Restoration of the norm and order in finite-dimensional spaces X and Y from the space L(X,Y) of operators

Novosibirsk Electrical Engineering Institute of Communications. Translated from Matematicheskie Zametki, Vol. 56, No. 5, pp. 108–116, November, 1994.     

The Equation f(X) = f(Y) in Rational Functions X = X(t), Y = Y(t)

We determine all the complex polynomials f(X) such that, for two suitable distinct, nonconstant rational functions g(t) and h(t), the equality f(g(t)) = f(h(t)) holds. This extends former results of Tverberg, and is a contribution to the more general question of determining the polynomials f(X) over a number field K such that f(X) – has at least two distinct K-rational roots for infinitely many K.     

Path connected components of the space of Volterra-type integral operators

We study the topological structure of the space of Volterra-type integral operators on Fock spaces endowed with the operator norm. We prove that the space has the same connected and path connected components which is the set of all those compact integral operators acting on the spaces. We also obtain a characterization of isolated points of the space of the operators and show that there exists no essentially isolated Volterra-type integral operator.     

Hereditary order convexity inL(X,Y)

LetX be a topological vector space,Y an ordered topological vector space andL(X,Y) the space of all linear and continuous mappings fromX intoY. The hereditary order-convex cover [K] ^h of a subsetK ofL(X,Y) is defined by [K] ^h ={A∈L(X,Y):Ax∈[Kx] for allx∈X}, where[Kx] is the order-convex ofKx. In this paper we study the hereditary order-convex cover of a subset ofL(X,Y). We show how this cover can be constructed in specific cases and investigate its structural and topological properties. Our results extend to the spaceL(X,Y) some of the known properties of the convex hull of subsets ofX ^*.     

3-连通(X,Y)-free图的Hamiltonian-连通性

一个图G称为(X,Y)-free图,如果G不含同构于子图X和Y的导出子图.本文证明了X=K_(1,3)、Y∈{D,W,B}的3-连通(X,Y)-free图是Hamiltonian-连通的.     

The Kuratowski convergence and connected components

We investigate the Kuratowski convergence of the connected components of the sections of a definable set applying the result obtained to semialgebraic approximation of subanalytic sets. We are led to some considerations concerning the connectedness of the limit set in general. We discuss also the behaviour of the dimension of converging sections and prove some general facts about the Kuratowski convergence in tame geometry.     

夹心半群T(X,Y,θ)上的最小真同余

本文讨论了夹心半群T(X，Y，θ)上的同余与集合Y上T^θ-等价关系之间的联系，建立了从夹心半群的同余格C(T(X，Y，θ))到Y上的T^θ-等价关系格吃(y)的满同态C和从Teq^θ(Y)到C(T(X，Y，θ))的单同态γ-讨论了Y上最小真T^θ-等价关系以及半群T(X，Y，θ)上最小真同余存在的条件．     

夹心半群S(X,Y,θ)上的α-同余

本文讨论了夹心半群T(X,Y,θ)上的α-同余与集合Y上Tθ-等价关系之间的联系,证明了对于每个Tθ-等价关系E,夹心半群同余格C(T(X,Y,θ))的完全子格γ-1(E)中的最大元是一个α-同余,并判明Symons同余l,d都是α-同余.对于某些拓扑空间X,Y,确定了夹心半群S(X,Y,θ)上的最小(最大)真α-同余.     

关于(X(+)Y)1的凸性

本文给出k-严格凸、(局部)k一致圆定义的几个等价形式.利用等价形式,讨论了(X(+)X)1的k-严格凸及(局部)k一致圆性.     

函数空间Cp(X)与有限乘积空间的弱Lindelof性

给出了空间 Ｘ 与 Ｘ 的有限乘积空间的两种弱 Ｌｉｎｄｅｌｏｆ 对偶性并证明了当 Ｘ是 Ｐ 空间时,对于 每个 ｎ ∈ ω, Ｘｎ 为拟 Ｌｉｎｄｅｌｏｆ空间当且仅当ｔ（ Ｃｐ （ Ｘ）） ＝ ω     

On the number of connected components in the space of M-polynomials in hyperbolic functions

We calculate the number of connected components in the space of the so-called M-polynomials of a given degree in hyperbolic functions. By definition an M-polynomial is characterized by the condition that all its critical values are real and distinct.     

Computation of the homology ofΩ(X∨Y)

Summary We compute the homology of Ω(X∨Y) (the loop space of the wedge of the spaces X and Y), in terms of the homogies of ΩX and ΩY. To do this we use the fact that our problem is equivalent to the computation of the homology of the free product of two topological groups in terms of the homologies of the topological groups. We establish a multiple Kunneth formula with coefficients over a Dedekind domain, which is used to prove a Kunneth like formula involves homologies over a Dedekind domain and generalizes similar results with integral or field coefficients. Over a principal ideal domain the formula for a free product is made more specific. Entrata in Redazione il 31 maggio 1978.