Numerical Solution of the Helmholtz Equation for the Superellipsoid via the Galerkin Method for the Dirichlet Problem

The objective of this work was to find the numerical solution of the Dirichlet problem for the Helmholtz equation fora smooth superellipsoid. The superellipsoid is a shape that is controlled by two parameters. There are some numerical issues in this type of an analysis; any integration method is affected by the wave number k, because of the oscillatory behavior of the fundamental solution. In this case we could only obtain good numerical results for super ellipsoids that were more shaped like super cones, which is a narrow range of super ellipsoids. The formula for these shapes was: x = cos(x)sin(y)n;y = sin(x)sin(y)n; z = cos(y) where n varied from 0.5 to 4. The Helmholtz equation, which is the modified wave equation, is used in many scattering problems. This project was funded by NASA RI Space Grant for testing of the Dirichlet boundary condition for the shape of the superellipsoid. One practical value of all these computations can be getting a shape for the engine nacelles in a ray tracing the space shuttle. We are researching the feasibility of obtaining good convergence results for the superellipsoid surface. It was our view that smaller and lighter wave numbers would reduce computational costs associated with obtaining Galerkin coefficients. In addition, we hoped to significantly reduce the number of terms in the infinite series needed to modify the original integral equation, all of which were achieved in the analysis of the superellipsoid in a finite range. We used the Green’s theorem to solve the integral equation for the boundary of the surface. Previously, multiple surfaces were used to test this method, such as the sphere, ellipsoid, and perturbation of the sphere, pseudosphere and the oval of Cassini Lin and Warnapala [9], Warnapala and Morgan [10]

Polarization of the Sri Lankan Polity: An Analysis of Presidential Elections (1982 – 2005)

Sri Lanka is a multi-ethnic, multi-religious developing country that has enjoyed continuous universal adult franchise since 1931. Under a new constitution enacted in 1978, Sri Lanka moved to a presidential system of government. Since 1982 five presidential elections were conducted. This paper analyzes voter behavior by looking at all the five presidential elections. This study shows that all the winners of the presidential elections (except in 2005) won them by appealing across racial and religious boundaries with a popular mandate. In 2005, there was a shift; the winner was able to secure victory by promoting a hard-line pro-Sinhala nationalistic platform. This signals a departure from the previous elections, as in the past it was understood that minority support is crucial to win the Presidency. The 2005 election sends a dangerous signal to a country that is ravaged by ethnic violence for over 20 years. Further, this study looks at the voter behavior in urban vs. rural areas. Similar to the red vs. blue states divide in the US, in Sri Lanka, there is a strong urban-rural division in voter behavior. Logistic regression was used to analyze the results of the elections

Numerical solution of the helmholtz equation for the superellipsoid via the galerkin method

The objective of this work was to find the numerical solution of the Dirichlet problem for the Helmholtz equation for a smooth superellipsoid. The superellipsoid is a shape that is controlled by two parameters. There are some numerical issues in this type of an analysis; any integration method is affected by the wave number k, because of the oscillatory behavior of the fundamental solution. In this case we could only obtain good numerical results for super ellipsoids that were more shaped like super cones, which is a narrow range of super ellipsoids. The formula for these shapes was: $x=cos(x)sin(y)^{n},y=sin(x)sin(y)^{n},z=cos(y)$ where $n$ varied from 0.5 to 4. The Helmholtz equation, which is the modified wave equation, is used in many scattering problems. This project was funded by NASA RI Space Grant for testing of the Dirichlet boundary condition for the shape of the superellipsoid. One practical value of all these computations can be getting a shape for the engine nacelles in a ray tracing the space shuttle. We are researching the feasibility of obtaining good convergence results for the superellipsoid surface. It was our view that smaller and lighter wave numbers would reduce computational costs associated with obtaining Galerkin coefficients. In addition, we hoped to significantly reduce the number of terms in the infinite series needed to modify the original integral equation, all of which were achieved in the analysis of the superellipsoid in a finite range. We used the Green's theorem to solve the integral equation for the boundary of the surface. Previously, multiple surfaces were used to test this method, such as the sphere, ellipsoid, and perturbation of the sphere, pseudosphere and the oval of Cassini Lin and Warnapala , Warnapala and Morgan

COVID-19 Pandemic Analysis by the Volterra Integral Equation Models: A Preliminary Study of Brazil, Italy, and South Africa

The COVID-19 pandemic has affected many people throughout the world. The objective of this research project was to find numerical solutions through the Gaussian Quadrature Method for the Volterra Integral Equation Model. The non-homogenous Volterra Integral Equation of the second kind is used to capture a broader range of disease distributions. Volterra Integral equation models are used in the context of applied mathematics, public health, and evolutionary biology. The mathematical models of this integral equation gave valid convergence results for the COVID-19 data for 3 countries Italy, South Africa and Brazil. The modeling of these countries was done using the Volterra Integral Equation, using the Gaussian Quadrature nodes. Inspired by the COVID-19 pandemic, the IRCD model included the number of initially infected individuals, the rate of infection, contact rate, death rate, fraction of recovered individuals, and the mean time an individual remains infected. This research investigated the feasibility of obtaining accurate convergence results for two models of the Volterra Integral Equation for the geographic locations of Italy, South Africa and Brazil. The IRCD model accounted for the infected rate, number of recovered, contact rate, and the death rate. The first 365 days of the pandemic were analyzed for the IRCD model. The ISR model accounted for the number of initially infected individuals, susceptible individuals, removed individuals, number of contacts per person, the recovery rate, and the total population. The ISR model specifically looked at COVID-19 in Brazil and South Africa for the first 300 days of the pandemic. Both models are mathematically and epidemiologically well posed

Note-Taking Mode and Academic Performance in Two Law School Courses

The use of laptops in law school classrooms has become fairly commonplace, especially in the last decade. Yet, studies in other higher education settings have found an association between note-taking mode and academic performance; specifically, using a laptop to take notes in the classroom is associated with negative academic performance outcomes. This study endeavors to assess the relationship between note-taking mode and academic performance in the law school setting. We compare the academic performance of handwriters to laptop users in two required, doctrinal courses as well as the effect of a randomly assigned treatment, exposing roughly half of the students in our analysis to a memorandum explaining the possible pitfalls of using a laptop to take class notes. We find that handwriting class notes has a strong positive association with academic performance in these two law school courses, supporting findings of the benefits of handwriting class notes in other academic settings

Minority electoral politics: A Sri Lankan case study

The two major parties in Sri Lanka are putting forward an increased number of minority candidates in order to prevent the minorities from voting for ethnically-/religiously-based minority parties. The proportional representative (PR) system that was introduced in 1989 was initially considered as a disadvantage to the minorities. However, the PR system has benefited the minorities and has not brought any expected benefits to the major parties. More minority candidates are getting elected, but the votes of the minorities are getting split, and therefore there is no net gain to the two major parties. In fact, there is evidence to say that parties that are taking a more nationalistic approach are doing better than the parties that are projecting a more inclusive image. Appealing to the minorities has alienated the majority, Sinhalese, so the major parties are moving towards more radical extremist nationalistic platforms. Wilcoxon Rank Sum test was used for this analysis, the samples that were taken were independent and no assumptions were made on the probability distributions other than the fact that they are continuous. © 2005 SAGE Publications

Tests of significance and effect size: Meaningful interpretation of statistical data in the health sciences

Learning to interpret and apply statistical principles is necessary for advanced study in the health professions world-wide. Because data play a critical role in a wide variety of biomedical and health-related studies, it is important to bring statistics to the forefront and discuss the implications of underlying data that often seem ambiguous to us. Case studies involving health-related statistics are often seen in a less than favorable light as the popular media, as well as researchers, misinterpret the data, collect certain data in large numbers yet miss certain critical measures, and do not always have replicable results to deliver. The focus of this paper is to discuss the concept of statistical significance with respect to meaningful effect size, citing examples of how experts reach different conclusions to national and global health science problems that are being analyzed. These insights should be helpful to researchers as they formulate and test hypotheses and draw appropriate conclusions

Z-Score Demystified: a Critical Analysis of the Sri Lankan University Admission Policy

In the year 2001, the University Grants Commission of Sri Lanka successfully appealed to change the method of determining the cut-off scores for university admissions from raw scores to standardized z-scores. This standardization allegedly eliminated the discrepancy caused due to the assumption of equal difficulty levels across all subjects. This paper analyzes the effectiveness of using z-score cut-offs for university admissions compared to raw score cut-offs. For the purpose of the analysis, only those students who were admitted to Sri Lankan universities on the basis of the district quota were considered. The Wilcoxon Rank Sum test was used for this analysis. The samples that were taken are independent and no assumptions were made on the probability distributions other than the fact that they are continuous

MS Word Export To Multiple PDF Files Software - Please purchase license.Tests of Significance and Effect Size: Meaningful Interpretation of Statistical Data in the Health Sciences

Learning to interpret and apply statistical principles is necessary for advanced study in the health professions world-wide. Because data play a critical role in a wide variety of biomedical and health-related studies, it is important to bring statistics to the forefront and discuss the implications of underlying data that often seem ambiguous to us. Case studies involving health–related statistics are often seen in a less than favorable light as the popular media, as well as researchers, misinterpret the data, collect certain data in large numbers yet miss certain critical measures, and do not always have replicable results to deliver. The focus of this paper is to discuss the concept of statistical significance with respect to meaningful effect size, citing examples of how experts reach different conclusions to national and global health science problems that are being analyzed. These insights should be helpful to researchers as they formulate and test hypotheses and draw appropriate conclusions

The Numerical Solution of the Exterior Boundary Value Problems for the Helmholtz\u27s Equation for the Pseudosphere

In this paper, the global Galerkin method is used to numerically solve the exterior Neumann and Dirichlet problems for the Helmholtz equation for the Pseudosphere in three dimensions based on Jones\u27 modified integral equation approach. Warnapala and Morgan have used this method for the Oval of Cassini and obtained good results. Theoretical and computational details of the method for small values of k for the pseudosphere are presented