We establish a criterion for the integrality of averages of roots of unity and apply it to settle a conjecture regarding the linearity of functions on $\mathbb {Z}_n$. Specifically, we prove that for any modulus $n \ge 1$, if $f: \mathbb {Z}_n \to \mathbb {Z}_n$ is a function whose averages $({1}{/n}) \sum _{x=0}^{n-1} \omega ^{f(x)+bx}$ (where $\omega =e^{2\pi i/n}$) are algebraic integers for all $b \in \mathbb {Z}_n$, then f is necessarily linear modulo n. This provides a short, elementary proof that works uniformly for all n and avoids the finite-field machinery used in previous partial results. Furthermore, when $n=p^r$, we use a local–global integrality argument to show that any normalised sum of $p^r$th roots of unity that is p-adically integral must be either $0$ or a single root of unity. As an application, we completely characterise the perfect isometries of the cyclic group $C_{p^r}$; they are precisely those induced by affine permutations $x \mapsto \alpha x + \beta $ with $\gcd (\alpha , p^r)=1$.

The goal of this Article is to contribute to the broader discussion on strengthening the resilience of the rule of law in Germany by focusing on how to protect trust in the judiciary and judicial independence against (possibly) radicalized judges. The Article provides an introduction to the current legal situation in Germany regarding judges’ active involvement in politics, as well as an overview of the historical background. It addresses the regulation governing the right to return to a post as a judge after having been a member of parliament, and disciplinary rules regarding appropriate behavior of judges. It illustrates the current legal status quo and considers possible consequences for judicial independence and public trust in the judiciary. The Article concludes with proposals to improve the resilience of the judiciary.

This article assesses whether recent Irish activation reforms for lone parents’ function as capacitating elements of a social investment (SI) strategy. Although activation is presented as evidence of Ireland’s shift towards SI, our analysis questions this claim using the concept of agentic capacity – the structured conditions enabling individuals to balance care, employment, and education. Drawing on two small-scale studies, we centre lived experiences of activation requirements and income supports and find that while access to education and training can strengthen autonomy and future planning, these benefits are constrained by conditionality, administrative burdens, childcare shortages, and age-based thresholds. The reforms thus represent a model in which coercive activation logics persist within SI rhetoric. By advancing agentic capacity as an evaluative criterion, the article contributes to debates on activation and SI, highlighting how institutional design, childcare provision, temporal alignment, and conditionality shape meaningful choice for lone parents.

Spectral broadening in guided-wave optics, an effect intrinsic to the interplay of dispersion and nonlinearity, presents a major obstacle to achieving the high spectral purity required for applications in nonlinear frequency conversion and precision spectroscopy. This effect is especially detrimental in high-power cascaded Raman fibre lasers (CRFLs). Leveraging a concept analogous to wavefront shaping but implemented passively, we introduce and validate a novel architecture for CRFLs incorporating random fibre grating-assisted spectral tailoring. A theoretical model is developed to show that the randomized reflectance profile of the grating induces wavelength-selective feedback, promoting intracavity mode competition and noise suppression through enhanced coherent enhancement of selected spectral components. Experimentally, we demonstrate a maximum output power exceeding 1 kW at 1185 nm from the cascaded Raman fibre oscillator. The random grating provides enhanced spectral control, achieving a 3-dB linewidth of approximately 1.0 nm. Notably, it also significantly suppresses the spectral wings, reducing the 10- and 30-dB linewidths to 2.9 and 9.5 nm, respectively. This work represents the first implementation of built-in passive spectral tailoring in an all-fibre nonlinear dynamical system, offering insights into the control of complex light fields in both temporal and frequency domains throughout cascaded nonlinear processes.

Herbicides are a critical tool for controlling aggressive invasive species, especially during large-scale restoration efforts, but their persistence in soils can hinder the establishment and success of desirable vegetation. Soil residence times differ among herbicides, and plant species exhibit variable tolerance to residues, causing difficulty in determining appropriate timelines for revegetation activities following herbicide application. Imazapyr is a commonly used herbicide to control highly aggressive species, such as giant reed (Arundo donax L.), with reported soil residence time ranging from three to twelve months. However, we observed herbicide damage on foundational tree species installed more than two years after imazapyr was applied to control expansive A. donax monocultures as part of a large-scale riparian forest restoration project. We monitored signs of herbicide damage and growth for Salix and Populus spp. individuals installed from cuttings to evaluate the relative susceptibility to and differential impacts of imazapyr residues on trees established using this standard propagation technique. Individuals from all species exhibited signs of herbicide damage, and the height of damaged individuals was significantly lower than that of healthy individuals across species. Salix Laevigata had the greatest percentage of damaged individuals (27%), but the smallest reduction in height due to herbicide damage (-13%). The frequency of herbicide damage was similar among Populus species (13%), but the fastest growing species, P. trichocarpa, experienced a much greater decline in growth (-67%) compared to P. fremontii (-32%) and P. x perryi (-31%). The study found that imazapyr, when used to control invasive monocultural species, can have lingering, previously undocumented effects on non-target riparian vegetation. Despite these negative impacts, imazapyr is still one of the most effective treatment options for some aggressive invaders. Thus, practitioners should be aware of the potential long-term effects of imazapyr and carefully consider species selection and planting schedule during revegetation.

We combine data from subscription lists in rights offers with church records and individual tax returns to study the demography of the Swedish stock market around 1920. The typical shareholder was a middle-aged married man with a high income and university education or business affiliation. Stock ownership was gender contingent and changed over the life cycle. Men became stock market investors before they married, while women became stockholders through marriage. Men continued investing in old age, while women depleted their stock portfolios. Women who failed to marry purchased stocks to provide for their financial future. We relate these life-cycle patterns to mortality risk, migration patterns, and different opportunities for men and women. We conclude that the stock market supported family formation and female emancipation.

The concept of ‘ecology of fear’ recognises the importance of non-consumptive effects of predators inspiring fear in their prey, but has subsequently been broadened to include the non-consumptive effects of parasites on their hosts. Here, we experimentally tested whether blue mussels (Mytilus edulis) exhibit species-specific avoidance behaviour in response to infective transmission stages (cercariae) of three trematode species: Himasthla elongata and Renicola roscovita, both of which use blue mussels as hosts, and Cryptocotyle lingua, which parasitises fish. Furthermore, we investigated whether avoidance behaviour could be triggered solely by chemical cues. The results show that mussels reduced clearance rates by 34% in the presence of H. elongata alone and of 51% in the presence of H. elongata and R. roscovita in sympatry, which was comparable to previously reported direct consumptive effects of infection on filtration activity. The presence of C. lingua and R. roscovita alone had no significant effect on clearance rate. The observed reduction in filtration activity resulted in lower infection rates of experimental mussels. Moreover, mussels reduced clearance rates by 42% in the mere presence of chemical cues from the first intermediate host Littorina littorea, while exudates of H. elongata had no additional negative effect. It is evident that a non-consumptive parasite effect can significantly reduce the filtration activity of experimental blue mussels. Considering the magnitude of this reduction and the widespread occurrence of parasitic trematodes, it is reasonable to expect that fear of parasitism has ecological consequences for the functioning role of mussels.

Reverse mortgage markets remain small internationally, with bequest motives frequently cited as a key reason. We develop a new two-generation lifecycle simulation model to study the role of reverse mortgages in intergenerational financial planning, particularly as a tool for families to bring forward bequests. The model incorporates parental altruism by assuming parents derive utility not only from their own consumption and housing but also from their child’s current and future financial well-being. This extends traditional bequest models, which typically consider only the wealth transferred at death. The model accounts for house price risk, interest rate risk, investment risk, wage growth, health shocks, long-term care costs, private pensions, and means-tested public pensions. Using this framework, calibrated to Australian economic and policy settings, we compare the welfare gains from bequests and early bequests (inter vivos gifts) for homeowning parents and adult children seeking to purchase their first home. The results suggest that families across a range of wealth levels can experience substantial welfare gains when the parent uses a reverse mortgage both for retirement income and to gift the adult child a first home deposit. Early bequests funded through reverse mortgages increase overall family welfare compared to preserving home equity for a bequest or using a reverse mortgage solely for the parent’s consumption, particularly for middle-wealth households. A policy experiment shows that gifting limits reduce welfare gains for some families, but have a small overall impact. These findings suggest that policies encouraging informed use of reverse mortgages could improve intergenerational financial security.

This report presents 50 AMS radiocarbon dates from wood charcoal samples extracted from newly uncovered Neolithic settlement layers at the archaeological site of Amzabegovo (Sveti Nikole, North Macedonia). It constitutes the largest radiocarbon dataset from a Neolithic site in North Macedonia. The calibrated dates span 6391–5635 cal BC (95.4%), falling within the Early and Middle Neolithic. With an inferred occupation onset potentially as early as 6391–6236 cal BC (95.4%), Amzabegovo provides the earliest securely dated Neolithic settlement layers in the Ovče Pole catchment area and serves as an important chronological reference point for Early Neolithic expansion in the Vardar River basin. The consideration of the radiocarbon dates from the American excavations led by Marija Gimbutas in 1969–1970 highlights both the possibilities and the limitations of juxtaposing the old and new datasets. Future analyses of the archaeological stratigraphy and material from Amzabegovo hold the potential to offer insights into the Neolithic period in the southern Balkans and beyond, with the new radiocarbon dates significantly enhancing our understanding of the region’s chronological developments.

We investigate plasma wake generation via Compton scattering from photon bursts in the Thomson regime, a non-ponderomotive process relevant when the photon wavelength is shorter than the interparticle distance. In this regime, electrons can reach relativistic velocities. We extend linear theory to the nonlinear regime, showing that plasma waves can reach the wave-breaking limit. Perfectly collimated drivers produce wakes propagating at the speed of light, allowing electron phase-locking (limited by driver depletion). Non-collimated drivers induce subluminal phase velocities, limiting acceleration via dephasing. Two-dimensional simulations reveal unique transverse fields compared with laser wakefields, with a DC magnetic field leading to consistent focusing. The work considers observational prospects in laboratory and astrophysical scenarios such as around highly luminous compact objects (e.g. pulsars, gamma-ray bursts) interacting with tenuous interstellar or intergalactic plasmas, where conditions favour Compton-dominated wakefield acceleration.

In 1985, Adams [‘The algebraic independence of certain Liouville continued fractions’, Proc. Amer. Math. Soc. 4 (1985), 512–516] proved that any finite number of Liouville series in different bases are algebraically independent. In this paper, we prove the equivalence between algebraic and linear independence when the bases are equal.

This research paper aimed to determine if bio-stimulation affects peri-oestrous behavioural patterns of buffalo heifers and can improve oestrous behavioural responses through fenceline and direct contact with bulls during the heifers’ first and second oestrous cycles. Buffalo heifers are known to have poor and less intense expression of oestrus behaviour. This leads to lower oestrous detection rates and ultimately results in poor reproductive efficiency. Providing social cues, such as bio-stimulation, is an effective technique for improving oestrous behaviours in cattle, pigs and sheep. For this study, 24 pre-pubertal Murrah heifers were allotted to 3 groups of 8 each; similar in age (15.08 ± 0.16 months) and body weight (199.5 ± 0.21 kg). In T0, heifers were not exposed to a bull; in T1, heifers had continuous exposure to a bull behind a fenceline; and in T2, heifers were directly exposed to a bull twice daily (6 hours total). Oestrous behaviours were recorded on days −3, −2 and −1 (prior to oestrus), day 0 (day of oestrus) and on days +1, +2 and +3 (post-oestrus) using 24-hour digital cameras. Oestrous behaviours (sniffing/licking, tail raising, micturition, chin resting and mounting attempts) commenced on day −3 or day −2, peaked on day 0 and waned by day +2 or day +1 during both oestrous cycles. Oestrous behaviour frequencies were higher for the heifers in T1 and T2 groups (P < 0.05) in comparison to the T0 group during both cycles. Receptive standing in response to mounting was only observed during the second oestrous cycle on the day of oestrus, and the frequency was significantly highest in T2, followed by T1 and T0 (P < 0.05). It was concluded that direct or fenceline bull exposure triggered more expression of oestrous behaviours in buffalo heifers, with direct exposure having a greater effect than fenceline exposure.

The Speaker of Parliament is potentially a highly influential actor in parliamentary politics. Yet, we know surprisingly little about how they make decisions. In many parliaments, they, for instance, choose who gets to speak in question time. By examining a unique data set from the Netherlands, which lists all proposed oral questions and which ones were selected to be asked in parliament, this paper examines the considerations of the Speaker when selecting questions. This gives a unique insight into their considerations in general. Our central question is as follows: under what conditions are Speakers more likely to select an oral question? We look at the following three options: the neutral arbiter that seeks to treat all parties fairly; the partisan Speaker who defends the interests of their own party; and the asset of the coalition that protects the unity of the governing coalition. Our analyses indicate that Speakers are more likely to block questions that would drive a wedge in the coalition and are more likely to accept questions that meet established norms of urgency and importance.

We present a detailed study of the evolution of the number of connected components in sub-critical multiplicative random graph processes. We consider a model where edges appear independently after an exponential time at a rate equal to the product of the sizes of the vertices. We provide an explicit expression for the fluid limit of the number of connected components normalized by its initial value, when the time is smaller than the inverse of the sum of the square of the initial vertex sizes. We also identify the diffusion limit of the rescaled fluctuations around the fluid limit. This is applied to several examples. In the particular setting of the Erdős–Rényi graph process, we give the fluid limit of the normalized number of connected components, and the diffusion limit of the scaled fluctuations in the sub-critical regime, where the mean degree is between zero and one.