Showing papers in "Journal of the Nigerian Association of Mathematical Physics in 2012"

Transformation of Groups

Abstract: In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30

Stability Results for Multistep Iteration Satisfying a General Contractive Condition of Integral Type in a Normed Linear Space

Abstract: In this paper, we prove some stability results for multistep iteration scheme by using maps satisfying contractive condition of integral type in a normed linear space. Our results are generalizations and extensions of some of the results of Berinde [1], Olatinwo [2], Osilike and Udomene [3], Rhoades [4, 5] and some other numerous results in the literature. Keywords : Multistep iteration, contractive condition of integral type, normed linear spaces.

Stability Analysis of Equilibrium States of an SEIR Tuberculosis Model

Abstract: We extend the tuberculosis model proposed by Blower etal. [1] by incorporating factors such as rates of detection and treatment of active tuberculosis (TB), proportions of recruited individuals due to immigration, rate at which susceptible individuals become infectious and the recovered class. We prove that the solution to the model is positive and bounded. We examine the stability and equilibrium states of the extended model with respect to the basic reproduction number R 0 . We show that the disease-free equilibrium (DFE) is globally asymptotically stable if R 0 ≤ 1 and that there exists at least one endemic equilibrium which is globally asymptotically stable if R 0 > 1. Finally, based on our results, we discuss optimum treatment strategies for tuberculosis epidemics. Keywords: Tuberculosis; Mathematical model; Global stability; Equilibrium; Epidemics; Basic reproduction number Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 119 – 124

Cayley-Hamilton theorem in rhotrix

Abstract: Cayley-Hamilton is one of the well-known theorems that is formulated and proved in linear algebra on matrices. In this paper we extend this theorem to the concept of rhotrix and also present some properties that are attached to it. Rhotrix is an object that lies in some way between n x n dimensional matrices and (2n -1) x (2n -1) dimensional matrices. Moreover, the representation of vectors in rhotrices is different from representation of vectors in matrices. Keywords: Cayley-Hamilton, rhotrix, determinant, characteristic polynomial, minimum polynomial Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 43 – 48

Uniform Order Zero- Stable K-Step Block Methods For Initial Value Problems Of Ordinary Differential Equations

Abstract: Efficient and accurate uniform order hybrid block methods are proposed in this paper. Specifically the schemes are of uniform order 4 for ki (i=2,3). The block schemes are adequate, and accurate and also yield good stability properties. The traditional implementation of single step method is avoided by new proposed method over block steps, which requires no starting value. Moreover the speed of integration is enhanced for k i (i=2,3) by 2k and (2k-1) successive steps movement at a time (block steps). Numerical examples are given to illustrate the performance of the new procedures. Keywords : Zero stability, Hybrid, k –step, Block methods, first order initial value problem

Augmented Lagrangian Method For Discretized Optimal Control Problems

Abstract: In this paper, we are concerned with one-dimensional time invariant optimal control problem, whose objective function is quadratic and the dynamical system is a differential equation with initial condition .Since most real life problems are nonlinear and their analytical solutions are not readily available, we resolve to approximate solutions. Our aim is to develop a numerical scheme to solve one dimensional optimal control problem. A discretization of the performance index using trapezoidal rule and the state equation using crank-Nicholson is adopted. By parameter optimization, this results into a sparse non-linear programming problem. With the aid of Augmented Lagrangian method, a quadratic function with a control operator (penalized matrix) amenable to conjugate gradient method is generated. Numerical experiments verify the efficiency of the proposed technique which compares much more favourably to the existing scheme. Keywords: Trapezoidal rule, Euler’s method, Augmented Lagrangian method, and Conjugate gradient method. Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 185 – 192

On a Numerical Comparison of the Proposed Randomized Response Technique with Hussain and Shabbir

Abstract: In this paper, we compared our newly developed Randomized Response Technique (RRT) with that of Hussain-Shabbir’s dichotomous Randomized Response Technique (RRT) when data are obtained through the randomized response technique (RRT) proposed by Hussain and Shabbir (2007) It was established that the variance of the proposed technique is less than that of the conventional technique for various orders of probabilities of each answer option Hence, the proposed technique is more efficient than the conventional technique Keywords : Probability of Selection, Privacy Protection, Sensitive Characters, Randomization Device

An EOQ Model for Delayed Deteriorating Items with Linear Time Dependent Holding Cost

Abstract: An EOQ model for delayed deteriorating items with linear time dependent holding cost is considered in this paper. This is a little deviation from most inventory models that consider the holding cost to be constant. In this paper, permissible delay in payment is not considered rather the payment is made immediately the replenishment of the inventory is made. The optimal cycle length that gives the minimum total inventory cost was at the end determined and seven numerical examples are given.

Multivariate Marshall and Olkin Exponential Minification Process

Abstract: A stationary bivariate minification process with bivariate Marshall-Olkin exponential distribution that was earlier studied by Miroslav et al [15]is in this paper extended to multivariate minification process with multivariate Marshall and Olkin exponential distribution as its stationary marginal distribution. The innovation and the joint distributions of random vectors ( X n (1) , X n (2) ,..., X n (k) ) and (X (1) n-j , (X (2) n-j ,..., X (k) n-j ), j >0, are presented. The autocovariance and the autocorrelation matrices are developed. Lastly, the unknown parameters are estimated and their asymptotic properties are also investigated in this research work. Keywords : Ergodic; Estimation; Minification process; Multivariate Marshall and Olkin Exponential Distribution; uniformly mixing.

Evaluation of Fractured Basement Complex Rock Porosity by Azimuthal Cross-Square Vertical Electrical Soundings

Abstract: This paper gave an overview on the use of Azimuthal Cross-Square direct-current resistivity sounding as complementary geophysical technique to Schlumberger vertical electrical sounding in characterizing fractured geologic systems. Previously, Schlumberger vertical electrical sounding was used to collect data. Interpretation of the generated data revealed the stratigraphic setting and identified anomalous feature at the downstream section of Tiga Dam, Northwestern Nigeria. The feature was large in size, appeared deep down and at an angle to the extended tip of the locality surface manifested fractures and increase in size on approaching the core of the dam. These prompted the use of Azimuthal Cross-Square Array direct-current resistivity soundings about eight strategically selected points in order to characterize the fractures and the anomalous feature. Some of the parameters deduced were depth of fracturing (76.00m – 126.02m), anisotropy (1.02 – 1.43), fracture swarthiness and secondary porosity (0.01 – 0.28). Specifically the anomalous body has porosity ranging from 0.16 to 0.28. Keyword : Anomalous, Azimuthal, Foliation, Porosity and Swarthiness.

The Tomonaga Model

Abstract: The Hamiltonian of the one-dimensional electron gas is not exactly solvable. Tomonaga upon recognizing that the excitations of the electron gas are approximately bosons, although the elementary particles, the electrons, are fermions, went on to assume that the excitations are exactly bosons. The resulting model, the Tomonaga model immediately becomes exactly solvable. This preliminary work reviews briefly what has been done so far with the Tomonaga model; details the Tomonaga model and all the equations it entails, as a starting point for subsequent studies. There is the likelihood that the model may, in the not too distant future, become as important as the Hubbard model.

Dynamics of a Perturbed Linear Chain of Atoms

Abstract: Since approximately 1950 an increasing portion of experimental solid state physics research has been concerned with studying defects in crystals. This work uses the code BORN written by Silsbee and Drager to simulate the dynamics of a perturbed linear chain of atoms. Specifically, the dispersion curves for pure and impure monatomic crystal has been obtain. Also the power law dependence of the impurity mode investigated. It has been found that the impurity atom does not alter the shape of the dispersion curve. The effect of the impurity occurs at higher wave vectors, that is, in the lower wavelength limit. As the ratio of the atomic mass of the impurity to the host mass decreases, the maximum angular frequency increases. Also, the power law dependence has been confirmed. Kaywords: Lattice vibration, Dispersion relation, Impurity mode, Defect Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 267 – 272

A note on Markovian manpower models

Abstract: In modelling manpower systems, most authors rely on Markov-based theoretic methodology as an analytic tool to unify the states of the system with the axiomatic foundation that there is a one-stage dependence of events. In this study, Markovian manpower models are surveyed. Specific areas are highlighted as future research directions. Keywords: Markov process; model; manpower systems; stochastic environment Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 369 – 378

Rhotrices and Elementary Row Operations

Abstract: In this note, elementary row operations on rhotrices are presented due to the vital roles they played in matrix theory. These operations can be used to determine rhotrix inverses and solve a system of n rhotrix equations. Keywords: rhotrices; echelon rhotrices; elementary row operations Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 37 – 42

Solution of Lie’ nard Equations using Modified Initial Guess Variational Iterative Method (MIGVIM)

Abstract: In this work, we obtained approximate solutions for Lie’ nard equations using modified initial guess variational iteration method (MIGVIM). This method proves to be very promising for obtaining approximate solutions for this kind of equations. We also demonstrate the superiority of MIGVIM over the decomposition method and the variational iteration method for this type of equations by providing numerical comparisons. Keywords: Variational Iteration, Lagrange multiplier, Lie’ nard equations, Adomian decomposition, Modified initial guess variational iteration Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 61 – 64

Scattering Phase Functions of Constituents of Mineral Dust Aerosols

Abstract: In this paper, a Mie-based numerical computation and visualization program has been used to compute the phase functions of typical constituents of mineral dust aerosols of Saharan origin in the visible range of the electromagnetic spectrum of wavelength 550nm. Computations were done for scattering angles ranging from 0 to 180 degrees and three radial sizes of 0.25, 0.5 and 1.0 microns. The results show that there is increasing asymmetry and complexity of the phase functions with increasing radial sizes for each of the selected constituents: Illite, Kaolinite, Montmorillonte, Hematite, Calcite and Quartz. The behaviour of these constituents as observed by their phase functions provide information on the optical properties and radiative effects of the mineral dust types and is therefore useful on regional and global scales in assessing radiative impacts of dust outbreak events. Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 331 – 342

Kronecker Product Analytical Approach to ANOVA of Surface Roughness Optimization

Abstract: The Fishers-Yates algorithm has remained the most widely used statistical approach that involves the use of sum of squares of treatments or blocks in the determination of mean square errors (MSEs) needful for the computation of F-statistic prior to the decision making based on the acceptance or rejection of the null hypothesis A review of literature on design of experiments shows a trend away from uncritical acceptance of the approach, thus confirming that sustained effort is being made to develop a new method As part of this effort, this paper attempts to develop a novel approach for determining the MSEs in designed experiment Using the new method, the combination of controllable variables that optimized most the surface finish of machined workpiece materials was determined with Kronecker product analysis which was enhanced by the use of MATLAB software package The response value for the surface roughness, X ijklmn , obtained from the model developed was 15368μm Residual analysis carried out indicates that the model output was adequate The analytical method explored can be used to develop a statistical software package that will be helpful in the computation of sums of squares of observation as well as make decision on the null hypothesis without recourse to Fisher’s table Keywords : Kronecker product, Sum of Squares, Mean sum of squares, Optimization

The Electronic Band Structure of Platinum Oxide (PtO)

Abstract: We have performed the electronic band structure of the bulk and monolayer of PtO using the full potential linear muffin-tin orbital and the projector augmented wave method with the density functional theory. We applied the LDA and LDA+U scheme to both methods. It was found out that the LDA calculation of bulk PtO predicted a metallic nature in agreement with previous LDA and GGA calculations but in disagreement with the semiconductor nature favored by experiment. Our LDA+U calculation for both methods predicted PtO to be a semiconductor with a band gap value of 1.4 eV which is in reasonable agreement with experiment the experimental band gap value of 1.2 eV. The band structure of the monolayer PtO was found to be the same with the bulk in qualitative term. The position of the valence band maximum and the conduction band minimum are same as the bulk. The indirect band gap value of 1.4 eV is in agreement with the experimental value of 1.5 eV.

Determination of Coefficients of Thermal Conductivity (CTC) of Plastics Using Lee’s Disc Method (LDM)

Abstract: Coefficients of thermal conductivity of different kinds of plastics were determined using the Lee’s Disc Method (LDM). Eight plastics such as Polyethylene Terephathalate (PET), High-Density Polyethylene (HDPE), Polyvinyl Chloride (PVC), Low-Density Polyethylene (LDPE), Polypropylene (PP), Polystyrene (PS), Acroylonitril Butadiene Stryrene (ABS) and Polycarbonate (PC) were sampled at the Wurukum and Modern Markets in Makurdi and their thermal conductivity coefficients were obtained. The coefficient of thermal conductivity of PET was calculated as 0.15±0.01 Wm-1K-1, HDPE as 0.42±0.01, PVC as 0.23±0.01, LDPE as 0.31±0.01, PP as 0.10±0.01, PS as 0.03±0.01, ABS as 0.17±0.01 and PC as 0.19±0.01. The results are in good agreement with the experiment. PP has the least coefficient of thermal conductivity and it is recommended for making of utensils handles while PVC recommended for making of ceilings and drain-waste pipes. Keywords: Coefficients of Thermal Conductivity, Plastics, Wurukum and Modern Markets, Makurdi Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 235 – 238

Simulation of the diffraction pattern of one dimensional quasicrystal

Abstract: The effects of the variation of atomic spacing ratio of a one dimensional quasicrystal material are investigated. The work involves the use of the solid state simulation code, Laue written by Silsbee and Drager. We are able to observe the general features of the diffraction pattern by a quasicrystal. In addition, it has been found that each golden mean produces a unique diffraction pattern and that the lower the golden mean the better the diffraction pattern resembles that of a periodic chain. Also the intensity of the central peak was found to decrease as the golden mean increases. However the value of golden mean has no effect on the spacing between the Bragg planes. Keywords: Quasicrystal Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 285 – 292

Linearity and Non-linearity of Photorefractive effect in Materials using the Band transport Model

Abstract: In this paper we have studied the Linearity and Non-linearity of Photorefractive effect in materials using the band transport model. For low light beam intensities the change in the refractive index is proportional to the electric field for linear optics while for non- linear optics the change in refractive index is directly proportional to the square of the electric field. From our studies, it can be seen that the diffusion field is inversely proportional to period of interference pattern while the space charge field is directly proportional to the period of interference pattern. In the absence of the external electric field the space-charge field is spatially phase shifted by /2 relative to the interference fringe pattern, which plays a key role in the energy exchange in two-wave mixing. In the presence of the external electric field the scaling factor is a complex quantity indicating that the externally applied electric field does not only change the magnitude of the space-charge field but also alters its spatial phase. Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 249 – 256

Prioritization of Programmer’s Productivity Using Analytic Hierarchy Process (AHP) Model

Abstract: This paper focuses on the application of Analytic Hierarchy Process (AHP) model in the context of prioritizing programmer’s productivity in University of Benin, Benin City Nigeria. This is achieved by evaluating the way in which the AHP model can be used to select the best programmer for the purpose of developing software and meeting the required time allocated. Keywords : Analytic Hierarchy process, AHP, Prioritizing Programmer’s Productivity, Cost Driver, Performance Measurement.

A Primal-Dual Augmented Lagrangian Method for Optimal Control of Linear-Quadratic Problem with Time Varying Systems

Abstract: In this paper we are concerned with time-varying optimal control problems whose cost is quadratic and whose state is a differential equation and with general boundary conditions. The basic new idea of this paper is to propose a primal-dual augmented Lagrangian method, embedded with a sequential quadratic programming(SQP) for the solution of such problems.The benefit of this approach is that the quality of the dual variables is monitored explicitly during the solution of the subproblem. Moreover, the formulation of a penalized matrix in the primal-dual variables with mesh-refinement strategy guarantees the reliability of the algorithm. Numerical experiments verify the efficiency of the proposed method. Keywords : Optimal control,primal-dual methods, augmented Lagrangian methods, conjugate gradient method, sequential quadratic programming.