TITLE:
Like a Sum Is Generalized into an Integral, a Product May Be Generalized into an Inteduct
AUTHORS:
Michael Grabinski, Galiya Klinkova
KEYWORDS:
Geometric Mean, Chaos, Finance, Ergodicity
JOURNAL NAME:
Applied Mathematics,
Vol.14 No.5,
May
29,
2023
ABSTRACT: It is well known that an integral is nothing but a continuous form of a sum. Is it possible to do the same thing with a product? The answer is yes and done for the first time in this publication. The new operator is called inteduct. As an integral is a proper tool to calculate the arithmetic mean of a function, the inteduct gives the geometric mean of a function. This defines a new branch of mathematics. Most applications may lay way ahead. Only some are discussed here. One is applying the inteduct to probability theory. There it is possible e.g., to determine a function for a life expectation rather than just a numerical value. Another application is to distinguish chaos from randomness within numerically given values. At least for the logistic map there exists a direct connection between Lyapunov exponent and inteduct. To distinguish between chaos and randomness is particularly important in finance. While randomness implies ergodicity, chaos is non-ergodic. And many fundamental financial theories from portfolio theory to market efficiency require ergodicity.