Risk Objectivism and Risk Subjectivism: When Are Risks Real

Typically, those who discuss Risk management envision a two-step process wherein, first, Risk is more or less objectively appraised and, second, the acceptability of those Risks is subjectively evaluated. This paper questions the philosophical foundations of that approach

Arithmetic functions at consecutive shifted primes

For each of the functions $f \in \{\phi, \sigma, \omega, \tau\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for $f(p_n-1) > f(p_{n+1}-1) > \dots > f(p_{n+k}-1)$. We also answer some questions of Sierpi\'nski on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length.Comment: Made some improvements in the organization and expositio

On the degrees of divisors of T^n-1

Fix a field $F$. In this paper, we study the sets \D_F(n) \subset [0,n] defined by [\D_F(n):= {0 \leq m \leq n: T^n-1\text{has a divisor of degree $m$ in} F[T]}.] When \D_F(n) consists of all integers $m$ with $0 \leq m \leq n$, so that $T^n-1$ has a divisor of every degree, we call $n$ an $F$-practical number. The terminology here is suggested by an analogy with the practical numbers of Srinivasan, which are numbers $n$ for which every integer $0 \leq m \leq \sigma(n)$ can be written as a sum of distinct divisors of $n$. Our first theorem states that, for any number field $F$ and any $x \geq 2$, [#{\text{$F$-practical $n\leq x$}} \asymp_{F} \frac{x}{\log{x}};] this extends work of the second author, who obtained this estimate when F=\Q. Suppose now that $x \geq 3$, and let $m$ be a natural number in $[3,x]$. We ask: For how many $n \leq x$ does $m$ belong to \D_F(n)? We prove upper bounds in this problem for both F=\Q and F=\F_p (with $p$ prime), the latter conditional on the Generalized Riemann Hypothesis. In both cases, we find that the number of such $n \leq x$ is $\ll_{F} x/(\log{m})^{2/35}$, uniformly in $m$

SUSTAINING ANIMAL AGRICULTURE AND ENVIRONMENTAL QUALITY IN THE SOUTH: WHAT HAPPENED AND WHY? DISCUSSION

Environmental Economics and Policy, Livestock Production/Industries,

Competing Conceptions of Risk

Recent literature is said to reflect growing acknowledgment of multiple conceptions of risk but often to obscure an important distinction. Building on work of Kristin Shrader-Frechette, the authors explore the potential for debate over competing philosophical conceptions of risk

Variations on a theorem of Davenport concerning abundant numbers

Let \sigma(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/\sigma(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq x,~n/\sigma(n) \leq u} 1 exists for all u \in [0,1] and varies continuously with u. We study the behavior of the sums \sum_{n \leq x,~n/\sigma(n) \leq u} f(n) for certain complex-valued multiplicative functions f. Our results cover many of the more frequently encountered functions, including \varphi(n), \tau(n), and \mu(n). They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport's result: For all u \in [0,1], the limit $$\tilde{D}(u):= \lim_{R\to\infty} \frac{1}{\pi R}\#\{(x,y) \in \Z^2: 0<x^2+y^2 \leq R \text{ and } \frac{x^2+y^2}{\sigma(x^2+y^2)} \leq u\}$$ exists, and \tilde{D}(u) is both continuous and strictly increasing on [0,1]

Content Analysis of General Practitioner Requested Lumbar Spine X-ray Reports

Aims and Background X-rays of patients with low back pain rarely show serious pathology but frequently reveal incidental age-related changes and always expose people to radiation. Patients who have X-rays are more satisfied but report worse pain and disability. Psychological factors such as illness beliefs,catastrophizing and fear avoidance have been shown to be predictors of chronicity/disability. Authorities suggest that the way X-ray information is transmitted and interpreted by patients may influence outcome, therefore this study was designed to determine the words used by radiologists to describe lumbar spine Xrays. Methods: 120 consecutive X-ray reports for patients referred by primary care physicians were anonymised. A formal summative content analysis was undertaken. The coded words were grouped into categories according to their perceived meaning, and the process was refined until there were only three mutually exclusive categories. Results: Half the sample was aged 60 years or younger. Three categories were identified: anatomical, pathological and descriptive. In the pathological category, 33% of words described normal appearances, 47% described age-related changes and 20% described other features. In only 2% of cases were pathological words used to describe conditions as being "normal for age". Overall, 89 (74%) of the 120 reports contained at least one phrase containing words indicating the presence of degenerative changes. Conclusions: Almost three-quarters of lumbar spine X-ray reports use pathological words such as 'degenerative changes' to describe age-related changes but rarely describe them as being "normal for age"

Reply to Valverde

Professor Thompson responds to Valverde\u27s argument, in the last issue, that his approach to Risk puts too much emphasis on the distinction between Risk subjectivism and Risk objectivism. In doing so, he asserts, inter alia, that anchoring Risk judgments in a probabilistic framework does not go far enough in rejecting reigning Risk-analysis notions of real Risk

AGAINST MECHANISM: METHODOLOGY FOR AN EVOLUTIONARY ECONOMICS

When the first economics departments were proposed at Cambridge and Oxford, the proponents thought acceptance would be improved if economics could be seen as incorporating the methods of physics. The enterprise was premised on the existence of economic laws that describe invariant relationships between events. These event regularities, like gravity, were not affected by human action. Humans could adapt and use them, but not change them. Thus the metaphor of "mechanism" seemed appropriate and became embedded in economists' language. It is common to use the term market mechanism to link prices and commodities. This suggests the economy is like turning a crank attached to a set of gears where there is a fixed relationship between the crank's motion and the last gear's motion. The gears have no ideas of their own, they don't get mad; there is no cognitive element between events and action.Institutional and Behavioral Economics,