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

(Z)-4-(2-Hy­droxy­benzyl­idene)-1-methyl-2-phenyl-1H-imidazol-5(4H)-one

In the title compound, C17H14N2O2, the asymmetric unit comprises two mol­ecules that are comformationally similar [the dihedral angles between the phenyl rings in each are 46.35 (2) and 48.04 (3)°], with the conformation stabilized by intra­molecular O—H⋯N hydrogen bonds, which generate S(7) rings. In the crystal, inversion-related mol­ecules are linked by pairs of weak C—H⋯O hydrogen bonds, forming dimers with an R 2 2(16) graph-set motif. Weak inter-ring π–π stacking is observed in the structure, the shortest centroid-to-centroid distance being 3.7480 (13) Å

(E)-4-[(4-Diethyl­amino-2-hy­droxy­benzyl­idene)amino]­benzonitrile

The title compound, C18H19N3O, displays an E conformation with respect to the C=N double bond. The dihedral angle between the mean planes of the two benzene rings is 24.49 (3)°. An intra­molecular O—H⋯N hydrogen bond generates an S(6) ring. In the crystal, mol­ecules are linked by nonclassical inter­molecular C—H⋯O hydrogen bonds to form an infinite one-dimensional chain along [010], generating a C(8) motif

Ultrafast all-optical switching via coherent modulation of metamaterial absorption

We report on the demonstration of a femtosecond all-optical modulator providing, without nonlinearity and therefore at arbitrarily low intensity, ultrafast light-by-light control. The device engages the coherent interaction of optical waves on a metamaterial nanostructure only 30 nm thick to efficiently control absorption of near-infrared (750-1040 nm) femtosecond pulses, providing switching contrast ratios approaching 3:1 with a modulation bandwidth in excess of 2 THz. The functional paradigm illustrated here opens the path to a family of novel meta-devices for ultra-fast optical data processing in coherent networks.Comment: 5 pages, 4 figure

Tinzaparin Provides Lower Lipid Profiles in Maintenance Hemodialysis Patients: A Cross-Sectional Observational Study

As a low-molecular-weight heparin, tinzaparin has effectively been used as an anticoagulant during hemodialysis sessions. However, the impact of different heparin types on dyslipidemia is still controversial. In our study, 434 chronic hemodialysis patients were evaluated. The mean age was 65 ± 13. Forty-eight patients (11%) and 386 patients (89%) were in the tinzaparin and unfractionated heparin (UFH) groups, respectively. Triglyceride had significant difference between the two groups (P=0.001) but total cholesterol, HDL, or LDL did not. In the univariate analysis, the triglyceride level was significantly associated with tinzaparin use [β: −39.9, 95% confidence interval (CI): −76.7 to −3.0], and this association remained following the multivariate analysis (β: −40.8, 95% CI: −75.1 to −6.5). The difference in serum total cholesterol level between tinzaparin and UFH became significant (β: −13, 95% CI: −24.5 to −1.56) after adjustment in the multivariate analysis. Moreover, in a subgroup analysis, male diabetic patients showed lower serum triglyceride levels with the use of tinzaparin, while older, nondiabetic, male patients showed significant advantages in total cholesterol levels with the use of tinzaparin. Based on our findings, tinzaparin shows a significant association with a lower lipid profile in patients with chronic hemodialysis when compared to UFH

Clinical Study Tinzaparin Provides Lower Lipid Profiles in Maintenance Hemodialysis Patients: A Cross-Sectional Observational Study

As a low-molecular-weight heparin, tinzaparin has effectively been used as an anticoagulant during hemodialysis sessions. However, the impact of different heparin types on dyslipidemia is still controversial. In our study, 434 chronic hemodialysis patients were evaluated. The mean age was 65 ± 13. Forty-eight patients (11%) and 386 patients (89%) were in the tinzaparin and unfractionated heparin (UFH) groups, respectively. Triglyceride had significant difference between the two groups ( = 0.001) but total cholesterol, HDL, or LDL did not. In the univariate analysis, the triglyceride level was significantly associated with tinzaparin use [ : −39.9, 95% confidence interval (CI): −76.7 to −3.0], and this association remained following the multivariate analysis ( : −40.8, 95% CI: −75.1 to −6.5). The difference in serum total cholesterol level between tinzaparin and UFH became significant ( : −13, 95% CI: −24.5 to −1.56) after adjustment in the multivariate analysis. Moreover, in a subgroup analysis, male diabetic patients showed lower serum triglyceride levels with the use of tinzaparin, while older, nondiabetic, male patients showed significant advantages in total cholesterol levels with the use of tinzaparin. Based on our findings, tinzaparin shows a significant association with a lower lipid profile in patients with chronic hemodialysis when compared to UFH