On Algebraic Polynomials with Random Coefficients

Operational matrices with respect to Hermite polynomials and their applications in solving linear differential equations with variable coefficients

In this paper, a new and efficient approach is applied for numerical approximation of the linear differential equations with variable coeffcients based on operational matrices with respect to Hermite polynomials. Explicit formulae which express the Hermite expansion coeffcients for the moments of derivatives of any differentiable function in terms of the original expansion coefficients of the f...

Bernoulli collocation method with residual correction for solving integral-algebraic equations

The principal aim of this paper is to serve the numerical solution of an integral-algebraic equation (IAE) by using the Bernoulli polynomials and the residual correction method. After implementation of our scheme, the main problem would be transformed into a system of algebraic equations such that its solutions are the unknown Bernoulli coefficients. This method gives an analytic solution when ...

On the average number of sharp crossings of certain Gaussian random polynomials

Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

The expected number of real zeros of an algebraic polynomial ao a1x a2x · · · anx with random coefficient aj , j 0, 1, 2, . . . , n 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 jth coefficient is var aj ( n j ) . It is sh...

Research Article On Zeros of Self-Reciprocal Random Algebraic Polynomials

This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN (θ)= ∑N−1 j=0 {αN− j cos( j +1/2)θ + βN− j sin( 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 w...

Real Almost Zeros of Random Polynomials with 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 c...