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Let $A, B$ be $ntimes n$ positive semidefinite matrices. Bhatia and Kittaneh asked whether it is true $$ sqrt{sigma_j(AB)}le frac{1}{2} lambda_j(A+B), qquad j=1, ldots, n$$ where $sigma_j(cdot)$, $lambda_j(cdot)$, are the $j$-th largest singular value, eigenvalue, respectively. The question was recently solved by Drury in the affirmative. This article revisits Drurys solution. In particular, we simplify the proof for a key auxiliary result in his solution.
An acute look at underbar{basic} facts concerning underbar{unbounded} subnormal operators is taken here. These operators have the richest structure and are the most exciting among the whole family of beneficiaries of the normal ones. Therefore, the l
These lecture notes are meant to accompany two lectures given at the CDM 2016 conference, about the Kadison-Singer Problem. They are meant to complement the survey by the same authors (along with Spielman) which appeared at the 2014 ICM. In the first
The Bhatia-Davis theorem provides a useful upper bound for the variance in mathematics, and in quantum mechanics, the variance of a Hamiltonian is naturally connected to the quantum speed limit due to the Mandelstam-Tamm bound. Inspired by this conne
The determinant of a lower Hessenberg matrix (Hessenbergian) is expressed as a sum of signed elementary products indexed by initial segments of nonnegative integers. A closed form alternative to the recurrence expression of Hessenbergians is thus obt
We make several improvements to methods for finding integer solutions to $x^3+y^3+z^3=k$ for small values of $k$. We implemented these improvements on Charity Engines global compute grid of 500,000 volunteer PCs and found new representations for seve