The arithmetic mean expectation model is not able to measure the income of portfolio investment correctly as bonds are a kind of successive investment,but the geometric average expectation can measure the income of successive investment.

Pricing of geometric average Asian options under time-dependent parameters;

The comparison between continuos random variables and power function distribution and some strong limit theorems of geometric average;

The pricing formulas of default geometric average asian option with stochastic liabilities;

Properties of weighted geometric means combination forecasting model based on L_1 norm;

Weighted geometric means combination forecasting method based on L_1 norm;

On a mixed arithmetic-geometric mean inequality;

Two-stage BCC model for cooperative efficiency evaluation using a geometric mean method;

For multi-component signals,Wigner distribution(WD) is well energy-concentrated but holds cross-terms,while spectrogram,the magnitude-squared STFT,has no cross-terms but bad energy-concentration,thus a time-frequency representation based on geometric mean of WD and Spectrogram(GMWS),which is simple and easy to be realized,was introduced to integrate both advantages.

With the boundary conditions,we derive the analytic formula of the rainbow geometric average Asian call option and call-put parity relationship on rainbow geometric average Asian options.

The practical usage of weighted mean in cotton had been expounded through calculated method of circular cone position in cotton bolls and mean of boll weight,and the practical usage of geometric mean,the growth per cent of plant height a day had been expounded through analysing exponential increasing phase of plant height in cotton.