Zero-Term Ranks of Real Matrices and Their Preservers

Zero-term rank of a matrix is the minimum number of lines (rows or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve zero-term rank of the m× nreal matrices. We also obtain combinatorial equivalent condition for the zero-term rank of a real matrix.     

Preservers of eigenvalue inclusion sets

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the Brauer region in terms of Cassini ovals, and the Ostrowski region. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)-Φ(B))=S(A-B) for all matrices A and B. From these results, one can deduce the structure of additive or (real) linear maps satisfying S(A)=S(Φ(A)) for every matrix A.     

Singularity Preservers over Local Domains

The classification of semilinear singularity preservers for matrices over noncommutative local rings is given. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.     

拟正则保正型过份函数的积分表示及h-结合过程的轨道性质

本文讨论拟正则保正型α-过份函数的积分表示,证明拟正则保正型（ε,D（ε））的任意α-过份函数u均可表示为εα（u,v）=∫vdμ,v∈D（ε）,μ是E上的σ有限测度。并证明（ε,D（ε））的h-结合过程是暂留的和非保守的。     

Preservers of eigenvalue inclusion sets of matrix products

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the union of Cassini ovals, and the Ostrowski’s set. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)Φ(B))=S(AB) for all matrices A and B.     

Preservers of pairs of bivectors with bounded distance

We extend Liu’s fundamental theorem of the geometry of alternate matrices to the second exterior power of an infinite dimensional vector space and also use her theorem to characterize surjective mappings T from the vector space V of all n×n alternate matrices over a field with at least three elements onto itself such that for any pair A, B in V, rank(A-B)?2k if and only if rank(T(A)-T(B))?2k, where k is a fixed positive integer such that n?2k+2 and k?2.     

Simultaneous Representation and Approximation Formulas and High-Order Sobolev Embedding Theorems on Stratified Groups

We give various integral representation formulas simultaneously for a function and its derivatives in terms of vector field gradients of the function of appropriately high order. When the function has compact support, simpler formulas can be derived. Many of the results proved here appear to be new even in the special case of classical Euclidean space. For instance, Theorem 2.2 below reduces to the following result in the usual Euclidean case: Let B be a ball in R^N with radius r(B), let m be a positive integer, and let f\in W^m,1(B). Then there is a polynomial P=_m(B,f) of degree m-1 such that for any integers i, j with 0 j < i m and a.e. x\in B, |\nabla^j(f-P)(x)| C\int_{B}\frac{|\nablaⁱf(y)|}{|x-y|^N-(i-j)} dy +Cr(B)^i-j-N\int_B|\nablaⁱf(y)| dy. Moreover, if 0 < i-j N, then for a.e. x \in B we have the more refined formula |\nabla^j(f-P)(x)| C\int_B\frac{|\nablaⁱf(y)|}{|x-y|^N-i+j} dy.     

Positivity and matrix semigroups

We study various aspects of how certain positivity assumptions on complex matrix semigroups affect their structure. Our main result is that every irreducible group of complex matrices with nonnegative diagonal entries is simultaneously similar to a group of weighted permutations. We also consider the corresponding question for semigroups and discuss the effect of the assumption that a fixed linear functional has nonnegative values when restricted to a given semigroup.     

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.