Mathematical Science

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Active leakage management with bayesian networks
(University of Zululand, 2021) Silwimba, Felix
Despite the existence of different active leakage management methods and models, their implementation in most developing countries’ Water distribution systems (WDS’s) remain limited. This is attributed to some limiting challenges to leakage management faced by these WDS. Some of these include; ﬁnancial constraints, poor record-keeping systems, and inadequate technicalskillsandtechnology. Thisleadsthesedistributionsystemstoadopt passive leakage management approaches that increase water losses and risk the destruction of neighboring infrastructure and water contamination. This study therefore presents a pipe leak monitoring and optimal maintenance sequencedeterminingmodellingframeworkthatisapplicableevenindistribution systems with limited data. The framework comprises of three models: a data adaptive Bayesian network(BN)modelforpredictingpipeleakprobabilitiesusedforleakagemonitoring, a water loss estimation model for estimating pipe leak water losses and a linear programming model in which water loss estimates and pipe leak predictions are used to determine the optimal pipe leak maintenance sequence. Fromtheassessmentandimplementationexampleofthemodelingframework which used simulated data, results indicate that the modeling framework is suitable for WDS with limited data.
BIANCHI-ICOSMOLOGICALMODEL IN f(R,T) GRAVITY
(University of Zululand, 2020) Jokweni, Siwaphiwe
Cosmology today is often described as being a ‘precision’ science, which reﬂects that cosmology had not always been seen as such. The enormous theoretical and experimental data from the cosmic microwave background (CMB), type Ia supernovae (SNe Ia), the Wilkinson Microwave Anisotropy Probe (WMAP), large scale structure (LSS), gravitational lensing, the Sloan Digital Sky Survey (SDSS), baryonic acoustic oscillations (BAO), and PLANCK, has drastically ameliorated cosmology; thus, providing a deeper understanding of the universe. These observations suggest that the universe is currently undergoing an accelerated expansion, where two thirds of its critical energy densityisreservedintheformofanenergycalleddarkenergy(DE).Thisenergyisusually associated with a cosmological constant, and the resulting standard cosmological model is called the ΛCDM model. This humongous episode confronted the fundamental theories of cosmology and astrophysics. Due to some shortcomings of the ΛCDM model, various alternatives have been proposed, which include modiﬁcations of general relativity itself, by imposing extra terms in the Einstein-Hilbert action (EH), or by considering dynamical candidates. These modiﬁed theories of gravity include GaussBonnet, f(G),higherderivative(HD)theories, f(R) theories, f(T) and f(R,T) gravity theories, while dynamical candidates include the cosmological constant, quintessence, phantom, quintom, k-essence, tachyon and Chaplygin gas, among others. Thoughthepresentuniverseishomogenousandisotropic,theoreticalstudiesandobservationaldatasupporttheexistenceofananisotropicphaseatearlyevolution,leading to the consideration of anisotropic- background models of the universe. Many authors have explored the features of modiﬁed theories of gravity in anisotropic background to studytheearlyuniverse. Amongstthevariousfamiliesofhomogeneous,butanisotropic geometries, the most well-known are the Bianchi type I -IX space-time line elements. However,earlierstudiesonthepossibleeffectsofanisotropicuniversemaketheBianchi type-I model a prime alternative. In particular, a locally-rotationally-symmetric (LRS) xii or a plane symmetric spacetime is the simplest version of Bianchi-I models. Most studies on standard gravity, as well as on modiﬁed gravity, assume the cosmic ﬂuid to be prefect, i.e. non-viscous. From a hydrodynamicist’s point of view, this is somewhat visionary, since there are several mechanisms in ﬂuid mechanics, even in homogeneous space without boundaries, it is where bulk viscous ﬂuid come into play. Dissipativeeffects, includingbothbulkandshearviscosity, aresupposedtoplayavery important role in the early evolution of the universe. The bulk viscous pressure term in the matter energy-momentum tensor may lead to an accelerating universe. This dissertation primarily investigates exact solutions of LRS Bianchi-I cosmological model with and without viscous matter in f(R,T) theory of gravity, where f(R,T) is an arbitrary function of the Ricci scalar R and the trace T of the energy-momentum tensor. In particular, we have studied f(R,T) = R+2f(T), where f(T) = λT with λ being an arbitrary constant. The function f(R,T) = R+2f(T) is used with two noninteracting ﬂuids: one the perfect ﬂuid, and the other from modiﬁed f(R,T) gravity. The characteristic of the dynamical evolution of each cosmological model has been performed. A number of viability criteria, such as the existence of exact real solutions and physical viability, have been taken care of from each cosmological model. This is a four-chapter dissertation comprising the ﬁrst introductory chapter; chapters two and three, being the actual research work, carried out by the authors; and the concluding chapter.
A data adaptive approach to the analysis of South African climate variability and its agricultural impacts
(University of Zululand, 2020) Zvarevashe, Willard
Climate variability, which is mostly physically identiﬁed through extreme rainfall and temperature, has a signiﬁcant impact on the daily livelihoods of people and other species. The temperature has been increasing at an average of 0.65ºC per decade since 1900, and this may be attributed to natural and human factors. Climate variability has more adverse eﬀects on Africa due to lack of resources, failure to adapt, and high dependence on the rain for agricultural purposes. Although Africa is the most aﬀected continent, there are few studies on the climate variability impact on the continent. Therefore, there is a need to carry out more studies using historical data to model the climate variability impact. The climate has become more variable recently, as can be observed by the increase in the occurrence of ﬂoods and droughts. Due to this high volatility, linear statistical methods are inconclusive. Some of the non-linear methods, such as Fourier and wavelet transform-based methods make several assumptions. In this study, a data-adaptive method, ensemble empirical mode decomposition (EEMD), is used to decompose the climate data into multiple time series called intrinsic mode functions (IMF). IMFs represents the variability of the time series at diﬀerent time scales. EEMD is a noise assisted method and there is a challenge on the amplitude of noise to be added. Therefore, in this study, a method to ﬁnd the optimal amplitude of the noise is proposed. To determine the physical meaning of the IMFs synchronisation is employed. Synchronisation is a method that can be used to ﬁnd the correlations between oscillating systems. Rainfall is synchronised with temperature, El iii Niño-Southern Oscillation (ENSO) and Quasi-Biennial Oscillation (QBO) to investigate their variability impact on rainfall at diﬀerent time scales. It has been shown that models, coupled with decomposition, perform better than single models; hence, for the ﬁrst time, the decomposed data (IMFs) are modelled using generalised extreme value distribution (GEVD). GEVD was found in previous studies to be the best suitable for many hydrological and meteorological problems. However, many studies assume stationarity of the data. In this study, the diagnostic plots (quantile-quantile plot, probability-probability plot and probability density plot) reveals that the IMFs GEVD model provided a better ﬁt. The goodness-of-ﬁt test using Kolmogorov-Smirnov (K-S) and Anderson-Darling (A-D) test shows that IMFs GEVD models are more robust, compared to the original rainfall time series GEVD model. EEMD is used to investigate the climate variability impact on agriculture using grapevines and sugarcane as case studies. The IMFs from the Normalised DiﬀerenceVegetativeIndex(NDVI)aresynchronisedtorainfallandtemperature. NDVI is used because it was found in previous studies to be positively correlated to the yield in many crops and has been used to gauge the expected yield for crops. The results show that an increase of temperature for the period under studydoesnothavenoticeableimpactongrapevines’NDVI.However, thisisnot the case with sugarcane where it is shown that temperature variability has an impactontheNDVIatseasonal,and26months(approximately2years)periods. ThisstudyistheﬁrsttousethedecompositionofNDVItoinvestigatetheclimate variability impact on plants. Furthermore, few statistical studies do not look at thegrowthphaseofcropsandplantationsbutuseyieldasabasisfortheanalysis of climate variability impact. The proposed model may be used for short and medium term planning to predict the future NDVI and expected yields.
The energy-momentum problem in general relativity
(2002) Xulu, Sibusiso S.; Virbhadra, K.S.; Dube, T.A.
Energy-momentum is an important conserved quantity whose definition has been a focus of many investigations in general relativity. Unfortunately, there is still no generally accepted definition of energ3r and momentum in general relativity. Attempts aimed at finding a quantity for describing distribution of energy-momentum due to matter, non-gravitational and gravitational fields only resulted in various energy-momentum complexes (these are nontensorial under general coordinate transformations) whose physical meaning have been questioned. The problems associated with energy-momentum complexes re¬sulted in some researchers even abandoning the concept of energy-momentum localization in favor of the alternative concept of quasi-localization. However, quasi-local masses have their inadequacies, while the remarkable work of Virbhadra and some others, and recent results of Cooperstock and Chang et ai have revived an interest in various energy-momentum complexes. Hence in this work we use energy-momentum complexes to obtain the energy dis¬tributions in various space-times. We elaborate on the problem of energy localization in general relativity and use energy-momentum prescriptions of Einstein, Landau and Lifshitz, Papapetrou, Weinberg, and Moller to investigate energy distributions in var¬ious space-times. It is shown that several of these energy-momentum com¬plexes give the same and acceptable results for a given space-time. This shows the importance of these energy-momentum complexes. Our results agree with Virbhadra's conclusion that the Einstein's energy-momentum complex is still the best tool for obtaining energy distribution in a given space-time. The Cooperstock hypothesis (that energy and momentum in a curved space-time are confined to the the regions of non-vanishing energy-momentum of matter and the non-gravitational field) is also supported.
Mathematical modeling for optimal control of breast cancer
(University of Zululand, 2019) Oke, Segun Isaac; Matadi, M.B.; Xulu, S.S.
Breast cancer, which often occurs in the inner lining of milk ducts, is the deadliest and most common form of invasive cancer among females according to a 2017 report of the World Health Organization. The purpose of this study was to develop a four compartmental mathematical model using a system of nonlinear Ordinary Di erential Equations (ODEs) which investigates the impact of anti-cancer drugs, ketogenic-diets and immune boosters on the dynamics of breast cancer. The study focused on the dynamical interaction of normal and tumor cells as well as the invasion of tumor cells during the metastasis stage of breast cancer. The systems of ODEs were analytically solved for the equilibria. Using the next generation matrix method, a threshold quantity called the treatment in- duced invasion reproduction number (R i ) was computed. Center manifold theory was used to investigate the possibility of the bifurcation analysis of R i being greater than unity. Using a suitable Lyapunov functions, the global stability of the tumor-free equilibrium was achieved in conjuction with LaSalle's invariance principle. Uncertainty and sensitivity analyses were performed on R i using Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coe cient (PRCC). R i was used as the response function while investigating the most signi cant parameters (such as: 1, 2, 1, d, and 1 ) that a ects disease progression and cell invasion. Optimal control theory was applied using the Pontryagins' Maximum Principle to investigate optimal strategies for controlling and eliminating tumor cells using time dependent controls such as u1(t) (anti-cancer drugs) and u2(t) (ketogenic diets). Numerical simulation results using a set of parameter values were provided to validate the analytical results. It was found that the tumor-free equilibrium points for ix breast cancer was locally asymptotically stable when the associated invasion reproduction number was less than unity and that it was otherwise unstable. The tumor-free equilibrium was found to be globally asymptotically stable if (Ri) < 1 . Sensitivity analysis showed that the natural death rate of normal cells has the most positive sensitivity index. However, increasing the death rate as a control measure is unreasonable biologically. The level of ketogenic diet rate was found to be most negatively sensitive to Ri. Therefore, the formulated model showed that reduction of the invasion reproduction number (R i ) below unity can be achieved by maintaining the level of ketogenic diet and by reducing tumor progression rate. It was shown from this study that the breast cancer model exhibited backward bifurcation with bifurcation parameter 1 which implies that the reduction R i below unity alone is not su cient to eradicate tumor cells from the body system while in the case of forward bifurcation, the reduction of R i above unity is su cient to eradicate tumor cells from the body system . The incremental cost-e ectiveness analysis of control strategies adapted in treating breast cancer has shown that the integration of ketogenic diet and anti-cancer drugs as intervention strategy is the most cost-e ective in ghting tumor cells. x
Some investigation into causal viscous cosmological models with varieble Lambda
(2001) Nkosi, Zakhele Thomas; Beesham, A.
In this thesis we investigate the evolution of viscous Friedmann-Lemaitre-Robertson-Walker models with a variable cosmological term. The numerical solutions are obtained and represented graphically. Each graph depends on the value of gamma. The present values of the deceleration parameter and density parameter were obtained by using Eckart theory and the truncated theory and given in the tabular form. Power law solutions for the Hubble parameter are shown to exist and we give the values of all the other cosmological variables. The behaviour of the temperature depends on the initial conditions. Furthermore, the equations are also transformed into a plane autonomous system by using dimensionless variables and a dimensionless equation of state, and the qualitative behaviour of the system is investigated. The sets of equilibrium points are determined and their behaviour discussed. The exact value of the Hubble parameter, cosmological term, bulk viscosity pressure and the energy density are obtained and discussed.

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