Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients
Joint Authors
Mahanti, Mina Ketan
Singh, Amandeep
Sahoo, Lokanath
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2015, Issue 2015 (31 Dec. 2015), pp.1-7, 7 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2015-08-09
Country of Publication
Egypt
No. of Pages
7
Main Subjects
Abstract EN
We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the form y = P n t = n 1 a 1 cosh t + n 2 a 2 cosh 2 t + ⋯ + n n a n cosh n t , where a 1 , … , a n is a sequence of standard Gaussian random variables, is n / 2 + o p ( 1 ) .
It is shown that the asymptotic value of expected number of times the polynomial crosses the level y = K is also n / 2 as long as K does not exceed 2 n e μ ( n ) , where μ ( n ) = o ( n ) .
The number of oscillations of P n ( t ) about y = K will be less than n / 2 asymptotically only if K = 2 n e μ ( n ) , where μ ( n ) = O ( n ) or n - 1 μ ( n ) → ∞ .
In the former case the number of oscillations continues to be a fraction of n and decreases with the increase in value of μ ( n ) .
In the latter case, the number of oscillations reduces to o p ( n ) and almost no trace of the curve is expected to be present above the level y = K if μ ( n ) / ( n log n ) → ∞ .
American Psychological Association (APA)
Mahanti, Mina Ketan& Singh, Amandeep& Sahoo, Lokanath. 2015. Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients. International Journal of Mathematics and Mathematical Sciences،Vol. 2015, no. 2015, pp.1-7.
https://search.emarefa.net/detail/BIM-1066191
Modern Language Association (MLA)
Mahanti, Mina Ketan…[et al.]. Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients. International Journal of Mathematics and Mathematical Sciences No. 2015 (2015), pp.1-7.
https://search.emarefa.net/detail/BIM-1066191
American Medical Association (AMA)
Mahanti, Mina Ketan& Singh, Amandeep& Sahoo, Lokanath. Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients. International Journal of Mathematics and Mathematical Sciences. 2015. Vol. 2015, no. 2015, pp.1-7.
https://search.emarefa.net/detail/BIM-1066191
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1066191