Measures are versatile objects which can represent how populations or supplies are distributed within a given space by assigning sizes to subregions (or subsets) of that space. To model how populations or supplies are shifted from one configuration to another, it is natural to use functions between measures, called transfunctions. Any measurable function can be identified with its push-forward transfunction. Other transfunctions exist such as convolution operators. In this manner, transfunctions are treated as generalized functions. This dissertation serves to build the theory of transfunctions and their connections to other mathematical fields. Transfunctions that identify with continuous or measurable push-forward operators are characterized, and transfunctions that map between measures concentrated in small balls -- called localized transfunctions -- can be spatially approximated with measurable functions or with continuous functions (depending on the setting). Some localized transfunctions have "fat graphs" in the product space and "fat continuous graphs" are necessarily formed by localized transfunctions. Any Markov transfunction -- a transfunction that is linear, variation-continuous, total-measure-preserving and positive -- corresponds to a family of Markov operators and a family of plans (indexed by their marginals) such that all objects have the same "instructions" of transportation between input and output marginals. An example of a Markov transfunction is a push-forward transfunction. In two settings (continuous and measurable), the definition and existence of adjoints of linear transfunctions are formed and simple transfunctions are implemented to approximate linear weakly-continuous transfunctions in the weak sense. Simple Markov transfunctions can be used both to approximate the optimal cost between two marginals with respect to a cost function and to approximate Markov transfunctions in the weak sense. These results suggest implementing future research to find more applications of transfunctions to optimal transport theory. Transfunction theory may have potential applications in mathematical biology. Several models are proposed for future research with an emphasis on local spatial factors that affect survivorship, reproducibility and other features. One model of tree population dynamics (without local factors) is presented with basic analysis. Some future directions include the use of multiple numerical implementations through software programs.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Bentley, Jason, "Transfunctions and Other Topics in Measure Theory" (2020). Electronic Theses and Dissertations, 2020-. 18.