Square Integer Heffter Arrays with Empty Cells

A Heffter array $H(m,n;s,t)$ is an $m \times n$ matrix with nonzero entries from $\mathbb{Z}_{2ms+1}$ such that $i)$ each row contains $s$ filled cells and each column contains $t$ filled cells, $ii)$ every row and column sum to 0, and $iii)$ no element from $\{x,-x\}$ appears twice. Heffter arrays are useful in embedding the complete graph $K_{2nm+1}$ on an orientable surface where the embedding has the property that each edge borders exactly one $s-$cycle and one $t-$cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be constructed in the case when $s=m$, i.e. every cell is filled. In this paper we concentrate on square arrays with empty cells where every row sum and every column sum is $0$ in $\mathbb{Z}$. We solve most of the instances of this case.Comment: 20 pages, including 2 figure

Finishing pigs for market

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Can You Hear us Now? Voices from the Margin: Using Indigenous Methodologies in Geographic Research

Indigenous methodologies are an alternative way of thinking about research processes. Although these methodologies vary according to the ways in which different Indigenous communities express their own unique knowledge systems, they do have common traits. This article argues that research on Indigenous issues should be carried out in a manner which is respectful and ethically sound from an Indigenous perspective. This naturally challenges Western research paradigms, yet it also affords opportunities to contribute to the body of knowledge about Indigenous peoples. It is further argued that providing a mechanism for Indigenous peoples to participate in and direct these research agendas ensures that their communal needs are met, and that geographers then learn how to build ethical research relationships with them. Indigenous methodologies do not privilege Indigenous researchers because of their Indigeneity, since there are many ‘insider’ views, and these are thus suitable for both Indigenous and non-Indigenous researchers. However, there is a difference between research done within an Indigenous context using Western methodologies and research done using Indig- enous methodologies which integrates Indigenous voices. This paper will discuss those differences while presenting a historical context of research on Indigenous peoples, providing further insights into what Indigenous methodologies entail, and proposing ways in which the academy can create space for this discourse