P-adic Mathematics Research Articles

Анализ инструментальных методов моделирования стохастических процессов в экономике

In this paper, the authors conduct a comparative analysis of instrumental methods used in modeling stochastic processes, namely, component analysis of time series, fractal modeling and modeling using p-adic mathematics. As an object of study, the authors chose the dynamics of the MICEX index. At the first step of the work, the authors carry out a detailed component analysis of the time series, which made it possible to identify the main development trend in the form of a quadratic function; periodic fluctuations with a period of six levels and a cyclical component describing fluctuations in the world economy with a period of fifty-five levels. At the second step of the work, the authors simulate the dynamics of the MICEX index using a fractal theory based on the self-similarity of the development of the economic process, which showed the ergodicity of the series under study with a stable influence of only the last twenty-four levels. The third step of the work was the p-adic modeling of the patterns existing in the series under study, which allowed the authors to reduce the model error to 6.8%. As a result of the work, a forecast of the dynamics of the MICEX exchange rate at four levels is presented, presented in three scenarios: optimistic, realistic and pessimistic. As conclusions of the work, an analysis was made of the possibility of using the considered methods for multiple, medium and long-term forecasts; the complexity of the methods and the need to use special software products are evaluated.

Formula omitted]-Adic mathematics and theoretical biology

The principal objective of this article is a brief overview of the main parts of p-adic mathematics, which have already had valuable applications and may have a significant impact in the near future on the further development of some fields of theoretical and mathematical biology. In particular, we present the basics of ultrametrics, p-adic numbers and p-adic analysis, as well as insight into their applications for modeling some cognitive processes, genetic code and protein dynamics. We also argue that ultrametric concepts and p-adic mathematics are natural tools for the viable description of biological systems and phenomena with a hierarchical structure.

Number theory as the ultimate physical theory

At the Planck scale doubt is cast on the usual notion of space-time and one cannot think about elementary particles. Thus, the fundamental entities of which we consider our Universe to be composed cannot be particles, fields or strings. In this paper the numbers are considered as the fundamental entities. We discuss the construction of the corresponding physical theory. A hypothesis on the quantum fluctuations of the number field is advanced for discussion. If these fluctuations actually take place then instead of the usual quantum mechanics over the complex number field a new quantum mechanics over an arbitrary field must be developed. Moreover, it is tempting to speculate that a principle of invariance of the fundamental physical laws under a change of the number field does hold. The fluctuations of the number field could appear on the Planck length, in particular in the gravitational collapse or near the cosmological singularity. These fluctuations can lead to the appearance of domains with non-Archimedean p-adic or finite geometry. We present a short review of the p-adic mathematics necessary, in this context.

P-Adic string theories

In this paper we study the p-adic string theory. Starting with the integral representation of the real string scattering amplitude, we obtain various p-adic string amplitudes. We show that it is possible to obtain a suitable p-adic theory starting with either the Koba-Nielsen or the Bardakçi-Ruegg formula. However, whereas in the real string case both representations of the amplitude correspond to the same theory, the p-adic generalizations lead to different theories. Furthermore, there is more than one p-adic generalization of each form of the amplitude. Each p-adic theory gives its own set of diagrammatic rules implying that it is describable by some nonlocal effective field theory. For completeness, in the first part of the paper we give an overview of the p-adic mathematics necessary for constructing our generalizations and for computing the resultant p-adic amplitudes.