Sums of Exponential Functions and their New Fundamental Properties,
with Applications to Natural Phenomena
This whole book is based upon one Theorem. We prove the Theorem in this paper and, in fact, the book content largely relates to the Theorem and introduction of methods and concepts used in the Theorem’s proof. However, the first chapter of the book introduces more general considerations which deal with the relationship of mathematical methods and philosophical concepts with Nature. The purpose of this chapter is to convince the reader in close relations of these three entities. If the reader is willing to explore only the mathematical content of this book he should skip this general introduction and go directly to chapter two, maybe with a short stop at section 1.7. The last section studies the relations of Fermat’s Last Theorem with the physical world. Fermat’s Last Theorem, in fact, is also a particular case of an equation defined by a sum of exponential functions that we study in this paper, although with integer variables.
Corollary 8 should be of interest to mathematicians working in the area of finance. This Corollary finally (at least we hope so) resolves the centuries old enigma about how many solutions the IRR equation has (IRR stands for Internal Rate of Return). This equation is a foundation and the main mathematical vehicle in a large area of mathematical finance dealing with mortgages, annuities, rates of return on investments, etc. It turns out that the IRR equation can have a maximum of three solutions. Details can be found in Section 3.4.8. However, we have to warn the reader that the proof is not trivial.
Large part of the book explores other Corollaries of the Theorem. They have very interesting practical applications and links with Nature, which go far beyond their mathematical subject matter. These results are related to many, if not all, areas of human activity and Nature in general. This may sound like an exaggeration. Of course, authors tend to overstate the significance of their discoveries at the beginning, being overwhelmed by their emotions. However, our assertion regarding the Theorem’s generality can be verified. This paper presents some arguments in support of this claim, and leaves it to the readers to draw their own conclusions.
The Theorem itself is stated as follows.
A sum of exponential functions can have a maximum of two points of abscissa intersection, two extremums and two inflection points. Each subsequent derivative of this sum can be equal to zero no more than two times.
So what? This is the question the reader may ask after reading these lines. At first glance, it looks like the subject that only mathematicians can be interested in. Well, this is not exactly true. For instance, let us take a look at Corollary 1 of the Theorem, which states:
Corollary 1 (One-time oscillation property)
Natural exponential phenomena created through the confluence of many exponential factors, which is the case for many natural phenomena and processes, have a “one-time oscillation (fluctuation)” property. Such phenomena can undergo a “set back” only once, after which they will continue to move in the previous direction.
Many natural phenomena are actually created through the confluence of several exponential factors. Knowledge of such deterministic behavior of the overall process allows forecasting the future with much more certainty, by immediately filtering away a myriad of invalid hypothetical scenarios. So, it turns out that there is much more certainty in Nature’s eternal process of change than thought before. This Corollary alone opens a whole new space to be populated by all sorts of discoveries, ranging from practical applications to general philosophical laws.
The Theorem has eight Corollaries and one Conjecture. It is not as if we could not prove the Conjecture. We did not even try. Maybe we will return to it later. This project consumed a lot of resources of different kinds, and we decided to stop and replenish them. All of the methods and concepts you will encounter in this paper have been developed from scratch. We could find nothing that would help us in solving the problem, and so, we invented methods to do this. The reader should be prepared to face some non-trivial concepts and transitions, but the good thing is that to understand what is written here is by no means impossible. In fact, knowledge of undergraduate calculus, an open mind, and scientific curiosity are sufficient to master the mathematical content of the book.
We present also more general considerations about the structure and inner workings of natural phenomena, for instance the cells’ growth and replication. Some of these considerations will resemble philosophical generalizations. However, readers may find practical value in them as well.
The book presents also an application of the discovered fundamental properties by considering a sum of exponential electrical signals. We show that the resulting electrical signal can only have one oscillation, regardless of how many exponential signals have been added. The author of this book cites his earlier article, which provides some experimental confirmation for this one-time oscillation property. The original purpose of the experiment was different. The author theoretically discovered a parametric effect in long electrical lines, and obtained experimental proof of this result. The one-time oscillation effect presented itself in the experiment, but it was not given enough attention. However, this effect was there, and has been documented using oscilloscope images, which the referenced article presents.
Of course, this experiment does not provide bullet-proof confirmation that sums of exponential electrical signals possess the one-time oscillation property, because of the limited scope of the obtained data. However, the experiment does support the result. Additional experiments that target this one-time oscillation property of sums of exponential electrical signals are required to perform scientifically valid verification of our result.
The primary goal of the book is to make the Theorem available for discussion by the community. The second goal is to attract the attention of specialists in other areas of science and industry to practical implications of the Theorem and its Corollaries. Any thoughtful feedback is welcome. The author is willing to clarify the issues, and address the questions which inevitably emerge when new results of this scale appear.
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