Using analytic methods the mean value of divisor function in the square-free number are studied,and a perfect asymptotic formula of this function is obtained.

The study of divisor and divisor function d(n) are the most basic and important in number theory.

It is proved that there exist infinitely many positive integers n satisfying δ(n)/n>(d(a0)+d(a1)+…+d(ak))/(k+1),where ai(i=0,1,…,k) are all digits of the decimal notation of n,and d(ai)(i=0,1,…,k) is the divisor function of ai.

Magnetorheological finishing tool and removal function;

Controllability of removal function in the ion beam figuring process for optics mirrors;

Influence of the parameters of ion beam on the removal function in IBF process;

The effect of the three kinds of material on the removal function is studied.

The plane motion is discussed based on Preston equation,the workpiece is analog computed with the motion removing function.

Kinematic principles were applied to deduce the removing function of polishing pad of dual-rotator mechanism according to the Preston assumption.

The base concept for computer controlled optical polishing (CCOP) is the deciding of material removing function of the subaperture polishing pad.

As a new and challenging type of imaging spectrometer,its successful launch implies IFTS will play an important role in hyperspectral remote sensing applications,in which apodization function is an important part of interferogram pre-processing,and it has a powerful effect on the accuracy of reconstructed spectra.

The performance of linearly chirped gratings is optimized by designing and changing the parameters of apodization functions,aiming at some issues such as the unexpectedly fluctuating group delay ripple(GDR) in chirped fiber Bragg grating(CFBG) which are applied to achieving all-optical dispersion compensation in 40 Gb/s optical communications.