Must a primitive non-deficient number have a component not much larger than its radical?

Let $n$ be a primitive non-deficient number, with $n= p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ where the $p_i$ are distinct primes. Let $R=p_1p_2 \cdots p_k.$ We prove that there must be an $i$ such that $p_i^{a_i+1} < 4R^2$. We conjecture that there is always an $i$ such that $p_i^{a_i+1} < kR$ and prove this stronger inequality in some cases.Comment: 10 page

Who Governed Yale? Kingman Brewster and Higher Education in the 1970s

Relying on archival material and oral history, this essay examines two committees at Yale in the 1970s as case studies in how University President Kingman Brewster reshaped the school after the student unrest of the long 1960s. The first committee, led by the political scientist Robert Dahl, endorsed the equal admission of female students in 1972. The second committee, chaired by historian C. Vann Woodward, composed a nationally renowned report on the importance of “unfettered” free expression at the university in 1974-5. I show how each of these committees was a carefully calibrated political tool that allowed Brewster to moderate extreme positions he had taken in the 1960s on the questions of coeducation and free speech. More broadly, this essay contributes to our understanding of the American academy in the 1970s, at a time after the civil rights and anti-Vietnam War fervor had waned on college campuses

Determinants of Monetary Poverty in the European Union

The main objective of the study is to analyze the most important determinants of monetary poverty (at macro-level) in the Western EU countries taking into account the effects of regional spillovers. According to the latest estimates over 16 per cent of the EU citizens are poor (based on monetary concept). Using Europe 2020 strategy indicator people at risk of poverty or social exclusion over 23 percent of EU citizens can be considered poor. In this study a spatial Durbin model (SDM) is employed. The sample includes 145 regions at NUTS-2 (in few cases at NUTS-1) level of 11 countries from the western part of the European Union. The at-risk-of-poverty rate (i.e. monetary poverty indicator) across western EU regions is the dependent variable, and four explanatory variables are employed in the study: disposable per capita income; long-term unemployment rate; education level and population density. All variables refer to observation year 2008. In order to quantify the impacts of explanatory variables the scalar summary measures are used. According to the results two non-spatially lagged explanatory variables (education and population density) and two spatially lagged explanatory variables (income and education) are not statistically significant. In terms of the scalar summary impact measures the following patterns can be observed: average direct impacts, as well as indirect and total impacts of income are negative. Average direct impacts of unemployment are positive, average indirect impacts are negative, and the average total effects are statistically insignificant. Average direct effects of population density are not statistically significant, but indirect and total effects are positive. Impacts of proxy for education level (defined as share of persons aged 25-64 with lower secondary education attainment) are statistically not significant. Such a result cannot not be interpreted in the sense that education has no impact on poverty levels. On the other hand we can assume that the given proxy measures only quantity, not the quality of education, and hence the variable is not significant