THE DRAZIN INVERSE OF MATRICES OF QUATERNIONS

Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R(?)C(?)H. Let H~(m×n) denote the set of all m×n matrices over H.If A=(a_(rs))∈H~(m×n),then there exist A_1 and A_2∈C~(m×n) such that A=A_1+A_2j.Let A_C denote the complexrepresentation of A,that is the 2m×2n complex matrix Ac=((A_1/A_2)(-A_2/A_1))(see[1,2]).We denote by A~D the Drazin inverse of A∈H~(m×n) which is the unique solution of the e-     

TO SOLVE MATRIX EQUATION sum (A~iXBi=c) BY THE SMITH NORMAL FORM

§ 1　IntroductionLet F be a field,F[λ] be the polynomial ring over F,Fm× n( or Fm× n[λ] ) be the setofall m×n matrices over F( or F[λ] ) .Let M(i) be the ith column of M∈Fm× m[λ] ,i=1 ,...,n.A g-inverse of M∈Fm× n will be denoted by M- and understood as a matrix for whichMM- M=M.In this paper,we discuss the linear matrix equation ki=0Ai XBi =C, ( 1 )where A∈Fm× m,Bi∈Fn× q,i=0 ,1 ,...,k,and C∈Fm× q.Equation( 1 ) is called universally solvable if ithas a solution f…     

两类惯量惟一的对称符号模式

§ 1　 IntroductionA sign pattern(matrix) A is a matrix whose entries are from the set{ +,-,0 } .De-note the setofall n× n sign patterns by Qn.Associated with each A=(aij)∈ Qnis a class ofreal matrices,called the qualitative class of A,defined byQ(A) ={ B =(bij)∈ Mn(R) |sign(bij) =aijfor all i and j} .　　 For a symmetric sign pattern A∈ Qn,by G(A) we mean the undirected graph of A,with vertex set { 1 ,...,n} and (i,j) is an edge if and only if aij≠ 0 .A sign pattern A∈ Qnis a do…     

交换环上上三角矩阵保对合与保立方幂等的模自同构

Suppose R is a commutative ring with 1, and 2 is a unit of R. Let Tn(R) be the n × n upper triangular matrix modular over R, and let (?)i(R) (i=2 or 3) be the set of all R-module automorphisms on Tn(R) that preserve involutory or tripotent. The main result in this paper is that f ∈ (?)i(R) if and only if there exists an invertible matrix U ∈ Tn(R) and orthogonal idempotent elements e1,e2,e3 ande4 in R with such that where     

The K2-groups over Fimite Commutative Rings

The present note determines the structure of the K2-group and of itssubgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 ≤ i ≤ m), where R ≌ m i=1 Ri and K2(R) ≌ m i=1 K2(Ri). We show that if charKi = p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.     

RESEARCH ANNOUNCEMENTS Necessary and Sufficient Convergence Conditions for Algebraic Image Reconstruction Algorithms

Many imaging systems can be modeled by the following linear system of equations Ax=b,(1) where the observed data is b=(b~1...b~M)~T∈K~M and the image is x=(x_1…x_N)~T∈K~N.The number field K can be the reals R or the complexes C.The system matrix A=(A_(i,j)) is nonzero and of the dimension M×N matrix.The image reconstruction problem is to reconstruct the     

GEOMETRY OF RECTANGULAR MATRICES OVER A SIMPLE ARTINIAN RING

Let R be a simple Artinian ring,M_(m×n)(R) be the set of all m×n matrices over R.GL_n(R) be the set of all n×n invertible matrices over R.Let A~T be the transpose matrix ofA∈M_(m×n)(R).By the Wedderburn-Artin theorem,R be isomorphic to a total matrix ringM_(s×s)(D) over a division ring D.Let α→(α)_D be an isomorphism of R onto M_(s×s)(D), if X=     

GENERALIZED HILBERT OPERATOR AND FEJ(E)R-RIESZ TYPE INEQUALITIES ON THE POLYDISC

Let ∫ be a holomorphic function on the unit polydisc Dn, with Taylor expansion ∫(z)= ∞∑|k|=0αkzk(=)∞∑k1+…+kn=0 αk1, …,knZk11…Zknn where k=(k1,…, kn)∈Zn+. The authors define generalized Hilbert operator on Dn by Hγ,n(f)(z)=∞ ∑|k|=0i1,…in≥0αi1,…,inn∏j=1Γ(γj+kj+1) Γ(kj+ij+1)/Γ(kj+1) Γ(kj+ij+γj+2)zk,where γ∈Cn, such that Rγj -1,j = 1,2,…,n. An upper bound for the norm of the operator on Hardy spaces Hp(Dn) is found. The authors also present a Fejér-Riesz type inequality on the weighted Bergman space on Dn and find an invariant space for the generalized Hilbert operator.     

BOUNDEDNESS OF COMPOSITION OPERATORS ON Dτ SPACES FOR τ>1

In this note, some conditions of composition operators on Dτspaces to be bounded are given by means of Carleson measures and pointwisemultipliers, for some ranges of τ. The authors prove that (i) Let 1<τ n+2 and 2k<τ 2k+1 (or 2k-1<τ 2k) for somepositive integer k. Suppose φ=(φ1,...,φn) be aunivalent mapping from B into itself, denote dμj(l)(z)=R(l)φj(z)2(k-l+2)(1-|z|2)2k-τ+1 dν(z) for l=1,2,...,k+1. If μj(l)φ-1 are (τ-2k+2l-4)-Carleson measures for all l, then the composition operator Cφon Dτis bounded; (ii) Let 1<τ n+2, φ=(φ1,...,φn)be univalent and the Fr echet derivativeof φ-1 be bounded onφ(B). If Rφj∈M(Dτ-2) forall j, then the composition operator Cφ on Dτis bounded; (iii) Let τ>n+2 and φ as in (ii).If φj∈Dτ for all j, then the compositionoperator Cφ on Dτ is bounded.     

Continuity of functors with respect to generalized inductive limits

Let(Ai,φi,i+1) be a generalized indue Live system of a sequeiiee (Ai) of unital separable C^*-algebras,with A =limi→∞(Ai,φi,i+1). Set φj,i=φi-1,i^0…0φj+1,j+2^0 φj,j+1 for all i>j. We prove that if φj,i are order zero completely positive contractions for all j and i>j, And L:=inf{λ|λ∈σ(φj,i(1Aj)) for all j uud i>j}>0, where σ(φj,i(1Aj)) is the speetrum of φj,i(1Aj),than limi→∞(Cu(Ai),Cu((φi,i+1))=Cu(A), where Cu(A) is a stable version of the Cuntz semigroup of C^*-algebra A. Let (An,φm,n) be a generalized inductive syfitem of C^*-algahrafl, with the ipmkn order zero completely positive contractions. We also prove that if the decomposition rank (nuclear dimension) of ,4n is no more t han some integer k for each n, then the decompostition rank (nuclear dimension) of A is also no more than k.     

Orbifold Gromov-Witten theory of weighted blowups

Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid (X, ω). Let X_a be the weight-a blowup of X along S, and D_a = PN_a be the exceptional divisor, where N is the normal bundle of S in X. In this paper we show that the absolute orbifold Gromov-Witten theory of X_α can be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X, S and D_α, the natural restriction homomorphism H*_CR(X) → H*_CR(S) and the first Chern class of the tautological line bundle over D_α. To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of (X_α | D_α) and (N_α | D_α). As applications of these results, we prove an orbifold version of a conjecture of Maulik and Pandharipande (Topology, 2006) on the Gromov-Witten theory of blowups along complete intersections, a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You (J Pure Appl Algebra, 2016).

     

Dlab groups

We argue that for any subgroup H of rank 1 in a multiplicative group of positive reals, among Dlab groups of the closed intervalI=[0],[1] on an extended set of reals, there exist groups D_H*(I) and D_H* which lack normal relatively convex subgroups, are not simple groups, and have just two distinct linear orders. The cardinality of a set of linear orders on Dlab groups is computed. It is established that every rigid l-group is Abelian if it belongs to a varietyD of l-groups groups generated by the linearly ordered groups D_H*(I) and D_H*. We prove that the quasivariety q(D_H*(I), D_H*) of groups generated by D_H*(I) and D_H* is distinct from a quasivarietyO of all orderable groups. Similar results are stated for a variety of l-groups and the quasivariety of groups that are not embeddable in D_H*(I) and D_H*. Supported by RFFR grant No. 96-01-00088. Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 531–548, September–October, 1999.     

Inertias and ranks of some Hermitian matrix functions with applications

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications, we derive the necessary and sufficient conditions for the existence of maximal matrices of

Central Extensions of *-ordered Skew Fields

A *-ordering of a skew field D induces an ordering of the field K of its central symmetric elements. Let F be an ordered field extension of K. We prove that the central extension of D by F exists and admits a *-ordering extending the given *-ordering of D and ordering of F. As a corollary, we show that every *-ordered skew field can be extended to a *-ordered skew field containing in its center.     

Inverse of the distance matrix of a block graph

A connected graph G, whose 2-connected blocks are all cliques (of possibly varying sizes) is called a block graph. Let D be its distance matrix. By a theorem of Graham, Hoffman and Hosoya, we have det(D)?≠?0. We give a formula for both the determinant and the inverse, D ^?1 of D.     

The Sum and Chain Rules for Maximal Monotone Operators

This paper is primarily concerned with the problem of maximality for the sum A + B and composition L* ML in non-reflexive Banach space settings under qualifications constraints involving the domains of A, B, M. Here X, Y are Banach spaces with duals X*, Y*, A, B: X ⇉ X*, M: Y ⇉ Y* are multi-valued maximal monotone operators, and L: X → Y is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as simpler proofs, improvements of previously known results, and several new results on the topic are presented.      

A General Theory of Almost Splitting Sets

Let * be a star-operation of finite type on an integral domain D. In this paper, we generalize and study the concept of almost splitting sets. We define a saturated multiplicative subset S of D to be an almost g*-splitting set of D if for each 0 ≠ d ∈ D, there exists an integer n = n(d) ≥1 such that d ⁿ  = st for some s ∈ S and t ∈ D with (t, s′)_* = D for all s′ ∈ S. Among other things, we prove that every saturated multiplicative subset of D is an almost g*-splitting set if and only if D is an almost weakly factorial domain (AWFD) with *-dim (D) = 1. We also give an example of an almost g*-splitting set which is not a g*-splitting set.     

On v-Domains and Star Operations

Let * be a star operation on an integral domain D. Let f (D) be the set of all nonzero finitely generated fractional ideals of D. Call D a *-Prüfer (respectively, (*, v)-Prüfer) domain if (FF ^?1)* = D (respectively, (F ^v F ^?1)* = D) for all F ∈  f (D). We establish that *-Prüfer domains (and (*, v)-Prüfer domains) for various star operations * span a major portion of the known generalizations of Prüfer domains inside the class of v-domains. We also use Theorem 6.6 of the Larsen and McCarthy book [30 Larsen , M. D. , McCarthy , P. J. ( 1971 ). Multiplicative Theory of Ideals . New York : Academic Press . [Google Scholar]], which gives several equivalent conditions for an integral domain to be a Prüfer domain, as a model, and we show which statements of that theorem on Prüfer domains can be generalized in a natural way and proved for *-Prüfer domains, and which cannot be. We also show that in a *-Prüfer domain, each pair of *-invertible *-ideals admits a GCD in the set of *-invertible *-ideals, obtaining a remarkable generalization of a property holding for the “classical” class of Prüfer v-multiplication domains. We also link D being *-Prüfer (or (*, v)-Prüfer) with the group Inv*(D) of *-invertible *-ideals (under *-multiplication) being lattice-ordered.     

Noncommutative potential theory and the sign of the curvature operator in Riemannian geometry

The aim of this work is to show that in any complete Riemannian manifold M, without boundary, the curvature operator is nonnegative if and only if the Dirac Laplacian D² generates a C*-Markovian semigroup (i.e. a strongly continuous, completely positive, contraction semigroup) on the Cliord C*-algebra of Mor, equivalently, if and only if the quadratic form $\mathcal{E}$_D of D² is a C*-Dirichlet form.     

Characterizations of *-Cancellation Ideals of an Integral Domain

Let D be an integral domain and * a star-operation on D. For a nonzero ideal I of D, let I ^{* _f} = ?{J* | (0) ≠ J ? I is finitely generated} and I ^{* _w} = ? _{P∈* f -Max(D)} ID _P . A nonzero ideal I of D is called a *-cancellation ideal if (IA)* = (IB)* for nonzero ideals A and B of D implies A* =B*. Let X be an indeterminate over D and N _* = {f ∈ D[X] | (c(f))* =D}. We show that I is a * _w -cancellation ideal if and only if I is * _f -locally principal, if and only if ID[X] _{N *} is a cancellation ideal. As a corollary, we have that each nonzero ideal of D is a * _w -cancellation ideal if and only if D _P is a principal ideal domain for all P ∈ * _f -Max(D), if and only if D[X] _{N *} is an almost Dedekind domain. We also show that if I is a * _w -cancellation ideal of D, then I ^{* _w}  = I ^{* _f}  = I _t , and I is * _w -invertible if and only if I ^{* _w}  = J _v for a nonzero finitely generated ideal J of D.