Proof of the Double Bubble Conjecture in R^n

The least-area hypersurface enclosing and separating two given volumes in R^n is the standard double bubble.Comment: 20 pages, 22 figure

Disorder-induced phonon self-energy of semiconductors with binary isotopic composition

Self-energy effects of Raman phonons in isotopically disordered semiconductors are deduced by perturbation theory and compared to experimental data. In contrast to the acoustic frequency region, higher-order terms contribute significantly to the self-energy at optical phonon frequencies. The asymmetric dependence of the self-energy of a binary isotope system $m_{1-x} M_x$ on the concentration of the heavier isotope mass x can be explained by taking into account second- and third-order perturbation terms. For elemental semiconductors, the maximum of the self-energy occurs at concentrations with $0.5<x<0.7$, depending on the strength of the third-order term. Reasonable approximations are imposed that allow us to derive explicit expressions for the ratio of successive perturbation terms of the real and the imaginary part of the self-energy. This basic theoretical approach is compatible with Raman spectroscopic results on diamond and silicon, with calculations based on the coherent potential approximation, and with theoretical results obtained using {\it ab initio} electronic theory. The extension of the formalism to binary compounds, by taking into account the eigenvectors at the individual sublattices, is straightforward. In this manner, we interpret recent experimental results on the disorder-induced broadening of the TO (folded) modes of SiC with a $^{13}{\rm C}$-enriched carbon sublattice. \cite{Rohmfeld00,Rohmfeld01}Comment: 29 pages, 9 figures, 2 tables, submitted to PR

The Computational Complexity of Knot and Link Problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur

A systems engineering framework for the design of bioprocess operator training simulators

Operator training simulators (OTS) are widely used in several industries including chemical processing, oil and gas, medicine, aircraft and nuclear facilities. However, developing a biorefinery OTS is a complex engineering design activity that requires a structured technique. This paper presents a structured methodology that applies design frameworks from other disciplines and a user-centred approach for biorefinery OTS design. These include the definition of end user requirements (operator training needs), and the analysis of these requirements using Quality Function Deployment (QFD). Furthermore, an algorithm for bioprocess optimisation and automatic adjustment of operating parameters is developed for integration into the OTS. This algorithm is based on the Nelder-Mead simplex method for multi-dimensional function minimisation. Identified user requirements were categorized into primary, secondary and tertiary training needs, with increasing levels of detail from primary to tertiary needs. The relationships between identified operator training needs and OTS technical and functional specifications were investigated, and a priority rating assigned to the most important OTS specifications. Identified OTS specifications were evaluated for robustness to ensure that important features were not omitted from the final design

Graphene formed on SiC under various environments: Comparison of Si-face and C-face

The morphology of graphene on SiC {0001} surfaces formed in various environments including ultra-high vacuum, 1 atm of argon, and 10^-6 to 10^-4 Torr of disilane is studied by atomic force microscopy, low-energy electron microscopy, and Raman spectroscopy. The graphene is formed by heating the surface to 1100 - 1600 C, which causes preferential sublimation of the Si atoms. The argon atmosphere or the background of disilane decreases the sublimation rate so that a higher graphitization temperature is required, thus improving the morphology of the films. For the (0001) surface, large areas of monolayer-thick graphene are formed in this way, with the size of these areas depending on the miscut of the sample. Results on the (000-1) surface are more complex. This surface graphitizes at a lower temperature than for the (0001) surface and consequently the growth is more three-dimensional. In an atmosphere of argon the morphology becomes even worse, with the surface displaying markedly inhomogeneous nucleation, an effect attributed to unintentional oxidation of the surface during graphitization. Use of a disilane environment for the (000-1) surface is found to produce improved morphology, with relatively large areas of monolayer-thick graphene.Comment: 22 pages, 11 figures, Proceedings of STEG-2 Conference; eliminated Figs. 4 and 7 from version 1, for brevity, and added Refs. 18, 29, 30, 31 together with associated discussio