Construction of functions with prescribed multiscale asymmetry. Asymmetric Weierstrass function

Abstract

components, and the interactions between those components within an organizational structure that exists in dynamic equilibrium. Furthermore, the organization of biologic systems is identified as necessarily hierarchical and recursive, stressing the importance of addressing both context and concurrency in any description. The resulting category-theoretical description is general enough to abstractly describe any biologic system: the general category of biology can be expressed as an abstracted biologic computational unit (ABCU) [1,2]. Mathematical descriptions of biologic systems can be constructed from instances of ABCUs and the processes that link them. Inversely, incomplete descriptions of aspects of biologic systems (such as biochemical pathways) do not fulfill the associative requirement of a category and are, therefore, not considered as members of the ABCU set. This “completeness” requirement places a constraint on the internal structure of a category-theoretical description of a biologic system and can be seen as analogous to the fact that biologic systems must follow basic physical and chemical laws. Conclusions: Biology is more than an aggregate of chemistry and physics. Although it incorporates and uses the laws of those domains, the richness of biology cannot be effectively characterized using merely the descriptive capacity of physics or chemistry. To develop the mathematics of biology, it is necessary to present a mathematical description of the fundamental properties that set biology apart from these other domains. Given the organizational complexity of biologic systems, this goal requires the use of more abstract and powerful mathematical tools that are used to delineate the laws of physics or chemistry. Category theory is particularly well suited to this goal, and it is further suggested that being able to produce a category-theoretical description may be both necessary and sufficient to define a complete biologic system, as distinguished from either a biophysical or biochemical one. Furthermore, because category theory is also being applied to various aspects of computer science, programming languages, and logic, it may provide a means of formalizing the transition from a biologic description to a computable one.