We present an explicit subset $A\subseteq \mathbb{N} = \{0,1,\ldots \}$ such that $A + A = \mathbb{N}$ and for all $\varepsilon \gt 0$,

\begin{equation*}\lim _{N\to \infty }\frac {\big |\big \{(n_1,n_2): n_1 + n_2 = N, (n_1,n_2)\in A^2\big \}\big |}{N^{\varepsilon }} = 0.\end{equation*}

This answers a question of Erdős.

The failure of the proposed Equal Rights Amendment (ERA) has been attributed to various organized interests, including the New Right and insurance companies. This study examines trends in lobby efforts regarding the amendment and correlations between lobby efforts and roll call votes among state legislators. Lobbyists active on the amendment appeared most often in states they perceived were most likely to approve. A second data set consisting of 6,952 votes reveals that explicitly pro- and anti-ERA lobby efforts were correlated with votes cast only by Republican state legislators. Lobby efforts by insurance companies were not correlated with any votes. The efforts of pro- and anti-amendment lobbyists, however, likely had no effect on the ultimate fate of the proposed amendment. Women and non-white legislators voted more often for the amendment, regardless of party. Moreover, changes in public support for the amendment led to partisan differences in legislators’ votes.

This paper explores the relationship between entrepreneurship, measured by the number of new firms per million inhabitants, and modern economic growth in Spain between 1886 and 2000. Following Audretsch and Keilbach’s methodology, our analysis seems to confirm that entrepreneurship has had a positive and statistically significant effect on GDP per capita and labor productivity. This finding challenges the traditional view that the entrepreneurial factor has hindered the country’s economic growth. Additionally, using data on the size and legal form of start-up firms, our results suggest that neither characteristic has been an important driver of Spain’s long-term economic growth. However, we find that the impact of both variables differs depending on the years studied. To our knowledge, this study is the first attempt to test econometrically the long-term contribution of entrepreneurship to Spain’s economic growth.

This article concerns the interpretation of refuse dumps discovered at three abandoned Soviet tactical nuclear bases in Poland and how their analysis prompted a reassessment of archaeological remote sensing results. The study employed a range of methods to document the remnants of these secret sites, including declassified spy satellite images, aerial photographs, airborne and terrestrial laser scanning, UAV prospection, and field surveys, supplemented by CIA reports and Warsaw Pact military documents. These data bridge significant gaps in archival records, offering valuable insights into the history of these sites. However, the discovery of Cold War-era refuse dumps near the bases containing materials that do not conform to other evidence present an interpretative challenge. It exposed ‘survivorship bias’ in the dataset, prompting a re-evaluation of earlier conclusions.

Ideas about morphological complexity have been used to classify languages and to link complexity to language age and social structure. Creoles and sign languages are often framed as younger and structurally simpler than other languages. Concurrently, sign language morphology has been described as paradoxical, as both simple and complex. This paper is a critical examination of claims about morphological complexity and its relationship to language age and social structure. We show that the theoretical and empirical foundations of claims that sign language morphology is paradoxical are flawed. Specifically, argumentation and evidence supporting analogies between creole and sign language complexity adopt theoretically contested and ideologically problematic assumptions about creoles and uncritically apply them to sign languages. We identify four flaws in argumentation: (i) use of limited morphological data to generate claims about global complexity, (ii) association of binary language categories with categorical complexity differences, (iii) use of language age to motivate predictions about morphological complexity, and (iv) extrapolating from creole complexity to sign language complexity. Based on these flaws, we develop nine theoretical and practical recommendations for working with morphological complexity and discuss uncritical cross-disciplinary transfer of ideas.

Commemorations of the Confederacy remain pervasive throughout the Southern U.S. Historians have long established that many of these symbols were erected during the Jim Crow era to reinforce white political dominance in public spaces. Yet, little is known about how these enduring symbols shape perceptions among people of different racial identities today. This study examines Confederate monuments where they are most prominently placed: courthouse grounds. Using an original survey experiment of Black, white, and Latino Southerners, it investigates whether the presence of a Confederate monument in front of a courthouse influences feelings of personal safety and welcomeness, as well as perceptions of the fairness of the court system. Findings reveal that a Confederate monument made Black and Latino Southerners feel less safe and welcome at the courthouse and led Black Southerners to perceive the court system as less fair toward people like them. In contrast, Confederate monuments had no overall effect on white Southerners’ perceptions of courthouses or the judicial system. These results underscore the role of contentious symbols in reinforcing inequalities in public spaces.

Bone and tissue fractions, obtained in 2017 following hip replacement surgery on a healthy Caucasian male, born in 1944, reflect in their 14C concentrations the integrated effect of the lifetime metabolic uptake and replacement of atmospheric bomb 14C at different tissue-specific turnover rates. The 14C content of hair and nails reflects recent carbon uptake. The 14C values in healthy cartilage and bone collagen/apatite correspond to those of the “local” atmosphere during the 2005–2009 Northern Hemisphere growing seasons, while those from damaged areas of the femur head correspond to the atmosphere in 2013–2014. A simple bone growth and regeneration model used in combination with the NH atmospheric 14C concentrations indicates remodelling rates around 9% per year in the healthy bone and a doubling to tripling in the damaged area depending on the model chosen. The differences in 14C concentration observed in the fractions provide both a caveat for sample selection for the 14C dating of archaeological bones and an indication of its potential in forensics and as a diagnostic tool for turnover rates in medical studies.

This article examines sovereign creditworthiness concerns and policies in a Latin American country that needed economic development and stabilization financing from bankers, the International Monetary Fund, and the World Bank during the early years of the Bretton Woods era. It underlines the significance for developing country foreign financing breakthroughs of applying sound, coherent, and sustainable macroeconomic policies; of credible and professionalized state institutions; of adhering to formal and informal rules of mainstream international finance; and the policymaking role of trustworthy economic teams coming from the local establishment who endorsed foreign financiers’ ideas and recipes. While written from the perspective of economic history, the analysis incorporates recent insights from earlier historical periods and worldwide case-studies, and of specialists in international political economy and credit rating studies.

Psychiatry education at both undergraduate and postgraduate level plays a critical role in shaping the future of psychiatry services. South Asia varies in the training offered and this article captures this aspect.

In this article, the authors contend that three blades, archaeometrically identified as made of obsidian from the Nemrut Dağ source in eastern Anatolia, were recovered from bona fide archaeological contexts at two sites in Poland. This is supported by somewhat contentious contextual evidence, which is thoroughly reviewed. If the findspots are accepted as genuine, these artefacts would mark the furthest western distribution of Nemrut Dağ obsidian, approximately 2200 km away from the source, more than three times the previously recorded western distribution of this material. The known history of recovery and curation of these artefacts, their techno-typological features, and their raw material source (based on EDXRF analysis) are assessed, and an interpretation of this unusual material is offered.

We provide a presymplectic characterization of Liouville sectors introduced by Ganatra–Pardon–Shende in [10, 12] in terms of the characteristic foliation of the boundary, which we call Liouville σ-sectors. We extend this definition to the case with corners using the presymplectic geometry of null foliations of the coisotropic intersections of transverse coisotropic collection of hypersurfaces, which appear in the definition of Liouville sectors with corners. We show that the set of Liouville σ-sectors with corners canonically forms a monoid that provides a natural framework for considering the Künneth-type functors in the wrapped Fukaya category. We identify its automorphism group that enables one to give a natural definition of bundles of Liouville sectors. As a byproduct, we affirmatively answer a question raised in [10, Question 2.6], which asks about the optimality of their definition of Liouville sectors in [10].

While the Common European Framework of Reference (CEFR) for languages was originally developed for the European context, its influence has extended to other regions. The present paper highlights this growing influence by reporting on revisions to grammar in South Korea’s National English Curriculum based on CEFR criteria and the related CEFR-based English Grammar Profile resource. Specifically focusing on Appendix 4 of the 2015 curriculum, which consists of example sentences of language forms for communication (e.g. Kate is from London), the revising process based on CEFR and the English Grammar Profile involved two steps: 1) adding grammatical categories for the example sentences, and 2) reorganizing the school level where the grammatical categories are recommended to be taught. The resulting changes were implemented in the 2022 Korean National English Curriculum, which began being applied nationwide to English education in 2024.