Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMinimum ranks of sign patterns via sign vectors and duality
128
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
A {it sign pattern matrix} is a matrix whose entries are from the set ${+,-, 0}$. The minimum rank of a sign pattern matrix $A$ is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of $A$. It is shown in this paper that for any $m times n$ sign pattern $A$ with minimum rank $n-2$, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer $ngeq 9$, there exists a nonnegative integer $m$ such that there exists an $ntimes m$ sign pattern matrix with minimum rank $n-3$ for which rational realization is not possible. A characterization of $mtimes n$ sign patterns $A$ with minimum rank $n-1$ is given (which solves an open problem in Brualdi et al. cite{Bru10}), along with a more general description of sign patterns with minimum rank $r$, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of $k$-dimensional subspaces of $mathbb R^n$ are obtained. In particular, it is shown that the maximum number of sign vectors of $2$-dimensional subspaces of $mathbb R^n$ is $4n+1$. Several related open problems are stated along the way.
rate research
Read More
A emph{sign pattern (matrix)} is a matrix whose entries are from the set ${+, -, 0}$. The emph{minimum rank} (respectively, emph{rational minimum rank}) of a sign pattern matrix $cal A$ is the minimum of the ranks of the real (respectively, rational) matrices whose entries have signs equal to the corresponding entries of $cal A$. A sign pattern $cal A$ is said to be emph{condensed} if $cal A$ has no zero row or column and no two rows or columns are identical or negatives of each other. In this paper, a new direct connection between condensed $m times n $ sign patterns with minimum rank $r$ and $m$ point--$n$ hyperplane configurations in ${mathbb R}^{r-1}$ is established. In particular, condensed sign patterns with minimum rank 3 are closed related to point--line configurations on the plane. It is proved that for any sign pattern $cal A$ with minimum rank $rgeq 3$, if the number of zero entries on each column of $cal A$ is at most $r-1$, then the rational minimum rank of $cal A$ is also $r$. Furthermore, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3.
A sign pattern matrix is a matrix whose entries are from the set ${+,-,0}$. If $A$ is an $mtimes n$ sign pattern matrix, the qualitative class of $A$, denoted $Q(A)$, is the set of all real $mtimes n$ matrices $B=[b_{i,j}]$ with $b_{i,j}$ positive (respectively, negative, zero) if $a_{i,j}$ is + (respectively, $-$, 0). The minimum rank of a sign pattern matrix $A$, denoted $mr(A)$, is the minimum of the ranks of the real matrices in $Q(A)$. Determination of the minimum rank of a sign pattern matrix is a longstanding open problem. For the case that the sign pattern matrix has a 1-separation, we present a formula to compute the minimum rank of a sign pattern matrix using the minimum ranks of certain generalized sign pattern matrices associated with the 1-separation.
The refined inertia of a square real matrix $A$ is the ordered $4$-tuple $(n_+, n_-, n_z, 2n_p)$, where $n_+$ (resp., $n_-$) is the number of eigenvalues of $A$ with positive (resp., negative) real part, $n_z$ is the number of zero eigenvalues of $A$, and $2n_p$ is the number of nonzero pure imaginary eigenvalues of $A$. For $n geq 3$, the set of refined inertias $mathbb{H}_n={(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)}$ is important for the onset of Hopf bifurcation in dynamical systems. We say that an $ntimes n$ sign pattern ${cal A}$ requires $mathbb{H}_n$ if $mathbb{H}_n={text{ri}(B) | B in Q({cal A})}$. Bodine et al. conjectured that no $ntimes n$ irreducible sign pattern that requires $mathbb{H}_n$ exists for $n$ sufficiently large, possibly $nge 8$. However, for each $n geq 4$, we identify three $ntimes n$ irreducible sign patterns that require $mathbb{H}_n$, which resolves this conjecture.
We consider various properties and manifestations of some sign-alternating univariate polynomials borne of right-triangular integer arrays related to certain generalizations of the Fibonacci sequence. Using a theory of the root geometry of polynomial sequences developed by J. L. Gross, T. Mansour, T. W. Tucker, and D. G. L. Wang, we show that the roots of these `sign-alternating Gibonacci polynomials are real and distinct, and we obtain explicit bounds on these roots. We also derive Binet-type closed expressions for the polynomials. Some of these results are applied to resolve finiteness questions pertaining to a one-player combinatorial game (or puzzle) modelled after a well-known puzzle we call the `Networked-numbers Game. Elsewhere, the first- and second-named authors, in collaboration with A. Nance, have found rank symmetric `diamond-colored distributive lattices naturally related to certain representations of the special linear Lie algebras. Those lattice cardinalities can be computed using sign-alternating Fibonacci polynomials, and the lattice rank generating functions correspond to the rows of some new and easily defined triangular integer arrays. Here, we present Gibonaccian, and in particular Lucasia
Sign language lexica are a useful resource for researchers and people learning sign languages. Current implementations allow a user to search a sign either by its gloss or by selecting its primary features such as handshape and location. This study focuses on exploring a reverse search functionality where a user can sign a query sign in front of a webcam and retrieve a set of matching signs. By extracting different body joints combinations (upper body, dominant hands arm and wrist) using the pose estimation framework OpenPose, we compare four techniques (PCA, UMAP, DTW and Euclidean distance) as distance metrics between 20 query signs, each performed by eight participants on a 1200 sign lexicon. The results show that UMAP and DTW can predict a matching sign with an 80% and 71% accuracy respectively at the top-20 retrieved signs using the movement of the dominant hand arm. Using DTW and adding more sign instances from other participants in the lexicon, the accuracy can be raised to 90% at the top-10 ranking. Our results suggest that our methodology can be used with no training in any sign language lexicon regardless of its size.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
