# Masters Degrees (Mathematics and Computer Science Education)

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Item A comparative study of collocation methods for the numerical solution of differential equations.(2008) Kajotoni, Margaret Modupe.; Parumasur, Nabendra.; Singh, Pravin.Show more The collocation method for solving ordinary differential equations is examined. A detailed comparison with other weighted residual methods is made. The orthogonal collocation method is compared to the collocation method and the advantage of the former is illustrated. The sensitivity of the orthogonal collocation method to different parameters is studied. Orthogonal collocation on finite elements is used to solve an ordinary differential equation and its superiority over the orthogonal collocation method is shown. The orthogonal collocation on finite elements is also used to solve a partial differential equation from chemical kinetics. The results agree remarkably with those from the literature.Show more Item Algebraic graph theoretic applications to cryptography.(2015) Mafunda, Sonwabile Templeton.; Amery, Gareth.; Mukwembi, Simon.; Swart, Christine Scott.Show more Abstract available in PDF file.Show more Item Analysis of co-infection of human immunodeficiency virus with human papillomavirus.(2014) Maregere, Bothwell.; Chirove, Faraimunashe.Show more We formulate a deterministic mathematical model for the co-infection of HPV with HIV without treatment. Mathematical techniques were used to analyze the stability of the models in terms of basic reproduction numbers for disease-free equilibrium point and fixed point theory used for analysis of the endemic equilibrium point. The model incorporating HIV and HPV co-infection sought to investigate the impact of HIV infection in the natural history of HPV infection, and the impact of HPV infection in the natural history of HIV infection, over a period of time. Numerical simulations were carried out to illustrate the trends of progression of HIV and HPV in the case of co-infection. The results from our study showed that when both HIV and HPV infected individuals are active in the system then co-infection grows faster compared to one infection which is active in the system. Our study also showed that when we started with HPV infection in the community and introduces HIV infection after sometime has more impact in the growth of co-infection population compared to start with HIV infection and introduces HPV infection after sometime in the community.Show more Item The application of artifacts in the teaching and learning of grade 9 geometry.(2005) Jojo, Zingiswa Mybert Monica.; Brijlall, Deonarain.; Maharaj, A.Show more The main focus of the study was to explore how the experiences that the learners went through in the Technology class during the construction and design of artifacts, could be used to inform the teaching of Geometry in the mainstream Mathematics classes. It was important to find out how the teaching of Geometry would allow the learners to both reflect and utilize the Geometry they know, as a starting point or springboard for further study of Geometry. Data was collected through observations, structured and semi-structured interviews of a sample of twenty grade 9 learners of Mashesha Junior Secondary School of Margate in KwaZulu Natal. It was collected through observation of drawings and completely constructed double-storey artifacts at different intervals of designing. Observations and notes on every activity done by the learners for example, measurements, comparisons, estimations, scaling, drawings use of symmetry and perspective drawing were kept and analyzed. Data for the interviews was collected in the form of drawings, photographs, transcriptions of video and audiotapes. The observations in particular were looking for the Geometry in finished artifacts. Interviews with the learners were directed at how each learner started drawing a house to the finish. When and how scale drawing, projections, angles made and length preservation were used by the learner, was of utmost importance. It is believed that grade 9 learners of Mashesha have Geometric experiences which can be used to inform the teaching of Geometry in mainstream mathematics. It was found that this experience brought by the learners from the Technology construction of artifacts could cause the learners to find mainstream mathematics interesting and challenging. It is also believed that the use of projective Geometry already employed by the learners can be incorporated in mainstream mathematics so as to improve how learners understand Euclidean Geometry. In this way, it is believed, that the teaching of Geometry will allow the learners to utilize and reflect the Geometry already known to them. This Geometry would therefore be used as a starting point for further study of Geometry. Suggestions for further research and recommendations for the improvement of Geometry teaching and learning have also been made.Show more Item Application of bivariate spectral quasilinearization method to second grade fluid flow equations.(2020) Dlongolo, Simphiwe Gloria.; Sibanda, Precious.; Goqo, Sicelo Praisegod.Show more In this study, the steady flow of a second grade magnetohydrodynamic fluid in a porous channel is investigated. We further investigate the hydromagnetic flow of a second grade fluid over a stretching sheet. The partial differential equations that describe the flows are solved numerically using the bivariate spectral quasilinearization method. The method is extended to a system of non-similar partial differential equations that model the steady two dimensional flow of Falkner-Skan flow of an incompressible second grade nano fluid. The work is also concerned with heat and the mass transfer from the electrically conducting second grade magnetohydrodynamic fluid over a stretching sheet. The sensitivity of the flow characteristics with respect to the second grade fluid parameter, magnetic field parameter, thermal radiation parameter, and the chemical reaction parameter are investigated. The accuracy of the numerical method is determined using the residual error analysis.Show more Item The application of the multigrid algorithm to the solution of stiff ordinary differential equations resulting from partial differential equations.(1992) Parumasur, Nabendra.; Mika, Janusz R.Show more We wish to apply the newly developed multigrid method [14] to the solution of ODEs resulting from the semi-discretization of time dependent PDEs by the method of lines. In particular, we consider the general form of two important PDE equations occuring in practice, viz. the nonlinear diffusion equation and the telegraph equation. Furthermore, we briefly examine a practical area, viz. atmospheric physics where we feel this method might be of significance. In order to offer the method to a wider range of PC users we present a computer program, called PDEMGS. The purpose of this program is to relieve the user of much of the expensive and time consuming effort involved in the solution of nonlinear PDEs. A wide variety of examples are given to demonstrate the usefulness of the multigrid method and the versatility of PDEMGS.Show more Item Applications of symmetry analysis to physically relevant differential equations.(2005) Kweyama, Mandelenkosi Christopher.; Govinder, Keshlan Sathasiva.; Maharaj, Sunil Dutt.Show more We investigate the role of Lie symmetries in generating solutions to differential equations that arise in particular physical systems. We first provide an overview of the Lie analysis and review the relevant symmetry analysis of differential equations in general. The Lie symmetries of some simple ordinary differential equations are found t. illustrate the general method. Then we study the properties of particular ordinary differential equations that arise in astrophysics and cosmology using the Lie analysis of differential equations. Firstly, a system of differential equations arising in the model of a relativistic star is generated and a governing nonlinear equation is obtained for a linear equation of state. A comprehensive symmetry analysis is performed on this equation. Secondly, a second order nonlinear ordinary differential equation arising in the model of the early universe is described and a detailed symmetry analysis of this equation is undertaken. Our objective in each case is to find explicit Lie symmetry generators that may help in analysing the model.Show more Item Aspects of spherically symmetric cosmological models.(1998) Moodley, Kavilan.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.Show more In this thesis we consider spherically symmetric cosmological models when the shear is nonzero and also cases when the shear is vanishing. We investigate the role of the Emden-Fowler equation which governs the behaviour of the gravitational field. The Einstein field equations are derived in comoving coordinates for a spherically symmetric line element and a perfect fluid source for charged and uncharged matter. It is possible to reduce the system of field equations under different assumptions to the solution of a particular Emden-Fowler equation. The situations in which the Emden-Fowler equation arises are identified and studied. We analyse the Emden-Fowler equation via the method of Lie point symmetries. The conditions under which this equation is reduced to quadratures are obtained. The Lie analysis is applied to the particular models of Herlt (1996), Govender (1996) and Maharaj et al (1996) and the role of the Emden-Fowler equation is highlighted. We establish the uniqueness of the solutions of Maharaj et al (1996). Some physical features of the Einstein-Maxwell system are noted which distinguishes charged solutions. A charged analogue of the Maharaj et al (1993) spherically symmetric solution is obtained. The Gutman-Bespal'ko (1967) solution is recovered as a special case within this class of solutions by fixing the parameters and setting the charge to zero. It is also demonstrated that, under the assumptions of vanishing acceleration and proper charge density, the Emden-Fowler equation arises as a governing equation in charged spherically symmetric models.Show more Item A case study: the use of GeoGebra to alleviate learner difficulty in learning the similarity of triangles in a South African grade 9 classroom.(2022) Mpanza, Nompumelelo.; Shongwe, Michael Bafana Mthembiseni.Show more The mixed methods study investigated the use of GeoGebra as a dynamic geometry software (DGS) to alleviate the learning similarity of triangles. A mixed methods philosophical framework in the form of an exploratory case study was used to conveniently and purposively select a sample of 60 Grade 9 learners enrolled at Sondelani Full-Service School (pseudonym), a township school in the Pinetown district in KwaZulu–Natal province, South Africa. During this research, GeoGebra and the concept of similarity of triangles were introduced to the participants. Then, participants answered several (Euclidean geometry) Similarity Achievement Test (SAT) questions prescribed by the National Mathematics pacesetter for Grade 9 and 10. A 10-item Likert scale questionnaire intended to elicit participants’ attitudes about GeoGebra and its impact on Euclidean geometry and mathematics was administered to these participants. The questionnaire also included four open-ended items, asking participants to reflect on the application of GeoGebra. The analysis of SAT data revealed that performance was higher after GeoGebra instruction (𝑀 = 22.50) than during traditional instruction, which did not feature GeoGebra (𝑀 = 11.65). Thus, it was found that the use of GeoGebra is an appropriate tool to increase achievement in learning geometry concepts; to promote accuracy, visualization; learner participation; and to create enjoyment and learner interest towards learning mathematics. It is recommended that mathematics teachers need to use GeoGebra for effective teaching and learning of similarity of triangles.Show more Item Character tables of the general linear group and some of its subgroups(2008) Basheer, Ayoub Basheer Mohammed.; Moori, Jamshid.Show more The aim of this dissertation is to describe the conjugacy classes and some of the ordinary irreducible characters of the nite general linear group GL(n, q); together with character tables of some of its subgroups. We study the structure of GL(n, q) and some of its important subgroups such as SL(n, q); UT(n, q); SUT(n, q); Z(GL(n, q)); Z(SL(n, q)); GL(n, q)0 ; SL(n, q)0 ; the Weyl group W and parabolic subgroups P : In addition, we also discuss the groups PGL(n, q); PSL(n, q) and the a ne group A (n, q); which are related to GL(n, q): The character tables of GL(2; q); SL(2; q); SUT(2; q) and UT(2; q) are constructed in this dissertation and examples in each case for q = 3 and q = 4 are supplied. A complete description for the conjugacy classes of GL(n, q) is given, where the theories of irreducible polynomials and partitions of i 2 f1; 2; ; ng form the atoms from where each conjugacy class of GL(n, q) is constructed. We give a special attention to some elements of GL(n, q); known as regular semisimple, where we count the number and orders of these elements. As an example we compute the conjugacy classes of GL(3; q): Characters of GL(n, q) appear in two series namely, principal and discrete series characters. The process of the parabolic induction is used to construct a large number of irreducible characters of GL(n, q) from characters of GL(n, q) for m < n: We study some particular characters such as Steinberg characters and cuspidal characters (characters of the discrete series). The latter ones are of particular interest since they form the atoms from where each character of GL(n, q) is constructed. These characters are parameterized in terms of the Galois orbits of non-decomposable characters of F q n: The values of the cuspidal characters on classes of GL(n, q) will be computed. We describe and list the full character table of GL(n, q): There exists a duality between the irreducible characters and conjugacy classes of GL(n, q); that is to each irreducible character, one can associate a conjugacy class of GL(n, q): Some aspects of this duality will be mentioned.Show more Item Chebyshev spectral pertutrbation based method for solving nonlinear fluid flow problems.(2014) Agbaje, Titilayo Morenike.; Motsa, Sandile Sydney.Show more In this dissertation, a modi cation of the classical perturbation techniques for solving nonlinear ordinary di erential equation (ODEs) and nonlinear partial di erential equations (PDEs) is presented. The method, called the Spectral perturbation method (SPM) is a series expansion based technique which extends the use of the standard perturbation scheme when combined with the Chebyshev spectral method. The SPM solves a sequence of equations generated by the perturbation series approximation using the Chebyshev spectral methods. This dissertation aims to demonstrate that, in contrast to the conclusions earlier drawn by researchers about perturbation techniques, a perturbation approach can be e ectively used to generate accurate solutions which are de ned under the Williams and Rhyne (1980) transformation. A quasi-linearisation technique, called the spectral quasilinearisation method (SQLM) is used for validation purpose. The SQLM employs the quasilinearisation approach to linearise nonlinear di erential equations and the resulting equations are solved using the spectral methods. Furthermore, a spectral relaxation method (SRM) which is a Chebyshev spectral collocation based method that decouples and rearrange a system of equations in a Gauss - Seidel manner is also presented. In the SRM, the di erential equations are decoupled, rearranged and the resulting sequence of equations are numerically integrated using the Chebyshev spectral collocation method. The techniques were used to solve mathematical models in uid dynamics. This study consists of an introductory chapter which gives the description of the methods and a brief overview of the techniques used in developing the SPM, SQLM and the SRM. In Chapter 2, the SPM is used to solve the equations that model magnetohydrodynamics (MHD) stagnation point ow and heat transfer problem from a stretching sheet in the presence of heat source/sink and suction/injection in porous media. Using similarity transformations, the governing partial differential equations are transformed into ordinary di erential equations. Series solutions for small velocity ratio and asymptotic solutions for large velocity ratio were generated and the results were also validated against those obtained using the SQLM. In Chapter 3, the SPM was used to solve the momentum, heat and mass transfer equations describing the unsteady MHD mixed convection ow over an impulsively stretched vertical surface in the presence of chemical reaction e ect. The governing partial di erential equations are reduced into a set of coupled non similar equations and then solved numerically using the SPM. In order to demonstrate the accuracy and e ciency of the SPM, the SPM numerical results are compared with numerical results generated using the SRM and a good agreement between the two methods was observed up to eight decimal digits which is a reasonable level of accuracy. Several simulation are conducted to ascertain the accuracy of the SPM and the SRM. The computational speed of the SPM is demonstrated by comparing the SPM computational time with the SRM computational time. A residual error analysis is also conducted for the SPM and the SRM, in order to further assess the accuracy of the SPM. In Chapter 4, the SPM was used to solve the equations modelling the unsteady three-dimensional MHD ow and mass transfer in a porous space previously reported in literature. E ciency and accuracy of the SPM is shown by validating the SPM results against the results obtained using the SRM and the results were found to be in good agreement. The computational speed of the SPM is demonstrated by comparing the SPM and the SRM computational time. In order to further assess the accuracy of the SPM, a residual error analysis is conducted for the SPM and the SRM. In Chapter 2, we show that the SPM can be used as an alternative to the standard perturbation methods to get numerical solutions for strongly nonlinear boundary value problems. Also, it is demonstrated in Chapter 2 that the SPM is e cient even in the case where the perturbation parameter is large, as the convergence rate is seen to improve with increase in the large parameter value. In Chapters 3 and 4, the study shows that SPM is more e cient in terms of computational speed when compared with the SRM. The study also highlighted that the SPM can be used as an e cient and reliable tool for solving strongly nonlinear partial di erential equations de ned under the Williams and Rhyne (1980) transformation. In addition, the study shows that accurate results can be obtained using the perturbation method and thus, the conclusions earlier drawn by researchers regarding the accuracy of the perturbation method is corrected.Show more Item A classical approach for the analysis of generalized linear mixed models.(2004) Hammujuddy, Mohammad Jahvaid. ; Matthews, Glenda Beverley.Show more Generalized linear mixed models (GLMMs) accommodate the study of overdispersion and correlation inherent in hierarchically structured data. These models are an extension of generalized linear models (GLMs) and linear mixed models (LMMs). The linear predictor of a GLM is extended to include an unobserved, albeit realized, vector of Gaussian distributed random effects. Conditional on these random effects, responses are assumed to be independent. The objective function for parameter estimation is an integrated quasi-likelihood (IQL) function which is often intractable since it may consist of high-dimensional integrals. Therefore, an exact maximum likelihood analysis is not feasible. The penalized quasi-likelihood (PQL) function, derived from a first-order Laplace expansion to the IQL about the optimum value of the random effects and under the assumption of slowly varying weights, is an approximate technique for statistical inference in GLMMs. Replacing the conditional weighted quasi-deviance function in the Laplace-approximated IQL by the generalized chi-squared statistic leads to a corrected profile quasilikelihood function for the restricted maximum likelihood (REML) estimation of dispersion components by Fisher scoring. Evaluation of mean parameters, for fixed dispersion components, by iterative weighted least squares (IWLS) yields joint estimates of fixed effects and random effects. Thus, the PQL criterion involves repeated fitting of a Gaussian LMM with a linked response vector and a conditional iterated weight matrix. In some instances, PQL estimates fail to converge to a neighbourhood of their true values. Bias-corrected PQL estimators (CPQL) have hence been proposed, using asymptotic analysis and simulation. The pseudo-likelihood algorithm is an alternative estimation procedure for GLMMs. Global score statistics for hypothesis testing of overdispersion, correlation and heterogeneity in GLMMs has been developed as well as individual score statistics for testing null dispersion components separately. A conditional mean squared error of prediction (CMSEP) has also been considered as a general measure of predictive uncertainty. Local influence measures for testing the robustness of parameter estimates, by inducing minor perturbations into GLMMs, are recent advances in the study of these models. Commercial statistical software is available for the analysis of GLMMs.Show more Item A classification of second order equations via nonlocal transformations.(2000) Edelstein, R. M.; Govinder, Keshlan Sathasiva.Show more The study of second order ordinary differential equations is vital given their proliferation in mechanics. The group theoretic approach devised by Lie is one of the most successful techniques available for solving these equations. However, many second order equations cannot be reduced to quadratures due to the lack of a sufficient number of point symmetries. We observe that increasing the order will result in a third order differential equation which, when reduced via an alternate symmetry, may result in a solvable second order equation. Thus the original second order equation can be solved. In this dissertation we will attempt to classify second order differential equations that can be solved in this manner. We also provide the nonlocal transformations between the original second order equations and the new solvable second order equations. Our starting point is third order differential equations. Here we concentrate on those invariant under two- and three-dimensional Lie algebras.Show more Item Closure operators on complete lattices with application to compactness.(1995) Brijlall, Deonarain.; Sturm, Teo.; Jordens, Olav.Show more No abstract available.Show more Item Cluster mass reconstruction via gravitational lensing.(2009) Musonda, Ededias.; Amery, Gareth.Show more The presence of massive objects is detectable in observations via the gravitational lensing effect on light from more distant sources. From this effect it is possible to reconstruct the masses of clusters, and the distribution of matter within the cluster. However, further theoretical work needs to be done to properly contextualize any proposed projects involving, for instance, SALT data sets. Observational lensing studies use one of two techniques to recover the lens mass distribution: parametric (model dependent) techniques; and, a more recent innovation, non-parametric methods. The latter deserves further study as a tool for cluster surveys. To this end, we provide a comprehensive analysis of existing non-parametric algorithms and software, as well as estimates on the likely errors to be expected when used as an astronomical tool.Show more Item Codes of designs and graphs from finite simple groups.(2002) Rodrigues, Bernardo Gabriel.; Moori, Jamshid.; Key, Jennifer Denise.Show more No abstract available.Show more Item Completion of uniform and metric frames.(1996) Murugan, Umesperan Goonaselan.; Baboolal, Deeva Lata.Show more The term "frame" was introduced by C H Dowker, who studied them in a long series of joint papers with D Papert Strauss. J R Isbell , in a path breaking paper [1972] pointed out the need to introduce separate terminology for the opposite of the category of Frames and coined the term "locale". He was the progenitor of the idea that the category of Locales is actually more convenient in many ways than the category of Frames. In fact, this proves to be the case in one of the approaches adopted in this thesis. Sublocales (quotient frames) have been studied by several authors, notably Dowker and Papert [1966] and Isbell [1972]. The term "sublocale" is due to Isbell, who also used "part " to mean approximately the same thing. The use of nuclei as a tool for studying sublocales (as is used in this thesis) and the term "nucleus" itself was initiated by H Simmons [1978] and his student D Macnab [1981]. Uniform spaces were introduced by Weil [1937]. Isbell [1958] studied algebras of uniformly continuous functions on uniform spaces. In this thesis, we introduce the concept of a uniform frame (locale) which has attracted much interest recently and here too Isbell [1972] has some results of interest. The notion of a metric frame was introduced by A Pultr [1984]. The main aim of his paper [11] was to prove metrization theorems for pointless uniformities. This thesis focuses on the construction of completions in Uniform Frames and Metric Frames. Isbell [6] showed the existence of completions using a frame of certain filters. We describe the completion of a frame L as a quotient of the uniformly regular ideals of L, as expounded by Banaschewski and Pultr[3]. Then we give a substantially more elegant construction of the completion of a uniform frame (locale) as a suitable quotient of the frame of all downsets of L. This approach is attributable to Kriz[9]. Finally, we show that every metric frame has a unique completion, as outlined by Banaschewski and Pultr[4]. In the main, this thesis is a standard exposition of known, but scattered material. Throughout the thesis, choice principles such as C.D.C (Countable Dependent Choice) are used and generally without mention. The treatment of category theory (which is used freely throughout this thesis) is not self-contained. Numbers in brackets refer to the bibliography at the end of the thesis. We will use 0 to indicate the end of proofs of lemmas, theorems and propositions. Chapter 1 covers some basic definitions on frames , which will be utilized in subsequent chapters. We will verify whatever we need in an endeavour to enhance clarity. We define the categories, Frm of frames and frame homomorphisms, and Lac the category of locales and frame morphisms. Then we explicate the adjoint situation that exists between Frm and Top , the category of topological spaces and continuous functions. This is followed by an introduction to the categories, RegFrm of all regular frames and frame homomorphisms, and KRegFrm the category of compact regular frames and their homomorphisms. We then present the proofs of two very important lemmas in these categories. Finally, we define the compactification of and a congruence on a frame. In Chapter 2 we recall some basic definitions of covers, refinements and star refinements of covers. We introduce the notion of a uniform frame and define certain mappings (morphisms) between uniform frames (locales) . In the terminology of Banaschewski and Kriz [9] we define a complete uniform frame and the completion of a uniform frame. The aim of Chapter 3 is twofold : first, to construct the compact regular coreflection of uniform frames , that is, the frame counterpart of the Samuel Compactification of uniform spaces [12] , and then to use it for a description of the completion of a uniform frame as an alternative to that previously given by Isbell[6]. The main purpose of Chapter 4 is to provide another description of uniform completion in frames (locales), which is in fact even more straightforward than the original topological construction. It simply consists of writing down generators and defining relations. We provide a detailed examination of the main result in this section, that is, a uniform frame L is complete of each uniform embedding f : (M,UM) -t (L,UL) is closed, where UM and UL denote the uniformities on the frames M and L respectively. Finally, in Chapter 5, we introduce the notions of a metric diameter and a metric frame. Using the fact that every metric frame is a uniform frame and hence has a uniform completion, we show that every metric frame L has a unique completion : CL - L.Show more Item Computer analysis of equations using Mathematica.(2001) Jugoo, Vikash Ramanand.; Govinder, Keshlan Sathasiva.; Maharaj, Sunil Dutt.Show more In this thesis we analyse particular differential equations that arise in physical situations. This is achieved with the aid of the computer software package called Mathematica. We first describe the basic features of Mathematica highlighting its capabilities in performing calculations in mathematics. Then we consider a first order Newtonian equation representing the trajectory of a particle around a spherical object. Mathematica is used to solve the Newtonian equation both analytically and numerically. Graphical plots of the trajectories of the planetary bodies Mercury, Earth and Jupiter are presented. We attempt a similar analysis for the corresponding relativistic equation governing the orbits of gravitational objects. Only numerical results are possible in this case. We also perform a perturbative analysis of the relativistic equation and determine the amount of perihelion shift. The second equation considered is the Emden-Fowler equation of order two which arises in many physical problems, including certain inhomogeneous cosmological applications. The analytical features of this equation are investigated using Mathematica and the Lie analysis of differential equations. Different cases of the related autonomous form of the Emden-Fowler equation are investigated and graphically represented. Thereafter, we generate a number of profiles of the energy density and the pressure for a particular solution which demonstrates that a numerical approach for studying inhomogeneity, in cosmological models in general relativity, is feasible.Show more Item Congestion control based on dynamics pricing scheme and service class-based joint call admission control in heterogeneous wireless networks.(2013) Mafuta, Armeline Dembo.; Adewumi, Aderemi Oluyinka.Show more Next Generation of Wireless Networks (NGWNs) are heterogeneous and consist of several Radio Access Technologies (RATs) that coexist in the same geographical area. This heterogeneity of wireless networks is supposed to support multiple mobile terminal calls coming simultaneously to the RATs. NGWNs have to handle the Quality of Services (QoS) of any incoming user calls and manage the in ow of calls into the RATs. Congestion problem arises wherever there are multiple incoming user calls; especially during peak hours of the day. Several attempts have been made, as extracted from literature, to control this problem. This research is also a study that seeks to proffer solutions to improve congestion control in Heterogeneous Wireless Networks (HWN). Recent techniques for solving the congestion control problem are the application of the dynamic pricing and the Joint Call Admission Control (JCAC) algorithm. Dynamic pricing proposes incentives to users by increasing or decreasing the price of calls to encourage users to make calls during the off peak period while discouraging users from making calls during the peak period in a day. The Service Class-based JCAC (SCJCAC) algorithm is a technique that admits calls into a suitable RAT, based on the classes of services in such a way that different RATs are optimized in order to support the different classes of services. These two methods are used together to reduce congestion in the HWN. In this research, two recent dynamic pricing for congestion control are investigated, these schemes are compared and furthermore, a SCJCAC algorithm is proposed and modelled by using the multi-dimensional Markov process model for controlling congestion during the peak hours of the day in the HWNs. The simulation evaluates the performance of the proposed SCJCAC algorithm, while the two dynamic pricing schemes are also compared to the at-pricing scheme during the peak hours.Show more Item Connectedness and the hyperspace of metric spaces.(2015) Rathilal, Cerene.; Matthews, Glenda Beverley.; Molenberghs, Geert.Show more One of the prime motivations for studying hyperspaces of a metric space is to understand the original space itself. The hyperspace of a metric space X is the space 2X of all non-empty closed bounded subsets of it, endowed with the Hausdorff metric. Our purpose is to study, in particular, connectedness properties of X and its hyperspace. We shall be concerned with knowing if a property P is extensional, that is, if X has property P then so does the hyperspace, or if a property is P is re ective, that is, if the hyperspace has property P then so does X itself. The hyperspace 2X and its subspace C(X) will be the focus of our study. First the Hau- dorff metric, p, is considered and introduced for the hyperspace 2X which is also inherited by C(X). As in (Nadler; [8]), when X is a continuum, the property of compactness is shown to be extensional to 2X and C(X). This is further generalised, when it is shown that each of 2X and C(X) is arcwise connected and hence are each arcwise connected continua, when X is a continuum. The classical results, the Boundary Bumping Theorems (due to Janiszewski [4]), which provide the required conditions under which the component of a set intersects its boundary, is proved using the Cut Wire Theorem (Whyburn; [13]). As an ap- plication, the Boundary Bumping Theorem (for open sets) is used to show the existence of continua arising out of convergence, in the Continuum of Convergence Theorem(Nadler; [8]). Using a construction of Whitney( [12]), the existence of a Whitney map, , for 2X and ! for C(X) are given. Using u, a special function o : [0; 1] -! 2X (due to Kelley [3]) called a segment is considered in the study of the arc structure of 2X and C(X). The equivalence of the existence of an order arc in 2X and the existence of a segment in 2X is also shown. A segment homotopy is then utilised to show that if one of 2X or C(X) is contractible then so is the other. This is presented in the Fundamental Theorem of Contractible Hyperspaces. The relationship between local connectedness and connectedness im kleinen is examined in order to understand the properties of Peano continua. Property S, introduced by Sierpin- ski( [10]), is considered and its connection to local connectedness is examined. Furthermore, a result of Wojdyslawski( [15]), which shows that local connectedness is an extensional prop- erty of a continuum X to the hyperspaces 2X and C(X), is given. Local connectedness is also re ective if either 2X or C(X) is a locally connected metric continuum. Lastly, Property K, by Kelley( [3]) is examined and is shown to be a sufficient condition for a continuum X to have its hyperspaces 2X and C(X) to be contractible. Consequently, if X is a Peano continuum then 2X and C(X) are contractible.Show more