Generalized Distances in Frameworks

  by   Martin Kovár, Alena Chernikava

Abstract and usage:

Framework is an algebraic structure whose purpose is capturing the topological or topology-like structure of a system from its externally observed properties. Formally, framework is a formal context whose incidence relation is the membership relation. The objects are called places and the attributes are collections of places connected by a possible presence of some physical object, for example, a particle. The set of attributes is called a framology. The structure may be used, for example, for investigation of topological properties of certain causal structures, motivated by quantum gravity, interactions of particles, Feynman diagrams, information systems and databases or solutions of certain differential equations and their global properties. The elements of framology are the places of the dual framework (for the definition of the dual framology, see, e.g., [1]) and they also may be considered as the abstract points of the original framework. The notion of a generalized distance between two abstract points of a framework is inspired by the notion of partial metrics due to S. Matthews. Our experimental application Generalized Distances checks if the original framework is correctly given by the input and then it calculates the tables of three modifications of the generalized distance between the abstract points of the given framework. The calculations are based on various set operations on the given framology.


The set of places:

The framology:


The original framework and the generalized distances of its abstract points:

The original framework:

The abstract points and their simplified notation:

The first generalized distance table:

The second generalized distance table:

The third generalized distance table:


The application Generalized Distances is written in Java powered by Wolfram webMathematica 3.1. The application is hosted at the server of the Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology. The results representing its theoretical background were presented by the authors on several scientific conferences as an integral part of their research. In case of interest in more detail, see [1] or contact the authors. For research and scientific activities the software is available free of charge. In all other cases, please contact RNDr. M. Novák, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, phone: +420 5 4114 3135. Acknowledgement: FEKT-S-11-2/921 "Vlastnosti řešení funkcionálních diferenciálních a diferenčních rovnic"

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