On the regularity of the roots of hyperbolic polynomials

We prove that a hyperbolic monic polynomial whose coefficients are functions of class $C^r$ of a parameter $t$ admits roots of class $C^1$ in $t$, if $r$ is the maximal multiplicity of the roots as $t$ varies. Moreover, if the coefficients are functions of $t$ of class $C^{2r}$, then the roots may be chosen two times differentiable at every point in $t$. This improves, among others, previous results of Bron\v ste\u \i n, Mandai, Wakabayashi and Kriegl, Losik and Michor

A mass spectrometric study on tannin degradation within dyed woolen yarns

Natural tannins from various plants have been used throughout human history in textile dyeing, often as mordant dyes. The ageing behavior of these dyes is a challenge in conservation science, requiring a thorough knowledge of the textile–mordant-dye system. In this work, we analyzed reference wool yarns dyed with natural tannins from oak gallnuts, walnut (Juglans regia), and catechu (Acacia catechu), after artificial ageing. To gain insights on the composition of the dyestuffs and on how they aged, an analytical procedure based on extraction with Na2EDTA/DMF (ethylenediaminetetraacetic acid/dimethylformamide) and high-performance liquid chromatography (HPLC) analysis using high-resolution mass spectrometry detection was used. Since conventional reversed-phase (RP) columns usually show poor retention efficiency of highly polar compounds such as tannins, an RP-amide embedded polar group stationary phase was used to achieve optimal retention of the most polar compounds. Tannins from oak gallnuts showed little degradation after ageing, while a significant increase in the content of hydroxybenzoic acids was observed for tannins from walnut and catechu. Finally, the analytical procedure was applied to characterize the tannin dyes in historical tapestries from the 15th to 16th century, and the results were discussed in comparison with the reference yarns

A novel microwave and induction heating applicator for metal making: Design and testing

The use of microwave heating in primary metallurgy is gaining an increasing interest due to the possibility to selectively process ores and to volumetrically heat large amounts of low-thermal conductivity minerals. In this paper the study, development and testing of a new applicator combining the use of microwave and induction heating for rapid reduction of metal containing oxides is described. Numerical simulation was used in order to achieve the proper control over heat generation, considering the use of microwave solid state generators. A prototype, with a capacity up to 5 liters of standard input feed but with the predisposition for continuous processing has been designed, built and tested on reference loads like iron oxide powders and pellets. Results on the microwave heating part of the applicator indicate that it allows to efficiently and rapidly process these kinds of loads, which change from dielectric to conductors as reduction proceeds. The use of variable frequency solid state microwave generators allows to maximize energy efficiency and to controllably change the heating pattern inside the load

The influence of oscillations on energy estimates for damped wave models with time-dependent propagation speed and dissipation

The aim of this paper is to derive higher order energy estimates for solutions to the Cauchy problem for damped wave models with time-dependent propagation speed and dissipation. The model of interest is \begin{equation*} u_{tt}-\lambda^2(t)\omega^2(t)\Delta u +\rho(t)\omega(t)u_t=0, \quad u(0,x)=u_0(x), \,\, u_t(0,x)=u_1(x). \end{equation*} The coefficients $\lambda=\lambda(t)$ and $\rho=\rho(t)$ are shape functions and $\omega=\omega(t)$ is an oscillating function. If $\omega(t)\equiv1$ and $\rho(t)u_t$ is an "effective" dissipation term, then $L^2-L^2$ energy estimates are proved in [2]. In contrast, the main goal of the present paper is to generalize the previous results to coefficients including an oscillating function in the time-dependent coefficients. We will explain how the interplay between the shape functions and oscillating behavior of the coefficient will influence energy estimates.Comment: 37 pages, 2 figure