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 .

  Objectives: This study aims to examine toxic parental behavior and its impact on children based on the attachment theory. It also explores how early parental relationships influence future relationships with partners and children. The research assesses types of toxic parenting (dysfunctional, harmful, and psychologically damaging parenting behaviors) and their resulting insecure attachment patterns—avoidant, ambivalent, and disorganized—through the characters in The Marriage of Bette and Boo. Methods: Toxic parents play a decisive role in children’s lives, affecting how they view themselves and impacting their behavior. The study uses a psychological evaluation method that relies on the attachment theory (1978), first prese

        Corruption is a phenomenon that exists in all times and places, they appear in the developing countries and in developed countries, and in all political systems. And its repercussions and economic repercussions on the society in which it is rampant, and lead to the undermining of economic development and obstruction, is also working on increasing poverty and unemployment, as well as its role in the worsening of social problems, and political instability. And corruption is deviant behavior is illegal and contrary to the teachings of the religious and moral values. The emergence of this phenomenon is due to several reasons, including political, economic, social and cultura

In this paper, an efficient image segmentation scheme is proposed of boundary based & geometric region features as an alternative way of utilizing statistical base only. The test results vary according to partitioning control parameters values and image details or characteristics, with preserving the segmented image edges.

Background: Sprite coding is a very effective technique for clarifying the background video object. The sprite generation is an open issue because of the foreground objects which prevent the precision of camera motion estimation and blurs the created sprite. Objective: In this paper, a quick and basic static method for sprite area detection in video data is presented. Two statistical methods are applied; the mean and standard deviation of every pixel (over all group of video frame) to determine whether the pixel is a piece of the selected static sprite range or not. A binary map array is built for demonstrating the allocated sprite (as 1) while the non-sprite (as 0) pixels valued. Likewise, holes and gaps filling strategy was utilized to re

In this study, Bis(4,4’-diaminophenoxy)ethane (compound C1) was synthesized via the reaction of p-hydroxyaniline with 1,2-dibromoethane. Schiff bases (compounds C2–C4) were subsequently obtained by condensing compound C1 with various aromatic aldehydes. These intermediates were further reacted with different anhydrides – namely phthalic anhydride and maleic anhydride – in order to yield the final derivatives (compounds C5–C10). All obtained compounds were characterized by using infrared spectroscopy and proton nuclear magnetic resonance, as well as through an assessment of their physical properties. Antimicrobial evaluation was conducted on some of the generated compounds using two bacterial strains (Escherichia coli and Staphyloc

Heterocyclic compounds are crucial for medicinal chemistry and the development of therapeutic agents like broad-spectrum antibiotics. This study devised a facile procedure to synthesize novel antimicrobial bicyclic heterocycles from 2-mercapto-3-phenylquinazolin-4(3H)-one. Advanced analytical techniques including 1 H and 13C NMR, elemental analysis, and FT-IR spectroscopy characterized the intricate chemical structures of the products. In vitro assays tested the heterocycles against aerobic and anaerobic bacterial strains using fluconazole and ciprofloxacin as antifungal and antibacterial controls. Results demonstrated the formidable broad-spectrum antibacterial and antifungal activities of the synthesized compounds, with growth inhibition

This research aimed to explore the use of foamed concrete (FC), a lightweight concrete, by replacing sand with crushed plastic from washing machines to improve environmental and thermal properties. The primary components included traditional materials like cement, water, and a foaming agent, along with the innovative use of waste plastic (WP). Utilizing Minitab software, the study applied the Response Surface Method to optimize mix designs, reducing the experimental mixtures to 31, which were then tested for dry density, porosity, water absorption, shrinkage, compressive strength, splitting tensile strength, and thermal conductivity. Results indicated a significant 24% reduction in dry density when using plastic as fine and coarse aggregate