In this paper, we introduce and investigate the concept of (A^{mathcal{I^{K}}})-summability as an extension of (A^{mathcal{I^{*}}})-summability which was recently (2021) introduced by O.H.H.~Edely, where (A=(a_{nk})_{n,k=1}^{infty}) is a non-negative regular matrix and (mathcal{I}) and (mathcal{K}) represent two non-trivial admissible ideals in (mathbb{N}). We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that (A^{mathcal{K}})-summability always implies (A^{mathcal{I^{K}}})-summability whereas (A^{mathcal{I}})-summability not necessarily implies (A^{mathcal{I^{K}}})-summability. Finally, we give a condition namely (AP(mathcal{I},mathcal{K})) (which is a natural generalization of the condition (AP)) under which (A^{mathcal{I}})-summability implies (A^{mathcal{I^{K}}})-summability.
{"title":"ON (A^{mathcal{I^{K}}})–SUMMABILITY","authors":"C. Choudhury, S. Debnath","doi":"10.15826/umj.2022.1.002","DOIUrl":"https://doi.org/10.15826/umj.2022.1.002","url":null,"abstract":"In this paper, we introduce and investigate the concept of (A^{mathcal{I^{K}}})-summability as an extension of (A^{mathcal{I^{*}}})-summability which was recently (2021) introduced by O.H.H.~Edely, where (A=(a_{nk})_{n,k=1}^{infty}) is a non-negative regular matrix and (mathcal{I}) and (mathcal{K}) represent two non-trivial admissible ideals in (mathbb{N}). We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that (A^{mathcal{K}})-summability always implies (A^{mathcal{I^{K}}})-summability whereas (A^{mathcal{I}})-summability not necessarily implies (A^{mathcal{I^{K}}})-summability. Finally, we give a condition namely (AP(mathcal{I},mathcal{K})) (which is a natural generalization of the condition (AP)) under which (A^{mathcal{I}})-summability implies (A^{mathcal{I^{K}}})-summability.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41501413","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}
We consider the problem of an enhanced evasion for linear discrete-time systems, where there are two conflicting bounded controls and the aim of one of them is to be guaranteed to avoid the trajectory hitting a given target set at a given final time and also at intermediate instants. First we outline a common solution scheme based on the construction of so called solvability tubes or repulsive tubes. Then a much more quick and simple for realization method based on the construction of the tubes with parallelepiped-valued cross-sections is presented under assumptions that the target set is a parallelepiped and parallelotope-valued constraints on controls are imposed. An example illustrating this method is considered.
{"title":"ON SOLVING AN ENHANCED EVASION PROBLEM FOR LINEAR DISCRETE–TIME SYSTEMS","authors":"E. K. Kostousova","doi":"10.15826/umj.2022.1.006","DOIUrl":"https://doi.org/10.15826/umj.2022.1.006","url":null,"abstract":"We consider the problem of an enhanced evasion for linear discrete-time systems, where there are two conflicting bounded controls and the aim of one of them is to be guaranteed to avoid the trajectory hitting a given target set at a given final time and also at intermediate instants. First we outline a common solution scheme based on the construction of so called solvability tubes or repulsive tubes. Then a much more quick and simple for realization method based on the construction of the tubes with parallelepiped-valued cross-sections is presented under assumptions that the target set is a parallelepiped and parallelotope-valued constraints on controls are imposed. An example illustrating this method is considered.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49540612","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}
Using the technique of generalized inequalities of the Hamilton--Jacobi--Bellman type, we study here the state estimation problem for a control system which operates under conditions of uncertainty and nonlinearity of a special kind, when the dynamic equations describing the studied system simultaneously contain the different forms of nonlinearity in state velocities. Namely, quadratic functions and uncertain matrices of state elocity coefficients are presented therein. The external ellipsoidal bounds for reachable sets are found, some approaches which may produce internal estimates for such sets are also mentioned. The example is included to illustrate the result.
{"title":"HJB-INEQUALITIES IN ESTIMATING REACHABLE SETS OF A CONTROL SYSTEM UNDER UNCERTAINTY","authors":"T. Filippova","doi":"10.15826/umj.2022.1.004","DOIUrl":"https://doi.org/10.15826/umj.2022.1.004","url":null,"abstract":"Using the technique of generalized inequalities of the Hamilton--Jacobi--Bellman type, we study here the state estimation problem for a control system which operates under conditions of uncertainty and nonlinearity of a special kind, when the dynamic equations describing the studied system simultaneously contain the different forms of nonlinearity in state velocities. Namely, quadratic functions and uncertain matrices of state elocity coefficients are presented therein. The external ellipsoidal bounds for reachable sets are found, some approaches which may produce internal estimates for such sets are also mentioned. The example is included to illustrate the result.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45938734","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}
In finite-dimensional Euclidean space, we analyze the problem of pursuit of a single evader by a group of pursuers, which is described by a system of differential equations with Caputo fractional derivatives of order (alpha.) The goal of the group of pursuers is the capture of the evader by at least (m) different pursuers (the instants of capture may or may not coincide). As a mathematical basis, we use matrix resolving functions that are generalizations of scalar resolving functions. We obtain sufficient conditions for multiple capture of a single evader in the class of quasi-strategies. We give examples illustrating the results obtained.
{"title":"MATRIX RESOLVING FUNCTIONS IN THE LINEAR GROUP PURSUIT PROBLEM WITH FRACTIONAL DERIVATIVES","authors":"A. I. Machtakova, N. Petrov","doi":"10.15826/umj.2022.1.008","DOIUrl":"https://doi.org/10.15826/umj.2022.1.008","url":null,"abstract":"In finite-dimensional Euclidean space, we analyze the problem of pursuit of a single evader by a group of pursuers, which is described by a system of differential equations with Caputo fractional derivatives of order (alpha.) The goal of the group of pursuers is the capture of the evader by at least (m) different pursuers (the instants of capture may or may not coincide). As a mathematical basis, we use matrix resolving functions that are generalizations of scalar resolving functions. We obtain sufficient conditions for multiple capture of a single evader in the class of quasi-strategies. We give examples illustrating the results obtained.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46417587","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}
Let (G=(V, E)) be a simple graph and (H) be a subgraph of (G). Then (G) admits an (H)-covering, if every edge in (E(G)) belongs to at least one subgraph of (G) that is isomorphic to (H). An ((a,d)-H)-antimagic total labeling of (G) is bijection (f:V(G)cup E(G)rightarrow {1, 2, 3,dots, |V(G)| + |E(G)|}) such that for all subgraphs (H') of (G) isomorphic to (H), the (H') weights (w(H') =sum_{vin V(H')} f (v) + sum_{ein E(H')} f (e)) constitute an arithmetic progression ({a, a + d, a + 2d, dots , a + (n- 1)d}), where (a) and (d) are positive integers and (n) is the number of subgraphs of (G) isomorphic to (H). The labeling (f) is called a super ((a, d)-H)-antimagic total labeling if (f(V(G))={1, 2, 3,dots, |V(G)|}.) In [5], David Laurence and Kathiresan posed a problem that characterizes the super ( (a, 1)-P_{3})-antimagic total labeling of Star (S_{n},) where (n=6,7,8,9.) In this paper, we completely solved this problem.
{"title":"NOTE ON SUPER ((a,1))–(P_3)–ANTIMAGIC TOTAL LABELING OF STAR (S_n)","authors":"S. Rajkumar, M. Nalliah, M. Venkataraman","doi":"10.15826/umj.2021.2.006","DOIUrl":"https://doi.org/10.15826/umj.2021.2.006","url":null,"abstract":"Let (G=(V, E)) be a simple graph and (H) be a subgraph of (G). Then (G) admits an (H)-covering, if every edge in (E(G)) belongs to at least one subgraph of (G) that is isomorphic to (H). An ((a,d)-H)-antimagic total labeling of (G) is bijection (f:V(G)cup E(G)rightarrow {1, 2, 3,dots, |V(G)| + |E(G)|}) such that for all subgraphs (H') of (G) isomorphic to (H), the (H') weights (w(H') =sum_{vin V(H')} f (v) + sum_{ein E(H')} f (e)) constitute an arithmetic progression ({a, a + d, a + 2d, dots , a + (n- 1)d}), where (a) and (d) are positive integers and (n) is the number of subgraphs of (G) isomorphic to (H). The labeling (f) is called a super ((a, d)-H)-antimagic total labeling if (f(V(G))={1, 2, 3,dots, |V(G)|}.) In [5], David Laurence and Kathiresan posed a problem that characterizes the super ( (a, 1)-P_{3})-antimagic total labeling of Star (S_{n},) where (n=6,7,8,9.) In this paper, we completely solved this problem.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49510173","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}
In this paper, we study the well-known problem of Isaacs called the "Life line" game when movements of players occur by acceleration vectors, that is, by inertia in Euclidean space. To solve this problem, we investigate the dynamics of the attainability domain of an evader through finding solvability conditions of the pursuit-evasion problems in favor of a pursuer or an evader. Here a pursuit problem is solved by a parallel pursuit strategy. To solve an evasion problem, we propose a strategy for the evader and show that the evasion is possible from given initial positions of players. Note that this work develops and continues studies of Isaacs, Petrosjan, Pshenichnii, Azamov, and others performed for the case of players' movements without inertia.
{"title":"DIFFERENTIAL GAME WITH A LIFELINE FOR THE INERTIAL MOVEMENTS OF PLAYERS","authors":"B. Samatov, U. B. Soyibboev","doi":"10.15826/umj.2021.2.007","DOIUrl":"https://doi.org/10.15826/umj.2021.2.007","url":null,"abstract":"In this paper, we study the well-known problem of Isaacs called the \"Life line\" game when movements of players occur by acceleration vectors, that is, by inertia in Euclidean space. To solve this problem, we investigate the dynamics of the attainability domain of an evader through finding solvability conditions of the pursuit-evasion problems in favor of a pursuer or an evader. Here a pursuit problem is solved by a parallel pursuit strategy. To solve an evasion problem, we propose a strategy for the evader and show that the evasion is possible from given initial positions of players. Note that this work develops and continues studies of Isaacs, Petrosjan, Pshenichnii, Azamov, and others performed for the case of players' movements without inertia.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46325131","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}
The paper deals with linearization problem of Poisson-Lie structures on the ((1+1)) Poincaré and (2D) Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.
{"title":"LINEARIZATION OF POISSON–LIE STRUCTURES ON THE 2D EUCLIDEAN AND (1 + 1) POINCARÉ GROUPS","authors":"Bousselham Ganbouri, Mohamed Wadia Mansouri","doi":"10.15826/umj.2021.2.002","DOIUrl":"https://doi.org/10.15826/umj.2021.2.002","url":null,"abstract":"The paper deals with linearization problem of Poisson-Lie structures on the ((1+1)) Poincaré and (2D) Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45346260","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}
The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition (c_2=1) (which means that every two vertices at distance 2 have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with (c_2=1) is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine (2)-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with (c_2=1) that admit an automorphism group acting (2)-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with (c_2=1) and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.
{"title":"ON A CLASS OF EDGE-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS","authors":"L. Tsiovkina","doi":"10.15826/umj.2021.2.010","DOIUrl":"https://doi.org/10.15826/umj.2021.2.010","url":null,"abstract":"The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition (c_2=1) (which means that every two vertices at distance 2 have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with (c_2=1) is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine (2)-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with (c_2=1) that admit an automorphism group acting (2)-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with (c_2=1) and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43903661","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}
This paper presents a new iterative algorithm for approximating the fixed points of multivalued generalized (alpha)–nonexpansive mappings. We study the stability result of our new iterative algorithm for a larger concept of stability known as weak (w^2)–stability. Weak and strong convergence results of the proposed iterative algorithm are also established. Furthermore, we show numerically that our new iterative algorithm outperforms several known iterative algorithms for multivalued generalized (alpha)–nonexpansive mappings. Again, as an application, we use our proposed iterative algorithm to find the solution of nonlinear Volterra delay integro-differential equations. Finally, we provide an illustrative example to validate the mild conditions used in the result of the application part of this study. Our results improve, generalize and unify several results in the existing literature.
{"title":"A ROBUST ITERATIVE APPROACH FOR SOLVING NONLINEAR VOLTERRA DELAY INTEGRO–DIFFERENTIAL EQUATIONS","authors":"A. Ofem, U. Udofia, D. Igbokwe","doi":"10.15826/umj.2021.2.005","DOIUrl":"https://doi.org/10.15826/umj.2021.2.005","url":null,"abstract":"This paper presents a new iterative algorithm for approximating the fixed points of multivalued generalized (alpha)–nonexpansive mappings. We study the stability result of our new iterative algorithm for a larger concept of stability known as weak (w^2)–stability. Weak and strong convergence results of the proposed iterative algorithm are also established. Furthermore, we show numerically that our new iterative algorithm outperforms several known iterative algorithms for multivalued generalized (alpha)–nonexpansive mappings. Again, as an application, we use our proposed iterative algorithm to find the solution of nonlinear Volterra delay integro-differential equations. Finally, we provide an illustrative example to validate the mild conditions used in the result of the application part of this study. Our results improve, generalize and unify several results in the existing literature.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43205922","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}
A subset ( H subseteq V (G) ) of a graph (G) is a hop dominating set (HDS) if for every ({vin (Vsetminus H)}) there is at least one vertex (uin H) such that (d(u,v)=2). The minimum cardinality of a hop dominating set of (G) is called the hop domination number of (G) and is denoted by (gamma_{h}(G)). In this paper, we compute the hop domination number for triangular and quadrilateral snakes. Also, we analyse the hop domination number of graph families such as generalized thorn path, generalized ciliates graphs, glued path graphs and generalized theta graphs.
图(G)的子集( H subseteq V (G) )是跳支配集(HDS)，如果对于每个({vin (Vsetminus H)})至少有一个顶点(uin H)使得(d(u,v)=2)。(G)的跳数支配集的最小基数称为(G)的跳数支配数，用(gamma_{h}(G))表示。本文计算了三角形蛇类和四边形蛇类的跳跃支配数。此外，我们还分析了图族的跳数支配数，如广义刺路图、广义睫状图、胶合路径图和广义θ图。
{"title":"ON HOP DOMINATION NUMBER OF SOME GENERALIZED GRAPH STRUCTURES","authors":"S. Shanmugavelan, C. Natarajan","doi":"10.15826/umj.2021.2.009","DOIUrl":"https://doi.org/10.15826/umj.2021.2.009","url":null,"abstract":"A subset ( H subseteq V (G) ) of a graph (G) is a hop dominating set (HDS) if for every ({vin (Vsetminus H)}) there is at least one vertex (uin H) such that (d(u,v)=2). The minimum cardinality of a hop dominating set of (G) is called the hop domination number of (G) and is denoted by (gamma_{h}(G)). In this paper, we compute the hop domination number for triangular and quadrilateral snakes. Also, we analyse the hop domination number of graph families such as generalized thorn path, generalized ciliates graphs, glued path graphs and generalized theta graphs.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43229894","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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