How Many Zeros of a Random Polynomial are Real? (1995)
. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coe#cients. We...
How many zeros of a random polynomial are real? (1994)
We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We...
How Many Zeros of a Random Polynomial are Real? (1994)
We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We...
The road from Kac's matrix to Kac's random polynomials (1994)
This paper tells the story of a matrix and a problem both of which are associated with the name Mark Kac, though most likely he never made the connection. This matrix appears as the Clement matrix in...
How Many Eigenvalues of a Random Matrix are Real? (1993)
Alan Edelman, Eric Kostlan, Michael Shub
Let A be an n \Theta n matrix whose elements are independent random variables with standard normal distributions. As n ! 1, the expected number of real eigenvalues is asymptotic to p 2n=ß. We obtain...
Matrix cubing is a gradient dynamical system (1993)
Consider the map from the real symmetric matrices of Frobenius norm one to themselves given by matrix cubing followed by rescaling. We show that this is a gradient dynamical system with potential...