluate the test conditions and those with the highest desirability value are utilized to separate acetylsalicylic acid, salicylic acid, and acetic acid. These conditions include using a 0.35 mM H₂SO₄ (pH 3.93) eluent at a flow rate of 1 mL min^?1 and an injection volume of 72 μL. To decrease the run time and improve the performance, a UHPLC C18 column is used after dynamic modification with sodium dodecyl sulfate. Using pure water as a mobile phase, a shorter analysis time and better resolution are achieved. In addition, the elution order is different from the IELC method which indicates the contribution of the reversed-phase mode to the separation mechanism.     

Minimum detectable activity, systematic uncertainties, and the ISO 11929 standard

The Currie formulation for minimum detectable activity (MDA) has served for decades as the standard method for estimating radiological detection limits-it is simple and statistically defendable. It does, however, lack a means to account for the effects of systematic uncertainties. In recent years we have seen various efforts to incorporate systematic uncertainties into an MDA framework. Perhaps most notable of these is the recent ISO standard 11929 for the determination of characteristic limits in ionizing radiation measurements. This standard brings a Bayesian perspective to the problem of characteristic limits in radiation measurements that are in many ways both welcome and long overdue. In this paper, however, we note some apparent drawbacks to the ISO 11929 approach. Namely, for small values of the systematic uncertainty the correction it makes to the Currie MDA is negligible, while for large systematic uncertainties the calculated MDA values can become infinite. In between these two extremes, the user has little basis for evaluating the reliability of the result. To address these issues, we consider the problem from a new approach, developing a straightforward phenomenological statistical model of the MDA that treats systematic uncertainties explicitly. We compare predictions from our model with results of the ISO 11929 formulation as well as the traditional Currie approach. Finally, some recommendations for alternative handling of the MDA in the face of significant systematic uncertainties are presented.