Optimal natural dualities. II: General theory

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set of finitary algebraic relations yields a duality on a class of algebras , those subsets of which yield optimal dualities are characterised. Further, the manner in which the relations in are constructed from those in is revealed in the important special case that generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on . Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.

     

Correction of the influence of baseline artefacts and electrode polarisation on dielectric spectra

The deconvolution of biological dielectric spectra can be difficult enough with artefact-free spectra but is more problematic when machine baseline artefacts and electrode polarisation are present as well. In addition, these two sources of anomalies can be responsible for significant interference with dielectric biomass measurements made using one- or two-spot frequencies. The aim of this paper is to develop mathematical models of baseline artefacts and electrode polarisation which can be used to remove these anomalies from dielectric spectra in a way that can be easily implemented on-line and in real-time on the Biomass Monitor (BM). We show that both artefacts can be successfully removed in solutions of organic and inorganic ions; in animal cell and microbial culture media; and in yeast suspensions of varying biomass. The high quality of the compensations achieved were independent of whether gold and platinum electrodes were used; the electrode geometry; electrode fouling; current density; the type of BM; and of whether electrolytic cleaning pulses had been applied. In addition, the calibration experiments required could be done off-line using a simple aqueous KCl dilution series with the calibration constants being automatically calculated by a computer without the need for user intervention. The calibration values remained valid for a minimum of 3 months for the baseline model and indefinitely for the electrode polarisation one. Importantly, application of baseline correction prior to polarisation correction allowed the latter's application to the whole conductance range of the BM. These techniques are therefore exceptionally convenient to use under practical conditions.     

Impact of molecular speciation on crystal nucleation in polymorphic systems: the conundrum of gamma glycine and molecular 'self poisoning'

The polymorphism of the simple amino acid glycine has been known for almost a century. It is also known that in aqueous solutions, at the isoelectric point (pI 5.9), the metastable alpha polymorph crystallizes, while the stable gamma form of glycine only nucleates at high and low pH. Despite the importance of understanding the process by which crystals nucleate, the solution and solid-state chemistry underlying this simple observation have never been explored. In this contribution, we have combined solution chemistry, crystallization, and crystallographic data to investigate the mechanisms by which this effect occurs. It is concluded that solution speciation and the consequent interactions between charged species and developing crystal nuclei determine the structural outcome of the crystallization process.     

Thermal models for copper-alloyed dies in pressure die casting

A recent innovation in pressure die casting is the use of copper-alloyed dies suitably protected with a thermally sprayed steel layer. The thermal response of copper-alloyed dies is dictated principally by the deposited layer, the cooling arrangement and the casting geometry. This paper is concerned with the development of efficient numerical models for the prediction of both steady-state and transient thermal behaviour of the new die designs. Die temperatures are cyclic but useful information is obtained from a steady-state model, which provides for time-averaged temperatures and energy fluxes. The modelling strategy presented in the paper involves the indirect determination of transient temperatures. A perturbation approach is adopted, where a model for the difference between transient and steady-state temperature is developed. It is shown that this approach can be utilised to determine transient temperature efficiently once steady-state information is available with the transient perturbation model only involving surfaces where a significant variation in temperature occurs.     

Dualisability versus residual character: A theorem and a counterexample

We show that a finite algebra must be inherently non-dualisable if the variety that it generates is both residually large and congruence meet-semidistributive. We also give the first example of a finite dualisable algebra that generates a variety that is residually large.     

From rectangular bands to k -primal algebras

We begin by giving a new proof that every finite rectangular band is naturally dualisable. Motivated by the dualising structure arising from this proof, we call an algebra k-primal if it is (isomorphic to) a product of k independent primal algebras. For each k \geq 2 we exhibit a strong duality between the quasi-variety generated by a k -primal algebra and the topological quasi-variety \lilcat D _k of Boolean topological k-dimensional diagonal algebras. The category \lilcat D ₂ is the category of compact, totally disconnected rectangular bands. This duality extends Hu’s duality for varieties generated by a primal algebra to the k -dimensional case. We find that Hu’s ``uniqueness principle’’ for such varieties also extends to the k -dimensional case, namely, we show that a quasi-variety is equivalent as a category to the quasi-variety generated by a k -primal algebra if and only if it is itself generated by a k -primal algebra.     

Neglected transport equations: extended Rankine–Hugoniot conditions and <Emphasis Type="Italic">J</Emphasis> -integrals for fracture

Transport equations in integral form are well established for analysis in continuum fluid dynamics but less so for solid mechanics. Four classical continuum mechanics transport equations exist, which describe the transport of mass, momentum, energy and entropy and thus describe the behaviour of density, velocity, temperature and disorder, respectively. However, one transport equation absent from the list is particularly pertinent to solid mechanics and that is a transport equation for movement, from which displacement is described. This paper introduces the fifth transport equation along with a transport equation for mechanical energy and explores some of the corollaries resulting from the existence of these equations. The general applicability of transport equations to discontinuous physics is discussed with particular focus on fracture mechanics. It is well established that bulk properties can be determined from transport equations by application of a control volume methodology. A control volume can be selected to be moving, stationary, mass tracking, part of, or enclosing the whole system domain. The flexibility of transport equations arises from their ability to tolerate discontinuities. It is insightful thus to explore the benefits derived from the displacement and mechanical energy transport equations, which are shown to be beneficial for capturing the physics of fracture arising from a displacement discontinuity. Extended forms of the Rankine–Hugoniot conditions for fracture are established along with extended forms of J -integrals.     

Methods for the efficient quantification of fruit provitamin A contents

As part of a screening program to identify micronutrient-rich banana and plantain (Musa) varieties, a simple, robust, and comparatively rapid protocol for the quantification of the provitamin A carotenoids contents of fruit pulp and peel tissues by HPLC and by spectrophotometry has been developed. Major points to note include the use lyophilisation and extensive tissue disruption procedures to ensure quantitative recoveries, and the avoidance of saponification and/or concentration steps which lead to significant losses of provitamin A carotenoids. The protocol showed excellent reproducibility between replicate extractions, without the need for an internal standard. Application of the methodology demonstrated that Musa fruit pulp has a relatively simple provitamin A carotenoids content, quite different from the overlying peel, and that the proportions of alpha- and beta-carotene are characteristic for each genotype. The protocol was also used to profile the provitamin A carotenoids of several other fruits.     

Absolute net charge and the biological activity of oligopeptides

Sequences of human proteins are frequently prepared as synthetic oligopeptides to assess their functional ability to act as compounds modulating pathways involving the parent protein. Our objective was to analyze a set of oligopeptides, to determine if their solubility or activity correlated with features of their primary sequence, or with features of properties inferred from three-dimensional structural models derived by conformational searches. We generated a conformational database for a set of 78 oligopeptides, derived from human proteins, and correlated their 3D structures with solubility and biological assay activity (as measured by platelet activation and inhibition). Parameters of these conformers (frequency of coil, frequency of turns, the degree of packing, and the energy) did not correlate with solubility, which was instead partly predicted by two measures obtained from primary sequence analysis, that is, the hydrophobic moment and the number of charges. The platelet activity of peptides was correlated with a parameter derived from the structural modeling; this was the second virial coefficient (a measure of the tendency for a structure to autoaggregate). This could be explained by an excess among the active peptides of those which had either a large number of positive charges or in some cases a large number of negative charges, with a corresponding deficit of peptides with a mixture of negative and positive charges. We subsequently determined that a panel of 523 commercially available (and biologically active) peptides shared this elevation of absolute net charge: there were significantly lower frequencies of peptides of mixed charges compared to expectations. We conclude that the design of biologically active peptides should consider favoring those with a higher absolute net charge.     

Flat unars: the primal, the semi-primal and the dualisable

A flat unar is a semilattice of height one with an additional unary operation. We characterise the primal, semi-primal and dualisable finite flat unars. Along the way, we give a useful characterisation of primality in general and use an array of techniques from the theory of natural dualities.