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1

Computational Classification of Tubular Algebras

Dowbor Piotr, Kim Yan

Fundamenta Informaticae

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2020

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Vol. 177, nr 1

39--67

EN

The effective method (based on Theorem 5.3) of classifying tubular algebras by the Cartan matrices of tilting sheaves over weighted projective lines with all indecomposable direct summands in some finite “fundamental domain” , by the reduction to the two elementary problems of discrete mathematics having algorithmic solutions is presented in details (see Problem A and B). The software package CART_TUB being an implementation of this method yields the precise classification of all up to isomorphism tubular algebras of a fixed tubular type p, by creating the complete lists of their Cartan matrices, and furnish their tilting realizations. In particular, the number of isomorphism classes of tubular algebras of the type p is determined (Theorem 2.3).

2

The least eigenvalue of the graphs whose complements are connected and have pendent paths

Wang C., Yu G., Sun W., Cao J.

Journal of Artificial Intelligence and Soft Computing Research

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2018

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Vol. 8, No. 4

303--308

EN

The adjacency matrix of a graph is a matrix which represents adjacent relation between the vertices of the graph. Its minimum eigenvalue is defined as the least eigenvalue of the graph. Let Gn be the set of the graphs of order n, whose complements are connected and have pendent paths. This paper investigates the least eigenvalue of the graphs and characterizes the unique graph which has the minimum least eigenvalue in Gn.

3

Applications of Bipartite Graphs and their Adjacency Matrices to Covering-based Rough Sets

Wang J., Zhu W.

Fundamenta Informaticae

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2017

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Vol. 156, nr 2

237--254

EN

In this paper, bipartite graphs and their adjacency matrices are applied to equivalently represent covering-based rough sets through three sides, which are approximation operators, properties and reducible elements. Firstly, a bipartite graph is constructed through a covering. According to the constructed bipartite graph, two equivalent representations of a pair of covering upper and lower approximation operators are presented. And an algorithm is designed for computing the pair of covering approximation operators from the viewpoint of these equivalent representations. Some properties and reducible elements of covering-based rough sets are also investigated through the constructed bipartite graph. Finally, an adjacency matrix of the constructed bipartite graph is proposed, and reducible elements in the covering are obtained through the proposed adjacency matrix. Moreover, an equivalent representation of the covering upper approximation operator is presented through the proposed adjacency matrix. In a word, these results show two interesting views, which are graphs and matrices, to investigate covering-based rough sets.

4

The notation of mechanism structure in structural research

Romaniak K.

Journal Biuletyn of Polish Society for Geometry and Engineering Graphics

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2013

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Vol. 25

3--10

EN

Different methods of notation are in use when describing the construction of mechanisms in structural research. This work deals with the issue of notation and presents: – constructional drawing in which the construction of the mechanism is detailed by the use of the Monge’s projection (views, projections), axonometric or perspective projections; – kinematic diagram in which kinematic pairs and links are described by symbols. Given dimensions of individual links allow to specify the positioning of specific parts of the mechanism depending on the movement of input links; – structural diagram in which the class of each kinematic pair is specified; – structural graph in a form of a polygon in which its sides and vertices correspond with kinematic pairs and links respectively; – adjacency matrix in a form of a square matrix in which the number of rows and columns corresponds with the number of links in the mechanism. Its elements are: 0 when links are not joined, or 1 when links are joined; – loop notation in which links and kinematic pairs that form individual closed contours of the mechanism are given; – author’s notation of the construction of spatial mechanisms, in which the classes of kinematic pairs are presented in a form of labels next to their graphic representation.

PL

Praca poświęcona jest różnym sposobom zapisu mechanizmów, używanych w badaniach strukturalnych. Przedstawiono: – zapis konstrukcyjny, w którym używając rzuty prostokątne (widoki, przekroje) oraz aksonometrię przedstawia się szczegółowo budowę mechanizmu;

5

Sprzętowa implementacja zachłannego algorytmu kolorowania grafu

Wyrwoł B.

Elektronika : konstrukcje, technologie, zastosowania

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2010

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Vol. 51, nr 9

21-23

PL

Kolorowanie grafów znajduje zastosowanie wszędzie tam, gdzie konieczny jest podział zbioru na rozłączne podzbiory wg określonego kryterium jakie spełniają lub nie elementy zbioru. Większość algorytmów kolorowania realizowana jest zwykle na drodze programowej. W sytuacji, kiedy dużą rolę odgrywają uwarunkowania czasowe, konieczna jest realizacja sprzętowa z wykorzystaniem dedykowanego układu. W artykule przedstawiony został zachłanny algorytm kolorowania wierzchołków grafu oraz jego sprzętowa implementacja w układzie programowalnym FPGA. Dodatkowo omówiona została metoda reprezentacji danych opisujących strukturę grafu i przykład wykorzystania sprzętowego modułu kolorowania grafu, wspomagającego proces dekompozycji lingwistycznej, w systemie wnioskowania przybliżonego.

EN

Graph coloring algorithms are used wherever it is necessary to divide set on disjoint subsets according to specified criteria or not they meet the elements of the set. Most of the coloring algorithms are usually implemented as a computer or microcontroller program. To reduce computing time of the coloring result it is necessary to implement hardware using a dedicated chip. The paper presents graph greedy vertex algorithm and its hardware implementation in an FPGA chip. It describes also a graph data structure and finally implementation of the graph coloring module in the fuzzy hierarchical inference system. It is used in linguistic decomposition process of the knowledge base in the stage of the partitioning the rule base.

6

Efficient Implementation of the Italiano Algorithms for Updating the Transitive Closure on Associative Parallel Processors

Nepomniaschay A.

Fundamenta Informaticae

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2008

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Vol. 89, nr 2-3

313-329

EN

We propose a simple data structure for an efficient implementation of the Italiano algorithms for the dynamic updating of the transitive closure of a directed graph represented as adjacency matrix on a model of associative (or content addressable) parallel processors with vertical processing (the STAR–machine). Associative versions of the Italiano algorithms are represented as procedures DeleteArc1 and InsertArc1. We prove the correctness of these procedures and evaluate their time and space complexity. We also present the main advantages of associative versions of the Italiano algorithms.