This article argues that two words in line 8.612 of the Aeneid, promissa and perfecta, have been transposed since the poem's composition, and that the restoration of their correct order yields a preferable sense. This corruption would have happened at an early stage in the poem's transmission, but there is some reason to believe that Servius’ comment on the verse reflects its original state.

The Frobenius–Schur indicators of characters in a real $2$-block with dihedral defect groups have been determined by Murray [‘Real subpairs and Frobenius–Schur indicators of characters in 2-blocks’, J. Algebra 322 (2009), 489–513]. We show that two infinite families described in his work do not exist and we construct examples for the remaining families. We further present some partial results on Frobenius–Schur indicators of characters in other tame blocks.

We developed a do-it-yourself test protocol using commercial Bacillus atrophaeus spores to assess the efficacy of ultraviolet-C (UV-C) light room-decontamination devices. Overall, 4 UV-C devices reduced B. atrophaeus by ≥3 log10 colony-forming units in 10 minutes, whereas a smaller device required 60 minutes. Of 10 in-use devices, only 1 was ineffective.

In many countries, the structure of wages and the labor law legislation are completely different for public and private sector employees. In this paper, we develop a general equilibrium overlapping generations model to study the effect of such differences on household savings and labor supply. To conduct our analysis, first we use microdata from two Brazilian household surveys to document that civil servants save and work significantly less than their counterparts in the private sector. Second, we use matched employer–employee microdata from Brazil (RAIS) to document differences between the two sectors in terms of wage and unemployment risk. Then, we calibrate the model to be consistent with micro and macro evidence for Brazil. Our counterfactual exercises show that differences in wages characteristics and labor law legislation account for nearly 70% of the gap in savings between civil servants and private sector workers, and 57% of the gap in labor supply. In addition, we find that eliminating those differences can produce sizable increase on aggregate savings, employment, and welfare.

In this paper, we consider the dynamical behaviour of a reaction–diffusion model for a population residing in a one-dimensional habit, with emphasis on the effects of boundary conditions and protection zone. We assume that the population is subjected to a strong Allee effect in its natural domain but obeys a monostable nonlinear growth in the protection zone $[L_1,\, L_2]$ with two constants satisfying $0\leq L_1< L_2$, and the general Robin condition is imposed on $x=0$ (i.e. $u(t,\,0)=bu_x(t,\,0)$ with $b\geq 0$). We show the existence of two critical values $0< L_*\leq L^*$, and prove that a vanishing–transition–spreading trichotomy result holds when the length of protection zone is smaller than $L_*$; a transition–spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$; only spreading happens when the length of protection zone is larger than $L^*$. Based on the properties of $L_*$, we obtain the precise strategies for an optimal protection zone: if $b$ is large (i.e. $b\geq 1/\sqrt {-g'(0)}$), the protection zone should start from somewhere near $0$; while if $b$ is small (i.e. $b< 1/\sqrt {-g'(0)}$), then the protection zone should start from somewhere away from $0$, and as far away from $0$ as possible.

Let $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$. In this paper, we introduce and characterize $\mathcal {F}$-(almost) greedy bases. Given such a family $\mathcal {F}$, a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb {N}$, and $G_m(x)$, we have

$$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$

$G_m(x)$

$\mathbb {K}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

Furthermore, we provide several examples of bases that are nontrivially $\mathcal {F}$-greedy. For a countable ordinal $\alpha $, we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$, where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $. We show that for each $\alpha $, there is a basis that is $\mathcal {S}_{\alpha }$-greedy but is not $\mathcal {S}_{\alpha +1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta $,

$$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$

In an influential recent study, Broockman and Skovron (2018) found that American politicians consistently overestimate the conservativeness of their constituents on a host of issues. Whether this conservative bias in politicians’ perceptions of public opinion is a uniquely American phenomenon is an open question with broad implications for the quality and nature of democratic representation. We investigate it in four democracies: Belgium, Canada, Germany, and Switzerland. Despite these countries having political systems that differ greatly, we document a strong and persistent conservative bias held by a majority of the 866 representatives interviewed. Our findings highlight the conservative bias in elites’ perception of public opinion as a widespread regularity and point toward a pressing need for further research on its sources and impacts.

While scholars have focused extensively on the consequences of partnerships between interest groups, less attention has been paid to the historical dynamics shaping when, how, and why such groups unite. This is especially true of “unexpected” partnerships, which unite groups with seemingly little in common. Such partnerships are important, as they can reshape to an unusual degree which actors, issues, and ideas “fit together” politically. We address the puzzle of how unexpected group partnerships form through case studies of previously non-existent alliances between labor unions and the gay/lesbian rights movement in the United States and United Kingdom in the 1970s and 1980s. Using these cases, we produce a theory arguing that unexpected partnerships are a product of a favorable political opportunity structure—present when each group experiences shared threats and mutual vulnerabilities—and the actions of entrepreneurial group leaders, who forge cross-group identities, accentuate shared ideological convictions, and build institutional ties.

To preserve the functioning of WTO dispute settlement following the blockage of the Appellate Body, a sub-set of WTO Members created the Multi-Party Interim Appeal Arbitration Arrangement (MPIA). In the wake of the first appeal award rendered under the MPIA, this contribution describes how the MPIA process works and lists some of the innovations that can be found in the first MPIA procedure. More innovations can be expected as arbitration under Article 25 of the Dispute Settlement Understanding (DSU) (be it ad hoc or under the MPIA) can be adjusted and molded case-by-case by the disputing parties in their appeal arbitration agreements. In this sense, the MPIA can serve not only as an interim stop-gap to preserve WTO dispute settlement. It can also function as a laboratory to explore and test new ways of making WTO dispute settlement more efficient and in line with WTO Members' goals and interests: experimental reform by doing, rather than one-off, formal DSU review.

In their logical analysis of theorems about disjoint rays in graphs, Barnes, Shore, and the author (hereafter BGS) introduced a weak choice scheme in second-order arithmetic, called the $\Sigma ^1_1$ axiom of finite choice (hereafter finite choice). This is a special case of the $\Sigma ^1_1$ axiom of choice ($\Sigma ^1_1\text {-}\mathsf {AC}_0$) introduced by Kreisel. BGS showed that $\Sigma ^1_1\text {-}\mathsf {AC}_0$ suffices for proving many of the aforementioned theorems in graph theory. While it is not known if these implications reverse, BGS also showed that those theorems imply finite choice (in some cases, with additional induction assumptions). This motivated us to study the proof-theoretic strength of finite choice. Using a variant of Steel forcing with tagged trees, we show that finite choice is not provable from the $\Delta ^1_1$-comprehension scheme (even over $\omega $-models). We also show that finite choice is a consequence of the arithmetic Bolzano–Weierstrass theorem (introduced by Friedman and studied by Conidis), assuming $\Sigma ^1_1$-induction. Our results were used by BGS to show that several theorems in graph theory cannot be proved using $\Delta ^1_1$-comprehension. Our results also strengthen results of Conidis.

Timely grazing decision-making requires routine information on the herbage mass (HM) and pasture growth rate (GR). The aim of this study was to compare the correspondence, cost and reliability of two indirect methods –the comparative yield method (COMPYLD) and the pasture-meter (CDAX)– to estimate HM and weekly GR of a 42 ha grazing area. Weekly assessments from April 2017 to October 2018 were made with both methods to estimate HM and GR of 13 individual paddocks. In addition, estimated GR were compared to aerial net primary productivity (ANPP) estimated using remote sensing (SAT). Estimated HM was 22% lower for COMPYLD than CDAX (HMCOMPYLD = 33 + 0.78*HMCDAX, R2 = 0.61, CV = 17%, RMSE = 291 kgDM/ha). The correspondence between methods of estimated weekly GR of individual paddocks was weak (GRCDAX = 0.18*GRCOMPYLD + 19.1, R2 = 0.05, CV = 73%, RMSE = 21.8 kgDM/ha/d). However, when integrated in three-week moving-averages, over the complete grazing area, COMPYLD and CDAX yielded similar GR up to 35 kg DM/ha/d. Accumulating GR of the grazing area over one year resulted similar to annual SAT-estimated ANPP. These results imply that, on one hand, decisions based on nominal HM, such as target HM and grazing strip size, would need to be adjusted depending on the method, but on the other hand, decisions based in temporal trends or GR, such as size and timing of set-aside areas for reserves, would be unaffected by method. Compared with COMPYLD, CDAX would be advantageous whenever high labour costs offset higher amortization, maintenance and fuel costs, provided there is an alternative in place to monitor during downtime periods.

The covid-19 pandemic has revived a longstanding, and understudied, trope in American politics: the association of immigrants with disease. There has been a great deal of scholarship on the economic, cultural, and criminal threat frames attached to immigrant groups in media coverage, but little to date has specifically examined how national and local sources have framed covid-19 in the context of immigrant communities. In this paper we analyze the prevalence of two different framings of the pandemic in national and local online news outlets over the first year of the pandemic: immigration as a public health threat to the nation, and covid-19 as a threat to immigrant communities within the nation. We find significant differences between national and local coverage, with the former more likely to frame immigration as a covid-19 threat, while local news outlets were more likely to discuss the threat the virus posed to already marginalized immigrant communities.