scholarly journals Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

2009 ◽  
Vol 2009 ◽  
pp. 1-6 ◽  
Author(s):  
K. Farahmand ◽  
M. Sambandham
Keyword(s):  
Algebraic Polynomial ◽  
Random Coefficients ◽  
Random Coefficient ◽  
Geometric Progression ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Geometric Progressions

The expected number of real zeros of an algebraic polynomial with random coefficient is known. The distribution of the coefficients is often assumed to be identical albeit allowed to have different classes of distributions. For the nonidentical case, there has been much interest where the variance of the th coefficient is . It is shown that this class of polynomials has significantly more zeros than the classical algebraic polynomials with identical coefficients. However, in the case of nonidentically distributed coefficients it is analytically necessary to assume that the means of coefficients are zero. In this work we study a case when the moments of the coefficients have both binomial and geometric progression elements. That is we assume and . We show how the above expected number of real zeros is dependent on values of and in various cases.

scholarly journals Real zeros of classes of random algebraic polynomials

2003 ◽  
Vol 16 (3) ◽  
pp. 249-255 ◽  
Author(s):  
K. Farahmand ◽  
M. Sambandham
Keyword(s):  
Asymptotic Formula ◽  
Algebraic Polynomial ◽  
Random Coefficients ◽  
Asymptotic Estimates ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Algebraic Polynomials

There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0+a1x+a2x2+⋯+an−1xn−1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients {aj}j=0n−1 it is shown that the above expected number is asymptotic to O(logn). This order for the expected number of zeros remains valid for the case when the coefficients are grouped into two, each group with a different variance. However, it was recently shown that if the coefficients are non-identically distributed such that the variance of the jth term is (nj) the expected number of zeros of the polynomial increases to O(n). The present paper provides the value for this asymptotic formula for the polynomials with the latter variances when they are grouped into three with different patterns for their variances.

scholarly journals REAL ZEROS OF ALGEBRAIC POLYNOMIALS WITH STABLE RANDOM COEFFICIENTS

2008 ◽  
Vol 85 (1) ◽  
pp. 81-86 ◽  
Author(s):  
K. FARAHMAND
Keyword(s):  
Stable Distribution ◽  
Random Coefficients ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Symmetric Stable Distribution ◽  
Number Of Real Zeros ◽  
Significant Interest ◽  
Special Case

AbstractWe consider a random algebraic polynomial of the form Pn,θ,α(t)=θ0ξ0+θ1ξ1t+⋯+θnξntn, where ξk, k=0,1,2,…,n have identical symmetric stable distribution with index α, 0<α≤2. First, for a general form of θk,α≡θk we derive the expected number of real zeros of Pn,θ,α(t). We then show that our results can be used for special choices of θk. In particular, we obtain the above expected number of zeros when $\theta _k={n\choose k}^{1/2}$. The latter generate a polynomial with binomial elements which has recently been of significant interest and has previously been studied only for Gaussian distributed coefficients. We see the effect of α on increasing the expected number of zeros compared with the special case of Gaussian coefficients.

scholarly journals Zeros of an algebraic polynomial with nonequal means random coefficients

2004 ◽  
Vol 2004 (63) ◽  
pp. 3389-3395
Author(s):  
K. Farahmand ◽  
P. Flood
Keyword(s):  
Algebraic Polynomial ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Coefficients ◽  
Random Polynomial ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
The Mean

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomiala0+a1x+a2x2+⋯+an−1xn−1. The coefficientsaj(j=0,1,2,…,n−1)are assumed to be independent normal random variables with nonidentical means. Previous results are mainly for identically distributed coefficients. Our result remains valid when the means of the coefficients are divided into many groups of equal sizes. We show that the behaviour of the random polynomial is dictated by the mean of the first group of the coefficients in the interval(−1,1)and the mean of the last group in(−∞,−1)∪(1,∞).

scholarly journals On Different Classes of Algebraic Polynomials with Random Coefficients

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
K. Farahmand ◽  
A. Grigorash ◽  
B. McGuinness
Keyword(s):  
Asymptotic Relation ◽  
Random Variables ◽  
Random Coefficients ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Level Crossings ◽  
Gaussian Random Variables ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Definition Of

The expected number of real zeros of the polynomial of the form , where is a sequence of standard Gaussian random variables, is known. For large it is shown that this expected number in is asymptotic to . In this paper, we show that this asymptotic value increases significantly to when we consider a polynomial in the form instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.

scholarly journals Real zeros of random algebraic polynomials with binomial elements

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
A. Nezakati ◽  
K. Farahmand
Keyword(s):  
Algebraic Polynomial ◽  
Asymptotic Estimate ◽  
General Theorem ◽  
Random Variables ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Algebraic Polynomials ◽  
Special Case

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a0+a1x+a2x2+…+an−1xn−1. The coefficients aj(j=0,1,2,…,n−1) are assumed to be independent normal random variables with mean zero. For integers m and k=O(log⁡n)2 the variances of the coefficients are assumed to have nonidentical value var⁡(aj)=(k−1j−ik), where n=k⋅m and i=0,1,2,…,m−1. Previous results are mainly for identically distributed coefficients or when var⁡(aj)=(nj). We show that the latter is a special case of our general theorem.

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

scholarly journals Covariance of the number of real zeros of a random trigonometric polynomial

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
K. Farahmand ◽  
M. Sambandham
Keyword(s):  
Trigonometric Polynomial ◽  
Random Coefficients ◽  
Expected Number ◽  
Asymptotic Value ◽  
Advanced Method ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial

For random coefficientsajandbjwe consider a random trigonometric polynomial defined asTn(θ)=∑j=0n{ajcos⁡jθ+bjsin⁡jθ}. The expected number of real zeros ofTn(θ)in the interval(0,2π)can be easily obtained. In this note we show that this number is in factn/3. However the variance of the above number is not known. This note presents a method which leads to the asymptotic value for the covariance of the number of real zeros of the above polynomial in intervals(0,π)and(π,2π). It can be seen that our method in fact remains valid to obtain the result for any two disjoint intervals. The applicability of our method to the classical random trigonometric polynomial, defined asPn(θ)=∑j=0naj(ω)cos⁡jθ, is also discussed.Tn(θ)has the advantage onPn(θ)of being stationary, with respect toθ, for which, therefore, a more advanced method developed could be used to yield the results.

Expected Number Of Real Zeros Of a Random Algebraic Polynomial

2018 ◽  
Vol 5 (1) ◽  
pp. 936-942
Author(s):  
Dipty Rani Dhal ◽  
DR.Prasana Kumar Mishra
Keyword(s):  
Algebraic Polynomial ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros

Expected Number of Real Zeros of Lévy and Harmonizable Stable Random Polynomials

2000 ◽  
Vol 7 (2) ◽  
pp. 379-386 ◽  
Author(s):  
S. Rezakhah ◽  
A. R. Soltani
Keyword(s):  
General Formula ◽  
Algebraic Polynomial ◽  
Stable Processes ◽  
Random Polynomials ◽  
Expected Number ◽  
Explicit Formulas ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Stable Random Vector ◽  
Stable Noise

Abstract Assuming (A 0, A 1, . . . , An ) is a jointly symmetric α-stable random vector, 0 < α ≤ 2, a general formula for the expected number of real zeros of the random algebraic polynomial was obtained by Rezakhah in 1997. Rezakhah's formula is applied to the case where (A 0, . . . , An ) is formed by consecutive observations from the Lévy stable noise or from certain harmonizable stable processes, and more explicit formulas are derived for the expected number of real zeros of .

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