Law of large numbers for branching symmetric Hunt processes with measure-valued branching rates

We establish weak and strong law of large numbers for a class of branching symmetric Hunt processes with the branching rate being a smooth measure with respect to the underlying Hunt process, and the branching mechanism being general and state-dependent. Our work is motivated by recent work on strong law of large numbers for branching symmetric Markov processes by Chen-Shiozawa [J. Funct. Anal., 250, 374--399, 2007] and for branching diffusions by Engl\"ander-Harris-Kyprianou [Ann. Inst. Henri Poincar\'e Probab. Stat., 46, 279--298, 2010]. Our results can be applied to some interesting examples that are covered by neither of these papers

Preparation, Characterization and Performance Study of Modified Titanium Dioxide Nanocrystals for the Lithium-Ion Battery

Title from PDF of title page, viewed on August 11, 2015Dissertation advisor: Xiaobo ChenVitaIncludes bibliographic references (pages 156-172)Thesis (Ph.D.)--Department of Chemistry and Department of Geosciences. University of Missouri--Kansas City, 2015The lithium-ion battery is one of the most widely used rechargeable batteries in today’s life. As an energy storage device, it can convert the stored chemical energy into electrical energy when it is being used. Titanium dioxide nanocrystals are well-known for the photocatalytic ability. However, benefited from the nanostructure and the electrochemical reactivity of lithium-ions, titanium dioxide nanocrystals are also investigated as a promising anode material used for the lithium-ion battery. It is safer than graphite as it can prevent the lithium deposition and formation of the solid electrolyte interphase; additionally, it is an environmentally friendly and economical material that can also provide good theoretical capacity. These superiorities have attracted many research interests and make it a target material in this dissertation. However, the battery made by titanium dioxide suffers poor battery performances that are caused by two major drawbacks of the material. The low electronic conductivity in the solid phase and the low diffusion coefficient of lithium-ions cause only a thin surface layer of the titanium dioxide particle to be effectively used for the intercalation and extraction of lithium-ions at high charge/discharge rates; thus, the actual application is hindered. In order to improve the battery performances, three modification methods were discussed: hydrogenation, vacuum treatment, and carbon coating. The structural and electronic properties of the pure titanium dioxide nanocrystals and the new modified titanium dioxide nanocrystals were studied with: transmission electron microscopy, x-ray diffraction patterns, Raman spectroscopy, Fourier transform infrared spectroscopy, thermal gravimetric analysis, ¹H magic-angle spinning solid state nuclear magnetic resonance spectroscopy, electron spin resonance spectroscopy, and x-ray photoelectron spectroscopy. The coin cells used titanium dioxide nanocrystals as electrode materials and were tested and analyzed in terms of discharge capacity, Coulombic efficiency, and rate performance. The electrochemical impedance spectroscopy was also studied in order to understand the electrochemical system. Compared with the pure titanium dioxide nanocrystals, the hydrogenated titanium dioxide nanocrystals, the vacuum-treated titanium dioxide nanocrystals, and the carbon-coated titanium dioxide nanocrystals showed improved battery performances. The structure and battery performances of three different titanium dioxide nanocrystals were related and discussed systematically.Introduction -- Experimental and analytical methods -- Design, preparation and properties of the hydrogenated titanium dioxide nanocrystals -- Design, preparation and properties of the vacuum-treated titanium dioxide nanocrystals -- Design, preparation and properties of the carbon-coated titanium dioxide nanocrystals -- Conclusio