Mutually catalytic branching at infinite rate

The purpose of this doctoral thesis is to prove existence for a mutually catalytic random walk with infinite branching rate on countably many sites. The process is defined as a weak limit of an approximating family of processes. An approximating process is constructed by adding jumps to a deterministic migration on an equidistant time grid. As law of jumps we need to choose the invariant probability measure of the mutually catalytic random walk with a finite branching rate in the recurrent regime. This model was introduced by Dawson and Perkins (1998) and this thesis relies heavily on their work. Due to the properties of this invariant distribution, which is in fact the exit distribution of planar Brownian motion from the first quadrant, it is possible to establish a martingale problem for the weak limit of any convergent sequence of approximating processes. We can prove a duality relation for the solution to the mentioned martingale problem, which goes back to Mytnik (1996) in the case of finite rate branching, and this duality gives rise to weak uniqueness for the solution to the martingale problem. Using standard arguments we can show that this solution is in fact a Feller process and it has the strong Markov property. For the case of only one site we prove that the model we have constructed is the limit of finite rate mutually catalytic branching processes as the branching rate approaches infinity. Therefore, it seems naturalto refer to the above model as an infinite rate branching process. However, a result for convergence on infinitely many sites remains open.

Designer synthesis of monodisperse heterodimer and ferrite nanoparticles

The work presented in this doctoral thesis is a facile procedure, thermal decomposition, forrnthe synthesis of different types of monodisperse heterodimer M@iron oxide (M= Cu, Co, Nirnand Pt) and single ferrites, MFe2O4 (M= Cu and Co), nanoparticles. In the following chapter,rnwe study the synthesis of these monodiperse nanoparticles with the similar iron precursorrn(iron pentacarbonyl) and different transition metal precursors such as metalrnacetate/acetylacetonate/formate precursors in the presence of various surfactants and solvents.rnAccording to their decomposition temperatures and reducing condition, a specific and suitablernroute was designed for the formation of Metal@Metal oxide or MFe2O4 nanoparticlesrn(Metal/M=transition metal).rnOne of the key purposes in the formation of nanocrystals is the development of syntheticrnpathways for designing and controlling the composition, shape and size of predictedrnnanostructures. The ability to arrange different nanosized domains of metallic and magneticrnmaterials into a single heterodimer nanostructure offers an interesting direction to engineerrnthem with multiple functionalities or enhanced properties of one domain. The presence andrnrole of surfactants and solvents in these reactions result in a variety of nanocrystal shapes. Therncrystalline phase, the growth rate and the orientation of growth parameters along certainrndirections of these structures can be chemically modulated by using suitable surfactants. In allrnnovel reported heterodimer nanostructures in this thesis, initially metals were preformed andrnthen by the injection of iron precursor in appropriate temperature, iron oxide nanoparticlesrnwere started to nucleate on the top or over the surfaces of metal nanoparticles. Ternary phasesrnof spherical CuxFe3-xO4 and CoFe2O4 ferrites nanoparticles were designed to synthesis just byrnlittle difference in diffusion step with the formation of mentioned phase separated heterodimerrnnanoparticles. In order to use these magnetic nanoparticles in biomedical and catalysisrnapplications, they should be transferred into the water phase solution, therefore they werernfunctionalized by a multifunctional polymeric ligand. These functionalized nanoparticles werernstable against aggregation and precipitation in aqueous media for a long time. Magneticrnresonance imaging and catalytic reactivities are two promising applications which have beenrnutilized for these magnetic nanoparticles in this thesis.rnThis synthetic method explained in the following chapters can be extended to the synthesis ofrnother heterostructured nanomaterials such as Ni@MnO or M@M@iron oxide (M=transitionrnmetal) or to use these multidomain particles as building blocks for higher order structures.