The aim of this paper is to prove some results for equivalence of moduli of smoothnes in approximation theory , we used a"non uniform" modulus of smoothness and the weighted Ditzian –Totik moduli of smoothness in by spline functions ,several results are obtained .For example , it shown that ,for any the inequality , is satisfied ,finally, similar result for chebyshev partition and weighted Ditzian –Totik moduli of smoothness are also obtained.

Pyrolysis of high density polyethylene (HDPE) was carried out in a 750 cm3 stainless steel autoclave reactor, with temperature ranging from 470 to 495° C and reaction times up to 90 minute. The influence of the operating conditions on the component yields was studied. It was found that the optimum cracking condition for HDPE that maximized the oil yield to 70 wt. % was 480°C and 20 minutes. The results show that for higher cracking temperature, and longer reaction times there was higher production of gas and coke. Furthermore, higher temperature increases the aromatics and produce lighter oil with lower viscosity.

In this paper , an efficient new procedure is proposed to modify third –order iterative method obtained by Rostom and Fuad [Saeed. R. K. and Khthr. F.W. New third –order iterative method for solving nonlinear equations. J. Appl. Sci .7(2011): 916-921] , using three steps based on Newton equation , finite difference method and linear interpolation. Analysis of convergence is given to show the efficiency and the performance of the new method for solving nonlinear equations. The efficiency of the new method is demonstrated by numerical examples.

The radial wave function R(r) and the radial distribution function P(r) as a function of (r), for the Hydrogen atom was calculated for several atomic state (1s,2s,2p,3s,3p,3d) The results were compared with Hydrogen like atom(He+,Li+2,Be+3).

In this paper, we introduce weak and strong forms of ω-perfect mappings, namely the -ω-perfect, weakly -ω-perfect and strongly-ω-perfect mappings. Also, we investigate the fundamental properties of these mappings. Finally, we focused on studying the relationship between weakly -ω-perfect and strongly -ω-perfect mappings.

Some researchers are interested in using the flexible and applicable properties of quadratic functions as activation functions for FNNs. We study the essential approximation rate of any Lebesgue-integrable monotone function by a neural network of quadratic activation functions. The simultaneous degree of essential approximation is also studied. Both estimates are proved to be within the second order of modulus of smoothness.

this paper presents a novel method for solving nonlinear optimal conrol problems of regular type via its equivalent two points boundary value problems using the non-classical