Fill an n × n matrix with independent complex Gaussians of variance 1/n. As n → ∞, the eigenvalues {zk} converge to a sum of an H1-noise on the unit disk and an independent H1/2-noise on the unit circle. More precisely, for C1 functions of suitable growth, the distribution of ∑n k=1(f(zk) − Ef(zk)) converges to that of a mean-zero Gaussian with variance given by the sum of the squares of the di...

We consider stochastic equations of the form Xk = φk(Xk+1)Zk, k ∈ N, where Xk and Zk are random variables taking values in a compact group Gk, φk : Gk+1 → Gk is a continuous homomorphism, and the noise (Zk)k∈N is a sequence of independent random variables. We take the sequence of homomorphisms and the sequence of noise distributions as given, and investigate what conditions on these objects res...

Antiangiogenic therapy is a promising new strategy of inhibiting tumor growthand formation of metastases. Recently, a number of compounds with different effects on tumor endothelial cells have entered clinical trials and revealed the need for diagnostic methods to detect their biological activity. Dynamic enhanced magnetic resonance imaging (dyMRI) is used in most clinical trials with antiangio...

Let I=〈f1,…,fm〉⊂Q[x1,…,xn] be a zero dimensional radical ideal defined by polynomials given with exact rational coefficients. Assume that we are approximations {z1,…,zk}⊂Cn for the common roots {ξ1,…,ξk}=V(I)⊆Cn. In this paper show how to construct and certify entries of Hermite matrices I from approximate {z1,…,zk}. When is non-radical, give methods roots. Furthermore, use signatures these cer...

In the present paper, we introduce concepts of Jacobi polynomials and intersection enumerators codes over Fq Zk for arbitrary genus g. We also discuss interrelation among them. Finally, give MacWilliams type identities polynomials.

We prove that for j ≥ 0 one has that inf |zk|=1 max ν=1,...,n2+j ∣

1. Let Zk~ be the manifold of all nondegenerate complex k × n-matrices, k < n , let Gkn be the Grassmannian of k-dimensional subspaces in C~', and let ~r: Zkn -~ Gk, be the na tura l projection (7~(z) for z E Zk~ is the k-dimensional subspace of C ~ generated by the rows of z). For a matr ix z E Zkn denote by px(z) = pil...ik(z) the minor of z composed by the columns with indices from the set I...

Let G be a graph whose each component has order at least 3. Let s : E(G) → Zk for some integer k ≥ 2 be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring c : V (G) → Zk defined by c(v) = ∑ e∈Ev s(e) in Zk, (where the indicated sum is computed in Zk and Ev denotes the set of all edges incident to v) results in a proper vertex col...