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Content available remote Minimal Base for Finite Topological Space by Matrix Method
EN
Topological base plays a foundational role in topology theory. However, few works have been done to find the minimal base, which would make us difficult to interpret the internal structure of topological spaces. To address this issue, we provide a method to convert the finite topological space into Boolean matrix and some properties of minimal base are investigated. According to the properties, an algorithm(URMB) is proposed. Subsequently, the relationship between topological space and its sub-space with respect to the base is concentrated on by Boolean matrix. Then, a fast algorithm(MMB) is presented, which can avoid a mass of redundant computations. Finally, some numerical experiments are given to show the advantage and the effectiveness of MMB compared with URMB.
EN
The paper presents two options of the parallel algorithm for finding the shortest covering of a large Boolean matrix, where the decomposition of the initial matrix into matrices of smaller sizes is based on the partitioning of rows. The parallel algorithm COVMB contains sequential algorithms for partitioning of initial m × n Boolean matrix on submatrices, building special matrices, summering of the shortest coverings of special Boolean matrices with smaller sizes as well as the sequential algorithm SECNOP for finding the shortest coverings of smaller Boolean matrices.
3
Content available remote Boolean Matrices and their Applications to Covering Reductions
EN
This paper proposes two different covering reduction algorithms by means of Boolean matrices. We define a dual notion of product on Boolean matrices and establish the relationship between characteristic matrix of a covering and relational matrices of two covering-induced relations, i.e. the minimum and the maximum relations in a covering. This paper shows that relational matrices of minimum and maximum relations in a covering can be written as intersection and union of many “elementary” matrices and each “elementary” matrix is corresponding to one element in the covering, respectively. Finally, as an application of this result, we propose two types of covering reduction algorithms.
EN
The paper presents results of investigations of the efficiency of a parallel program for the solution of the NP-complete problem of constructing of the shortest covering of a Boolean matrix. It was explored dependencies of the time of constructing of the shortest covering from the number processing processors of a cluster and from the dimension of a Boolean matrix.
PL
Przedstawiono sposób wyznaczenia testu diagnostycznego metodą macierzy boolowskich. Zaprezentowano przykład wykorzystania testu diagnostycznego stworzonego metodą macierzy boolowskiej. Przedstawiono sposób tworzenia grafów wiązań i równań stanu, które mogą być narzędziem oceny stanu technicznego urządzeń.
EN
This paper presents basic information on the structure of diagnostic tests used for the assessment of the technical state of parts and groups of technical systems. The Boolean matrix method is also demonstrated. The construction of bond graphs and state equations, which can be tools for the assessment of technical condition of machines has been presented.
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