Jean Muhirwa - Academia.edu (original) (raw)

Papers by Jean Muhirwa

African Institute for Mathematical Sciences Tanzania, Jul 12, 2015

Finite field is a wide topic in mathematics. Consequently, none can talk about the whole contents... more Finite field is a wide topic in mathematics. Consequently, none can talk about the whole contents of finite fields. That is why this research focuses on small content of finite fields such as polynomials computational, ring of integers modulo p where p is prime or a power of prime. Most of the times, books which talk about finite fields are rarely to be found, therefore one can know how arithmetic computational on small finite fields works and be able to extend to the higher order. This means how integer and polynomial arithmetic operations are done for Z p such as addition, subtraction, division and multiplication in Z p followed by reduction of p (modulo p). Since addition is the same as subtraction and division is treated as the inverse of the multiplication, thus in this paper, only addition and multiplication arithmetic operations are applied for the considered small finite fields (Z 2 − Z 17 ). With polynomials, one can learn from this paper how arithmetic computational throug...

This paper deals with the formulation and the ident ifiability of the variablevolume deterministi... more This paper deals with the formulation and the ident ifiability of the variablevolume deterministic model for the endothermic cont inuously stirred tank reactor (CSTR). The identifiability of physical parameters of the formulated model is done by using the least squares and the delayed rejection a daptive algorithm version of the Markov chain Monte Carlo (MCMC) method. The least s quare estimates are used as prior information for the MCMC method. To measure the mod el output associated with the perturbed model parameters, we use global sensitivi ty analysis implemented in Latin Hypercube Sampling method. The obtained results fro m partial rank correlation coefficients show that six parameters are very sens itive and correlated with the model outputs. Finally, we show that the least square and the MCMC numerical results impart the model to be realistic, reliable and worthwhile to describe the dynamics of CSTR processes as physical parameters of the model are w ell identified and their...

Applied Mathematical Sciences, 2017

Continuously stirred tank reactors (CSTRs) play a big role in chemical engineering and in industr... more Continuously stirred tank reactors (CSTRs) play a big role in chemical engineering and in industries nowadays. Therefore, it is important and necessary to make sure CSTRs work successfully by avoiding disturbance from external factors. In this paper, we look at the impact of different kind of uncertainties (errors) to the model fitting and the parameter estimation of the CSTR's, for both cases of exothermic and endothermic reactors by taking into consideration of high and low variances of noise into concentration and temperature of the model. Two parameters, activation energy E and pre-exponential Arrhenius reaction rate k 0 have been estimated and investigated, due to their sensitivities to the system facilitated by the guessed noisy data and by using Minimization Assimilation Method (MAM). Results showed that, the increase of noise in measurements affects activation energy E much more than the pre-exponential Arrhenius reaction rate k 0 for all types of noise as k 0 converges to its guessed values. Also, from the results obtained, the endothermic CSTR is more sensitive than the exothermic CSTR for any kind of noise according to much deviation observed in its estimated activation energy values. Furthermore, quasi random noise has provided closer values for parameter estimation compared to the remaining types of noise but with negative activation energies for both cases (exothermic and endothermic).

Rwanda Journal of Engineering, Science, Technology and Environment, 2019

Predator prey interactions are important in ecology and most of time in the analysis, the two ant... more Predator prey interactions are important in ecology and most of time in the analysis, the two antagonists are assumed to be in a closed system. The aim of this study is to model the unclosed predator-prey system. The model is built and simulated data are computed by adding noise on deterministic solution. Therefore, model parameters are estimated using least square method. We compute the two critical points and the stability analysis is carried out and results show that the population is stable at one critical point and unstable at (0,0). The model fits the synthetic data with coefficient of determination R2 = 0.9693 equivalent to 96.93%. Using the residual analysis to test the validity of the model, it is shown that there is no pattern between residuals. To strengthen the validity of the model, the Markov Chain Monte Carlo algorithms are used as an alternative method in parameters estimation. Diagnostics prove the chains’ convergence which is the sign of an accurate model. As concl...

African Institute for Mathematical Sciences Tanzania, Jul 12, 2015

Finite field is a wide topic in mathematics. Consequently, none can talk about the whole contents... more Finite field is a wide topic in mathematics. Consequently, none can talk about the whole contents of finite fields. That is why this research focuses on small content of finite fields such as polynomials computational, ring of integers modulo p where p is prime or a power of prime. Most of the times, books which talk about finite fields are rarely to be found, therefore one can know how arithmetic computational on small finite fields works and be able to extend to the higher order. This means how integer and polynomial arithmetic operations are done for Z p such as addition, subtraction, division and multiplication in Z p followed by reduction of p (modulo p). Since addition is the same as subtraction and division is treated as the inverse of the multiplication, thus in this paper, only addition and multiplication arithmetic operations are applied for the considered small finite fields (Z 2 − Z 17 ). With polynomials, one can learn from this paper how arithmetic computational throug...

This paper deals with the formulation and the ident ifiability of the variablevolume deterministi... more This paper deals with the formulation and the ident ifiability of the variablevolume deterministic model for the endothermic cont inuously stirred tank reactor (CSTR). The identifiability of physical parameters of the formulated model is done by using the least squares and the delayed rejection a daptive algorithm version of the Markov chain Monte Carlo (MCMC) method. The least s quare estimates are used as prior information for the MCMC method. To measure the mod el output associated with the perturbed model parameters, we use global sensitivi ty analysis implemented in Latin Hypercube Sampling method. The obtained results fro m partial rank correlation coefficients show that six parameters are very sens itive and correlated with the model outputs. Finally, we show that the least square and the MCMC numerical results impart the model to be realistic, reliable and worthwhile to describe the dynamics of CSTR processes as physical parameters of the model are w ell identified and their...

Applied Mathematical Sciences, 2017

Continuously stirred tank reactors (CSTRs) play a big role in chemical engineering and in industr... more Continuously stirred tank reactors (CSTRs) play a big role in chemical engineering and in industries nowadays. Therefore, it is important and necessary to make sure CSTRs work successfully by avoiding disturbance from external factors. In this paper, we look at the impact of different kind of uncertainties (errors) to the model fitting and the parameter estimation of the CSTR's, for both cases of exothermic and endothermic reactors by taking into consideration of high and low variances of noise into concentration and temperature of the model. Two parameters, activation energy E and pre-exponential Arrhenius reaction rate k 0 have been estimated and investigated, due to their sensitivities to the system facilitated by the guessed noisy data and by using Minimization Assimilation Method (MAM). Results showed that, the increase of noise in measurements affects activation energy E much more than the pre-exponential Arrhenius reaction rate k 0 for all types of noise as k 0 converges to its guessed values. Also, from the results obtained, the endothermic CSTR is more sensitive than the exothermic CSTR for any kind of noise according to much deviation observed in its estimated activation energy values. Furthermore, quasi random noise has provided closer values for parameter estimation compared to the remaining types of noise but with negative activation energies for both cases (exothermic and endothermic).

Rwanda Journal of Engineering, Science, Technology and Environment, 2019

Predator prey interactions are important in ecology and most of time in the analysis, the two ant... more Predator prey interactions are important in ecology and most of time in the analysis, the two antagonists are assumed to be in a closed system. The aim of this study is to model the unclosed predator-prey system. The model is built and simulated data are computed by adding noise on deterministic solution. Therefore, model parameters are estimated using least square method. We compute the two critical points and the stability analysis is carried out and results show that the population is stable at one critical point and unstable at (0,0). The model fits the synthetic data with coefficient of determination R2 = 0.9693 equivalent to 96.93%. Using the residual analysis to test the validity of the model, it is shown that there is no pattern between residuals. To strengthen the validity of the model, the Markov Chain Monte Carlo algorithms are used as an alternative method in parameters estimation. Diagnostics prove the chains’ convergence which is the sign of an accurate model. As concl...