Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine for computing the function ${{\rm R}}^{m}_{-\frac{1}{2}+i\tau}(x)$, a real-valued numerically satisfactory companion of the function ${\rm P}^m_{-\tfrac12+i\tau}(x)$ for $x>1$. In this way, a natural basis for solving Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication

Uniform asymptotic expansions for the zeros of Bessel functions

Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but not their derivatives, unlike traditional asymptotic expansions. From these, asymptotic expansions are derived for the zeros of Bessel functions that are valid for large positive values of the order, uniformly valid for all the zeros. The coefficients in the expansions are explicitly given elementary functions, and similar expansions are derived for the zeros of the derivatives of Bessel functions

Developing a methodology for the non-destructive analysis of British soft-paste porcelain

Soft-paste porcelain was produced in Britain in great quantities between the mid-18th and early 19th centuries. Due to industrial secrecy and the complexities of creating a product that would survive high-temperature firing, a range of paste recipes was employed by dozens of factories. This has resulted in an array of porcelains which vary in their elemental composition and mineralogy. This research carries out a meta-analysis of the published data for porcelain bodies and glazes and concludes that some discrimination can be achieved using the major and minor elemental composition of the bodies, and that for the glazes intra-factory variation is often greater than inter-factory variation in composition. A pilot investigation of the trace elemental composition of British porcelain is carried out using Laser Ablation Inductively Coupled Plasma Mass Spectroscopy, which finds compositional groups corresponding to different sources of clay and silica raw materials. In the interests of preserving intact objects, there is recognised a need for a non-destructive method for analysing British porcelain, in order to provenance and date objects. Such a method would rely on data from the surface of the object, which is typically covered by glaze and over-glaze coloured enamels, and this research demonstrates that the formulae used for the glaze and enamels are in some cases characteristic of the factory, or workshop, and period at which they were created. Hand-Held XRF analysis is used to analyse the glaze, underglaze blue and polychrome enamels on a selection of porcelain objects from different factories, and compositional traits are identified that allow some factories and periods to be distinguished. Glass standards are developed, which are representative of the glaze and enamel composition, and which could allow X-ray fluorescence (XRF) data to be calibrated for fully quantitative results

Large Parameter Cases of the Gauss Hypergeometric Function

We consider the asymptotic behaviour of the Gauss hypergeometric function when several of the parameters a, b, c are large. We indicate which cases are of interest for orthogonal polynomials (Jacobi, but also Krawtchouk, Meixner, etc.), which results are already available and which cases need more attention. We also consider a few examples of 3F2-functions of unit argument, to explain which difficulties arise in these cases, when standard integrals or differential equations are not available.Comment: 21 pages, 4 figure

The abyss of Calvino's deconstructive writing: an apology for non-foundational theology

This thesis argues that the later works of Italo Calvino function as an apologetic for a non-foundational theology. Calvino’s two late novels Invisible Cities and If on a winter’s night a traveller are viewed as embodiments of self-deconstructing rhetoric. Calvino’s essays offer a literary theory which is practised in these novels to lead to an expression of aporetic epistemology. This technique of ironic reflexivity is described as a rhetoric of the abyss. This becomes an apologetic for a theology of postmodernism when If on a winter’s night a traveller is viewed in terms of its treatment of the death of the author, because its application to theology is shown in a parallel to the historical process of secularisation. Similarly, Invisible Cities is offered as an apologetic for non-foundational theology because by embodying an exercise in rhetorical self criticism it becomes, for the reader, an experience of epistemological de-stabilisation. Reader-response criticism is the framework which makes this possible. An allegorical model of reading Invisible Cities is offered in viewing Marco Polo as a Heideggerean counsellor who supports his interlocutor, Kublai Khan, to explore the inadequacy of a metaphysical mastery in terms of western concepts of imperialism and philosophical absolutism. The counsel offered is compared to the concept of gift as being without Being, and grace as icon, in Jean-Luc Marion’s God Without Being. The equation of the rhetoric of abyss with Tillich’s God as ground and abyss is viewed as inadequate. The argument is made that instead, a careful reading of Calvino’s rhetoric creates an appreciation of the reasons for accepting the radical a-theology of faith put forward by the work of Thomas J.J. Altizer and John Caputo. Its ethical consequences are aligned to the argument made in Mark C. Taylor’s After God, for religion as a complex adaptive system with ethical and social responsibilities. Finally, following on from the pastoral concern of the Polo-Khan dialogue and briefly discussing Calvino’s novel Mr Palomar, links are made to the issue of autism as a spectrum of universal human problem and opportunity, and to the model of the 12 step recovery program of Alcoholics Anonymous as a secular spiritual praxis. The idea of nomadic theology is suggested as a possible response.