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Normal forms of linear second order partial differential equations on the plane

Science China Mathematics 2018年第11期61卷 1947-1962页

作者： alexey davydov Department of Mathematics The National University of Science and Technology MISiS Moscow 119049 Russia Department of Theory of Dynamical Systems Lomonosov Moscow State University Moscow 119991 Russia International Institute for Applied Systems Analysis Laxenburg A-2361 Austria

The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the *** discuss the results obtained in the last decades and some problems,which are important both for the development of this theory and the *** reduction theorem,which was used to obtain many of recent results in the theory,is included in the paper in the parametric form together with *** is a feeling that the theorem still has potential to get progress in the solution of open problems in the theory.

关键词： normal form mixed type partial differential equation main symbol

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Experimental and ab initio study of Ba_(2)Na_(3)(B_(3)O_(6))_(2)F stability in the pressure range of 0–10 GPa

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International Journal of Minerals,Metallurgy and Materials 2023年第9期30卷 1846-1854页

作者： Nursultan E.Sagatov Tatyana B.Bekker Yulia G.Vinogradova alexey V.davydov Ivan V.Podborodnikov Konstantin D.Litasov Sobolev Institute of Geology and Mineralogy Siberian Branch of Russian Academy of SciencesNovosibirsk 630090Russia Department of Geology and Geophysics Novosibirsk State UniversityNovosibirsk 630090Russia Research and Development Department Novosibirsk State University of ArchitectureDesign and Arts(NSUADA)Novosibirsk 630099Russia Vereshchagin Institute for High Pressure Physics Russian Academy of SciencesTroitsk 108840Russia Fersman Mineralogical Museum Russian Academy of SciencesMoscow 119071Russia

Both numerical and experimental studies of the stability and electronic properties of barium–sodium metaborate Ba_(2)Na_(3)(B_(3)O_(6))_(2)F(P63/m) at pressures up to 10 GPa have been carried out. Electronic-structure calculations with HSE06 hybrid functional showed that Ba_(2)Na_(3)(B_(3)O_(6))_(2)F has an indirect band gap of 6.289 eV. A numerical study revealed the decomposition of Ba_(2)Na_(3)(B_(3)O_(6))_(2)F into the BaB_(2)O_(4),Na BO_(2), and NaF phases above 3.4 GPa at 300 K. Subsequent high-pressure high-temperature experiments performed using ‘Discoverer-1500’DIA-type apparatus at pressures of 3 and 6 GPa and temperature of 1173 K confirmed the stability of Ba_(2)Na_(3)(B_(3)O_(6))_(2)F at 3 GPa and its decomposition into BaB_(2)O_(4), NaBO_(2), and NaF at 6 GPa, which was verified by energy-dispersive X-ray analysis and Raman spectroscopy. The observed Raman bands of the Ba_(2)Na_(3)(B_(3)O_(6))_(2)F phase were assigned by comparing the experimental and calculated spectra. The experimental Raman spectra of decomposition reaction products obtained at 6 GPa suggest the origin of a new high-pressure modification of barium metaborate BaB_(2)O_(4).

关键词： density functional theory phase stability borate high pressure Raman spectroscopy

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