title: Decomposing complete tripartite graphs into 5-cycles when the partite sets have similar size

authors: Billington, Elizabeth J.

subjects: graph decomposition

abstract: The problem of finding necessary and sufficient conditions to decompose a complete tripartite graph into 5-cycles was first posed at a conference in 1994 (Mahmoodian and Mirzakhani in Combinatorics Advances, 1995). Since then, many cases of the problem have been solved by various authors; however the case when the partite sets have odd and distinct sizes remains open. In this note, we introduce a new approach to the problem by embedding previously known decompositions into larger ones. Via this approach, we show that when the partite sets have asymptotically similar sizes, the conjectured necessary conditions for a decomposition are also sufficient.

text: Decomposing complete tripartite graphs into 5-cycles when the partite sets have similar size The problem of finding necessary and sufficient conditions to decompose a complete tripartite graph into 5-cycles was first posed at a conference in 1994 (Mahmoodian and Mirzakhani in Combinatorics Advances, 1995). Since then, many cases of the problem have been solved by various authors; however the case when the partite sets have odd and distinct sizes remains open. In this note, we introduce a new approach to the problem by embedding previously known decompositions into larger ones. Via this approach, we show that when the partite sets have asymptotically similar sizes, the conjectured necessary conditions for a decomposition are also sufficient.

year: 2011

title: Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials

authors: Boyer, C.P.

subjects: Mathematics

abstract: As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.

text: Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.

year: 1975-03

title: Kia whakatōmuri te haere whakamua – A Māori framework for coastal climate change adaptation

authors: Bailey-Winiata, Akuhata

subjects: Climate Change Adaptation

abstract: Current climate change adaptation frameworks do not sufficiently represent or acknowledge the intricacies and complex nature of climate adaptation for all communities. This thesis contributes a Māori (Indigenous peoples of Aotearoa New Zealand (A-NZ)) climate adaptation framework that is informed by Te Ao Māori world views, mātauranga Māori (Māori knowledge), Te Tiriti o Waitangi (Treaty of Waitangi) and research at the interface of Western knowledge and mātauranga Māori. This is needed to promote a more equitable and culturally sensitive approach to the current climate change adaptation process for hapū (sub-tribes) and iwi (tribes). To achieve this, this thesis is rooted in the whakataukī (proverb) “Kia whakatōmuri te haere whakamua – I walk backwards into the future with my eyes fixed on my past”, which exemplifies the power of acknowledging Māori history and intergenerational knowledge, while moving forward to a prosperous future. This thesis had four key aims. First, I looked to the past to understand ways Māori ancestors adapted and relocated in response to natural hazards to inform a future climate adaptation or relocation process. Second, I took stock of the contemporary risk to coastal marae (Māori meeting grounds) from sea level rise. Third, I looked internationally at other Indigenous communities' adaptation challenges and how they have overcome them as a mutual learning opportunity. And finally, I developed a conceptual Māori climate adaptation framework based on existing models and informed by a wānanga (workshop) with over 100 Māori participants. This two-part framework emphasises the importance of fostering Te Tiriti-centric relationships and partnerships that are essential for climate adaptation with hapū and iwi. In addition, it serves as a guide to support adaptation decision-making for hapū and iwi Māori. This thesis took a multidisciplinary approach to achieve these aims, incorporating historical and archival analyses, semi-structured interviews, a systematic review, coastal flooding and exposure analyses using spatial information, and wānanga. The findings highlighted that relocation due to natural hazards is not uncommon for Māori, as this thesis identified 51 examples of where relocation was completed or discussed in response to natural hazards. Furthermore, in a case study with Tūhourangi and Ngāti Rangitihi, this thesis identified vital aspects that enabled their relocation following the 1886 Tarawera eruption, the most influential being tuku whenua – the gifting of land to relocate to. This thesis also identified that by 2150, under shared socio-economic pathway (SSP) 5-8.5, 27% and 28% of coastal marae (total of 186 nationally) are exposed to a 100–year and 1000–year annual recurrence interval (ARI) extreme sea level events, respectively. Overall, this thesis contributes to our understanding of the requirements to support hapū and iwi Māori in climate adaptation, with parallels to other Indigenous communities. Requirements include resourcing, partnerships, autonomy, intergenerational planning, recognition, respect and redress. In Chapter 8 – the discussion of this thesis, we focus on four overall contributions this thesis has made, which are: 1) Hapū and iwi Māori are at risk of climate change, 2) We’ve done it before we can do it again, 3) Adaptation approach cannot be one-size-fits-all, and 4) Adaptation in Aotearoa needs a Te Tiriti-centric approach. Ultimately, this thesis underscores the urgent need for inclusive climate adaptation strategies that honour Indigenous knowledge systems, ensuring that hapū and iwi Māori are not only heard but are actively supported in navigating the impacts of climate change.

text: Kia whakatōmuri te haere whakamua – A Māori framework for coastal climate change adaptation Current climate change adaptation frameworks do not sufficiently represent or acknowledge the intricacies and complex nature of climate adaptation for all communities. This thesis contributes a Māori (Indigenous peoples of Aotearoa New Zealand (A-NZ)) climate adaptation framework that is informed by Te Ao Māori world views, mātauranga Māori (Māori knowledge), Te Tiriti o Waitangi (Treaty of Waitangi) and research at the interface of Western knowledge and mātauranga Māori. This is needed to promote a more equitable and culturally sensitive approach to the current climate change adaptation process for hapū (sub-tribes) and iwi (tribes). To achieve this, this thesis is rooted in the whakataukī (proverb) “Kia whakatōmuri te haere whakamua – I walk backwards into the future with my eyes fixed on my past”, which exemplifies the power of acknowledging Māori history and intergenerational knowledge, while moving forward to a prosperous future. This thesis had four key aims. First, I looked to the past to understand ways Māori ancestors adapted and relocated in response to natural hazards to inform a future climate adaptation or relocation process. Second, I took stock of the contemporary risk to coastal marae (Māori meeting grounds) from sea level rise. Third, I looked internationally at other Indigenous communities' adaptation challenges and how they have overcome them as a mutual learning opportunity. And finally, I developed a conceptual Māori climate adaptation framework based on existing models and informed by a wānanga (workshop) with over 100 Māori participants. This two-part framework emphasises the importance of fostering Te Tiriti-centric relationships and partnerships that are essential for climate adaptation with hapū and iwi. In addition, it serves as a guide to support adaptation decision-making for hapū and iwi Māori. This thesis took a multidisciplinary approach to achieve these aims, incorporating historical and archival analyses, semi-structured interviews, a systematic review, coastal flooding and exposure analyses using spatial information, and wānanga. The findings highlighted that relocation due to natural hazards is not uncommon for Māori, as this thesis identified 51 examples of where relocation was completed or discussed in response to natural hazards. Furthermore, in a case study with Tūhourangi and Ngāti Rangitihi, this thesis identified vital aspects that enabled their relocation following the 1886 Tarawera eruption, the most influential being tuku whenua – the gifting of land to relocate to. This thesis also identified that by 2150, under shared socio-economic pathway (SSP) 5-8.5, 27% and 28% of coastal marae (total of 186 nationally) are exposed to a 100–year and 1000–year annual recurrence interval (ARI) extreme sea level events, respectively. Overall, this thesis contributes to our understanding of the requirements to support hapū and iwi Māori in climate adaptation, with parallels to other Indigenous communities. Requirements include resourcing, partnerships, autonomy, intergenerational planning, recognition, respect and redress. In Chapter 8 – the discussion of this thesis, we focus on four overall contributions this thesis has made, which are: 1) Hapū and iwi Māori are at risk of climate change, 2) We’ve done it before we can do it again, 3) Adaptation approach cannot be one-size-fits-all, and 4) Adaptation in Aotearoa needs a Te Tiriti-centric approach. Ultimately, this thesis underscores the urgent need for inclusive climate adaptation strategies that honour Indigenous knowledge systems, ensuring that hapū and iwi Māori are not only heard but are actively supported in navigating the impacts of climate change.

year: 2024-05-28