Historical and mathematical aspects of iterative solutions for Monte Carlo simulations

Over the last 25 years Monte Carlo programs were being developed in Cracow in the group guided by Prof. Stanislaw Jadach. Many of those programs became standard in their application domains. In the following let us review some aspects of such projects which were probably at the foundation of their success. We will concentrate on mathematical aspects of their design and history of their construction. It is rather difficult to cover 25 years of the research in a single talk. That is why, I have organised my presentation around Monte Carlo PHOTOS but stressing its relation to other activities and projects often realized together with Prof. Jadach. Many of omitted aspects will find their way into other presentations collected in this volume. 1 will concentrate on issues related to phase-space parametrisation and spin amplitudes as used in our Monte Carlo programs such as MUSTRAAL, TAUOLA or KKMC and their similarities and differences with respect to solution used in PHOTOS.