Projectile fragmentation cross sections are calculated for reactions of 20 Ne on the C, Al, Cu, Sn and Pb targets at 400 A MeV by using the improved quantum molecular dynamics model together with the statistical model code GEMINI. The improved quantum molecular dynamics model is applied to describe the dynamical process, and the GEMINI model is used to simulate the de-excitation process of excited primary fragments. It is found that the total cross sections increase as a function of the target mass, which is in good agreement with the experimental results and other theoretical predictions. The odd-even effect of the partial cross sections observed in experiments is well reproduced, which appears in the de-excitation process of the excited primary fragments as a result of pairing effect and is mainly formed in the grazing collisions. The results of isospin distributions demonstrate that the odd-even effect of partial cross sections mainly comes from the fragments with T Z = 0, ±0.5.

The total charge-changing cross sections and partial cross sections for the production of projectile fragments are measured in the interactions of 400 A MeV 20Ne with aluminum, carbon and polyethylene targets sandwiched with CR-39 plastic nuclear track detectors. The measured total charge-changing cross sections are compared with the predictions using the Bradt-Peters semi-empirical formula, and the NUCFRAG2 and PHITS models. It is shown that the measured experimental results are in good agreement with the theoretical model prediction and other experimental results, and it can be clearly seen that the partial cross sections for fragment production show obvious odd-even effects.

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In this paper, we introduce a time-changed geometric Brownian motion and investigate the corresponding martingale properties and fractional Fokker-Planck type equation. As an application, we prove that the market model considered is arbitrage-free and gives pricing formulae for the prices of European call options when the underlying asset price follows the time-changed geometric Brownian motion.

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