scholarly journals Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns

2006 ◽  
Vol 418 (2-3) ◽  
pp. 394-415 ◽  
Author(s):  
Luz Maria DeAlba ◽  
Timothy L. Hardy ◽  
Irvin Roy Hentzel ◽  
Leslie Hogben ◽  
Amy Wangsness
Keyword(s):  
Maximum Eigenvalue ◽  
Minimum Rank ◽  
Sign Patterns ◽  
Symmetric Tree

scholarly journals Rational realization of maximum eigenvalue multiplicity of symmetric tree sign patterns

2006 ◽  
Vol 418 (2-3) ◽  
pp. 380-393 ◽  
Author(s):  
Atoshi Chowdhury ◽  
Leslie Hogben ◽  
Jude Melancon ◽  
Rana Mikkelson
Keyword(s):  
Maximum Eigenvalue ◽  
Sign Patterns ◽  
Symmetric Tree

Sign patterns with minimum rank 2 and upper bounds on minimum ranks

2013 ◽  
Vol 61 (7) ◽  
pp. 895-908 ◽  
Author(s):  
Zhongshan Li ◽  
Yubin Gao ◽  
Marina Arav ◽  
Fei Gong ◽  
Wei Gao ◽  
...  
Keyword(s):  
Upper Bounds ◽  
Minimum Rank ◽  
Sign Patterns ◽  
Rank 2

scholarly journals Minimum Rank of Matrices Described by a Graph or Pattern over the Rational, Real and Complex Numbers

10.37236/749 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Avi Berman ◽  
Shmuel Friedland ◽  
Leslie Hogben ◽  
Uriel G. Rothblum ◽  
Bryan Shader
Keyword(s):  
Computational Complexity ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Real Numbers ◽  
Minimum Rank ◽  
The Real ◽  
Sign Patterns ◽  
Symmetric Patterns

We use a technique based on matroids to construct two nonzero patterns $Z_1$ and $Z_2$ such that the minimum rank of matrices described by $Z_1$ is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by $Z_2$ is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture by Arav, Hall, Koyucu, Li and Rao about rational realization of minimum rank of sign patterns. Using $Z_1$ and $Z_2$, we construct symmetric patterns, equivalent to graphs $G_1$ and $G_2$, with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.

scholarly journals Essential sign change numbers of full sign pattern matrices

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Xiaofeng Chen ◽  
Wei Fang ◽  
Wei Gao ◽  
Yubin Gao ◽  
Guangming Jing ◽  
...  
Keyword(s):  
Sign Pattern ◽  
Minimum Rank ◽  
Open Problems ◽  
Sign Change ◽  
Real Polynomials ◽  
Sign Patterns ◽  
Sign Vector ◽  
Pattern Matrix ◽  
Point Line ◽  
Real Matrices

AbstractA sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. 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. The notions of essential row sign change number and essential column sign change number are introduced for full sign patterns and condensed sign patterns. By inspecting the sign vectors realized by a list of real polynomials in one variable, a lower bound on the essential row and column sign change numbers is obtained. Using point-line confiurations on the plane, it is shown that even for full sign patterns with minimum rank 3, the essential row and column sign change numbers can differ greatly and can be much bigger than the minimum rank. Some open problems concerning square full sign patterns with large minimum ranks are discussed.

scholarly journals Minimum ranks of sign patterns via sign vectors and duality

2015 ◽  
Vol 30 ◽  
Author(s):  
Marina Arav ◽  
Frank Hall ◽  
Zhongshan Li ◽  
Hein Van der Holst ◽  
John Sinkovic ◽  
...  
Keyword(s):  
Final Report ◽  
Research Station ◽  
General Description ◽  
Sign Pattern ◽  
Minimum Rank ◽  
Open Problems ◽  
New Approach ◽  
Sign Patterns ◽  
Pattern Matrix

A 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×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 n ≥ 9, there exists a nonnegative integer m such that there exists an m × n sign pattern matrix with minimum rank n − 3 for which rational realization is not possible. A characterization of m × n sign patterns A with minimum rank n − 1 is given (which solves an open problem in Brualdi et al. [R. Brualdi, S. Fallat, L. Hogben, B. Shader, and P. van den Driessche. Final report: Workshop on Theory and Applications of Matrices Described by Patterns. Banff International Research Station, Jan. 31 – Feb. 5, 2010.]), along with a more general description of sign patterns with minimum rank r, in terms of sign vectors of certain subspaces. Several related open problems are stated along the way.

scholarly journals Sarnak’s conjecture for sequences of almost quadratic word growth

2020 ◽  
pp. 1-56
Author(s):  
REDMOND MCNAMARA
Keyword(s):  
Box Dimension ◽  
Multiplicative Functions ◽  
Logarithmic Density ◽  
Liouville Function ◽  
Sign Patterns

Abstract We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the ( $\kappa -1$ )-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon })$ many words of length n where $t = \kappa (\kappa +1)/2$ . We prove a variant of the $1$ -Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than $1$ .

Analysis of grid operation state based on improved maximum eigenvalue sample covariance matrix algorithm

2021 ◽  
pp. 095965182110012
Author(s):  
Dinghui Wu ◽  
Juan Zhang ◽  
Bo Wang ◽  
Tinglong Pan
Keyword(s):  
Covariance Matrix ◽  
Signal To Noise Ratio ◽  
Poor Performance ◽  
Sample Covariance Matrix ◽  
Maximum Eigenvalue ◽  
Signal To Noise ◽  
Matrix Algorithm ◽  
Sample Covariance ◽  
Noise Ratio ◽  
Application Fields

Traditional static threshold–based state analysis methods can be applied to specific signal-to-noise ratio situations but may present poor performance in the presence of large sizes and complexity of power system. In this article, an improved maximum eigenvalue sample covariance matrix algorithm is proposed, where a Marchenko–Pastur law–based dynamic threshold is introduced by taking all the eigenvalues exceeding the supremum into account for different signal-to-noise ratio situations, to improve the calculation efficiency and widen the application fields of existing methods. The comparison analysis based on IEEE 39-Bus system shows that the proposed algorithm outperforms the existing solutions in terms of calculation speed, anti-interference ability, and universality to different signal-to-noise ratio situations.

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