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Search Results (3)

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Keywords = Frobenius inequality

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6 pages, 231 KiB  
Article
Three New Proofs of the Theorem rank f(M) + rank g(M) = rank (f,g)(M) + rank [f,g](M)
by Vasile Pop and Alexandru Negrescu
Mathematics 2024, 12(3), 360; https://doi.org/10.3390/math12030360 - 23 Jan 2024
Cited by 1 | Viewed by 1206
Abstract
It is well known that in C[X], the product of two polynomials is equal to the product of their greatest common divisor and their least common multiple. In a recent paper, we proved a similar relation between the ranks [...] Read more.
It is well known that in C[X], the product of two polynomials is equal to the product of their greatest common divisor and their least common multiple. In a recent paper, we proved a similar relation between the ranks of matrix polynomials. More precisely, the sum of the ranks of two matrix polynomials is equal to the sum of the rank of the greatest common divisor of the polynomials applied to the respective matrix and the rank of the least common multiple of the polynomials applied to the respective matrix. In this paper, we present three new proofs for this result. In addition to these, we present two more applications. Full article
(This article belongs to the Section A: Algebra and Logic)
18 pages, 359 KiB  
Article
Bounding the Zeros of Polynomials Using the Frobenius Companion Matrix Partitioned by the Cartesian Decomposition
by Mohammad W. Alomari and Christophe Chesneau
Algorithms 2022, 15(6), 184; https://doi.org/10.3390/a15060184 - 26 May 2022
Cited by 3 | Viewed by 1932
Abstract
In this work, some new inequalities for the numerical radius of block n-by-n matrices are presented. As an application, the bounding of zeros of polynomials using the Frobenius companion matrix partitioned by the Cartesian decomposition method is proved. We highlight several [...] Read more.
In this work, some new inequalities for the numerical radius of block n-by-n matrices are presented. As an application, the bounding of zeros of polynomials using the Frobenius companion matrix partitioned by the Cartesian decomposition method is proved. We highlight several numerical examples showing that our approach to bounding zeros of polynomials could be very effective in comparison with the most famous results as well as some recent results presented in the field. Finally, observations, a discussion, and a conclusion regarding our proposed bound of zeros are considered. Namely, it is proved that our proposed bound is more efficient than any other bound under some conditions; this is supported with many polynomial examples explaining our choice of restrictions. Full article
22 pages, 3546 KiB  
Article
On Geometry of Information Flow for Causal Inference
by Sudam Surasinghe and Erik M. Bollt
Entropy 2020, 22(4), 396; https://doi.org/10.3390/e22040396 - 30 Mar 2020
Cited by 5 | Viewed by 6585
Abstract
Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other fields. [...] Read more.
Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other fields. This paper takes the perspective of information flow, which includes the Nobel prize winning work on Granger-causality, and the recently highly popular transfer entropy, these being probabilistic in nature. Our main contribution will be to develop analysis tools that will allow a geometric interpretation of information flow as a causal inference indicated by positive transfer entropy. We will describe the effective dimensionality of an underlying manifold as projected into the outcome space that summarizes information flow. Therefore, contrasting the probabilistic and geometric perspectives, we will introduce a new measure of causal inference based on the fractal correlation dimension conditionally applied to competing explanations of future forecasts, which we will write G e o C y → x . This avoids some of the boundedness issues that we show exist for the transfer entropy, T y → x . We will highlight our discussions with data developed from synthetic models of successively more complex nature: these include the Hénon map example, and finally a real physiological example relating breathing and heart rate function. Full article
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