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Tricritical domination
Doost Ali Modjeh and Parisa Firoozi
Department of Mathematics

University of Mazandaran

Babolsar, IRAN, P.O. Box 47416-1467

A graph G is domination tricritical if the removal of any three of vertices decreases the domination number. Properties of tricritical graphs are studied. We show that a connected tricritical graph has minimum degree at least 4. Ways of constructing a tricritical graph from smaller tricritical graphs are presented.


The Total Edge-Irregular Strengths of Gears
Mathematics Department

Faculty of Mathematics and Natural Sciences

Universitas Hasanudin (UNHAS)

Jl. Perintis Kemerdekaan.

KM 10 Tamalanrea,

Makassar, Indonesia.

For a simple graph with the vertex set and the edge set , a labeling is called an edge-irregular total k-labeling of G if for any two different edges and in we have where . The total edge-irregular strength, denoted by , is the smallest positive integer k for which G has an edge-irregular total k-labeling. In this paper, we determine the total edge-irregular strength of gears.


Graphs with Exponent 3
Didi Febrian & Saib Suwilo
Department of Mathematics,

University of Sumatera Utara,

Medan 20155, Indonesia.

A connected graph G is primitive provided there is a positive integer k such that for each pair of vertices u and v there is a walk of length k connecting u and v. The smallest of such integer k is the exponent of G, denoted by exp(G). This paper discusses a necessary and/or sufficient condition for a primitive graph to have exponent 3 and determines the smallest number of edges for such graphs.


On The Basis Number and the Minimum Cycle Bases of the Wreath Product of Some Graphs
1M.M.M. Jaradat & 2M.K. Al-Qeyyam
1Department of Mathematics and Physices, Qatar Universtiy,

2Department of Mathematics, Yarmouk University


A cycle basis of a graph is called a fold if each edge of occurs in at most of the cycles in . The basis number, , of is the least non-negative integer such that has a fold basis. The length of a cycle basis is the sum of the lengths of its elements: . A minimum cycle basis is a cycle basis with minimum length. A construction of a minimum cycle bases for the wreath product of some classes of graphs is presented. Moreover, the basis numbers for the wreath product of the same classes are determined.


2-Exponents of Two-Colored Lollipops
Saib Suwilo
Department of Mathematics,

University of Sumatera Utara,

Medan 20155, Indonesia.

A two-colored digraphs is a digraph in which each of its arcs is colored by either red or blue. A two colored diagraph D is primitive provided there are nonnegative integers h and k such that for each pair of vertices u and v one can find a walk from u to v consisting of h red arcs and k blue arcs. The smallest positive integer h+k among all such nonnegative integers h and k is the 2-exponent of D and is denoted by exp2(D). An (n,s)-lollipop is a symmetric connected digraphs on n vertices consisting of a cycle of length s and a path of length (n-s). In this paper we give an upper bound for 2-exponent of two-colored (n,s)-lollipops in terms of s and n, and determine two-colored (n,s)-lollipop whose 2-exponent attains the upper bound.

Computers-Assisted Student Learning in Engineering Mathematics
Maya Pundoor & Ramadas Narayanan
1Lecturer in Mathematics, Curtin University of Technology, Sarawak Campus, Malaysia

2Lecturer in Mechanical Engineering, Curtin University of Technology, Sarawak Campus, Malaysia.

Engineering Mathematics is compulsory subject in foundation year of all engineering programs. This will be one of the staring subjects they will be taking in this university. Their paces of learning have very wide range. In addition, the current educational environment does not allow them to remedy their deficiencies in mathematics at their own pace. Another problem is that some students who do decide to stay with engineering programs achieve only minimal proficiency in mathematics. This is later a serious handicap to their educational and is the cause of endless problems for them and for their instructors. Computer-assisted student learning in Engineering Mathematics have will have immense scope in the area.

In this paper, an aspect of Engineering Mathematics that is challenging for the students and mastered poorly by some students is identified. This aspect is analyzed to determine the reasons for getting a less than optimal outcomes. An instructional approach that draws on some of the principles of adult learning developed and the factors that it will lead to improve student outcomes are identified. The key elements of your approach are implemented with a group of students and the effectiveness of the strategies is concluded.
Keywords: Computer Based Assessment; Engineering Mathematics.


Lattice Boltzmann Simulation for the Permeability of Reconstructed Porous Media
Irwan Ary Dharmawan
Department of Physics, Universitas Padjadjaran,

Jl. Raya Bandung-Sumedang KM.21, 45363 Sumedang, Indonesia.

Relating the transport properties of rocks to their pore structure has been a long-term goal of great interest to petroleum engineers, hydrologists, and other earth scientists. This problem can be addressed at various levels of detail, with the resulting models requiring varying amounts of microstructural data. The empirical permeability models such as Kozeny-Carman predict values of the permeability using knowledge only of porosity and a characteristic length such as the mean pore diameter, mean grain size, or specific surface. At the other approach of complexity lie those models that attempt to reconstruct the pore space of a rock, and then numerically solve the Navier-Stokes equations in the pore space. We present results for predicting permeability of the full three-dimensional porous media. The method consists of two key components, reconstruction of three-dimensional porous rock from two-dimensional thin sections and three-dimensional flow simulation using the Lattice-Boltzmann technique. We construct three-dimensional porous rock using serial sectioning method and Monte Carlo method. The last method is used with conditional data and input two-point correlation functions from thin sections. Permeability is then estimated through flow simulation on the reconstructed porous media by solving the Navier-Stokes Equations. The result shows that the reconstructed porous media have anisotropy permeability properties and agreed very well with empirical statement from Cozeny Karman law.


The unsteady Power Law blood flow through a multi-irregular stenosed artery

1Norzieha Mustapha, 2Norsarahaida Amin
1,2 Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia.

The non-Newtonian pulsatile model of blood flow through a multiple stenoses with irregular surface is considered. The generalized Power Law model of blood viscosity is chose to study the characteristics of blood flow in a multi-irregular stenosed artery. The flow is assumed to be unsteady, laminar, two-dimensional and axisymmetric. The calculation of the governing equations of motion in terms of the viscous shear stress in the cylindrical coordinate system is employed using a finite difference scheme based on the non-uniform grids. The numerical results obtained for a multi-irregular stenoses on flow velocity, wall shear stress, resistance to flow and flow rate are represented through some graphs. The results obtained show that the axial velocity, flow rate and wall shear stress produced lower values, while the resistance to flow presented higher values than a Newtonian model.

Keywords: Generalised Power-law model; Multi-irregular stenosed artery; Blood flow.


Mathematical Modeling of Boundary Layer Flow over a Moving Thin Needle with Prescribed Wall Temperature
1Syakila Ahmad, 1Norihan Md Arifin, 2Roslinda Mohd Nazar, 3Abdul Aziz Jaafar, 4Ioan Pop
1Institute for Mathematical Research & Department of Mathematics,

Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.
2School of Mathematical Sciences, Universiti Kebangssan Malaysia,

43600 Bangi, Selangor, Malaysia.
3Department of Aerospace Engineering, Faculty of Engineering,

Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.
4Faculty of Mathematics, University of Cluj,

R-3400 Cluj, CP 253, Romania.

The steady laminar forced convection boundary layer flow of an incompressible viscous fluid over a moving thin needle with variable wall temperature is considered. The governing boundary layer equations are first transformed into no-dimensional forms. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The numerical solutions are obtained for a blunt-nosed needle (m=0). Numerical computations are carried out for various values of the dimensionless parameter of the problem, namely the parameter a representing the needle size, with Prandtl number, Pr = 0.7 (air) and 6.8 (water at room temperature). It has been found that the heat transfer characteristics are significantly influenced by these parameters. However, the Prandtl number has no effect on the flow characteristics due to the decoupled boundary layer equations.


Effect of Body Acceleration on a Micropolar Blood Flow through a Mild Stenosed Artery
Ilyani Abdullah, Norsarahaida Amin
Department of Mathematics, Faculty of Science

Universiti Teknologi Malaysia

The pulsatile flow of blood under the influence of externally imposed body acceleration is considered. The situation like riding in vehicles, flying in airplanes and fast body movements during sport activities can lead to serious health problems in the cardiovascular system. A mathematical model is developed by treating blood as a micropolar fluid which takes into account blood rheology, as blood consists of microelements suspended in plasma. The governing equation involving unsteady nonlinear two-dimensional partial differential equations are solved employing finite difference scheme. Computational results on the velocity profiles and the flow characteristic are presented.


Wireless Sensor Network Deployment in Water Retention Problem
Shaharuddin Salleh, Ruzana Ishak, and Shazirawati Muhd Puzi
Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia,

81310 UTM Skudai, Johor, Malaysia
A sensor network is a deployment of massive numbers of small, inexpensive, self-powered devices that can sense, compute, and communicate with other devices for the purpose of gathering local information to make global decisions about a physical environment. This self-organized network has a powerful node as its sink which has computational capability, and transmitter and receiver for communicating with all other nodes in the network. One useful application of the wireless sensor network is in its deployment for collecting and disseminating information from a body of water. This paper discusses the computational model of a single-sink network for computing water retention in a medium such as a lake or water reservoir. We discuss a possible deployment of sensor nodes for various problems in water retention. They include computing the volume of water in an arbitrary size lake, estimating its surface area, generating an approximated lake floor and detecting the presence of certain harmful chemicals. In most cases, the most relevant mathematical technique for modeling is the finite element method. The domain in the problem consists of a mesh of triangles formed from the sensor nodes through Delauney triangulation. The model has a lot of potential especially in tackling environmental issues in the water retention problem.


Verification of Mathematical Model of A Splicing System
1Nor Haniza Sarmin , 2Noor Aini Abdul Rashid, 3Fong Wan Heng, 4Mohd Firdaus Abdul Wahab
1,3Department of Mathematics,

Faculty of Science,

Universiti Teknologi Malaysia,

81310 Skudai, Johor.
2,4Department of Biology,

Faculty of Science,

Universiti Teknologi Malaysia,

81310 Skudai, Johor.

A splicing system describes the action of sets of restriction enzymes and a ligase that acts on DNA molecules in order to produce further molecules. The language generated by a splicing system is called a splicing language, which can then be analyzed using concepts in formal language theory. It consists of the strings in the initial set and all strings in the closure of initial set under the operation of splicing. Adult language and limit language are subsets of the slicing language. This research initiates the connection between formal language theory and the study of informational macromolecules. A laboratory verification of the mathematical model of the actual wet-lab procedure is discussed, where the adult language and limit language are distinct in this case.


2-Dimensional Fuzzy Number in Multi-Stage Dynamical System: An Improved Algorithm.
Normah Maan & Tahir Ahmad
Department of Mathematics, Faculty of Science


The concept of one-dimensional fuzzy number is successfully used in many industrial applications such as in modelling of microstrip lines and furnace systems. But most applications in real world problems involved multivariable systems. Therefore, in this paper, 2-dimensional fuzzy number concept, specifically pyramidal fuzzy number is discussed. The verification of its properties and the definition of alpha-level set are also given. This concept is then employed in modelling the mass transfer process of multi-stage dynamical system.


An Introduction to Mathematical Models of Linguistic Theories
Tengku Muhammad Andri
Beg Berkunci 101, 86400 Parit Raja

Batu Pahat, Johor Darul Takzim

Grammatical rules of natural languages can be built in non mathematical ways as well as mathematical. Linguistic theories, Chomskyian or non Chomskyan, tried to explain the language phenomenon to understand the works of mind, therefore it is very interesting to see how mathematics can contribute to those linguistic theories. This paper will try to describe how mathematics can contribute and what obstacles and limitations it has.


Stable Self Similar and Locally Self Similar Processes
S. Rezakhah
Faculty of Mathematica and Computer Sciences,

Amirkabir University of Technology, Tehran – Iran

This paper considers wavelet based estimates for the Self Similar parameters of the Linear Fractional Stable Motions. The Consistency of the estimators is also studied. We obtain some statistical results for the Hurst parameter estimation of Fractionally Integrated Auto Regressive Moving Average, FARIMA, time Series with Stable innovations. We also consider a class of Locally Self Similar processes called linear linear Multifractional Stable Motions, which extends Multifractional Brownian Motion and provide where the distributions can have Heavy tail and be non-symmetric. New results for Multifractional Brownian Motion are obtained.


Metaheuristics for Solving Facility Layout Problems: Concepts and Trends
Nadia Nurul Nordin, Zaitul Marlizawati Zainuddin 2 & Kuan Yew Wong3
1,2Faculty of Science, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia.

3Faculty of Mechanical Engineering, Universiti Teknologi Malaysia,

81310 Skudai, Johor,

Facility Layout Problems (FLPs) are combinational optimization problems, which are relevant to both manufacturing and service sectors. FLPs are known to be NP-hard. They usually involve the arrangement of departments to minimize the distance traveled by units of flow, people, material, information, and other supporting services in the safest and most effective manner. Due to the practical importance of FLPs, many approximate algorithms i.e. metaheuristics have been developed to tackle them. This paper provide a review on the fundamental of some metaheuristics commonly used to solve FLPs, as well as the previous and current research trends in this area. Discussion and comparison of the metaheuristics will be made in terms of their formulations, the solutions obtained and the types of layout involved. It is hoped that this paper will provide a new perspective for research in FLPs.


Optimization Investment Models With a Single Stochastic Factor
Sugiyarto Surono1 & Ismail Mohd2
1Universitas Alunad Dahlan Yogyakarta, Indonesia

2Department of Mathematics, Faculty of Science and Technology, Universiti Malaysia Terengganu, Mengabang Telipot, 21030 Kuala Terengganu, Terengganu, Malaysia.

This paper will discuss about a class of stochastic optimization models of expected utility in markets with stochastically changing investment opportunities. The prices of the primitive assets are modeled as diffusion processes whose coefficients evolve according to correlated diffusion factors. Under certain assumptions on the individual preferences, we are able to produce reduced form solutions. Employing a power transformation, we express the value function in terms of the solution of a linear parabolic equation, with the power exponent depending only on the coefficient of correlation and risk aversion. The new result demonstrate an interesting connection with valuation techniques using stochastic differential utilities and also, with distorted measures in a dynamic setting.

Keywords: Stochastic Differential Equation; Investment Model; HJB Equation


A Solution of Optimal Control Problem of Continuous Interconnected Nonlinear System Using DISOPE Approach
Nor Hazadura Hamzah, Hazadura Hamzah & Mohd Ismail Mohd Aziz
Institute of Mathematical Engineering, Universiti Malaysia Perlis (UNIMAP),Perlis.

The main concerns of this study is to investigate and advance the knowledge of a hierarchical algorithm for solving continuous – time optimal control of interconnected nonlinear dynamical system, known as Hierarchical Dynamics Integrated System Optimization and Parameter Estimation (HDISOPE) algorithm. HDISOPE algorithm is developed by extending Dynamics Integrated System Optimization and Parameter Estimation (DISOPE) approach to a hierarchical structure of optimal control problem of interconnected system, to take into account model-reality difference that may have been deliberately introduced to facilitate the solution of the complex nonlinear problem or due to the uncertainty in the model used for computation. In this study, HDISOPE algorithm based on a linear quadratic model is implemented in MATLAB software. Two simulation examples with different levels of nonlinearities are carried out to investigate the effectiveness and the convergence properties of the algorithm.

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