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 .

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

The rotor dynamics generally deals with vibration of rotating structures. For designing rotors of a high speeds, basically its important to take into account the rotor dynamics characteristics. The modeling features for rotor and bearings support flexibility are described in this paper, by taking these characteristics of rotor dynamics features into standard Finite Element Approach (FEA) model. Transient and harmonic analysis procedures have been found by ANSYS, the idea has been presented to deal with critical speed calculation. This papers shows how elements BEAM188 and COMBI214 are used to represent the shaft and bearings, the dynamic stiffness and damping coefficients of journal bearings as a matrices have been found

In this paper, the system of the power plant has been investigated as a special type of industrial systems, which has a significant role in improving societies since the electrical energy has entered all kinds of industries, and it is considered as the artery of modern life.

   The aim of this research is to construct a programming system, which could be used to identify the most important failure modes that are occur in a steam type of power plants. Also the effects and reasons of each failure mode could be analyzed through the usage of this programming system reaching to the basic events (main reasons) that causing each failure mode. The construction of this system for FMEA is dependi

     In this article, we aim to define a universal set consisting of the subscripts of the fuzzy differential equation (5) except the two elements  and , subsets of that universal set are defined according to certain conditions. Then, we use the constructed universal set with its subsets for suggesting an analytical method which facilitates solving fuzzy initial value problems of any order by using the strongly generalized H-differentiability. Also, valid sets with graphs for solutions of fuzzy initial value problems of higher orders are found.

The Frequency-hopping Spread Spectrum (FHSS) systems and techniques are using in military and civilianradar recently and in the communication system for securing the information on wireless communications link channels, for example in the Wi-Fi 8.02.X IEEE using multiple number bandwidth and frequencies in the wireless channel in order to hopping on them for increasing the security level during the broadcast, but nowadays FHSS problem, which is, any Smart Software Defined Radio (S-SDR) can easily detect a wireless signal at the transmitter and the receiver for the hopping sequence in both of these, then duplicate this sequence in order to hack the signal on both transmitter and receiver messages using the order of the se

sensor sampling rate (SSR) may be an effective and crucial field in networked control systems.  Changing sensor sampling period after designing the networked control system is a critical matter for the stability of the system. In this article, a wireless networked control system with multi-rate sensor sampling is proposed to control the temperature of a multi-zone greenhouse. Here, a behavior based Mamdany fuzzy system is used in three approaches, first is to design the fuzzy temperature controller, second is to design a fuzzy gain selector and third is to design a fuzzy error handler. The main approach of the control system design is to control the input gain of the fuzzy temperature controller depending on the cur