Mathematics
Permanent URI for this collectionhttp://repository.kln.ac.lk/handle/123456789/2408
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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 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.