In both physical and social sciences, we usually use controlled differential equation to model various continuous evolving system; describing how a response y relates to another process x called control. For regular controls x, the unique existence of the response y is guaranteed while it would never be the case for non-smooth controls via the classical approach. Besides, uniform closeness of controls may not imply closeness of their corresponding responses. Theory of rough paths provides a solution to both concerns. Since the creation of rough path theory, it enjoys a fruitful development and finds wide applications in stochastic analysis. In particular, rough path theory provides an effective method to study irregularity of curves and its geometric consequences in relation to integration of differential forms. In the chapter 2, we demonstrate the power of rough path theory in classical complex analysis by showing the rough path nature of the boundaries of a class of Holder's domains; as an immediate application, we extend the classical Gauss-Green's theorem. Until recently, there has been only limited research on applications of theory of rough paths to high dimensional geometry. It is clear to us that many geometric objects, in some senses appearing as solids, are actually comprised of filaments. In the chapter 3, two basic results in the theory of rough paths which will motivate later development of my thesis has been included. In the chapters 4 and 5, we identify a sensible way to do geometric calculus via those filaments (more precisely, space-filling rough paths) in dimension 3. In a recent joint work of Hambly and Lyons, they have shown that every rectifiable path can be completely characterized, up to tree-like deformation, by an algebraic object called the signature, tensor of all iterated integrals, of the path. It is clear that all tree-like deformation of the path would not change its topological features. For instance, the number of times a planar loop of finite length winds around a point (not lying on the path) is unaltered if one deforms the path in tree-like ways. Therefore, it should be plausible to extract this topological information out from the signature of the loop since the signature is a complete algebraic invariant. In the chapter 6, we express the winding number of a nice loop (respectively linking number of a pair of nice loops) as a linear functional of the signature of the loop (respectively signatures of the pair of loops).

published_or_final_version / abstract / toc / Electrical and Electronic Engineering / Master / Master of Philosophy

published_or_final_version / abstract / toc / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy

published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy

published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy

published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy

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The Picture Exchange Communication System (PECS) is an augmentative and alternative communication program (Frost & Bondy, 2002). Although PECS has been effectively used to teach functional requesting skills for children with autism, mental retardation, visual impairment, and physical disabilities (e.g., Anderson, Moore, & Bourne, 2007; Chambers & Rehfeldt, 2003), there are limited studies examining the effectiveness of PECS to teach requesting skills for children with multiple disabilities including visual impairment (Lund & Troha, 2008).This study explored the effectiveness of combining tangible symbols and other adaptations with PECS to teach requesting skills to students with multiple disabilities. Specifically, the participants were four students with multiple disabilities including visual impairment who also had many challenges in communication skills. The research design was the multiple probe design across subjects, a variation of a multiple baseline design. A notable benefit to this design is that there was no need to collect continuous recordings of baseline measures, because a strong a priori assumption of stability and the possibility of causing strong participant reactions existed before introducing the intervention. Instead, the researcher made periodic recordings of baseline levels to insure no significant changes have occurred before introducing the intervention.The study included four parts: (a) the assessment of reinforcers, (b) baseline, (c) the implementation of the intervention, which was teaching the three PECS phases and conducting generalization sessions, and (d) the maintenance condition. Three important research questions were posed:1. Can students with multiple disabilities including visual impairment learn to make requests for preferred items using adapted PECS materials and procedures?2. Can students with multiple disabilities including visual impairment generalize requesting skills for preferred items using adapted PECS from training rooms to classrooms?3. Can students with multiple disabilities including visual impairment maintain requesting skills after training?The results indicated that all four participants learned requesting skills using adapted PECS, generalized the newly acquired skills to their classrooms, and maintained the requesting skills after training. Results of this study provided preliminary evidence that PECS with adaptations could be used effectively to teach requesting skills for students with multiple disabilities including visual impairment.