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Minimal indices and minimal bases via filtrations
No full textA new way to formulate the notions of minimal basis and minimal indices is developed in this paper, based on the concept of a filtration of a vector space. The goal is to provide useful new tools for working with these important concepts, as well as to gain deeper insight into their fundamental nature. This approach also readily reveals a strong minimality property of minimal indices, from which follows a characterization of the vector polynomial bases in rational vector spaces. The effectiveness of this new formulation is further illustrated by proving several fundamental properties: the invariance of the minimal indices of a matrix polynomial under field extension, the direct sum property of minimal indices, the polynomial linear combination property, and the predictable degree property
A projective approach to nonnegative matrix factorization
No full textIn data science and machine learning, the method of nonnegative matrix factorization (NMF) is a powerful tool that enjoys great popularity. Depending on the concrete application, there exist several subclasses each of which performs a NMF under certain constraints. Consider a given square matrix . The symmetric NMF aims for a nonnegative low-rank approximation to , where is entrywise nonnegative and of given order. Considering a rectangular input matrix , the general NMF again aims for a nonnegative low-rank approximation to which is now of the type for entrywise nonnegative matrices of given order. In this paper, we introduce a new heuristic method to tackle the exact nonnegative matrix factorization problem (of type ), based on projection approaches to solve a certain feasibility problem
On tensor GMRES and Golub-Kahan methods via the T-product for color image processing
No full textThe present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the T-product for two tensors to define tensor tubal global Arnoldi and tensor tubal global Golub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based global approaches can be exploited to solve ill-posed problems arising from recovering blurry multichannel (color) images and videos, using the so-called Tikhonov regularization technique, to provide computable approximate regularized solutions. We also review a generalized cross-validation and discrepancy principle type of criterion for the selection of the regularization parameter in the Tikhonov regularization. Applications to image sequence processing are given to demonstrate the efficiency of the algorithms
Line graphs of complex unit gain graphs with least eigenvalue -2
No full textLet be the multiplicative group of complex units, and let denote a line graph of a -gain graph . Similarly to what happens in the context of signed graphs, the real number , that is, the smallest eigenvalue of the adjacency matrix of , is not less than . The structural conditions on ensuring that are identified. When such conditions are fulfilled, bases of the -eigenspace are constructed with the aid of the star complement technique
Beyond the Fan-Hoffman inequality
No full textWe establish a singular value inequality inspired by the Fan-Hoffman inequality and resolve the -conjecture of Yang and Zhang
The algebra generated by nilpotent elements in a matrix centralizer
No full textFor an arbitrary square matrix , denote by the centralizer of , and by the set of all nilpotent elements in . In this paper, we use the Weyr canonical form to study the subalgebra of generated by . We determine conditions on such that is a subalgebra of . We also determine conditions on such that the subalgebra generated by is $C(S).
Benefits for Child Care Workers: How the State Could Help through a Medicaid Waiver
No full textChild care is expensive, and many parents struggle to afford care; furthermore, even though child care costs are high, child care providers in the United States (US) are not making a living wage. Child care professionals (ages 0-5 in child care homes or centers) earn less income than Kindergarten teachers, pre-K teachers, non-farm animal caretakers, and the US estimate of all workers’ annual median salary (Bureau of Labor Statistics, 2020a, 2020b). Workers in comparable professions are also usually offered benefits for their labor, which child care professionals are not (Kwon, 2019; National Survey of Early Care and Education Project Team, 2020; Otten et al., 2019; Whitebook, McLean, Austin, & Edwards, 2018). This often necessitates use of public assistance. Because many child care workers are not provided access to health insurance or other health-related benefits through their employers, they must seek access to health care in other ways. Additionally, turnover rates among child care workers are high, and wages and benefits are a large part of the reason why child care professionals leave their jobs (McDougald Scott, 2021a). This policy analysis (a) reviewed the current struggle (as of May 2021) that child care workers in the United States (in general) and South Carolina (in particular) experience compared with employees in other fields; and (b) explore options (particularly a Medicaid waiver option) that might improve the situation. South Carolina (SC) is one of the 13 states that have not expanded Medicaid; most of the 13 states are in Southern United States (US) region, which makes an extrapolation of SC research reasonable. Lessons learned from SC childcare data should reflect closely what may be found in other non-expansion states, but research from the literature review will not be SC-specific. Relevant peer-reviewed, government documents, state and national data, and grey literature were reviewed and analyzed. There have been ongoing efforts (although insufficient even in more successful efforts) with mixed results to improve the pay for child care workers for decades. Progress for earning a living wage will require a systems overhaul for early education, but child care providers cannot wait for workforce environmental improvements. Action must be taken now to augment the shortage of healthcare access for child care providers. In SC, Medicaid helps some child care workers receive access to health care, but expansion through Medicaid waiver 1115 would include many more child care workers who do not currently have access. 
