For billiards in two dimensional domains with boundaries containing only focusing and neutral regular components and satisfacting some geometrical conditions ^-property is proved. Some examples of three and more dimensional domains with billiards obeying this property are also considered.

Respiratory motion limits the potential of modern high-precision radiotherapy techniques such as IMRT and particle therapy. Due to the uncertainty of tumour localization, the ability of achieving dose conformation often cannot be exploited sufficiently, especially in the case of lung tumours. Various methods have been proposed to track the position of tumours using external signals, e.g. with the help of a respiratory belt or by observing external markers. Retrospectively gated time-resolved x-ray computed tomography (4D CT) studies prior to therapy can be used to register the external signals with the tumour motion. However, during treatment the actual motion of internal structures may be different. Direct control of tissue motion by online imaging during treatment promises more precise information. On the other hand, it is more complex, since a larger amount of data must be processed in order to determine the motion. Three major questions arise from this issue. Firstly, can the motion that has occurred be precisely determined in the images? Secondly, how large must, respectively how small can, the observed region be chosen to get a reliable signal? Finally, is it possible to predict the proximate tumour location within sufficiently short acquisition times to make this information available for gating irradiation? Based on multiple studies on a porcine lung phantom, we have tried to examine these questions carefully. We found a basic characteristic of the breathing cycle in images using the image similarity method normalized mutual information. Moreover, we examined the performance of the calculations and proposed an image-based gating technique. In this paper, we present the results and validation performed with a real patient data set. This allows for the conclusion that it is possible to build up a gating system based on image data, solely, or (at least in avoidance of an exceeding exposure dose) to verify gates proposed by the various external systems.

In our previous paper Markov partitions for some classes of dispersed billiards were constructed. Using these partitions we estimate the decay of velocity auto-correlation function and prove the central limit theorem of probability theory and Donsker's Invariance Principle for Lorentz Gas with periodic configuration of scatterers.

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We present the first natural and visible examples of Hamiltonian systems with divided phase space allowing a rigorous mathematical analysis. The simplest such family (mushrooms) demonstrates a continuous transition from a completely chaotic system (stadium) to a completely integrable one (circle). In the course of this transition, an integrable island appears, grows and finally occupies the entire phase space. We also give the first examples of billiards with a "chaotic sea" (one ergodic component) and an arbitrary (finite or infinite) number of KAM islands and the examples with arbitrary (finite or infinite) number of chaotic (ergodic) components with positive measure coexisting with an arbitrary number of islands. Among other results is the first example of completely understood (rigorously studied) billiards in domains with a fractal boundary. (c) 2001 American Institute of Physics.