Rough sets in graphs using similarity relations

<abstract><p>In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $ \mathcal{G} $ preserve the indiscernibility partitions. Further, we prove that for any graph $ \mathcal{G} $ with $ k $ orbits, any reduct $ \mathcal{R} $ consists of one element from $ k-1 $ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits of the graph.</p></abstract>

Metrical Books as one the Most Critical Important Historical Source in the Scholarly Research of Rizaeddin Fakhreddin

Metric books are one of the essential sets of sources that provide valuable material for historical research. Unfortunately, metric books' level of use and involvement as historical sources in scientific and scholarly works is still mediocre. At the beginning of the XX century, Rizaeddin Fakhreddin paid attention to those primary sources: he actively referenced them in his scientific research and contributed to their preservation. Thanks to Qadi, the mufti of the USSR Rizaeddin Fakhreddin, the extensive collections of Muslim metric books were preserved. The key sources for his famous works as «Famous Men», «Famous Women», «Asar» were metrical books. It is worth highlighting the bibliographic work, «Asar» There Fakhreddin restored the biography, exact dates of birth and death, and lineage of prominent people, theologians, and imams based on the materials from metrical books. This article aims to demonstrate the role and importance of metric books in the scientific research of Fakhreddin. This article set several tasks, namely: to analyze material the metric books as sources of historical research, demonstrate the degree of usage of metric books by Fakhreddin in his works, and to evaluate the possibilities of studying these documents in the disclosure of several scholarly topics. For the first time in this publication, the scientific legacy of Fakhreddin is studied from that perspective. The author also draws attention to the fact that the biography of R. Fakhreddin, written by his hand, contains many extracts from the metrical books of his family members and close relatives. It worth mentioning that the author in his article draws attention to such discussions, which have recently become quite relevant, for example, the question of whether a person registered in the metric books belongs to a particular nationality. As the article will demonstrate below, a person's social affiliation in metric books cannot be interpreted as his nationality. According to the rules of registration of metric books, there was no indication specifying a person's nationality; besides, a person's nationality did not have any significance. Fakhreddin's reverential attitude and appeal to metric books emphasize their essential place in Tatar-written sources' complex. The scientific study of a voluminous complex of metric books as valuable historical sources should be an important research area in historical science.

Diagrams and Essential Sets for Signed Permutations

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

On Stability of Fixed Points for Multi-Valued Mappings with an Application

This paper studies the stability of fixed points for multi-valued mappings in relation to selections. For multi-valued mappings admitting Michael selections, some examples are given to show that the fixed point mapping of these mappings are neither upper semi-continuous nor almost lower semi-continuous. Though the set of fixed points may be not compact for multi-valued mappings admitting Lipschitz selections, by finding sub-mappings of such mappings, the existence of minimal essential sets of fixed points is proved, and we show that there exists at least an essentially stable fixed point for almost all these mappings. As an application, we deduce an essentially stable result for differential inclusion problems.