Further refinements of the Heinz inequality

The celebrated Heinz inequality asserts that $2|||A^{1/2}XB^{1/2}|||\leq |||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||\leq |||AX+XB|||$ for $X \in \mathbb{B}(\mathscr{H})$, A,B\in \+, every unitarily invariant norm $|||\cdot|||$ and $\nu \in [0,1]$. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function $F(\nu)=|||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||$, some integration techniques and various refinements of the Hermite--Hadamard inequality. In the setting of matrices we prove that \begin{eqnarray*} &&\hspace{-0.5cm}\left|\left|\left|A^{\frac{\alpha+\beta}{2}}XB^{1-\frac{\alpha+\beta}{2}}+A^{1-\frac{\alpha+\beta}{2}}XB^{\frac{\alpha+\beta}{2}}\right|\right|\right|\leq\frac{1}{|\beta-\alpha|} \left|\left|\left|\int_{\alpha}^{\beta}\left(A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\right)d\nu\right|\right|\right|\nonumber\\ &&\qquad\qquad\leq \frac{1}{2}\left|\left|\left|A^{\alpha}XB^{1-\alpha}+A^{1-\alpha}XB^{\alpha}+A^{\beta}XB^{1-\beta}+A^{1-\beta}XB^{\beta}\right|\right|\right|\,, \end{eqnarray*} for real numbers $\alpha, \beta$.Comment: 15 pages, to appear in Linear Algebra Appl. (LAA

Knowlesi malaria in Vietnam

The simian malaria parasite Plasmodium knowlesi is transmitted in the forests of Southeast Asia. Symptomatic zoonotic knowlesi malaria in humans is widespread in the region and is associated with a history of spending time in the jungle. However, there are many settings where knowlesi transmission to humans would be expected but is not found. A recent report on the Ra-glai population of southern central Vietnam is taken as an example to help explain why this may be so

X-ray spectrum of the high polarization quasar PKS 1510-089

We present results on the X-ray spectra of the radio-loud, high-polarization quasar, PKS 1510-089, based on new data obtained using ASCA, and from archival ROSAT data. The X-ray spectrum obtained by ASCA is unusually hard, with the photon index=1.30+-0.06, while the (non-simultaneous) ROSAT data indicate a steeper spectrum (1.9+-0.3). The X-ray flux at 1 keV is within 10% during both observations. A break in the underlying continuum at about 0.7 keV is suggested. Flat X-ray spectra seem to be the characteristic of high polarization quasars, and their spectra also appear to be harder than that of the other radio-loud but low-polarization quasars. The multiwavelength spectrum of PKS 1510-089 is similar to many other gamma-ray blazars, suggesting the emission is dominated by that from a relativistic jet. A big blue-bump is also seen in its multiwavelength spectrum, suggesting the presence of a strong thermal component as well.Comment: 19 pages (Latex + 5 ps figures), Accpeted for publication in the Astrophysical Journal, December 20, 199