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Nonlocal Helmholtz Decompositions And Connections To Classical Counterparts, Andrew Haar, Petronela Radu
Nonlocal Helmholtz Decompositions And Connections To Classical Counterparts, Andrew Haar, Petronela Radu
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In recent years nonlocal models have been successfully introduced in a variety of applications, such as dynamic fracture, nonlocal diffusion, flocking, and image processing. Thus, the development of a nonlocal calculus theory, together with the study of nonlocal operators has become the focus of many theoretical investigations. Our work focuses on a Helmholtz decomposition in the nonlocal (integral) framework. In the classical (differential) setting the Helmholtz decomposition states that we can decompose a three dimensional vector field as a sum of an irrotational function and a solenoidal function. We will define new nonlocal gradient and curl operators that allow us …
Conjugacy Geodesics In Coxeter Groups, Aaron Calderon
Conjugacy Geodesics In Coxeter Groups, Aaron Calderon
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Take a square and flip it over the vertical axis, rotate it 90 degrees counterclockwise and then flip it again over the vertical axis. This sequence is the same as a 90 degree clockwise rotation but takes more steps to demonstrate the same symmetry. In general, the question of when a sequence of symmetries has minimal length is hard to answer and is dependent on the chosen generating set (in our toy example, rotation by 90 degrees and reflection). By realizing sequences of symmetries as paths in a group's Cayley graph, the problem becomes one about the set of shortest …
The History And Applications Of Fibonacci Numbers, Cashous W. Bortner, Allan C. Peterson
The History And Applications Of Fibonacci Numbers, Cashous W. Bortner, Allan C. Peterson
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The Fibonacci sequence is arguably the most observed sequence not only in mathematics, but also in nature. As we begin to learn more and more about the Fibonacci sequence and the numbers that make the sequence, many new and interesting applications of the have risen from different areas of algebra to market trading strategies. This poster analyzes not only the history of Leonardo Bonacci, but also the elegant sequence that is now his namesake and its appearance in nature as well as some of its current mathematical and non-mathematical applications.