Pub Date : 2022-04-14DOI: 10.33581/2520-6508-2022-1-46-65
J. T. Eghwerido
An accurate mathematical inference depends on the experimental design and the model adopted in the process. Thus, in this study Marshall – Olkin Teissier generated distribution was used to present the distribution of the true nature of lifetime data. The characteristics of the proposed model were examined in a closed form. The behaviour of the new model indicated that the hazard rate of the submodels could be J- and U-shaped, decreasing and increasing. Monte Carlo simulations were presented for different configurations of parameters with varying sizes. The results of the simulation and goodness-of-fit of the real lifetime data show that the Marshall – Olkin Teissier generated model is flexible, tractable and applicable when compared to some classical two parameters distributions.
{"title":"The Marshall – Olkin Teissier generated model for lifetime data","authors":"J. T. Eghwerido","doi":"10.33581/2520-6508-2022-1-46-65","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-46-65","url":null,"abstract":"An accurate mathematical inference depends on the experimental design and the model adopted in the process. Thus, in this study Marshall – Olkin Teissier generated distribution was used to present the distribution of the true nature of lifetime data. The characteristics of the proposed model were examined in a closed form. The behaviour of the new model indicated that the hazard rate of the submodels could be J- and U-shaped, decreasing and increasing. Monte Carlo simulations were presented for different configurations of parameters with varying sizes. The results of the simulation and goodness-of-fit of the real lifetime data show that the Marshall – Olkin Teissier generated model is flexible, tractable and applicable when compared to some classical two parameters distributions.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42243309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-14DOI: 10.33581/2520-6508-2022-1-97-102
M. Vatkin, D. A. Vorobey
Transaction data are the most popular data type of bank domain, they are often represented as sparse vectors with a large number of features. Using sparse vectors in deep learning tasks is computationally inefficient and may lead to overfitting. Аutoencoders are widely applied to extract new useful features in a lower dimensional space. In this paper we propose to use a novel loss function based on the metric that estimates the quality of mapping the semantic structure of the original tabular data to the embedded space. The proposed loss function allows preserving the item relation structure of the original space during the dimension reduction transformation. The obtained results show the improvement of the resulting embedding properties while using the combination of the new loss function and the traditional mean squared error one.
{"title":"Space semantic aware loss function for embedding creation in case of transaction data","authors":"M. Vatkin, D. A. Vorobey","doi":"10.33581/2520-6508-2022-1-97-102","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-97-102","url":null,"abstract":"Transaction data are the most popular data type of bank domain, they are often represented as sparse vectors with a large number of features. Using sparse vectors in deep learning tasks is computationally inefficient and may lead to overfitting. Аutoencoders are widely applied to extract new useful features in a lower dimensional space. In this paper we propose to use a novel loss function based on the metric that estimates the quality of mapping the semantic structure of the original tabular data to the embedded space. The proposed loss function allows preserving the item relation structure of the original space during the dimension reduction transformation. The obtained results show the improvement of the resulting embedding properties while using the combination of the new loss function and the traditional mean squared error one.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47716606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-06DOI: 10.33581/2520-6508-2022-1-75-82
A. Nikitsin, G. Mikhasev
The objective of study is to determine the effective Young’s modulus of open cell porous titanium based on the Gibson – Ashby model. Two novel models are proposed in the form of 3D Gibson – Ashby cell arrays with two variants for connecting vertical and horizontal beams – hinged support and rigid clamping. Calculations made on the basis of the developed models are compared with results of known models and literature data. It is proved the assumption that at high porosity, the deformation of samples occurs to a greater extent due to the deflection of horizontal beams, and with a decrease in porosity, the compressive deformation of vertical beams is playing an important role.
{"title":"Estimation of the effective Young’s modulus for open cell porous titanium based on 3D Gibson – Ashby cell array","authors":"A. Nikitsin, G. Mikhasev","doi":"10.33581/2520-6508-2022-1-75-82","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-75-82","url":null,"abstract":"The objective of study is to determine the effective Young’s modulus of open cell porous titanium based on the Gibson – Ashby model. Two novel models are proposed in the form of 3D Gibson – Ashby cell arrays with two variants for connecting vertical and horizontal beams – hinged support and rigid clamping. Calculations made on the basis of the developed models are compared with results of known models and literature data. It is proved the assumption that at high porosity, the deformation of samples occurs to a greater extent due to the deflection of horizontal beams, and with a decrease in porosity, the compressive deformation of vertical beams is playing an important role.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41351194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-06DOI: 10.33581/2520-6508-2022-1-83-96
O. Senko, A. Kuznetsova, E. M. Voronin, O. Kravtsova, Ludmila R. Borisova, I. L. Kirilyuk, V. Akimkin
The paper presents an original method for solving the problem of finding a connection between the course of the epidemic and socio-economic, demographic and climatic factors. The method was applied to solve this problem for 110 countries of the world using a set of corresponding curves of the COVID-19 growth rate for the period from January 2020 to August 2021. Hierarchical agglomerative clustering was applied. Four large clusters with uniform curves were identified – 11, 39, 17 and 13 countries, respectively. Another 30 countries were not included in any cluster. Using machine learning methods, we identified the differences in socio-economic, demographic and geographical and climatic indicators in the selected clusters of countries of the world. The most important indicators by which the clusters differ from each other are amplitude of temperatures throughout the year, high-tech exports, Gini coefficient, size of the urban population and the general population, index of net barter terms of trade, population growth, average January temperature, territory (land area), number of deaths due to natural disasters, birth rate, coastline length, oil reserves, population in urban agglomerations with a population of more than 1 million etc. This approach (the use of clustering in combination with classification by methods of logical-statistical analysis) has not been used by anyone before. The found patterns will make it possible to more accurately predict the epidemiological process in countries belonging to different clusters. Supplementing this approach with autoregressive models will automate the forecast and improve its accuracy.
{"title":"Methods of intellectual data analysis in COVID-19 research","authors":"O. Senko, A. Kuznetsova, E. M. Voronin, O. Kravtsova, Ludmila R. Borisova, I. L. Kirilyuk, V. Akimkin","doi":"10.33581/2520-6508-2022-1-83-96","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-83-96","url":null,"abstract":"The paper presents an original method for solving the problem of finding a connection between the course of the epidemic and socio-economic, demographic and climatic factors. The method was applied to solve this problem for 110 countries of the world using a set of corresponding curves of the COVID-19 growth rate for the period from January 2020 to August 2021. Hierarchical agglomerative clustering was applied. Four large clusters with uniform curves were identified – 11, 39, 17 and 13 countries, respectively. Another 30 countries were not included in any cluster. Using machine learning methods, we identified the differences in socio-economic, demographic and geographical and climatic indicators in the selected clusters of countries of the world. The most important indicators by which the clusters differ from each other are amplitude of temperatures throughout the year, high-tech exports, Gini coefficient, size of the urban population and the general population, index of net barter terms of trade, population growth, average January temperature, territory (land area), number of deaths due to natural disasters, birth rate, coastline length, oil reserves, population in urban agglomerations with a population of more than 1 million etc. This approach (the use of clustering in combination with classification by methods of logical-statistical analysis) has not been used by anyone before. The found patterns will make it possible to more accurately predict the epidemiological process in countries belonging to different clusters. Supplementing this approach with autoregressive models will automate the forecast and improve its accuracy.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49283918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.33581/2520-6508-2022-1-21-25
Aliaksandr S. Biadrytski, V. L. Timokhovich
The countably-compactification of a topological space X is such its extension Y, that Y is a completely regular and countably-compact space, and any closed countably-compact subset of X is closed in Y. But this extension does not always exist. Due to this, the concept of a saturation of a topological space appeared, which is a generalisation of the countably-compactification: instead of the condition of the countably-compactness of Y, it is necessary that any infinite subset of X has a limit point in Y. Meanwhile, the second condition remains unchanged. Such an extension is already defined for any T1-space. In this paper we consider a specific construction of saturation named as Ω-saturation. It is proved that under some additional (necessary and sufficient) condition to the separation of the initial space X, its Ω-saturation is canonically embedded in the Stone – Čech compactification βX. An analogous result is obtained for the countably-compactification by K. Morita.
{"title":"On the embedding of the Ω-saturation of a topological space","authors":"Aliaksandr S. Biadrytski, V. L. Timokhovich","doi":"10.33581/2520-6508-2022-1-21-25","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-21-25","url":null,"abstract":"The countably-compactification of a topological space X is such its extension Y, that Y is a completely regular and countably-compact space, and any closed countably-compact subset of X is closed in Y. But this extension does not always exist. Due to this, the concept of a saturation of a topological space appeared, which is a generalisation of the countably-compactification: instead of the condition of the countably-compactness of Y, it is necessary that any infinite subset of X has a limit point in Y. Meanwhile, the second condition remains unchanged. Such an extension is already defined for any T1-space. In this paper we consider a specific construction of saturation named as Ω-saturation. It is proved that under some additional (necessary and sufficient) condition to the separation of the initial space X, its Ω-saturation is canonically embedded in the Stone – Čech compactification βX. An analogous result is obtained for the countably-compactification by K. Morita.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47981187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.33581/2520-6508-2022-1-14-20
A. A. Bondarenko
Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n respectively over a field K, charK ≠ 2. Herein, the problem of the birational composition of f(X) and g(Y) is considered, namely, the condition is established when the product f(X)g(Y) is birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The main result of this paper is the complete solution of the problem of the birational composition for quadratic forms f(X) and g(Y) over a field K when m = 2. The sufficient and necessary conditions for the existence of birational composition h(Z) for quadratic forms f(X) and g(Y) over a field K for m = 2 are obtained. The set of quadratic forms is described which can be considered as h(Z) in this case.
{"title":"The birational composition of arbitrary quadratic form with binary quadratic form","authors":"A. A. Bondarenko","doi":"10.33581/2520-6508-2022-1-14-20","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-14-20","url":null,"abstract":"Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n respectively over a field K, charK ≠ 2. Herein, the problem of the birational composition of f(X) and g(Y) is considered, namely, the condition is established when the product f(X)g(Y) is birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The main result of this paper is the complete solution of the problem of the birational composition for quadratic forms f(X) and g(Y) over a field K when m = 2. The sufficient and necessary conditions for the existence of birational composition h(Z) for quadratic forms f(X) and g(Y) over a field K for m = 2 are obtained. The set of quadratic forms is described which can be considered as h(Z) in this case.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42285377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.33581/2520-6508-2022-1-103-107
V. A. Pavlovsky, I. L. Vasiliev
The theory of functions of an h-complex variable is an alternative to the usual theory of functions of a complex variable, obtained by replacing the rules of multiplication. This change leads to the appearance of zero divisors on the set of h-complex numbers. Such numbers form a commutative ring that is not a field. h-Holomorphic functions are solutions of systems of equations of hyperbolic type, in comparison with classical holomorphic functions, which are solutions of systems of equations of elliptic type. A consequence of this is a significant difference between the properties of h-holomorphic functions and the classical ones. Interest in studying the properties of functions of an h-complex variable is associated with the need to search for new methods for solving problems in mechanics and the plane theory of relativity. The paper presents a theorem on the local invertibility of h-holomorphic functions, formulates the principles of preserving the domain and maximum of the norm.
{"title":"On local invertibility of functions of an h-complex variable","authors":"V. A. Pavlovsky, I. L. Vasiliev","doi":"10.33581/2520-6508-2022-1-103-107","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-103-107","url":null,"abstract":"The theory of functions of an h-complex variable is an alternative to the usual theory of functions of a complex variable, obtained by replacing the rules of multiplication. This change leads to the appearance of zero divisors on the set of h-complex numbers. Such numbers form a commutative ring that is not a field. h-Holomorphic functions are solutions of systems of equations of hyperbolic type, in comparison with classical holomorphic functions, which are solutions of systems of equations of elliptic type. A consequence of this is a significant difference between the properties of h-holomorphic functions and the classical ones. Interest in studying the properties of functions of an h-complex variable is associated with the need to search for new methods for solving problems in mechanics and the plane theory of relativity. The paper presents a theorem on the local invertibility of h-holomorphic functions, formulates the principles of preserving the domain and maximum of the norm.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43160548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.33581/2520-6508-2022-1-38-45
Anna O. Zadorozhnuyk
In this paper two ways to order the nodes of a graph with respect to an arbitrary node are considered, both connected to random walks on the graph. The first one is the order according to probabilities of states of a random walk of fixed length started in that arbitrary node. The walks considered here are lazy walks – instead of making a step they are allowed to stay in the same node. A class of graphs, where such order the corresponds to the weak order by geodesic distances, was found. Square and toric n-dimensional grids are shown to be instances of this class. The second way of ordering is resistance distance to a fixed node. For another class of graphs, a pair of vertices with maximal resistance distance between them is established. Grids are again shown to be an example of graphs belonging to this class.
{"title":"Monotonicity of random walks’ states on finite grids","authors":"Anna O. Zadorozhnuyk","doi":"10.33581/2520-6508-2022-1-38-45","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-38-45","url":null,"abstract":"In this paper two ways to order the nodes of a graph with respect to an arbitrary node are considered, both connected to random walks on the graph. The first one is the order according to probabilities of states of a random walk of fixed length started in that arbitrary node. The walks considered here are lazy walks – instead of making a step they are allowed to stay in the same node. A class of graphs, where such order the corresponds to the weak order by geodesic distances, was found. Square and toric n-dimensional grids are shown to be instances of this class. The second way of ordering is resistance distance to a fixed node. For another class of graphs, a pair of vertices with maximal resistance distance between them is established. Grids are again shown to be an example of graphs belonging to this class.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48943568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-30DOI: 10.33581/2520-6508-2022-1-26-37
V. Voloshko, Anton I. Trubey
In this paper, we obtained the asymptotic power values for the statistical tests of multidimensional discrete uniformity under conditions of contiguous convergence of alternatives. Two versions of the test are considered, namely, with overlapping blocks (included in the NIST SP 800-22 test suit) and with non-overlapping blocks. The null hypothesis H0 is related to the so-called pure randomness of the observed sequence, i. e. independence and the same uniform distribution of its elements. An alternative H1 is assumed to be a Markov chain of some arbitrary fixed finite order.
{"title":"On the power of tests of multidimensional discrete uniformity used for statistical analysis of random number generators","authors":"V. Voloshko, Anton I. Trubey","doi":"10.33581/2520-6508-2022-1-26-37","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-26-37","url":null,"abstract":"In this paper, we obtained the asymptotic power values for the statistical tests of multidimensional discrete uniformity under conditions of contiguous convergence of alternatives. Two versions of the test are considered, namely, with overlapping blocks (included in the NIST SP 800-22 test suit) and with non-overlapping blocks. The null hypothesis H0 is related to the so-called pure randomness of the observed sequence, i. e. independence and the same uniform distribution of its elements. An alternative H1 is assumed to be a Markov chain of some arbitrary fixed finite order.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46889877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-16DOI: 10.33581/2520-6508-2021-3-25-33
V. Zakrevskaya
Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G) ≠ ∅. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ such that AH x = H xA for all H ∈ ℋ and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.
{"title":"Finite groups with given systems of generalised σ-permutable subgroups","authors":"V. Zakrevskaya","doi":"10.33581/2520-6508-2021-3-25-33","DOIUrl":"https://doi.org/10.33581/2520-6508-2021-3-25-33","url":null,"abstract":"Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G) ≠ ∅. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ such that AH x = H xA for all H ∈ ℋ and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45462923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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