2,501 research outputs found

    A block Hankel generalized confluent Vandermonde matrix

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    Vandermonde matrices are well known. They have a number of interesting properties and play a role in (Lagrange) interpolation problems, partial fraction expansions, and finding solutions to linear ordinary differential equations, to mention just a few applications. Usually, one takes these matrices square, q×qq\times q say, in which case the ii-th column is given by u(zi)u(z_i), where we write u(z)=(1,z,...,zq−1)⊤u(z)=(1,z,...,z^{q-1})^\top. If all the ziz_i (i=1,...,qi=1,...,q) are different, the Vandermonde matrix is non-singular, otherwise not. The latter case obviously takes place when all ziz_i are the same, zz say, in which case one could speak of a confluent Vandermonde matrix. Non-singularity is obtained if one considers the matrix V(z)V(z) whose ii-th column (i=1,...,qi=1,...,q) is given by the (i−1)(i-1)-th derivative u(i−1)(z)⊤u^{(i-1)}(z)^\top. We will consider generalizations of the confluent Vandermonde matrix V(z)V(z) by considering matrices obtained by using as building blocks the matrices M(z)=u(z)w(z)M(z)=u(z)w(z), with u(z)u(z) as above and w(z)=(1,z,...,zr−1)w(z)=(1,z,...,z^{r-1}), together with its derivatives M(k)(z)M^{(k)}(z). Specifically, we will look at matrices whose ijij-th block is given by M(i+j)(z)M^{(i+j)}(z), where the indices i,ji,j by convention have initial value zero. These in general non-square matrices exhibit a block-Hankel structure. We will answer a number of elementary questions for this matrix. What is the rank? What is the null-space? Can the latter be parametrized in a simple way? Does it depend on zz? What are left or right inverses? It turns out that answers can be obtained by factorizing the matrix into a product of other matrix polynomials having a simple structure. The answers depend on the size of the matrix M(z)M(z) and the number of derivatives M(k)(z)M^{(k)}(z) that is involved. The results are obtained by mostly elementary methods, no specific knowledge of the theory of matrix polynomials is needed

    Africa Capacity Report 2014: Capacity Imperatives for Regional Integration in Africa

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    The Africa Capacity Report (ACR) and its supporting indicators offer inputs for decisions on what to finance to develop capacity. Most countries are doing well on their policy environments and having processes in place to implement policies. Countries are doing less well on achieving development results and least on capacity development outcomes. The Report and its indicators also point to the regulatory and institutional reforms needed to better support public–private partnerships in capacity investment and building—and to the investments needed to further strengthen public administration. And they spotlight the importance of political will to enhance social inclusion and development. Each Report showcases an annual theme of key importance to Africa's development agenda. This year the focus is on the capacity imperatives for regional integration, a core mandate of the ACBF, and on the capacities of the regional economic communities (RECs). The Report outlines what is needed to strengthen the RECs. Integrate capacity building in wider efforts to achieve sustainable development. Assure adequate administrative and financial resources. Emphasize the retention and use of skills, not just their acquisition. And monitor and evaluate all efforts to develop capacity. The capacity dimensions and imperatives for regional integration are crucial today as countries, RECs, specialized regional institutions, and regional development organizations, are developing strategic regional frameworks and building capacity to pursue regional integration across the continent. The ACBF's many regionally oriented interventions help move the regional integration agenda forward by strengthening the RECs as platforms for harmonizing policy and enhancing trade among member countries

    Hamara Healthy Living Centre - an evaluation

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    Hamara is a Healthy Living Centre which aims to improve health and well-being through providing a range of culturally appropriate activities and services. Hamara has a vision of 'bringing communities together' and since it was established in 2004, the Centre has provided a valuable community resource in South Leeds. Partnership work between Hamara and Leeds Met goes back to 2002. In 2007, the Centre for Health Promotion Research carried out an evaluation of Hamara in partnership with Hamara staff and Leeds Met Community Partnerships and Volunteering. This was followed by a highly successful community cohesion conference 'One Community' which was held at Hamara on 10th October 2008, and was supported through a Leeds Met public engagement grant. The event attracted over a hundred people from diverse communities and organisations across Leeds. A packed audience heard Hilary Benn, local MP and Patron of Hamara, talk about the importance of working in collaboration around community cohesion. Jane South, Centre for Health Promotion Research, presented the main evaluation results and set out the some challenges for the future. The proceedings concluded with the presentation of awards to a number of for local community champions who work to bring people together and make a real difference in the city of Leeds

    Spartan Daily, February 12, 2019

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    Volume 152, Issue 8https://scholarworks.sjsu.edu/spartan_daily_2019/1007/thumbnail.jp

    Scattering Equations and a new Factorization for Amplitudes I: Gauge Theories

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    In this work we show how a double-cover (DC) extension of the Cachazo, He and Yuan formalism (CHY) can be used to provide a new realization for the factorization of the amplitudes involving gluons and scalar fields. First, we propose a graphic representation for a color-ordered Yang-Mills (YM) and special Yang-Mills-Scalar (YMS) amplitudes within the scattering equation formalism. Using the DC prescription, we are able to obtain an algorithm (integration-rules) which decomposes amplitudes in terms of three-point building-blocks. It is important to remark that the pole structure of this method is totally different to ordinary factorization (which is a consequence of the scattering equations). Finally, as a byproduct, we show that the soft limit in the CHY approach, at leading order, becomes trivial by using the technology described in this paper.Comment: 50+7 pages and typos fixed. Some modifications were made to improve the tex

    Planning Law and Democratic Living

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