RP-Department of Mathematics

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Action of the Cyclic Group Cn Acting on the Diagonals of a Regular N–Gon
(iiste, 2016) Olum, Fredrick Odondo; Ireri, Kamuti; Mutie, Kavila; Ochieng’, Raymond Calvin
The main objective of this paper is to investigate the act ion of the cyclic group G=Cn on set,X , the diagonals of a regular - gon. We will first d iscuss the transitivity and primitivity of this action after which we will give useful results regarding the suborbits, subdegrees an d ranks of this action. It is worth mention ing that most of the results here have been given as Lemmas and Theorems.
Adomian Decomposition Method On Nonlinear Singular Cauchy Problem of Euler-Poisson- Darbuox equation
(Islamic Azad University, Rasht Branch, 2010) Iyaya, Wanjala C. C.
In this paper, we apply Picard’s Iteration Method followed by Adomian Decomposition Method to solve a nonlinear Singular Cauchy Problem of Euler- Poisson- Darboux Equation. The solution of the problem is much simplified and shorter to arriving at the solution as compared to the technique applied by Carroll and Showalter (1976)in the solution of Singular Cauchy Problem.
Analysis of the Dynamics of SI-SI-SEIR Avian In uenza A(H7N9) Epidemic Model with Re-infection
(Earthline Journal of Mathematical Sciences, 2020) Bada, Oluwafemi I.; Oke, Abayomi S.; Mutuku, Winfred N.; Aye, Patrick O.
The spread of Avian in uenza in Asia, Europe and Africa ever since its emergence in 2003, has been endemic in many countries. In this study, a non-linear SI-SI-SEIR Mathematical model with re-infection as a result of continuous contact with both infected poultry from farm and market is proposed. Local and global stability of the three equilibrium points are established and numerical simulations are used to validate the results.
Analysis of Water, Ethylene and Propylene Glycol-Based Nanofluids for Optimal Radiator Coolant
(International Journal of Fluid Mechanics & Thermal Sciences, 2023-09) Kisengese, Hilder Mary; Mutuku, Winifred Nduku; Makau, Kimulu Ancent
Convection is the spontaneous movement of fluid phases, either single or multiple, driven by interactions with heterogeneous material properties and body forces such as density and gravity. This movement of heated fluid facilitates heat transfer within a system. Natural convection finds applications in heat dissipation, air conditioning, and microelectronics. However, industrial fluids commonly used for heat transfer, such as minerals, oil, water, and ethylene glycol (EG), face limitations due to their low thermal conductivities, hindering heat exchange efficiency. The production of efficient cost-effective cooling systems for automotive engines is a significant challenge in the automobile industry. Most engines depend on fluid for cooling and therefore use liquid coolants such as ethylene glycol and water, but with poor heat transmission properties. Nanoparticles, which have been shown to improve thermal conductivity, enhance the thermal properties of the fluids. This study compares six different radiator coolants; water-CuO, Propylene-glycol-CuO, ethylene-glycol-CuO, water-MgO, Propylene-glycol-MgO, and ethylene-glycol-MgO. Nanoparticles exhibit improved thermophysical qualities and therefore nanofluids are used as coolants in various mechanical and engineering contexts, including, but not limited to electronics, vehicles, transformers, computers, and electrical devices. The similarity transformation is utilized to non-dimensionalise the governing equations. The resulting equations are solved using a numerical method with the Runge-Kutta fourth-order method. The results show that water-based nanofluids provide the best coolant. However, when the radiator is close to the magnetic field emerging from the automobile engines, copper oxide or Magnesium oxide nanoparticles should be used with water as base fluid.
The applicability of mathematics
(Taylor & Francis, 1984) Uppal, S. M.
Asymptotic linear estimation of the quantile function of a location-scale family of distributions based on selected order statistics
(AFRICAN RESEARCH, 2005) Odongo, L. O.; Okango, A. A
Some general asymptotic methods of estimating the quantile function, Q(ξ), 0>ξ>1 of location-scale families of distributions based on a few selected order statistics are considered, with applications to some non-regular distributions. Specific results are discussed for the ABLUE of Qξ for the location-scale exponential and double exponential distributions. As a further application of the exponential results, we discuss the asymptotically best optimal spacings for the location-scale logistic distribution.
A Bayesian Test for Equality of Scale Parameters of Several Exponential Distributions
(2002) Odongo, L. O.; Samanta, M.
This article develops a Bayesian test for equality of scale parameters of several exponential distributions. The null distribution of the test statistic is approximated by the chi-square distribution using heuristic reasoning in conjunction with the Wilson-Hilferty transfomation for the chi-square random variable. The accuracy of the chi-square approximation of the test statistic and the modified likelihood ratio statistic is examined and their powers compared using Monte Carlo simulation. The proposed test is found to be comparable to Bartlett's modified likelihood ratio test in terms of accuracy and power. A numerical example is included to illustrate the applications of these tests
Buoyancy-Induced MHD Stagnation Point Flow of Williamson Fluid with Thermal Radiation
(Journal of Engineering Research and Reports., 2020) Ouru, J. O.; Mutuku, W. N.; Oke, A. S.
Flow of fluids subjected to thermal radiation has enormous application in polymer processing, glass blowing, cooling of nuclear reactant and harvesting solar energy. This paper considers the MHD stagnation point flow of non-Newtonian pseudoplastic Williamson fluid induced by buoyancy in the presence of thermal radiation. A system of nonlinear partial differential equations suitable to describe the MHD stagnation point flow of Williamson fluid over a stretching sheet is formulated and then transformed using similarity variables to boundary value ordinary differential equations. The graphs depicting the effect of thermal radiation parameter, buoyancy and electromagnetic force on the fluid velocity and temperature of the stagnation point flow are given and the results revealed that increase in buoyancy leads to an increase in the overall velocity of the flow but a decrease in the temperature of the flow.
C-Loops Obtained From Hypercomplex Numbers
(IJRIAS, 2020) Kioko, Eugene Moses; Njuguna, Lydia
Hypercomplex numbers have played a notable and critical role in the study and exploration of Loop Theory. Researchers have made numerous studies in this area especially in the investigation and construction of different loops. This paper has extended the research to C-loops where we are investigating the formation of C-loops obtained from hypercomplex numbers of dimension 2n ; 1 ≥ n ≤ 4. We are specifically working with the 24- dimensional algebra, called the sedenions. In constructing the C-loops, we have used the frame multiplication of hypercomplex numbers using the Cayley- Dickson construction. We have tested the satisfaction of the left, (x x) (y z) = (x (x y)) z and right, x ((y z) z) = (x y) (z z) C-loop identities by the sedenions. We have also formed split extension of sedenions and equally tested the satisfaction of the C-loop identities on them. We have found that the sedenions satisfy the C-loop identities hence forming C-loops. However, the split extension of sedenions satisfies the right C-loop identity only. AMS Subject Classification: 20B05
Car Antifreeze and Coolant: Comparing Water and Ethylene Glycol as Nano Fluid Base Fluid
(Sretech Journal of publication, 2018) Kimulu, Ancent Makau; Mutuku, Winfred Nduku; Mutua, Nicholas Muthama
Most internal combustion engines are fluid cooled using liquid coolants mostly water and ethylene glycol, whose heat transfer capabilities are limited. An alternative for improving their thermophysical properties is the addition of metal oxides so as to make Nanofluids. This study investigates car antifreeze and coolants by comparing water-based and ethylene glycol based Nanofluids to determine the Nanofluid which provides optimum cooling. The governing equations are non–dimensionalised using similarity transformation and the resulting equations solved numerically using the Runge–Kutta fourth order scheme. Graphical results of pertinent parameters on fluid velocity, temperature, skin friction and rate of heat transfer are presented and discussed in depth putting into consideration the industrial application. It was observed that the magnetic field slows the fluid flow while increasing the fluid temperature. Also ethylene glycol Nanofluid offers better cooling capabilities.
Classical Approach to Zero-Inflated Dynamic Panel Ordered Probit Model with an Application in Drug Abuse
(science publishing group, 2022) Kung’u, John; Odongo, Leo; Kube, Ananda
The Zero inflated ordered categorical data with time series structure are often a characteristic of behavioral research attributed to non-participation decision and zero consumption of substance such as drugs among the participants. The existing Semi-parametric zero inflated dynamic panel probit model with selectivity have exhibited biasness and inconsistency in estimators as a result of poor treatment of initial condition and exclusion of selectivity in the unobserved individual effects respectively. The model assumed that the cut points are known to address heaping in the data and therefore cannot be used when the cut points are unknown. In this paper, a Zero inflated dynamic panel ordered probit models have been developed to address the above challenges. Average partial effects that presents the impacts on the specific probabilities per unit change in the covariates are also given. Since the solutions are not of closed form, Maximum likelihood estimation based on Newton Raphson algorithm was used to estimate the parameters of the model. A Monte Carlo study was carried out to investigate some theoretical properties of the estimators in the models. The study found that the Zero inflated dynamic panel ordered probit models with independent and correlated error terms produced consistent estimators. The Zero inflated dynamic panel ordered probit models with independent and correlated error terms had more accurate estimators than the Dynamic panel ordered probit model. The Zero inflated dynamic panel ordered probit model with correlated error terms fitted the National Longitudinal Survey of Youth 1997 better than Zero inflated dynamic panel ordered probit model with independent error terms and Dynamic panel ordered probit model. The Zero inflated dynamic panel ordered probit model with independent error terms fitted the National Longitudinal Survey of Youth 1997 better the Dynamic panel ordered probit model.
Combined effect of Buoyancy force and Navier slip on MHD flow of a nanofluid over a convectively heated vertical porous plate
(Scientific World, 2013) Mutuku-Njane, W.N.; Makinde, O. D.
We examine the effect of magnetic field on boundary layer flow of an incompressible electrically conducting water-based nanofluids past a convectively heated vertical porous plate with Navier slip boundary condition. A suitable similarity transformation is employed to reduce the governing partial differential equations into nonlinear ordinary differential equations, which are solved numerically by employing fourth-order Runge-Kutta with a shooting technique. Three different water-based nanofluids containing copper (Cu), aluminium oxide (Al2O3), and titanium dioxide (TiO2) are taken into consideration. Graphical results are presented and discussed quantitatively with respect to the influence of pertinent parameters, such as solid volume fraction of nanoparticles (φ), magnetic field parameter (Ha), buoyancy effect (Gr), Eckert number (Ec), suction/injection parameter (f w ), Biot number (Bi), and slip parameter ( β ), on the dimensionless velocity, temperature, skin friction coefficient, and heat transfer rate.
Comparison of crisp and fuzzy classification trees using gini index impurity measure on simulated data
(European Scientific Institute, 2015-02-09) Muchai, Eunice; Odongo, Leo
Crisp classification trees have been used to model many situations such as disease classification. With the introduction of fuzzy theory, fuzzy classification trees are gaining popularity especially in data mining. Very little work has been done in comparing crisp and fuzzy classification trees. This paper compares crisp classification trees and fuzzy classification trees using Gini index as the impurity measure. The objective is to determine which of the two classification trees gives fewer errors of classification. The data used consisted of two sets of observations from multivariate normal distributions. The first set of data were from two 3-variate normal populations with different mean vectors and common dispersion matrix. From each of the two populations 5000 samples were generated. 1000 samples out of the 5000 were used to create the trees. The remaining 4000 samples from each population were used to test the trees. The second set of data were from three 4-variate normal populations with different mean vectors and common dispersion matrix. A similar sampling and testing procedure as for the case of first set of data was employed. Computations were implemented using R statistical package. The results from the test showed that fuzzy classification trees allocated observations to the correct population with fewer errors than did crisp classification tree.
Comparison of the Ratio Estimate to the Local Linear Polynomial Estimate of Finite Population Totals
(International Journal of Engineering, Science and Mathematics, 2013-03) Odongo, L. O.; Maua, M. Z.; Orwa, G.O.; Otieno, Romanus Odhiambo
In this paper, attempt to study effects of extreme observations on two estimators of finite population total theoretically and by simulation is made. We compare the ratio estimate with the local linear polynomial estimate of finite population total given different finite populations. Both classical and the non-parametric estimator based on the local linear polynomial produce good results when the auxiliary and the study variables are highly correlated. It is however noted that in the presence of outlying observations the local linear polynomial performs better with respect to design mean square error (MSE) in all the artificial populations generated.
Computational Fluid Dynamics (CFD) for Blood Flow in Cardiovascular Medical Devices and Blood Damage Prediction
(NCM, 2023) Niyonkuru, Venant
Background: The hemodynamic performance of cardiovascular medical devices and their potential to cause blood damage are critical factors in ensuring patient safety and device efficacy. Computational Fluid Dynamics (CFD) has emerged as a valuable tool for simulating blood flow within these devices and predicting the risk of blood damage. Objectives: This study aims to utilize CFD simulations to evaluate the local hemodynamic performance of a particular implantable device and to provide precise predictions about likely adverse clinical effects, cutting-edge techniques like laser doppler anemometry (LDA) or particle image velocimetry (PIV) must be accessible. Methods: A patient-specific CFD model of the cardiovascular system and medical devices was developed based on medical imaging data. Hemodynamic parameters such as shear stress and flow recirculation were computed to identify regions of potential blood damage. The simulations were validated against data. Results: The CFD simulations revealed intricate flow patterns and areas of concern within the medical devices. Elevated shear stresses and prolonged residence times were identified in certain regions, indicating a risk of blood damage. By quantifying these parameters, the study provided a comprehensive assessment of potential blood damage locations and severity levels. Conclusion: CFD proved to be a robust approach for evaluating blood flow within cardiovascular devices and predicting potential blood damage. The study highlighted specific design modifications that could mitigate the risk of blood damage, thus contributing to the improvement of device safety. The integration of CFD with patient-specific data offers clinicians and engineers a powerful tool for optimizing cardiovascular device design and minimizing patient risk
Cost-Effectiveness Analysis of Optimal Control Strategies for Malaria Transmission in Bubanza Province, Burundi.
(Elsevier, 2022) Niyonkuru, Venant; Mutuku, Winifred
Malaria is a parasitic infection ranked among the leading causes of mortality and morbidity in Sub-Sahara African countries. If recommended interventions measures are well applied, malaria can be prevented and controlled. In many cases, the budget allocated to malaria prevention and treatment project is not enough, using malaria intervention measures properly will guarantee the reduction of infected population while the intervention costs is minimized. This saves the budget and produces the results in economical way. The aim of this article is to understand the cost so that decision makers are well informed when they determine budget allocated to malaria interventions. After ordering different possible strategies from the smallest to the highest, utilizing Incremental Cost-Effectiveness Ratio (ICER), we studied the Cost-effectiveness of each strategy. This study analyses the cost-effectiveness of all possible optimal control measures to identify which is the intervention strategy is going to save available resources and cost-effective. After analysis, this study shows that malaria can be minimized in Bubanza using preventive measures at the most cost effective way.
Cycle Index of Internal Direct Product Groups
(International Mathematical Forum, 2011-12) Kamuti, I.N.
If M and H are permutation groups with cycle indices ZM and ZH respectively, and if * is some binary operation on permutation groups, then a fundamental problem in enumerative combinatorics is the determination of a formula for ZM *H in terms of ZM and ZH. To this end, a number of results have already been obtained (cf. Harary [1], [2], [3]; Harary and Palmer [6]; Harrison and High [7]; Pόlya [10]). This paper may be viewed as a continuation of a previous paper (Kamuti [8]) in which I have shown how the cycle index of a semidirect product group G= M×H can be expressed in terms of the cycle indices of M and H by considering semidirect products called Frobenius groups. Thus if G=M×H (internal direct product), the aim of this paper is to express the cycle index of G in terms of the cycle indices of M and H when G acts on the cosets of H in G.
Description of Minimal Entropy Hellinger Sigma Martingale Density of Order One, Order q and Order Zero
(Scientific Research Publishing Inc, 2021) Mwigilwa, Winfrida Felix; Aduda, Jane; Kube, Ananda Omutokoh
Generally in this paper, we show how the new version of parameter 2 U ∈  in Jacod decomposition will change an expression of entropy-Hellinger process of order one, order q and order zero and consequently an equation of minimal entropy Hellinger sigma martingale density for all orders. This is because even the measurable function W ∈ which is an important parameter of an equation of minimal martingale density changes. In order to get a required parameter W ∈ P , we introduce the function 1 t m f = − ∈  during our calculation for all orders. The result is different to order zero because we failed to get an equation of minimal entropy-Hellinger sigma martingale density of order zero.
Development of an Advanced Optimization and Optimal Control Mathematical Model for Energy Efficient Operations of Renewable Energy.
(IJRS, 2023) Niyonkuru, Venant
The potential for developing energy-efficient operations for renewable energy systems has increased with the expansion of renewable energy sources. As a result, creating advanced optimization and optimum control mathematical models will be necessary to handle the difficulties related to their integration into the grid. This study develops an enhanced optimization and optimal control model for the energy-efficient operation of renewable energy systems based on photovoltaic (PV) and hydropower. The created model takes into account the economic element while integrating an optimization method to address the energy management issue of PV systems. PV system operation is optimized taking into account a variety of operational restrictions using economic objectives like profit maximization and cost minimization. To further simplify energy balancing and lower risk, a hydropower energy system is also taken into consideration. Furthermore, by forecasting the ideal PV output and hydropower generation for various economic purposes, a dynamic optimum control approach is created to discover the best operational strategy for the systems. Case studies are used to evaluate this sophisticated optimization and optimum control model, and the findings show how effective it is at lowering operating expenses for the operation of renewable energy systems in an energy-efficient manner.
Different Types of Structure Conditions of Semimartingale with Jacod Decomposition
(Scientific Research Publishing Inc, 2022) Mwigilwa, Winfrida; Aduda, Jane; Le Doux Mbele, Bidima Martin; Kube, Ananda
The objective of this article is to use Jacod decomposition to develop different types of semimartingale structure conditions. We make the following contributions to that end: When a continuous semimartingale meets the structure condition (SC), we prove that there is a minimal martingale density and a predictable variation part. When a special semimartingale meets the minimal structure condition (MSC) and the natural structure condition (NSC), we derive a Radon-Nikodym decomposition and a Natural Kunita-Watanabe decomposition from a given sigma martingale density, which is written under the Jacod decomposition.

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