title: Biembeddings of cycle systems using integer Heffter arrays

authors: Cavenagh, Nicholas J.

subjects: mathematics

abstract: In this paper, we use constructions of Heffter arrays to verify the existence of face 2‐colorable embeddings of cycle decompositions of the complete graph. Specifically, for n ≡ 1 (mod 4) and k ≡3(mod 4), n k ≫ ⩾ 7 and when n ≡ 0(mod 3) then k ≡ 7(mod 12), there exist face 2-colorable embeddings of the complete graph K₂ₙₖ₊₁ onto an orientable surface where each face is a cycle of a fixed length k. In these embeddings the vertices of K₂ₙₖ₊₁ will be labeled with the elements of Z₂ₙₖ₊₁ in such a way that the group, (Z₂ₙₖ₊₁, +) acts sharply transitively on the vertices of the embedding. This result is achieved by verifying the existence of nonequivalent Heffter arrays, H (n ; k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo 2nk + 1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arrays H (n ; k) that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, for k ⩾ 11, that there are at least (n − 2)[((k − 11)/4)!/ ]² such nonequivalent H (n ; k) that satisfy both conditions (1) and (2).

text: Biembeddings of cycle systems using integer Heffter arrays In this paper, we use constructions of Heffter arrays to verify the existence of face 2‐colorable embeddings of cycle decompositions of the complete graph. Specifically, for n ≡ 1 (mod 4) and k ≡3(mod 4), n k ≫ ⩾ 7 and when n ≡ 0(mod 3) then k ≡ 7(mod 12), there exist face 2-colorable embeddings of the complete graph K₂ₙₖ₊₁ onto an orientable surface where each face is a cycle of a fixed length k. In these embeddings the vertices of K₂ₙₖ₊₁ will be labeled with the elements of Z₂ₙₖ₊₁ in such a way that the group, (Z₂ₙₖ₊₁, +) acts sharply transitively on the vertices of the embedding. This result is achieved by verifying the existence of nonequivalent Heffter arrays, H (n ; k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo 2nk + 1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arrays H (n ; k) that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, for k ⩾ 11, that there are at least (n − 2)[((k − 11)/4)!/ ]² such nonequivalent H (n ; k) that satisfy both conditions (1) and (2).

year: 2020

title: Exposing the hidden politics of housing provision in Aotearoa New Zealand: The complex governance landscape of new housing in Hamilton

authors: Dodd, Francesca Jane

abstract: Delivery of sufficient affordable, quality new housing in New Zealand has been high on the political agenda since the 1990s and, despite considerable government activity and policy attention, achieving that goal remains elusive. Access to an affordable house that is fit-for-purpose is, therefore, becoming increasingly remote for many New Zealanders, including those who earn above average incomes. In fact, it has become common to refer to the problem as a housing crisis. This research aims to analyse the effectiveness of the governance arrangements involved in delivering new housing in Aotearoa New Zealand, by examining the logics and discourses of public and private actors within that process and the decisions that emerge from those interactions. In approaching this question, it begins by exploring past approaches to governing new housing provision, before reporting on a case study of planning for new housing in the rapidly growing city of Hamilton. The in-depth qualitative study approaches decision-making for the provision of new housing in Hamilton through a network governance lens. It involved reviewing the local District and Structure Plans and other planning documents prepared by Hamilton City Council, including strategic documents about the future growth of the city. It also included interviewing 31 participants involved in making decisions about new housing for the city. These were from the public sector, largely council planners but also practitioners from the Building Unit and Growth and Analytics Unit. They were also from the private sector, including developers, architects, consultant planners and lawyers. One participant from an NGO was interviewed as well, to get a national perspective of the problems with new housing delivery.
The study examined their interactions, and the ideas and logics that informed the decisions about new housing that emerged from the governance network. It finds that entrenched beliefs and ideas about what constitutes a normal house in terms of who it is built for, the type of property it is, and the way that property should be delivered profoundly shape the decisions of actors within the governance network, although new ideas are emerging. Underpinning these ideas and beliefs, the market was consistently referred to as a credible source of information about the types of houses and communities householders needed and desired. Reference to the market was therefore an accepted way of explaining decisions about the size, design, location, type and density of new housing. The analysis also provides insights into the forces of inertia that continue to mean suboptimal outcomes are delivered. These included the complexity of the network, with multiple agencies involved in decision-making. Conflicting interests and imperatives within the governance network was also a barrier to change. The concentration of resources in the hands of developers was found to underpin conservative attitudes to development. Prospects for change were found within the network, such as examples of local authorities exercising leadership over built outcomes. Also, developers with innovative perspectives, and the resources to implement housing developments inspired by such perspectives, were found to exist in Hamilton.
In taking a governance approach to understanding decision making for new housing provision in New Zealand, the study extends traditional research approaches that have focused on policy solutions. It develops our understanding of the housing crisis, and why resolving it remains difficult.

text: Exposing the hidden politics of housing provision in Aotearoa New Zealand: The complex governance landscape of new housing in Hamilton Delivery of sufficient affordable, quality new housing in New Zealand has been high on the political agenda since the 1990s and, despite considerable government activity and policy attention, achieving that goal remains elusive. Access to an affordable house that is fit-for-purpose is, therefore, becoming increasingly remote for many New Zealanders, including those who earn above average incomes. In fact, it has become common to refer to the problem as a housing crisis. This research aims to analyse the effectiveness of the governance arrangements involved in delivering new housing in Aotearoa New Zealand, by examining the logics and discourses of public and private actors within that process and the decisions that emerge from those interactions. In approaching this question, it begins by exploring past approaches to governing new housing provision, before reporting on a case study of planning for new housing in the rapidly growing city of Hamilton. The in-depth qualitative study approaches decision-making for the provision of new housing in Hamilton through a network governance lens. It involved reviewing the local District and Structure Plans and other planning documents prepared by Hamilton City Council, including strategic documents about the future growth of the city. It also included interviewing 31 participants involved in making decisions about new housing for the city. These were from the public sector, largely council planners but also practitioners from the Building Unit and Growth and Analytics Unit. They were also from the private sector, including developers, architects, consultant planners and lawyers. One participant from an NGO was interviewed as well, to get a national perspective of the problems with new housing delivery.
The study examined their interactions, and the ideas and logics that informed the decisions about new housing that emerged from the governance network. It finds that entrenched beliefs and ideas about what constitutes a normal house in terms of who it is built for, the type of property it is, and the way that property should be delivered profoundly shape the decisions of actors within the governance network, although new ideas are emerging. Underpinning these ideas and beliefs, the market was consistently referred to as a credible source of information about the types of houses and communities householders needed and desired. Reference to the market was therefore an accepted way of explaining decisions about the size, design, location, type and density of new housing. The analysis also provides insights into the forces of inertia that continue to mean suboptimal outcomes are delivered. These included the complexity of the network, with multiple agencies involved in decision-making. Conflicting interests and imperatives within the governance network was also a barrier to change. The concentration of resources in the hands of developers was found to underpin conservative attitudes to development. Prospects for change were found within the network, such as examples of local authorities exercising leadership over built outcomes. Also, developers with innovative perspectives, and the resources to implement housing developments inspired by such perspectives, were found to exist in Hamilton.
In taking a governance approach to understanding decision making for new housing provision in New Zealand, the study extends traditional research approaches that have focused on policy solutions. It develops our understanding of the housing crisis, and why resolving it remains difficult.

year: 2020

title: What does it take to make us listen? The experiences of those leading whole school reform to improve outcomes for indigenous minority students

authors: Buckley, Louise Jane

subjects: Education

abstract: This thesis examines how the Kia Eke Panuku: Building on Success school reform initiative shaped the experiences of groups of educators within mainstream New Zealand secondary schools. These experiences are analysed against a setting of colonisation and historic educational policies of assimilation and integration that have created a situation of education debt. The lack of power for Māori in education is evident. This study highlights the role of the Strategic Change Leadership Team in deconstructing power hierarchies within schools so that educational outcomes can be improved for Māori students. It presents the Strategic Change Leadership Team as the vehicle for conscientisation, resistance and transformative praxis within schools. It shows that in schools where power hierarchies were deconstructed, Māori students were able to enjoy educational success as Māori.

text: What does it take to make us listen? The experiences of those leading whole school reform to improve outcomes for indigenous minority students This thesis examines how the Kia Eke Panuku: Building on Success school reform initiative shaped the experiences of groups of educators within mainstream New Zealand secondary schools. These experiences are analysed against a setting of colonisation and historic educational policies of assimilation and integration that have created a situation of education debt. The lack of power for Māori in education is evident. This study highlights the role of the Strategic Change Leadership Team in deconstructing power hierarchies within schools so that educational outcomes can be improved for Māori students. It presents the Strategic Change Leadership Team as the vehicle for conscientisation, resistance and transformative praxis within schools. It shows that in schools where power hierarchies were deconstructed, Māori students were able to enjoy educational success as Māori.

year: 2020