title: Critical sets of full Latin squares

authors: Raass, Petelo Vaipuna

abstract: This thesis explores the properties of critical sets of the full n-Latin square and related combinatorial structures including full designs, (m,n,2)-balanced Latin rectangles and n-Latin cubes.

In Chapter 3 we study known results on designs and the analogies between critical sets of the full n-Latin square and minimal defining sets of the full designs.

Next in Chapter 4 we fully classify the critical sets of the full (m,n,2)-balanced Latin square, by describing the precise structures of these critical sets from the smallest to the largest.

Properties of different types of critical sets of the full n-Latin square are investigated in Chapter 5. We fully classify the structure of any saturated critical set of the full n-Latin square. We show in Theorem 5.8 that such a critical set has size exactly equal to n³ - 2n² - n. In Section 5.2 we give a construction which provides an upper bound for the size of the smallest critical set of the full n-Latin square. Similarly in Section 5.4, another construction gives a lower bound for the size of the largest non-saturated critical set. We conjecture that these bounds are best possible.

Using the results from Chapter 5, we obtain spectrum results on critical sets of the full n-Latin square in Chapter 6. In particular, we show that a critical set of each size between (n - 1)³ + 1 and n(n - 1)² + n - 2 exists.

In Chapter 7, we turn our focus to the completability of partial k-Latin squares. The relationship between partial k-Latin squares and semi-k-Latin squares is used to show that any partial k-Latin square of order n with at most (n - 1) non-empty cells is completable.

As Latin cubes generalize Latin squares, we attempt to generalize some of the results we have established on k-Latin squares so that they apply to k-Latin cubes. These results are presented in Chapter 8.

text: Critical sets of full Latin squares This thesis explores the properties of critical sets of the full n-Latin square and related combinatorial structures including full designs, (m,n,2)-balanced Latin rectangles and n-Latin cubes.

In Chapter 3 we study known results on designs and the analogies between critical sets of the full n-Latin square and minimal defining sets of the full designs.

Next in Chapter 4 we fully classify the critical sets of the full (m,n,2)-balanced Latin square, by describing the precise structures of these critical sets from the smallest to the largest.

Properties of different types of critical sets of the full n-Latin square are investigated in Chapter 5. We fully classify the structure of any saturated critical set of the full n-Latin square. We show in Theorem 5.8 that such a critical set has size exactly equal to n³ - 2n² - n. In Section 5.2 we give a construction which provides an upper bound for the size of the smallest critical set of the full n-Latin square. Similarly in Section 5.4, another construction gives a lower bound for the size of the largest non-saturated critical set. We conjecture that these bounds are best possible.

Using the results from Chapter 5, we obtain spectrum results on critical sets of the full n-Latin square in Chapter 6. In particular, we show that a critical set of each size between (n - 1)³ + 1 and n(n - 1)² + n - 2 exists.

In Chapter 7, we turn our focus to the completability of partial k-Latin squares. The relationship between partial k-Latin squares and semi-k-Latin squares is used to show that any partial k-Latin square of order n with at most (n - 1) non-empty cells is completable.

As Latin cubes generalize Latin squares, we attempt to generalize some of the results we have established on k-Latin squares so that they apply to k-Latin cubes. These results are presented in Chapter 8.

year: 2016

title: Supreme Court stories: Narrating violence in Suva streets and homes

authors: Stevens, Kate

abstract: This chapter looks at the establishment and limits of the colonial legal system in the early decades of colonial Suva, focusing on the cases of rape and sexual violence brought before the Supreme Court between 1875 and 1920.

text: Supreme Court stories: Narrating violence in Suva streets and homes This chapter looks at the establishment and limits of the colonial legal system in the early decades of colonial Suva, focusing on the cases of rape and sexual violence brought before the Supreme Court between 1875 and 1920.

year: 2022-09-08

title: He Tangata Kei Tua: Guidelines for biobanking with Māori.

authors: Hudson, Maui

abstract: Māori ethical frameworks recognise that all research in New Zealand is of interest to Māori and outline community expectations of appropriate behavior in research to deliver the best outcomes for Māori. Research contributes to the broader development objectives of society and this endeavor is being supported by biobanking infrastructure. Ethics has a specific role in guiding key behaviours, processes and methodologies used in research.
This document outlines a framework for addressing Māori ethical issues within the context of biobanking. It draws on a foundation of mātauranga (Indigenous knowledge) and tikanga Māori (Māori protocols and practices) and will be useful for researchers, ethics committee members and those who engage in consultation or advice about biobanking with Māori in local, regional, national or international settings.

text: He Tangata Kei Tua: Guidelines for biobanking with Māori. Māori ethical frameworks recognise that all research in New Zealand is of interest to Māori and outline community expectations of appropriate behavior in research to deliver the best outcomes for Māori. Research contributes to the broader development objectives of society and this endeavor is being supported by biobanking infrastructure. Ethics has a specific role in guiding key behaviours, processes and methodologies used in research.
This document outlines a framework for addressing Māori ethical issues within the context of biobanking. It draws on a foundation of mātauranga (Indigenous knowledge) and tikanga Māori (Māori protocols and practices) and will be useful for researchers, ethics committee members and those who engage in consultation or advice about biobanking with Māori in local, regional, national or international settings.

year: 2016