The Geometry of Niggli Reduction I: The Boundary Polytopes of the Niggli Cone

Correct identification of the Bravais lattice of a crystal is an important step in structure solution. Niggli reduction is a commonly used technique. We investigate the boundary polytopes of the Niggli-reduced cone in the six-dimensional space G6 by algebraic analysis and organized random probing of regions near 1- through 8-fold boundary polytope intersections. We limit consideration of boundary polytopes to those avoiding the mathematically interesting but crystallographically impossible cases of 0 length cell edges. Combinations of boundary polytopes without a valid intersection in the closure of the Niggli cone or with an intersection that would force a cell edge to 0 or without neighboring probe points are eliminated. 216 boundary polytopes are found: 15 5-D boundary polytopes of the full G6 Niggli cone, 53 4-D boundary polytopes resulting from intersections of pairs of the 15 5-D boundary polytopes, 79 3-D boundary polytopes resulting from 2-fold, 3-fold and 4-fold intersections of the 15 5-D boundary polytopes, 55 2-D boundary polytopes resulting from 2-fold, 3-fold, 4-fold and higher intersections of the 15 5-D boundary polytopes, 14 1-D boundary polytopes resulting from 3-fold and higher intersections of the 15 5-D boundary polytopes. All primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes. All non-primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes and of the 7 special-position subspaces of the 5-D boundary polytopes. This study provides a new, simpler and arguably more intuitive basis set for the classification of lattice characters and helps to illuminate some of the complexities in Bravais lattice identification. The classification is intended to help in organizing database searches and in understanding which lattice symmetries are "close" to a given experimentally determined cell

The Geometry of Niggli Reduction II: BGAOL -- Embedding Niggli Reduction

Niggli reduction can be viewed as a series of operations in a six-dimensional space derived from the metric tensor. An implicit embedding of the space of Niggli-reduced cells in a higher dimensional space to facilitate calculation of distances between cells is described. This distance metric is used to create a program, BGAOL, for Bravais lattice determination. Results from BGAOL are compared to the results from other metric-based Bravais lattice determination algorithms

Converting three-space matrices to equivalent six-space matrices for Delone scalars in S6.

The transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as three-by-three matrices that transform three-space lattice vectors. Using those three-by-three matrices when working in the six-dimensional space of lattices represented as Selling scalars as used in Delone (Delaunay) reduction, one could transform to the three-space representation, apply the three-by-three matrices and then back-transform to the six-space representation, but it is much simpler to have the equivalent six-by-six matrices and apply them directly. The general form of the transformation from the three-space matrix to the corresponding matrix operating on Selling scalars (expressed in space S6) is derived, and the particular S6matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)

A Paraconsistent Higher Order Logic

Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.Comment: Originally in the proceedings of PCL 2002, editors Hendrik Decker, Joergen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/). Correcte

Structural and elastic characterization of Cu-implanted SiO₂ films on Si(100) substrates

Cu-implanted SiO₂ films on Si(100) have been studied and compared to unimplanted SiO₂ on Si(100) using x-ray methods, transmission electron microscopy, Rutherford backscattering, and Brillouin spectroscopy. The x-ray results indicate the preferred orientation of Cu {111} planes parallel to the Si substrate surface without any directional orientation for Cu-implanted SiO₂∕Si(100) and for Cu-implanted and annealedSiO₂∕Si(100). In the latter case, transmission electron microscopy reveals the presence of spherical nanocrystallites with an average size of ∼2.5 nm. Rutherford backscattering shows that these crystallites (and the Cu in the as-implanted film) are largely confined to depths of 0.4−1.2 μm below the film surface. Brillouin spectra contain peaks due to surface, film-guided and bulk acoustic modes. Surface (longitudinal) acoustic wave velocities for the implanted films were ∼7% lower (∼2% higher) than for unimplanted SiO₂∕Si(100). Elastic constants were estimated from the acoustic wave velocities and film densities. C₁₁ (C₄₄) for the implanted films was ∼10% higher (lower) than that for the unimplanted film. The differences in acoustic velocities and elastic moduli are ascribed to implantation-induced compaction and/or the presence of Cu in the SiO₂ film.B.J. and M.C.R. are grateful for financial support from the Australian Synchrotron Research Program, funded by the Commonwealth of Australia. M.C.R. would also like to thank the Australian Research Council for their financial support. The financial support of the Natural Sciences and Engineering Research Council of Canada NSERC is gratefully acknowledged by G.T.A. and J.S

Optimising strategy selection for the management of railway assets

In the railway industry a main concern is how to manage efficiently and effectively the railway assets under budget constraints. Optimal asset management involves decision making and selection of the best inspection, maintenance and renewal interventions for each asset along the network. This paper presents an optimisation method for supporting the decision making process. The method is based on a two-level approach. At a lower level, asset models combining degradation and intervention processes are used in order to evaluate the effects of different intervention strategies on the evolution of the asset state over time. On a system-level, a Knapsack-type optimisation model is developed to selects the optimal combination of intervention strategies to apply to all assets in the network in order to deliver the required level of performance while minimising the whole lifecycle costs

Internal electrostatic discharge hazard risk assessment to the Galileo orbiter

A worst case assessment was performed on the Command Data System (CDS) multilayer printed circuit board and an output power transformer module in the power subsystem. An estimate of the Jovian environment during the 35 hour orbit insertion was supplied by JPL and used as an input to calculate the electron transport into the Galileo components. A radiation shielding analysis computer code, CHARGE, calculated the electron transport deposition trapped in the anticipated sensitive areas of the multilayer board and transformer module. Based on these trapped charge calculations electric fields were calculated between the identified isolated areas and the spacecraft ground. The results of the assessment of electrostatic discharge (DSD) in the CDS multilayer printed circuit board indicate that the probability of ESD in the FR4 is low. The probability of ESD in the components attached to the multilayer board, however, is uncertain based on a lack of prior experimental data