Ph.D / M.Sc - 2004/2005

1.	Tesis : Ph.D Tajuk : Characterizations Of Combinatorially Homogeneous Digraphs Calon : KOK WAI KEONG Penyelia : Profesor Dr. Chia Gek Ling Abstract The major aims of this thesis are to classify all the combinatorially - homogeneous digraphs and to see how combinatorially -homogeneous digraphs can be constructed from other combinatorially -homogeneous digraphs which are simpler. We also investigate the structure of commutative association scheme since every combinatorially 2-homogeneous digraph corresponds to a commutative association scheme of class less than or equal to 4. In this thesis, · a set of feasibility conditions (Theorem 3.3.2) and a set of realizability conditions (Theorem 3.3.3) for a symmetric association scheme to be split into a non-symmetric commutative association scheme with exactly one pair of non-symmetric relations are introduced; · by applying our feasibility conditions to the list of feasible parameter sets of all the class 3 primitive symmetric association schemes with the cardinality of the scheme less than or equal to 100 given in [eD 96, A.4], we find a list of parameter sets corresponding to class 3 primitive symmetric association schemes of which the splitting into class 4 non-symmetric commutative association schemes are feasible; · we find that for every , it is impossible to split the Hamming scheme into a class 4 non-symmetric commutative association scheme if q ≢3 (mod 4); · we find that for every it is impossible to split the Johnson scheme into a class 4 non-symmetric commutative association scheme if q≢17 (mod 4); · we present a classification into six categories of the class 4 non-symmetric fission schemes of group-divisible 3-schemes, and provide complete solutions for three of the six categories and partial results for the remaining cases; · we discover the relationship between some class 4 non-symmetric commutative association schemes and some generalized Hadamard matrices with a special block form; · all the combinatorially homogeneous antisymmetric digraphs are completely classified; · we characterize the combinatorially homogeneous mixed digraphs by assuming the structure of the out-neighborhood of a vertex.
2.	Tesis : Ph.D Tajuk : Construction Of Almost Harmonic Matrix-Vector Pairs Calon : CHEN HUEY VOON Penyelia : Profesor Madya Dr. Thomas Bier Abstract Let A be real symmetric matrix and ℝ be a nonzero real n-vector. Let for integer and . The pair is said to be harmonic if and only if the expressions for all sufficiently large k. In particular, if u is an eigenvector of A then is a harmonic pair. The pair as above is almost harmonic if and only if the expressions for all (sufficiently large) values of even k. We say that u has m principal eigenvalues if and only if the orthogonal projection of u into precisely m different eigenspaces is nonzero. Then clearly . We study the real symmetric matrices that admit vectors forming almost harmonic pairs with the maximal number of principal eigenvalues. In this thesis, we will construct for each integer a real symmetric matrix A and a n-vector ℝ where u has n distinct principal eigenvalues of A such that the pair is almost harmonic. The existence of such real n-vector is related to the existence of solutions of quadratic equations defined by quadratic operators that arise form certain labellings. Hence, some combinatorial background using additive and multiplicative labellings of the complete graphs is also developed.
3.	Tesis : Ph.D Tajuk : On Graphs Determined By Their Chromatic Polynomials Calon : HO CHEE KIT Penyelia : Profesor Dr. Chia Gek Ling Abstract The chromatic polynomial of a graph G is the number of ways to colour the graph using at most colours, so that no two adjacent vertices have the same colour. Two graphs are said to be chromatic equivalent if they share the same chromatic polynomial. A graph G is said to be chromatically unique if for any graph Y which is chromatically equivalent to G, Y is isomorphic to G. All graphs having the same chromatic polynomial are said to form a chromatic equivalence class. Chapter 1 provides some preliminaries and definitions on graphs, followed by an introduction to chromatic polynomials and a brief survey of results on chromatic uniqueness as well as chromatic equivalence classes of graphs. In Chapter 2, we discuss the behaviour of the coefficients of chromatic polynomial of a graph. Some necessary conditions for two graphs to be chromatically equivalent are then deduced from these coefficients. In Chapter 3, we prove that the edge-gluing of , and the edge-gluing of , are chromatically unique for any . In Chapter 4, we consider those families of complete tripartite graphs in which any two of the partite sets differ by at most 3 in cardinality and show that these graphs are all chromatically unique. Also, it is shown that the graph is chromatically unique for all integers m and n such that . This answers a conjecture of Chia, Goh and Koh ( raised in [17] ) in the affirmative. Finally, in Chapter 5, we obtain the chromatic equivalence class for the join of a graph consisting of m copies of and n copies of .
4.	Tesis : M.Sc Tajuk : Run Length Distributions For Two-Sided Cusum Schemes Calon : CHONG FOOK SENG Penyelia : Profesor Dr. Pooi Ah Hin Abstract Cumulative sum (CUSUM) control charts are very effective in detecting small and moderate parameter changes. In judging the performance of a CUSUM procedure, it is important to know its run length distribution. Iterative formulas are already available for finding the run length distribution of two-sided CUSUM. Very often, the iterative formulas involve double integration and it is not easy to evaluate the probability that the process is declared to be out of control at time when 7. But when the parameters and of the two-sided CUSUM are such that 2, the iterative formulas will only involve one-dimensional integration and it is possible to evaluate for as large as 125. The evaluation of is illustrated using the normal, exponential and inverse Gaussian distributions. When 2, simulation is used to estimate the run length distribution. As the results based on simulation usually deviate randomly from the actual results, we fit the simulated results with a smooth curve. If is found that the curve given by the right tail of a translated gamma distribution gives a good fit to the simulated results. Finally, optimal two-sided CUSUM charts are found for the case when the distribution of the observations is normal, exponential or inverse Gaussian.
5.	Tesis : M.Sc Tajuk : Pengaturcaraan Matematik Dan Peruntukan (Umpukan) Optimum Sumber Tenaga Calon : JAMALIAH BINTI MOKHTARUDDIN Penyelia : Profesor Madya Dr. Nordin Hj. Mohamad Abstract Energy is a vital input for the economic growth: therefore it is essential to develop a sustainable development for the greater benefit of all and to ensure that it does sufficiently meet the demand. The objective of this research is to determine the optimal allocation of the main commercial energy resources, which are the natural gas, electricity, coal, and the petroleum products in Malaysia to four consuming sectors, namely industrial, residential, commercial, and transportation. The energy model consists of a macroeconometric and goal programming model. Three different scenarios have been employed for forecasting the future energy allocation until the year 2020. Scenario I, minimizes the total energy consumption, for Scenario II, maximizes the total gross domestic production. Scenario III is the combination of both of the objective that is then separated into two cases based priority. The MPG solutions indicate that our energy consumption for each scenario from 1985 to 1999 is in the optimal range, but occupied a different pattern of distribution compared to the actual one. The results suggest the consumptions of petroleum products and natural gas should be reduced, and the electricity and coal consumption should be increased. The projection optimal solution for the year 2000 to 2020 indicated that the allocations of natural gaseous and coal should be increased to compensate the decreasing of the petroleum products allocations. Future energy projection could help the policy and the decision makers to allocate the amount of energy usage for each of the commodities to the corresponding consuming sectors efficiently.
6.	Tesis : M.Sc. Tajuk : Optimal Inventory Control In Production System Calon : SITI SUZLIN BINTI SUPADI Penyelia : Profesor Madya Dr. Mohd Bin Omar Abstract In manufacturing systems, the quantity of raw materials needed in production is dependent on the production size. In this research we consider a manufacturing system which procures raw materials from suppliers and processes them to make a finished product. The problems are to determine an ordering policy for raw materials and a production policy for the finished product to satisfy a deterministic time-varying demand process. We present two models whereby the first model is a lot-for-lot model with multiple instalment and the second model is one lot of raw material for the whole lots of products. We find an optimal solution for these models by using the Solver of Microsoft Excel. We present some numerical examples for a discussion.
7.	Tesis : M.Sc Tajuk : On Crossing Numbers And Removal Numbers Of Graphs Calon : LEE CHAN LYE Penyelia : Profesor Dr. Chia Gek Ling Abstract Finding the crossing numbers of graphs is a challenging problem in graph theory. The crossing numbers of only few families of graphs are known. In this thesis, we provide a survey on crossing numbers of graphs and introduce the idea of the removal numbers of graphs. Some new results on removal numbers are then presented. The main results of this thesis are presented in the last chapter. It is shown that the removal number of the generalized Petersen graph P(3k,k) is where k ≥ 4.
8.	Tesis : M.Sc Tajuk : Solving Convection-Dispersion Equation By Eulerian-Lagrangian Method Calon : RAJNI SELVARAJ Penyelia : Encik Md Abu Omar Awang Abstract The analysis of transport phenomenon of a dilute solute has an ever increasing importance in the field of fluid mechanics and has long intrigued fluid dynamists and engineers. For example, how rapid a contaminant will travel or spread out in water system is the concern towards preservation and, in some cases, the restoration of our environment. It is also vital estimate or measure the concentration of a suspended material at a certain distance at a particular time. The transport processes i.e. the movement or distribution of the concentration in rivers, lakes, estuaries and oceans are governed by an equation called the convection-dispersion equation. I am going to study the properties of convection-dispersion in one and two dimensions. My main objective here is to compute the concentration using numerical methods on the equation mentioned above. I am going to use a method known as the Eulerian-Lagrangian Method (ELM) which is a family of finite-difference method. Two numerical methods under the Eulerian-Lagrangian Method (ELM) used are the Bilinear Interpolation and the Interpolation by Lagrangian Polynomials: Higher-Order of ELM. The programming language used in this project is FORTRAN 90. These two numerical methods have been tested in several different situations and from here it is seen that the Interpolation by Lagrangian Polynomials : Higher–Order of ELM performs better as it free form artificial numerical dispersion. The motivation in this resrch is to see if the free form artificial numberrcal dispersion. The motivation in this research is to see if the Interpolation by Lagrangian Polynomials : Higher-Order of ELM still performs at its best under more complex situation of the convection-dispersion equation.
9.	Tesis : M.Sc Tajuk : Some Properties Of Subsets Of Finite Groups Calon : JOSHUA TAN JUAT HUAN Penyelia : Profesor Madya Dr. Angelina Chin Yan Mui Abstract This thesis is mainly concerned with various properties of subsets of finite groups. We begin by investigating embeddings of generalized Latin squares of order in finite groups. First we consider squares which are commutative and obtain a complete list of all such squares. We determine the isomorphism classes of these squares and obtain a complete list of all the squares which can be embedded in groups. We also investigate the non-commutative generalized Latin squares of order 4; in particular, the squares with exactly 4, 15 and 16 distinct elements. Next we study finite groups with non-commuting elements such that and show that all even ordered groups with a non-central cyclic normal subgroup of order 4 have such elements. We also obtain some relationships between the conjugacy classes of a group and its product structure. We then study bases of finite groups. In particular, we obtain some bounds of 2-bases of certain finite groups and show that these bounds are sharper than some known bounds. Closely related to bases of groups are the exhaustion numbers of subsets of groups. We investigate the exhaustion numbers of certain non-minimal generating sets of some finite non-Abelian groups. Finally, we investigate complete decompositions of the cyclic groups ℤn (). Let be an integer. For an Abelian group and nonempty subsets of , we say that is a complete decomposition of of order if and Ø for (). We determine the values of for which complete decompositions of ℤn () of order exist.
10.	Tesis : M.Sc Tajuk : A New Generalization Of The Logarithmic Distribution Calon : KHANG TSUNG FEI Penyelia : Profesor Dr. Ong Seng Huat Abstract The present work deals with a new generalization of the logarithmic distribution (GLD-VI), which is the latest addition to the family of generalized logarithmic distributions. Major properties of this distribution are derived and studied. Two methods of parameter estimation: the method of first moment and frequency of one and maximum likelihood estimation, are given. A means of deciding whether a data set deviates sufficiently from the logarithmic distribution to justify the use of GLD-VI is given in the form of an hypothesis test using Rao’s score test statistic. Examples of data-fitting are given and a connection between GLD-VI and the inverse Gaussian distribution is demonstrated. Finally some indices of species diversity based on GLD-VI are proposed for use in ecology