A comparison of nonnegative real ranks and their preservers

This paper concerns three notions of rank of matrices over semirings; real rank, semiring rank and column rank. These three rank functions are the same over subfields of reals but differ for matrices over subsemirings of nonnegative reals. We investigate the largest values of r for which the real rank and semiring rank, real rank and column rank of all m×n matrices over a given semiring are both r, respectively. We also characterize the linear operators which preserve the column rank of matrices over certain subsemirings of the nonnegative reals.     

Rank properties of the semigroup of endomorphisms over Brandt semigroup

We investigate small rank, lower rank, intermediate rank, upper rank, and large rank of the semigroup of endomorphisms over Brandt semigroup.     

A comparison of nonnegative real ranks and their preservers

This paper concerns three notions of rank of matrices over semirings; real rank, semiring rank and column rank. These three rank functions are the same over subfields of reals but differ for matrices over subsemirings of nonnegative reals. We investigate the largest values of r for which the real rank and semiring rank, real rank and column rank of all m×n matrices over a given semiring are both r, respectively. We also characterize the linear operators which preserve the column rank of matrices over certain subsemirings of the nonnegative reals.     

Continuity and separation in symmetric topologies

The well-known Frobenius rank inequality established by Frobenius in 1911 states that the rank of the product ABC of three matrices satisfies the inequality rank(ABC) U rank(AB) + rank(BC)- rank(B) A new necessary and sufficient condition for equality to hold is presented and then some interesting consequences and applications are discussed.     

Bijective linear rank preservers for spaces of matrices over antinegative semirings

We classify the bijective linear operators on spaces of matrices over antinegative commutative semirings with no zero divisors which preserve certain rank functions such as the symmetric rank, the factor rank and the tropical rank. We also classify the bijective linear operators on spaces of matrices over the max-plus semiring which preserve the Gondran-Minoux row rank or the Gondran-Minoux column rank.     

Lower Bounds in Minimum Rank Problems

The minimum rank of a graph is the smallest possible rank among all real symmetric matrices with the given graph. The minimum semidefinite rank of a graph is the minimum rank among Hermitian positive semidefinite matrices with the given graph. We explore connections between OS-sets and a lower bound for minimum rank related to zero forcing sets as well as exhibit graphs for which the difference between the minimum semidefinite rank and these lower bounds can be arbitrarily large.     

迹稳定秩一的C~*-代数

本文引入了一类迹稳定秩一的C*-代数,证明了迹稳定秩一的C*-代数与AF-代数的张量积是迹稳定秩一的,得到了一个可分的单的有单位元的迹稳定秩一的,并且具有SP性质的C*-代数是稳定秩一的.同时,还讨论了迹稳定秩一的C*-代数的K-群的某些性质.     

集团序及其应用

提出了集团序的概念和方法.首先,定义了方案集的集团序,定义了方案的集团序指标和方案的集团优先关系.其次,对方案集团序进行了稳定性分析,给出了方案集团序稳定的充分条件.最后给出一个学科评估应用分析.     

On the Kapranov ranks of tropical matrices

We investigate the Kapranov rank functions of tropical matrices for different ground fields. For any infinite ground field we show that the rank-product inequality holds for Kapranov rank, and we prove that the Kapranov rank respects Green’s preorders on the semigroup of tropical n-by-n matrices. The rank-product inequality is shown to fail for Kapranov rank over any finite ground field. We provide an example of a 7-by-7 01-matrix whose Kapranov rank is independent of a ground field, equals 6, and exceeds tropical rank.     

Inequalities between rank moments of overpartitions

F.G. Garvan proved an inequality between crank moments and rank moments of partitions which utilizes an inequality for symmetrized rank and crank moments. In this paper, we study two symmetrized rank moments of overpartitions and prove an inequality between two rank moments of overpartitions.     

Linear operators that preserve maximal column rank of boolean matrices

This paper concerns two notions of column rank of matrices over semirings; column rank and maximal column rank. These two notions are the same over fields but differ for matrices over certain semirings. We determine how much the maximal column rank is different from the column ran for all m×n matrices over many semirings. We also characterize the linear operators which preserve the maximal column rank of Boolean matrices.     

关于矩阵(A B C 0)的g-逆中子块的独立性

In this paper, we -proved a interesting fact that the rank additivity condition rank (A B C 0) = rank (A C) + rank (B 0) = rank(A,B)+ rank(C, 0)is a necessary and sufficient condition for block independence in g-inverse of (A B C 0),and moreover for that the Moore-Penrose inverse of (A B C 0) is(?) where Q=[(I- BB⁺)A(I-C⁺C)]⁺.     

Existence of rank 3 vector bundles with given chern classes on homogeneous rational 3-folds

One proves that any rank 3 topological vector bundle on a homogeneous rational 3-fold has an algebraic structure. The proof uses extensions of ideals by rank 2 vector bundles. The paper also contains a construction of rank 3 vector bundles on 3-folds using extensions of ideals by rank 2 reflexive sheaves.     

Unitary stable ranks and norm-one ranks

In the context of commutative C*-algebras, we solve a problem related to a question of M. Rieffel by showing that the all-units rank and the norm-one rank coincide with the topological stable rank. We also introduce the notion of unitary M-stable rank for an arbitrary commutative unital ring and compare it with the Bass stable rank. In case of uniform algebras, a su?cient condition for norm-one reducibility is given.     

Column Ranks and Their Preservers of Matrices Over Max Algebra

The column rank of an m by n matrix A over max algebra is the weak dimension of the column space of A. We compare the column rank with rank of matrices over max algebra. We also characterize the linear operators which preserve the column rank of matrices over max algebra.     

Linear operators that preserve pairs of matrices which satisfy extreme rank properties

A pair of m×n matrices (A,B) is called rank-sum-maximal if rank(A+B)=rank(A)+rank(B), and rank-sum-minimal if rank(A+B)=|rank(A)−rank(B)|. We characterize the linear operators that preserve the set of rank-sum-minimal matrix pairs, and the linear operators that preserve the set of rank-sum-maximal matrix pairs over any field with at least min(m,n)+2 elements and of characteristic not 2.     

On fixed subgroups of maximal rank

We show that, in the free group F of rank n, n is the maximal length of strictly ascending chains of maximal rank fixed subgroups, that is, rank n subgroups of the form Fix^{^} for some 4> L Aut(F). We further show that, in the rank two case, if the intersection of an arbitrary family of proper maximal rank fixed subgroups has rank two then all those subgroups are equal. In particular, every G < Aut(F) with r(FixG) = 2 is either trivial or infinite cyclic.     

Rank Reducing Matrix Norms

We consider approximation numbers for some norms on matrices, and look at the question when a closest rank h p approximant can be chosen to reduce the rank of a matrix by p . If the latter is always possible, we call the norm rank p reducing. It is easily seen that any unitarily invariant norm is rank p reducing. We show that any absolute norm on $\shadC^{n \times m}$ is rank n m 1 reducing and that the numerical radius norm on $ \shadC^{n\times n}$ is rank n m 1 reducing as well. Non-examples and computations of approximation numbers are also presented.     

New scaling of Itzykson-Zuber integrals

We study asymptotics of the Itzykson-Zuber integrals in the scaling when one of the matrices has a small rank compared to the full rank. We show that the result is basically the same as in the case when one of the matrices has a fixed rank. In this way we extend the recent results of Guionnet and Maïda who showed that for the fixed rank scaling, the Itzykson-Zuber integral is given in terms of the Voiculescu's R-transform of the full rank matrix.     

Rank numbers for some trees and unicyclic graphs

A ranking on a graph is an assignment of positive integers to its vertices such that any path between two vertices of the same rank contains a vertex of strictly larger rank. The rank number of a graph is the fewest number of labels that can be used in a ranking. In this paper we determine rank numbers for some trees and unicyclic graphs.