Level crossings and turning points of random hyperbolic polynomials
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...
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| Format: | Article |
| Language: | English |
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Wiley
1999-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171299225793 |
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| _version_ | 1832562961358520320 |
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| author | K. Farahmand P. Hannigan |
| author_facet | K. Farahmand P. Hannigan |
| author_sort | K. Farahmand |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-8f6cd8acb8ad4dcdad87cf7585264d95 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1999-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-8f6cd8acb8ad4dcdad87cf7585264d952025-02-03T01:21:20ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251999-01-0122357958610.1155/S0161171299225793Level crossings and turning points of random hyperbolic polynomialsK. Farahmand0P. Hannigan1Department of Mathematics, University of Ulster, Jordastown, Co. Antrim, BT37 0QB, UKDepartment of Mathematics, University of Ulster, Jordastown, Co. Antrim, BT37 0QB, UKIn 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.http://dx.doi.org/10.1155/S0161171299225793Gaussian processnumber of real rootsKac-Rice formulanormal densitycovariance matrix. |
| spellingShingle | K. Farahmand P. Hannigan Level crossings and turning points of random hyperbolic polynomials International Journal of Mathematics and Mathematical Sciences Gaussian process number of real roots Kac-Rice formula normal density covariance matrix. |
| title | Level crossings and turning points of random hyperbolic polynomials |
| title_full | Level crossings and turning points of random hyperbolic polynomials |
| title_fullStr | Level crossings and turning points of random hyperbolic polynomials |
| title_full_unstemmed | Level crossings and turning points of random hyperbolic polynomials |
| title_short | Level crossings and turning points of random hyperbolic polynomials |
| title_sort | level crossings and turning points of random hyperbolic polynomials |
| topic | Gaussian process number of real roots Kac-Rice formula normal density covariance matrix. |
| url | http://dx.doi.org/10.1155/S0161171299225793 |
| work_keys_str_mv | AT kfarahmand levelcrossingsandturningpointsofrandomhyperbolicpolynomials AT phannigan levelcrossingsandturningpointsofrandomhyperbolicpolynomials |