A proof of Schur's conjecture and an improvement.
Ţincu, Ioan (2008)
General Mathematics
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Ţincu, Ioan (2008)
General Mathematics
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Love, E.R. (1998)
Journal of Inequalities and Applications [electronic only]
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Wilkins, J.Ernest jun., Souter, Shantay A. (1995)
Journal of Applied Mathematics and Stochastic Analysis
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Sendov, Bl. (2002)
Serdica Mathematical Journal
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In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by...
Javier Chavarriga, Jaume Giné (1996)
Publicacions Matemàtiques
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In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.
Javier Chavarriga, Jaume Giné (1997)
Publicacions Matemàtiques
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In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.
Peretz, R. (1993)
International Journal of Mathematics and Mathematical Sciences
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Farahmand, K., Li, T. (2010)
International Journal of Stochastic Analysis
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Ivanov, K., Sharma, A. (1996)
Serdica Mathematical Journal
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* Dedicated to the memory of Prof. N. Obreshkoff A Schoenberg conjecture connecting quadratic mean radii of a polynomial and its derivative is verified for some kinds of polynomials, including fourth degree ones.
Ren, Hongshan, Li, Hongyan (2002)
Applied Mathematics E-Notes [electronic only]
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Gregorac, R.J. (1989)
International Journal of Mathematics and Mathematical Sciences
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Javier Chavarriga (1994)
Applicationes Mathematicae
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We study integrability of two-dimensional autonomous systems in the plane with center type linear part. For quadratic and homogeneous cubic systems we give a simple characterization for integrable cases, and we find explicitly all first integrals for these cases. Finally, two large integrable system classes are determined in the most general nonhomogeneous cases.