Nor'Aini Aris

University of Technology Malaysia | UTM · Department of Mathematics

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the syst...

In elimination theory, the matrix method of computing the resultant remains the most popular method due to its lower computational complexity compared to Groebner-based and set characteristics approaches. However, for the matrix method to be effective, the size and nature of the elements of the matrix play an important role. If the resultant is not...

This paper revisits the comrade matrix approach in finding the greatest common divisor (GCD) of two orthogonal polynomials. The present work investigates on the applications of the QR decomposition with iterative refinement (QRIR) to solve certain systems of linear equations which is generated from the comrade matrix. Besides iterative refinement,...

Designing and implementing a procedure for computing the polynomial resultant provides an avenue for analyzing both the computational complexity and performance of such construction. In this paper a new Maple procedure called Sturmfelmethod for computing the Sturmfel-Salmon resultant method is proposed based on existing methods and assumptions. Exa...

Availability of computer algebra systems (CAS) lead to the resurrection of the resultant method for eliminating one or more variables from the polynomials system. The resultant matrix method has advantages over the Groebner basis and Ritt-Wu method due to their high complexity and storage requirement. This paper focuses on the current resultant mat...

This research investigates on the numerical methods for computing the greatest common divisors (GCD) of two polynomials in the orthogonal basis without having to convert to the power series form. Previous implementations were conducted using the Gauss Elimination with partial pivoting (GEPP) and QR Householder algorithms, respectively. This work pr...

Objective: To evaluate and compare different hybrid resultant formulations in relation to computational complexity, performance and optimality condition. Methods/Statistical Analysis: Hybrid matrices are evaluated using computer algebra system. Findings: we have shown that, none of the hybrid formulation works well with the exception of. However, a...

In this research, the method for computing the GCD of two polynomials in the orthogonal basis, using the comrade matrix approach is further investigated. Generally, polynomials in the orthogonal basis may be better conditioned than that of the power series form when finding polynomial roots. However when transforming the GCD problem into a linear a...

The resultant matrix constructed in this paper is based on the Sylvester-Bèzout formulation which gives a hybrid resultant matrix consisting of the Sylvester and Bèzout blocks. In this work, a new algorithm for the construction has been developed and implemented as a computer package written in C++. The empirical results of the implementation on ce...

The construction of the Bézout matrix in the hybrid resultant formulation involves theories from algebraic geometry. The underlying theory on toric varieties has very nice properties such as the properties of fan (or cones), homogeneous coordinate ring, normality, and Zariski closure are related to the structure of the lattice polytopes in R. This...

The resultant matrix of a polynomial system depends on the geometry of its input Newton polytopes. Therefore for sparse inputs, the matrix is lower in dimension. The aim of the study is to infer conditions on the class of polynomial systems that can give a resultant matrix whose size is minimized, that is an optimal or Sylvester-type sparse resulta...

This paper discusses the existence of solution for a fuzzy delay predator-prey system. First, we prove the existence theorem of at least one solution of fuzzy predator-prey system where the initial condition is also described by a fuzzy number on E2 space. The results are obtained by the fixed point principles. We also determine the necessary and s...

In this paper, the existing symmetric method for finding solutions of fuzzy linear system is extended. This extended method is proposed in order to solve a non singular (n × n) matrix with real coefficients and the right hand side is trapezoidal fuzzy number with non-zero spread. In this method we use probability density function to solve the 1-cut...

In this paper, a fuzzy delay predator-prey (FDPP) system is proposed by adopting fuzzy parameter in a delay predator-prey (DPP) system. The steady state and linear stability of FDPP system are determined and analyzed. Here, we show that the trivial steady state is unstable for all value of delays. Mean while the semi trivial steady state is locally...

The permanent of a matrix has many applications in many fields. Its computation is #P-complete. The computation of exact permanent large-scale matrices is very costly in terms of memory and time. There is a real need for an efficient method to deal well with such situations. This study designs a general algorithm for estimating the permanents of th...

In all polynomial zerofinding algorithms, a good convergence requires a very good initial approximation of the exact roots. The objective of the work is to study the conditions for determining the initial approximations for an iterative matrix zerofinding method. The investigation is based on the Newbery's matrix construction which is similar to Fi...

In this paper, we present a method of solving polynomial equations known as the extended Newbery's Method. The method generates initial values for a companion matrix with a characteristic polynomial, after which a sequence iterative procedure, the real and complex roots of the equation can be determined effectively. A good approximation to these in...

In this paper we examine the characterizations of multihomogeneous or multigraded systems that can give optimal resultant matrices by determining a bijection with the permutations of {1,…, r}. These characteristics are related to the study of the determinant of a resultant complex. Since multihomogeneous resultant has been studied in several areas...

Recently, fuzzy linear regression and fuzzy polynomial regression is considered by Mosleh et al., (2010; 2011). In this paper, a novel hybrid method based on fuzzy neural network for approximate fuzzy coefficients (parameters) of fuzzy hyperbolic regression models with fuzzy output and crisp inputs, is presented. Here a neural network is considered...

The method of finding the solutions of a system of non-linear polynomial equations has received a lot of attention since ancient times. Recent active ongoing research related to solving such equations is on the construction and implementations of the method of sparse resultant. The aim of this study is to investigate on the mechanization of the mul...

The comrade matrix of a polynomial is an analogue of the companion matrix when the matrix is expressed in terms of a general basis such that the basis is a set of orthogonal polynomials satisfying the three-term recurrence relation. We present the algorithms for computing the comrade matrix, and the coefficient matrix of the corresponding linear sy...

The objective of this work is to design an efficient algorithm for the division of two polynomials represented with respect to a given orthogonal basis. The theoretical developments toward achieving this goal are investigated, extended and presented. From the theories developed, the algorithm for determining whether a polynomial divides another pol...

Symmetry groups are powerful tools for describing the structure in physical system. The symmetry group which is composed of reflections and rotations is known as a point group. The space group of a crystal or crystallographic group is a mathematical description of the symmetry inherent in the structure. A crystallographic group G is a group extensi...