The 'algebra as object' analogy: a view from school

Treating algebraic symbols as objects (eg. “‘a’ means ‘apple’”) is a means of introducing elementary simplification of algebra, but causes problems further on. This current school-based research included an examination of texts still in use in the mathematics department, and interviews with mathematics teachers, year 7 pupils and then year 10 pupils asking them how they would explain, “3a + 2a = 5a” to year 7 pupils. Results included the notion that the ‘algebra as object’ analogy can be found in textbooks in current usage, including those recently published. Teachers knew that they were not ‘supposed’ to use the analogy but not always clear why, nevertheless stating methods of teaching consistent with an‘algebra as object’ approach. Year 7 pupils did not explicitly refer to ‘algebra as object’, although some of their responses could be so interpreted. In the main, year 10 pupils used ‘algebra as object’ to explain simplification of algebra, with some complicated attempts to get round the limitations. Further research would look to establish whether the appearance of ‘algebra as object’ in pupils’ thinking between year 7 and 10 is consistent and, if so, where it arises. Implications also are for on-going teacher training with alternatives to introducing such simplification.

A genetic algorithm for the design of a fuzzy controller for active queue management

Active queue management (AQM) policies are those policies of router queue management that allow for the detection of network congestion, the notification of such occurrences to the hosts on the network borders, and the adoption of a suitable control policy. This paper proposes the adoption of a fuzzy proportional integral (FPI) controller as an active queue manager for Internet routers. The analytical design of the proposed FPI controller is carried out in analogy with a proportional integral (PI) controller, which recently has been proposed for AQM. A genetic algorithm is proposed for tuning of the FPI controller parameters with respect to optimal disturbance rejection. In the paper the FPI controller design metodology is described and the results of the comparison with random early detection (RED), tail drop, and PI controller are presented.

Approximations to the scattering of water waves by steep topography

A new method is developed for approximating the scattering of linear surface gravity waves on water of varying quiescent depth in two dimensions. A conformal mapping of the fluid domain onto a uniform rectangular strip transforms steep and discontinuous bed profiles into relatively slowly varying, smooth functions in the transformed free-surface condition. By analogy with the mild-slope approach used extensively in unmapped domains, an approximate solution of the transformed problem is sought in the form of a modulated propagating wave which is determined by solving a second-order ordinary differential equation. This can be achieved numerically, but an analytic solution in the form of a rapidly convergent infinite series is also derived and provides simple explicit formulae for the scattered wave amplitudes. Small-amplitude and slow variations in the bedform that are excluded from the mapping procedure are incorporated in the approximation by a straightforward extension of the theory. The error incurred in using the method is established by means of a rigorous numerical investigation and it is found that remarkably accurate estimates of the scattered wave amplitudes are given for a wide range of bedforms and frequencies.

High performance blocking filters for the region 1µm to 20µm

A set of filters based on the sequence of semiconductor edges is described which offers continuity of short-wave infrared blocking. The rejection throughout the stop region is greater than 103 for each filter and the transmission better than 70% through one octave with a square cutoff. The cutoff points are located at intervals of about two-thirds of an octave. Filters at 2.6 ,µm, 5.5 µm, and 12 µm which use a low-passing multilayer in combination with a semiconductor absorption edge are described in detail. The design of multilayers for optimum performance is discussed by analogy with the synthesis of electric circuit filters.

Fidelity in interpretation: Lord Hoffmann and the adventure of the empty house

This paper explores Dworkin's 'law as a chain novel' analogy and considers the recent work of Dworkin and MacCormick through close scrutiny of two recent judgments of Lord Hoffmann, in Barlow Clowes v Eurotrust International [2005] UKPC 37 and Barker v Corus [2006] UKHL 20. The aim is to examine Dworkin's theory in the context of recent English private law decisions and determine whether Lord Hoffmann's approach to interpretation is consistent with that of Dworkin (as his Lordship has contended in the past). It is argued that Lord Hoffmann's treatment of recent decisions on which he himself sat raises significant questions regarding fidelity, coherence and the institutional structure of the House of Lords.

The elliptic sine-Gordon equation in a half plane

We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case. We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation. We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions.

Sequence-selective assembly of tweezer-molecules on linear templates enables frameshift-reading of sequence information

Monomer-sequence information in synthetic copolyimides can be recognised by tweezer-type molecules binding to adjacent triplet-sequences on the polymer chains. In the present paper different tweezer-molecules are found to have different sequence-selectivities, as demonstrated in solution by 1H NMR spectroscopy and in the solid state by single crystal X-ray analyses of tweezer-complexes with linear and macrocyclic oligo-imides. This work provides clear-cut confirmation of polyimide chain-folding and adjacent-tweezer-binding. It also reveals a new and entirely unexpected mechanism for sequence-recognition which, by analogy with a related process in biomolecular information processing, may be termed "frameshift-reading". The ability of one particular tweezer-molecule to detect, with exceptionally high sensitivity, long-range sequence-information in chain-folding aromatic copolyimides, is readily explained by this novel process.

Systolic random number generation for genetic algorithms

A parallel hardware random number generator for use with a VLSI genetic algorithm processing device is proposed. The design uses an systolic array of mixed congruential random number generators. The generators are constantly reseeded with the outputs of the proceeding generators to avoid significant biasing of the randomness of the array which would result in longer times for the algorithm to converge to a solution. 1 Introduction In recent years there has been a growing interest in developing hardware genetic algorithm devices [1, 2, 3]. A genetic algorithm (GA) is a stochastic search and optimization technique which attempts to capture the power of natural selection by evolving a population of candidate solutions by a process of selection and reproduction [4]. In keeping with the evolutionary analogy, the solutions are called chromosomes with each chromosome containing a number of genes. Chromosomes are commonly simple binary strings, the bits being the genes.

Mathematical modelling of decision making processes in construction projects

Many different individuals, who have their own expertise and criteria for decision making, are involved in making decisions on construction projects. Decision-making processes are thus significantly affected by communication, in which a dynamic performance of human intentions leads to unpredictable outcomes. In order to theorise the decision making processes including communication, it is argued here that the decision making processes resemble evolutionary dynamics in terms of both selection and mutation, which can be expressed by the replicator-mutator equation. To support this argument, a mathematical model of decision making has been made from an analogy with evolutionary dynamics, in which there are three variables: initial support rate, business hierarchy, and power of persuasion. On the other hand, a survey of patterns in decision making in construction projects has also been performed through self-administered mail questionnaire to construction practitioners. Consequently, comparison between the numerical analysis of mathematical model and the statistical analysis of empirical data has shown a significant potential of the replicator-mutator equation as a tool to study dynamic properties of intentions in communication.

Identification of a novel mammalian post-translational modification, phosphocholine, on placental secretory polypeptides

Placental neurokinin B appears to be post-translationally modified by phosphocholine (PC) attached to the aspartyl side chain at residue 4 of the mature peptide. Corticotrophin releasing factor (CRF) was found to be expressed by the rat placenta with the main secreted forms being phosphocholinated proCRF+/- one or two polysaccharide moieties. A combination of high-pressure liquid chromatography (HPLC) and two-site immunometric analysis suggested that PC was also attached to the placental precursors of adrenocorticotrophin, hemokinin, activin and follistatin. However, the fully processed forms of rat placental activin and CRF were free of PC. Formerly, the parasitic filarial nematodes have used PC as a post-translational modification, attached via the polysaccharicle moiety of certain secretory glycoproteins to attenuate the host immune system allowing parasite survival, but it is the PC group itself which endows the carrier with the biological activity. The fact that treatment of proCRF peptides with phospholipase C but not endoglycosidase destroyed PC immunoreactivity suggested a simpler mode of attachment of PC to placental peptides than that used by nematodes. Thus, it is possible that by analogy the placenta uses its secreted phosphocholinated hormones to modulate the mother's immune system and help protect the placenta from rejection.

Aziridine synthesis via nucleophilic attack of carbene equivalents on imines: the aza-Darzens reaction

The development of new methods for the efficient synthesis of aziridines has been of considerable interest to researchers for more than 60 years, but no single method has yet emerged as uniformly applicable, especially for asymmetric synthesis of chiral aziridines. One method which has been intensely examined and expanded of late involves the nucleophilic addition to imines by anions bearing a-leaving groups; by analogy with the glycidate epoxide synthesis, these processes are often described as "aza-Darzens reactions". This Microreview gives a summary of the area, with a focus on contemporary developments. ((C) Wiley-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2009)