Mathematics
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Item Advanced plane geometry research – I(Sri Lanka Association for the Advancement of Science, 2010) Amarasinghe, G.W.I.S.Item Advanced plane geometry research – II(Sri Lanka Association for the Advancement of Science, 2010) Amarasinghe, G.W.I.S.Item Advanced plane geometry research – III: Alternative proofs for the standard theorems in plane geometry(Sri Lanka Association for the Advancement of Science, 2010) Amarasinghe, G.W.I.S.Item Analytical solutions of the time-fractional non- linear Schrodinger equation with zero and non-zero trapping potential through the sumudu decomposition method(Journal of Science of the University of Kelaniya Volume:10, 2019) Mathanaranjan, K.; Himalini, K.Sumudu decomposition method is used to construct the approximate analytical solutions of time-fractional nonlinear Schrodinger equations with zero and nonzero trapping potential. The Sumudu decomposition method is a combined form of the Sumudu transform and the Adomian decomposition method. The fractional derivatives are defined in the Caputo sense. The exact solutions of some nonlinear Schrodinger equations are given as a special case of our approximate analytical solutions. The computations show that the described method is easy to apply, and it needs smaller size of computation as compared to the other existing methods. Further, the solutions are derived in a convergent series form which shows the effectiveness of the method for solving a wide variety of nonlinear fractional differential equations.Item An application of graph theory in asymmetric key cryptography(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Fernando, K.K.N.; Wijesiri, G.S.The development of digital communication in many facets of our everyday lives significantly impacts the evolving world we live in today. The rapid growth and evolution of digital communication have become the backbone of how we interact with other people. Therefore, it is imperative to protect information and data from unauthorised activities, such as accessing, using, exposing, damaging, modifying, copying, or deleting them. Safeguarding data from these invalid operations is crucial to ensure its integrity, confidentiality, and availability. Cryptography plays a pivotal role in ensuring the security and privacy of information in various contexts, such as online banking, e-commerce transactions, and communication between governments, military organisations, and businesses. Various types of mathematical techniques are available for application in modern cryptology. The application of graph theory is widely utilised in the field of cryptography due to its straightforward representation in computers as a matrix. In this study, we propose a novel asymmetric key cryptography scheme for secure message transmission using graph theory and matrices. The proposed scheme consists of four algorithms. The key generation algorithm on the receiver side is based on the properties of matrices, which enables us to establish the relationship between private key and public key through matrix operations. On the sender-side graph generation algorithm, a graph theory approach is applied to encrypt the original message, and the message is converted into a splitting graph and its minimum spanning tree. Then, the sender-side encryption algorithm is used to generate a complex final ciphertext using the receiver’s public key. The decryption algorithm follows the same process in reverse order, employing the receiver’s private key. This system will provide better security while storing data in the financial retail industry and sharing passwords in transactions.Item Approximate for the null distribution of a statistic caused by random combinations(Applied Statistics Association of Sri Lanka, 2000) Kulatunga, D.D.S.; Kudo, A.; Azuma, S.Item Asymptotic dynamics of a periodic toxin-producing phytoplankton model(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Dilshani, P.P.; De Silva, T.M.M.Harmful algal blooms (HABs) caused by Toxin-Producing Phytoplankton (TPP) have become increasingly common worldwide. Understanding the complex interactions between TPP and other organisms in the ecosystem is crucial. This study focuses on asymptotic dynamics of the plankton interaction between TPP and zooplankton, with an additional mortality factor accounting for zooplankton ingestion of TPP. One of the proposed models assumes constant toxin production by TPP, and the populations being studied are distributed uniformly throughout the space. This assumption results in an autonomous system of ordinary differential equations. To incorporate natural day and night, tidal, or seasonal cycles, the model is extended to a periodic system by including the toxin liberation rate y as a periodic function of the fonn y(t) = + Aq(t)), where A represents the magnitude of the periodicity, and q(t) is a —periodic function with > O. The study examines the existence of steady states and trivial periodic solutions and analyses the stability of both models. Moreover, using the concept of uniform persistence, we derive sufficient conditions for the coexistence of the periodic system based on the model parameters. The findings of this study demonstrate that increasing the avoidance of TPP by zooplankton enhances the coexistence of plankton populations in both the constant and periodic toxin production systems. The study also discusses Hopfbifurcations in the constant toxin-producing model, providing insights into the system's dynamic behaviour. Numerical simulations are performed to validate the analytical findings of proposed models and their implications.Item Centrality Measures to Identify Traffic Congestion on Road Networks: A Case Study of Sri Lanka.(IOSR Journal of Mathematics (IOSR-JM), Department of Mathematics, Faculty of Science, University of Kelaniya, Sri Lanka., 2017) Jayaweera, I.M.L.N.; Perera, K.K.K.R.; Munasinghe, J.This study presents a graph theoretical approach to identify the traffic congestion on a road network. Problem address on a city called Kiribathgoda situated in the western province of Sri Lanka. In the analysis of social networks, centrality measures played a vital role to identify the central nodes in a given network. We look at the applicability of centrality and betweenness measures in order to identify the most important locations which directly affect to the traffic congestion in road networks in Sri Lanka. Using the graph theoretical approach a traffic network for a selected area was constructed and several centrality measures were calculated. According to our simulation results, it was noted that the practically identified locations could be identified from the simulations carried out using the centrality measures.Item Confirmation of earth’s closed loop orbit using tidal waves.(Sri Lanka Association for the Advancement of Science Proceedings of the 73rd Annual Sessions - 2017 , Department of Mathematics, University of Kelaniya,Sri Lanka., 2017) Munasinghe, J; Kekulawala, K.I.S.The word 'tide' is a prevalent term used to define the alternating rise and fall in the sea level wit respect to the land, produced by the gravitational attraction of the moon and the sun. To a bett understanding of the tide,-it is necessary to study each astronomical motion, together with it associated tide producing forces, separately. The present study was carried out with an attempt t confirm the Earth’s closed loop orbit using tidal height changes due to the motion of the Sun, Moo and Earth in the Trincomalee coastal area of Sri Lanka. Tidal heights from the Mean Sea Level (MS were measured every fifteen minutes throughout the year 2015 (365 days) using the tide pole installs in the sea in the Trincomalee coastal area, which was built by the Hydrography Survey Unit of Sri Land Navy. Using the data obtained, the behavior of tidal waves was identified. The main tidal constituen were obtained using Tidal Analysis Tool (TAT). The tidal constituent ’Principal solar semidiurn constituent’ (52), which is the consideration of the tidal effect caused by the sun, was then chosen fro TAT application because the Earth‘s orbit around the Sun is caused only by the forces between t Earth and the Sun. Fast Fourier Transformation (FFT) and Interpolation methods were used to analyz the chosen tidal constituent, Sz, together with the obtained tidal data to confirm the Earth’s close loop orbit around the Sun. Meteorological factors and human errors can occur while collectin data and hence there are eighteen peaks towards the inside of the loop. The following ellipti shaped orbit was obtained at the confidence level of 50% after removing such data. Figure 1: Earth’s closed loop orbitItem Effect of long-range part of the potential on the elastic S-matrix element(Sri Lanka Association for the Advancement of Science, 2011) Shadini, A.M.D.M.; Munasingha, J.The quantum mechanical three-body Schrödinger equation can be reduced to a set of coupled differential equations when the projectile is easily breakable into two fragments and when scattering is a heavy stable nucleus. It has been found that the diagonal coupling potentials in this model take the inverse square form at sufficiently large radial distances and non-diagonal part of coupling potentials can be treated as sufficiently short-range to guarantee that numeral calculations are feasible. We will show that this long-range part of the potential has a small contribution to the elastic S-matrix element.Item Enumeration of heterotrophic, iron-precipitating bacteria in the soil samples collected from urban waste dumping sites, Matara district, Sri Lanka(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Maduwanthi, D.M.S.K.; Geeganage, S.D.L.; Perera, T. W.N.K.Heterotrophic, iron-precipitating bacteria are capable of using organic radicals from soluble organic iron salts. This is widely applied in the removal of iron from organic solutions and during the decomposition of organic iron compounds, ionic iron is released. The current study was designed to enumerate heterotrophic ironprecipitating bacteria in soil samples, collected from urban waste dumping sites in eight locations, in the Matara district, Sri Lanka. Generally, the identified area was almost flat and there were no complex topographic features. Surface soil, separately sampled from three sampling sites at each location, were used as the test samples. The pH value of each collected sample was recorded. Each soil sample (1.00 g) was added to sterilized water (9.0 mL), followed by tenfold serial dilutions. For the enumeration of the total viable heterotrophic iron-precipitating bacteria, serially diluted samples were pour-plated with Ferric Ammonium Citrate Nitrate Agar. All soil samples were analysed in duplicated agar plate-based assays and the number of colonies was counted after incubation of the plates at room temperature for 2-3 days. Quantitative determinations were made based on colonyforming units per gram (CFU g-l) of soil and expressed with 95% confidence interval limits. Further, the bacterial colony counts per gram of each soil sample were arranged in a completely randomized design and One-way analysis of variance was applied with Tukey's multiple comparison test. The results showed that the counts were significantly different among locations. The significantly highest counts were reported for the dumping sites at Walgama (pH 7.15) and Walpala (pH 6.90) areas and the relevant counts were recorded as 3.492><105 CFU gl and CFU g-l , respectively. The lowest count was recorded at the dumping site near Dikwella lagoon (pH 8.95). The study demonstrates the dispersion of heterotrophic, iron-precipitating bacteria in urban waste dumping sites, within the selected region, indicating that high counts were reported in nearneutral soil environments. The current findings would serve as a baseline for the further expansion of the research topic towards the application of these bacteria, for the removal of iron from accumulated organic waste.Item The equality of Schrödinger’s Theory and Heisenberg’s S-matrix Theory(Sri Lanka Association for the Advancement of Science, 2011) Silva, H.I.R.U.The main aim of this work is to show that the energy discrete eigen values given by the Schrödinger’s theory and Heisenberg’s theory are the same. To obtain this result, we have used Parabolic co-ordinates to solve the Schrödinger’s equation for the Hydrogen Atom. By using the Hyper Geometric Confluent functions we have expressed the S-matrix element using Gamma functions; ( ) ( ) (l in) l in S k n l G + − G + + = 1 1 where k e n 2 2 h μ = − By the definition of Gamma function, ( ) ( )Õ ¥ = − + + = 1 2 2 1 1 p p in in l e p z p z e z z S n g Then it is apparent that the S-matrix element contains infinite number of poles and zeros. Considering the relevant simple pole, we have derived an equation for the energy eigen values of the form 2 2 4 2 n e En h μ = − This shows that it is the same as the equation we have obtained in Schrödinger’s theory. Therefore Heisenberg’s S-matrix theory and Schrödinger’s wave mechanics give exactly the same eigen values in the cases we have examined.Item The identity of Fermat equation and simple proof of Fermat’s last theorem for n=5(Sri Lanka Association for the Advancement of Science, 2013) Pallewatta, P.G.M.O.; Piyadasa, R.A.D.Item Improving Sinhalese Personal Name Authority Control in Sri Lankan Online Public Access Catalogues (OPACs)(Department of Library and Information Science, University of Kelaniya, Sri Lanka, 2023) Wanigasooriya, PriyanwadaPersonal names play a crucial role in locating books systematically and identifying individuals. Previous studies have emphasized that searching by an author's name is the most common method used by library users in automated systems. The objective of name authority control is to enhance searching and browsing capabilities by ensuring the accurate retrieval of all works associated with an individual, including variations of personal identity such as pseudonyms and family names. Unfortunately, Sinhalese personal names are quite common. These variations negatively impact information retrieval quality, the linking of related resources, and copyright management. Rather than that misspelled names and duplicated same name in data entries can be easily detected by machines. However, other variations, such as an author using their full name or an abbreviated form, one person having different name forms, or multiple people sharing the same name, cannot be easily identified by machines. Therefore, libraries employ authority control files for personal names, using a single, distinct spelling of the personal name (known as the heading) for each library material. The objective of this survey was to develop an automated authority file capable of accurately identifying books based on specific personal names. In pursuit of this goal, a statistical approach was utilized, which entailed compiling an extensive list of author names extracted from the university libraries of Kelaniya and Moratuwa. The dataset of author names (n=76,592 names) was obtained from the Tag 100 and Tag 700 data fields present within these libraries. To ensure data accuracy, a series of initial data cleaning steps were implemented to remove redundant, irrelevant, and incomplete entries from the dataset. It has been observed that libraries follow inconsistent patterns of name forms, without considering proper authority forms. Establishing a comprehensive automated author name authority file is essential to minimize confusion in these cases. Furthermore, adopting a proper transliteration process is important for effective authority control functions. To minimize errors in searching personal names, as identified by the survey, suitable name forms were automatically detected and separately compiled for authors. This compilation can help facilitate easier information searching across libraries.Item Inadequacy of Zernike Polynomials for Accounting the Variability of Wavefront Measurements(Postgraduate Institute of Science Research Congress, Sri Lanka., 2024) Lakshmi, M.P.D.S.; Karunarathna, S.P.D.S.S.K.; De Silva, T.M.M.Wavefront measurements have been crucial in vision science and ophthalmology, especially for wavefront-driven corrections like laser refractive surgeries and customized contact lens design. However, the accuracy of these corrections hinges on precise measurements. In clinical practice, multiple measurements are taken from a patient’s eye during each session to avoid ambiguous conclusions from a single measurement. Despite these efforts, variations in measurements occur due to aberrometer misalignments, pupil size, lens position, and eye accommodation. These variations directly impact custom vision corrections, leading to errors. However, comprehensive studies on these dynamics are lacking, but understanding them is crucial in vision science. Previous studies have found that the variability in repeated measurements is significant, as indicated by changes in Zernike coefficients. However, these studies have only examined this variability through the Zernike coefficients themselves. In contrast, this study uniquely investigated their variability using the raw local slope data from wavefront measurements. The least squares estimation was implemented to analyse data from patients with astigmatism, myopia, keratoconus, and keratoplasty. Further, measurement noise was simulated using normally distributed random numbers, with signal-to-noise ratios (SNR) varied between 20 dB and 80 dB. The SNRs for individual Zernike coefficients have been determined. The results from patients showed that most Zernike coefficients had SNRs above 10, indicating minimal measurement variation impact. A few Zernike modes had SNRs below 2 for some patients. The study concludes that it is crucial to balance the addition of more Zernike modes with measurement variability to avoid introducing errors in custom vision corrections. Moreover, the results challenge the existing findings and suggest opportunities for deeper analysis and new insights.Item Investigation of a best fitting mathematical model for the frequency of occurrence of Trichoderma harzianum in Hakgala Montane Forest in Sri Lanka(International Conference on Applied and Pure Sciences, 2020 Faculty of Science, University of Kelaniya, Sri Lanka, 2020) Munasinghe, J.,; Jayalath, T. D.,; Kannangara, B. T. S. D. P.,Trichoderma is a genus commonly found in the soils of all climatic zones. All most all the species of Trichoderma can produce antimicrobial antibiotics and are good competitors of fungal pathogens, which promote plant growth, enhance stress resistance and induce disease resistance in plants. Interactions between plants and Trichoderma are ecologically important. Moreover, this genus is economically much important because Trichoderma has been used as a biofertilizer and bio pesticide. In the present study, the attention is given to Trichoderma species: Trichoderma harzianum. The aim of this study was to detect a proper mathematical model to investigate the frequency of occurrence of fungus; Trichoderma harzianum in Hakgala Montane Forest in Sri Lanka at any period of time. Data for the frequency of occurrence of Trichoderma harzianum were collected at once in three months intervals from the decomposing leaf litter of Hakgala Montane Forest in a previous study. Significance of the data was checked using the ANOVA test. Data were tested with five mathematical models (Exponential, Logistic, Gompertz, Brody, Von Bertalanffy) and parameters estimated using the nonlinear least square method in R Studio software. The models were tested for goodness of fit using the adjusted coefficient of determination (R2), Akaike’s information criterion (AIC) and Bayesian information criterion (BIC). The logistic model provided the best fit of the data due to the highest value of R2, lower values of AIC and BIC than other models. The developed logistic model revealed 0.549% for the growth rate of Trichoderma harzianum in Hakgala Montane Forest. Since the Hakgala Montane Forest is an undisturbed natural ecosystem with its equilibrium stage this proposed model can be used to investigate the frequency of Trichoderma harzianum at any time period even for future predictions.Item Investigation of the mixed cation effect and the irradiance level dependence on the efficiency of dye‑sensitized solar cells(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Wickramasinghe, H. M. N.; Karunathilaka, N. G. A.; Gnanarathne, D. M. T.; DeSilva, L. Ajith; Bandara, K. M. S. P.; Bandara, T. M. W. J.The development of photoelectrochemical energy conversion devices holds immense significance in addressing the escalating demand for renewable and environmentally benign energy. By harnessing the synergistic effects of salt mixtures that encompass both large and small counter ions, notable advancements in dye-sensitized solar cell (DSSC) performance have been realized. The investigated DSSCs with a novel organic electrolyte complex that contains LiI and (tetrahexylammonium) Hex4NI exhibited significant efficiency enhancement compared to that of their individual salt end components. The ionic conductivity variations and frequency-dependent AC conductivity in the electrolyte and dielectric properties were analyzed using complex impedance data. The conductivity in the electrolyte at room temperature is 11.44 mS cm−1. The investigated DSSCs are comprised of improved TiO2 multilayer photoelectrodes and Pt counter electrodes. Under an irradiance of 1000 W m−2, the energy conversion efficiency of the mixed salt system reached 8.37%, marking an impressive enhancement of 86.83% and 76.21% compared to the Hex4NI and LiI-based single salt counterparts, respectively. Additionally, an impressive efficiency of 10.57% is shown when the light intensity drops to 400 W m−2. The cells exhibited commendable short-term stability, likely attributed to the elimination of volatile solvents in the electrolyte. This study underscores the pivotal role played by mixed counter ions in the electrolyte, as they elicit synergistic effects that amplify DSSC performance enhancements, effectively overshadowing the effects imposed by conductivity variation.Item Kauffman bracket versus Jones polynomial skein modules(arXivLabs, 2022) Almeida, Shamon; Gelca, RazvanThis paper resolves the problem of comparing the skein modules defined using the skein relations discovered by P. Melvin and R. Kirby that underlie the quantum group based Reshetikhin-Turaev model for SU(2) Chern-Simons theory to the Kauffman bracket skein modules. Several applications and examples are presented.Item Mathematical Model for Dengue Transmission Dynamics(Department of Mathematics, Faculty of Science, University of Kelaniya, 2024) Munasinghe, Jayantha; De Silva, Mihiri M; Jayasooriya, Lakna KaumadeeDengue is a rapidly emerging pandemic disease in many parts of the world, especially in tropical and non-tropical areas. The dengue outbreak has a multisectoral impact on the medical, societal, economical, and political sectors. Dengue incidence has increased in Sri Lanka over the past 20 years, with deaths and illnesses. Almost all the districts in Sri Lanka have reported cases and posed a threat to the health of the people. Dengue fever is caused by dengue virus, first recorded in the 1960s in Sri Lanka. In this study, we propose a mathematical model to describe the transmission of dengue with a standard incidence rate for both human and vector populations. The impact of treatment capacity in the case of an epidemic scenario has been studied by using a constant treatment function. The equilibrium points and the basic reproduction number are computed. The conditions leading to the diseasefree and endemic equilibrium are determined. We observed that the reproduction number affects the asymptotic stability for both disease-free and endemic equilibrium points. The Lyapunov function theory is used to discuss the global stability. Based on actual data of infective population gathered from the Institute of Epidemiology Unit Ministry of Health in Sri Lanka, the parameters for infection and disease-related death rates are estimated. Numerical simulations of various compartments are used to investigate the impact of the key parameters affecting the disease’s transmission.Item Maximal embedding genus of 3-edge connected harary graphs(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Withanaarachchi, W.A.K.D.H.; Almeida, S.V.A.; Wijesiri, G.S.One of the most prominent problems of topological graph theory is to determine the type of surface a nonplanar graph can be embedded. Almost complete results have been obtained for 4-edge connected graphs. The methods that were used to obtain specific results (finding the maximum and minimum genus embedding) for 4-edge connected graphs do not generalise for 3-edge connected graphs. Graph embedding is an important representational technique that aims to maintain the structure of a graph while learning low-dimensional representations of its vertices. The aim of this research project was to study the embedding of 3-edge connected Harary graphs H3,n. Specifically to complete the problem of maximal embeddings of 3-edge connected Harary graphs. The result is proved using Jungerman’s study, which showed that for any graph, is upper-embeddable if and only if it has a spanning tree T such that has at most one component with an odd number of edges. More specifically, a spanning tree for each graph was observed by dividing all 3-edge connected Harary graphs into two groups: odd number of vertices and even number of vertices. The pattern of a set of deleting edges and corresponding spanning trees was generalised in both cases. It was proved that H3,n is upper-embeddable, and the maximum genus of H3,n is given by for each n, by analysing the odd components of the complement of the corresponding spanning trees.