The independence polynomial of n-th central graph of dihedral groups

An independent set of a graph is a set of pairwise non-adjacent vertices while the independence number is the maximum cardinality of an independent set in the graph. The independence polynomial of a graph is defined as a polynomial in which the coefficient is the number of the independent set in the graph. Meanwhile, a graph of a group G is called n-th central if the vertices are elements of G and two distinct vertices are adjacent if they are elements in the n-th term of the upper central series of G. In this research, the independence polynomial of the n-th central graph is found for some dihedral groups

An integral equation method for solving neumann problems on simply and multiply connected regions with smooth boundaries

This research presents several new boundary integral equations for the solution of Laplace’s equation with the Neumann boundary condition on both bounded and unbounded multiply connected regions. The integral equations are uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. The complete discussion of the solvability of the integral equations is also presented. Numerical results obtained show the efficiency of the proposed method when the boundaries of the regions are sufficiently smooth

On the generalized commuting and non-commuting graphs for metacyclic 3-groups

Let be a metacyclic 3-group and let be a non-empty subset of such that . The generalized commuting and non-commuting graphs of a group is denoted by and respectively. The vertex set of the generalized commuting and non-commuting graphs are the non-central elements in the set such that where Two vertices in are joined by an edge if they commute, meanwhile, the vertices in are joined by an edge if they do not commute

Integral equation approach for computing green’s function on doubly connected regions via the generalized Neumann kernel

This research is about computing the Green’s function on doubly connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nystrӧm method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gauss elimination method. Mathematica plots of Green’s functions for several test regions are also presente

Predicting growth of gold nanoparticle by photochemical reduction process associated with mathematical modelling

Nowadays, to develop new generation of nanodevices, most researchers such as chemists, physicists, biologists, even the engineers are focusing their studies towards the uniqueness as well as the chemical properties of metal nanoparticles. Even, the mathematician also has governed the appropriate mathematical modelling regarding of properties of nanoparticles such as gold nanoparticles. In this paper, some experiment regarding the photochemical and photocatalytic processes for predicting the growth of gold nanoparticles from previous studies has been reviewed. Besides that, in observing the growth rate of gold nanoparticles, a mathematical modelling has been governed. Where, ultraviolet, UV radiation with wavelength of 366 nm and 253.7 nm has been fixed as the constant parameters. The governing equation is then solved numerically using some iterative method known as Jacobi and Gauss Seidel. The convergence of both methods is discussed in detail and the numerical analysis is presented in table form to justify and validate the convergence as well as the performance of the proposed iterative methods

GEOGIRA II : Pendekatan ke arah rekabentuk berbantukan komputer

The Computer Aided Design (CAD) is a branch of applied mathematics dealing with the study and development of algorithms for the generation of curves and surfaces using computer graphics. Its diverse applications, especially for the generation of geometrical models has brought research activities in thi s area to the forefront. Thi s paper describes an effort to develop a mathematical model using the computer in a project code-named GEOGIRA II. The algorithms used are based on the least squares technique, Hermite and cubic spline interpolations

Sequential algorithm and numerical analysis on mathematical model for thermal control curing process of thermoset composite material

To reproduce and improve the efficiency of waste composite materials with consistence and high quality, it is important to tailor and control their temperature profile during curing process. Due to this phenomenon, temperature profile during curing process between two layers of composite materials, which are, resin and carbon fibre are visualized in this paper. Thus, mathematical model of 2D convection-diffusion of the heat equation of thick thermoset composite during its curing process is employed for this study. Sequential algorithms for some numerical approximation such as Jacobi and Gauss Seidel are investigated. Finite difference method schemes such as forward, backward and central methods are used to discretize the mathematical modelling in visualizing the temperature behavior of composite materials. While, the physical and thermal properties of materials used from previous studies are fully employed. The comparisons of numerical analysis between Jacobi and Gauss Seidel methods are investigated in terms of time execution, iteration numbers, maximum error, computational and complexity, as well as root means square error (RMSE). The Fourth-order Runge-Kutta scheme is applied to obtain the degree of cure for curing process of composite materials. From the numerical analysis, Gauss Seidel method gives much better output compared to Jacobi method

An example on computing the irreducible representation of finite metacyclic groups by using great orthogonality theorem method

Representation theory is a study of real realizations of the axiomatic systems of abstract algebra. For any group, the number of possible representative sets of matrices is infinite, but they can all be reduced to a single fundamental set, called the irreducible representations of the group. This paper focuses on an example of finite metacyclic groups of class two of order 16. The irreducible representation of that group is found by using Great Orthogonality Theorem Metho

Computing irreducible representation of finite metacyclic groups of order 16 using burnside method

Irreducible representation is the nucleus of a character table and is of great importance in chemistry. This paper focuses on finite metacyclic groups and their irreducible representation. This study aims to find out the irreducible representation of finite metacyclic groups of class two and finite metacyclic group of class at least three of negative type that can have order 16 by using Burnside method

An integral equation method for solving neumann problems in simply connected regions with smooth boundaries

This paper presents a new method for solving the interior Neumann problem in a simply connected region. The method is based on recent investigations on the interplay of Riemann-Hilbert (RH) problems and Fredholm integral equations with generalized Neumann kernel. In the first step the Neumann problem is reformulated as a uniquely solvable interior RH problem for the derivative of an auxiliary function which is analytic in the region. In the second step, the RH problem is transformed to a uniquely solvable Fredholm integral equation on the boundary of the region. Once this equation is solved, the auxiliary function and the solution of the Neumann problem can be obtained. The efficiency of the method is illustrated by some numerical examples