Non-uniqueness, Envelopes and the Edge of Regression

A piece of dogma current in the physical and engineering sciences is the idea that a theory is of no use if, given initial data, a unique outcome can NOT be predicted. The concept of an envelope is precisely when uniqueness fails, yet the useful concept of a set of Huygen wavelets building a wave front is precisely the idea of an envelope. The concept of a discontinuity is also evidence of non-uniqueness in the real non-ideal world.

From another point of view, the Froenius integrability theorem states the conditions required such that a Pfaffian expression (ordinary differential equation) has a unique direction field solution. The requirement in simple cases is precisely, the topological torsion A^dA, must vanish. Hence, somehow the concept that A^dA <> 0 is to be associated with non-uniqueness and existence of an envelope. This idea is exploited in the enclosed pdf file below.

pdf.gif Envelopes Notes 1

image As I have time I will replace this under construction
symbol with a link to a more detailed pdf or tex file download.

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Last update 07/02/2001
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