Positivity of Partitioned Hermitian Matrices with Unitarily Invariant Norms

We give a short proof of a recent result of Drury on the positivity of a $3\times 3$ matrix of the form $(\|R_i^*R_j\|_{\rm tr})_{1 \le i, j \le 3}$ for any rectangular complex (or real) matrices $R_1, R_2, R_3$ so that the multiplication $R_i^*R_j$ is compatible for all $i, j$, where $\|\cdot\|_{\rm tr}$ denotes the trace norm. We then give a complete analysis of the problem when the trace norm is replaced by other unitarily invariant norms.Comment: 6 page

On the ℓ\ell-adic valuation of certain Jacobi sums

Fix distinct primes $\ell$ and $f$, a finite field $\mathbf{F}_{q}$ such that $q \equiv 1 \pmod{\ell f}$, and a character $\chi : \mathbf{F}_{q}^{\times} \to \mathbf{C}^{\times}$ of exact order $\ell f$. We present a new $\ell$-adic congruence for the Jacobi sum $J(\chi^{\ell}, \chi^{f})$. These Jacobi sums are Frobenius eigenvalues of the curve $y^{\ell} = x^{f} + 1$.Comment: 18 pages; improved notation, reorganized sections accordingly, clarified a small subtlety with ell=2 cas

Chi Epsilon

Members of Chi Epsilon are pictured from left to right beginning on the front row: J. Kent Johnson, Patricia Jones, Marian Snowden, John Farr, Mark A. Turner, John Michael Saines, James Eannon, Mark Malone, Dr. Victor Zitta, Charles S. Jones, and Frank Hines, III.https://scholarsjunction.msstate.edu/ua-photo-collection/4648/thumbnail.jp

An improved form of the artificial diffusion parameter-X

Artificial diffusion mechanisms used to damp post-shock oscillations or to stabilize the computations contain a factor chi which determines the amount of diffusion provided. This paper suggests an improved procedure which consists of (1) varying chi from point to point depending on the local Courant number and (2) employing a value of local chi sub i which is determined from a monotonicity analysis. Results obtained by employing a locally dependent value of chi are compared with those for a global value of ch