scholarly journals On the average number of real zeros of a class of random algebraic equations

1990 ◽  
Vol 49 (1) ◽  
pp. 149-160 ◽  
Author(s):  
N. N. Nayak ◽  
S. Bagh
Keyword(s):  
Mathematical Expectation ◽  
Random Variables ◽  
Constant Independent ◽  
Algebraic Equations ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Mutually Independent ◽  
Normally Distributed

AbstractLet g1, g2, …, gn be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity. In this work, we obtain the average number of real zeros of the random algebraic equations Σnk=1 Kσ gk(ω)tk = C, where C is a constant independent of t and not necessarily zero. This average is (1/π) log n, when n is large and σ is non-negative.

The average number of real zeros of a random trigonometric polynomial

1968 ◽  
Vol 64 (3) ◽  
pp. 721-730 ◽  
Author(s):  
Minaketan Das
Keyword(s):  
Trigonometric Polynomial ◽  
Mathematical Expectation ◽  
Random Variables ◽  
Trigonometric Polynomials ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
Image Position ◽  
Mutually Independent

AbstractLet a1, a2,… be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity; let b1, b2,… be a set of positive constants. In this work, we obtain the average number of zeros in the interval (0, 2π) of trigonometric polynomials of the formfor large n. The case when bk = kσ (σ > − 3/2;) is studied in detail. Here the required average is (2σ + 1/2σ + 3)½.2n + o(n) for σ ≥ − ½ and of order n3/2; + σ in the remaining cases.

scholarly journals Mean number of real zeros of a random trigonometric polynomial IV

1997 ◽  
Vol 10 (1) ◽  
pp. 67-70 ◽  
Author(s):  
J. Ernest Wilkins
Keyword(s):  
Positive Integer ◽  
Trigonometric Polynomial ◽  
Random Variables ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
The Mean ◽  
Normally Distributed

If aj(j=1,2,…,n) are independent, normally distributed random variables with mean 0 and variance 1, if p is one half of any odd positive integer except one, and if vnp is the mean number of zeros on (0,2π) of the trigonometric polynomial a1cosx+2pa2cos2x+…+npancosnx, then vnp=μp{(2n+1)+D1p+(2n+1)−1D2p+(2n+1)−2D3p}+O{(2n+1)−3}, in which μp={(2p+1)/(2p+3)}½, and D1p, D2p and D3p are explicitly stated constants.

scholarly journals Mean number of real zeros of a random trigonometric polynomial. III

1995 ◽  
Vol 8 (3) ◽  
pp. 299-317
Author(s):  
J. Ernest Wilkins ◽  
Shantay A. Souter
Keyword(s):  
Trigonometric Polynomial ◽  
Approximate Formula ◽  
Error Term ◽  
Random Variables ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
The Mean ◽  
Normally Distributed

If a1,a2,…,an are independent, normally distributed random variables with mean 0 and variance 1, and if vn is the mean number of zeros on the interval (0,2π) of the trigonometric polynomial a1cosx+2½a2cos2x+…+n½ancosnx, then vn=2−½{(2n+1)+D1+(2n+1)−1D2+(2n+1)−2D3}+O{(2n+1)−3}, in which D1=−0.378124, D2=−12, D3=0.5523. After tabulation of 5D values of vn when n=1(1)40, we find that the approximate formula for vn, obtained from the above result when the error term is neglected, produces 5D values that are in error by at most 10−5 when n≥8, and by only about 0.1% when n=2.

scholarly journals Mean number of real zeros of a random hyperbolic polynomial

2000 ◽  
Vol 23 (5) ◽  
pp. 335-342 ◽  
Author(s):  
J. Ernest Wilkins
Keyword(s):  
Random Variables ◽  
Hyperbolic Polynomial ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
The Mean ◽  
Normally Distributed

Consider the random hyperbolic polynomial,f(x)=1pa1coshx+⋯+np×ancoshnx, in whichnandpare integers such thatn≥2,   p≥0, and the coefficientsak(k=1,2,…,n)are independent, standard normally distributed random variables. Ifνnpis the mean number of real zeros off(x), then we prove thatνnp=π−1 logn+O{(logn)1/2}.

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.

The average number of maxima of a random algebraic curve

1969 ◽  
Vol 65 (3) ◽  
pp. 741-753 ◽  
Author(s):  
Minaketan Das
Keyword(s):  
Mathematical Expectation ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Random Variables ◽  
Image Position ◽  
Mutually Independent ◽  
Normally Distributed

AbstractLet g0, gl, g2,…be a sequence of mutually independent, normally distributed random variables with mathematical expectation zero and variance unity. In this work, we obtain the average number of maxima (minima) of the random algebraic curves with the equationsThis average is (½(3½ + 1)) log N + O((log N)⅔ (log log N)½), when N is large.

scholarly journals Random Trigonometric Polynomials with Nonidentically Distributed Coefficients

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
K. Farahmand ◽  
T. Li
Keyword(s):  
Closed Form ◽  
Random Variables ◽  
Expected Value ◽  
Trigonometric Polynomials ◽  
Random Polynomials ◽  
Asymptotic Estimates ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Normally Distributed

This paper provides asymptotic estimates for the expected number of real zeros of two different forms of random trigonometric polynomials, where the coefficients of polynomials are normally distributed random variables with different means and variances. For the polynomials in the form of and we give a closed form for the above expected value. With some mild assumptions on the coefficients we allow the means and variances of the coefficients to differ from each others. A case of reciprocal random polynomials for both above cases is studied.

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 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.

Sign in / Sign up

Export Citation Format

Share Document