In a previous article [1] I showed that the probability of a difference of t units arising when n numbers from a uniform distribution are rounded before, as opposed to after, they are added is(1)Here n is a non-negative integer and t is a positive or negative integer with, the maximum possible error.

Amid declining public standing, many political parties seek to regain disaffected voters through various institutional strategies. One key approach is democratizing legislative candidate selection to grant party members or voters greater influence and signal improved responsiveness, transparency, and legitimacy. Yet does this strategy pay off electorally? The growing literature on this topic provides conflicting answers and limited evidence. We argue that more inclusive candidate selection does not have meaningful effects at the polls despite its merits. Whereas voters favor such procedures in principle, as some suggest, they underprioritize them in favor of other considerations when electing parties. We support this argument with observational and experimental data, including a matching difference-in-differences estimation of party performance across thirty-four democracies and a survey democracies and a survey experiment in three countries. This article contributes to our understanding of the relationship between party institutions and voter behavior in an age of eroding public trust and rising anti-establishment sentiment.

Who do we blame when bad things happen? Has division in American society made us less sympathetic to victims of tragedies? In previous trying times (e.g., 9/11 and Columbine), Americans rallied together to support victims and seek government solutions. In a highly polarized era, however, we have witnessed further division rather than unity. In this paper, I leverage original, pre-registered survey experiments to examine how much Americans blame and sympathize with someone who has tragically died from COVID-19. The studies find consistent evidence that partisans blame victims who hold an anti-vaccine perspective, regardless of partisanship. Less consistent evidence suggests that Democrats also blame victims who were Republican, but less than they do victims who held anti-vaccination views. Further, partisans are less sympathetic when the victim was anti-vaccine, but Democrats and Republicans are also less sympathetic when the person who died was an outpartisan. These results indicate that animosity towards outpartisans persists even through tragedy, but demonstrates limits to affective partisan polarization paired with evidence of rational blame and sympathy responses.

Algal beds are biodiversity hotspots in coastal ecosystems, forming complex food webs based on seaweeds. Epifauna serve as important indicators of algal productivity and environmental change, interacting closely with their host seaweeds. This study compared the composition of epifaunal assemblages associated with different seaweed species in the Noto Peninsula, Japan, to elucidate their characteristics. A survey of epifaunal communities on 13 seaweed species around Tsukumo Bay revealed significant differences in abundance and composition among species. Epifaunal abundance was particularly high in Sargassum siliquastrum and Sargassum patens. Differences in assemblage structures were observed among seaweed species and sampling sites. Fucales seaweeds harboured more diverse epifaunal communities compared with Laminariales species. Variations in epifaunal composition among taxa suggest that interactions with host seaweed species play a crucial role in shaping assemblage structure.

The article examines how the Czech courts addressed restrictions on the right to conduct business during the COVID-19 pandemic. Although grounded in the Czech authorities’ response to the health crisis, the analysis extends to broader conclusions about the necessity of upholding rule-of-law principles. The article therefore highlights how constitutional imperatives of proportionality, equality and the justification of government actions must be preserved in judicial review, even in times of crises.

A Rascal triangle is a Pascal-type numeric triangle developed in 2010 by three middle-school students, Alif Anggoro, Eddy Liu and Angus Tulloch [1]. During their mathematics classes they were challenged to provide the next row for the following sequence:The expected answer was the one that matches Pascal’s triangle, i.e. 1 4 6 4 1, obtained by applying the recurrence rule of binomial coefficients South = East + West. Instead, the young students suggested that the next row should be 1 4 5 4 1. In contrast to Pascal’s triangle rule South = East + West, they produced the new row using the relation they called the diamond formulaBy applying the diamond formula (1), the students produced an original triangular sequence known as the Rascal triangle, see Table 1.

The use of rumūz – symbols or ciphers – is pervasive across the Islamicate alchemical corpus. Historians have long debated how such symbols should be interpreted, with some emphasizing the spiritual or mystical idiom often employed by alchemists, and others seeking to decrypt alchemical literature for its technical or proto-scientific content. This paper proposes that one fruitful way to approach rumūz is to compare depictions of alchemy drawn from three distinct genres of source material: alchemical treatises, literary works and pedagogical encyclopedias. By tracing how each treats the purpose, form and interpretation of rumūz, I show how these differences illuminate the scholarly practices, pedagogical methods and intellectual positioning of alchemists in the Mamlūk period. More broadly, the study demonstrates the value of widening the evidentiary base beyond specialist treatises, revealing how alchemical discourse intersected with wider literary culture, reading practices and the transmission of esoteric knowledge in the later medieval Islamicate world.

Mathematics and physics are fascinatingly connected with each other. Since [1] first pointed out the relationship between the number π and the total number of collisions between two blocks and a wall, it has generated wide interest among the general public [2, 3, 4, 5]. A carefully made video presents an elegant and crystal clear geometric explanation [6], which is followed by a slightly more advanced explanation based on linear algebra, matrix formalism in particular [7]. References [2, 3, 6] attribute the geometric approach to different individuals, but the relationship actually appeared much earlier in a physics journal which focused on the number of collisions rather than its connection with the number π [8].

Non-perturbing high resolution spectroscopic diagnostic methods have been developed to reliably measure the temporal and spatial distributions of physical quantities in the strong-field region and in the plasmas in pulsed-power devices. The methods were employed to investigate the properties of the acceleration gap and the behavior of the highly dynamic nonequilibrium anode plasma in intense ion diodes. Conclusions on the electron density and current density in the diode gap, the magnetic field induced by the current flow, the plasma conductivity, plasma heating, plasma expansion, particle fluxes and velocity distributions in the plasma, and possible use in other pulsed-power configurations are discussed.

This article explores the early modern scholarly debate over the cock’s crow in the New Testament account of Peter’s denial, focusing on theologians and savants such as Johann Georg Altmann (1697–1758), John Lightfoot (1602–75) and Adriaan Reland (1676–1718). What began as a narrow philological puzzle, whether the text referred to a rooster or a human herald, expanded into a broader debate over scriptural authority. Set within the intellectual context of the republic of letters, the article shows how efforts to reconcile Scripture with ancient Jewish law and classical sources could unwittingly sow the seeds of doubt and unbelief.

We investigated variations in biomass among oil palm progenies and quantified K and Mg uptake by assessing the distribution pattern of these minerals throughout plant organs, including roots and bunches. A split-plot trial was set up in which potassium chloride (KCl: 60% K2O) and kieserite (MgSO4: 27% MgO) were applied as main factors, each with three levels: 0, 1.5, 3.0 kg KCl palm˗1 year˗1; and 0, 0.75, 1.5 kg MgSO4 palm˗1 year˗1. Each fertiliser combination was applied to subplots containing four oil palm progenies (C1 to C4). Samples of oil palm organs were collected to assess their dry matter (DM) and mineral contents. Oil palm biomass was significantly different among progenies, with C3 having the highest biomass (268 kg DM palm˗1). K and Mg contents in the entire palm also varied among progenies. C2 and C3 progenies exhibited the highest amounts of K (3.81 and 3.86 kg K palm˗1, respectively), whereas C1 and C4 displayed 3.35 and 3.31 kg K palm˗1, respectively. However, C4 progeny showed the highest leaflet K concentration, revealing the dilution phenomenon and the inequal mineral distribution among palm organs. Progeny C3 had the highest Mg content (0.54 kg palm˗1) and was the most productive, exporting more K and Mg through harvest than other progenies. Our results indicate genetic variability in K and Mg uptake by oil palm, and planting materials with high K and Mg uptake efficiency would be valuable for farmers and breeders.

This study applies the scaling patch approach to investigate the influence of pressure gradients on the mean-momentum balance in turbulent boundary layers (TBLs). Under strong pressure gradients, the force balance in the outer region is dominated by advective and pressure forces, with gradients of Reynolds stresses playing a minimal role. To retain the relevance of Reynolds stress gradients within the scaling patch framework, we propose a redistribution of the component $U_e \textrm {d}U_e/\textrm {d}x$ from the advective term to the pressure-gradient term. Here, $U_e$ is the mean streamwise velocity at the boundary layer edge. This reformulation enhances the outer-scaling framework of Wei & Knopp (2023 J. Fluid Mech. 958, 1–21), ensuring consistency across a wide range of pressure gradients, including those involving flow separation. Remarkably, the new outer-scaled gradient of Reynolds shear stress in TBLs under a pressure gradient closely resembles that observed in zero-pressure-gradient TBLs. In the inner region, the impact of pressure gradient is well captured by the Stratford–Mellor parameter $\beta _{\textit{in}}$. For weak pressure gradients ($|\beta _{\textit{in}}| \ll 0.07$), traditional inner scaling remains valid. However, for stronger pressure gradients $|\beta _{\textit{in}}| \gtrsim 0.07$, the near-wall dynamics is governed by a balance between pressure gradient and viscous force, as described by Stratford (1959 J. Fluid Mech. 5, 1–16) and Mellor (1966 J. Fluid Mech. 24, 255–274). In this sub-layer, viscosity and the imposed wall pressure gradient dictate the relevant velocity and length scales. Moreover, when $|\beta _{\textit{in}}| \gtrsim 0.7$ and the wall pressure $P_{w\textit{all}}$ gradient $\textrm { d}P_{w\textit{all}}/\textrm {d}x \gt 0$, a distinct sub-layer emerges outside the pressure–viscous balance region, characterised by a dominant balance between the imposed pressure gradient and the gradient of the Reynolds shear stress. In this region, the Reynolds shear stress increases linearly with distance from the wall. These findings provide new insights into the structure of TBLs under pressure gradients and establish a refined framework for modelling their dynamics.

We consider the problem of predicting the next bit in an infinite binary sequence sampled from the Cantor space with an unknown computable measure. We propose a new theoretical framework to investigate the properties of good computable predictions, focusing on such predictions’ convergence rate.

Since no computable prediction can be the best, we first define a better prediction as one that dominates the other measure. We then prove that this is equivalent to the condition that the sum of the KL divergence errors of its predictions is smaller than that of the other prediction for more computable measures. We call that such a computable prediction is more general than the other.

We further show that the sum of any sufficiently general prediction errors is a finite left-c.e. Martin-Löf random real. This means the errors converge to zero more slowly than any computable function.