Browsing by Author "Mallawa Arachchi, D.K."
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Item Analysis of a stochastic predator-prey model(Faculty of Science, University of Kelaniya, Sri Lanka, 2016) Prasadini, K.D.S.; Mallawa Arachchi, D.K.In biological systems Lotka-Volterra predator-prey model describes the population dynamics of two interacting species of predators and its preys. Classical predatorprey model is a primitive deterministic model governed by the two differential equations, namely, = ( − ) and = ( − ) where and denote prey and predator respectively, and , , and are parameters. This model can be improved by introducing stochasticity that accounts for the random fluctuations of a realistic predator-prey dynamical system. In this research work, we use Stochastic Differential Equation (SDE) approach. There are various ways, based on various assumptions, to incorporate SDE. One common approach is to use equations of the following form: = ( − ) + ( + ) = ( − ) + ( + ) These types of Stochastic Differential Equations (SDE) can be simulated in Matlab using numerical methods such as Euler-Maruyama method. Phase planes of the deterministic and stochastic models are carried out to demonstrate the behavior of this modified model. Our initial goal is to compare different stochastic models with the original deterministic model through simulations. The deterministic model has a positive equilibrium which is globally stable for positive values of the parameters. Nevertheless, in the stochastic model, the predator and prey populations may tend to extinction. Extinction percentages of predator or prey population are summarized and analyzed through this research work.Item Analysis of a stochastic predator-prey model(Faculty of Science, University of Kelaniya, Sri Lanka, 2016) Prasadini, K.D.S.; Mallawa Arachchi, D.K.In biological systems Lotka-Volterra predator-prey model describes the population dynamics of two interacting species of predators and its preys. Classical predatorprey model is a primitive deterministic model governed by the two differential equations, namely, ���� = (������ − ��������) ���� and ���� = (�������� − ������) ���� where �� and �� denote prey and predator respectively, and ����, ����, ���� and ���� are parameters. This model can be improved by introducing stochasticity that accounts for the random fluctuations of a realistic predator-prey dynamical system. In this research work, we use Stochastic Differential Equation (SDE) approach. There are various ways, based on various assumptions, to incorporate SDE. One common approach is to use equations of the following form: ���� = (������ − ��������) ���� + ��(���� + ����)�� ������ ���� = (�������� − ������) ���� + ��(���� + ����)�� ������ These types of Stochastic Differential Equations (SDE) can be simulated in Matlab using numerical methods such as Euler-Maruyama method. Phase planes of the deterministic and stochastic models are carried out to demonstrate the behavior of this modified model. Our initial goal is to compare different stochastic models with the original deterministic model through simulations. The deterministic model has a positive equilibrium which is globally stable for positive values of the parameters. Nevertheless, in the stochastic model, the predator and prey populations may tend to extinction. Extinction percentages of predator or prey population are summarized and analyzed through this research work.Item Categorizing T20 cricket grounds(Faculty of Science, University of Kelaniya, Sri Lanka, 2016) Pathirana, O.D.R.; Mallawa Arachchi, D.K.T20 cricket matches are played by all cricket playing countries. There are more than 80 grounds in various countries on which these games are played. It is hypothesized that some of these grounds favor batsmen while others favor bowlers, or some grounds are high-scoring while others are low-scoring. In this research work, we perform a statistical analysis to determine whether those grounds can be categorized based on the past data. Numerous factors can be considered for the analysis. Main factors we have been considering are the total runs scored in both innings, humidity level, gust, wind, air pressure and the temperature at the grounds when the matches are played. Cluster analysis was used in investigating and determining the number of categories. This study helps identify the behavior of the T20 cricket grounds all over the world and thus enables one to predict the winning possibilities. Data were collected through Cricinfo website from 84 cricket grounds throughout the world. Ward’s method of Hierarchical cluster analysis, which is a major statistical method used in determining the relatively homogeneous clusters, was used. We found that grounds can be clustered into 3 clusters according to the coefficients of the Wards linkage table. When we consider the countries in which these grounds are located, there is no evidence to conclude that grounds in some specific countries are belonging to a particular category. For example there are grounds in India belonging to all three categories. SPSS statistical software was used in this analysis to categorize the grounds. The research work is being carried out to identify how cluster changes with different factors.Item Derivation and application of a Stochastic Differential Equation (SDE) Model for cricket(University of Kelaniya, 2013) Weerasinghe, W.M.H.N.; Mallawa Arachchi, D.K.Stochastic Differential Equations (SDE) can be used to model many dynamical systems in place of the Markov chain approach. In this research, the SDE model that accounts for the score of a limited over cricket match is formulated based on the assumption that the runs scored and the number of wickets fallen within a single delivery follow a stochastic process. The peculiarity of this model is that a discrete process is modeled by a continuous-time continuous-space stochastic process, which is called a SDE model. Numerical simulations are performed using Euler-Maruyama method. Parameter estimation is carried out using the data available online for teams and individual players. Parameters were estimated for the players in the Indian and Sri Lanka teams considering ten ODI and fifteen T20 matches played between 2009 and 2012. Simulated results give evidence to the validity of the model. Some statistical tests were used to investigate the significance of the results. The model may be used for forecasting purposes. It can also be used to find a suitable batting order that optimizes the total score. The model can also be improved by taking into consideration the other factors that affect the scoring of an innings. For example, the pitch, weather condition and home-ground advantage can be taken into considerationItem A Different Look at the Primitive Integral Triads of z n = y n + xn (n = 2) and a Conjecture on z n - xn for any n(¹ 2)(University of Kelaniya, 2005) Piyadasa, R.A.D.; Mallawa Arachchi, D.K.; Munasinghe, J.The primitive Pythagorean triples (x, y, z) are now well understood [1]. However, we believe that a closer look at the solution is needed along new directions to understand the terrible difficulty in giving a simple proof for the Fermat’s last theorem. Keeping this fact in mind we look at the solutions of z 2 = y 2 + x2 , (x, y) = 1 in the following manner. (x, y, z) is a primitive Pythagorean triple if and only if x2 + y 2 = z 2 , (x, y) = 1 (1)It is obvious that one of (x, y, z) is even and it can be shown that z is never even by using (1) and substituting z = y + p, p ³ 1, in it. Now either x or y is even. If we suppose that y is even, z 2 - x2 = y 2 and then it follows that z - x = 22b -1 or z - x = 22b -1a 2 where a ,b ³ 1 and are integers. The following are examples for the justification of our point. 2 2 2 3 2 2 2 13 12 5 , 2 , 2 17 15 8 , 1 , 2 = + = - = = + = - = z x z x b b 1132 = 1122 +152 ,b = 1,a = 7 z - x = 2´ 72 Now we apply the mean value theorem of the form a2 - b2 = 2(a - b)x where a n for any prime n ³ 3 . Then, z n - xn = n (z - x)x n-1 by the mean value theorem and we conjecture that x is irrational when z - x =a nnbn-1 .Item Exact Formula for the Sum of the Squares of the Bessel Function and the Neumann Function of the Same Order of Half-Odd Integer(University of Kelaniya, 2008) Piyadasa, R.A.D.; Mallawa Arachchi, D.K.Sum of the squares of the Bessel function and the Neumann function of the same order of half-odd integer has been found to be very useful in addressing a puzzle in nuclear physics. One approximate formula available in the literature is valid for the complex argument whose real part is greater than zero, and the absolute value of error term is undefined for half-odd integers. Another approximate formula which is valid for all complex arguments has been obtained using sophisticated mathematical method called Barnes' method. However, the error in the formula is very difficult to calculate. We have obtained exact formula for the sum of the squares of Bessel and Neumann functions of the same order of half-odd integers which is valid for all complex arguments, and its proof is also given.Item Higher order Markov chain approach in modeling Cricket Scores(Faculty of Graduate Studies, University of Kelaniya, 2015) Jayamaha, J.H.R.K.; Mallawa Arachchi, D.K.The Markov chain models are applied in a wide range of topics such as physics, chemistry, medicine, music, game theory and sports. In this research work, higher order Markov chain model that accounts for scores of an innings of a limited-over cricket match (T20) which is based on the assumption that the runs scored and wicket occurred follow a higher order Markov chain. Therefore two models, first order and second order Markov chains are considered in this research. Parameter estimation is carried out using the data available online for T20 innings. Parameters were estimated for Sri Lanka and India teams considering fifteen T20 matches for each team between 15/06/2006 to 07/09/2014. The probabilities depend on the batsmen, the bowler, the number of wickets lost, the number of balls bowled and the innings. Simulated results give evidence to the validity of the model. Some statistical tests were used to investigate the significance of the results. The model may be used for forecasting purposes and find the effect when order of Markov chain increased. The model can be improved by taking into considerations the other factors that affect the scoring of an innings. For example, home-ground advantage, weather condition, pitch, team which they are playing against, batsman‘s performance, bowler‘s performance etc. And also the model can be extended to find the effect between the wicket occurring rate and scoring rate.Item New Interpretation Of Primitive Pythagorean Triples And A Conjecture Related To Fermat’s Last Theorem(University of Kelaniya, 2007) Piyadasa, R.A.D.; Munasinghe, J.; Mallawa Arachchi, D.K.; Kumara, K.H.In this study primitive Pythagorean triples have been carefully examined and found that all of them satisfy a simple rule related to mean value theorem. It is pointed out that integral triples satisfying the equation on Fermat’s Last Theorem should satisfy a special rule related to mean value theorem. A conjecture is proposed which may lead to find a simple proof of Fermat’s Last Theorem.Item Residual properties of groups(University of Kelaniya, 2000) Mallawa Arachchi, D.K.Item Stochasticity in SIRV Models for the transmission of epidemic diseases(University of Kelaniya, 2013) Hasanthika, N.H.E.; Kulathunga, D.D.S.; Mallawa Arachchi, D.K.