On k-spaces of real matrices

It is shown that if W is a linear subspace of real n × n matrices, such that rank (A) = k for all 0 ≠ A ∈ W, then dim W≤ n. If dim W = n.5≤ n is prime, and 2 is primitive modulo n then k =1.     

关于kr-空间与k-空间

本文给出了具有局部可数闭ｋ－网的ｋｒ－空间成为ｋ－空间的一个充要条件,此结果给林寿提出的问题作了一个回答。     

On k-spaces of real matrices

It is shown that if W is a linear subspace of real n × n matrices, such that rank (A) = k for all 0 ≠ A ∈ W, then dim W≤ n. If dim W = n.5≤ n is prime, and 2 is primitive modulo n then k =1.     

Expansions of k-spaces

Simple expansions and expansions by point finite and locally finite collections are studied for particular classes of k-spaces. All such expansions of Fréchet spaces are shown to be Fréchet, and sufficient conditions for the preservation of property P ? {k¹, sequential, k} under simple and locally finite expansions are established.     

Analytic k-spaces

We study sequential convergence in spaces with analytic topologies avoiding thus a number of standard pathologies. For example, we identify bisequentiality of an analytic space as the Fréchet property of its square. We show that a countable Fréchet group is metrizable if and only if its topology is analytic. We also investigate the diagonal sequence properties and show their productiveness in the class of analytic spaces.     

Resolution property in generalized k-spaces

We show that a T₁ space X is resolvable if the set of limit points λ (X) of various simultaneously separated subsets of X is dense in X. Moreover, if λ (X) is open also, then X is ω-resolvable. It follows that a self-dense, Hausdroff space satisfying a generalized k-space (sequential space) condition is resolvable (respectively, ω-resolvable).     

Bitopological and topological ordered k-spaces

Domain theory, in theoretical computer science, needs to be able to handle function spaces easily. It also requires asymmetric spaces, and these are necessarily not T₁. At the same time, techniques used with the higher separation axioms are useful there (see [Topology Appl. 199 (2002) 241]). In order to handle all these requirements, we develop a theory of k-bispaces using bitopological spaces, which results in a Cartesian closed category. The other well-known way to combine asymmetry and separation is ordered topological spaces [Nachbin, Topology and Order, Van Nostrand, 1965]; we define the category of ordered k-spaces, which is isomorphic to that found among bitopological spaces.     

On the sunflower bound for k-spaces,pairwise intersecting in a point

Designs, Codes and Cryptography - A t-intersecting constant dimension subspace code C is a set of k-dimensional subspaces in a projective space $$mathrm {PG}(n,q)$$ , where distinct subspaces...     

A problem of Westwick on k-spaces

Sets n×n matrices linear span contains only matrices of rank n?1 and 0 are investigated. To within a natural equivalence they are characterised for n ? 6. Partial results are obtained for general n.     

On box products

We prove two theorems about box products. The first theorem says that the box product of countable spaces is pseudonormal, i.e. any two disjoint closed sets one of which is countable can be separated by open sets. The second theorem says that assuming CH a certain uncountable box product is normal (i.e. <_ω1?□_α<ω1X_α where each Xα is a compact metric space).     

On products of experiments

Summary An experiment in the sense of Blackwell is a finite parameterized set of probability distributions on a sample space. The product of a parameterized set of experiments is the experiment which describes the process in which independent elements from all of the sample spaces in the set are observed. In the paper, methods are developed for computing the properties of a product of experiments from the properties of the components. In particular, asymptotic properties (as N ) of the experiment describing N independent observations from a single sample space are investigated.Research partially supported by National Science Foundation Grant (GP-5227).     

On products of integers

The main objective of this paper is to investigate the relation between the number of integers in a given subset $\text{A}$ of the integers 1, 2,…, n and the number of integers that can be chosen from 1, 2,…, n so that their pairwise products all appear in $\text{A}$. Other related problems are also considered.     

On products of matroids

We show, by means of counterexamples, that products with rank rk(M)rk(N) of a matroid M by a matroid N do not exist in general, and that there is no free-est product of M by N. We prove that a canonical product of M by N (having rank rk(M)+rk(N)?1) is a free-est product in a certain (weaker) sense.     

On products of unbounded operators

We first investigate the lattice structure of the partially ordered group of all derivations on a lattice ordered ring (l-ring) R. Then, using the inner derivations on R, we prove some results regarding the commutativity of R. Finally, we study homo and anti-homo derivations on quasi semi-prime, semi-prime, reduced, and l-prime rings and l-rings; and, among other things, we generalize some of the results of Bell and Kappe [1] to l-rings. This revised version was published online in June 2006 with corrections to the Cover Date.     

On pseudoinverses of operator products

The analytical structure of the Moore-Penrose pseudoinverse of the product ab of any two operators over finite-dimensional unitary spaces is studied. The existence of the unique representation of the form (ab)⁺=b⁺(h+g)a⁺ is proved. Here h:= (a⁺abb⁺)⁺ is an (oblique) projector and g is an operator with a number of special properties. In particular, h+g is a projector, g is orthogonal to h in some metric, and g³=0. A necessary and sufficient condition for the case (ab)⁺=b⁺ha⁺ is established. This case contains the classical one (ab)⁺=b⁺a⁺ (the reverse-order law). For the latter a new necessary and sufficient condition is given.     

On deformations of crossed products

Let A∗Γ be a crossed product algebra, where A is semisimple, finitely generated over its center and Γ is a finite group. We give a necessary and sufficient condition in terms of the outer action of Γ on A for the existence of a multi-parametric semisimple deformation of the form A((t₁,…,t_n))∗Γ (with the induced outer action). The main tool in the proof is the solution of the so-called twisting problem. We also give an example which shows that the condition is not sufficient if one drops the condition on the finite generation of A over its center.