Necessary and sufficient conditions for the operator equation I AXAX n*, to have a real positive definite solution X are given. Based on these conditions, some properties of the operator A as well as relation between the solutions X andAare given.

This research Sheds highlights the procedural protections that must be enjoyed by the consumer in the face of the product, which is the protection of no less dangerous than the substantive protection of our obligations and duties delivered by the legislature upon the product of consumer interest, what is the benefit of the right if the access road to him complicated, so know The consumer has a right to the face of the product, but leaves the claim, either to ignorance For access to this right either to the difficulty of connecting to him.
That this research modest attempt we tried through which to focus on the way to the consumer behavior of arrived right, as we tried to highlight the weaknesses and the complexity of the procedure to

New speaker identification test’s feature, extracted from the differentiated form of the wave file, is presented. Differentiation operation is performed by an operator similar to the Laplacian operator. From the differentiated record’s, two parametric measures have been extracted and used as identifiers for the speaker; i.e. mean-value and number of zero-crossing points.

This research deals with an important grammatical section of the Qur'anic grammar, which is the working names the work of acts in the Quranic grammar in the studies of Iraqi researchers from 1968 to 200 AD.
    The study of working names working verb in the Koran of the important studies, especially among Iraqi researchers, the Iraqi researcher has presented detailed studies related to working names particles action verb in the Koran, and my research is studying this important grammatical section of the Koranic grammar, which is the working names working verb in The Holy Quran in the books of Iraqi researchers and their theses from 1968-2000. I studied in the preface working names of the act, and what the Iraqi res

We study in this paper the composition operator of induced by the function ?(z)=sz+t where , and We characterize the normal composition operator C? on Hardy space H2 and other related classes of operators. In addition to that we study the essential normality of C? and give some other partial results which are new to the best of our knowledge.

In this paper, we introduce a new type of Drazin invertible operator on Hilbert spaces, which is called D-operator. Then, some properties of the class of D-operators are studied. We prove that the D-operator preserves the scalar product, the unitary equivalent property, the product and sum of two D-operators are not D-operator in general but the direct product and tenser product is also D-operator.

          We obtain the coefficient estimates, extreme points, distortion and growth boundaries, radii of starlikeness, convexity, and close-to-convexity, according to the main purpose of this paper.

       In this paper, the classical continuous triple optimal control problem (CCTOCP) for the triple nonlinear parabolic boundary value problem (TNLPBVP) with state vector constraints (SVCs) is studied.  The solvability theorem for the classical continuous triple optimal control vector CCTOCV with the SVCs is stated and proved. This is done under suitable conditions. The mathematical formulation of the adjoint triple boundary value problem (ATHBVP) associated with TNLPBVP is discovered. The Fréchet derivative of the Hamiltonian" is derived.  Under suitable conditions, theorems of necessary  and sufficient conditions for the optimality of the TNLPBVP with the SVCs are stated and proved.    

     This paper aims at studying the illocutionary speech acts: direct and indirect to show the most dominant ones in a presidential speech delivered by the USA president. The speech is about the most critical health issue in the world, COVID-19 outbreak.  A descriptive qualitative study was conducted by observing the first speech delivered by president Trump concerning coronavirus outbreak and surveying the illocutionary acts: directive, declarative, commissive, expressive, and representative. Searle's (1985) classification of illocutionary speech acts is adopted in the analysis.

     What are the main types of the illocutionary speech acts performed by Trump in his speech?; Why does

Complex-valued regular functions that are normalized in the open unit disk are vastly studied. The current study introduces a new fractional integrodifferential (non-linear) operator. Based on the pre-Schwarzian derivative, certain appropriate stipulations on the parameters included in this con-structed operator to be univalent and bounded are investigated and determined.