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Open access

Roman Frič and Martin Papčo

Abstract

The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

Open access

Martin Bednárik, Juraj Papčo, Vladimír Pohánka, Vladimír Bezák, Igor Kohút and Ladislav Brimich

Abstract

The surface deformation of the Tatra Mountains region in Western Carpathians can nowadays be studied directly thanks to precise geodetic measurements using the GNSS. The strain or stress tensor field is, however, a rather complex “data structure” difficult to present legibly and with sufficient resolution in the form of a classical map. A novel and promising approach to the solution of this problem is coding the three principal strain or stress values into the three colour channels (red, green, blue) of an RGB colour. In our previous study, the colour depended on the stress tensor shape descriptors. In the current study, the adapted colouring scheme uses a subset of shape descriptors common to stress and strain, which differ only in the scaling factor. In this manner, we generate the colour map of the surface strain rate field, where the colour of each grid point carries the information about the shape of the strain rate tensor at that point. The resulting strain rate colour map can be displayed simultaneously with the map of the faults or elevations and be easily checked for the data or interpolation method errors and incompatibility with the geophysical and geological expectations.

Open access

Roman Pašteka, Pavol Zahorec, David Kušnirák, Marián Bošanský, Juraj Papčo, Viktória Szalaiová, Martin Krajňák, Marušiak Ivan, Ján Mikuška and Miroslav Bielik

Abstract

The paper deals with the revision and enrichment of the present gravimetric database of the Slovak Republic. The output of this process is a new version of the complete Bouguer anomaly (CBA) field on our territory. Thanks to the taking into account of more accurate terrain corrections, this field has significantly higher quality and higher resolution capabilities. The excellent features of this map will allow us to re-evaluate and improve the qualitative interpretation of the gravity field when researching the structural and tectonic geology of the Western Carpathian lithosphere. In the contribution we also analyse the field of the new CBA based on the properties of various transformed fields – in particular the horizontal gradient, which by its local maximums defines important density boundaries in the lateral direction. All original and new transformed maps make a significant contribution to improving the geological interpretation of the CBA field. Except for the horizontal gradient field, we are also interested in a new special transformation of TDXAS, which excellently separates various detected anomalies of gravity field and improves their lateral delimitation.