On the Question of Statistical Research in the School Course of Mathematics

Some elements of probability theory and statistics have been only recently included in high school curriculum. However, there so far have been no well-established textbooks and no clear consensus on the depth and breadth of the topics to be taught. As a result, the course gives students an incomplete picture of the subject. The article discusses some possible additions to the existing curriculum. The article also provides examples of assignments where mathematical statistics methods are used to study distribution laws and to solve geometry and extremum problems.

Averaged Optimization and Finite-Time Thermodynamics

The paper considers typical extremum problems that contain mean values of control variables or some functions of these variables. Relationships between such problems and cyclic modes of dynamical systems are explained and optimality conditions for these modes are found. The paper shows how these problems are linked to the field of finite-time thermodynamics.

The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities

In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings.