A Cauchy-Dirac delta function

The Dirac delta function has solid roots in 19th century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his infinitesimal definition of delta.Comment: 24 pages, 2 figures; Foundations of Science, 201

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page

Stevin numbers and reality

We explore the potential of Simon Stevin's numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with arXiv:1104.0375, arXiv:1108.2885, arXiv:1108.420

The Sight Loss and Vision Priority Setting Partnership (SLV-PSP): overview and results of the research prioritisation survey process

Objectives: The Sight Loss and Vision Priority Setting Partnership aimed to identify research priorities relating to sight loss and vision through consultation with patients, carers and clinicians. These priorities can be used to inform funding bodies’ decisions and enhance the case for additional research funding. Design: Prospective survey with support from the James Lind Alliance. Setting: UK-wide National Health Service (NHS) and non-NHS. Participants: Patients, carers and eye health professionals. Academic researchers were excluded solely from the prioritisation process. The survey was disseminated by patient groups, professional bodies, at conferences and through the media, and was available for completion online, by phone, by post and by alternative formats (Braille and audio). Outcome measure: People were asked to submit the questions about prevention, diagnosis and treatment of sight loss and eye conditions that they most wanted to see answered by research. Returned survey questions were reviewed by a data assessment group. Priorities were established across eye disease categories at final workshops. Results: 2220 people responded generating 4461 submissions. Sixty-five per cent of respondents had sight loss and/or an eye condition. Following initial data analysis, 686 submissions remained which were circulated for interim prioritisation (excluding cataract and ocular cancer questions) to 446 patients/carers and 218 professionals. The remaining 346 questions were discussed at final prioritisation workshops to reach agreement of top questions per category. Conclusions: The exercise engaged a diverse community of stakeholders generating a wide range of conditions and research questions. Top priority questions were established across 12 eye disease categories. This is an Open Access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 3.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited and the use is non-commercial

A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint

Ten Misconceptions from the History of Analysis and Their Debunking

The widespread idea that infinitesimals were "eliminated" by the "great triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note: text overlap with arXiv:1108.2885 and arXiv:1110.545

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.Comment: 69 pages, 3 figure

Does Gender Influence Colour Choice in the Treatment of Visual Stress?

Purpose Visual Stress (VS) is a condition in which words appear blurred, in motion, or otherwise distorted when reading. Some people diagnosed with VS find that viewing black text on white paper through coloured overlays or precision tinted lenses (PTLs) reduces symptoms attributed to VS. The aim of the present study is to determine whether the choice of colour of overlays or PTLs is influenced by a patient’s gender. Methods Records of all patients attending a VS assessment in two optometry practices between 2009 and 2014 were reviewed retrospectively. Patients who reported a significant and consistent reduction in symptoms with either overlay and or PTL were included in the analysis. Overlays and PTLs were categorized as stereotypical male, female or neutral colours based on gender preferences as described in the literature. Chi-square analysis was carried out to determine whether gender (across all ages or within age groups) was associated with overlay or PTL colour choice. Results 279 patients (133 males and 146 females, mean age 17 years) consistently showed a reduction in symptoms with an overlay and were included. Chi-square analysis revealed no significant association between the colour of overlay chosen and male or female gender (Chi-square 0.788, p = 0.674). 244 patients (120 males and 124 females, mean age 24.5 years) consistently showed a reduction in symptoms with PTLs and were included. Chi-square analysis revealed a significant association between stereotypical male/female/neutral colours of PTLs chosen and male/female gender (Chi-square 6.46, p = 0.040). More males preferred stereotypical male colour PTLs including blue and green while more females preferred stereotypical female colour PTLs including pink and purple. Conclusions For some VS patients, the choice of PTL colour is influenced not only by the alleviation of symptoms but also by other non-visual factors such as gender

Available evidence and outcome of off-label use of rituximab in clinical practice

Purpose: To analyze the therapeutic indications for off-label use of rituximab, the available evidence for its use, the outcomes, and the cost. Methods: This was a retrospective analysis of patients treated with rituximab for off-label indications from January 2007 to December 2009 in two tertiary hospitals. Information on patient characteristics, medical conditions, and therapeutic responses was collected from medical records. Available evidence for the efficacy of rituximab in each condition was reviewed, and the cost of treatment was calculated. Results: A total of 101 cases of off-label rituximab use were analyzed. The median age of the patients involved was 53 [interquartile range (IQR) 37.5-68.0] years; 55.4 % were women. The indications for prescribing rituximab were primarily hematological diseases (46 %), systemic connective tissue disorders (27 %), and kidney diseases (20 %). Available evidence supporting rituximab treatment for these indications mainly came from individual cohort studies (53.5 % of cases) and case series (25.7 %). The short-term outcome (median 3 months, IQR 2-4 months) was a complete response in 38 % of cases and partial response in 32.6 %. The highest short-term responses were observed for systemic lupus erythematosus and membranous glomerulonephritis, and the lowest was for neuromyelitis optica, idiopathic thrombocytopenic purpura, and miscellaneous indications. Some response was maintained in long-term follow-up (median 23 months IQR 12-30months) in 69.2%of patients showing a short-term response. Median cost per patient was 5,187.5 (IQR 5,187.5-7,781.3). Conclusions: In our study, off-label rituximab was mainly used for the treatment of hematological, kidney, and systemic connective tissue disorders, and the response among our patient cohort was variable depending on the specific disease. The level of evidence supporting the use of rituximab for these indications was low and the cost was very high. We conclude that more clinical trials on the off-label use of rituximab are needed, although these may be difficult to conduct in some rare diseases. Data from observational studies may provide useful information to assist prescribing in clinical practice

Probing the Hofmeister Effect with Ultrafast Core Hole Spectroscopy

In the current work, X-ray emission spectra of aqueous solutions of different inorganic salts within the Hofmeister series are presented. The results reflect the direct interaction of the ions with the water molecules and therefore, reveal general properties of the salt-water interactions. Within the experimental precision a significant effect of the ions on the water structure has been observed but no ordering according to the structure maker/structure breaker concept could be mirrored in the results indicating that the Hofmeister effect-if existent-may be caused by more complex interactions