Quasi - Fractal Probabilistic Logic. Application to Brownian Motion

The chapter presents the following results in an overview order: The chapter deals with issues related with the questions of describing Brownian motion. In Chap. 7 , a model of Brownian motion is proposed, which is based on the approach developed in Serdyukova and Serdyukov (Algebraic Formalization of Smart Systems. Theory and Practice, Smart Innovation, Systems and Technologies, vol. 91, Springer Nature, Switzerland, 2018), Serdyukova and Serdyukov (Algebraic Identification of Smart Systems. Theory and Practice, Intelligent Systems Reference Library, vol. 191, Springer Nature, Switzerland, 2021), and based on the use of models in the form of algebraic systems to represent a closed system S Since Brownian motion is chaotic, it is proposed to use a free non-Abelian group (a free algebraic system) for its modeling. The fact is that there are no defining relations in a free non-Abelian group, that is, it is initially assumed that there are no connections between the elements of the system S. The following questions are considered: Algebra and Logic nowdays are the key points in central areas of Digital transformation research, and these areas can be considered as the most important mechanisms in enabling strategic, fundamental advances in digital transformations of almost all spheres, which are provided and connected the social society functioning. One of the most difficult and interesting issues in the field of digital transformation is the study of the emergence of a system from a chaotic state. In Chap. 6 we introduced the concept of a presystem. Here, in Chap. 7 , we present the following results: