Domination of the rectangular queen's graph

The queen's graph $Q_{m \times n}$ has the squares of the $m \times n$ chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal of the board. A set $D$ of squares of $Q_{m \times n}$ is a dominating set for $Q_{m \times n}$ if every square of $Q_{m \times n}$ is either in $D$ or adjacent to a square in $D$. The minimum size of a dominating set of $Q_{m \times n}$ is the domination number, denoted by $\gamma(Q_{m \times n})$. Values of $\gamma(Q_{m \times n}), \, 4 \leq m \leq n \leq 18, \,$ are given here, in each case with a file of minimum dominating sets (often all of them, up to symmetry) in an online appendix at https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i4p45/HTML. In these ranges for $m$ and $n$, monotonicity fails once: $\gamma(Q_{8 \times 11}) = 6 > 5 = \gamma(Q_{9 \times 11}) = \gamma(Q_{10 \times 11}) = \gamma(Q_{11 \times 11})$. Lower bounds on $\gamma(Q_{m \times n})$ are given. In particular, if $m \leq n$ then $\gamma(Q_{m \times n}) \geq \min \{ m, \lceil (m+n-2)/4 \rceil \}$. A set of squares is independent if no two of its squares are adjacent. The minimum size of an independent dominating set of $Q_{m \times n}$ is the independent domination number, denoted by $i(Q_{m \times n})$. Values of $i(Q_{m \times n}), \, 4 \leq m \leq n \leq 18, \,$ are given here, in each case with some minimum dominating sets. In these ranges for $m$ and $n$, monotonicity fails twice: $i(Q_{8 \times 11}) = 6 > 5 = i(Q_{9 \times 11}) = i(Q_{10 \times 11}) = i(Q_{11 \times 11})$, and $i(Q_{11 \times 18}) = 9 > 8 = i(Q_{12 \times 18})$

Domination of the rectangular queen's graph

The queens graph Qm×n has the squares of the m × n chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal of the board. A set D of squares of Qm×n is a dominating set for Qm×n if every square of Qm×n is either in D or adjacent to a square in D. The minimum size of a dominating set of Qm×n is the domination number, denoted by γ(Qm×n). Values of γ(Qm×n), 4 6 m 6 n 6 18, are given here, in each case with a file of minimum dominating sets (often all of them, up to symmetry) in an online appendix. In these ranges for m and n, monotonicity fails once: γ(Q8×11) = 6 > 5 = γ(Q9×11) = γ(Q10×11) = γ(Q11×11). Let g(m) [respectively g ∗ (m)] be the largest integer such that m queens suffice to dominate the (m+1)×g(m) board [respectively, to dominate the (m+1)×g ∗ (m) board with no two queens in a row]. Starting from the elementary bound g(m) 6 3m, domination when the board is far from square is investigated. It is shown (Theorem 2) that g(m) = 3m can only occur when m ≡ 0, 1, 2, 3, or 4 (mod 9), with the online appendix showing that this does occur for m 6 40, m 6= 3. Also (Theorem 4), if m ≡ 5, 6, or 7 (mod 9) then g ∗ (m) 6 3m − 2, and if m ≡ 8 (mod 9) then g ∗ (m) 6 3m − 4. It is shown that equality holds in these bounds for m 6 40. Lower bounds on γ(Qm×n) are given. In particular, if m 6 n then γ(Qm×n) > min{m, d(m + n − 2)/4e}. Two types of dominating sets (orthodox covers and centrally strong sets) are developed; each type is shown to give good upper bounds of γ(Qm×n) in several cases. Three questions are posed: whether monotonicity of γ(Qm×n) holds (other than from (m, n) = (8, 11) to (9, 11)), whether γ(Qm×n) = (m + n − 2)/4 occurs with m 6 n < 3m+ 2 (other than for (m, n) = (3, 3) and (11, 11)), and whether the lower bound given above can be improved. A set of squares is independent if no two of its squares are adjacent. The minimum size of an independent dominating set of Qm×n is the independent domination number, denoted by i(Qm×n). Values of i(Qm×n), 4 6 m 6 n 6 18, are given here, in each case with some minimum dominating sets. In these ranges for m and n, monotonicity fails twice: i(Q8×11) = 6 > 5 = i(Q9×11) = i(Q10×11) = i(Q11×11), and i(Q11×18) = 9 > 8 = i(Q12×18)

Abstraction And Libidinal Nationalism In The Works Of John Boyle And Diana Thorneycroft

This thesis examines the work of Canadian artists John Boyle and Diana Thorneycroft. It analyzes their imagery in aesthetic, political, and strictly materialist terms using the theoretical work of Wilhelm Worringer, Wyndham Lewis and Gilles Deleuze. Both artists have appropriated elements of Canadian history (historical figures, cultural archetypes and aspects of material culture) to create art works that are alienating and abstract. With Boyle, this has involved constructing a wall of noise to operate as a territorial marker, one that both parodies and reinforces the traditions of the nation. His art has been one devoted to nonsense, nihilism and a deliberate short-circuiting of communication. This is invested in advocacy for a Canadian nationalism that is both mournful and absurd. For Thorneycroft, it has meant a shift away from the production of absurd tableau dealing with Canadian nationalism, to an attempt to encounter the crimes of the country\u27s history with empathy. However, her formal means have consistently short-circuited her attempt to create a morally therapeutic art

Arctic sojourn : a teacher's reflections

The research describes the experience of a southern white teacher who lived and worked in a remote community in Canada's Far North. The impact of physical relocation and culture shock are discussed, as well as problems encountered when conflicting views of education and life goals meet in a cross-cultural setting. The thesis explores some of the difficulties facing mainstream teachers of Indigenous students when issues of past colonialism and present injustices come into play. Inuit community literacies (visual, kinesic and oral traditions) are explored and contrasted with traditional definitions of literacy, which center on the paramount importance of the printed word. Power issues are discussed, including the role played by literacy education in maintaining control in the hands of the dominant culture. The research is qualitative and phenomenological in nature. The teaching experience is viewed through a critical lens, and attempts to better understand the writer's southern white middle-class background as it relates to differing worldviews. The author recounts the process of re-examining assumptions of her own culture, and describes her personal and professional journey of coming to grips with its impact on her teaching

The Two Row Wampum : historic fiction, modern reality

Québec Université Laval, Bibliothèque 201

North America’s metropolitan imaginaries

Scholars of modernity have taken a particular interest in processes of urbanization and—thinking of Simmel, Benjamin, Mumford and Weber—the character of different varieties of city. From a different angle, notions of urban imaginary have gained greater purchase in the field of contemporary urban studies in comparative analysis of varieties of city. This essay begins with notes on both classical accounts of the city in social theory and current concepts of urban imaginaries. The notes revolve around the essay’s main topic: the institution of cities of New World, specifically those of the United States and Canada. Paralleling Castoriadis’ conception of the imaginary institution, the present author argues for a more exact notion of metropolitan imaginaries, differentiated from the broader subject of urban imaginaries. ‘Metropolitan imaginaries’ denotes processes of urbanization at the heart of networks of migration, transport, and flows of capital and culture. As part of larger imaginaries, metropolises generate immigrant cities. The specific kind of creation in question produces creativity also by concentrating intellectual and creative schools of design in architecture and visual culture. In sum, metropolises are not merely part of networks of connection and creation; they produce networks and act as the hubs of interaction and creativity within larger social imaginaries. The essay explores this argument in the contexts of US and Canadian modernity and state formation, with specific foci on New York, Chicago, and Toronto. The conclusion notes two limitations to the case presented here and sketches planned directions for future research

Nationalism in Official Tourism Websites of Balkan Countries: A Multimodal Analysis

This paper investigates the official tourism websites for the Balkan countries of Albania, Bosnia and Herzegovina, Bulgaria, Croatia, Greece, Montenegro, North Macedonia, Romania, Serbia, Slovenia, and Turkey to learn about their depictions of their nations for an international tourism market. The research combines Pauwels\u27 (2012) multimodal discourse analysis method designed for cultural websites with Smith\u27s (1998) six main institutional dimensions to seek out potential nationalistic patterns involving the state, territory, language, religion, history, and rites and ceremonies. The findings mostly involve verbal and visual signifiers that have a historical context to them such as antiquity, communism, Yugoslavia, religion, irredentism, the Ottoman Empire, and national identity. The findings illustrate that official websites, while being sensitive not to alienate international tourists, portray a sense of nationalism but do so in a different way, based upon the historical experiences and unique features of each country surveyed

Publicness of urban spaces.

Luk Wing Lun William."Architecture Department, Chinese University of Hong Kong, Master of Architecture Programme 2008-2009, design report."Includes bibliographical references.Chapter 1.0 --- BackgroundChapter 1a --- PhenmenonChapter 1b --- Thesis StatementChapter 1c --- CatagoriesChapter 1d --- MethodologyChapter 2.0 --- DefinitionChapter 2a --- Publicness OpennessChapter 2b --- """Public"""Chapter 2c --- HistoryChapter 2d --- TimelineChapter 3.0 --- Hong Kong SituationChapter 3a --- RegulationChapter 3b --- Cases StudyChapter 3c --- Urban ScaleChapter 4.0 --- New York SituationChapter 4a --- RegulationChapter 4b --- Cases StudyChapter 4c --- Urban ScaleChapter 5.0 --- Grid City and SiteChapter 5a --- Grid FabricChapter 5b --- SiteChapter 6.0 --- Policy and DesignChapter 6a --- ProcessChapter 6b --- PlazasChapter 6c --- Master Plannin