Automorphisms of Chevalley groups of different types over commutative rings

In this paper we prove that every automorphism of (elementary) adjoint Chevalley group with root system of rank $>1$ over a commutative ring (with 1/2 for the systems $A_2$, $F_4$, $B_l$, $C_l$; with 1/2 and 1/3 for the system $G_2$) is standard, i.\,e., it is a composition of ring, inner, central and graph automorphisms.Comment: 23 page

Automorphisms of Chevalley groups over commutative rings

In this paper we prove that every automorphism of (elementary) Chevalley group with root system of rank $>1$ over a commutative ring (with $1/2$ for the systems $A_2$, $F_4$, $B_l$, $C_l$; with $1/2$ and $1/3$ for the system~$G_2$) is standard, i.\,e., it is a composition of ring, inner, central and graph automorphisms.Comment: 20 pages, 1 figure. arXiv admin note: text overlap with arXiv:1108.0529, arXiv:2304.0625

Sha-rigidity of Chevalley groups over local rings

We prove that every locally inner endomorphism of a Chevalley group (or its elementary subgroup) over a local ring with an irreducible root system of rank >1 (with 1/2 for the systems A_2, F_4, B_l, C_l and with 1/3 for the system G_2) is inner, so that all these groups are Sha-rigid.Comment: 14 page

Automorphisms of a Chevalley group of type G_2 over a commutative ring R with 1/3 generated by the all invertible elements and 2R

In this paper we prove that every automorphism of a Chevalley group with the root system G_2 over a commutative ring R with 1/3, generated by all its invertible elements and the ideal 2R is a composition of ring and inner automorphisms.Comment: 14 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1108.0529, arXiv:2304.1344

The Diophantine problem in Chevalley groups

In this paper we study the Diophantine problem in Chevalley groups $G_\pi (\Phi,R)$, where $\Phi$ is an indecomposable root system of rank $> 1$, $R$ is an arbitrary commutative ring with $1$. We establish a variant of double centralizer theorem for elementary unipotents $x_\alpha(1)$. This theorem is valid for arbitrary commutative rings with $1$. The result is principle to show that any one-parametric subgroup $X_\alpha$, $\alpha \in \Phi$, is Diophantine in $G$. Then we prove that the Diophantine problem in $G_\pi (\Phi,R)$ is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in $R$. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.Comment: 44 page

Order logistics based on collaborative filtering

Modern approaches to the organization and management of logistics operators include issues directly related to the forecasting of the dynamics of the processes of the company under study, as well as the development of recommendations for further functioning in the market of goods and services. Services integrated into the order and delivery logistics system are an integral part of it and accompany the flow processes throughout the supply chain. Nowadays, the complex of information technologies, communications and technological solutions allows implementing the most daring offers of logistics companies. Innovative solutions developed using information technologies develop the complex “design-practical implementation” in logistics systems and form a digital space for managing material flows. The calculation of the forecast values of the indicators that characterize the order requires research from the point of view of taking into account the already known preferences of customers. We are talking about recommendation systems that offer to form a package of services in the delivery and assembly of a batch of orders based on the established preferences of other customers and users of the services. The well-known preferences of customers when accessing logistics services form the basis for developing recommendations that are close to them. The obtained data of forecasts of the state of the system under study should provide an opportunity to develop an individual package of solutions, based on the already known preferences of other users. The key value is the information that is used in this model to build forecasts. The practice of finding solutions in the management of order procedures in logistics is an analysis of the key problems and “problem points” that affect the overall performance indicator, taking into account the geography of transport systems, established trade practice, the characteristics of the markets studied and the location of each, specifically selected company