arXiv subjects

Alexander E. Guterman

Publications and source records attributed to Alexander E. Guterman.

4 recordsLinked to original sources

arXiv2026

Poset-refined majorization relations

Several classical majorization relations for sums or products of matrices involve a majorizing vector of perfectly aligned eigenvalues or singular values. By relaxing the order of alignment to a partial order, we show that the majorization can be strengthened, provided the change-of-basis matrices admit an LU-approximation with respect to this partial order. In this way, we obtain refined versions of Ky Fan's majorization relations, Horn's log-majorization relation, and von Neumann's trace inequality. As an application, we give a short proof of the separable Ky Fan majorization relation for an arbitrary number of tensor factors and extend it to a sum of tensor products of arbitrary matrices. Further applications concern majorization relations for sums of (anti-)symmetric powers and for products of Kronecker sums.

math.FA↗

arXiv2019

Upper bounds for the length of non-associative algebras

We obtain a sharp upper bound for the length of arbitrary non-associative algebra and present an example demonstrating the sharpness of our bound. To show this we introduce a new method of characteristic sequences based on linear algebra technique. This method provides an efficient tool for computing the length function in non-associative case. Then we apply the introduced method to obtain an upper bound for the length of an arbitrary locally complex algebra. We also show that the obtained bound is sharp. In the last case the length is bounded in terms of Fibonacci sequence.

math.CO↗

arXiv2018

Kräuter conjecture on permanents is true

In this paper we investigate the permanent of $(-1,1)$-matrices over fields of zero characteristics and our main goal is to provide a sharp upper bound for the value of the permanent of such matrices depending on matrix rank, solving Wang's problem posed in 1974 by confirming Kräuter conjecture formulated in 1985.

math.CO↗

arXiv2010

On the Polya permanent problem over finite fields

Let $\FF$ be a finite field of characteristics different from two. We show that no bijective map transforms permanent into determinant when the cardinality of $\FF$ is sufficiently large. We also give an example of non-bijective map when $\FF$ is arbitrary and an example of a bijective map when $\FF$ is infinite which do transform permanent into determinant. The developed technique allows us to estimate the probability of the permanent and the determinant of matrices over finite fields to have a given value. Our results are also true over finite rings without zero divisors.

math.CO↗