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.     

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.     

On products of k-spaces

As is well known, k-spaces are characterized as quotient spaces of locally compact spaces. For a certain k-space X, we give characterizations on X for X×Y to be a k-space for every space Y in some class of k-spaces. Also, for certation k-spaces X and Y, we give characterizations on X and Y for X×Y to be a k-space.     

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。     

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 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.     

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).     

k-spaces and Borel filters on the set of integers

We say that a countable, Hausdorff, topological space with one and only one accumulation point is a point-space. For such a space, we give several properties which are equivalent to the property of being a k-space. We study some free filters on the set of integers and we determine if the associated point-spaces are k-spaces or not. We show that the filters of Lutzer-van Mill-Pol are k-filters. We deduce that, for each countable ordinal , there exists a free filter of true additive class (Baire's classification) and a free filter of true multiplicative class for which the associated point-spaces are k-spaces but not , the existence being true in the additive case for . In particular, we answer negatively a question raised in J. Calbrix, C. R. Acad. Sci. Paris 305 (1987), 109--111.

     

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 on bands

In this paper, we consider an open problem proposed by Petrich and Reilly: What are necessary and sufficient conditions on a completely regular semigroup S in order that the trace relation T on the lattice of congruences on S is equal to the identity relation? By constructing some special congruences on S, we prove that T=ε if and only if S is a band.     

A problem on triples

If a system H of triples (3-uniform hypergraph) on n vertices has the following property: for every 3-coloring of the vertex-set there exists a 3-colored triple, what is the minimum size (S(n)) of H? The first values of S(n) are computed and the asymptotic behaviour of this function is studied.     

A problem on brachistochrone as invariant variational problem

Based on the modified theory of invariant variational problems developed by the author, we describe a theoretical group approach to the problem of brachistochrone.     

A problem on the conventional well-posedness of a multipoint problem

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 4, pp. 40–43, July–August, 1989.     

A note on Fermat's problem

The General Fermat Problem asks for the minimum of the weighted sum of distances fromm points inn-space. Dozens of papers have been written on variants of this problem and most of them have merely reproduced known results. This note calls attention to the work of Weiszfeld in 1937, who may have been the first to propose an iterative algorithm. Although the same algorithm has been rediscovered at least three times, there seems to be no completely correct treatment of its properties in the literature. Such a treatment, including a proof of convergence, is the sole object of this note. Other aspects of the problem are given scant attention.This paper was written while the author was Science Faculty Fellow of the National Science Foundation at the London School of Economics.     

A problem on convex functions

A conjecture is made on convex functions. It leads to the problem of characterizing a class of convex functions, which is of interest both from the theoretical point of view and in the field of minimization methods.     

A problem on random trees

The problem is considered of determining the distribution of the number of nodes in any given component of a forest obtained by removing edges from a random tree.     

A remark on Nehari's problem

An example of a strickly contractive Hankel matrix is given such that the central contractive extensionf _c of satisfies f _c =1. This way we answer a problem raised by Ciprian Foias.Partially supported by a Georgia State University Research Grant.     

A note on Chern-Yamabe problem

We propose a flow to study the Chern-Yamabe problem and discuss the long time existence of the flow. In the balanced case we show that the Chern-Yamabe problem is the Euler-Lagrange equation of some functional. The monotonicity of the functional along the flow is derived. We also show that the functional is not bounded from below.