On stable polynomials

Miloslav Nekvinda

Aplikace matematiky (1989)

  • Volume: 34, Issue: 3, page 177-196
  • ISSN: 0862-7940

Abstract

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The article is a survey on problem of the theorem of Hurwitz. The starting point of explanations is Schur's decomposition theorem for polynomials. It is showed how to obtain the well-known criteria on the distribution of roots of polynomials. The theorem on uniqueness of constants in Schur's decomposition seems to be new.

How to cite

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Nekvinda, Miloslav. "On stable polynomials." Aplikace matematiky 34.3 (1989): 177-196. <http://eudml.org/doc/15575>.

@article{Nekvinda1989,
abstract = {The article is a survey on problem of the theorem of Hurwitz. The starting point of explanations is Schur's decomposition theorem for polynomials. It is showed how to obtain the well-known criteria on the distribution of roots of polynomials. The theorem on uniqueness of constants in Schur's decomposition seems to be new.},
author = {Nekvinda, Miloslav},
journal = {Aplikace matematiky},
keywords = {Hurwitz polynomials; expository paper; Hurwitz-Routh criterion; stable polynomial; Hermite; decompositon of Schur; Hurwitz polynomials; expository paper; Hurwitz-Routh criterion},
language = {eng},
number = {3},
pages = {177-196},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On stable polynomials},
url = {http://eudml.org/doc/15575},
volume = {34},
year = {1989},
}

TY - JOUR
AU - Nekvinda, Miloslav
TI - On stable polynomials
JO - Aplikace matematiky
PY - 1989
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 34
IS - 3
SP - 177
EP - 196
AB - The article is a survey on problem of the theorem of Hurwitz. The starting point of explanations is Schur's decomposition theorem for polynomials. It is showed how to obtain the well-known criteria on the distribution of roots of polynomials. The theorem on uniqueness of constants in Schur's decomposition seems to be new.
LA - eng
KW - Hurwitz polynomials; expository paper; Hurwitz-Routh criterion; stable polynomial; Hermite; decompositon of Schur; Hurwitz polynomials; expository paper; Hurwitz-Routh criterion
UR - http://eudml.org/doc/15575
ER -

References

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  1. Ch. Hermite, Sur le nombre des racines d'une équation algébrique comprises entre des limites données, Crelles J. 52, 39 (1856). 
  2. J. Routh, A treatise on the stability of a given state of motion, London 1877. Zbl17.0315.02
  3. A. Hurwitz, 10.1007/BF01446812, Math. Ann. 46, 273 (1895). Zbl0962.01500MR1510884DOI10.1007/BF01446812
  4. J. Schur, 10.1002/zamm.19210010405, Z. angew. Math. Mech, 1, 307 (1921). (1921) Zbl48.0082.03DOI10.1002/zamm.19210010405
  5. L. S. Pontryagin, On the zeros of some elementary transcendental functions, (Russian) Izv. Ak. Nauk SSSR, Ser. Mat. 6 (1942), 115-134. English Translation: Amer. Math. Soc. Transl. (2) 1 (1955), 95-110. (1942) Zbl0068.05803MR0073686
  6. H. Cremer F. H. Effertz, 10.1007/BF01360969, Math. Ann. 137 (1959), 328-350. (1959) Zbl0092.01604MR0104684DOI10.1007/BF01360969
  7. R. Bellman, Introduction to Matrix Analysis, Mc Graw-Нill Book Company, New York 1960. (1960) Zbl0124.01001MR0122820
  8. F. R. Gantmacher, Theory of Matrices, (Russian) Izd. Nauka, Moskva 1966. (1966) Zbl0136.00410MR0202725
  9. B. P. Demidowich, Lectures on the Mathematical Theory of Stability, (Russian) Izd. Nauka, Moskva 1967. (1967) MR0226126
  10. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York 1977. (1977) Zbl0352.34001MR0508721

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