Backward stochastic differential equations with unbounded generators

In this paper we consider two classes of backward stochastic differential equations. Firstly, under a Lipschitz-type condition on the generator of the equation, which can also be unbounded, we give sufficient conditions for the existence of a unique solution pair. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Secondly, under the linear growth and continuity assumptions on the possibly unbounded generator, we prove the existence of the solution pair. This class of equations is more general than the existing ones

Characterization and Performance Improvement of Chitosan Films as Affected by Preparation Method, Synthetic Polymers, and Blend Ratios

Chitosan films prepared with addition of other polymers have been widely studied for their modified properties. In this dissertation, poly (ethylene oxide) (PEO) and poly (N-vinyl-2- pyrrolidone) (PVP) were blended with chitosan. The objectives of the study were (1) to investigate the the effects of film thickness, blend ratios, and preparation methods on the physical, and mechanical properties and functional performance of chitosan/PEO films, and (2) to compare characteristics and functional properties of chitosan/PVP and chitosan/PEO films. The results demonstrated that regular cast chitosan/PEO films have altered properties than films produced from either polymer alone. Regardless of molecular weight, chitosan decreased the tendency to spherulitic crystallization of PEO. Production of ultra-thin chitosan and chitosan/PEO films with thickness below 80 nm was possible by spin-coating on silicon wafers. The increase of PEO content did not affect thickness of the films but the surface of corresponding films became rougher probably due to formation of PEO crystallites. Comparing the functional properties of thick, thin and ultra-thin chitosan/PEO films, the latter showed a significantly higher chromium binding capacity compared to the regular cast films. However, ultra-thin chitosan/PEO films did not show significant antibacterial properties due to their extremely low weight. A decreased film-forming time, especially in the spin-coating method, greatly reduced extent of film crystallization. Incorporation of PVP or PEO into chitosan films reduced the yellowish color and made films easier to puncture and tear. Although chitosan/PEO blend films showed lower water vapor permeability (WVP) values than chitosan/PVP films, blending chitosan with hydrophilic polymers was not an effective way to significantly improve the WVP. Replacing even 50% of chitosan with PVP or PEO in chitosan films did not significantly decrease the metal-binding and antibacterial properties of the films. Since synthetic polymers are less expensive than biopolymer chitosan, blending chitosan and synthetic polymers could reduce the amount of chitosan and lower the production cost with no effect on functionality of the films. Chitosan/PVP and chitosan/PEO blend films have the potential to be used in the food industry as active packaging materials to inhibit food borne pathogens and as absorbent to bind heavy metal from the environment

Bootstrapping Witten diagrams via differential representation in Mellin space

We explore the use of the differential representation of AdS amplitudes to compute Witten diagrams. The differential representation expresses AdS amplitudes in terms of conformal generators acting on contact Witten diagrams, which allows us to construct differential equations for Witten diagrams. These differential equations can then be transformed into difference equations in Mellin space, which can be solved recursively. Using this method, we efficiently re-computed scalar four-point amplitudes and obtained new results for scalar six-point amplitudes mediated by gluons and scalars, as well as two examples of scalar eight-point amplitudes from gluon exchange