The α-power geometric mean gα(a,b) and generalized spectral geometric mean Eα(a,b) of two positive definite elements a and b in a C*-algebra is introduced.

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.

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.