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

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

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

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

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