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[2] viXra:2101.0167 [pdf] submitted on 2021-01-27 11:31:34
Authors: Yaroslav Shitov
Comments: 24 Pages. a full version of arXiv:1606.09068
Let A be a nonnegative matrix, that is, a matrix with nonnegative real entries. A nonnegative factorization of size k is a representation of A as a sum of k nonnegative rank-one matrices. The space of all such factorizations is a bounded semialgebraic set, and we prove that spaces arising in this way are universal. More presicely, we show that every bounded semialgebraic set U is rationally equivalent to the set of nonnegative size-k factorizations of some matrix A up to a permutation of matrices in the factorization. Our construction is effective, and we can compute a pair (A, k) in polynomial time from a given description of U as a system of polynomial inequalities with coefficients in Q. This result gives a complete description of the algorithmic complexity of several important problems, including the nonnegative matrix factorization, completely positive rank, nested polytope problem, and it also leads to a complete resolution of the problem of Cohen and Rothblum on nonnegative factorizations over different ordered fields.
Category: Algebra
[1] viXra:2101.0091 [pdf] replaced on 2021-01-18 20:10:29
Authors: Theophilus Agama
Comments: 20 Pages. An application added
In this paper we launch an extension program for single variable expansivity theory. We study this notion under tuples of polynomials belonging to the ring $\mathbb{R}[x_1,x_2,\ldots,x_n]$. As an application we show that
\begin{align}\mathrm{min}\{\mathrm{max}\{\mathrm{Ind}_{f_k}(x_{\sigma(i)})\}_{k=1}^{s}+1\}_{i=1}^{l}&<\frac{1}{l}\sum \limits_{i=1}^{l}\mathrm{max}\{\mathrm{Ind}_{f_k}(x_{\sigma(i)})\}_{k=1}^{s}+2+\mathcal{J}\nonumber
\end{align}where $\mathcal{J}:=\mathcal{J}(l)\geq 0$ and $\mathrm{Ind}_{f_k}(x_j)$ is the largest power of $x_j$~($1\leq j\leq n$) in the polynomial $f_k\in \mathbb{R}[x_1,x_2,\ldots,x_n]$.
Category: Algebra
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