On the Issue of Research and Scientific-Methodological Reporting and Oversight

2026-03-11T18:31:50+00:00

Science is created by people, and people are educated by people together forming the system of Education and Science funded by public resources. The budget is formed by the taxpayers of the Republic of Kazakhstan and administered by the State. For these reasons, the effective use of public funds allocated for the development of human capital becomes one of the central intellectual
challenges, one that cannot be resolved through purely speculative reasoning. Based on the author’s experience of 24 breakthroughs of varying scope in Mathematics and Computer Science, which resulted in the development of approximately three thousand pages of mutually coherent school and university mathematics textbooks, the article proposes a unified system of research and scientific methodological reporting and oversight.

On Weak Convergence of Empirical Measures for Sets Minimizing the Hausdorff Distance

2026-04-21T11:35:35+00:00

This paper investigates the asymptotic behavior of finite sets that minimize the Hausdorff distance to a compact metric space. The primary objective is to establish conditions under which the empirical measures supported on these optimal sets converge weakly to the normalized Hausdorff measure. To facilitate this, we introduce the class of uniformly asymptotically open convex Euclidean metrics. These metrics characterize spaces that exhibit a local Euclidean structure in an asymptotic sense, allowing for a rigorous analysis of the geometry of Voronoi cells. The paper provides a sufficient reformulation of weak convergence based on the concept of sets uniformly eating Voronoi boundaries with respect to a regular measure. A complete proof of weak convergence to the normalized one-dimensional Hausdorff measure is presented for the specific case of a connected one-dimensional manifold (the circle). While the one-dimensional case yields a clear result due to the simple structure of Voronoi boundaries, the paper concludes by noting that higher-dimensional cases remain an open challenge due to the increased geometric complexity of the resulting Voronoi cells.

Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series

On the Issue of Research and Scientific-Methodological Reporting and Oversight

On Weak Convergence of Empirical Measures for Sets Minimizing the Hausdorff Distance