The main motivation behind this line of work is that some networks are not planar but conform to some generalization of the concept of planarity, like being embeddable on a surface of bounded genus. Furthermore, by Kuratowski’s famous theorem [Kur30], planarity can be characterized be excluding K5 and K3,3 as minors. Hence, it seems natural to study further classes with excluded minors as a gen...

This paper continues an earlier research of the authors on universal quadratic identities (QIs) minors quantum matrices. We demonstrate situations when QIs are provided, in a sense, by ones four special types (Plűcker, co-Plűcker, Dodgson and quasi-commutation relations flag co-flag interval minors).

A generalized tournament matrix $M$ is a nonnegative that satisfies $M+M^{t}=J-I$, where $J$ the all ones and $I$ identity matrix. In this paper, characterization of matrices with same principal minors orders $2$, $3$, $4$ given. particular, it proven determine rest minors.

We prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymour’s result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on ma...

Some of these methods are appropriate to special systems, as for example to the axisymmetric " normal equations " of Least Squares. In many applications, however, as in problems of statistical correlation of many variables, it may be desired not merely to solve a given set of equations but to obtain as much knowledge as possible about the system or matrix of coefficients; perhaps to evaluate it...

Genetic testing in children raises many important ethical, legal, and social issues. One of the main concerns is the ethically inappropriate genetic testing of minors. Various European countries established professional guidelines which reflect the different countries perspectives regarding the main ethical issues involved. In this paper, we analyze the Italian and the British guidelines by hig...

It is well-known that the Plücker relations generate the ideal of relations of the maximal minors of a generic m× n matrix. In this paper we discuss the relations of t-minors for t < min(m,n). We will exhibit minimal relations in degrees 2 (non-Plücker in general) and 3, and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is...

It is shown that for any positive integers k and w there exists a constant N = N(k, w) such that every 7-connected graph of tree-width less than w and of order at least N contains K3,k as a minor. Similar result is proved for Ka,k minors where a is an arbitrary fixed integer and the required connectivity depends only on a. These are the first results of this type where fixed connectivity forces...

Stanford University For a random matrix following a Wishart distribution, we derive formulas for the expectation and the covariance matrix of compound matrices. The compound matrix of order m is populated by all m×mminors of the Wishart matrix. Our results yield first and second moments of the minors of the sample covariance matrix for multivariate normal observations. This work is motivated by...

Let IF be field, and let A and B be n X n matrices with elements in IF. Suppose that A is completely reducible and that B is symmetric. If the principal minors of A determined by the 1and 2-circuits of the graph of B and by the chordless circuits of the graph of A are equal to the corresponding principal minors of B, then A is diagonally similar to B; and conversely.