Pure and Applied Mathematics, Cosmology
Mathematical studies relate to properties of some elementary functions, such as polynomial, logarithmic, exponential and power functions. There is some material that considers properties of multidimensional spaces with regard to discrete geometry. In particular, notion of rarefaction of a discrete space has been introduced, which helped to better understand the geometry of the real space. The results are quite logically confirm, to some extent indirectly, the theory of quantum gravity and the uniqueness and singularity of Universe. Combination of mathematical notion of rarefaction and its properties and some dialectical considerations allowed to make an inference that Universe periodically goes through drastic transformations, essentially wiping out (but not necessarily completely) the traces of its previous existence, so that we do not have a single "Big Bang", but many such events coming in sequence. This is the way the Universe exists. The theory of quantum gravity says about the same, although its development and approaches are more sound than my more delicate and subtle considerations.
Another set of studies discovered some interesting properties of sums of exponential and other elementary functions. In particular, I found the generalization of Descartes Rule of Signs and proved it for sums of exponential and power functions. These and other mathematical studies have been published in books, some articles were submitted to journals, but they are still in review process. The books and articles are are listed below.
Yuri K. Shestopaloff, "Properties and interrelationships of polynomial, exponential, logarithmic and power functions with applications to modeling natural phenomena", AKVY Press, 2010, 230 p.
Shestopaloff Yu. K. "Sums of exponential functions and their new fundamental properties, with applications to natural phenomena", Second Edition, 2010, AKVY PRESS, 196 p.
Shestopaloff, Yu. K., 2011, Properties of sums of some elementary functions and their application to computational and modeling problems, Journal of Computational Mathematics and Mathematical Physics, Vol. 51, No. 5, p. 699-712. DOI: 10.1134/S0965542511050162