scholarly journals MASALAH EIGEN DAN EIGENMODE MATRIKS ATAS ALJABAR MIN-PLUS

2021 ◽  
Vol 15 (4) ◽  
pp. 659-666
Author(s):  
Eka Widia Rahayu ◽  
Siswanto Siswanto ◽  
Santoso Budi Wiyono
Keyword(s):  
Connected Graph ◽  
Irreducible Matrix ◽  
Eigenvalues And Eigenvectors ◽  
Communication Graph ◽  
Strongly Connected ◽  
Square Matrix

Eigen problems and eigenmode are important components related to square matrices. In max-plus algebra, a square matrix can be represented in the form of a graph called a communication graph. The communication graph can be strongly connected graph and a not strongly connected graph. The representation matrix of a strongly connected graph is called an irreducible matrix, while the representation matrix of a graph that is not strongly connected is called a reduced matrix. The purpose of this research is set the steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and also eigenmode of the regular reduced matrix over min-plus algebra. Min-plus algebra has an ispmorphic structure with max-plus algebra. Therefore, eigen problems and eigenmode matrices over min-plus algebra can be determined based on the theory of eigenvalues, eigenvectors and eigenmode matrices over max-plus algebra. The results of this research obtained steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and eigenmode algorithm of the regular reduced matrix over min-plus algebra

scholarly journals Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

2020 ◽  
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.

scholarly journals ALGORITMA POLINOMIAL MINIMUM UNTUK MEMBENTUK MATRIKS DIAGONAL DARI MATRIKS PERSEGI

2017 ◽  
Vol 6 (2) ◽  
pp. 282
Author(s):  
Himmatul Mursyidah
Keyword(s):  
Polynomial Algorithm ◽  
Matrix Theory ◽  
Equation System ◽  
Diagonal Matrix ◽  
Eigenvalues And Eigenvectors ◽  
State Classification ◽  
Linear Equation System ◽  
Special Matrix ◽  
Minimum Polynomial ◽  
Square Matrix

In mathematics, matrices have many uses, they are finding solutions of a linear equation system, looking for specific solutions of differential equations, determining state classification on Markov chains, and so on. There is a special matrix in matrix theory, that is a diagonal matrix. The diagonal matrix is a matrix whose all non-diagonal entries are primarily zero so that the product of the diagonal matrix can be computed by considering only the components along the main diagonal. A square matrix can sometimes be formed into a diagonal matrix. If a non-diagonal square matrix A can be conjugated with a diagonal matrix, then there is an invertible matrix P so PAP-1=D, where D is a diagonal matrix and P is said to diagonalize A. To find a square matrix diagonalizable or not, many researchers usually use eigenvalues and eigenvectors evaluation. In this study, we discuss that the other way to form a diagonal matrix by using Minimum Polynomial Algorithm.

LOCAL GROUPS IN FREE GROUPOIDS SATISFYING CERTAIN MONOID IDENTITIES

2002 ◽  
Vol 12 (01n02) ◽  
pp. 357-369
Author(s):  
ALAIR PEREIRA DO LAGO
Keyword(s):  
Free Group ◽  
Connected Graph ◽  
Generating Set ◽  
Burnside Groups ◽  
Cyclomatic Number ◽  
Strongly Connected

Let G be a (possibly infinite) strongly connected graph and let [Formula: see text] be a set of monoid identities such that any monoid satisfying [Formula: see text] is also a group. Let ℬ be the free groupoid on G satisfying [Formula: see text]. Then, the local groups ℬv, for v ∈ V (G), are all isomorphic to a free group satisfying [Formula: see text]. Furthermore, it is free over a generating set which can be effectively characterized and whose cardinality is the cyclomatic number of the graph G. We also show applications that establish an important connection between free Burnside groups and free Burnside semigroups.

Routing Flow Through a Strongly Connected Graph

Algorithmica ◽  
2002 ◽  
Vol 32 (3) ◽  
pp. 467-473 ◽  
Author(s):  
Erlebach ◽  
Hagerup
Keyword(s):  
Connected Graph ◽  
Strongly Connected ◽  
Flow Through

scholarly journals Discrete Random Walks on One-Sided ``Periodic'' Graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michael Drmota
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Generating Functions ◽  
Connected Graph ◽  
Infinite Graphs ◽  
Reflected Brownian Motion ◽  
Strongly Connected ◽  
International Audience ◽  
Null Recurrence

International audience In this paper we consider discrete random walks on infinite graphs that are generated by copying and shifting one finite (strongly connected) graph into one direction and connecting successive copies always in the same way. With help of generating functions it is shown that there are only three types for the asymptotic behaviour of the random walk. It either converges to the stationary distribution or it can be approximated in terms of a reflected Brownian motion or by a Brownian motion. In terms of Markov chains these cases correspond to positive recurrence, to null recurrence, and to non recurrence.

scholarly journals On Differentiating Eigenvalues and Eigenvectors

1985 ◽  
Vol 1 (2) ◽  
pp. 179-191 ◽  
Author(s):  
Jan R. Magnus
Keyword(s):  
Eigenvalues And Eigenvectors ◽  
First Derivative ◽  
Differentiable Functions ◽  
Second Derivatives ◽  
Derivatives Of ◽  
Square Matrix

Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λo of X0, so that the triple (X0, u0, λ0) satisfies the equations Xu = λu, . We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ(X0) = λO, u(X0) = u0, Xu = λu, and . We obtain the first and second derivatives of λ(X) and the first derivative of u(X). Two alternative expressions for the first derivative of λ(X) are also presented.

scholarly journals Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

Algorithms ◽  
2020 ◽  
Vol 13 (12) ◽  
pp. 321
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

In a previous paper we defined the black and white SAT problem which has exactly two solutions, where each variable is either true or false. We showed that black and white 2-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonboglár model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a black and white SAT problem. We prove a powerful theorem, the so called transitions theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as black and white SAT problems. We show that the Balatonboglár model is between the strong and the weak model, and it generates 3-SAT problems, so the Balatonboglár model represents strongly connected communication graphs as black and white 3-SAT problems. Our motivation to study these models is the following: The strong model generates a 2-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonboglár model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonboglár model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: the SAT problem and directed graphs.

scholarly journals Multi-Eulerian Tours of Directed Graphs

10.37236/5588 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Matthew Farrell ◽  
Lionel Levine
Keyword(s):  
Connected Graph ◽  
Directed Graphs ◽  
Minimal Length ◽  
Closed Path ◽  
Directed Edge ◽  
Simple Generalization ◽  
Strongly Connected ◽  
Eulerian Tours ◽  
Eulerian Tour

Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least once, and uses edges e and f the same number of times whenever tail(e)=tail(f). This definition leads to a simple generalization of the BEST Theorem. We then show that the minimal length of a multi-Eulerian tour is bounded in terms of the Pham index, a measure of 'Eulerianness'.

The length of a circuit in a strongly connected graph

Cybernetics ◽  
1972 ◽  
Vol 5 (1-2) ◽  
pp. 95-98
Author(s):  
V. G. Vizing ◽  
M. K. Gol'dberg
Keyword(s):  
Connected Graph ◽  
Strongly Connected
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