Procedures of Leibnizian infinitesimal calculus: An account in three modern frameworks

Recent Leibniz scholarship has sought to gauge which foundational framework provides the most successful account of the procedures of the Leibnizian calculus (LC). While many scholars (e.g., Ishiguro, Levey) opt for a default Weierstrassian framework, Arthur compares LC to a non-Archimedean framework SIA (Smooth Infinitesimal Analysis) of Lawvere-Kock-Bell. We analyze Arthur's comparison and find it rife with equivocations and misunderstandings on issues including the non-punctiform nature of the continuum, infinite-sided polygons, and the fictionality of infinitesimals. Rabouin and Arthur claim that Leibniz considers infinities as contradictory, and that Leibniz' definition of incomparables should be understood as nominal rather than as semantic. However, such claims hinge upon a conflation of Leibnizian notions of bounded infinity and unbounded infinity, a distinction emphasized by early Knobloch. The most faithful account of LC is arguably provided by Robinson's framework. We exploit an axiomatic framework for infinitesimal analysis called SPOT (conservative over ZF) to provide a formalisation of LC, including the bounded/unbounded dichotomy, the assignable/inassignable dichotomy, the generalized relation of equality up to negligible terms, and the law of continuity.Comment: 52 pages, to appear in British Journal for the History of Mathematic

Relationship between air pollution and metal levels in cancerous and non-cancerous lung tissues

We aimed to check the relationships between levels of metals (Ca, Cd, Cu, Fe, Hg and Zn) in cancerous and non-cancerous lung tissues and their link to air pollution, expressed as particulate matter (PM) concentrations. The study also examines the influence on metal concentration in the lung tissue of patients' sex and the distance of their homes from the nearest emitter. We found that the general pattern of ascending concentrations in tumor tissue was as follows: Hg < Cd < Cu < Ca < Zn < Fe. In non-affected lung tissue the order of concentrations of Ca and Fe was reversed. With the exception of Cd and Cu, levels of metals were found in higher accumulations in non-cancerous tissue (e.g., Fe 326.423 and Ca 302.730 μg/g d.w) than in tumorous tissue (Fe 150.735 and Ca 15.025 μg/g d.w). Neither the PM10 (PM of a diameter of 10 μm) concentration nor sex revealed any connection with metal concentrations. The shorter the distance from the emitter, the higher the metal concentrations that tended to be observed for almost all metals, but a statistically significant (but weak) relationship was noted only for Cu in tumor tissue (rs: -0.4869)

Cauchy, infinitesimals and ghosts of departed quantifiers

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Keywords: historiography; infinitesimal; Latin model; butterfly modelComment: 45 pages, published in Mat. Stu

The Long-Baseline Neutrino Experiment: Exploring Fundamental Symmetries of the Universe

The preponderance of matter over antimatter in the early Universe, the dynamics of the supernova bursts that produced the heavy elements necessary for life and whether protons eventually decay --- these mysteries at the forefront of particle physics and astrophysics are key to understanding the early evolution of our Universe, its current state and its eventual fate. The Long-Baseline Neutrino Experiment (LBNE) represents an extensively developed plan for a world-class experiment dedicated to addressing these questions. LBNE is conceived around three central components: (1) a new, high-intensity neutrino source generated from a megawatt-class proton accelerator at Fermi National Accelerator Laboratory, (2) a near neutrino detector just downstream of the source, and (3) a massive liquid argon time-projection chamber deployed as a far detector deep underground at the Sanford Underground Research Facility. This facility, located at the site of the former Homestake Mine in Lead, South Dakota, is approximately 1,300 km from the neutrino source at Fermilab -- a distance (baseline) that delivers optimal sensitivity to neutrino charge-parity symmetry violation and mass ordering effects. This ambitious yet cost-effective design incorporates scalability and flexibility and can accommodate a variety of upgrades and contributions. With its exceptional combination of experimental configuration, technical capabilities, and potential for transformative discoveries, LBNE promises to be a vital facility for the field of particle physics worldwide, providing physicists from around the globe with opportunities to collaborate in a twenty to thirty year program of exciting science. In this document we provide a comprehensive overview of LBNE's scientific objectives, its place in the landscape of neutrino physics worldwide, the technologies it will incorporate and the capabilities it will possess.Comment: Major update of previous version. This is the reference document for LBNE science program and current status. Chapters 1, 3, and 9 provide a comprehensive overview of LBNE's scientific objectives, its place in the landscape of neutrino physics worldwide, the technologies it will incorporate and the capabilities it will possess. 288 pages, 116 figure

Precise Measurement of the Neutrino Mixing Parameter theta(23) from Muon Neutrino Disappearance in an Off-Axis Beam

New data from the T2K neutrino oscillation experiment produce the most precise measurement of the neutrino mixing parameter theta_{23}. Using an off-axis neutrino beam with a peak energy of 0.6 GeV and a data set corresponding to 6.57 x 10^{20} protons on target, T2K has fit the energy-dependent nu_mu oscillation probability to determine oscillation parameters. Marginalizing over the values of other oscillation parameters yields sin^2 (theta_{23}) = 0.514 +0.055/-0.056 (0.511 +- 0.055), assuming normal (inverted) mass hierarchy. The best-fit mass-squared splitting for normal hierarchy is Delta m^2_{32} = (2.51 +- 0.10) x 10^{-3} eV^2/c^4 (inverted hierarchy: Delta m^2_{13} = (2.48 +- 0.10) x 10^{-3} eV^2/c^4). Adding a model of multinucleon interactions that affect neutrino energy reconstruction is found to produce only small biases in neutrino oscillation parameter extraction at current levels of statistical uncertainty

Measurement of the intrinsic electron neutrino component in the T2K neutrino beam with the ND280 detector

The T2K experiment has reported the first observation of the appearance of electron neutrinos in a muon neutrino beam. The main and irreducible background to the appearance signal comes from the presence in the neutrino beam of a small intrinsic component of electron neutrinos originating from muon and kaon decays. In T2K, this component is expected to represent 1.2% of the total neutrino flux. A measurement of this component using the near detector (ND280), located 280 m from the target, is presented. The charged current interactions of electron neutrinos are selected by combining the particle identification capabilities of both the time projection chambers and electromagnetic calorimeters of ND280. The measured ratio between the observed electron neutrino beam component and the prediction is 1.01 +/- 0.10 providing a direct confirmation of the neutrino fluxes and neutrino cross section modeling used for T2K neutrino oscillation analyses. Electron neutrinos coming from muons and kaons decay are also separately measured, resulting in a ratio with respect to the prediction of 0.68 +/- 0.30 and 1.10 +/- 0.14, respectively

Measurement of the neutrino-oxygen neutral-current interaction cross section by observing nuclear deexcitation gamma rays

We report the first measurement of the neutrino-oxygen neutral-current quasielastic (NCQE) cross section gamma It is obtained by observing nuclear deexcitation. rays which follow neutrino-oxygen interactions at the Super-Kamiokande water Cherenkov detector. We use T2K data corresponding to 3.01 x 10(20) protons on target. By selecting only events during the T2K beam window and with well-reconstructed vertices in the fiducial volume, the large background rate from natural radioactivity is dramatically reduced. We observe 43 events in the 4-30 MeV reconstructed energy window, compared with an expectation of 51.0, which includes an estimated 16.2 background events. The background is primarily nonquasielastic neutral-current interactions and has only 1.2 events from natural radioactivity. The flux-averaged NCQE cross section we measure is 1.55 x 10(-38) cm(2) with a 68% confidence interval of (1.22, 2.20) x 10(-38) cm(2) at a median neutrino energy of 630 MeV, compared with the theoretical prediction of 2.01 x 10(-38) cm(2)