When are Multiples of Polygonal Numbers again Polygonal Numbers?

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

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,

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

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

Automated, Parallel Optimization Algorithms for Stochastic Functions

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

Discursive Analysis of the Accounts of Victims of Racist Incidents

In this study I focus on identifying power and processes of ‘Othering’ in the accounts of victims of racist incidents by identifying discursive repertoires and using Critical Discourse Analysis (CDA). By re-analysing in-depth interviews with victims of racist incidents I have been able to identify discursive repertoires that occur in the speech of those who are othered through their lived experience of racism. Practitioners of CDA have given limited attention to exploring the lived experiences of minority groups but this study has shown distinctive patters of self and other positioning in their discourse. Racist markers, actions that could only be explained in the context of racism, served as a powerful tool to exclude and act as a reminder of the non-acceptance of minority groups. These markers were constructed through discourses of the Other, repertoires such as disruption and disapproval acted through name-calling, gendered racism, a continuum of violence with consequences that included emotional (shame and embarrassment), psychological harm and spirit murder (or total harm) arising from the accumulation of everyday racist experiences. This study suggests that revisiting qualitative data is a productive activity and sheds new light on existing social issues and that by focusing on a reanalysis new perspectives can be drawn or examined that were not central to the original research. The study also suggests that more research may be required in exploring discourses of the Other as identified by the Other

Catastrophe squared: COVID-19 vaccine inequity in humanitarian crises

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

A generic framework for hybrid simulation in healthcare

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University, 11/01/2010.Healthcare problems are complex; they exhibit both detail and dynamic complexity. It has been argued that Discrete Event Simulation (DES), with its ability to capture detail, is ideal for problems exhibiting this type of complexity. On the other hand, System Dynamics (SD) with its focus on feedback and nonlinear relationships lends itself naturally to comprehend dynamic complexity. Although these modelling paradigms provide valuable insights, neither of them are proficient in capturing both detail and dynamic complexity to the same extent. It has been argued in literature that a hybrid approach, wherein SD and DES are integrated symbiotically, will provide more realistic picture of complex systems with fewer assumptions and less complexity. In spite of wide recognition of healthcare as a complex multi- dimensional system, there has not been any reported study which utilises hybrid simulation. This could be attributed to the fact that due to fundamental differences, the mixing of methodologies is quite challenging. In order to overcome these challenges a generic theoretical framework for hybrid simulation is required. However, there is presently no such generic framework which provides guidance about integration of SD and DES to form hybrid models. This research has attempted to provide such a framework for hybrid simulation which can be utilised in healthcare domain. On the basis of knowledge induced from literature, three requirements for the generic framework have been established. It is argued that the framework for hybrid simulation should be able to provide answers to Why (why hybrid simulation is required), What (what information is exchanged between SD and DES models) and How (how SD and DES models are going to interact with each other over the time to exchange information) within the context of implementation of hybrid simulation to different problem scenarios. In order to meet these requirements, a three-phase generic framework for hybrid simulation has been proposed. Each phase of the framework is mapped to an established requirement and provides guidelines for addressing that requirement. The proposed framework is then evaluated theoretically based on its ability to meet these requirements by using multiple cases, and accordingly modified. It is further evaluated empirically with a single case study comprising of Accident and Emergency department of a London district general hospital. The purpose of this empirical evaluation is to identify the limitations of the framework with regard to the implementation of hybrid models. It is realised during implementation that the modified framework has certain limitations pertaining to the exchange of information between SD and DES models. These limitations are reflected upon and addressed in the final framework. The main contribution of this thesis is the generic framework for hybrid simulation which has been applied within healthcare context. Through an extensive review of existing literature in hybrid simulation, the thesis has also contributed to knowledge in multi-method approaches. A further contribution is that this research has attempted to quantify the impact of intangible benefits of information systems into tangible business process improvements. It is expected that this work will encourage those engaged in simulation (e.g., researchers, practitioners, decision makers) to realise the potential of cross-fertilisation of the two simulation paradigms

The Meaning of the Colour Blue

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