Local maxima of a random algebraic polynomial
We present a useful formula for the expected number of maxima of a normal process ξ(t) that occur below a level u. In the derivation we assume chiefly that ξ(t),ξ′(t), and ξ′′(t) have, with probability one, continuous 1 dimensional distributions and expected values of zero. The formula referred to a...
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| Format: | Article |
| Language: | English |
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Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S016117120100391X |
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| author | K. Farahmand P. Hannigan |
| author_facet | K. Farahmand P. Hannigan |
| author_sort | K. Farahmand |
| collection | DOAJ |
| description | We present a useful formula for the expected number of maxima of a
normal process ξ(t) that occur below a level u. In the
derivation we assume chiefly that ξ(t),ξ′(t), and ξ′′(t) have, with probability one, continuous 1 dimensional
distributions and expected values of zero. The formula referred to
above is then used to find the expected number of maxima below the
level u for the random algebraic polynomial. This result
highlights the very pronounced difference in the behaviour of the
random algebraic polynomial on the interval (−1,1) compared with
the intervals (−∞,−1) and (1,∞). It is also shown
that the number of maxima below the zero level is no longer O(logn) on the intervals (−∞,−1) and (1,∞). |
| format | Article |
| id | doaj-art-54b64b9391c5463bbd93948c2bfaf505 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-54b64b9391c5463bbd93948c2bfaf5052025-02-03T06:05:18ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252001-01-0125533134310.1155/S016117120100391XLocal maxima of a random algebraic polynomialK. Farahmand0P. Hannigan1Department of Mathematics, University of Ulster, Jordanstown, Co. Antrim, BT37 0QB, UKDepartment of Mathematics, University of Ulster, Jordanstown, Co. Antrim, BT37 0QB, UKWe present a useful formula for the expected number of maxima of a normal process ξ(t) that occur below a level u. In the derivation we assume chiefly that ξ(t),ξ′(t), and ξ′′(t) have, with probability one, continuous 1 dimensional distributions and expected values of zero. The formula referred to above is then used to find the expected number of maxima below the level u for the random algebraic polynomial. This result highlights the very pronounced difference in the behaviour of the random algebraic polynomial on the interval (−1,1) compared with the intervals (−∞,−1) and (1,∞). It is also shown that the number of maxima below the zero level is no longer O(logn) on the intervals (−∞,−1) and (1,∞).http://dx.doi.org/10.1155/S016117120100391X |
| spellingShingle | K. Farahmand P. Hannigan Local maxima of a random algebraic polynomial International Journal of Mathematics and Mathematical Sciences |
| title | Local maxima of a random algebraic polynomial |
| title_full | Local maxima of a random algebraic polynomial |
| title_fullStr | Local maxima of a random algebraic polynomial |
| title_full_unstemmed | Local maxima of a random algebraic polynomial |
| title_short | Local maxima of a random algebraic polynomial |
| title_sort | local maxima of a random algebraic polynomial |
| url | http://dx.doi.org/10.1155/S016117120100391X |
| work_keys_str_mv | AT kfarahmand localmaximaofarandomalgebraicpolynomial AT phannigan localmaximaofarandomalgebraicpolynomial |