Some effective methods for unconstrained optimization based on the solution of systems of ordinary differential equations

In this paper, we review briefly some methods for minimizing a functionF(x), which proceed by follwoing the solution curve of a system of ordinary differential equations. Such methods have often been thought to be unacceptably expensive; but we show, by means of extensive numerical tests, using a variety of algorithms, that the ODE approach can in fact be implemented in such a way as to be more than competitive with currently available conventional techniques.This work was supported by a SERC research studentship for the first author. Both authors are indebted to Dr. J. J. McKeown and Dr. K. D. Patel of SCICON Ltd, the collaborating establishment, for their advice and encouragement.     

Kinetic model for hydrocarbon-assisted particulate boron combustion

A kinetic model is presented to describe the high temperature (1800 K < T < 3000 K) surface oxidation of particulate boron in a hydrocarbon combustion environment. The model includes a homogeneous gas-phase B/O/H/C oxidation mechanism consisting of 19 chemical species and 58 forward and reverse elementary reactions, multi-component gas-phase diffusion, and a heterogeneous surface oxidation mechanism consisting of ‘elementary’ adsorption and desorption reaction steps. Thermochemical and kinetic parameters for the surface reactions are estimated from available experimental data and/or elementary transition state arguments. The kinetic processes are treated using a generalized kinetics code, with embedded sensitivity analysis, for the combustion of a one-dimensional (particle radius), spherical particle. Model results are presented for the oxidation of a 200 μm boron particle in a JP-4/air mixture at ambient temperatures of 1400 K and 2000 K. These results include temperature and gas-phase species profiles as a function of radial distance and particle burning rates. © 1994 John Wiley & Sons, Inc.     

Determination of a double Lie groupoid by its core diagram

In a double groupoid S, we show that there is a canonical groupoid structure on the set of those squares of S for which the two source edges are identities; we call this the core groupoid of S. The target maps from the core groupoid to the groupoids of horizontal and vertical edges of S are now base-preserving morphisms whose kernels commute, and we call the diagram consisting of the core groupoid and these two morphisms the core diagram of S. If S is a double Lie groupoid, and each groupoid structure on S satisfies a natural double form of local triviality, we show that the core diagram determines S and, conversely, that a locally trivial double Lie groupoid may be constructed from an abstractly given core diagram satisfying some natural additional conditions.

In the algebraic case, the corresponding result includes the known equivalences between crossed modules, special double groupoids with special connection (Brown and Spencer), and cat¹-groups (Loday). These cases correspond to core diagrams for which both target morphisms are (compatibly) split surjections.     

Purification of domoic acid from toxic blue mussels (Mytilus edulis) and phytoplankton.

Domoic acid was the primary neurotoxin in blue mussel (Mytilus edulis) that caused poisoning in humans. Further research showed that the algae, Nitzschia pungens, was the source of this toxin. In this study, a method for the extraction and purification of domoic acid from contaminated mussels and phytoplankton was developed. Domoic acid was extracted from these sources by treatment with a mixture of chloroform and methanol (1:2, v/v). The resulting extract was subjected to ultrafiltration through a PM1 Millipore filter, followed by repeated high-performance liquid chromatography on a reversed-phase column. The purity and yield of domoic acid prepared by this method are compared with two previously described methods of extraction. The current method is relatively simple, rapid, and results in improved recovery with comparable purity of domoic acid.     

Determination of colloidally-associated polycyclic aromatic hydrocarbons (PAHs) in fresh water using C18 solid phase extraction disks

A new application of reversed-phase octadecyl (C₁₈) solid phase extraction disks has been developed to separate the colloidally-associated polycyclic aromatic hydrocarbons (PAHs) from those that were truly dissolved in the samples of fresh water. A correction for the retention of small amounts of colloidal material on the C₁₈ disks was required, which would have otherwise lead to minor underestimates in the degree of partitioning between the two phases. Using the humic substance Aldrich Humic Acid (AHA) as a model colloid and the 16 PAHs on the US Enrivonmental Protection Agency priority pollutant list, the partitioning coefficients of the PAHs between the colloidal and truly dissolved phases were shown to be proportional to the hydrophobicity of the PAHs, as measured by their octanol water partition coefficients (K_ow). The values for the partition coefficients obtained (cK_doc′) were similar to those previously reported in the literature using alternative methods, confirming that the technique was producing acceptable results. The technique allows the in situ partitioning of PAHs between the truly dissolved and colloidal phases in fresh water bodies to be determined. It will provide an invaluable cross-check of the laboratory-based methods which often require substantial manipulation of the sample and potentially alter the partitioning between the phases.     

The six semifield planes associated with a semifield flock

In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array (a_ijk) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions (f,g) where f and g are functions from GF(q) to GF(q) with the property that f and g are linear over some subfield and g(x)²+4xf(x) is a non-square for all x∈GF(q)∗, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4,q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f,g) are of linear type or of Dickson-Kantor-Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations.     

Asymmetric synthesis of 6-(2′,3′,4′,5′,6′-pentafluorophenyl)-δ-lactones via “allyl”boranes: application for the synthesis of fluorinated analog of key pharmacophore of statin drugs

Asymmetric “allyl”boration of pentafluorobenzaldehyde with various α-pinene based “allyl”boranes provides homoallylic alcohols in high de and ee; the alcohols have been converted into δ-lactones via acryloylation, ring-closing metathesis and hydrogenation. Pentafluorophenyl analog of key pharmacophore of statin drugs has been synthesized using diastereoselective epoxidation and regioselective reduction as key steps.     

Bis[3‐(trimethyl­silyl)prop­yl] ethyl­enebisphospho­nate,H2DTMSP[EBP]

The title compound (H₂DTMSP[EBP]), C₁₄H₃₆O₆P₂Si₂, was crystallized by the slow evaporation of a solution in a 20:1 mixture of pentane and acetone. The H₂DTMSP[EBP] mole­cule lies about an inversion center. In the solid state, the mol­ecule exists in an anti configuration, with the mol­ecular backbone C—C bond located on an inversion center. The compound exists in the solid state as hydrogen‐bonded infinite sheets in the ab plane, unlike the methyl­ene analogue, which exists as hydrogen‐bonded infinite chains, demonstrating an `even–odd' effect of the length of the backbone alkyl chain.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.