Second-order terms for the variances and covariances of the number of prime factors—Including the square free case

AbstractWe obtain second-order terms for the variance and covariance of Ω(n) and ω(n), the number of prime divisors counted with and without multiplicity, and connect these results to a formula of Renyi. We discuss the heuristic connection with the Landau-Sathe extension of the prime number theorem and develop new expansions for the mean and variance of ω(n) in the square free case

Duhemian Themes in Expected Utility Theory

This monographic chapter explains how expected utility (EU) theory arose in von Neumann and Morgenstern, how it was called into question by Allais and others, and how it gave way to non-EU theories, at least among the specialized quarters of decion theory. I organize the narrative around the idea that the successive theoretical moves amounted to resolving Duhem-Quine underdetermination problems, so they can be assessed in terms of the philosophical recommendations made to overcome these problems. I actually follow Duhem's recommendation, which was essentially to rely on the passing of time to make many experiments and arguments available, and evebntually strike a balance between competing theories on the basis of this improved knowledge. Although Duhem's solution seems disappointingly vague, relying as it does on "bon sens" to bring an end to the temporal process, I do not think there is any better one in the philosophical literature, and I apply it here for what it is worth. In this perspective, EU theorists were justified in resisting the first attempts at refuting their theory, including Allais's in the 50s, but they would have lacked "bon sens" in not acknowledging their defeat in the 80s, after the long process of pros and cons had sufficiently matured. This primary Duhemian theme is actually combined with a secondary theme - normativity. I suggest that EU theory was normative at its very beginning and has remained so all along, and I express dissatisfaction with the orthodox view that it could be treated as a straightforward descriptive theory for purposes of prediction and scientific test. This view is usually accompanied with a faulty historical reconstruction, according to which EU theorists initially formulated the VNM axioms descriptively and retreated to a normative construal once they fell threatened by empirical refutation. From my historical study, things did not evolve in this way, and the theory was both proposed and rebutted on the basis of normative arguments already in the 1950s. The ensuing, major problem was to make choice experiments compatible with this inherently normative feature of theory. Compability was obtained in some experiments, but implicitly and somewhat confusingly, for instance by excluding overtly incoherent subjects or by creating strong incentives for the subjects to reflect on the questions and provide answers they would be able to defend. I also claim that Allais had an intuition of how to combine testability and normativity, unlike most later experimenters, and that it would have been more fruitful to work from his intuition than to make choice experiments of the naively empirical style that flourished after him. In sum, it can be said that the underdetermination process accompanying EUT was resolved in a Duhemian way, but this was not without major inefficiencies. To embody explicit rationality considerations into experimental schemes right from the beginning would have limited the scope of empirical research, avoided wasting resources to get only minor findings, and speeded up the Duhemian process of groping towards a choice among competing theories

Invisible interpretations: reflections on the digital humanities and intellectual history

Much has been made of the digital humanities, yet it remains an underexplored field in relation to intellectual history. This paper aims to add to the little literature which does exist by offering a survey of the ideas and issues facing would-be practitioners. This includes: an overview of what the digital humanities are; reflections on what they offer intellectual history and how they may be problematic in regard to, first, accessing texts, and second, analysing source material; a conclusion with three reflections on future best practices – to be sceptical of digital sources, to be reflective of methodologies and how they may need to be modified when engaging with the digital humanities, and to embrace more directly the methodological, statistical, and technical aspects behind digital humanities. The aim is not to provide all the answers – at this stage that is impossible – but to be part of an emerging and ongoing discussion

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