Mathematics as a tool in the Life Sciences

Teaching mathematics to students in the biological sciences is often fraught with difficulty. Students often discover mathematics to be a very 'dry' subject in which it is difficult to see the motivation of learning it given its often abstract application. In this paper I advocate the use of mathematical modelling as a method for engaging students in understanding the use of mathematics in helping to solve problems in the Biological Sciences. The concept of mathematics as a laboratory tool is introduced and the importance of presenting students with relevant, real-world examples of applying mathematics in the Biological Sciences is discussed.

On the universal TRO of a JC*-triple, ideals and tensor products

In a development from material introduced in recent work, we discuss the interconnections between ternary rings of operators (TROs) and right C*-algebras generated by JC*-triples, deducing that every JC*-triple possesses a largest universally reversible ideal, that the universal TRO commutes with appropriate tensor products and establishing a reversibility criterion for type I JW*-triples.

Spectral theory of Toeplitz and Hankel operators on the Bergman space A1

The Fredholm properties of Toeplitz operators on the Bergman space A2 have been well-known for continuous symbols since the 1970s. We investigate the case p=1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on Ap that arise when we no longer have 1of Hankel operators on A1.

Mathematics applied to the climate system: outstanding challenges and recent progress

The societal need for reliable climate predictions and a proper assessment of their uncertainties is pressing. Uncertainties arise not only from initial conditions and forcing scenarios, but also from model formulation. Here, we identify and document three broad classes of problems, each representing what we regard to be an outstanding challenge in the area of mathematics applied to the climate system. First, there is the problem of the development and evaluation of simple physically based models of the global climate. Second, there is the problem of the development and evaluation of the components of complex models such as general circulation models. Third, there is the problem of the development and evaluation of appropriate statistical frameworks. We discuss these problems in turn, emphasizing the recent progress made by the papers presented in this Theme Issue. Many pressing challenges in climate science require closer collaboration between climate scientists, mathematicians and statisticians. We hope the papers contained in this Theme Issue will act as inspiration for such collaborations and for setting future research directions.

Linguistic experiments and ordinary language philosophy

J.L. Austin is regarded as having an especially acute ear for fine distinctions of meaning overlooked by other philosophers. Austin employs an informal experimental approach to gathering evidence in support of these fine distinctions in meaning, an approach that has become a standard technique for investigating meaning in both philosophy and linguistics. In this paper, we subject Austin's methods to formal experimental investigation. His methods produce mixed results: We find support for his most famous distinction, drawn on the basis of his `donkey stories', that `mistake' and `accident' apply to different cases, but not for some of his other attempts to distinguish the meaning of philosophically significant terms (such as `intentionally' and `deliberately'). We critically examine the methodology of informal experiments employed in ordinary language philosophy and much of contemporary philosophy of language and linguistics, and discuss the role that experimenter bias can play in influencing judgments about informal and formal linguistic experiments.

Chronic limitations of neo-liberal capitalism and oligopolistic markets: an urgent case for socialized capital

Purpose – The purpose of this paper is to highlight the serious limitations of neo-liberal capitalism and urge for a shift to socialized capital before further economic deterioration leads to a succession of global conflicts. Design/methodology/approach – This conceptual paper adopts a macro perspective in presenting argument on how global, financial markets integration and capital flow liberalization have led to inadequate market and corporate governance measures. The argument is couched in a selected literature and is preceded by a proposed solution – the requirement for socialized capital. An analysis of the nature of socialized capital is outlined and the questions that require attention identified if a paradigm shift from neo-liberal capitalism is to take place. Findings – The need to urgently shift to a new philosophy of capitalism is overwhelming. Emphasized is that capital needs to adopt a socialised identity and is supported by investment horizons of 30 years or more. It is argued that non-market (e.g. state, NGOs, civil society) intervention is critical in setting appropriate frameworks within which socialized capital can operate. Research limitations/implications – This is a theoretical paper, in which questions are raised which require transparent, public debate. Originality/value – The paper presents the case for a fundamental reconsideration of present day markets, the role of capital and the influence of elites in determining the public good.

Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilization waves on eggs

Wave solutions to a mechanochemical model for cytoskeletal activity are studied and the results applied to the waves of chemical and mechanical activity that sweep over an egg shortly after fertilization. The model takes into account the calcium-controlled presence of actively contractile units in the cytoplasm, and consists of a viscoelastic force equilibrium equation and a conservation equation for calcium. Using piecewise linear caricatures, we obtain analytic solutions for travelling waves on a strip and demonstrate uiat the full nonlinear system behaves as predicted by the analytic solutions. The equations are solved on a sphere and the numerical results are similar to the analytic solutions. We indicate how the speed of the waves can be used as a diagnostic tool with which the chemical reactivity of the egg surface can be measured.

The rise of Western rationalism: Paul Feyerabend’s story

I summarise certain aspects of Paul Feyerabend’s account of the development of Western rationalism, show the ways in which that account is supposed to run up against an alternative, that of Karl Popper, and then try to give a preliminary comparison of the two. My interest is primarily in whether what Feyerabend called his ‘story’ constitutes a possible history of our epistemic concepts and their trajectory. I express some grave reservations about that story, and about Feyerabend’s framework, finding Popper’s views less problematic here. However, I also suggest that one important aspect of Feyerabend’s material, his treatment of religious belief, can be given an interpretation which makes it tenable, and perhaps preferable to a Popperian approach.

Quasi-sure existence of Gaussian rough paths and large deviation principles for capacities

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process.

Conditioning and preconditioning of the optimal state estimation problem

Optimal state estimation is a method that requires minimising a weighted, nonlinear, least-squares objective function in order to obtain the best estimate of the current state of a dynamical system. Often the minimisation is non-trivial due to the large scale of the problem, the relative sparsity of the observations and the nonlinearity of the objective function. To simplify the problem the solution is often found via a sequence of linearised objective functions. The condition number of the Hessian of the linearised problem is an important indicator of the convergence rate of the minimisation and the expected accuracy of the solution. In the standard formulation the convergence is slow, indicating an ill-conditioned objective function. A transformation to different variables is often used to ameliorate the conditioning of the Hessian by changing, or preconditioning, the Hessian. There is only sparse information in the literature for describing the causes of ill-conditioning of the optimal state estimation problem and explaining the effect of preconditioning on the condition number. This paper derives descriptive theoretical bounds on the condition number of both the unpreconditioned and preconditioned system in order to better understand the conditioning of the problem. We use these bounds to explain why the standard objective function is often ill-conditioned and why a standard preconditioning reduces the condition number. We also use the bounds on the preconditioned Hessian to understand the main factors that affect the conditioning of the system. We illustrate the results with simple numerical experiments.

An example in Beurling's theory of generalised primes

In this paper we prove some connections between the growth of a function and its Mellin transform and apply these to study an explicit example in the theory of Beurling primes.

Science learning and graphic symbols: an exploration of early years teachers’ views and use of graphic symbols when teaching science

The study investigated early years teachers’ understanding and use of graphic symbols, defined as the visual representation(s) used to communicate one or more “linguistic” concepts, which can be used to facilitate science learning. The study was conducted in Cyprus where six early years teachers were observed and interviewed. The results indicate that the teachers had a good understanding of the role of symbols, but demonstrated a lack of understanding in regards to graphic symbols specifically. None of the teachers employed them in their observed science lesson, although some of them claimed that they did so. Findings suggest a gap in participants’ acquaintance with the terminology regarding different types of symbols and a lack of awareness about the use and availability of graphic symbols for the support of learning. There is a need to inform and train early years teachers about graphic symbols and their potential applications in supporting children’s learning.

Alternative diagnostic diagrams and the `forgotten` population of weak line galaxies in the SDSS

A numerous population of weak line galaxies (WLGs) is often left out of statistical studies on emission-line galaxies (ELGs) due to the absence of an adequate classification scheme, since classical diagnostic diagrams, such as [O iii]/H beta versus [N ii]/H alpha (the BPT diagram), require the measurement of at least four emission lines. This paper aims to remedy this situation by transposing the usual divisory lines between star-forming (SF) galaxies and active galactic nuclei (AGN) hosts and between Seyferts and LINERs to diagrams that are more economical in terms of line quality requirements. By doing this, we rescue from the classification limbo a substantial number of sources and modify the global census of ELGs. More specifically, (1) we use the Sloan Digital Sky Survey Data Release 7 to constitute a suitable sample of 280 000 ELGs, one-third of which are WLGs. (2) Galaxies with strong emission lines are classified using the widely applied criteria of Kewley et al., Kauffmann et al. and Stasinska et al. to distinguish SF galaxies and AGN hosts and Kewley et al. to distinguish Seyferts from LINERs. (3) We transpose these classification schemes to alternative diagrams keeping [N ii]/H alpha as a horizontal axis, but replacing H beta by a stronger line (H alpha or [O ii]), or substituting the ionization-level sensitive [O iii]/H beta ratio with the equivalent width of H alpha (W(H alpha)). Optimized equations for the transposed divisory lines are provided. (4) We show that nothing significant is lost in the translation, but that the new diagrams allow one to classify up to 50 per cent more ELGs. (5) Introducing WLGs in the census of galaxies in the local Universe increases the proportion of metal-rich SF galaxies and especially LINERs. In the course of this analysis, we were led to make the following points. (i) The Kewley et al. BPT line for galaxy classification is generally ill-used. (ii) Replacing [O iii]/H beta by W(H alpha) in the classification introduces a change in the philosophy of the distinction between LINERs and Seyferts, but not in its results. Because the W(H alpha) versus [N ii]/H alpha diagram can be applied to the largest sample of ELGs without loss of discriminating power between Seyferts and LINERs, we recommend its use in further studies. (iii) The dichotomy between Seyferts and LINERs is washed out by WLGs in the BPT plane, but it subsists in other diagnostic diagrams. This suggests that the right wing in the BPT diagram is indeed populated by at least two classes, tentatively identified with bona fide AGN and `retired` galaxies that have stopped forming stars and are ionized by their old stellar populations.