Structured low-rank approximation is the problem of minimizing a weighted Frobenius distance to a given matrix among all matrices of fixed rank in a linear space of matrices. We study the critical points of this optimization problem using algebraic geometry. A particular focus lies on Hankel matrices, Sylvester matrices and generic linear spaces.

This paper addresses two coding schemes which can handle emerging errors with crisscross patterns. First, a code with maximum rank distance, so-called Rank-Codes, is described and a modified Berlekamp-Massey algorithm is provided. Secondly, a Permutation Code based coding scheme for crisscross error patterns is presented. The influence of different types of noise are also discussed.

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants suc...

Background: Osteoarthritis is a common chronic disease of the musculoskeletal system in older adults. Aim: This study aimed to compare the effects of in-person and distance exercise training on the outcomes of knee injury and osteoarthritis among the elderly with limited literacy. Method: In this two-group randomized clinical trial with a pretest-posttest design, 60 elderly patients with knee i...

In this article, several theorems on perturbations of a complex matrix by a matrix of a given rank are presented. These theorems may be divided into two groups. The first group is about spectral properties of a matrix under such perturbations; the second is about almost-near relations with respect to the rank distance.

One of important components in an image retrieval system is selecting a distance measure to compute rank between two objects. In this paper, several distance measures were researched to implement a foliage plant retrieval system. Sixty kinds of foliage plants with various leaf color and shape were used to test the performance of 7 different kinds of distance measures: city block distance, Eucli...

Skew cyclic codes play the same role as cyclic codes in the theory of errorcorrecting codes for the rank metric. In this paper, we give descriptions of these codes by idempotent generators, root spaces and cyclotomic spaces. We prove that the lattice of skew cyclic codes is anti-isomorphic to the lattice of root spaces and extend the rank-BCH bound on their minimum rank distance to rank-metric ...

We introduce the relative rank differential statistic which is a non-parametric approach to document and dialog analysis based on word frequency rank-statistics. We also present a simple method to establish semantic saliency in dialog, documents, and dialog segments using these word frequency rank statistics. Applications of our technique include the dynamic tracking of topic and semantic evolu...

Let p be a prime such that p ≡ 2 or 3 (mod 5). Linear block codes over the non-commutative matrix ring M2(Fp) endowed with the Bachoc weight are derived as isometric images of linear block codes over the Galois field Fp2 endowed with the Hamming metric. When seen as rank metric codes, this family of matrix codes satisfies the Singleton bound and thus are maximum rank distance codes, which are t...

The data of this study was collected from 252 teachers, principals and vice-principals offemales’ high schools in district one through one researcher-administered questionnaire, and was then analyzed. The validity of the questionnaire was approved by seven university professors. Cronbach’s alpha was used in order to measure the reliability of the questionnaire. The total calculated alpha was 0....