# Department of Mathematics

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Item Generalizing the Hilton–Mislin genus group(Elsevier, 2001) Witbooi, Peter J.Show more For any group H, let H be the set of all isomorphism classes of groups K such that K H . For a finitely generated group H having finite commu- Ž .tator subgroup H, H , we define a group structure on H in terms of embed- dings of K into H, for groups K of which the isomorphism classes belong to Ž . H . If H is nilpotent, then the group we obtain coincides with the genus group Ž .GG H defined by Hilton and Mislin. We obtain some new results on Hilton Mislin genus groups as well as generalizations of known results.Show more Item Graphs, designs and codes related to the n-cube(Elsivier, 2009) Fish, W; Key, J D; Mwambene, EShow more For integers n 1; k 0, and k n, the graph k n has vertices the 2n vectors of Fn 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular 1 n is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k D 1; 2; 3, following the results obtained for the binary codes of the n-cube in Fish [Washiela Fish, Codes from uniform subset graphs and cyclic products, Ph.D. Thesis, University of the Western Cape, 2007] and Key and Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph Qn where n is even, in: T. Shaska, W. C Huffman, D. Joyner, V. Ustimenko (Eds.), Advances in Coding Theory and Cryptology, in: Series on Coding Theory and Cryptology, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, pp. 152 159 ]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k D 1; 2; 3, and the dimensions of the ternary codes for k D 1; 2. We also obtain 3-PD-sets for the self-dual binary codes from 2 n when n 0 .mod 4/, n 8.Show more Item Numerical treatment of Kap's equation using a class of fourth order method(Academic Journals, 2011) Akanbi, M.A.; Okunuga, S.A.; Sofoluwe, A.B.Show more Kap's equation is a stiff initial value problem. This paper deals with the treatment of Kap's equation using a class of 4th order explicit Runge-Kutta method. Numerical computation was carried out using Microsoft Visual C++. The results of the computation were found to be highly accurate and consistent with minima errors. A comparison of the results generated from the scheme was also carried out vis-a-vis some other conventional explicit Runge-Kutta formulae. The proposed class of method was found to compare favourably well.Show more Item An assessment of the age reporting in Tanzania population census(Academic Research Publishing, 2012) Mwambene, Eric; Appunni, Sathiya Susuman; Hamisi, Hamisi F.; Lougue, Siaka; Regassa, Nigatu; Ogujiuba, KanayoShow more The objective of this paper is to provide data users with a worldwide assessment of the age reporting in the Tanzania Population Census 2012 data. Many demographic and socio-economic data are age-sex attributed. However, a variety of irregularities and misstatements are noted with respect to age-related data and sex ratio data because of its biological differences between the genders. Noting the misstatement / misreporting, inconsistence of age data regardless of its significant importance in demographic and epidemiological studies, this study assess the quality of the 2012 Tanzania Population and Housing Census data relative to age. Data were downloaded from Tanzania National Bureau of Statistics. Age heaping and digit preference were measured using summary indices viz., Whipple‟s index, Myers‟ blended index, and Age-Sex Accuracy index. The recorded Whipple‟s index for both sexes was 154.43, where males had the lower index of about 152.65 while females had the higher index of about 156.07. For Myers‟ blended index, the prefrences were at digits „0‟ and „5‟ while avoidance were at digits „1‟ and „3‟ for both sexes. Finally, the age-sex index stood at 59.8 where the sex ratio score was 5.82, and the age ratio scores were 20.89 and 21.4 for males and female respectively. The evaluation of the 2012 Population Housing Censes data using the demographic techniques has qualified the data as of poor quality as a result of systematic heaping and digit preferences/avoidances in recorded age. Thus, innovative methods in data collection along with measuring and minimizing errors using statistical techniques should be used to ensure accuracy of age data.Show more Item Optimal strategy for controlling the spread of HIV/AIDS disease: A case study of South Africa(Taylor and Francis Group, 2012) Yusuf, Tunde T.; Benyah, FrancisShow more HIV/AIDS disease continues to spread alarmingly despite the huge amounts of resources invested infighting it. There is a need to integrate the series of control measures available to ensure a consistentreduction in the incidence of the disease pending the discovery of its cure. We present a deterministic modelfor controlling the spread of the disease using change in sexual habits and antiretroviral (ARV) therapyas control measures. We formulate a fixed time optimal control problem subject to the model dynamicswith the goal of finding the optimal combination of the two control measures that will minimize the costof the control efforts as well as the incidence of the disease. We estimate the model state initial conditionsand parameter values from the demographic and HIV/AIDS data of South Africa. We use Pontryagin’smaximum principle to derive the optimality system and solve the system numerically. Compared withthe practice in most resource-limited settings where ARV treatment is given only to patients with full-blown AIDS, our simulation results suggest that starting the treatment as soon as the patients progress tothe pre-AIDS stage of the disease coupled with appreciable change in the susceptible individuals’ sexualhabits reduces both the incidence and prevalence of the disease faster. In fact, the results predict thatthe implementation of the proposed strategy would drive new cases of the disease towards eradication in10 years.Show more Item Fifth order two-stage explicit Runge-Kutta-Nystrom method for the direct integration of second order ordinary differential equations(Academic Journals, 2012) Okunuga, S.A.; Sofoluwe, A.B.; Ehigie, J.O.; Akanbi, M.A.Show more In this paper a direct integration of second-order Ordinary Differential Equations (ODEs) of the form using the Explicit Runge-Kutta-Nystrom method with higher derivatives is presented. Various numerical schemes are derived and tested on standard problems. The higher-order explicit Runge-Kutta-Nystrom (HERKN) method given in this paper is compared with the conventional Explicit Runge Kutta (ERK) schemes. Due to the limitation of ERK schemes in handling stiff problems, the extension to higher order derivative is considered. The results obtained show an improvement on ERK schemes.Show more Item Modeling the dynamics of an epidemic under vaccination in two interacting populations(Hindawi, 2012) Ahmed, Ibrahim H. I.; Witbooi, Peter J.; Patidar, KailashShow more Mathematical modeling of the numerical evolution of infectious diseases has become an important tool for disease control and eradication when possible. Much work has been done on the problem of how a given population is affected by another population when there is mutual interaction. The mere presence of migrant people poses a challenge to whatever health systems are in place in a particular region. Such epidemiological phenomena have been studied extensively, described by mathematical models with suggestions for intervention strategies. The epidemiological effect of migration within the population itself was modeled for sleeping sickness in a paper 1 by Chalvet-Monfray et al. In the case of malaria, there is for instance a study 2 by Tumwiine et al. on the effect of migrating people on a fixed population. The latter two diseases are vector borne. Diseases that propagate without a vector spread perhaps more easily when introduced into a new region. Various studies of models with immigration of infectives have been undertaken for tuberculosis, see for instance 3 by Zhou et al., or the work 4 of Jia et al., and for HIV, see the paper 5 of Naresh et al.Show more Item Stability of an SEIR epidemic model with indepenent stochastic perturbations(Elsevier, 2013) Witbooi, Peter J.Show more For an epidemic model of the type mentioned, we prove a theorem on almost sure exponential stability of the disease-free equilibrium. For small values of the diffusion parameter, σ, we describe the stability of the disease free equilibrium point in terms of an appropriate analogue, Rσ , of the basic reproduction number R0 of the deterministic special case. Whenever σ > 0 then Rσ < R0. For small values of σ, the stability theorem guarantees almost sure exponential stability whenever Rσ < 1. We also discuss the effect of increasing σ.Show more Item Contour integral method for European options with jumps(Elsevier, 2013) Ngounda, Edgard; Patidar, Kailash C.; Pindza, EdsonShow more We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points.Show more Item The fibre of a pinch map in a model category(Springer Verlag, 2013) Hardie, Keith A.; Witbooi, Peter J.Show more In the category of pointed topological spaces, let F be the homotopy fibre of the pinching map X ∪ CA → X ∪ CA/ X from the mapping cone on a cofibration A → X onto the suspension of A. Gray (Proc Lond Math Soc (3) 26:497–520, 1973) proved that F is weakly homotopy equivalent to the reduced product (X, A)∞. In this paper we prove an analogue of this phenomenon in a model category, under suitable conditions including a cube axiom.Show more Item A model for control of HIV/AIDS with parental care(World Scientific Publishing, 2013) Abiodun, Gbenga Jacob; Marcus, Nizar; Okosun, Kazeem Oare; Witbooi, Peter J.Show more In this study we investigate the HIV/AIDS epidemic in a population which experiences a significant flow of immigrants. We derive and analyze a mathematical model that describes the dynamics of HIV infection among the immigrant youths and how parental care can minimize or prevent the spread of the disease in the population. We analyze the model with both screening control and parental care, then investigate its stability and sensitivity behavior. We also conduct both qualitative and quantitative analyses. It is observed that in the absence of infected youths, disease-free equilibrium is achievable and is globally asymptotically stable. We establish optimal strategies for the control of the disease with screening and parental care, and provide numerical simulations to illustrate the analytic results.Show more Item A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems(Spring Verlag, 2013) Munyakazi, Justin B.; Patidar, Kailash C.Show more This paper treats a time-dependent singularly perturbed reaction-diffusion problem. We semidiscretize the problem in time by means of the classical backward Euler method. We develop a fitted operator finite difference method (FOFDM) to solve the resulting set of linear problems (one at each time level). We prove that the underlying fitted operator satisfies the maximum principle. This result is then used in the error analysis of the FOFDM. The method is shown to be first order convergent in time and second order convergent in space, uniformly with respect to the perturbation parameter. We test the method on several numerical examples to confirm our theoretical findings.Show more Item An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection(De Gruyter Open, 2013) Obaid, Hasim; Ouifki, Rachid; Patidar, Kailash C.Show more We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.Show more Item Multiplication of Crowns(Instituto de Matematica de Bahia Blanca (INMABB-CONICET), 2013) Witbooi, Peter J.Show more It is known that the only nite topological spaces that are H-spaces are the discrete spaces. For a nite poset which is weakly equivalent to an H-space, a generalized multiplication may be found after suitable sub-division. In this paper we construct minimal models of the k-fold generalised multiplications of circles in the category of relational structures, including poset models. In particular, we obtain higher dimensional analogues of a cer-tain ternary multiplication of crownsShow more Item Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options(Kent State University, 2013) Pindza, Edson; Patidar, Kailash C.; Ngounda, EdgardShow more In this paper we present a robust numerical method to solve several types of European style option pricing problems. The governing equations are described by variants of Black-Scholes partial differential equations (BS-PDEs) of the reaction-diffusion-advection type. To discretise these BS-PDEs numerically, we use the spectral methods in the asset (spatial) direction and couple them with a third-order implicit-explicit predictor-corrector (IMEX-PC) method for the discretisation in the time direction. The use of this high-order time integration scheme sustains the better accuracy of the spectral methods for which they are well-known. Our spectral method consists of a pseudospectral formulation of the BS-PDEs by means of an improved Lagrange formula. On the other hand, in the IMEX-PC methods, we integrate the diffusion terms implicitly whereas the reaction and advection terms are integrated explicitly. Using this combined approach, we first solve the equations for standard European options and then extend this approach to digital options, butterfly spread options, and European calls in the Heston model. Numerical experiments illustrate that our approach is highly accurate and very efficient for pricing financial options such as those described above.Show more Item Performance of Richardson extrapolation on some numerical methods for a singularly perturbed turning point problem whose solution has boundary layers(KOREAN MATHEMATICAL SOC, 2014) Munyakazi, Justin B.; Patidar, Kailash C.Show more Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used.Show more Item A robust spectral method for solving Heston’s model(Springer Verlag, 2014) Ngounda, E.; Patidar, Kailash C.; Pindza, E.Show more In this paper, we consider the Heston’s volatility model (Heston in Rev. Financ. Stud. 6: 327–343, 1993]. We simulate this model using a combination of the spectral collocation method and the Laplace transforms method. To approximate the two dimensional PDE, we construct a grid which is the tensor product of the two grids, each of which is based on the Chebyshev points in the two spacial directions. The resulting semi-discrete problem is then solved by applying the Laplace transform method based on Talbot’s idea of deformation of the contour integral (Talbot in IMA J. Appl. Math. 23(1): 97–120, 1979).Show more Item Protein interaction networks as metric spaces: A novel perspective on distribution of hubs(BMC, 2014) Fadhal, Emad; Gamieldien, Junaid; Mwambene, Eric CShow more In the post-genomic era, a central and overarching question in the analysis of protein-protein interaction networks continues to be whether biological characteristics and functions of proteins such as lethality, physiological malfunctions and malignancy are intimately linked to the topological role proteins play in the network as a mathematical structure. One of the key features that have implicitly been presumed is the existence of hubs, highly connected proteins considered to play a crucial role in biological networks. We explore the structure of protein interaction networks of a number of organisms as metric spaces and show that hubs are non randomly positioned and, from a distance point of view, centrally located.Show more Item An optimal portfolio and capital management strategy for basel III compliant commercial banks(Hindawi Publishing Corporation, 2014) Muller, Grant E.; Witbooi, Peter J.Show more We model a Basel III compliant commercial bank that operates in a financial market consisting of a treasury security, a marketable security, and a loan and we regard the interest rate in the market as being stochastic. We find the investment strategy that maximizes an expected utility of the bank’s asset portfolio at a future date. This entails obtaining formulas for the optimal amounts of bank capital invested in different assets. Based on the optimal investment strategy, we derive a model for the Capital Adequacy Ratio (CAR), which the Basel Committee on Banking Supervision (BCBS) introduced as a measure against banks’ susceptibility to failure. Furthermore, we consider the optimal investment strategy subject to a constant CAR at the minimum prescribed level. We derive a formula for the bank’s asset portfolio at constant (minimum) CAR value and present numerical simulations on different scenarios. Under the optimal investment strategy, the CAR is above the minimum prescribed level. The value of the asset portfolio is improved if the CAR is at its (constant) minimum value.Show more Item Binary codes and partial permutation decoding sets from the odd graphs(Walter de Gruyter, 2014) Fish, Washiela; Fray, Roland; Mwambene, EricShow more For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ω{k} , the set of all k-subsets of Ω = {1, 2, . . . , 2k + 1}, and any two of its vertices u and v constitute an edge [u, v] if and only if u ∩ v = ∅. In this paper the binary code generated by the adjacency matrix of O(k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k 4 is determined. Lastly, the relationship between the dual code from O(k) and the code from its graph-theoretical complement O(k), is investigated.Show more