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Shape as an Information‐Thermodynamical Concept and the Pattern Recognition Problem
Author(s) -
Ingarden Roman S.,
Górniewicz Lech
Publication year - 1990
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19901450108
Subject(s) - mathematics , countable set , entropy (arrow of time) , compact space , topological entropy , pure mathematics , topology (electrical circuits) , combinatorics , physics , quantum mechanics
A hierarchical classification of different concepts of shape of compact connected sets in Rn (topological, Lipshitz, homotopic, Borsuk an homological shapes) is given. The most general among them is the homological shape. There is only a countable number of homological shapes for connected compact sets in Rn . In the case n = 2 even the number of different Borsuk shapes for connected compact sets is countable. Giving a probability distribution of shapes we can define a shape entropy, a mean shape and shape fluctuations. This enables a formulation of information thermodynamics of shape and its applications to different fields (physics – small systems, chemistry, biophysics, pattern recognition). The paper does not develop yet these applications, its aim is to clear the basic notions.