title: Multi-latin squares

authors: Cavenagh, Nicholas J.

subjects: latin square

abstract: A multi-latin square of order n and index k is an n×n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square.
In this note we show that any partially filled-in k-latin square of order m embeds in a k-latin square of order n, for each n≥2m, thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable k-latin squares of order n for each n≥k+2. We also show that for each n≥1, there exists some finite value g(n) such that for all k≥g(n), every k-latin square of order n is separable.
We discuss the connection between k-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k-latin trades. We also enumerate and classify k-latin squares of small orders.

text: Multi-latin squares A multi-latin square of order n and index k is an n×n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square.
In this note we show that any partially filled-in k-latin square of order m embeds in a k-latin square of order n, for each n≥2m, thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable k-latin squares of order n for each n≥k+2. We also show that for each n≥1, there exists some finite value g(n) such that for all k≥g(n), every k-latin square of order n is separable.
We discuss the connection between k-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k-latin trades. We also enumerate and classify k-latin squares of small orders.

year: 2010

title: Business networks and the uptake of sustainability practices: the case of New Zealand

authors: Collins, Eva Marie

subjects: networks

abstract: This paper reports on the adoption of sustainability practices by a sample of over 800 firms in New Zealand, contrasting results from members of the Sustainable Business Network (SBN) with non-members. The study also examined differences related to business size. Not surprisingly, large firms were found to be more actively engaged in sustainability. On the other hand, of the sample approximately one-third cited the existence of barriers to adoption of sustainable practices. Of these, larger companies were more likely to state that costs or other priorities inhibited the adoption of such practices. Generally speaking members of an SBN were more likely than non-members to adopt environmental initiatives, citing in particular the importance of reputation and brand, but in terms of total waste and energy waste there were no significant differences between members and non-members. There were few differences between the two groups on the adoption of social practices. In assessing membership of an SBN and company size as determinants of good environmental practices, company size was found to be the more significant factor.

text: Business networks and the uptake of sustainability practices: the case of New Zealand This paper reports on the adoption of sustainability practices by a sample of over 800 firms in New Zealand, contrasting results from members of the Sustainable Business Network (SBN) with non-members. The study also examined differences related to business size. Not surprisingly, large firms were found to be more actively engaged in sustainability. On the other hand, of the sample approximately one-third cited the existence of barriers to adoption of sustainable practices. Of these, larger companies were more likely to state that costs or other priorities inhibited the adoption of such practices. Generally speaking members of an SBN were more likely than non-members to adopt environmental initiatives, citing in particular the importance of reputation and brand, but in terms of total waste and energy waste there were no significant differences between members and non-members. There were few differences between the two groups on the adoption of social practices. In assessing membership of an SBN and company size as determinants of good environmental practices, company size was found to be the more significant factor.

year: 2007

title: Te Karauna Hou: the senior Ngati Rahiri rangatira

authors: Hart, Philip

abstract: Te Karauna Hou, the principal Ngati Rahiri rangatira living at Te Aroha in 1880, had a distinguished whakapapa linking him to several hapu. Before settling permanently at Te Aroha in the 1870s he lived in several places, especially at Kaitawa, on the southern outskirts of Thames. One of the principal rangatira in Hauraki, he was loyal to the Crown during the Waikato War, and later assisted Pakeha settlement. To emphasize his mana and that of his hapu, he held big festivities at Kaitawa and at Omahu pa at Te Aroha. 
In 1871, when Ngati Haua won a (temporary) victory in the land court over the ownership of the Aroha Block, Karauna took control of it on behalf of the Marutuahu confederation, and subsequently kept Ngati Haua at bay. For a time he opposed road-making on this block, but later agreed to it, for financial reasons. Like all rangatira, he sought to maximize his ownership of as many blocks of land as possible, sometimes having his lies exposed through his contradicting his earlier evidence. He also denied receiving money for land when his denials were easily disproved. Despite leasing and selling land, selling timber and gum, and opening his land to miners, he often struggled financially.  
Karauna claimed to have found gold in Hauraki in 1852, and was willing to open his land at Thames to miners, but did not invest in any claims before the Te Aroha rush, when he attempted to extract a bonus of £1,000 from the government for opening the field. After his death in 1885, Pakeha remembered with gratitude his friendly attitude to them, but they also remembered his drunkenness, which meant he lost the respect of both Maori and Pakeha in his latter years.

text: Te Karauna Hou: the senior Ngati Rahiri rangatira Te Karauna Hou, the principal Ngati Rahiri rangatira living at Te Aroha in 1880, had a distinguished whakapapa linking him to several hapu. Before settling permanently at Te Aroha in the 1870s he lived in several places, especially at Kaitawa, on the southern outskirts of Thames. One of the principal rangatira in Hauraki, he was loyal to the Crown during the Waikato War, and later assisted Pakeha settlement. To emphasize his mana and that of his hapu, he held big festivities at Kaitawa and at Omahu pa at Te Aroha. 
In 1871, when Ngati Haua won a (temporary) victory in the land court over the ownership of the Aroha Block, Karauna took control of it on behalf of the Marutuahu confederation, and subsequently kept Ngati Haua at bay. For a time he opposed road-making on this block, but later agreed to it, for financial reasons. Like all rangatira, he sought to maximize his ownership of as many blocks of land as possible, sometimes having his lies exposed through his contradicting his earlier evidence. He also denied receiving money for land when his denials were easily disproved. Despite leasing and selling land, selling timber and gum, and opening his land to miners, he often struggled financially.  
Karauna claimed to have found gold in Hauraki in 1852, and was willing to open his land at Thames to miners, but did not invest in any claims before the Te Aroha rush, when he attempted to extract a bonus of £1,000 from the government for opening the field. After his death in 1885, Pakeha remembered with gratitude his friendly attitude to them, but they also remembered his drunkenness, which meant he lost the respect of both Maori and Pakeha in his latter years.

year: 2016