Level crossings and turning points of random hyperbolic polynomials

Farahmand, K.; Hannigan, P.

doi:https://doi.org/10.1155/S0161171299225793

International Journal of Mathematics and Mathematical Sciences

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Volume 22 |Article ID 957474 | https://doi.org/10.1155/S0161171299225793

K. Farahmand, P. Hannigan, "Level crossings and turning points of random hyperbolic polynomials", International Journal of Mathematics and Mathematical Sciences, vol. 22, Article ID 957474, 8 pages, 1999. https://doi.org/10.1155/S0161171299225793

Level crossings and turning points of random hyperbolic polynomials

K. Farahmand¹ and P. Hannigan¹

¹Department of Mathematics, University of Ulster, Jordastown, Co. Antrim, UK

Received01 Oct 1997

Abstract

In this paper, we show that the asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial a1sinhx+a2sinh2x+⋯+ansinhnx, where aj(j=1,2,…,n) are independent normally distributed random variables with mean zero and variance one, is (1/π)logn. This result is true for all K independent of x, provided K≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomial a1coshx+a2cosh2x+⋯+ancoshnx, with aj(j=1,2,…,n) as before, is also (1/π)logn.

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Copyright © 1999 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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