This paper presents the application of non-polynomi al Exponential spline method for finding the numerical solution of singularly pe rturbed boundary value problems. Two numerical examples are considered to demonstrate th e usefulness of the method and to show that the method converges with sufficient accu ra y to the exact solutions.
{"title":"Numerical Solution of Singularly Perturbed Two Point Boundary Value Problems by Using Non-Polynomial Exponential Spline Functions","authors":"Ahmed R. Khlefha","doi":"10.22457/jmi.v19a11184","DOIUrl":"https://doi.org/10.22457/jmi.v19a11184","url":null,"abstract":"This paper presents the application of non-polynomi al Exponential spline method for finding the numerical solution of singularly pe rturbed boundary value problems. Two numerical examples are considered to demonstrate th e usefulness of the method and to show that the method converges with sufficient accu ra y to the exact solutions.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"19 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89074238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Corruption is a worldwide problem that affects ma ny countries where by individuals loses their rights, lower community con fidence in public authorities, absence of peace and security, misallocation of resources a nd termination of employment. Despite various measures which have been taken by various c ntries to control corruption, the problem still exists. In this paper, we formulate a nd analyze a mathematical model for the dynamics of corruption in the presence of control m easures. Analysis of the model shows that both Corruption Free Equilibrium (CFE) and Cor ruption Endemic Equilibrium (CEE) exist. The next generation matrix method was used to compute the effective reproduction number ( ) which is used to study the corruption dynamics. T he results indicate that CFE is both locally and globally asym ptotically stable when < 1 whereas CEE is globally asymptotically stable when > 1. The normalized forward sensitivity method was used to describe the most sensitive para meters for the spread of corruption. The most positive sensitive parameters are κ and ν while the most negative sensitive parameters are α and β . Therefore, the parameters of mass education α and religious teaching β are the best parameters for control of corruption. The model was simulated using Runge-Kutta fourth order method in MATLAB and the results indicate that the combination of mass education and religious teachin g is effective to corruption control within short time compared to when each control str ategy is used separately. Therefore, this study recommends that more efforts in providin g both mass education and religious teaching should be applied at the same time to cont rol corruption.
{"title":"Mathematical Modelling and Analysis of Corruption Dynamics with Control Measures in Tanzania","authors":"Oscar Danford, M. Kimathi, S. Mirau","doi":"10.22457/jmi.v19a07179","DOIUrl":"https://doi.org/10.22457/jmi.v19a07179","url":null,"abstract":"Corruption is a worldwide problem that affects ma ny countries where by individuals loses their rights, lower community con fidence in public authorities, absence of peace and security, misallocation of resources a nd termination of employment. Despite various measures which have been taken by various c ntries to control corruption, the problem still exists. In this paper, we formulate a nd analyze a mathematical model for the dynamics of corruption in the presence of control m easures. Analysis of the model shows that both Corruption Free Equilibrium (CFE) and Cor ruption Endemic Equilibrium (CEE) exist. The next generation matrix method was used to compute the effective reproduction number ( ) which is used to study the corruption dynamics. T he results indicate that CFE is both locally and globally asym ptotically stable when < 1 whereas CEE is globally asymptotically stable when > 1. The normalized forward sensitivity method was used to describe the most sensitive para meters for the spread of corruption. The most positive sensitive parameters are κ and ν while the most negative sensitive parameters are α and β . Therefore, the parameters of mass education α and religious teaching β are the best parameters for control of corruption. The model was simulated using Runge-Kutta fourth order method in MATLAB and the results indicate that the combination of mass education and religious teachin g is effective to corruption control within short time compared to when each control str ategy is used separately. Therefore, this study recommends that more efforts in providin g both mass education and religious teaching should be applied at the same time to cont rol corruption.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"31 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85519805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-02DOI: 10.1101/2020.08.30.20184598
H. Baruah
There are standard techniques of forecasting the spread of pandemics. Uncertainty however is always associated with such forecasts. In this article, we are going to discuss the uncertain situation currently prevailing in the COVID-19 spread in India. For statistical analysis, we have considered the total number of cases for 60 consecutive days, from June 23 to August 21. We have seen that instead of taking data of all 60 days together, a better picture of uncertainty can be observed if we consider the data separately in three equal parts from June 23 to July 12, from July 13 to August 1, and from August 2 to August 21. For that we would first need to ascertain that the current spread pattern in India is almost exponential. Thereafter we shall show that the data regarding the total number of cases in India are not really behaving in an expected way, making forecasting the time to peak very difficult. We have found that the pandemic would perhaps change its pattern of growth from nearly exponential to nearly logarithmic, which we have earlier observed in the case of Italy, in less than 78 days starting from August 2.
{"title":"The Uncertain COVID-19 Spread Pattern in India: A Statistical Analysis of the Current Situation","authors":"H. Baruah","doi":"10.1101/2020.08.30.20184598","DOIUrl":"https://doi.org/10.1101/2020.08.30.20184598","url":null,"abstract":"There are standard techniques of forecasting the spread of pandemics. Uncertainty however is always associated with such forecasts. In this article, we are going to discuss the uncertain situation currently prevailing in the COVID-19 spread in India. For statistical analysis, we have considered the total number of cases for 60 consecutive days, from June 23 to August 21. We have seen that instead of taking data of all 60 days together, a better picture of uncertainty can be observed if we consider the data separately in three equal parts from June 23 to July 12, from July 13 to August 1, and from August 2 to August 21. For that we would first need to ascertain that the current spread pattern in India is almost exponential. Thereafter we shall show that the data regarding the total number of cases in India are not really behaving in an expected way, making forecasting the time to peak very difficult. We have found that the pandemic would perhaps change its pattern of growth from nearly exponential to nearly logarithmic, which we have earlier observed in the case of Italy, in less than 78 days starting from August 2.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"18 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78017949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).
{"title":"The ruin problem for a Wiener process with state-dependent jumps","authors":"M. Lefebvre","doi":"10.2478/jamsi-2020-0002","DOIUrl":"https://doi.org/10.2478/jamsi-2020-0002","url":null,"abstract":"Abstract Let X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"16 1","pages":"13 - 23"},"PeriodicalIF":0.3,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44028677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian structure defined on the space of parameters that characterize a family of probabilities. In this paper, we consider the finite dimensional case but the basic ideas can be extended similarly to the infinite-dimensional case. Our aim is to understand exponential families of probabilities on a finite set from an intrinsic geometrical point of view and not through the parameters that characterize some given family of probabilities. For that purpose, we consider a Riemannian geometry defined on the set of positive vectors in a finite-dimensional space. In this space, the probabilities on a finite set comprise a submanifold in which exponential families correspond to geodesic surfaces. We shall also obtain a geometric/dynamic interpretation of Jaynes’ method of maximum entropy.
{"title":"Geometry of the probability simplex and its connection to the maximum entropy method","authors":"H. Gzyl, F. Nielsen","doi":"10.2478/jamsi-2020-0003","DOIUrl":"https://doi.org/10.2478/jamsi-2020-0003","url":null,"abstract":"Abstract The use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian structure defined on the space of parameters that characterize a family of probabilities. In this paper, we consider the finite dimensional case but the basic ideas can be extended similarly to the infinite-dimensional case. Our aim is to understand exponential families of probabilities on a finite set from an intrinsic geometrical point of view and not through the parameters that characterize some given family of probabilities. For that purpose, we consider a Riemannian geometry defined on the set of positive vectors in a finite-dimensional space. In this space, the probabilities on a finite set comprise a submanifold in which exponential families correspond to geodesic surfaces. We shall also obtain a geometric/dynamic interpretation of Jaynes’ method of maximum entropy.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"16 1","pages":"25 - 35"},"PeriodicalIF":0.3,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46526102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The present paper introduces an advanced five parameter lifetime model which is obtained by compounding exponentiated quasi power Lindley distribution with power series family of distributions. The EQPLPS family of distributions contains several lifetime sub-classes such as quasi power Lindley power series, power Lindley power series, quasi Lindley power series and Lindley power series. The proposed distribution exhibits decreasing, increasing and bathtub shaped hazard rate functions depending on its parameters. It is more flexible as it can generate new lifetime distributions as well as some existing distributions. Various statistical properties including closed form expressions for density function, cumulative function, limiting behaviour, moment generating function and moments of order statistics are brought forefront. The capability of the quantile measures in terms of Lambert W function is also discussed. Ultimately, the potentiality and the flexibility of the new class of distributions has been demonstrated by taking three real life data sets by comparing its sub-models.
{"title":"Exponentiated quasi power Lindley power series distribution with applications in medical science","authors":"A. Hassan, A. Rashid, N. Akhtar","doi":"10.2478/jamsi-2020-0004","DOIUrl":"https://doi.org/10.2478/jamsi-2020-0004","url":null,"abstract":"Abstract The present paper introduces an advanced five parameter lifetime model which is obtained by compounding exponentiated quasi power Lindley distribution with power series family of distributions. The EQPLPS family of distributions contains several lifetime sub-classes such as quasi power Lindley power series, power Lindley power series, quasi Lindley power series and Lindley power series. The proposed distribution exhibits decreasing, increasing and bathtub shaped hazard rate functions depending on its parameters. It is more flexible as it can generate new lifetime distributions as well as some existing distributions. Various statistical properties including closed form expressions for density function, cumulative function, limiting behaviour, moment generating function and moments of order statistics are brought forefront. The capability of the quantile measures in terms of Lambert W function is also discussed. Ultimately, the potentiality and the flexibility of the new class of distributions has been demonstrated by taking three real life data sets by comparing its sub-models.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"16 1","pages":"37 - 60"},"PeriodicalIF":0.3,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44392607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The present paper provides a remedy for improved estimation of population mean of a study variable, using the information related to an auxiliary variable in the situations under Simple Random Sampling Scheme. We suggest a new class of estimators of population mean and the Bias and MSE of the class are derived upto the first order of approximation. The least value of the MSE for the suggested class of estimators is also obtained for the optimum value of the characterizing scaler. The MSE has also been compared with the considered existing competing estimators both theoretically and empirically. The theoretical conditions for the increased efficiency of the proposed class, compared to the competing estimators, is verified using a natural population.
{"title":"Restructured class of estimators for population mean using an auxiliary variable under simple random sampling scheme","authors":"S. Baghel, S. Yadav","doi":"10.2478/jamsi-2020-0005","DOIUrl":"https://doi.org/10.2478/jamsi-2020-0005","url":null,"abstract":"Abstract The present paper provides a remedy for improved estimation of population mean of a study variable, using the information related to an auxiliary variable in the situations under Simple Random Sampling Scheme. We suggest a new class of estimators of population mean and the Bias and MSE of the class are derived upto the first order of approximation. The least value of the MSE for the suggested class of estimators is also obtained for the optimum value of the characterizing scaler. The MSE has also been compared with the considered existing competing estimators both theoretically and empirically. The theoretical conditions for the increased efficiency of the proposed class, compared to the competing estimators, is verified using a natural population.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"16 1","pages":"61 - 75"},"PeriodicalIF":0.3,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45693383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
L. Edward, Mashaka James Mkandawile, Verdiana Grace Masanja
In this study, a model has been developed to find the minimum cost in distributing clean water. Linear Programming (LP) technique was used to formulate the model for Dodoma city. The developed model consists of both hydraulic and water treatment parameters. The model was then tested with real data collected from Ihumwa water network of Dodoma city and other treatment cost data from the literature to test the workability of the model. Hydraulic parameters such as head loss of the pipes, flow velocity and pipe pressure are calculated using water flow software. The resulted model was solved using lingo software by testing different intermediate values of pressure and velocity to obtain the minimum cost of distributing clean water. As a result, the values 650 N/m 2 and 700 N/m 2 as a maximum and minimum pressure and 0.5m/s and 2m/s as minimum and maximum velocity give the minimum cost of distributing clean water. Consequently, the objective value of resulted optimization model shows that the original cost of distributing clean water was reduced by 3.48%.
{"title":"Modeling and Optimization of Clean Water Distribution Networks","authors":"L. Edward, Mashaka James Mkandawile, Verdiana Grace Masanja","doi":"10.22457/jmi.v20a07190","DOIUrl":"https://doi.org/10.22457/jmi.v20a07190","url":null,"abstract":"In this study, a model has been developed to find the minimum cost in distributing clean water. Linear Programming (LP) technique was used to formulate the model for Dodoma city. The developed model consists of both hydraulic and water treatment parameters. The model was then tested with real data collected from Ihumwa water network of Dodoma city and other treatment cost data from the literature to test the workability of the model. Hydraulic parameters such as head loss of the pipes, flow velocity and pipe pressure are calculated using water flow software. The resulted model was solved using lingo software by testing different intermediate values of pressure and velocity to obtain the minimum cost of distributing clean water. As a result, the values 650 N/m 2 and 700 N/m 2 as a maximum and minimum pressure and 0.5m/s and 2m/s as minimum and maximum velocity give the minimum cost of distributing clean water. Consequently, the objective value of resulted optimization model shows that the original cost of distributing clean water was reduced by 3.48%.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90039873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the equation p + q = z in which p, q assume distinct odd primes and z is a positive integer. Then, for all possible in tegers y > 3, the equation p + q = z has no solutions.
{"title":"All the Solutions of the Diophantine Equation p3 + qy = z3 with Distinct Odd Primes p, q when y > 3","authors":"N. Burshtein","doi":"10.22457/jmi.v20a01188","DOIUrl":"https://doi.org/10.22457/jmi.v20a01188","url":null,"abstract":"In this paper, we consider the equation p + q = z in which p, q assume distinct odd primes and z is a positive integer. Then, for all possible in tegers y > 3, the equation p + q = z has no solutions.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"75 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78649048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the L2 distance between a point and a random variable. Similarly, the median is the same concept but when the distance is measured by the L1 norm. Also, a geodesic distance can be defined on the cone of strictly positive vectors in ℝn in such a way that, the minimizer of the distance between a point and a collection of points is their geometric mean. That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the logarithmic distance.
{"title":"Best predictors in logarithmic distance between positive random variables","authors":"H. Gzyl","doi":"10.2478/jamsi-2019-0006","DOIUrl":"https://doi.org/10.2478/jamsi-2019-0006","url":null,"abstract":"Abstract The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the L2 distance between a point and a random variable. Similarly, the median is the same concept but when the distance is measured by the L1 norm. Also, a geodesic distance can be defined on the cone of strictly positive vectors in ℝn in such a way that, the minimizer of the distance between a point and a collection of points is their geometric mean. That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the logarithmic distance.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"15 1","pages":"15 - 28"},"PeriodicalIF":0.3,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49260733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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