Sums of Exponential Functions and Their New Fundamental Properties With Applications to Natural Phenomena

Exponential functions constitute the foundation of numerous mathematical models, which adequately describe physical, biological and other types of real phenomena. The author discovered and proved new fundamental properties of sums of exponential functions, and illustrated application of these properties to different kinds of natural phenomena. In particular, he considered applications in biology, in the area of cells' growth and replication; he investigated also sums of transitional electrical signals, and analyzed other natural phenomena with relationship to their mathematical models, with some philosophical generalizations. The discovered properties of sums of exponential functions are formulated in the form of a mathematical Theorem, added with its eight corollaries and one conjecture. In particular, using the Theorem, the author solved a century old enigma how many solutions the IRR equation has. (This equation, which abbreviation stands for Internal Rate of Return, is a foundation of a large area of financial mathematics studying mortgages, annuities, other lending instruments, and investment performance measurement.) The author introduced also a new mathematical concepts of pair functions, pair functions synchronization and other interesting concepts in order to prove the Theorem. However, these new concepts have a much broader scope of application, and can be used in other areas of applied and pure mathematics, and in practical applications as a valuable tool for modeling of natural phenomena, data interpretation, etc. In general, the Theorem and its Corollaries represent very useful quantitative and qualitative instruments for the discovery and understanding of different aspects of Nature, and can be applied in numerous areas of science and technology.