Tits polygons

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank 2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least 4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank 1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

Constitutive relations for materials with strain state dependent properties

Many materials demonstrate a dependence of mechanical properties on the type of stressed or deformed states. This is most noticeable in the dependence of the processes of shear and bulk deformation. Such materials include rocks, structural graphite, concrete, some grades of steel, cast iron, and aluminum. The main properties of these materials are an absence of a "single curve" relationship between the intensity of stresses and the intensity of deformations. Under shear conditions, bulk deformations can occur. Such materials can be described by constitutive equations that depend on the parameter of the type of a stress state, which is the ratio of the first invariant of the stress tensor to the stress intensity. Thus, these defining relations give the dependence of the strain tensor components on the stress tensor components. Such defining relations can be quite cumbersome, and therefore do not allow an analytical treatment to obtain defining relations that give the dependence of the components of the stress tensor on the components of the strain tensor. The paper proposes the constitutive relations obtained from the analysis of test results of various materials, which properties depend on the type of deformed state. Conditions are derived for material constants that ensure the uniqueness of the solution of boundary value problems. Based on experimental data obtained under the conditions of the proportional loading of various rocks: limestone and talcochlorite, as well as the results of mechanical tests of several grades of concrete, the constants of the mathematical model are determined. The results of the experimental studies are compared with theoretical dependencies predicted by the model. The limited applicability of the proposed constitutive relations is established.

Cultural Codes of Society: Scientific and Practical Significance in Modern African Studies

The article is devoted to the topic of searching for cultural codes. It presents the author’s approach to understanding the place and role of the problem of cultural codes in modern studies of Africa. The article presents an overview of the main options for understanding and defining cultural codes from the standpoint of both theoretical-rational and empirical-intuitive approaches. The relevance of the topic of cultural codes is considered in two aspects: from the point of view of contribution to the scientific development of the problems of cultural codes as an interdisciplinary problem and from the point of view of solving the practical task of increasing the effectiveness of the system of Russian-African relations based on mutual understanding and mutual respect of partners. The problems of cultural codes are considered in the context of the global transformations of the 20th to 21st centuries with their great and tragic events that changed the essence and configuration of the regions of the world. Two world wars, the October Revolution, the collapse of the colonial system, the collapse of empires, the destruction of the USSR, the globalization of the world, the whole spectrum of postcolonial events raised the question of understanding countries and regions of each other around the world. As conclusions, the article defines the basic principles of studying African realities in the problem area of cultural codes: 1. Cultural codes of various cultures are value-semantic complexes formed on the basis of social experience and the information received in it. 2. Social experience is characterized by a number of parameters: geographical; geopolitical – defining relations with the outside world and the formation of historical memory, which in Africa is based on a generalized concept of the West, the image of the slave trade, colonization, racism, apartheid; and cultural and historical parameter. 3. Cultural codes form a semantic picture of the world, characteristic of each culture, on the basis of which people evaluate events and make choices that determine their activities and relationships. 4. Cultural codes evolve and change depending on social experience. 5. Cultural codes of different cultures interact in the forms of mutual influence or unilateral aggressive influence. Africa is characterized by the aggressive influence of Western culture, which has led to the emergence of border codes, the essence of which is the forced combination of traditional African codes with alien Western components. Hermeneutics, understood as the interpretation of the meanings of cultural texts, is considered as a research method. Two possible scenarios for the development of the border situation are considered: negative and positive. The negative scenario assumes the onset of acute social cataclysms, up to social chaos and the destruction of basic values. A positive scenario involves the renewal of society and the growth of its creative potential. It is necessary to predict the vector of development of the situation according to one of the above scenarios. The author believes that the deformation of African cultural codes and the “hacking” of Russia’s cultural codes of the late twentieth – early twenty-first century can become one of the points of contact that can consolidate Russian and African partners in the realization of the national interests of both sides.

AS-REGULARITY OF GEOMETRIC ALGEBRAS OF PLANE CUBIC CURVES

In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

Strong 1-boundedness of unimodular orthogonal free quantum groups

Recently, Brannan and Vergnioux showed that the orthogonal free quantum group factors [Formula: see text] have Jung’s strong [Formula: see text]-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in [Formula: see text] dimensions [Formula: see text]. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in [Formula: see text]-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a [Formula: see text]-bounded set without losing [Formula: see text]-boundedness. In particular, this allows us to include the character of the fundamental representation, proving strong [Formula: see text]-boundedness.

Evaluation of Human Resources in Transportation Companies Using Multi-Criteria Model for Ranking Alternatives by Defining Relations between Ideal and Anti-Ideal Alternative (RADERIA)

Multi-criteria decision-making methods (MCDM) represent a very powerful tool for making decisions in different areas. Making a rational and reliable decision, while respecting different factors, is a challenging and difficult task; MCDM models have a great impact on achieving this goal. In this paper, a new MCDM technique is presented—ranking alternatives by defining relations between the ideal and anti-ideal alternative (RADERIA), which was tested for the evaluation of human resources (HR) in a transportation company. The RADERIA model has three key advantages that recommend it for future use: (1) the RADERIA model has a new approach for data normalization that enables defining the normalization interval according to the judgments of a decision-maker; (2) an adaptive model for data normalization of the RADERIA model allows tough conversion into various forms of decreasing functions (linear, quadratic equation, etc.); and (3) the resistance of the RADERIA model to the rank reversal problem. Furthermore, in many simulations, the RADERIA method has shown stability when processing a larger number of datasets. This was also confirmed by a case study with 36 alternatives, as considered in this paper. The results and verification of the proposed new method were acquired through a comprehensive verification of the complexity of the results. The complexity of the results was executed through (1) comparison with four other multi-criteria methods, (2) checking the resistance of the RADERIA model to the rank reversal problem, and (3) the analysis of the impact of changes in the measurement scale on the ranking results.

Metric properties of the canonical defining pair for determined graphs

The aim of this paper is to study the representation of deterministic graphs (D-graphs) by sets of words over the vertex labels alphabet and to find metric properties of this representation. Vertex-labeled graphs are widely used in various computational processes modeling in programming, robotics, model checking, etc. In such models graphs playing the role of an information environment of single or several mobile agents. Walks of agents on a graph determines the sequence of vertices labels or words in the alphabet of labels. A vertex-labeled graph is said to be D-graph if all vertices in the neighborhood of every its vertex have different labels. For D-graphs in case when the graph as a whole and the initial vertex (i.e. the vertex from which the agent started walking) are known there exists the one-to-one correspondence between the sequence of vertices visited by the agent and the trajectory of its walks on the graph. In case when the D-graph is not known as a whole, agent walks on it can be arranged in such way that an observer obtains information about the structure of the graph sufficient to solve the problems of graph recognizing, finding optimal path between vertices, comparison between current graph and etalon graph etc. This paper specifies the representation of D-graphs by the defining pair of sets of words (the first describes cycles of the graph and the second -- all its vertices of degree 1). This representation is an analogue of the system of defining relations for everywhere defined automata. The structure of the so-called canonical defining pair, which is minimal in terms of the number of words, is also considered. An algorithm for building such pair is developed and described in detail. For D-graphs with a given number of vertices and edges, the exact number of words in the first component of its canonical defining pair and the minimum and maximum attainable bounds for the the number of words in the second component of this pair are obtained. This representation allows us to use new methods and algorithms to solve the problems of analyzing vertex-labeled graphs.

Pro-$p$ groups with few relations and universal Koszulity

Let $p$ be a prime. We show that if a pro-$p$ group with at most $2$ defining relations has quadratic $\mathbb{F}_p$-cohomology algebra, then this algebra is universally Koszul. This proves the “Universal Koszulity Conjecture” formulated by J. Miná{č} et al. in the case of maximal pro-$p$ Galois groups of fields with at most $2$ defining relations.