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 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 .

Laser is a powerful device that has a wide range of applications in fields ranging from materials science and manufacturing to medicine and fibre optic communications. One remarkable

Air pollution is one of the important problems facing Iraq. Air pollution is the result of uncontrolled emissions from factories, car exhaust electric generators, and oil refineries and often reaches unacceptable limits by international standards. These pollutants can greatly affect human health and regular population activities. For this reason, there is an urgent need for effective devices to monitor the molecular concentration of air pollutants in cities and urban areas. In this research, an optical system has been built consisting of aHelium-Neonlaser,5mWand at 632.8 nm, a glass cell with a defined size, and a power meter(Gentec-E-model: uno) where a scattering of the laser beam occurs due to air pollution. Two pollutants were examin

Linear Feedback Shift Register (LFSR) systems are used  widely in stream cipher systems field. Any system of LFSR's which wauldn't be attacked must first construct the system of linear equations of the LFSR unit. In this paper methods are developed to construct a system of linear/nonlinear equations of key generator (a LFSR's system) where the effect of combining (Boolean) function of LFSR is obvious. Before solving the system of linear/nonlinear equations by using one of the known classical methods, we have to test the uniqueness of the solution. Finding the solution to these systems mean finding the initial values of the LFSR's of the generator. Two known generators are used to test and apply the ideas of the paper,