### Main Subjects : Mathematics

##### Zero Divisor Graph Of ZpM qr with Applications

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 2, Pages 13-23

DOI:
10.33899/csmj.2019.167334

In this paper, we study zero-divisor graph of the ring Z_{p}^{m}_{qr} and give some properties of this graph. Also, we find the chromatic number, Hosoya polynomial and Wiener index of this graph.

##### Some Properties of Algebra of Quotients with Bounded Evaluation of a Norm Ideal on Complex Banach Space

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 2, Pages 25-31

DOI:
10.33899/csmj.2020.167335

Cabrera-Mohammed proved that the imbedding of a norm ideal on Hilbert space in algebra of quotients with bounded evaluation is continuous with other properties. In this paper we improve this result by using complex Banach space instated of Hilbert space.

##### Hosoya Polynomial, Wiener Index, Coloring and Planar of Annihilator Graph of Zn

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 2, Pages 41-52

DOI:
10.33899/csmj.2020.167337

Let R be a commutative ring with identity. We consider Γ_{B}(R) an annihilator graph of the commutative ring R. In this paper, we find Hosoya polynomial, Wiener index, Coloring, and Planar annihilator graph of Z_{n} denote Γ_{B}(Z_{n}) , with n= p^{m} or n=p^{m}q, where p, q are distinct prime numbers and m is an integer with m ≥ 1 .

##### ii- Open Set in Bitopological Spaces

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 2, Pages 33-39

DOI:
10.33899/csmj.2020.167336

** **In this paper, we define *ii*-open set in bitopological space as follows: Let *(* *, * * , * *) *be a bitopological space, a subset *A* of ** **is said to be ( *– ii-* open set ) if there exist *U,V ≠* and *U,V * * *such that:

*A=int ^{1}(U) or A=int^{2} (V) *

*A*

*or A*

We study some characterizations and properties of this class. Also, we explain the relation between

*ii*- open sets and open sets,

*i*-open sets and

*α*-open sets in bitopological space. Furthermore, we define

*ii-*continuous mapping on bitopological spaces with some properties.

##### Classification of the Financial Reality of the National Insurance Company for the Period (2008-2015) Using Cluster Analysis

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 2, Pages 25-37

DOI:
10.33899/csmj.2020.167344

This study aimed to use the cluster analysis to find the difference in the level of financial indicators of the National Insurance Company for the period (2008-2015) and to determine which financial indicators contributed significantly to this difference in the financial position of the company during the period under study. A cluster analysis method was used, represented by the hierarchical method (single link) and the method of averages (k-means). Where was concluded that the year (2008) had a low financial position for the company, while the year (2015) had an increase in the financial position of the company. And that the most important indicators that were the reason for the decline in the financial situation in the years (2008 and 2009) is the percentage of the company's contribution in other companies, while the rate of expenses and the rate of growth of the life insurance portfolio and the percentage of the company's contribution in other companies' were the most important factors that led to improvement The financial position of the company in the period (2010-2012) compared to previous years.

##### Design Simulation System to Simplifying Boolean Equation by using Karnaugh Map

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 1, Pages 97-115

DOI:
10.33899/csmj.2020.164680

Simulation is one of most important technique used for learning, it makes learning possible without cost and provides best way to improve the practical skills for learners. The purpose of this research was to design program to simulate processing of simplifying Boolean expression by using kranaugh- map depending on rules and procedures applied to Boolean equation in order minimize it to obtain final optimal expression with minimum number of variables ,and reduction in equipment that leads to reduce cost, and this research recommend to use modern methods in education which Simulation programs is one of this method to improve E-learning to keep up with universities which care to use E-learning with traditional education and make student more interactive with education progress.

##### On MP-Rings

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 1, Pages 35-40

DOI:
10.33899/csmj.2020.164797

An ideal I of a ring R is said to be right (left) Pure if for every , there is such that . A ring R is said to be right (left) MP-ring, if every maximal right (left) ideal of R is a left (right) pure. In this paper have been studied some new properties of MP-rings, there connections with strongly regular rings.

Some of the main result of the present work are as follows:

1- Let R be aright MP-ring, r(a) is a W-ideal for all then

*a*- Every essential ideal is a direct summand.

*b*- R is strongly regular ring.

2- Let R be aright MP-ring. If R is right almost abelian left NBF ring, then R is strongly regular.

##### A New Formula for Conjugate Gradient in Unconstrained Optimization

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 1, Pages 41-52

DOI:
10.33899/csmj.2020.164798

The conjugate gradient method is an important part of the methods of optimization that are not constrained by local convergence characteristics. In this research, a new formula for the conjugated coefficient is derived depending on the linear structure. The new method fulfills the regression requirement. In addition, using the Wolff search line terms, the overall convergence of the new method has been demonstrated. At the end of the research were presented numerical results that show the effectiveness of the proposed method.

##### The Restricted Detour Polynomial of the Theta Graph

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 1, Pages 13-20

DOI:
10.33899/csmj.2020.164664

The restricted detour distance D*(u,v) between two vertices u and v of a connected graph G is the length of a longest u - v path P in G such that <V(P)> = P. The main goal of this paper is to obtain the restricted detour polynomial of the theta graph. Moreover, the restricted detour index of the theta graph will also be obtained.

##### The Homotopy Analysis Method in Turning Point Problems

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 1, Pages 51-65

DOI:
10.33899/csmj.2020.164676

In this paper, we used the homotopy analysis method to ordinary differential equations of type boundary value problems with a parameter representing turning points."To show the high accuracy of the solution results, we compare the numerical results applying the standard homotopy analysis method with the integral equation and the numerical solution of the Simpson and Trapezoidal rules."Also, we give the estimated order of convergence (local) and the global estimated order of convergence along the interval.

##### Weiner Polynomials for Generalization of Distance for Some Special Graphs

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 2, Pages 103-120

DOI:
10.33899/csmj.2006.164061

The minimum distance of a vertex v to an set of vertices of a graph G is defined as :

.

The n-Wiener polynomial for this distance of a graph G is defined as

,

where is the number of order pairs (v,S), , such that

,

and is the diameter for this minimum n-distance.

In this paper, the n-Wiener polynomials for some types of graphs such as complete graphs, bipartite graphs, star graphs, wheel graphs, path and cycle graphs are obtained .The n-Wiener index for each of these special graphs is given. Moreover, some properties of the coefficients of are established.

##### Stability Analysis of Fisher Equation Using Numerical Galerkin Techniques

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 2, Pages 29-42

DOI:
10.33899/csmj.2006.164057

We studied the stability of the steady state solutions for Fisher Equation in two cases, the First one with constant amplitude and we show that the steady state solution is always stable under any condition, but the other two solutions and are conditionally stable.

In the Second case, we studied the steady state solutions for various amplitude by using two Methods. The First is analytically by direct Method and the second is numerical method using Galerkin technique which shows the same results, that is the steady state solution is always stable under any conditions, but the other two solutions and are conditionally stable.

##### Numerical Solution of the Problem of Heat Transfer by Convection

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 2, Pages 71-88

DOI:
10.33899/csmj.2006.164059

In this paper we have presated a heat transfer by convection in rectangular cavity filled with static fluid. Differentially heated end vertical walls. Two-dimensional motions are assumed. The governing vorticity and energy transport equations are solved by an alternating direction implicit finite- difference method. We transference heat equation into two finite- difference equations. The time interval has been deviled into two equal halves, alternating to compute an intermediate point in the first step and final value at T time. We get by result analysis, that we can reach the steady – state from Un steady –state after some iteration.

##### New Initial Parameter for the Constrained Optimization Method

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 1, Pages 61-68

DOI:
10.33899/csmj.2006.164036

In this paper, we have investigated a new initial parameter in the nonlinear constrained optimization method. The aim of this new method is to make a balance between interior and exterior method for constrained optimization. The new technique has been programmed to solve some of standard problems in the non-linear optimization. The results are too effective when compared with other standard optimization methods like interior and exterior methods.

##### On Singular and non Singular Rings

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 1, Pages 111-116

DOI:
10.33899/csmj.2006.164040

In this work, we study singular and non singular rings and we give some new basic properties of such rings and its relation with other rings. Finally, we consider rings for which R/Y(R) is regular

##### On an Approximate Solution to Rodriguez Conjecture

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 1, Pages 43-53

DOI:
10.33899/csmj.2006.164044

Rickart Theorem ensures the automatic continuity of a dense range homomorphism from a Banach algebra into a strongly Semisimple Banach algebra. Rodriguez conjecture is an extension of Rickart theorem in order to include the nonassociative algebras as follows:

Rodriguez conjecture:Every densely valued homomorphism from a complete normed nonassociative algebra into another one with zero strong radical is continuous.

There is an affirmative answer of Rodriguez conjecture in particular case of power-associative algebra’s. In this work, we give an approximate solution of Rodriguez conjecture:

If A and B are complete normed nonassociative algebras and if f is a dense range homomorphism from A into B such that M(A) (the multiplication algebra of A) is full and B is strongly Semisimple, then f is continuous.

Finally, we give a Gelfand theorem on automatic continuity as a corollary and as an applied example of our approximate solution of Rodriguez conjecture.

##### Numerical Analysis of Fisher Equation

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 1, Pages 85-100

DOI:
10.33899/csmj.2006.164046

The Fisher Equation had been solved numerically by using two Methods of Finite Differences Methods. The First is Explicit Scheme Method and the Second is Crank–Nicholson Method. A Comparison had been made between these two methods and we find that the Crank–Nicholson Method converges towards saturation state u=1 faster than the Explicit Scheme Method (Table 1). Also the numerical stability for both Methods had been made, the Explicit Scheme Method is conditionally stable and the condition is , while Crank–Nicholson Method has the condition for step size, but time step is unconditionally stable.

##### A Rational Triangle Function as a Model for a Conjugate Gradient Optimization Method.

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 1, Pages 43-54

DOI:
10.33899/csmj.2006.164034

This paper presents the development and implementation of a new numberical based on a non-quadratic Triangular rational function model. For solving non-linear optimization problem .The algorithm is implemented in one version, employing exact line search. This version is compared numberically against versions of the CG-method. The results indicate that in general the new algorithm is superior to the previon algorithm.

##### On Weakly Regular Rings and SSF-rings

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 1, Pages 55-59

DOI:
10.33899/csmj.2006.164035

In this work we consider weakly regular rings whose simple singular right R-Modules are flat. We also consider the condition (*): R satisfies L(a)Ír(a) for any aÎR. We prove that if R satisfies(*) and whose simple singular right R-module are flat, then Z (R) the center of R is a von Neunann regular ring. We also show that a ring R either satisfies (*) or a strongly right bounded ring in which every simple singular right R-module is flat, then R is reduced weakly regular rings.

** **

##### On the Rings of Differential Operators

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 1, Pages 117-126

DOI:
10.33899/csmj.2006.164041

The rings of differential operators have been studied by many mathematicians like Musson [5], Smith and Stafford [7]. Jones in [2] and [3] introduced new ideas for such kind of rings and he found a new line.

In this work, we generalize many of the relations of Jones in the first part, and we found a new proof for some relations of Jones.

##### On ERT and MERT-Rings

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2004,* Volume 1, Issue 1, Pages 28-33

DOI:
10.33899/csmj.2004.164095

The main purpose of this paper is to study ERT and MERT rings, in order to study the connection between such rings and II-regular rings.

##### Rings in which Every Simple Right R-Module is Flat

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2004,* Volume 1, Issue 1, Pages 88-95

DOI:
10.33899/csmj.2004.164099

The objective of this paper is to initiate the study of rings in which each simple right R-module is flat, such rings will be called right SF-rings. Some important properties of right SF-rings are obtained. Among other results we prove that: If R is a semi prime ERT right SF-ring with zero socle, then R is a strongly regular ring.

##### Automatic Self-Scaling Strategies for VM Updates

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2004,* Volume 1, Issue 1, Pages 54-64

DOI:
10.33899/csmj.2004.164097

In this paper a class of self-scaling VM-algorithms for unconstrained optimization is investigated. Some theoretical results are given on the scaling strategies that guarantee the global convergence of the new proposed algorithm.

##### On Dual Rings

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2004,* Volume 1, Issue 1, Pages 20-26

DOI:
10.33899/csmj.2004.164094

A ring R is called a right dual ring if rl(T) = T for all right ideals T of R. The main purpose of this paper is to develop some basic properties of dual rings and to give the connection between dual rings, regular rings and strongly regular rings.

##### Stability Analysis for Fluid Flow between Two Infinite Parallel Plates I

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2004,* Volume 1, Issue 1, Pages 8-19

DOI:
10.33899/csmj.2004.164093

A model of fluid flow with heat transfer by conduction, convection and radiation has been discussed for stability with respect to restricted parameters (k,a,r,T^{*}) which are proportional to: wave numbers, thermal expansion coefficient, combination of many numbers (Re,Pr,Ec,Bo,W,) and the ratio of walls temperatures, respectively using numerical technique which illustrate that the stability of the system depends on the parameters T* and a. A clear picture of the flow is shown by using an analytical method.

##### In Accurate CG-Algorithm for Unconstrained Optimization Problems

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2004,* Volume 1, Issue 1, Pages 34-53

DOI:
10.33899/csmj.2004.164096

An algorithm for unconstrained minimization is proposed which is invariant to a non-linear scaling of a strictly convex quadratic function and which generates mutually conjugate directions for extended quadratic function. It is derived for inexact line searches and is designed for general use, it compares favorably numerical tests [over eight test functions and dimensionally up to (2-100) with the H/S, DX, F/R, P/R, and A/B algorithms on which this new algorithm is based.

** **