Remarks on an operator Wielandt inequality

Let $A$ be a positive operator on a Hilbert space $\mathcal{H}$ with $0<m\leq A\leq M$ and $X$ and $Y$ are two isometries on $\mathcal{H}$ such that $X^{*}Y=0$. For every 2-positive linear map $\Phi$, define $$\Gamma=\left(\Phi(X^{*}AY)\Phi(Y^{*}AY)^{-1}\Phi(Y^{*}AX)\right)^{p}\Phi(X^{*}AX)^{-p}, \, \, \, p>0.$$ We consider several upper bounds for $\frac{1}{2}|\Gamma+\Gamma^{*}|$. These bounds complement a recent result on operator Wielandt inequality.Comment: 6page

Properties of the narrow line Seyfert 1 galaxies revisited

There is growing evidence to suggest that the black hole mass has been previously underestimated with the H$\beta$ line width for certain active galactic nuclei (AGN). With the assumption of the flatter rather than isotropic velocity distribution of gases in the broad-line region of AGN, we investigated the properties of narrow line Seyfert 1 (NLS1) galaxies, like the black hole mass and the Eddington ratio, and compared with broad line Seyfert 1 (BLS1) galaxies. Since gamma-rays detected in a few NLS1s which favor a smaller viewing angle in NLS1s than BLS1s, with the projection effect we estimated the relative black hole mass and Eddington ratio for NLS1s and BLS1s. The result implies that the NLS1s and BLS1s have similar black hole masses and Eddington ratios, peaked at a larger black hole mass and lower Eddington ratio for the NLS1s than thought before. Furthermore, with applying the correction factor 6 of average black hole mass as derived from the modelling of both optical and UV data in radio-loud NLS1s by Calderone et al., to the Xu et al. sample, we find that the NLS1s and BLS1s also show similar black hole masses and Eddington ratios, peaked at $2.0\times10^{7}M_{\odot}$ and 0.12 (Eddington ratio) for the NLS1s. The $M_{BH}-\sigma$ relation due to the enhanced black hole masses of NLS1s is discussed. In addition, there seems to show a linear correlation between jet power and disk luminosity for the flat spectrum radio-loud NLS1 sample, which implies an accretion dominated rather than black hole spin dominated jet.Comment: 9 pages, 6 figures, accepted for publication in International Journal of Astronomy and Astrophysic

A new electrochemical method for the determination of chondroitin sulfate based on its supramolecular interaction with cupferron-lead(II) complex

In this paper, the interaction of cupferron (Cup) and lead (II) complex [Cup-Pb (II)] with chondroitin sulfate (CS) was investigated by linear sweep voltammetric method. In the selected medium of pH 5.5 (acetic acid-hexamine buffer solution), Cup can interact with Pb (II) to form a stable complex of [Cup-Pb(II)], which has a sensitive second order derivative polarographic reductive peak at -0.64V (vs.SCE). After the addition of CS into Cup-Pb (II) complex solution, the reductive peak current decreased without any shift of the peak potential and no new peak appeared, which indicated that an unelectroactive supramolecular complex of CS with [Cup-Pb(II)] was formed. The binding reaction conditions were carefully investigated. Under the optimal conditions, the interaction mechanism was discussed. The decrease of reductive peak current was directly proportional to the CS concentration, thus a new quantitative determination method for CS was established with the linear regression equation as ΔIp″(nA)=36.97 C/mg L-1+12.45 (n=10, γ=0.995). The effects of other substances on the determination were carefully investigated and three synthetic samples were determined with satisfactory results. The binding constant (βs) and the binding number (m) of CS with [Cup-Pb(II)] complex were calculated from the voltammetric data with the results as βs=1.89×1010 and m≈2.5