Research Articles (Mathematics)
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Item Automorphism groups of graph covers and uniform subset graphs(Taylor and Francis Group, 2018) Mumba, Nephtale; Mwambene, EricHofmeister considered the automorphism groups of antipodal graphs through the exploration of graph covers. In this note weextend the exploration of automorphism groups of distance preserving graph covers. We apply the technique of graph covers todetermine the automorphism groups of uniform subset graphsΓ(2k,k,k−1) andΓ(2k,k,1).The determination of automorphismgroups answers a conjecture posed by Mark Ramras and Elizabeth Donovan. They conjectured that Aut(Γ(2k,k,k−1))∼=S2k×,whereTis the complementation mapX↦→T(X)=Xc={1,2,...,2k}\X,andXis ak-subset ofΩ={1,2,...,2k}.Item Binary codes and partial permutation decoding sets from the odd graphs(Walter de Gruyter, 2014) Fish, Washiela; Fray, Roland; Mwambene, EricFor 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.Item A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer(Wiley, 2019) Munyakazi, Justin B.; Patidar, Kailash C.; Sayi, Mbani T.The objective of this paper is to construct and analyzea fitted operator finite difference method (FOFDM) forthe family of time-dependent singularly perturbed parabolicconvection–diffusion problems. The solution to the problemswe consider exhibits an interior layer due to the presence ofa turning point. We first establish sharp bounds on the solu-tion and its derivatives. Then, we discretize the time variableusing the classical Euler method. This results in a system ofsingularly perturbed interior layer two-point boundary valueproblems. We propose a FOFDM to solve the system above.Item Modeling the dynamics of an epidemic under vaccination in two interacting populations(Hindawi, 2012) Ahmed, Ibrahim H. I.; Witbooi, Peter J.; Patidar, KailashMathematical 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.Item A NSFD method for the singularly perturbed Burgers-Huxley equation(Frontiers Media, 2023) Derzie, Eshetu B.; Munyakazi, Justin B.; Dinka, Tekle G.This article focuses on a numerical solution of the singularly perturbed Burgers-Huxley equation. The simultaneous presence of a singular perturbation parameter and the nonlinearity raise the challenge of finding a reliable and efficient numerical solution for this equation via the classical numerical methods. To overcome this challenge, a nonstandard finite difference (NSFD) scheme is developed in the following manner. The time variable is discretized using the backward Euler method. This gives rise to a system of nonlinear ordinary differential equations which are then dealt with using the concept of nonlocal approximation. Through a rigorous error analysis, the proposed scheme has been shown to be parameter-uniform convergent. Simulations conducted on two numerical examples confirm the theoretical result. A comparison with other methods in terms of accuracy and computational cost reveals the superiority of the proposed scheme.Item A NSFD method for the singularly perturbed Burgers-Huxley equation(Frontiers Media, 2023) Derzie, Eshetu B.; Munyakazi, Justin B.; Dinka, Tekle G.This article focuses on a numerical solution of the singularly perturbed Burgers-Huxley equation. The simultaneous presence of a singular perturbation parameter and the nonlinearity raise the challenge of finding a reliable and e cient numerical solution for this equation via the classical numericalmethods. To overcome this challenge, a nonstandard finite dierence (NSFD) scheme is developed in the following manner. The time variable is discretized using the backward Euler method. This gives rise to a system of nonlinear ordinary dierential equations which are then dealt with using the concept of nonlocal approximation. Through a rigorous error analysis, the proposed scheme has been shown to be parameter-uniformconvergent. Simulations conducted on two numerical examples confirm the theoretical result. A comparison with other methods in terms of accuracy and computational cost reveals the superiority of the proposed scheme.Item On maximal inequalities via comparison principle(SpringerOpen, 2015) Makasu, CloudUnder certain conditions, we prove a new class of one-sided, weighted, maximal inequalities for a standard Brownian motion. Our method of proof is mainly based on a comparison principle for solutions of a system of nonlinear first-order differential equations.Item On the exact constants in one-sided maximal inequalitiesfor Bessel processes(Taylor and Francis Group, 2023) Makasu, CloudIn this paper, we establish a one-sided maximal moment inequalitywith exact constants for Bessel processes. As a consequence, weobtain an exact constant in the Burkholder-Gundy inequality. Theproof of our main result is based on a pure optimal stopping prob-lem of the running maximum process for a Bessel process. The pre-sent results extend and complement a number of related resultspreviously known in the literature.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, FrancisHIV/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.Item Protein interaction networks as metric spaces: A novel perspective on distribution of hubs(BMC, 2014) Fadhal, Emad; Gamieldien, Junaid; Mwambene, Eric CIn 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.Item Relative homotopy in relational structures(Cambridge University Press, 2018) Witbooi, PeterThe homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by B. Larose and C. Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs (X, A) where X is a poset and A is a subposet of X. We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif.Item A robust spectral method for pricing of American put options on zero-coupon bonds(Global-Science Press, 2018) Pindza, Edson; Patidar, Kailash C.American put options on a zero-coupon bond problem is reformulated as a linear complementarity problem of the option value and approximated by a nonlinear partial differential equation. The equation is solved by an exponential time differencing method combined with a barycentric Legendre interpolation and the Krylov projection algorithm. Numerical examples shows the stability and good accuracy of the method. A bond is a financial instrument which allows an investor to loan money to an entity (a corporate or governmental) that borrows the funds for a period of time at a fixed interest rate (the coupon) and agrees to pay a fixed amount (the principal) to the investor at maturity. A zero-coupon bond is a bond that makes no periodic interest payments.