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The paper explores the formation of a cluster partition using a system of bundles of vectors. Each object is represented by a vector with coordinates corresponding to its characteristic factors. The study focuses on the principles of changes in the total density of these bundles when objects move from one cluster to another. Based on the findings, an algorithm is proposed to reduce the total intracluster variance of the initial cluster partition.
The proposed algorithm is based on a modern technique for analyzing latent classes. This technique emphasizes that the forming factors of objects within each cluster should be highly correlated. Since the number of forming factors is arbitrary, the algorithm replaces this requirement with the maximum possible proximity of object factor vectors to the average vector within each cluster. The degree of this closeness is referred to as the tightness of the cluster vector bundle.
The paper also introduces a new method for quantifying clusters in the improvable clustering constructed using the proposed algorithm. A practical example of the algorithm's application to process medical data is presented. The study discusses the reasons for the dependence of the algorithm's outcome on the choice of initial cluster partition.

Dronov Aleksandr M.VI Conference as Part of the Interdisciplinary Central European Seminar «A Woman in the Heart of Europe: Non-Obvious Aspects of Gender Issues in the History and Culture of Central Europe and Neighboring Regions»

The review analyzes the monograph by Croatian historian Branko Ostajmer Narodna stranka u Slavoniji i Srijemu 1883.–1903. [The People's Party in Slavonia and Srem in 1883–1903] and considers the historiography, source base, structure, and content of the book. Not only does the author of the monograph contribute to the development of Croatian historiography, but he also polemicizes with his predecessors and contemporaries. The extant archival sources of the period studied by B. Ostajmer are not plentiful, so the book is largely based on periodicals. A specific feature of his work is the author's focus on the outstanding personalities of the era associated with the activities of the People's Party. In addition, it is necessary to highlight the author's analysis of Slavonism, a concept that is important for understanding the period and the region under study. B. Ostajmer studies the internal organization of the party and its electorate, including the interaction with individual social and ethnic groups, as well as the activities of the party in the context of Croat-Hungarian relations. Much space in the narrative is given to the figure of Croatian Ban K. Khuen-Héderváry, whose reign was taken as the chronological framework of the work. The book can be of interest to historians of parties and social movements of the Modern Era as well as specialists in Croatia, the Balkans, and Central Europe.

Processing large amounts of data can be greatly simplified if this data is divided into approximately homogeneous groups. Splitting into such groups is the task of cluster analysis. However, the question of constructing an objective, natural partition into clusters remains open. The paper considers a modern approach to the search for such an objective cluster structure by highlighting the indicator of a common essential part from the set of characteristics that define objects (we call them the forming ones). When this indicator is fixed, the remains of the forming characteristics become independent or close to such. The resulting independent residuals are interpreted as a kind of information noise, and the latent cluster variable, the common fixed part that provides such a transformation, can be a reason for the objective integration of objects into clusters. A new algorithm for the formation of a cluster partition based on the proximity or coincidence of the values of a latent cluster variable with the simultaneous quantification of its values is proposed. The algorithm is based on the targeted search of partitions, the transition from the start one to the partition, more close to the objective. The algorithm proposed in the paper can be easily modified to the case of non-numeric categorized characteristics.

This paper studies the mutual arrangements of multidimensional vectors with coordinates that are consecutive natural numbers and the maximum of which coincides with the dimension of the corresponding Euclidean space. The set of all such vectors is called an ensemble of integer vectors. Since the membership of a vector in an ensemble means that its coordinates are some permutation of coordinates of any other vectors of the ensemble, all these vectors have the same length and, when they are placed at the origin, all their ends are lying on a sphere with a radius equal to this length. The vector (1, 2, ..., n) is considered as the main vector of the ensemble. All possible angles that can be formed by the vectors of an integer ensemble are studied. In particular, authors are looking for the number of those that form angles of equal magnitude with the main vector. Obtained results can be used in the construction of the exact distribution of the rank correlation coefficient. In conclusion, the application of this distribution to one of the problems of differential diagnosis in the analysis of deep vein thrombosis and pulmonary artery thromboembolism is considered.