Is this project an undergraduate, graduate, or faculty project?
Undergraduate
Project Type
group
Campus
Daytona Beach
Authors' Class Standing
Ibrahim Arnous, Junior
Lead Presenter's Name
Ibrahim Arnous
Lead Presenter's College
DB College of Arts and Sciences
Faculty Mentor Name
Dr. Keshav Acharya
Abstract
Classical limits describe asymptotic behavior through convergence to a single value, but many important oscillatory functions do not converge in this sense. Standard examples such as sin(π₯) as π₯ββ and sin(1/x) as xβ0 instead display stable distributions of values over time. This project examines how such behavior can be described using a measure-theoretic framework, particularly through occupation measures and, in sequence-based settings, Young measures. The objective is to present this perspective in a clear and accessible way while introducing the Ansatz representation, a compact notation for recording the support and density of an asymptotic value distribution when the limiting measure is absolutely continuous. The project does not propose a new foundational definition of limit; rather, it develops a concise descriptor for a class of oscillatory, nonconvergent behaviors already recognized in the literature. Preliminary results clarify both the usefulness and the limits of this representation by showing that it captures one-point asymptotic value statistics, while a fuller reconstruction of temporal behavior would require additional structural information. More broadly, this work highlights how measure-theoretic tools can extend the language of asymptotic analysis beyond classical pointwise convergence.
Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?
No
Included in
Analysis Commons, Dynamical Systems Commons, Probability Commons
A Compact Representation of Oscillatory Limits via Asymptotic Value Distributions
Classical limits describe asymptotic behavior through convergence to a single value, but many important oscillatory functions do not converge in this sense. Standard examples such as sin(π₯) as π₯ββ and sin(1/x) as xβ0 instead display stable distributions of values over time. This project examines how such behavior can be described using a measure-theoretic framework, particularly through occupation measures and, in sequence-based settings, Young measures. The objective is to present this perspective in a clear and accessible way while introducing the Ansatz representation, a compact notation for recording the support and density of an asymptotic value distribution when the limiting measure is absolutely continuous. The project does not propose a new foundational definition of limit; rather, it develops a concise descriptor for a class of oscillatory, nonconvergent behaviors already recognized in the literature. Preliminary results clarify both the usefulness and the limits of this representation by showing that it captures one-point asymptotic value statistics, while a fuller reconstruction of temporal behavior would require additional structural information. More broadly, this work highlights how measure-theoretic tools can extend the language of asymptotic analysis beyond classical pointwise convergence.