Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

The point-source method for 3D reconstructions for the Helmholtz and Maxwell equations

We use the point-source method (PSM) to reconstruct a scattered field from its associated far field pattern. The reconstruction scheme is described and numerical results are presented for three-dimensional acoustic and electromagnetic scattering problems. We give new proofs of the algorithms, based on the Green and Stratton-Chu formulae, which are more general than with the former use of the reciprocity relation. This allows us to handle the case of limited aperture data and arbitrary incident fields. Both for 3D acoustics and electromagnetics, numerical reconstructions of the field for different settings and with noisy data are shown. For shape reconstruction in acoustics, we develop an appropriate strategy to identify areas with good reconstruction quality and combine different such regions into one joint function. Then, we show how shapes of unknown sound-soft scatterers are found as level curves of the total reconstructed field.

Perspex machine IX: transreal analysis

We introduce transreal analysis as a generalisation of real analysis. We find that the generalisation of the real exponential and logarithmic functions is well defined for all transreal numbers. Hence, we derive well defined values of all transreal powers of all non-negative transreal numbers. In particular, we find a well defined value for zero to the power of zero. We also note that the computation of products via the transreal logarithm is identical to the transreal product, as expected. We then generalise all of the common, real, trigonometric functions to transreal functions and show that transreal (sin x)/x is well defined everywhere. This raises the possibility that transreal analysis is total, in other words, that every function and every limit is everywhere well defined. If so, transreal analysis should be an adequate mathematical basis for analysing the perspex machine - a theoretical, super-Turing machine that operates on a total geometry. We go on to dispel all of the standard counter "proofs" that purport to show that division by zero is impossible. This is done simply by carrying the proof through in transreal arithmetic or transreal analysis. We find that either the supposed counter proof has no content or else that it supports the contention that division by zero is possible. The supposed counter proofs rely on extending the standard systems in arbitrary and inconsistent ways and then showing, tautologously, that the chosen extensions are not consistent. This shows only that the chosen extensions are inconsistent and does not bear on the question of whether division by zero is logically possible. By contrast, transreal arithmetic is total and consistent so it defeats any possible "straw man" argument. Finally, we show how to arrange that a function has finite or else unmeasurable (nullity) values, but no infinite values. This arithmetical arrangement might prove useful in mathematical physics because it outlaws naked singularities in all equations.

Visualizing a logic of dependability arguments

This paper offers general guidelines for the development of effective visual languages. That is, languages for constructing diagrams that can be easily and readily interpreted and manipulated by the human reader. We use these guidelines first to examine classical AND/OR trees as a representation of logical proofs, and second to design and evaluate a visual language for representing proofs in LofA: a Logic of Dependability Arguments, for which we provide a brief motivation and overview.

Euler's theorem under the microscope

In this article Geoff Tennant summarises the first half of Imre Lakatos's seminal 1976 book, "Proofs and refutations: the logic of mathematical discovery". Implications are drawn for the classroom treatment of proof.

Stability results for the time-harmonic Maxwell equations with impedance boundary conditions

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators.

CUP Intermediate Greek-English Lexicon

Document design and typeface design: A typographic specification for a new Intermediate Greek-English Lexicon by CUP, accompanied by typefaces modified for the specific typographic requirements of the text. The Lexicon is a substantial (over 1400 pages) publication for HE students and academics intended to complement Liddell-Scott (the standard reference for classical Greek since the 1850s), and has been in preparation for over a decade. The typographic appearance of such works has changed very little since the original editions, largely to the lack of suitable typefaces: early digital proofs of the Lexicon utilised directly digitised versions of historical typefaces, making the entries difficult to navigate, and the document uneven in typographic texture. Close collaboration with the editors of the Lexicon, and discussion of the historical precedents for such documents informed the design at all typographic levels to achieve a highly reader-friendly results that propose a model for this kind of typography. Uniquely for a work of this kind, typeface design decisions were integrated into the wider document design specification. A rethinking of the complex typography for Greek and English based on historical editions as well as equivalent bilingual reference works at this level (from OUP, CUP, Brill, Mondadori, and other publishers) led a redefinition of multi-script typeface pairing for the specific context, taking into account recent developments in typeface design. Specifically, the relevant weighting of elements within each entry were redefined, as well as the typographic texture of type styles across the two scripts. In details, Greek typefaces were modified to emphasise clarity and readability, particularly of diacritics, at very small sizes. The relative weights of typefaces typeset side-by-side were fine-tuned so that the visual hierarchy of the entires was unambiguous despite the dense typesetting.

Solutions of the fully compressible semi-geostrophic system

The fully compressible semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alternative proof of the earlier result on the existence of weak solutions of this system expressed in the so-called geostrophic, or dual, coordinates. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measures.

Financial viability appraisals for site-specific planning decisions in England

In England, appraisals of the financial viability of development schemes have become an integral part of planning policy-making, initially in determining the amount of planning obligations that might be obtained via legal agreements (known as Section 106 agreements) and latterly as a basis for establishing charging schedules for the Community Infrastructure Levy (CIL). Local planning authorities set these policies on an area-wide basis but ultimately development proposals require consent on a site-by-site basis. It is at this site-specific level that issues of viability are hotly contested. This paper examines case documents, proofs of evidence and decisions from a sample of planning disputes in order to address major issues within development viability, the application of the models and the distribution of the development gain between the developer, landowner and community. The results have specific application to viability assessment in England and should impact on future policy and practice guidance in this field. They also have relevance to other countries that incorporate assessments of economic viability in their planning systems.

Change in the archive: Blanchot's 'L'Entretien infini'

Today we are faced with the problem of how to relate the archives of deconstructionist thinkers to their thought, which seems opposed to historically-oriented or genetic criticism. This article is the first to look comprehensively at the gestation of Maurice Blanchot's L'Entretien infini, including the work's page-proofs which were made available in 2009. I thus address the multiple processes of change at work throughout Blanchot's writing, with change being a process whose importance goes beyond any single new form of writing resulting from it. My presentation of the archival material is contextualized via a discussion of the notion of L'Absence de livre (the project's working title). Rather than being a straightfoward incompleteness or fragmentation, this notion establishes a fraught relationship between such ideas and what it calls ‘le Livre’.

Transreal logical space of all propositions

Transreal numbers provide a total semantics containing classical truth values, dialetheaic, fuzzy and gap values. A paraconsistent Sheffer Stroke generalises all classical logics to a paraconsistent form. We introduce logical spaces of all possible worlds and all propositions. We operate on a proposition, in all possible worlds, at the same time. We define logical transformations, possibility and necessity relations, in proposition space, and give a criterion to determine whether a proposition is classical. We show that proofs, based on the conditional, infer gaps only from gaps and that negative and positive infinity operate as bottom and top values.