of exponential functions and their new fundamental properties, with Applications to
Midwest Book Reviews (Oregon,
... 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
Professor Dr. B. N.
Chichkov, Head of Nanotechnology Department, Laser Zentrum,
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, ...
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?
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
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.
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.
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.
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
It remains only to add that the book is well published and reads easily,
which is also important for the reader.