Let A be a lattice-ordered group. Gusi? showed that A can be equipped with a C-topology which makes A into a topological group. We give a generalization of Gusi?’s theorem, and reveal the very nature of a “C-group”...Let A be a lattice-ordered group. Gusi? showed that A can be equipped with a C-topology which makes A into a topological group. We give a generalization of Gusi?’s theorem, and reveal the very nature of a “C-group” of Gusi? in this paper. Moreover, we show that the C-topological groups are topological lattice-ordered groups, and prove that every archimedean lattice-ordered vector space is a T 2 topological lattice-ordered vector space under the C-topology. An easy example shows that a C-group need not be T 2. A further example demonstrates that a T 2 topological archimedean lattice-ordered group need not be C-archimedean, either.展开更多
This is a short review article in which we discuss and summarize the works of various researchers over past four decades on Zeeman topology and Zeeman-like topologies, which occur in special and general theory of rela...This is a short review article in which we discuss and summarize the works of various researchers over past four decades on Zeeman topology and Zeeman-like topologies, which occur in special and general theory of relativity. We also discuss various properties and inter-relationship of these topologies.展开更多
Two non-discrete Hausdorff group topologiesτandδon a group G are called transversal if the least upper boundτ⋁δofτandδis the discrete topology.In this paper,we discuss the existence of transversal group topologi...Two non-discrete Hausdorff group topologiesτandδon a group G are called transversal if the least upper boundτ⋁δofτandδis the discrete topology.In this paper,we discuss the existence of transversal group topologies on locally pseudocompact,locally precompact,or locally compact groups.We prove that each locally pseudocompact,connected topological group satisfies central subgroup paradigm,which gives an affirmative answer to a problem posed by Dikranjan,Tkachenko,and Yaschenko[Topology Appl.,2006,153:3338–3354].For a compact normal subgroup K of a locally compact totally disconnected group G,if G admits a transversal group topology,then G/K admits a transversal group topology,which gives a partial answer again to a problem posed by Dikranjan,Tkachenko,and Yaschenko in 2006.Moreover,we characterize some classes of locally compact groups that admit transversal group topologies.展开更多
A Riesz type product as Pn=nЛj=1（1＋awj＋bwj＋1）is studied, where a, b are two real numbers with ｜a｜ ＋ ｜b｜ 〈 1, and {wj} are indepen- dent random variables taking values in （-1, 1} with equal probability. Le...A Riesz type product as Pn=nЛj=1（1＋awj＋bwj＋1）is studied, where a, b are two real numbers with ｜a｜ ＋ ｜b｜ 〈 1, and {wj} are indepen- dent random variables taking values in （-1, 1} with equal probability. Let dw be the normalized Haar measure on the Cantor group Ω = （-1, 1}^N. The sequence of P,~dw 1 probability measures {Pndw/E（Pn） } is showed to converge weakly to a unique continuous measure on/2, and the obtained measure is singular with respect to dw.展开更多
For a commutative ring R, its related characterizations are given by investigating the structure and properties of H0R. Furthermore, by virtue of H0structure, some important characterizations of CPF properties and con...For a commutative ring R, its related characterizations are given by investigating the structure and properties of H0R. Furthermore, by virtue of H0structure, some important characterizations of CPF properties and connected properties on K0R are obtained from related rings.展开更多
The existing containment control has been widely developed for several years, but ignores the case for large-scale cooperation. The strong coupling of large-scale networks will increase the costs of system detection a...The existing containment control has been widely developed for several years, but ignores the case for large-scale cooperation. The strong coupling of large-scale networks will increase the costs of system detection and maintenance. Therefore, this paper is concerned with an extensional containment control issue, hierarchical containment control. It aims to enable a multitude of followers achieving a novel cooperation in the convex hull shaped by multiple leaders. Firstly, by constructing the three-layer topology, large-scale networks are decoupled. Then,under the condition of directed spanning group-tree, a class of dynamic hierarchical containment control protocol is designed such that the novel group-consensus behavior in the convex hull can be realized. Moreover, the definitions of coupling strength coefficients and the group-consensus parameter in the proposed dynamic hierarchical control protocol enhance the adjustability of systems. Compared with the existing containment control strategy, the proposed hierarchical containment control strategy improves dynamic control performance. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed hierarchical control protocol.展开更多
基金supported by the Fund of Elitist Development of Beijing (Grant No. 20071D1600600412)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘Let A be a lattice-ordered group. Gusi? showed that A can be equipped with a C-topology which makes A into a topological group. We give a generalization of Gusi?’s theorem, and reveal the very nature of a “C-group” of Gusi? in this paper. Moreover, we show that the C-topological groups are topological lattice-ordered groups, and prove that every archimedean lattice-ordered vector space is a T 2 topological lattice-ordered vector space under the C-topology. An easy example shows that a C-group need not be T 2. A further example demonstrates that a T 2 topological archimedean lattice-ordered group need not be C-archimedean, either.
文摘This is a short review article in which we discuss and summarize the works of various researchers over past four decades on Zeeman topology and Zeeman-like topologies, which occur in special and general theory of relativity. We also discuss various properties and inter-relationship of these topologies.
基金This work was supported by the Key Program of the Natural Science Foundation of Fujian Province (No. 2020J02043)the National Natural Science Foundation of China (Grant No. 11571158)the Institute of Meteorological Big Data-Digital Fujian, and Fujian Key Laboratory of Data Science and Statistics.
文摘Two non-discrete Hausdorff group topologiesτandδon a group G are called transversal if the least upper boundτ⋁δofτandδis the discrete topology.In this paper,we discuss the existence of transversal group topologies on locally pseudocompact,locally precompact,or locally compact groups.We prove that each locally pseudocompact,connected topological group satisfies central subgroup paradigm,which gives an affirmative answer to a problem posed by Dikranjan,Tkachenko,and Yaschenko[Topology Appl.,2006,153:3338–3354].For a compact normal subgroup K of a locally compact totally disconnected group G,if G admits a transversal group topology,then G/K admits a transversal group topology,which gives a partial answer again to a problem posed by Dikranjan,Tkachenko,and Yaschenko in 2006.Moreover,we characterize some classes of locally compact groups that admit transversal group topologies.
文摘A Riesz type product as Pn=nЛj=1（1＋awj＋bwj＋1）is studied, where a, b are two real numbers with ｜a｜ ＋ ｜b｜ 〈 1, and {wj} are indepen- dent random variables taking values in （-1, 1} with equal probability. Let dw be the normalized Haar measure on the Cantor group Ω = （-1, 1}^N. The sequence of P,~dw 1 probability measures {Pndw/E（Pn） } is showed to converge weakly to a unique continuous measure on/2, and the obtained measure is singular with respect to dw.
文摘For a commutative ring R, its related characterizations are given by investigating the structure and properties of H0R. Furthermore, by virtue of H0structure, some important characterizations of CPF properties and connected properties on K0R are obtained from related rings.
基金supported in part by the National Natural Science Foundation of China(U22A20221,62073064)in part by the Fundamental Research Funds for the Central Universities in China(N2204007)。
文摘The existing containment control has been widely developed for several years, but ignores the case for large-scale cooperation. The strong coupling of large-scale networks will increase the costs of system detection and maintenance. Therefore, this paper is concerned with an extensional containment control issue, hierarchical containment control. It aims to enable a multitude of followers achieving a novel cooperation in the convex hull shaped by multiple leaders. Firstly, by constructing the three-layer topology, large-scale networks are decoupled. Then,under the condition of directed spanning group-tree, a class of dynamic hierarchical containment control protocol is designed such that the novel group-consensus behavior in the convex hull can be realized. Moreover, the definitions of coupling strength coefficients and the group-consensus parameter in the proposed dynamic hierarchical control protocol enhance the adjustability of systems. Compared with the existing containment control strategy, the proposed hierarchical containment control strategy improves dynamic control performance. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed hierarchical control protocol.
