On the zeros of a polynomial and its derivatives

If is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of to a nearest zero of ? We obtain bounds for this distance depending on degree. We also show that this distance is equal to for polynomials of degree 3 and polynomials with real zeros.

     

Approximation of the Hausdorff distance by the distance of continuous surjections

The aim of this paper is to answer the following question: let (X,?) and (Y,d) be metric spaces, let A,B⊂Y be continuous images of the space X and let be a fixed continuous surjection. When is the inequality

     

Mechanism of 4‐methyl‐1,2,4‐triazol‐3‐thiole reaction with formaldehyde. A DFT study

Contrary to the typical nucleophilic substitution, occurring on the sulfur atom of 4‐methyl‐1,2,4‐ triazol‐3‐thiole, the reaction with formaldehyde leads to the formation of the N? C bond rather than the S? C bond. The mechanism of this reaction has been characterized theoretically. Calculations indicate that the reaction proceeds via a cyclic transition state involving one solvent molecule with the Gibbs free activation energy of only 2 kcal/mol. The alternative pathway that leads to the S? C bond formation is about 5 kcal/mol more energetically demanding. Copyright © 2007 John Wiley & Sons, Ltd.     

Sub‐microsecond‐resolved multi‐speckle X‐ray photon correlation spectroscopy with a pixel array detector

Small‐angle X‐ray photon correlation spectroscopy (XPCS) measurements spanning delay times from 826 ns to 52.8 s were performed using a photon‐counting pixel array detector with a dynamic range of 0–3 (2 bits). Fine resolution and a wide dynamic range of time scales was achieved by combining two modes of operation of the detector: (i) continuous mode, where data acquisition and data readout are performed in parallel with a frame acquisition time of 19.36 µs, and (ii) burst mode, where 12 frames are acquired with frame integration times of either 2.56 µs frame^?1 or 826 ns frame^?1 followed by 3.49 ms or 1.16 ms, respectively, for readout. The applicability of the detector for performing multi‐speckle XPCS was demonstrated by measuring the Brownian dynamics of 10 nm‐radius gold and 57 nm‐radius silica colloids in water at room temperature. In addition, the capability of the detector to faithfully record one‐ and two‐photon counts was examined by comparing the statistical distribution of photon counts with expected probabilities from the negative binomial distribution. It was found that in burst mode the ratio of 2 s to 1 s is markedly smaller than predicted and that this is attributable to pixel‐response dead‐time.     

Calculation method of spin accumulations and spin signals in nanostructures using spin resistors

Determination of spin accumulations and spin currents is essential for a deep understanding of spin transport in nanostructures and further optimization of spintronic devices. So far, they are easily obtained using different approaches in nanostructures composed of few elements; however their calculation becomes complicated as the number of elements increases. Here, we propose a 1-D spin resistor approach to calculate analytically spin accumulations, spin currents and magneto-resistances in heterostructures. Our method, particularly applied to multi-terminal metallic nanostructures, provides a fast and systematic mean to determine such spin properties in structures where conventional methods remain complex.     

Biodegradable polycarbonates containing side carboxyl groups—synthesis,properties, and degradation study

The main objective of the presented research was to synthesise biodegradable aliphatic polycarbonates containing reactive carboxyl pendant groups and to examine the influence of the copolymer chain microstructure and composition on the process of their hydrolytic degradation and cytocompatibility. The work describes copolymerization of cyclic trimethylene carbonate derivative containing benzyl‐ester pendant group (benzyl 5‐methyl‐2‐oxo‐1,3‐dioxane‐5‐carboxylate) with trimethylene carbonate. The copolymerization was conducted with the use of zinc (II) and lanthanum (III) acetylacetonates as ring‐opening polymerization coordination initiators. Detailed NMR analysis allowed to define the microstructure of the obtained copolymers, which depended on the composition and type of used initiator. The final tapered chain microstructure of the obtained copolymers was related to huge differences in comonomers reactivity and evidenced low level of transesterification of the main copolymer backbone. Chosen copolymers, with unprotected carbonyl groups, were subjected to in vitro degradation test and cytocompatibility studies. It was found that high concentration of carboxyl groups resulted in copolymers which formed hydrogels and were very prone to hydrolytic degradation; they were also cytotoxic toward osteoblast‐like MG 63 cells. Copolymers with lower content of carboxyl groups were found less susceptible to degradation and cytocompatible with studied cells. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2017 , 55, 2756–2769