"Sums of exponential functions and their new fundamental properties, with Applications to Natural Phenomena"


Midwest Book Reviews (Oregon, WI, USA)

... very highly recommended reading for advanced mathematics students and academia is Yuri Shestopaloff's 152-page treatise, "Sums of Exponential Functions and their new Fundamental Properties, with Applications to Natural Phenomena"

Professor Dr. B. N. Chichkov, Head of Nanotechnology Department, Laser Zentrum, Hannover

The book studies an interesting and practical area of adequate description of natural phenomena by mathematical models. In the first chapter the author “plows” unexpectedly fertile soil, with regard to practical applications in physics, biology, and cosmology, of seemingly pure classical mathematical constructs. He considers exponential and logarithmic functions as the most adequate representatives of general growth, destruction and transitional processes, ...  more

Alexander Kyriako, PhD in Elecrtrophysics

The book is dedicated to the insufficiently investigated field of mathematical modeling of complex natural phenomena such as boundary processes. It is shown that for reasons not completely known, identical mathematics, in particular, sums of exponential functions, describe many different phenomena. ...  more

V. Agranat, Ph.D in mathematical methods, modeling of thermodynamical processes

The author discovered fundamental properties of exponentials functions in the classical area of mathematics studied for centuries. What else to say?






....and supports his findings by convincing mathematical proofs. In particular, he considers growth of biological cells; he discovers also very interesting properties of transitional electrical signals.

These are certainly high level and valuable scientific results on their own, but the author goes to the next level of generalization and suggests that all these natural phenomena are closely and inherently interconnected through the boundary functions, separating the areas of mathematical convergence and divergence of sums of infinite series. The author provides mathematical support of this hypothesis. However, this important problem, which he discovers and formulates, has to be studied on a larger scale. If his finding is true, then it will have significant implications in many areas of science and technology, because in this case the introduced boundary functions become reference points for many natural processes, and this is the result of a fundamental importance.

Considering relationship of Fermat’s Last Theorem and physical reality, the author, beside the mathematical considerations from the area of number theory, includes some philosophical considerations. This part is certainly thought provoking and interesting; some new concepts, such as the space rarefaction, can be used as a foundation for the following research. Overall, it is more an illustration how the pure mathematical concepts are connected to the physical world, and how the mathematical modeling should be approached to create models that are adequate to natural phenomena.

The second part of the book introduces and proves a Theorem about properties of sums of exponential functions, in particular “one time oscillation property”. The author switches to strict mathematical proof of his Theorem, and this is a mathematically challenging part of the book that requires laborious efforts to get through. On the other hand, its understanding does not need knowledge of advanced mathematical subjects, and we agree with the author that familiarity with calculus is a sufficient prerequisite. Some new mathematical concepts, which the author developed, have a broader meaning for calculus in general; in particular, the notion of pair functions and associated operations certainly deserve close attention of mathematicians.

With regard to the Theorem itself, we think that it is very important in all aspects. Judging from the mathematical perspective, it is amazing that the author managed to discover such fundamental properties of exponential functions that were studied for centuries by an army of mathematicians. With regard to possible practical applications of the Theorem, they are numerous, and the Theorem can be used beneficially virtually in any area of science due to the adequacy of models based on exponential functions. The author himself provides several examples of such applications in the form of corollaries, which compose the last part of the book.

Our impression is that the first part of the book should evolve into a separate volume in the future editions, because it appeals to a wide audience of scientists from different areas, while the second part should be of great interest to mathematicians. Overall, the book presents high level, and in many instances fundamental scientific discoveries that certainly have a great value for science, as well as for different technological and social disciplines. The general public may be also interested in the content presented in the first part of this book.

.... It is well known that the exponential function and its inverse, the logarithmic function, describe a large number of Nature’s phenomena. In the first chapter of the book, several examples of such phenomena in different areas of science and technology are given. In particular, exponential and logarithmic processes are used to describe a cell’s replication and growth process.

In order to understand the importance of the aforementioned example, it is necessary to note that the discussion deals with the processes of evolutionary development of any kind of animal populations, beginning from the simplest amoebas and ending with humans. Here, it is also worthwhile to mention the basic psychophysical law of the perception of information - the Weber-Fechner law (“Sensation is proportional to the logarithm of excitation”), which relates to different systems (detectors) of perception, which plays an important role in the vital activity of humans and animals.

Furthermore, an example examined in the book that deals with the description of electrical processes of charging and discharging of a capacitor, without which not one device that generates electromagnetic waves (these are radio, television, mobile telephones and the like) can exist, gives an idea about the important role of these functions in inanimate nature.

Exponential functions are also considered in relation to Fermat's Last Theorem. This touches on another very interesting subject where these functions appear in an unexpected manner, in such a way that they can be used for the interpretation of complex natural phenomena.

Conditionally speaking, a “deficiency” of the first chapter of the book is the fact that the author did not decide to make this chapter into an independent book. The series of many interesting questions, which the author considers here, can be of serious independent interest for many readers.

For the author – a professional mathematician and physicist, purely scientific problems have a higher priority: namely, the general theorem discovered and proven by him, which has to do with the special features of sums of exponential functions of a real variable. To this theorem and to its consequences the author devotes the second chapter of the book. In order to see the significance of this theorem, it suffices to say that it makes possible the description and analysis of the majority of the processes mentioned in the first part of the book, as well as many others.

It remains only to add that the book is well published and reads easily, which is also important for the reader.







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