Given a nonzero integer n, Gupta and Saha [‘Integer solutions of the generalised polynomial Pell equations and their finiteness: the quadratic case’, Canad. Math. Bull., to appear] classified all polynomials $x^2+ax+b\in {\mathbb {Z}}[x]$ for which the polynomial Pell equation $P^2-(x^2+ax+b)Q^2=n$ has solutions ${P,Q\in {\mathbb {Z}}[x]}$ with $Q\neq 0$. We generalise their work to the equation $P^2-(f^2+af+b)Q^2=nR$, where f is a fixed polynomial in ${\mathbb {Z}}[x]$. As an application of our results, we study the equation $P^2-D(f)Q^2=n$, where D is a monic, quartic and non square-free polynomial in ${\mathbb {Z}}[x]$. This extends Theorem 1.4 of Scherr and Thompson [‘Quartic integral polynomial Pell equations’, J. Number Theory 259 (2024), 38–56].

Gaming disorder (GD) is increasingly recognized as a clinically significant condition, yet its implications in first-episode psychosis (FEP) remain largely unexplored. This perspective article focuses on the intersection of GD and FEP, highlighting key diagnostic and treatment challenges, including symptom overlap that complicates differential diagnosis, the absence of validated screening tools, and difficulties in sustained patient engagement. Drawing insights from substance use disorder management in FEP, we propose a preliminary clinical framework for integrating GD assessment and intervention into early intervention in psychosis programs. This approach prioritizes comprehensive evaluation, patient-centered care, and a harm-reduction model that supports digital well-being. Addressing GD inFEP populations is crucial for optimizing functional recovery and promoting a holistic, recovery-oriented approach to psychiatric care. Further research is needed to refine screening tools and validate tailored interventions in this population.

The spatial and temporal distribution of species is influenced by multiple processes operating at various scales. Beta diversity, which describes variation in species composition among sites, helps to understand community assembly mechanisms in spatial dimensions. Taxonomic beta diversity reflects differences in species composition, while functional beta diversity accounts for variations in ecological roles and traits among species. Both provide insights into the processes influencing the patterns of composition of communities. Accordingly, this study explores taxonomic and functional beta diversity patterns in anuran communities across different vegetation types in the Atlantic Forest of southeastern Brazil, focusing on two components of variation in species composition: turnover and nestedness. Field surveys were conducted in 16 sites in northern Rio de Janeiro state, covering five vegetation types. Our findings show that species turnover predominantly drives beta diversity, both taxonomically and functionally. Montane and Submontane Ombrophilous Forests contribute significantly to both taxonomic and functional beta diversity by hosting unique assemblages of species, including microendemic taxa and rare functional traits not found in other vegetation types. Nevertheless, other vegetation types also harbour exclusive species and contribute to overall diversity. Rather than focusing solely on conservation recommendations, these findings provide novel insights into how turnover and nestedness contribute differently to taxonomic and functional beta diversity, revealing the distinct ecological processes and habitat characteristics that shape anuran community composition across the Atlantic Forest.

Flower colour is a key trait shaping pollination, reproduction and plant–environment interactions. In arid ecosystems, it may also signal adaptations to heat and (Ultraviolet) UV stress. Tecomella undulata, a threatened keystone tree of the Indian Desert, exhibits striking flower colour polymorphism with yellow, orange and red morphs. This study tested whether artificial intelligence (AI) can reliably classify these morphs, thereby supporting conservation efforts. Field surveys were conducted across natural populations in the Thar Desert. An accessible no-code AI platform (Google Teachable Machine) was used for supervised classification of flower and tree images, with unsupervised clustering applied for validation. The AI classifier achieved high accuracy in distinguishing morphs at both flower and tree scales. Morphs showed consistent separation, with orange functioning as an intermediate form. Despite red morphs being more frequent, the presence of yellow and orange morphs contributes essential functional diversity important for pollinator interactions and reproductive resilience. This study demonstrates that no-code AI provides an effective, scalable approach to documenting intraspecific variation in threatened species. By enabling rapid and reliable identification of flower colour morphs, the approach offers practical applications for ex situ conservation, restoration and morph-aware biodiversity management in T. undulata and other arid-zone trees.

The author’s primary goal in this paper is to characterize $\omega$-Rudin sets and $\omega$-Rudin spaces via sequence convergence and give some important applications of such characterizations. For an irreducible closed set $A$ of a $T_0$-space $X$, we prove that the following four conditions are equivalent: (1) $A$ is an $\omega$-Rudin set; (2) there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(a_n)_{n\in \mathbb{N}}$ simultaneously converges to all points of $A$; (3) there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(\overline {\{a_n\}})_{n\in \mathbb{N}}$ converges to $A$ in the Hoare power space of $X$; (4) there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(\overline {\{a_n\}})_{n\in \mathbb{N}}$ converges to $A$ in the sobrification of $X$. Based on these characterizations, we obtain some characterizations of $\omega$-Rudin spaces and sober spaces. In particular, we show that for a complete lattice $L$, its Scott space $\Sigma L$ is sober iff for any nonempty Scott irreducible closed set $A$ of $L$, there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(a_n)_{n\in \mathbb{N}}$ simultaneously Scott converges to all points of $A$ or, equivalently, the sequence $(\overline {\{a_n\}})_{n\in \mathbb{N}}$ converges to $A$ in the sobrification of $\Sigma L$. Several related examples are presented. We also investigate some basic properties of $\omega$-Rudin spaces. It is proved that the property of being an $\omega$-Rudin space is retractive, productive, and closed-hereditary. We give two examples to show that it is not saturated-hereditary and the category $\boldsymbol{\omega }$-$\mathbf{Rud}$ of $\omega$-Rudin spaces does not have equalizers, and hence, $\boldsymbol{\omega }$-$\mathbf{Rud}$ is not reflective in the category $\mathbf{Top}_{0}$ of all $T_0$-spaces. Finally, we discuss the Smyth power spaces of $\omega$-Rudin spaces. It is shown that if the Smyth power space of a $T_0$-space $X$ is an $\omega$-Rudin space, then $X$ is an $\omega$-Rudin space. The question naturally arises whether the Smyth power space of an $\omega$-Rudin space is still an $\omega$-Rudin space.

Ideal Contractualism views principles of justice as corresponding to what rational, mutually disinterested persons would collectively choose behind a veil of ignorance. It is well-known that Ideal Contractualism faces profound challenges in accounting for justice between generations. We present a unified solution to these problems that involves rejecting the assumption that the parties conceive of their choices as causally efficacious and assumes instead that the parties choose in light of the news value of their decision. And we explore what concrete principles would be chosen by the parties as governing intergenerational justice against the backdrop of this assumption.

In “Parental Love and Filial Equality,” Giacomo Floris and Riccardo Spotorno offer an explanation for why parents should treat their children as equals. The authors argue that this moral obligation is grounded in parents’ duty to love their children in an attitudinal, though not necessarily emotional, sense. This duty, they contend, requires them to disregard variations in their children’s status-conferring properties, as long as those properties meet a minimum threshold. This article argues that this account of filial equal treatment has serious shortcomings. In its place, it proposes a more outcome-oriented or consequentialist account.

Thinking about desire has been integral to radical feminism. My goal is to revive a critical politics of desire informed by the history of radical feminist thought: one sensitive to social determinants of romantic and sexual attraction and open to the possibility that our desires can be radically transformed outside of oppressive environments. To do this, I reconstruct radical feminist strategies for navigating politically problematic desires, including demonstrating that recent scepticism toward this project has underestimated its available resources. In particular, I build upon attempts to reconfigure the social contexts in which romantic and sexual desires are formed, including recommending cultural and economic interventions which influence who is seen as desirable. Radical feminists also recognized potential harms of questioning desire, including the problem of intense sexual moralism. In dialogue with this history, I propose that changes in the infrastructure of desire-formation are often better placed to avoid the unproductive shame and defensiveness associated with a critique of desire. So too, I suggest that attempts to remake our public sexual culture ought to incorporate feminist insights about the importance of imagination, experimentation, and open discussion.