A well-known result in spectral graph theory states that if a graph has an equitable partition then the eigenvalues of the associated divisor graph are a subset of the graph's eigenvalues. A natural question question is whether it is possible to recover the remaining eigenvalues of the graph. Here we show that if a graph has a Hermitian adjacency matrix then the spectrum of the graph can be decomposed into a collection of smaller graphs whose eigenvalues are collectively the remaining eigenvalues of the graph. This we refer to as a complete equitable decomposition of the graph.

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Univ., Diss.--Aix-Marseille.

Awareness of the importance of policy regarding the equitable distribution of global water resources is increasing. The purpose of this research study was to investigate the effects of allocation rules and drought conditions on allocation decisions through a decision experiment based on an idealized river basin that simulated conditions in the Saskatchewan River Basin. Participants took on the roles of water managers responsible for allocating water resources to four competing sectors in Alberta, Saskatchewan, and Manitoba. Three variations of allocation rules were tested: (a) status-quo allocation rules mimicking the current governance structure involving prearranged allocation rules across regions, (b) no predetermined rules, and (c) no predetermined rules but communication among participants. Each allocation rule was tested under two potential water levels: (a) drier-than-average conditions today reflecting 81% of historical flows, and (b) severe drought conditions with a 45% reduction from today’s flows. Results showed that policy had a significant effect on how participants allocated water resources, indicating that the absence of defined minimal flow rules and the lack of communication among riparian users resulted in less equitable distribution of water, with negative ramifications for downstream users. Additionally, results showed that drought-induced water scarcity significantly affected allocation patterns, with participants choosing to protect municipal water use at the expense of industry and agriculture in the face of water shortage conditions. For decision makers and water stakeholders in the Saskatchewan River Basin, these findings provide insight into the effectiveness of the 1969 Master Agreement on Apportionment to ensure the equitable distribution of water through defined minimum flows and the problem of fragmented governance, which prevents effective communication between upstream and downstream users. The findings also highlighted the importance of having a formal rule structure to oversee allocations or ongoing communication processes to facilitate problem solving in preparation for drought conditions.

Students with Visual Impairment (VI) s5ll experience barriers to educa5on despite the right to educa5on s5pulated in the United Na5ons Conven5on on the Rights of Persons with Disabili5es (UNCRPD). Challenges such as delays in the conversion of curriculum content to accessible formats, inaccessible online course sites and teaching and learning that is mostly visual. With the University of Cape Town (UCT) going fully online due to COVID-19 pandemic, it became necessary to explore how equitable access to the curriculum is understood. The research topic is: How do staff and students at UCT understand equitable access to the curriculum for students with VIs? Four conceptual framework components were used. The hidden and enacted curriculum was used to explore hidden curriculum aspects and their effect on the enacted curriculum. Universal Design for Learning (UDL) framework was used to explore enablers such as assis5ve technology (AT) and challenges such as inaccessible content. Eight elements of digital literacies were used to explore access to opportuni5es to acquire digital literacies and the UNCRPD to ensure alignment with the right to educa5on. A Q methodology study was conducted which is a hybrid of both quan5ta5ve and qualita5ve methods. It sta5s5cally groups viewpoints that are significantly similar to or dis5nct from each other, quan5ta5vely into factors, then qualita5vely interprets these factors thema5cally to reveal par5cipant views about the research topic. Data was collected from students with VI, lecturers, staff from Disability Services, ICT Services, Library Services, and the Centre for Higher Educa5on Development using Q sor5ng where par5cipants ranked sixty statements into disagree, neutral and agree. Focus group discussions were used to support the interpreta5on of the factors. Findings revealed that: accessible curriculum is also a technical issue which is not priori5sed at UCT, and lecturers struggle with compe5ng demands such lack of 5me, need for promo5on and research. Accessibility design from the start both for curriculum development and support services is not valued. The right to educa5on for students with VI is par5al, varying their experience of the curriculum. Tes5ng of a course site for accessibility and lack of AT nega5vely affects right to educa5on. This study argues that students with VI do not yet enjoy full par5cipa5on in the curriculum due to lack of understanding of the complexity involved. UDL can help academics move from a deficit view to an asset view of students with VI. UCT should change its opera5onal model to accessibility from the start. Then UCT will move closer to equitable access to the curriculum for students with VI.

The tension between the Equidistance-Special Circumstances rule (articulated in the 1958 Geneva Convention on the Continental Shelf) and the Equitable Principles-Relevant Circumstances rule (declared in the 1969 North Sea Continental Shelf cases of the International Court of Justice) prevented the 1982 United Nations Convention on the Law of the Sea from stipulating a specific method for the delimitation of the EEZ or the continental shelf. For this reason, the role or status in maritime delimitation of the equidistance method, equitable principles, and relevant circumstances must generally be determined by reviewing the decisions of international courts and tribunals. The equidistance method has been employed in international case-law as a means of constructing a ‘provisional’ line. Analysis of international case-law also shows that the current rule governing maritime delimitation is the ‘achievement of an equitable solution’ in and of itself. The concept of equity in maritime delimitation is therefore only relevant to the equitability of the ‘result’ to be reached. Thus, the argument that equitable principles applicable to the examination of relevant circumstances are ascertainable and meaningful in maritime delimitation should be dismissed. Guaranteeing the equitability of principles or processes does not necessarily lead to the achievement of an equitable result. The remaining concept of significance to the delimitation process is that of ‘relevant circumstances’, which must be taken into account in order to reach an equitable result. The patterns or processes involved when the delimitation rule is applied by taking account of relevant circumstances are not predictable. The achievement of an equitable solution in maritime delimitation is preceded by the flexible consideration of relevant circumstances. The increasing importance of relevant circumstances to the delimitation of maritime boundaries is proven by two practical instances. One is the establishment of a single maritime boundary, which constitutes one typical maritime boundary; the other is the delimitation of the outer continental shelf, which has recently been included in international case-law. In brief, this thesis will show how best to arrive at an equitable solution by drawing attention to the demise of equitable principles and the dominant role of relevant circumstances. The clarification of taking into account relevant circumstances enables us more clearly to understand what is entailed in the task of achieving an equitable solution.

Let $V$ be a finite-dimensional vector space over a field $\mathbb{K}$, and let \text{End}$(V)$ be the set of all $\mathbb{K}$-linear transformations from $V$ to $V$. A {\em Leonard system} on $V$ is a sequence \[(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d),\] where $\A$ and $\B $ are multiplicity-free elements of \text{End}$(V)$; $\lbrace E_i\rbrace_{i=0}^d$ and $\lbrace E^*_i\rbrace_{i=0}^d$ are orderings of the primitive idempotents of $\A $ and $\B$, respectively; and for $0\leq i, j\leq d$, the expressions $E_i\B E_j$ and $E^*_i\A E^*_j$ are zero when $\vert i-j\vert > 1$ and nonzero when $\vert i-j \vert = 1$. % Leonard systems arise in connection with orthogonal polynomials, representations of many nice algebras, and the study of some highly regular combinatorial objects. We shall use the construction of Leonard pairs of classical type from finite-dimensional modules of $\mathit{sl}_2$ and the construction of Leonard pairs of basic type from finite-dimensional modules of $U_q(\mathit{sl}_2)$. Suppose $\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$ is a Leonard system. For $0 \leq i \leq d$, let \[ U_i = (E^*_0V+E^*_1V+\cdots + E^*_iV)\cap (E_iV+E_{i+1}V+\cdots + E_dV). \] Then $U_0$, $U_1$, \ldots, $U_d$ is the {\em split decomposition of $V$ for $\Phi$}. % The split decomposition of $V$ for $\Phi$ gives rise to canonical matrix representations of $\A$ and $\B$ in terms of useful parameters for the Leonard system. %These canonical matrix representations for $\A$, $\B$ are respectively lower bidiagonal and upper bidiagonal. In this thesis, we consider when certain Leonard systems share a split decomposition. We say that Leonard systems $\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$ and $\hat{\Phi}:=(\hat{\A} ;\hat{\B}; \lbrace \hat{E}_i\rbrace_{i=0}^d; \lbrace \hat{E^*}_i\rbrace_{i=0}^d)$ are {\em friends} when $\A = \hat{\A}$ and $\Phi$, $\hat{\Phi}$ have the same split decomposition. % We obtain Leonard systems which share a split decomposition by constructing them from closely related module structures for either $\mathit{sl}_2$ or $U_q(\mathit{sl}_2)$ on $V$. We then describe friends by a parametric classification. In this manner we describe all pairs of friends of classical and basic types. In particular, friendship is not entirely a property of isomorphism classes.

Water resources located in more than one country are complex systems governed by customary international law embodied in a rule known as equitable and reasonable utilisation, a recent development in international law not yet been applied by an international tribunal to resolve a dispute or to allocate transboundary water resources between countries. Water scarcity on a global scale has reached critical proportions with 1.1 billion people without access to sufficient safe water for personal and domestic use; over half that number depend on transboundary watercourses that will disappear over the next century. Conflicts of use over shared water resource have the potential to escalate into armed conflict; certainty in the peaceful means to avoid and resolve disputes is needed. The thesis examines international procedural practice and jurisprudence applying equitable principles in a case study to illustrate how equitable and reasonable utilisation may be applied by an international tribunal. The survey of international practice will inform States about procedural options for dispute avoidance and resolution in disputes over the use of transboundary water resources.

Thesis (Ph. D.)--University of Pennsylvania, 1937. / Bibliography: p. 153-159.