Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix

The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace \X, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$ with respect to \X. If the subspace \X is $A$-invariant then the Ritz values are some of the eigenvalues of $A$. If the $A$-invariant subspace \X is perturbed to give rise to another subspace \Y, then the vector of absolute values of changes in Ritz values of $A$ represents the absolute eigenvalue approximation error using \Y. We bound the error in terms of principal angles between \X and \Y. We capitalize on ideas from a recent paper [DOI: 10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute values of differences between Ritz values for subspaces \X and \Y was weakly (sub-)majorized by a constant times the sine of the vector of principal angles between \X and \Y, the constant being the spread of the spectrum of $A$. In that result no assumption was made on either subspace being $A$-invariant. It was conjectured there that if one of the trial subspaces is $A$-invariant then an analogous weak majorization bound should only involve terms of the order of sine squared. Here we confirm this conjecture. Specifically we prove that the absolute eigenvalue error is weakly majorized by a constant times the sine squared of the vector of principal angles between the subspaces \X and \Y, where the constant is proportional to the spread of the spectrum of $A$. For many practical cases we show that the proportionality factor is simply one, and that this bound is sharp. For the general case we can only prove the result with a slightly larger constant, which we believe is artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and Applications (SIMAX

Tight Regulation of Mechanotransducer Proteins Distinguishes the Response of Adult Multipotent Mesenchymal Cells on PBCE-Derivative Polymer Films with Different Hydrophilicity and Stiffness

: Mechanotransduction is a molecular process by which cells translate physical stimuli exerted by the external environment into biochemical pathways to orchestrate the cellular shape and function. Even with the advancements in the field, the molecular events leading to the signal cascade are still unclear. The current biotechnology of tissue engineering offers the opportunity to study in vitro the effect of the physical stimuli exerted by biomaterial on stem cells and the mechanotransduction pathway involved in the process. Here, we cultured multipotent human mesenchymal/stromal cells (hMSCs) isolated from bone marrow (hBM-MSCs) and adipose tissue (hASCs) on films of poly(butylene 1,4-cyclohexane dicarboxylate) (PBCE) and a PBCE-based copolymer containing 50 mol% of butylene diglycolate co-units (BDG50), to intentionally tune the surface hydrophilicity and the stiffness (PBCE = 560 Mpa; BDG50 = 94 MPa). We demonstrated the activated distinctive mechanotransduction pathways, resulting in the acquisition of an elongated shape in hBM-MSCs on the BDG50 film and in maintaining the canonical morphology on the PBCE film. Notably, hASCs acquired a new, elongated morphology on both the PBCE and BDG50 films. We found that these events were mainly due to the differences in the expression of Cofilin1, Vimentin, Filamin A, and Talin, which established highly sensitive machinery by which, rather than hASCs, hBM-MSCs distinguished PBCE from BDG50 films

A comparison of lysosomal enzymes expression levels in peripheral blood of mild- and severe-Alzheimer’s disease and MCI patients: implications for regenerative medicine approaches

The association of lysosomal dysfunction and neurodegeneration has been documented in several neurodegenerative diseases, including Alzheimer’s Disease (AD). Herein, we investigate the association of lysosomal enzymes with AD at different stages of progression of the disease (mild and severe) or with mild cognitive impairment (MCI). We conducted a screening of two classes of lysosomal enzymes: glycohydrolases (β-Hexosaminidase, β-Galctosidase, β-Galactosylcerebrosidase, β-Glucuronidase) and proteases (Cathepsins S, D, B, L) in peripheral blood samples (blood plasma and PBMCs) from mild AD, severe AD, MCI and healthy control subjects. We confirmed the lysosomal dysfunction in severe AD patients and added new findings enhancing the association of abnormal levels of specific lysosomal enzymes with the mild AD or severe AD, and highlighting the difference of AD from MCI. Herein, we showed for the first time the specific alteration of β-Galctosidase (Gal), β-Galactosylcerebrosidase (GALC) in MCI patients. It is notable that in above peripheral biological samples the lysosomes are more sensitive to AD cellular metabolic alteration when compared to levels of Aβ-peptide or Tau proteins, similar in both AD groups analyzed. Collectively, our findings support the role of lysosomal enzymes as potential peripheral molecules that vary with the progression of AD, and make them useful for monitoring regenerative medicine approaches for AD