Stress matrices and global rigidity of frameworks on surfaces

In 2005, Bob Connelly showed that a generic framework in \bR^d is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation. His results gave a key step in the characterisation of generic global rigidity in the plane. We extend these results to frameworks on surfaces in \bR^3. For a framework on a family of concentric cylinders, cones or ellipsoids, we show that there is a natural surface stress matrix arising from assigning edge and vertex weights to the framework, in equilibrium at each vertex. In the case of cylinders and ellipsoids, we show that having a maximum rank stress matrix is sufficient to guarantee generic global rigidity on the surface. We then show that this sufficient condition for generic global rigidity is preserved under 1-extension and use this to make progress on the problem of characterising generic global rigidity on the cylinder.Comment: Significant changes due to an error in the proof of Theorem 5.1 in the previous version which we have only been able to resolve for 'generic' surface

Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jord\'an, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in $\mathbb{R}^2$. The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in $\mathbb{R}^3$ when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson's necessary conditions for the global rigidity of generic frameworks on the cylinder, cone and ellipsoid.Comment: 13 page

Water management in New Zealand: a road map for understanding water value

With New Zealand facing poorer water quality, and shortages, this paper examines the water situation and the policy challenges it presents. Introduction Better water management will benefit all New Zealanders. The focus on water policy has been driven by poorer water quality, shortages, and unease at the costs with poor water allocation. This paper examines the characteristics of the water situation in New Zealand and the policy challenges it presents. We develop a multi-faceted framework for examining those challenges, with a view to helping stakeholders think about what needs to be done for freshwater policy to more accurately reflect society’s preferences. It has been prepared as part of the NZIER’s public good programme to provide an independent and succinct review of what we know and do not know about the topic. In preparing this report, we found that a lot has already been written. We did not want to repeat it – we refer to various publications that the reader may seek out. Instead, we wanted to produce a way of thinking about water, water policy, water management and water allocation that would allow us to reflect on current problems and work towards solutions. The goal is to use water wisely for the benefit of New Zealanders. How we do that depends on a wide range of factors, as this report describes. Most importantly, it depends on how we think about water resources. Thus, we focus on producing some mental models that we hope others will find helpful. This report does not consider any issues arising from the Treaty of Waitangi. We recognise that those issues are important, but we are not experts on them. In addition, the focus of the report is on better water management to improve the wellbeings of all New Zealanders, including tangata whenua. Water policy in New Zealand has been in a state of flux for quite some time. The uncertainty is driven by increased competition for water, a lack of understanding of society’s preferences, a lack of scientific information about water quality and inertia on the part of some users and institutions. It is clear that this situation is changing. We hope this report contributes something positive to that change

Global rigidity of generic frameworks on the cylinder

We show that a generic framework (G,p) on the cylinder is globally rigid if and only if G is a complete graph on at most four vertices or G is both redundantly rigid and 2-connected. To prove the theorem we also derive a new recursive construction of circuits in the simple (2,2)-sparse matroid, and a characterisation of rigidity for generic frameworks on the cylinder when a single designated vertex is allowed to move off the cylinder

Developing a National Design Scoreboard

Recognising the growing importance of design, this paper reports on the development of an approach to measuring design at a national level. A series of measures is proposed, that are based around a simplified model of design as a system at a national level. This model was developed though insights from literature and a workshop with government, industry and design sector representatives. Detailed data on design in the UK is presented to highlight the difficulties in collecting reliable and robust data. Evidence is compared with four countries (Spain, Canada, Korea and Sweden). This comparison highlights the inherent difficulties in comparing performance and a revised set of measures is proposed. Finally, an approach to capturing design spend at a firm level is proposed, based on insights from literature and case studies. Keywords: National Design System, Design Performance</p

Rigid Cylindrical Frameworks with Two Coincident Points

We develop a rigidity theory for graphs whose vertices are constrained to lie on a cylinder and in which two given vertices are coincident. We apply our result to show that the vertex splitting operation preserves the global rigidity of generic frameworks on the cylinder, whenever it satisfies the necessary condition that the deletion of the edge joining the split vertices preserves generic rigidity

Rigid cylindrical frameworks with two coincident points

We develop a rigidity theory for graphs whose vertices are constrained to lie on a cylinder and in which two given vertices are coincident. We apply our result to show that the vertex splitting operation preserves the global rigidity of generic frameworks on the cylinder, whenever it satisfies the necessary condition that the deletion of the edge joining the split vertices preserves generic rigidity

An improved bound for the rigidity of linearly constrained frameworks

We consider the problem of characterising the generic rigidity of bar-joint frameworks in R d in which each vertex is constrained to lie in a given affine subspace. The special case when d = 2 was previously solved by I. Streinu and L. Theran in 2010 and the case when each vertex is constrained to lie in an affine subspace of dimension t, and d ≥ t(t − 1) was solved by Cruickshank, Guler and the first two authors in 2019. We extend the latter result by showing that the given characterisation holds whenever d ≥ 2t