We present a solution method for a general Riccati differential equation by imposing relationships between

the coefficients of the Riccati differential equation and explaining them through proofs and examples, we can find the

general solution to the various forms of Riccati's equation by integration directly after transforming it into a separable

differential equation.

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Harmonic function is one of an important branches of complex analysis. The first study of complex – valued, harmonic mappings defined on a domain ꓓ ⊂ C was given by Clunie and Sheil-Small [1]. Harmonic functions have been studied by different researchers such as Silverman [6]. In the present paper, a new class of harmonic univalent functions will be introduced. Various properties of functions belong to this class which include coefficient bounds, growth bounds, a closure property, extreme points, neighborhood and a convex combination will be obtained.

Harmonic function is one of an important branches of complex analysis. The first study of complex valued, harmonic mappings defined on a domain D was given by ClunIe and SheIl-Small [1]. Harmonic functions have been studied by different researchers such as Silverman [6]. In the present paper, a new class of harmonic univalent functions will be introduced. Various properties of functions belong to this class which include coefficient bounds, growth bounds, a closure property, extreme points, neighborhood and a convex combination will be obtained.

The Hill cipher is considered the first encryption technology that has some

advantages in encrypting data. The aim of this research is to modify the Hill

cipher to process letters of the Arabic alphabet. We have proven the success of

the Hill encryption on letters of the Arabic alphabet and the research was

supported by a number of different examples which were then processed using

MATLAB.

Hill Cipher is one of the symmetric key algorithms in cryptography, we discussed Encryption

and decryption for the alphabet Arabic by Hill cipher encryption and descriptions using the

invertible matrix as a key. In this study, we introduce a new method on the Hill Cipher for the

alphabet Arabic using a rectangular matrix as a key.

The division algorithm for multivariate polynomials over fields has been introduced not very long ago, in connection with algorithmic and computational problems in the rings since the work of Buchberger. The well-known fact is a term order must be a well-ordering and the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not leading term, but is freely chosen. In this paper, we will show the original geometric approach to the classification of all possible orders on the ring of polynomials in two variables is given. The connection between this classification and the well-known classification of Robiano is exposed in details.

The theory of operators is an important subject in analytic function theory, geometric function theory and univalent function theory. It also is an important subject in applied sciences. It is still an active field of research and various types of problems, which can be solved by generalising operators. Recently, many researchers have shown great interests in the study of differential operators in the theory of univalent functions and various subclasses of analytic functions defined in the open unit disc. In this paper a generalised derivative operator will be used to derive some results concerning the subordination and superordination of analytic function in the open unit disc.

The division algorithm for multivariate polynomials over fields has been introduced not very long ago, in connection with algorithmic and computational problems in the rings since the work of Buchberger. The well-known fact is a term order must be a well-ordering and the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not leading term, but is freely chosen. In this paper, we will show the original geometric approach to the classification of all possible orders on the ring of polynomials in two variables is given. The connection between this classification and the well-known classification of Robiano is exposed in details

The theory of operators is an important subject in analytic function theory, geometric function theory and univalent function theory. It also is an important subject in applied sciences. It is still an active field of research and various types of problems, which can be solved by generalising operators. Recently, many researchers have shown great interests in the study of differential operators in the theory of univalent functions and various subclasses of analytic functions defined in the open unit disc. In this paper a generalised derivative operator will be used to derive some results concerning the subordination and superordination of analytic function in the open unit disc.