To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibili...To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibility and an involution.This paper begins by outlining the conditions necessary for the existence of the m-weak group inverse of a.Moreover,it explores the criteria under which a can be considered pseudo core invertible and weak group invertible.In the context of a weak proper*-ring,it is proved that a is weak group invertible if,and only if,a D can serve as the weak group inverse of au,where u represents a specially invertible element closely associated with a D.The paper also introduces a counterexample to illustrate that a D cannot universally serve as the pseudo core inverse of another element.This distinction underscores the nuanced differences between pseudo core inverses and weak group inverses.Ultimately,the discussion expands to include the commuting properties of weak group inverses,extending these considerations to m-weak group inverses.Several new conditions on commuting properties of generalized inverses are given.These results show that pseudo core inverses,weak group inverses,and m-weak group inverses are not only closely linked but also have significant differences that set them apart.展开更多
The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-ci...The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-circulant matrices are also given.展开更多
We study a new class of group inverses determined by right c-regular elements.The new concept of right c-group inverses is introduced and studied.It is shown that every right c-group invertible element is group invert...We study a new class of group inverses determined by right c-regular elements.The new concept of right c-group inverses is introduced and studied.It is shown that every right c-group invertible element is group invertible,and an example is given to show that group invertible elements need not be right c-group invertible.The conditions that right c-group invertible elements are precisely group invertible elements are investigated.We also study the strongly clean decompositions of right c-group invertible elements.As applications,we give some new characterizations of abelian rings and directly finite rings from the point of view of right c-groupinverses.展开更多
Abstract We first consider the group inverses of the block matrices（AB0C）over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group invers...Abstract We first consider the group inverses of the block matrices（AB0C）over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group inverses of the block matrices（ABCD）over a ring with unity 1 under the following conditions respectively： （i） B = C, D = 0,B#and（BπA0#both exist; （ii） B is invertible and m = n;（iii）A#and （D - CA#B）# both exist, C = CAA#, where A and D are m × m and n × n matrices, respectively.展开更多
Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the...Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the displacement structure of the iteration matrix, the matrix-vector multiplication involved in Newton＇s iteration can be done efficiently. We show that the convergence of the modified Newton iteration is still very fast. Numerical results are presented to demonstrate the fast convergence of the proposed method.展开更多
The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP ...The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R , then a is EP if and only if aa a^# = a^#aa . At the same time, the equivalent characterizations of a regular element to be EP are discussed.Let a∈R, then there exist b∈R such that a = aba and a is EP if and only if a∈R , a = a ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R , then a is normal if and only if a^＊a = a a^＊. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R ∩R^#, n∈N, then a is normal if and only if a^＊ a^＋（ a^#） n = a^# a＊（ a^＋） ^n. The results generalize the conclusions of Mosiet al.展开更多
Harish-Chandra have got a Fourier inversion formula for C-c(infinity)(SL (2, R)). In this paper, we give a discussion on approximation identity kernels on SL(2, R) and get some properties of their Fourier transforms, ...Harish-Chandra have got a Fourier inversion formula for C-c(infinity)(SL (2, R)). In this paper, we give a discussion on approximation identity kernels on SL(2, R) and get some properties of their Fourier transforms, and then, making use of these properties and Harish-Chandra's result, we prove that the Fourier inversion formula obtained by Harish-Chandra is also valid for C-o(3)(SL,2, R)).展开更多
Let K<sup>n×n</sup> be the set of all n×n matrices and K<sub>r</sub><sup>n×n</sup> the set {A∈K<sup>n×n</sup>|rankA=r} on askew field K. Zhuang [1] ...Let K<sup>n×n</sup> be the set of all n×n matrices and K<sub>r</sub><sup>n×n</sup> the set {A∈K<sup>n×n</sup>|rankA=r} on askew field K. Zhuang [1] denotes by A<sup>#</sup> the group inverse of A∈K<sup>n×n</sup> which is the solu-tion of the euqations:AXA=A, XAX=X, AX=AX.展开更多
Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element o...Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean.First,some characterizations of centrally clean elements are given.Furthermore,some properties of centrally clean rings,as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated.Centrally clean rings are closely related to the central Drazin inverses.Then,in terms of centrally clean decomposition,the necessary and sufficient conditions for the existence of central Drazin inverses are presented.Moreover,the central cleanness of special rings,such as corner rings,the ring of formal power series over ring R,and a direct product ∏ R_(α) of ring R_(α),is analyzed.Furthermore,the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated.Finally,as an application,an example that lists all invertible,central group invertible,group invertible,central Drazin invertible elements,and centrally clean elements of the group ring Z_(2)S_(3) is given.展开更多
In this paper, we present a method how to get the expression for the group inverse of 2×2 block matrix and get the explicit expressions of the block matrix (A C B D) under some conditions.
In this paper, some conditions for the nonsingularity and group inverses of linear combinations of generalized and hypergeneralized projectors are established. Moreover, some formulae for the inverses and group invers...In this paper, some conditions for the nonsingularity and group inverses of linear combinations of generalized and hypergeneralized projectors are established. Moreover, some formulae for the inverses and group inverses of them are derived. The work of this paper extends some previous results.展开更多
This paper proposes a novel inverse synthetic aperture radar（ISAR） imaging method based on second-order keystone transform（KT） and Sandglass transform for group targets flying in a formation with constant accelera...This paper proposes a novel inverse synthetic aperture radar（ISAR） imaging method based on second-order keystone transform（KT） and Sandglass transform for group targets flying in a formation with constant accelerated rectilinear motion in the same radar beam. First, range curvature and range walk of each sub-target among group targets are corrected by the second-order KT combined with the quadratic phase term compensation. After range alignment, the signals in each range frequency cell can be modelled as multiple chirp signals and then the Sandglass transform is utilized to cross-range imaging, which transforms the time–frequency distribution of the signals in each range frequency cell into beelines parallel to the slow time axis simultaneously. Finally, cross-range profiles of group targets in each range frequency cell are obtained via a projection of the perk of every scatterer in the two-dimensional accumulation plane onto the frequency axis. The advantage of the proposed method is that it can align range profiles of each sub-target simultaneously and image cross-range profiles directly without separating the returned signals, which simplifies the operation procedure. Simulation results are used to demonstrate the effectiveness of the proposed method.展开更多
Necessary and sufficient conditions for the existence of the group inverse of the block matrix in Minkowski Space are studied, where are both square and . The representation of this group inverse and some related addi...Necessary and sufficient conditions for the existence of the group inverse of the block matrix in Minkowski Space are studied, where are both square and . The representation of this group inverse and some related additive results are also given.展开更多
A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of count...A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of countable free groups on the rational line Q, which generalizes their results. As applications,we obtain the highly ordered transitive representation for the direct product of countable free groups,and the inverse limit of countable free groups must be an action on the set Q.展开更多
In order to study the Drazin invertibility of a matrix with the generalized factorization over an arbitrary ring, the necessary and sufficient conditions for the existence of the Drazin inverse of a matrix are given b...In order to study the Drazin invertibility of a matrix with the generalized factorization over an arbitrary ring, the necessary and sufficient conditions for the existence of the Drazin inverse of a matrix are given by the properties of the generalized factorization. Let T = PAQ be a square matrix with the generalized factorization, then T has Drazin index k if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible. The formulae to compute the Drazin inverse are also obtained. These results generalize recent results obtained for the Drazin inverse of a matrix with a universal factorization.展开更多
In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related t...In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related to AT,S^（2）.展开更多
This note is to present some results on the group invertibility of linear combina- tions of idempotents when the difference of two idempotents is group invertible.
Let φ be a pre-additive category. Assume that φ： X→X is a morphism of φ. In this paper, we give the necessary and sufficient conditions for φ to have the Drazin inverse by using the von Neumann regular inverse f...Let φ be a pre-additive category. Assume that φ： X→X is a morphism of φ. In this paper, we give the necessary and sufficient conditions for φ to have the Drazin inverse by using the von Neumann regular inverse for the φ^k, and extend a result by Puystjens and Hartwig from the group inverse to Drazin inverse.展开更多
Let R be a unitary ring and a,b∈R with ab=0.We find the 2/3 property of Drazin invertibility:if any two of a,b and a+b are Drazin invertible,then so is the third one.Then,we combine the 2/3 property of Drazin inverti...Let R be a unitary ring and a,b∈R with ab=0.We find the 2/3 property of Drazin invertibility:if any two of a,b and a+b are Drazin invertible,then so is the third one.Then,we combine the 2/3 property of Drazin invertibility to characterize the existence of generalized inverses by means of units.As applications,the need for two invertible morphisms used by You and Chen to characterize the group invertibility of a sum of morphisms is reduced to that for one invertible morphism,and the existence and expression of the inverse along a product of two regular elements are obtained,which generalizes the main result of Mary and Patricio(2016)about the group inverse of a product.展开更多
In this paper, we first give two equalities in the operation of determinant. Using the expression of group inverse with full-rank factorization Ag = F（GF）^-2G and the Cramer rule of the nonsingular linear system Ax ...In this paper, we first give two equalities in the operation of determinant. Using the expression of group inverse with full-rank factorization Ag = F（GF）^-2G and the Cramer rule of the nonsingular linear system Ax = b, we present a new method to prove the representation of group inverse with arlene combinationAg=∑（I,J）∈N（A） 1/υ^2det（A）IJ ajd AJI.A numerical example is given to demonstrate that the formula is efficient.展开更多
基金The National Natural Science Foundation of China(No.12171083,12071070)Qing Lan Project of Jiangsu Province and the Postgraduate Research and Practice Innovation Program of Jiangsu Province(No.KYCX22_0231).
文摘To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibility and an involution.This paper begins by outlining the conditions necessary for the existence of the m-weak group inverse of a.Moreover,it explores the criteria under which a can be considered pseudo core invertible and weak group invertible.In the context of a weak proper*-ring,it is proved that a is weak group invertible if,and only if,a D can serve as the weak group inverse of au,where u represents a specially invertible element closely associated with a D.The paper also introduces a counterexample to illustrate that a D cannot universally serve as the pseudo core inverse of another element.This distinction underscores the nuanced differences between pseudo core inverses and weak group inverses.Ultimately,the discussion expands to include the commuting properties of weak group inverses,extending these considerations to m-weak group inverses.Several new conditions on commuting properties of generalized inverses are given.These results show that pseudo core inverses,weak group inverses,and m-weak group inverses are not only closely linked but also have significant differences that set them apart.
文摘The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-circulant matrices are also given.
基金Supported by the National Natural Science Foundation of China(Grant No.12161049).
文摘We study a new class of group inverses determined by right c-regular elements.The new concept of right c-group inverses is introduced and studied.It is shown that every right c-group invertible element is group invertible,and an example is given to show that group invertible elements need not be right c-group invertible.The conditions that right c-group invertible elements are precisely group invertible elements are investigated.We also study the strongly clean decompositions of right c-group invertible elements.As applications,we give some new characterizations of abelian rings and directly finite rings from the point of view of right c-groupinverses.
基金Acknowledgements The authors were grateful to the referees for their constructive comments and suggestions. This work was supported by the National Natural Science Foundation of China （Grant No. 11371109） and the Education Department of Heilongjiang Province of China （No. 12541605）.
文摘Abstract We first consider the group inverses of the block matrices（AB0C）over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group inverses of the block matrices（ABCD）over a ring with unity 1 under the following conditions respectively： （i） B = C, D = 0,B#and（BπA0#both exist; （ii） B is invertible and m = n;（iii）A#and （D - CA#B）# both exist, C = CAA#, where A and D are m × m and n × n matrices, respectively.
基金Research supported in part by the National Natural Science Foundation of China under grant 10471027 and Shanghai Education Committee, RGC 7046/03P, 7035/04P, 7045/05P and HKBU FRGs.The authors would like to thank the referees for their useful suggestions.
文摘Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the displacement structure of the iteration matrix, the matrix-vector multiplication involved in Newton＇s iteration can be done efficiently. We show that the convergence of the modified Newton iteration is still very fast. Numerical results are presented to demonstrate the fast convergence of the proposed method.
基金The National Natural Science Foundation of China(No.11371089)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20120092110020)the Natural Science Foundation of Jiangsu Province(No.BK20141327)
文摘The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R , then a is EP if and only if aa a^# = a^#aa . At the same time, the equivalent characterizations of a regular element to be EP are discussed.Let a∈R, then there exist b∈R such that a = aba and a is EP if and only if a∈R , a = a ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R , then a is normal if and only if a^＊a = a a^＊. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R ∩R^#, n∈N, then a is normal if and only if a^＊ a^＋（ a^#） n = a^# a＊（ a^＋） ^n. The results generalize the conclusions of Mosiet al.
文摘Harish-Chandra have got a Fourier inversion formula for C-c(infinity)(SL (2, R)). In this paper, we give a discussion on approximation identity kernels on SL(2, R) and get some properties of their Fourier transforms, and then, making use of these properties and Harish-Chandra's result, we prove that the Fourier inversion formula obtained by Harish-Chandra is also valid for C-o(3)(SL,2, R)).
基金This work is Supported by NSF of Heilongjiang Provice
文摘Let K<sup>n×n</sup> be the set of all n×n matrices and K<sub>r</sub><sup>n×n</sup> the set {A∈K<sup>n×n</sup>|rankA=r} on askew field K. Zhuang [1] denotes by A<sup>#</sup> the group inverse of A∈K<sup>n×n</sup> which is the solu-tion of the euqations:AXA=A, XAX=X, AX=AX.
基金The National Natural Science Foundation of China(No.12171083,11871145,12071070)the Qing Lan Project of Jiangsu Province。
文摘Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean.First,some characterizations of centrally clean elements are given.Furthermore,some properties of centrally clean rings,as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated.Centrally clean rings are closely related to the central Drazin inverses.Then,in terms of centrally clean decomposition,the necessary and sufficient conditions for the existence of central Drazin inverses are presented.Moreover,the central cleanness of special rings,such as corner rings,the ring of formal power series over ring R,and a direct product ∏ R_(α) of ring R_(α),is analyzed.Furthermore,the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated.Finally,as an application,an example that lists all invertible,central group invertible,group invertible,central Drazin invertible elements,and centrally clean elements of the group ring Z_(2)S_(3) is given.
基金Supported by the Fund for Postdoctoral of China(2015M581688)Supported by the National Natural Science Foundation of China(11371089)+2 种基金Supported by the Specialized Research Fund for the Doctoral Program of Higher Education(20120092110020)Supported by the Natural Science Foundation of Jiangsu Province(BK20141327)Supported by the Foundation of Xuzhou Institute of Technology(XKY2014207)
文摘In this paper, we present a method how to get the expression for the group inverse of 2×2 block matrix and get the explicit expressions of the block matrix (A C B D) under some conditions.
基金Supported by the National Natural Science Foundation of China(11271105)the Key Research Project of Educational Department of Hubei Province(D20122202)Youth Research Project of Educational Department of Hubei Province(B20122203)
文摘In this paper, some conditions for the nonsingularity and group inverses of linear combinations of generalized and hypergeneralized projectors are established. Moreover, some formulae for the inverses and group inverses of them are derived. The work of this paper extends some previous results.
基金supported by the National Natural Science Foundation of China (No. 61372159)
文摘This paper proposes a novel inverse synthetic aperture radar（ISAR） imaging method based on second-order keystone transform（KT） and Sandglass transform for group targets flying in a formation with constant accelerated rectilinear motion in the same radar beam. First, range curvature and range walk of each sub-target among group targets are corrected by the second-order KT combined with the quadratic phase term compensation. After range alignment, the signals in each range frequency cell can be modelled as multiple chirp signals and then the Sandglass transform is utilized to cross-range imaging, which transforms the time–frequency distribution of the signals in each range frequency cell into beelines parallel to the slow time axis simultaneously. Finally, cross-range profiles of group targets in each range frequency cell are obtained via a projection of the perk of every scatterer in the two-dimensional accumulation plane onto the frequency axis. The advantage of the proposed method is that it can align range profiles of each sub-target simultaneously and image cross-range profiles directly without separating the returned signals, which simplifies the operation procedure. Simulation results are used to demonstrate the effectiveness of the proposed method.
文摘Necessary and sufficient conditions for the existence of the group inverse of the block matrix in Minkowski Space are studied, where are both square and . The representation of this group inverse and some related additive results are also given.
文摘A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of countable free groups on the rational line Q, which generalizes their results. As applications,we obtain the highly ordered transitive representation for the direct product of countable free groups,and the inverse limit of countable free groups must be an action on the set Q.
基金The National Natural Science Foundation of China(No.10571026,10871051)Specialized Research Fund for the Doctoral Pro-gram of Higher Education(No.20060286006,200802860024)
文摘In order to study the Drazin invertibility of a matrix with the generalized factorization over an arbitrary ring, the necessary and sufficient conditions for the existence of the Drazin inverse of a matrix are given by the properties of the generalized factorization. Let T = PAQ be a square matrix with the generalized factorization, then T has Drazin index k if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible. The formulae to compute the Drazin inverse are also obtained. These results generalize recent results obtained for the Drazin inverse of a matrix with a universal factorization.
基金Supported by the National Natural Science Foundation of China(11271105)the Key Research Project of Educational Department of Hubei Province(D20122202)Youth Research Project of Educational Department of Hubei Province(B20122203)
文摘In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related to AT,S^（2）.
基金supported by the National Natural Science Foundation of China under grant No.11171222the Doctoral Program of the Ministry of Education under grant No.20094407120001
文摘This note is to present some results on the group invertibility of linear combina- tions of idempotents when the difference of two idempotents is group invertible.
基金The first author is supported by the NNSF (10571026) of China the NSF (BK 2005207) of Jiangsu Province in ChinaThe second author is supported by the NNSF (10471027) of China
文摘Let φ be a pre-additive category. Assume that φ： X→X is a morphism of φ. In this paper, we give the necessary and sufficient conditions for φ to have the Drazin inverse by using the von Neumann regular inverse for the φ^k, and extend a result by Puystjens and Hartwig from the group inverse to Drazin inverse.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12171083,11871145,12071070)the Qing Lan Project of Jiangsu Provincethe Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX22-0231)。
文摘Let R be a unitary ring and a,b∈R with ab=0.We find the 2/3 property of Drazin invertibility:if any two of a,b and a+b are Drazin invertible,then so is the third one.Then,we combine the 2/3 property of Drazin invertibility to characterize the existence of generalized inverses by means of units.As applications,the need for two invertible morphisms used by You and Chen to characterize the group invertibility of a sum of morphisms is reduced to that for one invertible morphism,and the existence and expression of the inverse along a product of two regular elements are obtained,which generalizes the main result of Mary and Patricio(2016)about the group inverse of a product.
基金the Shanghai Science and Technology Committee (No.062112065)Shanghai Prior-ity Academic Discipline Foundation+1 种基金the University Young Teacher Sciences Foundation of Anhui Province(No.2006jq1220zd)the PhD Program Scholarship Fund of ECNU 2007.
文摘In this paper, we first give two equalities in the operation of determinant. Using the expression of group inverse with full-rank factorization Ag = F（GF）^-2G and the Cramer rule of the nonsingular linear system Ax = b, we present a new method to prove the representation of group inverse with arlene combinationAg=∑（I,J）∈N（A） 1/υ^2det（A）IJ ajd AJI.A numerical example is given to demonstrate that the formula is efficient.
