scholarly journals On Zeros of Self-Reciprocal Random Algebraic Polynomials

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Matrix Theory ◽  
Random Coefficients ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Level Crossings ◽  
Random Algebraic Polynomials ◽  
The Relationship ◽  
Normal Standard ◽  
Independent Identically Distributed

This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN(θ)=∑j=0N−1{αN−jcos(j+1/2)θ+βN−jsin(j+1/2)θ}, where αj and βj, j=0,1,2,…, N−1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials with complex variables and complex random coefficients, with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos(N+θ/2). We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.

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 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.

On the Expected Number of Level Crossings of Random Trigonometric Polynomials

2005 ◽  
Vol 23 (6) ◽  
pp. 1141-1147
Author(s):  
K. Farahmand ◽  
A. Shaposhnikov
Keyword(s):  
Trigonometric Polynomials ◽  
Expected Number ◽  
Level Crossings

scholarly journals Algebraic polynomials with random coefficients

2002 ◽  
Vol 15 (1) ◽  
pp. 83-88
Author(s):  
K. Farahmand
Keyword(s):  
Probability Space ◽  
Random Variables ◽  
Random Coefficients ◽  
Random Polynomial ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Asymptotic Value ◽  
Fixed Probability ◽  
Special Case ◽  
Normally Distributed

This paper provides an asymptotic value for the mathematical expected number of points of inflections of a random polynomial of the form a0(ω)+a1(ω)(n1)1/2x+a2(ω)(n2)1/2x2+…an(ω)(nn)1/2xn when n is large. The coefficients {aj(w)}j=0n, w∈Ω are assumed to be a sequence of independent normally distributed random variables with means zero and variance one, each defined on a fixed probability space (A,Ω,Pr). A special case of dependent coefficients is also studied.

scholarly journals The expected number of level crossings of random trigonometric polynomials

2004 ◽  
Vol 17 (9) ◽  
pp. 1085-1089
Author(s):  
K. Farahmand ◽  
A. Shaposhnikov
Keyword(s):  
Trigonometric Polynomials ◽  
Expected Number ◽  
Level Crossings

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 Real almost zeros of random polynomials with complex coefficients

2005 ◽  
Vol 2005 (2) ◽  
pp. 195-209
Author(s):  
K. Farahmand ◽  
A. Grigorash ◽  
P. Flood
Keyword(s):  
Stochastic Process ◽  
Random Polynomials ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Gaussian Stochastic Process ◽  
Absolute Value ◽  
Mild Conditions ◽  
Random Algebraic Polynomials ◽  
Complex Valued ◽  
Complex Coefficients

We present a simple formula for the expected number of times that a complex-valued Gaussian stochastic process has a zero imaginary part and the absolute value of its real part is bounded by a constant value M. We show that only some mild conditions on the stochastic process are needed for our formula to remain valid. We further apply this formula to a random algebraic polynomial with complex coefficients. We show how the above expected value in the case of random algebraic polynomials varies for different behaviour of M.

scholarly journals Extrema of random algebraic polynomials with non-identically distributed normal coefficients

2001 ◽  
Vol 70 (2) ◽  
pp. 225-234
Author(s):  
K. Farahmand ◽  
A. Grigorash
Keyword(s):  
Asymptotic Estimate ◽  
Random Polynomials ◽  
Asymptotic Estimates ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Mean Values ◽  
Random Algebraic Polynomials ◽  
Analytical Device

AbstractAn asymptotic estimate is derived for the expected number of extrema of a polynomial whose independent normal coefficients possess non-equal non-zero mean values. A result is presented that generalizes in terms of normal processes the analytical device used for construction of similar asymptotic estimates for random polynomials with normal coefficients.

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