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 .

Wellbore instability and sand production onset modeling are very affected by Sonic Shear Wave Time (SSW). In any field, SSW is not available for all wells due to the high cost of measuring. Many authors developed empirical correlations using information from selected worldwide fields for SSW prediction. Recently, researchers have used different Artificial Intelligence methods for estimating SSW. Three existing empirical correlations of Carroll, Freund, and Brocher are used to estimate SSW in this paper, while a fourth new empirical correlation is established. For comparing with the empirical correlation results, another study's Artificial Neural Network (ANN) was used. The same data t

Background: Large amounts of oily wastewater and its derivatives are discharged annually from several industries to the environment. Objective: The present study aims to investigate the ability to remove oil content and turbidity from real oily wastewater discharged from the wet oil's unit (West Qurna 1-Crude Oil Location/ Basra-Iraq) by using an innovated electrocoagulation reactor containing concentric aluminum tubes in a monopolar mode. Methods: The influences of the operational variables (current density (1.77-7.07 mA/cm2) and electrolysis time (10-40 min)) were studied using response surface methodology (RSM) and Minitab-17 statistical program. The agitation speed was taken as 200 rpm. Energy and electrodes consumption had been studi

A novel design and implementation of a cognitive methodology for the on-line auto-tuning robust PID controller in a real heating system is presented in this paper. The aim of the proposed work is to construct a cognitive control methodology that gives optimal control signal to the heating system, which achieve the following objectives: fast and precise search efficiency in finding the on- line optimal PID controller parameters in order to find the optimal output temperature response for the heating system. The cognitive methodology (CM) consists of three engines: breeding engine based Routh-Hurwitz criterion stability, search engine based particle
swarm optimization (PSO) and aggregation knowledge engine based cultural algorithm (CA)

In today's world, the science of bioinformatics is developing rapidly, especially with regard to the analysis and study of biological networks. Scientists have used various nature-inspired algorithms to find protein complexes in protein-protein interaction (PPI) networks. These networks help scientists guess the molecular function of unknown proteins and show how cells work regularly. It is very common in PPI networks for a protein to participate in multiple functions and belong to many complexes, and as a result, complexes may overlap in the PPI networks. However, developing an efficient and reliable method to address the problem of detecting overlapping protein complexes remains a challenge since it is considered a complex and har

Early detection of eye diseases can forestall visual deficiency and vision loss. There are several types of human eye diseases, for example, diabetic retinopathy, glaucoma, arteriosclerosis, and hypertension. Diabetic retinopathy (DR) which is brought about by diabetes causes the retinal vessels harmed and blood leakage in the retina. Retinal blood vessels have a huge job in the detection and treatment of different retinal diseases. Thus, retinal vasculature extraction is significant to help experts for the finding and treatment of systematic diseases. Accordingly, early detection and consequent treatment are fundamental for influenced patients to protect their vision. The aim of this paper is to detect blood vessels from