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 .

Abstract

The current research aims to examine the effect of the Adi and Shayer model on the achievement of fifth-grade students and their attitudes toward history. To achieve the research objective, the researcher has adopted two null hypotheses. 1) there is no statistically significant difference at the level of (0.05) between the average score of students of the experimental group who study the history of Europe and modern American history according to the model of Addie and Shayer, and the average scores of the students of the control group who study the same subjects according to the traditional method in the test of post-achievement. 2) There was no statistically significant difference at the level (

تعد الطوابع احدى وسائل الاتصال المهمة في المجتمع لما تتميز به من مستوى في القيمة الفنية والفكرية، فضلاً عن كونها وسيلة من وسائل الخطاب الاعلامي وذلك لتعدد وظائفها التي من ابرزها:التوثيق لمرحلة معينة وحدث ما.التداولية الرسمية للمراسلات المحلية والعالمية.الاعلانية كونها تعطي انطباعاً عن مراحل التطور والتقدم الاجتماعي والحضاري والسياسي للمجتمع.الدعائية لما تتضمنه من مفاهيم وأفكار وتوجهات سياسية واقتصادية

Suffer most of the facilities of the high cost of inventory , which affects the high cost of the product and thus affects many administrative decisions , as well as suffer the facilities of the systems developed by the provisions of inventory control , and this problem is exacerbated in the construction sector where the inventory in the form of Construction spin of the Year for another it becomes difficult to control the cost effectively , and is the research problem in question follows: What are the implications of the use of the system in time inventory accounting system for the contracting company does kills Alrkaah to the provisions of the cost of inventory and what is the optimal approach to inventory control ? Find assumed

   Chaotic features of nuclear energy spectrum in ⁶⁸Ge nucleus are investigated by nuclear shell model. The energies are calculated through doing shell model calculations employing the OXBASH computer code with effective interaction of F5PVH. The ⁶⁸Ge nucleus is supposed to have an inert core of ⁵⁶Ni with 12 nucleons (4 protons and 8 neutrons) move in the f5p-model space ( and ). The nuclear level density of considered classes of states is seen to have a Gaussian form, which is in accord with the prediction of other theoretical studies. The statistical fluctuations of the energy spectrum (the level spacing P(s) and the Dyson-Mehta (or statistics) are well described by the Gaussian orthogonal ens

The present study deals with the effect of self -observation on EFL University students` achievement in conversation classes. The process of self-observation helps the teacher to understand one’s own actions and the reactions in the process of teaching. The sample of this study is EFL students in the third stage at the Department of English Language, morning studies, College of Education /Ibn-Rushd .The sample of the study consists of (84) students distributed on experimental group(A) includes (42) students, and (42) students as control group(B). In order to achieve the aim of the study ,and to gain a closer idea about the impact of reflective teaching technique(self-observation) on the students achievement in conversation classes, a chec

A field-pilot scale slow sand filter (SSF) was constructed at Al-Rustamiya Sewage Treatment Plant (STP) in Baghdad city to investigate the removal efficiency in terms of Biochemical Oxygen Demand (BOD5), Chemical oxygen demand (COD), Total Suspended Solids (TSS) and Chloride concentrations for achieving better secondary effluent quality from this treatment plant. The SSF was designed at a 0.2 m/h filtration rate with filter area 1 m2 and total filter depth of 2.3 m. A filter sand media 0.35 mm in size and 1 m depth was supported by 0.2 m layer of gravel of size 5 mm. The secondary effluent from Al-Rustamiya STP was used as the influent to the slow sand filter. The results showed that the removal of BOD5, COD, TSS, and Chloride were

The aim of this article is to study the dynamical behavior of an eco-epidemiological model. A prey-predator model comprising infectious disease in prey species and stage structure in predator species is suggested and studied. Presumed that the prey species growing logistically in the absence of predator and the ferocity process happened by Lotka-Volterra functional response. The existence, uniqueness, and boundedness of the solution of the model are investigated. The stability constraints of all equilibrium points are determined. The constraints of persistence of the model are established. The local bifurcation near every equilibrium point is analyzed. The global dynamics of the model are investigated numerically and confronted with the obt

This paper discusses reliability R of the (2+1) Cascade model of inverse Weibull distribution. Reliability is to be found when strength-stress distributed is inverse Weibull random variables with unknown scale parameter and known shape parameter. Six estimation methods (Maximum likelihood, Moment, Least Square, Weighted Least Square, Regression and Percentile) are used to estimate reliability. There is a comparison between six different estimation methods by the simulation study by MATLAB 2016, using two statistical criteria Mean square error and Mean Absolute Percentage Error, where it is found that best estimator between the six estimators is Maximum likelihood estimation method.