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

Bilateral linear scleroderma "en coup de sabre" associated with facial atrophy and neurological complications

BACKGROUND: Linear scleroderma "en coup de sabre" (LSCS) usually affects one side of the face and head in the frontoparietal area with band-like indurated skin lesions. The disease may be associated with facial hemiatrophy. Various ophthalmological and neurological abnormalities have been observed in patients with LSCS. We describe an unusual case of LSC. CASE PRESENTATION: A 23 year old woman presented bilateral LSCS and facial atrophy. The patient had epileptic seizures as well as oculomotor and facial nerve palsy on the left side which also had pronounced skin involvement. Clinical features of different stages of the disease are presented. CONCLUSIONS: The findings of the presented patient with bilateral LSCS and facial atrophy provide further evidence for a neurological etiology of the disease and may also indicate that classic progressive facial hemiatrophy (Parry-Romberg syndrome) and LSCS actually represent different spectra of the same disease

Subclinical myocardial injury in patients with Facioscapulohumeral muscular dystrophy 1 and preserved ejection fraction - assessment by cardiovascular magnetic resonance

Background: Facioscapulohumeral muscular dystrophy type 1 (FSHD1) is an autosomal dominant and the third most common inherited muscle disease. Cardiac involvement is currently described in several muscular dystrophies (MD), but there are conflicting reports in FSHD1. Mostly, FSHD1 is recognized as MD with infrequent cardiac involvement, but sudden cardiac deaths are reported in single cases. The aim of this study is to investigate whether subclinical cardiac involvement in FSHD1 patients is detectable in preserved left ventricular systolic function applying cardiovascular magnetic resonance (CMR). Methods: We prospectively included patients with genetically confirmed FSHD1 (n = 52, 48 ± 15 years) and compared them with 29 healthy age-matched controls using a 1.5 T CMR scanner. Myocardial tissue differentiation was performed qualitatively using focal fibrosis imaging (late gadolinium enhancement (LGE)), fat imaging (multi-echo sequence for fat/water-separation) and parametric T2- and T1-mapping for quantifying inflammation and diffuse fibrosis. Extracellular volume fraction was calculated. A 12-lead electrocardiogram and 24-h Holter were performed for the assessment of MD-specific Groh-criteria and arrhythmia. Results: Focal fibrosis by LGE was present in 13 patients (25%,10 men), fat infiltration in 7 patients (13%,5 men). T2 values did not differ between FSHD1 and healthy controls. Native T1 mapping revealed significantly higher values in patients (global native myocardial T1 values basal: FSHD1: 1012 ± 26 ms vs. controls: 985 ± 28 ms, p < 0.01, medial FSHD1: 994 ± 37 ms vs. controls: 982 ± 28 ms, p = 0.028). This was also evident in regions adjacent to focal fibrosis, indicating diffuse fibrosis. Groh-criteria were positive in 1 patient. In Holter, arrhythmic events were recorded in 10/43 subjects (23%). Conclusions: Patients with FSHD1 and preserved left ventricular ejection fraction present focal and diffuse myocardial injury. Longitudinal multi-center trials are needed to define the impact of myocardial changes as well as a relation between myocardial injury and arrhythmias on long-term prognosis and therapeutic decision-making. Trial Registration: ISRCTN registry with study ID ISRCTN13744381

A Bayesian analysis of pentaquark signals from CLAS data

We examine the results of two measurements by the CLAS collaboration, one of which claimed evidence for a $\Theta^{+}$ pentaquark, whilst the other found no such evidence. The unique feature of these two experiments was that they were performed with the same experimental setup. Using a Bayesian analysis we find that the results of the two experiments are in fact compatible with each other, but that the first measurement did not contain sufficient information to determine unambiguously the existence of a $\Theta^{+}$. Further, we suggest a means by which the existence of a new candidate particle can be tested in a rigorous manner.Comment: 5 pages, 3 figure

Search for medium modification of the ρ\rho meson

The photoproduction of vector mesons on various nuclei has been studied using the CEBAF Large Acceptance Spectrometer (CLAS) at Jefferson Laboratory. The vector mesons, $\rho$, $\omega$, and $\phi$, are observed via their decay to $e^+e^-$, in order to reduce the effects of final state interactions in the nucleus. Of particular interest are possible in-medium effects on the properties of the $\rho$ meson. The $\rho$ spectral function is extracted from the data on various nuclei, carbon, iron, and titanium, and compared to the spectrum from liquid deuterium, which is relatively free of nuclear effects. We observe no significant mass shift for the $\rho$ meson; however, there is some widening of the resonance in titanium and iron, which is consistent with expected collisional broadening.Comment: 8 pages, 4 figure

Light Vector Mesons in the Nuclear Medium

The light vector mesons ($\rho$, $\omega$, and $\phi$) were produced in deuterium, carbon, titanium, and iron targets in a search for possible in-medium modifications to the properties of the $\rho$ meson at normal nuclear densities and zero temperature. The vector mesons were detected with the CEBAF Large Acceptance Spectrometer (CLAS) via their decays to $e^{+}e^{-}$. The rare leptonic decay was chosen to reduce final-state interactions. A combinatorial background was subtracted from the invariant mass spectra using a well-established event-mixing technique. The $\rho$ meson mass spectrum was extracted after the $\omega$ and $\phi$ signals were removed in a nearly model-independent way. Comparisons were made between the $\rho$ mass spectra from the heavy targets ($A > 2$) with the mass spectrum extracted from the deuterium target. With respect to the $\rho$-meson mass, we obtain a small shift compatible with zero. Also, we measure widths consistent with standard nuclear many-body effects such as collisional broadening and Fermi motion.Comment: 15 pages, 18 figures, 3 table

First measurement of coherent ϕ\phi-meson photoproduction on deuteron at low energies

The cross section and decay angular distributions for the coherent \phi meson photoproduction on the deuteron have been measured for the first time up to a squared four-momentum transfer t =(p_{\gamma}-p_{\phi})^2 =-2 GeV^2/c^2, using the CLAS detector at the Thomas Jefferson National Accelerator Facility. The cross sections are compared with predictions from a re-scattering model. In a framework of vector meson dominance, the data are consistent with the total \phi-N cross section \sigma_{\phi N} at about 10 mb. If vector meson dominance is violated, a larger \sigma_{\phi N} is possible by introducing larger t-slope for the \phi N \to \phi N process than that for the \gamma N \to \phi N process. The decay angular distributions of the \phi are consistent with helicity conservation.Comment: 6 page

Modern microwave methods in solid state inorganic materials chemistry: from fundamentals to manufacturing

No abstract available