Some equitably 3-colourable cycle decompositions

Let G be a graph in which each vertex has been coloured using one of k colours, say c(1), c(2),..., c(k). If an m-cycle C in G has n(i) vertices coloured c(i), i = 1, 2,..., k, and (i) - n(j) less than or equal to 1 for any i, j is an element of {1, 2,..., k}, then C is equitably k-coloured. An m-cycle decomposition C of a graph G is equitably k-colourable if the vertices of G can be coloured so that every m-cycle in C is equitably k-coloured. For m = 4,5 and 6, we completely settle the existence problem for equitably 3-colourable m-cycle decompositions of complete graphs and complete graphs with the edges of a 1-factor removed. (C) 2004 Elsevier B.V. All rights reserved.

Sustainable and equitable retirement in a life course perspective

We argue that long term sustainability of social security systems requires not only better equilibrium between the proportion in retirement and in employment but also an equitable distribution of the additional financial burden that aging inevitably will require. We examine how a proportional fixed ratios model of burden sharing between the aged and non-aged will establish inter-generational equity. Additionally we address the question of intra-generational equity and argue that the positive association between lifetime income and longevity requires more progressive financing of pensions and of care for the elderly.

Towards Equitable Access to Postsecondary Education: Learning From the Voices of First Generation Students

This qualitative, phenomenological study investigated first generation students’ perceptions of the challenges they experienced in the process of accessing higher education and the type of school-based support that was received. Particular emphasis was placed on the impact of parental education level on access to postsecondary education (PSE) and how differences in support at the primary and secondary levels of schooling influenced access. Purposeful, homogenous sampling was used to select 6 first generation students attending a postsecondary institution located in Ontario. Analysis of the data revealed that several interrelated factors impact first generation students’ access to postsecondary education. These include familial experiences and expectations, school streaming practices, secondary school teachers’ and guidance counselors’ representations of postsecondary education, and the nature of school-based support that participants received. The implications for theory, research, and practice are discussed and recommendations for enhancing school-based support to ensure equitable access to postsecondary education for first generation students are provided.

Equitable and Inclusive Education for All? Public Funding for Denominational Schools in Ontario

Roman Catholic separate schools’ denominational right to receive public funding is a contentious issue in Ontario’s educational system. Ontario’s publicly funded denominational schools historically served a purpose at Confederation; however, in light of Ontario’s evolving demographics, publicly funding denominational schools today may no longer serve the needs of Ontario. The research problem in this study is expressed through growing problems reconciling Roman Catholic schools with diversity and current public views. Additionally, recent tensions, public views, and political consensus suggest it is time to revisit the existing policy. In order to understand both the history of denominational schools and the present context, this study conducted II policy analyses as its research design by completing 2 policy cycles. The first policy cycle determined that based upon Upper and Lower Canada’s pre-Confederation diversity, extending public funding to denominational schools at Confederation was an effective way of protecting minority rights; however, the analysis in the second policy cycle; which examined how equitable and inclusive denominational schools are today, concluded that the denominational school system no longer serves the diversity and equity needs of contemporary Ontario. Building on these findings, this study then explored two viable alternative educational arrangements for Ontario’s future educational system: publicly funding all faith-based schools, or publicly financing a one-school-system. To address the diversity issue in Ontario, transitioning toward publicly funding a one-school-system is found to be the most viable option.

Alternative Development in Ecuador: Prospects for Equitable Growth

Realiza un análisis de la economía ecuatoriana y de sus políticas de desarrollo económico como la sustitución de importaciones e impulso a la industrialización interna y los impactos en la economía de los ajustes estructurales y las reformas neolibarales. Estudia las estrategias de desarrollo en el Ecuador incluyendo la dolarización y expone la precaria situación de vida de la sociedad. Analiza la forma en que se distribuye la riqueza en el país y compara su situación económica con otros países de la región, dando a conocer los coeficientes Gini y los niveles de inequidad. Analiza modelos alternativos de desarrollo más equitativos y sustentables para el país. Finalmente estudia el contexto político del país bajo la presidencia de Rafael Correa, su política económica y social, particularmente el desarrollo de la economía solidaria.

Equitability revisited: why the “equitable threat score” is not equitable

In the forecasting of binary events, verification measures that are “equitable” were defined by Gandin and Murphy to satisfy two requirements: 1) they award all random forecasting systems, including those that always issue the same forecast, the same expected score (typically zero), and 2) they are expressible as the linear weighted sum of the elements of the contingency table, where the weights are independent of the entries in the table, apart from the base rate. The authors demonstrate that the widely used “equitable threat score” (ETS), as well as numerous others, satisfies neither of these requirements and only satisfies the first requirement in the limit of an infinite sample size. Such measures are referred to as “asymptotically equitable.” In the case of ETS, the expected score of a random forecasting system is always positive and only falls below 0.01 when the number of samples is greater than around 30. Two other asymptotically equitable measures are the odds ratio skill score and the symmetric extreme dependency score, which are more strongly inequitable than ETS, particularly for rare events; for example, when the base rate is 2% and the sample size is 1000, random but unbiased forecasting systems yield an expected score of around −0.5, reducing in magnitude to −0.01 or smaller only for sample sizes exceeding 25 000. This presents a problem since these nonlinear measures have other desirable properties, in particular being reliable indicators of skill for rare events (provided that the sample size is large enough). A potential way to reconcile these properties with equitability is to recognize that Gandin and Murphy’s two requirements are independent, and the second can be safely discarded without losing the key advantages of equitability that are embodied in the first. This enables inequitable and asymptotically equitable measures to be scaled to make them equitable, while retaining their nonlinearity and other properties such as being reliable indicators of skill for rare events. It also opens up the possibility of designing new equitable verification measures.

Sociodemographic vulnerability in the Caribbean: an examination of the social and demographic impediments to equitable development with participatory citizenship in the Caribbean at the dawn of the twenty-first century

Includes bibliography

Edge Colourings of Graphs Avoiding Monochromatic Matchings of a Given Size

Let k and l be positive integers. With a graph G, we associate the quantity c(k,l)(G), the number of k-colourings of the edge set of G with no monochromatic matching of size l. Consider the function c(k,l) : N --> N given by c(k,l)(n) = max {c(k,l)(G): vertical bar V(G)vertical bar = n}, the maximum of c(k,l)(G) over all graphs G on n vertices. In this paper, we determine c(k,l)(n) and the corresponding extremal graphs for all large n and all fixed values of k and l.