In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schrödinger equation (NLS),

\begin{equation*}\text{i } \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \mathbb{R} \times \mathbb{R}^d,\end{equation*}

where $d \geq 1$, $\mu \in \mathbb{R}$ and $0 \lt \sigma \lt \infty$ if $1 \leq d \leq 4$ and $0 \lt \sigma \lt 4/(d-4)$ if $d \geq 5$. In the mass critical and supercritical cases, we establish the existence of blowup solutions to the problem for cylindrically symmetric data. The result extends the known ones with respect to blowup of solutions to the problem for radially symmetric data.

Age-related changes can affect mental health, but aging-focused mental health research is limited. The objective was to identify the top 10 unanswered research questions on aging and mental health according to what matters most to aging Canadians. A steering group of experts-by-experience (e.g., older adults, caregivers, health and social care providers) guided three phases of a modified James Lind Alliance priority-setting partnership: (1) a broad national survey (n = 305) and a rapid literature scan; (2) a follow-up national survey (n = 703); and (3) four online workshops (n = 52) with a nominal group technique. Forty-two unique questions on aging and mental health resulted, of which 18 were determined to be answered by existing evidence. Of the 25 partially and unanswered questions, 10 were ranked as top priority. Findings can be used to prioritize future research, knowledge mobilization, and funding decisions, and to promote and support collaboration between longstanding siloed research and care fields.

Let Mn and Tn denote the nth Motzkin number and the nth central trinomial coefficient respectively. We prove that for any prime $p\ge 5$,

\begin{equation*}\begin{aligned}\\[-22pt]&\sum_{k=0}^{p-1}M_k^2\equiv \left(\frac{p}{3}\right)\left(2-6p\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}kM_k^2\equiv \left(\frac{p}{3}\right)\left(9p-1\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}T_kM_k\equiv \frac{4}{3}\left(\frac{p}{3}\right)+\frac{p}{6}\left(1-9\left(\frac{p}{3}\right)\right)\pmod{p^2},\\[-6pt]\end{aligned}\end{equation*}

where $\left(-\right)$ is the Legendre symbol. These results confirm three supercongruences conjectured by Z.-W. Sun in 2010.

Iguaçu National Park, a UNESCO World Heritage Site, faces threats from proposed legislation aimed at downgrading its protection to allow reopening of the Caminho-do-Colono road, which would fragment the National Park. This study assesses the National Park’s importance in preserving native vegetation and its economic significance via payments for environmental services. We evaluated different scenarios, demonstrating how these resources are vital for certain municipalities. By comparing economic performance before and after the road’s closure, our findings reveal that, contrary to local beliefs, the closure did not negatively impact the local economy. Additionally, these legislative bills are legally infeasible and fail to offer environmental or economic benefits, favouring small interest groups over the collective good. This research highlights that the Caminho-do-Colono road is not a viable approach to municipal development. Instead, maintaining the National Park’s integrity is crucial for both ecological preservation and economic stability in the region. Our analysis underscores the broader implications of environmental conservation efforts and the necessity of rejecting projects that compromise protected areas. This study provides a comprehensive evaluation of the National Park’s role, emphasizing the need for sustainable development that aligns with environmental and community well-being.

A numerical method is proposed for a class of one-dimensional stochastic control problems with unbounded state space. This method solves an infinite-dimensional linear program, equivalent to the original formulation based on a stochastic differential equation, using a finite element approximation. The discretization scheme itself and the necessary assumptions are discussed, and a convergence argument for the method is presented. Its performance is illustrated by examples featuring long-term average and infinite horizon discounted costs, and additional optimization constraints.

This paper calls for the lawyering profession – which is often viewed as unabridged – to be reframed into two distinct occupations: legal aid practice and private practice, to better incorporate the divisions in labour. In order to better understand contemporary legal aid work and its workers, the hidden realities must be unveiled from behind their private counterparts, which opposingly signify wealth, professionalism, autonomy and privilege. Set within a context of crumbling professional identities, a shrinking industry and financial constraints, the paper draws on ethnographic and interview data. It finds that those working in legal aid undoubtedly face a more stagnated, under-resourced and precarious working environment, which means that their professional experience is vastly different from their private counterparts. Likewise, those in the field face toxic narratives from the government, the media, the public, and their private counterparts alike, resulting in persistent discourse of vilification. Ultimately, it calls for a refocus of legal aid work as a separate vocation due to its altruistic underpinnings, unique ‘professional’ identity, and values.

Associations between adversity and youth psychopathology likely vary based on the types and timing of experiences. Major theories suggest that the impact of childhood adversity may either be cumulative in type (the more types of adversity, the worse outcomes) or in timing (the longer exposure, the worse outcomes) or, alternatively, specific concerning the type (e.g., parenting, home, neighborhood) or the timing of adversity (e.g., specific developmental periods). In a longitudinal sample from the Future of Families and Wellbeing Study (N = 4,210), we evaluated these competing hypotheses using a data-driven structured life-course modeling approach using risk factors examined at child age 1 (infancy), 3 (toddlerhood), 5 (early childhood), and 9 (middle childhood). Results showed that exposures to more types of adversity for longer durations (i.e., cumulative in both type and timing) best predicted youth psychopathology. Adversities that occurred at age 9 were better predictors of youth psychopathology as compared to those experienced earlier, except for neglect, which was predictive of internalizing symptoms when experienced at age 3. Throughout childhood (across ages 1–9), aside from the accumulation of all adversities, parental stress and low collective efficacy were the strongest predictors of internalizing symptoms, whereas psychological aggression was predictive of externalizing symptoms.

The Qing dynasty enforced a policy of separate governance for the people of the Eight Banners, ruling that bannermen were neither to be administered under the regular civilian administrative system, nor listed on the civilian register. Institutionally and legally, the labels “Banner” (qi 旗) and “civilian” (min 民) marked a fundamental divide between different social groups in the Qing. However, in actual practice, the boundary between the two was less rigid. An ambiguous area existed within the seemingly strict legal and administrative regime, providing opportunists with an abundance of loopholes to exploit. Some changed their status from “civilian commoner” to “bannermen” to acquire land, while others moved from “bannermen” status to “civilian commoner” status to pursue promotion in the civil service. Shedding light on the everyday lives of these people, this article delves into the intricate Banner–civilian classification of the Qing dynasty, with a focus on the overlapping area between the parallel systems. It aims to rectify the conventional binary perspective that strictly dichotomizes Banner and civilian status. By doing so, it highlights the multifaceted nature and diversity inherent in Qing ethnic relations and local society.

We investigate some aspects of the problem of the estimation of birth distributions (BDs) in multi-type Galton–Watson trees (MGWs) with unobserved types. More precisely, we consider two-type MGWs called spinal-structured trees. This kind of tree is characterized by a spine of special individuals whose BD $\nu$ is different from the other individuals in the tree (called normal, and whose BD is denoted by $\mu$). In this work, we show that even in such a very structured two-type population, our ability to distinguish the two types and estimate $\mu$ and $\nu$ is constrained by a trade-off between the growth-rate of the population and the similarity of $\mu$ and $\nu$. Indeed, if the growth-rate is too large, large deviation events are likely to be observed in the sampling of the normal individuals, preventing us from distinguishing them from special ones. Roughly speaking, our approach succeeds if $r\lt \mathfrak{D}(\mu,\nu)$, where r is the exponential growth-rate of the population and $\mathfrak{D}$ is a divergence measuring the dissimilarity between $\mu$ and $\nu$.

The closed material procedure (CMP) – ever since its introduction to English law – has been subjected to a very significant amount of academic criticism. But over time, the CMP has become increasingly settled as a fixture in English law. Whilst the existence of the CMP per se in English law seems settled, the extent of its deployment is not. Given this development, it seems important and constructive to examine whether – and the conditions under which – a CMP can ever be normatively justified, all things considered. Two propositions will be made. First, a common argument for the CMP – the maximising argument – does not demonstrate that the CMP is normatively justified, all things considered, for it does not sufficiently mitigate the two main objections to the CMP, based respectively on the principles of natural justice and open justice. Secondly, where the clear advantage variation is deployed – ie when a CMP allows the excluded party to make use of material that: (a) clearly advantages him; and (b) would otherwise be unavailable for the court's consideration – both objections are sufficiently mitigated. In such a case, the CMP is normatively justified, all things considered. This constitutes a limited normative case for the CMP.

Globally, companies are developing and implementing their strategies to reduce greenhouse gas emissions to limit the global temperature increase to 1.5 degrees Celsius above pre-industrial levels, in line with the Paris Agreement. However, there is also growing recognition and awareness of the potential negative impacts of these activities on human rights. Recent pressure from international normative standards and ESG expectations, emerging legislative developments, and legal precedents are driving companies to consider human rights impacts across their climate action. This piece explores some of the human rights risks associated with the transition to renewable energy; the transformation to regenerative agriculture; the scaling up of the circular economy; and the implementation of nature-based solutions. It then explores the challenges of conducting effective human rights due diligence given the scale and scope of the transitions needed and provides examples of how companies are seeking to respect human rights in their climate action.