For valued fields K of rank higher than 1, we describe how elements in the henselization K of K can be approximated from within K; our result is a handy generalization of the well-known fact that in rank 1, all of these elements lie in the completion of K. We apply the result to show that if an element z algebraic over K can be approximated from within K in the same way as an element in K, then...

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over L, where L is a finite extension of a field K. This is a generalization of the case where K = Fq and L = Fqm of Gabidulin codes to arbitrary characteristic. We show equivalence to previous definitions, in particular the ones by K...

Low-rank tensor approximations are plagued by a well-known problem — a tensor may fail to have a best rank-r approximation. Over R, it is known that such failures can occur with positive probability, sometimes with certainty: in R2×2×2, every tensor of rank 3 fails to have a best rank-2 approximation. We will show that while such failures still occur over C, they happen with zero probability. I...

For a square matrix A, the smallest nonnegative integer k such that rank (A(k)) =rank (A(k+1)) is called the Drazin index of A. In this paper, we give some results on the Drazin indices of sum and product of square matrices.

Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C *-algebra, C * (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C * (Λ). The K-groups of C * (Λ) for k > 2 can be calculated under certain circumstances. We state that for all k, the torsion-free rank of K 0 (C * (Λ)) and K 1 (C * (Λ)) are equal when C * (Λ) is u...

For a minor-closed class M of matroids, we let h(k) denote the maximum number of elements of a simple rank-k matroid in M. We prove that, if M does not contain all simple rank-2 matroids, then h(k) grows either linearly, quadratically, or exponentially.

We define rank-width of graphs to investigate clique-width. Rank-width is a complexity measure of decomposing a graph in a kind of tree-structure, called a rankdecomposition. We show that graphs have bounded rank-width if and only if they have bounded clique-width. It is unknown how to recognize graphs of clique-width at most k for fixed k > 3 in polynomial time. However, we find an algorithm r...

Let $\mathds{k}$ be a real quadratic number field. Denote by $\mathrm{Cl}_2(\mathds{k})$ its $2$-class group and $\mathds{k}_2^{(1)}$ (resp. $\mathds{k}_2^{(2)}$) first second) Hilbert The aim of this paper is to study, for field whose discriminant divisible one prime congruent $3$ modulo 4, the metacyclicity $G=\mathrm{Gal}(\mathds{k}_2^{(2)}/\mathds{k})$ cyclicity $\mathrm{Gal}(\mathds{k}_2^{...

Let $mathcal {A} $ and $mathcal {B} $ be C$^*$-algebras. Assume that $mathcal {A}$ is of real rank zero and unital with unit $I$ and $k>0$ is a real number. It is shown that if $Phi:mathcal{A} tomathcal{B}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $Phi(|A|^k)=|Phi(A)|^k $ for all normal elements $Ainmathcal A$, $Phi(I)$ is a projection, and there exists a posit...