Let $p(n)$ and $M(m,L;n)$ be the number of partitions of n and the number of partitions of n with crank congruent to m modulo L, respectively, and let

$$ \begin{align*}M(a,b,c;L;n):= M(a,L;n) + M(b,L;n) + M(c,L;n).\end{align*} $$

In this paper, we focus on some relations between $M(m,L;n)$ and $p(n),$ which Dyson, Andrews, and Garvan etc. have studied previously. By applying a modification of the circle method to estimate the Fourier coefficients of generating functions, we establish the following inequalities between $M(a,b,c;L;n)$ and $p(n):$ for $n \geq 467$,

$$ \begin{align*} M(0,1,1;9;n)> \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 0,1,5,8 ~ (\mathrm{mod}~9), \\ M(0,1,1;9;n) < \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 2,3,4,6,7 ~ (\mathrm{mod}~9), \\ M(2,3,4;9;n) < \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 0,1,5,8 ~ (\mathrm{mod}~9), \\ M(2,3,4;9;n)> \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 2,3,4,6,7 ~ (\mathrm{mod}~9). \end{align*} $$

In the proof of these inequalities, an inequality for the logarithm of the generating function for $p(n)$ is derived and applied. Our method reduces the last possible counterexamples to $467 \leq n \leq 22471$, and it will produce more effective estimates when proving inequalities of such types.

Law & Society scholars often dismiss Law & Economics (L&E) as insoluble with our core beliefs about distributive justice, culture, and social solidarity. This reaction has yielded missed opportunities for new theory emergent between the fields. One such opportunity came in 1978, when Guido Calabresi and Philip Bobbitt argued that societies make “tragic choices” about scarce resource allocations so as to reconcile such choices with core culture, ethics, and values. In Calabresi’s later words, their book was a “more or less explicit appeal to anthropology for help.”1 Today, sociolegal studies remain well-poised to answer this appeal. Taking theory about moral costs from Calabresi in L&E and adding anthropological thought on the meaning of “value,” this essay presents situated valuation – a contextualized notion of value that accounts for the moral costs of inequalities while supporting principled scrutiny of redistributive policies meant to reduce inequality but sometimes worsening it. This discussion highlights the importance of interpretive social science in the study of distributive inequality, while showcasing a neglected but generative link between mutually imbricated interdisciplinary communities.

Early childhood education and care (ECEC) is among the most important services for children and their parents as it promotes children’s development and enables mothers’ employment. Previous research has shown that there is an educational gradient as children of mothers with a low education level participate less in ECEC services, but less is known about the development of this inequality. This study, using EU-SILC survey data, focuses on the development of inequality in ECEC use of children under 3 years of age during 2004–2019, and on disparities between three categories of education levels among mothers. The results show that, together with increasing ECEC participation rates, overall inequality has increased in Europe. Inequality has increased between low- and other education levels, whereas in a few cases, a decrease has happened between medium- and high-educated mothers. It is important to pay attention to socioeconomic disparities with rising participation rates.

For $r\in(0,1)$, let $\mu \left( r\right) $ be the modulus of the plane Grötzsch ring $\mathbb{B}^2\setminus[0,r]$, where $\mathbb{B}^2$ is the unit disk. In this paper, we prove that

\begin{equation*}\mu \left( r\right) =\ln \frac{4}{r}-\sum_{n=1}^{\infty }\frac{\theta _{n}}{2n}r^{2n},\end{equation*}

with $\theta _{n}\in \left( 0,1\right)$. Employing this series expansion, we obtain several absolutely monotonic and (logarithmically) completely monotonic functions involving $\mu \left( r\right) $, which yields some new results and extend certain known ones. Moreover, we give an affirmative answer to the conjecture proposed by Alzer and Richards in H. Alzer and K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl. 432(1): (2015), 134–141, DOI 10.1016/j.jmaa.2015.06.057. As applications, several new sharp bounds and functional inequalities for $\mu \left( r\right) $ are established.

Gender equity and authorship diversity are believed to be the essential parts of building a dynamic scientific atmosphere. The purpose of the present study was to determine the status of gender equity in research on echinococcosis and the editorial diversity in major parasitology journals over the past four decades. All articles were retrieved from major databases from the years 1980, 2000, 2010, 2015, and 2020. Journals belonging to the four quartiles of parasitology journals listed in the Journal Citation Report were selected, and the gender and region of each editorial member were identified. Among the 3583 first authors of the articles published in all selected years, 2236 (62.4%) were men, whereas 1040 (29%) were women. There was a significant increase in women’s contributions as the first author, from 6.8% in 1980 to 35.8% in 2020 (P < .001). A greater gender gap was found for the senior authors, showing 2391 (66.7%) men and 837 (23.4%) women. The gender gap has been narrowed in most of the six regions of the world, particularly for the Western Pacific region, where the gender inequity had almost diminished in 2020; i.e. the man-woman ratios of the first and last authors from this region were 2.25 and 1.75 in 1980, reaching 1.04 and 0.97, in 2020, respectively. Our findings also indicated that articles authored by men received 2.5 to 3.1 times more citations than women authors. Gender distribution of the editors-in-chief, associate editors, and editorial board members across all quartiles showed that 78.7%, 69.5%, and 72.7% were men, respectively, and mostly affiliated with the European and American regions. Findings of the present study showed that gender inequity is still present and women researchers continue to be the minority in the field of parasitology, particularly in the research on echinococcosis.

In this paper, we construct an elaborate general equilibrium model with a continuum of production fragments for an intermediate good, then incorporate it in a growth model to address the effects of global production fragmentation, vertical specialization, and trade on growth and inequality for a small developing country. Among other things, we show that a small developing economy grows faster than the rest of the world as a result of global fragmentation and trade in intermediates if it is skilled-labor scarce. We further address the effects of such a trade opening on wage inequality.

During the last decades, political distrust has seemingly become a common trend across Latin American democracies, however, differences in the levels of confidence among groups have also been identified. This article considers the potential effects of ethno-racial structures and their interactions with other forms of socioeconomic inequalities on political trust. Building on data from four waves of the Latinobarometer project and contextual measures from different sources, we analyze these relations and find that both socioeconomic and ethno-racial inequalities affect political trust and impact on the formation of different relations with the political system across Latin America. Furthermore, in particular it is found that at the individual-level interactions between inequalities shape political trust differently depending on the particular ethno-racial identification. These findings contribute to the understanding of ethnicity and race and its associations with other structural inequalities in shaping mass political culture.

What are the consequences of including a “don't know” (DK) response option to attitudinal survey questions? Existing research, based on traditional survey modes, argues that it reduces the effective sample size without improving the quality of responses. We contend that it can have important effects not only on estimates of aggregate public opinion, but also on estimates of opinion differences between subgroups of the population who have different levels of political information. Through a pre-registered online survey experiment conducted in the United States, we find that the DK response option has consequences for opinion estimates in the present day, where most organizations rely on online panels, but mainly for respondents with low levels of political information and on low salience issues. These findings imply that the exclusion of a DK option can matter, with implications for assessments of preference differences and our understanding of their impacts on politics and policy.

This paper studies the process of labour market formation in the tourism industry in Spain. Results show that tourism regions diverged in their capacity to attract local labour, a factor that led to different compositions of the workforce. In the most dynamic regions, circular migration became a key factor as a result of housing shortages, seasonality and labour policy. Tourism agents promoted these flows by different mechanisms such as recruitment at origin and temporary accommodation. Migration benefited growth of firms, natives' upward mobility and migrants' accumulation of capital. However, inequality in the regional labour market and host society increased.