A new definition of random set

A new definition of random sets is proposed in the presented paper. It is based on a special distance in a measurable space and uses negative definite kernels for continuation from the initial space to the one of the random sets. Motivation for introducing the new definition is that the classical approach deals with Hausdorff distance between realisations of the random sets, which is not satisfactory for statistical analysis in many cases. We place the realisations of the random sets in a complete Boolean algebra (B.A.) endowed with a positive finite measure intended to capture important characteristics of the realisations. A distance on B.A. is introduced as a square root of measure of symmetric difference between its two elements. The distance is then used to define a class of Borel subsets of B.A. Consequently, random sets are defined as measurable mappings taking values in the B.A. This approach enables us to use more general family of distances between realisations of random sets which allows us to make new statistical tests concerning equality of some characteristics of random set distributions. As an extra result, the notion of stability of newly defined random sets with respect to intersections is proposed and limit theorems are obtained

Quantitative characteristics of inhomogeneous microstructure in UFG copper

The ultrafine-grained microstructure of pure copper processed by equal-channel angular pressing and its temperature-induced changes were evaluated in order to characterize heterogeneous distribution of fine- and larger-sized grains in the microstructure. ECAP was conducted at room temperature with a die that had an internal angle of 90° between the two parts of the channel. The subsequent extrusion passes were performed by route Bc up to 8 ECAP passes and tested under constant load. Creep test was performed on the samples processed by 8 ECAP passes in tension at 373 K under an applied stress 260 MPa. The analyses of microstructure were performed by 3 dimensional electron-back scatter diffraction (3D EBSD) technique. The volume characteristics of microstructure and its inhomogeneity were evaluated and the relationships microstructure/creep behaviour of UFG copper was discussed. © Published under licence by IOP Publishing Ltd

Modely pro náhodné sjednocení kruhů se vzájemnými interakcemi

Katedra pravděpodobnosti a matematické statistikyDepartment of Probability and Mathematical StatisticsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Models for Random Union of Interacting Discs

Katedra pravděpodobnosti a matematické statistikyDepartment of Probability and Mathematical StatisticsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Quermass-interaction process with convex compact grains

summary:The paper concerns an extension of random disc Quermass-interaction process, i.e. the model of discs with mutual interactions, to the process of interacting objects of more general shapes. Based on the results for the random disc process and the process with polygonal grains, theoretical results for the generalized process are derived. Further, a simulation method, its advantages and the corresponding complications are described, and some examples are introduced. Finally, a short comparison to the random disc process is given

ASSESSING DISSIMILARITY OF RANDOM SETS THROUGH CONVEX COMPACT APPROXIMATIONS, SUPPORT FUNCTIONS AND ENVELOPE TESTS

In recent years random sets were recognized as a valuable tool in modelling different processes from fields like biology, biomedicine or material sciences. Nevertheless, the full potential of applications has not still been reached and one of the main problems in advancement is the usual inability to correctly differentiate between underlying processes generating real world realisations. This paper presents a measure of dissimilarity of stationary and isotropic random sets through a heuristic based on convex compact approximations, support functions and envelope tests. The choice is justified through simulation studies of common random models like Boolean and Quermass-interaction processes