Saturday, May 10, 2008

[GR 1] Formalizing qualitative properties

Sometimes the human mind intuitively recognizes a new property P of some object X, which can initially be stated in vague, qualitative terms only. For example: "This trajectory is marked by subsequent periods of hopping and stalling".

One would then like to define a mathematical function p(X) which returns for each suitable object X a number p that measures to which extend this property is really present - the "p-ness" of X.
  • In order for this to work, the objects X must be parametrizable by a set of N numbers, so that each concrete sample object occupies a point in an N-dimensional space. The function p(X) is a scalar field over that space.
  • The p-function should reflect our intuitive property as closely as possible.
  • By comparing p(X) with p(Y), one can order X and Y with respect to the p-property.
  • Any monotoneous function q(p) would also allow for the ordering, so there is some ambiguity in the definition.
  • If the values of p are known for some "standard objects", one can even make qualitative statements: "The p-temperature is larger than that of freezing water, but smaller than that of boiling water".
  • If the property is "binary" (as in the case of "subdiffusive/superdiffusive") one must provide a standard object defining the border line ("just diffusive").

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