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 .

The impact of a proposal for the curriculum dictated by some of the barriers to the development of motor skillsA. M. D. Huda Ibrahim RezoukiM. M. Susan Salim DawoodIs the childhood of the most important and most fertile stage of basic education because of their significant impact in building a base on which the stated origin of sound in the early stages older age because the baby at this stage be very vulnerability to environmental factors different surroundings in which to leave their mark on his life in the stages of life other.The Gymnastics of sports long-term training, which begins between the ages very early relative to the rest of other games to reach the stage of the tournament from the age of 13-17 years almost to the women, this a

Heuristic approaches are traditionally applied to find the optimal size and optimal location of Flexible AC Transmission Systems (FACTS) devices in power systems. Genetic Algorithm (GA) technique has been applied to solve power engineering optimization problems giving better results than classical methods. This paper shows the application of GA for optimal sizing and allocation of a Static Compensator (STATCOM) in a power system. STATCOM devices used to increase transmission systems capacity and enhance voltage stability by regulate the voltages at its terminal by controlling the amount of reactive power injected into or absorbed from the power system. IEEE 5-bus standard system is used as an example to illustrate the te

In high-dimensional semiparametric regression, balancing accuracy and interpretability often requires combining dimension reduction with variable selection. This study intro- duces two novel methods for dimension reduction in additive partial linear models: (i) minimum average variance estimation (MAVE) combined with the adaptive least abso- lute shrinkage and selection operator (MAVE-ALASSO) and (ii) MAVE with smoothly clipped absolute deviation (MAVE-SCAD). These methods leverage the flexibility of MAVE for sufficient dimension reduction while incorporating adaptive penalties to en- sure sparse and interpretable models. The performance of both methods is evaluated through simulations using the mean squared error and variable selection cri

In this paper the variable structure control theory is utilized to derive a discontinuous controller to the magnetic levitation system. The magnetic levitation system model is considered uncertain, which subjected to the uncertainty in system parameters, also it is open-loop unstable and strongly nonlinear. The proposed variable structure control to magnetic levitation system is proved, and the area of attraction is determined. Additionally, the chattering, which induced due to the discontinuity in control law, is attenuated by using a non-smooth approximate. With this approximation the resulted controller is a continuous variable structure controller with a determined steady state error according to the selected control

Inefficient wastewater disposal and wastewater discharge problems in water bodies have led to increasing pollution in water bodies.  Pollutants in the river contribute to increasing the biological oxygen demand (BOD), total suspended solids (SS), total dissolved solids (TDS), chemical oxygen demand (COD), and toxic metals render this water unsuitable for consumption and even pose a significant risk to human health. Over the last few years, water conservation has been the subject of growing awareness and concern throughout the world, so this research focused on review studies of researches that studied the importance of water quality of wastewater treated disposal in water bodies and modern technology to management w

Spraying pesticides is one of the most common procedures that is conducted to control pests. However, excessive use of these chemicals inversely affects the surrounding environments including the soil, plants, animals, and the operator itself. Therefore, researchers have been encouraged to...

This paper proposes a self organizing fuzzy controller as an enhancement level of the fuzzy controller. The adjustment mechanism provides explicit adaptation to tune and update the position of the output membership functions of the fuzzy controller. Simulation results show that this controller is capable of controlling a non-linear time varying system so that the performance of the system improves so as to reach the desired state in a less number of samples.