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In this paper, we establish some new conditions for the existence and the representations for the $$(P, Q)$$ generalized reflexive and anti-reflexive solutions to matrix equation $$AX = B$$ with respect to the generalized reflection matrix dual $$(P, Q)$$ . Moreover, in corresponding solution sets of the equation, the explicit expressions of the nearest matrix to a given matrix in the Frobenius norm have been provided. [ABSTRACT FROM AUTHOR]

In this paper we prove a quantitative version of the classical isoperimetric inequality in the hyperbolic space $$\mathbb {H}^n$$ . The constant only depends on the dimension and an upper bound for the volume of the set. [ABSTRACT FROM AUTHOR]

In this paper, we present a new approach to proximity structures based on the recognition of many of the entities important in the theory of ideals. So, we give a characterization of the basic proximity using ideals. Also, we introduce the concept of g-proximities and we show that for different choice of "g" one can obtain many of the known types of generalized proximities. Also, characterizations of some types of these proximities - (g0,h0) - are obtained. [ABSTRACT FROM AUTHOR]

In some previous papers we studiedgeneralizednearness by supernear spaces in connection withunificationsand topologicalextensions, respectively. Now, this last part of series deals with those supernear spaces, whichcoverclosely LODATO spaces, b-supertopologies and hence EF-proximities as well. Moreover, we will show that some special kind of these spaces have alocaltopological extension iff they aresuperbunchspaces. Consequently, thisfundamentalresult leading us to a furthergeneralizationof LODATO's theorem, and in addition Doîtchinov's achievement can be dealt with, but in a modified fashion. [ABSTRACT FROM AUTHOR]

In this paper, we prove a related fixed point theorem for single-valued mappings in two Menger spaces. [ABSTRACT FROM AUTHOR]

Completeness for metric spaces is traditionally presented in terms of convergence of Cauchy sequences, and for uniform spaces in terms of Cauchy filters. Somewhat more abstractly, a uniform space is complete if and only if it is closed in every uniform space in which it is embedded, and so isomorphic to any space in which it is densely embedded. This is the approach to completeness used in the point-free setting, that is, for uniform and nearness frames: a nearness frame is said to be complete if every strict surjection onto it is an isomorphism. Quasi-uniformities and quasi-nearnesses on biframes provide appropriate structures with which to investigate uniform and nearness ideas in the asymmetric context. In [9] a notion of completeness (called “quasi-completeness”) was presented for quasi-nearness biframes in terms of suitable strict surjections being isomorphisms, and a quasi-completion was constructed for any quasi-nearness biframe. In this paper we show that quasi-completeness can indeed be viewed in terms of the convergence of certain filters, namely, the regular Cauchy bifilters. We use the notion of aT-valued bifilter, which generalizes the characteristic function of a filter. An important tool is an appropriate composition for such bifilters. We show that the right adjoint of the quasi-completion is the universal regular Cauchy bifilter and use it to prove this characterization of quasi-completeness. We also construct the so-called Cauchy filter quotient for a biframe using a quotient of the downset biframe that involves only the Cauchy, and not the regularity, condition. Like the quasi-completion, this provides a universal Cauchy bifilter; unlike the quasi-completion, this construction is functorial. [ABSTRACT FROM AUTHOR]

Abstract: We extend Bishopʼs concept of function spaces to the concept of pre-function spaces. We show that there is an adjunction between the category of neighbourhood spaces and the category of Φ-closed pre-function spaces. We also show that there is an adjunction between the category of uniform spaces and the category of Ψ-closed pre-function spaces. [Copyright &y& Elsevier]

Abstract: We give a predicative construction of a completion of a uniform space in the constructive Zermelo–Fraenkel set theory. [Copyright &y& Elsevier]

In this paper we show that the problem of discrete duality can be extended beyond the clasical setting of duality between a class of algebras and a class of relational structures. Namely, for some classes of algebras, the relevant dual structures are the structures with multirelations. Several applications of multirelations will be described. [ABSTRACT FROM AUTHOR]

We say that $X=[x_{ij}]_{i,j=1}^n$ is symmetric centrosymmetric if x ij = x ji and x n − j + 1, n − i + 1, 1 ≤ i, j ≤ n. In this paper we present an efficient algorithm for minimizing || AXA T − B|| where ||·|| is the Frobenius norm, A ∈ ℝ m× n, B ∈ ℝ m× m and X ∈ ℝ n× n is symmetric centrosymmetric with a specified central submatrix [ x ij] p ≤ i, j ≤ n − p. Our algorithm produces a suitable X such that AXA T = B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim. [ABSTRACT FROM AUTHOR]