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 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 On random orthogonal polynomials

2001 ◽  
Vol 14 (3) ◽  
pp. 265-274 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Orthogonal Polynomials ◽  
Legendre Polynomials ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Dependent Random Variables ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

Let T0∗(x),T1∗(x),…,Tn∗(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (−1,1). The asymptotic estimate of the expected number of real zeros of the random polynomial g0T0∗(x)+g1T1∗(x)+…+gnTn∗(x) where gj, j=1,2,…,n are independent identically and normally distributed random variables is known. In this paper, we first present the asymptotic value for the above expected number when coefficients are dependent random variables. Further, for the case of independent coefficients, we define the expected number of zero up-crossings with slope greater than u or zero down-crossings with slope less than −u. Promoted by the graphical interpretation, we define these crossings as u-sharp. For the above polynomial, we provide the expected number of such crossings.

scholarly journals Real zeros of a random polynomial with Legendre elements

1997 ◽  
Vol 10 (3) ◽  
pp. 257-264
Author(s):  
K. Farahmand
Keyword(s):  
Legendre Polynomials ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Normally Distributed

Let T0∗(x),T1∗(x),…,Tn∗(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (−1,1). The asymptotic estimate of the expected number of real zeros of the random polynomial g0T0∗(x)+g1T1∗(x)+…+gnTn∗(x) where gj, j=1,2,…,n are independent identically and normally distributed random variables with mean zero and variance one is known. The present paper considers the case when the means and variances of the coefficients are not all necessarily equal. It is shown that in general this expected number of real zeros is only dependent on variances and is independent of the means.

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 On real zeros of random polynomials with hyperbolic elements

1998 ◽  
Vol 21 (2) ◽  
pp. 347-350
Author(s):  
K. Farahmand ◽  
M. Jahangiri
Keyword(s):  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Polynomials ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Normally Distributed

This paper provides the asymptotic estimate for the expected number of real zeros of a random hyperbolic polynomialg1coshx+2g2cosh2x+…+ngncoshnxwheregj,(j=1,2,…,n)are independent normally distributed random variables with mean zero and variance one. It is shown that for sufficiently largenthis asymptotic value is(1/π)logn.

scholarly journals On the variance of the number of real zeros of a random trigonometric polynomial

1997 ◽  
Vol 10 (1) ◽  
pp. 57-66 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Trigonometric Polynomial ◽  
General Formula ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Normal Process ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
Upper Estimate

The asymptotic estimate of the expected number of real zeros of the polynomial T(θ)=g1cosθ+g2cos2θ+…+gncosnθ where gj(j=1,2,…,n) is a sequence of independent normally distributed random variables is known. The present paper provides an upper estimate for the variance of such a number. To achieve this result we first present a general formula for the covariance of the number of real zeros of any normal process, ξ(t), occurring in any two disjoint intervals. A formula for the variance of the number of real zeros of ξ(t) follows from this result.

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 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 Level crossings and turning points of random hyperbolic polynomials

1999 ◽  
Vol 22 (3) ◽  
pp. 579-586
Author(s):  
K. Farahmand ◽  
P. Hannigan
Keyword(s):  
Asymptotic Estimate ◽  
Turning Points ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Level Crossings ◽  
Hyperbolic Polynomials ◽  
Normally Distributed

In this paper, we show that the asymptotic estimate for the expected number ofK-level crossings of a random hyperbolic polynomiala1sinhx+a2sinh2x+⋯+ansinhnx, whereaj(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 allKindependent ofx, providedK≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomiala1coshx+a2cosh2x+⋯+ancoshnx, withaj(j=1,2,…,n)as before, is also(1/π)logn.

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