5,592 research outputs found

    When are Multiples of Polygonal Numbers again Polygonal Numbers?

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    Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions.Comment: 17 pages, 1 figure, 2 tables. New version added a table of solutions to the second proble

    Economics of Production of Green Peas (Pisum sativum L.) in Punjab

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    The economics of production and factors influencing the productivity of green peas in Punjab have been studied using the primary data from pea growers. The data have been analysed using simple tabular and functional analyses. The results have revealed that the green peas and wheat are the main crops in rabi season. It has been noticed that 75.85 per cent of the farmers purchase pea seeds from dealers. The yield of green peas has been found the highest on small farms among all the farm-size categories. The total cost incurred has been higher in large than small and medium farmers due to more use of inputs by the former. The gross and net returns have been found higher in large than small and medium farmers due to realization of higher prices by them and exploring of other markets due to their higher marketable surpluses. The functional analysis has revealed that the fertilizers, irrigation and machinery are the impact variables, influencing the productivity of green peas positively. The returns over variable costs in the case of peas have been higher by 129 per cent than those in wheat (main competing crop). It has been argued that the farmers be advised by the agricultural extension experts to adopt green pea cultivation for improving the efficiency of the farms through increased income per unit of land. Moreover, it will provide impetus to the diversification program of the state government and improve the soil health.Agricultural and Food Policy,

    Automated, Parallel Optimization Algorithms for Stochastic Functions

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    The optimization algorithms for stochastic functions are desired specifically for real-world and simulation applications where results are obtained from sampling, and contain experimental error or random noise. We have developed a series of stochastic optimization algorithms based on the well-known classical down hill simplex algorithm. Our parallel implementation of these optimization algorithms, using a framework called MW, is based on a master-worker architecture where each worker runs a massively parallel program. This parallel implementation allows the sampling to proceed independently on many processors as demonstrated by scaling up to more than 100 vertices and 300 cores. This framework is highly suitable for clusters with an ever increasing number of cores per node. The new algorithms have been successfully applied to the reparameterization of a model for liquid water, achieving thermodynamic and structural results for liquid water that are better than a standard model used in molecular simulations, with the the advantage of a fully automated parameterization process

    Trump’s appeal is built around giving a public voice for private trouble

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    How can we explain Donald Trump’s appeal to so many Americans? During and soon after the 2016 Election campaign Kusminder Chahal and Harris Beider interviewed more than 400 Americans in five US cities who considered themselves to be white and working class

    Catastrophe squared: COVID-19 vaccine inequity in humanitarian crises

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    The COVID-19 pandemic has caused physical, social, and economic devastation all around the world. While more manageable case numbers and immunisation efforts seem to indicate that the world has come a long way in controlling the virus, there is great inequity in vaccination numbers around the world. Low-income countries have only received 14 doses per 100 people—13-fold lower than the 182 doses per 100 people in upper-middle income and high-income countries. This paper highlights the disparity of COVID-19 vaccination rates in high-income countries versus those afflicted with crises and raises the need for linking pandemic response with humanitarian assistance

    The Meaning of the Colour Blue

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    The language of the birds outside my bedroom window has to compete with the sound of traffic seeping through the brick walls that becomes a part of my day, like the canned music that fills my ear when I'm once again waiting for a real voice on the telephone instead of a “press three” for hours of operation

    Review: Mediactive (2010)

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