Expected Characteristic Polynomials

Domain of attraction of normal law and zeros of random polynomials

Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...

Buckling and vibration analysis of angle -ply symmetric laminated composite plates with fully elastic boundaries

The main focus of this paper is on efficiency analysis of two kinds of approximating functions (characteristic orthogonal polynomials and characteristic beam functions) that have been applied in the Rayleigh-Ritz method to determine the non-dimensional buckling and frequency parameters of an angle ply symmetric laminated composite plate with fully elastic boundaries. It has been observed that o...

The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix

A t × n random matrix A can be formed by sampling n independent random column vectors, each containing t components. The random Gram matrix of size n, Gn = A A, contains the dot products between all pairs of column vectors in the randomly generated matrix A, and has characteristic roots coinciding with the singular values of A. Furthermore, the sequences det (Gi) and perm(Gi) (for i = 0, 1, . ....

A pr 2 01 5 Finite free convolutions of polynomials ∗

We study three convolutions of polynomials that are inspired by free probability. We define these to be the expected characteristic polynomials of certain random matrices. The symmetric additive and multiplicative convolutions have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, appears new. We prove that these convolution...

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

2 9 Ju n 20 06 Average volume , curvatures , and Euler characteristic of random real algebraic varieties

We determine the expected curvature polynomial of random real projec-tive varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective varieties is found. This considerably extends previously known results on t...