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Mathematical Foundations of Non-Classical Extensions of Similarity Theory

  • Stephan Rudolph
Conference paper
Part of the Iutam Bookseries book series (IUTAMBOOK, volume 10)

Abstract

Similarity theory in form of the Pi-Theorem guarantees for any dimensionally homogeneous function the existence of a dimensionless similarity function of dimensionless parameters. Similarity theory is known for its successful applications in science and engineering. Classical applications of similarity theory in engineering mostly exploit the fact that two distinct objects or processes are said to be completely similar if their dimensionless parameters are identical. Similarity theory is used extensively in mathematics for the invariance analysis of differential equations and the derivation of exact and/or approximate solutions. In this work, an extension of the classical applications of similarity theory to artificial intelligence, notably in the fields of case-based and rule-based reasoning, neural networks, data mining, pattern recognition and sound classification, is presented. It is shown that the validity of the extension can be guaranteed by a so-called embedding theorem based on the property of dimensional homogeneity.

Keywords

Similarity theory Pi-theorem embedding theorem artificial intelligence 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Stephan Rudolph
    • 1
  1. 1.Priv.-Doz. Dr.-Ing., Similarity Mechanics Group HeadInstitute for Statics and Dynamics of Aerospace Structures University of StuttgartGermany

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