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 .

Praise be to God, Lord of the worlds, and prayers and peace be upon the most honorable of God’s creation, Muhammad bin Abdullah, the illiterate Arab prophet, the faithful, and his family and companions.
And yet ...

     The realistic view is that Islamic jurisprudence is not exaggerated in the judgment of compensation, so it is judged for everything that has been lost by the victim of a profit or his loss, but he takes into account the bearing of the guarantee between profit and loss or the principle on which the theory of bearing liability is found, and it is clear through research that Islam The principle of respecting money, souls and rights is considered a public order, but it may exclude some ca

The accretion circumstellar disk of young stars and the Brown dwarf plays an essential role in the formation and evaluation of the planet. Our main work in this paper is to investigate the geometrical shape model for the protoplanetary disk around one of the Brown Dwarfs. The photometric measurements for the brown dwarf CFHT-BD-Tau 4 were extracted from the Vizier archive. We used a numerical simulation to build a model of the spectral energy distribution of our target CFHT-BD-Tau 4. The spectral energy distribution model was fitted with observational data for the brown dwarf CFHT-BD-Tau 4. A transitional disk has been assumed around CFHT-BD-Tau 4. We obtained physical properties of the two disks and the size of the gap between them

This study investigates the stomach morphology and histochemistry of Clarias gariepinus. Grossly, the stomach is a J-shaped organ with three distinct regions: cardiac, fundic, and pyloric. Histologically, its wall comprises four layers: mucosa, submucosa, muscularis externa, and serosa. The mucosa exhibits broad longitudinal folds lined by high columnar cells with basal oval nuclei. These cells contain apical mucosubstances that react positively with Periodic Acid Schiff (PAS) stain and negatively with Alcian Blue (AB). Gastric pits result from mucosal invaginations. Glands are present in the fundic and cardiac regions but absent in the pyloric. Oxynticopeptic cells exclusively line the fundic glands. Enteroendocrine cells are distr

HR Ghanim, GA Abdulhassan, International Journal of Early Childhood Special Education, 2022

This study investigates the stomach morphology and histochemistry of Clarias gariepinus. Grossly, the stomach is a J-shaped organ with three distinct regions: cardiac, fundic, and pyloric. Histologically, its wall comprises four layers: mucosa, submucosa, muscularis externa, and serosa. The mucosa exhibits broad longitudinal folds lined by high columnar cells with basal oval nuclei. These cells contain apical mucosubstances that react positively with Periodic Acid Schiff (PAS) stain and negatively with Alcian Blue (AB). Gastric pits result from mucosal invaginations. Glands are present in the fundic and cardiac regions but absent in the pyloric. Oxynticopeptic cells exclusively line the fundic glands. Enteroendocrine cells are distr

This research aims to know the effect of adopting IFRS 9 on the relevance of the value of the accounting information of the companies in the Iraqi Stock Exchange. Researchers relied on analyzing the financial statements of 10 listed companies for years 2016 – 2019. Researchers used the Ohlson price model to test the relationship between accounting information and value relevance. The research indicated that there is a significant relationship between the adoption of IFRS 9 and the relevance of the value of the earnings and the book value, but the earnings information is more relevance than the book value information, it is due to the interest of investors in the income statement in making investment decisions.

Objective: Schiff’s and Mannich bases of isatins are an important group of heterocyclic compounds which are of great importance in medicinal chemistry as antimicrobial agents. In the vision of these facts, new bis-Schiff bases and Mannich bases of isatins were synthesized. Methods: Three different bis-Schiff bases (3a-c) have been synthesized by reacting isatin, 5-fluoroisatin and 5-methoxy isatin with thiophene-2- carboxaldehyde using hydrazine hydrate to link between the carbonyl compounds, and then these bis- Schiff bases were condensed with two different secondary amines (piperidine and morpholine) separately, and formaldehyde to form the Mannich bases (4a-c and 5a-f), respectively. Results: The structures of the newly synthesized com