Fiedler–Pták scaling in max algebra

Sergeev, Sergĕı

doi:10.1016/j.laa.2011.12.039

Cited by 5 publications

(5 citation statements)

References 29 publications

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“…For a short survey on the use of visualization scaling in max algebra see [21]. Let us conclude with the following observation concerning the visualization of max-algebraic powers.…”

Section: Definition 227 (Visualization)mentioning

confidence: 99%

Generalizations of bounds on the index of convergence to weighted digraphs

Merlet

Nowak

Schneider

et al. 2014

53rd IEEE Conference on Decision and Control

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We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply to the weights of optimal walks when one of their ends is a critical node.

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“…For a short survey on the use of visualization scaling in max algebra see [21]. Let us conclude with the following observation concerning the visualization of max-algebraic powers.…”

Section: Definition 227 (Visualization)mentioning

confidence: 99%

Generalizations of bounds on the index of convergence to weighted digraphs

Merlet

Nowak

Schneider

et al. 2014

53rd IEEE Conference on Decision and Control

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“…. (21) t−1 341, which is −301. For this observe that the walk 621 has an even length and therefore we need to use one of the three-cycles to make it odd, and using the southern three-cycle in the end of the walk is the most profitable way to do so.…”

Section: More General Casementioning

confidence: 99%

“…. (21) t−1 5625, where we have to use the northern triangle in the end of the walk to create a walk of even walk and minimise the loss.…”

Section: Counterexamplesmentioning

confidence: 99%

Extending CSR decomposition to tropical inhomogeneous matrix products

Kennedy-Cochran-Patrick

Sergeev

2023

ELA

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This article presents an attempt to extend the CSR decomposition, previously introduced for tropical matrix powers, to tropical inhomogeneous matrix products. The CSR terms for inhomogeneous matrix products are introduced, and then, a case is described where an inhomogeneous product admits such CSR decomposition after some length and a bound on this length is given. In the last part of the paper, a number of counterexamples are presented to show that inhomogeneous products do not admit CSR decomposition under more general conditions.

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“…The final assumption below is inspired by the visualisation scaling studied in Sergeev et al [23], see also [21] and references therein for more background on this scaling. Definition 2.10 (Visualisation).…”

Section: Main Assumptionsmentioning

confidence: 99%

Extending CSR Decomposition to Tropical Inhomogeneous Matrix Products

Kennedy-Cochran-Patrick¹,

Sergeev²

2020

Preprint

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This article presents an attempt to extend the CSR decomposition, previously introduced for tropical matrix powers, to tropical inhomogeneous matrix products. The CSR terms for inhomogeneous matrix products are introduced, then a case is described where an inhomogenous product admits such CSR decomposition after some length and give a bound on this length. In the last part of the paper a number of counterexamples are presented to show that inhomogenous products do not admit CSR decomposition under more general conditions.

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Fiedler–Pták scaling in max algebra

Abstract: This is essentially the text of my talk on Fiedler-Pták scaling in max algebra delivered in the invited minisymposium in honor of Miroslav Fiedler at the 17th ILAS Conference in Braunschweig, Germany.

Cited by 5 publications

References 29 publications

Generalizations of bounds on the index of convergence to weighted digraphs

Generalizations of bounds on the index of convergence to weighted digraphs

Extending CSR decomposition to tropical inhomogeneous matrix products

Extending CSR Decomposition to Tropical Inhomogeneous Matrix Products

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