The primitive solutions to x^3+y^9=z^2

We determine the rational integers x,y,z such that x^3+y^9=z^2 and gcd(x,y,z)=1. First we determine a finite set of curves of genus 10 such that any primitive solution to x^3+y^9=z^2 corresponds to a rational point on one of those curves. We observe that each of these genus 10 curves covers an elliptic curve over some extension of Q. We use this cover to apply a Chabauty-like method to an embedding of the curve in the Weil restriction of the elliptic curve. This enables us to find all rational points and therefore deduce the primitive solutions to the original equation.Comment: 8 page

Wetting and drying of a rigid substrate under variation of the microscopic details

Wetting and drying of a rigid substrate by a Lennard-Jones fluid in molecular dynamics simulations is reported. The size of the substrate particles, being smaller than the fluid particles in former simulations, is now taken to be equal to, respectively larger than, that of the fluid particles. Recently, for the latter type of system a first order drying transition has been reported. Like before we find a continuous-like transition for all systems considered. This also holds for substrates with incompletely-filled top layers, the so-called molecularly rough surfaces. All systems studied behave qualitatively alike, but inconsistencies are found in the solid-vapour surface tension on approach of the wetting transition and for the solid-fluid surface tension in general.Comment: 17 pages Latex, 6 figures, to appear in Physica

The arithmetic of Prym varieties in genus 3

Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym-variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to do Chabauty- and Brauer-Manin type calculations for curves of genus 5 with an unramified involution. As an application, we determine the rational points on a smooth plane quartic with no particular geometric properties and give examples of curves of genus 3 and 5 violating the Hasse-principle. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over Q(t).Comment: 21 page

Computing in Jacobians of projective curves over finite fields

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations can be done efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps and Kummer maps; and finding a basis for the $l$-torsion of the Picard group, where $l$ is a prime number different from the characteristic of the base field.Comment: 42 page

Visualising Sha[2] in Abelian Surfaces

Given an elliptic curve E1 over a number field and an element s in its 2-Selmer group, we give two different ways to construct infinitely many Abelian surfaces A such that the homogeneous space representing s occurs as a fibre of A over another elliptic curve E2. We show that by comparing the 2-Selmer groups of E1, E2 and A, we can obtain information about Sha(E1/K)[2] and we give examples where we use this to obtain a sharp bound on the Mordell-Weil rank of an elliptic curve. As a tool, we give a precise description of the m-Selmer group of an Abelian surface A that is m-isogenous to a product of elliptic curves E1 x E2. One of the constructions can be applied iteratively to obtain information about Sha(E1/K)[2^n]. We give an example where we use this iterated application to exhibit an element of order 4 in Sha(E1/Q).Comment: 17 page

Sowing in the autumn season : exploring benefits of green care farms for dementia patients

In the Netherlands an increasing number of farms combine agricultural production with care services for people with care needs. It is generally believed that these green care farms (GCFs) have beneficial effects on the health status of a diversity of target groups. At present, empirical studies testing this hypothesis are scarce. The main objective of the studies described in this thesis was to gain insight into the potential benefits of day care at GCFs for community‐dwelling older dementia patients. Day care at GCFs was therefore compared with day care at regular day care facilities (RDCFs). In view of the differences between both day care types regarding the day care setting and day care program it was hypothesized that they would differ in their effects on the health status of dementia patients. In two cross‐sectional studies it was tested to what extent the day program of dementia patients at GCFs differed from those at RDCFs. It appeared that at GCFs, dementia patients were (physically) more active, participated in more diverse activities, were more outdoors, and had more opportunities to perform activities in smaller groups than those at RDCFs. It was tested whether these differences resulted into different effects for five domains of health: dietary intake, cognition, emotional well‐being, behaviour, and functional performance. In a comparative cross‐sectional study dietary intake of dementia patients attending day care at GCFs or RDCFs was recorded both at home and during their time at the day care facility. The study showed that dementia patients attending day care at GCFs had significantly higher intakes of energy, carbohydrate, and fluid than their counterparts attending day care at RDCFs. In a cohort study, rates of change during 1 year in cognitive functioning, emotional well‐being, behavioural symptoms, and functional performance were compared between dementia patients attending day care at GCFs and RDCFs. Functioning in these domains remained rather stable and no differences were observed between subjects from GCFs and RDCFs. In the cohort study, also caregiver burden of family caregivers of these dementia patients was assessed. Caregivers’ quality of life, emotional distress, and feelings of competence remained rather stable in family caregivers of dementia patients from both day care settings. In conclusion, the present work has shown that GCFs exceeded RDCFs in offering older dementia patients a diverse day program and in stimulating their dietary intake. The latter may result into a better preserved nutritional status in dementia patients attending day care at GCFs than in those attending day care at RDCFs. GCFs and RDCFs were equally effective in preventing significant decrease of cognitive functioning, emotional well‐being, and functional performance and in preventing significant increase of the number of behavioural symptoms. Both day care types further prevented significant increase of caregiver burden. Day care at GCFs is a new and valuable addition to the present care modalities for community‐dwelling older dementia patients and their caregiver