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: Gender and the Outdoors: An International Conversation

authors: Pinch, Katherine J.

subjects: gender

abstract: Over the past two to three decades in outdoor education circles, there has been a gradual swell of interest in bringing a gender lens to the examination of issues, theory and practice within the field. Although feminist theory has been subjected to lively debate and undergone sophisticated shifts in the ways of conceptualizing and analyzing gender, much of the literature coming out of the outdoor field and much of our practice is still centered on women and difference. As Bell (1997) so clearly asked, “Has the dialogue on the nature of gender and associated social issues not changed in the past decade?” A group of international researchers, educators and practitioners discussed gender theory and practice in their “neck of the woods” and challenged attendees to ask that question again today.

text: Gender and the Outdoors: An International Conversation Over the past two to three decades in outdoor education circles, there has been a gradual swell of interest in bringing a gender lens to the examination of issues, theory and practice within the field. Although feminist theory has been subjected to lively debate and undergone sophisticated shifts in the ways of conceptualizing and analyzing gender, much of the literature coming out of the outdoor field and much of our practice is still centered on women and difference. As Bell (1997) so clearly asked, “Has the dialogue on the nature of gender and associated social issues not changed in the past decade?” A group of international researchers, educators and practitioners discussed gender theory and practice in their “neck of the woods” and challenged attendees to ask that question again today.

year: 2008

title: Foresight in hindsight: Visions of the future from Julius Vogel, Āpirana Ngata, Truby King, and Ettie Rout, 1889-1923

authors: Lemberg, Nicola Johanna

subjects: history

abstract: This thesis examines seven future-focused texts authored by four New Zealanders: Julius Vogel, Āpirana Ngata, Truby King, and Ettie Rout. ‘The future’ is a neglected lens for historical research and this thesis explores the value of such an approach, using New Zealand society from the 1880s and to the 1920s as a case study. A methodological combination of textual analysis, biography, and historical contexts, including consideration of print culture, allows for nuanced examination of depictions of the future, and suggests such connections are vital to fully understanding histories of the future. This research serves as a framework to inform further exploration of historical future imaginings, either in New Zealand or in other geographic or temporal contexts.

From Vogel’s novel Anno Domini 2000 (1889) to Ngata’s 1892 essay The Past and Future of the Maori to King’s and Rout’s twentieth-century guidebooks on the care of babies and the use of contraception, the four authors utilize various textual forms to communicate their imagined future, and commonly frame their ideas within concepts of utopia and dystopia. This thesis compares the futures depicted by male and female authors, and those emerging from Māori and Pākehā cultural contexts. It examines futures envisioned by politicians, community leaders, and ostracized individuals. Shaped by preoccupations with empire, gender, and identity, these future-focused texts highlight relationships between national and global spaces. This in turn allows for a fresh exploration of New Zealand’s past, and especially the country’s colonial histories.

text: Foresight in hindsight: Visions of the future from Julius Vogel, Āpirana Ngata, Truby King, and Ettie Rout, 1889-1923 This thesis examines seven future-focused texts authored by four New Zealanders: Julius Vogel, Āpirana Ngata, Truby King, and Ettie Rout. ‘The future’ is a neglected lens for historical research and this thesis explores the value of such an approach, using New Zealand society from the 1880s and to the 1920s as a case study. A methodological combination of textual analysis, biography, and historical contexts, including consideration of print culture, allows for nuanced examination of depictions of the future, and suggests such connections are vital to fully understanding histories of the future. This research serves as a framework to inform further exploration of historical future imaginings, either in New Zealand or in other geographic or temporal contexts.

From Vogel’s novel Anno Domini 2000 (1889) to Ngata’s 1892 essay The Past and Future of the Maori to King’s and Rout’s twentieth-century guidebooks on the care of babies and the use of contraception, the four authors utilize various textual forms to communicate their imagined future, and commonly frame their ideas within concepts of utopia and dystopia. This thesis compares the futures depicted by male and female authors, and those emerging from Māori and Pākehā cultural contexts. It examines futures envisioned by politicians, community leaders, and ostracized individuals. Shaped by preoccupations with empire, gender, and identity, these future-focused texts highlight relationships between national and global spaces. This in turn allows for a fresh exploration of New Zealand’s past, and especially the country’s colonial histories.

year: 2024