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 .

Varied uses of international rivers in the past few decades dramatically, resulting in this multiplicity of uses and all associated with it for the occurrence of freshwater scarcity activities, and thus an increase in conflicts and disputes around on the rights of each of the riparian countries to benefit from the waters of the river at various purposes, particularly the establishment of dams on some of them as is the case (Renaissance Ethiopian) big impact on downstream countries Dam (Egypt and Sudan), due to the Oukuahma at the end of the Nile Valley made them vulnerable to environmental fluctuations, political crises facing the Nile basin countries, and any reduction in the proportion of water is not only the Nile River, but for all r

Accurate computation of the roughness coefficient is important in the studies of open channel flow. To measure and identify the hydraulic characteristics of the flow system, the model simulation is necessary to study and get the results of the hydraulic properties to specify Manning coefficient of the Euphrates River. In this study, the reach is extended along the Euphrates River from Haditha Dam to Ramadi Barrage with a distance of 169km. The HEC-RAS model was implemented to simulate the flow within the study reach. The geometry of the river was represented by more than two hundred cross-sections surveyed in 2013 and 2021. The model was calibrated using some observed discharges at the Heet gage station for records of th

In this paper, we characterize the percolation condition for a continuum secondary cognitive radio network under the SINR model. We show that the well-established condition for continuum percolation does not hold true in the SINR regime. Thus, we find the condition under which a cognitive radio network percolates. We argue that due to the SINR requirements of the secondaries along with the interference tolerance of the primaries, not all the deployed secondary nodes necessarily contribute towards the percolation process- even though they might participate in the communication process. We model the invisibility of such nodes using the concept of Poisson thinning, both in the presence and absence of primaries. Invisibility occurs due to nodes

This study is conducted in order to, investigate the trophic state of Duhok Lake Dam located within Duhok city, Iraq. Water samples are collected seasonally from three monitored sites during 2011. The parameters used for assessing water quality and trophic status level include: water temperature, pH, EC, TDS, DO, BOD5, nutrients, Secchi disk transparency, and chlorophyll a. The results reveal that DO is above 5 mg.l-1 in all sites, BOD5 value is within permissible level for domestic uses. Water quality considered as a hard type. High sulfate concentration is recorded during the study period. Trophic state shows that water type is classified as mesotrophic during autumn season, while it is regarded as eutrophic in other seasons. TDN/TDP rati

The present study reports the effect of temperature and liquid hourly space velocity (LHSV) on the cumene cracking reaction rate and selectivity by using a laboratory continuous flow unit with fixed bed reactor operating at atmospheric pressure. The prepared HX zeolite was made from Iraqi kaolin with good crystallinity .The activity and selectivity of prepared HX-zeolite was compared with standard HY zeolite and HX zeolite catalysts in the temperature range of 673-823K and LHSV of 0.7-2.5 h-1 . It was found that the cumene conversion increases with increasing temperature and decreasing LHSV at 823K and LHSV of 0.7 h-1 the conversions 65.32, 42.88 and 59.42 mol% for HY, HX and prepared HX catalysts respectively and at LHSV of 2.5 h-1 and th