The investigation of determining solutions for the Diophantine equation  over the Gaussian integer ring for the specific case of  is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

Abstract

      The current research aims to find out the role of administrative leadership in the implementation of public policies and their effectiveness and their ability to do so, As well as analysis and testing of influence and correlations between research variables, The researcher has the descriptive and analytical approach, And several tools used to gather information consisted of personal interviews and field visits, While distributed questionnaire tool that consists of (35) items on a sample consisting of (147) individuals formed of staff of the Baghdad Provincial Council.

The research was based on a set of assum

The theatrical field has witnessed a huge development that included the semantic and formal sides, considering that the theatrical experience has gone through a long period of description and the dominance of literature and linguistics, thus the theatrical experience has been open to many workings and uses of high quality in forming and composing the visual system. The semiotics had a role in presenting transformed sign forms and features generated in the theatrical space. Thus (costumes, décor, makeup, lighting, and actor) are considered elements with sign forms and attributes that together constitute integrated semiotic unit that include all the layouts in the scenographic space. The two researchers determined on semiotics due to its

This study deals with the intellectual representations whose intellectual systems are incarnated in the Jewish personality, which is considered one of the complex intellectual systems that has caused controversy throughout the ages because of the ambiguity due to the religious and psychological factors that were reflected directly and strongly on the intellectual structure of the Jewish community in general and the Jewish - Zionist personality in particular, in an attempt (to create new dimensions embodied by intellectual representations of a human nature embodied by what that - peaceful - religiously oppressed - psychologically and socially isolated character presents, because of the curse of the peoples of the world that was a cause fo

The study aims to identify the educational research obstacles as perceived by the faculty members at the universities of south in the west bank. As for study population, it included all (60) faculty-member in the colleges of education (bait lahem, alahliyah, al-khalil, and al-Quds almaftoha). To collect study data, the researcher used a questionnaire that consisted of (43) item; it has categorized into seven-domains: academic working conditions, academic management, resources and information, faculty members, publication, planning, and funding educational research. The findings revealed that Educational research obstacles were high with an average of (4, 39), no significant differences among sample averages and stander deviations on the

The paper discusses the structural and optical properties of In 2 O 3 and In 2 O 3-SnO 2 gas sensor thin films were deposited on glass and silicon substrates and grown by irradiation of assistant microwave on seeded layer nucleated using spin coating technique. The X-ray diffraction revealed a polycrystalline nature of the cubic structure. Atomic Force Microscopy (AFM) used for morphology analysis that shown the grain size of the prepared thin film is less than 100 nm, surface roughness and root mean square for In 2 O 3 where increased after loading SnO 2 , this addition is a challenge in gas sensing application. Sensitivity of In 2 O 3 thin film against NO 2 toxic gas is 35% at 300 o C. Sensing properties were improved after adding Tin Oxi

Learning the vocabulary of a language has great impact on acquiring that language. Many scholars in the field of language learning emphasize the importance of vocabulary as part of the learner's communicative competence, considering it the heart of language. One of the best methods of learning vocabulary is to focus on those words of high frequency. The present article is a corpus based approach to the study of vocabulary whereby the research data are analyzed quantitatively using the software program "AntWordprofiler". This program analyses new input research data in terms of already stored reliable corpora. The aim of this article is to find out whether the vocabularies used in the English textbook for Intermediate Schools in Iraq are con

This work describes the enhancement of phenol red decolorization through immobilizing of laccase in chitosan and enzyme recycling. Commercial laccase from white rot fungus, Trametesversicolor (Tvlac), was immobilizedin to freshly prepared chitosan beads by using glutaraldehyde as a cross linker. Characterization of prepared chitosan was confirmed by FTIR and scanning electron microscope (SEM). Tvlac (46.2 U/mL) immobilized into chitosan beads at 0.8 % glutaraldehyde (v/v) within 24 hrs. Synthetic (HBT) and natural (vanillin) mediators were used to enhance dye decolorizoation. It was found that 89 % of phenol red was decolorized by chitosan beads within 180 min. in the absence of enzyme and mediator, while decolorization percenta