The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation

We establish the existence of a real solution y(x,T) with no poles on the real line of the following fourth order analogue of the Painleve I equation, x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for y(x,T) as x\to\pm\infty.Comment: 27 pages, 5 figure

The Hamiltonian Structure of the Second Painleve Hierarchy

In this paper we study the Hamiltonian structure of the second Painleve hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The n-th element of the hierarchy is a non linear ODE of order 2n in the independent variable $z$ depending on n parameters denoted by ${t}_1,...,{t}_{n-1}$ and $\alpha_n$. We introduce new canonical coordinates and obtain Hamiltonians for the $z$ and $t_1,...,t_{n-1}$ evolutions. We give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates

Exposure to green spaces may strengthen resilience and support mental health in the face of the covid-19 pandemic

Mughal and colleagues provide a useful overview of mental health support during the covid-19&nbsp;pandemic. Outdoor recreation in green spaces has emerged during the present pandemic as an essential tool to combat distress.&nbsp;The mental health benefits of exposure to green spaces has been documented extensively before.&nbsp;Here we briefly demonstrate that such exposure may restore and improve aspects of resilience and complement other means of mental health support, such as that offered by GPs. Firstly, exposure to green spaces facilitates recovery from physiological stress, restoration of directed attention, and improvement of cognitive performance. This aids in strengthening mental resilience by dampening the body’s stress response, allowing for a more mindful thinking style and supporting adaptive thinking styles.&nbsp;Secondly, green spaces may stimulate physical activity and improve physical resilience. Thirdly, purposeful or pro-environmental behaviour and the anticipation of seeing interesting species (birds or orchids, for example) may activate positive emotions and improve emotional resilience. And finally, gathering outdoors, even at a safe distance, improves social cohesion and helps people to feel connected to the outside world, thus reinforcing social&nbsp;resilience. To unlock the potential of green spaces for human health, it is recommended that they are more explicitly included in public health and spatial planning policies, and that prescribed exposure to green spaces is more often used to complement mental health management in primary care, now and post-covid-19. In highly urbanised and densely populated regions, however, natural areas are usually too small to accommodate increasing recreational pressure without jeopardising their biodiversity. As the mental health benefits of green spaces often depend more on perceived biodiversity than true species richness,&nbsp;we argue that pressure on nature could be alleviated by greening and improving recreation infrastructure in cities and rural landscapes. Investing in natural resilience is investing in human&nbsp;resilience.</p

Residential green space, gardening, and subjective well-being: A cross-sectional study of garden owners in northern Belgium

Urban green spaces and the biodiversity therein have been associated with human health and well-being benefits, but the contribution of domestic gardens to those benefits is insufficiently&nbsp;known. Using data from a cross-sectional sample (n&nbsp;=&nbsp;587) of domestic garden owners in Flanders and Brussels (northern Belgium), associations between residential green space quality in and around domestic gardens, green space related activities and socioeconomic background variables of the&nbsp;gardeners, and self-reported health (stress and depression) were investigated with structural equation&nbsp;models. Socioeconomic security was associated with lower stress and depression. Nature relatedness and green space in the neighbourhood of the house were associated with higher exposure to green space, which was in turn negatively associated with stress and depression. Garden quality, indicated by biodiversity values and size, and nature relatedness were associated with being active in the garden, which was in turn associated with lower values of depression, but not&nbsp;stress. Nature relatedness seems to play a key role in the pathway linking gardens to improved health. Improving biodiversity and ecosystems services in gardens may increase exposure to green space and help to restore and enhance nature relatedness. This, in turn, could potentially improve human health and well-being, and contribute to the&nbsp;conservation of biodiversity&nbsp;in urban&nbsp;environments.</p