The Framework Topological Models

  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, e.g., 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. A topological space (X, τ) is called an open model for a framework (P, π) if there exists a framework (S, σ) and a subset X' ⊂ X such that S ⊂ τ and σ={{A | A ∈ S, x ∈ A} | x ∈ X' }. The closed model of a framework with respect to a topological space is defined analogously, for more detail, the reader is referred to [1]. Our application Topological Models checks if the original framework is correctly defined by the input and then it generates the set of all open and closed models of the framework with respect to a given topological space.

Input:
The framework data:

The set of places:

The framology:



The modeling topological space:

The set of points:

The topology:

  Selected open model:  
   
  Selected closed model:  

 
Results:
The original framework and its topological models:

The original framework:



The set of open models:

The number of open models:

The selected open model:

The modeling framework:

The list of possible modeling izomorphisms:

The used points:



The set of closed models:

The number of closed models:


The selected closed model:

The modeling framework:

The list of possible modeling izomorphisms:

The used points:


 

The application Topological Models 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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