In this paper we generalize Jacobsons results by proving that any integer  in   is a square-free integer), belong to . All units of  are generated by the fundamental unit  having the forms

our generalization build on using the conditions

This leads us to classify the real quadratic fields  into the sets  Jacobsons results shows that  and Sliwa confirm that  and  are the only real quadratic fields in .

ZnS nanoparticles were prepared by a simple microwave irradiation method under mild condition. The starting materials for the synthesis of ZnS quantum dots were zinc acetate (R & M Chemical) as zinc source, thioacetamide as a sulfur source and ethylene glycol as a solvent. All chemicals were analytical grade products and used without further purification. The quantum dots of ZnS with cubic structure were characterized by X-ray powder diffraction (XRD), the morphology of the film is seen by scanning electron microscopy (SEM). The particle size is determined by field effect scanning electron microscopy (FESEM), UV-Visible absorption spectroscopy and XRD. UV-Visible absorption spectroscopy analysis shows that the absorption peak of the as-prep

The aim of the research was to prepare the fields and items for safe environment scale form professional basketball players and coaches’ point of view. The researchers used the descriptive method on professional athletes of players and coaches from (11) clubs; for the pilot study (21) athletes were selected and for the building sample (103) athletes, and standardization sample (101) athletes from the participants in the Iraqi league (2018-2019). The scale of the safe environment was concluded with (6) fields and (52) items. The researchers came up with the safe environment scale from the point of view of professional basketball players and coaches. Finally the researchers recommended paying attention to providing safe environment for athl

A mathematical model has been formulated to predict the influence of high outdoor air temperature on the performance of small scale air - conditioning system using R22 and alternative refrigerants R290, R407C, R410A. All refrigerants were investigated in the cooling mode operation. The mathematical model results have been validated with experimental data extracted from split type air conditioner of 2 TR capacity. This entailed the construction of an experimental test rig which consists of four main parts. They are, the refrigeration system, psychrometric test facility, measuring instrumentation, and auxiliary systems. The conditioned air was maintained at 25 0C dry bulb and 19 0C wet bulb for all tests. The outdoor ambient air temperatur

Background: Semen contamination is a detrimental factor in decreasing fertility. Seasonal changes may affect the contamination, too. Objectives: This study was designed to detect semen contamination in ovine and caprine during different seasons. Methods: Six fully mature male sheep and goats were subjected to electro-ejaculator collection twice monthly from February 1, 2022, to January 31, 2023 (Spring, February 1, 2022-April 30, 2022; Summer, May 1, 2022, July 31, 2022; Autumn August 1, 2022, October 31, 2022; Winter November 1, 2022, January 31, 2023), for studying the seasonal effect. A total of 288 semen samples were collected from both species (36 samples from each per season). All samples were subjected to bacterial isolatio

Many international companies have branches or agencies in Iraq and have large market shares, and this has contributed to the increase in the import of these companies' products. This process created an additional problem of handling a large number of expired or defective products. There are many methods that contribute significantly to reducing these problems. The most prominent of these is the activation of reverse logistics options in Iraq by international and local companies. This research aims to conduct a survey of the reverse logistics options approved by some local and international companies and compare between them. This paper found that Amaron, Hitachi, Apple, Samsung and Huawei have limited options for reverse logistics and that

The research aims to monitor and analyze the visual and symbolic features, derived from the Omani heritage, and inspired by the concept of local identity, for a selection of contemporary Omani graphic art works, which represent a mixed, multi-category of research sample, comprising three levels: the works of professional Omani artists, and the works of young artists and specialized students, as well as the work of a number of active academics in the field of graphic theorizing and teaching. The sample is also divided - in terms of technical classification - into a category of works executed using traditional engraving and printing media, and a category of works executed using modern and digital graphic printing media.
Through the da

In this research we present An idea of setting up same split plots experiments in many locations and many periods by Latin Square Design. This cases represents a modest contribution in area of design and analysis of experiments. we had written (theoretically)  the general plans, the mathematical models for these experiments, and finding the derivations of EMS for each component (source) of sources of variation of the analysis of variance tables which uses for the statistical analysis for these expirements