Browsing by Author "Munasinghe, J."

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Anomalous Absorption of Deuteron Partial Waves by Nuclear Optical Potential
(University of Kelaniya, 2006) Piyadasa, R.A.D.; Kawai, M.; Munasinghe, J.
M.Kawai and Y. Iseri (2), (3) found an interesting phenomenon in nuclear physics, motivated by the work of (1), in case of nucleon-nucleus elastic scattering. In the following this phenomenon is discussed in case of neutron (n)-nucleus (A) elastic scattering. In elastic scattering of neutron on (A), the elastic S – matrix element for a particular combination of j l E A cm , , , becomes very small (almost zero), and they called this phenomenon anomalous absorption of neutron partial waves by nuclear optical potential, where j is the total angular momentum, l is the angular momentum, cm E the centre of mass energy and A the mass of the nucleus. The striking feature of this phenomenon is systematic in various parameter ( f ,l, E , A) cm planes. Among them, systematic in ⎟⎠ ⎞ ⎜⎝ ⎛ 3 1 , A k l plane is actually remarkable, which consists of straight lines. All straight lines correspond to a definite node of wave functions associated with A k E l j cm , , , . It is quite interesting to examine whether this phenomenon occurs in case of composite projectiles such as d , He , etc. Now, it has been shown (4) that this phenomenon is universal. The main purpose of this paper is to report results of the case d − A after being rescrutinized by us. It is striking that the systematic in ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + 3 1 2 , ( 1) A k η η l l plane is remarkably clearer than the case of neutron. Here, k η + η 2 + l (l +1) , the closest approach is physically meaningful in case of the presence of the Coulomb potential.
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.
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 .
Effect of a long-ranged part of potential on elastic S-matrix element
(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Shadini, A.M.D.M.; Piyadasa, R.A.D.; Munasinghe, J.
It has been found that quantum mechanical three-body Schrödinger equation can be reduced to a set of coupled differential equations when the projectile can be easily breakable into two fragments when it is scattering on a heavy stable nucleus [1]. This coupled set of differential equations is solved under appropriate boundary conditions, and this method, called CDCC, has been found to be a very successful model in high energy quantum mechanical three body calculations [2]. It can be shown, however, that the coupling potentials in the coupled differential equations are actually long-range [3],[4] and asymptotic out going boundary condition, which is used to obtain elastic and breakup S-matrix elements is not mathematically justifiable. It has been found that the diagonal coupling potentials in this model takes the inverse square form at sufficiently large radial distances [3]and non-diagonal part of coupling potentials can be treated as sufficiently short-range to guarantee numeral calculations are feasible. Therefore one has to justify that the long range part of diagonal potential has a very small effect on elastic and breakup S-matrix elements to show that CDCC is mathematically sound .Although the CDCC method has been successful in many cases, recent numerical calculations[5],[6]indicate its unsatisfactory features as well. Therefore inclusion of the long range part in the calculation is also essential. The main objective of this contribution is to show that the effect of the long range part of the potentials on S-matrix elements is small.
Empirical computation of changes in Mean Sea Level around Sri Lanka Vertical Datum
(4th International Research Symposium on Pure and Applied Sciences, Faculty of Science, University of Kelaniya, Sri Lanka, 2019) Dilakshi, K. A. L.; Munasinghe, J.
Tides occurs as the combined effects of the gravitational forces exerted by the moon and the sun and rotation of the Earth. Mean Sea Level (MSL) is the mean height of the surface of the sea and vertical datum of Sri Lanka is the base measurement point or set of points from which elevations are determined. Tidal constituents are the net result of multiple influences impacting tidal changes over certain time period. The aims of this study are to update the MSL of Sri Lanka, to find the eight main tidal constituents and to fit a model for variations of tidal heights. Tidal heights in every 15 minutes from 1st of January 2015 to 31st of December 2017 in Colombo, Galle, Trincomalee and Kankesanthurai coastal areas in Sri Lanka were used to analyse the behaviour of the tidal waves. For this analysis, The Mean High Water (MHW), Mean Low Water (MLW) and the average value of the heights of tides of each month in each coastal areas were calculated and graphs were plotted using MATLAB programming. These data have been used to update the MSL in each coastal areas. A Tide Analysing Tool which has been generated earlier using MATLAB programming, was used to find the main tidal constituents which are very important for national planning and naval operations. values of MSL in areas like Colombo, Galle, Trincomlee and Kankesanthurai have increased by 16.26 cm, 14.36 cm, 20.84 cm and 14.73 cm, respectively and the value of MSL in Sri Lanka has increased approximately by 16.55 cm. Values of main eight tidal constituents like Principal Lunar semi diurnal constituent (M2) in Colombo, Galle, Trincomalee and Kankesanthurai were identified as 0.1333m, 0.0767m, 0.0882m and 0.1133m respectively. After analysing the behaviour of weekly tidal heights in January and February in 2016, their variations were fitted into the Fourier trigonometric series. This series was later used to predict weekly tidal heights in January and February in 2017. This method can also be applied for tidal analysis and predictions of any given period of time in any coastal area
Fixed low-dose triple combination Antihypertensive Medication vs usual care for blood pressure control in patients with mild to moderate hypertension in Sri Lanka: A Randomized Clinical Trial
(American Medical Association, 2018) Webster, R.; Salam, A.; de Silva, H.A.; Selak, V.; Stepien, S.; Rajapakse, S.; Amarasekara, N.; Amarasena, N.; Billotm, L.; de Silva, A.P.; Fernando, M.; Guggilla, R.; Jan, S.; Jayawardena, J.; Maulik, P.K.; Mendis, S.; Mendis, S.; Munasinghe, J.; Naik, N.; Prabhakaran, D.; Ranasinghe, G.; Thom, S.; Thisserra, N.; Senaratne, V.; Wijekoon, S.; Wijeyasingham, S.; Rodgers, A.; Patel, A.; TRIUMPH Study Group
IMPORTANCE: Poorly controlled hypertension is a leading global public health problem requiring new treatment strategies. OBJECTIVE: To assess whether a low-dose triple combination antihypertensive medication would achieve better blood pressure (BP) control vs usual care. DESIGN, SETTING, AND PARTICIPANTS: Randomized, open-label trial of a low-dose triple BP therapy vs usual care for adults with hypertension (systolic BP >140 mm Hg and/or diastolic BP >90 mm Hg; or in patients with diabetes or chronic kidney disease: >130 mm Hg and/or >80 mm Hg) requiring initiation (untreated patients) or escalation (patients receiving monotherapy) of antihypertensive therapy. Patients were enrolled from 11 urban hospital clinics in Sri Lanka from February 2016 to May 2017; follow-up ended in October 2017. INTERVENTIONS: A once-daily fixed-dose triple combination pill (20 mg of telmisartan, 2.5 mg of amlodipine, and 12.5 mg of chlorthalidone) therapy (n = 349) or usual care (n = 351). MAIN OUTCOMES AND MEASURES: The primary outcome was the proportion achieving target systolic/diastolic BP (<140/90 mm Hg or <130/80 mm Hg in patients with diabetes or chronic kidney disease) at 6 months. Secondary outcomes included mean systolic/diastolic BP difference during follow-up and withdrawal of BP medications due to an adverse event. RESULTS: Among 700 randomized patients (mean age, 56 years; 58% women; 29% had diabetes; mean baseline systolic/diastolic BP, 154/90 mm Hg), 675 (96%) completed the trial. The triple combination pill increased the proportion achieving target BP vs usual care at 6 months (70% vs 55%, respectively; risk difference, 12.7% [95% CI, 3.2% to 22.0%]; P < .001). Mean systolic/diastolic BP at 6 months was 125/76 mm Hg for the triple combination pill vs 134/81 mm Hg for usual care (adjusted difference in postrandomization BP over the entire follow-up: systolic BP, -9.8 [95% CI, -7.9 to -11.6] mm Hg; diastolic BP, -5.0 [95% CI, -3.9 to -6.1] mm Hg; P < .001 for both comparisons). Overall, 419 adverse events were reported in 255 patients (38.1% for triple combination pill vs 34.8% for usual care) with the most common being musculoskeletal pain (6.0% and 8.0%, respectively) and dizziness, presyncope, or syncope (5.2% and 2.8%). There were no significant between-group differences in the proportion of patient withdrawal from BP-lowering therapy due to adverse events (6.6% for triple combination pill vs 6.8% for usual care). CONCLUSIONS AND RELEVANCE: Among patients with mild to moderate hypertension, treatment with a pill containing low doses of 3 antihypertensive drugs led to an increased proportion of patients achieving their target BP goal vs usual care. Use of such medication as initial therapy or to replace monotherapy may be an effective way to improve BP control.
Investigation of a best fitting mathematical model for the frequency of occurrence of Trichoderma harzianum in Hakgala Montane Forest in Sri Lanka
(Faculty of Science, University of Kelaniya, Sri Lanka, 2020) Munasinghe, J.; Jayalath, J.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 (R 2 ), 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 R 2 , 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.
Mathematical model for kinematics of basketball free-throw shooting
(Faculty of Science, University of Kelaniya, Sri Lanka, 2020) Munasinghe, J.; Gunawardhana, G.H.B.C.
Basketball shooting is a basic and essential practice for a basketball player. Points are scored in a game of basketball, by throwing the ball over a hoop. This can be done by Free throws and field goals. A free throw is a special shot granted when a player is fouled while shooting a basket. Any player of the opposition team who is on the court at that time can be awarded a free throw. Hence, all the members of a team should be competent in scoring baskets. The angle and the velocity of the ball are important mathematical parameters in scoring goals. If the players are aware of the correct angle and velocity for a successful free throw then players can practice accordingly and aim perfect shots. Hence a model for the players to know their perfect successful angles would be helpful. Therefore, the main purpose of this research is to obtain a model to calculate the best releasing angle and the releasing velocity for the free throw shots. The air resistance is considered as zero, magnus effect is negligible and having zero spin of the ball were the assumptions in developing the model. Equations of motions for horizontal and vertical directions of the velocity was used and simplified them. An expression was developed for minimum angles by examining the descends of the ball to the hoop. Tables were constructed for the release angles and velocity for various heights using MATLAB(R2018a) to plot the graphs to find the feasible regions for the angles and velocity. Finally, the minimum angle for the longest shot was calculated. A physical model was developed to find the releasing angles with their release velocities. Basketball players could use the free throw shot using the best angles developed in the current study. Measuring velocity is not practical in basketball games, hence the players can practice with the best angle for a free throw and manage their velocities with the angle according to the velocities given in the table. Based on the tables, an increase in height of the player including the height to the releasing point will decrease the releasing angle and releasing velocity. The derivation of the model for optimal angle a quadratic equation was solved to develop the model. Therefore, there were ranges of releasing angles for some releasing velocities. The effect of air resistance was small for the motion of the basketball shot was observed.
A mathematical model for the description of the drug distribution in the human body
(Faculty of Science, University of Kelaniya, Sri Lanka, 2021) Madhushika, M. A. G.; Munasinghe, J.
In pharmaceutical industry, various researches have been carried to discover new drugs and to find effective ways of transporting drugs into target parts of the human body. Oral administration, intravenous and inhalations are the commonly used methods in administering drugs. This research study mainly considers oral administration of a drug in tablet form. Determining the absorption processes of a drug after oral administration to the body is a complicated problem. But mathematical modeling can provide the optimal solutions to various complex problems. The compartment model is the mathematical representation of human organs, created to study the drug distribution. This study describes a four-compartment model for drug distribution in the oral mode. In this study, stomach, small intestine, blood and tissues were used as four compartments. In oral consumption of a drug, absorption into the bloodstream occurs in the stomach and small intestine. The blood carries medicine from the site of absorption to the targeted sites and also to sites of metabolism or excretion, such as the kidneys, the liver. Finally, the drugs are excreted through urine after the metabolic process. Drug absorption rate and drug flow rate were used in developing this mathematical model. The rate of drug movement between compartments was described by first-order kinetics. The drug is absorbed through the stomach when drugs are taken orally but has been neglected in this study because of its small amount of absorption compared with other absorption constants. The formulation of this model is based on the principle of continuity and first-order kinetics. First the mathematical interrelationships between the drug concentration were derived with distribution time and rate constants. Then the first-order differential equation system was derived. This first-order differential equation system was solved using the Laplace transform method to avoid the complication raised while finding the eigenvector in the eigenvalue method. Existing rate constants in the mathematical model take different values for different drugs. The rate constants needed for this study were taken from the clinical trial conducted for paracetamol. The drug used in the current mathematical model was paracetamol. The mathematical model for paracetamol distribution was solved using the 4th order Runge-Kutta method and the solutions were plotted using MATLAB. Here, with higher absorption rate constants, the drug is absorbed faster into the body and with lower absorption rate constants, the drug will be retained within the body as a residue, thereby causing side effects. Hence, when the second dose is administered after six hours, the residual drugs from the first dose were also calculated. The present study helps to estimate the relationship between drug absorption, distribution, and the elimination processes.
A mathematical model to investigate the antimicrobial activity of Ceylon high- grown green tea and black tea against human pathogenic bacteria and yeast species
(Faculty of Science, University of Kelaniya, Sri Lanka, 2021) Samaraweera, S. A.; Munasinghe, J.; Gunasena, G. D. D. K.
Ceylon tea has a great demand worldwide and considered as one of the finest tea in the world. Green and black tea are the widely consumed types of tea in general. Tea has many biologically active compounds, and these antimicrobial agents can inhibit the growth of human pathogenic microorganisms. This research focused to derive an appropriate mathematical model to investigate the antimicrobial activity of high-grown green tea and black tea against selected human pathogenic bacteria (Staphylococcus aureus, Pseudomonas aeruginosa, Bacillussubtilis) and three yeast (Candida) species. The behaviour of the model solutions was analysed graphically and the antimicrobial activity of tea for a given concentration can be found using this model. First, descriptive analysis was carried out to analyse the basic characteristics of the experimental data. Then, using the Interpretative/ Breakpoint criteria provided in the Performance standards for antimicrobial disk susceptibility tests; CLSI, the antimicrobial effects were further discussed. Those results revealed that there was a difference between the two tea varieties and the antimicrobial activity was higher in green tea. Interaction between the two independent variables (tea type, concentration) on the dependent variable (diameter of the inhibition zone) was analysed using the two-way ANOVA test. The results revealed that interaction between two independent variables on the dependent variable was significant in all tested bacteria and yeast species except in Candida tropicalis. Then the most suitable mathematical model was developed, model parameters were calculated using experimental data, and values were predicted for the inhibition area of each species of microorganism for both tea varieties independent of each other. The model accuracy was examined using MAD (mean absolute derivation), RMSE (root mean square derivation), and MAPE (mean absolute percentage error) values and using histograms for residuals. The findings showed that the experimental data fitted and agreed with the model. The analytical output of the model was implemented using MATLAB (2018a) and it demonstrated experimentally observed antimicrobial activity. The model can be used to describe the antimicrobial activity of high-grown green tea and black tea at a given concentration against the tested bacteria and yeast species. Further, results revealed that the concentration of tea is directly proportional to the antimicrobial activity up to a certain point, respective to each microorganism, and thereafter no such correlation was observed.
Mathematical modeling of diabetes mellitus
(Faculty of Science, University of Kelaniya Sri Lanka, 2024) Subawickrama, H. D. K. M.; Munasinghe, J.; De Silva, T. M. M.
Diabetes is becoming a silent epidemic that is endangering public health worldwide. For instance, the majority of the 422 million diabetics globally reside in low- and middle-income nations, and the illness directly causes 1.5 million fatalities per year. In recent decades, the prevalence and complexity of the chronic medical condition diabetes have raised serious concerns about global health. As a result, studying and developing mathematical models becomes increasingly important since they provide an advanced perspective for realizing the complex nature of the disease and creating realistic care with preventive measures. Furthermore, it is a useful tool for tracking the rising incidence of the disease and creating affordable control measures for both its incidence and consequences. Thus, the main focus of this study is to propose a mathematical model that is employed to forecast changes in the prevalence of diabetes using a saturated incidence rate, which reflects a diminishing rate of new infections as the number of affected individuals increases, by extending the Diabetes Complication (DC) and Susceptible Diabetes Complication (SDC) models. It has been established which factors lead to both endemic and disease-free equilibriums. The conditions that contribute to the equilibrium between endemic and disease-free states are identified. Based on the eigenvalues of the Jacobian matrix, the local stability of the equilibrium points has been determined. Since diabetes is not a transmissible disease, there is no disease-free equilibrium and when the constant term 𝑅0 < 1, the endemic equilibrium point is locally asymptotically stable. Additionally, the Lyapunov function theory has been utilized to study global stability. We observe the effect of the term saturated incidence rate under some parameter conditions. Based on data gathered from annual mortality reports and health bulletins that have been published by the Ministry of Health, Sri Lanka and the United Nations’ World Population Prospects the parameters for complications related mortality rate, the natural mortality rate, and the birth rate are estimated. Numerical simulations using MATLAB’s ODE 45 technique are conducted to validate the analytical findings of the proposed model and assess its approach. This model also monitors the number of susceptible people as well as the population with and without diabetes complications. We predict that the 2.29 million will be the approximate Sri Lankan total prevalence of diabetes in 2024 and additionally, it notes an average annual increase of about 0.2199% in diabetes prevalence. The model’s accuracy is highlighted by its minimal average relative error, making it effective for forecasting and valuable for informing disease management strategies.
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.
Physical Interpretation of Anomalous Absorption of Partial Waves by Nuclear Optical Potential,
(University of Kelaniya, 2008) Piyadasa, R.A.D.; Munasinghe, J.; Karunatileke, N.G.J.
Anomalous absorption of light ion partial waves by the nuclear optical potential is an interesting phenomenon showing striking systematic in various parameter planes. However, the theoretical description of this phenomenon is extremely difficult. In this contribution, we address the problem of its physical interpretation. It is shown that the zero of the S-matrix element associated with the anomalous absorption of a partial wave is due to the destructive interference of the reflected waves, from the innermost turning point and the outermost turning point, in the asymptotic region when WKB approximation is feasible.
Qualitative and quantitative analysis of the dengue fever model with reference to the data obtained from Sri Lanka
(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Jayasooriya, J. A. D. L. K.; Munasinghe, J.; De Silva, T. M. M.
Dengue 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, economic, and political sectors. The main economic impact of dengue is due to production costs. Lower-income groups may be more vulnerable to dengue and may also bear a higher financial cost as a result of it. Moreover, once a family’s primary wage worker contracts the sickness, the lost productivity that results from the illness puts a financial strain on the family. This affects the household’s ability to pay for treatment. Resource-poor countries are particularly hard hit because of inadequate public health infrastructure, lack of resources to combat the vector, and limited health care services to manage cases. Dengue incidence has increased in Sri Lanka over the past 20 years, with deaths and illnesses increasing disproportionately among adults compared to children. Dengue fever is caused by Dengue virus, first recorded in the 1960s in Sri Lanka. Aedes aegypti and Aedes albopictus are both mosquito species native to Sri Lanka. In this study, a SIR model for the human population and SI model for the vector (mosquito) population with a constant treatment function is considered to describe dengue transmission. The equilibrium points and the basis reproduction number (𝑅0) are computed. It is emphasized that reproduction number affects the asymptotic stability for both endemic and disease-free equilibrium points. The conditions leading to the disease-free and endemic equilibrium are determined. The eigenvalues of the Jacobian matrix corresponding to the reduced system are used to demonstrate the local stability for the equilibrium points, and the Lyapunov function theory is used to assess the global stability. When 𝑅0 ≤ 1, the disease-free equilibrium point exhibits global asymptotic stability. But as 𝑅0 > 1, the endemic equilibrium point becomes globally asymptotically stable. Based on actual data gathered from the Institute of Epidemiology Unit Ministry of Health in Sri Lanka, the parameters for infection and disease-related death rates are estimated. The numerical simulation is used to validate the findings of the analytical results. It is important to determine a suitable capacity for treating a disease. We have observed that the treatment function affects the infected compartment. That is, the increased rate of treatment function reduces the infection. This shows that to eliminate the disease adequate treatment facilities must be provided.
Roots of a cubic and simple proof of Fermat’s last theorem for n=3
(University of Kelaniya, 2013) Ubeynarayana, C.U.; Piyadasa, R.A.D.; Munasinghe, J.; Ekanayake, E.M.P.
Introduction: Fermat’s last theorem (FLT) which was written in 1637, became public in 1670, without proof. It has not only evoked the interest of mathematicians but baffled many for over three hundred and fifty years [1],[2]. It is well known that FLT despite its rather simple statement had been difficult to prove even for the small prime exponent .The main objective of this paper is to provide simple proof of the theorem for this special exponent using the Method of Taragalia and Cardom of solving a cubic which is much older than Fermat’s last theorem. Proof of Fermat’s last theorem for n=3 Fermat’s last theorem for 3  n can be stated as that the equation , ( , ) 1 3 3 3 z  y  x x y  (1.1) is not satisfied by non-trivial integer triples x, y, z . Assume that the equation is satisfied by non-trivial integer triples x, y, z . If let x  3m1, y  3k 1, z  3s 1, then we have (2,0, 2)(mod3 ) 3 3 2 y  x   However 1(mod3 ) 3 2 z   , therefore our assumption is wrong and we conclude that xyz  0(mod3) . Since we consider the equation (1.1) for positive and negative integer values, without loss of generality, we can assume that y  0(mod3) ,and let 0(mod3 ) m y  . Then 3 1 3 3 3 z x 3 u , z y h , x y g m        due to Barlow relations, where are respectively the factors of . Now, 3 2( ) 3 3 1 3 3 g u h x y z m       (1.2) Since x  y  z  x  (z  y)  (x  y)  z  y  (z  x) , x y z 0(3 ugh)It can be shown that [1],   3 3 3 3 3 x y z 3(x y)(z x)(z y) 3 h u g m        (1.3) and therefore 3 2 3 0 3 3 3 1 3       g h u ugh m m (1.4) This necessary condition must be satisfied by the integer parameters , of respectively. We will first fix the parameters of and show that (1.4) is not satisfied by integer for any integer using the Method of Tartagalia and Cardoon[4] of finding a roots of a cubic. The equation (1.4) is of the form 3 0 3 3 3 g  vwg  v  w  (1.5) where 3 3 3 1 3 3 v w 3 u h m     , uh vw m 1 3. 2   , and its roots can be written as 2 ,   w v w v   and +w (1.6) with the cube root of unity. Now 3 3 , w v are the roots of the equation 0 2 3    H Gt t (1.7) where (3 ) 3 1 3 3 G u h m     , . 3. 2 1uh H m   Moreover real and distinct and this equation has only one real root, namely, g  v  w, since the discriminant of (1.5) is ,where ( 4 ) 2 3   G  H is the discriminant of (1.7),and it is negative. Therefore, it has only one real root [4]. Note that 2 3 6 2 6 3 3 3 3 6 3 1 3 3 2 3 3 3 3 (G 4H ) 3 u 14.3 u h h (3 u h ) 4.3 u h m m m m          which is positive when uh  0. On the other hand . 3 1 3 3 3 3 3 3 (3 u h ) 32.3 u h m m     which is positive when If     1 2 1 2 1 1 1 1 v  w ,v  w ,v  w (1.8) is another representation of the roots, we must have 2 2 1 1 1 1 v  w  v  w,v  w  v  w , v  w  v  w 2 1 2 1 or v  w  v  w v  w  v  w 2 2 1 1 1 1 , , 2 1 2 1 v  w  v  w In other words , ( ) ( ) 1 1 1 1 v  v  w w v  v  w w or ( ) ( ), 1 1 v  w  v  w ( ) ( ) 1 1 v  w  v  w This means that v  v w  w 1 1 , or , . 1 1 v  w w  v Therefore, roots must be unique. In particular, we must have a unique real root. From (1.4), it follows that the real root can be expressed in the form g h j m 1 3    or g j h m   1 3 where j is an integer satisfying (3, j) 1. This is due to the fact that ( )[( ) 3 ] 0(mod3 ). 3 3 2 m g  h  g  h g  h  gh  It is now clear that 3 h satisfies the equation 0 2 3 t Gt  H  , This means that (3 ) 8.3 0 6 3 1 3 3 3 3 3 3 3       h u h h u h m m implying that 0  u or , 0  h that is y  0 or x  0 in Fermat’s equation. Hence there is no non-trivial integral triple satisfying the Fermat equation (1.1). We have shown that Fermat’s Last Theorem for 3  n can be proved without depending on the method of infinite descent or complex analysis. The proof given above is short and simple, and simpler than proving [3] the theorem using the method of infinite descent.
Selecting a best fitting mathematical model for fermentation kinetics of spontaneous wine fermentation process
(Faculty of Science, University of Kelaniya Sri Lanka, 2024) Madhuhansi, H. A. R.; Munasinghe, J.; Undugoda, L.
Spontaneous wine fermentation involves fermenting “grape must” using indigenous yeasts inhabiting the grapes’ skin. Consistent alcohol level is crucial for producing a final product that meets quality standards in the wine industry. Therefore, this study aims to identify the most suitable mathematical model to characterize the alcohol production dynamics of indigenous yeast during spontaneous wine fermentation, with a specific focus on Sri Lanka’s wine industry. The research highlights the contributions of five major indigenous yeast species: P. kudriavzevii, H. guilliermondii, H. opuntiae, H. uvarum, and S. bacillaris. Grape samples collected from Urumpirai, Jaffna, Sri Lanka, underwent separate fermentation processes with each yeast species, and alcohol levels were measured at regular intervals. All the experimental data sets were statistically analyzed using SPSS (version 23). The box plot analysis revealed that no outliers were detected from an initial set of 120 data points. One-way ANOVA test revealed statistically significant differences in alcohol levels across various time intervals. The kinetic parameters of the wine fermentation process were analyzed using several models, including the logistic model, the Gompertz model, the Richards model, and their respective modified versions. Among them, the modified Gompertz model became the best-fit mathematical model to describe the alcohol production patterns of each yeast species separately. The model’s accuracy was evaluated using several statistical measures, such as the coefficient of determination (R²), adjusted chi-square value, residual sum of squares (RSS), and F-value. The results demonstrated a significant variation in alcohol levels over time for each yeast species, highlighting their distinct fermentation profiles. Key parameters of the modified Gompertz model, maximum alcohol production rate (%𝑣/𝑣 ℎ-1), lag period (ℎ), and upper asymptote for product formation (%𝑣/𝑣) were estimated numerically by using the Originlab (2024) software, for each yeast species and, their alcohol production abilities were studied separately. These findings pave the way for industrial scale quality wine production using the concept of spontaneous fermentation.
Selection of best fitting mathematical model to analyse the anthracene degradation ability of Bacillus velezensis
(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Madushika, S. A. R. R.; Munasinghe, J.; Undugoda, L.
Living things frequently come into contact with a wide range of harmful pollutants, including polyaromatic hydrocarbons (PAHs) like anthracene, which are highly carcinogenic and genotoxic. Microbial degradation is the most promising approach for removing anthracene from the environment, using microorganisms to transform it into nontoxic compounds, rather than using physical and chemical methods. The study aimed to assess the kinetic approach in anthracene degradation by Bacillus velezensis, isolated from the phyllosphere of leaf samples. Leaf samples (Ixora chinensis, Ervatamia divaricate, and Plumeria sp.) were collected by using random sampling techniques from the most polluted urban areas (Maradana, Orugodawatta, Pettah, Panchikawatta, Sapugaskanda, and Colombo Fort) in Sri Lanka. Then, in vitro, anthracene degradation patterns were analysed using HPLC analysis at different anthracene concentrations (100 ppm, 200 ppm, 300ppm, 400 ppm, 500 ppm, and 600 ppm) as a model of limited substrates while monitoring the growth patterns based on fluctuations in dry cell biomass. Anthracene was completely degraded in six days of incubation for low initial anthracene concentrations. Dry cell biomass and degradation percentages were measured with time for different initial liquid substrate concentrations of PAHs. The specific growth rates and degradation rates were calculated. The study used box plot analysis to identify and eliminate outliers from 65 data points, ensuring the validity and reliability of the findings. After eliminating four outliers, 61 usable data points were obtained, and the ANOVA was used to compare average specific growth rates between anthracene concentration groups, revealing all specific growth rates were statistically significant. The kinetic parameters of anthracene degradation were analysed by using various models including, Monod's, Haldane’s, Wayman and Tseng's models because they focused on their convergence with experimental data. The models were evaluated by determining the parameter’s significant value, Root Mean Square Error (RMSE), and the Adj.R2 value. Among these models, Wayman and Tseng's model demonstrated the better fit for the experiment with Adj.R2 - 0.95191 and RMSE – 0.0019. The experimental kinetics data of anthracene degradation closely followed Wayman and Tseng’s model, indicating that this model provided an accurate representation of the interaction between the PAH substrate (anthracene) and the growth kinetics of the microorganisms. The half saturation constant Ks, the maximum specific growth rate “max, the threshold substrate concentration “, and the inhibition coefficient i were defined as Wayman and Tseng’s model parameters and were estimated in their numerical values.
Selection of the best fitting mathematical models to investigate the growth inhibition of selected plant pathogenic fungi by Trichoderma harzianum
(Faculty of Science, University of Kelaniya Sri Lanka, 2022) Poornima, V. A.; Munasinghe, J.; Kannangara, S. D.
Plant diseases can cause a significant impact on agricultural productivity. One of the main causative factors for plant diseases is pathogens. Plant pathogens can be fungi, bacteria, viruses, or nematodes. Plant diseases can be prevented, mitigated, or controlled by using a variety of methods. Among these methods, bio-controlling is more effective and environmentally friendly. Trichoderema species are the most commonly used fungal biocontrol agents against various plant pathogens. Many researchers observed that the potential of Trichoderma harzianum in controlling various pathogens, but there were no proper mathematical models to understand how this fungus inhibits those pathogens. This research is focused on selecting the most suitable mathematical models to investigate the growth inhibition of pathogenic fungi; Fusarium oxysporum, Colletotrichum gloeosporioides, Lasiodiplodia theobromae, and Xylaria spp. by Trichoderma harzianum. For this purpose, five existing growth models, namely Exponential, Logistic, Brody, Von Bertalanffy, and Gompertz were used to investigate the growth inhibition. The data had been collected using the dual culture method to test the antagonistic properties of Trichoderma against the fungal pathogens. The data set consisted of 50 data points for six consecutive days. First, statistical analysis was performed to identify the distribution and characteristics of the data and detect outliers. Then one-way Analysis if Variance (ANOVA) test was done under the 95% confidence level. The results revealed that all the mean values are statistically significant. Tukey test was then conducted to check which specific group means were different. Then the most suitable growth models were identified for each fungus separately under two conditions: in the absence of T. harzianum and in the presence of T. harzianum. In the absence of T. harzianum, for the fungi; F. oxysporum, C. gloeosporioides, L. theobromae, and Xylaria spp. the best fitted models were given by Exponential, Gompertz, Exponential, and Exponential, respectively. In the presence of T. harzianum for the fungi; F. oxysporum, C. gloeosporioides, L. theobromae, and Xylaria spp. the best fitted models were given by Brody, Exponential, Brody, and Von Bertalanffy respectively. The goodness of fit was tested using the Coefficient of Determination (R2), Root Mean Square Error (RMSE), Sum Squared Error (SSE), and residual plots. Then the comparison of the growth in the absence of T. harzianum and the presence of T. harzianum was made graphically using the above best fitted models. Hence the simulation results indicated significant growth controls of all the pathogenic fungi tested by T. harzianum.
Tidal variation m the west coastal area of Sri Lanka.
(International Conference on Computational Modelling and Simulation-2017 (ICCMS 117), Department of Mathematics, Faculty of Science, University of Kelaniya, Sri Lanka., 2017) Munasinghe, J.; Gunasekera, H.D.S.
The present study was carried out in an attempt to observe and analyze the tidal height changes due to the motion of the sun, moon and earth, in the west coastal area of Sri Lanka. Tidal height deviations from the Mean Sea Level (MSL) were measured every 15 minutes throughout the year 2015 using the tide pole installed in the sea, 100m away from Colombo Fort, which was built by the Hydrography Survey Unit of the Sri Lanka Navy. Using the obtained data, the behaviour of tidal waves was identiﬁed. The main tidal constituents were obtained using the Fast Fourier Transformation (FFT) and the Interpolation method. The mean value of the High Water Level (MHWL) and the mean value of the Low Water Level (MLWL) of the tides were then calculated for each month of the year. These mean values were used to update the Mean Sea Level (MSL). The main tidal constituents for each month were then used to identify the behaviour of tidal waves.
Viscous airflow connected to a compressed Air-shoe
(4th International Research Symposium on Pure and Applied Sciences, Faculty of Science, University of Kelaniya, Sri Lanka, 2019) Munasinghe, J.
Viscous concentric airflow in between two parallel circular discs has been considered. Since the gap in between the plates is small, Prandtl boundary-layer equations were used to explain the flow. In order to solve these equations, the method analogous to Karman-Pohlhausen was used and also a polynomial of the fourth order was assumed for the velocity profile. The coefficients of this polynomial were determined via boundary and symmetry conditions, the integral continuity and momentum equations. The latter is derived from the Prandtl boundary-layer equations.