Piotr Bartłomiejczyk (2007)
Annales Polonici Mathematici
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We introduce simple connection matrices. We prove the existence of simple connection matrices for filtered differential vector spaces and Morse decompositions of compact metric spaces.
Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka (1999)
Banach Center Publications
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We present a generalization of topological transition matrices introduced in [6].
Konstantin Mischaikow, Marian Mrozek, Piotr Zgliczyński (1999)
Banach Center Publications
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Piotr Bartłomiejczyk, Zdzisław Dzedzej (1999)
Banach Center Publications
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In [C] and [F1] the connection matrix theory for Morse decomposition is developed in the case of continuous dynamical systems. Our purpose is to study the case of discrete time dynamical systems. The connection matrices are matrices between the homology indices of the sets in the Morse decomposition. They provide information about the structure of the Morse decomposition; in particular, they give an algebraic condition for the existence of connecting orbit set between different Morse...
David Richeson (1999)
Banach Center Publications
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We discuss the ideas of Morse decompositions and index filtrations for isolated invariant sets for both single-valued and multi-valued maps. We introduce the definition of connection matrix pairs and present the theorem of their existence. Connection matrix pair theory for multi-valued maps is used to show that connection matrix pairs obey the continuation property. We conclude by addressing applications to numerical analysis. This paper is primarily an overview of the papers [R1] and...
Rafael Bru, Ljiljana Cvetković, Vladimir Kostić, Francisco Pedroche (2010)
Open Mathematics
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This paper deals with some properties of α1-matrices and α2-matrices which are subclasses of nonsingular H-matrices. In particular, new characterizations of these two subclasses are given, and then used for proving algebraic properties related to subdirect sums and Hadamard products.
Michael Cox, Kevin N. Vander Meulen, Adam Van Tuyl, Joseph Voskamp (2024)
Czechoslovak Mathematical Journal
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The Fiedler matrices are a large class of companion matrices that include the well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class of companion matrices that can be characterized by a Hessenberg form. We demonstrate that the Hessenberg form of the Fiedler companion matrices provides a straight-forward way to compare the condition numbers of these matrices. We also show that there are other companion matrices which can provide a much smaller condition...
Luis Verde-Star (2015)
Special Matrices
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We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its...
Konstantin Mischaikow (1999)
Banach Center Publications
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A brief introduction to the Conley index theory is presented. The emphasis is the fundamental ideas of Conley's approach to dynamical systems and how it avoids some of the difficulties inherent in the study of nonlinear systems.
Brydon Eastman, Kevin N. Vander Meulen (2016)
Special Matrices
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The class of sparse companion matrices was recently characterized in terms of unit Hessenberg matrices. We determine which sparse companion matrices have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix and describe how to find the permutation involved. In the process, we determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to find...