Well-posedness of the cauchy problem in gevrey classes for some weakly hyperbolic equations of higher order

This article is devoted to the study of the Cauchy problem in Gevrey classes for some higher order weakly hyperbolic equations with time-dependent coefficients and without lower order terms.     

Novel application of liquid chromatography/mass spectrometry for the characterization of drying oils in art: Elucidation on the composition of original paint materials used by Edvard Munch (1863–1944)

Modern oil paints, introduced at the beginning of the 20th century, differ from those classically used in antiquity in their chemical and compositional features. The main ingredients were still traditional drying oils, often used in mixtures with less expensive oils and added with several classes of additives. Consequently, detailed lipid profiling, together with the study of lipid degradation processes, is essential for the knowledge and the conservation of paint materials used in modern and contemporary art.     

Oxalate films and red stains on Carrara marble

The analytical studies carried out during two different diagnostic surveys, respectively in 1983 and 2003, offered the opportunity to control decay phenomena development on stones facing Certosa of Pavia (Italy). Calcium oxalate films and red stains, present on Carrara marble surface, have been particularly focused; these are the only decay phenomena which apparently have remained unchanged during a period of twenty years. More sensitive and in-depth analytical studies (FTIR equipped with diamond cell, GC-MS, SEM-EDS and optical microscopy) achieved a better knowledge about their composition. Results allowed a critical evaluation of the role of oxalate films on the external marble surface and to suggest new hypotheses about the formation of red stains.     

The Cauchy problem for weakly hyperbolic systems

We consider the well-posedness of the Cauchy problem in Gevrey spaces for N×N first-order weakly hyperbolic systems. The question is to know whether the general results of Bron?tein [1 Bron?tein, M.D. (1982). The Cauchy Problem for hyperbolic operators with characteristic of variable multiplicity. Trudy Moskov. Mat. Obshch. 41:83–99. [Translation: Trans. Moscow. Math. Soc. 41:87–103]. [Google Scholar]] and Kajitani [9 Kajitani, K. (1986). The Cauchy Problem for Uniformly Diagonalizable Hyperbolic Systems in Gevrey Classes, in Hyperbolic Equations and Related Topics. Proceedings of the Taniguchi International Symposium, Katata and Kyoto, 1984. Boston: Academic Press, pp. 101–123. [Google Scholar]] can be improved when the coefficients depend only on time and are smooth, as it has been done for the scalar wave equation in [3 Colombini, F., Jannelli, E., Spagnolo, S. (1983). Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10:291–312. [Google Scholar]]. The answer is no for general systems, and yes when the system is uniformly diagonalizable: in this case, we show that the Cauchy problem is well posed in all Gevrey classes G^s when the coefficients are C^∞. Moreover, for 2×2 systems and some other special cases, we prove that the Cauchy problem is well posed in G^s for s<1+k when the coefficients are C^k, which is sharp following the counterexamples of Tarama [12 Tarama, S. (1994). Une note sur les Systèmes Hyperboliques Uniformément Diagonalisables. Mem. Fac. Eng. Kyoto Univ. 56:9–18. [Google Scholar]]. The main new ingredient is the construction, for all hyperbolic matrix A, of a family of approximate symmetrizers, S_𝜀, the coefficients of which are polynomials of 𝜀 and the coefficients of A and A*.     

On the uniqueness in gevrey spaces for degenerate elliptic operators

The uniqueness and the non–uniqueness of the Gevrey ultradifferentiable solutions to the Cauchy problem for a class of second order degenerate elliptic operators is studied. Some uniqueness results are proved and the necessity of the hypotheses is discussed by the construction of some counter-examples.     

Analysis of lignin from archaeological waterlogged wood by direct exposure mass spectrometry (DE-MS) and PCA evaluation of mass spectral data

The chemical characterisation of waterlogged archaeological wood is of fundamental importance to understand the degradation processes undergone by wooden objects and consequently to develop suitable consolidation and conservation procedures. Lignin extracted from archaeological waterlogged wood samples was characterized using direct exposure electron ionisation mass spectrometry (DE-MS). DE-MS achieves a mass spectral fingerprint of the sample in a few minutes, avoiding any chemical pre-treatment and requiring only few micrograms of material.Mass spectral data were put in relation to the chemical composition of lignin and evaluated by means of principal component analysis (PCA). The preliminary results, presented in this study, demonstrate the feasibility and the potential of DE-MS as a reproducible and rapid screening method for archaeological waterlogged wood samples.     

Py-GC/MS, GC/MS and FTIR investigations on LATE Roman-Egyptian adhesives from opus sectile: New insights into ancient recipes and technologies

An analytical protocol based on optical microscopy, Fourier transforms infrared spectroscopy (FTIR), analytical pyrolysis in the presence of hexamethyldisilazane followed by gas chromatographic/mass spectrometric analysis (Py-GC/MS) and gas chromatography/mass spectrometry after alkaline hydrolysis, solvent extraction and trimethylsilylation (GC/MS) was used in the chemical characterisation of the original adhesives used to fix monochrome and mosaic glass and stone plaques coming from the Late Roman archaeological site of Antinoopolis (Egypt).FTIR analysis demonstrated the presence of calcite fragments, and Py-GC/MS and GC/MS analyses provided detailed molecular compositions, highlighting the presence of a wide range of compound classes including diterpenoid acids, tricyclic abietanes with a high degree of aromatisation, mid- and long-chain monocarboxylic fatty acids, mono- and di-hydroxy acids, α,ω-dicaboxylic fatty acids, n-alkanols, and n-alkanes. Characteristic biomarkers and their distribution patterns indicated the presence of pine pitch in all the adhesives, which in some cases was admixed with beeswax and brassicaceae seed oil.The results provided new insights into the complex recipes used by artisans in ancient Egypt in the production of adhesives and in the sophisticated manufacture of opus sectile decorations.     

The development of a gas chromatographic–mass spectrometric analytical procedure for the determination of lipids,proteins and resins in the same paint micro-sample avoiding interferences from inorganic media

This paper presents a GC–MS analytical procedure for determining proteinaceous materials, glycerolipids, natural waxes and terpenoid resins in the same paint micro-sample. The procedure is also reliable when high amounts of interfering inorganic pigments, dryers and charges are present. The characterisation of proteinaceous binders in a paint sample can be subject to analytical interferences by inorganic materials. Such materials may form complexes with functional groups of proteins, thus preventing their efficient derivatisation, which is necessary prior to GC analysis. For this reason an analytical procedure has been developed based on two extractions and a clean-up step, in order to obtain two fractions: a lipid-resinous fraction and a proteinaceous fraction. The lipid-resinous fraction is subjected to salification/saponification assisted by microwaves, followed by acidification, extraction, derivatisation and GC–MS analysis. The proteinaceous fraction is analysed by GC–MS after hydrolysis and derivatisation of the freed amino acids. The desalting step is applied before the hydrolysis, and is based on the use of the monolithic sorbent tip technology with a C4 stationary phase. Reference paint replicas of egg, casein and animal glue were prepared with and without several metals containing pigments, and used to develop and validate the analytical procedure. The procedure proved to be efficient in desalting the proteinaceous materials both from cations and anions. Although non quantitative, it is reliable in the analysis of samples whose content of extractable proteins is <1 μg, thus showing it to be suitable for the characterisation of paint samples. An example of how the analytical procedure was used to characterise a sample from a 15th century panel painting is also discussed.     

Exponential growth for the wave equation with compact time‐periodic positive potential

We prove the existence of smooth positive potentials V(t, x), periodic in time and with compact support in x, for which the Cauchy problem for the wave equation u_tt ? Δ_xu + V(t, x)u = 0 has solutions with exponentially growing global and local energy. Moreover, we show that there are resonances, z ∈ ?, |z| > 1, associated to V(t, x). © 2008 Wiley Periodicals, Inc.