Weighted Shift Matrices: Unitary Equivalence, Reducibility and Numerical Ranges

An $n$-by-$n$ ($n\ge 3$) weighted shift matrix $A$ is one of the form [{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & & 0{array}], where the $a_j$'s, called the weights of $A$, are complex numbers. Assume that all $a_j$'s are nonzero and $B$ is an $n$-by-$n$ weighted shift matrix with weights $b_1,..., b_n$. We show that $B$ is unitarily equivalent to $A$ if and only if $b_1... b_n=a_1...a_n$ and, for some fixed $k$, $1\le k \le n$, $|b_j| = |a_{k+j}|$ ($a_{n+j}\equiv a_j$) for all $j$. Next, we show that $A$ is reducible if and only if $A$ has periodic weights, that is, for some fixed $k$, $1\le k \le \lfloor n/2\rfloor$, $n$ is divisible by $k$, and $|a_j|=|a_{k+j}|$ for all $1\le j\le n-k$. Finally, we prove that $A$ and $B$ have the same numerical range if and only if $a_1...a_n=b_1...b_n$ and $S_r(|a_1|^2,..., |a_n|^2)=S_r(|b_1|^2,..., |b_n|^2)$ for all $1\le r\le \lfloor n/2\rfloor$, where $S_r$'s are the circularly symmetric functions.Comment: 27 page

Cutaneous Mycobacterium intracellulare infection presenting as multiple asymptomatic papulonodules in an immunocompetent adult: A case report and review of the literature

AbstractDisseminated cutaneous nontuberculous mycobacteria infection is rare in immunocompetent hosts. We report a case of Mycobacterium intracellulare infection in an immunocompetent patient presenting with simultaneously developing multiple asymptomatic cutaneous papulonodules. The possibility of lung lesions as the primary focus is suspected. We review the literature for other cases of multiple cutaneous M avium complex infections in immunocompetent hosts. There are differences in the virulence of M avium and M intracellulare, and hence in the underlying immune status of the hosts

Kappa-Opioid Receptors in the Caudal Nucleus Tractus Solitarius Mediate 100 Hz Electroacupuncture-Induced Sleep Activities in Rats

Previous results demonstrated that 10 Hz electroacupuncture (EA) of Anmian acupoints in rats during the dark period enhances slow wave sleep (SWS), which involves the induction of cholinergic activity in the caudal nucleus tractus solitarius (NTS) and subsequent activation of opioidergic neurons and μ-receptors. Studies have shown that different kinds of endogenous opiate peptides and receptors may mediate the consequences of EA with different frequencies. Herein, we further elucidated that high-frequency (100 Hz)-EA of Anmian enhanced SWS during the dark period but exhibited no direct effect on rapid eye movement (REM) sleep. High-frequency EA-induced SWS enhancement was dose-dependently blocked by microinjection of naloxone or κ-receptor antagonist (nor-binaltorphimine) into the caudal NTS, but was affected neither by μ- (naloxonazine) nor δ-receptor antagonists (natatrindole), suggesting the role of NTS κ-receptors in the high-frequency EA-induced SWS enhancement. Current and previous results depict the opioid mechanisms of EA-induced sleep

TGF-β inhibits IL-1β-activated PAR-2 expression through multiple pathways in human primary synovial cells

To investigate the mechanism how Transforming growth factor-β(TGF-β) represses Interleukin-1β (IL-1β)-induced Proteinase-Activated Receptor-2 (PAR-2) expression in human primary synovial cells (hPSCs). Human chondrocytes and hPSCs isolated from cartilages and synovium of Osteoarthritis (OA) patients were cultured with 10% fetal bovine serum media or serum free media before treatment with IL-1β, TGF-β1, or Connective tissue growth factor (CTGF). The expression of PAR-2 was detected using reverse transcriptase-polymerase chain reaction (RT-PCR) and western blotting. Collagen zymography was performed to assess the activity of Matrix metalloproteinases-13 (MMP-13). It was demonstrated that IL-1β induces PAR-2 expression via p38 pathway in hPSCs. This induction can be repressed by TGF-β and was observed to persist for at least 48 hrs, suggesting that TGF-β inhibits PAR-2 expression through multiple pathways. First of all, TGF-β was able to inhibit PAR-2 activity by inhibiting IL-1β-induced p38 signal transduction and secondly the inhibition was also indirectly due to MMP-13 inactivation. Finally, TGF-β was able to induce CTGF, and in turn CTGF represses PAR-2 expression by inhibiting IL-1β-induced phospho-p38 level. TGF-β could prevent OA from progression with the anabolic ability to induce CTGF production to maintain extracellular matrix (ECM) integrity and to down regulate PAR-2 expression, and the anti-catabolic ability to induce Tissue inhibitors of metalloproteinase-3 (TIMP-3) production to inhibit MMPs leading to avoid PAR-2 over-expression. Because IL-1β-induced PAR-2 expressed in hPSCs might play a significantly important role in early phase of OA, PAR-2 repression by exogenous TGF-β or other agents might be an ideal therapeutic target to prevent OA from progression

Ameliorate Effects of Li-Fu Formula on IL-6-Mediated Cardiac Hypertrophy in Hamsters Fed with a Hyper-Cholesterol Diet

Hypercholesterolemia diets are considered as major sources to cause cardiac hypertrophy. This study intends to evaluate the effects of Li-Fu formula on cardiac hypertrophy induced by hypercholesterolemia diet. Twenty-four male Golden Syrian hamsters were randomly divided into control, cholesterol and Li-Fu formula groups and fed with different experimental diets for 2 months. Histopathological analysis and western blotting were performed to measure the myocardial architecture, and various cardiac hypertrophy-associated molecules in the excised left ventricle from hamsters. The ratios of whole heart weight/body weight (BW) and left ventricle weight/BW were significantly higher in the cholesterol group but significantly lower in the Li-Fu formula group. The protein levels of both atrial natriuretic peptide and brain natriuretic peptide were significantly increased in the cholesterol group but significantly reduced in the Li-Fu formula group. Additionally, significantly increased interleukin-6, STAT3, MEK5, p-ERK5 and non-cardiomyocyte proliferate signal molecules such as p-MEK and p-ERK, were detected in the cholesterol group but significantly reduced in the Li-Fu formula group. Notably, no significant variations of inflammatory signaling molecules, including p-P38 and p-JNK, were detected in all groups. Our experimental results demonstrated the significant reductions of cardiac hypertrophy and related eccentric hypertrophy signaling, non-cardiomyocyte proliferate signaling in the excised left ventricle of hamsters from the Li-Fu formula. We suggested the protective effects of Li-Fu formula on cardiac hypertrophy that may be useful in prevention or treatment of hypertrophy-associated cardiovascular diseases