How to Play M 13?

A puzzle called “M ₁₃” J. H. Conway has described recently is explained. We report an implementation of the puzzle in the programming language Java. The program allows the human user to “play M ₁₃” interactively (and to cheat by solving it automatically). The program is an example on how to bring to life a nice piece of discrete mathematics. In this sense it presents not only a didactical way of seeing “mathematics at work”, but also displays the stabilizer chain method developed by C. Sims to solve group theoretic puzzles, the most famous of which being Rubik's cube.     

Structural studies on alkylisocyanate polymers by thermal degradation tandem mass spectrometry

Homopolymers and copolymers of alkylisocyanates having n-hexyl, 2,6-dimethylheptyl, 3,7-dimethyloctyl, and (2,2-dimethyl-1,3-dioxolan-4-yl)methyl substituents underwent thermal degradation in the course of desorption electron ionization to yield trimers and monomers that were characterized in situ by tandem mass spectrometry. The trimers were trisubstituted cyanuric acids, the protonated molecules displaying a characteristic series of alkene eliminations on collision-induced dissociation to yield protonated cyanuric acid, m/ z 130. Confirmation of the identity of the pyrolysates was obtained by using two types of MS3 experiments: the reaction intermediate scan and the two-dimensional familial scan. The ion chemistry of the trimers and of the protonated monomer, the alkylisocyanate, was elucidated. Among the many interesting fragmentation processes undergone by the ionized trimers were a and 3 C-C bond cleavages and charge-remote fragmentations, which provided information on branching in the alkyl substituent. The dioxolane-containing substituent showed unique ion chemistry. The monomer distribution in the copolymers was deduced from the abundances of the various protonated trimers. The distribution was found to be random in all copolymers except that containing the dioxolane substituent.     

Investigations of praseodymium oxide electrodes in lithium concentration cells

Praseodymium oxide is one of the group of binary oxides that show a wide range of stoichiometry, but in which the deviation from the ideal stoichiometry is not continuous, as is expected from the classical model of randomly distributed isolated point defects. Instead, there is a number of phases of unique structures and stoichiometries with no appreciable concentrations of point defects, and thus essentially no compositional width. These structures are closely related to each other, and to a simple parent structure. In the case of praseodymium oxide this is the fluorite structure. Transitions between these phases upon heating and cooling, as well as upon changes in the oxygen activity of the environment, occur very rapidly, indicating unusually high oxygen mobility. The equilibrium phase at ambient temperature in air is not PrO₂, but Pr₆O₁₁. These phases have been shown to have cation sublattices essentially identical to that of fluorite-type PrO₂. The stoichiometric changes are accommodated by the presence of topotactically ordered arrangements of large localized concentrations of oxide ion vacancies which divide the structure up into microdomains. Experiments have shown that Pr₆O₁₁ can absorb significant amounts of water, even at ambient temperature, apparently by the electrically neutral mechanism proposed by Stotz and Wagner. This involves the absorption of oxide ions at sites where there were previously oxide ion vacancies. Protons are also absorbed, and reside in interstitial sites. Consistent with the high mobility of oxygen in this structure, this water is irreversibly desorbed at relatively low temperatures. Electrochemical experiments will be reported that have been undertaken to investigate the insertion behavior of lithium into these unusual structures, both with and without the initial presence of protons as the result of the absorption of water. Paper presented at the 6th Euroconference on Solid State Ionics, Cetraro, Calabria, Italy, Sept. 12–19, 1999.     

Comparing distance metrics for rotation using the k‐nearest neighbors algorithm for entropy estimation

Distance metrics facilitate a number of methods for statistical analysis. For statistical mechanical applications, it is useful to be able to compute the distance between two different orientations of a molecule. However, a number of distance metrics for rotation have been employed, and in this study, we consider different distance metrics and their utility in entropy estimation using the k‐nearest neighbors (KNN) algorithm. This approach shows a number of advantages over entropy estimation using a histogram method, and the different approaches are assessed using uniform randomly generated data, biased randomly generated data, and data from a molecular dynamics (MD) simulation of bulk water. The results identify quaternion metrics as superior to a metric based on the Euler angles. However, it is demonstrated that samples from MD simulation must be independent for effective use of the KNN algorithm and this finding impacts any application to time series data. © 2013 Wiley Periodicals, Inc.     

MC-Smooth: A mass-conserving,smooth vorticity–velocity formulation for multi-dimensional flows

A novel vorticity–velocity formulation of the Navier–Stokes equations – the Mass-Conserving, Smooth (MC-Smooth) vorticity–velocity formulation – is developed in this work. The governing equations of the MC-Smooth formulation include a new second-order Poisson-like elliptic velocity equation, along with the vorticity transport equation, the energy conservation equation, and N_spec species mass balance equations. In this study, the MC-Smooth formulation is compared to two pre-existing vorticity–velocity formulations by applying each formulation to confined and unconfined axisymmetric laminar diffusion flame problems. For both applications, very good to excellent agreement for the simulation results of the three formulations has been obtained. The MC-Smooth formulation requires the least CPU time and can overcome the limitations of the other two pre-existing vorticity–velocity formulations by ensuring mass conservation and solution smoothness over a broader range of flow conditions. In addition to these benefits, other important features of the MC-Smooth formulation include: (1) it does not require the use of a staggered grid, and (2) it does not require excessive grid refinement to ensure mass conservation. The MC-Smooth formulation is a computationally attractive approach that can effectively extend the applicability of the vorticity–velocity formulation.     

Mixed-conductors with either cation or anion insertion

It has been known for some 25 years that charge can be stored in some important battery electrodes by the insertion of ionic species from the electrolyte. Insertion reactions play an especially important role in current versions of lithium batteries, where lithium cations are typically the inserted species in both electrodes. Hydrogen cations (protons) are the guest species during the operation of both the Ni(OH)₂/NiOOH electrode and the “MnO₂” electrode in aqueous systems. Whereas most attention has been given to materials in which the guest species are cations, it is also possible to have anion insertion into some crystal structures. Materials in which the structure can accommodate either cations or anions are especially interesting. The hexacyanometallates, with variations of the cubic ReO₃ type structure with rather large intercell windows, represent an interesting example. They can accommodate a wide variety of guest ions of both charges. Cations can be inserted into the structure at relatively low potentials, and anions can be inserted at more positive potentials. A number of interesting features and properties of this family of materials are discussed. Paper presented at the 3rd Euroconference on Solid State Ionics, Teulada, Sardinia, Italy, Sept. 15–22, 1996     

One-Dimensional Models for Atoms in Strong Magnetic Fields,II: Anti-Symmetry in the Landau Levels

Electrons in strong magnetic fields can be described by one-dimensional models in which the Coulomb potential and interactions are replaced by regularizations associated with the lowest Landau band. For a large class of models of this type, we show that the maximum number of electrons that can be bound is less than aZ+Zf(Z). The function f(Z) represents a small non-linear growth which reduces to A _p Z(logZ)²when the magnetic field B=O(Z ^p ) grows polynomially with the nuclear charge Z. In contrast to earlier work, the models considered here include those arising from realistic cases in which the full trial wave function for N-electrons is the product of an N-electron trial function in one-dimension and an antisymmetric product of states in the lowest Landau level.