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 .

In this work, two cone-inverted cylindrical and cross-hybrid dielectric resonator antennas are stacked and excited by the coaxial probe method with an operating standard resonant frequency of 5.438 GHz. A drawback of these standard Dielectric Resonator Antennas (DRAs) is their narrow bandwidth. For good antenna performance, a stacked DR geometry and a thick dielectric substrate having a low dielectric constant are desired since this provides large bandwidth, better radiation power, reduces conductor loss and nonappearance of surface waves. Many approaches, such as changing the shape of the dielectric resonator, have been used to enhance bandwidth. Using DRA, having the lowest dielectric constant, increases the bandwidth and the electroma

paper

Substance use disorders are a widely recognized problem among hepatitis C-infected patients; moreover, substance abuse by intravenous injection is a common mode of transmission of the hepatitis C virus worldwide. The frequency of substance use disorders and their relation to hepatitis C infection are still unknown in Iraq. This cross-sectional study, conducted among a sample of hepatitis C- infected patients attending the Gastrointestinal Tract Center in Baghdad Medical City, aimed to examine the prevalence of substance use disorders, the sociodemographic characteristics of the abusers, and the relation between intravenous

Smoking and central obesity have both been linked to periodontitis, but their combined relationship with periodontal disease may be influenced by demographic and behavioral factors. This cross-sectional study analyzed records of 420 adult dental patients attending the College of Dentistry at the University of Baghdad. Data included demographic characteristics, smoking status, periodontal clinical findings, body mass index (BMI), and waist-to-height ratio (WHtR). Periodontitis was defined according to the 2018 classification framework, and logistic regression models were used to examine the associations of smoking and obesity-related indicators with periodontitis. The overall prevalence of periodontitis was 36.4%. Participants with p

The ground state proton, neutron and matter densities of exotic ¹¹Be and ¹⁵C nuclei are studied by means of the TFSM and BCM. In TFSM, the calculations are based on using different model spaces for the core and the valence (halo) neutron. Besides single particle harmonic oscillator wave functions are employed with two different size parameters  B_c and B_v.  In BCM, the halo nucleus is considered as a composite projectile consisting of core and valence clusters bounded in a state of relative motion. The internal densities of the clusters are described by single particle Gaussian wave functions.

 Elastic electron scattering proton f

لمقدمة

     الحمد لله رب العالمين والصلاة والسلام على سيد الأنبياء والمرسلين نبينا محمد صلى الله عليه وسلم وعلى  واصحابه أجمعين  ومن تبعهم وأهتدى بهداهم الى يوم الدين اما بعد :

        فوظيفة القضاء وظيفة سامية يراد منها اقامة العدل ولا يستقيم حالهم الا به دفعاّ للظلم ، ولقد اولى النبي صلى الله عليه وآله وسلم ومن بعده الخلفاء الراشدون