Cauchy's infinitesimals, his sum theorem, and foundational paradigms

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy's proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy's proof closely and show that it finds closer proxies in a different modern framework. Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc

Influence of Matrix Polarity on the Properties of Ethylene Vinyl Acetate–Carbon Nanofiller Nanocomposites

A series of ethylene vinyl acetate (EVA) nanocomposites using four kinds of EVA with 40, 50, 60, and 70 wt% vinyl acetate (VA) contents and three different carbon-based nanofillers—expanded graphite (EG), multi-walled carbon nanotube (MWCNT), and carbon nanofiber (CNF) have been prepared via solution blending. The influence of the matrix polarity and the nature of nanofillers on the morphology and properties of EVA nanocomposites have been investigated. It is observed that the sample with lowest vinyl acetate content exhibits highest mechanical properties. However, the enhancement in mechanical properties with the incorporation of various nanofillers is the highest for EVA with high VA content. This trend has been followed in both dynamic mechanical properties and thermal conductivity of the nanocomposites. EVA copolymer undergoes a transition from partial to complete amorphousness between 40 and 50 wt% VA content, and this changes the dispersion of the nanofillers. The high VA-containing polymers show more affinity toward fillers due to the large free volume available and allow easy dispersion of nanofillers in the amorphous rubbery phase, as confirmed from the morphological studies. The thermal stability of the nanocomposites is also influenced by the type of nanofiller

Transmissibility of Atypical Scrapie in Ovine Transgenic Mice: Major Effects of Host Prion Protein Expression and Donor Prion Genotype

Atypical scrapie or Nor98 has been identified as a transmissible spongiform encephalopathy (TSE) that is clearly distinguishable from classical scrapie and BSE, notably regarding the biochemical features of the protease-resistant prion protein PrPres and the genetic factors involved in susceptibility to the disease. In this study we transmitted the disease from a series of 12 French atypical scrapie isolates in a transgenic mouse model (TgOvPrP4) overexpressing in the brain ∼0.25, 1.5 or 6× the levels of the PrPARQ ovine prion protein under the control of the neuron-specific enolase promoter. We used an approach based on serum PrPc measurements that appeared to reflect the different PrPc expression levels in the central nervous system. We found that transmission of atypical scrapie, much more than in classical scrapie or BSE, was strongly influenced by the PrPc expression levels of TgOvPrP4 inoculated mice. Whereas TgOvPrP4 mice overexpressing ∼6× the normal PrPc level died after a survival periods of 400 days, those with ∼1.5× the normal PrPc level died at around 700 days. The transmission of atypical scrapie in TgOvPrP4 mouse line was also strongly influenced by the prnp genotypes of the animal source of atypical scrapie. Isolates carrying the AF141RQ or AHQ alleles, associated with increased disease susceptibility in the natural host, showed a higher transmissibility in TgOvPrP4 mice. The biochemical analysis of PrPres in TgOvPrP4 mouse brains showed a fully conserved pattern, compared to that in the natural host, with three distinct PrPres products. Our results throw light on the transmission features of atypical scrapie and suggest that the risk of transmission is intrinsically lower than that of classical scrapie or BSE, especially in relation to the expression level of the prion protein

Attack Interference: A Path to Defending Security Protocols

Traditionally security protocol analysis relies on a single Dolev-Yao attacker. This type of attacker is so powerful that overall attack power does not change if additional attackers cooperate. In this paper, we take a fundamentally different approach and investigate the case of multiple non-collaborating attackers. We show how non-collaboration between attackers gives rise to interference between ongoing attacks and that it is possible to actively exploit attack interference to mitigate security breaches and provide partial protection to weak protocols

Principled teaching for deep progress: improving mathematical learning beyond methods and materials

This paper contributes to knowledge about principled action which makes a difference to learners attainment. We report on the Improving Attainment in Mathematics Project, a project focusing on low-attaining secondary students. The purpose of the project was to introduce innovations in practice through action research with 10 teachers over 2 years, and evaluate the effect on students learning using national test scores, teachers reports, non-routine tasks and other performance indicators. However, this is not a study which shows how certain methods lead to better results. While it was found that learning improved, the methods and strategies the teachers used were not always generalisable across the project, indeed some were contradictory. Continued searching led to the identification of common underlying principles of teaching which different teachers manifested in different ways. Overt methods were less important than the collection of beliefs and commitments which underpinned teachers choices. There was, however, a convergence of practice around a focus on long-term development, the process of becoming a learner of mathematics, rather than short-term gains. In addition, we had to deal with some of the realities of authentic collaborative research with practitioners