Showing papers by "Albrecht Böttcher published in 2020"
01 Dec 2020
TL;DR: In this article, the authors give sufficient conditions on F to ensure that for every integer b there exists an integer vector such that F(\varvec{a}) = b, and the vector can be found in a finite number of steps.
Abstract: Let
$$F(\varvec{x})$$
be a homogeneous polynomial in
$$n \ge 1$$
variables of degree
$$1 \le d \le n$$
with integer coefficients so that its degree in every variable is equal to 1. We give some sufficient conditions on F to ensure that for every integer b there exists an integer vector
$$\varvec{a}$$
such that
$$F(\varvec{a}) = b$$
. The conditions provided also guarantee that the vector
$$\varvec{a}$$
can be found in a finite number of steps.
2 citations
TL;DR: In this paper, the eigenvalues and singular values of operators composed by a positive definite linear operator on a finite-dimensional Hilbert space are connected to those composed by the same operator in the same way as in this paper.
Abstract: The paper is devoted to results connecting the eigenvalues and singular values of operators composed by $P^\ast G P$ with those composed in the same way by $QG^{−1}Q^\ast$. Here $P +Q = I$ are skew complementary projections on a finite-dimensional Hilbert space and $G$ is a positive definite linear operator on this space. Also discussed are graph theoretic interpretations of one of the results.
1 citations