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The paper concerns estimation of anisotropy of planar fibre systems using the relation between the rose of directions and the rose of intersections. The discussion about the properties of the Steiner compact estimator is based on both theoretical and simulation results. The approach based on the distribution of the Prokhorov distance between the estimated and true rose of directions is developed. Finally the curved test systems are investigated in both Fourier and Steiner compact analysis of anisotropy....
Suppose that red and blue points occur as independent homogeneous Poisson processes in ℝd. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d=1, 2, the matching distance X from a typical point to its partner must have infinite d/2th moment, while in dimensions d≥3 there exist schemes where X has finite exponential moments. The Gale–Shapley stable marriage is one natural matching scheme, obtained by iteratively...
Článek se věnuje známé matematické úloze, která nese název problém čtyř bodů. Její řešení není jednoznačné, liší se podle volby pravděpodobnostního rozdělení bodů. Zde uvádíme řešení při volbě rovnoměrného rozdělení na množinách různých tvarů a porovnáváme jednotlivé výsledky.
Functionals of spatial point process often satisfy a weak spatial dependence
condition known as stabilization. In this paper we prove process level
moderate deviation principles (MDP) for such functionals, which is
a level-3 result for empirical point fields as well as a level-2 result
for empirical point measures. The level-3 rate function coincides with
the so-called specific information. We show that the general result
can be applied to prove MDPs for various particular functionals,
including...
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