title: Complete sets of metamorphoses: Twofold 4-cycle systems into twofold 6-cycle systems

authors: Billington, Elizabeth J.

subjects: Cycle decomposition

abstract: Let (X,C) denote a twofold k-cycle system with an even number of cycles. If these k-cycles can be paired together so that: (i) each pair contains a common edge; (ii) removal of the repeated common edge from each pair leaves a (2k-2)-cycle; (iii) all the repeated edges, once removed, can be rearranged exactly into a collection of further (2k-2)-cycles; then this is a metamorphosis of a twofold k-cycle system into a twofold (2k-2)-cycle system. The existence of such metamorphoses has been dealt with for the case of 3-cycles (Gionfriddo and Lindner, 2003) [3] and 4-cycles (Yazc, 2005) [7]. If a twofold k-cycle system (X,C) of order n exists, which has not just one but has k different metamorphoses, from k different pairings of its cycles, into twofold (2k-2)-cycle systems, such that the collection of all removed double edges from all k metamorphoses precisely covers 2 Kn, we call this a complete set of twofold paired k-cycle metamorphoses into twofold (2k-2)-cycle systems. In this paper, we show that there exists a twofold 4-cycle system (X,C) of order n with a complete set of metamorphoses into twofold 6-cycle systems if and only if n≡0,1,9,16 (mod 24), n≠9.

text: Complete sets of metamorphoses: Twofold 4-cycle systems into twofold 6-cycle systems Let (X,C) denote a twofold k-cycle system with an even number of cycles. If these k-cycles can be paired together so that: (i) each pair contains a common edge; (ii) removal of the repeated common edge from each pair leaves a (2k-2)-cycle; (iii) all the repeated edges, once removed, can be rearranged exactly into a collection of further (2k-2)-cycles; then this is a metamorphosis of a twofold k-cycle system into a twofold (2k-2)-cycle system. The existence of such metamorphoses has been dealt with for the case of 3-cycles (Gionfriddo and Lindner, 2003) [3] and 4-cycles (Yazc, 2005) [7]. If a twofold k-cycle system (X,C) of order n exists, which has not just one but has k different metamorphoses, from k different pairings of its cycles, into twofold (2k-2)-cycle systems, such that the collection of all removed double edges from all k metamorphoses precisely covers 2 Kn, we call this a complete set of twofold paired k-cycle metamorphoses into twofold (2k-2)-cycle systems. In this paper, we show that there exists a twofold 4-cycle system (X,C) of order n with a complete set of metamorphoses into twofold 6-cycle systems if and only if n≡0,1,9,16 (mod 24), n≠9.

year: 2012

title: Managing rising residual flood risk: A national survey of Aotearoa-New Zealand

authors: Fu, Xinyu

abstract: Flooding is one of the most frequent and costliest natural hazards around the world. Traditionally, flood risk management has relied on building protective structures such as levees and dams to protect assets based on historical data (e.g., 1% AEP flood), which encourages development on floodplains, provides a potentially false sense of security to communities, particularly in the context of climate change, and increases residual flood risk (the risk remaining after implementing risk reduction measures). This article aims to contribute to the management of residual risk by drawing on empirical findings from a survey carried out with flood risk practitioners from the public and private sector in Aotearoa New Zealand. Findings indicate fundamental concerns relating to how to best manage residual flood risk, despite its long profile and integration in current policy and practice. Other issues revolve around the use of outdated information to guide decisions and the lack of regulatory power to restrict developments in flood-prone areas protected by hard defense structures. Identified barriers to improving current practice include the lack of national guidance and support, financial resources, public awareness, and some technical constraints such as uncertainties in flood modeling, staff expertise, and data availability.

text: Managing rising residual flood risk: A national survey of Aotearoa-New Zealand Flooding is one of the most frequent and costliest natural hazards around the world. Traditionally, flood risk management has relied on building protective structures such as levees and dams to protect assets based on historical data (e.g., 1% AEP flood), which encourages development on floodplains, provides a potentially false sense of security to communities, particularly in the context of climate change, and increases residual flood risk (the risk remaining after implementing risk reduction measures). This article aims to contribute to the management of residual risk by drawing on empirical findings from a survey carried out with flood risk practitioners from the public and private sector in Aotearoa New Zealand. Findings indicate fundamental concerns relating to how to best manage residual flood risk, despite its long profile and integration in current policy and practice. Other issues revolve around the use of outdated information to guide decisions and the lack of regulatory power to restrict developments in flood-prone areas protected by hard defense structures. Identified barriers to improving current practice include the lack of national guidance and support, financial resources, public awareness, and some technical constraints such as uncertainties in flood modeling, staff expertise, and data availability.

year: 2023-01-01

title: Mōteatea: He pātaka whakairinga kōrero nō Ngāti Rangiwewehi

authors: Paterson, Awatea

subjects: Ngāti Rangiwewehi

abstract: Mai rā anō a Ngāi Tāua, te iwi Māori, e tuku iho ai i ngā mātauranga mā te ārero tupuna mai i tētehi whakatupuranga, ki tētehi whakatupuranga. Ko ngā mātauranga ka tukua iho ko ngā pūrākau, ko ngā karakia, ko ngā whakataukī, ā, ko ngā waiata hoki. Ko te pūtake o te rangahau nei he titiro pū ki te āhua o te noho o ngā mōteatea hei pātaka whakairinga kōrero mō Ngāti Rangiwewehi ake, nō reira ka āta tirohia ngā momo puna mōhiohio, ngā momo kura aweko ka whakairia ki te pakitara o te pātaka kōrero o Ngāti Rangiwewehi. Nā ngā mōteatea i tohu te wā, i kōrero mō te wāhi, i whakapūkete ngā mahi a ngā tūpuna o te wā i titoa ai te mōteatea. Nā, ko ngā waiata, ko ngā mōteatea koirā te ara whakapūkete kōrero ka wānangahia e te tuhinga nei. Mā te āta wetewete i tētehi mōteatea pū o Ngāti Rangiwewehi, ka hora i ngā kura āuki, i ngā māpihi pounamu, i ngā puna mātauranga o te iwi kua whakaemi mai ki roto i te mōteatea hei kōrero tuku iho mō te iwi o Ngāti Rangiwewehi, hei whakatauira ake hoki i te āhua o te noho o ngā mōteatea hei ara whakapūkete mātauranga.

text: Mōteatea: He pātaka whakairinga kōrero nō Ngāti Rangiwewehi Mai rā anō a Ngāi Tāua, te iwi Māori, e tuku iho ai i ngā mātauranga mā te ārero tupuna mai i tētehi whakatupuranga, ki tētehi whakatupuranga. Ko ngā mātauranga ka tukua iho ko ngā pūrākau, ko ngā karakia, ko ngā whakataukī, ā, ko ngā waiata hoki. Ko te pūtake o te rangahau nei he titiro pū ki te āhua o te noho o ngā mōteatea hei pātaka whakairinga kōrero mō Ngāti Rangiwewehi ake, nō reira ka āta tirohia ngā momo puna mōhiohio, ngā momo kura aweko ka whakairia ki te pakitara o te pātaka kōrero o Ngāti Rangiwewehi. Nā ngā mōteatea i tohu te wā, i kōrero mō te wāhi, i whakapūkete ngā mahi a ngā tūpuna o te wā i titoa ai te mōteatea. Nā, ko ngā waiata, ko ngā mōteatea koirā te ara whakapūkete kōrero ka wānangahia e te tuhinga nei. Mā te āta wetewete i tētehi mōteatea pū o Ngāti Rangiwewehi, ka hora i ngā kura āuki, i ngā māpihi pounamu, i ngā puna mātauranga o te iwi kua whakaemi mai ki roto i te mōteatea hei kōrero tuku iho mō te iwi o Ngāti Rangiwewehi, hei whakatauira ake hoki i te āhua o te noho o ngā mōteatea hei ara whakapūkete mātauranga.

year: 2013