An ultradiscrete matrix version of the fourth Painleve equation

We establish a matrix generalization of the ultradiscrete fourth Painlev\'e equation (ud-PIV). Well-defined multicomponent systems that permit ultradiscretization are obtained using an approach that relies on a group defined by constraints imposed by the requirement of a consistent evolution of the systems. The ultradiscrete limit of these systems yields coupled multicomponent ultradiscrete systems that generalize ud-PIV. The dynamics, irreducibility, and integrability of the matrix valued ultradiscrete systems are studied.Comment: 12 pages, 12 figures, Latex2e, Submitted to J. Phys. A, corrections mad

An Elliptic Garnier System

We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at 4m + 12 points for m ≥ 1, which appear in pairs due to a symmetry condition. We parameterize this linear system in terms of a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which m = 1 with the elliptic Painlevé equation; hence, this work provides an explicit form and Lax pair for the elliptic Painlevé equation

People need freshwater biodiversity

Freshwater biodiversity, from fish to frogs and microbes to macrophytes, provides a vast array of services to people. Mounting concerns focus on the accelerating pace of biodiversity loss and declining ecological function within freshwater ecosystems that continue to threaten these natural benefits. Here, we catalog nine fundamental ecosystem services that the biotic components of indigenous freshwater biodiversity provide to people, organized into three categories: material (food; health and genetic resources; material goods), nonmaterial (culture; education and science; recreation), and regulating (catchment integrity; climate regulation; water purification and nutrient cycling). If freshwater biodiversity is protected, conserved, and restored in an integrated manner, as well as more broadly appreciated by humanity, it will continue to contribute to human well-being and our sustainable future via this wide range of services and associated nature-based solutions to our sustainable future

Key considerations for patient and public involvement and engagement in health research

Patient and public involvement and engagement (PPIE) can provide valuable insights into the experiences of those living with and affected by a disease or health condition. Inclusive collaboration between patients, the public and researchers can lead to productive relationships, ensuring that health research addresses patient needs. Guidelines are available to support effective PPIE; however, evaluation of the impact of PPIE strategies in health research is limited. In this Review, we evaluate the impact of PPIE in the ‘Therapies for Long COVID in non-hospitalised individuals’ (TLC) Study, using a combination of group discussions and interviews with patient partners and researchers. We identify areas of good practice and reflect on areas for improvement. Using these insights and the results of a survey, we synthesize two checklists of considerations for PPIE, and we propose that research teams use these checklists to optimize the impact of PPIE for both patients and researchers in future studies.</p

Using large climate ensembles to plan for the hydrological impact of climate change in the freshwater environment

We explore the use of large ensembles of climate scenarios to inform climate change adaptation in response to hydrological impacts on the freshwater environment, using a sensitive chalk river in south east England to illustrate the approach. The climateprediction.net experiment provides large ensembles of transient climate series from 1920 to 2080. We use 246 transient climate series in the CATCHMOD rainfall-run-off model to develop large ensembles of plausible river flows for the River Itchen. This transient ensemble allows the exploration of how flows may change through the twenty-first century, and demonstrates the range of possible consequences for freshwater ecosystems, based on invertebrate community impacts. Hydrological modelling of flow sequences including abstraction allows the continued effectiveness of river support from groundwater to be assessed. A new environmental impact matrix considers the response of the freshwater ecosystem in the Itchen, concentrating particularly on macro-invertebrates. Through the century increasing numbers of models fail the flow targets, with a minority of models suggesting flows that would lead to irreversible change to the invertebrate community. The large ensemble provides a richer picture of the range of possible change, allowing managers to explore a range of different responses. The approach used is illustrative, but demonstrates that large ensembles may be of great value in improving the understanding of the possible impact of climate change, provided that they can be communicated effectively to decision-makers