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Item ALGEBRAIC AND COMBINATORIAL RESULTS OF ORDER-PRESERVING FULL CONTRACTION TRANSFORMATION SEMIGROUP(UNIVERSITY OF ILORIN, 2017-02) IBRAHIM-GARBA, RisqotLet be a finite set, the semigroup of full contraction transformation and the subsemigroup of of all order-preserving full contraction transformation semigroup. Several works have been done on algebraic properties of semigroups and results were obtained among these are generating set, the structure of starred Green’s relations in , the local and global U-depth of singular self and strictly partial one-one mappings but the combinatorics properties of has not be considered. Therefore, this study focuses on combinatorics properties of using the classes of the starred Green’s relations and other algebraic properties such as the local U-depth, status of which as not be investigated were examined. The structure of Green’s relations of were also examined, which extended some results in the literature. The aim of this study is to develop the algebraic and combinatorics properties of order-preserving full contraction transformation semigroup and objectives are to: (i) determine the local and global U-depth of , where is the generating set; (ii) obtain the status of using the global U-depth; (iii) examine the number of , , and classes of height r ; (iv) determine the total number of , , and classes of ; (v) investigate the number of elements in each , , and classes within ; and (vi) characterize Green’s relations of . The following procedures were used to obtained the results of the study: the elements of the semigroup were arranged based on their height, within each height by their images sets and their kernel sets; from the table obtained triangular array and sequences were formed; the patterns of the arrangement were studied; formulas were deduced in each case through the combinatorial principles. The gap software were used to confirmed the total number of elements. Also, the minimum length of factorisation that gives were obtained from the known generating set, for all The findings of the study were: • for each the local U-depth of is equal to its defect and global U-depth is . ; • the status of satisfies the property ; • the number of , , and classes of height r are : , ; ; and respectively. • the total number of , , and classes in are: ; ; and respectively. • the number of elements in each , and within -class are: ; ; and respectively. • the equivalence classes of Green’s relations were also characterized based on their image set and kernel classes. In conclusion, some algebraic and combinatorics properties results of subsemigroup OCTn were obtained with relevant examples. This research work and its findings are expected to be beneficial in the areas such as: computational theory, automata theory and formal languages. It can assist in sorting data and designing better network and also, new in.Item ASPECTS OF ORGANIC PETROGRAPHY AND GEOCHEMISTRY OF MAASTRICHTIAN COALS AND SHALES AT GOMBE, GONGOLA BASIN, NIGERIA(UNIVERSITY OF ILORIN, 2019-10) JIMOH, YUSUF AYOOLAThe investigated source rock interval of the Maastrichtian Gombe Formation is located at the Maiganga coal mine, near Gombe, Gongola Basin, Nigeria. Samples were collected from a borehole and the exposed part of the mine (Phase I and II). The sections at Maiganga coarsen upward with the basal part made up of coal and shale interbeds while the upper part consists of siltstone, ironstone and sandstone. The coals and interbedded shales appear most prospective source rock facies in the formation and were evaluated with respect to their hydrocarbon source rock potential including the kinetics, paleodepositional environments, source, type and maturity of the organic matter in the organofacies. Geochemical techniques employed includes TOC, Rock-Eval Pyrolysis, Soxhlet Extraction, Gas Chromatography, Medium Performance Liquid Chromatography, Gas Chromatography - Mass Spectroscopy, Isotope Ratio – Mass Spectroscopy, Palynological technique, Maceral and Visual kerogen analysis were employed in this study. Inorganic study includes Elemental and Proximate analysis. The log motif of the sections at Maiganga mine and other outcrop sections at Zange, Liji, Mallam Inna, Maiduku and Garin Magaji revealed coarsening upward sequence. Field observations in these locations indicate fluvial depositional settings. The poorly sorted sandstones with sub angular to angular grains observed in the locations represent channel facies while the fine grained clayey siltstones and shale represent overbank deposits reflecting low energy condition. The occurrence of coal at Maiganga suggests development of incoaling condition in a deltaic setting. The studied samples (coal, shale and coaly shale) were rich in organic carbon, the TOC of the coals and shale range between (32.40-65.29wt %) and (0.64-4.84 wt %) with an average of 38.03wt% and 3.14 wt% respectively while the coaly shale has an average of 12.1wt%. The average Genetic potential (GP) value for the coals, shales and coaly shales are 70.7mg/g, 5.04mg/g and 12.6mg/g respectively indicating an excellent source potential. HIs is less than 200mgHC/gTOC for all the samples. The organic matter in the coaly source rocks suggests Type III kerogen and plant contributions from terrestrial sources probably from plant (Vicia villossa), peat vegetation (Sphagnum L) or mangrove. This is corroborated by broad activation energy (Ea) distribution (41-65kcal/mol) reflecting heterogeneous composition. Consequently, gaseous hydrocarbon potential is exhibited in the source beds. xxi The major phase and peak of hydrocarbon generation occurs at 460 – 465oC for the coal and shale facies corresponding to Ea (54-55kcal/mol) defining up to 70% of the petroleum potential. The coals and the shales were characterized by high molecular weight n-alkanes which range from C10-C41 for the coal facies and C9- C41 in the shaly facies. The n-alkanes were the dominant component of the aliphatic hydrocarbon fraction and they show variation in quantity and distribution. The extracts were characterized by predominance of polar compounds indicating low maturity level of kerogen cracking. Saturated fractions comprises of tricyclic and pentacyclic terpanes, steranes, diasteranes and tetracyclic polyprenoids while aromatic fractions comprises mono, triaromatic steroids and triaromatic methylsteroids. The n-alkanes distributions in the coaly source rocks maximizes at C29. This pattern indicates organic matter derived mainly from terrestrial organic matter while the shaly facies maximizes at C16 and C29 indicating mixed source. Pr/Ph ratio for the coaly facies ranges from 2.15-3.51 (avg. 2.79) while the shaly facies ranges between 0.95-2.37 (avg. 1.82) indicating fluvio-deltaic depositional environment in sub-oxic to oxic setting. The predominance of Pristane (n-C17) over Phytane (n-C18) suggests humic origin of the organic matter for the coaly source rocks in a mixed/transitional depositional environment. The coals were rich in C29 steranes (> 50%) and (~40%) calculated from isosterane αββS and αααR configuration. This is an indication of significant land plant contribution as evidenced from the very low C27 steranes. This is corroborated by the long chain length and isotopic value of -27.7 δ13‰ PDB and -31.0 δ13‰ PDB for the coal and shale respectively. The shale also have high C29 steranes but with appreciable quantity of C27 and C28 αββS sterane (26% and 29%) and (34% and 24%) in C27, C28αααR sterane confirming probable marine contribution for the organic matter. The sterane/hopane ratio values range from 0.06-0.35 (<0.6) suggesting incorporation of high level of bacterial inputs commonly associated with terrigenous organic matter in coals. The dibenzothiophene/phenanthrene (DBT/PH) ratio range from 0.03 to 0.19 for the coal and 0.08-0.14 for the shaly source rocks respectively indicating appreciable quantity of terrestrial organic matter. The average Tmax for the coal and shaly facies were less than 435oC, and the vitrinite reflectance less than 0.55Ro. The carbon preference index (CPI) is greater than unity and in conjunctions with other saturate and aromatic maturity indices indicates that the samples were immature to low maturity status. Petrographic examination revealed vitrinite is the most abundant maceral followed by inertinite and liptinite, although there were absence of alginate and amorphous organic matter. The organic matter in the organic facies consists of opaque to non opaque biostructured phytoclasts and cuticles associated with common amounts of spores and pollen such as Arecidites sp., Ephedripites sp., Monocolpites marginatus, Cyathidites sp., Foveotriletes margaritae, Longapertites marginatus, Longapertites vaneendenburgi, Proxapertites cursus, Proxapertites operculatus, Cingulatisporites ornatus Constructipollenites ineffectus, Retibrevitricolpites triangulatus, Echitriporites trianguliformis, Nyssapollenites sp., Araucariadites australis, and Fovoetriletes margaritae. The coal and shales facies were dated Early- Late Maastrichtian and assemblage zone of Arecipites – Dictyophyllidites harrisii were assigned. xxii The proximate analysis of the coals revealed sub-bituminous rank, non-coking but suitable for fuelling, combustion and power generation due to high calorific value, high volatiles and low ash content. The average concentrations of the trace elements are lower when compared to world coals and indicate lesser negative impact to the environments, plants, animals and humans. The positive correlations of the trace elements such (Na, Ba, Th, and Co) with SiO2 and Al2O3 in the coals suggests primarily aluminosilicate affinities. In conclusion, the Maastrichtian coal and shale of Gombe Formation, Gongola Basin constitute good to excellent source rocks with potential to generate mainly gas and will generate oil if higher thermal maturity level is attained. Also, the coal facies were formed from organic matter derived from mainly terrestrial/terrigenous organic matter under oxidizing condition and deposited in the fluvial/deltaic environments, while the shaly facies were from mixed sources deposited under reducing lacustrine environments.Item Bayesian Inference from Discretely Observed Epidemic Model Using Multidimensional Diffusion Approximation Approach(UNIVERSITY OF ILORIN, 2018-08) Aliu, Abbas HassanMost epidemic data discretely observed are subjected to environmental influence. The dynamic of such data can well be described by discretised-version of Stochastic Differential Equations (SDEs). However, estimating the parameter for such a process proves to be challenging in practice. The difficulty that underlies this was the general intractability of the transition density, resulting into the likelihood functions that are not in closed forms. A direct implementation of Bayesian method of estimation results into convergence problems. Indeed, the literature has not adequately addressed these challenges for multi-dimensional cases. The aim of the study was to find the estimation procedure which would not suffer from these challenges. The objectives of these study were to: (i) derive epidemic models for explaining the behaviour of SDEs models, (ii) propose an improved diffusion bridge sampler capable of overcoming convergence problems, (iii) propose Bayesian data-augmentation method of parameters estimation which would not suffer from convergence, and (iv) compare the performance of the proposed methods with existing methods using simulated and real-life datasets. The exact likelihood functions of SDEs are very rare in practice. A numerical approximation approach through discretization of the process using Euler-Maruyama scheme was adopted. Data were simulated by introducing augmented values between every two consecutive time points to allow Euler scheme converge to the true continuous-time SDEs. Then, Bayesian data-augmentation method of estimation was performed for such problems via Markov Chain Monte Carlo (MCMC) method of sampling. Posterior means (PM), posterior credible interval (PCI), acceptance probability, trace plot, Autocorrelation function (ACF) plot and the density plot were use to assess the performance of the estimators under different number of intermediate sub-interval data point. The findings of the study were that: (i) the evolution of SDEs (diffusion process) was found to be the most suitable trajectory that mirrored the exact dynamic epidemic models’ influence; (ii) amongst the diffusion sampler considered, the proposed diffusion bridge sampler was found to be better when the number of augmented values tend to infinity; (iii) the proposed method of parameter estimate was found not to suffer from convergence problem; and (iv) results of assessment from epidemic outbreak in the simulation study and in two different real-life datasets revealed that the proposed method of parameter estimate was better than existing methods. The study concluded that modeling from discretely-observed Stochastic Differential Equations (SDEs) using Bayesian data-augmentation approach provided alternative method of estimation to obtained parameter estimates of any natural dynamic phenomena that experience random variations. It is therefore recommended that the epidemiologists should be encouraged to apply diffusion approximation approaches whenever epidemic models are subjected to environmental influence.Item BLOCK UNIFICATION AND BOUNDARY VALUE NUMERICAL INTEGRATION SCHEMES FOR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS(UNIVERSITY OF ILORIN, 2020-02) MODEBEI, MARK IFEANYIMany physical problems can be modelled as Differential Equations (DEs). DEs require solutions, either in closed form (analytic form) or in approximate form. Closed form solutions are rare to obtain for some DEs of high order. Hence, the need for numeri- cal methods/techniques. The difficulties in obtaining closed form solutions for higher order Boundary Value Problems (BVPs) in Ordinary Differential Equation (ODEs) and Initial-Boundary Value Problems (IBVPs) in Partial Differental Equations (PDEs) are the motivations for this study. This study was aimed at developing new classes of continuous block integrators for solution of fourth and fifth order BVPs and IBVPs. The objectives were to: (i) develop new classes of continuous block integrators for step number k = 5; (ii) analyse the developed integrators for consistency and convergence; (iii) apply the de- rived block integrators to solve the IBVPs; and (iv) compare the efficiency of the new methods with some existing methods in literature. The fourth and fifth order ODEs of the form: y(m) = f (x, y, yJ, . . . , y(m−1)), a ≤ x ≤ b) (1) where m = 4, 5 with appropriate boundary conditions were considered. The assumed trial solution is of the form k p(x) = ρiTi∗(x) ∼= y(x) (2) i=0 where ρi’s are real constants, y ∈ Cm(a, b), and Ti∗(x) is the ith degree shifted Chebyshev polynomial of the first kind over the interval [0, k]. In order to determine the unknown coefficients ρi’s of (2) the process of collocation was adopted. This yielded the desired xv continuous block integrators. In order to extend the scope of application of the methods to IBVPs, the Method of Lines was adopted to semi-discretize PDEs The findings of this study were: (i) two new classes of continuous implicit five-step methods namely: Block Unification Method (BUM) and Boundary Value Method (BVM) for direct solution of general fourth and fifth order BVPs were derived; (ii) the new classes of methods were proved to be consistent and convergent; (iii) new class of continuous implicit five-step methods were extended to solve fourth and fifth order IBVPs; (iv) the efficiency of the BUM and BVM compare favourably with some existing methodsin literature in terms of accuracy; (v) the BUMs and BVMs are self starting, and admit easy change of step-size and function evaluation at off-step points. The study concluded that the proposed two new classes of methods are efficient, and when compared with some existing methods in literature are accurate. The proposed methods are therefore recommended for the solution of fourth and fifth order BVPs and IBVPs.Item BOUNDARY VALUE METHODS FOR NUMERICAL SOLUTION OF FOURTH ORDER PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS(UNIVERSITY OF ILORIN, 2018-02) SALAMI, Adesina JimohMany real life situations can be modelled mathematically as Ordinary Di er-ential Equations (ODEs) or Partial Di erential Equations (PDEs). Problems arising from these are either of Initial Value Problem (IVP) or Boundary Value Problem (BVP) types in the case of ODEs and Initial Boundary Value Problems (IBVPs) in the case of PDEs. As PDEs are more di cult to solve, techniques have been developed to transform them into ODEs of BVP form. For higher order di erential equations, their solutions are mainly obtained numerically by reducing them to equivalent rst order systems. However, this is not considered e cient if direct method could be found for their so-lution. The main focus of the present study is therefore the development, analysis and application of a class of Boundary Value Methods (BVMs) for direct solution of Fourth Order ODEs of both Initial Value and Boundary Value types. A BVM is a Linear Multistep Method (LMM) coupled with boundary conditions. Also BVMs have the advantage of being self - starting viz-a-viz many other numerical methods. Therefore, the objectives of this study were to: (i) develop polynomial- tted BVMs for solving fourth order ODEs; (ii) develop trigonometrically- tted block BVMs for solving oscilla-tory fourth order ODEs; (iii) analyse the basic properties of the methods developed and these include zero- stability, consistency and convergence of the methods; (iv) implement the methods on speci c fourth order ODEs; and compare the performance of the proposed methods with those of existing ones. The problems considered in this work were the IVP : v yiv= f(x; y; y0; y00; y000) a x b y(a) = y0; y0(a) = y00; y00(a) = y000; y000(a) = y0000 and the BVP : yiv= f(x; y; y0; y00; y000) a x b y(a) = A1; y0(a) = A2; y00(b) = B1; y000(b) = B2 Trial solution of the type: p+q 1 Xr U(x) = arxr' y(x) =0 for the polynomial - tted BVMs and p+q 3 Xr arxr+ ak+3cos wx + ak+4sin wx ' y(x) U(x) = =0 for trigonometric - tted BVMs, where ar are uniquely determined coe cients,! is the frequency, and p and q are the number of interpolation and collocation points, respectively. Interpolation and collocation were achieved through the sets of equations: u(xn) = yn; u(xn+1) = yn+1 u(xn+k) = yn+k uiv(xn) = fn; uiv(xn+1) = fn+1; uiv(xn+k) = fn+k: This system of equations was solved by Mathematica 8.0 for ar; r = 0(1)(k+ The substitution of the resulting a0rs into the trial solution yielded the desired BVMs. The ndings of the study were the: vi derivation of BVMs with polynomial and basis; derivation of BVMs with trigonometric basis ; derived methods are consistent, zero stable and convergent; proposed BVMs that can handle both initial and boundary value prob-lems; and methods compare favourably well in terms of accuracy with existing methods . The study concluded that a class of BVMs with polynomial and trigonomet-ric bases derived was developed and successfully implemented on sti and non- sti problems in fourth order ODEs. It is therefore recommended for application in determining the solution of real life problems leading to IVPs and BVPs in Fourth Order ODEs.Item COLLOCATION TECHNIQUES FOR SOLVING MULTI-ORDER FRACTIONAL DIFFERENTIAL AND INTEGRO-DIFFERENTIAL EQUATIONS.(UNIVERSITY OF ILORIN, 2018-06) UWAHEREN, Ohigweren AirenoniMulti-order fractional differential and integro-differential equations are increasingly being used to model many physical phenomena which occur in fluid mechanics, engineering and other areas of science. Unfortunately, exact solutions do not exist for most of this class of problems in closed form. The necessity to nd approximate solutions to such problems thus arises. Therefore, an approximate solutions through numerical treatment is desired. The Multi-order fractional integro-differential equations considered in this work are generally in the form: n Xi D ((y(x)) + piy(i)(x) + J (y(x)) = f(x) =0 subject to the conditions; yk(0) = k; k = 0; 1; 2:::n 1 where D and J denote the fractional order derivative and integral of the function y(x) respectively. The objectives of the study were to: (i) construct new orthogonal polynomials for the general class of Multi-order Fractional Differential Equations (MFDEs) and Multi-order Fractional Integro-differential Equations (MFIDEs); (ii) use Standard and Perturbed Collocation methods to solve the class of problems in (i); (iii) examine the absolute estimated errors; and (iv) examine the accuracy of the proposed methods by comparing the results obtained with exact solutions where such exist. The use of Standard Collocation Method (SCM) and Perturbed Collocation Method (PCM) to nd the numerical solution of multi-order fractional differential and integro-differential equations using the constructed orthogonal polynomials as basis function in the approximation used were tested. Some numerical examples were considered to illustrate the accuracy of the methods. From the computational view, the proposed methods are reliable and in total agreement with some known results in the literature. The results obtained include: (i) new orthogonal polynomials basis functions were constructed; (ii) Standard Collocation Method (SCM) successfully solved linear and nonlinear multi-order fractional differential and integro-differential equations; (iii) Perturbed Collocation Method (PCM) successfully solved linear and nonlinear multi-order fractional differential and integro-differential equations; (iv) the absolute estimated errors closely approximate the exact solution. The study con-cluded that Standard and Perturbed Collocation Methods serve as alternative and additional ways of nding numerical solutions of multi-order fractional differential and integro-differential vi equations. The use of newly constructed orthogonal polynomial as basis function for solving multi-order fractional differential and integro-differential equations is recommended, and also, the proposed methods can be extended to solve integral and partial fractional order differential equations.Item CONSTRUCTION OF BALANCED INCOMPLETE SEQUENCE CROSSOVER DESIGNS FOR FIRST ORDER RESIDUAL EFFECT(UNIVERSITY OF ILORIN, 2021-04) ADEBARA, LanreCrossover design is a repeated measurements design in which individual subjects receives different sequences of treatments (t) during the different time period one at a time with object of studying differences between individual treatments. It has been found useful in several fields of research, among which are clinical trials and nutrition experiments involving dairy cattle .A noticeable feature of crossover design is that it leaves direct and carryover effects which is the effect of a treatment from previous time period on the response at the current time period in the succeeding periods. A design method that can generate designs for a large number of treatments is a challenge to researchers. Therefore, the aim of this study is to obtain method of construction of designs for a greater number of treatments. The objectives of this study were to: (i) determine prime numbers that satisfy primitive roots, x = 2 and 3 ; (ii) modify a construction method for balanced incomplete sequence crossover design for prime numbers for primitive root 2 = x ; (iii) construct a new class of balanced incomplete sequence crossover design for primitive roots 2 = x and 3; and (iv) use data set to validate the modified construction method. A universally optimal Balanced Incomplete Sequence Crossover Design (BISCOD) for first order residual effect with parameter n (experimental units), t (number of treatments) and k (number of treatment in each experimental units or number of periods) was constructed in this study. This is to enhance the estimation of residual effect over class of design in which no treatment was allocated more than once on each subject, equal replications for all treatments in the first p-1 period and each treatment preceded each other treatment (k-1) times. A method of design that satisfies primitive root 2 = x and a proposed method that satisfies primitive roots x = 2 and 3 were obtained through balanced incomplete block design. The findings of the study were that: i. prime numbers that satisfied primitive roots 2 = x and 3 included 5, 7,11, 13,17, 19, 37, 61 ii. design using a modified construction method were generated for all prime number of treatments for primitive root x = 2; iii. a construction method generates all prime number of treatments that satisfied primitive roots x = 2 and 3 was developed; and iv. modified design construction method was validated The study concluded that the modified method performed better in terms of generated designs for a greater number of treatments than existing method. The set of data from experiment of the crossover type was validated. The study recommended that the modified construction generated be used for prime numbers that satisfy primitive roots x = 2 and 3Item CONSTRUCTION OF OPTIMAL MINIMUM REPLICATE DESIGNS(UNIVERSITY OF ILORIN, 2018) AIYELABEGAN, Adijat BukolaA binary block design of size D(t,b,k,r) has ttreatments, set out in b blocks, each block having kunits, and each treatment occurring once or not at all in each block and r=bk/t the number of replicates. The focus in every experimental situation is to search for a design that provides maximum information but utilizes minimum experimental material. A design which can guarantee high precision in estimating parameters with small probabilities of committing Type I and Type II errors is considered useful to the experimenter. Therefore, the aim of this research work was to construct a new class of optimal designs with minimum replication. The specific objectives were to; (i) develop an algorithm for the optimal binary block designs, (ii) assess the binary designs constructed using a new optimality criterion, Minimum Variance MV={∑_(i=1)^(t-1)▒(e_i-e ̅ )^2 } where e_iis the i^th eigenvalue of the information matrix D(t,b,kr)and e ̅ the mean of the eigenvalues, (iii) classify the obtained designs as partially balanced incomplete block design PBIBD with massociate classes, with m=1,2,. . .. The best designs being cases withm=1 or 2, namely PBIBD/1 and PBIBD/2, respectively. Classes of optimal designs were constructed using the developed algorithm for the following parameter combinations, namely, 2 ≤r≤t, 6≤t≤15, and 2 ≤k≤t/2. The information matrix was obtained for each design D(t,b,kr)and the one with minimum MV was selected. Designs with equal eigenvalues were classified as the PBIBD/1, those with only two distinct eigenvalues as PBIBD/2. The following were the major findings from this study: The algorithm developed had been able to search for the optimal design amongst a class of designs of same sizeD(t,b,kr). The MV criterion used in design assessment outperformed the well known A-, D-, or the E-Optimality criteria and it was found to be less cumbersome in computation, A new class of optimal designs were constructed and results obtained for parameters combinations 2 ≤r≤t, 6≤t≤15, and 2 ≤k≤t/2. The study concluded that the algorithm for constructing small size replicates optimal experiments was effective. The study also confirmed that increasing the number of replications or block sizes would lead to improved efficiency but higher cost for the experiment. Thus, the research has opened up efficient designs for pilot survey and for easier computation with less cost. For an experimenter to obtain optimal design for the replication and unit constraint problem, construction of designs and their complementary designs should be considered. Optimal minimum replicate design is recommended for the construction of a better design for a particular anti-symmetric design with an improved frequency distribution of concurrence matrix.Item DERIVATION OF A GENERALIZED DISTRIBUTION FOR MATERNAL MORTALITY RATIO(UNIVERSITY OF ILORIN, 2018-05) OMEKAM, IFEYINWA VIVIANMaternal mortality is an outcome of adverse health and remains a major challenge in some societies. Study on distribution fitting for global Maternal Mortality Ratio (MMR) is therefore needful. Analysis of data without a pre-knowledge of the distribution that describes data may lead to misleading or irrelevant results. Distribution fitting to data provides the best fitting distribution for data analysis. Limitations in characteristics of existing distributions motivate generalization of distributions in order to improve goodness of fit and introduce more flexibility. The aim of this study was to derive a generalized distribution for MMR while the specific objectives were to: (i) generate families of generalized distributions; (ii) illustrate the flexibility of generalized distributions; (iii) fit some existing distributions to MMR to determine the best fitting distribution; (iv) derive generalized distributions for the best fitted distribution; and (v) fit and assess the derived generalized distributions to MMR. Five existing parameter induction methods were applied to generate families of generalized distributions with an additional parameter. Methods in permutations of these five existing parameter induction methods, namely: Lehmann Alternative 1; Lehmann Alternative 2, Marshall and Olkin Method, Power Transformation Method and α-Power Transformation Method taking two methods at a time were applied sequentially to obtain generalized families with two additional shape parameters. Same methods were also applied twice. Exponentiated families of Generalized Pareto Distribution (GPD) were generated and flexibility of generalized families was illustrated by showing effects of introduced parameters on the Probability Density and Hazard Function shapes. Histogram plot, Kolmogorov-Smirnov (K-S) distances and Akaike Information Criterion (AIC) were employed in determining the best fitting distribution for MMR. Using best fitted distribution as base distribution, its generalized distributions were derived from families generated and subsequently fitted to MMR using AIC and K-S distances for selection. Findings of the study were that: (i) seventeen distinct generalized families of distributions having two additional parameters were generated for commutativity and idempotency of some methods. Five families with one additional parameter were also generated; (ii) Lehmann type I GPD improved flexibility by introducing the unimodal and bathtub shapes in both Probability Density and Hazard Functions shapes; (iii) Frechet distribution provided the best fit for MMR amongst plausible distributions studied; (iv) four generalized Frechet distributions with an additional shape parameter and two generalized Frechet distributions with two additional shape parameters were generated; and (v) generalized Frechet distributions improved goodness of fit based on K-S distances with Lehman type II generalized Frechet giving the best fit while AIC selected Marshall-Olkin generalized Frechet distribution as a good fitting but parsimonious distribution for MMR. In conclusion, families of generalized distribution introducing n parameters may be generated by sequentially applying methods in permutations of s (s≥n) distinct parameter induction methods taking n methods at a time. Lehmann type I GPD introduced flexibility thereby illustrating flexibility of generalized distributions. Generalized Frechet distributions improved goodness of fit but for parsimony, Marshal-Olkin generalized Frechet distribution is recommended as an alternative reference distribution to the Frechet distribution for modelling MMR.Item DEVELOPMENT AND APPLICATION OF ZERO-TRUNCATED PROBABILITY MASS FUNCTION OF THE COM-BINOMIAL DISTRIBUTION(UNIVERSITY OF ILORIN, 2018-05) ADEROJU, SAMUEL ADEWALEThe major problem in modeling count data is over-dispersion, which occurs as a result of long tail, too many occurrences of zeros or none occurrence than would normally be expected by Poisson and binomial distributions. The zero-truncated distributions arise in those situations where there is no zero by nature of the data (structurally).The existing zero-truncated distributions are not flexible enough to capture peculiar characteristics such as long tail as well as over-dispersion at the same time. The aim of this research was to develop and implement a more generalized zero-truncated distribution that come from the mixture of distributions for non-zero count data to include most of the characteristics which existing ones do not have. The specific objectives were to: (i) examine the issues zero-truncated count data; (ii) investigate the effectiveness of selected binomial-based distributions for count data with various dispersions; (iii) develop zero-truncated Com-binomial distribution; (iv) identify some properties of the newly developed distribution and (v) demonstrate the use and assess the performance of the distribution against some existing ones using real life datasets. Following mixture distribution methodology, the Zero-Truncated Com-Binomial (ZTCB) was derived from the mixture of Conway-Maxwell-Poisson generalization distribution to the Binomial distribution. The first two moments via probability generating function were also derived. The Maximum Likelihood Estimation of the parameters were also obtained by direct maximization of the log-likelihood function using “optim” routine in R software. The findings of the study were: (i) the challenge in modeling count data, with a long tail and the problem arising from zeros (truncated or excess) were resolved; (ii) the Com-Binomial (CB) distribution outperformed other mixture distributions in handling either under-dispersion or over-dispersion situation; (iii) the probability mass function (pmf) of ZTCB distribution was derived and the first two moments were derived via probability generating function; (iv) the ZTCB distribution peaks around the mean to have two tails to resemble the normal distribution or peaks around 1 or n to resemble exponential distribution for carefully chosen parameter v . It is the generalization of the Zero-Truncated Com-Poisson, Zero-Truncated Poisson and Zero-Truncated Binomial distributions; and (v) the ZTCB distribution is more flexible to handle all levels of dispersion than Zero-Truncated Multiplicative Binomial distribution. The study concluded that the ZTCB distribution which is characterized by two parameters is more flexible than other distributions. . It is recommended that when modeling structurally zero-truncated data, ZTCB distribution be used to obtain a robust result. This study therefore provides useful alternative to the existing distributions.Item DEVELOPMENT OF DETERMINISTIC MODELS OF TYPHOID FEVER WITH PREVENTION AND CONTROL STRATEGIES(UNIVERSITY OF ILORIN, 2021-04) PETER, OLUMUYIWA JAMESTyphoid fever is a serious bacterial infection caused by Salmonella typhi. Typhoid fever is usually contracted by the ingestion of water or food contaminated with urine or faeces of infected carriers and is, therefore, common in areas with poor sanitation. In developing countries, refugee camps and crowded areas, typhoid fever outbreaks are frequent. In most existing literature on typhoid fever models, the effects of human to human and environment to human transmission of typhoid fever are not properly addressed, this study therefore aimed at extending the existing typhoid fever models to address the effects of some control strategies (educational campaign and vaccination) on human to human and environment to human transmission in typhoid fever prone communities. The objectives were to (i) obtain the existence, uniqueness and positivity solution of all the models; ( ii ) derive the disease free and endemic equilibra that govern the spread of typhoid fever; (iii) perform a sensitivity analysis of the models; (iv) determine the optimal strategy for implementing combine control strategies. The method employed included the application of Next Generational Matrix (NGM) technique to obtain the basic reproduction number, local and global stability analyses of the models using Lyapunov function and Lasalle’s invariance principle, series solutions of the models by using Differential Transform Method (DTM) and xvi Variational Iteration Method (VIM), optimal control analysis to establish the conditions necessary for eradication of the disease. The findings of the study were that: (i) the existence, uniqueness and positivity solutions of all the models established that the models were well posed; (ii) the disease free and endemic equilibra that govern the spread of typhoid fever were found to be stable; (iii) the most sensitive parameter to typhoid fever management was education; and (iv) control strategies become more effective when vaccination and educational campaign program are combined, prevention methods become more effective in preventing the disease transmission. The study concluded that possible optimal control strategies become more effective in the control and elimination of typhoid fever when vaccination and educational campaign are combined. Since educational campaign was the most sensitive to typhoid fever management, it is therefore recommended that any measure directed towards achieving typhoid fever-free society should include educational campaign.Item DEVELOPMENT OF EXPONENTIATED GENERALIZED EXPONENTIATED EXPONENTIAL DISTRIBUTION(UNIVERSITY OF ILORIN, 2019-12) BUKOYE, ABDULWASIUProbability distributions have been used in describing and predicting real life phenomena. The inability of existing distributions to fit diverse nature of the data sets has necessitated the development of more flexible distributions for real life data set. Recent studies on developing new families of distributions from existing ones have helped in describing data better. The inability of exponential distributions to properly fit skewed data has been a serious concern. This study therefore aimed at developing a new tractable distribution that will fit data that are positively or negatively skewed. The objectives of the study were to: (i) derive a new distribution and its statistical properties. (ii) examine the effect of the shape and scale parameters on the distribution; and (iii) compare the performance of the proposed distribution with the existing ones using simulated and real-life data sets. The generator employed in developing the proposed probability distribution function is the Exponentiated Generalized Family of distribution using the Exponentiated Exponential as the baseline distribution. The properties of the distribution such as mean, median and their deviations, quantile, Renyon entropy, skweness and kurtosis were obtained using analytical and simulation method. Simulated and real life data were used to assess and compare the performances of the proposed distribution with the existing ones using Akaike information criterion (AIC), Bayesian information criterion (BIC), log-likelihood function and likelihood ratio test. The findings of the study were that: (i) the probability density function (pdf) and cumulative distribution functions(cdf) of the Exponentiated Generalized Exponentiated Exponential (EGEE) were derived together with their properties such as moments, quantiles, skewness, kurtosis, etc.; (ii) Exponential (E), Exponentiated Exponential (EE) and Exponentiated Generalized Exponential (EGE) distributions were obtain by setting the shape parameters value to one; (iii) the mean, standard deviation and median are increasing functions of the scale and shape parameters of the proposed distribution while the skewness and kurtosis were unaffected by scale parameter but only by the shape parameters; and (iv) the proposed EGEE competed very well with other existing distributions for small sample sizes and even performed better as the sample size increases. The study therefore concluded that EGEE distribution can serve a better alternative distribution to the existing distributions (E, EE and EGE) in fitting skewed data. Therefore, the proposed EGEE distribution is preferred and recommended for fitting a skewed data.Item EFFECTS OF PERTURBATIONS IN THE CORIOLIS AND CENTRIFUGAL FORCES ON THE STABILITY OF GENERALIZED PHOTO-GRAVITATIONAL RESTRICTED THREE-BODY PROBLEM(UNIVERSITY OF ILORIN, 2017-02) JAIYEOLA, Sefinat BolaThe study of classical Restricted Three-body Problem (RTBP) and its generalizations have been of major interest to researchers over the years. This is due to the rise in the need for accuracy in determining astrometric positions which would help to reveal some peculiarities of components of motion and draw conclusions on the stability of space vehicles to be launched. This has led to the necessity of considering all possible physical properties (oblateness/triaxiality, radiation pressure, Poynting-Robertson (PR) drag, perturbing forces etc.) that affect the motion of particles in space. The effect of perturbations in the coriolis and centrifugal forces on the stability of the generalized photo-gravitational RTBP has been a major focus of investigations. However, the effect under the influence of the PR-drag from both oblate bodies has received little or no attention. Therefore, the aim of this research work was to investigate how perturbations in the coriolis and centrifugal forces affect the stability of the triangular libration points of the RTBP when the primaries were considered to be oblate, radiating with PR-drag effects. The objectives of this study were to: determine the effect of PR-drag on the stability of the libration of the generalized RTBP; investigate the effects of and on the stability of the generalized RTBP in the linear sense; establish the periodic orbit: period of oscillation, orientation and semi-axes of the proposed system; and verify the results obtained using astrophysical data for the Kruger 60 and RXJ0450, 1-5658 binary systems. The Hamiltonian and Lagrangian methods were employed to establish the relevant equations of motion, obtain the triangular libration points and investigate their stability using Murray’s and Routh Hurwitz’s criteria and verifying the results for the two binary systems using, MATLAB and Microsoft Excel Mathematical softwares. The findings from this study showed that the: • generalized system was unstable around the triangular libration points due to the presence of the PR-drag effect from both bodies; • presence of the parameter of the stabilizing factor, , in the roots of the characteristics equation does not change the instability of the system around the libration points; • period for the growth of the particle oscillation is dependent on the PR-drag parameter only, in the linear sense; • orientation and length of semi- axes are dependent on all the perturbing parameters; and • change in the values of and affects the values of the libration points and roots of the characteristics equations computed for the two binary but does not satisfy the criteria for stability. The study concluded that the system remained unstable even with the significant influence of perturbations due to the strong destabilizing effect of the PR-drag force. This work as a generalization of the classical case and the work of others, is therefore recommended to serve as a form of reference to achieving more interesting and vital results in Space Dynamics and also an added value to designers of space crafts and aerospace agencies.Item EXPONENTIALLY-FITTED NUMERICAL INTEGRATION SCHEMES FOR FIRST AND SECOND ORDER DIFFERENTIAL EQUATIONS(UNIVERSITY OF ILORIN, 2021-04) FANIYI, Olugbade EzekielMathematical models are developed to help in the understanding of physical phenomena. These models often yield equations that contain some derivatives of an unknown function of one or several variables. Such equations are called Differential Equations (DEs). A DE in which the unknown function is a function of two or more independent variables is called a Partial Differential Equation (PDE). Those in which the unknown function is a function of only one independent variable are called Ordinary Differential Equations (ODEs). This work concerns the study of numerical solutions of both types. Since most DEs do not have closed form solution, the development of numerical techniques becomes necessary. Methods used for solving Initial Value Problems ( IVPs ) of general ODEs have been reported to have setbacks as regards to handing certain attributes like singularity of stiffness and oscillatory nature of problems. Furthermore, implementation via predictor-corrector mode in Linear Multistep Methods (LMMs) have been found to be exhausting in terms of the number of function evaluations per step. K-step exponentially-fitted block methods circumvent these challenges associated with LMMs, for solving IVPs, Boundary Value Problems (BVPs) and Initial Boundary Value Problems (IBVPs) of special nature; hence the motivation of this work. This study is therefore aimed at developing a class of efficient numerical integration schemes, for direct solution of first order IVPs, second order BVPs and IBVPs. To this end, the objectives were to: (i) construct trial solution with power series fitted with exponential function as basis function; (ii) develop new classes of k-step continuous x exponentially-fitted block methods; (iii) develop continuous 2-step hybrid exponentiallyfitted block method; (iv) analyze the methods for zero stability, consistency and convergence; and (v) compare the accuracy of the proposed schemes with those of existing methods using some existing problems. The first order IVP of the form y0 = f(x,y), a < x < b (1) and the second order BVPs of the form y00 = f(x,y,y0), a < x < b (2) with appropriate boundary conditions were considered. The trial solution of the form k p(x) = Xaixi + ak+1ewx i=0 (3) ∼= y(x) was assumed where the ai’s are unknown constants and ω is the frequency. In order to determine the unknown constants, interpolation and collocation methods were employed. The resulting values were later substituted back into (3), which after some algebraic manipulations yielded the desired continuous exponentially fitted block method. The findings of this study were; (i) new classes of k-step exponentially fitted block methods for direct solution of first order IVPs and second order BVPs were derived; (ii) new 2-step Continuous hybrid exponentially fitted block method for solving second order IBVPs were developed; (iii) the methods are zero stable, consistent and convergent; xi (iv) the methods have small global error and less computation time compared to existing methods; and (v) the methods are self-starting and admit easy change of step-sizes and function evaluation at off grid points. The study concluded that an efficient class of continuous and discrete numerical integration schemes of implicit form for first and second order problems were developed and successfully implemented. These schemes are therefore recommended for solution of first and second order IVPs, second order BVPs and IBVPs.Item FLOW BEHAVIOUR OF NON-DARCIAN MAGNETOHYDRODYNAMIC CASSON NANOFLUID OVER A STRETCHING OR SHRINKING SHEET WITH SLIPS AND CONVECTIVE HEATING(UNIVERSITY OF ILORIN, 2021-04) ADEOSUN, ADESHINA TAOFEEQInvestigations of electrically conducting Magnetohydrodynamic (MHD) Casson nanofluid flow are important areas of research due to their applications in industries and engineering for industrial coolant, brake fluid, MHD generator, nanodrug delivery, and others. Several studies concerning heat and mass transfer of MHD nanofluids have been reported in literature. However, most of the existing related works do not consider the combined effects of nonlinear radiation and non-Darcian porous medium. If the fluid velocity is high and the temperature differences within the fluid are sufficiently large, both Darcy and linearized radiative heat flux models may not accurately determine the fluid flow and thermal structure. Therefore, this study was aimed at analysing the effect of nonlinear radiation, binary chemical reaction, Arrhenius activation energy, variable viscosity and thermal conductivity on convective non-Darcian MHD dissipative Casson nanofluid flow past a stretching or shrinking sheet with slip and convective heating. The objectives of the study were to: (i) determine the flow behaviour of a non-Darcian MHD dissipative Casson nanofluid over a stretching or shrinking sheet with multiple slip boundary conditions; (ii) examine the influence of variable viscosity and variable thermal conductivity on natural convective flow of non-Darcian MHD Casson nanofluid flow with velocity slip and convective heating; and (iii) analyse the effect of binary chemical reaction and Arrhenius activation energy on forced convective flow of non-Darcian MHD Casson nanofluid in the presence of non-Navier velocity slip condition. The mathematical equations governing the flow, heat and mass transfer of Casson nanofluid in a non-Darcian porous medium are: = 0, u v x y + 2 2 * 2 0 2 1 ( ) 1 = 1 1 , B B f f f p p u u u B x b u v u u u x y y k k + + − − + − 2 2 2 2 1 = 1 ( ) T B B p f T T T C T D T u u v D and x y y y y T y C y + + + + + 2 2 2 2 = . T B C C C D T u v D x y y T y + + Where 0 ( , ) , , , , , , , , , , , , , , B f p B p T u v B k b T D C C D are respectively velocity components in x and y directions, dynamic viscosity, fluid density, Casson parameter, electric conductivity, magnetic field, porous medium permeability, Forchhiemer inertial coefficient, fluid temperature, fluid thermal diffusivity, heat capacity ratio, Brownian diffusion coefficient, nanoparticle concentration, heat capacity and thermophoretic diffusion coefficient. The above governing equations were reduced to nonlinear ordinary differential equations using similarity transformation and then solved by employing weighted residual method. The computational procedure was implemented by writing codes in MATHEMATICA Software. The findings of the study were that: [label=()] 1. an increase in non-Darcian porous medium parameter resulted into a reduction in velocity, while it enhanced both temperature and nanoparticle volume fraction; 2. a hike in temperature dependent viscosity and thermal conductivity parameters contributed to a decrease in both temperature and nanoparticle volume fraction profiles and a rise in velocity profile; 3. nanoparticle volume fraction profile was reduced as a result of an increase in chemical reaction parameter, while increasing activation energy parameter led to an increase in nanoparticle volume fraction profile; and 4. an increase in radiation parameter resulted into a decrease in temperature profile. The study concluded that the thermal radiation, Forchheimer, variable viscosity, variable thermal conductivity, chemical reaction and activation energy parameters have significant effects on the velocity, temperature, nanoparticle volume fraction, Nusselt number and the Sherwood number of Casson nanofluid. It is therefore recommended that those parameters are necessary and should be considered when carrying out a design involving Casson nanofluid flow.Item GREEN SYNTHESIS, CHARACTERIZATION, MODIFICATIONS AND APPLICATIONS OF ZINC OXIDE NANORODS(UNIVERSITY OF ILORIN, 2018) ADELEKE, JOSHUA TOYINThe demand for clean energy and clean water is becoming challenging issues having been greatly influenced by human’s lifestyle and the growing world population. Nanotechnology offers promising solution through the applications of nanomaterials. The orthodox methods of synthesizing nanomaterials are expensive and generate hazardous toxic chemicals. Green method route of synthesis has become an alternative to overcome such limitations. This research work therefore aimed at synthesising a multifunctional zinc oxide nanorod through green principle route of synthesis. The objectives were to: (i) synthesize Mangifra indicaleave extract-mediated zinc oxide nanorods (ZnO NR); (ii) synthesize spinel nickel ferrite nanorods (NF NR); (iii) modify the ZnO NR with NF NR by calcination to obtain a nickel ferrite-zinc oxide nanocomposite; (iv) characterize the synthesized nanostructures for structural, elemental composition, functional groups,morphological and optical studies using X – Ray Diffractometer XRD, Fourier Transform Infrared (FTIR), Energy Dispersive X-ray Spectrometer (EDX), Scanning Electron Microscope (SEM), UV-VIS Spectrophotometer, and Photoluminescence (PL) spectroscopy respectively; (v) examine the applications of the synthesized nanostructures for dielectric studies, solar cell nanocoating, biosensing, photodegradation and treatment of both domestic and industry waste water. Green principle route of synthesis was used for the synthesis of zinc oxide nanorods. The zinc oxide nanorods (ZnO NR) were modified with spinel nickel ferrite nanorods (NF NR) by calcination both at 850 0C for ten hours to obtain zinc oxide-nickel ferrite (ZnO/NF) nanocomposite. Both unmodified and modified ZnO NR were characterized and examined for applications in dielectrics, solar cell nanocoating, glucose sensing, photodegradation and waste water treatment. The findings of this research work showed that: (i) ZnO NR have been synthesized by green synthesis route and had been successfully modified with nickel ferrite nanorods synthesized by coprecipitation method. (ii) the dielectric constantε/ of the green synthesized ZnO NR is very high (750) with capacitance value of 0.78 pF. Its dielectric loss Ԑ//, dielectric loss angle, σdecreases with increasing frequency and then reaches a constant value, hence a good dielectric. (iii) the fill factor of a silver solar cell coated with the green synthesized ZnO NR by drop-casting method was enhanced showing improved solar cell efficiency up to 5%. (iv) the modified ZnO NR was successfully used as a biosensor for glucose. (v) the photocatalytic studies reveal that the ZnO/NF nanocomposite exhibits high degradation efficiency under UV light and that •OH and photoproduced holes h+ are the main species in the removal of the organic dyes solution. (vi) for the treatment of domestic and industrial waste water, the green synthesized ZnO NR showed 93.95% and 93.95% of chemical oxygen demand (COD) reduction in 75 minutes. This research work concluded that green synthesis route is suitable for the synthesis of ZnO NR, appropriate modification of which makes it multifunctional. It is recommended that, green synthesis method should be adopted for the synthesis of ecofriendly, cost effective and multifunctional ZnO NR.Item HYBRID PLAN DESIGNED AS POSSIBLE PENSION PLAN FOR NIGERIA(UNIVERSITY OF ILORIN, 2016-09) AKALONU, RAPHAEL ONYECHEFULEThe history of pensionschemes in Nigeria started with the 1951 Pensions Ordinance. In 1979, the then Military Government established a Defined Benefit Scheme (Old) for civil servants. The scheme failed to meet its objective due to maladministration and lack of funding, resulting in non¬payment of benefits to workers on retirement. Consequently, the government established the Pension Reform Act 2004 (current), a Defined Contribution (DC) plan, aimed at remedying the shortcomings of the old one. Shortly, complaints adorned the media that the 15 percent contribution of the current plan was not enough to provide meaningful benefit after 35 years of service.The main objective of this Study is to design Hybrid Plan as possible Pension Plan for Nigeria. Specifically, the Study focused on: (i) to compare the monetary benefits between the old and the current pension schemes; (ii) to compare the adequacy of the current scheme with those of eight other developed and developing countries’ pension schemes selected from five continents; (iii) to determine effects of the Pension Risk Factors (Mortality and Interest Rate Volatility); and (iv) to design three new pension plans. The population for the study was Nigerian employees grouped into seven by Nigeria’s National Salaries, Incomes and Wages Commission (NSIWC) as at July 2010. Four groups were randomly selected for the study. Replacement Ratio data for the eight countries whose pension plans were to be compared with that of Nigeria were obtained from 2012 publications of International Monetary Fund and Organization for Economic Cooperation and Development. The schedule for calculating pension benefits under the old scheme was obtained. Such schedule does not exist for the current scheme. Actuarial method for estimating such benefits was used. The pension benefits of the two pension schemes were then compared. Replacement Ratio was calculated for the current scheme to compare its adequacy with those of the eight countries whose Pension Replacement Ratios had been obtained. Assumed simulated interest rates, salary incremental and annuity values were used and combined with mortality functions. The results showed that: (i) the ratio of gratuity paid by the old and current schemes was 3.5 to 1, while that of pension benefits was 2.3 to 1, implying that the old scheme paid at least twice as the current; (ii) the eight other countries had adopted the World Bank's (1994) three-pillar pension models in their respective reforms while Nigeria has the mandatory Defined Contribution plan for workers and no social security or voluntary plan for either formal or informal sectors; (iii) increase in interest rate increased the amount available for purchase of annuity while decrease in mortality rate improved life expectancy and hence annuity rate resulting in decrease in amount of pension receivable; (iv) three pension plans, namely, Minimum Guaranteed Money Purchase Plan, Cash Balance Plan and Hybrid Plan were designed for the formal sector and a Mandatory Collective Personal Plan was also proposed for the informal sector. The study concluded that the Hybrid Plan had higher replacement ratio, even at 20% volatility, than others. It was therefore recommended that the Hybrid Plan with higher replacement ratio be adopted by Nigerian Government for its workers.Item IDENTIFICATION OF TRAIT MARKERS USING MARKOV CHAIN PROPERTIES(UNIVERSITY OF ILORIN, 2021-03) OLARINOYE Segun Bamidele, Segun BamideleLineage is characterized by some traits whose genetic marker is passed from any of the parents' genes to their children. Some of these traits may be physical. For example, hair, eye and/or skin colours. Laboratory tests some of which are costly and non accessible have been used for screening such traits whereas a non-laboratory Markov chain methodology which is cheaper and faster was proposed as an alternative. The study aimed at establishing a non-laboratory procedure using Markov Chain to identify useful markers to link traits. The objectives were to: (i) identify traits, the state spaces and classify them using probability transition matrix; (ii) determine the properties of the state space for each trait; (iii) determine which of the traits are the markers; and (iv) demonstrate the use of the procedure using real life data. The process of transmitting traits from one generation to another was conceptualized as a Markov Chain. Markov chain methodology was employed to examine presence or absence of a given trait in an individual. The corresponding transition probability Matrix was studied to identify properties of the state spaces. The nature of Markov chain was examined to establish ergodicity and obtain the mean recurrence time. A test of homogeneity of the Markov chain was developed when at least three successive time points existed. These properties were used to characterize traits of Albinism as a real life example. The findings of the study were that: i. Markov chain concepts and techniques were successfully applied to classify various states of the available traits; ii. all transition probability matrices were found to be recurrent with different periodicity; iii. a best marker was identified to have the shortest period when state spaces are recurrent; and iv. in the case of real life data involving albinism, the colour of the eye was the best marker. The study concluded that Markov Chain is a vital tool to identify appropriate trait markers as an alternative to laboratory test. The study recommended that where information is available a Markov chain can be used to identify trait markers.Item MAGNETOHYDRODYNAMIC HEAT AND MASS TRANSFER OF NON-NEWTONIAN FLUIDS FLOW THROUGH VERTICAL PLATE IN THE PRESENCE OF THERMO-PHYSICAL PARAMETERS(UNIVERSITY OF ILORIN, 2021-04) FALODUN, Bidemi OlumideFluid is a substance that deforms continuously when subjected to shear stress. Such fluid can either be Newtonian or non-Newtonian. The Newtonian fluid obeys the law of viscosity while non-Newtonian fluid does not obey the law of viscosity. The present study is concentrated on non-Newtonian fluids. The study of non-Newtonian fluids attracted the attention of numerous researchers in the field of fluid dynamics due to its rheological applications in mechanical and chemical engineering processes. Fluids of this type are generally complex and are considered under science of deformation and flow. Non-Newtonian fluids are applicable in the movement of biological liquids, food processing, liquid cosmetics, dyes, lubricants and puncturing sludge are few examples of rheological fluids. This study aimed at examining magnetohydromagnetic (MHD) heat and mass transfer of non-Newtonian fluids flow through vertical plate (both porous and non-porous) in the presence of thermo-physical parameters using spectral methods. The equations governing the study are: = 0 (1) (4) together with the boundary conditions u = Bx ,v = −ν(x) ,T = Tw ,C = Cw ,at y = 0 (5) u −→ 0 ,T −→ T∞ ,C −→ C∞ ,as y −→∞ (6) viii u and v represents the relations u = ∂ψ∂y and v = −∂ψ∂x . In the definition of u and v, ψ(x,y) is the stream function which automatically satisfies the continuity equation (1). The objectives of the study are: (i) to examine the physics of the problem of heat and mass transfer non-Newtonian fluids flow in a semi-infinite vertical plate and vertical porous plate; (ii) to examine the influence of pertinent flow parameters such as magnetic parameter, radiation parameter, heat generation parameter and so on, on the flow of non-Newtonian fluids within the boundary layer regime; (iii) to determine the physical quantities of engineering interest such as skin friction, Nusselt number and Sherwood number on all flow parameters; and (iv) to test the accuracy and validity of the spectral relaxation method and spectral homotopy analysis method. Scholars in the field of fluid dynamics now consider spectral methods as essential tools in solving highly coupled non-linear differential equations. Spectral methods involve approximating the unknown functions using truncated series of orthogonal functions or polynomials. The spectral relaxation method (SRM) employed the concept of Gauss-Seidel method to decouple system of differential equations. The spectral homotopy analysis method (SHAM) is based on a blend of the Chebyshev pseudospectral method with the homotopy analysis method. To apply SHAM on differential equations, the domain [0,L] of the problem is first transformed to the domain [−1,1]. The partial differential equations which govern the model were simplified with the help of appropriate similarity variables and non-dimensional quantities. The transformed non-linear coupled ordinary differential equations along with the boundary conditions were solved numerically using spectral methods. The results obtained are as follows: (i) It was discovered that as the value of the viscoelastic parameter increases, the velocity profile close to the plate gradually decrease while far away from the plate, it slightly increase; (ii) The result revealed that variable viscosity and thermal conductivity greatly affects the fluids within the boundary layer as it enhances velocity and temperature respectively; (iii) It was found that the effects of Soret and Dufour on the temperature and concentration profiles are opposite; ix (iv)It was found that the results obtained were useful in food processing, drilling operations and bioengineering. Also, Soret parameter on the fluid flow is significant in isotope separation; (v) It was found that increase in the thermal Grashof number drastically increase the hydrodynamic boundary layer thickness; and (vi) The numerical methods used in this study were found useful in solving highly non-linear differential equations in science and engineering. In this study, it can be concluded that the momentum and thermal boundary layer thickness drastically increase with increase in Casson parameter and viscoelastic non-Newtonian fluid. Also, it can be concluded that Soret and Dufour parameter simultaneously affect the hydrodynamic boundary layer thickness. It is recommended that the findings of this investigation be used in plasma studies, geothermal energy extractions, generators and control of boundary layer in the field of aerodynamics. The method used in the investigation is recommended for solving highly non-linear differential equations in engineering.Item MIXED VARIABLE DISCRIMINANT ANALYSIS WITH ZERO CELL FREQUENCY(UNIVERSITY OF ILORIN, 2018-06) MBAEYI, GEORGE CHINANUWhen available data for discriminant analysis is the mixed variables type, the common procedure assigns codes to the possible states of the discrete variables and proceeds with analysis as if all data are continuous. This may lead to loss of information. The Location Model (LM) proposed by Olkin and Tate and developed by Krzanowski has combined these two data types. The problems of large number of parameters to be estimated when discrete variables are many, inability to perform analysis when one or more cell has zero frequency and limitation of number of discrete variables to be handled are some of the disadvantages of LM. Therefore, the aim of this study is to propose a modification to the LM, called Modified Location Model (MLM), particularly for when one or more cells of the resulting contingency table have zero frequency. The objectives were to: (i) propose a model, MLM, for developing a discrimination procedure for mixed variables; (ii) derive an estimator of the variance covariance matrix for the mixed variable case; (iii) derive an estimator of the vector of means and cell probabilities in the presence of empty cell(s); and (iv) compare the proposed procedure with two existing methods, namely, the LM and Fisher Linear Discriminant Function (FLDF). The MLM procedure was obtained by first estimating the variance covariance matrices of each cell of the two groups. When one or more cells have zero frequency, it uses the Independent Binary Model (IBM) to estimate cell probabilities, vector of means and variance covariance matrix for the empty cell(s). Simulated data were analyzed for several combinations of number of discrete and continuous variables including states within variables that were cross classified yielding some empty cells. Error rates, sensitivity and specificity measures were used as performance criteria. Three sets of real life data were used to validate the results obtained from simulation study. Findings from this study were that: i. MLM procedure for mixed variables in discriminant analysis was obtained; ii. a procedure for estimating variance covariance matrix when there are empty cells was obtained; iii. IBM, for estimating cell probabilities and mean vector when there are empty cells was obtained; iv. proposed MLM gave higher classification accuracy than the LM over all cases considered; v. MLM was better than both the FLDF procedure and the LM for small sample sizes in terms of classification accuracy; vi. MLM and FLDF procedure performed closely and better than LM in terms of specificity and sensitivity; and vii. MLM performed better than both the LM and the FLDF procedure when validated with real life data. The study concluded that the proposed Modified Location Model was feasible, applicable and performed better than both FLDF and LM procedures based on error rate, when available data for discriminant analysis is of the mixed variables type. The MLM is useful when one or more unique response patterns of discrete variable have empty/zero frequency. It is therefore recommended for discriminant analysis of mixed variables especially when there are many discrete variables