The bending energy of a thin, naturally straight, homogeneous and isotropic elastic rod of length L is given byis the arclength parametrisation and κ = γ (s) the curvature vector. Consider the following boundary value problem: Given points P , Q ∈ R m and unit vectors v, w ∈ S m−1 find the shapes of static elastic curves with clamped ends and fixed length. Defining the spacethis can be reformulated to find the minimizers of F : C → R.

The bending energy of a thin, naturally straight, homogeneous and isotropic elastic rod of length L is given byis the arclength parametrisation and κ = γ (s) the curvature vector. Consider the following boundary value problem: Given points P , Q ∈ R m and unit vectors v, w ∈ S m−1 find the shapes of static elastic curves with clamped ends and fixed length. Defining the spacethis can be reformulated to find the minimizers of F : C → R.

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ObjectivesPatient and public involvement (PPI) is argued to lead to higher quality health research, which is more relatable to and helps empower the public. We synthesised the evidence to look for examples of PPI in health research in low/middle-income countries (LMICs), looking at levels of involvement and impact. Additionally, we considered the impact of who was undertaking the research on the level of involvement and reported impact.DesignSystematic review.Data sourcesEMBASE, Medline and PsychINFO, along with hand-searching references, grey literature, Google search and expert advice.Eligibility criteriaAny health research with evidence of patient or public involvement, with no language restrictions dated from 1978 to 1 Dec 2017.Data extraction and synthesisData relating to stage and level of involvement, as well as impact, were extracted by one researcher (NC), and a coding framework was developed using an inductive approach to examine the impact of PPI on research. Extracted data were then independently coded by a second lay researcher (RK) to validate the data being collected. Discrepancies were referred to a third independent reviewer (MT) for review and consensus reached.ResultsSixty-two studies met the inclusion criteria. The review revealed the most common stage for PPI was in research planning, and the most common level of involvement was collaboration. Most studies did not provide evidence of effectiveness or elaborate on the impact of PPI, and they tended to report impact from the researcher’s perspective. Where impact was mentioned, this generally related to increased relevance to the community, empowerment of participants and alterations in study design.ConclusionsThe literature describing approaches to and impact of PPI on LMIC health research is sparse. As PPI is essential to conducting high-quality research, it should be fully reported and evaluated at the end of the research project.

Building on and from the work of Brydges and Imbrie, we give an elementary calculation of the volume of the space of branched polymers of order n in the plane and in 3-space. Our development reveals some more general identities, and allows exact random sampling. In particular we show that a random 3-dimensional branched polymer of order n has diameter of order √ n.