MST-Department of Mathematics

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Action of Symmetric Group S5 on The Cosets of Some of Its Non-Maximal Subgroups
(Kenyatta University, 2020-07) Omori, Kerubo Lydiah
The Action of Symmetry Groups of Platonic Solids on their Respective Vertices
(Kenyatta University, 2020) Waweru, R. Brian
Platonic solids are 3-dimensional regular, convex polyhedrons. Each of the faces are equidistant and equiangular to each other in any of the solids. They derive their name from the ancient Greek philosopher, P lato who wrote about them in his dialogue, the Timaeus as reported by Cornford (2014). The solids features have fascinated mathematicians for decades including the renown geometer, Euclid: In his Book XIII of the Elements, as rewrote by Heath et al. (1956), he successfully determined the exact number of solids that qualify to be Platonic Solids; tetrahedron, cube, octahedron, dodecahedron and icosahedron. In group theory, the symmetry group of an object is the group of all transformations under which the object remains unchanged, endowed with the group operation of composition. Due to their inherent symmetry of these solids many mathematicians have attempted to derive their symmetry groups. For instant, Foster (1990) who successfully enumerated the symmetry groups of the dodecahedron and recently Morandi (2004) attempted to compute these symmetric groups of the solids using a computer program called Maple. Although such contributions are noteworthy, a few attempts have been made to explore other features such as the symmetry groups of the platonic solids. Thus, this project investigates the properties of the group action of the symmetry groups of these platonic solids acting on their respective vertices. We embark on constructing the symmetry groups of each of the solids then employ the orbit-stabilizer and other theorems to determine the ranks and sub-degrees of each solid. The action of G on V shows that tetrahedron has a rank of 2, the octahedron has a rank of 3, dodecahedron has a rank of 6 while the cube and icosahedron have a rank of 4.
Analysis of a 3D Heat Transfer of Magneto hydrodynamics Cu-H20 and Al2O3-H20 Nonoflluid over an Exponationally Stretching Plate
(Kenyatta University, 2023-08) Ruto, Celestine C.
Analysis of flow in regular bend pipe
(2012-02-01) Chirchir, Julius Kibet
A fluid is a substance that deforms continuously on application of slight shear stress. The study of fluid mechanics deals with motion of fluids and conditions that support or after motion. Fluid mechanics can be divided into fluid kinematics and fluid dynamics. Fluid kinematics deals with forces involved in motion of fluid, whereas fluid dynamics deals with the state of fluid in motion caused by imbalances of forces acting on it. Kinematics of fluid motion uses vector quantities such as; velocity, acceleration and rate of discharge which is defined in terms of scalar quantities; length and time in some specified coding system. These depend on some boundaries of a particular system under in investigation. Fluid dynamics involves the applications of Newton's second law of motion that states:- rate of change of momentum is directly proportional to the applied force; Transfer of fluids through pipes involves changes in potentials/pressure energy between two ends of the pipes under study. This potential difference can either be caused by: - (a) Gravitational differences (g) (b) Differences in height (h) (Potential energy) (c) Density differences, (p), for example that caused by concentration differences, Beek (1985). These three factors are related by equation (i) on the next page. P=phg .........................................................(i) The flow of fluids in channels can either be turbulent or laminar. Flow is where the streamlines do not intersect such that there is no mixing up of fluids. Laminar flow occurs when fluid has; 1. Low Reynold's number (Re<2000) 2. Low viscosity (internal friction of fluid particals 3.Friction between the channel and the fluid negligible. Turbulent flow occurs where there is continuous mixing up of the fluid in motion. This type of flow occurs when the fluid has; 1. High Reinhold’s number 2. High viscosity. 3. Boundary of channel and fluid have got high friction. Fluids transfer can either occur in open or closed channel depending on the geometry of the channel. Open channels are like surface run-off, rivers and streams, dug-out drains, of which all have got a free surface while confined (closed) flow occurs in porous media, closed cracks and in pipes. Fluid flow in both types of channels have got its uses, but highlighted here are only importance of closed channel flow. That is the used both in domestic and industrial sectors. In domestic side flow in closed channels are important in areas, for example in passage of fluids for example water from one point to other. It is important to have piped water at our vicinity especially at this age of development. On the industrial side, flow in closed channels at factories is important and to mention a few examples, are in areas like; - 1. Sewage collection and disposal. 2. Passage of oil, for example the oil pipeline from Mombasa to Eldoret 3. Passage of raw materials in chemical processes, for example breweries 4. Passage of gases in geothermal plants like in Olkaria In this work, only confined type of flow will be considered in a uniform circular cross sectional bend of a pipe. The fluid here is a liquid, for example water. And since the fluid delivered need not necessarily follow a straight path definitely there will be bends in pipes. Shapes such as this might be met in area like: Sprinkler nozzles. Nozzles of sprayers and spouts of various utensils like kettles. Detailed knowledge on fluid flow and the prevailing suitable conditions for laminar flow must be well defined as given earlier. In this work, however. We seek to give a clear picture on: - 1. Pressure distribution. 2. Velocity distribution and profile where necessary. 3. Forces acting on the pipe at certain parts. We therefore consider in detail how flow factors are affected by the shape of the region of the pipe considered. The proper interaction balance of these factors play a role in flow mechanism. Earlier, various analyses on straight pipe of some geometry Douglas (1995). The writer has analysed flows by application of methods of differential calculus and coupled with a well-defined boundary conditions have given practically near accurate results. On basis of the results obtained by this method it has become evident that complete, continuity, momentum and energy (Bernoulli) equations have to be well understood. In most of flows in pipes, velocity distribution, pressure distribution and force analysis are investigated by use of calculus modified by trigonometric ratios with successful results. Differential models has been powerful in predicting a number of flow problems in diverse fields of flow in the past decade.
Analysis of Heat and Mass Transfer on Magnetohydrodynamics (Mhd) Nanofluids with Thermal Radiation and Brownian Motion Over A Heated Vertical Plate
(Kenyatta University, 2019-04) Ndungu, Elizabeth Wambui
This study investigates the effect of heat and mass transfer on Magnetohydrodynamics (MHD) nanofluid with thermal radiation and brownian motion over a heated vertical plate. The Magnetohydrodynamics (MHD) nanofluid flow have different electrical conductivities and behave differently in presence of thermal radiation, magnetic field, thermophoresis and brownian motion. The rate of heat and mass transfer on nanofluid along the vertical plate under the influence of a magnetic field with thermal radiation and Brownian motion leads to change in the fluid motion. The diverse applications of nanofluids in engineering and industries it is of this great importance hence the need to investigate the effects of thermal radiation, thermophoresis and Brownian motion on nanofluids and magneto hydrodynamics. Nanofluids are considered as potential working fluids to be used in high heat flux systems such as electronic cooling systems, solar applications, heat pipes, and nuclear reactors. As secondary fluids, they can be applied in commercial refrigeration, chiller and solar panels in absorption systems. They provide much more energy for a given weight of fuel than any technology in use , at the same time reducing thermal pollution. The governing non-linear boundary layer equations are formulated and transformed into ordinary differential equations using the similarity transformation. The resulting ordinary differential equations are solved numerically using the fourth order Runge-Kutta method. The numerical results for dimensionless parameters as well as the skin-friction coefficient and nusselt number, are presented graphically and analysed quantitatively. We note that increasing magnetic field, radiation, thermophoresis and Brownian motion parameters leads to an increase in the fluid temperature resulting in a reduction in the Nusselt number and Sherwood number
Analysis of Magnetohydrodynamic Convective Heat Transfer of Casson Nanofluid Flow Over a Heated Stretching Vertical Plate
(Kenyatta University, 2022) Kigio, John Kinyanjui; Winifred Nduku Mutuku
Casson fluid has many industrial and engineering applications. However, Casson nanofluid possesses superior electrical and thermal conductivities compared with the original Casson fluid. Hence, the Casson nanofluid is considered in this project. The equation governing the natural convective magnetohydrodynamics flow of Casson nanofluid across a a convectively heated vertical plate is formulated. The equation is reformulated into a system of ordinary differential equations (ODEs) using similarity transformations. Thereafter, a numerical solution of the resulting ODEs is obtained using the Runge-Kutta-Gills method. It is found that flow temperature profiles increase with increasing Eckert number, Biot number and magnetic field strength while flow velocity decreases as the flow becomes Newtonian and as nanoparticle volume fraction increases.
Analysis of Magnetohydrodynamic Heat and Mass Transfer with Carbon Nanotubes-Graphene Casson Hybrid Nanofluid
(Kenyatta University, 2022) Juma, Belindar Atieno; W.N.Mutuku
Fluids such as liquids and gases are amassed into either Newtonian or non-Newtonian groups. Most fluids fall under the non-Newtonian category and, as a result, various models like Casson fluid and Williamson fluid model have been proposed to deal with the non-Newtonian fluid behaviour. Due to its ability to model the flow of blood, Casson fluid model is of major medical importance. A development on ordinary fluids are nanofluids, which posses enhanced thermophysical properties. Hybrid nanofluid, obtained when two non-identical nanoparticles are dispersed in a fluid, is an improvement on the novel nanofluids. It has superior thermal conductivity when compared with nanofluids or ordinary fluids. Multiple research have shown that the shape of the nanoparticles used during the hybridization process has significant impact on the thermal properties of the hybrid formed. Applications of hybrid nanofluid include refrigerators, electronic devices, and cancer treatments. In the study of hybrid nanofluids, the focus has been placed on the dynamic properties and heat transfer rate. In consensus, the superiority of the hybrid’s properties are emphasized. Carbon being the most abundant product, hardest, strongest and stable known compound, it is an excellent thermal conductor. CNTs and graphene are allotropes of Carbon. In the HAMT research, no researcher has explored the impact of suspending a combination of CNTs and graphene nanoparticles on a Casson base fluid. To bridge this gap, this study is designed to analyse the HAMT rate of a 2-D magnetohydrodynamic hybrid Casson nanofluid. The nanoparticles are Carbon nanotubes and Graphene. The flow is across a surface stretching exponentially. Volume fraction, nanoparticle size and other pertinent parameters are investigated on the HAMT rate.The governing equations are converted to their non - dimensional form using similarity variables, and subsequently to an ODEs. The RK4 with Shooting Technique is adopted as a method of solution. Simulation of the model and investigation of the HAMT rate is carried out using MATLAB bvp4c. The primary velocity is reduced with Casson fluid parameter but enhanced with the radiation parameter. The temperature profiles boost with Casson fluid parameter, magnetic and radiation parameters. The local skin friction increases with Casson fluid parameter and radiation parameter but decreases with magnetic field strength. HAMT rate is enhanced with increasing Grashof number but decreases with Casson fluid parameter and magnetic field strength.
Analysis of Magnetohydrodynamic Stagnation Point Flow Due to a Fluid towards a Convectively Heated Permeable Stretching Sheet
(Kenyatta University, 2020-10) Mwangi, Njoroge Kelvin
Stagnation-point flow of an electrically conducting fluid over a continuously stretching surface in presence of magnetic fields is significant in many industrial processes such as the metallurgy, polymer processing, glass blowing, filaments drawn through quiescent electrically conducting fluid subject to magnetic fields, cooling of metallic plate, hot rolling, wire drawing, aerodynamic extrusion of plastic sheets, crystal growing. In these applications of stagnation point flow, the desired output depends largely on the rate of heating and the velocity of the fluid on the surface. The required rate will be achieved by variation of various thermophysical parameters such as suction parameter, Grashoff number, Hartmann number and buoyancy parameter. The resulting nonlinear partial differential equations governing this flow were reduced to nonlinear ordinary differential equations using similarity transformations and the resulting equations solved numerically using the fourth order Runge-Kutta scheme with a shooting technique. Graphical results were presented and discussed quantatively with respect to the effects of thermophysical parameters on both velocity and temperature profiles of the fluid. From the study we note that an increase in Grashoff number (Gr), Hartmann number (Ha) and suction parameter resulted to a corresponding increase in fluid velocity. The fluid temperature also increased with increase in Gr and Ha but decreased with increase in suction parameter and buoyancy parameter.
Analysis of seasonal time series with missing observations
(2012-04-11) Kihoro, John Mwaniki
In the past, Kenya's Tourist data has been analysed by among others, Mutiso (1982) who carried out Spectral Analysis to the monthly data for a series covering 10 years (1971-1980). Mutiso came up with general conclusion on cycles experience by the tourism industry in Kenya. Onyango (1993) fitted non-linear model to the tourist data before testing it for linearity. In an effort to confirm or disapprove their findings we decided to analyse a longer series, that is, to collect and use data for the years 1971 to 1990. It is in this process that we encountered the problem of missing data. Monthly data for the year 1984 could not be located in the Kenya central Bureau of Statistics (CBS) records, but quarterly totals were available. Our immediate problem was to fill in the block of missing values and as such most of the work in this project point to this direction. In this dissertation we have used the two known segments [(1971 - 1983) and (1985 - 1990)] of tourist data to estimate the block of missing values. We have considered two methods under regression (indirect) imputation methods and imputed the missing values from eventual forecasts. We have showed how two forecasts arising from the two regression methods can be combined to come up with more accurate estimates. We have also suggested a method of adjusting the estimates to incorporate the information from the already known quarterly totals. We have proposed simpler and direct method of imputing the missing values and we have showed that the final adjusted estimates from the direct and the indirect methods are similar. Finally we modelled the two data sets arising from the direct and the indirect imputation and generated eventual forecasts, which further showed that the two methods give almost similar estimates.
Analysis of Vertical Transmission Dynamics of Infectious Hepatitis B Virus: Mathematical Model Involving Vaccination And Treatment in Turkana County, Kenya
(Kenyatta University, 2022) Ghai Kuei, Geu; Gatoto
Hepatitis B has been a major global health menace for it is a potentially life-threatening liver disease. Close to 0.25 billion persons are living with this infectious disease across the world. It’s transmitted by infected individual to uninfected person either vertically (transmission before or during birth by carrier mother to the baby) or horizontally (transmission when the bodily fluid of an infected person comes into contact with the hepatitis B virus-free person). This can happen through the sharing of non-sterilized injection syringes, tattooing objects and through sexual intercourse. This particular project studied a mathematical model that combined both vaccination and treatment as a means to controlling the hepatitis B virus (HBV). In our mathematical model, equations are derived from the flow chart representing the HBV transmission dynamics. The determination of the disease-free equilibrium state (DFE), the endemic equilibrium state (EE) and the basic reproduction number 𝑅0 were made. The stability of these points were determined and the results show that the disease-free equilibrium was both locally and globally asymptotically stable. In other words, 𝑅0 < 1. The stability analysis of endemic equilibrium point also reveals that the point is locally and globally asymptotically stable, that is, 𝑅0 > 1. The basic reproduction number 𝑅0 is computed using the next generation matrix method. The system of ordinary differential equations (ODEs), which is non-linear are solved by numerical simulation. This was achieved by use of Runge-kutta method of order four with the help of MATLAB software. These results show that either of the method, treatment or vaccination, administered is effective in alleviating the spread of HBV disease. However, when both control strategies are combined, the disease is quickly controlled and ultimately brought to eradication.
Application of canonical correlation analysis ;a comparative study on academic perfomance
(1990-06) Makambi, Kepher Henry
Aspects of controllability of autonomous systems in locally convex topological spaces
(2012-05-16) Oduor, Shem Richard
In this project, two theorems in (6) on controllability of autonomous systems defined with the help of ordinary Differential Equations (ODE) in finite dimensional vector spaces is genralized to infinite dimensional locally convex topological spaces. This has been achieved with the help of the idea of a derivative, much more general than Frechet's derivative and an integral both defined in (12)
Bioconvection of a Newtonian Nanofluid Over a Vertical Plate in Presence of Gyrotactic Microorganisms
(Kenyatta University, 2019-12) Kibe, John Ngugi
Bioconvection induced by gyrotactic microorganisms in a Newtonian nanofluid past a permeable vertical plate is studied. Addition of motile microorganisms to a suspension of nanoparticles in a basefluid enhances mass transfer and mixing in most microsystems in addition to the enhancement of the convectional properties of the nanofluid. This concept has solved many heating problems in various areas including civil engineering, chemical engineering and mechanical engineering. The present study looks into the movement of motile microorganisms present in a nanofluid over a vertical plate for purposes of saving energy through the enhancement of heat transfer. The objectives are to formulate and solve the mathematical equations governing bioconvection of a Newtonian Nanofluid flow along a vertical plate, investigate the effects of gyrotactic microorganisms on the temperature and velocity and temperature profiles of the nanofluid and to analyse the effects of nanoparticles on microorganisms concentration, temperature and velocity profile. Ordinary differential equations are obtained from the governing partial differential equations by use of similarity variables. To numerically solve the ODE’s, the Runge-Kutta Ferhlberg method is used with the shooting technique. Further, an investigation on the effects of controlling parameters on several numbers and dimensionless quantities of our interest. It is found that the Nusselt Number, the Sherwood Number and Skin Friction are strongly affected by nanofluid and bioconvection parameters
Buoyancy-Induced Mhd Stagnation Point Flow of Williamson Fluid with Thermal Radiation Over a Stretching Surface
(Kenyatta University, 2020) Ouru, John Olum; Winfred N. Mutuku
The current study investigates the buoyancy-induced MHD stagnation point flow of Williamson fluid with thermal radiation over a stretching sheet. A system of nonlinear partial differential together with the boundary conditions governing the fluid flow are formulated, then transformed to ordinary differential equations using appropriately identified similarity variables. The resulting equations are solved numerically by combination of shooting technique and Runge-Kutta fourth order integration scheme. The results are depicted graphically to illustrate the effects of thermal radiation parameter, buoyancy and electromagnetic forces on both the fluid velocity and temperature. The results revealed that an increase in buoyancy leads to an increase in the overall velocity of the flow but a decrease in the temperature of the flow.
Comparison of the performance of estimators in estimation of finite population total
(2012-02-01) Muema, Benjamin K.
In this project we compare the performance of two different estimators of the population total. One estimator is model-based and the other one is model assisted. We look at model-based properties of the two estimators. We observed that under the general model, the biases of the two estimators are different.
Coriolis Effects on Air, Nanofluid and Casson Fluid Flow Over a Surface with Nonuniform Thickness
(Kenyatta University, 2022) Samuel, Oke Abayomi; Winifred N. Mutuku
The trajectories of moving objects in a rotating frame is usually deflected by Coriolis force and this leads to many of the natural phenomena on Earth. Some of the natural outcomes of Coriolis effect include the tropical storms, cyclones, and hurricanes. Understanding the Coriolis effect on Newtonian and non-Newtonian fluid flow is therefore vital in determining the right actions needed to forestall these natural disasters. The invention of mass-flow meter to measure the density of fluid and the rate of mass-flow in a tube and the invention of bioreactors are some of the many practical uses of Coriolis force. The path followed by the air (Newtonian fluid) and nanofluid (fluid suspending some nanoparticles) heated by the sun to reach the Earth is usually deflected by the Coriolis force. It becomes worthwhile to investigate the influence of Coriolis force on the flow of these fluids. Also, the effect of a rotating frame of reference on the Brownian diffusivity and thermophoretic diffusivity of nanofluids is important in industries for its role in cooling/heating processes. Casson fluid possesses characteristics that closely agree with the nature of blood; with applications in bioengineering. More so, the Earth’s surface is neither horizontal nor vertical, and it is neither inclined. The Earth surface is geoid and so, a more appropriate surface to model it is one with non-uniform thickness. This thesis investigates and gives a comprehensive insight into the significance of rotational force on the flow of Newtonian and non-Newtonian fluids over a surface with non-uniform thickness. A suitable set of equations for the flow is obtained by introducing the Coriolis force as a body force into the Navier-Stokes equations. Appropriate similarity variables are used to reduce the set of governing partial differential equations to a system of non-linear ordinary differential equations. The transformed set of equations is numerically solved via shooting technique alongside Runge-Kutta algorithm. The results are depicted graphically and in tables to show the effects of pertinent parameters on the flow. It is observed that rotation enhances primary flow velocity but impedes secondary flow velocity. Rotation also has an increasing effect on temperature profiles but decreasing effects on the heat transfer rate of Casson and Newtonian fluids.
Cycle Indices of Cyclic and Dihedral Groups Acting on Diagonals of a Regular N-Gon and R-Element Subsets of the Set X = * +.
(Kenyatta University, 2022) Paul, Doris Kiende; Ireri. N. Kamuti
Polya (1937) used the cycle index in counting configurations. Several authors have computed cycle index formula of several group actions which have been used globally in chemistry and mathematics, in counting of chemical compounds and graphs. However dihedral group Dn acting on the set of diagonals of a regular n - gon, 4-elements and r-elements subsets of the set X = ( 1,2,…n) has not been done. The project shall come up with the formulae for Z. / and Z. /= [Σ ( ) ]. The project seek to extend Muthoka’s findings on the formula for Z( , -) and Z( , -) to derive the formulae for Z( , -) and Z( , -) = [Σ ( ) ( ) ( ) . ( ) ( )/]. This shall be done by re-visiting some of the results by Harary and Palmer (1973) and also some of the contributions by Kamuti and Njuguna (2004) in the study shall be used.
Derivation of Cycle Index Formulas of Semidirect Product Groups
(Kenyatta University, 2019-12) Muthoka, Geo-Rey Ngovi
The concept of the cycle index formulas of a permutation group was discovered in the year 1937. Since then cycle index formulas of several groups have been studied by di erent scholars. For instance the cycle index of the dihedral group Dn acting on the set of vertices of a regular n􀀀gon is known and has been applied in enumeration of di erent mathematical structures. In this study the relationship between the cycle index formula of a semidirect product group and the cycle index formulas of the two subgroups which the group is a semidirect product of was established. In particular the cycle index formula of the dihedral group Dn of order 2n is expressed in terms of the cycle index formula of a cyclic group of order two C2 and the cycle index formula of the cyclic group of order n, Cn; the cycle index formula of the symmetric group Sn is expressed in terms of the cycle index formula of the alternating group An and the cycle index formula of a group generated by a cycle of length two, h(ab)i. The cycle index formula of an a ne(p) group has been derived by considering the di erent cycle types of elements of the group and expressed in terms of the cycle index formula of Cp = fx + b; where b 2 Zpg and the cycle index formula of Cp􀀀1 = fax; where 0 6= a 2 Zpg. We further extend this to a ne(q) where q is a power of a prime p and to the a ne square(p) and a ne square(q) groups. Finally, the cycle index formula of a Frobenius group is expressed in terms of the cycle index formula of the Frobenius complement H and the cycle index formula of the Frobenius kernel M. The cycle index formulas which are known such as that of the dihedral group and the symmetric group were used and the groups whose cycle index formulas are not known such as the a ne(p), a ne square(p); a ne(q) and a ne square(q) group were rst derived as part of the research. It was noted that for semidirect groups which are Frobenius such as the dihedral group Dn with an odd value of n, the a ne groups and the a ne square groups, we can fully express the cycle index of the group in terms of the cycle index formulas of the subgroups which the group is a semidirect of. However, for semidirect product groups which are not Frobenius such as the dihedral group Dn with an even value of n and the symmetric group Sn, the cycle index formula of the group cannot be expressed fully in terms of the cycle index formulas of the subgroups the group is a semidirect product of.
Determination of the Effects of Applying Variable Pressure Gradient to a Magnetohydrodynamic Fluid Flowing Between Plates with Inclined Magnetic Field
(Kenyatta University, 2018-11) Malia, Miriam
In this study the effects of applying variable pressure gradient to a MHD fluid flowing between two plates in an inclined magnetic field was carried out. It involved an unsteady hydromagnetic fluid flowing through two plates where the upper plate was porous and moving with a constant velocity in a direction that is contrary to fluid flow direction. The lower plate was non-porous and stationary. In the past years, research studies relating to MHD fluid flow through plates have been done. The information obtained from these studies have been implemented in various industrial systems for instance designing of electromagnetic meters and MHD pumps. However these research studies have been carried out when the pressure gradient was a constant and none when the pressure gradient was a variable, hence the reason why we carried out this investigation. The objective of this project was to determine the effect of applying variable pressure gradient to a MHD flow in inclined magnetic field. Also the outcome of varying Hartmann number, Suction number, Reynolds number and Eckert number on velocity and temperature profiles have been determined. The governing equations used to carry out analysis were continuity equation, Navier stokes equations, electromagnetic equations and energy equations. These equations were then non-dimensionalised. The resulting equations were non-linear thus were solved using the finite difference method. MATLAB software was then inculcated in the resulting equations which were expressed in finite difference form and resulting solutions illustrated graphically. The results obtained showed that the application of a variable pressure gradient in the presence of an inclined magnetic field resulted to an increase in velocity of the fluid and a decrease in the fluid temperature, increasing the suction number decreased the fluid velocity while the temperature increased; increasing Reynold’s number yielded to a reduction in both velocity and temperature profiles; increase in Hartmann number resulted to decrease in velocity and an increase in temperature. These velocity and thermal profiles are important in the dyeing industries and modelling of systems that help in cooling of automobile moving parts
Determining the Socio-Economic Classes Using Principal Component Analysis Based on Binary Data
(Kenyatta University, 2023-01) Nasokho, Ruth Simuli; Leo Odongo; Anthony Gichangi
Abstract

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