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Full-Text Articles in Physical Sciences and Mathematics
Homeomorphisms Of The Sierpinski Carpet, Karuna S. Sangam
Homeomorphisms Of The Sierpinski Carpet, Karuna S. Sangam
Senior Projects Spring 2018
The Sierpinski carpet is a fractal formed by dividing the unit square into nine congruent squares, removing the center one, and repeating the process for each of the eight remaining squares, continuing infinitely many times. It is a well-known fractal with many fascinating topological properties that appears in a variety of different contexts, including as rational Julia sets. In this project, we study self-homeomorphisms of the Sierpinski carpet. We investigate the structure of the homeomorphism group, identify its finite subgroups, and attempt to define a transducer homeomorphism of the carpet. In particular, we find that the symmetry groups of platonic …
Graph Replacement Systems For Julia Sets Of Quadratic Polynomials, Yuan Jessica Liu
Graph Replacement Systems For Julia Sets Of Quadratic Polynomials, Yuan Jessica Liu
Senior Projects Spring 2018
Belk and Forrest construct a specific class of graph replacement systems that give sequences of graphs that converge to fractals. Given a polynomial, we have an algorithm that gives a replacement system that leads to a graph sequence which we conjecture converges to the Julia set. We prove the conjecture for the quadratic polynomial $z^2+c$ where $c$ is a real number and the critical point is in a three cycle. We present some additional results and observations on replacement systems obtained from certain polynomials.
Stationary Automata, Anaam Mutar Bidhan
Stationary Automata, Anaam Mutar Bidhan
Graduate Theses, Dissertations, and Problem Reports
In this dissertation, we investigate new automata, we call it stationary automata or ST-automata. This concept is based on the definition of TF-automaton by Wojciechowski [Woj2]. What is new in our approach is that we incorporate stationary subsets of limit ordinals of uncountable cofinality. The first objective of the thesis is to motivate the new construction of automata. This concept of ST-automata allows us to make a connection with infinite graph theory. Aharoni, Nash-Williams, and Shelah [AhNaSh] formulated a condition that is necessary and sufficient for a bipartite graph to have a matching. For a bipartite …