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 study aims to identify university students' attitudes towards reading and its relationship to some demographic variables in the universities of the Sultanate of Oman. The study sample consisted of (1434) male and female university students from various Omani public and private universities affiliated with the Ministry of Higher Education. The study covered all (11) governorates of Oman. The researcher adopted the descriptive analytical approach. The researcher employed a scale of reading attitudes to collect the needed data. The study results showed that university students' reading attitudes recorded a high degree. The results also showed there are statistically significant differences at th

This article examines the usage features of the phraseological units in Russian political discourse and the peculiarities of the phraseological units’ usage by politicians in their speeches.  The main focus is on the expressive and pragmatic functions of using phraseological units in political texts due to their evaluative and expressive nature.  The present article also examines the structural and stylistic changes of the phraseological units in political discourse.  Moreover, the work provides examples from the speeches of the Minister of Foreign Affairs of the Russian Federation S.V.  Lavrov.  As a result, the author comes to the conclusion that phraseological units make it possible to establish a contact be

This research aims to study the methods of reduction of dimensions that overcome the problem curse of dimensionality when traditional methods fail to provide a good estimation of the parameters So this problem must be dealt with directly . Two methods were used to solve the problem of high dimensional data, The first method is the non-classical method Slice inverse regression ( SIR ) method and the proposed weight standard Sir (WSIR) method and principal components (PCA) which is the general method used in reducing dimensions,    (SIR ) and (PCA) is based on the work of linear combinations of a subset of the original explanatory variables, which may suffer from the problem of heterogeneity and the problem of linear

Background: Complete seal of the root canal system following its chemo-mechanical debridement plays a pivotal role for achieving successful endodontic treatment. This can be established by reducing the gaps between the core filling material and root canal wall. Aim: To assess and compare the dislocation resistance of root canals obturated with GuttaFusion® and TotalFill BC sealer versus single cone obturation technique and TotalFill BC sealer after instrumentation of the canals with WaveOne, ProTaper Next and ProTaper Universal system. Material and Method: Sixty extracted human permanent mandibular premolars were conducted in the current study. The teeth were decorated and left the root with 15mm length; the roots were divided randomly i

The Vulnerable Indian Roofed Turtle Pangshura tecta (Gray, 1831) (Testudines: Geoemydidae) occurs in the Sub-Himalayan lowlands of India, Nepal, Bangladesh, and Pakistan. Little is known about its natural history, no studies have been conducted revealing its natural predators. In this study, a group of Large-billed Crow Corvus macrorhynchos Wagler, 1827 (Passeriformes: Corvidae) was observed hunting and predating on an Indian Roofed Turtle carcass in the bank of river Kuakhai, Bhubaneswar, India. The first record of this predation behaviour is reported and substantiated by photographic evidence.

The Na Bop-Pu Sap Pb-Zn ore bodies represent a typical vein-type lead-zinc deposit situated in the Cho Don area and are currently being extracted for their lead and zinc resources. This deposit is characterized by its significant scale and quality and is considered one of the prominent lead-zinc deposits in the Cho Don area. Despite its significance, this deposit has not received adequate attention, resulting in limited knowledge of its geology, mineralization, and deposit genesis model. To address this knowledge gap, our study utilized several methodologies, including field surveying, ore mineral analysis under a microscope, and S and Pb isotopic geochemistry. By employing these approaches, we were able to obtain specific insights

ان شركات التأمين وبسبب طبيعة نشاطها التأميني تتجمع لديها أموال ضخمة عن الأقساط التي تحصلها من المؤمن لهم, ولغرض عدم ترك هذه الاموال عاطلة, فأنه يتم استثمارها في مجالات وأدوات استثمار متنوعة لتحقيق عوائد منها. هذا ولغرض تحديد وتوجيه هذه الاموال في مجال أو أداة الاستثمار المناسبة أو تحديد مدى جدوى هذه الاستثمارات, فانه يتم تقييمها, وهناك عدة نماذج رياضية لتقييم بدائل الاستثمار في شركات التأمين, إلا أنه لا