On faces and W-irreducible operators in finite-dimensional linear spaces

This paper clarifies the theoretical relations between the concept of W-irreducibility (Schröder [11]) and those statements in Vandergraft [14] which rely on the notion of faces and the noninvariance properties of irreducible operators. Simple applications involve operators of the form M=I?A with positive A and matrices “of positive type”.     

On some combinatorial questions in finite-dimensional spaces

Bounds are given for supinf∥Σ^p_i=1ν_i∥, where sup is taken over all set systems V₁,…, V_p of Rⁿ with 0∈∩^p_i=1convV_i and sup_v∈Vi∥ν∥?1 for i=1,…, p, and inf is taken over all possible choices ν_i∈V_i for i=1,…, p. Another similar problem is considered. The bounds are sharp.     

On a generalization of perfectly normal spaces

The notion of quasi-perfect normality which generalizes the notion of usual perfect normality is introduced and the behavior of the dimension functions in this class of spaces is investigated.     

On a new generalization of metric spaces

In this paper, we introduce the \({\mathcal {F}}\)-metric space concept, which generalizes the metric space notion. We define a natural topology \(\tau _{{\mathcal {F}}}\) in such spaces and we study their topological properties. Moreover, we establish a new version of the Banach contraction principle in the setting of \({\mathcal {F}}\)-metric spaces. Several examples are presented to illustrate our study.     

Relaxed linear spaces and a generalization of the Cayley-Dickson process

In this paper first we introduce a new generalization of vector spaces and linear nonassociative algebras, and then we apply these new concepts to produce new structures related to the classical real division algebras but with dimensions other than 1, 2, 4 and 8.     

Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces

Let BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite-dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection such that P(BY)⊂A. The main results of the paper: (1) Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. (2) If a finite-dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.     

On multi-orthogonal bases in finite-dimensional non-Archimedean normed spaces

The paper deals with the problem of the existence multi-orthogonal bases in finite-dimensional normed spaces over K, where K is a non-Archimedean complete valued field.     

Parametrization of the varieties of symmetrizers for operators in finite-dimensional vector spaces

We determine the space that parametrizes the cone of symmetrizers of a semisimple operator with real spectrum in a finite-dimensional vector space. It is proved that the set that determines the cone of symmetrizers is parametrized by the portion of the unit sphere of the vector space lying the first octant.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 1–3.     

On posets and independence spaces

By constructing a correspondence relationship between independence spaces and posets, under isomorphism, this paper characterizes loopless independence spaces and applies this characterization to reformulate certain results on independence spaces in poset frameworks. These state that the idea provided in this paper is a new approach for the study of independence spaces. We outline our future work finally.     

On the stability of the first-order linear recurrence in topological vector spaces

Suppose that X is a sequentially complete Hausdorff locally convex space over a scalar field K, V is a bounded subset of X, (a_n)_n≥0 is a sequence in K?{0} with the property lim inf_n→∞|a_n|>1, and (b_n)_n≥0 is a sequence in X. We show that for every sequence (x_n)_n≥0 in X satisfying

     

Characterization of best approximations in normed linear spaces of matrices by elements of finite-dimensional linear subspaces

A general characterization theorem of best approximations in normed linear spaces is specialized to the linear space of real n × n matrices endowed with the spectral norm.     

Extremal problems in finite-dimensional spaces

A theory of the first variation in the finite-dimensional case is presented using the concepts of essential covering.