Inverse spread limit of a nonnegative matrix.

Authors

  • Atif Abueida University of Dayton.
  • Mark Nielsen University of Dayton.
  • Tin-Yau Tam Auburn University.

DOI:

https://doi.org/10.4067/S0716-09172010000200004

Keywords:

Nonnegative matrices, inverse spread, evolutionary biology, DNA, matrices no negativas, dispersión inversa, biología evolucionaria, ADN.

Abstract

For a given nonnegative n × n matrix A consider the following quantity img02.JPGas long as the denominator is positive. It is simply the ratio between the smallest and the largest entries of Am. We call s(Am) the inverse spread of Am which is interpreted as a measure of the maximum variation among the entries of Am in the multiplicative and reciprocal sense. Smaller s(Am) means a larger variation for Am. Clearly 0 = s(Am) = 1 for all m = 1, 2, . . . We study the asymptotic behavior of s(Am), that is, the behavior of s(Am) as m → ∞. The study arises from evolutionary biology.

Author Biographies

Atif Abueida, University of Dayton.

Department of Mathematics.

Mark Nielsen, University of Dayton.

Department of Biology.

Tin-Yau Tam, Auburn University.

Department of Mathematics and Statistics.

References

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9, SIAM, Philadelphia, (1994).

F. R. Gantmacher, The Theory of Matrices, Volume I, Chelsea Publishing Company, New York, (1959).

F. R. Gantmacher, The Theory of Matrices, Volume II, Chelsea Publishing Company, New York, (1959).

J. Z. Hearon, Compartmental matrices with single root and nonnegative nilpotent matrices, Math. Biosci., 14, pp. 135—142; (1972).

R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, (1985)

D.T. Jones, W.R. Taylor, and J.M. Thornton, The rapid generation of mutation data matrices from protein sequences, Comput. Appl. Biosci. 8, pp. 275-282, (1992).

L.H. Rieseberg, Chromosomal rearrangements and speciation, Trends in Ecology and Evolution, 16, pp. 351-358, (2001).

J.S. Taylor and J. Raes, Duplication and Divergence, The Evolution of New Genes and Old Ideas, Annual Review of Genetics, 38, pp. 615-643, (2004).

J. C. Tiernan, An Efficient Algorithm to Find the Elementry Circuit of a Graph, Comm. ACM, 13 No. 12, pp. 722-726, (1970).

Z. Yang, R. Nielsen, Hasegawa M., Models of Amino Acid Substitution and Applications to Mitochondrial Protein Evolution. Mol. Biol. Evol. 15(12), pp. 1600-1611, (1998).

Z. Yang, R. Nielsen, Estimating synonymous and nonsynonymous substitution rates under realistic evolutionary models. Mol Biol Evol. 17(1), pp. 32-43, (2000).

Published

2011-01-07

How to Cite

[1]
A. Abueida, M. Nielsen, and T.-Y. Tam, “Inverse spread limit of a nonnegative matrix.”, Proyecciones (Antofagasta, On line), vol. 29, no. 2, pp. 109-122, Jan. 2011.

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