Mathematics
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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 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 Phylloremediation of pyrene and anthracene by endophytic fungi inhabiting tea leaves (Camellia sinensis (L.) Kuntze) in Sri Lanka(Tylor & Francis, 2023) Undugoda, Lanka; Thambugala, Kasun; Kannangara, Sagarika; Munasinghe, Jayantha; Premarathna, Nadeeka; Dharmasiri, NadeemaPolyaromatic hydrocarbons (PAHs) released from vehicular emissions and oil refineries deposit on the phyllosphere, compromising the quality of leaf-based food products by posing many health issues. Nevertheless, the tea phyllosphere harbours a variety of endophytes that are highly effective at degrading polyaromatic hydrocarbons, anthracene, and pyrene. The present study attempts to analyse the pyrene and anthracene degrading capability of phyllosphere endophytic fungi that inhabit Camellia sinensis (L.) Kuntze leaves. The frequency of occurrence of endophytic fungi in different leaf tissue layers was examined using light and scanning electron microscopy (SEM). The best pyrene and anthracene degrading strains were selected based on the High-Performance Liquid Chromatography (HPLC) results, and further kinetic assays. Light microscopy and SEM observations highlighted a heterogeneous endophytic fungal distribution among leaf tissue layers; the upper epidermis had the highest fungal distribution compared to other leaf layers. HPLC results revealed that Phyllosticta capitalensis, Colletotrichum gloeosporioides, Colletotrichum siamense, Pseudopestalotiopsis chinensis, and Daldinia eschscholtzii, have higher pyrene and anthracene degradation respectively and their PAH degradation kinetics follow the first-order kinetic model. The best anthracene and pyrene degrader, P. capitalensis showed the lowest half-life. The present investigation highlights the potential of P. capitalensis, the best pyrene and anthracene degrader that can remediate PAHs deposited on the phyllosphere of tea leaves.Item A Study on Electronic Services Provided by the National Library of Sri Lanka(Department of Library and Information Science, University of Kelaniya, Sri Lanka, 2023) Thilakarathna, M.D.B.C.M.; Wanigasooriya, PriyanwadaElectronic services are information services in electronic format that are handled by a computer device and encoded for handling. The National Library of the world provides a wide range of electronic library services to users. Today, the National Library of Sri Lanka provides e-library services in addition to traditional library services. With the COVID pandemic, there has been a rapid increase in the number of Electronic library services provided by the National Library of Sri Lanka. Today, there are 33 Electronic services provided by the National Library. It remains to be seen whether users will benefit from these services. For this research, numerous studies have studied users' behavior, usage of print resources, evaluation of electronic library services, and evaluation of library services using serverQUAL and LibQUAL, analyzing national and global research publications and printed and electronic media. The primary objectives of this study are four-fold. Firstly, it aims to identify and catalog the various electronic services that have been introduced by the National Library. Secondly, the study seeks to conduct an in-depth analysis of the current usage and adoption of these E-services among library users. Thirdly, it aims to identify any potential vulnerabilities or shortcomings in the existing E-services to ensure their optimal functionality and security. Lastly, the study endeavors to propose practical and effective measures that can be implemented to expand and enhance the range and quality of E-services provided by the National Library. By pursuing these objectives, the study aims to contribute to the overall improvement and advancement of library services in the digital age. This study has used a web-based survey methodology to study what electronic library services are available to readers, using a list of 59 items. In addition, two online questionnaires have been submitted to National Library users and National Library staff to study the status and use of Electronic services provided by the National Library. Based on the web-based survey methodology and the dual questionnaires, the National Library identifies the weaknesses in the electronic services provided and presents the results in simple frequencies and percentages. The research revealed that the National Library of Sri Lanka provides 69.49% of Electronic library services. Research has shown that 72% of respondents use the Electronic services provided by the National Library, and 28% of respondents do not use the electronic library services provided by the National Library. The research proposes the introduction of state-of-the-art electronic library services, features for the website, publicity of library services among the people of the country, and workshops and projects for staff IT knowledge and development.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 User perception of Electronic Services (E-Services) in university libraries Sri Lanka(Department of Library and Information Science, University of Kelaniya, Sri Lanka, 2023) Caldera, T.K.I.M.; Wanigasooriya, A.W.A.P.L.Electronic services in university libraries refer to the digital and electronic resources, tools, and services provided by the library to support teaching, learning, research, and information access in an online environment. These services aim to enhance access to information, facilitate efficient research processes, and provide a seamless user experience for users. This study investigates how users perceive E-services in university libraries and examine their satisfaction levels when utilizing various information services to fulfill their information requirements. Furthermore, the study aims to identify any difficulties faced by users while utilizing E-services. The study aims to provide recommendations for improving the current state of E-services in the researched universities. These recommendations will focus on enhancing service quality, improving accessibility, and implementing new initiatives to serve the information needs of users. The survey research method was utilized, involving dividing the population into two distinct groups. The first group consisted of 500 randomly selected users who frequently visited the main libraries. The second group comprised randomly selected permanent library staff members. It was observed that there is a lack of adequate infrastructure for E-services in the university libraries. Most users surveyed, reported using E-services for accessing e-journals, utilizing online databases, using e-tutorials, and preparing assignments. The majority of users reported their limited knowledge about E-services. Based on these findings, several recommendations are proposed. Firstly, it is crucial to ensure regular updates to the library website to keep it current and user-friendly. Additionally, it is essential to provide a broader range of E-resources to cater to the diverse information needs of users. Conducting user training programs focused on effectively utilizing E-services can enhance user experience and promote optimal utilization of available resources. Introducing innovative E-services is also crucial to meet user expectations in university libraries.Item Relationship between the kauffman bracket polynomials of torus knots: (T3,3n+1) and (T3,3n+2)(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Dissanayake, A.A.A.W.N.; Almeida, S.V.A.Knot theory is a branch of topology that studies mathematical knots. New knot invariants were the foundation for the work of many mathematicians. A knot invariant is a property for a knot , where is the same for any projection of . A knot polynomial is one such knot invariant. Knot polynomials are polynomials that are assigned to knot projections based on the mathematical properties of the knots. This study is restricted to polynomials of torus knots, knots that lie on an unknotted torus, without crossing over or under themselves as they lie on the torus. Every torus knot is a (, )-torus knot, where and are two relatively prime integers that are represented by the symbol ,. Most of the research done on this particular area of Knot Theory, has focused on finding polynomial representations such as Kauffman Bracket polynomial, the Bracket polynomial for the (2, )-torus knot and polynomial representations such as Alexander polynomial, Conway polynomial and Jones polynomial for the (, )-torus knot. With the exception of the complete solution to the Alexander, Conway, and Jones polynomials of ,, the problem of determining the polynomial for , is almost solved. The study is an attempt to solve the computation problem for the Kaufmann Bracket Polynomial of ,. The work will provide a relationship between Kauffman Bracket polynomials of the torus knots’ , and , for .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.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 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.